CORNELL UNIVERSITY LIBRARY Gradations in Euclid 3 1924 031 273 083 olin.anx Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31 924031 273083 GKADATIONS IN EUCLID: BOOKS I. AND II. M INTRODUCTION TO PLANE GEOMETRY, ITS USE AND APPLICATION; WITH SEBMARKS ON GEOMETEICAL REASONING, AND ON ARITHMETIC AND ALGEBRA APPLIED TO GEOMETRY; 'PRACTICAL RESULTS AND EXERCISES. HENRY GREEN, A.M. FIFTS THOUSAND. "GHOMSmT TBf FERHAPSf OF *t.t. ^HZ FABI8 OF Jli^ISSamnCS, TEAT WHICH OUGHT TO BB TATIGHT FlBflT : IT APPEABS TO HE TiaiT PKOPEE TO nTTESBST CHIIJ>BEN, PSOTIOBD IT IS PBK- SEMTBD TO THEM PEINCIPAXLT IN EELATIOH TO ITS APFUCAHONS, -WHZTHBE ON PAPER OR ON Laaroix, p. SOff. JOKH HEywooD, BxoELSiOB BtritDiNGS, Jomf' Balt^oit S'rM^ET, Makohestee; And 18, Patersoster Sqtjase, London, B.p. Agents: Simpkin, MAjbhali., & Co. .\ " Geometrie, Tbroiigh tvliiuh a man hath the sleiizhi OJ length, of brede, of depth, of height." GowEB, Con. A., b, viii,. PEEEAOE, EBSPEOTING THE QBADATIOHS IN EUOUD S PLAUE QEOMETET, SKELETON PROPOSITIONS, ETC. The chief aim of the Author or Compiler of the Gradations in Euclid, with Skeleton Propositions, &c., for Written Examinations, has been to furnish a useful hooh to those who have not much time for the pursuit of Geometrical Studies, and to whom, therefore, the Practical Application of whatever they learn is of great importance. He is, however, persuaded that those who have hoth time and full opportunity, either in Public Schools or in Colleges, for attaining proficiency in the Higher Mathematics, wfll find an Introduction, such as is given in this work, very suitable to prepare them more thoroughly to appreciate Geometrical Truths, and to take an interest in them as the ground-work of accurate science. The Inteodttotion is of general use to all Students of Geometry : it contains — a brief account of the Gradual Growth of Geometry and of the Elements of Euclid ; the Signs and Contratftions that may be employed ; and some Eemarks on the Nature of Geometrical Reasoning — on the Application of Arithmetic and Algebra to Geometry — on Incommensurable Quantities — and on Written and Oral Examinations. Several of the subjects treated of presuppose, indeed, that the Learner has a clear under- standing of Fractions, common and decimal — of the extraction of the Square Eoot — and of the inti'oduotory principles of Algebra : but this knowledge is indispensable for those who would really master the Elements of Geometry. The Editions of Euclid by Potts and Blakelook have shown the advantages of printing separately and distinctly the parts of a Proposition and of its Demonstration : it is >• plab which undoubtedly gives very valuable help to Learners in attaining a more exact acquaintance with the Principles on which Geometry, as a science, is founded. No argument is here needed to prove the importance of being able to estimate the force IV PREFACE. and certainty of demonstrable truths : — it is the first condition of snn- cess, and the sure means of proficiency in geometrical and in all other studies. The Gradations in Euclid endeavour to carry out the Plaji to a greater extent, and with increased distinctness. The Propositions through- out are separated into successive steps ; and in the margin, between the vertical lines, direct references are made to the Seasons — the Definitions, Axioms, or preceding Propositiona — on which such Construction or Demon- stration depends. The method of printing, which has been adopted, also gives a clearer view both of the whole Proposition and of its parts ; and familiarises the mind to an orderly and systematic arrangement — so important an auxiliary to all sound progress. By following out a plan of this kind. Learners can scarcely fail to form a distinct conception of what they have to do or to prove, and of the means by which their purpose is to be accomplished. The Explanatory Notes direct the Learner's attention to several points of interest connected with the Definitions and Propositiona ; and to many of the Propositiona is appended an account of the chief PeachcaIi Uses to which they may be applied. This is valuable for many reasons, but principally that the Learner may at once see, not simply the theoretical and abstract truths of Geometry, but their direct utility in various ways. There are very many persons who, from studying only the Common Editions of Euclid, which treat exclusively of the Theory of Geometry, never attain to a perception of its importance, and never reaGse the full advantages of geometrical studies. Indeed, they have read and demonstrated the whole work without perceiving any value in their laborious pursuit except as an exercise of the memory and mental powers in a very dry coarse of reasoning. It is the main object of our Gradations in Euclid to conibine Theory and Practice, and, as soon aa a geometrical truth has been established, to point out its use and application. The Author is thoroughly persuaded that this invmediate Combination of Theory and its Application, not only awakens and maintains a livelier interest, but, in fact, leads to a more scholar-like understanding of both, than when they are studied separately, or at wide intervals of time. The Uses to which the Propositions may be applied are very numerous; and we have given only a porti''n of them — more in the way of example, and to point out in some instances the progress of geometrical discovery. PEEPACB. V than with the view o£ exhausting the subject. The yajious works on Practical Mathematics ■will supply what may be wanting in this respect. For the full developement of the Uses and Applications of the First and Second Books of Euclid, geometrical principles not worked out in those books must occasionally be introduced ; and though it is not strictly logical to employ truths that have not really been established, as the ground-work of further reasoning — now and then, in this part of the work. Lemmas, or truths borrowed from another part of the subject, will be adopted as the foundation of new truths. The Peaotioai, Results will, it is presumed, be instructive to the Learner in various ways, but especially as exhibiting a synoptical view of all the Problems contained in Euclid's Elements of Plane Geometry He win not, indeed, have arrived at the means of demonstratiug the Prob- lems which lie out of the First and Second Books ; but, inasmuch as their construction depends almost entirely on those two books, he wil possess, for practical purposes, a knowledge of the methods by which, the regular Geometrical Figures, not being sections of a Cone, are to be constructed. The Appendix, containing Gbometmoal Analysis and Geometbioal EXEKOISES, appeared to the Writer needed for the completion of his plan — i. e., for comprising a systematic teaching of Geometry, as far as the First and Second Books furnish the means of doing it. The Appendix, and a Key to the Exercises, will each be published separately from the Gradations. The Skeleton Peopositions, &o., for penrand-imJc examinations, are arranged and will be published in two Series — one with the references in the margin — the other without those references. The first Series is intended for beginners ; the second, for those who may be reasonably suppose d to be prepared for a strict examination. The two Series will ba found- well adapted to test the Progress of the Learner, and to ascertain how far his knowledge of geometrical principles, and his power to apply them, really extend. The object is, in the first Series, to furnish the Learner', step by step, with the truths from which other truths are to bel evolved, but to leave him to work out the results, and from the results, as. they arise, to aim at more advanced conclusions : in the second Series, where there are no references supplied in the margin, the object is to make the examinations strict and thorough, yet so as to be conducted on one uniform plan. This uniformity wiU be found greatly to assist Examiners' VI PREFACE. when they compare the examination papers together for the purpose of deciding on their respective merits. The Skeleton Propositions may be used either simultaneously with the Gradations in Euclid, or after the First and Second Books have been read in any of the usual editions of Simsoh's Ettclid, as a recapitulation of the ground already gone over; if used sinmltaneoiisly, the Learner must first study the Definitions and Propositions in their order, and then, laying the printed book aside, reduce his knowledge to a written form, as the references indicate which are given in the vertical columns of the Skeleton Propositions ; but if used as a recapiUilatory exercise, a course in some respects different is recommended. In the recapitulatory exercite, the following plan is recommended for adoption : — flrst, that the Learner should give in writing a statement of the meaning of various Geometrical Terms, of the nature of Geometrical Seasoning, and of the application of Algebra and Aiithmetic to Geometry ; secondly, that he should fill in — not hy copying from am/ iooh, tut from the stores of his ovm mind and thought, trained by previous study of the Grada- tions in Euclid, or of some similar work — ^the Definitions, Postulates, and Axioms of which the leading words are printed ; and thirdly, that he should proceed to take the Propositions in order, and write out the proofs at largo, as the printed forms and references in the margin indicate : this should be done systematically in all the Propositions, beginning with those truths already tetablished which are reqvured for the Construction and Demon- stration, and then taking in order the Exposition, the Data and Queesita, or the Hypothesis and Conclusion, the Construction with its methods, and the Demonstration with its proofs, separated from each other, and given, step by step, in regular progression. For the thorough Examinations, that Series of the Skeleton Propositions must be used which contains the General Enunciation only, without any references printed in the margin. The Spaces for the Exposition, Construc- tion, and Demonstration are retained, and also the vertical lines within which the Learners themselves are to place the references ; but this is done simply for the purpose of securing a uniformity of plan in the written examinations, and for the convenience of Examiners. The Student imder examination should also be required to write out the Propositions, &c.,* needed in the Construction and Demonstration, and to supply the * This 1b required because the repetition of truths and principles gains for them a more permanent residence in the mind. PBBPAOB. Vll references to the various geometrical truths by which the steps of the Proposition are established. No figures, or' diagrams, are given in either of the Series of Skeleton Propositions ; as it is more conducive to the Learner's sound progress that he be left entirely to himself to construct these. The Uses and AppUoations of the Propositions, at least i/n a brief wwy, — and where requisite, the Algebraical and Arithmetical niustrations, — should not be neglected: it is in these that the practical advantage of ibstraot' truths is rendered apparent. It is imperative th^t the Teacher should revise each Proposition after it has been written out, and note the misapprehensions and inaccuracies before the Learner proceeds to the following Proposition. In Self-Tuition, the Learner must consult the Gradations, and by them correct the already £lled-up Skeleton ; but he must be faithful to himself, and to his own improvement, by not consulting the Gradations as a Key, until he has first worked out and written down his own conception of what the Demonstra- "tion demands. He will thus build up for himself and of himself ; he will make the dead bones of the Skeleton Propositions Uve, elothe them with £esh and sinews, and round them off in all their proper proportions. A course of this kind followed faithfully through two books of the Elements of Geometry, will scarcely fail to render the Student competent laj himself to master the other books of Euclid, and, should he desire it, by the same means. He will have learned the value of method and exact- ness ; and, expert in these, he wiU attain a solid and durable knowledge of geometrical principles. , At the present day nearly every edition of Euclid's Elements must be, more or less, a compilation, in which the Author draws freely on the labours of his predecessors. The Gradations are, in a great degree, of this character ; and an open acknowledgment wiU suffice, once for all, to xepel any charge of intentionally claiming what belongs to others. It is affectation to pretend to great originality on a subject which has, like ■Geometry, for so many centuries exercised men's minds. If by the methods employed in the following pages, the Study of Geometry among all classes of the community be rendered more interesting and more practi- cally useful, the objects of the Editor will be accomplished. He desires no worthier calling than to be a fellow-labourer with the many excellent VUl PREFACE. and talented Masters to whom the reaponeibility is entrusted of trainingf the young in sound learning. In commendation of the study of Geometry, Principal Hill, of Harvard University, in his First Lessons on the subject, p. 8, declares " Geometry is the most useful of all the Sciences. To understand Geometry wUl be a great help in learning all other Sciences ; and no other Science can be learned unless you know something of Geometry. To study it will make your eye quicker in seeing things, and your hand steadier in doing things. You can draw better, write better, cut out clothes, make boots and shoes, work at any mechanical trade, or learn any art the better for understanding Geometry." The principal editions of Euclid to which the Editor is under obligation, are those of Potts and Laedner, and of an old Writer who professes to give "the uses of each Proposition in all the parts of the Mathematicks." An exemplification and recommendation of the plan pursued in the Grada- tions, and in the Skeleton Propositions, may be found in the preface to Labdner's Euclid, and in a Treatise on the Study cmd DiffimUies of Mathematics, p. 74, attributed to Professsor De Mobqaw. He is also indebted to various persons — Schoolmasters and others — ^for valuable sug- gestions, which he takes this opportvmity to acknowledge. The work is longer than the Author at first contemplated ; but ha trust* that the additions — especially the Practical Results and the Exer- oises — ^will add considerably to its usefulness and value. INTKODUCTION. SECTION I, GKADUAL GROWTH OF QEOMETET AND OF THE ELEMENTS OF EUCLID. Geometry, land-measuring, as the word denotes (from gee, eartk or land, and metron, a measure), was in its origin an Art, and not a Science : it embraced a system of rules, more or less complete, for performing the simpler operations of land-surveying ; but these rules rested on no regularly-demonstrated principles, — ^they were the oifspring rather of experiment and individual skill, than of scientific research. In the same way poems — even some of the noblest — were composed before the principles of poetry had been collected into a system j languages were spoken, long before a grammar ■ had been compiled ; and men reasoned and debated before they possessed either a logic or a rhetoric : so measure- ments were made, while as yet there was no accurate theory of measuring — no abstract speculations concerning space and its properties. The points and lines of such a Geometry were necessarily visible quantities. A mark, which men could see, would be their point ; - a measuring-rod, or string, which they could handle, their line ; a wall, or a hedge, or a mound of earth, their boundary. The first advance beyond this would be to identify the instruments which they vised in measuring, with the lines and boundaries themselves : the finger's breadth, the cubit, the foot, and the, pace, would be- come representatives of a certain length without reference to the shape. It was only as the ideas and perceptions of those who cultivated the art of measuring grew more refined and subtile, that an Abstract Geometry would be evolved, such as Mathematicians understand by the term, in which a point marks only position ; a line, extension from point to point ; and a surface, space enclosed, by mathematical lines. Geometry, thus understood, has been defined in general terms to be — "the Science of Space," or "the Science of Form." It^ 2 GEOWTH OF GEOMETRY. investigates the properties of lines, surfaces, and solids, and the relations which exist between them. Plane Geometry investigates the properties of space under the two aspects of length and breadth ; iSolid Geometry, under the three — of length, breadth, and thick- ness. It is the consideration of the Elements of Plane Geometry on which we are about to enter. The Truths of Geometry, as a science, regularly as they are laid down and deduced in the Elements of Euchd, were not worked out by one mind, nor established in any systematic order. Some were discovered in one age, some in another ; two or three propositions by one philosopher, and two or three by some one else. The collection of geometrical truths had thus a gradual growth, imtil it received comparative completion at the hands of Euclid of Alexandria. Thales, who predicted the eclipse that happened b.o. 609, is said to have brought Geometry from Egypt, and to have established by demonstration Propositions 5, 15, and 26, of bk. i. ; 31, iii. ; and 2, 3, 4, and 5, of bk. iv. Pythagoras, born about 570 b.o., was the first who gave to Geometry a scientific form, and discovered Propositions 32 and 47 of bk. i. ; Oenopides, a follower of Pytha- goras, added the 12th and 23rd of bk. i. ; and Eudoxas, B.C. 366, a friend of Plato, wrote the doctrine of proportion as developed in the fifth book of the Elements. These assertions may not rest on the firmest authority, yet they show, even if they are only sur- mises, that Geometry was regarded by the Greeks as a science of very gradual formation, receiving accessions from age to age, and from various countries. It was at first a set of rules, until philo- sophy investigated the principles on which the rules were founded, and out of the chaos created knowledge. According to Proclus, Euclid of Alexandria flourished in the reign of the first Ptolemy, b.o. 323-283. To him belongs the glory, for such it is, of having collected into a well-arranged system the scattered principles and truths of Geometry, and of having produced a work, which, after standing the test of above twenty centuries, seems destined to remain the Standard Geometry for ages to come. Euclid's work comprises thirteen books, of which the first four and the sixth treat of Plane Geometry; the fifth, of the Theory of Proportion, applicable to magnitude in general; the seventh, eighth, and ninth are on Arithmetic; the tenth, on the Arithmetical Euclid's elements. S ■Charactoristios of the divisions of a straight line ; the eleventh and twelfth, on the Elements of Solids ; and the thirteenth, on the Eegular Solids. To the thirteen , books by Euclid, Hypsicles of Alexandria, about a.d. 170, added the fourteenth and fifteenth books — also on the Eegular or Platonic Solids. In modern times it is not usual to read more than six books of Euclid's Elements. The seventh, eighth, ninth, and tenth books treat of Arithmetic and of the Doctrine of Incommensurables, and have no proper connection with the first six books ; and the eleventh, and twelfth books, comprehending the First Principles of Solid Geometry, are to a considerable degree superseded by other Treatises. Of the Six Books, the^r«< may be described in general terms as treating of the Geometry of Plane Triangles ; the second, of Kect- ' .angles upon the parts into which a straight line may be divided ; the third book, of those Properties of the Circle which can be ■deduced from the preceding books ; the fourth book, of such regular and straight-lined figures as can be described in or about a circle ; the jfifth, of Proportion with regard to magnitude in general ; and the sixth, of similar figures, and of Proportion as applied to Geometry. Our proposed limits confine us, for the present at least, to the First and Second Books. The First Book, besides the Definitions, Postulates, and Axioms, contains forty-eight Propositions, of which, fourteen are problems for giving power to construct various lines, angles, and figures; and thirty-four are theorems, being the ex- positions of new geometrical truths. Of these theorems, some may he regarded as simply subsidiary to the proof of others that are more important, and of wider and more general application. The Propositions to be ranked among those of high importance, are Props, i, 8, and 26, containing the criteria of the equality of triangles ; Prop. 32, proving that the three interior angles of every triangle are together equal to two right angles; Prop. 41, declaring that a parallelogram on the same base and of the same altitude as a triangle, is double of the triangle ; Prop. 47, demonstrating that the square on the hypotenuse of a right-angled triangle, is equal to the sum of the squares on the base and perpendicular. It is on these Propositions of the first book — namely, 4, 8, and 26, 32, 41, and 47 — that Geometry in its after applications mainly depends, and, therefore, nmst they be most thoroughly understood and mastered. The Second Book treats of the properties of Eigsht-Anglbd PaeallelogeamS; contained by the parts of divided straight lines. 4 BaCLID's ELEMENTS. There are fourteen Propositions, of which Props. 11 and 14 are problems — the other twelve are theorems. Props. 12 and 13 give the Elements of Trigonometrical Analysis, or the Arithmetic of Sines, and are of great use in the Higher Geometry : the other Propositions may be classified according to the mode of dividing the line or lines ; Prop. 1 relating to the rectangles formed by one undivided line and the parts of a ditided line ; Props. 2, 3, 4, 7, and 8, to the rectangles formed by a line and any two parts into which it may be divided ; Props. 5 and 9, to the rectangles on a line divided equally and unequally; and Props. 6 and 10, to the rectangles formed on a line bisected and produced. The English Translation of Euclid, published by Dr. Kobert Simson, of Glasgow, in 1756, has nearly, in some form or other, superseded aU others, and is considered the standard text of an English Euclid. As containing " the Elements of Geometry," it is "imexceptionable, but is not calculated to give the scholar a proper idea of the Elements of Euclid," as Euclid himself left them. Various alterations, additions, and improvements were made by Simson ; but, " with the exception of the editorial fancy about the perfect restoration of Euclid, there is little to object to in this celebrated edition. It might, indeed, have been expected that some notice would have been taken of various points on which Euclid has evidently fallen short, of that formality of rigour which is tacitly claimed for him. We prefer," says De Morgan, " this edition very much to many which have been fashioned upon it — particularly to those which have introduced algebraical symbols into the demonstrations in such a manner as to confuse geometrical demonstration with algebraical demonstration." — (See the Article Hucleides, by De Morgan, in Smith's Dictionary of Greek and Roman Biography, Vol. II., pp. 63-74.) In the face of such authority, it may seem bold to advocate the use of a Symbolical Notation ; but, within certain limitations, the symbols of Arithmetic and of Algebra have a universal meaning, and may therefore be employed without any disadvantage,* and * See Lacroix' Essaig, ed. 1838, p. 227. " L'histoire des Mathdmatiques prouve ausBi que c'est I'usage de plus en plus ^tendu des symboles arbitraires imagines dans la vue d'abr^ger lea expressions ou de mettre en Evidence leur analogie, qui a contiibu^ le plus i, ravancement de la science ; en soulageaut la m^moire et facilitant les combinaisous des relations donu^es et des raisonuemens." SYMBOLICAL NOTATION. 5 certainly without confusion in our ideas. The precaution needed is, that we take care not to depart from the strictly geometrical application. For an outline of the origin and progress of the science of Geometry, the learner should consult the Introduction to the Elements of Euclid, edited by Robert Potts, M.A., Trinity College, greater than. ^ not greater than. < less than. «^ not less than. pliis, more, increased by. minus, less, lessened by. difference between. ratio. equality of ratios. proportion. progression. ABBREVIATIONS. II. — Representative, or Geometrical Signs. ' a point. I straight line. II parallel to. IJs paraUela* L angle. A triangle. D perpendicular to, or at right angles, parallelogram, square, rectangle, circle. 0ce circumference. A single capital letter, in reference to a diagram, as A, or B^ denotes the point A, or the point B.t Two capital letters, also in reference to a diagram, as A B, or CD, denote the straight line A B, or C D, or the side of a triangle, or other figure. Two capital letters, with the figure " just above to the right hand, as A B^, denote, not the square of A B, but the square on the straight line A B. Capital letters, with a point between them, as A B . C D, denote, not the product of A B multiplied by C D, but the rectangle formed by two of its sides meeting in a common point. III. — Abbreviations. Add. Addenda, by adding. App. Application. At. Axiom. Cone Conclusion, inference. Cor. Corollary. Cor Cons. .Construction. C. 1 &c. ...Step 1 &c. of the Con- struction. Dot. Datum, or data. Sef. Definition. D. or DeuL Demonstration. D. 1 &o, ...Step 1 &c. of the Dem. E. or Exp. .Exposition, or Particu- lar enunciation. Gen General enunciation. H. or Hyp. .Hypothesis. H. 1 &c. ...Step 1 &o. of the Hyp. Prob Problem. Pr. or Prs. Proposition, or Proposi- tions. Pst. or Pste.PostuIate, or Postulates Rec Recapitulation. Bemk. Remark to be made. * When an s is added to the sign, the pteal is denoted. + N.B. A single capital letter may also denote an angle, when the sign z , or word angle is used. Also, a parellelogram may be named by naming the single letters at the opposite angles. GEOMETRICAL TERMS. Sch SchoUvm, or Scholia. \ Sup Suppose. ...Svpei-ponendo, by su- perposition. Theor Theorfem. Sim Similarly. S. or Sol. ...Solution. Svib; Suitrahendo, by taking away. Q.E.D., guod erat demonstra/ad/ma, whioh was the thing to be proved. Q.E.F., gmd erat fadmdum, whioh was the thing to be done. fig figure. int interior. a fort a fortiori, hj 3, stronger argument or reason. alt altitude. altr. alternate. assum assumendo, by assuming, adopting, or taking. bis bisect. ad imvposs. ad i/mpossUdU, reduced to an impossibility. bisd bisected. bisg. bisecting. com common. con. sup. ...contrary supposition. c. sc circumscribe. d. sc describe. ec[. ang. ...equiangular. eq. lat equilateral. ea. o6s. ex absurdo, by an ab- surdity, ext exterior. magn magnitude. opp opposite. par parallel. parlm parallelogram:. pos position. qu. ang. ...quadrangular. qu. lat quadrilateiaL rad radius. rect rectangle. rectil. rectilineal reotang. ...rectangular. rem remaining. rt right. sq. square. St. straight. uneq. unequal. vert vertex. SECTION III. EXPLANATION OF SOME GEOMETRICAL TERMS. A Befinition (from definire, to set bounds^to) is a short descrip*- tion of a thing by such of its properties as serve to distinguish it from all other things of the same kind. A Postulate (posttdatum, a thing demanded) is a self-evident problem, the admission of whioh is demanded -without formal proof. GEOMETRICAL TERMS. An Axiom (axioma, a thing of worth) is a self-evident theorem, or the assertion of a tinith, which does not need demonstration : it is worthy of credit as soon as stated. A Proposition (proponere, to set forth) is something proposed to te done, as a problem ; or to be proved, as a theorem. A Problem (probleema, a thing proposed) is a proposal to do a thing, to construct a figure, or to solve a question. A Theorem (theoreema, a subject of contemplation) is the assertion of a geometrical truth, and requires demonstration. The Data (datum, a thing granted) are the things granted in a problem ; The QucEsita (qusesitum, a thing sought) are the things sought for in it; The Hypothesis (hypothesis, a supposition) is the supposition made in a theorem ; The Conclusion (concludere, to infer) is the consequence or inference deduced from it. The General Enunciation (enunciare, to speak out, declare) of a proposition sets forth in general terms the conditions of the problem, or theorem, with what has to be done, or with what is inferred or concluded. A Diagram (diagramma, a drawing of lines) is the drawing which represents a geometrical figure. The Exposition (exponere, to set forth), or Particular Unundation, sets forth the same conditions with an especial reference to a figure that has been drawn. The Solution (solutio, an unloosening, an explaining) of a problem shows how the thing proposed may be done. The Construction (constructio, a putting together) prepares, by the drawing of lines, &c., for the demonstration of a proposition. The Demonstration (demonstrare, to point out) proves that the process indicated in the solution is sound, or that the conclusion deduced from an hypothesis is true — i.e., in accordance with geometrical principles. The Pecapitulation (recapitulare, to go over the muin points again), or Conclusion, is simply the repetition of the proposition, or general enunciation, as a fact, or as a truth, with the declaration Q.E.F., or Q.E.D. GEOMETRICAL TERMS. 9 A Corollary (corolla, a little wreath, a deduction) is an inference ma de immediately from a proposition. A Scholium (scholion, a comment) is a note or explanatory observation. A Lemma (leema, a thing taken) is a preparatory proposition borrowed from another part of the same subject, and introduced for the purpose of establishing a more important proposition. The Converse (conversum, a thing turned round) of a proposition is when the hypothesis of a former proposition becomes the con- clusion, or predicate, of the latter proposition; as in Props. 5 and 6, 18 and 19, 21 and 25, bk. i. The Contrary of a proposition is when that which the proposition assumes is denied. Direct Demonstration is when the very thing asserted is proved to be true. Indirect Demonstration is when all other cases, or conditions, except the one in question, are proved not to be true, and the inference is made : therefore, the very thing in question must bo true ; the assumption being that one out of several, or many, must be right. The Position only of a line is meant, wheix the line is said to be given. The Length only of a line is meant, when the Hue is said to be finite. The Base of a figure (basis, a foundation) is the side on which it appeai-s to stand ; but each side, in turn, with the position of the figure changed, may become the base.. The Vertex (vertex, the top or crown of the head) is the highest angular point of a figure ; with a change of position in the figure, each angle may be named the vertical angle. The Subtend (subtendere, to stretch under) of an angle is the side stretching across opposite to the angle. The Hypotenuse (hupotenousa, that which stretches under) is the subtend to a right angle. The Perpendicular (perpendiculum, a plumb-line) is the line forming with the base a right angle : lines are perpendicular to each other when at the point of junction they form a right angle. 10 GEOMETRICAL BEASONINO. A Figure is applied to a straight line when the line forms one of its boundaries. The Altitude of a figure (altitudo, height) is the perpendicular distance from the side or angle opposite to the base, to the base itself, or to the base produced. A Diagonal (diagonios, from corner to corner) is a straight line joining two opposite angular points. The Complement of an angle (complementum, that which fills up) is what is wanted to make an acute angle equal to a right angle, or to 90°. The Supplement of an angle (supplementum, a filling up) is what is wanted to make an angle equal to two right angles, or to 180°. The Eocplement of an angle (explementum, a filling) is what is wanted to make an angle equal to four right angles, or to 360°. The Complements of a Parallelogram (see Fig. to Definition A.), when the parallelogram is bisected by its diagonal, and subsi- diary parallelograms are formed by two lines — one, parallel to one side, and the other, parallel to the other side, and both intersecting the diagonal, — the complements of the parallelogram are those sub- sidiary parallelograms through which the diagonal does not pass ; and these, with the subsidiary parallelograms through which the diagonal does pass, fill up or complete the whole parallelogram. The Area of a figure (area, an open space) is the quantity of surface contained in it, reckoned in square units, as square inches, square feet, &c. A locus (locus, place) in Plane Geometry is a straight line, or a plane curve, every point of which, and none else, satisfies a certain condition. SECTION IV. NATURE OP GEOMETRICAL REASONING. In Mechanics, Chemistry, and the kindred sciences we are so accustomed to experimental proof, and find it so very useful, that we readily suppose that experiment may be applied to Geometry equally well. Thus we draw a triangle, and at some GEOMBTEIOAL EEASONING. 11 little distance from, the angular points we out off the angles; then, by experiment, we place the three angles together so that two of the sides of different angles may form a straight line, and with the third angle we fill up the unoccupied space between the other two, : it will thus appear that the three angles which we cut off from the original triangle are together equal to two right angles. But the appearance is not a perfect and absolute proof. " Mathematical proofs," says Locke, vol. iv., p. 428, " like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning." Now the Demonstrations in Euclid's Elements of Geometry consist of strict arguments or reasonings by means of which the assertions made in the propositions are proved to be absolutely true. Thus, in the 15th Prop., bk. i., the assertion is made that " the opposite, or vertical angles, formed by two intersecting lines, are equal ; " and the demonstration shows by strict argument or reasoning, founded upon truths already admitted or proved, that the assertion must be received as true, — ^we cannot disbelieve it, if we would. When fully stated, each argument contains both the thing which is proved, and the means by which the proof is established ; and, as in the -arrangement of the parts of an argument the means of proof usually precede the thing proved, they are named the premisses (premisse, a thing which precedes or goes before); and the thing proved is named the conclusion, or inference. Thus; in Prop. 1, bk. i., the premisses are^lst, things equal to the same thing are equal to each other ; 2nd, the line A C, and also the line B C, are each equal to the same line A B ; and 3rd, the inference, con- clusion, or thing proved, is, that the line A C equals the line B 0, or, adhering more strictly to the forms of reasoning, the two lines A C and B C are equal to each other. Here in the premisses two things are laid down, or granted to be true : as — " things equal to the same thing are equal to each other," — ^this is one truth; " the line A C equals the line A B, and the line B C also equals the same line A B," — ^this is another truth ; and from the two things thus declared to be true, thsre is made in the conclusion the necessary and unavoidable inference, — there- fore, A C equals B C — i. e., the two lines A C and'B C are equal to each other. The subject of the conclusion, " the two lines A C and B C," is is called the minor term ; the predicate of the conclusion (predi- 12 GEOMETRICAL REASONING. catum, thing declared) " are equal to each other," is called the major term. There are three things in the premisses, — the major and the minor terms, and the third term with which as with a standard the major and the minor terms are compared. This third term, being the medium of the comparison, is named the middle term : it enters into the major premiss as the subject, and into the minor premiss as the predicate. In the example given, " things equal to the same," is the middle term, being the subject of the major premiss — "things equal to the same are equal to each other," and the predicate of the minor premiss — " the lines A and B C are each equal to the same line A B." The premiss in which the major term — i. «., the predicate of the conclusion — appears, is called the major premiss; that in which the minor term, or subject of the conclusion, appears, is the minor premiss. Thus, in the argument already given : — Major premiss, — because things equal to the same are equal to each other; Minor premies,— and because the two lines A C and B C are each equal to the same line A B ; Conclusion, — therefore, the two lines A and B C are equal to each other. Here, " tlie two lines A 0, S 0" is the subject, and " equal to each ot/ier" the predicate of the conclusion ; "The two lines AC, BC" is the eubject, and '^ each equal to the sam£ line A -S," the predicate of the minor premiss ; " Tilings equal to the same line A B" the subject, and "equal to each other" the predicate of the major premiss. Thus, " equal to each other " is the major term ; and " t?ie two lines A C, B C," tlie minor term ; and " things equal to the same line A B," the middle term of the argument. In this mode of reasoning, it is seen that assertions are broadly made ; and we may ask, on what evidence are these assertions themselves to be received as true ? The jirst kind of evidence is from the definition of the thing : thus, we define a triangle to be a figure bounded by three sides ; and if, of any figure placed before us, we can aifirm that it has three sides exactly, the conclusion is inevitable, that this figure aldo is a triau"le. GEOMETRICAL REASONING. 13 The second kind of evidence is from the axioms, or truths so plain that they need no proof : for example — we receive as undeniable, that, if equals be added to equals, the wholes are equal ; and we argue, if to the line AD, or to its equal the line BC, we add another line E F, then the whole line made up of A D + E F, will equal the whole line made up of B C + E i\ The third kind of evidence is from the hypothesis, or supposition, which we make as the condition of our assertion : we declare, " in an isosceles txiangle, the angles at the base are equal ;" the very words, though not in the exact form of an hypothesis, directly imply the supposition, " if a triangle is isosceles," " then the angles at the base are equal." An isosceles triangle is here taken as the starting point of the reasoning ; and though, for the demonstra- tion of the inference, " the angles at the base are equal," it is necessary to draw various lines which are not mentioned in the hypothesis, the conclusion at which we arrive is altogether depen- dent on the hypothesis. The fourth kind of evidence is from proof already given ; for what has once been established, may afterwards be taken for granted. For instance, when we have once established the truth, that "the interior angles of every triangle are together equal to two right angles," and we afterwards come to a proposition in the demonstration of which we need this established truth, we do not aaain go through all the steps by which the equality of the sum of the interior angles of a triafigle to two right angles has been proved J but, without going down again to the bottom of the ladder before we make a step higher, we start from the step we had already gained, and at once take up our position on a more advanced truth. But the foundation, the Principle of Geometrical Reasoning is, hat from two propositions established or received as true, a third proposition, or inference, shall be made. Now, that this may be done, there must be something in common contained in both the propositions, with which common thing, the other two things are compared : we say, for example — AU the triangle is in the circle. All the square is in the triangle; therefore, All the square is in the circle : the common term of comparison here is " the triangle," and our inference is coireot. 14 GEOMETRICAL REASONING. But if we say — All the triangle is in the circle, All the square is in the circle, and infer. All the square is in the triangle, — this may be, or may not be true; for it may happen that only a part of the square is in the triangle. The fault of the apparent argu- ment is, there is no proper term of comparison — no middle term which is at the same time the subject of the major premiss, and the predicate of the minor premiss. To have the argument sound, we must say — Major P. All the triangle is in the circle. Minor P. All the square is in the triangle ; therefore, All the square is in the circle. "When a connection is thus declared to exist between the pre- misses and the conclusion — that is, when reasons are stated and an inference made — this mode of argument receives the name of a Syllogism (syllogismos, a collection); for a Syllogism is a bringing together or collecting into one view the two steps of the reasoning on which a truth depends, and the truth itself; or as Whately, in his Elements of Logic, p. 52, defines a Syllogism, it is "an argu- ment so expressed, that the conclusiveness of it is manifest from the mere force of the expression, i,e., without considering the meaning of the terms : e. g., in this Syllogism :■ — " Every Y is X, or, Every man is mortal j ,, Z is Y, John is a man ; therefore, Z is X, John is mortal ; "the conclusion is inevitable, whatever terms, X, Y, and Z, respectively, are understood to stand for. And to this form all legitimate Arguments may ultimately be brought." Every Syllogism is made up of three propositions, or assertions ; of which, two are named the premisses, and the third, the conclusion. The Proposition which contains the predicate of the conclusion, is called the mxijor premiss ; and that which contains the sulifeci of the conclusion, the minor premiss. The premisses are usually introduced by the word because, or by some similar word, and the conclusion by the word therefore. The Syllogism is exhibited in four forrtis, ordures, distinguished from each other by the situation of the Middle Term, or Standard GEOMETRICAL SEASONING. 15 of Comparison, with respect to the extremes of the conclusion — that is, with respect to the major or minor terms. " The proper order is to place 1 he Major premiss ^wi, and the Minor, second; hut this does not constitute the Major and Minor premisses ; for that ptemiss (wherever placed) is the Major which contains ilie major-term, and the Minor, the minor." — {Elements of Logic, p. 57.) The major term, as we have before said, is the predicate, and the minor term the subject of the conclusion. Taking X to represent the Major term, Z the Minor term, and Y the Middle term, we may now exhibit the four forms of the Syllogism, of which four forms one or the other is used in all legitimate reasoning. I. The First Form, or figure, of the Syllogism, which is also the clearest and most natural, makes the Middle term the subject of tlie major premiss, and the predicate of tlie minor. Thus, in Prop. 1, bk. i., of Euclid's Elements : — , Major P. Because Y=5, or, teoaiise the line A E is equal to the line A C, Minor P. and Z = Y ; and the line B C is equal to the Ime A B ; Concl. therefore, Z = X. therefore, the line B C is equal to the Kne A C. Here we may observe that BC, or Z, is the subject of the conclusion, and AC, or X, the predicate ; B C, or Z, the subject, and A B, or Y, the predicate of the minor premiss ; A B, or Y, the subject, and A C, or X, the predicate of the major premiss ; ; AC, or X, the major term; BC, or Z, the minor term; and A B, or Y, the middle term, or the standard of comparison. II. In the Second Form of the Syllogism, the Middle term is the predicate of both premisses; as in, Prop. 47, bk. i., of the Elements : — Major P. Because X=Y, or, because the angle D B C is a right angle, Minor P. and Z=Y; and the angle F B A is a right angle ; Concl. theref orp, Z = X. therefore, the angle F B A = the angle D B C. Here ^FBA, or Z, is the subject, and Z.DBC, or X, the predicate of the conclusion ; 16 GEOMETBICAL REASONING. ^FBA, or Z, is the subject, and rt. angle, or Y, the predicate of the minor premiss ; ^DBC, or X, is the subject, and rt. angle, or Y, the predicate of the major premiss ; ^ D B C, or X, is the major term ; l¥BA, or Z, the minor term ; and right angle, the middle term. III. In the Third Form of the Syllogism, the Middle term .s the subject of both premisses ; as in Prop. 28, bk. i., of the Elements : — •P. Because Y=X, or, because ^ E GB is equal to z QHD, Minor P. and Y=Z; and z EGB is equal to z AGH; Cond. therefore, Z=X. therefore, ^ AGH is equal to z GHD. Here Z, or z. A G H, is the subject, and X, or z. G H D, the predicate of the conclusion ; Y, or Z.EGB, is the subject, and Z, or ^AGH, the predicate of the minor premiss ; Y, or Z.EGB, is the subject, and X, or Z.GHD, the predicate of the major premiss ; X, or z. G H D, represents the major term; Z, or lAGH, the minor term, and Y, or z. E G B, the middle term. IV. The Fourth Form of the Syllogism is the reverse of the first, and is the most unnatural of aU : it places the Middle term the predicate of the major premiss, and the subject of the minor. Thus, in Prop. 26, case 2, bk. i., of the Elements : — Major P. Because X=T, or, because ^ B H A is equal to zEPD, Minor P. and Y=Z; and ^ EFD is equal to ^ BC A; Concl. therefore, Z=X. therefore, /i B C A is equal to ^BHA. Here Z, or a B C A, is the subject, and X, or z. B H A, the predicate of the conclusion ; Y, or z. E F D, is the subject, and Z, or z. B C A, the predicate of the minor premiss ; X, or Z.BHA, is the subject, and Y, or Z.EFD, the predicate of the major premiss ; and X, or z. B H A, is the major term ; Z, or z. B A, the minor ; and Y, or i E F D, the middle term. GEOMETitlCAL BEASONIN& 17 N.B. — This fourth form is never employed except by some accidental awkwardness of expression. Prop. 26, case 2, bk. i., ■would be much better arranged if we reduced it to the second form, and said : — Major P. Because X=Y, Minor P. and Z=Y; Concl. therefore, Z=X. ■, because z B H A equals z E F D, and ^ B C A equals z E F D ; therefore, z B C A equals z B H A. All legitimate arguments may be brought to one or the other of these four forms of the Syllogism ; and in reading Euclid's Elements it ■will be an improving exercise, occasionally, to present the argu- ments in the regular Syllogistic form. We subjoin an example from The Study and Difficulties of Mathematics, pp. 73, 74 : it is Prop. 47, bk. i., of the Elements. It -will serve as a model for putting other propositions into the syllogistic form. Let ABC be a rt. angled triangle, BAG being the rt. angle ; the squares on AB and A C, sides about the rt. angle, together equal the square on B 0, the side opposite the rt. E angle. Describe the squares on BCandBA; produce D B to meet E F, produced, if necessary, in.G; Exp. 1 Hyp/ 2 Concl. CONS.l hy 46. 1. 2 Pst. 2. 3 31.1. Dem. 1 6yDef.30. C. 1. 2 Sim. B H and dra^w H K parallel to B D and through . A. I. Conterminous sides of a square are at rt. angles to one another: E B and B A are conterminous sides of a square ; .'. E B and B A are at rt. angles, II. By a similar Syllogism we prove that sides D B and B C are at rt. angles, and also that sides G B and B C are at rt. angles. 13 GEOMETRICAL REASONING. D. 1 & 2, D. 1 & 2. Def. 30. C. 1. Def. 10. C.l&Hyp 26.1. D.3,4,5 D. 6. Def. 30. Def. A. C. 1. 35.1. C. 3. III. Two rt. lines perpendi- cular to two other rt. lines make the same angle as those others: the lines E B and B G, AB and B C, are two rt. lines perpen- dicular, &c. E"^ ,'. the angle E B G is equal to the angle ABC. IV. All the sides of a square are equal ; A B and B E are sides of a square ; .'. A B and B E are equal sides. V. All riffht angles are equal : /.BEG and z. B A G are rt. angles ; .'. z. B E G and z. B A C are equal angles. VI. Two triangles having each two angles and the interjacent side equal, are equal in all respects : BEG and BAG are two triangles having the angles BEG and E B G respectively- equal to the angles BAG and ABC, and the side E B equal to the side B A ; .*. the triangles BEG and BAG are equal in all respects. VII. The side B G is equal to the side B C, and the side B C is equal to the side B D • .'. the side B G is equal to the side B D. VIII. A four-sided figure of which the opposite sides are parallel, is a parallelogram : B G H A and B P K D are four-sided figures of which the opposite sides are parallel ; .'. BGHA and BPKD are parallelograms. IX. Parallelograms upon the same base and between the same parallels, are equal : 'EB A F and B G H A are parallelograms on the same base, &c. ; .'. E B A F and BGHA are equal roaral- lelograms. GEOMETEIOAL REASONING. 19 X. Parallelograms on equal bases and between the same parallels, are equal : B G H A and B D K P are parallelograms on equal bases, namely, B G- = B D ; .'. the parallelograms B G H A and B D K P are equal. XI. The parallelogram E B A F is equal to pa- rallelogram B G H A ; parallelogram B G H A is equal to paral- lelogram B D K P ; .*. E B A F, the square on A B, is equal to parallelogram B D K P. XII. A similar argument, from the commencement, proves that the square on A C is equal to the rectangle C P K. XIII. The rectangles B K and C K are together equal to the square on A B ; the squares on sides BA and AC are to- gether equal to the rectangles B K and CK; ,'. the squares on sides B A and A C are together equal to the square on A B. Tnduotion is a, species of argument in -which that is inferred respecting a whole class which has been ascertained respecting several individuals of the class, Cq-rried out completely, Inductive reasoning is that in which a universal proposition is proved by proving separately each of its particular cases : thus, the angles A, B, C, and D are all the angles ia a, certain figure A B G D : we prove that A is a right angle, B a right angle, C a right angle, and D a right angle ; and we say — therefore, all the jingles of the figure A B C D are right angles, The ai'gument " a fortiori," by the stronger reason, proves that a given predicate belongs in a greater degree to one subject than to another : as, A is greater thaii B ; B greater than C ; much more, a fortiori, is A greater than C. An example of this kind of ai'gument occurs in Prop. 21, bk. i., of the Elements: thus — The angle B D C is greater than the angle C E D, ?ind angle C E D is greater than the angle BAG; much more .'. is angle B D C greater than angle BAG, [.10 36. I. C. 3. 11 D. 9. D. 10. 12 Sim. 13 Ax. 8. D.11&12 20 GEOMETRICAL REASONING. The " reductio ad impossibile," the reduction to an impossibility, is when the argument shows that any given assertion is impossible : as in the 14th Prop., bk. i., where it is supposed that both the line B E and the line B D are continuations of another line C B : the demonstration conducts to the conclusion that the less angle ABE equals the greater angle A B D ; but this is an impossibility, for the less cannot equal the greater. There is also the " reductio ad adsurdicni," the reduction of an argument to an absurdity : it takes place when the conclusion, at which we arrive, involves something foolish, or utterly unrea- sonable : thus, in Prop. 7, bk. i., the angle B D C is proved, by the course of argument adopted, to be, first, equal to the angle BCD, and next, greater than the same angle BCD; but this is an absurdity. There is little real diiFerence in Geometry between the impossible and the absurd. " The Validity of an Argument depends upon two distinct considerations : — 1, the truth of the relations assumed, or repre- sented to have been proved before ; 2, the manner in which these facts are combined so as to produce new relations ; in which last, the reasoning properly consists. If either of these be incorrect in any single point, the result is certainly false." " The same thing holds good in every species of reasoning ; and it must be observed, however different geometrical argument may be in form from that which we employ daily, it is not difierent in reality." " The com- monest actions of our lives are directed by processes exactly identical with those which enable us to pass from one proposition of geometry to another. A porter, for example, who being directed to carry a parcel from the City to a Bti;eet which he has never heard of, and who, on enquiry, finding it is in the Borough, con- cludes that he must cross the water to get at it, has performed an act of reasoning, differing nothing in kind from those by a series of which, did he know the previous propositions, he might be con- vinced that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the sides," — On the Studies and Difficulties of Mathematics, p. 76. APPLICATION OP ALGEBRA, ETC. 21 SECTION V. THE APPLICATION OF ALGEBRA AND ARITHMETIC TO GEOMETRY. Algebraical Geometry, or the Application of Algebra to Geometry, has two provinces : the one, when Algebra is employed for the investigation of geometrical theorems and problems, and as the great instrument of mathematical Analysis ; the other, when the Notation and Methods of operation usually appropriated to Algebra are introduced to give expression to geometrical truths, and to furnish the formulas according to which the Practical Geometrician must solve the questions which come before him. For success in the use of it, the first kind requires a more extended acquaintance both with Geometry and with Algebra than beginners in either science can have attained : the second kind, therefore, although it is not fitted to enlarge the boundaries of strictly s;eometrical knowledge, is that of which we proceed to treat, Arithmetical Geometry, or the Application of Arithmetic to Geometry, is never employed for purposes of investigation or of analysis : its object is, in cases of particular lines, surfaces, and solids, to express by numbers their properties — properties, the truth of which Geometry has already demonstrated, and for the numerical statement of which Algebra has given the requisite formula, or method of operation. AU questions, as they are called, in Mensuration, are instances of the appUoation of Arithmetical Geometry. A special example will show the respective provinces of Geometry, Algebra, and Arithmetio j we assert, that the square on the hypo- tenuse (A) of a right-angled triangle is equal to the sum of the squares on the base (6) and on the perpendicular (a) : Geometry demon- strates the truth of the assertion ; Algebra either investigates and analyses that truth, or simply expresses it in the form of an equar tion, thus, h^ = a? + b^ ; and Arithmetic assuming a particular case, as that h = 5, a = 4, and 6 =; 3, shows, that a square on the line of 5 equals the sum of the squares on the lines of 4 and 3 respectively ; or that the square of 5 equals the sum of the squares of 4 and 3 ; '5x5,, or 25, being equal to 4 x 4, or 16, added to 3 x 3, or 9. We shali afterwards see that no numbers, except 5, 4, and 3, or their equ multiples, can with perfect accuracy give an arithmetical expression to the geometrical truth. The main Axioms of Geometry, Algebra, and Arithmetic are the same : thus, in Geometiy we say, — Magnitudes equal to the same 22 APPLICATION OP ALGEBRA, ETC. magnitude are equal to each other; in Algebra, Quantities are equal when each is equal to the same quantity ; and in Arithmetic, Numbers are equal to one another when they are equal to the same number : but the three variations in the mode of stating the axiom are expressions for one and the same universal truth, — Things equal to the same thing are equal to one another. On the ground that a perfectly accurate expression of all geomet- rical truths cannot be given by numbers, it is oj great importance in reasonings strictly geometrical, where we seek for absolute truth, that we should not confound a Geometrical Demonstration with the Algebraical or Arithmetical representation of it. Being accustomed to speak of one magnitude as containing another smaller magnitude a certain number of times, we hesitate not to say, that a line contains 10 linear units ; or a rectangle, 25 square inches ; or a solid, 6 cubic inches : but Plane Geometry, having to do with space generally, does not reason respecting any definitely-assigned quantity as expressed by concrete numbers, but about its universal properties. A point, considered theoretically or mathematically, marks posi- tion, and not magnitude, and no succession or series of such points could make up a line ; and a line mathematical, having extension only in one direction through space, and not being a part of space, no succession of lines — that is, of lengths without breadths — could form a surface ; and a surface having extension only in two direc- tions, length and breadth, no supposed piling up of surfaces could form a solid ; for the first surface in the imaginary series being without thickness, and the second and the third, &c., being equally destitute of that property, no number of such surfaces could form a solid. Solids, therefore, though bounded by surfaces, are not made up of surfaces ; surfaces, though bounded by lines, are not made up of lines; neither are lines, though they have «»ountless points in them, made up of an aggregation of points. But a point and a line considered practically, and as they were considered before Geometry, or land-measuring, became a theoretical Science, have in their very elements the property of extension ; they do, and they must, occupy space. On the Atomic theory of Chemistry, there are ultimate atoms beyond which no actual division of an elementary body can proceed ; so on the supposition that the points and lines employed in Mensuration possess visible properties, there are ultimate points and lines beyond which we ETC. 23 cannot carry our process of refinement. In practice tlie point and the line are both visible, and whatever is visible ceases to be a mathematical point, or a mathematical line : it has length and breadth to make an impression on the optic nerve, and is therefore a surface. The word monads, applied to denote the elementary atoms by the aggregation of which mineral, vegetable, or animal substances are formed, may be introduced, not disadvantageously, into Prac- tical Geometry. A point is the monad, or element of a line : we may make it as small as we please — the thousandth part, or the ten-millionth part of an inch ; but however small, it really possesses extension : and so a line may become thinner and thinner, and finer and finer, but it possesses visible properties of breadth — in fact, it is a surface ; and the monad of a line, or its least elemen- tary part, if visible at all, must also be a surface. We need not continue an argument of this kind. When Arith- metic is applied to Geometry, the li"ne is made up of points, or parts — the point or part being some generally-recognised monad of length, or first and least element in the line : thus, when a line is made up of parts, each part containing one-tenth of an inch, then the unit, or monad of extension, is one-tenth of an inch. Again, for Surfaces made up of lines having visible breadth, the line that measures a surface in its length is made up of the units or monads of length ; and the line which measures the same surface in its breadth is made up of the same units or monads, in this case called breadth. The monad of length and the monad of breadth, placed together at right angles to each other, constitute the monad or unit of Surface : it may be a square inch, or a square foot, or a square mile. Thus, 100th of an inch, taken in one direction, may represent the monad, or elementary part of length; And 100th of an inch, taken in a direction at right angles from one extremity of the monad of length, will represent the moiiad of hreadth ; ^ And the monad of length and the monad of breadth, depth, or height — for in a plane they are three words for one thing — ^thus placed, constitute the monad, or unit of swrface. Also, 100th of an inch, taken in a direction at right angles from the common point of junction of the two lines which represent the monad of surface, will represent the monad of thickness ; And the monad of length, the monad of breadth, and the monad 24 APPLICATION OP ALGEBRA, ETC. of thickness thus placed, and constituting extension in three direc- tions, originate the monad, or unit of solidity. Though, strictly speaking, the Monads, or least elementary parts of a line, are themselves lines-^nay, from being visible, are actually surfaces, — yet, if such a monad be set in motion, it generates or traces out a longer line ; and a line thus formed and sot in motion generates a surface ; and a surface also being moved, traces out a solid. It is on this principle that a visible point may cover from view the line mad^ up of such points ; the visible line cover the surface made up of such lines ; and the surface cover from view the solid made up of such surfaces. In Plane Geometry, however, we have to do only with surfaces and with lines as constituting the boundaries of surfaces : conse- quently, we have to consider only two dimensions — the measure- ment of length and the measurement of breadth, and the combination of the two as indicating the measurement of surface. Lines and Surfaces, as well as all otiier xnagnihbdes, may he expressed by numbers; indeed, without numbers they can only be very imperfectly expressed. To do this, as we have seen, some Standard of length, or of surface, is assumed ; as, a linear inch, or a square inch : the number of such linear inches, or square inches, expresses the magnitude of the line or of the surface : Thus, the symbolical expression J'i denotes the diagonal of a square of which the side is 1 : 4x3, or 12, the number of square units in a rectangle of which the sides contain 4 and 3 linear units respectively ; . And generally, if we designate the number of linear units in one line by the algebraical symbol a, and the number in another line by b, the algebraical symbol a b will denote the area of the rectangle formed by the lines a and 5, We proceed, therefore, to consider the meaning which we should attach to algebraical and numerical symbols when applied to lines and surfaces. First, — of a Line. Any straight line whatever, as A B, or a, or 1, may be taken to represent the unit of length ; to any longer line, as C D, or 6, or 3, we apply A B, or a, or 1, as the measuring line or unit ; and if APPLICATION OP ALGEBRA, ETC. 26 A B is contained an exact number of times, as 3 times, in C D, then we say, C D is equal to 3 times A B ; thus, CD = 3AB = 3 terior angles on the same side of it taken together less than two right angles, these two straight lines, being continually pro- E duoed, shall at length meet upon that side on which the angles are less than two right angles. This is almost equivalent to saying, that */ two straight lines are parallel, all the 'perpendiculars enclosed between them shall ie egual; for if CD is parsdlel to EF, and the perpendicular GH less than the perpendicular AB, the lines CD and EF are nearer together towards D and P than towards C and E, contrary to the definition of parallel lines. An illustration, by figures I. and II., will make the O <-./A, axiom plainer, I. The angles DGH and GHF are less than two right angles, :uid st. lines CD, EF meet in . K. II. The^ angles CGH and GHE,- being less than two right i angles, the st. lines CD and EF meet in. L — i. e., on the side of AB on which the angles are less than two right angles. Straight lines, like CD and EF (in fig. I.), converge, when they approach nearer and at last meet in a point, K: but st. lines, like LCD and LEF (in II.), diverge, when setting out from a point, L, they recede more and more. When one st. line, AB, meets two other st. lines, CD and EF, there are four angles formed on the right hand of AB, and four on the left ; of these angles, the interior angles are CGH, GHE, DGH, and GHF; the exterim- angles are AGC, AGD, BHE, and BHF ; the opposite angles are AGC to AHE, or BHF to BGD, Sec. ; the adjacent angles, AGC and AGD, or AGD and D GH, &o. ; the vertical angles, those of which the vertex is at the same point, as -^ s CGH and AGD ; and the alternate angles, every other one, as z s C GH and GHF, or ^sDGH and GHE. Instead of the twelfth axiom, Playfair adopts the following : — "Two sti-aight lines which intersect one another, cannot be both paa-alkl to the same siraigM line." PSOP. I. — BOOK I. 51 PEOPOSITIONS. Prop 1, — Prob. To describe an equilateral triangle upon a given finite straigM line. SoLOTlON. — Pst. 3. A circle may be described from any centre at any distance from that centre. Pst. 1. A straight line may be drawn from any one point to any other point. Dekonsteation. — Def. 15. A circle is a plane figure contained by one line which is called the circumference, and is such that ajl straight lines drawn from a certain point within the figure (called the centre) to the circumferenee are equal to one another. Ax. 1. Magnitudes which are equal to the same magnitude, are equal to one another. Exp. 1 3 CONS.l 2 3 4 Dem. 1 2 3 4 5 Datum. Qucss. hy Pst. 3. Sol. 6yC.l,Def.l5 C.2,Def.l5 D.l,2,Ax.l D.' 1, 2, 3 Recap. Given, the st. line ABj on it to desc. an equil. A /■ From centre A -with rad. / ;^ ABdesc. BCD; [D A| and from centre B with \ rad. BA desc. © \v._ ACE; from the point C where the circles cut, draw St. lines C A and CB ; then A ABC shall be the equil. A required. •.• . A is centre of BCD, .-. AC = AB; V .B „ ACE, .-. BC = BA: But AC, BC eaoh = AB, .-. AC = BC. Thus AB, BC, and AC, sides of a triangle, are equal to one another. Therefore, the A ABC is equil. and cm the given st. line AB. Q.E.F. Scholium. — ^A second equil. triangle AFB, may be drawn on the other side ■of the given st. line AB. Use AMD Application. — T. The only use to which Euclid applies this proposition, is in solving the next two problems, and problems 9, 10, and 11. 2. For merely practical purposes, it is sufficient, in describing an equU- •triangle, to draw arcs intersecting in the common point C, or F, and to join 52 GRADATIONS ,IN EUCLID. the points A, B, C. By aji accommodation of thia method, an isosceles triangle may be made : thus— from the extremities of the base draw arcs with a common radius equal to the equal sides of the triangle, and jom the points A, B, and C. 3. A figure approximatiry to an oval may be y^"^ A. ^S. drawn by describing, from the extremities of a / / ^ \ \ given St. line AB, equal circles intersecting in / / \ \ .8 C and F, and by taking the diapieter AE, jD ! l E|, or BD, and from .s C and F drawing the arcs I ^i jS I GH and IK to meet the arcs IQ and KH ; \ \ J J the figure DGHEKID will approximate to an \ \P./' y oval. ^^.-_..---'---.__---^ 4. By drawing an equilateral triangle on wood or brass, an instrument, BDE, or CFG, maybe made, with which an inaccessible distance can be measured. Let A be an object on the other side of a river ; at / \ station B place the instrument so that A can be seen / 'i along the at. edge BD ; then, without changing the / A^ position of the instrument, look along the other st. •^ •' y^ edge BE, and set out a st. line of indefinite length as /\ /\ a continuation of BE ; carry the instrument along the L — ^ L — J^ St. line BC, with its edge BE upon the line BC, until BE (r C by looking along the edge D E brought to coincide with CF, the same object A will appear in a straight line with CF : then A, B,. and C will be the angular points of an equilateral triangle ; and as the sides - are equal, by measuring from the station B to the station C, the distance from . B to A, or from . C to A, will be ascertained. Prop. 2. — Pros. From a given point to draw a straight line equal to a given straight' line. Sol. — Pst. 1. A st. line may be drawn from any point to any other point. P. 1. On a given st. line to deso. an equil. triangle. Pst. 2. A terminated st. Une may be produced to any length in a st. line. Pst. .3. A circle may be drawn from any centre at any distance fromi that centre. Dem. — Def. 15. All st. lines from the centre of the circle to the cii'cum ference, are equal. Ax. 3. If equals be taken from equals, the remainders are equal. Ax. 1. Things which are equal to the same thing, are equal each to the others. PROP. II. — BOOK I. 53 'Exp. 1 Bala. 2 Qwxs. 'CONS.1 hy Pst. 1. 2 P. 1. 3 Pst. 2. 4 Pst. 3. 5 9) 6 Sol. "Dem. 1 6yC.4,5. 2 3 4 Def. 15. C. 2. Stii. 5 6 7 Ax. 3. D.2,ATr,l Ree. Given the point A, and the St. line BC ; from . A to draw a st. Ime = BC. From . B the extremity of BC, draw a st. line to the point A ; on AB construct an eq. lat. A BDA.^ lengthen st. lines DB and DA indefinitely to E and F ; from centre B with rad. BC desc. the CHG; and from centre D with rad. DG desc. GLK ; then the st. line AL from A = the given st. line BC. • • . B is the centre of CGH, and . D of GLK ; .-. BG = BC, and DL = DG. But the St. line D A = DB ; and taking away these equals from the equals DL and DG, the rem. AL = the rem. BG : But BG being = BC, .". the rem. AL = BC. Where/ore, from the given point A has been drawn a St. line Ali = the given st. line BC. SoH. — ^When the given point is out of the rgiven line, or of the given line produced, this problem admits of eigJit cases, each of which is a solution of the problem ; but if the given point is in the given line, or in the given line produced, of • only/o«r cases. It is very conducive to the learner's improve- • ment, when the proposition admits of it, to vary the mode of solution : of the eight cases men- tioned, we will take another, in which the given point A is joined to C, the other extremity of the :Bt. lineBC. The same method will be pursued in the Solu- tion : join .s A and C, and on st. line AC con- iStruct an equ.il. triangle ADC; produce its sides ,-''K i>i GRADATIONS IN EUCLID. to E and F ; and with CB as radius, desc. the circle GBH, and with DO a» radius, the circle GLK ; the st. line AL will be drawn equal to CB. The .Demonstbation foUows the same course as in the first case given above. The learner may solve some of the other cases for himself. Use AMD App. — Practically the given distance BC will be taken in th& compasses, or measured by a string, or some instrument, as a foot, or a yard^ and a st. line of the req^uired length be marked off from . A, as AL. Prop. 3 — Prob. From the greater of two given st. lines to cut off a pari equal to the less. SOL.- -P. 2. line. Prom a given point to draw a st. line equal to a. given st. Pst. 3. A © may be described from any centre at any distance from that centre. Dem. — Def. 15. AU straight lines from the centre to the circumference of a, ©, are equal. Ax. 1. Magnitudes which are equal to the same magnitude, are equal to each other. Exp. 1 CONS.1 2 3 Dem. 1 2 3 4 Data. Quaes. hy P. 2. Pst. 3. Sol. hy C. 2. Def. 15. C. 1. Ax. 1. 5 I Rec. Given the st. lines AB and C, AB being the greater of the two ; it is required to cut off from AB a part = the less C. c ir lE From the point A \ '■■•■..£....■•■' draw a St. line AlD "x f(j = toC; and from centre A with radius AD desc. the DEF ; then the st. line AE cut off from AB, = C the less of the two given st. lines. V A is the centre of the FED, .'. the st. line AE = the st. line AD : But the st. line AD = the given st. line C ; consequently, the st. line AE, cut off from AB, = the St. line C. Therefore, from AB, the greater has been cut off, &0. Q.E.F. PROP. III. — BOOK I. 55' ScH. — ^A less line AD may be made equal to a greater AB, by describing a circle GBH with the radius AB, and producing AD until it meets the circle in H ; then AH will equal AB. Use and App. — This problem is performed practically, by transferring the distance C from . A on st. line AB. 2. The Problems 2 and 3 are of very frequent application, for in GeiSmetry we are continually required to draw a st. line equal to a given st. line ; or to take away from a greater line a part equal to the less ; or to lengthen the less BO that with the part produced it may equal the greater. 3. The 3rd Problem furnishes the means of constructing a Scale of Equal Parts; thus, Take a st. line AB of indefinite length towards B, and let C be the given st. line or part that is to be out off from AB ; from AB cut off a part equal to C, as AE ; then again from EB another part equal to C, as EF ; and so on ; the parts in AB are each equal to C and to one another ; and AB is a scale of equal parts ; for the radii AD, ED', FD", 6D'", being each equal to C,— AE, EE, FG, and GB are equal. A B F G B J K lu . fl ,,,, ? ■ ^1 ^1 ^1 -^ 1 \ On the same principle we take a st. line KL, and from one extremity K set off on the line ten equal parts, as in KO ; then from . set along the st. line parts each equal to KO : if the parts between K and are tenths, the parts 1, 2, 3, 4, 5 will be units; but if the parts between K and are units, then the parts numbered 1, 2, 3, 4, 5 wiU be tens, namely, 10, 20, 30, 40, 50.. By a scale of this kind the comparative lengtJis of st. lines may be readily measured. For the advantageous use of a Scale of Equal Parts, we should understand the nature of Representative Values. A miniature, of not more than a square inch in surface, is representative of the human face ; and a map, on a square of 12 inches, may be representative of a tract in the heavens that takes in distances which we can scarcely conceive. The lines in the miniature and in the map, are in due proportion to those in the face and in the expanse of heaven ; and thus they possess a representative value, — they are not the actual distances, but they stamd for them. The inch on a scale may indicate a mile, or a thousand miles of distance ; but if each portion of a mile, if each mile, or thousand miles, is given of the same relative size, then the map is a true representation ; stretch out all its parts in an equal degree, and at l&^ly superposition it would actually cover every point of the wide surface of which it is but the mark or outline. 56 GRADATIONS IN EUCLID. By such a use of the Scale of Equal Parts, and of the Semicircle, the Scale for Ajigular Magnitude, we can construct figures that are a true index of the positions and real distances of eitiss, mountains, and seas, and in some respects of the constellations of heaven. For instance, if by actual observation and measurement it is ascertained that there are tiro towns each distant thirty- five miles from a third, aad that the two, in reference to the third, are at an angle of 21° apart, a plan may easily be drawn which shall correctly shew their situation with respect to m each other. Prom L draw a st. ■" line LM of 35 from a scale of equal parts ; at . L with a semi- circle, or protractor, make an angle of 21°; and along LT from the same scale set another 35 : the points L M T will represent the situations and distances of the three towns. Prop. 4. — Thbob. — (Important.) If two triangles have two sides of the one equal to two sides of the other, each to each, and have likevnse the angles contained hy those sides equal to each other, they shall likewise have their bases, or third sides, equal ; and the two triangles shall he equal, and their angles shall he equal, each to each, viz., those to which the equal sides are opposite. Dem. — Ax. 10. Two straight lines cannot enclose a space. Ax. 8. Magnitudes which coincide with one another are equal. Exp. 1 2 4 d Dbm. 1 2 3 by Hyp. 1. Concl. 1. „ 2. » 3. hy Superp. D. 1, H. 1. Ax. 8. In the As ABC, DEF, let side AB = side DE, and side AC = side DF; Also let the included L BAG = the inclu- ded L EDF; then the base BC = the base EF; and A ABC = ADEF; also I. ABC = L DEF; and L ACB = l DFE. For, applying A ABC to A DEF, so that. is on . D, and st. line AB on DE; •.• AB coincides with and is equal to DE ; .•. the . B shall coincide with the . E : PROP. IV. — BOOK I. 57 Again, •.* AB coincides with DE, and^ BAC= Z.EDF, .". the St. line AC shall fall on the st. lino DF: But AC being = DF, the . shall fall on the . F, and . B falling on ^ E,. and . C on . F, the St. line BC, falls on the st. line EF ; For if, though . B falls on . E, and . C on . F, base BC does not fall on base EF, then two St. lines will enclose a space ; But this is impossible ; .".the base BC does coincide with and = the base EF : And AB falling on and being equal to DE; ACtoDF;, andBCtoEF, .'. A ABC coincides with and = A DEF. Also, since side DE coincides with side AB, and side EF with BC, the L ABC shall coincide with and equal L. DEF. And in a similar way L ACB = L DFE. Wherefore,, if. two triangles have two sides, ' &C. Q.E.D. SOH. — 1. This being, the firat Theorem in. the Elemettts, it is exclusively proved, by means of the Axioros^ 2. The converse of the 8th Axiom iB assumed' ; naxaelj,tha,t if magnitudca are equal, not merely if they sixe equivalent, tJiey will also coincide. 3. The equality spoken of in this Proposition, is equality of the sides and of yie angles. Triangles may be equal in a/rea, though the sides and angles of the one are not equal to the sides and angles of the other. When the sides and angles mutually coincide, the triangles are named eqxMl tri- angles ; when their aresa only are equal, such triangles are called egydvaUnt triangles. 4. Some have taken the 4th Proposition for an axiom. We perceive its truth indeed, almost without demonstration ; but the number of axioms in any science should not be needlessly increased ; and as this proposition can be naturally established by means of the received axioms of Geometry, it holds its proper place when classed with truths to be demonstrated. It may be more briefly enunciated, thus — " If two triangles have each two tides and their ivxluded angle eoual, the triangles are eaual in every resvect." Dem. 4 D.2,3, H.2. 5 Conol. 6 H. 1 & Con. 7 D. 3, 6. 8 Con. sup. 9 10 Ax. 10. Ax. 8. 11 D. 2, 5, 6, 10 12 13 Ax. 8. Hyp.l,D.10 U Ax. 8, Conol. 15 16 Hyp. 1, D. 7. Hecap. 58 GRADATIONS IN EUCLID. Use Aim App. — This Proposition contains the first of the critena by which to infer the equality of triangles, and is applied to various uses ; as — • 1st. Very frequently in all parts of Geometry to eatablUh tlue eamUty of t riangles. 2nd. To ascertain an maccessihle diadmce, as AB, the breadth of a lake. With an instrument for mea- suring angles, as the Theodolite, p. ii, take the angle at C formed j^Jj by the st. lines A C, B C, from the jg^itegg-- ^^- y3 E^ extremities A and B of the inac- cessible distance ; and with a chain or other measure of length, find the distances CA and CB. The Representative Values of these measurements must now be taken p JJ from a Scale of Equal Parts, p. 55, and drawn on paper, or on any plane surface : thus, draw a st. line DF of an indefinite length, and at . D, by aid of the graduated semicircle, form an angle FDE equal to the angle BCA : from a scale of equal parts the distance from A to C is represented in proper proportion by the line DE, and the distance from B to C by DF ; consequently, on a principle established in the Sixth Book, Prop, i, and which we now assume aa a Lemma, — tfuit the sides ahout simila/r triangles are proportional, — the st. line EF will represent in the due proportion the distance from A to B ; and if to the same scale we apply the st. line EF, that distance on the scale will be the re'preaesniatiaie measurement of the actual distance AB. N.B. — If the ground near the lake was level enough to admit of setting out the triangle DEF in the actual measurements of CA and CB, we should have EF of the very length of AB, on the principle that two triangles having two sides and their included angle in each equal, have their third sides equal ; and if we measure one, as EF, we ascertain the other, AB ; but as it is seldom we find the actual surface of a country smooth enough for our purpose, we use the method of Hepretentative Vahies, or of Geometrical Construction. Prop. 5. — Theor. TJte av^les at the hose of an isosceles triangle are equal to eath other ; and if the equal sides he prodiuxd, the angles on the other side of the hose shall be equal. Cons. — Pst. 2. A terminated st. line may be produced to any length in a st. line. P. 3. From the greater st. line to cut oflf a part equal to the less. Pat. 1. A St. line may be drawn from any one point to any other point. PilOP. V. — BOOK I. 59 Dbm. — p. i. If two triangles hav» two sides and their included angle of one triangle equal to two sides and their included euigle of anotbcr triangle, the two triangles axo equed in every respect. Ax. 3. If equals are taken from equals, the remainders are equal. Exp. 1 3 4 CONS.1 2 Dem. 1 2 iy Hyp. 1. H. 2 & Pst; 2. Concl. 1. .. 2. by P. 3. Pat. 1. 6yC.l&Hyp.l, P. 4. D 2. Let fig>. ABC be an isoso. A, liaving the sides AB and AC equal ; and let the equal sides be produced inde- finitely to .8 D and E ; then the L s ABC, ACB, at the base are equal; and the L s DBC, ECB, on the other sid? of the base are equal. On BD take any . F, and make AG = AF; join the .s F and C, G and B, by the st. lines FC and GB. •_• AF = AG, AG = AB, and z. A is common, .'•.the base FC = base GB, and A AFC = A AGB; also L ACF = L. ABG, and l AFC = z. AGB. 60 GEADATIONS IN EUCLID. A Dem. 4 10 11 12 13 14 Cl&Hyp.l Sub. Ax. 3. D. 5 & 2. Again v the whole A F = whole A G, and part A B = part A C, on taking away the equals A B and A C, the rem. B F = the rem. C G. Butin AsBCF, BCG, BF = CG, andFC = GB, and/.BFCorAFC=^/.CGBorAGB; .-.ABFC = ACGB, z,FBC = lGCB, and^BGF= ^CBG, * Now, the whole ^ A B G = whole i, A C F, and the part z. C B G = the part z. B C F ; .■. on taking away 4 s C B G and B C F, the rem. /. A B C =; the rem. z. A C B, and these are angles at the base. And ^ F B C was proved to be equal to L GOB, and these are the angles below, or on the other side of the beise. Wherefore, the angles at the base. <&c. Q.B.D. ScH. — To assist the learner, the o^inal figure 1 is separated into its parts, 2, 2, and 3, 3 ; and the equality of the triangles proved by Prop. 4. D. 3 & 8. Sub. & Ax. 3, RemL D. 8. Remk. Recap. PROP. VI. BOOK I. CI Cob. — Every equilateral triangle is also equiangular. Exp. 1 2 Dem. 1 2 3 4 % Hyp. Conol. hy Def. 24, P 5. Hyp. P. 5. Ax. 1. ConcL Recap. Let ABC be an equil. A sideAB = BC-=ACj then shall its angles be equal, ^ Ato /L B to -i C. Since side AB = side AC, .-. ^B=/.G, B and since side CA = side CB, .•. ^A= /.Bj consequently, z. C = Lk. Hence, the angles are all equal, L A to B, z. B to C, and iL C to A. Wherefore, if a l\he equilateral, die. Q.B.D, Prop. 6.--THfeoll. If two angles of a triangle he equal to one another^ the sides also which subtend, or are opposite to the equal angles, shall he equal to one another. GONS. — P. 3. From the greater st. line to cut oflF a part equal to the lesa. iPst. 1. A Bt. line may be drawn from one point to another. DeMv — P. 4. If two triangles have two eides and the included angle equal in each) the triangles a^e in all respects equal. Exp. Suv. 1 CONS> Dem. 1 2 3 6y Hyp. Concl. 6yP.3, and join DC. \'DB is made = AC, and BC is common, and V I. DBC= ^ACB; .'. base DC = baaeAB, and ADBC = A A.BC. 62 Dem. 4 5 6 GRADATIONS IN EUCLID. ex dbs. Concl. Kecap. Thus the less is declared equal to the greater, which is absurd ; .'. AB is not =1= AC— that is, AB = AC. Wherefore, if two angles of a triangle, &c. Q.BI.D. Cor. — Every equiangular triangle shall also he equilateral. Exp. 1 2 Dem. 1 2 3 4 by Hyp. Concl. 6yH.&P.6, H.&P.6 Ax. 1. Concl. Becap. Let A ABC have the L s equal, A to B, B to C, and C to A ; then the sides are all equal, AB to BC, EC to CA, and CA to AB. •: lB^ lC, .', side AC = side AB ; and '; lB= L A., .'. side AC = side BC ; and .-. AB = BC. Hence, the side AB = BC, side BC = CA, and side CA = AB. Wherefore, every equiangular triangle, &o. Q.E.D. SoH.^1. Thia proposition is named the converse of the 5th. OeomeiricaX conversion takes place when the hypothesis of the former proposition is made the predicate of the latter, and vice versd, as in Props. 5 and 6, 18 and 19, 24 and 25, of this book. 2. Converse theorems are not universally true ; for instance, the following direct proposition is universally true, — " If two triangles have their three sides respectively equal, the three angles of each shall lie respectively equal ;" but the converse is not universally true, namely, — " // two triangles h/i/ve the three angles in each respectively equal, the three sides are respectively equal." — Pott^ Euclid, p. 48. To the equality of triangles it is always indispensable that one side at least of the one triangle should be given equal to one side of the other triangle. 3. In Geometry there are two modes of Demonstration : the dmvct, showing why a thiiig is so ; and the indirect, proving that it must be so ; the former being the usual method. Direct Demonstration, as in Prop. 5, is that in which we find intermediate steps which proceed regularly to prove the truth of the proposition : the Indirect Method is only employed, as in Prop. 6, when the predicate of it admits of an alternative, and one of them niMst be true, because they exhaust every case that can possibly exist. We prove that the alternative cannot be true, and infer, therefore, the predicate must be true. With respect to equality between magnitudes, there are two alternatives — equal or unequal; and if we prove that inequality cannot or does not exist, of necessity equality must exist. PROP. Vn. — BOOK I. Use and App. — Hieronymus, the historian, records that Thales of Miletus, who was living 546 b.c., measured the height of the pyramids of Egypt, fry vbservinfftjie shadows which they cast when the shadows were as long as the pyramids were high. This would be the case when the altitude of a pyramid, . of any other object, and the perpendicular length of the shadow, were (lal. The height of an object and the length of its shadow are the same, when the light S, which the object AB intercepts, is at an elevation of 45° : this condition being observed, the shadow BC is equal to the height BA; because the angles BCA, BAG, being each half a rt. angle, the sides which subtend them are equal. Thus, by measuring the shadow CB, we obtain the height BA. The height would also be obtained by making an observation with a quadrant of altitude: thus — walk away in a line perpendicillar to the altitude of the object, until, at the station C, the quadrant Qi shows A to have an elevation of 45°: the distance gone over from B to C, or BC, will equal BA, the altitude. Peop. 7..— Theor. Upon ike same hose and upon the same side of it there cannot he two triangles t/iat have their sides, which are terminated in one extremity of the base, equal to one another, and likewise those eqiud which are terminated in the other extremity. Cons. — Pst. 1. A st. line may be drawn from one point to another. Fst. 2. A St. line may be produced to any length in a st. line. Dbu. — P. 5. The angles at the base of an isosceles triangle are equal, and if the equal sides be produced, the angles upon the other side of the base are equal. Ax. 9. The whole is greater than its part. Exp. 1 6yHyp.l. Concl. StJP. On base AB let there be two As, ACB, ADB; and let the side C A = the side DA: then it is impossible that side CB should equal side DB. If possible, let side CA = DA, and side CB = DB. G4 GRADATIONS IN EUCLID. Case I. — Let the vertices C and D he wUhotit each other. Cons. 1 1 hy Pst. 1. Dem. 1 i6yHyp.&P.5 i 2 i C. & Ax. 9 3 j hfort. 4 i H. & P. 5. 5 ! D. 3. 6 j D. 3, 4. Join .8 C and D. V AC = ADin AACD, .-. Z.ACD = Z.ADC: But ^ ACD > L BCD, .-. :L ADC>z.BCD: and mucli more is L BDC>z.BCD. Again, •.• side BC = BD in A BCD, .-. z.BDC= ^BCD; but L BDC is also > L BCD ; .•. L BDC is both > and = ^ BCD, wMoh is impossible. Case II. — Let the vertex J) of ^ ADB, he within the A ACB. Cons. 1 Dem. 6y Pst. 142 6yHy.l&P.5 C.&Ax. 9, d, fort. H. & P. 5. D. 3. D. 4, 3. Join C to D, and produce AC, AD to E and P. • • AC = AD in A ACD, :.L ECD= l¥T>C. But L ECD is greater than L BCD; .'. L FDC is greater than ^ ^BCD; and much more is z. BDC > /L BCD. Again, •.• BD = BC in A BDC, .-. ^BDC= ^BCD. But ^ BDC is also > l BCD ; .-. ^BDC is both > and = .iBCD, which is impossible. Case III. — When the vertex D is on the side BC, no demonstration is required. 7 I Eecap. I Therefore, upon, the same I I hase, &c. Q.E.D. ScH. — The argument made use of in this proposition, is the Dilemma, or double Antecedent, in which the truth of the one is impossible, if we admit the truth of the other. The argument called the dilemma may, however, have PROP. VIII. — BOOK I. Co more than two antecedents ; and Whately, p. 72, defines the true dilemma as " a concHUonal Syllogism, with several antecedents in the major, amd a disjunctive in the minor." Use. — The only purpose for which this proposition is employed, is to prove Prop. 8. Prop. 8. — Theor. — (Importanl.) If two triangles have two sides of the one equal to two sides of the otlier, each to each, and have likewise their bases equal, the angle which is contained by the two sides of the one, shall be equal to the angle contained by the two sides eqxud io them of the other. Dem. — Pr. 7. On the same side of thft same hase there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those which are terminated in the other extremity. Ax. 8. Magnitudes which coincide are equal to one another. i:xF. 1 Dem. 1 2 3 by Hyp. Conol. bySuperp. Hyp- D.2,Hyp. 4 Sup. 1. i „ 2. Let the As ABC, DEF, have side AB = DE, side AC = DF, and also base BC = EF ; then ^BAC shall equal iL.EDF. Apply AABC to ADEF, with . B on . E, and St. liueBConEFj •.■ BC = EF, .•. . C coincides with . F. Wherefore, \' BC coincides with EF, BA and CA shall coincide with ED, FD. For, suppose that base B C coinoid eswithbaseEF, but sides BA and CA not with sides ED, FD, but with other st. lines EG, FG, 66 GRADATIONS IN EUCLID. Dem. 5 Concl. 6 7 P. 7. D. 2, 3. 8 D. 7. 9 10 Ax 8. Recap. then on this sup,, in As EDF, EGF, on the same side of EF, side ED shall = EG, and also side FD = FG j which is impossible. .'. since base BO coincides with base EF, the sides BA, CA, coincide with sides ED, FD ; Wherefore, z. BAG must coincide with L. EDF; and .-. ^BACis= 4. EDF, Therefore, if two triangles have two sides, &o. Q E.D. ScH. — The Equality established is that of the angles, — ^but the sides being equal, the triangles also must be equal. This is the second criterion of the equality of triangles. Use. — 1. By the aid of this proposition, and of Prop. 22, the amgle at a given point C, made by st. lines from two objects, as A and B, may be determined without a tkeodiolite. Measure the distances AB, BC, CA, — and from a scale of equal parts construct a trian. gle D E F, the sides of which, DE, EF, and FD, will be re- presentative of the distances AB, BC, and CA : then with the semicircle find the number of degrees in angle F ; and as the triangles ACB, DEF, are similar, that number of de» grees will also be the measure of angle C. 2. When the instruments for angular magnitude cannot be employed, by reason of the inequalities of surface, or the difficulty of placing the instruments, this proposition is useful for measuring and cutting angles in a solid body, as in a block of stone, or for bevelling, i.e., for giving th# desired shape to the PEOP. IX. — BOOK I. 67 angular edges of timber, &o. For instance, a groove of the same triangular shape and size wiUi the triangle ABC, is to be cut in a block of marble D E F Q H. At fKe point in the edge DG of the block, where the groove is to commence, set off a Une ac equal to AC ; and on the plane surface DEFG, with ac for one side, con- struct a triangle aic, with sides equal to the sides in triangle ABC; ahc will be the end of the groove, and if the guidance of a 5 c be followed, the whole groove when finished will be of the same angular magnitude with ABC. On the same principles, a beam of ^ A ^ B .fc>f HI K- ,w M '< timber, the end of which is represented by fig, KLMN', may be bevelled so that the bevelled edge shall be of the same angle with a given angle NPM ; for, on the end of the beam draw a triangle, the sides of which shall equal those of the triangle NMP ; then, by Prop. 8, the bevelled edge NQM is equal to the given angle TSFUL Prop. 9, — Pbob, To bisect a given rectilineal angle, that is, to divide it into two equal parts. Sol. — P. 3. From the greater st. line to out off a part ec[ual to the less. P. 1. On a St. line to draw an equil. triangle. Pst. 1. A line may be drawn from one point to another, Dem. — P. 8. If two triangles have two sides of the one equal to two sides of the other, each to each, and have Ukewise their bases equal, the angle which is contained by the two sides of the one, shall be equal to the angle contained by the two sides equal to them of the other. Exp. 1 2 CONS.I 2 3 Datum. Qtcees. hy Assum. P.3&Pst.l P.l^Pst,!. Sol. Let the given l be BAG; it is required to bisect it. Tate any point D in AB; on AC make AE = AD, and join DE, and on side I)E,renlote from 4 A, construct an equil. A, DFE and join AF ; then L BAF shall = l CAP, the z. B AC being bisected by St. line AF. G8 GRADATIONS IN EUCLID. Dem. 1 2 3 62/C.2&P.I P. 8. Recap. •.• in As DAF and EAF, AD = AE, DF = EF, and AF common, .-. L DAF is = z. EAF. Wherefore, the L BAG is bisected hy the st. line AF. Q.E.r. ScH. — 1. The bisection of the arc which measures an angle is also effected by the bisection of the angle. The arc DGf, on fig. to Prop. 9, ia the measure of the i BAG, or DAE ; the .; DAG is one half, and BAG the other half, of z DAE; and halves of the same being equal, the arc D G is equal to the arc GE. 2. An isosceles triangle would serve equally well for the solution and demonstration. 3. By successive hisectiom an angle may be divided into any number of equal parts, indicated by a power of two, as into four, eight, sixteen, thirty.two, &c., equal parts. 4. Hitherto no method has been discovered of geometrically trisecting an angle, so that the division of the quadrant of 90° into single degrees is in part effected mechanically : by simple bisection, we divide 90° into two 45° ; by setting off a semi-diameter from one extremity of the quadrant, we out off an arc of 60°; 60° bisected gives 30°; and 30° bisected gives 15°: but for the division of the 15° we require to have the means of trisecting an angle, which means Geometry does not supply. But having mechanically divided an arc of 1S° into three 5°, and from one arc of 5° set off an arc of 3", the simple bisec- tion of the remaining arc of 2° gives an arc of 1° — a unit in the measure of the cii'cumferenoe. Of course this is only a practical, not a theoretical proof. If, however, instead of taking an arbitrary quantity, 360, as the measure of the equal parts in the circumference of a circle, those who first made such division had followed the strictly geometrical process of this 9th Proposition, they would have arrived at a unit for the degrees in a given circumference with as much absolute certainty as they do how at the unit for a scale of equal parts. By making use of the powers of 2, and by their aid dividing the circle, the unit of the division is demonstrahly accurate. Suppose the number of equal parts into which the circle had been divided, had been repre- sented by the 9th power of 2, or 512, the bisection would have given 256 for the semicircle ; 128 for the quadrant ; and 64 for the octant : and 64 by successive bisection, would have given 32, l6, 8, 4, 2, and 1 equal parts. Thus, every step in the division would have been strictly in accordance with geome- trical verities. Again — the unit of such degrees represented by 64, equal parts, would in the same way have been divisible into 32, 16, 8, i, 2, and 1 minutes, and so on, to whatever extent of minuteness we might wish to carry our bisections. Probably no fact in geometrical measurements more clearly shows the unscientific nature of the early geometry, than the division of the circle into 360 equal parts. It is now, however, too late to attempt aja alteration on purely geometrical grounds ; and, fortunately, there is no real inconvenience or inaccuracy in the received method ; for an arc of the 360th part of a circle is in practice as readily obtained as the arc of the 512th part of the same circle would be. We have only to bear in mind that Plane Geometry does not supply the means for any division of a circle, except by the method of bisections, i. «., by using in regular series the powers of 2. PEOP. IX. — BOOK I. 69 Use akd Appiication.— 1. Practically the angle BAG, in tbe figure to Prop. 9, p. 67, would be bisected by drawing the arc D GE, and with any radius from D and B drawing arcs intersecting in . F ; A P is the bisecting line. 2. By Prop. 9, we show that the angles at, the haw of an isosceles triangle are eqitat ; for bisecting ^ ACB by CD, CA is equal to CB, CD common, and z ACD equal to /; BCD ; therefore, by P. 4 L CAD equals .; CBD : 3. Also that the line vMch bisects the vertical amgle of an isosceles triangle, bisects the haseperpen- dicndwrly; for AC equalling B C, D C being common, and z AC D, by construction equalling zBCD, by Prop. 4, AD equals DB; and A ADC equals A BDC, and ^ ADC equals z BDC : conse- quently, by Def. 1,0, st. line DC is perpendicular to AB. 4. The Manner^! Compass is divided into its 32 parts or points by Props. 8 and 10. The bisection of the diameter by another diameter at rt. angles, gives the cardinal points N., E., S., W. ; the quadrants, being bisected, give the points N.E., S.E., S.W., N.W. ; and these bisections are continued until the number of equal divisions of the circle amounts to 32 — ^the arc at each pair of points enclosing an angle of 1 1° 15'. This instrument is used for taking the bearings or directions of places from some central station of observation ; and the correctness of the method depends on the physical law that a magnetised 'U GBABATIONS OP EUCLID. needle or in4ex, working freely on a, pivot, does, aIlo?ring for deviation, from the same place, point in the same direction. To be quite accurate in making the observation, it is requisite to know the amount of deviation from the true North, at any given place. For Great Britain, the deviation amounts to about 24° west of north ; so that when the magnetised steel index, or needle, points 24° west, the pointer on the compass card marked N. indicates the true North. Prop. 10. — Peob, To bisect a given finite straight line. Sol. — P. 1. On a given st. line to construct an equilateral triangle. F. 9. To bisect a given rectilineal angle. Dem. — P. 4. Two triangles are equal in every respect when two sides and the included angle of one are equal to two sides and the included angle of the other triangl?. Exp. CONS.I 2 3 Dem. 1 Datum. Qiices. %P. 1 P. 9 Sol. 6yC.l(fc2. P. 4. Eecap. Given the st. line AB ; it is required to bisect it. On AB make ati equil. A ABC; let St. line CD make z.ACD= ^BCD; then AD = BD, i.e, AB is bisected in D- '.• in As ACD, BCD, AC = BC, CD is common, and /L ACD = I.BCD; .*. base AD = base DB. Where/ore, AB is bisected in D, Q.E.P. SoH. — By successive bisections, a line may thus be divided into any number of equal parts indicated by a power of 2 ; aa 4, 8, 16, 32, 64, &c. Use. — 1. In practice, the st. line AB would be divided into equal parts by drawing with equal radii arcs intersecting in . C and . E ; the st. line CE will bisect the st. line AB. 2. Also, if the length of the line AB be ascertained by means of a scale of equal parts, the division into any required number of equal parts, as 2, PBOP. XI. — BOOK I. 71 8, i, &c., will be effected by dividing the numerical value by 2, 3, i, ftc, as the case may be. 3. But for a ready and infallible method of bisecting a line, Prop. 10 cannot be dispensed vrith. Prop. 11. — Peob. To draw a si. line at right angles to a given st. line from a given point in the same. Sol. — P. 3. From the greater of two st. lines to cut off a part equal to the lees. P. 1. On a given st. line to construoii an equilateral triangle. Pst. 1. Two points may be joined by a line. Dbm. — P. 8. In two triangles, if two sides and the base, of one triangle be equal to the two sides and the base of andther triangle, the angle between the two equal sides of the one is equal to the angle between the two equal sides of the other. Def. 10. When a st. line on another st. line makes the adjacent angles equal, each of the angles is a right angle. Az. 1. Things equal to the same thing are equal to one another. Exp. 1 CONS.1 Bt-M. 1 2 3 4 5 Data. Quaes. hy P. 3. P.l&Pst.l Sol. 62^0. 1&2. P. 8. C. 2. Def. 10. Recap. Given the st. lineAB,aiid C a point in it J to draw from . C a st. line at ± AB. In AC take „ J A D C E B any . D, and makeCE = CD; on DE construct the eqnil. A EDE, and join EC. then /. sDCE and ECE are equal, and conse- quently rt. angles. •• DC = EC, EC common to the As DCE, ECE, andDF = EF, .-. z.DCF= L ECE: And by construction they are adjacent angles; .•. the /L s DCE and ECE are rt. angles. Wherefore, from the given point C, &c. Q.e.f. 72 GBADATIOKS IN EUCLID. Cor. — Herux it may he seffment. shoum that two St. lines cannot have a common Exp. 1 CONS.l Dem. 1 2 3 If possible, let the segment AB be common to the st. lines ABC and ABD. E by P. 11. 6y Hyp. Def. 10. Hyp. Def. 10. D.2,4,Ax, Recap. C At . B draw BE, making the L. s with AB A rt. angles. " ■.• St. lines AB and BC make one st. line ABC and L ABE is a rt. angle ; .\ ^ ABE = Z.EBC: Again, •/ st. lines AB and ED form one st. line ABD, and L ABE is a rt. angle ; .-. L ABE = L EBD :. Wherefore, l EBD = L EBC; which is impossible. Therefore, two st. lines cannot have a common, segment. Q.E.D. SCH. — 1. This corollary Bhould be ranked among tte axioms ; for it is assumed in Prop. 4, where it is taken for granted that if certain st. lines, placed on one another, coincide for any portion of their length, they must coincide throughout. It is also assumed in Prop. 8. 2. When the given point is at the extreijiity of a given st. line, the line from that extremity should be produced,, and the rt. angle be then con- Btructed ; 3. Or the following method may bo . r adopted :-^Let AB be the giveast. line,. ..■-■'•'' — --. and . B the extremity from which the perpendicular is to be raised. Take any point Q above the line AB, and with radius 6B describe a circle cut- ting AB in . H ; join 6H and produce it to M ; or from . H set the same radius GB on HK, KL, and LM ; M is the point from which if a st. line be drawn A - to B, that st. line MB will be perpeA- dicular to AB. The radius equals ths ~- ;.,-'^ chord of C0° ; three times 60° equal (^ 180° ; and 180° is the semicircle ; and by P. 31, bk. iii., the anjle HBM in a temicircle i^ art. a/nglt. ,/ K/ ,.■■:■' T Cr.. \ y ■•-,, » ,' \ y ^ ■'" *--. H\ (M B PROP. XII. — ^BOOK I. 75 Use and App. — 1. By this proposition the Square is constructed, an iustni- ment employed for ascertaining the perpendicular to an horizontal line, and for all purposes for ^hich right angles ore needed. 2. 0» a given st. line AB, to describe an iioteeles triangle of which the per- pendicular height CD, is equal to the hase AB. Com. 1 2 3 Dem. 1 2 3 JyP. 10,11,&3 Pst. 1. Sol. ly C. 1. P. 4. C. 1. Bisect AB iu.D, and make st. line DC perpen. and equal to AB ; join AC andBC; the figure ABC is the isopc. A required. V in AS ADO, BDC, AD=DB, DC is common, and i A.IiC= i BDC; .-. AC=BC. And the perp. DC has heen made equal to AB. Q.S.V. Prop. 12. — Pros. To draw a perpendicular to a given st. from a given point without it. line of unlimited length Sol. — Pst. S. A circle may be drawn from any centre at any distance from that centre. P. 10. To bisect a given st. line. Pst. 1. A line may be drawn from one point to another. Dem. — Def. 15. The radii of the same circle are all equal. P. 8. If two triangles have two sides in one eqiwl to two sides in the other, and the base equal to the base, the angles contained by the two pair of equal sides are equal. Def. 10. A St. line at rt. angles to another st. line, is perpendicular to it. 1* GRADATIONS IN EUC3LID. Ekp. CONS.I 2 3 4 Dbm. 1 X/ H A E" \ ..E Data. QlKBS. hy Aasum. Pst. 3. P.10,Pst.l. SoL 6yC.3,Defl6 Givea the st. line AB, and the . C out of it j required from . C a perpendicular to AB. Take any . D on the other side of AB j from.C with rad. CD draw the arc EGF; cutting AB in . F and . G ; bisect FG in H, and join C and H, C and F^ and C and G ; then the st. line CH is perpendicular to AB. • in As FHC, GHC, FH = HG, OF = CG, and HC common, 2 P. 8. .-. z.CHF = z.CHG: 3 C. Remk. Now these are adjacent angles ; 4 Def 10. .'. the st. line CH is perpendicular to AB. 5 Recap. Therefore, from the given point, &c. Q.B.r. ScH. — 1. The properties of the circle form the subject of the third book, but in the constructiou of the 12th Prop., the Lemma is borrowed from Prop. 2.^ bk. iii, that the circle will intersect the line in two points. 2. If the given point C is over, or nearly over, the extremity of the given st. Kne AB; — In AB take any point 0, and with rad. OC describe the arc CLM ; and from N another point in AB, with rad. NC, the arc CBM : join the points of intersec- tion C and M, CD is the perpendicular xequired. 7M PROP. XIII. — BOOK I. 75 In the triimglea NOO, NMO, — tlie three sides of the one are equal to the three sides of the other, and by P. 8, the angle CN equals the angle M NO : thus in the triangles NDC, NMD, two sides and the included angle of one are equal to two sides and the included angle of the other : — therefore, by P. 4, ang. ADC equals ang. ADM, and they are adjacent angles ; hence St. line CD is perpendicular to st. line AB. Use add Afp. — I. In practice, the problem wiU be solved by drawing, as in the figure to P. 12, the arc PD E, and' from the points P and &, with equal radii, describing arcs intersecting in . K ; by joining C K, the perpendicular to AB will be d!rawn. 2. This problem is indispensable to all Artificers, Surveyors, and "aeers. i'BOP. 13. — Theoe. Tvse angles which one st. line makes with another upon one side of it are either right angles, or together equal to two right angles. CoMS. —P. 11. To draw a st. line at right angles to a given st. liiie from a given point in the same. - Deh. — Def. 10. When a st. line standing, on another st. line makes the adjacent angles equal, each of the angles is a rt. angle. Ax. 8. Magnitudes which exactly fill the same space are equal. Ax. 2. If equals be added to equals, the sums will be equal. Ax. 1. Magnitudes which are equal to the same, are equal to each other. Exp. 1 ! &y Hyp. Let the st. line AB make angles with the St. line DC, Concl. 1. then the /. s C B A, ABD, are two rt. angles, j, or are together = two rt. angles. Case 1,— -Suppose that the ang. CBA is equal to ang. ABD, as in the 1st fig. ; i Def. 10. I then each of the angles I I is a rt. angle. 13 _ B 76 GRADATIONS IN EUCLID. Case II. — Bid suppose that the angle CBA is not equal to angle ABD, as in the 2ndjigure below ; Cons. 1 2 Dem. 1 by P. 11. Def. 10. bi/Ax.8,Add Ax. 2. 3 Ax. 8, Add. i Ax. 2. D. 2. Ax. 1. C. Ax. 1. Recap. At . B in DC draw BE at rt. angles to CD, then Z.S CBE, EBD are tworfc. angles. z.CBE= Z.CBA+ Z.ABE, to each equal add .lEBD, /.s CBE, EBD, = z.sCBA,ABE, and EBD. Again, •/ l DBA = n. B Z.S DBE & EBA, to both equals add aABC; ;, LB DBA + ABC = AS DBE, EBA and ABC. But /.sCBE, EBD = AS DBE, EBA, and ABC ; .-. Z.S CBE, EBD= AS DBA, ABO; Now AS CBE, EBD are two rt. angles, .". AS DBA, ABC together = two rt. angles. Wherefore, the angles which one line, &c. Q.E.D. ScH, — 1. A rt. angle FBG is formed by bisecting the angles ABD, ABC. 2. If one angle be a, rt. angle, the other is a rt. angle ; if on€ be obtuse, the ether is acute ; and if one be acute, the other is obtuse. 3. A semicircle is the measure of two right amgles ; and all the angles formed by any number of lines converging to one point, on one side of another line, are together equal to two right angles. 4 The awpplement to an angle is what it is deficient of too rt. angles ; thus, z ABD is the supplement of angle ABC : the complement, what is wamting to make up one rt. angle ; as, z ABE is the complement of ang. ABC. Use and App. — This theorem is of frequent use in Trigonometiy and Asironomy. When we know one of the angles which a st. line meeting another St. line makes, at a point in it, we in fact know the other, for the two amglet are atmays egual to 180° ; and if we subtract the given arc from 180° we have the other angle : thus, let ang. ABC equal 70', ^ ABD equals 180—70, or 110. PHOP. XIV. — BOOK I. 77 Prop. 14. — ^Theoe. If ait a point in a st. line, two other lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two lines shall he in one and the same line. Cons. — Pat. 2. A straiglit line may be produced to any length in a st. line. Dbm. — P. 13. The angles made by one st. line with another on the same side of it, are equal to two rt. angles. Ax. 1. Tlungs equal to the same thing are equal to one another, p Ay . 3. If equals be taken from equals, the remainders are equaU Exp. 1 Hyp. Cons. 1 2 Dem. 1 2 3 4 5 6 7 8 9 Concl. hySupp Pst. 2. 6yHyp. P. 13. Hyp. Ax. 1. SiO). Ax. 3. ex ahs. Concl. Sim. At . B in St. line AB, let two st. lines BC and BD make the adja- cent Z.S ABC, ABD = two rt. angles ; then BC and BD wiU form one and the same St. line CD. "5 10 ConcL 11 'Hecap. Should BD not be a production of the st. line CB, make BE a continuation of st. line CB. AB with the st. line CBE makes the ^s ABC, ABE, these adjacent z.sABC, ABE = two rt. angles ; But L s ABC and ABD also = two rt. angles, Z.S ABCandABE= as ABC and ABD; taking away the common l ABC, then the rem. l ABE = therem. z. ABD, and the less l ABE = the greater /. ABD, which is impossible ; BE is not in the same st. line with CB. In the same way no line except BD is a loontinua- tion of the st. line CB ; .'. BD is in one and the same st. line with CB. Wherefore, if at a point in a line, &c, q.b.d 78 GRADATIONS IN BUCLTD. SoH. — 1. The worda "upon the opposite sides of it" are of essentisS importamoe ; for two lines may make with a third, two angles equal to two rt. angles, and yet the two lines not be in one st. line. 2. The fourteenth proposition is the converse of the thirteenth. Prop. 15. — Theor. If two St. lines cut one another, the opposite or vertical angles shall be equal. Deu. — P. 13. The angles which one st. line makes with another st. line upon one side of it, are either two rt. angles, or equal to two rt. angles. Az. 1. Things equal to the same are equal to each other. Ax. 3. If equals be taken from equals, the remainders are equal. Exp. 1 iHyp. Let the st. lines AB, CD intersect in the point Ej A 2 Concl. then z. AEC = l ^^^"-v^ DEB, ^ and L CEB= Z.AED. Dem. 1 by Hyp. ".• st. line AE makes with CD at the . E the ^sCEAand AED, 2 P. 13. .'. these cs CEA and AED = two rt. angles. 3 Hyp. And '.• St. line DE makes with AB at . E the AsAED, DEB; 4 P. 13. .-. these (_8 AED and DEB = two jrt. angles; 5 Ax. 1. .-. also L s CEA, AED = ^ s AED, DEB : 6 Siib. Take away the common angle AED, 7 Ax. 3. and rem. /.CEA wiU = rem. L DEB. 8 Sim. In like manner ang. CEB is .proved equal to j ang. AED. 9 Recap. Therefore, if two st. lines cut one wnother, &c. I Q.B.D. CoE. 1. — The angles formed by two st. lines, AB and CD, cutting each other in one point E, are together equal to four right angles. Dem. 16yP. 13. V ^s AEC and CEB = two rt. angles, 2 Pr. 13. and ^ s AED and DEB also = two rt. angles ; Z Add. by adding the equals to the equals, 4 As. 2. .-. the l s AEC, CEB, AED, and DEB = four rt. angles. PROP. XV. — BOOK I. 79 Cor. 2. — And all the anffles formed iy any number of st. lines, AC, BC, DC, EC, FC, <&c., div&rging from a common centre C, oire together eqiwd to four rt. angles. Cons. 1 Dem. 1 2 3 4 hyVsi. 1. 6yCorl,15 P. 13. P. 13. Conol. Produce any two of the given st. lines, as AC to G, BC toH. A •.• st. lines AG and B H intersect in the point C, .-. Z.S ACB, BCG, ACH, and HCG = four rt. Z.S. But Ls ACB, BCD, DCG = ls ACB and BCG; also ^s ACF, FCE, and ECG = z.s ACH and HCG; .'. all the angles diverging from . C are together equal to four rt. angles. Scs. — 1. This proposition might be briefly proved by saying that the ifppodte wnghs are equaZ, because they ha/ve the same supplement. 2. " This Prop, is the development of the definition of an angle. If the St. lines at the angular point he produced, the produced lines have the same inclination to one another as the original lines have." — Pott's Euclid, p. 49. 3. The converse of this proposition is, " If fowr lines meeting in a point make the vertical amgles egual, each alternate pair oj lines shall te in one and the same st. line." Use aud App. — 1. By an easy application of Prop. 15, we find the ireadth J' fa lake, or the distance between two inaccessible objects, A and B. From any station C, accessible both to A and B, measure the st. lines CA and C B, and produce them until C E is equal to CA, and CD to CB ; join .s D and E, the distance DE will equal the distance AB. For the st. lines CA, CB equal the fit. lines CE, CD; and the angle ACB equals the angle DOE; therefore, by Prop, i, the St. lines AB and DE are equal. 80 GRADATIONS IN EUCLID. 2. On the principle that the angle ofrejlectim equals the angle of incidence, this proposition is useful to the billiard player to enable him with a ball at A to strike by reflection another ball at B. Let D Q be one side of the table ; imagine a St. line from . B to be perpendicular to DG, and produced unlal DC equals BD : if now the baJl A be driven in the St. line A EC, so that if it was not for the side DG it would reach . C, the ball A on striking E will be reflected from its course BO as to reach the ball B ; i.e., the angle of reflection BED will be equal to the angle of gU''''' incidence AEG. In triangles BDE, CDE, the side BD is equal to side DC, DE common, and the included angles at . D equal : therefore, by Prop. 4, the ang. BED is equal to the ang. CED ; but by Prop. 15, ang. CED is equal to ang. AEG : consequently, the ang. BED equals the ang. AEG. 3. Another use of this proposition is, to determine the number and hind of polygons which may he joined to career a given space. The circle, consisting of 360°, is the measwre for the svm of all the angles that cam, posmbU/ be drawn from a point ; and no regular right-lined figures can fill up the space around a point, xinless the angle contained by any two of the sides of the figure be a measure of 360°. The only regida/r right-lined figwres of which the amgles contained by two sides a/re measures of a circle, are the equilateral triangle, the square, and the hexagon; for 60°, 90°, and 120°, the respective angles of these figures, are measures of 360°. The angle formed by two conterminous sides of the pentagon contains 108° ; of Vba heptagon, 128f° ; of the octagon, 135°, &c, ; but none of these numbers are measures of 360°. Prop. 16. — ^Thbor. If one side of a triangle he produced, tlie exterior angle is greater iJian either of the interior opposite angles. CoKS. — ^P. 10. To bisect a given st. Une. P. 3. From the greater st. line to take a part equal to the less. Pst. 1. A St. line may be drawn from one point to another. Fst. 2. A st. line may be produced to any length in a st. line. Dbm. — P. 15. If two st. lines cut one another, the vertical angles shall be equal. P. 4. If two triangles have in each two sides aad the included angle equal, they are equal in all respects. As. 9. The whole is greator than its part. Exp. 1 Hyp- 2 Conol. CoKs.l SyPlO&Pstl 2 Pst2,P.3, ) Pst. 1. / Dem. 1 2 3 4 5 6 JyC. 1&2. P. 15. P. 4. C. Ax. 9. /Sim. P. 10. 7 D. 1—5. 8 Kecap. PROP. XVI. — BOOK I. Let BC, a side of A ABC, be produced toD; then the ext. z. ACD > L CBA, or than L CAB. Make CE = AE, and join BE : produce BE so that BE=EF, and join PC. 81 B "CT \G V AE = EC, and BE = EF, in As ABE, CFE,- and aAEB= l CEF; .•.AAEB= ACEF, and ^BAE= l FCE. But L ECD or ACD > l ECF ; and .-. L ACD > l BAC. In like manner produce side AC to • G, and bisect BC, and L BCG, i.e., l ACD, wiU be shewn to be > z. ABC. Therefore, if one side of a triangle he produced, &C. Q.E.D. SoH. — I. We may illustrate Prop. 16 in the folio-wing way :— Let triangle- ABC slide along the at. line BD until the point B covers the point C ; it is obvious that vertex A, or rather F, will be at the right hand of the point A, aud that the line CF must be witUn the ang. ACD ; the ang. FCD or ABC will therefore be less than the ext. ang. ACD. 2. Sack angle of a triangle is leas tJum tTie supplement of either of the other angles. Use aud Aep. — Among the conclusions to be derived from this proposition are — 1. Only one perpmdicular AB can le drawn from a point A to a st. line FD. Let AB be perp. to BC, another st. line, as AC, is not perpendicular. For ext. ang. ABD by P. 16, is greater than ACB, and ABC also being a rt. angle tmd greater than ACB, ACB is not a rt. angle, nor is St. line AC perp. to FD. 2. If a St. line AC make the angle ACB acute, and ^ ACF obtuse, the perpendicular AB from ■ A shall fall on the same side as the acute angle does ; for if we suppose AK, the line- on the side of the obtuse angle to be perpendicular, and AEF to be a rt. angle,, then the rt. angle AEF would be greater than the obtuse angle ACE. fi2 GRADATIONS IN EUCLID. 3. In meaauring triaagles, parallelograms, and trapeziums, and in reducing them to rectangular figures, these .and similar conclusions are of great use. Prop. 17. — Theob. Any two angles of a triangle are together less than two right ■angles. Cons. — Pst. 2. A st. line may be produced to any length in a st. line. Dem.— P. 16. K one side of a triangle be produced, the ext. angle is greater than either of the int. opposite angles. Ax. 4. If equals be added to unequals, the wholes' are unequal. P. 13. The angles formed by one line on another, are together equal to two rt. angles. Exp. 1 Hyp. 2 Concl. CONS.1 by Pst. 2. Dem. 1 by Cons. 2 P. 16. 3 Add. 4 Ax 4. 5 P. 13. 6 D. 4. 7 Sim. 8 .Sim. 9 Recap. Let ABC be a triangle; then the z. s A and B are < two rt. angles ; also /. s B and C, and C and A. Produce any side, as BC to D. V z. ACD is the exterior angle of A ABC, .'. z. ACD > z. B or z. A int. and opposite C a. To each of these unequals add the z. ACB. Then A s ACD and ACB > z.s ABC and ACB. But z. s ACD and ACB = two rt. angles. .'. i-s ABC, ACB < two rt. ajigles. In like mamier z. s BAC, ACB < two rt. angles. and z-s BAC, ABC < two rt. angles. Therefore, any two angles of a triangle, &c. Q.E.D. SCH. H. — 1. This Proposition is explanatory of the tmelfth Axiom, and the con- ofit. 2. Both the sixteenth and the seventeenth propositions will be included in iihe thirty-second, in which it will be proved that the three angles of a triangle togctJier equal two right angles. 3. The seventeenth Prop, is useful for demonstrating some of those that follow. peop. xix. — book i. Pbop. 18.^Theor. 83 Tlie greater side of every triangle is opposite to ilie greater angle. CoNa. — P. 3. From the great» line to take a part equal to the less. Pst. 1. A St. line may be drawn from one point to another Dem. — P. 5. The angles at the base of an isosceles triangle are equal. P. 16 If one side of a triangle be produced, the ext. angle is greater than either of the int. and opposite angles. Exp. 1 2 Cons. 1 Dem. 1 2 3 4 5 6 Hyp. Concl. 6yP. 3&Pst.l ly C. & P. 5. P. 16. D. 1. D. 2, 3. d, fort. Recap. In A ABC, let side AC be greater than side AB; then i ABC is > aACB. Make side AD = side AB, ^ and join BD. V AD = AB, in the A ABD, .-. L ADB= L. ABD: But ext. A ADB > int. and opp, ^ DCB, and L ADB= l ABD; .-. z.ABD> L DCBj much more is l ABC > l ACB. Tlherefore, the greater side of eivery triangle, <£'C. 1 Q.E.D. SoH. — The argument on which the conclusion depends is named " d, fortiori" hy the stronger reason, and proves that a given predicate belongs in a greater degree to one subject than to another ; as in the Syllogism, — Y is greater than Z, and X greater than Y ; much more is X greater than Z. Use AMD App, — For the demonstration of other propositions, as Prop. 19. Pkop. 19. — Theoe. The greater angle of every triangle is subtended hy the greater side, or has the greater side opposite to it. Dem. — P. 5. The angles at the base of an isosceles triangle ai-e equal. P. 18. The greater side of every triangle is opposite to the greater angle. 84 Exp. 1 Sup.- AB. Hyp. Concl. GBADATIONS IN EUCLID. In A ABC let L ABC >L ACB; then side AC is > side AB -The side AC must be greater than, equal to, or less than Dem. bySup.&'P.5. Hyp. D. 1 & 2. Sup.&V. 18. Hyp. D. 4 & 5, D. 3 & 6. Concl. Eecap. If AC = AB, then z. ABC = l ACB: but i. ABC 4= Z.ACB; .-. side AC 4= AB. Again, if AC is < AB, then ^ ABC is < ^ACB: but I. ABC is <|; l ACBj .". side AC is <|; AB. Now, AC is neither equal to, nor less than, AB ; .'. side AC is > side AB. Wherefore, the greater angle of every triangle, &C. Q.B.D. SoH. — 1. This proposition is the convene of the eighteenth, and bears the game relation to it as the 6th does to the 5th Prop. 2. Propositions 5, 6, 18, and 19, may be combined into one proposition, thus, — "One amgle of a triangle is greater than, equal to, or Um <%a»- another angle, as the side opposed to the one is greater than, equal to, or less than, the side opposed to the other ; and vice versd." 3. By aid of Props. 17, 18, and 19, we may prove that from the same point there can be Srairni but one perpendicular to a given line, and that this perpendicular is the shortest of all the lines from the given point to the given, Une. 4. As from a given point only three lines can be drawn perpendicular to each other, it is impossible to imagine that there are more than three species of quantity, — a Line, a Surface, and a Solid. Use and App. — 1. The perpendicular AD is tJie shortest line from a point A to a given St. line B C. Because ang. ADB is a rt. ang., the ang. formed by any other line from . A, as z ABC, is acute, by P. 17; and by P. 19 the Bide AD is less than AB : and in the same way we can prove that no other line is less than AD : therefore, th£ perpendicular is the shortest Une from a point to a, given St. line. For this reason the Perpendicular is made use of in measuring : and irregular figures are reduced to those of which the angles are rt. angles. A,. V / B 1 ) C £ PROP. XX. BOOK I. 85 2. From the same point A only two equal st. lines, A B and A C can he drawn to a given st. line B E. See the last figure. Suppose another st. Ime AE also equal to AC. Since AC is equal AB, by P. 5, ang. ACB is equal to ang. ABC ; but in triangle A EC, the ext. ang. A C B,by P. 16, is greater than the int. ang. A EC, and ^ ABC, being equal to ^ A C B, is greater than ^ AEC ; therefore, by P. 19, the side AE is greater than BA, or than CA : thus it is proved that another at. line besides AB and AC cannot be drawn from . A to BE equal to those lines. 3. AUheavyhodies free to move, continually descend, or seek the nearest to the earth's centre. Let HALM represent the earth's cir- cumference, HCL the rational horizon of station A, and AC a perpendicular to the centre ; also let A B be a plane parallel to HL, — suppose it the channel of a canal ; then water poured in at B will flow towards A, because at A, by P. 19, the distance AC is less than the distance BC. A canal •thus constructed might be full at A, and almost empty at B. In the same way a sphere placed at B would roll towards A, and finally settle or come to rest at A, because AC is the shortest line to the earth's centre. 4. By aid of the 19th and the ith Propositions we can construct a triangle when the hase A B, the less angle, A, at the base, and the cUfferenee, AD, of the sides, are given. On the st. line AD, produced indefinitely, and forming with AB the ang. A, take AD, the difference of the sides ; join . s D and B, and bisect D B in . E ; at . E raise a perpendicular till it meets AC in . C, and join CB ; the triangle ABC is the triangle re- ^Jui^ed. By Hyp. and P. 19, BC is the least side, being opposite to the least ang. A. By Const., DE is equal to EB, EC common, and the angles at . E equal ; therefore, by P. 4, the side D C is equal to the aide BC ; and CAis greater than CB by the given difference AD. ABC, therefore, is the triangle required. The figure Prop 20.— Thbor. Any two sides of a triangle are together greater tlian, the third side. Cons. — Pst 2. A st. line may be lengthened out in a st. line. P. 3. From the greater st. line to cut off a part equal to the less, Pst. 1. A st. line may be drawn from one point to another. 86 GRADATIONS IN EUCLID. Dem. — P. 5. The angles at the base of an isosceles triangle are equal Az. 9. The whole is greater than its part. P. 19. The greater angle of every triangle is subtended by the greater side. Exp. 1 2 CONS.1 2 Dem. 1 2 3 4 5 6 7 8 9 Cons. Djsh. 1 2 3 4 Hyp. Concl. 1. „ 2. „ 3. 6yPst.2,P.3 Pst. 1. 6yC.l&P.5, C. 1 & 2. Ai. 9. P. 19. C. 1. D. 4 & 5. Sim. Concl. & E. Or, by P. 9. by P 16. P. 19. Concl. Recap. A Let ABC be a triangle ; then the sides AB and AC are > BC ; sides AB and BC > AC ; „ BCandCA>AB. Produce side BA to . D, and make AD = AC ; join D and C. •.•AD = AC, i.ADC=^ACD; but z. BCD is >z.ACD; .-. .^BCDis> ^ADCorBDC; and .'.also side BD is > side BC. But side BD = BA and AC together ; .'. sides BA and AC together are >BC In like manner it may be proved that sides AB and BC are >AC, andBCandAC>AB Therefore, any two sides of a triangle, &o. Q.E.D. Let St. line AE bisect z. BAC. • • L BEA is > ^ EAC, and L CEA than L EAB; .". side BAis > BE, and side AC than CE : And BA and AC > BE and CE, which are equal to BC. Therefore, any two sides, &c. q.b.d. Cor. — The difference of any two sides of a triangle is less than the remaining side. Dem. 1 2 by P. 20. Sub. Ax. 5. The sides AC and BC are>AB ; take away AC both from (AC + BC), and also from AB; .'.side BC is >AB diminished by AC, or than the difference between AB and AC. PBOP. XX. — BOOK I. 87 N.B. — In the proof of this Corollary, says Lardner, p. 33, " we assume something more than is expressed in the fifth Axiom. For we take for granted, that if one quantity (a) be greater than another quantity (6), and that equals be taken from both, the remainder of the former (a) will be greater than the remainder of the latter (5). This is a principle which is frequently used, though not expressed in the Axiom." SoH. — " Let the beginner remember that the object of this proposition is not to convince him of the truth stated, but to show how it may be connected with and deduced from the fundamental axioms and definitions." — Mason, p. 56. Use and App. — 1. Of all st. limes that can he d/ra/um from, one point, A, to mwther, E, and reflected to a tJii/rd point, B, those are the shortest, AE, EB, which make the angle ofinddenee, AEQy equal to the angle of reflection, BED. ci/-- Cons. 1 2 3 Dem. 1 2 3 4 5 6 7 8 9 P. 12 & Pst. 2j From . B draw BD perp. to D G, and produce BD indefinitely ; P. 3. make St. line DC = DB, and join EC. Assum., Pst. 1. Also in st. line DG assume another point F, and joinAF, BF, andCF hy C. 1, 2. P. 4. Sim. Hyp.&C.&D.2 C. D.5&C.3,P.20 D. 2, 3. Concl. Becap. In triangles BED, CED, side DE is common, BD equal to D C, and the angles at D equal. .-. side BE is=CE, and ^ BED = z CED. In the same way we prove side BF to be = side CF. Now z s BED, DEC, and AEG are all equal ; and AEC is a st. line. Also AFC is a A ; .-. sides AF and FC > A C • but AC = AE and EB ; and FB = PC ; .-. AF, FC, or AF, PB, > AE and EB. Wherefore, of all st. limes that cam, he drawn, &o. Q.E.B. 2. We may observe that natural causes act by the shortest lines ; therefore, all reflections are made by the lines which cause the angle of reflection to equal the angle of incidence. Hence, hy means of a mirror placed horizontally, we may construct a triangle, the perpmdnicuka- of which shall he represenioMve of the height of any object. 88 GRADATIONS IN EUCLID. Let A B be the height of a tower, AC an so horizontal line, with a mirror at C. BCA will be the ang. of incidence of the light from . B on the mirror at ■ C : if now an ohserrer at D measures the angle of reflection ECD, and the distance AC, he wiU have the means of constructing a triangle representative of ACB : for make the ang. ECD the angle of reflection, let C D contain the equal parts from a scale that represents the yards or feet in AC, and at D let a perpendicular DE be raised, — D E applied to the same scale of equal parts, will give the height of AB. ? D 3. Were we to take as a lemma, prop. 4, bk. vi. that " the sides about the equal angles of eqwiangvlwr triamgles are proportiondls," we should have from the foregoing construction the proportion, and, consequently, the equation, which would determine the height AB. For triangles BAG and E D C being similar, the base of the one is to the base of the other as the perpendicular x>f the one to the perpendicular of the other : or, DC :AC ::ED -BA, and the product of the extremes egualUng the prodiKt of the means. „„ . AC . ED, 50 X 6 DC . BA = AC . ED; whence AB = — ^ — ^^ or, =30; when AC=50, ED=6, and DC=10. Prop. 21. — Theob. If from the ends of the side of a triangle there he dmvm two si. iines to a point within the triangle, these lines shall be less than the other two sides of the triangle, but shall contain a greater angle. Cons. — Pst. 2. A terminated st. line may be produced. Dem. — P. 20. Any two sides of a triangle are together greater than the third side. Ax. 4. If equals be added to unequals, the sums will be unequaL P. 16. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles. «9 Exp. 1 Cons. Dem. 1 2 3 6 6 7 8 9 10 11 12 Hyp. Concl. 1. hy Pst. 2 hy P. 20. Add. Ax. 4. P. 20. Add. Ax. 4. D. 3. ctfort. P. 16. P. 16. ctfort. Recap. PROP. XXI. — BOOK I. Given the A ABC, and from . s B and C the st. lines BD, CD meeting within the A in-the pt. then BD and CD shall be < BA and CA, the other two sides. but the ^BDCshall be >^BAC. Produce BD to meet side CA in the . E. In A ABE the sides AB and AE are > BE ; to each of these add the line EC ; then the lines AB, AE, and EC, are > BE and EC. Again, in A CED, the sides CE and ED are >CD; to each of these add the line DB ; then sides CE, ED, and DB together are > CD and DB together. But AB and AC are > BE and EC ; much more are AB and AC > CD and DB. Again, in A CDE the est. L BDC is > the int. ^ CED, and in A ABE the ext. L CED is > the int. L BACj much more is L BDC > l BAC. Therefore, if from the ends of a side, &c. Q.E.D. Use and Application. — In Optics this Proposition is used to prove that if irom . A we could see the st. line S C, and also from D a point nearer to the line, the basQ B C would appear less from A than from D : it does this on the principle that quantities seen under a greater angle appear greater. For this reason the apparent diameter of the sun measures more when the earth is ia perihelion, than when it is in apJtelion. And thus — according to Vitruvius, who com- posed his workvon Architecture about 15 B.c. — ^the tops of very high piUstrs should be made but little tapering, because they will, from the distance, of themsalTes seem less. 90 GRADATIONS IN EUOUD. Peop. 22. — Prob. To make a triangle, of which the sides sludl he equal to three given St. lines, but any two whatever of these must be greater than tlie third. Sol. — P. 3. From the greater st. line to out off a part equal to the less. Pst. 3. A circle may be drawn with any radius. Pat. 1. A line may be drawn from point to point. Dem. Def. 15. A circle is a figure bounded by one continued line, called its circumference, and having a certain point within it from which all lines drawn to the circumference are equal. Ax. 1. Magnitudes which are equal to the same magnitude, are equal to each other. Exp. 1 Data. 2 Quaes. CONS.I 2 3 4 5 6 hy Assum P. 3. Pst. 3. „ 3. „ 1. SoL DsiM. 1 2 6yC3,Def.l5 C.2&AX. 1. Given the st. lines A, B, and C, of which A and B > C, A and C > B, and B and C > A ; to make a A with sides = A, B, and C. Assume a st. line DE unlimited towards E; makeDr = A, FG = B, andGH = C; from centre F with rad. D F draw the D KL, and from centre G with rad. GH the© HLK; join the .s F, K and G, K ; then the fig. KFG is a A, and side FK = side A, FG = B, and GK = C. V . F is the centre of DKL, FK = FD ; but F D = St. Une A, :. F K also = A. Dem. 3 4 5 6 C.3, Def. 15 C.2&AX.1. C 2 D. i, 4, &5. Becap. ' PEOP. XXIT.: — BOOK I. 91 Again, •/ G is the centre of HKL, GH=GK; butGH = C; .-. alsoGK = C, and FG is equal to B ; .•. St. lines FK, FG, GK are respectively equal to St. lines A, B, and C. And tMrefore FK,FH>the rad. of DLK. 2 .-. the . H is outside of the D L K. 3 C&Dat. Sm5. Also, -.• sides FK and GK are > FG, takfe away GK orGH; i :. FM is < FK, and .M is within the DLK. 5 D. 2 & 4. Now, H being without, and M within, the ©s, 6 Gonol. therefore, the ©s intersect. 2. If two of the given lines were together equal to the third, the circles viovM touch externally ; if the two were together less than the third, the circles would not touch at all ; in either case no triangle could be drawn. Use asb App. — -l. All rectilineal figures being divisible into triangles, this Proposition is of very extensive use in the construction of Geometrical figures, - — either for making one rectilineal figvire equal to another, or on the theory of Kepresentative Values making one figure li%e to another: in the first case, the triangles into which the rectilineal figure has been divided are repeated, side for side, in another rectilineal figure of exactly the same linear dimensions : and when the construction is completed, if the one figure were placed on the other, the two would correspond, angle to angle, line to line, and point to point : in the second case, that of making one rectilineal figure similar to another ; the sides and angles of the first must be measured, and from a scale of equal parts, lines drawii in the second, representative of those in the first, and angles in the second equal to those in the first ; — for equality of angles, according to Definition 1, book vi., is essential to simila/rity of figure. 2. Practically, a triangle with sides equal to three given st. lines, will be drawn, by describing arcs, with radii equal to the sides, intersecting in .G, and joining CA, and CB ; thus, Given, three lines. AB equal to 20, B C to 25, G-it- and CA to 15 ; to form a 1/riwngle. From the scale of equal parts set oflF a line AB equal to 20 ; at A with the distance CA, 15, from the same scale, de6o. ah arc ; and at B with the distance B C, desc. another arc : ' the arcs both intersect in C, and joining the points, A B C is the triangle required. 92 GRADATIONS IN EUCLID. 'o. Ona given et. line, AB, to describe an isosceles triangle, ACB, having tach tfthe equaZ sides, AC, BC, double of the base, AB, or eiiml fJ BO. Biaect AB in D, by the perp. DE, pro- duced indefinitely toE. From . A with rad. equal to twice AB, deso. the arc FCG ; and from B with the Bame rad. desc. the aroHCK; join the points C A and CB. The fig. ACB is the isosc. A required. V side AD = side DB, "-■ D C common, O and ^ADC= ^BDC, .". the St. line AC equals the st. line BC, and they are each double of AB. Wherefore, on a given line AB, &c. Cons. 1 by P. 10, 11, Pst. 2. 2 Pst. 3, P. 3. 3 » 3, „ 3 4 „ 1- 5 Sol. Dem. 1 by C. 1. 2 3 4 P. 4. C. 2 & 3. Concl. Q.B.F. 4. On a given st. line, AB, and with a given side equal to BO, to describe an isosceles triangle. The same construction being made, and the same demonstration followed, we arrive at the conclusion — that side BC equals side CA, and that each of them is equal to the given side BO. Prop. 23.— Prob. At a given point in a given st. line to make a rectilineal angle equal to a given rectilineal angle. Sol. — Pst. 1. A st. line may be drawn from one point to another. P. 22. With three given st. lines, any two of which are greater than the tliird, to make a triangle. Dem. — P. 8. If two triangles have the three sides of one equal to the three sides of another, each to each, the angle contained by two equal and conterminous sides of the one, shall be equal to the angle con- tained by any two equal and conterminous sides of the other. Exp. 1 Data, Quoss. CONS.1 2 3 Dem. 1 2 3 hyAmim, &Pst.l P. 22. Sol. hy C. 2. P. 8. Becap. PROP. XXIII. — BOOK I. 93 Given the . A in the st. line A B, and l DCE; required at . A in AB to make an :L = DCE. In St. lines CD, L J ^f. DE take .s »^ ^ D and E, and /B join DE ; with side AF = CD, AG = CE, and FG = DE, make the A APG j then the l. FAG shall be = :L DCE. V FA = DC, AG = CE, and DE = FG; .-. the L. FAG= L DCE, Therefore, at . Kin AB, &c. ' q.b.f. ScH. — This Proposition is an extension, or rather a generalisation, of the eleventh : by the eleventh we dram an cmgle of a particiMar species — a right angle ; but by the twenty-third we draw any angle wliatever. UsB AHD App. — 1. This Proposition is of the widest use, in Surveying, Engineering, Perspective, and indeed in all the other parts of Practical Mathe- matics. 2. Nest to the use of the Semicircle for measuring and making angles of a determinate magnitude, that of a Mne of Chords is most important. Its c<€ ^\ D h- iJ Pbob. II. — An angle being given, CAB, to find the measure of it in degree* of a circle. With the chord of 60°, from the line of chords, for radius, describe the arc BC ; t&le the distance, i.e., the chord, BC in the compasses, and apply the distance to the same Une of chords : the number of degrees in the arc B C will be ascertained. PnOB. III. — From, the extremity D, of a st. line AD, to draw a perpendicular. From . D, with the chord of 60°, from the line of chords, for radius, desoribe the arc FG ; take the chord of 90° for a radius, and from F describe another arc intersecting FG in P ; draw PD, — and it is the perpendicular required. Peob. IV. — To construct a triamgle of which the hose AB contains 30 equal parts, the angle at A 40°, and the side AC 25 equal parts ; and to find the angles C and, B, and the otlwr side BC. Take 30 from the scale of equal parts, and by P. 3, draw st. line AB ; at . A, with the chord of 60°, desc. the arc DE, and from . D, with the chord of 40°, or of ang. A, cut the arc in . E ; join AE, — and EAB is the required angle. Produce AE, and set on it 25, or AC ; join CB, — and the triangle is completed. To measure the st. line CB, take the distance from C to B, and apply it to the same scale, which will give the equal parts 19 : and to measure the angles,— with the chord of 60°, describe from . s C and B the arcs GH and DP; take in the compasses the distances G to H, and D to F : these distances applied to the line of chords will give the angle C 85°, and the angle B 55°. PROP. XXIV. BOOK I. Prop. 24 — Thboe. , If two triangles Ivave two sides of the one equal to the two sides of the other, each to each, but the angle contained hy the two sides of the one greater than the- angle contained hy the two sides of the other, the hose of that which has the greater angle shall he ' greater than the hose of the other. Cons. — P. 23. To make an iEtngle equal to a given angle. P. 3. From the greater st, line to take a part eqiml to the less. Pst. 1. Any two points may be joined by a st. line. Dem. — P. 4. When two sides and the included angle of one triangle are equal to the two sides and the included angle of another triangle, the triangles are equal in every respect. P. 5. The angles at the base of an isosceles triangle are equal. Axl 9. The whole is greater than its part. P. 16. The greater angle of every triangle is subtended by the greater side. Exp. 1 Hyp. 1. 2 3 „ 2. Conol. CONS.l 6yHyp.&P.23 2 P. 3&Pst.l. Dem. 1 2 6yHyp.&C.l. P. 4. 3 C. 2, & P. 5. ^ Ax. 9. 5 ctfort. In A s ABC, DEF, let AB = DE, and AC = DF; buttBAC> Z.EDF; then the base BC > the base EF. Let side DFEGF, and iDFG>z.EGF; much more is .i. EFG> z. EGF. 96 GRADATIONS IX EUCLID. Dem. 6 P. 19. But the greater angle is opposite to th» greater side ; 7 D.5 &P.19. .-. the side EG is > side EF. 8 D. 2. But EG = BC, and .*. BC is > EF. 9 Recap. Tlierefore, if two triangles have two sides, base EF ; then the l BAC is > z. EDF. If Z.BAC is > Z.EDF, it is either equal or less : Suppose that z.BAC= Z.EDF; then base BC = base EF : PROP. XXVI. BOOK I. 97 ButBCis^rEF; .-. ^BACis4=/.EDF. Again, suppose Z.BAC < Z.EDF; then base B C is < base E F : ButBCis <(; EF; .-. L BAG is L EDF. Therefore, if two triaiigles have two sides, &a. Q.E.D. -Propositions 24 and 25 have the same relation to each other as Props. 4 and 8, and the foxir may be combined thus : — If two trianghs have two sides of the one respectively equal to two sides of the other, the remaining side of the one will he greater or less than, or equal to, the remaimmg side of the other, according as the amgle opposed to it in the one is greater or less than, or equal to, the angle opposed to it in the otlier ; or vice versd. — Laedner's Mtclid, p. 66. Use and App. — The principal use of Preps. 24 and 25 is to assist in the demonstration of the following propositions. Dem. 4 5 Hyp. Concl. 6 7 Swp. H.&P.24 8 9 Hyp. Concl. 10 D. 5 & 9. 11 Concl. 12 Recap. SOH.- Pbop. 26. — Tn^OB.— .(Important.) If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side — viz., either the sides adjacent to the equal angles in each, or the sides opposite to them — thfn shall the other sides be equal, each to each, and also the third angle of the one to t/ie third angle of the other. Cons. — P. 3. From the greater of two st. lines to cut off a part equal to the less. Pst. 1. A St. line may be drawn from one point to another. Deu. — P. 4. Two triangles are equal, when two sides and the included angle in one are equal to two sides and the included angle in the other. Ax. 1. Magnitudes which are equal to the same, are equal to each other. P. 16. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles. Exp. 1 2 3 4 Hyp. 1. „ 2. Concl. 1. „ 2. In As ABC, DEF, let l ABC = l D£F, and L ACB= aDFE; also let one side = one side ; then the other sides shall = the other sides, and the third L of the one = the third l of the • other. 98 GRADATIONS IN EUCLID. Case I. — Let ike equal sides be adjacent to the equal angles. Exp. 1 2 3 OONS. 1 2 3 Dem. 1 2 3 4 5 6 Hyp. Conol. 1. by Sup. Sup.. P.3&Pst.l 6yC.3&Hyp P. 4. Hyp. Ax. 1. Concl. D.5&Hyp. P. 4. Let side B C = side E F ; then AB = DE, ^ andAC = DF, and third l BAC = third z. E D F. For if A B is r::^ D E, one is > the other. LetABbe>DE:B ^ Make B G = D E, and join G C. V inAs,GBC,DEF,BG = ED,BC = EF, and^GBC= Z.DEF; .-. baseGC = baseDF, AGBC=A DEF, aDdiBCG= Z.DFE: ButiL'DFE= Z.BCA; .•. z.BCG= aBCA, the less = the greater, which is impossible. .-. A B is not 4= D E, i.e., AB does = D E ; Hencein As ABC, DEF, V AB = DE, BC = EF, andz.ABC= /lDEF; .-. base Ac = base DF, and z. BAC = Z.EDF Case II. — Let the equal sides be opposite the equal angles in each. Exp. 1 2 3 Cons. 1 2 3 Dem. 1 Hyp. Concl. 1. ,, 2. by Sup. Sup. P. 3, Pst. 1 by C. & Hyp Let AB = DE, iB= z.E, and/.ACB= the' int. ^ E FG ; but ^ AEP = i. EFG; .". z. AEF ia both > and = ^ EF G , which is impossible : :. AB, CD do not meet towards B, D. In the same way it may be shown, they do not meet towards A, C : .". AB is parallel to GD. Wherefore, if a si. line falling, &c. Q.B.D. SOH. — Since, however, there are some curved lines which are not parallels, though they never intersect, as in the Introduction, § vi., p. 30, another demonstration may he made. Let EH fall on AB, CD, and make the altr. c s AFG and FGD equal ; then the lines AB and CD are parallel. Dem. 1 2 3 4 6 7 8 hy Sup. Def 35. Vs,t.2&,Sup. 6yC.3&P.16 Hyp. D. 1 & 2. ad vm/poss. ConcL Sim. Def 35. Kecap. Exp.l Hyp. 2 Coucl. CONS-l by P. 12. 2 P. 3,P8t.l Dem.1 53;C2 2 3 4 5 6 7 8 9 C. 1. Def. 10. Hyp. D. 3. Ax. 12, n. ConcL . At G draw GA perpend, to f AB; ^ take GD = AF, and join FD. H In A s AGF, DFG, side GD=side AF, GF common, andzAFG=^FGD; /. side AG=side DF, andz GDF= z GAF : But z GAF is a rt. angle. .•. z GD F is a rt. angle, and D P perpendicular to C D. Moreover, the par. to C D is to be drawn from P ; and since the perpendiculars AG and DF are equal, the parallel to C D must pass through . A : .-. at. line AB is parallel to st. line CD. Q.E.D. ScH. — Of the angles which two st. lines, one at each extremity of a third st. line, make with it, the altemaU angles are those which are on opposite sides and at opposite extremities of the third Une. pbop. xxviii. book i. Prop. 28. — Theob. 103 KXP. 1 Hyp. 1. „ 2. „ 3. 2 ConoL Dem. 1 2 by, Hyp. P. 15. 3 At 1. 4 P. 27. 5 6 Hyp. P. 13. 7 Ax. 1. 8 Sub. 9 Ax. 3. 10 Cons. 11 P, 27. 12 Eeoap. If a si. line falling upon two other st. Unes makes the exterior angle equal to the interior and cpposite angle upon the same side of the line; or mahes the interior angles upon the same side together equal to two rt. angles ; the tioo St. lines shall be parallel. Dem. — P. 15. If two st. lines cut one anothei;, the vertiijal angles shall be equal. Ax. 1. Magnitudes equal to the same are equal to one another. P. 27. If a St. line failing upon two other st. lines makes the alternate angles equal, these two st. lines shall be parallel. ' ' P. 13. The angles which one st. line makes with another upon one side- bf it, are equal to two rt. angles. Ax. 3. If equals oe taken from equals, the remainders are equal. Let the st. line EF fall on the two st. lines AB, CD; make the ext. l EGB = the int. andopp. L GHD; and the int. ls BGH + GHD = two rt. angles ; then the st. line AB is || the st. line CD. For, V L EGB =.i. GHD, and L EGB = z. AGH; .'. z. AGH = GHD j and they are altr. i. s ; and .". also st. line AB is || CD. Again, •." z.sBGH,+ GHD = two rt. angles, and LS AGH + BGH = two rt. angles; .:. /.sAGH + BGH = Z.SBGH + GHD: Take away the common z. BGH, .". the rem. z. AGH = the rem. z. GHD: But Z.S AGH and GHD are altr. angles; .". the St. line AB is || the st. line CD. Therefore, if a st. line falling upon the other, the other: 2 „ 2. Let L AGH be > z. GHD ; 3 Add. and to each angle let the i. BGH De added; 4 Ax. 4. then z.s AGH, BGH together are > ^s BGH, GHD: 5 P. 13. But Z.S AGH, BGH together are = two rt. angles ; 6 Ax. 1. ; .". ^s BGH, GHD together are < two rt. I angles. 7 Ax. 12. iNow on this condition st. lines AB and CD, j \ being produced, wiU meet. PROP. XXIX. — BOOK I. 105 Dem. 8 Hyp.Def.35 9 Coiicl. 1. 10 P.15&AX.1 11 Add. 12 Ax. 2. 13 P. 13. 14 Ax. 1. 15 Recap. but St. line AB is parallel to st. line CD ; .'. L AGH is not njr, i.e., is =, L GHD : But A. AGH = EGB, .-. l EGB-= Z.GHD; to each angle add /. BGH: then ^sEGB + BGH= as GHD+BGH: But L.S EGB + BGH = two rt. angles; .•. /.s GHD + BGH = two rt. angles. Wherefore, if a st. line fall on two parallel sf. - The equality of AC and BD ; 6°- The equality of the angles A and D ; 6°- The equality of the angles B and G ; 7">- The bisection of AD by BC ; 8°- The bisection of BC by AD ; 9°' The bisection of the area by AD ; IQo- The bisection of the area by BC. These ten data, being combined in pairs, will give 45 distinct pairs ; with each of these pairs it may be required to establish any of the eight other properties, and thus 360 questions, respecting such quadrilaterals, may be raised. These questions will furnish to the student useful geometrical exercises. " The 9th and *Oth data require the aid of subsequent propositions." — Labdneb's Euclid, p. 49. Use ahd App. — 1. The construction and accuracy of the parallel ruUr depends on Prop. 84. 2. A finite St. line may he divided into any given number of equal parts. £xp. 1 Cons. 1 Dem. Data, Quces. ' Pst. 1. P.3&Pst. 1. P. 31. Sol. P. 31. -"■1/ Given a st. line AL, ^; X and the number of /■;' parts, as four ; ~r'' "• to divide AL into Q ,,--'; \ those parts. -p ,.-■; \ \ From one extremity . ..-•* '■■--'"' ''. '• of AL, as A, draw ^ * ' «* an indefinite st. line AX, take AB, and make BC, CD, DE each =AB, and join E L ; through . s D, C, B draw Dd, Cc, B5, ||s to EL ; the line AL is so divided that A.h=hc, bc=cd, and od=fdlj. Also through h draw Jm || AX. 1 6iyC.3&5,P.34 •.•B5Cmiaaa, .-. 5m=BCorAB; 2 P. 29. also i A= /c5m, and/! A6B= z Jem : P. 26. therefore A 5 is equal to he. Sim. And in the same way dc=hc, and dL=cd. Q.E.D. 3. On the same principle the Sliding Scale, called from the inventors the Vernier or Nonius, is constructed. This scale is very useful, as in the t\G GRzlDATIONS IN EUCIjID. Barometer, for measuring t'iie hundredth part of an inch ; or in the Theodolite, for measuring the minutes into which a degree is divided. 10. p^r meaiurirv; the hundredth part of an inch, we may take, because of its clearness, Ritchie't description, in his Geometry, p. 32. " Suppose an inch AB, .A c 6 1 a iR 1 1 1 1 II 1 I / 'r 1 1 1 Hd ii i divided into 10 equal parts, each part will be ^ of an inch. Again, suppose a St. line CD equal to 9 of these parts to be also divided into 10 equal parts, each of these will be y^r °^ tV of ^^ inch, which is y^. But the length of one of the first divisions being iV, or .}^, and that of the second yly, one of the first divisions is ^^ of an inch longer than one of the second. If the st. line CD slide along parallel to AB till the two divisions marked 1, 1, form a con- tinuous St. line, the sliding scale will have moved tttt P*'^' of ^'^ i""^ towards B. If it slide along till the next two divisions coincide, it will have moved yfu of an inch, &c." 20. Pq^, Tneasuring the minutes into which a degree on a circle is divided. " If 69 degrees on the circumference or limb of the instrument be divided into 60 equal parts, the diflPerenoe between the length of one degree and one of the latter divisions will obviously be -^ of a degree, or one minute. This kind of vernier, on account of its gieat length, is seldom employed. If each degree on the arc A B of 15° be divided into two equal parts or half degrees, and if 29 of these be taken and divided into 30 equal parts, as in CD, the difference between the length of half a degree and one of the new divisions in C D, will obvioufely be •jV of half a degree, or -^ of a degree, that is, one minute. This is a very common vernier." If the limb CD move along the arc AB, so that the first division on CD and the first on AB are in one and the same st. line, the vernier will have moved -I of ^\f, or ^'j of a degree, i.e., one minute ; if the second division on CD coincide with the second on the arc AB, the vernier will have moved -^ of a degree, or two minutes, &c. 4. When two objeeti, A and B, are inaccessible from . n each other, with an instrument to set out perpendiculars — from them to a line CD, parallel to the line joining A and B, the distance may be obtained ; for, by Prop. 34, CD is equal to AB ; if therefore CD be measured, we learn the distance from A to B. c D 5. Also, when tlure intervenes an obstacle, as a house, or a lake, to the con- tinuation of a St. line AB, beyond the obstacle H ; the direction of AB may be ascertained by the method of taking perpendiculars, and of drawing parallel St. lines. PROP. XXXV. — BOOK I. 117 At B raise a perpendicular BE, and at E another per- pendicular EF ; at F, a point beyond the obstacle, also draw a perpendicular FC ; and set out PC of the same length as B B ; then at C draw auother perpendicular CD, and the st. line CD will be in the same direction with the St. line AB : and were H the obstacle removed, and C and B joined, ABCD would be in one and the same st. line. By construction BEPC is a rectangle, and the angles at C and B rt. angles ; therefore, by Prop. 14, A B a^d CD will be in the same st. line with BC. 6. A field of the shape of a parallelogram, ABCD, may he divided into two equal parts by the diagonal AD ; but if it has to be divided from the point E, bisect the diagonal in F, join E F, and produce EP to G ; the st. lin« E G will divide the field into two equal portions. For in the triangles A F E, D F G, the angles EAP, AEF, are respectively equal to the angles C ' FDG, FGD, by Prop, 29, and AF equal to FD ; therefore, by Prop. 26, the triangles APE and FDG are equal. And since the trapezium BEPD and the triangle APE, by Prop. 30, together make up half the field ABD ; the same trapezium with the triangle FGD, which is equal to the triangle APE, will also make up half the field : therefore the, at. line EG divides the field into two equal portions. Prop. 35. — Theoh. Parallelofframs 'upon the same base and between the same parallels are equal, or rather eguivaient, to one another. Dem. — P. 34. The opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them. Ax. 6, Doubles of the same magnitude, equal each other. Ax. 1. Things equal to the same, are equal to one another. Ax. 3. If equals are taken from equals, the remainders are equal. P. 29. A st. line falling on two parallel st. lines makes the exterior angle equal to the int. and opp. angle. P. 24. Triangles with two sides and their included angle in each equal, , are equal in all other respects. 118 Exp. ] Dem. 1 2 CasbII. Deu. 1 2 3 4 Hyp.l. Concl. Sup. by P. 34. Ax. 6. Sup. 6yH. ^^ V^'^ °" ^^ ''®^*' I™^ A> *°-" — Eitchue's Gcom., p. 29. In the Diagonal Scale- KB GO thus constracted^the' distance AB represents 100, and eaeh of the divisions between A and B 10 ; apd on the diagonal line diverging from A, the first distance from the perpendicular AD to the diagonal will be ^V of 10, or 1 ; the second distance from AD to the same diagonal, ^ of 10, or 2 ; the third distance, ^ of 10, or three ; and so on. Thus the spaces between E and A, are hundreds ; between A and B, tens ; and between B and C, units. If, however, the spaces between E and A are tens, those between A and B are uniis, and between B and C, teviks .- indeed, the values depend on what we call the spaofis between E and A. The extent from / on the perpendicular EL, to A on the seventh- diagonaltovth» seventh parallel, may be taken for 277, 27'7, or 2"77, according as we consider B A hundreds, or tens, or units. The Diagonal Scale is of very extensive use in the construction and mea- surement of Geometrical Kgures. 122 GRADATIONS IN EUCLID. Pbop. 37. — Theok. Triangle) 'upon the same base and between the same parallels are eqiial to one another. Cons. — Pst 2. A st. line may be produced in a et. line. P. 31. Through a given point to draw a st. line parallel to a given st. line. Def. A, A parallelogram is a four-sided figure of vfhioh the opposit* sides are parallel, and the diagonal joins opposite angles. Deu. — P. 35. Parallelograms on the same base and between the same parallels are equal. P. 34. The opposite sides and angles of parallelograms are equal, and the diameter bisects them> Ax 7. Halves of the same magnitude are equal. Exp. 1 Hjp. 1. „ 2. 2 Con, 2. CONS.1 by Pst. 2 2 P. 31. 3 Def. A. Dem. 1 byC.Z&YL 2 3 P. 35. P. 34. 4 5, Ax. 7. Recap. Let the As ABC, DBC, be on the same base EC, and between the same lis AD, BC; then the A ABC = the A DBC. Produce AD both ways indefinitely, through B draw BE || CA, and through C, CF || BD; then EBCA and DBCF are parallelograms. '.' thecz? s are on the same base BC, and between the same ||s BC, EF ; .-. thec:^ EBCA = the c=7 DBCF: But a I 7 is bisected by its diameter, and .-. A ABC = half the c=7 EBCA, and A DBC = half the £Z3' DBCF j .-. also A ABC = A DBC. Wherefore, triangles upon the same base, &c. Q.B.D. ScH. — On the principle that triangles are the halves of parallelograms, the areas of triangles are obtained ; the product of the base and altitude gives the area of a parallelogram : consequently, half the product of the hose and aUitude ijives the area oj the triangle. PROP. XXXVIII. BOOK I. Prop. 38.— Theor. Triangles upon equal bases and between the same parallels are equal to one another. Con. — Pst. 2. A st. line may be produced indefinitely. P. 81. To draw a st. line through a given point parallel to a given st. line. Def. A. A parallelogram is a four-sided figure of which the opposite sides are parallel, and the diagonal joins opposite angles. Dem. — P, 36. Parallelograms upon equal bases and between the same parallels are equal. P. 34. The opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them. Ax. 7. The halves of the same or of equal magnitudes are equal. B ..._]Fr Hyp. 1. Let the As ABC, DEF, be on equal %. bases BC,EP, \ and between the \ same II s BF, AD J \ Concl. thentheAABC= 'i the A DEF. by Pst. 2. Produce AD both ways indefinitely ; P. 31. through B draw BG || CA, and through F, FH 1| ED ; Def. A. then GBCA and DEFH are parallelograms. 6yH.&C.l. ■ ■ base BC = base EF, and BF || GH; P. 36. .-.theCTJ GBCA = the CZJDEFH: P. 34. But a parallelogram is bisected by its diagonal; .-. As ABC, DEF, each = half GBCA, DEFH; Ax. 7. and .-. A ABC = A DEF. Eecap. Wherefore, triangles upon equal bases, &o. Q.E.D. ScH. — 1. The bases of the triangles are placed so as to form portions of the same st. line. 2. The Area of a triangle may ie iisecied, or divided into any nuniber of equal parts : in the one case, tn/ Tdsecting the iase ; in the other, by dividing the base into equal parts, and joining the points of division and the vertex. Use and App. — 1. By the last two propositions we arrive at a, practical way of dividing a triangular space, as A.B C, into two equal parts ; for if the base BC be bisected from the vertex A by AK, then, baoause the two triangles ABK and ACK are on equal bases BK and CK, and between the same pai-allels, AD and BC, those triangles are of equal areas. Exp. 1 2 CoNs;l 2 3 Dem 1 2 3 1 4 6 124 GRADATIONS IN EUCLID, 2. When any point whatever, as H in the side AC of a triangle ABC, ts tahen, the triangle may be divided into two equal parts from that point. Cons. 1 Dem. 1 2 3 4 5 by P. 31. P.lO&Psil. Pst.l&P.31. Sol, byC.2 P. 38. C. 3. P. 37. Sub. Ax. 3. D. 2, 6. Ax. 3. Add. Aj!:.2. Recap. &1. Sub. Through the vertex C draw CFIIAB; Bisect AB in D, and join CD ; Join HD, and from C draw CE || HD, and join HE ; then the A AHE = the trapezium ,\ CHEB. V AD = DB, and FC is || AB; .-. A ACD = A BCD. Again, V HD is || CE, and HD the common base, .-. the A CHD = the A DHE ; from each of these equals take away the common partHGD; and the rem. A CHG = therem. A ED 6. Next, from the equals ACD, BCD, take the equals CHG, EDG; and the remainder AHGD equals the remainder CGEB: to each of the eqiials AH CD, CGEB, add one of the equals EDG, CHG, and the whole AEH equals the whole CHEB. Thus the triangle ABC ha^ been bisected from the point H bij the st. line HE. ta square. At . A and . B draw perpendiculars AEandBP; bisect the it. angles by AC and BC meeting in C ; through .3 B and A draw BD {| AC, and ADHCB; then AB C D is the square required. •_■ the z s C AB and C B A are equal, side AC = side CB ; and V AGED is an, AD = CB, and BD = CA; .-. AC = CB = BD = D A, and the fig. is equilateral. Again, the ^ s CAB, CBA being together one rt. auyle, the angle C is a rt. angle ; K CONS.l P. U. 2 P. 9. 3 P. 31. 4 Sol. Dem.1 JyC.2,&P.6. 2 3 4 5 P. 34. ConcL C. P. 32. 133 Deh.6 7 8 GRADATIONS IN EUCLID. P.i6,Cor.2. But in a a, as AC BD, when one angle is a rt. angle, all the angles are rt. angles ; C.3,Concl. /. also ACBD has its angles rt. angles ; Concl, And therefore, the figure being equilateral and rectan- gular, ACBD is the square requirecl. Use aot) App. — The Geometrical Square is an instrument 5y means of which, and of the property of similar triangles that the sides about the equal angles ai* proportional, the height of an inaccessible oiject can he cucertamied, pro- vided a measurement to the perpendicular from the object can be made. The edges of the square are each divided into 100 equal parts, and from one comer a plummet is suspended ; when the object is seen along one edge of the instrument, the plummet cuts another edge, and forms a triangle similar to the triangle formed by lines representing the perpendicular from the object, a parallel to the horizontal line at its base, and the hypotenuse, or distance (i-om the point of observation to the object itself. In the adjoining figures, | AB represents the horizon ; pD a parallel to the horizon ; D B the height of the instrument ; C D the height of the object C above the parallel to the horizon ; sp the edge along which the object is to be seen ; sr, rn, and pn gradu- ated edges each of 100 parts ; and p the point of suspension for the plummet. From the place of observation measure the distance p D, and the height of the in- strument DB ; direct the edge sp towards the object C, and note the number of parts in sr or in rn. 1°- When the plummet outs sr in 0, the triangle ^so is similar to the triangle COp ; and we have the proportion so : sp :: pD : CD ; whence CD = 'P-P^ so and CB = CD + D B. 2°' When the plum- met cuts rn in o, the triangles onp and CD^ are similar ; and we liave the propor- tion pn : no '.: p'p : DC; whence CD = ""■P" ; andCB = CD-^DB. pn For example, let ^D = 60 ft. ; CB. Here 50 ;100 :: 00 : 120 Tate's Oeom/^j, pp. 49-51. so = 50 eq. pts. ; and DB = 8 ft. ; required = CD, and 120 -(- 6 = 126 ft. = CB.— Sec PEOP. XLVII. — BOOK I. 139 Prop. 47. — Theoe. — {Most Important.) In any right-angled triangle, tlie square which is described upon the tide subtending, or opposite to, the right angle, is equal to the squares described upon the sides containing the right angle. Cons. — P. 46. To describe a square on a given straight line. P. 31. Through a point to draw a st. line parallel to a given st. line, Pst. 1. Any two points may be joined by a st. line. Dem. — Def. 30. A square has all its sides equal, and its angles rt. angles. P. 14. If at a point in a st. line two other st. lines on the opposite sides of it make the adjacent angles together equEd to two irt. angles, these two st. lines shall be in one and the same st. line. Ax. 1. Things equal to the same, are equal to each other. Ax. 2. If equals be added to equals, the wholes are eqiial. P. 4. If two triangles have each two sides and their included angle equal, the triangles are in every other respect equal. P. 41. If a parallelogram and a triangle be upon the same base and between the same parallels, the parallelogram is double of the triangle. Ax. 6. Things double of the same, are equal to each other. Exp. 1 2 CONS.l 2 3 Dem. 1 2 Hyp. Concl. 6yP 46. P. 31. Pst. 1. 6yH&Def.30 P. 14. H.Def.30\ &P, 14./ Let ABC be a A and BAG a rt. angle ; then the sqnare on BC = the squares on AB and AC. Draw on BC a square BE; on BA, a square BGjandonAC, a square HC ; through . A draw AL|| BDortoCE; join AD and EC ; also AE and BK. •/ z. s B AC and BAG are each a rt. angle ; .-. the lines AC, AG on opp. sides of AB inake the adj. z. s = two rt. angles, and .*, CA is in the same st. line with AG. Also AB is on the same st. line with AH. I-IU GRADATIONS IN EUCLID. Dem. 4 C.l, Def. 30 Axl, Add. Ax. 3. C.l,4D6f,30 •.• the LB DBC and FBA are each a rt. angle ; f .".the L DBC = the L FBA : To each of the equals add the L ABC; .'. the whole l DBA = the whole ^ FBC : Hence, /. AB = FB, BD = BG, and ^ DBA = ^FBC; P. 4-, .'.baseAD = baseFC,andAABD = AFBC. C. 2. Now the crrj BL and the A ABD are both on the same base BD, and between the same parallels BD and AL ; P. 41. .', the c=y BL is double of the A ABD. C. 2. Also the square GB and the A FBC are both on the same base FB, and between the same parallels FB and G C ; P. 41. .', the / — 7 Of square GB, is double of the AFBC; D. 9. Butthe A ABD = the AFBC; Ax. 6. .'. the I — J BL =? the square GB. C. 3. Also, after joining AE, BK, the izzr CL = the square HC : Ax. 2. .'. the whole square BDIEC — the two squares GB and HC ; C I. Now the squares are BE on side BC, BG on side BA, and CH on side C A ; Concl .'. the square on side BC = the two squares on sides BA and CA. Becap, Therefore, in any right-angled-triangle, 4 Tin ATI2 A T.a A Tja AK^ AL« AD» AB» area AEF _ area AHC area AKE area ALH AD» AB« and AD» area ADK . AD« ■ AB» area ABL AB» • Ui GRADATIONS TN EUCHD. Adding these equals, we have, areaADKEF _ areaABLHC . areaADKEF _ AD AB AD» AB" aieaABLHC -^See Tate's Oeometry, p. 92. 2°' To make a circle the double or the half of another circle. Let AB be the diameter of ADBC ; at . A raise a perpendicular AE, and at.B make the angle ABE equal to half a rt. angle ; produce | B C until it cuts AE : the square on BE will be double of the square on AB, and the circle of which BE is the diameter double of the circle of which AB is the diameter. Again, let B E be the diameter of a circle ; at , s E and B make angles each equal to half a rt. angle ; and the square on AB will be one-half of, the square on BE ; and the circle of which AB is the diameter one-half of the circle of which BE is the diameter. 2. By this 47th Proposition, the Chords, Natteral Sines, Tangents, and Secants of Trigonometrical Tables wre constructed. With St. line A C as radius G desc. an arc C G, and from . A P\ ~^^ i^,., ^-"'H raise a perpendicular AG; the arc CBKG being the measure of a rt. angle, is equal to 90° : join C and G, CG is tJie line of I chords ; — on which the chords of f C B 30°, C K 60°, being inflected, C6 is the chord of 30°, Ch the chord of 60°. The sine of arc CB is BD, the tangent CE, the secant AE. The co-sine is FB, the co-tangent Q H, the co-secant ■"■ AH. Let it be supposed that the radius AB is divided into 1,000,000 parts, and that the arc BC is 30°. Since the chord C4 of 60° is equal to the radius AC ; BD the sine of 30° shall be equal to the half of AC, or 500000, in the rt. angled triangle ADB. Now, AB' = AD^ + BD'' ; and AB« - BD' = AD" or BF', the sine of the complement: substituting the numbers we have V (1000000' - 600000') = 866025 = FB. Next, as the triangles ABD, AEC are equi-angular, we have the proportion AD : BD :: AC : CE; therefore the tangent of 30° CE - BD . AC, AD „^ 500000 X 1000000 50000000000 cTwocn mt. »,,. ^r,o , x,„ "'' 866025 = -866025- = ^^^^^"^ Then AC + CE' = AE', and AE is the secant of 30° ;— therefore V (1000000' -t- 577350') = 1154703, the Natural Secant for an arc of 30°. 3. Ai-ithnuticaUy, when the given numbers are 8, 4, 5, or their equi- multiples, the sum of the squares of the two less is equal to the squares of the greater, as (6 X 6) -I- (8 X 8) = 10 x 10 = 100, and when any two are given PUOP. XLVII. — BOOK X. 145 we can find the third exactly : but with respect to all other numbers, though the sum of the squares of any two numbers always equals or constitutes the square of a number greater than either, we cannot attain that number with perfect accuracy ; excepting in the case of right triangular nmiibers, all we can do is to approach its value by increasing the number of decimal places in theroot. Thus (5 X 5) + (8 x 8) == 25 + 64 = 89 ; but the square root of 89 is 9'433981, &c. — See the Introduction, § vi., on Incommenawable Quan- tities. Sight triangula/r numten may he found thus ; 2 4^- 2 X 25 Put n any odd number, then the third number : = the second number, 1 „- ._^49+ 1 aud- + 1 = 25. -Now = 24, and-\- 625 ; the rt. triangular numbers thus, take 7, then (24 X 24) + (7 X 7) = 576 + 49 =25 idng 7, 24, and 25. 4. The height of any elevation on the earth's surface is so small when compared with the earth's diameter, that for practical purposes, as levelling, and ascertaining the height of mountains, we may consider the earth's actual dja,meter, and the dia- meter + the elevation, as the same quantity, i.e., B E and L E not sensibly to differ ; nor the arc AL from the horizontal level AB. W^ assume LE to be 7960 miles, or that we may have an easier number, 8000 miles. If we take AB one mile, then BL=Tinnr P^i'* "f * mile, or nearly 8 inches ; i.e., for every mile of survey, the swface or curvature of the earth is 8 inches below the horizontal level. b. Heights and Distances from the curvature of the earth are computed by Prop. 47, from the principle established in Prop. 16, bk. iii., that the tangent AB is perpendi cular to the radius C A of the arc AL. Then if AB be required, we have \/ (LC + LB)' — AC = AB : if LB, the formula is VAB»+AC' = BC, and BC-LC or AC = BL. Example 1. Oiven B L, the height of the Peak of Teneriffe ; what will be the radius of its horizon, or the distance at which it may be seen ? Here CB=CL + LB. And VCB'-AC = A B the horizontal radius. Or, 4002» - 4000'' = 16016004 - 16000000 = 16004. And ^16004 = 126 miles = A B, distance at which visible. Ex. 2. A meteor B is seen over a distance from X to D of 200 miles ; required its height. Here BL=BC-LC or AC. An d v'AC''+AB»=CB. Or, V^OOO' + 100« = V16010000 = 4001-24 miles. And 4001'24-4000 = 1'24 miles, height of the meteor, Ex. 3. A fountain B one mile from A, is observed from, A to have the same apparent level : horn much is B above A. 1 L e., how much is B further from the f/iffth's centre than L or A ? Here BC - LC = BL. And V (4000" + 1") = 4000-0001255 = B C : then 4000-001255 - 4000 = ■ 0001255 of a inile = curvature 8 inches nearly. 146 GRADATIONS IN EUCLID. By Prop. 36, bk. iii., the square of the tangent AB equals the rectangle of BL into BE ; and as in levelling the distajices are usiuiXly small, AB' = BL X EL nearly. When AB is 1 mile, BL is § of 1 foot, or 8 inches ; „ AB is 2 miles, BL is § of 4 feet, or 32 inches ; „ AB is 3 miles, BL is | of 9 feet, or 6 feet ; „ AB is 4 miles, BL is | of 16 feet, or 10"6 feet. Thus two-thirds of the square of the number of miles that the level is long, gives the height of B above A in feet, or what the horizontal level differs from, the level of the earth' i earvatwre. Prop. 48. — Thbor. If the squares described upon one of the sides of a triangle he equal to the squares described vpon the other two sides of it, the angle contained hy the two sides is a rt. angle. Cons. — P. 11. To draw a st. line at rt. angles to a given st. line from a given point in it. P. 3. From the greater st, line to cut off a part equal to the less. Pst. 1. A st. line may be drawn from one point to another. Dem. — Ax. 2. If equals be added to equals, the wholes are equal. ~ P. 47. In a rt. angled triangle the square on the side subtending the rt. angle is equal to the sum of the squares of the sides containing the rt. angle. Ax. 1. Magnitudes equal to the same magnitude, are equal to each other. P. 8. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal, the angle which is contained by the two sides of the one shall be equal to the smgle contained by the two sides equal to them of the other. Exp. 1 2 CONS.I 2 Hyp. ConcL ,6yP. 11. P.3&Pst.l In triang. ABC let the square on BC = the sum of the squares on AB and AC; then the l BAG is a rt. angle. At . A draw A D at right angles with AC; make AD = AB, and join DC, A / B PKOP. XLVIII. — BOOK I. 147 Dem. 1 2 3 4 r. 9 10 11 12 ly C. 2. Add. Ax. 2. C. 1. P. 47. Hyp. 1. Ax. 1. C.2&D. P. 8. C. 1. Ax. 1. Eecap. ".■ D A = AB, .". square on D A = square on AB; Let the square on AC be added to each ; then the squares on DA and AC = the squares on AB and AC : But L DAC is a rt. angle ; .". the square on DC = the squares on AD and AC. Also the square on BC == the squares on AB and AC j .'. the square on DC = square on BC, and side DC = side BC ; Thus in As DAC, BAC, side AD = side AB, DC = BC, and AC is common; .-. L DAC = L BAC: But L DAC is a rt. angle ; ,'. z. BAC is a rt. angle. Therefore, if the squares described, &o. Q.E.D. SoH. — The 48th is the converse of the 47th Proposition, and may be extended thus : — The vertical angle of a tricmgle is less than, equal to, or greater than, a rt. angle, as the square on the base is less tham, egual to, or greater than, the sum of the squares of the sides. EEMAEKS ON BOOK I. 1. It will have been seen that the First Book is founded entirely on the Definitions, Postulates, and Axioms : — ^the Definitions fixing the meaning of the terms employed ; the Postulates assigning the instruments that may be used ; and the Axioms setting forth the principles on which the comparisons and arguments are conducted. In a few instances, for the illustration of certain propositions, other principles, not belonging to the first book, have been assumed ; but these are to be regarded in their proper light, not as strict proofs, hut as methods of explanation. 14!j GRADATIONS IN EUCLID. 2. A few only of the properties of the circle are mentioned : those of the straight line and rectilineal angle are subservient to the proof of the properties of the triangle ; and all rectilineal figures are either triangles, or may be resolved into triangles. The Fitst Book therefore may in general terms be described as treating of the Geometry of Plane Triangles. 3. Excluding the Definitions, Postulates, and Axioms, it is not iinusual to make a three-fold division of the contents of this Book. IhiS first part, extending from the 1st Prop, to the 26th, unfolds the properties of triangles ; the second, from Prop. 27 to 32, those of parallel lines ; and the third, from Prop. 33 to 48, those of parallelograms, of course including the square. 4. The most important Propositions are, — three, — namely, Props. 4, 8, and 26, containing the criteria, or conditions of equality between triangles ; one. Prop. 32, the equality of the exterior angle to the two interior and opposite angles, and of the three interior angles of every triangle to two right angles ; oiie. Prop. 41, the proportion of the parallelogram to the triangle on the same base and between the same parallels ; and one, Prop. 47, the relation between the hypotenuse and the sides about a right angle. These propositions at least must be thoroughly mastered, not by committing them to memory,* but by becoming so perfectly familiar with the principles contained in them, and with the con- nexions which exist between the arguments or reasonings employed, as never to feel at a loss for the demonstration, however diversified may be the figures constructed, nor even though no figure at all be drawn. The great aim should be to understand, and as a means to this, to follow up each proposition regularly through all its gradations, and verify it by its appropriate proofs. * To check the practice of committing to memory, and to induce exact familiarity with the principles and reasons of propositions, tke Utters in the figures used shemld ofte/a le vwHed, GEADATIONS IN EUCLID. BOOK II. CONTAINING THE PEOPERTIES OF RIGHT-ANGLED PARALLELOGRAMS, OR RECTANGLES. In this Book, the relatioas will be investigated between the rec- tangles formed by the segments of straight lines^ or of st. lines produced. When a st. line is out or divided at any point, the segments are the portions between the point and the extremities of the line ; when that point is within the extremities, the line is out internally ; when the poiijt assumed is without the given line, and the line has to be lengthened, it is cut extemallyif- — ^the pro- duction of the line in this case containing the point of section. If a st. line is cut internally, the line is the sum, of the segments ; but if cut externally, the st. line is their d$fevence^ The subject of Geometry being magnitude, and not number, it is necessary, as we have said (p. 22), to discriminate between the Geometrical conception of a rectangle, and the Algebraical or Arithmetical representation of it : yet the latter, as illustrative of the Geometrical truth, wilt materially assist the former,— our ideas, of number being inore definite than our ideas of space. Accord- ingly, to each of the Propositions will be appended, what some have named, though loosely, the Algebraical or Arithmetical proof. The numerical area of a rectangle is obtained by supposing the two sides containing the rectangle to be divided into a number of linear units of the same kind, as inches, feet, &o., and then multiplying the units in one side by the units in the other ; the proohict represents the, J.rea or enclosed space. Of the two sides, one is considered as the base, the other as the altitude ; and they may be represented by the letters 6' and a .• thus the formula for the area of a rectangle will be ab ; and for that of a triangle "^ or J ab; and for a square a^ or 6^, according to the side taken, — ^the sides in this case being equal. 150 GRADATIONS IN EUCLID. Definitions. 1. Every right-angled parallelogram, or rectangle, is said to be contained by any two of the st. lines which contain one of the right angles. The rectangle is contained by any two conterminous sides. 2. In every parallelogram, any of the parallelograms about a diameter, together ■^ with the two complements, is called the Gnomon. Thus the parallelogram HG, together with the complements AF, FC.. is the gnomon; which is more briefly expressed by the letters AGKorEHC, __ which are at the opposite angles of the 13 parallelograms which make the gnomon. Axiom. D 7 K a " The leading idea, which runs through the demonstrations of the first eight propositions of book ii., is the obvious axiom, founded on the 8th Axiom, bk. i., that the whole area of every figure, m each ease, is equal to all the parts of it taken together." — Potts' Euclid, p. 68. N.B, — The propositions, &c., required for the Construction and Demonstra- tion will not in every instance be given. The learner is supposed to be familiar with most of them. Prop. 1. — Theorem. // there he two st. lines, one of which is divided into any mtmhsr of parts, the rectangle contained by the two st. lines is equal to the rectangles contained by the undivided st. line and the several parts of the divided st. line. Cons. — 11. I. At a point in a st. line to draw a right angle. 3. I. From the greater of two st. lines to cut off a part equal to the less. 31. 1. Through a point to draw a st. line parallel to a given st. line. Dm — 34. I. The opposite sides and angles of parallelograms are equal. Ax. 8. Magnitudes which coincide are equal : i.e., the whole area of every figure, in each case, is equal to all the parts of it taken together. Exp. 1' Hyp. 2 Concl. CONS.1 2 3 Dem. 1 2 3 4 5 6 7 8 9 COE. SCH.- hy 11. I. 3. T. 31.1. Concl. 6yDef.l&C. Concl. Def. 1 & C. Concl. C. & 34. I. Concl. Sim, Ax. 8. Eecap. PROP. I. — BOOK II. 151 Let A and BC be A B the two St. lines, BC being divided in . s D and E ; then qA.BC = a A . BD + n A . DE + en A. EC. At . B draw BF at rt. angles to BC ; make BG = A : through D, E, and C draw DK, EL, and CH lis BG, and through G, GH || BC ; then □ BH = as BK + DL + EH. .• en BH is contained by the lines GB, BC, of which GB = A ; •. cdBH = i=iA.BC: Also •.• a BK is contained by GB, BD, of which GB = A ; .• aBK = □ A.BD: And •.■ □ DL is contained by DK, DE, of which DK = GB = A; •. nDL = nA.DE: In like manner cnEH = cziA.EC: □ A.BC = asA.BD + A.DE + A. EC. Wherefore, if there he twost. lines, one of which, &c. Q.B.D. 2 A. ^BC; or3 A. ^BC; of 4 A. ^BC, &o. = A.BC. -The propositions of this Book may be verified by Algebra and by Arithmetic ; and in doing this we shall first state the Hypothesis algebraically and numerically, and then separately give, what are denominated, the Algebraic and Arithmetical Proofs. Alg. & Arith. Hyp.— Let A = a = 6;BC = 5 = 10; BD+DE+EC = to + n+^ = 5 + 3 + 2 = 10. Alg. 5 = m + n + p Arith. 10 = 5 + 3 + 2 (x o) al = am + an + ap (x6) 6 x 10 = (6 x 5) + (6 x3) + (6x2) or, 60 = 30 + 18 + 12 Use ahd App. — One of the methods of Dmumstrati'ng the Rule for*the Multiplication of numbers depends on this proposition. In the last figure, let A represent 8, and BG 54. We out' or separate the number 54 into as many parts as there are digits : for example, 50 + 4 ; each 152 GHA3ATI0NS IN EUCLID. part is multiplied by 8 : the one part 4 x 8 = 32, and the other part 50 x 8 = 400, Now, all the partial products taake up the whole product ; therefore (4 X 8) -J (50 X 8) = 6 4 X 8 i or 32 + 40Q = 432. Prop. 2. — ^Thboh, If a St. line be divided into any two^ parts,, the rectangles contained iy the whole st. line and each of the paxts^ are together equal to the square of the whole st. line. Cons. — 46. I. On a given st. line to describe a square, 31. Through a given point to draw a parallel to a given st. line, Dem. — Def. 30. I. Of four-sided figures, a square is, that, which has aU its sides equal, and all its angles rt. angles. Ax. 1. Magwtudes which are equal, &c. Exp. 1 2 CONS.l Dbu. 1 Alg. 5 Hyp.. ConcL by 46. I. 31.1. ConcL Let St. line AB be di- j) ■vided into any two parts in . C ; then the as AB . AC, + AB.CB=< the sij^, onAB. On AB describe the square ADFB ; and through C draw A CE||ADorBF; then as DC + EB = B' the square DB. ^ . P- T B y by Cons. •.• a AF = a s AE + CF, and A is the square on AB : Def. 80. 1, also ".• a AE is contained by AD, AC, of which AD = AB ; Ax. 1. .-. aAE = AB.AC: Cons. And •; □ CF is contained by BF, CB, of which BF = AB; Ax. 1. .-. aCF = AB.CB: D. 3 & 5. Therefore as AB . AC + AB . CB = the square on AB. Kecap. If, therefore, a st. line be divided into any two parts, (fee. Q.E.D. ?.;'. Ij^ AB and CB = twice AB.BC + AC^- On AB make a square ADEB, and join D F E DB; through . C draw C F || B E, and „ .G „ HK||AB; then the squares AE + CK = twice AK + HF,, •." the compl. A G = the compl. G E ; on adding □ C K to each, mAK =iiiiCE; .•. a AK + a CE = twice □ AK ; But AK + CE make up gnom. AKP + D CK j .". AKF and CK together = twice AK: but|BK = |BC, twice □ AK = twice AB . BK, and twice AB . BK = twice AB.BC; .'. the gnom. AKF + Q CK = twice the' rect. AB.BC; Adding to both the equals HF, i.e., the square on HG or AC, then AKF + CK + HF = twice AB. BC And AC squared : But AKP + CK + HP make up the figures ADEB and CK, and ADEB and CK are the squares on AB' andCB, .-. AB" +BC2= twice AB.BC + ACl Wherefore, if a st. line be divided into any: two parts, &c. Q.B.D. PROP. VIII. — BOOK II. 165 Alg. Ji Anth. Byp. — Let AB = a linear units = 16 ; AC = m = 9 ; and CB =«=7. ^Ig. Then a=m + »; Squai-ing, a"=m' + 2m» + »'. (Addm") o"+ji«=m=' + 2m«+2»2. But 2mre+2m''=2 (m+n) n=2an, therefore a'+n' — 2an + m^. Arith. 16 = 9 + 7; Squaring, 256 = 81 + 126 + 49. (Add 49) 256 + 49 = 81 + 126 + 98. But 126 + 98 = 2 (9 + 7) 7 = 2 (16x7)=224, therefore, 256 + 49 = 222 + 81 = 305. Another form of stating the same result is, LetAB=a=16; AC=6 = 9; and BC=a-i = 16-9=7. thenAB'>=a= Aud2AB BC = 2a'-2ab BC==o*-2a6 + 5'' AC= 6^ Sum2o''-2a6 + 6» = 2a'-2ab + h' CoE. 1. — I^ AB and BC be considered as two independent lines, AC being their difference, " the sum of the sqwxres of any two St. lines is equal to twice the rectangle under them together with the square of the difference;" i.«., AB^ + BC'' = 2 AB . BC + AC; or 100 +64 = (2x80) + 4. CoE. 2. — Hence and (by 4. II.) the square of the sum of two st. iines, the sum of their squares, and the square of their difference, are in arithmetical progression — the common different^ being twice the rectangle under the sum. By 4. 11. (AB + BC)^ = AB^ + BC^ + 2 AB . BC ; and by Cor. 7. II. AB^ + BC^ = (AB-BCf + 2AB.BC; or AC^ + 2 AB. BC. Thus the common difference is 2 AB . BC ; therefore the quanti- ties (AB + BC)^ (AB^ + BC*), and AC are in arithmetical progres- sion; as 324, 164, and 4, — the com. dif. being 160. Peop. 8. — Theoe. If a St. line be divided into any two parts, four times the rec- tangle contained by the whole line and one of ths parts together with the square of the other part, is equal to the square of the st. line which is made up of the whole and that part. 166 GRADATIONS IN EUCLID. CoKS. — Fst. 2. A terminated st. line may be produced in a st. line. 3. 1. From the greater st. line to cut off a part equal to the less. 46. 1. On a given st. line to describe a square. 81. I. Through a given point to draw a st. line parallel to a given at, line. Deu. — 34. 1. The opposite sides and angles of parallelograms are equal to- one another, and the diameter bisects them. Ax 1. Magnitudes equal, &c. 36. I. Parallelograms upon equal bases and between the same parallels are equal. 43. 1. The complements of the parallelogram which are about the diameter of any parallelogram, are equal to one another. 4. II. If a at. line be divided into any two parts, the square of the whole line equals the square of the two parts together with twice the rectangle contained by the parts. Def. 30. 1. A square is a four-sided figure having all its sides equal, and its angles rt. angles. Cor. 4. II. The parallelograms about the diameter of a square are also- squares. Ax. 2. If equals be added, &c. Ax. 8. Magnitudes which coincide are equal. Ax. 6. Things double of the same are equal. Exp. 1 CONS.1 Hyp. ConcL 6yPst.2&3.I 46. 1. Pst. 1 31.1. Concl. Let I AB be di- vided in the point Cj M then four times X AB. BC with the square on AC added = the square of (AB + BC). C B D to Gi.Ki:i.iN B LR. hQ ! H L F Produce | AB . D, so that BD = BC; On St. line AD describe the square AEFD, and join ED; through . s B and C draw ||s BL, CH, to AE or DF, and cutting ED in the points K and P ; also through . s K and P, draw I MGKN and jXPRO||stoADorEF; then AK + M R -f (HR -I- BN) + NL -I- XH, fill up the figure AEFD. PROP. VIII. — BOOK II. ler hyC.l.&SiJ. Ax. 1. C. l&D. 2. 36. I. C. & 43. 1. D. 4. Ax. 1. f} Ax. 6. C.l,Def.30 & 31. 1. 34.I.Cor.4.II Ax. 1. D. 12 & 2. C. 3 & 4. 36. I. - Const. 43.I.&AX.1. D. 15 & 17. Ax. 1. D. 9. Add. &Ax.2. Const. D. 21. Ax. 1. .Id Cor. 4. II. 34. 1. & Ax. 2. Ax. 8. Concl. Becap. V CB = BD, CB = GK, and BD = KN ; .•.GK = KN, In like manner PR = RO : And •.• CB = BD, and GK = KN ; .-. n CK = BN, and GR = dRN : But as CK, RN are bompls. of the paral- lelogram CO, and therefore CK = RN : also rect. BN = reot. GR ; .-. the four as BN, CK, GR, RN, equal one- another ; and therefore the sum of these four = four times CK. Again, •.• CB = BD, BD = BK, and BK = CG; and also •.• CB = GK, and GK ^ GP : .-. CG = GP. And •.• CG = GP, and PR = RO j and AX || CP, and PO || HF ; ' .•. reot. AG = MP, and reot. PL = RF : But •.• as MP, PL are the compls. of the: parallelogram MKLE, .'. □ MP= PL, and also □ AG = RF ; .•. ens AG, MP, PL, and RF are all equal to one another ; and the sum of the four = four times any one, as AG. But (BN + CK + GR + RN) = four times CK: .•. the eight rectangles in the gnomon AOH = four times AK. Now rect. AK is contained by AB . BK, and AB.BK = AB.BC; .*. four times AK = four times AB . BC : But four times AK = the gnomon AOH; .-. four times AB . BC = AOH : Add to each XH, that is the square on AC f then four times AB . BC + AC = AOH + XH: But AOH + XH make up AEFD, tha square on AD ; .-. 4AB . BC + AC' = AD' = (AB + BC)«. Wherefore, if a st. line be divided, &o. Q.E.D. 168 GRADATIONS IN EUCLID. Alg. ikAritli. Sjfp.— Given AB = o=16 ; AC=m=10, aiidCB = Alg. ThenTO + n. = ffl; taking » from jlWiA. 10 + 6 = 16 ; (-6) and 10 = 16 each, m=a-n ; Squaring, m' =a' - 2 a»+ »" ( + 4 an) then 4 am + m' = a' + ian+n' ; Buta= + 2o»+»== (a+m)", Therefore 4a»+TO' = (o + m)'. We obtain the same result in another form : LetAB = o=16, BC = i = 6, and AC=a- 6 = 16-6 = 10 ; then4 AB.BC=4a6y And (AB + BC)^ =a'' + 2 a6 + 6' AC2=a"-2a5 + i'' Squaring, 100=256-192 + 36, ( + 4x96) then 384 + 100 = 256 + 192 + 36; But 256 + 192 + 36= (16 + 6)', therefore 384 + 100= (22)^ = 484. Sum a^ + iah + l^ Sum a'+2db + b' SoH. — 1. The Proposition may be otherwise expressed : — " The square of the swm of two St. lines is equal to four times the rectangle under them together with tlie square of their difference ;" thus (AB + CB)' = 4AB . CB+ (AB - CB)'' = (16 + 10)'' = 4(16xl0) + (16-10)'' ; or 676 = 640 + 36. 2. Or, "fowr times the square of half the sum is equal to four times the rectangle under the lines together with fowr times the square of half the difference ; " iihus, 4 ^"t^^^' =4AB.CB + 4 <-^^~^°)'' =4xl69 = (4xl60) + (4x9) = 676. Use akd App. — The principles established in Props. 6, 7, and 8 are applied to Algebra and to various operations connected with the extraction of the Square root. Peop. 9. — Thboe. If a St. line he divided into two equal parts, and also into two miequal parts, the squares of the two unequal parts are together double of the square of the half line and of the square of the line between tlie points of section. Cons. — 11. I. To draw a st. line at rt. angles to a given st. line at a given point in it. 3. I. From the greater of two st. lines to cut off a part equal to the less. Pst. 1. A st. line may be drawn, &o. 31. I. Through a given point to draw a St. line parallel to a given st. line. Dbm. — 5. I. The angles at the base of an isosc. triangle are equal. 32. I. If a side of any triangle be produced, the ext. angle equals the two int. and opp. angles ; and the three int. angles of every triangle equal two rt. angles. PROP. IX. — BOOK II. 109 '29. I. A St. line falling on two parallel st. lines makes the alt. angles equal, and the est. angle equals the int. and opp. angles upon the same side, and the two int. angles upon the same side equal two right angles. *. I. If two angles of a triangle be equal to one another, the sides opposite the equal angles shaU be equal to one another. 47. I. In every rt. angled triangle the square on the side subtending the rt. angle is equal to the sum of the squares on the sides con- taining the rt. angle. 34. I. The opposite sides and angles of parallelograms equal one another, and the diagonal bisects them. Ax. 1. Magnitudes equal to the same are equal. & G-- xF -.-a: :exp. 1 2 »CONS.l 2 3 5 Dem. 1 2 3 4 5 6 7 Hyp.] » 5 Concl. 6y 11. I. 3. I. & Pst. 1 31.1. 31. 1. & Pst. 1 ConcL 6y C. 2 & 5. 1. C. 1 & 32. I. D. 1 & 32. I. D.4,29.LC.4 32. I. 6.1. A C D B Let St. line AB be divided into, two equal parts at . C, and into two unequal parts at . D ; then AD'' + DB== 2 (AC + CD"). From . C draw CE at rt. angles to AB : make CE = AC or CB, and join E A, EB ; through. D draw DF || CE, and meetin!' EBinF; and through . F draw FG 11 AB, and ioia AF; then the squares on AD and DB = twice the squares on AC and CD. V AC = CE, .-. L EAC== A. AEC; and •.• ACE is a rt. angle, .*. i s AEC and EAC = a rt. angle ; and z. AEC being = i. EAC, each is half a rt. angle. So, V L CEB = L EBC half a rt. angle, .-. (AEC and BEC) or the whole z. AEB is a rt. angle : And •.• /. GEF is half a rt. angle, and z. EGF = ECB a rt. angle ; .•. rem. z. EFG = half a rt. angle, and z.GEF= Z.EFG; and .•. the side EG = the side GF, n 171) GRADATIONS IN EUCLID. A, D B D :m. 8 9 10 11 12 D. 4. 32. I. 6.1. C. 2. Concl 29. I. 13 C. 1. '&47. I 14 D. 7. Again, •.• z., FED = half art. angle, and FDB = the int. z. ECB a rt. angle ; .". rem. l BFD is half a rt. angle, and Z1.FBD = Z.BFD; and .'. the side DF = the side DB And V AC = CE, .". AC" = CE%- and .-. AC + OE^ = twice ACl But /.ACE being a rt. angle, AE^ = A C= + CE^ or twice AC^ : Again, since EG = GF, the square on EG = that on GF ; and .-. EG" + GF'' = twice GF" : butEF^ = EG^andGF"; .-. EF" = twice GF" which = twice CD*j and from above AE" = twice AC"; .-. AE" + EF" = twice (AC" + CD"): But /. AEF being a rt. angle, AP" = AE" + EF"; and .-. AF" = twice (AC" + CD") : But L AD¥ being a rt. angle, AF" = AD" + DF"; and .-. AD" + DF" = twice (AC" + CD"); and side DF equalling side DB, AD" + DB" = twice (AC" + CD"). Wherefore, if a st. line be divided into twOf &0. Q.B.D. Alg. & Arith. Byp. — Let AC, CB each = <»=10 ; AB = 2a = 20; and C D = m = 4. Also, let AD = o + to = 10 + 4 andDB =a-m = 10-4. 24 15 47. I. 16 34. I. 17 D. 13. 18 D. 13 & 15. 19 D. 4. & 47. 1. 20 Ax. 1 & D. 18 21 C. & 47. I. 22 At 1. 23 D. 10. Eeoap. ji^. Then(a + TO)^=a2+2am+m2 smd (a-m)^ = a^ -iam + mj Adding (a + m)= + (a-m)^ =20^^ + 2m' Anth. Then (10 + 4)" =100 + 80 + 16 = 196 and a0-4)" = 100-80 + 16 = 36 Adding (10 + 4)« + (10 - 4)" = 200 + 32=232 PROP. X. — BOOK U. 171 Another form of the same Algebraical result is, AD2= a^ + 2am+ m? And2AC==2a2 DB''^ a'-2am+ m' 2CD''= liii^ AD«+DB2 = 2o" + 2m2=2AC' + 2CD=i = 2a^ + 2mi' ScH.— 1. The Proposition may be expressed, "the. sum of the aqmans of any tioo lines is equal to twice the square of half their sum togetlier with twice the square of half their tUference." Because AD2 = (AC 4- CD)!' =AC'' + 2AC. CD + CD'' AndBD2 = (BC-CD)^ = {AC-CD)''=AC''-2AC.CD + CD'' By addition AD^' + BD" =2AC'' + 2CD2 But AC= ^P + PS and 00= ^°-°^ 2 2 Therefore, tTie sum of the squares of any two lines, &c. q.e.d. 2. Or, the sum of the squares is equal to half the square of the sum together with half the square of the difference; thus AP' + BP"-^^^'*"^^^'' + ( AD-DB)" . ^j. j^ ^„„ji,gj.3_ U^ + 6^ i.e., 196 + 36 = .!^^ + — J.e., 200 + 32=232. Pkop. 10. — Theoe. If a St. line he bisected and produced to any point, the stivmre of the whole line thus prodticed, and the square of the part of it produced, are together double of the square of half the line and of the square of tlie line made up of the half and the part produced. Cons.— 11. 1., 3. 1., 31. 1., Pst. 1. Dem.— 29. 1., Ax. 9. Ax. 12. If a St. line meet two st. lines, so as to make the two int. angles on the same side of it taken together less than two rt. angles, these two st. lines, being continually produced, shall at length meet on that side on which the angles are less than two rt. angles. 6. I., 32. I. 15. I. If two st. lines cut one another, the opp. or vertical angles shall be equal. 6. I., 34. I. 36 . I. Parallelograms upon equal bases and between the same parallels are equal 47.1, 172 GRADATIONS IN EUCLID. Exp. 1 1 Hyp. Concl. CONS.1 2 3 4 Syll. I. 3.1. & Pst. 1. 31.1. Pst. 2 & 1. 5 ConcL Dbm. 1 byC.3&29.I. 2 3 Ax. 9. Ax.l2,Pst.l. 4 5 6 7 C. 1 But (2a+m)"+m=' = 4a'' + 4a«i + 2m" therefore {ia + mY+m' =2 a" +2 (o+to)» ArUh. ThenAB + BD=AD = 20 + 2 = 22 andCB + BD=CD = 10 + 2 = 12 therefore (20 + 2)" = (4 x 100) + (4 X 20) + 4 ( + 4)and(20 + 2)"+4 = (4xl00) + (4x20) + {2x4) Again (10 + 2)" = 100 + 40 + 4 174 GRADATIONS IN EUCLID. ( + 100) and (10 + 2)'= +100 = 200 + 40 + 4 (x2)and 2 (10 + 2)^+200 = 400 + 80 + 8 But (20 + 2)» + 4 = 400 + 80 + 8 therefore (20 + 2) = + 4 = 200 + 2 (10 + 2)' or 484 + 4 = 200 + 288 = 488 In another form the Algebraical and Arithmetical illustration may be stated. LetACorCB = a = 10;BD=m = 2;CD = a + )7i=:10 + 2 = 12; and AD = 2 o + TO = 20 + 2 = 22 Then AD'' =ia^ + iam + m' And 2 AC = 2 a» DB' = m^ 2 CD^ = 2 g' + iam + 'im'' AD" + DB= =4a'' + 4am+2m'' =2 AC^ + 2CD= = 4 a'' + 4aTO + 2m» Or, 22 X 22 = 484 And 2 x 10 x 10 = 200 2x 2= 4 2x12x12 = 288 Sum 488 = 488 ScH. — 1. Propositions 9 and 10 are applicable to Algebra, and like Prop. 6 and 6 are identical ; for the different enunciations arise from the two ways of representing the differences between two lines. Gen. Use and App. op Prop. I.-X, Bk. II. — These ten Propositions contain the whole theory of the relations of rectangles and squares formed by lines and their parts. As we have shown, all these relations may be expressed Algebraically, and may also be applied equally well to numbers. " When lines are expressed numei-ically, various problems may be proposed respecting them, the solution of which may be derived from the preceding propotir • tions. We shall here subjoin some of those problems, which will probably be sufficient to familiarise the student with such investigations." 1°' Given the sum. S, and difference D, of two magnitudes, A B *Ae greater and CD the less; to find the magnitudes themselves. The formulas are, — QJ T\ S D —+-^=AB the greater ; aud-^ ^ = CD the less. 2°' Since the area of a rectangle expressed in numbers is equal to the product of its sides, — if the Area be divided by one side, the quotient will give the other : thus, let the sides be AB and BC ; the rectangle is expressed by AB . BC • ,AB.BC .„ AB.BC __ , _, and — g-g — = AB ; — -— — = BC ; or let the area m numbers be 144, 144 144 and the sides 9 and 16 ; then — = 16 ; and —= = 9. 30. II rphere are five quantities depending on a, rectangle, any two of which being given, the sides of the rectangle can be found : 1°- The sum of the sides. 2°- The difference of the sides. 3°- The area. 4°' The sum of the squares of the sides. 5°' The difference of the squares of the sides. These being combined, 1° and 2° ; 1° and 3° ; 1° and 4° ; 1° and 6° ; 2° and 3° ; 2° and 4° ; 2° and 5° ; 3° and 4° ; 3° and 5° ; and 4° and 5°, produce ten Problems, — of which the combination 3° and 5° alone presents any diffi- culty. The solutions will form useful exercises for the learner." — Lardnek'S Euclid, pp. 79 and 80. PROP. XI. BOOK II. 175 Prop. 11. — Prob. To divide a given st. line into two parts, so that the rectangle eoTb- tained by the whole and one of the parts shall he equal to the square ■o/ the otlier part. Sol. — 46. I. To describe a square on a given st. line. 10. I. To bisect a given finite st. line. 3. I. Prom the greater st. line to cut off a part equal to the less. Psts. 1 and 2. Dem. — 6. II. If a st. line be bisected and produced to any point, the rectangle contained by the whole lioe thus produced and the part of it produced, together with the square of half the hue bisected, ia equal to the square of the st. line which is made up of the half and of the part produced. 47. I. In any rt. angled triangle, the square of the hypotenuse is equal to the sum of the squares of the base and perpendicular. As. 3 and 1. Def. 30. I. A square has aU its sides equal, and its angles rt. angles. Exp. 1 2 CONS.1 2 3 4 5 6 Dem. 1 2 3 Datum. Quass. by 46. I. 10.I.&Pst.l Pst.2,&3.1. 46. I. Pst. 2. Sol. 6yC. 2&3. 6. II. C. 2. 47.1. Let AB be th.e given St. line ; ■" to divide it into two parts, so that rect. AB. BH = A H square. A On A B describe square ABDC; bisect AC in E, making AE = EC, ' and join EB ; produce CA to F, making EF = EB ; and on AF describe the square AHPG; Produce GH to K; the line AB is divided in H", so that AB . BH ^ the square on AH. '.• st. line AC is bisected in E, and produced so that EP ^ EB ; .-.aCF.AF + AE^ = EF^andEF^ = EB»; also the square on EB = the squares on AB and AE : 176 Dem. 4 1 Sub. Ax. 3. I Const. Ax. 1. Sub. Ax. 3. C. 4. GRADATIONS IN EUCLID, Take away AE^ from each; then * □ CF.AF = AB^ But □ FK = OF . FA, and AD^ = AB^ A .'. the square AT)^ the rect. FK : Take away A K, and the square FH j;i' = therect. HD: 8 C. (feDef, 30 But en HD is con- tained by HB . BD, orHB.AB; and F H is the C square on AH ; 10 i Concl. .-. the □ AB . BH = AH^ 11 ! Recap. Wherefore, the given line has been dividedy &C. Q.E.F. N.B. — A line thus divided is (in 30. vi.) said to he cut in' extreme and- mean ratio. The Algetraical Solution may be given thus : Take AB=a, — required the point H, so that AB ■ BH shall equal AH''. LetAH=a!; then HB=a-a;. Now by the prob. a (a-x)=x'' ; or x''=a^ -ax; i.e., x^ + ax=a^, « ■ . .1 1 i. 1 o "■' ^ a\ So" Solving the quadratic, we have «^ + ax + — =:a^ +"2" ~ ~Z" extracting the root, x + - ot\/5 i Hence x - ^a\/5 — a 2 Thua AH the one part = x ='^^ — • AB, and HB the other part=a-x=a- AH= — ^^.AB. Or, "required how far a given line must be produced so that the rectangle contained by the given line and the line made up of the given line and the part produced, may be equal to the square of the part prodMced." — Pott'» Euclid, p. 73. Or, "to find two lines having a, given difference, such that the recta/ngle contained by the difference and one of them may be equal to the square of the other." The Arithmetical Solution cannot be expressed in whole numbers; we can, however, approximate to the values of AH suid HB by extending the root of PROP. XII. — BOOK II. 177" 5 to any number of decimal places desired. Thus, supposing AB = a = 10,, 2 9 1st. V^l.a=2-236^7l x 10 = i~ xlO = 6-I8034=AH ; 2nd. ?^. „^3- 2-236068 ^ ^^ ^ -763932 ^ ^^ ^ 3.81966 = HB ; 2 2 2 And AH + HB=AB=6-18034 + 3.81966 = 10. 00000. CoE. I. — To cut a St. line in extreme and mean ratio, it must first he produced in extreme and mean ratio ; that is, CF . FA must equal AB^ Cor. II. — When a st. line EF, or its equal, is cut in extreme and mean ratio, the rectangle AC . (AC — AF) is equal to the square of AH, or AF, or AC . HB = kW. Hence, if a st. line CF be cut in extreme and mean ratio, the- greater segment AC will be cut in the same manner, by taking in it a part equal to the less AF ; and the less AF will be simi- larly cut by taking in a part equal to the difference (AC — AF)' or HB ; and so on. CoE. III. — A St. line CF being cut in extreme and mean ratio,, therect. AC. AF under its segments = AC^ — AH^ the difference between their squares; thus, 10 x 6-18034 = 100 - 38-1966 = 61-8034. ScH. — Let A be a st. line cut in extreme and mean ratio, G the greater ■ segment, L the less, and D the diflerenoe : then 1st. A'' + L^ = 3 G^ ; 2ud.. (A + L)==5G» ; 3rd. A.D=G.L; and 4th. L2 = G.D. Use and App. — This 11th Prop, is applied in 10. iv., to the drawing of an isosceles triangle, of which each of the angles at the base is double of the third angle ; and in 30. vi., to the cutting of a st. line in extreme and mean ratio ; — that is, so that the whole st. line is to the greater segment as the greater is to the less. The construction of pentagons in bk. iv. also depends on this problem^ and of regular bodies, as described in bks. xiii., xiv., and xv., called also the Platonic Solids. Prop. 12. — Theoe. — (Important.) In obtuse angled'triangles, if a perpendicular be dravfn, from either- of the acitte angles to the opposite side produced, the square of the side- subtending the obtuse angle is greater than the squares of the sides 178 GRADATIONS IN EUCLID. ■coniaining the obtuse aiigle hy twice the rectangle contained hy tlie side ■upon which when produced the perpendicular falls, and the st. line intercepted without the triangle between the perpendicular and the obtuse angle. Cons. — 12. 1. To draw a st. line perpendicular to a, given st. line of an unlimited length from a given point without it. Pst. 2. A terminated st. line may be produced to any length in a st. line. Deh. — 4. II. If a St. line be divided into any two parts, the square of the whole line is equal to the square of the two parts, together with twice the rectangle contained by the parts. Ax. 2. If equals be added to equals, the wholes are equal. 47. I. In any rt. angled triangle, the square described upon the side subtending the rt. angle is equal to the square described upon the side containing the right angle. Let ABC a triangle have an obtuse i. ACB; From . A to BC produced draw a perp. A D ; then AB» > (AC= + BC^) bytwicethe rect.BC. CD. For, since | BD is divided into two parts at . C ; .-. BD^ = BC^ + CD^ + 2BC.CD: adding AD^ to each, BD''+AD^=BC^+CD> + AD''+ 2BC. But L ADB being a rt. ang., AB'' = BD^ + AD^ and also AC^ = CD» + AD''; therefore AB^ = BC^ + AC^ + 2 BC . CD ; i.e.,AB' > AC' + BC' by twice BC . CD. Where/ore, in obtuse angled triangles, if a perpendicular, &c. Q.B.D. Exp. 1 byRjp. Cons. 2 byl2.I.&Fst.2 3 Concl. Dbm. 1 byRjp. 2 4. II. 3 Add. & Ax. 2. 4 C. 2 & 47.1. 5 6 7 8 47. I. Concl. Explan. Recap. .CD. ATff. & Arifh. ^j^p.— Let BC=-a=5 ; CA=,6 = 6-708 ; AB=c=10; let CD=m=3 ; DA=n=6 j then BD=a+m=5 + 5 = 8. the triangles ABD andACD are rt. angles, viz., at ADB and ADC. -Alg. Sincec'' = (a + m)''+»' ; and52=m^+»2; therefore c'' -h'^ = {a,+'mY - m'' =a^ + 2 am + m^ - m'' = a' + 2 am. and c' =b' +a' +2 am ; i.e., c- is greater than h^ +a^ by twice am. ^d PROP. XIII. — BOOK II. 179 Ariih. Since 10'' = (5 + 3)2 + 62=64 + 36; and 6-708' or 45 = 3" +6'' =9 + 36 ; therefore 10" - 6-708", or 100 - 45 = (5 + 3)" - 3" =64- 9 =25 + 30 ; and 10» = 6-708=+52 + (2x5x3) = 45 + 25 + 30 : i.e., 10= or 100 greater than 45 + 25 by 30. Use aud App. — 1. By this Proposition the Area of a triangle may he -ascertained when the three sides ore hnovm. ' Let the sides AB = 20, AC = 13, and BC = 11. Now (12. 1.) AB" =AC''+BC2 + 2BC.CD ; -therefore AB» - AC - BC" ; or AB" - (AC" +BC") = 2 BC . CD 5 i.e., 2 B C . CD = 400 - (169 + 121) = 400 - 290 = 110 ; therefore BC . CD = ^^ = 55, and ?%^ = CD = ^ = 5 : But (47. 1.) AC" -CD" =AD"; i.e., 169-25 = 144 ; andAD = Vl44 = 12. TV^e have now ascertained AD, the altitude of the triangle ABC, and BC, the base, is given : Again (41. I.) the Area of a triangle equals half that of a parallelogram on the same base and of the same altitude : •n, *v A **• 1 iBr. AD.BC 12x11 132 Thus the Area of triangle ABC = 5 = — ^ — = —^ = 60, Or, by bisecting the line BC in G, we obtain the same result ; thus, _. AB"-AC" „„ _,_, ,, . BC.DG ^„ Smce ;- = BC . D G ; therefore — ^,^-=1 — = DG ; And DG+^ = DB; then (47. 1.) AB" -BD" = AD" ; whence wefind AD itself. Numerioally, = = 115.5, therefore -^r^ = 10.5 = DG ; And 10.5 + ^ = 16 ; then (47. I.) 400 - 256 = 144 ; and V144 = AD as before. Peop. 13. — Theob; In every triangle, the square of the side subtending either of the acute angles is less than the sqitares of the sides containing that acute angle hy twice the rectangle contained by either of these^ sides and the straight line intercepted between the perpendicular let fall upon it from the opposite angle and the acute angle. Cons. 12. I. Pst. 2. 180 GRADATIONS IN EUCLID. Dem. — 7. II. If a St. line be divided into any two parts, the squares of the: whole line and of one of the parts are equal to twice the rectangle contained by the whole trnd that part, together with the square of the other part. Ax. 2. 47. 1. 16. I. If one side of a triangle be produced, the exterior angle is greater than either of the int. and opposite angles. 12. II. In obtuse angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than the squares of the sides containing the obtuse angle, by twice the rectangle con- tained by the side upon which, when produced, the perpendicxUar falls, and the straight line intercepted without the triangle between the perpendicular and the obtuse angle. 3. II. If a St. line be divided into any two parts, the rectangle contained by the whole and one of the parts equals the rectangle contained by the two parts together with the square of the aforesaid part. Ax. 6. Doubles of the same are equal. Exp. 1 Cons. 2 3 Case I. Dbm. 1 Hyp. hi/ 12. I. Concl. Let A ABC have the L B acute, On BO one of the sides let the perp. AD fall. from L BAC; then AC < (AB» + BC^) by twice BC . BD. —Let the perp. kH fall within the triangle ABC. 6yC.2&7.n. 3 4 5i Case II, Dem. 1 2 3 Add.&kTL.% C. 2&47.I. D. 2 & 3. Concl. Since | BC is divided at D, CB^ + BD^=CD^ + 2BC.BD; On adding AD^ CB'' + BD^ + AD" = CD" + AD" +2BC.BD: ^ But z. s at D being rt. l s AB" = BD" + AD", and AC" = CD" + AD"; CB" + AB" = AC" + 2B0.BD: that is, A C" < (B C" + A B") by twice B C . — Let the perp. AD fall without the A ABC. BD. by C.&16.I. 12. II. Add. Ax. 2. C. & 3. II. Since z. D is a rt. z. , z. ACB >art. z.. V AB" = AC" + BC" + twice BC. CD; (?, add BC" to each, n n and AB" + BC" = '^ ^ AC" + twice (BC" + BC . CD). But St. line BD being divided in . the □ BD . BC = BC" + a BC .CD; PROP. XIII. BOOK II. 181 Dbm. 5 6 7 8 Ax. 6. Ax. 6. D. 3 & 6. Concl. Now the doubles of equals being equal, twice a BD . BC ^ twice en BG . CD + twice BC^: .-. AB= + BC = AC^ + twice □ BD . BC ; ,-. AC^ alone < AB^ + BC by 2 BD . BC. -Case III. — Lastly, let the side AG be perpendicular to BC. Dem. 1 9 47. I. Add& Ax.2. Recap. HereAB»=AC'' + BC^; and adding BC^ then AB" + BC^= AC^ + 2 BCS or 2BC.BC. Wherefore, in any triangle, the square subtend- ing, &C. Q.B.D. ■€ase l.—Alg. & Arith. Hyp. LetBC = a = 10; AB = c = V^l = 7'8102 ; AC = J = V« = 6-4031 ; BD = m = 6 ; AD = » =5 ; andDC = a-m = 10 - 6 = 4. ^Ig. (47. I.) c^ =n'+m' and i^ = m" + (a-my Subtracting, c^ -l^ =m^~(a-m)^ =m^ -(a^ -2am+m) = 2am-a' Transpose, a' + c^ =i'' + 2 a m y or 6" + 2 am = o^ + c^ therefore, 5^ is less than a'^ + c^ \>y 1 am. jLrith. (47. I.) 61 = 25 + 36, and 41 = 25 + 16 Subtracting, 61 - 41 = (2 x 10 x 6) - 100 = 120 - 100 = 20 Transposing, 100 + 61 = 41 + 120 ; or 41 + 120 = 100 + 61 therefore, 41 is less than 100 + 61 by 120. Oasb II. — The perpendicular AD here passes out of the triangle, and the positions of C and D are changed ; so that we have BC = a = 2;AB = c = V61; AC = 6=^*1; BD=m=6; AD=» = 5; and DC = m-a=.6-2 = 4. -Alg. (47. I.) c= = m= +n^ and &« = {m-a)''+n^ Subtracting c^ - J'' =m,^ -(m-a)'' = m^ -m' + '2am-a' =2am~a* Transpose a' + c' = b' +2 am; or 6= + 2 o«i = a= + c^ therefore b' is less than a" + c^ by twice am Aritk. (47. I.) 61 = 36 + 25, and 41 = 16 + 25 Subtracting 61 -41 =36-16 = 36-36 + 24-4 = (2 x 12)-4 Transpose 4 + 61 = 41 + (2 x 12) ; or 41 + 24 = 4 + 61 therefore 41 is less than 4i + 61 by 24. Case III.— The perpendicular AD aud the side AC coincide, and the points DandC; sothat we have BC = a = 4 ; AD = AC = 6 = 5; andAB" = c=V«. 182 GRADATIONS IN EUCLID. Alg. (47.I.)S'+o» = C» Add a' and 5= + 2 a» = c» +0" %.€., 6' is lesa than c' + o« by 2 a" or 2aa -4W«A. (47.1.) 25 + 16 = 41 Add 16 and 25 +'32 =41 + 16 i. e., 25 is less than 41 + 16 by 2 xl6 Cob. I. — If in the figure to Case II. a perpendicular CG be drawn from the angle G to AB, the rectangle under the side AB and the part GB intercepted between the perpendicular and B is equal to the rectangle of BC . DB. The two rectangles AB . BG Smd BC . DB eacli equal half the difference between the A^ J .1. K-o A-on ■ (AB'' + BC^)-AC» square AC and the squares AB and BC ; ».«., 5 ^— ^ = BC.BD = AB.BG. As (^LtllliU = ^^ = 12 = 2 + 6- = 7-8012 + 1-5364. SoH. — The Propositions 12 and 13 are of high importance,— /w «Acy contain the elements of Trigonometrical Analysis, or ike Arithmetic of Sines. TTsE AHD App. — 1. In finding the Area of a Triangle, it is of the greatest advantage to obtain the perpendicular either by calculation or by measurement. When the three sides of a triangle are given, the perpendicular ma,y be obtained^ on the foregoing principles, in either of the ways following : PniST Method. — When the perpendicula/r, AT}, falls mthin the base. By 12. 1., from ^ A draw AD perpen. to B C ; \ 47. 1., in triangle ABD, AB" = AD^^ + BD" 47. I.,andintriang.ACD, AC = AD« + CD" Subtract, and AB" - AC" = BD" - CD" By 3 & 10. I. From BC cut off BE equal to DC, , and bisect ED in G ; ED is evidently bisected in G, so that 2 GD ^ -s — , — — . = ED. B E (i D Now, since, by Cor. 5. II., the difference of the squares of two lines equals th» rectangle under their sum and difference ; AB"-AC" = (BA+AC) (BA-AC) = (BD + DC) (BD-DC)=BCx 2GD = 2BCxlGD. Dividing now both sides of the equation by 2 B C, we have „^ (BA + AC) (BA-AC) , . T,„ BC ^^ BC , „-. 6D = ^ ^g^ '; and smce BG= -^ .-. BD = — + GO,. and AD = >/^B" - BD" ^^^ perpendicular required. Ex. Qiee/n BC = 5, AB = 4, amd AC = S ; required the perpendieular AD. Bisect BC in G ; then GD =|^ = ,-^ ; and DB = 5 + .^ = 3J therefore, by 47. 1., AD = VI6 - "^» =VVtf - "^ = VW = ¥ = ^ \ PROP. XIV. BOOK II. 18^ Second Method.— WAe» the perpendicular, AD, falls without the base. By ]2. I., from z A draw AD perpendicular to BC produced ; 47. I., intriangle ABD, AB''=AD2+BD2 47. I., andintriangleACD, AC2=AD2+CD» Subtract, and AB^ -A.G' ^BT)^ -CB" By 10. I, let BC be bisected in G ; then BD + CD = 2GD, because BD = 2GB +CD, and BD + 2CD =2GB + 2CD=2GD. AlaoBD-CD = BC. But, by Cor. 5. II., AB« -AC''=(AB + AC) (AB -AC)=(BD + CD) (BD-CD) TliusBCx2GD, or2BCx GD=(AB + AC) (AB -AC) ^ Divide by 9Br!— ..n^ftn- (AB + AC) (AB-AC) 2BC lirowCD=GD-GC; and (47. I.) AD^VAC -(GD-GCi) Ex. Given the three sides BA=5, AC=4, and BC=2 ; required the perp. AD.. By the formula Q-D^^+R^zH- ' -oj 2x2 " 4 * CD=2i-l=li and AD=^'i^IJ5 _V256r25^V.231 16 16 4 = 15-198684^3.y33gy^ 4 2. When the intercepts between the foot of the perpendicular and the angles on the base are ascertained, the Area of the Triangle is by the First Method 2 x AD ; and by the Second Method ^ — x AD ; that is, half the rectangle of the base multiplied by the perpendicular. Prop. 14. — Pros. To describe a square that shall be equal to a given rectilinear Jigure. Sol. — 45. I. To describe a parallelogram equal to a given rectilineal figure^ and having an angle equal to a given angle. Def. 30. A square, &c. Psts. 2 and 3 ; 3. I. 10. I. To bisect a given finite at. line. Pst. 1. Dem. — 5. II. If a St. line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts- together with the square of the line between the points of section, equals the square of half the line. "184 GRADATIONS IN EUCLID. Def. 15 and 16. A circle is a plane figure contained by one line called the oircumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. And this point is called the centre of the circle. 47. I. The square of the hypotenuse, &c. Ax. 3. If equals be taken, &;c. Ax. 1. Things equal, &c. ;Exp. 1 ; Datum. 2 I Quces. •CONS.l 7 Dem. 1 by 45. I. Def. 30. Sup. Pst.2&3.1. lO.I.Pst.3. Pst. 2 & 1. Sol. iy C. 5. be the reotil. -1. B K-- G ■I E -f Let A given figure, to make a square equal to the fig. A. Describe a right angled en BCDE = A; "' " "' If the side BE = ED, the fig. is a square ; but if BE 4= ED, produce BE to -F, and make EF = ED : Bisect BF in G, and from centre G wiSi distance GB describe the semicircle BHF; [produce DE to meet circumference in H, and join GH : the square on EH = the figure A. .■ BF is bisected in G, and cut unequally iuE; 2 5.II.Def.l6 ••□BE.EF + EG^= GF^andGF2 = Gff: 3 47. I. But the square on GH = EH^ + EG"* ; 4 *S'm6.&Ax.3 taking away EG", then cd BE.EF, = EH''; 5 C. l.Ax. 1. ButnBD = BE.ED, orBE.EF; 6 Concl. .'. C3 BD = the square on EH ; 7 C. 1&D.6 Buti=iBD = thefig. A: .-. EH" = fig. A. 8 Recap. Wherefore, a square has been made equal, &a. Q.E.F. Alff. & Arith. Byp. — Given the fig. A = a6 = 36, in area ; required the side EH ^a; of a square equal in area. Let the lines EH= a; ; BE one side of the rectangle = 6 = 9, and E D the other side = a = 4. ■Alg. By the problem, oJ = a; '^ i Ariih. 4x9 = a!^ therefore, (V) Vo6"= x the side | and \/36 = sc = 6 the side of the of the square. | square. Use anb App. — 1. By this proposition we may find a mean, proportional la two given lines, as demonstrated in Prop. 13, bk, vi. EEMABKS ON BOOK II. 185 Given the liaes BE and EF ; their sum = 2r = 13 ; BE = x=S ; EF = 2 r - K = 1 3 - 9 = 4 : required to find ^, a mean proportional to a;, and 2 »• - ae ; ».«., BO that X -.y ;: y -.^r — x, : (9x4) +6-25=6-52 = 42-25 42-25 = 3(!' +6-25 9x4 = 36=jr» Bisect BE in G; GF=r=6-5 By 5. 11. x.{2r-x) + (x-r)^=r^ SutGF' = r' = y'' + {x-ry Subtract {x-r)' And a;. {2r-x) Extract the \J ; y = \/ S6 = 6 the mean proportional between 9 and 4. Briefly, Multiply the two numiers, and take the sguare root of their prochict for the mean proportional. 2. The same Prop. 14 serves also for approximating to the square of curve- lined figures, and even of the circle itself ; for the circle may he regarded (as in 41. I., Use i) as a polygon with an infinite nwnbcr of sides; — and if we reduce this polygon to a square, we nearly obtain the square of the circle. 3. The Areas of all plane figures a/re calculated in squares ; and by means of reducing any right-lined figure, first into a rectangle, and then i»to a square, we obtain the ready means of finding the Area in the units of surfaee. N.B. — A selection of the most useful and remarkable problems and theorems, which may be inferred from the principles developed in the second book, will be found in the Appendix to the Gradations in Euclid, Semes II., Probleim and Theorems^ bk. ii. EEMAEKS ON BOOK II. 1. Of the fourteen Propositions of this book, the. ten first con- tain the theory of the relations of the rectangles and squares on divided lines ; the twelfth and thirteenth, the theory of the rela- tion between the square of any one side of a triangle, and the squares of the other two sides, whatever be the angles, — and constitute the foundation of the Arithmetic of Sines. 2. Lines cut into any two parts are considered in Propositions 2, 3, 4, 7, and 8. 3. Lines euit into two equal and two unequal parts, — ia Proposi- tions 5, 6, 9, and 10. 18G GRADATIONS IN EUOIiID. SYNOPSIS OF BOOK II. Case \.-~-€riven two st. lines, as A undivided = 10, and BC = 60 divided into the parU B D = 30, DE = 20, a»d E C = 10. A. 1 2___^_c Prop. I. A.BC = A.BD + A.DE + A. EC; or 10x60 = (10 x 30) + (10x 20) + (10xl0) = 600. Cor. 2A.iBC; or 3 A.J BC; or 1 A. iBC.&c. =A.BC. 20 X 30 ; 30 X 20 ; 40 x 15 = 10 x 60 = 600. Case II. — Let a st. line be divided into any two parts ; AB = 10, representing the whole line / A D = 7, and D B = 3, the unequal segmenis. A P B Prop. II. AB.AD + AB.DB = AB» ; or (10 x 7) + (10 x 3) = 10 x 10 = 100. Peop. III. AB . BD = AD . DB + DB'' ; or (10 x 3) = (7 x 3) + (3 x 3) = 30. Cor. 1. AB'' - DB= = (AB + DB) (AB - DB) ; or 100 - 9 = 13 x 7 = 91. 2. AD" - DB'' > (AD - DB)" by2DB. (AD-DB); or 49 - 9 > 16 by 2 v 3 X 4 or by 24. Peop. IV. AB» = AD" + DB" + 2 AD . DB ; or 100 = 49 + 9 + (2 X 7 X 3) = 49 + 9 + 42. Cor. 1. Parallelograms about the diameter of a square are also squares. 2. AB" = ii^Y; or 100 = 4 X 25. S.^^ =2{^iy; or5|2 = 2x25. i. When a line is divided into any number of parts, as a; = 5 ,- y = 3 s = 2; then AB" = a;" + y" + «" + 2 (a. y + a; . z + j . ?) / JOO = 25 + 9 + 4 + 2 (15 + 10 + 6) SYNOPSIS. — OASB III. 187 Prop. VII. AB« + BD« = 2 AB.BD + AD«, or 100 + 9 = (2 x 10 x 3) + 49 = 109. Cor. 1. AB2 +BD'' =2 AB.BD + (AB - BD)« ; or 100 + 9 = 60 + (10 - 8)= = 109. 2. (AB + BD)'' ; (AB»; - BD«) ; and (AB - BD)'' are in Arith- metical Progression, the common difference being 2 A B . B D ; or, 169, 109, and 49 ; the common difference being 2 x 10 x 3 = 60. Prop. VIII. 4 AB.BD + ADi* = (AB + B D)!" ; or 4 (10 x 3) + 49 = (10 x 3)'' = 169. Cor. 1. (AD + DB)2 = 4AD. DB + (AD -DB)'=; or (7 + 3)'' 4 (7 X 3) + (7 - 3)« = 100. „ . /AD + DB\a . ._ TM, , t /AII-DB\2 . -. 2. 4 ( ^ j =4AD.DB + 4 \ 5 j ; or 4 x 25 = (4 X 21) + (4 X 4) = 84 + 16. Case III. — Let a st. line he divided equally and unequally, A B = 10 representing the whole line; AC or CB = 5, the Italf line; and AD = 7; DB= 3, the unequal parts. A 9 ^ -R Prop. V. A D . D B + C D^ = C B=i ; or (7 x 3) + (2 x 2) = 5 x 5 ; I.e., 21 + 4 = 25. N.B.— The arith. mean = ^° + °" = AC ; the com. dif. = ^°~°^ = CD. . ^AD + DB)» = AD.DB + C^^y ; or 25 = 21 + 4. Or, Let A D and B D be regarded as two independent lines ; thenAD.DB + (^^f^> = (^^±^)^or(7x3) + 4 = (^T = 25. Again, (AD + D B) .(AD-DB) + DB'' =AD»; or (10x4) + 9 = 7x7 = 49. or, AD'' - DB» = (AD + DB) . (AD - D B) ; or 49-9 = 10 x 4 = 40. Cor. AC - CD* = (AC + CD) (AC - CD); or 25 - 4 = 7 x 3 = 21. Labdneb's Cor. 1. The rectangle is a maximum when AB is bisected by D ; its maximum value = ("S"/ 188 , GRADATIONS IN EUCLID. Cor. 2. The sum of the squares of the parts is a minmum when A B ia bisected ; the mini'm.'u.'m value being 2 ( -5- ) 3. Of all rectangles having the same perimeter, the square contains the greatest area. 4. Of all rectangles equal in area, the square is contafned by the least perimeter. 5. If a perpendicular be drawn from the vertex of a triangle, as in Prop. 13, to the base (A B + A C) . (AB~ A C) = (BD + D C) . (DC~BD); or (7-8102 + 6-4031( X (7-8102 - 6-4031) = (6 + 4) X (6 - 4) = 20. 6. The difference between the squares 6f the sides of a triangle, as in Prop. 13, is equal to twice the rectangle under the base and the distance of the perpendicular from the middle point, k, of B C : ».e., AB^i -AC'' = 2BC.Da!;or61-41 = 2xl0xl =20; or, 61 - 41 = (2 X 2) X 5 = 20. N.B. — If the perpendicular A D fall within the base, Dg= ^° ;;'P^ ; andBG= BD + DC: a but if the perp. A D fall without the base, Dg=r !" ;andBC=BD-DC. Prop. IX. AD'^ + D B^ = 2 (A C» + C D)' ; or 49 + 9 = 2 (25 + 4) = 68. Or AD^ + DB« = 2 {h^S^^ + 2 (^J^«)= ; or (2 x 25) + (2x4)= 58. Or AD^ + DB» = (A£±£B)» + (AD-D_B). „, 1|0 + ^6 ^ 50 ^ 8_ Case IV. — Let a st. line be bisected and produced, AB = 10 repre- tenting the original line ; AC or CB = 6, its half; and BD = Z,the part produced. A C B I> "■ 1 ■ i- Peop. VI. AD,DB + CB" = CD" ; or (13 X 3) + (5 X 5) = 64. Cor. Let A B = 10, the base of an isosceles triangle, be -E; bisected in C, and produced to D ; join E D = 11-7898; then ED'-EB» = AD.DB; or 139 - 100 = / 13 X 3 = 39. / A C BD SYNOPSIS. CASE VII. 189 Prop. X. AD' + DB' = 2 (AC + CD=); or 169 + 9 = 2 x (25 + 64) = 2 X 89 = 178. Case V. — A st. line is cut in extreme and mean ratio, when the rectangle of the whole line and one segment equals the sqimre of the other segment. A It Peop. XI. AB so cut in H that AB . HB = AH» 10 X 3-81966 = 381966 = 618034 x 618034. Cor. 1. To cut a line in extreme and mean ratio, it must first be produced in extreme and mean ratio ; i.e., in fig. Prop. 11, CF . FA must equal A B*. 2. When a line C F, or its equal, is cut in extreme C iv JJ and mean ratio, the rectangle AC.(AC-^AF) = AH° or AF2; or AC. HB = AH». 3. A line C F being cut in extreme and mean ratio, A C . A F = AC - AH^; i.e., 10 x 6'18034 = 100 - 38-1966 = 61-8034. Case VI. — The measure of the square of the side of an obtuse- angled triangle opposite to the obtuse angle. Pkop.XII. AB= = 80" + AC + 2BG.CD, A 100 = 25 + 45 + (2 X 15). //, or.AB' >BC + ACby2BC.CO, y/' 100 > 25 + 45 by 30. N.B. — The Area of a triangle may be ascertained when the . , AD. BO BCD three sides are known ; tor Area = — ^ — . Case VII. — Tlie measure of the square of the side subtending an acute angle. A i» A Prop. XIII. 1«- CB'' + AB» = AC + 2 BC . BD ; 100 + 61 = 41 + 120. or, AC= < (CB» + AB'') by 2 BC . BD ; 41 < 161 Iw 120. 190 GBADATIONS IN EUCLID. 2'- AB» + BC«=AC» + 2BD.BC; 61 + 4 = 41 + (2 X 12); or AC2 < (AB» + BC) by 2 BD . B C ; 41 < 65 by 2Ci S"- A C2 < (A B= + B C')' by 2 B C . B C ; 25 < 41 + 16 by ? X 16. If in 2"-' a perpendicular C G be drawn from z C to AB, the C AB . GB = B C . D B ; or 7-8012 ^ 1-6364 = 12 = 2 x 6. Case VIII. — A square is found equal to a given rectilineal figure. --vH Q IE Prop. XIV. By Const., n B C D E = A ; BE.EF + EG" = GF* = GH=' ; 9 x 4 + 6-25 = 42-28 j C3BD = BE.ED, orBE. EF; 9 x 4 = 36; C3BD = EH» = A. v'36 = 6 = EHBideof thesq;!!*-:. PEiCTIOAL EESULTS. The Practical Results at which we arrive from studying the principles demonstrated in the first and second Books of Euclid, are among the most useful and important of any belonging to Geometry : when united to a few leading theorems of the third and sixth books, they constitute by far the most fruitful source of scientific progress ; they are, in fact, the foundation on which our after reasoinings depend respecting all magnitudes, space, and number, — the forms and motions of the heavenly bodies, — and the mechanical laws by which the material universe is governed. Nearly all the Problems in Euclid's Six Books of Plane Geometry may be derived directly and immediately from the first and second books. Indeed, for the simple construction of Geometrical Figures, scarcely any principles are demanded which are not to be found in, or naturally inferred from, those books. More fully, therefore, to give the learner an idea of the wide application of the know- ledge he may have acquired, I will briefly exhibit the Pbactical Eesults to which we have attained : they may be arranged under four leading divisions : — 1st. The Problems for the Oonstriiction of Geometrical Figures hoth stated and demonstrated in Books I. and II. ; 2nd. Problems in the other hooks which, though demonstrated under their respective Propositions, are yet most intimately connected with the first and second, for their construction and solution; 3rd. Principles of Construction for Geometrical Instruments to measure st. lines and angles ; and for Geometrical Figures to exhibit the representative values of actual magnitude and space ; and 4th. Principles which wiihout requiring that we should measure all the boundaries of a Surface, yet enable us accurately to calculate distances, magnitudes, and areas. 192 PRAOTICAI. RESULTS. I. — The Problems for the Construction op Geometrical Figures both stated and demonstrated in Books I. and II. Problem 1. On a given st. line AB to describe an equilateral triwngle. From the extremities of A B, with the distance _ _ Q AB for radius, describe two circles intersecting in C ' ' and F ; join the point C to the points A and B, and / the required triangle is drawn. {1. I.) .' Practically, it is sufSoient to draw the arcs in- !D tersecting in C, and to join the points A, B, and \ C. An isosceles triangle may be drawn by taking " the length of the equal sides, and using it as ths radius from A and B. -*-^ Pi'.OB. 2. From a gioen point A to d/tam a at. line egy,al to a given St. line C B. Join A and B, and on AB construct an equU. triangle; produce its sides DA and DB indefinitely ; from B, with the radius B C, describe a circle HCG ; and from D, with DG, the circle 6KL ; AL is the line of the required length. (2. I.) Practically, the distance CB will be trans- ferred by the compassses or by some other instrument. Prob. 3. Prom to C, the less. •■ greater, AB, of two given St. lines to cut off a part From A, with a, radius equal to C, cut the line AB in E : AE will be equal to C. (3. I.) A less line may be made equal to a greater, by describing an arc with the radius of ^e greater, and by producing the less vintil it touches the circumference. -S's Cj X lE-p ''.. TS ./ / •-..R • Pkob. 4. To bisect a given rectilineal angle, BAC | that is, to divide it mio two equal parts. SECTION I. — PROB. VII. 193 From the angulai point A, describe an are, cutting the sides A B and AC in D and E ; from D and E, with the same radius, draw arcs inter- secting in F : on joining A and F, the angle BAG is bisected. (9. I.) By following the same method, the bisected parts may also be further bisected. Practically, the division of an angle into any of the powers of 2 will be effected by measuring the angle BAG with a protractor, and dividing the angular distance by the required power of 2 ; as, 4, 8, 16, &c. Proe. 5. To Insection of the a given finite straigM line, AB, and to continue &0 'parts. From A and B, with the same radius, de- scribe arcs intersecting both above and below the line in C and E ; join GE ; and the line AB is bisected in D. Follow the same method vrith AD or DB, and for all the subsequent bisections. (10. 1.) As in the angle, the division by 2, or by any power of 2, will be performed practically by measuring the given line, and by dividing the lineal measurement by 2, or the required power of 2, 4, 8, 16, &e. Pboe. 6. To draw a perpendicular, i. e., a straigM line at right angles to a given straight line AB, from a given point G, in the same. By 3. I., take CD equal to CE ; p and from D and E, with the same radius, describe arcs intersecting in F : the st. line from F to G is the perpendicular required. (11. I.) •The drawing of a perpendicular from the extremity of a line is shown in § II., Prob. 5, Prac. Ees. The practical method is by means of an instrument called a square ; or by placing a protractor with its edge along the given line AB, and its A. D C B 5 centre at G, the angular point ; and // A. D C then joining C with 90° on the graduated edge of the protractor. Peob. 7. To d/ram a perpendicular to a given unlimited length, from a given point, C, without it. line, AB, •)/ an 194 PRACTICAL KESULTS. \ ..E D :^x G B From C, with a radius CD, C extending below or beyond the /\^ line AB, intersect AB in P and / G ; and from P and G, with the / same radius, describe arcs inter- / secting in K ; join C K : and C H / is the perpendicular required. / (12.1.) .. / „ The easiest way is to use ike x / TT square, or the protractor: lay one & E'^-'""- edge of the square along the given ^- """" line, and slide the square, keeping that edge on the given line, untU the other edge just covers the given point: — a line along the edge will be the perpendicular. The protractor is used in a similar way. Pros. 8. To make a triangle of which the sides shall he equal to three given straight lines, A, B, and C, but of which any two whatever must be greater than the third. Draw a line of an indefinite length DE, and on DE, from D, in succession lay lines, by 3. I., DP equal to A, FG to B, and GH to C : from F, with radius PD, describe a circle : and from G, with radius GH, another circle ; join their point of intersection K, with F and G ; and the figure P G K is the triangle required. (22. I.) In the same way one tri- angle may be made equal to another. In practice, a line FG is taken equal to B ; and '"^ from P and G, arcs, with radii '*v equal to A and C, are drawn intersecting in K, and K joined to F and 6. Prob. 9. At a given point A, in a given st. line AB, to make a rectilineal angle equal to a given rectilineal angle, DCE. From C and A, with the same radius, describe arcs D E and P G ; take D £ as a radius, and with it from F describe another arc intersecting arc P 6 in G ; join G A : and BAG is the angle required. (23. I.) Or, the given angle DOE may be measured by the protractor; and by pla- cing the centre of the protractor at the ■ gjven point A, with its edge along A B, ai il noting the same number of degrees SECTION I. PBOP. XII. 195 from AB as equal the measurement of angle DC£, and joining that note or mark with A, the required angle mil be drawn. The same may also be done by the line of chords. Peob. 10. To drmo a straight line throvffh u, given point A, parallel to a given st/raight line B C. Join A and any point D, in BC ; from D r; . A with DA describe the arc Kg; and from A, E y\''' ' "^'v '-^F with the same radius AD, describe the arc ' V / \ DA; by 23. I., make the angle DAE equal to \ / '. ^ the angle ADC : the line E A will be parallel to D Ti ■^ the given line BC. (31.1.) In practice, the parallel ruler, or the triangular square, is used. Prob. 11. To dessribe a parallelogram tJiat shall he equal to a given triangle ABC, and have one of its angles equal to a given rectilineal angle D. * By 10. I. bisect B C in E ; by 23. I. at E make an angle CEF equal to D ; through C and A, by 31. I., draw CG parallel to EF, and AG parallel to BC : the figure FECG is the paraUelograjn required. (42. If it were required to describe a triangh equal to a given parallelogram FECG, and having an angle equal to a given angle D, the method would be, — Produce the parallels GF and CE ; and, by 3. I., make EB equal to E C ; and by 23. I., make the angle CBA equal to D ; join C A ; — and the triangle' ABC will equal the parallelogram FECG, and have an angle equal toD. Peob. 12. To a given line, AB, to a^ly a parallelogram which shall ie 'equal to a given triangle C, and have one of its angles equal to a given angle D. By 42. I. make a parallelogram BEFG flqual to the triangle C, and having the angle EBG equal to the angle D ; produce EB ; uid by 3. I. make BA equal to the given fine ; produce P G to H, and, by 31. I,, through A draw HL parallel to QB or FE ; join HB, and produce it until it meets FE produced in K ; through K draw K L parallel to EA or FH ; and produce GB to M : the figure ABML is the parallelogram required. (44. I.) 196 PRACTICAL RESULTS. Pros. 33. To describe a paraMelogram equal to a given rectilineal figure ABCD, and having an angle equal to a given rectilineal angle E. Divide +he given figure into triangles, and draw an indefinite line KM ; and at K, by 23. I., make the angle FKM equal to angle E; to FK apply, by 42. I., a parallelogram FH equal to the triangle ADB ; and to GH the parallelogram GM equal to the triangle DEC : the whole figure FKML having an angle K equal to £, is a parallelogram equal in area to the rectilineal figure ABCD. (45. I.) Prob. 14. To desa-ibe a square 074 a given straight lime, A B, From A, by 11. I., draw the perpendicular AC, and on it make, by 3. I., AD equal to AB ; through B and D, by 31. I., draw the parallels, BE to AD, and D E to A B : the figure ABED is the square required. (46. I.) Pbob. 15. To divide a given at. line, AB, irdo two parts, so that the rectangle eontained by the whole lime and one of its parts, AB . BH, shall be equal to the square of the otjier pa/rt, AH. On AB, by 46. I., describe the square AD ; bisect, 10. I., CA in E, and join EB ; produce CA indefinitely, and, by 3. I., make EF equal to EB ; on AF, by 46. I., construct the square FH, and produce GH to K: the line AB is divided at H, so that A B . B H equals the square on AH. (11.11.) N.B. — In Prop. 30, bk. vi., a line thus divided k said to be cut in extreme and meam ratio ; i.e. . so that the whole line AB shall be to the greater segment AH as the greater segment AH to the less segment HB. C 'K D Pbob. 16. To describe a square that shall be equal to a gi/een rcctUinetA A. SECTION I. PROB. XVHI. 197 Describe, by 45. I., a, rectangle B C D E equal in area to the given figure A : if the sides B E and E D are equal, the conatruotiou ia finished ; but if not, produce B E until, \ by 3. I., E P equals E D ; by 10. I. bisect B F in G, and with 6 B as radius, describe the semicircle B H F ; produce D E to meet the semicircle in H ; the square on E H is the square equal to the rectangle B I) and to the figure A. (14. II.) -ij- G F Subsidiary Problems. — Books I. and IL Pbob. 17. To construct a Scale of Equal Parts. A/\/\/\ A E F G B o □xt 4, ^ Take a st. line AB of indefinite length towards B ; and let C be the given St. line that is to be cut off from A B ; from A B, by 3. I., cut off a part equal to C, as A E ; then again from E B another part equal to C, as E F ; and so on : the parts in AB are each equal to C and to one another ; and A B is a scale of equal parts. (3. I.) On the same principle the line KL is divided. Each of the parts num- bered 1, 2, 3, 4, is representative of 10 equal parts, and those between and K of units ; or if the smaller divisions be tenths, the larger wiU be units. By such a scale the comparative lengths of lines are ascertained. Pkob. 18. Given AB the tase, angle A ths less a/ngle at the base, and AD tht di^erence of the two sides of a triangle, to construct it. On AD the line forming with AB the angle A, by 3. I., set the difference AD of the two sides; join DB; and, by 10 and 11. I., bisect DB by the perpendicular EC ; and produce EC and AD to meet in C : ABC is the triangle required; (19 and 4. I.) 198 PRACTICAL RESULTS. ADB Peop. 19. On a given st. line A B, to describe an isosceles triangle, having each of the equal sides double of the base. Produce AB both ways; and by 3. I., make BC, AO each equal to AB : OB and AC each equals twice AB. With AC from A, and with BO from B, describe arcs cutting in C ; join C to A and B ; ABC is the isosceles triangle with sides each the double of the base. (22. I.) Prob. 20. To describe an isosceles triangle, the base AB being given, and P a st. line eq;iutl to each of tlie tides. Bisect the base in D, and raise a perpendicular at D ; from A with radius equal to F, draw an arc cutting the perpendicular in C ; join C A and CB : the required isosceles triangle is ACB. (22. I.) Prob. 21. To construct a, regular polygon, as ABDEFG. By 32. I. the formula for the angular mag- nitude of each angle in a regular polygon is, . (S X 180) - 360 ^, , , ^ <; A = ^ r; , the angle between any two sides ; z = -=-, for the angle at the centre between two radii to the angular points. (iV^oie, 32. I., p. 112.) l"- When the side AB is not given. With any radius, as C A, describe a circle ; at C, the 360 centre, make an angle equal to S and pro- duce C B to the circumference ; join A B : the line A B will, if set round the circumference, divide it into as many parts as there are sides to the polygon ; join the points, and the polygon is made. 2"' When, the side A.S is given. By formula, z A = ^ =^ ; ascer- tain the angles GAB, ABD, and construct them ; by 9. I., bisect each of the angles ; and the lines of bisection, AC, BC, intersect in C ; from C, with radius CA, describe a circle ; if now, with AB as radius, successive arcs be cut off from the circumference, there will be as many arcs as sides, and the drawing of the chords to those arcs will complete the polygon. ( Use and App. 32. I., p. 112.) Prob. 22. To divide a finite straight line AL, into any given number of equal parts, as four. At A, draw a line of indefinite length, AX, making with AL the angle SAL ; take a line AB on AX ; and, by 3. L, set along AX three other lines BC, CD, DE, Q . each equal to AB ; join EL ; and through the points D, C, and B, by 31. I., draw T)d, Gc, Sb, each parallel to EL : the line AL will be divided into four equal parts in the points b, c, d. (Use and App. 34. 1., p. 115.) A^ — ir E..-X ■•'■\ 199 urn, to divide tht SECTION I. — PROB. XXVII. Peob. 23. From a pmnt E in the aide XB of a par parallelogram into two equal portious. Draw the diagonal A D, and, by 10. I., bisect it in F ; join E P, and produce E F to G : the line E G makea the portion A E G C equal to the portion EGDB. {UseandApp. 6,Si.I.,p.m.) Prob. 2i. To convert a parallelogram,, A.'BBF, into a rectangle, ABCE, o/ equal area. C E F D 1 b£=:;;7^ 3 y. — .^^ S 7^::™ —^^ A li Produce indefinitely the parallel DF ; at B and A, the extremities of the other parallel A B, by 11. I., raise the perpendi- culars B E and A C to meet the parallel DC ; then ABCE is a rectangle equal in area to the parallelogram ABDP. {JJie and App. 35. I., p. 119.) Pkob. 25. Cfiven a triangle, ABC, and a point, H, in one side, AC; from ikat point to bisect the triangle. Through C the vertex, by 31. I., draw CP parallel to AB; by 10. I. bisect AB in D, and join D C ; join also D H ; and from C draw CE parallel to HD, and join HE : then the triangle AHE will equal the tra- pezium CHEB. {Use and App. 2, 38. I., p. 124.) E B equal to a given Pbob. 26. On a given line, E K, to dram a pa/rallelogram, KD. Produce PK, and by 3. I. from . K set off a distance equal to the given line ; produce indefinitely the sides DP, HK, and DH ; and by 31. I. draw through E, AB parallel to HG or DC; draw the diagonal AK, and produce it until it cuts D F produced in C ; through C draw a parallel to DA or PE, meeting HK and AE produced in G and B : the parallelogram BK will be equal to the given parallelogram KD. (Use ,and App. 43. I., p. 131.) Pbob. 27. To change any right-lined figure, ABODE, first into a triangle, dnd then into a rectangle of equal area. 200 PRACTICAL RESULTS. Divide the figure into triangles by joining DA, DB ; produce AB indefinitely ; through E, by 31. I., draw EH parallel to DA, and through C, CF par. to DB ; join DH and DF : then the triangle DHP is equal in area to ABODE. Next, draw DN parallel to HF; by 10. I. bisect HF in L ; at L, 11. I., raise the perpen- dicular LM, and draw FN parallel to LM : the fig. LMNF ia a rectangle equal in area to DHF, which is equal to ABCDE. (Use and App. 2, 45. I. Pbob. 28. To make straigJU a crooked boundary, ABCDE, ietvieen two fields, M and N. Draw AC, the subtend to angle B ; and through B, 31. 1., FB parallel to AC, and join CP ■ the crooked boundary AB, BC is now converted into the single boundary CF. In a similar way FC and CD will be converted into one boundary ; and this last and DE into a single boundary ; and thus the crooked boundries A B, B C, CD, D E, will be changed into one straight botmdary without affecting the areas of ' the iwo fields. (Use and App. 3,45. I. p. 136.) Peoe. 29. Given the diagonal, AB, to construct a square. At A and B, by II. I., draw the perpendiculars AE, BF ; by 9. I., bisect the right angles by AC and B C meeting in C ; and by 31. I., draw through A and B, parallels, AD to BC, and BD to AG : then the figure A B C D is the square required. (46. I., p. 137.) Pros. 30. To find a square equal to the sum of any nuw^r of given squares, OS om A B, B C, C D, D E ; or a squa/re tltat is a multiple o/ any given squa/re, A B ; or a square that is equal to the difference of two squares. V' Set the lines AB, BC, representative of the two squares or AB, B[C, at right angles, and join AC ; then AC^" = AB'' + BC^ ; at C place CD, repre- sentative of the square on C D, at light angles to A C ; and A D^ = A C" + CD'' ; and at D, forming a right angle with A D, place D E representative of the square on D E ; join A E : and the square on A E equals the squares on DE, DC, CB, and BA. (flor. 8, 47. L, p. 141.) 2°' Supposing A B to be representative of the line on which the given square is constructed, its multiple square will be obtained in a similar way; SECTION II. BOOK HI. PIIOB. Hi 201 for in this case, BC, CD, DE, being each equal to AB, the square onAE -is the multiple of the square on AB. (Oor. 3, 47. I., p. 141.) S"- Let AB be the less line, and AC the greater ; at B, the extremity of -the less, raise a perpendicular B G, and from A at the other extremity, witli AC as radius, inflect on BG the greater line ; the square of the intercept 'CB will equal the difference of the squares on AC and AB. {Cor. 3, 47. L, p. 141.) Pros. 31. To make a square that shall he the half, fourth, (fee, of a givea ■tquare on AB. At A and B make the angles each equal to half a _, "right angle ; C being a right angle, the square on AC y!^ ■will be one half of the square on AB. Again, at A ' and C make the angles CAD, ACD each equal to half a right angle: then the square on CD will be haU that on AC, or one-fourth of the square on AB. By a similar process a square may be obtained that is " '^ ii tV) A> &o., of the original square. (Cor. 3, 47. I., p. 142.) Il, — The Problems in Book 3, 4, and 6, which though •DEMONSTRATED UNDER THEIR RESPECTIVE PROPOSITIONS, ARE, FOR THEIR Construction and Solution, most intimately connectbd 'WITH THE First and Second Books. BOOK III. Pbob. 1. In a given circle, ABC, to find the centre. Draw any chord AB ; and, 10. I., bisect it at D ; -from D, 11. I., raise a perpendicular produced both ways to meet the circumference in C and E ; bisect C E in P ; and F is the centre of the circle. (1. III.) Pbob. 2. To draw a tangent to a given circle BCD, from a given point, ^iher without as A, or in the circumference as D. Find the centre E of the circle BCD, and join AE ; from E with EA describe the circle AFG , ■from D, 11. I., draw DF at right angles to EA, and join EBF and AB : then AB shall touch the -cii-de in B, and D F in D. (17. III.) 202 PRACTICAL RESULTS. Peob. 3. To describe a circle through three gvoen points, A, B, D, not in the tame st. line ; or to complete the circle of which a segment, ABD, i^ given. Join A, B, and B, D ; 10 and 11. 1., bisect AB and SD by perpendiculars GC, FC, meeting in C : C is the centre of the circle, and a radius from C to either A, B, or D, will be the means of completing the circle. (25. III.) Peob. 4. To hisect a given circwmference, AS)'&, Join the extremities of the arc A, B ; 10 and 11. I., and bisect AB by a perpendicular from C : the line CD, ■wUl bisect "■-- -'— " -' ' ■'- ^"- — "'• " (30. 111.) the given circumference in the point D. Lemma.— 31. III. Tlie angle in a semicircle is a right angle. Pkob. 5. To (Iraw a perpencHculwr, X"- from point H in a, Une CB, or at tfe extremity ; 2°- from a point D out of a line ; and, 3°- so as to he a tangent to a given circle HBE. A,.-- c'-^ l"- Take any point A, above the line, and with a radius equal to AB describe an arc to cut the given line in C ; join C and A, and produce CA to meet the arc in D : DB is the required perpendicular. 2°- Join D and C : and, 10. I., bisect CD in A ; and from A with AC describe the semicircle CBD, and join DB: DB is the perpendicular. S"- Join the given point A, out of the circle, and C the centre of the circle ; 10. I., bisect AC in D ; and with DA describe a semicircle: where the semicircle cuts the circle, B, is the tangent point, and AB the tangent from A. Or, from B, a point in the circle, draw a line BC to the centre ; and at B, 11. I., draw a, right angle : BA is the tangent from B. Peob. 6. Upon a given straight Une AB, to describe a, segment of a circll containing an angle egual to C, a given rectilineal amgle. At A, 23. I., make the angle BAD equal to angle C ; and 11. I., at A make AE at right angles with DA ; 10 and 11. I., bisect AB in P by the per- pendicular GF ; with GA or GB describe a circle: *nd in the segment AHB the angle AEB equals BAD, which also equals angle C. (33. III.) ^ .^TFT^E D SECTION II. — BOOK III. — PROB. X. 203 Pecs. 7. Given the angle D equal to the vertical angle of a triangle, and tht tase AB, to find tlie loous of the vertex. At A, by 23. I., make the angle BAF equal to angle D, and, 11. I., angle FAH equal to a right angle ; bisect. 10 and 11. 1., AB in G by the perpen- dicular G H ; and from H, with radius HA, or H B describe a circle ABC: the locus of the vertex will be at any point in the arc of the segment ACB. (33. III.) Pbob. 8. Given AB the hase, angle ACD the vertical angle, and AD the per- pendicular from the extremity of the hase A on the opposite side B C, to construct the tiiangle. Make on AB, by Prob. 6, a segment containing the given angle ACD; bisect A B, 10. I., in E ; and with rad, E A or E B describe the semicircle A D B ; and from A, inflect the perpendicular AD upon the semicircle in the point D : B D A is a right angle, 81. III., and B D produced to C, and CA joined, give the triangle required. (33. III.) Pbob. 9. To divide a given circle of which the diameter m A B, into any number of equal parts, of which the perimeters also are equal. By Use and App. 34. I., divide the diameter A B into the required equal parts at C, D, E, F ; then on one side, from A describe the semicircles 1, 2, 3, 4, 5, &o. ; and on the other side from B, the semicircles 7, 8, 9, 10, 11. &c., — of which the Aj diameters are B P, BE, BD, BC ; so shall the parts 1 and 11, 2 and 10, 3 and 9, 4 and 8, and 5 and 7, be equal each to the other, both in Area and Perimeter. — Leslie's Geometry. Prob. 10. From a given circle, ABG, to cut off a segment which sluM con- tain an angle equal to a given rectilineal angle D. Draw any radius CB; and at B, 11. I., draw EP maltin); ri^ht angles with CB ; at B, 23: I., make the anglfl FBG equal to angle D : the segment BAG con- tains an angle BAG equal to the angle QBF, which equals angle D. (34. III.) 204 PRACTICAL RESULTS. BOOK IV. Pkob. \. Ina given circle, ABC, to place a st. line equal to a given st. line D, ntt greater than the diameter of the circle. Draw BC the diameter of ABC : 3. I., / make CE equal to D ; and with CE describe ; the circle AEP: CA is the line required. ', (X. IV.) D_ 1*E0B. 2. I. In a given circle, ABC, to inicribe a triangle equiangular to a given triangle, DEF. Draw OA any radius ; and at A, 11. I., GH a tangent ; also at A, 23. 1., make the angles HAC and GAB equal to the angles DEF and DFE, and join BC : the triangle ABC is equiangular with the given triangle DEF. (2. IV.) Pbob. 3. About a given circle, ABC, io deicriie a triangle equiangular to a given tria/ngle, DEF. Produce EP both ways to G and H ; find K the centre of the circle, and draw a radius KB ; at K, 23. I., make the angle BKC equal to angle DFH, and angle BKA equal to angle DEQ ; at the points A, B, and C draw, 11. I., lines at right angles to AK, BK, CK, and produce them both ways until they meet : the fig. LMN is equiangular with the triangle DEF, and described about the circle ABD. (3. IV) Peob. i. Jn a given triamgle ABC, to inscribe a circle. By lines B D, CD, meeting in D, bisect 9. I., the angles B and C ; from D, 12. I., drop per- pendiculars D E, D F, D G, to A B, B C, and C A ; with radius D E, describe a circle : the circle EFG is inscribed in the given triangle. (4. IV.) Peob. 5. About a given triangle, ABC, to describe a cir. a SECTION U. — ^BOOK IV. — PROB. XI. 205 -^ A F By 10 and 11. I., bisect AB and AC by perpendi- culars meetiiig In F, and join FA ; with FA as radius ^ieaoribe the circle ACGB ; it is described about the given triangle. (5. IV.) This is the same as Frob. 8, from bk. III., to describe a circle through three given pointis, A, B, 'and C. Peob. 6. In a given eircU, ABCD, to iriseribe a square. Draw BD and AC two diameters at right angles to each other, and join A, B, C, and D : the figure Q- ABCD is a square in the circle. (6. IV.) Peob. 7. Alovta given circle, ABCD, to descrSie a sgvare. Draw two diameters, AC s^d BD, mating right angles at E ; at the extremities A, B, C, D, draw lines 11. 1., at right angles to EA, EB, EC, and ED, and produce them until they meet in G, H, K, F : the fig. GHKF is a square about the circle. (7. IV.) Peob. 8. To inscribe a circle in a given square, ABCD. By 10 and 11. I., bisect each of the sides by the perpendi- culars EG, FG, HG, KG, intersecting in G ; and with GE as radius describe the circle E F H K in the square. (8. IV.) Peob. 9. To describe a drcleaiout a given square, ABCD. Draw the diagonals AC, BD, and from their point of intersection Q, with GA, describe a circle : the circle ABCD is about the given square. (9. IV.) Peob. 10. To describe an isosceles triangle having each of the angles at Hie hase double of the third angle. Take any line AB, and, by 11. II., divide it so that the rectangle AB . BC shall equal the square on CA ; from A, ■with radius AB, describe the circle BDE, and in it, by 1. IV., place the line BD equal to AC ; and join DA: ABD is the isosceles triangle required. (10. IV.) For the proof, join DC, and about the triangle ADC, 3. III., describe the circle A CD. Prob. 11. In a given circle, ABCDE, to inscribe an «( gidcur pentagon. Draw two diameters, AF and GH, at right angles ; 10. 1., bisect the radius OH in L ; and with A L as radius describe an arc cutting OG in K : the distance AK is equal to one side of the pentagon ; and if set round the circle will complete the figure. If the aroq are bisected, and the points of bisection joined, a decagon will be drawn. (P. 10, bk. XIII.) E X 206 PBACTIOAL RESULTS. Or, 10. IV., describe an isoaceles triangle, FGH, having each of the angles at the base double of tluit at the vertex ; and, 2. IV. , in the given circle inscribe a triangle A CD, equi- angular with FGH ; 9. I., bisect the angles A CD, ADC, and let the bisecting lines be pro- duced to meet the circumference in B and E : the points A, B, C, D, E, are the angular points of the required pentagon. (11. IV.) Prob. 12. Ahout a given circle, ABCDE, to describe an eguilateral angulwr pentagon. By 11. IV., let A. B, C, D, E, be the angular points of a pentagon inscribed in the circle ABCDE ; and from F, the centre of the circle, let lines be drawn to those points ; at A, B, C, D, and E draw lines, 11. 1., at right angles to FA, FB, FC, FD, and FE : the fig. HGMLK is the pentagon about the given circle. (12. IV.) egwi- Pbob. 13. In a given equilateral and equiangular pentagon, ABCDE, to inscribe a circle. By 9. I. bisect the angles BCD, CDE, Sec, by lines CP, p F, meeting in F ; from F, 12. I., draw perpendiculars FH, FK, PL, to BC, CD, DE ; and with any one, as F H, as radius, describe a circle : this circle GHKLM will be inscribed in the penta- gon ABCDE. (13. IV.) Peob. 14. To describe a, circle about a given equilateral and equiangular pentagon, ABCDE. By 9. I. bisect the angles ABC and BCD by the lines BF and CF ; and from their intersection in F draw FD, FE, FA ; with either of these aa radius, describe a circle, as ABCDE : this circle vrill be about the given pentagon. (14. IV.) SECTION II. BOOK IV. PROB. XVIII. 207 Prob. 15. To inscribe a regulwr Tiexagon in a given circle, ABCDEF. Draw a diameter of the circle, as AD, and bisect it in G ; from A with a distance equal to the radius AG, draw arcs cutting the circle in F and B ; and from D with the same radius, arcs cutting in E and C ; join the points A, B, C, D, E, and F,— and the hexa- gon is inscribed in the circle. (16. IV.) N.B. By joining the alternate points, A, C, E, an ■eguilateral triangle A C E is inscribed ; and by bisect- ing each of the arcs of the hexagon, and joining the points, a dodecagon is formed and inscribed. Prob. 16. In a given circle, ABCD, to inscribe an equiangular and equilateral By 2 and 11. IV. inscribe in the circle an equi- lateral triangle ACD, and u regular pentagon ; AB being a side of the pentagon, and AC a side of the triangle : 30. III., bisect BC in E, and BE or EC will be the fifteenth part of the ciroum- E ference, — ^from which the quindecagon may be con- f\\ fitructed. (16. IV.) ^ Prob. 17. To trisect a quadranU From A and B, with the radius of the circle, describe arcs cutting the quadrant in D and E ; join EC, DC, and the quadrant is trisected. Prob. 18. In a given circle, ABP, to inscribe any regular polygon; or iff divide the cirevimference of a circle into any assigned number of equal parts. By Use and App. 34. I., divide the diameter AB into the same number of equal parts as the figure has sides, suppose 9 ; from the centre C, 11. I., draw a perpendicular Oh ; divide the radius Cy into four equal parts ; and set off three of those parts from y to Jc; join i: and z the second of the divisions from A, and produce Jz to the circum- ference in P ; the line AP wiU be the side of the required polygon. — See a Treatise on Mensuration, Irish National Schools, p. 19 ; Demonstrationa, p. 53. 208 PRACTICAL RESULTS. BOOK VI. Lemma. — 2. VI. Jfa straight line be drawn parallel to amy side of a t^-ia^yle, it divides the other sides, or those sides produced, into proportional segments. Pbob. 1. To cut off from a given st. line, AB, any part required. A At A place any line AC, making with AB the angle BAG; in AC take any point D, and in succession, by 3. I. cut off from AC as many parts each equal to A D as there are to be parts in A B ; join B C ; and, by 31. I., through D draw D E parallel to B C ; whatever part AD is of A C, the same part is AE of AB. (9. VI.) Pbob. 2. To divide a given straight line, AB, similarly to a given divided iC line, AC. a Set A B and A C so as to make an angle BAG, and join y? B C ; through D and E, the points of division on AC, by /\ 3i; I., draw D F and E G parallel to B C ; in the points F and -c. / \ t» G, AB is similarly divided to AC. y ' }^ N.B. When one line is cut similarly to another, the several G/ — 7^'^E segments of the one are proportional to the several segments of I 1 \ the other. /.... !. \ Pbob. 3. To find a third proportional to two given straight lines, A B omC AC. Place the two lines so as to form an angle BAG ; pro- A duco AC indefinitely, and A B so that B D, 3. I., is equal A to A C ; and join B C ; through D, 31. I., draw D E parallel / \ to BC ; CE is the third proportional ; i.e., AB : AC :; ■d/...AC AC : GE. (11. VI.) / \ N.B. A repetition of the same construction will avail for / V continuing the progression. yj \ ._ Prob. 4. To find a fourth proportional to three given straight lines, E, F, anil 6. E— Draw any two lines from A so as to make the / \ p angle BAD; by 3, 1., on A B make A G equal to / \ JJ E, C B equal to F, and AH equal to G ; and, 31. ft a "X G I., thiough B draw BD parallel to CH ; HD is ,.' \ the fourth proportional. (12. VI.) / \ / I> SECTION II. — BOOK VI. — PROB. VIII. 20» Pbob. 5. To fitid a mean proportional AB and AC. between two given straight linei^. -D Set the.giveu lines so as to form one straight line AC ; by 11. I., bisect A C in E, and, with E A or E C as radius describe a semicircle ; at B, by 12. I., raise a perpendicular B D to D : D B is the mean proportional : i. &, A B : B D : : BDtoBC. (13. VI.) Or, let AC = 2?-=13; AB=a!=9; and BC=2?-— a:: then B D^ =jf» =2 »■ K— k". _ Pbob. 6. On a given line, A B, to construct a rectilineal figure similar aniR mmilarly sitaated to a given rectilineal figure, G D E F G. ^ £ B C To divide the given figure into triangles, draw C E and C F ; by 23. I., at B and A make the angles A B H, B AH, H AI, and lAK equal to the angles E D C, D C E, E C F, and / j F C Gr ; at H make angle A H I equal to angle ^/ j C E F ; and at I, angle A I K equal to angle ^ i CFG: the figure A B H I EC is similar and ' similarly situated to the given figure C D E F G (18. VI.) Pkob. 7. (The most extensive problem in the Elemenis of Geotnetry, ) To describe a rectilineal figure similar to one given rectil. figure, ABC, oniJ egmil to another rectil. figure, B. By Cor. 45. 1., on B C describe the par- allelogram B E equal the figure ABC; and on C E a parallelogram equal to D ; and ■with the angle F C E equal to the angle C B L, by 13. VI., or 14 II., find G H a meauB, proportional between B C and C F ; and on GH, describe by 18. VI., GHK similar ^- toABC; GHK is the figure required. ^^ (25. VI.) Prob. 8. To a given st. line, A B, to apply a parallelogram equal to a girett- rectilineal figure C, and d^cient by a figure similar to a given parallelogram D.. But the rectilineal figure mwst not be greater than the parallelogram applied tof- half the given st. line, whose defect is similar to the given pwrallelogram D. By 10. I., bisect A B in E ; and by rr 18. VI., on EB construct EBFG "■ similar to D ; and by 31. I., complete the parallelogram A G, making it either equal to C, or greater than C, If A G is equal to C, the problem is solved, — for on A B has been applied the paral- lelogram AG equivalent to C, and deficient by the parallelogram EP similar to D. But if the area of A G be greater than that of C, by 25. VI., make th» parallelogram KM equal to tile excess of parallelogram EF above C, and. :210 PRACTICAL RESULTS. similar to D. Make X G, 3. I., equal ^ G O I" Xi M ■to K L, and Q to L M ; and complete — — — — T---1 i — > the parallelogram G P X ; X and £ F have the same diagonal ; draw that -diagonal G P B, and produce X P to R, and P to S, and complete S R : then t;he parallelogram T S equal to the given figure C is applied to the given line A B deficient by the parallelogram S R, which is similar to the given par- allelogram D. (28. VI.) Pbob. 9. To a gwien, straight line, A B, to a/pply a pwrallelogram equal to a ■given rectilineal figure Z, and exceeding hy a parallelogram BX similwrto a/mther j/iven parallelogram X. By 10. 1., bisect A B in E ; and by 18 VI. on E B construct E B L P similar to X, and K G H, (equalling parallelogram E L, and figure Z) similar to X. Produce F L and F E, making, 3. 1., F L M equal to K H, and FEN to KG; complete the par- allelogram M N ; EL and M N have the same diagonal F X ; draw that diagonal, -and produce E B to 0, L B to P, MO and N" P to X : then the parallelogram A X equal to the given figure Z, and exceed- ing by the parallelogram P which is similar to X, has been applied to the straight line A B. (29. VI.) Pbob. 10. To cut a given straight line, A B, in extreme and inean ratio, i.e., so thai, the whole line is to tlie greater segment as the greater segiMnt to tht lest. On AB, by 46. I., describe the square B C ; and, by 29. VI., apply to A C the paral- lelogram C Q equal to the square B C, and •exceeding by the square A G similar to the square B : the line A B is cut in H in ex- treme and mean ratio. (30. VI.) Or, by 11. II., divide AB in H, so that A B . H B equals the square on A H ■ then also A B is cut in extreme and mean ratio in the point H, f Er ii ™B K S Pbob. 1 1 . To make a sywme a^oproximaling in area to the a/rea of a giim cirell tm the diameter A B. SECTION III. — GEOM. INSTEUM. 211 In Cor. 8. VI., it is proved that AB . AS=AC'. Divide AB the diameter into fourteen equal parts, and of these set off eleven from A to S ; at S, by 11. I., raise the perpendicular SC terminated in the oir- •cumference ; join AC, and the square on AC will closely approximate to the area of the cii-ole of which AB is the diameter. — Mensv/ration, Irish NatiorMl Schools, p. 20. III. — Principles of Construction. § 1. For Geombteical Insteumbnts to Mbasuee Straight Lines, i.e.. Distances, — and Angles. 1. Instruments for measvnng st. lines are constructed by following out -the Applications given in pages 55 and 115, 116, when treating of Propositions 3, and 34, bk. I. Ani/ St. line may he assumed as the unit: if it is the tenth of an inch, then, l)y the repitition of that tenth ten times, an instrument to measure an inch, "will be made ; if an inch be the unit, and that inch be repeated tweVoe times, ■we have the measure for a foot ; if a foot, and there be three repetitions of it, ■we have a yard ; and if a yard be taken 1760 times, we have a mile. A very usual unit is tie! ink, containing 7'92 inches, and 100 of these being taken, the Gunteir's chain equal to 66 feet is formed. The instruments commonly employed for the measuring of Lines, are the Inch, Foot, &o. ; the Offset Staff and Chain ; the Perambulator, — a wheel of ■which the circumference is ^ of a chain, or 8"25 feet ; and Screws of various sizes. The threads on the screw are at exactly equal distances, and a single -tiim of the screw measures one of those equal distances. The spaces between -the threads are generally small ; if there are 100 threads in the lineal inch, each turn of the screw represents ytb °^ ^^ inch, and the hundredth part of a turn TffTT of TW or ^w^ of an inch. 2. Instruments for measuring angles are only in part constructed on strictly Geometrical principles. At p. 79, Cor. 1 and 2, 15. I., we find that if a circle be round a point, all the possible angles from the centre to the circumference taken together equal four right angles : the circumference being always divided into 360°, each quadrant contains 90°; the quadrant may be trisected — I of the quadrant, or 60°, measures the sextant ; the 30° being bisected give- two fifteens ; but the division of 15° into fifteen equal parts can only be effected mechanically ; — see p. 68, where it is shown that if the method of successive bisections had been followed, the degrees of the circle would be in strict accordance with the principles established in Plane Geometry. The instruments generally employed for measuring angles, are the Ciioss Staff for right angles, — ^the Theodolite for any angle, — ^the Qdadba nt f?r an angle not exceeding 90°, — and the Sextant for an angle not exceeding \^°. The Goniometer is used for measuring the angles of crystals, &c.; and Halley's 212 PRACTICAL RESULTS. Ekfleoting Quadbant for taking the angles between luminous points, as the stars, &c. Spirit levels and a plummet enable us to find a perpendioulai-, or a line at right angles to the horizontal line ; and a Graduated or Oeonietrical Square fastened on a staff and placed in the true vertical position, to measure angles of elevation. An angle may also be measured by any line of equal parts, by simply mea- suring the same distance along the lines that form the angle, and from the extremities of the distances thus measured, measuring the subtend to the- angle. The Magnetic Compass, fixed on a circle called a Circumferenter, is often used for surveying mines or large tracts of land where much accuracy is not required. The knowledge and use of such instruments are only to be attained by a. practical acquaintance with them. 3. There is a great difference hetween Lineal Magnitude and Angular Magni- tude. The unit in Lineal Magnitude is a fixed quantity, — three feet is always the same amount of space ; but the unit in Angular Magnitude is •^ variable quantity, changing with the circumference of the circle. Whatever may b.i the size of the circle, its circumference is divided into 360 degrees ; so that a degree may at one tim« represent the tenth of an inch, — at another time com- prise ten million miles. This is familiar in the degrees of longitude : at tht> equator the degree of longitude measures 60 geographical miles; at Cairo, lat. 30° 6', about 51 miles ; at Peteraburgh, lat. 60°, about 30 miles ; and at^ the Magnetic Pole, lat. 70° 5', only 20 miles : — thus gradually diminishing, until at the Pole itself, the circle having dwindled to a point, all magnitude ha» ceased. § 2. Foe Geometbioal Figures to exhibit the representative values of actual inagnitude and space. When lines have been measured and the angles taken which are formed W the lines drawn from any point or station to two or more objects in the fiell of view, we may enter the measurements and the angular magnitudes in ,i book : but besides thjs, we need some method of representing the relativij positions and distances of object. Geometrical Drawing, — as in the plans of the Architect and of the Engineer, or in the Maps of an Estate, a Parish, or a Kingdom, — is the means by which such a representation is effected. For the actual truth of a Demonstration in Geometry, it is of very little consequence whether we employ any figure at all, or whether the figure we do construct is perfectly accurate ; but for the representation of distances and of forms His ot great importance that we should draw well all the lines and angles, in their due proportion, and in their proper positions ; — indeed, without this, a map is a deception, and a plan given to a workman for his guidance, would only mislead and perplex him. To ensure the accuracy so essential for all maps, plans, and working drawings. Scales of equal parts, and especially the Diagonal Scale, are con- structed and employed ; see pp. 55, 121. The principle acted on is a very simple one, — that in the same Map or Plan, whatever scale is made use of for any one line or distance, shall be adopted for all the lines and distances in that particular Map or Plan. Let the scale fixed on be an inch to represent a mile ; then every inch of space on the plan or map, to which the scale is attached, will represent the value or distance of a mile : and again in a working drawin" given to a mechanic for his guidance, let the scale agreed on be haJtan-incS SECTION III. — GEOM. FIGURES. 213 to represent a foot, then every half -inch in the drawing will denote a foot in the thing to be constructed. The Lines in a Map or Plan are representative only of the values of the real •distances of objects ; — but the Angles in a Map or Plan, if correctly drawn, awe identical with the angles of the objects themselves ; and the reason is, that the «ize of an angle does not depend on the length of the lines which form it, but on the narrowness or width of the opening between them. The instruments for transferring to a drawing the angles measured between any two objects and the angular point or station of observation, are principally the Circle or Pro- tractor, and the Line of Chords ; pp. 40 and 89. I'he Methods for drawing nearly all Qeometrieal Figv/res are contained in the several Problems, pp. 192 — 211, which constitute the first and second of the lading divisions of the Practical Besdlts ; but in the Application of them vfe must remember tJtat lines and angles of a definite numerical value are given when we are rehired to constmct Figures tJutt shall show tJie positions, forms, and ■distances of objects. We subjoin therefore a few examples M the wat in which SUCH FIGURES ARE TO BE DRAWN. Pros. 1. Given the two angles of a triangular field, equal respectively to 50' ~^md 45°, and the interjacent side equal to 645 Knhs ; required by Construction the ■ exact figure of the triangle, the a/mount of the other angle, a/ad the value in links oftlie other two sides. Take from the diagonal scale 645 equal parts, Tepresentative of 645 links, and of that length C draw a line AB ; at one extremity. A, make an /' ^»^ angle of 50° ; at the other extremity, B, an angle ^ '' N^ of 45° ; produce the lines from A and B until /' *«^ they meet in C ; ABC is the exact Cy. required. /, .^»^ If the angle at C be measured, it will be found l^^ 1 . ^g to equal 86° ; and from the same scale of equal parts, AC will be found to equal 458 links nearly, and EC 496 links nearly. Prob. 2. The distance K'Bfrom the foot of a precipice is 288 feet ; at A, tht angle made by a string C A, reaching to the top, with AB, is 53° 8' j required by ■ construction the height of the precipice and the length of the string. c A Draw a line AB containing 288 equal parts ; at B /' •erect a perpendicular, and at A draw an angle of 53° 8' ; / produce the lines AC, BC to meet in C : — AC, repre- senting the length of the string, will measure 480 feet ; /' ■and BC, the height of the precipice, 384 feet. / Azr: . JB 214 PRACTICAL KESULTS. Prob. 3. The breadth AD or BC, of the slutft of a well, of which the sides are: parallel, is 10 feet; tlie angle of depression, CAB, 15°: regui>-ed by construction, AC, the depth of the well. D A Take A D equal to 10 equal parts ; and at A and D draw perpendicvdars to AD ; make the angle CAB 15° ; and the point B, where AB and DB intersect, will be at the bottom of the shaft ; through B draw BC parallel to DA ; and CA equals DB ; apply CA or DB to the same scale as AD, and the depth will be found 37-3 feet. Peob. 4. Two ships of war iniending to cannonade a fort at C, separate fromt each other 500 yards ; the a/ngles between each ship and the fort and the other ship- are A, 38° 16'. and B, 37° 9': 300 yardsbeimg a convenient distance for a cannonade^ have the two ships taken up their stations at a proper distance ! y C \ A^ :^ The line A B being drawn equal to 500, angle A to 38° 16', and angle B to 37° 9', and the tri- angle ACB completed ; it will be found from the same scale, that A.C measures 312, and BC 320 yards : the ships therefore are nearly at the exact distance required. Prob. 5. In surveying afield, theamgles which the aides of the field made with the magnetic meridian, were observed at Hve stations, and the lengths of the mie»- foeaswred ; required to construct the figwre of the field. Station 1. North 17° West 262 Unks. „ 2. North 62° East 324 „ „ 3. South 51° East 221 „ „ 4. South 28° West 296 „ 5. West 242 „ At Station 1 make an angle 17° West of North, and set oflf 262 equal parts ;■ at Station 2, an angle of 62° East of North, and set off 324 ; at Station 3, an angle 51° East of South, and set off 221 ; at Station 4, an angle 28° West of South, and set off 296 ; and at Station 5, a Une due West, with a set off of 242 ; the figure will be completed : if now the diagonals and perpendiculars were- drawn, the area of the field might be computed. SECTION IV. — ^DISTANOBS. 215^ Pros. 6. For more extensive surveys two stations A and B are taken, the line A B is measured, and from a scale is set on the Plan ; all the angles, which are formed by lines from the stations A and B to the objects at 1, 2, 3, 4, 5, 6, 7, and 9, are also measureed and drawn ; and the angular points being joined, an out- line of the figure is obtained in the rela- tive size or magnitude of its parts ; and by taking any of the distances in the plan, and applying them to the aams- Boale as that from which A B was set out, their equivalent, or representative- values will also be obtained. Thus by drawing the diagonals, and dropping^ perpendiculars from the angular points which the diagonals subtend, the area^ of the figure may be computed from the scale. It must, however, be observed, that this method of computation is very liable to error, for it is difficult to • construct an intricate figure so as to make it perfect in all its parts. Oalcula- iions from data attained hy actual measurements, are the only reliable methods of ascertaining with accuracy the areas of figures. The foregoing examples will be sufficient to show how by the construction of figures we may exhibit the representative values of actual magnitudes and spaces. Other instances are given at pp. 55, 56, 58, 66, 87, 91, 94, 99, 100, 108, 111, 112,. 121, 133, &c, TV. — Principles which without hbquibing that we should^ MEASURE ALL THE BOUNDARIES OF A SURFACE, YET ENABLE US" ACCURATELY TO CALCULATE DiSTAMCES, MAGNITUDES, AND ArBAS. § 1. Lines, or Distances. 1. The sides of an equilateral triangle, by Def. 24. I., of a square, by Def- 30. 1., and of a rhombus, by Def. 32. I., are all equal ; the TMosurement of one aide in each, therefore, is sufficient for ascertaining the other sides, 2. In an isosceles triangle, by Def. 25. I., two sides are equal ; therefore the- measurement of one of those sides will give the other. 3. In parallelograms, by 34. I., the opposite sides are equal ; it is necessary, therefore, only to measure any two conterminous sides. ' 4. Of the three sides, H hypotenuse, B base, and P perpendicular ; amy *joa> being measured, the third may be calcul ated ; for, by 47. I. H = Vb" + P' ;. B = VH» - P» ; and P = VH» - B^ 5. In a similar way, in a rectangle,vof the two conterminous sides, AB, BC, and the diagonal, AC, any two b eing given, the third is found by 47. I. ; for- AC = VaB« + BC" ; AB = VaC" - BC""; and BC = VaC» - AB". 6. If C, the circumference of a circle, or D, the diameter, be measured, n the other may be found ; for by 41, I. Use 4, = D x 3-1416, and D = 7. When, of the three parts of a regular polygon, the side A B, the per- pendicular on the side P, and the radius B, or distance from the centre to 216 PRACTICAL RE3CLT3. the angular points, any two are measured, the other is found ; for, 41 and 47. I., ^ = VR' - P' ; P = VR" - ^AB\= ; and R = VP + " /AB\» 8. In any right-lined figure, if diagonals are drawn dividing it into triangles, and from the angular points perpendiculars are drawn to the diagonals, — on Toeamring the perpendiculars and tJie segments of each diagonal, then, by 47. I., tJie sides muy he calculated. 9. In the case of inaccessible distances, the amount or calculation is obtained in various ways : as in 1. I., Use 4 ; 3. I., Use 3 ; 4. I., Use 2 ; 6. I., Use ; 15. I., Use 1 ; 20. I., Use 2 ; 26. I., Use 1, 2, 3 ; 29. I., Use ; 31. I., Use 2 ; 33. 1. ; 34. I., Use 4 ; 46. 1., Use ; 47. 1., Use 4, 5 ; 6. II. Use. § 2. Angles. 1. In equilateral triangles, rectangles, squares, and regular polygons, the measurement of one angle is equivalent to the measurement of all ; for, by Cor. 5. I., Def. 31, Def, 30, such figures have all their angles equal. 2. If we measure any two angles in a triangle, the third angle by 32. I. will equal the supplement of 180°, or 180° — the sum of the two angles. 3. In a right-angled triangle, with one acute angle known, the other acute angle is equal to the complement of 90°, or 90° — the given acute angle. 4. When two st. lines intersect, by 15. I., the measure of any one angle gives the opposite vertical angle. 5. By measuring any two adjacent angles of a parallelogram, 34. I., we obtain the other two angles. 6. When a st. line cuts two parallel lines, the exterior angle equals the interior opposite angle, and the alternate angles are equal, 29. I, ; by measuring -one angle, therefore, we know the other. 7. At the base of an isosceles triangle the angles are equal, 5. I. ; one being measured, the other is known. 8. By approaching to, or receding from, a horizontal mirror, on which a given point from an object is reflected, we can ascertain the acute angle and the base of the right-angled triangle, 20. L , Use 2 ; and consequently we can cal- culate the vertical angle. 9. The parallax of a heavenly body is ascertained, 32. I., Use 1, by sub- tracting the zenith distance at the earth's centre from the zenith distance at the earth's surface. 10. When, by 4. I. : 8. 1. ; or 26. 1. ; two or more triangles are proved to have equal angles, the measurement or calculation of the angles of any one triangle is equivalent to tne measurement or calculation of the angles of the others. § 3. Magnitudes or Areax, 1. In squares, the measurement of OTie tide is sufficient, — for the altitude and base are equal ; and the Area = the square of the side : 40. I. 3. 2. In all 'parallelograms, measure the base and altitude ; for, by 35. 1., Use, -and 41. I. Use, the Area = the base x the altitude. 3. In aU triangles, also measure the base and altitude, and take half the ibase ; for, by 37. I., and 41. I., Use 1, the Area = ^ the base x the altitude. RBMABKS. 217 4. All nght-Uned figwa may be divided into triangles ;-r-then the Area of the figure =iAe jmm of the areas of the triangles : 41. I., Use 2. 5. Lines from the centre of a regular polygon divide it into equal triangles ; its Area, therefore = the Area of one triangle X the number of sides ; or, = the perpendicular X J the perimeter ; 41. 1., Use 3. 6. The Area of a trapezium is found, 40. I., Use 2, by taking half the sum of the parallel sides and the altitude, and multiplying the two quantities together. 7. In ci/rdes, measure the diameter D, and ascertain the circumference C ; or measure the circumference, and ascertain the diameter ; then, by 41. I.,Use4.theArea = 5MJiLB^,„r, = ,-3^;or, = ?4i-«;or, = the radius X the semi-ciroumfereuoe. By the use of other Geometrical Truths, the Student might have a much more extended view of the Principles which assist Mathematical Calculations; but many of those truths lie out of the limits of an elementary work; and enough has been advanced to show the wide application and Utility of the First and Second Books of Euclid's Plane Geometry. Indeed, if any defence were required for confining the Exami- nations of Pupil Teachers, and of Scholars generally, in Elementary Schools, to the two books referred to, it is furnished by the very valuable Practical Eesults which have just been exhibited. "Whoever has mastered and retains his familiarity with the Geo- metrical Principles now set before him, will possess sufficient knowledge of the subject for aU the usual purposes of life, — and, what is more, will possess the means, whenever he chooses to ■employ them, of advancing with comparative facility to the higher and more abstruse mathematical learning. The right foundation, has been laid; and the calls of professional duties and employ- ments may be left to determine, whether the student should remain satisfied with th'ei mark attained, or go beyond it and labour in a wider field. If he is called, or prompted, to try the p 218 PRACTICAL RESTOTS. more difficult paths, he will never regret that his attention in youth was chiefly confined to the Introductory Books of Plane Geometry. It is the accuracy and the thoroughness of the early training, — and not the wide extent of the subjects, traversed indeed, but not known, — which increase the power of the mind ; and the true aim of the teacher is to strengthen power by a smaller quantity well done, than to waste it on a midtitude of projects, to none of which it is able to do justice. The steam, that sounds a thousand jerking whistles does not perform half so much useful work as that which keeps in steady motion a single loom. " By what steps and ways we are to advance in knowledge," observes John Locke, Of Human Understanding, Book IV., chap. 12, § 7, " is to be learned in the schools of the mathematicians, who, from very plain and easy beginnings, by gentle degrees, and a continued chain of reasonings, proceed to the discovery and demonstration of truths that appear at first sight beyond human capacity. The art of finding proofs, and the admirable method^ they have invented for the singling out, and laying in order, those intermediate ideas that demonstratively show the equality or inequality of unapplicable quantities, is that which has carried them so far, and produced such wonderful and unexpected discoveries : but whether something like this, in respect of other ideas, as well as those of magnitude, mav not in time be found out, I will not determin'}-," APPENDIX. I— GEOMETRICAL SYNTHESIS AND ANALYSIS. The Principles of Plane Geometry, as taught in Euclid's Elements, are established, by proceeding in a regular series from Definitions, Postijlates, and Axioms already known, to the consequences ■which flow from, and which are dependent upon, the Definitions-^ Postulates, and Axioms. This method is entirely one of building up, or of putting together, and is therefore named Synthesis : " it commences with what is given, and ends with what is sought," — the materials being furnished, out of them it fashions a garment ; it takes elementary substances, and forms a compound. Analysis pursues an opposite course : it takes the compound, and resolves it into its constituent parts ; the garment entire and completed is given for examination, and the aim is to discover of what it is made : Analysis begins with the thing sought, as a thing perfected and accomplished, and ends with whatever may have been supplied for the construction. No aids except those derived from Geometry were admitted by the Ancients in conducting an Analysis j and therefore the term Geometrical Analysis is employed. In analyzing a Problem, the solution is assumed to have been effected; and in analyzing a Theorem, the truth of the assertion contained in it is first of all admitted. When a problem is analyzed, the object is — ^to discover something which, if done, would of necessity lead to the solution of the problem ; and when a Theorem is analyzed, the object is — to determine whether the assertion is true or false. 220 APPENDIX. An example will more clearly show the diflference between the analytical and the synthetical Methods : for this purpose we take the problem — In a given st. line, D E, to find a point, C, lehich shall 5e at the same distanci from two other given points, A a/nd B. By Analysis. — ^We assume what was required, , namely, that C, in DE is the point equally distant ■"■V ?-... from the given poiats A and B. \ ; -,^ The St. line CA = the st. line CB ; and AB \ ,' ^.''' being joined, the figure ACB is an isosc. A. If \ ' y' not the St. Ime AB is bisected in F, we have, in \ ; ,,-' the two triangles, CA = CB, AF = BF, and FC j) '^ .JJ common ; ."., by 8. I., A AFC = A BFC, and ^ '^ AFC = ^ BFC. But when two points, as A and B, are given, the line joining them, AB, is given ; and the line being given, its middle point F, is also given ; at F, that middle point, a perpendicular, F C, is given ; and consequently, if produced, its point of intersection, C, with the given line DE. But C was the required point, and the production of the perpendicular from F determines it. By Stkthesis. — Join the points A and B, and bisect the st. line AB in the point F ; at F raise a perpendicular, and produce it to intersect DE in the point C ; — ^that point C is equi-distant from A and B. V AP = FB, FC common, and ^ AFC = ^ BFC. .". the side CA = the side CB : which was required. In the Analysis, we have taken the Problem to pieces ; in the Synthesis, we have put the parts together, and completed the purpose at which we aimed. It is by following similar Methods that other Geometrical Propositions may be analyzed and estab- lished. Analysis, however, is more suited for Problems; — Synthesis for Theorems. The Eules for conducting an Analysis are few ; — for the mode of procedure depends in a great measure on the knowledge and skiU of the Student ; and the greater these are, the greater facility and clearness will be manifest in making an analysis. It is a process which calls forth aU the resources of his mind, — and therefore a very improving exercise for the young Geometrician. The suggestions which are contained in Ritchie's Geometry, p. 50, may be of service to the Learner, and therefore are recom- mended to his attention. " 1st. As in every other study, endeavour to ascertain what it is you have to do ; examine into the nature and meaning of the Proposition, and form a clear, well-defined idea of the quantities concerned in the investigation. 2nd. Construct every figure with exactness, that the eye may aid the judgment. ANALYSIS. — STSTHESIS. 221 _ 3rd. It will often be necessary to join certain points so as to form equal triangles, isosceles or equilateral triangles, and other right-lined figures. 4th. Often, also, to lengthen lines, to draw perpendiculars from certain points in a line, or to let f^ perpendiculars from certain points on straight lines. Sth. Often, straight lines or angles must he bisected, one angle added to another, so as to get the sum of two angles ; or a Une drawn within an ajigle, so as to get the diBference of two angles. 6th. Also, it will often be necessary to draw st. lines parallel to certain lines through remarkable points, which may be either given or required. 7th. In short, the Learner must form such a combination of lines, angles, and circles, as will, in his judgment, lead to the discovery of the object required. If, after trial, he finds he cannot reach the required point, and take the citadel by the path he has sketched out, he must commence the attack anew by following a different road, and by adopting » different system of tactics." " Bemare. — 1. A point is said to be given, when its position is either given, or may be determined. 2. A line is given in position when its direction is given ; in magnitvde, when its length is given. 3. A line is given in position and magnitude when both its direction and magnitude are given. 4. The Position of a point can be found only— Jierst, by a st. Une cutting another st. line ; second, by a st. line cutting the circumference of a circle ; or, third, by the intersection of the arcs of two circle. 5. The Position of a line is found, when any two points in it are found ; and its Magnitude, when the extreme points are found . A few Examples, selected from various sources, and restricted to the First and Second Books of the Elements, will show the Learner how an Analysis, or a Synthesis may be conducted. Ex. 1. — Pbob. — Given an amgle, A ; a side, B 0, opposite to it, and the sum, B D, of the other two sides of a triamgU : — to construct the triangle. Analysis. — The figure B A C is the required A , z A the given angle, B C the side opposite to ji A, and B D tide sum of the other two sides. Join D C : the sides of the triangle B A and A C = the sum B D. Take away the common part BA, and the rem. AO=therem. AD; .-. z ACD= z ADC. Buttheext. .;: BAC= z ACD+ -2 AD C, or.^ BAG = 2.iBDC; thus/: BD = 4/: BAG. Hence the method of construction. Stmthesib. — ^At D, one extremity of BD, make an angle = 4 /; BAG ; and from B, the other extremity, draw B C, the given side, to meet D in C J at C, make ^ A D = .^ A D C, so that G A may meet B D in A : the triangle BAG will have its sides B A + AG=B D the sum ; its /: .=the gireu A , and its side = the given side. Ex. 2. — Pbob. — Given B C the iase, aB atthe hose, and B D the sum of the other two sides of a triangle : — to construct the triangle itself. 222 APPENDIX. Ahaltbib. — The triangle ABC is the triangle re- quired ; f or ^ B equals the given angle, B C is the base, and BD=BA + AC, the sum of the sides. On joining D C, ACD becomes an isosceles A , and z ACD=zADC. Stkthesis. — Make BC=the base, ;^B=the given I. at the base, and B D=the sum of the sides. Join C D, and at C mate ^ D C A= z A D C ; let C A meet BD in A ; then the figure CAB is the tri- angle required. Ex. 3. — Peob. — Pram tiDo given points, A and B, on the same side of a sS. line given in position, C D, to draw two st. lines which shall meet in that line C D, and make equal amgUs with it. Analysis. — The point sought is G ; and ^ B Q D = ^ A G C. A^ Produce B G untU G E = G A, and join A E. The zEffC= zBGD=^AGC; /-■ .-. /EGF=^AGF, and A G=GE, and F G^- is common, .•. /: A F G= ^ E F G ; and they are rt. z s, and AF.= FE. But since A and C D are given, the perpendicular AF is given ; and the point E is known, and G is determined by the inter- section of C I) and E B. Stnthbsis. — From A draw a perpendicular to C D, and produce it, till F E = A F ; join E, B, cutting C D in. G ; and draw A G : the lines A G and B G are the lines required. For A F=FE,PGia common, and zAFG=^EFG; .-. zAGF= ^EGP= ^BGD ; and .". ^AGC= zBGD. Ex. 4. — Peob. — To divide a given St. line, A B, into two such paHs thai thfi rectamgle contained by them may be th/ree-fourths of the greatest which the case ml' admit of. Analysis. — Let AB be the given line, and bisected in D ; then □ A D . D B, or the square on D B, is the greatest possible rectangle. A ssume C, in A B, as the point required, so that AC.CB=|of D B«. On A B describe a semicircle, and draw C E perpendicular from A B ; then the square onCE=aAC.CB. Again, bisect D B in F ; on F B describe the rfc. DB; then B G is the line the square of which =| of D B'. Hence, G B = E C ; join G and E ; .-. the points E and C are found. Synthesis.— Let A B be bisected in D, and D B in F ; and on F B let the rt. ^ d. A F B 6 be made, having its hypotenuse F G = D B. On A B describe a semicircle ; through G draw G E 11 A B, and through E. E C II G B ; then C is the point required, the n A C . C B being equal to f of the sq. on D B, which is half the line A B. ■^ A O D J- 3 ^d. A in which FG= VARIETY OP PATHS. 223 Some Problems admit of only one definite solution, and they are called determinate ; and some admit of two or more definite solu- tions, and these are called indetermmate. A Problem is determinate, when it relates to the determination of a dngle point, and the data are sufficient to determine the position of that point : it is indeterminate, if, on omitting one or more of the conditions, the data which remain may be sufficient for the determination of more tlian mie point, each of which satisfies the conditions of the problem. In general, such points are found to be situated in some line ; and hence such line is called the locus of the point which satisfies the conditions of the problem. Some Problems also are possible within certain limits ; and cer- tain magnitudes increase, while others decrease, w;ithin those limits, and after having reached a certain value, the former begin to decrease, while the latter increase. This circumstance gives rise to questions of maxima and minima, or the greatest and least values which certain magnitudes may admit of inindeterminate problems. — Potts' Euclid, p. 289-292. It may be remarked that Synthesis is well adapted for commu- nicating a knowledge of the first principles of Geometry, or of any Science ; but that Analysis is the great instrument or means of discovery, — it enables the Student to find out the solution of a problem for himself, and makes him more and more self-reliant in his search for Geometrical truth; — a most valuable power of mind- worth many a struggle to attain. II.— VARIETY OF PATHS. Until some practice and self-work have given familiarity with the Processes of Geometrical Eeasoning and of Geometrical Analysis, students are apt to suppose there is but one unvarying Method of conducting a Demonstration and of arriving at a sound conclusion. Fixed, rigid, and unalterable indeed is the law, that, beyond Postulates and Axioms, nothing is to be taken for granted, and that each step must rest upon mathematical certainty before another is attempted ; but a variety of paths, all free from error, may in many instances be followed. Such a variety of paths is admirably shown in the Second Booh of Geometry; 8vo, 1863, by Thomas HiU. of Harvard College, Cambridge, Mass., U.S.N. A ; and 224 APPENDIX. with this candid acknowledgment that chapter of his work is here made use of: — " 1. As there are usually many paths by which we may ascend a hill, so there are usually many modes ly which we maiy demonstrate a proposition. In the case of a simple proposition, it is not usually worth while to try more than one mode. But with more difficult problems, it is sometimes worth while to spend a great deal of labour in discovering the simplest mode of demonstration. There are geometrical truths which can be demonstrated in so simple a manner as to require only twenty lines to write down the demonstration : and yet some writers, from ignorance of this simple mode, have written more than twenty pages to prove the same truths. " 2. It will therefore be useful to you, to show you by a simple example, such as that of the equality of the sum of the angles in a triangle to two right angles, the great variety of methods by which a single proposition can be proved. " 3.' In the proofs of this proposition, which I will now give you, I shall not be careful to follow out every step. It will be enough, for the pwrpose we now have im, view, simply to show you the general line of the paths, without taMng you through every step of the way. " 4. The line D E might be drawn so as to coincide in direction with one of the other sides of the triangle, as A B, which would -j.. give us the figure in the margin. And by imagining \^^ the dotted line AP parallel to CB, we should have p : --^>\A ^ PAC equal to / ACB, and i PAD equal to ^^'"^^^ z C B A, which would make the three angles at A ^^ ^^E equal to the three angles of the triangle, as in the •^- ^' proof of Prop. 32, bk. i. CO "5. By prolonging the lines C A and B A through the point A, D E being parallel to B C, JT'v j we shall have z NAD equal to i. ABC, N. a ^^ /. JAE equal to i. ACB, and l NA J equal _^ J*^^^ j; to z CAB. So that the three angles of the " ^,»^\^^ triangle ABC will be equal to the three angles ^^^ N. DA N, NAJ, and JAE, which are manifestly ^ ^^ ^-^B equal to two right angles. " 6. The three methods of proving this proposition that have now been given are strictly Qeometrical. Others might be given, that are some thing more like Algebraical reasoning. " 7. Let us, for instance, imagine each of the angles of the triangle ABC prolonged at its right-hand end, as in the figure, and also a st. line CG drawn from one vertex, as C, parallel to the opposite side BA. Now, the external angles, PCB, D B A, EAF, are plainly equal to the three angles PCB, B C 6, G C F, and amount in magnitude to four right angles. But each external angle is plainly the supplement of one of the angles of the triangles ; that is, it is equal to the difierenoe between two right angles and one angle of the triangle. The sum of the three external angles PCB, VABIETT OP PATHS. 22& DBA, and E A F must therefore be equal to the difference between six right angles and the angles of the triangle. But as this diffei'enoe amounts to four right angles, the three angles of the triangle must be equivalent to two^ rijjht angles. " 8. If we introduce the idea ofmoUon, we can devise quite a different sort of demonstration. Suppose, for instance, that I stand at the point A, with my face towards the point C. Let me now C turn to the right until I face towards the point B. I have now changed the direction of my face by an amount which is equal to the angle at A. Suppose that I now walk to B, without turning, I shall have my back towardti A, and if standing still, I turn to the right, until I have changed my direction by an amount equal to the angle A B C, I shall have my back towards C. Let me now walk back- wards, without turning, until I reach C, and I shall have my face towards B. I will now turn a third time to the right, until I face the point A. My thres- turnings, o? changes of direction, have been equal to the three angles of the triangle ; they have all been to the right ; therefore my whole change of direc- tion is equal to the sum of these angles ; I am now looking in exactly the opposite direction to that from which I started ; I am lookmg from C to A, instead of from A to C ; I have turned half way round ; that is, through two right ngles. Whence, the sum of tht three angles of the triangle is eguiialent to tes right angles. " 9. Another demonstration, iy means of motion, may be obtained as f ol" lows : — Suppose an arrow, longer than either side of the triangle, to be laid upon the side AC, pointing in the direction from A to C. Taking hold of the pointed end beyond C, turn the arrow round upon the point A, as a pivot, until the arrow lies upon ibe line AB. Taking now hold of the further end, beyond A, turn the arrow upon B as a pivot, imtil the arrow lies upon the line BC. Using C as a pivot, turn it now until the point of the arrow is over A. The arrow has thus been reversed iu direction, turned half-way round, or through two right angles, It has been turned successively through the three angles of a triangle, and every time in the same direction, like the hands of a watch ; so that the total change of direction, two right angles, is equivalent to the sum of the three angles. " 10. You have thus seen how a single proposition, and that a simple one, may be proved in a variety of ways. We have shown, in .six different ways, what is the value of the sum of tiie angles in a triangle ; in three, by what is called rigid Geometry j in one, by a parfly Algebraical process ; and in two, by introducing the idea of motion. And I wish you to observe that every one of" the six ways is satisfactory. They are aU proofs that are certain, because they lead you from self-evident truths by self-evident steps. One is not more certain than the other, because they are all absolutely certain. The only choice- between them is, that some are more purely geometrical ; some are better adapted to the peculiar tastes of different students ; and some are neuter, and more quickly perceived by xmtaught persons. " 226 APPENDIX, IIL^GEOMETRICAL EXERCISES. The Skeleton Propositions, which foi-m a continuation or com- pletion of the plan pursued in the Gradations of Euclid, furnish a most useful and improving series of exercises ; and, on the ascer- tained fact in the art of teaching, that repetition is a most important auxiliary, they are recommended to the notice and adoption of the teachers of mathematics. At any rate, where employed, they will accustom the student to bp systematic and exact, and not to advance a step without a reason ; and progress in this way, though it may chance to be less rapid, will be on a solid basis, and bring into play some of the most valuable qualities of the mind. Potts, Colbnso, Coolet, Chambers, and others, have each published Collections of Geometrical Exercises ; but as teachers may not wish to go beyond the limits of the present work, two Series of Exercises are now appended ; the First Series consisting of Problems and Theorems which are inserted in the Gradations, and of which the General Enunciations may be given out to learners, for analysis, solution, or proof ; and the Second Series containing Propositions which are not fully proved or not inserted in the ■ Gradations. SERIES I. Problems in Boole I. 1. By means of an equilateral triangle to meastire an inaccessible distance. 2. To construct a scale of equal parts. 3. To ascertain an inaccessible distance when two sides and their included angles have been measured. 4. To show, by observations on the shadows which objects east, the perpen- dicular heights of the objects. 5. To determine without a theodolite the angle at a given point made by a St. line from two objects meeting in that point. 6. To construct the Mariner's Compass-Card. 7. From a given point at the end of a st. line to raise a perpendicular. 8. On a given line to describe an isoceles triangle of which the perpendi- cular height is equal to the base. 9. From a given point over the end of a st. line to let fall a perpendicular to the line. 10. Given an angle of 73°, required its complement ; an angle of 96°, re- quired its supplement. SERIES I. — BBOB. 36. I. 227 11. By the application of the principle that vertical angles are equal, to find the distance between two objects. 12. From a given point, A, to direct a ray of light against a mirror, so that the ray shall be reflected to another given point. IS. To determine the number and kind of polygons which may be joined so as to cover a given space. 14. To construct a triangle when the base, the less angle at the base, and the difference of the sides, are given. 15. By means of a mirror placed horizontally, to construct a triangle, the perpendicular of which shall be representative of the height of any object. 16. Given three st. lines respectively equal to 40, 60, and 30 equal parts, to form a triangle. 17. On a given st. line to describe an isosceles triangle having each of the equal sides double of the base. 18. On a given st. line and with a given side to construct an isosceles triangle. 19. To construct a Line or Scale of Chords. 20. By aid of a line of Chords,— 1°' to make an angle containing a certain number of degrees, as 40° ; 2°' to measure a given angle ; 3°' from the extremity of a line to raise a perpendicular ; and 4°" to construct a triangle of which the base contains 40 equal parts, one of the angles at the base 40°, and its other adjacent side 35 equal parts ; and to measure the other side and the other angles. 21. To measure an inaccessible distance, A B, only A being approachable. 22. By the theory of Representative VaJiies to find the distance between two stations. 23. To measure an inaccessible distance, A B, neither A nor B being approachable. 24. Given the vertical angle and the perpendicular height of an isosceles triangle, to construct it. 25. On the principle that two rays of light proceeding from the centre of the sun to two points on the earth are physically parallel, to ascertain the earth's circumference. 26. By the method of parallel lines to ascertain the distance of an inaccessible object. 27. To determine the Parallax of a heavenly body. 28. To construct a. figure which will give the representative value of the perpendicular height of a mountain. 29. To construct a regular Polygon, — 1°' when the side is given ; 2*" when the side is not given. 30. To ascertain both the perpendicular height of a mountain, and the hori- zontal distance from the foot of the mountain to the foot of the per- pendicular. 31. To divide a finite st. line into any given number of equal parts. 32. To construct a SUding Scale to measure the hundreth part of an inch. 33. To construct a Sliding Scale for measuring the minutes into which a degree on a circle is divided. 34. To ascertain the continuation of a st. line when an obstacle intervenes. 35. From a given point in the side of a parallelogram to bisect . the parallelogram. 36. To convert a parallelogram into an equivalent raatangle. 228 APPENDIX. 87. By the method of Indivisibles to explain the equality of parallelograms' - on the same base and between the same parallels. 88. To construct a Diagonal Scale. 39. To divide a triangular space into two equal parts. 40. From any point in the side of a triangle to divide the triangle into two equal parts. 41. To find the Area of a trapezium. 42. To find the Area of a square. 43. To find the Area of a triangle. 44. To find the Area of any right-lined figure. 45. To find the Area of a regular polygon. 46. To find the Area of a circle. 47. Of the Diameter, Circumference, Area, and Ratio of the diameter and circumference, any two being given, to ascertain the others. 48. To describe a triangle equal to a given parallelogram, and having an angle equal to a given angle. 49. A parallelogram being given, to find another parallelogram equal to it, and having one side equal to a given st. line. 50. Given the area of one figure, and the side of another which is to be a parallelogram equal to the given figure, to find the other side of that parallelogram. 51. To change any right-lined figure, first into a triangle, and then into a rectangle of equal area. 62. To straighten a crooked boundary without changing the relative size of two fields. 63. Given the diagonal to construct a square. 64. To ascertain the height of an inaccessible object by aid of the Geometrical Square. 55. Given in numbers the sides of a right-angled triangle, to find the hypotenuse. 56. Given in numbers the hypotenuse and one side, to find the other side. 57. To find a square equal to any number of squares ; or a square that is the multiple of a given square ; or a square that equals the difference of two squares ; or a square that is the half, the fourth, &c., of a given- square. , 58. To make a rectilineal figure similar to a given rectilineal figure. 69. To make a circle the double, or the half, of another circle. 60. To construct the Chords, Natural Sines, Tangents, and Secants, of Trigonometrical Tables. 61. To find right triangular numbers. 62. To compute Heights and Distances from the curvature of the earth. Theorems in Booh I. By the bisection of the vertical angle of an isosceles triangle to show that the angles of the base are equal ; and also that the bisecting line bisects the base at right angles. Only one perpendicular can be drawn from a given poiat to a given st. line. SERIES I. — PBOB. 9. IL 229 -S. The perpendicular is the shortest line from a given point to a given Bt. line. 4. From the same point only two equal st. lines can be drawn to a given st. line. 5. All heav^ bodies free to move continually descend, or seek the point which is nearest to the earth's centre. ■6. Of all st. lines that can be drawn from one point to a plane surface, and reflected to a third point, those are the shortest which make the angle of incidence equal to the angle of reflection. 7. The chord of 60° is equal to the radius of the circle. 8. A st. line which is perpendicular to one parallel, is also perpendicular to the other. 9. If a st. line falling on two other st. lines make the interior angles on tho same side less than two rt. angles, those two lines on being produced shall intersect. 10. A parallel to the base of a triangle through the point of bisection of one side, will bisect the other side. 11. The st. lines which join the middle points of the three sides of a triangle,' divide it into four triangles which are equal in every respect. 12. The St. line joining the points of bisection of each pair of sides of a triangle, is equal to half the third side. 13. A trapezium is equal in area to a parallelogram of the same altitude, and of which the base is half the sum of the parallel sides. 14. The squares of equal st. lines are equal ; and if the squares are equal, the lines are equal. 15. Every parallelogram having one rt. angle, has all its angles rt. angles. 16. If a perpendicular be drawn from the vertex of a triangle to the base, the difference of the squares of the sides is equal to the difference between the squares of the segments. i7. If a perpendicular be drawn from the vertex of a triangle to the base, or to the base produced, the sums of the squares of the sides and of the alternate angles are equal. Problems in Book II. 1. From Propositions 1, 2, and 3, deduce various methods for the Multiplicar tion of Numbers, and demonstrate the rule. 2. From Prop. 4, point out a practical way of extracting the Square root of a number, and prove the correctness of the formula. 3. To find the difference between the squares of two unequal numbers without squaring them. 4. To find Quantities in Arithmetical Progression. 5. To find the value of an Adfected Quadratic Equation in Algebra. 6. By aid of Prop. 6, to ascertain the diameter of the earth. 7. Given the sum and the difference of two magnitudes, to find the magni- tudes themselves. H. From the Area of a rectangle and one side given, to obtain the other side. S. To divide a given Line a, so that its parts x and oe - a; may make a(a-x) = x^. Let the solution be given both algebraically and arithmetically. 230 APPENDIX. 10. To ascertain the Area of a triangle when the three sides are known. 11. From the three sides of a triangle given, to obtain the perpendicular : — 1°. when the perp. falls withm the base ; and 2° when it falls without the base. 12. To find a mean proportional to two given lines. 13. To approximate to the square of any curve-lined figure. 14. To calculate the Area of any right-lined figure. Theorems in Booh II. 1. The difference of the squares of two quantities, equals the rectangle of their sum ajid difierence. 2. The difference of the squares of two quantities is greater than the square of their difference, by twice the rectangle of the less and their difference. 3. The square of the sum of two lines is equal to four times the rectangle under them, together with the square of their difierence. 4. Four times the square of half the sum is equal to four times the rectangle under the lines, together with four times the square of half the difference. 6. The sum of the squares of any two st. lines is equal to twice the square of half their sum, together with twice the square of half their difference. 6, The sum of the squares is equal to half the square of the sum, together with half the square of the difference. SERIES II, PROPOSITIONS NOT FULLY PROVED, OR NOT INSERTED IN THB GRADATIONS. Problems. — Booh I. 1. To find a point which is equidistant from the three vertical paints of a triangle. 2. To bisect a triangle by a st. line drawn from a given point in one of its sides. 3. Describe a circle which shall pass through two given points, and have it» centre in a given line. 4. Through a given point to draw a st. line that shall be equally inclined to two given st. Imes. 6. Given a triangle ABC, and a point D in AB ; to construct another triangle ADE equal to the former, and having the common angle A. 6. To change a triangle into another equal triangle of a given altitude. 7. To draw a at. line which, if produced, would bisect the angle between two. given st. lines, without producing them to meet. 8. To trisect a right angle. SEBIE3 II. — THEOR. 15. I. 231 *. To trisect a given st. line. 10. Given the sum of the sides of a triangle, and the angles at the base, to construct it. 11. Given the diagonal of a square, to construct the square of which it is the diagonal. 12. Given the sum and difference of the hypotenuse and a side of a, right- angled triangle, and also the remaining side, to construct it. 13. To find the loeua of all points which are equidistant from two given points. Theorems. — £ook I. 1. In an isosceles triangle, the right line which bisects the vertical angle also bisects the base, and is perpendicular to the base. 2. If four straight lines meet at a' point, and make the opposite vertical angles equal, each alternate pair of st. lines will be in the same st. line. 3. The difference of any two sides of >•. triangle is less than the remaining side. 4. Each angle of an equilateral triangle is equal to one-third of two right angles, or to two thirds of one right angle. S> The vertical angle of a triangle is right, acute, or obtuse, according as the st. line from the vertex bisecting the base is equal to, greater, or less than half the base. 6. If the opposite sides or opposite angles of a quadrilateral be equal, the figure is a parallelogram. 7 If the four sides of a quadrilateral are bisected, and the middle points of each pair of conterminous sides joined by st. lines, those joining lines will form a parallelogram the area of which is equal to half that of the given quadrilateral. 8. If two opposite sides of a parallelogram be bisected, and two st. lines be drawn from the points of bisection to the opposite angles, these two St. lines trisect the diagonal. 9. In any right-angled triangle, the middle point of the hypotenuse is equally distant from the three angles. 10. The square of a st. line is equal to four times the square of its half. 11. The st. line which bisects two sides of a triangle, is parallel to the third side, and equal to one half of it. 12. If two sides of a. triangle be given, its area will be greatest when they contain a rt. angle. 13. Of equal parallelograms that which has the least perimeter is the square. li. The area of any two parallelograms described on the two sides of a triangle, is equal to that of a parallelogram on the base, whose side is equal and parallel to the line drawn from the vertex of the triangle to the intersection of the two sides of the former parallelograms pro- duced to meet. 15. The vertical angle of a triangle is acute, rt. angled, or obtuse, according as the square of the base is less than, equal to, or greater than, the sum of the squares of the sides. 232 APPBNDIX. Problems. — Booh II. i. The sum and difference of two magnitudes being given, to find the magnitudes themselves. 2. To describe a square equal to the difference of two given squares. 3. To divide a given st. line into two parts, such that the squares of the whole line and of one of the parts shall be equal to twice the square of the other part. 4. To divide a given st. line into two such parts that the rectangle contained by them may be three-fourths of the greatest of which the case admits. 6. Given the area of a right-angled triangle, and its altitude or perpendicular from the vertex of the rt. angle to the opposite side, to find the sides. 6. Given the segments of the hypotenuse made by the perp. from the rt. angle, to find the sides. 7. To divide a st. line internally, so that the rectangle under its segments shall be of a given magnitude. 8. To cut a st. line externally, so that the rectangle under the segments shall be equal to a given magnitude, as the square on A. 9. Given the difference of the squares of two st. lines and the rectangle under them, to find the lines. 10. There are five quantities depending on a, rectangle, — 1°' the sum of the sides ; 2°" the difference of the sides ; 3°" the area ; 4°' the sum of the squares of the sides ; and 6°' the difference of the squares of the sides ; — by combining any two of these five quantities, find the sides of the rectangle. ' Theorems. — Booh II. 1. The square of the perpendicular upon the hypotenuse of a right-angled triangle drawn from the opposite angle, is equal to the rectangle vinder the segments of the hypotenuse. 2. The squares of the sum and of the difference of two st. lines, are together double of the squares of these lines. 3. In any triangle the squares of the two sides are together double of the squares of half the base, and of the st. line joining its middle point with the opposite angle. 4. The square of the excess of one st. line above another, is less than the squares of the two st. lines by twice their rectangle. 6. The squares of the diagonals of a parallelogram are together equal to the squares of the four sides. '6. If a st. line be divided into two equal and also into two unequal parts, the squares of the two unequal parts are together equal to twice the rectangle contained by these parts, together with four times the square of the St. line between the points of section. 7. If a St. line be drawn from the vertex of a triangle to the middle point of the opposite side, the sum of the squares of the other sides is equal SERIES II. — THOB. 15. II. 233 to twice the sum of the squares of the bisector and half of the bisected Bide. 8. The sum of the squares of the sides of a quadrilateral figure is equal to the sum of the squares of the diagonals, together with four times the square of the st. line joining their points of bisection. 9, If St. lines be drawn from each angle of a triangle bisecting the opposite side, four times the sum of the squares of these lines is equal to three times the sum of the squares of the side of the triangle. 10. The square of either of the sides of the rt. angle of a rt. angled triangle, is equal to the rectangle contained by the sum and difference of the hypo- tenuse and the other side. 11. If from the middle point C, of a st. line AB, a circle be described, the sums of the squares of the distances of all points in this circle from the ends of the st. line A.B. are the same ; and those sums are equal to twice the sum of the squares of the radius and of half the given Ime. 12. Prove that the sum of the squares of two st. lines is never less than twice their rectangle ; and that the difference of their squares is equal to the rectangle of their sum and difference. 13. If, within or without a rectangle, a point be assumed, the sum of the squares of st. lines drawn from it to two opposite angles, is' equal to the sum of the squares of the st. lines, drawn to the other two opposite angles. 14. If the sides of a triangle be as i, 8, and 10, the angle which the side 10 subtends will be obtuse. 15. If in a rt. angled triangle a perpendicular be drawn from the rt. angle to the hypotenuse, the rectangle of one side and of the non-adjacent seg- ment of the hypotenuse, shall equal the rectangle of the other side and of the other non-adjacent segment of the hypotenuse. TABLE OF CONTENTS. PEEFACE. PAGES. Respecting the Gradations in Euclid's Plane Geometry, Skeleton Propositions, &c iii. -viii. INTRODUCTION. Section I. Gradual Growth of Geometry and of the Elements of Euclid 1-5 Section II. Symbolical Notation and Abbreviations that may be used 5-7 Section III. Explanation of some Geometrical Terms 7-10 Section IV. Nature of Geometrical Reasoning 10-21 Section "V. Application of Arithmetic fmd Algebra to Geometry ... 21-27 Section VI. On Incommensurable Quantities 28-30 Section VII. On Written and Oral Examinations and Means of Progress 30-35 BOOK I. Geometry of Plain Triangles 39-148 Definitions 40-46 Postulates and Axioms 47-49 Propositions i. - xl viii 51-147 Remarks 148 BOOK II. The Properties of Right-angled Parallelograms or Rectangles 149 Definations and Axiom 150 Propositions i. - xiv 150-185 Remarks 185 Synopsis of Book II 186-190 PRACTICAL RESULTS. I. Problems 5 - 16, for the Construction of Geometrical Figures stated, and proved in Books I and 11 191 -197 Subsidiary Problems, 17 - 31, in Books L and II 197-201 II. Problems in Books III., IV., and VI. most intimately connected with Books I. and II. : — Book III., Problems 1-10 201-203 „ IV., „ 1-18 204-207 „ VI., „ 1-11 208-211 CONTENTS. 235 III. Principles of Construction : — Section 1. For Geometrical Instruments to Measure Distances and Angles 211-212 Section 2. For Geometrical Figures to exhibit representative values of Magnitude and Spaced 212-215 IV. Principles for accurately calculating Distances, Magnitudes, and Areas : — Section 1. Lines or Distances 215 Section 2. Angles .'..'.' 216 Sections. Magnitudes or Areas 216 Eemarka .....217-218 APPENDIX. I. Geometrical Synthesis and Analysis 219-223 II. Variety of Paths 223-225 III. Geometrical Exercises 226-233 Series l"- Problems in Book I. 1-62 226 Theorems in Book I. 1-17 228 Problems in Book n. 1-14 229 Theorems in Book 11. 1-6 230 Seeies 2°" Propositions not fully proved, or not inserted in the Gradations : — Problems.— Book I. 1-13 230 Theorems. — Bookl. 1-15 231 Problems.— Book II. 1-10 232 Theorems.- Book II. 1-15 232 " Even the Sleeping Geometrician," says old Kalph Codwoeth, p. 160, "hath at that time, all his Geometrical Theorems and Knowledges some way on him : as also the Sleeping Musician, all his Musical Skill and Songs : and therefore why may it not be possible for the Soul to have likewise some Actual Energie on it, which it is not Expressly Conscious of ? " John Hetwood, Excelsior Printing and Stationery Works, Hulmo Hall Boad, Manchester. EUCLID'S PLANE GEOMETRY, BOOKS m— VI., PEACTICALLY APPLIED; GRADATIONS IN EUCLID, PART IL, ALGEBRAICAL AND AKITH5IETICAL ILLUSTRATIONS, EXPLANATORY NOTES, AND A SYNOPTICAL INDEX TO THE SIX BOOKS, SHEWING THE USES OF THE PROPOSITIONS, Sic, BY HENKY GEEEN, A.M. " THERE IS (GENTLE HEADER) NOTHING (THE WORD OP GOD ONELT SET APART) WHICH SO UUCH BEADTEFIETH AND ADORNETH THE SODLE AND UINDE OF HAN, AS DOTH THE KNOWLEDGE or GOOD ARTES AND SCIENCES. " BttHngsUy't Euclid, A. D. 1670. MANCHESTER: JOHN HEYWOOD, 143, DEANSGATE. LONDON : SIMPDN, MARSHALL, Si CO. • 18 6 1. TaWIILIAM FAIEBAIEN, C.E,LLD., M,S, F.G.S., CORREgPONDINO UEIiIBER OP THE NATIONAL INSTITUTB OF FRANCE, AND OF TIIE HOTAL ^CADEMT OP TUBIN, CHEVALIER OP THE LEGION OF nOKOUH, ETC., ETC. ; ■ WHOSE AIJIS IN LIFE HAVE LED HIST TO ENTERPRISES OF PUBLIC UTILITY, AND WHOSE WRITINGS AND EXAMPLE SHOW THE PRACTICAL USES OF SCIENTIFIC KNOWLEDGE; THIS WORE, BT HIS FEIOIISSION, \S INSCRIBED; AND WITH EVERY SENTIMENT OF HCSPECT- PREFACE., The Preface to Bks. I and II of the " Gradations in Euclid" contains most of the observations which can be required from the Author. One of them he now repeats ; — " At the present day nearly every edition of Euclid's Elements must be more or less a compilation, in which the Author draws freely on the labours of his predecessors. ' The Gradations' are, in a great degree, of this character ; and an open acknowledgment will suffice, once for all, to repel any charge of intentionally claiming what belongs to others. It is affectation to pretend to great originality on a subject ■(yhich has, like Geometry, for so many centuries exfircised m.en's minds." Originality has not been the Author's aim, but usefulness. With this purpose before him, he has endeavoured to show how few, if any. Geometrical Truths are destitute of a practical appli- cation. The stigma which some have attempted to fasten on what they name, " mere Mathematical Theories," is thus removed, and the Science, instead of being repulsive from its dry abstractions, is invested with the ever-abiding charm of being both the foundation and the builder-up of very many most important practical results. In fact, there is scarcely a branch of human knowledge, from the art of sketching an outline, to that of spanning and measuring the heavens themselves, which does not depend for its vigour ai\d coni- prehensiveness on the aids which Geometry furnishes. Step by step man ascends the Himalayas and compasses the earth; step by step is the Mathematician's course. As in the First Part, so in this Second Part of the " Gradations,'' the same method has been pursued. Prefixed to the Proof are references to the principles that have to be employed, and often quotations of the very words in which Euclid embodies those principles. These are not so fully given, indeed, as in the First Part, for it is presumed that familiarity with leading principles and propositions has been already attained. Occasionally too, in the steps of the Construc- tion and Demonstration, the special reference is not made in the margin to the evidence on which an argument or a conclusion rests ; but the Learner will scarcely find this any obstacle, if he has mastered what he has read. It will be well for the Learner thoroughly to consider the refer- ences, before he proceeds to' the Particular Enunciation, the Con- struction and the Demonstration of the Proposition ; — indeed, ■were he of himself to'put together the truths with which he is supplied) and to see how the new truth is to be deduced from them, he would derive the best assistance, that from the reasoning of his own mind, to understand and appreciate the fuller proof of the formal demonstration. The Memory, no doubt, is a most valuable power in acquiring any kind of knowledge, but in Mathematics es- pecially it is the understanding and the reasoning faculty that are employed to most advantage and developed with greater exactness. As an instance of the method recommended, let the Learner take that important Proposition, 35, III, " If two st. lines cut one another within a circle, the rectangle contained hy the segments of one of them is equal to the rectangle contained hy the segments of the other." After weighing these words of the General Enunciation, let him call to PEEFAOB. VU, mind the several truths contained in the references under the heads, "Con.," construction, and "Dem.," demonstration; and if he has forgotten any of them let him turn back to the very propositions numbered, as 10, I., 3, III., 47, I., &c., and carefully think them over. He thus burnishes up his old weapons; and now let him try to trace out the connexion between the propositions referred to, and to ascertain how they lead to the new proposition which he seeks to establish. He will say to himself, here are several undoubted truths and facts presented to me; — I have already accepted thein as prin- ciples of Geometrical Eeasoning, — and they are now given that I may demonstrate some other truth, or solve some other problem. Can I not, with the implements provided, build this new hou se and see how, like the others, it is composed of indestructible materials ? He may rely, that by thus exercising his judgment, he will do more, than any mere effort of memory can do, for really under- standing and retaining mathematical truths. A very full Table of signs and abbreviations is given, and this should be consulted untU they have become well known. Some of the Demonstrations, as in 8, V, and 15, V, have been shortened. By the time the Learner has mastered so much of the Geometry, he will readily perceive the connexion of the argument, and not require the entire fulness of which it is capable ; or, if he should, he may be expected to supply it from his own resources. The Index was a subject of some consideration. An alphabetical Index had been prepared, but it was rejected, because it would have occupied too much space. The advantage of learners appeared to be more promoted by having the whole of the General Enunciations TlU. PHEFAOE. of Euclid's Geometry brought together under the heads of Problems and Theorems, with their respective illustrations, applications, and uses. An alphabetical Index would have facilitated references to particular truths, but the consecutive or synoptical Index conduces more to the understanding of the whole work, and to the tracing out of the connexions of its parts. A word or two to those who, from inexperience, do not under- stand the difficulty of avoiding errors of the press in a work where, many signs, abbreviations, and references are used. As a most justly celebrated mathematician* has observed, — Tlie Table of Corrigenda, at the end of the volume, "may convey an impression that the work is incorrectly printed, which is not the case ;" and he adds, " If every mathematical work, at its completion, had the fruits of some years of examination presented to the reader, I know of none which would not have lists as large in proportion to their size and the number of symbols contained in them as the present one." On this subject the author will simply remark, that should " the Euclid Practically applied" attain a second edition, these and some other faults vrill be carefully amended. In his Mathematical Preface, " written at his poor House at Mortlake, Anno 1570, February 9," " John Dee, of London," addressed himself " to the vnfained Lovers of Truthe and constant Studentes of Noble Sciences ;" " he hartely wisheth them grace from heaven and most prosperous successe in all their honest attemptes and exercises." • So, with him, I say to all who value good learn- ing, " I commit you vnto God's MeroyfuU direction for the rest; hartily beseechyng hym, to prosper your Studyes and honest Intentes to His Glory and the Commodity of our Country." October 1, 1861. ♦ Db Morgan, in his Differential and Integral Calcuhu, a.d. 1842. SYMBOLICAL NOTATION. SYMBOLICAL NOTATION AND ABJiRBVIATIONS. I. — Signs common to Arithmetic, Algebra, and Geometry. > > < because. therefore. wherefore. equals, or equal. not equal to, or unequal. greater than. not greater than. less than. + pliis, add, together with. — minus, subtract, take away. ~ difference between. X into, multiply. -~ by, divide. ^/ root. : ratio. : = : equality of ratios. : : : : proportion. : : : Numbers or Quantities in Progression The signs >, >, <, <, between ratios, as A : B > C: D, or A : B > C : D, or A : B < C : D, or A : B <' C : D, denote that the one ratio is greater than, or not greater than, ^ less than, or not less than, the other ratio, according to the sign* * <|; not less than. II. — Geometrical Signs. m A triangle. / 7 parallelogram. D square, I I rectangle. © circle. • 0ce circumference. • a point. * I straight Hne. {{ parallel, parallel to. /. angle. _L perpendicular to, at rt. /_ s. * When an « is added to a sign, or to an abbreviation, the plural is denoted A single capital letter, as A, or B, denotes the point A, or the point B ; but sometimes, as in Bts. V and VI, the quantity, or magnitude. A, B, C, &c. Two capital letters, as A B, or G,D, denote the straight line A B, or C D ; but when the letters indicate opposite angles, they denote a parallelogram, or a rectangle, or a polygon, as the figure wUl show. A capital letter, or two capital letters, with the wMmeraZ ^ just above to the right hand, as A^, or A B^, denote not the square of A, A B, but the square ow A or A B. X. ALGEBRAIC ESPEESSIONS, &C. Capital letters, with a point between them, as AB.CD, denote, not the prodiict of A B multiplied by C D, but the rectangle formed by two of its sides meeting in a common point. III. — Additional Algebraic Expressions. M Magnitude. n orp m multiple. m + n m A &c. multiple of A &c. »i A, m B, &c. equimultiples of A,B mn &c. mn A m (A + B) multiple of (A+B) (m + n) A m(A-^B) multiple of (A-B). m (A+B-C) multiple of the excess pt. of (A+B) above C. sub-m another multiple. the sum of the quanti- ties m Sin. the product of m X n. a multiple of A by mn a multiple of A by ' m + n. " part. submultiple. , III. — Ahhreviations. Add Addenda, by adding. App Application of a Prop. Appl .^Mplicando, by applying 0. or Con. .(instruction. C. 1 &c. ..Step 1 &o. of the Con- struction. Cone Conclusion, inference. Cor CoroUaiy. Dat Datum, or data. T>. orDem. Demonstration. D. 1 &c. . .Step 1 &c. of the Dem. E. or Exp. .Exposition, or Particular Enunciation of a Prop. Ex Example. Gen General Enunciation. H. or Hyp.Hypothesis of a Prop. H. 1 &c.. .Step 1 &c. of the Hyp. L Line. JI. or Mag. Magnitude. P. or Prop. Proposition. Pan Ponendo, byplacing, by position. Prob Problem. Proced ...Precedendo,hy going on. Prel Preliminary. Prod Product. Pst. orPsts.Postulate, or Postulatei Quaes Quaesitum or Qumsita. E Ratio. Eec Hecapitulation. Eemk. . . .Remark to be made. Sch Scholium or Scholia- Sim., So, similarly, by similar 1^, reasoning. S. or Sol. . .Solution of a Problem. Sum Sume, take away. > Sup .Suppose, or Let. Super Superponendo, by super- dosition. Theor Theorem. ABBREVIATIONS Q. E. D. quod erat . demonstrandum, wMoli was the tMng to be proved. Q. E. F. quod erat faciendum, wliich was the thing to be done. adj adjacent. ad imposs. . ad impossib^le, to an im- possibility. a fort. . .. .a fortiori, by a stronger reason. alt altitud_e. altr .'alternate. antec . .. .antecedent. ang angular. ^j^ assum assumendo,hy adopting. bis ; .bisects, or bisect. bisd.. .. ;. .bisected. bisg bisecting. cen centre. ch chord. com common. comp- . .. .compound. compl complement. con. sup contrary supposition. c. scr circumscribe. c. scg circumscribing. conseq consequent. cont continued.. conterm ...conterminous. contn contain, or contained. descr describe, or described. descg describing. drag diagonal. diam diameter. diff. difference. dist distance, or distant. div divide, or divided. dupl duplicate. eq equal or equally. eq. ang.. ..equiangular. eq. lat equilateral. ex. ab ex absurdo, by an ab- surdity. ex gr, exempli gratia, for example's sake. ext; .exterior, or exteriorly. extn extenially. extr. .; .. .extremity, or extremities. fig figure. gr greater. homol. . .homologous.' hyp hypotenuse. incl include, included. indef. . . . .indefinitely. inscr inscribe, inscrib.ed. int interior. inters intersect, intersection. intr internal, internally. mul--}-'^«P'« magn magnitude. meas.- . .. .measure. mid?. .. ,. .middle. ob obtuse. opp opposite. par parallel. parlm parallelogram. pent pentagon. perp perpendicular. ABBREVIATIONS, pos position. prod produce, prodaced. propl proportional. pt part. qu. ang. . . qnadrangol&r. qu. lat. .. .quadrilateral. rad radins. rat ratio. recip reciprocal. rect rectangle. recti rectilineal. rectr rectangular. reg regular. rem remaining. resp respective. rt right. sect sector. eeg .segment. sem. c. . .semicircle. sem. cirf....semiclrcumference. sim similar to, similarly. sim. sit. .. .similarly situated. sq. square. St. straight. suppl supplemental. tang tangent. ^rap trapezium. undiv undivided. nneq. unequal, or unequally. vert vertex, vertical. GRADATIONS IN EUCLID. BOOK III. TREATING OF THOSE PROPERTIES OP THE CIRCLE, AND OP STRAIGHT LINES IN AND ABOUT IT, WHICH CAN BE DEDUCED PROM THE PIEST AND SECOND BOOKS. A circle, strictly speaking, signifies the space bounded by a •circumference, but in this book the term is employed sometimes to denote that space, and at other times, the circumference itself. Euclid, too, occasionally assumes from experimental knowledge, certain preppies of the circle, which a more rigid and exact method of' reasoning would hare established before, using them. This is the case in the first Proposition itself, where it is taken for granted that the perpendicular to the chord of an arc will meet tiie circle in two points. In some instances also the method of indirect demonstration is adopted, when the more satisfactory '"method of direct proof is available ;* examples of this occur in Props. 2, 13, i6 and 36. , By restricting the meaning of the term angle to an opening formed by two conterminous lines, and less than two right angles, Z GEADATIONB IN EUCLID. Euclid renders some of Ms demonstrations, as that of Prop, 21, more cumbersome than they need be. The Properties of the right-angled triangle, of the circle, and of certain lines in and about a circle, as the radius, the sine, the tangent and the secant, have laid the foundations of by far the most extensive branch of Mathematics. Trigonometry, Plane and Spherical, resting on these properties and at first "confined to the solution of one general problem, has now spread its uses over the whole of the immense domains of the mathematical and physical sciences." — Laednbe's Trigonometry, p. 3. The Learner may therefore enter on the study of this Third Book with the assurance, that he is about to cross the threshold of one of the most important parts of Plane Geometry. "The influence, indeed, of the properties of the circle upon abstract mathematical analysis has been so great that an attempt to describe the manner in which the means of expression derived from this figure has been used, would fill a volume." The quaint English Editio Princeps of Euclid, published in 1570, thus opens to the Eeader the Summary of Bk. III. "This third booke of Euclide entreateth of the most perfect figure, which is a circle. Wherefore it is much more to be estemed then the two bookes goyng before, in which he did set forth the most simple proprieties of rightKned figures. For sciences take their dignities of the worthynes of the matter that they entreat of. But of al figures the circle is of most absolute perfection, whose proprieties and passions are here set forth, and most certainly demostrated. Here also is entreated of right lines subtended to arkes in circles: also of angles set both at the circumference and at the centre of a circle, and of the varietie and difference of them. Wherfore the readyng of this booke, is very profitable to the attayning to the knowledge of chqrdes and arkes. It teacheth moreover which are circles continget, and whicli are catting tte one the other: and also that the angle of contingence is the least of all acute rightliaed angles : and that the diameter in. a circle is the longest line that can be drawen in a circle. Farther in it may we leame how, three pointes beyng geuen how soever (so that they be not set in a right line) may be drawen a circle passing by them all three. Agayne, how in a solide body, as in a Sphere, Cube, or such lyke, may be found the two opposite pointes. Whiche is a thyng" very necessary and commodious, chiefly for those that shall Saafce instrumentes seruyng to Astronomy and other artes." — BiLLiNGSLBy's Euclid, fol. 81. Definitions. 1. Equal circles are those of which the diameters are equal, or, from the centres of which the straight lines to the circiunferences are equal. The criterion of the eqnaKty of circles is that their ffiameteis or their radii aa'e equal ; — but this is neither a Sefinitioa nor an Axiom ; properly it is a Theorem, the truth of which may be proved by super- position.; — for if centre be placed on centre and the eqiial radii or diameters on each other, the circumference of the one -will in each point coincide with the circnmference of the other, — and thus the space included by one circumference will equal the space included by the other. 2. A sixaight line is said to touch a circle, i. e. is a Tangent, when it meets the circle, and being produced does not cut it; as AB tan- gent to EFC in G. The point in which the straight line meets the cirde is the point of contact; the straight line "does not cut," i. e. does not pass into the circle. A Secant is a straight line which, when it meets the A. circle and is produced, passes iato the circle,' i,e., cuts or crosses the circnmference; as BHD, secantjto- EFCinH. 4 DEFINITIONS. The terms Tangent and Secant are often restricted in meaning j — the first to the line which by one extremity touches an extremity of tlie diameter &l right angles to it, and which has its other extremity terminated by a straight line, the Secant, from the centre of the circle across the circum- ference ; the second to the line from the centre, across the circumference, and terminated by the tangent. Thus 6 C is the tangent and D B the secant of the arc CH, or of the angle CDB measured by the arc; the name Cosine being givln to the space D K cut off between the centre D and the sine ; and Yersed Sine to the space K C between the sine and the tangent point C. The term Sine denotes a perpendicular to a diameter from the point where the secant crosses the circumference ; as HK. The terms Tangent, Secant, Sine, &c., thus restricted, were of continual use in Trigonometry j and with a widely extended meaning are now con- stantly employed. The process is instructiTe, by which extension has been given to Trigonometrical Synibols, and may thus be briefly stated j 1. Sine, cosine, &c., at first denoted lines so named drawn in and about a circle, with reference to an angle at the centre, and measured by its arc, each angle having a diifereut sine, &c., according as the radius of the circle was increased or diminished in length. 2. To avoid these continual diversities, that radius was supposed always to be a unit, or rather the »nil of measurement for the other lines ; aind the secant, sines, &c., to be multiples or fractional parts of that unit ; thus the sine of 60°, being equal to the radius, was unity, or 1, and the sine of 30° was i, or • 5. The names sines, cosines, &c., in this way lost their first meaning ; they denoted, not lines, but the numerical ratios of those lines to the radius, and were abstract numbers. 3. Another step was to represent the angle itself by an abstract number; Degrees and minutes had been the measure of the central angle, that angle was measured by its arc, and the arc bore a numerical ratio to tha unit of measurement, the radius. 4. A fourth step made the process perfect. Hitherto the sum of the angles could not exceed four right angles, but this limit also was to be passed. The idea of a line revolving round a point, and continuing its rotation after a revolution had been completed, originated the method of using angles consisting of more than four right angles. Thus angles, sines, &c., were all represented by numbers ; and though the old names were retained, Trigonometiy which at first was a simple application of Geometrical truths, and which still rests on Geometry for its foundation, became a branch of the higher Arithmetic, and has its operations conducted on arithmetical and algebraical principles. 8. Circles are said to touch one another, which meet, but do not cut one another. Thus the circle of vrhich L is the centre, touches E F in E, and circle G touches it in F. DEFINITIONS. 4. Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to jg^ ^ ^ .^ them from the centre are equal; thus EF and GH are equally distant from C, when perp. CA= perp. CB. 5. And the straight line on which the greater perpendicular falls, is said to be farther from the centre; thus IK is farther from C than GH is, because the perp. CD > As the distaace from the vertex of a triangle to its hase is measnred by a perpendicnlar, so the distance of a straight line from the centre of a circle is the perpendicular drawn to it from the centi-e. A Proposition analagons to Props. 7 and 8, Book m., would explain "why the per- pendicnlar fi-ora a point on a straight line is called the distance from that line." perp, 6. A segment of a circle is the figure con- tained by a straight Hne, and the circumference which it cuts off; as the fig. AB CA. "A figure included by an arc and its chord is j. a segment." — Labdneb. The straight line of a segment, as AB, is named the chord, and the circumference it cuts off the arc, as ACB. Every chord, except a diameter, — ^.-■ divides the circle into unequal portions. The F division of the circle by a diameter makes *emi-circfes, DCE, and DFE- by two diameters, bisecting at right angles, quadrants, as DP, EE.' Two arcs ACB, APB, having the same chord AB, evidently 'make np the whole circle. 7. The angle of a segment is that which is contained by the straight line and the circumference; as angle ABC contained by AB and the arc BOA. 8. An angle in a segment is the angle con- tained by two straight lines drawn from any point of the circumference of the segment, to the extremities of the straight line which is the base of the segment; as /_ ACB, or /_ ADB. 6 DEFINITIONS AND AXIOM A. 9. An angle is said to insist, or, stand nipon the circmnference intercepted between the straight lines that contain the angle; as Z ACBonarc AEB. 10. A sector of a circle is the figure contained by two straight lines drawn from the centre, and by the circmnference between them; as ACB. ,/' \ Sectors are equal when they have eqnal radii and equal angles ; for by superposition their boun- daries coincide in every respect. Certain Sectors of a definite size are known by definite names ; as the Quadrant, a sector of which the arc is 90° ; the Sextant, of 60° ; the Octant, of 45°. 11. Similar segments of circles are those in which the angles are equal, or which contain equal angles ;aszACB=zDFE. The idea of similarity here introduced belongs to all regular figures : aU squares, equilateral triangles, hexagons, &c. are similar, though not equal, — the similarity depending on the equality of the angles. Figures become identical when their sides as well as their angles are respectively equal each to each. 12. Concentric circles are such as have a common centre; thus, drcle A and circle B hare the same centre C. Axiom A. " If the distance of a point from the centre of a circle be less than the radius of the circle, the point is within the circle; and if the distance of a point from the centre of a circle is greater than the radius, the point is without the circle." — Hose, p. 300. See also SoH. 2, Pr. 1, III. PROP. I. — ^BOOK III. PEOPOSITIONS. Pbop. 1. — Peob. Tof/nd the centre of a given, circle. Sol. — Pst. 1. Let it be granted that a st. line may be drawn from any- one point to any other point. 10, 1. To bisect a given finite st. line. 11, I. To draw a st. line at rt.^s to a given st line from a given p 2. The rigour of the reasoning would have been greatly promoted, if Euclid, previously to the above Problem, had established the following proposition ; Any point, T>,ftg. I, F, fig. 2. being assumed within aQ, a rt. line, HD or HE, drawn through it and produced indefinitely in both directions f tpill meet the Q in two points, and not in more ; and every point of the line between these two points of intersection will be within the ©, and every point beyond them without it, — IiAbdneb's Euclid, p. 91 I. Let H D through D,. also pass through the cen. C. ElG. C 1 & 2 1) 3 i 5 6 Pst. 3. Cor. 3, 1. Ax. A. ni. 3.1. Ax. A.m. 3.1. Def. 15, 1. Prod. H X) indef. in both directions j & from C make C K, C L each > C B, CA; .". the ■ s K & L are without the 0. Also fi'om C make C D, C H each < CB,CA; .*. the • s H & D are within the ©. lastly, from C make C B, C A each = rad. C E ; .". the • s A & B o» the ©ce. n. Let H E through E mit pass through the cen. C. C 1 & 2 D 3 4 5 11 L 19, L a fort Cor.3. 47,L 3, L Pst. 1. D.3.«547, L Erom C draw CD X E H ; •.•CD rad. C A j .". the • s G & H are vtithout the 0. Hence the st. line H F meets the 0ce only in two Q. E. D. •, Any point being assumed, S;c. TISB. — 1. Practically the centre of a is found, by bisecting any chord, A B with a perp. CE, terminated in the 0ce ; and C E being bisd. in F, J" is the centre. 2. The First Prop. bk. m. is applicable to all cases in which the centi'e of any circular object, as of the hori- /^ zontal section of a tree, may be required. A circular disk of metal, a wheel, ' a flower-bed, any object possessing the circular form will have its centre found in the same way. Peop. 2. — Theok. If any two points he taken in the circwmference of a circle, the straight line which joins them, shall fall within the circle. Con. 1, m. Pst. 1 & 2. Dem. Def. 15, 1. 5, 1. The angles at the base of an isosc.A are equal to each other; and if the equal sides be produced, the /.s on the other side of the base shall be equal. 16, 1. If one side of a A be produced the ext. /_ is greater than either of the int. opp. angles. 19, 1. The gr. /_ of every A is subtended by the gr. side, or has the gr. side opposite to it. El 2 Hyp. Cone. Let A B C be 0, and A B any two • s in the ©ce; tte st. line from A to B within the 0.' 10 GRADATIONS IN EUCLID. Sdp. — If not vAthin, and it is possible, let it be with/mt, as AEB. EindDcen.of© ABC, &joinDB, DA ; in ©ce A F B take any join D F, and prod, it to meet A B in • E. Then •/ D A = D B, .-. Z DAB =Z DBA; & V AEin A DAE is prod. toB; .-. ext. I. DEB > int. & opp. Z DAE. But Z. DAE = Z DBF; ,-, Z DEB >Z DBF; and .-. DB > DF. Now DB = DF .-, Dr> DE; an impossibility ; .*. the liae from A to B not withowtihE.Q. So, AB does not fall upon the 0ce; .•. AB is within the 0. •, If any two points be taken §-c. Q. E. D. CoK. 1. — A St. line, A B, cannot cut the Qce of a Q in more points than two ; for, eyery st. line joining any two points in the Qce falls within the ©, neither co-ineiding with any other points in the ©ce, nor meeting it except ia the two given points. CoE. 2. — A St. line which touches a circle meets it only in one point. CoE. 3. — ^A circle is concave towards its centre. C. 1 2 3 D. 1 2 3 4 5 6 7 1. III. Pst. 1. Assum. Pst. 1 & 2. Def. 15, I. 5, I. C. & 16. 1. D. 1. 19.1. Def. 15, 1, ad imp Cone. Sim. Cone. Eecap. ' ScH. — ^Instead of the ex dbsurdo demonstration of this Proposition, a direct method of proof, founded on Axiom A, bk. in. was given by CoMMANnnrai, who lived between AD. 1509 and 157S ; he applied himself to mathematics at Verona, and in 1572, at Fesaro, published Euclid's Elements in fifteen books, inliatin. PROP. in. — ^BOOK III. 11 C. 1 2 D. 1 2 S 4 5 6 7 Asstati. l,m.Est.I. Def. 15jl. 5,1 C. 10,1. D. 1. D. 4. 19, 1. Bemk. Ax. A. m. Sim. Cono. In the giren line AB, taJke any • E; find D cen. of tiie © j and join DA, DE, DB. VinADAB, DA = 1)B; .: Z. DAB=Z.DBA; and *.' in A A ED, AE is prod. toB, /. ext Z_ DEB > int. and opp. A ZDAE; but Z. DAE=Z DBE; .-, 2IDEB > /.DBE. , But Z D EB > /. D B E .-. D B > D.B ; i. e. DE the dist. of E from D < D B the rad. .*. the • E is within the circle. So is every . between A and B ; ,". the line A B, joining A and B falls within the 0. Q. £. D. TJsB. — On this proposition are grounded those which show, that a circle touches a st. line in only one point; for if the st. line touched two points of the ©ce the st. line would be &awn from one point of the ©ce to the other, and consequently would fall within the circle, contrary to the very definition of such a line, that it does not cut the circumference. Theodosius of Tripolis, a jnathematician who lived some time after the reign of Trajan, compiled a work on the Properties of the Sphere and of tlie circles described on its surface, an edition of which was published at Oxford in 1675 : he used Prop. 2. bk. in. to demonstrate that a Globe resting on a plane surface cannot touch the plane in any but a single point ; otherwise the plane would enter the globe. Prop. 3. — Theob. If a straight line drawn through the centre of a circle bisect a straight line in it which does not pass through the centre, it shall cut it at right angles; and conversely/, if it cuts it at right angtes, it shall bisect it. Con. 1, ni. Pst. 1. Dem. Def. 15, L 8, 1. Def. 10, 1. 5. 1. 26, 1. If two As have two Z_s of the one equal to two /.s of the other, each to each, and one side equal to one side, viz., either the sides adj. to the equal /.s in each, or the sides opp. to them, then shall the other sides be equal, each to each, and also the third /_ of the one to the third /_ of ^^ other. 12 GKADATIONS IN EtJCLID. E. 1 2 3 4 C.l Hyp. 1. „ 2 Cone. 1. „ 2. 1, III. Pst. 1. Let AB C be a circle ; & D a straight line through E the can. bisg. A B a st. line not through E ; then CDcutsAB± inF; & if J. in P, A P = PB. Take E cen. of 0, &join E A, EB. I. The St. line C D slmll cut AB at rt. /_ s. D. 1 2 3 H 2. & C. Def. 15, 1. 8, I. Eemk. Def. 10, 1. Cone. •.• A F = P B, and PE com. to As A P E, B P E ; & •.• EA=EB.-. ZAPE=ZBPE; & being adj. Zs .-. Z.s APE & BPE rt. Zs. •. CD through E, bisg. AB not throughi E, cuts AB at rt. /_a. II. Also CD cutting AB at rt. Z«, Us. AB, i.e. AF=PB. D. 1 2 3 4 5 Def. 15, 1. 5, 1. Def. 10, I. D. 1, 2 & C. 26,1. Eec. V EA=EB .-. Z EAP= ZEBP; andrt. Z APE = rt. Z BPE; Thus Zs EAF, AFE= Zs EBP, BPE, & EP com.; .-. AP = PB, i.e. AB is bisd. ■. If a St. line he drawn, SfC. q.e.d. CoE. 1. — A St. line E P, bisecting any chord, A B, at rt. Z*> passes through the centre E of the circle. CoE. 2. — All chords, A B, G H par. to the tang. C K, at either extremity of the diam. CD, are bisected by the diam. CoE. 3. — The St. line which bisects AB, the com. chord of two circles, ACB, ALB, at rt. Z s, passes throiigh E ^ D, the centres of both circles. PROP. III. BOOK III. 13 Cor. 4. — When in a circle there are several chords, AB, GH par. to each other the locus of their points of bisection is in that diam. CD, which is at rt. /_s to them; and if the line, CD, which bisects one ahord be aperp. that line bisects all the par. chords at rt. /.s. ScH. — This Prop, might be more briefly given ; "If a diam. cut any other chord at rt. /.sj it shall bisect it ; and conversely; if a diam. bisect any other chord, it shall cut it at rt. /.s." Use AND App.^The use of this Prop, extends to the various cases, in which arcs, or circles, have to be drawn, and their centres ascertained j as, — ' I. Given a circle, ABC, to find its centre, as in the last iigure. S. 1 2 3 D. 1 a Pst. 1. 10,1. 11,1. Pst. 2. 10. I. C 1. Cor. 3, in. C. 2., Def. 15, 1. Draw any ch. AB ; bis. it in P by the perp. EP ; prod. E F to . s C, D, fprming diam. CD j ■ bis. CD in E ; and E is the centre. ".• CD bis. AB J_ .'. D C through the centre ; *.• CD is a diam. .•. its mid. • E cen. of © Q. B. F. n. CHven an arc ABD, to find the cen. of the of which it is an arc. s. 1 Assum. Take any ■ B, between the ex- tremities of the arc ; . , 2 Pst. 1. from B draw chords to A & D ; -' 3 10,1. & 11, 1. bis. AB & B D by perps. from P & G, meeting in C ; 4 Sol. then, the intersecting • C is the centre. D 1 Cor. 3, in. •/ the cen. lies in P C & aJso inGC; 2 Cone. .•. it must be in the intersect- ing . C. N.B. With rad. C A, or C D, the circle, AB D B, may be completed. m. Through three points,^ A, B, D, not in a st. line, to draw a circle. Pst. 1. 10, 1. 11. I. Cor. 3, m. Pst. 3. Join AB, B D, and bis. them in P, G; at P & G raise perps. intersecting in G; C is the centre of through A; B, D: and with rad. A C, or BC the circle may be drawn. N.B. The Dem. is given in Prop. 9, m. 14 GRADATIONS IN EUCLID. rV. In Trigonometry we show 1st, by this Prop., in the last figure bnt one, that the half chord F B or F A of an arc A H B is perp. to the semldiam. C H, and conseqnently is the Sine of the half arc H B, or H A ; 2nd, that the sides of a triangle (4, VI.) have the same ratio as the sines of their opp. angles. V. In the last figure but one. That part of the perp. to the chord which passes through the centre and is intercepted between the centre and the chord, namely, CP, is called the versed sine (see note to Def, 2, p. 4) ; and the radius, semichord, and versed sine form respectively the hypotenuse, base and perp. of a rt. angled triangle, and by 47, I., when any two are measured, or given, the third may be found ; — for, rad. =: \/semich.^ + vers. Mne* ; and vers. «ne = ^rad.^ — semich.^ semich. = ^rad.^ — vers, mae? j Prop. 4. — Theor. If in a circle two straight lines cut one (mother, which do not loth pass through the centre, they do not bisect each other. Con.— 10, 1. 1, in. Pst. 1. Deh.— Def. 15, 1. 3, m. Ax. 11. AH rt. /_s are equal. Ax. 9. The whole is greater than its part. E.1 2 Hyp.l. ( ,,2. Coac. Let AB be a circle ; & A C, B D two St. lines cuttmg in E, but not both through cen. P; then AC, B D do not bis. one another ; i. e. E not mid. • both of A C & B D. Sup. L Let BD pass through the centre atid AG not. C. B. 10, 1. Def. 15, 1. Def. 15, 1. Bis. BD in F, then F cesa. of 0. V FB = FD .-. B not mid. . ofBD; i. e. B D not bisected in E bj AC. PEOP. V. — BOOK. III. 15 Sup. II. Zet neither AC nor BD pass through the, centre. E.l C. D.l 2 3 4 5 6 7 8 Hyp. 1, in. Pst. H. 3. III. H. 3, ni. , Ax. 11. Eemk. As. Coae. Kee. 9. If possible let both A E = EC and BE = ED. Find P the cen. and join PE- ■.• FE through cen. P, bis. .( AC not through F, cV .-. FE cuts A0±, and FB A ^ is a rt. Z • Again, •.• FE through cen. F bis. BD not through P, .-. FEcuts BD X, and FEB is art. £. .: PEA a rt. Z = FEB a rt. Z, i. e. a part = the Trhole; — an impossibility; " AC and BD do not bisect eadi other. {. • -d-vy oiiiu AJXJ uu ilUL uist^uu am. '. If in. a circle two st lines, ^c. Q.E.D. CoR. — JVo parallelogram except a rectangle can be inscribed in a cirde. D.l 2 3 C. & 34. 1. Def. 15, 1. 8, I. 34, 1. Cone. *.• the diags. are diams. »'. the diags. bis. each other in their centres. .*. the diags. are equal. And •.• the suppl. Z s are equal ; .'. each Z is a rt. Z , and all the Z s are rt. Z s. i. e. the inscribed fig. must be a rectangle. Use. — The fotirth Prop, has been employed to determine the eccentricity of the Sun's apparent path, or of the Earti's orbit described in a year. In an eccentric wheel the distance of the fixed point, or centre of rotation, E, round whidi the revolution is performed, ftom I", the centre of the wheel, will be found in the same way. Prop. 5. — Theor. If two cirdes cut one another, they shall not have the same centre. Con.— Pst. 1. Dem.— De£ 15, 1. Ass. 1 & 9, IG GRADATIONS IN EUCLID. E.l 2 Hyp. Cone Let the two 0s ABC, CDG cut in • s B & C ; then, they hare not the same centre. Strp. — If possible let E be the cen. of both circles. C.l D.l 2 3 4 6 6 Pst. 1. Sup. Def. 15, 1. D. & Ax. 1. Remk. Ax. 9. Cone. Rec. Join EC, & from E draw a st. line EFGH, meeting the ©s in • s F & Gr. V Ecen. of © ABC .-. EC = EF. V Ecen. of © CDG .-. EC = EG. But EC.= EF .-. EP = EG, i.e. the less = gr.; an impossibility; .-. E not the com. cen. of ©s ABC, CDG. •. If circles cut one another, ^c. Q.E.D. ScH. — "This proposition may be better an- nounced thus : 'Concentric circles cannot meet, and that which has the lesser radius will be included within the other.'" — ^Laednee, p. 94. For •.• C A < CB .'. © A within © B. Prop. 6. — Theor. If one circle touch another internally, they shall not have the same centre. Con — ^Pst. 1. Dem.— Def. 15, 1. Axa. 1 & 9. E.l 2 Hyp. Cone. Let © CDE touch © ABC inter- nally in C ; they hiPPi not the same cen. PROP. VII. BOOK III. 17 Sup. — If they have the same centre let it he • F. C. D.l 2 3 4 5 6 Pst. 1. Sup. Def. 15, I. D. 1. & Ax. 1. Eemk. Ax. 9. Gone. Eec. Join F G, and from F draw a st. line FEB, meeting the ©s in ■ s E & B. •.• F cen. of ©ABC .-. F C = F B; •.• F cen. of © DE .-. F C = F E; ButFC = FB .-. FE = FB, i.e. the less = gr.; which is impossible;, .-. F not the com •.cen. of.©s ABO, ODE. •. If one circle touch another internally, ^-c. Q.B.D. Scs. — ^Props. 5 & 6 may be combined into one; "circles with a common centre do not touch either externally or interaally :" for the circle with tlie less radius will have every point within the circumference of the other, and con- sequently does not meet the other in any point whatever. Peop. 7. — Theok. If any point which is not the centre he tahen in the diameter of a circle, then, 1st, of all the straight lines which can he drawn from it to the circumference, the greatest is that in which the centre is, and the other part of that diameter is the least; and, 2nd, of any other st. lines, that which is nearer to the line whichpasses through the centre is always greater than the one more remote; also Zrd, those lines which make equal angles with the diameter are equal ; and, 4 FC more remote and FC > FG; Third, lines FB, FI making eqttal Z s with diam. AD. the line FB = the line PI. and Fourth, of lines from the same • P to the Qoe, only two eq. Hnes, F G, P H, one on each side of diam. AD. Join BE, CE, GE. I. — A lithe PA through een.. E > an^ other line as B F. D.1 2 20.1. Def.l5, 1. V two sides of a A > third side /. BE + EF > BF, bnt AE = BE ,-. AE + EF i.e, PA > FB. II. — The other part of the diam., FD < aray other lime FG.. D.l 2 3 4 5 6 Def. 15, I. C. Ax. 9. 24, I Sim. 20. 1, Def. 15, 1 Sub. Ax. 5. Cone. Again V BE = CE & FE com. to asBEF, CEF, but ZBEF > ZCEP .-. BF > CP; So CF > GPandGF > DP. Also V GF + PE > GE, and EG = ED; .-. GF + FE > ED. take away com. pt. F E .v rem. GP > rem. FD. .", Of all st Unes from F a ■ not the cen. to the 0ce, FA through cen. E is the greatest, andPD the least; and PROP. VII. ^BOOK III. 19 III. — Line "EF nearer diam. > CPmore remote ^ CF > GF. lY.— The lines PB, FI, making with AD Z BFA = ZiFA, are equal. Suppose one to be gr. i. e. FB > FI. Make FL = FI and join EL and EI. C B.l 2 3 3, I. Pst. 1. C. & H. 4, I. Def.l5,lJLx.I. Ax. 9. Cone. In AS FLE, FIE v PE com., FL = FI, & Z BFA= Z IFA, .-. EL = EI; but EI = EC A EC = EL. i. e. a part =: tbe •vrbole, wMcb is absurd. /. Neitber F B nor F I the gr. ; I. e. F B = P I. V. — Also only two equal lines, FG, FH, /rom • F to ike Qcc. C. D.l 2 3 4 5 6 7 8 23. L Pst. 1. Def. 15, 1. C. 4, I. Eemk. Sup. C. Ax. 1. Eemk. Case III. Bee. Z FEH = Z PEG, EFcom. and Z GEF = At E in EP ,ma]ie and join PH. V GE = HE, Z HEP, .-. PG = FH, And from P to ©ce. no other line ^ FG, If possible let FK = FG; V FK = PG = PH .-. FK = FH, i. e. a line nearer the diam.=a line more remote, which has been proved to be impossible. Therefore, If amy point which is not the centre, SfC. Q. E. D. ScH. — If from a point, within a circle, not the centre, as I", a St. line of indef. length, as FX, revolve so as in each part of its revolution to be terminated or cut off hy the (arcumference, as in A, B, C, G, D ; its maximum length FA, will he attained when it coincides with that part of the diam. AD, in which E the centre ia ; and its minimum FD, when it coincides with the other part of the diam. ; and the nearer F X, is to the maximum the greater it is, as FB, and the nearer to the minimum, the less, as F Gr. UaB. — ^Theodosius, mentioned p. 11, by aid of this proposition proves that, if from the pole of the world, which is not the pole of the horizon, (for the zenith is its pole) several arcs of great circles he drawn to the circum- ference of the horizorr, the greatest arc shafi be that part of the meridian which passes through the zenith. By this proposition we may also prove that the Earth being in Aphelion is at the greatest distance from the Sun, in Perihelion, at the least ; and so for all the other planets. 20 G2ADATI0NS IN EUCLID. Peop. 8. — Theoh. If any point be taken without a circle, and straight lines be drawn from it to the circumference, whereof one passes through the centre; 1st, those which malce equal angles with the line passing through the centre are equal ; 2rarf, of those which fall upon the concave circum- ference, the greatest is that which passes through the centre; and of the rest, that which is nearer to the one passing through the centre is always greater than one more remote; but, 3rd, of those which fall upon the convex circumference, the least is that, between the given point without the circle and the diameter, and of the rest, that which is nearer to the least is always less than one more remote; and, ith, only two equal lines can be drawn from the same point to the circumference^ one upon each side of the line which passes through the centre. Con. 1, m., Pst. 1. 3, 1. 23, 1. Dem. 4, I. Def. 15, 1. Ax. 1 & 2. 20, 1. Ax. 9. 24, I. Ax. 5. 21. I. If from the ends of the side of a A there be drawn two St. lines to a, • within the A, these lines shall be less than the other two sides of the A, butjhall contain a gr. /.. Let ABO be a and D any • Tvithout it ; from • D let st. lines D A, DE, DF, DC, and DO be drawn to the ©ce; Of these let D A pass through cen. M, and DE, DO make with DA, Z ADE= Z ADO; the line DE = line D O ; 1st. Of lines incident on AEFC the concave Qce, the greatest is D A passing through cen. M ; and any line nearer D A is > a line more remote, z. e. DE > DPandDF > DC: E.l Hyp. 1. 2 ,, 2. 3 „ 3. 4 „ 4. 5 Cone. 1. G 7 „ 2. „ 3. PROP. Vlil. BOOK III. 21 E1.8 9 10 G.l 2 .Cone. 1, III. Pst. 1. 2nd. But of lines incident on HLKG the convex ©ce, tie least is D G, between • D and diam. G A ; and any line nearer D G < a line more remote, i. e. DK < DL and D L < DH. Also only two equal lines, D B and D K, from • D to 0ce, can be drawn, one upon eacb side of DA. Take M the cen. of AB C and join MO, ME, MF, MC, MH, ML, and MG. I. The St. lines DE and DO are equal, making equal /_ s with DA. If possible let D O > DE. C. D.l 2 3 4 Sup. 3, I. Pst. 1. C. and H. 4,1. Def.l5,I.Ax.l Ax. 9. Cone. Make DP = D E, and join M E, MP. In AsMPD, MED •.• MDcom. DP = DE and Z ADE = Z AJ>Oor ADP; .-. MP = ME. But M Q = ME .-. MP = MQ, !. e. a part = the whole, which is absurd ; ,-. Neither D O nor D E gr., «. e. D = D E. II. Of lines incident on AEFC the concave Q ce, DA through the cen . is greatest, and DE, nearer DA is > DP more remote, and DP > DC ; D.l 2 3 4 5 6 7 Def. lb,I.Add. Ax. 2. 20, 1. Def.l5,I.Ax.9 24,1. Sim. Cone. •.• AM = ME, to each add MD; .-. AM + MD, i.e. AD = EM + MD; but EM + MD > ED .-. AD > ED. Again •.• ME = MP, MD com. & Z EMD > Z PMD, .-. ED > PD. In like manner PD > CD. .-. DA through cen. Mis the greatest, DE> DP &DP>DC. III. Of lines incident on HLKG iAe convex Qce, DG, between D and tJie diam. AG, is the least, DK MD, & MK = MG; . rem. KD > rem. GD; i.e. GD < KD, : •.' M L D is a a , and from • s M, D, extrs. of one side MD, MK & DK arc drawn to • K within the A ; 22 ghadations iir euclid. DA 5 6 7 21, I. Def.l5,I.Ax.5 Sim. Cone. .-. MK + KD < ML + LD; but MK = ML .-. rem. KD < rem. LD. Li like mamier D L < D H : .-. DG the least, DK < DL & DL < DH. IV. Also only two equal sf. lines can be drawn from the • D to the 0ce, one on each side of AD the line which passes through cen. M. C. D.l 2 3 4 5 6 7 8 9 23, I. Pst. 1, C. 4,L Eemk. Sup. D.4&3,Ax.l Eemk. D. IL 7. Cone. Eec. At M in MD make /_ DMB = z DMK, & joinDB V in AS KMD, BMD, MK = MB, MD com. & Z KMD = Z BMD: .-. DK = DB. But besides D B no other line from - D to the 0ce = DK; If there can, let it be D N ; V DK = DN & also = DB .-. DB = DN i.e. a line nearer the least = one more remote- but this has been proved to be impossible; .-. No Kne but DB = DK. Therefore, If any ■point he taken, &c. q. e. d. ScH. 1. — The concave and convex parts of a circumference are determined by the tangents from a point external to the circumference. All the parts of the circumference which aie farther from the external point than the tangent points of the circle, are concave with respect to that point ; and those nearer to the external point than the tangent points are convex. ScH. 2. — ^A proposition, analagous to Prop. 7 & 8, explains why in Def. 4, m. and in 14 & 15, III. the perp. from a point out of a st. line to the line is called the distance of the point from or to the line. — ^Hose, p. 302. If any • C be taken out of a st. Zine ABj tlien of allst. lines, CD, CE, C F, C G, from that ■ to the st. line, the least, C I), shall be that which is perp. to the St. line ; and of the rest, C E, C F, C G-, that which is nearer, as C E, to the perp., shall always be less than that, CE or CXr, which is more remote ; and from this • C there can .be drawn to the st. line AB, only one st. line =: a givm St. line, CE, drawn from the same • C, to the St. line, which shaU be on the opp. side of the perp. EKOP. Vnr. BOOK III. 23 I. The perp. CD is the least line from C to AB: CE>CD,CI'>CE and C G > C F. V Z CDEisaTt Z. .". Z CED Z CED .-. CE > CD. Also •.• Z CED < art. Z .". Z CEF > art. Z- and in like manner Z CED < a rt. Z- .-. Z CEE > Z CFE, and CE > CE. SoalsoCG>CE; Hence the perp. CD is the least ; CE > CD, CE > CE, & CG>CF. n. From • C only one St. line, CK = CEi a given st. line from C to AS? on the opp. side of the perp. CD. D.l H 2 H. & Sup. 3 17,1. 4 D.&19,I. 5 D. 4. 6 Sim. 7 19,1. 8 Sim. 9 Gone C.1 s 3 D.l S 4 23,1. Pat. 2. Cone. Ax.ll, C.l 26,1. Kemls. Sot. H.D.2AX.1 Cone. At C in D C malie ZDCH=ZDCE; ^nd let C H out AD in K ; ThenCK=CE. V rt. Z CDK;= rt. Z CDE. Z DCK = Z HCE, and C D com. .-. CK=:CE. Also no line from C to A B, but C K = C E ; If there can be, let it be C L ; Then •.■ CL = CE & CK = CE .-. CD = CK; i. e. oil, more remote from CD than CK, equals CK; which is impossible. Hence from C only one St. line to AB, namelj- CK ^ CE on the opg. Bide of perp. CD. SCH. 3. — ^^Thys Proposition," says the Translator of 1570, "is called 'Commonly in old booUes amongest the barbarous Cauda Pauonis, that is, the Eeacookes taile."— Pol. 88. Use and App. — ^I. The Proposition 8 Bk. HE. is employed to show, that ii a tangent and secant be drawn to the same point, the tang. < sec., but > external part of the secant : Thus in fig. to Prop. 8, Jet D H represent a tang, and M D a sec. to aic HG, or Z HMG, then D H < D M but > D G. IL By aid of Props. 7 & 8, and Ax. A, bk. HI., we may demonstrate ; 1st. "When one circle A is contained within another, B, without touching it, the distance between the centres, DE < (DE oj EG), the difference of the radii': and conversely, when the distance between the centres, D E < (DE oo EG), the difference of the radii, the lesser circle will be within (be greater without meeting it. 24 GRADATIONS IN EUCLID. 2ud. When two circles, B, C, lie each without the other and do not meet the distance between the centres, DH > (DF + HK,) the sum of the radii, and conversely, when the distance between the centres, DH > (DP + HK,) the sum of the radii, the circles lie each without the other and do not meet. Prop. 9. — Thkor. If a point he taken within a circle, from, which there fall more than two equal straight lines to the circumference, that point is the centre of the circle. Cos. Pst. 1 & 2. Dem. Def. 17, 1. A diam. of a is any st. line through the cen., and ter- minated both ways by the 0ce. — 7, IH. Cone. E.llHyp.l 2 „ 2 3 C. D.l 2 3 4 5 6 In ABO let ■ D be taken, & from D to 0ce more than two equal w lines, asDA = DB = DO; ' then • D shall be the cen. of the 0. Stjp. For if not, let • E he the centre. Join DE, & prod, it to the 0ce in P & G. in P G is taken Pst. 1 & 2. Def. 17, I. 7, III. H. ad imp. Oonc. Sim. Cone. Kec. •.* P G is the diam., & not the cen. • .•. DG greatest line from D to 0ce, DO >DB, andDB > DA; but they are also equal ; — an impossibility ; .-. - E not the cen. of A B 0. So no point but D the cen. /. • D is the cen. •. If a point he taken, ^c. q.e.d. CoK. — 1. From any other point than the centre only two equal St. lines can he drawn to the Qce, whether the point (7, III.) he within,' or (8, III.) without the circle. Cor. — 2. From three point's given not in the sam^e st. line, the circumference of the circle may he found. PROP. X. BOOK III. 25 ScH. — This Prop, gives the criterion for determining the centre of a cii'cle ; it is, that from a point, supposed to be the centre, to the circumference more than two points in the circumference shall be equally distant from the centre. Use and App. — 3j this Proposition the Problems may be demonstrated ; 1st, To draw a circle through three given points, A, B, D ; 2nd, To find the centre of a given circle A B D B ; and 3rd. To determine the centre of A. B D, an arc of a circle. The demonstration of the first; is equivalent to the demonstration of the other two. Join • s A,B,D and bis. AB and BD ; at F and G raise perps. PC, GO meeting in C j join CA, CB, CD and with either as rad. from C desc. a the passes through A,B and D. •.• AF = FB, FC com. and Z AFC= ZBFC, .-. AC = CB In like manner CD = CB .•. C A = C B = C D & .-. C cen. of through • s A, BD, Q. B. F. CI Pst. 1. 10, 1. 2 11, I. 3 Psts. 1. & 3. 4 Sol. Dl. C. 1,&2 2 3 4 4, I. Sim Ax.l.&9.in. Peop. 10. — Theor. One circumference of a circle cannot cut another in more than two points. Con. 1, in. Pst. 1. dbm. Def. 15, 1. 9, rn. 5. in. If two circles cut one another they shall not have the same centre. Sup. — If possible let Qce FAB cut Qce DEF ire more than two • s, as in B, G, F. C D.l 2 .1, III. Pst. 1 Take K cen. of A B G, and join KB, KG, KF 0. Def. 15, 1. •.• K cen. of ABC .-, KB = KG = KF: u C. and •.- within DEF tiere fall from K to © more than tjfo equal St. lines. S6 OEADATIOUB IN EUCLID. D.3 4 5 9, III. .-. .Xiscen.ofQDEF; 0. but . K also cen. of © ABC; Cone. .-. the same • is cen. of two ©s cutting each other ; 5, m. which is impossihle ; Eec. J ■fcerefore, One circumfermoe of a circle, 4^. Q. E. D. Son. — " Two circles cannot hare more than two points in , common j" if they coincide in three points they will coincide in every point ; or, " onlyiOne circle can he drawn through three given points." Peop. 11. — Theoe. If one circle touch another internally in any point, the straight line which joins their centres being produced shall pass through the point of contact. Con. Psts. 1, 2. Dem. 20, 1. Dcf. 15, 1. Ax. 5. E.l 2 3 Hyp. 1. „ 2 Cone. Let AD E touch ©ABC in- ternally in A, , and let P he cen. of © ABC, G h/^, cen. of© AD E; then the st. line joining P, G, being produced passes through A, the • of contact. Sup. — If P Gr produced do not pass through A, let it, if possible, fallasFQB'H.. C. Pst. 1. Join AP and AG. D.l . 2 3 20,I.Def.l5,I Sub. Ax. 5. •.• PG + GA > PA, but PA = PH, .-. PG + gA> FH, take away com. pt. FG .•. rem. AG > rem. GH; PROP. XI. BOOK III. 27 D.4: 5 6 7 8 Def. 15, 1. Remk.adimp. Cone. Henik. Eec. but AG = GD .-. GD > GH; i. e. the less > the greatei:; — an impossibility. .■. F G joiBing P and G, being produced cannot fall except upon • A ; i.e. FG pirod. must pass througb A, tlie • of contaet. Therefore, If one circle touch, ^c. Q. E. D. c. Pst. 1. D.l 2 3 4 20,I.I>ef.l5,I. Sim. Cone. ScH. — ^When the distance, P K, between F and K, the centres of the two cirdes ADE and ABC is equal to the difference of the radii, AP and AK, the circles touch intemcMy. Por Take L, a • in ADE, and join KL, PL ; then •.• in A PKL, PL < PK + KL, but KA = KL ; .-. PL < PK +KA i. e. < PA .'. L within ABC. So all other ■ s in AD E except • A lie within ABC. yVhen. the distance between the centres, Sfc. Use and App. — ^I. By aid of this proposition an oval may be described on any given major axis, o^ A B. Bis. AB in C by a perp. CD, CE the minor axis is OP, i. e. DE produced. If on C with rad. CA or CB a line revolve a circle will be traced ; but div. AB into threeeq.pts.AP = PG = GB; fromP and G with PAandGBdesc.©sAEGDandBEPD; from . s of inliprs. D,E, draw DE, DG, EP, EG, prod, to H, I, K, L ; next from D with rad. D H or D I desc. arc M H O N, and from E with rad. E K or E L desc arc MK P N ; the arcs AH, HI, IB, B L, LK, and K A touch in • sH,I,E,L, and the curve AHCIB LPKA will form an oval. CI 10,L11, 1. 2 Bemk. 3 Pst. 3 4 XJse2,34,I 5 6 Pst. 3. Pst. 1, 2. 7 8 9 10 Pst. 3. Pst. 3. 11,111. Sol. If the major axis be divided into four or more equal parts, by a similar method ovals more elongated, or with the minor axis in less proportion to the major, may be described. N.B.— The oval thus described is only an approach to the true ellipse, the method hwg practically useful, not theoretically correct. 28 GRADATIOKS I2f EUCLID. n. It is on the same principle that a Spiral is described, by suc- cessive semicircles taken alternately from two common centres A and B; for thelineABwhich joins them being produced, passes through the points of contact of the successive semicircles, I, m, n, o, p, q, r, a. From A, the eye of the spiral,s ■with rad. AB describe the semi- circle C; from B, with B/, the scmic. D ; from A, with Am, semic. E; from B, with B«, semic. F; from A, with Ao, G ; from B, with Bp, H ; from A, with Ag, I ; and from B, with B r, K. The spii-al may be continued to any extent in the same way. A Spiral is a curve line making revolutions round the centre or eye ,of the curve, which do not retuvn into themselves as the revolutions of a circle do. A Plane Spiral is generated by a continually increasing radius, and according to the law or rate of increase spirals differ in their curves. Our older English writers considered the circle and the ellipse to be spirals ; but they are excluded by the above definition, inasmuch as their revolutions return into themselves, or to the very point from which they started. Besides the above, the principal plane Spirals are, the Spiral of Archimedes or Conon, — the Hyperbolic or Reciprocal Spiral, — the Lituus, and the Logo- rithirtic Spiral ; but they can only be noticed hereby way of definition. 1st, When from a given point any number of lines are drawn forming equal angles nt that point, and the length of each line increases in succession by an equal quantity, the curve which passes through tliese points is named the Spiral of Archimedes. 2nd, The Hyperbolic or reciprocal Spiral is a cui-ve passing through the extremities of any number of arcs of circles of equal length measured from a given st. line. 3rd, The Lituus, so named trom the crooked staff of the Roman augurs, is a Spiral to be thus described •,— " Let a variable circular sector always have its centre at one fixed point, and one of its terminal radii in a given direction. Let the area of the sector always remain the same, then the extremity of the other terminal radiu% as it revolves describes the Lituus." 4th, The Logarithmic Spiral, in which the radii make equal angles, and the spiral cuts them all at an equal angle, the length of the successive radii increasing in geometrical progression. It may be observed that curves are infinite in variety, though only about thirty have received specific names. The Parabola, Hyperbola, Cycloid, Watt's Parallel motion curve, &c., are among them j but it would be out of place to explain them here. PROP. XII. BOOK III. 29 Pkop. 12. — Theok. If two circles toucTi each other externally in any point, the straight line which joins their centres shall pass through that point of contact. Con. Pst. 1. Dem. Ax. 2. 9. Def. 15, I. 20, I. E.l 2 3 Hyp.l „ 2, Cone. Let tile two 0s ABC, ADE touch extern, in A ; and let F be the cen. of ABC, G of ADE; then F G shall pass through A, the • of contact. Sup. If not, let it pass through C and D. C D.l 2 3 4 5 Pst. 1. H.2.Def.l5,I, H.2.Def.l5,I Ax. 2. 9. 20, 1, ad imp. Cone. Ree. Join FA and GA. •.• F is cen. of ABC .-. FA = FC ; and •.• G is een. of ADE .-. GA = GD; .-. FA + GA = FC + GD, and .-. FG > FC + DG. But FG < FA + AG; which is impossible. .•. the st. line FG cannot pass except through A the • of contact; i. e. FG must pass through A. Therefore, Jf two circles touch, 4rc. Q. e. d. SoH, — Should/i/, the distance between the centres of two circles, be equal to /A + A Jf, the sum of the I'adii, the circles touch each other externally. Use and App. — The drawing of a Sarpentim Line, or coma recta, between two given points, as A, B, depends on this 12th Prop. For join A B, and bisect it in C ; again bis. A C by perp. D E and B C by perp. FG ; make DE = FG ; and from E with rad. B C desc. arc AH C, and from G with< rad. G C, the arc C K B ; the arcs touch in the • C, and the two arcs form the serpentine AHCKB ; which might be continued to any extent by following up the same process. 3& GRADATIONS IN EtTOLID. Prop. 13. — Theoe. One circle cannot touch another in more points than one, whether it touches it on the inside or the outside. Con. Pst. 1, 11, 1. dem. 2, in. 11, m. Cor. 1, m. E.1 C. D.l 2 3 4 5 Hyp. If possible let EBP touch ABC in more • s than one. I. — On the inside in the • s B, D. PBt.l.ll,I. H. 2, III. Cor.1,111. 11, in. H. Cone. Eec. Join B D, and draw GH bisecting BD J.. •/ . s B, D are in the _ 0ce of each .". BD falls within them ; .", their centres are in G, G H which bis. BDi.; .•. GH passes through the • of contact; but • B B, D, are without the st.line GH, /. GH does not pass though the • of contact, which is absurd. .". one cannot touch another internally in more • s than one. U. — If possible, let ACK touch ABC externally in A and 0. c. Pst. 1. D.l Hyp. 2 2, ni. 3 H. Join A C. •.• A& are in the 0ce the A C KT, .", AC which joins them falls within ACE; But © A C K is without ABC, without ABC; '^ \^_y^^^ AC is PROP XUI. BOQK III. 31 D.4 H. but •/ A & C are • s in the 0ce of the ©ABC, 5 2, III. ,-. A C must also be within the same ; an absurdity ; 6 Cone. .'. one © cannot touch another on the outside in more • s than one. 7 Case I. And ©s cannot touch on the inside in more • s than one; 8 Eee. Therefore one circle cannot touch, ^c. q. e. d. ScH. — It has been by assuming two points of contact between cirdea,. and by showing the assumption to be impossible/ that the proposition has been proved, that two circles cannot touch in more parts than one. But for this in- direct method of demonstration, it is better to substitute tbe direct method; thus 1st. If the circles touch internally, as at I), each point in the Qce of the less ©, except the common point of contact, D, throi^h which the hne, AB, joining their centres, A, B, passes, must be within the ©ce of the other Q. C. Pst. 1. Eor take C, a • on the ©ce of D C I', & join centres & C. D.l 7, in. D. 1. Ax.A.ni. Sim. Here A D > A C. And •.■AC < A D, the rad. of© DEG; /.the . Ciswithin©DEG; So every • , except D, of © D C F is •within the ©DBG. 2nd. If lihe ©s D C F and DEG toueh extemaBi/,. evevy point of the one, except the common point of contact is without the other. C, D.l 2 Pst. 1. H. & 8, in. Ax.A,III. Sim. Join AB, the centres, and draw ACE. •.* the ©s touch externally, .-.AB > ACorAD; .'. the • E lies out of the © DCF; thus eveiy • of © D E G, ex- cept the • D, lies out of the © D C !F, UsEandApp. — The four Propositionai 10, 11, 12 and IS, are employed by astronomers to explain the motion of the Planets in Epicycles. An Epicycle is a circle, ABD, the centre of wluch, C, is carried round upon another circle / ABE. " Ptolemt of Alexandria, the, celebrated astronomer, ' A.D. 139, in explaining the motions of the planets on what is called the Ptolemaic System, employs the theory of Epicycles : but the common notion is erroneous that th& use of the Epicycle is peculiar to the Ptolemaic Astronomy: "the modem astronomer to this day resolves the same motions into epicydic ones. "When the latter expresses a result by series of sines and cosines (especially when the angle is a mean motion or a multiple of it) he uses epicycles ? and for one which Ptolemy scribbled on the heavens, to use Milton's pirase, he scribbles twenty."— A. De M. Gk.. and Horn. Biography^ Vol. lH., p. 576. 32 GRADATIONS IN EUCLID. Prop. 14. — Theor. Equal straight lines in a circle are equally distant from the centre; and conversely, those which are equally distant from the centre, are equal to one another. Cox 1, ni. 12, I. To draw a perp. to a st. lino from a point without it. Pst. 1. Dem. 3, ni. Ax. 7. Things that aro halves of the same thing, or of equals, are equal. Def. 15, 1. 47, I. In any rt. /_i A the squareon the side opp. to the rt. /. shall be equal to the squares on the sides including the rt. ^. Axs. 1, 3. Def. 4, in. Ax. 6. The doubles of the same thing, or of equal things, are equal. E.l 2 C.l 2 D.l 3 4 5 6 7 8 9 10 11 12 Hyp. Gone. 1, III. 12,I.Pst.l 0.2 3, III. Sim. H.&AX.7 Def. 15, 1. C.2&47,I. Sim. Ax. 1. D.4. Ax. 3. Def.4,III. Cone. I. Let AB = CD in ABCD; then AB & CD shall be equally distant froin cen. . E. Find E the cen. of AB CD. from E draw EP, EG J_ AB, CD & join E A, EC. A/ •.• EP through cen. E cuts AB y not through cen. at rt. Z. s ; .-. EP bis. AB, & AP = PB, and AB = 2AP; For the same reason CD = 2 C G; but AB = CD .-. AF= CG. And •.• AE = EC .-. AE^ = EC^; but Z APE being a rt. Z .-. AF^ + FE2=AE«; SoEG^ +GC2 = EC2; /, AF2 + FE2 = EG2 + GC2; but AP = CG .-, AP2 = CG2; .-. rem. FE^ = rem. EG^, & .-, FE = EG; but lines are equally dist. from cen. when the perps. to them from the cen. are equal, .'. AB and CD are equally dist. from E. I'BOP. XV.— BOOK HI- SS E.l c. D.l 2 3 4 5 6 H.Df.d.m Cone. Sim. C.&3,III. 47, I. H. Ax. 3.. D.i.Ax.er. Eec. II. Let AB & CD be equally dist. from E ; i. e. FB = EG; • theaAB = CD. Let the same construction be made. V EF bis. AB and EG bis. CD; .-. AB = 2AFandCD = 2CG; and EPs + PA^ = EG^ + GC^S; and V PE = EG, /. FE^ = EG^; .-. rem. APs = rem. CG^, and .-. AF =CG. But AB = 2AF, and CD = 2CG; .-. AB= CD. Therefore, Equal St. lines are equally, ^c. q £. o. ScH. — A principle employed in this and the next proposition is — if two quantities A + B = C + D, ,thon if A=C, B = D ; il'A > C, B D. Pkop. 15. — Theoe. The diameter is the greatest straight line in a circle ; and of all others that which is nearer to the centre is ahoays greater than one more remote : and the greater is nearer to the centre than the less. Con. 12, 1. Pst. 1. Dem. Def. 15, 1. Ax. 1. 20, 1. U, III. 47, 1. Ax. 5. Def. 5, m. The st. line on which the gr. perp. from the cen. falls is farther from the cen. E.l 2 3 C.l Hyp. 1. „ 2 Cone. 12, L Pst. 1 L LetABCDbea 0,AD its diam., and E its cen. and let B C be nearer to E than PGis; then AD > any line BC not a diam., and B C > PG. From E draw EH, EK perp. to BC, PG; and joinEB, EC, EP. 84 GRADATIONS IN EUCLID. D.l 2 3 4 5 6 7 E.l 2 D.l 2 3 4 Def.l5,I.Ax.2 20,1. XT 2 Def.5,III. 14, III. D. 47, I. D. 3. Cone. H. Cone. H.&D.7.CaseI, D. 5. Case I. Ax. 5. Def. 5, III. Eec. V AE = EB, and ED = EC; .-. AD=EB + EC; but V EB + EC > BO, .-. AD > BC. And •.• B C is nearer cen. E than F G ; .-.EH < EK; butBC = 2BH, FG = 2FK, and EHs + HB^ = EK" + KF^; and V EH < EK .-. EH^ < EK^; .-. BH2 > PK2 and BH > FK; and .-, BC > FG. II. Next, let BC > FG; then BC is nearer cen. E.than FG is ; i.e. EH < FK. •.• BC > FG, BH > KF, and BH^ > PKS; and V BH2 + HE^ = EK^ + KF^; .-. rem. HE^ < EK^, and HE < EK; and.-. BC is nearer the cen. E than F G is. •. The diam. is the greatest st. line, Sfc. Q. E. D. ScH. 1. — In a circle the longest chord is the diam., as A D ; the shortest, through a given point, as M, is that, L C, which is perp, to the longest. Through M, a given • draw a diam. AD, and any other chord, as B E ; and to AD, through M, a perp. L C also a chord ; and from the cen. of the draw OPpei-p. toBE. By 47, 1. OM > OP ; and, 15, m., BE > LC ; and the same being true of any other chord through M, .•, LC at rt. /.s to AD is the least chord, and, 15, IDC., AD the longest, through • M. 2. The less the /. which a given chord through a given . , as M, makes with the diam. the greater will the chord be ; !For, it is evident, that as the /. AMB diminishes, the perp. OP, marking the distance of the chord from the cen., will also diminish, and the chord increase. Laednbk's BucUd, p. 104. Use and Apt. — ^I. The 14th and 15th Props, were employed by Theo- Dosius to demonstrate that in a Sphere the centres of the least circles on the sphere are the most distant from the centre of the sphere itself; or, in other words, that the circles of latitude diminish as the poles of the earth are approached. PROP. XV. — BOOK III. II. In the Astrolabe, a circular instroment for taking the stars, i. e. for observations on the heavenly bodies, the same propositions are serviceable. HrppAECHUs, the greatest of Greek astronomers, B.C. 160, was probably the first who constnicted an instrument of this kind. Its general nature may be seen from the following representation, in which the approach of the chord BF to the diam. NS may be measured, either by the perp. CD, or the arc BQ. Let NESQ be a circle fixed to one position, vertical or horizontal; and let A B he a line or tube moveable round the cen. C ; through the tube, along the line A B, any object, as P, may be seen; and, by turning the tube round its centre C, another object, as O, may also be seen ; the £_ QCB, subtended by the two objects is their angular distance, which may be measured in degrees when the circle NES Q is graduated. If E Q represent the equator, and if the plane of the circle E S Q N pass through the poles N and S ; then the /_ B CN is the north poZor distance, and the /_ BCQ, the declination of the star or planet P. Again, if the cu-cle ESQN be ftsed in the plane of the equator, and EQ point to the vernal equinox at the instant that the tube, AB, points to the starP, then the /.BCQ will be the right ascension of the star. m. It has been said that Aeistotle, b.c. 340, propounded a question in mechanics Similar to the following : — "At what part of a gaUey, EE, with rounded sides, EAE, EBF, does -p an oar, handled fl'om a seat, or sta- tion, just above the line of the keel, EE, produce the greatest efiect in moving the galley ? " If we examine the conditions of the rounded sides, we shall find that AB, the diam. of the ACBD, is the greatest width ; then, by 15, in., AB being the gi-eatest Une in the circle, all other chords, GH, IK, are less than A B. When the oar is applied at A, the leverage pfi of the rower, in proportion to w/i , _2 the leverage of tiie weight or re- «7"'^ sistance, is greater than when the oar is applied at L or at M ; where again it is greater than when applied at N or O. The best position there- fore for the oars is in tiie line of the greatest width, AB. 36 GKADAHONS XN BCCLID. Prop. 16. — Theok. The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle , and no straight line can be drawn from the extremity between that straight line and the cir- cumference, so as not to cut the circle ; or, which is the same thing, no straight line can make so great an acute angle with the diameter at its extremity, or so small an angle with the straight line which is at right angles to it, as not to cut the circle. Con. Pst. 1. 12, 1. Dbm. Def. 15, I. 5, I. The l_B at the base of an isosc. A are equal. 17, I. Any two /.s of a A are together leas than two rt. /.s. Ax. 9. 19, I. The gi-. ^ of any A shall he subtended by the gr. side. Def. 2, in. A St. line is said to touch a 0, when it meets the 0, and being produced, does not cut it. 2, in. If any two • s be taken in the 0ce of a 0, the st. Bne which joins them shall fall within the 0. E.l Hyp. 1 Let ABC be a 0, fig. 3, D the cen., and AB the diam. ; 2 „ 2 and the st. line AE be J_ to AB, at its extremity A ; 3 Conc.l then AE shall fall without the AB C ; 4 „ 2 & no st. line can be drawn from A, between AE and the 0ce, unless it cuts the 0. I. The St. line A 'Et,fig. 3, shall fall without the ©ABC. Sup. 1 & 2. — If not, it must fall either within or on the ©ce. C. D.l 2 3 4 Sup. 1. Pst. 1. Def.15,1. 5,1 H.2. D.l &2, 17, 1. Cone. 1st, Let AE fall ivithin the 0. Join D and C, the • where AE cuts the ©ce. VDA = DC Bl .-. Z DAC = Z DCA; but Z D A C is a rt. z , .-. Z DCA is a rt. Z ■ .'. in A ADC,, the Z s ' D AC & DCA = 2 rt. Z s ; which is impossible .•, AE if J_ to DA does not fall within the ©. PROP. XVI. BOOK III. C. D.l Sup. 2. Pst. 1. D.l,2&3,lst. 17,1. Cone. 2iid, If possible let AE fall on the 0ce. In 0ce take a • C and join DC. As befoji-e in A ADC, Z DCA&DAO = 2rt. Zs. ■wHch is impossible ; .*. AE cannot fall owthe 0ce II. No si. line can be drawn from A, between AE and the Qce, unless it cuts the circle. Sup. 1 and 2. — If sncli a line can be drawn, as A P, it falls either without or on the 0. 1st. liBtAE Ml ivithout the © ABC. FroniDsup.DGj.AF, andmeeting ©cein H. • Z DAG < Z DAE, but Z DAE a rt. Z ; . Z DAG< art. Z. But Z DGA is a rt. Z , .-. Z DGA > Z DAG; and .-. DA > DG; now D A = DH, .-. DH > DG, a part > the whole ; an impossibility ; . AP cannot fall witlwut the circle. c. D.l 2 8 6 C D.l Sup. 1. 12,1, Ax. 9. H. 2. C. 19, I. Def.l5,I.Ax.9. Cone. Sup. 2. 12,1. D.l,2,3CaseII. Def. 15,1. Cone. Eee. CoE. I. — If a St. I 0, AB, from its 2nd. If possible then, let AP fall on the ©ce. DrawDG X AF, G being in the supposed ©ce. Then as before DA > DG; , But DA = DG; wHchis^\ impossible ; .*. AP cannot fall ore the ©ce. Hence, The st. line drawn at rt. z«) ^c. Q. E. D. le, A E, be drawn at rt. ^s to any diam. of extremity, A, it shall touch th« © of. the 38 GRADATIONS IN ETJOLID. ity : and a St. line touching the © at one • shall touch it at no other point. D.l 2 3 4 5 6 Sup.&2,III. 16,III.pt.I. Def.2,III. Remli:. Sup.&2,III. 16,III.pt.I For, if AE cut the , part would fall within the 0; which has been proTed to be impossible : .*. AE touches the at A, the extremity of BA. And AE touches the at no other point except A ; for if it could, as before, part would fall within the . which has been proved to be impossible. Cor. II. — By 28, I. st. lines at rt. /_s to the extremities of the same diam. are parallel. Cor. III. — Tangents to a circle from the same point are equal. ScH. — 1. Thus, a Tangent to a circle is a rt. line perpendicularly raised on the extiemity of a diam., or of a radius. 2. Tlie 16th Prop, might have been proved directly ; thus I. From D draw any line DI to meet AE j *.• DAI is a rt. /., .-. ZDIAAI; &VDA FG: butPC = FB .-. PB > FG, the less than the gr. ; which is impossible ; .-. FG is not perp. to DE. So, no line except FC is perp. to DE ; .-. FCisperp. toDE. .*, If a St. line touches, SfC. q. e. d. ScH. — ^Propositions 16 and 18 maybe regai'ded as the converse of each ether ; in the one we prove, that a line perpendicular to the extremity of the radius is a tangent ; in the other, that, if a line be a tangent to a circle, the radius from the point of contact will be perpendicular to it. Thus in Prop. 16, AE is proved to he perp. to AB, and in Pi-op. 18, PC perp. to CE : in tlie one case, the tang, is perp. to the rad.; in the other, the rad. is perp. to the tang, ; two expressions for the same thing. ).l C. 17, I. .,2 19, I. 3 Def. 15, : Ax. 9. 4 Cone. 5 Sim. 6 Cone'. 7 Rec. 42 GRADATIONS IN EUCLID. Use and App. — To draw a tang, to each of two given Qa ABC, DEF. E B C.l 2 3 4 5 6 D.l 2 .3 4 S Pst.l.3,I. Pst. 3. 17, nr. Pst. 1,2. 31,I.Pst.l Sol: Join the centres A, D, and make F Gr = A C j from D with rad. T> G desc. D GH ;' and from A draw AH, tang, to D GH ; join DH, and prod, it to E in © DFE ; from A draw AB || DE, and join EB ; then EB is tang, to each 0, ABC and DEF. C3,18,III V AHistang.inHto©DGH, .-. /.DHAisart. ^; 13, 1. C. and Z. EH A is art. /_; and AB EH is a / — 7 ; C.& 46,1. and •.• /_ EH A is a rt. /. .'. all the Z_a in ABEH ara rt.^s, D. 3 then •.' D E from cen. D, & A B from cen. A, meet B E at rt./.s, Cor 16,nil /. EB is tang, to both ©s, AB C and D E F. q. b. p. Prop. 19. — Thboe. If a St. line touches a circle, and from the point of contact a St. line be drawn at right angles to the touching line, the centre of the circle shall be in that line. Con. Pst. 1. Dbu. 18, m. Ax. 11. All rt. ^s. are eqnal tp one another.— Ax. 9, Let the st. line DE touch the © ABO in C; & from let OA be at rt. Z8 to B DE; then the cen. of the shall be inCA. D E.1 Hyp.l 2 ,, 2 8 Cone. PROP. XIX. BOOK III. 43 Sup. — For if not, and it he possible, letF be the cen. and join PC. D.l 8up. 18, III. H. 2, Ax.ll Eemk. Ax.9. Cone. Sim. Cone. Rec. then •,• D E is a tang, and F C a rad. from the • of contact ; /. FO is pei-p. to DE, and Z FCE a rt. Z. ; but ACEisart.Z, A Z FCE = Z ACE; i. e. the part = the whole, which is impossible. .". the • F is not the cen. of the ABC. And thus no other • , except it be in C A, is the cen.; .■. the cen. of the © is in CA. Therefore, If a st. line touches, ^c. Q. e. d. ScH. — 1. If in two concentric circles ABC, DET, a chord AB of the greater meet the less, as in D and E, the parts AD andEB, intercepted be- tween the two 0s are equal ; and all chords of the greater 0, as G C, which touch the leas, are bisected at the • s of contact, as F ; and are equal. C. D.l 2 3 12,1. S,IlI.Ax.3. c. 3, ni, Sim. i 14, in. From O the cen. draw O H OF J. GO. . AB, and I. VAH = BHandDH = EH .-. AD?=BE. II. V ZOFC=ZOEG; .-, GF = FC. Thus all chords of ABC tangs, to DEF, are bisd. at the • of contact, and m. such chords are equal. 2. If any number of equal chords, G C, K L, be drawn in a circle, the loeiis of their points of bisection, as F, I, &c., is a circle of which the rad.' = OG" minus G'E'. Use and App.— I. In Optics the properties of tangent lines are employed, among other purposes, for determining the part of a globe which may be enlight- ened by a luminous body, ashy a lighthouse, a volcano, a meteor, or the sun, II. Bv a tangent line lepresentingthe limit or e^ictent of vision, the diameter of the earth may be ascertained, as in Prop, 6, Book II. in. Tangent lines, too, serve to explain the Theory of the Phases of the Moon, and were employed by HiPPASOHtrs, B.C. 160 to 146, to ascertain the distance of the sun, IV. In Navigation the dip of the horizon corresponds with the tangent line from the point of observation ; and were the tangent line to reVolVe round that point it would trace out the circle of the physicd horizon. An approxi- 44 GRADATIONS IN EUCLID. mate rule for ascertaining the distance of the verge of the natural horizon is to take the square root of the height of the spectator's eye in feet and multiply it hy 1'3 ; — the product in feet wUl nearly give the distance, or rad. of the horizpu, in miles ; thus from the mast of a ship 81 feet above the sea the horizon is 11'7 miles off; fory'Sl X 1-3 = 9 X 1-9 = 11-7 j or, How distant is the tangent point from the highest peak of Teneiiffe, whidi has a n elevat ion of about 1 1,946 feet ? Vll,946 X 1-3 = 109 X 1-3 = 1417 miles. y. In Dialling, also, or Gnomonics, tangents are employed for calculating the hour lines and ascertaining their exact position ; but Dialling is now com- paratively of little importance. Prop. 20.~Theou. The angle at the centre of a circle is double of the angle at the cireumfererice upon the same hase, that is, upon the same part of the circumference. Con. Pst. 1 & 2. Dem. Def. 15, 1, 5, 1 32, 1. In every A, if any side be produced, then the est. /. == tbe two int. and. opp. /.s ; and the three int. /.s = two rt. ^s. E.l Hyp. 1 la a © ABC let E be the cen.; Z BEC the /L at the cen. and Z. BAG the Z. at the ©ce; 2 „ 2 and let Zs BAG, BEG have for base the same arc BC; 3 Cone. then the Z BEG = twice Z BAG. C. Pst. 1. 2. Join AE and produce it to the ©ce P. (fig, 2.) Case I. Let E the cen. be on AG, one of the st. lines AB, AC. A D.l Def.l5,I.5,I 2 C. 32. 1. • 3 Cone. V EA = EB .-. A EAB=: -Z EBA; and /_S EAB + EBA= 2 Z EAB; but V in A AEB, AEisprod. to G, .-. ext. Z BEG= Z EAB + EBA V z BEG = twice z BAG, rnor. xx. — book hi. 45 D.l . 2 3 4 5 Case II. Let E the con Def.l5,I.5,I, Remk. ! -within the ^ B A C C. 32, I. Cone. Sim. Cone. •.• EA = EB, .-. Z EAB = Z EBA, and zs EAB + EBA = twice Z EAB; but z BEF= Z EAB + z' EBA; .-. Z BEF = twice Z EAB: Thus also Z PEC = twice z EAC; .-. the whole Z B E C = twice the whole / BAC. Case III. Lastly let E le without the Z BA C. D.l 32, I. Rec then V Z PEC=: 2 Z EAC, & Z FEB = 2 Z FAB; .*. rem. Z BEC = twice rem. Z BAC. Therefore in every case, the angle, 6rc. Q. E, D Cor. — Any angle, as Z CAB, ai the circumference is measured by half the arc C B, on which it stands. ScH. — I. The reasoning employed anticipates Prop. 5, Book V., and assumes, " that among four magnitudes, if the^r«f = twice the second, and the third =r twice the fourth, then the Jirst + the third = twice (the second + the fourth') ; and also, that if one magnitude = twice another, and apart from the first =: twice a part from the second ; then the rem. of the first = twice the rem. of the second." This principle is neither sufiiciently self-evident to Ije received as an axiom, neither has it heen demonstrated ; another method of proof for Case III. has therefore to be adopted. C.l Pst.land2. D.l 32,1. Def.l5,I.5,I. Cone. 32,1. Def.l5,I.5,I. Join AE and prod, it to F ; and let AB, E C inters, in D. + and in A BED ext. Z BDC = Z I>EB 4- Z EBD ; . EB = EA, .-. Z EBD = ZEAD; .-, ZBDC=ZI>EB+ ZEAD. Again, ".• in A ADC ext. Z BDC = ZDAC+ ZCCA; and V EA = EC .-. ZI>CA=Z EAC = Z DAC + ZDAE; 46 GRADATIONS IN EUCLID. D. 4 and 5. D.3. Ax. 1. Sub. Ax. 3. .•. Z_ BDC = twice (^DAC+ /_ DAE); but/. BDC= Z DEB+ /.EAD; ,-. /_ DEB + Z EAD = tvrioe(Z DAC+ Z.DAE); take away the com. /. E A D ; .■. rem 2. I> EB = twice rem. /. D A C ; i. e. the /. at cen, = twice the Z. ^i ©oe. n. If Euclid's definition of an angle (Def. 9, Book I.) be strictly adhered to, the 20th Prop, is geometrically true only when the angle at the centre is less than a rt. angle ; but if an angle may be regarded as any angular magnitude less than four right angles, the proposition is uniTersally true. The relation between the angles at the centre and at the circumference, subtended by the same arc, includes the cases in which the angle at the centre is greater than two rt. angles. Take, for instance, the re-entrant angle BEG, made up of ^s BEE, EEC. C. D.l Fst. 1 and 2. 20, ni. Cone. Join A E and prod, it to E, the re-entrant ZBEC=ZsBEF,EEO. •/ Z BEE = twice Z BAF, B, and Z EE C = twice Z C AE ; .■. re-entrant Z BBC = twice Z BAC. In this way the proposition is universally true. m. The demonstrations often appear plainer when aiTanged exactly as a Simple Equation, and worked by the same rules ; thus, in reference to the figures Pr. 20, III. " se 2. ZBEE = 2Z BAF, or 12= 6 + and ZEEC = 2ZEAC, 14= 7 + .Add. ZBEC = 2ZBAC, 26 = 13 + 13 Case 3.. Z EEC = 2Z E AC, or 20 = 10 -|- 10 and ZEEB = 2ZEAB, 12: 6+6 Sub. ZBBC = 2ZBAC, 8= 4+ 4 Thus, the sum or difierence of the angles at the centre = twice the sum or difference of the angles at the circumference. Use inv App. — 1. This proposition is applied in Trigonometry, of which examples will be found in the TJse and Application of Prop. 21, Book III. 2. It was employed by Ptolemy to determine the eccentricity of the sun and the epicycle of the moon. 3. In ascertaining the earth's aphelion by three observations the angle at the centre of the orbit is taken double the angle at the circumference. PROP. XXI. — BOOK HI. 47 Prop. 21. — Thbor. The angles in the same segment of a circle are equal to one another. ' Cost. 1, HI. Pst. 1 and 2. Dem. 20, in. Ax. 7. Ax. 2, E.l 2 0. D.l 2 3 i Hyp. Cone. In ABCD, let BAD, BED be Z.s in the same sag. BAED; then Z BAD = Z. BED. Case I, — Let the seg. BAED be greater than a semicircle. l,III.PBt.l C. 20, III. Sim. Ax. 7. Take P the cen. of ABCD and joinBF, FD. V Z BPD is at F, the cen. and Z BAD at the ©ce, and •.• each Z has the same base, arc BOD; .-. Z BFD = twice Z BAD. For the same reason Z BFD ]^ = twice Z BED; .-. z BAD = Z BED. Case II. — Let the seg. BAED be not gr. than a semicircle. Find F the cen., join A P, and prod, it to 0, and join GE, CB, CD. fi '.• AC is a diam. .•, seg. ABDC is a semic. also segs. BAEDC, DEABC each > a semic. and •.• BAEDC > a semic. .-. Z BAC = Z BEC; and '.• DEABC > a semic. ,-. z CAD= z CED; Hence the whole Z BAD = whole Z BED. •. The angles in the same segment, ^c. Q. E. D, c. l,III.Pst.l,2. D.l C. 2 0. 3 C. and Case I. 4 C. and Case I. 5 6 Add. Ax. 2. Eec. 48 ca;AUA'riuxs in bl'i;hd. K.li. This ri'op. may bs veiy simply proved ; for tlio angles of tlie segments arc the halves of the same central angle standing on the same common base. Cor. — If, as in fig. to Case I, upm the base, BD, of a A, there he described a segment, BAED of a circle, the vertex of the A shall fall without, as at G, or within, as at F, or upon the arc, as at E according as the vertical angle is <, >, or = the /_ A, in the segment. For, if the vertex fall loii/joMi, as at G, then, 16, I., / BED = Z BAD and is > /_ BGD; if within, as Z. F, then, 21, I, Z F > z BAD; and if on, as E, then 21, III., z BED = Z BAD. SoH. 1. — Any number of triangles whatever, upon the same base, and with their vertical angles equal, have the locusot their vertices in the arc of the same circle ; hence, by constracting innnmerablo triangles iulfllling these two con- ditions, the vertices would form the segment of a circle. 2. From this Proposition it follows that the angle at the circumference has for its measure one half of the arc on which it stands. Use akd App. —I. This proposition may be employed for constructing a building in which all the spectators shall see any object under the same angle. A circular amphitheatre is of this kind, for let BD, fig. Case I., be the stage and BAED the circle in which the seats are placed, then from every one of the seats the stage subtends the same angle of vision ; and it is a law in Optics that a line seen under the same angle appears of the same size. C.1 2 3 4 D.t 2 3 II. To bisect an angle, as /^D AC Pst. 3. From A with AB desc. an arc BC ; 3,I.Pst.3 3,I.Pst.3. Pst. 1. Sol. C. 20, ni 8,1. Ax. 1. Cone. take BD == BA, and with BD desc. the arc DP A I and from D with rad. = B C cut DFA in Fj and join BF, AF, BC, and DF, then the line A F bisects Z ■^• •.• Z DBF is at the ceu., and Z D AF at the 0ce ; .-. ZDBF = 2 ZDAF; but ZBAC=Z DBF, .-. Z BAC= 2 ZDAF. /. Z D AC is bisected by AF. m. To construct a figure representative of the distance of the place of observation from an tibjeet. Ex. — Three objects, A,B, C, are distant, AB = 8 miles, BC = 7'2, and AC = 12 miles, from the place of observation D ; the Z CDB = 25°, and Z BD A = 19° ; required the distance from D to B. PKOI'. XXII. — BOOK III. 49 From a scale of eq. parts, with the given dis- tances, construct the A ABC ; at C make £_ ACE = 19° = /. BD A, and at A, l_ CAE = 25° = Z C D B ; and through the three points A, E, and C, (Prop. 3. Use 4, m.) draw a circle ADCE; join BE, and produce it to intersect the circle in D; then B D is representative of the distance from D to B, and is equal to about 15 miles, taken from the scale. For /.s ACE, ADE, being in the same segment, ADCE, are equal, by 21, III., and also /.s CAE, CDE, are equal, being in the same segment, E ADC. . Tatbs' Geometet, p. 56. IV. To draw the arc of any circle, especially on a large scale, by meana of two St. pieces of wood fastened so as to form a certain angle, A CB. Kx into the gi'ound, or on a floor, two pins, A and B, at any required distance, not greater than that of the extremities, A and B, of the angular frame ACB. Move the angular frame round, keeping the sides press- ing close to the pins, and a tracing point or pencil at C will mark out the arc of a circle on the ground, or on a floor. The base, AB, remains the same, and the vertices of the triangles have their locus in the arc of the same cu'cle. This method of dravring, or tracing a circle, or the arc of a circle, without having its centre, may be employed for giving a spherical figure to metal cauldrons, or to optical glasses ; also for making large Astrolabes, or for marking out the meridian lines, and the lines of latitude on lai'ge maps ; indeed for every pm-pose which requires an arc of great piagnitude. Prop. 22.— Theor. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles. Con. Pst. 1. Dem. 32, 1. 21, ni. Ax. 2. Ax. 1. E.l Hyp. Let ABOD tea qu. lat. ill the ABC. 2Conc.l. then Zs ABC + A^DC = 2rt. Zs. 3 „ 2. & ZsBAD + BCD=2rt.'zs. C. Pst. 1. Join AC, BD. 50 GKADATIONS IN EUCLIIJ. Case l.—The op;?; Z.s ABC + ADO = 2 rt. /_s. D.l 2 3 4 5 6 32, I. 21, III, Ax. 2. D. 1. Ax. 1. In A CABj the zs ABC + BCA + CAB = 2 rt. / s • but Z CAB = z CDB, and Z ACB = Z ADB. .-. the whole Z ADC = Z CAB, +Z BCA; to each add Z ABC, and Zs ADC + ABC = zs CxVB + ABC + BCA; but zs ABC + BCA + CAB = 2 rt. Zs; .-. zs ABC + ADC = 2 rt. Zs. Case II. — The other pair of opp. Z«, BAD, BCD, also = 2 rt. Z«- D.l 32, I. Form A BAD, ZS BAD + ADB + DBA = 2rt. Zs. 2D.2,4. Case! as before Zs BAD + BCD = Zs BAD + ADB + DBA; 3 Cone. .-. Zs BAD + BCD = 2 rt. zs. i Eec. Therefore, The opposite angles of any qiiadril., ^c. Q. E. D. Cor. I. — And conversely, if the opposite angles of a quadrilateral he together equal to two rt. angles, a circle way he described about the quadrilateral. C.l 2 D.l 9, III. Use. Ass. Pst. 1. 22,111., & H. Siih. Ax. 3. C. Sch. 21, IJI. Cone. B Through A, B, D, the vertices of three Zs,drawa0 ABD; the 0ce will also pass through the fourth vertex, C. -A, For take any other point E in the seg. and join AE, BE. V Z E + z D = 2rt. zs, and Z C + Z D = 2 rt. ZS. .•. taking away com. Z D, Z E = Z C ; and these angles have a com. baSe, A B ; .*. Z s E aiid C, are both in the 0oe. and .•, ©may be described about the ^u.lat. ADBC. PROP. XXIlIi — ^BOOK m. 51 CoK. II. — If any side of a quadril. in a circle, as AB, in fl^ 1, he produced ia E, the ext. ^1 EEC = int. opp. Z ADC; for they have a com. supplement Z C B A, or each of them together #ith the int. adj. Z = 2 rt. Zs. GoE. III. — " If two chords cut off similar segments from the same or different cirbles, tlie other seghienis will also be similar, Since the angles which they contain ate supplemental to thofee in the former segments." CoK. IV. — " If opp. angles of a quadril. be equal, they must be both rt. angles, rt. angles heing the only equal angles which are supplemental." Lardner's Euclid, p. 110. Use ahd App. — Ptolemy availedhimselfof this proposition to construct the Tables of Chords ; and in Trigonometry it inay be appUed to proVe that the sides Of an obtuse angled trianglfe have the same ratio to one another as the sines of their opposite dhgles ; tod hence, if any three be given, the fourth inay be found; Prop. 23.— Theor. Upon the same straight line and upon the same sidi of it there cannot he two similar segments of circles, not coinciding with one another. Con. Pst. 1, 2. Dem. 10, ni. Def. 11, Itl. 16, 1. E. D.l 2 3 Sup. Sup. 10, III. Cone. If possible, on AB and on the same side of it let there be two sim. segs. ACB, ADB, not coin- ciding. V ACB cuts ADB in A and B ; .". these 0s ACB, ADB cut in no other points ; .■. one of the segs. falls entirely within the other : 52 GHAPATIOMS IN EUCLID. C.l 2 D.l 2 3 4 , 5 Sup. — Let seg. A C B fall entirely within seg. A D B ; In arc A C B take any • C, and join B produce BC to D, and join CA, DA, P$t. 1 Pst. 2 H. Def.ll,III Cone. 16,1. Rec. •.• seg. ACB is sim. to seg. ADB; and sim. segs. of 0s contain equal ^s : ,-. Z ACB = Z ADB, i. e. the ext. /. = tlie int. Z. ; but this is impossible ; Therefore, there cannot he two similar, ^c. Q. E. D. ScH. — This proposition, relating to two similar segments of a circle, is the same in principle with the 7th of Book I., which says, that, "on the same- base and on the same side of it, there cannot be two triangles which have their sides tenninated in one extremity of the base equal, and likewise those termi- nated in the other extremity equal, not coinciding with one another :" and as the only purpose for which 7, 1., was employed was to prove 8, 1. ; so the purpose to which 23, HI. is applied is the demonstration of 24, III. We throw such propositions away as soon as we have used them, yet they are needful links in the chain of geometrical argument. Pkop, 24.— Theoe. Similar segments of circles upon equal straight lines are equal to one another. _ Dem. 23, ni. Ax. 8. Magnitudes which coincide are equal. G E.l D.l Hyp. Cone. Super. Let AEB, CFD, be sim. segs, of ©s on equal St. lines, AB = CD; then seg. AEB = seg. CFD. Suppose seg. AIJB placed on seg. CFD, so that Abe on C, AB on CD, and arcs AEB, CFD, on the same side of C D ; PROP. XXIV.— BOOK HI. H. D. 2 and 23, III. Sup. Eemk. 23, III. Cone. Ax.i Eec. 1. then •/ AB = CD, .•. B shall coincide with D. and AB coi'iieiding with CD, and the SfegSj being on the same side of C D, .■. the seg. AEB must coincide with seg. CFD. If not, the arc AEB would take another direction, asCGD;-' ^ thus on C I), and on the same side of it, there would, be two similar segs. of ©s not coinciding, CGD and OPD; which is impossible. .•, seg. AEB coincides with seg. CFD;' and seg. AEB = seg. CFD. ■. Similar segments of cirples, ^c. q. b. d.» . Cob. I. — Similar segments having equal cliords have also equal arcs ; to be established hj the same principle of superposition. CoE. II. — Similar segments having equal chords are parts of equal circles ; for circles which agree in more than two points agree in every point. Oon. m.-^'lf the radii, AB, EP, and angles, BAG, PEG, of sectors BACD, PEGH, are equal, th^ sector's themselves are equal. C.l 2 3 D.l 2 Pst. 1. Pst. 2, 1, Pst.. 1. 4,1. 20, III. Draw the chords BC and PG; Produce B A to L, PE to M, and join LC, MG,BD,DC, PH, HG. Then v a BAC= a PEG; and •.• Z BAC = /_ PEG, and Z BLCia Z FMG: 54 GRADATIONS IN EUCLID. D.3 5 22, III. Pef. 10, III. D. 1.24, III. Add. Ax. 2. also •.• /_ BDC = z fUQ, .: seg. B D C is sim. tq seg. P H G. Again ■.• EC, = FU /. seg.' B D C = seg. FH G ; To each seg. add equal ^sBAC, FEG; .-, EDO + BAG = FHG + FEG, i. e. sect. BACD == sect. FEGH. Use and App. — Curve lined figures, asADB,CEA) are often reduced to rectilineal figures by this proposition. For, if ^o like segments, A DP and AEC of circles are described on the equal sides, A B, A C, of an equilateral A A B C, it is etident that by transposing the seg. AEC on A D B, the A A B C = the curve lined fig. AD B, CE A. B Prop. 25. — Pnon. A segment of a circle being given, to describe the circle of which it in the segment, Coif. 10, I. 11, 1. Pst, 1. 23, 1. Pst. 2. Dem. 6, 1. If a A have two of its /.s equal, then the sides opp. the equal /.s shall be equal. Ax I. 9, m, Dcf. 10, 1, 4, 1. E.1 2 C.l 2 3 Dat, Quaes, 10,1.11,1 Pst. 1. Eemk. Let A E C be the seg. of a circle, to desc. the of which it is the seg. Bis. AC by the perp. DB cutting arc AEC in B; and join AB; there will be two cases according as the Z. s BAD, ABD are = or ^. Case I. Let z. B AT) == / A Bp. D.l 2 H. G, I. C.&AX.1 V Z DBA = z DA!;. .-, DA = DB; and '.• DO = DA, .-. DA = DB=DCi \ A^ PKOP. xxy. — BOQ:p iii,j 55 D.3 C#;n,2. And v ff pm J) to % ©oe, DA, DB, pC are equal ; 4 9, III. .'. D is the cen. of the of which ACB is an arc. 5 Sol. Hence, if from D with rad. DAorDBorDCa be desc; it will be that of which AB in an arc. Eemk. And •.• the cen. is in AO, .•, the seg. ABC is a semicircle. Casb II, let L BAD t^ L ABD. 0.1 % D.l 2 8 4 5 6 B 23,1. Psts. 2&1, C. 1. 6, I. O.Def.10,1. 4, L D.l, Ax. 1. D. 3,4.9,111 Sol. Eemk. Bemk, 9 Rec. F P At A in AB make Z BAE = Z ABD; prod., if necessary, BD to meet AE in E, and join E C. V L ABE =: L BAB, .-, BE = AE; and •.• AD = DO, DE com. and Z ADE = ZCDE, /, in AS ADE, ODE, baseAE = base EC; t)ut AE = EB, .•. BE also = EC ; and •/ AE = EB = EC, .-. E is the cen. of the ■ Hence, if from E with rad. EA, EB, or EC, a be described, it wjll be that of which ABC is an arc. IfZ ABD> ZBAD, thecen. E falls v^ithmt the seg., which therefore is less thdn a semi- circle ; but, if z ABD < Z BAD, the cen. E falls within the seg., which therefore is greater than a semicircle. •. A segment of a circle heing given, ^c. q. e. f. ScH. — This problem might be proposed in another way, — as, to inscribe a triangle in a circle ; or, to make a circle pass through three given points, pro- vided they are not in a st. line. The mode of doing this has been pointed out in Use and App. II., III., and IV., of Prop. 3, Book HI., and also in the App. aG GRADATIONS IN EUCLID. of Prop. 9, Book III. If, for making a circle pass through three given points, A. B, C, not in a st. line, the method pursued in Prop. 25, Book' III., bo followed, the process will be — C.l 2 3 4 D.l Pst. 1. Join the three given • s A, B, C ; 10,1. U, l.i bis. AC in D by a perp. from D, andAB ^f in E by a perp. from E ; / the • G where the pei'pendiculars inters, is the cen. of the required 0. Join G A, GC, GB. Sol. Pst. 1. C. 4, 1. •.• AD = DC, DG com. and /.s at D equal, .•. GA = GC. ■Sim. for a like reason GB = G C, s! Ax.l.9,ni. .•. G A = GB = GC ; and with either as rad. a may be drawn through A, B, C. CoE. — In like manner the remainder of the 02o of a segment of a circle may be completed. Use and App. — The proposition is of frequent use in aU cases when a circle, or an arc, has to be drawn through any three points. Thus : 1. Por constructing an arch, ol which AB the span and D H the peip. height to the centre, are given : 2. Por di'awing the plan of a Gothic arch, in which the span AB, and the radii of the arcs intersecting in C arc the same : When AB, the span of the arch, or distance of two of the three points, A, B, is great, the method may. be adopted, which is given in Use and App. IV., Prop. 21, Book Ht., p. 49. 3. For cutting stone, wood, or metal, so that a circle shall pass through three given points : orbit A B 4. For finding the apogee of the moon, and the eccentricity of the earth's Prop. 26.— Theob. In equal bircles, equal angles stand upon equtit arCB, ivhether they be at the centres or ctrcumferencesi Coif. Pst. 1, Dem. Def. 1,111, 4,1. Def, n, Jll. 24, m, As,- 3, PROP XXVI. — BOOK III. E.l 2 C. Hyp. 1. „ 2. Cone. Pst. 1. Let © ABC = DBF; and /_ BGC= Z BHFat their centres and z. BAC,= Z PDE at their ©ces; then arc BKC = arc ELF. JoinBC, andEF. A D D.l 2 H.Def. 1,111 H. 4, I. H.Def. 11, m, D. 2. 24, III. H. Ax. 3. Cone. Eec. V ABC = DEF, .-. BG = GC = E H = H P each to each ; and Z G=:: Z H, .-. BC = EF. ' And •.• Z at A = Z at D, .-. seg. B A C is sim. to seg. E D P ; but V BC = EF,.-, seg. BAC = seg. EDF; Now © ABC = DEF, .-. rem. seg. BKC = rem. seg. ELF; and arc BKC = arc ELF. ". In equal circles, equal angles; ^e. Q. E. D. Cor. 1. — Since by Cor. 4, Prop. 22, bk. Ill, if the opp. Zs of a qu. lat. in a circle are equal, those Z s are rt. Z s, it follows from Prop. 26, if the opp. Z.sl>e equal their opp. diagonal must he a diameter, and, the segment a semicircle. CoE. 2. — Inthesame or equal circles one central or circumferential angle is less than, equal to, or greater ijhan another, as the arc of the one is less than, equal to, or greater than, the arc of the other. CoE. 3. — The diameters which intersect at rt. angles divide the circumference into four equal arcs, or 'the circle ihto four equal parts. CoR 4. — When the sum of the central angles equals four right angles, the sum of their arcs equals the whole circtimference, 58 GRADATIONS IN EUCLID. Cor. 5. — When the smn of the angle^ at the circumference equals two rt. {ingles, the sum of their arcs also equals the whole circumference. Cor. 6. — Similar arcs of equal circles are equal. CoR. 7. — Parallel chords, AE, QJ), of a circle intercept equal arcs, AC ED. CoR. 8. — If two chords, AB, CD intersect ■mi\\m a circle, ABE, the sum of the intercepted arcs, JfC + D B is equal to the arc which an angle would intercept at the circumference, B A E, that is equal to the angle, BED, under the chords. C. D.l 31,1. C.29, 1.26, III Add. D.I. 15, I. Ax. 1. Cone. Draw A E 11 CD. y'B" AE II CD .-. Z EAF = Z AFC, and .-, arc ED = arc AC; To arc ED add arc. DB, .". arc EB = arc AC + arc DB; but Z BAE i. e. PAE = Z AFC; and Z AFC = Z BFD, .-. Z BAE = zBFD. Q. E. d, Cor 9. — If two chords, AB, CD, intersect at a point F, without a circle the difference of the arcs, B D ~ AC, which they intercept is equal to the arc, BE, which an angle BAE, would intercept at the circumference that is equal to the angle BFD, under the chords. C. 31, T. DmvAE||CDorFD, D.l as in Cor. 8. then arc AC = arc DE 2 Sub. Ax. 3. from arc B D take arc D B, .-. arc BE=arc BD— arcAC. 29, I. and Z EAB = z BFD. Lardnbr's Euclid p. 113. Us|! AND Apj>. 1.— Since, by Oat. 8, qf the preceding Pvop. Z BFP = Z BAE, and by Cor. 20, ni, an Z at the '0ce, BAE, on an arc EB is Bill measui'ed by halftlio arc, BE, on which it stands, namely by — this property PROr. JfXVII.^— BOOK 111. 59 is avgiJitle forJm4ing the t^ite central angle of .an irr^erfeej^ cflnstracted theodolite, or of any sinlilar instrument, in which tjie revolving liiibs AF, !pB, in the last figure but one, are not at the centre of the circle ; thus, if the arc AC, as shown by one limb FA, is 50° 20' and the arc BD, as shown by tijie other limb ertO of\' j_ 4.QQ 1 o' Q8^ 32^ FB, is 48° 12' the true central angle = '- ~^^2 ~ ~^~o — ^ *®° ^®'' 2. T/ye may also apply Cor. 9 to determine the l_ DFB, when the arcs AC and BI) are given ; for if arc AC = 48°, and arc BD = 100°, then Z. DFB = 100° - 48° = .52°. PboI'. 27. — ThEoe. In equal circles the angles iJ)hich stand upon equal arcs are equal to one another, whether they be at the centres or the circumferences. Con. 23, 1. Dem, 20, ni. Ax. 7. 26, III. Ax. 1. Ax. 9. E.l 2 3 Hypas Hyp. 2. Cone. Let ABC, DEF, bp eq. 0s, G and H bei^g the pentres ; and let zs BGC, fl^F, j^t GandH, and Zs BAC, E D F, at the ces, stajid upon the eq. arcs B C, E P ; then z BaC = ZEUFand Z BAC = Z EDF. Sup.— If Z BGC - Z EHP, tto (20, HI, and Ax. 7) Z B A C = Z E P F ; but if not, oi^e Hiust be the greater. C.l 2 Slip. 23, I. Let z BGC > Z EHF, At GinBGmake Z BGK = Z EHF. 60 QEADA*10NS IN EUCtlf). D.l C.26,III. 2 H.2.AX.1 3 Ax. 9. 4 Eemk. 5 20, III. G Ax. 7. 7 Rec. Then •.• L BGK = L EHF, .-. arc BK = arc PE; but arc EF = arc BC, .-. arc BK = arc BC ; i. e. the less = the gr., whicli is impossible ; .-, l_ BGCnot:^ ZEHF, i. e. L BGC =/:EHF; but Z at A = 1 /: BGC,and £ at D = J Z. EHF; .-, Z at A = "z at D, •- In equal circles, the angles, 4rC. Q. ^- d. Cor. I. — In the same or in equal circles, ABC, DEF, the sectors BGC, EHF, which stand upon equal arcs, BC, EF, are equal, and conversely. c. Pst. 1. D.l Def.l5,I.27,III 2 4,1. 3 Sub. 4 5 Ax. 3. Def. 11, III. 6 D. 2 & 24, III. 7 Add. 8 Ax. 2. .•r.*» '■■■■■; Join B C, EF; and from • s K, L, draw KB, KC, LE, LF. V lines BG, GC = EH, HF, and z BGG = z EHF. .-. base BC = base EF, and ^ BGC = aEHF. From eq. ©s A B C, D E F, take equal arcs B C, E F ; .". rem. arc B AC = rem. arc EDFj .-. Z BKC= Z ELF, and seg. BKC is sim. to seg. ELF; bui base B C = base E F, ,-. seg. BKC = seg. ELF, To the cq. as BGC, EHF, add the equal segs. BKC, ELF; .'. the sector BGCK = the sector EHFL. N.B. The Converse may be left for the Student to demonstrate. PKor. xxvai. — book hi. 61 Cor. II.— As in Cor. 7, P. 26. III., if the chorda AE, CD, oj a circle are parallel, they intercept equal arcs, and vice versa. C. D.l 2 8 Pst. 1. 29, I. 26, III. D. 1. 27, III. 27,1. Join EC. V Z AEC = z ECD, .*. arc ED = arc AC. Again •.• arc AC = arc ED, \ .-•• \v .-. Z ECD= z AEC; '^^'v—S; .-. AE II CD. ^ 'D ScH. — 1. Propositions 26 and 27 are converse propositions; and what is tiTio of equal circles is true of equal ares in tlie same circle. 2. As in Cor. 2, Prop. 15, 1., all the angles formed by any number of lines diverging from a common centre are together equal to four rt. angles, so the sum of the angles at the centre of a circle subtended by arcs, which together make up the whole circumference, is equal to four rt. angles. Also the sum of the angles at the circumference subtended by those same arcs is equal to two rt. angles, 3. And, since eq. arcs of cq. circles subtend eq. angles, such cq. arcs contain similar segments. Use and App. — 1. By Cor. II. of this Prop., a parallel through a given point E to a given st. line CD may readily be drawn ; for join E C, and malve Z CEA= Z DCE, and AE is parallel to CD. 2. The principlo on which the area of a sector is ascertained may be developed from Cor. I. of this proposition, for the area of the triangle B G C, added to the area of the segment BKC gives the area of the sector B GEK. , Or, when the rad. and Z ^ Cr C are given, by principles hereafter to be ;, .v . r.r. c . Areaof© X ZBGC, proved, the Ai-ea of the Sector = go Pkop. 28.— Theoe, In equal circles, equal straight lines cut off equal arcs, the greater equal to the greater, and the less to the less. Con. 1, in. Pst. 1. Dem. Def. I, m. 8, I. 26, HI. Ax. 3 62 GRADAtlONS IN EUCLlll. E.l 2 3 C. Hyp. 1. „ 2. Cone. l,III.Pst.l. BAG, Let ABC and DBF be eq. 0s, and BC, EF eq. St. lines in them ; and let B C, E F cut off two gr. arcs EDF, and two less BGCj EHF ; then the gr. arc BAC = the fer. EDF; alid. the less BGO = the less EHF. Take K, L, centl-es of the 0s, and join KB, KC, LE, LF. D.l 2 3 4 H. Dei: 1,111. H. 8, I. 26, III. H. Ax. 3. Rec. V © ABC = DEF, .-. KB, KG = LE, LF, each to cachj and BC = EF, .-. L BKC = L ELF; .-. the arc BGC = the arc EHF ; but ABC = DEF; .•. rem. arc BAC = rem. arc EDF. Therefore, in equal circles, eqiial si. lines, ^c. Q. B. D. ScH. As in other instances, the prihcijite bf the prbposition extends to equal st. lines in the same circle, in which also such equal st. lines cut off equal arcs. Pitop. 29.— THEon. In equal circles equal arcs are subiended by equal stmighl lines. Con. 1, rn. Pst. 1. Dem. 27, m. Def. 1, ni. 4, L PROP. XXIX. BOOK III. 68 E.l 2 G. Hyp. Cone. l,III.Pst.l. Let © ABC = © DEF, and arc BGC = arc EHF, then on joining B C, and E F, ctord B C = chord EF, Find K, L centres of the .©s, and ioin KB, KC, LE, LF. D.l 3 3 4 H. 27, III. H.Def.l,III. D.l and 4, 1. Rec. V arc BGC = arc EHF, .-. Z BKC = L ELF; and V ABC = © DEF, .-. BK, KC = EL, LF each to each; and Z BKC = Z ELF, .-, chord BC = chord EFs Therefore, in equal circles eqkal arcs ^-t. y. e. d. CoK. L— By the same Idnd of dfelnbnstration it may be shown that, in the same or in equal circles, equal hectors stand upon equal arcs ; and converseli/. CbR. it. — Prom Prop. 26, 27 and 29, straight lines which inter- cept equal arcs are parallel ; and parallel st. lines intercept equal arcs ; for the alternate atigles are equal. Use aIid App. — 1. We may declare genetally that whatever has been proved with respect to equal circles is also true when applied to the sameckcle. 2. In Spherical Trigonotiietry, Props. 26, 27, 28, fand 29 are of continual use. By means of I'rops. 27 and 28, TSEODOSiiis dembnstrated that the arcs of the circles of the Italian aiid Babylonian hours, comprehended between two parcels, are equal ; and, in the same way, it may be proved that the.arCs of circles of the astroiloidical hoiirs, compreheitded between the two pdl-aUels to the equator, are fequftl. 61 GIUDATIONS IN EUCWD. Prop. 30.— Prob. To hiaeot a, given arc of a circle, i.e., to divide it into two equal parts. , Con. 10, 1. 11, 1. Pst. 1. Dem. Def, 10, 1. 4, 1. 28, HI. Cor. 1, UI. E.l 2 0.1 2 3 D.l 2 3 4 5 Datum. Quajs. Pst.l& 10.11,1. Sol. Pst. 1. C.l.Def.10,1. 4,1. 28, III. Cor. 1, III. Gone. Lot ADB be the given arc ; it is required .to bisect it. Join DB, and bis. AB in C by perp. CD ; then the arc AD B is bis. in D, i. e. arc AD = arc BD. Join AD, and D B. d ■.• AC = BC, CD com. // and Z ACD = Z BCD, A.^ .; base AD = baseBD. But eq. St. lines cut off eq. arcs, the gr. = the gr., and the less = the less ; and ■.•' DC passes through the ccn., arcs AD, D B, each < a semicircle. ,% arc AD = arc D B, and ADB is bis. in D. Q. K. F. Con. — Hence, by successive bisections, as in Sch. 1 and 3, Prop. 9, I., a given arc may be divided into any number of equal parts that are the powers of two, as 4, 8, 16, 32, &c. Sch. — 1. The bisection of a given rectil. angle, ADB.Prop. 9, 1., implies a bisection of ADB, an arc of a circle ; but, as by Plane Geometry, a rectil. angle, except in the case of a right ang!e, D CB, cannot be divided into 3, 5. 6, &c., equal parts, so an arc of a circle, except in the case of a quadi-ant, D B, which U the circular measure of a right angle, DCB, cannot be cut into 3, 5, 6, &c., equal parts. N.B. If, as is mentioned below, other plane curves, besides the circle, had been admitted by Euclid, any angle could also be divided into 3, 5, 6, &c., eiual parts. PROP. XXX. BOOK III. 65 to dmdo a rt. /., ACB, into three 2. To trisect, a quadrant, AB, i. e. OQ[ual ports. 1, 1. On CB const, an eqnil. A BE C, 9, 1. and bis. its /. B CB by CB, Sol. tlie rt. Z. ACB will be trisected, 1. e. ZBCD = /.DCE = /_ ECA. Cor.32,I.C2. •.• /_ BCE = f of a rt. /.. and is bis. by CD J ■. Z.BCD= ZDCE. 1. Also •.• ACB is art. Z. andBCE = f of a rt. /. ; •. Z_ ACE = 4 of art. /_ ; ■. /_ ACE = Z ECD = /. DCB, i. e. the quadrant is trisected. C.l D.l 2 3' Cone. H. &D. 4i Ax. 3. 5' Cone. Or, — ^From A and B, with the rad. of the circle, describe arcs cutting the quadrant in D and E | join EC, DC, and the quadrant is trisected. 3. By successive bisections of the one-third of art./., the l-6th, l-12th, l-24th, &c., of a rt. /.> otc of a quadrant is obtained. 4. The division of a quadrant into five equal parts depends on P. 10, IV. 5. Since Euclid confines himself to straight lines and circles, the trisection of an angle, or. of an arc cannot be effected hy his Geometry '; if, however, other curves, formed by the sections of the Cone, were admitted among the curves of our Plane Geometry, the problem could readily be solved. Por, " if with two- thirds of any given line, A, as a major axis, an hyperbola be described whose asymptotes," or incoincident lines, "make ah angle of 120°; and if with A as a base, and a point on the branch of the hyperbola adjacent to the single thu'd of A as a vertex, a triangle be described, the larger of the angles adjacent to A •will always be double of the smaller. Consequently, one of the external angles will be triple of one of its internal and opposite angles ; so that by des- cribing on a straight line A a segment of a circle containing the supplement of any given angle less than 180°, that circle will cut the branch of the hyperbola in a point which, being jomed with the further extremity of A, will give an angle equal to the given angle." — ^Pjbnnt Cycl., XXV., p. 260. 6. What is required for the ti'isection of an arc or of an angle is the solution of tlie following prob- lem J " from a given point, as A, in the cireum- ierenoe of a ABD, to draw a st. line, A XT, such that the part X Y, between the circumference and a given diameter BD produced, shall be equal to the radius CA." D.l 2 3 4 H. 6, 1. 32,1. Gone. V CX = XY,.-. Z XC Y = Z. XYC ; but Z AXC or CAX = 2Z XCD, and Z ACB = Z XCD -f- Z AXC; .-. Z ACB = 3 Z XCD, .-. arc AB= SareXD. Thus Z X CD = J of Z A CB, and .-. arc XD = J arc A B, 66 GRADATIONS IN EUCLID. 7. One of the Trochoidal curves, known by the name of the triseetrix, is peculiarly possessed of the property of dividing any arc into three eqnal parts. UsB AND A pp. — ^By this problem, the semicircle is divided into quadrants, and the quadrant into arcs of 45°, 22^°, &c. ; and the Mariner's Compass, as inUse4, 9, 1., into 32 equal parts called Bhnmbs ; but the division intoi single degrees cannot be performed by Enclid's Geometry. Prop. 31. — Theoe. In a circle, the angle in a semicircle is a right angle, hut the angle in a segment greater than a semicircle is less than a right angle f and the angle in a segment less than a semicircle is greater than a right angle. Con. 10, I. Psts. 1 & 2. Dbjt. Def. 15, 1. 5, 1. Axs. 1, 2. Def. 10, 1. 22, IH. 32, 1. If a side of a A be produced, the ext. /. =: the two int. and opp. /.s ; and the three int. /.s of every A are together equal to two rt. /.s. 17, 1. Any two angles of a A shall together be less than two rt. angles. E.l 2 Hyp. 1. „ 2. Cone. 1. 4 5 „ 2. „ 3. C.l 2 3 10,1. Pst. 1&2 Pst. 1. Let ABC be a 0, BCits diam. and E its cen. and from C let C A divide the into segs. ABC, ADC, ofo wMch seg. A B C is > a semic. and ADC < . a semic. Then Z. B AC in tie semic. is a rt. Z; Z_ ABC in seg. ABC is < a rt. Z ; and Z ADC in seg. ADC is > art. /_. Bisect diam. B C in E ; E is the cen. join E A, B A ; and produce B A to F ; in' arc AD C take any • D, and join AD, DO- PRWP. XXXI.— BOOK III. Case I. — The /_ BAG in the semieircle shall be a rt. /,. 67 D.l 2 8 4 5 Def. 15,1.5,1. Add. Ax. 2. 32,1. Ax. 1. Def. 10,1. Cone. Now, V EB = EA= EC; .-. Z EBA = Z BA:I4 and Z EAC= Z ECA; add the eqs. and the whole Z BAC =the two Zs ABO, ACB; but in A ABC, the ext. Z FAC = the two int. zs ABO, ACB; .-. Z BAC = Z FAC, and .-. each is a rt. Z • ■. the Z BAC in the semic. is a rt. Z • Case II. — The Z -^BO, in a se(f. ABC r/r. than a semic. ,^ shall less than a rt. Z ■ And •.• in a ABC the Z s ABC, BAC are< 2 rt. Z s. and that Z BAC is a rt. Z ; .•. Z ABC must be < a rt. /_. •. the Z , in 8' seg. > a semic., is less than a rt. Z • Case III. — The Z ADC, in a seg. ADC < a semic, shall he gr. than a rt. Z ■ D.l 17,1. 2 Case I. 3 Cone. 4 Rec. D.l 2 3 4 0.22,111. Case II. Cone. Rec. •.• ABCD is a quadrilateral in a © ; .-. ZS ABC, ADO = 2rt. Zs; and ■.• Z ABC is < a rt. Z ; .•, the other Z ADC is > a rt. Z. •. In a circle the angle, ^c. CoK. — If one angle of a triangle he eq. to the other two, it is a rt. angle. For in A ABO, let Z BAC = Zs ABC + AOB; then Z BAC is a rt. Z - E.l 2 C. D.l 2 Hyp- Cone. Pst. 32, I. Def.10,1. Pi-oduceEAtoF. V z FAC = Zs ABC + AOB; .-. Z FAC = Z BAG, .-, Z BAC is art. Z- Sen. — The converse of Tiop. 31, is,, " the segment which contains an acute angle is greater than a eemiciicle, and that which contains an obtuse angle is less than a semicircle." G8 GRADATIONS IN EUCLID. 2. The Demonstration which Labdnek gives of the 31st Prop, is remark- able for its elegance and brevity i it is founded on Prop. 20, Book >p. IS ren In.;- '.• the central /. on a semicircle = two rt. /.s ; .'. the circumferential /_ = one rt. /Jl Again, '.• the cen. ^ on an arc less than a semic. is less than two rt./.s ; .". the circumferential /. is < a rt. /.. And *.• the cen. £_ on an arc gi\ than a semic. is gr. than two rt. /.s ; .•. the circumferential /. is > a rt. /.. Use and App. — I. From the property, Case I., P. 31, III., that the angle in a semicircle is a rt. angle, the following Problems are derived : Pkob. 1. From, a point B, in a line, or at the extremity of a line, to draw a ■perpendicular. Fi'om any • A, out of the line, with the distance A B, ,,. .^ tlescribe a semic. meeting C A produced in D, and joiuBD; /' then BD is the perp. / N.B. Pelitakius, a mathematician, often quoted by J A-''' BiLLiKGSLET, givcs this Cor. to Pr. 31. — "If in a circle be \ ,,--'' ~'-.., inscribed a rectangle triangle, the side opposite unto the C '■f^- '■ right angle shall be the diameter of the circle." '•>,. •? 'B PfiOB. 2. — From a point D, mthout CB o line, to draw a perp. Join D, C, and bisect DC in A ; and from A with AC describe a semic. and join AB ; DB is the peq). required. tnon. 3. — From a point A, without a circle, to draw a tangent. ' Join A and the cen. C ; bis. A C iri E ; and . from E with E A desc. a circle ; and draw °- A B and A D ; then A B and A D are both tangs, from A to the BCD. The Demonstration may be left to the Student. Or, join A, C ; 10, 1., bis. AC in D, and with DA y desc. a semicircle ABC; where the semic. cuts <; the B, is the tangent point ; and A B the A."' tangent from A. /pj. II. By means of a square the centre of a circle mat/ be easily found; For, if B the angular point of the square, CBD (as in the fig. to Prob. 1. above) touch any point in the 0ce ; and if also the sides nf the square, BD, BC, fall upon two other points, C, D of the 0ce ; then the line C D is a diam. and its middle point the centre of the circle. PROP. XXXII. — BOOK III. 69 in. In this proposition workmen possess a very simple v>ay of trying if their squares be exact; For on the hypotenuse D B desc. a semicircle DAB, and apply the angular point A of the square on the ©ce of the circle, and one of the sides of the sq. AB, so that the point B of the square may fall upon B the extremity of the diam ; then if D, the extremity of the other side of the square, fall upon D the other extremity of the diam. BD, the square is correct ;. but if that other side, as A E, falls within the circle, the /_ BAB is < a rt. Z. ; if without, as AF, the /_ FAB is > a rt. /_. Prop. 32. — Thbor. If a straight line touches a circle, and from the point of contact a straight line he drawn cutting the circle ; the angles which this line makes with the line touching the circle, shall be equal to the angles ivhich are in the alternate segments of the circle. Con. 11, 1. Pst. 1. dem. 19, ni. 31, nr. 32, i. Axs. 1, 2 & 3. 22, in. 13, I. The adj. /_s which one st. line makes with another on the same side of it shall either be two rt. /.s, or be together equal to two rt. /.s. E.l 2 3 Hyp. 1. „ 2. Conc.l. 4 C.l 2 Con. 2 11, I Pst. 1 Let EP touch AB CD in B; and from B let BD cut the ; then Z PBD = Z BAD in the altr. seg. DAB. and Z DBE = Z BCD in the altr. seg. DCB. From B draw BA J_ EF, cutting the in A; in arc. D B take any • C, andjoinAD, DC, CB. Case I.— The Z PBD = Z BAD m the altr. seg. DAB. D.l 2 H. and C. 1. 19,111.31,111 ,* EF touches the in B ; and BA J_ fromB ; '. the cen. of the is in BA, and Z ADB is a rt. Z ; 70 GRADATIONS IN EUCLID. D.3 32, II. C. Ax. 1. Suh. Ax. 3. aad .-. also /.s BAD, ABD, = a rt- ^ ; but ABF is art. /., .'. z ABF = zs BAD, ABD; tabe away /. ABD; .-, rem. Z DBF = rem. Z BAD, in the altr. seg of the ©. Case II.— TAe z. DBE = Z BCD in the altr. seg. DCB. D.l C. •.• fig. ABCD is a quadrilateral in a ; 2 22, III. .-. the opp. Zs BAD + BOD = 2 rt. Zs; 3 13,1. but'Zs DBF + DBE = 2rt. zs; 4Ax.l.5,I. .-. Zs DBF + DBE = Zs BAD + BCD, and Z DBF = Z BAD; 5 Sub. take away the equals DBF and BAD; 6 Ax. 3. .-. rem. Z DBE = rem. Z BCD in the altr. seg. of the 0. 7 Eec. '■ If a st. line touches a circle, (J-c. q. e. d. CoH. I. — Also, conversely, " if from the end, B, of a line, DB, cutting the circle, there be drawn a st. line, EF, such that the /_s F B D, E B D, which it makes with the cutting line, are equal to the Z_s BAD, BCD, in, the altr. seg. of the 0, that st. line must touch the circle." CoH. II. — If two or. more circles, ABC, DEC, FGC, touch each other, either externally or internally, and through the point ef contact, 0, two st. lines, AG, BF, be drawn, meeting their Qces, the chords AB, DE, FG, of -the intercepted arcs will be parallel. C. D.l 2 3 4 5 17, III 11,12,111. 18, III. 16, III. 32, III. 15,1. Ax.l Through C, draw H T a tangent. •.• a line joining the centres passes through C ; and •.' HT makes rt. Zs with H that line ; .-. HT is a tang, to the 0s AB C. DEC. ,^, But Z HCF = z CGFin altr. \" seg. of0FCG; ^ and Z HCF = Z TCB; .-, Z CGF= Z TCB; PROP. XXXII. BOOK III. 71 D.6 32, III. also /_ TCB = Z CAB in the altr. seg. of ABC; 7 Ax. 1. .-. z. CGF = L CAB, and they are alt. Zs; 8 27, I. .-, FG and AB are parallel. 9 8irn. So it may be shown that FG || DE. 10 Eeo. Therefore, if two or more circles, ^c. q. e. d. Cor. III. — Also, if two or more circles tmich each otherr either internally or externally, at a common • of contact, C, any line, as KG:, passing through the • q/ contact, will mt off sim. segments . from each; i. e., the segs. ACA, DCD, and GCG, are similar. Cor. IV. — In an equil a, ABC, if the sides he bisected in D,E,F, and St. lines he drawn joining the points of bisection, of those lines, two FD, FE, loill be tangents to the circle DEC, which passes through D,E, the ends of the other line, and C the angular point opp. to that line. D.l C. Sch.4,32,1. C. Ax. 1. C. Cor.32,ni. •.• D E F is an eq. lat. A ; .-. Z FDE=§ofart.Z; also •.• ABC is an eq.lat. a ; .-. ZACB=§ofart.Z; .-. Z C = Z PDE; but Z DCE is in the altr. seg. of CDE ; * .-. DF, and also EF, touch the in D & E. Cor. Y. — On the same principle, tangents, as FD, FE, through the extremities, D E, of the same chord, D E, make the angles on the same side of DE equal; i.e. Z FDE = Z FED. Cor. VI. — ^^Also, if tangents, as DF, GH, are parallel, the line J) G joining the points of contact, D,G, is a diameter; for Z FDG = Z HGD, and 29, I., each is a rt. Z ; and 19, III., the line D G, from the points of contact, D and G, passes through the centre ; /. Def. 17, I., D G is a diameter. ScH. — The 32nd Prop, is sometimes enunciated thus — " If a tangent be drawn to a circle, and from the point of contact a line be drawn cutting off an arc, the angle, between the tangent and the line cutting the circle, will be equal to an angle at the circumference of the circle standing on the arc cut off." Use and App. — The Proposition is preliminaiy to the proof of Prop. 33, and is required for the demonstration of all Theorems dependent on the equality ot the angles formed by a tangent and secant, and the anglesnn the alternate segments of the circle. 72 gradations in euclid. Prop. 33.— Prob. Upon a given straight line to describe a segment of a circle, loliich shall contain an angle equal to a give •: rectilineal angle. Sol. 10,1. 23,1. 11,1. Pst. 1 & 3. DEJf. 31, in. Def. 10, 1. Def. 15, 1. 4. 1. Cor. 16, ni. 32, m. Ax. 1. &1 2 C.l Data. Quaes. Given A B a st. line, and C a rectil. Z ; on AB to desc. a seg. of a © ■with an Z. = /. C. Case I. — Let the given angle C be a rt. Z. • D. 10,I.Pst.3. Sum.Fst.l. Sol. C. 31, III H Bis. AB in F ; and from F, with FA, desc. the semic. AHB ; Take a • H in the arc, and join HA, PIB; then AHB is the seg. required. •.• AHB is a semic. : .-. Z AHB = rt. Z C. Case II. — But if the given A G be not a rt. Z. C.l 2 3 4 D.l 23,1. 11,1. 10.11,1. Pst.l. Sol. C. 3. Def. 10, 1. 4, 1. Def. 15, 1. Sum. At A in AB make Z BAD=ZO; from A draw A E ± AD; bis. AB in F byperp. FG; and join GB; then the seg. AHB is the seg. required. V AF = FB, FG com., and 'Z AFG = Z BFG; .-. AG = GB,and0 from G, with GA, passes through B. Let this be AHB; in seg. AHB the Z = Z C. PROP. XXXm. BOOK III. D.4 5 6 7 8 C.2.Cor.lG,III. C. 32, III. C. 1. Ax. 1. Eec. •.• at A, AD is J_ to AE a diam., .*. AD is a tang.; and •." AB from the ■ of contact, A, cuts the ©, .-. L DAB = Z in the altr. sag. AHB ; bntZDAB=ZC; .-. zC=Zintheseg.AHB. •. upon the given line is described, ^c. Q. E. F. ScH. — ^Though not belonging to tliis Proposition, yet from relating to the flivision of a circle into segments bounded by carved lines, it is convenient here to introduce the following Problem : — To divide a given circle, of which the diameter is A B, into any nuraher of equal pjirts, ofwhicli the perimeters also are equal. By Use and App. 2, P. 34, I,, divide the diam. AB into the required number of eq. parts, as C, I), E, F, B ; then on one side of the diam., beginning ii-om the extremity A, desc. the semicu'cles numbered 1, 2, 3, 4, 5, &c., of which the joinCA, EA, andEB; then AC or AE =: the distance required; applied to the scale AC ^ 3'5 and AE= 19. N.B. The distance =: 19, cuts the seg. in two parts, C and E, and thus the problem has two solutions. in. Given the base AB and vertical Z ^ Z. T AB, to find the locus of the vertex. D.l 2 33, m. 21, in. Cone. On AB desc. a seg. with L = L TAB. •/ all the /js. in seg. A CDB are equal ; .". the locus of the vertex is in that seg. IV. Given the vert. /_, the base AB, und the area to construct a triangle. C.l. (41, 1.) 2' 33, m. D. 1 2 3 Pst. 1. Sol. 41,1. C. D. 1, 2. On AB const, rect. ADCB =: twice the area; and on AB a, seg. AEB with /_ = the given /. ; join EA, EB; and AEB is the req. A. ■.' A AEB = f rect. AC ; .".A AEB is of the areagiven ; and /. AEB = the vert. /_, and AB is the given base ; .*. A AEB is the A required. PROP. XXXIII. ®0OK III. 75 v. Through three given points, A, B, C, to draw three st. liises, so as to make an equilateral triangle. C. 1 4 D.l 33, m. Pst. 1. Psts. 1, 2. Sol. C. Ax. 1. 3 32, 1. D. 1. 4| Ax. 3. C. 5' Cor. 6, 1. Join A C, BC, and on each desc. a seg. wiflx an /. ^ f of a rt. /. , asseg. ADC, BEC; thvough C draw any line ED, cut- ting the 0s in D, E ; join D A, E B ; and prod, them to meet in P; then A DEP is the eq. lat. A required. ■.• in seg. ADC /_!> = % of s, rt./.; and in seg. B E C /. E = f of a rt. /. ; .'. Z. I> = Z E- 4 And-." ^sD4-E + P = 2rt./.s;and/.sD+E = ^rt. /.s; .-. /^ P = f of a rt. /.; and .-. A DBF is eq. angular ; and /. also it is equilateral. Q. e. r. VI. Given the /_ D equal to the vertical /_ofa triangle, and the base A B, to find the locus 0/ the vertex. At A, by 23, I., make the l_ BAP = /. D; and 11, 1., Z_ PAH — a rt. /. ; bisect, 10 and 1 1, 1., AB \ / S^I^B in G by the perp. GH; and from H, with rad. HA or \ t'J^''' ^.-fT HB, desc. a ABC ; the locus of the vertex will be E""^*-iQ — "^-ii. at any pointy the arc of the segment ACB. A. "' — ..^ Vn. Given AB the base, ^ ACD the vertical L, «»<' ^^ ''^ V^P- from the extremiti/ of the base A on the opp. side B C, to construct the trangle. On AB, by 33, IH., make a segment containing the giyen Z A CD j bisect AB, 10, 1., in E ; and with rad. EA or EB desc. the semicircle ADB; and from A inflect the perp. AD upon the semicircle in the point D ; BDA is a rt. /., 31, III.; and B D produced to C, and OA joined, give the triangle required. The Demonstrations may serve as Exercises. 7G GKADATIONS IN EUCLID. Pkop. 34. — Pbob. From a given circle to cut off a segment, lohich shall contain an angle equal to a given rectilineal angle. Con. 17, m. 23,1. Dem. 32, m. Ax. 1. E. 1 2 C. 1 D. 1 2 3 4 Data. Quajs. 17, III, 23, I. Sol.' C. 1. C. 2. 32, III C. 2. Cone. Given the ABC and the rectil. Z U; to cut from the a seg. with an Z = D. Draw E F a tang, in B to the ABC; at B in BE make Z FBC ^ ■□ = ZD; ^ ^ then seg. BAG contains an Z = Z D. •.• the St. line E F touches the ABC, and •.• B C is di-awn from B the • of contact ; .-. Z FBC = the Z in the altr. seg. B AC ; but Z FBC = Z D; .-.in the altr. seg. the Z =ZD. •. from the ABC, the seg. BAC is cut off, con- taining an Z = the given Z D. q. e. k. Sen. — The following method, on exactly the same principles, and having the same demonstration, gives tha Jull construction from Problems 11 and 23, Book I. A^ From a given 0, ABG (o cut off a segment which shall contain an /_ equal to a given rectil, /_ D. Draw anv Rad. CB ; and at B, 11, 1., draw EF at rt. /^s with CB; at B, the tangent-point of the rad., make the Z ^ B G = Z D ; the segment BAG con- tains an/.KAG=Z.GBF=ZI'- Use ajtd App. — I. By the last two Problems, if three observations be taken, the eccentricity uf the animal orbit of the earth and its aphelion may be found. PROP XXXV. BOOK III. It n. Also in Optics, if t>«o unequal lines, AB, BC, are given, the point jnai/ be ascertained where they vntl appear equal, that is, under equal angles. Place A B, B C so as to be contcvm. in B; on AB, and BC, within the /. ABC, construct 0s, each containing eq. p| /.s in their respective segs. ADB, BCE ; from r, the ■ of inters, of the 0s, draw FA, FB, FC ; Z_ AFB = Z. BFC, and the lines AB and BC appear equal from • F. *.■ the /_ in scg. ADB = aocrtain /_; and ".• the /. in seg. BCE = the same /_ ; .: l_ AFB = /_ BFC, and AB appears = to BC. Q. e. s. CI 2 & 3, I. 2 33, in. 3 Pst. 1. 4 Cone. D.l 2 C. C. 3 Ax. 1. Prop. 35. — Theoe. (Very Important.) If two St. lines cut one another within a circle, the rectangle con- tained hy the segments of one of them is equal to the rectangle contained hy the segments of the other. Cos. 10,1. 12,1. 1,111. Pst. 1. Dbm. Def. 15, 1. 3, m. 47, 1. Axs. 1, 2, 3, I. 36, 1. Parhns. .upon eq. bases and between the same ||s are equal. 5, II. If a straight line be divided into two equal parts, and also into ' two uneq. parts, the rect. contained by the uneq. parts, together with the square of the line between the • s of section, is equal to the square of half the line. E.l 2 Hyp. Cone. Within ABCD let AC and BD cut in E ; then AE. EC = BE -ED. There are /our Cases of this proposition, according as the intersecting lines pass throughme centre Of the or not. ^ A "\D Case I. — Let both lines, AC, BD, pass through the centre E. D.l 2 Def.15,1. 36, I. AE =EC = BE = ED; AE - IJC = BE . ED. 78 GRADATIONS IN EUCLID. Case II. — Let BD, passing through the cen., cut AC not passing through the cen. at rt. Ls in the point E. C. D.l 10,1. C.& 3,111. Bis. BD in F, the centre; joinFA. '.• BD through F cuts AC not passing through F, and at rt. Zs inE; .-. AE = EC; C. and •.• BD is div. equally in F, and uneq. in E ; 3 5, n. .-. BE . ED + EF2 = FB^, i. e. = FA^ ; 4 47, I. but AE2 + EF2 = FA« ; 5 Ax. 1. .-. BE . ED + EF« = AE^ + EF^; 6 Sub. take away the com. sq. E F, 7 Ax. 3. and rem. BE • ED = rem. AE^ ; i. e. AE • EC. Case III. — Let B D passing through the centre cut A C, not passing through the centre, but not at rt. As in the point E. C.l 2 D.l 5 47, I. 6Ax.l.Def.l5,I. 71 5, II. SL Ax. 1. 9 Sub. Ax. 3. Case IV.- through the centre. 10, 1. Bis. BD as before, & join AF. 12, 1. and from F, draw F G, perp. to AC C. 2. 3, III. •.• FG through cen. F cuts AC J.; .-. AG = GC; C. and ■.• AC is divided eq. in G and uneq. in E ; 5,11. .-. AE .EC + GE2 = AG2 or GC2; AeM. Ax. 2. Add GF^, .-. AE . EC + GE^ + GF^ = AG^ + GF2; but GE2 + GF2 = EF2, & AG2 + GF^ = AF^ ; .-. AE . EC + EF2 = AF2, i.e. = FB^; but FB2 = BE .ED + EF2; .-. AE . EC + EF2 = BE . ED + EF=; take away EF^ ; and AE • EC = BE • ED. Let neither A C nor B D pass C.l D.l i,ni.Pst.i&2. Case III. Ax. 1. Bee. Find the cen. F ; join E F ; and prod, it to meet the in H and G. . , V AE . EC = GE - EH- ^ and BE . ED=GE • EH .-. AE . EC = BE • ED. •- If two St. lines cut one another, SfC. q. e. d. PROP. XXXV. BOOK UI. 7ft OoE. — Conversely, if the rectangles he equal, contained hy two lines intersecting within a circle, the extremities of the lines, A, C, B, D, shall be in the circumference of the same circle. ScH. — 1. The Proposition is sometimes enunciated thus — " If two chords of a circle cut one another, the rectangles under their segments terminating in the point of section shall he equal." 2. The rectangles are each equal to Sad.' — E ^.'. 3. When, as in Case I., the fig. B A B is a semicircle, and A E, the ordinate, is perp. to B D, the abscissa, then the square of the ordinate is equal to the pro- duct of the abscissae, t. e. A E^ = B E • E D. Note. — The terms ordinate and abscissa may require some explanation When any two st. lines, A B, CD, in the same plane, meet in a common point O, that point may be considered as the origin of the lines from which they diverge, and the lines them- selves as axes. To know the position of any point, as P, in the plane of the axes A B, CD, we must know — 1st, between which of the angles P is, whether between ^s A O C, A O D, B O C, or /.BOD; and, 2nd, how far P is from each aadls, the distances being measured on parallels to the axes, i. e. on the sides of the parallelogram M P N. Either of the parallels ON or O M is named the abscissa, and the other parallels, P N or P M, the ordinates, the abscissa being the line cut off, the ordinate the line which determines the point of section ; the two are named co-ordinates, because together they order or deter- mine the position of the point. Thus, with respect to the point P, iiO M be the abscissa,^ P M is the ordinate. AU points in P M have the same abscissa, all in P ifr the same ordinate. It is usual to denote the length of the abscissa by the lett& X, and that of the ordinate by the letter y. Use and App. — ^I. If of two equal circles, ABC, DBE, the centres A and D he each on the crrcumfer6nce of the other, and a com. chord P GHI be drawn parallel to A D, the line joining the centres, then- the lines AF, A H, D G, D I, !F G, and H I, joining the points P, G, H, I, where the com. chord cuts the circles, and A, D, the extremities of the hne joining the centres, form parallelograms ; and if A H be produced to meet the circumferences in K and L, 'GH = HLandPI = KL. C.l 2 3 4 Pst. 2. Pst. 1, 2. Pst. 1. Cone. 1. Cone 2. Prod. AD to the 0ces in E and C ; join AH, and prod toK andL ; join AP, D G, and DI ; r then figs. ADIH and ADGP * QXQ I / S ■■ andGH = IIL,andPI=KL. R' 80 GKADATIONS IN EUCLID. D.l 2 3 4 5 6 H.Cor.2.27,in. Cor. 2. 27, III. 28, 1. Dcf. A. Sim. 34,1. 35, III. Ax. 3 & 2. Cone. •.• FI II EC ; .*. arc FE = arc DH = arc IC ; and .■. also Z. IDC = Z HAD ; now AH II DI ; .-. fig. AD HI is a / — 7 ; and in like manner fig. AD GF is a / 7 . HenceHI = AD = rG; but GH . HI = AH . HL ; and HI = AH .-. GH = HL ; and FI = 2 AD + GH ; also 2 AD + GH = KH + HL = KL, .•. FI = KL. Q. E.D. 2. By this Prop, we arrive at a practical way of findinrj a linii which is the fourth proportional to three given lines, or tlie third propon'tional to two given lines. Ex. 1. Let there be three lines, AC ^ 2 eq. pts.. B C = 3, and CD = 4; I'equired a fourth line in proportion to the other throe. C.l Pst. 2. 2 3,1. 3 3, 1. Pst. 2. 4 Use 9, in. 5 Sol. D.l C] 2 35, HI. From any • B draw an indef. lino ; on it set the 2nd and 3rd terms, BC = 3, and CD = 4j from C set CA = 2 ; and prod. AC indef.; through the three • s. A, B, D, dose, a _, cutting A C produced in E ; Jj\ the distance CE is the 4th proportional ; and measured, CE = 6 eq. pts. ■.• within a the lines BD and AE cut each other. .-. BC • CD = AC • CE, and CE is the 4tU proportional Jf of the four segments any three he qiven, the fourth mow be found: for BC.CD_„„.EC.CD_ ACCE _;. , AC . CE ^_„ —Air -^^' ^CE- -^^'— BC- = ^^' ''"'1 -C"D"~'=^^- Ex. 2. Next, let there he two lines, AB = 3 cq. pts. and BC = 6; required a third line in proportion to the other two. C.l 2 3 D.l 2 11, 1. Pst.2&31 Use 9, HL Sol. C. 35, m. Set the two lines so as to fonn a rt. Z at B ; prod. AB, CB indef. ; and make BD = BC i through the three • s, C, A, D desc. a cutting AB in E; the dist. B E is the 3rd proportional j and measured, BE =: 12 eq. pts. , •.• CD and AE, within » 0, cut each other in B ; .-. BC . BD, or BC^ = AB • BE, and BE is the- 3rd proporortional. If of the three segments any two be given, the third may be found; for, VaB . be = B C; J§° = BE; and ^ = E A. PROP. XXXVI. BOOK III. ^1 Pkop. 36. — Thecr. (Important.) If ffom any point witkout a circle two st. Ipies be drawn, one of which cuts the circle, and the other touches it; the rectangle con' tained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it. Cojsr. 10, 1. 1, nr. 12, 1. Dem. 18, ni. 47, 1. Axs. 1, 2, 3, 1. 3, HI. 17, IH. 6, n. If a St. line be bisd. and produced to any point ; the root, contained by the whole line thus produced, and the pt. of it produced, together with the square of half the line bisd., is equal to the square of the St. line which is made up of the half and the part produced. E.l 2 Hyp. 1 » 2 Cone. Let D be any • -wittout ABC; and DC A, DB two st. lines from • D, of which D C A cuts and D B touches the ; then AD . DC = DB^ Beme. — Of this Proposition there are two Cases, according as, D C A, the cutting line, passes through the centre of the circle, or not. C. 1 2 D. 1 Case I. — Let D G A pass through 10, I. Pst. 1. H. & C. 2 18, III. C. 1. 6, II. Def. 15, I. D.2&47.I. Ax. 1. Sub. Ax. 3. Bis. A C in E, and E is the cen, of ABC; join BE. centre of the ABC. B •.• D B is a tang., and B E a line jj from E to B; .-. Z EBDisart. /_. And".' AC is bisd. in E and prod, to D; .-. AD . DC + E02 = ED2: but EC = EB; ,.-. AD . DC + EB2 = ED2; and EBD being a rt. Z, .: ED2 = EB^ + BD=; .-. AD . DC + EB2 = EB2, + BD^; take away EB2; and AD . DC = DB2, . i. e. = the square on the tang. 82 GRADATIONS IST EUCLID. Case II. — Let DC A not pass through the centre of the ABC. C.l 2 D.l 2 9 10 11 12 13 14 1, III. & 12, I. Pst. 1. H. & 18, III. C.l. 3, III. D. 3. 6, III. Add. Ax. 2. D. 1. & 47, I. D. 1. & 47, I. D. 6. 7. 8. Ax. 1. Def. 15. I. H. & 47, I. Sub. Ax. 3. Kec. Find the cen. E, and draw EFperp. to AC; andjoinEB, EC, ED. •.• as before, Z EBDis a rt. Z ; and •.• EF, through the cen., cuts at rt. /_s AC, -"i'-- not through the cen. ; .-. EF bisects AC, 2.e. AF = FC: A> And •.• AC is bisd. in F, and prod, to D; .-. AD . DC + PC2 =.FD2; to each equal add F E^ ; .-. AD . DC + FC2 + PE2 = FD2 + FE«: but EFD being a rt. Z, .-. BD2 = DF2 + FE2; andEC^ = CF^ + FE^; .-. AD . DO + EC2 = ED2; but CE = EB, .-. AD . DC + EB2=ED»; and E B D being a rt. Z , .-. AD - DC + EB2 = EB2 + BD^j take away EB^, .-. AD • DC = BD^; i. e. = the square on tang. ■. If from any point without a circte, ^o. Q. B. D. CoR I. — If from any point A, without a circle, BDC, there le drawn two st. lines, A B, AC, cutting it in E and F ; then the rectangles contained hy the whole lines and the parts of them without the circle shall he equal to one another; j. e. BA - AE = CA • AF. C.l 17, III. From AD draw a tang, to the 0. D, D.l . 2 36, III. Ax. 1. ,-. BA . AE = AD2, and CA - AP = AD^? •. BA . AE = CA . AF. PROP. XXXVI. BOOK III. 83 Cor. II. — If from the same point, A, two tangents, AD, AG, he drawn to the same circle, they are equal ; for the squares on them are each equal to the same reotangle. ScH. — Prop. 36 may also be enunciated thus ; " If any chord of a circle he produced to cut a tangent to the same circte, the square of the tangent shall bo equal to the rectangle under the segments of the chord so produced." Use AND App. — I. In any semicircle, ABDC, if from A, B, the ex tremities of the diameter, chords be drawn,' AD, BC, intersecting within the semicircle, the sum of the rectangles formed by any two of such chords, AD, BC, intersecting within it, and by the sections of the chords, AP, B P, between the extremities of the diameter AB and the intersecting point P, shall equal the square on the diameter, AB. C.l D.l 2 3 4 12,1. Use 9, m, Cor.36,in. Cor.36,ni 2,n. Cone. From P draw PE perp. to AB ; and draw 0s through A, P, E, and B, P, E. Now in O APE, AB . BE = CB . BP ; and in BPE, BA • AB = DA • AP; Also AB2 = AB . BE + B A . AE ; .-, AB- = CB . BP + DA . AP. n. Erom this Prop, is deduced the Art of taking a hue Level on the earth's surface. 1°. In tlie more considerable problems of Levelling, it is usual to employ a telescopic sight, and staves with sliding vanes, the distances and observations being entered in a field-book. Thus BACK SIGHTS. station. Distance. Height. Corrected. 1 1420 links lft.5in. lft.4Jin. 2 2030 ,, 6 2 6 l| FOKE SIGHTS. Distance. Height. Corrected. 2448 links. 6ft. Sin. 6ft. Tin. 2870 „ 7 9 7 7| 7 5f 14 2f Then 14ft. 2fin. minus 7it SJin = 6ft. 9in. the height of A above B. 84 GkADATIONS in EUCLID. 2° In measuring an Ascent, AD, the perpendicular and horizontal lines may be found by means of a spirit level and a staff ; For a^ + r D = AB ; and Ao + j3r ='BT>. 3°. But if the horizontal distance shotdd exceed 100 yards, it will be necessary to correct it for the curvature; the horizontal line AB being a tangent, whilst the line of the true level, that of the earth's curvature, AL, is at every point equally distant from the earth's centre. The deviation of the horizontal fi-om the true level = B L ; A 1^2 irnd, by 36, HI., BE • BL= AB=,.-.BL = ^=r jiiii. N.B. The Greek letters of the diagram a, r/, have b een changed to a, r. Example, — A fountain, B, one mile distant trom A, is observed from A, to be on the same horizontal level with the point A ; how much is B above A ? J. e. how much is B farther from the earth's centre C, than tlie points A or L ? CL or C A being 3956 miles. HereBC — LC = BLj For, by 47, 1, BC = v'3956" + 1^ = ;^15649937 = 3956-0012639, Then, BL = 3956-00012639 minus 3956 = -00012639 of a mile = 8-00808 inches. 4°. By following the same method it will be found that the deviation, of the horizontal fi-om the ti-ue level, — Por 1 mile = 8 inches ; 2 miles = 32 inches ; 3 miles = 6 feet ; and for 4 miles = 10-6 feet, &c. Or, — Two-thirds of the square of the horizontal level in miles gives the deviation in feet. 5°. The distance within which an object may be seen at sea, or the dis- tance of the boundary of the horizon from the spectator is ascertained on the same principle ; for the rad. of the horizon A B = ;^BK X BL. Ex. — The Peak of Teneriffe, BL, is 2-5 miles above the sea level ; what will be the radius of its horizon, or the greatest distance fi-om which it is visible ? > Here, B E = 7912 + 2-5 = 7914-5 m iles .•. AB = V7914-5 X 2-5 = v'19786-25 = 140-66 miles. 6°. From an elevation, as the top-mast of a ship, an object will be soon at a greater distance -, in this case the sum of the horizons of the object and of the elevation will give the whole horizon ; ;. e. AB + AS =: SB. Ex. — The top-mast of a ship, BS, is 100 feet above the surface of the sea ; at what distance to a spectator standing there will the summit of Teneriffe be visible ? Here SF = 7912 + -01893 = 79 12-01893 miles. /, AS = v'7912-01893 X -01893 = v'149'7745183449 = 12-23824 miles. As above AB = 140-66, /. SB = 140-66 -|- 12-23824 = 152-89824 miles rad. of the whole horizon. PROP. XXXVII. BOOK III. 85 7°. The earth's diameter may be ascertained from Ivnowing the horizon AB, AB'' and the height, BL, of an object ; for LE = BE — BL, and BE = -^^^ ' Ex. — ^From the summit of Teneriffe the radius, AB, of the horizon is 140'66 miles, and the height of BL above the sea level is 2-5 miles ; required tixe eai'th's diameter. Here BE = 140-66 X 1-0-66 19785-2356 = 7914-098 ; 2-5 2-5 And LE ^ 7914-098 minus 2-5 = 7911-598 miles, earth's diameter. Prop. 37.— Theoe.. If from a point without a circle there be drawn two lines, one of which cuts the circle, and the other meets it ; if the rectangle con- tained by the whole line which cuts the circle, and the part of it without the circle, be .equal to the square of the line which meets it^ the line which meets shall touch the circle. Con. 17, IH. 1, IH. Pst. 1. Dem. 18, HI. 36, in. Ax. 1. Def. 15, 1. Cor. 16, HI. 8, 1. If two As have the three sides of the one respectively equal to the three sides of the other ; then these triangles shall be equal in every respect. E.l Hyp. 1. Let D be any • without the ABC; 2 „ 2. and from D two st. lines be drawn, of which DC A cuts the , and DB meets it, / so that AD . DC = DB^; b/ 3 Cone. then DB shall be a tang, p.? to ABC. \' C.l 17, III. Draw D E a tang, in E to \^ 0ABC; A 2 1, III. find F the cen. of the ABC; 3 Pst. 1. and join PB, ED and PE. 86 GRADATIONS IN KUOLID. D.l 2 S 4 5 6 7 8 9 10 c. 1 & 18, in. H. 2 & C. 1 36, III. H. & Ax. 1. Def. 15,1. D. 4. 8,1. D. 1. Ax. 1. 0. Cor. 16, III. Eec. - •/ DE is a tang. ; .-. /. FED is a rt ^ : and •.' D C A cuts, and D E touches the ; .-. AD . DC = DW; but AD . DC = DB2, .-. DE2 = DBS, and DE = DB. Also •.• FE = FB, FD com. and DE = DB .-. in AS DEF, DBF, £ DEF = z DBF but DEF is a rt. A', .'. L DBF is a rt. /_ thus BD is J_ to the rad. FB, at its extr. B .•. D B touches the at B. . If from any 'point without a circle, ifc. Q. E. D. CoR. — Taxigents, as D B, D E, from the same point, D, without a circle are equal. D.l 36, III. V DB2 = AD . DC and DE^ also = AD . DC; 2 Ax. 1. .-. DB2 = DE2, and DB = DE. Use and App.— I. By Uie 36th and 37th Propositions the Problem is solved, " through two given pointt, A,B, to describe a circle touching a given circle CDB." C. 1 2 3 4 5 6 Assum. Use 9, m. Pst. 1. 2. 17, HI. 25, in. Sol. In CD E assume any point T) of the 0ce ; and through, A, B, D desc. a cutting CDE in D, C ; join D C, and prod. AB, D C to meet in F ; from F draw FG, FG' tangs, to CDE; and desc. a through • s A, B, G ; thea the ABH triH touch CDE in G.' 87 D.I C.4&36,ra. 2 H. » Om.-. 36, in. 4 Ax. 1. 5 Bemk. 6 Cone. '.■FGistang. to0CDE; .-, FG^ = I'D • FC; but A, B, C, D iire all in the ©ce of ABCD; .-. FD . FC = FA . FB. .-. alsoFA •FB = F(P; and.-.FGig tang.of ABH ; Now F G is a tang, common to both 0s ; .•, the 0s ABH, CDE touch in the • G. n. The last three Propositions, 35, 36, and 37, are amongst important in Plane Geometiy. It was by their aid that Mauholico, of Messina, in the sixteenth century, calculated tlie diameter of the earth ; for by the method which has been shown, having ascertained AD the vertical height of R a mountain ; the £_ BAG made by the vertical line, and /\. AB the line fi-om the summit A to B Ihe boundaiy of sight ; he found the length of AB, by Trigonometry ; then •.• AB" = AE . AD, .-. AE = -£j= and AE — AD = BD the diameter of the eai-th. EEMAEKB. 1. lu classifying tte Propositions of tlie Third Book, it will be iisefiil to cQosider tli«m under five general lieads ; — • 1° "the propositions which relate to the Centre of a circle; 1 — 15 and 20. 2°. To the Tangents of a circle ; 16—19. 3°. To the Segments of circles ; 21 — 25. 4°. To the Angles in eii'cles, or in their segments ; 26 — 34. 5°. To the Equality of the rectangles contained by the segments of lines intersecting each other within, or without the circle; 35-3". 2. Of the 37 Propositions which are found in this book only six are Problems, namely : — Pr. 1. To find the Centre of a circle. Pr. 17. To draw a Tangent to and from a circle. Pr. 25. To complete the circle of which a Segment is given. Pr. 30. To bisect a given Circumference. 88 GRADATIONS IN EUCLID. Pr. 33. On a given line to draw the segment of a circle containing an angle of a given magnitude ; and Pr. 34. From a circle to cut off a segment which shall contain an angle of a given magnitude. 3. From the Properties of Plane Figures demonstrated in this and in the preceding books, various other problems, however, jnay be deduced; as 1. To draw a circle through three given points not in a st. line i. Use 9, and 23, III. 2. To describe an oval on any given major axis, or a spiral with the radius of the eye given j Use 11, IH. 3. To join two given points by a serpentine line, or cima recta; Use 12, III. 4. To draw a tangent to each of two given circles ; Use 18, III. 5. To draw the arc of a chord without knowing the centre of the circle ; Use 21, HI. • 6. To reduce curve-lined figuies to rectilineal of equal areas ; Use 24, m. 7. Through a given point to di-aw a line parallel to another line ; Use 27, in. 8. To trisect a quadrant ; Use 30, III. 9. Prom a point in a line, or from the extremity of a line to raise a per- pendicular 5 and from a point without a line to drop a perpen- dicular ; Use 31, III. 10. Given the base and vertical /. of a A to find the locus of the vertex; given the vertical angle, the base, and the area of a A to con- struct the triangle ; and given any three points not in a st. hne to describe through them an equilateral triangle ; Use 33, III. 11. To determine the point where two unequal lines wiU appear equal,, i. e. under the same angle ; Use 34, HI. ) 2. To find a line proportional to two given lines ; and also a line pro- portional to three given lines ; Use 35, HI. 4. Were it required in a work like the present, problems might be -introduced which show how circles may be drawn which are taii'Tents to two or three given circles, or to two st. lines and a circle ■ or to two circles and a st. line, &c. ; but for these reference may be made to " Geometry, Plane, Solid, and Spherical," JBook III., § 8. EEMAEKS. BOOK III. 89 5. Several of tlie principles on -wMcli the Levelling and Sur- veying of Land, and Geographical and 'Astronomical Observations depend, have been given in the Third Book, — such as the Methods : — 1. Of computing tlie distances and heights of objects when situated on the verge of the natural horizon ; Use 1 6, III. 2. Of determining the part of a globe which may be enlightened by a ' luminous body, as by a meteor, volcano, lighthouse, &c. ; of ex- plaining the theory of the phases of the moon ; of ascertaining the distance of the sun ; and of obtaining the dip of the horizon ; Use 19, ni. 3. Of constructing a figure representative of the distance of the place of observatioii from an object ; and of tracing the are of a circle for giving a spherical figure to optical glasses ; Use 21, III. 4. Of finding the true centre of an imperfectly constructed theodolite, or similar instrument ; Use 26, III. 5. In Coast Surveying, for noting soundings, bearings, &c.; Use 33, III. 6. For taking a true level on the earth's surface j Use 36, III.; and 7. For calculating the earth's diameter ; Use 37, III. These are but Examples of the many useful purposes to which geometrical science may be aijplied ; and they may serve to redeem geometry from the prejudiced objection that it is a system of theoretical reasoning without practical results. The practical results are really most important, and in the actual business and occxipations of life are an everyday's demand. GEADATIONS IN EUCLID. BOOK IV. CONTAINING THE METHODS 0¥ CONSTRUCTING HEGDLAK STRAIGHT- LINED FIGURES IN AND ABOUT A CIRCLE, AND CIRCLES IN AND ABOUT REGULAR STRAIGHT-LINED FIGURES. Exeeptiag Prop. A, Tbeor, tie fourtii book of Euclid's Plane Geometry consists entirely of Problems ; it is in fact tbe Application of the third book to the purposes of inscribing and circumscribing triangles and other regular straight-lined figures in and about circles, or circles in aaad about such regular figures. The former books supplied the means of drawing regular plane figures of 3, 4, 5, and 15 sides ; and by continued bisections of making them of 6, 12, 24, &c., or 8, 16, 32, &c.; or 10, 20, 30, &c., sides. The employment of those means constitutes the object of the book on which we are now entering. In Trigonometry, Astronomy, and the various departments of Civil and MiEtary Engineering, the fourth book is found of essential service ; we also deduce from it the method of obtaining, with suffi- cient exactness, the quadrature of the circle, and of proving that circles are to one another in the proportion of the squares of their diameters. GRADATIONS IN EUCLID. Definitions. 1. A rectilineal figure is said to be inscribed in another recti- lineal figure when the angular points of the . inscribed figure touch the sides of the figure in ^ •which it is inscribed, each upon each. Thus, the fig. ABCD is inscribed in the fig. EFGH. 2. In like manner, a figure is said to be pL described about another figure when the sides of the circumscribed figure touch the angular points of the figure about which it is described, each upon each ; Thus the fig. EFGH is circumscribed about the fig. ABCB. It is noteworthy that BucuD gives no example of one rectilineal figure being inscribed in another rectilineal figm-e, or circumscribed about it. 3. A rectilineal figure is said to be inscribed in a circle when each angular point of the inscribed figure touches the circumference of the circle ; Thus, the qu. lat. ACBD is inscribed in the circle ADBC. E 4. A rectilineal figm-e is said to be described about a circle when each side of the circumscribed figure touches the circumference of the circle ; Thus, the qu. lat. EFGH is described about the circle ABCD. p 5. In like manner, a circle is said to be inscribed in a rectilineal figure when the circumference of the circle touches each side of the figure ; Thus, the circle ABCD is inscribed in the quadrilateral EFGH. 6. A circle is said to be described about a rectilineal figure when the circumference of a circle touches each corner of the figure about which it is described ; Thus, the circle ABCD is described about the figure ADBC. DEFINITIONS. BOOK IV. 93 7. A straight line is said to be fitted exactly into a circle, or to be applied in it, when the extremities of it are on the circum- ference of the circle ; Thus, the lines AC'and AD are applied to the cirelc ABCD. Definitions Additional to those of Euclid. 8. A circle is said to be exscribed to a triangle when, having for centre the point of intersection of any two straight lines that bisect . . the exterior angles of the triangle, ^ ^\\*« /^ the circle touches a side of that triangle. Thus, EFG from cen. D, (whore the lines CD, BD, inter- sect, which divide the ext. /.s CBK, BCL, each into two equal parts), and touching in IT the side B C of the A AB C, is exscribed to that triangle ; as also are the 0s from the centres H and I. 9. " Any rectilineal figure, of five sides and angles, is called a pentagon ; of six sides and angles, a hexagon ; of seven sides and angles, a heptagon; of eight sides and angles, an octagon; of nine sides and angles, a nonagon , of ten sides and angles, a decagon; of twelve sides and angles, a duodecagon ; of fifteen sides and angles, a quindecagon," &c. 10. " These figures are included rmder the general name of polygons ; and are called equilateral when their sides are equal ; and equiangular when their angles are equal. Also, when both, their sides and angles are equal they are called regular polygons." Potts' Euclid, p. 124. N.B. — The force of the propositions in Simpson's Edition is often lessened by not rendering the Greek original into English corresponding, as far as differences of idiom will admit, more closely with Euclid's text. To avoid this, Galbraith and Haughton's rendering of the general enunciation is often followed, though they have not been so thoroughly exact as is desirable. PEOPOSITIONS. Prop. 1. — Prob. Into a given circle to fit exactly a right line equal to a given right line, which is not greater than tJie diameter of tlie circle. Sol. 1, III. To find the centre of a given circle. 3, 1. From the greater of two given lines to cut oS a part equal to the less. Pst. 3 and 1. A circle may he described fi-om any centre at any dis- tance fi'om that centre. A st. line may he drawn from any one point to any other point. Dem. 15,1. A circle is a plane figure contained by one line, which is called the circumference, and is such that all st. lines drawn from a certain point within the figure to the circumference, are equal to one another. Ax. 1. Things which are equal to the same thing, arc equal to one another. Def. 7, IV. A St. line is said to be fitted exactly into a circle, or to he applied in it, when the extremities of it are on the circumference of the circle. E.l 2 3 C.l Dat. 1. „ 2. 1, III. Sup. H. 3,1. Pst. 3 & 1. Sol. Let ABC be the given©; and D the st. line > C B diam. of© ABC; in ABC to place st. line =: D. Find E the cen. of ABO, and draw any diam.,B C, through it ; if B C = D, the required thing is done ; hut if not, and B C is > D ; from CB cut oJBfCP = D; and from C, with OF, deso. © GFA, join C A; then C A is the line required. and PROP. II. BOOK IV. 95 D.l 2 3 C.5.Def.l5,I C. i. Ax. 1. Def. 7, IV. V C is cen. of GPA ,■, C A = CF ; but OF = D, .-. D = CA. •. in ABC, a st. line has been placed, CA = the given st. line D. q. e. p. Use and Am*. — I. Within a given Q, ABC, to place a line of a given length, D, not greater than the diam. of the given Q, which line shaltpass through A, a given point in the Qce. C.l 2 3 4 5 D.l 2 1,IV. i,in. 12,1. Pst. 3. 17, nr. Pst. 2. SoL Def.l5,I.&14,in. C. 1. Ax. 1. In the ABC place BC ^ D the given Ime ; find cen. of ABC, and draw O P perp. to B C ; F A—- with OP, from cen. 0, draw „ 0PGH; -K from A draw AH tang, to P GH, and prod. AH to _ cut ABd in E; and A join OH ; I) — then AE = B C = 1). . V OH = OP; .-. AE = BC; butBC = D; .-. AE=:D. Q. E. r. n. To draw that diam. of aQ which shall pass at a given distance, N, from a given point A C.l 2 D.l 2 Pst. 3. & 1,IV. Sol. Def. 16,1. &C. Cone. With OA = N, desc. a ABC, and in it place AK= N ; then K O produced to L is the diam. required. •.' KL is a diam., and AK = AO = TS ; .•. KL, a diam., passes at the given distance from A. Prop. 2. — Pkob. In a given circle to inscribe a triangle equiangular to a given triangle. Coif. 17, ni. To draw a st. line from a given point, either without or in the circumference, which shall touch a given circle. 23, 1. At a given point in a given line to make a rectU. Z. equal to a given rectil. /_. Pst. 1. DlSM. 32, Hf. If a St. line touches a 0, and from the point of contact a st. line be drawn cutting the circle, the /.s which this line makes with the line touching the 0, shall be equal to the /_s which are in the altr. segs. of the 0. Ax. 1. 96 GRADATIONS IN EUCLID. E.l C.l D.l 2 3 4 5 6 Cor. 3, 82, 1. If two As have two Z.S of the one respectively equal to two /.s of the other, then the third /. of the one shall be equal to the third l_ of the other. Def. 3, IV. A rectil. figure is said to be inscribed in a circle when each angular point of the inscribed figure touches the 0ce of the cirelo. Data. Quaes. 17, III. 23, I. 23,1. Pst. 1. Sol. C. 1, 2, 32, III. C. 2.Ax. 1, Sim. Cor. 3. 32,1 Dcf. 3, IV. Let ABC be the given©, and DEF the given a ; ill A B C to insc. a A eq. ang. to A DEF. To any point. A, in p.^. the 0ce, draw a tang. GAH; at A, in AH, make L HAG = L DEF; and at A, in AH, make Z GAB = L DFE ; join B G ; then A A B G is the A required. •.' HAG is a tang., and AC from A cuts the ; .•. Z. HAC = Z ABC in the alternate seg. ; hut L HAG = L DEF ; .-. L ABC = L DEF; In like manner /_ ACB = L DFE; .-. rem. /_ BAG = rem. Z EDP; Hence a ABC is eq. ang. with A EDF, and is inscribed in the 0. q. e. f. ScH. — The Analysis of a problem is a very usofal exercise, and, that the learner may become accustomed to the method, some examples will be given. Thus, of Prop. 2, setting out with the admission that the A ABC has its angles respectively equal to the Zs D, E, F, the Analysis will be- Through the point A draw A GH, a tangent to the ; then, •.• Z CAH = Z -A-BC = Z B, ,*. the line AC is given in position ; and, being cut by the 0ce, the point C is given. In the same way it will appear that the point B also is given 5 and •.■ the three points, A, B, C, are given, /. their junction forms A AB C, inscribed in the circle. Use and App. — An eq. lat. A ABC, being inscribed in a circle, and through the angular points A, B, C, tangents, DE, EF, FD, being drawn, these tangents will also form an eq. lat. A, DEF, the area of which is four times that of the inscribed eq. lat. A. PROP. III.— BOOK IV. 97 C. 1 2 D.i 2 6 7 8 9 10 H 12 13 14 15 16 17 i,ni. Pst. 1. 18, in. 47,1. Def. 15, 1. D. 3. and H. 4,1. 26,1. 3,m. Sim. Cor. 5, 1.32, 1. Sim. Cor. 6, 1. Sim. Cone. D. 11, 12. Cone. Knd O the cen. of the ^■ given eircle, and join OA, OB, OC, OD, OE, OF. '.• Z.SOBD, OCD, are rt. Z.S, .-. 0D= = 0B= + BD» = 0C= + CD»; but OB = OC, .-.BD* = CD»andBD = DC; and V 0B= OC, BD = DC, and / OBD = ZOCD; .-. A OBD = A OCD, Z. BOD = Z. COD, andZ.BDO= Z CDO; I. e. Z.S B C and BDC are each bisected by DO; also DO bisects the line BC; .', DO bis. BC at rt. ^s, and passes through the vertex A, So, EO bis. AC, and passes to the vertex B; and FO bis. AB, and passes to the vertex C. Now Z. OBC = J of a rt. Z. ; .'. Z JDBC = f of a rt./ j So Z.S DCB, BDCj each =f of a rt. £, .•. A BD C is eq. lat. and = A ABC; So As ACE, ABE are each eq. lat. and =: A ABC; .-. ADEE = 4 A ABC. AlsoDE = 2DC = 2BC; EF = 2 AE = 2 AC= 2BC; andED = 2FB=:2 AB = 2BC; .". A DEE is equilateral. q. e. f. Peop. S.-^Prob. About a given circle to draumscrihe a triangle equiangular to a given triangle. Sot. Pst. 2. 1, HI. 23, 1. 17, lit. Dem. 18, III. If a St. line touches a 0, the St. line drawn from the centre to the point of contact, shall be perpendicular to the line touching the circle. Cor. 1. 32, 1. All the interior Z.s of any rectil. fig., together with four rt. /.s, are equal to twice as many rt. 2.S as the figure has sjdes. Ax. 3, 1. If equals b^ taken from equals the remainders are equal. 13, I. The Z.S which one St. line makes with another upon one si4e of ■ it are either rt. /.s, or together equal to two rt. /.s. Ax. 1. Def. 4, IV. A rectil. fig. is said to be described about a 0, when each side of the circumscribed fig. touches the 0ee of the Q, H 98 GRADATIONS IN EUCUD. E4 C.l 2 3 Data. Qua»s. Pst. 2. l.III.Pst.l 23,1. 17, III. Sol. Let ABC be the given0,&DEF the given A ; about the ABC to desc. a a eq. ang. to the A DEF. Produce EF both ways to Gr, H ; D.l 2 .._^ ,„, _ B N find K tiie cen.'of ABC, and draw KB; at K in BK make L BKA = L DEG, and Z BKC = L DPH; and through A,B,C, draw LM, MN, NL, tangs. to ABC; then A LMN shall be the A required. C. 4. •.• LM, MN, NL, are tangs to ABC; C. 2, 3. and KA, KB, KC, lines from the cen. to A,B,C ; 18, III. /. the Zs at A, B, and C are all rt. Zs. Cor.1,32,1. And v the 4 Zs of the qu. lat. AMBK = 4rt. Zs; D 2. and two of the four, RAM, KB M, are rt. Zs; Ax. 3. .-. the other two Zs, AKB, AMB = 2 rt. zs ; 13, L but ZsDEG, DEF = 2rt. zs; Ax. 1. .-. zs AKB, AMB, = ZsDEG, DEF; 9 C.3. Ax.3. and Z AKB = L DEG, .-, Z AMB=: Z DEF. 10 Sim. In like manner Z LNM = z DFE ; 11 32,I.Ax.3. .-. rem. Z MLN = rem. Z EDF; 12 Def.4.IV. :. a LMN is eq. ang. with A DEF; and is de- scribed about the ABC. q. e. f. Sea. — Analysis ! We suppose tho proUem to have teen solved, the A LMN being described about the given ABC, so that Z L = Z. I*' Z M = Z E, and Z N = Z F. Join K the cen. of the to the tang, points A,B, C. In the qu. lat. BKCN, the four Zs = four rt. Zs ; and '.• Zs KBN, KCN, arc 2 rt. Zs ; .: ZsBKC, BNC = 2 rt. ^s. But Z ^ being given, its supplement Z BKC is also given ; consequently, KB and KC, the two radii, are given in position. Thus, it may be shown that the Z jS-KB is given, and the line KA given in position. The inters. ofKA, KB, KC, with the 0cc, or the points A, B, C, are given; .•, the tangs. MN, NL, LM, are given in position. Thus the A LMN is given. 99 Peop. 4.— Pros. To inscribe a circle in a given triangle. Con. 9, 1. To bisect a given tectil. /., i. e. to divide it into two equal parts 12, I. To draw a perp. to a given.st. line of unlimited lengtli from a given point without it. Pst. 3. Dem. Ax. 11. All rt. /_s are equal to one another. 26,. I. If two As have two /.s of the one equal to two /.s of the other, each to each, and one side equal to one Side ; then shall the other sidts be* equal, each to each, and also the third /_ of the one to the third /. of the other. Ax. 1. , 9,, ni. If a point be taJien within a circle, A'om which there fall more than two equal st. lines to the 0ce, that point is the cen. of the 0. Cor. 1, 16, m. If a st. line be drawn at rt. /.s to any diam. of a 0, from its exti'emity, it shaU touch the at the extremity ; and a st. line touching the at one point shall touch it at no other point. Def. 5, IV. A is said to be inscribed in a rpetil. flg. when the 0ee .of the touches each side of the figure. E.l 2 C.l 2 3 4 D.l Datum. QuiES. 9,1. 12,1. Pst. 3. Sol. C. 1,2. LetABObethegivenA; in it to inscribe a 0. Bis. Za ABC,ACB,by BD, CD,meetinginD; fromD di-awDE, DF, DG, perps. to AB, BC, ,CA. with DF as rad. draw a EFG ; then the EFG is' the reqtdred, .• zEBD = z FED, ^DEi^ = z DPB, and BD com.; 2 26, I. .-. DE = DP; 3 Sim. Ax. 1. So DG = DF; .-. DE = DG = DP; 4 9, III. /. thepoints E, F, G, are in the 0cfe of the ©; 5 C. 2. And '.* the Zs at E, F, G, are rt. Zs; 6 Cor. 1.1 6, III. ..-. AB, BC, C A touch the EFG, Def. 5. IV". and .', EFG is inscr. in A ABC. Q. E. V. ScH. — 1. In other words, the 4th Fro^. ii-r'To describe aQ which slwll touch three giben st. lines not parallels. ' ' , 2. In the Analysi^ of this problem we asgiima thjitthe © EP G has been inscribed in the A AB C. 100 GRADATIONS IN KUCLin. The cen. of the being D, join D A, DB, DC. Now •.• DE = DF, DB, com. to As BFD, BED, and the £s at E and F ai-e equal, being rt. ^s ; .-. A BFD = A BED, Z. DBF = I, DBE, and thus BD is given in position. By a similar argument it may be shown that D A and D C arc given in position : — .*. their point of int<;vs., D, is also given. Use AM) App.— I. Connected with this problem is the following theorem : If three £s. A, B, C, of a A ABC be bisected by at. lines, BD, CD, AD, tluae lines will intersect in the same point. C. D.l 2 3 4 12,1. C. and H. 26,1. Sim. Ax. I. Cone. From D, the inters, of BD and CD, draw' DE, DF, DG, perps. to AB, BC, and CA. •.• /.s DFC, DCF = /_s DGC, DOG, and DC is com. ; .-, FD = GD. SoED = GD; .-. ED = FD; .'. the lines BD, CD, AD, have a com. point of inters. D. II. An expression for the Area of a Triangle, and for the Radius of the inscribed circle may be deduced from this theorem. First. — For the Area of the Triangle, the sides AB = a, BC = A, and CA = c, and the radius, DF = r, of the inscribed being given i The A ABC = AS ABD + BDC + CDA; the area of ABD = a x ^ ; of BDC= 6 x ^and of CDAxzc X — ' .-. the area of A AB C = (a + 6 + c) ^• Second. — For the Radius of the inscribed circle, the area and sides being given ; 2 area of A ~ a + b + c • Ex. 1. The sides of a A are 52, 5C, and 60 yards, and the rad. of the inscribed 16 yards, required the area. Here (52 + 56 + 60) x -^ = 168 X 8 = 1,344 .square yards. Ex. 2. The area of a a is 216 square feet, and the sides 18, 24, and 80 feet, required the radius of the inscribed circle. „ 2 X 216 _ 432 _ „ . ^'''' 18 + 24 + 30 - 12- = « «=<='• nX The Corollaries to Prop. 4, given by Galbraith ttlld Haughton, in their "Manual of Euclid," Bk. IV., pp. 6, 7, open up an interesting subject of inquiry, namely— The Properties of Circles exseribed to a Triangle. From those Corollaries we select the following, referring to Def. 8, IV, :•— rnop. IV. — BOOK iv. '101 l-.The bisectors of any internal /_, as DA of / BAG, and pf the remaining two ext. Zs, as DB and DC of /s CBK, BCL, interBect" in the same pomt, D, which is the cen. of the (3 EPG, touching the side, BC, opposite the given /. BAG and the productions, BK, GL, of the two sides AB, AC, containing it. 2. The radius, as BF, of the ex- scribed 0, EFG, may be found in numbers, by dividing the area of the. A ABC, by half the difference be-, tween the sum of the two sides and the base ; because the whole triangle is the difference between the sum of the two tiiangles ACD and ABD, whose bases are the sides AB, AC, and common altitude the radius of the exscribed circle, and the A BCD, whose base is the third side B C, and altitude the same radius D F. 3. From 13, II., a Fonnnla is deduced for calculating the .<4;'ca ()/■« Triangle in the terms of the sides ; 2s being the perimeter or sum of the sideSj and s tlie semi-perimeter, and a, h, c, the sides of the triangle : — Area of A =«/« (« — o) (« — 6) (« — c). 4. By aid of this Formula and Cor. 2, given above, we can readily express the radii of the three exscribed circles ; r denoting the radius of the inscribed circle, and r', r', r", the radii of the circles exscribed to the sides opposite A, B, C, respectively ; Then, r" =^ > r" =V and by Use 2, P. 4, IV. r: o-f- 6 -f- c s — a • s— Cj orr=:;^' r"=V-- ~a • s — b • s — c. 5. By taking the product of the four radii, r r' r' r", the following remark able expression is obtained for the area of a tiiangle : — Area of A = >/»• r' r" r". Ex. 1. Find the radii of the inscribed and exscribed circles in the triangle of which the sides are o = 100, b = 86, and c = 72. „ 100 -I- 86 -f 72 258 ,„„ „„ , ,„ , Here« = r — ' =~S-= 129,«-o = 29,i— 6 = 43,ands-c=:.17. , J 2 Area ,s—a • s—b • s—c. And r = — ; — j— r~ or '" =\/ a + b + c ^ s 102 GRADATIONS IN EUOLID. Now area = v'129 X 2i» X 43 X 5 7=«/9l69I91 = 3028-06; and r='Ji^= 23-473, or. = V?i^^^^= ^^ = 23473 ^129J<«_XJ7 ^ ^316179 ^ ^^^^^ ^ ^^^.^^^ ^mxJZl! = V^= v/l^ = 70-420 ,129 X 29 X 43 , 160863_ r'" = V" g^ = ,/— —=^^2822-1579 = 53-124 Ex. 2.- From the two foi-mulas given above, find the area of the foregoing ti-iancle : — 1°. Area = ^V29 X 29 x 43 X 57 = ^9l6yi\)i. == 3028-06. 2°. Area = v'23-473 X 104-416 X 70420 X 53-124 = ^9168919-3231896 = 3028-0223. 6. In a rt. angled A the diam. of the inscribed Q is equal to the difference between the sum of the sides and the hypotenuse ; and the diam. of the exscribed to the hypotenuse is equal to the perimeter of the triangle. N.B. For other properties of the inscribed j circumscribed, and exscribed circles of a given triangle, consult the Appendix to Galbraith and Haughton's Manual of Euclid, Bk. IV., p.p. 20-25. Prop. 5. — Pbob. To nircumscribe a circle aboytt a given trimigle. Con., 10, 1. To bisect a given finite st. line. Pst. 1 and 3. 11,1. To draw a St. line at rt. ^s to a given st. line from a given point in the same. Dem. Ax. 9. The whole is greater than its part. Ax. 12. If a st. line meets two St. lines, so as to make the two int. /_& on the same side of it taken together less than two rt. /^s, these hvo st. lines, being continually produced, shall at length meet upon that side on which the /_s are less than two rt. /_s. Def. 10, 1. When a st. line standing on another st. line makes the adj. /.s equal to each other, each of these /_s is called art. /_; and the St. line which stands on the other is called a perp. to it. 4, 1. If two As have two sides and the incl. /_ of one equal to two sides and the incl. £_ of the other, the A^ are equal in every respect. Ax. 1. Dcf. 6, rV. A circle is said to be ieac. about a rcctil. fig. when the 0ce of the touches each comer of the fig. about which it is described. 31, m. In a the /. in a semi is a rt. /^ ; but the /. in a seg > a semi is less than a rt. /. ; and the /. in a seg. < semi. 0, is greater than a rt. /.. . . PKOP, V. — BOOK IV. 103 E. 1 2 C. 1 2 D. 1 2 3 4 5 6 7 Datum. Quees. 10, I. 11, I. Post. 1, C. 2. Ax. 9 Ax. 12. Sup. Pst. 1. Sup. Pst. 3. Sol. C.1.2.Def.lO,I 4,1. Sim. . Ax. 1. D. 2, 3. • Eem. Def. 6, IV. Sol. Let A B C be- the given A ; To desc. a about tbe A ABC. Bisect A B, ACinD, E; and from D, E, draw DF, EF _Ls to AB, AC and join DE; tben V Z.S ADF, AEF are rt. Z.S, .-. Zs EDP, DEF are < two rt. Zs; and .". DF, EF being prod, shall meet; let them meet in F, and join FA; If F be not on BC, join also FB and FC; with PA or PB desc. a © ; the ABC is c. scr. about the a ABC. '.• AD = DB, DP com., and Z ADF = Z BDP; .-. A ADF = A BDP, and AP = PB. So CF = PA, .-. BF = PC; thus PA = FB = PC; .-. a © from F, with rad. PA or PB, or PC, will pass through the extrs. of the other two; •.• the © passes through the angr. • s A, B, C; and .-. the A ABC is c. so. by the © ABC. Q. B. F. QoE. 1. When F, the centre of the 0, falls within the A, each angle is an acute Z, (31, III.); when on a side of the A, the z'opp. that side is a rt. Z, (31, HI-); and J5| when without the Ai ^^^ ^ OPP- t^^ side nearest the centre is an obt. Z, (31, HI.); and conversely. If the given A ^^ acute angled; the cen. of the © falls within it; if rt. angled, the centre is on the side bpp. the i-t. Z; and if obtuse angled, the centre falls without the triangle, g beyond the side opp. the obtuse angle. 2. The perps. bisecting the sides of a A ™eet at the centre of the circumscribing circle. lOi GKAPATIOKS IN EUCLID. 3. Perpendiculars from each L on the opp. side intersect in the same point. ScH. 1. This proposition is identical ; 1° with that (in 3, HI. and 9, III.) of describing a circle through three given points, A, B, C, not in the same st. line i 2° through two given points. A, B, and touching a given St. line ; or 3° through a given point A, and touching two given st. lines. 2. It is also identical with describing a circle; 1° through two given points and touching a given circle ; 2° through a given point and touching two given cu'cles ; or 3° touching three given circles. 3. And it may be farther extended to solve the problems of describing a circle ; 1° through a given point, touching a given st. line, and also touching a given circle ; 2° touching two given st. lines and also a given circle ; or 3° touching a given st. line, and two given circles. See Geom. Plane, Solid, and Spherical, pp. 114, 118. Use and App. — "If one circle EDF is inscribed in an equil. A ABC and another K AH circumscribed about it, ike circles EDP, K AH are concen- tric, and the diam OG of one is double tlie diam. 0X>, of the other. Bis, the /_s A, B, C by the st. lines AG, BF,CE; and join BG, GC. '.• BA := CA. AG com., and ZBAD=/.CAD; .•.BD = DC,andZ.satDrt. /.s. So AE = EB, and AF = EC, and /.s at E, E, are rt. /,s. .". AG, CE, BE, pass through O thecen. of© ABC; and O, the cen. of AG a diam., is thecen. of© ABC. Again •.■ EB = BD, OB com., and /_ EBO : .-. OE = OD; SoOE = OD; .-. 0E = OE = OD; .'. O is also the cen. of the © EDE. Next, •.• AKG and AH G arc sem. cs. ; .-. Z ABG=4 ACG; and •/ /\ ABC is eq. ang.; /. each Z = f of a rt. Z ! .-. alsoz DBG = Jit. /_ = /_ DCG. Now Z BDO = Z CDG, Z DBO=ZDCG,and BD = DC: .-. CD = DG, or 2 CD = OG = O A. -. the diam. of © A K G H = 2 diam. of E D E. q. e. u. C.l 9,1. 2 Pst. 1. JD.l H. and C, 1. 2 4,l.Dcf. 10,1 3 Sim. 4 Cor. 1, m. 5 i,in. 6 H. and C. 7 4.1 8 Sim. Ax. 1. q Cor. 1, m. 10 Cor. 31. in. 11 Ax. 11, 12 H. 13 14 D. 2. 15 26,1. 16 Cone. ZUBO; PKOr, A, — BOOK IV, 105 Pkop. a. — Thkos. A circle may be described about any regular polygon, or inscribed ivithin it ; and conversely. E. 1 2 3 C. 1 3 D. 1 2 3 4 5 6 7 8 Hyp. Con. 1. „ 2. Use 9, III. 10,1, Pst. C. 1, 2, & 3, III. 4,1. Sim. Def.6, IV. C. 1, 2. 4, I." and 3, III. Sim. Def. Kec. Def. 5, IV, Let tlie reg. polygon be ABODE; the A B D may be circum- scribed ; or GLI may „ be inscribed. Througli A, B, C desc. a 0, being the cen- tre; bis. the sides AB, BO, &c., in • G, H, &c.; and join OA, OB, GO, &c.; OG, OH, 01, &c. •.• CI = ID, 10 com. and z CIO = Z DIO; .•. OD = 00; and passes through D; .•. also passes through E; and .•. a is descr. about ABODE. Next, •.• AG = A L, A com. andz OAG = Z OAL; .-. OG = OL, and the touches AB and AE. So the touches H, I, and K. .*. GLI is inscribed in the polygon. •. A circle may be described, ^c. q. e. d. Use akb App. 1. — In a reg. polygon the angles at the centre, opposite to the sides, are all equal, and together make up four rt. angles ; therefore in OgQO constructing a polygon, ^ . , ■ ^ each Z at the centre ; and the chord of that Z = <* *"'* >■ *". e. v. Cor. — The square on the radius, EC=, of an inscribed ©, is half the square inscribed in the circle^ and one-fonrth the square on its diam., AC^; i. e. tC"~ = ^^."^ - ^. E.l Datum 2 Quas. C.l 1, III. 11, I 2 Pst. 3. 3 Sol. D.l Dcf. 1.5, I. 2 D.lC.lAx.11 3 4, 1, and Sim. 4 Cone. 5 C.l. 31, III. 6 Sim. 7 Gone. 8 D.4.7.Def.30 9 Def. 3, IV. PEOP. VII. BOOK IV. 107 ScH.— In other words, the inscribed square ABCD = 2EC==iAC2; for the /.s at E being rt. /.s, we have, by 47, 1, BC^ = BE^ + CE'' = 2r^ ; .•.BC = r^/2; And AC2 = 4EC2 = 2BC='. E.v. 1. — The radius being 8, required the area of the inscribed square. 2X8X8 = 128. Ex. 2. — When the area of the insc. square is 288, wliat is the diain. of tlic circle ? V" ( 2 X 238 ) = ^ 676 = 24. Use and Am-. —By bisecting the arcs AD, DC, CB, BA, and joining the points F, G, &c., an octagon will bo formed ; and by continnfaig the bisection,'!, other regular polygons of 16, 32, C4, &c., sides. Prop. 7. — Pros. 2'o circumscribe a square about a given circle. Con. 1, in. II, I. Pst. 1. 17, III. Di:jr. 18, ni. Def. 15-17, I. Def. 30. Dof. 4, IV. 28, I. If a St. line falUng upon two other st. hues makes tlie ext. /_ =z. the int. and opp. /. upon the same side of the line ; or makes the int. /_s upon the same side together equal to two rt. /_s ; the two St. lines shall be pai-atlcl. 30, 1. Straight lines parallel to the same straight line are parallel to each other. Def. A. I. A parallelogi'am is a fom'-sided flg., of which the opp. sides are parallel. 34, 1. The opp. sides and /.s of / 7 S are equal to one another, and the diara. bisects them ; i. e. divides them into two equal parts. Cor. 2. 46, 1. Every / 7 having one rt. /_ has all its ^s rt. /,s. E.l Datum. 2 QuEBS. C.l 1, III. 2 11,1. 17, III. Sol. Givena ABCD; to descr. a square about it. ^^ Find B the cen. of the ©; . at B draw two diams. g AC, ED, atrt. / s to each other ; , and through A, E, C, D, L tangents forming iig. ^^ , GHKB; then GHK F is the D recjuired. 108 GRADATIONS IN EUCLID. D.l 2 3 4 5 6 7 8 9 10 11 12 13 U 15 16 C. 3 and 2. 18, III. Sim. C. 2 & 3. 28, I. Sim. 30, I. Sim. Def. A.I. 34 I. Def.15-17. 31,1. Cone. D. 7. C. 2. 34, I. Cor. 2. 46, I. Cone. Def.lO,14.Def.30 Def. 4, IV. ■.• GFis a tang, at A, and AE a st. line from A to cen. E ; .*. the /. s at A are rt. /_s; So the Z.S at B, C, D, are rt. Z.s. And •.■ the Zs AEB, EBG, are rt. Zs, .-. GH|| AC; also PK II AC, andGHII FK; andGP, FK, each || BD; .-. the figures GK,GC, AK, FB, BK, arezz::7s ; and .-. GP = HK and GH = PK; and V AC = BD; AC = GH =PK; andBD= GP =HK; .-. GH = PK = GP=HK, and PGHK is equilateral. Again, •.• GBE A is a/ — 7, and Z AEB a rt. Z ; .-. Z AGBisart. Z; and .". Zs at H, K, and P are rt Zs ; .". F G H K is rectangular ; .•. the fig. PGHK is a square, and it is ciremnscribed about the ABC. Q. E. F. CoH. — In the same circle the circumscribed square GHPKis double of the inscribed square ; i. e. B D^ or A C^ = 2 A B^ t= 4 AE2. Use ahd App. 1. — ^By bisecting the arcs and drawing cliords a regular octagon niay be inscribed, AI, IB, &c. ; and by drawing tangents through the angulai' points of the inscribed octagon, a reg. octagon may be circumscribed. 2. — A reg. octagon, A IB, &c., inscribed in a circle, ABCD, is eqnal to the rectangle under the sides, AB, ]3D, of the inscribed and circtimscribing squares. D.l 2 3 4 5 6, 7, IV. 28, m. 41,1. Add. elKemk. For BD and AE are at rt. Zs ; and BD |i GF || HK side of circumscribing n ; also AI ^ IB, and arc AI;''= arc IB j and EI J_ AB bisects it in I ; .-. 2A EAI = IE . AL ; and 2 A EBI = IE • BL. .•. 2 the qu. lat. BEAI = IE • (AL + BL) = IE • AB ; Or 4 BEAI = 2 BE . BA = BD . B A i Now the octagon is 4 B E A I ; and B D = side of circumscrg. □ , and AB = side of inscribed square. rnop. VIII. — BOOK iv. 109 3.— If a qnadril. AB CD be circumscribed about a n circle EFGH, any two of its opp. sides AB+CD, or '" AD+BC = half its perimeter, i.e. AB+BC+CD+AD. D.l Cor. 37, m, 2 Ax. 1. •.' tangs. AE = AH, BE = BF, CF = CG,andDG = DH; .-. AE + BE + CF + DG- = AH +BE + CG + DH; i. e. AB + CD = AD + BC, half the perimeter. Pbop. 8. — PltOB. To inscribe a circle in a given sqti.are. Con. 10, 1, 31, 1. Through a given point to draw a parallel to a giren St, line. Dem. Def. SO, I. Cor. 2. 46, 1. 34, 1. Def. 5, IV. Ax. 7. Things which are halves of the same are equal to one another. 29, I. If a line fall upon two par. st. lines, it makes the alt., /_s equal to one another ; and the est. /_ = int. and opp. /. upon the same side; and likewise tlie two int. /.s upon the same side together = two rt. ^s. Cor. 1. 16, HI. If a St. line be di-awn at rt. /.s to any diam. of a from its extremity, it shall touch the at the extremity ; and a St. line touching the at one point shall touch it at no other point. E.l 2 C.l 2 Datum. Qujes. 10,1. 31,1. Pst. 3. Sol. Let GHKP be the given „ sqaaxe, to inscr. a circle in that D . Bis. the sides GH, GFin^ B and A ; through A and B draw AO IIGHorFK, andBDIl GF H orHK; with rad. E A, from E, draw a ; the © ABC D is inscr. in the sq. GHKF. 110 GRADATIONS IN EUCLID. D.l C. 2. Each of the figs. GD, DH, GC, CF, GE, EK, HE, EFis a n ; 2 De^. 30. and •/ each contains an Z. of the sq. GHKF ; 3 Cor. 2. 46, I. .•. each of those figures is rectangular; 4 34, I. and .•. of each, the opp. sides are equal. 5 Def. 30. El. Now v GP = GH, and GA = JGF, andGB = ^GH; 6Ax. 7. D. 3. .-. GA = GB, andBE = EA; 1 Sim. ThusEA = ED = EC = EB; 8 Cone. /. the from E, with rad. E A, passes thi;ough the . s B, C, D. 9 29, I. And •.• the Zs at A, B, C, D, are rt. Zs; 10 C. and the st. lines GP, FK, KH, HGareat the ends of diams. BD, AC ; 11 Cor. 16, III. .•. each of those st. lines is a tang to the ; 12 Def. 5, IV. ■ . the A B C D is inscribed in the D G H K P. Q. K. F. N.B. — The diagram will illustrate the Cor. to Pr. 7, IV. ScH. — Euclid confines himself in this book to the inscription and circum- scription of circles and regular rt. lined figui-es, — ^but circles may be inscribed in segments and sectors ; for example, To inscribe a circle in a given quadrant ABC. C.l 3 4 D.l 9, 1. 31, 1. 9, 1. 31, 1. 31,1. 11,1 Sol. C. 1. H. 6,1. .3 34, 1. Ax. 1. 11 C. 2. 29, I. Ax. 1.6,1. D. 3. 9,ni. Cor. 46, 1. 16, in. 11, UL 10 C. 3 and D. 6. 11, in. With AD bis. Z CAB and , draw DE II AC J C with DF bis. Z ADE, and draw FG || AC and meeting AD in G ; also draw GH || AB and KL J. AD in D ; H the cen. of the required is G; and GD or GF its rad. •.■ Z DAB = JZ CAB a rt. Z. and GFA is a i-t. Z ; .-. Z AGF = 4 art. Z, and AF = FG; butAF = HG; .-. FG = HG. Again •.• Z GDF = Z BDF, and Z EDF = Z DFG; ,-, Z GDF = Z DFG, and GF = GD = HG ; .•. the HFD passes through the points, D, F, H. Also '.' the /_s at H and F are rt. Zs ; , .■. the touches AC and AB in H and F ; and .". AG, joining the centres A and G, passes through D ; and •.' LD or KD, a perp. to AD at D, is a tang, to arc CDB and to the DFH ; ' ' /, the DFH touches the arc CDB. o. e. p. PEOP. IX. — BOOK IV. Ill Prop. 9. — Peob. To circumscribe a circle about a given square. Con. Pst. 1 and S. Dem. Def. 30, 1. Axs. 11 & 7. Def. 6, IV. 8, 1. If the As liave two sides of tlie one equal to two sjdes of the pthev, each to each, and have lifcewise .their h^es equal, the Z. which is contained by the two sides of the one, shall be equal to the /. contaiped by the two sjdes equal to them of the other. 6, 1. If two /.s of a A be equal to one another, the sides opp. to tlie eqinal /.s shall be equal to one another. E.l 2 C.l 2 D.l .2 3 4 5 6 7 8 9 Quaes. Pst. 1. 3.' Sol. Def. 30, 1. 8,1. Sim. Def. 30, I. D, 3. Ax. 7 & 6. I. Sim. Ax. 1. 9, III. Let ABCD be the- given a ; to circumsc. a about ABCD. A Join AC, BD cutting inG; ] and with. GA desc. a ; that ABCD is tlie g<< required. •.• in As ABC, ADC, DA = AB, CB = CD, and A com. ; .-. Z. DAC = /_ B AC, i. e. L BAD is bisd. by AC; So Z.S ABC, BCD and CD A are bisected by BD and AC. Now V L DAB = L ABC; and L GAB = iL DAB, and L GBA = UABC: .-. L GAB = L GBA, and GA = GB. So GA = GB, and GC = GB = GD ; .-. GA = GB = GC = GD. Hence a from G, with ,rad. GA, will pass through B, C, and D. And •.' the ABC passes through the angu- lar • s of the sq. ABCD, .-, the ABC is c. scrd. about the n ABCD. Q. B. F. 10 D. 8 11 Def. 6, IV. N3. In the diagram a circle i^ also inscribed in the square ABCD. 112 GnAt>A?ldNs IS eWclid. Prop, 10. — PnoB. To construct an. isosceles triangle, having each of the angles at the base double of the third, or vertical angle. Con. Pst. 3. 1. IV. Pst. 1. 5, IT. 11, II. To divide a given line into two parts, so that the rect. contained by the whole and one of the parts shall equal the sq. of the other part. Dem. Ax. ]. 32,111. 6,1. 37, m. If from tt • W diam. of BDE, and join Ad ; then in A ABD, Z. ABD = Z. BDA = 2 Z BAD. Join DC, and ahout A AD Q desc. © A CD. V AB . BC = ACS and A = BD; .-. AB . BC = BD2; and •.• from B, a • out of A CD, are drawi* BCA, BD, one cutting the in C, the other meeting it in D ; and •.• AB . BC = BD«, ,-, BD touches the ACDinD. PROP. X. BOOK IV. 113 D.4 6 7 8 9 10 11 12 13 14 15 16 D. 3. & C. 32, III. Add. Ax. 3. 32, I. Ax. 1. C.5, LAx.l. Ax. 1. D.7.,9. 6, I. 0.3. Ax.l.5,I Ax. 2. D. 7. 32, I. D. 9. Kec. Again •.■ BD touches the ACD, and DC a St. line from the • of contact D, exits that ; .-. Z. BDC = Z DAG in the alt. seg.; Add Z CD A to each; .-. Z BDA = Z CD A + Z DAC; But •.• ext. Z BCD = Zs CDA, DAC, .-. Z BDA= Z BCD; and •.• AD = AB, and Z BDA = Z CBD; .-. Z CBD or DBA = Z BCD; .-. Z BDA = z DBA = z BCD. Again V Z DBC = Z BCD; .-. BD = DC; but BD = CA, .-. AC = CD and I CDA = Z DAC; .-. zs CDA + DAC = 2 Z DAC; but z BCD = z CDA + Z DAC; .-. Z BCD = 2Z DAC. Now Z BCD = Z BDA = Z DBA; .-. Z BDA = DBA = 2z DAB. .'. the A ABD is the isosc. A required. Q. E. F. Use akd App. — The following are some of the various problems which bear a close relation to the 10th : — 1°. 77(6 side AC inscribed in the smaller Q ACD equals the side of a j-cjufar pentagon in that circle, and also equals the side of a regular decagon in the larger QBDK Prel. 1| 2 Cor. 15,1. •.• the Zs formed by lines from a central point I = 4 rt. Zs ; 2| and •.• in a regular polygon the central /_s arc all equal ; 3 ,*. the central Z of »' pent. =: 4-5ths of a rt. Z ; 4' and .*. the central Z of a decagon = .4-lOths = 2-5ths of ait. Z. ^.-^- * / jFiVs* — ^For the decagon in D E F., D.l 10,rV.Prel.4.i-.- AC = BD ; and Z A = 2-5ths j of art. Z ; Cone. I.'. AC orBD = the side of a reg. I decagon in BEF. 114 eRAPATiosrs in buclid. Second,— Foi the pentagon in A CD. P.l lo.rv., &c. 20, m. 3! Ax. 6. 4' Cone. '.• CD = BD = AC ; and AC subtends Z. ADC at the 0ce ofaQ ACD. and '.• also Z. ADC = 2-5ths of a rt. /. = i the /. at the centre of© ACD ; .•. the Z. at the. cen. of© ACD = 4-5ths of a rt. /. ; and .•. A C = side of a reg. pent, in the ©ACD. 2°. On the side D C being produced to meet tlic circle B D E in F, and F B being joined, the Z. ABIT = three times /. BFD. D.l 2 3 4 6 C. aad 32, L Ax. 1. 20, III. Sum. Ax. 3. •." Z BAD = Z ADC, and V /. BCD = Z-SBAD + ADC; .-. ZBCD = 2 /.BAD. Now, /_ BAD = 2 Z BFD; and ext. /. BCD = 4 l_ BFD= /.sABF + BFD; &om each side take away /. EFD ; .-, Z. ABF = 3 Z. Bl'D. 3°. To quinguitect, i.e., divide a quadrant B A C into five equal partt. Construct an isosc. A ,■ '' '""■ BAE, having /_ ■' BAE = Z BEA = 2 Z. ABE ; ,^ \ prod. AE to E, and div. Z. FAB into / 4 equal parts ; .' then Z. CAE = Z. FAG = Z_ GAH =Z.HAI=ZIAB = l-5th /_ BAC. * of a decagon the central Z. ABE = 2-5ths of a rt. /_ i and •.• Z BAF = 2Z.ABE = 4-5ths of a rt. Z. ; but Z. BAG is a rt. Z; .-. Z CAE = l-5th of a rt. Z,and /_ FAB = 4-5th3 of art.Z. and Z FAB is div. into four equal parts ; .-. the quadrarft BAC has been divided into five equal paits. Q. E. F. It in ev.'dent that, by the proces"! of continual bisection, the quadrant mav now be divided into 10, 20, 40, 80, &c., equal parts. C.l 10,1V. ( 2 9, 1. Soh. 6 Sol. D.1 C. & Def. S C, 3 Datum. 4 5 6 Ax. C. 2. Core. PROP. XI.-— BOOK IV. 115 Prop. 11. — Pkob. To inscribe an equilateral an^ equiangular pentagon in a given circle. Con. 10, IV. , 2, IV. 9, 1. Pst. 1. Dem. 26, III. In eq. circles, eq. angles stand upon eq. arcs, whether they be at tlie centres or circumferences. 29, in. In oq. circles eq, arcs are subtended by eq. sL lines. Ax. 2, 27, III. In eq. circles, the angles which stand upon eq. arcs are equal to one another, whether they be at the centres or the 0ces. Def. 3, IV. Dat. Quffls. 10, IV. 2, IV. 9, I, Pst. 1. Sol. G. 2, & 1 xVx. 1, 0. S. Ax. 7. 26, III. Cone. 29, III. Gone. Cone. 1. AM. LetABGDBbe the given ; to inscr. therein a reg. pent. Desc. an isosc. a FGH, with Z G = Z H = 2 Z F; in ABGDEinsc. A AGD equ. ang. to a FGH; so that Z CAQ = Z F, and Zs ACD, CDA each = Z G = ZH; Bis. ZsAOD, ADC by CE, OB cutting the 0ce in B and E ; joinAB, BG, DEandEA; then the fig. A B ODE is the pent. rec[uii'ed. V A AGD is eq. ang. to A FGH, and Z G = Z H = 2 Z F; .-. z AGD = Z ADC = 2 Z CAD. and •/ the eq. / s ACD, ADC are bisected by CE, DB; .-. z DAC= Z AGE = / EGD = Z CDB = Z BDA; but eq. Zs at the 0ce stand on eq. arcs; .-. the arcs, AB,-BC, CD, DE, EA are all equal; and eq. arcs have equal chords ; .-. the chords' AB, BO, CD, DE, EA are all equal; •. the pentagon is eqidkteral. Again, to each of the eq. arcs AE, DE, add arc BOD; 116 GK.\DATIONS IN EUCLID. 11 Ax. 2. .-. the ^iliole arc ABCD = llic wliole nrc EDOB; 12C.&D.10&11 but Z. AED stands on arc ABCD, and t BAE on arc ED CB; 13 27, III .-, Z BAE = Z AED; 14 Sim. So Zs ABO, BCD, CDE, each = Z BAE, or Z AED; 15 Cone. 2. .•. the pent, is equiangular. 16 D. 9. And it has been shown to be equilateral; 17 C. and ■.• the angular . s A, B, C, D, E arc in the 0ce, 18 Def.3, IV. .-. the pent, is inscribed in the circle. tj. E. F. SciT. I. — In a reg. pentagon we may remark ; 1st., that each diagonal, as A C, is parallel to the side, as ED, to which it is, not conterminous ; 2nd., that triangles, B C 6, ED e,' AE d, &c., are isosc. As, equiangular with A C A 1\ &c., and having the Z at the base = twice the vert, angle ; 3rd., that the fv^. ABcE is a lozenge, also BCDc, &c.; and 4th., that fig. abode is a rog, pentagon. I/ABdner's Euclid, p. 130. II. Practically, a pentagon is inscribe din a circle, ■•-,-. by drawing two perpendicular diameters, AB, CD, l'\ and bisecting the rad. FD in E ; from E with E A desc. AK, and from A with AK desc. KG ; then A G is the side of the pentagon ; and if the arc A G, GiT , be bisected in H, the chord A H is a side of a decagon. Cf- Scc Euclid 10, xiii. III. It is generally hy aid of isosc. As, with the Zs at the base equimultiples of the vert. /_, that reg. recti], figm-es which have an odd number of sides are inscribed in cu'cles. Thus, /or the pentagon ; By 10, iv., desc. an isosc. A F G H, having each of the Zs, Cr and H at tlie base double of Z F, the Z at the vertex. 2, iv., in the given insc. a A A C D eq. ang. with A FGH; 9, I, bis. the Zs A C D, A D C ; and let the bisecting lines bo produced to meet the 0ce in B and B ; Then the points. A, B, C, D, E, are the angular points of the required pentagon, rV. Eeg. rectil. polygons with an even number of sides may also bo inscribed in circles by aid of isosc As ; but in such isosc. As, the Zs at the base are multiples sesquialter, as it is termed, of the vert. Z ; '• ^-j ^ach angle at the base contains IJ times, or 2^ times, or SJ times, &c., the magnitude" of he vert /_. Thus /or the octagon ; PKOP. XI. — BOOK lY. 117 Construct an isosc. A ABC, of which each /. at the base, /_ B or /. C = 2J times the vert. /_ A. By 2, iv., in the given D G H inso. a aDEF eq. ang. with A AB C ; then the base EP, the chord of the arc EF, will be a side of the octagon to be inscribed. in D GH. V. Fortmla for determining the relative magnitudes of the angles of isosc. As, to be used in the con- struction of regular polygons. The vert. /. of such isosc As = A. \ number of sides sides — 1 The multiplier of that vert. /^ 1 __ for the /.s at the base -j 2. Thiis in a regular polygon of 5 sides, the vert. /. = 36° ; the multiplier is 2 ; the /_ s at the base each ■ = 72° 7 9 n 13 = 25°-f; = 20°j = 16°iV; : 13°44-; And in a regular polygon of 6 sides, the vert. Z. = 30°; the multiplier is 2 J 8 ,, =22°4; 10 „ = 18°; 12 „ =15°; 14 „ = 12°-f ; 16 „ =n°J; 18 „ =10°; 20 „ = 9°; App. — I. To draw a triangle equal ABODE see on page 118. 0.1 2 3 4 D.l 2 3 3; jy = n°\ 4; = 80° 5; 6; j> = 81°f, = 83°i, 2J; the Z_s Hi 4i; 5i; at the base each = 75° = 78°-i = 81° . =82°i eh Hi »» = 83°4 = 84°| = 85° = 85°j: al in area to a given pentagon, Pst.l&2 31, I. Pst. 1. Sol. 37,1. Add. Ax. 2. Join AC, AD, and prod. CD ; through E and E draw BF || and join AF, AG; then A AJFG = pent. A B C D E. '.-A ACP= A ACB, and A ADO . add A A CD to the equals ; .-. A AFG = pent. ABODE. AC, EG II AD; A ADE ; II. The lines AC, BD, OE, D A, 'E,'B,joinifi.g the alternate angles, A,C ; B, D ; C, E ; D, A ; E, B ; of a reg. pen tagon, . A B C D E, will form another reg. pentagon,a be de; and the points of intersection, A', B', C, D', E', of the alternate sides o/" ABODE produced, namely, AB and CD, or DE; BO and AE, or ED; CD and AB, or AE; DE anrfAB, or BO; EA (Wf?3BC, or CD j will alsofom another reg. pentagon, A' B' C D' B', 118 GRADATIONS IN EUCLID. First — The figure abode is a regular pentagon. n.i C. & H. D.l 27, m. Ax. 3. H.15,I.&D.3. 6,1 26,1. 32, 1. Hemk. 5,1. D. 9 & 12. •/ arc BCDE = arcAEDC, and arc B C = arc A E ; ,-. Z. BAE = Z A B C, and /_ BAG = /. ABE; .-.remg. ZCAE xj'X" =remg./.EBC. ^ ^ and ■.• BC = AE, ZB aC = l_ A a E, and l_ CAE = Z.EBC; ,", A. a = B a, and Z ABE = Z BAG; /. Z Aae = 2 Z ABE = 2/. AEB=Z.Aea; and ,•. Aa^Ac = Ba=Es. Thus the As Aac, B a 6, G6c, &c., aie isosc. as ; they are also equal ; .'. ae = ab = bc ^=ed = dc. Now, in isosc. As the l_a on the other side of the bass are equal ; .'. /_ cah = /_abc =. /_b c d, &c. .". also the fig. ahcde\s equiangul.ir. '. abcde\s,a, regular pentagon. Q. e. r. Second., — ^Also A' B' C' D' E' is a reg. pentagon. H. 13, 1. Ax. 3. Sim, H. 26, 1. Sim. D. 1-4. 4, 1. & Sim. Gone. D. Cone. D. 7, & 9. V Z- ABC = Z. BAE, and /_s A'BA + ABC = /_s A'AB + BAE; .-. Z. A'AB = Z_ A'BA; and so = / B'GB = /.B'BC, &c. : AlsoAB=BC; .-. A ABA' = aBCB'; So As CDC, BED' and EAE' are equal. And •.• BB', BA' = E'A, A A', & Z. B'BA' = /.E'AA'i /. B' A' = A'E' ; and so = E'D' = D'C' = C'B' ; .-. the fig. A'B'C'D'E' is equilateral. Also the 3 Z.S at A' = the 3 /.s at B' = 3 z.s at C, &c. .•, the fig. A'B'C'D'E' is equiangular ; ,•, it is a reg. pentagon. q. e. f. Piiop. 12.— PaoB. To circuinseribe an equilateral and equiangular pentagm. niout a given circle. Cm, 11,1V, 17,111, 1,111, Pst. 1 PHOP. XII. B OK IV. 119 Dem. Ax. 1. Def. 16,1. Ax. 3. 8,1. 4,1. 27,111. Ax. 7. Dof. 10,1 26,1. Ax. 6. Bef. 4, IV. 28, III. In equal circles, equal st. lines cut off eq. arcs, the gr. = the gi-., and the less to the Ites. , 18, III. If a St. line touehes a ©, the st. line drawn from tire centre to the point of contact, shall be perpendicular to the line tonehing the circle. 47, 1. In any rt. angled A the square on the side opp. to the rt. /_ is equal to the squares on the sides containing the rt. l_. Dat. Quses. 11, IV. 17, III. Sol. 1, III. Pst. 1. C. 1. 28, III. C. 2, 4. 18, III. Sim, D. 4 & 5. 47,1. Ax. 1. Def. 15. Ax. C. and D. 9. 8,1. Eemk. Let A B D be the given. ; to desc. a reg. pent. about that 0. PI^ In ABD insc. areg. pent., of which the angular • s are A,B, 0,D,E, and the arcs, AB, BC, CD, DE, EA, are equal ; through A, B, 0, &c., draw GH, H K, KL, &c., tangents to the ABD ; then the fig. GHKLM is the pent required. Find P the centre of the ABD ; and join PB, FK, PC, PL, and PD. •.• A, B, C, D, E, are angular • s of the reg. pent. .-. arcs AB, BO, CD, &c., are equal ; and ".• KL is a tang.; and from cen. P, P C is drawn to C ; ,-. PC ± KL. and ^s PCK, PCL, rt.Z.s. So zs PBK, PDL, &c.,arert zs; Now •.• Z s PCK and PBK are rt. /_ s ; .-. PC2 + CK2 = PK2, and PB« + BK^ = PK^; .-. FC2 + CK2 = PB2 + BK2; but F C* = P B2 ; .-. C K2 = B K2, and C K = B K. Also •.' PC = PB, PKcom., and CKrsBE: .-. Z CFK = Z BPK, and Z CKP == I BKP; thus PK bisects Zs BFC and B KO, and z BPC = 2 Z BFK, and z BKC «:2 ZBKF. 120 GRADATIONS IN EUCI-ID. 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Sim. C.1.27,III.Ax.7 Def. 10. D. 14. 26,1. Sim. D.9.D.l7.Ax.6. Sim. Cone. D. 16. 12. 13. Ax. 6. Sim. Cone. D. 20&24. C 2 Def.'4, IV. So PL bisects z.s CFD, CLD. Again •.• arc B C = arc C D, .-. z BFC = Z CFD, and Z KFC=: Z CPL. Also V Z KCF = Z LCF, FC com., and Z K F C = Z C F L ; V KC = CL, and Z FKC= Z FLC. Now V KC = CL ; .-. K L = 2 KC, andsoHK = 2BK; and V BK = KC, KL = 2 KC, andHK = 2BK; .-. HK = KL. So GH, GM, ML, each = HK, or KL; .". the pent. G H K L M is equilateral. Lastly, .-, z F K C = ZF L C, Z H K L = 2 Z FKC, and zKLM = 2 Z FLC; .-. Z HKL = Z KLM; arid so zs KHG, HGM, G M L, each = Z HKLor Z KLM; .•. the pent. G H K L M is equiangijar ; .-. the fig. GHKLM is a reg. pentagon. And each side touches the giren A B D ; .•. the pent. GHKLM is circumscribed about the ABD. Q. E. F. ScH. — It is a general truth, that, " If the circiimfoi-ence of a circle bo divided into any mimher of parts, the chords joining the points o division shall include a regular polygon, inscribed in the circle ; and the tangents drawn through those points shall include a regular polygon of the same number of sides circumscribed aboixt the circle."' Prop. 13.— Prob, To inscribe a circle in, a given 'equilateral awl equiangular pentagon. Con.— 9, 1. I?, I. Pst. 3. Dem.— 4j I. Ax. 1. Ax. 11, All rt. Zs live equ.al to one another. 26, L Cor. 16. III. Def. 5, tV. E. 1 2 Datum, Qu«s, Let ABODE be the given reg. pentagon ; in that pent, to inscribe a circle. PROP. XIV. — BOOK IV. 121 C. 1 9,1. 2 12,1. 3 4 Pst. 1.' Pst. 3. 5 Sol. D. 1 H. C. 1. 2 3 4,1. C. 1. • 4 5 Ax. 1. D. 2 Ax. 1 6 7 Sim. C. 1, 2. 8 9 10 11 26,1. Sim. D. 8 & 9. Eemk. 12 13 Cor. 16, III Def. 5, IV Bis. zs BCD, CDE by CF, DF cutting in F, from F draw FG, FH, FK, &c.,.perps. to AB, BO, &c.; joinFA, FB,FC, FD, FE; with any of the perps. from F, desc. a GHL; then the GHL is the circle required. •.• in AS BCF, DCF, BC = CD, CF, com., and Z BCF = Z DCF; .-. BF = FD, and Z CBF = Z CDF; But Z C D F = ^ Z C D E, and-Z CDE = Z CBA; .-. Z CDF = J Z CBA; and Z CBF ="z CDF; .-. Z CBF = i z CBA, i. e., BF bisects Z CBA; So Zs B AE, AED are bisected by AP, FE. Now in AS PHC, FKC, •.• Z HCF = Z KCF, Z PHC = Z FKC and PC com.; .-. perp. PH = perp. FK; alsoFK = PL = FM = FG; .*. the five St. lines are equal ; and a from F at the distance of one of the fire lines, F G, will pass through H, K, L, M ; that GHKLM will also touch the st. lines; .". GHL is inscribed in the given pent. ABCDE. ' q. e. v. Prop. 14. — Peob. To circumscrihe a circle ahout a given equilateral and equiangular pentagon. Cos. ■ 9, 1. Psts. 1 and 3. \ Dkm. 13, v. Ax, 7. 6,1, Def, 6, IV. 122 GRADATIOKS IN EUCLID. E.l 2 Datum. Quaes. C.l 9,1. 2 Pst. 1. 3 4 Pst. 3. Sol. D.l 13, IV. Sim 2 H. & C. 1 3 4 5 6 AX.7&6.I Sim. Ax. 1. Eemk. 7 D. 6. 8 Def. 6, IV. Let AB CD E be a reg. pent.; to cii'cumso. a about that pent. Bis. the ZsBCD, CDE,by CF, FD; from F, the point in which CF, FD, meet, draw FB, FA,FE,tothepoints B,A,E; •with any one, as FA, desc. a ACE ; then the ACE is circumsc. about the pent. ABCDE. As in 13, IV., Zs CBA, BAE, AED, are bis. by FB, FA, FE; and ■.■ Z. BCD = z_ CDE, arid Z FCD = J Z BCD, and Z CDF= i Z CDE; .-. Z~ FC D = Z FD C, and CF = DF. So FB, FA, FE, each = FC = FD; .*. FA = FB = FC = FD = FE. Hence, a from F, with any one of these sides, as FA for a rad., will pass through the ex- tremities A, B, C, D, E ; and .-. the ACE passes through the angr. • s A, B, C, D, E ; .*. the A C E is circumscd. about the reg. pent. ABCDE. Q. E. F. ScH. — I. Gcenerally the method of bisecting two adjoining angles of any reg. polygon may be applied to the circiimscribinp; and inscribing of a circle about, or in that polygon ; for, "If any two adjoining angles, as BCD, CDE, of a regular polygon, ABCDE, be bisected, the intersection F of the bisecting lines, C F, D F, shall be the common centre' of two circles, the one circum- scribing the polygon, the other inscribed in it." n. Propositions 12, 13, and 14, may be generahsed so as to include a'.l regular polygons, and will then embrace the three problems : — Phob. 1. — To circumscribe a regular polygon about a given circle. To be done by drawing tangents through the angular points of the inscribed- polygon. Pbob. 2. — To inscribe a circle in a regular polygon. By bisecting two adjacent angles, and irom the point where the angle- bisecting lines meet, drawing a pei-pendicular to a side of the polygon, and, with that perpendicular for radius, and this point as centre, by describing a circle. PeoB. 3.— To circumscribe a circle about a given polygon. By bisecting two adjacent, angles, and from the point where the angle- bisecting lines meet, with one of those lines for radius, describing a circle. — Sec Hose's Euclid, p, 1.53. ADDENDA TO 14, ly. 123 Addenda to 14, IV. Before dismissing the regular pentagon and its intimately allied figure, the regular decagon, it will be useftil to state some of the principles of their con- struction. I. To analyze the conditions on which the drawing of those figures depends, First — Of the Decagon : At the centre there are ten equal angles, each = 1-lOth ot 360°, or 4-lOths = 2-5ths of 90°. At C draw an /_ equal 2-5ths of a rt. /_ ; and from C, with any rad., make CA == CB, and join A B ; then AB the chord of 1-lOth of 360° will measure the circle into ten equal parts, and consequently is the side of a regular decagon. Since Z C = l-5th of 180°, and /. A = /.B, .-. Z A and Z B of A AB C each = 2-6ths of 180°, and each is double of Z C. Bis. /_ A'BC, and the A B D C is an isose. A and the ext. Z. ADB = 2 /_ C = ^ A. The A ABD is also isosc, and its /s are equal to the Zs of A A CB. Hence AS A B D, A CB, are similar, and (by 4, VI.) the sides about the equal /_s are proportional ; i. e. A C : A B : : A B : A D ; and A B = B D = CD; /.AC : CD :: CD . AD. Thus when C A, the rad. of a cu'cle, is divided, at D, into extreme and . mean ratio (U, 11.) the greater segment, CD, will be the side of the decagon inscribed in that circle. Hence, Pkob. 1. — To find the side of an inscribed decagon, when the rad,, C A, o/" a circle is given. By 11, n., divide the rad. C A into extreme and mean ratio in D, and the greater segment C D equals the side of the decagon required. Pkob. 2. — To find the radius of the circumscribing circle when the side' A B, of the regular decagon is given. By 11, H., produce B A, so that the rectangle contained by the whole lino produced BE, and the part produced, A E, i. e. B E • A B = A B" ; the whole line thus produced, B E, is the radius of the circumscribing circle. Second. Oi the Pentagon. By joining the alternate angular points of the decagon a corresponding Pentagon will be drawn. II. To demonstrate, that the square on the side AB of a reg. pentagon inscribed in a circle equals the sum of the squares of the rad. AC, and of the tide A D, of the inscribed decagon ; i. e. A B» = A C' + A D'. 124 GRADATIONS IN EUCLID. Let the sides of the inscribed decagon .■-' i ^ heAD, DB; join'A, B; then AB is / i the side of the inscribed|pent. Bis. ^ B C D by C B, and join ED. Then V DC,CE = BC, CE, and / DCE = 2IBCE; • (4, I.) DE = EB, and A CDE = A CBE. And in As BED, ADB, •;• /. EBD is com. /_ EDB = Z BAD, and Z_ BED = /.ADB, .". the As BED, ADB, are similar. Hence (4, VI.) AB : BD : : BD : BE, orBD» =AB.BE (1). Again, '.• the rad. CA bisects the /_ BAF, of a regular figm-e, /. 2. C A E = f /. B A r of the inscribed pentagon ; And ".• thejSue angles together = 6 rt. /_s, .'. the half of one of the five, , namely /. C A E = 3-5ths of a rt. /.. But, V Z ACD = 2-5thsof art. /., and Z I>CE= l-5th of art. Z. ,•. Z A C E = 3-5ths of a rt. Z- Thus, Z CAE = Z ACE, and A AEG is isosceles and similar to aacb. Hence (4, TL) AB : AC : : AC : AE ; or AC" = A B • AE (2), Adding Equations (1) and (2); BD^ + AC= = AB • BE + AB. AE = (2, n.)AB^. CoE. — Hence a Practical Method for in.?cribing a pentagon in a given circle A D B E. Draw diam. GH at rt. Zs to diam A P \ bisect the rad. O H in L, and join L A ; make L K = L A, and join K A ; K A is the length of the side of the pentagon required. N.B.— The rad. O G is cut' in extreme and mean ratio in the point K ; and K is the length of the side of an inscribed decagon. Prop. 15.!— PeoS. To inscribe cm equilateral and equiangular hexagon in a given tircle. Cox. !, HI, P.?tf. 3. 1. 2, piior. XV. — BOOK IV, 126 Dem. Def. )5, I. Ax. 1. •32,1. 26, in. 29, IH. Ax. 2. i7, III. Def. 3, IV. Cor. 5, 1. Every eqiiil. A is also equiangular. 13, 1. The /.s irhich cue St. line makes with another upon one side of it are either rt. /_s, or together equal to two rt. /_s. 15, 1. If two St. lines cut one another, the opp. or vert, /.s shall bo equal. E.l 2 Datum. Quaes. C.l 1, III. 2 Pst. 3. 3 Psts. 1 & -2. 4 Pst. 1. 5 Sol. D.l 2 3 C. 1 & 2. Def.l5,I.Ax.l. D.2. Oor.5,1. 4 32,1. 5 6 Sim. 13,1. 7 8 9 Cone. D. 4, 5, 7. 15, I. 10 D. 8. 9. 11 12 26, III. Gone. 18 14 15 29, III. Con D13. Add. LetACEbea given circle; to inscribe in it a reg. hexagon. F, Find Gthecen.of© ACE, and draw the diam. AD ; from D, with D G, desc. 0ECH;. join EG, C G, and prod. them to B, F ; join also A B, B C, C D, DE, EF, FA; «... A^^....6. the fig. A B C D E F is the hexagon required. •.• G is cen. of ACE, and D of ECH; .-. GE = GD, and DE =: GD ; /. GE = ED; and •/ A E G D is equil. ; .-. zEGD=zGDE = zDEG; but the 3 Zs of a A = 2 rt. Z.s ; .-. Z EGD = Jof2rt. /.s. So Z DGC = Jof 2i-t. zs; and •.• G C with E B makes Z s E G C, C GB, = 2 rt. z s ; ' .-. remg. Z CGB = Jof 2rt. Zs; .-. Z,EGD=zDGC=zCGB; < to these Z s arc equal the vert. Z s B G A, AGF, FGE; .-. the six Zs are eq. ; EGD = DGC = CGB = BGA = AGF = FGE. But eq. Z s stand on eq. arcs, .". the six arcs are eq. ; AB = BC = CD = DE = EF = FA; and eq. arcs have eq. chords ; .■. the six chords are eq.; and ABCDEF is equil. Again, •.• arc AF = arc E D, to each add arc ABCD; 126 6RADATI0NS IN EUOLID. 16 Ax. 2. .-. the whole arc FA BCD = the whole arcEDCBA. 17 C. Now Z FED stands on arc FAB CD, and Z A F E = ou arc F D C B A ; .-. Z AFE = Z FED; So Zs AB C, BOD, CDE each = z AFE = Z FED; .'. the hexagon is eq. lat. and eq. angular, and it is inscribed in A C E. ' Q. B. F. Cob. 1. — Tlie side of a regular hexagon inscribed in a circle is equal to the radius, or semi-diameter, of the circle ; or, in other words, tJie chord of 60° is equal to the radius. 18 19 20 27, III. Sim. D. 14 and 10. 2ll Def. 3, IV. D.l 15,IV.D.2. Cone. • GE = ED; ED being the side of the hexagon, and GE the rad. ; . E D the side, or chord of 60° = G E the radius. CoR. 2. — An equilateral triangle would he inscribed by joining the points A, E, and 0, alternate points in the hexagon. CoR. 3. — Every equil. figure inscribed in a Q is also equiangular; for •.• its /_ s are contained bi/ the chords of equal arcs, ared(28, III.) stand on equal arcs, .". its Z s 'W'^ <^^i equal, each to each. ScH. — 1. The opposite sides, AB and D E, or B C and EF, or CD and E A, of a hexagon arc parallel ; for, •.' A D meeting A B and D E makes the alt. Z BAD = the alt. Z ADE, .-. (27, 1.) AB || DE. 2. If, through the angular points of the inscribed hexagon, A, B, C, D, E, F, tangents to the circle be drawn, ab, be, cd, de, ef,fa, these will form a regular hexagon circumscribing the circle. The Proof is the same as in Prop. 12 and 13, IV. Use akd App. — 1. On a given line, AB, to describe a regular hexagon. C.l 2 3 D.l 2 3 1,1. Pst. 3. Sol. C.l. i,l. kSim. Cone. On AB draw an cq. lat. A A G B ; from G, with GA, or GB, deso. a 0ACE; then AB will exactly divide the 0ce into six equal parts. V GA, GB = GC, GB, and Z AGB= ZBGC; .-. AB = BC; and BC=CD = DE = EF = FA. .•, fig. ABCDEF is a regular hexagon. PROP, XV, BOOK IV. 127 2. The inscribed hexagon, A B C D E F, m a circle is three fourths of the area of the circumscribed hexagon, abcdef. E.1 CI 3 4 D.l 12 Dat. Pst. 1. Pst. 1. 10,1. Cono. 3,1. 4,I.Def.lO,I, Cor.l,15,IV. 18, in. C. 2. done. C. 2. 38, 1. D. 6. 38, 1, Remk. Cone. Sim. Cone. Let circamsc. hex., abcdef, toueh the circle ACE, at the Z.S A,B,C,D,E,F, of the inscribed hex. ABCDEF, Join G, and any angular • as a of the circumscribing hex. ; join also GA, GB, adjacent points of contact ; Bis. G a in K, and join AK j then A AGH = f of A AGo. ^s GHA, GHB, are rt. /_». A GAa = A GB a; .-. /_ AGa = Z BGa ; and •.• A GAH = A GBH ; .-, _ But G a = G/ = fa ; and G A is perp. to/a, , also GA bisects /a. . and •.• A K bisects Go, AIv = Ka = Aa; A A K o is. eq. lat. ; and K a bisected by A H ; but-.- GK = Ka, .-. A GAK= A aAK; and •.■ KH = H a, .-. A KAH = A H Aa ; thus A KAa = J AGAa, and A HAa = -J GAa ; .-. A GAH= S A GAo. So, by drawing lines from G to the angr. • s of the inscribed and circumscribed hexagons, each A of the inscrd. hex. ^ \ of each A of the circumscrd. hex. ■. the inscr. hex.ABCDEF= J of the c. scr. hex.aicde/. Q. E. D. 3. The area of a reg. hex. ABCDEF is six ftmcs the area ol the eq. lat. A A GB described on the same st. line, AB. • 4. Because the side of a reg. hexagon is equal to the chord of 60°; and the chord of 60° equals the rad.; .", \ rad. = sine of 30°. 5. The inscribed hexagon and the successive bisections of its arcs, have been employed as the groundwork for finding the approximate ratio of the circumference of a circle to its diameter. By forming Polygons of any number of sides that are successive bisections of the original 6, as 12, 24, 48, &c., we obtain the apothem, i. e., the perpendicular from the centre on one of the sides, which continually approaches the radius in length ; and from a polygon of 1536 sides we deduce iie approximate values of the cu-cumference and radius to be 6-283185 and 1. When a square is taken as the groundwork of the process, and the apothems and sides of successively inscribed polygons of 8, 16, 32, &c., sides are employed ; on arriving at the polygon with 32768 sides, the Areas of the inscribed and circumscribed Polygons agree to the seventh place of decimals, 3'1415926, and since the area of the circle is intermediate, this value, 31415926, as far as it goes, must also be the ai-ea of the corresponding circle. Now, by Use 4, Pr. 41, Bk. I., the Area of a circle is equal to the rectangle under its circum- ference and semi-radius, or under its radius and semi-circumference ; therefore the radius being 1, the semi-circumference is 3'1415926 ; and since the diameter is twice the radius, 3"1415926 to 1 is also the Ratio of' the circimiference to the diameter. — See Chambers's Euclid, pp. 202-221, and Penny Cyclopcedia, Vol. XIX., pp. 186, 187. ' m GKADATIONS IN EDCLID. Peop. 16. — Phob. To inscribe an equilateral and equiangular quinchcagon in a given circle. CoK. 1,1. 2, IV. ■ Cor. 6, 1. 11, IV. Cor. 5, I. Every eq. lat. A is also eq. ang. 28, III. To bisect a given arc of a circle. Dem. 28,111. 1,IV. 11, IV. E.l 2 C.l 2 3 4 6 7 D.l 2 3 4 5 Dat. Quffis. l,I.Cor.5,I. 2, IV. Cor. 6, I. II, V. Sup. 30, III. Sol. 28, III. Cone. 1. Cone 2. Sim. D. 3. 4. Ax. 3. C. 6. 1. IV. Cone. lU. U, IV. Let A B F D given ; be the in that to inscr. a reg. quindecagon. Dese. an eq. ang. and eq. lat. a ; and in AB F Dinser. A A C D eq. ang. to it; then A ACD will be eq. lat. Also insc. in A B FD a reg. pent! ABFGH; and let one angr. • A be com. to the A and the pent. Bis arc B C in E, and join EC, EB ; then E C is a side of the quindecagon, round the ABFD. •.• equal chords cut off equal arcs, .'. arc .AC = arc CD = arc DA; and /. arc AC = J 0ce. So also arc AB = ^ 0ce. Thus of the 15 eq. ares contains 5 and arc A B, 3 ; /. arc BC, equalling arc AC minus arc AB, contains 2 of the 15 ; .-. arc BE = arc EC = ^^ of the 0ce. Hence, if st. lines each = chord BE, be placed in succession round the ABFD; • then a reg. quindecagon will be inscribed in the given 0. Th3 pi-oof will be similar to that for the pentagon. to be set in the 0ce, arc AC I'r.op. XVI.— BOOK IV. 129 Cor. — The only reg. st. lined figures which can be placed, side by side, so as to make a continuous plane surface, are the equi- lateral triangle, the square, and the hexagon, — See Use 3. Pr. 15, Bh.I. Sen. — 1. As in tlie Pentagon, 12, IV., if th'rovigh the angular points B, E, C, P, &o., of the inscribed quindecagon, tangents to tlio circle be drawn, a reaular quindecagon will be circumscribed ; and, .is ill. the Pentagon, 13 and 14, I v., a circle may be inscribed in a given quindecagon, or circumscribed about it. 2. Tha are subtemding a side of a regular thirty-sided figure may be readily found, by placing in a circle, ABCD, from the same point A, the line AB = the side of an inscribed hexagon, and AC = the side of an , ins,cribed pentagon ; theUj the ai"c BC = the arc of a reg. thirty- sided figure ; for l-5th of a© minus l-6lh of a0 = l-30th of the same circle. Also twice BC = B E = 1-1 5th the' arc of a regular quindecagon, UsB. — This proposition opens the way for the construction of other polygons ; for, if we obtain the common measures of 360°, as 2, 3, 4, 5, C, S, 9, 10, &c., and divide 360° by aiiy two successive common measures, the difference of their quotients will give an arc of the circle, ty which the circle will be exactly measured ; consequently, a regular polygon may be inscribed, the sides of which are equal in nuinber to 360 divided by the difference of the quotients arising from the division of 360° by the two successive common measures : Thus, l-9th of 360 = 40 ; 1-lOthof 360 = 36; and 40 — 36 = <; and = 90, the number of parts into which the circle would be 4 divided, and the number of the sides of the polygon. Again, l-36th of 360 = 10 ; l-4bth of 360 =9 ; and 10 — 9 = 1, the difference of the arcs ; and 1 will give a polygon of 360 sides. 130 GRADATIONS IN EUCLID. OBSERVATIONS ON POLYGONS. Obs. I. — The only known ways of dividing a circle geome- trically are — 1st, into 3, 6, 12, 24, &c., parts— by Prop. 15, IV., & Prop. 9, 1. 2nd, „ 4, 8, 16, 32, &c. „ „ 6, IV., „ 9, 1. 3rd, „ 5, 10, 20, 40, &c. „ „ 11, IV., „ 9, 1. 4th, „ 15,30,60,120, &c. „ „ 16, IV., „ 9,1. Obs. II. — Conseciuently there are many polygons which cannot be inscribed in a circle, except by mechanical contrivances ; but the following methods approximate so near to the truth, that they may be adopted without sensible error. 1. In a given circle, as ABP, to inscribe any regular rt. lined figure ; or to divide the circumference of a dtcU info an number of equal parts. C.l 2 3 4 5 6 7 8 U. & A. 34, I. 11,1. U. & A. 34, 1. 3,1. Pst. 1. Pst. 2, & 1. SoL Bemk. Divide the diam. A B, into the assigned num- ber of eq. parts, as 9 ; from cen. C raise a perp. C^; div. fad. y into 4 eq. parts ; and set off 3 of those parts from y %o h; join h and z, the second of thedivisions from A; prod. Ic z to 0ce, P; and join PC, A P; the line A P = the side of required figure ; and A P set round the will divide it into the assigned number of parts. N.B. It is usual to denote the circujnference, nieasuriDg four rt. angles, by 2 tt; the semi-circumference, by tt; and the number of sides of the figure, or equal parts of the circle, by n. OBS. II. oiJ POLYOaKB. 131 Anahjsis.—rko: L ACp" is given = _ n also /I C A P or z G P A :;= 90° — 2 AC ^ being given, Q,z= (^ — 2^ is gif en ; ,". P« can te feund; as also Z. A« P, or its fequal / C «^; hence the fcpmpl. of the Z i: 1z, namely Z C A /,• can be found. Am QiZ teiiig given, C k can te found ; wA hence yfcy can be found ; and hy = | Qy nearly.— ^/S'te TretMise On Mensuration^. Irish National Schools, p. 19. DeinonstratiortSj p. 53. 2. Though no exact geometrical rule has been discovered for inscribing many figures, as the heptagon, enneagon, or nonagon, hendecagon, &c., figui-es of 7, 9, li, l3, &c., sideSj — yet a high degree of accuracy may be obtained by the following apprOieirriative process, the real errors of which are far less than thosfe which th^ impferfections of our best instruments entail in all geometrical constructions. By 9, I. continue the series of bisections of the circle and of its arcs 2, 4, 8, 16, 32, 64, &c., until a number be, found greater or less by dhe thaih a multiple of the nuirfber of sides in the required polygon. Of the equal parts thus found, take as many as constitute a multiple part of the required polygon, stnd let one more of the equal parts obtained by successive bisections be also bi^tscted suc- cessively, until its parts are one more than the number of the sides of the polygon ; — then the first multiple part plus the second multiple part will, with stifflcierii acctiraCy, be equal to the side of the required polygon. Thus, for the Heptagon : The continued bisection of the 0ce, until it is divided into 64 equal psirts, gives a nuinber greater by one than 9x7; or 63, a multiple of 7. Now by bisecting the quadrantal arc A B in D, arc D B in E, arc D E in F, and arc D P lii &, (A G beiiig equsl iti. 9 out of the 64 parts); the arc A Gr #111 be less than a seventh part of the circumference by a seventh part of one of them, D G. 182 GRADATIONS IN KUCLUi. But the arc D G being small, a seventh part of its chord may, without any considerable error, be assumed for the seventh part of the arc itself, being somewhat less than the latter ;■■ — and if the chord of A a be taken equal to this approximate seventh part, the error of assuming for it the arc A a, which is some- what greater than its chord, will be still less,— so that arc G a will be equal, very nearly, to one-seventh of the circumferenGe,-T-and the chord of G a, very nearly equal to the side of a regular heptagon inscribed in a circle. — Geometry, Plane, Solid and Splierical, p. 121. Obs. III. — In the construction of polygons it is useful to ascertain the magnitude either of the angle, --. AOB at the centre, or of the angle ABC, x-^^^^nT^ formed by two adjacent sides. /y^ ^X As before, in the formulas, let it denote 2 rt. ^s, \ Y / or the semi-circumference ; 27r, 4 rt. /.s, or the whole \ \ /V^, / / circumference ; u, the number of sides ; and 9 the \ \ _.'' ! '\ / / magnitude of one angle, either at the centre, or at the Ni^^ ' 'J/ concave boundary. A.^~~-?-^\B Is*. To find the magnitude of an ^ at the centre, as l_ AOB of a polygon. n d = sum of all the £s at the centre, or 2 tt = 4 rt. Z.s, (Cor. 2. 15,, I.) .'. — = e, the magnitude of one Z, as ^i AOB at the n centre. 2nd. To find the magnitude of an angle, as /. ABC, formed hy two adj. sides of a polygon. n d = the §um of the magnitudes of all the interior /. s ; But, Cor. 1. 32, I., ?i e -h 2 TT = ra 7r; and by trans, n 6 = n tt — 2 tt = (» — 2). t- .: e = "'~ ' . TT = the mag. of one /. , as /. AB C, fonncd by two adj. sides. Note. — For n substitute the number of sides, 3, 4, 5, 6, &c.\ and for 2 it, 360°, and the magnitude of the angles is found in numbers. OBS. IV. ON -POLYGONS. 133 Obb. IV. — On a given rt. line, AB, to construct a regl ■polygbn, 1. By diricMng 2 ir = 360° by n the number of sides, we obtain the arc AB, or the measure of the /_ at the cen. Z. AOB ; and A AOB being isosc, Zb OAB, OB A are equal. Now 180°— z AOB = Z OAB = z OBA; and •.• Z ABO= ^ ZABC, .-. 180° - Z AOB = Z ABC, one of the angles of the polygon to which all the others are equal. Hence the Rule, or Formula ; IT — — = Z ABC, one of the eq. Zs of the polygon; at A and B make Zs OAB, OBA each = =^-— ; then from 0, with OA, or OB, desc. a © ; and in this circle place the st. line AB continually; a polygon of the assigned number of sides will be drawn. 2. By means of Trigonometry the radius of the circumscribing circle is calculated, assuming the side of the polygon to be 1, and' the following Table constructed. No. of Sides. Name of Polygon. Radius of Qirciunscribing Circle. Angle OAB or OBA. , 3 Trigon •5773503 30° 4 5 6 7 8 9 Tetragon Pentagon Hexagon Heptagon Octagon Nonagon •7071068 -8506508 rSide ^ radius 1-1523825 1-3065630 • 1-4619022 45° 54: 60 64f° 67J° 70° 10 11 12 Decagon . Undecagon Dodecagon 1-6186340 1-7747329 1-9318516 72° 73^T° 75° Then the units of length in AB x tabular rad. = units of length in OA or OB, the rad. of the actual circle, in which the st. line A B is to be placed, step by step, so that the polygon may be formed. 134 GRiJU^flQSS IN EBCLjp, Obs. V. — To palpjiJatiB numerically tjie areq, pf c^ny regular polygon, it is sufficient to find the area of one of the triangles into which the polygon may be divided, and to multiply that area by the number of. triangles ; Or, given a side, as AB, and OP the perpendicular, frmn the centre, on AB; then, - — :- X P = the area of the polygon. To facilitate the calculation when only a side, as AB, is given, the following Table has Jjeen forined on trigonometrical principles — the side of the polygon being 1. No. of sides. Badins of Inscribed CSrcle. Aiea of Polygon, or Multiplier. Trigonometrical Expression. 3 0-2886751 0-4330127 = i tang. 30° = ^^3- 4 0-5 1- = |tang. 45° =1 X 1. 5 G-?881910 1-7204774 = itang.54° =-fV(l+|v/5). 6 0-8660254 2-5980762 = 1 tang. 60° =6V3. = Itang. 64f . 7 1-0382617 3-6339124 8 1-2071068 4-8284271 = «tang.67i°=2x(l+^2). 9 1-8737387 6-1818242 = f tang. 70° 10 i -5388418 7-6942088 = 1^ tang. 72° =1^ (5+2^5). 11 1-7028437 9-3656^04 = V tang. 73 V = L2tang.75'^=3x(2+^3). 12 1-8660254 11-1961524 By using the Table we find, A B* X tabular area = area of the reg. polygon ; or, AB X rad. of inscr. © X — — = area. OBS. VI. ON POLYGONS. 135 Obs. VI. — DoDsos, in his " Calculator," supplies the following Tables for the calculation and cgnstruction of any rgg. polygOB haying nc^ more than 12 sides. 1° When the length of the side = 1. mdjusof ■ Sides. Dircumecribed Eadijis of A^fa. Circle. Inscribed. 3 0-5773503 0-2886751 0-4330127 4 0-7071068 0-5 1- . 5 0-8506508 0-6881910 1-7204774 6 1- 0-8660254 2-5980762 7 1-1523825 1-0382617 3-6339124 8 1-3065630 1-2071068 4-8284271 9 1-4619022 1-3737387 6-18l8g42 10 1-6180340 . 1-5388418 7-6948088 11 1-7747329 1-7028437 9-3656404 12 1-9318516 1-8660254 11-1961524 2° yVhen radius of circufiia(H'ibed circle =, 1. Sides. Length pf Side. Radius of Inscribed Circle Area. 3 1-7320508 0-5 1-2990381 4 1-414?136 0-7071068 2- 5 1-1755705 0-8090170 2-3776412 6 1- 0-8660254 2-5980762 7 0-8677674 0-9009689 2-7364102 8 0-7653668 0-9238795 2-8284271 9 0-6840403 0-9396926 2-8925437 10 0-6180340 0-9510565 2-9389263 11 0-5634651 0-^594931 2-973525Q 12 0-5176381 0-ag59259 3- 136 GRADATIONS IN EUCLID. 3° Whiii 7'(idtits of uiscribscl circle ^= 1. Sidts. Length of Side. ■ I'"-' '<' Rad. of Ch'cumscribeiJ Circle. Area. 3 3-4641016 2- 5-1961524 4 2- 1-4142136 4- 5 1-4530851 1-2360680 3-6327128 6 1-1547005 1-1547005 3-4641016 7 0-9631491 1-1099160 3-3710222 8 0-8284271 1-0823919 3-3137084 9 0-7279405 1-0641776 3-2757315 10 0-6498394 1-0514622 3-2491970 11 0-5872521 1-0422172 3-2298913 12 0-5358984 1-0352760 3-2153904 . . ■ . — i^ 4° When the area = 1. Bad. of Circumscribed Radius of Sides. Length of Side, Circle. Inscribed Circle. 3 1-5196716 0-8773827 0-4386912 4 1- 0-7071068 0-5 5 0-7623870 0-6485251 0-5246678 6 0-6204033 0-6204033 0-5372849 7 0-5245813 0-6045183 0-5446520 8 0-4550899 0-5946034 0-5493420 9 0-4201996 0-5879764 0-5525172 10 0-3605106 0-5833184 0-5547687 11 0-3267617 5-5799148 0-5564242 12 .0-2988585 " 0-5773503 0-5576775 Use and App. — By the help of these tables, and of the compasses, and a scale of equal parts, the construction of any regular st. lined figure is reduced to a simple calculation, to the drawing of a circle, and to the setting oif of equal chords on that circle. OBS. VII. ON POLYGONS. 137 Ex. — To construct a reg. heptagon with an area 144 times greater tlian the square on any one division of the scale of eq. parts. Increase the side and rad., in Table 4. in the proportion of ^ 144 to «/ 1, or of 12 to 1; then •5245813 X 12 = 6-2949756 = length of a side of the hfipt. •6045183 X 12 = 7-2542196 = „ of rad. of circumscribed Q, -5446520 X 12 = 6-5358240 = „ of rad. of inscribed ©, rrom the same centre with the two rtuiii, draw two cii'cles ; the side found above will measure the larger circle into seven steps; and chords, joining the • s in which the 0ce is cut, will touch the inner circle. Thus, a reg. heptagon will be inscribed, or circumscribed. Obs. VII. — In Arithmetic polygonal numbers are such as are the sums of a series of numbers beginning with imity and so increasing as to be repre- sentative of the figiire of a polygon. These F f numbers are subdivided into triangular, quad- Er rangular, pentagonal, &c., and may be ex- -nl plained by taking a pentagonal number. _|\ 1. Construct a set of pentagons A c C, A d D 1 &c., doublej treble, &c. of A fi B, in lineal B dimension ; . I Divide the sides of each pent, into parts each .-A. = corresponding side A 6 B ; Then, beginning with A, one point, and taking in all the points of pent. A i B, we have 1 + 4^5 points ; - Add all the • s of pent. A c C that are not in pent. A 4 B, and we have 1 -|- 4 + 7 = 12 points ; So for pent. A d D, 1 -|- 4 + 7 + 10 = 22 points ; For pent. Ac E, 1 + 4-1-7+ 10 + 13 = 35 points ; And pent. A/F, 1+4 + 7 + 10 + 13 + 16 ^51 points &c. Li_M ...1 ■I c d e ^ This series 1, 5, 12, 22, 35, 51, &c., is a series of pentagonal numbers, in a way similar to that in which 1, 4, 9, &c., are a series of square numbers. By aid of the Square on A/, the series of square numbers will be readily formed. 2. To find the ^liumbers wliicli bear the name of an n-sided figure. The mth number of the w-sided fig. = 1+ mn -^ (m—l) - 138 GRADATIONS IN EUCLIII. Form a series pf terijis beginpipg wit}; 1, and, by 3. com. difference = n — 2, increasing in Ai'iihmetical progressiou ; then the sums of the terms of the arithmetical series form the series of polygonal numbers. Thus, for Decagonal numbers, in which n — 2 =: 8 ; the series 1, 9, 17, 25, 33, 41, 49 &c. in arithmetical progression j gives 1, 10, 27, 52, 85, 126, 175 &c. Decagonal nupihers. Some of the pplygonal numbers are, Triangular, 1, 3, 6, 10, 15, 21, &c. Quadrangular, 1, 4, 9, 16, 25, 36, &c. Pentagonal, 1, 5, 12, 22, 3b, 51, &c. Hexagonal, 1, 6, 15, 2^,45, 66, &c. Heptagonal, 1, 7, 18, 34, 55, 81, &c. For a fm-ther accouiit, see Penny Cyclopedia, Vol. 16, p 364. Obs. VIII. As it has been obserTed, Euclid treats only of regular convex polygons; though, according to his definition of a reg. St. -lined figure, they are both equilateral and equiangular, he regular polygons with Figures of this liind are does iiot name Star-shaped polygons, ?.( re-entrant angles, as a, h, c, d, e, /, g, h, i. described, by first constructing any regular convex polygon, as might be at the points 1,2,3,4,5,6,7,8,9; and by then drawing successive diagonals, as 1 — 3, 2 — 4, 3—5, &c., so as to cut off that number of sides which is prime to the number of sides of the assumed convex polygon ; or what leads to the same results, so as to cut off a number of Sides which neither measures the number in the given convex polygon, norismeasured by any of its comrnon measures. To 5 the no. of sides in a pentagonj 2 and 3 are the prime numbers ; 7 „ ' heptagon, 4 and 3, and 5 and 2 „ 9 „ noiiagon, 2 and 7, and 4 and 5 „ 12 „ dodecagon, 5 and 7 „ 13 „ tridecagon 2 and 11, 3 and 10, 4 and 9, 5 and 8,, 6 and 7 ; KEMARKB BOOK IV. 139 And diagonals cutting off successively in a Pentagon, 2 and 3 sides, -will form ... 1 star-shaped polygon; Heptagon, 4 and 3, or 5 and 2 will form ... 2 „ polygons ; Nonagon, 2 and 7, or 4 and 5 will form . . 2 „ polygons ; Dodecagon, 5 and 7, will form... 1 „ polygon; Tridecagon, 2 and 11, 3 and 10, i and 9, 5 and 8, or 6 and 7 sides will form ... 5 „ polygons. A similar process will form other star-shaped polygons. N.B.— The figure as C at the centre of a star-shaped polygon, formed by the intersection of lines from the angular points, 1, 2, 3, &c., is also a regular convex polygon, with the same number of sides as there are re-entrant angles to the star-shaped polygon. Remaiucs on Book IV. 1. The -Problems of which the Fourth Book of Euclid is entirely composed require little classification ; they may be brought under the general heads ; — 1st, Of rt.-lined regular figures inscribed in circles; Pr. 2, 6, 11, 15 and 16; 2nd, Of rt.-lined regular figures, circumscribed about circles ; ■ Pr. 3, 7, 12 ; 3rd, Of circles inscribed in regular rt.-lined figm-es ; Pr. 4, 8, 13; 4th, Of circles circumscribed about rt.-lined Agues ; Pr. 5, 9, 14; 5th, Of constructing an isosceles triangle, having each angle at the base double of the vertical angle, Pr. 10. 2. The Use and Application of the Propositions of this Book have been given so much in extenso, that it would be superfluous to add to these remarks, except again to challenge attention to the value of theoretical reasoning as the guide in Geometry to most important Practical Results. GEADATIONS IN EUCLID. BOOK V. THE THKORY OF FROPOETION, OR OF THE COMPARATIVE MAGNITUDES OF PLANE FIGURES. This Book is entirely independent of the four books which preceded it. In the main they relate to the Properties of Figures on a plane surface, hut the fifth book introduces Properties of a more general kind, and though restricted by Euclid to the comparative magnitudes of right lines, really extends to all kinds of magnitude. It is no longer absolute equality, or inequality, which we have to consider, but the ratio, or mode of estimating the relative lengths of lines and the magnitudes of figures, and the proportion, or setting forth of those relative lengths or magnitudes. As in the Second Book material assistance for illustrating the properties of Eectangles, was derived from Algebra and Arithmetic, so in this book similar help will be obtained from the same sources. Indeed very many of the terms employed will be already familiar from their use in Arithmetic ; and it will be seen that the estimating of Eatios, and the setting forth of proportions, rest so entirely on a numerical basis, that to a very high degree the Fifth Book, or the Theory of Proportion, is an application of Numbers to the purposes of Plane Geometry. For this reason it will be of advantage to the Learner to be presented with a brief view of the Principles on which are esta- blished the Properties of Proportional Numbers and Quantities. If however, he has already mastered the subject, he may pass over the next few pages, and at once enter on the Theory of Geometrical Proportion. 142 GEADATIOHS IN EUCWD. SOME PROPEUTIES OP PROPORTIONAL NUMBERS, INTRODUCTORY TO EUOLId's THEORY OF GEOMKTRICAIj PROPORTION. We may comj)rtre two numbers together, eitllet by their diffSrertce, or by their quotient, i. e. by the number of times which the greater contains the less, or the less measures the greater. When we say 12 — 9 = 3, we compare 12 and 9 by their difference; but when we say 12 contains 9, one and one quarter times, we compare them by a division of 12 into 9 and a part of 9. The proportion which one number or quantity bears to another is often called its Hatio ; the ratio, measured by the difference is named an arithmetical ratio, — that measured by the quotient, — a geometrical ratio. Proportion is applied, either to an identity of difference between three or more numbers, as 12, 9, 6, where the common difference or the Arithmetical ratio is 3; or, to identity of relative magnitudes, as 12 : 9 ::.8 : 6, — where 12 contains 9 just as often as 8 contains 6, — the common quotient or ratio teingl J. When the differences are identical, the numbers are in Arithmetical Proportion; ■yhen the contents of each pair of terms are identical, the numbers are in Geometrical Proportion. The term projportion, taken by itself, is usually restricted to numbers in geometrical proportion; and of these we nave how to treat. Identity in the quotients of successive pairs of numbers constitutes PropoHiahi Take for an example, 15 : 5 : : 36 : 12; the quotient obtained on dividing 15 by 5 is the same as that obtained by dividing 36 by 12; -and these four numbers, 15, 5, 36, and 12, — or any othetfo'ur numbers fulfilling the condition of ec(t(ality' of qubtients iil each sticcessite pair, form a Proportion, Or set6f Propteioiials; The extremes arc the first and last terms in tie series; the terms placed between the first and last terms, the means; the tmi^ckdent is the first tefrii of a ratio; the consequent, the Second term. The ratio may be expressed, either by a whole number or by a fraction; thus in the proportionals, 18 : 6 :: 24 : 8, the constant ratio is 3; in the propor- tionals 12 : 9 :: 36 : 27 that ratio is ^ or IJ. If we take two sets of quantities in direct proportion, a : h :: c : d (which may also be written in a fractional fonn -r=~, ) we Can readily exhibit varibUS rules that are employed iii modifying ^ Proportion : the^ are all dependent on the principle that reiUltinc/ e^uittibhs art eqUtilh/ true wheneter the thihg which is dme on one aide of an equatidn is also dont on the other side. Take as an example — = — ' b d Rule 1. Multiply each side of the equation by h X d; we obtain — = -j-' h a or ad =: cb; .". the product of the extremes = the predict of the means ; and conversely, if the product of the extremes = the product of the means, the quantities are proportional; for, dividing each side by b d, we have — = — ' or - = —'i.e., a : b Si c : J are proportionals. oa od d PROPIi. NUM. — BOOK V. 143 KbLE a MnltiJ)!^ each side by *-!-the sides became ^>ss£iot ^=h, " ■ he do c d i.e.a : c := : d. Rule 3. Add unily to each side, and the equation is — + 1 = — -f- 1 . or 6 d ■ ' reducing the niixed quantity, i+_* = 1+J i.e.,a+ b:b = e-\,d:d. Rule 4. Subtract uniti/ from each side, we have the equatien— — i =S . b d —1 ! or —J- =^^°i.e., a — b:b = c~d:d. RnLB 5. Any common factor, as m, may be expunged, except from the two extremes, or from the two means ; for, if ^ =^M dividingboth sides . by m, the resultant is ^-?= .2J.. or, o_<; r,r If ^ — ^ • ni, multiplyingby»!,wehave;^"'=^,. or, again ?=-^. Or,if-^ ^ bm md' ' •^"^ i, — j- "> ^j c ma c .ma a ma c.., .,.,a c. -J ; .'. r-j — :) >■ . • • — -T = T s . .■ • — r = j >s identical withT-= -_ d m a mob m b d b d Rule 6. In_ two or 'esnltan:twillbeil=— ^' " f d h bf dh Three or more series of proportionals are compounded in a similar way. The Essential Property, or Criterion, of numbers in true proportion is, that ike product qf the extremes is equal to the product of the means; and reciprocally, if the product of any t*o numbers equa,ls the products of any other two numbers, the series of numbers constitutes a proportion; thus, in 15 : 5 : : 36 : 12 ;— 15 X 12 = 180 = 5 X 36; .•. a true proportion. in 14 : 5 :: 35 : 12 ;— 14 X 12 = 168; — 5 X 35 = 175; .-. afalse propoiiion. From the Essential Property of numbers in true proportion it follows, that when any three terms are given the fourth may be found; for on dividing the product of the means by *he given extreme, or that of the extremes by the given mean, the other exti-eme, or the other mean, will be ohtained; thus in 6 : 8:: 24 : a:; ,• 6 X = 8 X 24; .•. * = — 4r— = 33, and thetermsai-e 6 : »= 24 : 32. ' 6 Or, in 7 : X : ; 14 : 6, •7 X 6 = 14- X, .•. = ^ ^ = 3; and the terms are 7 : 3 = 14 : 6. ' ' 14 If each of the means is the same numjier, as in 3 : 6 :: 6 : 12, the product of the extremes eqiials the square of one of the means; the value of one of the 144: GRADATIONS IN EUCHD. meuns is therefore the square root of the product of the means; thus in i: X :: X : 16,. ,*. 4X16 = x'', .•. a; = ,y 64 = 8; and the terms ai-e 4 : 8 :: 8 : 16. When the value of one of the means equals the square root of the product ■of the extremes, that number is named a mean proportional between two given numbers. Stated algebraically, when a ■■ x =: x : b, then x''=. ab; or j: = ^ab; ^ being the mean proportional bet\veen a and 4. There are other important properties of numbers in proportion resting on the A C equation -which may be formed from four proportionals, -=j=—. For instance as shown above,— add 1 to both sides; ^ + 1 =-^ + 1, or:^ + ^= ^ + "i which is the theorem known by the word componendo, putting together : Or, taking 1 from both sides; A _ i = C _j . ^ A-B __C-p . ^.^^.j^ j^ ^,^j. theorem indicated by diuidendo, taking apart. Again, on dividing unity, ih'st, by g, andnext, by ^, wo have-| -H - =-^, and L- ^ ^ =_|? j thus- giving. invertendo, by inversion, _ =_ ; And on multipljing each side by^^;, we have :^ X -g = -q' and --X -p' =-rj ''■ <>., alternando, by taking each other one, A_ B C ~D- These Properties, with their variations and combinations we now mention in their order. 1°. Multiplicando, or dividendo, by multiplying, or dividing by the same number, the two first terms or the two last, — the two antecedents or the two consequents,— the equation and consequently the proportion is nndisturbed. Take the Proportionals 9 : 12 :: 16 : 24; Multiplicando, by 3 ; Dividendo, by .",. 1 & 2 tenns, 27 : 36 : : 18 24; 3 4 : : 18 24; 3&4 „ 9:12: : 54 72; 9- 12 : G S; 1 &3 „ 27 : 12 : : 54 24; 3 12: : 6 24; 2&4 „ 9 : 36 : : 18 72; 9 4 : : 18 8; In all these changes the product of the extremes remains equal to the product of the means, and the numbers resulting are therefore still proportionals. In &ct the ratio of the two first terms ^^, or that of the two last W^,is the same fractional number |, — and fractions undergo no change of value when the terms of the fraction are both multiplied or both divided by the same number. Also when the product of the extremes and that of the means are each multiplied or divided by the same number, the resulting products or quotients are also equal, and the proportion remains. 2°. Alternanito, or invertendo, by taking the tenns alternately, or by inverting the terms, the proportion also remains undistm'bed; I'KOP. NUM. BOOK V. 145 Time in the proportions 12 = |4, 1st, exchange the extremes, S.5 = 1^. the ratio being ^, 2nd, „ means, ^ = V; „ „ ^, Srd, „ means & extremes, t. = SLA. „ „ i- But the proportions continue unbroken though the ratio in each case if> diiferent; for after the various exchanges the criterion is satisfied, — the product of the extremes doe* equal that of the means. 3°. Componendo vel dividendo, by putting the two first terms of the proportion together, and the two last; orb7separatingthetwo,firstand the two last,— there arises the proposition, that the sum or the difference of the two first terms is to the second, as the iiumor difference of the two other terms is to the fourth: for example. In the proportion 24 : 8 :: 45 : 15, the com. ratio is 3; componendo by adding 24 + 8 : 8 :: 45 + 15 : 15, „ „ 4; dividendo by subtracting 24 — 8 : 8 :: 45 — 15 : 15, „ „ 2; Tlie explanation to be given is, that when we increase or diminish each antecedent by its consequent, we do nothing except increase or diminish by unity each of the two ratios ; and since these ratios were equal at first, they remain equal after such an increase or diminution: for changing the means, 24 + 8 : 45 + 15 :: 8 : 15; (1) but 24: 8:: 45: 15; (2) or, 24: 45 :: 8: 15; (3) & •.• two ratios equal to a third are equal to each other, .-. 24 + 8 : 45 + I5 :: 24 : 45; or rather 24 jf- 8 : 24 :: 45 + 15 : 45- Thus the sum, or the difference of the two first terms, is to the first term, as Me sum, or difference of the two other terms, is to the third term. 4°. Generally, addendo vel subtrahendo, by increasing, or diminishing the sum, or the difference of the antecedents is to the sum, or the difference of the 'Consequents, as any one of the antecedents is to its consequent Take the proportion, 24 : 8 : : 45 : 15 ; exchanging the means, 24 : 45 :: 8 : 15; by the former property 8°, 24 + 45 : 45 :: 8 + 15 : 15; again exchanging the means, 24 + 45 : 8 + 15 :: 45 : 15; whence by reason of the common ratio 45 : 15, 24 + 45 : 8 + 15 :: 24 — 45 : 8 _ 15; or, rather exchanging the means, 24 + 45 : 24 — 45 : : 8 + 15 : 8 — 15. i. e., the sum of the antecedents is to their difference, as the sum of the consequents is to their difference. CoH. Let there be a succession of numbers, forming, two by two, equal ratios, tlie sum of all the antecedents is to the sum of all the_ consequents, as any one antecedent is to its consequent. Assume the series, 8 : 12 :: 2 : 3 :: 4 : 6 &c., or, a : b z= c : d=: e :f, &c. the ratios 8 : 12 :: 2 : 3, a : b :: c : d; by 4° give 8 + 2 : 12 + 3 :: 2 : 3; a + c : b + d= c : d; but 2 : 3 :: 4 : 6, c ■.d= e:f; 8 + 2: 12 + 3:: 4 : 6 a + c : b + d :: e -.f; applying 4°.8 + 2 + 4: 12 + 3 + 6:: 4: 6 a + c + e-.b + d+f=e:f. and so on, whatever may be the number of equal ratios. L 146 GRADATIONS IN EOCLID. If there be any number of fractions equal in value, as -^, -J, || &c., if the sum ov the difference of the numerators axii that of the denominators be taken the resulting fraction is equal in value to each of the yiven fractions : thus, in P- 4 4 8 + 4 + 2 = 14 or 2,and 12 + 6 + 3 = 21 or 3, the resulting fraction being ^'f or -J, The equation ^ = f leads to the proportion 8 : 12 •: 2 : ,!; 8-4-2 whence by the fljregoing property, ^^_ = 14 = .0. = .3 5°, COMPOUPTD KATIO; Cmponendo, by placing together or combining. In any number of proportions, if all the corresponding antecedents and consequents he respectively multiplied together, the resulting products will be in proportion. For example, 3 : 8 :: 12 : 32, or, a : b :: c : d; 7: 15:: 28: 60, «:/::«: A: 40 : 12 :: 50 : 15, J:A::/:m, ^c. being proportionals they may be represented by A = 1^ or, « = 1. 8 32, b d' -L — ?? £ = 3- 15 ~ 6O' / ft' 40 _ 50 i—ls), 12~"15' A"™'*"-' on multiplying the corresponding sides of these equations, there will result the equal products, 3 X 7 X 40 _ 12 X 28 X 50 iae _c g I _^ 8 X 15 X 12 32 X 60 X 15 ' hfk dhm' i. c, -^ = 1?800 g^Q . J44Q .. 15800 ; 28800, 1440 28800 satisfying the criterion of proportion that 840 X 28800 =; 1440 X 16800. If we divide by 20, we have = — — , an identical proportion, 28800 ■' ' 1440 1440 the teiins of the two ratios being the same. N.B. — The co««toBtm«8o of the preceding proportion, namely —- is equal to the product of the three constant ratios of the given proportions. Thus the throe constant ratios being 4 JL. A.%, or 10 ^ wo have for their product |.Jg; or suppressing Hie common factor 30, X, to which the fraction 8JU> may be reduced by suppressing the common factpr 120. The ratio ^, thus arising from thq multiplication of sevarftl other ratios, is named the compoutid ratio. 6°. From the theory of compound proportion it follows, when four numbers are i^ proportion, let, their squares, cubes, and other like powers are also in proportion, ani 2nd, their square roots, cube roots, and other like roots are in orlm&rtion. PROP. NUM.— BOOK V. 147 Ut, Take a : ^ :: c ; d, or, 4; 6 ;; 8 : 12; squai-ing; ai-.b'r.e^:^, Ifj : SQ ; 64:144; cubing, as: hi: c«: da, 64 : 216 : 5lg : l?2S. Thus there results a series of proportions which, multiplied in the order of the antecedents and consequehts, will give products also in proportion. 2nd. Take a" : b^ :-. c'' : d% or, 4 : 9 :: 16 : 36 ; extraoting'the^, o: 6 :: c :rf, or, 2:3:: 4: 6; u^ c^ i 16 '.• tlie ratio r^ =?: ^^ or - =: — , .•. tiio ratios of the roots are equal. Put to extract the square root of a fraction, as -J or 4^^ wc extract tjje square root of the mimerator and that of tlie denominator, which gives v? 9 3 v' 36 a 7°, Irrational numbers, or numbers without a perfect root, and Incommett- surMbks. When quantities as a, b, c, d, or numbers, as 8, 3, 8, 12, are mit perfect squares, the quantities ^ a, ^ b, ^ c, J d^ or >J i, J 3' V 8, V 12, are named irrational quantities or Rvunbers ; wd when quantities or numbers are so related, that, although one of them may be represented iu the terms of a certain unit, the other cannot be so represented, such quantities or numbers ai'e named incommensurable. The diameter of the circle in relation tfl tjie circumference, and the diagonal of a square of whjcli the side is unity in relation to that side, aj-e femiliar examples. Also in numbers, f^ 2, ^ S,i^ 15, &c., are incommensurable with unity, for there is no mixed nuniber, nor fraction, exactly equivalent to them. Proportion however exists ' between incommensurable numbers, for ^ 2 : v^ 3 = /^ 8 : v' 12; and we we led tP consider ratios, which must in general be regarded ?« irrational, because be- tween numbers without a perfect root : but by ratio we must bear in mind that we mean the proportion between quantities or numbers ; — and by irrational, the non-existence of an exact root. Now the question arises, can we apply to proportions of this kind, i.e., between irrational and incommensurable quwtJtJISi or numbers, all the properties which have just been established f The answer is affirmative, if we remember that an irrational ntmiber may always be replaced in the mind by an exact fractional number which only differs from the proposed number by a quantity so small tbat, when we neglect it, we need not have aiiy regard to the mistake committed j and it is between the commensura,ble numbers substituted for the irrational magnitudes that the ratios are judged to be established. As to the ratios between exact fi'aetional numbers, it is easy to understand, that according to the rule for the division of fractions, they may always be re- placed by the ratios between whole numbers ; thus, the ratio of -2. to s being the quotient aiising from the division of A by _5_ is equal to 3 X U , or to 4-?-, that is to say, to the ratio of S3 to 35. Agfiin, the ratiq of J tq |_^ is equal to | x f f' W to the ratip of iqi to 120. 148 GllADATIONS IN EUCMD. Thus, all the properties of proportion are true with respect to numbers, whatever may be tiie numbers about which we reason. Similar truth and cer- tainty will appear even in a higher degi'ce when we consider ; THEOEY OP GEOMETRICAL PROPORTION. By some writers this fifth book of Euclid has been named the Elements of Mathematical Logic. The other books have shown us the absolute properties, the equality or inequality of Plane Geo- metrical figures ; — this book enables us to institute a comparison between them. The Definitions and Propositions indeed are so expressed, as if they applied only to such plane figures ; but they extend equally to lines, surfaces, amd solids, and to every species of quantity. Whatever be the science or branch of Imowledge that depends upon Proportion, it is founded on the Principles con- tained ill the fifth book of Euclid. We take, for instance, Practical Geometry; — nearly all its measurements are calculated by the Doc- trines of Proportion, — the survey of an estate, or of a whole king- dom, is carried out by the application of those doctrines : and in Arithmetic, Astronomy, Statics, &c., the use of this book is indis- pensable. " In fine," says an old writer, " one may affirm, that if one should take away the knowledge of the Propositions that this Book giveth us, the remainder would be of little use." The Definitions and Propositions of the fifth book may be ex- tended to every species of quantity and magnitude ; and may be easily applied to number ; but though our clearest notions of ratio and proportion are derived from, numbers, Number, as we have said before, is only to a small extent the actual subject of Geometry. N.B. — The text of the Definitions and Propositions is from Simson's Edition; — ^variations are given in the notes, and if between brackets, or " " they are from the Greek text of Eiichd. Definitions. l.—rA less magnitude is said to be a part of a greater magnitude, when the less measures the greater ; that is when the less is con- tained a certain number of times exactly in the greater. " A part is a magnitude of a magnitude,— the less of the greater, when it exactly measures the greater," — Eucun, DEP. II.^BOOIt Y. 149 " For the clearer vnderstandyng of a parte, it is to be noted, that a part is taken iu the Mathematical! Sciences two maner of wayes. One way a part is a Icsse quantitie iu respect of a greater, whether it measure the greater or no. The second way, a part is onely that lesse quantitie in respect of the greater which measuretli the greater." — ^Billingslet, fol. 126. By part is meant an aliquot ■pent, oi submultiple,— not any portion of a whole or greater magnitude, but the portion, which if repeated will exactly make up that whole, or greater magnitude ; thus, if A represents a line = 3 inches, ajid B a line ^ 9 inches, A = 3 is an aliquot part or submultiple of B ^ 9 ; — A repeated exactly measures B, and B is the multiple of A. In numbers 4 and 6 are submultiples of 24 ; and tlie quantities, a, b, c, d, of a b c d. Aliquant part has been used for the less magnitude, which though repeated does not exactly make up the greater ; thus a measure of 3 feet is an aliquant part of 7 feet. The magnitudes compared must be of the same kind, — Unes, or surfaces, or Bolids, — ^weight, or time, SCC. Lines bound surfaces, but are no part of them ; an hour cannot be measm-ed by an ounce, — and a penny is no submultiple of a mile. ■ , , One magnitude measures another when it is contained in that other mag- nitude an exact number of times. And a magnitude which is a measure of two or more magnitudes is . named the common measure of those magnitudes. II. — A greater magnitude is said to be a- multiple of a less, when, the greater is measured by the less, that is, " when the greater contains the less a certain number of times exactly." " The greater magnitude is a multiple of the less, when it is exactly measured by the less." — ^Euclid. " The Multiplex is a great quantity compared with a less which it contains precisely some number of times." — ^De Chales. " By this worde multiplex," says IBillinoslet, fol. 127, " which is a terme proper to. arithmitike and number, it is easy to consider that there can be no exact knowledge of proportion and proportionaJitie, and so of this fifth booke, wyth all the other bookes followyng, without the ayde and knowledge of num- bers." If excuse be needed, this is the reason why' we have prefixed, " Some jProperties of Proportional Numbers to'EucLiD's otherwise matchless "Theory of Geometrical Proportion." In things of the same kind every greater magnitude contains the less ; thus a vessel of 27 cubic inches contains another vessel of 13 cubic inches — but the greater is a multiple of the less only when it is measured exactly by the leas, — !. e., by its submultiple ; thus a piece of cloth 27 yards in length is the multiple of another piece 9 yards in length, — 3 repetitions of the less exactly making up the greater. Equimultiples are magnitudes containing their respective aliquot parts the same number of times ; thus one yard and one barrel are equimultiples of an inch and a galion ; — for the yard contains the inch 36 times, and the' barrel also contains the gaUon the same number of times, ISO OKADATtONR TN EUCLID. AU magnitndes for which an exact common measure, or submtlltit)Ie, can be found are Commensurahle magnitudes, as the itiditts and diameter of a circle, and the angle of an eqnilateral triangle, of a square, ol- of a reg. hexagon, and the space round any point : and those magnitudes are Jncommensurable which have no common measure ; thus the diagonal of a square represented in nm^bers, and the side of that square have no common measure ; — for if the side contains 100 units, the diagonal will contain less than 142 and more than 141 ; — what is the submultiple of tlie one is not the submultiple of the other. The s^imo is also trae, — 1°. of the diameter and circumference of a circle ; 2°. of the diagonal and side of a cube ; 3°. of the segments of a line cut in extreme and mean ratio. !None of these have a common measure, neither have they a common multiple. III. — Batio is a mutual relation of two magnitudes of tlie same ktod to one another in respect of quantity. A mistake in translating Euclid's x-ara, TuXixoTwa " in respect oif quan- tity," has tended to confuse this definition. The how great one thing is when compared with another is the hinge on which the definition turns. Euclid is spealdng of two magnitudes with respect to the spaces which they occupy, whether lengtji, or area, or billk, and his meaning therefore is better expressed by saying that " BatiO is the relative size which two magnitudes of the same idnd have to one another with respect to the space which they occupy." A square of six acres in area, though greater than a Square of foiir acres in area, is, when compared with foiu- acres, a less magnitude than a square of three acres is, when compared with a square of one acre ; the space which the six acres occupy is only one and a half times larger than the space which the four acres occupy,-^but the space comprised in three acres is three times larger than the space com- prised in one acre. Thus the how ffreat one tiling is, when compared With another, is the essential idea which belongs to Euclid's definition of ratio. Certainly the how great is best expressed by numbers ; an algebraical notation, m times, or n times, or — times, may denote generally that one magnitude B is m times A, or — timesAjbUt "the particular ratio of two given magnitudes. Whether commensurable or otherwise, can be " expressed or "conceived only by means of the numbers which denote how often the same m^nitude.is contained, or nearly contained, in each." With- out these numbers wc form no idea of the relative taBgnitnde of the two given magnitudes, — for numbers conetitute, eith« its exact, or its proximate measure. See Geom. Pl. Sot. & Spher. p. 32. A itttid is expressed by two terms, as A : B, or 6 : 12 ; the foundation from which the comparison proceeds. A, is named the antecedent, and the - lerm to which the comparifeon extends, B, the consequent. Ih twd dOtnmefisurible magnitudes the numerical ratio of one to tlie other, " is ft certain number, Wbolo or fractional, which expresses ?iow many, and what parts of the second .-ive eontnined in the first ; for example, if DEP. IV-V BOOK V. 151 the common measm-e of A and B be contaioed in AfiM times, and in B six times, or, which is the same thing, if A contains Sths of B, then A is said to have to B the numerical ratio ' 5 to 6,' which is written 5 : 6, or, in the fractional form, 4." The mfflrare of a ratio, or how great one magnitude is when Compared with another of the same kind, is determined, by ascertaining Jtoie often the first magnitude is cont^ned in the second, or what part the first magnitude is of the second j if one line, A Bj contains 12 units of length, and another line, C D, 4 of the same units, the measure of their yelative magnitudes = 12 -^ 4, or 3 ; and if one line, E P, contains 3 units, and anotlier G H, 12 units, the measure = 12 -^ 3, or j. When A = B, the ratio A : B is one of equality ; when A is > B, the ratio is of greater inequality; and when A is < B, it is of less inequality. The inverse or reciprocal ratio arises from changing the order of its two terms; as, instead of A : B, or 5 : 6, making the ratio B i A; 6 ; 5 or 4. IV. — Magnitudes are said t& have a ratid to one another, when the less can be multiplied so as to exceed the other. " Magnitudes are said to h*ve a ratio to one another, which are able on being multiplied to exceed one another." — ^Euclid. " In Geometry, multiplication is only a repeated addition of the same Biag- nitude, and division is only a repealed subtraction." — Potts. The successive foldings up of a stting. Or of a piece of cloth, in folds of the same length, is geometrical multiplication ; the successive unfoldings, or cuttings off, of pieces of the same leng&, is geometrical division. By this definition is excluded the cotaparison between any two magnitudes of which onfe is finite and the other infinite ; — for no addition of finite things can ever equal, much less exceed, the infinite ; and for the same reason we cannot institute a comparison between two magnitudes of which one is infinitely small and the other infinitely large. V^ — Definition, of Proportion. — The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and foUrth ; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth; or, if the niultiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth. " Magnitudes are said to he in the same ratio, the first to the second, and * the third to the fourth, — ^when the equimultiples of the first and third, being at the same time compared wjth the equimultiples of the second and fourth, each with each, are, whatever the multiplication may be, together greater, or together equal, or together less."— EtFCSHD. A. B. C. D. 2. 4. 3. 6. E. F. G. H. 10. 8. 15. 12. K. L. M. N. 8. 8. 12. 12. 0. P. Q. B. 6. 16. 9. 24. 15!? GRADATIONS IN EITCLID. The whole reasoning of the fiftli book rests on this celebrated definition as its foundation ; — ^it is indeed, the definition which supplies a criterion for determining the equality of two ratios. We shall exemplify its meaning by taking four magnitudes A, B, C, D ; and of A and C equimultiples E, G, five times A, C ; and of B and D equimulti- ples F,H, twice B, D : also K, M, four times A, C ; and L, N, double of B, D ; finally, O, Q, thrice A, C ; and P, E, fom- times B, D ; now, VE>r, andG>H; K=L, andM = N; < P, and Q < B ; .-. A : B = C : D. Or, Let a : b = e : d; and let a, c, each be taken m times, J and d each n times ; then, 1. If »i a > n 6 ; also m c > n d; 2. Ifma = 7i6; „ mc = n d ; 3. Ifma indicating the diviijion of the aptecedent by the consequent. DEF. VII-X. BOOK V. 153 VII. — When of the equimultiples of four magnitudes, (taken as in the fifth definition), the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth ; then the first is said to have to the second a greater ratio than the third magnitude has to the fourth ; and, on the contrary, the third is said to have to the fourth a less ratio than the first has to the second. A greater ratio exists, if the first magnitude contains any aliquot part of the second a greater number of times than the third contains the like aliquot part of the fourth ; for example, in the four quantities, a, b, c, d, let jna > n 6, but m e < n d, then a : i > c : d ; or, let m a < n 4 but mc> ndthena : b <, c : d. Or, take four numbers 101, 10, 200, 20 ; then, ".' 101 contains one hundred and one times the tenth part of 10, — but 200 contains only one hundred times the tenth part of 20, .•. the ratio 101 : 10 > the ratio 200 : 20 ; and .•. 101 is a greater number with respect to 10 than 200 is with respect to 20. VIII. — Analogy or Proportion is the similitude of the ratios. " Analogy is the sameness (identity) of the ratios." — Buclid. Analogy is a reasoning out of two sets of comparisons and a declaring of their identity ; thus, the admiral and the fleet are one set of relations, — the general and the aimy another, — and our reasoning is an analogy, when we declare, as an admiral to his fleet, so is a general to his army. Katio is the comparison instituted between two magnitudes, — Analogy, the comparison between two ratios, — the analogy being complete or perfect when the ratios are identical. Take four magnitudes A, B, C, D, and from them form the ratios A : B and C : D ; if the one ratio equals the other, there exists a proportion between the magnitudes,— A: B :: C : D. This proportion ' does not establish the absolute, but the relative magnitudes. IX. — Proportion consists in three terms at least.. The middle term being repeated, there are in reality four teims, without which a proportion cannot be established; thus, n : 6 : ; 4 : c, or 2 : 4': : 4 X. — When three magnitudes are proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second. A duplicate, or two-fold ratio is a ratio compotmded of two equal ratios ; thus if a : 6 : c, then a:c=T-X-;;; i or2:4;8; then ■§■ = s x J- = _a- = 1. 3S 4." 154- ORADATIONS IN EVCUII. The duplicate ratio is expressed algebraically and arithmetical^ by tlie ratio of their squares ; thus, a : 6 : o;— a : c = o^ : i", or 2 : 4:8; 2 : 8 = 4 : 16. Such magnitudes are in continued proportion ;— for magnitudes ai'e con- tinued prbportionals, when every two terms have always the same ratio; or when the first has the same ratio to the second as the second has to the thirdj-^and the second to the third the same as the third to tlie fourth; thus, a : h : c : d : e, &c., or 2 : 4 : 8 : 16 : 32, &c,, the oonimon ratio being 2. In continued proportionals the first and last terms are the extremes, the intermediate are the means. A mean proportional is a magnitude between two other Magnitudes, forming with them a continued proportion ; and the third proportional is the magnitude in continued proportion with the two other magnitudes. A double Ratio and a duplicate Ratio must not be confounded. A double, triple. Sec, Batio is So called when the antecedent is double, triple, &c., of the consequent ; a duplicate Ratio is a ratio compounileci of two equal ratios, as in the proportionals, 2 : 4 : 8, or 3 : 9 : 27; — 2 : 8, or 3 : 27 being compounded of 2 : 4, and 4 : 8; — or of 3 : 9, and : 27 ;— in the one instance it is the half of a half, — in the other the third of a third. The half of a half is the square of ^, — the third of a third is the square of J ; 2 is J of the \, or i of 8, — and 3 is the i of the i, or i (if 27.' In like manner 8 : 2 is a duplicate ratio of 8 : 4 because 8 is the double of 4. Duplicate ratio is a species of compound ratio, of which instances oCCnr in propositions 19 and 20, bk. vi. XI. — When four magnitudes are continual proportionals, the first is said to have to the fourth, the triplicate ratio of that which it has to the second, and so on, quadruplicate, &C., increasing the denomination still by unity in any number of proportionals. "When four magnitudes are proportional, the first is said to have to the fourth a ratio triplicate of that which it has to the second ; and so on sucpessively in order, as far as the analogy (or proportion) mav extend." — Buca-tD. Triplicate ratio, compounded of three equal ratios, is the ratio of the Cubes; thus, ifA:B:C:D, or 2:4:8: 16;— then A : D = Aa : B3, or 2 : 16 =8 : 64. The ratio 2 : 16 is triplicate of the ratio of 2 • 4c for 2 is * of 5- of i, or i of 16. ^ Triplicate ratio is a species of compound ratio, instances of wliich are found in the 11th .and 12th books of Euclid on the Geometry of Solids. A immerous list of the kind.s of proportion might be given; as, Duple pro- portion, triple, quadruple, quintuple,—Sesquialto,r, sesqujtertia, sesqiii- qUarta, — Super-partiens, super-bipartiens, super-tripartiens, &o — Multiplex, super-particular, &c., &c. ;— but most of them arc rather ingenious puzzles than of practical utility. DEF. A-B. BOOK V. 158 Def. a — Of Compound iZaiw.— When there is any number of magnitudes of the same kind, the first is said to have to the last of these the ratio compounded of the ratio which the first has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and so on unto the last magnitude. for exMnple) if A, B, C, t), be four magnitudes of the same kind, tlie first, A, is Bttid to have to the last, D, the ratio compounded of the ratio of A to B, and of the ratio B to C, and of C to D ; or the ratio of A -. D is said to be/;ompounded of the ratios of A : B, B : C, and C : D. So 4, 5, 3, 11 are four numbers, and 4 : 11 = A ^ 4 X -t- — ^'^ • l^'''- And if A : B = B : P, B : C = G : H, G : D = K i L; then A : D is a ratio compoundfifl of ratios, which are the same with the ratios of E : F, G : H, and k : L. And the same thing is to be understood when it is more briefly expressed by fiaying. A' has toD the ratio compounded of the ratios of B to F, Cr to H, aiid K to L j thus, If A:B = E:F; B:C = G:H; &C:D = K:L; tlien, A : D = E iF, ofG:H, ofK:L. Or, if 3 : 4 ±= 6 : 8 ; 4 : 5 s: 8 : 10 ; and 5 ; 7 = 10 : 14 ; then, A = 6 of A of i_» = 6 : 14. . IT I p 1.4 In like maimer, the same things being supposed, if M hap to N the same ratio which A has to D; then, for shortness, sate, M is said to have to ST the ratio compounded of the ratios of E to F, G to Hj and K to L. Thus If A : D s= M : N; tiienM : N = E : P, of G : H, of K : L. Or,,if 3 : ? =21 : 49j then 2l : 49 = | X A X >» = -A^ = 3:7. It may he observed, 1°., that the product of the fractions which represent numerical ratios corresponds to tlie compound ratios of magnitudes; 2°. in ratios compounded by this definition, the second term of each ratio is the same as the first term of the following ratio,— the consequent of the former becoming the antecedent of the latter. Dei?. B— There is a specks oi Progression in the lengths of chords •of the same thiclmess and degree of tension, which produce the mu- sical sounds of a certain note, of its fifth], and of its octave. Tims, if a musical String, C 0> be divided so that its parts are in the pi'oportiofi to one another as the numbers, i, % f, i, j, f , ^, }, the 15C GP.ADATIOXS IN EUC'tlD. vibrations of the respective parts, C O, D 0, E 0, P 0, G 0, A 0, B 0, c O, will jdeld the eight sounds 8 ■* s 2 5 0. 1 9 5 4 3 5 15 2 CDEFCrABc O to which musicians give the names, C, D, E, F, G, A, B, c. Of these parts of a musical string, thus divided, it is the property, that the first is to the third, as the difference between the first and second is to the difference between the second and third. Thus 1 : -| = -^ : -Ju. Three straight lines, therefore, are said to be in Harmonical Progression, when the first is to the third as the difference of the first and second is to the difference of the second and third. And when three lines, C 0, D O, and E O are in harmonical Progression, D O is named a liarmonical mean between C O and E O ; and E is named the third hdrmonyiol prOg^essional to C O and D 0. In the same way magnitlides 6f any other kind are in harmonical pro- gression, when the first : thirtl : : the dif. between the first and second : the difference between the second and third ; thus, And any number of lines or magnitudes are in harmonical progression, when evciy successive three are in hannonical progression. XII. — In proportionals the antecedent terms are called homo- logous to one another, — as also the consequents to one anothel*. " Magnitudes are said to be homologous, — the antecedents to the ante- cedents, and the consequents to the consequents." — Euclid. Homologous magnitudes are those which correspond in the proportion ; thus in A : B = C : D, A and C being antecedents, are homologotis ; and B and D, being consequents ; but in A : B : C : D, A and C, Band C, B and D are homologous, — for the full statement of the progressionals is, A : B=B : C = C : D, — ^where B and C are at one time consequents, and at another time antecedents. TEOHNICAti WORDS TO DENOTE CHANCiES IN TUB ORDEU OP PEOPORTIONALS. 1. FOR FOUR PROPORTIONALS. XIII. — Permutando, or altemando, by permutation, or alter- nately. This word is used when there are four proportionals, and it DBF. XVI. BOOK V. 157 is inferred that the first has the same ratio to the third which the second has to the fourth ; or that the first is to the third as the second to the fourth ; as is shown in Prop. 16 of this fifth book. Thus, IfA:B = C:D, or 1:2 = 3:6; then altemando A:C = B:D; or 1:3 = 2:6. "Alternate ratio is the comparison of the antecedent to the antecedent and of the consequent to the consequent." — Euclid. XIV. — Invertendo, by inversion ; when there are four propor- tionals, and it is inferred that the second is to the first, as the fourth to the third. Prop. B. Book V. Thus, AC B T) If =j- = = , or J = f ; then invertendo, — = _i or ? = 4 ±5 JJ A U " Inverse ratio is the taking or comparing of the consequent as antecedent with the antecedent as consequent." — Euclid. In all changes of the order of Proportionals it is essential, that if one ol' the means be exchanged for an extreme, the other mean must also change places with the other extreme ; and that if one of the extremes be placed as a mean, the other extreme must be placed as the othev mean ; thus, if A : B = C : D, or I : 2 = 3 : 6; then A : C i D • B or \^l XV. — Componendo, by composition ; when there are four pro- portionals, and it is inferred that the first together with the second is to the second as the third together with the fourth is to the fourth. Prop. 18. Book V. Thus, IfA:B = C:D; thenA+B:B = C + D-D • Or, 1:2 = 3:6; then 1+2:2 = 3-1-6:6. " The Synthesis, or Composition of ratio, is the taking of the antecedent along with the consequent as one term, in compai-ison with the conw quent."— Euclid. Generally, the sum of the first and second is to the first or second, as the sum of the third and fom-th to the third or fourth ; thus if :^ =5 ^''^ A^= §^' "■' '^1- = l> *^" t = i2, and I = j_o_ XVI. — Dividendo, by division; when there are four propor- tionals, and it is inferred that the excess of the first above the 158 OBADATIONti IN BDOLID. second is to the second, as the excess of the third above the fourth is to the fourth. Prop. 17, Bk. V. Thus, If A : B = C : D, A being greater than B, and C than D; then A-B:B= C-D:D; or, if 3 : 1 = 6 : 2 ; then, 3-1:1 = 6-2:2. " The Diceresis or Division of ratio is the taking of the excess whereby the antecedent exceeds the consequent in comparison with the consequent." Euclid. Geometrical Division, as we have seen, is the successiro subti'action of a less magnitude from a greater. Generally also, — the difiFerence of the fcst and second is to the first or second, as the difference of the third and fourth to the third or fowth ; thus, if -^= — or I- = -JL • then, — Or, B B'^ -I' 'AorB C or D 2 oj 6 8 (VJ 24 2 or 6 8 or 24 XVII. — Convertendo by conversion ; when there are four pro- portionals, and it is inferred that the first is to the excess above the second, as the third to its excess above the fourth. Prop. E., Bk. V. Thus, if A: B = C : D, or 2 : 1 = 6 ; 3; then A: A - B = C: C- D, or 2: 2 — 1 =6: 6-3. " An anastrophe or reversion of the ratio is the taking of the antecedent in comparison with the excess whereby the antecedent exceeds the conse- quent." — Edclid. Combining several definitions into one, — conjumendo, the sum or diffi>j;ence of the first and second is to the first or second, as the sum or diflferenoe of the third and fourth is to the third or fourth ; — or the sum of the first and second is to their difference, as the sum of the third and fpxu'th to their difference. 2°. FOB ANY NDMBKE OF PROPORTIONALS ABOVE TWO. XVIII. — JEx oequali (sc. distantid) or ex cequo, from equality of distance; when there is any number of magnitudes more than two, and as many others such that they are proportionals when taken two and two of each rank ; and it is inferred that the first is to the last of the first rank of magnitudes, as th« first is to the last of the others. Of this there are the two kinds, in definitions 19 and 20, which arise from the different order in which the magnitudes are taken two and two. DE1-. XX. BOOK V. 159 " The ratio of equal distance, (or of intervals) is wben there are several magnitudes and others equal to them in number, taken two and two in the same ratio, and it arises that, as in the first s«t 6f magnitudes the iirat is to the last, so ii) the second set of magnitudes the first is to the last." "Otherwise, a taking of the extremes fcr comparisoii by a removal of the means." — ^Eitomd. Thus, assuming a : b = d : c; b : c = e -.f; r : s == It : y; s : t ==y : z; 1st Series. a,b,c ... r,d,t, or 2,4,8, ... 6,12,24; 2ud „ d, e,/, ... k,y,z, 8,6,12, ... 9,18,36; then a: t = d : z, or 2 : 24 = 8: 36. XIX. — Ex cequali, or c.c aquo ordinate. This term is used simply by itself, vfhen the iirst magnitude is to the second of the fii'st rank, as the first to the second of the other rank ; and as the second is to the thii'd of the first rank: so is the second to the third of the other ; and so on in order ; and the inference is as mentioned in the preceding definition ; hence this is called ordinate proportion. It is demonstrated in Prop. 22, Bk. V. " Arranged Analogy, or Proportion is when antecedent is to consequent, as antecedent to consequent ; and also as consequent to some one thing, so is consequent to some other thing." — Euclid. Let A : B : C : D, 12 : 6 : 18 : 36; and E : F : G : H, 6 : 3 : y ; 18; then A : D :: E ; H; 12 : 36 = 6 : 18. XX. — Ex cequali in proportione pertwhata sen inordinata, or ex wquo perturhate, from equality inperturbate or disorderly proportion. Archimedes de sphmrd et cylindro. Prop. IV, Lib. 2. "Hxis term is used when the first magnitude is to second of the first rank, as the last but one is to the last of the second rank ; and as the second is to the third of the first rank, so is the last but two to the last but one of the second rank; and as the third is to the fourth of the first rank so is the third from the last to the last but two of the second rank ; and so on in a cross order ; and the inference is as in the 18th defi- nition. It is demonstrated in Prop. 23, Bk. V. " Disturbed Analogy, or Proportion is when, on there being three magnitudes and others equal to them inmunher, it comes to pass, that, as in the first magnitudes, antecedent is to consequent, as in the second magnitudes antecedent to consequent ; so, in the first magnitudes, as the consequent is to some other thing, so, in the second magnitudes some other thing is to the antecedent."— Euclid. 160 GRADATIONS IN EUULID. 4 :.8 3 : 6; 8 2 =z 12 3; 4 2 = 12 6; Let one series of m^nitndes be' A, B, C, 4, 8, 2; and another D, E, F, 12,3,6; so that A ; B = B : F, or and B : C = D : E, and then A : C i D : F ; then i.e„ the magnitudes being taken in a cross order are therefore said to be in disturbed or disordered Proportion, though in reality the Proportion is as exact as in any other case of proportion. " Both this and the former inference come under one general principle, sail., that ratios which are compounded of equal ratios arc equal."— Labdnee. The definitions, ex aquo ordinate and ex mquo perturbate, may readily be extended to any number of magnitudes, compared with an equal munber of other magnitudes. AXIOMS. BOOK V. 161 POSTULATES, Let it be granted, 1. — That a given magnitude may bo so increased that any re- quired multiple of it may be taken. 2. — That any given multiple of a magnitude may be divided into parts, each of which is ec[ual to that magnitude. AXIOMS. 1. — Equimultiples of the same, or of equal magnitudes, are equal to one another. 2. — Those magnitudes of which the same or equal magnitudes are equimultiples are equal to one another. 3. — A multiple of a greater magnitude is greater than the same multiple of a less. 4. — That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude. Additional Algebbaic Exfuessioks, &c. N.B^— The capital letters AB, CD, EF, &c; or A, B, C, D &c., denote either lines, or other magnitudes of a like kind. An JM denotes magnitude,-— ilfs, magnitudes. m multiple. m A &c. nrultiple of A, &c. m A, m B, &c., equimultiples of A,B, &c. m (A + B) multiple of (A + B). m (A — B) multiple of (A— B). m (A+B — C) multiple of the excess of (A+B) above C. n another multiple, m + B the sum of the quanti- ties m & R. m B the product of m x n. mnA. a multiple of A by m n. (m + B)A „ „ A by (m+ n). pt. part. su6-m. submultiple. The signs >, >, <, <," between ratios,as'A : B > C : D, or A: B > C : D, or A : B < C : D, or A: B < C : D, denote that he one ratio is "less than, or not less than, greater or not greater than the other, according to the sign, 162 GRADATIONS IN EOOLID. Prop. I, — Theok. If any nwrnber of magnitudes be equimultiples of as many, each of each ; what mnltipVi soever any one of them is of its part, the same shall all the first magnitudes be of all the other. Con. Pst. 2, v.— Any giTen multiple of a magnitude may be divided into parts each of which is equal to that magnitude. Dem. Ax. 2, V. — Those magnitudes of which the same or equal magni- tudes are equimultiples are equal to one another. Def. 2, v. — A greater magnitude is said to be a multiple of a less, when the greater is measured by the less. Case I. — Let the number of magnitudes in each set be two. A G Hyp. ^ Let A B, CD, &c., be equims. of, E, F, &c., each of each, then mitlt. A B of E = miilt. (AB+ CD;of (E+ P). V AB, CDequims. of E, F; /, mags in A B, each = E, equal Ms, in C D, each = F. Divide A B into AG, G B, each = E, and D into H, HD, each = P. .-. no. of Ms C H, H D = no. of other Ms A G, G B. •.• A G = E, and C H = F, li .-. A G + C H = E + P ; 18 and •.• G B = E, and H D = F, .-. G B + H D = E + P; .■. no. of Ms in AB, each = E equal no. of Ms in A B + CD, ieach = E + F. .-. the mult. A B is of E, that same mult A B + CD is of E+ E. Case Ti.^-Let the number of magnitudes in each set be more titah two ; the same demonstration, ■vrhicU hftS been applied to tw<), holds for any number of magnitudes. E.l 2 D.l 2 C. D.3 4 5 6 7 — J r • Cone. Hyp. Ax. 2. Pst. 2, V. H. C.&Ax.2,V. C.&Ax.2,V. Cone. Def. 2, V. 'J3 24 I? IS PnOP. I. — THEOR. 163 Cone. \ ■. If any number of magnitudes. &(i. Q. E. D. Alg. §• Arith. Hyp.— Let A = 24, B = 21, C = 18, &c., be equimnlts. say »i times, or 3, of a = 8, i =i 7, & c = 6. Alg.—Thm. ',' A =• ma = a + a + a ; B = )rei = J + fi4-6; & C= mo = c + c + c ; + the Equals, A + B + C = mo + wi + ma = m (a + i + c) ; .•. A + B + C same mult, m, of a + i + e, es A, B, C, are respectively of a, b, c. Arith.— •.• 24 = 3 X 8 = 8 +8 + 8; 21 = 3X7 = 7 + 7 +7; & 18 =3x6 = 6 + 6 + 6; + the EquaJsi 24 + 21 + 18 = (3 X 8) + (3 X 7) + (3 X 6) = 3(8 + 7 + 6); .-. 24 + 21 + 18, or 63, the same mult, of 8 + 7 + 6, or 21, that 24 is of 8, 21 of 7, and 18 of 6. CoK. — Hence, if m beany number, mA + mB + mC = m (A+ B + C), i.e., the sum of the equimultiples = the equimultiple of the sum. ScH. — If to a multiple of a magnitude by any number a multiple of tite same magnitude by any number be (idded, the sum uiill be the same multiple of that magnitude that the sum of the two numbers is of unity. E. 1 2 D. ] 2 8 4 Hyp. Cone. H. 1, V. H. 1, V. Add. Ax. 2, 1. Cone. Let A = m C and B = n C ; then A + B = (m + n) C. •.* A = m C, .*. A =: C + C + C &c. repeated* timeg ; and •.• B = ?i C, .-. B = C + C + C &;c, „ n „ ; Adding equals, A + B = C taken m + it times; i. e. A + B = ()» + k) C ; .*, A + B contains C as often as there nre ijnitp in m + n. Cob. 1— Thus, if there bo any number of multiples whatever, as A = m E, B = reE, C==;>E &o., it is shown that A+B + C=(»B+n + p)E. Cob. 2.— Hence also, •.•A + B + C=:(m+B + p)E; and •.• A = j« E, B = n B, and C = p B ; .-. m E + Ji E + /) E = (m + n + p) E. 164 GKADATIONS IS EUOLID. Pkop. II. — Theok. If the first magnitude he the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; then shall the first, together ivith thefifthhe the same multiple of the second, that the third together with the sixth is of the fourth. 10 1 — ^, 1 A 1'.' ik 15,- D 9 £ QTM H C sp r *'." E.l 2 3 D.l 2 3 4 5 6 Hyp. 1 Let A B the 1st = m C the 2nd, & D E the 3rd = m F the 4th ; „ 2 & B G the 5th = M C the 2nd, and E H the 6th =reF the4th; Cone. then AG = AB + BC=(to + m)C; & DH = DE + EH=(ot +«)F. Hyp. 1 •.• AB = TO C, and D E = mP; Def. 2, V. ohs. .-. magns. in A B (each = C) = magns. in DB (each=,F); Sim So magns. in B G (each = C) = magns. in E H '(each = P) ; Ax. 2 .-. magns. in A G (each = C) = magns. in DH (each = F) ; Def. 2, V. .-. A G same mult, of C that D H is of P; Kemk. i i. e. A G (the 1st + 5th) same mult, of C the 2nd, that DH fthe 3rd + 6th) is of P the 4th. Eec. \ If therefore the first be the same, Src, Q. E.D. CoK. — Hence, if any no. of magns. AB, BG, GH, bo mults. of C, and as many, DE, EK, KL, the same ms of F, each of each ; then, AH i.e. (AB + BG + GH) the samemof C that PL, i. c, (DE+ EK + KL) is of P. PROP, III. ^THEOE. 165 16 »- H 24 J, 3C Alg. §• Arith, Ityp.—Taks 6 quantities A = 2 = 16,1"= 14, m = 3,aiidn==2. B = 8, C = 21,D = 7, Alg. •.' A the 1st a= m B, and C tha 3rd = m D. E the 5th = B B, and F tile 6th = n D ; + equals, then A + E = (m + n) B and C + 1? = (Mi + «) D- Now A + E contains B, (m + n) times, & C + E contains D, (m 4- n) times ; /. Def. 2, V. A + E is the eame mult, of B that C + E is of D.. Aritk. '.• 24 = s' X 8, and 21 =. 3 X 7 ; 16 = 2 X 8, „ 14=»2 X 7 : + equals, 24+ 16 =(3 + 2)8, and 21 + 14 ==(3 + 2)7. But 40 contains 8, five times, and 35 contains 7, five timesi, .■. Def. 2, V. 40 the same mult, of 8 that 35 is of 7. SoHOL.— Allied to this Proposition is the Theorem" : If the first of three magnitudes contain the second as often as there are units in a certain number; — ana if the second contain the third also as oft^n as there are units in a certain num- ber, the first will contain tlie third as often as there are units in the product of these two numbers." E.1 2 D. 1 2 3 4 S Hyp. Cone. H. 2.V. Kemk. Ax. 1, V. H. As. 1, 1, Let A = )» B, and B = » C; then A = mnC. •.• B = n C, /. ?n B = m (« C + « C &c.)) but m (n C + n C &o.) = m -f (ji + « + &C.) C \ , and n X M a=! (n n ; .•. m B = mn C. But A =« jkB, .". A s= mn Ct PS6i>. Ill,— TSEOE-; If the first he the same multiple of the SicoMi Vsfdch the third is of the fourth ; and if of the first and third there hj& taken equimultiples; these shall he equimultiples, the one of the sfCotld; itn§ the other of the fourth. 166 GRADATIONS IN EDCLID. CoK. Pst. 2, v.— Dem. Def. 2, V.— Cor. 2, V. E. 1 2 3 D. 1 2 C. 1 2 D. 3 4 5 6 7 8 9 10 11 Hyp. 1. „ 2. Cone. Hyp. 2. Def. 2, V. Pst. 2, V. OODC. Hyp. 1. C. 1. Def. 2, V. Sim. 1. „ 2. Oor, 2, V. 2,V. Cone. H ,h s B A B G D Let A the 1st = ?n B the 2nd; and C tlie 3rd = m D the 4th ; Also, let E F = n A, and GH=jiC; then E F the same to of B as -G H is of D. •.• EPsa/ime«iof AasGH ofC; .•. as many magns. in E F, each = A, as in GH, :' PSiCD. ^•~ G Divide EP into EK,KF, .aBr^GHttoGL,LH,^°- i^.^-^*- 12. 4. each, = C; the no. of Jfs in EK, KF = no. oi Ms in GL, LH. •.• A ssHtie m of B, that C is of D; ana •.• EK = A, ana'GL = C; .". EK same m of B, that GL is of D ; and .-. KF same m of B, -that LH is of D. and BO, if more Ms in EP & GH each = A, C. Hence •.• -fhe 1st EK same m of the 2nd B, as 3raGL, of 4th D; and •.• the 5th KF same m of the 2nd B, as 6th LH of 4th D; .-. EF (1st + 5fth) same m of the 2na B, as GH (3rd + 6th) of 4th D. If, therefore the first he the same, ^c, Q. E. D. CoR. If A, A' be equimults. of B, B' and also of C, C; and if B he a m of 0, the other B' shall be the flame m of C A^.-^Arith.Uyp.—'iake. « = 12, i = 4, c s-= 15, ril=5j »i = g&M = 2. Alg. — Let a = mh, and c <= md ; then na ae mn.l>, and nc =^ mnd ; i. e., the eqnims. na & n c of the 1st and Srrl. are mnlta, of the 2nd «nd 4th. PROP. IV. THEOE. 167 Arith. ■/ 12 = 3 X 4, and 15 = 3 X 5, .•. 2 X 12 = 6 X 4, & 2 X 15 = 6X5; i. e. the equims. 24 & 30, of the let 12, and 3rd 15, are equims. of the 2nd 4, and of the 4th 5. SoH, — "If any equimultiples «iA, mC, be taken of the antecedents of an analogy, A : B : : C : D, and any equimultiples, n B, b D, of the conse- quents, these multiples, taken in the order of the terms, are proportional," i. e. niA:n'B::mC:nJ). C. 1 D. 1 2 3 4 5 6 Pst. 1, V- 3, V. Cone. Sim H. and D. 2. D. 3. Def. 5, V. 7 D.2. 8 D. 5. 9| Def. 5, V. Of ?n A, m C take equims. p times, and of n B, n D, equims. q times j ... then •.' m A, m C, contain A and C,pm units of tim,es ; .*, equims. m A, m C by p are equims. of A and C, an& equal pm A, pm C. So » B X ? a,nd nT> X q = qn'B and ? n D. Since A : B : : C : D, an^ equims. of A and C are p m A, pm C, and •.• equims. of B, D are g b B, j n D, .". if J) IB A > qn'BtpmC > qnX> ; i£=, =, a^A if But pmA,pmC are also equims. pf m A and m C, and q n 'B, g n D also equims. of n B and n D ; /. m A : n B : : m C : n D j Cor. When n= 1, then m A : B = m C : D. Pros IV.— Theor. If the ^d of four mdn^mfiudes has the same ratio to the seeond tuhioh the third has to tkefmr&i ; then any equal multiples whenever of the first and third shall have the same ratio to any equimultiples of the second and fourth; viz.) "tjhe equimultiple of ;tji.e fept ^all li»v^ the same ratio to tjiat of the second, which the equimultipl* of th^ thii'd has to that of the fourth,, Dem. 3, V. Def, 6, V. Let A: B = ; D; and B, P be any equims. of A and C, and G, H ^ny equims. of B and D ; thenE; Q; = FiH. Con.— Pst. 1, V E. 1 2 3 4 Hyp. 1. „ 3. Cone. 168 GRADATIONS IN EUCLID. 9 __,^ ICIB 5 'R 6 1_A2 2 BS 7 GlO 11 M20 S6li ir 12F_ c 4 0_ _s iOD_ 4 20 H_ a 40 N- C. 1 2 D. 1 2 3 5 5 6 7 8 9 10 11 12 Pst. 1, V. Hyp'.' 2 C. 1. 3, V. Sim. Hyp. 1. D. 3. D. 4. Def. 5, V. C. 1. C. 2. Def. 5, V. Gone. Of E and F take any equims. K, L; and of G and H „ „ M, N. "/ E is the same m of A as F of C, and K, L are equims. of E and P ; .■. K same m of A, that L is of C; So M same m of B, that N is of D. And '.• A : B = : D ; and K, L are equims. of A and C ; and M, N equims. of B and D ; .-. K, >, = or < M, so L >, = or < l!f- But K, L are equims of E and P ; and M, N equims. of G and H ; .-. E : G = P : H ; Therefore if the first of four magnitudes Sfc. Q. E. D. Cor. 1. — Likewise, if the first has the same ratio to the seeoitct,- which the third has to the fourth, then also any equimultiples whatever' of the first and third shall have the sam^ ratio to the second and fourth ; and in like man0er, the first and third shall have the same ratio' to any equimultiples whatever of the second and fourth." Or, " li 4 Ms be pi'oportional, then 1°, any eqttlma. being tstken of the 1st and 3rd, the m of the Ist i 2nd = ot of 3rd i 4th) ^nd 11*, any eqtsims, being taken of 2nd and 4th, the ISt i m of 2nd = Srd i m of the 4th," E. 1 2. 3 C. Hyp. 1.- Hyp. 2. Cone. Pst. 1, V, Let A : B =2 : D and let E, P be j,ny eqfiiftlg. bf A & C ; then E : B =3 P : D. take of E, P a ny equims, K, L, and B,DeqmTjfa. G, B. of D. 1 2 3 4 5 6 7 8 4,V. Hyp. 1. C. C. Def. 5, V. 0. Def. 5, V. Sim. As before K same m of A, that L is of C. And .-, A : B = C : D, and K, L are equims, of A & C, and G, H are equims. of B & D ; ,-. K > = or < G, BO L > = or < H. But K,L are equims of E,F, & G,H any of B,D; .-. E : B = P : D. In the same way, A : G = C : H Cor. 2. •.• in Dem. 8, Pr. 4, V., if K > = or < M, L > =a or < N ; .'. if M > = or < K, N > = or < L. Hence G : E = H : F. Therefore, if four magnitudes are proportionals, they will be proper' tional by inversion. N.B. — ^This Cor. is not in its proper places it correctly forms Prop. B, V. , CoE. 3. If A : B = C : D, — and if any like parts of A and C he taken, as _, , and also any like parts of B and D, as _, — , these like parts will also he proportional; 'i. e., -.:-- = -:—' 2 2 3 o Alg. §• Arith. Ilyp.—Ut a^2, Jssfl, e = 4, d=10, m = 2&n = 3. Alg. — '.' a : b ^ c : d, thetl ma : nh ^ tnc : nd. For, 3, V. equims. ot ma and mc are equims. of a & c ; and equims of nb, nd also equims. of b & d. But, Def. 5, v., a §■ c < = or > J & rf, .•, equims. of o & c < = or > b& d, and also < = or > equims. of 6 & d. Hence, Def. 5, V. m « : nb = mc : nd, Arith. '.• 2 : S = 4 : 10, then 2X2!3x5!=2X4i3Xl0. Now equims. of 4 & S are equims. of 2 & 4j end equims of 15 & 30 also equims. of 5 & 10. But, Def. 5, V. •.' 2 & 4 < t= Or > 5 & 10 .*, equifflg of 2 & 4 < = or > 5 & 10, and also < = > equims. of 3 & 10. Hencei Def. 5, V. 4 s 15 = 8 : 30. Appl.— From Coi*. 3, arises the i-ule in simple proportion in taithnietic, of dividing the 1st and 2nd terms by any common mea- sure, and using the resulting instead of the original numbers. 170 ORADilTIONS IN EUCLID. Prop. V. — Theoe. If one magnitude he the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other ; the re- mainder is the same multiple of the remainder, that thewhoU is of.th^ whole. Con. — Pst. 1, V. Dbm. 1, V. Ax. 1,V. Equimultiples of the | Q same or of equal magnitudes are equal to one another. I Ax. 3. 1. — If equals be taken fijQjn equals the remainders j 8 are equal. A, E. 1 C D. 1 Uyp.i Cone. Pst. 1, V. C. 1,V. 3 H. 4 1,V. 5 Ax. 1, V. 6 Sub&Ax.^,I 7 C. 8 D, 6 9 1,V. 10 H. 11 1,V. 12 Cone. Let A B be the same m of C D that A E, taten from A B, is of C F, taken from CD; then the rem. E B is the same m of rem. F D, as the whole A B of the whole CD. Take A G same m of P D, that A E is of C P. 2-4 8 eqiiims. of 82 6 8 V AG, AE are P D and C P, .-. A G, A E, i. e., E G, same m of C F, P D, i.e., C D, that A E is of C F. but A E same m of C F, that A B is of C D; .-. E G same w of C D that A B is of C D, .-. EG=: AB; from each take A E ; /, rem. A G = rem. E B. V AEsamemof CEthat AGisof FD, andAG = EB; .-, A E same m of CP that E B is of PD; But A E same m of C P that A B is of C D ; .-. E B same m of F D that A B is of CD. Therefore if one magnitude be the same, ^c. Q.ED. PROP. VI. THEOR. 171 Alg. §• Ariih. Hyp.— Lei A (a part of B) = 6, m = 4. Mg. Let A = m B. C= mD; : 32, B = 8, (a part of A) = 24, and D Arith, Let 32 = 4X8. 24 = 4X6. Subt. A— C = ni(B — D), Subt. Thus A— C is m times (,B— D, Thus as A ism „ B. as 8 = 4 (8—6); 8 is 4 times 2; 32 was 4 „ 8. So the rem. is the same m of the rem. as the whole is of the whole. O/-, Let A — B = D; to both sides add B,— then A = B + D ; .'. (1, v.) mA = »!B+ mD; Subtract mB, andmA — mB = mD5 butD = A — B; .-. wA — »»B = m(A — B); Thus the rem. is the same m of the rem. that m A is of A. SCH. " If from a multiple of a magnitude hy any number, a multiple qf the same magnitude by a less number be taken away, the remainder will be the same multiple of that magnitude that the difference qfthe numbers is of unity. Let ffi A, B A be mults. of A, m being > n; then JB A — n A = (m — n) A. Let m — n =: q; thenm =: n -\- q. Here mA:=MA+ q A; from both take n A; then m A — n A = q A; ,'. m A — n A = (m =^ k) A. CoK. When the difference of the two numbers is equal to unity, or m—n = 1, then mA=nA = A; or2A — A = A. E. 1 Hyp. 2 Cone. C. Sum. D. 1 C & 2, V. 2 Sub. 3 Cone. Prop. VI. — Thssob. ' Tftioo Msgnitudea be equimultiples of two others, and ifequintul- tiples of these be taken from the first two; the remainders are either equal to these ethers, or equimultiples of them. Con. Pst. 2, 1. — 3, 1. From the gr. of two lines to cut off a part equal to the less. Pst. 1, V. E. Dem. Ax. 1, v.— Ax. 8, L Hyp Ax. 1, L-2, V. 1. 2. ;, 3. Cone. Let A B, C D be equims. of E F ; and let A G taken from A B be an equita. of E, and C H from C D an equim. of P ; then rems. G B, H D either = E P, or are eqnims. of them. 172 GBADATIONS IN EUCLID. Case. I. —Let GB = E, then HD shall equal F. c. D. 1 Pst. 2 I & 3 I H. 2 2 H. &C. 3 Cone. 4 H. 5 Cone. 6 7 Ax. 1 V. Suh Ax. 3 1. 8 C. Ax. 1 I. Make C K = P. •.• A G same m of E, that C H isof F; and vGB=E,andCK = .•. A B same m of E, that A K H is of F. But A B same m of E, that C D is of F. .". K H same m of F that C D is of F. .-, K H = C D, take away C H, .•.rem.KC=rem.HD; Si ?6 f butKC=F, .-. HD = F. ^ Case II. — Let G B bo a «i of E, then H D same m of F. c. Pst. 1, Y. Of P take C K the same m that G B is of E. K D. 1 H. '.• AG same m of Ethat C H is of F. 18 2 C. and G B same ?n of E that #1 KHisof F; C„ 3 2, V. /. A B same m of B that At K H is of F. 18 4 H. But A B same m of E that 14 C D is of F, « -H 5 Cone. .". K H same m of F that C D is of F, Gr 13 6 Ax. 1, V. .-. KH = CD. H T 7 Sub. Ax. 3, 1. Take away CH, 1 1 .-, rem. KC, = rem. HD. B D E 8 C. And .'. G B same to of E that K C is of F, 28 3e r 9 D. 7 andKC=HD; 10 Cone. .-. H D same moi'B that G B is of E. 17 Eec. If, therefore, two magnitudes, ^c Q. E.D. PROP A. — XHKOR. 173 Aly. ^ Arith. Hyp. — Let m> n express any integers, as 4 and 3; A = 28, B = 36, C = 7,D = 9. Alg. — ^Let A, B be cquims. of C & D ArithA Let 28 &36 be equims. of 7 &9, A = mC, B=:mD, 28= X 7; 36=4X9 A>bC, B>bD; 28>3x7; ' 36>3X9; Subt.A— BC = mC-reC = (ra— b)C; B— ji D =mD-BD =(»t-B)D. Suppose m—n = 1. 1°. A— 7!C=C, B — nD = D, 2°. Or, equimsofC&D im—n) C and (m—n) D. Subt. 28—21 = 4x7 — 3 X 7 = (4-3)7. 36— 27 = 4X9— 3 X 9 = (-4-3)9. Here 4 — 3 = 1. 1°. 28 — 21 = 7. 36 — 27 = 9, 2°. Or,, equims. of 7 & 9. as (5 — 2) 7, and (5-2) 9 ScH. — The six preceding Propositions are chiefly useful for establishing, by the method of Equimultiples, the Propositions which follow^.^ "WTi^ this method 13 not employed some have adopted the Postulate, — Thf^Maghitndes, A, B, C, being, given, let it be granted that there is a ith magnitude, we- may call it X, to which C has the same ratio, as A to B; i.e. A : B ;= C' : a:. Prop. A, — Theoh. If the first of four magnitudes has the same ratio to the second which the third has to the fourth; then if the first he greater than the second the third is greater than the fourth; and if equal, equal ; if less, less. Con.— Pst. 1, V. Dem.— Dcf. 5, V. Let A : B = C : D, then if A > = or < B, C> = or < D. Take any equims., as 2 A, 2 B, 2 C, 2 D. •.• A : B = : D, and of the 1st and 3rd equims. 2 A, 2 C are taken, and of the 2.nd and 4th eqnims. 2 B, 2 D ; .-. 2 C is > = or < 2 D, as 2 A is > = or < 2B.' But 2 C iS: > = or < .2 D as C is. > = or < D ; ^^ and so, 2 A is > == or < 2 B, as A ig > = or < B; .-. 0. is > = or < D, as A is > = or < B. Q. E. D. B.l Hyp. 2 Cone. C Pst. 1, V D.l H. 2 C. Def. 5, V. Sim. Cone. 174 GRADATIONS IN EOOLID. Or, more briefly ; D.l 2 3 4 Def. 5, V. >> Cone. Sim. If2A>2B, 2C>2D; but if A > B, 2 A > 2 B ; .-. 20 > 2D, and .-. > D. So if A = or < B, = or < D. Q. E. D. Use. — SiMSON added this and the next three Propositions. Prop. A is required for the demonstration of.25, V; 21, VI ; 34, XI, and 15, XII, and is often employed by Geometers. .i Prop. B. — Thkor. invertendo, by inverting! — If four magnitudes are propoHidnals, •they are proportionals also when taken inversely. Con. Pst. 1, V. Dem. Def. 5, V. 24 27 G -A .B .E H_ C. D. F. 52 h12 56 E. Hyp. Cone. Pst. 1, V. Let A : B = C : D ; then B : A = D : C. Of B,D, take any equims. E,F ; and of A,C, any equims. G,H. First. — Let E > G. i. e., G < E. D. 1 2 "4 H. C. 2. C. 1. Def. 5. Cone. •.• A : B = C : D, and of the 1st A, and 3rd C, are equims. G & H, and of the 2nd B, and 4th D, „ E & F ; .-. H > = or < F, as G > = or < E ; If .-, E > G, F,is > E. Second.— ^0, if E = or < G, P = < H. C. Def. 5, Kec. But E,r any equims. of B,D ; & G,H of A,C ; .-. B : A = D : C. Therefore, if four magnitudes, (j-c. Q. E. D, PROP C. — THEOR. 175 Alg. Sf Arith. Hyp. Let A = 6, B = 9, C = 8 ; D = 12; m = 4, » = 3. Alg. Let m A & mC be equims. of A & C. nB&nD „ B&D; •.• A : B = C : D Def. 5, V. .•, m A, m C> = or < bB, nD. If m A, ffiC > JiB, n D ; nB, nD< m A, mC; IfmA,mC < ten.itDD ; nB, /}D > BiA, mC; .% Anyequims. ofB.D > =oi- < A.D. Def. 5, V. .-. B : A = D : C. ^n'M. Let 4x6,&4x8 be eqiiims. Of 6 X 8, 3X9, &3,X12 „ of 9 & 12. •/ 6 : 9 = 8 : 12. .■. 24, 32 < 2? & 36. i. e. 27 & 36 > 24 & 32. .-. equims. of 9 & 12 > those of 6 & 8. /. 9 : 6 = 12 : 8. ScH. 1. The Proposition may be stated, — '' The reciprocals of equal ratios are equal to one another." 2. By an inaccuracy Prop. B has been pliaced by some as a Corollaiy to Pr 4. V. Peop. C. — Theor. If the first he the same multiple of the second, or the same part (i. e. measure, or measure submultiple) of it that the third is of the fourth, the first is to the second, aS the third is to the fourth. Con. Pst. 1 V. Dem. 3 V.— Def. 5 V.— Def. 1 V.— Def. 2 V., P. B. V. There are two Cases of this Prop, according as A and C are multiples or parts of B and D. Case I. —Let Ac} Che multiples ofB ^ D. E. 1 2 C. 1 2 Hyp. Let A 1st be same m of B 2nd, that C 3rd is of D 4th, , Cone. then A : B r= : D. Pst. 1. V. Take of A, C any eqaims. E, F; „ aud of B, I) any equimB. Cr, H. 17G OUAPATIOKS IS KVW-ID. D. 8 10^ _A _B _C .D E_ 24 -.10 so H. & C. 3,V. C. Cone. Eemk. Sim. C. Def. 5, V. •.• A, C equims. of B, D and E, F of A, C ; .-. E, P equims. of B, D. But G, H equims. of B, D ; .-. if E> ni of B than G is of B, F is a > m of D than H of D; i. e. if E > G, F > H. So, if E = or < G, F = or < H. But E, F are equims. of A, C, and G,H of B,D ; .-. A : B = C : D. Q. E. D. Case II. Let A^- G he parts, or ml -multiples, of B ^ D. E. 1 H. Cone. Def. 1, V. Def. 2, V. Case 1. B. V. Invert. Eec. Let A the 1st, be the same part of B the 2nd, that C the 3rd, is of D (16 8 ABC 10 SO Q.E.D the 4th; then A : B = C : D. D. 1 Def. 1, V. •.• A the same part of B that C is of D, .', the same m of A that D is of C ; .-. B : A = D : C. .-. A : B = C : D. Tlierefore, if the first he the same multiple, ^c. Alg.^ Arith. Hyp. Tal£eA=2B = 8, and C= 2D = 10; a= 3& /) =4 Alg. Take n A, re C any equims. of A, C ; pB,pD „ ofB,D, Then ■.• A same m of B, that C isof D ; and n A „ „ of A, „ n C is ol C ; .". n A „ „ of B. ,. n C is of D, i.e. re A, re C are equims of B, D. But p B, p D „ „ of B, D ; .•. If n A > m of B, than p B is of B, reOmofD, „ pDofD; i. e., if nA >pB, nC > pD. So, if re A = or m of 4, than 16 is of 4, 30 > m of 5, „ 20 of 5, i. e., if 24 > 16, 30 > 20. So, if 27= or < 16, 30 = or < 20. ' But 24 & 30 are 'equims. of 8 & 10, 16 & 20 „ 4 & 5. .•. by Def. .■), V. 8 : 4 = 10 : 5. ScH. — The 7th, 8th, 9th, and 10th boolcs of Euclid's Entire Work treat ol Arithmetic and the doctrine of Incommensurables; and the 20th Def Book Vn., giTes'a definition of quantities which are proportional ; but "most of the commentators" says Sbhson", and Potts repeats thewordsj "judge it difBcuIt to prove that four magnitudes which are proportionals according to the 20th def. of the 7th book; are, also proportionals according to the 5th def. of the 5th book' The fcfemonstration, however, is as follows; from Simson's Kotes, page 317. 1". As to four magnitudes in proportion according to Def. 5, V. Case 1.— iet A, B, C, D be four Ms, such that A = m B, or - ; andC= P m D,or ■^, ; then, by Pr. C, bk. Y; A : S = C : D. P Cask 2. — In A B let there be the same parts of C D as there are of G H in E F, then also AB : CD = £1" : GH ; For F C.1 2 B.l 2 3 Let C K be of C D the same pt. that G Lis of GH, and AB same m of CK that EFis ofGL; .-. AB :CK = EF :GL; And •.' C D, G H are equims. of OK, GL, the 2nd and 4t.h, Cor.4,Y. .-. AB:CD = Er:GH Pst.2,V. Pr. C,V, C. B 24 12 A C 30 30 10 H -L » P. G M 178 GRADATIONS IN ETTCLID. 11°. — And, if four magnitudes are proportionals according to the 5tli def. of Book V, they are also proportionals aceormng to the 20th def. of Book VII. Case 1.— Let A : B = C : D. then, by Pr. D, book V, if A is any m, or pt of B, C is the same m or pt of D. Case 2. — Let AB : CD := EF : GH, as in the foregoing iigm-e : then if A B = any p« of C D, E F = the same pt of G A. Take CKajo a, a^d o" be a terminate no. between a & a. a" a'M ~b = or > ^''^ ^"^ > <^^) 01' ^ ; a'M A n" a' ITM. > B ; 5" > i' . and .-. a' > a' But a° also < n', an impossibility, So a < a, .-. 5 = "3 a' >a. Case 4. Let a be terminate and 5 interminate. ^ D. I ySim. I By a demonstration similar to tbat of Case 3. Case 5. Let a and b be both interminate.- D. 1 3 Sup. Case 4. Sim. _„ A a A a' ^* S" ^ i", let g- = J- ; let a' > a & a' be a terminate no. between a & a a" a'M The rest of the dem. as in Case 3. A a ■'• W — ~i Prop. VIL— Theoii. Equal magnitudes have the same ratio to the same magnitudef and conTersely, the sanie has the same ratio to equal magnitudes. CoK. Pst 1, V. A given M. may be so increased that any required m of it may be taken. ^ Dem. Ax. 1, V. Eqnims, of the same or of equal Ms. or equims. are equal to one another. Dcf. 5, V. The first of four magnitudes is said to have the a-meratii-, 182 GRADATIONS IN EUCLID. E. 1 2 C. 1 Hyp. 1 Cone. 1 & 2. Pst. 1, V. Pst. 1, V. Let A = B, & C be a 3rd M. of the same Idnd then A : C = B : ; and C : A = C : B. Of A, B, take any equims. D, E ; and of C, take any l^ eqnim. F. 20 First. A and B shall each have the same ratio to C. D. 1 0. 2 H. Ax. 1, V 3 4 C. 1 & 2. Def5, V. 10 16 •/ D & E are equims. ofA&B; and A ^ !]^, .-. D = E ; .-. if D > '= or > F, E > = or > F ; but D, E are equims. of A,B, and F is a m of C; .-. A : C = B : C. Second. C shall have the same ratio to A that it has to B. D. 1 O £J 3 4 5 Sim. , C. 2 & 1. Def. 4, V. Rec. As before, D ^ E ; .-. if F > = or > D, F > = or > E; but F any m of C, and D, E any m of A, B ; .-. C : A = C : B. Therefore, equal magnitudes have the same ratio, ^c. • 0. E. D. Cor. — If a ratio A : C which is compounded of two ratios, A : B, and B : C, be a ratio of equality, one of these must be the inverse, or reciprocal of the other; i.e.. A : B is the inverse, or reciprocal of B: 0. ScH. — Tlie second part, C : A = C : B, follows from the corollary of Pr. 4. Bk. V. PRQP 8, THEOR. 183 Prop. VIII.— Theok. Of two unequal magnitudes ■, the greater has a greater ratio to another magnitude than the less has ; and conversely, the same mag- nitude has a greater ratio to the less of two other magnitudes, than it has to the greater. COH. Pst. 1,. V. Dem. 1, V, Djef. 7, V. — When of the equims. of fo»r 5Is., (taken as in the fifth def.) the m oi the 1st is > that of the 2nd, but the m of the 3id ]J> than the m of the 4th ; then the first is said to have to the 2nd a greater ratio than the 3rd M lias to the 4th ; and on the contrary, the 3rd is said to have to the fourth a less ratio than the 1st has to -the second. E. 1 2 8 Hyp. Gone. 1. Let A B* > B C, and D any other M ; then A B : D > BO :D ; and D : B > D : A B. First.— AB : D > B : D. C. 1 Sap. 1. I If of the two Ms., A C, B, the not gr. of the two be not less than D ; ?st. 1, v.! take E F = 2 A C, aadi P G, = 2 C B. D- A fi C 6^8 _B H .10 JE 12 F 16 _G K 15 L 20 C.3j Sup. 2. j ButifofthetwoJfs., thewoi^r. Mbe < D; 4 Pst. 1, V.l take of AC, CB, eqwims. EF, F G each > D. A 4 C E 8 16 T>. 5 B H —10 G K 15 L 2Q 18i GRADATIONS IN EUCLID. C. 5 Pst.l,V. 6 Pst.l,V. 7 J? D. ] C. 2 8 c. 4 1,V. - 5 Eemk. fi D.2,&C. 7 8 C. 9 C. D. 5. 10 C. 11 Def.7.V In all cases take H = 2 U, K = 3 U, &c., till the mof D be that Tvhicli is first > FG. Let L, a m of D, be found > F G ; and K them ofD next less to L; i.e., FG<;K. •.• L is the m of D which first is > F G ; /. K, the 771 of D next preceding, is > F G, i. e., F G < K. And V EFsamemof AG, thatFGisof CB; .-, FG the same m of CB thatE G is of AB; i. e., EG and FG are equims. of AB and CB. And •.• F G < K, and EF > D ; .-. the whole EG = FG + FE>K+D; butK+ D = L, .-. EG > L. But FG > L, and EG, FG equimsof AB, CB ; and L is a i» of D ; .-. AB: D > BC: D. Q. E. D. Second.— D : BC > D : AB. c. I). 1 2 Sim. Sim. C. As in the first part. As before, L > FG but > E G ; and L a tti of D ; 3 D. 5. pt. 1. also FG, EG were proved equimsof CB, AB; 4 Def. 7, V. .-. D: BC > D: AB. •5 Kec. •. Of two unegual magnitudes, (f-c. Q. E. D. Otkerviise. Let A + B = 5 + 4, and A = 5, be two unequal Ms; and C = 10 a third M ; then A + B : C> A : C; i. 5 : 10. PEOP IX. THEOK. 185 Al(/. Tate mA, niB, each > C; nC the least nralt. > mA + mB. Hence (re — 1) C < tbA + niB, or m (A + B); m(A + B) > (re— 1) C, ornC— C; '.• n C > (mA +mB), and C < mB, .-. nC-C > mA, ormA < (n— 1) C. Hence m (A + B) a m of A + B > (n— 1) C, a m of C. but jmA, is a m of A 3> (n — 1) C, a mof C, .-. byDef.7,V. A+B:C>A:C. Arith. 3X5, and 3X4, each > 10 ; • 3 X 10 least m >(3 X5) + (3 X 4). Hence (3—1) 10 < (3 x 5) + (3 X 4)or 3 (5+4). 3 (5+4) >(3— 1) 10, or (3x10)— 10. •.• 3 X 10 > (3X5) + (3X4), and 10 < 3X4. .-. (3X10)— 10 > 3X5, or 3 X 5 < (3—1) 10. Hence 3 (5+4) > (3-1) 10. but3 X 5> (3—1) 10, .-. 5 + 4 : 10 > 5 : 10. Pkoi'. IX. — Theok. Magnitudes which have the same ratio to the same magnitude arc equal to one another ;- and those to lohich the same magnitude has the same ratio are equal to one another^ Con. Pst. 1, V. Dkm. 8, V.— Def. 7. V.— Def. 5, V. E. 1] Hyp. 1. 2 „ 2. 3 Cone. Let A : C = B : C ; or C : A = C : B ; then A = B. First. Let A : C = B : C, then A = B. I). 1 Sup. 8, V. Def. 7, V. If A^t B, let A >,B; then the gr. M, A : C > the less M, B : 0. Now equims. of A,B, mK, jnB, and &m of C, (n — 1) C, may betaken, so that mA> , and mB> («— 1) C ; 13 Vi. 11 ISC GRADATIOKS IN EDCLID. Pst. 1, V Hyp. D. 4. Def. 5, Y D. 4. Cone. Second. Of A & B take equims. D,E ; and of C am, F, so that D > P, but E .> F ; then •.• A : C = B : C; and of A & B equims. D, E, and of a in, F, amlD > F; .-. E > F ; Bat E also > F; — ^two things, of which one is impossible ; .•. A not ^t B, i. e. A = B. D. Sup. 8, V. Def. '. Let C : A = C : B,— then A = B. If A ijfc B, let A > B. then C : B the less M > C : A the gr. M. V. •." there may be taken a m of C the 1st and 3rd, and equims. of A & B the 2nd and 4th ; so that m of C > mB, but '> m A ; Pst. 1, V. Take F, a m of C, and E,D, equims. of B, A, so that F > E, but > D. Hyp. And ■.• : B == C : A, D. 4. and F, m of C> E, m-of B ; Def. 5. V. .-, F, a m of C > D, a M of A ; H. Cone. but F also > D ; an impossibility, .•, A = B. Eec. .'. Magnitudes ivhich have the same ratio, ^x. q. E. D. Otlierwisc. Hyp.— T^t A : C =: B : C, or C : A= C ; B ; then A = B ; or let 4 : 6 = 4 : 6 and 6:4 = 6:4, then 4 = 4 Case I.— If A :^ B, let A > B. Assume, (8, V.) m A > n C, but m B 3> « C : •.• A : C = B : C, .-. m A, m B > = or < n C ; but m A > n C, and m B < n C, an absurdity; .-. A not ^t B, i. e., A = B. Case IL— If C : A = B : A, then A = B. Invertendo. A : C = B : C ; .".by Case 1, A =; B. CoR. — A ratio compounded of two, ratios, of which one is the reciprocal of the other,, is a ratio ofequaliin. For in A, B, C, magnitudes ef the same kind, — if B : C = B: A, A= C; ■ i. e., the ratio, A : C compcamdad of A : B and of B. : C, one the recipioeal of the othes, is a ratio of equality. Note Def. 3, V. Def. A, V. PEOP X. THEOR 187 Prop. X. — Theoe. That magnitude lohich has a greater ratio than another has unto the same inagnitude is the greater of the two ; and that magnitude to which the same has a greater ratio than it has unto another magnitude, is the less of the two. Con.— Pst. 1. V. Dem. Def. 7, V. Ax. 4, V. That magnitude, of whicli a multiple is greater than the same rBiiltiiilo of another, is greater than that other magnitude. E. Ij Hyp. 1. 2! Cone. 1. -; Hyp. 2. 41 Cone. 2. Let A : C > B : C ; then A > B. Let C : B > G : A; then B < A. Case I.— Let A : C > E : C, then A > B. 1). i; Hvp. 1. 2 Def. 7, V. 'Pst. 1, V. 12 •.• A : C> B ; C ; .•. of A 1st and B Srd can be taken some eqainas., and of C the 2nd and 4th a m so that m A > TO C and m B ^ m C. 5, jd Take D, E, such equims., of A, B, find F a 7b of C, so that D > F but E > F ; > .-. D > E. And .". D and E are equims. of A and B, and if the 1st < the 2nd, the m of the 1st < the same m of the 2nd ; and D > E ; .-. A > B. D. 3. Ax. 4, T. I D. 4. Cone Case II.— Let C : B > C : A, then B < A D. Hyp., Pst.l.V. D. 3, 4. Ax. 4.V. llec. For .-. 0: B >C: A; .". of C 1st and Srd can be taken some ,);, apd of B, A, 2nd and 4th eqiiims, so that ji C > 5)1 B,, but > ??j A. Of C take such m, P, and of B, A equims. E, D so that P > E but > D; .-. E < D. and •..• Ev D are equims. of B, A, and E < D- .-. B < A. Therefore, that Magnitude: tvhich Has a greater ^'c. Q. E. D. 188 GRADATIONS IN EUCLID. Oilierwise. Hyp. 1°. Let A : C> B : A, then A > B ; or 6 : 5 > 4 : 5, then 6 > 4. 2°. Let C : B > C : A, then B5:6, then 4 < C. Alg. \° Def. 7, V. Take m A, > n C, and m B < » C. .-. TO A > m B, and Ax. 4, V. A > B. 2°. Def. 7, V. Take » C> m B, and n C < m A. '.* m B < n C, and ni A > n C ; ' .". m B < m A, and .'. B < A. Arith. 1°. 3 X 6 > 3 X 5 ; but 3 X 4 < 3 X 5. .•. 3 X 6 > 3 X 4, and /. 6 > 4. 2°. :! X 5 > 3 X 4, but 3 X 5 < 3 X C. ".• 3 X 4 < and 3 X 6 >, 3 X 5. .-. 3 X 4 < 3 X 6 ; .-. 4 < 6. Prop. XI.— TfiEOR. Ratios that are the same to the same ratio, are the same to one another. CoN.-Pst. 1, V.' Deji. Def. 5, V. Definition of Proportion. Let A : B = C : D, & C : D = E : F; E. 1| Hyp. 21 Cone. C. 1). G A- B- 1 2 1 2 3 4 5 6 7 8 9 10 11 then A : B = E G 6 12 H C_l D_- ! -M F. E- F- -4N Pst.l.V. Hyp. Def.5,V. Hyp. Def.5,V. D, 3. C. 1. Def. 5,V. Eec. 16 Of A,C,E, take any equims. G,H,K; and of B,I), E, take any eqnims. L.M.N. •.• A : B = C ; D, . and G,H are equims. of A,C ; L,M equims. of B,D; .-. if G > L, H > M ; if =, = and if less, less. Again, •.• C : D = E : P, and H, K are equims. of C, E ; M, N, of D, F ; .-. if H > = or < M, K > = or < N. but if p^ > L, H > M, if =, =, and if less, less; .-. if G > = or> L, K > = or < N. And G, K, are equims. of A, F, and L, N, of B,F; .-, A : B = E : F. •. Ratios are the same, ^c. Q. E. D PROP XII. THKOR. 189 Otherwise. Hyp.. If A : B = C : D, and C : D = E : F; then A : B = E : F; or, if 3 : 6 = 1 : 2; and 1 : 2 = 4 : 8, then 3:6 = 4:8. , Alij. Of antecedents take mA, mC, niE; and of consequents „ nB, ».B, «]?. -.• A :B = C : D; ifm A>=or < nB, .". m C > = or < »D. Again, •.• C : D ^ E : F; if »i G > = ov < n 1), .'. m E > = or < » F J NowmA.niE eqnims. of A, E ; nB, nF, of B, Fj .•.byDef.5,V A:B = E:F. Aritli. Take them* 2X3, 2X1,2X4; and 2 X6, 2 X2, 2 X 8; •.•3 : 6 = 1 : 2, if 2 X 3 < 2 X 6, .-. 2X1< 2X2. Again, '.• 1 : 2 = 4 : 8, if 2 X 1 < 2 X 2, .-. 2X4 < 2X8. Now 2 X 3, 2 X 4 are equims; also2 X 6 and 2X8; ' .-.3:6 = 4:8. ScH. Tliis Proposition is to Eatios. 'what Prop. . 30, Bk. 1, is to Parallel lines; Ax. 1, Bk. I, to Magnitudes;' and Ax. 1, Bk. Y'. to Equimultiples. Cor. 1,. If A : B = C : D, but C : D > or < E : F, then A : B > or < E : F. For, whatever part of D be contained in C, a greater or less munber of times than the like part of F is contained in E, the like part of B must be contained in A the same greater or leS^ number of times. ' CoE. 2 Thus also, if A : B > or < d : D, and C : D = E ; F, then A : B > or < E : F. ' Pkop.' Xll. — Thbob, If any number of Tnagmiudes be proportionals, as one of the ante- cedents is to its eonsequent, so shall allthe atitecedents taken together be to all the consequents. Cos. Pst 1, V. Dem. Def. 5, V. Definition of Proportion. 1, V. If any no. of jSf» be equims of as many, each of each, what m soever any one of them is of its pt,tlie same m ahall all the first Ms be of all the others. 199 GRADATIONS IX EUCLID. E; li Hyp. ]LetA:B = C:D = E:F; 2 i Cone. tbenA : B = A+C + E : B + D + F. G — -2 A- 1 B 3 L 6 C- B- M K- E- r . -12 N- -18 c. D. 9 10 11 12 Pst. 1, V. Hyp. C. 1. C. 2. Def. 5, V. 0. 1.' 1, V. D. 7 & 8. Sim. Def. 5, Kec. V Of A,C,E, tate equims. of G,H,K; and of B,D,F, eqiiims. L,M,]Sr. then ■.• A : B = C : D = E : F, and G,H,K, are equims of A,C,E, and L,M,N, equims. of B,D,P; .-. G > = or < L, H also > = or < M, and K > = or > N ; wherefore if G > = or < L, then G + H + K > = or < L + M + N. !Bnt G, and G + H + 15, are equims. of A, and A + C + E. for whatever to of A, G is, the same m are all, G + H + K of A + C + E; i. e. G, and G + H + K are equims. of A, and A+C + E. So L, and L + ,M + Ntere equims. of B, and B + D+F. .-. A : B = A + C + E : B + D + F. Wherefore, if any number of magnitudes, ^c. Q. E. D. Otherwise. Hyp, A : B + D + F; or 1 : 3 = or 1 : 3 = 6 : 18. , i B = C : D = E : F; then A:B=zA + C + B : 2 : 6 = 3 : 9, then 1:3=1+2+3:3 + 6+9, Alg.I. OftheAntecs. take mA,.mC, »iE, Of the Conseqs. „ bB, nD, mE; V A:B= C : D, .•. if mA > = or<«B, mC, > = or < nD; •.- C : D =:E : E, .-. ifm C> = or < nD,m'E > =.or< «E: .-. IfmA > = or < reB, mA+ otC + ibE > = or = or < 2 X 3, 3 X 2 > = or < 2X6; and .-. 2 : 6 = 3 : 9, .-. if 3 X 2 > = or > 2 X 6, 3 X 3 > = or > 2 X 9. ) Now (3 X 1) + (3 X 2) -(- (3 X 3) = 3 (1 + 2 + 3), SO that 3 X 1, (3 X 1) + (3 X 2) + (3 X 3) are equinis. of 1 and (1 + 2+3) and 2 X 3, (2 X 3) + (2 X 6) + (2 X 9) equims. of 3 and (3+ 6 + 9); .•. 1 : 3 = (1 + 2 + 3) : (3 + 6 + 9), or 1 : 3 = 6 : 18. Alg. II. For Antecs. take a, and a, b, c, d, e, &c. For Conseqs. „ a', and a', b', c', d', ef, &c. , BygivenHyp. _r=-^^ = ^ = ^ = ^=&c. Take product of extremes and means; ab' = a'b, a c' =. a' c,ad' =. a' d, a e' = a' e, &c. ' Add Antecs. for numerator, Conseq.s. for deiifeminator, a + i4;c + rf + e + & c. _ a' + h' + d H-d' + e' + &c. ' Divide by — , and we nate, ^-i ,— -i -J- — -^^ — ; — = — a aa -{■ ab -{- ac + a d -\- a. e &c. Now — = 1 '& °°' + a'i. + .g'c + a'd+ a' e & c. _ j_ a ■■ ' , aa' + ab' -j- ac' + ad' + ae' &c. And the quotients id each case being unity, .". a: a':^a-\-b + c + d+e &c. :. a' + 6' + c' + d' + c' &c. Prop. XIII.— Theok. If the first has to the second the same ratio lohich the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth; the first shall also ham to the second a greater ratio than the fifth has to the sixth. Con. Pst. 1, V. Dem. Def. 7, V. Definition of greater an:l less ratio Def. 5, V. Definition of Prorortion. J'J2 GRADATIONS IN EUCLID. E. 1 Hyp. I Let A : B = C : D, but C : D > E : F; 2 Cone. I then A : B > E : F. 1.5 1 2 10 1 c ' 5t A « N' 50 24 12 IS 18 j. 1 2 H. Def.7,Y. 3 Pst. 1,V. 4 Pst.l.V. >. I H&C.3,4 2 3 4 Def. 5, V. C. 3. C. 3 & 4. 5 Def,7,V. Rec. •.• C : D > E : F, .". somejeqiiims of C tlie 1st and E the tliird, and some of D the 2nd and F the 4th ; so that »» of C > moi D. but ??iof E 3> "jof F; Take such equims. G & H of C & E, K & Lof D & F, so ttiat G > K, but H > L ; Whatever m, G is of C, take M of A ; and wliatever m, K is of G, take N of B. •.• A : B = C : D, aftd •.■ M, G, cqiiims. of of A,C;N, Kof B,D; ' ' . .-. if M > = or < N, G > = 6r < K. but G > K .-, M > N. ButH > L, and M, H equims of A, E ;— N, L of B, F ; ■ .-. A : B > E : F. Wherefore (/ 1 he /first has to the second 4'C. Q. E> D. Cor. If A : B > or < B : E, Ijut C : D = E : F, it may also be demonstrated that A : B > or < E F. Alg.SrArith Syp. I<5t A 5 : B 6 = C 10 : D 12 ; but C 10 : D 12 > E 5 : F 9 ; then A 5 : B 6 > E 5 : F 9. Let m = 3, & n = 2. Alg. '.' C : D > E : F, Assume m C, n D, m E, n F, so that if m C> n D, m E < n F. PROP. XIV. THEOK, 193 Now.- A :B = C :!>,— ifmOnD, m A> nB; ,•. m A > « B, and mE < nlT, .•. A : B > E : F. Arith. ;• 10 : 12 > 5 : 9, Assume 3 X 10 > 2 X 12, but 3 X 5 < 2 X 9. Now •.• 5 : 6 = 10 : 12, if 3 X 10 > 2X12; andS X 5 >2X 6 .-. 3 X 5 > 2 X 6, and 3 X 5 < 2 X 9 ; .-. 5 : 6 > 5 : 9. ScH. "This proposition is equivalent to stating) 1°. that if any ratio be greater than another, every ratio which is equal to the former will also be greater than the latter; 2°. Also, that if one ratio be greater than another, every ratio which is greater than the forniBr is also greater than the latter." IiAKDJTEK. Pkop. XIV.— Theok. If the first has the same ratio to the second which the third has to the fourth; then, if the first be greater than the third, the second shall ie greater than the fourth ; and if equal, equal ; and if less, less. E. Dem. 8, V- The gr. M. a gr. ratio. 13, V.— 10, V. one M with a gr. ratio to a third. 9, V. Two Ms. each with the same ratio to a third. 1| Hyp. 2! Cone. Let A : B = C : D, If A> = or < C, B > = or < D. 18 18 18 1 2 ] 2 12 12 9 « i 6 6 ABOD ABCD ABCl) Case I. Let A > C, B will be > D. 194 GRADATIONS IN EUCLID. D. 1 2 3 4 Hyp. 8,V. 13, V. 10, V. Cone. •.• A : B = C : D, or C : D = A : B ; andif A > 0, A: B > C: B; •. C : B > C : B. But the M to which the same has the gr. ratio is the less, .-. D < B, i. e., B > D. Case II. If A == C, B = D. 1). 1 I Hyp. & 9,V. I •.• A : B = C, i e. A : D, .-. B = D. Case III. If A < C, B < D. D.l 2 3' Hyp. Case 1. Bee. •.• C > A, and C : D = A : B ; .-. D > B, l e. B < D. Therefore, if the first has thesame ratio,^c. Q.E.D. Briefly. 1°. Let A > C; then,' by 8, V., A : B > C : B; but A : B = C : D, .-. by 13, v., C : D > C : B, and .-. by 10, V., B > D. Similarly, 2°. if A = C, by 9, V., B = D, and 3°. if A < C, B < D. CoR. Hence also, if A : B = C : D, and if the 2nd B > = or < the 4th D, then the 1st A >= or < the 3rd C. Prop. XV.^Thbor. Magnitucks have the same ratio to one another whiah their equi- multiples have. Con. Pst. 2, v. Any given m of a M may be divided into parts, each of ■which is equal to that M. Dem. 1, v. Equal Ms, with the same ratio. 12, V. Anteoejents to con- seqnents. PROP. XV.— TSEOE. 195 E. 11 Hyp. I Let A B be the same m of C, that D E is of F ; 2 Cone. then C : F = A B ; D E. G 6 H -B 18 C K - 6 4 L- E 12 c. 1 Pst. 2, V 2 J) >) D. 1 C. 2 3 4 C. 1 & 2. 7,V. 12, V. 6 7 C. 1 & 2. Cone. Rec. Dmde A B into Ms each = C, Le., AG=GH=HB = C. & „ D B „ each = F, I. e., DK = KL=LE = F. Now A G, G -H, H B in number = D K, K L, &LE; and A G = G H = HB, and D K=KL=L E,. .-. AG: DK=GH: KL = HB : L E. And •.• A G : D K = A G + G H + H B : D K + K L + L E, but A G = C, and D K = F; .-. C : F = A B : D E. Wherefore, Magnitudes have the same ratio, &c. Q. E. D. Otherwiser Take m any number, A & B two:iftagnitndes ; then A : B = m A : ?n B. ■.• A : B?= A : B, .". A : B=A+ A : B+B, or2 A = 2B; and •.• A ;B= 2A :2B, .-. A :B = 2A+ A :2B + B»ot3A= 3B. And so on for alt eqnims. of A and B ; .". A : B =: mA : mB. CoR. 1. Magnitudes have the same ratio to one anotlter which- their equal suhmultipks or like parts hteoe. ^ A H 12 Ifi thus, A : B = ^ : g-; or, 12 : 18 = -g- : -3 ,2. e,, 12 : 18 = 1 :6. CoR. 2- In PEoportiouals the eqtHins. of the Ist and 2nd have the same ratio as the eqnims. of the Ssd and 4th. If A : B = C : D. then m K : m'R = nC : nJ) ; D. 1 7,V.&12,V. D. 1, 3 12, V. 4 Sim. 5 Cone. 19G GRADATIONS IN EUCLID. Arlrh. If 2 : 3 = 4 : 6, then 2x2:2x3=3x4:3x6; i. e., 4 : 6 = 12 : 18. Cor. 3. In Proportionals also, any like parts of the 1st and 2nd, and also of the 3rd and 4th are proportional; as A B C^ . 5_ 2 '• 2 — 3 ■ 3 • Sell. Tlic Proportion is sometimes explained, " one magnitude shall have the same ratio to another magnitude of the same kind wliich any multiple ot the fii'st has to the same multiple of the second." — Hose. Prop. XVI.— Theoh. Alternando, or Permiitando. If four Magnitudes of the same kind he proportionals, they shall also be proportionals when taken alternately. CoK. Pst. 1, V. Dem. 15, v.— 11, v.— 14, V. E. 1 2 Hyp. Cone. 18 — Let A : B = C : D ; then A : C = B : D. 12- -B G- C- D- H- -12 -16 -24 c. D.I 2 3 4 5 6 7 Pst. 1, V. Take E, P, equims. of A, B, and G, H, equims. ofC, D. C. •.' E same m of A, that F is of B, 15, V. .-. A:B = E:P; H. & 11, V. but y A : B = C : D, .-. C : D = E : F. C. Again •/ G & H are equimg. of C & D 15, V. •/ C : D = G : H. D.3,^U,V but C : D = E : F, /. E : F = G: H. 14, V. In 4 proportionals, if 1st, > = or < 3rd, the 2nd is > = or < 4th ; PEOP. XV. — THEOE. 197 9 10 11 0. Def. 5, V. Eec. Or, c. D. 1 2 3 4 Pst. 1, V. 15,V. &n,v, i5,v.&n,v. 14, V. Def. 5, V. Alq. §• Arith. Hyp Alg. •.• if E > = or < G, F > = or < H; Now E, F, are equiras. of A, B ; & G, H, of C.D; .-. A : C = B : D. Ifih.ea.four magnitudes of the same kind, ^c. Q. E. D. Take m A, m B, n C, ti D, eqnims. of A, B, C, D. •.• mA :mB = A : B, .-. m A : mB = C : D. also, -.• raC : ™D = C : D, .-. m A : mB = kC : reD; now mB > = or < nT>, as m A > = or < nC; .•.A:C=B:D. Let -- ; then _- 6 c ' Let -T- = -7 . d then ad = be. ^ . t, j% , ad _bc' (.-hy cd), and-^-^. or. — = -f Arith. ifA = then 2X6 = 4X3. ^ 2 X S 4X3 (-^by 3 X 6,) and ^^^ ^Tx'&' or — = 4 6. Use & App L From principles established, especially, from Def. 7, Vr— and from Props. 7, 11, 14, 85 15, bk. V, the following Theorem, Chambers Ex. p. 56, may be demonstrated ; Jfto the terms of a ratio, A : B, the same magnitude, C, be added, the ratio will be unchanged, increased, or diminished, according as it is a ratio of eixiiality, o/less inequality, or of greater inequality. Case L If A = B, then A+C:B+C=A:B. D. lAx. 2,L |VA = B,.-.A+C = B+C; 7,V.&16,V. .•,A+C:C = B + C:C,andA+C:B + C=C:C. Sm. &H,V.lSoA:B = C:C. and.-.A+ C :B + C= A :B. Case IL D.l H. Sup. If A < B.thenA +C:B + C>A:B. ■ A < B,if B — A = D; then B = A + D. D. 2, 3. Sup. r>. 2, 5, & 1. Let p A, and » D the least m of D exccedii g p A, so thatp A < n D ; , hence ;jA + A>hD, or=nD; & ,-. ;) A+ ;) raD, or jo (A + C) > nD. Next, let ra = n + P) orp = m—n ; then '.• p A < « D ; p = m—n, and D = B— A ; .-.(»!— ri) A < H CB— A) ; or m A—n A< n li—n A: 198 GRADATIONS IX EUCLID. 13; T). 8. Def. Case HI. D.l. 8 Add. To both sidqs add n A, .•. »» A < n B. 9 D. 4. Again. •.• p (A + C) < n D, 10 D. 6. .'.pA + pOn (B— A);ormA— re A+ mC—n C >nB-»A. 11 Add. Add » A + « C to each side of the equation ; 12 Ax. 4,1. /, m A + m C> n B + » C i or m (A + C) I >re (B + C) ; 7, V. but m A < reB, .-. A + C : B + C> A : E. If A >B, then A + C : B + C < A : B. H. •.• A > B, if A-B = D, then A = B + !>• Sup. Let /) C > B ; and m D the least m of D, so that jt> B .< m D ; D. 2. hencepB + B > or = m D; D. 2 & 3, &,-.;> B + p C> m D, or p (B + C) > m D. Sup. Next, let re = m + p J orp = n—vi ; D. 2, 5, & 1. then •.• p B < »i D, p = n— m, and D = A— B, .*. (re— m)B<>n(A— B);"orreB— mB re B. D. 4. Again •/ p (B + C) > m D, D. 6. /. pB + p C> m(A-B); orreB-mB + reC— m C >»i A — mB ; Add. Add m B + )» C to each side of the equation ; Ax.4,I. .-. B(B+C)>m(A+C); orm(A +C) <»(B + C). D.S, Def. 7, V. But m A > re B, /. A + C : B + C < A : B. Eec. Wherefore, If to the terms of a ratio Ifc. n. Props. 11, 15 & 16, bk. v., also furnish the principles by which to establish the useful Theorem, — Chambers' Ex. p. 54; — that, " If all the terms, or any, two homalogpus terms, or the terms of either of the ratios of a proportion, hemmtipliedor divided by the same number, the resulting magnitudes will remain proportional. E. 1 I Hyp. 1 let A : B = C : D, and m, n be any two ntimhers; then, Case I. m A : m B =: m C : ml>; D. 11 15, V. I •.• A :B = mA :7«B; andC :D=mC : mD; a] 11, V. 1 .-. mA : )mB=:»iC : )kD. Case n. 1b = Ic Id. D. 1 H. ! 2 15, v. i 3 Sim. I 4 11, V. A & B are mults. of -*■ X m, and of J-B x m ; m m 1*1 .-. A : B = ±A m In like manner C : P :^ B. m __1_ m U C : -1-D. m Ld. PROP XVtI.— THEOSw 199 CAsb Ur. m A : B =: w C r D. D. 11 H. & Sfch. 3, V.l •.> A :B = C:D, .-. mA :B=±:»nC :fi; 2| -S'm. I In like manner, A : jtB =z C : jj D. Case IV. i-A : B = J-C : D. 1). 1 2 H. & 16, V. Oaae 2, 3i 16, V. •.• A : B = C : D, and A : C = B : D; ,-. L A : i- C = B t B, and J-A :B = -Lc :D. Ca&S V. m A : ?re B = C i D. D. 11 H. & 15, V. I •.• A : B = C : D, and A : B = »« A : mB; 2|n, V. 1.-. »iA: mB = C :I). Case VI. i-A : i- = C : D. D. 1 2 3 4 H. 15, V. 11,V. Roc. •.• A : B = C : D, & A,B are equims. ol —A, -B by m; ^ mm .-. A ; B = La . Lb. m m and. .-. La : Lb = C : D. m m , Wherefore, If all the terms, 8fc. Q. E. IX Prop. XVII ,— Tbeok. Uividendo. If magnitudes taken, jointly, be proportionals^ tkei/ sJiall also be proportionals when taken separately ; that is, if two ■magnitudes together have to one of them, the ■ same ratio which two others have to one of these, the remaining one of the first tioo shall have to the other the same ratio which the remaining one. of the last, two has to the other of these. N.B. The General Enunciation of this 17th Proposition is variously given ; " If magnitudes be proportional, they will also be proportional by divi- sion." — Euclid. 200 GRADATIONS IN EUCLID. " If four magnitudes. A, B, C, D, be pi-oportionale, they shall also bo pro- portionals, when taken dividedly : that is, the difference of the first and second -shall be to the second as the difeence of the third and fourth to the fourth ; or dividendo, A<>jB;B = C(viD: D." — ^De Mosgak. "If four magnitudes be proportional, the first being greater than the second and the third greater than the fourth ; then the excess of the first above the second shall be to the second, as the excess of the third above the fourth is to the fourth." — Hose. Con. Pst. 1, V. Dem. 1, V. — If any number of Ms be equims. of as many, &c., 2, V. If the Ist M be the same tb of the 2nd, &c. Dbp. S.V. Criterion of the equality of two ratios. Ax. 4, 1. If equals be added to unequals the wholes arc unequal. Ax. 5, 1. If equals be taken fi'om tin- equals the remainders are unequal. El Hyp. 1 Cone. Let A B, B E, C D, D F, be Msiak&a jointly and proportionals, i.e., AB: BE = CD : DEi-AB > BEand CD > DF; and let A E be the excess of A B above B E, and C'F the excess of C D above D F ; then A E : E B = C F : F D. G 10 H 6 K G X 22 r i A 5 E3 B 8 I C 10 F G n 16 X 20 M 12 N 12 P44 C.l Pst. I, V. 2 jj ?) D.l C. f 2 1, V. 3 c.l. 4 5 C.l. 6 1, V. 7 D. 4. 8 D. G & 7- 9 C. 1. 10 C. 2. 31 -2V. Of A E, E B, C F, F D take equims. G H, H K, LM, MN; and of E B, F D any equims. X K, N P. •.• G H, H K are equims. of A E, B E ; .•. G H same m of A E that G K is of A B ; But G H same m of AE that L M is of C F ; .-. GK „ AB „ LM „ CF. Again •.• L M same m of C F that M N is of FD; LM „ ofCF „ LN „ CD; but L M same mof C F that G K is of AB , .•. G K, L N are equims of A B, D. Next, ■.- H K, M N are equims of E B, F D ; and •.• KX, NP equims, of E B, F D, .-. HK + KX&MN-t- NPequims.of EB,FD; i. e. nX and MP are equims of EB, FD; rnoi'. XVII. — THEOR. 201 12 13 14 15 10 17 18 19 20 21 22 23 Hyp. D. 8. D. 9. Def. 5, V. Sup. Add. Ax., 5 I. Sill). Def. 5, V. C. 1. C. 2. Def. 5, V. Eec. And V AB: BE = CD: PD; / and •.• GK, LN are equims. of AB, CD, , and HX, M P equims. of B E, F D ; - .-. if GK > = or < HX, then LN > = or KX, add H K to both, .-. GK>HX; .-. also LN > MP; take M N from both ; and L M > N P ; .-. if GH > = or < KH, LM > = or < NP. But GH, LM are equims. of A E, CP, and KX, NP are equims. of EB. PD; .-. AE : EB = CP: PD. Wherefore, If Magnitudes taken jointly, 4'c. Q. E. D. C.l D.I C + D : D, then by division A , B = C : D; Take mA, nB,miilts.of A,B. Let »! A, > nB. To both sid?s add jnB; /. tbA + m B > raB + nB, or (m + n) B, buf.-A + B:B = C+D:D, .-. if m (A +B) >(m +«)B, m (C +D) >(m + «) 1>; thus mC + m D >mD + nB; from both sides take m D, /. mC > mD; i.e., if mA > nB,mC >nD. If m A = n B, then m C = n D, and if m A < n B, then m C < nB. .-. A : B = C : D, by division Def. 16, V. Wherefore, ifmagnitudes, S[c. Alg. and Arith. Hyp. Tace a, 8 : 6, 2 : : c, 12 : rf, 3. a £, It ~ d «_! = «, _l. Ollta-wise. If A + B :B Ps^. 1, V. Sujx. 1. Add.A^.i,J, H. Sub. Ax. .5,1. Sup. 2. Sim. Sup. 3. Sim. Def. 5, V. llec. Alg. Hero Sub 1 from each ; 6 ' d a—b _ c—d , b d or a—b : b = c—d i d. i. e. Arith. Kub. 1 -|— 1 8-2 3 ' = J^— 1, 3 ' 12—3. -2 :2 = 12— 3 :3,or 6 : 2 = 9 : 3. 202 GRADATIONS IN EUCLID. Con. 1. Convertendo. — If four Ms, A,B,C,D bo proportionals A : B = C : D, they shall also be proportionals by conversion, i. e., the 1st : lst~2nd = 3rd : 3rd~4tb,— or A : A~B = C : C~D. For, invertendo, Pr. B,V. dividendo, 17, V. invertendo, B, V. B: A = D:C; B~A: A = D~C: C; A : A~B = C : C~D; or convertendo. Cor. 2. If four Ms of the same kind, A,B,G,D, be proportionabj A : B = C : D, tlien the greatest + the least are together greater than the other tivo. D. r Sup. 1. ! 2' 14, V. i H. & 17, V 4| D. 2, 51 Add. Sujj. 2. V Pr. B, V, 8 Sim. Let A one of the extremes be the greatest; as 12 : 8 = 3 : 2. V BD, .-. A-B >C-D. To each add B + D ; /. A + D>C + B; i. e. the greatest 12 + the least 2 > the other two 8+3. Let B one of the means be greatest ; as4 : 16 = 3 : 12; invert. B : A = D : C. So, as before, B + C > A -f D, i. e. the greatest B 16 + the least C 3 > tlie other two 4 + 12. .•. the sum of the greatest and least > the sum of the other two. N. B. — This Corollary is identical with Prop. 25, Bk. V. CoE. 3. In three Proportionals, A : B : C, as 2 : 4 : 8, or 9 : 6, 4, the sum of the extremes, A + G is greater than tivice the mean, 2 B ; and therefore half the mean, B. For, 1) Kec. A+C of the extremes is greater than \ if the mean B > C, then B < A; and if B < C, then B > A. .•. the extremes A & C are the greatest M, and the least, and as before, in Cor. 2, A + C > B + B, or 2 B ; A+C and >B. PROP XVIII. THEOR 203 UaE & App. When half the sum of two Ms, ^ + ^ , is as much gi'eater than the one, as it is less than the other, that half sum is an arithmetieal mean between the two; thus, in ^-—^ = 5, 8—5 = 3, & 5—2=3 ; and 2 : 5 : 8 are in arithmetical progression. Except when the magnitudes are equal, the arithmetical mean between two magnitudes, A and C, is therefore greater than A _i_ p the geometrical mean, {. e, the arith. mean, — ^^—^ > B the geom. mean; 8 -^ 2 ">■ — — i. <:., 5 > 4. Geom. Plane, Sol §• Sph. pp i\ S,- 42. Peop. XVIII.— Theoe. Componendo. If magnitudes taken separately, be proportionals, they shall also be proportionals when taken jointly, by composition; that is, if the first he to the second, as the third is to the fourth, the first and second together shall he to the second, as the third and fourth tof/ether to the fourth. " If magnitudes be proportional, they will also be proportional bi/ compo- sition." — EtrcLiD- " Xhe terms of an analogy are proportional by composition." — Sell Con. Pst. 1, V. Dem. Def 5, V. Criterion of the equality of ratios. Ax. 3, V. A IK of a gr. M is gr. than the same m of a less. 5, V. If one M be the same m of another which a M taken from the 1st is of a M taken from the other ; the rem. is the same m of the rem. that the whole is of the whole. 6, V. If two Ms be eqnims. of two others, and if equims. of these [be taken from the first two ; the rems. either= these otliei-s, or are equims. Cor. 4, V. If 1st : 2nd ^ 3rd : 4th, then any equims. of 1st and 3rd the same ratio to 2nd and 4th ; and 1st : 3rd = equim. 2nd : equim 4th. Fr. A, V. If 1st : 2nd = 3rd : 4th ; then if 1st > 2nd, 3rd > '4th, and if i=, = ; if <, <. Ax. 2, V. Those Ms of which the same or equal Ms arc equims. are equal. Ax. 4, V. That M of which a m is greater than the same m of another is gr. than that other M. 204 GRADATIONS IN EUCLIlJ. E. 1 2 C. 1 2 Hyp. Cone. Pst. 1, V. Let AE : EB = CF : FD. then AE + EB = : BE = CF + PD : DF; ;. e., AB : BE = CD : DF. Of AB, BE, CD, DF take any equims. GH, HK, LM, MN; and of BE, DF any equims. KO, NP. 1 fi H O 6 32 MP N I'lG. 1. ] D16 6 - BS' 3 10 E- 10 12 M32 6 FIG. 2. H16 3 20 E B8 F Dig 6 N a A C L G 10 12 H K 14 PIG. 3. O 19 3 Vii P38 6 M N - B8 10 E DIG 6 10 C L G A C M 12 22 6 II 20 K N DIG 6 B 8 F 3 10^ 10 12 12 20 GAG D. 1 2 C. 2. Def..5,V. •.• KO, NP are equims of BE, DF, & KU, NM also equims of BE. DP ; .-. if KO > = or < KH, then NP > = or > NM. Case I. Let KO > KH, & .-. NP > NM. Fig. 1 & 2. D. 1 C. 1. •.• GH, HK equims of AB, BE, AB being gr. than BE; 2 Ax. 3, V. .-. GH > HK, 3 H.Ax.3,V. but KO > KH, .-. GH > KO. ■i Sim. In the same manner LM > NP ; 5 H. ••• KO > KH, .-. GH, a w of AB always > KO the same m of BE ; I .-, also LM the m of CD > NP the m of DP. PROP. XVm. THEOR. 205 Case II. Let KO > KH, and .-. NP > NM. Fig. 3. D. 1 C. 1. 5,V. Sim. 5,V. d:2. 6,V. C. 2. Pst. 1, Y, 6, 5. And ■.• the whole GH is the same m of the whole AB as HK of BE, .•. rem. GK same m of rem. AE as GH of AB. or as LM of CD. '.• LM the same m of CD, as MN of DP, .•, rem. LN same m of rem. CF as the whole LM of the whole CD. But LM same m of CD, as GK of AE ; .-. GK same m of AE, as LN of CF, i. e. GK and LN are equims. of AE & CF. And ■.• KO, NP are equims. of BE, DF ; & •.• from KO, NP may be taken KH, NM also equims. of BE, DP, .-. rems. HO, MP either = rems. BE, DF, or are equims. of BE, DF. Subdivision 1 Let HO, MP = BE, DF. Fig. 3. D. 1 2 3 H&Case2,D.6. Cor. 4, V. H. Pr. A, V. •.• AE : EB = CF : PD, & GK, LN are equims. of AE, CF ; V GK: EB = LN: FD; But H0= EB & MP = DF, /. GK-: HO = LN : MP ; .-. if GK > = or < HO, LN > = or < MP. Sued. 2. Let HO, MP be equims. of EB, FD. Fig. 4. D. 1 2 3 H.&Case2,D.6 H. Def. 5, V. Suh. & Ax. 5, 1. •/AE : EB = CF : FD, and of AE, CF, the equims are GK, LN, & of EB. FD the equimg-are HO, MP; .-. if GK > = or < HO, LN > = or > MP; but if GH > KO, from each taldng KH, then GK > HO ; 206 GRADAMOSS IN EUCLID. Add. Ax. 4, I. Sim. Case I. D. 3, 4. 5,V. .• also LN > MP ; to both add NM .-. LM > NP; .-. if GH > KO, LM > NP. So, if GH = or < KO, LM = or < NP; And when KO > KH, then GH > KO & LM > NP; but GH, LM equims of AB, CD, & KO, NP of BE, DE ; .-. A B : BE = CD : DF, i. e., AE + EB : BE=CP + PL : DF, Wherefore, if magnitudes taken separateb/ #c. . Q. E. D. ' Otherwise. li A : B = C : t) ; — and of B, D any like pts, as the nth, lie contained in A. C, m times exactly, or with like remainders ; i. e. if m B = A, & m D = C, — then those pai-ts will be contained in A + B & C + D, tlie same number of times exactly, m + re with the same remainders ; .-. by Def. 5,V. A + B : B = C + D : D. Cob, A:A + B = C:C + I'- ;) 6 7 8 9 10 11 12 C. Def. Rec. By indirect Proof. E. n. IjHyp. 2| Cone. 1' Sup. 2 3 4 H. 17, V. H. II, V. 9, V. Cone. If A : B = C : », then A + B : B=C + D : D. IfC+ D : D^t A + B :B, let e + d : d = A + B : B, and (i! i^t D. •.• A + B : B = C + d : rf, .-. A : B = C but A : B = C : D, .-. C : D = C : d. .•, D := dy contrary to the hypothesis; J .-. A + B : B = C + D : D. A!g. ^Arith.Hifp. Alg. (+ 1) (+1) o5 :63 = c- 10 : d C Let a : i = = c d, «+! = d + - ,e.l^- e ¥' (X, a + b :b — c + d: Let 5 : 3 = 10 6, A + 1 = 10 6 + 1, .5 + 3 _ 10+6 5 6 8 : 3= 16 : 6. PROP. XIX. THEOR. 207 Use & Appl. This method of reasoning is often employed. By an extension of it, as indicated in the Corollary, we find that the terms of a pro- portion ai'e also proportional by addition. For, Let A : B = C : D, — then, addenda, A:A + B=C:C+D; Invertendo, B : A=D : C; Componendo, 18, V. .-. invert. (A, V.) A : A + B = C : C + D. A+B:A=C-|-D:C; Peop. XIX.— Thbor. If a whole magnitude he to a whole, as a magnitude taken from the first is to a magnitude taken from the other ; the remainder shall he to the remainder as the whole to the whole. " If a whole be to a whole as a part taken away to a part taken away; the part left is to the part left, as the whole is to the whole." Euclid. " If four magnitudes, which are all of the same kind, be proportional, the first being greater than the third, and the second than the fourth ; then the excess of the first above the third, shall be to thait of the second above the fourth as the second is to the first." — Hose. Dem, 16, V. AUernando; — ■ 17; V. Dividvndo; 11, "V. Ratios the same to the same ratio are the same to one another. E. 1 Hyp. 2 Cone. D. 1 H. & 16, V 2 17, V. 3 16, V. 4 H. 5 11, V. 6 Eec Let the whole AB : the whole CD = a pt. from AB : CE a pt.from CD; b then rem. EB : rem. ED = the whole AB : the whole CD. •.• AB : CD = AE : GF, .-. «?*, BA: AE=DC: CE; & •.• Ms jointly are propl., — separately they are also; .-. BE : EA = DF : FC, & alt. BE : DF = EA :FC ; but A E : CF = AB : CD; /, rem. BE : rem. DF = in- teger AB : integer CD. .'. If a whole magnitude, o C ; and so on. CoK. 3. And conversely, any number of Magnitudes,, A, B, C, I) &c. in geometrical progression, A : B : C : D &c,, may he considered as the differences of other magnitudes A, B' C' D' &c., forming a geometrical progression in which the first term A' is to A as A to A (vj B, A' to B' as A to B &c., and the successive terms have the same ratio with the successive terns of the foi-mer. For let a progression be taken in which A' : A = A : A r>o B, and A' : B' = A : B. PROP XIX. BOOK V. Cono. 209 B; D. 1 H. Cor. 1, 17, V. Then •.• A' : B' = A : B, /. convert. A' : A' ~ B' =A : A 2 H. 9, V. but A' : A = A : A ~ B, .-. A' : A' ~ B' = A' : A. 3 D. 2. 14, V. and •.• 'the 1st A' is the same with the third A', .•.■A*~ B' = A. 4 Cor. 2, 19, V. but A' ~ B', B' -. C, C' ~ D'&c, form a progression in which A ~ B' : B' ~ C = A' : B', i. e. := A : B • 5 14, V. .-. F ~ C = B, C ~ D' = C &c. 6 Eemk. But the progressions A, B, C, D &c. and A' ~ B', B' ~ C, C ~ D', D' ~ E' &c., have the same first terms and the same common ratio. .". those progressions cannot but be identical. Geom.Pl. Sol. ^ Sph. p. 42, 53. Alg. §• Arith. Hyp. Let a = 15, 4 ^ 10 be two Ms; A = 6 & y = 4 the respective pai-ts. Alg. By Hyp. .*. ay =z b X, we have to prove that clearing fractions, subtract ab, .". bx ^ ay as before. Because of this identity of products the quantities must be in proportion ; .'. a — X : b — y ^ a : b. a — X a ■. b—y ~ b ab — bx = ab — by ; Cor. Let :?. = - •. ay = b X ( !')• a X : Arith. b y To prove that ,- ^ a — X — y clearing fractions ax—ay'= ax — bx; Subtract ax, ay = bx, an identity with (1); a : a—b =: x : x — y. 15 _ 6 ; 10 ~ 4 15 X 4 = 10 X 6. to be proved 15-6 _ l-'i' 10—4 10' P 210 ' GKADAXIONS IN EUCLID. clearing, (15 X 10) — (6 X 10) = (10 X 15) — (4 X 15), take away 15 X 10 .*. 6 X 10 = 4 x 15. Now 6 X 10 & 4 X 15 give identical products, /. 15 — e : 10 — 4='15 : 10. I. e., , 9 : 6 = 15 : 10. .-.15 X 4 = la X 6 (1>. Let i5 = 10 15 To prove that :- ; ^ 15 — 10 6 — 4 ' » clearing, (16 X 6) - (15 X 4) =i= (15 X 6)-(.10 X 6)j Subtract 15 X 6, and 15x4 = 10x6, an identity; ,-. 15 : 15-10 = 6 : 6-4. t. e. 15 : 5 = 6 : 2. Prop. E. — Theok. ConTertenda. If four magnitudes he prdportionctls, they are also p'- oportionala by conversion : that is, the first is to its excess above the second, as the third to its excess above the fourth. " If four raagnittides be proportional, the first being greater than the second, and the third than the fourth ; then the first shall be to its excess above the second, as the thu'd is to its excess above the fourth."— Hose. " The' terms of an analogy are proportional by conversion." — ^BeIX. Dem. ITjV. dividendo; B, V. invertendo; 18, V. componendo. E. 1 2 B. 1 Hyp. Cone. H. 17, V. Pr. B, V. 18, V. Kec< Let AB : BE = CD : DF; B 12 then B A : A E = D C : C F. •.• AB : BE = CD : EF, AB being>BE, & CD >EF, .-. div. AE : EB = CF : PD. and invert BE : EA = DF : CF, wherefore comp. BE + AE : AE =F D + CF : CF, i. e. BA : AE = DC : CF. If therefore four magnitudes, SfC,. Q. E. T)i PEOP E. BOOK V. 211 Or, If A : B = C : D, by conversion, A : A — B ss C : C ■ D. 1 2 3 H. &17, V. Pr. B, V. 18, V. •.• A ; B = C : Di /. Aw. A — B : B = C — D : D. and tnt). B : A — B = D : C — D ; .-, comp. A=A — B = C:C — D. Alg. S[ Arith. Hyp. Let a 12 ; J 9 = c 8 : A and D > C; And '.• ( A + B + C + D)— (B + C + D + E) = A (v; E, .■. A(>wiB:A:=AooE:A + B + C-j-D. 212 GRADATIONS IN EUCLID. 2. In a series of continued proportidnah, A : B : C : D : E, &c., the differences of the successive terms A(vjB, BfviC.CtvjD, &c., are also in con- tinued proportion, — AooB: Br^C: C r^Tt, and B ro C : C (v) D = C (v; D : D :vj E. Case 1. Let A > B, — the series is continually decreeing ; then A — B : B — C = B — C:C — D, andB — C:C-D = C — D:D — E. D. 1 a 4 H. 17, V. •.■A:B = B:C, /.div. A — B : B = B — C : Cj 16,V. andoZfern. A— B : B — Cj=B : C, Sim. So, •.• B : C = C : D, .-. B-C : C-D = C : D. H. 11, V. bntB:C = C:D, .-. A — B : B — C = B — C :C — D. 5| Sim. SoB — C:C — D = C — D :D — Ej el Rec .". A — B : B — C : C — D : D — B, are continned proportionals. Case II. If A < B, the series is continually increasing. By a like method this case is also proved^ 3- In an infinitely decreasing series of Magnitudes in continued proportion, ttie first term. A, is a mean proportional between its excess above the secmid, B, and the sum of the series. Let A, B, denote the 1st and 2nd tenns, — Z the last term, and S the sum of the Series, then, as in the last example but one, Use 1, E, V., A— B : A=Z : S— Z. But the last term Z may be less than any magnitude or quantity we fix on, however small ; and hence to the values of A— Z and S—Z there will be limits namely A and S. • A-B:A = A:S. Prop XX— Theoe.. If there be three magnitudes, and other three, which, taken two and two, have the same ratio ; then if the first be greater than the third, thi fourth shall he greater than the sixth, and if equal, equal ; and if "1 Dem. 8, V. The gr. M a gr. ratio; 13, V. If 1st : 2nd = 3rd : 4th, but 3rd : 4th > 5th ! 6th, the Ut ; 2nd > 5th : 6th. PROP XX. BOOK V. 2ia Cor. 13, V. If 1st : 2nd > 3rd ; 4th, but 3rd : 4th = 5th : 6th, the l3t : 2nd > 5th : 6th. 10, V. The M with gr. ratio, the gr. of two Ms. 7, V. Equal Ms. the same ratio to the sameM &c. M, V. Ratios the same to the same r. the same to one another. Pr. B, V. Invertendo. 9, V. Ms. with the same ratio equal &c. K I a 3 Hyp. 1. Hyp- 2. Cone Let there be three Ms. A, B, C, & three other Ms. D, E, F ■ let A : B = D : E, & B : C = E : F ; if A > = or < C, thenD > = or < F. 14 Fig. 1. Fig. 2. Fig. 3 12 8 I 6 ! i 6 i 5 ' 1 I 3 4 4 4 4 ; < i A I i c I ) 1 D p 'i I I ! ( 3 I ) I i i J i. 1 i ( 3 I 1 » E F Case I. Let A > C,then, D > F. Fig. i. . 1 2 3 H. 8,V. H&13,V. H&B,V. D. 3, Cor. 13, V. 10, V. •.• A > C,& Bis another M; .-. A : B > C : B. but D : E = A:B, .-. D : E > C : B. A. B. C. D. E. F. 4 5 6 And vB:C =E:P, .: im and •.• D: E> C: B, .-. D .-. D > F. ;.C:B = F:E; : E >F: E; Ca8e II. Let A == 0, then D = F. Fig. 2. D. 1 2 3 H. & 7, V. H. 11,V.B,V.9,V. V A = C, .-. A : B = C : B; butA:B=D:E, &C:B = F:JE; .-. D : E = F : E, & .-, D = F. 214 SRADATIONS IN EUCLID. D. Case III. Let A < C, then then D < F. Fig. 3. H. & Case 1. Case 1. Eec. vC>A,andC:B = F:E,&B:A=E:D; .-. F>D, i.e., D 3rd, 4, & 5th, 4 > 6th2 Case 2. If § = 4-, and f = A ; then the 1st, 8=3rd 8, & 4th 4=6th 4. Case 3. If f = \^, and .^ = i| j then the 1st 6 < 3rd, 7, &4th 10 < 6th 14. ScH. The Proposition is also ennnciated, -with reference to the formula annexed, "If the first magnitude. A, be to the second B, as the third, C, is to the fourth, D j and, if the second B, be to the fifth E, as the fourth, D, is to the sixth, F ; then the third, C, shall be gieater than, equal to, or less than the sixth, F, according as the first, A, is greater than, equal to, or less than the fifth E."— Hose. There are of course three Cases, A > E, A = E, and A < B, to be proved as in the foregoiag proposition fi;om Smsoir's text. A. C. B. D. E. F. Peop. XXI.— Theoe. If there he three magnitudes and other three which have the same ratio taken two and two, hut in a cross order, i. e. in proportione perturbata, in disturbed proportion ; then if the first Tnagnitude be greater than the third, the fourth shall he greater than the sixth ; and if equal, equal; and if less, less. Dem. 8, V. 13, V. Pr. B, V. invertendo; Cor. 13, V. 10, V. 7, V. 11, V. and 9, V. PROP XXI. BOOK V. 215 E. 1 Hyp.l 2, Cone. Fig. 1. Let there be two series of Ms, A, B, C and D, B, F : and let A : B = E : P, and B: C = D: B; then, if A >, =, or < 0, D >,=, or < F. A. D. B, O. 12 Fig. 2. Fiff. 3. 10 A B 2 3 7.5 D E P A B D B F Case. I. Let A > C, then D > F. Ficf. 1 A B B B D. 1 2 3 4 H. & 8, V. H. & 13, V. H. Pr. B, V. D. 2. Cor. 13, V, 10, V. .". A > C, and B another M, .-. A : B > C : B; but E : P = A : B, /. B : F > C :B ? & '.• B : C=D : E, ,-. inv. C : B =E : D; & ... E : P > C : B, .-. E : P > E : D; .-. P < D, i.e., D > P. Case II. Let A = C, then D = P. Fig. 2. D. 1 2 3 4 H & 7, V. H. D.3,CaseI 11, V. 9,V. V A= C, .-, A:B= C: B; but A : B = E : F, & : B = E : D, .-. E : P = E : D ; and .\ D = P. D. 1 2 3 4 Case III. Let A < C, then D < P. Fig. 3 H&D.3, Casel Case I. Bee, V C > A, .-. C:B = E:D, andB: A = P: E; ,-. P > D, i. e. D < P. Therefore, if there be three magnitudes, ^o. Q. E. D. 216 GRADATIONS IN EUCLID. Arith. Hyp. Let the two series of three magnitudes each be represented by numbers,— Cbse 1. by 8, 6, 4, 6. 4, 3; Cose 2. by 4, 6, 4, 3, 2, 3 : and C(Ue 3. by 5,6,8,7.5,10,12. Case I. If f = f, & S = If *lien if Ist 8 > 3rd 4, the 4th 6 > 6th, 3. Case 2. If |- = f & f = 4; then if 1st 4 = 3rd 4, the 4th 3 = 6th, 3. Case 3. If f =!■&,& I = 4^ ; then if 1st 5 < 3rd 8, the 4th 7.5 < 6th 8. SoH. The following is a variation of Prop. 21, having reference to the annexed formula ; " If the first magnitude, A, he to the second B, as the third C is to the fourth D ; and if the second, B, be to the fifth, E; as the sixth, F, to the third. C; then the sixth, F, shall be greater than, equal to, or less than the fourth, D, according as the first. A, is greater than, equal to, or less than the fifth E." — HosB. Again, there are three Cases, A>E, A^E, & A = or < 3rd, the 4th > = or < 6th. Def. 5, V. Criterion of the equality of ratios. PKOP XXII. BOOK V. 217 Case I. Let there be two series of three Ms, A, B, C, and B, E, F. E. 1 2 Hyp. Cone. Let A : B = D : E, & B: C = E : F ; then A : C = D : F. 16- 12 ^G 8 -K 6 — M 4- -A -B -C D 4 H. E — 3 L- F — 2 N- C. 1 D. 1 2 Pst. 1, V. H&C. 4,V D. 2. 20, V. 0. Of A,Dtake equims G,H; of B,E eqnims K,L; and of C,F, equims. MjN. •-.• A : B = D : E ; & G,H equims of A,D, and K,L, equims of B,B; .-. G:K = H:L; &K:M = L:N; and •.• G ,K,M are 3 mags.. & H,L,N, 3 others, which taken two and two have the same ratio ; .-. if G >, = or < M, H >, =, or < N". But G, H are equims. of A,D ; M,N equims of C,F; Def. 5, V. .-. A : G = D : F. Case II, Let there he two series of four Ms eacA, A,B,C,D and E,F,G,H. E 1. Hyp. Cone. Let A : B = E : F ; B : C = F : G, and C : D = G : H; then A : D = E : H. A. 12- -B 6 — C. 18- - D. 36. E. 6- -F. 3 — G 9.- ■ H. Is. Then A 12 : D36 = E6 :H 18. D. 1 H. Case 1, H. & Case 1, Sim, Eec. •.• A, B, C, are 3 Ms, and E, F, G, three others, which taken two and two have the same ratio; .-. A : C = E : G ; but C : D = G : H, .-. A : D = E : H. And in a similar way, whatever be the num- ber of Ms. Therefore, if there be any number ^c. Q. E. D. 218 GEADATIOSrs IN EUCLID. Alg. ^ Arith. Hyp. Take the two series of quantities, a 12, i 6, c 18, and a' 6, 6' 3, -^'9. Alg. BjUjv. ^=|:.and| = i:, Multiply. 2. X A, and -?: x -^j We have -,— , or — = -n— n o"^ -7 ; ■6c c be' c i. e. a : c =r o' : c'. Arith. ByH. 1^2=1, &^ = ^; Mult. 1^ X T^, &| X f; We have A%, o' 11- & M' " I- i. e. 12 : 18 = 6 : 9. Cor. Whatever be the number of Analogies, — for instance, three, z. e. A : B = C : D;, B : E = D : F, and E : G = F : H; if they are so constituted that the second and fourth terms of each, as B & D, form respectively the first and third of the next, as B & D ; then A, the first term of the next proportion shall be to the second of the last, aS the third C of the first proportion to the fourth of the last. 3 = A2 :L9 = G4 ■Mis In other words, — " Ratios compounded of any number of equal ratios in the same order, are equal to one another." For, by Hyp. '.• A : B = C : D, and B : E = D : F, ,-. A : E = C : F. And again, •.• A : E = : F, and E : G = F : H ; /. A : G = : H. And so on, whatever be the number of proportions. PROP. XXIII. BOOK V. 219 SoH. The 22nd Prop, may be tlins varied ; " If the first Mag. A, he to the second B, as the third C, is to the fourth D ; and if the second Mag. B, he to the fifth E, as the fourth D is to the sixth P ; then the first A shall be to the fifth E, as the third C, is to the sixth E."— Hose. A similar process of reasoning is to he followed, as in Case 1 and 2, Pr. 32 G. A. C. H. K. B. D. L. M. E. E. N. Use and App. Bycomhining the principles contained in Props. 18, 17 and 22, i. e. componendo. and ex , we arrive at the further truth, that Proportionals remain proportional miscendo, by mixing, or as it is some- times named, — by using the sum and difference. D. 1 Hyp. Cone. 18, r. 17, V. Pr. B, V. D. 1. 22, V. Let A : B = C : D J then A-|-B:Aiv;B = + D:C~D; adopting A — B, or B — A and C — D or D— C, as B < or > A, and C < or > D. Comp. A4-B:B = e + D:D; Biv. A — B:B = C— D:D; Inv. B : A — B = D : C — D ; but A + B:B=C + D:D; Ex aequo A + B : A-B = C + D : C — D. Prop. XXIII.— Thbok. Ex aequo perturbato. If there he any number of magnitudes, and as many others, which taken two and two m a cross order, have the sam^ ratio ; the first shall have to the last of the first magnitudes the same ratio which the first has to the last of the others. " If there he any number of magnitudes, and as many others, which taken, two and two, have the same ratio, and their proportion be disturbed, they shall be in the same ratio by equality." — ^Euclid. N. B. This Proposition is usually cited by the words, " ex isquali in pro- portione pcrfarJaf a, by equality in perturbate proportion-, or "ex aquo pertur- bato, by perturbate equality. 220 GRADATIONS IN EUCLID. Coif. Pst. 1, V. Dem. 15, V. Ms hare the same ratio as their eqnimg, 11, V. Katios the eame to the same ratio, the same to one another. 4, v. Equims of the 1st and 3rd have the same ratio to eqniins of 2nd and 4th, 21, V, Two seriesof 3 Ms. in each havingthesame ratio but in a cross order. Def. 5, V. Criterion of the equality of ratios. Case I, Let there be two series of 3 Ms, each, A,B,C,D,E,F, which taken two and two in cross order have the same ratio. E. C. Hyp. Cone. Pst. 1, V, LetA:B = E:F, &B:C = D:E; then A : C = D : F. Take of A,B, D any eqnims. G,H,K ; and of C,E,F any eqnims. L,M,N. 12 16 12 LHG ABC DEF K M N D. 1 2 C.&15,V. Sim. H&ll, V. H. 4,V. D.3. 21, V. C. 1&?. Def. 5, V. •.' G,Hare equims. of A,B; .". A : B=G : H, •.• M,N are equims of E,F, .-. E : F = M : N; but A : B = E : F, .-. G : H = M : N; And •.• B : C=D : E, & H,K are eqnims of B,D, & L,M equims. of G,E ; .-. H: L = K: M; now G : H = M : H, and G,H,L — K,M,N, are two series of 3 Ms. each, and taken in cross order two and two, they have the same ratio ; .-, if G >, =,or , =or < N; -but G,K, any equims. of A,D. & L,]Sr any equims. of 0, F ; .-. A : C = D : F. PROP XXIII. BOOK V. 221 Case II. Next, let there he two series of four Ms. each, A, B,.C, D, and E, F, G, H which taken two and two in a cross order also have the same ratio ; E. 2 Hyp. Cone. And let A : B = G : H ; B : = F : G; andC :D= E : F; then A : D = E : H. A2 E5 B3 ( F15 34 D12 G20 H 30 (::-4^Di2 A2 : D12 = E5r H30- D. 1 2 3 4 5 H. Case 1. H.&Casel Sim. Eec. •.• A,B,C & F,G,H, are two series of 3 Ms each, and taken 2 & 2 in a cross order have the same ratio. .• A: C = F: H; but C : D = E : F, .-. A : D = E : H. And so on, whateTer be the no. of Ms. Therefore, if there he any number, &c. Q. E. D. Alg. §• Arith. Hyp. Let a = 8, 5 = 6, c = 4; o' = 6, J' = 4, c'= 3. Alg. Arith. By Hyp. Multiply ; i. e. 1. e. T = — , . and — = b a b a ^ c '• b'' b' a' a __ a' 'VT" 7 - V : c'. . *- = t &5- — 1 6 3' 4 4' 8X6 6X4 3X4 8:4 = 6:3. 4X6 „ 8 _ 6 or — ^ — ; 4 3 • Illustration. If A 12 : B 6 ,^-> C 3 : D 9, E 4 :F12^%. G6:H3, then,'A12:D9 E4:H3. 222 GRADATIONS IN EUCLID. ScH. There are different ways of announcmg Prop. 23. 1. Eatios, compounded of any number of equal ratios, but in reverse order, are equal to one another ; For let there be the two series, A,B,C,D and E,F,G,H ; As before, Pr. 23 Case 2, if A : B = G : H, B : C = P : 6, and C : D ^ E : F ; then, ex cequo perturbato, A : D =: E i H. 2. In any two Series of Magnitudes, A, B, C & D, E, F, which taken two and two in a cross order, have the same ratio, if A, the first M, be to B, the second, as C the third is to D the fourth ; and if B, the second, he to E the fifth, as F the sixth, to C the third ; then the first A is to the fifth E, as the sixth F to the fourth D. And if there be any number oi Analogies ; for instance three : and if the . second and third terms of each Analogy form respectively the first and foui-th terms of the next ; then the first term, A of the first Ana- logy, shall be to the second, G, ot the last, as the third tenn H, of the last, is to D the fourth of the first. The same conclusion would follow were the number of Analogies four or more. A 12 : B 6 = 06: D 3 B 6 : E 4 = F 9 C 6 E 4 : G 2 = H 18 F 9 H 12 : G 2 = H 18 : D 3 Vnop. XXIY.— Theor. If the first has to the second the same ratio which the third hai to the fourth ; and the fifth to the second the same ratio which the sixth has to the fourth ; the first and fifth together shall have to the second, the same ratio which the third and sixth together have, to the fourth. DeMi Pr. B. V. Four Ms being proportionals are proportionals inveradyt 22, V. Ex mguo. 18, V. Componendo. Pr, A, V. If 1st : 2nd = 3rd 4th,— then if 1st > = or < 2nd, tho 3rd> = or < 4th. 17, "V. Oividendo, PROP XXIV. BOOK V. 228 E, 1 2 D. 1 2 3 4 5 6 Hyp. Cone. H. & Pr. B, V. H. & D. 1. 22, V. 18, V. H. & 22, V. Eec. 14 H G B B 2 Let AB : C=DE : P, and BG : C=EH : F ; then AB+,BC, I.e., AG : 0=DE + EH, i.e. BR-: F. :• BG : C=EH : F, and inv. C : BG = P : EH; and •.• AB : C=DE : P, and C : BG=P : EH; ,-. ex aq. AB : BG = DE : EH; .-. comp. AG : BG=DH : HE ; but GB: C=HB : P; .-. ex esq. AG : = DH : P. •. If the first has to the second, ^c. ' Q. E. D 21 C D 1- 12 Ariih. Illust. If A B 6 D E 9. C2, B G 8. E P 12. P3. Then AB + BG : C =DE + EH:P; Or, A G 14 : C 2 = D H 21 : P 3. Coil. 1. On the same Hypothesis being made, the first how- ever, being greater than the fifth and the third than the sixth it follows, that "the excess, AG, of the first, AB, above the fifth, BG, shall be to the second G, as the excess, DH, of the third D E, abovb the siotth, E H, is I the fourth F ; or, in other Words, the difference between the first and fifth shall be to the second, as the difference between the third and sixth is to the fourth. B 15 E. 1 Hyp. D, 3, Pr. A. V. Let AB : C = DE : P, and BG : C = . G EH : P, AB being >BG; and .-. AB : BG = j DE:EH,&DE>EHi E 10 s ' H 4 D 224 GKADATI0N8 IN EtJCLID. 3 Cone. D. 1 H. 2 17, V. 3 H. & 22, V. 4 Eec. then AG, i. e. AB-BG : C = DH, i. e. DE— EH : P. •.• AB : BG = DE : EH, AB being > BG and DE > EH ; .-. div. -AG: BG = DH; EH; and V BG: = EH: P; .-. ex. mq. AG: C = DH : P. •. the excess af the first, ^c. Q. E. D. Cor. 2. The Proposition holds true of two ranks of magni- tudes whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to the fourth magnitude ; Or, in any number of Proportions, if the second term, B, is the same throughout, and also the fourth term, D, the same ; tben the sum of all the first- terms, A + E + G, is to the common second term, B, as the sum of all the third terms, C + P + H, is to the common fourth term, D. A4 : B2 = C6 : D3, E6 : B2 = F12 D3, GIO • B2 = H15 D3. A+E+G 22 :B2 = C+F+H33 D3. E. 1 2 D. 1 2 3 4 5 Hyp. Cone. H. 18, V. D. 2, & H. 18, V. Sim. Suppose three proportions so constituted, that A:B = C:D; E:B = P:D, and G : B= H : D ; then A + E + G:B=C + P + H:D. •.• A : B = C : D, and E : B = P : D ; .-. A + E : B = C + P : D. And •.• A + E : B = C + P : D, and G : B= H : D ; .-. A + E + G:B = C + P + H:D. In the same way for any number of proportions. PROP XXIV. BOOK V. 225 Sea*' Prop. 24 may also be expressed, — " If two series of Proportionals have the same consequents, B, D, the sum of the first antecedents, A + E, shall be to their common consequent, B, as the sum of the second antecedents, C + F, D; inv. B:A— B = D:C — D; ex(Bq. A + B : A — B=C + D : C— D. D. I 18. T. 2 17, V. 3 B. V. 4 22, V. " Cor. 3. In any number of magnitudes of the same kind forming two series, one A, B, C, D, E, F, &c.; and the other as many, G, H, I, K, L, M, &c . — if the ratios of the first to the second, A : B, or G : H; of the second to the third, B : C, or H : Ij of the third to the fourth, C : D, or I : K; and so on, be the same in the two series; then, any two combinations whatever, miscendo, i. c, 6y the sum and difference of the magnitudes of the first series, A + C — E, and B — C + D, shall be to one another as two similar combinations of the corresponding magnitudes of the second series, G + I — L, and H — I -j- K, See Geom. Plane, Sol. Sj' Sph. p. 55. 226 GRADATIOKB IN EUOLID. Pkop. XXV.— Theoe. If four magnitudes of the same hind are proportionals, the greatest and least of them, together are greater than the other two together. Con. 2 & 3, 1. To draw a line equal to a given at. line, and from a gi-. line to cut off a part equal to the less. Deh. U, Y. Batios the same to the same ratio are the same to one another. 7, V. Equal Ms the same ratio to the same M, and conyersely. 19, v. If a M : a M = a pt, : a pt. — the rem : the rem. =: the H : theM. A, V. If Ist : 2nd = 3rd : 4th, then, if 1st > = or < 2nc(, 3rd, > = or < 4th. Ax. 2. I. If equals be added to equals, the wholes are equal. Ax. 4, 1, If equals he added to unequals the wholes are unequal. E. 1 2 3 C. D. 1 2 3 10 Hyp. 14,V.&A,V. Cone. 2 & 3, I. H. & C. 11 & 7, V. D. 2. 19, V, H. & A, V. C. A^. 2, I. Add. Ax. 4, I. Eec- Let AB : CD = E : P; and let A B be the greatest, jj 12 consequently F the least ; then AB + F > CD + E. D 9 Take in AB, AG = E, and in CD, CH = F. V AB : CD = E : P, and AG=E, & CH=P ; .-. AB : CD = AG : CH. but •.• the whole AB : the whole CD = pt. AG : pt. A C E ^ : CH ; the rem. GB : rem. HD = AB : CD, but AB > CD, .-, GB > HD. And •.• AG = E, and CH = P ; .-. AG + P = CH + E. To the unequal Ms, . BG; HD, BG being > HD, add AG + P, and CH + E respectively, .-. BG + AG + P > HD + CH + E, ;. e., AB + P > CD + E. Therefore, if four magnitudes of the suTne kind, ^c. Q. E. D. PROP F. BOOK V. 227 Arith. must. If AB 12 : CD 9 = E 4 : F 3 ; Then A + F, 12 + 3 > CD + E, 9 +4. N.B. This Proposition has been inserted as Corollarjr 2 to Pr. 17, bic. V. Cor. If three magnitudes he proportionals, A : B: C, the sum of the extremes A+C, will he greater than twice the mean, 2 B, and A+C therefore half the sum — ^ greater than the mean B. For the proof see Cor. 3, Pr. 17, bk. V. Thus the arithmetical mean between two magnitudes is greater than the geometrical mean between them, the case excepted in which the magnitudes are equal to one another. SUPPLEMENTARY PROPOSITIONS. It has been customary from yery early times, perhaps from the days of Euclid himself, to add several Propositions to this Fifth Book. The first English Edition, which added Nine, announces them in this way, — " Here follow certayne propositions added by Gampane which are not to be contemned, and are cited even of the best learned, namely of Johannes Begio Montanus, in the Epitome which he writeth vpon Ptolome." — Billingsley's Euclid, fol. 150. The Supplementary Propositions here given are chiefly from SiMSON with some from other sources. Prop. P. — Theor. liatios which are compounded of the same ratios are the same to one another. * Dem. 22. V. Ex aqvo by equality. 23, V, Ex aqm perturbaio by per- turbate equality. 228 GRADATIONS IN EUCLID. E. 1 2 Hyp. Cone. D. II H. i I LetA:B = D:E&B:C = E:F; then the ratio compounded of A : B & B : C, — (by Def. A, V.) the ratio A : C,— shall be the same with the ratio D : F, which (by Def. A, V.) is compounded of D : E & E ; F. •.• in the 2 series of Ms, A,B,C, & D,E,F, A:B=D:E, &B:C = E:F; A 8. D 3. B 6. E 4. C 8. F 3. A 8 :C ;8=D 3 : F3 2( 22, V. 3i H. 4! 23, V. .5' Sim. tii Eec. .-. ex (Eg. A : C = D : F. Next, •.• A : B = E : F, and B : C = D : E ; .". ex ceq. pert. A : C = D : F. i. e., ratio A : C compd. of A : B & B : C is the- same with ratio D : F „ D : E & B : P. In like manner for any number of ratios. Therefore, Eatios which are compounded, &c. ScH. Two cases only are demonstrated in the above proposition ; one, of ratios compounded of the same ratios in the same order, as in Pr. 22, V: the other of ratios compounded of the same ratios in a reverse order as in 23, V. There remains the Case of Ratios compounded in any other order ; which may be demonsti'ated in a similar way. " For if K,L,M represent the three ratios in one order, in whatever other order they may be arranged, two of them will be found which are contignons in both arrangements ; commencing with which two, the demonstration will differ little from the above." Also, " ratios which are compounded of the same four ratios, K,L,M,N in jvhatsoever orders, are the same with one another ; as for instance, in the orders K,L,M,N, and M,K,N,L ; — for the latter ratio is the same with the ratio which is compounded of the same ratios in the order M,K,L,N, because the ratio which is compounded of K,N,Ii, is the same with that which is compounded of K,L,N ; and for a similar reason, the ratio which is compounded of M,K,L,N, is the same with that which is compounded of K,L,M,N." " And the same reasoning may be extended to five, six, or any other num- ber of ratios." — Geom. Plane, Sol. & Sph. p. 55, 56. PROP Q. — BOOK V. 229 Pkop. G. — Theoe. If several ratios he the same to several ratios, each to each ; the ratio which is compounded of ratios which are the same to the first ratios, each lo each, shall he the same to the ratio compounded of ratios which arethe same to the other ratios, each to each. Dem. Def. A, v. In any nmnber of Ms of the same kind, the 1st has to the last the ratio compd. of the ratio which the 1st has to the 2nd, and of the ratio which the 2nd has to the 3rd, and of the ratio which the 3rd has to the 4th, and so on unto the last magnitude. 22 ; V. Ex aequo bj equality. E. 1 2 3 4 Hyp. 1. Cone. 1. by Def. A, V. Hyp. 2. Cone. 2, by Def. A, V. Cone. 3. Let A: B=E: F, &C:D = G:H; also A:B=K:L, & C:D = L:M; then, K : M is compounded of K ; L & L : M ; andK: L & L : M the same with A : B& 0:D; AgainasE:Flet]Sr:0, &asG: HletO : P; then ratio N : P is compd. of ratios, N : & : P; and N : O, & : P are the same with E : P and G : H; And it is to be shewn that K : M = N : P. A 6. E3. B 4. C 8. D 6. F 2, G 4. H 3. K 12. L 8. K 6. N 18. 12. P 9. K 12 : M 6 = 18 : P 9. D. 1 2 H. 1 & 2. H. 1 & 2. 3 4 22, V. Eetf. •.• K : L, as ( A : B, as E : F as) N : ; and •.• as L : M so ;is (C : D, and so is G : H, and so is) : P; .■. ex mquali. K : M = N : P; Therefore, if several ratios he the same, Src. Q. E. D. Otherwise 230 GRADATIONS IN EUCLID. E. 1 2 3 4 6 D. 1 2 3 4 5 Hyp. 1. Hyp. 2. Cone. 1. Cone. 2. Cone. H. 2, 1. 11, V. H. 2&3 11, V. 22. v.. In two series of Ms, A,B,C,D; and A', B', C, D', let A : B = A' : B'j and C : D =C' : D'j Also in two other series, K, L, M, and K', L', M', let K : L = A : B; and L : M = C : D; Also K' : L' = A' : B', and L' : M' = C : D'; Then, (by Def. A, V.) ratio K : M is compd. of K : L and of L : H; and ratios K : L and L : M are equal to ratios A ; B and C : D. Again, (by TJef. A, V.) ratio K' : M' and of K' : L and of L' : M'; and ratios K' : L' and L' : M' are equal to ratios A' : B' and C : D'; It is then to be proved that ratio K : M = ratio K' : M', ■.• K : L = A : B, and A : B = A' : B' = K' : L'; .-. K : L = K' : L'. Again, •.• L : M = C : D, and C : D= C' : D', add C : D' = L' : M'; •/ L : M = L' : M'. /, in series K, L, M, and K', L', M', ex teg. K : M = K' : M'. Peop. H. — Thkor. If a ratio which is compounded of several ratios be the same to a ratio which is compounded of several other ratios ; and if one of the first ratios, or the ratio which is compounded of several of them, be the same to one of the last ratios, or to the ratio which is com- pounded of several of them; — then the remaining ratio of the first, or, if there be more than one, the ratio compounded of the remaining ratios, shall he the same to the remaining ratio of the last, or, if there be more than one, to the ratio compounded of these remaining ratios. Dem. Pr. B, V. Invertendo. 22, V. Ex tequo. E. 1 2 Hyp. 1. „ 2. Let the first ratios be A : B, B : C, C : D, D : E E: F, and tlie other ratios G : H, H : K, K : L, L:M; PROP k. — BOOK V. 231 Cone. Also let A : P, compel, of the 1st Mios, = G : M, oompd. of the other ratios ; and let A : D, compd. of A : B, B : 0, C : D, be the same with the ratio G : K, compd. of G : H, and H : K, then the ratio D : F, compd. of D : E, and E : F shall equal K : M, compd. of K : L, and L : M. A 12 B 8. C 6. t> 4. E 3. F 2. G 18. H 12. K 6. L 6. M 3. D4 I" 2 = K 6 M 3. D. 1 2 3 H. Pr. B V. H. & 22, V. Eec. ■ •.• A : D = G : K ; .-. (inv.) D : A = K: G. and •/ A : F = G : M; .-. ex cequo. D : F ' = K : M. Therefore, if a ratio, which is compounded, ^c. Q. E. D. Pkop. K. — Theor. If there he any number of ratios, and any number of other ratios such, that the ratio which is com,pound.ed of ratios which are the same to the first ratios, each to each, is the same to the ratio which is com- pounded of ratios which are the same, each to each, to the last ratios ; and if one of the first ratios, or the ratio which is compounded of ratios which are the same to several of the first ratios, each to each, be 232 GRADATIONS IN EUCLID. the same to one of the last ratios, or to the ratio which is compounded of ratios which are the same, each to each, to severafof the last ratios ; then, the remaining ratio of the first, or, if there he more than one, the ratio which is compounded of ratios which are the same, each to each, to the remaining ratios of the first, shall be the same to the remaining ratio of the last, or, if there be more than one, to the ratio which is compounded of ratios which are the same each to each, to these remain- ing ratios. Dem. Def. A, 5. Compound Ratio. 22, V. Ex tsquo. 11, y. Hatios the same to the same ratio are the same to one another. Pr. B, V. Jnvertendo. E, 1 2 3 4 5 10 11 Hyp. 1. „ 2. „ 3. Cone. Hyp. 4. Cone. 2. Cone. 3. Hyp. 5. Hyp. 6. Hyp. Cone. 4. Let th.e first ratios be A : B, C : D, E : F; & the others, G : H, K : L, M : N, : P, Qt K. and let A : B = S : T; C : D = T : V ; and E : F = V: X; then (Def. A, V.) S : X is compd. of S : T, T : V, and V : X, which are the same to A : B, C : D, and E : P, each to each. ; Also let G : H = Y : Z, and K : L = Z : a ; M:N = a:S; 0:P = J:c, andQ:R = c : d; then, (Def. A, Y.)Y : d is compd. of Y : Z, Z : a, a : b, b : c, and c : d; which are the same, each to each, to the ratios of G : H, K : L, M : N, : P, and Q : R; .-. S : X = Y : d. Also, let A : B, i. e., S : T =,e : ^, compd. of e : /, and/ : g, the same as G : H and K : L; and let h : I be compd. of h: k and k : I; which are the same to the remaining first ratios, namely, to those of C : D and E : F, Also let m: p be compd. oi m : n, n : o, & o : p; which are the same, each to each, to the remaining other ratios, of M : N, : P and Q : R. then h : I, = m : p. PROP. K. BODE V, 233 h, k, I. 4, 14, 10. A,B; C.Dj^ E,F. s, T, V. X. 18, 9. 8, 28.*' 21, 15. , 4, 2, 7. 5. G,H; K, L; M, Nj 0, P; Q,B. Y, Z, a, b, c, d. 9, 6. 12, 9. 6, 9. 13, 20. 24, 30. 12, 8, 6, 9, 12, 15. e, f, g, m, ii, 0, p. , h . I ^ m . p, 6, 4, 3, 8, 12, 16, 20. " ' 4 : 10 = 8 : 20. 1 E. 8 & 5. 3 E. 8 & 5. 2 22, V. 4 H. 3 & 5. 5 11, V. 6 Pr. B, V. 7 E. 7 & 22, 8 E. 8 & 3. 9 E. 8 & 3. 10 22, V. 11 Sim. 12 Con. 11, V 13 Kec. D. IE. 8&5. V e:/= G: H = Y: Z; and/: g == K : L = Z^: a ; .•. ex esq. e : g = Y : a. And •.■ A : B .-. S : T = e : 5- ; .-.. S : X = Y : a ; and .•. inv. T : S = o : Y.. But S : X = Y : d, .-. ex mq. T : X=a : d. Also •.• A : A = 0: D = T: V; and A : Z = E : F = V : X ; .•, ex mq. h : I = T : X. So m : p = a : d, and T : X = a : d ; .', h : 1=: m : p ; Therefore, if there be any number of ratios, ^c. Q. E. D. * SpH, Propositions F, G, fl, and K "are annexed to the 5th book," says -SiMSON in his Notes," because they are fluently made use of by both ancient and modem geometers. And in many cases, compound, ratios cannot Be brought into demonstration, without malang use of them." And SiMSON adds, "Whoever desires to see the doctrine of ratios delivered in this 5th book solidly defended, and the argument brought against it bj;, And. .Tacquet, Alph. Bprellus, and others, fully refuted, may read Dr. Barrow's mathematical lectures, yiz., the 7th and 8th of the year 1666." Fuller information, if desired, may be obtained from Geometry, Plane, Solid and Spherical, Bk. II., pp. 31—78. De Mokga^st's Study and Biffim^ ties of Mathematics, — On Proportion, ch. XVI, pp. 79 — 86. Ariihmetic and Algebra, pp. 33—39. Connection of Number and Magnitude and Proportion and Ratio. Penny Cyclapadia, Vol. XIX., p. 49 & 307. 234 GRADATIONS IN EUCLID. Pbop. L. — Theos. A compound Ratio is equal to the product of its component simple ratios. Dem. Def. A, V. In any nnmber of magnitudes of the same kind, the first to the last has the ratio compounded of the ratio of the 1st to the 2qd, of the 2nd to the 3id, of the 3rd to the 4th, and so on to the ^t magnitude. Pr. F, V. Batios compounded of the same ratios are the same to one another. 7 A, V. The ratio of two lines is the same as that of the numbers which expi;ess the number of ^ones that any third line is contained in them respectively. ^. 15, V. Magnitudes hare the' same ratio to one another which their equimultiples bare. E. 1 2 C. D. 1 2 3 4 5 6 7 Hyp. Cone. Sup. Def. A, V. C. H. Pr. F, V. Pr.7A,V.&15,V, D. 4. Cone. Let the ratio- A : B = the ratios com- pounded' of. C : D and of E : F ; ,, A C • E then B" = D ■ !■• ■ Let ratio C jJD = ratio Q : H, andL ratip. E : F = ratio H : K. '.• G : K is^ompd. of G : H and H : K; and •.• G : H and H : K = C : D and E : F ; and •.• also the ratio conipotmded of C : D and E : P = A : B ; , /. the ratio G : K = ratio A : B. G a . m gn £ m c ^ K = k = But and ratio -g- = m h ratio C the camp, ratio 3"= p" X p"- Q. E. /' D. PROF. M. — BOOK. V. 235 Arith. Illustration. Let — be compounded of — and — ; 64 *^ 4 16 then L6= 2 A 64 * ^ 16 Take-|- = |,andfj = ^, o ^ R 9 A *.• j^ is compounded of — and — , or of — and-— 12 6 12 4 16 ■"12 64 ' Bnt ^ — ^ — 3 X 1 _ 1 X 3 _ J. y 1. 12 4 3jX 4 ~ 3 X 4 2 8 ' And ^« = ^ • 16 _ 2 X 8 64 12 ' • ' 64 4 X 16 , Prop. M. — Theor. If there be two fixed magnitudes, A and B, which are the limits of two others, P and Q, (that is, to which P and Q, by increasing together, or hy diminishing together, rtiay be made to approach more nearly than by any the same given difference'), and ifP be to Q always in the same given ratio of G tQ D; then A shall be to B in the same ratio. Coir. N. B. In the first case, that of commensurable proportion, the obvious principle is assumed, iiat to two given magnitudes of the same kind, and a third there is some magnitude which is a fourth proportional; but in the second or other case, that of incommensurable proportioii, we can only approximate to the fourth proportional, as we approximate to the ratio of the two magnitudes numerically; since, however, such approximation may be contained without limit, it is presumed, that there is some magnitude between them, which is" to the given third magnitude in the same ratio which the second has to the first ; that is, some mag- nitude which is a fourth proportional to the three . Dem. 11, V. Ratios that are the same to the same ratio, are the same to one another; or Magnitudes A, B and C, D, which have the same ratio with .the same magnitudes P, Q, have the same ratio with one another. 236 GRADATIONS IN EUCLID. 14, v. If four magcitadea of the same kind be proportionals, then if the first be greater than the third, the second shall be greater than the fourth ; if equal, equal ; and if less, less. First. Let P and Q, by a continual increase, approach to A and B, respectively, so that P and Q can never equal, much less exceed, A and B, but may be made to approach A afld B more nearly than by any the same difference. As above Take a magnitude B', such that A : B' = C: D Svp. If B' 9^: B', then B' either < B, or > B. 1°. Let B be < B, or B' = B—b. D. 1 2 3 4 H. & C. 11, V. 14, V. D. 3, & H. 5 Cone. 6 7 Eemk. Cone. •.• P : Q, = C : D, and A : B' = C : D; .-. A:B' = P:Q; But A always > P, /. B' always > Q. Now '.• Q < B' and B' < B by the differ- ence b, ' .". Q cannot approach B within the differ- ence } ; , — but this is contrary to the hypothesis ; .•. B' cannot be < B. 2°. Let B' be > B; and take A' such that A' : B := A : E. D. 1 2 3 4 5 7 8 9 10 14, V. 11, V. 18, V. D. 4. Cone. Kemk. Cone. D. 7, I. & D. 8. Cone. Then .'. B < B', A' is < A, as by the difference a; And •.■ A' : B=A : B' & P : Q =A : B ; .-. A' : B' = P : Q. but B always > Q, .". A' always > P. Wherefore, •.' P is always < A, and A' < A by a; .•. P cannot approach A within the differ- ence a, but this is contrary to the hypothesis ; .•, B' cannot be > B ; And ".• B' neither < nor > B ; .-. B' = B, t. e., A : B = C : D. PROP. N. — BOOK T. 237 D. 1 Second.. Let P and Q approach to A and B respectirelf by a continual decrease. Sim. Cone. Eec. In the same manner, by substituting " greater " for "less," and "less" for "greater," we demonstrate, that B' also is neither > nor < B ; again .-. B' = B, i. «., A : B = C : D. Therefore, If there he two fixed magnitudes, 4rc. Q. E. D. UsB & Appl. The Author of Geometry, Plane, Sol. and Spher., p. 46, says of this Proposition, — " it will be found of very extensive application in Geometry. By help of it, the lengths of plane curves, and the areas bounded by them, the cui-ved surfaces of solids, and the contents they envelope, may in many instances be brought into comparison with little greater difficulty than right lines, rectilineal areas, and solids bounded by planes." " But the use of the proposition is by no means confined to these. It may be regarded as ons of the first steps to what is called the higher Geometry, and in tiiia view like- wise, is well worth the attention of the student." Prop. N. — Prob. To find a common measure of two lines. Sol. & Dem. Def. 1, V. — Note. One magnitude measures another when it is contained in that other magnitude an exact number of times. And a magnitude which is a measure of two or more magnitudes is named the common measure of those Magnitudes. E. 1 2 S. 1 Dat. Quaes. Sup. 1. Def. 1, V. Let A'B and C D be two lines, or magnitudes; to find a line which will be contained exactly both in A B and in C D ; If C D is contained exactly in A B, then CD measles A B; — and then also any aliquot part of C D will be the common measure both of C D and A B ; or '.' m C D^AB, .". CD is a measure of A B; and •.■ m C F = C D, ^•. C F a com. meae. of C D and A B, 238 GRADATIONS IN EUCLID. 5 -JD 5 10 15 B 17 10 11 12 13 14 15 16 Sup. 2 Def. 1, V. Cone. Sup. 3. Hyp. S, 8. 8.8 Cone. Sup. 4. Cone. Sim. ButifmCD=AB — EB, andnEB=CD; then « E B also measures A E, a mult, of CD, and .•. E B will measure A B, and .•. is a com. measure of A B and C D. But if E B does not measure C D, let 2 E B = CD — C F, i. «., = D P. Then •.■ E B measures D P, if C P measures ■ E B, C P also will measure D P ; and D P is a mult, of E B ; .*, C P a measure of C D, and of A E and also of A B. Let 2 C P = E B ; then CD = 2EB + CP = 4CF + CF = 5CP; andAB = 3CD + EB = 15CP + 2CP = 17 CP. .•. C P is contained in D Jive tivaes, and in A B seventeen times ; and C P is the com,: meas. of C D and A B. Thus may be found the com. meas. of any other two commensurable lines. CoE. 1. " The greatest common measure of the remainder and lesser magnitude is also the greatest common measure of the two magnitudes," Por '.• every com. nfeas. of A and B is also a com. meas. of B and R — the remainder ; ' .•. the greatest com. meas. of A and R will be found among the com. measures of R and E. Now every one of the latter measures both A and B ; .■. the greatest among them-, is the greatest com. meas. of A and B. PROP. N.— BOOK V. 23^^ CoE. 2. Any aliquot part or submultiple of a common, measure, is also a common measure. Cor. 3. By repeating the process with the remainder and lesser magnitude, and again with the new remainder (if there be one) and the preceding, and so on, the greatest common measure of two given commenanj^ahle •magnitudes, A am,d B, may he found. S. Hyp. Cor. 1, N, V. Hyp. Let2B = A— R; 3R=B-R2=R-R3 and 5 Eg = Rj exactly. Then ".• the greatest com. meas. of A & B, is also the greatest for B & R; and the greatest for B & R, the greatest for R & R^ and so on ; and •.' Rg is contained in itself and also in ^ R2 exactly ; .•. Rj is the greatest common measure of R2 & R3, — and .*, also the greatest common measure of A&B. GoR. 4. Any two commensurahte lines are to one another as the numbers denoting the number of times that they respectively con- tain their common measure ; thus, if the com. meas. q/" A B 6e E F, taken 5 times ; and that of C D, the same E F, taken 7 times, then the ratio A B : C D = AC, or2 AB>ACi .-. AC> AB, but < 2 AB. And •.* the same is true of every square ; PROP. P. BOOK V. 241 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Def. 15, I & H Cor. 16, III. 37, III. C. 5, I. & 6, I. Kemk. Sim. D. 12-16 Cone. Cone. .'. AC — AB = AD, and AD is < AB. Also CD = CB = AB, and AD < AB ; and •/ ED and EB are tangts. from tlie same • E ; .-. ED= EB. • And •.• ADE is aA, and ^ ADE a rt.Z; and /. A = ^ a i-t. Z. ; .-. ZDEA = I a rt.Z, and .-. AD = DE = EB. Now when AD the first rem., or its equal E B is taken from A B, then the rem. AE is the diag. of a sq., of T^hich A D, D E are the sides. The same process as before will then have to be followed out ; and when A D as side has been taken from A E as diag. ; then the rem. lines will again be side and diag.. , But •." a diag. — a, side always V,leaTes a remainder. .•. in this process there will ever be a rem. ; .". the process will never terminate ; and .•, AC the diag. of a sq., and CB a side, are incommensurable. Q. E. D. Prop. P. — Theor. If four strai^U.lines, A,'B,C, J), be proportionals, (whether commensurable, or incommensurable,^ the rectangle under the extremes A'D will be equal to the rectangle under the means B. C. Book n. p. 145. The numerical area of a rectangle is obtained by sup- posing the two sides containing the rectangle to be divided into a number of linear units of the same kind, as inches, feet, &c., and then multiplying the units on one side by the units oii ithe other: the product represeutii the area or enclosed space. 242 GRADATIONS IN EUCLID. Cor. 1. Pr. 29, 1. § 4. p. 18. Gem. Plane, Sol. §• Spher. " Ji there be two St, lines, one of which is contained an exact number of times in one side of a rectangle, and the other an exact number of times in the side adjoin- ing it; then, the rectangle under those two st. lines shall be contained as often in the given rectangle, ag is denoted by the product of the two numbers which denote how often the lines themselves are contained in the two sides." Def. 5, V. Criterion of the Equality of Batios. Pr. M, Bk. V. First. Let A & B be commensurable, and .\ also C & D. C. 1 2 3 D. 1 2 Assum. Pst. 2, V. C. 3 Sim. 4 Ax. 1, I. Take any com. ratio whatever, as 7 : 5 ; and for com. measures, M and N ; let M be contained in A seven times,in Bfive times; & N be contained in C seven times, in Dfive times. '.• A = 7 m, and D = 5n, .: rect. A - D = 7 X 5 (m.n) ^ 35 m.n. So, rect. B • C = 5 x 7 (n.m) := 35 m.n; .'. rect. A - D = rect. B • C Second. Let A & B Je incommensurable, and .: C & D. C. 1 2 D. 1 Pr. M, V. Pr. M, V. Pr. M, V. Rec. A M B C N D Find St. lines P & Q which approach nearer A& C than any assigned difference, and let P& Q contain like parts of B & D, so that P . D = Q . B. Now, •.• by taking like parts of B & D, con- tinually less and less, P & Q, increase towards A & C within any assigned dif- ference ; .•. P.D and Q. B, by increasing together, approach more nearly A . D and C . B than any assigned difference ; .-, rect. A , D = rect. . B or B.C. Therefore, Jffour st. lines, ^c, Q. E. D. EXAMPLES OP REASONING. BOOK V. 243 UsB & Appl. The Theory of Proportion in Arithmetic and Algebra is founded on a similar truth; namely, If four magnitudes be proportionals, and if A, B, C, D, represent those magnitudes numerically, i.e., t^'A and B represent the numbers of times, the unit of their kind is contained in the two first, and if C andD represent the numbers of tim£s, the unit of their kind is contained in the two last, then the quotient or fraction^ shall be equal toJ2 ; and conversely," See Geam. Plane, Sol. and Spher. p. 46 & 47. EXAMPLES OF EBASONING BY PROPORTION. 1". IfA2:B4:C8; then A2:C8 = A»4: BUG, byDef. 10,V. 2° If A2 : B4: C8; D16; then A 2 : D 16 = A' 8 : B* 64, by Def. 11, V. 3°. If A6: B»= C8: D12; invertendo, B9:A6 = D12:C8; 4° If A 9 : B 6 = C 12 : D 8 ; alternando, A 9 : C 12 = B 6 : D 8, by 16, V. 5°. If A5 : B4= CIO: D8; dividendo, A5— B4:B4 = C 10— D 8 : D 8, by 17, V. 6° If A 5 : B 3 = G 10 : D 6 ; componmdo, A5 + B3:B3 = C10+D6: D6,by 18,V. 7= If A5 : B4 = C10 : D8; convertendo, A5 : A 5— B 4 = C 10 : C 10— D 8. by Pr. E,V- 244 GRADATIONS IN EUCLID. 8°. If A 12 B 6 C 18 D 36 E6 P3 G9 His' ex (squali, A 12 : D 36 = E 6 : H 18. by 22, V. 9°. IfA 12 : B6 , ^, C3: D9 E4 ■.'^12^^ G6: H3 ex. aq. perturbato, A 12 : D 9 = £ 4 : H 8. by 23, V. 10°. IfA12:B6 = C6:D3; miscendo. A 12 +B 6 : A 12 — B 6 == C 6 + D 3 : C 6 — D 3. Use and App. 22, V. IP. IfA5:B4=C10;D8, permutando, A 5 : C 10 = B 4 : D 8, by 16, V. invertendo, C 10 : A 5 = D 8 : B 4, by B, V. eomponendo, C10 + A5 : A 5 = D8 + B 4 : B4, by 18, V. EEMAKKS ON BOOK V. To the Notes and Observations gathered from various sources we simply add the commendation of Billingsley, fol. 126. ' " This fifth booke of Euclidb is of very great commoditie and vse in all Geometry, and much diligence ought to be bestowed therin. It ought of all other to be throughly and most perfectly and readily knownes For nothyng .in the bookes foUowyng can be vnderstood without it : the knowledge of them all depende of it. KEMARKS ON BOOK V. 245 « And not onely they and other writinges of Geometry, but all other Sciences and also artes : as Musihe, Astronomy, Perspective, Arithme- tique, the arte of accomptes and reckoning, with other such like. This booke therefore is as it were a chiefe treasure, and a peculiar iuell much to be accompted of. It entreateth of proportion and Analogic, or proportionalitie, which pertayneth not onely vnto Unes, figures, and bodies in Geometry ; but also vnto sounder & voyces, of which Musike entreateth, as'witnesseth ^oeiiMS and others, which write of Musike. Also the whole arte of Astronomy teacheth to measure proportions of tymes and mouings. Archimedes and lordan, with other, writing of waightes, affirme, that there is proportion betwene waight and waight, and also betwene place and place. Ye see therefore how large is the vse of this fift booke. Wherfore the definitions also thereof are common, although here, of Euclid^ they be accommodate and applied onely to Geometry. The first author of this booke was, as it is affirmed of many, one Eudoams, who was Platos scholer, but it was afterwards framed and put in order by Euclide.'" GEADATIONS IN EUCLID. BOOK VI. THE THEORY OF PROPORTION APPLIED, FOR COMPARING THE SIDES AND AREAS OF PLANE RECTILINEAL FIGURES. " This sixth Booke is for vse and practise a most speciall booke. In it are taught the proportions of one figure to an other figure, and of their sides the one to the other, and of the sides of one to the sides of an other, likwise of the angles of one to the angles of the other. Moreover it teaoheth the description of figures like to figures geuen and marueilous applications of figures to lines, euenly, or with decrease or excesse, with many other theoremes, not onely of the Proportions of right lined figures, but also, of sectors of circles, with their angles. On the Theoremes and Problemes of this Booke depend for the most part the compositions of all instrumentes of measuring length, breadth, or deepenes, and also the reason of the vse of the same instrumentes, as of the Geometri- cal square, the Scale of the Astrolabe, the quadrant, the staffe, and 248 GRADATIONS IN EUCLID. such others. The vse of which instrumentes, besides all other mechanicall instrumentes of raysing up, of mouing, and drawing huge things incredible to the ignorant, and infinite other ginnes - (which likewise haue their groundes out of this Booke) are of wonderfuU and vnspeakeable profite, besides the inestimable pleasure which is in them." — Billingslby, fol. 153. i The Theory of Proportion, exhibited in the fifth book, is in tHe sixth applied to determining the proportions which exist be- tween both the sides and the areas of similar plane rectilineal figures. The basis of the comparisons instituted is not identity of size, but identity of form ; and when this cannot be predicated, or clearly inferred, no true Geometrical proportion can be established. The sixth book however advances further than this, and enables us to construct a figure, which shall possess the form of a first given figure and the size of a second. By the second book we may de- cribe a square equal to a given rectilineal figure ;-r-by the sixth we may make any right-lined figure which we choose, equal in size to a given rectilineal figure. We are also empowered, to find Lines and to draw rectilineal figures in proportion the one to the other ; and to increase or diminish any figure according to a given Eatio. Prom this book we derive the principles of what is termed the Eule of Three, and the geometrical form for the solution of a quadratic equation; it extends also to a much wider application the fertile truth, that the square of the hypotenuse equals the sum of the squares of the sides of a right-angled triangle ; and it supplies the easiest and most certain rules by which to conduct Measurements of all kinds. These will be seen when we show the Uses of various Propo- sitions. DEFINITIONS. BOOK VI. 249 In general terms it may be said that the sixt book estab- lishes ; 1st, the proportion between the sides of similar triangles ; and, 2nd, the proportion existing between the areas of similar recti- lineal figures : it also lays down the methods, either of finding magnitudes proportional to other magnitudes, or of describing figures similar to other figures, or equal to them. Definitions. I. Similar rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals : thus the A s ABC, DEF, are similar, if Z A = Z D, Z B = Z E, Z C = Z F, and if AB : AC, = DE ; DF, and AB : BC == DE : EF. ' This definition, like some others to be found in the Elements; is excessire. To contain no more than is strictly necessary (or, indeed, than as yet has appeared to be probable), it should be modified as follows: — Two rectilineal figures are said to be similar wlien the first has all its sides but one pro- portional to the sides of the other, and the angles included by those sides equal to the angles included by the corresponding sides of the other." Geum. PL, Sol. If Sph. p. 57. C B According to the definition, for one rectilineal figure to be similar to another, the conditions to be fillfiUed are equal to twice the number of 250 GRADATIONS IN EUCLID. sides, or rather to the sum of the number of sides and of the number of angles. Thus in the pentagons ABODE, FGHKLj \°. /_ A = /. F; 2°. Z B = Z G; 3°. Z C = Z H; 4°. Z D =ZK; and 5°. /_ ^ = /_"L. Also 6°. EA:AB = LF:FG; 7°. AB :BC=FG : GH; 8°. BC:CD = GH:HK; 9°. CD :DE = HK:KL; andlO°-DE :EA = KL:LF. p. 198. See Hosb's Euclid, II. " Eeciprocal 'figures, viz., triangles and parallelograms, are such as have their sides about two of their angles propor- tionals in such a manner, that a side of the first figure is to a side of the other, as the remaining side of this other is to the remaining side of A. the first;" thus, AB : C D = D E : B F;^the analogy beginning in one figure and ending in the same. B FD E " Kgures are reciprocal when the antfecedents and the consequents of ratios are in each of the figures." EncLiD. Another way of putting the definition is:— "The sides of two figmres, ABE, CDS, are reciprocatti/ proportional, when the extremes of the pro- portion are sides of one figure, and the means are sides of the other; " as AB.BE = CD.DE. The sides are directly proportional, when in each figure the two sides compared are one an extreme and the other a mean: thus, if AB : BE = CD : EF, the proportion is direct. III. A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment, as the greater segment is to the less ; thus in the line AB and its parts, . C t? AB:AC = AC:CB. For Euclid's definition, Lahdnee substitutes, — "when the whole line is to one segment as that segment to the remaining one." DEFINITIONS. BOOK VI. 251 A line thus divided is also said to be divided medially; and the ratio t)f its se^ents is named the medial ratio. — ^Prop. 30, 'VX is the problem by which the segments are made, and is but another form of Prop. 11, 11. ; to divide a line so that the rectangle under the whole line and one seg- ment shall equal the sqnare of the other segment. IV. The altitude of any figure is the straight line drawn from its yertex perpendicular to the base ; thus A D is the altitude of the triangle ABC. ^^ A, Whichever side of a figure is assumed as the base, the altitude is the perpendicular distance from the base, or the base jaroduoed, to the point or » line most distant from the base. The altitude of the same figure may vary with the change in position of its base. SUBSIDIAKY DEFINITIONS. Def. a. a straight line, O C, divided into three parts, is said to be harmonically divided, when the whole line C is to d b one of its extreme segments ' O E, as the other extreme C D is to the middle part D E; z. e., C : E = C D : DE. Four St. lines are said to be harmonicals, when they pass through the same point, and divide any one st. line harmonically. Def. B. "A figure is given in species, when its several angles and the ratios of lie sides about them are ^Ven." Def. 0. "A figure is given in magnitude^ when its area, or any figure equal to it in area, is given." Def. D. "A parallelogram is said to be applied to a straight line, when it is described upon it as one of its sides ; ex. gr. the parallelogram A C is applied to the straight line A B. A. D C B 252 GRADATIONS , IN EUCLID. Dep. E. " But a parallelogram A E, is said to be applied to a straight line A B, deficient by a parallelogram, when A D, the base of the parallelogram A E, is less than A B, the base of parallelogram A C ; and because the / — 7 A E is less than the / — 7 A C, (described upon A B, with the same Z A, and between the same parallels AB, EC,) by the / — 7 DO ; — therefore the/ — zD C is called the defect of / 7 A E. Dep. p. And a parallelogram A G is said to be applied to a straight line A B exceeding by a parallelogram, when A F, the base of / 7 A G is greater than the base A B of / — 7 A C; and because the OZJ A G exceeds the / — 7 A C, (described upon A B, with the same /. A, and between the same parallels A P, E G,) by the , B G ; theretore the £Z7 B G is named the excess of / — 7 A C. Propositions. Prop. 1. — Theor. Triangles and parallelograms of the same altitude are one to another as their bases. Con. Pst 1,1. A st. L. may be drawn from one • to any other point. Pat. 2, I. A terminated st. L. may be produced in a St. L. 3, I. From the gr. of two st. Ls. to cut off a pt. = the less. Bem. 38, I. Triangles upon equal bases and between the same parallels are equal to one another. 1, V. Def. 5, V. The 1st of four Ms is said to have the same ratio to the 2nd which the 3rd has to the 4th, when any equims. whatsoever of the 1st and 3rd being taken, and any equims whatsoever of the 2nd and 4th; if the m of 1st be < = or > that ot the 2nd, the m of the 3rd is also < = or > that of the 4th ; 41, 1. Ka / 7 and a A be upon the same base and between the same |[s, the / 7 shall be double of the A. 15, V. Ms have the same ratio to one another which their equims. have. 11. V. Batios that are the same to the same ratio, are- the same to one another. PHfiP. I. BOOK VI. 253 28, 1. If a St. line falling upon two other St. lines makes the ext. /. = the int. and opp. /_ upon the same side of the line; or makes the /j& upon the same side together =: 2 rt. /.s; the two St. lines shall be parallel. 33, 1. The St. lines which join the extrs. of two eq. and parallel St. lines towards the same parts, are also ^ and ||. 36, I. I 7 s on eq. bases and between the same || s are equal to one another. Case I. Triangles of the same alt. are to one another as their E. 1 2 Hyp. Cone. Let As ABC & ADE have the same alt. AH; i. e., let BE || ST ; then BC : DE = A ABC : A ADE. K O JM MB C HD E PQ L c. 1 2 3 4 Pst. 2, 1. 3,1. 3,1. Pst. 1, 1. D 1 2 3 C.2&H. 38, I. 1, v.. 4 5 ,6 Remk. Sim. 38,1. 7 D. 4. • Produce BE indef. to K & L; from BK cut off, BM, MN, NO, each = B C ; and from EL cut off EP, PQ, each = DE ; join AM, AN, AG ; AP, AQ. Then •.• BC = BM = MN = NO, & KL || ST ; .-. AS ABC, ABM, AMN & ANO are equal ; .-. the m which CO is of CB, a ACO is of aACB; ;. e. OC & A ACO are equims. of BC & a ABC. So, DQ & A ADQ are equims. of DE & a ADE; andifOC > = or < DQ, a ACO > = or < A ADQ. Now, V C0& AACOareeqnims.ofCB& aACB; 254 GEADATIOSrS IN Bi^CLID. 10 D. 5. I & V DQ& A ADQ are equims. of DE & A ADE ; D. 6. I & V A AOC, the m of A ACB > = or < A ADQ, them of A ADE, as 00, the TO of BO, is > =or< DQ,theOTof DE; Def.5,V.| .-. BO : DE = A ABO : A ADE. Q.E.D. Otherwise. Let B C contain the subm. M. 4 times, and let CD contain it 3 times; and let Be = M; then D. 1 2 3 4 38, I. 15, V. H. 11, V. AABC = 4AABe and A ACD = 3 A ABe; hence A ABC : aACD = 4 : 3, -o— g ^ ButBC :CD = 4 :3) M~ ^ /. A ABC : A ACD = BC:CD. Oase II. Parallelograms of the same alt. are also to one another as their bases. E. 1 0. D. 1 2 3 4 5 6 7 Hyp. Cone. Pst. 1, I. 0. & 41, 1. 15, V. Oase 1. D.3. 11, V. Rec. Let / — 7 s PC & GD have the same alt. AH ; t. e., let FAG || BHE ; ^ A G then BC : DE : = C=JFG:CD GD. Join AB & AE. VCZ7FC = 2 aABO, B CH D E &£Z7GD=2aADE; & •.• Ms hare the same R. as their equims ; ,-, A ABO: A ADE = /=7F0: /CZ7 GD. But V BO : DE = aA^O : A ADE; & V aABO : aADE = zny FC : djGD; .-. BO : DE = ^ZJ FO : £=7 GD, .■. Triangles and parallelograms, 4rc. Q. E. D. OoK. 1. From this it is evident that, triangles and parallelo- grams that have equal altitudes are one to another as thdr bases ; and having equal bases are as their altitudes. PROP. I. — BOOK. VI. 255 Case I. — they are their Jases.— See fig. 1. C. 1 2 D. 1 2 Pon. 1. „ 2 28,1. 33,1. Sim. 1, IV. » Place the given figures on the same st. line, as KL, or BE ; & let the triangles be on the same side of the line. Then •/ the perps. are equal and parallel; .*. the line joining the vertices will be || the base. .*. as in Case I, 1, VI, the As having the same alts, are as their bases ; and as in Case II, 1, VI., the / — 7 s having the same alts, are also as their bases. Case II.- — they are as their altitudes. D. 1 2 3 4 38,1 36,1. Sim. ;• the AS = rt /_d as on eq: bases & alts. ; & •.• the / 7 s = rectangles on eq. bases & alts. ; .•. the alt. as the base, and vice versd. So, the dem. becomes an instance of Case I, Cor.. 1. CoK. 2. Any two triangles, or paraUelogratns,. T,,T,' are to one another in the ratio oompcfunded, of th^ ratios of their alts., a, a' and of their hases, h, V . C. D. 1 Sup. Cor. 1,1, VI Cone. Tate M a A or I 7 with alt. a and base 6' •/ T : M = J : 6', and M : T' = a : a' ; .-. T : T- = I ^ I *^' I , a E. compounded of the Es of the bases and altitudes. CoR. 3. The reptangle under two lines, A ^ B, is a mean proportional between their squares. D. 1 1 H. •.• the square on A & the rect. A-B have the same alt. A; 2 1, VI. .-. A= : A-B = A : B. 3 H. And •/ the sq. and the rect. have the same base B ; 4 1, VI. .-. A ■ B : B' = A : B. 256 GRADATIONS IN EUCLID. Cor. 4. Generally, " if two triangles or two parallelograms be as their bases, they have equal altitudes ; and if they be as their alti- tudes they have equal bases." ScH. 1 . From the principle that rectangles of the same altitude are to one another as their bases, the first Proposition might be directly inferred; for / 7s are equal to rectangles on the same base and with the same alt. ; and AS are one-half of the Area of the respective i 2. When, as in Case n, a first figure, FC, is to a second GD, as the base EC, or alt. AH, &c.,of the first tothe base DE, or alt. AH, &c., of the second. — they vary exactly as the base, or altitude, &c., varies. Propositions of this kind constitute a very numerous class, and are distinguished by the general name of Variants. Under the commercial law of supply and demand, the price of a commodity varies as these vary, — the proportion being inverse ; i.e., as the supply increases so the price or estimated value diminishes. But under the mechanical law of power and work done, as the power increases so the work done increases also, the proportion being direct. We use the words as and so to avoid the longer and ftiUer enunciation; thus, the work done in a given time by a machine of one degree of power, is to the work done in the same time by another machine of a different degree of power, as the power of the first machine to the power of the second. 3. We shall do well to remember that " one quantity does not vary as another, because it varies with that other. A square and its side or root vary together, but the square does not vary as the side or root; for instance, if the side or root be doubled, the square is not doubled, but quadrupled ; " with a side of 2 the square is 4, but with a side of 2 x 2, or 4, the square is 16, or four tunes larger. Penny Cyclop., ■0)1. xxvii, p. 137. Use & App. This Proposition is very frequently referred to for the demonstration of other propositions. It may also be employed for dividing a rt. lined surface : thus. From a Trapezium A B C D, with A D || B C, to cut off a third part. C. I 2 & 3, 1. 2C. D. 1 2 3 4 5 H. C. 1,&26,L Ax. 2, I, C. & 1, VI. Ax. 1, 1. Take CE = AD ; and BG = i BE, and join AG, AE; then A A B G = i of trap. ABCD. •.• AD II BE, .-. AS ADF and FCE are eq. ang. ; and ■.• A D = CE, .-. A ADE=: A ECE; /. A ABE= trap. ABCD. Now A ABG = iof A ABE; .-. A A B G = i of trap. ABCD. Q. E. D. PROP II. — BOOK VI. 257 Prop. 2. — Theob. If a St. line he drawn parallel to one of the sides of a triangle, it shall cut the other sides, or these produced, proportionally ; and conversely, if the sides, or sides produced, be cut proportionally, thi St. line which joins the points of section shall he parallel to the re- maining side of the triangle. Sgm. .37, 1. As upon the same base and between the same ||s are equal to one another.. 7, v. Eq. Ms hare the same R to the same M; and conversely. .1, TI.— 11,V. B, V. If 4 Ms are proportionals, they are proportionals also when taken inversely. 9, V. Ms which have the same B to the same M are eq. to one another; and those to which the same M has the same E are eq. to one another. 39, 1. Eq. AS upon the same base and upon the same side of it are between the same parallels. Case I. Let DE be || BO, one of tkfr sides of A ABC; and let DE cut the other sides, AB, AC, in D,E ; then BD : DA = CE : EA ; or AD : DB = AE : EC. Join BE & CD. E. 1 Hyp.l. 2 ?) 3 Cone. C. c: 258 OEADATIONS IN BUOUD. D. 1 H. &C. 2 37,1. 3 Eemk. 4 7, V. 5 C. &H. 6 1,VI. 7 Sim. 8 11, V.BV E. 1 Hyp. 2. •.• On a com. base DE, and between the \\ s BC, DE, .-.aBDE = aCDE. But ADE is another A : ,-. aBDE : A ADE = aCDE : A ADE. Now, •." A s BDE & ADE have the same alt, ; .-. A BDE : A ADE = BD : DA. So, aCDE : A ADE = CE : EA ; .-. BD : DA = CE : EA, & inv. AD ■- DB =AE : EC. Case II. Next, in A ABC let AB, AC, or AB, AC produced, be cut in D & E ; and so that BD : DA = CE : EA; then DE || BC. Make the same construction •.• BD ■■ DA = CE : EA; & BD : DA = aBDE: aADE; &.V CE : EA = aCDE : a ADE; .-. aBDE : aADE= aCDE : a ADE; .-. A BDE = aCDE; And these a s are on the same base : .-. DEjl BC. • ; If a St. line be drawn parallel, ^c. Q.E.D. Cor. In the same manner it may be shown that, if the sides AB, AG, of an angle be cut by any number of parallels BC, DE, FG, HI, any two parts of the one 'will have the same ratio to one another, as the corresponding parts of the other , i. e., the sides will be similarly divided ; and every pair of corresponding segments in each side will be proportional ; BH : HF = CI : IG; FD : DB = GE : EC, &c. ScH. In the first part of the Proposition the Enunciation is not snificiently exact, and in the other part it is " stnctly speaking false, inasmuch as a line may cut two sides proportional!!/, and yet not be parallel to the third side." The Enunciation should be, " 1°. If a line be drawn parallel to any side of a triangle, it divides the other sides, or those sides produced; so that their seg- ments between the parallel and the third side shall have the same ratio to their segments between the parallel and the vertex of the opp. angle; and 2°. if a line cut the two sides in this manner, it will be parallel to the third side/' Labdnee's Euclid, p. 178. 2 3 Cone. c. Pst. 1, 1. D. 1 H.2, 1,VI 2 1, VI. 3 11, V. 4 9,V. 5 C. 6 39,1. 7 Eec. PROP II. BOOK VI. 259 ^a. 2. The Theory of Transversal Lines, that is, of lines drawn across seyeral others so as to cut them all either internally or externally, is intimately connected with this second Proposition: where, how- ever, in the last figure, the transversal D E, cutting two of the sides AB, AC, propor- tionally, does not cut the third side B C, except on the idea that the point of external section is at an infinite distance, the seg- ments being equal. In triangle ABC in the mar^n, the line the a c 6 is the transversal, cutting A C, AB internally in the points b, c, and C B externally in the point a. Here and in all similar oases, Ac x B a x C6 = BcxCoxA6. In the Projection of figures, and in Surveying, especially, when inacces- sible points are requbed by the aid only of signal poles and a measuring line, the theory of Transversals is very useful; for its leading principles, however, we refer to Appendix III. pp. 324 — 332, of Lakdner's Euclid. 3. The ways in which a st. hue A B, may be cut in a given ratio, A D : D B, belong to the full consideration of Prop. 2, Bk. VT. Here one point of section wiU be D. Take C, the point in which A B is bisected, and make C E=CD. Q Thus, BE = AD and AE = i BD; .-. BE :EA is the given ratio, and E another required point of section. are thus two points of internal section. In the same way, ifAF:BF& AG:BGbe each the given ratio, E & G are twopoints of external section, cutting the line A B produced in the given ratio. There are, therefore, four points, two internal and two external, at which a line may be cut in a given ratio. 4. This Proposition shows, moreover, that parallel lines, as E D, e H Set CD and EF, so that A may be seen in the same line with E, C, the tops of the two poles ; measure EF & CD, ED and DB, i.e.,Eh& AG; AG X AC then EA : AG = A C : GA; i. e. GA = < and AB = GA + EF. EA Ex. By measurement EF= 5ft ; CD = 10 ft ; ED = E A =r 6 ft ; and pB = A G = 24 ft i required the height BA. 24 X 5 Here 6 : 24 = (10-5) : GA ; .-. GA = = 20 ft ; 6 & 20 + 5 = 25 ft = BA. , Pkob. IV. By means of one pole DE, placed perpendicularly, to ascertain the altitude BA of a perpendicular object. 272 GBADATIOITB IK EUCLID. Let DE be set npright, where A can be eeen in a at lino with its top from C ; measure CE, DE & EBj then; CE : EB = DE : AB ; ^ EB ■ DE y/y whence AB ; CE £ Ex. At a. distance CE of 10 ft from DE = 7 ft, A can be seen in a line with point D ; and the distance from E toB is 21 feet ; what is the altitado BA? 21 X 7 Here 10 : 21 = 7 : BA ; i. e. BA = = U • 7 feet. 10 Peob. V. By means of a Geometrical Square to measure the height of an object, CB, the foot only of which we can reach. Let AB represent the horizontal Bne; p D a parallel to it ; DB the height of the instrument above AB ; CD the height of the object above ;> D ; ip the edge of the geometrical sqnaift along which the top, C, of the object is seen ; tr, rn, and pn graduated edges, each of 100 eq. parts, andp the point of suspen- sion for the plummet. AL. - 0^ / * D A^- D B PROP. V. BOOK VI. 273 From the place of observation measure the distance p D, and the height of the instrument, DB; direct sp towards the object C, so that s, p, and C may be points in one st. line j and note the number of parts in sr, or m, cut off by the plummet line. * 1°. When the plummet line cuts sr. in o, the triangles pso and C Dp are eq. ang. ; for ".• p o 1| C B, /^ s p o= ^ p CD and Z * = Z I^i both being rt. As; -sp • pT> .-. so : sp = pl) : CD; whence CD= ; and CB = CD + DB. 2°. When the plummet line cuts r n in o, the As, o np and C J)p, are eq. ang.; for p o || CB, /. op n = Z Cp D, and Z_pon=: /_pCD; .', pn. no ^ pt) : D C; n • pH whence D C = — , andCB = CD + D B. pn Ex. 1. The distance p D = 80 ft.; ,«o = 60 eq. pts., and D B = 6 ft.; required C B. 80 X 100 8000 Here 60 : 100 = 80 : CD; .-. CD = = = 133f ft, 60 60 And 133f ft. + 6 5= 139f = CB. Ex. 2. The plummet line p o cuts offno = 20 eq. pts.; DB = 5 ft.; and pD = 90 ft.; required CB. ' Here 100 : 20 ; and 18 + 5 : : 90 : DC ; whence DC = ; 23 ft. = CB. 20X90 100 = 18 ft.; Pbod. VI. By means of a geometrical square, with an index to point out] the extremities, whether altitude or otherwise, of two Objects, one of which is in the vertex of a right angle; to measure the distance. Place the square, each side of which is divided , into 100 eq. parts, on a Une, CB, which is at rt. /_s toAB; and along the moveable index CD, observe the object A, and note the number of eq. parts cut off by the index on the edge of the square, DE or CE; then. •/ EC || AB, and ED || CB, .-. Z E = / B, / ECD = / CAB, and Z. EDC = Z ACB; thus the As, ABC and ^ ' ^ "- EC . CB CED are eq. ang., and /. DE : EG = CB : BA; •. BA = •. DE t 274 GRADATIONS IS E0CLID. Ex. 1. Prom the foot, B, of the perp. AB, to C, I measure BC, — it is 49 feet; the index CD, cuts off 100 eq. pts., as ED;— EC also being 100 eq. pts.} required AB. Here 100 : 100 = 40 : BA; ■. BA = 100 X 40 ■ = 40 feet. 100 Ex. 2. The index cuts ED at GO; as before CB =: 40 feet, EC or DE = 100 eq. pts.; required AB. 100 X 40 Here 60 : 100 = 40 : AB; •. AB = ■ = 66} feet. 60 Pbob Vll. To find, by aid of the cross staff, or theodolite, the distance of A from B without approaching A. At rt. Z_B to AB, lay down the lino BD ; and in BD take a • C, at which place a staff; from D, but on the other side of BD, lay down DE perp. toBD, and measure along DE, until E, C & A are in one st. line ; then, ".• the /_s at C are equal, /.D = /.B, and /.E =/. A ; .•. the As ABC, CDE are equiangular ; CB-DE .-. CD : CB = DE : AB ; i. e. AB = CD • Ex. The measurement of BC = 200 links ; that of CD = 60 links, and of DE = 50 ; how many links are there between the • s B and A f Here 60 : 200 = 50 ; BA ; i.. e., BA : 200 X 50 10000 60 60 ■ = 166? links PaoB. Vlll. — By means of a line, DE, of which the length is known, ta find the length of its parallel, AB, one end of which, B, only can be approached/ Set out the line BE, and on it ascertain the point C, where a line joining A and D would cross BE; measure BC, CE; •.* the AS ACB and DCE are eq. ang,, .-. CE . CB = DE : AB; CB.DE i. e., AB = CE Ex. The parallel DE measures gOO links j the side EC, 900 links, and CB, 1800 links; required the links in AB. PROP. VI. — BOOK VI. 275 1800X800 Here 900 : 1800 = 800 : AB; .*. AB = = 1600 links. 900 Obs. An instrument in common use, — the Proportional Compasses, is an example of tlie last Problem; in this instrument, the common centre C, about which.tbelegs turn, is changed at pleasure, yet so as to preserve any given proportion between EC and CB, and consequently between ED and AB. These compasses give great facility in enlarging or diminishing a plan or map; and with sufficient accuracy for many purposes, — the principle being perfect, . but the application liable to fail for want of the requisite care. The Pentagraph and Eidograph are also instruments, of which the princi- ple is, that the arms move parallel to each other, and consequently that the triangles formed are always similar, being exemplifications of Euclid's Prop. 4, Bk. VI. Eor accuracy and precision the Eidograph is far superior to the Pentagraph, See Bbadlbt's Pract. Geom. p. 59. Prop. 6.-^Theoe. If two triangles have one angle of the one equal to the one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular, and shall have those angles equal which •are opposite to the homologous sides. Con. 23, I.— 32, 1, Dem.4, VI. 11, V. 9,V. 4,1. Ax. 1, 1. 32,1, E. 1 2 3 Hyp. 1 » 2 Cone. Let AsABC,DEP have A BAG = A EDF, and BA : AC = ED : DP ; then AsABC, DEP are eq. ang. ; i.e., ZB = ZE, and ZACB ZDPE, 276 GRADATIONS IN EUCLID. c. 1 23,1. 2 32, I. D I C, 4, VI. 2 3 4 H. 2. 11, V. 9, V. C. 5 4,1. 6 C.2.AX. 1,1 7 H. 1. 32, I. 8 9 Cone. Rcc. At • s D, F in DP make zFDG = ZBAO, or zEDF; & ZDPG; = ZACB; .•.rem. Z_B = rem. Z.G. •.• A DGrF is eq. ang. with A ABC ; .-. BA : AC =: GD : DF ; but BA : AC = ED : DF ; .-. ED: DF = GD : DF; & .-. ED = DG. Aad •.• DF com., ED =GD, & zEDP = ZGDP; .-. EF= FG, A EDF = aGDF, zDFG = ZDPE, & ZG= ZE; but zDFG = ZACB, .-. ZACB = DFE; & •.• ZBAC = ZEDF, .-. rem. ZB = rem. zE. .-. A ABC is equiangular to A DEF ; •. If two triangles have one angle ^c. Q.E.D. CoR. 1. — It may be added, that the sides also about each pair of equal angles shall be proportional; i. e., by 4, VI, AB : BC = DE : EP ; & BC : CA = EP : FD. CoH. 2. If through any points, b, c, d, &c., of a straight line NM, parallels J B, c C, rf D &c., be drawn, which are proportional to the distances A b, b c, cd &c.,from any point A on the 1st line, then their extremities B, C, D &c., will be on the rt. line DA passing through A. HJL V A6:5B = Ac:cC;&zA6B = AcC; .". aA6 Bis similar to aAc C; & .: ZBAJ= zCAc; and •/ A b coincides with Ac; /. AB & AC, being on the same side of NM, also coincide. PROP. VI. — BOOK vr. 277 N. B. — The equation of a rt. line in Analytical Geometry depends on this principle. See Laednee's Euelid, p. 184. Use & App. 1. Since all rectilineal figures may be divided into triangles, if any two rectilineal figures, ABODE I" abode/, thus divided have all their angles but two equal in order, ^ B ^ Z. e, and the corresponding sides about the equal angles proportionals ; then their remaining angles shall be equal each to each, ZA= /.a, ZF= Z/, and their remaining sides Ar, af, in the same ratio with any other two corresponding sides AB, ab. Thus the one figure shall be similar to the other by Def. 1,VI. E. 1 Pst. 1, 1. D. 1 H. 2 6, VI. 3 H. & D. 2. 4 Ax. 3, 1. 5 D. 2. & H. 6 22, V. 7 6, VI. 8 Sim. 9 Cone. iO H. 11 22, V. 16, V. Join AC, AD, AE; ac, ad, ae. * , •.• AS ABC, o 6 c, have /JB = /_b, & AB : BC =oi : Jc; .•. l_ ACB = /_ acb,s>BA AC : CB = ac : cb; and •/ l_ BCD = /_bcd, and ZACB = /_acb; .: Z ACD. = /_acd. Also, ■.• AC : CB = oc : c b; and CB : CD = cb : cd; .'. ex, ZDEF; and makeZ ABG = Z DEP. And •.• Z A = Z D, Z ABG = ZDEP, and Z AGB = Z DPS ; .-. A ABG is eq. ang. to A DEP; and ,-. AB: BG = DE : EF; but as DE : EP so AB : BC, .-. AB : BC = AB : B G; and •.■ AB : BO = AB : BG, .-. BC= BG; and .-. Z BGC = z BCG. But Z BCG rt. Z . Now zAGB = ZP, .-. ZP'> rt. z ; but Z P < rt. z ; — which is absurd. .-. Z ABCnot# DEP, i.e. Z ABC = Z DEP, and Z A = Z D; .', rem. Z C = rem. Z P. .-. A ABC is eq. ang. to a DEP. Case II. Let Zs C ^ P be each < a rt. Z ; Cone. D. 1 D. 1—5 Case I H. 17,1. Cone. then A ABC, is eq. ang. to A DEP. As before BC = BG, .-. Z C = ZBGC; but. Z C < rt. Z, .-. z BGC < rt. Z ; .•. two zsof A BGC, are < 2rt.zs; which is impossible ; and .•. A ABC is eq. ang. to A DEP, as in Case I. 280 GRADATIONS IK EUCLID. Case III. Let one of the /_s, C, F, namely /^C, be a rt^^ D. 1 Cone. Then also A ABC . is eq. -ang. to yA y^ A DEF. X yC\^ If not, at B in /S--^ E^F Sup. 23, I. AB,make^ABG, 13^^^=—- ^ = zDBF; Case 1. 5, I. then BG = BC, and ,-, /_ BCG = z BGC ; H. Ax. 1, I. but i_ BCG is a rt; /, .-, i_ BGC is a rt. /_ ; 17,1. .: two /_8 of A BGC are < 2 rt. ^s; which is an impossibility ; Cone. .-. A ABO is eq. to ang. A DEF. Eec. ■. If two triangles have one angle, 4rc. Q. E. D. ScH. When two angles are both greater, or both less than right angles they are both said to be of the same affection ; and in enunciating this Prop, instead of " both greater or both less than right angles," it is not unusual to say, " both of the same affection." Use and App. — In Book L Propositions 4, 8 and 26 contain the criteria oi the equality of two triangles ; and in Book Yl. Propositions 4, 5, 6 and 7 may be classed together as giving the conditions on which we declare the simi- laiity of two triangles. Equality in triangles is absolute, not in area only j — but the similarity i^ likeness of shape, not identity of size. The criteria of similarity are ; 1°, — The equality of the three angles, 4, VL ; 2°. — The identity of the ratios of the respective sides, 5, VI. ; 3°. — The equality of two angles, one in each triangle, — ^and the identity of the ratios of the containing sides, 6, VL ; 4°. — The identity of the ratios of two sides in each triangle, — the equality of an angle in each opposite one pair of homologous sides; — and each of the remaining angles opposite the other pair of homologous sides less than a right angle, or one of them a right angle. PROP. Till. BOOK VI. 281" the liyeraally, if triangles fiilfil any one of these four conditions of similarity, about the equal angles are proportional. Pkop. 8. — Theor. (Important.) In a rt. angled, triangle, if a perpendicular he drawn from the right angle to the base ; the triangles-on each side of it are similar to the whole triangle and to one another. Dem. Ax. 11,1. All rt. As are equal to one another. 32,1. 4, VI. Def. 1, VI. E. 1 2 D. 1 2 3 4 5 Hyp. I. Cone. Ax. 11,1. 32,1. 4, VI. Def. I, VI. Sim. D. 2 & 4. Eec. Let A ABC have BAG a rt. /_; & from A let AD be J_ BC the hypote- nuse; then A s A'feD, ADO, are sim. to A ABC, and to each other. •.• ZBAC = ZADB, & zBcom.; .-. rem, ZACB = rem. zBAD .". A ABC is eq. ang. to a ABD, and the sides propls.; .", A ABC is similar to A ABD. So, A ADC is eq. ang. and sim. to A ABC ; and .•. A ABD is eq." ang. & sim. to A ACD, which is eq. ang. and sim. to a ABC. Therefore)! in a rt. angled triangle, ^c. Q. E. D. 282 GRADATIONS IN EUCLID. CoK. 1. The perpendicular, A J), from, the vertex of the ri,f l_ BAG to the opposite side BO, is a mean proportional between' the segments BD, DC of this side; and also each of the sides, BA, AC, including the rt. /^ is a mean proportional hetween the opposite side BC, and the segment of it, BD, or DC adjacent to that side, BA or AC. D. 1 H. Def. 1, VI. Obs. Def. 10, V H. 4, VI. H. 4, VI. '.• A ADB is sim. to A ADC, .-. BD : DA = DA : DC ; i. e. DA is a mean proportional between BD and DC. Also, ■.• A ABC is sim. to A DBA, .-. BC : BA = BA : BD ; i. e. A B is a mean proportional between BC and BD. And •/ A ABC is sim. to A ACD, .-. BC^: CA = CA.: CD; i. e. AC is a mean proportional between BC and CD. CoR. 2. The segments BD, CD, of the hypotenuse, made hy the perp. AD, are to one another as the squares on the. sides of the rt. L, BA2 : AC^. •.• by D. 2 & 3 of Cor. 1. 8, VI. BD. BC : CD . BC = BA« : AC2; if we divide the 1st & 2nd terms of the analogy by BC, then BD : X3D = BA^ : CA^ CoR. 3. The squares on the sides about the rt. /_ and on the hypotenuse are to each other as the segments of the hypotenuse, made by the perp.. AD, and the hypotenuse itself. BD.BC : CD.BC : BC.BC = AB^ : CA^ on dividing the three terms by* BC, BD : CD : BC = ABs : CA^ : BC^. BC^i PEOP. VIII. BOOK VI. 283 N. B. — Several other guisidiory Corollaries might be added, — ^but the most important deductions have been given, and we subjoin only ; Cor. 4. If the base of a triangle, BC, the two sides, AB, ^G, and the perpendicular KD, he four proportionals, the triangle must be right angled. D. 1 Hyp. 2 » 3 !) 4 7, VI. Cone. ".• in A ABC, and in one of the component A s ABD, two sides are proportionals ; and tlie Zs opp. onepair of homologous, sides are equal ; and also •.■ of the Zs opp. the other pair of homol. sides one is a rt. Z ; .*. the whole A ABC is sim. to the component A ABD; and .*, also A ABC is rt. angled. ScsH. 1. The 8th Proposition, and the deductions that may be made from it are particular cases of a more general principle; namely, If from the vertex, ^, of a A. ABC, two lines BD, BE. he drawn to the base, AC, making the angles at the base, BDA. and BEC, or their supplements each equal to the vertical angle ABC ; then the A* BDA, BEC, formed by those lines, and by the segment DA or EC which each cuts off, shall be similar to the whole triangle and to one another. E. 1 D. 1 Hyp. Cone. Bemk. Let ABC be a A, and from vertex B, let there be drawn BD, BE, making Z BDA = Z BEC = Z ABC; then A BED is isosc. ; and A BDA is eq. ang. to A BEC which is eq. ang. to A ABC. When Z ABC > rt. Z. Zs BDA and BEC are est. Zs. at the base of the i§osc. A BDE; 284 GRADATIONS IN EDOLID. Eemk. 2. but when /_ ABC < rt. /_, then those /.s ai-e internal. 3. As the obtuse /, ABC decreases, the base DE diminishes and the sides BD, BE approach; 4. when /. ABC becomes a rt. /_, BD and BE coincide; 5. and after l_ ABC becomes < a rt. ^, BD and BE change sides. In the general proposition, this isosc. A DBE, is what the perpendicular is, when the given A is rt. angled. Cone. /. the sides of this A PBE, and the triangles under them, and the sides of the given A ABC, possess many of the properties already proved in the case of a rt. /.d A ; for instance, as in Cor. 1, Pr. 8, VI,; 1 . AC : AB = AB : AD, where AB is a mean proportional. 2. AC : CB = CB : CE, where CB is a mean proportional. 3. AC : AB = BC : BE, 4. AD : BD = BD : CE, where BD is a mean proportional. or since BD = BE ; AD . BE = BE : CE, where BE is a mean propl. 5. •.• AD : BD = AB : BC, .*. the segments AD and EC are in the duplicate ratio of the sides AB and BC. 2. " Hence in a right-angled triangle the segments of the hypotenuse by the perpendicular, are in the duplicate ratio of the sides." Labdner's Euclid, pp. 187, 188. Use &~ App. 1. The first Corollary of Pr. 8, bk. VL supplies the princi- ple on which a very clear and brief demonstration may be given of Prop. 47, bk. L ' BC X BD = BA=, and BC X CD = CA'; , Add. BC . BD -(- BC . CD = BA= + CA=. Or, BC X (BD + CD) =BA» -f CA^ i. e. BC X BC, or BC^ — BA= -|- CA^. 2. Also according to this Proposition, and by aid of a square, inaccessible distances, as DB, may be measured, * PROP IX. BOOK VI. 285 At D raise the perp. DA, and measure it; And at A place a square, so that by looking along one of its sides Ab, the point B m,SLj be seen in the same st. line with Ab; And along the other side Ac, the point O may be seen ; and measure DC; DA^ Then, •.• CD : DA = DA : DB; .-. DB =- Ex. Suppose AD = 3, and CD = 2-25; then DB = ■ CD 9 2.25 • = 4. 3. In a circle any chord, as BA, is a mean ■proportional between the diameter 3C, and that segment of the diameter SD, which is drawn from one extremity of the chord, B, and cut off by a perpendicular, AD, let fall from A the other extremity of the chord. D. 1 2 31, m. & c. 4, VI. 8, VI. .• AS BAG and ADB have each a rt. /_, and l_ B common; ■. the A BAG is eq. ang. to A ADB, and .-. BC : BA = BA : BD; i. e. the chord BA is a mean propl. to BC & BD. And.'.* AS BAC, BDA and ADC are similar, the seg"- ments of the hypotenuse are in the duplicate ratio of the sides. From a given st. measure or submultiple Prop. 9. — Prob. line to cut off any part required, i. «., any Con.— 8, 1. 31, 1. 286 OnADATIONS IN EDCLID. Dem. — 2, VI. 18, v. Componendo. If Ms, taken separately, be propls, they shall also be propls when taken jointly ; i. e, if first : 2nd = 3rd ; 4th ; then 1st + 2nd : 2nd = 3rd + 4th : 4th. D, V— If 1st : 2nd = 3rd : 4th, if 1st a m or pi of the 2nd, the 3rd is the same m or pt of the 4th. Def. 1, V. — A less M is a p(of a greater when the less measures the greater. E. 1 Dat. Let AB be a given st. line ; 2 Quaes. to cut off from it any pt. required. C. 1 Pst. 1, I. From. A draw AC making any £, /—AD Z with AB ; 3, I. in AC take any . D, and make AC the same m of AD that AB is of AE the pt to be cut off. 3 31, I. join BC, and draw ED |J BC ; 4 Sol. then AE is the submultiple required ; D. 1 C. 3. •/ ED II BO one of the sides of a ABC; 2 2, VI. .-. CD : DA = BE : EA; 3 18, V. and compon., CA : AD = BA : AE ; 4 C. D, V. but CA is a m of AD ; .-, BA the same m of AE ; 5 Def. 1, V. .-. AE the same pt of AB, that AD is of AC. Becap. ; • From a given st. line, ^e. Q. E. P. ScH. Prop. 10, Bk. I. by which a rt. line may be bisected, and its bisec- tions also bisected, is a particular case of this Problem. Use & App. A simple extension of the Problem enables us; Ist, to divide a given line, AIL, into any number of equal parts. !&. 1 Dat. & Quaes, C. 1 Pst. 1, I. 2 3, I. Let it be required to divide AL, into four eq. pts. Draw AX, making any /_ with AL; from A on AX set off four eq. spaces, t. e. AB = BC = CD = DE i D,.' A-:A^ 9<'- E...-X -'■L PEOP IX.— BOOK VI 287 3 i D. 1 31, 1. Sol. 9, VI. Sim. Cone. Join EL, and through D, C, B draw ||s to EL and cut- ting AL; tlien AL is divided into four eq. pts. in b, e, d, L. '.' A6 is the same pi. of AL that AB is of A E; i. e. the 4th part; and •.' A6, = be ^ cd = dL; .•. AL is divided into four eq. parts. 2. To divide a triangle, ABC, into any number ofeq. parts, say four, 6y lines fiom a given point, P, in one of the sides, as BC. ' C, 1 2 3 4 Sch. 1. 9, VI. 31, L Pst. 1, 1. Sol. Divide BC into four eq. pta. in D, B, E, and join AP ; and through D, E, E draw DG, EH, EK each || AP; then join AD, AE, AE, and PG, PH, PK; and the As BPG, GPH, HPK, KPC, each = i aABC j i. e. by lines fromP, A ABC is divided into 4 eq. pta D, 1 3?, L ".• A DPG = A GAD, both being on GD and between the same ||3 GD, AP; to each add A BGD, ,\ A B]^G = A ADB. Next, •.■ BD = DE = EF = EC and the altitude com, .-. AS ABD, ADE, AEE, AEG, ate equal; and each is J A ABCj but A BPG = A ADB, .-. BPG = J A ABC. So A GPA = A DP A, and A HPA = A APE, take A HPA from A GPA, and A APE from A DPAj .-. rem. A GPH = rem. A ABD = J A ABC. Now fig. PHAK = APH + APK; A APH = A APE and A APK = A APE; .-. fig. PHAK = A APE + A APE = i A ABC. Now fig. ABPK = J of A ABC; .•. rem. fig KPC also = i of A ABC; •. A ABC is divided into 4 eq. pta. by lines from ■ P. 3. Given the nth part of a line AS, tofind the (n + 1) th part. 2 3 4 Add. Ax. 2, L C 1. 38,1. 5 6 7 8 9 D. 3 Sim. , Sub. Ax. 3, 1. D. 4 C. 10 11 12 13 Ax. 1, L D. 5, 8, 10, Ax. 8, 1. Cone. 288 GRADATIONS IN EUCLID. E. 1 2 C. 1 Dat. Qoaes. 46, 1. Pet. 1, 1. . 2 3 V. 1 2 3 31,1. So}. c. 4, VI. Cone. AB Given AC = , or AB =r n . AC; n AB to find . n+1 On AB desc. a eq. ABEF, and join AF, EC cutting in G ; E K A. HC through G draw HGK 11 AE, cutting AB, EE, in H & K; AB then AH, or BK = ; or (n+ 1) AH = AB. n+1 '.' As AHG, EKG, are eq. ang. ; and As AGO & EGF; .-, AH : FK,;or BH, = AG : GF = AC : EF, or AB. and .•, BH = n. AH: and AB = (n+1) AH; AB •••^H^^TY- Q.E.F. Prop. 10.— Prob. I'o divide a given st. line similarly, i. e., proportionally, to a given divided st. line ; or to divide a given st. line into parts that shall have the same ratios to one another which the parts of the divided given st. line have. Or, " To divide a given undivided line similarly to a given divided line." Euclid. Cow. Pst. 1, 1. 31, L 1, 1. PROP. X. BOOK VI. 289 Dbm. 34, 1. The opp. sides and /.s of / 7 3 are eq. to one another, and the diam. bis. them. 2, VI. 7, V. 6, VI. 28, I E. 1 2 C. 1 2 3 D. 1 2 3 4 5 6 7 Dat. 1. „ 2. Quaes. Pon. Pst. 1, 1. 31, I. Sol. C. 2. 34, I. C. C. 2. ,VI. 7,V. D. 2. 2, VI. D. 4& 5. D. 4&4. Given AB to be divided ; &"AC a line divided at the • s D, E ; to divide AB siinilarly to AC. Let AB and AC form an /. at A ; & join BO ; through D,E, draw DF,EG, each II BC ; and also through D, DHK |1 AB ; then AB is div. in P,G, as ACis in D,E. •.• FH,.HBareEZ7s; .-. DH = PG, and HK = GB ; and •.• HE || KC, a side of aDKC ; .-. CE : ED = KH : HD ; but KH = BG, & HD = GF ; .-. OE : ED = BG :GF. Again, •.• FD || GE in a AGE,; .-. ED : DA = GF : FA ; and '.• CE : ED = BG : GF, andED : DA = GF: PA; .•. AB is divided in the same proportion as AC. Q. ^- F. C. 1 2 3 4 D. 1 Or, 1,1. 3,1. Pst. 1, I. Sol., C. 2. 6, IV. On AO desc. an eq. lat. aAHO; and from HA, HO, cut HK, HL; each = AB; & join KL ; from H draw lines to D,E, the divisions of AC ; then KL = AB is div. in F,G, similarly to AO. •.• HK = HL, & HA = AC, .-. A HKL is eq. an. to a HAC ; 290 GRADATIONS IN EUCLIIV. 2 C. 1. 3 Sim. 28, I. 4 Cone. but AC = HA, .-. KL = KH = AB, So ^HKL = /I HAG; .-. KL || AC; and •.• KL = AB .'. AB is div. similarly to AC. Q. E. F. Use ahd App.— By aid of this p^podtion several useful Probtems may be solved. Prob. 1. To divide a given st. line, AB, internally, or externaily, in a given ratio, ax o/M. : IS. C. 1 2 3 Pst. 1, 1. 3,1. 31,1. Sol. From A draw AX at any /_ with AB ; take AC = M, and CD = Nj and join DB; Through C draw CE || DB, and meeting AB, or AB prod^iced. Then AB is divided in the ratio M : N. ' D. 1 C. 3. 2 2, VI. 3 Cone ■ in AS AEC and ABD, CB |1 DB; . AE : EB = AC : CD = M : N; , AB is divided in E in the given ratio. N. B. A line cannot be cut externally in a ratio of equality. Pbob. 2. To find a harmonical mean between two given st, lines, AB and AC. C. 1 2 3 D. I Pon. Prob. 1. Sol. 2, VI. Def. A. VI. ConcL Place the lines so as to form one st. line, AC being; set off on AB. divide BC in D in the ratio AB : AC; then AD is the harmonical mean. A CD B AB : AC = BD : DC; .'. AB, AD and AC are in harm, progression ; , AD is a harmonical mean between AB and AC. PEOP. X. — BOOK VI. 291 PsoB, 3, Tojinda third harmonical progressiondl to two given st. lines, AB and AC. C. 1 3 D. 1 Pon. Prob. 1. SoL 2, VI. & 16, V. Def. A, VI. Cpncl. Set off A B on AC, and produce ACj divide AC produced, in I),iu the ratio AB : BC; tlien AD is the harm, progressional required. B_C .D •.■AD:CD=AB:BC; and cdt. AD : AB = CD : BC, .•. AB, AC and AD, are in harm, progression, and AD is the third haim. progress, to AB, AC. Pbob. 4. — To construct a triangle of which one side AB,the angle ADB opposite to it, and the ratio of the other sides are given. c. 1 33, HL 2 i,in. 3 10, VI. 4 Pst. 1. 2, L 5 Sol. D 1 c. 2. 3, m. 4,1. 2 3 4 28, m., 27, ni. 3, VI, Cone. On AB desc. a © in which the /. = the given /_ ; draw a diam. EC, at rt. /.s to xj AB: divide AB so that AF : FB = ratio of the sides; join CF, and produce CP to D in the 0cej join AD, DB ; and ADB is the A required. •.• /.s at G- rt- ^s; .•. AG = GB, and chord AC = CB. Now arc AC = arc CB; .•, ^ ADC = Z.CDB, and CD bisects Z ADB; .'. AD : DB = AP : FB, which is the given ratio: .'. A ADB is on AB, its vert. /_ = a given /_, and its sides, DA : DB = a given iratio. Pbob. 5. Through a given ■ A, to draw a line, which, on being produced, would pass through the' points of intersection of two given lines, HI, KL, without their being produced to meet. Through A draw any Kne BC, cutting HI in B, & KL in C ; draw Die || BC, and divide DE so that BF . PD = CA : AB; then AF is the line required. 292 GRADATIONS IN EUCLID. D. 1 2 3 4 Sup. c. 18, V. Cone. For should HI, KL, have met in G, and GF cnt BC in A'; then •.■ DP : S"E = BA' : A'C; .-. BA' : A'C = BA : AC; and comp. BC : A'C = BC : AC; .•. A' coincides with A; and AF is the line required. N.B. This operation is frequently called for; and the Centrolinead is an instrument, invented by Mr. P. Nicholson, senior, for the purpose of drawing lines tending to the inaccessible point where two given lines, if produced, would meet. Another instrument for the same purpose is the invention of Mr. John Fary. Bradley's Pract. Geom. p. 42. There is, however, a simpler instru- ment used by draughtsmen, consisting of three rulers stiffly moveable about a common joint; but it is less convenient in form, and less accurate in its results, though depending on the same principle. Prop. 11. — Prob. To find a third proportional to two given st. lines ^ Cos. Pst. 2, I. 31, 1. Dem. 2, VI, 7, V. E. 1 2 C. 1 ; 2 3 Data. Quaes. Pon. Pst. 2, 1. 3,1. 31, I. Sol. Given the two lines AB and AC ; to find a third proportional to them. Place AB and AC to form an /_ BAG, and pro- duce AB and AC ; take BD = AC, join BC, and draw DE || BC; then CE is a third proportional : i. e., AB : AC : CE. PROP XI. BOOK VI. 293 D. 1 2 3 C.2, 2, VI. C.2, 7,V. Rec. '.• BC II DE, a side ol A ADE ; .-. AB : BD = AC : CE ; but BD = AC ; .-. AB : AC = AC : CE. •. to the given lines AB, AC, a 3rd prop. CE, has been found. Q. E. F. ScH. 1. There are various constructions by means of which this Problem may be solved: we select one which in practice requires the use of the com- passes alone;— the given lines being AB and CD. C. 1 3 4 D. 1 Pst. 3, 1. Pst. 1, I. Sol. Con. 8, 1. 5, 1; 4,1. D. 2. 31, ni. 28,1 C. 1. 29, 1. 4, VI. C. 7,V. Cone. From A with AB desc. 0BEF; and from , B with DC, ©EPGj from E on aire EEG, set off CD three times toG; join EF, AF, BF, BG, and GF; the chord FG is the third proportional. •.• AE, EB = AF, FB, and AB com. ; .-. t. EBA = Z, FBA; also l_ BEF = L BFE, and ^s at H rt. Zs; - and •." /.s at H are rt. /.s, and /_ EFG is a rt. .-. AB II FG. and •.• AS ABE, BFG, are isoso. ; and /BFG = .'. A ABF is eq. ang. to A BFG; .-. AB : BF = BF : FG; but BF = DC, .-. AB : CD = CD : FG; .•. FG is the third proportional. Z ABF; Use and App. — 1. By repeating the same construction we solve the P-roblem, toctmlinue a series of ratios in progression, AB : BC being the given antecedent and consequent. C. 1 2 3 D. 3, 1. 31, 1. 2,VI."7,V. Take A 6 = BC, and draw C c || B 6 ; „ CD = ic „ D d II C c ; „ DE = cd „ E e II D d, &e. It is evident AB : BC : CD : DB, EF, &c. 294 GRADATIONS IN EtTCLID. BCD EH* BC • BC Arithmetically. Since CD := AB the third pioportiooal to two given numbers is fonnd by dividing the square of the consequent by the antece- 6X6 dent ; thus, the 3rd proportional of 9 : 6 = = 4; i. e., 9 : 6 == 6 : 4. 2. From Labsntib's Notes on Prop. 11, Bk. VI. we take several most nsEi'tii, Theobems allied to the last Problem. Theob. I. " If a series of magnitudes A, B, C, D, be in continued pro- portion, their successive differences, a, o, c, d, are also in continued prdportion, and in the same ratio. D. 1 H. 2 E. V. 3 16, V. 4 Sim. 5 D. 3, 4. 6 Proced. V A : B : C; .'. conv. A : a = B : i. and alt. A : B = a : 6. So, •.• B : C = i : c; .', a : b : c. And a : b : c : d, &c. Arith.- 2 : 6 : 18; 2 : 4 = 6 : 12. 2 : 6 = 4 ; 12. So, 6 : 18 = 12 : 36; 4 : 12 : 36. and 4 : 12 : 36 : 108 &c. Theob. EL — If a series in continued proportion, A : B : C : Dlfc., be an increasing one, there is no limit to the increase of its terms. Let o = B — A ; 6 = C — B ; c = D — C &c. -, and L be the last term. '." no magn. so great that we cannot obtain a greater, .•. let M. be a magn. however great we please. Pind what multiple of a, M is; M and continue the series to a gr. number of terms than •— ; .•.Ii> Abyo + 6 +c+d&c. +( the no. of terms = — ) But '.' a : b : c : d &c. is an increasing series ; .". each snrcessive term > a ; 8' .". their sum + A > M. N.B. — The different steps tof this demonstration rest rather on truths that may be implied &om Euclid's Elements, than on those expressly taught and proved ; they are, however, so plain that, perhaps, a formal proof is not required. The same remark should be extended to tiie Theorems 3 and 4 which follow. E. Hyp. D. 1 2 3 t 4 5 6 7 Hyp. PEOP. XI.— BOOK VI. 295 Thbor. in. If a series in continued proportion, A : B : C ; D, Sfc, be a •decreasing one, there is no limit to the diminution of its terms. C. 1 2 3 4 5 6 D. 1 2 3 4 Use U, VI. H. 22, V. 16, V. C. 6. 14, V. Proced. Contiane the series until a term be found less than m, any assigned magnitude however small; let m : 1 = B : A; and let the ratio, m : Z, be continued in a series; 'l' B < a, .". m < I, and the series m : I increases; and ,•, a teim may be found, a > A, Let the series be continued until a > A, and let the series A : B : C &c., be continued the same no. of terms; then its last term M wiU be < m, the assigned magnitude. For, V A : B : C : D : L : M, & a : b : c : d : I : m; .*. ex. aquo. A : M = o : m ; and alt. A : a : but A < o, .*, M < m. And so on by continuing a like process. : M : »t; Thboe. rV. " If a series of magnitudes, decreasing in continued propor- tion, be continued, or imagined to be continued, to an infinite number of terms ; the sum of all the terms, or the sum of the series, will be a finite and determinate magnitude. C. 1 Use 11, Vt Pst. 1, 1. 31,1. 31, L Cone. On the line AZ let the decreasing series be set, AB : BC : CD : DE, &c., and draw a line ZM, making an /_ with AZ; at A, B, C, D, E &c. raise parallels cutting MZ in the ° s M, N, O, P, Q; draw through • N, NL || AZ, and cutting AM; the Sum of the series S = AZ. A BODE Sup. IfAZijtS, S is either gr. or less than AZ. Cash. I. S is not gr. than AZ. IS D. 1 2 3 4, VI. 14, V. For each parallel DP, EQ is less than DZ, EZ, from which it is to be taien to determine the • for the next parallel; and AM : AZ = DP : DZ = EQ : EZ; and •/ AM < AZ ; .-. EQ < EZ. 296 GRADATIONS IN EUCLID. Case II. Though the series is unlimited, S is not less than AZ. Sup. If S were < AZ, let S = AT. Now '.■ the parallels AM, ^N, CO, &c., continnally decrease proportionally; and •.■ AZ, BZ, CZ &c. are in proportion to them ; .•. AZ, BZ, CZ &c., are in decreasing continued proportion. Continue this series through a determinate number of terms ; at last a term will he found < YZ. Thus the sum of the corresponding parallels must be greater than AT; i. e., 'AM + BN + CO + &c. are > AT. Hence the Sum of a limited no. of terms is > AT, the sum of an unlimited number; .•. a part is > the whole, which is absurd; .•. AZ ]J> S the sum of an infinite number of terms: nor i£ AZ < S, .-, AZ = S. D. 1 H. 2 4, VI. 3 Use 11, VI 4 5 6 7 8 Ax. Cone. Case 1. 3. Ceetain Problems also may be deduced from the foregoing principleft. Pkob. I. On the last Theorem, Theor. IV. Gbegobt's Problem, as it is named, is to be solved; — from the two first terms in a series AB : BC, to obtain S, the sum of the series. C. 1 2 D. 1 2 3 4 31, I. SoL 2, VI. C. Usell.VI Cone. Bemk. Draw NL II AZ, and meeting AM the parallel' to BN &c. then AZ = S, the sum Of the series. ForML :LN = MA : AZ; but LN = MA = AB; BC = BN; & ML = MA-LA, or AB-NB, or AB-BC; .-. AB-BC : AB : AZ; i. e., S the sum of the series is a third proportional to the difference of the 1st & 2nd terms, and the 1st term. Pkob. II. By anticipating Propositions 13 & 16, bk. VI., it follows, that of the three quantities, the first and second terms, AB, BC, ^ S the sum of the series, if any two be given the remaining one may be found. 1°. Giyen AB, BC:— then AZ = 64 AB' AB-BC; as 8 and 4: then = 16 8-4 PROP. XI. BOOK VI. 297 2°. Given AZ 85 ABj then AB — BC = as 16 and 8; then 8—4 = 5i-=4. 16 AB' AZ = BC. ,/1X 3°. Given AZ & AB-BCj then AB = v<'AZ .(AB-BC); as 16 & 8-4; then ^^16 X 4= v'64= 8 = ABj and BC = '8—4 = 4. 4°. Given AZ andBC; (ien AB : AZ = BC : BZ; .-.16, VI. AB.BZ = AZ.BO; Thns, AZ divided at B, so that reot. AB . BZ = rect. AZ . BC will give the solution. Let AZ, Cor 14, ll., be so divided; aind the ifirsi; term of the series will be either segment, AB, or ZB, the Problem having two solutions; f6r C. 1 2 D. 1 Pst 1, 2, 1. 2, VI. 16, V. Cone. Join AN, and produce it to meet the perp. from Z, in the . X. Then BZ = ZX. •.• AB : BN = AZ : ZX; and AB : AZ = BN : BZ, i. e., BC : BZ; and •.' BZ = ZX; .-. alt. AB : BN = AZ : BZ. ■. ZA =: S, the sum of the series, of which the first term is BZ, and the second term, BK, or its equal BC. 298 6RADATI0NB IN EUCLID. Peop. 12.— Prob. To find a fourth proportional to three given St. linee. Con. 3, 1. 31, L Dem. 2, VI 7- V. E. 1 2 C. 1 D. 1 Data QtiseSj Pst. 1, I. 3,1. 31, I. Sol. C. 3, 2, VI, C. 2. Cone. Let E, P, G, be the three given st. lines; to find a fourth proportional. CA vH E- F- Prom a com. • A draw AB, AD, forming /. A PAD; A, on AB make AC = E, / ' CB=P; andonAD, AH=G; joinCH: and draw BD || CH ; / \ then HD is the 4th /. D proportional to " E, P, G. V CH II BD, a side of a ABD; .-. AC : (JB = AH : HD ; but AC = E, CB = P and AH = G, .-. E : P = G : HD; •- to E, P, G, a 4th proportional HD, is found. Q. E. P. SoH.— Among several other constractionB for the Solution of this Problem there is one, for which the compasses alone axe sufficient, From any cen. 0, with A ._ & B, desc. two 0s ; & from I), a . in the outer 0, setoffDE = C; from D,E, with DO cut the inner in G, F ; then FG is the 4th pro- portional required, •/ DF = EG ; OE = OD, & OG = OF ; .'. aODF has its sides = the sides of A OEG J C. 1 Pst. 3, 1. 2 tt 3 4 Sol. D. 1 C. 3, 1. 2 D. 1. PROP. XII. BOOK VI. 299 8, L AddyOrSub. Ax. 3, L C. Def. 1, VI. 4, VI. Cone. .'. AODF = A OEG, and /. DOF = Z EOG; In Zs DOF EOG add or take away com. ^DOG ; in thjg case, /. EOD = /.GOE. And •.• AS GOE, DOB axe isosc. & ZDEO = Z.GFO; .•. AGOE is eq. ang. to aDOB j .-.BO :OG = ED :EG; .'. EG is a 4th prop, to A, B, C. Q. E. F. Use and App. 1— The Sector is an instrument invented by Gunter ■who lived hetweeu A.D. 1581 and 1626 ; it consists of two eqnal legs; or rather rulers, turning like a Carpenter's rule, on a pivot at the centre. The Scales on it OB, OA, converging to the pivot are those which properly belong to the Sector, though in the vacant spaces, it is usual to insert various other scales. This instrument has not inaptly been termed, " a large number of pairs of compasses packed up into one." Eor explanation we will take one pair of these compasses or scales, OA, and that which corresponds to it, OBj on each is laid down a scale of chords, OA = OB = 90° ; OP = OQ being radii, or chords of 60° ; and if OP be 4 inches, or 8 inches, and the proper divisions inserted, we have a scale of chords, with a radius of 4, or 8 inches, laid down. 2. — Bt aid op a pais op Compasses and the Sectoe, the following Problems, among others, may be solved. Peob. X. — To find a fourth proportional to three given lints K a, Li, M 6 Take OD, DC each = K 3, and open the instrument till DC = L 4 : then take OQ, OP, eadi = M 6, and the distance QP = the 4th proportional = 8. For ".■ DC II QP, a side of aOQP ; .-. OD 3 : DC 4 = OQ 6 : QP 8. Peob. II. — To find a chord, Baj of 40°, to a radius, as of 5 inches, from a Sector the rod. of the chord of which OP = 4 inches. 300 GRADATIONS IN EUCLID. From a scale ol inches take 5 in the compasses, and open the Sector go that QP = 5 inches ; fix now one leg of the compasses at C, the extremity of OC = 40°, and the other leg at D, of OD = 40° ; the distance DC will gire the chord of 40°. when the radius is 5 instead of 4 inches. For, h7 similar As, OP 4 : PQ 5 = OC 40° : DC 50°, when the rad. is 5 inches. Peob. in.— To divide agiven line L, =: 120, into two parts, x and y, which shall be to each other, as two lines, or numbers OD_= 3, and OQ ^ 5. Take OE, OF each = (OD 3 + OQ 5) = 8 ; open the sector so that EF = L = 120 ; then DC = :!:, and QP = y. For, on drayring QG || OF, •.• OD = QE, Z.EQG = /.DOC, & Z.QEG = ^ODC, .-. aODC = aQEG, DC = EG, & QP = GF. Now DC : QP = 3 : 5 ; .-. EG : GF = 3 : 5. 120 X 120 . Hence the parts will be i = 3 ( ) = 45 ; & y = 5 ( ) = 75. 3+5 ^3+5 3.— Vabious other Lines are marked on the Sector, and for their use depend on the same principles as those which affect the Line of Chords and Line of Numbers. 1.— .4 Line of Sines, marked S, of which the rad. ^ 90°. 2.— Two Lines of Tangents, one marked fi-om 0° to 45°, the radius being the sine of 90° ;— the other, on a smaller scale, marked from 45° to 75°, the rad. being the tangent of 45°, or distance to the beginning of the scale. 3. — A Line of Secants, the rad. being the secant of 0° extending to about 75° 4. — A Line of Polygons, marked POL. showing the length of the sides of regular polygons inscribed in circles. 5. — A Line of equal parts, by which a fourth proportional to three given numbers may be found ; the line being divided into 100 parts, — that number is the limit of the accurate use of this line. 6. — ^A link of chords, marked C, the radius being the chord of 60°. The extent of this line is from to 60° ; — and if a chord for a greater arc than 60° be required, — a circle, with a rad. equal to the chord of 60°, must be drawn, in which the chord of 60° must he set off; and then from the point where the chord of 60° cuts the circle, must also be set off the amgunt of the given arc above the 60°; — the distance in the circle from the extremities of the sum of the two arcs will be the chord required. PROP XII. BOOK VI. 301 Similar Lines to these, with some additional, are marked on Gumtek's Scale ; — ^but as this Scale is sometimes of two feet in length, and sometimes of one, it will be found that the Magnitudes, though proportional to the corres- ponding lines on the six inch Sector are not identical. Besides the Lines of Numbers, Sine Rhumbs. Tangent Ehumbs, &c., on Gtmter's Scale, are Logarith- mic Lines, the uses of which depends on the principle that the logarithms of the terms of equal ratios are eguidifferent. . Tor an explanation, of the Logarithmic Lines on Gnnter's Scale, their Construction and Use, the Student may-consult Keith's Plane and Spherical Trigonometry, pp. 18 — 24. The General Rule given is this ; " the extent pf the compasses from the first term to the second, will reach, in the same direction, from the third to the fourth tei-m. Or, the extent of the compasses from the first term to the third, wiU reach, in the same direction, from the second to the fourth." 4. — An Example or two will show use op the Seotoe. Ex. 1. In a rt. /.d. aAI^C, as in fig. to 13, VI, to find the side DC op- posite Z_A, when /_A = 40° and AC = 4 inches. Take 4 inches in the compasses, and open the Sector, until 90° and 90° on the two identical lines of Sines are 4 inches apart ; then the distance from 40° on one of those lines of Sines to 40° on the other will equal the side DC. Ex. 2. Given in a A two Z_s, of 59° and 38° respectively, and the side opposite to /^69° equal to 76 equal parts ;— ^required the length of the side opposite to /_38°. Sine 38!' X 76 The analogy is,— Sine 59° : Sine 38° : : 76 : * ; i. e., x = : . Sine 59° Open the Sector until the counterpart lines of sines, 59° and 59°, are is far apart as the distance from to Sine 38° ; then the distance from 76 to 76, on the corresponding lines of equal parts, will show the value of x. Ex. 3. To inscribe a regular polygon of 10 sides in a circle of which the 'adius is 3 inches. Open the Sector until 6 and 6, on the counterpart lines of Polygons, are J inches apart, then the distance from 10 to 10, on the same counterpart lines, vill equal the length of one side. 5. The Sector may also be applied, to extract the Square and Cube Eoots, md to double the Cube ; to measure Triangles, — to find the Areas of Figures, md the Contents of Solids,— and to increase or diminish any Figure according io any given Proportion. NB. It must however be remembered, that no great accuracy, is to be obtained Tom the small Sectors in common use ; and they are the less to be trusted in jroportion to the greater opening of the sides or legs. It is only^ by considera- te practice that a pex-son can become expert and exact jn the use^pf this in- itrnmentr 302 GRADATIONS IN EUCLID . Prop. 13.— Prob. To find a mean proportional between two given St. lines. CoK. Pst. 3, 1, 11, 1. Dem. 31, in. CoE. 8, VI. E. 1 2 C. 1 2 3 Sol D. 1 Data Qaaes. Pst. 3, I. 11,1. ol. 31,111.0.2. Cor. 8, VI. Cone. Let the given st. lines be AB and AC ; to find a mean proportional between them. Place AB, BC, in a st. line AC; D A E B C & on AC descr. a semicircle ADC ; from B draw BD _L AB and join AD, DC; then BD is a mean proportional to AB & AC. / zADCisart. Z,&inthert. Z'dAABC, BD is _L the base ; .■. DB a mean propl. to the segments of .AC, namely, AB, BC ; .•. between AB, BG, a mean proportional DB, is found. Q, E. P. ScH. There are other conatmctions hy the aid of which a mean propor- tional between two given lines can be obtained, — but none simpler or easier than the above. For numerical calculations however the following formula may be used: LetAC = 2r; AB = j:; BC = 2r — z; andBD=y then I :y : 2 r— x ■,0T ,g'=2rx — r", & y = ^2 rx — x*, ory = -s/AB-BC. Ex. Find a mean proportional to 9 and 4- ^ i X 9 = ^36 = 6 the mean propl ; t. e., 9 : 6 = 6 : 4. XTsE and Appl. L — Any ' rectangular parallelogram may he reduced to an equivalent square by this proposition ; — for if AB & BC represent the two sides, then the square on DB = AB ■ BC ; — the demonstration of which fol- lows from R:op 17, VI. TT. Combining Propositions 11 and 13, we are able, when, of three lines in continued proportion, any two are given, to find the unknown line ; See Use Bud Appl. 11, VI Problem n. PROP. XIII. — BOOK VI. 303 ni. Given one of three terms and the sum of the other two; to find the two unknown terms. 1°. Given the mean BD, and the sum AC = AB+ BC. C. 1 2 3 4 D. 1 2 Pst. 1, L U,I. 31,1. 12,1. Sol. C. 4, VI. Cone. On AC desc. a eemi. 0; and at C draw CG,=BDJ_ACj Through G draw GF || AC, cuttmg the arc in • s F, D ; and from I) drop DB J_ AC; then the other two terms are AB and BC. . ?^-^l E B O "-■ AS ABD and DBC are similar, •/ AB : BD = BD : BC; Thus BD being the mean, AB & BC are the extremes. Q. E. F. 2°. Given AB, one eitreme, and the sum of the two terms, BD = BC + CD; required the mean, and the other extreme. 2.1. 23,1. 33, in. 3,1. 23,1, 23,1, Sol. Set the sum of the three terms, AB -|- BD, in one St. line AD ; at the extremity A draw an /., DAH =:ext. /. of an eq. lat. A ; and on AD desc. a seg. of a with /. = /I DAH; then set on AD the given extreme AB; at B make /. -ABB = ^ DAH; prod. BE, and at E make l_ FEC = /_ DAH; BC is the mean, and CD the other extreme. ,'V jrw \.^ D. 1 C. 1 •.' EBC is an eq. lat. A, and BC = BE; Sch. ,1. .-. AB : BC = BC : CD. 8, VI. Cone, I -, BC is the mean, and CD the other extreme. Q. E, F. 304 GRADATIONS IN BUOLID. rV. When three lines are in continued proportion, if one be given and the difference of the other two, those other two may be found. 1° Given the mean BD, and AC the difference of the extremes. C. 1 33, m. D. 1 On AC descr. any seg. of a© ADC; To the at A draw a tang. AH , = given mean; join H and cen. of the 0, and with OH desc. concentric HBG ;* Psts. 2 &' produce AC to B; 17, m. Psts 1 & 3,1. 3,1. 23,1. Sol. 32, in. C.&D.2 DeflVI 4, VI. 6 Cone TT^'-..- and from B with rad. ^ AH cut the inner circle inD; join DA and DC; then AB and BC are the two terms required; '.• BD touches the ©ACD in D, and DC cuts it in C; .-. jL BDC = Z DAC. In AS ADB, DCB we have /.BDC = /.DAC, and /B com. .". AS ADB, DCB, are eq. ang,, and their sides proportional; .-. AB : DB = DB : BC. .•. AB and BC are the other two terms. 2°. Given one extreme, BC, and the difference between the mean and the other extreme, AB— BD; to find both the mean and that other extreme. D. 1 2 Theo. LXJse 11, m. 16, VL E«mk. D3. & H. Use 3, 10, II. Bemlc. Use 1, 10, n. '.• the differences of successive terms are as the terms them- selves, .". the rect. under one extr. and the dif. of the mean and the other extr. = the rect. under the mean and the difference between it and the given extreme. i. e., BC. (AB-DBJ = DB. (DB-BC). Since the Area of this rect. and the difference of the sides are given ; .•. (fie sides themselves may be found. Forv''('difference\2 , ^, Sum. ^ V 2 ■' + ™® ^^^ = — 2 — Lakdner's Euclid, p. 79. and -^ +_ ' = the greater, or the less magnitudes. V. To find two St. lines to contain u- rectangle equal to a given rectangle AB . BC, and to have a given ratio one to the other. ADDENDA. PE. XIII. BOOK VI. 305 A r^-.. F ^.\\. B ■■• At M, the middle • of AG, a side of the A, or of the part between A & the circumference, fix a fine thread ; and attach the thread to E ; So, fix a thread at G, and keep it extended through C. Now, if Ihe eq. lat. A move round A till the intersection in D of the two threads falls on the line AB, then AE wUl cut off j of /.CAB. •.• fig. AGE, is an eq. lat. A and /.GAE = j of 2 rt. Zs; .-. /.GAE = i of the Zs EAG, GAE, EAC ; .-. ZGAF = i of Zs BAG + EAC. But ZEAG at centre = 2 ZACG at 0ce. & •/ ZACp = ZE"AB, .-. ZEAG = 2 ZI"AB, ;. e., Z^AB = i ZEAG j Now ■.■ ZEAB = i ZEAC : .-. are BE = ^ arc, EC ; .-. arc BE = i arcBC, & ZBAE = i ZBAC ; .-. ZBAC is trisected by AE and AH. Q. E. E. N.B. — Either of these methods solves the Problem, and may be adopted in practice when the Trisection of an angle is required. 310 GRADATIONS IN EUCUD'. Prop. 14. — Thkoe. Equal Parallelograms, which have one angle of the one equal to one angle of the other, have their sides about the equal angles recipro- cally proportional; and conyersely, parallelograms that have one angle of the one equal to one angle of the other, and the sides about the equal angles reciprocally proportional, are equal to one another. " When / 7 s are eq. ang. the sides which are ahout the eq. /_s, are recip- rocally proportional ; and when the / 7 s ai-e eq. ang., and the sides, which arc about the eq. /.s, are reciprocally proportional, those / 7 s ai'e equal." — Edclid. Con. 14, 1 If at a ■ in a St. line, two other St. lines, upon the opp. sides of it, make the adj. /_s together eq. to two rt. ^s, these two St. lines shall he in one and the same St. line. 31,1. To draw a st. line through a given • || to a given st. line. Dem. Def. 2, VI. Reciprocal As and / 7 3 have their sides ahout two of their /.s proportionals, so that a side of the 1st fig : a side of the other : : flie rem. side of the other : the rem. side of the 1st. 7, V. Eq. Ms have the same ratio to the same M ; and the same has the same ratio to eq. Ms. 1) VI. As and / 7 s of the same alt. are to one another as their bases. 11, V. Ratios that are the same to the same ratio, are the same to one another. i 9, V. Ms which have the same ratio to the same M are eq. to one another : and those to which the same M has the same ratio are eq. to one another. E. 1 2 C. 1 H. Case I. Cone. Def. 2. VI. App. 14,1. 31, I. Let £17 AB = £=7 BO, & zDBP = Z EBG tlien DB, BF and GB,BE are reciprocally proportionals. i.e., DB : BE =i GB : BP. k .F D Apply / — 7 AB to BC so that DB, form one st. line ; and PB, BG also one line: and complete the / 7 PE, EB St. ■ / / / tKOP. XV. BOOK VI. 311 D. 1 2 E. D. 1 2 3 4 5 H. &C. 7. V. 1,V. 11, V. Cone. H. Case II, Cone. H. & 1, VI, 1,VI. 11, V. 9,V. Rec. •.■ CZJ AB = m BC, & EFis another , .-. CZ7 AB : /ZZ7 FE = CZJ BC :.£Z7 FE ; but /rus AB : PE = bases DB : BE, and I — 7 s BC : FE = bases GB : BF ; .-. sides DB : BE = sides GB : BP ; .". the sides of i 7 s' AB, BC, about the eq. L s are reciprocally proportionals. Let sides DB : BE = GB : BP ; thenci; AB = £=7 BC. •.• DB : BE = GB : BF ; and DB : BE = ^Z7 AB : cm EP; and •.- GB : BP = ,CZ7 BO : £137 EP ; AB : ,^=7 FE = £Z7 BC : PE; AB = £117 BC. Therefore, Equal parallelograms, ^c. Q.E.D. Sc!H. 1. There is a third case, not introduced in the proposition, and de- pending for its proof on Prop. 16, bk. VT j — it is, " When two I 7 s have eq. areas and their sides' are reciprocally propl., they will be eg. ang." It wiU be sufficient to prove that one /. in the one is eq. to the corresponding /_ in the other /~~7 ; for from tlxe proof it follows, that each separate /_ ™ t^e one is eq. to each corresponding /_ in the other. 2. Inasmuch as As =: the halves of their equiangular / 7 s. — what is proved respeotihg eq. / 7 s might be proved respecting bq; As. (Ax, 5, 1); and the following Proposition, the 15th', form a corollary of the 14th ; for if ADBF= a EBG, and /. DBF, = /.EBG, then also DB : BE = GB : BF. The parallelogram and triangle are united into one Theorem, Prop. 1, VI, and might also be united here. Pbop. 15. — Theoe. Equal angles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally pro- 312 GRADATIONS IN EUCLID. portional: and conversely, triangles which have one angle in the one equal to one angle in the other, and the sides about the equal angles reciprocally proportional, are equal to one another. Con. U, Pst. I, I. Dem. 7, V. 1, VI. 11, V. Del. 2, VI. 9, V Hyp. Concl. Def.;2, VI Appl. 14,1. Pst. 1, I. H. &C. 7,V. I, VI. II, V. Cone. Case I.— Let a ABC = aADE, and'zBAC = Z BAE; then CA, AB, EA, AD are reciprocally propor- tional. i. e., CA : AD = EA : AB. Place pA & AD in one St. line CD ; ■. EA, AB also in one St. line EB ; join DB. •/ A ABO= A ADE, & fig. ABD ia another a ; .-. aABC : A ABD = A AED : A ABD. but A ABC : A ABD = bases CA : AD, and A AED : a ABD = bases EA : AB ; .-. CA : AD = EA : AB. .", the sides about the eq. Z. s are reciprocally proportional. Hyp. Cone. 14,1 H. 1,VL 11,1. D. 3. 9, V, Rec. Case II.-Let CA : AD = EA : AB ; then A ABC = A ADE. Make the same construction as in Case I. •.• CA : AD = EA : AB ; and '.• CA : AD = A ABC : A ABD, and EA : AB = A AJ)E : a ABD ; /, A ABC : A ABD = A ADE : a ABD; and •.• A ABD is com., .-. a ABC = A ADE. Therefore, equal triangles which have one angle, ^c. Q. E. D. PROP. XV.— BOOK VI. 313 Or, for Case II. C. 1 Appl. Pl^e as before, CA, AE in one st. line, and DA, AB in another. 2 31,1. and complete the / /s - AP, AG, AH. D. 1 H. •.• BA : AD = EA : AC, 2 1,VI. .-. BA : AD = r-7 AG : £ and BA : AC = ^— 7 AF 3 9,V. .-. dJ A.Q = C=l AF, 41,1. and A ABC = A ADE. N. B, Case I, may be proyed on the same construotion. ScHOL. 1. Asa criterion for the equality of tri3iigles this Proposition may he classed with Props. 4, 8, & 26, bk. I. 2. When in each triangle the egual angle, included by the reciprocal sides has the same supplement, t£e sides are also reciprocally proportional. Use' & App. To construct an isosceles tfiangleeqiial to a given scalejie tri- angle ABC, and with the same vertical angle B A C. C. 1 2 3 4 D. 1 2 3 4 3,1. Pst. 3, 1. 11. 1, 3,1. Sol. Cor.8,VI. C. 3, Def.25,1. D. 1, 2. 15, TL Produce BA, and make AD == AC; on BD desc. a Q DEB; and draw AE_LBD; • in AB prod., take AF =: AE, and in AC, AG = AE; join GE; then A AGE is tiie isosc. A required. •.• BA : AE = AE : AD, .-, BA : AE = AE : AC; but AF, AG, each = AE; .•. AGP is an isosc. A and .-. BA : AG = AF : AC. i.e., the sides aboUjt the vert, /.are regip. props. .". the isosc. A AGE = the scalene A ABC. Q. E. F. 314 GRADATIONS IN EUCLID. Prop. 16. — Theor. {Very Important.) If four St. lines he proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means ; and con- versely, if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four St. lines are proportionals. Con. 11,1. 3, I. 31,1. Dbm. 7, V. 14, VI. Def. 1, II. A rect. is contained by any two of its conterminous sides. Ax. 1, 1. E. 1 Hyp. Casb I.— Let AB : CD = = E: F; •2 Cone. then 1 — 1 AB . F - 1 — i CD . E. C. 1 11,1. From A, C, draw AG i. AB, and CH J. CD, in 2 3, I. 31, !.■ makeAG = F, CH = E, P - B and eomplete | |s EG, DH. G;.. ; " D. 1 C. 1 & 2. •.• AB : CD = E : P, and •.• E = CH and F = AG. b 8 2 7, V. .-. AB: CD = CH: AG; k . 4 Rr 3 i. e., sides about the eq. Z.S artf recip. propl. ; ' 3 14, VI. .-. [ZDBG = C^DH; 4 C. 2, but •.• AG = P, Def. 1, II. .*. 1 1 BG is contained by AB and F ; 5 and ■.• CH = E, .•. 1 1 DH is contained by CD and E ; 6 Cone. .-. CZl AB.F = cIDCD .E. jb E. 1 2 C. D. 1 Hyp. Cone. Sim. H. &C. Def. 1, II. Case II.— Let CZI AB . F =[zm CD . E then AB : CD = E : F. Using the same construction ; ■.• AB . F = CD . E, and •.• AG = F and CH = E ; .". CD BG is contained by AB . P, andczi DH by CD.E; PROP XVI. BOOK VI. 315 Ax. 1. C. 14, VI. C. 7, V. Eec. .-, CZ] BG = CZ: DH. and •.• Z A = Z 0, /. AB : CD = CH : AG, i.e., they are reoiprooally proportional; but CH = E, and AG = F ; .-. AB : CD = E : F. •. If four St. lines he proportionals SfC. Q. E. D. Alg. §■ Arith. Hyp. Let a4:63 = c8:d6. Alg. (Xbxd) conversely, Arith. (X 3 X 6) conversely (4-3X6) d, bed. J , ^ — J— , 1. e., aa ^ oc; a . . a • T abd b •.' ad = be. ad be JI ~ bd' .. 4 _ 8 •T ~1"' 4X3X6 i.e., ^ = 8X3X6 3 6 - •.•4X6 = 3X8. 4X6 _ 3 X 8 .-'.i.e., 4 X 6 = 3 X 3X6 3X6^ t. e., - = CoE. Rectangles which have their sides about the right angles reciprocally proportional are equal ; and if the rectangles are equal, the sides about the right angles are reciprocally proportional. • ScH. I. Definition. Two ratios A : B and a : J are said to he reciprocally proportionals, when the antecedent is to the conse- quent in the one as the consequent to the antecedent in the other ;, as A : B = J : a. n. As a necessary consequence, the 16th Prop., Bk. VT, may be deduced! from the principlee resting on this definition;— 1°. That a ratio compounded, of reciprocal ratios is a ratio of equality; and 2°. That if a ratio of aqtlaJity be componnded of two ratios they must be reciprocals. 1°. A ratio ratio of equality. of reciprocal ratios, as, A : B and a :b, is a '*!■ r 316 GRADATIONS IN BUCLID. D. 1 H. Cone. ..A.Ti_i„ .A:B)_A:B . A:B=6:«; .. „. j } = b : A' .'. A : i = A : A; and A : A is a ratio of equality. N. B. The form A :B1 a : ij denotes a ratio componnded of A : B and a : b. 2°. If a ratio of equality, A : A, 6e compounded of two ratios, u : b, and c : d, they must be reciprocals. D. 1 2 3 4 H. Cone. Let c : d= b : . bat a : :i; is a ratio of equality; i. e. x = a, .', b : X is the the reeip- of a -.b; and c : dia the reeip. of a : &. 3°. The sides of equal rectangles are four proportionals. D. 1 Cor. 2, 1,VI. 2 Theor. 2 Sch. n. 3 Theor. 2 Sch. n. 4 Eemk. 5 6 7 Cone. 8 16, VI '.' As and / 7 s are in a ratio compounded of the bases and altitudes. ,". if the As andZ3I7s are equal, those bases and alts, are reeip. propls., and also those As or / 7 s are equal. Now, '.' the means are sides of one, and the extremes the sides of the other; .•. those sides of the eq. 7 7 8 are propls.; and •.• the sides are propls. .". the I I under the means = I I under the extremes; and .•. the sides of those eq. I l a are proportionals. " Thus the sixteenth Proposition," says Laedkeb, p. 203, ' diately from the first." follows imme- ni. By the principle, that if four st. lines are proportional, the rectangle of the extremes is equal to the rectangle of the means, we convert the equality of the two ratios, or the proportion of the four lines, into the equality of two rect- angles. In numerical proportion the expression, the product of the extremes equals the product of the means, points out a very similar, if not an identical process; and the phrase, " product of two lines," is sometimes used instead of " rec^ai^le under two lines." The word product however, supposes the four magnitudes which are proportionals to be commensurable, and really denotes the result by multiplication of the numbers which stand for the magnitudes: but in generail, it may be demonstrated both of commensurable and of incommeTir surable magnitudes, that, if four st. lines, A, B, C, D ie proportionals, A : B : : C -.Dtthe rectangles under the extremes A.D =: the rectangle under the means, B.C. PROP. XVI. — BOOK VI. 317 IV. Tl^e doctrine of Limits, however, is needful far the full demonstra- tion of this Proposition, when incommensurable magnitudes are intrq^qed. Briefly stated, by a, limit we mean a.^aeei magnitude to which anothfe* and a variable magnitude may approach as nearly as we please j and ye't the variable mag^ide is never able to attain an exact equality with the fixed magnitude. OTua, the circle is the limit of an inscribed |iolyg6n ; the polygon may approach in area, as near as we choose, to the area of the circle, but never actually attains that area. Take any fixed quantity. A, and a variable quantity P ; — ^in order that A may be named the limit of P, the two conditions have to be ftilfilled — lat, that P never attains equality to A ; — and 2nd, that P shall be capable of being made as nearly as we please on an equality with A. Again, " Two fixed magnitudes, A and B, are the limits of two others, P and Q, when P and Q by increasing together, or by diminishing together, may be made to approach more nearly to A and B respectively, than by any the same given difiference, but can never become equal to, inuch'lesa pass A and B." — Geou. Plane, Sol. §• Spher. p. 248. V. For the proper elucidation of this subject two leading Theorems are needed, given as M, v . p. 235 ; and P. V. p. 241 . Theoe. I. If there be two fixed magnitudes A and B which are the limits of two others, P and Q, and ifP be to Q always in the same given ratio of C to b; then A shall be to Bin the same ratio. E. 1 2 3 4 S Hyp. 1. .. 2. Hyp. 3. Case 1. Let P and Q, approach A and "B bp a continual increase. Let P and Q never = A and B, much less exceed A and B, but P & Q approach A and B more nearly than by any given difference. Let now a M, B' be taken, so that A : B' ^ C : D; if B' ^t B, B' > or < B. Sop. 1°. Let B' be < B,by any difference, as b; D. 1 2 3 4 5 H. ll,V. H. D.2,&Sup.l°. .-. P : Q = C :DJ & A : B' = C : D ; .-. A:,B' = P : Q ; but A always > P, .•. B' always > Q. •. . •.• Q always < B', and B' < B by S ; .'. Q cannot approach B within the difference b ; hut this is against the Hyp, j /. B' B; and take A' so that A' !S = A: B'. 318 GRADATIONS IN EUCLID. JJ .1 2 H. 14, V. Sup.2°..H.ll,V 3 4 5 6 7 H. Cone. E. H. D. 1 Sim. Eec. •.• B < B' .". A' < A, as by some difference a. And ■.• A' : B = A : B' & P : Q = A : B'j .•.A':B = P:Q; but B always > Q ; .•. A' always > P ; •.,•.• P always < A', & A' < A by o ; .•, P cannot approach A within the diSexencea; but this is contrary to the Hyp. ; .". B' j> B ; .". B' cannot but = B, i. e. A : B = C : D Case n. Let P & Q approach A & B respectively by a continual decrease. In this case the same demonstration may be given, taking care to substitute the word " greater" for "less" and "less" for " greater." Therefore; iftherebe twofixedmagnitdues, Sfc. Q. E. D. Theoe. II. If four St. lines, A B C D, is proportionals, (whether com- mensurable or otherwise) the rectangle under the extremes will be equal to the rectangle under the means. Case I. Let A and B be commensurable, & .'. also C & D. Let their com. E. be 7 : 5, and their com. meas. M and N; E. 1 Hyp. 2 Pst. 2, V. 3 D. 1 E. 2, 3. 2 E. 2, 3. 3 Ax. 1. 1. i. e. let M. be contained 7 times in A, and 5 times in B, and N be contained 7 times in C, and 5 times in D ; •.• A = 7M, andr) = 5N; .-. A.D = 7 X 5 timeaM.N; and •.• B = 5 M, and C = 7 N; .-. B . C =■ 5 X 7 times N. M; .-. A.D = B.C. 2t 3 14 a I C 3? 10 Case H. Let A & B be incommensurable, and .". also C and D. D. 1 H 2 Pr M,V. Pr.M,V. •.• A :B = C :D; and P, Q may approach nearer A, C than any assigrifed difference, P and Q also containing like parts of B and D ; ..-. P.D=Q.B. And •.• there may be taken like parts of B, D continually less and less, and •/ P and Q increase towards A and D within any assigned difference; PROP. XVI. BOOK VI. 319 Theor. I. and also •,' P. D and Q . B, by increasing together, approach nearer to A . D and C . I) than any assigned difference ; A.D= C.B,orB..C. Rec. ] Therefor^^t/'/oMr st. lines be proportionals, Sfc. Q. E. D. Use & App. I. The Theory of Limits may be applied to establish Prop. 1, VI., both for bases that are commensurable, and for those which are Incom- mensarable; thus, The Rectangles AC and EG, having the same alt. AD = EH,. ore to each other as their bases, AB and EF. Case I. Let AB, EF be commensurable, i. e,, exact multiples of A6 = Ey. c. 1 2 Sup. 11, L D. 1 2 C. 2. 3 CI. 4 5 Sim. Let AB = 5 AJ, and EF = 3 E/; and from each . of the divisions of the bases erect perpendicu- lars, making 5 eq. I I s on AB, and 8 eq. .' I s on EF. •.• CZl AC = 5 m, & CZi EG = 3 H/; _ „ „ = 5:3; and '.• AB = 5 AS and EF = 3 E/; .-. IZZl AC : I 1 EG = AB : EF. In like manner, whatever may be the ratio. A B EJ' F Case H. Let the bases, AB, EF, be incommensurable, i. e., not exact multiples of Afi and E_/! AT UA B^ J" IB C. 1 2 D. 1 2 3 4 5 Sup. C. 1 &2 Sch. 4. 16, VI. Let AB, EF form one st. line, AF = 8 Afi; and let e be the ■ in the division nearest to B or E. •.• AF and Ae are commensm-able, AF = 8 AS & Ae =r 5 AS- .-. AehD : AFGD = Ae : AFj AEHD EeAH _ AE Ee "'"'AFGD "^ AFGD ~ A F + AF' But E e A H and E e may be reduced as small as we please, while AFGD and AF remain unchanged; .*, by the theory of limits, AEHD : AFGD = AE : AF. 320 GRADATIONS IN EUCLID. n. The Beetanglea AC and AF are to each other as the product of their bases by their altitudes ; i. e., as AB . AD : AE . AI.. C. D. 1 Pst 2,L LastTh. X the equat. Sup. Bk.n. p. 145. Produce the line IF to cut BC in H. ABHI AB„ABCD_AD Al' AEFI ABHIxABCD AB ABCD AE ABHl' I> and ABxAD AEFI X ABHI AExAI ABCD AB . AD \ Hi 32 B AEFI AE.AI' Let AX = AE ^ unittj/; then Q AEFI is a unit of surface. .•. Units of surface in r I AC = AB . AD; {. e. = the linear units in AB X the linear units in AD. N. B. In this sense the Area of a I 7 = product of base and perpendi-. cnlar. And '.■ o A ^ 4 I — 1 on the same base and of the same altitude; /. Area ofaA.:=\ the product of the base and perpelidicnlar. TTT. Among other Theorems the following is easily dedncible; The rect. contained by AB, AC, any two sides of a A, ABC, is equal to the rect. contained by AF, its alt, or perp. to the third side, BC, from the opp. /. BAG, and by the diam. AD, of the circumscribing BACD. D. 1 31, rn, &H. 2 21, in. 3 Cor. 33, 32,1. 4 4, VI. 16, VX. 5 Eec. •.• /. ABD is a rt. ^ and also /_ AFC; and /_ ADB = ^ ACF; .■, AS ADB and ACF are equi- and .-. AB : AD = AF : AC; andAB. AC = AD.AF. Therefore, the rect. contained hj any two sides, &c. Q. E. D. ly ■■ PROP yvil. BOpK VI 321 Prop.. 17, — Theoe. Cob. to Prop. 16. If three st. lines be proportionals, the rec- tangle contained by the extremes is equal to the square of the! mean ; and conversely, if the rectangle contained by the extremes be equfll to the square of th^ mean, the three st, lines are proportionals. Con. 3, 1. Dem. 7, V. 16, VI. E. 1, 2 C. JD. 1 2 3 Hyp. Gone. 3,1. H & 7, V. 16, VI. C. Cone. Case I. Let A : B = B: C; then A . C = B . B, orB'. Take D = B 8 A B B cl C •.• A : B = B : C, & B = D; .-. A:B= D: C; & ■•■ IZn A.C = I — iB.D. but •.• B = D .'. en B . D = B''; & .-. czii A . c = B--*. Q E 1. 2 D. 1 2 3 4 Hyp. Cone. H. 16, V. C. Cone. Eec. Case II. And if CZD A . C = B" then A : B = B : C. .-, A.C=Bs&B= = B.D; .-. A.C=B.D; & .-. A : B = D : C; but B = D ; .-. A . B = B : C. Theiefore, If threest. lines 'be proportionals ^■c. Q. E. D. Alg. S; Arith. Hyp. Alg. X (b Xc) Conrersely, H- (4 X c) Let a 2 :J4 = J4 :c8. If^-A; b c bbc abc ~b ~ ~' : ac = 4=; . ac b' i. e. ae = i°; he be' b c 322 GRADATIONS IN EUCLID. Arith. X (4 X 8) Conv. H- (4 X 8) If 1 = ±; 4 8 2 X 4 X 8 _ 4X4X8 i 8 ' •.■ 2 X 8, = 4 X 4; . 2 X 8 _ 4 X 4 i.e., 2X8 = 4X4- 4X8 4X8 j 1. e., __ = -_ UsB & App. I. The Demonstration of Proportion in Arithmetic, conunonl'y nnmcd " The Bule of Three," depends on the last four Props, viz., 14, 15, 16 & 17, bk. VI.; and from that demonstration we dedube the Rule tor finding a fourth Proportional, D, to three given terms, A 8, B 6, and C 4. E. D. 1 Hyp. I Suppose the 4th propL D to be found. 16, VI. •.■A.D=B.C; .-, D = — '■ — Deduction, 1. B.C_ 6X4 _ 24 _ A 8 8 2, B.C 6X4_24_j, . D. 33 3. A.D_8X3_ 24_ „ B 6 6 4. A.D_8X3_24_ „ C 4 4 X) AS B8 c* II. The Properties of Equiangular triangles established in Prop. 4 and 5, ,and those of Proportionals in Prop. 16 & 17, lead to a very clear demonstration of 47, 1; that in a rt. /_ d A ABC, the square of the hypotenuse AC, equals^ the sum of the squares of the other tteo sides, AB & BC. C. 12, 1. D. I 8, VI. ^ 4, VI. sl 17, VI. Ax. 2, 1. From the vert of the rt. /_ B, draw BD J_ AC. '.' A ADB is eq. ang. to A ABC which is eq. ang. to A BDC. .-. AC : AB = AB ; AD, B and AC : BC = BC : DC; .-. AB'i= ACAD; and BC» = AC . DC. ^^. JA + equals-, .-. AB^ + BC* = AC , AD + AC . DC. PROP. XVII. BOOK VI. 323 D. 5 6 7 c. 1, n. Cone. Rec. Now '.• side AC is com. .-. AB2 + BC2= AC . (.AD + DC); but AD + DC = ACj .-. AB= + BC= = AC . AC = AC ■. In art. l_d triangle, Sfc. Q. E. D. Thus the most important peopekties of Plane Geometiy are reduced to the single property; that "in similar triangles the sides about the equal anglei are proportional. ni. Some of the deductions, made in connection with this demonstratioDj may be stated in other words, yet still in accordance with Prop. 16 & 17 ; thus, instead of the inference, " if a perp. be drawn from the rt. l_ to the hypotenuse the square of either side is equal to the rect. under the hypotenuse and seg. adj. to that sidi, wo say, " either side, as AB, is a' mean proportional between the whole hypotenuse AC, and the seg. AD adj. to that side." Also,instiead of " the square of the perp. BO, on the hyp. A C, will equal the rect, under the segs. of the hyp. AD .DC," we declare, — "the pei-p- BD, is a mean propl. between the segs. of the hyp. AD & DC." And in place of the deduction, " ere every A if a perp. BD be drawn from ■Ae vertex B, to the base AC, fig. 1, or base produced AD, fig. 2, the difference of the squares of the sides BA, BC, shall he equal io the difference of the squares of the segs of base, AD, CD, or of the base produced," we substitute, " the base is to the sura of the sides as the difference of the sides is to the difference of the segments of the base, or sum of the segments of the base produced. IV. As instances in which the use of Props. 16 & 17 very considerably shortens the demonstrations of other propositions," let us take ; v' Ex. U If from a point A, there be drawn two st. lines one AC, a tangent to a circle, the other, AB,, a secant, then the tang, will be a mean propl. between the whole secant .^B, and its ext. seg- ment, AF. D. 1 2 H. & 36, III. 17, VI. '.' AC touches and AB cuts the 0, .-. AC = AB . AF ; ..-, BA : AC = AC : AE ; i. e., the tang, is a mean propl. to AB & AF. 324 GKADATIONS IN EUCLID. Ex. 2. If from a . A there be drawn several st. lines AS, AG cutting the circle, then the whole secants, AB, AG, will be one to another inversely as their external segments., AF, AD. D. I 2 3 3fi, nr. Ax. 1,1. 16, VI. ••• AC = BA . AF & also = GA . AD ; .-. BA . AF = GA . AD ; & .-. BA : GA = AD : AF. N.B. For additional Examples of the TTse and Application of Proposi- tions 16 & 17, the long series may he consulted given in XiABdKbb's Euclid pp. 204 — 208. Prop. 18. — Pkob. (Important.) Upon a given St. line to describe a rectilineal figure similar, and similarly situated, to a given rectilineal figure. Con, 23, I. At a given . in a given st. line to make a recti. /_ equal to a given recti. /.. 32, I. If a side of a A be prod, theext. A = twoint.&opp. /_s ; and the three int. /.s of every A together = two rt. ^s. Cor. 3, 32, 1. In As if two /.s in each be eq. the third /_b also eq. Dem. 4, VI. 22, V. If there be any no. of Ms, and as many others, which taken two and two in order have the same K ; the first shall have to the last of the first Ms, the same E, which the first has to the last of the others. Def. 1, VI. Ax. 2, 3, L Dei. 12, V, Homologous sides. Case I. Griven AB and A CDF; on AB to desc. a A sim. to A CDF, & AB homo- gous to CD. At A make /. A = Z 0, & atB, ZB, = ZD; then rem. Z G = rem. Z F, & A ABGr is eq. ang. t© A DF '.• A ABG is eq. ang. to A CDF, .•. those A s aresimilar ; .-. BA : AG = CD : CF ; & AB, CD are homolog. sides. Hence on the given line AB &c. E. 1 Dat. 2 Quses. C. 1 23, 1. 2 Cor. 3, 32, 1 D. 1 C. 2,4,VI 2 Def. 12, Y 3 Hence PROP. XVIII.— BOOK VI. 325 E. 1 Dat. Quaes. C. 1 2 3 4 D. 1 2 3 4 5 6 7 8 9 10 Pst. 1. 23,1. 23, 1. Ax. 3. 32, I. 32,1. Ax. 3. C. 2 & 3. Ax. 2. Sim. Cor. 3,32, 1. D. 3.4,VI. C. 4. 4, VI. 22, V. Sim. 4, VI, Eec. 11 Def. 1, VI, Case II. Given AB & recti fig. CDEF of four sides ; on AB to desc. a recti fig. similar, and similarly situated to fig. CDEP. Join DP; at A mate Z_A' = Z.G, and at 6, ZABG = zCDF; then rem. Z AGB = rem /_ CFD ; and A PCD is eq. arg. to a GAB. Again at G, make zBGH = zDPE ; & at B, zGBH = zFDE ; then rem. Z H ^ rem. Z E, & A GBH is eq. ang. to a PDE. V ZAGB = zOPD,& zBGH = Z DPE; .-. the whole zAGH = the whole ZCPE. Thus zABH = ZODE, ZA = zC, & ZH = ZE: ,-. recti, fig. ABHGiseq.ang.torectl.fig.CDEP. Also /. A GAB is eq: ang. to a PCD, /, BA : AG = DO : CP ; & •/ aBGH is eq. ang. to A DPE; .-, AG : GB=CP : PD;.& GB : GH=PD : PE; /. ex mq. AG : GH = CP : PE. So AB:BH= CD : DE, &GH: HB =PE:ED. Now, '.* recti, fig. on AB. is eq. ang. Avith the recti, fig. on CD,, and iheir sides pro.por- tional ; .-. figures, ABHG & CDEP are similar. E, 1 2 Dat. Quaes. Case III.— Given AB and recti, fig. CDKE F of five sides ; on AB to desc. a recti. %, similar, and similarly situated to fig. CDKEP. 326 GRADATIONS IN EUCLID. C. 1 2 3 D. 1 .9 10 11 Pst. 1, I, & Case II. 23,1. 32, I. Ax. 3 C.Def.l.V] C. 2. Sim. C: Def. l,Vl 4, VI. 22, V. SAn. 0. 2, 4, VI. D. 3. Cone. Sim. Join DE, and on AB desc. recti, fig. ABHG sim- and similarly situated with fig. CDEF; at B made Z HBL = EDK, and at H, /.BHL = Z DEK; then rem. Z L = rem. Z K. and • fig. AH is sim. to fig. CE, Z GHB = Z FED ; Z BHL = Z DEK, .-. whole Z GHL = whole Z FEK. So Z ABL = Z CDK; .• pent. GL is eq. ang. with pent. FK. And •.• fig. AH is sim. to fig. CE ; .-. GH : HB = FE : ED ; but HB : HL = ED : EK ; .-. ex. mq. GH : HL = FE : EK So AB : BL = CD : DK. And •.■ A BLH is eq. ang. with a DKE ; .-. BL : LH = DK : KE. ■." pent. GL is eq. ang. with pent. FK, and their sides proportional. .*. pent. GL is similar to pent. FK. By a like process a hexagon &c., may be described on AB. Therefore, Upon a given st. line, ^o. Rec. Therefore, Upon a given st. line, SfO. Q. E.. F. ScH. I. A more simple way far making a rectil. Jig., as ADKEF, similar to a rectil. Jig., ABLHC, would be; D. 1 2 3,1. Pst. 1, HI. 31,1. Sol. C. 3. .29, I. 4, VI. Method 1. On AB or AB produced set AD: From the centre of similarity A, divide fig. ABLHC, into As, by indefinite lines AK, ATI; make DK |1 BL, KE || LH, andEE||HC; then fig. ADKEF is similar to fig. ABLHC. *.' the respective lines, are parallel; .-. ZsatD= ZsatB; ZsatK = /_B at L, &c. and Z^ at A com. i. e., the respective As arft eij. ang. ; .•. the sides about the eq. Zs are propl. ■ .-, AK : AL = DK : BL,^ and AK ; AL = KE : LH; PUOP XVIII. — BOOK VI. .327 /. DK : BL = KK : LH. So KB : LH = EF : HC, and so on; ,•, the sides about the eq. /.s are proportional and the figijrfii similar. Q. E. F. On AB to place a %. similar to the recti, fig. CDKEF. At a convenient distance from CD place AB || CD; join AG and BD, if AB ; > CD, AC and B1) will meet on being produced^ let them meet iii the . F; join PF, PE, PK, arid produce them indef ; \ through . A draw AG || GF, through;. G, GH||FE, , and through . H, HL j| BK; and join BL; then fig. AL on AB is sim. to fig. CK on CD. *.• the respective lines CF, AG, FE and GH, EK and HL, &c. are parallel ; .*, the /_s about the resp. points C & Aj F & G, B: & equa:l ; & .', the resp. As are equiangular and similar ; .-. CD : DP = AB : BP : & DP : DK = BP : BL .-. ex ceq. CD : DK = AB : BL. So the sides about the other eq. /.s are proportional. .", fig. AL on AB is similar to fig CK on CD. N.B. — Should AB = CD the demonstration miist be derived trom 32, L II. To construct a recti. Jig., as in the last diagram, similar to a given recti. Jig. AL on AB, and having its perimeter equal to a given st. line CQ. C. I , 2 3 4 5 5 6 7 Sim. Cone Method 2. c. 1 2 31, L Pst. 1, 1 3 4 Psts. 1, 2,1. 31, L 5 Cone. D 1 C. 1,4. 2 29; L 3 4 5 6 7 4, VI 22, V. Sim. Cone. H &c. ar« Q.E.D. D. 1 2 3 Pst. 2, 1 3, L 12, VL 3,L 18, VL Cone. C. 2 C. 1, & . Cone. Prod. AB until AR ^ the perimeter of the given fig. AL to AE, CQ & AB take a 4th prop!. CD ; set the 4th propl CD on CQ ; ■'•■--. and on CD desc. a recti fig. CK sim. to fig. AL ; then the perimeter of fig. CK = the st. line CQ. Now per. AE : per. CQ = AB : CD ; but per. of fig. AL = AR, and per. of fig. CK ^ CQ ; .•. a fig. has been drawn, similar to AL & with peiimeter = CQ. III. "As many figures of the same species with different areas can be constructed on the same right line as the figure of the proposed species has side* of different lengths." 328 GBADATIONS IN EUCLID IV. Anticipating tlie next two propositions, and using the same con- struction as in the last figure but one, we may also prove, that Similar triangle* and polygons are to one another as the squares of their homologous sides. „ Area AEF _ AE^ _ AJ^ _ AD^* . ' AreaAHC AH= ~ AL» ~ AB^' " .-. Area AEF : AD» = Area AHC : AB». In like manner, Area AKE^ area AXH ^ areaADK _ area ABL ' AD2 AB" AD2 AB= Adding equals, Area AD KEF : AD= = area ABLHC : AB=; .-, Area ADKEF : area ABLHC = AD» : ATP Use & App. Nearly all the practical methods of taking a Plan of any building or place, or of drawing a Map of a field, of an estate, or of a whole country, are dependent on this Proposition. The lines made use of in lie Plan or Map are proportionals to the existing lines in a huilding, estate, or country, and Representatives of the same ralues. The object of the Surveyor is to lay down a figure, as ABHIK, exactly like p v in shape to the Prototype, or original, A as CDEFG, possessing similar angles /s and proportional sides; and this he ai> / | / complisnes by actual measurements or Q/ l / calculations of the sides and angles, and ^^ ' / then by reducing the lengths of the ^ v!-' lines in his drawing or plaA in exact C proportion to the original Features or Outlines. The very same angles are given in the Map that exist in the field or the country that is surveyed ; thus the figure ABHIK is a reduced copy of the fig. CDEFG, preserving the /.s at A,B, &c., equal to those atC, D, &c., — ^but giving the lines AB, BH, &c. in proportion only to the lines CD, DE, &c. Whenever we have to mate a Design, or a Model, we use this Proposition either actually or virtually, so that its application extends to all the Problems of Geodoesia, or the Art of Measuring and representing surfaces, — to Plans, Maps, and Geometrical Drawings of every kind. , Prop. 19. — Theoe. {Very important.) Similar triangles are to one another in the duplicate ratio of their homologous sides ; i. e., as the squares of their like sides. PEOP. XIX. — BOOK VI. 329 E. Con. 11, VI. Eem. Def 12, v.— 16, V. J£ iom Us of of tiie same kind be propls., they shall also be propls. when taken alternately. 11, y, 15, VI., Def. 10, V. When three Ms are propls. the 1st is said to hare to the 3rd the duplicate B. of that TChich it has to the 2nd. 1, VI. Hyp. I. „ 2. Def. 12, V. Cono. Let AABObesim. to At)EF, & zB = zB; & let AB.: BO = DB : EF, the side BC being homol. to EP ; then A ABC : a DEF = BC^ : EF^. C. 1 D. 1 2 3 4 5 6 7 10 11, VI. H.2.16,V. G. 11, V. So.l,16,VI 15, VI. C. Def. 10, V. 1, VI. D. 6, 7. D. 4. Take BG a 3rd propl. to BC & EP, i.e., BC : EP = EP : BG ; and join GA. •.• AB : BC = DE : EP ; .-. altem. AB : DE = BC : EP ; but BC: EP = EP: BG; .-, AB: DE= EP: BG; .•. the sides about the eq. Z_s. are recip. propl., .-, A ABG = A DBF. And V BO: EP= EF: BG; .-. BO : BG has the duplicate B of BC : EP;, but BC : EF or BG = A ABC : a ABG ; .-. A ABC : A ABG = BC« : EF^ ; but A ABG= A DBF; .-, A ABC : A DBF = BC^ : Bps. Therefore, Similar triangles are ^c. Q. E. D. CoR. If three st. lines are proportionals, as the first is to the third, so is any > triangle itpon the first to a similar and similarly described triangle upon the second. . SchI. The duplicate ratio of two st. lines is the same with- the. ratio of their squares ; thus similar triangles are to one another as the squares of their homologous sides. 330 GKADATIONS IN EUOLID. II. This Proposition might be deduced as a particular case of Prop. 1 , VI, by the 2nd Cor. " that triangles are to one another in the ratio compounded of the ratios of their bases and altitttdes ;" for, when triangles are similar, their altitudes are in the proportion of their bases, and by Def. 12, Vi this compound ratio is the duplicate ratio of their bases. Use and Appl. L The 19th Prop, supplies a general method of reasoning oonc'eming all similar geometrical figures ; for we declare from it that the areas c/timilar triangles are to one another as the squares/or the corresponding sides. E. 1 2 3 D. 1 2 3 Hyp. 1. Cor. 3,32,1. 4, VI. Use 2, 16, VI. Use 2, 16, VI. 6t Seh. 1. 19, VI. Sim. Let As ABC, AEF be similar ; & AD L BC; and AD produced, or AG,J.EFi & AS ADB, AGE equiangular, then Area of A ABC : Area of A AEF = BC : EE», or AB= : AE=; ■.• AD : AG = BC : EF ; & BC : EF = BC : EF ; on multiplying these equals together. AD . BC : AG. EF = BC» : EF'. But AD . BC = Area of /ZZ7 on BC with alt. AD ; and AG . EF = Area of /CZ7 on EF „ AG. And Area of As = J Area of r~7 s on the same base and altitude, .-, Ai-ea of A ABC : Area of A AEF = BC" : EF^ Thus are the Areas as the squares of any other pair of cor- responding sides; tor Area of A ABC : Area of A AEF = AB« : AE». Ex. 1. From a given A AEF to cut offa part ABC = J of aAEF in area, by a line BC II EF. Since, Area aAEF : Area aABC = EF" : BC« ; or 1 : i = EF» : BC, ,.BC= = lI%rBC = ^- Ex. 2. If the Area of a triangle be 36000 sq. feet, of which the base is 900 feet, what wiU be the area of a similar triangle of which the base is 450 feet. Take x as representative of the unknown. area ; ■/ 36000 ; x= 900' : 450's /, 810000 x = 7290000000i and X = 7290000000 -~ 810000 = 9000 sq. feet. Ex, 3. The base EF, of a triangle is 3240 Unks, and its perpendicular 840 links; required the distance, from the base, of the parallel line, BC, whjcb shall divide the biangle into two equal paits. PROP. XX.— BOOK VI. 331 I Here, Area of A AEF = ^^ . "9^?*° _ 272I6OO sq. links; andArea of A ABC = i^?Mi°— 1360800 „ „ Now AG- AD = DG distance of parallels BC & EF; To fiad AD; Area A AEF : Area A ABC = AG' : AD', T\,c„ Ana 1360800X840= „,„„„„ ^'"'"'^ = 2721600 =352800; .-, AD = v/352800 = 593 . 978 links. And 840-593.978 = 246.022 = DG, distance of parallel BC from El\ II. When in similar triangles, any side of the one, as AE, is double that of the other, the area of the one triangle is four times that of the other; for the Areas are as the squares of the sides; and 2' : 1^ = 4 : 1. Pkop. 20. — Theor. Similar i'olygons may he divided into the same number of similar triangles, having the same ratio to one another that the polygons have, and the polygons have to one another, the duplicate ratio of that which their homologous sides have. Similar polygons are divisible into similar triangles equal in number and homologous to the whole polygons ; and the polygon has to the polygon the duplicate ratio which the homologous side has- to the homologous side." — Euclid. Con, Pst. 1. 1. Dem. Def. 1, VI. 6, VI. 4, VI. 32, 1. Ax. 3, 1. 22, V. 11 V. 12, V. If any number of Ms be propls, as one of the antece- dents is to its consequent, so shall all the antecedents taken together be to all the consequents. 19, VI. E. 1 2 Hyp. 1. „ 2. Cone. 1. Let pol. AD be similar to pol. FI, and the Zs at A, B, C,D, E respectirely = / s atP, G,H,I, L; and let the sides AB, BO, CD.,DE, BA be respectively homol. to the sides FG, GH, HI, IL, LF; • then pols. AD and FI, are divisible into a s of which each contains the same number of sim, as; 332 GRADATIONS IN BtTCLID. 2. 3. of these sim. a s each has to each the same ratio as the pol. AD to the pol. FI. and pol. AD : pol FI has the duplicate ratio of the E, AB : FG. C. D. 1 2 3 4 5 6 7 9 10 11 12 13 Pst. 1, I. Join E with the Z. . s B, C ; and L with the Z. ; s G,H. Cask I. The pols. AD & FI are diyisible into L Sj &c. H.l,Def.l,VI. H. 1, 2. 6, VI. 4, VI. Hyp. Def. 1, VI. 32, 1. Ax. 3. D. 3. 4, VI. Def. 1, VI. 22, V. D9. 6, VI. 4, VI. Sim. Cone. •.• pol. AD is sim. to pol FI /. /. BAE= ZGFL, & BA : AE = GF : FL; And •.• A s ABE, FGL have Z A = Z F, and their sides proportionals ; .". A ABE is eq. ang. & sim. to A FGL, .-. zABE= ZFGL. And •/ pol. AD is sim. to pol. FI, .-. whole ZABG = whole ZFGH; & .-, rem. ZEBG = rem. zLGH. And •/ A ABE is sim to A FGL, .-. EB : BA = LG : GF ; Also V of sim. pols. ,-. AB:BC=FG:GH; .-. ex. ceq. EB : BC = LG ; GH. And '." the sides about eq. Z s are propls, .•. A EBC is eq. ang. & sim to A LGH. So aECD is eq. ang. and sim to aLHI. .■. Sim. pols. AD, FI are divided into the same number of sim. a s. Cask IL Also a ABE : a FGL A EBC: A LGH aECD: aIHL = pol. AD : pol. FL PEOP. XX. — BOOK VI. 333 Case III, And pol. AD : pel. PI has the duplicate K of AB : FG. D. 1 2 3 4 ,5 6 7 8 10 11 12 OaseI.19,VI Sim. 11, V. CaseI.19,VI Sim. 11, V. D. 3. 12, V. 19, VI. Eec. ••. A ABB is sim. to a FGL, .-. A ABE : A FGL = BE^ ; GL^ So A BEG : aGLH = = BE^ : GL^. .-. a ABE : A FGL = A BEG : a GLH. Again •.• a EBC is sim. to A LGH, .-. A EEC : A LGH = EC^ : LH^; So A EGD : A LHI = EC« ; LHs, .-. A EBC : A LGH = A ECD : A LHI; but A EBC : A LGH = a ABE : A FGL, .-. AS ABE : FGL = EBC : LGH = ECD : LHI ; .'. antec. : conseq. = all the antecs. : all the conseqs. i. e. A ABE : A FGL = pol. AD : pol. FI. But A ABE : A FGL = AB^ : FG«, .-. pol. AD : pol. FI = AB2 : FG«. Wherefore, Similar Polygons ^o. Q. E. D. CoR. I. Similar figures of four sides, or of any number of sides, as abeady proved of triangles 19, VI. ; are to one another in the duplicate ratio of their homologous sides. CoR. II. If three lines ie proportionals, then, as was proved for triangles, Cor. 19, VI., the first shall he to the third as any polygon on the first is to the similar, and similarly described polygon on the second. C. D. 1 2 3 4 11, VL Def, 10. V. Cor. 1,20, VI. 11, V. Bee, To AB, FG two homoi sides take M, a third proportional. Then AB : M has the dupl. E of AB : FG ; but pol. AD : pol. FI = AB2 : FG^; ,-. AB : M = fig. on AB : fig. on FG. •. If three lines he proportionals, ^c. Q.E.D. 334 GRADATIONS IN EUCLID. Cob. III. Because all squares are similar figures, the ratio of any two squares to one another is the same with the duplicate ratio of the sides; and hence also, oray two similar rectilineal figures are to one another as the squares of the homologous sides. CoK. IV. In similar figures, their perimeters are to one another as the ratio of the homologous sides. :• AB,BC &c. : FG,GH &c. = AB : PG =BC : GH ; .-. AB + BO + CD &c. : FG + GH + HI &c. = AB : FG. «. e., Perim. of fig. AD ; perim. of fig. PI = AB : PG. CoR. V. The homologous diagonals being sides of similar triangles are also homologous sides ; and theretoie, perimeters of simi- lar figures are as their homologous diagonals. CoR, VI. A Circle and its inscribed Polygoirof an infinite number of sides do not differ from each other in any degree however small, but that a smaller difference might be assigned ; and practically, they may be considered of identical values. As there- fore it has been predicated of similar polygons, 20, VI, that "they are to each other as the squares of their corresponding sides," whether diagonals or sides ; — so it may be predicated of all circles, that being similar figures, they are to each other as the squares of their respective diameters, radii, and circumferences. Cor. VII. If onthethree sides, AB, AG, CB, of art. Z_d triangle ACB, similar figures be described as semicircles, AeCg-B, AdC and C/ B ; the figure on the hypotenuse,, AB, will be equal to the sum of the similar figures on the two other sides, AC, CB. D. ] Cor. 6, 20, VI. 47,1. • Semicircles, or any sim. fig., on the three sides, are as the squares on those three sides ; & •.•AB2 = AC= + CB2, . Semicircle on AB ^= Semic. on AC + Semic. p&op. xt. — BOOK VI. 335 HipPOCTUTES, of Chios, a Pythagorean philosopher, who lived about 460 B.C., is said to have made this appUoatiou of the universal principle, tyhicn under different forms appears in 47, I., and 81, VI., that "if three similar figures be described upon the sides of a rt. angled triangle, the contents of that which is described upon the hypotenuse will be equal to the sum of the contents of the figures described upon the sides," Hence, was deduced, CoR. VIII. That the tunes Ae Cd, Cg B/, formed by- describing semicircles, as Ae Cg' B, A d & C/B, on the sides of art. /_i triangle, are equal in area to the right angled triangle ACB. D. 1 2 3 4 5 G Cor. 7. 20, YI. Sum. Sum. Ax. 4, VI. Semic. on AB = semic. on AC+ semic. onCB; from semic. on CB take away the shaded seg- ments A e C & C g' B ; The remainder is the rt. /.d A ACB ; From semicircles on AB, CB take away the same segments ; The remainders are the two lunes AeOd, B^C/; .-. the Lunes AeCd + BgCf = A ACB." N. B. This is the first known instance in which a curvilinear space ivas reduced to an equivalent rectilinear space. ScH. 1. Further elucidation of Prop. .20, will be found iaLAKDUEB's Euclid, pp. 210—214, and in pp. 127, 134 of this wofk. 2. In circles, which are all similar,, when the segments are similar, the radii and the chords become homologous sides. Use & App. I. By this proposition, «• rt. lined figure may be increased or diminished in any ratio Thus, to make a pentagon five times the size of the pent, on DC, (in last figure but one.) ; Take a side AE = 2; and bet ween AE and ,5 times AE, i. e., 10, find the mean proportional;— it is v'2x 10 = ^W — 4.472136; A Bim. fig. on a side of 4,472136 will be fire times the pent, on AE. 336 GRADATIONS IN EUCLID. II. When the homologous sides are known the Proportion qf one figure to another will be obtained by finding a third proportional to any two con-esponding sides. III. The increase or diminution of Circles is effected in the same way. Thus one circle is on a diam. of 1,— another is to be constructed 4 times larger; required the diam. of the second circle. The mean propL to 1 & 4 is s/lx4 = ^ i z= 2. .". the diam. is 2 for the required circle. Or, Given the diameters 1 & 2, required the magnitude of the second circle when compared with the first. Here, z2L- = _— = 4, — the second circle is four times larger than the first. rV. In two similar figures, if, of the areas and corresponding sides, any three be given, the fourth may readily be found. The principle employed is, that the areas of similar figures are to one another as the squares of their homologous sides, and vice versa. , Tor, (fig. 20, VI.) A A3E : A FGL = AB= : FG=; A BCE : A GHL = BC* : GBP; and A CDE : A HIL = CD» : HI^. Hence, As ABB + BCE + CDE : As FGL + GHL + HIL = AB' : FG^. i.e. fig. ABODE : fig. EGHIL = AB^ : FG=;. _,, „„, FGHILxAB^ .-r., _ ABCDExEG' . T^^^^^ = -ABODE— ' ^^ IGIHL .^^^_ FGHI LxAB' Area ABCDB = ^^^^ ; ABODExFG' and Area FGHIL = AB2 Ex. 1. Two similar hexagons are respectiyely of the Areas of 2500 sq. yards and 3000 sq. yards; a side of the first is 5 lineal yards; required the length of the corresponding side of the other hexagon. Here, the mJmown side x = ./ 3600 x 5X5 = ^/ 36 = 6 Uneal yards. 2500 Ex. 2. One of the sides of the base of a pyramid measures 50 yards, and the area'of the base 7500 sq. yards;— in a similar pyramid, what is the area of the base when the corresponding side is 60 yards ? rr^ . 7500 X 60X60 _ 270000 ,„^„„ , The Area x = — ^q ^ ^q -^^ = 10800 sq. yards. PKOP. XXI.— BOOK VI. 337 Prop. 21.— Theoe. Rectilineal figures which are similar to the same rectilineal figure, are also similar to one another. Dem. Def 1, VI. Ax. 1,1. U, V. E. 1 2 D. 1 Hyp. Cone. H.Def.l,VI H.Def.l,VI Ax. 1, I. 11, V. Def. 1, VI. Rec. Let figures A & B be each sim. to fig. C. then fig. A is sim. to fig. B, •.• A is sim. to C, .•. A is eq. ang. with C, & their homol. sides are proportionals. '.• B is sim. to C, /. B is eq. ang. with C, arid their homol. sides, propls. .•. A & B are each eq. ang. with 0, and their homol. sides propls. .•. recti, fig. A is sim. to recti, fig. B. • Therefore, Bectilinealfi^ures which are similar, ^e. Q. E. D. ScH. This proposition follows evidently from Def. 1, VI., of similar recti- lineal figures; it is equivalent to an Axiom, and in this respect agrees with Prop. 30, 1. St. Lines parallel to the same st. line are parallel to each other. „ 11, VI. Ratios that are the same to the same ratio are the same to one another; variations of the General Principle,— Things equal to the same thing are equal to one another. 338 GRADATIONS IN EUCLID. Prop. 22.— Theor. If four St. lines he proportionals, the similar rectilineal figures sirmlarly described upon them shall also he proportionals ; and coa- yer|,ely, if the similar rectilineal figures similarly described upon four St. lines he proportionals, those st. lines shall also be propor- tionals. Con. 11, VI. 12, VI. 18, VI. Deji. U, V. 22, V. Cor. 2, 20, VI. 9, V. Magnitudes which hare the same E to the same M are eq. to one another; and those to which the same M have the same E are eq. to one another. 7, V. Eq. Ms have the same E to the same M ; and the same has the same E. to eq. Ms. Case I. Let four st. lines be proportionals, &c. Let AB : CD = JP EF : GH ; on AB & Crriet sim. rect. figs. KAB &LCD, be similarly described; ^i^ " ^ <;" & on EF & GH sim. rect. figs. MF & NH, be also similarly described ; then KAB: LCD = MF : NH. ^J E. 1 Hyp. 1. 2 „ 2. 3 „ 3. 4 Cone. C. 11, VL D. 1 2 3 H. 1. 11, V. 22, V. Cor. 2, 20, VI. 11, V. To AB, CD take a 3rd propl. X, & to EP,GH a 3rd propl. Q. •.• AB : CD = EF : GH, .-. CD: X= GH: O; •. ex esq. AB : X = EF : 0. but AB : X = fig, -KAB : fig. LCD ; and EF : O = fig. MF : fig. NH ; .-. fig. KAB : fig LCD = fig. MF : fig. NH. PROP. XXII. BOOK VI. 339 Case II. If the sim. rectilineal, figures, ^c. E. 1 2 Hyp. Cone. C. 1 12, VI. 2 18, VI. D. 1 C. 1. 2 C. 2. 3 C. 2. 4 5 6 Hyp. 9, V. 7 C. 2. 8 C. 1. D. 7 9 7, V. 10 Eec. Let fig. KAB : fig. LCD = fig. MP : fig. NH; thenAB: CD = EF: GH. Make AB : CD = EF : PE, and on PR descr. fig. SR sim. and similarly situated to MP or NH. •.• AB: CD = EP: PR; and on AB, CD are sim. and sim. desc. figs. KAB, LCD ; and on EP, PR „ „ figs. MP and 8R. .-. figs. KAB : LCD = MF : SR. But KAB : LCD = MF : NH; .-.rect. fig. NH = recti, fig, SR. and •.• they are sim. and sim. situated, .-. GH = PR. And •.• AB : CD = EP : PR, and PR = GH; .-. AB : CD = EP : GH. If therefore four st. lines be proportionals, ^c. Q .E. D. Cor. As a particular case, if four st. lines A,B,C,D, be pro- portionals their squares A^, B^, C^, D", shall also be proportionals ; and conTersely ; i.e., if A : B = C : D; then A^ : B^ = C^ : D^; and if A^ : B^ = C^ : D^; then A : B =.C : D. ScH. Thn8, this proposition ie equivalent to the theoreip, that, "if two ratios be equal, their duplicates and subduplicates will also be equal," or in other words their powers and roots. Use & App. The principle contained in this,Theorem is often employed in Arithmetic and Algebra; for if o 3 : 4 4 z= c6 : dS ; then a^9 : li" 16 =: <;' 36 : d'6i. If four quantities or numbers are in proportion, their like powers or roots are c' 340 GRADATIONS IN EUCLID Lemma. — Theor. If rectilineal figures he equal and similar their homologous sides are equal. Dem. 16, V. 20, VI. E. 1 2 3 D. 1 2 Hyp. 1. Cone. Sup. H. 2.16, V. D. 1.20, VI H. 1. Cone. Let fig. AC be =& sim. to fig. DF & let BC : BA = EF : ED ; ^-^G then side EF = side BO. .r-""^^; Di If EF ^ BC, let EF > BC. V EP: ED = BC : BA; .-. altern. EF : BO =ED : BA ; bntEF>BO .-. ED>BA; .-. fig.DF>fig. AC; nowfig, DFalso = fig. AC; — ^which is impossible ; .-. EP not # BC ; i. e., EF = BC. Q. E. D. Prop. 23.— Theok. Equiangular parallelograms have to one another the ratio which is compounded of the ratio of their sides. Con. 14, 1. 31, L 12, VI. Dem. Def. A. V. of Compound Eatio. When there are any no. of Ms of the same kind, the 1st is said to hare to the last of them the K compounded of the R. which the 1st has to the 2nd, and of the E which the 2nd has to the 3rd, and of the E. which the 3rd has to the 4th, and so on imto the last magnitude. 1, VI. 11, V. 23, Y. PROP XXIII. BOOK VI. 341 E. 1 Hyp.' 2 Cone. 0. 1 Pon. 14,1. 2 31,1. 3 12, VI. D. 1 C. 1., 2 3 4 5 6 D f.A.V. 1,VI, C. 3. 11, V. C. 3 7 8 9 10 11 11, V. D.5&7, 22, V. Eemk. 12 Eec. Let CHJ AC be eq. ang. to i — 7 CF, having ZBCD = ZECG; then I 7 AC, : / 7 CP = E compounded of the ratio of the sides, BC : CG & DO : CE. Place BC & CG in a st. line and also DC & CE ; complete the / 7 DG; and take a st. line K; make BC : CG = K : L ; & DC : CE = L : M. A 8 » BE 4 B C .5 G • 7 K X.M E iP Now K : L & L : M = BC : CG & DC ; CE ; but K : M is compounded of K : L & L : M ; .•. K : M is a E compounded of the E of the sides. And •.• BC : CG = ^Z7 AC : ,^=7 CH, but BC : GC = K : L; .-. K : L = £ZI7 AC : ^=7 CH. Again •.• DC : CE = £1=7 OH : ^117 CF, but DC : CE = L : M; ■•■ L : M = r^ OH : ^^ CF. And V K: L = C=7 AC -.zZZ? OH; .-. ea;. ffi?. K : M = C=7 AC : CZ7 OF. But K : M is compounded of the E of the sides ; .•. I 7 AC : I 7 OF is compounded of the Es of the sides ; , i. e. of BO : CG and DC : CE. •. Equiangular parallelograms have ^c. I Q. E. D. N, B. This 23rd proposition would follow as a Corollary from the Theorem, " that any two rectangles are to one another in the ratio which is compounded of the ratios of the sides." A more brief .demonstration would he, E. 0. Hyp. 31, I. Let AC & CF be two eq. ang. parallelograms ; Complete the / 7 OH. 342 GRADATIONS IN EUCLID. D. 1 2 3 Def.A,V, Cone. *.• I 7 AC : I 7 CF has the E compoimded of the ratio of AC : CH, and of CH : CF ; but z::^ AC : CZ7 CH = BC : CG, and i^7 CH : ,CZ7 CF = DC : CE ; .". I 7 AC : I 7 CF is a ratio compounded of Rs which are the same with the Rs of the sides. Arith. Hyp. Let BC = 8, CG = 5, DC = 4j and CE = 7. then 4:7 = 5: 8.75; and of the three numbers 8, 5, & 8.75, 8:5 = OU 4X8 : / 7 4X5; ■ and 5 : 8.75 = / 7 4x5 : I 7 4 X 8.75; thns 8 : 8.75 is the ratio compounded of both; or 32 : 35. Cor. 1. If the terms of two analogies are lines, the rectangles under their corresponding terms are proportional. CoR. 2. Hence, also. Rectangles whose bases are proportional and also their altitudes, are themselves proportionals. Use & App. To describe a rhombus equal to a given rectilineal figure R, and having an angle equal to a qiven angle L. C. 1 45,1. 23, I. Pst. 2 I. 3, I. Pat. 3 ,1 11,1. 3,1. Sol. Draw / 7 AC = given figure E. and with /.BAD = given /.L; Produce BA, DA, indefinitely to G & F; Make AF = AD; and on BF desc. a semicircle; from A raise a perp. AE cutting the semicircle in E. make AG, AH each = AE; and AG, AH are sides of the required rhombus. PROP XXIV. — rBOOK VI 343 D. 1 2 3 4 5 6 C. 1, 2. &c. 23, VI. Kemk. 17, VI. Bemk. Cone. *.' the I 7 b AC, AK are equiangular; .•. I — 7 AC : CZJ AK has a ratio compounded of the ratio of the sides; Thus AB : AH or AE = AG or AE : AD; .-. AG^ == AB . AD. Now AG is a side of the rhoinbns AE; .*, tiie rhombus AK = the rhomboid AC. Q. E. J". Prop. 24.^ — ThSior. ; Pdrallelograms about the diameter of any parallelogram,, are similar to the whole and to one another. Dem. 29, 1. If a St. line fall upon two f] st. lines, it mates the alternate /_s =. io one another; and the ext. /_ = the int. and opp. /. upoli the same side; and likewise the two int. /.s upon the same side = 2 rt. Zs. 4- VI. 34, I. The opp. sides and /.s of' / 7 s are eq. to one another, and the diam. bisects them. 7. V. Def. 1, VI. 21, VI. Eecil. figures sim. to the same, are similar to each other. E. 1 2 D. 1 2 3 4 Hyp. 1. Cone. H. 29, I. H. 29, 1. 34, I Let BD be a I — 7 of whick AC is the diam ; & EH, GF £:^7s about the diam.-; then I — 7 EH is sim. to ',-it^GF; & each sim. to / 7 ED. •.• BC II EP, .•./ABC=ZAEF; & •.• DC II HK, or KG, .-. ZADC=ZAHK; And v.zDOB = ZBAD, & zHKE = zBAD; ,-, ZDGB = HKE = zBAD; 344 GRADATIONS IN EUCLID. 10 11 12 13 14 Cone. D. 2. 4, VI. 34,1. 7,V. Def.l,VI Sim. 21, VI. Kec. '. I 7 s BD & EH are equiangular. Aod V ZADC= zAHK,&zDACcom.; .•. A DAG is eq. ang. to aHAK; & .-. AD : DO = AS : HK. And •/ opp. sides of / — 7 s are equal ; .-. AD : AB = AH : AE ; BO: CD = EK: KH; & BO : BA = KE : EA; .-. sides of dJs BD, & EH about eq Z s are proportionals. ; & ory BD sim. to CZD EH. So czj BD is sim to CZJ GF. •• on& EH & GP are each sim to ^zyBD ; .•. dj EH is sim to IZU GF. WkextiovQ parallelograms about, ^c. Q. E. D. ScH. Proposition 24 should have changed places with Prop. 25. Use & App. In Perspective this proposition is available to show that a cop7 is drawn like the original, by the help of a parallelogram. Prop. 25. — Prob. ( Of extensive use.') To describe a rectilineal figure which shall be similar to one and equal to another given rectilineal figure, i. e. in area. ' Con. Cor. 45, L To desc. a I 7 ee^ual to a given rectil. fig. and having an /_ equal to a given rectil. /.• 29, 1. 14, 1. 13, VI. 18, VI. Dem. Cor. 2, 20, VI. ' 1, VI. 11, V. 14, V. If the 1st has the same B to the 2nd which the 3rd has to the 4th; then, if the Ist be > = or < the 3rd, the 2nd shall be > = or < the 4th. E. 1 Data. 2 Quaes. I Given rectil. fig. ABC, "and f ectil. fig. D I to desc. a recti, fisr. sim. to ABO. and eo ea. to D. PEOP. XXV. BOOK YI. 345 C. 1 2 4 5 6 D. 1 2 3 4 Cor. 45, I. 29, 1. 14, 1. 13, YI. 18, VI. Sol. 0.4. Cor.2.20,VI 1,VI. 11, V. 0. 1.14,V, 0. 7 Sol. On BO descr. ,CI7 BE = fig. ABC; and on CE, / — 7 OM = fig. D, B and having Z FOE = Z CBL. f Now BO, OP are ^ in a st. line BF, and LE, EM, in st. line LM ; find then GH a mean propl. to BC, and OF ; and on GH descr. a recti, fig. KGH, sim. to ABO, and similarly situated, then the fig. KGH is the recti- fig. required. •.• BO : GH = GH : OF ; .-, BO : OF = fig. ABO : fig KGH; EF; BE : criEF. but BO : OF = ^^ BE .-. fig. ABO : fig. KGH, = , And •.• fig. ABO = OH BE, .-, fig.MKGH = ,C=7EF; but fig. EF = fig. D, .-. KGH = D and is sim. to ABO ; Therefore fig. KGH is sim. to ABO and equal to D. Q. E. P. Or, varying the figures. 1, 14, n. 2 12, VI. al 18, VI. Find L,M, the sides of squares eq. to the figures ABC & D ; take GHa 4th propl. to L, M, & BC; and on GH desc. the fig. KGH sim. to ABC. D. 1 2 3 4 20, VI. C. 1. C. 3. Cone. ■.' fig. ABC : fig. KGH = BC^ : GH» ; i. e., as L« : M'; and ■.■ fig. ABC = 1?, and fig. KGH = M^ = fig. D; and *.• also KGH is sim. to fig. ABC. .•, KGH is the fig. required. 346 GRADATIONS IN EUCLID. SoH. The chief point of this problem is to find, as in the last fig. but one, a mean proportional to BC, a side of the given figure to which a similar one is to be constructed, and to CF, a side of the rectangle made equal to fig. D; — that mean proportional is GH, which becomes the side of a figure similar to ABC, and equal to D. Use & App. By this proposition, while we keep always the same area, we may change the form or shape of the figure, — a procisss of great use in- Practical Geometry; and especially convenient, if we do, what has very often to be done; i.e., reduce an irregular rectilineal figure to its equivalent square. Prop. 26.— Theor. If two similar parallelograms have a common angle, and be similarly situated ; they are about the same diameter ; "If from a parallelogram a parallelogram be taken away similar to the whole, and similarly placed and having a common angle vrith it, — it is about the same diameter with the whole."— Etjolid. CoK. Pst. 2, 1. 31, 1. Dem. 24, VI. Def. l,Vt 11, V. 9, V. Ax. 9, 1- E. 1 C. 1 2 3 4 Hyp. 1. Cone. Sup. 1. „ 2. Pst. 2, 1, 31, I. Let £Z7 BD be sim. to dZJ EG, having Z A com.; & let AE, AG, sides of CTJ EG fall on AB,AD, homologous sides of CZJ BD; '4- then diags. AF,AO are in "Kl '^"•;-|h! one line, Ef i. e. f — n BD & EG are about the same diam. : > B I. If not, and it be possible, let AHC be the diag. of czn BD ; & AHC be different from AP the diag. of I — 7 EG; let GE meet AHC in H, & hiough . H draw HK || AD or BC. PROP. XXVII. BOOK VI. 347 D. 1 2 3 4 5 6 0. 1—4. 24, VI. Bef.l,VI. H. Def. 1, VI. 11, V. 9,V.Ax.9 Cone. Eec. ".' /~~7 "s BD & KG are about the same diag. AHC; .*. I — 7 KG is sim. to / — 7 BD ; & DA : AB = GA : AK ; but •.• I — 7 BD is sim. to / — 7 EG, .-. DA : AB = GA : AE ; .-. GA : AE = GA : AK; & .*, AK = AEj — ^which is impossible: .•, diag. of I — 7 BD through, £_ A cannot be otherwise than on AF, i. e., AP & AC are in the same st- line. Therefore, If two similar parallelograms Sfc. Q. E. D ScH. — This Proposition is the converse of Prop. 24, and properly should follow it immediately, or be incorporated with it. Prop. 27.— Theok. Of all parallelograms applied to the same St. line, and deficient by parallelograms similar and similarly situated to that which is described upon the half of the line ; that which is applied to the half and is similar to the defect is the greatest. N. B. More easily to understand this and th4^two following propositions, attention must be given to the Subsidiary Definitions D,E, & F. p. 251, 2. Con. Pst. 1. 1. 31, 1. Dbm. 43, 1. The complements of the / — 7 s which are about the diam. of any 7 7 . are eq. to one another. 26. VI. \ Axs. 1, 2, 9. 1. 34,1. 36,1. £l^s upon &[. bases, and between the same {|6 are eq. Ax, 4, 1. 348 GRADATIONS IN EUCLID. E. 1 Hyp. 1. 2 „ 2. 3 Cone. 1. 4 Hyp. 3. 5 Cone 2. M G 3> I> JI A^ V \ c K B J9E ^ Let the st. line AB be bisd. in C ; & on the half AC let a /— 7 AD be applied, deficient from / — 7 AB on the whole line AB, by the / — 7 CE on the other half line CB; then, of all / — 7 s applied to any other pts. of AB, deficient by / — 7 8 sim. &similarly situated to / — 7 CE, / — 7 AD shall be the greatest. Let any / — 7 AF be applied to AK, a pt. of AB and ^ AC or CB, so as to be deficient from ^^ AH on AB by z=j KH,— / — 7 KH being sim. and similarly situated to / — 7 CE; then r-^ AD on AC is > £—7 AF. AB; O. D. 1 2 Case I. Suppose AK, Sas« 0/ ZTIZ AF, > AC the half of and •.• by Hyp. 8, dj CE is sim, to ZZZ7 HK, .". by 26, VI., they are about the same diam. DB. 31, L 43,1. Ax 2, I. H. 1. Ax. 1 Add. Ax. 2, Ax. 9. Case II. Let AK, base of C. D. 31, L H. 1. 34, L 36, L Ax. 9. 43, L Add. Ax. 4. Draw diag. DB, and complete the scheme. •/ £Zy CF = /ZZ7 FE, to each add dJ KH, .-. the whole dJ CH = the whole EZ7 KE ; But ■.• AC = CB, .-. £Z:7 CH = ^=7 CG = KE ; to each add CF, •.• £ZI/ AF = gnomon CHL ; .-. CU CE or AD > IZZ? AF. AF, be < AC, the half of AB. ■(Sf -r \ A similar construction is to be made. V«C = CA, .•;HM = MG; and •.• CTJ DH = ^C^DG. .-. Cd/'DB.> dn LG. but dD DH == dJ DK, and dr/ DK > /d7 LG; to each ad /ZZ7 AL, .-. dZ/ AD > dJ AF I ■kS- PEOP. XXVII.— BOOK VI. 349 D. 5 6 Cone. Eec. Thus in both cases r — 7 AD > i — 7 AF. Therefore, of all l 7 s applied to AB, and deficient hy I i s each sim. and similarly situated to AB, the 00 AD is the greatest. Q. E. D. Otherwise the Proposition may be enunciated ; — Of all the rectangles contained hy the segments of a given st. line, the greatest is the square which is described on half the IvKe. E. 1 D. 1 2 3 Hyp. Cone. H. 5,11. Cone. c — *- — K— Given AB bisected in and unequally j^ divided in D ; then AC2 > |ZI] AD . DB. •.• AB is div. equally in C and uneq. in D ; .-. AD . DB + CD2 = AC2 .-. A.& > AD . DB. B ScH. De Chales, Labdnek, and some other geometricians recommendt that Propositions 27, 28, and 29 should be omitted as unnecessary ; hut they' were frequently employed by the ancient mathematicians, and are required es- pecially for the solution of several problems. Use and App. — In a given aABZ to inscribe the greatest parallelogram possible, having an angle, A, in common ivith the triangle. C. 1 31,1. 2 10,1. 3 31, I. 4 *) 5 SoL Complete the / 7 AE of which LB is the diag. bis. AB in C, AC in K ; AL in N and LN in (J ; and through . s K, C, I, draw ||s to AIi, and through . s G, N, P draw ||s to AB. then paraUelog. AD, with /_A com. tp aABL, is the greatest parallelog. possible in aABL. ll B a ^ ^.«. jfl N ^^D M P l^iJ- «„ -s^|VJ ^ L 1 t C 3 i B 350 GRADATIONS IN EnCLID. D. 1 24, VI. 26, VI. C. 2-4. 27, VI. *.' the defects of parallelogs AO, AD &c. are parallelogs. sim. to parallelog. CM ; .'. those parallelogs. are about the same diag. BL ; .". pargs. AO, AD, AJP &c. are inscribed in the given A ABL; And *.* parallelog. AD is described on AC half the base, .". parallelog. AD is the greatest parallelog. in the A ABL, and the /_A is com. to the parallelog. & the A. Q. E. F. Prop. 28.— Peob. To a given st. line to apply a parallelogram equal to a given rectilineal figure, and deficient by a parallelogram similar to a given parallelogram ; but the given rectilineal Jigure to which the parallelo- gram to be applied is to be equal, must not be greater than the parallelogram applied to half of the given line, having its defect similar to the defect of that which is to be applied; that is, to the given parallelogram. " To given rt. line to apply a parallelogram eqnal to a given rectilineal figm'e and deficient by a figure similar to the given parallelogram." — Euclid. COH. 10, I. To bis. a given finite st. line, 18, VL 31, 1. 25, VI. 3, 1. Dem. 21, VI. 26, VI. Ax. 3. 43, 1. 36, I. Ax. 1. 24, VI. o r L M H G. Tl ^.. ^\\ fR K"""'N E SB .0 PROP. XXVIII.— HOOK VI. 351 E. 1 Dat. 1. 2 „ 2. 3 „ 3. 4 Qusesi 0. 1 2 10,1. 18, VI. 31, I. Let AB be the given st. line, and C the given fig. to which the parallelogram is to be equal; let fig. be > the on-o^ the half line AE, of which the defect is sim. to the defect of that / 7 which is to be applied ; and let fig. D be the / 7 to which this defecfr is to be similar. To apply to AB a I — 7 = C, and deficient from the I 7 on the line AB, by a / 7 sim. to fig. D. Bis. AB in E ; and on EB desc. I — 7 EP sim. and similarly situated to fig. D ; complete the / 7 AG, either = C, or > C. Case I. .Suppose IZZI AG =fig. G ;' D. I Oonc. I then on AB k applied / 7 'A(t = fig. C, I I and deficiently [ZJ EF sim to fig. D. Q.E.D. Case II. Suppose IZZI AG ^ fig. C, but >C ; and by 36, I, CUW =CZJ AG, & .-. [ZD EP.also > fig. 0. c. 25, VI. D. 1 C.1.21,VI 2 3 4 3 Sup. C. 3. 3,1.31,1. 6 C. D. 1. 7 8 9 26, VI. Sup. 31, 1. 10 D. 3, 6. Make / — 7 KM = excess of / — 7 EF above fig. C, & sim. and simly situated to fig. D. •.• fig D is sim. to £ZZ7 EF, .-. on KM is sim. to/ — 7 EP. Let KL be homol. to EG, & LM to GF. And '.• ^^ EP = + KM, .-. EF > KM, .-. Hne EG > KL, & GP > LM. Make GX = LK, GO = LM; and complete / 7 XO. •.• /—? XO is = & sim. to £ZJ KM, & KM sim. toCZjEP, .-. CZJ XO is sim. to ,CZ7EF, & .-. CZS XO & EP both are about diag. GB. Let GPB be their diag. ; and complete the scheme, then •.• ZZZ7 EF = C + KM, and a part / — 7 XO = / — 7 KM a part of the other ; 352 GRADATIONS IN EUCLID. D.ll 12 Ax. 3. 43, I. 13 14 C. 1,36,1. Ax.l. Add. 15 16 D.ll. Eec. 24, VI. .•. rem. gnomon EEO= rem. C. And .-.OZ/ OE = parlm. XS, /.on addingZIIjSR to each, / — 7 OB = parlm. XB ; but AE = EB; .-. parlm. XB = parlm. TE & parlm. TE = parlm. OB ; to each add parlm. XS ; /. the whole CZl TS = the gnomon EEO; but EEO = fig. C ; .-. also parlm. TS = fig. C. *. parlm, TS = fig. C is applied to st. line AB, deficient by parlm. SB sim. to fig. D, because SR is sim,. to EF. Q. E. F. SoH. 1. The Proposition may be thus enunciated ; To divide a given at line A B, so that the rectangle contained by the segments may be equal to a given space, as the square onC; but that given space must not be- greater than the square of half the given line. E. 1 Dat. Quffls. C. D. 1 2 3 C. 1 2 3 4 D. 1 2 ,3 4 5 10,1 Sup. 1. Sup. 2. Sim. 3,1.11,1 3,1. Pst.3&l,I Sol. C. 5, n. 47,1. Ax.l. 1. Sub. As.. 3 Given st. line AB, & C= fte space to which the s^. of AB must be equal, butc=>(fr to divide AB so that the r I contained by the segs = C^ E A ■ • « v^ F -G iB Bis. AB InD. If AD2 = C^ the Problem is solved j but if AD2 i^t CS & AD > C ; then the Solution though differing in form wiU be like the foregoing Prop. 28 in substance. At rt. /.s to AB draw DE= C ; & prod. ED so that EF = AD or DB ; fi-om Ewith rad, EF cut AB in G, and join EG; then AB is div. in G so that AG . GB = C •.• AB is divided equallv in D and unequally in G ; .-. [ZZI AG . GB + DG=' = DB^ = EF^ = EG" ; butED2 + DG''=: EG"; .-. CZl AG . GB + DG^ = ED= + DG' ; take away DG^, & CZl AG . GB =^ ^D'= C. Q. E. F. ayja. ^. Prop. 28. bk. VI. is equivalent to the Problem;—" To inscribe I a 0ven A a parlm. equal to a given figure not greater than the maximum inscribed parlm. and having an /. in common with the A." Manual of Euclid, Pt. II,p. 98. SCH. 2. PROP. XXVIII. BOOK VI. 353 The Demonstration is similar to that in Use and App.'27, VI. Use & Appl. There are two cases of this Proposition which are not imfreqnently employed by geometers. 1°. A variation of Sch. 1, 28, VI. To a given st. line AB to apply a rect. AH, deficient by a square GK, which rect. shall be equal to a given square, that on line C; but the given- square on C must not be greater than the Square onthehalf AD of line AB. 1 10, 1. "2 Sup. 11, I. 3, 1. 3,1. 5tPst.3&l,I 6 7 D. 1 46,1.31,1, Sol. C. 5, n. 47, 1, Sub. Ax. 3. C. 6. Cone. ^ r-^' ..--?-.., L ^,'' D /Cr' B ■c ■E ,' Bis. AB in D; and if AD" = Xi B. C^ the prob; is solved. But if AD^ at C* and AD . >C; draw DE X -^j and make DB = C; produce ED so that EF = ADtrriB; from cen. E, with EF desc, arc meeting AB in G; and i join EG; on GB desc. the square GK, and coinplete rect. GL'; then I I AH = C^, and deficient by GK, has been apj)lied to AB. ".' AB is div. equally in D, and unequally in G; .-. AG.GB + DG^ = DB2 = EF' = EG^ = ED" DG=. Take DG" from each, .-. AG . GB = ED=, i. e. = C". But. AG . GB is IZZ] AH, .-, GH = GB; .-. dZl AH = 0=". •. I 1 AH = C^ has been applied to AB, deficient by the sq. GK. 0. E. B. ' + 2°. To a given st. line to apply a rectangle, which shall be equal to a given rectangle, and be deficient by a square ; but the given rectangle must not be greater than the square upon half the given line. E. 2 G. 1 2 3 Dat. Quaes. 11,1. 2, I. 10, I. Pst. 3, I. Let AB be the given line, and on lines C & D a given I | not greater than ( ) ; to apply to AB a I I = C . D> ,--•''' ""•'.. deficient by a square. On' the same side of AB, at A & B, draw AE, BE each _L AB; make AE = C and BE = D; join EF and bisect it in G; from G, with rad. GE, desc. a meeting AE in H; F O 354 GRADATIONS IN EUCLID. D. 1 2 3 4 31,1. 31,111. Ax. 11,1. 28,1. 34,1. Kern. C. 2. C. 2, 4. 34,1. 3, ni. Cone. Add. 47, 1. Def. 2, m. 36, in. Snp. Sup. 46, 1. 31, 1. 3, in. Ax. 3 C. c. Eec. join HF ; and draw GK || HF, and GL || AE, meeting AB in L. •.' Z_ EHF in a sem. c. i^ a rt. /., and = ^ EAB ; /, AB II HP and AH || BF, . AH = BF; and CZ3 EA . AH = CZl EA . BF, i. c, CD. and ■.• EG = GF, and AE || LG 1| BF; .-. AL = LB, & EK = KH; and C . D > AL" or (^)'f -. EA . AH > AL^, i. e. KG"; add KE^ .-. AK" > EK" + KG", i. e. EG"; and ,-. AK or GL > GE. Now if GE = GL, EHF touches AB in L; .-, AL" = EA.AH, ;. e. CD. But if EG T^ GL, and EG > GL; .-. EHF cuts AB ; Let the cut AB in . s M and N; on NB desc. NB" and complete the I I AP. •/ LM = LN, and AL = LB, .-. AM = NB, and .-. AN . NB = NA . AM, i. e. EA . AH or C . D. But AN.NBisCI] AP; .-. PN = NB; /, I I AP = C. D, and AP has been applied to AB, deficient by BN". Q. E. F. This last Problem may be thus enunciated ; " To cut a given line AB in the point N so as to make the rectangle AN . NB equal to a given space.'^ Or, which is the same thing, " Having AB the sum of the sides of a rectangle given, and also its magnitude, or area, to find the sides." Prop. 29.— Prob. To a given st line to , apply a parallelogram equal to a given rectilineal figure, and exceed ingby a parallelogram similar to another giv'ji parallelogram. Con. 10,1 18, VL 25, VI. 21, VI. 2, L 31, L Dem. 26, VL 36, I. 43, L 24; VL PROP. XXIX, BOOK VI. 355 E. 1 2 Dat. Quees. Given AB, £Z7 X, & rectil. fig. Z, to apply to AB a I — 7 = Z, and exceeding by a / — 7 sim. to / — 7 X. C. 1 2 3 4 5 6 7 8 9 10 11 12 D. 1 2 3 4 5 6 H ■-1 10, 1. 18, VI 25, VI. 21; VI. C. 2. Pst.2,1.3,1. 31,1. 0,6,7. C. 3. 26, VI. 31,1. Sol. C.2,8.Ax. 1 Sub. Ax. 3. C. 1, 31, 1. 43, I. Add. Ax. 2. D. 2. Cone. Bis; AB in E, and on EL descr. / — 7 EL sim. & sim. sit. to I 7 X ; make I — 7 GH = EL + Z, & sim. ■& sim. to^^yX. •. OZJ GH is sim. to CZJ EL. Let KH be homol. to PL, & KG to FE ; then •.■ Oiy GH > OU EL, .-. HK > PL & KG > PE ; Produce PL & PE, & make PLM = KH, & PEN = KG ; ■ & complete the I — 7 MN ; .-, CTJ MN is eq. & sim. to nZD GH ; but GH sim to EL, . • MN is sim to EL ; & .*. EL & MN are 1 — 7 s a^out the same diag. Draw their diag. FX, & complete the scheme, then to AB is applied / — 7 AX = Z, & ex- > ceeding by / 7 PO sim. to / 7 X. •.• GH=EL + Z,&^3'GH=zr=7MN, .-. ^^ MN = ^uEL + Z=^ Z ; take away / 7 EL ; .'. rem. gnomon NOL= Z. And •.• AE = EB .-. OZJ AN= £Z7 NB £^ BM. Add / — 7 NO, .■. the whole / — 7 AX = the whole gnomon. NOL ; but NOL = Z, .-. /:^7 AX = fig. Z. •- to AB is applied a 1 7 AX = fig. Z, exceeding by / 7 PO, sim. to fig. X ; PO is sim. to £Z7 EL. Q. E. F. 356 GRADATIONS IN EUCLID. N.B. — In the diagram, X by mistake of the engraver occurs twice. SoH. — Of the thirteen books of Elements written by Euclid, the tenth bears evident ti-aces of the greatest attention having been bestowed to render it complete. The doctrine of Incommensurables there receives its developement, and is ti-eated with great exactness. "The most conspicuous propositions of elementary geometry," says an eminent writer, "which are applied in the tenth book, are the 27th, 28th, & 29th of the sixth book, of which it may be useiiil to give the algebraical signification. The first of these (the 27th) amounts to shewing that 2 x — r' has its greatest value when a; = 1, and contains a limi- tation necessary to the conditions of the two which follow. The 28th propo- sition is a solution of the equation ax — x'' = b, upon a condition derived from the preceding proposition, namely, that J a' shall exceed b. It might appear more correct to say that the solution of this equation is one particular case of the proposition, namely, where the given parallelogram is a square ; but never- theless, the assertion applies equally to all cases. Euclid, however, did not detect the two solutions of the question ; though if the diagonal of a parallelo- gram in his' construction be produced to meet the production of a line which it does not cut, the second solution may he readily obtained. This is a strong presumption against his having anything like algebra ; since it is almost im- possible to imagine that the propositions of the tenth book, deduced from any algebra, however imperfect, could have been put together without the discovery of the second root. The remaining proposition (the 29th) is equivalent to a solution of az -|- a' =r J ; but the case of x' — ax =: b is wanting, which is another argument against EucUd having known any aJgebraical reasoning. — Penny Ctc. XII, p. 38. Use and App. — Several Problems of a like kind to Prop. 29, and in some respects equivalent to it may be here advantageously introduced ; Peob. 1°. e:tscribeTo to a given triangle ABC, a varallelogTam equal to a given rectilineal figure, AX, and having an angle equal to one of the angles of the given triangle. A, Here AX is the inscribed I 7 & AT the exscribed N.B. This Prob. is equivalent to to the foregoing Prop. 29. Manual of Euclid, p. 100. Phob. 2°. To a given st. line ABto apply a rectangle which shall be equal ■ to a given square, that on C and exceeding by a square.. PROP. XXIX. BOOK VI. 357 C. 1 : 2 .3 4 D. 10,1.11,1. iPst. 1, 3, I. 36, 1, 31, 1. Sol c. 6,11 Sub. Ax. 3,1. C. 3. Cone. i / \ 1 ^' i 1. B B '^ Bis. AB in D, and draw BE J_ to AB so that' BE = C ; join ED, and from D with DE desc. a Q meeting AB produced in G- ; on BG deso. the sq. BH, and complete I I GL then I I AH = C^ & exceeding by GK is applied to AB. ■.• AB is diyided equally in D and produced to G, .-. AG . GB 4- DB2 = D6^ = DE.^ = EB=' + BC^ ; from each take I^B^, .-. rem. AG . GB = BW= C^ but ■.■ GH =: GB, .'. AG, GB is rectangle AH ; .-. I 1 AH = C' .■. AH = C^, and exceeding by GK, is applied to AB. Q. E. F. N.B. — This Problem is the same as, " To produce a given st line, AB, so that the rectangle contained by the external segments of the given line may be ■ tgual to a given space, as C." The foregoing Con. and Dem. may be used, and then it will appear that the problem is, to produce AS so that AG. GB C. 1 2 3 4 a 6 7 D. 1 2 3 4 5 2,1.11,1. Pst. 1, 1. 31,1. Pet. 3, 1. Pst. 1,31,1. 46, 1, 31, I Sol. C. 1, 28, 1. 34,1 3,ni.Ax ,1. 35, III. Ax. 1, 1. Uonc. ii-'i). Ml'vA XG- Draw AE = C, & BP = D, _|_s AB on contrary sides join EP & bisect EP in G ; &om G with GB desc. ""' meeting AE in H ; join HP, & draw GL || AE. Let the meet AB produced in M,N ; on NB desc. sq. NO, & complete I I NQ. to AB is applied a i I AP = C . D & exceeding by square liV BP. •.• Z.EHF a rt. Z = Z EAB, .-, AB || HP; .-. AH = BF; &EA.AH = EA.BP=C.D, And •.• ML = LN, & AL= LB, .-. MA = BN j & .-. AN'.NB = MA.AN=EA.AH = C.D : .-. AN-.NB,;.e.,AP = C.D. •. to AB is applied AP = C . D & exceeding by sq. BP. ' Q. E. r. OP/ N.B. This Problem is the same' as, — " To find a point N in a given st. line AB produced, so as to make the rectangle AN. NB ^ a given space." '* Or, which is the same thing, — " Saving given AB the difference of the sides of a rectangle and the magnitude of it, to find the sides." 358 GRADATIONS IN EUCLID. Prop. 30.— Peob. To cut a given st. line in extreme and mean ratio ; i. e., bo that the whole line shall be to the greater segment as the greater segment to the less. CoK. 46, I. To describe a sq. on a given St. line. 11, n. To divide a given st. line into two pts, so that the rect. con- tained by the whole and one of the pts, shall be equal to the sq. of the other part Dem. Ax. 3, 1. 14, VX. 34, 1. Def. 30, 1. A sq. has all its sides eq. and all its Z.S rt. ZS- 14. VI. 17, YI. Def. 3, VI. E, 11 Dat. 1 Giyen AB a st, line ; 21 Qa%s. 1 to cut it in extreme and mean ratio. 0. 1 2 3 D. 1 2 45,1. 29, VI. Sol. 0.1. C. 2. Ax. 3,1. 0.14, VI. On AB, construct a square AD ; to AC apply CU OG = nzi AD, exceeding by- fig. AG, sim. to AD ; then AB is cut in extreme & mean ratio in . H. ".' AD is a square, .". AG is a sq. •.• sq. AT) = 03 OG, /. rem. BK = rem. AG-; & •.• BKiseq, ang. to AG, .". KH: HG = AH: HB; PROP. XXX. — BOOK Vlr 359 34,1. Def.30,1. Ax. 9, I. 14, V. Con.Def. 3, VI. Otherwise, but KH = AC = AB ; &HG==AH; .-. BA : AH = AH : HB ; but AB > AH ; AH > HB. .'. AB is cut in extreme and mean ratio, i. e., AB : AH = AH : HB. Q. E. F C. D. 1 2 ii,n. C. 17,VI. Def.3,VI. H Divide AB in H so that AB . BH = Am '.• AB . BH = Aff, .-. AB : AH = AH : HB ; ,'. AB is cut in extreme and mean ratio in . H. Sot:. 1. A St. line thus divided is also said to be divided medialh/; an the ratio of its segments is called the medial ratio. The same division take place in Prop. 11, bk. H, when the rectangle of the whole and one of its parts is equal to the square on the other part. 2. The dividing a line into extreme and mean ratio belongs to a class of Problems which relate to inconmiensurable magnitudes. The following is the General Theorem respecting them. " Let there be two Magnitudes of the same kind, P& Q ; and let P be con- tained in Q a certain number of times which is to ¥ as F is to Q; then the Magnitudes P ^ Q shall be incommensurable. C. I Sup. C. 1 Hyp. 15, V. 2 16,V. 17,V. 3!l6,V.23,V. A,V. Sim. \ Sch. N, V. »i,Conc. Let 5 P = Q — R a remainder. Then •/ R : P = P : Q, and 5 E : 5 P = P : Q; .-, aft. 5R :P = 5P : Q, and div.SlHr^P: P = 5P rvj Q : Q. But P : E = Q : P, .-.ex ffi?. 5 K ~ P : E = 5 P (>J Q or R : P. Now R < P, .-. 5 E := or < the 2nd, the Srd also is > =: or < the 4th. E. 1 2 C. D. 1 Hyp. 1. „ 2. Cone. 12,1. C. 8, VI. D. 2. 4, VI. Cor.2,20, VI. Let ABO be a rt. Z d A , the rt. Z being Z BAG; and let fig. o on BO Be sim. and similarly desc. to figures b ^ c on AC, AB; then fig. a = fig. b + fig. e. Draw from . A, AD _L BC. •.• from A, the ver- tex of the rt. Z AD ± base BC, .•. A s ABD, ADO, are sim. to a ABC and to each other. And '.• a ABC is sim. to aADB, & to a ADC, .-. CB : BA= BA : BD, & OB : CA = CA : DC; and ".• in 3 propls, as. 1st : 3rd so is fig. on 1st : fig. on the 2nd, 364 GRADATIONS IN EUCLID. 10 n 12i B. V. Sim. 24, V. Ax. 8, I. A. V. Eec. Or, D. 1 21 Or Since .•. CB : BD = fig. a on CB : sim. and similaily desc. fig. c on BA ; and inv. DB : BC = fig. c on BA : fig. a on CB ; So DC : CB = fig. J on CA : fig. a on CB ; .-. BD + DC : BC= fig. c on BA + fig. b on AC : fig. a on BO but BD + DC = BC. .•. fig. a on BC = bim. and similarly desc. figs. c + J on BA and AC. Wherefore, in rt. /_ d triangles, the rectilineal fig. ^c. Q. E. D. 23, VI. Cor .1.20, VI 11, V. Sim. 24, V. 47,1. Cone. .•. sim. figs. : one another in the duplicate E. of homol. sides, .•. fig. a on BC : fig. c on BA, in the duplicate K of CB to AB ; but BC2 : BA2 = BC : BA ; .-. fig. a on CB : fig. c on BA = CB^ • AB^. So fig, a on BC : fig. 6 on CA = BC^ : CA^; .-. fig. a onBC : %s. c + 6 on BA, AC = BC^ : AB2 + AC2; but BC2 = BA2 + AC2 ; /. fig. a on BC = fig. c on BA + fig J on AC. Q. B. D. DB cm on BA : EC = : IZD on BC DO CZZl on CA, DB + DC : BC = CZZI on B A + CD on Therefore, 24, V. CA : CZl on BC. ScH. 1. Proposition 31, bk. VI, is very comprehensive and renders Prop. 47, bk. I only a particular case. There is however a theorem still more general; it is given in his Mathematical Collections by Pappus, one of the later of the Greek Geometricians, who flourished at Alexandria during the reign of Theodosius, AD. 379—395 ; PROP. XXXI. — BOOK VI. 365 " If any C^s AF,BE, be described on two sides, AC, BC, of any £i.ACB; and if the sides of the / 7 ^. J> E. GF, be produced to meet, as inH, and if that point of intersection B. and the vertex C of the triangle be joined, and the li ne HC produced to N; then these I 7 ^ AF, BE are equal in area to a I l AL. described on the .base AB, and having two of its sides AK, BL parallel to CN the line produced through the point of intersection H and the vertex C. and limited by the sides, DE, GF, of the two £117*. .X vC iS; V //^ y \'' // ■ / ^ \ A \l^ 1 V,\ A. N B' D. 1 2 3 4 5 6 7 C. 33, 1. C. 31, I. Ax.l,I.C. 35, X. 35, 1. Ax. 2, L Eec. •.• fig AKHC is a / — 7 . '.; AK = CH ; & .-. fig. ABLK is a / — 7 . /. AKL= I;B ; .-. LB = CH ; «:; AL is a Now £ZJ NL = CZ7BH = £3700 ; & /zr/ NK = /ZZ/ AH = CZ2 AF ; .-. nU NL + izn NK, z. e., I — 7AL=/ — 7CD + I — 7Ag ," If any i is be described on the two sides; S^c. Q. e. d. N.B. When ^ABC is a rt. /., the s CD, AF &. AL become squares, and the 47, 1 occurs, that the square on the hypotenuse is eonal to the squares on the sides including the rt. angle. 2. Circles, as well as squares, are similar figures, and it described with the sides, AB, AC, BC, of a rt,/.d A for ^^ their diameters, then the circle with the hypo- tentise AB for its diameter is equal in Area to the two circles, that have AC, BC, the sides about the rt. /., for their diameters. See Cor. 5 & 6 Pr. 20, VI. 366 GRADATIONS IN EUCLID. Prop. 32.— Theor. Ij two triangles which have two sides of the one proportional to two sides of the other, he joined at one angle, so as to have their homo- logous sides parallel to one another ; the remaining sides shall be in a St. line. E. 1 2 3 D. 1 2 3 4 5 6 7 8 9 10 11 12 Hyp. 1. Cone. H. 29, I. Sim. Ax. 1, 1, D. 3. & H. 6, VI. D.2. Ax. 2,1 Add. 32,1. D. 8, 9. 14,1. Eec. Let AS ABO, DOE A\ have BA : AC = CD : DE ; and let AB be || DC and AC II DE; Then BO, CE form one st. line BE. •.• AB II DO and AQ mieets them, .-. Z BAC = z ACD. So z ODE = z ACD, . .-. Z BAC = Z ODE. And •.• AS ABC, DOE have Z A = Z D, and BA: AC = CD : DE ; .•. A ABC is eq. ang. to A DOE, and ZB = Z DOE ; and •.• ZA = Z ACD, .-. whole Z ACE = Z ABO + Z BAO ; add ZACB; then ZsACE + ACB= ZsA + B + ACB. But Zs A + B + ACB = 2 rt. Zs ; .-. ZS ACE + ACB = 2 rt. zs; and "." at in AC, the lines BC and CE on opp. sides of C make adj. Zs = 2 rt. Zs. .•. the lines BO, CE are in one st. line BE. Therefore, if two triangles which have two sides ^c. Q. E. D. ScH. The position of the given sides AC, DC, which are not homologous should be]|such as to form an angle, ACD, at the point of junction C; otherwise BC and CE may not be in one and the same st. line BE. PROP ZXXIII. BOOK VI. 367 Pbop. 33.— Theor. In equal circles angles, whether at tHe centres, or eireumferenees have the same ratio which the circumferences j on which they stand, have to one another; so also have the sectors Con. 1, IV. In a given to place a st. line eqnaJ to a given st. line which is not greater than the diam of the 0. Pst. 1, 1. Dem. 28, in. In eq. 0s, eq. St. lines cut off eq. arcs. 27, III. In eq. 0s the /.s on eq. arcs are eqnal, whether they be at the centres or 0ce3. Def. 5,V. 20,111. The /. at the ceh..is double of the /. at the 0ce upon the same base. ' . 15, V. Magnitudes have the same E; to one another which their equi- multiples have. 4, 1. Def. 11, in. Sim. segs. of 0s are those in which the /^s are eqnal. 24, III. Sim. segs. of 0s upon eq. St. lines are eq. to one another. Ax. 3, I. E. 1 2i Hyp.l. ,/ 2. Gone. 1. Hyp3. Cone. Let ABC = DEF, having G & Has centres. 1°. On arcs BC, EF let there be at the centres :• Z_s BGC EHP, and at the 0ces Zs BAC» EDF;- V .then ai-c BC : are EP = z BGC' : Z. EHF, and Z BAC : Z EDF. : ■ 2°. Also on arcs BC, EP let there be sectors, BGC and EHF ; then arc BC : arc EP = sect. BGC : sect. EHF. -««^ 368 GRADATIONS IN EUCLID. C. 1 2 3 1,IV. Pst. 1, 1 D. 1 2 28, III. 27, III. 3 Sim. 4 27, III. 5 H. 1. 6 D. 2. 7 D. 3. 8 D. 4. 9 10 Def.5,V. 20, III. 11 12 15, V. Cone. Of angles at the centres or Qces of equal circles. In ABC take any eq. arcs BC = CK = KL, and in DE any eq. arcs, EF = FM = MN ; and join GK, GL, HM, HN. •.■ arcs BO = CK = KL, .-. ZsBGC = CGK = KGL; .■. the mult, which arc BL is of arc BC, that same mult, is I. BGL of BGC. So the mult, which arc EN is of arc EF, that same mult, is Z EHN of Z EHP ; and if arc BL = > or < arc EN, Z BGL = > or < Z EHN. Now '.■ there are 4 Ms, arcs BC, EF, and /. s BGC and EHF ; and of arc BC and Z BGC equims, arc BL and Z BGL are taken; and of arc EF and Z EHF equims, arc EN and ZEHN; and •.• if arc BL > = or < arc EN, Z BGL > = < Z EHN; .-. arc BC : arc EF = Z BGC : Z EHF. and •.• Z BGC = 2 / BAC, and Z EHF = 2 Z EDF, .-. Z BGC : Z EHF = Z BAG : z EDF ; .-. arc BC : arc EF = Z BGC : z EHF = Z BAC : Z EDF. Case II. C. 1 Pst. 1, 1. 2 Pst. 1, 1. D. 1 H. 2 4,L Of angles at the vertex of equal sectors. Join BC, CK, and in arcs BC, CK take any . i3 X,0; and join BX, XC, CO, OK, •.• in A GBC, GCK, BG, GC = CG, GK, and Z BGC = zCGK. .-. base BC = base CK, and a GBC = A GCK. PROP XXXIII. BOOK VI. 369 3C.1.AX.3, 1. 4 5 6 7 8 9 10 11 12 13 14 15 16 17 27, III, Def.ll, III. D. 2. 24, III. D. 2. Sim. Sim. Cone. Sim. Hyp. D. 9. D. 10. D. 11. Def.5,V. Rec. And arc BO = arc CK, .•. rem. arc BLO ^ rem. arc CLK; .•. Z X = Z. O, and seg. BXC is sim. to seg. CKO. and •.■ EC = CK, .-. seg. BXC = seg. CKO. And A BOO = A OGK, .•. the whole sect. BGO = the whole sect. OGK. So sectors KGL = BGO = OGK, and sectors EHF = FHM = MHN; .". the mult, which arc BL is of arc BO, that same mult. sect. BGL is of sect BGO ; and the mult, which arc EN is of arc EF, that mult. sect. EHNis of sect EHF; and if arc BL = > or < arc EN, the ' sect. BGL = > or < sect. EHN. Now •.• there are 4 Ms, the arcs, BC, EF, and the sects. BGO, EHF ; and of arc BC and sect. BGO, arc BL and sect. BGL are equims.; and of arc EF and sect. EHF, arc EN and sect. EHN are equims.; and •.• if arc BL > = or < arc EN, the sect. BGL > = or < sect. EHN; .-, arc BC : arc EF = sect. BGC : sect,, EHF. ", In equal circles, angles, Sfc. Q. E. D. CoK. 1. The sectors are to each other as their angles. For, if arc BC : arc EF = Z BGC : zEHF; and arc BO: arc EF = sect. BGO : sect. EHF, then 11, V. sect. BGC : sect EHF = Z BGC : Z EHF. CoK. II. 'Similar sectors df the same or equal circles are equal. Cor. III. An angle at the centre of a circle is to four rt. angles cs the arc on which it stands to the circumference of the circle. 370 GRADATIONS IN EUCLID. For, the L at the centre : one rt. /. = arc subtending central /. : arc subtending a rt. Z_ or quadrant ; Then, 4, V. Z at cen. : 4 rt. Z. s = arc of central Z. : whole 0ce. CoR. IV. In different circles the arcs of equal angles at the centres or circumferences are similar. CoR. V. Hence, similar segments are contained hy similar arcs and vic^ versd. Son. 1. If the arcs and sector had all been in one circle the proof wouM have been the saTne,-^for H would have coincided with Gr, and I),E,F,M,N would have been points in the circumference of circle ABL. 2. The second part of the Proposition was added by Theok, the Ptolemaist, and father of the renowned but unfortunate Hypatia, of Alexan- dria, in the time of the elder Theodosius ; — it is given in the Commentaiy on Ptoi.emy 's Almagest. a. That the angles at the circumferences are as the arcs on which they stand, follows also as a CoroUai-y from Prop. 20, bk. lU. Prop. 33 . u. In the same or in equal circles AGS, CHD angles, whether atihe centres, as /_s AEB, CFD, or at the circumferences, ^s AGB, CHD, have the same ratio as the arcs, AB, CD. on which they stand. E&C.l 2 Bef. 1, V. Pst. 1, 1 Let AM or CN be a com. meas. of arcs AB, CD ; & arc AMB = 4 arc AM, and arc CND = 3 CN ; then join EM and FN. PEOP XXXIII. BOOK VI. 371 D. 27, III. 7aV. U,V. 20,111, 15, V. •.' Z.AEM = ^CTN, & Z.AEB = 4 /.ABM, &ZCFD = 3 /CFN ; .-. ZAEB : ZOFD = 4:3; i. e., arc AMB: arc CND = 4:3; ,-. ZAEB : /_ CJFJ) = arc AMB : arc CND. And ".' Z.S AEB, CED are mults. of /.s G & H by 2, .'. Z.Gr : Z H = arc AMB : arc CND q. e. d. Cob. Since the Z^s and arcs are proportional when oommensurabls, .•, tltei/ are also proportional when incommensurable. See Sch. 3 — 5 & Use 1, 16, VI. Prop. 33 b. In the same circle, ABK, or in equal circles, the sectors BGC,,GCK, that stand on equal arcs, BC, CK, are equal. Take any . s X,0, in the arcs BC, CK, & join BX, XC, CO, OK. Prom ABK take separately arcs BC & CK, , rem. arc. BAC = rem. are CAK; , ZBXC is Sim. to /.COK, and seg. BXC to seg. CKO. Again •.• BG = GC = GK, &ZBGC=ZCGK, .•. A BGC = aCGK, & base BC = base CK. And •.• seg. BXC is sim. to seg. CKO, & their bases, BC, CK equal. .-. seg. BXC = seg CKO, & sect, BGC = sect CGK. So, if sect. CGK were in a which is equal to ABC, sect. CGK =sect. BGC. C. D 1 2 I Pst. 1, 1. Sub. 2 Ax. a, I. 3 27, m. 4 27, III. 5 6 4,1. D. 3. 5. 7 24, ni. Sim- , Phop 33 V. In the same circle ABK, as in fig. to 33 i. or in equal circles, ABG, CHD, as in fig. to 33-0, sectors AJEB, CFD, have pie same ratio as tht arcs, AMB, CND, ore which the ' ' E.&C. 1 2 3 D. 1 2 3 4 Eig. 33, a. Pst. 1,1. 33, b. VI. C. 2. 7 a V. 11, V. Let arc AM = arc CN, & be a com. measure of arcs AB, CD ; * & let arc AB = 4 arc AM, & arc CD = 3 arc CN, then join EM and EN. •.• sect. AEM = sect CEN ; & ".• sect. AEB = 4 sect. AEM, & sect. CED = 3 sect. CEN ; .-. sect. AEB : sect. CED = 4:3; and arc AB : arc CD = 4 : 3 ; .-. sect. AEB : sect. CFD = arc AB ; arc CD. 372 GRADATIONS IN EUCLID. CoE. Since the sires and sectors are thus proportional when commensu- rable, .'.they must also be proportional when incommensurable. 4. From this 33rd Proposition it results, that the angle at the centre of a circle is said to be measured by the arc on which it stands. Use and Appl. 1. If arcs ACB, AEB, of different circles have a com- mon chord, AB, the lines, AC, AT), diverging from one of its extremities, A, will cut the arcs proportionally, i. e. BF : FE = BD ; DC. D. 1 2 3 4 Ax. 8, I. 33, VI. 11, VI. •.* /. s B AF, BAD are identical, and also /_& EAF, CAD ; .-, arc BF : arc FB = /. BAF : ^EAF, also arc BD : arc DC — /.BAF : ZEAF ; .•. arc BF : ai'c FE = arc BD : arc DC. Q. E.D. 2. The arcs, A, A' of unequal circles are in a ratio compounded of their central angles a, a' and their radii, R, E'. C. I D. 1 2 3 H. & C. 2 C. &H. Cor. 5, 20, VI. D. 1 & 2. With rad. = E deso. an Z = a', and let the subtending arc be rn. '.' arcs A ^nd m havoan eq. E, .*, A : m = o :o '; and '.■ m and A' have eq. central /. s ! ,-. m : A', = E : E' But A ; A' is compounded of •! ." w or of the equivalent, ratios, u, : a' and E : R'. 3. Central angles a, a', are in a ratio compounded of the direct ratio of their arcs A, A', and the inverse ratio of their radii E, E'. D. 1 2 3 4 5 ITs»2, 33, VI. For A : A' = I ^ : -^'• Let each of the equal ratios be compounded with B' : B; But E : R' I . ,. „ ,., j^, . -g MS a ratio of equality, .-, A : A'l R' : B J = " = " See Lardnee's Euclid, p. 224. D. 1 and 2 Sch. 1, 16, VI. Cone. PROP. B.— BOOK VI. 373, Obs. "And herewith," remarks Captain Thomas Eudd, Chiefe Engineer to Charles I., "is the first six Books of EtrcLiDE ended. There be hereafter added certain Propositions, which although they be not Euclides, yet because they are both witty and nsefuU, I thought it good not to omit." Eudd's Euclides Elements, A.D. 1651, p. 253. SUBSIDIAEY PEOPOSITIONS. Prop. B. — Theor. If an angle of a triangle he bisected by a st. line which lihewist cvLts the base; the rectangle contained by the sides of the triangle is iqual to the rectangle contained by the segments of the base, together with the square of the st. line which bisects the angle. Con. 5, rV. To desc. a about a given A . Pst 2, 1, 1. Dem. 21, HE. The /.s in the same seg. of a are equal. 3, n. If a st. line be divided into any two pts, the I I contained by the whole and one of the pts is eq. to the 1"^ contained by the two parts together with the square of the aforesaid part. 35, in. If two st. lines cut one another within a the rect. contained by the segs. of one is eq. to the rect. contained by the segs. of the other. 32,1. 4, VI. 16, VI. E. 1 C. 1 2 Hyp. Gone. 5, IV. Pst. 2 1, I. In A ABC let /. A be bisected by AD; then BA . AC = BD . DC + AD^. About A ABC describe ACB; prod. AD to 0ce E, and join EC; 374 GRADATIONS IF EUCLID. :-.-.. (J D. 1 H. 21, III. 2 32, I. 4, VI. 3 16, VI. 3, I 4 5 6 35, III. Cone. Eec. E Then V zBAD= / CAE, & z.ABD = ZAEC; /. A ABD is eq. ang. with a AEG, & .'. BA: AD =EA:AC; & .-, BA . AC = EA . AD, i. e., = ED . DA + AD2. But ED . DA = BD . DC. .-. AB . AC = BD . DC + AD^. /. If an angle of a triangle, ^c. Q. E. D. Cor. Prom the hypothesis, that the exterior Tertical angle, ^C AF.is bisected by GA, which also cuts the base produced BCGin Q ami the circle in H, the conclusion follows, that the rectangle of the sides together with the square of the line which bisects the exterior angle is equal to the rectangle of the whole line produced and the ex- treme segment; i. e BA . AC + AG^ = BG.GO. ScH. Pfcp B and its Corollary combined, may be thus enunciated ; " if the vertical or ext. vertical /. of a A be bisected by a St. line, which cuts the base, or the base produced, the square of the st. line shall be equal to the differ^ ence of, the rectangles under the two sides and under the segments of the base or of the base produced." Tn the case of the bisection of the vert. /.BAC, AD' = BA . AC— BD . DC. And " ext. vert. ZCAF, AG» = BG . GC ~ BA . AC Prop. C. — Theok. If from any angle of a triangle a st. line be drawn perpendi- cular to the base ; the rectangle contained by the aides of the triangla is equal to the rectangle contained by the perpendicular and the dia- meter of the circle described about the triangle. 375 Con. 5, IV. Dem. 31, m. 21, ni. 4, VI. 16, VI. E. 1 Hyp. c. D. 1 2 3 4 Cone. 4, VI. 31,111. & 21 III. 4, VI. 16, VL Rec. In A ABC, from /_ A let AD be JL base BC, then BA . AC = AD . AE. About A ABC desc. ACB andits diam. AE, & join EC. •.• rt. Z BDA = Z ECA, & L ABD = L AEC; .•. A BDAis eq. ang. with A ACE; & .-. BA: AD = EA : AC; .-. BA.AC .= EA.AD. If therefore from any angle, S[c. Q. E. D. CoB.< — If two -A s, ABC, ACE, he inscribed in the same or in, equal 0s, the red. under the two sides of the one, BA . AC, shall be to the rect. und^ the two sides of the other, EC. CA, as the perp. AD, which is drawn from the vertex A, to the base, BC, of the one, is to the perp.which is drawn from the vertex C to the base, AE, of the other : i. e. BA . AC : EC . CA = AD : perp. from C. Prop. D. — Thbor. The rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle, is equal to both the rectangles contained by its opposite sides. Cos. 23, 1. Dem. 21, m. 4, VI. 16, VI. 1, n. if there be two st. lines, one of which is divided into any number of pts. ; the rect. con- tained by the two st. lines is equal to the rectangles contained by the undiT. Ime and the several pts. of the divided line. 376 GRADATIONS IN EUCLID. E. 1 Hyp. 2 Cone. C. 23,1. 1). 1 Add. 2 21, III. 3 4 4, VI. 16, VI. C. 21, III 5 4, VI. 6 7 8 9 16, VI. D. 3. 1,11. Eeo. In a let a qu. lat. ABCD be insc with diags. AC, BD ; then AC.BD = AB. CD + AD . BC. Make Z ABE=zDBC. Add Z EBD to the equals, .-. ZABD = ZEBC. and •.• ZBDA=ZBGE .•, aABD, is eq. ang. with aEBC; •. BC : CE = BD : DA, and .-. BC . DA = BD . CE. Again, '.• zABE = Z DBC, and Z BAE = Z BDC, .■. A ABE is eq. ang with a BCD, .-. BA : AE = BD : DC; •. BA.DC= BD.AE. ButBC.DA = BD.CE; .-. the whole cn AC . BD = AB . CD + AD.BC. Therefore, the rectangle contained, ^-c. Q. E. D. ScH. This Proposition is named Ptolemy's Theoreni, and is a Lemma at p. 9 of his Almagest, or MeyaXn Suvrafi; of which it is said, "Ptolemy appears as a splendid mathematician, and an (at least) indifferent observer." It is curious to note how editor after editor of Euclid has followed the iden- tical diagrams of the earliest printed editions. Ptolejiy's thirteen books " MagiKB Constructionis, Id est. Perfects cmlestium motuum, pertractationis," published at Basle hj Symon Grynoeus in the year 1538, contain several dia- grams, and the one to Prop. D, bk. VI, among the number, which have scarcely undergone any change siricp that time. BUlingsley's English Euclid also fur- nishes very many examples from which to prove the imitativenesa of succeeding Editors, and by which to justify, if need be, the continuation of the practice. PEOP E. BOOK VI. 377 Pkop. E. — Theor. The diagonals, AB, CD, ■ of a (Quadrilateral ACBD, inscribed in a circle, ABO are to one another as the sums of the rectangles under the sides adjacent to the ^extremities of those diagonals : L e., AB: CD: = CA.AD,+ CB.BD: AO.CB+ AD.DB. Con. 12, 1. Dem. Cor. C, VI. Sch. 3, 24, V. In any mrniber of mag- nitudes of the same kind forming two series, if the ratios of the 1st to the 2ud, of^the 2nd to the 3rd, of the 3rd to the 4th, and soon,j3e the same in the two series ; then any two combinations whatever, shall be to one ano- ther as two similar combinations of the eorresponding magnitudes of the second series. 4, 5, VI. 11, V. C. I Sup. I Let AB, CD, cut one anotlifer in'the . E. Case - 1. Suppose that AB cuts CD at rt. L s- ..- — A V AS ACD, BCD,CAB, DAB are in the same ; •.'perps. AE, BE CE and DE are to one another as thei— IS AC. AD,.BC .BD, CA.CB, DA.DB; .-, AE + EB, or AB : CE + ED, or CD, = AC. AD + BC . BD : CA . OB + DA . DB. Case II. Or, Suppose that AB cuts GJD not at rt. /_s. D. 1 Hyp. ''. 2 Cor. 0, VI 3 Sch. 3, 24, V. c. D. 1 2 12,1. Sim. C. Draw the perps. Aa, B6, Cc, "Bd. As before, Aa + Bb: Cc + Bd = ACAD + BC : BD : CA . OB + DA . DB ; . but •.• AS AEffl, BE5, OEc, and DEcZ areeq. ang.; 878 GRADATIOifS IN EUCLID. D. 3 4 5 6 4, 5, VI. 11, V. Eec. /. Aa, Bb, Ce, Dd, are to one another, as AE, BB, CE and DE; .-. Aa, + Bb : Cc, + D(^ = AE + BE : CE + DE = AB : CD. .-. AB : CD = AC . AD + BO . BD : CA . CB + DA.DB. Therefore, the diagonals of a quadrilateral, ^c. Q. E. D. Use and App. By aid of the last two Propositiona, D & E, bk. VI, the following Problem may be solred. Given four st. lines, any three of which are together greater than the fourth, to construct a quadrilateral, of which the sides shall be equal to those four given ft. lines, in a given order, each to each, and of which also its angular points lie in the circumference of a circle. By E, VI. The ratio of the diagonals is given; they are as the eimis of the rectangles under the sides ; By D, VI. The rectangle of the diagonals is given, being equal to the sum of the rectangles under the opp. sides ; By Prob. 5 Use and App. 13, VI., we find the two st. lines; in this case, they are the two diagonals equal to a given rectangle ; And by 22, 1, having now the sides and diagonals of the quadrilateral, we construct two triangles the sum of which will equal the required quadrilateral Prop. F. — Theor. If AB, a segment of a circle, ABD, be bisected in C, and from the extremities, A, B, of the base of the segment, and from C the point of bisection, St. lines be dawn to any point D in the circumference; the sum of the two lines AD + DB drawn from the extremities of the base, ivill have to the line DC drawn from the point of bisection the same ratio which the base B A, of the segment has to AC the base of half tKc segment. PROP. G. BOOK VI. 379 Dem. D, VI. 1, n. U, VI. D. 1 a D. VI. H. D, 2, 3. 1,11. 14, VI. Cone Kec. '.' ADBC is a qu. lat. inscribed in a 0, of which AB and CD are diagonals; .-. AD.CB + DB.AC = AB . CD. but •/ AC = CB, .-. AD . CB = AD, AC .-, AD . AC + BD . AC = AB . CD. But AD . AC, & BD . AC are the tectanglea contained by AC & AD + DB ; .-, rect. AC . (AD + DB) = rect. AB C. D; & ".• the sides of eq. rectangles are reciprocally propl; .-. AD + DB : DC = AB : AC. Therefore, if a segment of a circle be "bisected ^c. Q. E. D, SCH. This and the foUoiving Subsidiary Propositions have been adopted, with some slight alteiations from Bell's Plane Geometry, p. 194 — 198. Prop G. — Theok. If two points, E & F, Je taken in the diameter AC of a circle, or of the diameter produced, CP, such that the rectatigle, ED.DP, eoniainal by the segments intercepted between them and D the centre of the circle be equal to the square of AD the semidiameter ; and if from these points two St. lines, EB, FB, he inflected to any point whatever, B, in the circumferenece of the circle, the ratio of the lines inflected, EB :BE, will be the same with the ratio of the segments, PA : AE, intercepted between the two first mentioned points and the circumference of the circle. 380 GRADATIONS IN EUCLID. Dem. 17, VI. 6, VI. 4, VI. 16, V. altern. 17, V, dividendo. U, V. C I Pst.l&2,I. 1 Join DB, CB, AB &prod. FB to G. D. 1 H. 2 17, YI 3 D. 2. 4 6, VI. 5 4, VI. 6 16, V. 7 D,2. 8 17, V. 9 16, V. 10 D. 6. 11 11, V. 12 Eec. •.• FD . DB = AD2 = DB2 ; V FD:DB= DB : DE; & •.• in A s FDB, BDE the sides about the common Z D are propl. ; .-. A FDB is eq. ang. with A BDE, Z DBF = Z DEB, & DBE = Z DFB. • ,•, FB : BD = BE : ED, & alt FB : BE = BD : ED, or AD : DE. But •.• FD : DB or DA = DA : DE ; .-. div. FA : DA = AE : ED. & alt. FA : AE = DA : ED. Now FB : BE = AD : DE, .-. FB : BE = FA : AE. ■ • If two points be taken in the diameter, Src- Q.E.p. CoE.l. V FB:BE = FA: AE, .-. enjoining AB, % 3, VI., Z. FBE is bisected by AB. Cor. 2. Also on joining BC, the ext. vert. I, EBG is bisected byBC; for •.• FD : DB or DC = DO : DB, .-. eomp. FC : DC = CB : ED ; & '.• FA : AD or DC = AB : ED, .-. ex. mq. FA : AB = FC : CE. But FB : BE = FA : AB, .-. FB : BE = FC : CB ; .-. ext, Z EBG is bisected by BC. Q . E. D. D. 1 17,VI.&18,V. 2 17, V. 3 G, VI. 4 A, VI. PROP. H. BOOK VI. 381 Puop. H. — Theoe. If from one extremity, A, of AC the diameter of a circle ABC, a chord AB he draivn, and a perpendicular DE, to the diameter, cut both the diameter and the chord either internal^ or externally in D and P, the rectangle, CA . AD, under the diameter and its segment reckoned from that extremity A, is equal to the rectangle BA . AF, under the chord and its corresponding segment. Dem. i,ni. 15,1. 4, VI. 16, VI. D. H. H. 31, III. 15,1. 4, VI. 16, VI. Rec. '." Z.ABCis in a semic, /. ZABCisart. /_; but ADF is also art. Z., &, zBAC = zDAF; .". A ABC is eq. ang. with a ADF; .-.BA: AO = AD:AF: .-. BA.AF= ACAD. '• If from one extremity of the diameter, ^c. Q.E.D. Peop. K. — Theoh. If the angles, A ^ B, ai the base AB of a triangle ABC, he bi- sected by two lines, AD, BD, that meet, as in D, and the exterior angles at the base, foi-med by producing the two sides CA,OB, be similarly bisected by AG & BG ; then the two points of concourse, D & G, and the vertex, C, shall be in one St. line, which shall bisect the vertical angle, 4-CB. 382 GRADATIONS IN EUCLID. Pbeuminart Theorem, that may be demonstrated by superposition, " If two AS have two sides of the one respectively equal to two sides of the other, and the /. opp. one of the sides in the firstequal to the Z.opp. to the equal sida in the second, these As are equal when they are of the same species or affec- tion, i. e., when they are both acute-angled, both right-angled, or both obtuse- angled. See Sch. P, 7, VI Con. 12, 1. Dem. 26, 1. & Prel. Theor. C. 1 2 3 12,1. Draw DE, DF, DL ± to the sides AC, BO & AB ; & GrM, GN _L to the sides produced AM, BN ; & GK J. to the side AB: h. 1 2 i) 4 5 6 7 8 <) 10 Hyp. 26,1. Sim. D, 2, 3. D. 5, 0. Prel. Theor. Sim. Cone. Eec. ■.■ in AS ADB, ADL, ZEAD = ^DAL, Z. B at E & L also equal, & AD common. .-. AL = AE & DL = DE. So BL = BF, & DL = DF ; also AM = AK, GM = GK, BN = BK, & GN = GK. & •.• DE & DF each = DL, .-. DE = DF, & Sim. GM = GN Again, -.-in as CED,CFD,CD,DE=CD,DF, & Z. s at E & P are rt. ^ s ; .-. A CED = aCFD, & ZECD = zFCD. So aCMG= aCNG,& zMCG -NCG; .', the lines CG & CD coincide. '. If the angles at the base, &c. Q .E. 1). PROP. 1,, BOOK VI. 383 Prop. L. — Theor. In a triangle AOB, as in the last proposition, the segments, CM or CN, of each side produced that are intercepted between the vertex, C and the external perpendiculars, GM, GN, are each equal to 8, the semiperimeter of the triangle; the segments, CF. or CE, of these sides next the vertex, C, are equal to 8 — AB, the excess of the semiperimeter above the base AB ; and the segment AB or BF, of each of these sides next the base is respectively equal to S — BG, or to S — AC, the excess of the semiperimeter above the other side. Dkm. 26, 1. Ax. 6 §• 7., I. Ax. 2, 1. D. 1 2 3 9 10 11 12 13 26, 1. 26,1. Ax. 6, I. D.5,K.D.2,K Ax. 6 & 7, I. Add. Ax. 2. D. 5 & 4. Cone. 1 D. 7, 6&K,4 Cone. 2. D.7,&K3.D.2 Cone, 3. Rec. •.• AK = AM, & BK = BN, /. CM + CN = perimeter; & CM = CN = S, the semiperimeter, & AM = S — AC. Also 2 CE + 2 AE + 2 AM = perimeter = 2 CF + 2 AL + 2 BL. Now CE = CF, & AE = AL ; .-. 2 AM = 2 BL, & AM or AK = BL; adding KL to both, .•, AL = BK. And ■.• AM = BL, & AE = AL, .-. ME = AB, & CE = S - AB, I. e., CE = excess of semiperimeter above the base. And •.• AE =AL = BK = BN; .-. AE = 8 — BC; & .-. AM = AK = BL=BP,&AM=S-iy3, ,-. BF = S-AC i. e., each seg. of the side next the base is equal to the excess of the semiperimeter abote the other side. .■ the segment of each side, ^c. Q. E. fj. 384 GKADATIONS IN EUCLID. E. 1 2 C Hyp. 1 Prop. M. — Theoe. 27(6 area of a triangle is a mean proportional between tvx) rectangles, the sides of one of which are equal to the semiperimeter and its excess above the base, and the sides of the other equal to the excesses! of the semiperimeter above the other two sides. Dem. K & L, VI 4, VI. 29, 1. Cor. 1, 23, VI. 41, I. 13, 1. 32 .1 16, VI. Let ABC be a A , of ■whicli in the sides n OA&CBproduced CM = CN^ = S, the Semiperimeter; S— AB =CE; S— AC = AM & S— BC = AE, i. e. the excesses of 5 above AB, AC 6 BC; then CM . CE : W aABC=aABC/ : AM . AE. / Cone. K & L, VI. 29, I. 4, VI Corl,23,VI. C. 41,1. 13, I. 32, 1. Ax. 3, I. C. The fig. being con- structed as in Pr. K & L, VI. •.• ED II MG, .-, CE ED = CM: MG; but CM : CM = ED : ED; .-. CE . CM : ED . CM=CM . ED : MG . ED. But A ABC = AS ADC, ADB, & CDB; & these AS = ^C^ ^ AB.DL ^ BC.DF = ED . CM. And •.• zs EAL + MAK = 2 rt. z a, .-. Zs EAD + MAG = 1 rt. Z= Zs AGM + MAG; .-. z EAD= z AGM; & •.■ Z s at E & M are. rt. Z s, .". A s AGM, EAD are eq. ang.; PEOP. M.TT-BOOK VI. S85 9 10 11 4, VI. 16, VI. D3. H. 1, 2. .-. AE : ED = MG : AM, & .-, AE.AM = MG.ED. But CM . CE : ED . CM =CM . ED: MG . ED becomes CM • OE : a ABC = A ABO :AE.AM; .-. S(S-AB) : ABC = ABC:rS-AC)(S— BC) Q. E. D." tlien CoE. Let the sides opp. /.s A, B, C, be deaoted by a, b,& t; s (s-^c) : A ABO = A ABC : (s—a) (s—h); .: Area of A ABO =?= a/ s (s — c) (s-r^-a) {s — b)'. Use and At?- From this Prop, the Solution is. obtained of the Prohlem Given the three sides of a triangle to'find' the Area ; for, as in Cor. M. VI, th? continual product of the Semiperimeter into the excesses of the semiper. above the .three sides is equal to the square of the Area ; -whence the extraction of the square root gives the Area. Ex. The sides of a triangle, ABC, are AB 221, BC 255 & AC 238 feet ; required the A.reft. Here 221 + 255 + 238 X i = 357 the Semiperimeter. And 357—221 = 136 j 357—255 = 102, & 357— 238 = 119, the Excesses. Then Area' = 357 X 136 X 102 X 119 = 589324176. .•. Area of given A = ^/5S932ins = 24276 square feet. REMARKS ON BOOK VI. 1 . In a general way it may be sa;id that the Sixth Book, feeing an ■application of the Theory of Proportion propounded in the Fifth, treats chiefly of similar rectilineal and curriliiiear figures, or of figures that differ in size, but not in form. 386 GRADATIONS IN EUCLID. 2. The Book contains 33 propositions by Euclid, of which 10 are Problems, and 23 Theorems ; to these have been added 13 Subsidiary Theorems. 3. The 11th, 12th, 13th, 18th and 25th Propositions are the most important among the Problems ; and the 4th, 5th, 8th, 16th, 19th, and 31st, among the theorems. 4. As an approximate Classification for the Sixth Book, it may he divided, or rather arranged ; 1°. — Into Propositions which treat of the Proportion existing between the sides of triangles ; as Prop. 2, 4, 5, 6, 7, 8. 2°. — Into Propositions showjng the Proportions between the surfaces oi rectilineal figures; as Prop 1, 14; 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27 and 31. 3°. — Into those which relate to lines in Harmonical Progi'ession j as Prop. 3 & A. 4°.— To the angles at the centres or circumferences of circles ; as Pr. 33. 5°. — To Eectangles under the segments of chords ; as Prop. B, C, D, E, F, G,H. 6°. — To certain Properties of the triangle when its angles are bisected, and its Semiperimeter compared with its sides ; as Prop. K, L, & M. 5. In addition to the 10 Problems which Euclid gives, there are several other very useful Problems evolved from the Principles established, as 1. To divide any rt. lined surface. Use P. I. 2. To measure the height of an inaccessible object which casts an acces- sible shadow. Use 2, P. 2. 3. To divide a given line into proportional parts. Use 3, P. 2. 4. — To find a harmonic mean. Use 1, P. A. Use 2, P. 10. 5. — Eight Problems for ascertaining heights, distances, &e. ■ Use P. 5. 6. — ^By aid of a square to measure inaccessible distances. Use 2, P. 8. 7. — To divide a given line into any number of equal parts. Use 1, P. 9. 8. — To divide a A into any number ef equal parts. Use 2, P. 9. 9.— To divide a given line in a given ratio. Use 1, P. 10. 10.— To find a third harmonical progressional. Use 3, P, 10. 11. — To construct a triangle, one side, the angle opposite to it, and the ratio of the other sides being given. Use 4, P. 10. REMARKS ON BOOK VI. 387 12. — To di'aw a line which, if two other Hues were produced, woiil Use P. E, 30 — Given the three sides of a triangle to find the Area. Use P. M. 6. — As there are several useful Problems, so are there many important Theorems referred to, or contained in the Notes to Bk. VI. 1. — The Theory of Transversal Lines. Sch. 2, P. 2. 2. The y,. ;,u!ition of a rt. hne in Analytical Geometry. Cor. 2, P. 6. 388 GRADATIONS IN EUCa,ID. 3. — The Criteria of similarity in Triangles. Use P. 7. 4— The General Principle, if from the vertex of a triangle two lines bo dxawn to the base making the angles at the base or their supplements each equal to the Tert. angle then the triangle formed by those lines and the segments shall be similar to the whole triangle and to one another. Sch. 1, P. 8. 5. — Series of Magnitudes in continufid projportion. Use, 2, P. 1 1- 6. — ^Parallelograms are equiangular when their areas and sides are reci- procally proportional. Sch. 1, P. 14. 7. — Reciprocal ratios, and ratios of equality, Sch. H, P. 16. 8. — The conversion of the equality of two ratios into the equality of two rectangles. Sch. HI, P. 16. 9.— The leading Theorems for the Doctrine of Limits. Sch. IV, P, 16. 10. — ^Various deductions and processes Stated in other words than the original propositions, as Use m, P. 17. Sch. P.,S2. Sch. P. 28. Use 2°, P. 28. Use 2° and 3°, P. 29. 11. — The General Theorem respecting incommensurable Magnitudes. Sch. 2, P. 30. 12. — The General Theorem, by Pappus, under given conditions, of the equality in Area of a parallelogram or a circle on the base of a triangle to the sum of the parallelograms or circles on the other two sides. Sch. 1, 2, P. 31. 13.— Three Propositions substituted for P. 33, VI. Sch. 4, P. 33. 14. — The cutting of arcs proportionally. Use, 1, P. 33. 15. — The ratio of arcs in unequal circles, 'and of central angles: Use 2, 3, P. 33. N. B. — Additional Corollaries will be found under tbeir respective Propositions. 7. — The importance of the sixth Book to the Mathematician can scarcely be over-estimated; it is the head and crown of Plane Geometry. Whoever has mastered it, has added no little to his amount of useful knowledge, and consequently to the power of hii mind. Whether or not he adopts the quaint Motto round the Effigies of an old English Geometer,— "Liefb is Deathe and Death is Liefe,'' certain it is thathe has planted his feet on one REMARKS ON BOOK VI. 389 of the summits of Human Wisdom, and as he looks around, either to survey the Country already traversed, or to take note of the Heights still rising before him, he may inscribe upon his work, in the spirit of devotion ; tms Mm, I thank God, and take courage. so »0W so EVER. EUCLID'S PLANE GEOMETRY PEACTICALLY APPLIED. Synoptical Index to Books I — VI. Part I. containing Bks. I. — II. INTKODtlCTION. Gradual Growth of Geometiy and of the Elements of Euclid. Symbolical Notation and Abbreviations. Explanation of some Geometrical Terms. Nature of Geometrical Reasoning. Application of Algebra and Arithmetic to Geometry. On Incommensurable Quantities. Written and Oral Examinations, and Plan of Examination. BOOK I. The Oeometry of Plane Triangles. pp. 37 — ^43 Explanatory Notes. 43 — 44 A Pai-allelogram. Obserration on Mag- nitudes. 44 EucMD added three other Postulates. 44—45 Applicable to number and quantity as well as to magnitude. n. 8—12 45 — 46 Applicable especially to magnitude. 45 — 46 Super-position. Illustration of Lines parallel and not parallel. Lines converging and diverging. Angles interior, exterior, opposite, adjacent, vertical and alternate. Section. Page. L 1—5 n. 5—6 in. 7-9 rv. 9 — 18 V. 19—26 VL 26—28 VIL 29—33 Definitions, Definition. Postulates. Axioms. I. Notes. 1—35 A. 1—3 1-7 FOCETEEN PROBLEMS : Prop. 1, 2, 3 ;-9, 10, 11, 12 •,—22, 23 ;— 31 ;— 42 ;— 44, 45, 46. Prop 1. p. 47. To describe an equil. A on a given st. Line. Use or App. 1°,— To solve 2, 3, 9, 10 & 11, 1 ; 2°. draw an isosc. A- ; 3°. approximate to an oval ; and 4° to measure an inaccessible distance. 2. 48. To draw from a . a st. Line = a given st. Line. SoH. — ^Eight Solutions of this Problem. 3. 50. From the gr. of two Lines to cut oif a part = the less. 392 INDEX. ScH. — To lengthen the less to eq. the greater ; Use. — to construct u Scale of eq. parte, and to apply the principle of Representative values. 9. 63. To bisect a rectil. /. , i. e. to divide it into two eq. parts. ScH. — To bisect an arc of a ; & hj successive bisections to divide an /_ into any parts indicated by a power of 2. Use. To bisect the base of an isosc A ; and to constract the Mariner's Compass. 10. 06. To bisect a give st. Line. Sen — By successive bisections to divide a L. into parts, indicated by a power of 2. 11. 67. From a . m a L. to draw a L. at rt. /.s to it. CoE. Two lines cannot have a Common segment. ScH.— To draw aPerp. from the extremity of a L. Use 1. — To construct a square ; 2. On a given Line to describe an isosc. A of which the perp. height = the base. 12. 69. From a . without a L. to draw a Perpendicular. ScH. — When the . is over the extremity of the L. Use. This Prop, indi- spensable to all artificers, &c. 22. 86. To make a a of which the sides shall be eq. to three given st. Ls;j any two being > than the third. ScH. — Assumed that two 0s will have at least one . of intersection. Use.— Of most extensive use, — to malte one rect. fig. ^ or similai' to another; And, on a given L. to describe an isosc. A with sides each = twice the base ; &c. 23. 88. At a given . to make a rectil. ^ == a given rectiL Z. • Ust: 1. — Of the widest use in Practical Mathematics j 2. To construct k Linfi of Chords ; and by it to make an /. of a certain magnitude ; At the end of a L. to draw a Perp. ; to find the measure of an /. j and to draw As with certain parts given. 31. 103. Through a . to draw a st. L. parallel to a given st. L. ScH.— Demonstration of the 12th Axiom. Ute. — Prop. 31 required in all branches of Practical Mathematics ; — ^it enables the Surveyor to ascertain inaccessible distances, 42. 125. To de'scr. a / — 7 = a given A, and having one Z = a given Z- ScH. — Or, a A = a given CZJj and having an Z = a given Z- 41. 127. To a given L. to apply a / — 7 = a given A , and hftving an Z = a given Z ■ Usti.-^Greometrical Division illustrated. BOOK I. 393 45, 130. To deScr. a r-~i = a giTen rectil. flg., and having an /. i= a given Z • Use. — To measure the superficial content of any recti!, fig. ; 2, 'To change any rectil. fig. into a A, and then into a rect. of eq. Area ; and 3. To straighten a crooked boundary without changing the dimension. 46. 132. To describe a Square on a given st. L. . CoE. 1. The squares on eq. Lines are eq. ; and conuersdy. 2. Every parallelograBo, with one ft. / has all its /_s rt /;.?.• ScH. — Given the diagonal to construct a square. Use. — The Geometrical Square ; its constructions its use in ascertaining in- accessible distances, as heights. THIKTT FOUE THEOEEMS. — ONE LEMMA ; Prop. 4, 5, 6, 7, 8 j-lS, 14, 15, 16, 17, 18, 19, 20, 21 ;— 24,25, 26 ;-Lemma ;— 27, 28, 29, 30,-32, 33, 34, 35, 36, 37, 38, 39, 40, 41, j —43 ; 47, 48. 4. 52. Important. — If Wo as have each two sides and their included /. eq. the bases and other /.s are eq, and the AS equal. ScH.'^'the equality is perfect, not in Area only. USe.^-TJiB first criterion for establishing the equality of As S very fi:equently applied ; and useful, along with the Theory of Eepresentative Values^ for ascertMning inacces- sible distances. 5. 54. The /.s at the base of an isosc A are eq. ; and if the eq. sides are produced, the Z s on the other side also are 6q^al-. Cor. Every equal triangle is also equiangular. 6. 57. Conversely. If two /.sofa a ate eq. th6 sides opp. the feq. Z.S also are equal. Cob. Every equiangular A shall be equilateral. ScH. 1, 2.'— Cbnterse Theorems not universally true. 3. Two modes of Demonstration, direct and indirect. Use. To determine the Height of an object by its shado'v^. 7i 59. On the saiae side of the same base there calinotbe two A s, with sides terminated in one extr^itiity equal, and also the sides equal tel:ininated to. the other ^tremity. . The Dilemma, or Double Antecedent, Prop. 7, used ofijy to prove 8, 1. 394 INDEX. 8. 61. Important. — If two as have each the three sides eq. the Z. contained by two equal sides in one A equals the /_ contained by the two corresponding eq. sides in the other A , and the A s are equal. ScH. — The second criterion for equality of As. Use 1. To determine without a theodolite the /^ at a given . , made by Lines from two objects. 2. To measm-e and cut angles in a solid body. 13. 71. The Z.S made by one st. L. with another on one side of it, are either rt. £ s, or together = two rt. /. s. ScH. — Any number of Lines converging to a , in a L on one side of it make the /.s together = two rt. /.s. Supplement and complement, of an /. explained. Use Pr. 13 of frequent Use in Trigonometry, to de- termine the third £, when two /_s are given. 14 73. Conversely. — If at a- in a st. L., two lines on the opp. sides of it, make the adj. /.s together = 2 rt. Zs, the two Lines form one and the same st. L. 15. 74. If two st. Lines cut one another, the opp. or vert. z. s shall be equal ; and conversely. Cob. 1. The /.s formed by two lines crossing each other are together = 4 rt. l_s. 2. All the /_& formed by any number of lines diverging from a com. centre are together = 4 rt. /.s. SoH. — A developement of the def of an /.. The Converse true. Use 1. — To 6nd the distance between two inaccessible objects ; 2. To make one elastic ball strike another by reflection ; and 3. to determine the number and kind of polygons which on being joined cover a given space. 16. 76. If one side of a A be produced, the ext. Z is > eitherof the int. opp. /_&. ScH.— Each Z of a A is < the supplement of either of the other /_s. Use 1 Among other conclusions, — only one perp. from a , to a given L. 2. Prop 16, of great use in reducing As and other rectil. figures to rectangles. 17. 78. Any two Zs of a A are together < 2 rt. Zs. Explanatoiy of Ax. 12. Both Pr. 16 & 17 included in Prop. 32. 18. 79. The gr. side of every a is opp. to the gr. Z ■ An instance of the argument " a fortiori." 19. 79. Conversely. — The gr. side of every A is subtended by the gr. Z- BOOK I. 395 SCH.— Prop. 5, 6, 18 & 19, combined, prove, "One /. of a A is, > =, or < another /_, as the side opposed is >, ^, or < the other side oppo- sed ; and " vice versa." Use 1. The Perp. is the shortest L from a . to a given L. 2. From one , only two eq. lines to a given L. can be drawn. 3. All heavy bodies free to move seek the . nearest the earth's centre. 4. To construct a A, having the base, the less /. at the base, and the diff. of the sides given. 20. 81. Any two sides of a a are together greater than the third side. Cos. The diff. of any two sides of a A is less than the remaining side. N.B. — ^More assumed in the Cor. than is expressed in Ax. 5. Use 1. — Of aU lines from one . to another and reflected to a third ; those the shortest which make the /. of incidence = the /. of reflection. 2. tNatiiral causes act by the shortest lines ; hence, by means of a mirror to construct a A of which the Perp. is representative of the height of an object. 21. 84. If from the ends of a side of a A two Ls be drawn to a . within the A , these lines are < the other two sides, but contain a greater / , Applied in Optics, Astronomy, and Architecture. 24. 91. If two AS have two sides of one = two sides of the other, but the /. contained by the two sides of the one > the Z. contained by the two sides of the other, the base of that which has the gr. Z shall be > the base of the other. 25. 92. Conversely. — If two as have two sides of one = two sides of the other, each to each, but the base of one > the base of the other,>'the /_ opp. the gr. base shall be > the /. opp. the less base. Pr. 4, 8, 24 and 25^ may be combined ; "If two As have each two sides =: two sides, the third side of the one will be >, <, or =: the third side of the other, as the /. opposed in one is >, <, or = the /. op- posed in the other ; and viae versa. 26. 93. Important. — If two A s have two /.sand a side of the one = two Zs and a side of the other; the other sides shall be eq. each to each, and the third Z of the one = the third Z of the other., ' .; ScH. — The third criterion of the equality of As. In two or more As, any three parts of which one must be a side, being given equal, the equality of the other parts will follow. 396 INDEX. Use 1. — Applied to measure iBaccessible distances ; — 2 & 3, by the Theory of RepresentatiTe Values to iind the distance of two stations j — 4, to construct an isosc. A , the vert. ^ and perp. height of the A being given. 96. Lemma. A L., perp. to one parallel, is also perp. to the other. 27. 97. If a L falling on two other lines makes the alternate /.s equal, these two lines are parallel. ScH. — Since some curved lines, though they never intersect, are not parallels, another demonstration is given. 28. 99. If a L falling on two other lines niakes the ext. /_ ■= the int. and opp. /. on the same side of the line ; or the int. /.s together on the same side = 2 rt. /is; the two lines shall be parallel. ScH — The principle in Ax. 12 really is, — that two st. lines intersecting cannot both be || to the same L. 29. 100. If a L fall on two || st. lines, it makes the alternate /_ s equal ; and the ext. £_ = the int. and opp. Z on the same side ; and the two. int. Z, s on the same side together = two rt. Zs. Converse of Pr. 27 & 2S. ScH. — Methods of expressing Ax. 12, Definition of Parallel Lines. Use.— Pr. 27, 28 and 29 are applied to determine the earth's cu'Cumference. 30. 102. Lines || the same L are parallel to each other. CoK. — Two lines || the same L cannot pass through the same point ' equivalent to Ax. 12. 32. 105. Very important. — If a side of a A be produced, the ext. Z. = the two int. & opp. /_ s ; and the three int. Z. s of every A togetl).er = 2 rt. /.s. CoK. 1.— All the int /_s of any rectil. fig., + 4 rt. /_s = twice as many rt. /.S as the fig. has sides. This Cor. is of universal extent. 2.— All the ext. /.s of any rectil. fig. are together = 4 rt. /_B, Applicable only to convea; figures ; not to figures with re-entrant /.s. 3.— K two As have two Z_s of the one = two /_s of the other; the third /. of the one = the third /_ of the other. ScH. — 'Lardner's Euclid gives twenty four corollaries. Use. — This The&rem employed, 1. To determine the Parallax of a heavenly body ; 2, To give the representative height of a mountain; and 3, to construct any regular right-lined figufe. 33. 109. The lines joining the extremities of eq. and parallel lines, towards the same pArtS, are also eq. and parallel. BOOK I. 397 Use. — To ascertaiu tlie perp. height of a mouBtain, as wall as the distance from the base to the foot of tl5 perp. 34. 110. The opp. sides and z.s of l — 7 6 are eq. to one another, and the diagonal bisects them; and conversely. ScH. — If a qnadril. fig. have any too of certain ten, data, it will also have the others. By combining the ten, 360 questions are raised, Use. 1. — The construction of the parallel ruler depends on this Pfop. It is also useful, 2, to diyide a line into any number of eq. parts ; 3, to con- struct the Sliding Scale, called the Vernier or Noniits for measuring minute parts ; 4, to obtain the distance between two objects ; 5, to con- tinue a St. line when an obstacle intervenes ; 5, to diyide a CZD into two eq. pts. from a ■ in one of the sides, &c. 35. 113. Parallelogfams on the same base and .between the same parallels are equal, or rather equlYalent, to one another. ScH. — The equality of I 7 s proved by the Method of indivisibles. Use. — The Prop. appUed to convert a I 7 into a rect. of eq. area. The lineal- units in the base mnltipUed by the linear units in the altitude of a I 7 gives the Area. 36. 116. Parallelograms on eq. bases, and between the same Ijs are equal. Use. — The Construction of the Deojonai Scofe, and its application. 37. 118, Triangles on*the same base and between the same ||s, are equal. Half the product of :the base and altitude of a A gives the Area. 38. 119. Triangles on eq. bases and between the same ,||s, are eq. to one another. ScH. — By4ividiHg the base into eq. pts., and joining -the .s of division to the vertex a A is divided into eq. parts. Use. — This Prop, also enables us from any . in a side of a A to divide it into two eq. parts. 39. 120. Eq. as on the same base and on the same side of it are between the same parallels. The loci of theTCrtices of eq. as on the same base, foi-m a st.line 40. 121. Eq. A son eq. bases in the same st.line, and towards t"he same parts, are between the same paraillels. ScH. Cob. 1. — A || to the (base of a A thioiigh the middle .of one side ■will bisect the other side. '2.— The lines joining the middle . s of the three sides divide the A into four eq. As. 398 INDEX. 3. — The L joining the points of hisection of each pair of sides is eq. to half the third side. 4. — A trapezium = a I 7 of the same alt. and of which the base is half the sum of the || sides. The Area of a trapezium = ^ the sum of the |{ sides X the altitude. The Area of a Square := the Square of the lineal units in one side. Man7 oflier corollaries may be derived from Prop. 40. 41. 123. Important. — If a / 7 and a A be on the same base and between the same ||s, the 1 — 7 shall be double of the A ; and conversely. Use. — The Area of any figure iresolvable into As depends on this Propo- sition ; which enables us to find, — 1, the ai-ea of a A ; 2, the area of any rectilineal figure ; 3, the area of a polygon ; and 4th, the area even of a circle. 43. 126. The complements of the £37s -which are about the diam. of any / 7 , are eq. to one another. CoK.— The I 7 3 about the diag. and their complements are equiangular with the whole I 7 . Use. — To find a I 7 = a given I 7 . and having one side = a given line. 47. 135. Most important. — In any rt. Zd-A, the square on the side opp. to the rt. /_ is eq. to the squares on the sides containing the rt, /. . Cor. 1. — Hence, if the sides of a rt. /.d A be given in numbers the hypotenuse may be found. 2. — If the hyp. and one side be given, the other side may be found. 3.— If any number of squares be given, a square may be found = their sum ; or the multiple of a sq. may be ascertained ; or the difference of two squares ; or a sq. may be made = the 4, J &c. of a given sq. 4.— If a perp. be drawn from 'the vert, of a A to the base, the cUfference of the squares of the sides = the difference of the squares of the segments. 5. — If a perp. be drawn from the vertex to the base or base produced, the sums of the squares of the sides and alternate segments are equal. 138. ScH — A Practical Illustration of Prop. 47. I. 139. Use 1. — Combined with other propositions, the 47. I. is applied 1°, to make a rectil. fig. similar to a given rectil. fig. ; 2°. to make a double, or the half of another ; 2. To construct the Chords, Natural Lines, Tangents and Secants of Tiigouometrical Tables;. 3. To find right triangular nmnbers ; & 4, 5, To ascertain heights and distances from the curvature of the earth. 48. 142. Conversely. — If the sq. on one of the sides of a a be eq. to the squares on the other two sides of it, the /_ contained by the two sides is a rt. /_.' BOOK I. 39& Extension of the Proposition, — the vert. ^ of a A is <, =, or > a rt. /. as the sq. on the base is <, =, or > the sum of the squares of the sides. Remarks. The First Book founded entirely on the Defs. Posts. & Axioms. Only a few of the propei-tieg of a mentioned. A threefold Division of the Book ; 1°. from Pr. 1 to Prop. 26 the properties of As unfolded ; 2°. from Pr. 27 to Pr. 32, those of parallel lines '; and 3°. from Pr. 33 to Pr. 48, those of parallelo- grams. 4. 144. The most important Propositions are Prop. 4, 8, 26, 32, 41 and 47. 1. 143. 2. 144. 3. 144. BOOK II. The Pkopbkties of kt. /.d. I 7 s. ok Kectangles. 145. A L may he cut internally or externally. Magnitude the subject of Geometry, — Algebra and Arithmetic furnish, not proofs, but illustrations. The Numerical Area of a Eectangle =r ab, the altitude being represented by a, the base by b ; that of a A =: 4 oi. Def. 1 & 2. p. 146. A rect is contained by any two conterminous side.s ; and in every i 7 , any / 7 about a diam. -f- the complements is called the Grnomon. Axiom, p. 146. The whole Area = the Areas of all the parts. TWO PKOBLEMS. — Prop. 11 and 14. 11. 171. To divide a L. into two parts, so that the rect. under the whole and one of the parts shall be eq. to the sq. of the other part; thus, in L — ! P the L. is cut in the . H so that AB . HB = AH*. Or, as in 30, VI. to divide a L. in extreme and mean ratio. The Algebraical and Arithmetical Solutions. ■too INDEX. CoE. I. To cut a L. as AB, in extreme and mean ratio, it must first be produced in extreme and mean ratio, i. e. CF . FA («'—., — ,G- must = AB'' II. — When a L. as AB, or its equal AC, is cut in extreme , and mean ratio, the rect under the whole L. and its "■ less segment = the sq. on the greater segments ; tlius, -, AC . ( AC— AF) = AF', or AC . HB = AIi= '^ III. — Also the rect. under the whole L. as AC or AB and its greater segment =: the difference between C K D their squares, or AC . AF or AH = AC — AW. ScH. — Let L. be a line cut in extreme and mean ratio, g, the greater seg. I the less, and d the difference : then 1° 1? + P= 3 g"; 2°.(1, + l)' = Sg"; 3°.'L.d= g.l; and4 °. P =g .d. Use. — This Proposition is of fi-equent application, as for the construction of pentagons and the regular bodies called the Platonic Solids. 14. 179. To describe a square that shall be eq. to a given rectil figure A; i. e. EH^ = rect. BD = A. Use. 1 To find a mean propl. between two given lines ; 2 to approximate to the sq. of curve-lined figures ; 3. and to calculate the Areas of all j'liine figures. TWELVE THEOSEMS. Prop. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ; ^-12, 13. 1. 146. If there be two st. lines, one A, undivided, the other BC, divided into any number of pts. as BD, ED. EC, then ■ A.BC = A.BD + A.DE + A.EO. N.B. The Alg. and Arith. illustrations are attached to each proposition. Use. Applied to the Demonstration of the Rule for the Multiplication of numbers. 2. 148. If a St. line AB, be divided into any two pts. AD, DB then the rectangles AB . AD, + AB . DB = the sq. on AB. Numerical Multiplication may also be proved by this Prop. 3. 149. If a line, AB, be divided into any two pts. AD, DB, then the rectangle AB . DB = AD . DB + DB^; or AB . AD = AD . DB + AD=. CoR. 1. AB=-DB= = (AB + DB) (AB-DB). 2._AD=— DB» > (AD— DB)'' by 2 DB . (AD^DB). Also applicable to the proof of numerical Multiplication. BOOK II. 401 4. 150. If a line AB, be divided into any two pts. AD, DB, theit AB2 = AD2 + DB2 + 2 AD . DB. Cos. 1. The parallelograms about the diam. of a sq. are also squares. -The square of a line is four times the square of its half ; thus AB' 2. — ^HaJf the sq. of a line = 2 the sq. of half the line ; thus AB» e- rAB\= 4.-nThe sq. of a line will be equal to the sum of the squares of the- parts -f- double the reot. under every distinct pair of parts. Use l.-^In Algebra the square of a binomial, as (x + yy := x' ,+ y^ + 2 xt/. 2.— This Prop, points out a practical way of extracting the Square Root of a number. C D -5. 154. If a line AB, A- ■ B, be divided into two eq. parts, AC, CB, and twoimeq. parts, AD, DB; then tte rectangle AD . DB + CD^ = OB^. CoE. The difference of the squares of two uneq. lines, AC, CD = the rect^ under their sum and diff.; i. e. AC^— CD^ = (AC + CD) (AC— CD). Zardner's; Corollaries to this Prop, are six. ScH.— The principal properties connected with the eq. and uneq. division of a line. Use. — ^We may apply this Prop. 1°. To find the diff. between the squares igftuio uneq. numbers without squaring them ■ 2°. To find quantities in Arithmetical Progression ; 3". To establish Prop. 35, III ; and 4°. To find tlie value of an Adfected Quadratic Equation in Algebra. 16. 157. If a L. AB be bisected in C, and produced to any.D; as A. _*^ 5 ? ' ^^"^ '^^ '^^'^^^ AD . DB + CB2 = CD2. jCok. If a line AD be drawn from the vert. A .of an isosc. A to the base, or its production, •the diff. between the squares of this line and the side of the A is the rect. under the segments of the base ; thus AD'' (v; AC= = BD . DC. ^ ,- . ED F CIS E CD* TTsB.— By aid of this Prop, the diam. of the earth may be measarecl^ 3! 2 402 INDEX. 7. 160. If a L. AB, be divided into any two parts, as AD, DR then ABs + DB^ = 2 AB . BD + AD«; or AB2 + AD2 = 2 AB . AD + DB^. CoE. 1. If AB and BD be considere d » jtwo independent lines, AD beinff their diff. ; then AB^ + BD' = 2 AB . BD + (AB— BDf 2.— Also(AB + BD)2 ; (AB^ + BD'); and (AB-BD)'arc ia Arith. Progression ; the com. diff. being 2 AB . BD. 8. 161. If a L. AB, be divided into any two parts, as AD, DB,. four times the rect. under the L. and one of the pts. + the sq. of the other pt. = the sq. of the L. made up of the whole and that part; i. e. 4 AB . BD + AD^^ = (AB + BD)2 ScH. 1. Otherwise the sq. of the sum of two lines = 4 lines the rect. nnder them + the sq. of Aeir diff. i. e. (AD + DBV = 4 AD . DB + (AD— DB)' 2. — Four times the sq. of half the sum = 4 times the rect. under the • lines + 4 sq. of half the diff.; i.e.4 (4:5Jl^)^=4AD DB- + 4 (^^r Use. — ^The above principles are applied to Algebra and to the extraction ot the sq. root. 9. 164. If a L. be divided into two eq. pts, and also into two un- C D equal parts, as A i- 1 f )B,- then the squaresof the twouneq. pts. together = doubleof the sq. of the .half line, and of the sq. of the line between the • a. of section ; i. e. AD^ + DB^ = 2 (AC^ + CD^ J SCH.-Qr,l. AD=+DB'=2 (AD+DBy^3 ^AD-DB^, 2. AD»+DB'= ( AD + D'B )' ^ (AD-MB y 10 167. If a L. be bisected and produced to any point, as, A C B D, then the sq. of the whole line thus produced + the sq. of the part produced = double ot the sq. of the half line + the sq. of the line made up of the half and the pt. produced ; thus AD^ + DB^ = 2 (AC^+CDs) ScH. — ^Props. 9 and 10 are applicable to Algebra. Use.— Prop. 1—10 contain the whole theory of the rdations of rectangles an^ squares foxsaei by lines ani their pwts. 403 1°. Given the srtm and diffi of two Ms. to find them. 2°, If the Area be divided by a side the quotient = the other side. 3°. Of the five quantities depending on a rect, any two being given, the sides can be found. 12. 173. Important. — In obtuse /_d as, if a perp. be drawn from either of the acute / s to the opp. side produced, the sq. of the side subtending the obt. /_ is > the squares of the sides containing the obt. /_ by twice the rect. under the side on -which fi C !> ' the perp. falls, and the line intercepted without the A between the perp. and the obt. / . ; as. AB2 = BC2 + ACS + 2 BC . CD. or AB» > BC^ + AC= by 2 IBC .CD. XJSE. — ^By this Prop, the Area of a A niJsy be ascertained when the three sides are known. 13. 175. In every A, the sq. of the -side subtending cither of the acute ./.sis < the squares containing that araite Z. by twice the rect. under either of these sides, and the L. intercepted be- tween the perp. let fall upon it&am the opp /_., and the ■.acute /_ ; i. e., 1°. AC2 < (AB2 H- B02; by 2 BC.BD; 2°. or, by 2 BD . BC; and 3°. by 2 BC. BC. Thus is obtained the measure of the sq, of the side subtending an acute /_. CoK. If in the fig. to Case 2°.. a perp. CG be drawn from./. C to AB, then the rect. AB . GB = the rect. BC . DB Sch;— Prop, 12 and 13 contain the Elements of Trigonometrical Analysis. Use.— To obtain the perp, when the throe sides ofa A are given; l°."when the perp. falls within the base ; 2°. without the base The Area of the A = ?P-±5C ^ AD ; or ^^-'"^ xAD. Kemabes. 181. Of the fouiKteBH Propositions, the Itea first contain i?he theory of the relations of the rectangles and squares on divided lines ; the twelfth and thtfteenUi the theory of the relation between the sq. ot any one side of a A, and the squares of the other two sides. 404: INDEX. 2. 181. Lines cut into any two parts, in Prop. 2, 3, 4, 7 and 8. 3. 181. Lines cut into two eg. and two uneg. parts, in Vrop. 5, 6, 9 and 10. S>/nopsis of Book II. Case I- VIII. Pages 182 - 186. Practical Eesults. I. 188—197 Problems 1—31 for the Construction of Geom. Figures bks. I and II. n. 197—199 Problems 1—10 „ „ bk. Ill 200 — 203 Problems 1 — 18 „ „ bk. IV. 204—207 Problems 1—11 „ „ in. 207—^11 Principles of Constmction ; 1°. Por Geom. Instruments to measure Distances and Angles ; 2°. Por Geom. Pigures to exhibit the representative values of actual magnitude and space. IV. 211 — 213 Principles which, 'without requiring that we should measure all the boundaries of a Surface, enable us accurately to calculate 1°, Lines or Distances ; 2°. Angles ; and 3°. Mag- nitudes, or Areas. APPENDIX. I. 215-219 Geometrical Analysis, Eules for Conducting, and Examples. 11.219 — 227 Geometrical Exercises ; Series 1 Problems and Theorems in Bk. I ; Problems and Theorems in'Bk. II. Series II Pro- positions, including Problems and Theorems, not fully proved, or not inserted in Bks. I. and II. PART n. Containing Books III, IV, V, §• VI. Preface Symbolical Notation and Abbreviations. BOOK HL PEOPBRTIES or THE CIECLE AND OP LINES IN AND ABOUT IT. 1 The word circle employed in two senses ; and certain Properties assumed from experimental knowledge. 2 The foundations of Trigonometry, Plane and Spherical. 3 Summaiy of Book HI from Billingsley's Euclid. Def. 1 — 12 pp. 3 — 6. Explanatory Notes. Ax. A, 6. 405 SIX PROBLEMS. Prop. 1 ;— 17i— 25;— 30i— 33, 34. Prop. 1. p. 7. To find tte centre of a circle. CoE. If in a one' L. bisects anotlier at it. /_s, the een. of the is in the bisecting line. Sen. — The rigour of the reasoning requii'es a previous proof of the conditions on which a point is within or without a cu'cle. 17. 39. To draw a L. from a giren point, either without or in the 0ce, which shall touch a given circle. ScH. — From the same . two eq. tangents may be drawn. Use.— Tangent Lines, how drawn practically ; — of frequent use in Trigo- nometry. 25. 54. A segment of a circle being given, to describe the circle of which it is the segment. ScH. This problem is equivalent to the following ; — to inscribe a A in a 0; or, to make a pass through three given . s not in the same line. Use. — Applied for eonstvucting an arch, — for cutting stone, wood, metals &c., and for finding the apogee of the moon, and the eccentricity of the earth's orbit. 30. 64. To bisect a given arc of a circle. Cor, To divide an arc into' any number of eq. pts. that are powers of 2. ScH. 1. — Except in the case of a quadrant an arc cannot be cut into 3, 5, &e., eq. pts. 2. To trisect a quadrant. The trisection of an /_ fiot effected by Euclid's Geometry; — ^what required for the trisection of an arc ;— the trisectrix one of the trochoidal cm-ves. 33. 72. On a given L. to deseribe a segment of a circle, which shall contain an /. = a given rectil. /_ - Sen. — To divide a into any number of eq, pts. of which the perimeters also are equal. Use — Of extensive Apphcation, as, 1. Given the distances of three land- marks to find their distances from the place of observation. 2. To ascertain the distance from two stations ; 3. Given the base and vert. /^ to find the locus of the vertex ; 4. Given the vert. /., the base, and the area to construct the A ; 5. Through three . s to draw lines so as to malce an equil. A ; 6. Given the /_ =p the vert. /., and the base, to find the locus of the vertex. 7. Given the base, the vert. /. and the pei-p. from one end of the base to the opp. side, to construct the A. 34. 76. Prom a given circle to cut off a segment, which shall contain an Z. = a given rectil. Z. ■ 406 INDEX. Use 1 . — By Prop. 33 and 34, if three observations be taken, the eccentricity of the earth's annual orbit, and its aphelion may be found . 2. Also iu optics, to ascertain the , where two uneq. lines will appear eqnal. THIKTT-ON-E THEOEEMS. Prop. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. 16;— 18, 19,20, 21, 22, 23, 24;— 26, 27, 28, 29;— 31, 32;- -35, 36, 37. 2. 9. If any two . s be taken in tte 0ce of a circle, the st. line which joins them shall fall within the circle. CoR 1 . A St. line cannot cut the 0ce of a Q in more than two points. 2. A St. lino which touches a circle meets it only in"one point. 3. A circle is concave towards its centre. ScH. — Commandine's direct proof of Prop. 2. Use. — A globe on a plane surface touches the plane iu only a single point. 3. 11. If a st. line through the cen. of a bisect a st. line not through the centre, it shall cut it at rt. /. s ; audconverselg CoE. 1. A L. bisecting a chord at rt. /_s passes through the cen. of tie circle, 2. AU chords || to the tang., at either extremity of the diam., are bisected by the diam. 3. The Line bisecting the com. chord of two 0s at rt. /_a passes through the centres of both 03. 4. 'VYhon in a there are several chords || to each other, the locus of their . s of bisection is in that diam. which is at rt ^s to them ; and if the L. which bisects one chord be a perp., that line bisects all the par. chords at rt /^s. Use. Various, as 1 . Given a to find its centre ; 2. Given an arc, to find the cen. of the of which it is an arc; 3. Through threes not iji a St. line to draw a ; and 4. In Trigonometry. 4. 14. If in a two st. lines cut one another, which do not pass" through the centre, they do not bisect each other. CoE. No / 7 except a rectangle can be inscribed in ar circle. Use. — Applied to detennine the eccentricity of the Sun's apparent path ; and the centre of rotation of an eccentric wheel. 5. 15. If two circles cut one another, they shall not have the same centre. Son, Or, concentric 0s cannot meet, — and that with the less rad. is within the other. 6. 16. If one circle touch another internally, they shall not have the same centre. ScH.— Props, and 6 may be combined into one. BOOK III. 407 7. 17. If any.wHch is notthecen. betaken inthediam. of a 0, then, 1st, of all St. lines from it to the 0ce, the greatest is that in •which the cen. is, and the other pt. of that diam. is the least ; and 2nd, of any other st. lines, that which is nearer to th^line through the centre is always greater than the one more remote ; also 3rd, those lines which make eq. Z s with the diam. are equal ; and, 4th, from the same . there can be drawn only two equal st. lines, one on each side of the diam. Son. — Maximum and minimum of a revolving line cut by a ©ce."" 0SE.— Of the arcs of great 0s &om the pole to the hcffizon the greatest is that part of the meridian which passes through the zenith. The Prop. may be used for showing aphelion and periheSon distance. S. 20. If amy . be taken without a 0, and st. Unes be drawn, from it to the 0ce, whereof one passes through the cen. ; 1st, those which make eq^i Zs with the line through the cen, are equal ; 2nd, of those which faU upon the concave Qce,tliegreatestis thatwhich passes through the cen. ; and of the rest that nearer to the one passing through the cen. is always greater than one more remote ; but, 3rd, of those which fall Upon the <;onvex 0ce, the least is that between the given , without the and the diam., and of the rest, that nearer to the least is always less than one more remote ; and 4th, only two eq. lines can be drawn from the same . to the 0ce, one upon each side of the line passing through the centre. ScK. 1. The eoiicave and convex pts of the0ce determined by tai^nts. 2. What is meant by the distance of a . &om or to a line; Of all st. lines from a . out of a line to a given line, the perp. is the least, andthe lines neaa'er to the perp. are less than those more remote ; and to a line on one side of the perp. there is only one eq. line on the other side of the peip. Use.— If a tang, and sec. be drawn to the same . , the tang, is < sec., but > ejrt. pt. of sec. 2. By Pr. 7 & 8, and Ax. A, HI, 1st, When one is within another, without touching, the distance of the centres, are < the diff. of the radii ; and conversely. 2nd, When two 0s lie each without the other and do not meet the distance of the centres is > the sum of the radii ; and conversely. S. 24. If a . be taken within a 0, from which there fall more than two eq. st. lines to the 0ce,. that point is the centre of the 0. 408 INDEX. CoE. 1. I'rora any other . than the cen. only two cq. st. lines can be drawn to the 0ce, whether the . bo within, or without the 0. 2. From thres . s given not in the same st. line the 0ce of the Q may be found. ScH.— This Pi op. gives the criterion tor determining the cen. of a 0. Uses. Applied, 1st, To draw a through three given . s ; 2nd, To find the cen. of a given ; and 3rd, To deteimine the centre of an arc of a©. 10. 25. One 0ce of a cannot cut anotlier in more than two parts. Scs. — If the 0ceB of 0s coincide in three . s they will coincide in every point. 11. 26. If one touch another internally in any . , the st. line joining their centres, being produced shall pass through the • of contact. ScH. — When the distance between the centres = the diii. of the radii the 0s touch internally. ^ XTsB 1. — A practically useful method for drawing an oval on any given major axis may be derived from this Prop. ■2. On the same principle a Spiral is described. A Spiral defined and some of its vai-ieties. 12. 29. If two 0s touch externally in any ., the st. line joining their centres shall pass through that . of contact. ' Use.— The drawing of a Serpentine Line, or cima recta, depends on this Prop. 13. 30. ' One cannot touch another in more points than one, whether it touches it on the inside or the outside. ScH. 1, 2. — A direct method of demonstration substituted for the indirect. Use. — Prop. 10, 11, 12 and 13, explain the motion of the Planets in Epicycles. 14. 32. Eq. st. lines in a are equally distant from the centre ; and conversely. ScH. — A principle employed in Prop. 14- & 15 is this ; if A + B := C + D, then if A = C, B = D ; if A > C, B < D j if A < C,B >D. 15. -33. The diam. is the greatest st. line in a ; and of all others that nearer to the cen. is always greater than one more remote : and the greater is nearer to the cen. than the less. ScH. The longest chord is the diam. ; the shortest through a given , , that perp. to the diam. BOOK III. 409 Use. Prop. 14 & 15 employed, 1°. To show that 0s of latitude diminish towards the poles ; 2°. In constructing the Astrolabe. 3°. In determi- ning the position of greatest effect for a given lever. 16. 36. The st. line at rt. Zs to the diam. of a 0, from the ex- tremity of it, falls without the ; and no st. line can be drawn from the extremity between that st. line and the 0ce, so as not to cnt the ; or, which is the same thing, no St. line can make so great an acute Z with the diam. at its extremity, or so small an /. with the st. line at rt. -Zs to it, as not to cut the 0. CoK. 1. If a St. line be drawn at rt /_& to any diam. of a 0, from its ex- tremity, it shall touch the at the extremity ; and a st. L. touching the at one , shall touch it at no other point. 2. By 28, 1, st lines at rt. Zs to the extremities of the same diam. are parallel. 3. Tangents to a from the same , are equal. ScH. The 16th iProp. may be proved directly. Use. By this Prop, we can prove, 1°. The infinite divisibility of hnear magnitudes ,• 2°. The distances and heights of objects on the earth's sur- face, when they are situated on the verge of the natural horizon. 18. 41. If a st. L. touches a 0, the st. L. fromthe cen. to the . of contact shall be perp. to the line touching the 0. ScH. Prop. 16 & 18 may be regarded as the converse of each other. Use. To draw a tang, to each of two given circles. 19. 42. If a st. L. touches a 0, and from the . of contact a st. L. be drawn at rt. Z s to the touching line, the cen. of the shall be in that line. ScH. In concentric 0s all chords of the greater touching the less are eq. and bisected at the . of contact. Use. Tangent lines are employed, 1°. in Optics to detei-mine the part of the eartli enlightened by a meteor, &c. 2°. To ascertain the earth's diam. ; 3°. To explain the Phases of the Moon j 4°. To trace the of the physical horizon, and 5°. in Dialling to calculate the Hour Lines. 20. 44. The Z at the cen. of a is double of the Z at the0ce on the same base, .i e., on the same part of the 0ce. CoK. Any Z at the 0ce is measured by half the arc on which it stands. ScH. 1. The reasoning assumes, that among 4 Ms, A, B, C, D, if A = 2 B, and C = 2 D, then A+ C = 2 (B -h D) ; also that if M = 2 M', and (M— o) := 2 (M— a) then the remainder of M. = 2 the remainder of M'. Another method of proof given for Case 3, Prop. 20. 2. If Euclid's def of an Z l^e stiictly adhered to, Prop. 20 not nniversaUy true. 410 INDEX. 3. Demonstrations may sometimes be arranged as a Simple Equation. tTsE. This Prop, applicable to Trigonometry and Astronomy. 21. 47. The /.s in tlie same segment of a are eq. to one another. Cob. If on the base of a A, there be described a seg of a 0, the vertex of the A sliallfall without, or within, or upon the arc, as the vert. /. is <, >, or =: the /_ in the segment. Sen. The /_ at the ©ce has for its measure one half of the arc on wliicli it stands. Use. Applied, 1°. To construct a building so that the spectators may all SCO an object under the same /. ; 2°. To bisect an /. ; 3°. To con- struct a fig, representative of the distance of the place of obser\'ation trom an object. 4°. To draw the arc. of any on a large scale. 22. 49. The opp. Z.S of any quadril. figure inscribed in a are together eq. to two rt. /is; and Cor. 1. conversely. CoE. 2. If any side of a quadril. in a be produced the ext. /_ = iut- opp. /_. 3. If two chords cut off sim. segments from the same or different 03, the other segments will also be similar. 4. If opp. /.s of a quadril. be equal they must be bath rt. ^s. Use. Applicable to the Construction of the Tables of Cords, and in. Trigo- nometiy. 23. 51. On the same st. L. and on the same side of it there can- not be two sim. segments of s, not coinciding with one another. Sen. Tliis Prop, the same in principle as the 7th bk. I. 24. 52. Similar segments of 0s on eq. st. lines are equal to one ' another. Con. 1. Sim. scgs. having eq chords have also eq, axes. 2. Sim. segs. having eq. choi'ds are parts of eq. circles. 3. If the radii, and Z_s of sectors are eq., the sectors themselves arc equal. Use. By this Prop, curved lined ffgnres are often reduced to rectilineals. 20. 56. In eq. 0s, eq. Zs stand upon eq. arcs, whether they be at the centres or the 0ces. CoR. 1. If the opp. /.s be eq. their opp. diagonal must be a diam., andthc seg a semicircle. 2. In the same or eq. 0s one central or circumferential Z. is <, =, or > another, as the arc of the one is <, :=, or > the arc of the other. 3. The diameters intersecting at rt. /.s divide the 0ce into four eq. arcs. 4. When the sum of the central /_s ^ 4 rt. Z.s, the sum of their arcs = the whole 0ce. BOOK III. 411 5. When the snm of tke /.s at the 0ce =2 rt. ^s, the som of thfiiii arcs, also =: the whole 0oe., 6. Sim. arcs, of eq. 0s are equal. 7. Par. chords of a intercept eq. arcs. 8. If two chords intersect within a 0, the sum of the intercepted arcs ^ the a»c which the /_ woiid intercept at the 0ce, that is eq. to the /. under the chords. 9. If two chords intersect a* a . without a 0, the difference of the arcs which they intercept is =: to the arOj which an £_ would imtercept at the 0ce that is eq. to the /_ under the chords. Use.. Applied to find, the true central /_ of an imperfect theodolites, 27. 59. In eq. 0s tte Zs which stand, upon eq. arcs are eq. to one another, whether they be at the centres or the 0oes. CoE. 1. In the same or eq. 0s,'the sectors on eq. arcs are equal ; and conversely. 2. If the chords of a are parallel, they intercept eq. arcs ; and vixie versa. SCH. 1. What iS' true. of eq, 0B, is true of eq. aica in the same 0. 2. The sum of the /_s at the cen. of a = 4 rt Z.3 ; and the sum of the /.s at the circumference = 2 rt /.s- Use. By this Prop.. a . parallel through agiren . may he drawn. 2. To ascertain the Area of a Sector. 28. 61. In eq. 0s, eq., st. lines cutoff eq. aicS,. the greater. =- the greater, and the less to the lessi 29. 62. In eq. 0s, eq. arcs are subtended by eq. st. lines. Cor. 1. In the same or eq. 0a, eq. sectors stand on eq. arcs; andcon- versely. 2. St. lines which intercept eq; arcs are parallel ; and par. St. lines intercept eqi arcs. Use. IJa Spherical Ki^onometryProp. 26^ 27, 28 and 29, are of continual use. 3L 66. In a 0, the Z in , a semicircle is a rt. Z, but the Z in a seg. > a semicircle iS' < a rt. Z ? and the Z in a seg,. < a semicircle is' >•■ a rt. Z • CoK. If one Z of a A be eq; tO' the other two, it is a rt Z • ScH. 1. Canversely, The seg. containing an acute Z is > a semicircle ; and the seg; containing an obtuse Z_ is < a semicircle. 2. Lardner's elegant and brief demonstration of 31. III. 68. Use. Problems derived ; 1°- From a, .in a, Ene or at its exti-emity to draw a pei-p. ; 2°. Prom a . without a line to draw a perp,,;, 3°. Prom a . without a to dra\^ a tangent ; 4°. By means, of a Square to find the centre of a ; 5°- Tb try if Squares are exact. 412 ISDEX. 32. 69. If a st. L touches a ©, and from the . of contact a st L. be drawn cutting the ; the /_ s which this line makes with the line touching the 0, shall be eq. to the /.s in the alternate segs of the 0. Con. 1. Conversely, If from the end of a line cutting' the 0, &c. 2. If two or more 0s touch each other, and through' the , of con- tact two St. lines be drawn meeting their 0ce3, the chorda of the inter- cepted arcs will be parallel. 3. If two or more 0s touch each other, at a com. point of contact, any line passing through the . of contact will cut oiF sim. segs. from each. 4. In an equil. A, if the sides be bisected, and st. lines be drawn joining the . s of bisection, of those linos, two will be tangents to the 0, which passes through ilie ends of the other line, and the ang. point opp. to that line. 4 0. Tangents through the extremities of the same chord, make the /.s on the same side equal. 6. If tangents are par., the line joining the . s of contact is a diara. Use. 1. This Prop, preliminary to the 33rd, and required for Tarious Theorems. 35. 77. Very important. If two st. lines cut one another within a 0, the rect. under the segments of one of them = the rect. under the segments of the other. CoE. Conversely, If the rectangles be equal &c. ScH. Or, If two chords of a cut one another, the rectangles under their segs. terminating in the . of section shall be eq. The terms ordinate and abscissa explained. Use. I. If of two eq. 0s, the centres be each on the 0ce of the other, and a com. chord be drawn || to the line joining the centres, then the lines joining the . s where the com. chord cuts the 0s and the extremities of the lines joining the centres, form I 7 s ; 2. To find a line which is a fouith propl. to three given lines, or a third propl. to two given lines. 36. 81. Important. If from any . without a Q two st. lines be drawn, one of which cuts the 0, and the other touches it; the rect. under the whole line which cuts the 0, and the part of it without the 0, shall be = the sq. of the L. which touches it. Cor. 1. If from any . without a there be drawn two St. lines cutting it ; then the rectangles trader the whole lines and the parts of them without the shall be = to one another. CoK. 2. If from, the same, two tangents be drawn to the same 0, they are equal. ScH, Or, If any chord of a be prod, to cut a tang, to the same 0, the sq. of the tang, shall be = tiie rect. under the segs. of the chord produced. BOOK lA'. 413 Use 1. In any semicircle if from the extremities of tbe diam. chords be drawn intersecting within the semicircle ; then the sum of the rectangles under any two such intersecting chords and the sections of the chords between the extremities of the diam. and the intersecting . , shall = the sq. on the diam. Use. 2. The Art of taking a true Level is deduced from this proposition ; 1° Plan ofa Field Book; 2° The measuring of an Ascent. 3° Correction for curvature. 4° Deviation of the horizontal from the true level. 5° Distances of the horizontal boundary, 6° Sum of the horizons wlien taken. 7° The earth's diam. ascertained. 37. 85. If from a . without a there be drawn two lines, one of which cuts the 0, and the other meets it^ if the rect. contained by the whole L. which cuts the 0, and the part of it without the 0, be eq. to the sq. of the line which meets it, the L. which meets shall touch the 0. CoK. Tangents from the same . without a ai*e equal. Use 1. Through two given . s to describe a touching a giten 0. 2. Prop. 35, 36 and 37 are amongst the most important in Plane Geometry. The earth's diam. calculated from them by Maurolico in the 16th century. Eemarks. 1. 87. The Propositions of Book IH, classified under jfiue general heads. 2. 87. Of the 37 Propositions six only are Problems. 3. 88. Twelve others have been deduced. 4. 88. Problems might be given for drawing 0s which are tangents to two or three given 0s, &c. 5. 89. Several of the Principles of Levelling and Surveying, and of making Geographical and Astronomical Observations laid down. BOOK IV. METHODS OF CONSTEUCTIKG EEGTJLAR STEAIOHT-LIKED EIGUEES IS AND ABOUT A CIRCLE &C. 91. Excepting Pr. A, Theor., the fourth book consists entirely of Problems. This book of essential service in Astronomy, and in Cvni. and Militaiy Engineering. Def. 1—7 pp. 92, 93. Euclid's Definitions. 8 — 10 97. Definitions additional to those oi Euclid. 414 SIXTEEN PKOBLEMS AND ONE THEOREM. &op. 1, 2, 3, 4, 5;-Prop. A;— Pi-op. 6,7,8,9,10,11,12,13,14, 15, 16. 1 . 94. In a given to fit exactly a rt. line = a given rt. line, which is not greater than the diam of the 0, Use 1°. 'Within a given to jilace a line of a given length, not greater than the diam. of the given 0, which line shall pass tlirough a given . in the 0ce. 2°. To draw that diam. of a which shall pass at a given distance from a given point. 2. 95. In a given to inscribe a a equiaiDgular to a given A . Sen. The analysis given. Use. An eqitil. A, being inscribed in a0, and through the angular . s "tangents being drawn, these tangents will also form an equil. A, the area of which is four times that of the inscribed equal. A. 3. 97. About a given to drcumscribe a A equiangular to a given A. Sen. The analysis of the Problem. 4. 99. To inscribe a in a given triangle. ScH. 1. Or, to describe a to touch three given lines not parallels. Analysis. TJsE. 1. IF three /.s of a A he bisected by st. lines, these lines will intei^ sect in the same point. 2°. An expression deduced for the Area of a A; and for the Mad. of tlie inscribed Q. 3° The Properties of Circles execribed to a triangle. 1. The bisectors of any int. /_ and of the remaining two ex. /..s ; 2. The rad. of the exscribed may he found in nixmbers; 3. A Formula for the Area of a A in the terms of the sides; 4. Expressions/or the radii of the three exscribed Qs; 5. Area of A ^\/ r r' r" r'"; 6. In a rt. /.d A the diam. of the inscribed = the diff. of the sum of the sides and the hyp.; and the diam. of the exscribed to the hyp. = the perimeter of the A. 5. 102. To circumscribe a circle about a given triangle. CoE. 1. Wben the cen. of the falis within "the A, each /_ is ;an acute /.; when on a side of the A, the /. opp. that side is a rt. /. ; and when without the A, the /. opp. the side nearest the cen. is an obt. /_; and conversely. CoE. 2. The perps. bisecting the sides of a A meet at the cen. of the circumscribing 0. CoE. 3. Perps. from each /_ on the opp. side intersect in the same point. Sen. With what tliis proposition is identical. BOOK IV. 415 Use. If one be inscribed in an equil. A and another circumscribed about it, the 0s are concentric, and the rad. or the diam of one is double the rad. or the diam. of the other. A. 105. Theor. a circle may be described about any • reg. polygon, or inscribed wibhinit; and conversely/. Use. I. The ODnstrnelaon of a reg. polygon ; 2. To inscribe a polygon, in. a given ; 3. To circumscribe a polygon ; 4. The Area of a reg. polygon ; 5. The Area of a 0. 6. 106. Prob. To inscribe a square in a giren circle. Con. The Sq. on the rad. of an inscribed is ^the sq. insmbed in a 0, and J the eq. on its diameter. Use. By bisecting the arcs, and joining the . is of bisection an octagon may be formed, &c. 7. 107. To circumscribe a square about a given circle. OoB. In the same the cirGiimscribed square = twice the insoribed square. Use 1. To inscribe and to circumscribe a rpg. octagon. ■ 2. A reg octagon inscribed in a = rect. under the sides ot the inscribed and circumscribing squares. S. If a quadril. be circum- scribed about a 0, any two of its opp. sides = § the peiiimeter. S. 109. To inscribe a circle in a given square. ScH. To inscribe a circle in a given qiaatomit. 9. 111. To circumscribe a circle about a given square. IQ. 112. To construct an isoso. A , baving each of the Z s at the base double of the third, or vertical angle. Use 1. In the iig. constructed for Prob. 10, the side AC inscribed in t!:e smaller 0ACfi = the side of a pentagon in that 0, and also = tljo side of a reg. decagon in-the larger BDE. 2. On the side DC being produced to meet the BBE in iF, and I'B being joined, the /_ ABF = three times /.EFD. ,3. To quinquisect, i- e., divide a quadrant into five equal pai-ts, &c. li. 115. To inscribe an equU. and equiangular pentagon in a given circle. ScH. 1 1. Remarks on reg. polygons. S. !FoHnnlffi.for determining the relative magnitudes of the /_s of isosc As to be Hsed anlJie constrnc- tion of reg. polygons. Use. 1. To draw a A = in area to a given polygon. 2. The lines joining the alternate y,s of a reg. pentagon, will form another reg. pentagon ; and the . s of intersection of the alternate sides produced will also form another reg. pentagon. 12. 118. To circumscribe an equil. and equiang. pentagon about a given circle. 416 IXDEX. ScH. If the 0ce of a bo divided into any number of parts, the chord joining the . s of division shall include a reg. polygon, inscribed in the ; and the tangents through those . s shall include a rcg. polygon of the same number of sides circumscribed about the 0. 13. 120. To inscribe a circle in a given equil. and equiau. peutagon. 14. 121. To circumscribe a circle about a given equil. audequiang. pentagon. Sen. 1. To. circumscribe a rcg. polygon about a given circle. 2. To inscribe a circle in a reg. polygon. 3. To circumscribe a circle about a given polygon. Addekda to 14, IV. 1. To analyze the conditions on ivhich the drawing of a reg. decagon and a reg. pentagon depends. 2. To demonsti-ate, that the sq. on the side of a reg. pentagon inscribed in a = the sum of the squares of the rad., and of the side of the inscribed decagon. 15. 124. To inscribe an equil. and equiang. hexagon in a given circle. CoE. 1. The side of a reg. hexagon inscribed in a is = the rad., or semidiam, of the circle ; or the chord of 60° = the rad. 2. An eqmil. A would he inscribed by joining the alternate .s in the hexagon. 3. Every equil. fig. inscribed in a is equiangular. SCH. 1. The opp. sides of a hexagon are parallel. Use 1. On a given L. to describe a reg. hexagon. 2. The inscribed hexagon in a is three-fourths of the ai'ea of the cir- cumscribed hexagon. V 4. Half the rad. = sine of 30°. 5. The inscribed hex. and the successive bisections of its arcs, the ground-work for finding the approximate ratio oftheQ;i.e.,bfaQ to its diameter. O ra Square may be taton as the ground-work of the process. 16. 128. To inscribe an equil and equiang. quindecagon in a given circle. CoK. The only reg. st. lined figures which can be placed, side by side,'so as to make a continuous plane surface, are the equil. A, the square, and the hexagon. ScH. 1. To circumscribe a reg. quindecagon about a circle. 2. To find the arc subtending a side of a reg. thirty-sided figure. Use. This Prop, opens the way for the construction of other polygons. Observations on Polygons. 1. 130. I. The /our known ways of dividing a geometrically. 2. 130. II. To many polygons we must apply an approximate process. LOOK IV. 417 1. In a given to inscribe any reg. rt. lined fig. ; or to divide tlie 0oe of a into any assigned nnmbsr of eq. parts. 2. Approximatiye Method,— for the Heptagon, Nonagon, &c. 3. 132. III. 1st. To find the inagnitade of an /_ at the centre of a reg. polygon. 2nd. To find the magnitude of an L formed by two adjacent sides of a polygon. 4. 133. ^IV. On a given rt. Line AB to construct a reg. polygon, 1. The Formula, ?r — _ = one of the eq. /__ s of the polygon. 2. The Line X tabular rad. ^ units of length in the rad. of the ciremn- scribing 0. 5. 134. V. The Area of a reg. polygon = " X the perp. from the centre. By n.3ing the Table, AB ^ X tabular Area = Area of reg. polygon. 6. 135. VI. Dodson's Tables for calculating and constructing^ Polygons of not more than 12 sides. 1°. When the length of the side = 1. 2°. When the radius of cireumscribed 0=1. 3°. When the radius of inscribed 0=1. 4°. When the area = 1. 7. 187. VII. Polygonal numbers, their nature and constructioH. Xafind the numbers which bear the name of an n — sided figure. 8. 13 s. VIII. The nature and Construction of o 2nd : 2nd (>J 3rd. Three st. lines when in Harmonical Progression. Harmonical Mean. 12. 156. Terms when homologous, or corresponding., TECHNICAL WORDS TO DENOTE OHATTGES IN THE OEDBR OF PEOPOETIONALS, 1°. — For Four Proportionals. 156 — 8. Def. 13. Permutando, or alternando ; Def. 14. Inver- tendo ; Def. 15. Componendo ; Def. 16. Dividendo ; Def. 17, Convertendo. Conjumgendow 2°. — For any number of Proportionals above Two. 158 — 160. Def. 18. Ex cequali (sc. distantia) or ex mquo ; Def. 19. Ex aquali, ot: ex mquo ordinate; Def. 20. Ex cequali in proportione perturbatd seu inordinatd, or ew mquo pertubate. Postulates 1 and 2 p. 161. 3: 173". Three Ms being given, A,B,C, there is a 4th M., as x, to which C has the same ratio, as A to B, i.e., A : B = C : x. 420 INDKX. Axioms. — 1 — 4 p. 161. Algebraical Expressions, &c., p. 161. OXE PKOBLEM. — PROP. N. N. 237. To find a common measure of two lines. • CoK. 1. The greatest com. meas. of the rem. and lesser M. is also the greatest com. measure of the two Ms. 2. Any aliquot part or subm. of a com. meas. is a com. measm'O. 3. By repeating the process with the rem. and the lesser M., and again with the new rem. (if there be one) and the preceding, and so on, the greatest com. meas. of two given commensurable Ms. may be found. 4. Any two commensurable Lines are to one another as the numbers denoting the no. of times that they respectively contain their com. mea- sure. 239. ScH. 'Whon Ms are incommensurable. UsB. To find the greatest com. measure of two numbers. ETTCUD'S THEOKEMS are TIYENTT-FIVE,— the subsidiary FOUHTEBSr. Prop. 1, 2, 3, 4, 5, 6 ;-A, B, C, D ;— 7 A j-7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 ;— E ;— 20, 21, 22, 23, 24, 25 ;— ]?, G, H, K, L, M ;— 0, P. 1. 162. If any no. of Ms be equims. of as many, each of each; what m soever any one of them is of its part, the same shall all the first Ms be of all the other. Cor. The sum of the equimultiples = the equimultiple of the sum. ScH. If to a TO of a M. by any number a m of the same M. by any num- ber be added, the sum will be the same m of that M that the sum of the two numbers is of unity. Cor. 3, Thus, as A = m E, B = n E, C = /> E, &c., A -|- B -1- C = (m + n + p) E. 2. Also m E -f- n E -I- ;) E = (m -1- M 4- p) E. 2 164. If the first M be the same m of the second that the third is of the fom-th, and the fifth M the same m of the second that the sixth is of the fourth ; then shall the fiirst + the fifth be the same m of the second, that the third -1- the sixth is of the fourth. €oK. If AB, BG, GH, &c., be mults. of C. and as many, DE, EK, KL, &c., the same ms of E, each of each, then AH, i. e., (AB + BG + GH &c.) the same ms of C, that DL, i. e., (DE -f BK -1- KL) is of E. BOOK V. 4il ScH. If the 1st of three Ms contain the 2nd as often as there are units in a certain number ; — and if the 2nd contain the 3rd also as often as there are units in a certain number, the 1st will contain the third as often as there are units in the product of these numbers. 3. 165. If the 1st be the same m of the 2r^d, which the 3rd is of the 4th; and if of thelst and 3rd there be taken equima; these shall be equims, the one of the 2ad, and the other of the 4th. , Cor. If A, A' be equims of B, B', and also of C, C ; and if B be a m of C, then B' is the same m of C ScH. If A ; B = C : D, and m A, m C, n B, n D be taken, then m A : nB := OT C : n D. CoK. When n = 1, then m A ; B = m C : D. -1. 167. If the 1st of four Ms has the same ratio to the 2nd which the 3rd has to the 4th; then any equal ms whatever of the 1st and 3rd shall have the same ratio to any equims ■ of the 2nd and 4th. CoR. 1. If the 1st has the same ratio to the 2ncl, which the 3rd has to the- 4th, tlien also any equims whatever of the 1st and 3rd shall have the- same ratio to the 2nd and 4th ; so, the 1st and 3rd shall have the same ratio to any equims. whatever of the 2nd and 4tb. 2. If 4 Ms are propls^ they will be propis by inversion. 3. If A : B = C : D, then A : ^ = 1 : H. 2 2 3 3 Appl. Hence in the rule for Simple Proportion in Arith., — divide the 1st and 2nd terms by any com. measm-e and make use of the results. 5. 170. If one M. be the same m of another, which a M. taken from the 1st is of a M. taken from the other; the rem. is the same m of the rem., that the whole is of the whole. ScH. If fi'om a m of a M. by any number.a m of the same M. by a less number be taken away, the rem. will be the same m of that M. lliat the diff. of . the numbers is of unity. CoR. When m — n = 1, then mA=»A= ; or2A — A = A. G. 171. If two Ms be equims of two others, and if equims of these - be taken from the first two ; the rems are either = these others, or equims of them. ScH. Pr. 1—6 chiefly nseftil to establish, by the method of equims. the Props, which follow. Postulate. Three Ms, A, B, C, being given, there is a 4th M., as x, to which C has the same ratio as A to B ; t. c, A ; B = C : x. 422 index". A. 173. If the 1st of four Ms has the same ratio to the 2nd -which the 3rd has to the 4th; then if the 1st be > the 2nd, the 3rd is > the 4th ; and if = , = ; if < < . Use. Fov 25, V ; 21, VI ; 34, XI, and 15, XII. Snisox added Props.A, B, C, V. B. 174. Invertendo. If 4 Ms are propls.they arepropls. also when taken inversely. ScH. Or the Pr. may bo stated, " The reciprocals of eq. ratios are eq. to one another." C. 175. If the 1st be the same m of the 2nd, or the same part, i. e., siibm. of it that the 3rd is of the 4th, the 1st : 2nd = 3rd : 4th. ScH. Four Ms in proportion by Def. 5, V. are also in proportion by Dcf. ' 20, Bk. VII. D. 178. If the 1st be to the 2nd as the 3rd to the 4th, and if the 1st be a m, or pt of the 2nd, the 3rd is the same m or the same pt of the 4th. 7. A. 179. The ratio of two lines is the same as that of the numbers which express the number of times that any third line is contained in them respectiTely. 7. 181. Equal magnitudes have the same ratio to the same M., and conversely. CoE. If a rat. A : C, compound of two ratios A : B, and B : C, be a rat. of equality one of them must bo the inrerse oi" reciprocal of the other ; i. -e. A : B is the inverse or reciprocal of B ; C. 8. ISS. Of two uneq. Ms, the gr. has a gr. ratio to another M. than the less has; and conversely. 9. 185. Ms which haye the same ratio to the same M. are eq. to one another; and those to which the same M. has the same ratio are eq. to one another. CoE. A ratio compounded of two ratios, of wliich one is the reciprocal of the other is a ratio of equality. 10. 187. That M. which has a gr. ratio than another has unto the same M. is the gr. of the two ; and that M. to which the same ratio has a gr. ratio than it has unto another M. is the less of the two. 11. 188. Ratios that are the same to the same ratio, are the same to one another. liooK V. 423 Cor. 1. If A : B = C : D, but C : D > or < E : T, then A :'B > or < 2. if A : B > or < C : D, and C : D = E : F, then A : B > or< E: F. 12. 189. If any number of Ms be propls, as one of the antecs is to its conseq., so shall all the anteceds taken together be to all the conseqs. 13. 191. If the 1st: 2nd = 3rd: 4th, but the 3rd : 4th > 5th: 6th ; the 1st : 2nd > 5th : 6th. Cob. If a • B > or < B : E, but C : D = E : E, then A : B > or < E: F. 14. 193. If the 1st: 2nd = 3rd : 4th; then if the 1st > 3rd, the 2nd > 4th; and if ==, =, and if <, <. CoK. If A : B =2 C : r», and if B > = or < D, then A > = or < C. 15. 194. Ms h&Te the same ratio to one another which their equims haye. Cos.. \. Ms have the same ratio to one another which their eq. snbmults. have. 2. If A ; B = C : D, then m A : mB = nC: nJ). 3. Also ^..^ = ^: ^. 2 2 3 3 16. 196. Alterncmdo. If 4 Ms of i/ie same feW be propls, they shall also be propls when taken alternately. 3 97. TJsB 1. If to the terms of a rat. the same M. be added, the rat. will be unchanged, increased, or diminished, according as it is a rat. of equality, of less iBeqnality, or of greater inequality. 2. If all the terms, or any twohomol. terms, or the terms of either of the ia,tios of proportion, be multiplied or divided by the same number, the resulting Ms will remain propl. 17. 199. Dividendo. If Ms taken jointly, be propls, they shaU also be propls when taken separately; L e.j if two Ms. together hare to one another the same ratio which two others have to one of these, the rem. one of the first two shall have to the other the same ratio which the rem. one of the last has to the other of these. Cob. 1. Convertmdo. If A: B = C: D, then A : A the other two. 3. If A : B : C, then A + C > 2 B i and A±-^ > B.' Use. The arith. mean betv?een 2 Ms is > the geom. mean. 424 INDEX. 18. 203. Componendo. If Ms taken separately be propls, they shall also be propls when taken jointly, by composition; i. e. if A : B = C : D, then A + B: B = + D : D. . Use. Addenda. If A ; B = C : D, then A:A + B = C:C + D. 19. 207. If a whole M. be to a whole, as a M. taken from the first is to a M. taken from the other ; the rem. shall be to the rem. as the whole to the whole. CoE. 1. Also the rem : rem. =. M. from the 1st : M. from the other. 2. If A : B : C : D &c. then Atv.B:BtvjC:Ct>oD&;c win form a geom. piibgresaion, the successive terms of which hare the same ratio with the successive terms of the former ; and CoK. 3, conversely. E. 210. Gonvert^ndo. If four Ms be propls, they are also propls by conversion ; i. e. the 1st is to its excess above the 2nd as the 3rd to its excess above the 4th. Use 1. If any no. of Ms be in contd. proportion, the diff. between the 1st and 2nd terms i,v to the 1st, as the diff. between the 1st and last is to the sum of all the terms, except the last. Use 2. In a series of coi>td. propls., the differences of the successive terms are also in contd. proportion. 3. In an infinitely decreasing series of Ms in contd. proportion, the 1 St term is a mean propl. between its excess above the 2nd, and the smu of the series. '20. 212. If there be three Ms, and other three, which, taken two and two, have the same ratio; then if the 1st be >, =, or < the 3rd, the 4th shall be >, =, or < the 6th. ScH. Also, if A :;B = C : D ; and B : E = D : F ; then C>, = or < F, as A >, =, or < E. 21. 214. If there be three Ms and other three which have the same ratio taken two and two, but in a cross order, i. e. in proportione perturbatd, in disturbed proportion ; then if ' the 1st M be >, =, or < the 3rd, the 4th shall be >, =, or < the 6th. Son. A variation of the Prop, If A : B = C : D ; and B : B = E : C; then !F shall be >, =, or < D, as A is >, = or < E. 22. 216. Ex mquali, or ex aquo, by equality. If there be any no. of Ms and as many others, which taken two and two in order, have the same E. ; the first shall have to the last of the first Ms the same R. which the first has to the last of the others. BOOK V. 425 CoK. Rs. compd. of any no. of eq. Ks. in the same order are cqnal to one another. .ScH. Varied thus ;— If the 1st M be to to the 2nd, as the 3rd to the 4th ; and if the second M be to the 3th, as the 4th to the 6th ; then the 1st shall be to the 5th, as the 3rd to the 6th. Use. Proportionals remain propl, miscendo, by mixing; i.e., by using their sum and difference. 23. 219. Ex asquo perturhato. If there be any no. of Ms, and as many others, which taken two and two in a cross order, have the same R. ; the first sliall have to the last of the first Ms the same R. which the fii-st has to the last of the others. ScH. Bifferent ways of announcing Prop. 23. 1 . Es compd of any no. of eq. Rs, but in reverse order, are eq. to one another ; 2. Also for two or more series of Ms. 24. 222. If the first has to the second the same R. which the third has to the fourth, and the fifth to the second the same R. which the sixth has to the fourth ; the first and fifth together shall have to the second, tho same R. which the third and sixth together have to the fourth. Cob. 1. The excess of the 1st above the 5th, shall be to the 2ad, as the excess of the 3rd above the 6th is to the 4th. 2. In any no. of Proportions, — if the 2nd is the same throughout and also the 4th ; then the sum of all the first terms is to the com. 2nd term, as the sum of all the third tei-ms is to tho com. 4th term. ScH. Or, " If two series of Propls. have- the same conscqs., the sum of the first antecs : com. conseq. = sum of the second antecs : their com. conseq," Cob. 1. If two proportions have the same consequents, as A : B = C : D, and E : B = F : D, then A-E:B = C — F:l). 2. If four Ms form a proportion, A : B =: G : D, miscendo, A + B : A — B = C + D :C— D. 3. In two series A, B, C, D, E, E, &c., and G, H, I, K, L M, &c.,— if the ratios, A : B, or G : H ; B : C, or, H : I, &c., be tho same in tho two series, then any two combinations of the lirst series shall be to one another as any two similar combinations of the second series. 25. 226. If four Ms of the same kind are propls, the greatest + the least are > the other two together. Cob. If three Ms be propls, the sum of the extremes > twice the mean, and half the sum > the mean. The Arith. mean is > the geom. mean. F. 227. Ratios compounded of the same ratios are the same to one another. 426 ixDEx. ScH. Eatios compounded in any order whateror are the same with one another. AREAS OT Plane eectilineal tigures. 247. Billingsley's Suimniairy. 248. Identity of Form, not of size, the basis of the Comparison. Tlie extension given to other truths. 249. Wliat the sixth Book establishes stated in general terms. BEFINITIONS. 1. 249. Similar rectil. figures, — -conditions to be fulfilled. 2. 250. Eeciprocal figures, — in what way the sides propl. S. 250. A st. L. Cut in extreme and mean ratio. A X. divided medially, or in medial ratio. 4. 251. The altitude of a figure. subsidiary def. a — E. A. 251. A L. harmonically divided. B. 251. A fig. given in species. C. 251. A fig. given in magnitude. D— F. 252,3. A CZJ applied to a st. L. 428 INDEX. TEN PKOBLEMS. Prob. 9, 10, 11, 12, 13 ;— 18 j— 25 ; 28, 29, 30. 9. 285. From a given st. L. to cut off any measure, or sub- multiple. ScH. Pr. 10, 1, a particular case of this Prob. TJsE 1. To divide a given L. into any no. of eq. parts. 2. To divide a A into any no. of eq. pts., by lines from a given . in one of the sides. 3. Given the nth part of a L to find the (n + l)th part. 10. 288. To divide a given st. L. similarly to a given divided st. L. ; or into parts that shall have the rat. to one ano- ther which the pts. of the divided given st. line have. Use 1. To divide a given St. L. internally or externally, in a given ratio, as of M : N. 2. To find a harmonical mean between two given st. lines. Z. To find a third harmonical progressional to two given st. lines. 4. To conetruct a A of which one side, the /. opp. and the E. of the other sides are given. 5. Through a given . to draw a L., which, on being produced, would pass through the . s of intersection of two given lines without their being produced to meet. The Centrolinead. 11. 292. To find a third proportional to two given st. lines. ScH. Construction requiring the compasses alone. Use. 1. To continue a series of Es. in progression, AJB : BC, being the given antec. and conseq. 2. Theorems allied to the last Prob. ; 1°, If a series of Ms. be in cont. proportion, their successive dilferences are also in cont. proportion, and in the same E. 2°. If a series in cont. proportion be an increasing one, there is no limit to the increase of its terms. 3°. If a series in cont. proportion be a decreasing one, there is no limit to the diminution of its terms. 4°. If a series of Ms decreasing in cont. proportion, be continued, or imagined to be continued, to an infinite no. of terms, the sum of all the terms, or the Sum of the Series, will be a finite and determinate M. 3. Problems from the foregoing principles. 1°. Gkeqokt's Prob. , from the two first terms in a series to obtain the sum of the terms. 2°. Of three quantities, the 1st and 2ud terms, and the Siim of the Series, if any two be given, the rem. one may be found. 12. 298. To find a fourth proportional to three given st. lines. SoH. Solution of the Problem by the Compasses alone. Use ] . The Sector,— '• A large number of pairs of compasses packed up one." BOOK VI. 429 2. Problems t»y aid of the compasses and sector ; 1°. To find a fourth propl. to three given lines. 2°. To find a chord to a rad., from a Sector the rad. of the chord of which equals 4 inches. 3°. To divide a given L. into two pts, x and i/, which shall be to each other as two lines or numbers. 3. The various Lines on the Sector, and on Guntee's Scale. 4. Examples of the use of the Sector. 5. Other applications, and caution. 13. 302. To find a mean proportional between two giren st. lines. Sen. Other constructions ; — foimula for numerical calculations. Use. 1. Any rectangular parallelogram may be reduced to an equivalent square. 2. Of three lines in cont. proportion, any two being given, to find the unknown. 3. Given one of three terms and the sum of the other two, to find the two unkno-wn terms. 4. Of three lines in cont. proportion, if one be given and the diff. of the other two, those other two may be found. 5. To find two St. lines to contain a rect.,=: a given rect. and to have a given ratio one to the other. 6. To find any number of means represented by a power of 2, minus 1. Addenda. I. To obtain two mean propls. between two given st. lines ; 1°, Plato's method of aperp. moveable along a side of a sq. 2°. Philo's method of a graduated ruler revolving rovmd the vertex of art. /.. 3°. The method of Des Caetes, with a collection of rulers. 308. II. The Trisection of a rectilineal /_. 1° The Trammel of NicoMEDES, a T square with a moveable ruler. 2°. By this instrument to trisect a given /_. 3°. Coolet's Tentative Method. 18. 324. Upon a given st. L. to describe a rectilineal figure, similar and similarly" situated to a given rectilineal figure. . ScH. 1. A more simple way for making arectil. fig. sim. to a given reetil. fig. 2. To construct a reetil. fig. sim. to a given recti], fig., and having its perimeter = a given st. L. 3. Kgures of the same species with different areas on the same rt. L. 4. Sim. AS and polygons are to one another as the squares of their honaol. sides. Use. Nearly all the practiccU methods of taking a Flan or Map, founded on this Prop. 25. 344. Of extensive use. To describe arectil. figure ■vrhicli shall be sim. to one and eq. to another given rectU. fig. ; i. e. eq. in area. 430 IKDEX. SoH. The chief point in this Prob. is to find a mean propl. Use. While ive keep the same area, we can change the form. 28. 350. To a given st. L. to apply a parallelogram = a given recto, fig., and deficient by a parallelogram sim. to a given / 7 , but the given rectil. fig. to whioli the i T to be applied is to be eq., must not be greater than the I 7 applied to half of the given L, having its defect sim. to the defect of that which is to be applied ; i. e. to the given I 7 . ScH. 1. To divide a given St. L., so that the rect. contained by the segs. may be eq. to a given space, as the sq. on C ; but that given space must not be gr. than the sq. of half the given L. 2. To inscribe in a given A a I 7 eq. to a given fig. not gr. than the maximum inscribed / 7 . and having an /. in com. with the A . Use 1. To a given st. L. to apply a rect., deficient by a Sq., which rect. shall be eq. to a given square, that on line C ; but the given sq. on C must not be gi-. than the sq. on the half of the given L. 2. To a given st. L. to apply a rect., which shall be eq. to a given rect., and be deficient by a square ; but the given rect. must not be gr. than the sq. upon half the given line. Other enunciations. 29. 354. To a given st. L. to apply a / 7 eq. to a given rectil. fig., and exceeding by a / 7 sim. to another given / 7 . ScH. The algebraical signification of Pr. 27, 28, & 29, bk. VI, Use 1°. To exscribe to a given A, a / 7 eq. to a given rectil. fig., and having an /. = to one of the /_s of the given A. 2°. To a given st. L. to apply a rect. which shall be eq. to a given square, and exceeding by a square. Or, To produce a given st. L. so that the rect. under its external segs may be eq. to a given space, as C. 3°. To a given L. to apply a rect. that shall be eq. to a given rect., and exceeding by a square, as on BN. Variations of the Prob. 30. 358. To cut a given st. L. in extreme and mean ratio ; /. e. so that the whole L, shall be to the gr. seg. as the gr. seg. to the less. ScH. 1. A L. thus divided, divided medially ; medial ratio. 2. This Prob. belongs to a class which relate to incommensurable Ms. Theok. Let there be two Ms of the same kind, P & Q; and let P be contained in Q a certain no. of times which is to P as P is to Q ; then the Ms, P and Q shall be incommensurable. 3. Or this 30th Prob. may be considered as a particular case of the Pkob. To divide a line so that the rect. under the whole L. and one part shall bear a given ratio, as m: n, to the square of the other part. 4. Two lines in ext. and mean R. are cut similarly ; and conversely. • 5.' Numerically to approximate to the ratio of incommensurables. BOOK VI. 431 TJsE. This Prop, employed in Euclid's 13th Bk. on the Platonic Solids. Pkob. ' 1°. On a given L. to construct a rt. /,d A, the sides of which shall be in cont. or geom. progression. Prob. 2°. The altitude of a rt. /_& A being given, of which the sides arc in a given R., to find the sides. PsOB. 3°. Given, «, b, two sides of a A, ajid the diam. d, of the circum- scribing 0, to iind the other side, t. EUCLID'S THEOKEMS AEB TWENTT-THBBE ; THE SUBSIDIAET TWELVE. Pi-op. 1, 2, 3 ;— A,4, 5, 6, 7, 8 ;— 14, 15, 16, 17 ;— 19, 20, 21, 22 ;— Lenuna. 23, 24, 25, 26, 27 ;— 31, 32, 33 ;— B, C. D ;— B, F, G, H, K, L, M. 1. 252. Triangles and parallelograms of the same altitude are to one another as their bases. Cob. 1. Triangles & / 7 s with eq. altitudes are one to another as their bases ; & conversely. 2. Any two As or / 7 s. are to one another in the R. compounded of the Es of their altitudes and of their bases. . 3. The rect. under two lines is a meanpropl. between their squares. 4. If two As or two I 7s be as their bases, they have eq. altitudes j and if they have eq. altitudes they have eq. bases. ScH. 1. This Proportion might have been directly inferred. 2. Propositions distinguished by the name Variant. 3. One quantity does not vary as another, because it varies withit. Use. Prom a trapezium to cut off a third part. 2. 257. If a St. L. be drawn || to one of the sides of a a , it shall cut the other sides, or these produced, proportionally ; and conversely. Cob. If the sides of an /. be cut by any no. of parallels, any two pts. of the one will have the same R. to one another, as the corresponding parts of the other. SCH. 1. The Enunciation not sufficiently explicit. 2. The theory of Transversal lines is connected with this Prop. 3. The ways in which a st. L. may be cut in a given Ratio. 4. Parallel lines cut diverging lines proportionally. Use 1. Por the measurement of the height of an inaccessible object which casts an accessible shadow. 2. To divide a given L. into parts propl, to those of another L. 3. 2&0. If the Z of a A be divided into two eq. /.s by a st. L. which also cuts the base ; the segs. of the base shall have the same E. which the other sides of the A have to one another ; and eonversely. 432 ixDEs. A. 262. If the outward L oi a. a, made by producing one of its sides, be divided into two eq. /_b, by a st. L., which, also cuts the base produced ; the segs. between the dividing line and the extremities of the base, have the same K. which the other sides of the A have to one another ; and conversely. CoE. 1. The segs. of the base produced made \>j the external bisector are propl. to the segs. of the base made by the internal bisector. 2. The two lines bisecting the vert. /. and its adj. ext. /., cut the base produced harmonicaUy. 3. Also tlie two sides of a A, and the lines which bisect the vert. & cxt. Tert /^s are harmonieah. 4. If BG, BC & BD, in the same st. L. be inharmonical progression, DC, DG & DB, will also be in harmonical progression. Son. Case in which there is no point of external bisection &c. Use 1. The harmonic mean obtained from the harmonic proportion. 2. Applied to Optics and Acoustics. 4. 2G5. The sides about the eq. Zs of eq. ang. as are propor- tionals; and those which are opp. to the eq. /.s are homol. sides. CoE. 1. If diverging Ls, cut par. Ls, the par. Ls will be cut proportionally. 2. If two par. st. Ls be cut by any number of diverging Ls, the par.iUols will be similarly cut in the . s of section. 3. In a A , a L from the vertex, bisecting the base also bisects the parallel to the base. 4. A par. to the base of a A cuts oif a sim. A. ScH. 1. Homologous sides, — homologous terms. 2. Only in As, if tlieir /.s are eq. their sides about the eq. ^a are propl. , 5. 268. Important. Conversely, — If the sides of two As, about each of their Z. s, be propls, the A s shall be eq. ang. ; and the eq. Z s shall be those which are opp. to the homol. sides. 269. ScH. Pr 47,47,1, and 4, 5, VI contain the principles of every kind of rectilineal Measurement. Use 1. The Theory of Representative Value,— the Practice of Triangu- lation, and various other methods of Practical Geom. depend on Pr. 4 & 5, bk. VL 2. For Practical Purposes Eight Problems and Formulas deduced ; 1°. Given the observed length of the shadows of two porp. objects, and- the alt. of one, to find tke alt. of the other. BOOK VI. 433 2°. By means of a mirror placed horizontally, the /_ of incidence being ' = the l_ of reflection, to ascertain the AeijAt of a perp. object. .3°. To find the height of a perp. object, by means of two imeq. rods or poles, placed perpendiCulaify on a horizontal line. 4°. By means of a pole placeC^erp., to ascertain the alt. of a perp. ob- ject. 5°. By a Geom. Square to measure the height of an object. 6°. By the same means, with an index to point to the extremities of two objects, of which one is in the vertex of a rt. ^ ; to measure the distance. : 7° To find, by aid of the cross staff, or theodolite, the distance between two objects. 8°. By means of a L. of which the length is known, to find the length of its parallel, one end of "which only can be approached. The Proportional Compasses, Pentagraph, and Eidograph. 6. 275. If two AS have one Z of the one = one /_ of the other, and the sides about the eq. /_ s propl., the A s. shall be eq." ang., and shall have those /.s. eq. -whicli are opp. to the homol. sides. CoE. 1. The sides also about each pair of eq. ^s shall be propl. 2. If through any . s of a rt. L. parallels be drawn, proportional to distances from any . A in that St. L., then their extremities will be on the rt. L. passing through A. Use 1. Conditions of similarity between one reotil. figure arid another. 2. Sim. rectil. figures may be divided into the same no. of sim. As. 7. 278. If two AS, have one /. of the one = one Z of the other, and the sides about two otiier /_s propls, ; then, if each of the rem. Z s be either < or <(; a rt. Z , or if one of them be a rt. Z > the A s shall be eq. ang. and shall have those Zs. eq. about which the sides are proportionals. ScH. Angles when of the same affection. Use. The criteria of the similarity of two As ; 1°. Equality of the three Zs. 4, VI. 2°. Identity of the Es of the respective sides, 5, VI. 3°. Equality of two ZS) one in each A, and the identity of the Rs of the containing sides, 6, VI ; 4°. Identity of the Es of two sides in each A ; the equality of an Z ™ ^^'^^ oPP- °^^ V^^ °^ homol. sides ; and each of the rem. Zs opp. the other pair of homol. sides < a rt. Z, or one of them a rt. Z- S. 281. Important. — In a rt. Z 347. Of all / 7 a applied to the same st. line, and deficient by I 7 a sim. and similarly situated to that which is described upon the half of the L. ; that which is applied to the half, and is sim. to the defect, is the greatest. Or, Of all the rectangles contd. by the segs. of a given st. L., the greatest is the square described on half the L. Use. In a given A to inscribe the greatest parallelogram possible, having an ^ in common with the A. 31. 363. Important. In rt. /.d as, the rectil. fig. described on the side opp. to the rt. Z , is = the sim. and similarly described figures on the sides containing the right Z. ScH. 1. A very comprehensive Prop., bttt one still more general is,— D any y 7 s be described on the two sides of any A, and if the sides of the/Z^s be produced to meet, and if that . of intersection and the vertex of the A be joined, and the L. produced ; then these I 7 s are eq. in area to a J~~l described on the base, and having two of its sides parallel to the Ii. produced through the . of intersection and the vertex, and limited by the sides of the two I 7s. , 2. A circle with the hyp. of a rt. Zd A for diam. is equal in area to the two 08, having the other two sides for diameters. 32. 366. If two AS vrhich have two sides of the one propl. to two sides of the other, be joined at one Z , so as to have their homol. sides par. to one another ; the rem. sides shall be in a st. L. ScH. The position of the given sides, not homo]., most form an Z at the . of junction. 33. 367. In eq. ©s, Zs, whether at the centres, or 0ces, have the same E. which the 0ces, on which they stand, have to one another; so also have the sectors. CoE. 1. The sectors are to each other as their /_a. 2. Sim. sectors of the same or eq. 0s are equal. 3. An Z at the centre of a is to.four rt Zs <"* tlie arc on wljich it stands to the 0oe of the 0. 4. In different 0a the arcs of eq. Zs at the centres or 0ces are similar. 5. Hence, sim. segments are contained by sim. arcs, and vice versa. ScH. Some Editors of Euclid substitute the three following Propositions applicable to the same or equal circles. 438 INDEX. Prop. 33. a. /_a, Whether a{ the cent, or at the ©oes, hare the same K. as the arcs on which they stand. CoE. The arcs are also propl. when incommensurable. Pkop. 33. 6. The sectors on eq. arcs are equal. Pkop. 33. c. Sectors hare the same R. as the arcs on which they stand. Cor. They must also be propl. when incommensurable. The /_ at the cen. of a is measured by the arc on which it stands. Use 1 . K arcs of different 0s have a com. chord, the Ls diverging from one of its extremities will cut the arcs proportionally. 2. The arcs of uneq. 0s are in a E, compd. of their central /.s and their radii. 3. Central /.s are in a R. compd. of the direct E. of their arcs, and the inverse E. of their radii. Subsidiary Propositions. B. 373. If an Z. of a* a be bisected by a st. L., whicli likewise cuts tbe base; the rect. contained by the sides of the A is eq. to the rect. contained by the segs. of the base + the sq. of the st. L. which bisects the base. CoE. The rect. of the sides + the sq. ,of the L. which bisects the ext. /. = the rect. of the whole L. produced and the ext. seg. ScH. Pr. B and its corollary may be combined. 0. 374. If from any Z of a A a st. L. be drawn perp. to the base; the rect. contained by the sides of the A = the rect. contained by the perp. and the diam. of the described about the A . CoE. If two AS be inscribed in the same or in eq. 0s, the rect. under the two sides of the one shall be to the rect. under the two sides of the other, as the perp. from the vertex to the base of the one is to the perp. from the vertex to the base of the other. D. 375. The rect. contained by the diags. of a quadrili fig. inscribed in a is eq. to both the rectangles contained by its opp. sides. Sen. Ptolemt's Theorem. Imitativeness of Euclid's editors. E. 377. The diagonals of a quadrU. inscribed in a are to one another as the sums of the rectangles under the sides adj. to the extremities of those diagonals. SooK VI. 439 tJsB. To solve the Prob. ;— Given four st. Ls, any three of which are together > the fourth, to construct a quadril., of which the sides shall be =: those four given st. Ls., in a given order, each to each, and of , which also its angular .s lie in the 0ce of a 0. F. 378. If a seg. of a be bisected, and from the extremities of the base of the seg., and from the . of bisection, st Ls be drawn to any . in the 0ce; the sum of the two Ls from the extremities of the base, will have to the L. from the . of bisection the same E. which the base of the seg. has to the base of half the seg. G. 379. If two . s be taken in the diam. of a ©, or of the diam. produced, such that the rect. contained by the segs. intercepted between them and the cen. be eq. to the sq. of the semidiam. ; and if from these . s two st. Ls be inflected to any . whatever in the 0ce of the ©, then the K. of the Ls inflected will be the same with the E. of the segs. intercepted between the two first-men- tioned .s and the ©ce of the ©. CoE. 1. In the fig. l_ FBEis bisected by AB. 2. Also the ext. vert. /. EBGr is bisected by BC. H. 381. If from one extr. of the diam. of a a chord be drawn, and aperp. cut both the diam. and the chord, either intr, or extr., the rect. under the diam. and its seg. reckoned from the extremity is eq. to the rect. under the chord and its corresponding seg. K. 381. If the Zs at the base of a A be bisected by two Ls that meet, and the ext. z s at the base, formed by producing the two sides, be sim. bisected; then the two . s of con- course, and the vertex shall be in one st. L. which shall bis. the vert. /. . L. 383. In a A , as with last Pr., K, the segs. of each side pro- duced that are intercepted between the vertex and the extn. perps. are each = the semiperimeter of the a ; the segs. of these sides next the rect. are = the excess of the semiperimeter above the base; and the seg. of each of these sides next the base is respectively = the excess of the semiperimeter above the other side. ■140 INDEX. M. 384. The area of a A is a mean propl. between two rectangles, the sides of which are eq. to the semiperimeter and its excess above the base, and the sides of the other eq. to the excesses of the semiperimeter above the other two sides. CoE. Let S denote semiperimeter, and a, b, c the sides opp. /_e A, B, C ; the Area of A = >/ s (s — c) (s — a) (s — b.) Use. The Sol. of the Prob. Given the three sides of a A to find the Area. Eemaeks. 1. 385. The 6th Book treats chiefly of similar rectil. and cnrvilineal figures. 2. 386. The Book contains 33 Prs. by Euclid, of which 10 are Probs. & 23 Theors ; there are 13 Subsidiary Theorems A— M. 3. The most important Problems and Theorems. 4. An approximate Classification of the 6th Book. 6. Thirty-three useful Problems additional to those of EucuD. 6. 387. Fifteen additional Theorems referred to in the notes. 7. 388. The sixth Book the Head and Crown of Plane Geometry. COBSIGEHDA. » The errors to be found in Mathematical works, even in those oi high repute prove the diflSculty of avoiding them, especially when signs and abbreviabons are Ireely used. Besides, the printing was undertaken by those who were hot accustomed to mathematical work; and there was also an unavoidable change of workmen ; hence without much real blame to any one, the mistakes are more than they Would have been ; few, however, extending to any error in the reasoning. Along with the remark in the Preface on the same subject, — will the Beader accept the qnadnt apology of an old Editor of Euclid ? " gome mistakes will stiU remain, which when thou chancest to meet with correct with the pen, so shalt thoa do right to the Author, and supply the defects of the Bevisor, and in both doe good to thyself."— Rudd's Euclidea Element^. N.B. — ^In Column 1, the page is given, P ; in Col. 2, the line, L, the arabic numeral on the left baud denoting the lino firam the top of the page, — that on the right, the line from the bottom ; in Col. 3, the correction, corr. ; and in Col, 4, the printed error, err. COERIGKNDA. 441 P. ■8 11 8 17 2 39 19 44 82 85 99 104 17 108 110 2 115 19 22 120 6 126 4 128 6 172 176 178 7 179 11 182 187 188 1 L. corr. for err. 20 Pst. 2. 8 ! 16,1. 10,1. PC; FG ; AC-CB;=CB; 1 .■. 4 A 1 I'D 12 DEB rad. 6 in.L ; €~2; AD ED DEE dioM. in I ; n ZCAD /.CAG DB CB 192 194 3 197 : 198 2 201 10 202 14 arcE&c. E&c. 3o,in ?8,in. 15 KC KH 20 .-.B • contains = GH GA C=E C=E 11 or< 01- > 14 > < :C, :A. 12 I-, E, 7 C:D, B:E, D> B> 20 2 2 T S _ >bD DE; CE . A, . by 214 21^ 8 DE(6;«) EK "i & 4th, 22 a' a+l) 'T' & 5th, 32 221 8 ^fL. c' 222 14 A 12 i' H12 230 10 is com pd. and 235 7 continued con- tained 236 11 ^tB gtB' Jl B'; ■15 f 10 B B' 8 A' A 238 5 B K 4 of B of R 250 8 DE, EF, 255 1 are as are 261 14 '.• • 264 21 GD, CD, 272 3,4 CB EB 6 C E 289 2 HC, AC, 302 5,11 BC ; AC; 12 ADC, ABC, P. 311 320 325 329 330 331 1 6 335 12 15 336 U L. 2 6 14 II 3 corr. for err. triangles angles Cor. 3. Cor.33. omit EE or 6480 3240 6480 .969 AB AC 2X2 1 3240 .973 CB AB 2X2 2 348 12 353 18 355 4 8 356 9 357 21 358 8 362 6 15 364 12 365 9 375 7 393 11 402 406 413 421 423 10 428 2 432 8 EB EL 36,1. 31,1.. To exscribe 10,1. 31,1. 46,1. 45,1. EC EB CE AB 33, 1. 31, 1. 5, IV. 4, VI. equil. eqnal 4 times 4 lines 5 and and 93 97 ^— A \ ^ ; C:D B :E up into up 47,48, 47, 47. Eekata, Involving an Error of Reasoning. P. D. P. I *• 46 1&3 remove the bracket. 332 3 208 16 BivjC, CroD, /orro M B :CCD 354 6 2SS 1 add .-. A^: A.B = A.B:B». 359 16 259 14 BG:AG/orAG:BG 430 7 270 17&20 GB/or CB 276 7 Ac, Ad for be, cd FIJ> ris. read in the duplicate rabp of AB : EG. The fiiil argument not given. {read) times with a re- mainder. BY THE SAME AUTHOR, Price 2s. 6d., pp. 228. Diagrams 226. EUCLID'S PLANE GEOMETRY, BOOKS I & II, PRACTICALLY APPLIED; OB, GRADATIONS IN EUCLID^ PART I, AN XNTRODUCTION TO PLANE GEOMETRY, ITS USE AND APPLICATION, WITH AN EXPLANATORY PREFACE, REMARKS ON GEOMETRICAL REASONING, PRACTICAL RESULTS AND EXERCISES. N,B. — The Six Books, i. e. Parts I and 11, may be had in one Volume, price 6s. 6d. CRITICAL NOTICES. " Mr. Green's * Gradations* is a raltiable Tvork, though it takes in only the first two books of Euclid. Without endorsing all the detail, which we have not had time to examine, we may unhesitatingly, speak in praise of the method which we have examined with care. No intro- ductory Euclid, which has fallen in our way, contains anything liktf the help, in the shape of notes and arrangement, that this does." " The whole of the propositions are arr^ged after the manner recommended by Professor De Morgan, in his ' Treatise on the Study £md Dlfflcultiea of Mathematics.* To many of the propositions is appended an account of the practical uses to which the proposition may be applied. An Appendix is added, which contains much that is useful in the way of notes and exercises. We feel sure that schoolmasters will largely patronise Mr. Green's ' Gradations in Euclid,' as a very complete apparatus for teaching the First and Second Books." — Papers for the /3choolmasier,Julyt 1858,p. 112, " We commend the work to the consideration of Teachers." — Athenceum, July 17, 1858. " By printing the distinct parts of the propositions separately, a very great help is gained, and the principles more thoroughly learned." — The Bookseller, July 24, 1858. 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