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 BY WM. FEWSM1TH, A.M., AND EDGAR A. SINGER. 
 
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Cornell University Library 
 eometr 
 
 arV19506 
 
 The normal elemental 
 
 3 1924 031 267 150 
 olin.anx 
 
Cornell University 
 Library 
 
 The original of this book is in 
 the 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 2671 50 
 
THE 
 
 NORMAL 
 
 ELEMENTARY GEOMETRY: 
 
 £MBRACING A BRIEF TREATISE ON 
 
 icitsuraEtrtt anlr wn$onomctt$. 
 
 DESIGNED FOR 
 
 ACADEMIES. SEMINARIES, HIGH SCHOOLS, NORMAL SCHOOLS, 
 AND ADVANCED CLASSES IN COMMON SCHOOLS. 
 
 Eevised Edition, 
 by 
 EDWAKD BROOKS, A.M., Ph.D., 
 
 LATH PRINCIPAL OF STATE NORMAL~"sCHOOL PENNSYLVANIA, AND AUTHOR OF THE NORMAL 
 
 PRIMARY ARITHMETIC, NORMAL MENTAL ARITHMETIC, NORMAL WRITTEN ARITHMETIC, 
 
 NORMAL UNION ARITHMETIC, PHILOSOPHY OF ARITHMETIC, METHODS OF 
 
 TEACHING, MENTAL SCIENCE AND CULTURE, ETC. 
 
 PHILADELPHIA: 
 SOWER, POTTS & CO., 
 
 630 MARKET ST., and 523 MINOR ST. 
 
 ® 
 
1" ' IOt F I T 
 
 /<UMj 
 
 CORNELL^ 
 UNIVERSITY 
 ^LIBRARY 
 
 Entered according to Act of Congress, in the year 1865, by 
 
 EDWARD BROOKS, 
 
 In the Clerk's Office of the District Court of the United States for the Eastern 
 
 District of Pennsylvania. 
 
 Recopyright, 1884, by EDWARD BROOKS. 
 
 XLECTROTYPfiD BY I. JOHNSON & CO. 
 PHILADELPHIA. 
 
 ^Sherman A, Co., 
 Printers, PhUa. 
 
PREFACE. 
 
 Progress in education is symbolized in the multiplication and 
 improvement of text-books. They are showered upon us like 
 the flowers of spring-time, until a new text-book is no longer 
 a novelty. To-day the author sends forth this little volume 
 on wha't it is hoped may be a mission of usefulness. It comes 
 modestly claiming a welcome from the public as an addition 
 to our educational literature, and in support of such • claim 
 the following statement of its object and peculiarities is pre- 
 sented. 
 
 Our text-books upon Geometry, thoush well adapted to our 
 higher institutions, are, for a large class of schools, toth too 
 voluminous and difficult. In many of our Academies, Semina- 
 ries, High and Normal Schools, the time allotted to Geometry 
 is too brief to allow the pupil to complete more than four or 
 five books of the ordinary text-book : in consequence of which, 
 all of that most important and practical part treating of the 
 measurement of the surface and volume of prisms, pyramids, 
 cylinders, cones, and spheres, must be omitted. To supply this 
 defect and enable the pupil to acquire a fuller knowledge of the 
 subject, the present volume has been written. 
 
 General Features. — In its adaptation to the class of pupils 
 designated, this work is characterized by four general features. 
 First, ' an abbreviation of the ordinary text-books ; second, a 
 simplification, so far as possible, of the methods of demonstra- 
 tion usually employed ; third, examples to impart the power of 
 making a practical application of the principles of 'the science; 
 fourth, undemonstrated theorems, to cultivate the power of 
 
4 PREFACE. 
 
 original thought and investigation. These general character 
 istics will be briefly noticed. 
 
 Abbreviation. — In the abbreviation of the subject, the object 
 has been to present the most valuable part of Geometry in 
 about one-half of the space usually devoted to it. This object 
 has been accomplished in two ways : — first, by an omission of 
 all that is not essential to the final results; and secondly, bj 
 such a modification of the remainder as to preserve the chain 
 of logic intact. The difficulty of this will be appreciated by thos6 
 who remember that many propositions, apparently of little im- 
 portance, are essential to the proof of others which follow them. 
 
 Simplification. — Much care has been taken to simplify the 
 subject as far as possible. The author has endeavored to give 
 the very simplest methods of treating special subjects, such as 
 parallels, areas, volumes, the circle, etc. ; and also the clearest 
 and most concise methods of demonstrating individual theorems. 
 The method of infinites, as applied to incommensurable quantities, 
 the circle, and the sphere, has contributed largely to this simplifica- 
 tion and abbreviation. This method is regarded by some as less sat- 
 isfactory than the method of reductio ad absurdum, or the method! 
 of limits; but it will be remembered that it is supported by the 
 authority of our most eminent mathematicians, and, being so much 
 more simple and concise, is believed to be preferable in a brief 
 work like this. 
 
 Applications. — A radical defect of most of our text-books upon 
 Geometry is that they present the subject so abstractly, that 
 when the pupil has completed his course he is often unable to 
 make any practical application of what he has learned. This 
 defect has been supplied by the presentation of a collection of 
 practical examples at the close of each book. With these, ths 
 pupil can see the application, the practical value of what he is 
 doing, and will not only be able to make use of his knowledge, 
 but will be incited to study the subject with more interest and 
 earnestness. 
 
 Theorems for Original Thought. — Another general defect or 
 
PREFACE. 5 
 
 our text-books upon Geometry is the lack of matter for original 
 thought, for the training of the inventive powers of the student. 
 The pupil is required to learn the demonstrations of the text- 
 book, but hefhas no undemonstrated theorems to test his own 
 geometrical powers, and to train him in reasoning independently 
 of the text-book. In view of this general defect, a collection of 
 theorems for original thought has been given at the close of each book. 
 
 Geometry, in both the respects mentioned, has been treated 
 quite differently from arithmetic and algebra. In these latter 
 works, we have generally a large class of problems both for the 
 application of the principles and the exercise of original thought. 
 It is proper to remark, too, that several authors have realized this 
 defect of Geometry, and have occasionally given some practical 
 problems, and, in one or two instances, a collection of undemon- 
 strated theorems. In the present work, such problems and 
 theorems are an essential and prominent part of the plan. 
 
 Special Features. — The attention of teachers is also respect- • 
 fully invited to the following special features of the work: 
 
 1. The Systematic Arrangement of the subject-matter, it is thought, 
 will be an acceptable feature of the work. 
 
 2. The Analysis at the beginning of each book is supposed to 
 be valuable in giving the pupil a general idea of the object of the 
 book, and thus often indicating the course of reasoning in its 
 development. 
 
 3. The Doctrine of Parallels, resulting from the modern definition 
 of an angle and parallel lines, now adopted by several authors, is 
 here presented in its most simple and concise form. 
 
 4. The Subject of Areas in Book III., and Volumes in Book VI., 
 are presented with great conciseness and simplicity by the use 
 of the doctrine of infinites and indivisibles. 
 
 5. The treatment of the circumference ana area of a circle, in their 
 
 relation to w, is much more simple and logical than any thing 
 
 the author has met. It is confidently believed that it will give 
 
 pupils a clearer view of the subject than they usually acquire in 
 
 the study of other text-books. 
 
 1* 
 
6 PREFACE 
 
 Mensuration. — A formal treatise upon Mensuration is also 
 appended, although the most of it is really presented in the 
 practical exercises. A few rules are given without demonstra- 
 tion ; such omissions may be supplied by the teacher. . 
 
 Trigonometry. — The little treatise on Trigonometry presentu 
 the elements of the subject briefly, and contains about as much 
 as the advanced pupils in our ordinary academies, seminaries, etc. 
 should be required to learn. A short treatise on Analytical 
 Trigonometry is appended for those who have time to study 
 this very interesting and useful subject. The Trigonometry and 
 Geometry will be bound together, and also separately, to accom- 
 modate schools of different grades. 
 
 A general acknowledgment of indebtedness is due to those 
 who have previously written upon the subject of Geometry, 
 especially to American and French authors, many of whose 
 works have been examined with great interest and profit. 
 
 Thanking my friends for the generous appreciation bestowed 
 upon my previous labors, I send forth this little volume, hoping 
 that it may be as kindly welcomed, and that, in its mission of 
 usefulness, it may aid in awakening a deeper interest in the 
 beautiful science of form, — a science over which the ancient 
 sages mused with such deep enthusiasm, and to which the 
 achievements of modern art and invention are so largely in- 
 debted. 
 
 EDWARD BROOKS. 
 
 State Normal School, Jan. 10, 1865. 
 
CONTENTS. 
 
 GEOMETRY. 
 
 Introduction. 
 
 Subject-Matter of Geometry 13 
 
 Reasoning of Geometry 15 
 
 Geometrical Language 18 
 
 BOOK I. 
 
 Angles, Triangles, and Polygons. 
 
 Definitions. 
 
 Geometry, etc 21 
 
 Plane Angles 22 
 
 Parallel Lines 22 
 
 Plane Figures 23 
 
 Triangles 24 
 
 Quadrilaterals 25 
 
 Circles 25 
 
 Axioms 26 
 
 Postulates 27 
 
 Theorems. 
 
 Of Angles 29 
 
 Parallel Lines 30 
 
 Triangles 33 
 
 Quadrilaterals 39 
 
 Angles of Polygons 42 
 
 Practical Examples 43 
 
 Theorems for Original Thought 44 
 
 BOOK II. 
 Ratio and Proportion. 
 
 Definitions 46 
 
 Theorems 48 
 
 Practical Examples 52 
 
 Theorems for Original Thought... 52 
 
 BOOK III. 
 
 Areas and Relations of Polygons. 
 
 page 
 
 Definitions.. 53 
 
 Theorems : 
 
 Area of Polygons 55 
 
 Squares on Lines 58 
 
 Similar Triangles 63 
 
 Relation of Polygons 67 
 
 Practical Examples 71 
 
 Exercises for Original Thought.... 73 
 
 BOOK IV. 
 Of the Circle. 
 
 Definitions 74 
 
 Theorems : 
 
 Nature of the Circle 76 
 
 Measure of Angles 80 
 
 Circumference and Area 86 
 
 Practical Examples 92 
 
 Theorems for Original Thought... 93 
 
 Problems in Construction 95 
 
 BOOK V. 
 Planes and their Angles. 
 
 Definitions 103 
 
 Theorems .104 
 
 Theorems for Original Thought... 115 
 
 Problems ". 116 
 
 BOOK VI. 
 Polyedrons. 
 
 Definitions 117 
 
 7 
 
8 
 
 CONTENTS. 
 
 Theorems : page 
 
 The Prism 120 
 
 The Pyramid 130 
 
 The Frustum 137 
 
 PracticaI*Examples 140 
 
 Theorems for Original Thought.. ..141 
 
 BOOK VII. 
 The Three Round Bodies. 
 
 Definitions 142 
 
 Theorems : 
 
 The Cylinder 145 
 
 The Cone 146 
 
 The Frustum 148 
 
 The Sphere 150 
 
 Examples and Theorems 156 
 
 BOOK VIII. 
 Spherical Geometry. 
 
 Definitions.. 160 
 
 Theorems 161 
 
 Practical Examples 173 
 
 Theorems for Original Thought. ...174 
 
 MENSURATION. 
 Lengths and Surfaces. 
 
 Definitions 175 
 
 The Triangle 176 
 
 The Quadrilateral 176 
 
 PAGH 
 
 Regular Polygons 177 
 
 Irregular Polygons 178 
 
 The Circle. 
 
 The Circumference 178 
 
 The Length of an Arc 178 
 
 The Area 179 
 
 Area of a Sector 180 
 
 Area of a Segment 180 
 
 Area of a Circular Ring 180 
 
 The Parabola 181 
 
 Volumes. 
 
 Definitions 182 
 
 The Prism 183 
 
 The Pyramid 183 
 
 The Cylinder 184 
 
 The Cone 184 
 
 The Frustum 185 
 
 The Sphere. 
 
 The Surface 186 
 
 Surface of a Zone 186 
 
 Contents of a Sphere ....186 
 
 Spherioal Segment 187 
 
 Cylindrical Rings 187 
 
 Side of Inscribed Cube 188 
 
 Volume of Irregular Body 188 
 
 Miscellaneous Problems 189 
 
 TRIGONOMETRY. 
 
 Logarithms. 
 
 Definitions 1 
 
 Principles 2 
 
 Table of Logarithms 4 
 
 Operation with Logarithms 7 
 
 Of Logarithms 11 
 
 Synthetic Trigonometry. 
 
 Definitions 13 
 
 Table of Sines and Cosines 16 
 
 Theorems of Trigonometry 20 
 
 Solution of Triangles 23 
 
 Practical Applications 29 
 
 Analytical Trigonometry. 
 
 .ols 3£ 
 
 Fundamental Formulas 35 
 
 Signs of Functions 37 
 
 Limiting Values 38 
 
 Functions of an Arc 39 
 
 Of Quadrants and an Are 39 
 
 Functions of Negative Arcs 42 
 
 Of Right-Angled Triangles 42 
 
 Sum and Difference of Arcs 43 
 
 Double and Half Arcs 44 
 
 Additional Formulas 46 
 
 Introduction of the Radius 47 
 
 Table of Natural Sines 48 
 
 Table of Logarithmic Sines 48 
 
 Theorems and Problems 49 
 
HISTOEY 
 
 GEOMETRY AND TRIGONOMETRY. 
 
 Qepmetkt is generally supposed to have had its origin in Egypt, where the 
 annual overflowing of the Nile obliterated the landmarks and rendered it 
 necessary to have reoourse to mathematical measurement to re-establish them. 
 This origin is indicated by the term itself, Geometry being from two Greek 
 words, yi, earth, and fitrpau, measure, signifying, literally, the measurement of 
 the earth. But, whatever may have been the origin of the term, the natural 
 tendency of the human mind to compare things in respect of their forms and 
 magnitudes is so universal, that a geometry more or less perfect must have 
 existed since the first dawn of civilization. 
 
 Geometry, originating in Egypt, is supposed to have been introduced into 
 Greece by Thales, who lived about the year 650 B.C. Pythagoras, who lived 
 about'570 B.C., was one of the earliest Greek geometers. He is supposed to 
 have discovered the following principles: — 1. Only three plane figures can 
 fill up the space about a point; 2. The sum of the angles of a triangle equals 
 two right angles; 3. The celebrated proposition of the square on the hypo- 
 thenuse. Some say that in honor of this Jast discovery he sacrificed one 
 hundred oxen. Plutarch says but one ox. Cicero doubts even that, as it was 
 in opposition to his doctrines to offer bloody sacrifices, and suggests that they 
 may have been images made of flour or clay. 
 
 The next geometer of eminence was Anaxa'goras, who composed a treatise 
 on the quadrature of the cirele. Plato, the "poetical philosopher," delighted 
 in the science, and cultivated it wilh great success, as is proved by his simple 
 and elegant solution of the duplication of the cube. About fifty years after 
 the time of Plato, Euclid collected the propositions which had been discovered 
 by his predecessors, and formed of them his famous " Elements,'' — a work of 
 such eminent excellence that by many it is regarded, even at the present day, 
 as -the best text-book npon the subject of Elementary Geometry. It consists 
 of fifteen books, thirteen of which are known-to have been written by Euclid; 
 but the fourteenth and fifteenth are supposed to have been added by Hypsicles 
 of Alexandria. 
 
 Apollonius of Perga, about 250 years B.C., composed a treatise on Conic 
 
10 HISTORY OF 
 
 Sections, in eight books. He is said to have given them their names, parabola, 
 etc. About the same time flourished Archimedes, who distinguished himself 
 in Geometry by the discovery of the beautiful relation between the sphere 
 and cylinder. See Theorem XI. Book VII. He also distinguished himself by 
 his work on conoids and spheroids, by his discovery of the exact quadrature 
 of the parabola, and his very ingenious approximation to that of the circle. 
 
 Other geometers of eminence followed, among whom the most illustrious, 
 t perhaps, were Pappus and Diophantus ; but the Greek Geometry, though it 
 was afterward enriched by many new theorems, may be said to have reached 
 its limits in the hands of Archimedes and Apollonius, and a long interval of 
 seventeen centuries elapsed before this limit was passed. In 1637, Descartes 
 published his Geometry, which contained the first systematic application of 
 algebra to the solution of geometrical propositions. Soon after this followed 
 the discovery of the infinitesimal calculus of Leibnitz and Newton j and from 
 that time to the present Geometry has shared in the general progress of all 
 mathematical sciences. 
 
 Trigonometry. — Trigonometry, it is generally believed, originated with 
 the Greek astronomers of Alexandria. The solutions of the most useful cases 
 of spherical triangles have been known from the time of Hipparchus, and the 
 fundamental formulas appear in the Analemma of Ptolemy. 
 
 The Greeks used the chords of the double arcs, instead of the sines. The 
 sines, or semi-chords, were introduced by the Arabians, probably by the astro- 
 nomer Albategnius. To the Arabs, who preserved and cultivated the sciences 
 during the dark ages, this science is indebted also for several other improve- 
 ments. - Regiomontanus introduced the tangents, which did much to simplify 
 the calculations. 
 
 The term sine seems to be derived from the Latin sinus, a bosom ; the arc is 
 supposed to represent a bow, and thus gets its name ; the string, half of which 
 represents the sine of half the arc, would come against the heart or bosom j 
 hence the name sine. The terms tangent and secant are naturally derived from 
 the old geometrical definitions. The cosine and co-secant of an arc mean the 
 sine and secant of the complement, the co being merely an abbreviation of 
 complement. They were first introduced by Gunter. 
 
 There are two methods of treating Trigonometry, known as the analytical 
 and synthetic methods. The synthetic method regards -the trigonometrical 
 functions as lines, or geometrical magnitudes, and develops the science ac- 
 cording to the laws of geometrical reasoningr The analytical method regards 
 these functions as ratios or numbers, and develops the science by means of 
 analytical formulas. , 
 
GEOMETRY AND TRIGONOMETRY. 11 
 
 The modern or analytical method is superseding the ancient or geometrical 
 method. This method is said to hare been first introduced by Dr. Peacock. 
 Professor De Morgan, however, one of the first English authorities, tells us 
 that " Rhetious, who gave the first complete trigonometrical table, and invented 
 the secant and co-secant to complete it, used the method of ratios." 
 
 Logarithms. — Logarithms were invented by Lord Napier, Baron of Mer- 
 chiston, in Scotland. ffis work upon them was first published in 1614; though 
 it is probable that he had commenced the investigation of them as early as 
 1594. The invention is regarded as one of the most useful ever made. It 
 gave the author so high a reputation that Kepler dedicated a work to him in 
 1617, and succeeding mathematicians have paid him the highest compliments. 
 
 Napier's system of logarithms was afterward improved by Henry Briggs, a 
 contemporary of the inventor, and Professor of Geometry in Gresham College. 
 Assuming 10 for the basis, he constructed a system of logarithms corresponding 
 to our system of numeration, which is much more convenient for the ordinary 
 purposes of calculation. The two systems are distinguished as the Napierian 
 and Briggean, or the Hyperbolic and Common logarithms. The former are 
 called Hyperbolic because they represent the area of a rectangular hyperbola 
 between its asymptotes ; the latter are called Common because they are those 
 in common use. 
 
 Briggs calculated the logarithms to 14 places, with the index of all numbers 
 between 1 and 20,000, and between 90,000 and 100,000, and published them in 
 1024. Adrian Vlaoq, a native of Holland, computed the logarithms of numbers 
 between 20,000 and 90,000, and thus completed what Briggs had begun : he 
 reduced the tables, however, to 10 decimal places. Vlacq's treatise was published 
 in 1628, and contained the logarithms of all numbers up to 100,000, and also 
 the logarithms of the sines, tangents, and secants of every minute of the 
 quadrant. In 1623, he published a work containing the logarithmic sines, 
 cosines, tangents, and cotangents for every ten seconds of the quadrant, cal- 
 culated from the natural sines, etc. of the Opus Palatinum of Rheticus. 
 
 In the same year the Trigonometrica Britannica was published at Gouda, 
 which contained the logarithmic sines and tangents for the 100th part of every 
 den-ree of the quadrant, together with a table of .natural sines, tangents^and 
 secants. These had been computed by Briggs. Since then, many different 
 tables have been published. The most complete are those of Vlacq; but these 
 are very scarce. Hutton's Logarithms and" Babbage's Logarithms of Numbers 
 are among the most accurate and convenient. For more information upon the 
 subject, see Brande's Encyclopedia, from which most of this history is collated. 
 
SUGGESTIONS TO TEACHERS. 
 
 The author desires to present the following suggestions to those who 
 may use this work : 
 
 1. Young pupils should have a preliminary drill upon concrete 
 Geometry before taking up the text-book. Let them be required to cut 
 out triangles, squares, etc. from paper, give their names, compare them, 
 and draw them upon the board. In this manner a general idea of the 
 subject, the figures treated of, and even the method of reasoning, may 
 be obtained, and the transition from this to the abstract will be simple 
 and easy. 
 
 2. In the recitation, the pupil should be required to construct his 
 diagram upon the blackboard without the aid of the text-book, and 
 then enunciate and demonstrate the theorem, care being taken that the 
 language and reasoning be accurate. At the close of the demonstration, 
 those of the class who have noticed errors, upon being called upon by 
 the teacher, should rise and point them out; after which the teacher 
 may make any criticisms or explanations he may think proper. 
 
 3. With quite young pupils, and those whose time for the study is 
 limited, the theorems for original thought may be omitted ; with others, 
 however, these exercises will be found to be of great value. They can 
 be given in connection with the demonstrations of the book, or lessons 
 may be assigned upon them after completing the book to which they 
 belong, or they may be omitted until review. The latter method will 
 be generally preferred. 
 
 The Practical Exercises should be solved by all classes. The easier 
 problems may be assigned in connection with the theorems which they 
 illustrate ; the others may be deferred until the book upon which they 
 depend is completed. The most difficult problems may. be omitted 
 until the whole Geometry is completed. 
 12 
 
ELEMENTARY GEOMETRY. 
 
 INTRODUCTION. 
 
 LESSON I. 
 
 SUBJECT-MATTER OF GEOMETRY. 
 
 Evert object that we can see occupies some poiiion 
 of space, and has extent and form. If we consider some 
 object, as this book, for instance, we will perceive that it 
 has length, breadth, and thickness. These are called the 
 dimensions of the book. 
 
 If, now, we remove the book from before us, we can 
 still imagine the space which it filled to be in the form 
 of a book. This space, of course, will not be a material 
 thing like the book, but it will have form and extent the 
 same as the book had. Such definite portions of space, 
 their forms and extent, are the things considered in 
 Geometry. 
 
 These limited portions of space are called Volumes. A 
 
 volume has length, breadth, and thickness, and these are 
 
 .called its dimensions. We should be careful to distinguish 
 
 the geometrical volume, which is a portion of space, from 
 
 the solid body, which occupies space. The one is material, 
 
 the other is immaterial ; the one is real body, the other is 
 
 ideal body or pure form. It is ideal body or pure form that 
 
 is treated of in Geometry. 
 
 2 13 
 
14 INTRODUCTION. 
 
 Let us now consider this ideal body or volume a little 
 more closely. First, we will notice that it is distinctly 
 separated : from the surrounding space.. That which so 
 separates it is called a Surface; and, since that which 
 bounds the volume forms no part of the volume, it will 
 bo seen that a surface has no thickness, and possesses, 
 therefore, but two dimensions, — length and breadth. 
 
 If we consider one of these bounding surfaces, we will 
 see that it also 1 is limited or bounded. That which limits 
 a surface is called a Line ; and since that which limits a 
 surface forms no part of the surface, it is seen that a line 
 has only one dimension, — length. 
 
 Again, if we examine one of these lines, we will see 
 that, its ends are limited. Thie limit is called a Point; 
 and, since the limit forms no part of the line, a point 
 has neither length, breadth, nor thickness, but position 
 only. 
 
 Now, although we have considered a point 'as the limit 
 of a line, a line as the limit of a surface, and a surface as 
 the limit of a volume, yet each of these may be regarded 
 in a purely abstract manner, distinct from each other. 
 Thus, we may consider points without regard to lines, 
 lines without reference to surfaces, and surfaces without 
 reference to volumes. 
 
 "We have now attained a conception of the ideas of 
 Geometry by passing from a body to an abstract volume, 
 from this volume to a surface, from a surface to a line, and 
 from a lino to a point. This is the method of analysis, 
 and is, without doubt, the method in which these ideas 
 were primarily attained. They may, however, also be 
 attained by synthesis, in the following manner. 
 
 Fix upon the idea of a point in space. Now, supposo 
 
LESSON II. 1ft 
 
 this point to move, and we have a, line ; suppose the line 
 to move in a particular manner, and we have a surface ; 
 suppose the surface to move, and we have a volume. 
 
 These lines, surfaces, and volumes, of which we have 
 attained the idea, are the fundamental quantities of 
 Geometry. A quantity, you will remember, is any thing 
 that can be measured. When one line crosses another, 
 their divergence may be measured : hence we have a fourth 
 kind of geometrical quantity, called angles. "We are now 
 prepared to define Geometry. 
 
 Geometry is that science which treats of the properties 
 and relations of geometrical magnitudes. Its subject- 
 matter are lines, surfaces, volumes, and angles. 
 
 LESSON II. 
 REASONING OF GEOMETRY. 
 
 The subject-matter of Geometry, we have seen, are lines, 
 surfaces, volumes, and angles. These general conceptions 
 give rise to many special forms; these special forms are 
 described, and such descriptions constitute the Definitions 
 of the science. 
 
 When we consider these special forms of quantity, as 
 weir as quantity in general, we perceive some' truths con, 
 cerning them that are self-evident, — that must be true, 
 since they cannot be conceived as untrue. These self- 
 evident truths are called Axioms. 
 
 The science of Geometry begins with these primary 
 ideas of space and the self-evident truths arising out of 
 them, and from these, as a basis, rises to the higher truths 
 by a process of reasoning. The axioms and definitions are, 
 
16 INTRODUCTION. 
 
 therefore, said to be the basis of the science of Geometry. 
 The definitions present the subjects upon which we reason; 
 the axioms give the laws which guide us in the reasoning 
 process. From these we trace our way, step by step, to 
 the loftiest and most beautiful truths of the science, by 
 the simple process of comparison. This process of com- 
 parison, is called reasoning; and to this we now call 
 attention. 
 
 Eeasoning. — All Eeasoning is comparison. A compari- 
 son requires a standard or basis, and this standard is the 
 simple, the axiomatic, the known. To these we bring the 
 complex, the theoretic, the unknown, and learn to understand 
 them by comparing the complex with the simple, the theoretic 
 with the axiomatic, the unknown with the known. 
 
 There are two distinct methods of geometrical reasoning, 
 which may be distinguished as the analytic and synthetic 
 methods. The analytic method is adapted to the dis- 
 covery of truth ; the synthetic method, to the proving of 
 a truth when it has already been discovered. 
 
 Synthetic Method. — The synthetic method, which is 
 generally employed in proving a truth which is already 
 known, is called demonstration. There are two distinct 
 methods of demonstration, called the Direct and the In- 
 direct Method. 
 
 The simplest form of the Direct Method is that in which 
 figures are directly compared by applying one to another. 
 This is called the method by superposition. The more 
 general form of the direct method is that in which truths 
 are proven by a reference to the definitions and axioms, 
 or some principle previously proven. 
 
 The Indirect Method, known as the reductio ad absurdum, 
 consists in supposing the proposition to be proven not to 
 
LESSON II. 17 
 
 be true, and then showing that such an hypothesis leads 
 to a contradiction of some known truth. This is fre- 
 quently used to prove the converse of a proposition, when 
 there is no good direct method ; it is also used in incom- 
 mensurable quantities. 
 
 There are two errors of demonstration into which young 
 pupils are liable to fall. The first is called Seasoning in a 
 Circle; the second is Begging the Question. We reason in a 
 circle when, in demonstrating a truth, we employ a second 
 truth which cannot be proven wit bout the aid of the first. 
 We are said to beg the question when, in order to establish 
 a proposition, we employ the proposition itself. 
 
 Analytic Method. — The analytic method begins with 
 the thing required, and by tracing the relations of the 
 various parts we arrive at some known truth. It is a 
 kind of going back from the result sought by a chain of 
 relations to what has been previously established. In a 
 demonstration, we pass through every step from the sim- 
 plest self-evident truth to the highest deductions of the 
 science ; in the process of analysis, we pass over every 
 step from the latter truths down to the simplest. 
 
 Analysis is the method of discovery; synthesis, of 
 demonstration. The one has for its object to find un- 
 known truths; the other, to prove known ones. Fre- 
 quently both methods are employed simultaneously, when 
 the object is to discover new relations, or the solution of 
 new problems ; but when we wish to prove to others the 
 truths we have discovered, the synthetical method is 
 
 usually preferred. 
 
 2* 
 
18 INTRODUCTION. 
 
 LESSON" III. 
 
 GEOMETRICAL LANGUAGE. 
 
 Language is the instrument of thought tod the medium 
 of expression. All thinking is by means of language ; and 
 the more concise and perfect the language, the more pro- 
 found and searching is our thought. The language of 
 mathematics differs somewhat from that of ordinary- 
 usage, in being more concise and more definite in its use. 
 
 Much of the language of mathematics is symbolical; 
 that is, a symbol is used in place of the written word. 
 There are three classes of symbols in Geometry: symbols 
 of quantity, symbols of operation, and symbols of relation. 
 
 The Symbols op Quantity are usually pictured repre- 
 sentations of the quantities considered. Sometimes, how- 
 ever, the letters of the alphabet are used to indicate them. 
 
 The Symbols of Operation are as follow : — 
 
 The Sign of Addition, -)-, called plus; thus, A -f B, denotes 
 that B is to be added to A. 
 
 The Sign of Subtraction, — , called minus; thus, A — B, 
 denotes that B is to be subtracted from A. 
 
 The Sign of Multiplication, X ; thus, A X B, denotes that 
 A is to be multiplied by B. 
 
 The Sign of Division, -h ; thus, A -s- B, denotes that A is 
 to be divided by B. 
 
 The Exponential Sign; thus, A i , denotes that A is used 
 four times as a factor, or is raised to the fourth power. 
 
 The Radical Sign, ^/; thus, y'A, fB, denotes that the 
 square root of A and the cube root of B are to be ex- 
 tracted. 
 
 The Parenthesis and Vinculum denote that the quantity 
 
LESSON III. 19 
 
 is to be operated upon as a whole ; thus, (A -f- B) X 0. or 
 A-\-B X C, denotes that the sum of A and B is to be mul- 
 tiplied by G. 
 ■ The Symbols op Eelation are as follow : — 
 
 The Sign of Equality, =', thus, A = B -J- G, denotes that 
 A is equal to the sum of B and G. 
 
 The expression of the equality of two quantities is ah 
 equation ; thusj A = B -p C, is an equation. The part on 
 the left of the sign of equality is the first member j that on 
 the right is the second member. 
 
 The Sign of Inequality, > or < ; thus, A > B, denotes 
 that A is greater than B. The greater quantity is at the 
 opening of the sign. 
 
 The Sign of Matio, : ; thus, A : B, denotes the ratio of 
 A to B. 
 
 The Sign of Equal Ratios, : : ; thus, A: B :: G: D, denotes 
 that the ratio of A to B equals the ratio of G to D. 
 
 We present also a few combinations of these symbols, 
 called formulas, which will be found valuable in some of 
 the demonstrations. 
 
 1.AXB+CXB = (A+C)XB- 
 2.iAxB—iBxC=i(A—G)XB. 
 
 3. (A + By = A* + 2AxB + B\ 
 
 4. (A — 5) 2 = J. 3 — 2AXB + B*. 
 
 5. (A -f B) X (A — -B) = 4 2 — B 1 . 
 
 DEFINITION OF TERMS. 
 
 An Axiom is a self-evident truth. 
 
 A Theorem is a truth to be demonstrate'd. 
 
 A Problem is a question to be solved. 
 
 A Postulate is a problem whose solution is self-evident. 
 
20 
 
 INTRODUCTION. 
 
 A Corollary is an obvious consequence of, or a theorem 
 suggested by, one or more propositions. 
 
 A Scholium is a remark upon one or more propositions. 
 
 Theorems, Axioms, Problems, and Postulates, are all 
 called Propositions. 
 
 An Hypothesis is a supposition made in the statement 
 of a proposition, or in its demonstration. 
 
 Note. — In making references, A. stands for Axiom; B. for Book; C. for 
 Corollary; D. for Definition; I. for Introduction; Th. for Theorem; P. for 
 Problem ; S. for Scholium. In referring to another Book, the number of the 
 book is given ; in referring to the same Book, the number of the Book is not 
 given. 
 
ELEMENTARY GEOMETRY. 
 
 BOOK I. 
 
 DEFINITIONS. 
 
 1 Geometry is the science which treats of the pre 
 ptrties and relations of geometrical magnitudes. 
 
 2. A Geometrical, Magnitude is some definite 
 element of space. It is a line, a surface, a volume, or an 
 angle. 
 
 3. A Point is that which has position, but no mag- 
 nitude. 
 
 4. A Line is that which has length, but no breadth 
 or thickness. Lines are straight or curved. 
 
 5. A Straight Line is one which 
 
 a — : B 
 
 has the same direction at every point : 
 as, AB. 
 
 6. A Curved Line is one which 
 changes its direction at every point: 
 as, OB. 
 
 The word line used alone, means a straight line; the 
 word curve, alone, means a curved line. 
 
 7. A Surface is that which has length and breadth, 
 without thickness. Surfaces are plane or curved. 
 
 8. A Plane is a surface such that if any two of its 
 points be joined by a straight line, every part of that line 
 will lie in the surface. 
 
 21 
 
22! 
 
 GEOMETRY. 
 
 9. A Volume is that which has length, breadth, and 
 thickness. 
 
 PLANE ANGLES. 
 
 10. An Angle is the difference of direction, or the 
 divergence, of two lines proceeding from 
 
 A 
 
 a common point. 
 
 The point from which the lines proceed 
 is called the vertex of the, angle ; the lines 
 themselves are the sides of the angle. 
 
 An angle is named by the letter at the vertex, or by 
 the three letters with the letter at the vertex in the 
 middle. Thus, we say the angle C, or the angle ACB. 
 
 11. Adjacent Angles are angles 
 which have a common vertex with a com- 
 mon -side between them ; thus, A CD and 
 BCD are adjacent angles. 
 
 12. A Eight Angle is an angle formed 
 by one straight line meeting another, 
 making the adjacent angles equal. The 
 first line is then said to be perpendicular to 
 the other. 
 
 13. An Obtuse Angle is one which is 
 greater than a right angle ; as, A CD. 
 
 An Acute Angle is one which is less 
 
 -a 1/ JS 
 
 than a right angle ; as, DOB. 
 
 Obtuse and acute angles are called oblique angles, in dis- 
 tinction from right angles. 
 
 PARALLEL LINES. 
 
 14. Parallel Lines are those 
 which have the same direction; as, a 
 A Band CD. 
 
 c- 
 
BOOK I. 23 
 
 When a straight line intersects 
 two parallel straight lines, the an- 
 gles formed take particular names. 
 Suppose the line EF to intersect j,_ 
 the parallels AB and GD; then — 
 
 1. Interior Angles on the 
 same side are those which lie within the parallels, on 
 the same side of the secant, or intersecting line; thus, 
 GGH and ARG; also, HGD and GHB; 
 
 2. Alternate Interior Angles lie within the 
 parallels, on different sides of the secant line, but not 
 adjacent; as, CGH &&&. GHB; 
 
 3. Exterior-interior Angles lie on the same 
 side of the secant line, one without and the other within the 
 parallels, but not adjacent ; as EGD and GHB. They are 
 also called corresponding angles. 
 
 PLANE FIGUBES. 
 
 15. A Plane Figure is a plane bounded by lines 
 either straight or curved. 
 
 16. A Polygon is a plane figure bounded 
 by straight lines. These lines are called sides 
 of the polygon ; taken together, they form 
 the perimeter of the polygon. 
 
 17. A Polygon of three sides is called a triangle; of 
 four sides, a quadrilateral; of five sides, a pentagon; of 
 six sides, a hexagon; of seven sides, a heptagon; of eight 
 sides, an octagon; of nine sides, a nonagon; of ten sides, a 
 decagan,~etc. 
 
 18. An Equilateral Polygon is one whose sides 
 are equal. An Equiangular Polygon is one whoso angles 
 are equal. 
 
 Two. polygons are mutually equilateral when their sides 
 
24 GEOMETRY. 
 
 are respectively equal. Two polygons are mutually equU 
 angular when their angles are respectively equal. 
 
 19. A Diagonal of a polygon is a line joining the 
 vertices of two angles, not consecutive. 
 
 TEIANGLES. 
 
 20. A Triangle is a polygon of three sides and three 
 angles. Triangles are classified by their sides and their 
 angles. 
 
 21. A Scalene Triangle is one in 
 which the three sides are unequal. 
 
 22. An Isosceles Triangle is one 
 which has two of its sides equal. 
 
 23. An Equilateral Triangle is 
 one which has its three sides equal. 
 
 24. A Eight-Angled Triangle 
 is one which has one right angle. The 
 side opposite the right angle is called the 
 kypothenuse. 
 
 25. An Acute-Angled Triangle is one in which 
 all the angles are acute. 
 
 26. An Obtuse-Angled Triangle is one which 
 has one obtuse angle. 
 
 Triangles are the simplest of all polygons, since three sides are the 
 least number that can bound a plane figure. The properties of poly* 
 gons are determined by analyzing them into triangles. 
 
BOOK I. 25 
 
 QUADRILATERALS. 
 
 27. A Quadrilateral is a polygon of four sides and 
 four angles. ' There are three classes : — 
 
 1. The Trapezium is a quadrilateral 
 having no two sides parallel. 
 
 2. The Trapezoid is a quadrilateral 
 having two of its opposite sides parallel. 
 
 - 3.- The Parallelogram is a quadri- 
 lateral having its opposite sides parallel. 
 
 28. Parallelograms are divided, from their angles, into 
 two classes,' — right-angled and oblique-angled parallelo- 
 grams. 
 
 1. A Eectangle is a parallelogram 
 whose angles are right angles. 
 
 ASQUAREisan equilateral rectangle. 
 
 2. A Ehomboid is' a parallelogram 
 whose angles are oblique. 
 
 AEHOMBUsisan equilateral rhomboid. 
 
 THE CIRCLE. 
 29. A Circle is a plane figure bounded by a curve 
 line, every point of which is equally distant from a point 
 within, called the centre. 
 
26 
 
 GEOMETRY. 
 
 The Circumference is the bound- 
 ing line ; any part of the circumference 
 is called an arc. A line through the 
 centre having its ends in the circum- 
 ference is a diameter ; a line from the 
 centre to the circumference is the radius. 
 
 30. The arcs of circles are used to measure angles. An 
 angle haying its vertex at the centre of a circle is measured 
 by the arc included between its sides ; thus, the arc BD meas- 
 ures the angle DOB. 
 
 To measure angles, the circumference is divided into 360 
 equal parts, called degrees. Bach degree 
 is 'divided into 60 equal parts, called 
 minutes; each minute into 60 equal 
 parts, called seconds. Degrees, minutes, 
 and Seconds are marked thus, ° ' " ; 16° 
 24' 32" are read, 16 degrees, 24 minutes, 
 and 32 seconds. 
 
 A right angle, it will be seen, is measured by 90 
 right angle, by 45° ; two right angles, by 180° ; four right 
 angles, by 360°. 
 
 °; half a 
 
 AXIOMS. 
 31. An Axiom is a self-evident truth. There are two 
 classes of axioms in Geometry. First, those which pertain 
 to quantity in general ; second, those which arise out of 
 the special forms of geometrical quantity. 
 
 FIRST CLASS. 
 
 1. Things which are equal to the same thing are equal 
 to each other. 
 
 2, If equals be added to equals, the sums will be equal. 
 
BOOK I. 27 
 
 3. If equals be subtracted from equals, the remainders 
 Will be equal. 
 
 4. If equals be added to or subtracted from unequals, 
 the results will be unequal. 
 
 5. If equals be multiplied by equals, the products will 
 be equal. 
 
 6. If equals be divided by equals, the quotients will be 
 «qual. 
 
 7. The whole is greater than any of its parts. 
 
 8. The whole equals the sum of all its parts. 
 
 SECOND CLASS. 
 
 9. Only one straight line can be drawn connecting two 
 given points. 
 
 10. A straight line is the shortest distance from one 
 point to another. 
 
 11. All right angles are equal to each other. 
 
 12. Parallel straight lines cannot meet each other when 
 produced. 
 
 13. Through a given point only one straight line can be 
 drawn parallel to a given line. 
 
 Corollary. From axiom 10, it is evident that any side of a 
 triangle is less than the sum of the other two sides. 
 
 POSTULATES. 
 32. The following postulates are self-evident problems 
 resulting from the preceding definitions : — 
 
 1. A straight line can be drawn from one point to 
 another. 
 
 2. A straight line may be prolonged to any length. 
 
 3. A line or an angle may be bisected. 
 
 4. An angle may be described equal to a given angle. 
 
28 GEOMETRY. 
 
 5. A line may be drawn through a given point parallel to a 
 given line. 
 
 6. A perpendicular may be drawn to a given line from a 
 point without the line or in the line. 
 
 Analysis of Book I. — Book I. treats mainly of angles, parallel lines, 
 triangles, and parallelograms. It treats of the angles formed by one 
 line meeting or cutting another, of the angles formed by one line cutting 
 two parallel lines, of the equality and inequality of triangles, of the sum 
 of the angles of a triangle, of the relation of the angles and sides of a 
 parallelogram, and of the exterior and interior angles of a polygon. It 
 is thus seen that the idea of angles is a prominent, if not the principal 
 »ne at the book. 
 
BOOK I. 
 
 29 
 
 OP ANGLES. 
 
 THEOREM I. 
 
 When one straight line meets another straight line, the sum of 
 the two adjacent angles equals two right angles. 
 
 Let the straight line D G meet the straight line AB at 
 the point 0; then will AGB -{- BCB = E 
 
 two right angles. 
 
 For, at the point' C, erect GE perpen- 
 dicular to AB; then (D. 12,) the angles j — n 
 AGE and EGB are both right angles. Now, 
 AGD =u right angle -f- EGB; and 
 DGB — a right angle — EGB ; hence, adding, 
 we have, A GB -f- B OB = two right angles. 
 
 Therefore, when a straight line meets another straight 
 line, the sum of the two adjacent angles equals two right 
 angles. 
 
 Gor. 1. If one of the angles A GB or BGB is a right angle, 
 the other is also a right angle. 
 
 Gor. 2. The sum of all the angles formed on the same 
 
 side of a straight line by drawing lines 
 
 to any point of that line, is equal to 
 
 two right angles. For, their sum is 
 
 equal to the sum of A GB and BGB, 
 
 which is equal to two right angles, 
 
 according to the proposition. 
 
 3* 
 
30 GEOMETRY. 
 
 THEOREM II. 
 
 When two straight lines intersect each other, the opposite or 
 vertical angles are equal. 
 
 Let the two straight lines AB c 
 and CD intersect each other at the 
 point E; then will AEG be equal to 
 BED. 
 
 .For, since GE meets AB, the angle 
 
 AEG + GEB = two right angles (Th. I.); and since BE 
 
 meets CD, the angle GEB -\- BED = two right angles ; but 
 
 things which are equal to the same thing are equal to 
 
 each other (A. 1) ; hence, 
 
 AEG + GEB = GEB + BED. 
 
 Taking from each sum the common angle GEB, there 
 
 remains (A. 3), 
 
 AEG • = BED. 
 
 In a similar manner it may be shown that the anglq 
 AED equals GEB. Therefore, etc. 
 
 Cor. 1. The sum of the four angles formed by the inter- 
 section of two lines is equal to four right angles. 
 
 Gor. 2. The sum of all the angles that can be formed 
 about a point is equal to four right angles. 
 
 PARALLEL LINES. 
 
 THEOREM III. 
 
 If a line intersect two parallel lines; 
 
 1. The exterior4nterior angles will be equal. 
 
 2. The alternate interior angles will be equal. 
 
 3. The sum of the interior angles on the same side will be 
 equal to two right angles. 
 
 Let the line EF intersect the two parallels AB and CD ; 
 
 then, 
 
a 
 
 
 */ „ 
 
 A- ' 
 
 
 /H 
 
 
 BOOK I. 31 
 
 First. The angle EGD is equal to / 
 
 GSB. For, since SB and GD are pa- 
 rallel, they have the same direction; 
 hence, they must diverge equally from 
 the line EF; therefore, the difference 
 of direction or divergence of GE and GD must be equal 
 to the divergence of SE and SB, or the angle EGD equal 
 to GSB. In the same way it may be shown that FSB = 
 SGD. 
 
 Second. The two alternate angles C(3\H~and.Eff.Z?will be 
 equal. For, GGS equals EGD (Th. II.) ; but EGD equals 
 ESB; therefore, CGS= GSB (A. 1); and in the same 
 manner it may be shown that ASG equals SGD. 
 
 Third. The sum of the two interior angles GSB and 
 SGD equals two right angles. For, EGD -f- DGS — 
 two right angles (Th. I.); but EGD '= ESB; hence, GSB 
 + SGD = two right angles. In the same way it may 
 be shown that ASG -\- GGS equals two right angles. 
 Therefore, etc. ..'.'■ ; ' 
 
 Cor. If a line is perpendicular to one of two parallels, it is 
 perpendicular to the other also. Far, if EGD were a right 
 a-ngle, its equal ESB would be a right angle also. 
 
 THEOREM IV. 
 
 Conversely. — If a straight line meets' two other straight lines, 
 these two lines will be parallel; 
 
 1. When the exterior-interior angles are equal. 
 
 2. When the alternate interior angles are equal. 
 
 3. When the sum of the two interior angles on the same side 
 is equal to two right angles. 
 
 Let the straight line EF meet the two straight lines AB 
 and CD; then, 
 
32 GEOMETRY. 
 
 First. If the angles EGD and EHB E 
 
 are equal, the lines are parallel. For, G / 
 
 since EGD and EHB are equal, the / 
 
 lines GD and SB must diverge equally,. "/^ 
 from EF; hence, they have the same F 
 direction, and are, therefore, parallel (D. 14). 
 
 Second. If the alternate angles CGHand GHB are equal, 
 the lines are parallel. For, since CGH equals EGD (Th. II.), 
 when CGH equals GHB, EGD equals GHB; but then the 
 lines are parallel, as has just been shown ; hence, the lines 
 are parallel when the alternate angles are equal. 
 
 Third. If the sum of the two interior angles GHB and 
 HGD equals two right angles, the' lines are parallel. For, 
 EGD + HGD = two right angles (Th. I.) ; hence, EGD + 
 HGD = HGD + GHB (A. 1). Taking HGD from each, 
 we have EGD = GHB; but then the lines are parallel, 
 according to the first part of the theorem; hence, they 
 are parallel when GHB -\- HGD equals two right angles. 
 
 Cor. 1. If two lines are perpendicular to the same line, they 
 are parallel. For, if EGD and GHB are both right angles, 
 they are equal, and the lines CD and AB are parallel. 
 
 Cor. 2. If two lines are parallel to the same line, they are 
 parallel to each other. 
 
 Cor. 3. If the sum of the two interior angles on the same 
 side is less than two right angles, the lines will meet. 
 
 THEOREM V. 
 
 Two angles which have their sides respectively parallel, and lying 
 in the same direction or in opposite directions, are equal. • 
 
 First. Let the angles ABC and DEF have their side* 
 parallel and lying in the same direction; then will ABC 
 equal DEF. For, prolong FE to G. Then, since AB and 
 
BOOK I. 33 
 
 DE are parallel, DEF equals AGE (Th. 
 III.); and since GF and BC are paral- 
 lel, AGE equals ABC (Th. III.) ; hence, 
 DEF equals ABC (A. 1). 
 
 Second. Let the angles ABC and D.E'i'' 
 have their sides parallel and lying in 
 opposite directions; then will ABC 
 equal DEF. For, prolong C5 to G. 
 Then ABG equals EGB.(Th. III.); 
 but EGB equals Di2F, being al- 
 ternate ; hence, ABC equals. DEF. 
 Therefore, etc. 
 
 TRIANGLES. 
 
 THEOREM VI. 
 
 If two triangles have two sides and the included angle of the 
 one equal to two sides and the included angle of the other, 
 each to each, the triangles will be equal in all their parts. 
 
 Let the triangles ABC 
 and DEF have the side AB 
 equal to DE, AC to DF, 
 and the angle A equal to 
 the angle Dj then will the 
 triangle ABC be equal to 
 the triangle DEF. 
 
 For, apply the triangle ABC to the triangle DEF, placing 
 the side AB upon the equal side DE; then, since the angle 
 A equals the angle J), the side AC will take the direction 
 of DF, and the point C wjll coincide with F, since the two 
 lineb are equal ; and the side CB will coincide with the side 
 
34 GEOMETRY. 
 
 FE (A. 9). Therefore, the triangles coincide and are equa^ 
 in all their parts. Therefore, etc. 
 
 THEOREM VII. 
 
 If two triangles have two angles and the included side of the 
 one equal to two angles and the included side of the other, 
 each to each, the triangles will be equal in all their farts. 
 
 Let ABC and DEF be 
 two triangles having the 
 angle A equal to the angle 
 D, the angle B equal to the 
 angle E, and the included 
 side AB equal to the in- 
 cluded side DE; then will the two triangles be equal in 
 all their parts. 
 
 Tor, apply the triangle ABC to the triangle DEF, placing 
 the side AB upon DE, the point A upon. D, and the point B 
 upon E; then, since the angle A equals the angle D, the 
 side A C will take the direction DF, and the point Cwill be 
 found somewhere in the line DFj and since the angle B 
 equals the angle E, the side B C will take the direction EF, 
 . and the point C will be found somewhere in EF. Hence, 
 the point C being in the two lines DF and EF, must be at 
 their intersection; consequently, the triangles coincide and 
 are equal in all their parts. Therefore, etc. 
 
 THEOREM VIII. 
 
 If two triangles have two sides of the one equal to two sides of 
 the other, each to each, and the included angles unequal, the 
 third side will be greater in the triangle having the greater 
 included angle. 
 
 Itet ABC and DEF be two triangles in which A C= DF. 
 
BOOK I. 35 
 
 BG= EF and A GB > BFE; then will AB be greater than 
 BE. 
 
 For, at the point G make" 
 the angle BGG = EFD, 
 make C(x = FB, and draw 
 !?(?; then will the triangle ■*' 
 GGB equal BFE and &.B 
 equal BE (Th. VI.). Draw. 
 OX", bisecting the angle A GH, and draw also GK ; the two 
 triangles AGK and XOG 1 are equal (Th. VI.), &n&AK= 
 KG. Now, KG-\-KB> GB; hence AK + 7t5, or J..B, is 
 greater than 6rj? or its equal BE. 
 
 The same demonstration will apply when the point G falls 
 within AB. If it falls on AB, the theorem is true by A. 7. 
 
 Cor. The converse of this theorem is also true. 
 
 THEOREM IX. 
 
 If two triangles have the three sides of the one equal to the, 
 three sides of the other, each to each, the triangles will be 
 equal in all their parts. 
 
 Let ABC and BEF be 
 two triangles, having AB 
 equal to BE, A G to BF, 
 and BG to EF; then will 
 the triangles be equal in 
 all their parts. 
 
 For, since AG and AB are respectively equal to BF and 
 BE, if the angle A were greater than B, BG would be 
 greater than EF (Th. VIII.) ; and if .1 were less than B, 
 BG would be less than EF, for the same reason. But BG 
 is equal to EF, therefore the angle A must be equal to B. 
 
3(j GEOMETRY. 
 
 In the same way it may be shown that the angle G equal* 
 F, and the angle B equals E. Therefore, etc. 
 
 THEOREM X. 
 
 In an isosceles triangle the angles opposite the equal sides an 
 
 equal. 
 
 . Let ABO be an isosceles triangle, having 
 the side A equal to the side B 0; then will 
 the angle A be equal to the angle B. 
 
 Join the vertex and the middle point of 
 the base ABj then in the two triangles ADG 
 and GDB, AG equals BG, DC is common, and AD equals 
 DB; hence, the two triangles are equal in all their parts 
 (Th. IX.), and the angle A is equal to the angle B. 
 
 Cor. 1. A line drawn from the vertex of an isosceles tri- 
 angle to the middle point of the base, bisects the vertical 
 angle and is perpendicular to the base ; also, a line bisecting 
 the vertical angle is perpendicular to the base and bisects it ; 
 also, a line drawn from the vertex perpendicular to the base 
 bisects both the base and the vertical angle. 
 
 Cor. 2. Hence, also, an equilateral triangle is equian 
 gular ; that is, it has all its angles equal. 
 THEOREM XI. 
 
 Conversely. — If two angles of a triangle are equal, the sid<% 
 opposite them are also equal, and the triangle is isosceles. 
 
 Let ABO be a triangle, having the angle 
 A equal to the angle B; then will the side 
 AC be equal to BG. 
 
 For, if AG and OB are not equal, sup- 
 pose one of them, as A 0, to be the greater. 
 Then, take AD equal to BO, and draw DB. 
 How. in the triangles ABO and ABD, we have the side A Z? 
 
BOOK I. 37 
 
 equal to BC, by construction, the side AB common, and the 
 included angle ABC equal to the included angle DAB, by 
 hypothesis; hence, the two triangles ABB and ABC are 
 equal (Th. VI.) ; that is, a part equal to the whole, which 
 is impossible (A. 7). Hence, the hypothesiB that A C and 
 BC are unequal is false ; therefore, they are equal, and the 
 triangle is isosceles. Therefore, etc. 
 
 THEOREM XII. 
 
 In any triangle the greater side is opposite the greater angle, 
 and, conversely, the greater angle is opposite the greater side. 
 
 In the triangle ABC, let the angle ABC be 
 greater than CAB; then will A C be greater 
 than BC. 
 
 For, draw BD, making the angle ABD = 
 BAB; then will AB = BB (Th. XI.). To. 
 each add BC and we have AB + BC= BB + BC; but 
 BB + B C > B C (A. 10) ; hence, AB + BC,orAC,is greater 
 than BC. 
 
 Conversely. Let the side A C> BC; then will the angle 
 ABC> CAB. For, if ABC < CAB, AC<BC, from what 
 has just been proved ; and if ABC= CAB, AC '— BC (Th. 
 XI.) ; but both of these results are contrary to the hypo- 
 thesis; hence, ABC must be greater than CAB. There- 
 fore, etc. 
 
 THEOREM XIII. 
 
 Jn every triangle the sum of the three angles is equal to two 
 
 right angles. ' 
 
 c 
 Let ABC be a triangle ; then will 
 
 the sum of its three angles, A, B, C, 
 
 be equal to two right angles. 
 
 For, prolong AB, and draw BB jr 
 
38 GEOMETRY. 
 
 parallel t^ AC; then, since the parallels A and BD&re cut 
 Oy AE, the angle A is equal to the opposite exterior angle 
 DBE (Th. III.). In like manner, since the parallels are 
 tut by BO, the alternate angles C and GBD are equal; 
 denee, the sum of the three angles of the triangle is equal 
 to the sum of the angles ABC, GBD, DBE; but this latter 
 Bum equals two right angles (Th. I. C. 2); therefore, the 
 sum of the three angles of the triangle equals two right 
 angles. Therefore, etc. 
 
 Cor. 1. If two angles of a triangle are given, the third 
 will be -found by subtracting their sum from two right 
 angles, or 180° ; hence, if two triangles have two angles of the 
 one equal to two angles of the other, the third angles will be equal. 
 
 Cor. 2. A triangle cannot have more than one right angle ; 
 for if there were two the third angle would be zero. 
 
 Cor. 3. A triangle can have only one obtuse angle, but 
 must have at least two acute angles. 
 
 Cor. 4. In a right-angled triangle the sum of the two acute 
 angles equals one right angle, or 90°. 
 
 Cor. 5. In every triangle the exterior angle is equal to the 
 sum of the two interior opposite angles. 
 
 Scholium. This theorem may be demonstrated by draw- 
 ing a line parallel to either of the other sides of the tri- 
 angle. Let the pupils be required to do it. 
 
 THEOREM XIV. ' 
 
 If from a point without a straight line a perpendicular be let 
 fall on the line and oblique lines be drawn ; 
 
 1. The perpendicular will be shorter than any oblique line. 
 
 2. Any two oblique lines which terminate at equal distances 
 from the foot of the perpendicular are equal. 
 
 3. The oblique line which terminates at the greater distance 
 from the foot of the perpendicular is the greater. 
 
BOOK I. 39 
 
 Let C be a given point, and AD a given line, CE a per- 
 pendicular, and GA, GB, and CD, oblique 
 lines ; then, 
 
 First. In the triangle AEG, the angle 
 AEG is a right angle, and, consequently, 
 greater than A; therefore, the side CE is 
 shorter -than GA (Th. XII.). 
 
 Second. Let AE=EB; then, since OE is common and 
 the angle AEG= GEB, the triangles AEG and GEB are 
 equal (Th. VI.), and A G equals BG. 
 
 Third. Let ED be greater than EB ; then, since GBEis 
 an acute angle, CBD must be obtuse and BD C acute, and in 
 the triangle CBD, CD is greater than B C, being opposite the 
 greater angle (Th. XII.). Therefore, etc. 
 
 Cor. 1. Only one perpendicular can be drawn from the 
 same point to the same straight line. 
 
 Cor. 2. Two equal oblique lines terminate at equal distances 
 from the foot of the perpendicular. 
 
 Con. 3. A line having two points, each equally distant from 
 the extremities of another line, is perpendicular to that line 
 and bisects it. 
 
 QUADRILATERALS.- 
 
 THEOREM XV. 
 
 In any parallelogram, the. opposite sides and angles are equal, 
 each to each. 
 
 Let AB GD be a parallelogram ; then will j> ^ 
 
 AB be equal to DC, and AD to BG. / \ / 
 
 For, draw the diagonal DB. Then, since / \/ 
 
 AB and DC 'are parallel, the alternate an- 
 gles ABD and BDG are equal (Th. III.) ; and since AD and 
 
40 GEOMETRY. 
 
 BG are parallel, the alternate angles ADB and DBG ar& 
 equal. Hence, the two triangles ABB and DBG have two 
 angles and the included side, DB, of one, equal to two 
 angles and the included side, DB, of the other, each to 
 each; therefore, the triangles are equal (Th.VIL); and 
 the side AB opposite the angle ADB is equal to the side 
 DG opposite the equal angle DBG: hence, also, the side 
 AD equals BG; therefore, the opposite sides of a parallel- 
 ogram are equal. 
 
 Again, since the triangles are equal, the angle A is equal 
 to the angle G; and the angle ADG, which is made up of 
 the two angles ADB and BDG, is equal to the angle ABG, 
 which is made up of the equal angles DBG and ABD- 
 Therefore, etc. 
 
 Cor. 1. The diagonal divides the parallelogram into two 
 equal triangles. 
 
 Cor. 2. Two parallels included between two other paral- 
 lels are equal. 
 
 Gor. 3. Two parallelograms are equal when they have 
 two sides and the included angle of one, equal to two sides 
 and included angle of the other. 
 
 THEOREM XVI. 
 
 If the opposite sides of a quadrilateral are equal, each to each, 
 the equal sides are parallel, and the figure is a parallelogram. 
 
 TjetABGD be a quadrilateral, in which 
 AB equals DG, and AD equals BG; then 
 will it be a parallelogram. 
 
 For, draw the diagonal DB. Then the 
 triangles ABD and DBCh&re all the sides 
 of the one equal to all the sides of the other, each to each ; 
 therefore the two triangles are equal (Th. IX.") ; and the 
 
BOOK I. 41 
 
 angle ABB opposite the side AB is equal to the angle BBC 
 opposite the equal side B C; therefore, the side AB is paral- 
 lel to the side DG (Th. IV.)- For a like reason, AB is 
 parallel to BC; therefore, the figure ABCB is a* paral- 
 lelogram. Therefore, etc. 
 
 THEOREM XVII. 
 
 If two sides of a quadrilateral are equal and parallel, the figure 
 is a parallelogram. , 
 
 Let ABCB be a quadrilateral, having the 
 sides AB and BO equal and parallel ; then 
 will ABCB be a parallelogram. 
 
 For, draw the diagonal BB. Then, since 
 AB is parallel to BO, the alternate angles 
 ABB and BBC are equal (Th. III.). Now, the triangles 
 ABB and BBC have the side AB equal to BC, by hypo- 
 thesis, the side BB common, and the included angles ABB 
 and BBC equal ; hence, the triangles are equal (Th. VI.), 
 and the alternate angles ABB and BBC are equal ; hence, 
 the sides AB and BC are parallel (Th. IV.), and the figure 
 is a parallelogram. Therefore, etc. 
 
 THEOREM XVIII. 
 
 The diagc nals of a parallelogram bisect each other ; that is, 
 divide each other into equal parts. 
 
 Let ABCB be a parallelogram, and^C %_ 
 and BB its diagonals ; then will AH be 
 equal to EG, and BE to EB. 
 
 For, since AB and BC are parallel, the A B 
 
 angle CBE equals ABE (Th. III.); and also BCE equals 
 EAB; and since AB equals BC, the triangles AEB and 
 BEC have two angles and the included side of the one 
 
 4* 
 
42 GEOMETRY. 
 
 equal to two angles and the included side of the other, 
 hence, the triangles are equal (Th. VII.), AE equals CE, and 
 DE equals BE; therefore, the diagonals are bisected at E. 
 
 ANGLES OF POLYGONS. 
 
 THEOREM XIX. 
 
 If each side of a convex polygon be produced so as to form one 
 
 exterior angle at each vertex, the sum of the exterior angles will 
 
 be equal to four right angles. 
 
 Let ABCDEF be a convex polygon, with each side pro- 
 duced so as to form one exterior angle at 
 each vertex; then will the sum of the ex- 
 terior angles be equal to four right angles. 
 
 For, from any point within the polygon, 
 draw lines respectively parallel to the sides 
 of the polygon ; the angles contained by the 
 lines about this point will be equal to the exterior angles 
 of the polygon (Th. V.). But the sum of the angles formed 
 about a point equals four right angles (Th. II. C. 2) ; hence, 
 the sum of the exterior angles of a polygon equals four 
 right angles. Therefore, etc. 
 
 Cor. 1. The sum of the interior angles of a polygon is 
 equal to twice as many right angles as the polygon has 
 sides, less four right angles. 
 
 The sum of each exterior and interior 
 angle equals two right angles, and there 
 are as many of each as the polygon has 
 sides ; hence, the sum of all the exterior 
 and interior angles equals two right angles 
 taken as many times as there are sides of 
 the polygon. But the sum of the exterior angles equals 
 
BOOK I. 43 
 
 four right angles ; hence, the sum of the interior angles 
 equals two right angles taken as many times as the poly- 
 gon has sides, minus four right angles. 
 
 Cor. 2. The sum of- the interior angles of a quadrilateral 
 equals 2 right angles multiplied by 4, minus 4 right angles, 
 which is 8 — 4, or 4 right angles. In a rectangle each angle 
 is a right angle. 
 
 Cor. 3. The sum of tne angles of a pentagon equals 2 X 
 
 5 — 4 = 6 right angles. Each angle of an equiangular pen- 
 tagon is i of 6 or f of a right angle, or 108°. 
 
 Cor. 4. The sum of the angles of a hexagon equals 2 X 
 
 6 — 4 = 8 right angles. Each angle of an equiangular 
 hexagon is | of a right angle, or 120°. 
 
 Cor. 5. In polygons of the same number of sides, the 
 sum of the angles is the same. In equiangular polygons, each 
 angle equals the sum -divided by the number of sides. 
 
 Scholium. This theorem is true at whichever extremity the 
 sides are produced. 
 
 PRACTICAL EXAMPLES. 
 A common deficiency of pupils in the study of Geometry, is their 
 inability to make a practical application of their knowledge. To remedy 
 this, practical examples should be given, either in connection with the 
 theorems or at the close of each book. The following problems may 
 be used in either of these ways which the teacher may prefer. 
 
 1. If one line meet another line at an angle of 60°, what is the value 
 of the adjacent angle ? 
 
 Solution. — If the line DC meets AB, making the 
 angle BOB equal to 60°, the angle AOD will equal 
 180 o _ 60°, or 120°, since ACV+DCB=180°. J 
 
 2. If two lines me«t a third at the same point, making angles equal 
 to 30° and 80° respectively, required the angle between the two lines. 
 
44 GEOMETRY. 
 
 3. How many degrees in each angle of a rectangle '' 
 
 4. How many degrees in each angle of an equilateral triangle ? 
 
 5. If two angles of a triangle are 43° and 75° respectively, what is 
 the other angle ? 
 
 6. If two angles of a triangle are each 45°, what is the other angle, 
 and what is the kind of triangle ? 
 
 7. If one angle of a triangle is 60°, what is each of the other two., 
 if equal, and what is the kind of triangle ? 
 
 8. If one of the two equal angles of a triangle is 30°, what is each 
 of the other angles ? 
 
 9. Required the number of degrees in each angle of an equiangular 
 pentagon. 
 
 10. Required the number of degrees in each angle of an equiangular 
 hexagon. 
 
 11. In a triangle whose angles are A, B, C, what is each angle if A is 
 twice and B three times C? 
 
 12. In the preceding problem, what is the kind of triangle ? 
 
 13. Required each angle of an isosceles triangle, if the unequal angle 
 equals twice the sum of the other two. 
 
 14. Required the value of each exterior angle of an equiangular 
 octagon. 
 
 EXERCISES FOR ORIGINAL THOUGHT. 
 
 We now give some theorems to exercise the pupil in original thought 
 The importance of such exercises cannot be overestimated. Much of 
 the discipline of Geometry is lost by the pupil memorizing the demon- 
 strations given in the book. One can become a good- geometer only by 
 trying his powers with new theorems and problems, and endeavoring to 
 find out demonstrations and solutions for himself. 
 
 These theorems may be given upon review, one of them in connection 
 with the regular lesson ; or, if the teacher prefer, the lesson may consist 
 wholly of them. With classes whose time for the study is limited, they 
 may be omitted. 
 
 1. If the equal sides of an isosceles triangle be produced, the two 
 obtuse angles below the base will be equal. 
 
BOOK I. 45 
 
 2. If the three sides of an equilateral triangle be produced, all the 
 external- acute angles will be equal, and all the obtuse angles will be 
 equal. 
 
 3. Either side of a triangle is greater than the difference between the 
 other two. 
 
 4. If a line be drawn bisecting an angle, any point of the bisecting 
 line is equally distant from the sides of the angle. 
 
 5. Prove that the diagonals of a rectangle are equal. 
 
 6. If the diagonals of a quadrilateral bisect each other at right angles, 
 the figure is a rhombus or square. 
 
 7. If a line joining two parallels be bisected, any other line through 
 the point of bisection and joining the two parallels, is also bisected at 
 that point. 
 
 8. If from any point within a triangle, two straight lines be drawnto 
 the extremities of any side, their sum will be less than the sum of the 
 other two sides of the triangle. 
 
 9. If a line is perpendicular to another line at its middle point, — . 
 1. Any point in the perpendicular will be equally distant from the ex 
 treraities. 2. Any point out of the perpendicular will be unequally dis- 
 tant from the extremities. 
 
BOOK II. 
 
 RATIO AND PROPORTION. 
 
 1. All reasoning is by comparison. In comparing two 
 quantities, we see that they bear a certain relation to each 
 other. 
 
 2. Eatio is the measure of the relation of two similar 
 quantities. It is found by dividing the first by the second; 
 
 thus, the ratio of 8 to 4 is |, or 2, the ratio of A to B is — 
 
 3. The two quantities compared are called the Terms of 
 the ratio. The first is called the Antecedent, the second 
 the Consequent, and the two constitute a CoicpleL 
 
 4. A ratio is indicated by placing a colon between the 
 quantities, or by writing the consequent under the ante- 
 cedent, as in division ; thus, the ratio of A to B is written, 
 
 A: B, or-. 
 B 
 
 5. A Proportion is an expression of equality between 
 equal ratios ; thus, the ratio of 8 to 4 equals the ratio of 
 6 to 3, and a formal comparison of these, as 8 : 4 = 6 : 3, is 
 a proportion. 
 
 6. The equality of ratios is usually indicated by a double 
 colon ; thus, 8 : 4 : : 6 : 3. This is read, the ratio of 8 to 4 
 equals the ratio of 6 to 3, or, 8 is to 4 as 6 is to 3. 
 
 - 7. There are four terms in a proportion ; the first and 
 fourth are called the extremes; the second and thirdj the 
 
 46 
 
BOOK II. 47 
 
 means. The first and second together are the first couplet'; 
 the third and fourth, the second cotivlet 
 
 8. Quantities are m proportion by Alternation, when ante- 
 cedent is compared with antecedent, and consequent with 
 consequent ; thus, if A : B : : C: D, by alternation we have 
 A : 0: : B : D. 
 
 9. XJ uail tities are in proportion by Inversion, when the 
 antecedents are made consequents and the consequents 
 antecedents ; thus, if A : B : : G : D, by inversion we have 
 B : A : : B : 0. 
 
 10. Quantities are in proportion by Composition, when the 
 sum of antecedent and consequent is compared with either 
 antecedent or consequent ; thus, if A : B : : 0:D, by com- 
 position we have, A : A -\- B : : C: C -\- D. 
 
 11. Quantities are in proportion by Division, when the 
 difference of antecedent and consequent is compared with 
 either antecedent or consequent ; thus, if A : B : : G:D, we 
 have, A: A - B : : C-.G—D. . 
 
 A Continued Proportion is a series of equal ratios; 
 as, A: B::C;D:;E;F::, etc. 
 
 Analysis.— The object of the theorems of this book is to derive the 
 principles of proportion. These principles are employed in the books 
 which follow. The method consists in regarding .a proportion as an 
 equation, which it really is, — an equality of ratios. Thus, the pupil 
 'should be taught to regard the proportion^! : B: : C: D as equivalent 
 to A -+- B =C-j- D, and as'soon as this idea is clearly fixed in the mind 
 the subject becomes simple and easy. The first proportion is the basis 
 of demonstration for the others, and may be used as a test of the truth 
 of all others. 
 
48 GEOMETRY. 
 
 THEOREM I. 
 
 If four quantities are in proportion, the product of the means 
 will equal the product of the extremes. 
 
 Take the proportion 
 
 A: B :: G: D; then we wish to prove 
 ihat 1XD = ^X G. 
 
 For, from the proportion we have 
 
 A C 
 
 b = 3 ; multi P 1 y in s bv B x -A 
 
 we have, Ay^D = BX G. 
 Therefore; if four quantities are, etc. 
 
 THEOREM II. 
 
 If the product of two quantities equals the product of two other 
 quantities, the quantities forming one product may be made the 
 means, and the other two the extremes of a proportion. 
 
 Suppose we have 
 
 A X D = B X 6; dividing by B X B, 
 
 A G 
 we have, ~f> = Y>> pl acm g this i n another form, 
 
 we have, A: B : : G:D. 
 Therefore, etc. 
 
 THEOREM III. 
 
 A mean proportional between two quantities equals the square 
 root of their product. 
 
 Let j? be a mean proportional between A and G; then 
 we have, 
 
 A : B : : B : G; 
 whence (Th. I.), B* = AxC, 
 
 or, B = y/AxG. 
 
 Therefore, etc. 
 
BOOK II. 49 
 
 THEOREM IV, 
 
 If four quantities are in proportion, they will be in proportion 
 by alternation. 
 
 Suppose A:B::G:D; from this (Th. I.) 
 
 we have, A X D = -B X G; dividing by D X G, 
 
 we have, 7y— 7i> whence, 
 
 A:G::B:D. 
 Therefore, etc. 
 
 Remark. — The proposition is evidently true, since we have the same 
 products when we take the product of the means and extremes as 
 before the change. This principle may he applied to several other 
 propositions. 
 
 THEOREM V. 
 
 If four quantities are in proportion, they will, be in proportion 
 by inversion. > 
 
 Suppose A : B : : G : D; from this 
 
 we have, — =r— ; taking the reciprocal, 
 
 we have, — = — ; whence, 
 
 B : A : : D : G. 
 Therefore, etc. 
 
 THEOREM VI. 
 
 If four quantities are in proportion, they will be in proportion 
 by composition. 
 
 Suppose A : B : : G: D; then 
 
 A 
 we have, •5 = "n- Adding qne to each 
 
 A G 
 
 we have, — + 1 = ■= + 1 ; reducing to a common denomi- 
 
50 GEOMETRY. 
 
 , A + B C+D , 
 
 nator, we have, — B — = — '=■ — ; whence, 
 
 B JJ 
 
 A + B: B:: C+D: D. 
 
 Therefore, etc. 
 
 THEOREM VII. 
 
 If four quantities are in proportion, they will be in proport, * 
 by division. 
 
 Suppose A: B :: C: D; then 
 
 we have, — = — ; subtracting 1; 
 
 A C 
 
 we have, — — \ = -= — 1 ; reducing, 
 
 A—B C—D 
 we have, = — = — =■ — ; whence, 
 
 A — B:B:: C— D : D. 
 Therefore, etc. 
 
 THEOREM VIII. 
 
 JQF two proportions have a couplet in each the same, the other 
 couplets will form a proportion. 
 
 Suppose A : B : : C: D; and 
 
 A: B::H:F;. then, 
 
 . A C ,A B , .. _ 
 
 B = -p and -^ = - ; hence (A. 1), 
 
 5 = |? ; whence C : D ::E: F. 
 JJ Jf 
 
 Cor. If two proportions have a couplet in each proportional, 
 
 the other couplets will form a proportion. 
 
 THEOREM IX. 
 
 Equimultiples of two quantities are proportional to the quan- 
 tities themselves. 
 
 Let A and B be any two quantities ; then 
 
 A A 
 
 j} = ~j>; multiply both terms of the first by m, 
 
BOOK II. 51 
 
 , mA A . 
 
 we nave, — n^ - ^; whence, 
 
 mB B 
 
 mA:mB::A:B. 
 
 Therefore, etc. 
 
 THEOREM X. 
 
 If four quantities are in proportion, any equimultiples of the 
 first couplet will be proportional to any equimultiples of the 
 second couplet. 
 
 Suppose A: B :: Q:D; then 
 
 — =. =: ; hence, also, 
 B JJ 
 
 , mA nC . 
 
 we have, — = — -^- ; whence 
 
 mB nD 
 
 we have, mA:mB::nC:nD. 
 
 Therefore, etc. 
 
 THEOREM XI. 
 
 The products of the corresponding terms of two or more propor- 
 tions are proportional. 
 
 Suppose A : B : : G: B, and 
 
 M-.N-.P: Q; then 
 we have, iX-D = -BX G, 
 
 MX Q = N X Pj taking their product, 
 we have, 
 
 A X M X & X Q = B X NX X P; whence (Th. II.) 
 we have, A X M : B XN: : Cx P-D X Q. 
 
 THEOREM XII. 
 
 In any continued proportion, any antecedent will be to its 
 consequent as the sum of the antecedents is to the sum of the 
 consequents. 
 
 Let A :B::C:D::E:F, etc. 
 
 Then, since A : B : : C : D, and 
 
52 GEOMETRY. 
 
 A: B:: E: F ; we have 
 Ay.D = BX C, and 
 
 4XJ ? =-BX-E; adding 
 to these, A X B = ^. X -B, we have, 
 
 -4 X -B + A X -D + A X -F = A X -B + B X C+ 5 X E, 
 
 or, ix(i + 2) + J)=21(l+0+J)i 
 
 tvhence, 4 : B: :A+ C+JE: B + D + F. 
 
 PKACTICAL EXEKCISES. 
 
 1. If the first three terms of a proportion are 12, 14, and 18, what ia 
 the fourth term ? Ans. 21. 
 
 2. Given the proportion 3 : 12 : : 5 : 20 ; what proportion have we by 
 composition ? 
 
 8. Find a mean proportional to 12 and 27 ; to m and n. 
 
 Ans. 18 ; ^/m X n. 
 
 4. If the ratio of A to B is $ , what is the ratio of 3 A to 2 B ? 
 
 Ans. |. 
 
 5. If the ratio of 3 A to 2 Z? is f , what is the ratio of ^1 to £ ? 
 
 ^4n«. J. 
 
 6. What proportion is deducible from the equation M X iV= A 2 — B\ 
 
 Ans. M:A + B::A — B:N. 
 
 7. What proportion is deducible from the equation ( C -|- Z)) X -^ = 
 ( L A + B)XC* Ans.A:B::C:D 
 
 THEOREMS FOR ORIGINAL THOUGHT. 
 
 1. If a : I : : c : d, prove that am : bn: : cm: dn. 
 
 2. If a : b : : c : d, prove that — :-::—:-. 
 
 m n m n 
 
 3. If a : b : : c : d, prove that a: a -\- b : : c: c -\- d. 
 
 4. If a : b : : c : d, prove that a -\-b : a — b: : c -\- d: c — d. 
 
 5. If a : b : : c : d and m : c : : n : d, prove that a : b : : m : n. 
 
BOOK III. 
 
 AREAS AND RELATIONS OF POLYGONS. 
 
 1. This book treats of the area of polygons and then 
 relation to each other. 
 
 2.. The Area of a polygon is its quantity of surface : it 
 is expressed by the number of times which the polygon 
 contains some other area assumed as a unit of measure. 
 
 3. The Altitude or a Triangle is ' 
 the perpendicular distance from the vertex 
 of either angle to ,the opposite side, or the 
 opposite side produced. 
 
 The vertex of the angle from which the 
 altitude is drawn is called the vertex of the Triangle; the 
 opposite side is called the base of the triangle. 
 
 4. The Altitude of a Parallelogram is the per- 
 pendicular distance between two opposite 
 
 sides. 
 
 These opposite sides are called bases, one 
 is the upper base, the other the lower base, i 
 
 5. The Altitude of a Trapezoid 
 is the perpendicular distance between its 
 
 i 
 
 parallel sides. / 
 
 These sides are called bases; one is 
 called the upper base, the other the lower base. 
 
 6. Similar Polygons are those which are mutually 
 equiangular, and in which the corresponding sides are pro- 
 portional. 
 
 5* 53 
 
54 GEOMETRY. 
 
 Corresponding sides or angles are those which are like 
 placed. They are sometimes called homologous. 
 
 7. Equivalent Polygons are those which are 
 equal in area. Polygons which, being applied to each other, 
 coincide throughout their whole extent, are said to be equal 
 in all their parts, or simply equal. 
 
 The term equal is often used in geometry for equivalent 
 meaning equal in area. The sign of equality, =, is used in 
 comparing equivalent figures, and is read " equals," or " is 
 equal to." 
 
 8. A Kegclab, Polygon is a polygon which is both 
 equilateral and equiangular. 
 
 Analysis. — The first object of this book is to find the area of poly- 
 gons. It begins with the area of a rectangle, assuming as a unit of 
 measure a square whose side is a measure of the sides of the given 
 rectangle. From the area of the rectangle we pass to tha area of any 
 parallelogram, thence to the area of a triangle, and from this to the 
 area of any plane figure. 
 
 The book also treats of the relations of the squares on the sides of 
 triangles, and the relation of the angles, sides, and area of similar poly- 
 gons, to each other. It is one of the most interesting and practical 
 books of Geometry. 
 
BOOK III. 55 
 
 
 
 
 
 
 
 E 
 
 
 
 
 
 
 
 
 
 
 
 AREA OF POLYGONS. 
 
 THEOREM I. 
 
 The area of a rectangle is equal to the product of its base and 
 altitude. ' 
 
 Let ABGB bo a rectangle ; then will its area be equalto 
 the product of its base and altitude. 
 
 For, let the line AB be a unit of 
 measure of the base and altitude, and 
 suppose it contained any number as 5 
 times in the base and 3 times in the 
 
 A B 
 
 altitude ; then, divide AB into 5 equal 
 parts and AB into 3 equal parts, and through the points of 
 division draw lines parallel, respectively, to the sides AB 
 and AD; then will the rectangle be divided into equal 
 , squares. For, their sides are equal (B. I. Th. XV. 0. 2) ; 
 their angles are right (B. I. Th. III.); hence, the figures 
 are equal squares (B. I. Th. XV. C. 3). 
 
 Now, the whole number of these squares is equal to the 
 number in one row multiplied by the number of rows, 
 which is the same as the number of. linear units in the 
 base multiplied by .the number of linear units in the alti- 
 tude ; and the same is evidently true for any other numbers 
 than 3 and 5. Hence, the area of ABGB equals AB X AB. 
 
 Since this is true when the linear unit of measure is any 
 length, it is true when it becomes exceedingly small, and 
 is, therefore, true when it becomes infinitely small, as it 
 must when the two sides are incommensurable. There 
 
56 GEOMETRY. 
 
 fore, the area of a rectangle is equal to the product of its 
 base and altitude. 
 
 Cor. 1. Bectangles are to each other as the products of their 
 bases and altitudes. For, let AB and AD represent the 
 base and altitude of one rectangle, and EF and EH the 
 base and altitude of another ; then we will have the iden- 
 tical proportion, ABOD : EFGH: :i£x^: EF X EH. 
 
 Cor. 2. Bectangles having equal bases are to each other as 
 their altitudes. For, suppose the bases AB and EF equal ; 
 then, cancelling the equal factor in the second couplet, we 
 have, ABCD : EFGH: : AB : EH. 
 
 Cor. 3. Bectangles having equal altitudes are to each other 
 as their bases. For, suppose the altitudes AB and EH are 
 equal ; then, by cancelling the equal factor in the second 
 couplet of Cor. 1, we have, ABCB : EFGH: : AB : EF. 
 
 THEOREM II. 
 
 The area of a parallelogram is equal to the product of its base 
 and altitude. 
 
 lietABCDbe a parallelogram, AB its base, and EB its 
 altitude ; then will its area be equal to 
 
 For, at the points A and B draw the ' 7 
 two ^perpendiculars AF and BE, and com- / 
 
 plete the rectangle ABEF. Then, the £ - 
 
 angle AD F equals the angle BCE, and 
 FAB equals CBE (B. I. Th. V.); hence, the two triangles 
 are equal (B. I. Th. VII.) ; therefore, ABEB + BCE is equal 
 to ABED -\- ADF, or the parallelogram ABCD is equal to 
 the rectangle ABEF. But the area of the rectangle is equal 
 to AB X -B-®; hence, the area of the parallelogram is equal 
 to AB X SE. Therefore, etc. 
 
BOOK III. 57 
 
 Cor. 1. Parallelograms are to each, other as the products 
 of their bases and altitudes^ 
 
 Cor. 2. Parallelograms having equal altitudes are to each 
 other as their bases ; and parallelograms having equal bases 
 are to each other as their altitudes. 
 
 THEOREM III. 
 
 The area of a triangle is equal to half the product of its base 
 and, altitude. 
 
 Let ABC be a triangle, AB its base, and CD its altitude ; 
 then will its area be equal to half the product of its base 
 and altitude. 
 
 For, draw BE parallel to AC, and 
 CE parallel to AB, completing the 
 parallelogram ABEC ; then will the 
 triangle A B C be one-half the paral- 
 lelogram ABEC (B. I. Tfi. XV. C. 1). 
 
 But the area of the parallelogram is equal to AB X CD; 
 hence, the area of the triangle is equal to £ AB X CD. 
 Therefore, etc. 
 
 Cor. 1. Triangles are to each other as the products of 
 their bases and altitudes. 
 
 Cor. 2. Triangles having equal altitudes are as their 
 bases ; having equal bases, they are as their altitudes. 
 
 THEOREM IV. 
 
 The area of a trapezoid is equal to one-half the sum of th6 
 parallel sides multiplied by the altitude. 
 
 Let ABCD be a trapezoid, A B and DC its parallel sides, 
 and DE its altitude ; then will its area equal \ (AP 4- DC) 
 XDE. 
 
58 
 
 GEOM-ETRY. 
 
 For, draw the diagonal AG, dividing 
 the trapezoid into the two triangles Ai? (7 
 and ADO, the altitude of each being DE. 
 The area of ABO is, \AB X DE, the area 
 of ABO is ^ DC X DE; hence, the area 
 of ABCB, the sum of these triangles, is 
 ^ABxDE plus ^BGX DE, which is £ (AB + DC) X 
 DE. Therefore, etc. 
 
 H 
 
 SQUARES ON LINES. 
 
 THEOREM V. 
 
 The square described on the sum of any two lines is equal to 
 the sum of the squares described on the lines, plus twice the 
 rectangle of the lines. 
 
 Let JL.Bandr.BObe two lines, and AG 
 their sum ; then will 
 
 AC 1 = AB* + BC* + 2 AB X BO. 
 
 ForjOn^ construct the square A CFD 
 and on AB construct the square ABHG; 
 prolong BH to E and GH to I. Now, 
 it is readily seen that HIFE is the 
 square of BO, also that BCIH equals the rectangle on AB 
 and BC, and GHED equals the rectangle on AB and BO; 
 therefore, the square A CFD consists of the square on the 
 two lines plus twice the rectangle of the two lines. 
 
 Cor. 1. The square of the difference of 
 two lines equals the sum of the squares of 
 the lines, minus twice the rectangle of the 
 lines. For, construct a square on A C 
 and on AB, prolong BH to E and HG 
 to N, making GN=BC, and con- 
 
 S B 
 
BOOK III. 
 
 59 
 
 
 
 
 ■T 
 
 
 D 
 
 
 
 A B K 
 
 EC\ 
 
 atruct the square GM ; then the rectangles BF and Hit are 
 each equal to A C X B 0. Now, A CFD + NGDM— B GFE 
 — HEMN==ABHG; or, 
 
 A~C 2 + B~C~* — 1AC X BC= A~B\ 
 Cor. 2. The rectangle contained by the sum and difference oj 
 two lines equals the difference of their squares. For, construct a 
 square on AB and on AC, take BK 
 = BC, and construct the rectangle AL ; 
 then AK=AB + BC, AC=AB — 
 B C, BKLI = D GFE, and AXLE = 
 {AB + BC) {AS — BC). ' Now, 
 AKLE = ABIE + DF, which equals 
 ABHF— DIHG; hence, 
 
 (AB + BC)(AB — BC) = A~B> ■ 
 THEOREM VI. 
 
 The square described on the hypothenuse of a right-angled tri- 
 angle is equal to the sum of the squares described on the other 
 two sides. 
 
 Let ABC be a triangle, right-angled at B; then will 
 4C* = AB* + BC\ 
 
 For, construct squares on each 
 of the sides, draw BD parallel to 
 AF and produce it to E, and draw B £ 
 the diagonals BF and HC. The 
 two triangles HA C and BAF are 
 equal; for, AC equals A F, being 
 sides of the same square, HA 
 equals AB, for the same reason, 
 and the angle HAC equals the 
 
 angle BAF, both being equal to a right angle -plus BA C; 
 hence, the triangle HA C equals BAF. 
 
 The triangle BAF is one-half of the rectangle AFED, 
 Since it has the same base and the same altitude (Th. III.) J 
 
60 GEOMETRY. 
 
 also, since IB is a straight line, the triangle HAG and 
 square ABIH have the same altitude ; hence, the triangle 
 is one-half of the square (Th. III.). But these two trian- 
 ' gles BAF and HAG are equal ; hence, the rectangle AFEB 
 is equal to the square ABIH. In the same manner we may 
 prove that the rectangle EGGD is equal to the square 
 BCLK; hence, the sum of the two rectangles, or the square 
 on AG is equal to the sum of the two squares HB and BL. 
 Therefore, etc. 
 
 Cor. 1. The square of either side about the right angle is equal 
 to the square of the hypothenuse diminished by the square of the 
 other side. 
 
 For, since AB 2 + BG 2 = AC 2 , we have, by transposing, 
 AB 2 = AC 2 — BG 2 . 
 
 Cor. 2. The square of the diagonal of a square is equal to 
 twice the square of the side of the square. 
 
 Let ABCD be a square, then will AC 2 = 
 2 AB 2 . For, we have, by the theorem, A G 2 = 
 AB 2 + BC 2 ; but AB 2 equals TfG 2 ; hence, by 
 substitution, we have AG 2 = AB 2 -f AB 2 , or, 
 AC 2 = 2AB\ 
 
 Cor. 3. The side of a square is to its diagonal as 1 is to the 
 square root of 2. 
 
 For, since 2 AB 2 = AC 2 , or, 2 X AB 2 = AC 2 X 1, we have 
 the proportion (B. II. Th. II.), 
 
 AB 3 : AC 2 :: 1:2; extracting the square root, 
 we have, AB : AC: : 1 : y'2. 
 
 Cor. 4. Two rightrangkd triangles are equal in all their part* 
 ■when they have two corresponding sides respectively equal. 
 
 Note. — This is the celebrated Pythagorean proposition, so called because 
 it was discovered by Pythagoras. It is also known as the 47th of Euclid 
 that being its number in the first book of Euclid's Elements. 
 
BOOK III. 
 
 61 
 
 THEOREM VII. 
 In any obtuse-angled triangle, the square of the side opposite 
 
 the obtuse angle is equal to the sum of the squares of the other 
 
 two sides, plus twice the product of the base into the distance 
 from the vertex of the obtuse angle to the foot of the perpen-, 
 
 dicular drawn from the vertex of the angle opposite the base to 
 
 the bnse produced. 
 
 Let ABC be a triangle, of which A is ail obtuse angle, 
 AB its base, and CD the perpendicular drawn to the base 
 produced ; then will 
 
 BG* = AC* -f AB* + 2 AB X AD. 
 
 For, in the right-angled -triangle DBG, 
 we have, BO 2 = DC* + DB* ; 
 but. DB =AB +AD; 
 
 hence, DB' = AB" + AD* + 2ABX-AD (Th. V.). 
 Hence, BO* = DO* -f AB* + AD* + 2ABxAD.^ 
 But, DC + AD* = AC*. 
 
 Hence, B 0* = AB* + AC* + 2 AB X AD. 
 
 Cor. 1. If the angle CAB becomes a right angle, AB 
 becomes zero, and we have, BC* = AB* -(- AC*- 
 
 THEOREM VIII. 
 In any triangle, the square of a side opposite an acute angle is 
 equal to the sum of the squares of the other two sides, minus 
 twice the product of the base and the distance from the vertex 
 of the acute angle to the foot of the perpendicular let fall 
 upon the base or the base produced. 
 
 Let AB C be any triangle, B an acute .angle, 
 il^its base, and CD the perpendicular; then 
 will 
 
 AC* = AB* + BC* — 2ABX BD. 
 
 For, in the right-angled triangle ADC, ■ 
 
62 
 
 GEOMETRY. 
 
 we have, AC* = DC* + AD*; 
 but AD =AB —DB; 
 
 hence, AD* = AB* + DB* — 2ABX&B (Th. V. C. 1). 
 Hence, AC* = DC* + AB* + DB' — 2 A.B X DB. 
 But, i)© 2 -f DB* = BC*, in 5Z> C. c 
 
 Hence, H? 2 =ZB 2 -f B~C* —2ABX DB. 
 The same may also be shown if the per- 
 pendicular meets the base produced, as in 
 the second figure. Therefore, etc. 
 
 Note.. — This 8th Proposition can be Tery prettily drawn from the 7th, by 
 transposing the terms of the 7th, and reducing. Let the pupil try it. 
 
 THEOREM IX. 
 
 In any triangle, a straight line drawn parallel to the base 
 divides the other sides proportionally. 
 
 Let ABC he a triangle, and DE a line parallel to the base; 
 
 then will 
 
 CD :DA::CE: EB. 
 
 For, draw AE and DB; then, since the 
 two triangles ADE and DEC have their 
 bases in the same line and their vertices 
 at the same point E, they have the same 
 altitude; hence, they are to each other as 
 their bases (Th. III. C. 2), or, 
 
 AED : DEC : : AD : DC. 
 
 For a similar reason, the triangles BED and DEC are to 
 each other as their bases ; hence, we have, 
 BED : DEC : : BE : EC. 
 
 But the triangles AED and BED have the same base DE 
 and the same altitude, since their vertices are in the line 
 (LB parallel to DE; hence, they are equal (Th. HI.), and 
 
BOOK III. 63 
 
 the two proportions have a couplet in each equal ; hence, 
 the remaining terms are proportional (B. II. Th. Till.), 
 and we have, 
 
 AD: DC:: BE: EC. 
 , Therefore, etc. 
 
 Cor. 1. By composition, we have, 
 
 AD + DC : AD : : BE + EC : BE, 
 or, AC : AD :: BC : BE; and, in the same way, 
 
 AC:DC::BC:EC. 
 
 Cor. 2. Conversely, If a line divides two sides of a triangle 
 proportionally, it will be parallel to the third side. 
 
 Let DE divide CA and CB proportionally ; then, if DE is 
 not parallel to AB, draw DE' parallel to AB. Now CA: CD:: 
 ■CB : CE' (Cor. 1), but CA : CD : : CB : CE byhypothesis ; 
 hence, CE' = CE, which is absurd. Therefore, etc. 
 
 Cor. 3. Since DEC : AEC : : DC : AC and AEC : ABC: : 
 EC-.BC, and also, DC : AC :: EC : BC; therefore, 
 DEC : AEC : : AEC : ABC. 
 
 That is, the triangle AEC is a mean proportional between 
 DEC and ABC. 
 
 SIMILAR. 'TRIANGLES. 
 
 THEOEEM X. 
 
 Triangles which are mutually equiangular are similar. 
 
 Let ABC and DEF be two tri- 
 angles having the angle A = D, 
 the angle i?:=.&, and C=F; then 
 will they be similar. 
 
 For, on AC take CG- equal to 
 FD, and on BC take CH equal to 
 
61 
 
 GEOMETRY. 
 
 FE, and draw GH; then the triangle GGH will be equal te 
 FDE (B. I. Th. VI.) and the angle GGH will equal FDA; 
 hence, the angle GGH equals GAB, and GH is parallel to 
 AB (B. I. Th. IV.). Hence, we have (Th. IX. C. 1). 
 
 AG: BO:: GO : HO, or,' 
 
 AO:BO::DF:EF; 
 and the same may be shown for the sides containing the 
 other equal angles ; hence, the triangles are similar (D. 
 6). Therefore, etc. 
 
 THEOREM XL 
 
 Triangles which have their corresponding sides proportional 
 are similar. 
 
 Let ABC and DEF be two triangles having their cor- 
 responding sides proportional; 
 then will they be similar. 
 
 For, if they are not similar, 
 suppose some other triangle, as 
 DEG, to be constructed upon 
 the side DE, similar to ABC. 
 Then, by the preceding the- 
 orem, we have, 
 
 AB:DE::AC:DG; 
 
 but, by hypothesis, 
 
 AB-.DE.-.AC: DF; hence, 
 
 we have, DG=DF. 
 
 In the same way, it may be shown that 
 EG = EF. 
 
 Hence, the triangles DEG and DEF must be equal in all 
 their parts (B. I. Th. IX.), and, therefore, mutually equi 
 angular; hence ABC and DEF are mutually equiangular. 
 and, consequently, similar. Therefore, etc. 
 
BOOK III. _ 65 
 
 THEOREM XII. 
 
 Triangles which have an angle in each equal, and the sides 
 including them proportional, are similar. 
 
 Let ABC and DEFbe two triangles having the angle G 
 equal to the angle F, and 
 
 AG:BG::DF: EF; 
 then will the triangles be similar. 
 
 For, apply the angle DFE to 
 AGB, and the triangle DFE, will 
 take the position GCH, and, from 
 the proportion above, we shall have T b d e 
 
 AO:BG::GG:HG; 
 hence, GR is parallel to AB (Th. IX. C. 2), and the trian- 
 gles OCH and ACB mutually equiangular, and therefore 
 similar. But, GCH is equal to DFE; therefore, A CB and 
 DFE are mutually equiangular, and similar. 
 
 THEOREM XIII. 
 
 Triangles which have their sides parallel, each to each, or per- 
 pendicular, each to each, are similar. 
 
 First. ~LetABC andDEF be two triangles having the 
 side AB parallel to DE, A G parallel 
 to DF, and GB parallel to FE; then 
 will they be similar. 
 
 For, sinceAC is parallel to DF 
 and AB to DE, the angle A is equal 
 to D (B. I. Th. V.) ; for a similar 
 reason G is equal to F and B to E; hence, the triangles arc 
 mutually equiangular, and, consequently, similar. 
 
 Second. Let ABG and DEFbe two triangles having theit 
 sides respectively perpendicular; then will they be similar, 
 
66 
 
 GEOMETRY. 
 
 For, produce the sides of DEF till they meet the sides of 
 ABC. In the trapezium GEIC, the sum of 
 the four angles equals four right angles 
 (B. I. Th. XIX. C. 2), and since two of the 
 angles are right angles, the sum of the 
 angles C and GEI equals two right an- 
 gles. But the sum of GEI and FED 
 equals two right angles (B. I. Th. I.) ; 
 hence, the angle FED equals the angle C. In the same way 
 it may he shown that FDE equals B, and DFE equals A; 
 hence, the two triangles are mutually equiangular, and, 
 consequently, similar. Therefore, etc. 
 
 THEOREM XIV. 
 
 If, in a right-angled triangle, a line be drawn from the vertex 
 of the right angle perpendicular to the hypothenuse ; 
 
 1. The two triangles thus formed will be similar to the given 
 triangle and to each other. 
 
 ' 2. Each side about the right angle will be a mean propor- 
 tional between the hypothenuse and adjacent segment. 
 
 3. The perpendicular will be a mean proportional between 
 th°, two segments of the hypothenuse. 
 
 TietABC be a right-angled triangle, the right angle, 
 and CD the perpendicular ; then, 
 
 First. The triangles A CD and 
 AB C have each a right angle, and 
 the angle A common ; hence, the 
 remaining angles are equal, and 
 the triangles are similar (Th. X.). 
 In the same manner, we show BCD and AB C equiangular 
 and similar; and then ADC and BDC, being both similar 
 to ABC, are similar to each other. 
 
BOOK III. 67 
 
 Second. The two triangles being similar to the given 
 
 one, we have, 
 
 AB: AC:: AC: AD, 
 
 and also, AB:BC::BC: BD. 
 
 Therefore, etc. 
 
 Third. The two triangles being similar, we have, 
 
 ' AD:DC::DC:DB, 
 
 Therefore, etc. 
 
 RELATION OF POLYGONS. 
 
 THEOREM XV. 
 
 Triangles which have an angle in each equal, are to each other 
 as the products of the sides including those equal angles. 
 
 Let ABC and DEF be two triangles having the angle 
 F equal to the angle C; then will 
 
 ABC : DEF ::ACXBC:DFXEF- 
 
 For, place the angle F on its 
 equal C, and the triangle DEF 
 will take the place GCH; then 
 draw-li?. Now, since the tri- 
 angles ARC and QHC have 
 their bases AC and GC in the 
 same line A C, and vertices at 
 H, they have the same altitude, and are to each other as 
 their bases ; hence, 
 
 AHC:GHC::AC:GC. 
 
 Also, since AHC and ABC have their bases SC and BC in 
 the same line, and vertices at the point A, they have the 
 same altitude, and are as their bases ; hence, 
 ABC:AHC::BC:HC; 
 
68 - 
 
 GEOMETRY. 
 
 multiplying the corresponding terms of these two propor. 
 
 tions together, and omitting the common factor AHC, we 
 
 have, 
 
 ABG : GHG ::AGxBG:GGxSO, 
 
 or, ABG:DEF::AGXB0:DFXFH. 
 
 Therefore, etc. 
 
 THEOREM XVI. 
 
 Similar triangles are to each other as the squares of their 
 homologous sides. 
 
 Let ABG and ABE be two similar triangles ; then will 
 they be to each other as the 
 squares of any two homolo- 
 gous sides. Draw the alti- 
 tudes AG and AF ; then, 
 since the triangles are as 
 the product of their bases 
 and altitudes (Th. III. C. 1), 
 
 we have, 
 
 ABO : ADE: : BO X AG:DE X AF. 
 
 But, by similar triangles, we have, 
 
 BO:DE::AB:AD, 
 and, AG:AF::AB:AD; 
 
 hence, BO X AG: DE X AF ::AB*:AD*. 
 Comparing this with the first proportion, we have, 
 
 ABO:ADE::AB":AB\ 
 
 F JB 
 
 THEOREM XVII. 
 Similar polygons may be divided into the same number of tri- 
 angles, similar each to each, and similarly situated. 
 Let AB ODE and FGSIKbe two sim ilar polygons, having 
 the angle A equal to the angle F, B to G, OtoH, etc. ; then 
 
BOOK III. 
 
 69. 
 
 can they be divided into the same number of similar tri- 
 angles similarly situated. 
 
 From tbe homologous 
 angles A and F draw the 
 diagonals AC, AD, and 
 FH,F1. Since the poly- 
 gons are similar, the tri- 
 angles ABC and FGH 
 have the angles B and G equal, and the sides about these an- 
 gles proportional; they are, therefore, similar (Th. XII.). 
 
 Since the triangles ABC and FGH are similar, the angle 
 AGB equals FUG, and the sides AG and FH are propor- 
 tional to BO and GH, and hence to CD and HI. If we take 
 the equal angles A CB and FHG from the equal angles BCD 
 and GHI, we have A CD equal to FHI; hence, the triangles 
 ^ CD and .KHZ have an angle in each equal, and the sideu 
 including these angles proportional ; they are, therefore, 
 similar (Th. XII.). In a similar manner, it may be shown 
 'that ADE and FIK are similar. Therefore, etc. 
 
 THEOREM XVIII. 
 
 The perimeters of similar polygons are to each other as any 
 two homologous sides) and the polygons are to each other as 
 -the squares of those sides. 
 
 lietABCDF and FGHIKbe two similar polygons; then 
 will their perimeters be 
 to each other as any 
 two homologous sides,, 
 and their areas be as 
 the squares of those 
 sides. 
 
 First. Since the polygons are similar, we have* 
 
70 GEOMETRY. 
 
 AB.FG-.-.BG: GH: : CD : HI, etc. ; 
 hence (B. II. Th. XII.), 
 
 AB + BC + CD+ etc. : FG + GH+HI + etc. : :AB:FG; 
 or, the perimeter of the first to the perimeter of the second 
 as any side of the first to the homologous side'of the 
 second. 
 
 Second. Since the triangles are respectively similar, we 
 have, ABG : FGH : :~AC* : FH*; 
 
 and also, A CD : FH1 : : ~AC* : FH*; 
 
 hence, we have, ABO : FGH ::AGD: FHI. 
 In a similar manner, we find, 
 
 AQD : FHI : : ADE : FIK. 
 Hence (B. II. Th. XII.), the sum of the antecedents, ABG-{- 
 ACD -j- ADE, is to the sum of the consequents, FGH-\- 
 FHI -\- FIK, as any antecedent ABG is to its consequent - 
 FGH; and, since ABG is to FGH as AB 2 to FG 3 , we have, 
 ABCDE : FGHIK ::A&: FG 1 . 
 
 Therefore, etc. 
 
 Gor. The perimeters are to each other as any two homo- 
 logous lines, and the polygons are as the squares of those 
 lineB. 
 
 THEOREM XIX. 
 
 Regular polygons of the same number of sides are similar 
 
 figures. 
 
 Let ABGDEF and GHIKLM he two regular polygons 
 of the same number of 
 sides ; then will they be 
 similar. 
 
 j?or, the correspond- 
 ing angles in each are 
 equal (B. I. Th. XIX. C. 
 5), and the corresponding sides are proportional, since thoy 
 
BOOK III. 71 
 
 are equal; hence, the polygons are similar (D. 6). There 
 fore, etc. 
 
 Cor. Since regular polygons of the same number of sides 
 are similar figures, their perimeters are proportional to 
 any homologous lines, and their areas are as the squares 
 of those lines. 
 
 PRACTICAL EXAMPLES. 
 
 1. Required the perimeter and area of a square whose sides are each 
 20 inches. 
 
 2. Required the perimeter and area of a rectangle whose sidea are 
 respectively 18 and 24 inches. 
 
 3. What is the area of a parallelogram whose base is 16 inches and 
 altitude 12 inches ? 
 
 4. A man has a hoard in the form of a triangle ; what is its area if 
 . the hase is 9 feet and the altitude 18 inches ? 
 
 5. A farmer has a field in the form of a trapezoid ; the two parallel 
 sides are 40 and 60, rods, and the distance between them 32 rods ; re- 
 quired its area. 
 
 *6. Required the hypothenuse of a right-angle triangle, the two sides 
 being 3 and 4 inches respectively. 
 
 7. The sides of a triangle are 18 and 21, and the base 24 ; what are 
 the sides of a similar triangle whose base is 8 ? 
 
 8. A man had a lot in the form of a right-angle triangle ; the hypo- 
 thenuse is 78 and one side 30 ; required the other side and the area. 
 
 9. A ladder 65 feet long is placed against a house, so that its foot is 
 25 feet from the house ; how high does it reach ? 
 
 10. A pole was broken 75 feet from the top, and fell so that the end 
 struck 60 feet from the foot ; required the length of the pole. 
 
 11. A has a triangular piece of ground, the base of the triangle being 
 
 * The numbers 3, 4, and 5 are the^emallest integers which can express the 
 relation of the three sides of a right-angle triangle. It is evident that we 
 may have an infinite number of right-angle triangles with their sides in this 
 ratio. Thus, 6, 8, 10 ; '9-12, 15, etc. Another integral relation of sides' is 5, 
 12, 13. 
 
72 GEOMETRY. 
 
 20 rods ; what is the base of a similarly-shaped lot containing 4 times.as 
 much land ' Ans. 40 rods. 
 
 12. A man has a lot 40 rods long and 23 rods wide ; what are the 
 dimensions of a. similar lot 9 times as large ? Ans. 120 ; 69. 
 
 13. A ladder, whose length is 91 feet, stands close against a building, 
 how far must it be drawn out at the bottom that the top may be low- 
 ered 7 feet? Ans. 35 feet. 
 
 14. A ladder 130 feet long, with its foot in the street, will reach on 
 one side to a window 78 feet high, and on the other .to a window 50 feet 
 high ; what is the width of the street ? Ans. 224 feet. 
 
 15. There is a rectangular field whose sides are 25 yards and 16 
 yards respectively ; what is the side of a square field of equal area ? 
 
 Ans. 20 yards. 
 
 16. If it cost $328 to put a fence around a farm 50 rods long and 32 
 rods wide, how much less will it cost to enclose a square farm of equal 
 area with the same kind of fence ? Ans. $8. 
 
 17. The gable ends of a house are each 48 feet wide, and the perpen- 
 dicular height of the ridge above the eaves is 10 feet ; how many feet of 
 boards will it take to board up both gables ? Ans. 480. 
 
 18. A man has a field in the form of a rectangle which contains 40 
 acres ; what are its dimensions if the length is twice the breadth ? 
 
 Ans. Length, 113.136 rods; width, 56.568 rods. 
 
 19. A cemetery containing 60 acres is laid out in such a manner that 
 its length is equal to three times its width ; required the dimensions of 
 the cemetery. Ans. Length, 169.704 rods; width, 56.568 rods. 
 
 20. A general wishing to draw up his corps in the form of a square, 
 found by the first trial he had 100 men over ; he then increased the 
 side of the square by 2 men,, and found he lacked 136 men to complete 
 the square ; how many men had he in the corps 1 Ans. 3464. 
 
 21. A man has a square yard containing ^ of an acre ; he makes a 
 gravel walk around it which occupies £J of the whole yard ; what is the 
 width of the walk ? Ans. 4 feet 1£ inches 
 
 22. In a triangle the two sides are 13 and 15, respectively, and the 
 perpendicular from the vertex of the angle which they form to the op 
 posite side, 12 ; required the third side. Ans. 14. 
 
BOOK III. 73 
 
 EXERCISES FOR ORIGINAL THOUGHT. 
 
 1. Two squares are to each other as the squares of their diagonals. 
 
 2. Two similar parallelograms are to each other as the squares of 
 their diagonals 
 
 3. Prove that the diagonals of a rectangle are equal to each other. 
 
 4. Prove that the greater diagonal of a parallelogram is opposite the 
 greater angle. 
 
 5. Show where a line from the vertex of a triangle must he drawn to 
 divide the triangle into two equal parts. 
 
 6. Prove that the ratio of the side of a square to its diagonal is as 1 
 to the square root of 2. 
 
 7. The straight line joining the middle points of the oblique sides of 
 a trapezoid will he parallel to the other sides, and equal to half their 
 sum. 
 
 8. The four lines joining the middle points of the adjacent sides of a 
 quadrilateral form a parallelogram. 
 
 9. The lines drawn from the vertices of the three angleB of an equi- 
 lateral triangle, perpendicular to the opposite sides of the triangle, will 
 intersect each other in the same point. 
 
 10. The line which bisects the vertical angle of a triangle divides 
 the base into two parts which are proportional to the adjacent sides. 
 
 11. If a lin« be drawn parallel to the. base of a triangle, and lines be 
 drawn from the vertex of the triangle to the base, these lines will divide 
 the base and parallel proportionally. 
 
 12. Triangles which have an angle in each equal, are to each other 
 &» the rectangles of the sides including those angles. 
 
BOOK IV. 
 
 OF THE CIRCLE. 
 
 DEFINITIONS. 
 
 1. ACiRCLEisa plane bounded by a curve line, eveiy 
 point of which is equally distant from a point within, 
 called the centre. 
 
 2. The Circumference is the bound- 
 ing line of a circle. An Arc is any a[ 
 
 part of the circumference; as, BD. 
 
 3. The Badius is a straight line 
 drawn from the centre to any point of the circumferenc e ; 
 thus, CD is a radius. 
 
 4. The Diameter is a straight line passing through 
 the centre and terminating at both extremities in the cir- 
 cumference ; as, AB. 
 
 5. A Chord is a straight line joining the extremities 
 of an arc ; thus, BD is a chord. 
 
 6. A Segment is a portion of the 
 circle included between an arc and its 
 chord ; as, DBE. 
 
 7. A Sector is a portion of the 
 circle included by an arc and the radii 
 drawn to its extremities; as, DCBE. 
 
 8. ATANGENTisa straight line which touches the eh\ 
 cumference in one point; thus, AB is a tangent. The 
 point E is called the point of tangency. 
 
 74 
 
BOOK IV- 
 
 75 
 
 9. A Secant is a straight line which cuts the circum- 
 ference in two points ; thus, CD is a 
 secant. 
 
 10. An Inscribed Angle is 
 an angle whose vertex is in the cir- 
 cumference and whose sides are 
 chords ; as, ABO in the next figure. 
 
 11. An Inscribed Polygon is a 
 polygon whose sides are chords, the 
 vertices of the angles being in the cir- 
 cumference; as, ABCDEF. 
 
 12. A Polygon is circumscribed 
 about a circle when all of its sides are 
 
 tangents to the circumference. The circle is at the same 
 time inscribed in a polygon. 
 
 AXIOMS. 
 
 1. The radii, and also the diameters, of a circle, or of 
 equal circles, are equal. 
 
 2. Every diameter is double the radius, or is equal to 
 the sum of two radii. 
 
 3. A straight line can cut a circumference in only two 
 points. 
 
 Analysis. — This book treats of the nature of the circle, the measure- 
 ment of angles, the finding of the circumference, the measurement of 
 the area of a circle, and the relation of the circumferences, and also 
 »f the areas of circles. The method of treatment in finding the cir- 
 iumference and area, and also their relations, is to regard the circle 
 is a polygon of an infinite number of sides, and derive the principles 
 from those of polygons. By a simplification of the subject, we embrace 
 in one book what is usually given in two. 
 
76 GEOMETEY 
 
 NATURE OF THE CIRCLE. 
 
 THEOEEM I. 
 
 The diameter of a circle is greater than any other chord. 
 
 Let AB be any chord; then will it be less than any 
 diameter. 
 
 For, from the point A draw the dia- 
 meter AD, and draw also the radius 
 CB. Then, in the triangle ACB, the 
 sum of the sides A C and CB is greater 
 than AB (B. I. A. 10. C). But A C + 
 CB equals AD (Ax. 2) ; hence, AD is greater than AB. 
 Therefore, etc; 
 
 THEOEEM II. 
 
 In the same circle or equal circles, equal angles at the centre 
 intercept equal arcs on the circumference. 
 
 In the equal circles ABC and DEF let the angle AOC 
 equal DOF; then will the 
 arc A C be equal to the arc 
 DF. 
 
 For, apply the circle ABC 
 to the circle DEF so that the 
 angle AOC shall coincide 
 with the angle DOF. Then, since OC=OF and OA = 07), 
 the point C will fall on F and the point A will fall on D, 
 and the arc A C will coincide with the arc DF, since every 
 point of each arc is equally distant from the centre of the 
 circle. Therefore, etc. 
 
BOOK IV. 
 
 77 
 
 Cor. Conversely. — In the same circle or equal circles, equal 
 arcs subtend equal angles at the centre. For, if we apply the 
 equal arcs AC and DF, placing the point C on F, they will 
 coincide, and the point A will fall on D; hence, the line OC 
 will coincide with OF and OA with CD, and the angle 
 AOC will be equal to DOF. 
 
 THEOREM III. 
 
 Any radius which is perpendicular to a chord bisects the chord 
 and also the arc subtended by the chord. 
 
 Let AB be the chord, and CD the radius perpendicular 
 to it; then will AD'=DB and AE = EB.^ 
 
 First. Draw the radii -CA and CB ; then 
 the angle A CD equals DCB (B. I. Th. X. 
 C. 1), and the triangles A CD and DCB 
 are equal (B. I. Th. VI.) ; hence, the side 
 AD equals DB. 
 
 Second. Since the triangles A CD and 
 DCB are equal, the angle ACE equals ECB ; hence, the arc 
 AE equals the arc EB (Th. II. C). 
 
 THEOREM IV. 
 
 Through three points not in the same straight line a circumference 
 may be made to pass. 
 
 Let A, B, and C be any three points not in the same 
 straight line; then may a circumference 
 be described through them. 
 
 Draw AB and BC, and at E and D, the 
 middle points of AB and BC, draw perpen- 
 diculars, and unite the points E and D. 
 Now r since OED + ODE is less than two 
 
 right angles, the perpendiculars will meet 
 
 7* 
 
78 GEOMETRY. 
 
 in some point, as (B. I. Th. IV. C. 3). Draw OA, OB, 
 and OC; then OA = OB (B. I. Th. XIV.), and, for the same 
 reason, OB = OC; hence, a circumference described from 
 as a centre will pass through the three points A, B, and C. 
 
 Cor. It may also be readily shown that but one circum- 
 ference can be made to pass through three points. 
 
 THEOREM V. 
 
 If a straight line is perpendicular to a radius at its extremity, 
 it will be tangent to the circle at that point. 
 
 Let the straight line AB be perpendicular to the radius 
 CD at D ; then will it be tangent to the 
 circle at the point D. 
 
 For, take any point of AB, as E, and 
 draw the line CE. Now, CE is greater 
 than CD (B. I. Th. XIV.); conse- 
 quently, the point E will be without the 
 circle, and hence the line AB touches 
 the circumference in only one point : it is therefore tangent to 
 it at the point D (D. 8). Therefore, etc. 
 
 Cor. Conversely. — A tangent to the circle is perpendicular 
 to the radius drawn to the 'point of contact. 
 
 For any line, as CE, is greater than CF, or its equal CD ; 
 hence, CD, being the shortest line from C to the tangent, 
 is perpendicular to the tangent at D (B. I. Th. XIV.). 
 Therefore, etc. 
 
 THEOREM VI. 
 
 Two parallel lines intercept equal arcs on the circumference. 
 
 " There may be three cases : first, when both lines are secants ; 
 
 second, when one is a secant and the other a tangent ; third, 
 
 when both are tangents. 
 
BOOK IV. 79 
 
 First. Let AB and CD be two lines cutting the circle ; 
 , then will the arcs MN and PQ be equal. 
 
 For, draw the radius OR per- 
 pendicular to the chord NQ; it 
 will be perpendicular to MP (B. I. 
 Th. III. C), and will bisect the 
 arcs NRQ and MRP at the point 
 JT(Th. III.) ; hence, NR equals RQ 
 and MR equals RP; and, therefore, 
 MR-NH=PR— QR, or MN equals PQ. Therefore, etc. 
 
 Second. If one of the lines, as C'U, is a tangent. Then 
 the radius OR, drawn to the point of contact, R, is perpen- 
 dicular to the tangent CD 1 (Th. V.), and consequently to its 
 parallel AB. Since OR is perpendicular to the chord MP, 
 it bisects its arc MRP (Th. III.) ; hence, arc MR equals arc 
 PR. 
 
 Third. If both lines, as CD 1 and A'B', are tangents. Draw 
 any secant, as AB, parallel to A'B' ; it will be parallel to 
 C'U (B. I. Th. IV. C. 2). By the second case we have arc 
 MK= arc PK, and arc MR= arc PR; adding, we have 
 MR+ MK= PR+ PK, or arc RMK= arc RPK. 
 
 Cor. 1. In the case of parallel tangents it is evident that 
 each arc is a semi-circumference. 
 
 Cor. 2. The straight line joining the points of contact of 
 two parallel tangents is a diameter. 
 
 Scholium. Regarding a tangent as a secant whose two points 
 of intersection coincide, the demonstration of the first case of 
 the theorem may be regarded as including the other two cases. 
 
80 GEOMETRY. 
 
 MEASUREMENT OF ANGLES. 
 
 THEOEEM VII. 
 
 In the same circle or in equal circles, two angles at the centre 
 have the same ratio as their intercepted ares. 
 
 Let ACB and DC'E be two angles at the centre" of equal 
 circles, and AB and DE their 
 intercepted arcs ; then will 
 ACB: DC'E: : AB : DE. 
 
 First. Suppose some com- 
 mon unit is contained 5 times 
 in the arc AB and 3 times 
 in the arc DE; then 
 
 arc AB : arc DE : : 5 : 3. 
 
 Draw radii to the several points of division of the arcs ; 
 the angles thus formed will he equal, since their arcs are equal 
 (Th. II. C.) ; hence the angle ACB will consist of 5 equal 
 parts and the angle DG'E of 3 such equal parts ; therefore 
 angle A CB : angle DC'E : : 5 : 3. 
 
 Comparing the two proportions, we have 
 
 angle A CB : angle DC'E : : arc AB : arc DE. 
 
 Second. Now this is true whatever the size of the unit of 
 measure ; hence it is true when the unit of measure becomes 
 indefinitely or infinitely small, as it must when the two arcs 
 are incommensurable. Therefore, any two angles at the centre 
 of the same or equal circles are to each other as their intercepted^ 
 ares. 
 
BOOK IV. 81 
 
 THEOREM VIII. 
 
 An angle having its vertex at the centre of a circle is measured 
 by the are intercepted between its sides. 
 
 Let A OB be an angle at the centre of the circle A CB, and 
 AB its intercepted arc ; then will AB be the measure of the 
 angle A OB. 
 
 For, let B OD be the unit of measure 
 of the angle A OB, and the arc BD be 
 the unit of measure of the arc BA ; then, 
 by Theorem VII., we have 
 
 AOB-.DOB: : AB : DB, 
 AOB = AB 
 ° r ' DOB~ DB' 
 
 Now, A OB divided by DOB equals the number of units in 
 the angle A OB, and AB divided by DB equals the number 
 of units in the arc AB; hence the number of units in the 
 angle is equal to the number of units in the arc ; therefore 
 the arc may be used as the measure of the angle. 
 
 Scholium 1. This theorem is usually expressed thus: An 
 angle at the centre is measured by its intercepted are. The 
 statement is, however, rather conventional, " measured by " 
 meaning " having the same numerical measure." Both angle 
 and arc have the same numerical measure; hence the arc 
 may be assumed as the measure of the angle. 
 
 Scholium 2. It would seem more natural to measure an 
 angle by a quantity of the same kind, and for this purpose 
 the right angle would naturally be taken as the unit of 
 measure. It has been found more convenient, however, to- 
 use the arc of a circle as the measure of an angle, and for 
 this purpose the circumference has been divided into degrees, 
 minutes, and seconds, as before explained. 
 
82 
 
 GEOMETRY. 
 
 THEOREM IX. 
 
 An angle having its vertex at the circwnference of a circle is 
 measured, by half the arc intercepted between its sides. 
 
 There may be three cases ; first, when the centre of the 
 circle is on one of the sides of the angle ; second, when it 
 is within the angle ; third, when it is without the angle. 
 
 First. Let ABC be the angle, having 
 its vertex at B, and be the centre of 
 the circle; then will ABC be measured 
 by one-half of A C. 
 
 For, draw the radius AO ; then the 
 exterior angle A OC is equal to the sum 
 of the opposite interior angles ABO and 
 OAB (B. I. Th. XIII. C. 5). But, the triangle A OB being 
 isosceles, the angles A and B are equal ; and, consequently, the 
 angle A OC is double the angle ABC. But AOC, being at 
 the centre, is measured by the arc A C (Th. VIII.) ; hence, 
 the angle ABC is measured by one-half of the arc AC. 
 
 Second. Let ABC be the angle, and the eentre of the 
 circle; then will ABC be measured by 
 one-half of ADC. 
 
 For, draw .the diameter BD ; then, 
 from what we have just shown, the 
 angle ABB is measured by one-half of 
 AD. and the angle DBC by one-half of 
 DC; hence, their sum, or the angle ABC, 
 is measured by one-half of the sum of AD and DC, or one- 
 half of ADC. 
 
 Third. Let ABC be the angle, and the centre being 
 without the angle; then will ABC be measured by one-half 
 of -AG 
 
BOOK IV. 
 
 83 
 
 For, draw the diameter BD ; then, ABD is measured by 
 one-half of AD, and GBD is measured 
 by one-half of CD; hence, ABC, their 
 difference, is measured by one-half 
 of AD minus CD, or one-half of AC. 
 Therefore, etc. 
 
 Cor. 1. All the angles ABC, ADC, 
 inscribed in a semicircle are right 
 angles, being measured by one-half 
 of the semi-circumference AEC (Th. 
 IX.). 
 
 Cor. 2. All the angles ABC, ADC, 
 etc., inscribed in a segment greater 
 than a semicircle are less than right 
 angles, being measured by less than 
 one-half of a semi-circumference. Any 
 angle AEC inscribed in less than a 
 semicircle is greater than a right 
 angle, being measured by more than 
 one-half of a semi-circumference. 
 
 Cor. 3. All the angles inscribed in 
 the same segment are equal, being 
 measured by one-half of the same arc. 
 
 Scholium. A right angle is measured by one-half a semi- 
 circumference, or a quadrant. 
 
 THEOREM X. 
 
 The angle formed by a tangent and a chord is measured by half 
 the are intercepted between its sides. 
 Let AB be a tangent to the circle at C, and CD a chord 
 meeting the tangent at C ; then will the angle A CD be meas^ 
 ured by one-half the arc CED. 
 
84 
 
 GEOMETEY. 
 
 For, draw the diameter CE. The angle ACE is a right 
 angle, and is measured by half the 
 semi-circumference CFE (Th. IX. S.) ; 
 the angle ECD is measured by half the 
 arc ED (Th. IX.); hence, the angle 
 ACD, which equals AGE + ECD, is 
 measured by half the sum of the arcs 
 CFE and ED, or by half the arc CFD. A 
 Therefore, etc. 
 
 THEOKEM XI. 
 
 An angle formed by two chords which intersect is measured by 
 
 half the sum of the intercepted arcs. 
 
 Let AEC be an angle formed by the intersection of the 
 
 chords AB and CD; then will it be measured by half the 
 
 sum of A C and DB. 
 
 For, draw DF parallel to AB; then the arc AF equals the 
 arc DB (Th. VI.), and the angle FDC 
 equals the angle AEC (B. I. Th. III.). 
 Now, the angle FDC is measured by one- 
 half the .arc FC (Th. VII.) ; hence, the 
 angle AEC is measured by one-half of 
 FC, or l(AC+AF), or i(AC+DB). 
 Therefore, etc. 
 
 THEOREM XII. 
 The angle formed by two secants is measured by half the dif- 
 ference of the intercepted arcs. 
 
 Let the angle ABC he formed by the two secants AB 
 and CB; then will it be measured by one-half the differ- 
 ence of the arcs A C and ED. 
 
BOOK IV. 
 
 85 
 
 For, draw DF parallel to AB ; then 
 the arc AF is equal to the arc ED, and 
 the angle FDC equal to ABC. Now, the 
 angle FDC is measured by one-half of 
 the arc FC; heace, ABC is measured by 
 one-half of FC; that is, by 
 
 i(AC-AF) or i(AC-ED). 
 
 THEOREM XIII. 
 
 The angle formed by a secant and a tangent is measured by 
 half the difference of the intercepted arcs. 
 
 Let AB be a secant cutting the circle in E, and AC a 
 tangent at the point D ; then will the 
 angle BA C be measured by one-half of 
 the difference of the arcs DB and DE. 
 
 For, draw EF parallel to A C; then 
 the angle FEB equals CAB, and the 
 arc DE equals arc DF. Now, the angle 
 FEB is measured by one-half of FB; 
 hence CAB is measured by one-half of FB; butFB=DB-DF 
 or DB - DE; therefore. CAB is measured by \(J)B — DE). 
 
 THEOREM XIV. 
 The angle formed by two tangents is measured by half the 
 
 difference of the intercepted arcs. 
 Let AB be a tangent at D, and AC a tangent at E; then 
 will the angle BA C, be measured by- 
 half the difference of the arcs DFE 
 and DE. 
 
 For, draw EF parallel to AB; then 
 the angle FEC equals BAC, and the 
 arc DE equals the arc DF. Now, the 
 angle FEC is measured by half the 
 
86 GEOMETRY. 
 
 arc FE (Th. X.) ; hence BA C is measured by half the arc 
 FE; but arc FE=DFE-DF, or DFE—DE; henceBAG 
 is measured by i(DFE — DE). Therefore, etc. 
 
 THE CIRCUMFERENCE AND AREA. 
 
 THEOREM XV. 
 
 The circumference of a circle may be circumscribed about a 
 regular polygon, and it may also be inscribed within it. 
 
 Let ABGD be a regular polygon ; then can the circum- 
 ference of a circle be circumscribed about it. 
 
 Through the three vertices A, B, and 
 C, describe a circumference ; its centre 
 will be in OK drawn perpendiculai 
 to BG at its middle point K. The tri- A 
 angle BOG being isoceles, the angles OBG 
 and OCB are equal, which, being subtracted 
 from the equal angles ABC and BCD, 
 leave ABO and OCD equal; hence, the triangles OB A and 
 OCB have two sides and an included angle respectively 
 equal, and are equal (B. I. Th. VI.), and OD equals OA; 
 hence, the circumference passing through A also passes through 
 D; and in the same way it may be shown to pass through all 
 the vertices. 
 
 Second. Since the triangles A OB, BOG, etc. are all equal, 
 their altitudes are equal ; hence, a circumference described 
 from as a centre with the radius OK will touch all the 
 chords at their: middle points, and, consequently, be in- 
 scribed within the polygon. Therefore, etc. 
 
BOOK IV. 
 
 87 
 
 THEOREM XVI. 
 
 The circumferences of circles are as their radii, and their areas 
 are as the squares of their radii. 
 
 Let C and be the centres of two circles whose radii 
 are CA and OM; then 
 
 ""'will their circumfer- 
 ences be to each other 
 as their radii, and their 
 areas as the squares of 
 their radii. 
 
 Inscribe in the cir- 
 cles regular polygons of the same number of sides. These 
 polygons being similar figures, their perimeters are to each 
 other as any two homologous, lines CA and OM, and their 
 areas are as the squares of those lines (B. III. Th. XVIII. 
 
 C); and this is true whatever -the number of sides; 
 hence, it is true if the number of sides is infinite, and the , 
 polygon becomes the circle. Hence, we have, 
 
 circ. CA : circ. OM :: CA : OM; and, also, 
 area CA : area OM : : HI* : OM\ 
 
 Cor. 1. Since the radii of circles are to each other as- 
 the diameters, we have the circumferences to each other 
 as the diameters, and the areas as the squares of the dia- 
 meters. 
 
 Cor. 2. From this we see that the circumference of a 
 circle is to its diameter as the circumference of another 
 circle to its diameter ; hence, the ratio of the circumfer- 
 ence to the diameter is a constant quantity. This con- 
 stant ratio mathematicians represent by jt, the .Greek letter 
 
88 GEOMETEY. 
 
 p, called pi. Letting C represent the circumference and L 
 
 Q 
 
 the diameter we have jt = -= . 
 
 Note. — This symbol ir is of great importance in mathematics : the pupil 
 should be very careful to thoroughly understand its signification and use. 
 
 THEOREM XVII. 
 
 The circumference of a circle equals the diameter multi- 
 plied by jr. 
 
 Since the ratio of the circumference to the diameter is 
 represented by jt, we have, 
 
 Q 
 
 — = tt; and, multiplying by D, 
 
 we have, G=t:.D. 
 
 Therefore, etc. 
 
 Cor. Since the diameter is twice the radius, if we sub- 
 stituted M for D, we will have, 
 
 C=5rX 2iJ, or C=2nU. 
 
 Hence, the circumference equals the radius multiplied 
 by 2 jr. 
 
 Remark. — The value of ir cannot be exactly expressed in numbers. 
 The number generally used is 3.1416, which is sufficiently accurate for 
 practical purposes. 
 
 THEOREM XVIII. 
 
 The area of a circle is equal to the circumference multiplied 
 by one-half the radius. 
 
 Let be the centre of a circle whose radius is OA, and 
 circumference AB GD, etc.; then will its area be equal to 
 circ. OAx? OA. 
 
BOOK IV. 
 
 Inscribe in the circle a regular poly- 
 gon ABOD, etc., and draw the radii 
 OA, OB, etc., and the perpendicular OJE. 
 The area of each triangle of the poly- 
 gon is equal to its base multiplied by 
 one-half its altitude, and since the alti- 
 tudes are equal being radii of the inscribed circle, the area 
 of the polygon is equal to the sum of the bases, or its peri- 
 meter multiplied by one-half of OE. iN"ow, this is true 
 whatever the number of sides ; hence, it is true when the 
 number of sides'is infinite and the polygon becomes a circle. 
 In this case the perimeter becomes the circumference, and 
 the line E, the radius. Therefore, the area of a circle is 
 equal to the circumference multiplied by one-half of the 
 radius. 
 
 Gor. The area of a circle is equal to the circumference 
 multiplied by one-fourth of the diameter. 
 
 THEOREM XIX. 
 
 The area of a circle equals the square of the radius multi- 
 plied by ic. 
 
 Let G be the centre of a circle ; denote its radius CA by 
 R, and its area by area GA; then from the previous theorem 
 we have, 
 
 area GA = circ. GAX h^> 
 but, circ. CA = 2«R (Th. XVII. C.) ; 
 
 hence, ' area GA = 2 it B X | -#, 
 
 which, reduced, gives, 
 
 area GA = n B 2 . 
 Therefore, etc. 
 
90 GEOMETEY. 
 
 Cor. In a similar manner, we find that area CA = x \D\ 
 or area CA = \-k U 1 . 
 
 Scholium. The finding the exact length of the circum- 
 ference of a circle is called the rectification of the circle. 
 The finding of the area of a circle is called the quadra- 
 ture of or squaring the circle. Both of these are celebrated 
 problems, and can only be solved approximately, as may 
 be shown by Calculus. 
 
 It was stated in Theorem XVII. that the value of n is 
 about 3.1416. This value is generally determined by find- 
 ing a numerical expression for the area of a circle whose 
 radius is unity, which area may be shown equal to the 
 r?tio of the circumference to the diameter. The solution 
 is given in the following proposition. 
 
 THEOEEM XX. 
 
 Problem. — To find the numerical value of jt, the ratio of the 
 circumference to the diameter. 
 
 The area of a circle equals tzT&; but when B = 1, the 
 arsa of the circle equals it\ hence, we may find the value 
 of 7t by finding the area of a circle whose radius is 1. As 
 a circle is a polygon of an infinite number of sides, by con- 
 structing successive similar inscribed and circumscribed 
 polygons of double the number of sides, two may be found 
 whose areas so nearly approach each other that either of 
 them may be taken for the area of the circle. 
 
 Let C be the centre of the circle, AB the side of an in- 
 scribed, and EF of a circumscribed, polygon. Draw the 
 chord AM, and the tangents AP and BQ; then AM will be 
 the side of an inscribed, and PQ of a circumscribed poly- 
 
BOOK IV. 
 
 91 
 
 gon of double the number of 
 sides. Draw CE, CP, CM and 
 CF. 
 
 Let P represent the area of 
 the given circumscribed poly- 
 gon ; p, the area of the given in- 
 scribed polygon ; P', the area of 
 a circumscribed polygon of dou- 
 ble the number of sides ; and p', the area of an inscribed poly- 
 gon of double the number of sides. Also, represent the tri- 
 angles CEM, CAD, CPQ and CAM, which are respectively 
 like parts of P, p, P and p', by T, t, T and f. 
 
 1. The triangle CAM is a mean proportional between CAD 
 and CEM (B. III. Th. IX. C. 3), hence, 
 
 T: H ::H :t; 
 whence, P:p' ::p':p (B. II. Th. X.) ; 
 
 therefore, p' = i/p X P. (1) 
 
 2. Because of a common altitude, CAM and CAD are to 
 each other as CM to CD; and CEM to CPM as .EM" to PM; 
 hence, t : t : : CM : CD, 
 
 and, T:hT::EM:PM, 
 
 by division, T— iTiiT:: EP-.PM, since EM— PM- EP. 
 The triangles CAD and AEP are similar ; hence, 
 AC : CD : : EP : AP, or, since AC= CM and XP = PJf, 
 Cif: CD:: EP-.PM. 
 Hence, from the first and third proportions, we have, 
 
 
 (:t:: T—IT 
 
 W; 
 
 whence, 
 
 p':p::2P—P 
 
 P. (B. II. Th. X.) 
 
 and 
 
 p'+p:p::2P:F; 
 
 (B. II. Th. VI.) 
 
 whence. 
 
 r 2pXP 
 
 (2) 
 
 p'+p 
 
92 GEOMETRY. 
 
 Now if' p and P are squares, the radius being 1, the area of 
 P is 4; and the side of p is t/2 (B. III. Th. VI. C. 3) ; hence 
 the area of p is 2 ; 
 then, from (1), p' = i/^= 2.8284271, 
 
 and, from (2), F = — ^— = 3.3137085 ; 
 
 which are the areas of the inscribed and circumscribed 
 octagons; and in the same manner we may find the areas 
 of polygons of 16, 32, etc. sides. For 8192 sides, the area 
 of the inscribed polygon is 3.1415923 -f , and of the circum- 
 scribed polygon, 3.1415928 -f-, either of which may be taken 
 for the area of the circle whose radius is 1 ; and, since we 
 have shown this to be the value of w, we have n = 3.14159-t-. 
 Scholium. The value of it is generally taken to be 3.1416. 
 
 Note. — We invite special "attention to the method of treating the circum- 
 ference and area of the circle, and also to the simple and concise method of ■ 
 presenting the derivation of the value of ir, as given in the last proposition. 
 
 PRACTICAL EXERCISES. 
 
 1. The radius of a circle is 6 inches ; what is its circumference ? 
 
 2. The diameter of a circle is 8 inches ; what is its area ? 
 
 3. The circumference of a circle is 50.2656 feet; required the radius. 
 
 Ans. 8 feet. 
 
 4. The area of a circle is 490.875 square inches; required the dia- 
 meter and circumference. 
 
 Ans. Diameter, 25 ; circumference, 78.54: 
 
 5. The distance around a circular park is 180 rods ; required the area 
 of the park. Ans. 16 A. 18.23 P. 
 
 6. What is the length of an arc of 75° on the circumference of a 
 .circle whose radius is 5 feet ? Ans. 6.545 feet. 
 
 7. How many degrees in an arc 18 inches Ions', on a circumference 
 Whose radius is 5 feet ? Ans. 17° IV 19". 
 
BOOK IV. 93 
 
 ' 8. A circle 20 feet in diameter is circumscribed by another- circle 30 
 feet in diameter; what is the area of the space included between them? 
 
 9. A has a circular garden whose diameter is 18 rods, and B has one 
 whose area is 2 J times as great ; what is the diameter of B's garden ? 
 
 Ans. 30 rods. 
 
 10. Find the side of a square inscribed in a circle whose diameter is 
 5 feet. Ans. 3.535 feet. 
 
 11. Within a circular park 160 rods in circumference is a circular 
 lake-80 rods in circumference; required the width of the ring of land 
 surrounding the lake. Ans. 12.732 rods. 
 
 12. Deborah has a circular garden and John a square one, and the 
 distance around each is 120 rods ; which contains the most land, and 
 how much? Ans. 245.95 square rods. 
 
 13. A man has a square garden and his wife a circular one, and each 
 garden contains one acre; how much further around is one than the 
 other? Ans. 5.756 rods. 
 
 14. The area of a circle is 344.16 ; if this circle be circumscribed by 
 a square, required the area of the part between the circumference and 
 the perimeter of the square. Ans. 85.84. 
 
 15. The area of a circle is 4 acres; required the side of the in- 
 scribed square, and the area of the part of the circle between the cir- 
 cumference and nerimeter of the square. Ans. 1 A. 1 R. 32 P. 
 
 THEOREMS FOE ORIGINAL THOUGHT. 
 
 1. If two circumferences intersect, the distance between their cen- 
 tres will be less than the sum of their radii and greater than the differ- 
 ence. 
 
 2. If two circumferences intersect, the points of intersection will lie 
 in a perpendicular to the line joining their centres, and at equal dis- 
 tances from it. 
 
 3. In equal circles the greater arc has the greater chord, and, con- 
 Tersely, the greater chord subtends the greater arc. 
 
 4. In equal circles, equal chords are equally distant from the centre, 
 and the greater chord is nearer the centre. 
 
 5. If we inscribe a square in a circle, the radius is to the side of the 
 inscribed square as 1 is to y/Z. 
 
94 GEOMETEY. 
 
 6. If a regular hexagon be inscribed in a circle, each side will be 
 equal to the radius of the circle. 
 
 7. The area of a triangle is equal to the perimeter multiplied by one- 
 half the radius of the inscribed circle. 
 
 8. In any inscribed quadrilateral, the sum of the opposite angles is 
 equal to two right angles. 
 
 9. When a quadrilateral circumscribes a circle, the sums of its oppo- 
 site sides are equal. 
 
 10. When the radius of a circle is unity, its area and semi-circum- 
 ference are numerically equal. 
 
PRACTICAL PROBLEMS IN GEOMETRY 
 CAL CONSTRUCTION, 
 
 INVOLVING THE PKINCIPLES OF BOOKS I., II., III., AND IV. 
 
 The following problems are solved by the principles of 
 tbe previous books. The solution of a few is given in 
 full ; in others, the construction is given, and the reason 
 for the solution indicated by referring to the theorem or 
 theorems upon which it depends. The pupil will give the 
 explanation in full. 
 
 The object of these is to teach the pupil to draw accu- 
 rately upon paper. They are of great use in drawing the 
 notes of a survey, or in representing any geometrical figure 
 upon paper. The pupils need two instruments, a rule and 
 compasses; with these all the following problems may be 
 readily solved 
 
 PROBLEM I. 
 To bisect a given straight line. 
 Let AB be the given straight line. From A and B, as 
 centres, with a radius greater than 
 one-half of AB, describe arcs inter- x/ 
 
 secting at E and F; draw the line 
 EF; then will G be the middle point 
 of AB. For, E and F are each equally 
 distant from A and B; hence, EG bi- 
 sects AB (B. I. Th. XIV. C. 3). 
 
 t 
 
 95 
 
96 
 
 GEOMETEY. 
 
 PROBLEM II. 
 
 From a given point without a straight line to draw a perpen- 
 dicular to the line. 
 
 Let AB be the given line, and G the given point. From 
 G as the centre, with a radius suffi- 
 ciently great, describe an arc cut- 
 ting the line AB in the two points 
 A and B; then from A and B as cen- 
 tres, with a radius greater than one- 
 half of AB, describe two arcs cut- 
 ting each other in D, and draw GD; 
 it will be the perpendicular required (B. I. Th. XIV. C. 3). 
 
 'V 
 
 V 
 
 A 
 
 D 
 
 -/* 
 
 PROBLEM III. 
 
 At a given point in a straight line to erect a perpendifiular-to 
 that like. 
 
 
 Let AB be the given line, and 6 the given point. Then, 
 in the line AB take- the points A and B 
 equally distant from G, and with A and 
 B as centres, and a radius greater than 
 one-half of AB, describe two arcs cut- 
 ting each other at D ; draw DC; it will 
 be the perpendicular required (B. I. Th. 
 XIV. C. 3). 
 
 PROBLEM IV. 
 
 At a point on a given straight line to make an angle equal to a 
 given angle. 
 
 _ Let A be the given point, AB 
 the given line, and EFG the 
 given angle. 
 
 From the point F as a centre, 
 
PEACTICAL PEOBLEMS. 97 
 
 with any radius FG, describe the arc EG. From A && & 
 centre, with the same radius, describe the arc CB; then, 
 with a radius equal to the chord EG, describe an arc from 
 B as a centre, cutting the arc CB in D, and draw AD ; 
 then will the angle DAB equal EFG (B. I. Th. IX.). 
 
 PROBLEM V. 
 To bisect a given arc, or a given angle. 
 
 First. Let ADB be the given arc, and C its centre. Draw 
 the chord AB, and from C draw CD per- 
 pendicular to AB (P. II.); then will CD 
 bisect AB(B. IY. Th. III.). 
 
 Second. Let ACB be the given angle. 
 Then, with C as a centre and any radius 
 CA, describe the arc AB, and bisect this 
 arc by the line CD, as in the previous case ; then will CD 
 bisect ACB (B. IV. Th. III.). 
 
 PROBLEM VI. 
 
 Through a given point to draw a straight line parallel to a 
 
 given straight line. 
 
 Let A be the given point and CD the given line. From 
 iasa centre, with a radius greater than the shortest dis- 
 tance from. A to CD, describe an 
 
 indefinite arc DE; from D as a (Jfi —^~ 
 
 centre, with the same radius, de- \ __, " * \ 
 
 scribe the arc AF; take DE equal -jgf 1 - — f B 
 
 to AF, and draw AB; AB will be 
 the parallel required. 
 
 For, drawing AD, we have ADF = DAE (Prob. IV.) ; 
 hence, AE and CD are parallel (B. I. Th. IV.). 
 
98 
 
 GEOMETRY. 
 
 PROBLEM VII. 
 Two angles of a triangle being given, to find the third. 
 Let M and N be the given angles. 
 Draw the indefinite line AB; at any 
 point, as C, construct the angle A CD equal 
 to M, apd the angle DGE equal to N; 
 then will EGB equaLthe third angle. 
 
 PROBLEM VIII. 
 
 Given two sides and the included angle of a triangle, to con- 
 struct the triangle. 
 
 Draw the indefinite line AD; take AB equal to one of 
 the given sides; at A construct the ^ 
 
 angle A equal to the given angle, and 
 take A equal to the other given side; 
 draw BG; then will ABC be the re- 
 quired triangle (B. I. Th. VI.). 
 
 PROBLEM IX. 
 
 Given one side and two angles of a triangle, to construct the 
 triangle. 
 
 If the angles are not adjacent, find the third angle by 
 P. VII. ; we then have two angles and 
 the included side, and proceed thus : — 
 
 Draw the indefinite line AD; take 
 AB equal to the given side ; at A make 
 the angle BA G equal to one of the an- 
 gles; at B make the angle AB G equal 
 the other angle; then produce A C and B C till they meet, 
 and ABO will be the required triangle (B. I. Th. VII.). 
 
PRACTICAL PROBLEMS. 99 
 
 PROBLEM X. 
 
 Given two adjacent sides of a parallelogram and the included 
 angle, to construct the parallelogram. 
 
 Draw the indefinite line AE, and upon it take AB equal 
 to one of the sides. At A construct 
 the angle BAB equal to the given 
 angle, and take AD equal to the other 
 given side. Draw DO parallel to 
 AB, and B G parallel to AD; then will 
 ABCDbe the parallelogram required (B. I. Th. XV. C. 3). 
 
 PROBLEM XL 
 To find the centre of a given circumference or arc. 
 Take any three points, A, B, and G, on- 
 the circumference or arc, and unite them 
 by the lines AB and BG. Bisect these 
 chords by the perpendiculars DO and 
 BIO j then will their intersection be 
 the centre of the circle (B. IV. Th. IV.). 
 
 PROBLEM XII. 
 
 Through a given point to draw a tangent to a given circle. 
 
 First. Suppose the given point P to be in the circum- 
 ference. 
 
 Find G, the centre of the circle (P. 
 XI.); draw the radius GP; and then 
 through P draw the perpendicular 
 DE; DE will be the tangent re- 
 quired (B. IV. Th. V). 
 
 Second. Suppose the given point P 
 
100 
 
 GEOMETRY. 
 
 to be without the circle. Join P and 
 the centre of the circle ; bisect PG 
 in D ; with Das a centre, and a ra- 
 dius DO, describe the circumference 
 intersecting the given circumference 
 in A and B; draw PA or PB; then 
 each of those will be the tangent required. 
 
 For, since GAP is a semicircle, the angle GAP is a right 
 angle (B. IV. Th. IX. C. 1); hence, AP is a tangent (B. 
 IV. Th. V). 
 
 PROBLEM XIII. 
 To divide a given line into any number of equal parts. 
 
 Let AB be the given line, and suppose we wish to divide 
 it into any number, say 5 equal parts. 
 
 Through A draw the indefinite 
 line AE, making any angle with 
 AB. Take AC of any convenient 
 length, and apply it 5 times to AE; 
 join B with the last point of the 
 division; and through the other 
 
 points of division draw lines parallel to EB; then will AB 
 be divided into 5 equal parts. 
 
 For, since DG and BE are parallel, we have (B. III. 
 
 Th. IX.), 
 
 AG : AE : : AD : AB. 
 
 But AC is one-fifth part of AE; hence, AD is one-fifth part 
 of AB. 
 
 PROBLEM XIV. 
 To divide a given line into parts proportional to given lines. 
 Let AB be the given line, to be divided into parts pro- 
 
PRACTICAL PROBLEMS 
 
 portional to the given lines P, Q, and E. 
 A G, making any angle with AB. On 
 AG lay off AG equal P, CE equal Q, 
 EG equal JR; draw BG, and from the 
 points G and E draw CD and EF 
 .parallel to GB; then will AD; DF, 
 and FB be proportional to AG, GE, and EG (B. III. 
 Th. IX.). 
 
 PROBLEM XV. 
 To construct a mean proportional to two given lines. 
 Let Pand Q be the two given lines. Draw an indefinite 
 line, and on it lay off -ID equal to P, and 
 DB equal to Q ; on AB as a diameter 
 describe a semicircle, and draw D per- 
 pendicular to AB; then, in the triangle 
 AOB, will DO be a mean proportional to 
 AD and D.B (B. III. Th. XIV.). 
 
 PROBLEM XVI. 
 To construct a square equal to a given triangle. 
 Let ABC be the given triangle, AB its base, and OD its 
 altitude. 
 
 Find a mean proportional 
 between CD and one-half of 
 AB (Prob. XV.). Let FG 
 be that mean proportional, 
 and on it, as a side, construct 
 
 the square FGHI; this will be the square required. For, 
 by the construction, we have FG 2 = ^ AB X' GD, which 
 equals the area of ABG. 
 
 9* 
 
102 GEOMETKY. 
 
 PROBLEM XVII. 
 To inscribe a regular hexagon in a circle. 
 Suppose the problem to be solved, and that ABCDEF 
 is a regular hexagon; draw the radii 
 " OB and OC. Now, the arc BC is one- 
 sixth of a circumference, or 60° ; hence,' 
 the angle BOG is 60°, and the other 
 angles OBCand BOO equal 180° minus 
 60°, or 120°, and, OB being equal to OC, 
 the angles OBC and BCO are equal; 
 hence, each is equal to one-half of 120°, or 60°. Con- 
 sequently, the triangle OBC is equiangular, and therefore 
 equilateral; hence, the side BC is equal to the radius OB. 
 Therefore, to inscribe a regular hexagon in a circle, w» 
 apply the radius six times as a chord to the circumference. 
 
 PROBLEMS FOE ORIGINAL THOUGHT. 
 1. Given the three sides of a triangle, to construct the triangle. 
 <2. v Given two sides of a triangle, and the angle opposite one of them, 
 to construct the triangle. 
 
 3. To inscribe a circle in a given triangle. 
 
 4. To inscribe a circle in a square, and a square in a circle. 
 
 5. To find the side of a square which shall be equal to the sum of 
 two given squares. 
 
 6. To find the side of a square which shall be equal to the difference 
 between two given squares. 
 
 7. To construct a rectangle equal in area to a given triangle. 
 
 8. To find a fourth, proportional to three given lines. 
 
 9. On a given line to construct a rectangle which shall be equal to 
 a given rectangle. 
 
 10. To construct a square that shall be equal in area to a given paral- 
 lelogram. 
 
BOOK V. 
 
 PLANES AND THEIR ANGLES. 
 DEFINITIONS. 
 
 1. APlane is a surface such that a straight line con. 
 necting any two of its points will lie entirely in the surface. 
 
 2. A straight line is perpendicular to a plane when it 
 is perpendicular to every line of the plane passing through 
 its foot. The foot is the point where the line meets the 
 plane. 
 
 ileciprocally, the plane is also perpendicular to the line. 
 
 3. A straight line is parallel to a plane when it can- 
 not meet the plane, however far both be produced. 
 
 Eeciprocally, the plane is also parallel to the line. 
 
 4. Two planes are parallel when they cannot meet, 
 however far both be produced. 
 
 5. "When two planes meet, they form a line, which is 
 called their line of intersection. 
 
 6. ADiEDRALANGLEis the divergence of two planes. 
 The line in which the planes intersect is 
 called the edge of the angle; the planes 
 are called the faces of the angle. 
 
 A diedral angle is measured by the an- 
 gle formed by two lines, one in each plane 
 and perpendicular to the edge at the same 
 point. Thus, the diedral angle in the margin is mea- 
 sured by the angle ACB. 103 
 
104 GEOMETEY. 
 
 7.. A Polyedral Angle is the divergence of three 
 or more planes proceeding from a common point. 
 
 The common point is called the vertex of the angle; the 
 planes are its faces; the intersection of the planes, its edges., 
 
 8. A Triedral Angle is a polyedral angle of three 
 faces. 
 
 9. Two planes are Perpendicular to each other when 
 their diedral angle is a right angle. 
 
 Anaiysis. — This Book treats of planes, the lines and angles formed 
 by their intersection. It is not so valuable in itself as the other Books 
 of Geometry, and much less interesting. Its object is to prepare for the 
 Book which immediately follows it. 
 
 THEOREM I. 
 
 Through three points not in the same straight line, one plane 
 can be passed, and but one. 
 
 Let A, B, and C be the three points ; then can one plane 
 be passed through them. 
 
 For, join two of the points, as A and 
 C, by the line AC. Pass a plane through 
 AC, and turn it around AC until it con- 
 tains the point B; it will then pass 
 through the three points A, C, and B. 
 If now the plane be turned about A C, 
 it will no longer contain the point Bj hence, only this one 
 plane can be passed through the three points. Therefore, 
 etc. 
 
 Cor. 1. Since only one plane can be passed through 
 three points, three points are said to determine the posi- 
 tion of a plane. 
 
 Cor. 2. Two lines which are parallel or which intersect de- 
 termine the position of a plane. 
 
BOOK V. 10 5 
 
 THEOREM II. 
 
 If two planes cut one another, their common section is a 
 straight line. 
 
 Let the, two planes AB and CD cut one another in the 
 points E and F ; then will their common 
 section be a straight line. 
 
 For, draw the line EF. uniting the two 
 common points E and F of the planes. 
 Now, this line, having two points in the 
 plane AB, will lie wholly in the plane 
 AB (B. I. Def.), and, having two points 
 in the plane CD, it will lie wholly in the plane CD; hence, 
 the line EF is common to both planes, and must therefore be 
 in their common intersection. Therefore, etc. 
 
 THEOREM III. 
 If from a point without a plane lines be drawn to the plane, 
 
 1. The perpendicular is the shortest distance from the point 
 to the plane ; 
 
 2. Oblique lines which meet the plane at equal distances from 
 the foot of the perpendicular are equal ; 
 
 3. Of two oblique lines which meet the plane at unequal dis- 
 tances from the foot of the perpendicular, the one which meets 
 it at the greater distance is the longer. 
 
 Let A be a point without the plane MN; let AB be a per- 
 pendicular to the plane, and let A C, AD, and AE be oblique 
 lines. 
 
 First. AB will be shorter than any oblique line A C. For, 
 through B draw the line BC; then in the triangle ABC, AB 
 is less than A C (B. I. Th. XIV.). 
 
 Seeond. Let A C and AD meet the plane at equal distances 
 
106 
 
 GEOMETRY. 
 
 from the point B; then A C will be equal to AD. For, draw 
 BC and BB; then the right-angled triangles ABC and ABD 
 will have BC equal to BD, and the 
 side AB common ; hence, the tri- 
 angles are equal, and AG equals 
 AD. 
 
 Third. Let AC and AE meet 
 the plane so that the distance BE 
 is greater than BC; then AE will 
 be greater than AC. For, take BF equal to BC and draw 
 AF; then AE>AF (B. I. Th. XIV) ; but AF=AC; hence, 
 AE>AC. 
 
 Cor. 1. Equal oblique lines drawn from a point to a plane 
 meet the plane at equal distances from the foot of the per- 
 pendicular; and of two unequal oblique lines, the greater 
 meets the plane at the greater distance from the foot of 
 the perpendicular. 
 
 Cor. 2. The equal oblique lines meet the plane in the cir- 
 cumference of a circle whose centre is B ; hence, to draw a 
 perpendicular from a point A to a given plane MN, find any 
 three points, C, D, and F, of the plane equally distant from 
 A, then find the centre of the circumference passing through 
 these points ; then AP will be the perpendicular required. 
 
 THEOEEM IV. 
 
 If a straight line is perpendicular to two straight lines of a plane 
 at the point of their intersection, it is perpendicular to the 
 plane of those lines. 
 
 Let AP be perpendicular to PB and PC at the point P; 
 
 theti will it be perpendicular to MN, the plane of those lines. 
 
 For, let PD be any other straight line of the plane MN 
 
BOOK V. 
 
 107 
 
 drawn through P. Draw BC, cutting PB, PD, and PC in 
 B, D, and C; produce AP making PA' = AP; and draw AB, 
 AD, AG, A'B, A'D and A'G. Then, 
 since BP and CP are perpendicular 
 to A A' at its middle point, AB equals 
 A'B, and A C equals A'C, and the tri- 
 angles ABC and A'BC are equal 
 (B. I. Th. IX.), and also AD equals 
 A'D ; whence PD is perpendicular to 
 AA' (B. I. Th. XIV. C. 3). Hence, 
 AP is perpendicular to any line passing 
 
 through its foot ; it is, therefore, perpendicular to the plane MN. 
 Cor. Only one perpendicular can be erected to a plane from 
 a point of the plane. 
 
 THEOEEM V. 
 
 If from the foot of a perpendicular to a plane a line is drawn 
 at right angles to any line of the plane, and the point of 
 intersection is joined with any point of the 'perpendicular, the 
 last line will be perpendicular to the line of the plane. 
 
 Let AP be a perpendicular to the plane MN, P its foot, 
 BC the given line, and Aja.nj point of 
 AP; draw PD perpendicular to BC, 
 and join the points A and D; then will 
 AD be perpendicular to BC. 
 
 For, lay off BD equal to DC, and 
 draw PB, PC, AB, and A C. Since PD M 
 is perpendicular to BC, and DB equals DC, PB equals PC 
 (B. I. Th. XIV.) ; hence, in the triangles APB and APC, 
 AB equals A C. Therefore the line AD, having two points, 
 A and D, equally distant from B and C, is perpendicular to 
 BC (B. I. Th. XIV. C. 3). 
 
108 GEOMETRY. 
 
 Cor. The line BC is perpendicular to the plane of the tri- 
 angle APD, because it is perpendicular to AD and PD at the 
 point D (Th. IV.). 
 
 THEOREM VI. 
 
 Ij one of two parallels is perpendicular to a plane, the other 
 is also perpendicular to the plane. 
 
 Let AB and CD be two parallel lines, and let AB be per- 
 pendicular to the plane MN; then will 
 CD also be perpendicular to MN. 
 
 For, pass a plane through the parallels 
 cutting MN in BD ; draw AD, and in 
 the plane MN draw EF perpendicular 
 to BD at the point D. Then, EF is 
 perpendicular to the plane A BD C (Th. 
 V. C.) ; henee, the angle EDC is a right angle ; but CDB is 
 a right angle, since CD is parallel to AB (B. I. Th. III. C.) ; 
 hence, CD is perpendicular to the two lines BD and EF at 
 their point of intersection ; it is, therefore, perpendicular to 
 the plane MN (Th. IV.). Therefore, etc. 
 
 Cor. 1. Conversely. ■-- Two lines which are perpendicular to 
 the same plane are parallel. For, suppose the two lines AB 
 and CD to be perpendicular to the plane MN; then, if they 
 are not parallel, draw from the point D a line which is parallel 
 to BA ; this line will be perpendicular to MN (Th. VI.) ; we 
 shall then have two perpendiculars to the plane MN from the 
 same point, which is impossible (Th. IV. C.) ; therefore, AB 
 and CD are parallel. 
 
 Cor. 2. Two lines parallel to a third line are parallel to each 
 Hher. Let the two lines A and B be parallel to a third line 
 
 ; pass a plane perpendicular to C,~it will be perpendicular 
 
BOOK V. 
 
 109 
 
 to both A and B (Th. VI.) ; hence, A and B, being perpen- 
 dicular to the same plane, are parallel (Th. VI. C. 1). 
 
 THEOKEM VII.- 
 
 If two planes are perpendicular to the same straight line, they 
 
 are parallel. 
 
 Let the two planes MN and PQ be- perpendicular to the 
 straight line ABj then will they be parallel. 
 
 For, if they are not parallel, they 
 will meet in some point 0. Prom 
 draw the lines OA and OB; then, 
 since OA lies in the plane MN, it 
 will be perpendicular to AB at A 
 (D. 2) ; and since OB lies in the plane 
 PQ, it will be perpendicular to AB 
 at B. Hence, we have two perpendiculars drawn from the 
 same point to the same straight line, which -is impossible (B. I. 
 Th. XIV. C. 1); consequently, the planes cannot meet, and 
 are, therefore, parallel. 
 
 THEOREM VIII. 
 
 If a plane meet two parallel planes, the lines of intersection 
 are parallel. 
 
 Let the plane AG intersect the two parallel planes MN 
 and PQ; then will AB and CD be parallel. 
 
 For, if the lines AB and CD are not 
 parallel, since they lie in the same plane, 
 they will meet if sufficiently produced, 
 and, consequently, the planes MN and PQ 
 will meet; but the planes cannot meet, 
 since they are parallel ; hence, the lines 
 AB and CD cannot meet ; they are, there- 
 fore, parallel. 
 
110 
 
 GEOMETRY. 
 
 ,/■ 
 
 t-r 
 
 
 D 
 
 I' 
 
 Kf 
 
 Gor. Parallel lines included between parallel planes are 
 equal. For, the opposite sides of the figure AG being 
 parallel, it is a parallelogram, and hence AD equals BG. 
 
 THEOREM IX. 
 
 If a straight line is perpendicular to one of two parallel planes, 
 it is perpendicular to the other also. 
 
 Let MN and PQ be two parallel planes, and let the line 
 AB be perpendicular to PQ; then will it 
 also be perpendicular to the plane MN. 
 
 For, pass any plane through AB; the 
 intersections A C and BD will be paral- 
 lel (Th. VIII.) ; since AB is perpendicular 
 to PQ, it will be perpendicular to BD 
 (D. 2), and since BD and A C are parallel, & a 
 
 it will be perpendicular to A G (B. I. Th. III. C.) ; hence, 
 BA, being perpendicular to any line of the plane MN 
 passing through its foot, is perpendicular to the plane MN. 
 Therefore, etc. 
 
 THEOREM X. 
 
 If two angles not in the same plane have their sides parallel 
 and lying in the same direction, the angles will be, equal a,nd 
 ' their planes parallel. 
 
 Let BAG and DEF be two angles not in the same plane, 
 having their sides respectively parallel e 
 
 and lying in the same direction ; then 
 will these angles be equal and their 
 planes parallel. 
 
 Take ED equal to AB, and EF equal 
 to AG, and draw BG, DF, AE, BD, and 
 GF. 
 
 First. The angles BA G and DEF will 
 be equal. 
 
 /< 
 
 D 
 
 1\ 
 
 ■1 
 
 p 
 
 
 N 
 
 1- 
 
 B 
 
 .1 
 
BOOK V. HI 
 
 For, sincere and EF are equal and parallel, the figure 
 AGFE is a parallelogram (B. I. Th. XVII.), and AE and GF 
 are equal and parallel. Since AB and ED are equal and 
 parallel, ABBE is a parallelogram, and AE and BD are 
 equal and parallel ; hence, BD and OF are equal and paral- 
 lel (Th. VI. C. 2), and, consequently, D-Fis equal and parallel 
 to B C. Hence, the triangles AB C and EDF have their cor- 
 responding sides equal; they are, therefore, equal, and the 
 angle DEF equals the angle BA G. 
 
 Second. The planes are parallel. 
 
 For, three lines which intersect determine the position - 
 of a plane ; and since the three sides of the triangles aro 
 respectively parallel, their planes must be parallel. 
 
 Cor. If three straight lines not in the same plane are equal 
 and parallel, the triangles formed by joining the extremities of 
 these lines will be equal, and their planes parallel. This is 
 readily proved ; let the pupil show it. . 
 
 THEOREM XI. 
 
 If two straight lines are cut by three parallel planes, they will 
 be divided proportionally. 
 
 Let the lines AB and GD be cut by the parallel planes 
 
 MN, PQ, and BS, in the points A, E, ' s 
 
 B, and G, G, D; then will [~^°~] 
 
 AE : EB : : GG : GD. I f\ 1 / 
 
 s l\ T Q 
 
 For, draw the line AD, meeting the /~j V I 7 
 
 plane PQ in F; draw also A G, EF, FG, I s j~iTT I " 
 
 and BD. Now, since the planes MN p — r \ i 
 
 and PQ are parallel, EF is parallel to f~j Vj 7 ( 
 
 BD (Th. VIII.) ; and since PQ and RS I L— _ ___J / 
 
 are parallel, AG is parallel to FG. L / 
 
 Hence (B. III. Th. IX.), we have, 
 
112 
 
 GEOMETEY. 
 
 AF:EB::AF: FD; and also, 
 AF-.FD:: GG: GD. 
 
 Hence, from the principles of proportion, we have, 
 
 AE:EB::CG: GD. 
 Therefore, etc. 
 
 THEOREM XII. 
 
 Hither angle of the three plane angles which form a triedral 
 angle, is less than the sum of the other two. 
 
 Let the triedral angle whose vertex is S be formed by the 
 three plane angles ASG, ASB, and GSB; then will any one 
 of these be less than the sum of the other two. 
 
 If the angle considered is less than 
 either of the other two, it is evidently 
 less than their sum. Suppose, how- 
 ever, the angle greater than either of 
 the other two, and let ASB be that an- 
 gle. In the plane ASB make the angle 
 BSD equal to BSG, draw the line AB 
 at pleasure, make SO equal to SD, and draw AG and BG. 
 
 In the two triangles BSG and BSD, BS is common, CS 
 
 equals DS, and the angle BSG equals BSD by construction ; 
 
 hence, the triangles are equal, and BD equals BG. Now 
 
 (B. I. A. 10, C), 
 
 AD + DB<AG + BG. 
 
 And, taking away the equals DB and BG, we have, 
 
 AD<AG. 
 
 Hence (B. I. Th. VIII. C), we have, 
 
 angle ASD < angle ASG; 
 
 and, adding the equal angles DSB and GSB, we have, 
 
 angle ASB < angle ASG + angle GSB. 
 
 Therefore, etc. 
 
BOOK V. 113 
 
 THEOREM XIII. 
 
 The sum of the plane angles which form any polyedral angle, 
 is less than four right angles. 
 
 Let S be the vertex of a polyedral 
 angle formed by the plane angles ASB, 
 BSO, OSD, etc. ; then will' the sum of 
 these plane angles be less than four 
 right angles. 
 
 For, pass a plane cutting the edges in 
 the points A, B, O, D, E, and F, and the 
 faces in -the lines AB, BG, etc. Prom 
 any point, 0, in the polygon thus formed, 
 draw the lines OA, OB, OC, etc. We then have two seta 
 of triangles, one set having their vertices at S, the other 
 at 0, and both having the common bases AB, BO, etc. 
 
 Now, the sum of the angles of the upper set of tri- 
 angles is equal to the sum of the angles of the lower set 
 of triangles, since both sets consist of the same number 
 of triangles. But the sum of the angles SB A and SBCis 
 greater than ABC, or ABO + OBC (Th. XII.); and also 
 SCB + SOD is greater than OCB + OCD ; and so on with 
 the other angles at D, E, etc. Hence, the sum of all the 
 angles at the bases of the upper set of triangles is greater 
 than the sum of all the angles at the bases of the lower 
 set of triangles; therefore, the sum of the angles at S 
 must be less than the sum of the angles at 0. But the 
 sum of the angles at is equal to four right angles (B. I. 
 Th. II. C. 2) ; hence, the sum of the angles at S is less than 
 four right angles. Therefore, etc. 
 
 Scholium. This proposition supposes that the polyedral 
 angle is convex ;_ if it were not, the sum of the plane an- 
 gles would be unlimited. 
 
 10 * 
 
114 GEOME.TEY. 
 
 THEOEEM XIV. 
 
 If the three face angles of a triedral angle are respectively equal, 
 the triedral angles are either equal or symmetrical. 
 
 Let S and &' be the vertices of two triedral angles in which 
 ASB equals DS'E, BSC = ES'F and ASC = DS'F; then 
 the diedral angles are respectively equal, and the triedral 
 angles are either equal or symmetrical. 
 
 For, on SB take any point B, and in the faces ASB and 
 BSC draw the lines BA and BC 
 perpendicular to SB ; then the angle 
 ABC will measure the diedral angle 
 of the faces ASB and BSQ (Def. 6). 
 On S'E lay off S'E equal to SB, and 
 on the faces DSE. and ES'F draw 
 BE and EF perpendicular to S'E; then the angle DEF mil 
 measure the diedral angle of the faces DS'E and ES'F . 
 
 The right-angled triangles SBA and S'ED are equal, since 
 SB = S'E and ASB = DS'E; hence, AB equals DE and J.S 
 equals ZW. In a similar way it may be shown that BC equals 
 EF and CSS equals Fff. Hence, the triangles ASG and DS'F 
 have their sides respectively equal ; and ASC equals DS'F by 
 hypothesis ; therefore A C equals DF. Hence, the triangles 
 ABC and DEF have their sides respectively equal, and con- 
 sequently their corresponding angles are equal. Hence the 
 angle ABC, which measures the diedral angle of the planes 
 ASB and BSC, is equal to the angle DEF, which measures 
 the diedral angle of the planes DS'E and ES'F, or the diedral 
 angles are equal. In the same way it may be shown that the 
 other diedral angles are respectively equal. 
 
 Now, if the face angles of these triedral angles are sim- 
 ilarly placed, the triedral angles may be applied to each other 
 
BOOK V. 115 
 
 and they will coincide ; if, however, the face angles are not 
 similarly placed, the triedral angles will not coincide, but are 
 then said to be symmetrical. Therefore, etc. 
 
 Scholium. Polyedral angles are said to be symmetrical when, 
 having the same number of face angles, these angles and the 
 successive diedral angles are respectively arranged in a reverse 
 order. 
 
 THEOREMS FOR ORIGINAL THOUGHT. 
 
 1. Prove that but one plane can be passed through a given point 
 perpendicular to a given line. 
 
 2. If a line is perpendicular to a plane, every plane passed through 
 the line is perpendicular to that plane. 
 
 3. If two planes are perpendicular to each other, a line drawn in 
 one of them perpendicular to their intersection is perpendicular to the 
 
 other. 
 
 4. If two planes which cut each other are both perpendicular to a 
 third plane, their intersection is perpendicular to that plane. 
 
 5. Prove that through a given line of a given plane, only one piano 
 perpendicular to the given plane can be passed. 
 
 6. Prove that through a line parallel to a given plane, only one plane 
 perpwndicular to the given plane can be passed. 
 
 7. If two planes which intersect contain two lines parallel to each 
 other, the intersection of the planes will be parallel to the lines. 
 
 8. If a line is parallel to one plane and perpendicular to another, 
 these two planes are perpendicular. 
 
 9. If two planes are parallel to a third, they are parallel to each 
 
 other. 
 
 10. Only one plane can be drawn through a given point parallel to a 
 
 given plane. 
 
 11. If two lines are parallel in space, and planes be passed through 
 them perpendicular to a third plane, the (wo planes will be parallel., 
 
116 GEOMETEY. 
 
 PROBLEMS. 
 
 The following problems are easily solved from the principles already 
 presented. 
 
 1. To erect a perpendicular to a given plane at a given point of the 
 plane. (See Prop. III.) 
 
 2. To construct a plane parallel to a given plane. 
 
 3. To construct a plane perpendicular to a given plane intersecting 
 it in a given straight line. 
 
 4. To draw a line from a given point of a plane making any given 
 angle with the plane. 
 
 5. To draw a plane intersecting a given plane and making any givea 
 angle with it. 
 
BOOK VI. 
 
 POLYEDRON S. 
 
 <LP 
 
 DEFINITIONS. 
 
 1. APoLYEDRONisa volume bounded by polygons. 
 The bounding polygons are called the faces of the polye- 
 
 dron ; the lines in which the faces meet are called edges; 
 and the points in which the edges meet are called vertices 
 of the polyedron. 
 
 2. A Prism is a polyedron, two of whose faces are equal 
 polygons, having their homologous sides pa- 
 rallel ; the other faces are parallelograms. 
 
 The equal polygons are called bases of the 
 prism; one, the upper base; the other, the 
 lower base. The parallelograms constitute 
 the lateral or convex surf 'ace of the prism; the 
 intersections of the lateral faces are called 
 lateral edges. 
 
 3. The Altitude of a prism is the perpendicular dis- 
 tance between its bases. 
 
 '4. AEightPrismIs one whose lateral edges are per- 
 pendicular to the bases. In a right prism, each lateral 
 edge is equal to the altitude. 
 
 5. An Oblique Prism is one whose lateral edges are 
 oblique to the bases. Each lateral edge is, consequently, 
 greater than the altitude. 
 
 6. A prism is named from its bases. A triangular prism is 
 
 one whose bases are triangles ; a quadrangular prism is one 
 
 whose bases are quadrilaterals • and so on. 
 
 117 
 
118 
 
 GEOMETKY. 
 
 7. A Parallelopipedon is a prism whose bases 
 are parallelograms. 
 
 A Rectangular Parallelopipe- 
 
 don is a right parallelopipedon with rect- 
 angular bases. A cube is a rectangular 
 parallelopipedon, all of whose faces are 
 equal squares. 
 
 8. APyRAMiDisa polyedron bounded 
 
 by a polygon, and by triangles meeting at a common point, 
 
 The polygon is called the base of the pyramid ; 
 the triangles, its lateral or convex surface; and 
 the point where the triangles meet, its vertex. 
 
 9. Pyramids are. named from their bases ; 
 thus, we have triangular, quadrangular, pent- 
 angular, etc. pyramids, as the -'bases are tri- 
 angles, quadrilaterals, pentagons, etc. 
 
 10. The Altitude of a pyramid is the perpendicular 
 distance from the vertex to the plane of the base. 
 
 11. A Eight Pyramid is one whose base is a regular 
 polygon, and in which a perpendicular from the vertex to 
 the base passes through the centre of the base. This per- 
 pendicular is called the axis of the pyramid. 
 
 12. The Slant Height of aright pyramid is the per- 
 pendicular distance from its vertex to any side of the base. 
 
 13. A Frustum of a Pyramid is the part of a 
 pyramid included between its base and a plane 
 
 cutting the pyramid parallel to the base. 
 
 14. The Altitude of a frustum of a pyra- 
 mid is the perpendicular distance between its 
 bases. 
 
 15. The Slant Height of a frustum of a 
 
BOOK VI. 119 
 
 right pyramid is that portion of the slant height of the 
 pyramid included between the bases of the frustum. 
 
 16. Similar Polyedrons are those which are 
 bounded by the same number of similar polygons, similarly 
 placed. Parts which are similarly placed are homologous, 
 whether faces, angles, or edges. 
 
 17. The Diagonal of a polyedron is a line joining the 
 vertices of any two polyedral angles not in the same face. 
 
 18. The Volume of a polyedron is its numerical value, 
 expressing how many times it contains some other polye- 
 dron as a unit. 
 
 19. A Eight Section of a prism is a section perpen- 
 dicular to its lateral edges. An oblique section is one oblique 
 to its lateral edges. 
 
 20. A Truncated Prism is a portion of the prism 
 included between either base and an oblique section of the 
 prism. 
 
 Analysis. — This book treats of prisms, pyramids, and frustums. 
 IThe object is to find the surface and volume of these polyedrons, 
 and the relation of those which are similar. Their surface is readily 
 determined by finding the area of the polygons which form their faces. 
 In finding their volumes, we begin with the rectangular parallelopipe- 
 don, assuming for a unit of measure a cube whose edge is a unit of 
 measure of the edges of the parallelopipedon. From the volume of -a 
 rectangular parallelopipedon we pass to that of any parallelopipedon, 
 thence to the volume of a triangular prism, and from this to that of 
 any prism. The division of a triangular prism into three equal parts 
 gives the volume of a triangular pyramid, from which we pass to the 
 volume of any pyramid, and also of any frustum. 
 
120 
 
 GEOMETRY. 
 
 K 
 
 THE PRISM. 
 
 THEOREM I. 
 
 The convex surface of a right prism is equal to the perimetet 
 of the base multiplied by the altitude. 
 
 Let ABODE — iTbe a right prism; then will its convex 
 surface be equal to 
 
 (AB + BC+CD + DE + EA) X AF. 
 
 For, the convex surface of the prism 
 is equal to the sum of all the rectangles 
 AG, BH, 01, etc. Now, the altitude of 
 each of these rectangles is equal to the 
 altitude of the prism, and the area of 
 each rectangle is equal to its base mul- 
 tiplied by its altitude ; hence, the convex 
 surface, which is the sum of the areas of these rectangles, 
 is equal to 
 
 (AB + BO + CD + BE + EA) X AF; 
 
 ^_P 
 
 or, the perimeter of the base multiplied by the altitude. 
 Therefore, etc. , 
 
 Cor. If two right prisms have the same altitude, their 
 convex surfaces are to each other as the perimeters of 
 their bases. 
 
 THEOREM II. 
 
 Tf a prism be cut by parallel planes, the sections formed will be 
 equal polygons. 
 
 Let the prism ABODE — iTbe cut by the parallel planes 
 
BOOK VI. 
 
 121 
 
 LO and QT; then will the sections I/O and QT be equal 
 polygons. 
 
 For, LM and QR are parallel, being the 
 intersections of the two parallel planes 
 with ABGF (B.V.Th. VIII.); these lines' 
 LM and QR are also equal, since they are 
 parallels included between the two paral- 
 lels AF and BO (B. I. Th. XV. C. 2.). For 
 alike reason, MN is equal and parallel to 
 RS, NO to ST,-OP to TU, etc. ; hence, the 
 angle LMN is equal to the angle QRS, 
 MNO to RST, etc. (B. V. Th. X.); there- 
 fore, the sections are equal polygons. 
 
 Gor. Every section of a prism parallel to the base is equal 
 to the base. 
 
 THEOKEM III. 
 
 Two prisms are equal if three faces, including a triedral angle 
 of one, are respectively equal to three faces similarly placed, 
 including a triedral of the other. 
 
 Let the three faces of the triedral angles A and L of the 
 prisms ABODE — F and LMNOP — § be equal and similarly 
 placed ; then will the prisms be 
 equal. 
 
 For, place the base ABODE 
 
 on its equal LMNOP; then, 
 
 since the triedral angles A and 
 
 L are equal (B. V. Th. XIV.), 
 
 they will coincide when applied 
 
 to each other, the face A Q will coincide with LR, and the 
 
 face AK with LS; hence, the sides FQ and FK of the upper 
 
 base of one prism will coincide with the sides QR and QS of 
 
 11 
 
122 GEOMETRY. 
 
 the upper base of the other prism ; hence also the planes of 
 their bases will coincide, and, since these bases are equal, they 
 will coincide throughout ; consequently, all the lateral faces 
 of the two prisms will coincide, each to each, and the prisms 
 will coincide throughout, and are therefore equal. There- 
 fore, etc. 
 
 Cor. 1. Two right prisms are equal if they have equal bases 
 and equal altitudes. For, if the faces are similarly placed, the 
 prisms will coincide when applied to each other. If the faces 
 are not similarly placed, by inverting one prism the faces will 
 be similarly placed, and the prisms may be applied to each 
 other and will coincide. 
 
 Cor. 2. Two truncated prisms are equal if three faces includ- 
 ing a triedral angle of one are respectively equal to three faces 
 including a triedral angle of the other. For, the above demon- 
 stration will apply whether the upper bases are parallel to the 
 lower bases or are inclined to them, as they are in a truncated 
 prism. 
 
 THEOREM IV. 
 
 An oblique prism is equivalent to a right prism whose base is 
 equal to a right section of the oblique prism, and whose alti- 
 tude is equal to a lateral edge of the oblique prism. 
 
 F' K' 
 
 T-ieiABCDE—A' be an oblique prism. 
 At any point F, in the edge A A', pass a 
 plane perpendicular to A A', forming the 
 right section FCHIK. Produce AA' 
 to F', making FF' equal to AA', and 
 through F' pass a second plane perpen- 
 dicular to the edge AA', intersecting 
 all the faces of the prism produced, 
 and forming another right section, 
 
BOOK VI. 
 
 123 
 
 F'O'H'I'K', parallel and equal to FOHIK. The prism 
 FOHIK — F' is a right prism whose base is the right sec- 
 tion FOHIK, and whose altitude, FF', is equal to the lateral 
 edge of the oblique prism. 
 
 Now, the" truncated prism ABODE — F is equal to the 
 truncated prism A'B'C'DE'-F' (Th. III. Cor. 2) ; if to each 
 of these equal prisms we add the volume FOHIK— A', we 
 shall have the oblique prism AB CDE — A', equivalent to the 
 right prism FGHIK— F'. Therefore, etc. 
 
 ■H — \$" 
 
 THEOKEM V. 
 
 Any parattelopipedon is equivalent to a rectangular parallel- 
 opipedon having the same altitude and an equivalent base. 
 
 Let ABCD — D 1 be any oblique parallelopipedon whose 
 base is ABCD and altitude B'P. Produce the edges, AB, 
 DC, A'B', and DC; 
 on AB produced take 
 EF equal to AB, and 
 through E and F pass 
 planes perpendicular 
 to the produced edges, 
 forming the parallelo- 
 pipedon EFGH-H'. 
 This parallelopipedon, regarding EE'H'H as the base' and 
 EF the altitude,- is a right prism, and is equivalent to the 
 oblique parallelopipedon ABCD — A' (Th. IV.). 
 
 Again, from E' and H' let fall the perpendiculars E'K 
 and H'N to EH produced, and from F' and G' let fall the 
 perpendiculars F'L and G'M to FQ produced, and draw LK 
 and MN; then KLMN— H' will be a rectangular parallelo- 
 pipedon whose base is KLMN and altitude E'K equal to B'P. 
 
124 GEOMETEY. 
 
 Thus, the rectangular parallelopipedon KLMN—H' is 
 equivalent to the oblique parallelopipedon EFF'E' — H 
 (Th. TV); but this parallelopipedon EFF'E' — H, which, 
 with EE'H'H as a base, is a right parallelopipedon, is equiv- 
 alent to the parallelopipedon ABCD — jy (Th. IV.) ; hence, 
 ABCD — If is equivalent to the right parallelopipedon 
 KLMN—H. Also, the base KM is equal to the base E'G', 
 and hence to its equal EG, which is equivalent to the base 
 AC; hence KM is equivalent to AC. Therefore, etc. 
 
 THEOEEM VI. 
 
 The plane passed through two diagonally opposite edges of a 
 parallelopipedon divides the parallelopipedon into two equiv- 
 alent triangular prisms. 
 
 Let ABCD — E be any parallelopipedon, and let a plane 
 be passed through its opposite edges BF and DH ; then will 
 the triangular prisms ABD — H and 
 BCD— H be equivalent. 
 
 For, let KLMN be any right sec- 
 tion of the parallelopipedon made by 
 a plane perpendicular to the edge 
 AE ; the intersection, LN, of this 
 plane with the plane BH is the diag- M 
 onal of the parallelogram KLMN, 
 and divides the parallelogram into 
 two equal triangles, LMN and KLN. The oblique prism 
 ABD — His equivalent to a right prism whose base is the 
 triangle KLN and whose altitude is AE (Th. IV.) ; and the 
 oblique prism BCD — H is equivalent to a right prism whose 
 base is the triangle ZJOfand altitude AE; but the two right 
 prisms are equal (Th. III. C. 1) ; hence, the two oblique 
 prisms are equivalent to each other. Therefore, etc. 
 
BOOK VI. 
 
 125 
 
 
 ^ 
 
 THEOREM VII. 
 
 The opposite faces of a parallelopipedon are equal and parallel. 
 TietABGD — H be a parallelopipedon ; then will its op- 
 posite faces be equal and parallel. 
 
 IP XT 
 
 For, the bases are equal and paral- 
 lel, by the definition of a parallelo- 
 pipedon. Also, BG is equal and pa- 
 rallel to AD, since the figure ABGD 
 is a parallelogram, and, for a similar 
 reason, BF and AF are equal and B 
 
 parallel ; consequently, the angles EAD and FB are 
 equal, and their planes parallel (B. V. Th. X.), and, 
 therefore, the parallelograms BG and AM are equal (B. I. 
 Th. XV. C. 3). In a similar manner it may be shown that 
 the faces AF and DG are equal and parallel. Therefore, 
 etc. 
 
 Cor. 1. Any two opposite faces of a parallelopipedon may 
 be taken as the bases. 
 
 Cor. 2. The diagonals of a parallelopipedon bisect each 
 other. 
 
 Draw the diagonals FD and BJS; draw also BD and 
 FH; then, since BF and HD are equal and parallel, the 
 figure BDHF is a parallelogram; 
 hence, the diagonals FD and BE. 
 bisect each other at (B. I. Th. 
 XVIII.). In the same manner it 
 may be shown that either of these 
 and any other diagonal bisect each 
 other ; hence, all the diagonals bisect 
 each other. 
 
 Cor. 3. In a rectangular parallelopipedon, the square of 
 
 either, diagonal equals the sum of the squares of the three 
 
 n* 
 
 :i: 
 
 >p 
 
126 
 
 GEOMETRY. 
 
 edges which meet at the same vertex. Let the pupil show 
 it ; that SB 1 = BC* + DC* + DR\ 
 
 THEOREM VIII. 
 
 The volume of a rectangular parallelopipedon is equal to the 
 product of its base and altitude. 
 
 lietABCB — H be a rectangular parallelopipedon; then 
 will its volume be equal to its base ABGD multiplied by 
 its altitude AB. 
 
 Suppose AK to be a common unit of 
 measure of the three sides AB, AD, 
 and AB, and suppose it to be contained 
 4 times in AB, 3 times in AD, and 5 
 times in AB; then divide AB into 4 
 equal parts, AD into 3, and AB into 5 
 equal parts, and pass planes through 
 the points of division parallel to the 
 faces of the parallelopipedon. The 
 parallelopipedon will thus be divided into equal cubes, 
 oqual since their sides are equal and their angles are 
 equal, all being right angles. 
 
 Now, the number of these little cubes upon the base is 
 equal to the number of surface units in the base, and the 
 whole number of cubes in the parallelopipedon is equal to 
 the number upon the base multiplied by the number of 
 layers, and the number of layers is the same as the number 
 of units in the- altitude ; hence, the number of cubic units 
 in the parallelopipedon is equal to the base multiplied by 
 the altitude. Now, this is evidently true whatever be the 
 size of the linear unit; hence, it is true when the linear 
 unit is exceedingly small, and, consequently, when it is 
 infinitely small, as it -must be when the three sides are 
 
 / 
 
 
 / 
 
 *£ft 
 
 
 1/ 
 
 1 / / 
 
 / 
 
 ti rr 
 
 / 
 
 rrtr 
 
 / 
 
BOOK VI. 127 
 
 incommensurable. Therefore, the volume of a rectangular 
 parallelopipedon is equal to the' product of its base and 
 altitude. 
 
 Cor. 1. It is evident that the number of cubic units upon 
 the base is equal to the number of rows multiplied by the 
 number in each row ; that is, the length of the base mul- 
 tiplied by its breadth; hence, the volume of a rectangular 
 parallelopipedon equals the product of its length, breadth, and 
 altitude, or the product of its three dimensions. 
 
 Cor. 2. Any two rectangular parallelopipedons are to 
 each other as the products of their bases and altitudes, or as 
 the products of their three dimensions. 
 
 Cor. 3. "When their bases are equal, they are to eacb 
 other as their altitudes; when their altitudes are equal, 
 they are to each other as their bases. 
 
 THEOEEM IX. 
 
 The volume of any parallelopipedon is equal to the product 
 of its base and altitude. 
 
 Let ABCD—E be any parallelopipedon whose base is 
 ABCD and altitude H; then will its 
 volume be equal to the base ABCD 
 multiplied by the altitude H. 
 
 For, the parallelopipedon ABCD—E 
 is equivalent to a rectangular parallelo- 
 pipedon having the same altitude and 
 an equivalent base (Th. V.) ; but the 
 volume of such a rectangular parallelo- 
 pipedon is equal to the product of its 
 base and altitude (Th. VIII.) ; hence, the volume of the 
 parallelopipedon ABCD — E is equal to the product of its 
 base and altitude. Therefore, etc.- 
 
128 
 
 GEOMETRY. 
 
 THEOEEM X. 
 
 The volume of any prism is equal to the product of its base 
 and altitude. 
 
 This prism is 
 
 1st. ~Let ABC — E be a triangular prism, 
 half the parallel opiped on constructed 
 on its edges AB, BC, and BF (Th. 
 VI.). The volume of this parallelo- 
 pipedon is equal to its base ABQD mul- 
 tiplied by its altitude (Th. IX.) ; hence, 
 the volume of the triangular prism 
 ABC— E is equal to its base ABC, the 
 half of ABCD, multiplied byits altitude. 
 
 2d. Let ABCDE — F be any prism. Divide it into tri- 
 angular prisms by passing planes through any lateral edge 
 BG; these prisms will have a common 
 altitude, the altitude of the prism. 
 
 The volume of any triangular prism,' 
 ABE — F, is equal to the product of its 
 base and altitude, as. just shown ; hence, 
 the volume of the prism ABCDE—F, 
 which is the sum of these triangular 
 prisms, is equal to its base, which is the 
 sum of the bases of the triangular prisms, 
 multiplied by its altitude. Therefore, etc. 
 
 Cor. 1. Prisms having equivalent bases and equal altitudes 
 are equivalent. 
 
 Cor. 2. Any two prisms are to each other as the products 
 of their bases and altitudes. 
 
 Cor. 3. Prisms having equal altitudes are to each other as 
 their bases. 
 
 Cor. 4. Prisms having equal bases are to each other as 
 their, altitudes. 
 
BOOK VI. 
 
 129 
 
 THEOEEM XI. 
 
 Similar triangular prisms are to each other as the cubes of 
 their homologous edges. 
 
 Let ABG—E and GBH—L be two similar triangular 
 prisms ; then will they be 
 to each other as the cube 
 of any two homologous 
 edges AB and GB. 
 
 For, since the two prisms 
 are similar, the faces con- 
 taining the triedral angles 
 B and B are respectively 
 
 similar ; therefore, the prism GBH—D being applied to 
 th p prism ABG—E will take the position GBJE—P. From 
 D draw DM perpendicular to the base, and from K draw 
 KN perpendicular to the base ; then the two triangles 
 JDMB and KNB must be similar, since they are mutually 
 equiangular. 
 
 Now, since the bases are similar, we have (B. III. Th. 
 XVL), 
 
 base ABC: base GBR: : AB 1 : GB 2 ; 
 
 and, since the triangles DMB and KNB are similar, and 
 also the parallelograms AD and GK, we' have, 
 
 DM: KN: : DB : KB : : AB : GB. 
 
 Multiplying together the corresponding terms of the 
 first and last couplets of these two proportions, we have, 
 " base ABC X DM: base GBSX KN: -.AB 3 : GB 3 . 
 
 But baseABGX DM is the volume of the prism ABG—E, 
 and base GBB X KN is the volume of the prism 
 GBH—L; hence, the prisms are to each other as AB 3 to 
 G£ 3 . Therefore, etc. 
 
130 
 
 GEOMETRY. 
 
 Cor. 1. Any two similar prisms are to each other as the 
 cubes of their homologous edges. 
 
 For, since the prisms are similar, their bases are similar, 
 and may, therefore, be divided into the same number of 
 similar triangles, similarly situated (B. III. Th. XVII.); 
 hence, each prism may be divided into the same number 
 of similar triangular prisms. But these triangular prisma 
 are to each other as the cubes of their homologous edges ; 
 hence, the polygonal prisms 'which are respectively the sum 
 of these triangular prisms must be to each other as the cubes 
 of their homologous edges. 
 
 Cor. 2. Similar prisms are to each other as the cubes of 
 their altitudes, or as the cubes of any other homologous 
 lines. 
 
 THE PYRAMID. 
 
 THEOEEM XII. 
 
 The convex surface of a right pyramid is equal to the perime- 
 ter of the base multiplied by one-half of the slant height. 
 
 Let ABCDE — S be a right pyramid, and SS the slant 
 height; then will the convex surface be 
 equal to the perimeter AB -\-BC-\- CD -\- 
 DE + EA multiplied by £ of SIT. 
 
 Draw SO perpendicular to the base; 
 then, from the definition of a right pyra- 
 mid. O is the centre of the base ; conse- 
 quently, the distances AO,BO, CO, etc. are 
 all equal, and tharefore the edges SA, SB, 
 SC, etc., are all equal (B. V. Th. III.) ; 
 and, since ,the sides AB, BC, etc., are 
 all equal, the triangles SAB, SBC, etc. are all equal, and 
 
BOOK VI. 
 
 131' 
 
 their altitudes, which is the slant height of the pyramid, 
 are equal. 
 
 Now, the area of each triangle is equal to its base multi- 
 plied by one-half of its altitude ; hence, the sum of the 
 areas of these triangles, which is the convex surface of 
 the pyramid, equals the sum of their bases into one-half 
 of the slant height SH; that is, the convex surface of the 
 pyramid equals 
 
 (AB + BC+CD + DE + EA)Xh ss - 
 Therefore, etc. 
 
 THEOREM XIII. 
 
 If a pyramid be cut by a plane parallel to the base; 
 
 1. The edges and altitude will be divided proportionally. - 
 
 2. The section will be a polygon similar to the base. 
 
 Let the pyramid S— ABODE be cut by a plane 
 GHIKL parallel to the base ; then will the edges SA, SB, 
 SO, etc., with the altitude SO, be divided 
 proportionally, and the section GHIKL 
 will be similar to the base. 
 
 First. Since the planes ABODE and 
 GHIKL are parallel, the intersections 
 AB and GH are parallel (B.V. Th.V-IIL); 
 for the same reason, BO is parallel to 
 HI, and BO to IIP. Hence, we have 
 (B. III. Th.IX. C. 1), 
 
 , SA:SG::SB:SH; 
 and also, SB :SS: : SO: SI; 
 
 and also, SB : SH : : SO : SP. 
 
 Hence, the edges and altitude are divided proportion- 
 ally. 
 
132 
 
 GEOMETKY. 
 
 Second. Since GH is parallel to AB, and HI to JBC, the 
 angle GHI is equal to ABC (B. V. Th. X.) ; and, for the 
 same reason, each angle of the polygon GHIKL is equal 
 to the corresponding angle of the hase ; hence, the two 
 polygons are mutually equiangular. 
 
 Again, since GH is parallel to AB, we have, 
 GH:AB::SH:SB; 
 and, since HI is parallel to BC, we have, 
 
 HI:BC::SH:SB. 
 Hence, from equal ratios, we have, 
 
 GH:AB::HI:BG. 
 In the same manner, it may he shown that all the sides 
 of the two polygons are proj)ortional; hence, the section 
 GHIKL is similar to the base ABGDE (B. III. D. 6). 
 
 THEOREM XIV. 
 
 If two pyramids have the 'same altitude, and their bases in the 
 same plane, the sections made by a plane parallel to their 
 bases are to each other as their bases. 
 
 Let S— ABCBE and S—MNO be two pyramids, having 
 the same altitude, and 
 their bases in the same 
 plane; and let GHIKL 
 and PQB, he sections 
 made by a plane parallel 
 to their bases; then will 
 these sections be to each 
 other as the bases. 
 
 For, the polygons 
 ABODE and GHIKL, 
 being similar, are to each other as the squares of their 
 sides AB and GH (B. III. Th. XVIII.) ; but 
 
BOOK VI. 
 
 133 
 
 AB:GH::SA:SG. 
 Hence, ABODE : GHIKL : : #Z 2 : SG\ 
 
 For a similar reason, 
 
 MNO : PQB : : SM> : SP 1 - 
 But (B. V. Th. XI.) we have, 
 
 8A:SG::SM:SP: 
 Hence, AB ODE : GHIKL : : MNO : PQB. 
 
 Therefore, etc. 
 
 Cor. 1. If the bases are equal, any two sections parallel to 
 the bases at equal distances from the vertices are equal. 
 
 Cor. 2. Any two sections parallel to the base are propor- 
 tional to the squares of their distances from the vertex. 
 
 THEOEEM XV. 
 
 Two triangular pyramids having equivalent bases and equal 
 altitudes are equal in volume. 
 
 Let S — ABC and S'—A'B'C be two triangular pyramids 
 having equivalent' bases ABC and A'B'C and equal altitudes 
 A T; then will these pyramids be equal in volume. 
 
 For, place the bases of the pyramids in the same plane, 
 
 divide the altitude, A T, into any number of equal parts, and 
 through the points of division pass planes parallel to the 
 plane of their bases, forming the sections DEF, D'E'F', etc., 
 
 12 
 
134 GEOMETRY. 
 
 and construct prisms in the two pyramids with these sections 
 as upper bases. Now, the corresponding sections DEF, D'E'F', 
 etc., are equivalent (Th. XIV.) ; hence, the corresponding 
 prisms, having equivalent bases and equal altitudes, are 
 equivalent (Th. X. C. 1). 
 
 Now, this is true whatever the equal number of inscribed 
 prisms ; hence, it is true when the number in each prism 
 becomes indefinitely or infinitely great, in which case they 
 will coincide respectively with the two pyramids ; therefore, 
 the pyramids are equal in volume. Therefore, etc. 
 
 THEOREM XVI. 
 
 A triangular prism may be divided into three equal triangular 
 
 pyramids. 
 
 Let ABC — F be a triangular 
 prism; then may it be divided 
 into three equal triangular pyra- 
 mids. 
 
 Pass a plane through the edge 
 AC and the point i?, cutting off 
 the pyramid ABO — E; pass an- 
 other plane through HE and the b 
 point 0, cutting off the pyramid DEF—G; there will re« 
 main a pyramid whose base may be regarded as ACD, 
 having its vertex at E. Now, the two pyramids ABC — E 
 and DEF — are equal in volume, since they have equal 
 bases and equal altitudes (Th. XV.). Regarding the pyra- 
 mid DEF — as having the base DCF and vertex &% E, 
 it is equal in volume to the pyramid A CD — E, since their 
 bases are equal, being halves of the parallelogram A CFD, 
 and their altitudes are equal, since their bases are in the 
 same plane and vertices at the same point. Hence, the 
 
BOOK VI. 
 
 135 
 
 three pyramids into which the prism is divided are all 
 equal in volume. Therefore, etc. 
 
 Cor. 1. A triangular pyramid is one'-third of a prism 
 having an equal base and an equal altitude. 
 
 Cor. 2. The volume of a triangular pyramid is one-third 
 of the product of its base and altitude. 
 
 ' THEOREM XVII. 
 
 The volume of a pyramid is equal to one-third of the product 
 of its base and altitude. 
 
 Let & — ABCDE be a pyramid, and SO the altitude ; then 
 will its volume be equal to ABCDE X 
 
 ISO. 
 
 Draw the diagonals A C and AD, and 
 pass the planes SA C and SAD through 
 these diagonals and the vertex S; the 
 pyramid will then be divided into tri- 
 angular pyramids, whose altitiides are 
 equal, being the altitude of the pyra- 
 mid. Now, the volume of each of 
 these triangular pyramids is equal to 
 its base by one-third of the altitude 
 (Th. XVI. C. 2) ; hence, the volume of the pyramid 
 S — ABCDE, which is the sum of these triangular pyra^ 
 mids, is equal to the sum of their bases into one-third of 
 the altitude ; that is, base ABCDE X i SO. Therefore, etc. 
 
 Cor. 1. The volume of a pyramid is one-third of the 
 volume of a prism having an equal base and an equal alti- 
 tude. 
 
 Cor. 2. Pyramids are to each other^ as the products of 
 their bases and altitudes. 
 
 Cor. 3. Pyramids having equal bases are to each other 
 
136 
 
 GEOMETEY. 
 
 as their altitudes ; pyramids having equal altitudes are to 
 each other as their bases. 
 
 Scholium. The volume of any polyedron may be found 
 by dividing it into triangular pyramids, by passing planes 
 through its vertices. 
 
 THEOEEM XVIII. 
 
 Similar pyramids are to each other as the cubes of their homo' 
 
 logons edges. 
 
 Let S— ABODE and S— FGHIK be two similar pyra- 
 mids ; then will they be to each other 
 as the cubes of any two homologous 
 sides AB and FG. 
 
 For, since the pyramids are similar, 
 they may be so placed that their ho- 
 mologous angles at the vertex will 
 coincide. Then, since the faces SAB 
 and SFG are similar, AB is parallel to 
 FG; and since SBC and SGH are A j, 
 similar, BO is parallel to GH; hence, 
 the planes of the bases are parallel 
 (B. Y. Th. X.). 
 
 Draw SO perpendicular to the base AB ODE; it will also 
 
 be perpendicular to the base FGHIK a,t some point, P; then 
 
 (Th. XIII.), 
 
 SO :SP:: SB. :SG::AB: FG; 
 
 and, consequently, 
 
 ISO : \ SP : : AB : FG. 
 
 But, the bases of the pyramids being similar, we have (B. 
 
 III. Th. XVIII.), 
 
 base ABODE : base FGHIK: : AB* : FG\ 
 
 Multiplying these two proportions, term by term, we have, 
 
 base ABODE X\SO: base FGHIK X%SP: -.IB 3 : FG 3 . 
 
BOOK VI. 
 
 137. 
 
 But, base ABODE X \ SO is "equal to the volume of the 
 pyramid 8— ABODE, and base FGHIK X i SPin equal to 
 the volume of the pyramid S— FGHIK ; hence, the two 
 pyramids are to each other as the cubes of the homologous 
 edges AB and FG-. Therefore, etc. 
 
 Oor. Similar pyramids are to each other as the cubes of 
 their altitudes, or as the cubes of any two homologous 
 lines. 
 
 FRUSTUM OP A PYRAMID. 
 
 THEOREM XIX. 
 The convex surface of a frustum of a right pyramid is equal 
 to one-half of the sum of the perimeters of the upper and lower 
 bases, multiplied by the slant height. 
 Let ABODE— K be the frustum of a 
 right pyramid, and NM its slant height ; 
 then will its convex surface be equal to 
 one-half of the sum of the perimeters of 
 its two bases) multiplied by NM. 
 
 The faces forming the convex surface 
 are equal trapezoids; for the faces of the n a 
 
 pyramid of which this frustum is a part are equal, and the 
 faces of the pyramid cut off are equal; hence, the figures 
 which remain are equal, and their upper and lower bases 
 being parallel, they are equal trapezoids, and have a com- 
 mon altitude NM, the slant height of the frustum. , 
 
 Now, the area of each trapezoid, as ABGF, is equal to 
 | (AB + FG) X NM (B. III. Th. IV.) ; hence, the area of 
 the convex surface, which is the sum of all the trapezoids, 
 is equal to one-half the sum of the perimeters of the upper 
 and lower bases multiplied by the slant height. There- 
 fore, etc. 
 
 12* 
 
138 GEOMETEY. 
 
 THEOREM XX. 
 
 The volume of a frustum of a triangular pyramid is equal to 
 the sum of the volumes of three pyramids, whose common 
 altitude is the altitude of the frustum, and whose bases are 
 the lower base of the frustum, the upper base of the frustum, 
 and a mean proportional between the two bases. 
 
 Let ABC — F be the frustum of a triangular pyramid. 
 Through the points A, E, C, pass a plane cutting off the 
 pyramid E — ABC. This pyramid has 
 the altitude of the frustum, and for its 
 base the lower base of the frustum. 
 Through the points D, E, C, pass a plane 
 cutting off the pyramid C—DEF This 
 pyramid has the altitude of the frus- 
 tum, and for its base the upper base 
 of the frustum. The remaining part 
 of the frustum is a pyramid whose 
 base is ACD, with its vertex at E. 
 
 Now, draw EG parallel to DA; draw also GD; then the 
 pyramid E — A CD is equal to the pyramid G — A CD, since 
 they have the same base and equal altitudes. But the pyramid 
 G — ACD may be regarded as having A GC for its base, and 
 its vertex at D; it will then have the altitude of the frus- 
 tum. We will now show that its base AGO is a mean pro- 
 portional between the two bases of the frustum. 
 
 Draw GH parallel to BC ; then the triangles A GH and 
 DEF, being similar to ABC, are similar to each other, and, 
 hence, equiangular; and since AG equals DE, the triangle 
 AGS equals DEF (B. I. Th. VII.). Now, AGC is a mean 
 proportional between A GH and AB C (B. III. Th. IX. C. 3) ; 
 hence, the base of the third pyramid is a mean propor- 
 tional between the upper and lower bases. Therefore, etc. 
 
BOOK VI. 139 
 
 Cor. This proposition is true for the frustum of any pyra- 
 mid. For, since any pyramid is equal to a triangular pyra> 
 mid having an equal base and equal altitude, by cutting 
 , the pyramids with a plane parallel to the base, and re- 
 moving the upper part, it may be shown that the frustum 
 of any pyramid is equal to the frustum of a triangular 
 pyramid having equal bases and the same altitude; hence, 
 if the proposition is true for triangular frustums, it is true 
 for all frustums. 
 
 REGULAR POLYEDRONS. 
 
 A Regulab Polyedkon is one whose faces are all 
 equal and regular polygons. 
 
 There can be five, and only five, regular polyedrons, 
 namely : 
 
 1. The Teteaedbon, or regular pyramid, a polyedron 
 bounded by four equal equilateral triangles. 
 
 2. The Hexaedeon, or cube, a polyedron bounded by 
 six equal squares. 
 
 3. The Octaedeon, a polyedron bounded by eight equal 
 equilateral triangles.^ 
 
 4. The DoDECAEDEOif, a polyedron bounded by twelve 
 equal regular pentagons. 
 
 5. The Icosaedeon, a polyedron bounded by twenty equal 
 equilateral triangles. 
 
 1st. In the tetraedron the polyedral angle is formed of 
 three equilateral triangles ; in the octaedron, of four such 
 triangles ; in the icosaedron, of five triangles. The combina- 
 tion of six such angles (each angle being f of a right angle) 
 gives four right angles, or a plane, and hence no polyedral 
 
140 GEOMETRY. 
 
 angle ; and the combination of more than six will not form 
 a convex angle; hence only three regular polyedrons can be 
 formed of triangles. 
 
 2d. In the hexaedron the polyedral angle is formed of 
 three squares. The combination of four squares gives a plane, 
 and a greater number would not give a convex angle ; hence, 
 but one regular polyedron can be formed of squares. 
 
 3d. In the dodecaedron the polyedral angle is formed of 
 three regular pentagons. The combination of more than three 
 such angles (each angle being f of a right angle) exceeds four 
 right angles, and will not give a convex angle ; hence, but 
 one regular polyedron can be formed of pentagons. 
 
 4th. Three or more angles of a regular hexagon (each 
 angle being f of a right angle) exceeds a right angle, and 
 cannot form a convex polyedral angle ; and the same is true 
 of the heptagon, octagon, etc. 
 
 Therefore, only the five regular polyedrons named above 
 
 PRACTICAL EXAMPLES. 
 
 1. Required the convex surface of a right prism whose altitude is 14 
 inches and perimeter of the base 16 inches. Ans. 224 square inches. 
 
 2. Required the contents of a prism the area of whose base is 24 square 
 feet and altitude 7 feet. Ans. 168 cubic feet. 
 
 3. .Required the convex surface of a right pentangular pyramid whose 
 Blant height is 18 inches and each Ride of the base 6 inches. 
 
 Ans. 270 square inches. 
 
 4. Required the volume of the frustum of a square pyramid, the sides 
 ef whose bases are 8 and 6 inches, and whose altitude is 12 inches. 
 
 Ans. 592 cubic inches. 
 
 5. Required the entire surface of a cube whose sides are each H 
 inches. Ans. 726 square inches. 
 
BOOK VI. 141 
 
 6. A man wishes to make a cubical cistern whose contents are 373248 
 cubic inches ; how many feet of inch boards will line it? 
 
 Arts. 180 square feet. 
 
 7. What is the side of a cube which contains as much as a yolume 20 
 feet 6 inches long, 10 feet 8 inches wide, and 6 feet. 9 inches high ? 
 
 Ans. 11.4 feet. 
 
 8. What is the depth of a cubical cistern which shall contain 1600 
 gallons, each 231 cubic inches of water! Ans. 5.98 feet. 
 
 9. Required the dimensions of a cube whose surface shall be nu- 
 merically equal to its contents. Ans. 6 units. 
 
 10. There are two similar prisms whose lengths are as 7 to 28 re- 
 spectively ; required the relation of their contents. Ans. 1 : 64. 
 
 11. Required the contents of a pyramid whose altitude is -20 inches 
 and whose base is a regular hexagon, each side being 6 inches. 
 
 Ans. 623.5386 cubic inches. 
 
 12. If we pass a plane parallel to the base of the pyramid of the 11th 
 problem, half-way between its vertex and base, required the convex 
 surface and contents of the frustum. 
 
 Ans. Vol. = 545.596 cubic inches. 
 
 13. A farmer wishes to know what must be the depth of a cubical box 
 which shall contain 100 bushels of grain, each bushel 2150.42 cubic 
 inches. Ans. 4.9 feet. 
 
 THEOREMS FOR ORIGINAL THOUGHT. 
 
 1. Parallelopipedons having equal bases and equal altitudes are equal 
 in volume. 
 
 2. The diagonals of a rectangular parallelopipedon are equal. 
 
 3. If a plane be passed through the opposite edges of a rectangular 
 parallelopipedon, the triangular prisms formed are equal. 
 
 4. Two prisms having the same base are to each other as their alti- 
 tudes. 
 
 5. Two similar pyramids are equal when the base and lateral edge of 
 the one equal the base and lateral edge of the other. 
 
 6. The surfaces of similar polyedrons are to each other as the squares 
 sf their homologous edges 
 
^ 
 
 BOOK VII. 
 
 THE CYLINDER, THE CONE, AND THE SPHERE. 
 
 1. A Cylinder is a volume which may be generated 
 by the revolution of a rectangle about one of its sides as 
 an axis. 
 
 Thus, if the rectangle ABCD be revolved around the side 
 AB as an axis, it will generate the cylinder ^-r—^. 
 in the margin. The line AB is called the 
 axis; the surface described by CD is called 
 the convex surface; the circle .BCis the lower 
 base ; the circle AD is the upper base. 
 
 It is evident that the circle described by 
 the line EF perpendicular to the axis is equal to either 
 base ; hence, if a cylinder be cut by a plane parallel to the 
 base, the section will be a circle equal to the base. 
 
 2. A Cone is a volume which may be generated by the 
 revolution of a right-angled triangle about one of its sides 
 adjacent to the right angle. 
 
 Thus, if the right-angled triangle SBA be revolved 
 around SB as an axis, it will generate 
 the cone ADE—S. The side SB is the 
 axis of the cone ; the circle described by 
 AB is the base ; the hypothenuse SA is 
 the slant height; the surface generated 
 by SA is the convex surface. 
 
 it is evident that the circle described 
 by any line MN< perpendicular to the 
 
 142 
 
BOOK VII. 
 
 143 
 
 axis is a circle ; hence, the section of a cone by a plane 
 parallel to the base is a circle. 
 
 3. A Frustum of a Cone is the part which remains 
 after cutting off the top with a plane parallel to the base. 
 
 Thus, ADC— G is the frustum of a cone; FB is its 
 altitude; FA is its slant height. The 
 frustum of a cone may be generated by 
 the revolution of the trapezoid ABFE. 
 
 4. Similar Cylinders or Cones 
 are those whose axes are proportional to 
 the radii or the diameters of their bases. 
 
 5. A prism may be inscribed in a cylinder 
 
 by inscribing similar polygons with their sides parallel m 
 each base, and uniting the vertices of the angles with 
 straight lines. The cylinder is then said to circumscribe - 
 the prism. 
 
 6. A pyramid may be inscribed in a cone; and a frustum 
 of a pyramid may be inscribed in the frustum of a cone. 
 
 7. ASpHEREisa volume bounded by a curved surface, 
 every point of which is equally dis- 
 tant from a point within, called the 
 centre. 
 
 The distance from the centre to 
 the circumference is called the , 
 radius. The diameter is a line pass- 
 ing through the centre and limited 
 at both extremities by the surface. 
 
 8. A Spherical Sector is a volume generated by the 
 revolution of a sector of a circle about the diameter. 
 Thus, the revolution of ACF will generate a spherical 
 sector. 
 
 9. A Z o n e is a portion of the surface of a sphere in- 
 
144 GEOMEi'KY. 
 
 eluded between two parallel planes. The bounding lines 
 of the zone are called its bases; the distance between the 
 planes is its altitude. 
 
 10. A Spherical Segment is a portion of the sphere 
 included between two parallel planes. 
 
 11. If a semi-circumference be divided into equal arcs, the 
 chords of these arcs form half of the perimeter of an inscribed 
 polygon. The half perimeter is called a regular semi-per- 
 imeter. 
 
 The figure bounded by the diameter of the semi-circle and 
 the regular semi-perimeter is called a regular semi-polygon. 
 The diameter is called the axis' of the se^ni-polygon. 
 
 12. The Cylinder, the Cone and the Sphere are the 
 Three Eound Bodies of Geometry. 
 
 Analysis. — This book treats of the cylinder, the cone, and the sphere. 
 Its object is to find the convex surface and volume of each of these 
 bodies, and also their relation to each other. 
 
 The method of treatment consists in regarding these volumes as 
 polyedrons of an infinite number of sides. Thus, the cylinder is re- 
 garded as a right prism of an infinite -number of sides, the cone as a 
 right pyramid, and the sphere as a polyedron having its centre at the 
 centre of the sphere. 
 
BOOK VII. 
 
 145 
 
 CYLINDER, CONE, AND FRUSTUM. 
 
 THEOREM I. 
 
 The convex surface of a cylinder is equal to the circumference 
 of its base multiplied by its altitude. 
 
 Let ABDE be the base of a cylinder whose altitude is 
 H; then will its convex surface be equal 
 to circumference CA X S. 
 
 For, inscribe in the cylinder a prism 
 whose base is a regular polygon. Now, 
 the convex surface of this prism will be 
 equal to the perimeter of its -base multi- 
 plied by its altitude (B. VI. Th. I.) ; and 
 this is true whatever the number of sides ; 
 hence, it is true when the number of sides is infinite. But 
 when the number of sides is infinite, the convex surface of 
 the prism becomes the convex surface of the cylinder, the 
 perimeter of the base of the prism becomes the circum- 
 ference of the base of the cylinder, and the altitudes being 
 the same, therefore, the convex surface of the cylinder 
 equals the circumference of its base multiplied by its alti- 
 tude. 
 
 Cor. 1. Since the circumference of the base is 2 nB, the 
 expression for the convex surface of a cylinder is 2 
 nBxS. 
 
 Cor. 2. The convex surfaces of cylinders which havs 
 equal altitudes are to each other as the circumferences of 
 their bases. 
 
 18 
 
146 
 
 GEOMETRY. 
 
 THEOREM II. 
 
 The volume of a cylinder is equal to the area of its base multi- 
 plied by the altitude. 
 
 Let ABD — F be a cylinder, whose altitude is Sj then 
 will its volume be equal to the area of 
 its base multiplied by its altitude. 
 
 For, inscribe in the cylinder a prism 
 whose base is a regular polygon. Now, 
 the volume of this prism is equal to its 
 base multiplied by its altitude (B. YI. 
 Th. IX.), and this is true whatever the 
 number of sides, and therefore true when 
 the number of sides is infinite. But when the number of 
 sides is infinite, the prism coincides with the cylinder in 
 every respect ; hence, the volume of the cylinder is equal 
 to its base multiplied by its altitude. Therefore, etc. 
 
 Cor. 1. Since the area of the base is xB*, the expression 
 for the volume of a cylinder is %B? X S. 
 
 Cor. 2. Cylinders are to each other, as the products of 
 their bases and altitudes. Cylinders having equal' bases 
 are to each other as their altitudes; cylinders having 
 equal altitudes are to each other as their bases. 
 
 Cor. 3. Similar cylinders are to each other as the cubes 
 of their altitudes, or of the radii of the bases. Let the 
 pupil prove it. 
 
 THEOREM III. 
 
 The convex surface of a cone is equal to the circumference of 
 its base multiplied by one-half of the slant height. 
 
 Let # — ABCD be a cone whose base is ABB and slant 
 tyeight SA; then will its convex surface be equal to the 
 
BOOK VII. 
 
 147 
 
 circumference of its base multiplied by one-half of its 
 slant height. 
 
 For, inscribe . in the cone a right 
 pyramid. The convex surface of this 
 pyramid is equal to the perimeter of 
 its base multiplied by one-half of the 
 slant height (B. VI. Th. XII.); and 
 this is true whatever the number of 
 sides of the base ; hence, it is true 
 when the number of sides is infinite. 
 But when the number of sides is infinite, the pyramid 
 coincides with the cone in every respect ; hence, the con- 
 vex surface of the cone is equal to the circumference of 
 its base multiplied by one-half of the slant height. 
 
 Cor. 1. If S represents the slant height, the expression for 
 the convex surface of a cone is 2 nB X iS, or -kR X S. 
 
 THEOREM IV. 
 
 The volume of a cone is equal to the base multiplied by one- 
 third of the altitude. 
 
 Let S — AJBCD be a cone whose base is ABQD and alti- 
 tude SO; then will its volume be equal 
 to its base multiplied by one-third of 
 its altitude; 
 
 For, inscribe in the cone a right 
 pyramid. The volume of this pyramid 
 is equal to the base ABGD multiplied 
 by one-third of its altitude 80 (B. VI. 
 Th. XVII.) ; and this is true whatever 
 the number of sides of the base ; hence, it is true when 
 the number of sides is infinite. But when the number of 
 
148 
 
 GEOMETRY. 
 
 sides of the base is infinite, the pyramid becomes tha 
 cone; hence, the volume of a cone is equal to its base 
 multiplied by one-third of its altitude. Therefore, etc. 
 
 Cor. 1. The expression for the volume of a cone is 
 wiJ 2 X { S, or, I *W X B.. 
 
 Cor. 2. A cone is one-third of a cylinder having an equal 
 base and altitude. 
 
 Cor. 3. Cones are to each other as the products of their 
 bases and altitudes ; cones having equal bases are to each 
 other as their altitudes ; cones having equal altitudes are 
 to each other as their bases. 
 
 THEOREM V. 
 
 The convex surface of a frustum of a cone is equal to one-half 
 of the sum of the circumferences of the upper and lower bases 
 multiplied by the slant height. 
 
 Let ABBE — H be a frustum of a cone, C its altitude, FA 
 its slant height; then will its convex 
 surface be equal to one-half the sum of 
 the circumferences of its two bases mul- 
 tiplied by its slant height. 
 
 For, inscribe within the frustum of 
 a cone the frustum of a right pyramid. 
 The convex surface of this frustum is 
 equal to one-half the sum of the peri- 
 meters of its bases multiplied by the slant height (B. YL 
 Th. XIX.) ; and this is true whatever the number of lateral 
 faces ; hence, it is true when the number of faces is infinite. 
 But when the number of faces is infinite, the frustum of a 
 pyramid becomes the frustum of a cone, the perimeters of 
 its bases become the circumferences of the bases of the 
 
BOOK VII. 149 
 
 frustum of the cone, and the slant height of the frustum 
 of a pyramid becomes the slant height of the frustum of a 
 cone ; hence, the convex surface of the frustum of a cone 
 equals one-half the sum of the circumferences of its bases 
 multiplied by the slant height.- „ 
 
 Gor. The expression for the convex surface of a frustum 
 of a cone is £ (2 nB -\- 2 tcB') X S, where B and B' repre- 
 sent the radii of the bases, and S the slant height. 
 
 Scholium. Through L, the middle point of HE, draw LK 
 parallel to EG, and HB and LS perpendicular to EG; 
 now BS= SE (Bk. III. Th. IX.) ; OH-= CB, and KL = CB 
 + BS (Bk. I. Th. XV. C. 2). But CB + BS=SE + OH; 
 hence, 
 
 KL = \ (CB + BS+ SE+ OH) or i (CE + OH). 
 
 Multiplying this by 2n, we have, 
 
 2 t:KL = 1(2 izGE + 2 nOH); 
 
 that is, circ. KL equals £ of the sum of the circumferences 
 of the two bases ; hence, the convex surface of the frustum, 
 of a cone, generated by the revolution of the line HE, is equal 
 to the circumference of a circle generated by its middle point 
 into the length of the line. 
 
 THEOREM VI. 
 
 The volume of the frustum of a cone is equal to the sum of the 
 volume of three cones, having for a common altitude the alti- 
 tude of the frustum, and for bases the two bases of the frustum 
 and a mean proportional between them. 
 Let ABDE—H be a frustum of a cone, OG its altitude; 
 then will its volume be equal to the sum of the volumes 
 of three cones whose common altitude is OG, and whose 
 bases are the two bases and a mean proportional between 
 them. 
 
 13* 
 
150 
 
 GEOMETRY. 
 
 f/y *\ rr 
 
 / / 
 / / 
 
 a \ 
 
 / / 
 / / 
 
 "^4 
 
 For, inscribe in the frustum the 
 frustum of a right pyramid. The 
 volume of this frustum is equal to the 
 sum of the volumes of three pyramids 
 having the common altitude of the 
 frustum, and whose bases are the two 
 bases of the frustum and a mean pro- 
 portional between them (B. VI. Th. 
 XX.) ; and this is true whatever the number of lateral 
 faces, and, hence, true when the number of faces is infi^ 
 nite. But when the number of lateral faces is infinite, the 
 frustum of the pyramid becomes the frustum of a cone, 
 and the three pyramids become cones; hence, the volume 
 of the frustum of a cone equals the sum of the volumes 
 of three cones, whose common altitude is the altitude of 
 the frustum, and whose bases are the two bases of the 
 frustum and a mean proportional between them. 
 
 Cor. The expression for the volume of a frustum of a 
 Vine is (nB? + nr 2 + tcR X r) X i H. 
 
 THE SPHERE. 
 
 THEOREM VII. 
 Every section of a sphere made by a plane is a circle. 
 Let C be the centre of a sphere 
 whose radius is GA, and ADB any 
 section made by a plane; then will 
 this section be a circle. 
 
 For, draw CO perpendicular to the 
 section ABB, and draw the lines OD 
 and OE to different points of the 
 
BOOK VII 151 
 
 curve ADB; draw also the radii CD and CE. Then, since 
 the radii CD and CE are equal, the lines OD and OE must 
 be equal (B. V. Th. III. C. 1); hence, the section ADB is a 
 circle. Therefore, etc. * 
 
 Cor. If the plane pass through the centre of the sphere, the 
 radius of the section will be equal to the radius of the sphere. 
 The section is then called a great circle. All other sections 
 are called small circles. 
 
 THEOKEM VIII. 
 
 If a regular semirpolygon be revolved about its axis, the surface 
 generated by the semisperimeter.urill be equal to the circumfer- 
 ence of the inscribed circle multiplied by the axis. 
 
 TjetABCDFF be a regular semi-polygon, AF the axis, 
 ON the radius of the inscribed circle ; then 
 will the surface generated by the revolution 
 of the semi-polygon be equal to circ. ON X 
 AF. 
 
 For, from the extremities of any side, as 
 BC, draw BGr and CH perpendicular to AF; 
 fromiV", the middle point of BC, draw NM 
 perpendicular to AF; draw also BL per- 
 pendicular to CH. Now, the surface de- 
 scribed by BC is equal to circ. MN X .BC (Th. V. S. ). But, 
 since the triangles BCL and NOM are similar, we have, 
 BC: BL or CH : : ON: NM : : circ. ON : circ. NM; 
 hence, circ. NM X BC = circ. ON X CH; 
 
 that is, the surface generated by BC is equal to the cir- 
 cumference of the inscribed circle multiplied by the alti- 
 tude CH; and the same may be shown for each of the 
 other sides; hence, the surface described by the entire 
 
152 GEOMETRY. 
 
 semi-perimeter is equal to the circumference of the inscribed 
 circle multiplied by the sum of AG, GR,Hl, etc., or the 
 axis AF. Therefore, etc. 
 
 Cor. The surface described by any portion of the peri- 
 meter, as BCD, is equal to circ. ON X Gtl. 
 
 THEOREM IX. 
 
 The surface of a sphere is equal to the circumference of a great 
 circle multiplied by the diameter. 
 
 Let ABCDJEF be a semicircle, its centre, and AF its 
 diameter; then will the surface of the A 
 
 sphere generated by the revolution of the B ^^ 
 
 semi-circumference about the diameter be /fa 
 
 equal to circ. OA X AF. c f~~^- 
 
 For, inscribe in the semi-circumference 
 
 a regular semi-polygon. The surface de- X- 
 
 scribed by the revolution of the polygon is \ 
 equal to circ. OH X AF (Th. Yin.); and ^^ 
 
 this is true whatever the number of sides ; 
 hence, it. is true when the number of sides is infinite, in 
 which case the volume becomes a sphere with the radius 
 OA; hence, the surface of a sphere is equal to circ. OA X 
 AF. Therefore, etc. 
 
 Cor. 1. The surface of a sphere is equal to four of its great 
 circles. For, sur. = circ. OA X 2 OA; but circ. OA = 2 kOA; 
 hence, sur. = 2 nOA X 2 OA; which gives sur. = 4 nOA 2 ; 
 but xOA 2 is the area of a great circle ; hence, 4 nOA 2 is the 
 area of four great circles. 
 
 Cor. 2. The expression for the surface of a sphere is 4 
 itR\ or 7rZ> 2 , in which B, is the radius and D the diameter. 
 
 Cor. 3. The surfaces of spheres are to each other as the 
 
BOOK VII. 153 
 
 squares of their radii or diameters. For, sur. S = 4 tcB?, and 
 sur. s == 4 tit 2 ; hence, 
 
 S: s : : 4 ttJB 2 : 4 Ttr 2 , oriZ 2 : r 2 . 
 
 Cor. 4. The surface of a zone is equal to the circumfe- 
 rence of a great circle multiplied by its altitude. 
 
 Cor. 5. Zones on the same sphere, or on equal spheres, 
 are to each other as their altitudes. A zone is to the 
 surface of a sphere as the altitude of the zone is to the 
 diameter of the sphere. 
 
 THEOREM X. 
 
 The volume of a sphere is equal to its surface multiplied by 
 one-third of its radius. 
 
 For, conceive a regular polyedron to be inscribed in a 
 sphere ; this polyedron may be conceived as consisting of 
 pyramids having their vertices at the centre of the sphere, 
 and for bases the taces of the polyedron. The volume of each 
 of these pyramids is equal to its base multiplied by one-thin? 
 of its altitude, and, their altitudes being equal, the volume of 
 the polyedron will be equal to the sum of all their bases, 
 which is the surface of the polyedron, multiplied by one-third 
 of the common altitude. Now, the sphere may be regarded as 
 a polyedron consisting of an infinite number of pyramids, hav- 
 ing their vertices at the centre of the sphere and their bases at 
 its surface, their altitudes being equal to the radius of the sphere ; 
 hence, the volume of a sphere is equal to its surface multiplied 
 by one-third of the radius. 
 
 Cor. 1. If we represent the volume of a sphere by vol. S, and 
 the surface by sur. S, we will have, 
 
 vol. 8 = sur. SXiRj and since sur. S= 4 n/J 1 
 we have, vol. S=i kB 2 X iB; which, reduced, 
 
154 
 
 GEOMETRY. 
 
 gives, (1) vol. S — i itR'. 
 
 But, B = £ B, or R* = £ B 3 , 
 
 Hence, (2) vol. 8=1*1?. 
 
 Cor. 2. Spheres are to each other as the cubes of tlnsir 
 radii, or diameters. 
 
 Cor. 3. The volume of a spherical sector or pyramid is 
 equal to its base multiplied by one-third of the radius. 
 
 For the sector or pyramid may be conceived as consist- 
 ing of an infinite number of pyramids having their ver- 
 tices at the centre of the sphere, and the volume of the 
 sum of these will be the sum of their bases multiplied 
 by one-third of the radius. 
 
 Cor. 4. The volume of a spherical segment of one base 
 and less than a hemisphere, as that gene- 
 rated by A CB revolving about AF, is equal 
 to the volume of the spherical sector AOC 
 minus the volume of the cone formed by 
 OCB. 
 
 The volume of a spherical segment of 
 one base and greater than a hemisphere, as 
 AEB, is equal to the volume of the spheri- 
 cal sector AOE plus the volume of the 
 cone formed by EBO. 
 
 The volume of a spherical segment of two bases, as that 
 generated by BCED, is equal to the volume of the sector, 
 formed by COE, plus the volume of the cones formed by 
 OCB and OED. If the points C and E fall on the same 
 side of the centre, the last cone must be subtracted. Ths 
 measure is as follows : 
 
 Segment BCED = zone CE X i OC + idffi 1 X i OB + 
 «nE 2 X I OB. 
 
BOOK VII. 
 
 155 
 
 THEOREM XI. 
 
 The surface of a sphere is to the entire surface of the, circum- 
 scribed cylinder, including its bases, as 2 to 3; and their 
 volumes are to each other in the same ratio. 
 
 Let AE be a cylinder circumscribed about a Bphere 
 whose centre is P; then, 
 
 First. The surface of the sphere is 
 to the entire surface of the cylinder 
 as 2 is to 3. 
 
 For, the surface of the cylinder 
 equals circumference AO X OG (Th. 
 I.) ; that is, the circumference of a 
 great circle of the sphere multiplied 
 by the diameter of the sphere ; but 
 this is equal to the surface of a 
 
 sphere (Th. IX.) ; hence, the surface of the cylinder equals 
 the surface of the sphere ; but the surface of the sphere 
 equals four great circles ; hence, the convex surface of the 
 cylinder equals four great circles, and adding the two 
 bases, we have the entire surface of the cylinder equal to 
 six great circles ; hence, the surface of the sphere is to the 
 surface *of the cylinder as 4 great circles is to 6 great 
 circles, or as 4 to 6, or 2 to 3. 
 
 Second. The volume of the sphere is to the volume of 
 the cylinder as 2 is to 3. 
 
 For, the volume of the sphere is | tt-B 3 (Th. X. C. I.), 
 
 and the volume of the cylinder is ttJK 2 X GO (Th. II.), or 
 
 xR* X 2 R = | t-R 3 ; hence, 
 
 or, 
 
 vol. 8. : vol. cyl. . 
 
 : 4 tzR 3 : | wifr 
 
 • 
 
 : 4 : 6, or 2 
 
 Therefore, etc. 
 
 
156 GEOMETRY. 
 
 PRACTICAL EXAMPLES. 
 
 1. Required the convex surface and contents of a cylinder whost 
 altitude is 16 inches, and diameter of the base 8 inches. 
 
 Am. 402.12; 804.25. 
 
 2. Required the convex surface and volume of a cone whose altitude 
 is 24 inches, and radius of the base 10 inches. 
 
 Ana. 816.816; 2513.28. 
 
 3. Required the convex surface of the frustum of a cone whose alti- 
 tude is 36 inches, the radius of the upper base 6 inches, and lower 
 base 21 inches. Am. 3308.1048. 
 
 4. Required the volume of a frustum of a cone whose altitude is 9 
 feet, diameter of lower base 4 feet, and of upper base 2 feet. 
 
 Am. 65.9736. 
 
 5. Required the surface and contents of a sphere whose diameter is 
 16 inches. Am. 804.2496 ; 2144.6656. 
 
 6. The surface of a sphere is 1809.5616 square inches; required its 
 diameter and its volume. Am. D. = 24 inches. 
 
 7. The volume of a sphere is 113.0976 cubic inches; required its 
 diameter and its surface. Am. D. = 6 inches. 
 
 8. Given the volume of a sphere 268.0832 cubic inches; required 
 the altitude of the circumscribing cylinder. Am. 8 inches. 
 
 9. What is the surface of a zone of a single base whose altitude is 
 10 feet, the diameter of the sphere being 100 feet? 
 
 Am. 3141.6 sq. ft. 
 
 10. Required the volume of a. spherical segment of one b«Se whose 
 altitude is 2 feet, the diameter of the sphere being 8 feet. 
 
 Am. 41.888 cubic feet. 
 
 11. Required the volume of a spherical segment whose greater dia- 
 meter is 24 inches, less diameter 20 inches, and distance of bases 4 
 inches. Am. 1566.6112 cubic inches. 
 
 THEOREMS TOR ORIGINAL THOUGHT. 
 
 1. Prove that two great circles of a sphere bisect each other. 
 
 2. Prove that every great circle divides the sphere into two equal 
 parts. 
 
BOOK VII. 157 
 
 3. Prove that the centres of a small circle and the sphere are in a 
 tine perpendicular to the small circle. 
 
 4. Prove that the radius of a small circle is less than the radius of 
 the sphere. 
 
 5. Prove that circles whose planes are equidistant from the centre 
 ire equal. 
 
 6. Prove that the intersection of two spheres is a circle. 
 
 7. Prove that the arc of a great circle may be made to pass through 
 any two points on the surface of a sphere. 
 
 8. Prove that if a cone and sphere he inscribed in a cylinder, that 
 these bodies are to each other as 1, 2, and 3. 
 
 MISCELLANEOUS PROBLEMS. — PLANE FIGURES. 
 
 1. How many bricks 8 inches long and 4 inches wide will it take to 
 pave a yard 20 feet by 16 feet? Ans. 1440. 
 
 2. How much will it cost to plaster a room whose length is 24 ft., width 
 18 ft., and height 12 ft., at 16 cts. a square yard? Arts. $25.60. 
 
 3. What is the difference in area between a rectangle 60 feet by 40 
 feet, and a square which has the same perimeter ? Ans. 100 sq. ft; 
 
 4. What is the diagonal of » square whose area is equal to the area 
 of a rectangle 16 inches by 25 inches ? Ans. 28.28 inches. 
 
 5. The diagonal of a square is y/50 inches ; required the side of the 
 square. Ans. 5 inches. 
 
 6. Required the diagonal of a room whose length is 48 feet, width 
 20 feet, and height 39 feet. Ans. 65 feet. 
 
 7. A vessel' sailed north 20 miles, then west 30 miles, then north 60 
 miles, then west 70 miles ; how far was it then from the point at which 
 it started ? . A™- 128.06 miles. 
 
 8. The gable ends of a house are 48 ft. wide, and the ridge-pole is 10 
 ft. above the eaves ; required the length of the rafters. Ans. 26 ft. 
 
 9. Required the area of an isosceles triangle whose base is 20,feet, 
 and each of its equal sides 15 feet. Ans. 111.803 square feet. 
 
 10. A flag-staff was broken, and fell, the broken part resting upon 
 the upright, so that the end struck 48 feet from the foot ; the upright 
 part measured 36 feet ; how long was the staff ? Ans. 96 feet. 
 
 14 
 
158 GEOMETRY. 
 
 11. I wish to enclose a square rod in the form of an equilateral tri 
 angle ; what must be the length of each side ? Ans. 25.076. 
 
 12. Given the area of a circle 19.635 square inches ; required the 
 diameter and circumference. Ans. D. = 5 inches. 
 
 13. The equal sides of an equilateral triangle are each 16 feet ; what 
 is the side of the inscribed square ? Am. 7A25. 
 
 14. I have a plank 12 feet long which contains 15 square feet ; what 
 is the width of each end, if they are as 2 to 3 ? Ans. 12 in. ; 18 in. 
 
 15. If th,e minute-hand of a clock is 6 inches long, over how much 
 space does it pass in 40 minutes ? Ans. 75.398 square inches. 
 
 16. What is the circumference of a circle whose diameter equals the 
 diagonal of a square which contains 25 sq. rds. ? Ans. 22.211112. 
 
 17. What is the diameter of a wheel which makes 200 revolutions in 
 a minute, when the cars are going 30 miles an hour ? Ans. 4£ -|- feet. 
 
 18. A horse is fastened in a meadow, by a halter 20 feet long, to the 
 top of a post 6 feet high ; what is the area of the circle over which he 
 can graze ? Ans. 127.06 square yards. 
 
 19. Required the area of a circle in which the number expressing 
 its. area equals the number expressing its circumference. 
 
 Ans. 12.5664. 
 
 20. The area of a circular park is 4 acres ; how long will it take to 
 , drive round it at the rate of 6 miles an hour ? Ans. 2 min. 48 sec. 
 
 21. A circular garden containing 2 acres is bordered by a gravel 
 walk of uniform width, which takes up \ of its area; required the 
 width of the walk. Ans. 22.308 feet. 
 
 22. If the hour-hand of a clock is 4 inches long, and the minute- 
 hand 6 inches, what is the difference of the surfaces over which they 
 travel in an hour? Am. 108.91 square inches. 
 
 MISCELLANEOUS PROBLEMS. — VOLUMES. 
 l._ Required the surface, of a brick 8 inches long, 4 inches wide, and 
 2 inches thicks Ans. 112 square inches. 
 
 2. Required the entire surface of a right pyramid whose base is a 
 square 4 in. long, and the slant height 12 inches. Ans. 112 sq. in. 
 
 3. Required the entire surface of a cylinder whose altitude is 16 
 in., the radius of the base being 6 inches. Ans. 829.3824 sq. in. 
 
BOOK VII. 159 
 
 4. Required the entire surface of a cone whose height is 16 feet, the 
 radius of the base being 12 feet. Am. 1206.3744 sq. ft. 
 
 5. Required the surface and contents of a sphere inscribed in a cube 
 whose edge is 20 inches ; and also the space between them. 
 
 6. The surface of a sphere is 6.305 square feet ; required its diameter 
 and volume. Ans. Vol. 1.48868 cubic feet. 
 
 7. The volume of a sphere is 1.2411 cubic feet ; required the diameter 
 •nd surface. Ana. D. 16 inches. 
 
 8. The convex surface of a cylinder whose altitude is 14 feet is 116.666 
 square feet; required the diameter of its base. Ans. 2.65 ft. 
 
 9. What is the volume of a cylinder whose height is 20 feet, and the 
 circumference of the base is 20 feet also 1 Ans. 636.64 feet. 
 
 10. The volume of a cylinder is 15.708 cubic feet; what is the alti- 
 tude, if the diameter of the base is 2 feet 1 Ans. 5 feet. 
 
 11. The convex surface of a cone is 141.372 square feet, and the dia- 
 meter of the base 4.5 feet; required the slant height and altitude. ' 
 
 Ans. 20 feet. 
 
 12. If a segment of 6 feet slant height be cut off of a cone whose 
 slant height is 30 feet, the circumference of the base being 10 feet, 
 what is the surface of the frustum ? Ans. 144 square feet. 
 
 13. The convex surface of a frustum is 376.992 square feet, the slant 
 height 20 feet, and the diameter of the less end 4 feet; what is the 
 diameter of the greater end ? Ans. 8 feet. 
 
 14. The volume of a cone is 8.83575 cubic feet, the altitude 15 feet; 
 what is the diameter of the base ? Ans. 18 inches. 
 
 15. The volume of a frustum of a cone is 65.9736 cubic feet, the 
 diameter of one end is 4 feet and of the other 2 feet; required the 
 altitude. Ans. 9 feet. 
 
 16. Required the entire surface of the frustum of a cone whose height 
 is 12 feet, the radius of the lower base being 9 feet and the upper base 
 4 feet. Ans. 266 jr. 
 
 17. Required the entire surface of the frustum of a pyramid whose 
 bases are squares, the lower 9 feet, the upper 4 feet, on a side, the alti- 
 tude being 12 feet. Ans. 415.68 sq. ft. 
 
 18. How far must a person ascend above the earth that he may see 
 one-third of the surface ? Ans. 2 times the radius. 
 
BOOK VIII. 
 
 SPHERICAL GEOMETRY. 
 
 1. Sphericjal Geometry has for its object the invest 
 tigation-of the properties and relations of those portions of the 
 surface of a sphere which are bounded by arcs of great circles. 
 
 2. A Spherical Angle is an angle included between 
 the arcs of two great circles meeting at a point. The arcs are 
 called sides, and the point at which they meet the vertex of 
 the angle. 
 
 The measure of a spherical angle is the same as that of the 
 diedral angle included between the planes of its sides. Spheri- 
 cal angles may be acute, right, or obtuse. 
 
 3. A Spherical Polygon is a portion of the surface 
 of a sphere bounded by arcs of great circles. These arcs are 
 called the sides of the polygon, and the points in which they 
 meet the vertices. Each arc is supposed to be less than a semi- 
 circumference. 
 
 4. A Spherical Triangle is a spherical polygon of 
 three sides. Spherical triangles are classified in the same man- 
 ner as plane triangles. 
 
 5. A Lune is a portion of the surface of a sphere bounded 
 by two semi-circumferences of great circles. 
 
 8. A Spherical "Wedge is a portion of a sphere bounded 
 oy a lune and the planes of its two sides. 
 
 7. A Spherical Pyramid is a portion of a sphere 
 bounded by a spherical polygon and the circular sectors formed 
 upon the sides of the polygon. The spherical polygon is called 
 the base of the pyramid, and the centre of the sphere is called 
 the vertex. 
 
 160 
 
BOOK VIII. 161 
 
 8. A Pole of a Circle is a point on the surface of the 
 sphere equally distant from all points of the circumference of 
 the circle. 
 
 9. A Diagonal of a spherical polygon is an arc of a great 
 circle joining the vertices of any two angles not consecutive. 
 
 10. The Supplement of an arc is what the arc lacks of 
 Being a semi-circumference. x 
 
 THEOEEM I. 
 Any side of a spherical triangle is less than the sum of the other two. 
 
 Let AB G be a spherical triangle, being the centre of the 
 sphere; then will any side, as AB, be less 
 than the sum of the sides A G and BG. 
 
 For, draw the radii OA, OB, and OG, 
 forming a triedral angle whose vertex is 0; 
 then the plane angle A OB is less than the 
 sum of the angles A OG and BOG (B. V. 
 Th. XII.) ; hence the arc AB, which measures A OB (B. IV. Th. 
 VIII.), is less than AG + BC, which measure AOC+BOG. 
 Therefore, etc. 
 
 THEOEEM II. 
 Any side of a spherical polygon is less than the sum of the oiher 
 
 'Let ABODE be a spherical polygon; then will any side, as 
 iE, be less than the sum of AB, BG, CD, 
 and BE. 
 
 For, draw the diagonals BE and CE, 
 dividing the polygon ABODE into tri- a( 
 angles. The arc AE is less than the sum 
 of AB and EB, EB is less than the sum 
 of BG and EC, and EC is less than the sum of ED and DO 
 
 14* 
 
162 GEOMETKY. 
 
 (Th. I.) ; hence AE is less than the sum of AB, BC, CD, and 
 
 DE. 
 
 Cor. The are of a great circle measures the shortest distance 
 
 between two points on the surface of a sphere. 
 
 For, if wS divide any arc of a small circle joining the two 
 points into equal parts, and through their extremities pass arcs 
 of a great circle, the arc of the great circle joining the twc 
 given points will be less than the sum of these arcs (Th. II.). 
 When the number of arcs becomes infinite, their sum is equal 
 to the arc of the small circle. Therefore, etc. 
 
 THEOKEM III. 
 
 Tlie sum of the sides of a spherical polygon is less than the cir- 
 cumference of a great circle. 
 
 Let ABCDE be a spherical polygon, and the centre of the 
 sphere ; then will the sum of its sides be less 
 than the circumference of a great circle. / 
 
 For, draw the radii OA, OB, OC, OD, and A 
 OE, forming a polyedral angle whose vertex is \ 
 0; then the sum of the plane angles AOE, 
 EOD, DOC, COB, and BOA is less than four 
 right angles (B. V. Th. XIII.) ; hence the sum 
 of the arcs which measure them is less than the circumference 
 of a great circle, which is the measure of four right angles. 
 Therefore, etc 
 
 THEOKEM IV. 
 
 If a diameter of a sphere be drawn perpendicular to the plane of 
 any circle of the sphere, its extremities will be poles of that circle. 
 
 Let CFD be any circle of a sphere, and AB a diameter of 
 the sphere perpendicular to the plane of CFD ; then will A 
 and B be the poles of the circle CFD. 
 
BOOK VIII. 
 
 163 
 
 The diameter AB, being perpendicular to the plan* of CFD, 
 must pass through the centre E, 
 since the diameter CD is a chord 
 of the great circle ADBC, and 
 must therefore be bisected by a 
 diameter perpendicular to it (B. 
 IV. Th. III.). If arcs of great cir- 
 cles A C, AF, and AD be drawn 
 from A to different points of the 
 circumference CFD, the chords of 
 those arcs will be equal (B. V. Th. III.) ; hence the arcs will 
 be equal. But these arcs are the shortest lines that can be 
 drawn from A to the points of the circumference (Th. II. Cor.) ; 
 hence A, being equally distant from all points of the circumfer- 
 ence, is a pole of the circle CFD (Def. 8). It may be proved 
 in like manner that B is also a pole of the circle. Therefore, etc. 
 
 Cor. 1. The poles of a great circle are at equal distances from 
 the circumference ; and those of a small circle are at unequal dis- 
 tances, the sum of the distances being equal to the semi-circumfer- 
 ence of a great circle. 
 
 For, let CHI be a great circle perpendicular to AB; then 
 will the angles A OH, B OH, etc., be right angles, and the arcs 
 AH, BH, etc., will be quadrants (B. V. Th. IX. C). The arc 
 A C is less than a quadrant, and BC is greater than a quadrant, 
 and their sum equals A CB. 
 
 Cor. 2. If any point in the circumference of a great circle is 
 joined with either pole by the arc of a great circle, the latter arc 
 will be perpendicular to the circumference of the given circle. 
 
 For, the line A being perpendicular to the plane CHI, any 
 plane, as A OH, passed through it, will also be perpendicular 
 to the plane CHI; hence the spherical angle AHC is a right 
 angle, and the arc AH is perpendicular to the circle GHI. 
 
164 
 
 GEOMETRY. 
 
 Cor. 3. A point on the surface of a sphere at the distance of a 
 quadrant from two points in the arc of a great circle, not at the 
 extremities of a diameter, is a pole of the arc. 
 
 For, since the arcs AG and AH are quadrants, the angles 
 AOG and A OH are right angles, and the line OA is perpen- 
 dicular to the straight lines G and OH; hence OA is a radius 
 perpendicular to the plane GHI (B. V. Th. IV.), and the 
 point A is therefore a pole of the arc HG. 
 
 /Scholium. By means of poles we may with facility describe 
 arcs of a circle on the surface of a sphere. For, revolving 
 the arc A C around the pole A, the extremity C will describe 
 the small circle CFD ; and by revolving the quadrant AG 
 around the pole, the extremity G will describe the great circle 
 'GHI. 
 
 THEOREM V. 
 
 The angle formed by the intersection of two arcs of great circles 
 is equal to the angle included between the tangents to these arcs 
 at the point of intersection, and is measured by the arc of a 
 great circle described from the vertex as a pole, and limited by 
 the sides, produced if necessary. 
 
 Let the angle BA G be formed by the intersection of the two 
 arcs AB and A G; then it is equal to the 
 angle HA G formed by the tangents AH 
 and A G, and is measured by the arc ED 
 of a great circle described from A as a 
 pole. 
 
 For, the tangent A G, drawn in the plane 
 of the arc AC, is perpendicular to the 
 radius AO; and the tangent AH, drawn 
 in the plane of the arc. AB, is also perpen- 
 dicular to the radius A 0; hence the angle 
 
BOOK VIII. 165 
 
 SAG is equal to the angle formed by the planes ABDF and 
 A CEF (B. V. Def. 6), which is the angle formed by the arcs 
 AB and A C. Now, if the arcs AD and AE are both quad- 
 rants, the lines OD and OE are perpendicular to A 0, and the 
 angle DOE is equal to the angle of the planes ABDF and 
 A CEF; hence the arc DE, which is the measure of D OE, is 
 also the measure of B A C. 
 
 Cor. The opposite or vertical angles formed by two arcs of 
 great circles intersecting each other are equal, and the sum of 
 any two adjacent angles is equal to two right angles. 
 
 Scholium,. The angles of spherical triangles are compared by 
 means of the arcs of great circles. described from their vertices 
 as poles and included between their sides. Thus a spherical 
 angle can always be constructed equal to a given angle. 
 
 THEOREM VI. 
 If from the vertices of the angles of a spherical triangle, as poles, 
 
 ares be described forming a spherical triangle, the vertices of 
 
 the angles of the second triangle will be respectively poles of the 
 
 sides of the first. 
 
 From the vertices A, B, C as poles describe the arcs DE, 
 EF, FD, forming the triangle EFD; then 
 will the vertices D, E, F be respectively 
 poles of the sides BC, AC, AB. 
 
 For, since the point A is the pole of the 
 arc EF, the distance AE is a quadrant ; 
 and since the point C is the pole of the arc 
 DE, the distance CE is a quadrant ; hence the point E is at 
 a quadrant's distance from A and C, and therefore it is the 
 pole of the arc AC (Th. IV. Cor. 3). In the same manner it 
 may be shown that D is the pole of CB, and F the pole of AB. 
 Therefore, etc. 
 
166 GEOMETKY. 
 
 Scholium. The triangle ABC may be described when DEF 
 
 is given, as DEF is described when ABC is given. Triangles 
 
 thus related are called polar triangles or supplemental triangles. 
 
 Since great circles intersect each other in 
 
 , , . , . e\- -.. d ,--- if 
 
 two points, three other triangles may be > % y ■ 
 
 formed by producing the arcs DE, FE '''-•{- \<'' 
 
 zndFD; but the central triangle only is > i 
 
 taken as a polar triangle, being the only \ / 
 
 one in which the vertices A and D are on «* 
 
 the same side of BC, the vertices B and E on the same side of 
 
 AC, and the vertices Cand F on the same side of AB. 
 
 THEOKEM VII. 
 
 In two polar triangles, any angle of one triangle is measured by 
 the supplement of the side lying opposite to it in the other. 
 
 Let AB G and DEF be two polar triangles ; then will any 
 angle of either triangle be measured by 
 the supplement of the side lying opposite 
 to it in the other. 
 
 For, produce AB and A C, if necessary, 
 till they meet EF in H and G. Since A 
 is the pole of EF, the angle A is meas- -EL 
 ured by the arc GH (Th. V.). But since 
 E is the pole of AG, the arc EG is a quadrant ; and since F 
 is the pole of AH, the arc FHia a quadrant, and the sum of 
 the arcs EG and FH is a semi-circumference. But EG + FH 
 = EF + GH; hence the arc GH, the measure of the angle 
 A, is equal to a semi-circumference minus the arc EF. In the 
 tame manner it may be proved that the measure of any other 
 angle in either triangle is the supplement of the side lying 
 opposite to it in the other. 
 
BOOK Vlir. 167 
 
 THEOREM VIII. 
 
 The sum of the angles of a spherical triangle is less than six right 
 angles and greater than two. 
 
 For, any angle, being measured by the supplement of the side 
 lying opposite to it hi -the polar triangle (Th. VII.), is less than 
 two right angles ; hence the sum of the three angles is lesa 
 than six right angles. Also, the measure of the sum of tho 
 three angles is equal to three semi-circumferences minus the 
 three sides of the polar triangle. But the three sides of a tri- 
 angle are less than a circumference (Th. III.) ; hence the meas- 
 ure of the sum of the three angles is greater than a semi-cir- 
 cumference, and the sum of the angles is greater than two right 
 angles.. Therefore, etc. 
 
 Cor. 1. A spherical triangle may have two, or even three, 
 right angles j'also two, or even three, obtuse angles. 
 
 Cor. 2. If the triangle ABC has two right angles, it is called 
 bi-rectangular. Since the arcs AB and A C are 
 perpendicular to BC, they must both pass through 
 the pole of BC; hence their point of intersection, 
 A, is the pole, and the arcs AB and A C are quad- 
 rants. 
 
 If the angle A is also a right angle, the triangle is tri-reet- 
 angular, the sides being quadrants. Four tri-rectangular tri- 
 angles make up the surface of a hemisphere, and eight that of 
 a sphere. 
 
 Scholium. The sum of the three angles of a spherical triangle 
 is not constant, like that of the angles of a plane triangle, but 
 varies between two right angles and six without ever reaching 
 either limit. Two angles, therefore, being given, do not serve 
 to determine the third. The excess of the sum of the angles 
 over two right angles is called the spherical excess, and taking 
 
.168 GEOMETBY. 
 
 the right angle as the unit may be represented thus : A -f- B 
 + C — 2 = spherical excess. 
 
 THEOREM IX. 
 
 Two spherical triangles on the same sphere or on equal spheres 
 are equal — 
 
 1. When two sides and the included angle of the one are equal 
 to two sides and the included angle of the other. 
 
 2. When two angles and the included side of the one are equal 
 to two angles and the included side of the other. 
 
 3. When the three sides are respectively equal. 
 
 The three cases of this theorem may be demonstrated, as in 
 plane triangles, by applying one of the given triangles to the 
 tther or to its symmetrical triangle. 
 
 Scholium. The symmetrical triangle is formed thus : 
 
 Let AB, BQ and AC be three arcs of great circles, from 
 which either of the two triangles ABC, ABC 
 may be formed. These two triangles, although 
 all their parts are equal, are not capable of su- 
 perposition, because in inverting one in order to 
 bring the corresponding parts together the con- 
 vex surfaces would be turned toward each other, 
 angles are called symmetrical triangles. 
 
 Cor. The circles passed through the vertices of two mutually 
 equilateral triangles on the same sphere or on equal spheres are 
 equal. 
 
 For, the plane triangles formed by the chords of the sides 
 of these spherical triangles must be equal ; hence, if the 
 spherical triangles are applied 'to each other, the vertices of 
 the spherical triangles will coincide, and the circles passing 
 through the vertices are equal. 
 
BOOK VIII. 169 
 
 THEOEEM X. 
 Two symmetrical spherical triangles are equal in area. 
 
 Let ABO and DEF be two symmetrical triangles, jn which 
 AB equals DF, A C equals DE, and CB a d 
 
 equals EF.. Then will the area of the /f\ JV\ 
 
 two triangles be equal. / \ / | \ 
 
 For, let G be the pole of the small \^~t/ \\ J 
 
 circle passing through the points A, B, B p 
 
 and C, and IT the pole of the circle passing through D, E, and 
 F; these circles will be equal (Th. IX., C). Join A, B, and C 
 with G, and D, E, and F with 27, by arcs of great circles ; 
 these arcs will all be equal, since -they measure the distances 
 from the circumferences of equal circles to their poles. The 
 triangles ACG and DEH, being isosceles and having equal 
 sides, may be applied to each other, and are equal in area ; so 
 also CB G is equal to EFH, and AB G to DFH. Hence A CG 
 + CBG — ABG = DEE + EFH— DFH, or, reducing, 
 ABG=: DEF. Therefore, .etc. 
 
 Scholium. If the point G fall within the triangle A CB the 
 point H will also fall within the triangle DEF, and the areas 
 of the triangles will equal the sum of the three isosceles tri- 
 angles. 
 
 THEOEEM XL 
 
 If two triangles on the same, or on equal spheres, are mutually 
 equiangular, they are also mutually equilateral. 
 
 Since the two given triangles are mutually equiangular, their 
 polar triangles must be mutually equilateral (Th. VII.), and 
 consequently mutually equiangular (Th. IX.). But if these 
 polar triangles are mutually equiangular, the given triangles 
 are mutually equilateral (Th. VII). 
 
 15 
 
170 
 
 GJEOMETKY. 
 
 Scholium. This proposition does not extend to plane triangles, 
 for similar plane triangles are not necessarily mutually equi- 
 lateral. But two spherical triangles on the same or equal 
 spheres cannot be similar without being equal. 
 
 r T 
 
 
 a \ 
 
 yK 
 
 £ 
 
 
 THEOEEM XII. 
 
 The surface of a lune is to the surfaee of a sphere as the angle of 
 the lune to four right angles, or as the are which measures that 
 angle is to the circumference of a great circle. 
 
 Let ABGDA be a lune on the surface of a spherej and BD 
 the arc of a great circle whose poles A 
 
 are A and G, the vertices of the 
 angles of the lune ; then will the sur- 
 face of the lune be to the surface of 
 
 ■Bl* 
 
 the sphere as the arc BD to the cir- 
 cumference BDEG. 
 
 For, if we divide the arc BD and 
 tfce circumference BDEG into equal 
 parts, BF being one of those parts, and* pass planes through 
 the diameter AG and each of the points of division, the surface 
 of the sphere will evidently be divided into equal lunes, of 
 which the given lune will contain as many as there are parts 
 in the arc-BD; hence, the lune ABGDA is to the whole sur- 
 face of the sphere as the arc BD is to the circumference BDEG ; 
 and the same may also be shown when the arc BD and the cir- 
 cumference are incommensurable. But BD is the measure- of 
 the angle of the lune, and the circumference is the measure of 
 four right angles. Therefore, etc. 
 
 Cor. 1. Lunes on the same sphere or on equal spheres are to 
 each other as their angles. 
 
 Ppr. 2. Taking the right angle as the unit of angles, and 
 
BOOK VIII. 171 
 
 denoting the angle of a lune by A, the area by L, and the 
 surface of a tri-rectangular triangle by T, we have 
 
 L:8T::A:4; 
 whence, L — T X 2 A. 
 
 Therefore the area of a lune is equal to the tri-rectangular 
 'riangle multiplied by twice the angle of the lune. 
 
 Cor. 3. A spherical wedge bears the same relation to the 
 entire sphere as the angle of the wedge to four right angles, as 
 may be shown by a similar course of reasoning to that em- 
 ployed in the theorem ; hence the volume of the wedge is equal 
 to the lune which forms Us base, multiplied by one-third of the 
 "adius. 
 
 THEOEEM XIII. 
 
 If two circumferences of great circles intersect on the surface of 
 o. hemisphere, the sum of either two of the opposite triangles 
 thus formed is equal to a lune whose angle is equal to that 
 formed by the circles. 
 
 Let the circumferences AEBF, CEDE intersect on the sur- 
 face of a hemisphere; then will the •» 
 sum of the opposite triangles A EG, S/f \\ 
 DEB be equal to the lune whose / / \ \ 
 
 / / \D \ 
 
 angle is AEG. v /---'/" ~"V, \ 
 
 For, since AEB and EBF are semi- \H^_ ^Tj 
 
 circumferences, if we take away the \ \ / / 
 
 common part EB we have AE equal \\< y'\/ 
 
 to BF. In the same way, we find f 
 
 CE equal to JDF and BD equal to A C; hence the two triangles 
 AEC and BFD, having their sides equal, must be symmetrical, 
 and therefore equal in area (Th. X.). But the sum of the tri- 
 angles DEB and BFD is equal to the lune EDFBE, whose 
 
172 GEOMETKY. 
 
 angle is AEC; hence the sum of AEC and DEB equals th« 
 lune whose angle is AEG. Therefore, etc. 
 
 Scholium. It is evident that the two spherical pyramids which 
 have the triangles A CE and BED for bases are together equal 
 to the spherical wedge, whose angle is AEC. 
 
 THEOEEM XIV. 
 
 The area of a spherical triangle is equal to its spherieal excesi 
 multiplied by the trvredangular triangle. 
 
 Let ABC be a spherical triangle; then, representing the 
 tri-rectangular triangle by T, the sur- 
 face of the given triangle will be equal 
 to (4 +B+ 0—2) X T. 
 
 For, produce the sides until they 
 meet the circumference of a great 
 circle, drawn without the triangle, 
 forming three sets of opposite tri- 
 angles. By the last theorem the area 
 of each of these sets is equal to the lune whose angle is the cor- 
 responding angle of the triangle. Hence (Th. XII., Cor. 2), 
 ADF + AHI = 2 A X T. 
 BEI + BFG = 2B X T. 
 CGH + CDE = 2 C X E 
 But the sum of these six triangles exceeds the hemisphere, or 
 4 T, by twice the triangle ABC; hence, by adding the equation: 
 and substituting in the first member its value, we have 
 
 4T+2ABC=2AX T + 2B X T + 2 C X T; 
 reducing, ABC = (A + B + C— 2) X T. 
 
 But A -\- B -\- C — 2 is the spherieal excess of the triangle 
 (Th. VIII. Seh.), and T is the tri-rectangular triangle. There- 
 fore, etc. 
 
BOOK VIII. 173 
 
 THEOEEM XV. 
 
 The area of a spherical polygon is equal to its spherical excess 
 multiplied by the tri-rectangular triangle. 
 
 Let ABODE be a spherical polygon. If we draw the diag» 
 aals A C and AD, the polygon will be divided 
 ;nto as many triangles as there are sides, less 
 two. Now, the area of each triangle is equal 
 to the sum of its angles minus two right 
 angles, multiplied by the tri-rectangular tri- 
 angle ; and the area of the polygon, or the 
 sum of all the triangles, is equal to the sum of all the angles of 
 the triangles, or the sum of the angles of the polygon, dimin- 
 ished by two right angles, taken as many times as the polygon 
 has sides, less two, and the difference multiplied by the tri- 
 rectangular triangle ; which is the spherical excess of the polygon 
 multiplied by the tri-rectangular triangle. Therefore, etc. 
 
 Cor. If we represent the sum of the angles by S, and the 
 number of the sides by n, we shall have 
 
 the area of a polygon = [S — 2 (n — 2)] X T; and re- 
 ducing have area of a polygon = (S — 2 n + 4) X T. 
 
 PRACTICAL EXAMPLES. 
 
 L Find the area of a spherical triangle each of whose angles is 70°. 
 
 Ans. \ 7r E 2 . 
 
 2. Find the area of a spherical polygon of six sides each of whose 
 Migles is 150°. Ans. it E 2 . 
 
 3. Given the spherical triangle whose angles are respectively 80°, 90°, 
 and 140°, to find the sides of its polar triangle. 
 
 Ans. 100°; 90°; 40°. 
 
 4. If the sides of a triangle" are respectively 75°, 110°, and 130°, what 
 
 «re the angles of its polar triangle ? Ans. 105° ; 70° ; 50°. 
 
 15* 
 
174 GEOMETRY. 
 
 5. "What is the area of a bi-rectangular triangle whose vertical angla ' 
 is 108° ? . " Ans. | ■k W. 
 
 6. Find the area of a spherical triangle whose angles are 60°, 90°, and 
 120°, the diameter of the sphere being 8. Arts. 8 t. 
 
 7. Find the area of a lune, its angle being 45°. Am. J ir R 2 . 
 
 8. Find the area of a lune, the angle being 54° and radius of the 
 sphere being 5. Ans. 15 ir. 
 
 9. Find the volume of a spherical wedge, the angle of the lune being 
 72°. Aiis. Jg tr R s . 
 
 10. Find the volume of a spherical wedge, the angle of the lune being 
 36° and the diameter of the sphere 10. Ans. lGf t. 
 
 11. Find the angles of an equilateral spherical triangle whose area is 
 equal to the surface of a great circle. Ans. 120°. 
 
 12. What must be the angles of an equilateral spherical triangle that 
 its area may be equal to an equilateral spherical hexagon, each of whose 
 angles is 130° ? Am. 80°. 
 
 THEOREMS FOE ORIGINAL THOUGHT. 
 
 1. Prove that in an isosceles spherical triangle the angles opposite 
 the equal sides are equal. 
 
 2. Prove that a spherical triangle having two equal angles is isopceles. 
 
 3. In any spherical triangle the greater side is opposite the greater 
 angle, and conversely. 
 
 4. If from any point of a hemisphere two arcs of great circles are 
 drawn perpendicular to its circumference, the' shorter of the two arcs is 
 the shortest arc that can be drawn from the given point to the circum- 
 ference. 
 
 5. Two oblique arcs drawn from the same point to points of the cir- 
 cumference at equal distances from the foot of the perpendicular are 
 equal. 
 
 6. Of two oblique arcs, that which meets the circumference at the 
 greatest distance from the foot of the perpendicular is the longer. 
 
 7. Prove that the area of a spherical triangle, each of whose angle* 
 is | it a right angle, is equal to the surface of a great circle. 
 
MENSURATION. 
 
 MENSURATION OF LENGTHS AND SURFACES. 
 
 1. Mensuration is the science which treats of the 
 measurement of geometrical magnitudes. 
 
 2. The Area of a figure is its quantity of surface; it is 
 expressed by the number of times which it contains the 
 unit of measure. 
 
 3. This Unit of Measure is a square whose side is some 
 known length ; as, an inch, a foot, etc. 
 
 4. The unit of surface has generally the same name as 
 the linear unit ; thus, if the linear unit is one foot, the sur- 
 face unit is one square foot, etc. 
 
 5. Some superficial units have no corresponding linear 
 unit of the same name ; as, the rood and acre. 
 
 6. To refresh the memory,, we give a few of the more 
 important measures of surfaces. 
 
 1 rood = 40 perches, or square rods. 
 
 1 acre = 4 roods. 
 
 1 square mile = 640 acres. 
 Also, 
 
 1 chain = 100 links = 4 rods. 
 10 chains = 1 furlong. 
 1 square chain — 100 X 100 = 10 s 000 square links. 
 1 acre = 10 square chains = 100,000 square links. 
 
 175 
 
176 MENSURATION. 
 
 THE TRIANGLE. 
 
 7. The area is found by the following rules : 
 
 Eule 1. — Multiply the base by one-half of the altitude) or, 
 Eule 2. — Take half the sum of the sides, subtract from it 
 each side separately, multiply the half sum and these remain- 
 ders together, and take the square root of the product. 
 
 1. What is the area of a triangular field whose base is 
 40 rods and altitude 16 rods ? Ans. 2 acres. 
 
 2. Bequired the area of a triangle whose sides are 20, 30, 
 and 40 chains respectively. Ans. 29 A. 8 P. 
 
 3. A man has a triangular garden whose sides are 150, 
 200, and 250 feet respectively ; required the area. 
 
 Ans. 1666.66 yards. 
 
 THE QUADRILATERAL. 
 
 8. Parallelogram. — The area is found as follows : 
 Eule. — Multiply the base by tht altitude. 
 
 1. What is the area of a parallelogram 9 feet long and 7 
 feet wide ? Ans. 63 square feet. 
 
 2. How many acres in a square field whose side is 70£ 
 chains ? Ans. 497 A. 4 P. 
 
 3. A man has a lot in the form of a rhombus, whose 
 length is 333 feet and altitude 33.35 feet; required its 
 area. Ans. 1233.95 square yards. 
 
 9. Trapezoid. — The area is found as follows : 
 
 Eule. — Multiply one-half of the sum of the parallel sides 
 by the altitude. 
 
 1. Eequired the area of a trapezoid, one side being 192 
 inches and the other 96 inches, and altitude 12 feet. 
 
 Ans. 144 square feet. 
 
GEOMETRY. 177- 
 
 2. What is the area of a plank 24 feet long, 18 inches 
 wide at one end and 12 inches at the other ? 
 
 Ans. 30 square feet. 
 
 3. A farmer has a field in the form of a trapezoid, whose 
 parallel sides are 95 and 75 rods respectively, and the per- 
 pendicular distance between them 65 rods; how much land 
 in the field ? Ana. 34 A. 2 E. 5 P. 
 
 10. Teapezium. — The area is found as follows : 
 
 Rule. — Divide the trapezium into two triangles by a diago- 
 nal, find the area of each triangle, and take their sum. 
 
 1. What is the area of a trapezium whose diagonal is 
 290 inches, and the altitudes of the triangles, the diagonal 
 being the base, are 60 and 80- inches respectively ? 
 
 Ans. 140 square feet, 140 square inches. 
 
 2. Required the area of a trapezium the lengths of whose 
 sides are respectively 40, 60, 50, and 70 chains, and the 
 diagonal 80 chains. Ans. 289 A. 1 R. 24 P. 
 
 POLYGONS OP ANY NUMBER OF SIDES. 
 
 11. Regular Polygons. — The area is found as follows: 
 Rule. — Multiply half the perimeter by the perpendicular let 
 
 fall from the centre on one of the sides. 
 
 1. What is the area of a regular hexagon whose side is 
 14.6 feet and perpendicular 12.64 feet? 
 
 Ans. 61.5147 square yards. 
 
 2. Required the area of an octagon whose sides are 9.941 
 feet and its perpendicular 12 feet. 
 
 Ans. 477.168 square feet. 
 
 12. The following table shows the areas of ten regular 
 polygons when the side is 1 : — 
 
178 MENSURATION. 
 
 Triangle 0.4330127 
 
 Square 1.0000000 
 
 Pentagon 1.7204774 
 
 Hexagon 2.5980762 
 
 Heptagon 3.6339124 
 
 Octagon 4.8284271 
 
 Nonagon 6.1818242 
 
 Decagon 7.6942088 
 
 Undecagon 9.3656404 
 
 Dodecagon 11.1961524 
 
 Now, since the areas of similar polygons are to each 
 other as the squares of their homologous sides, to find the 
 area of a regular polygon we have the following 
 
 Eule. — Square the side of the polygon, and multiply by the 
 tabular area set opposite the polygon. 
 
 3. What is the area of a regular hexagon whose side is 
 5 inches long ? Ans. 64.9519 square inches. 
 
 4. Eequired the area of an octagon whose sides are each 
 3 feet 4 inches. Ans. 53.649 square feet. 
 
 13. Irregular Polygon. — The area is found as follows : 
 Rule. — Draw diagonals dividing the polygon into triangles, 
 
 find the area of these triangles, and take the sum. 
 
 1. In the irregular pentagon ABCDE, the diagonal AG 
 is 24 inches, the diagonal AD is 18 inches, the altitude of 
 the triangle ABC is 8 inches, of ACD is 10 inches, and of 
 AED 6 inches ; required the area. 
 
 Ans. 240 square feet. 
 
 2. In the irregular hexagon ABCDEF, the side AB is 
 268, BG 249, CD 310, DE 290, EF199, and AF 24S links, 
 and the diagonals AC 459, CE 524, and AE 326 links; 
 required the area. Ans. 1 A. 2 E. 22 P. 13 yd. 47 ft. 
 
 THE CIRCLE. 
 
 14. The circumference is found by the following 
 Eule. — Multiply the diameter by 3.1416. 
 
 Note. — Hence, the diameter equals the circumference divided by 
 8.1416, or multiplied by .31831. 
 
GEOMETRY. 179 
 
 1. What is the circumference of a circle whose diameter 
 is 50 inches ? Ans. 157.08 inches. 
 
 2. A man has a circular fish-pond 32 rods in diameter ; 
 what is the distance around it? Ans. 100.5312 rods. 
 
 3. Eequired the diameter of a water-wheel whose cir- 
 cumference is 78.54 feet. Ans. 25 feet. 
 
 4. A man has a garden in the form of a circle, the dia. 
 meter being 45 rods ; what is the distance around it ? 
 
 Ans. 141.372 rods. 
 
 15. The length of an arc, when its degrees and radiun 
 are given, is found as follows : 
 
 Eule. — Multiply the number of degrees by the decimal 
 .01745, and the product by the radiui. 
 
 1. The degrees in an arc are 45, and the radius 10; 
 what is the length of the arc ? Ans. 7.852. 
 
 2. What is the length of an arc of 32° 38' 42", the radius 
 being 25 inches? Ans. 14.2414 inches. 
 
 16. When the chord and chord of the half arc are given. 
 Eule. — From 8 times the chord of half the arc, subtract tht 
 
 shord of the whole arc, and divide the remainder by 3. 
 
 1. The chord of an arc is 96 inches, and the chord of 
 half the arc is 60 inches ; what is the length of the arc ? 
 
 Ans. 128 inches. 
 
 2. The chord of an arc is 16 inches, and the diametei 
 of the circle is 20 inches; what is the length of the 
 arc ? Ans. 18.5178 inehes. 
 
 17. The area of A circle is found .as follows : 
 
 Eule I. — Multiply the circumference by one-fourth of the 
 diameter, or the square of. the radius by 3.1416. 
 
 Eule II. — Multiply the square of the diameter by .7854, or 
 the square of the circumference by .07958. 
 
 (Let the pupil prove the last rule from the previous principles.) 
 
180 MENSURATION. 
 
 1. What is the area of a circle whose diameter is 50 
 inches and circumference 157.08 inches? 
 
 Ans. 1963^ square inches. 
 
 2. Eequired the area of a circle whose diameter is 18 
 inches. Ans. 254.4696 square inches. 
 
 3. "What is the area of a circular garden whose circum 
 ference is 90 rods ? Ans. 644.598 square rods. 
 
 18. The area op a sector is found as follows : 
 Eule. — I. Multiply the arc by one-half the radius; or, 
 
 II. The sector is to the circle as the number of degrees in 
 the sector is to 360°- 
 
 1. What is the area of a circular sector whose arc con- 
 ' tains 18°, the diameter of the circle being 6 feet ? 
 
 Ans. 1.4137 square feet. 
 
 2. Eequired the area of a sector, the chord of half the 
 arc being 30 inches, and the radius 50 inches. 
 
 Ans. 1523.45 square inches. 
 
 19. The area or A segment is found as follows : 
 Eule. — Find the area of the sector having the same arc, 
 
 and also the area of the triangle formed by the chord of the 
 segment and the radii of the sector. 
 
 If the segment is greater than a semicircle, add the two 
 areas ; if less, subtract them. 
 
 1. Eequired the area of a segment whose height is 2 
 inches, and chord 20 inches. Ans. 26.864 square inches. 
 
 2. What is the area of a segment whose height is 18 
 " inches, the diameter df the circle being 50 inches ? 
 
 Ans. 632 sq. in. 
 
 3. Eequired the area of a segment whose arc is 1S0°, and 
 radius of circle 12 feet. Ans. 226.1952 
 
 20. The area op A circular ring is foand as follows: 
 
GEOMETRY. 181 
 
 Btjle. — Find the difference of the squares of the radii, and 
 multiply it by 3.1416. 
 
 Demonstration. — Let the figure represent two circles having a com. 
 mon centre 0; then the difference between them 
 will be a circular ring. The area of circle OA 
 is ■kOA\ and of OB is nOB 1 ; the difference is 
 nOA* — n-OB* = k ( OA* — OB*), which proves 
 the rule. 
 
 1. What is the area of the circular 
 ring when the diameters are 20 and 30 ? 
 
 Ans. 392.70. 
 
 2. A circular park 400 feet in diameter has a carriage- 
 way around it 24 feet wide; required the area of the 
 carriage-way. Ans. 3149.9776 square yards. 
 
 21. The side of an inscribed square is found thus : 
 Eule. — Multiply the diameter by .7071, or multiply the cir- 
 cumference by .2251. 
 
 1. What is the side of a square that can he cut out of a 
 circular board whose diameter is 14 inches ? 
 
 Ans. 9.899 inches. 
 
 2. How large a square can be cut out of a circular 
 board whose circumference is 400 inches ? 
 
 Ans. 90.04 inches. 
 
 THE ELLIPSE. 
 
 22. An Ellipse is a plane figure bounded by a curve, 
 the sum of the distances from every point of which to 
 two fixed points is equal to the line drawn through those 
 points and terminated by the curve. 
 
 The two points are called foci; the line through the 
 foci is the transverse axis; a line perpendicular to this 
 through the centre is the conjugate axis. 
 
 16 
 
182 
 
 MENSURATION. 
 
 23. The area is found by the follow- 
 ing 
 
 Rule. — Multiply half of the two axes ■& (— -* 
 together, and multiply that product by 
 8.1416. 
 
 1. "What is the area of an ellipse whose transverse axis) 
 is 20 inches and conjugate axis 16 inches? 
 
 Ans. 251.328. 
 
 2. Required the area of an elliptical mirror whose 
 length is 6 feet and breadth 5 feet. 
 
 Ans. 23.562 square feet. 
 
 MENSURATION OF VOLUMES. 
 
 24. Mensuration or Volumes is the process of deter- 
 mining their surface and contents. 
 
 25. The Contents of a volume is the number of times it 
 contains a given unit of measure. 
 
 26. The Unit or Measure of a volume is a smal 1 cube 
 whose dimensions are known. 
 
 MEASURES OP VOLUMES. 
 
 1 cubic foot = 1728 
 
 cubic inches. 
 
 1 " yard = 27 
 
 CI 
 
 feet. 
 
 1 " rod = 4492^ 
 
 fc 
 
 feet. 
 
 1 wine gallon = 231 
 
 « 
 
 inches. 
 
 1 ale gallon = 282 
 
 11 
 
 inches. 
 
 1 bushel = 2150.42 
 
 tc 
 
 inches. 
 
 1 cord = 128 
 
 CC 
 
 feet. 
 
GEOMETRY. 183 
 
 THE PRISM. 
 
 27. The convex surface or a right prism is found thus: 
 Eule. — Multiply the perimeter of the base by the altitude. 
 To find the entire surface, we add the bases. 
 
 1. What is the convex surface of a triangular prism, the 
 three sides of whose base are respectively 6, 7, and 8 
 inches, and the height 50 inches ? 
 
 Ans. 1050 square inches. 
 
 2. "What is the entire surface of a cube, the length of 
 each side being 16 inches ? Ans, 10| square feet. 
 
 3. What is the entire surface of the triangular prism 
 given in the first problem? Ans. .1090.66 square inches. 
 
 28. The contents oe. a prism are found thus : 
 
 Eule. — Multiply the area of the base by the altitude of the 
 prism. 
 
 1. Eequired the contents of a cube whose sides are 30 
 inches. Ans. 15.625 cubic feet. 
 
 2. Eequired the contents of a square prism whose alti- 
 tude is 27 feet, and the side of the base 4 feet ? 
 
 Ans. 432 cubic feet. 
 
 3. Eequired the contents of a triangular prism whose 
 altitude is 24 feet, the sides of the base being 3, 4, and 5 
 feet respectively. Ans. 144 cubic feet. 
 
 THE PYRAMID. 
 
 29. The convex, surface of a right pyramid is found 
 f.hus : 
 
 Eule. — Multiply the perimeter of the base by one-half the 
 slant height. 
 
 1. What is the convex surface of a triangular pyramid 
 whose sides are 3, 4, and 5 feet, and slant hefght 20 feet? 
 
 Ans. 120 square feet. 
 
184 MENSURATION. 
 
 2. Eequired the convex surface of a pentangular pyra- 
 mid whose sides are each 5 feet, and slant height 60 feet. 
 
 Ans. 750 square feet. 
 
 30. The contents or A pyramid are found thus : 
 Eule. — Multiply the base by one-third of the altitude. 
 
 1. Eequired the contents of a pyramid whose base is a 
 hexagon, each side being 5 feet, and whose altitude is 20 
 feet. Ans. 433.013; 
 
 2. The pyramid of Cheops is 480 feet high, and the base 
 is a square 763.4 feet on a side ; required its solid contents. 
 
 Ans. 93244729| cubic feet. 
 
 THE CYLINDER. 
 
 31. The convex surface and contents are found thus : 
 Eule 1. — The surface equals the circumference of the base 
 
 multiplied by the altitude. 
 
 Eule 2. — The contents equal the area of the base multiplied 
 by the altitude. 
 
 1. What is the convex surface of a cylinder 12 feet loDg 
 and 6 feet in diameter? Ans. 226.1952 square feet. 
 
 2. Eequired the convex surface of a cylinder whose length 
 is 20 feet and the diameter of the base 8 feet. 
 
 Ans. 502.656 square feet. 
 
 3. A man has a log 12 feet long and about 6f feet in 
 diameter ; required its contents. Ans. 418.88 cubic feet. 
 
 4. The "Winchester bushel is a cylinder containing 2150.42 
 cubic inches, its height being 8 inches; what is its dia- 
 meter? Ans. 18£ inches. 
 
 THE CONE. 
 
 32. The convex surface and contents are found thus : 
 Eule 1. — The surface equals the circumference of the base 
 
 into one-half of the slant height. 
 
GEOMETRY. 185 
 
 -Rule 2. — The contents equal the area of the base into one* 
 third of the altitude. 
 
 1. Find the convex surface and contents of a cone, the 
 diameter of the base being 6 ft. and altitude 4 ft. 
 
 Ans. Sur. = 47.124. 
 
 2. Find the surface and contents of a cone whose slant 
 neight is 26 in. and radius of the base 10 in. 
 
 Ans. Yol. = 2513.28. 
 
 THE FRUSTUM OF A PYRAMID AND CONE." 
 
 33. The convex surface is found by the following 
 Eule. — Find the sum of the perimeters or circumferences of 
 
 the two bases, and multiply it by one-half of the slant height. 
 
 1. Eequired the convex surface of the frustum of a square 
 pyramid whose slant height is 24 ft., the side of the 'lower 
 base 12 ft., and of the upper base 8 ft. Ans. 960 sq. ft. 
 
 2. Eequired the surface of a frustum of a cone whose 
 slant height is 20 ft., the diameter of the lower base being 
 12 ft., and of the upper base 8 ft. Ans. 628.32 sq. ft. 
 
 34. The contents of a frustum are found as follows : 
 Etji.e. — Find the sum of the two bases and the square root 
 
 of their product, and multiply this sum by one-third of the alti- 
 tude of the frustum. 
 Note. — In a frustum of a cone the following formula giveB a shorter 
 
 rule :— V= f (R 1 + J- 2 + R . r) X *■ 
 3 
 
 1. "What is the amount of timber in a log which mea- 
 sures 40 feet in length, the radius of one base being 6 feet 
 and of the other 3 feet 1 Ans. 2638.944 cubic feet. 
 
 2. Eequired the contents of the. frustum of a regular 
 hexagonal pyramid, the side of the greater end being 3 
 feet, that of the less 2 feet, the height being 24 feet. 
 
 Ans. 394.9075 cubic feet. 
 
 16* 
 
186 MENSURATION. 
 
 3. A cask, consisting of two equal conic frustums joined 
 at their larger ends, has its bung diameter 30 inches, and 
 its head diameter 20 inches; how many gallon of wipe 
 will it hold if 3£ feet long ? Ans. 90.44 gallons. 
 
 THE SPHERE. 
 
 35. The surface op a sphere is found as follows : 
 Eule. — Multiply the diameter by the circumference; or, 
 
 Square the radius, and multiply it by 4 and 3.1416. 
 
 1. Eequired the surface of a sphere whose diameter is 
 17 inches. Ans. 6.305 square feet. 
 
 2. How many square miles on the surface of the earth, 
 the diameter being about 7-912 miles ? 
 
 Ans. 196,663,355 square miles. 
 
 36. The surface of a zone is found as follows : 
 
 Eule. — Multiply the height of a zone by the circumference 
 of a great circle of the sphere. 
 
 1. The diameter of a sphere is 25 feet, and the height of 
 the zone 6 feet ; what is the surface of the zone ? 
 
 Ans. 471.24 square feet. 
 
 2. Eequired the surface of the torrid zone, the diameter 
 of the earth being 7912 miles. 
 
 Ans. 78,419,272 square miles. 
 
 Note. — This is to be solved after the pupil has completed Trigonometry. 
 
 37. The contents of a sphere are found as follows : 
 Eule. — Multiply the surface by one-third of the radius ; or, 
 
 Multiply the cube of the diameter by \ of 3.1416. 
 
 1. Eequired the contents of a sphere whose diameter is 
 17 inches. Ans. 2572.4468 cubic inches. 
 
 2. Eequired the contents of the planet Mars, the diameter 
 being about 4500 miles. Ans. 47713050000. 
 
GEOMETRY. 187 
 
 38. The contents of a spherical segment of one base 
 are found thus : 
 
 Rule. — Add the square of the height to three times the square 
 of the radius of the base; multiply this sum by the height, and 
 the product by .5236; or, see Th. X. B. VII. 
 
 Note. — For the volume of a segment of two bases, see B. VII. Th. 
 X. C. 4. 
 
 1. Required the contents of the segment of a sphere 
 whose height is 4 inches, and radius of the base 8 inches. 
 
 Ans. 435.635 cubic inches. 
 
 2. Find the volume of either temperate zone, the dia- 
 meter of the earth being 7912 miles. 
 
 Ans. 54,919,403,678 cubic miles. 
 
 Note. — The pupil will solve this after completing Trigonometry. 
 
 CYLINDRICAL RINGS. 
 
 39. A cylindrical ring is formed by bending a cylinder 
 until the two ends meet. We find its surface by the fol- 
 lowing , 
 
 Etjle. — To the thickness of the ring add the inner diameter; 
 multiply this sum by the thickness of the ring, and the product 
 by 9.8696. 
 
 Note. — For contents, multiply the sum by the square of \ the thickness, 
 instead of the thickness, the other part of the rule being the same as 
 for surface. 
 
 1. The thickness of a cylindrical ring is 4 inches, and 
 the inner diameter 18 inches ; what is the convex surface ? 
 
 Ans. 868.52 square inches. 
 
 2. The thickness of a cylindrical ring is 2 inches, and 
 the inner diameter 1 foot ; required its contents. 
 
 Ans. 138.1744 cubic inches. 
 
188 
 
 MENSURATION. 
 
 \ 
 
 \ 
 
 ,/\ 
 
 
 / 
 
 a 
 
 \z\ 
 
 40. The side of an inscribed cube is found thus : 
 Eule. — Multiply the diameter by .57736, or the radius by 
 
 1.15472. E . 
 
 Demonstration. — Let the cube in the mar- 
 gin represent an inscribed cube ; then will 
 AE be the diameter of the sphere. Let the 
 diameter be denoted by D, and the radius by 
 R. Now, AE* or &* = AC 1 + TLB 2 ; but 
 AC = AB* +UC 2 ; hence, D* = AB* + 
 BC 1 -f~ CE 2 , or, since the sides are equal, 
 _D*= 3AB*, or, D = AB X^ = ^X * 
 
 1.73205, and, consequently, AB = D X y.rtW = z) X -57736, or .45= 
 B X 1-16472. 
 
 1. Eequired the side of a cube that can be cut out of a 
 Bphere whose diameter is 16 inches. Ans. 9.23776. 
 
 2. Eequired the volume of a cube inscribed in a sphere 
 whose circumference is 18.849552 inches. 
 
 Ans. 41.571219 cubic inches. 
 
 41. The volume of an irregular body is found thus -. 
 Eule. — Immerse the body in a vessel of known dimensions, 
 
 containing water; note the rise in the water, and calculate 
 accordingly. 
 
 1. A stone immersed in a cylindrical vessel 10 inches in 
 diameter, raised the water 5 inches ; required the volume 
 of the stone. Ans. 392.70. cubic inches. 
 
 2. A man put a stone into a vessel 14 cubic feet in capa- 
 city, and it then required 2-1 quarts of water to fill the 
 vessel ; required the volume of the stone. 
 
 Ans. 13.9164 cubic feet. 
 
GEOMETRY'. 1§9 
 
 MISCELLANEOUS PROBLEMS — PLANE FIGURES. 
 
 1. How many yards of paper that is 30 inches wide will it require to 
 cover the wall of a room 15J feet long, 11J feet wide, and 7f feet high ? 
 
 Ans. 55.2833 yards. 
 
 2. A ladder 130 feet long, with its foot in the street, will reach on 
 one side to" a window 78 feet high, and on the other to a window 50 feet 
 high ; what was the width of the street ? Ans. 224 feet. 
 
 3. The diameter of a circle is 4 feet; required the area of the inscribed 
 equilateral triangle. Ans. 3^3 square feet. 
 
 4. From a plank 16 inches broad, 6 square ft. are to be sawed off ; at 
 what distance from the end must the line be struck 1 Ans. 4J feet. 
 
 5. The ball on the top of a church is 6 feet in diameter ; what did 
 the gilding of it cost, at 8 cents per square inch ? Ans. $1302.884. 
 
 6. The area of an equilateral triangle, whose base falls on the dia- 
 meter and its vertex in the middle of the are of a semicircle, is 100 
 square feet ; what is the diameter of the semicircle ? Ans. 26.32148. 
 
 7. The cost of paving a semicircular plot of ground, at 20 cents a 
 square foot, amounted to $20 ; required its diameter. Ans. 15.9576. 
 
 8. A gentleman has a garden 80 feet long and 60 feet wide ; what 
 must be the width of a walk extending around the garden, which shall 
 occupy one-half of the ground 1 Ans. 10 feet. 
 
 9. Required the perimeter of a regular dodecagon which shall con 
 tain the same area as a circle whose circumference is 1000 feet. 
 
 Ans. 1011.67 feet. 
 
 10. If a horse tied to a post in the centre of a field by a rope 1 chain 
 78 links can graze upon an acre, what length of rope would allow it to 
 graze upon 5$ acres ? Ans. 4 chains 15£ links. 
 
 11. A has a circular garden which is 20 rods, and B has a circular 
 garden whose area is 6J times as great ; what is the diameter of B's 
 garden? Ans. 50 rods. 
 
 12. A has a circular garden, and B a square one ; the distance around 
 each is 64 rods ; which contains the most land, and how much ? 
 
 Ans. 69.948 square rods. . 
 
 1 3. Atherton has a circular garden and Fell has a square one, and 
 
190 MENSURATION. 
 
 they contain 4 acres ; how much farther around is one than the 
 other? Ans. 11.512 rods. 
 
 14. Mr. Thompson has a square yard containing ^ of an acre ; he 
 makes a gravel walk around it which occupies £J of the whole yard ; 
 what is the width of the walk ? Arts. 4 feet 1 J inches. 
 
 15. A general, attempting to draw up his division in the form of a 
 square, found he lacked 100 men to complete the square ; he then re- 
 ceived a reinforcement of five companies of 100 men each, and found 
 he could increase the side of the square by 3 men and have 1 man 
 remaining ; how many men had he at first ? Arts. 4125 men. 
 
 VOLUMES. 
 
 1. The volume of a sphere is 606.132 cubic feet; required its diar 
 meter. Ans. 10.5 feet. 
 
 2. The edge of a cube is 6 feet ; what is the volume of a sphere that 
 may be inscribed within it ? Ans. 113.0976 cubic feet. 
 
 3. I have a cistern in the form of the frustum of a cone, its top dia- 
 meter being 12 feet, its bottom diameter 9 feet, and its depth 5 feet ; 
 how many barrels of water will it contain? Ans. 103.515 barrels. 
 
 4. Bunker Hill Monument is 220 feet high, 30 feet square at the base, 
 and 15 feet at the vertex ; what is its volume ? Ans. 115500 cubic ft. 
 
 5. Mr. Wilson has a pond which covers 100 acres, the average depth 
 being 10 feet ; how many cubic feet of water does it contain 1 
 
 Ans. 43560000 cubic feet. 
 
 6. A man has a log of wood 20 ft. long, the larger end being 3 ft. in dia- 
 meter, and the smaller 2 ft. ; required the contents of the largest square 
 stick, 20 ft. long, that can be sawed out of it. Ans. 63J cubic feet. 
 
 7. A bushel measure is 18£ inches in diameter and 8 inches deep ; 
 what should be the dimensions of a measure of similar form to contain 
 64 bushels ? Ans. Diameter, 74 inches ; depth, 32 inches. 
 
 8. Mr. Benson can dig a shaft 5 feet each way in one day ; how long 
 will it take him to dig a shaft 20 feet each way ? Ans. 64 days. 
 
 9. A man has a square garden 100 feet long, and wishes to make a 
 gravel walk half-way around it ; what will be the width of the walk if 
 >t takes up one-half of the garden ? Ans. 29.289 feet. 
 
GEOMETRY. 191 
 
 10. A wishes to enclose his garden, which is 100 feet long and 80 feet 
 wide, with a ditch 4 feet wide ; how deep must it be dug that the soil 
 taken out may raise the surface 1 foot ? Ans. 5.319 feet. 
 
 11. A cubic foot of brass is to be drawn into a wire fa of an inch in 
 diameter ; required the length of the wire, supposing there is no loss 
 of metal in the process. Arte. 31.252 miles. 
 
 12. Mr. Bonnycastle mentions a globe whose volume and surface are 
 represented by the same number ; what was the diameter of this globe ? 
 
 Ans. 6. 
 
 13. Mr. Haswell requires the weight of an iron shell 4 inches in dia- 
 meter, the thickness of the metal being 1 inch, estimating a cubic inch 
 of iron at \ of a pound. Ans. 7.3304 pounds. 
 
 14. Bunker Hill Monument is 220 feet high, the lower base being 30 
 feet square, the upper 15 feet square ; through its centre runs a cylin- 
 drical opening 15 feet in diameter at the bottom and 11 feet at the top ; 
 how many cubic feet of material in the monument ? 
 
 Ans. 86068.444 cubio feet. 
 
 15. A gentleman has a bowling-green 300 feet long and 200 feet 
 broad, which he wishes to raise 1 foot higher by means of the earth 
 that is to be taken from a ditch that is to go around it ; to what depth 
 must the ditch be dug, supposing its breadth to be 8 feet ? 
 
 Ans. 7 feet 3.21 inches. 
 
 16. A man having a garden 100 feet long and 80 feet broad, wishes 
 to make a gravel walk half-way around it ; what will be the width of 
 the walk if it takes up one-half of the garden ? Ans. 25.9688 feet. 
 
 17. Three persons having bought a sugar loaf, want to divide it equally 
 among them by sections parallel to the base; what is the altitude of 
 each person's share, supposing the loaf is a cone 20 inches high ? 
 
 Ans. 13.867 upper part ; 3.604 middle; 2.528 lower. 
 
 Suggestion. — Solve it by the principle of similar cones being to each other 
 as the cubes of their altitudes. 
 
 Note. — Several of these problems are from the old writers on Mensuration. 
 For more methods and exercises, see Bonhycastle's and Haswell's works on 
 Mensuration. 
 
ELEMENTS OF TRIGONOMETRY. 
 
 INTRODUCTION. 
 
 LOGARITHMS. 
 
 1. Logarithms are a. species of numbers used to abbreviate 
 Multiplication, Division, Involution, and Evolution. 
 
 2. The logarithm of a number is the exponent denoting the jpower 
 to which a fixed number must be raised in order to produce the 
 first number. 
 
 3. This fixed number is called the base of the system. The base 
 of the common system is 10. 
 
 4. Eaising 10 to different powers, we have, 
 
 10° = 1 ; hence, is the log of 1 ; 
 10!= 10 " 1 " 10; 
 
 10 2 = 100 " 2 " 1'00; 
 
 10 3 =1000 " 3 " 1000; 
 
 etc. 
 
 5. From this we have the following principles: 
 
 Prin. 1. The logarithm of a number between 1 and 10 is between and 
 1, and is, therefore, a decimal. 
 
 Prin. 2. The logarithm of a number between 10 and 100 is between 1 
 and 2, and is, therefore, 1 and a decimal. Thus, it has been found 
 that the log. of 76 is 1.880814. 
 
 Prin. 3. The logarithm of a number between 100 and 1000 is between 
 2 and 3, and is, therefore, 2 and a decimal. Thus, the log. of 458 is 
 2.660865. 
 
 6. When the logarithm consists of an integer and a decimal, 
 
 16 1 
 
2 TRIGONOMETRY. 
 
 the integer is called the characteristic, and the decimal part the 
 mantissa. Thus, in 2.660865 the 2 is the characteristic, and .660865 
 is the mantissa. 
 
 PROPERTIES OF LOGARITHMS. 
 Prist. 1. — The characteristic is always one less than the number qf in- 
 tegral places in the number. 
 
 For, from Art. 4, we see that the log. of 100 is 2, the log. of 
 1000 is 3, and of any number between 100 and 1000 it is 2 and a 
 decimal ; hence, the characteristic is one less than the number 
 of integral places. 
 
 ' Pkin. 2. — The logarithm qf the base is 1, and the logarithm qf 1 is 
 zero. 
 
 For, since 10 1 = 10, the log. of 10 is 1 ; and since 10° = 1, the 
 logarithm of 1 is 0. 
 
 Pein. 3. — The characteristic of the logarithm of a decimal is negative, 
 and is numerically one greater than the number qf ciphers between the 
 decimal point and the first significant figure. 
 
 For, if we raise the base, 10, to powers which give decimals, we 
 will have, 
 
 10° = 1 ; hence, log 1 = ; 
 10- ^.l " log.l = — 1; 
 
 10T Z =.01 " log .01 = — 2; 
 
 10- 8 =.001 " log .001 = — 3; 
 etc. etc. 
 
 which proves the principle. Thus, the log. of .458 is 1.660865. 
 
 Prin. 4. — The logarithm of the product of two numbers is equal to the 
 sum of the logarithms of those numbers. 
 
 For, let M and N be any two numbers, and m and n their loga- 
 rithms ; then we shall have, according to the definition, 
 
 10»> = M, 10" = JV. 
 
 Multiplying these equations, member by member, we have, 
 
 10 m +"=ilf XiV". 
 Ifence, log (M X N) = m + n; or, = log M-\- log N. 
 
INTRODUCTION. 3 
 
 Erin. 5. — The logarithm of the quotient of two numbers equals the dif- 
 ference of the logarithms of those numbers. 
 
 For, from the definition, we have, 
 
 lO™ = M, 10" = N. 
 
 Dividing the first by the second, we have, 
 
 Hence, log ( — 1 = m — n, or, = log M — log N. 
 
 Prin. 6. — The logarithm of any power of a number is equal to the 
 logarithm of the number multiplied by the exponent of the power. 
 For, since 
 
 10">=Jf, 
 if we raise both members to the nth power, we have, 
 
 10 mn = M n . 
 Hence, log M" = mn, or, = log M X n. 
 
 Prin. 7. — The logarithm of the root of any number is equal to the 
 logarithm of the number divided by the index of the root. 
 For, since 
 
 10 m = M, 
 
 if we take the nth root of both members, we have, • 
 
 Hence", log y/W= -, or, log M-i- n. 
 
 Prin. 8. — The logarithm of the product of any number multiplied by 
 10 is equal to the logarithm of the number increased by 1. 
 
 Suppose log M = m; then, by Prin. 4, 
 
 log (MX 10) = log M+ log 10. But log 10 = 1; 
 Hence, log {MX 10) = m + 1- 
 Thus, log (76 X 10) = 1.880814 + 1 ; or, log 760 = 2.880814. 
 
 Prin. 9. — -The logarithm of the quotient of any number divided by 10 
 is equal to the logarithm of the number diminished by 1. 
 
4 TRIGONOMETRY. 
 
 Suppose log M=m; then, by Prin. 5, 
 
 log (M h- 10) = log M— log 10; from which 
 log(Jf-i-'lO)=m — 1. 
 
 Thus, log (458 -s- 10) =2.660865 — 1; 
 
 or, log 45.8 = 1.660865. 
 
 7. The following examples will' illustrate Principles 1, 3, 8, 
 and 9. 
 
 log 234 is 2.369216, 
 log 23.4 " 1.369216, 
 log 2.34 " 0.369216, 
 log .234 " L369216, 
 log .0234" 2.369216. 
 
 From this, we see that when we change the place of the de- 
 cimal point we change the characteristic, but do not change the 
 decimal part of the logarithm. 
 
 The minus sign is written over the characteristic, showing that 
 it only is negative. 
 
 TABLE OF LOGARITHMS. 
 
 8. A Table op Logarithms is a table by means of which we can 
 find the logarithms of numbers, or the numbers corresponding 
 to given logarithms. 
 
 9. In the annexed table the entire logarithms of the numbers 
 up to 100 are given. For numbers greater than 100 the mantissa 
 alone is given ; the characteristic being found by Prin. 1. 
 
 10. The numbers are placed in the column on the left, headed 
 N ; their logarithms are opposite, on the same line. The first two 
 figures of the mantissa are found in the first column of loga- 
 rithms. 
 
 11. The column headed D shows the average differences of 
 the ten logarithms in the same horizontal line. This difference 
 is found by subtracting the logarithm in column 4 from that in 
 column 5, and is very nearly the mean or average difference. 
 
INTRODUCTION. 5 
 
 TO FIND THE LOGARITHM OF ANY NUMBER. 
 
 12. To find the logarithm of a number o/one or two figures. 
 
 Look on the first page of the table, in the column headed N, 
 and opposite the given number will be found its logarithm. 
 Thus, 
 
 the logarithm of 25 is 1.397940, 
 87 is 1.939519. 
 
 13. To find the logarithm, of a number of thkee figures. 
 
 Look in the table for the given number; opposite this, in 
 column headed 0, will be found the decimal part of the loga- 
 rithm, to which we prefix the characteristic 2, Prin. 1. Thus, 
 the logarithm of 325 is 2.511883, 
 876 is 2.942504. 
 
 14. To find the logarithm of a number of Form figures. 
 
 Find the three left-hand figures in the column headed N, and 
 opposite to these, in the column headed by the fourth figure, 
 will be found four figures of the logarithm, to which two figures 
 from the column headed are to be prefixed. The character- 
 istic is 3, Prin. 1. Thus, 
 
 the logarithm of 3456 is 3.538574, 
 " " 7438 is 3.871456. 
 
 15. In some of the columns, small dots are found in the place 
 of figures : these dots mean zeros, and should be written zeros. 
 If the four figures of the logarithm fall where zeros occur, or if, 
 in passing back from the four figures found to the zero column, 
 any of these dots are passed over, the two figures to be prefixed 
 must be taken from the line just below. Thus, 
 
 the logarithm of 1738 is 3.240050, 
 2638 is 3.421275. 
 
 16. To find the logarithm of a number of more than tow. figures. 
 Place a decimal point after the fourth figure from the left hand, 
 
 thus changing the number into an integer and a decimal. Find 
 
 the mantissa of the entire part by the method just given. Then 
 
 16* 
 
6 - TRIGONOMETRY. 
 
 from the column headed D take the corresponding tabular differ- 
 ence, multiply it by the decimal part, and add the product to the 
 mantissa already found ; the result will be the mantissa of the 
 given number. The characteristic is determined by Prin. 1. 
 
 If the decimal part of the product exceeds .5, we add 1 to the 
 entire part ; if less than .5, it is omitted. 
 
 EXAMPLES. 
 
 1. Find the logarithm of 234567. 
 
 Solution. — The characteristic is 5, Prin. 1. Placing a decimal 
 point after the fourth figure from the left, we have 2345.67. The 
 decimal part of the logarithm of 2345 is .370143 ; the number in 
 column D is 185 ; and 185 X -67 = 123.95, and since .95 exceeds .5, 
 we have 124, which, added to .370143, gives .370267 ; hence, log 
 234567 = 5.370267. 
 
 2. Find the logarithm of 4567. Ans. 3.659631. 
 
 3. Find the logarithm of 3586. Ans. 3.554610. 
 
 4. Find the logarithm of 11806. Ans. 4.072102. 
 
 5. Find the logarithm of .4729. Ans. 1.674769. 
 
 6. Find the logarithm of 29.337. Ans. 1.467416. 
 
 17. To find the number corresponding to a given logarithm. 
 
 1. Find the two left-hand figures of the mantissa in the column 
 headed 0, and the other four, if possible, in the same or some 
 other column, on the sanu line ; then, in column N, opposite to 
 these latter figures, will be found the three left-hand figures, and at 
 the top of the page the other figure of the required number. 
 
 2. When the exact mantissa is not given in the table, take out the four 
 figures corresponding to the next less mantissa in the table ; sub- 
 tract this mantissa from the given one; divide the remainder, 
 with ciphers annexed, by the number in column D, and annex 
 the quotient to the four figures already found. 
 
 3. Make the number thus obtained correspond with the cha- 
 racteristic of the given logarithm, by pointing off decimals oi 
 annexing ciphers. 
 
INTRODUCTION. 7 
 
 EXAMPLES. 
 
 1. Find the number whose logarithm is 5.370267. 
 Solution.— The mantissa of the given logarithm is . . .370267 
 
 The mantissa of the next less logarithm of the table is . .370143 
 
 and its corresponding number is 2345. ■ I 
 
 Their difference is 124 
 
 The tabular difference is 185 
 
 The quotient is . 185)124.00(.67 
 
 Hence, the required number is . 234567. 
 
 Note. — If the characteristic had been 2, the number would have been 
 234.567; if it had been 7, the number would have been 23456700; if it had 
 been 2, the number .would have been .0234567, etc. 
 
 2. Find the number whose logarithm is 3.659631. 
 
 ■Ans. 4567. 
 
 3. Find the number whose logarithm is 2.554610. 
 
 Ans. 358.6. 
 
 4. Find the number whose logarithm is 1.072102. 
 
 Ans. 11.806. 
 
 5. Find the number whose logarithm is 2.674769. 
 
 Ans. .04729. 
 
 6. Find the number whose logarithm is 3.065463. 
 
 Ans. .0011627. 
 
 MULTIPLICATION BY LOGARITHMS. 
 
 18. From Prin. 4, for the multiplication of numbers by means 
 of logarithms, we have the following 
 
 Eule. — Find the logarithms of the factors, take their sum, and find the 
 number corresponding to the result; this number will be the required product. 
 
 Note. — The term sum is used in its algebraic sense. Hence, when any of 
 the characteristics are negative, — the mantissa is always positive, — we take 
 the difference between the sums of the positive and negative characteristics, 
 and prefix to it the sign of the greater. If any thing is to be carried from 
 the addition of the mantissas, it must be added to a positive characteristic, 
 or subtracted from a negative one. 
 
8 ' TRIGONOMETRY. 
 
 EXAMPLES. 
 
 1. Multiply 35.16 by 8.15. 
 Solution. log 35.16 = 1.546049 
 
 log 8.15 = 0.911158 
 
 2.457207 
 
 457125 
 
 Product, 286.554 
 
 152)82.00(.54 
 
 2. Find the product of .7856, 31.42. Ans. 24.6835, 
 
 3. Find the product of 31.42, 56.13, and 516.78. 
 
 Ans. 911393.7. 
 
 4. Find the product of 31.462, .05673, and .006785. 
 
 Ans. 01211168. 
 
 5. Product of .06517, 2.16725, .000317, and 42.1234 
 
 Ans. .001886. 
 
 6. Product of 2.3456, .00314, 123.789, .00078, and 67.105. 
 
 Ans. .04772076. 
 
 DIVISION BY LOGARITHMS. 
 
 19. From Prin. 5, t6 divide by means of logarithms, we have 
 the following 
 
 Rule. — Find the logarithms of the dividend and divisor, subtract the 
 latter from the former, and find the number corresponding to the result ; 
 this number will be the required quotient. 
 
 Note. — The term subtract is here used in its algebraic sense ; hence, we 
 must subtract according to the principles of algebra. 
 
 1. 
 
 Divide 783.5 by 
 
 6.25. 
 
 EXAMPLES. 
 
 Solution. 
 
 log 
 
 783.5 
 
 = 2.894039 
 
 
 
 log 
 
 6.25 
 
 = 0.795880 
 
 
 2.098159 
 
 o 
 
 nA+.id»Ti+. 
 
 V. 
 
 >fi Sfi 
 
 .097951 
 
 346)208(6 
 
INTRODUCTION. 9 
 
 2. Divide 272.636 by 6.37. Ans. 42.8. 
 
 3. Divide 50.38218 by 67.8. Ans. .7431. 
 
 4. Divide 155 by .0625. Ans. 2480. 
 
 ARITHMETICAL COMPLEMENT. 
 
 20. The operation of division when combined with multipli- 
 cation is somewhat simplified by using the principle of the arith- 
 metical complement. 
 
 21. The Arithmetical Complement of a logarithm is the result 
 arising from subtracting the logarithm from 10. Thus, the arith- 
 metical complement of the logarithm 5.623427 is 10 — 5.623427, 
 or 4.376573. 
 
 22. The arithmetical complement may be written directly from 
 the table, by subtracting each figure of the logarithm from 9, except the 
 right-hand figure, which must be taken from 10. This is the same as 
 subtracting the logarithm from 10. 
 
 . 23. We will now prove that the difference between two logarithms is 
 equal to the first logarithm, plus the arithmetical complement of the second, 
 minus 10. 
 
 Let a = the first logarithm, 
 
 b = the second logarithm, 
 and c = 10 — b = arith. comp. of b. 
 
 The difference is a — b. 
 But, — b = c — 10. 
 
 Hence, a — b = a + c — 10, 
 
 which proves the principle. 
 
 24. Hence, to divide by means of the arithmetical complement, 
 we have the following 
 
 Rule. — Add the arithmetical complement of the logarithm of the divisor 
 to the logarithm of the dividend, subtract 10, and find the -number corre- 
 sponding to the difference, this number will be the required quotient. 
 
 EXAMPLES. 
 
 1. Divide 856.3 by 45.32. 
 
10 TKIGONOMETK5T. 
 
 Solution. log 856.3 
 
 2.932626 
 
 (a. o.) log 45.32 
 
 8.343710 
 
 Quotient, 18.8945 
 
 1.276336 
 
 2. Divide 0.3156 by 78.35. 
 
 
 log 0.3156 . 
 
 . 1.499137 
 
 (a. c.) log 78.35 
 
 . 8.105961 
 
 Quotient, .004028 £605098 
 
 3. Divide 3.7521 by 18.346. Ans. .204519. 
 
 4. Divide 483.72 by .30751. Ans. 1573.02. 
 
 5. Multiply 32.16 by 7.856, and divide the product by 45.327. 
 
 Ans. 5.574. 
 
 6. Divide the product of 31.57 and 123.4 by the product of 
 316.2 and .0316. Ans. 389.8884. 
 
 7. Find by logarithms the first term of the proportion, 
 x : 73.15 : : 48.16 : 3167. Ans. 1.11237. 
 
 INVOLUTION BY LOGARITHMS. 
 
 25. From Prin. 6, to raise a number to any power, we have the 
 following 
 
 Kule. — Find the logarithm of the number, multiply it by the exponent 
 of the power, and find the number corresponding to the result. 
 
 EXAMPLES. 
 
 1. Find the 4th power of 45. 
 Solution. 
 
 log 45 = 1.653213 
 4 
 
 Power, 4100625 6.612852 
 
 2. Find the cube of 0.65. Ans. 0.2746. 
 
 3. Find the 6th power of 1.037. Ans. 1.243. 
 
 4. Find the 7th power of .4797. Ans. 0.005846. 
 
INTRODUCTION. 11 
 
 EVOLUTION BY LOGARITHMS. 
 
 26. From Prin. 7, to extract any root of a number, we have 
 the following 
 
 Rule. — I. Find the logarithm of the number, divide it by the index of 
 the root, and find the number corresponding to the result. 
 
 II. If the characteristic is negative and not divisible by the index of the 
 root, add to it the smallest negative number that will make it divisible, 
 prefixing the same number with a plus sign to the mantissa. 
 
 EXAMPLES. 
 
 1. Find the square root of 576. 
 Solution. log 576 = 2.760422 
 
 2.760422-^2=1.380211 
 Hence, the root is 24. 
 
 2. Find the fourth root of .325. 
 
 Solution. log .325 = 1.511883 = 1 + 3.511883. 
 
 Then (I + 3.511883) -s- 4 = 1.877971 
 
 Hence, the quotient is,. .75504. 
 
 3. Find the fifth root of .0625. Ans. .574348. 
 
 4. Find the cube root of 7. . Ans. 1.9129. 
 
 5. Find the fifth root of 5. Ans. 1.3797. 
 
 6. Find the tenth root of 8764.5. Ans. 2.479. 
 
 CALCULATION OF LOGARITHMS. 
 
 The pupil will by this time naturally inquire how these logarithms 
 are calculated. This we have not room to explain here ; in fact, an 
 explanation of the modern methods would be almost too difficult for the 
 majority of pupils who study this book. Only a general idea can here 
 be given. 
 
 In computing logarithms, it is only necessary to calculate the loga- 
 rithms of prime numbers, since the logarithms of composite numbers 
 may be obtained by adding the logarithms of their prime factors. 
 
 The logarithms of the prime numbers were first computed by com- 
 
12 TRIGONOMETRY. 
 
 paring the geometrical and arithmetical series, 1, 10, 100, etc., and 0, 
 1, 2, etc., and finding, geometrical and arithmetical means ; the 
 arithmetical mean being the logarithm of the corresponding geome- 
 trical mean. This method was exceedingly laborious, involving so 
 many multiplications and extractions of roots. 
 
 The method now generally used is that of series, by which the com- 
 putations are much more easily made. The following formula is de- 
 rived by algebraic reasoning. 
 
 ■ log (1 + ,) = A (f _|+| 3 _|V|-etc.) 
 
 In this the quantity A is called the modulus, which in the Napierian 
 system is unity. The series, when A is one, put in a more convenient 
 form, becomes, 
 
 •etc. 
 
 log. (. + 1) - log. , = 2 ( ir _L_ + w ± fW + gfarpijr 
 
 From which, knowing the logarithm of any number, we readily find 
 the logarithm of the next larger number. The pupil will be interested 
 in finding logarithms by this formula. Begin with 2, in which 2 = 1. 
 
 The logarithm found will be the Napierian logarithm, and this mul- 
 tiplied by 0.434294 will give the common logarithm. 
 
PLANE TRIGONOMETRY. 
 
 DEFINITIONS AND PRIMARY PRINCIPLES. 
 
 1. Plane Trigonometry is the science which treats of tha 
 solution of plane triangles. 
 
 2. The Solution of a triangle is the operation of finding the 
 unknown partsTvhen a sufficient number of the known parts are 
 given. 
 
 3. In every triangle there are six parts ; three sides and three 
 angles. These parts are so related that when three of the parts 
 are given, one being a side, the other parts may be found. 
 
 4. An angle is measured, as we have previously seen, by the 
 arc included between its sides, the centre of the circumference 
 being at the vertex of the angle. 
 
 5. For measuring angles, as has already been explained, the 
 circumference is divided into 360 equal parts, called degrees; 
 each degree into 60 equal parts, called minutes, etc. 
 
 6. AQuADRANTis one-fourth of the circumference of a circle; 
 hence, if two lines be drawn through the 
 
 centre of a circle at right angles to each 
 other, they will divide the circumference 
 into four quadrants. Each quadrant con- 
 tains 90°. 
 
 7. The Complement of an arc is 90° 
 minus the arc ; thus, DC is the complement 
 of BC ; also, the angle DOOis the comple- 
 ment of BOO. 
 
 8. The Supplement of an arc is 180° minus the arc; thus, 
 
 17 13 
 
14 
 
 PLANE TRIGONOMETRY. 
 
 AE is the supplement of the arc BDE; also, the angle AOE is 
 the supplement of the angle BOE. 
 
 9. In Trigonometry, instead of comparing the angles of tri- 
 angles, or the arcs which measure them, we compare certain 
 lines, called functions of the arcs. A function of a quantity is 
 something depending upon the quantity for its value. These 
 functions are the sine, cosine, tangent, cotangent, secant, and cosecant. 
 
 10. Thus, instead of reasoning with the angle ACB, or the arc 
 AB, which measures it, we draw the per- 
 pendicular AD, and use the lines AD and 
 ■ CD. The line AD is called the sine of the 
 arc or angle; the line CD is called the cosine 
 of the arc or angle. 
 
 11. If we draw BE perpendicular to CB, 
 meeting CA produced in E, the line BE is 
 called the tangent of the angle, and the line 
 
 - CE is called the secant. 
 
 12. In comparing the sides and angles, 
 these lines, we say, are used instead of the 
 
 i angles or the arcs. The necessity for such 
 lines is evident, since we could not compare the sides, which are 
 straight lines, with the angles, or the curve lines, which measure 
 them; 
 
 We will now represent these lines in the first and second 
 quadrants. 
 
 13. The Sine of an arc is the per- 
 pendicular let fall from one extremity 
 
 -of the arc on the diameter which 
 passes through the other extremity. 
 Thus, CD is the sine of the arc AC. 
 
 14. The Cosine of an arc is the 
 sine of its complement ; or it is the 
 distance between the foot of the sine 
 
 and the centre of the circle ; thus, CE or OD is the cosine of the 
 arc A C. 
 
PLANE TRIGONOMETRY. 15 
 
 15. The Tangent ofan arc is a line which is perpendicular 
 to the radius at one extremity of the arc, and limited by a line 
 passing through the centre of the circle and the other extremity; 
 thus, AT is the tangent of AC. 
 
 16. The Cotangent of an arc is equal to the tangent of the 
 complement of the arc ; thus, BT' is the cotangent of AC. 
 
 17. The Secant of an arc is a line drawn from the centre of 
 the circle through one extremity of the arc, and limited by a 
 tangent at the other extremity ; thus, OT is the secant of AC. 
 
 18. The Cosecant of an arc is the secant of the complement 
 of the arc; thus, OT' is the cosecant of AC. 
 
 19. The sine, cosine, tangent, cotangent, etc. of an arc are in- 
 dicated as follows : 
 
 sin A C; tan A C; sec A C; 
 
 cos A C; cot A C; cosec A C. 
 
 20. Principles. — From the definitions now given, we can readily 
 derive the following simple principles. 
 
 1. The sine of an arc equals the sine of its supplement, and also the 
 cosine of an arc equals the cosine of its supplement. 
 
 Dem.— Take the arc ABF; its sine is FG, its supplement is 
 FH, and the sine of its supplement is FG. Hence, its sine 
 equals the sine of its supplement. Jts cosine is GO, which is 
 also the cosine of FH. Hence, etc. 
 
 2. The tangent and cotangent of an arc are respectively equal to the 
 tangent and cotangent of the supplement of the arc. 
 
 Dem.— The tangent of the arc ABF is AT'", and the tangent 
 of its supplement FH is HT", and, by similar triangles, it may 
 be shown that AT'" equals HT" ; therefore, etc. 
 
 3. The secant and cosecant of an arc are respectively equal to the secant 
 and cosecant of the supplement of the arc. 
 
 This may be demonstrated in a manner quite similar to those 
 above. Let the pupil be required to show it. 
 
 4. If a equals any arc or angle, then we shall have, from the 
 definitions, 
 
16 PLANE TRIGONOMETRY 
 
 sin a = cos (90° — a) 
 tan a = cot (90° — a) 
 sec a = cosec (90° — a) 
 
 NATURAL SINES, COSINES, ETC. 
 
 21. The length of these trigonometrical lines may be expressed 
 in numbers, differing, of course, as the radius of the circle is 
 lai'ger or smaller. If the radius is regarded as unity, or 1, we 
 have what are called natural sines, cosines, etc. The method of 
 calculating these sines, cosines, etc. will be explained hereafter. 
 
 The operation of multiplying and dividing by these natural 
 sines being long and tedious, it has been found more convenient 
 to use logarithmic sines, which we will now explain. 
 
 TABLE OP LOGARITHMIC SINES. 
 
 22. A Logarithmic Sine, Cosine, Tangent, or Cotangent is' the 
 logarithm of the sine, cosine, tangent, or cotangent of an arc of a 
 circle whose radius is 10,000,000,000. 
 
 23. A Table of Logarithmic Sines is a table containing the 
 logarithmic sine, cosine, tangent, and cotangent of arcs. 
 
 24. The table of logarithmic sines may be calculated from a 
 table of natural sines, as will be explained hereafter. In the 
 table, the degrees are given at the top and bottom of the page, 
 and the minutes at the sides, in the column headed M. 
 
 25. The column headed D contains the increase or decrease for 
 1 second. This is found by subtracting the logarithmic sine, etc. 
 of an arc from that next exceeding it by 1 minute, and dividing 
 the difference by 60. 
 
 26. To find the logarithmic sines, cosines, etc. of arcs or angles. 
 
 1. When the arc is expressed in degrees, or in degrees and minutes. If 
 the angle is less than 45°, look for the degrees at the top of the 
 page, and for the minutes in the &/i!-hand column ; then, opposite 
 to the minutes, on the same horizontal line, in the column headed 
 
PLANE TRIGONOMETRY. 
 
 17 
 
 Sine, will be found the logarithmic sine ; in that headed Cosine 
 will be found the logarithmic cosine, etc. Thus, 
 
 log sin 23° 35' 
 log tan 23° 35' 
 
 9.602150 
 9.640027 
 
 If the angle exceeds 45°, look for the degrees at the bottom of the 
 page, and for the minutes in the right-hand column ; then, oppo- 
 site to the minutes, in the same horizontal line, in the column 
 marked at the bottom Sine, will be found the logarithmic sine, etc. 
 
 Thus, 
 
 log cos 65° 24' 9.619386 
 
 log tan 65° 24' 10.339290 
 
 2. When the arc contains seconds. — Find the logarithmic sine, etc. 
 as before; then multiply the corresponding number found in 
 column D by the number of seconds, and add the product to the 
 preceding logarithm for the sines or tangents, and subtract it for 
 cosines or cotangents. 
 
 "We subtract for cosine and cotangent, because the greater the 
 arc the less the cosine or cotangent. In multiplying the tabular 
 difference by the number of seconds, we observe the same rule 
 for the decimal point as in logarithms. If the arc is greater tlmn 
 90°, we find the sine, cosine, etc. of its supplement. 
 
 EXAMPLES. 
 
 1. Find the logarithmic sine of 36° 24' 42". 
 
 log sin 36° 24', 
 Tabular difference, 
 No. of seconds, 
 
 Product, 
 
 log sin 36° 24' 42", 
 
 2. Find the logarithmic cosine of 64° 30' 30". 
 
 17* 
 
 2.85 
 42 
 
 119.70 to be added, 
 
 9.773361 
 
 120 
 9.773481 
 
L8 
 
 PLANE TRIGONOMETRY. 
 
 4.41 
 30 
 
 log cos 64° 30', 
 • Tabular difference, 
 No. of seconds, 
 
 Product, 
 
 log cos 64° 30' 30", 
 
 3. Find the logarithmic tangent of 120° 15' 24". 
 
 9.633984 
 
 132.30 to be subtracted, 
 
 132 
 9.633852 
 
 The given arc, 
 
 sqL 
 
 ice, 4.84 
 36 
 
 24", 
 
 thmic sine 
 
 UTION. 
 
 180° 00' 00" 
 120 15 24 
 
 
 Supplement, 
 log tan 59° 44', 
 Tabular differer 
 No. of seconds, 
 
 59 44 36 
 
 to be added, 
 of 40° 40' 40". 
 
 10.233905 
 174.24 
 
 log tan 120° 15' 
 Find the logari 
 
 10.334079 
 Ans. 9.814117. 
 
 5. Find the logarithmic cosine of 140° 30' 20". 
 
 'Ans. 9.887441. 
 
 6. Find the logarithmic tangent of 85° 25' 45". 
 
 Ans. 11.097200. 
 
 7. Find the logarithmic cotangent of 144° 44' 28". 
 
 Ana. 10.150603. 
 27. To find the arc corresponding to any logarithmic sine, cosine, tangent, 
 or cotangent. 
 
 1. Look in the proper column of the table for the given loga- 
 rithm ; if found there, and the name of the function be at the 
 head of the column, take the degrees at the top, and the minutes 
 on the left; but if the name of the function is at the fool of the 
 column, take the degrees at the bottom, and the minutes on the 
 right. t 
 
 2. If the given logarithm is not exactly given in the table, 
 
PLANE TRIGONOMETRY. 19 
 
 chen take the next less logarithm, subtract it from the given 
 logarithm, and divide the remainder by the corresponding tabular 
 difference ; the quotient will be seconds, which must be added to 
 the degrees and minutes corresponding to the logarithm taken 
 from the table, for sines and tangents, and subtracted for cosines and 
 cotangents. 
 
 EXAMPLES. 
 
 1. Find the arc whose logarithmic sine is 9.617033. 
 
 SOLUTION. 
 
 Given log sine, 9.617033 
 
 Next less in table, 9.616894 
 
 Tabular difference, 4.63) 139.00(30, to be added. 
 
 Hence, the arc or angle is 24° 27' 30"- 
 
 2. Find the arc whose logarithmic cosine is 9.704682. 
 
 SOLUTION. 
 
 Given log cosine, 9.704682 
 
 Next less in table, 9.704610 
 
 Tabular difference, 3.58) 72.00(20, to be subtracted. 
 
 Hence, the arc or angle is 59° 33' 40"- 
 
 3. Find the arc whose logarithmic sine is 9.438672. 
 
 Ans. 15° 56' 14". 
 
 4. Find the arc whose logarithmic cosine is 9.634520. 
 
 Ans. 64° 27' 47". 
 
 5. Find the arc whose logarithmic tangent is 10.753246. 
 
 Ans. 79° 59' 24". 
 
 6. Find the arc whose logarithmic cotangent is 11.449852. 
 
 Ans. 2° V 40"- 
 
 28. Having learned how to find logarithmic sines, cosines, etc., 
 we will next demonstrate some theorems for the solution of tri- 
 angles. 
 
20 PLANE TRIGONOMETRY. 
 
 THE THEOREMS OF TRIGONOMETRY. 
 
 29. The Theorems of Trigonometry express the relation between 
 the sides and trigonometrical functions of the angles of triangles, 
 
 30. We give five theorems, the first three relating to triangles 
 in general, the others to right-angled triangles. 
 
 THEOREM I. 
 
 31. In any plane triangle, the sides are proportional to the sines of thi 
 opposite angles. 
 
 Let ABC be a plane triangle ; then will 
 CB:C4::sin4:sin.B. 
 
 For, with A as a centre, and a radius AE 
 equal to BC, describe the arc EG, and draw 
 the perpendicular EF. With B as a cen tre, 
 and the equal radius BC, describe the arc CH, and draw the per- 
 pendicular CD; then will CD be the sine of the angle i?, and EF 
 be the sine of the angle A, to the same radius. Now, by similar 
 triangles (B. III. Th. X.), 
 
 AE : AC : : EF : CD. 
 BvAAE equals CB, EF is sin A, and CD is sin B. 
 Hence, BC : AC: :sin-4 : sin B. 
 
 In a similar manner, it may be shown that 
 
 ^lC:AB::sin.B::sin C. 
 Therefore, etc. 
 
 THEOREM II. 
 
 32. In any plane triangle, the sum of any two sides is to their difference 
 an the tangent of half the sum of the opposite angles is to the tangent of 
 half their difference. 
 
 Let ABC be any plane angle ; then will 
 
 BC + A C : BC — A C : : tan £ (A + B) : tan £ (A — B). 
 
PLANE TRIGONOMETRY 
 
 For, produce AC to D, making CD equal 
 to CB, and draw BD; take CE equal to 
 AC, dx&vrAE, and produce it toF; then 
 AD is the sum and BE the difference of the 
 two sides A C and £C. 
 
 The sum of the angles CAE and AEC 
 equals the sum of CAB and C.RA, both 
 sums being equal to 180° minus ACB (B. 
 I. Th. XIII.); but the angle CAE equals 
 AEC (B. I. Th. X.) ; hence, CAE or CAF is the half sum of CAB 
 and CRA; also, BAF is, the Aa^ difference of the angles C4-B and 
 ABC; since it equals the half sum CAE, subtracted from the greater 
 angle CAB* 
 
 The angle CDF equals CBD, since CB equals CD; also, C4.E. 
 which equals AEC, is equal to the vertical angle FEB; hence, 
 the third angles of the triangles, AFD and EFB, are equal, and, 
 therefore, AFis perpendicular to BD; consequently, if then we 
 regard AF as, the radius, FD will be the tangent of DAF, and FB 
 will be the tangent of FAB. Now, by similar triangles, 
 AD:EB::FD:FB; or, 
 CB + AC-.CB — AC:: tan i (A + B): tan i(A — B). 
 
 THEOREM III. 
 
 33.' In any plane triangle, if a line is drawn from the vertical angle per- 
 pendicular to the base, then the whole base will be to the sum of the other 
 two sides as the difference of those sides is to the difference of the segments 
 of the base. 
 
 Let ABC be a triangle, and CD perpendicular to the base ; then 
 
 will 
 
 AB:AC + BC::AC — BC-.AD — DB. 
 
 * This principle is thus proven : — Let a and 6 be any two quantities ; then the half sum Js 
 — - — , and the half difference is — - — ; and 
 minus the half mm equals the half difference. 
 
 — i— t and tk e half difference is — - — ; and a — = — ^— ; that ii, the greater 
 
22 PLANE TRIGONOMETRY. 
 
 For, from Th. VI. Book III., 
 AC 2 = lD 2 +l)C : ', 
 and ~BC 2 4= ~RD* + T)C 2 . 
 
 Subtracting, Zc* — £C 2 = 'A& — ~BD\ 
 Hence (B. III. Th. V. C. 2), 
 
 {AC+ BC) X(iC- BO) = [AD+BD) X (AD— BD) • 
 therefore, AD + DB: AC+BC : : AC—BC : AD—DB. 
 Therefore, etc. 
 
 THEOREM IV. 
 
 34. In any right-angled plane triangle, radius is to the sine of either 
 angle as the hypothenuse is to the side opposite. 
 
 Let C-4-Bbe a triangle right-angled at A, and denote the radius 
 by It : then will 
 
 M : sin C-.-.CB: AB. 
 
 For, from the point O as a centre and any 
 radius, as CE, describe the arc EF, and draw 
 ED perpendicular to CA ; then will ED be 
 the sine of the angle C. The two triangles CED and CAB are 
 similar; hence, we have (B. III. Th. X.), 
 
 CE : ED : : CB : BA, 
 or, B : sin C : : CB : BA. 
 
 Therefore, etc. 
 
 Cor. It may also be shown that radius is to the cosine of either acute 
 ingle as the hypothenuse is to the side adjacent. 
 
 THEOREM V. 
 
 35. In any right-angled plane triangle, radius is to the tangent of either 
 acute angle as the side adjacent is to the side opposite. 
 
 Let CAB be a triangle right-angled at A; then will 
 
 R : tan C : : CA : AB. 
 For, with C as a centre and any radius CD, describe the aro 
 
PLANE TRIGONOMETRY. 23 
 
 DE, and draw DF perpendicular to CA; FD 
 will be the tangent of the angle C. The 
 triangles CDF and CAB are similar ; hence, 
 
 CD : DF-.-.CA: AB, 
 or, S : tan C : : CA : AB. 
 
 Therefore, etc. 
 
 Cor. It may also be shown that radius is to cotangent of either angle, 
 as side opposite is to side adjacent. 
 
 SOLUTION OF TRIANGLES. 
 
 36. The Solution, of a Triangle is the process of finding the 
 anknown parts when a sufficient number of the parts are given. 
 
 37. There are six parts in a plane triangle, and three of these — 
 one of the three being a side — must be given to find the other 
 parts. 
 
 38. If the angles alone were given, it is clear that the sides 
 could not be determined, since there could be an indefinite 
 number of triangles having their angles respectively equal. 
 
 39. There are four cases, as follows : 
 
 i 1. -When two angles and a side are given. 
 
 2. When two sides and an angle are given. 
 
 3. When two sides and the included angle are given. 
 
 4. When the three sides are given. 
 
 CASE I. 
 
 40. Given two angles and one side, to find the remaining parts. 
 Method. — We subtract the sum of the - given angles from 180° 
 
 to find the third angle, and then find the sides by Theorem I. 
 
 EXAMPLES. 
 
 1. In a triangle ABC, there are given the angle A — 32° 24', 
 the angle B — 40° 32', and the side AB = 240 ; required the other 
 parts. 
 
24 PLANE TRIGONOMETRY. 
 
 Solution. — Let AB C represent the triangle; then the sum of 
 A and B = 72° 56', and C= 180° — 72° 56' = 
 
 107° 04'. Then, to find 4 C, we have, 
 AC : AB:: sin B : sin C. 
 Hence, 4C = ABX sin.B-=-sin C. 
 
 From which .4 C is readily found by multi- 
 plying 240 by the natural sine of B, and dividing by the natural 
 sine of C. It is simpler, however, to use logarithms. To find 
 AC, we add the log of AB and log sin B, and subtract log sin 
 C, or add the arith. comp. of log sin C. 
 
 a. c. log sin C (107° 04'), 
 
 0.019558 
 
 
 log sin B (40° 32'), 
 
 9.812840 
 
 
 log AB (240), 
 
 2.380211 
 
 
 log^C, 
 
 2.212609 . 
 
 -.4(7=163.158 
 
 To find the side BC, we have, 
 
 
 BC-.AB:: 
 
 sin A : sin 
 
 C: 
 
 or, by logarithms, 
 
 
 
 a. clog sin C (107° 04'), 
 
 0.019558 
 
 
 log sin A (32° 24'), 
 
 9.729024 
 
 
 iog AB (240), 
 
 2.380211 
 
 
 log -BC, 
 
 2.128793 . 
 
 :BC = 134.522 
 
 2. In the triangle ABC, there are given the angle A = 27° 40', 
 the angle C= 65° 45', and the side AB = 625, to find the other 
 parts. . Am. B = 86° 35' ; BC = 318.29 ; AC = 684.266. 
 
 CASE II. 
 
 41. Given two sides and an angle opposite one of them, to find the 
 remaining parts. 
 
 Method. — One of the required angles is found by Theorem I. 
 The third angle is found by subtracting the sum of the two from 
 180° ; the third side is found by Case I. 
 
PLANE TRIGONOMETRY. 25 
 
 EXAMPLES. 
 
 1. In the triangle ABC, there are given AC = 200, CB=150, 
 and the angle A = 44° 26', to find, the other parts. 
 
 Solution. — Let ABC be a triangle in which c 
 
 A = 44° 26', 4C = 200, and 50 = 150 ; then, 
 to find the angle B, we have, 
 
 sin .B : sin ;!:: .4 C:.BC, 
 
 or, BC (150) a. c. 7.823009 
 
 : AC (200) ■ 2.301030 
 
 i: sin4 (44° 26 r ) 9.845147 
 
 : sinB ( ) 9.969186 
 
 .-. B = 68° 40' 16", or, 111° 19' 44" 
 In this problem, if the side BC, opposite the given angle .4, 
 Is shorter than>the other given side AC, the solution will be am- 
 biguous; for two triangles, ACB and ACB', may be formed, each of 
 which will satisfy the conditions of the problem. Hence, the 
 angle B found above may be either ABC or B'. But these, it will . 
 be seen, are supplements of each other; hence, in finding the 
 angle corresponding to sin B, we take the angle or its supplement. 
 In practice, there is often some circumstance to determine 
 whether the angle is acute or obtuse. If the angle given is obtuse, 
 the other angles must be acute, and there will be but one solu- 
 tion. If the side BC is equal to or greater than AC, there will be but 
 one triangle. 
 
 In the given diagram above, the angle ABC= 111° 19' 44", 
 and AB'C= 68° 40' 16"; hence, the angle ACB = C 2A° 13' 16", 
 and the angle ACB' = 113° 6' 16". 
 To find the side AB, we have, 
 
 AB-.CB:: sin ACB : sin A; 
 from which, by logarithms, we find AB = 88.085. 
 To find the side AB', we have, 
 
 AB'-.CB':: sin A CB' : sin A ; 
 from which, by logarithms, we find AB' = 197.484. 
 
 is 
 
26 PLANE -TRIGONOMETRY. 
 
 2. In a triangle ABC there are given AB 45.96, BC 62.50, and 
 the angle A 79° 21'; find the remaining parts. 
 
 Ans. C = 46° W 38" ; B = 54° 22' 22" ; AC = 51.69. 
 (There is no ambiguity, since the side _SC is greater than^C.) 
 
 3. In a. triangle ^JJCthere are given BC= 15.71, AC= 21.12. 
 and the angle A = 27° 50' ; find the other parts. 
 
 Ans. C= 113° 17' 13"; 5= 38° 52' 47"; AB= 30.906. 
 or, C= 11° 2' 47"; -B= 141° 7' 13"; ^-B= 6.447. 
 
 CASE III. 
 
 42. Given two sides and the included angle, to find the remaining parts. 
 
 Method. — We find the sum of the two angles by subtracting 
 the given angle from 180°, and divide this by 2 for the half sum. 
 We then find the half difference, by Theorem II. Having found 
 the half sum and half difference of the two angles, we find the 
 greater angle by adding the half difference to the'half sum ; and 
 the less by subtracting the half difference from the half sum. 
 The third side is found by Theorem I. 
 
 EXAMPLES. 
 
 1. In the triangle ABO, let BG= 680, AC= 460, and the in 
 eluded angle 84° ; required the other parts. 
 
 Solution. — Let ABO represent the triangle, .4C=460, 
 BC= 680, and the angle C = 84°. Then, 
 AC+ 50=460 + 680= 1140; BC—AC= 
 680 — 460 =220. A + B = 180° — 84° = 
 96°; hence, half sum = 48°. The half dif- 
 ference we find by the following proportion. 
 
 BO+AG 1140 ar. co. 6.943095 
 
 : BC—AO 220 . . 2.342423 
 
 : : tan J (A + B) 48° . . 10.045563 
 
 : tanj (A — B) 12° 5' 49" 9.331081 
 
 Hence, A = 60° 5' 49" ; and B = 35° 54' 11' 
 
 The other side, found by Theorem I., equals 783.733. 
 
PLANE TRIGONOMETRY. 
 
 27 
 
 2. Given two sides of a plane triangle 240 and 360, and the 
 included angle 68° 36' ; required the other parts. 
 
 Ans. 72° 02' 26 ; 39° 21' 34"; 352.349. 
 
 CASE IV. 
 
 43. Given the three sides of a plane triangle, to find the angles. 
 
 Method. — Let fall a perpendicular upon the greater side from 
 the angle opposite, dividing the triangle into two right-angled 
 triangles. Find the difference of the segments of the base by- 
 Theorem III. ; half this difference added to half the base gives 
 the greater segment, and subtracted from half the base gives the 
 less. We will then have two sides and the right angle of two 
 right-angled triangles, from which we can find the acute angles 
 /by Theorem I. 
 
 EXAMPLES. 
 
 1. In a triangle ABC, given AB = 60, .4(7=50, and BC= 40, 
 to find the angles. 
 
 Solution. — Let ABC represent the tri- 
 angle; then AB = m, AC— 50, BC = 40; 
 then, by Th. III., 
 
 AB : AC + BC: : AC— BC: AD — BD, 
 or, 60: 90 :: 10 .AD — BD. 
 hence, AD — BD= 90 X 10-4- 60 = 15; 
 
 then, XD = i(60+15) = 37.5 
 
 and BD = i(60 — 15) =22.5 
 
 Then, in the triangle ACD, to find the angle ACD, 
 a. o. AC (50) 8.301030 
 
 AD (37.5) 
 : sin D (90°) 
 
 sin A CD 48° 35' 25" 
 
 1.574031 
 10.000000 
 
 9.875061 
 
28 PLANE TRIGONOMETRY. 
 
 Then, in the triangle BCD, to find the angle BCD, ■ 
 a. c. BC (40) 8.397940 
 
 : BD (22.5) 1.352183 
 
 ::sinD (90°) 10.000000 
 
 : sin BCD 34° 13' 44" 9.750123 
 ■ Hence, A = 90° — 48° 35' 25" = 41° 24' 35", 
 and .B = 90° — 34° 13' 44" = 55° 46' 16", 
 
 and C = 48° 35' 25" + 34° 13' 44" = 82° 49' 09". 
 
 2. In a plane triangle the sides are 1005, 1210, and 1368 ; re- 
 quired the angles. 
 
 Ana. 45° 22' 35"; 58° 58' 18"; 75° 39' 7". 
 
 SOLUTION OF RIGHT-ANGLED TRIANGLES. 
 44. In the solution of right-angled triangles we have the four 
 following cases: 
 
 1. When the hypothenuse and an acute angle are given. 
 
 2. When the hypothenuse and a side are given. 
 
 3. When one side and the angles are given. 
 
 4. When the two sides about the right angle are given. 
 Method. — The first three cases are readily solved by Theorem 
 
 ... ; remembering that the sine of 90° is radius, the log. sin. being 
 10. The fourth case may be solved by Theorem V. ; or we may 
 find the hypothenuse by B. III. Th. VI., and then find the angles 
 by Theorem I. 
 
 These four cases may also be solved by Theorems IV. and V. ; 
 but the method suggested above is preferred, since it is simpler 
 and more easily remembered. 
 
 EXAMPLES. 
 
 1. In a right-angled triangle, given the hypothenuse 475 and 
 the angle at the base 36° 34' ; find the other parts. 
 
PLANE TRIGONOMETRY. 29 
 
 Solution. —Let CAB represent the triangle, BC being equal 
 to 475 and the angle C = 36° 34' ; then, to find 
 AB, we have, 
 
 sin A 90° a. c. 0.000000 
 
 : sin C 36° 34' 9.775070 
 
 : : CB 475 2.676694 
 
 : AB 282.985 2.451764 
 
 The angle B = 90° — (36° 34') = 53° 26 ; then, by a similar 
 proportion, we can find the side CA = 381.503. 
 
 2. Given the hypothenuse 45.36 and the angle at the base 45° 
 36' ; required the other parts. Ans. 
 
 3. Given the hypothenuse 396 and the base 218, to find the 
 other parts. - Ans. 330.59; 33° 24' 05"; 56° 35' 55". 
 
 4. Given the two sides 58.75 and 74.58, to find the remaining 
 parts. Ans. 94.94; 38° 13' 45"; 51° 46 / 15". 
 
 PRACTICAL APPLICATIONS. 
 
 HEIGHTS AND DISTANCES. 
 
 45. A Horizontal Plank is one which is parallel to the 
 plane of the horizon. 
 
 ''46. AVertical Plane is one which is perpendicular to a 
 horizontal plane. 
 
 47. A Horizontal Line is any line in a horizontal plane. A 
 vertical line is a line perpendicular to a horizontal plane. 
 -" 48. A Horizontal Angle is an angle in a horizontal 
 plane. 
 
 49. A Vertical AHGLiis ar* angle in a vertical plane. 
 
 50. An Angle op Elevation is a vertical angle having one 
 
 18* 
 
30 
 
 PLANE TRIGONOMETRY. 
 
 c 
 
 V 
 
 
 4 
 
 side horizontal, and the inclined side above 
 the horizontal side ; as BAD. 
 
 51. An Angle of Depression is a ver- 
 tical angle having one side horizontal, and 
 the inclined side under the horizontal side; 
 as CDA. 
 
 52. Distances upon the ground are usually 
 measured by a chain, called Gunter's Chain. 
 Dr 66 feet long, and consists of 100 links 
 chain is used, consisting of 50 links. 
 
 53. Angles are measured by various instruments. Horizontal 
 angles are measured by an instrument called The Compass. Hori- 
 zontal and vertical angles are both measured by the Theodolite, or, 
 what is still better for general use, a Transit-Theodolite. 
 
 This chain is 4 rods 
 Sometimes a half 
 
 CASE I. 
 
 • 
 
 54. To determine the height of a vertical object standing upon a hori- 
 zontal plane. 
 
 Method. — Measure from the foot of the a 
 
 object any convenient horizontal distance 
 AB; at the point A take the angle of ele- 
 vation BAC; then, in the triangle ABC we 
 have a side and an acute angle ; hence, we 
 can readily find the altitude. 
 
 1. From the foot of a tower I measure a 
 horizontal line 120 feet, and at its extremity find the angle of 
 elevation to be 48°. 36'; what was the height of the tower ? 
 
 Ans. 136.113 feet. 
 
 
 
 CASE II. 
 55. To find the distance of a vertical object whose height is known. 
 Method. — Measure the angle of elevation to the top of the 
 object, as before; we will then have a right-angled triangle in 
 
PLANE TRIGONOMETRY. 
 
 31 
 
 which we know the perpendicular and an acute angle ; hence, wa 
 can "readily find the base. 
 
 D ' C 
 
 1. I took the angle of elevation to the 
 
 top of a flagstaff whose height I knew to 
 
 be 160 feet, and found it to be 20°; how far 
 
 was I from the staff? 
 
 Ans. 439.60 feet. 
 
 CASE III. 
 
 56. To find the distance of an inaccessible object.. 
 
 Method. — Measure a horizontal base-line AB, and then take 
 the angles formed by this line and lines 
 from the object to the extremities of this 
 base-line, as CAB and ABC; the distance 
 .4 Cor BC can then be readily found. 
 
 1. I am on one side of a river, and wish 
 to know the distance to a tree on the 
 other side. I measure 300 yards by the 
 side of the river, and find that' the two 
 
 angles formed by this line and the lines from its extremities to 
 the tree are 72° 40' and 45° 36'; required the distance from 
 each extremity of the base-line to the tree. 
 
 Ans. 243.362 yards ; 325.15 yards. 
 
 CASE IV. 
 
 57. To find the distance between two objects separated by an impassable 
 barrier. 
 
 Method. — Select any convenient station, 
 as C, and measure the distance from it to 
 each of the objects A anrf-B, and the angle 
 C included between these lines. We can 
 then readily find the distance AB. 
 
 1. The distance between two trees can- 
 riot be directly measured : I therefore take a third position from 
 
32 
 
 PLANE TRIGONOMETRY. 
 
 which each of the trees can be seen, and find the distances from 
 it to the trees to be 300 and 250 yards, and the included angle 
 43° 16' ; required the distance between the trees. 
 
 Ans. 208.02 yards. 
 
 CASE V. 
 
 58. To find the height of a vertical object standing upon an inclined 
 plane. 
 
 Method. — Measure any convenient ' js^ 
 
 distance AD on a line from the foot 
 of the object, and at the point D mea- 
 sure the angles of elevation, EDA and / / -^E 
 .EZXB, to foot and top of the tower By 
 means of the two triangles DEA and 
 DEB, we can find the height of AB. 
 
 1. Wishing to determine the height of a tower situated upon 
 a hill, I measured a distance down the slope of the hill 400 feet, 
 and found the angles of elevation to the foot of the tower 42'' 28', 
 and to the top of the tower 68° 42' ; required the height of the 
 tower. Ans. 486.747. 
 
 CASE VI. 
 
 59. To find the height of an inaccessible object above a horizontal plane. 
 
 First Method. — Measure any con- 
 venient horizontal line AB directly 
 toward the object, and take the an- 
 gles of elevation at A and B; we will 
 then have conditions sufficient to 
 find DC. 
 
 1. "Wishing to find the altitude of 
 a hill, I measured the, angle of elevation at the bottom 60° 37', 
 and 460 feet from the foot in a right line of the top of the hill 
 and the point at the foot, and in the same horizontal plane as the 
 foot, I measured the angle of elevation 36° 52'; required the 
 height of the hill. Ans. 597.092. 
 
 Second Method. — If it is not convenient to measure a horizontal base'. 
 
PLANE TRIGONOMETRY. 
 
 33 
 
 tine towards the object, w& measure any 
 line AB, and also measure the hori- 
 zontal angles BAD, ABD, and the 
 angle of elevation DBC.\ Then, by 
 means of the two triangles ABD and 
 CBD, the height CD can be found. 
 
 CASE VII. 
 
 60. To find the distance between two inaccessible objects when points can 
 be found at which both objects can be seen. 
 
 Method. — The method of measurement 
 is indicated in the following problem. 
 The method of solution we prefer leaving 
 to the ingenuity of the pupil, that he may 
 learn to think for himself. 
 
 1. "Wishing to know the horizontal dis- 
 tance between a tree and house on the 
 opposite side- of a river, I took the fol- 
 lowing measurements : 
 
 AB = 400 ; CAD = 56° 30', 
 BAD = 42° 24' ; ABC= 44° 36', 
 and DBC= 68° 50'. 
 Eequired the distance CD. Ans. 747.913. 
 
 CASE VIII. 
 
 61. To find the distance between two inaccessible objects when no points 
 can be found from which both objects can be seen. 
 
 Method. — The method is in- 
 dicated in the following pro- 
 blem and figure. This and 
 the following case may be 
 omitted with young pupils. 
 
 1. "Wishing to know the hori- 
 zontal distance between two in- 
 
34 
 
 PLANE TRIGONOMETRY. 
 
 accessible objects when no point can be found from which both 
 objects can be seen, two objects Cand D are taken, 600 feet apart, 
 from the former of which A can be seen, from the latter -B.. From 
 .Cwe measure the distance CF, not in the direction DC, equal to 
 600 feet, and from D a distance DE equal to 600 feet. We then 
 measure the following angles : 
 
 CFA = 80° 16', BED = 86° 25', 
 
 ACF= 52° 24', BDE = 60° 24', 
 
 ACD= 56° 36', BDO= 150° 30'. 
 Required the distance AB. Ans. 1117.44 feet. 
 
 CASE IX. 
 
 62. To find the distances from a given point to three objects whose dis- 
 tances from each other are known. 
 
 Method. — The method is indicated in 
 the problem and figure. 
 
 1. I wish to locate three buoys, A, B, and 
 C, in a harbor, so that the distance between 
 A and B is 800 yards, between A and C 
 600 yards, between B and C 400 yards, and 
 from a fixed point on shore, the angle. 
 APC shall equal 33° 45', and BPC 22° 
 30' ; required the distances PA, PC, and 
 PB. 
 
 Ans. PA = 710.193 ; PC= 1042.522 ; PB = 934.291. 
 
 Note. — This last problem is given by quite a number of authors, and seem? 
 to ba general property. 
 
ANALYTICAL TRIGONOMETRY. 35 
 
 ANALYTICAL TRIGONOMETRY. 
 
 63. Analytical Trigonometry is that branch of Mathe- 
 matics which treats of the properties and relations of trigono- 
 metrical functions. 
 
 64. Trigonometry,' in its origin, was confined to triangles, the 
 method of reasoning being geometrical. After the invention of 
 
 - analysis, mathematicians began to apply it to trigonometry, and, 
 in course of time, developed the general properties of trigono- 
 metrical functions. This has enlarged the science and greatly 
 increased its power as an instrument of investigation and dis- 
 covery. 
 
 DEFINITIONS. 
 
 63. A circumference consists of four quadrants. AB is the first 
 quadrant; BG is the second quadrant, etc. 
 
 66. The origin of arcs is at A, all arcs being 
 generally supposed to begin at A. 
 
 67. The extremity of an arc is where it ends. 
 An aro is said to be in that quadrant where 
 its exti-emity is situated. 
 
 68. The sine, cosine, tangent, cotangent, 
 etc. of an arc have already been defined, 
 
 and need not be repeated here. ■ The versed sine of an arc is the 
 distance from the foot of the sine to the origin of the arc. The 
 co-versed sine is the versed sine of the complement. 
 
 The sines, cosines, etc. are called the circular functions of the 
 arcs. 
 
 69. Fundamental formulas expressing the relation between 
 the circular functions of any arc. 
 
 1. Let a represent the measuring arc of any angle. Draw the 
 lines represented in the figure. Then, from the definitions/ 
 
36 ANALYTICAL TRIGONOMETRY. 
 
 AB =1, BE = tan a, 
 
 CD = sin a, AE = sec a, 
 
 AD = cos a, DB = ver sin a. 
 
 In the right-angled triangle ADC, we have, 
 CD 2 + AD 1 = AC 2 , or, by substitution, 
 sin 2 a + cos 2 a = 1. (1) 
 
 Hence, sin 2 a=l — cos 2 a; (2) cos 2 a = l — sin 2 a. (3) 
 
 2. From the figure, we also have, 
 
 DB = AB — AD; that is, 
 
 ver sin a = 1 — cos a. " (4) 
 
 Since this is true for any value of a, it is true for 90° — a; 
 hence, ver sin (90° — a) = 1 — cos (90° — a), 
 
 or, co-ver sin a = 1 — sin a. (5) 
 
 3. Again, the triangles ADC and ABE being similar, 
 
 EB : AB : : CD : ^LD, 
 or, tan a : 1 : : sin a : cos a; 
 
 , sin a 
 
 hence, tan a = ■ (6) 
 
 cos a v ' 
 
 Substituting 90° — a for a, we have, 
 
 ,„™ , sin (90° — a) 
 
 tan (90° — a = ~ {, 
 
 K ' cos (90 — a) 
 
 cos a 
 or, cot a = — — . f 7 ) 
 
 sin a v ' 
 
 4. Again, multiplying equations (6) and (7), we have, 
 
 tan a cot a = 1 ; (8) 
 
 hence, tan a = — r— (9), and cot a = . (10) 
 
 cot a v " tan a v ' 
 
 5. Again, from the same triangles, we have, 
 
 AE : AB : : AC : AD, 
 or, sec a : 1 : : 1 : cos a; 
 
 hence, seea = • (11) 
 
 ' cos a * ' 
 
 Substituting 90° — a for a, 
 
 sec(90°-a) = co8(9( ; o _ g) , 
 
 or. cosec a = — — « (121 
 
ANALYTICAL TRIGONOMETRY. 
 
 37 
 
 6. Again, from the triangle ABE, we have, 
 
 sec 2 a = 1 + tan 2 a ; (13) 
 
 hence, cosec 2 a = 1 + cot 2 a. (14) 
 
 70. These are the fundamental formulas of trigonometry, and 
 should be committed to memory. We will collect thern, forming 
 the following table : 
 
 1. Sin 2 a -\- cos 2 a = 
 
 2. Sin 2 a 
 
 3. Cos 2 a 
 
 4. Ver sin a 
 
 5. Co-ver sin a 
 
 6. Tan a 
 
 7. Cot a 
 
 8. Tan a cot a 
 
 = 
 
 
 1 
 
 = 
 
 1 
 
 — cos 2 a 
 
 = 
 
 1 
 
 — sin 2 a 
 
 = 
 
 1 
 
 — cos ffi 
 
 = 
 
 1 
 
 — sin a 
 
 = 
 
 
 sin a 
 cos a 
 
 
 
 cos a 
 
 
 
 sin a 
 
 
 
 1 
 
 
 
 9. Tan o = 
 
 10. Cot a = 
 
 11. Sec a = 
 
 12. Co-sec a = 
 
 13. Sec 2 a = 
 
 14. Co-sec 2 a = 
 
 cot a 
 
 1 
 tan a 
 
 1 
 cos a 
 
 1 
 sin a 
 l + tan 2 a 
 1 -f- cot 2 a 
 
 ALGEBRAIC SIGNS OF- THE CIRCULAR FUNCTIONS. 
 
 71. In analytical trigonometry, we regard the algebraic signs of 
 the functions as well as their numerical value. The sign of a 
 function is determined by the following principles. 
 
 1. All lines estimated upward from the horizontal diameter are positive; 
 all lines estimated downward from it are negative. 
 
 2. All lines estimated from the vertical diameter towards the right are 
 positive ; all lines estimated toward the left are negative. 
 
 Thus, the sines NP and WP' are 
 positive, while N"P" and N"'P'" 
 are negative ; so also the cosines OP 
 and OP'" are positive, while OP' and 
 OP" are negative. 
 
 72. The simplest way to determine 
 
 the algebraic signs of the different 
 
 functions is to derive those of the 
 
 sine and cosine from'the figure, and 
 
 the others from the formulas. 
 
 18 
 
38 ANALYTICAL TRIGONOMETRY. 
 
 1. The Sine is positive in the first and second quadrants, being 
 measured above, and negative in the third and/ourtA quadrants. 
 
 2. The Cosine is positive in theirs* and fourth quadrants, and 
 negative in the second and third quadrants. 
 
 3. The Tangent is positive in the first and third quadrants, and 
 
 negative in the second and fourth. 
 
 For, from formula (6), 
 
 sin a 
 
 tan a = > 
 
 cos a 
 
 and this is positive when sine and cosine have like signs, and 
 
 negative when they have unlike signs. In the first quadrant, 
 
 both sine and cosine are plus, in the third both are minus, in 
 
 the second and fourth one is plus and the other minus ; hence, 
 
 the tangent is positive in the first and third quadrants and nega' 
 
 tive in the second and fourth. , 
 
 4. The Cotangent is positive in the first and third quadrants, 
 and negative in the second and fourth; as is readily shown from the 
 
 formula, 
 
 cos a 
 
 cot a = —. . 
 
 sin a 
 
 5. The Secant is positive in the first ondfourth quadrants, and 
 negative in the second and third. For, from formula (11), 
 
 1 
 
 sec a = ; 
 
 cos a 
 
 hence, the secant has the same sign as the cosine. 
 
 6. The Co-secant is positive in the first and second quadrants, 
 and negative in the third and fourth, as may be shown from For. (12). 
 
 Note. — Some of these may also be readily shown from the figure. In the 
 seaant, when the distance is estimated toward the extremity of the arc, it is 
 plus ; when from the extremity, minus. 
 
 LIMITING VALUES OF THE CIRCULAR FUNCTIONS. 
 
 73. The limiting values of the circular functions are their values 
 at the beginning and end of the different quadrants. 
 
 These values are determined by the principle that the value of a 
 variable quantity up to the limit is its value at the limit. 
 
ANALYTICAL TRIGONOMETRY. 
 
 39 
 
 Beginning at the origin, we see that the value of sin is 0, and the 
 cos is the radius, or 1. As the arc increases, the sine increases 
 and the cosine decreases, until at 90° the sine is 1 and the cosine 
 0. As the are increases from 90° to 180°, the sine decreases and 
 cosine increases numerically (diminishes algebraically), until at 
 180° the sine is + and cosine — 1. In the same way we see that 
 sin 270° = — 1, and cos 270° = — ; also, sin 360° =— 0, and co- 
 sine 360° =1. 
 Now, since, by formula (6), 
 
 sin a 
 
 cos a 
 
 sin 
 
 cos " 
 
 cosO 
 
 substituting for a, 
 and, also, 
 
 tan a = 
 tanO : 
 cot0 = 
 
 if- 
 
 1 
 
 sin " 
 
 74. By a similar examination of the limiting values of all tho 
 functions, we have the following table : 
 
 TABLE II. 
 
 Arc = 
 
 Arc = 90° 
 
 Arc = 
 
 = 180° 
 
 
 Arc = 
 
 = 270° 
 
 Arc = 
 
 = 360° 
 
 Bin = 
 
 
 
 sin = 
 
 1 
 
 Bin 
 
 = 
 
 
 
 sin 
 
 = — 1 
 
 sin 
 
 = — 
 
 cos = 
 
 1 
 
 cos =■ 
 
 
 
 cos. 
 
 — — 
 
 1 
 
 cos 
 
 — — o 
 
 cos 
 
 = 1 
 
 v-sin = 
 
 
 
 v-sm = 
 
 1 
 
 v-sin 
 
 = 
 
 2 
 
 v-sin 
 
 = 1 
 
 v-sin 
 
 = o 
 
 co-v-sin = 
 
 1 
 
 co-T-sm = 
 
 
 
 co-v-sin 
 
 = • 
 
 1 
 
 co-v-sin 
 
 = 2 
 
 co-v-Bin 
 
 = 1 
 
 tan = 
 
 
 
 tan = 
 
 OO 
 
 tan 
 
 = — 
 
 
 
 tan 
 
 = OO 
 
 tan 
 
 = _ o 
 
 cot = 
 
 00 
 
 cot — 
 
 
 
 cot 
 
 = — 
 
 OO 
 
 cot 
 
 = 
 
 cot 
 
 = CO 
 
 sec = 
 
 1 
 
 see = 
 
 OO 
 
 sec 
 
 = — 
 
 1 
 
 sec 
 
 = OO 
 
 sec 
 
 = 1 
 
 cosec = 
 
 00 
 
 COB6C = 
 
 1 
 
 cosec 
 
 = 
 
 OO 
 
 cosec 
 
 = — 1 
 
 cosec 
 
 = CO 
 
 FUNCTIONS OF THE SUM OR DIFFERENCE OF AN ARC AND 
 ANY NUMBER OF QUADRANTS. 
 
 75. The trigonometrical function of any arc formed by adding 
 an arc to or subtracting it from any number of quadrants, may be 
 expressed in functions of the arc which is added to or subtracted 
 from. 
 
 1. Let a represent any arc less than 90°; then, from the defini- 
 tions, we have, 
 
4C ANALYTICAL TRIGONOMETRY. 
 
 sin (90° — a) = cos a, cot (90° -a) = tan a, 
 
 cos (90° — a) = sin a, sec (90° — a) = cosec a, 
 
 tan (90° — a) — cot a, cosec (90° — a) = sec a. 
 
 2. Now, let a represent the arc -BJV', then will ABW =^90° + a. 
 From the figure, Art. 71, we see that 
 
 WM' = sin a, ilf'O =cosa, 
 
 P'O = cos (90° + a), JV'P' = sin (90° + a). 
 Hence, remembering that ABW, being in the second quadrant, its 
 cosine is negative, we have, 
 
 sin (90° + a) = cos a, and cos (90° + a) = — sin a. 
 Substituting these values in the formulas for tan, cot, etc. found 
 in Table L, we have, 
 
 tan (90° + a) — — cot a, sec (90° + a) = — cosec a, 
 cot (90° + a) = — tan a, cosec (90° -|- a) = sec a. 
 
 3. Again, let a represent the arc CW, then will ABW = 180 — a. 
 From the figure, we have, 
 
 _ZV" / P / = sin a, P'0= cos a, 
 
 2^' = sin (180 — a), P / 0= cos (180 — a). 
 
 Hence, remembering that the cosine of ABW ending' in tl*e 
 second quadrant is negative, we have, 
 
 sin (180° — a) = sin a, and cos (180° — a) = — cos a. 
 Substituting these values in the formulas for tan, cot, etc. in Table 
 I., we have, 
 
 tan (180° — a) = — tana, sec (180° — a) = — seca, 
 cot (180° — a) = — cot a, cosec (180° — a) = cosec a. 
 From the above, we see that the sine of an arc equals the sine of its 
 tupplement, and the cosine of an arc equals minus the cosine of its sup- 
 plement, etc. 
 
 76. In a similar manner, by deriving the values of the sines 
 and cosines from the figure and making the substitutions in ths 
 proper formulas, we may obtain the functions of 180°+ a, 270° — a, 
 270° + a, and 360° — a. All of these, with the above, are exhi' 
 bited in the following table : 
 
ANALYTICAL TRIGONOMETRY. 
 
 41 
 
 Bin = 
 
 COS: 
 
 tan: 
 
 sin = 
 
 COS: 
 
 tan: 
 
 sin: 
 
 COS: 
 
 tan: 
 
 TABLE III. 
 
 Arc = 90° + a. 
 cos a, cot = — tana, 
 -sin a, sec = — cosec a, cos=- 
 -cota, cosec = sec a. tan = 
 
 Arc =180° — a. 
 
 sin a, cot = — cot a, 
 -cos a, sec = — sec a, 
 -tan a, cosec = cosec a. 
 
 Arc = 180° + a. 
 
 - siu a, cot = cot a, 
 
 - C9S a, sec = — sec a, 
 tan a, cosec = — cosec a. 
 
 Arc = 270°-* a. 
 - cos a, cot = tan a, 
 ■ sin a, sec = — cosec a, 
 
 cot a, cosec = — sec a. 
 
 sm = 
 cos = 
 tan: 
 
 SID : 
 
 cos = 
 tan: 
 
 Arc = 270° -fa 
 -cos a, cot = — 
 
 sin d, sec = 
 -cot a, cosec = — 
 
 tan a, 
 cosec a,, 
 sec a. 
 
 Arc = 360° — a. 
 -sin a, cot = — cot a, 
 
 cos a, sec = sec a, 
 -tana, cosec = — cosec a. 
 
 77. This table can easily be committed to memory, by observing 
 that when the arc is connected with 180° or 360°, the functions in 
 both columns have the same name; but when connected with 90° 
 or 270°, the functions in the two columns have different names. 
 
 78. The principles of this table are of great value. By their 
 means the functions of any arc may be expressed in functions of an arc 
 lest than 90°. Thus, 
 
 sin. 120° = sin ( 90° + 30°) = cos 30°, 
 tan 243° = tan (180° + 63°) = tan 63°, 
 cot 304° = cot (270° + 34°) = — tan 34°. 
 
 79. When the arc is greater than 360°, we may subtract 360° 
 one or more times until we obtain an arc less than 360°; the 
 remainder will have the same origin and extremity: hence, the cir- 
 cular function of the remainder will be the same as of the given 
 arc, and this remainder being less than 360°, its functions can be 
 expressed in functions of an arc less than 90°. Hence, the functions 
 of any arc can be expressed in functions of an arc less than 90°. 
 
 19* 
 
42 ANALYTICAL TRIGONOMETRY. 
 
 CIRCULAR FUNCTIONS OF NEGATIVE ARCS. 
 
 80. Suppose AB to be any arc, and AC, esti- 
 mated from the origin downward, be numeri- 
 cally «3qual to AB; then, if the arc AB be J 
 denoted by a, the arc AC will be denoted by 
 — a; and CD will be the sine, and OD the co- 
 sine, of — a. 
 
 Now, since BD = CD and OD is the cosine of both a and — a, 
 
 we have, 
 
 sin ( — a) = — sin a, and cos ( — a) = cos a. 
 
 Substituting these in the formulas of Table I., we will have, 
 
 ver sin ( — a) = ver sin a, cot ( — a) = — pot a, 
 co-ver sin ( — a) =± 1 + sin a, sec ( — a) = sec a, 
 
 tan ( — a) = — tan a, co-sec ( — a) = — co-sec a. 
 
 81. From what has now been presented, we see that the circular 
 functions .of all arcs, whether positive or negative, may be ex- 
 pressed in functions of arcs less than 90°; hence, in the tablet of 
 sines, cosines, etc., we have only positive arcs and those less than 
 90°. 
 
 RELATION OF THE SIDES AND FUNCTIONS OF RIGHT-ANGLED 
 TRIANGLES. 
 
 82. Let ACB be a right-angled triangle, the right angle being 
 at A. Represent the angles by A, B, C, and 
 
 their opposite sides by a, b, c. - With a radius 
 \CE= 1, describe the arc EF, and draw the 
 perpendicular ED; then ED = sin C, and 
 GD = cosC. 
 Now, from the figure, we readily obtain, 
 1 : sir C ::a:c, 
 
 and, also, 
 
 1 : cob C :: a:b; 
 
 hence, 
 
 sin C^~ a (1), cosC=- (2), 
 
 or, 
 
 c = asih C (3), b =acos C (4). 
 
ANALYTICAL TRIGONOMETRY. 
 
 43 
 
 Dividing (1) by (2) and then (2) by (1), we have, 
 
 tan C = ■ 
 
 (5), cotC = - 
 
 (6), 
 
 ' b c 
 
 or, c = 6tan C(7), and J = ccot C (8). 
 
 83. These the pupil will commit to memory, and also translate 
 into common language. The first, thus translated, is as follows : 
 
 1. The sine of either acute angle of a right-angled triangle is. equal to 
 the opposite side divided by the hypothenuse. 
 
 84. General formulas relating to the sum and difference 
 of arcs, double arcs, etc. 
 
 1. Let AB and BG be two arcs having the common radius 
 OA or OC= 1 ; denote AB by b and BC by a. 
 From G draw CD perpendicular to OA, and 
 CN perpendicular to OB; from N draw NE 
 perpendicular to OA, and NM parallel to 
 OA. Then, 
 
 CD = sin (a + b), CN=sma, ON = coua. 
 Now, CD=CM+NE. 
 
 In the triangle OEN, 
 
 NE = ON sin B = cos a sin b; 
 since CMN and NOE are similar, and the angle MCN= NOE= 5, 
 
 CM = CN cos b = sin a cos b. 
 Substituting these values in equation (1), we have, 
 
 sin (a + b) = sin a cos b + cos a sin b. (A) ~ 
 
 This formula expresses the value of the sine of the sum of two 
 arcs in terms of the sine and cosine of the single arcs. It is 
 enunciated as follows : 
 
 The sine of the sum of two arcs or angles is equal to the sine of the first 
 into the cosine of the second, plus the cosine of thefltst into the sine of the 
 
 D E A 
 
 2. If in formula (A) we substitute — b for b, we have, 
 
 sin (a — b) = sin a cos ( — b) + cos a sin ( — b) ; 
 but (Art. 80) cos( — b) =cosi, and sin (— &)= — sin6; 
 hence 1 , sin (a — 5) = sinacos6 — cosasinJ. (B) 
 
44- ANALYTICAL TRIGONOMETRY. 
 
 3. If in formula (B) we substitute 90° — a for a, we have, 
 
 sin (90°— a — b) =sin (90°— a) cosb — cos (90° — a) sini,- 
 but, sin (90°— a — &) = sin(90°— (a + b))=00B{a + b), 
 and, sin (90° : — a) = cos a, and cos (90 — a) = sin a; 
 hence, cos (a + b ) = cos a cos & — sin a sin 6. (C) 
 
 4. Substituting — b for 6 in formula (C), we have, 
 
 cos (a — b) = cos a cos ( — 6) — sin a sin ( — b), 
 or, cos (a — b) = cos a cos b-\- sin a sin J. (D) 
 
 5. From Table I., For. (6), and formulas (A) and (C), we have, 
 
 . sin (a + 6) sin a cos b + cos a sin b 
 
 tan (a + 6) = -. — ttt= 1 : ; — v 
 
 x ' cos(a-)-o) cos a cos 6 — sin a sin 6 
 
 Dividing both terms of the last member by cos a cos b, we have, 
 
 sin a cos b cos a sin b 
 
 , , . . cos a cos b cos a cos b 
 
 tan (a + o) = ; : — ; . 
 
 sin a sin 6 
 
 cos a cos b 
 
 Cancelling common factors, and reducing, we have, 
 
 tan a 4- tan b ,„ 
 
 tan (a + 5) = 1 _ tanatan& . (E) 
 
 6. Substituting — 5 for b in formula (E), and reducing, we have, 
 
 , ,. tan a — tan b ,„, 
 
 tan( a -6) = 1 + tanatan& . (F) 
 
 7. Dividing formula (C) by (A), and reducing as in (5), we have, 
 
 it , i\ cotacoti — 1 
 
 cot (a + 6) = — n~-, 1 — • (Cr) 
 
 ^ ' i cot b + cot a v ' 
 
 8. Substituting — b for 6 in formula (Gr), and reducing, we have, 
 
 cot a cot b 4- 1 ,„, 
 
 cot ("-*)= cot*-.coL - ( H ) 
 
 85. Formulas tor double and half arcs. 
 
 1. Making a = b in formulas (A), (C), (E), and (Gr), we have, 
 sin 2 a = 2 sin a cos a; (A') 
 
 cos 2 a = cos 2 a — sin 2 a, ((7) 
 
ALYTICAL TRIGONOMETRY. 
 
 45 
 
 2 tan a 
 
 [W) 
 
 1 — taira 
 
 cot 2 a — 1 
 cot2a = — s — i — •. 
 
 (G')' 
 
 2. If now in (C) we put 1 — sin 2 a for cos 2 a, and then 1 -^ cos 2 a 
 
 for sin 2 a, we have, 
 
 cos2<z = l — 2 sin 2 a, (1) 
 
 cos 2 a = 2 cos 2 a — 1, (2) 
 
 from which we have, 
 
 /l — cos2a 
 sin a = -y g , ( A") 
 
 / l + cos 2 a 
 cosa = ^/ 2 ■ (C") 
 
 Dividing (A") by (C") and then (C") by ( A"), multiplying nume. 
 rator and denominator by the denominator, and reducing, 
 
 sin 2 a 
 tana = 1 + cos2a , (E") 
 
 cota^ /" 2 " ■ (G".). 
 
 1 — cos 2 a \ t 
 
 3. Now, substituting J a for a in (A 7 '), (C"), (E"), and (G"), 
 
 . / ! — cos a 
 sin A. a = -yj ^ , ( Aj) 
 
 /1 + cos a 
 cos i a = -^1 g- , (d) 
 
 sin a 
 
 tan A <z = , — ; , (Ei) 
 
 * 1 + cos a' K ' 
 
 sin a 
 
 cot ia= = . (&!) 
 
 2 1 — cos a v ' 
 
 Taking the reciprocals of (Ei) and (Gi), we have, 
 
 1 + cos a 
 a sin a v ' 
 
 1 — cos a 
 tan A a = — : ; (G») 
 
46 ANALYTICAL TRIGONOMETRY. 
 
 86. Additional formulas. 
 
 1. Adding and subtracting formulas (A) and (B), and doing 
 the same with (C) and (D), we have, 
 
 sin (a + b) + sin (a — b) = 2 sin a cos b, (1) 
 
 sin (a + £) — sin (a — b) = 2 cos a sin b, (2) 
 
 cos (a + b) + cos (a — b) =2 cos a cos b, (3) 
 
 cos (a — b) — cos (a + 6) =2 sin a sin b. (4) 
 
 2. Now, making 
 
 a-\-b =p and a — b = q, 
 -whence, a = i (p + g)and6 = £ (p — g) ; 
 
 and substituting these in the above, and we have, 
 
 sin^ + sin g = 2 sin i(p + q) cos i{p — q), (K) 
 
 sinp — sin q = 2 cos i{p + q) sin £ (^> — q), (L) 
 
 cos^> + cosg = 2cos£ C?> + g) cosij (p — q), (M) . 
 cosq — cos^> = 2sin£ (.p + g) sin £ (^> — g). (N) 
 
 3. Now, dividing (K) by (L), 
 
 sin p + sin q ^ sin £ (p + g) cos £ (ff — g) _ tanj (p+ q) ^ 
 sharp — sin q cos £{p + q) sin J {p — q) tan £ (p — q)' ^ ' 
 In a similar manner, we obtain 
 
 sinp + sing 2sin£(p + g)cos£b-g) 
 
 cosp + cosg 2 cos £ (p + g) cos i [p — q) * ^ *" v ^' 
 
 sin y -sin ? = 2smj(^- g )cosj(p + ? ) = ta 
 eos^>+cosg 2 cos i (p-f-q) cos £ (^ — q) * ^ 3 ' v ' 
 
 sin p + sin q _ 2smj [p + q) cos j [p — q) = cos jjp — q ) 
 sin(j» + g) 2sin£(p + g) cos£ Qo + g) cosi(p + g)' *• 
 sinp— sing _ 2 sin £(jp — g) cos£ (p + q ) _ sin £ (p — q) 
 sin (p + g) 2 sin i (j» + g) cos £ (p + q) sin i (_p + q)' * ' 
 sin (p — q) _ 2sml(p — q)cosi{p — q) _ cosb{p — q) 
 sins — sing 2sini(p — g)cos£(^? + g) cos \ [p + g)' * 
 
 These formulas may be enunciated in propositions ; thus for- 
 mula (P) gives, 
 
 The sum of the sines of two arcs is to the difference of their sines as the 
 tangent of one-half of the sum of the arcs is to the tangent of one-half of 
 their difference. 
 
ANALYTICAL TRIGONOMETRY. 47 
 
 Comparing (8) and (U), we have, 
 
 sin (p — q) sin p -\- sin q 
 smp — sing* sin (p + q) ' 
 
 Hence, the sine of the difference of two arcs is to the difference of their 
 sines as the sum of the sines is to the sine of the sum. 
 
 INTRODUCTION OF THE RADIUS. 
 
 87. In the preceding formulas, the radius, being unity, does not 
 appear in any of the terms. When the radius is other than a 
 unit, it should appear in these formulas, and we will now show 
 how it may be introduced. 
 
 Let a be an arc whose radius is 1, and a' 2> 
 
 be an arc whose radius is R; then, by simi- 
 lar triangles, 
 
 sin a: sin a': : 1 : R; _^ 
 
 „ . . sin a' *A E B 
 
 hence, sina / = JcX sma; sma=— =5— ; 
 
 and the same may be shown for the other circular functions. 
 
 Therefore, any circular function whose radius is R is equal to the cir- 
 cular function whose radius is 1, multiplied by R. 
 
 Also, any circular function whose radius is 1 is equal to the circular 
 function whose radius is R, divided by R. 
 
 Now, if we substitute these in any of the formulas, we will find 
 that R will be introduced in such a manner as to make the for- 
 mulas homogeneous. Thus, For. 6, Tab. I., gives, 
 
 sin n? 
 
 tan a' R R sin a' 
 
 — — = .; or, tana' = - r . 
 
 R cos a' cos a' 
 
 R 
 Here, tan a' is a line, and R sin a' -h- cos a' is a surface divided by a 
 line, which is also a line; hence, the formula is homogeneous. And 
 since the same is generally true, therefore, we can introduce the 
 radius in any formula by multiplying or dividing by R, so as to 
 make the formula homogeneous. 
 
48 ANALYTICAL TRIGONOMETRY. 
 
 CALCULATION OF A TABLE OF NATURAL SINES. 
 
 88. The circumference of a circle whose diameter is 1 is 3.14159 . . . .; 
 hence, when the radius ia 1, the semi-circumference is 3.14159 .....; and 
 if we divide this by 10800, the number of minutes in 180°, the quotient,, 
 
 .000290888 , will be the length of an arc of one minute. Now, 
 
 this arc is so small that it does not differ materially from its sine ; hence, 
 we may assume .000290888 as the sine of one minute. 
 
 We then find the cosine of V by For. 3, Table I. Thus, 
 
 cos V = y/\ — sin* V = .999999957 (1) 
 
 To find the sine of other arcs, we take the formula under Art. 86, - 
 putting it in the form, 
 
 sin (a -f- °) = 2 sin a cos b — sin (a — b). 
 Now, make b = 1', and then in succession, a equal to V, 2', 3', etc., 
 and we have, 
 
 sin 2' = 2 sin V cos V — sin = .0005817764 
 
 sin 3' = 2 sin 2' cos V — sin V = .0008726646 
 
 sin 4' = etc. 
 
 We may thus obtain the sines of any number of degrees and minutes 
 up to 45°, the corresponding cosines being obtained from equation (1). 
 
 Then, since the sine of an arc equals the cosine of its complement, 
 etc., the sines and cosines of arcs between 45° and 90° are immediately 
 derived from those between 0° and 45°. 
 
 The tangents are found by dividing the sines by the cosines; the 
 cotangents are found by dividing the cosines by the sines, or by dividing 
 1 by the tangents. 
 
 CALCULATION OF A TABLE OF LOGARITHMIC SINES. 
 89. A table of logarithmic sines is computed from a table of natural 
 sines. The process is as follows : 
 
 For the logarithmic sine, take the logarithm of the natural sine, and 
 add 10. 
 
 For, let sin a represent the natural sine, and let Sin a represent the 
 sine to a radius of 10,000,000,000 ; then, Art. 87, 
 Sin a = sin ay^M; 
 
ANALYTICAL TRIGONOMETRY. 49 
 
 taking logarithms, we have, 
 
 log Sin a == log sin a -(- log R. 
 But log R = log 10,000,000,000 = 10 
 Hence, log Sin a = log sin a -)- 10. 
 
 In the same manner, we find the log cosine ; and in a similar manner, 
 from the formulas of Table I., we can find all the other logarithmic cir- 
 cular functions. 
 
 THEOREMS AND PROBLEMS. 
 We now present a few exercises for original thought. The first and 
 third are derived from a diagram ; the 5th by For. 2. Art. 84 ; several 
 which follow, by substituting values from Table L, obtaining an equation 
 involving but one unknown quantity, which can then readily be found ; 
 the others, by judicious substitutions and reductions 
 
 1. Prove that sin 60° = \ y 3, and cos 60° = \. 
 
 2. Prove that sin 30° = £, and cos 30° = \ i/3. 
 
 3. Prove that sin and cos of 45° equal J yl. 
 
 4. Prove that tan 45° = 1, and sec 45° = -j/2. 
 
 t/3 — 1 r/3 + 1 
 
 5. Provesml5°,orsin(60°— 45°)=-^y- 7 2-,andcosl5° = J: -2-^— . 
 
 6. Prove tan 15° = 2 — yS, and cot 15° = 2 + yS. 
 
 7. If sin a cos a = \ j/3 find sin a and cos a. 
 
 Ans. sin os = J y"& ; cos a = \. 
 
 8. If 3 sin a -\- 5 yS X cos a = 9, find sin a. Ans. sin a = \ or \. 
 
 9. If sin a (sin a — cos a) = ^, find sin a. Ans. sin a = -|. 
 10. If tan a = f , find sin a and cos a. Ans. sin a = £ ; cos a = J. 
 
 - 11. If tan a -f- cot a = 2, find tan a. Ans. tan a = 1. 
 
 12. Prove that tan 2 a — sin 2 a = tan 2 a sin 2 , a. 
 
 13. Prove that sec 2 a cosec 2 a = sec 2 a + cosec 2 a. 
 
 4 
 
 14. Prove that sin (30° + a) + sin (30° — a) = cos a. 
 
 15. Prove that cos (60° + a) + cos (60° — a) = cos a. 
 
 16. If a + b + c = 180°, prove that 
 
 tan a -\- tan J + tan c = tan a tan J tan c. 
 
 17. If a -f J + c = 90°, prove that 
 
 cot a + cot b -\- cot c = cot a cot 5 cot c. 
 
 Suggestion.— In 16th, tan (a + b) = tan (180° — 6), develop and simplify; 
 
 and similarly in 17th. 
 
 20 
 
A TABLE 
 
 OF 
 
 LOGARITHMS OF NUMBERS 
 
 From 1 to 10,000. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 N. 
 
 Log. 
 
 i 
 
 o-oooooo 
 
 26 
 
 1-414973 
 
 5i 
 
 1-707570 
 
 76 
 
 1-880814 
 
 2 
 
 o-3oio3o 
 
 11 
 
 i-43i364 
 
 52 
 
 I -716003 
 
 77 
 
 1-886491 
 1-892085 
 
 3 
 
 o • 477 1"2 1 
 
 1 -447158 
 
 53 
 
 1-724276 
 
 78 
 
 4 
 
 o- 602060 
 
 V) 
 
 i-4'>2398 
 
 54 
 
 1-732394 
 
 79 
 
 1-897627 
 
 5 
 
 0-698970 
 
 3o 
 
 1-477121 
 
 55 
 
 1 • 74o363 
 
 80 
 
 1 -90J090 
 1-908485 
 
 6 
 
 0-778151 
 
 3i 
 
 1-491362 
 
 56 
 
 1-748188 
 
 81 
 
 I 
 
 0-845098 
 
 32 
 
 1 -5o5i5o 
 
 u 
 
 1-755875 
 
 82 
 
 1-913814 
 
 ■ 903090 
 
 33 
 
 i-5i85i4 
 
 1-763428 
 
 83 
 
 1-919078 
 
 9 
 
 ■ 954243 
 
 34 
 
 1 -53 1479 
 
 5 9 
 
 1-770852 
 
 84 
 
 1-924279 
 
 10 
 
 I -oooooo 
 
 35 
 
 1 ■ 544068 
 
 60 
 
 1 -7-78151 
 
 85 
 
 1 -929410 
 
 II 
 
 i ; o4i393 
 
 36 
 
 1 • 5563o3 
 
 61 
 
 1 -78533o 
 
 86 
 
 1-934498 
 
 12 
 
 1-079181 
 1 - 1 1J943 
 
 U 
 
 1-568202 
 
 62 
 
 1-792392 
 
 87 
 
 1-939519 
 
 i3 
 
 1-579784 
 
 63 
 
 1-799341 
 
 88 
 
 1 -944483 
 
 U 
 
 1-146128 
 
 3 9 
 
 1 -591065 
 
 64 
 
 1-806181 
 
 89 
 
 1.949390 
 
 i5 
 
 1-176091 
 
 40 
 
 1 -602060 
 
 65 
 
 1-812913 
 
 90 
 
 1 -954243 
 
 16 
 
 1 -204120 
 
 41 
 
 1-612784 
 
 66 
 
 1-819544 
 
 9' 
 
 1 -959041 
 
 \l 
 
 1 > 23o44o 
 1-255273 
 
 42 
 
 1-623249 
 
 tl 
 
 1-826075 
 
 92 
 
 1-963788 
 
 43 
 
 1-633468 
 
 1-832509 
 
 93 
 
 1-968483 
 
 19 
 
 1-278754 
 
 44 
 
 1-643453 
 
 69 
 
 1-838840 
 
 94 
 
 1 -973128 
 
 20 
 
 1 -3oio3o 
 
 -45 
 
 1 -6532 1 3 
 
 70 
 
 1-845098 
 
 9 5 
 
 1.977724 
 
 21 
 
 I-32221D 
 
 1-342423 
 
 46 
 
 1-662753 
 
 71 
 
 1 -85i258 
 
 96 
 
 1 -982271 
 
 22 
 
 47 
 
 1 -672098 
 
 72 
 
 i-85 7 333 
 
 u 
 
 1-986772 
 
 23 
 
 1-361728 
 
 48 
 
 1-681241 
 
 73 
 
 1-863323 
 
 1 -991226 
 
 24 
 
 I- 38021 1 
 
 49 
 
 1 -690196 
 
 74 
 
 1-869232 
 
 99 
 
 1 -995635 
 
 25 
 
 * 
 
 1.397940 
 
 5o 
 
 1 -698970 
 
 75 
 
 1-870061 
 
 100 
 
 2 - oooooo 
 
 Eemabk. — In the following table, in the nine right- 
 hand columns of each page, where the first or lead- 
 ing figures change from 9's to O's, points or dots are 
 introduced instead of the O's, to catch the eye^ and to 
 indicate that from thence the two figures of the Log- 
 arithm to be taken from the second column, stand in 
 the next line below. 
 
2 
 
 A TABLE 
 
 OF 
 
 LOGARITHMS 'PROM ] 
 
 TO 
 
 10,000. 
 
 
 N. 
 
 100 
 
 
 
 1 
 
 2 
 
 3 
 
 4 | 3 | 6 | 7 
 
 8 
 
 9 
 
 1). 
 
 000000 
 
 0434 
 
 0868 
 
 l3oi| 17341 2166 2598I 3029! 3461 
 
 38 9 i 
 
 432 
 
 101 
 
 4321 
 
 475i 
 
 5i8i 
 
 5609' 6o38J 6466' 6894I 7321 7748 
 
 8174 
 
 428 
 
 102 
 
 8600 
 
 9026 
 
 945i 
 
 0876! »3oo' »724 : Ii47! «5lo 1993 
 
 241 5 
 
 424 
 
 io3 
 
 012837 
 
 3259 368o 4100 4521 
 
 4940 536ol 5779 
 
 6197 
 
 6616 
 
 419 
 
 104 
 
 7033 
 
 745 1 
 
 7868' 8284 
 
 8700 
 
 9116 9532 9947 
 3252 3V64 1 4075 
 
 •36 1 
 
 •775 
 4896 
 
 416 
 
 Io5 
 
 021 189 
 
 i6o3 
 
 2016 2428 
 
 2841 
 
 4486 
 
 412 
 
 106 
 
 53o6 
 
 5715 
 
 6i25 6533 
 
 6o42 
 
 735o| 7-571 8164 8571 
 
 8978 
 
 408 
 
 108 
 
 9384 
 
 9789 
 3826 
 
 •195 *6oo 
 
 1004 
 
 14081 18121 2216 
 
 2619 
 
 3021 
 
 404 
 
 o3c(424 
 
 4227 4628 
 
 5029 
 
 543oi 583o 
 
 6?3o 
 
 6629 
 
 7026 
 
 4"0 
 
 109 
 
 7426 
 
 7825 
 
 822! 8620 
 
 90(7 
 
 9414 981 1 
 3362! 3755 
 
 •207 
 4U8 
 
 •602 
 
 •998 
 
 4932 
 
 883o 
 
 3 9 6 
 
 110 
 
 041 3q3 
 
 1787 
 
 21S2 2576 
 
 Xl 
 
 4540 
 
 3 9 3 
 38 9 
 
 1 ' 1 
 
 5323 
 
 5714 
 
 6io5 6495 
 
 7275! 7664 
 
 8o53 
 
 8442 
 
 113 
 
 921S 
 05307S 
 
 9606 
 3463 
 
 oyy3 »38o 
 
 •766 
 
 1 1-53 
 
 i538i 1924 
 
 2309 
 
 2694 
 
 386 
 
 1 1 3 
 
 5 ->46 423o 
 
 461 3 
 
 4996 
 88o5 
 
 5378 5760 
 9i85i oo63 
 2 9 58j 3333 
 
 6142 
 
 6524 
 
 382 
 
 H4 
 
 6903 
 
 72R6 
 
 7666' 8046 
 
 8426 
 
 9942 
 
 •320 
 
 379 
 
 n5 
 
 060O9S 
 4458 
 
 1075 
 
 14J2 1825 
 
 2206 
 
 2582 
 
 3709 
 7443 
 
 4o83 
 
 376 
 
 n6 
 
 4«32 
 
 52o6 : 558o 
 
 5953 
 
 6326 
 
 6699 7°7 ' 
 
 78i5 
 
 3 7 2 
 
 "7 
 
 8180 
 
 8557 
 
 8928, 9298 
 2617' 2985 
 
 9668 
 3352 
 
 ••38 
 
 •407 
 
 •776 
 
 ii45 
 
 i5i4 
 
 36 9 
 
 nS 
 
 071882 
 
 225o 
 
 3 7 i8 
 
 4o85 
 
 445 1 
 
 4816 
 
 5i82 
 
 366 
 
 119 
 
 5547 
 
 5 VA 
 9543 
 
 3i44 
 
 6276 6640 
 
 7004 
 
 7368 
 
 773 1 
 
 8094 
 
 8457 
 
 8819 
 
 363 
 
 120 
 
 079181 
 
 9904 
 3oo3 
 
 •266 
 
 •626 
 
 •987 
 
 i347 
 
 1707 
 
 2067 
 
 2426 
 
 36o 
 
 121 
 
 082780 
 
 386 1 
 
 4219 
 
 4576 
 
 4934 
 
 5291 
 
 5647 
 9198 
 
 6004 
 
 357 
 
 122 
 
 636o i 6716 
 
 7071 
 
 7426 
 
 7781 
 
 8i36 
 
 8490 
 
 8845 
 
 9552 
 3071 
 
 355 
 
 i:3 
 
 0905 
 093422 
 
 •258 
 
 •611 
 
 • 9 63 
 
 i3i5 
 
 1667 
 
 2018 
 
 2370 
 
 2721 
 
 35i 
 
 124 
 
 3772 
 
 4122 
 
 4471 
 
 4820 
 
 5169 
 
 55i8 
 
 5866 
 
 62i5 
 
 6562 
 
 349 
 346 
 
 125 
 
 6910 
 100371 
 
 7257 
 
 7604 
 
 ; 9 5i 
 
 8298 
 
 8644 
 
 8990 
 2434 
 
 9335 
 
 9681 
 3i 19 
 
 ••26 
 
 126 
 
 0715 
 
 1059 
 
 i4o3 
 
 1747 
 
 2091 
 
 2777 
 
 3462 
 
 343 
 
 III 
 
 38o4 
 
 4U6 
 
 4487 
 
 4828 
 
 5i6g 
 8565 
 
 55io 
 
 585i 
 
 6191 
 
 653 1 
 
 6871 
 
 340 
 
 7210 
 
 7549 
 
 7888 
 
 8227 
 
 8903 
 
 9241 
 
 9 5 79 
 
 9916 
 3275 
 
 •253 
 
 338 
 
 129 
 
 110590 
 
 0926 
 
 1263 
 
 1 599 
 
 i 9 34 
 
 2270 
 
 26o5 
 
 2940 
 
 3609 
 
 335 
 
 i3o 
 
 1 13943 
 
 4277 
 
 46u 
 
 4944 
 
 6278 
 
 56u 
 
 5943 
 
 6276 
 9 586 
 
 6608 
 
 6940 
 
 333 
 
 i3i 
 
 7271 
 
 7603 
 
 ;g34 
 
 8265 
 
 85 9 5 
 
 8926 
 
 9256 
 
 9915 
 3i 9 8 
 6456 
 
 •245 
 
 33o~ 
 
 |32 
 
 120574 
 
 0903 
 
 I23l 
 
 i56o 
 
 1888 
 
 2216 
 
 2544 
 
 2871 
 
 3525 
 
 328 
 
 i33 
 
 3852 
 
 4178 
 
 45o4 
 
 483o 
 
 5i56 
 
 548! 
 
 58o6 
 
 6i3i 
 
 6781 
 
 325 
 
 134 
 
 7io5 
 
 7429 
 o655 
 
 7753 
 
 8076 
 
 8399 
 
 8722 
 
 9045 
 
 9 368 
 
 9690 
 
 ••12 
 
 323 
 
 i35 
 
 i3o334 
 
 0977 
 
 1298 
 
 1619 
 
 5$ 
 
 2260 
 
 258o 
 
 2900 
 
 3219 
 64o3 
 
 321 
 
 1 36 
 
 3539 
 
 3858 
 
 4ni 
 
 4496 
 
 4814 
 
 545 1 
 
 5769 
 
 6086 
 
 3i8 
 
 III 
 
 6721 
 
 0879 
 !43oi5 
 
 7037 
 
 7354 
 
 7671 
 
 7987 
 
 83o3 
 
 8618 
 
 8 9 34 
 
 9249 
 
 9D64 
 
 3i5 
 
 •194 
 
 •5o8 
 
 •822 
 
 n36 
 
 U5o 
 
 1763 
 
 2076 
 
 2389 
 
 2702 
 
 3i4 
 
 139 
 
 3327 
 6438 
 
 363g 
 6748 
 9835 
 
 3951 
 
 4263 
 
 4574 
 
 4885 
 
 5196 
 
 55o7 
 
 58 18 
 
 3n 
 
 140 
 
 146128 
 
 7058 
 
 7367 
 
 7676 
 
 7 9 85 
 
 8294 
 
 86o3 
 
 8911 
 
 309 
 
 141 
 
 9219 
 152288 
 
 9 52 7 
 
 •142 
 
 •449 
 
 • 7 56 
 38i5 
 
 io63 
 
 1370 
 
 1676 
 
 1982 
 
 307 
 
 142 
 
 2594 
 
 2900 
 
 32o5 
 
 35io 
 
 4120 
 
 4424 
 
 4728 
 
 5o32 
 
 3o5 
 
 143 
 
 5336 
 
 6640 
 
 5o43 
 
 6246 
 
 6549 
 
 6852 
 
 7154 
 
 745 7 
 
 •77 5 9 
 
 8061 
 
 3o3 
 
 144 
 
 8362 
 
 8664 
 
 8965 
 
 9266 
 
 9567 
 
 9868 
 
 •168 
 
 •469 
 
 •769 
 ,3758 
 
 1068 
 
 3oi 
 
 145 
 
 i6i368 
 
 1667 
 
 1967 
 
 2266 
 
 2564 
 
 2863 
 
 3i6i 
 
 3460 
 
 4o55 
 
 299 
 
 146 
 
 4353 
 
 465o 
 
 4947 
 
 5244 
 
 5541 
 
 5838 
 
 6i34 
 
 643o 
 
 6726 
 
 7022 
 
 297 
 
 147 
 
 7317 
 
 76i3 
 
 7908 
 0848 
 
 82o3 
 
 8497 
 U34 
 
 8792 
 
 9086 
 
 9 38o 
 
 967* 
 
 9968 
 2895 
 
 295 
 
 148 
 
 170262 
 
 o555 
 
 1141 
 
 1726 
 
 20.19 
 
 23 1 1 
 
 26o3 
 
 293 
 
 149 
 
 3i86 
 
 3478 
 
 3769 
 
 4060 
 
 43 5 1 
 
 4641 
 
 4o32 
 
 5222 
 
 55i2 
 
 58o2 
 
 2^9 
 
 i5o 
 
 176091 
 
 638 1 
 
 6670 6959 
 
 7248 
 
 7536 
 
 7825 
 
 8n3 
 
 8401 
 
 8689 
 1 558 
 
 i5i 
 
 8977 
 
 9264 
 
 9552 
 
 9839 
 
 •126 
 
 •4i3 
 
 •699! »985 
 3553, 383 9 
 
 1272 
 
 287 
 
 285 
 
 I 52 
 
 181844 
 
 2120 
 4973 
 78o3 
 
 241 5 
 
 2700 
 
 29H5 
 
 3270 
 
 4123 
 
 4407 
 
 153 
 
 4691 
 
 5259 
 
 55421 5825 
 
 6108 
 
 6391. 6674 
 
 6 9 56 
 
 7239 
 
 -283 
 
 154 
 
 7521 
 
 8084 
 
 8366! 8647 
 
 8928 
 
 92091 9490 
 2010 2289 
 
 9771 
 
 ^••5i 
 
 281 
 
 i55 
 
 190332 
 
 0612 
 
 0892 
 
 1171 i45i 
 
 1730 
 
 2567 
 
 2846 
 
 279 
 278 
 
 156 
 
 3i25 
 
 34o3 
 
 368 1 
 
 3959! 4237 
 
 45i4 
 
 4792 
 
 5069 
 
 5346 
 
 5623 
 
 i58 
 
 58991 6176 
 
 8607 8o32 
 
 201397] 1670 
 
 6453 6729 7005 
 
 7281 
 
 7556 
 
 7832 
 
 8107 
 
 8382 
 
 276 
 
 9206 9481! 9755 ••29 
 
 •3o3 
 
 •577 
 
 •85o 
 
 1 124 
 
 274 
 
 159 
 
 1943' 2216' 2488 2761 
 
 3o33' 33o5 3577 
 
 3848 
 
 272 
 
 N. 
 
 1 . 
 
 2 | 3 i 4 | 5 | 6 | 7 I 8 9 
 
 D. 
 
A TABLE OF LOGARITHMS FROM 1 TO' 10,000. 
 
 N. 
 
 1, i 
 
 2 " 
 
 3' 
 
 . i 
 
 -5; 
 
 6 
 
 7 
 
 8 J 9 
 
 D.| 
 
 160 
 
 204120 4391 
 
 4663 
 
 4g34 
 
 5204 
 
 5475 
 
 5746 
 
 6016 
 
 6286J 6556 
 
 371 
 
 161 
 
 6826 1 7096 
 96 1 5 9783 
 
 7365 
 
 •7634 
 
 7904J 8173 
 •586 ! »853 
 
 8441 
 
 8710 
 
 8979I 9347 
 1654 1921 
 
 269 
 
 162 
 
 ••5 1 
 
 •3i9 
 
 112! 
 
 1 388 
 
 267 
 266 
 
 163 
 
 212188, 2454 
 
 2720 
 
 2986 
 
 3252 
 
 35i8 
 
 3783 
 
 4049 
 
 43 14 
 
 4579 
 
 164 
 
 4844 5109 
 
 5373 
 
 5638 
 
 5go2 
 
 6166 
 
 643o 
 
 6694 
 
 6o57 
 9 585 
 
 7221 
 
 264 
 
 i65 
 
 748| ; 7747 
 
 8010 
 
 8273 
 
 8536 
 
 8798 
 
 9060 
 
 9 323 
 
 9846 
 
 263 
 
 166 
 
 220108 0370 
 
 o63i 
 
 0892 
 
 n53 
 
 Ui4 
 
 l6 7 5 
 
 1036 
 4533 
 
 2196 
 
 2456 
 
 261 
 
 i6 7 ; 
 
 2716 2976; 3236 
 
 34g6 
 
 3755 
 
 4oi5 
 
 4274 
 
 4792 5o5i 
 
 259 
 258 
 
 168 
 
 53o 9 5568 5826i 6084 
 
 6342 
 
 6600 6858, 71 1 5 
 
 7372 7630 
 
 169 
 
 78871 8144 
 
 84oo 
 
 8657 
 
 8913 9170 9426! 9682 
 
 gg38 »ig3 
 
 256 
 
 170 
 
 230449 0704 
 
 0960 
 35o4 
 
 1 2 1 5 
 
 I47<>| 1724 
 
 1979J 2234 
 
 24881 2742 
 
 254 
 
 •7« 
 
 2996. 325o 
 
 37D7 
 
 401 1 1 4264 
 
 4517 
 
 477° 
 
 5o23 5276 
 
 253 
 
 172 
 
 5028 
 
 5 7 8i 
 
 6o33 
 
 6285- 6537 6789 
 
 7041 
 
 7292 
 
 7544' 7795 
 
 252 
 
 i 7 3 
 
 8046 
 
 8297 
 
 8548i 8799! 9049 1 9299! 955o 
 
 9800 
 
 ••5o # 3oo 
 
 25o 
 
 '74 
 
 240549 
 
 0799 1 048 
 3286, 3534 
 
 1297 
 
 3782 
 
 1 546 1795 
 
 2044 
 
 2293 
 
 2541 2790 
 
 249 
 
 l 7 5 
 
 3o38 
 
 4o3o 
 
 4277 
 
 4525 
 
 4772 
 
 6019 6266 
 
 248 
 
 176 
 
 55i3 
 
 5759' 6006 
 
 6252 
 
 6499 
 
 6745 
 
 6991 
 
 Vl 1 
 
 7482, 7728 
 
 246 
 
 >77 
 
 ' 7973 
 
 8219' 8464 
 
 8709 
 
 8o54 
 1395 
 
 9198 
 i638 
 
 9443 
 
 9687 
 
 9932 '176 
 
 -245 
 
 178 
 
 25o42o 
 
 0664 
 
 0908 
 3338 
 
 ii5i 
 
 1881 
 
 2125 
 
 2368 
 
 2610 
 
 243 
 
 IE 
 
 2853 
 
 3096 
 
 358o 
 
 3822 
 
 4064 
 
 43o6 
 
 4548 
 
 4790 
 
 5o3i 
 
 242 
 
 255273 
 
 55i4 
 
 5 7 55 
 
 5996 
 83g8 
 
 6237 
 
 6477 
 
 6718 
 
 6n58 
 9355 
 
 7198 
 
 7439 
 
 241 
 
 181 
 
 7679 
 
 7918 
 
 81 58 
 
 8637 
 
 8877 
 
 9116 
 
 9594 
 
 g833 
 
 23g 
 
 238 
 
 182 
 
 26007 1 
 
 o3io 
 
 o548 
 
 0787 
 
 1025 
 
 1263 
 
 i5oi 
 
 1739 
 
 1976 
 4346 
 
 2214 
 
 i83 
 
 245 1 
 
 2688 
 
 2925 
 
 3i62 
 
 3399 
 
 3636 
 
 3873 
 
 4109 
 
 4582 
 
 23 7 
 
 184 
 
 4818 
 
 5o54 
 
 5290 
 
 552,5 
 
 5761 
 
 5996 
 
 6232 
 
 6467 
 
 6702 
 
 6937 
 
 235 
 
 i85 
 
 7172 
 
 7406 
 
 7641 
 
 7875 
 
 81 10 
 
 8344 
 
 8378 
 
 8812 
 
 9046 
 
 9279 
 
 234 
 
 186 
 
 90i3 
 
 9746 
 
 9980 
 
 •2l3 
 
 •446 
 
 •679 
 
 •912 
 
 H44 
 
 i3 77 
 
 i6og 
 
 233 
 
 188 
 
 271842 
 
 2074 
 438 9 
 
 2J06 
 
 2538 
 
 2770 
 
 5o8i 
 
 3ooi 
 
 3233 
 
 3464 
 
 3696 
 
 3927 
 
 232 
 
 41 58 
 
 4620 
 
 485o 
 
 53u 
 
 5542 
 
 5772 
 
 6002 
 
 6232 
 
 23o 
 
 189 
 
 6462 
 
 6692 
 
 6921 
 
 7i5i 
 
 73So 
 
 7609 
 
 7838 
 
 8067 
 
 8296 
 
 8525 
 
 229 
 
 228 
 
 190 
 
 278754 
 
 8982 
 
 •9211 
 
 9439 
 
 9667 
 
 9896 
 
 •123 
 
 •35i 
 
 •378 
 
 •806 
 
 191 
 
 28io33 
 
 1261 
 
 1488 
 
 1715 
 
 1942 
 
 2169 
 
 23g6 
 
 2622 
 
 2849 
 
 3075 
 
 227 
 
 192 
 
 ~ 33oi 
 
 3527 
 
 3753 
 
 3979 
 
 42o5 
 
 443i 
 
 4656 
 
 4882 
 
 5107 
 
 5332 
 
 226 
 
 193 
 
 555 7 
 
 5782 
 
 6007 
 
 6232 
 
 6456 
 
 6681 
 
 6905 
 
 7i3o 
 
 7354 
 
 7 5 7 8 
 
 225 
 
 194 
 
 7802 
 
 8026 
 
 8249 
 
 8473 
 
 8696 
 
 8926 
 
 9143 
 
 g366 
 
 o58g 
 
 9812 
 
 223 
 
 195 
 
 290035 
 
 0257 
 
 0480 
 
 0702 
 
 0925 
 
 1 147 
 
 i36g 
 
 i5gi 
 
 i8i3 
 
 2o34 
 
 222 
 
 196 
 
 2256 
 
 2478 
 
 2699 
 
 292a 
 
 3i4i 
 
 3363 
 
 3584 
 
 38o4 
 
 4025 
 
 4246 
 
 221 
 
 ■\u 
 
 4466 
 
 4687 
 
 4907 
 
 5i 27 
 
 ?5347 
 
 5567 
 
 5787 
 
 6007 
 
 6226 
 
 6446 
 
 220 
 
 6665 
 
 6884 
 
 7104 
 
 7323 
 
 7542 
 
 7761 
 
 7979 
 
 8198 
 
 8416 
 
 8635 
 
 219 
 2l8 
 
 199 
 
 8853 
 
 9071 
 
 9289 
 
 9507 
 
 9725 
 
 9943 
 
 •161 
 
 •378 
 
 •5g5 
 
 •8i3 
 
 200 
 
 3oio3o 
 
 1247 
 
 1464 
 
 1681 
 
 1898 
 4009 
 
 2114 
 
 233 r 
 
 2547 
 
 2764 
 
 2980 
 
 217 
 
 201 
 
 3196 
 
 3412 
 
 3628 
 
 .3844 
 
 ' 4275 
 
 4491 
 
 4706 
 
 4g2i 
 
 5i36 
 
 2l6 
 
 202 
 
 535i 
 
 5566 
 
 5 7 8i 
 
 5oq6 
 
 62 1 1 
 
 6425 
 
 663g 
 
 6854 
 
 7068 
 
 7282 
 
 2l5 
 
 303 
 
 7496 
 
 7710 
 9843 
 
 7924 
 
 8i37 
 
 835i 
 
 8564 
 
 8778 
 
 8991 
 
 g204 
 
 9417 
 
 2l3 
 
 204 
 
 963o 
 
 ••56 
 
 •268 
 
 •481 
 
 •6 9 3 
 
 •906 
 
 1118 
 
 i33o 
 
 1 542 
 
 212 
 
 2o5 
 
 3in54 
 
 1966 
 
 2177 
 4289 
 
 2389 
 
 2600 
 
 28 1.2 
 
 3o23 
 
 3234 
 
 3445 
 
 3656 
 
 211 
 
 206 
 
 386 7 
 
 4078 
 
 4499 
 
 47<° 
 
 4920 
 
 5i3o 
 
 5340 
 
 555 1 
 
 5760 
 
 210 
 
 207 
 
 5970 
 
 6180 
 
 63 9 o 
 
 65g( 
 8689 
 
 6809 
 
 7018 
 
 7227 
 
 7436 
 
 7646 
 
 7854 
 
 209 
 208 
 
 208 
 
 8o63 
 
 8272 
 
 8481 
 
 8898 
 
 9106 
 
 93i4 
 
 9 522 
 
 9730 
 
 9938 
 
 1 20 9 
 
 320146 
 
 o354 
 
 o562 
 
 0769 
 
 0977 
 
 1 184 
 
 i3oi 
 
 • 3458 
 
 1D90 
 
 180D 
 
 2012 
 
 207 
 
 ; 210 
 
 322219 
 
 2426 
 
 2633 
 
 283? 
 
 3o46 
 
 3252 
 
 3665 
 
 3871 
 
 4077 
 
 206 
 
 1. ( 
 1 211 
 
 4282 
 
 4488 
 
 4694 
 
 4899 
 695c 
 
 5io5 
 
 53 10 
 
 55i6 
 
 5721 
 
 5926 
 
 6i3i 
 
 205 
 
 212 
 
 6336 
 
 6541 
 
 6745 
 
 7i55 
 
 735g 
 
 7563 
 
 7767 
 
 7972 
 
 8176 
 
 204 
 
 113 
 
 838o 
 
 8583 
 
 8787 
 
 8991 
 
 9194 
 
 9398 
 
 9601 
 
 9805 
 
 •••£ 
 
 •211 
 
 203 
 
 314 
 
 33o4i4 
 
 061" 
 
 0819 
 
 1022 
 
 1225 
 
 1427 
 
 i63o 
 
 1 832 
 
 2o34 
 
 2236 
 
 202 
 
 2l5 
 
 2438 
 
 2640 
 
 2842 
 
 3o44 
 
 3246 
 
 3447 
 
 3649 
 5658 
 
 385o 
 
 4o5i 
 
 4253 
 
 203 
 
 216 
 
 4454 
 
 4655 
 
 4856 
 
 5o57 
 
 5257 
 
 5458 
 
 . 5859 
 7 858 
 
 6o5g 
 8o58 
 
 6260 201 
 
 217 
 1 218 
 
 ' "»9 
 
 6460 
 
 6660 
 
 6860 
 
 706c 
 
 7260 
 
 7459 
 
 7659 
 
 820- 
 
 200 
 
 8456 
 340444 
 
 8656 
 C642 
 
 8855 
 0841 
 
 go54 
 Io3$ 
 
 9253 
 1237 
 
 945 1 
 1435 
 
 g65c 
 1632 
 
 9849 
 i83o 
 
 ••47 
 2028 
 
 •246 
 
 2225 
 9 
 
 igo 
 I98 
 
 N. 
 
 ! | 1 1 2 J 3 
 
 ! A 
 
 | 5 1 6 1 7 
 
 8 
 
 D. 
 
 20* 
 
i 
 
 A TABLE 
 
 OF 
 
 LOGARITHMS FROM ] 
 
 TO 
 
 10,000. 
 
 
 N. 
 
 
 
 1 
 
 ' 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 320 
 
 342423 
 
 2620 
 
 2817 
 
 3oi4 
 
 3212 
 
 3409 
 
 36o6 
 
 38o2 
 
 3999 
 
 4196 
 61 5 7 
 
 197 
 
 221 
 
 4392 
 
 4589 
 
 4785 
 
 4981 
 
 5i 7 8 
 
 5374 
 
 5570 
 
 5766 
 
 0962 
 
 196 
 
 222 
 
 6353 
 
 6549 
 
 6744 
 
 6939 
 
 7i35 
 
 733o 
 
 7525 
 
 7720 
 
 7915 
 
 81 10 
 
 i 9 5 
 
 223 
 
 83o5 
 
 85oo 
 
 8694 
 o636 
 
 8889 
 
 9083 
 
 9278 
 
 9472 
 
 9666 
 
 9860 
 
 ••54 
 
 '9-i 
 
 224 
 
 350248 
 
 0442 
 
 0829 
 
 1023 
 
 1216 
 
 1410 
 
 i6o3 
 
 1796 
 
 1989 
 
 i 9 3 
 
 225 
 
 2i83 
 
 2375 
 
 2568 
 
 2761 
 
 2o54 
 
 3i47 
 5o68 
 
 333 9 
 
 3532 
 
 3724 
 
 3916 
 5834 
 
 i 9 3 
 
 226 
 
 4108 
 
 43oi 
 
 4493 
 
 4685 
 
 4876 
 
 5260 
 
 5452 
 
 5643 
 
 192 
 
 127 
 
 328 
 
 6026 
 
 6217 
 
 6408 
 
 6599 
 
 6790 
 
 6981 7172 
 
 7363 
 
 7554 
 
 7744 
 
 .191 
 
 7 9 35 
 
 8l2D 
 
 83i6 
 
 85o6 
 
 8696 
 
 8886 
 
 9076 
 
 9266 
 
 9456 
 
 9646 
 
 190 
 
 229 
 
 9 835 
 
 ••25 
 
 •2l5 
 
 •404 
 
 •5o3 
 
 •783 
 
 •972 
 
 J 161 
 
 i35o 
 
 1 539 
 
 it 
 
 lio 
 
 361728 
 
 1917 
 
 2io5 
 
 2294 
 
 2482 
 
 2671 
 
 2859 
 
 3o48 
 
 3236 
 
 3424 
 
 23l 
 
 36i2 
 
 38oo 
 
 3 9 88 
 5862 
 
 4176 
 
 .4363 
 
 455i 
 
 4739 
 
 4926 
 
 5u3 
 
 53oi 
 
 188 
 
 232 
 
 5488 
 
 56 7 5 
 
 6049 
 7915 
 
 6236 
 
 6423 
 
 6610 
 
 6796 
 
 6 9 83 
 8845 
 
 7169 
 
 ft 
 
 233 
 
 7356 
 
 7542 
 
 7729 
 
 8101 
 
 8287 
 •i43 
 
 8473 
 
 865 9 
 •5i3 
 
 9o3o 
 
 234 
 
 9216 
 
 9401 
 
 9 58 7 
 
 9772 
 
 9958 
 
 •328 
 
 •698 
 
 •883 
 
 i85 
 
 235 
 
 371068 
 
 1253 
 
 U37 
 
 1622 
 
 1806 
 
 '99i 
 383i 
 
 2175 
 
 236o 
 
 2544 
 
 2728 
 
 184 
 
 236 
 
 2912 
 
 3oo6 
 49J2 
 
 3280 
 
 3464 
 
 3647 
 
 401 5 
 
 4198 
 
 4382 
 
 4565 
 
 184 
 
 23t 
 238 
 
 4748 
 
 5n5 
 
 5298 
 
 5481 
 
 5664 
 
 5846 
 
 6029 
 
 6212 
 
 63g4 
 
 i83 
 
 65 7 7 
 83 9 8 
 
 6759 
 
 6942 
 
 7124 
 8943 
 
 73o6 
 
 7488 
 
 7670 
 9487 
 
 7862 
 
 8o34 
 
 8216 
 
 182 
 
 239 
 
 858o 
 
 8761 
 0073 
 
 9124 
 
 9306 
 
 9668 
 
 9849 
 
 ••3o 
 
 181 
 
 240 
 
 3802 1 1 
 
 0392 
 
 0754 
 
 0934 
 
 Hi5 
 
 1296 
 
 1476 
 
 1656 
 
 1837 
 3636 
 
 181 
 
 241 
 
 2017 
 
 2197 
 
 2377 
 
 2557 
 
 2737 
 
 2917 
 
 3o 97 
 
 3277 
 
 3456 
 
 180 
 
 242 
 
 38i5 
 
 3995 
 5 7 85 
 
 4174 
 5964 
 
 4353 
 
 4533 
 
 4712 
 
 4891 
 
 5070 
 
 5249 
 
 5428 
 
 % 
 
 243 
 
 56o6 
 
 6142 
 
 6321 
 
 6499 
 
 6677 
 
 6856 
 
 l£i 
 
 7212 
 8989 
 
 244 
 
 7 3 9° 
 
 7568 
 
 7746 
 
 7 ? 23 
 9698 
 
 8101 
 
 8279 
 
 8456 
 
 8634 
 
 i 7 8 
 
 245 
 
 9166 
 
 9343 
 
 9D20 
 
 9 8 7 5 
 
 ••5 1 
 
 •228 
 
 •4o5 
 
 •582 
 
 • 7 5 9 
 
 177 
 
 246 
 
 390935 
 
 1112 
 
 1288 
 
 1464 
 
 1641 
 
 1817 
 
 1993 
 
 2169 
 
 2345 
 
 2521 
 
 176. 
 
 248 
 
 2697 
 4452 
 
 2873 
 
 3o48 
 
 3224 
 
 34oo 
 
 35 7 5 
 
 3?5i 
 
 3926 
 
 4101 
 
 4277 
 
 176 
 
 4627 
 
 4802 
 
 4977 
 
 5i52 
 
 5326 
 
 55oi 
 
 56 7 6 
 
 585o 
 
 6025 
 
 i 7 5 
 
 249 
 
 6199 
 
 6374 
 
 6548 
 
 6722 
 
 6896 
 8634 
 
 7071 
 8808 
 
 7245 
 8981 
 
 7419 
 
 7592 
 
 7766 
 95ol 
 
 174 
 
 25o 
 
 397940 
 
 8114 
 
 8287 
 
 8461 
 
 9154 
 
 9328 
 
 i 7 3 
 
 25l 
 
 9614 
 
 9847 
 
 ••20 
 
 •192 
 
 •365 
 
 •538 
 
 •711 
 
 •883 
 
 Io56 
 
 1228 
 
 i 7 3 
 
 2D2 
 
 401401 
 
 i5 7 3 
 
 1745 
 
 1917 
 3635 
 
 2089 
 
 2261 
 
 2433 
 
 26o5 
 
 2777 
 
 2949 
 
 172 
 
 253 
 
 3 1 2 1 
 
 3292 
 
 3464 
 
 3807 
 
 3978 
 
 4149 
 
 4320 
 
 4492 
 
 4663 
 
 I7 1 
 
 254 
 
 4834 
 
 5oo5 
 
 5i 7 6 
 6881 
 
 5346 
 
 55i7 
 
 5688 
 
 5858 
 
 6029 
 
 6199 
 
 6370 
 
 171 
 
 255 
 
 6540 
 
 Bijio 
 
 7o5i 
 8749 
 
 7221 
 8918 
 
 73m 
 
 756i 
 
 7731 
 
 7901 
 
 8070 
 9764 
 
 !S 
 
 256 
 
 8240 
 
 8410 
 
 85 79 
 
 9087 
 
 9 25 7 
 
 9426 
 
 9&,5 
 
 25t 
 258 
 
 9933 
 
 ' *102 
 
 •271 
 
 •44o 
 
 •609 
 229J 
 
 •773 
 2461 
 
 •946 
 
 1114 
 
 1283 
 
 I45i 
 
 IS' 
 
 41 1620 
 
 I78B 
 
 1956 
 
 2124 
 
 2629 
 43o5 
 
 2796 
 
 2964 
 
 3i32 
 
 259 
 
 33oo 
 
 3467 
 
 3635 
 
 38o3 
 
 3970 
 
 4i3 7 
 58o8 
 
 4472 
 
 463o 
 63o8 
 
 4806 
 
 167 
 
 260 
 
 414973 
 
 5i4o 
 
 5307 
 
 5474 
 
 5641 
 
 5974 
 7638 
 
 6141 
 
 6474 
 
 167 
 
 261 
 
 6641 
 
 6807 
 
 6973 
 
 •45 1 
 
 t3p6 
 8964 
 
 7472 
 
 7804 
 
 7970 
 
 8(35 
 
 166 
 
 262 
 
 83oi 
 
 8467 
 
 8633 
 
 9129 
 
 9 2 9 5 
 
 9460 
 
 9625 
 
 9791 
 
 i65 
 
 '..63 
 
 9956 
 
 •121 
 
 •286 
 
 •616 
 
 •781 
 
 •945 
 2090 
 
 mo 
 
 1278 
 
 U39 
 
 i65 
 
 264 
 
 421604 
 
 1788 
 
 iq33 
 
 3$ 
 
 2261 
 
 2426 
 
 2754 
 
 2918 
 
 3o82 
 
 164 
 
 265 
 
 3246 
 
 3410 
 
 35 7 4 
 
 3901 
 5534 
 
 4o65 
 
 4228 
 
 4392 
 
 4555 
 
 4718 
 
 164 
 
 266 
 
 4882 
 
 5o45 
 
 5208 
 
 53 7 i 
 
 6697 
 
 586o 
 
 6023 
 
 6186 
 
 6340 
 79 7 3 
 9591 
 
 i63 
 
 267 
 '268 
 
 65n 
 
 6674 
 
 6836 
 
 •6999 
 8621 
 
 7161 
 8 7 83 
 
 7324 
 8944 
 •55g 
 
 7486 
 
 7648 
 
 7811 
 
 162 
 
 8i35 
 
 8297 
 
 845a 
 
 9I06 
 
 9268 
 
 9429 
 
 162 
 
 269 
 
 9752 
 43 1 364 
 
 9914 
 
 ••7D 
 i685 
 
 •236 
 
 •3 9 8 
 
 •720 
 
 •881 
 
 1042 
 
 1203 
 
 161 
 
 270 
 
 l523 
 
 1846 
 345o 
 
 2007 
 
 2167 
 
 2328 
 
 2488 
 
 2649 
 
 2809 
 
 161 
 
 2 7I 
 
 2969 
 
 3i3o 
 
 3290 
 4888 
 
 36io 
 
 3770 
 
 3930 
 5526 
 
 4090 
 
 4249 
 
 4409 
 
 160 
 
 272 
 
 4669 
 6i63 
 
 4729 
 
 5o48 
 
 5207 
 6798 
 8384 
 
 536 7 
 
 5685 
 
 5844 
 
 6004 
 
 1 59 
 
 273 
 
 6322 
 
 6481 
 
 6640 
 
 6957 
 8542 
 
 7116 
 
 72-5 7433 
 8859; 9<"7 
 
 7592 
 
 m 
 
 274 
 
 7751 
 9 333 
 
 7909 
 
 8067 
 9648 
 
 8226 
 
 8701 
 
 9173 
 
 275 
 
 9491 
 
 9806 
 
 9964 
 1 538 
 
 •122 
 
 •279 
 
 •437 »5o4 
 
 •752 
 
 2323 
 
 1 58 
 
 276 
 
 440909 
 
 1066 
 
 1224 
 
 i38i 
 
 1695 
 
 i852 
 
 2009 
 
 2166 
 
 i5 7 
 
 277 
 ?78 
 
 2480 
 
 2637 
 
 2793 
 
 2g5o 
 
 3io6 
 
 3263 
 
 3419 
 
 3576 
 
 3 7 32 
 
 3889 
 
 1 5 7 ' 
 
 4045 
 
 4201 
 
 4357 
 
 4Di3 4669 4825 
 
 4981 
 
 5i3 7 
 
 5293 
 
 5449 
 
 7003 
 
 1 56 
 
 »79 
 
 56o4 
 
 5760 
 I 
 
 5gi5 
 
 2 
 
 6071 
 3 
 
 6226 6382 
 
 6537 
 
 6092 
 
 6848 
 
 1 55 
 D, 
 
 \N. 
 
 
 
 4 1 5 
 
 6 
 
 7 
 
 8 
 
 9 
 

 A TABLE 
 
 OP 
 
 LOGARITHMS FROM ! 
 
 ro 
 
 10,000. 
 
 5 
 
 N. 
 
 
 
 1 " 
 
 . I | 3 
 
 4 
 
 5 | 6 1 7 
 
 8 
 
 9 
 
 8552 
 
 D. 
 
 1 55 
 
 380 
 
 447i58 
 
 8706 
 
 73i3 
 8861 
 
 7468 7623 
 
 7778 
 
 7933 8088 8242 
 
 8397 
 
 281 
 
 9013 
 
 9170 
 
 9324 
 
 9478 9d33 
 
 9787 
 
 9941 
 
 ••95 
 1 633 
 
 1 54 
 
 282 
 
 450249 
 
 04 ^3 
 
 o557 
 
 071 1 
 
 o865 
 
 1018 
 
 1172 
 
 1326 
 
 1479 
 
 1 54 
 
 283 
 
 1786 
 
 1940 
 
 2093 
 
 2247 
 
 2400 
 
 2553 
 
 2706 
 
 2859 
 
 3oi2: 3 1 65 
 
 153 
 
 284 
 
 33i8 
 
 k 34 7 i 
 
 3624 
 
 3 777 
 
 3930 
 
 4082 
 
 4235 
 
 438 7 
 
 4540 4692 
 
 153 
 
 285 
 
 4848 
 
 '4997 
 
 5i5o 
 
 53o2 
 
 5454 
 
 56o6 
 
 5758 
 
 5910 
 
 6062 
 
 6214 
 
 152 
 
 286 
 
 "6366 
 
 65i8 
 
 6670 
 
 6821 
 
 6973 
 
 7125 
 8638 
 
 7276 
 
 7428 
 8940 
 
 7 5 79 
 
 77 3i 
 
 l52 
 
 28 1 
 288 
 
 7882 
 
 8o33 
 
 8184 
 
 8336 
 
 8487 
 
 8789 
 
 9091 
 
 9242 
 
 i5i 
 
 9392 
 
 9543 
 
 9694 
 
 9845 
 
 9995 
 
 •146 
 
 •296 
 
 •447 
 1948 
 
 •597 
 2098 
 
 •748 
 
 i5i 
 
 289 
 
 460898 
 
 1048 
 
 1 198 
 
 i348 
 
 1499 
 
 1649 
 
 '799 
 
 2248 
 
 no 
 
 290 
 
 462398 
 
 2548 
 
 2697 
 
 2847 
 
 2 997 
 
 3i46 
 
 3296 
 
 3445 
 
 35 9 4 
 
 3744 
 
 i5o 
 
 291 
 
 38 9 3 
 5383 
 
 4042 
 
 4191 
 568o 
 
 434o 
 
 4490 
 
 463o 
 6126 
 
 4788 
 
 4936 
 
 5o85 
 
 5234 
 
 149 
 
 292 
 
 5532 
 
 5829 
 
 5 977 
 
 6274 
 
 6423 
 
 6571 
 
 6719 
 
 \% 
 
 293 
 
 6868 
 
 7016 
 8495 
 
 7164' 7312 
 8643 8790 
 
 746o 
 
 7608 
 
 77 56 
 
 7904 
 
 8o52 
 
 8200 
 
 294 
 
 8347 
 
 8 9 38 
 
 9085 
 
 9233 
 
 938o 
 
 9527 
 «998 
 
 9675 
 
 148 
 
 295 
 
 9822 
 
 9969 
 1438 
 
 •116! »263 
 
 •410 
 
 •55 7 
 
 •704 
 
 •85 r 
 
 1 U5 
 
 "47 
 
 296 
 
 47 1 292 
 
 2756 
 
 i585 1732 1878 
 
 2025 
 
 2171 
 
 23i8 
 
 2464 
 
 2610 
 
 146 
 
 297 
 298 
 
 2oo3 
 4362 
 
 3o4o! 3ro5| 33.41 
 45o8 4653 4799 
 5962! 61071 6232 
 
 3487 
 
 3633 
 
 3779 
 
 3925 
 538i 
 
 4071 
 
 146 
 
 4216 
 
 4944 
 6397 
 
 5090 
 
 5235 
 
 5526 
 
 146 
 
 299 
 
 5671 
 
 58i6 
 
 6542 
 
 6687 
 
 6832 
 
 6976 
 
 145 
 
 3oo 
 
 477121 
 8566 
 
 7266 
 871 1 
 
 7411 
 8855 
 
 7555i 7700 
 8999 1 9143 
 0438 o582 
 
 7844 
 
 7989 
 
 8i33 
 
 8278 
 
 8422 
 
 145 
 
 3ol 
 
 9287 
 
 943 1 
 
 9 5 7 5 
 
 97'9 
 
 9 863 
 
 144 
 
 302 
 
 480007 
 
 oiSi 
 
 0294 
 
 0725 
 
 0869 
 
 1012 
 
 n56 
 
 1299 
 2731 
 
 144 
 
 3o3 
 
 1443 
 
 1586 
 
 1729 
 
 1872 
 
 2016 
 
 2159 
 
 2302 
 
 2445 
 
 2588 
 
 143 
 
 3o4 
 
 2874 
 
 3oi6 
 
 3i5o 
 4585 
 
 33o2 
 
 3445 
 
 3587 
 
 3730 
 
 3872 
 
 4oi5 
 
 4i57 
 
 143 
 
 3o5 
 
 43oo 
 
 4442 
 
 47 2 7 
 
 4669 
 
 5on 
 
 5i53 
 
 5295 
 
 5437 
 
 5579 142 
 
 3o6 
 
 5721 
 
 5863 
 
 6oo5 
 
 6147 
 
 6289 
 
 643o 
 
 65 7 2 
 
 6714 
 
 6855 
 
 6997 142 
 
 307 
 
 7i38 
 855i 
 
 7280 
 
 7421 
 
 7 563 
 
 7794 
 
 7845 
 
 7986 8127 
 
 8269 
 
 8410 
 
 141 
 
 3o8 
 
 8692 
 
 8833 
 
 8974 
 
 9114 
 
 9255' 9396 9537 
 
 9°77 
 
 9818 
 
 141 
 
 309 
 
 9958 
 491362 
 
 ••99 
 
 •23g 
 
 •38o 
 
 •520 
 
 •66 1 1 «8oi 
 
 •941 
 2341 
 
 1081 
 
 1222 
 
 140 
 
 3io 
 
 l502 
 
 1642 
 
 1782 
 
 1922 
 
 3319 
 
 2062' 2201 
 
 2481 
 
 2621 
 
 140 
 
 3n 
 
 2760 
 
 2900 
 
 3o4o 
 
 3179 
 
 3458 
 
 35g7 
 
 3 7 3 7 
 
 3876 
 
 4oi5 
 
 13, 
 
 3l2 
 
 4r55 
 
 4294 
 
 4433 
 
 4572 
 
 471 1 
 
 485o 
 
 4989 5i28 
 
 5267 
 
 5406 
 
 1 3 9 
 
 3i3 
 
 5544 
 
 5683 
 
 5822 
 
 5960 
 
 6099 
 
 6238 
 
 63 7 6 
 
 65i5 
 
 6653 
 
 6791 
 
 i3 9 
 
 3i4 
 
 6930 
 83n 
 
 7068 
 8448 
 
 7206 
 
 7344 
 
 7483 
 8862 
 
 7621 
 •8999 
 
 77 59 
 
 7897 
 
 8o35 
 
 8i 7 3 
 
 i38 
 
 3i5 
 
 8586 
 
 8724 
 
 9137 
 
 9275 
 
 9412 
 
 955o 
 
 138 
 
 3i6 
 
 9687 
 
 9824 
 
 9962 
 
 ••99 
 
 •236 
 
 •374 
 
 •oil 
 
 •648 
 2017* 
 
 •783 
 
 •922 
 
 '37 
 
 3i8 
 
 001059 
 
 1 196 
 
 1 333 
 
 1470 
 
 1607 
 
 1744 
 
 1S80 
 
 2 1 54 
 
 2291 
 3655 
 
 \u 
 
 2427 
 
 2564 
 
 2700 
 
 2837 
 
 2973 
 4335 
 
 3 109 
 
 3246 
 
 3382 
 
 35i8 
 
 3 19 
 
 5odi5o 
 
 3927 
 
 4o63 
 
 4199 
 555 7 
 
 4471 
 
 4607 
 
 4743 
 
 4878 
 
 5oi4 
 
 1 36 
 
 320 
 
 5286 
 
 5421 
 
 56 9 3 
 
 5828 
 
 5964 
 
 6099 
 
 6234 
 
 6370 
 
 136 
 
 321 
 
 65o5 
 
 6640 
 
 6776 
 
 6911 
 
 7046 
 
 7181 
 
 73i6 
 
 745 1 
 
 7586 
 
 7721 
 
 135 
 
 322 
 
 7856 
 
 799' 
 9-"7 
 
 8126 
 
 8260 
 
 83g5 
 
 853o 
 
 8664 
 
 8799 
 
 8934 
 
 9068 
 
 i35 
 
 323 
 
 9203 
 
 9471 
 
 960& 
 
 9740 
 
 9874 
 
 •••9 
 
 •143 
 
 •277 
 
 •411 
 
 1 34 
 
 324 
 
 5io545 
 
 0679 
 
 o8i3 
 
 0947 
 
 1081 
 
 I2l5 
 
 1 349 
 
 1482 
 
 1616 
 
 1750 
 
 134 
 
 355 
 
 i883 
 
 2017 
 
 2l5l 
 
 2284 
 
 2418 
 
 255i 
 
 2684 
 
 2818 
 
 2951 
 
 3o84 
 
 i33 
 
 326 
 
 32i8 
 
 335i 
 
 3484 
 
 36i7 
 
 3700 
 
 3883 
 
 4016 
 
 4149 
 
 4282 
 
 44i4 
 
 133 
 
 327 
 
 4548 
 
 4681 
 
 48i3 
 
 4946 
 
 5079 
 
 52 1 1 
 
 5344 
 
 5476 
 
 5609 
 
 574i 
 
 133 
 
 328 
 
 58 7 4 
 
 6006 
 
 6139 
 
 6271 
 
 64o3 
 
 6535 
 
 6668 
 
 6800 
 
 6932 
 
 7064 
 
 132 
 
 3io 
 
 7196 
 
 7328 
 
 7460 
 
 7592 
 8909 
 
 7724 
 
 7855 
 
 7987 
 
 81 19 
 
 825i 
 
 8382 
 
 132 
 
 33o 
 
 5i85i4 
 
 8646 
 
 8777 
 
 9040 
 
 9171 
 
 93o3 
 
 9434 
 
 9 566 
 
 9697 
 
 i3i 
 
 33 1 
 
 9828 
 
 ogSg 
 
 ••90 
 
 •221 
 
 •353 
 
 •48.' 
 
 •6i5 
 
 •745 
 
 •876 
 
 1007 
 
 i3i 
 
 332 
 
 521138, 1269 1 1460 
 
 i53o 
 
 1661 
 
 1792 
 
 1922 
 
 2t>53 
 
 2i83 
 
 23i4 
 
 r3i 
 
 333 
 
 2444! 2575 
 
 2705 
 
 2835 
 
 2966 
 
 3096 
 
 3226 
 
 3356 
 
 3486 
 
 36i6 
 
 i3o 
 
 334 
 
 3746| 3876 
 
 4006 
 
 4 1 36 
 
 4266 
 
 4396 
 
 4526 
 
 4656 
 
 4785 
 
 4915' l3o 
 
 335 
 
 5o45 5 1 74 
 
 53o4 
 
 5434 
 
 5563 
 
 56 9 3 
 
 5822 
 
 5g5i 
 
 6081 
 
 6210' 129 
 
 336 
 
 6339 6460 
 
 65q8 
 
 6727 
 
 6856 
 
 6 9 85 
 
 ,7114 
 
 7243 
 
 7372 
 
 tSoi 129 
 8788 I2g 
 
 337 
 
 763o 
 
 7709' 78881 8016 
 
 8i45 
 
 8274 
 
 8402 
 
 853 1 
 
 8660 
 
 338 
 
 8917 
 
 9045 91741 93o2 
 o328! o456| o584 
 
 943o 
 
 9559 
 
 9687 
 
 9815 
 
 9943 
 
 ••72 ia8 
 
 33 9 
 
 530200 
 
 0712 
 
 0840 
 
 oo68| 1 096 
 
 1223 
 
 1 3 5 1 ' 128 
 
 " N. 
 
 
 
 I 
 
 2 f 3 
 
 4 
 
 5 
 
 6 
 
 1 
 
 8 
 
 9 1 rs. 
 
6 
 
 A TABLE 
 
 OF LOGARITHMS FROM ] 
 
 TO 
 
 10,000. 
 
 
 N. 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 | 8 
 
 9 
 
 D. 
 
 34o 
 
 531479 
 
 2734 
 
 1607 
 
 1734 
 
 1862 
 
 J99P 
 3'264 
 
 2117 
 
 2245 
 
 2372I 25oo 
 
 2627 
 
 128 
 
 341 
 
 2882 
 
 3009 
 
 3i36 
 
 3391 
 
 35.8 
 
 3645 1 3772 
 
 3899 
 
 127 
 
 m 
 
 4026 
 
 41 53 
 
 4280 
 
 4407 
 
 4534 
 
 4661 
 
 booi 
 
 4914 5o4i 
 
 5167 
 
 .'2 
 
 343 
 
 5294 
 6558 
 7819 
 
 5421 
 
 5547 
 
 56 7 4 
 
 58oo 
 
 5927 
 
 6180' 63o6 
 
 6432 
 
 344 
 345 
 
 6685 
 7945 
 
 68u 
 8071 
 
 6 9 3 7 
 8197 
 
 7063 
 
 8322 
 
 8448 
 
 73i5 
 85 7 4 
 
 7441 1 7567 
 8699' 8825 
 
 3$ 
 
 126 
 126 
 
 346 
 
 9076 
 
 9202 
 
 9 32 7 
 
 9432 
 
 o83o 
 
 9703 
 
 9829 
 
 9934 ••79 
 
 •204 
 
 125 
 
 347 
 348 
 
 640329 
 
 0455 
 
 o58o 
 
 0705 
 
 0955 
 
 1080 
 
 1203 i33o 
 
 1454 
 
 125 
 
 1579 
 
 1704 
 
 1829 
 
 1953 
 
 2078 
 
 2203 
 
 2327 
 
 2452, 2576 
 
 2701 
 
 125 
 
 349 
 
 2823 
 
 2930 
 
 3074 
 
 3199 
 
 3323 
 
 3447 
 
 3571 
 
 36g6 38so 
 4936, 5o6o 
 
 3944 
 
 124 
 
 330 
 
 544068 
 
 4192 
 543 1 
 
 43i6 
 
 4440 
 
 4564 
 
 4688 
 
 4812 
 
 5i83 
 
 124 
 
 331 
 
 5307 
 
 5555 
 
 56 7 8 
 
 5802 
 
 6925 
 
 ■6049 
 
 6172' 6296 
 
 6419 
 
 124 
 
 332 
 
 6543 
 
 6666 
 
 6789 
 
 6913 
 
 7036 
 8267 
 
 7i5 9 
 838 9 
 
 7282 
 85i2 
 
 74o5, 7629 
 8635,; 8758 
 
 7652 
 8881 
 
 123 
 
 353 
 
 7775 
 
 7898 
 
 8021 
 
 8144 
 
 123 
 
 354 
 
 goo3 
 
 9126 
 
 9249 
 0473 
 
 9 3 7 i 
 
 9494 
 
 9616 
 
 9739 
 
 9861 | 9984 
 
 •106 
 
 123 
 
 355 
 
 550228 
 
 o35i 
 
 0595 
 
 ° 7 ^ 
 1938 
 
 0840 
 
 0962 
 
 1084 1206 
 
 1328 
 
 122 
 
 356 
 
 i45o 
 
 1072 
 
 1694 
 
 1816 
 
 2060 
 
 2181 
 
 23o3 
 
 2425 
 
 2547 
 
 122 
 
 35t 
 358 
 
 2668 
 
 2790 
 
 2911 
 
 3o33 
 
 3i55 
 
 3276 
 4489 
 
 33 9 8 
 
 35i9 
 
 3640 
 
 3762 
 
 121 
 
 3883 
 
 4004 
 
 4126 
 
 4247 
 
 4368 
 
 4610 
 
 473i 
 
 4852 
 
 4973 
 6182 
 
 121 
 
 35 9 
 
 5094 
 
 521 5 
 
 5336 
 
 5457 
 
 55 7 8 
 6785 
 
 5699 
 6903 
 
 5820 
 
 5940 
 
 6061 
 
 121 
 
 36o 
 
 5563o3 
 
 6423 
 
 6544 
 
 6664 
 
 7026 
 
 7146 
 
 7267 
 
 738 7 
 858 9 
 
 120 
 
 36i 
 
 7507 
 8709 
 
 7627 
 8829 
 
 7748 
 8948 
 
 7868 
 
 7988 
 
 8108 
 
 8228 
 
 8349 8469 
 
 120 
 
 36a 
 
 9068 
 
 9188 
 
 93o8 
 
 9428 
 
 9548 
 
 9667 
 
 9787 
 
 120 
 
 363 
 
 9907 
 
 ••26 
 
 •146 
 
 •265 
 
 •385 
 
 •5o4 
 
 •624 
 
 •743 
 
 •863 
 
 •982 
 
 119 
 
 364 
 
 56noi 
 
 1221 
 
 i34o 
 
 1459 
 
 1578 
 
 1698 
 
 1817 
 
 1 936 
 
 2o55 
 
 2174 
 
 II9 
 
 365 
 
 2293 
 3481 
 
 2412 
 
 253i 
 
 265o 
 
 2769 
 
 2887 
 
 3oo6 
 
 3i25 
 
 3244 
 
 3362 
 
 119 
 
 366 
 
 36oo 
 
 3 7 i8 
 
 3837 
 
 3953 
 
 4074 
 5237 
 
 4192 
 
 43n 
 
 4429 
 
 4548 
 
 JB 
 
 361 
 368 
 
 4666 
 
 4784 
 
 4903 
 
 5021 
 
 5i3g 
 
 5376 
 
 5494 56 1 2 
 
 6730 
 
 .5848 
 
 5 9 66 
 
 6084 
 
 6202 
 
 6320 
 
 6437 
 
 6555 
 
 6673 
 
 679 r 
 
 6909 
 
 :i8 
 
 36g 
 •370 
 
 7026 
 568202 
 
 7144 
 83i9 
 
 7262 
 8436 
 
 8554 
 
 7497 
 8671 
 
 7614 
 8788 
 
 7732 
 8 9 o5 
 
 7849 
 9023 
 
 7967 
 9140 
 
 8084 
 9257 
 
 118 
 117 
 
 3 7 i 
 
 9374 
 
 9491 
 
 9608 
 
 9725 
 
 9842 
 
 9959 
 
 ••76 
 
 •193 
 1 35 9 
 
 •309 
 
 •426 
 
 117 
 
 3 ^ 
 
 570543 
 
 0660 
 
 0776 
 
 0893 
 
 IOIO 
 
 1 1 26 
 
 1243 
 
 1476 
 
 l5o2 
 
 2765 
 
 "7 
 
 3 7 3 
 
 1709 
 2872 
 
 i825 
 
 1942 
 
 2038 
 
 2174 
 
 2291 
 3452 
 
 2407 
 3568 
 
 2523 
 
 2639 
 
 116 
 
 3 si 
 
 2988 
 
 3io4 
 
 3220 
 
 3336 
 
 3684 
 
 38oo 
 
 3 9 i5 
 
 116 
 
 Vi 
 
 4o3i 
 
 4i 47 
 
 4263 
 
 4379 
 5534 
 
 4494 
 565o 
 
 4610 
 
 4726 
 588o 
 
 4841 
 
 4957 
 
 5072 
 
 116 
 
 376 
 
 5i88 
 
 53o3 
 
 5419 
 
 5 7 65 
 
 5996 
 
 6111 
 
 6226 
 
 n5 
 
 m 
 
 6341 
 
 6457 
 
 6372 
 
 6687 
 
 6802 
 
 6917 
 
 7o32 
 
 7147 
 82 9 5 
 
 7262 
 8410 
 
 7377 
 8525 
 
 ii5 
 
 j! 4 ? 2 
 
 7607 
 8754 
 
 7722 
 8868 
 
 7836 
 8983 
 
 795 1 
 
 8066 
 
 8181 
 
 n5 
 
 379 
 
 863 9 
 
 9°97 
 
 9212 
 
 9326 
 
 9441 
 
 9 555 
 
 9669 
 
 114 
 
 38o 
 
 579784 
 580925 
 
 9898 
 1039 
 
 ••12 
 
 •126 
 
 •241 
 
 •355 
 
 •469 
 
 •583 
 
 •607 
 1 836 
 
 •811 
 
 114 
 
 38i 
 
 u53 
 
 1267 
 
 i38i 
 
 1495 
 263 1 
 
 1608 
 
 1722 
 
 1960 
 
 114 
 
 382 
 
 2063 
 
 2177 
 
 2291 
 
 2404 
 
 25i8 
 
 2745 
 
 2858 2972 
 
 3o85 
 
 114 
 
 383 
 
 3 J?9 
 433 1 
 
 33i2 
 
 3426 
 
 3539 
 
 3652 
 
 3765 
 
 3870 
 
 3992, 4io5 
 
 4218 
 
 1.3 
 
 384 
 
 4444 
 
 4557 
 
 4670 
 
 4783 
 
 4896 j 5009 
 60241 6137 
 
 5122 1 5235 
 
 5348 
 
 n3 
 
 385 
 
 546i 
 
 -5574 
 
 5686 
 
 5799 
 
 5912 
 
 625o 6362 
 
 6476 
 
 n3 
 
 386 
 
 658 7 
 
 6700 
 7823 
 8944 
 
 6812 
 
 6925 
 
 7037 
 
 7-149! 7 262 
 8272 8384 
 
 7374 1 7486 
 8496 8608 
 
 75 9 o 
 
 8720 
 
 ■9838 
 
 112 
 
 387 
 
 77ii 
 
 8832 
 
 7 9 35 
 
 8047 
 
 8160 
 
 112 
 
 388 
 
 9056 
 
 9167 
 
 9279 
 
 9 3 9' 
 
 95o3 
 
 96i5 9726 
 
 112 
 
 389 
 
 9950 
 
 ••61 
 
 •173 
 1287 
 
 •284 
 
 •396 
 
 •5o7 
 
 •619 
 
 •730 »842 
 1843 i 9 55 
 
 • 9 53 
 
 112 
 
 390 
 
 591065 
 
 1176 
 2288 
 
 l3 99 
 
 i5io 
 
 1621 
 
 1732 
 2843 
 
 2066 
 
 in 
 
 391 
 
 2177 
 
 2399 
 
 25io 
 
 2621 
 
 2732 
 
 2954 1 3o64 
 
 3i 7 5 
 
 111 
 
 3ol 
 
 3286 
 
 33 97 
 
 35o8 
 
 36i8 
 
 3729 3840 3950 
 4834: 4945; 5o55 
 
 4061 1 4171 
 
 4282 
 
 111 
 
 393 
 
 4393i 45o3 
 
 4614 
 
 4724 
 
 5i65 5276 
 
 ■5386 
 
 110 
 
 3 9 4 
 
 5496 1 56o6 
 
 0717 
 
 5827 
 
 5937. .6047 j 6157 
 
 6267 6377 
 
 6487 
 
 no 
 
 393 
 3g6 
 
 6597 6707 
 nt<)5\ 7805 
 8701 , 8900 
 9883| 9992 
 6009731 1082 
 
 6817 
 79'4 
 
 6927 
 8024 
 
 7037, 71461 7236 
 8i34 8243, 8353 
 
 7366, 7476 7586 
 8462 8572 8681 
 
 no 
 no 
 
 III 
 
 9009 
 
 9"9 
 
 9228, 9337 9446 
 •3io »428 »537 
 
 9556 9665 9774 
 •646 *755| «864 
 I734 ! i843! i 9 5i 
 
 109 
 
 •101 
 
 •210 
 
 109 
 
 399 
 
 1 191 I2Q9 
 
 1408 1517 1625 
 
 109 
 
 K. 
 
 ll » ! 3 ! 4 i 5 '| 6 
 
 7 ! 8 | 9 
 
 B. 
 
A TABLE OF LOGARITHMS FROM 1 TO 10,000. 
 
 N. 
 
 | 1 | 2 
 
 3 
 
 4 | 5 
 
 6 | 
 
 7 | 
 
 8 
 
 9 1 
 
 D. 
 
 400 
 
 602060J 2169' 2277 
 3i44! 3253. 336i 
 
 2386 
 
 2494 26o3 
 
 27111 2819' 2928! 3o36 108 
 
 401 
 
 3469 
 
 3577: 3686' 3794 
 4658' 4766' 4874 
 
 3902 4oiO| 41 18, 108 
 
 402 
 
 42261 4334' 4442 
 
 455o 
 
 4982J 5089' 5197! 108 
 
 4o3 
 
 53o5 54i3 5521. 
 
 5628 
 
 3736' 5844! 5g5 1 
 
 6o5g 6i66i 6274' 108 
 7 i33i 7241! 7348, 10.7 
 82o5; 83 1 2 ' S419! 107 
 
 404 
 
 638i 
 
 6489 6096 
 
 6704 
 
 6811 
 
 6919 
 
 7026 
 
 40D 
 
 7455 
 
 t562 7669 
 8633; 8740 
 9701. 9808 
 
 7777 
 
 7884 
 8 9 54 
 
 7991 
 
 8098 
 
 .406 
 
 8526 
 
 8847 
 
 9061 
 
 9167 
 
 9274 
 
 g38l | 9488I 107 
 
 407 
 
 9 5 94 
 
 9914 
 
 ••21 
 
 •128 
 
 •234 
 
 •341 
 
 •447 
 
 •334i 107 
 
 408 
 
 610660 
 
 07671 0873 
 
 0979 
 
 1086 
 
 1 192 
 
 1298 
 
 i4o5 
 
 i5ii 
 
 1617 106 
 
 409 
 
 1723 
 
 1829' 1936 
 
 2042 
 
 2148 
 
 2234 
 
 236o 
 
 2466 
 
 2572 
 
 2678 
 
 106 
 
 410 
 
 612784 
 
 2890! 2996 
 
 3 102 
 
 3207 
 
 33i3 
 
 3419 
 4475 
 
 3525 
 
 363o 
 
 3736 
 
 106 
 
 411 
 
 3842 
 
 3947! 4033 
 
 4l5g 
 
 52i3 
 
 4264 
 
 4370 
 
 458 1 
 
 4686 
 
 4792 
 
 106 
 
 412 
 
 4897 
 
 5oo3 5 108 
 
 5319 
 
 5424 
 
 5529 
 
 5634 
 
 5740 
 
 5845 
 
 io5 
 
 4i3 
 
 59^0 
 
 6o55i 6160 
 
 6265 
 
 6370 
 
 6476 
 
 658 1 
 
 6686 
 
 6790 
 
 68g5 
 
 103 
 
 414 
 
 7000 
 
 7io5 7210 
 8 1 53 825 7 
 
 73i5 
 8362 
 
 7420 
 
 7525 
 
 7629 
 8676 
 
 7734 
 8780 
 
 783 9 
 
 7943 
 
 lo5 
 
 41 5 
 
 8048 
 
 8466 
 
 85 7 i 
 
 8884 
 
 8989 
 
 io5 
 
 416 
 
 9093 
 
 9198 
 
 9302 
 
 9406 
 
 90 1 1 
 
 9615 
 
 97'9 
 
 9824 
 
 9928 
 
 ••32 
 
 104 
 
 418 
 
 620106 
 
 0240 
 
 0344 
 
 0448 
 
 o552 
 
 0656 
 
 0760 
 
 0864 
 
 0968 
 
 1072 
 
 104 
 
 1.76 
 
 12S0 
 
 1384 
 
 1488 
 
 1592 
 
 i6o5 
 27J2 
 
 '799 
 2835 
 
 1903 
 
 2007 
 
 2110 
 
 104 
 
 419 
 
 2214 
 
 23i8 
 
 2421 
 
 5525 
 
 2628 
 
 2939 
 3 9 73 
 
 3o42 
 
 3i46 
 
 104 
 
 420 
 
 623249 
 
 3353 
 
 3456 
 
 3559 
 
 3663 
 
 3 7 66 
 
 386g 
 
 4076 
 
 4179 
 
 io3 
 
 421 
 
 4282 
 
 4385 
 
 4488 
 
 4591 
 
 4695 
 
 4798 
 
 4goi 
 
 5oo4 
 
 5 107 
 
 5210 
 
 io3 
 
 422 
 
 53i2 
 
 54i5 
 
 55i8 
 
 5621 
 
 5724 
 
 5827 
 
 5929 
 
 6o32 
 
 6i35 
 
 6238 
 
 io3 
 
 423 
 
 6340 
 
 6443 
 
 6546 
 
 6648 
 
 6751 
 
 6853 
 
 6956 
 
 7o58 
 8082 
 
 7161 
 8i85 
 
 7263 
 
 io3 
 
 424 
 
 7366 
 838g 
 
 7468 
 
 7571 
 85 9 3 
 
 7673 
 
 8693 
 
 7775 
 8797 
 
 j8 7 8 
 8900 
 
 7980 
 
 8287 
 93o8 
 
 102 
 
 425 
 
 8491 
 
 9002 
 
 9104 
 
 9206 
 
 102 
 
 426 
 
 9410 
 
 9512 
 
 9613 
 
 9715 
 
 98.7 
 
 9919 
 
 ••21 
 
 •123 
 
 •224 
 
 •326 
 
 102 
 
 428 
 
 630428 
 
 o53o 
 
 o63i 
 
 0733 
 
 o835 
 
 0936 
 
 io38 
 
 1139 
 
 2153 
 
 1 241 
 
 1342 
 
 102 
 
 1444 
 
 1 545 
 
 1647 
 
 1748 
 
 1849 
 
 1951 
 
 2052 
 
 2255 
 
 2356 
 
 101 
 
 429 
 
 2457 
 633468 
 
 2559 
 
 2660 
 
 2761 
 
 2862 
 
 2963 
 
 3064 
 
 3i65 
 
 3266 
 
 -3367 
 
 101 
 
 43o 
 
 3569 
 
 3670 
 
 3771 
 
 38 7 2 
 
 3973 
 
 4074 
 
 4n5 
 5i82 
 
 4276 
 
 4376 
 
 100 
 
 43 1 
 
 4477 
 
 45 7 8 
 
 4679 
 
 4779 
 5785 
 
 4880 
 
 4981 
 
 5o8i 
 
 5283 
 
 5383 
 
 100 
 
 432 
 
 5484 
 
 5584 
 
 5683 
 
 5886 
 
 5 9 86 
 
 6087 
 
 6187 
 
 6287 
 
 6388 
 
 100 
 
 433 
 
 6488 
 
 6588 
 
 6688 
 
 6789 
 
 6889 
 
 6989 
 
 7089 
 8090 
 9088 
 
 7189 
 8190 
 9188 
 
 7290 
 8290 
 9287 
 
 7390 
 838g 
 
 93»7 
 
 lco 
 
 434 
 435 
 
 7490 
 8489 
 
 8589 
 
 7690 
 8689 
 
 779° 
 8789 
 
 8888 
 
 7990 
 8988 
 
 99 
 99 
 
 436 
 
 94S6 
 
 9 586 
 
 9686 
 
 9783 
 
 9 S85 
 
 99 8 4 
 
 ••84 
 
 •i83 
 
 •283 
 
 •38a 
 
 99 
 
 43t 
 438 
 
 6404S1 
 
 o58i 
 
 06S0 
 
 0779 
 
 0879 
 
 0978 
 
 1077 
 
 "77 
 
 1276 
 
 1375 
 
 99 
 
 1474 
 
 i573 
 
 1672 
 
 ■771 
 
 1871 
 
 1970 
 
 2069 
 
 2168 
 
 2267 
 
 2366 
 
 99 
 
 43g 
 
 2465 
 
 2563 
 
 2662 
 
 2761 
 
 2860 
 
 2959 
 
 3o58 
 
 3i56 
 
 3255 
 
 3354 
 
 99 
 
 44o 
 
 643453 
 
 355i 
 
 365o 
 
 3749 
 
 3847 
 
 3g46 
 
 4044 
 
 4i43 
 
 4242 
 
 434o 
 
 9 o 
 
 44 1 
 
 4439 
 
 4537 
 
 4636 
 
 4734 
 
 4832 
 
 493i 
 
 5029 
 
 5127 
 
 5226 
 
 5324 
 
 98 
 
 442 
 
 5422 
 
 5521 
 
 56 1 9 
 
 6717 
 
 58i5 
 
 5qi 3 
 
 6894 
 
 601 1 
 
 61 10 
 
 6208 
 
 63o6 
 
 ■ 9 2 
 
 443 
 
 6404 
 
 65o2 
 
 6600 
 
 6698 
 
 6796 
 
 6992 
 
 7089 
 
 7187 
 
 7285 
 
 9 « 
 
 444 
 
 7 .383 
 
 836o 
 
 7481 
 8458 
 
 7579 
 
 7676 
 
 7774 
 
 7872 
 8848 
 
 7969 
 
 8067 
 
 8i65 
 
 8262 
 
 98 
 
 445 
 
 8553 
 
 8653 
 
 8750 
 
 8943 
 
 9043 
 
 914a 
 
 9237 
 
 97 
 
 446 
 
 9 335 
 
 9432 
 
 953o 
 
 9627 
 
 9724 
 
 9821 
 
 9?'9 
 0890 
 
 ••16 
 
 •n3 
 
 •210 
 
 97 
 
 447 
 
 65o3o8 
 
 0403 
 
 0502 
 
 0599 
 
 0696 
 
 0793 
 
 0987 
 
 1084 
 
 1181 
 
 97 
 
 448 
 
 1278 
 
 i375 
 
 1472 
 
 1 569 
 
 1666 
 
 1762 
 
 1839 
 
 1936 
 
 2o53 
 
 2i5o 
 
 97 
 
 449 
 45o 
 
 2246 
 
 2343 
 
 2440 
 
 2536 
 
 2633 
 
 273o 
 
 2826 
 
 2923 
 3888 
 
 3oig 
 
 3i:6 
 
 9 l 
 
 9 2 , 
 
 653213 
 
 3309 
 4273 
 
 3403 
 
 3 302 
 
 35 9 8 
 
 3695 
 
 3791 
 4734 
 
 3984 
 
 4080 
 
 45i 
 
 4177 
 
 436 9 
 
 4465 
 
 4562 
 
 4658 
 
 485o 
 
 4946 
 
 5o42 
 
 96 ' 
 
 432 
 
 5i38 
 
 5235 
 
 533 1 
 
 5427 
 
 5523 
 
 5619 
 
 5 7 i5 
 
 58io 
 
 5go6 
 6864 
 
 6002 
 
 96 
 
 453 
 
 6098 
 70D6 
 
 6194 
 
 6290 
 
 6386 
 
 6482 
 
 6577 
 
 66 7 3 
 
 6769 
 
 6960 
 
 96 
 
 454 
 
 7162 
 
 7247 
 
 7343 
 
 7438 
 83g3 
 
 7534 
 
 7629 
 8584 
 
 7720 
 8679 
 
 78201 701b 
 87741 8870 
 
 96 
 
 455 
 
 801 1 
 
 8107 
 
 8202 
 
 8298 
 9230 
 •201 
 
 8488 
 
 95 
 
 456 
 457 
 
 8 9 65 
 9916 
 
 9060 
 ••11 
 
 9i55 
 °io6 
 
 9346 
 •296 
 
 9441 
 •3oi 
 
 9 536 
 •486 
 
 963i 
 •58i 
 
 9726 9821 
 •676] 0771 
 
 9 l 
 9 l 
 
 458 
 
 66o865 
 
 0960 
 
 io55 
 
 n5o 
 
 1245 
 
 l33g 1434 
 
 1 S29 
 
 10231 I7lf 
 
 95 
 
 409 
 
 i8i3 
 
 1907 
 
 .2002; 2096I 2191I 2286' 238o 
 
 2473 
 
 2369 266; 
 
 95 
 
 N. 
 
 
 
 I 
 
 '2 | 3 i 4 1 5 | 6 i 7 | 8 j 9 | D. 
 
8 
 
 A TABLE 
 
 OP 
 
 LOGARITHMS FROM ] 
 
 TO 
 
 10,000. 
 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 1 5 j 6 | 7 j 8 
 
 9 
 
 D. 
 
 460 
 
 662758 
 
 2852 
 
 2947 
 
 3o4i 
 
 3i35i 323o| 3324! 34i8 
 
 4078' 4172? 4266! 4360 
 
 35i2 
 
 3607 
 
 94 
 
 461 
 
 3701 
 
 3795 
 
 4736 
 
 388g 
 
 3783 
 
 4454 
 
 4548 
 
 94 
 
 462 
 
 4642 
 
 483 
 
 4)24 
 5362 
 
 5oi8j 5u2 
 
 3206 
 
 5299 
 
 5393 
 633 1 
 
 5487 
 
 94 
 
 463 
 
 558 1 
 
 56 7 5 
 
 5769 
 
 5956' 6o5o 
 
 6l43 
 
 623^ 
 
 6424 
 
 94 
 
 464 
 465 
 
 65i8 
 7453 
 8386 
 
 6612 
 7546 
 
 670C 
 
 7640 
 
 85 7 2 
 
 b M 
 
 6892 
 7826 
 8759 
 
 6986 
 8852 
 
 80IJ 
 
 7173 
 8106 
 
 7266 
 8199 
 
 7360 
 8293 
 
 94 
 9^ 
 
 466 
 
 8479 
 
 8665 
 
 8945 
 
 9 o38 
 
 9i3i 
 
 9224 
 
 93 
 
 467 
 468 
 
 9 3ij 
 
 9410 
 
 95o3 
 
 9596 
 
 9689 
 
 9782 
 
 9 8 7 5 
 
 9967 
 
 ••60 
 
 •i53 
 
 9f 
 
 J70246 
 
 o33 9 
 
 043i 
 
 o524 
 
 061-; 
 i543 
 
 0710 
 1636 
 
 0802 
 
 0,895 
 
 0988 
 
 1080 
 
 93 
 
 469 
 
 1173 
 
 1265 
 
 1358 
 
 I45i 
 
 I728 
 
 1821 
 
 iqi3 
 
 2005 
 
 9 3 
 
 470 
 
 672098 
 
 2190 
 
 2283 
 
 2375 
 
 2467 
 
 256o 
 
 2652 
 
 2744 
 
 2836 
 
 2929 
 
 -385o 
 
 92 
 
 471 
 
 3o2I 
 
 3n3 
 
 32o5 
 
 3297 
 42i8 
 
 3390 
 
 3482 
 
 3574 
 
 3666 
 
 3758 
 
 92 
 
 472 
 
 3942 
 4861 
 
 4o34 
 
 4126 
 
 43io 
 
 4402 
 
 4494 
 
 4586 
 
 4677 
 
 4769 
 
 92 
 
 473 
 
 4q53 
 
 5o45 
 
 5i37 
 
 5228 
 
 5320 
 
 5412 
 
 55o3 
 
 55 9 5 
 
 568 7 
 
 92 
 
 474 
 
 5 77 8 
 6694 
 
 5870 
 6 7 85 
 
 5o62 
 
 bo53 
 
 6145 
 
 6236 
 
 6328 
 
 6419 
 7333 
 8245 
 
 65u 
 
 6602 
 
 92 
 
 475 
 
 6876 
 7789 
 8700 
 
 6968 
 
 7039 
 
 7 1 5i 
 
 8o63 
 
 7242 
 
 7424 
 8336 
 
 75i6 
 8427 
 
 91 
 
 476 
 
 itu 
 
 7698 
 8609 
 
 788, 
 8791 
 
 m 
 
 8i54 
 
 9> 
 
 477 
 478 
 
 8973 
 9882 
 
 9064 
 
 9 i55 
 
 9246 
 
 9337 
 
 9' 
 
 9428 
 
 9 5l 9 
 
 9610 
 
 9700 
 
 9791 
 
 9973 
 
 ••63 
 
 •i54 
 
 •245 
 
 9' 
 
 479 
 
 68o336 
 
 0426 
 
 o5i7 
 
 0607 
 i5i3 
 
 0698 
 
 0789 
 
 0879 
 
 0970 
 1874 
 
 1060 
 
 ii5i 
 
 9" 
 
 480 
 
 681241 
 
 1332 
 
 1422 
 
 i6o3 
 
 1693 1784 
 
 1964 
 2867 
 
 2055 
 
 90 
 
 481 
 
 SI45 
 
 2235 
 
 2326 
 
 2416 
 
 25o6 
 
 2596 2686 
 
 2777 
 
 2957 
 
 90 
 
 482 
 
 3047 
 
 3i37 
 
 3227 
 
 33i7 
 
 3407 
 
 3497 
 
 358 7 
 
 36 77 
 
 3767 
 4666 
 
 385 7 
 
 90 
 
 483 
 
 3 9 47 
 
 4o37 
 
 4127 
 
 4217 
 
 43o7 
 
 4396 
 
 4486 
 
 4576 
 
 4?56 
 5652 
 
 90 
 
 484 
 
 4845 
 
 4935 
 
 5o25 
 
 5ii4 
 
 5204 
 
 5294 
 6189 
 7083 
 
 5383 
 
 5473 
 
 5563 
 
 P 
 
 485 
 
 6742 
 
 583 1 
 
 5921 
 68i5 
 
 6010 
 
 6100 
 
 6279 
 
 6368 
 
 6458 
 
 6547 
 
 486 
 
 6636 
 
 6726 
 
 6904 
 
 6994 
 
 7172 
 8064 
 
 7261 
 81 53 
 
 735 1 
 8242 
 
 7440 
 833 1 
 
 a 9 
 
 487 
 488 
 
 7629 
 8420 
 
 7618 
 85oo 
 
 7707 
 85o8 
 
 7796 
 8687 
 
 7886 
 8776 
 
 I97 5 
 8865 
 
 £ 9 
 
 8o53 
 
 9042 
 
 9i3i 
 
 9220 
 
 o 9 
 
 489 
 
 9309 
 
 9 3n8 
 
 0280 
 
 9486 
 
 9 5 7 5 
 
 9664 
 
 9753 
 
 9841 
 
 9930 
 
 ••19 
 090D 
 
 •107 
 "W93 
 1877 
 
 %> 
 
 490 
 
 690196 
 1 08 1 
 
 o3 7 3 
 
 0462 
 
 o55o 
 
 0639 
 
 0728 
 
 0816 
 
 89 
 
 491 
 
 1170 
 
 1258 
 
 1347 
 
 1435 
 
 1 524 
 
 1612 
 
 1700 
 
 1789 
 
 88 
 
 492 
 
 1965 
 
 2o53 
 
 2142 
 
 2230 
 
 23i8 
 
 2406 
 
 2494 
 
 2583 
 
 2671 
 
 2759 
 
 88 
 
 493 
 
 1847 
 
 2935 
 38i5 
 
 3o23 
 
 3 1 11 
 
 3199 
 
 3287 
 
 3375 
 
 3463 
 
 355i 
 
 363 9 
 
 88 
 
 494 
 
 3727 
 
 3903 
 
 3991 
 
 4078 
 
 4166 
 
 4254 
 
 4342 
 
 443 
 
 45i7 
 
 88 
 
 495 
 
 46o5 
 
 4693 
 
 4781 
 
 4868 
 
 4g56 
 
 5o44 
 
 5i3i 
 
 5219 
 
 53o7 
 
 53g4 
 
 -88 
 
 496 
 
 548 2 
 
 556 9 
 
 5657 
 
 5744. 
 
 5832 
 
 5919 
 
 6007 
 
 6094 
 
 6182 
 
 6269 
 
 a 7 
 
 497 
 498 
 
 6356 
 
 6444 
 
 653 1 
 
 6618 
 
 6706 
 
 6 79 3 
 
 6880 
 
 6968 
 
 7o55 
 
 7142 
 80 14 
 
 ? 7 
 
 7229 
 
 7317 
 8188 
 
 7404 
 8275 
 
 8362 
 
 75 7 8 
 8449 
 
 7660 
 8535 
 
 775-2- 
 8622 
 
 783 9 
 8709 
 
 7926 
 
 £ 7 
 
 499 
 
 8101 
 
 8796 
 
 8883 
 
 I 1 
 
 5oo 
 
 698970 
 
 9057 
 
 9144 
 
 9231 
 
 9317 
 
 9404 
 
 9491 
 
 9 5 7 8 
 
 9664 
 
 9 75i 
 
 a 7 
 
 5oi 
 
 9838 
 
 9924 
 
 ••II 
 
 ••98 
 
 •184 
 
 •271 
 
 •358 
 
 •444 
 
 •53 1 
 
 •617 
 
 ?7 
 
 502 
 
 700704 
 
 0790 
 
 0877 
 
 0963 
 
 io5o 
 
 Ii36 
 
 1222 
 
 1 309 
 
 1395 
 
 1482 
 
 86 
 
 5o3 
 
 i568 
 
 16J4 
 
 1741 
 
 1827 
 
 lgl3 
 
 i?99 
 
 2086 
 
 2172J 2258 
 
 2344 
 
 86 
 
 5o4 
 
 243 1 
 
 25i7 
 
 26o3 
 
 2689 
 
 2 77 5 
 
 2861 
 
 2947 
 
 3o33 3i ig 
 
 32o5 
 
 86 
 
 5o5 
 
 32gi 
 4i5i 
 
 3377 
 
 3463 
 
 3549 
 
 3635 
 
 3721 
 
 3807 
 
 38o3 3979 
 
 4o65 
 
 86 
 
 . 5o6 
 
 4=36 
 
 4322 
 
 4408 
 
 4494 
 53 5o 
 
 4579 
 
 4665 
 
 4751 [ 4837 
 
 4922 
 
 86 
 
 507 
 5o8 
 
 5oo8 
 
 5094 
 
 5179 
 6o35 
 
 5265 
 
 5436 
 
 5522 
 
 5607! 5693 
 
 5 77 8 
 
 86 
 
 5S64 
 
 5o4q 
 68o3 
 
 6120 
 
 6206 
 
 6291 
 
 63 7 6 
 
 6462, 6547 
 
 6632 
 
 85 
 
 509. 
 
 6718 
 
 6888 
 
 6974 
 
 7o5g 
 
 7144 
 
 7229 
 8081 
 
 73i5| 7400 
 8166! 825i 
 
 7485 
 
 85 
 
 5io 
 
 707570 
 
 7655 
 
 7740 
 8591 
 
 7826 
 8676 
 
 791 1 
 8761 
 
 TP 6 
 8846 
 
 8336 
 
 85 
 
 5u 
 
 8421 
 
 85o6 
 
 8g3l 
 
 9015 9100 
 
 9185 
 
 85 
 
 5l2 
 
 • 9270 
 
 9355 
 
 9440 
 
 9D24 
 
 .9609 
 
 9694 
 
 9779 
 0625 
 
 9863 j 9948 
 
 ••33 
 
 85 
 
 5i3 
 
 710117 
 
 0202 
 
 0287 
 
 0371 
 
 0456 
 
 0540 
 
 0710' 0794 
 1 554; 1W9 
 
 0879 
 
 1 7 23 
 
 85 
 
 5U 
 
 0963 
 1807 
 
 1048 
 
 Il32 
 
 1217 
 
 >3oi 
 
 1 385 
 
 1470 
 
 84 
 
 5 1 5 
 
 1892 
 2734 
 
 1076 
 
 2060 
 
 2144 
 
 2229 
 
 23 13 
 
 2397 2481 
 
 2566 
 
 84 
 
 5i6 
 
 265o 
 
 2818 
 
 2902 
 
 2986 
 
 3070 
 
 3i54 
 
 3238 
 
 3323 
 
 3407 
 
 84 
 
 5, 1 
 
 5i8 
 
 3491 
 
 3575 
 
 365g 
 
 3742 
 
 3826 
 
 3910 
 
 3g94 
 4833 
 
 4078 
 
 4162 
 
 4246 
 
 84 
 
 433o 
 
 44i4 
 
 4497 
 
 458 1 
 
 4665 
 
 4749 
 
 4916 
 
 5ooo 
 
 5o84 
 
 84 
 
 5l9 
 
 5167 
 
 525i 
 
 5335 
 
 54i8 
 
 55o2 5586 
 
 566 9 
 
 5^53 5836 5920 
 
 84 
 
 W. 
 
 
 
 I 
 
 2 
 
 3 
 
 4 | 5 
 
 6 
 
 7 1 8 I 9 j 
 
A TABLE OF LOGARITHMS FROM 1 TO 10,000. 
 
 N. 
 
 530 
 521 
 522 
 
 5j3 
 
 bJ4 
 525 
 526 
 521 
 528 
 529 
 '53o 
 ,53i 
 532 
 533 
 534 
 "535 
 536 
 53 7 
 538 
 53 9 
 54o 
 541 
 542 
 543 
 544 
 545 
 546 
 54t 
 548 
 549 
 55o 
 55i 
 502 
 553 
 554 
 555 
 556 
 557 
 558 
 55g 
 56o 
 56i 
 . 562 
 563 
 564 
 565 
 566 
 56 7 
 568 
 56 9 
 570 
 5 7 i 
 572 
 5 7 3 
 574 
 575 
 576 
 
 ill 
 
 J>79 
 N. 
 
 716003, 6087 6170 
 683S, 6921 j 7004 
 7671 7754' 7837 
 85o2l 8585 1 8668 
 
 9 33 4, 
 
 720159 
 0986 
 1811 
 2634 
 3456 
 
 724276 
 6095 
 5912 
 6727 
 7541 
 8354 
 9i65 
 9974 
 
 730782 
 1 58 9 
 
 732394 
 3197 
 3999 
 4800 
 5599 
 6397 
 7 ig3 
 
 I^ 1 
 8781 
 
 9072 
 
 74o363 
 
 -Il52 
 
 i 9 3 ? 
 2720 
 35io 
 4293 
 5075 
 5855 
 6634 
 7412 
 -48188 
 8963 
 
 70o5o8 
 1271 
 204! 
 2816 
 3583 
 4348 
 5i 1 2 
 
 755875 
 6636 
 
 $5 
 8912 
 9668 
 760422 
 1176 
 1928 
 2679 
 
 94Ui 
 0242 
 •1068 
 l8 9 3 
 2716 
 3538 
 4358 
 6176 
 5993 
 6809 
 7623 
 8435 
 9246 
 ••55 
 o863 
 1669 
 2474 
 
 6254! 633} 
 7088, 7171 
 
 9497 
 
 o323 
 
 n5i 
 
 " 97 8 
 2798 
 
 3620 
 
 4440 
 
 5258 
 
 6073 
 
 6890 
 
 7704 
 
 85i6 
 
 9327 
 
 •i36 
 
 0944 
 
 1750 
 
 2555 
 
 3278| 3358 
 4160 
 4960 
 
 4 S 9 
 4880 
 
 6679 
 
 6476 
 
 7272 
 
 8067 
 
 8860 
 
 9651 
 
 0442 
 
 1230 
 
 2018 
 2804 
 
 3588 
 4371 
 5i53 
 5 9 33 
 6712 
 
 I' ■ 
 8266 
 
 9040 
 9814 
 0586 
 i356 
 2125 
 2893 
 366o 
 4425 
 5189 
 SgSl 
 6712 
 7472 
 823o 
 8988 
 9743 
 0498 
 
 I25l 
 
 2oo3 
 
 2754 
 
 792c 
 
 875 1 
 9580 
 
 1233 
 2o58 
 2881 
 3702 
 
 8oo3 
 8834 
 9 663 
 0490 
 i3i6 
 2140 
 2963 
 3i84 
 
 4.6221 4604 
 5340 5422 
 
 5 7 59 
 6556 
 7352 
 8146 
 8 9 39 
 973" 
 
 0521 
 
 i3o9 
 2096 
 2882 
 3667 
 4449 
 5231 
 601 1 
 6790 
 
 9118 
 989 
 
 6663 
 1433 
 2202 
 2970 
 3736 
 45ol 
 5265 
 6027 
 6788 
 7548 
 83o6 
 9063 
 9811 
 057. 
 1326 
 2078 
 2829 
 
 6i56 
 6972 
 
 ll 9 n 
 9408 
 •217 
 1024 
 1 83o 
 2635 
 3438 
 4240 
 5o4o 
 5838 
 6635 
 743i 
 8225 
 9018 
 9810 
 0600 
 1388 
 2175 
 2961 
 3 7 45 
 4528 
 53o9 
 6089 
 '6868 
 7645 
 8421 
 
 s# 
 
 0740 
 i5io 
 2279 
 3o47 
 38i3 
 45 7 8 
 5341 
 6io3 
 6864 
 7624 
 8382 
 9i3 9 
 9894 
 0649 
 1402 
 2i53 
 2904! 
 
 6238 
 7053 
 7866 
 8678 
 9481 
 •29) 
 no5 
 191 1 
 271 5 
 35i8 
 4320 
 
 5l20 
 
 5 9 i8 
 6715 
 7311 
 8.3o5 
 909 
 
 067& 
 1467 
 2254 
 3o3q 
 3823 
 4606 
 5387 
 6167 
 6945 
 7722 
 8498 
 
 nil 
 
 0817 
 
 1 587 
 2356 
 3i23 
 3889 
 4654 
 54i7 
 6180 
 6940 
 7700 
 8458 
 9214 
 9970 
 0724 
 
 1477 
 22;8 
 2978 
 
 6421 
 
 7204 
 
 8086 
 
 8917 
 
 9745 
 
 0673 
 
 i3g8 
 
 2222 
 
 3043 
 
 3866 
 
 4685 
 
 55o3 
 
 6320 
 
 7i34 
 
 7' 
 
 8759 
 
 9570 
 
 •378 
 
 1 186 
 
 1991 
 
 2796 
 
 35 9 8 
 
 4400 
 
 5200 
 
 5998 
 6795 
 7590 
 8384 
 9177 
 9968 
 0757 
 1546 
 
 2332 
 
 3n8 
 
 3902 
 
 4684 
 
 5465 
 
 6245 
 
 7023 
 
 7800 
 
 85 7 6 
 
 935o 
 
 •i23. 
 
 089 
 
 1664 
 
 2433 
 
 3200 
 
 3966 
 4730 
 5494 
 
 6256 
 7016 
 
 7775 
 8533 
 9290 
 ••45 
 •0799 
 1 552 
 23o3 
 3o53 
 
 65o4 
 7338 
 8169 
 9000 
 9828 
 o655 
 1481 
 23o5 
 3127 
 3948 
 476] 
 5585 
 6401 
 7216 
 8029 
 8841 
 9651 
 •45o 
 1266 
 2072 
 2876 
 3679 
 4480 
 5279 
 6078 
 6874 
 7670 
 8463 
 9256 
 
 ••47 
 o836 
 
 1624 
 241 1 
 3i 9 6 
 3g8o 
 4762 
 5543 
 6323 
 7101 
 7878 
 8653 
 
 9427 
 •200 
 0971 
 1741 
 25o9 
 3277 
 4042 
 4807 
 5570 
 6332 
 7092 
 7801 
 8609 
 9 366 
 •121 
 0875 
 1627 
 2378 
 3i28 
 
 9 I 
 
 6588 
 7421 
 8253 
 9 o83 
 
 99" 
 o 7 38 
 1 563 
 2387 
 3209 
 4o3o 
 4849 
 5667 
 6483 
 7297 
 81 10 
 8922 
 97 32 
 •540 
 1347 
 
 2 1 52 
 2g56 
 
 3 7 59 
 
 456o 
 
 5359 
 
 61 57 
 
 6 9 54 
 
 774< 
 
 854: 
 
 9 335 
 
 S126 
 
 0915 
 
 !7o3 
 
 2489 
 
 32 7 5 
 
 4058 
 
 4840 
 
 562 
 
 6401 
 
 7'7' 
 
 79 5: 
 
 8731 
 
 9604 
 
 •277 
 
 1048 
 
 isr 
 
 2586 
 3353 
 4119 
 4883 
 5646 
 6408 
 7168 
 7927 
 8685 
 9441 
 •196 
 0900 
 1702 
 2453 
 32o3 
 
 D. 
 
 6671! 6 7 54! 83 
 
 75o4' 7587I 83 
 
 8336! 8419; 83 
 
 9160, 9248 
 
 9994 
 0821 
 1646 
 2469 
 3291 
 4112 
 
 '77 
 0903 
 1728 
 
 2552 
 
 3374 
 
 4194 
 
 493i| 5oi3 
 5748 583o 
 65641 6646 
 
 7 3 79 
 8191 
 9003 
 9813 
 •621 
 1428 
 
 2233 
 
 3o37 
 
 3839 
 
 4640 
 
 5439 
 
 6237 
 
 7o3. 
 
 7829 
 
 8622 
 
 94U 
 
 •2o5 
 
 0994 
 
 1782 
 
 2568 
 
 3353 
 
 4i36 
 
 5699 
 
 6479 
 7256 
 8o33 
 8808 
 g582 
 •354 
 1 1 25 
 1895 
 2663 
 343o 
 4195 
 4960 
 6722 
 6484 
 7244 
 8oo3 
 8761 
 9 51 7 
 •272 
 
 1025 
 
 1778 
 
 2529 
 
 3278 
 
 460 
 8273 
 9084 
 9893 
 
 •702 
 i5o8 
 23i3 
 3117 
 3919 
 4720 
 55i9 
 63i7 
 7ii3 
 7908 
 8701 
 9493 
 •284 
 1073 
 i860 
 2647 
 343 1 
 42i5 
 4997 
 3777 
 6556 
 
 7334 
 8110 
 
 9659 
 •43 1 
 1202 
 1972 
 2740 
 35o6 
 4272 
 5o36 
 5799 
 656o 
 7320 
 8079 
 8836 
 9 5 9 2 
 
 •347 
 1:01 
 1 853 
 2604 
 
 3353| 75 
 
10 
 
 A TABLE OP LOGARITHMS FROM 1 TO 10,000. 
 
 N. 
 THo" 
 
 OI! 
 
 3 -| 4 
 
 5 
 
 6 | 7 8 
 
 9 
 
 D. 
 
 763428 35o3 3578 36531 3727 
 
 38o2 
 
 3877! 3 9 52i 4027 
 
 4101 
 
 75 
 
 58i 
 
 4176 4261 
 
 1 4326| 44oo 4475 
 
 455o 
 
 4624 46 99 ' 477/ 
 
 4848 
 
 75 
 
 5S2 
 
 4923 
 
 4998 
 
 50721 5i4- 
 
 5221 
 
 52 9 6! 5370! 5445j 552c 
 
 55 9 4 
 
 75 
 
 583 
 
 566 9 
 64i3 
 
 5743 
 
 58i8 
 
 58 9 2 
 6636 
 
 5 9 66 
 
 6041 
 
 6110! 6igo 6264 
 
 6338 
 
 74 
 
 584 
 
 6487 
 
 6562 
 
 6710 
 
 6 7 85 
 
 685 9 
 
 6 9 33[ 700' 
 
 7082 
 
 74 
 
 585 
 
 7i56 
 
 723o 
 
 73o4 
 
 8120 
 
 7453 
 8104 
 8o3< 
 
 7527 
 
 8268 
 
 7601 
 8342 
 
 7675; 7749 
 8416! 8490 
 91 56j 9230 
 
 7823 
 
 74 
 
 586 
 
 7898 
 8638 
 
 7972 
 8712 
 
 8046 
 
 856; 
 
 74, 
 
 58 7 
 
 8786 
 
 8860 
 
 9 ooE 
 
 9082 
 
 9 3oc 
 
 74 
 
 588 
 
 93?7 
 
 9 45i 
 
 9 525 
 
 9599 
 
 o336 
 
 9073 
 
 9746 
 
 9820 
 
 9894: 9 9 6S 
 
 ••42 
 
 74 
 
 58 9 
 
 7701 13 
 
 oi8 9 
 
 0263 
 
 0410 
 
 0484 
 
 o55] 
 1293 
 
 o63i 
 
 0705 
 
 0778 
 
 74 
 
 590 
 
 770852 
 
 9 20 
 
 "999 
 1734 
 
 1073 
 
 1 146 
 
 1220 
 
 1367 
 
 1440 
 
 i5i4 
 
 74 
 
 591 
 
 1387 
 
 1661 
 
 1808 
 
 1881 
 
 i 9 55 
 
 2028 
 
 2102 
 
 217J 
 
 2248 
 
 73 
 
 5g2 
 
 2322 
 
 23 9 5 
 
 2468 
 
 2542 
 
 26i5 
 
 2688 
 
 2762 
 
 2835 
 
 2 9 o8 
 
 2 9 8l 
 
 73 
 
 5 9 3 
 
 3o55 
 
 3i28 
 
 3201 
 
 3274 
 
 3348 
 
 3421 
 
 3494 
 
 3567 
 
 3640 
 
 37i3 
 
 73 
 
 ^i 
 
 3786 
 
 386o 
 
 3 9 33 
 
 4006 
 
 4079 
 
 4i52 
 
 4225 
 
 42 9 8 
 
 4371 
 
 4444 
 
 73 
 
 5 9 5 
 
 45i7 
 
 4590 
 
 4663 
 
 4736 
 
 48o 9 
 
 4882 
 
 4 9 55 
 
 5o28 
 
 5ioo 
 
 5n3 
 
 73 
 
 5 9 6 
 
 52/,6 
 
 5319 
 
 53 9 2 
 
 5465 
 
 5538 
 
 56io 
 
 5683 
 
 5 7 56 
 
 582 9 
 
 5 9 02 
 
 73 
 
 III 
 
 5 97 4 
 
 6047 
 
 6120 
 
 6 I9 3 
 
 6260 
 
 6338 
 
 641 1 
 
 6483 
 
 6556 
 
 6629 
 
 73 
 
 6701 
 
 6774 
 
 6846 
 
 6 9 i 9 
 
 6 99 2 
 
 7064 
 
 7'37 
 
 720 9 
 
 7282 
 8006 
 
 7354 
 
 ' 73 
 
 5 99 
 
 7427 
 778151 
 
 74 99 
 
 7572 
 8296 
 
 7644 
 8368 
 
 77H 
 8441 
 
 7789 
 85i3 
 
 7862 
 8585 
 
 7 9 34 
 8658 
 
 8079 
 
 72 
 
 POO 
 
 8224 
 
 8 7 3o 
 
 8802 
 
 72 
 
 601 
 
 8874 
 
 8047 
 
 9019 
 
 9 S 9 I 
 
 9 i63 
 
 9 236 
 
 9 3o8 
 
 9 38o 
 
 9452 
 
 9 524 
 
 72 
 
 602 
 
 9 5 9 6 
 
 9 66 9 
 
 974i 
 
 9 8i3 
 
 9 885 
 
 99 5 7 
 
 "2 9 
 
 •101 
 
 •173 
 
 •245 
 
 72 
 
 6o3 
 
 780317 
 
 o38 9 
 
 0461 
 
 o533 
 
 o6o5 
 
 0677 
 i3go 
 
 °74 9 
 
 0821 
 
 o8 9 3 
 
 o 9 65 
 
 72 
 
 604 
 
 1037 
 
 II0 9 
 
 1181 
 
 1253 
 
 1324 
 
 1468 
 
 1 54o 
 
 1612 
 
 1684 
 
 72 
 
 6o5 
 
 1755 
 
 1827 
 
 i8 99 
 
 1931 
 
 2688 
 
 2042 
 
 2114 
 
 2186 
 
 2258 
 
 2329 
 
 2401 
 
 72 
 
 606 
 
 2473 
 3i8 9 
 
 2544 
 
 2616 
 
 27 5g 
 3476 
 4189 
 
 283 1 
 
 2902 
 36i8 
 
 2 97 4 
 
 368o 
 44o3 
 
 3o46 
 
 3i 17 
 
 72 
 
 607 
 
 3260 
 
 3332 
 
 34o3 
 
 3546 
 
 3 7 6i 
 
 3832 
 
 7' 
 
 608 
 
 3 9 o4 
 
 3 9 75 
 
 4046 
 
 4i 18 
 
 4261 
 
 4332 
 
 4475 
 
 4546 
 
 7' 
 
 609 
 
 4617 
 
 468 9 
 
 4760 
 
 483 1 
 
 "4g02 
 
 4 9 74 
 5686 
 
 5o45 
 
 5n6 
 
 5187 
 
 5259 
 
 7i 
 
 610 
 
 78533o 
 
 54oi 
 
 5472 
 6i83 
 
 5543 
 
 56i5 
 
 5 7 5 7 
 
 6828 
 
 58 99 
 
 5970 
 6680 
 
 7« 
 
 611 
 
 6041 
 
 6112 
 
 6254 
 
 6325 
 
 63 9 6 
 
 6467 
 
 6538 
 
 6609 
 
 7i 
 
 612 
 
 6751 
 
 6822 
 
 68 9 3 
 
 6 9 64 
 
 7o35 
 
 7106 
 
 788^ 
 85 9 3 
 
 7248 
 
 7319 
 
 8098 
 
 71 
 
 6i3 
 
 7460 
 8168 
 
 7?3 1 
 823 9 
 
 7602 
 
 M 
 
 7744 
 845 1 
 
 78i5 
 
 8522 
 
 7 9 56 
 8663 
 
 8027 
 
 7« 
 
 014 
 
 83io 
 
 8734 
 
 8804 
 
 7' 
 
 6i5 
 
 8875 
 9 58i 
 
 8946 
 
 9016 
 
 9 o8 7 
 
 9 'Jl 
 
 9 228 
 
 9299 
 
 9369 
 
 9 44o 
 
 9 5io 
 
 7i 
 
 616 
 
 9 65i 
 
 9722 
 
 9792 
 
 9 863 
 
 9933 
 
 •••4 
 
 "7,4 
 
 •144 
 
 •2l5 
 
 70 
 
 6, l 
 618 
 
 790285 
 
 o356 
 
 0426 
 
 04 9 6 
 
 0567 
 
 0637 
 
 0707 
 
 o 7 |8 
 1480 
 
 0848 
 
 0918 
 
 70 
 
 0988 
 
 1059 
 
 1 1 29 
 
 1199 
 
 I26 9 
 
 1340 
 
 1410 
 
 i55o 
 
 1620 
 
 70 
 
 6*9 
 
 1691 
 
 1 761 
 
 i83i 
 
 1901 
 
 1071 
 2672 
 
 2041 
 
 2111 
 
 2181 
 
 2252 
 
 2322 
 
 7" 
 
 620 
 
 792392 
 
 2462 
 
 2532 
 
 2602 
 
 2742 
 
 2812 
 
 2882 
 
 2 9 52 
 
 3022 
 
 70 
 
 621 
 
 3o 9 2 
 
 3i62 
 
 323 1 
 
 33oi 
 
 33 7 i 
 
 3441 
 
 35u 
 
 358i 
 
 365r 
 
 3721 
 
 70 
 
 622 
 
 3790! 386o 
 
 3o3o 
 4027 
 
 4000 
 
 4070 
 4767 
 
 4139 
 
 420 9 
 
 4279 
 
 4349 
 5o45 
 
 44l8 
 
 70 
 
 623 
 
 4488 
 
 4558 
 
 4697 
 53 9 3 
 
 4836 
 
 4906 
 56o2 
 
 4976 
 
 5u5 
 
 70 
 
 624 
 
 5i85 
 
 5254 
 
 5324 
 
 5463 
 
 5532 
 
 5672 
 
 5741 
 
 58n 
 
 70 
 
 625 
 
 588o 
 
 5 9 4 9 
 
 6019 
 6 7 i3 
 
 6088 
 
 6i58 
 
 6227 
 
 62 9 7 
 
 6366 
 
 6436 
 
 65o5 
 
 °9 
 
 626 
 
 65 7 4 
 
 6644 
 
 6782 
 
 6852 
 
 6021 
 7614 
 
 6990 
 
 7060 
 
 7I2 9 
 
 7198 
 
 69 
 
 627 
 
 7268, 7337 
 
 7406 
 8098 
 
 7475 
 8167 
 8858 
 
 7545 
 8236 
 
 7683 
 83 7 4 
 
 7752 
 
 7821 
 
 78 9 o 
 
 69 
 
 628 
 
 7 9 6o| 8029 
 865i 8720 
 
 83o5 
 
 8443, 85 1 3 
 
 8582 
 
 69 
 
 629 
 
 8789 
 
 8 9 27 
 
 8 99 6 
 
 9065 
 
 9 i34' 9203 
 
 9 3 72 
 
 69 
 
 63o 
 
 799341 | 9400 
 
 O3002 9 | 0008 
 
 9478 
 
 9547 
 
 9616 
 
 9 685 
 
 9754 
 
 98231 9802 
 
 99° ' 
 
 69 
 
 63 1 
 
 0167 
 
 0236 
 
 o3o5 
 
 0373 0442 
 
 o5 n J o58o 
 
 0648 
 
 69 
 
 632 
 
 0717 O786 
 
 0854 
 
 0923 j 
 
 0992 
 
 1 061 
 
 1 1 29 
 
 n 9 8 1266 
 
 ■ 335 
 
 69 
 
 633 
 
 1404 1472 
 
 1 541 
 
 1609 1 1678 
 
 1747 
 
 i8j5 
 
 1884 I 9 52 
 2568 2637 
 
 2021 
 
 69 
 
 634 
 
 2089 2 1 58 
 
 2226 
 
 2295 2363 
 
 2432 
 
 25oo 
 
 2705 
 
 69 
 
 635 
 
 2774 2842 
 
 2910 
 35 9 4 
 
 29791 3o47 
 
 3n6 
 
 3 184 
 
 3252 3321 
 
 338o 
 
 68 
 
 636 
 
 3407 3525 
 
 3662 3730 
 
 3 79 8 
 
 386 7 
 4548 
 
 3o35 
 
 4&16 
 
 4oo3 407-1 
 
 68 
 
 637 
 638 
 
 4i3 9 4208 
 
 4276 
 
 4344! 4412 4480 
 
 46851 4 7 53 
 5365! 5433 
 
 68 
 
 4821 4889 
 
 4o57 
 563 7 
 
 5o25 5o 9 3 5i6i 
 
 5220 
 
 5 9 o8 
 
 52 9 7 
 
 68 
 
 63 9 
 
 55oi, 556 9 | 
 
 5705J 5773 584i 
 
 5 97 6 
 
 6o44| 6112 
 
 68 
 
 N. 
 
 i 1 | 2 
 
 3 j 4 | 5 
 
 6 
 
 7 
 
 8 | 9 
 
 D 
 

 A TADLE 
 
 OF LOGARITHMS FROM 1 
 
 TO 
 
 10,000. 
 
 11 
 
 N. 
 T40H 
 
 
 
 1 
 
 2 
 
 3 1 4 
 
 5 
 
 6 
 
 7 
 
 8 | 9 
 
 D. 
 
 806180 6248 
 
 63 16 
 
 6384 6431 
 
 6519 
 
 6587 
 
 6655 
 
 6723J 6790 
 
 68 
 
 641 
 
 6858 
 
 6926 
 
 6994 
 
 7061 
 
 7129 
 
 7197 
 
 7264 
 
 7332 
 
 7400 7467 
 
 68 
 
 642 
 
 7535 
 
 7603! 7670 
 8279 8346 
 
 7738 
 84>4 
 
 7806 
 8481 
 
 78 7 3 
 8549 
 
 7941 
 8616 
 
 8008 
 
 8076 
 
 8143 
 
 68 
 
 643 
 
 8211 
 
 8684 
 
 8751 
 
 8818 
 
 67 
 
 644 
 
 8886| 
 
 8o53 
 
 9021 
 
 9088 
 
 9 1 56. 
 
 9223 
 
 9290 
 
 9 358 
 
 9425 
 
 9492 
 
 F 
 
 645 
 
 9360' 9627 
 
 9694 
 
 9762 
 
 9829 
 
 9896 
 
 9964 
 
 ••3j 
 
 ••98 
 
 •i65 
 
 67 
 
 646 
 
 8ios33, o3oo 
 
 0367 
 
 0434 
 
 o5oi 
 
 0569 
 
 o636 
 
 0703 
 
 0770 
 
 0837 
 i5o8 
 
 67 
 
 647 
 
 0904I 0971 
 iD75i 1642 
 
 1039 
 
 1 1 06 
 
 1 173 
 
 1240 
 
 1307 
 
 1374 
 
 1441 
 
 67 
 
 64S 
 
 1709 
 
 1776. 
 
 i843 
 
 1910 
 
 ■977 
 
 2044 
 
 21 1 1 
 
 2178 
 
 67 
 
 649 
 
 2245! 23l2 
 
 2379 
 
 2445 
 
 25 1 2 
 
 ?5 79 
 
 2646 
 
 2713 
 
 2780 
 
 2847 
 
 67 
 
 65o 
 
 8l2gl3i 2980 
 
 3047 
 
 3 1 14 
 
 3i8i 
 
 3247 
 
 33i4 
 
 338 1 
 
 3448 
 
 35i4 
 
 &7 
 
 65i 
 
 3 58 1 1 3648 
 
 3714 
 
 3781 
 
 3848 
 
 39U 
 
 3 9 8i 
 
 4048 
 
 4114 
 
 4181 
 
 67 
 
 6J2 
 
 4248! 43 1 4 
 
 438 1 
 
 4447 
 
 45i4 
 
 458i 
 
 4647 
 
 47U 
 
 4780 
 
 ■4847 
 
 b J. 
 
 653 
 
 49 1 3 4o8o 
 
 5o46 
 
 5u3 
 
 f/75 
 5843 
 
 5246 
 
 53 1 2 
 
 5378 
 
 5445 
 
 55n 
 
 66 
 
 654 
 
 5578 
 
 5644 
 
 5711 
 
 5-777 
 
 5910 
 6573 
 
 5976 
 
 6042 
 
 . 6109 
 
 6i 7 5 
 
 66 
 
 655 
 
 6241 
 
 63o8 
 
 63 7 4 
 
 644o 
 
 65o6 
 
 663 9 
 
 670s 
 
 6 77 . 
 
 6838 
 
 66 
 
 656 
 
 6904 
 
 6970 
 
 7036 
 
 7102 
 
 7169, 
 
 V o 3 l 
 
 73oi 
 
 8028 
 
 7433 
 '8094 
 
 7499 
 
 66 
 
 607 
 
 7365 
 8226 
 
 7631 
 8292 
 8961 
 
 8338 
 
 7764 
 8424 
 
 783o 
 
 7896 
 8'556 
 
 7962 
 8622 
 
 8160 
 
 66 
 
 658 
 
 8490 
 
 8688 
 
 8754 
 
 8820 
 
 66 
 
 65g 
 
 8885 
 
 9017 
 
 9083 
 
 9149 
 
 9215 
 
 9281 
 
 9346 
 
 9412 
 
 9478 
 
 66 
 
 660 
 
 8i9544 
 
 9610 
 
 9676 
 
 974i 
 
 9807 
 
 9873 
 
 ■9939 
 0595 
 
 1231 
 
 •••4 
 
 ••70 
 
 •i36 
 
 66 
 
 661 
 
 820201 
 
 0267 
 
 o333 
 
 0399 
 10O0 
 
 0464 
 
 o53o 
 
 0661 
 
 0727 
 
 0792 
 
 66 
 
 662 
 
 o858 
 
 0924 
 
 0989 
 
 1120 
 
 1 186 
 
 l3i7 
 
 1382 
 
 1448 
 
 66 
 
 663 
 
 i5i4 
 
 1 579 
 
 2233 
 
 1645 
 
 1710 
 
 I77 5 
 
 1841 
 
 I906 
 
 1972 
 
 2037 
 
 2103 
 
 65 
 
 664 
 
 2168 
 
 2299 
 
 2364 
 
 243o 
 
 2495 
 
 2360 
 
 2626 
 
 2691 
 
 2756 
 
 65' 
 
 665 
 
 2&22 
 
 -2887 
 
 2932 
 
 3oi8 
 
 3o83 
 
 ~3i48 
 
 32i3 
 
 3279 
 
 3344 
 
 3409 
 
 65 
 
 666 
 
 3474 
 
 3539 
 
 36o5 
 
 3670 
 
 3 7 35 
 
 38oo 
 
 3865 
 
 3930 
 
 3996 
 
 4061 
 
 65 
 
 667 
 668 
 
 4126 
 
 4191 
 
 4256 
 
 432i 
 
 4386 
 
 445 1 
 
 45i6 
 
 438i 
 
 4646 
 
 47 1 1 
 
 65 
 
 4776 
 
 4841 
 
 4906 
 
 4971 
 
 5o36 
 
 5ioi 
 
 5x66 
 
 523 1 
 
 5296 
 
 536i 
 
 65 
 
 669 
 
 5426 
 
 549 1 
 
 5556 
 
 5621 
 
 5686 
 
 575i 
 
 58i5 
 
 588o 
 
 5 9 45 
 65 9 3 
 
 6010 
 
 65 
 
 670 
 
 826075 
 
 6140 
 
 6204 
 
 6269 
 
 6334 
 
 6399 
 
 6464 
 
 6528 
 
 6658 
 
 65 
 
 671 
 
 6723 
 
 6787 
 
 6852 
 
 6917 
 
 6981 
 
 7046 
 
 7111 
 
 T75 
 
 7240 
 
 73o5 
 
 65 
 
 672 
 673 
 
 7 36g 
 8oi5 
 
 7434 
 8080 
 
 7499 
 8144 
 
 7363 
 8209 
 
 7628 
 8273 
 
 7692 
 8338 
 
 77^7 
 8402 
 
 7821 
 8467 
 
 7886 
 853 1 
 
 7961 
 85 9 5 
 
 65 
 64 
 
 674 
 
 8660 
 
 8724 
 
 8789 
 
 8853 
 
 8918 
 9361 
 
 8982 
 
 9046 
 
 9111 
 
 91^ 
 
 9^9 
 
 64 
 
 675 
 
 93o4 
 
 9 368 
 
 9442 
 
 9497 
 •U9 
 
 9625 
 
 9690 
 
 9754 
 
 9818 
 
 9882 
 
 64 
 
 676 
 
 9947 
 830089 
 
 ••11 
 
 ••75 
 
 •204 
 
 •268 
 
 •332 
 
 •396 
 
 •460 
 
 •525 
 
 64 
 
 %l 
 
 o653 
 
 0717 
 i358 
 
 0781 
 
 o845 
 
 0909 
 i55o 
 
 0973 
 
 io3- 
 
 1 102 
 
 1 166 
 
 64 
 
 I23o 
 
 1294 
 2373 
 
 1422 
 
 •i486 
 
 1614 
 
 1678 
 
 1742 
 
 1806 
 
 64 
 
 679. 
 
 1870 
 
 1998 
 2637 
 
 2062 
 
 2126 
 
 2189 
 2828 
 
 2253 23 17 
 
 238i 
 
 2445 
 
 64 
 
 680 
 
 83 2309 
 
 2700 
 
 2764 
 
 2892 
 
 2956 
 
 302C 
 
 3i«S3 
 
 " 64 
 
 681 
 
 3 1 47 
 
 3211 
 
 3275 
 
 3338 
 
 3402 
 
 3466 
 
 353o 
 
 35o3 
 
 365- 
 
 . 3721 
 
 64 
 
 682 
 
 3784 
 
 3848 
 
 3gl2 
 4548 
 
 3975 
 
 4039 
 
 4io3 
 
 4166 423c 
 
 429. 
 
 4357 
 
 64 
 
 683 
 
 4421 
 
 4484 
 
 461 1 
 
 4673 
 
 4739 
 
 4802 ' 4866 
 
 4925 
 5564 
 
 4993 
 
 64 
 
 684 
 
 5o56 
 
 5 120 
 
 5i8'3 
 
 5247 
 
 53 10 
 
 5373 
 
 5437' 55oc 
 
 5627 
 
 63 
 
 685 
 
 5691 
 6324 
 
 5754 
 
 58i7 
 
 588 1 
 
 5944 
 
 6007 
 
 607 1 | 6i34 
 
 619- 
 
 6261 
 
 63 
 
 686 
 
 6387 
 
 645i 
 
 65i4 
 
 6077 
 
 6641 
 
 6704 1 6767 
 
 683c 
 
 6894 
 
 63 
 
 687 
 688 
 
 8219 
 
 ' 7020 
 
 7o83 
 
 7146 
 
 7210 
 
 7273 
 
 7336 7J99 7462 ; 75a5 
 7967 8o3o! 8093 81 56 
 
 63 
 
 7652 
 8282 
 
 77i5 
 
 7778 
 
 7841 
 8471 
 
 7904 
 8334 
 
 63 
 
 689 
 
 8345 
 
 8408 
 
 85g7 866o| 8723 8786 
 
 63 
 
 690 
 
 838849 
 
 8912 
 
 8975 
 
 9 o38 
 
 9101 
 
 9164 
 
 9227 9289 1 9352' 94t5 
 
 63 
 
 691 
 
 9470 
 
 9341! 9604 
 
 9667 
 
 9729 
 
 9792 
 
 9855 9918 9981 » # 43 
 
 63 
 
 692 
 
 840106 
 
 O169' 0232 
 
 0294 
 
 o357 
 
 0420 
 
 0482 0345| 0608 0671 
 
 63 
 
 693 
 694 
 
 0733 
 
 ,35 9 
 1980 
 2609 
 
 0790! o85c 
 1422! 1485 
 
 0921 
 
 1 547 
 
 0984 
 1610 
 
 1046 
 1672 
 
 1 109 
 1735 
 
 1 172 1234 1297 
 
 I797 i860' IQ22 
 2422! 2484 23,47 
 
 63 
 63 
 
 1 693 
 
 2047I 2110 2172 
 
 2235 2297 
 
 236o 
 
 62 
 
 696 
 
 26721 2734 2796 
 
 2869' 2921 
 
 2 9 83 
 
 3046I 3io8 3170 
 
 62 
 
 697 
 698 
 
 32331 329s 1 3357 3420 
 
 3482 3544 
 
 36o6 3669I 3j3i. J79J 
 
 62 
 
 3855 3qi8! 3q8o 4042 
 
 4so4 4166 
 
 -4229' 4291 1 4J53 44i5 
 
 6a 
 
 699 
 
 4477 
 
 4339 
 
 | 4601 4664' 4726 4788 
 
 4850-4912 4974 5o36 62 
 
 
 
 
 
 1 
 
 2 3 | 4 | 5 
 
 6 | 7 U8 | 9 1 D. 
 
12 
 
 A TABLE 
 
 OF 
 
 LOGARITHMS FROM I 
 
 TO 
 
 10,000. 
 
 
 «. 
 
 
 
 I | 2 
 
 3 | 4 
 
 5 
 
 6 | 7 I 8 
 
 9 
 
 D. 
 
 700 
 
 845098 
 
 5l6o' 5222 
 
 5284 5346 5408 
 
 5470 5532 1 5594 
 
 5656 
 
 62 
 
 701 
 
 5718 5780' 5842 
 
 5904 5966 60281 6090 6131 62i3 
 
 6275 
 
 62 
 
 702 
 
 6337 
 
 6399 
 
 6461 
 
 6323 6585: 6646] 6708 
 
 6770 
 
 6832 
 
 6894 62 
 
 703 
 
 6o55 
 
 7017 
 
 7079 
 
 7141 7202 
 
 7264 
 
 7326 
 
 7388 
 8004 
 8620 
 
 7449 
 
 75ii| 62 
 8128 6z 
 8743 62 
 
 704 
 
 7o5 
 
 88o5 
 
 7634 
 
 8231 
 
 7696 
 83i2 
 
 7738 7819, 
 83-74 8435 
 8989 9031 
 
 7881 
 9497 
 
 It 
 
 8066 
 8682 
 
 706 
 
 8866 
 
 8928 
 
 91 1 2 
 
 9174 
 9788 
 
 9235 
 
 9297 
 
 9358! 61 
 
 707 
 708 
 
 85?o3? 
 
 9481 
 
 9042 
 
 9604 9665 
 
 9726 
 
 9849 
 
 9911 
 
 m 
 
 61 
 
 0095 
 
 oi56; 0217 0279 
 
 o34o 
 
 0401 
 
 0462 
 
 0324 
 
 6i 
 
 709 
 
 0646 
 
 0707 
 
 0769 
 
 o83o 0891 
 
 0932 
 1364 
 
 1014 
 
 1075 
 
 n36 
 
 1:97 
 
 61 
 
 710 
 
 85i258 
 
 1320 
 
 i38i 
 
 1442 i5o3 
 
 i625 
 
 1686 
 
 1747 
 2358 
 
 1809 
 
 6i 
 
 7" 
 
 l8 l° 
 2480 
 
 19.3 1 
 2D4I 
 
 1992 
 
 2o53 2 1 14 
 
 2175 
 
 2236 
 
 2297 
 
 2419 
 
 61 
 
 712 
 
 2602 
 
 2663' 2724 
 
 2785 
 
 2846 
 
 2907 
 35i6 
 
 It* 
 
 3o2o 
 
 61 
 
 7.3 
 
 3090 
 
 3i5o 
 
 3211 
 
 3272, 3333 
 
 33 9 4 
 
 3455 
 
 4i85 
 
 3637 
 
 61 
 
 Vi 
 
 36g8 
 
 3759 
 
 3820 
 
 3881 
 
 3941 
 4349 
 
 4002 
 
 4o63 
 
 4124 
 
 4245 
 
 61 
 
 7 .5 
 
 43o6 
 
 4367 
 
 4428 
 
 4488 
 
 4610 
 
 4670 
 
 473i 
 
 4792 
 
 4852 
 
 61 
 
 716 
 
 4oi3 
 5519 
 
 4974 
 
 5o34 
 
 5095 
 
 5i56 
 
 52i6 
 
 5277 
 
 5337 
 
 53 9 8 
 
 5459 
 
 61 
 
 717 
 718 
 
 558o 
 
 564o 
 
 6701 
 
 5761 
 
 5822 
 
 5882 
 
 5943 
 6548 
 
 6oo3 
 
 6064 
 
 61 
 
 6124 
 
 6i85 
 
 6245 
 
 63o6 
 
 6366 
 
 6427 
 
 6487 
 
 6608 
 
 6668 
 
 6o- 
 
 719 
 
 6729 
 
 6789 
 73 9 3 
 
 685o 
 
 6910 
 7 5i3 
 
 6070 
 7374 
 
 7o3i 
 
 7091 
 
 702 
 
 7212 
 
 7272 
 
 60 
 
 720 
 
 857332 
 
 7453 
 
 7634 
 
 7694 
 
 8357 
 8958 
 
 78i5 
 8417 
 9018 
 
 •7875 
 &t77 
 
 60 
 
 721 
 
 It 
 9 ,38 
 
 7995 
 
 8o56 
 
 8116 
 
 8176 
 
 ■ 8236 8297 
 
 60 
 
 722 
 
 85 9 7 
 9198 
 
 865 7 
 
 8718 
 
 8778 
 
 8838 
 
 8898 
 
 9078 
 
 60 
 
 723 
 
 9258 
 
 9318 
 
 9379 
 
 9439 
 
 9499 
 
 $8 
 
 9619 
 •218 
 
 9679 
 
 60 
 
 724 
 
 8blht 
 
 9799 
 0398 
 
 9 85 5 
 
 9918 
 
 9978 
 
 ••38 
 
 ••98 
 
 •278 
 
 60 
 
 725 
 
 0458 
 
 0318 
 
 0378 
 
 0637 
 
 0697 
 
 0757 
 
 0817 
 
 0877 
 
 60 
 
 726 
 
 0937 
 
 0996 
 
 io56 
 
 1116 
 
 u 7 6 
 
 1236 
 
 1293 
 
 i355 
 
 Ui5 
 
 U75 
 
 60 
 
 727 
 728 
 
 1 534 
 
 1394 
 
 1634 
 
 1714 
 
 i 77 3 
 
 1 833 
 
 i8 9 3 
 2489 
 
 1932 
 2349 
 
 2012 
 
 2072 
 
 60 
 
 2l3l 
 
 2191 
 
 2787 
 
 225l 
 
 23 10 
 
 2370 
 
 243o 
 
 2608 
 
 2668 
 
 60 
 
 729 
 
 2728 
 863323 
 
 2847 
 
 2906 
 
 2966 
 356i 
 
 3o25 
 
 3o83 
 
 3i44 
 
 3204 
 
 3263 
 
 60 
 
 73o 
 
 3382 
 
 3442 
 
 3301 
 
 3620 
 
 368o 
 
 3 7 3o 
 4333 
 
 3 799 
 
 3858 
 
 5, 
 
 731 
 
 3917 
 
 3977 
 
 4o36 
 
 4096 
 
 4i55 
 
 4214 
 
 42 -i 
 
 4392 
 
 4452 
 
 5 9 
 
 732 
 
 4DII 
 
 4570 
 
 463o 
 
 4689 
 
 4748 
 
 4808 
 
 4867 
 
 4926 
 55i 9 
 
 4985 
 
 5o45 
 
 59 
 
 733 
 
 5io4 
 
 5i63 
 
 5222 
 
 5282 
 
 5341 
 
 5400 
 
 5459 
 
 5578 
 
 5637 
 
 5 9 
 
 734 
 
 56 9 6 
 
 5755 
 
 58i4 
 
 5874 
 
 5g33 
 
 6324 
 
 5992 
 6583 
 
 6o5i 
 
 6110 
 
 6169 
 
 62,8 
 
 5 9 
 
 735 
 
 6287 
 6878 
 
 6346 
 
 64o5 
 
 6463 
 
 6642 
 
 6701 
 
 .6760 
 
 6819 
 
 5 9 
 
 736 
 
 6 § 37 
 
 8174 
 
 7055 
 
 7114 
 
 7173 
 
 7232 
 
 7880 
 8468 
 
 ''735o 
 
 7409 
 
 £9 
 
 $" 
 
 7467 
 8o56 
 
 7526 
 8n5 
 
 7644 7703 
 8233 8292 
 
 7762 
 835o 
 
 7821 
 8409 
 
 » 3 ° 
 8327 
 
 9 
 5 9 
 
 739 
 
 8644 
 
 8703 
 
 8762 
 
 88211 8870 
 
 9408, 9466 
 
 8o38 
 
 8997 
 
 9036 
 
 9ii4 
 
 9173 
 
 5 9 
 
 740 
 
 ' 869232 
 
 9290 
 
 9349 
 9935 
 
 9525 
 
 9384 
 
 9642 
 
 9701 
 
 9760 59 
 
 741 
 
 9818 
 
 9877 
 
 9994 "53 
 
 •111 
 
 • 170 
 
 •228 
 
 •287 
 
 •345 5g 
 og3o| 58 
 
 742 
 
 870404 
 
 0462 
 
 0321 
 
 0379 o63£ 
 
 0696 
 
 0755 
 
 o8i3 
 
 0872 
 
 743 
 
 098c 
 
 1047 
 
 1 106 
 
 U64 
 
 1223 
 
 1 281 
 
 1 339 
 1920 
 
 i3o8 
 
 1456 
 
 13l5 
 
 58 
 
 744 
 
 i5 7 3 
 
 i63i 
 
 1690 
 
 1748 
 
 1806 
 
 1 865 
 
 1981 
 
 2040 
 
 2098 
 2681 
 
 58 
 
 745 
 
 2i 56 
 
 22 1 5 
 
 2273 
 
 2855 
 
 233 1 
 
 2389 
 
 2448 
 
 23o6 
 
 2364 
 
 2622 
 
 58 
 
 746 
 
 2739 2797 
 
 2913 
 
 2972 
 
 3o3o 3o88 
 
 3i46 
 
 3204 
 
 3262 
 
 58 
 
 74 Z 
 748 
 
 332 1 i 3379 3437 
 
 3495J 3553 
 
 36u| 3669 
 
 2727 
 
 3785 
 
 3844 
 
 58 
 
 3 9 02 
 
 3o6o; 4018 
 
 4076, 4i34 
 
 4192 4230 
 
 43o8 
 
 4366 
 
 4424 
 
 58 
 
 749 
 
 4482 
 
 4540! 4598 4656' 47U 
 
 4772 483o 
 
 4888 
 
 4945 
 5324 
 
 5oo3 
 
 58 
 
 75o 
 
 875061 
 
 5iio' 5177 
 5698 5756 
 
 5235; 5293 
 
 535 1 j 5409 
 
 5466 
 
 5582 
 
 58 
 
 7 5i 
 
 5640 
 
 58i3| 5871 
 
 5929 5987 
 
 6o45 
 
 6102 
 
 6160 58 
 
 lit 
 
 6218 
 
 6276! 6333 
 
 639^ 6449' 65°7 6564 
 
 6622 
 
 6680 
 
 6737 
 
 58 
 
 753 
 
 6793 
 
 6853 6910 
 
 6968J 7026 70S3, 7141 
 
 7199 
 
 7256 
 
 7314 
 
 58 
 
 754 
 
 7371 
 
 7429' 7487 
 8oo4| 8062 
 
 7644J 76021 7659 1 7717 
 8119 1 8177! 8234-1 8292 
 
 8349 
 
 7832 
 
 7889 
 8464 
 
 58 
 
 755 
 
 7947 
 
 8322 
 
 8407 
 
 57 
 
 756 
 
 8079' 8637 
 
 8694 
 
 8i52 8809 8866 
 9325 9383 9440 
 
 8924 
 
 8981 
 
 9o3q 
 
 57 
 
 $ 
 
 9096. 9153, 921 1 
 
 9268 
 
 9497 
 
 9555 
 
 9612 
 
 57 
 
 9669 9726 9784 
 
 9841 
 
 9898 9936 »»i3 
 
 ••70 
 
 •127 
 
 •i85 
 
 57 
 
 7 5 9 
 
 880242 0299 o356, o4i3; 0471 0028 o585 
 
 0642 
 
 0699 
 
 0756 
 
 57 
 
 N. 
 
 1 I | 2 j 3 4 |- 5 | 6 
 
 7 8 
 
 9 
 
 r>. 
 
TABLE OP LOGARITHMS FROM 1 TO 10,000 
 
 13 
 
 N. 
 
 760' 
 761 
 762 
 763 
 764 
 763 
 766 
 
 1 b l 
 768 
 
 769 
 
 770 
 
 77' 
 772 
 773 
 .774 
 
 775 
 
 776 
 
 777 
 778 
 779 
 780 
 
 V£ l 
 782 
 
 783 
 
 784 
 
 785 
 
 786 
 
 787 
 
 788 
 
 789 
 
 790 
 
 79' 
 792 
 7 9 3 
 794 
 7 9 5 
 796 
 
 797 
 798 
 
 799 
 800 
 801 
 802 
 8o3 
 804 
 8o5 
 806 
 807 
 808 
 809 
 810 
 811 
 812 
 8i3 
 814 
 8i5 
 816 
 
 Sl l 
 818 
 
 819 
 
 • N. 
 
 880814 
 1 385 
 io55 
 2325 
 3093 
 3661 
 4229 
 479S 
 536i 
 5926 
 
 886491 
 7034 
 7617 
 8179 
 8741 
 9S02 
 9862 
 
 890421 
 0980 
 i53t 
 
 892095 
 ?65 1 
 3207 
 3762 
 43 16 
 4870 
 5423 
 5g75 
 6526 
 7077 
 
 S97627 
 8176 
 8725 
 9273 
 9821 
 
 900367 
 0913 
 U58 
 
 2003 
 
 2547 
 903090 
 3633 
 4n4 
 4716 
 5256 
 5796 
 6335 
 6874 
 74' 
 7949 
 
 0871 
 1442 
 
 2012! 2061 
 
 358 1 1 203i 
 
 3i5o| 3207 
 
 3718, 3775 
 
 42851 4342 
 
 4852 4909 
 
 54i8 
 
 5o83 
 
 6547 
 
 7111 
 
 7674 
 
 8236 
 
 87. 
 
 93: 
 
 9918 
 
 io35 
 i5o3 
 
 2 1 DO 
 2707 
 3262 
 3817 
 4371 
 
 4 9 25 
 
 5478 
 
 6o3o 
 658i 
 7i32 
 7682 
 823i 
 8780 
 9328 
 9 8 7 5 
 0422 
 0968 
 
 I3l3 
 
 2057 
 2601 
 3i44 
 3687 
 4229 
 477° 
 53io 
 585o 
 6389 
 6927 
 7465 
 
 0928' 0985 
 1499I '556 
 2126 
 269a 
 3264 
 3832 
 4399 
 496a 
 553i 
 6096 
 6660 
 7223 
 
 B r 
 8909 
 9470 
 
 ••3o 
 0689 
 1147 
 i7o5 
 2262 
 
 5474 
 
 6039 
 
 6604 
 
 7167 
 
 773o 
 
 8292 
 
 8853 
 
 94U 
 
 9974 
 
 o533 
 
 1091 
 
 1649 
 
 2206 
 
 2762 
 
 33 " 
 
 3873 
 
 4427 
 
 49S0 
 
 5533 
 
 6o85 
 
 6636 
 
 7187 
 
 7737 
 
 8286 
 
 8835 
 
 9 383 
 
 9930 
 
 0476 
 
 1022 
 
 1 56 7 
 
 2112 
 
 2655 
 
 3199 
 
 3 7 4i 
 
 4283 
 
 4824 
 
 5364 
 
 5904 
 
 6443 
 
 698 1 
 
 7019 
 
 4 
 
 3 
 
 9021 
 
 9556 
 
 9IOO9I 
 
 0624 
 
 n58 
 1690 
 2222 
 -' 2753 
 3a84 
 
 8002 1 8o56 
 853g] 8592 
 9074 9128 
 9610 9663 
 0144 0197 
 0678 0731 
 1211 1 264 
 1743 1797 
 227D 2J28 
 2806 285g 
 33371 3390 
 
 3373 
 3928 
 4482 
 5o36 
 5588 
 6140 
 6692 
 7242 
 
 lit 
 8890 
 9437 
 9g85 
 o53i 
 1077 
 1622 
 2166 
 2710 
 ■ 3253 
 3 7g 5 
 4337 
 4878 
 5418 
 5 9 58 
 
 6497 
 7 o35 
 75 7 3 
 8110 
 8646 
 9181 
 97 16 
 025i 
 078: 
 i3i7 
 i85o 
 238i 
 2913 
 3443 
 
 1042 
 
 i6i3 
 2183 
 
 27P2 
 
 3321 
 3888 
 4455 
 
 30.22 
 5587 
 
 6i52 
 6716 
 7280 
 7842 
 8404 
 8n65 
 9626 
 ••86 
 0645 
 
 1203 
 
 1760 
 2317 
 
 2873 
 3429 
 3984 
 
 4538 
 509 1 
 5644 
 6195 
 
 6747 
 
 7297 
 
 7847 
 
 83 9 6 
 
 89. 
 
 94 
 
 ••39 
 
 o586 
 
 ii3i 
 
 1676 
 
 2221 
 
 2764 
 
 3307 
 
 3849 
 
 4391 
 
 4932 
 
 5472 
 
 6012 
 
 655i 
 
 7089 
 
 7626 
 
 8i63 
 
 8699 
 
 9233 
 
 9770 
 
 o3o4 
 
 o838 
 
 1371 
 
 1903 
 
 2435 
 
 2066 
 
 3496 
 
 1099 
 1670 
 
 2,240 
 2809 
 
 33 77 
 3g43 
 4512 
 5078 
 5644 
 6209 
 6p3 
 7 336 
 
 8460 
 
 9021 
 
 9582 
 
 •141 
 
 0700 
 
 I25g 
 
 1816 
 
 23 7 3 
 
 2929 
 
 3484 
 
 4039 
 
 45 9 3 
 
 5U6 
 
 5699 
 
 625 
 
 6802 
 
 7352 
 
 7902 
 
 845 1 
 
 8999 
 
 g547 
 
 ••g4 
 
 0640 
 
 1 186 
 
 , 7 3. 
 
 227D 
 
 2818 
 
 336 
 
 3904 
 
 4445 
 
 4986 
 
 5326 
 
 6066 
 
 6604 
 
 7143 
 
 7680 
 
 8217 
 
 8753 
 
 u56 
 
 1727 
 2297 
 2866 
 3434 
 4002 
 4569 
 5i3d 
 5700 
 6265 
 6829 
 7392 
 
 Itl 
 
 9"77 
 9 038 
 
 •'97 
 
 0756 
 i3i4 
 1872 
 24:9 
 2980 
 3d 40 
 4094 
 4648. 
 
 5201 
 
 57D4 
 63o6 
 6807 
 7407 
 
 79 5 .7 
 
 85o6 
 
 905 
 
 9602 
 
 •140 
 
 069D 
 
 1240 
 
 i 7 85 
 
 2320 
 
 I2i3 1271 
 
 I784] l84I 
 
 2354 ! 241 1 
 2923. 2980 
 3491 3D48 
 40D9 41 15 
 4625 4682 
 
 5192 
 
 57D7 
 6321 
 6885 
 
 7449 
 801 1 
 8273 
 9134 
 9694 
 
 •233 
 
 0812 
 
 i3 7 o 
 
 1928 
 
 2484 
 
 3o4o 
 
 35o5 
 
 4i5o 
 
 470. 
 
 525 7 
 
 58og 
 
 636i 
 
 6912 
 
 7462 
 
 8012 
 
 856i 
 
 9109 
 
 9 656 
 
 •2o3 
 
 0741 
 
 129 
 
 1840 
 
 2384 
 
 9 
 "l328 
 
 D. 
 
 28731 2927 
 3416 3470 
 3968 1 4012 
 4499: 4553 
 5o4o! 0094 
 558o. 5634 
 6119! 6173 
 6658| 6712 
 7196I 7250 
 7734, 7787 
 8270I 832 
 88oti 8860 
 
 9289] 93421 9396 
 9823; 9877J 9930 
 
 o358 
 0891 
 1424 
 1936 
 2488 
 3019 
 3549 
 
 5248 
 
 58i3 
 
 6378 
 
 6942 
 
 73o5 
 
 8067 
 
 8629 
 
 9190 
 
 9730 
 
 •3oi 
 
 086 
 
 1421 
 
 1983 
 
 2340 
 
 3096 
 
 36di 
 
 42o5 
 
 4759 
 
 53l2 
 
 5864 
 6416 
 6967 
 
 I, 1 
 8067 
 
 86i5 
 
 9164 
 
 WW 
 •258 
 
 0804 
 
 1 349 
 
 1894 
 
 2438 
 
 2981 
 
 3324 
 
 4066 
 
 460 
 
 5i* 
 
 5688 
 
 6227 
 
 6766 
 
 73o4 
 
 f 
 8378 
 
 89U 
 
 9449 
 
 998 
 
 o5i 
 
 041 1, 0464 1 
 
 0944 0998 io5i 
 
 1477] ' 53 °! ,58 4 
 
 2009' 2o63 
 
 2341 
 
 3072 
 3602 
 
 2594 
 3i25 
 3655 
 
 2468 
 3o37 
 36o5 
 4172 
 
 Hoi 
 5870 
 6434 
 6998 
 7561 
 8n3 
 8685 
 .9246 
 9806 
 •365 
 0924 
 1482 
 2039 
 2593 
 3i5] 
 3706 
 4261 
 4614 
 5367 
 5920 
 
 6471 
 7022 
 7572 
 8122 
 8670 
 9218 
 9766 
 
 •3l2 
 
 0859 
 
 1404 
 
 1948 
 
 2492 
 
 3o36 
 
 3578 
 
 4120 
 
 4661 
 
 5202 
 
 5742 
 
 6281 
 
 6820 
 
 7358 
 
 7895 
 
 #43 1 
 
 8967 
 
 9303 
 
 ••37 
 
 0571 
 
 1 104 
 
 1637 
 
 2169 
 
 2700 
 
 323i 
 
 2116 
 2647 
 3178 
 3708 3761 
 
 5i 
 57 
 
 5 , T 
 
 « 7 
 
 l 1 
 
 ll 
 
 56 
 
 56 
 
 56 
 
 56 
 
 56 
 
 56 
 
 56 
 
 56 
 
 56 
 
 56 
 
 56 
 
 56 
 
 56 
 
 55 
 
 55 
 
 55 
 
 55 
 
 55 
 
 55 
 
 55 
 
 55 
 
 55 
 
 55 
 
 55 
 
 55 
 
 55 
 
 55 
 
 54 
 
 54 
 
 54 
 
 54 
 
 54 
 
 54 
 
 54 
 
 54 
 
 54- 
 
 54 
 
 54 
 
 54 
 
 54 
 
 54 
 
 54 
 
 53 
 
 53 
 
 53 
 
 53 
 
 53 
 
 53 
 
 53 
 
 53 
 
 D- 
 
14 
 
 A TABLE OF LOGARITHMS FROM 1 TO 10,000. 
 
 N. 
 
 82Q 
 
 
 
 . | 2 j 3 | 4 | 5 
 
 6 j 7 
 
 8 9 
 
 D. 
 
 9i38i4 
 
 3867J 3920 3973 4026, 4079 
 
 4i3a| 4184 
 
 42371 4290 
 
 53 
 
 821 
 
 4343 
 
 4396 4449! 4302 
 
 4555 4608 
 
 4660! 471.3 
 
 4766; 4819 
 
 '3 
 
 822 
 
 ,4872 
 
 4925 49771 5o3o 
 
 5o83' 5i36 
 
 6189' 5241 
 
 5294 5347 
 
 53 
 
 823 
 
 5400 
 
 5453 55o5| 5558 
 
 56n! 5664 
 
 57161 5769 
 
 5822 5875 
 
 53 
 
 824 
 
 5927 
 
 5980 6o33 
 
 6o85 
 
 61 38 6191 
 
 6243 6296 
 
 6349 64° ' 
 
 53 
 
 823 
 
 6434 
 
 65o7 655c, 
 
 6612 
 
 6664 67 1 7 
 
 6770 6822 
 
 68731 6927 
 
 53 
 
 826 
 
 6980 
 
 7033 708: 
 
 7i38 
 
 7190 7243 
 
 7295 
 
 7348 
 
 7400 1 7453 
 
 53 
 
 8 n 
 828 
 
 > 7D06 
 8o3o 
 
 7D08 7611 
 8o83 8i35 
 
 7663 
 8188 
 
 7716 7768 
 
 7820 
 
 78 7 3 
 .8397 
 
 7925 7978 
 845o! 85o2 
 
 52 
 
 8240 8293 
 
 8345 
 
 52 
 
 829 
 
 8555 
 
 8607 865c 
 9i3o 918J 
 
 8712 
 
 8764 8816 
 
 8869 
 
 8921 
 
 89735 9026 
 
 52 
 
 83o 
 
 919078 
 
 9235 
 
 9287, 9340 
 
 9 3 9 2 
 
 9444 
 
 9496 j 9549 
 
 52 
 
 83 1 
 
 0601 
 926*123 
 
 9653 9706 
 
 9738 
 
 98 1 o, 9862 
 
 9914 
 
 9967 
 
 ••19 "71 
 
 52 
 
 832 
 
 0176, 0226 
 
 0280 
 
 o332' o384 
 
 ■0436 
 
 0489 
 
 o54i 
 
 OD93 
 
 52 
 
 833 
 
 o645 
 
 0697! 0749 
 1218J 1270 
 
 0801 
 
 o853, 0906 
 
 0908 
 
 1010 
 
 1062 
 
 1114 
 
 52 
 
 834 
 
 1166 
 
 1322 
 
 1374! 1426 
 
 1478 
 
 l53o 
 
 i58a 
 
 1634 
 
 52 
 
 835 
 
 1686 
 
 I738j 1790 
 
 1842 
 
 1894 1946 
 
 1998 
 
 2o5o 
 
 2102 
 
 21 54 
 
 52 
 
 836 
 
 2206 
 
 2258, 23 [0 
 
 2362 
 
 2414 
 
 2466 
 
 23 18 
 
 2570 
 
 2622 
 
 2674 
 
 52 
 
 83t 
 838 
 
 272D 
 
 2777 2829 
 
 2881 
 
 2933 
 
 2985 
 
 3o37 
 
 3o8g 
 
 3 1 40 
 
 3192 
 
 52 
 
 3244 
 
 32g6 ! 3348 
 
 3399 
 
 345 1 
 
 35o3 
 
 3555 
 
 3607 
 
 3658 
 
 3710 
 
 52 
 
 839 
 
 3;62 
 
 38i4' 3865 
 
 3917 
 
 3969 
 
 4021 
 
 4072 
 
 4124 
 
 4176 
 
 4228 
 
 52 
 
 8jo 
 
 924279 
 
 433i, 4383 4434 
 
 4486 
 
 4538 
 
 4589 
 
 ■464i 
 
 4693 
 
 4744 
 
 52 
 
 841 
 
 4796 
 
 4848 4S09 
 
 49)1 
 
 5oo3 5o54 5 1 06 
 
 5i57 
 
 5209 
 
 6261 
 
 52 
 
 842 
 
 53i2 
 
 5364 54i 5 
 
 5467 
 
 55i8 5570 1 5621 
 
 5673 
 
 5723, 5776 
 
 52 
 
 8 ',3 
 
 5328 
 
 5S79 593 1 
 
 5982 
 
 6o34 6o85j 6i37; 6188 
 
 6240! 6201 
 
 5! 
 
 844 
 
 6342 
 
 6394 6445 
 
 6497 
 
 6548 6600! 665 1 j 6702 
 
 6 7 54 
 
 68o5 
 
 5i 
 
 845 
 
 6357 
 
 6908 6959 
 7422' 7473 
 
 701 1 
 
 7062 71 14 7i65, 7216 
 
 7268 
 
 73i9 
 
 5i 
 
 846 
 
 73 7 o 
 
 7524 
 
 8o3; 
 
 7576, 7627 
 8088 8140 
 
 7678, 7 7 3o 
 91911 8242 
 8703J 8754 
 
 7781 
 8293 
 
 7832 
 
 8345 
 
 5i 
 
 847 
 848 
 
 7883 7, r. •;,/-,, 
 
 5i 
 
 8396 
 
 8447 8498 
 
 8549 
 
 8601 8652 
 
 88o5 
 
 885 7 
 
 5i 
 
 849 
 
 8908 
 
 8959 9010 
 
 9061 
 
 9112 9163 
 
 9.2 1 5| 9266 
 
 9317 
 
 9 368 
 
 5i 
 
 85o 
 
 929419 
 
 9470 9521 
 
 9572 
 ••83 
 
 9623 9674 
 •i34 °i85 
 
 9725j 9776 
 •236, 0287 
 
 9827 
 
 9879 
 •38 9 
 
 5i 
 
 85i 
 
 99 3o 
 
 9981 »">32 
 
 •338 
 
 5! 
 
 852 
 
 900440 
 
 0491 0542 
 
 0592 
 
 0643 0694 
 
 0745, 0796 
 
 0847 
 
 0898 
 
 5i 
 
 853 
 
 0949 
 
 1000 io5i 
 
 1102 
 
 1 1 53 1204 
 
 1254 1 i3o5 
 
 1356 
 
 1407 
 
 5i 
 
 854 
 
 1458 
 
 1 609 i56o 
 
 1610 
 
 1661 1712 
 
 I763i 1814 
 
 1 865 
 
 1915 
 
 5i 
 
 855 
 
 1966 
 
 2017 2068 
 
 2118 
 
 2169 2220 
 
 227I 2jB2 
 
 2372 
 
 2423 
 
 5i 
 
 856 
 
 2474 
 
 2524 2575 
 
 2626 
 
 2677 
 
 2727 
 
 27781 282g 
 
 2S79 
 
 2930 
 
 5i 
 
 807 
 858 
 
 2981 
 
 3o3i : 3o'-i2 
 
 3i33 
 
 3i83 
 
 3234 
 
 3285 
 
 3335 
 
 3386 
 
 3437 
 
 5i 
 
 Mil 
 
 35381 3589 
 
 363 9 
 
 3690 
 
 3740 
 
 379' 
 
 3841 
 
 38 9 2 
 
 3 9 43 
 
 5i 
 
 8J9 
 
 3993 
 
 4044 4094 
 
 4i45 
 
 4195 
 
 4246 
 
 4296 
 
 4347 
 
 4397 
 
 4448 
 
 5i 
 
 860 
 
 934498 
 
 4049' 4599 
 
 465o 
 
 4700 
 
 475 1 
 
 4801 
 
 4852 
 
 4902 
 
 4953 
 
 5o 
 
 861 
 
 5oo3 
 
 5o54 5 1 04 
 
 5 1 54 
 
 52o5 
 
 5255 
 
 53 06 
 
 5356 
 
 54o6 
 
 5437 
 
 5o 
 
 862 
 
 5507 
 
 5558, 56o8 
 
 5658 
 
 5709 
 
 5759 
 
 5809 
 
 586o 
 
 5910 
 
 6960 
 
 5o 
 
 863 
 
 601 1 6061 ' 61 1 1 
 
 6162 
 
 6212 
 
 6262 
 
 63i3j 6363 
 
 64i3 
 
 6463 
 
 5o 
 
 864 
 
 65 14 
 
 6564! 66U 
 
 6665 
 
 6715, 6765 
 
 68i5 6865 
 
 6916 
 
 6966 
 
 5o 
 
 865 
 
 7016 
 
 7066' 7117 
 7568. 7618 
 8069 1 81 19 
 
 7167 
 
 7217 7267 
 
 7317 
 
 7367 
 
 74i8 
 
 7468 
 
 5o 
 
 866 
 
 7518 
 8019 
 
 7668 
 8169 
 
 77181 7769 
 
 7819 
 
 7869 
 
 7919 
 
 7969 
 
 5o 
 
 867 
 868 
 
 8219' 8269 
 
 8320 
 
 8370 
 
 8420; 8470 
 
 5o 
 
 8520 
 
 8570, 8620 
 
 8670 
 
 8720J 8770 
 
 8820 
 
 8870 
 
 8920 
 
 8970 
 
 5o 
 
 869 
 
 9020 
 
 9070! 9120 
 
 9170 
 
 9220 9270 
 
 9320 
 
 9369 
 
 9419 
 9918 
 
 9460 
 
 5o 
 
 870 
 
 939519 9569 1 9619 
 
 9669 
 
 9719 1 9769 
 
 9819 
 
 9869 
 
 9968 
 
 5o 
 
 871 
 
 940018 
 
 0068 0110 
 
 0168 
 
 0218; 0267 
 
 o3i7 
 
 0)67 
 
 0417 
 
 0467 
 
 5o 
 
 872 
 
 o5i6 
 
 o566 0616 
 
 0666 
 
 07i6i 0760 
 
 081 5 
 
 o865 
 
 0915 
 
 0964 
 
 5o 
 
 873 
 
 1014 
 
 1064! mj 
 
 1 [63 
 
 I2i3| 1263 
 
 l3i3 
 
 1 362 
 
 1412 
 
 1462 
 
 5o 
 
 874 
 
 i5i i 
 
 i56i| 161 1 
 
 , 1660 
 
 1710! 1760 
 
 1809 
 
 1839 
 
 1909 
 
 1958 
 
 5o 
 
 87 a 
 
 200S, 20381 2107 
 
 2157 
 2653 
 
 2207I 2256 
 
 23o6 
 
 2355 
 
 2403 
 
 2455 
 
 5o 
 
 876 
 
 25o4J 2554 26o3 
 
 2702: 2762 
 
 2801 
 
 285 1 
 
 2901 2o5o 
 
 5o 
 
 87? 
 879 
 
 3oooj 3040: 3oq9 
 
 3148 
 
 3198' 3247 
 
 3297 
 
 ■ 3346 
 
 33 9 6 
 
 3445 
 
 5 9 
 
 34g5| 3544, 35o3 3643 
 3989 4o38 4oH8| 4i3-> 
 
 36o2 3742 
 4186 4236 
 
 3 7 oi 
 428a 
 
 3841 
 4335 
 
 38 9 o 
 4384 
 
 3939 
 4433 
 
 5 9 
 
 If. 
 
 | 1 j 2' . 
 
 3 |-4 | 5 
 
 i 7 
 
 8 
 
 9 
 
 D. 
 

 A TABLE 
 
 OF LOGARITHMS FUOM 1 
 
 TO 
 
 10,000. 
 
 15 
 
 N. 
 
 
 
 I 
 
 a 
 
 3 1 4 | 5 | 6 | 7 
 
 8 
 
 9 
 
 D. 
 
 880 
 
 944483 
 
 4532 
 
 458 1 
 
 463 1 1 4680! 4729I 4779! 4828: 4877J 4927 
 
 49 
 
 881 
 
 4976 
 
 5o25 
 
 D074 
 
 5i24l 5i73i 5222, 5272; 532i' 5370: 54iQ 
 
 49 
 
 882 
 
 5469 
 
 55i8 
 
 5567 
 
 56i6 5665 1 '5 7 i5j 5764 58i3 
 6108; 6157! 6207 6256: 63o5 
 6600! 6649' 6698! 6747! 6796 
 
 5862 5912 
 
 49 
 
 883 
 
 5961 
 
 6010 
 
 6059 
 
 6354 64o3 
 
 49 
 
 884 
 
 6452 
 
 65oi 
 
 655i 
 
 6845 6894 
 
 49 
 
 885 
 
 6943 
 
 6992 
 7483 
 
 7041 
 
 7090J 7140, 7189J 7238 
 
 7287 
 
 7336 : 7385 
 
 49 
 
 886 
 
 7434 
 
 7532 
 8022 
 
 758i 763oj 7679, 7728 
 
 7777 
 
 7826 7875 
 
 49 
 
 887 
 888 
 
 7924 
 
 7973 
 
 8070' 81 19 1 8168 
 
 8217 
 
 8266, 83 1 5 8364 
 
 49 
 
 84i3 8462 
 
 85 n 
 
 856o; 8609 
 
 8657 
 
 8706 
 
 8 7 55 8804 8853 
 
 49 
 
 889 
 
 8902 
 949J90 
 
 8931 
 
 8999 
 
 9048; -9097 
 
 9146 
 
 9 rn5 
 9 683 
 
 9244 1 9292 g34i 
 
 49 
 
 890 
 
 9439 
 
 9488 
 
 9536 
 
 9583 
 
 9634 
 
 973 1 
 
 9780 
 
 9829 
 
 49 
 
 891 
 
 9S78 
 
 9926 
 
 9975 
 
 ••24 
 
 ••73 
 
 •121 
 
 • 170 
 
 •219 
 
 •267 
 
 •3 16 
 
 49 
 
 892 
 
 9Do365 
 
 0414 
 
 0462 
 
 o5u 
 
 o56o 
 
 0608 
 
 06D7 
 
 0706 
 
 0754 
 
 o8o3 
 
 49 
 
 89J 
 
 o85i 
 
 0900 
 
 0949 
 
 0997 
 1483 
 
 1046 
 
 1095 
 
 1143 
 
 it 92 
 
 1240 
 
 1289 
 1773 
 
 49 
 
 894 
 
 1338 
 
 1386 
 
 i43o 
 
 1532 
 
 1 58o 
 
 1629 
 
 1677 
 
 1726 
 
 4o 
 
 895 
 
 1823 
 
 1872 
 
 1920 
 
 1969 
 
 2017 
 
 2066' 21 14 
 
 2i63 
 
 2211 
 
 2260 
 
 48 
 
 896 
 
 23o8 
 
 2356 
 
 24o5 
 
 2453 25o2 
 
 255o 2599 
 
 2647 
 
 2696 
 
 2744 
 
 48 
 
 ! 9 2 
 
 2792 
 
 2841 
 
 28S9 
 
 lo38 
 
 2986 
 
 3o34 
 
 3o83 
 
 3i3i 
 
 3i8o 
 
 3228 
 
 48 
 
 898 
 
 3276 
 
 3323 
 
 3373 
 
 342 1 
 
 3470 
 
 35i8 
 
 3566 
 
 36 1 5 
 
 3663 
 
 3711 
 
 48 
 
 899 
 
 3760 
 
 3808 
 
 3856 
 
 3905 
 
 3 9 53 
 
 4001 
 
 4049 
 
 4098 
 
 4146 
 
 4194 
 
 48 
 
 900 
 
 954243 
 
 4291 
 
 4339 
 
 438 7 
 
 4435 
 
 4484 
 
 4532 
 
 458o 
 
 4628 
 
 4677 
 5i58 
 
 48 
 
 901 
 
 4725 
 
 4773 
 
 4821 
 
 4S69 
 
 4918 
 
 4966 
 
 5oi4 
 
 6062 
 
 5iio 
 
 48 
 
 902 
 
 5207 
 
 5255 
 
 53o3 
 
 535i 
 
 5399 
 58So 
 
 5447 
 5928 
 
 5495 
 
 5543 
 
 5592 
 
 5640 
 
 48 
 
 903 
 
 5688 
 
 5736 
 
 5 7 84 
 
 5832 
 
 5976 
 
 6024 
 
 6072 
 
 6120 
 
 48 
 
 904 
 
 6168 
 
 6216 
 
 6265 
 
 63i3 
 
 636i 
 
 6409 
 
 6457 
 
 65o5 
 
 6553 
 
 6601 
 
 48 
 
 905 
 
 6649 
 
 6697 
 
 6745 
 
 6793 
 
 6840 
 
 6SS8 
 
 6 9 36 
 
 6984 
 
 7032 
 
 7080 
 
 48 
 
 906 
 
 7128 
 
 7176 
 
 7224 
 
 7272 
 
 7320 
 
 7368 
 
 7416 
 
 7464 
 
 7512 
 
 7559 
 8o33 
 
 48 
 
 ■907 
 
 .. 908 
 
 7607 
 8086 
 
 7655 
 
 7703 
 8181 
 
 775i 
 
 7799 
 
 7847 
 8325 
 
 7894 
 
 7942 
 8421 
 
 7990 
 
 48 
 
 8i34 
 
 82*9 
 
 8277 
 
 8373 
 
 8468 
 
 85i6 
 
 - 48 
 
 9"9 
 
 8564 
 
 8612 
 
 865g 
 
 8707 
 
 8 7 55 
 
 88o3 
 
 885o 
 
 8898 
 
 8946 
 
 8994 
 
 48 
 
 910 
 
 959041 
 
 9089 
 
 9 l3 7 
 
 9 i85 
 
 9232 
 
 9280 
 
 9328 
 
 9 3 7 5 
 
 9423 
 
 94T 
 
 48. 
 
 9 u 
 
 95i8 
 
 9 566 
 
 9614 
 
 9661 
 
 97°9 
 
 97 5 7 
 
 9804 
 
 Q 852 
 
 9900 
 
 9947 
 
 48 
 
 912 
 
 9995 
 
 ••42 
 
 " a qo 
 
 •i38 
 
 •180 
 
 •233 
 
 •280 
 
 •328 
 
 •376 
 
 •423 
 
 . 48 
 
 913 
 
 96047 1 
 
 o5i8 
 
 o566 
 
 o6i3 
 
 o66t 
 
 0709 
 
 0756 
 
 0804 
 
 0831 
 
 0899 
 
 48 
 
 914 
 
 0946 
 
 0994 
 
 1041 
 
 1089 
 1 563 
 
 "n36 
 
 1 184 
 
 1 23 1 
 
 1279 
 1753 
 
 i326 
 
 1374 
 
 47 
 
 9i5 
 
 1421 
 
 1469 
 1943 
 
 i5i6 
 
 1611 
 
 i658 
 
 1706 
 
 1801 
 
 1848 
 
 47 
 
 916 
 
 i8 9 5 
 
 1990 
 
 2o38 
 
 2o85. 
 
 2l32 
 
 21S0 
 
 2227 
 
 2275 
 
 2322 
 
 47 
 
 917 
 918 
 
 2 l b 9 
 
 2417 
 
 2464 
 
 25ll 
 
 255g 
 
 2606 
 
 2653 
 
 2701 
 
 2748 
 
 2795 
 
 Al \ 
 
 284J 
 
 2890 
 
 2937 
 
 2985 
 
 3o32 
 
 3079 
 
 3i26 
 
 3i74 
 
 3221 
 
 3268 
 
 47 
 
 919 
 
 33i6 
 
 3363 
 
 34io 
 
 3457 
 
 3 304 
 
 3552 
 
 3599 
 
 3646 
 
 36 9 3 
 
 3741 
 
 47 
 
 920 
 
 963788 
 
 3835 
 
 3882 
 
 3929 
 
 3977 
 
 4024 
 
 4071 
 
 4118 
 
 4i65 
 
 4212 
 
 47 
 
 921 
 
 4260 
 
 4307 
 
 4354 
 
 4401 
 
 4448 
 
 4495 
 
 4542 
 
 4590 
 
 4637 
 
 4634 
 
 47 
 
 922 
 
 473i 
 
 4778 
 
 4825 
 
 4872 
 
 4919 
 
 4966 
 
 5oi3 
 
 5o6i 
 
 5io8 
 
 5i55 
 
 47 
 
 923 
 
 5202 
 
 5249 
 
 5296 
 
 5343 
 
 5390 
 5S6o 
 
 543 7 
 
 6484 
 
 553 1 
 
 55 7 8 
 
 5623 
 
 47 
 
 924 
 
 5672 
 
 5719 
 
 5 7 66 
 
 58i3 
 
 5907 
 
 5954 
 
 6001 
 
 6048 
 
 6095 
 
 4,7 
 
 923 
 
 6l42 
 
 6189 
 
 6236 
 
 6283 
 
 6329 
 
 63 7 6 
 
 6423 
 
 6470 
 
 65i 7 
 
 6564 
 
 47 
 
 926 
 
 66l I 
 
 6658 
 
 6705 
 
 6752 
 
 6799 
 
 6845 
 
 6892 
 
 6939 
 
 6986 
 
 7o33 
 
 47 
 
 927 
 
 7080 
 
 7127 
 
 7173 
 
 7220 
 
 7267 
 
 73i4 
 
 736i 
 
 7408 
 
 7454 
 
 75oi 
 
 47 
 
 .928 
 
 7548 
 
 7595 
 
 7642 
 8109 
 
 7688 
 8i56 
 
 P 3 2 
 8203 
 
 7782 
 
 7829 
 
 7875 
 
 7922 
 ,83 9 o 
 
 7969 
 
 47 
 
 9 2 9 
 
 8016 
 
 8062 
 
 8249 
 
 8296 
 
 8343 
 
 8436 
 
 47 
 
 9J0 
 
 968483 
 
 853o 
 
 85 7 6 
 
 8623 
 
 8670 
 
 8716 
 
 8 7 63 
 
 8810 
 
 8836 
 
 8903 
 9369 
 
 47 
 
 931 
 
 8 9 5o 
 
 8996 
 
 9043 
 
 9090 
 
 9i36 
 
 9i83 
 
 9229 
 
 9276 
 
 9 323 
 
 47 
 
 932 
 
 9416 
 
 9463 
 
 9009 
 
 9336 
 
 9602 
 
 9649 
 
 9695 
 
 9742 
 
 9789 
 
 9 S35 
 
 47 
 
 933 
 
 9882 
 
 9928 
 
 9973 
 
 ••21 
 
 ••68 
 
 •114 
 
 •161 
 
 •207 
 
 •254 
 
 •3oo 
 
 47 
 
 934 
 
 970347 
 
 o3g3 
 
 0440 
 
 0486 
 
 o533 
 
 0379 
 
 0626 
 
 0672 
 
 0719 
 
 0765 
 
 46 
 
 9 35 
 
 0S12 
 
 0808 
 
 0904 
 1369 
 
 0951 
 
 0997 
 
 1044 
 
 1090 
 
 u3 7 
 
 n83 
 
 1229 
 
 46 
 
 936 
 
 1276 
 
 1322 
 
 I4i5 
 
 1461 
 
 i5o8 
 
 1534 
 
 1601 
 
 1647 
 
 i6g3 
 2137 
 
 46 
 
 III 
 
 1740 
 
 ' 17S6 
 
 i832 
 
 1879 
 
 I 9 2 S 
 
 1 971 
 
 201!: 
 
 2064 
 
 :iio 
 
 46 
 
 2203 
 
 2249 
 
 2295 
 
 2342 
 
 2388 
 
 2434 
 
 248t 
 
 2527 
 
 2373 26'9 
 
 46 
 
 9 3 9 
 
 2666 
 
 2712 
 
 2738 
 
 2804 
 
 285i 
 
 2897 
 
 294J 
 
 2989 
 
 3o35i 3o82 
 
 46 
 
 K. 
 
 
 
 • 1 
 
 2 3 
 
 4 
 
 5 1 6 
 
 7 
 
 8 | 9 
 
 D. 
 
 21 ■■ 
 
6 
 
 A TABLE 
 
 OF 
 
 LOGAltlTHMS FROM 1 TO 
 
 10,000. 
 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 | 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 940 
 
 973128 
 
 3i74 
 
 3220 
 
 3266 
 
 33i3| 3359 
 
 34o5 
 
 345 1 
 
 3497 
 3o5o 
 
 3543 
 
 46 
 
 94i 
 
 35oo 
 4o5i 
 
 3636 
 
 3682 
 
 3728 
 
 3774I 3820 
 
 3866 
 
 3oi3 
 4374 
 
 4oo5 
 
 46 
 
 942 
 
 4.097 
 4558 
 
 4U3 
 
 4189 
 
 4235| 4281 
 
 4327 
 4788 
 
 4420 
 
 4466 
 
 46 
 
 943 
 
 45i2 
 
 4604 
 
 465o 
 
 4696 4742 
 
 4834 
 
 4880 
 
 4926 
 5386 
 
 46 
 
 944 
 
 4972 
 
 5oi8 
 
 5064 
 
 5iio 
 
 5n6 
 
 5202 
 
 5248 
 
 5294 
 5753 
 
 5340 
 
 46 
 
 945 
 
 5432 
 
 5478 
 
 5524 
 
 5570 
 
 56i6 
 
 5662 
 
 5707 
 
 5799 
 
 5845 
 
 46 
 
 946 
 
 5891 
 635o 
 
 6854 
 
 5983 
 
 6029 
 
 6075 
 
 6121 
 
 6167 
 6625 
 
 6212 
 
 6258 
 
 6304 
 
 46 
 
 947 
 948 
 
 6442 
 
 6488 
 
 6533 
 
 6579 
 
 6671 
 
 6717 
 
 6 7 63 
 
 46 
 
 6808 
 
 6900 
 
 6946 
 
 6992 
 
 7037 
 
 7o83 
 
 7129 
 
 7175 
 
 7220 
 
 46. 
 
 949 
 
 7266 
 
 7 3l2 
 
 7358 
 
 74o3 
 
 7449 
 
 7495 
 
 7541 
 
 7586 
 8043 
 
 7632 
 8089 
 
 7678 
 8i35 
 
 46 
 
 9_5o 
 
 977724 
 81 Si 
 
 7769 
 8226 
 
 7 8i5 
 8272 
 
 7861 
 
 7906 7952 
 
 7998 
 
 46 
 
 931 
 
 83i 7 
 
 8363, 8409 8454 
 
 85oo 
 
 8546 
 
 85 9 i 
 
 46 
 
 952 
 
 863t 
 9093 
 
 8683 
 
 8728 
 
 8774 
 
 8819 1 8865 
 927D 9321 
 
 8911 
 
 8 9 56 
 
 9002 
 
 9047 
 
 46 
 
 9 53 
 
 9i38 
 
 9184 
 
 9230 
 
 9366 
 
 9412 
 
 9457 
 
 95o3 
 
 46 
 
 9 54 
 
 9548 
 
 9394 
 
 9639 
 
 9 685 
 
 97 3 °' 9776 
 
 9821 
 
 9867 
 
 99 ! 2 
 
 9953 
 
 46 
 
 9 55 
 
 980003 
 
 0049 
 o5o3 
 
 0094 
 
 0140 
 
 oi85 o23i 
 
 0276 
 0730 
 
 0322 
 
 0367 
 
 0412 
 
 45 
 
 9 56 
 
 0458 
 
 o549 
 
 0594 
 
 0640! o6S5 
 
 O776 
 
 0821 
 
 0867 
 
 45 
 
 lU 
 
 0912 
 1 366 
 
 0957 
 
 1 oo 3 
 
 1048 
 
 1093: 1 1 39 1184 
 
 I229 
 
 1275 
 
 1320 
 
 45 
 
 1411 
 
 1456 
 
 i5oi 
 
 . i547 1692 
 
 1637 
 
 i6S3! 1728 
 
 i 77 3 
 
 45 
 
 9 5 9 
 
 1819 
 
 1864 
 
 1909 
 
 1954 
 
 2000, 2045 
 
 2090 2i35 2181 
 
 2226 
 
 45 
 
 960 
 
 982271 
 
 23i6 
 
 2362 
 
 2407 
 
 2402 1 2497 
 
 2543, 2588 
 
 2633 
 
 2678 
 
 45 
 
 961 
 
 2723 
 
 2769 
 
 2814! 2839 
 
 2904 2949 
 
 2994 3 040 
 
 3o85 
 
 3i3o 
 
 45 
 
 962 
 
 3i75 
 
 3220 
 
 3265i 33io 
 3716 3762 
 4167! 4212 
 
 3356 
 
 34oi 
 
 3446 
 
 3491 
 
 3536 
 
 358i 
 
 45 
 
 9 63 
 
 3626 
 
 36 7 i 
 
 3807 
 
 3852 
 
 38 97 
 
 3o42 
 4J92 
 
 3987 
 
 4o32 
 
 45 
 
 964 
 
 4077 
 
 4122 
 
 4257 
 
 4302 
 
 4347 
 
 4437 
 
 4482 
 
 45 
 
 9 65 
 
 4527 
 
 4572 
 
 4617 4662 
 
 4707 
 
 4752 
 
 4797 
 
 4842 
 
 4887 
 
 4932 
 
 45 
 
 966 
 
 4977 
 
 3022 
 
 5067 
 
 5lI2 
 
 5i5 7 
 56o6 
 
 5202 
 
 5247 
 
 5292 
 
 5337 
 5 7 86 
 
 5382 
 
 45 
 
 967 
 968 
 
 5426 
 
 5471 
 
 55i6 
 
 556i 
 
 565 1 
 
 56 9 6 
 
 574. 
 
 583o 
 
 45 
 
 58 7 5 
 
 5920 
 
 636 9 
 
 5965 
 
 6010 
 
 6o55 
 
 6100 
 
 6144 
 
 6189 
 
 6234 
 
 6279 
 
 45 
 
 969 
 
 6324 
 
 64i3 
 
 6458 
 
 65o3 
 
 6548 
 
 65 9 3 
 
 6637 
 
 6682 
 
 6727 
 7175 
 
 45 
 
 , 97° 
 
 .986772 
 
 6817 
 
 6861 
 
 6906 
 7353 
 
 6951 
 7 3 9 8 
 
 6996 
 
 7040 
 
 7o85 
 
 7i3o 
 
 45 
 
 1 97' 
 
 7219 
 
 7264 
 
 7 3o 9 
 
 7443 
 
 7488 
 
 7532 
 
 7577 
 8024 
 8470 
 
 7622 
 8068 
 85i4 
 
 45 
 
 97? 
 973 
 
 7666 
 8u3 
 
 VI 1 
 8i5 7 
 
 77^6 
 8202 
 
 7800 
 8247 
 
 7845 
 8291 
 87J7 
 
 83^6 
 
 lt\ 
 
 lilt 
 
 43 
 45 
 
 974 
 
 855o 
 9003 
 
 8604 
 
 8648 
 
 8693 
 9i38 
 
 8782 
 
 8826 
 
 8871 
 
 8916 
 9361 
 
 8960 
 
 45 
 
 97 5 
 
 9049 
 
 9094 
 9 53o 
 
 gi83 
 
 9227 
 
 9272 
 
 93i6 
 
 94o5 
 
 45 
 
 976 
 
 g45o 
 
 9404 
 993o 
 o3b3 
 
 9 583 
 
 9628 
 
 9672 
 
 97'7 
 
 9761 
 
 9S06 
 
 985o 
 
 44 
 
 977 
 978 
 
 9895 
 
 9983 
 
 ••28 
 
 ••72 
 
 •117 
 
 •161 
 
 •206 
 
 •25o 
 
 •294 
 07J8 
 
 44 
 
 990339 
 
 0428 
 
 0472 
 
 o5i6 
 
 o56i 
 
 o6o5 
 
 o65o 
 
 0694 
 1137 
 
 44 
 
 979 
 
 o 7 83 
 
 0827 
 
 0871 
 
 0916 
 1309 
 
 0960 
 
 1004 
 
 1049 
 
 1093 
 i536 
 
 1 182 
 
 44 
 
 980 
 
 991226 
 
 1270 
 
 i3i5 
 
 i4o3 
 
 1448 
 
 1492 
 io35 
 2377 
 
 i58o 
 
 1625 
 
 44 
 
 981 
 
 1669 
 
 1713 
 
 1758 
 
 1802 
 
 1846 
 
 1890 
 2333 
 
 '979 
 
 2023 2067 
 
 44 
 
 '2? 
 
 21 11 
 
 2i56 
 
 2200 
 
 2244 
 
 2288 
 
 2421 
 
 2465 
 
 25og 
 
 44 
 
 983 
 
 2554 
 
 25o8 
 3o39 
 
 2642 
 
 2686 
 
 2730 
 
 2774 
 
 2819 
 
 2863 
 
 2907 
 
 2q5i 
 3J92 
 3833 
 
 44 
 
 984 
 
 2995 
 3436 
 
 3o83 
 
 3127 
 3568 
 
 3172 
 
 ' 32i6 
 
 326o 
 
 33o4 
 
 3348 
 
 44 
 
 '^ 
 
 3480 
 
 3524 
 
 36i3 
 
 365 7 
 
 3701 
 
 3745 
 
 3 7 8 9 
 
 44 
 
 986 
 
 3877 
 
 3921 
 436 1 
 
 3965 
 
 4009 
 
 4o53 
 
 4097 
 4537 
 
 4Ui 
 
 4i85 
 
 4229 
 
 4273 
 
 44 
 
 ? 
 
 43i7 
 
 44o5 
 
 4449 
 
 4493 
 
 458 1 
 
 4625 
 
 4669 
 5io8 
 
 47'3 
 
 44 
 
 5io6 
 995635 
 
 4801 
 
 4845 
 
 4889 
 
 4933 
 53 7 2 
 
 4977 
 
 5021 
 
 5o65 
 
 5i52 
 
 44 
 
 989 
 
 5240 
 
 5284 
 
 5328 
 
 5416 
 
 5460 
 
 55o4 
 
 5547 
 5 9 86 
 
 55of 
 6o3o 
 
 44 
 
 991 
 
 K79 
 
 5723 
 
 5767 
 
 58n 
 
 5854 
 
 58 9 8 
 6337 
 
 5 9 42 
 638o 
 
 44 
 
 991 
 
 6074 
 
 6„ 7 
 
 6161 
 
 6205 
 
 6249 
 
 6293 
 6 7 3i 
 
 6424 
 
 6468 
 
 44 
 
 992 
 
 65i2 
 
 6555 65o 9 
 6993, 7037 
 743o 7474 
 
 6643 
 
 6687 
 
 '6774 
 
 6818 
 
 6862 
 
 6906 
 7343 
 
 44 
 
 99 3 
 
 6949 
 7386 
 
 7080 
 
 7124 
 
 7168 
 
 7212 
 
 7255 
 
 7299 
 7736 
 
 8172 
 
 44 
 
 994 
 
 7517 
 
 7 56i 
 
 76o5 
 8041 
 
 7648 
 
 7692 
 8129 
 
 7779 
 8216 
 
 44 
 
 99? 
 
 7823 
 8259 
 86o5 
 91J1 
 
 7867! 7910 
 83o3, 8347 
 
 8390 
 
 7998 
 8434 
 
 8o85 
 
 44 
 
 996 
 
 8477 
 
 8521 
 
 8564 
 
 8608 
 
 8652 
 
 44 
 
 997 
 
 8 7 3 9 ' 8782 
 
 8826 
 
 8869 
 
 8 9 i3 
 
 8o56 
 9392 
 
 9000 
 
 ■9043 
 
 9087 
 
 44 
 
 998 
 
 9174' 9218 9261 
 
 93oo| 9348 
 
 9435 
 
 9479 0522 
 
 44 
 
 999 
 
 N. 
 
 9565 
 
 9609 9652 
 
 ' 9696 
 _ 3~ 
 
 9739 
 i 
 
 9783 
 
 9826 
 
 9870 
 
 _99i3 
 8 
 
 9957 
 
 43 
 
 
 
 I 
 
 2 
 
 5 
 
 6 
 
 7 
 
 9 ! D„ 
 
A TABLE 
 
 oir 
 
 LOGAKITHMIC 
 SINES AND TANGENTS 
 
 FOB EVERY 
 
 DEGREE AND MINUTE 
 
 OF THE QUADEANT. 
 
 REMARK. The minutes in the left-hand column of 
 each page, increasing downwards, belong to the de- 
 grees at the top ; and those increasing upwards, in the 
 right-hand column, belong to the degrees below. 
 
18 
 
 (0 
 
 DEGREES.) A TABU 
 
 OF IOGARITHMIO 
 
 
 M. 
 
 
 
 Sine 
 
 0' 000000 
 
 ]>. 
 
 Cosine 
 
 D. 
 
 Tang. 
 
 1). 
 
 Cptang. 
 
 1 ' 
 
 
 10- 000000 
 
 
 ■ onoooo 
 
 
 Infinite. 
 
 60 
 
 I 
 
 6-463726 
 
 5017.17 
 
 000000 
 
 • 00 
 
 6-463726 
 
 5017 
 
 S3 7 
 
 i3 536274 
 
 235244 
 
 tl 
 
 a 
 
 764736^ 2934 
 
 85 
 
 000000 
 
 • 00 
 
 764706 
 
 . 2934 
 
 3 
 
 940847 2082 
 
 3i 
 
 000000 
 
 ■ 00 
 
 940847 
 
 2082 
 
 3i 
 
 o59i53 
 
 57 
 
 4 
 
 7065786. i6i5 
 
 u 
 
 000000 
 
 • 00 
 
 7.060786 
 
 i6i5 
 
 .'7 
 
 !2-o342i4 
 
 56 
 
 5 
 
 162696 1319 
 
 000000 
 
 • 00 
 
 162696 
 
 1319 
 
 69 
 
 837304 
 758122 
 
 55 
 
 6 
 
 241877 in5 
 
 75 
 
 9.999999 
 
 .0. 
 
 24187.8 
 
 HID 
 
 i 
 54 
 
 54 
 
 I 
 
 308824 
 3668i6 
 
 966 
 852 
 
 53 
 54 
 
 999999 
 999999 
 
 • 01 
 
 •0! 
 
 3o8825 
 366817 
 
 852 
 
 691 175 
 633 1 83 
 
 53 
 
 52 
 
 9 
 
 - 417968 
 
 762 
 
 63 
 
 999999 
 
 •01 
 
 417970 
 
 762 
 
 63 
 
 582o3o' 5i 
 
 10 
 
 463725 
 
 689 
 
 88 
 
 999998 
 
 ■01 
 
 . 463727 
 
 689 
 
 88 
 
 536273! 5o 
 
 12-494880; 49 
 
 407091: 40 
 
 ii 
 
 7-5otyi8 
 
 629 
 
 81 
 
 9-999998 
 
 •01 
 
 7-5o5i20 
 
 629 
 
 81 
 
 12 
 
 542906 
 
 579 
 
 36 
 
 999997 
 
 •01 
 
 542909 
 
 5 79 
 
 33 
 
 i3 
 
 577668 
 
 536 
 
 41 
 
 999997 
 
 •01 
 
 577672 
 
 536 
 
 42 
 
 422328 
 
 s 
 
 U 
 
 6og853 
 
 499 
 
 38 
 
 999996 
 
 '01 
 
 609857 
 
 499 
 
 3o 
 
 3ooi43 
 
 i5 
 
 639816 
 
 467 
 438 
 
 14 
 
 999996 
 
 •01 
 
 639820 
 
 467 
 438 
 
 13 
 
 36oi8o 
 
 45 
 
 16 
 
 667845 
 
 81 
 
 999995 
 
 ■01 
 
 667849 
 
 82 
 
 332i5i 
 
 44 
 
 12 
 
 694173 
 
 4i3 
 
 72 
 
 99999 s 
 
 ■01 
 
 694179 
 
 4i3 
 
 73 
 
 3o582i 
 
 43 
 
 718997 
 
 391 
 
 35 
 
 999994 
 
 •01 
 
 719004 
 
 3gi 
 
 36 
 
 280997: 42 
 
 19 
 
 742477 
 
 3 7 i 
 
 27 
 
 999993 
 
 ■01 
 
 742484 
 
 3 7 i 
 
 28 
 
 257516! 41 
 
 20 
 
 764754 
 
 353 
 
 i5 
 
 999993 
 
 •01 
 
 764761 
 
 35i 
 
 36 
 
 235239! 40 
 
 21 
 
 7 -785 9 43 
 806146 
 
 336 
 
 7 2 
 
 9-999992 
 
 •01 
 
 7-785951 
 8061 55 
 
 336 
 
 73 
 
 12 •214049! 39 
 
 22' 
 
 321 
 
 75 
 
 999991 
 
 •01 
 
 321 
 
 76 
 
 I93845J 38 
 
 23 
 
 82545i 
 
 3o8 
 
 o5 
 
 99999° 
 
 •01 
 
 820460 
 
 3o8 
 
 06 
 
 I74540I 37 
 i56o56] 36 
 
 24 
 
 843934 
 
 •295 
 283 
 
 u 
 
 999989 
 
 •02 
 
 843944 
 
 295 
 283 
 
 49 
 
 25 
 
 861662 
 
 999988 
 
 •02 
 
 861674 
 
 90 
 
 i38326i 35 
 
 26 
 
 8786951 273 
 
 17 
 
 999988 
 
 •02 
 
 878708 
 
 2 7 3 
 
 18 
 
 121292 
 
 34 
 
 -3 
 
 890085 
 
 263 
 
 23 
 
 999987 
 
 •02 
 
 895099 
 
 263 
 
 25 
 
 1 0490 1 
 
 33 
 
 910879 
 
 .253 
 
 $ 
 
 999986 
 
 •02 
 
 910894 
 926134 
 
 254 
 
 01 
 
 089106 
 073866 
 
 32 
 
 29 
 
 9261 19 
 
 . 245 
 
 999985 
 
 •02 
 
 245 
 
 40 
 
 3i 
 
 3o 
 
 940842 
 
 237 
 
 33 
 
 999983 
 
 ■02 
 
 94o858 
 
 23 7 
 
 35 
 
 069142 
 
 3o 
 
 3i 
 
 7-955082 
 
 229 
 
 80 
 
 9-999982 
 
 •02 
 
 7'955ioo 
 
 229 
 
 81 
 
 12-044900 
 
 29 
 
 32 
 
 968870 
 982233 
 
 222 
 
 73 
 
 ■ 9999 81 
 
 ■02 
 
 968889 
 982253 
 
 222 
 
 75 
 
 o3 1 1 1 1 
 
 28 
 
 33 
 
 216 
 
 08 
 
 999980 
 
 •02 
 
 216 
 
 10 
 
 017747 
 
 2 
 
 34 
 
 995198 
 
 209 
 20J 
 
 81 
 
 999979 
 
 •02 
 
 995219 
 
 209 
 
 203 
 
 83 
 
 004781 
 
 .35, 
 
 8-007787 
 
 90 
 
 999977 
 
 ■02 
 
 8-007809 
 020045 
 
 n 
 
 n-992101 
 979955 
 
 2D 
 
 36 
 
 020021 
 
 198 
 
 3i 
 
 999976 
 
 ■02 
 
 198 
 
 24 
 
 u 
 
 031919 
 043601 
 
 i 9 3 
 
 02 
 
 999975 
 
 ■02 
 
 o3io45 
 043027 
 
 1 93 
 
 o5 
 
 968055 
 
 23 
 
 188 
 
 01 
 
 999973 
 
 •02 
 
 188 
 
 o3 
 
 956473 
 
 22 
 
 3 9 
 
 054781 
 
 1 83 
 
 25 
 
 999972 
 
 ■02 
 
 054809 
 o65Ko6 
 
 i«3 
 
 27 
 
 945191 
 
 21 
 
 40 
 
 065776 
 
 178 
 
 72 
 
 99997' 
 
 ■02 
 
 178 
 
 74 
 
 934194 
 
 20 
 
 41 
 
 8-076500 
 086965 
 
 174 
 
 41 
 
 9-999969 
 
 ■02 
 
 8-07653i 
 
 174 
 
 44 
 
 11-923469 
 9>3oo3 
 
 3 
 
 42 
 
 170 
 166 
 
 3i 
 
 9999°° 
 
 ■02 
 
 086997 
 
 170 
 
 34 
 
 43 
 
 097183 
 
 3 9 
 
 999966 
 
 •02 
 
 097217 
 
 166 
 
 42 
 
 , 902783 
 
 n 
 
 44 
 
 107167 
 1 16926 
 
 162 
 
 65 
 
 999964 
 
 •o3 
 
 107202 
 
 162 
 
 68 
 
 fe 7 2 7 
 
 883o37 
 
 16 
 
 45 
 
 1 5 9 
 
 08 
 
 999963 
 
 ■ o3 
 
 116963 
 126310 
 
 1 5 9 
 
 10 
 
 i5 
 
 46 
 
 126471 
 
 i55 
 
 66 
 
 999961 
 
 ■o3 
 
 155 
 
 68 
 
 873490 
 
 14 
 
 % 
 
 i358io 
 
 1 52 
 
 38 
 
 999; 5o 
 
 ■o3 
 
 i3585i 
 
 1 52 
 
 41 
 
 864149 
 
 i3 
 
 144953 
 
 149 
 
 24 
 
 999958 
 
 •o3 
 
 144996 
 
 149 
 
 27 
 
 855oo4 
 
 12 
 
 i 9 
 
 153907 
 
 146- 
 
 22 
 
 999956 
 
 • o3 
 
 1539:12 
 
 146 
 
 27 
 
 846048 
 
 11 
 
 5o 
 
 162681 
 
 143 
 
 33 
 
 999954 
 
 • o3 
 
 1627:17 
 8-171328 
 
 143 36 
 
 837273 
 11 828672 
 
 10 
 
 5i 
 
 8-171280 
 
 140 
 
 54 
 
 9-999952 
 
 ■o3 
 
 140 67 
 
 1 
 
 52 
 
 179713 
 
 i3 7 
 
 86 
 
 99995o 
 
 • o3 
 
 179763 
 i88o36 
 
 137-90 
 135-32 
 
 820237 
 
 53 
 
 187985 
 
 135 
 
 29 
 
 999948 
 
 • o3 
 
 81 1964 
 
 I 
 
 54 
 
 196102 
 
 132 
 
 80 
 
 999946 
 
 • o3 
 
 196156 
 
 132-84 
 
 8o3844 
 
 55 
 
 204070 
 
 l3o 
 
 41 
 
 999944 
 
 • o3 
 
 204126 
 
 i3o-44 
 
 7o58 7 4 
 
 5 
 
 56 
 
 2ii8o5 
 219581 
 
 128 
 
 10 
 
 999942 
 
 • 04 
 
 21 1953 
 
 128-14 
 
 788047 
 
 4 
 
 n 
 
 125 
 
 8 7 
 
 999940 
 
 • 04 
 
 219641 
 
 125-90 
 
 78o35o 
 
 3 
 
 227134 
 
 123 
 
 72 
 
 999938 
 
 ■ 04 
 
 227195 
 
 123-76 
 121-68 
 
 772803 
 765379 
 
 2 
 
 5 9 
 
 234557 
 
 121 
 
 64 
 
 999936 
 
 ■04 
 
 234621 
 
 1 
 
 6o 
 
 24i855 
 
 1 1 9 • 63 
 
 999934 
 
 • 04 
 
 241921 
 
 119-67 
 
 758079 
 
 
 
 ( 
 
 Cosine 
 
 D. 
 
 Sine 
 
 89° 
 
 Cotang. 
 
 D. - 
 
 Tang. 
 
 M. 
 
SINES AND TANGENTS. (1 DEQREE.) 
 
 l'b 
 
 M. Sine i D. \ 
 
 Cosine 
 
 IX. | Tang. | 
 
 D. 
 
 Cotang. 1 
 
 
 
 3-24i855 
 
 119-63 
 
 9-999934 
 
 •04 8-241921 
 
 119-67 
 
 11-758079 
 
 60 
 
 i 
 
 249033 
 
 117-68 
 
 999932 
 
 .04! 249102 
 
 117-72 
 
 760898 
 
 11 
 
 3 
 
 256094 
 
 u5-8o 
 
 999929 
 
 ■04j 256i65 
 
 HD-84 
 
 743835 
 
 3 
 
 263042 
 
 n3-g8 
 
 999927 
 
 ■04 263n5 
 
 114-02 
 
 736885 
 
 iz 
 
 4 
 
 269881 112-21 
 
 999925 
 
 -04, 269906 
 
 112-20 
 
 730044 
 
 5 
 
 2766I4I IIO-5o 
 
 999922 
 
 •04 1 276691 
 
 UO-54 
 
 723309 
 
 55 
 
 6 
 
 283243 io8-83 
 
 999920 
 
 ■04 283323 
 
 108-87 
 
 716677 
 
 54 
 
 I 
 
 289773 107-21 
 
 999918 
 
 ■o4j 289866 
 
 107-26 
 
 710144 
 
 53 
 
 290207 io5-65 
 
 999910 
 
 •.04' 296292 
 
 105.70 
 
 703708 
 697366 
 
 52 
 
 9 
 
 302046. 104- i3 
 
 999913 
 
 •04 302634 
 
 io4- 18 
 
 5i 
 
 10 
 
 308794 
 
 102-66 
 
 999910 
 
 -04) 3o8884 
 
 102-70 
 
 691 1 16 
 
 5o 
 
 ii 
 
 8-3i49o4 
 
 101 - fj 
 
 9-999907 
 
 •04 8-3i5o46 
 
 101-26 
 
 11-684954 
 678878 
 ■672886 
 
 % 
 
 12 
 
 321027 
 
 90-82 
 98-47 
 
 999905 
 
 •04 
 
 32112 2 
 
 99.87 
 
 13 
 
 327016 
 
 999902 
 
 .04 
 
 327114 
 
 98 -5i 
 
 47 
 
 U 
 
 332924 
 
 97-14 
 
 999899 
 
 ■o5 
 
 333o25 
 
 97-19 
 
 666975 
 
 46 
 
 i5 
 
 338753 
 
 g5-86 
 
 999897 
 
 -o5 
 
 338856 
 
 95.90 
 
 661 144 
 
 45- 
 
 16 
 
 344504 
 
 94-60 
 
 999894 
 
 • o5 
 
 344610 
 
 94-65 
 
 655390 
 
 44 
 
 \l 
 
 35oi8i 
 
 9 3 -38 
 
 999891 
 
 ■o5 
 
 350289 
 355895 
 36i43o 
 
 9 3-43 
 
 6497 1 ! 
 
 43 
 
 355783 
 
 92-19 
 91 -o3 
 89-90 
 88- 80 
 
 999888 
 
 ■o5 
 
 92-24 
 
 644 1 o5 
 
 42 
 
 ■9 
 
 36i3i5 
 
 999885 
 
 -o5 
 
 91-08 
 
 638570 
 
 41 
 
 20 
 
 366777 
 
 999882 
 
 •o5 
 
 3'668 9 5 
 
 88* ■ $5 
 
 633io5 
 
 40 
 
 21 
 
 8-372171 
 
 9-999879 
 
 •o5 
 
 8-372292 
 
 11-627708 
 622378 
 
 ll 
 
 22 
 
 377499 
 382762 
 
 87-72 
 
 999876 
 
 ■o5 
 
 377622 
 382889 
 
 87-77 
 
 23 
 
 86-67 
 
 . 999873 
 
 • o5 
 
 86-72 
 
 617111 
 
 37 
 
 24 
 
 38796a 
 39J101 
 
 85-64 
 
 999870 
 
 •o5 
 
 388oo2 
 3932J4 
 
 .85-70 
 
 611908 
 
 36 
 
 25 
 
 84-64 
 
 999867 
 
 ■o5 
 
 84-70 
 
 606766 
 
 35 
 
 26 
 
 398179 
 
 83-66 
 
 999864 
 
 -o5 
 
 3 9 83i5 
 
 83-71 
 
 601685 
 
 34 
 
 2 
 
 403199 
 
 82-71 
 
 999861 
 
 -o5 
 
 4o3338 
 
 82-76 
 81-82 
 
 596662 
 
 33 
 
 408161 
 
 80 -ll 
 
 999858 
 
 •o5 
 
 4o83o4 
 
 691696 
 5867B7 
 
 32 
 
 2 9 
 
 4i3o68 
 
 999854 
 
 •o5 
 
 4i32i3 
 
 80-91 
 
 3i 
 
 3o 
 
 417919 
 
 79-96 
 
 999851 
 
 -06 
 
 418068 
 
 80 -02 
 
 58-1 9 32 
 
 3o ' 
 
 3i 
 
 8-422717 
 
 ]n 
 
 9.999848 
 
 • 06 
 
 8-422869 
 
 79-14 
 
 n-577i3f 
 
 3 
 
 32 
 
 427462 
 
 999844 
 
 .06 
 
 427618 
 
 78.30 
 
 572382 
 
 33 
 
 432i56 
 
 77-40 
 
 999841 
 
 •06 
 
 4323 i5 
 
 77-45 
 
 56 7 685 
 
 2 7 
 
 34 
 
 4368oo 
 
 76-57 
 
 999838 
 
 •06 
 
 436n62 
 44i56o 
 
 76-63 
 
 563o38 
 
 26 
 
 35 
 
 44i394 
 
 73-77 
 
 999834 
 
 .06 
 
 75-83 
 
 558440 
 
 25 
 
 36- 
 
 445941 
 
 74-99 
 
 999831 
 
 ■06 
 
 4461 10 
 
 7 5. o5 
 
 553850 
 549387 
 
 24 
 
 ll 
 
 45o44o 
 
 74-22 
 
 999827 
 
 ■06 
 
 45o6i3 
 
 74-28 
 
 23 
 
 454893 
 
 73.46 
 
 999823 
 
 .06 
 
 455070 
 459481 
 
 73-52 
 
 544o3o 
 540619 
 
 22 
 
 3 9 
 
 459301 
 463665 
 
 72-73 
 
 999820 
 
 ■06 
 
 72-79 
 
 21 
 
 4o 
 
 72-00 
 
 999816 
 
 • 06 
 
 463849 
 
 72-06 
 
 536i5i 
 
 20 
 
 41 
 
 8-467985 
 
 71-29 
 
 9-999812 
 
 ■ 06 
 
 8-468172 
 
 71-35 
 
 n-53i828 
 
 \l 
 
 42 
 
 472263 
 
 70-60 
 
 999809 
 999805 
 
 ■ 06 
 
 472454 
 
 .70-66 
 
 527546 
 
 43 
 
 476498 
 480693 
 
 69-91 
 
 ■ 06 
 
 476693 
 480892 
 
 69-98 
 
 523307 
 519108 
 
 'I 
 
 44 
 
 69-24 
 
 999801 
 
 .06 
 
 69-31 
 68-65 
 
 l6 
 
 45 
 
 484848 
 
 68 -5g 
 
 999797 
 
 .07 
 
 485o5o 
 
 5i4o5o 
 5io83o 
 
 i5 
 
 46 
 
 488963 
 
 67.94 
 67-31 
 66-69 
 66- 08 
 
 999793 
 
 .07 
 
 489170 
 
 68-01 
 
 14 
 
 47 
 
 493040 
 
 999790 
 
 .07 
 
 49J 2 Do 
 
 67-38 
 
 506750 
 
 i3 
 
 48 
 
 497078 
 5oio8o 
 
 999786 
 
 .07 
 
 497293 
 
 66-76 
 
 502707 
 
 12 
 
 49 
 
 999782 
 
 ■ 07 
 
 501298 
 
 66-i5 
 
 498702 
 
 11 
 
 5o 
 
 5o5o45 
 
 65-48 
 
 999778 
 
 ■07 
 
 5o5267 
 
 65-55 
 
 494733 
 
 10 
 
 5i 
 
 8-508974 
 512867 
 516726 
 
 64-89 
 
 9-999774 
 
 .07 
 
 8 - 509200 
 
 64-96 
 
 11-490800 
 
 9 
 
 52 
 
 64-3i 
 
 999769 
 
 .07 
 
 513098 
 
 64-39 
 
 486902 
 
 8 
 
 53 
 
 63-75 
 
 999765 
 
 .07 
 
 5 1 696 1 
 
 63-82 
 
 483o39 
 
 7 
 
 54 
 
 52o55i 
 
 63-19 
 
 999761 
 
 ■07 
 
 620790 
 
 63-26 
 
 479210 
 475414 
 
 6 
 
 55 
 
 524343 
 
 62-64 
 
 999757 
 
 •°7 
 
 524586 
 
 62-72 
 
 5 
 
 56 
 
 528102 
 
 62 -u 
 
 999753 
 
 •°7 
 
 528349 
 
 62-18 
 
 47i65i 
 
 4 
 
 -ll 
 
 53i828 
 
 61 -58 
 
 999748 
 
 •07 
 
 532o8o 
 
 6i-65 
 
 467920 
 
 3 
 
 535523 
 
 61 -06 
 
 999744 
 
 .07 
 
 535779 
 
 6i-i3 
 
 464221 
 
 2 
 
 5 9 
 
 539186 
 
 6o-55 
 
 999740 
 
 ■ 07 
 
 539447 
 
 60-62 
 
 46o553 
 
 1 
 
 6o 
 
 542819 
 
 60 -04 
 
 9997 35 
 Sine < 
 
 .07 
 88° 
 
 543o84 
 
 60-12 
 
 456916 
 
 
 
 I | OOSJQO 
 
 Cotang. 
 
 D. 
 
 Tang 
 
 : 
 
20 
 
 (2 
 
 DEGREES.) A TABLE 
 
 OF LOGARITHMIC 
 
 
 M. Sine 
 
 D. 
 
 Cosine ] D. 
 
 Tang. D. 
 
 Cotang. | 
 
 o f 
 
 i-5428i9 
 
 60-04 
 
 9-999735 -07 
 
 8- 543o84 
 
 60-12 
 
 11-4569161 60 
 
 453309! So 
 44973 2 1 58 
 446i83| Si 
 442664I 56 . 
 
 i 
 
 546422 
 
 59.55 
 
 999731 
 
 .07 
 
 546691 
 
 59-62 
 
 2 
 
 3 
 
 549995 
 553 oiq 
 
 5g-o6 
 58-58 
 
 999726 
 999722 
 
 •07 
 •08 
 
 550268 
 553817 
 557336 
 
 59-14 
 58-66 
 
 4 
 
 557054 
 
 58-u 
 
 999717 
 
 .08 
 
 58- 19 
 57-73, 
 
 5 
 
 56o54o 
 
 5 7 -65 
 
 9997 1 3 
 
 .08 
 
 560828 
 
 439172, 55 
 43 5709 ' 54 
 
 6 
 
 563999 
 567431 
 
 57-19 
 
 999708 
 
 ■08 
 
 564291 
 
 \vv 
 
 I 
 
 56-74 
 
 9997°4 
 999699 
 
 ■08 
 
 567727 
 
 56-82 
 
 432273 
 428863 
 
 Oi 
 
 670836 
 
 ■56-3o 
 
 .08 
 
 5 7 1 1 3 7 
 
 56-38 
 
 52 
 
 9 
 
 574214 
 
 55.87 
 
 999694 
 999689 
 
 •08 
 
 574020 
 
 55- g5 
 
 420480 
 
 5i 
 
 ID 
 
 577566 
 8-580892 
 
 55-44 
 
 .08 
 
 577877 
 8- 581208 
 
 55-52 
 
 422123 
 
 5o 
 
 II 
 
 55-02 
 
 9-999685 
 
 .08 
 
 55-10 
 
 11-418792 
 4i5486 
 
 % 
 
 12 
 
 584193 
 
 54- 60 
 
 999680 
 
 •08 
 
 5845 U 
 
 54-68 
 
 i3 
 
 587469 
 
 54-19 
 
 999675 
 
 •08 
 
 587795 
 591031 
 
 54-27 
 
 4l22o5 
 
 47 
 
 - 14 
 
 59072 1 
 
 53-79 
 
 999670 
 
 .08 
 
 53-87 
 
 408949! 46 
 
 i5 
 
 593948 
 
 53- 3 9 
 
 999665 
 
 .08 
 
 5 9 4283 
 
 53-47 
 53- 08 
 
 . 4067 1 jl 45 
 
 16 
 
 597152 
 
 53-00 
 
 999660 
 
 ■08 
 
 597492 
 
 402508 
 
 44 
 
 \l 
 
 6oo33a 
 
 52-6i 
 
 999655 
 
 •08 
 
 600611 
 
 52-70 
 
 399323 
 
 43 
 
 603485 
 60662,) 
 
 52-23 
 
 ■ ggyuSo 
 
 .08 
 
 6o383o 
 606978 
 
 52-32 
 
 396161 
 
 42 
 
 '9 
 
 5i-86 
 
 9996^5 -og 
 
 5i -g4 
 
 5i-58 
 
 5l-2I 
 
 3o3o22 41 
 
 2q 
 
 21 
 
 609734 
 8.612823 
 
 51-49 
 
 5l-I2 
 
 99g640| -og 
 9-999635 -og 
 
 610094 
 8-613189 
 
 389906 
 
 n-38681 1 
 
 40 
 
 22 
 
 6i58qi 
 
 618937 
 
 5o-7& 
 
 999629 
 
 •og 
 
 616262 
 
 5o-85 
 
 383738 
 
 23 
 
 5o-4i 
 
 999624 
 
 • 09 
 
 6i93i3 
 
 5o-5o 
 
 380687 
 
 17 
 
 H 
 
 621962 
 
 5o-o6 
 
 999619 
 
 .09 
 
 622343 
 
 5o-i5 
 
 377657 
 374648 
 
 36 
 
 25 
 
 624965 
 
 49-72 
 
 999614 
 
 -09 
 
 623352 
 
 49-81 
 
 35 
 
 26 
 
 627948 
 
 49-38 
 
 999608 
 
 -og 
 
 628340 
 
 49-47 
 49 -i3 
 
 371660 
 
 34 
 
 u 
 
 63ogii 
 
 49-04 
 48-71 
 48 -3g 
 
 99g6o3 
 
 ■ 09 
 
 63i3o8 
 
 368692 
 
 33 
 
 633854 
 
 - 99g5g7 
 
 -og 
 
 634256 
 
 48-80 
 
 365744 
 362816 
 
 32 
 
 29 
 
 636776 
 639680 
 
 9995g2 
 
 -og 
 
 637184 
 
 48-48 
 
 3i 
 
 3o 
 
 48-06 
 
 999586 
 
 -og 
 
 64ooo3 
 
 8 •642982 
 
 645853 
 
 48-16 
 
 359907 
 11-357018 
 
 3o 
 
 3i 
 
 8-642563 
 
 47-75 
 
 9-999581 
 
 •09 
 
 47-84 
 
 2I 
 
 32 
 
 645428 
 
 47-43 
 
 999 5 7 5 
 
 •09 
 
 47-53 
 
 354U7 
 
 33- 
 
 648274 
 
 47-U 
 
 999570 
 
 .09 
 
 648704 
 65i537 
 
 47-22 
 
 351296 
 
 27 
 
 34 
 
 65iio2 
 
 46-82 
 
 999564 
 
 .09 
 
 46-91 
 46-61 
 
 348463 
 
 26 
 
 35 
 
 65391 1 
 
 46-52 
 
 999558 
 
 • 10 
 
 654352 
 
 345648 
 
 25 
 
 36 
 
 n 
 
 656702 
 659475 
 662230 
 
 46-22 
 
 45-92 
 
 999553 
 999547 
 
 ■10 
 ■ 10 
 
 66714c 
 65g92i 
 
 46-3i 
 46-02 
 
 ,34285i 
 340072 
 
 24 
 
 23 
 
 45-63 
 
 99g54i 
 
 • 10 
 
 662689 
 665433 
 
 45-73 
 
 3373 1 1 
 
 22 
 
 3 9 
 
 664968 
 
 45-35 
 
 ggg535 
 
 • 10 
 
 45-44 
 
 334567 
 
 21 
 
 40 
 
 667689 . 
 
 8-67o3o3 
 
 673080 
 
 45- 06 
 
 9gg52g 
 
 ■ 10 
 
 668160 
 
 45-26 
 
 33i84o 
 
 20 
 
 41 
 
 44-79 
 
 9-999324 
 
 • 10 
 
 8-670870 
 
 44-88 
 
 n-32gi3o 
 326437 
 
 \l 
 
 42 
 
 44-5t 
 
 9gg5i8 
 
 • 10 
 
 673563 
 
 44-6i 
 
 43 
 
 676751 
 
 44-24 
 
 99g5i2 
 
 • 10 
 
 676239 
 
 44.34 
 
 323761 
 
 17 
 
 44 
 
 678405 
 681043 
 
 43-97 
 
 99g5o6 
 
 .-10 
 
 678qo< 
 
 44-17 
 
 321 100 
 
 16 
 
 45 
 
 43-70 
 
 99g5oo 
 
 ■ 10 
 
 681644 
 
 43- 80 
 
 3 1 8456 
 
 i5 
 
 46 
 
 683665 
 
 43-44 
 
 999493 
 
 ■ 10 
 
 68417! 
 686784 
 
 43-54 
 
 3i5828 
 
 14 
 
 % 
 
 686272 
 
 43- 18 
 
 999487 
 
 • 10 
 
 43-28 
 
 3i32i6 
 
 i3 
 
 688863 
 
 42-92 
 
 99948i 
 
 • 10 
 
 68g38i 
 
 43-o3 
 
 3 1 06 19 
 
 12 
 
 49 
 
 691438 
 
 42-67 
 
 999475 
 
 •IC 
 
 69196- 
 694321; 
 
 42-77 
 
 42-52 
 
 3o8o3- 
 
 11 
 
 5o 
 
 693998 
 8-696043 
 
 42-42 
 
 999469 
 9-99g46c 
 
 ■15 
 
 305471 
 
 10 
 
 5i 
 
 42-17 
 
 ■II 
 
 8-697081 
 
 42-28 
 
 11-302910 
 3oo383 
 
 8 
 
 5s 
 
 699073 
 701589 
 
 41-92 
 41-68 
 
 999456 
 
 ■II 
 
 6gg6i- 
 
 42 -o3 
 
 53 
 
 99945o 
 
 • II 
 
 702 13< 
 
 41-70 
 4i-55 
 
 297861 
 
 I 
 
 54 
 
 704090 
 
 41-44 
 
 ggg443 
 
 -II 
 
 70464c 1 
 
 ' 295354 
 
 55 
 
 706577 
 
 41-21 
 
 999437 
 
 -II 
 
 70714c 
 
 41-32 
 
 292860 
 
 5 
 
 56 
 
 709049 
 
 40-97 
 
 99943i 
 
 •II 
 
 7096 1 1 
 
 4i- 08 
 
 290382 
 287917 
 
 4 
 
 u 
 
 711507 
 
 40-74 
 
 99942/ 
 
 • 11 
 
 71208; 
 
 40-85 
 
 3 ' 
 
 7i3a52 
 716383 
 
 40-31 
 
 999418 
 
 • 11 
 
 71453/ 
 
 40-62' 
 
 285465 
 
 2 
 
 59 
 
 40-29 
 
 9994 1 1 
 
 • 11 
 
 71697; 
 7ig3gf 
 
 4o-4o 
 
 283028 
 
 1 
 
 60 
 
 718800 
 
 4o 06 
 
 999404 
 
 •II 
 
 40-17 
 
 28060/ 
 
 
 
 
 Cosine 
 
 D. 
 
 Sine 
 
 87 1 *! Cotang. 
 
 D. 
 
 Tang. 
 
 M. 
 

 SINES AND TANGENTS 
 
 (3 DEGREES.) 
 
 
 21 
 
 M. 
 
 Sine 
 
 D. 
 
 Cosine 
 
 D. 
 
 Tang. 
 
 D. 
 
 Cotang. { 
 
 o 
 
 8-718800 
 
 4o- 06 
 
 9-999404 
 
 -11 
 
 8-719396 
 
 4o 
 
 17 
 
 11-280604 60 
 
 , > 
 
 721204 
 
 39-84 
 
 999398 
 
 • 11 
 
 721806 
 
 ?9 
 
 9 5 
 
 278194' So 
 
 ) 2 
 
 723595 
 
 39*62 . 
 
 999391 
 999384 
 
 ■11 
 
 724204 
 
 3g 
 
 74 
 
 275796 58 
 
 3 
 
 725972 
 728337 " 
 
 39-41 
 
 -11 
 
 726588 
 
 39 
 
 52 
 
 273412! 57. 
 
 4 
 
 3l-o? 
 
 999378 
 
 •11 
 
 728959 
 73i3i7 
 
 39 
 
 3o 
 
 271041I 56 
 
 5 
 
 73o688 
 
 999371 
 
 •11 
 
 39 
 
 09 
 
 268683! 55 
 
 6 
 
 733027 
 
 38-77 
 
 999364 
 
 • 12 
 
 733663 
 
 38 
 
 89 
 
 266337] 54 
 
 I 
 
 735354 
 
 38-57 
 
 999357 
 
 •12 
 
 735996 
 
 38 
 
 68 
 
 264004! 53 
 
 737667 
 
 38-36 
 
 999350 
 
 •12 
 
 738317 
 
 38 
 
 48 
 
 26l683 52 
 
 9 
 
 73.9969 
 
 38-i6 
 
 999343 
 
 •12 
 
 740626 
 
 38 
 
 27 
 
 259374 5i 
 
 <o 
 
 742256 
 8- 744536 
 
 37-96 
 
 999336 
 
 •12 
 
 742922 
 
 38 
 
 n 1 
 
 257078 5o 
 
 ii 
 
 37-76 
 
 9 -999329 
 
 •12 
 
 8-745207 
 
 3 7 
 
 87 
 
 1 1 • 254793 49 
 
 12 
 
 746802 
 
 37-56 
 
 999322 
 
 •12 
 
 747479 
 
 37 
 
 68 
 
 252521 48 
 
 i3 
 
 749055 
 
 37-37 
 
 9993 1 5 
 
 ■12 
 
 749740 
 
 37 
 
 49 
 
 250260 47 
 
 14 
 
 751297 
 753528 
 
 37-17 
 36- 9 8- 
 
 999308 
 
 ■ 12 
 
 751989 
 
 37 
 
 29 
 
 2480II^ 46 
 
 i5 
 
 999301 
 
 ■12 
 
 754227 
 
 37 
 
 10 
 
 i45773| 45 
 
 16 
 
 755747 
 
 36-79 
 
 999294 
 
 ■12 
 
 756453 
 
 36 
 
 92 
 
 243547 44 
 
 \l 
 
 757955 
 
 36-6i 
 
 999286 
 
 •12 
 
 758668 
 
 36 
 
 73 
 
 . 24i332. 43 
 
 760151 
 
 36-42 
 
 999279 
 
 • 12 
 
 760872 
 
 36 
 
 55 
 
 239128, 42 
 
 ■9 
 
 762337 
 
 36-24 
 
 999272 
 
 • 12 
 
 763o65 
 
 36 
 
 36 
 
 2369351 41 
 
 20 
 
 7645 1 1 
 
 36- 06 
 
 999266 
 
 ■12 
 
 765246 
 
 36 
 
 i3 
 
 > 234754 40 
 
 21 
 
 8-766675 
 
 35-88 
 
 9-999257 
 
 ■12 
 
 8-767417 
 769578 
 
 36 
 
 00 
 
 n-232583, 3o 
 23o422J 38 
 
 22 
 
 768828 
 
 35.70 
 
 999250 
 
 • i3 
 
 35 
 
 83 
 
 23 
 
 770970 
 
 35-53 
 
 999242 
 
 •i3 
 
 771727 
 
 35 
 
 65 
 
 2282731 37 
 
 24 
 
 773ioi 
 
 . 35-35 
 
 999235 
 
 •i3 
 
 773866 
 
 35 
 
 48 
 
 226134 36 
 
 25 
 
 775223 
 
 35.i8 
 
 999227 
 
 •i3 
 
 775995 
 
 35 
 
 3i 
 
 224oo5 35 
 
 26 
 
 777333 
 
 35-oi 
 
 999220 
 
 ■i3 
 
 778114 
 
 35 
 
 14 
 
 221886 
 
 34 
 
 ■ 27 
 
 779434 
 
 34-84 
 
 999212 
 
 •i3 
 
 780222 
 
 34 
 
 V, 
 
 219778 
 
 33 
 
 28 
 
 78i524 
 
 34-67 
 
 999205 
 
 •i3 
 
 782320 
 
 34 
 
 2 1 7680 
 
 32 
 
 29 
 
 ~7836o5 
 
 34-5i 
 
 999197 
 999189 
 
 •i3 
 
 784408 
 
 34 
 
 64 
 
 215592 
 
 3i 
 
 3o 
 
 785675 
 
 34-3i 
 
 -i3 
 
 786486 
 
 34 
 
 i 1 
 
 2i35i4 
 
 3o 
 
 3i 
 
 8-787736 
 
 34-i8 
 
 9-999181 
 
 ■i3 
 
 8-788554 
 
 34 
 
 3i 
 
 11-211446 
 
 29 
 
 32 
 
 » 
 
 34-02 
 
 999 '74 
 
 •i3 
 
 790613 
 
 34 
 
 i5 
 
 209387 
 207338 
 
 28 
 
 33 
 
 33-86 
 
 999166 
 
 ■i3 
 
 792662 
 
 33 
 
 8 
 
 27 
 
 34- 
 
 793859 
 
 33-70 
 
 999158 
 
 -i3 
 
 794701 
 
 33 
 
 205299 
 
 26 
 
 35 
 
 795881 
 
 33-54 
 
 999150 
 
 -i3 
 
 796731 
 
 33 
 
 68 
 
 203269 
 
 25 
 
 36 
 
 797894 
 
 33- 3 9 
 33-23 
 
 999142 
 
 ■i3 
 
 798752 
 800763 
 
 33 
 
 52 
 
 201248 
 
 24 
 
 12 
 
 799897 
 801892 
 
 999134 
 
 •i3 
 
 33 
 
 37 
 
 199237 
 
 23 
 
 33- 08 
 
 999126 
 
 • i3 
 
 802765 
 
 33 
 
 22 
 
 197235 
 
 22 
 
 3 9 
 
 803876 
 
 32-g3 
 
 9991 18 
 
 -i3 
 
 8o4758 
 
 33 
 
 °7 
 
 195242 
 
 21 
 
 40 
 
 8o5852 
 
 32-78 
 
 999110 
 
 ■i3 
 
 806742 
 
 32 
 
 9? 
 
 193258 
 
 20 
 
 41 
 
 8-807819 
 
 32-63' 
 
 9-999102 
 
 • i3 
 
 8-808717 
 
 32 
 
 t 
 
 11-191283 
 
 '9 
 
 42 
 
 809777 
 
 _32-49 
 
 999094 
 
 •14 
 
 8io683 
 
 32 
 
 1893171 18 
 187359 17 ' 
 
 43 
 
 811726 
 
 32-34 
 
 999086 
 
 •14 
 
 81 2641 
 
 32 
 
 48 
 
 44 
 
 8i366 7 
 
 32-IO 
 
 999077 
 
 ■14 
 
 814589 
 
 32 
 
 33 
 
 189411 
 
 16 ' 
 
 45 
 
 815599 
 
 32-65 
 
 999069 
 
 •14 
 
 816529 
 
 32 
 
 19 
 
 1 8347 1 
 
 15 ; 
 
 46 
 
 817522 
 
 3: -91 
 
 999061 
 
 •14 
 
 818461 
 
 32 
 
 OD 
 
 181539 
 
 14 ' 
 
 % 
 
 819436 
 
 3r-77 
 
 999053 
 
 •14 
 
 820384 
 
 3i 
 
 9' 
 
 179616 
 
 i3 ; 
 
 821343 
 
 3i-63 
 
 999044 
 
 •14 
 
 82229S 
 
 3i 
 
 B- 
 
 177702 
 
 12 
 
 49 
 
 823240 
 
 3i-49 
 31.35 
 
 999036 
 
 •14 
 
 824205 
 
 3i 
 
 175795 
 
 273897 
 
 11-172008 
 
 11 * 
 
 5.6 
 
 825i3o 
 
 999027 
 
 •14 
 
 826103 
 
 3i 
 
 5o 
 
 10 
 
 5i 
 
 8-827011 
 
 3l-22 
 
 9-999019 
 
 •14 
 
 8-827992 
 
 3i 
 
 36 
 
 I 
 
 52 
 
 828884 
 
 3i-o8 
 
 999010 
 
 •14 
 
 829874 
 
 3i 
 
 23 
 
 170126 
 
 53 
 
 830749 
 
 3o-g5 
 3o-82 
 
 999002 
 998993 
 
 •14 
 
 83i748 
 
 3i 
 
 10 
 
 168252 
 
 n 
 
 54 
 
 832607 
 
 •14 
 
 8336i3 
 
 3o 
 
 96 
 
 166387 
 
 6 
 
 55 
 
 834456 
 
 3o.6g 
 
 998984 
 
 •14 
 
 835471 
 
 3o 
 
 83 
 
 i645»o 
 
 5 
 
 56 
 
 836297 
 838i3o 
 
 3o-56 
 
 998976 
 
 •14 
 
 837321 
 
 3o 
 
 70 
 
 162679 
 
 4 
 
 % 
 
 3o-43 
 
 998967 
 
 • i5 
 
 839163 
 
 3o 
 
 57 
 
 160837 
 
 3 
 
 839956 
 
 3o-3o 
 
 998958 
 
 -i5 
 
 840998 
 
 3o 
 
 45 
 
 159002 
 
 2 
 
 ■ 59 
 
 841774 
 843585 
 
 30-17 
 
 998950 
 
 ■ i5 
 
 842825 
 
 3o 
 
 32 
 
 157175 
 
 1 
 
 60 
 
 3o-oo 
 
 998941 
 
 • i5 
 
 844644 
 
 30-19 
 
 155356 
 
 
 
 Cosine 
 
 D 
 
 Sine 
 
 36° 
 
 Cotang. 
 
 D. 
 
 Tang. 
 
 M. _ 
 
!2 
 
 (4 DEGREES 
 
 .) A TABLE 
 
 Or LOGARITHMIC 
 
 
 M. 
 
 Sine | 
 
 D. | 
 
 Cosine 
 
 n. 
 
 Tang. 
 
 D. 
 
 Cotang. | 
 
 
 
 3,843585 
 
 3o-o5 
 
 9-998941 
 
 •i5 
 
 8-844644 
 
 3o-i9 
 
 11 -155356 
 
 60 
 
 i 
 
 845387 
 
 29-92 
 
 998932 
 
 ■ i5 
 
 846455 
 
 3o-o7 
 
 153545 
 
 & 
 
 l 
 
 847183 
 
 29-00 
 
 998923 
 
 ■i5 
 
 848260 
 
 29«o5 
 29-82 
 
 151740 
 
 3 
 
 848971 
 
 29-67 
 
 998914 
 
 •i5 
 
 830057 
 
 149943 
 148154 
 
 n 
 
 4 
 
 85o75i 
 
 29-55 
 
 998905 
 
 • i5 
 
 85i846 
 
 29-70 
 
 5 
 
 852525 
 
 29-43 
 
 998896 
 998887 
 
 • i5 
 
 853628 
 
 29-58 
 
 146372 
 
 55 
 
 6 
 
 854291 
 
 29-31 
 
 ■i5 
 
 855403 
 
 29-46 
 
 144597 
 
 54 
 
 I 
 
 856o49 
 
 29-19 
 
 998878 
 
 •i5 
 
 867171 
 858932 
 
 29-35 
 
 142829 
 141068 
 
 53 
 
 857801 
 
 29-07 
 
 998869 
 
 •i5 
 
 29-23 
 
 52 
 
 9 
 
 859546 
 
 28-96 
 28-84 
 
 998860 
 
 -i5 
 
 860686 
 
 29-11 
 
 i3g3i4 
 
 5i 
 
 10 
 
 861283 
 
 998801 
 
 •i5 862433 
 
 29-00 
 28-88 
 
 137567 
 
 5o 
 
 II 
 
 8-863014 
 
 28-73 
 
 9-998841 
 
 • i5, 8-864173 
 
 11 -135827 
 
 % 
 
 12 
 
 864738 
 
 28-61 
 
 998832 
 
 ■i5, 865906 
 
 28-77 
 28-66 
 
 134094 
 
 i3 
 
 866455 
 
 28 -5o 
 
 998823 
 
 •16 86 7 632 
 
 132368 
 
 % 
 
 14 
 
 868i65 
 
 28 -3o 
 28-28 
 
 998813 
 
 .16 
 
 86 9 35i 
 
 23-54 
 
 130649 
 
 i5 
 
 869868 ' 
 
 998804 
 
 ■16 
 
 871064 
 
 28-43 
 
 128936 
 
 45 
 
 16 
 
 871565 
 
 28-17 
 28-06 
 
 998795 
 
 •16 
 
 872770 
 
 28-32 
 
 127230 
 
 44 
 
 \l 
 
 ,873255 
 
 998785 
 
 ■16 
 
 874469 
 
 28-21 
 
 1 2553 1 
 
 43 
 
 874938 
 876616 
 
 27-95 
 
 998776 
 
 ■16 
 
 876162 
 
 28-11 
 
 123838 
 
 42 
 
 19 
 
 27-86 
 
 998766 
 
 ■16 
 
 877849 
 
 28.00- 
 
 I22l5l 
 
 4i 
 
 2l< 
 
 878285 
 
 27-73 
 
 998757 
 
 ■16 879529 
 
 27.89 
 
 1 2047 1 
 
 40 
 
 21 
 
 8-879949 
 881607 
 
 27-63 
 
 9-998747 
 
 ■16 8-881202 
 
 27.79 
 
 II 1 1 8708 
 
 ll 
 
 22 
 
 27-52 
 
 998738 
 
 ■16 
 
 882869 
 
 27.68 
 
 i i 71J1 
 
 23 
 
 883258 
 
 27-42 
 
 998728 
 
 •16 
 
 88453o 
 
 27.58 
 
 1 1 5470 
 
 ll 
 
 24 
 
 834oo3 
 886542 
 
 27-3i 
 
 998718 
 
 ■16 
 
 886 1 85 
 
 27-47 
 
 u38i5 
 
 25 
 
 2.7-21 
 
 998708 
 
 ■16 
 
 887833 
 
 27-37 
 
 112167 
 
 35 
 
 26 
 
 888174 
 
 27-11 
 
 998699 
 
 ■16 
 
 889476 
 
 27-27. 
 
 no524 
 
 34 
 
 U 
 
 889801 
 
 27-00 
 
 998689 
 
 •16 
 
 891112 
 
 27-17 
 
 108888 
 
 33 
 
 891421 
 
 26; 00 
 
 998679 
 
 •16 
 
 892742 
 
 27-07 
 
 107268 
 
 32 
 
 2 9 
 
 893o35 
 
 26-80 
 
 998669 
 
 •17 
 
 894366 
 
 -26-97 
 26-87 
 
 io5634 
 
 3i 
 
 3o 
 
 894643 
 
 26-70 
 
 998659 
 
 ■17 
 
 896984 
 8-897596 
 
 104016 
 
 3o 
 
 3i 
 
 8-896246 
 
 26-60 
 
 9-998649 
 
 ■17 
 
 26-77 
 
 11-102404 
 
 29 
 
 32 
 
 807842 
 
 26-5i 
 
 998639 
 
 •17 
 
 899203 
 
 26-67 
 
 100797 
 
 28 
 
 33 
 
 899432 
 
 26-41 
 
 998629 
 
 •17 
 
 900803 
 
 26-58 
 
 099197 
 
 27 
 
 34 
 
 901017 
 
 a6-3i 
 
 998619 
 
 •'7 
 
 902398 
 903987 
 906070 
 
 26-48 
 
 097602 
 
 26 
 
 35 
 
 902596 
 
 26-22 
 
 998609 
 
 •■7 
 
 26-38 
 
 096013 
 
 25 
 
 36 
 
 904169 
 
 26-12 
 
 998599 
 
 ■■" 
 
 26-29 
 
 094430 
 
 24 
 
 ll 
 
 905736 
 
 26 -o3 
 
 998589 
 
 •17 
 
 907147 
 908719 
 910285 
 
 26-20 
 
 092853 
 
 23 
 
 907297 
 908853 
 
 25-g3 
 
 998678 
 
 •17 
 
 26-10 
 
 091281 
 089715 
 0881 54 
 
 22 
 
 3 9 
 
 25-84 
 
 9 9 8568 
 
 ■n 
 
 26-01 
 
 21 
 
 4o 
 
 9 r 0404 
 
 25-76 
 
 998558 
 
 ■17 
 
 91 1846 
 
 25.92 
 25-83 
 
 20 
 
 41 
 
 8-911949 
 913488 
 
 25-66 
 
 9-998548 
 
 •17, 8-913401 
 
 11-086599 
 
 \i 
 
 42 
 
 25-56 
 
 998537 
 
 •'7 
 
 914951 
 
 25-74 
 
 085049 
 o835o5 
 
 43 
 
 915022 
 
 25-47 
 
 998527 
 
 •17 
 
 916495 
 918034 
 
 25-65 
 
 \l 
 
 44 
 
 9i^55o 
 
 25-38 
 
 998516 
 
 ■18 
 
 25-56 
 
 081966 
 
 45 
 
 918073 
 
 ' 25-29 
 
 998506 
 
 •18 
 
 919568 
 
 25-47 
 
 o8o432 
 
 i5 
 
 46 
 
 919091 
 
 25-20 
 
 998495 
 998485 
 
 •18 
 
 921096 
 
 25-38 
 
 078904 
 077381 
 
 14 
 
 8 
 
 921103 
 
 25-12 
 
 •18 
 
 922611; 
 
 25 -3o 
 
 i3 
 
 922610 
 
 25-o3 
 
 99S474 
 
 ■18 
 
 924136 
 
 25-21 
 
 075864 
 
 12 
 
 49 
 
 9241 1 2 
 
 24-9* 
 
 998464 
 
 .18 
 
 926649 
 
 25-12 
 
 07435i 
 
 11 
 
 ■5o 
 
 925609 
 
 24-86 
 
 998453 
 
 •18 
 
 927166 
 
 25 -o3 
 
 072844 
 
 10 
 
 5i 
 
 8-927100 
 
 24-77 
 
 9 •99 8 442 
 
 ■18 
 
 8- 9 28658 
 
 24-95 
 
 24-86 
 
 11-071342 
 
 o 
 
 52 
 
 9 28587 
 93oofS 
 
 24-69 
 
 998431 
 
 ■18 
 
 93oi 55 
 
 069845 
 
 53 
 
 24- 60 
 
 998421 
 
 •18 
 
 93i64" 
 
 24-78 
 
 068353 
 
 I 
 
 54 
 
 93i 544 
 
 24-52 
 
 998410 
 
 •18 
 
 933 i3^ 
 
 24-70 
 
 066866 
 
 55 
 
 933oi5 
 
 24-43 
 
 998399 
 998388 
 
 -18 
 
 934616 
 
 24-61 
 
 065384 
 
 5 
 
 56 
 
 934481 
 
 24-35 
 
 •18 
 
 93609: 
 9 3 7 565 
 
 24-53 
 
 063907 
 
 4 
 
 .3 
 
 935o42 
 937398 
 938850 
 
 24.27 
 
 998377 
 
 .18 
 
 24-45 
 
 062435 
 
 3 
 
 24-19 
 
 998366 
 
 ■18 
 
 939032 
 
 24.37 
 
 060968 
 069606 
 068048 
 
 2 
 
 59 
 
 24-11 
 
 998355 
 
 ■18 
 
 940494 
 941962 
 
 24-3o 
 
 1 
 
 6o 
 
 940296 
 
 ?4>o3 
 
 998344 
 
 .18 
 
 24-21 
 
 
 
 " 
 
 Cosine 
 
 D. 
 
 Sins 
 
 85 c 
 
 Cotang. 
 
 D. 
 
 Tang. 
 
 M. 
 

 8INES AND TANGENTS 
 
 (5 DEGREES.) 
 
 2) 
 
 TT-.I 
 
 Sine J 
 
 D. 
 
 Cosine j D. 
 
 Tang. 
 
 E 
 
 • 
 21 
 
 Cotang. | - 
 
 o £.940296 
 
 24- o3 
 
 9.998344 
 
 ■ 19 
 
 8-941952 
 
 24 
 
 n-o58o48' 6a 
 
 I 
 
 94H38 
 
 23o4 
 23 §7 
 
 99H333 
 
 • 19 
 
 943404 
 
 24 
 
 i3 
 
 056596! ~5o 
 o55i48' 58 
 
 2 
 
 g43 i 74 
 
 998322 
 
 -19 
 
 944852 
 
 24 
 
 o5 
 
 3 
 
 944606 
 
 23-79 
 
 998311 
 
 • 19 
 
 946295 
 947734 
 
 23 
 
 97 
 
 003705 57 
 
 4 
 
 946034 
 
 23-71 
 23-63 
 
 998300 
 
 • 19 
 
 23 
 
 90 
 
 032266 56 
 
 5 
 
 947K6 
 
 998289 
 
 • 19 
 
 949168 
 
 - 13 
 
 82 
 
 o5o832 55 
 
 a> 
 
 948874 
 950287 
 951696 
 
 23-55 
 
 998277 
 
 ■ 19 
 
 950597 
 
 23 
 
 -A 
 
 049463 54 
 
 7 
 
 23-48 
 
 998266 
 
 • 19 
 
 952021 
 
 • 23 
 
 66 
 
 047979 53 
 
 8 
 
 23-4o 
 
 998255 
 
 ■ 19 
 
 953441 
 
 23 
 
 60 
 
 046559 52 
 
 9 
 
 953ioo 
 
 23-32 
 
 998243 
 
 ■ 19 
 
 9 54856 
 
 23 
 
 5i 
 
 o45 1 44' 5 1 
 
 10 
 
 954499 
 
 23-20 
 
 998232 
 
 ■'9 
 
 956267 
 
 23 
 
 44 
 
 043733 5o 
 
 ii 
 
 8-9558 9 4 
 957284 
 958670 
 
 23-17 
 
 9.998220 
 
 ■19 
 
 8-957674 
 
 23 
 
 37 
 
 11 -042326' 49 
 040925* 48 
 
 12 
 
 23- 10 
 
 998209 
 
 •19 
 
 909075 
 
 23 
 
 23 
 
 i3 
 
 23-02 
 
 998197 
 998186 
 
 ■ 19 
 
 960473 
 
 23 
 
 039D27. 47 
 
 U 
 
 960062 
 
 22-95 
 
 22-88 
 
 •19 
 
 961866 
 
 23 
 
 14 
 
 o3oi34' 46 
 
 i5 
 
 961429 
 
 998174 
 
 •'9 
 
 963255 
 
 23 
 
 °7 
 
 036745; 45 
 
 16 
 
 962801 
 
 22-8o 
 
 998163 
 
 .19 
 
 J364639 
 
 23 
 
 00 
 
 o3536i 44 
 
 \l 
 
 964170 
 
 22-73 
 
 99S 1 5 1 
 
 ■19 
 
 966019 
 
 22 
 
 t 
 
 o33 9 8i 43 
 
 965534 
 
 22-66 
 
 998139 
 998128 
 
 -20 
 
 967394 
 968766 
 
 22 
 
 032606 42 
 
 >9 
 
 966893 
 
 22-59 
 
 -20 
 
 22 
 
 79 
 
 o3i234' 4i 
 
 20 
 
 968249 
 
 22-52 
 
 9981 16 
 
 •20 
 
 970133 
 
 22 
 
 7< 
 
 029867 40 
 
 2! 
 
 8-969600 
 
 22-44 
 
 9-998104 
 
 ■20 
 
 8-071496 
 972855 
 
 22 
 
 65 
 
 Ii-0285o4! 3a 
 027145. 38 
 
 22 
 
 970947 
 
 22-38 
 
 998092 
 ■ 998080 
 
 ■ 20 
 
 22 
 
 t 1 
 
 23 
 
 972289 
 
 22-3l 
 
 • 20 
 
 974209 
 
 22 
 
 5i 
 
 020791 37 
 
 24 
 
 973623 
 
 22-24 
 
 998068 
 
 • 20 
 
 97556o 
 
 22 
 
 44 
 
 024440 36 
 
 25 
 
 974962 
 
 22-17 
 
 998056 
 
 •20 
 
 976906 
 
 22 
 
 37 
 
 023094 35 
 021702 34 
 
 26 
 
 976293 
 
 22-10 
 
 998044 
 
 •20 
 
 978248 
 
 22 
 
 3o 
 
 23 
 
 977619 
 978941 
 
 22 o3 
 
 998032 
 
 .20 
 
 97 9 586 
 98092 1 
 
 22 
 
 23 
 
 020414! 33 
 
 21-97 
 
 998020 
 
 •20 
 
 22 
 
 '7 
 
 0190791 32 
 
 2 9 
 
 980259 
 
 9 8i5 7 3 
 
 8-982883 
 
 21-90 
 
 998008 
 
 -20 
 
 982201 
 
 22 
 
 10 
 
 - 017749: 3i 
 01 6423 1 3o 
 
 3o 
 
 21-83 
 
 997996 
 
 • 20 
 
 • 983577 
 
 22 
 
 04 
 
 3i 
 
 21-77 
 
 9-997985 
 
 •20 
 
 8-984899 
 
 21 
 
 97 
 
 ii-ciSioii 20 
 oi3 7 83! 28 
 
 32 
 
 984189 
 
 21-70 
 
 997972 
 
 •20 
 
 986217 
 
 21 
 
 11 
 
 33 
 
 983491 
 986789 
 988083 
 
 21-63 
 
 997959 
 
 ■20 
 
 987532 
 988842 
 
 21, 
 
 oi2468i 27 
 
 34 
 
 21-57 
 
 997947 
 
 • 20 
 
 21 
 
 78 
 
 oin58| 26 
 
 35 
 
 21-50 
 
 997935 
 
 ■21 
 
 990 ! 49 
 
 21 
 
 71 
 
 ooo85 I i 25 
 
 008549! 24 
 
 36 
 
 989374 
 
 21-44 
 
 997922 
 
 •21 
 
 99I45I 
 
 21 
 
 65 
 
 Si 
 
 990660 
 
 21-38 
 
 997910 
 
 -21 
 
 992750 
 
 2i 
 
 58 
 
 0072501 23 
 
 991943 
 
 2I-3l 
 
 997897 
 
 •21 
 
 994045 
 
 21 
 
 52 
 
 005955| 22 
 
 3 9 
 
 993222 
 
 21-25 
 
 997885 
 
 • 21 
 
 995337 
 
 21 
 
 46 
 
 oo4663i 21 
 
 4o 
 
 994497 
 8-995768 
 
 21-19 
 
 997872 
 
 • 21 
 
 ■996624 
 
 21 
 
 40 
 
 0033761 20 
 
 41 
 
 21-12 
 
 9-997860 
 
 •2! 
 
 8-997908 
 
 21 
 
 34 
 
 11-002092 19 
 000812! 10 
 
 42 
 
 997036 
 998299 
 
 2i-o6 
 
 997847 
 
 -21 
 
 999188 
 
 21 
 
 27 
 
 43 
 
 21-00 
 
 997835 
 
 •21 
 
 g. OO046O 
 
 21 
 
 21 
 
 30-999535' 17 
 
 44 
 
 999560 
 
 20-94 
 
 20- 87 
 
 997822 
 
 ■21 
 
 OOI738 
 
 - 21 
 
 i5 
 
 998262' 16 
 
 45 
 
 9.000816 
 
 ■997809 
 
 -21 
 
 003007 
 
 21 
 
 ol 
 
 996993; i5 
 
 46 
 
 002069 
 
 20-82 
 
 997797 
 997784 
 
 -21 
 
 004272 
 
 21 
 
 995728! 14 
 
 % 
 
 oo33i8 
 
 20-76 
 
 •21 
 
 oo5534 
 
 1 20 
 
 9- 1 
 
 9944661 i3 
 
 004563 
 
 20--"> 
 
 997771 
 
 •21 
 
 006792 
 
 20 
 
 u 
 
 9932081 12 
 
 49 
 
 oo58o5 
 
 20-6/t 
 
 997758 
 
 •21 
 
 008047 
 009298 
 
 20 
 
 991953 11 
 
 56 
 
 007044 
 9-008278 
 
 20-58 
 
 997745 
 
 -21 
 
 20 
 
 80 
 
 990702 10 
 
 5i 
 
 20-52 
 
 9-997732 
 
 •21 
 
 9-010046 
 
 20 
 
 74 
 
 10-989454; 9 
 988210 8 
 
 52 
 
 009510 
 
 20-46 
 
 9977 '9 
 
 •21 
 
 011790 
 
 20 
 
 68 
 
 53 
 
 010737 
 
 20-40 
 
 997106 
 997693 
 997680 
 
 •21 
 
 oi3o3i 
 
 20 
 
 62 
 
 986969 7 
 
 54 
 
 011962' 
 
 20-34 
 
 ■22 
 
 014268 
 
 20 
 
 56 
 
 985732 6 
 
 55 
 
 oi3i82 
 
 20-20 
 20-23 
 
 •22 
 
 oi55o2 
 
 20 
 
 5i 
 
 984498 5 
 
 56 
 
 014400 
 
 997667 
 
 ■22 
 
 016732 
 
 20 
 
 45 
 
 983268 4 
 
 u 
 
 oi56i3 
 
 20-17 
 
 997654 
 
 ■22 
 
 017950 
 019183 
 
 20 
 
 40 
 
 982041 3 
 
 016824 
 
 20-12 
 
 997641 
 
 ■22 
 
 20 
 
 33 
 
 980817 2 
 
 5 9 
 
 oi8o3i 
 
 20- 06 
 
 997628 
 
 ■22 
 
 ,020403 
 
 20 
 
 28 
 
 978380 
 
 6o 
 
 019235 
 
 20-00 
 
 997614! 
 
 ■22 
 
 021620 
 
 - — : 
 
 20-23 
 
 1 
 L 
 
 Cosine . 
 
 d: 
 
 Sine !84° 
 
 Cotang. 
 
 1 D. 
 
 22 
 
24 
 
 (6 DEGREES.) A TABLE OV LOGARITHMIC 
 
 M. I Sine 
 
 D. 1 
 
 Cosine 
 
 D. 
 
 Tang. 
 
 D. 
 
 Cotang. 
 0-978380 
 977166 
 
 60 
 
 'q-oiq'235 
 
 20-00 
 
 9-997614 
 
 ■22 
 
 9-021620 
 
 20-23 
 
 i 
 
 020435 
 
 i 9 -o5 
 19-89 
 
 997601 
 
 ■22 
 
 022834 
 
 20-17 
 
 2 
 
 02l632 
 
 997388 
 
 •22 
 
 024044 
 
 20-11 
 
 975956 
 
 3 
 
 022825 
 
 19-84 
 
 997574 
 
 •22 
 
 02D25l 
 
 20- 06 
 
 974749 
 973545 
 
 tl 
 
 4 
 
 024016 
 
 19-78 
 
 997561 
 
 -22 
 
 026455 
 
 20-00 
 
 56 
 55 
 
 5 
 
 025203 
 
 19-73 
 
 997547 
 
 ■22 
 
 027655 
 028852 
 
 I9-95 
 
 972345 
 
 6 i 
 
 026386 
 
 19-67 
 
 997534 
 
 •23 
 
 \ 9 ai 
 
 971148 
 
 54* 
 
 2 
 
 027567 
 028744 
 
 19-62. 
 
 997520 
 
 •23 
 
 o3o<>46 
 
 969954 
 
 53 
 
 19-57 
 
 997507 
 
 ■23 
 
 o3 1 237 
 
 19.79 
 
 968163 
 
 5a 
 
 9 
 
 029918 
 
 19-5i 
 
 997493 
 997480 
 
 •23 
 
 032425 
 
 19-74 
 
 967575 
 
 5i 
 
 10 
 
 031089 
 
 19-47 
 
 •23 
 
 033609 
 
 19-69 
 
 96639 1 
 
 5o 
 
 II 
 
 9-o32257 
 
 19-41 
 
 9-997466 
 
 •23 
 
 9- o3479 1 
 035909, 
 
 19.64 
 
 10-965209 
 
 3 
 
 12 
 
 o3342i 
 
 i 9 -36 
 
 997452 
 
 •23 
 
 19-58 
 
 96403 1 
 
 i3 
 
 034582 
 
 19-30 
 
 997439 
 
 ■23 
 
 037144 
 
 i 9 -53 
 
 962856 
 
 %'. 
 
 14 
 
 035741 
 036896 
 
 19-25 
 
 997425 
 
 •23 
 
 q383i6 
 
 19-48 
 
 961684 
 
 i5 
 
 19-20 
 
 99741 1 
 
 •23 
 
 039485 
 
 19-43 
 
 9605 1 5 
 
 -45 
 
 16 
 
 o38o48 
 
 I9-I5 
 
 997383 
 
 •23 
 
 o4o65i 
 
 ig-38 
 
 969349 
 958187 
 
 44 
 
 11 
 
 039197 
 
 19-10 
 
 •23 
 
 04i8i3 
 
 19-33 
 
 43 
 
 18 
 
 64o342 
 
 ig-o5 
 18-99 
 
 997369 
 997355 
 
 •23 
 
 042973 
 
 19-28 
 
 957027 
 
 42 
 
 '9 
 
 041485 
 
 •23 
 
 o44i3o 
 
 19-23 
 
 955870 
 
 4i 
 
 20 
 
 042625 
 
 18-94 
 18-89 
 
 997341 
 
 •23 
 
 045284 
 
 19-18 
 
 954716 
 
 40 
 
 21 
 
 9*043762 
 044895 
 
 9.997327 
 
 •24 
 
 9-046434 
 
 19-13 
 
 10*953566 
 
 31 
 
 22 
 
 18-84 
 
 9973 1 3 
 
 •24 
 
 047582 
 048727 
 049869 
 o5ioo8 
 
 19-08 
 
 952418 
 
 23 
 
 046026 
 
 18.79 
 18-75 
 
 697299 
 
 .24 
 
 19 -o3 
 18-98 
 
 961273 
 95oi3i 
 
 37 
 
 24 
 
 047i54 
 048279 
 
 997285 
 
 • 24 
 
 36 
 35 
 
 25 
 
 18-70 
 
 997271 
 
 •24 
 
 18-93 
 18-89 
 
 948992 
 947856 
 
 26' 
 
 049400 
 
 18-65 
 
 997257 
 
 •24 
 
 032144 
 
 34 
 
 11 
 
 o5o5ig 
 o5i635 
 
 18-60 
 
 997242 
 
 ■24 
 
 053277 
 
 18-84 
 
 946723 
 
 33 
 
 18-55 
 
 997228 
 
 ■24 
 
 054407 
 
 18.79. 
 
 945693 
 
 32 
 
 29 
 
 052749 
 
 o^385o 
 
 9-054966 
 
 18 -5o 
 
 997214 
 
 ■24 
 
 055535 
 
 18-74 
 
 944465 
 
 3i 
 
 3o 
 
 18-45 
 
 99T99 
 9.997185 
 
 •24 
 
 o56659 
 
 18-70 
 
 943341 
 
 3o 
 
 3i 
 
 18-41 
 
 ■24 
 
 9-057781 
 058900 
 
 i8-65 
 
 10-942219 
 
 3 
 
 32 
 
 056671 
 
 18-36 
 
 997170 
 997166 
 
 • 24 
 
 18-69 
 18.55 
 
 941 100 
 
 33 
 
 057172 
 058271 
 
 i8-3i 
 
 •24 
 
 060016 
 
 939984 
 938870 
 
 2 I 
 
 34* 
 
 18*27 
 
 997 141 
 
 •24 
 
 061 i3o 
 
 i8-5i 
 
 26 
 
 35 
 
 059367 
 
 18-22 
 
 997 I2 7 
 
 •24 
 
 062240 
 
 ' 18-46 
 
 937760 
 
 25 
 
 36 
 
 060460 
 
 18-17 
 
 997112 
 
 ■24 
 
 063348 
 
 18-42 
 
 936652 
 
 3 
 
 n 
 
 - 06,1 55 1 
 
 i8-i3 
 
 997008 
 99708J 
 
 •24 
 
 064453 
 
 18-37 
 
 935547 
 
 062639 
 
 18-08 
 
 ■25 
 
 065556 
 
 i8-33 
 
 934444 
 
 22 
 
 39 
 
 063724 
 064806 
 
 18-04 
 
 997066 
 
 •25 
 
 066655 
 
 18-28 
 
 933345 
 
 21 
 
 40 
 
 17-99 
 
 997052 
 
 -25 
 
 067752 
 9- 068846 
 
 18-24 
 
 932248 
 
 20 
 
 41 
 
 9-o65885 
 
 17-94 
 
 9.997039 
 
 ■25 
 
 18-19 
 
 10.931154 
 
 w 
 
 42- 
 
 066962 
 
 ' n-jo 
 17-86 
 
 997024 
 
 •25 
 
 069938 
 
 i8-i5 
 
 930062 
 
 43 
 
 o68o36 
 
 997009 
 
 ■25 
 
 071027 
 072113 
 
 iS-IO 
 
 928973 
 927887 
 
 17 
 
 Ai 
 
 069107 
 
 17-8, 
 
 996994 
 
 ■25 
 
 18-06 
 
 16 
 
 45 
 
 070176 
 
 17-77 
 
 996979 
 
 ■25 
 
 073197 
 074278 
 
 18-02 
 
 92680.: 
 
 i5 
 
 46 
 
 071242 
 
 17-72 
 
 996964 
 
 •25 
 
 '7-97 
 
 925722 
 
 14 
 
 % 
 
 072306 
 
 17-68 
 
 996949 
 
 •25 
 
 075356 
 
 17-93 
 17. §9 
 
 924644 
 
 i3 
 
 073366 
 
 17-63 
 
 996934 
 
 •25 
 
 076432 
 
 923568 
 
 12 
 
 49 
 
 074424 
 
 17.59 
 
 996919 
 
 •25 
 
 07t5o£ 
 078576 
 
 17-84 
 
 922495 
 
 I] 
 
 5o 
 
 075480 
 
 i7-5o 
 
 996904 
 9.996889 
 
 •25 
 
 17-80 
 
 921424 
 
 10 
 
 5i 
 
 9-076533 
 
 i7-5o 
 
 ■25 
 
 9-079644 
 
 17-76 
 
 10.920356 
 
 \ 
 
 52 
 
 077583 
 078631 
 
 17.46 
 
 99687; 
 996868 
 
 •25 
 
 080710 
 
 17.72 
 
 919290 
 918227 
 
 53 
 
 17-42 
 
 ■25 
 
 of-:833 
 
 17-67 
 
 I 
 
 54 
 
 079676 
 080719 
 
 17-38 
 
 996843 
 
 ■25 
 
 17-63 
 
 9JJ167 
 
 ' 55 
 
 17-33 
 
 996826 
 
 ■25 
 
 o838gi 
 
 I^ 
 
 91610c 
 9i5o53 
 
 5 
 
 56 
 
 081759 
 
 17-29 
 17-25 
 
 996812 
 
 • 26 084947 
 
 4 
 
 57 
 
 082797 
 o8383a 
 
 9967971 .26| 086000 
 
 17 -5i 
 
 914000 3 
 
 58 
 
 17-21' 
 
 996782! -26! 087050 
 
 17-47 
 
 912950 2 
 
 5 9 
 
 084864 
 
 17-17 
 i 7 -ii 
 
 996766 -26 088098 
 
 17-43 
 
 911- J2 I 
 
 [ 66 
 
 085894 
 
 996751] -26! 089144 
 
 17-38 
 
 oio856| 
 
 1_ 
 
 Conine 
 
 D. 
 
 Sine l83°i Cotang. 
 
 1 D. 
 
 Tang. |m. 
 
8INM8 AND TANGENTS. (7 DEGREES.) 
 
 2fi 
 
 Jt. 
 
 o 
 
 Sine 
 
 D. 
 
 Cosine 
 
 D. 
 
 Tang. 
 
 D. 
 
 Co tang. 
 
 1 
 
 9-085894 
 
 17- 13 
 
 9-996751 
 
 ■ 26 
 
 9-089144 17-38 
 
 10-910856 
 
 60 
 
 I 
 
 086922 
 
 17-09 
 
 996735 
 
 ■26 
 
 09018- 
 
 • 7-34 
 
 9008 iC 
 
 5 9 
 
 2 
 
 ' 087947 
 088970 
 
 17-04 
 
 996720 
 
 •26 
 
 1 09122s 
 
 i7-3o 
 
 908775 
 
 58 
 
 3 
 
 17-00 
 
 996704 
 
 •26 
 
 092266 
 
 17.27 
 
 907734 
 
 57 
 
 4 
 
 089990 
 
 16-96 
 
 996688 
 
 •26 
 
 09330s 
 
 17-22 
 
 90669s 
 
 5b 
 
 5 
 
 091008 
 
 16-92 
 
 996673 
 
 •26 
 
 094336 
 
 17-19 
 17-13 
 
 oo566/ 
 
 55 
 
 6 
 
 092024 
 
 16-88 
 
 996607 
 
 ■26 
 
 09536- 
 
 90463; 
 
 54 
 
 2 
 
 093037 
 
 16-84 
 
 996641 
 
 •26 
 
 09639! 
 
 17- ii 
 
 90360! 
 
 53 
 
 094047 
 
 16-80 
 
 996625 
 
 ■26 
 
 097422 
 
 17-07 
 
 90257! 
 
 52 
 
 9 
 
 095o56 
 
 16-76 
 
 996610 
 
 ■26 
 
 098446 
 
 I7*o3 
 
 901554 
 
 5i 
 
 10 
 
 096062 
 
 16.73 
 
 996694 
 
 ■26 
 
 099468 
 
 16-99 
 
 900S3; 
 
 5o 
 
 II 
 
 9- 097065 
 09,8066 
 
 16.68 
 
 9*996578 
 
 •27 
 
 9.100487 
 
 16-95 
 
 10-89951; 
 
 t 
 
 12 
 
 16-65 
 
 996562 
 
 •27 
 
 I0l5o! 
 
 16-91 
 16-87 
 
 898496 
 897481 
 
 i3 
 
 099065 
 
 16-61 
 
 996546 
 
 •27 
 
 I025lC 
 
 47 
 
 '.4 
 
 100062 
 
 16-67 
 
 996530 
 
 ■27 
 
 io3532 
 
 16-84 
 
 896468 
 
 46 
 
 i5 
 
 ioio56 
 
 i6-53 
 
 996514 
 
 •27 
 
 104542 
 
 16-80 
 
 8 9 5458 
 
 45 
 
 16 
 
 102048 
 
 16-49 
 
 .996498 
 
 ■27 
 
 io555o 
 
 16-76 
 
 894450 
 
 44 
 
 \\l 
 
 io3o37 
 
 i6-45 
 
 996482 
 
 ■27 
 
 Io6556 
 
 16-72 
 
 893444 
 
 43 
 
 104025" 
 
 16-41 
 
 996465 
 
 •27 
 
 107559 
 
 16-69 
 16-65 
 
 892441 
 
 42 
 
 J '9 
 
 ioSoio 
 
 16-38 
 
 996449 
 996433 
 
 ■27 
 
 io856o 
 
 891440 
 
 41 
 
 20 
 
 106992 
 
 16-34 
 
 ■27 
 
 109559 
 
 16-61 
 
 800441 
 
 40 
 
 21 
 
 9' 106973 
 
 i6-3o 
 
 9-996417 
 
 •27 
 
 9-iio556 
 
 16-58 
 
 io- 889444 
 888449 
 
 3 9 
 
 22 
 
 107951 
 
 16-27 
 
 996400 
 
 ■27 
 
 1 1 1 55i 
 
 i6-54 
 
 38 
 
 23 
 
 108927 
 
 16-23 
 
 996384 
 
 •27 
 
 1 12543 
 
 16 -5o 
 
 887467 
 
 37 
 
 24 
 
 1 0990 1 
 
 16-19 
 
 996368 
 
 ■27 
 
 1 13533 
 
 16-46 
 
 - 886467 36 
 
 25 
 
 1 10873 
 
 16-16 
 
 996351 
 
 •27 
 
 1 1452 1 
 
 16-43 
 
 885479! 35 
 8844931 34, 
 
 26 
 
 1 1 1842 
 
 16-12 
 
 996335 
 
 • 27 
 
 n55o7 
 
 i6-3 9 
 
 3 
 
 I 1 2809 
 
 16-08 
 
 996318 
 
 :ll 
 
 116491 
 
 16-36 
 
 883509! 33 
 
 1 13774 
 
 i6-o5 
 
 996302 
 
 1 17472 
 1 18452 
 
 16-32 
 
 88252o| 32 
 
 39 
 
 1U737 
 1 1 56n8 
 
 16-01 
 
 996285 
 
 • 28 
 
 16-29 
 
 16-25 
 
 88i548i, 3i 
 
 3o 
 
 i5-97 
 
 996269 
 
 • 28 
 
 ■ 1 1 94^9 
 
 880671 1 3o 
 
 3i 
 
 9-116656 
 
 15-94 
 
 9.996252 
 
 .28 
 
 9 • 1 20404 
 
 16-22 
 
 10-879596! 29 
 8786231 28 
 
 32 
 
 117613 
 
 - i5-9o 
 
 996235 
 
 • 28 
 
 121377 
 
 16-18 
 
 33 
 
 1 18567 
 
 i5-o 7 
 
 996219 
 
 ■ 28 
 
 122348 
 
 i6-i5 
 
 877652! 27" 
 
 34 
 
 119519 
 
 15-83 
 
 996202 
 
 -28 
 
 I233I7 
 
 16-n 
 
 876683; 26 
 
 35 
 
 120469' 
 
 i5-8b 
 
 99&I85 
 
 ■ 28 
 
 124284 
 
 16-07 
 
 875716, 25 
 
 36 
 
 121417 
 
 i5- 7 6 
 
 J996168 
 
 •28 
 
 125249 
 
 i6'-o4 
 
 874751 1 24 
 
 u 
 
 122362 
 
 i5- 7 3 
 
 996151 
 
 • 28 
 
 12620 
 
 16-01 
 
 873789' 23 
 872828, 22 
 
 i233o6 
 
 15-69 
 
 996134 
 
 • 28 
 
 ' 127172 
 I28i3o 
 
 1-5-97 
 
 3 9 
 
 124248 
 
 15-66 
 
 9961 17 
 
 •28 
 
 15-94 
 
 87 1 870 21 
 
 40 
 
 125187 
 
 15-62 
 
 996100 
 
 ■28 
 
 1 29087 
 
 i5-gi 
 
 870913 20 
 
 4i 
 
 9-126125 
 
 i5-59 
 
 9.996083 
 
 • 29 
 
 9-i3oo4i 
 
 i5-»7 
 
 10-869959 1 
 869006 IB 
 
 868o56' 17 
 867107 16 
 
 . 42 
 
 127060 
 
 15-56 
 
 996066 
 
 ■ 2 9 
 
 j 3,0994 
 
 15-84 
 
 43 . 
 
 127993 
 I23q25 
 
 15-52 
 
 996049 
 
 •29 
 
 131944 
 
 . i328 9 3 
 
 i3383 9 
 
 i5-8i 
 
 i 44 
 
 i5-49 
 i5-45 
 
 996032 ' 
 
 •29 
 
 15.77 
 
 45 
 
 1 29854 
 
 996016 
 
 ■ 29 
 
 15-74 
 
 866161 
 
 i5 
 
 46 
 
 130781 
 
 15-42 
 
 99599S 
 
 •29 
 
 134784 
 
 I5-7I 
 
 i5-6 7 
 
 8652i6 
 
 14 
 
 48 
 
 131706 
 
 i5-3 9 
 15-35 
 
 995980 
 
 •29 
 
 535726 
 
 864274 
 
 i3 
 
 I32WO 
 
 995963 
 
 •29 
 
 136667 
 
 15-64 
 
 863333 
 
 12 
 
 49 
 
 l3355i 
 
 15-32 
 
 995946 
 
 •29 
 
 137605 
 138542 
 
 i5-6i 
 
 8623951 11 
 86U58 10 
 
 56 
 
 134470 
 
 ■ 5-29 
 
 991928 
 
 • 2 9 
 
 15-58 
 
 5i 
 
 9-I35337 
 
 15-25 
 
 9-995911 
 995894 
 
 ■29 
 
 9-139476 
 
 15-55 
 
 io-86o524 
 
 8 
 
 52 
 
 i363o3 
 
 15-22 
 
 ■29 
 
 140409 
 
 i5-5i 
 
 8595gi 
 85866o 
 
 53 
 
 i3t2i6 
 
 i5- 19 
 
 995876 
 
 •29 
 
 1 41 34o 
 
 i5-48 
 
 7 
 
 54 
 
 i33i2'-i 
 
 i5-i6 
 
 995859 
 
 ■ 2 9i 
 
 142269 
 
 i5-45 
 
 857731 
 856804 
 
 6 
 
 55 
 
 139037 
 
 1 5- 1 2 
 
 996841 
 
 •29! 
 
 143196 
 
 i5-42 
 
 5 
 
 56 
 
 139944 
 
 i5-09 
 
 995823 
 
 •29 
 
 I44I2I 
 
 15-39 
 
 855879 4 
 
 11 
 
 140850 
 
 i5-ob 
 
 995806 
 
 •29 
 
 I45o44; 
 
 15-35 
 
 854966; 3 
 
 141754 
 
 i5.o3 
 
 995-788 
 
 • 29 
 
 I45g66 
 
 15-32 
 
 854o34 ■ 2 
 
 5 9 
 
 142655 
 
 15-00 
 
 996771 
 
 ■29 
 
 146885 
 
 I5.20 
 
 853u5 
 
 I 
 
 ;.™_ 
 
 143555 
 
 14-96 
 
 995753I 
 
 •2 9 | 
 
 l47So3| 
 
 15-26 
 
 862197 
 
 |, 
 
 Cosine J 
 
 D. 
 
 Sine i82°l 
 
 Cotang. 1 
 
 x>. 
 
 Tang. 
 
 M. | 
 
26 
 
 TIT 
 
 (8 DEGREES.) A TABLE OF LOGARITHMIC 
 
 "Sine 
 
 o !y 
 
 D. 
 
 20 
 
 21 
 22 
 23 
 21 
 25 
 26 
 
 11 
 
 3 9 o 
 3i 
 32 
 
 33 
 31 
 35 
 36 
 
 ll 
 
 39 
 4o 
 H 
 it 
 
 a 
 
 44 
 45 
 46 
 
 % 
 
 i 9 
 
 5o 
 5i 
 52 
 53 
 54 
 55 
 56 
 
 ll 
 
 I 43555 
 144453 
 145349 
 146213 
 147'36 
 148026 
 1 48915 
 149802 
 1 5o686 
 i5i 369 
 1 5245 1 
 l5333o 
 1 54208 
 i55o83 
 155957 
 1 5683o 
 157700 
 i5856o 
 159435 
 i6o3oi 
 161164 
 9-162025 
 162885 
 163.713 
 164600 
 i65l54 
 166307 
 167159 
 168008 
 168856 
 169702 
 170047 
 171389 
 172230 
 173070 
 173908 
 171714 
 175578 
 176411 
 177242 
 178072 
 •178900 
 179726 
 i8o55i 
 i8[37l 
 182196 
 i83oi6 
 183831 
 1 8465 1 
 185466 
 186280 
 • 187092 
 187903 
 188712 
 189519 
 190325 
 191 i3o 
 191933 
 192734 
 193534 
 19433a 
 
 Cosine D. I Tang. 
 
 Oos'me 
 
 14-9° 
 14-93 
 14-90 
 14-87 
 U-84 
 14-81 
 14-78 
 14-75 
 U-72 
 14-69 
 U-66 
 U-63 
 14-60 
 U-57 
 14-54 
 i4-5i 
 14-48 
 14-45 
 14-42 
 14-39 
 U-36 
 14-33 
 i4-3o 
 14-27 
 1 1-24 
 
 U-22 
 14-19 
 
 14-16 
 U-i3 
 14- 10 
 14-07 
 l4-o5 
 14-02 
 13-99 
 13-96 
 i3-9l 
 i3-gi 
 13-88 
 13-86 
 13-83 
 i3-8o 
 i3-77 
 i3-7l 
 
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 13-03 
 
 780290 
 
 35 
 
 26 
 
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 779508 
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 34 
 
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 33 
 
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 12-90 
 
 10-775618 
 
 3 
 
 32 
 
 219116 
 
 12-53 
 
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 225i56 
 
 12-88 
 
 774844 
 
 33 
 
 219868 
 
 1:2 -50 
 
 99 3 9 3 9 
 993918 
 
 •35 
 
 225929 
 
 12-86 
 
 774071 
 
 27 
 
 34 
 
 220618 
 
 12-48 
 
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 226700 
 
 12-84 
 
 7733oo 
 
 26 
 
 35 
 
 22l367 
 222115 
 
 12-46 
 
 993896 
 
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 227471 
 228239 
 
 12-8l 
 
 772529 
 
 25 
 
 36 
 
 12-44 
 
 993875 
 
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 12-79 
 
 771761 
 
 24 
 
 ll 
 
 222861 
 
 12-42 
 
 993854 
 
 ■ 36 
 
 229007 
 
 12-77 
 
 770993 
 
 23 
 
 2236o6 
 
 12-39 
 
 993832 
 
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 229773 
 
 12-75 
 
 770227 
 
 22 
 
 3 9 
 
 224349 
 
 12-37 
 
 99381 1 
 
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 23oo39 
 
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 769461 
 
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 21 
 
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 225092 
 
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 12-71 
 
 20 
 
 41 
 
 12-33 
 
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 12-69 
 
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 42 
 
 226573 
 
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 993746 
 
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 232826 
 
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 767174 
 
 43 
 
 22?3i 1 
 
 12-28 
 
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 233586 
 
 766414 
 
 \l 
 
 44 
 
 228048 
 
 12-26 
 
 993703 
 
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 234345 
 
 12-62 
 
 7 65655 
 
 45 
 
 228784 
 
 12-24 
 
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 235io3 
 
 I2-6o 
 
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 i5 
 
 46 
 
 229518 
 
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 14 
 
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 12-56 
 
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 237368 
 
 238120 
 
 12-54 
 
 762632 
 
 12 
 
 49 
 
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 I2-l6 
 
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 12-52 
 
 761880 
 
 11 
 
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 238872 
 
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 10 
 
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 12-12 
 
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 52 
 
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 53 
 
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 54 
 
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 55 
 
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 242610 
 
 12-40 
 
 757390 
 
 5 
 
 56 
 
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 12-01 
 
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 243354 
 
 12-38 
 
 756646 
 
 4 
 
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 11-99 
 
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 3 
 
 238235 
 
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 2 
 
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 238o53 
 239670 
 
 1195 
 
 993374 
 
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 245579 
 
 12-32 
 
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 1 
 
 66 
 
 11-93 
 
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 246319 
 
 12-30 
 
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 b 
 
 
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 Sine 
 
 80° 
 
 Co tans- 
 
 D. 
 
 Tang. 
 
 il-J 
 
 
 r— 
 
 
 
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28 
 
 (10 
 
 DEGREES.) A 
 
 TABLE OF LOGARITHMIC 
 
 
 M. 
 
 Bine 
 
 D. 
 
 Cosine 
 
 D. 
 
 Tang. 
 
 D. 
 
 Cotang. 
 10-753687 
 
 «l 
 
 
 
 9-239670 
 240386 
 
 I.. 9 3 
 
 9-99335i 
 
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 9-246319 
 
 12 -3o 
 
 i 
 
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 993329 
 
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 247057 
 
 12-28 
 
 752943 
 
 n 
 
 2 
 
 241 101 
 
 993307 
 
 ■? 7 
 
 247794 
 
 12-26 
 
 752206 
 
 3 
 
 241814 
 
 II.87 
 
 993285 
 
 •37 
 
 24853o 
 
 12-24 
 
 751470 
 750736 
 
 57 
 
 4 
 
 242526 
 
 11 -85 
 
 993262 
 
 •37 
 
 249264 
 
 12-22 
 
 56 
 
 5 
 
 243237 
 
 H-83 
 
 993240 
 
 :ll 
 
 24999 s 
 25o-j3o 
 
 12-20 
 
 750002 
 
 55 
 
 6 
 
 243947 
 
 11. 81 
 
 993217 
 
 I2-I8 
 
 749270 
 
 54 
 
 I 
 
 244.656 
 
 H-79 
 
 993195 
 
 • 38 
 
 25i46i 
 
 12-17 
 
 748539, 53 
 
 245363 
 
 W.]l 
 
 993172 
 
 ■ 38 
 
 252191 
 
 I2-l5 
 
 747809 
 
 5a 
 
 9 
 
 246069 
 24677^5 
 
 993149 
 
 ■38 
 
 252920 
 
 I2-I3 
 
 747080 
 
 5i 
 
 10 
 
 n-73 
 
 993127 
 
 • 38 
 
 253648 
 
 12-11 
 
 746352 
 
 5o 
 
 ii 
 
 q- 247478 
 248181 
 
 11-71 
 
 9-993104 
 
 •38 
 
 9-254374 
 
 12 09 
 
 10-745626 
 
 49 
 
 12 
 
 11-69 
 
 993081 
 
 •38 
 
 255ioo 
 
 12-07 
 
 744900' 48 
 
 -i3 
 
 248883 
 
 11-67 
 
 993059 
 
 ■38 
 
 255824 
 
 12 o5 
 
 744176 
 743453 
 
 47 
 
 14 
 
 249583 
 
 u-65 
 
 993o36 
 
 •38 
 
 256547 
 
 1203 
 
 46 
 
 o 
 
 2502L2 
 
 n-63 
 
 9930 1 3 
 
 •38 
 
 257269 
 
 12-01 
 
 742731 
 
 45 
 
 16 
 
 250980 
 
 n-6i 
 
 992990 
 
 ■38 
 
 257990 
 258710 
 
 12-00 
 
 742010 
 
 44 
 
 17 
 
 251677 
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 n-5g 
 
 992967 
 
 •38 
 
 II.98 
 
 7412901 43 
 
 i3 
 
 n-58 
 
 992044 
 
 •38 
 
 259429 
 
 II.96 
 
 740571 42 
 
 '9 
 
 253067 
 
 u-56 
 
 992921 
 
 • 38 
 
 260146 
 
 n-94 
 
 739854 
 
 41 
 
 20 
 
 253761 
 
 n-54 
 
 992898 
 
 ■ 38 
 
 26o863 
 
 11-92 
 
 739i3 7 
 
 40 
 
 21 
 
 9-254453 
 
 n-52 
 
 9-992875 
 
 •38 
 
 9-261578 
 
 11 -90 
 
 10-738422 
 
 39 
 
 22 
 
 255i44 
 
 11 5o 
 
 9928D2 
 
 •38 
 
 262292 
 
 11-89 
 
 737708 
 
 38 
 
 23 
 
 255S34 
 
 n-48 
 
 992829 
 
 • 3g 
 
 263oo5 
 
 11.87 
 
 736995 
 
 ll 
 
 24 
 
 256523 
 
 n-46 
 
 992806 
 
 •39 
 
 263717 
 264428 
 
 n-85 
 
 736283 
 
 23 
 
 257211 
 
 n-44 
 
 9927S3 
 
 -3 9 
 
 n-83 
 
 735572 
 
 35 
 
 26 
 
 257898 
 258583 
 
 11-42 
 
 992759 
 
 -3 9 
 
 2 65i38 
 
 11-81 
 
 734862 
 
 34 
 
 11 
 
 U-4I 
 
 992736 
 
 -3 9 
 
 265847 
 
 Il:?8 
 
 734153 33 
 
 25 9 268 
 
 ii-3 9 
 
 992713 
 
 -3 9 
 
 266555 
 
 733445 
 
 32 
 
 ? 
 
 25 99 5i 
 
 1. -3 7 
 
 992690 
 
 •39 
 
 267261 
 
 n.76 
 
 732739 
 732033 
 
 3i 
 
 3o 
 
 26o633 
 
 n-35 
 
 992666 
 
 •3g 
 
 267967 
 9-268671 
 
 n-74 
 
 3o 
 
 3i 
 
 9-261314 
 
 n-33 
 
 0-99^043 
 
 -3 9 
 
 11-72 
 
 io-73i32o 
 730623 
 
 3 
 
 32 
 
 261994 
 
 u-3i 
 
 992619 
 
 -3 9 
 
 269375 
 
 11-70 
 
 33 
 
 262673 
 263351 
 
 n-3o 
 
 992596 
 
 ■39 
 
 270077 
 
 11-69 
 
 729923 27 
 
 34 
 
 11-28 
 
 992572 
 
 •39 
 
 270779 
 
 11.67. 
 
 II-OJ 
 
 7202_2l| 26 
 
 728321! 25 
 
 35 
 
 264027 
 26470J 
 
 11-26 
 
 992549 
 
 •39 
 
 271479 
 
 36 
 
 11-24 
 
 992320 
 
 •39 
 
 272178 
 
 n-64 
 
 727822! 24 
 
 ll 
 
 265377 
 
 11-22 
 
 9925oI 
 
 -3g 
 
 272876 
 
 11-62 
 
 727124 23 
 
 266o5 1 
 
 11-20 
 
 992478 
 
 •40 
 
 273573 
 
 11 -6o 
 
 726427 22 
 
 3 9 
 
 266723 
 
 11-19 
 
 992454 
 
 ■40 
 
 274269 
 
 n-58 
 
 72573l 
 
 21 
 
 4o 
 
 267395 
 9.268065 
 
 WM 
 
 992430 
 
 •40 
 
 274964 
 
 n-5 7 
 
 725o36 
 
 20 
 
 41 
 
 9-992406 
 
 •40 
 
 9-275658 
 
 11-55 
 
 10-724342 
 
 IS 
 
 42 
 
 268734 
 
 11 - 13 
 
 9923S2 
 
 •40 
 
 27635i 
 
 n-53 
 
 723649 
 
 43 
 
 269402 
 
 1 1 - 1 1 
 
 992359 
 
 •40 
 
 2-7043 
 
 n-5i 
 
 722957 
 
 17 
 
 4> 
 
 270069 
 270735 
 
 11-10 
 
 992333 
 
 •40 
 
 277734 
 278424 
 
 n-5o 
 
 722266 
 
 16 
 
 45 
 
 11-08 
 
 9923l 1 
 
 • 40 
 
 11,48 
 
 721576 
 
 720887 
 
 i5 
 
 46 
 
 27:400 
 
 nc6 
 
 992287 
 992263 
 
 • 4o 
 
 279113 
 
 n-47 
 
 14 
 
 % 
 
 272064 
 
 n-o5 
 
 • 40 
 
 279801 
 280488 
 
 u-45 
 
 720199 
 
 i3 
 
 272726 
 
 11 -o3 
 
 992239 
 
 •40 
 
 n-43 
 
 719612 
 
 12 
 
 i 9 
 
 273388 
 
 11-01 
 
 992214 
 
 •40 
 
 281174 
 
 1 1 -4i 
 
 718826 
 
 11 
 
 5o 
 
 274049 
 
 9 274708 
 
 275367 
 
 10.99 
 10-98 
 
 992190 
 
 •40 
 
 28i858 
 
 ii-4o 
 
 718142 
 
 10" 
 
 5i 
 
 9-992166 
 
 ■40 
 
 9-282542 
 
 n-38 
 
 10 717458 
 
 I 
 
 5: 
 
 10-96 
 
 992142 
 
 ■40 
 
 283225 
 
 n-36 
 
 716775 
 
 53 
 
 276024 
 
 •10-94 
 
 992117 
 
 •41 
 
 283907 
 184588 
 
 1 1 -35 
 
 716093 
 
 7 
 
 54 
 
 276681 
 
 10-92 
 
 992093 
 
 •41 
 
 n-33 
 
 715412 
 
 6 
 
 55 
 
 277337 
 
 IO-QI 
 
 992069 
 
 •41 
 
 285268 
 
 ii-3i 
 
 714732 
 
 5 
 
 56 
 
 277991 
 278644 
 
 IO-8 9 
 
 992044 
 
 •41 
 
 385947 
 286624 
 
 n-3o 
 
 7i4o53 
 
 4 
 
 5 7 
 
 IO-87 
 
 992020 
 
 ■41 
 
 11-28 
 
 713376 
 
 3 
 
 5B 
 
 279297 
 
 279948 
 
 280599 
 
 CoBJr.s 
 
 10-86 
 
 99I996 
 
 ■41 
 
 287301 
 
 II-26 
 
 7 1 2693 
 
 3 
 
 !5 
 
 1084 
 
 991971 
 
 ■41 
 
 287977 
 
 11-23 
 
 7 1 202 j 
 
 1 
 
 60 
 
 10-82 
 
 991947 
 
 Siiio 
 
 ■41 
 793 
 
 388652 
 Cotimg." 
 
 n-23 
 
 7II348 
 
 
 
 a 
 
 Tnug. 
 
 ». 
 

 8IKES AND TANGENTS 
 
 (11 DEGUEES. 
 
 ) 
 
 ■'M. 
 
 
 
 Sine 
 
 D. 
 
 1 Cosine | D. 
 
 1 Tang. 
 
 D. 
 
 Cotang. 
 
 9 ■ 280599 
 
 10-82 
 
 9-991917 -4i 
 
 9-288652 
 
 11-23 
 
 10-711348 
 
 I 
 
 281248 
 
 10-81 
 
 991922 
 
 ■41 
 
 289326 
 
 11-22 
 
 710674 
 
 2 
 3 
 
 281897 
 
 10-79 
 
 991897 
 
 ■41 
 
 289999 
 
 11-20 
 
 710001 
 
 282544 
 
 10.77 
 
 991873 -4i 
 
 290671 
 
 11-18 
 
 709329 
 
 4 
 5 
 6 
 
 283190 
 283836 
 
 10-76 
 
 991S48I -4i 
 
 291342 
 
 11-17 
 
 708658 
 
 10-74 
 
 991823I -4i 
 
 2920K 
 
 ii-i5 
 
 707987 
 
 284480 
 
 10-72 
 
 991799 -4i 
 
 292682 
 
 11-14 
 
 707318 
 
 I 
 
 285i24 
 
 10-71 
 
 991774] -42 
 
 293350 
 
 II -12 
 
 7o665o 
 
 285766 
 
 10-69 
 
 991749 -42 
 
 29 10 1 7 
 
 II-II 
 
 705983 
 7o53 16 
 
 9 
 
 286408 
 
 10-67 
 
 991724, -42 
 
 29 ',684 
 
 11-09 
 
 10 
 
 287048 
 
 10-66 
 
 991699 
 
 ■42 
 
 29534c 
 9-29601^ 
 
 11-07 
 
 704651 
 
 ii 
 
 9-287687 
 
 10-64 
 
 9-99167! 
 
 ■42 
 
 11 -06 
 
 10-703987 
 703323 
 
 12 
 
 288326 
 
 10-63 
 
 991649 
 
 ■42 
 
 296677 
 
 u-o4 
 
 i3 
 
 288964 
 
 10-61 
 
 991624 
 
 •42 
 
 297339 
 
 n-o3 
 
 702661 
 
 14 
 
 * 289600 
 
 io-5n 
 
 991599 
 
 •42 
 
 291001 
 
 11-01 
 
 701999 
 
 i5 
 16 
 
 2Q0236 
 
 io-58 
 
 991674 
 
 •42 
 
 2g%62 
 
 I 1 -00 
 
 701338 
 
 .290870 
 
 io-56 
 
 991549 
 
 •42 
 
 299322 
 
 10-98 
 
 700678 
 
 \l 
 
 291504 
 
 10-54 
 
 991524 
 
 ■42 
 
 299980 
 
 10-96 
 
 700020 
 
 292137 
 292768 
 
 10-53 
 
 99149? 
 
 ■42 
 
 - 3oo638 
 
 io-t;5 
 
 699362 
 
 ig 
 
 IO-?I 
 
 991473 
 
 ■42 
 
 3oi2o5 
 301901 
 
 10-93 
 
 698705 
 
 20 
 
 293399 
 
 I0-5o 
 
 991448 
 
 ■42 
 
 10-92 
 
 698049 
 
 :o -697393 
 
 696739 
 
 21 
 
 9-294029 
 294658 
 
 10-48 
 
 9-991422 
 
 ■42 
 
 9-302607 
 
 16-90 
 
 22 
 
 10-46 
 
 991397 
 
 ■42 
 
 3o326i 
 
 10-89 
 
 23i 
 
 295286 
 
 / io-45 
 
 991372 
 
 ■43 
 
 " 3o3ni4 
 
 jo- 87 
 
 696086 
 
 24 
 
 295qi3 
 296039 
 
 10-43 
 
 991346 
 
 ■43 
 
 3o4»7 
 3o52i8 
 
 10-86 
 
 695433 
 
 25 
 
 10-42 
 
 991321 
 
 ■43 
 
 10-84 
 
 694782 
 
 26 
 
 297164 
 
 10-40 
 
 991295 
 
 •43 
 
 3o586 9 
 
 10-83 
 
 694i3r 
 
 11 
 
 297788 
 
 10-39 
 
 991270 
 
 ■43 
 
 3o65iq 
 
 io-8i 
 
 6 9 348i 
 
 298412 
 
 10-37 
 
 991244 
 
 ■43 
 
 307168 
 
 10-80 
 
 692832 
 
 I 9 
 
 299034 
 
 10-36 
 
 991218 
 
 ■43 
 
 3078 1 5 
 3o8463 
 
 10-78 
 
 692185 
 
 3o 
 3i 
 
 299655 
 
 10-34 
 
 991 193 
 
 ■4? 
 
 10-77 
 
 691537 
 
 9-300276 
 
 10-32 
 
 9-991167 
 
 ■43 
 
 9-309109 
 
 10-75 
 
 10-690891 
 
 32 
 
 300895 
 
 io-3i 
 
 991 141 
 
 ■43 
 
 309754 
 
 10-74 
 
 600246 
 
 ' 33 
 
 3oi5i4 
 
 I0-2O 
 
 991 1 i5 
 
 ■43 
 
 3 1 0398 
 
 10-73 
 
 689602 
 
 34 
 
 302132 
 
 10-28 
 
 991090 
 
 •43 
 
 3uo42 
 
 10-71 
 
 688958 
 6883 i5 
 
 35 
 
 302748 
 
 10-26 
 
 991064 
 
 •43 
 
 3n685 
 
 10-70 
 
 36 
 
 3o3364 
 
 10-25 
 
 99io38 
 
 •43 
 
 3i2327 
 
 10.68 
 
 687673 
 
 12 
 
 - 3o3o79 
 
 10-23 
 
 991012 
 
 • 43 
 
 312967 
 
 10-67 
 
 687033 
 
 ■304093 
 
 10-22 
 
 990986 
 
 •43 
 
 3i36o8 
 
 10-65 
 
 6863 9 2 
 
 3 9 
 
 3o5207 
 
 10-20 
 
 990960 
 
 ■43 
 
 • 3 14247 
 
 10-64 
 
 685 7 53 
 
 40 
 
 3o58i9 
 
 10-19 
 
 990934 
 
 ■44 
 
 3 14885 
 
 16-62 
 
 685u5 
 
 ' 41 
 
 9"3o643o 
 
 10-17 
 
 9 • 990908 
 
 •44 
 
 9-3i5523 
 
 10-61 
 
 10-684477 
 
 42 
 
 307041 
 
 10-10 
 
 990882 
 
 ■44 
 
 3i6i5q 
 316795 
 3 17430 
 
 Io-6o 
 
 683841 
 
 43 
 
 307650 
 308259 
 
 10-14 
 
 99oS55 
 
 •44 
 
 10-58 
 
 683205 
 
 44 
 
 io-i3 
 
 990829 
 990803 
 
 •44 
 
 10-57 
 
 682570 
 
 45 
 
 308867 
 
 IO-II 
 
 •44 
 
 318064 
 
 10-55 
 
 681936 
 
 46 
 
 309474 
 3 1 0080 
 
 10-10 
 
 990777 
 
 •44 
 
 318697 
 
 10-54 
 
 68i3o3 
 
 2 
 
 10-08 
 
 990750 
 
 •44 
 
 319329 
 
 io-53 
 
 680671 
 
 3 io685 
 
 10-07 
 
 990724 
 
 ■44 
 
 319061 
 
 320092 
 
 io-5i 
 
 68oo3o 
 679408 
 678778 
 
 i 9 
 
 311289 
 3 1 1893 
 
 io-o5 
 
 990697 
 
 •44 
 
 io-5o 
 
 5o 
 
 10-04 
 
 99067 1 
 
 •44 
 
 321222 
 
 10-48 
 
 5i 
 
 9-3 12495 
 
 10 -o3 
 
 9-990644 
 
 ■44 
 
 9-32i85i 
 
 10-47 
 
 10-678149 
 
 52 
 
 313097 
 313698 
 
 10-01 
 
 990618 
 
 •44 
 
 322479 
 
 10-45 
 
 677521 
 
 53 
 
 10-00 
 
 990591 
 
 ■44 
 
 323 106 
 
 10-44 
 
 676894 
 
 54 
 
 314297 1 
 
 9-98 
 
 990565 -44 
 990538 .44 
 99o5ii| -45 
 
 323733 
 
 10-43 
 
 676267 
 
 55 
 
 314897 
 
 9'97 
 
 324358 
 
 io-4i 
 
 675642 
 
 56 
 
 31549D 
 
 9-96 
 
 32jqS3 
 
 10-40 
 
 675017 
 
 57 
 
 316092 
 
 9-94 
 
 990485! -45 
 
 3256o-7 
 
 10-39 
 
 674393 
 673769 
 
 58 
 
 316689 
 
 9-93 
 
 990458] -45 
 
 32623! 
 
 io-37 
 
 5 9 
 
 317284 
 
 991 
 
 99043 1 : -45 
 
 3268^3 
 
 10-36 
 
 673147 
 
 60 
 
 317879 
 
 9-90 
 
 990404 -45 
 
 327475 
 
 io-35 
 
 672525 
 
 
 Cosine 
 
 D. 
 
 Sine 78° 
 
 Cotang. 
 
 L>. 
 
 Tang. 
 
 29 
 
 1 60 
 
 5o 
 
 Mr. ' 
 
30 
 
 (12 DEGREES.) A 
 
 TABLE OF LOGARITHMIC 
 
 
 M. 
 
 i Sine 
 
 D. 
 
 Cosine .| D. 1 Tung-. 
 
 D. 
 
 Cotang 
 
 . 1 
 
 
 
 9-317879 
 3i8473 
 
 9.90 
 9-88 
 
 9-990404 -45 9-32747 
 
 i 10-35 
 
 10672526 60 
 
 i 
 
 990378 1 -45 32S095 io-33 
 
 671906! 5o 
 671285 58 
 
 2 
 
 319066 
 
 9-87 
 
 99035 
 
 -45 32871 
 
 ) 10-32 
 
 3 
 
 3 19638 
 
 9:86 
 
 990324 -45 329334 io-3o 
 
 670666 57 
 
 4 
 
 320249 
 
 9.84 
 
 990207 -4! 
 
 32995. 
 33o5tc 
 
 10-20 
 
 67004- 
 
 56 
 
 5 
 
 320840 
 
 9-83 
 
 99027c 
 
 -4; 
 
 10 28 
 
 66943c 
 66881 3 
 
 55 
 
 6 
 
 32i43o 
 
 9-82 
 
 99024; 
 
 •4! 
 
 3311ft- 
 
 ios6 
 
 54 
 
 I 
 
 322019 
 
 9-80 
 
 99021' 
 
 •4' 
 
 33i8oC 
 
 10-25 
 
 66819- 
 66 7 58s 
 
 53 
 
 322607 
 
 9-79 
 
 99018? 
 
 •4c 
 
 33241? 
 
 10-24 
 
 52 
 
 1 9 
 
 323ro4 
 
 9-77 
 
 990161 
 
 ■45 333o3^ 
 
 10-23 
 
 666967 
 666354 
 
 5i 
 
 1 10 
 
 323780 
 
 9-76 
 
 990 1 3^ 
 
 •45 33364C 
 
 10-21 
 
 So 
 
 • ii 
 
 9-324366 
 
 9.75 
 
 9-99010- 
 
 ■46 9-334255 
 
 10-20 
 
 10-665741 
 
 % 
 
 12 
 
 32495o 
 
 9- 7 3 
 
 990079 
 
 •46 
 
 334871 
 
 lo- 19 
 
 665 12c 
 
 i3 
 
 325034 
 
 9-72 
 
 990652 
 
 •46 
 
 335482 
 
 10-17 
 
 6645i( 
 
 s 
 
 14 
 
 3261 17 
 
 9-70 
 
 • 990025 
 989997 
 
 ■46 
 
 33609c 
 
 10- 16 
 
 66390- 
 663 298 
 662689 
 
 i3 
 
 326700 
 
 5.69 
 
 ■46 
 
 336702 
 
 1 • 1 5 
 
 45 
 
 16 
 
 327281 
 
 9. 6ft 
 
 989970 
 
 ■46 
 
 3373 1 1 
 
 io-i3 
 
 44 
 
 \l 
 
 327862 
 
 9-66 
 
 989942 
 
 •46 
 
 337919 
 338527 
 
 10-12 
 
 662081 
 
 43 
 
 328442 
 
 9-65 
 
 989915 
 
 •46 
 
 10-11 
 
 66i473 
 
 4» 
 
 >9 
 
 32902! 
 
 9.64 
 
 989887 
 
 ■46 
 
 33gi33 
 
 10-10 
 
 660867 
 
 41 
 
 20 
 
 329599 
 
 9-62 
 
 989860 
 
 ■46 
 
 339735 
 
 10-08 
 
 660261 
 
 40 
 
 21 
 
 9-330176 
 
 9-61 
 
 9-989832 
 
 •46 
 
 9-34o344 
 
 10-07 
 
 10-659656 
 
 18 
 
 :2 
 
 33oi53 
 
 o- 60 
 
 989804 
 
 •46 
 
 340948 
 
 io-o6 
 
 65oo52 
 
 23 
 
 33i3:o 
 33190J 
 
 9-58 
 
 989777 
 
 ■46 
 
 341552 
 
 10-04 
 
 658448 
 
 37- 
 
 2i 
 
 9 -5 7 
 
 989749 
 
 ■47 
 
 342155 
 
 io-o3 
 
 657845 
 
 36 ' 
 
 2J 
 
 332478 
 
 9-56 
 
 989721 
 
 •47 
 
 342757 
 343358 
 
 10-02 
 
 657243 
 
 35 . 
 
 26 
 
 333o5i 
 
 9-54 
 
 989693 
 
 '•47 
 
 10-00 
 
 656642 
 
 34 
 
 3 
 
 333624 
 
 9-53 
 
 989665 
 
 ■47 
 
 343o58 
 344558 
 
 9'99 
 
 656o42 
 
 33 
 
 334195 
 
 9-52 
 
 989637 
 
 ■47 
 
 9-98 
 
 655442 
 
 32 
 
 ?9 
 
 334766 
 
 9-5o 
 
 989609 
 
 •47 
 
 345i57 
 
 9-97 
 
 654843 
 
 3i 
 
 3o 
 
 33533j 
 9-335906 
 
 9.49 
 
 989582 
 
 ■47 
 
 345 7 55 
 
 •9-96 
 
 654245 
 
 3o 
 
 3i 
 
 9.48 
 
 9-989553 
 
 ■47 
 
 9-346353 
 
 994 
 
 10-653647 
 
 29 
 
 32 
 
 336475 
 
 9.46 
 
 989525 
 
 •47 
 
 346949 
 347545 
 
 9-93 
 
 653o5i 
 
 .28 
 
 33 
 
 337043 
 
 9-45 
 
 989497 
 
 •47 
 
 9-92 
 
 652455 
 
 27 
 
 34 
 
 33i6io 
 
 9-44 
 
 989469 
 
 •47 
 
 348141 
 
 9-91 
 
 65 1 859 
 65i265 
 
 26 
 
 35. 
 
 338176 
 
 9-43 
 
 989441 
 
 •47 
 
 348735 
 
 9-90 
 
 25 
 
 36 
 
 338742 
 
 9-41 
 
 9 8 9 4i3 
 
 ■47 
 
 349J29 
 
 9-88 
 
 660671 
 
 24 
 
 u 
 
 339306 
 
 9-4o 
 
 9 8 9 384 
 
 ■47 
 
 349922 
 
 9- §7 
 
 9-86 
 
 ■ 660078 
 
 23 
 
 339871 
 
 9-3g 
 
 9 8 g 356 
 
 •47 
 
 35o5i4 
 
 649486 
 648894 
 
 22 
 
 3 9 
 
 340434 
 
 9-37 
 
 989328 
 
 •47 
 
 35no6 
 
 9-85 
 
 21 
 
 40 
 
 340996 
 9 -34i 558 
 
 9-36 
 
 989300 
 
 •47 
 
 35i6o7 
 
 9-83 
 
 6483o3 
 
 20 
 
 41 
 
 9-35 
 
 9-989271 
 
 •47 
 
 9-352287 
 352876 
 
 9-82 
 
 10-647713 
 
 !o 
 
 42 
 
 342119 
 
 9-34 
 
 989243 
 
 •47 
 
 9-81 
 
 647124 
 
 43 
 
 342679 
 
 9-32 
 
 989214 
 
 •47 
 
 353465 
 
 9-80 
 
 646535 
 
 \l 
 
 44 
 
 3432J9 
 
 9-3i 
 
 989186 
 
 •47 
 
 354033 
 
 9-79 
 
 645947 
 645360 
 
 45 
 
 343797 
 344355 
 
 9-3o 
 
 989157 
 989128 
 
 •47 
 
 354640 
 
 9'77 
 
 ■i5 
 
 46 
 
 9-29 
 
 •48 
 
 355227 
 
 9-76 
 
 644773 
 
 14 
 
 2 
 
 344912 
 
 9-27 
 
 989100 
 
 •48 
 
 3558i3 
 
 9-75 
 
 644187 
 
 i3 
 
 345469 
 
 9-26 
 
 989071 
 
 •48 
 
 3563o8 
 356982 
 357566 
 
 9-74 
 
 643602 
 
 12 
 
 £> 
 
 346024 
 
 9-25 
 
 989042 
 
 •48 
 
 9-73 
 
 643oi8 
 
 11 
 
 5o 
 
 346579 
 
 9-24 
 
 989014 
 9-988985 
 
 •48 
 
 9-71 
 
 642434 
 
 10 
 
 5i 
 
 j-347134 
 
 9 22 
 
 -48 
 
 9-358i49 
 
 9-70 
 
 io-64i85i 
 
 ? 
 
 52 
 
 347687 
 348240 
 
 9.. 21 
 
 9 88 9 56 
 
 ■48 
 
 35873i 
 
 o-6o 
 
 641269 
 
 53 
 
 9-20 
 
 988927 
 988898 
 
 •48 
 
 35931 3 
 
 9.6ft 
 
 640687 
 
 7 
 
 54 
 
 348792 
 
 9-19 
 
 ■48 
 
 35 9 8g3 
 
 9-67 
 
 640107 
 
 5 
 
 55 
 
 349343 
 
 9.17 
 9.16 
 
 988869 
 
 •48 
 
 360474 
 
 o-66 
 
 639526 
 638947 
 638368 
 
 5 
 
 56 
 
 349893 
 
 988840 
 
 •48 
 
 36io53 
 
 9-65 
 
 i 
 
 n 
 
 350443 
 
 9- 15 
 
 98881 1 
 
 •49 
 
 36i632 
 
 9-63 
 
 3 
 
 350092 
 35 1 540 
 
 9-14 
 
 988782 
 
 ■49 
 
 362210 
 
 9-62 
 
 637790 
 
 1 
 
 5 9 
 
 9- 13 
 
 988753 
 
 ■49 
 
 362787 
 
 9.61 
 
 637213 
 
 1 
 
 6o 
 
 352088 
 
 9-n 
 
 988724! 
 
 •49' 
 
 363364 
 
 9.60 . 
 
 636636 
 
 
 Coaini: | 
 
 T>. 
 
 Sine 
 
 J JO 
 
 Coianp;. ! 
 
 D. 
 
 Tang. I 
 
 M. 
 

 SIXES AND TANGENTS 
 
 (13 DEGREES 
 
 ) 
 
 3} 
 
 I'm. 
 
 
 
 Sine 
 
 r>. 
 
 Cosine 
 
 D. 
 T49 
 
 Tang. 
 
 D. 
 
 Cotang. 
 
 ■ 
 
 9-352o88 
 
 9-n 
 
 9-988724 
 
 9-363364 
 
 -9.60 
 
 I0-63663C 
 
 60 
 
 I 
 
 352635 
 
 9-10 
 
 988695 
 
 • 49 
 
 36394a 
 36451 5 
 
 9- 5a 
 9-58 
 
 636o6c 
 
 5 9 
 
 2 
 
 353i8i 
 
 9.09 
 
 988666 
 
 .49 
 
 635485 
 
 58 
 
 3 
 
 .353726 
 
 9-08 
 
 9 88636 
 
 ■49 
 
 365ooa 
 
 9.57 
 
 634910 
 634336 
 
 57 
 
 4 
 
 354271 
 
 9-07 
 
 988607 
 
 •49 
 
 365664 
 
 9-55 
 
 56 
 
 ..5 
 
 354*i5 
 
 9-o5 
 
 9 885 7 8 
 
 •49 
 
 366237 
 
 9-54 
 
 633763 
 
 55 
 
 6 
 
 355358 
 
 9-04 
 
 988548 
 
 •49 
 
 3668io 
 
 9-53 
 
 633190 
 
 54 
 
 I 
 
 355901 
 
 9-o3 
 
 988519 
 
 ■49 
 
 367382 
 
 9-52 
 
 632618 
 
 53 
 
 356443 
 
 9-02 
 
 988489 
 
 .49 
 
 36 79 53 
 
 9-5i 
 
 63204" 
 
 52 
 
 9 
 
 356984 
 
 O-OI 
 
 988460 
 
 •49 
 
 368024 
 
 9-5o 
 
 631476 
 
 5i 
 
 10 
 
 35732: 
 
 8-99 
 
 988430 
 
 •49 
 
 369094 
 
 9-49 
 n 48 
 
 63ooo6 
 io-63o337 
 
 5o- 
 
 ii 
 
 9- 358o64 
 
 8- 98 
 
 9.988401 
 
 ■49 
 
 9.369663 
 
 % 
 
 12 
 
 3586o3 
 
 8-97 
 
 988371 
 
 ■49 
 
 370232 
 
 9.46 
 
 629768 
 
 i3 
 
 3D9141 
 
 8-96 
 
 988342 
 
 •49 
 
 370799 
 
 9-45 
 
 629201 
 628633 
 
 47 
 
 14 
 
 359678 
 
 8-g5 
 
 9 883 1 2 
 
 .50 
 
 371367 
 
 9-44 
 
 46 
 
 l5 
 
 36o2i5 
 
 8-93 
 
 988282 
 
 .50 
 
 371933 
 
 9-43 
 
 628067 
 
 45 
 
 16 
 
 360752 
 
 8-92 
 
 988252 
 
 .50 
 
 372499 
 
 9-42 
 
 627501 
 
 U 
 
 \l 
 
 361287 
 
 8-91 
 
 988223 
 
 .50 
 
 373o64 
 
 9-4r 
 
 626o36 
 626371 
 
 43 
 
 361822 
 
 h; 
 
 988193 
 
 -50 
 
 373629 
 
 940 
 
 42 
 
 '9 
 
 362356 
 
 988163 
 
 -50 
 
 374193 
 
 Hi 
 
 625807 
 
 41 
 
 20 
 
 362889 
 
 8-88 
 
 988i33 
 
 • 50 
 
 374716 
 
 626244 
 
 40 
 
 21 
 
 9-363422 
 
 8-87 
 
 9.988103 
 
 .50 
 
 9.370319 
 
 9- 3 7 
 
 10-624681 
 
 ll 
 
 22 
 
 363 9 54 
 
 8-85 
 
 988073 
 
 •50 
 
 373881 
 
 9-35 
 
 6241 19 
 623558 
 
 23 
 
 364485 
 
 8-84 
 
 988043 
 
 •5o 
 
 376442 
 
 9-34 
 
 37 
 
 24 
 25 
 
 365cn6 
 365546 
 
 8-83 
 8-82 
 
 988013 
 987983 
 
 -50 
 •50 
 
 377003 
 377563 
 
 9-33 
 9-32 
 
 622997 
 6224.37 
 621878 
 
 36 
 35 
 
 26 
 
 366075 
 
 8- 81 
 
 987953 
 
 •50 
 
 378122 
 
 9-3i 
 
 34 
 
 ll 
 
 366604 
 
 8-8o 
 
 987922 
 
 -50 
 
 378681 
 
 9-3o 
 
 621319 
 
 33 
 
 367131 
 
 8-79 
 
 987892 
 
 .50 
 
 379239 
 
 9-29 
 
 620761 
 
 32 
 
 ?9 
 
 367659 
 368i85 
 
 8-77 
 
 987862 
 
 •50 
 
 379797 
 .38o354 
 
 9-28 
 
 620203 
 
 3i 
 
 3o 
 
 8-76 
 
 987832 
 
 •51 
 
 9-27 
 9-26 
 
 619646 
 
 3o 
 
 3i 
 
 9-368711 
 
 8-75 
 
 9.987801 
 
 ■51 
 
 9.380910 
 
 10-619090 
 6i8534 
 
 29 
 
 32 
 
 369236 
 
 8-74 
 
 987771 
 
 •51 
 
 38i466 
 
 9-25 
 
 28 
 
 33 
 
 369761 
 
 8-73 
 
 987740 
 
 •5i 
 
 382020 
 
 9-24 
 
 617980 
 
 ll 
 
 34 
 
 370285 
 
 8-72 
 
 987710 
 
 ■51 
 
 382575 
 
 9-23 
 
 617425 
 
 35 
 
 370808 
 
 8-71 
 
 987679 
 
 ■51 
 
 383129 
 
 9-22 
 
 616871 
 
 25 
 
 36 
 
 37i33o 
 
 8-70 
 
 987649 
 
 •51 
 
 383682 
 
 9-21 
 
 6i63i8 
 
 24 
 
 ll 
 
 371852 
 
 869 
 
 987618 
 
 ■51 
 
 384234 
 
 9-20 
 
 61 5 7 66 
 
 23 
 
 372373 
 
 8.67 
 
 987588 
 
 •5i 
 
 384786 
 
 9-io 
 
 610214 
 
 22 
 
 39 
 
 372894 
 
 8-66 
 
 987557 
 
 • 51 
 
 385337 
 385888 
 
 9-i8 
 
 614663 
 
 21 
 
 4o 
 
 373414 
 
 8-65 
 
 987526 
 
 ■5i 
 
 9-17 
 
 614112 
 
 20 
 
 4i 
 
 9.373933 
 
 8-64 
 
 9-987496 
 
 •51 
 
 9-386438 
 
 9-i5 
 
 io-6i3562 
 
 s 
 
 42 
 
 - 374452 
 
 8-63 
 
 987465 
 
 •51 
 
 ,386987 
 387M6 
 388o84 
 
 9-14 
 
 6i3oi3 
 
 43 
 
 374970 
 375487 
 376003 
 
 8-62 
 
 937434 ! 
 
 ■51 
 
 9- 13 
 
 612464 
 
 \l 
 
 44 
 
 8-6i 
 
 987403 
 
 ■ 52 
 
 9- 12 
 
 611916 
 611J69 
 
 45 
 
 8-6o 
 
 987372, 
 
 •52 
 
 388631 
 
 9-n 
 
 i5 
 
 46 
 
 3 7 65io 
 
 8-5 9 
 
 98^341 
 
 •52 
 
 389118 
 
 9-10 
 
 610822 
 
 14 
 
 47 
 48 
 
 377035 
 
 8-58 
 
 987310 
 
 •52 
 
 389724 
 
 9.09 
 
 610276 
 
 i3 
 
 377549 
 378063 
 
 8-57 
 8-56 
 
 987279 
 987248, 
 
 ■52 
 
 • 52 
 
 390270 
 390815 
 
 9-08 
 9-07 
 
 _ 609730 
 '609183 
 
 12 
 11 
 
 5o 
 
 378577 
 
 8-54 
 
 987217 
 
 •52 
 
 391360 
 
 9.06 
 
 608640 
 
 IC 
 
 5i 
 
 9-379089 
 
 8-53 
 
 9-987186 
 
 •52 
 
 9*391903 
 
 9-o5 
 
 10-608097 
 
 Q 
 
 52 
 
 379601 
 
 8-52 
 
 987155. 
 
 •52 
 
 392447 
 
 9.04 
 
 607533 
 
 8 
 
 53 
 
 3Son3 8-5i 1 
 
 987124 
 
 ■52 
 
 392989 
 39353i 
 
 9-o3 
 
 60701 1 
 606469 
 
 7 
 
 54 
 
 380624 
 
 8-5o 
 
 987092. 
 
 ■52 
 
 9-02 
 
 6 
 
 55 
 
 38n34 
 
 8-40 
 
 987061 
 
 •52 
 
 394073 
 
 9-01 
 
 605927 
 
 5 
 
 56 
 
 38i643 
 
 8-48 
 
 987030I 
 . 986998; 
 
 •52 
 
 394614 
 
 9 00 
 8.99 
 
 6o5386 
 
 4 
 
 ll 
 
 382152 
 
 8-47 
 
 •52 
 
 395i54 
 
 604846 
 
 3 
 
 382661 
 
 8-46 
 
 986967, 
 986936 ' 
 
 ■52 
 
 395694 
 
 8-98 
 
 604306 
 
 2 , 
 
 ?9 
 
 383 168 
 
 8-45 
 
 • 5a| . 3cp&!33 
 
 8-07 
 
 603767 
 
 1 
 
 00 
 
 383675 
 
 8-44 
 
 j -,986904 
 
 •52J 396771 
 
 8-96 
 
 6o322g| 
 
 
 
 Cosine 
 
 D. 
 
 BSpe 76° 1 Cotaiig. 
 
 D. 
 
 Tanj2 ll.^ 
 
82 
 
 (14 DEGREES.) A 
 
 TABLE OP LOGARITHMIC 
 
 
 M. 
 
 Bine 
 
 D. 
 
 Cosine j D- 
 
 Tsui£. 
 
 D. 
 
 Cotnng. 
 
 
 
 
 9-3836-5 
 
 8-44 
 
 9-986904 
 
 •52 
 
 9-396771 
 
 8-96 
 
 10-603229 
 
 60 
 
 I 
 
 384ib2 
 
 *8 
 
 43 
 
 986873 
 
 •53 
 
 397309 
 
 8- 9 6 
 
 602691 
 602134 
 
 u 
 
 2 
 
 384687 
 
 8 
 
 42 
 
 986841 
 
 •53 
 
 397846 
 
 8- 9 5 
 
 3 
 
 385ig2 
 
 8 
 
 41 
 
 986809 
 
 •53 
 
 3 9 8383 
 
 8-94 
 
 601617 
 
 % 
 
 4 
 
 3856 9 7 
 
 8 
 
 40 
 
 986778 
 
 ■53 
 
 398919 
 399455 
 
 8- 9 3 
 
 601081 
 
 56 
 
 5 
 
 386201 
 
 8 
 
 \l 
 
 986746 
 
 •53 
 
 8-92 
 
 6oo545 
 
 55 
 
 6 
 
 386704 
 
 8 
 
 986714 
 
 ■53 
 
 399990 
 4oo524 
 
 8-91 
 
 600010 
 
 54 
 
 I 
 
 387207 
 
 8 
 
 37 
 
 986683 
 
 ■53 
 
 8-90 
 8-89 
 8-88 
 
 599476 
 
 53 
 
 387709 
 388210 
 
 8 
 
 36 
 
 986651 
 
 ■53 
 
 4oio58 
 
 598942 
 
 52 
 
 9 
 
 8 
 
 35 
 
 986619 
 
 •53 
 
 40 1 591 
 
 598409 
 597876 
 
 5i 
 
 10 
 
 3887 1 1 
 
 8 
 
 34 
 
 ■ 986587 
 
 •53 
 
 402 1 24 
 
 8-87 
 
 5o 
 
 II 
 
 9-389211 
 
 8 
 
 33 
 
 9-9%555 
 
 ■53 
 
 9-402656 
 
 8-86 
 
 10-597344 
 
 % 
 
 12 
 
 3897 1 j 
 
 8 
 
 32 
 
 9 86523 
 
 •53 
 
 4o3i87 
 403718 
 
 -8-85 
 
 596813 
 
 ■3 
 
 390210 
 
 8 
 
 3i 
 
 986491 
 986459 
 
 •53 
 
 8-84 
 
 596282 
 
 % 
 
 14 
 
 390708 
 
 8 
 
 3o 
 
 •53 
 
 404249 
 404778 
 
 8-83 
 
 695751 
 
 i5 
 
 391206 
 
 8 
 
 28 
 
 , 86427 
 
 ■53 
 
 8-82 
 
 595222 
 
 45 
 
 16 
 
 391703 
 
 8 
 
 27 
 
 986395 
 
 •53 
 
 4o53o8 
 
 8-8i 
 
 594692 
 
 44 
 
 ;; 
 
 392199 
 392693 
 
 8 
 
 26 
 
 9 86363 
 
 ■54 
 
 4o5836 
 
 8-8o 
 
 564164 
 
 43 
 
 8 
 
 25 
 
 98633i 
 
 •54 
 
 4o6364 
 
 8-70 
 8-78 
 
 5 9 3636 
 
 42 
 
 "9 
 
 3g3iqi 
 
 8 
 
 24 
 
 % 986299 
 
 ■54 
 
 406892 
 
 593108 
 
 41 
 
 20 
 
 3g3685 
 
 8 
 
 23 
 
 986266 
 
 ■54 
 
 407419 
 
 9-407945 
 
 408471 
 
 8-77 
 
 592581 
 
 40 
 
 21 
 
 9-394170 
 394673 
 
 8 
 
 22 
 
 9-986234 
 
 ■54 
 
 8-76 
 
 10-592055 
 
 3o 
 
 22 
 
 8 
 
 21 
 
 9%202 
 
 •54 
 
 8-75 
 
 591529 
 59100J 
 
 38 
 
 23 
 
 3o5i66 
 
 8 
 
 20 
 
 986169 
 
 •54 
 
 408997 
 409521 
 
 8-74 
 
 37 
 
 24 
 
 3 9 5658 
 
 8 
 
 19 
 
 986137 
 
 ■54 
 
 %'H 
 
 590479 
 589955 
 
 36 
 
 25 
 
 396150 
 
 8 
 
 18 
 
 9% 1 04 
 9%072 
 
 ■54 
 
 410045 
 
 8-73 
 
 35 
 
 26 
 
 396641 
 
 8 
 
 17 
 
 • 54 
 
 410569 
 
 8-72 
 
 58o43 1 
 
 34 
 
 3 
 
 397132 
 
 8 
 
 \l 
 
 986039 
 
 ■ 54 
 
 41 1092 
 
 8.71 
 
 588908 
 
 33 
 
 397621 
 3981 1 1 
 
 8 
 
 986007 
 
 ■54 
 
 4n6i5 
 
 8-70 
 
 588385 
 
 32 
 
 2 9 
 
 8 
 
 i5 
 
 983974 
 
 • 54 
 
 4i2i3t 
 412658 
 
 8-69 
 
 58 7 863 
 
 3i 
 
 3o 
 
 39S600 
 
 8 
 
 14 
 
 9S5942 
 
 •54 
 
 8-68 
 
 58t342 
 
 3o 
 
 3i 
 
 9-399088 
 
 8 
 
 13 
 
 9-9S5909 
 985876 
 
 • 55 
 
 9-4i3i79 
 
 8-67 
 
 10-586821 
 
 It 
 
 32 
 
 399575 
 
 8 
 
 12 
 
 • 55 
 
 413699 
 
 8-66 
 
 5863oi 
 
 33 
 
 400062 
 
 8 
 
 11 " 
 
 985843 
 
 .55 
 
 414219 
 
 8-65 
 
 585 7 8i 
 
 11 
 
 34 
 
 400549 
 4oio35 
 
 8 
 
 10 
 
 9 858i 1 
 
 • 55 
 
 414738 
 
 8-64 
 
 585262 
 
 35 
 
 8 
 
 09 
 
 985778 
 
 •55 
 
 4i5257 
 
 8-64 
 
 584743 
 
 25 
 
 36 
 
 40 1 520 
 
 8 
 
 08 
 
 985745 
 
 • 55 
 
 415775 
 
 8-63 
 
 584225 
 
 24 
 
 u 
 
 40200D 
 
 8 
 
 07 
 
 985712 
 
 • 55 
 
 416293 
 
 8-62 
 
 583707 
 
 23 
 
 402489 
 
 8 
 
 06 
 
 985679 
 
 • 55 
 
 416810 
 
 8-61 
 
 583 1 90 
 
 22 
 
 3 9 
 
 402972 
 
 8 
 
 o5 
 
 9 85646 
 
 • 55 
 
 417326 
 
 8-6o 
 
 582674 
 
 21 
 
 40 
 
 403455 
 
 8 
 
 04 
 
 980613 
 
 ■ 55 
 
 417842 
 9-418358 
 
 8-5 9 
 8-58 
 
 582i58 
 
 20 
 
 41 
 
 9-4o3938 
 
 8 
 
 o3 
 
 9-985580 
 
 • 55 
 
 io-58i642 
 
 \l 
 
 42 
 
 404420 
 
 8 
 
 02 
 
 985547 
 
 ■ 55 
 
 418873 
 
 8- 5 7 
 
 581127 
 
 43 
 
 404901 
 
 8 
 
 01 
 
 985014 
 
 ■55 
 
 419387 
 
 8-56 
 
 58o6i3 
 
 \l 
 
 44 
 
 4o5382 
 
 8 
 
 00 
 
 985480 
 
 •55 
 
 419901 
 
 8-55 
 
 580099 
 
 45 
 
 4o5862 
 
 7 
 
 3 
 
 935447 
 
 •55 
 
 420415 
 
 8-55 
 
 579585 
 
 i5 
 
 46 
 
 4o634i 
 
 7 
 
 985414 
 
 • 56 
 
 420927 
 
 8-54 
 
 579073 
 578560 
 
 14 
 
 % 
 
 406820 
 
 7 
 
 97 
 
 98538o 
 
 •56 
 
 421440 
 
 8-,53 
 
 13 
 
 407299 
 
 7 
 
 96 
 
 ■ 985347 
 
 -56 
 
 421962 
 
 8-52 
 
 578048 
 
 12 
 
 & 
 
 407777 
 
 7 
 
 9 3 
 
 9853 1 4 
 
 ■56 
 
 422463 
 
 8-5i 
 
 577537 
 
 11 
 
 5o 
 
 408254 
 
 7 
 
 94 
 
 985280 
 
 • 56 
 
 422974 
 
 8-5o 
 
 577026 
 
 10 
 
 5i 
 
 9 408731 
 
 7 
 
 94 
 
 9-985247 
 985213 
 
 • 56 
 
 9-423484 
 
 8.40 
 
 10-576516 
 
 I 
 
 53 
 
 409207 
 
 7 
 
 9 3 
 
 •56 
 
 423993 
 4245o3 
 
 8-48 
 
 576007 
 
 53 
 
 409682 
 
 7 
 
 92 
 
 985i8o 
 
 ■56 
 
 8 48 
 
 575497 
 574989 
 
 1 
 
 54 
 
 410157 
 
 7 
 
 9 1 
 
 985146 
 
 ■56 
 
 425ou 
 
 8-47 
 8-46 
 
 6 
 
 55 
 
 4io632 
 
 7 
 
 90 
 
 gS5n3 
 
 •56 
 
 425519 
 
 574481 
 
 5 
 
 56 
 
 411106 
 
 7 
 
 89 
 
 985079 
 985045 
 
 •56 
 
 426027 
 
 8-45 
 
 573973 
 
 4 
 
 §2 
 
 411579 
 
 7 
 
 88 
 
 ■56 
 
 426534 
 
 8- 44 
 
 573466 
 
 3 
 
 412052 
 
 7 
 
 87 
 
 98501 1 
 
 ■56 
 
 427041 
 
 8-43 
 
 572959 
 572453 
 
 2 
 
 5, 
 
 412524 
 
 7 
 
 86 
 
 984978 
 
 ■56 
 
 427647 
 428o52 
 
 8-43 
 
 1 
 
 60 
 
 412996 
 
 7 
 
 85 
 
 984944 
 
 56 
 
 8-42 
 
 5719481 
 
 _- 
 
 Oosine 
 
 D. 
 
 Sine IT 5° 
 
 Cotang. 
 
 I). 
 
 "fiii^n m. 
 

 SINES 
 
 AND TANOENTS. 
 
 (15 DEGREES. 
 
 ) 
 
 9? 
 
 k 
 
 Sine 
 
 D. I 
 
 Cosine [ D. 
 
 Tang. 
 
 D. 
 
 Cotang 
 
 
 
 
 9-412996 
 
 7-85 
 
 9-984944] 
 
 •57 
 
 9-428032 
 
 8 
 
 42 
 
 10-571948 
 
 60 
 
 I 
 
 413467 
 
 V 
 
 84 
 
 984910 
 984876 
 
 .37 
 
 428557 
 
 8 
 
 41 
 
 571443 
 
 18 
 
 2. 
 
 4i3938 
 
 7 
 
 83 
 
 'V 
 
 429062 
 
 8 
 
 40 
 
 570938 
 
 3 
 
 414408 
 
 7 
 
 83 
 
 984842 - 
 
 •57 
 
 429366 
 
 8 
 
 39 
 
 570434 
 
 ll 
 
 4 
 
 414878 
 
 7 
 
 82 
 
 984808 
 
 'V 
 
 430070 
 
 '8 
 
 38 
 
 569930 
 
 5 
 
 416347 
 4i58i5 
 
 7 
 
 81 
 
 984774 
 
 I 7 
 
 43o573 
 
 8 
 
 38 
 
 569427 
 
 55 
 
 6 
 
 7 
 
 80. 
 
 984740 
 
 •57 
 
 431075 
 
 8 
 
 ll 
 
 568925 
 
 54 
 
 I 
 
 416283 
 
 7 
 
 71 
 
 , 084706 
 
 'V 
 
 43i577 
 
 8" 
 
 568423 
 
 53 
 
 416751 
 
 7 
 
 984672 
 
 9846J7 
 
 ' 98460J 
 
 ■57 
 
 432079 
 43258o 
 
 8 
 
 35 
 
 567921 
 
 52 
 
 9 
 
 417217 
 
 7 
 
 77 
 
 •37 
 
 8 
 
 34 
 
 567420 
 
 5i 
 
 10 
 
 417684 
 
 7 
 
 76 
 
 ■57 
 
 433o8o 
 
 8 
 
 33 
 
 566920 
 
 5o 
 
 It 
 
 9'4ioi5o 
 
 7 
 
 75 
 
 9.984569 
 
 ■37 
 
 9-433586 
 
 8 
 
 32 
 
 10-566420 
 
 % 
 
 12 
 
 4i86i5 
 
 7 
 
 74 
 
 984530 
 
 •37 
 
 434o8o 
 
 8 
 
 32 
 
 565920 
 
 i3 
 
 419079 
 
 7 
 
 73 
 
 984500 
 
 i 7 
 
 434579 
 435078 
 
 8 
 
 3i 
 
 565421 
 
 s-. 
 
 14 
 
 419544 
 
 7 
 
 73 
 
 984466 
 
 :U 
 
 8 
 
 3o 
 
 564922 
 
 i5 
 
 420007 
 
 7 
 
 7^ 
 
 084432 
 
 435576 
 
 8 
 
 29 
 
 564424 
 
 45 
 
 16 
 
 420470 
 
 7 
 
 71 
 
 984397 
 
 ■58 
 
 436073 
 
 8 
 
 28 
 
 563927 
 
 44 
 
 !3 
 
 42oq33 
 42i3o5 
 
 7 
 
 70 
 
 984363 
 
 ■58 
 
 436570 
 
 8 
 
 28 
 
 56343o 
 
 43 
 
 7 
 
 & 
 
 984328 
 
 ■58 
 
 437067 
 437563 
 
 8 
 
 ll 
 
 562933 
 
 42 
 
 '9 
 
 421837 
 4223i8 
 
 7 
 
 984294 
 
 ■58 
 
 8 
 
 562437 
 
 41 
 
 20 
 
 7 
 
 67 
 
 984259 
 
 ■58 
 
 438o59 
 
 8 
 
 25 
 
 561941 
 
 40 
 
 21 
 
 9-422778 
 
 7 
 
 67 
 
 9-984224 
 
 • 58 
 
 9.438554 
 
 8 
 
 24 
 
 io-56i446 
 
 ll 
 
 22 
 
 423238 
 
 7 
 
 66 
 
 9«4i9? 
 
 •58 
 
 439048 
 
 8 
 
 23 
 
 560952 
 
 23 
 
 42%97 
 
 7 
 
 65 
 
 984155 
 
 ■58 
 
 439543 
 
 8 
 
 23 
 
 56o457 
 
 ll 
 
 24 
 
 424106 
 
 7 
 
 64 
 
 984120 
 
 • 58 
 
 44oo36 
 
 8 
 
 22 
 
 559964 
 
 25 
 
 4246i5 
 
 7 
 
 63 
 
 984085 
 
 •58 
 
 440529 
 
 8 
 
 21 
 
 559471 
 558978 
 558486 
 
 35 
 
 26 
 
 426073 
 
 7 
 
 62 
 
 984050 
 
 •58 
 
 441022 
 
 8 
 
 20 
 
 3-C 
 
 3 
 
 42553o 
 
 7 
 
 61 
 
 984015 
 
 • 58 
 
 44i5i4 
 
 8 
 
 '9 
 
 33 
 
 420987 
 
 7 
 
 60 
 
 9 83 9 8i 
 
 • 58 
 
 442006 
 
 8 
 
 19 
 
 557994 
 557Do3 
 
 .32 
 
 29 
 
 426443 
 
 7 
 
 60 
 
 983946 
 
 ■58 
 
 442497 
 442988 
 
 8 
 
 iJ 
 
 3l 
 
 3o 
 
 426899 
 9-427334 
 
 7 
 
 3 
 
 98391 1 
 
 ■58 
 
 8 
 
 17 
 
 557012 
 io- 556521 
 
 3o 
 
 3i 
 
 7 
 
 9-983875 
 
 • 58 
 
 9-443479 
 
 .8 
 
 16 
 
 ll 
 
 32 
 
 427809 
 428263 
 
 7 
 
 57 
 
 983840 
 
 I 9 
 
 443968 
 
 8 
 
 16 
 
 556o32 
 
 33 
 
 7 
 
 56 
 
 9 838o5 
 
 I 9 
 
 444458 
 
 8 
 
 i5 
 
 555542 
 
 21 
 
 34 
 
 428717 
 
 7 
 
 55 
 
 983770 
 9 83 7 35 
 
 .59 
 
 444947 
 
 - 8 
 
 14 
 
 555o53 
 
 26 
 
 ,35 
 
 429170 
 
 7 
 
 54 
 
 I 9 
 
 445435 
 
 8 
 
 i3 
 
 554565 
 
 25 
 
 36 
 
 429623 
 
 7 
 
 53 • 
 
 983700 
 9 83664 
 
 I 9 
 
 446923 
 
 8 
 
 12 
 
 554077 
 553589 
 
 24 
 
 33 
 
 43oo75 
 
 7 
 
 52 
 
 I 9 
 
 44641 1 
 
 8 
 
 12 
 
 23 
 
 43o527 
 430978 
 
 7 
 
 52 
 
 983629 
 
 I 9 
 
 446898 
 
 8 
 
 11 
 
 553io2 
 
 22 
 
 3 9 
 
 7 
 
 5i 
 
 9 835 9 4 
 
 I 9 
 
 447384 
 
 8 
 
 10 
 
 562616 
 
 21 
 
 40 
 
 '431429 
 
 7 
 
 5o 
 
 9 83558 
 
 I 9 
 
 447870 
 
 8 
 
 09 
 
 552i3o 
 
 20 
 
 41 
 
 9-431879 
 
 7 
 
 ■49 
 
 9.983523 
 
 ■ 5 9 
 
 9-448356 
 
 8 
 
 09 
 
 io-55i644 
 
 18 
 
 42 
 
 432329 
 
 7 
 
 •49 
 
 983487 
 
 P 
 
 448841 
 
 8 
 
 08 
 
 55u59 
 
 43 
 
 432778 
 
 7 
 
 .48 
 
 983452 
 
 I 9 
 
 449326 
 
 8 
 
 07 
 
 550674 
 
 17 
 
 44 
 
 433226 
 
 7 
 
 :% 
 
 983416 
 
 I 9 
 
 449S10 
 
 8 
 
 06 
 
 550190 
 
 16 
 
 45 
 
 433675 
 
 7 
 
 9 8338i 
 
 -5 9 
 
 430294 
 
 8 
 
 06 
 
 549706 
 
 i5 
 
 46 
 
 434122 
 
 7 
 
 ■45 
 
 983345 
 
 I 9 
 
 400777 
 
 8 
 
 o5 
 
 540223 
 
 14 
 
 % 
 
 434569 
 
 7 
 
 •44 
 
 983309 
 
 • 5 9 
 
 461260 
 
 8 
 
 04 
 
 548740 
 
 13 
 
 435oio 
 
 7 
 
 .44 
 
 9 832 7 3 
 
 •60 
 
 45i743 
 
 8 
 
 o3 
 
 648267 
 547776 
 
 12 
 
 49 
 
 435462 
 
 7 
 
 •43 
 
 9H3238 
 
 .60 
 
 452225 
 
 8 
 
 62 
 
 11 
 
 5c 
 
 435oo8 
 
 9-436353 
 
 436798 
 
 7 
 
 ■42 
 
 983202 
 
 •60 
 
 452706 
 
 8 
 
 02 
 
 547294 
 
 10 
 
 5i 
 
 7 
 
 41 
 
 9-983166 
 
 • 60 
 
 9-453187 
 453668 
 
 8 
 
 01 
 
 io-5468i3 
 
 8 
 
 52 
 
 7 
 
 ■40 
 
 983 i3o 
 
 • 60 
 
 8 
 
 00 
 
 546332 
 
 , 53 
 
 437242 
 
 7 
 
 •40 
 
 983094 
 983o58 
 
 ■60 
 
 454148 
 
 7 
 
 99 
 
 545852 
 
 7 
 
 54 
 
 43 7686 
 438129 
 438572 
 
 7 
 
 •39 
 
 • 60 
 
 454628 
 
 7 
 
 3 
 
 545372 
 
 6 
 
 55 
 
 7 
 
 ■ 38 
 
 983022 
 
 • 60 
 
 455i07 
 
 7 
 
 544893 
 
 5 
 
 56 
 
 7 
 
 ■37 
 
 982986 
 
 • 6o 
 
 455586 
 
 7 
 
 97 
 
 5444i4 
 
 4 
 
 % 
 
 439014 
 439456 
 
 7 
 
 ■36 
 
 982950 
 
 •60 
 
 456064 
 
 7 
 
 9 6 
 
 543936 
 
 ,3 
 
 7 
 
 • 36 
 
 982914 
 982878 
 
 ■ 60 
 
 456542 
 
 7 
 
 t 
 
 543458 
 
 2 
 
 ss 
 
 439897 
 44o338 
 
 7 
 
 -35 
 
 ■60 
 
 457019 
 457496 
 
 7 
 
 9 5 
 
 542981 
 542604 
 
 _ TangT 
 
 1 
 
 7 
 
 •34 
 
 982841 
 
 ■ 60 
 
 7-9* 
 
 
 M. 
 
 
 Cosine 
 
 D. 
 
 Sine 
 
 74° 
 
 Cotang. 
 
 ■ J 
 
 D. 
 
u 
 
 (16 
 
 DEGREES.) A TABLE OF LOGARITHMIC 
 
 
 M. 
 
 Sine I D. 
 
 Cosine 
 
 I). 
 
 Tung. 
 
 D- 
 
 Coltmg. 
 
 
 
 
 9-44o338| 7-34 
 
 9-982842 
 
 -6o 
 
 9-457496 
 
 7-94 
 
 Io-54?5o4 
 
 60 
 
 I 
 
 440778I 7-33 
 
 982805 
 
 -60 
 
 , 457973 
 
 7- 9 3 
 
 542027 
 
 5 9 
 
 2 
 
 441218 7-32 
 
 982769 
 982733 
 
 ■ 61 
 
 458449 
 458925 
 
 7- 9 3 
 
 54i55i 
 
 58 
 
 3 
 
 441658 7-3i 
 
 • 61 
 
 7.92 
 
 541075 
 
 57 
 
 4 
 
 442096; 7 - 3 1 
 
 982696 
 
 • bi 
 
 459400 
 
 7-91 . 
 
 540600 
 
 56 
 
 5 
 
 442535 7'3o 
 
 982660 
 
 • 61 
 
 459875 
 
 7-90 
 
 540125 55 
 
 6 
 
 442973 
 
 ]:% 
 
 982624 
 
 ■61 
 
 46o349 
 
 7.90 
 
 ?:8 
 
 53 9 65i 
 
 54 
 
 'I 
 
 443410 
 
 982587 
 
 ■61 
 
 460823 
 
 530177 
 53o7o3 
 
 53 
 
 443847 
 
 7-27 
 
 982551 
 
 •61 
 
 461297 
 
 52 
 
 i 9 
 
 444284 
 
 7-27 
 
 982514 
 
 •61 
 
 461770 
 
 7.88 
 
 53823o 
 
 5i 
 
 10 
 
 444720 
 
 7-26 
 
 982477 
 
 •61 
 
 462242 
 
 7-87 
 
 537758 
 
 5o 
 
 n 
 
 9'445i55 
 
 7-25 
 
 0-982441 
 
 •61 
 
 9-462714 
 
 7.86 
 
 10-537286 
 
 49 
 
 12 
 
 445590 
 
 7-24 
 
 982404 
 
 •61 
 
 463 1 86 
 
 7-85 
 
 5368i4 
 
 48 
 
 13 
 
 446025 
 
 7-23 
 
 982367 
 
 •61 
 
 463658 
 
 7-85 
 
 536342 
 
 47 
 
 U 
 
 44645o 
 446893 
 
 7-23 
 
 982331 
 
 •61 
 
 464129 
 
 7.84 
 
 5358 7 i 
 
 46 
 
 i5 
 
 7-22 
 
 982294 
 982257 
 
 •61 
 
 464599 
 
 7-83 
 
 5354oi 
 
 45 
 
 16 
 
 447326 
 
 7-21 
 
 •61 
 
 465069 
 
 7-83 
 
 534931 
 
 44 
 
 \l 
 
 447759 
 448191 
 
 7-20 
 
 982220 
 
 •62 
 
 46553o 
 466008 
 
 7.82 
 
 53446i 
 
 43 
 
 7-20 
 
 982 i83- 
 
 •62 
 
 7.81 
 
 533992 
 533524 
 
 42 
 
 '9 
 
 448623 7- 19 
 
 982146 
 
 ■62 
 
 466476 
 
 7.80 
 
 41 
 
 20 
 
 449054 7- 18 
 
 982109 
 
 ■62 
 
 466945 
 
 7-80 
 
 533o55 
 
 4o 
 
 21 
 
 9-449485 
 
 7-ij 
 
 9-982672 
 
 ■62 
 
 9-467413 
 
 7-79 
 
 io-53258 7 
 
 IS 
 
 22 
 
 449915 
 45o345 
 
 7.10 
 
 982035 
 
 ■62 
 
 467880 
 
 7-78 
 
 532120 
 
 23 
 
 7. .6 
 
 981998 
 
 •62 
 
 468347 
 
 7-78 
 
 53i653 
 
 ll 
 
 U 
 
 450775 
 
 7 -i5 
 
 981961 
 
 • 62 
 
 468814 
 
 7-77 
 
 53n86 
 
 25 
 
 45 1 204 
 
 7-U 
 
 981924 
 981886 
 
 ■62 
 
 469280 
 
 7-76 
 
 530730 
 
 35 
 
 26 
 
 45i632 
 
 7. .3 
 
 ■62 
 
 469746 
 
 7.75 
 
 530254 
 
 34 
 
 27 
 
 452o6o 
 
 7-13 
 
 981849 
 
 •62 
 
 4702 1 1 
 
 7-75 
 
 - 529789 
 
 33 
 
 28 
 
 452488 
 
 7-12 
 
 981812 
 
 •62 
 
 470676 
 
 7-74 
 
 529324 
 
 32 
 
 I 9 ' 
 
 452915 
 
 7-n 
 
 981774 
 
 ■62 
 
 471141 
 
 7-73 
 
 52885 9 
 
 3i 
 
 3o- 
 
 453342 
 
 7-10 
 
 981737 
 
 ■62 
 
 471605 
 
 7.73 
 
 5283o5 
 10-527932 
 
 3o 
 
 3i 
 
 9-453768 
 
 7-10 
 
 9-981699 
 
 ■63 
 
 9-472068 
 
 7.72 
 
 3 
 
 32 
 
 454194 
 
 7.09 
 7-08 
 
 981662 
 
 •63 
 
 472532 
 
 7-71 
 
 527468 
 
 33 
 
 454619 
 
 981625 
 
 •63 
 
 472995 
 473457 
 
 7.71 
 
 527005 
 
 27 
 
 34 
 
 455o44 
 
 7-07 
 
 '981587 
 
 ■63 
 
 7-70 
 
 '526543 
 
 26 
 
 35 
 
 455469 
 455893 
 
 7-07 
 
 981549 
 
 ■63 
 
 473919 
 
 7-69 
 
 526081 
 
 25 
 
 36 
 
 7-06 
 
 981512 
 
 •63 
 
 47438i 
 
 7.69 
 
 5256iq 
 
 24 
 
 jiz 
 
 4563 1 6 
 
 7-o5 
 
 981474 
 
 •63 
 
 474842 
 
 r bl 
 
 5a5i58 
 
 23 
 
 '456739 
 
 7-04 
 
 981436 
 
 •63 
 
 4753o3 
 
 7-67 
 
 524697 
 5242J7 
 
 22 
 
 1 39 
 
 457162. 
 
 7-04 
 
 981399 
 
 •63 
 
 475763 
 
 7*7 
 
 2C 
 
 40 
 
 45J584 
 9- 458oo6 
 
 7-o3 
 
 981361 
 
 ■63 
 
 476223 
 
 7-66 
 
 523777 
 
 20 
 
 41 
 
 7-02 
 
 9-98i323 
 
 •63 
 
 9-476683 
 
 7-65 
 
 io-52.33i7 
 
 3 
 
 42 
 
 458427 
 458848 
 
 7-01 
 
 .'981285 
 
 •63 
 
 477142 
 
 7-65 
 
 522858 
 
 43 
 
 7-01 
 
 981247 
 
 •63 
 
 477601 
 
 7-64 
 
 522399 
 
 n 
 
 44 
 
 459268 
 
 7-00 
 
 981209 
 
 ■63 
 
 478059 
 
 7-63 
 
 521941 
 
 16 
 
 45 
 
 459688 
 
 6.90 
 
 981 171 
 
 ■63 
 
 478517 
 
 7-63 
 
 521483 
 
 i5 
 
 46 
 
 460108 
 
 6.9I 
 
 981 i33 
 
 •64 
 
 478975 
 
 7-62 
 
 521025 
 
 14 
 
 2 
 
 460527 
 460946 
 -461J64 
 
 6-98 
 
 981095 
 98i'o57 
 
 ■64 
 
 479432 
 
 7-61 
 
 520568 
 
 i3 
 
 6-97 
 6-96 
 
 •64 
 
 479889 
 480346 
 
 7.61 
 
 5201 11 
 
 12 
 
 49 
 
 981019 
 
 .64 
 
 7-60 
 
 519655 
 
 11 
 
 5o 
 
 461782 
 
 6- 9 5 
 
 980981 
 
 •64 
 
 480801 
 
 7-69 
 
 10-518743 
 
 10 
 
 5i 
 
 9-462199 
 
 6- 9 5 
 
 9-980942 
 
 • 64 
 
 9-481257 
 
 ]il 
 
 8 
 
 52 
 
 462616 
 
 6-94 
 
 980904 
 
 •64 
 
 481712 
 
 518288 
 
 53 
 
 463o32 
 
 6-o3 
 
 980866 
 
 ■64 
 
 482167 
 
 7-5 7 
 
 5i 7 833 
 
 7 
 
 54 
 
 463448 
 
 6- 9 3 
 
 980827 
 
 -64 
 
 482621 
 
 7-57 
 
 517379 
 516925 
 
 6 
 
 55 
 
 463864 
 
 6-92 
 
 980789 
 
 •64 
 
 483075 
 
 7-56 
 
 5 
 
 56 
 
 464279 
 
 6-91 
 
 980750 
 
 -64 
 
 483529 
 
 7-55 
 
 5 i 647 1 
 
 4 
 
 u 
 
 464694 
 
 6-90 
 
 980712 
 
 -64 
 
 483982 
 
 7-55 
 
 5i6oi8 
 
 3 
 
 465 1 08 
 
 6-90 
 
 .980673 
 
 ■64 
 
 484435 
 
 7-54 
 
 5i5565 
 
 a 
 
 & 
 
 465522 
 
 6.89 
 6-88 
 
 980635 
 
 •64 
 
 484887 
 
 7-53 
 
 5i5u3 
 
 1 
 
 60 
 
 463935 
 
 980596 
 
 •64 
 
 485339 
 
 t-53 
 
 5i466i 
 
 a 
 
 . 
 
 Oosiue D. 
 
 Sine ■ 
 
 '3° 
 
 Cotang. 
 
 tt 
 
 , Tang. 
 
 mT 
 

 SINKS AND TANGENTS. 
 
 (17 DEGREES.' 
 
 
 3a 
 
 f5T 
 
 o i 
 
 Sine 
 
 D. 
 
 Cosine 
 
 D. 
 
 -64 
 
 Tang. 
 : 9 -485339 
 
 D. 
 
 Cotang. 1 
 10-014661 
 
 
 
 9-465o35 
 466348 
 
 6-88 
 
 9-980596 
 980558 
 
 7-55 
 
 60 
 
 11 
 
 I 
 
 6-88 
 
 ■64 
 
 485791 
 
 7-52 
 
 514209 
 
 2 
 
 466761 
 
 6-87 
 
 980519 
 
 •65 
 
 486242 
 
 7 .5. 
 
 5i3758 
 
 3 
 
 467173 
 
 6-86 
 
 980480 
 
 ■ 65 
 
 -, 486693 
 
 7-5i 
 
 5 1 33o7 
 
 5o 
 
 4 
 
 467585 
 
 6-85 
 
 980442 
 
 • 65 
 
 487143 
 
 7 00 
 
 512857 
 
 
 
 467996 
 468407 
 
 6-85 
 
 980403 
 
 • 65 
 
 487593 
 488043 
 
 7-49 
 
 512407 
 
 55 
 
 6 
 
 6-84 
 
 980364 
 
 -65 
 
 7.4a 
 7.48 
 
 51195-7 
 5ii5o8 
 
 54 
 53 
 
 52 
 
 5i 
 5o 
 
 2 
 
 468817 
 
 6-83 
 
 980325 
 
 ■65 
 
 488492 
 
 469227 
 
 6-83 
 
 980286 
 
 • 65 
 
 488941 
 
 7-47 
 
 5no59 
 
 9 
 
 469637 
 
 6-82 
 
 980247 
 980208 
 
 ■65 
 
 489390 
 489838 
 
 7-47 
 
 5io6io 
 
 10 
 
 470046 
 
 . 6-8i 
 
 • 65 
 
 7.46 
 
 5ioi62 
 
 ii , 
 
 9-470455 
 
 6-8o 
 
 9-980169 
 
 ■ 65 
 
 9-490286 
 
 7.46 
 
 10*509714 
 
 49 
 
 12 
 
 470863 
 
 6-8o 
 
 980130 
 
 ■ 65 
 
 490733 
 
 7.45 
 
 609267 
 
 48 
 
 13 
 
 471271 
 
 6-79 
 
 980091 
 
 • 65 
 
 491180 
 
 7-44 
 
 5o882o 
 
 47 
 
 U 
 
 471679 
 
 6-78 
 
 980002 
 
 •65 
 
 491627 
 
 7.44 
 
 5o8373 
 
 46 
 45 
 
 i5 
 
 472086 
 
 6-78 
 
 980012 
 
 • 65 
 
 492073 
 
 7-43 
 
 507927 
 
 16 
 
 472492 
 
 6-77 
 
 979973 
 
 •65 
 
 492519 
 
 7-43 
 
 507481 
 
 44 
 43 
 
 15 
 
 472898 
 
 6-76 
 
 979934 
 979895 
 97 9 855 
 
 ■66 
 
 492965 
 
 7-42 
 
 5o7o35 
 
 473304 
 
 6-76 
 
 ■ 66 
 
 493410 
 
 7-41 
 
 506590 
 
 42 
 
 '9 
 
 473710 
 
 6-75 
 
 ■66 
 
 493854 
 
 7-40 
 
 5o6i46 
 
 41 
 
 20 
 
 4741 1 5 
 
 6-74 
 
 979816 
 
 ■ 66 
 
 494299 
 
 7.40 
 
 5o570i 
 
 40 
 
 39 
 38 
 
 21 
 
 9-474510 
 474923 
 475327 
 
 6-74 
 
 9-979776 
 
 .66 
 
 9-494743 
 
 7-40 
 
 Io-5o5257 
 
 22 
 
 6-73 
 
 979737 
 
 • 66 
 
 490186 
 
 7-3g 
 
 504814 
 
 23 
 
 6-72 
 
 979608 
 
 ■ 66 
 
 4g563o 
 
 7-38 
 
 504370 
 
 ll 
 35 
 
 24 
 
 475730 
 
 6-72 
 
 ■ 66 
 
 496973 
 
 • 7-37 
 
 503927 
 
 25 
 
 476i33 
 
 6-71 
 
 979618 
 
 ■ 66 
 
 4965 1 5 
 
 7-37 
 
 5o3485 
 
 26 
 
 476536 
 
 6-70 
 6-69 
 
 979579 
 
 ■ 66 
 
 496957 
 
 7-36 
 
 5o3o43 
 
 34 
 33 
 32 
 3i 
 3o 
 
 a 
 
 476o38 
 
 979539 
 
 ■ 66 
 
 497399 
 
 7-36 
 
 5o26oi 
 
 477340 
 
 6-69 
 6-68 
 
 979^99 
 979459 
 
 ■ 66 
 
 497841 
 
 7-35 
 
 5o2i5o 
 
 ?9 
 
 477741 
 478142 
 
 ■ 66 
 
 498282 
 
 7-34 
 
 501718 
 
 3o 
 
 6-67 
 
 979420 
 
 ■ 66 
 
 498722 
 
 7-34 
 
 501278 
 
 3i 
 
 9-478542 
 
 6-67 
 
 9-979380 
 
 ■ 66 
 
 9-499163 
 
 7-33 
 
 io-5oo837 
 
 2I 
 
 32 
 
 478942 
 479342 
 
 6-66 
 
 979340 
 
 • 66 
 
 499603 
 
 7-33 
 
 5oo3g-7 
 499908 
 499019 
 
 33 
 
 6-65 
 
 979300 
 
 .67 
 
 5ooo42 
 
 7.32 
 
 % 
 
 34 
 
 479741 
 480140 
 
 6-65 
 
 979260 
 
 -67 
 
 5oo48i 
 
 7-3i 
 
 35 
 
 6-64 
 
 979220 
 
 •67 
 
 500920 
 5oi359 
 
 7-3i 
 
 499080 
 
 2D 
 
 36 
 
 48o539 
 
 6-63 
 
 979180 
 
 •67 
 
 7-3o 
 
 498641 
 
 24 
 
 23' 
 22 
 
 12 
 
 486-937 
 48i334 
 
 6-63 
 6-62 
 
 979140 
 979 1 00 
 
 .67 
 .67 
 
 501797 
 5o2235 
 
 7-3o 
 7-29 
 
 498203 
 
 497765 
 
 3 9 
 
 481731 
 
 6-6i 
 
 979059 
 
 •67 
 
 502672 
 
 7.28 
 
 497328 
 
 21 
 
 40 
 
 482128 
 
 6-6i 
 
 979019 
 9-978979 
 
 • 67 
 
 5o3l09 
 
 7-28 
 
 496801 
 10-496404 
 
 20 
 
 41 
 
 9-482525 
 
 6-6o 
 
 .67 
 
 9 -5o3546 
 
 7-27 
 
 \% 
 
 42 
 
 482921 
 4833 1 6 
 
 6-59 
 
 978935 
 
 ■ 67 
 
 ■ 503982 
 
 7 " 2 I 
 
 496018 
 
 ••43 
 
 6-5 9 
 6-58 
 
 97889^ 
 978858 
 
 •67 
 
 5o44i8 
 
 7.26 
 
 495582 
 
 17 
 16 
 
 15 
 
 44 
 
 483712 
 
 .67 
 
 5o4854 
 
 7 . 2 5 
 
 495l46 
 
 45 
 
 484107 
 
 6-57 
 
 978817 
 
 .67 
 
 505289 
 
 7-25 
 
 49471 1 
 
 46 
 
 4845oi 
 
 6-57 
 
 978777 
 
 .67 
 
 5x>5724 
 
 7-24 
 
 494276 
 
 14 
 i3 
 
 % 
 
 484895 
 
 6-56 
 
 978736 
 
 ■67 
 
 5o6i5o 
 506590 
 
 7-24 
 
 493841 
 
 485289 
 
 6-55 
 
 978696 
 
 m 
 
 7-23 
 
 493407 
 
 12 
 
 49 
 
 485682 
 
 6-55 
 
 978655 
 
 • 68 
 
 507027 
 
 ■ 7-22 
 
 492973 
 492040 
 
 11 
 
 5o 
 
 486075 
 
 6-54 
 
 978615 
 
 ■ 68 
 
 507460 
 
 7-?2 
 
 10 
 
 5i 
 
 9-486467 
 
 6-53 
 
 9-9785-74 
 
 • 68 
 
 9-507893 
 5o8326 
 
 7-2! 
 
 10-492107 
 
 % 
 
 52 
 
 486860 
 
 6-53 
 
 , 97853: 
 
 • 68 
 
 7-21 
 
 491674 
 
 53 
 
 487251 
 
 6-52 
 
 97849.3 
 978452 
 
 •68 
 
 508759 
 
 7-20 
 
 491 241 
 
 I 
 5 
 
 54 
 
 487643 
 488o34 
 
 6-5i 
 
 ■•68 
 
 509191 
 
 7-19 
 
 49080c 
 
 55 
 
 6-5i 
 
 97841 1 
 
 • 68 
 
 509622 
 
 7-18 
 7-18 
 
 49037* 
 489946 
 4890 1 5 
 
 56 
 
 488424 
 488814 
 
 6-5o 
 6-5o 
 
 97837c 
 97832c 
 
 ■ 68 
 -68 
 
 5ioo54 
 5 1 o485 
 
 4 
 3 
 
 48920^ 
 489093 
 489982 
 
 6-49 
 
 6-48 
 
 -6-48 
 
 97828* 
 ■97824-, 
 .97820c 
 
 ■68 
 ■ 68 
 ■68 
 
 5iooi6 
 5n346 
 011776 
 
 ■7.17 
 
 7-16 
 7-16 
 
 489084 
 4«8654 
 488224 
 
 2 
 1 
 
 
 
 Cosine 
 
 D. 
 
 Sine IT2° 
 
 CoUmg. 
 
 ' ll 
 
 1 Tang- 
 
 _M. 
 
36 
 
 (18 DEGREES.) A 
 
 TABLE OF LOGARITHMIC 
 
 1 m i| 
 
 M. 
 
 
 
 Sine 
 
 D. 
 
 Cosine i 1>- 
 
 Tang. 
 
 D. 
 
 Cotauft. 
 
 
 9-489982 
 49037 i 
 
 6-48 
 
 9-918206! -68 
 
 9-511776 
 
 7 
 
 16 
 
 10-488224 
 
 60 
 
 
 6 
 
 48 
 
 978165 
 
 68 
 
 512206 
 
 7 
 
 16 
 
 487794 
 
 11 
 
 n 
 
 i 
 
 490739 
 
 6 
 
 40 
 
 -978124; 
 
 68 
 
 512635 
 
 7 
 
 i5 
 
 487365 
 
 3 
 
 491 i47 
 
 491535 
 
 6 
 
 978083; 
 
 69 
 
 5 1 3064 
 
 7 
 
 14 
 
 486o36 
 
 4 
 
 6 
 
 46 
 
 978042! 
 
 69 
 
 5 13493 
 
 7 
 
 14 
 
 486607 
 
 5 
 
 491922 
 492308 
 
 6 
 
 45 
 
 978001 1 
 
 69 
 
 5i3o2i 
 514349 
 
 7 
 
 i3 
 
 486079 55 
 
 6' 
 
 6 
 
 44 
 
 977959 
 
 69 
 
 7 
 
 i3 
 
 48565i 
 
 54 
 
 3 
 
 492695 
 
 6 
 
 44 
 
 977918 
 977077 
 
 
 69 
 
 514777 
 
 7 
 
 12 
 
 485223 
 
 53 
 
 493081 
 
 6 
 
 43 
 
 
 69 
 
 516204 
 
 7 
 
 12 
 
 484796 
 484369 
 483943 
 
 52 
 
 9 
 
 493466 
 
 6 
 
 42 
 
 977835 
 
 
 69 
 
 5i563i 
 
 7 
 
 11 
 
 5i 
 
 10 
 
 49385i 
 
 6 
 
 42 
 
 977794 
 9.977752 
 
 
 69 
 
 5 1 6057 
 
 7 
 
 10 
 
 5o 
 
 ii 
 
 9 '494236 
 
 6 
 
 41 
 
 
 69 
 
 9.516484 
 
 7 
 
 10 
 
 io-4835i6 
 
 49 
 
 12 
 
 494621 
 
 6 
 
 41 
 
 9777" 
 
 
 69 
 
 516910 
 5i7335 
 
 7 
 
 09 
 
 483090 
 
 48 
 
 i3 
 
 495oo5 
 
 6 
 
 40 
 
 977669 
 977628 
 
 
 69 
 
 7 
 
 00 
 
 482666 
 
 47 
 
 U 
 
 495388 
 
 6 
 
 3 9 
 
 
 69 
 
 517761 
 . 5i8i85 
 
 7 
 
 482239 
 
 46 
 
 i5 
 
 495772 
 
 6 
 
 ll 
 
 977586 
 
 
 69 
 
 7 
 
 08 
 
 48i8i5 
 
 45 
 
 16 
 
 496154 
 
 6 
 
 977544 
 
 
 70 
 
 5i86i 
 
 7 
 
 07 
 
 481390 
 
 44 
 
 ■7 
 
 496537 
 
 6 
 
 37 
 
 9775o3 
 
 
 70 
 
 519034 
 
 7 
 
 06 
 
 480966 
 480042 
 
 43 
 
 18 
 
 496919 
 '4973o 1 
 
 6 
 
 37, 
 
 977461 
 
 
 7° 
 
 5 1 9458 
 
 7 
 
 06 
 
 42 
 
 '9 
 
 6 
 
 36 
 
 977419 
 
 
 70 
 
 519882 
 
 7 
 
 o5 
 
 480118 
 
 4i 
 
 20 
 
 . 497682 
 
 6 
 
 36 
 
 977 3 77 
 
 
 70 
 
 52o3o5 
 
 7 
 
 o5 
 
 479695 
 
 40 
 
 21 
 
 9-498064 
 
 6 
 
 35 
 
 9-977335 
 
 
 70 
 
 9.520728 
 
 7 
 
 04 
 
 10-470272 
 4-18849 
 
 3 9 
 
 22 
 
 498444 
 
 6 
 
 34 
 
 977293 
 
 
 70 
 
 521 I 5 T 
 
 7 
 
 o3 
 
 38 
 
 l3 
 
 498825 
 
 6 
 
 34 
 
 977201 
 
 
 70 
 
 021573 
 
 7 
 
 o3 
 
 478427 
 
 37 
 
 14 
 
 499204 
 
 6 
 
 33 
 
 977209 
 
 
 70 
 
 521995 
 
 7 
 
 o3 
 
 478006 
 
 36 
 
 i5 
 
 499584 
 
 6 
 
 32 
 
 977167 
 977125 
 
 
 70 
 
 622417 
 
 7 
 
 02 
 
 477583 
 
 35 
 
 16 
 
 499963 
 
 6 
 
 32 
 
 
 70 
 
 522838 
 
 7 
 
 02 
 
 477162 
 
 34 
 
 s 
 
 5oo342 
 
 6 
 
 3! 
 
 977083 
 
 
 70 
 
 52325 Q 
 
 7 
 
 01 
 
 476741 
 
 33 
 
 500721 
 
 6 
 
 3i 
 
 977041 
 
 
 70 
 
 523686 
 
 7 
 
 01 
 
 476320 
 
 32 
 
 ! 9 
 
 501099 
 
 6 
 
 3o 
 
 976999 
 976957 
 
 
 7° 
 
 524100 
 
 7 
 
 00 
 
 476900 
 
 3i 
 
 lo 
 
 501476 
 
 6 
 
 29 
 
 
 70 
 
 52452o 
 
 6 
 
 99 
 
 475480 
 
 3o 
 
 Ii 
 
 9-5oi854 
 
 6 
 
 3 
 
 9-976914 
 976872 
 976830 
 
 
 70 
 
 9-524939 
 52535q 
 525778 
 
 6 
 
 3 
 
 10-475061 
 
 20 
 
 12 
 
 502231 
 
 6 
 
 
 71 
 
 6 
 
 474641 
 
 !3 
 
 502607 
 
 6 
 
 28 
 
 
 7 1 
 
 6 
 
 98 
 
 4742'22 
 
 27 
 
 U 
 
 502984 
 
 6 
 
 27 
 
 976787 
 976745 
 
 
 T 
 
 526197 
 
 6 
 
 97 
 
 4738o3 
 
 26 
 
 \5 
 
 5o336o 
 
 6 
 
 26 
 
 
 7> 
 
 5266i5 
 
 6 
 
 9 Z 
 
 96 
 
 473385 
 
 25 
 
 16 
 
 5o3735 
 
 6 
 
 26 
 
 976702 
 
 
 71 
 
 527033 
 
 6 
 
 472967 
 472649 
 
 24 
 
 ii 
 
 5o4no 
 
 6 
 
 25 
 
 976660 
 
 
 71 
 
 52745i 
 
 6 
 
 t 
 
 23 
 
 5o4485 
 
 6 
 
 25 
 
 976617 
 
 
 71 
 
 527868 
 528285 
 
 6 
 
 9 5 
 
 472132 
 
 22 
 
 i? 
 
 504860 
 
 6 
 
 24 
 
 976574 
 
 
 7 1 
 
 6 
 
 9 5 
 
 471716 
 
 21 
 
 to 
 
 5o5234 
 
 6 
 
 23 
 
 976532 
 
 
 7 1 
 
 528702 
 
 6 
 
 94 
 
 471298 
 
 20 
 
 ii 
 
 9-5o56o8 
 
 6 
 
 23 
 
 9-976489 
 
 
 7> 
 
 9-529119 
 52 9 535 
 
 6 
 
 9 3 
 
 10-470881 
 
 lo 
 
 (2 
 
 5o5o8i 
 5o6354 
 
 6 
 
 22 
 
 976446 
 
 
 7' 
 
 6 
 
 9 3 
 
 470465 
 
 10 
 
 43 
 
 6 
 
 22 
 
 976404 
 
 
 71 
 
 529950 
 53o366 
 
 6 
 
 9 3 
 
 47oo5o 
 
 'I 
 
 44 
 
 606727 
 
 6 
 
 21 
 
 976361 
 
 
 '/' 
 
 6 
 
 92 - 
 
 469634 
 
 16 
 
 45 
 
 507099 
 
 6 
 
 20 
 
 6 7 63i8 
 
 
 T 
 
 530781 
 
 6 
 
 9' 
 
 469219 
 468804 
 
 i5 
 
 46 
 
 507471 
 
 6 
 
 20 
 
 976275 
 
 
 7' 
 
 531196 
 
 6 
 
 9' 
 
 14 
 
 % 
 
 607843 
 5o82i4 
 
 6 
 
 '9 
 
 9762J2 
 
 
 72 
 
 53i6u 
 
 6 
 
 90 
 
 46838o 
 467970 
 467661 
 
 i3 
 
 6 
 
 
 976189 
 976146 
 
 
 72 
 
 532025 
 
 6 
 
 90 
 
 12 
 
 49 
 
 5o8585 
 
 6 
 
 l8 
 
 
 72 
 
 532439 
 532853 
 
 6 
 
 §9 
 
 11 
 
 5o 
 
 5o8o56 
 9.509.326 
 
 6 
 
 18 
 
 976103 
 
 
 72 
 
 6 
 
 §8 
 
 467147 
 
 10 
 
 5i 
 
 6 
 
 17 
 
 9.976060 
 
 
 72 
 
 9-533266 
 
 6 
 
 10-466734 
 
 
 
 52 
 
 509696 
 
 6 
 
 16 
 
 976017 
 
 
 72 
 
 533679 
 
 6 
 
 88 
 
 466321 
 
 53 
 
 5ioo65 
 
 6 
 
 16 
 
 975974 
 
 
 72 
 
 534092 
 
 6 
 
 |7 
 
 465908 
 
 7 
 
 54 
 
 5 1 o434 
 
 6 
 
 i5 
 
 975o3o 
 
 
 72 
 
 534304 
 
 6 
 
 oo 
 
 466496 
 465o84 
 
 6 
 
 55 
 
 5io8o3 
 
 6 
 
 i5 
 
 975887 
 
 
 72 
 
 534916 
 535328 
 
 6 
 
 5 
 
 56 
 
 511172 
 
 6 
 
 14 
 
 975844 
 
 
 72 
 
 6 
 
 86 
 
 464672 
 
 4 
 
 u 
 
 5u54o 
 
 6 
 
 i3 
 
 975800 
 
 
 72 
 
 535739 
 
 6 
 
 85 
 
 464261 
 
 3 
 
 511907 
 
 6 
 
 i3 
 
 975757 
 
 
 72 
 
 536 1 5o 
 
 6 
 
 S5 
 
 46385o 
 
 2 
 
 5 9 
 
 5i2275 
 
 6 
 
 12 
 
 975714 
 
 
 72 
 
 53656i 
 
 6 
 
 84 
 
 46343o 
 463028 
 
 1 
 
 66 
 
 5i:642 
 
 6 
 
 12 
 
 975670 
 
 
 72 
 
 536972 
 
 6 
 
 84 
 
 
 
 r 
 
 Cosine 
 
 D. 
 
 Bine 
 
 710 
 
 Cotang. 
 
 D. 
 
 Tang. 
 
 M. 
 
BINES AND TANGENTS. (19 DEGRESS.) 
 
 37 
 
 M. 
 
 
 
 Sine 
 
 D. 
 
 Cosine 
 
 D. 
 
 Tang. 
 
 D. 
 
 Cotang. 1 
 
 
 9.512642 
 
 6-12 
 
 9.975670 
 
 ■73 
 
 9.536972 
 537382 
 
 6-84 
 
 10-463028 
 
 60 
 
 i 
 
 5i3oog 
 5i33p 
 
 6- 1 1 
 
 975627 
 
 •73 
 
 6-83 
 
 462618 
 
 5 9 
 
 2 
 
 6-U 
 
 975583 
 
 ■73 
 
 537792 
 538202 
 
 6-83 
 
 462208 
 
 58 
 
 3 
 
 5i374i 
 
 6-io 
 
 975539 
 
 •73 
 
 -6-82 
 
 461798 
 461389 
 
 h 
 
 4 
 
 514107 
 
 6-09 
 
 975496 
 973452 
 
 ■73 
 
 5386i 1 
 
 6-82 
 
 56 
 
 5 
 
 514472 6-09 
 
 ■73 
 
 539020 
 
 6-8i 
 
 460980 
 460671 
 
 55 
 
 6 
 
 5i4837 
 
 6-08 
 
 975408 
 
 ■73 
 
 539429 
 
 6-8i 
 
 54 
 
 I 
 
 5l5202 
 
 6-o8 
 
 9 7 5365 
 
 •73 
 
 539837 
 540245 
 
 6-8o 
 
 460163 
 
 53 
 
 5i5566 
 
 6-07 
 
 975321 
 
 •73 
 
 6-8o 
 
 /i59755 
 
 52 
 
 9 
 
 5i593o 
 
 6-07 
 ' 6-06 
 
 975277 
 975233 
 
 ■73 
 
 540653 
 
 6-79 
 
 45q347 
 
 5i 
 
 IO 
 
 5 1 6294 
 
 9.516667 
 
 517020 
 
 ■73 
 
 54 1 06 1 
 
 6-70 
 
 458939 
 10 -458532 
 
 5o 
 
 II 
 
 6-o5 
 
 9.975189 
 
 •73 
 
 9-541468 
 
 6-78 
 
 % 
 
 12 
 
 6-o5 
 
 9l5i45 
 
 •73 
 
 541875 
 542281 
 
 6-78 
 
 458125 
 
 i3 
 
 517382 
 
 6-o4 
 
 975101 
 
 •73 
 
 6-77 
 
 457719 
 
 47 
 
 14 
 
 517745 
 518107 
 518468 
 
 6-o4 
 
 975o57 
 
 •73 
 
 542688 
 
 6-77 
 
 4573i2 
 
 46 
 
 i5 
 
 6-o3 
 
 975oi3 
 
 ■73 
 
 543094 
 
 6-76 
 
 456906 
 4565oi 
 
 45 
 
 16 
 
 6-o3 
 
 974969 
 
 •74 
 
 543499 
 
 6-76 
 
 44 
 
 \l 
 
 518829 
 
 6-02. 
 
 974925 
 
 ■74 
 
 543900 
 5443io 
 
 6-75 
 
 406095 
 
 43 
 
 519190 
 5i955i 
 
 6-oi 
 
 974880 
 
 •74 
 
 6-75 
 
 4556oo 
 
 42 
 
 i 9 
 
 6-oi 
 
 974836 
 
 •74 
 
 5447i5 
 
 6-74 
 
 455285 
 
 41 
 
 20 
 
 519911 
 
 6-oo 
 
 97479 2 
 
 •74 
 
 545119 
 
 6-74 
 
 45488i 
 
 4o 
 
 21 
 
 9.520271 
 52o63i 
 
 6.oo 
 
 9'-974748 
 
 •74 
 
 9.545524 
 
 b -l\ 
 
 10-454476 
 
 38 
 
 22 
 
 5-99 ) 
 
 9747^3 
 
 •74 
 
 545928 
 
 6-73 
 
 454072 
 
 23 
 
 .520990 
 
 5-99 
 
 974609 
 
 •74 
 
 54633 1 
 
 6-72 
 
 45366n 
 453265 
 
 ll 
 
 * 24 
 
 521349 
 
 5-98 
 
 974614 
 
 •74 
 
 546735 
 
 6-72 
 
 25 
 
 521707 
 
 5- 9 8 
 
 974570 
 
 •74 
 
 547i3S 
 
 6-71 
 
 452862 
 
 35 
 
 26 
 
 522066 
 
 5-97 
 
 974D25 
 
 ■74 
 
 547540 
 
 6-71 
 
 452460 
 
 34 
 
 11 
 
 522424 
 
 5-96 
 
 974481 
 
 ■74 
 
 547943 
 548345 
 
 6-70 
 
 452057 
 
 33 
 
 622781 
 
 5- 9 6 
 
 974436 
 
 ■74 
 
 6-70 
 
 45 i 655 
 
 32 
 
 20 
 
 523i38 
 
 5- 9 5 
 
 974391 
 
 ■74 
 
 548747 
 
 6-69 
 
 45 1 253 
 
 3i 
 
 3o 
 
 5234o5 
 
 9.523832 
 
 524208 
 
 5- 9 5 
 
 974347 
 
 ■75 
 
 549 1 49 
 
 6-69 
 
 45o85i 
 
 3o 
 
 3i 
 
 5-94 
 
 9-974302 
 
 ■75 
 
 9 ■ 549550 
 
 6-68 
 
 io«45o45o 
 
 18 
 
 32 
 
 5- 9 4 ' 
 
 - 974 2 57 
 
 •75 
 
 549o5 1 
 55o352 
 
 6-68 
 
 45ao4o 
 449648 
 
 33 
 
 524564 
 
 5- 9 3 
 
 974212 
 
 •75 
 
 6-67 
 
 27 
 
 34 
 35 
 
 524920 
 52J275 
 
 5- 9 3 
 5-92 
 
 974167 
 974122 
 
 •75 
 •75 
 
 550752 
 55n52 
 
 6-67 
 6-66 
 
 449248 
 448848 
 
 26 
 
 25 
 
 36 
 
 52563o 
 
 5-91 
 
 974077 
 
 •75 
 
 55 1 552 
 
 6-66 
 
 448448 
 
 24 
 
 12 
 
 525o84 
 52633o 
 526693 
 
 5-91 
 5-90 
 
 974032 
 973987 
 
 ■75 
 ■75 
 
 55io52 
 55235i 
 
 6-65 
 6-65 
 
 448048 
 4*47649 
 
 23 
 22 
 
 3 9 
 
 5.90 
 5-8 9 
 
 973942 
 
 ■75 
 
 552750 
 
 6-65 
 
 44725o 
 
 21 
 
 40 
 
 527046 
 
 973897 
 
 ■75 
 
 553 1 49 
 
 9-553548 
 
 553o46 
 
 554344 
 
 6-64 
 
 446851 
 
 20 
 
 42 
 
 43 
 
 9.527400 
 527753 
 528io5 
 
 5-8 9 
 5-88 
 5-88 
 
 9-973852 
 973807 
 973761 
 
 ■75 
 ■75 
 •75 
 
 6-64 
 6-63 
 6-63 
 
 IO-446452 
 446o54 
 445656 
 
 !8 
 
 '1 
 
 44 
 
 528458 
 
 5-87 
 
 973716 
 
 •76 
 
 554741 
 
 6-62 
 
 445259 
 
 16 
 
 45 
 
 528810 
 
 5-8 7 
 
 973671 
 
 •76 
 
 555 1 39 
 
 6-62 
 
 444861 
 
 i5 
 
 46 
 
 529161 
 5295i3 
 529864 
 
 5-86 
 
 973625 
 
 •76 
 
 555536 
 
 6-61 
 
 444464 
 
 14 
 
 % 
 
 5-86 
 5-85 
 
 973580 
 973535 
 
 ■76 
 .76 
 
 555o33 
 55632c 
 556721 
 
 6-6i 
 6-6o . 
 
 444067 
 443671 
 
 i3 
 12 
 
 40 
 
 53o2i5 
 
 5-85 
 
 973489 
 
 .76 
 
 6-6o 
 
 443275 
 
 11 
 
 5o 
 
 53o565 
 
 5-84 
 
 973444 
 
 ■76 
 
 557121 
 
 6-59 
 
 442870 
 
 10 
 
 5i 
 
 9. 53o9i5 
 531265 
 
 5.84 
 
 9-973398 
 
 •76 
 
 9.557517 
 
 6-59 
 
 IO-442483 
 
 8 
 
 52 
 
 5-83 
 
 97 3352 
 
 .76 
 
 557913 
 5583o8 
 
 6-5o 
 6-58 
 
 442087 
 
 53 
 
 53i6i4 
 
 5.82 
 
 973307 
 
 .76 
 
 441692 
 
 
 
 54 
 
 53io63 
 
 532312 
 
 5-82 • 
 
 973261 
 
 .76 
 
 ■ 558702 
 
 6-58 
 
 441298 
 
 55 
 
 5-8i 
 
 9732i5 
 
 -76 
 
 M9097 
 
 6-57 
 
 440903 
 
 5 
 
 56, 
 
 532661 
 
 5.8i 
 
 973169 
 
 .76 
 
 559491 
 55 9 885 
 
 6-57 
 
 440009 
 4401 15 
 
 4 
 3 
 
 5? 
 
 533009 
 533357 
 533704 
 534o52 
 
 . 5- 80 
 
 973124. 
 
 .76 
 
 6-56 
 
 58 
 60 
 
 5. So 
 5.78 
 
 973078 
 973o32 
 972986 
 
 .76 
 ■77 
 •77 
 
 66027c 
 56067' 
 
 56io66 
 
 6-56 
 6-55 
 6-55 
 
 439721 
 43o327 
 438934 
 
 2 
 1 
 
 
 
 
 Cosine 
 
 ]>. 
 
 Sine 
 
 T0° 
 
 Cotang. 
 
 D. 
 
 Tang. M. 
 
38 
 
 (20 DEGREES.) A TAB1E OF LOGARITHMIC 
 
 M. 
 
 Sine 
 
 D. 
 
 12 
 
 i3 
 U 
 
 yl5 
 16 
 
 19 
 20 
 21 
 22 
 23 
 24 
 
 25 
 
 26 
 
 ?9 
 3o 
 3i 
 3a 
 .33 
 .34 
 35 
 36 
 
 §3 
 3 9 
 40 
 •n 
 42 
 43 
 44 
 45 
 
 46 
 % 
 
 £ 
 
 5i 
 
 52 
 53 
 54 
 55 
 56 
 
 33 
 
 2S 
 
 )• 534032 
 534399 
 034745 
 535og2 
 53543<i 
 535i83 
 536129 
 536474 
 5368.8 
 ■537163 
 537507 
 )-53j85i 
 538194 
 538538 
 53888o 
 539223 
 539565 
 539907 
 540249 
 54o5oo 
 5409J 1 
 9-541272 
 54i6i3 
 541953 
 5422o3 
 54263s 
 542971 
 5433io 
 543649 
 543987 
 544325 
 q- 544663 
 545ooo 
 545338 
 545674 
 546oi 1 
 546347 
 546683 
 547019 
 V7354 
 547689 
 9.548024 
 54835 9 
 548693 
 549027 
 549360 
 549693 
 55oo26 
 55o359 
 550692 
 55io24 
 9-55i356 
 55i68 7 
 5520i8 
 552349 
 55268o 
 553oio 
 553341 
 553670 
 554ooo 
 554329 
 
 5.78 
 5-77 
 5-77 
 
 111 
 
 5-76 
 
 5- 7 5 
 
 5-74 
 
 5-74 
 
 5-73 
 
 5-73 
 
 5-72 
 
 5-72 
 
 5-71 
 
 5-71 
 
 5-70 
 
 5-70 
 
 5-6 9 
 
 5.69 
 
 5-68 
 
 5-68 
 
 5-67 
 
 5-67 
 
 5-66 
 
 5-66 
 
 5-65 
 
 5-65 
 
 5-64 
 
 5-64 
 
 5-63 
 
 5-63 
 
 5-62 
 
 5-62 
 
 5-6i 
 
 5-6l 
 
 5-6o 
 
 5-6o 
 
 5-5 9 
 
 5.5 9 
 
 5-58 
 
 5-58 
 
 5.57 
 
 5.57 
 
 5-56 
 
 5-56 
 
 5-55 
 
 5-55 
 
 5-54 
 
 5-54 
 
 5-53 
 
 5-53 
 
 5-52 
 
 5-52 
 
 5-52 
 
 5-5i 
 
 5-5i 
 
 5-5o 
 
 5-5o 
 
 5-49 
 
 5-49 
 
 5.48 
 
 Cosine I O. 
 
 9-972986 -77 
 972940 -77 
 
 972894 -n 
 
 972848 -77 
 •77 
 
 972802 
 
 972 7 55 -77 
 •77 
 ■77 
 •77 
 •77 
 •77 
 
 972709 
 972663 
 972617 
 972570 
 972524 
 
 9-972478 
 972431 
 972385 
 972338 
 972291 
 972245 
 972108 
 9721S1 
 972105 
 972o58 
 
 9-972011 
 97 1964 
 
 97'9'7 
 971870 
 971823 
 971776 
 971729 
 971682 
 971635 
 971588 
 
 9-971540 
 97 • 493 
 971446 
 971398 
 971351 
 97i3o3 
 971256 
 971208 
 971161 
 97iu3 
 
 9-971066 
 971018 
 970970 
 970922 
 970874- 
 970827 
 970779 
 970731 
 970683 
 970635 
 
 9-970586 
 970538 
 970490 
 970442 
 970394 
 970345 
 970297, 
 970249 
 970200 
 970152 
 
 Cosine 
 
 Tang. 
 
 ■11 
 .78 
 -78 
 -78 
 .78 
 •78 
 -78 
 .78 
 ■78 
 .78 
 
 ■^ 
 •78 
 -78 
 -78 
 ■78 
 •79 
 •79 
 ■79 
 ■79 
 •79 
 •79 
 ■79 
 •79 
 ■79 
 •79 
 •79 
 ■79 
 ■79 
 
 :ll 
 .80 
 .80 
 -80 
 
 •81 
 
 D. 
 
 9.561066 
 561459 
 56n 85 1 
 562244 
 562636 
 563028 
 563419 
 5638i 1 
 564202 
 564592 
 564Q83 
 
 9-565373 
 565763 
 566i 53 
 566542 
 566o32 
 567J20 
 567709 
 568oo8 
 568486 
 568873 
 569261 
 
 570035 
 570422 
 570801 
 57119 
 '5 7 i58i 
 571967 
 572352 
 572738 
 
 )-573i23 
 573507 
 573892 
 574276 
 574660 
 575o44 
 575427 
 573810 
 576193 
 576576 
 
 5-576958 
 577341 
 577723 
 578104 
 
 578867 
 579248 
 579629 
 580009 
 58o38 9 
 9-580769 
 58n49 
 58i528 
 581907 
 582286 
 582665 
 583043 
 583422 
 5838oo 
 584H7 
 
 Sine |6a°j Cotang. 
 
 6-55 
 
 6-54 
 
 6-54 
 
 6-53 
 
 6-53 
 
 6-53 
 
 6-52 
 
 6-52 
 
 6-51 
 
 6-5i 
 
 6-5o 
 
 6-5o 
 
 6-49 
 
 6-49 
 
 6-4Q 
 
 6-48 
 
 6.48 
 
 6-47 
 
 6-47 
 
 6-46 
 
 6-46 
 
 6-45 
 
 6-45 
 
 6-45 
 
 6-44 
 
 6-44 
 
 6-43 
 
 6-43 
 
 6-42 
 
 6-42 
 
 6-42 
 
 6-41 
 
 6-4i 
 
 6-40 
 
 6-40 
 
 6-3 9 
 
 6-39 
 
 6-3 9 
 
 6-38 
 
 6-38 
 
 6-37 
 
 6-37 
 
 6-36 
 
 6-36 
 
 6-36 
 
 6-35 
 
 6-35 
 
 6-34 
 
 6-34 
 
 6-34 
 
 6-33 
 
 6-33 
 
 6-32 
 
 6-32 
 
 6-32 
 
 6-3i 
 
 6-3i 
 
 6-3o 
 
 6-3o 
 
 6-29 
 
 6- 29 
 
 Cotang. 
 
 o-438o34 
 438541 
 438i49 
 437756 
 43l364 
 436972 
 436581 
 436189 
 435798 
 4354o8 
 435017 
 
 10-434627 
 434237 
 433847 
 433458 
 433o68 
 432680 
 432291 
 431902 
 43 1 5 1 4 
 431127 
 
 10-430739 
 43o352 
 429965 
 429378 
 429(91 
 4288o5 
 421" 
 . 428o33 
 427648 
 427262 
 
 10-426877 
 426493 
 426108 
 425724 
 425340 
 424956 
 4245 7 3 
 424190 
 423807 
 423424 
 
 10-42 3o4i 
 422659 
 422277 
 421896 
 
 421314 
 
 421133 
 420752 
 420371 
 419991 
 41961 1 
 419231 
 4i885i 
 
 418472 
 418093 
 417714 
 417335 
 416957 
 416678 
 416200 
 415823 
 
 \ 
 
 5 9 
 58 
 
 u 
 
 55 
 54 
 53 
 52 
 5i 
 5o 
 49 
 
 1). 
 
 Tang. I M. 
 

 SINES AND TANGENTS. 
 
 " (21 DEUBEKS.) 
 
 
 39 
 
 -Mil 
 
 o 
 
 Sine 
 
 9-5543a9 
 
 554658 
 
 1). 
 
 Cjshie 
 
 D. 
 
 Tang. | 
 
 P. | Cotang. | 
 
 
 5-48 
 
 9970152 
 
 ~tii 
 
 9-584177, 
 584555' 
 
 6-29 
 
 16-415823 
 
 nr 
 
 
 5-48 
 
 970 io3 
 
 .81 
 
 6-29 
 6-28 
 
 415445 
 
 is 
 
 2 
 
 554987 
 5553 1 5 
 
 5.47 
 
 970055 
 
 -81 
 
 584932| 
 585309! 
 
 41 5o68 
 
 3 
 
 5-47 
 
 970006 
 
 • 81 
 
 6-28 
 
 414691 
 
 ti 
 
 4 
 
 555J343 
 
 5-46 
 
 969957 
 
 .81 
 
 585686 
 
 6-27 
 
 4143U 
 
 ;56 
 
 5 
 
 555*97 1 
 
 5-46 
 
 969909 
 
 .81 
 
 586o62 
 
 6-27 
 
 413938 
 
 55 
 
 6 
 
 556299 
 
 5-45 
 
 969860 
 
 -81 
 
 58643o 
 5868i5 
 
 6-27 
 
 4i356i 
 
 54 
 
 7 
 8 
 
 556626 
 
 5-45 
 
 969811 
 
 • 81 
 
 6-26 
 
 4i3i85 
 
 53 
 
 556953 
 
 5-44 
 
 969762 
 
 • 81 
 
 587100 
 587666 
 
 6-26 
 
 412810 
 
 52 
 
 9 
 
 10 
 
 557280 
 
 5-44 
 
 969714 
 
 • 81 
 
 6-25 
 
 412434 
 
 5i 
 
 557606 
 
 5-43 
 
 969665 
 
 .81 
 
 587941 
 
 6-25 
 
 412069 
 
 5o 
 
 ii 
 
 9-557932 
 
 5-43 
 
 9-969616 
 
 .82 9-588316 
 
 6-25 
 
 10-411684 
 
 4 3 
 
 12 
 
 558258 
 
 5-43 
 
 969667 
 
 .82 
 
 588691 
 
 6-24 
 
 41 1 309 
 
 48 
 
 13 
 
 553583 
 
 5-42 
 
 969018 
 
 ■ 82 
 
 589066 
 
 6-24 
 
 410934 
 
 47 
 
 • U 
 
 558909 
 
 5-42 
 
 969469 
 
 •82 
 
 589440 
 
 6-23 
 
 4io56o 
 
 46 
 
 i5 
 
 55n234 
 
 5-41 
 
 $69420 
 
 ■82 
 
 589814 
 
 6-23 
 
 410186 
 
 45 
 
 16 
 
 55 9 558 
 
 5-4i 
 
 969370 
 
 .82 
 
 .690188 
 
 6-23 
 
 409812 
 
 44 
 
 » 
 
 55 9 883 
 
 5 .'40 
 
 969321 
 
 •82 
 
 590562 
 
 6-22 
 
 409438 
 
 43 
 
 560207 
 
 5-4o 
 
 969272 
 
 ■82 
 
 590930 
 
 6-22 
 
 409065 
 
 42 
 
 19 
 
 20 
 
 6o53i 
 
 5-3 9 
 
 969223 
 
 .82 
 
 591308 
 
 6-22 
 
 408692 
 
 4i 
 
 56o855 
 
 5-39 
 
 969173 
 
 • 82 
 
 591681 
 
 6-21 
 
 408319 
 
 40 
 
 21 
 
 9-56ii78 
 
 5-38 
 
 9-969124 
 
 .82 
 
 9-592064 
 
 6-21 
 
 10-407946 
 
 3 9 
 
 22 
 
 56i5oi 
 
 5-38 
 
 ."969076 
 
 •82 
 
 592426 
 
 6-20 
 
 407674 
 
 38 
 
 23 
 
 56i824 
 
 5.37 
 
 969025 
 
 •82 
 
 592798 
 
 6-20 
 
 407202 
 
 37 
 
 24 
 
 562146 
 
 5-37 
 5-36 
 
 968976 
 
 ■82 
 
 593170 
 
 6- 19 
 
 406829 
 
 36 
 
 25 
 
 562468 
 
 968926 
 
 . -83 
 
 693542 
 
 6-19 
 
 406458 
 
 35 
 
 26 
 
 662790 
 
 5-36 
 
 968877 
 
 •83 
 
 593914 
 
 6-i8 
 
 406086 
 
 34 
 
 22 
 
 563U2 
 
 5-36 
 
 968827 
 
 •83 
 
 5g4285 
 
 618 
 
 405716 
 
 33 
 
 563433 
 
 5-35. 
 
 968777 
 968728 
 
 ■83 
 
 5 9 4656 
 
 6- 18 
 
 4o5344 
 
 32 
 
 £ 
 
 563755 
 
 5-35 
 
 ■83 
 
 595027 
 59D398 
 
 6-17 
 
 404973 
 
 3[ 
 
 564075 
 
 5-34 
 
 968678 
 
 •83 
 
 6-n 
 
 404602 
 
 3o 
 
 3i 
 
 9-564396 
 
 5-34 
 
 9-968628 
 
 ■83 
 
 9-596768 
 
 6-17 
 6- 16 
 
 10-404232 
 
 ll 
 
 32 
 
 5647i6 
 
 5-33 
 
 968678 
 
 ■83 
 
 5 9 6i38 
 
 4o3862 
 
 33 
 
 565o36 
 
 5-33 
 
 96S528 
 
 •83 
 
 5g65o8 
 
 6- 16 
 
 403492 
 
 27 
 
 34 
 
 565356 
 
 5-32 
 
 968479 
 
 ■83 
 
 596878 
 
 6-i6 
 
 4o3i22 
 
 26 
 
 35 
 
 565676 
 
 5-32 
 
 968429 
 
 •83 
 
 597247 
 
 6-i5 
 
 402753 
 
 25 
 
 36 
 
 566995 
 5663i4 
 
 5-3i 
 
 968379 
 
 •83 
 
 597616 
 
 6-1 5 
 
 402384 
 
 24 
 
 ll 
 
 5-3i 
 
 968329 
 
 •83 
 
 697986 
 5 9 3354 
 
 6-i5 
 
 4o20i5 
 
 23 
 
 566632 
 
 5-3i 
 
 968278 
 
 •83 
 
 6-14 
 
 ' 401646 
 
 22 
 
 3, 
 4o 
 
 56695i 
 
 5-3o 
 
 963228 
 
 •84 
 
 698722 
 
 614 
 
 401278 
 
 21 
 
 567269 
 
 5-3o 
 
 968178 
 
 ■84 
 
 599091 
 9-599459 
 
 6-i3 
 
 400909 
 io- 400041 
 
 -20 
 
 4i 
 
 9-567687 
 
 5-29 
 
 9-968128 
 
 .84 
 
 6-i3 
 
 18 
 
 42 
 
 567904 
 
 568222 
 
 5-29 
 
 968078 
 
 ■84 
 
 599827 
 
 6-i3 
 
 400173 
 
 43 
 
 5-28 
 
 968027 
 
 .84 
 
 600194 
 
 6-12 
 
 399806 
 
 \l 
 
 44 
 
 56853g 
 
 5-28 
 
 967977 
 
 •84 
 
 600662 
 
 6-12 
 
 3 99 438 
 
 45 
 
 568856 
 
 5-28 
 
 967927 
 
 ■ 84 
 
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 939883 
 
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 653663 
 
 5-62 
 
 346337 
 
 45 
 
 16 
 
 6i3825 
 
 4-67 
 4-66 
 
 93982!: 
 
 • 9 5 
 
 654ooo 
 
 5-62 
 
 346000 
 
 44 
 
 17 
 
 6i4io5 
 
 939768 
 
 •gS 
 
 634337 
 
 5-6i 
 
 345663 
 
 43 
 
 18 
 
 6i4385 
 
 4-66 
 
 9397 1 1 
 
 ■95 
 
 654674 
 
 5-6i 
 
 345326 
 
 42 
 
 '9 
 
 6 1 4665 
 
 4-66 
 
 95054 
 
 ■95 
 
 655on 
 
 5-6i 
 
 344989 
 
 41 
 
 20 
 
 61494-! 
 
 4-65 
 
 939596 
 
 .95 
 
 655348 
 
 5-6i 
 
 344652 
 
 40 
 
 21 
 
 9-613223 
 
 4-65 
 
 9-959539 
 
 • 9 5 
 
 9-655684 
 
 5-6o 
 
 io-3443i6 
 
 3 9 
 
 22 
 
 6i."5o2 
 
 4-65 
 
 959482 
 
 .95 
 
 656o2o 
 
 5-6o 
 
 343980 
 
 38 
 
 23 
 
 615781 
 
 4-64 
 
 939425 
 
 .95 
 
 656356 
 
 5-6a 
 
 343644 
 
 ll 
 
 24 
 
 616060 
 
 4-64 
 
 939368 
 
 • 9 5 
 
 656692 
 
 5.59 
 
 3433o8 
 
 25 
 
 616338 
 
 4-64 
 
 939310 
 
 •96 
 
 657028 
 
 5-5 9 
 
 342972 
 
 35 
 
 26 
 
 616616 
 
 4-63 
 
 939253 
 
 .96 
 
 657364 
 
 5-5 9 
 
 342636 
 
 34 
 
 s 
 
 616894 
 
 4-63 
 
 959195 
 95913s 
 
 .96 
 
 ' 657699 
 
 5-5 9 
 5-58 
 
 3423oi 
 
 33 
 
 617172 
 
 4-62 
 
 .96 
 
 658o34 
 
 341966 
 
 32 
 
 ?9 
 
 617400 
 
 4-62 
 
 959081 
 
 .96 
 
 65836 9 
 
 5-58 
 
 34i63i 
 
 3i 
 
 3o 
 
 617727 
 
 4-62 
 
 939023 
 
 .96 
 
 658704 
 
 5-58 
 
 341296 
 
 3o 
 
 3i 
 
 9.618004 
 
 4-61 
 
 9-958963 
 
 .96 
 
 9"659o3o 
 639373 
 
 5-58 
 
 10-340961 
 
 3 
 
 3; 
 
 6182^1 
 
 4-6i 
 
 938908 
 95885o 
 
 .96 
 
 5-57 
 
 340627 
 
 33 
 
 6 1 8558 
 
 4-6i 
 
 .96 
 
 659708 
 
 5-57 
 
 340292 
 339968 
 
 27 
 
 34 
 
 6i8834 
 
 - 4-6o 
 
 958792 
 
 .96 
 
 660042 
 
 5.57 
 
 26 
 
 35 
 
 619110 
 
 4-6o 
 
 938734 
 
 .96 
 
 660376 
 
 5-5 7 
 5.56 
 
 339624 
 
 25 
 
 36 
 
 619386 
 
 4-'o 
 
 938677 
 
 .96 
 
 660710 
 
 339290 
 338957 
 
 24 
 
 12 
 
 619662 
 
 4- 5g 
 
 938619 
 
 .96 
 
 661043 
 
 5-56 
 
 23 
 
 619938 
 
 4-59 
 
 958561 
 
 .96 
 
 661377 
 
 5-56 
 
 338623 
 
 22 
 
 3 9 
 
 620213 
 
 4-5o- 
 
 9385o3 
 
 •97 
 
 661710 
 
 5-55 
 
 . 3382O0 
 
 21 
 
 40 
 
 620488 
 
 4-58 
 
 958445 
 
 ■97 
 
 662043 
 
 5-55 
 
 337907 
 
 20 
 
 4i 
 
 9-620763 
 
 4-58 
 
 9-938387 
 
 ■97 
 
 9-662376 
 
 5-55 
 
 10-337624 
 
 10 
 
 42 
 
 62io38 
 
 4-57 
 
 938329 
 
 •07 
 
 662709 
 
 5-54 
 
 337291 
 336958 
 
 l8 
 
 43 
 
 62i3i3 
 
 4-57 
 
 958271I .97 
 
 663o42 
 
 5-54 
 
 n 
 
 44 
 
 621587 
 
 4-37 
 4-56 
 
 95821 3 -o7 
 
 6633 7 5 
 
 5-54 
 
 336625 
 
 16 
 
 '45 
 
 621861 
 
 958 1 54 
 
 ■97 
 
 663707 
 
 5-54 
 
 336293 
 
 i5 
 
 46 
 
 622i35 
 
 4-56 
 
 958096 
 
 •97 
 
 664039 
 
 5-53 
 
 335961 1 14 
 
 % 
 
 62^409 
 
 4-56 
 
 958o38 
 
 •97 
 
 664371 
 
 5-53 
 
 335629] l3 
 
 622682 
 
 4-55 
 
 937979 
 
 •97 
 
 664703 
 
 5-53 
 
 335297 
 
 12 
 
 P 
 
 6229561 4-35 
 623229 4 '-^ 
 
 957921 
 
 ■97 
 
 665o35 
 
 v 5-53 
 
 334965 
 
 11 
 
 5o 
 
 957863 
 
 •97 
 
 665366 
 
 5-52 
 
 334634 
 
 10 
 
 5i 
 
 9-6235o2 
 
 4-54 
 
 9-957804 
 
 •97 
 
 9 -6656 9 7 
 
 5-52 
 
 io-3343o3 
 
 2 
 
 52 
 
 623774 
 
 4-54 
 
 957746 
 
 .98 
 
 666029 
 
 5-52 
 
 333971 
 
 53 
 
 624047 
 
 4-54 
 
 937687 
 
 .98 
 
 66636o 
 
 5-5i 
 
 333640 
 
 I 
 
 54 
 
 624319 
 
 4-53 
 
 957628 
 
 ■98 66669I 
 
 5-5i 
 
 333309 
 
 55 
 
 624591 4-3J 
 
 '957570 
 
 ■98 667021 
 
 5-5i ' 
 
 332979 
 . 332643 
 
 5 
 
 56 
 
 624863 
 
 4-33 
 
 95751 1 
 
 .98 
 
 667352 
 
 5-5i 
 
 4 
 
 tl 
 
 625i35 
 
 4-52 
 
 937452 
 
 .98 
 
 667682 
 
 5-5o 
 
 3323i8 
 
 3 
 
 625406 
 
 4-52 
 
 957393 
 957335 
 
 .98 
 
 66801.3 
 
 5-5o 
 
 331987 
 
 2 ■ 
 
 5, 
 
 625677 
 625948 
 
 4-52 
 
 .98 
 
 668343 
 
 5-5o 
 
 33 1657 
 
 1 
 
 6o 
 
 4-5i 
 
 957276 
 
 •98 
 
 668672 
 
 5-5o 
 
 33i328 
 
 
 
 
 Cosine 
 
 Sine 
 
 ea° 
 
 Uotiuitf. 
 
 D. ' 
 
 .JP»fc_J 
 
 M. 
 

 SIN'ES 
 
 V,*D TANGENTS. 
 
 (25 DEGREES.' 
 
 
 43 
 
 It. 
 
 
 
 Sine 
 
 p. 
 
 Cosine | D. 
 
 Tan ff . 
 
 D. 
 
 Cotang. 
 
 
 9-625948 
 
 4 
 
 5l 
 
 0-957276 -98 
 
 9-668673 
 
 5-5o 
 
 Io-33i327 
 
 60 
 
 I 
 
 626219 
 
 4 
 
 5l 
 
 937217 
 957158, 
 
 .98 
 
 669002 
 
 5 
 
 49 
 
 330998 
 
 n 
 
 a 
 
 626490 
 
 4 
 
 5i 
 
 .98 
 
 669332 
 
 5 
 
 49 
 
 33o668 
 
 3 
 
 626760 
 
 4 
 
 5o 
 
 957099 
 
 .98 
 
 669661 
 
 5 
 
 49 
 
 33o33 9 
 
 p 
 
 4 
 
 627030 
 
 4 
 
 5o 
 
 957040 
 
 .98 
 
 669991 
 670320 
 
 5 
 
 48 
 
 330009 
 
 56 
 
 5 
 
 627300 
 
 4 
 
 5o 
 
 936981 
 
 • 98 
 
 5 
 
 48 
 
 329680 
 
 55 
 
 • 6 
 
 627570 
 
 4 
 
 49 
 
 936921 
 9 56862 
 
 •99 
 
 670649 
 
 5 
 
 48 
 
 329351 
 
 54 
 
 8 
 
 627840 
 628109 
 
 4 
 
 49 
 
 •99 
 
 670977 
 671306 
 
 5 
 
 48 
 
 329023 
 
 53 
 
 4 
 
 49 
 
 9 568o3 
 
 •99 
 
 5 
 
 47 
 
 328694 
 
 52 
 
 9 
 
 628378 
 
 4 
 
 48 
 
 956744 
 956684 
 
 ■99 
 
 671634 
 
 5 
 
 47 
 
 328366 
 
 5i 
 
 10 
 
 628647 
 9-628916 
 
 4 
 
 48 
 
 ■99 
 
 671963 
 
 5 
 
 47 
 
 328037 
 
 5o 
 
 ii 
 
 4 
 
 47 
 
 n-956625 
 
 ■99 
 
 9-672291 
 
 5 
 
 4 I 
 
 ,0-327709 
 
 49 
 
 12 
 
 629185 
 
 4 
 
 47 
 
 9 56566 
 
 ■99 
 
 672619 
 
 5 
 
 46 
 
 327381 
 
 48 
 
 i3 
 
 629453 
 
 4 
 
 a. 
 
 9565o6 
 
 •99 
 
 672947 
 
 5 
 
 46 
 
 32io53 
 326726 
 326398 
 
 47 
 
 14 
 
 629721 
 
 4 
 
 956447 
 
 ■99 
 
 673274 
 
 5 
 
 46 
 
 46 
 
 i5 
 
 629989 
 
 4 
 
 46' 
 
 936387 
 
 ■99 
 
 673602 
 
 5 
 
 46 
 
 45 
 
 16 
 
 630237 
 
 4 
 
 46 
 
 956327 
 936268 
 
 ■99 
 
 673929 
 
 5 
 
 45 
 
 326071 
 
 44 
 
 >7 
 
 63o524 
 
 4 
 
 46 
 
 ■99 
 
 674257 
 
 5 
 
 45 
 
 325743 
 
 43 
 
 18 
 
 630792 
 
 4 
 
 45 
 
 956208 
 
 •00 
 
 6 7 4584 
 
 5 
 
 45 
 
 3254i6 
 
 42 
 
 '9 
 
 63 1009 
 
 4 
 
 45 
 
 956148 
 
 •00 
 
 674910 
 
 5 
 
 44 
 
 325090 
 
 41 
 
 20 
 
 63i326 
 
 4 
 
 45 
 
 956089 
 
 •00 
 
 675237 
 
 5 
 
 44 
 
 324763 
 
 40 
 
 21 
 
 9-63i5q3 
 63i85 9 
 632125 
 
 4 
 
 44 
 
 g- 956029 
 
 •00 
 
 9.675564 
 
 5 
 
 44 
 
 ,o.324436 
 
 It 
 
 22 
 
 4 
 
 44 
 
 955969 
 
 •00 
 
 675890 
 
 5 
 
 44 
 
 324110 
 
 23 
 
 4 
 
 44 
 
 955909 
 
 955849 
 
 •00 
 
 676216 
 
 5 
 
 43 
 
 323784 
 
 37 
 
 24 
 
 6323o2 
 632658 
 
 4 
 
 43 
 
 •00 
 
 676543 
 
 5 
 
 43 
 
 323457 
 
 36" 
 
 25 
 
 4 
 
 43 
 
 955789 
 
 -00 
 
 676869 
 
 5 
 
 43 
 
 323i3i 
 322806 
 
 35 
 
 26 
 
 632923 
 
 4 
 
 43 
 
 955-729 
 
 •00 
 
 677194 
 
 5 
 
 43 
 
 34 
 
 2 
 
 633189 
 
 4 
 
 42 
 
 955669 
 
 •00 
 
 677520 
 
 5 
 
 42 
 
 322480 
 
 33 
 
 633454 
 
 4 
 
 42 
 
 955609 
 
 • 00 
 
 677846 
 678171 
 
 5 
 
 42 
 
 322)54 
 
 32 
 
 29 
 
 633719 
 
 4 
 
 42 
 
 955548. 
 
 ■00 
 
 5 
 
 42 
 
 321829 
 
 3i 
 
 3o 
 
 633984 
 
 4 
 
 41 
 
 955488 
 
 •00 
 
 678496 
 
 5 
 
 42 
 
 32i5o4 
 
 3o 
 
 3i 
 
 9-634249 
 
 4 
 
 41 
 
 0-955428, 
 
 ■01 
 
 9.678821 
 
 5 
 
 4i 
 
 ,0-321179 
 
 29 
 
 32 
 
 6345 14 
 
 4 
 
 40 
 
 9 55368 
 
 ■01 
 
 679146 
 
 5 
 
 41 
 
 320854 
 
 28 
 
 33 
 
 634778 
 
 4 
 
 40 
 
 955307 1 
 
 ■01 
 
 679471 
 
 5 
 
 41 
 
 32o52q 
 
 320203 
 
 27 
 
 34 
 
 635o42 
 
 4 
 
 40 
 
 955247 
 
 •01 
 
 679795 
 6S0120 
 
 5 
 
 41 
 
 26 
 
 35 
 
 6353o6 
 
 4 
 
 3 9 
 
 955186 
 
 •01 
 
 5 
 
 40 
 
 319880 
 
 25 
 
 36 
 
 635570 
 
 4 
 
 3 9 
 
 955126 
 
 ■01 
 
 680444 
 
 5 
 
 40 
 
 .319536 
 
 24 
 
 ■ Ii 
 
 635834 
 
 4 
 
 3 9 
 
 955o65 
 
 • 01 
 
 680768 
 
 5 
 
 40 
 
 319232 
 
 23 
 
 '636097 
 
 4 
 
 38 
 
 o55oo5 
 
 ■ 01 
 
 681092 
 
 5 
 
 40 
 
 318908 
 
 3 1 8584 
 
 22 
 
 3 9 
 
 63636o 
 
 4 
 
 38 
 
 954944 
 
 ■01 
 
 681416 
 
 5 
 
 3 9 
 
 ,21 
 
 4o 
 
 636623 
 
 4 
 
 38 
 
 954883 
 
 • 01 
 
 681740 
 
 5 
 
 3 9 
 
 318260 
 
 20 
 
 41 
 
 9-636886 
 
 4 
 
 37 
 
 9-954823 
 
 • 01 
 
 q. 682o63 
 
 5 
 
 3 9 
 
 I0.3I7937 
 
 IO 
 
 42 
 
 637148 
 
 4 
 
 37 
 
 954762 
 
 • 01 
 
 682387 
 
 5 
 
 ll 
 
 317613 
 
 IS 
 
 43 
 
 637411 
 
 4 
 
 37 
 
 954701 
 
 ■ 01 
 
 682710 
 
 5 
 
 3n 2 9 
 
 17 
 
 44 
 
 63 7 6 7 3 
 
 4 
 
 37 
 
 954640 
 
 • 01 
 
 683o33 
 
 5 
 
 38 
 
 3 1 6967 
 
 16 
 
 45 
 
 637935 
 
 4 
 
 36 
 
 954579 
 
 •c: 
 
 683356 
 
 5 
 
 38 
 
 316644 
 
 i5 
 
 46 
 
 638197 
 638458 
 
 4 
 
 36 
 
 954518 
 
 • 02 
 
 683679 
 
 5 
 
 38 
 
 ,3i632i 
 
 14 
 
 47 
 
 4 
 
 36 
 
 954457 
 
 •02 
 
 684001 
 
 5 
 
 37 
 
 3 1 3999 
 
 i3 
 
 48 
 
 638720 
 
 4 
 
 35 
 
 934396 
 954335 
 
 ■02 
 
 684324 
 
 5 
 
 37 
 
 -3 1 2676 
 
 12 
 
 49 
 
 638981 
 
 4 
 
 35 
 
 ■02 
 
 684646 
 
 5 
 
 37 
 
 3i5354 
 
 11 
 
 5o 
 5i 
 
 639242 
 9 -639303 
 
 4 
 4 
 
 35 
 34 
 
 954274 
 9-954213 
 
 •02 
 ■02 
 
 684968 
 0-685290 
 
 5 
 5 
 
 37 
 36 
 
 3 1 5o32 
 lo-3i47i° 
 
 10 
 I 
 
 52 
 
 639764 
 
 4 
 
 34 
 
 954i52 
 
 •02 
 
 685612 
 
 5 
 
 36 
 
 3U388 
 
 53 
 
 640024 
 
 4 
 
 34 
 
 954090 
 
 ■02 
 
 685934 
 
 5 
 
 36 
 
 314066 
 
 
 
 54 
 
 640284 
 
 4 
 
 33 
 
 954029 
 953968 
 
 •02 
 
 686255 
 
 5 
 
 36 
 
 3 1 3745 
 
 55 
 
 640544 
 
 4 
 
 33 
 
 ■02 
 
 686577 
 
 5 
 
 35 
 
 3 1 3423 
 
 5 
 
 56 
 
 640804 
 
 4 
 
 33 
 
 953906 
 
 ■02 
 
 • 686898 
 
 5 
 
 35 
 
 3i3i02 
 
 4 
 
 .57 
 
 641064 
 
 4 
 
 32 
 
 953845 
 
 •02 
 
 687219 
 
 5 
 
 35 
 
 312781 
 
 3 
 
 58 
 
 64i324 
 
 4 
 
 32 
 
 953783 
 
 •02 
 
 687540 
 
 5 
 
 35 
 
 3 1 2460 
 
 2 
 
 5 9 
 
 64 1 584 
 
 4 32 
 
 933722 1 
 
 ■03 
 
 687861 
 
 5 
 
 34 
 
 3 1 2 1 39 
 3n8i8 
 
 1 
 
 60 
 
 641842 
 
 4-3i 
 
 953660 
 
 ■o3 
 
 688182 
 
 5-34 
 
 
 
 
 Cosine 
 
 D. 
 
 Sine 1 
 
 !4° 
 
 Cotanfr. 
 
 D. 
 
 Tang. 
 
 M. 
 
44 
 
 (2G 
 
 DEGREES.) A TABLE OF LOGARITHMIC 
 
 
 M. 
 a 
 
 Sine 
 
 D. 
 
 Cosine 
 
 D. 
 
 i-o3 
 
 Tang. 
 
 I). 
 
 Cotruig. 
 
 L- 
 
 9 -64 1842 
 
 4-3i 
 
 9-953660 
 
 9-688182 
 
 5-34 
 
 io-3ii8i8| 60 
 
 i 
 
 642101 
 
 4-3i 
 
 953599 
 
 i-o3 
 
 6885o2 
 
 5 
 
 ■34 
 
 311498 5g 
 
 2 
 
 642360 
 
 4-3i 
 
 953537 
 95347D 
 
 i-o3 
 
 688823 
 
 5 
 
 34 
 
 311177 
 
 58 
 
 3 
 
 642618 
 
 4-3o 
 
 i-o3 
 
 689143 
 
 5 
 
 •33 
 
 3 1 0857 
 
 57 
 
 4 
 
 642877 
 
 4-3o 
 
 9534i3 
 
 i-o3 
 
 68 9 463 
 
 5 
 
 ■33 
 
 3 1 o537 
 
 56 
 
 5 
 
 643i 35 
 
 4-3o 
 
 953352 
 
 i-o3 
 
 689783 
 
 5 
 
 •33 
 
 310217 
 
 55 
 
 6 
 
 643393 
 
 4-3o 
 
 953290 
 
 i-o3 
 
 690103 
 
 5 
 
 33 
 
 309897 
 
 54 
 
 I 
 
 643630 
 
 4-29 
 
 q53228 
 
 i-o3 
 
 690423 
 
 5 
 
 33 
 
 309577 
 309238 
 
 53 
 
 643908 
 
 4-29 
 
 933166 
 
 i-o3 
 
 690742 
 
 5 
 
 32 
 
 52 
 
 9 
 
 644i65 
 
 4-29 
 
 933104 
 
 i-o3 
 
 691062 
 
 5 
 
 32 
 
 3o8 9 38 
 
 5! 
 
 10 
 
 644423 
 
 4-28 
 
 933042 
 
 i-oj 
 
 6 9 i38i 
 
 5 
 
 32 
 
 308619 
 
 5o 
 
 ii 
 
 9-6446* 
 
 4-28 
 
 9-952980 
 
 1-04 
 
 9-691700 
 
 5 
 
 3i 
 
 io-3o83oo 
 
 % 
 
 12 
 
 644936 
 
 4-28 
 
 952918 
 
 1-04 
 
 692019 
 
 5 
 
 3i 
 
 307981 
 
 l3 
 
 640193 
 
 4-27 
 
 952855 
 
 1-04 
 
 69233a 
 
 5 
 
 3i 
 
 307662 
 
 47 
 
 U 
 
 645430 
 
 4-27 
 
 932793 
 
 1-04 
 
 692656 
 
 5 
 
 3i 
 
 307344 
 
 46 
 
 i5 
 
 645706 
 
 4-2T 
 
 9 32 7 3 1 
 
 1-04 
 
 692975 
 
 5 
 
 3i 
 
 307025 
 
 45 
 
 16 
 
 645962 
 
 4-26 
 
 952669 
 
 1-04 
 
 693293 
 
 5 
 
 3o 
 
 306707 
 3o6388 
 
 44 
 
 \l 
 
 646218 
 
 4-26 
 
 952606 
 
 1-04 
 
 693612 
 
 5 
 
 3o 
 
 43 
 
 646474 
 
 4-26 
 
 952544 1 
 
 1-04 
 
 693930 
 
 5 
 
 3o 
 
 306070 
 
 42 
 
 19 
 
 646729 
 
 4-25 . 
 
 952481 
 
 1-04 
 
 694248 
 
 5 
 
 3o 
 
 3o5752 
 
 41 
 
 20 
 
 646984 
 
 4-20 
 
 952419' 
 
 1-04 
 
 694566 
 
 5 
 
 29 
 
 3o5434 
 
 40 
 
 21 
 
 9-647240 
 
 4-25 
 
 9-952356 
 
 1-04 
 
 9-694883 
 
 5 
 
 29 
 
 io-3o5ii7 
 
 11 
 
 22 
 
 647494 
 
 4-24 
 
 952294 
 
 1-04 
 
 695201 
 
 5 
 
 29 
 
 304799 
 3o44B2 
 
 23 
 
 647749 
 
 4-24 
 
 952231 
 
 1-04 
 
 6g55i8 
 
 5 
 
 29 
 
 37 
 
 24 
 
 648004 
 
 4-24 
 
 952168 
 
 i-o5 
 
 6 9 5836 
 
 5 
 
 29 
 
 304164 
 
 36 
 
 25 
 
 648258 
 
 4-24 
 
 932106 
 
 i-o5 
 
 696153 
 
 5 
 
 28 
 
 3o3847 
 
 35 
 
 26 
 
 648512 
 
 4- 23 
 
 952043 
 
 i-o5 
 
 696470 
 696787 
 
 5 
 
 28 
 
 3o353o 
 
 34 
 
 11 
 
 648766 
 
 4-23 
 
 951980 
 
 i-o5 
 
 5 
 
 28 
 
 3o32r3. 
 
 33 
 
 649020 
 
 4-23 
 
 951917 
 95 1 854 
 
 i-o5 
 
 697103 
 
 5 
 
 28 
 
 302807 
 
 32 
 
 9 
 
 649274 
 
 4-'22 
 
 i-o5 
 
 697420 
 
 5 
 
 27 
 
 3o258o 
 
 3,i 
 
 3o 
 
 . 649527 
 
 4-22 
 
 951791 
 
 i-o5 
 
 697736 
 
 5 
 
 27 
 
 302264 
 
 3o 
 
 3i 
 
 9-649781 
 
 4-22 
 
 9.951728 
 
 i-o5 
 
 9-698053 
 
 5 
 
 2 7 
 
 10-301947 
 
 3 
 
 32 
 
 65oo34 
 
 4-22 
 
 961665 
 
 i-o5 
 
 6n836 9 
 .698685 
 
 5 
 
 27 
 
 3oi63i 
 
 33 
 
 630287 
 
 4-21 
 
 95 1 602 
 
 i.o5 
 
 5 
 
 26 
 
 3oi3i5 
 
 11 
 
 34 
 
 65o53 9 
 
 4-21 
 
 95i539 
 
 i-o5 
 
 699001 
 
 5 
 
 26 
 
 300999 
 
 35 
 
 650792 
 
 4-21 
 
 95 147.6 
 
 i-o5 
 
 699316 
 
 5 
 
 26 
 
 3oo684 
 
 25 
 
 36 
 
 65io44 
 
 4-20 
 
 * 951412 
 
 i-o5 
 
 699632 
 
 5 
 
 26 
 
 3oo368 
 
 24 
 
 32 
 
 631297 
 
 4-20 
 
 9 5i 349 
 
 1-06 
 
 699947 
 
 5 
 
 26 
 
 36oo53 
 
 23 
 
 65 1 549 
 
 4-20 
 
 , 9 5i286 
 
 1-06 
 
 700263 
 
 5 
 
 25 
 
 299737 
 
 22 
 
 3 9 
 
 65 1 800 
 
 4-19 
 
 951222 
 
 1-06 
 
 700578 
 
 5 
 
 25 
 
 299422 
 
 21 
 
 4o 
 
 652052 
 
 4-19 
 
 95i t5g 
 
 1-06 
 
 700893 
 
 5 
 
 25 
 
 299107 
 
 20 
 
 4i 
 
 9-6523o4 
 
 4-19 
 
 9-951096 
 
 1-06 
 
 9.701 2G8 
 
 5 
 
 24 
 
 10-298792 
 
 in 
 
 42 
 
 652555 
 
 4- 18 
 
 95io32 
 
 [■06 
 
 701 523 
 
 5 
 
 24 
 
 298477 
 
 10 
 
 43 
 
 652806 
 
 4- 18 
 
 950968 
 
 t.06 
 
 701837 
 
 5 
 
 24 
 
 298163 
 
 ■7 
 
 44 
 
 653o57 
 6533o8 
 
 •4- 18 
 
 95ooo5 
 
 [■06 
 
 702152 
 
 5 
 
 24 
 
 297848 
 
 16 
 
 45 
 
 4- 18 
 
 950841 
 
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 5 
 
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 Tang. 
 
 D. 
 
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 ,-9-94988i : i-07 
 
 9-707166 
 
 5-20 
 
 10-292834 
 
 
 I 
 
 4-i3 
 
 949816'! -0" 
 
 707478 
 
 5-20 
 
 292522 
 
 5o 
 
 
 2 
 
 657542 
 
 4-12 
 
 949752 1 .07 
 
 707790 
 708102 
 
 5- 20 
 
 292210 
 
 58 
 
 
 3 
 
 657790 
 
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 57 
 
 
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 56 
 
 
 5 
 
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 708726 
 
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 55 
 
 
 6 
 
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 94Q4Q4 1 -o£ 
 
 709037 
 
 5-19 
 
 290963 
 
 54 
 
 
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 53 
 
 
 659025 
 
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 709971 
 
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 23 
 
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 285686 
 
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 26 
 
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 284140 
 
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 33 
 
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 27 
 
 
 34 
 
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 26 
 
 
 35 
 
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 36 
 
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 947533 
 
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 23 
 
 
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 718940 
 
 5-12 
 
 ,281060 
 
 22 
 
 
 3 9 
 
 666583 
 
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 719248 
 
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 43 
 
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 17 
 
 
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 it 
 
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 277379 
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 53 
 
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 54 
 
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 723844 
 
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 55 
 
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 946270.1 -12 
 
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 275851 
 
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 56 
 
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 3-97 
 
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 Cosine | 
 
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 720674 
 Cotang. 
 
 5-o8 
 
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 r 
 
 1 
 
 D. 
 
 Sine ! 
 
 D. 
 
 Tang. J 
 
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46 
 
 (28 DEGItEES.) A TABLE OP LOGARITHMIC 
 
 
 M. 
 
 Sine 
 
 D. 
 
 Cosine | D. 
 
 Tang. | D. 
 
 Cotang. 
 
 > 60 
 
 
 
 9-67160$ 
 
 3.96 
 
 o.-945§35'i.n 
 945868 1 - 1 - 
 
 9-725674! 5- 08 
 
 10-27432* 
 
 i 
 
 67184- 
 
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 725979' 5-o8 
 
 27402 
 
 59 
 
 3 
 
 67208/ 
 
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 94.5800 1 • 1 - 
 
 726284 1 5-07 
 
 273716 58 
 
 3 
 
 672321 
 
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 945733 1 - 1 1 
 
 726588, 5-07 
 
 27341 ' 
 
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 56 
 
 4 
 
 67255E 
 
 3.95 
 
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 726892; 5-07 
 
 27310! 
 
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 672795 
 673032 
 
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 7271971 5-07 
 
 27280C 
 
 55 
 
 6 
 
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 272499 54 
 
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 673268 
 
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 272190! 53 
 
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 271891I 52 
 
 9 
 
 673741 
 
 3.93 
 
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 271588 
 
 ! 3| 
 
 10 
 
 673977 
 
 3. 9 3 
 
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 271284 
 
 5o 
 
 ii 
 
 9-67421; 
 
 3- 9 3 
 
 9-945I93 I-K 
 
 g. 729020 5-o6 
 
 10-270980 
 
 49 
 
 48 
 
 12 
 
 674448 
 
 3-92 
 
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 27067- 
 
 13 
 
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 47 
 
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 46 
 
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 45 
 
 16 
 
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 43 
 
 6 7 585 9 
 
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 73 1 141 
 
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 20 
 
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 21 
 
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 10-267952 
 
 ll 
 
 52 
 
 676796 
 677030 
 
 3.90 
 
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 73235i 
 
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 267649 
 
 23 
 
 3- 9 o 
 
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 267347 
 
 37 
 
 24 
 
 677264 
 
 lb 
 
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 267045 
 
 36 
 
 25 
 
 677498 
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 733257! 5-o3 
 733558, 5-o3 
 
 266743 
 
 35 
 
 26 
 
 3-8 9 
 
 9441721-14 
 
 266442 
 
 34 
 
 IS 
 
 677964 
 
 , 3-8§ 
 
 944104 1-14 
 
 733860 5-02 
 
 266140 
 
 33 ' 
 
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 944o36 1 • 14 
 
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 265838 
 
 32 
 
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 3-88 
 
 943967^.14 
 
 734463| 5-02 
 
 26553 7 
 
 3i 
 
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 673663 
 
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 9438991-14 
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 265236 
 
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 9-678895 
 
 Ml 7 
 
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 10-264934 
 
 29 
 
 32 
 
 679128 
 
 3-87 
 
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 264633 
 
 28 
 
 33 
 
 679360 
 
 3-8 7 
 
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 264332 
 
 27 
 
 34 
 
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 735969! 5-oi 
 
 26403 1 
 
 26 t 
 
 35 
 
 ■ 679824 
 68oo56 
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 736269 
 
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 263731 
 
 25 l 
 
 16 
 
 u 
 
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 943417 1 - 15 
 943348i-i5 
 
 736570 
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 5-oi 
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 263430 
 263 1 29 
 
 24 
 
 23 
 
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 3-85 
 
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 262829 
 
 22 
 
 3 9 
 
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 737471 
 
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 262529 
 
 21 
 
 40 
 
 680982 
 
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 737771 
 
 5-oo 
 
 262229 
 
 20 
 
 41 
 
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 3-85 
 
 9-943U1 1 - 1 5 
 
 9-738071 
 
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 10-261929 
 
 in 
 
 42 
 
 681443 
 
 3-84 
 
 943072 i-i5 
 
 738371 
 
 5-oo 
 
 261629 
 
 IB 
 
 43 
 
 681674 
 
 3-84 
 
 943oo3 i-i5 
 
 738671 
 
 4.99 
 
 261329 
 
 >7 
 
 44 
 
 681905 
 
 384 
 
 942934 1 - 1 5 
 
 738971 
 
 4-99 
 
 261029 
 
 16 
 
 45 
 
 682135 
 
 3-84 
 
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 739271 
 
 4.99 
 
 260729 
 
 i5 
 
 46 
 
 682365 
 
 3-83 
 
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 739570 
 
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 260430 
 
 14 
 
 % 
 
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 3-83 
 
 942726 i- 16 
 
 739870 
 
 4-99 
 
 260 i3o 
 
 i3 
 
 682823 
 
 3-83 
 
 942656 1-16 
 
 740169 1 4-99 
 
 2598.31 
 
 12 
 
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 683o55 
 
 3-83 
 
 942587, 1 • 16 
 
 740468 
 
 4.98 
 
 209532 
 
 11 
 
 5o 
 
 683284 
 
 3-82 
 
 9425ij'i . 16 
 
 9-942448 I- 16 
 
 740767 
 
 4.98 
 
 s5o233 
 
 10 
 
 5i 
 
 9-6835U 
 
 3-82 
 
 9-741066 
 
 4-98 
 
 10-258934 
 
 I 
 
 52 
 
 683743 
 
 3-82 
 
 942378 i- 16 
 
 74-a65 
 
 4-98 
 
 208635 
 
 53 
 
 683972 
 
 3-82 
 
 9423o8 I- 16 
 
 741664 
 
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 258336 
 
 I 
 
 54 
 
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 942239^1 • 16 
 
 741962 
 
 4-97 
 
 258o38 
 
 55 
 
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 94216911-16 
 
 742261 
 
 4-97 
 
 257739 
 
 5 
 
 56 
 
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 257441 
 
 4 
 
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 942029 1 -16 
 
 742858 
 
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 941959 I- 16 
 9418891-17 
 
 743 1 56 
 
 4-97 
 
 256844 2 
 
 I 9 
 
 685343 
 
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 743454 
 
 4-97 
 4-96 
 
 D. " 
 
 256546| 1 
 
 6o 
 
 685571 
 
 3-8o 
 
 941819 1-17 
 
 743752 
 Cotang. 
 
 256248] 
 
 Cosine 
 
 D. 
 
 Sine 61° 
 
 Tang. 1" 
 
 M. 
 

 SINKS AND TANGENTS. 
 
 (29 DEGREES. 
 
 1 
 
 il 
 
 M. 
 
 Sine 
 
 I). 
 
 Cosine | D. 
 
 Tang. 
 
 D. 
 
 Cotang. 
 
 
 
 
 9-685571 
 
 3-8o 
 
 9-941819 1-17 
 
 9-743752 
 
 4-96 
 
 10 -256248 
 
 60 1 
 
 I 
 
 685799 
 
 3-79 
 
 94H49 I- 17 
 
 744o5o 
 
 4.96 
 
 255950 
 
 5 9 
 
 I 
 
 68602^ 
 
 3-79 
 
 941679 1 -17 
 
 744348 
 
 4-96 
 
 255652 
 
 58, 
 
 3 
 
 686254 
 
 3-79 
 
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 4-96 
 
 255355 
 
 n 
 
 4 
 
 686482 
 
 3-79 
 
 94i53g i- 17 
 
 744943 
 
 4.96 
 
 255657 
 
 5 
 
 686709 
 
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 941469 1 -17 
 
 745240 
 
 4.96 
 
 254760 
 
 55 
 
 6 
 
 686g36 
 
 3-78 
 
 941398 1 -17 
 
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 4-95 
 
 254462 
 
 54 
 
 I 
 
 687163 
 
 3- 7 8 
 
 94i328 1 - 17 
 
 745835 
 
 4-95 
 
 254i65, 53 
 
 687389 
 
 3-78 
 
 941258,1-17 
 
 746i32 
 
 4-95 
 
 253868 
 
 52 
 
 9 
 
 687616 
 
 3-77 
 
 ' 941187J1-17 
 
 746429 
 
 4-95 
 
 253571 
 
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 10 
 
 687843 
 
 3.77 
 
 941117,1-17 
 
 746726 
 
 4-95 
 
 253274 
 
 5o , 
 
 II 
 
 9-688069 
 688295 
 
 3-77 
 
 9-94104611 -iE 
 
 9-747023 
 
 4.94 
 
 10-252977 
 252681 
 
 t 
 
 12 
 
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 940975 I- 1£ 
 
 747319 
 
 4.94 
 
 i3 
 
 688521 
 
 g4ooo5|i-iE 
 
 747616 
 
 4-94 
 
 252384 
 
 47 
 
 14 
 
 688747 
 
 3-76 
 
 940834! i- 18 
 
 747913 
 
 4-94 
 
 252087 
 
 46 
 
 15 
 
 688972 
 
 3-76 
 
 940763 I- 1£ 
 
 748209 
 
 4-94 
 
 251791 
 
 45 
 
 16 
 
 689198 
 
 3- 7 6 
 
 940693 I- 18 
 
 7485o5 
 
 4-93 
 
 25i495 
 
 44 
 
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 689423 
 
 3- 7 5 
 
 940622 I- iS 
 
 748801 
 
 4-93 
 
 251199 
 250903 
 
 43 
 
 689648 
 
 |-75 
 
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 749097 
 
 4- 9 3 
 
 42 
 
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 68987; 
 
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 7493t)3 
 
 4-93 
 
 250607 
 
 41 
 
 20 
 
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 749689 
 9-749985 
 
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 25o3n 
 
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 21 
 
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 22 
 
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 750281 
 
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 23 
 
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 750576 
 
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 249424 
 
 37 
 
 24 
 
 690996 
 
 3-74 
 
 940125:1-19 
 
 750872 
 
 4-92 
 
 249128 
 
 36 
 
 23 
 
 691220 
 
 3-73 
 
 94oo54|i-i9 
 
 761167 
 
 4-92 
 
 248833 
 
 35 
 
 26 
 
 691444 
 
 3- 7 3 
 
 939982 1 • 19 
 
 751462 
 
 4-92 
 
 248538 
 
 34 
 
 27 
 
 691668 
 
 3- 7 3 
 
 9 3 9 9i 1 
 
 1-19 
 
 751757 
 
 4-92 
 
 248243 
 
 33 
 
 28 
 
 691892 
 
 3- 7 3 
 
 939840 
 
 1-19 
 
 752052 
 
 4-91 
 
 247948 
 
 32 
 
 29 
 
 6921 i5 
 
 3-72 
 
 939768 
 
 1-19 
 
 752347 
 
 4-91 
 
 247653 
 
 3i 
 
 3o 
 
 692339 
 
 3-72 
 
 939697 
 
 1-19 
 
 ;52642 
 
 4-91 
 
 247358 
 
 3o' 
 
 3i 
 
 9-69256: 
 
 3-72 
 
 9-939625 
 
 1-19 
 
 9-752937 
 
 4-91 
 
 io-247o63 
 
 2Sl 
 
 32 
 
 69278s 
 
 3-71 
 
 939554 
 
 1 -19 
 
 75323i 
 
 491 
 
 246769 
 
 33 
 
 6g3oof 
 
 3.71 
 
 939482 
 
 1-19 
 
 753526 
 
 4-91 
 
 . 2464741 27 
 
 34 
 
 6g3 23 1 
 
 3- 7 i 
 
 939410 
 
 1-19 
 
 753820 
 
 4-9° 
 
 246180 
 
 26 
 
 35 
 
 693453 
 
 3-71 
 
 939339 
 
 1-19 
 
 7541 i5 
 
 4.90 
 
 245885 
 
 25 
 
 36 
 
 693676 
 
 3, -70 
 
 9J9267 
 
 1-20 
 
 754409 
 754703 
 
 4.90 
 
 245591 
 
 24 
 
 3 7 
 
 693898 
 
 3-70 
 
 939195 
 
 1-20 
 
 4-9° 
 
 245297 
 
 23 
 
 38 
 
 694120 
 
 3-70 
 
 939123 
 
 1-20 
 
 75*997 
 
 4.90 
 
 245oo3 
 
 22 
 
 3 9 
 
 694342 
 
 3-70 
 
 939052 
 938980 
 
 1-20 
 
 755291 
 7 55585 
 
 4.90 
 4-89 
 
 244709 
 
 21 
 
 40 
 
 694564 
 
 3-6 9 
 
 1-20 
 
 244413 
 
 20 
 
 41 
 
 9 • 694786 
 
 3-6 9 
 
 9.938908 
 
 1-20 
 
 9-755878 
 
 4.89 
 
 10-244122 
 
 !8 
 
 .42 
 
 695007 
 
 3-6 9 
 
 ■ 9 38836 
 
 1-20 
 
 756172 
 756465 
 
 4-8 9 
 
 243828 
 
 43 
 
 695229 
 
 3-6 9 
 
 ■ 938763 
 
 1-20 
 
 4-89 
 
 243535 
 
 n 
 
 44 
 
 695450 
 
 3-68 
 
 938691 
 
 1-20 
 
 756759 
 
 4-89 
 
 243241 
 
 16 
 
 45 
 
 69567 1 
 
 3-68 
 
 938619 
 
 1-20 
 
 757052 
 
 4-8o 
 
 242948 
 
 i5 
 
 46 
 
 695892 
 
 3-68 
 
 938547 
 
 1-20 
 
 757345 
 
 4-88 
 
 242655 
 
 14 
 
 % 
 
 6961 1 3 
 
 3-68 
 
 938475 
 
 1-20 
 
 757638 
 
 4-88 
 
 242362 
 
 i3 
 
 696334 
 
 3-6 7 
 
 938402 
 
 I-2I 
 
 757931 
 
 4-88 
 
 242069 
 
 12 
 
 49 
 
 696554 
 
 3-67 
 
 9 3833o 
 
 I • 21 
 
 758224 
 
 4-88 
 
 241776 
 
 11 
 
 5o 
 
 696775 
 
 3-67 
 
 ■ 938238 
 
 I-2I 
 
 7585i7 
 
 4-88 
 
 24U83 
 
 10 
 
 5i 
 
 9-696995 
 
 3-67 
 
 9- 9 38i85 
 
 1-21 
 
 9-758810 
 
 4-88 
 
 10-241190 
 
 
 
 6 
 
 52 
 
 697215 
 
 3-66 
 
 938n3 
 
 1 -21 
 
 759102 
 
 4-87 
 
 240898 
 
 53 
 
 697435 
 
 3-66 
 
 938040 
 
 1 .21 
 
 759395 
 
 4-87 
 
 24o6o5 
 
 7 
 
 54 
 
 697654 
 
 3-66 
 
 937967 
 
 I -21 
 
 759687 
 
 4.87 
 
 24o3i3 
 
 6 
 
 55 
 
 697874 
 
 3 =-66 
 
 937895 
 
 I -21 
 
 759979 
 
 4^-87 
 
 240021 
 
 5 
 
 56 
 
 698094 
 
 3-65 
 
 937822 
 
 I - 21 
 
 760272 
 
 4-87 
 
 239728 
 
 4 
 
 5 7 
 
 6 9 83 1 3 
 
 3-65 
 
 937749 
 
 1-21 
 
 760564 
 
 4-87 
 
 239436 
 
 3 
 
 58 
 
 698532 
 
 3-65 
 
 937670 
 
 I - 21 
 
 76o856 
 
 4-86 
 
 239144 
 238852 
 
 2 
 
 5 9 
 
 698751 
 
 3-65 
 
 937604 
 
 I-2I 
 
 761 148 
 
 4-86 ' 
 
 I 
 
 66 
 
 698970 
 
 3-6,4 
 
 937531 
 
 I -21 
 
 761439 
 
 4-86 
 
 238561 
 
 
 
 
 Cosine 
 
 D. 
 
 Sine 
 
 80° 
 
 Co tang. 
 
 D. 
 
 Tang^ 
 
 J*> 
 
 24 
 
(8 
 
 (30 
 
 DEGREES.) A TABLE OF LOGARITHMIC 
 
 
 M. 
 
 Sine 
 
 D. 
 
 Cosine | D. 
 
 Tang. 
 
 D. 
 
 Cotang. 
 
 
 
 
 9.698970 
 
 3-64 
 
 9-937531 1 .21 
 
 9-761439 
 
 4-86 
 
 10. 23856i 
 
 60 
 
 i 
 
 699189 
 
 3 
 
 64 
 
 937458 
 
 1-22 
 
 761731 
 
 4 
 
 86 " 
 
 238269 
 
 S8 
 
 57 
 
 3 
 
 3 
 
 699407 
 699626 
 
 3 
 3 
 
 64 
 64 
 
 937385, 
 987312' 
 
 ■22 
 1-22 
 
 762023 
 762314 
 
 4 
 4 
 
 86 
 86 
 
 =37977 
 237686 
 
 4 
 
 699844 
 
 3 
 
 63 
 
 937238! 
 937165' 
 
 1-22 
 
 762606 
 
 4 
 
 85 
 
 237394 
 
 56 
 
 5 
 
 700062 
 
 3 
 
 63 
 
 1-22 
 
 762897 
 . 7 63 1 88 
 
 4 
 
 85 
 
 237io3 
 
 55 
 
 6 
 
 700280 
 
 3 
 
 63 
 
 937092J 
 
 '22 
 
 4 
 
 85 
 
 236812 
 
 54 
 
 I 
 
 700498 
 
 3 
 
 63 
 
 9370191 
 
 •22 
 
 7634T9 
 
 4 
 
 85 
 
 236521 
 
 53 
 
 700716 
 
 3 
 
 63 
 
 936946 
 
 •22 
 
 763770 
 
 4 
 
 85 
 
 23623o 
 
 52 
 
 9 
 
 700933 
 
 3 
 
 62 
 
 936872! 
 
 •22 
 
 764061 
 
 4 
 
 85 
 
 235939 
 235648 
 
 5i 
 
 10 
 
 70ii5i 
 
 3 
 
 62 
 
 936799 
 9-936725 
 
 ■22 
 
 764352 
 
 4 
 
 84 
 
 5o 
 
 ii 
 
 9-701868 
 
 3 
 
 62 
 
 ■22 
 
 9 ■764643 
 
 4 
 
 84 
 
 10-235357 
 
 **2 
 
 12 
 
 701585 
 
 3 
 
 62 
 
 936652 
 
 •23 
 
 764933 
 
 4 
 
 84 
 
 2J5067 
 
 48 
 
 i3 
 
 701802 
 702019 
 
 3 
 3 
 
 61 
 61 
 
 936578 
 9365o5 
 
 ■23 
 •23 
 
 765224 
 7655i4 
 
 4 
 4 
 
 84 
 84 
 
 234776 
 234486 
 
 % 
 
 15 
 
 702236 
 
 3 
 
 61 
 
 93643i 
 
 •23 
 
 7658o5 
 
 4 
 
 84 
 
 234195 
 
 45 
 
 16 
 
 702452 
 
 3 
 
 61 
 
 936357 
 
 •23 
 
 766095 
 766385 
 
 4 
 
 84 
 
 233905 
 
 44 
 
 \l 
 
 702669 
 702885 
 
 3 
 
 60 
 
 936284 
 
 ■23 
 
 4 
 
 83 
 
 2336 1 5 
 
 43 
 
 3 
 
 60 
 
 936210 
 
 •23 
 
 766675 
 
 4 
 
 83 
 
 233325 
 
 42 
 
 "9 
 
 7o3ioi 
 
 3 
 
 60 
 
 936 1 36 
 
 ■23 
 
 766965 
 
 4 
 
 83 
 
 233o35 
 
 41 
 
 20 
 
 703317 
 
 3 
 
 60 
 
 936062 
 
 •23 
 
 767255 
 
 4 
 
 83 
 
 232745 
 
 40 
 
 21 
 
 9-703 533 
 
 3 
 
 5 9 
 
 9-935988 
 
 •23 
 
 9-767545 
 
 4 
 
 83 
 
 10-232455 
 
 \l 
 
 22 
 
 703749 
 
 3 
 
 5 9 
 
 935914 
 935840 
 
 ■23 
 
 767834 
 768124 
 
 4 
 
 83 
 
 232166 
 
 23 
 
 703964 
 
 3 
 
 5 9 
 
 ■23 
 
 4 
 
 82 
 
 231876 
 
 37 
 
 24 
 
 704179 
 ' 704395 
 
 3 
 
 5 9 
 
 935766 
 
 ■24 
 
 768413 
 
 4 
 
 82 
 
 23 1 587 
 
 36 
 
 25 
 
 3 
 
 5 9 
 
 935692 
 
 •24 
 
 768703 
 
 4 
 
 82 
 
 231297 
 
 35 
 
 26 
 
 704610 
 
 3 
 
 58 
 
 9356i8 
 
 • 24 
 
 768992 
 
 4 
 
 82 
 
 23 1008 
 
 34 
 
 3 
 
 704825 
 
 3 
 
 58 
 
 935543 
 
 ■24 
 
 769201 
 
 4 
 
 82 
 
 230719 
 
 33 
 
 705040 
 
 3 
 
 58 
 
 935469 
 
 ■ 24 
 
 769570 
 
 4 
 
 82 
 
 23o43o 
 
 32 
 
 n 9 
 
 705254 
 
 3 
 
 58 
 
 935390 
 
 ■24 
 
 769860 
 
 4. 
 
 81 
 
 23oi4o 
 
 3i 
 
 3o 
 
 705469 
 9 •.7o5683 
 
 3 
 
 57 
 
 935320 
 
 •24 
 
 770148 
 
 4 
 
 81 
 
 229852 
 
 3o 
 
 3i 
 
 3 
 
 57 
 
 9-935246 
 
 •24 
 
 9.770437 
 
 4 
 
 81 
 
 10-229563 
 
 29- 
 
 32 
 
 705898 
 
 3 
 
 5 7 
 
 935171 
 
 • 24 
 
 770726 
 
 4 
 
 81 
 
 229274 
 228985 
 228697 
 
 28 
 
 33 
 
 7061 12 
 
 3 
 
 57 
 
 935097 
 
 ■24 
 
 771015 
 
 4 
 
 81 
 
 11 
 
 34 
 
 706326 
 
 3 
 
 56 
 
 935022 
 
 ■24 
 
 77i3o3 
 
 4 
 
 81 
 
 35 
 
 706539 
 706753 
 
 3 
 
 56 
 
 934948 
 
 • 24 
 
 771592 
 771880 
 
 4 
 
 81 
 
 228408 
 
 25 
 
 36 
 
 3 
 
 56 
 
 934873 
 
 ■ 24 
 
 4 
 
 80 
 
 228120 
 
 24 
 
 u 
 
 706967 
 
 3 
 
 56 
 
 934798 
 
 ■25 
 
 772168 
 
 4 
 
 80 
 
 227832 
 
 23 
 
 707180 
 
 3 
 
 55 
 
 934723 
 
 ■25 
 
 772457 
 
 4 
 
 80 
 
 . 227543 
 
 22 
 
 39 
 
 707393 
 
 3 
 
 55 
 
 934649 
 
 •25 
 
 772745 
 
 4 
 
 80 
 
 227265 
 
 21 
 
 40 
 
 707606 
 
 3 
 
 55 
 
 934574 
 
 •25 
 
 773o33 
 
 4 
 
 80 
 
 226967 
 
 20 
 
 41 
 
 9-707819 
 
 3 
 
 55 
 
 9-934499 
 
 •25 
 
 9-773321 
 
 4 
 
 80 
 
 10.226679 
 
 !3 
 
 42 
 
 708032 
 
 3 
 
 54 
 
 934424 
 
 •25 
 
 7 7 36o8 
 
 4 
 
 79 
 
 226392 
 
 43 
 
 708245 
 
 3 
 
 54 
 
 934349 
 
 •25 
 
 7738o6 
 
 4 
 
 79 
 
 226104 
 
 \l 
 
 44 
 
 708458 
 
 3 
 
 54 
 
 934274 
 
 •25 
 
 774184 
 
 4 
 
 79 
 
 2258i6 
 
 45 
 
 708670 
 708882 
 
 3 
 
 54 
 
 934199 
 934123 
 
 •25 
 
 774471 
 
 4 
 
 79 
 
 225529 
 
 i5 
 
 46 
 
 3 
 
 53 
 
 ■25 
 
 774759 
 
 4 
 
 79 
 
 225241 
 
 14 
 
 % 
 
 709094 
 
 3 
 
 53 
 
 934048 
 
 •25 
 
 775046 
 
 4 
 
 79 
 
 224954 
 
 i3 
 
 709306 
 
 3 
 
 53 
 
 933973 
 
 •25 
 
 775333 
 
 4 
 
 7? 
 
 224667 
 
 12 
 
 49 
 
 709518 
 
 3 
 
 53 
 
 933898 
 
 • 26 
 
 775621 
 
 4 
 
 78 
 
 224379 
 
 11 
 
 5o 
 
 709730 
 
 3 
 
 53 
 
 933822 
 
 • 26 
 
 775908 
 
 4 
 
 78 
 
 224092 
 
 10 
 
 5i 
 
 9-709941 
 
 3 
 
 52 
 
 9-933747, 
 
 • 26 
 
 9-776195 
 
 4 
 
 78 
 
 io-2238o5 
 
 I 
 
 52 
 
 7ioi53 
 
 3 
 
 52 
 
 933671 
 
 • 26 
 
 776482 
 
 4 
 
 78 
 
 2235i8 
 
 53 
 
 710364 
 
 3 
 
 52 
 
 9 335 9 6 
 
 ■ 26 
 
 776769 
 777055 
 
 4 
 
 78 
 
 22323l 
 
 7 
 
 54 
 
 710575 
 
 3 
 
 52 
 
 93352o! 
 
 ■ 26 
 
 4 
 
 78 
 
 222945 
 
 6 . 
 
 55 
 
 710786 
 
 3 
 
 5i 
 
 933445 
 
 • 26 
 
 777342 
 
 4 
 
 .78 
 
 222658 
 
 5 
 
 56 
 
 710997 
 711208 
 
 3 
 
 5i 
 
 933369! 
 933293] 
 
 • 26 
 
 777628 
 
 4 
 
 77 
 
 22^372 
 
 4 
 
 u 
 
 3 
 
 5i 
 
 ■ 26 
 
 777915 
 
 4 
 
 77 
 
 222085 
 
 3 
 
 711419 
 
 3 
 
 5i 
 
 933217 
 
 ■ 26 
 
 778201 
 
 4 
 
 77 
 
 221799 2 
 
 5 9 
 
 711629 
 
 3 
 
 5o 
 
 933i4i 
 
 ■ 26 
 
 778487 
 
 4 
 
 77 
 
 22l5l2J I 
 
 60 
 
 71 1839 
 
 3-5o 
 
 933066] 
 
 • 26 
 
 778774 
 
 4-77 
 
 221226] 
 
 Cosine 
 
 D. 
 
 Sine 1 
 
 5a° 
 
 Cotang. 
 
 D. 
 
 Tang. I M. 
 

 StNTES 
 
 AND TANGENTS. 
 
 (31 DEGREES. 
 
 ) 
 
 49 
 
 a. 
 
 o 
 
 Sine 
 9-7ii83^ 
 
 D. 
 
 Cosine 1 D. 
 
 Tang. 
 
 D. 
 
 Cotang. 
 
 
 3 
 
 5o 
 
 9.933066 
 
 1-26 
 
 9-778774 
 
 4 
 
 77 
 
 10-221226 
 
 60 
 
 ' I 
 
 712060 
 
 3 
 
 5o 
 
 932990 
 
 1-27 
 
 779060 
 
 4 
 
 76- 
 
 220940 
 220654 
 
 59 
 
 2 
 
 712263 
 
 3 
 
 5o 
 
 932914 
 
 ■ 27 
 
 779346 
 
 4 
 
 58 
 
 3 
 
 712469 
 
 3 
 
 49 
 
 932838, 
 
 • 27 
 
 779632 
 
 4 
 
 t 
 
 220368 
 
 l r 
 
 4 
 
 712679 
 
 3 
 
 49 
 
 932762 
 
 ■27 
 
 779918 
 
 4 
 
 7& 
 
 220082 
 
 56 
 
 5 
 
 712889 
 713098 
 
 3 
 
 49 
 
 932685 
 
 1-27 
 
 780203 
 
 4 
 
 t 
 
 219797 
 
 55 
 
 6 
 
 3 
 
 49 
 
 932609 
 932533 
 
 [-27 
 
 780489 
 780776 
 
 4 
 
 76 
 
 21931 1 
 
 54 
 
 J 
 
 7i33o8 
 
 3 
 
 49- 
 
 1-27 
 
 4 
 
 76 
 
 219225 
 
 2 1 8940 
 
 53 
 
 713517 
 
 3 
 
 48 
 
 932457 
 
 1-27 
 
 781060 
 
 4 
 
 76 
 
 52- 
 
 9 
 
 713726 
 
 3 
 
 48 
 
 93238o 
 
 1.27 
 
 781346 
 
 4 
 
 ~>\ 
 
 218654 
 
 5i 
 
 10 
 
 713935 
 
 3 
 
 48 
 
 932304 
 
 1-27 
 
 78i63i 
 
 4 
 
 75 
 
 218369 
 
 5o 
 
 ii 
 
 9-714144 
 
 3 
 
 48 
 
 9-932228 
 
 1-27 
 
 9-781916 
 
 4 
 
 7 5 
 
 10-218084 
 
 49 
 
 12 
 
 714352 
 
 3 
 
 47 
 
 g32l5i 
 
 •27 
 
 782201 
 
 4 
 
 75 
 
 2 '7799 
 
 48 
 
 i3 
 
 1456 1 
 
 3 
 
 47 
 
 932075 
 
 •28 
 
 782486 
 
 4 
 
 75 
 
 217514 
 
 47 
 
 !4 
 
 714769 
 
 3 
 
 47 
 
 931998 
 
 •28 
 
 782771 
 
 4 
 
 75 
 
 217229 
 
 46 
 
 i5 
 
 714978 
 
 3 
 
 47 
 
 931921 
 93i845 
 
 •28 
 
 783o56 
 
 4 
 
 75 
 
 2 i 6944 
 
 45 
 
 16 
 
 7i5i86 
 
 3 
 
 47 
 
 • 28 
 
 783341 
 
 4 
 
 7 5 
 
 216659 
 
 44 
 
 17 
 
 7 1 53 9 4 
 
 3 
 
 46- 
 
 931768 
 
 -28 
 
 783626 
 
 4 
 
 74 
 
 216374 
 
 43 
 
 1 8 
 
 715602 
 
 3 
 
 46 
 
 931691 
 
 ■28 
 
 783910 
 
 4 
 
 74 
 
 216090 
 
 42 
 
 '9 
 
 7 1 5809 
 
 3 
 
 46 
 
 931614 
 
 ■28 
 
 784195 
 
 4 
 
 74 
 
 2i58o5 
 
 41 
 
 20 
 
 716017 
 
 3 
 
 46 
 
 9^1537, 
 
 1-28 
 
 784479 
 
 4 
 
 74 
 
 2 1 552 1 
 
 40 
 
 21 
 
 9-716224 
 
 3 
 
 45 
 
 9.931460! 
 
 ■ 28 
 
 9-784764 
 
 4 
 
 74 
 
 io-2i5236 
 
 3o 
 
 22 
 
 716432 
 
 3 
 
 45 
 
 9 3 1 383 
 
 .28 
 
 785048 
 
 4 
 
 74 
 
 214952 
 
 38 
 
 23 
 
 716639 
 
 3 
 
 45 
 
 93i3o6 
 
 •28 
 
 785332 
 
 4 
 
 73 
 
 214668 
 
 37 
 
 '24 
 
 716846 
 
 3 
 
 45 
 
 931229 
 
 ■29 
 
 7 856i6 
 
 4 
 
 73 
 
 214384 
 
 36 
 
 25 
 
 717053 
 
 3 
 
 45 
 
 93 1 1 52 
 
 ■ 29 
 
 785900 
 
 4 
 
 73 
 
 214100 
 
 35 
 
 20 
 
 717259 
 
 3 
 
 44 
 
 931075 
 
 •29 
 
 786184 
 
 4 
 
 73 
 
 2i38i6 
 
 34 
 
 .27 
 
 717466 
 
 3 
 
 44 
 
 930998 
 
 •29 
 
 786468 
 
 4 
 
 73 
 
 2 1 3532 
 
 33 
 
 28 
 
 717673 
 
 3 
 
 44 
 
 930921 
 93o843 
 
 ■ 29 
 
 786752 
 
 4 
 
 73 
 
 2i3248 
 
 32 
 
 29 
 
 717879 
 718085 
 
 3 
 
 44. 
 
 ■29 
 
 787036 
 
 4 
 
 73 
 
 2 1 2964 
 
 3i 
 
 3o 
 
 3 
 
 43 
 
 930766 
 
 • 29 
 
 787319 
 
 4 
 
 72 
 
 212681 
 
 3o 
 
 3i 
 
 9-718291 
 
 3 
 
 43 
 
 9,930688 
 
 •29 
 
 9-787603 
 
 4 
 
 72 
 
 10-212397 
 
 u 
 
 32 
 
 718497 
 
 3 
 
 43 
 
 93o6i 1 
 
 •29 
 
 787886 
 
 4 
 
 72 
 
 212114 
 
 33 
 
 718703 
 
 3 
 
 43 
 
 93o533i 
 
 •29 
 
 788170 
 
 4 
 
 72 
 
 2ii83o 
 
 27 
 
 34 
 
 718909 
 
 3 
 
 43 
 
 93o456 
 
 .29 
 
 788453 
 
 4 
 
 72 
 
 2 1 1 547 
 
 26 
 
 35 
 
 719114 
 
 3 
 
 42 
 
 93o378 
 
 •29 
 
 788736 
 
 4 
 
 72 
 
 2 1 1 264 
 
 25 
 
 36 
 
 719320 
 
 3 
 
 42 
 
 93o3oo 
 
 -3o 
 
 789019 
 
 4 
 
 72 
 
 2 1 O98 1 
 
 24 
 
 n 
 
 719525 
 
 3 
 
 42 
 
 93o223 
 
 -3o 
 
 789302 
 
 4 
 
 7' 
 
 2 [ 0698 
 
 23 
 
 719730 
 
 3 
 
 42 
 
 93oi45 
 
 •3o 
 
 789585 
 
 4 
 
 71 
 
 2I04l5 
 
 22 
 
 3 9 
 
 719935 
 
 3 
 
 41 
 
 930067 
 
 -3o 
 
 789868 
 
 4 
 
 71 
 
 2IOI32 
 
 21 
 
 4o 
 
 720140 
 
 3 
 
 41 - 
 
 929989 
 
 • 3o 
 
 790i5i 
 
 4 
 
 71 
 
 209849 
 
 20 
 
 &\ 
 
 9-720345 
 
 3 
 
 41 
 
 9.92991 1 
 929833 
 
 ■ 3o 
 
 9-790433 
 
 4 
 
 71 
 
 10-209567 
 
 19 
 
 4? 
 
 720549 
 
 3 
 
 41 
 
 -3o 
 
 790716 
 
 4 
 
 71 
 
 ■ 209284 
 
 18 
 
 43 
 
 720754 
 
 3 
 
 40 
 
 929755 
 
 -3o 
 
 - 790999 
 
 4 
 
 7f 
 
 20O0O1 
 
 17 
 
 44 
 
 720958 
 
 3 
 
 40 
 
 929677 
 
 -3o 
 
 791281 
 
 4 
 
 71 
 
 208719 
 
 -16- 
 
 45 
 
 721162 
 
 3 
 
 4o 
 
 929599 
 
 -3o 
 
 79 1 563 
 
 4 
 
 70 
 
 208437 
 
 i5 
 
 46 
 
 72 1 366 
 
 3 
 
 40 
 
 929521 
 
 -3o 
 
 791846 
 
 4 
 
 70 
 
 2o8i54 
 
 14 
 
 47 
 
 721570 
 
 3 
 
 40 
 
 929442 
 
 -3o 
 
 792128 
 
 4 
 
 70 
 
 207872 
 
 i3 
 
 48 
 
 721774 
 
 3 
 
 3 9 
 
 929364 
 
 • 3i 
 
 792410 
 
 4 
 
 70 
 
 207590 
 
 12 
 
 49 
 
 721978 
 
 3 
 
 3g 
 
 929286 
 
 • 3i 
 
 792692 
 
 4 
 
 70 
 
 207308 
 
 11 
 
 56 
 
 722181 
 
 3 
 
 3 9 
 
 929207 
 
 • 3i 
 
 792974 
 
 4 
 
 70 
 
 207026 
 
 10 
 
 5i 
 
 9-722385 
 
 3 
 
 3 9 
 
 9-929129 
 
 •3i 
 
 9 ' 7 '^^ 
 
 4 
 
 70 
 
 10-206744 
 
 9 
 
 52 
 
 722588 
 722791 
 
 3 
 
 3o 
 
 92oo5o 
 
 ■3i 
 
 793538 
 
 4 
 
 69 
 
 206462 
 
 8 
 
 53 
 
 3 
 
 38 
 
 928972 
 
 • 3i 
 
 793819 
 
 4 
 
 69 
 
 206181 
 
 I 
 
 54 
 
 722994 
 
 3 
 
 38 
 
 928893 
 
 • 3i 
 
 794101 
 
 4 
 
 69 
 
 205899 
 
 55 
 
 723197 
 
 3 
 
 38 
 
 928816 
 
 • 3i 
 
 794383 
 
 4 
 
 69 
 
 205617 
 
 5 
 
 56 
 
 723400 
 
 3 
 
 38 
 
 928736 
 
 • 3i 
 
 794664 
 
 4 
 
 69 
 
 205336 
 
 4 
 
 57 
 
 7236o3 
 
 3 
 
 37 
 
 928657 
 
 • 3i 
 
 794945 
 
 4 
 
 2 9 
 
 2o5o55 
 
 3 
 
 58 
 
 7238o5 
 
 3 
 
 37 
 
 928578 
 928499 
 
 ■ 3i 
 
 795227 
 
 4 
 
 69 
 
 204773 
 
 1 
 
 59 
 
 724007 
 
 3 
 
 37 
 
 ■3i 
 
 795508 
 
 4 
 
 68 
 
 204492 
 
 1 
 
 60 
 
 724210 
 
 3 
 
 37 
 
 . 928420 
 
 •3i 
 
 795789 
 
 4 68 
 
 2042 m 
 
 
 Cosine 
 
 D- 
 
 Sine 
 
 58° 
 
 Cotang. 
 
 D. 
 
 Tang. 
 
 M. | 
 
50 
 
 (32 DEGREES.) A TABLE OF LOGARITHMIC 
 
 It. 
 
 Sine 
 
 D. 
 
 Cosine | D. 
 
 Tang. 
 
 D. 
 
 Cotang. 
 
 
 
 
 9-724210 
 
 3-37 
 
 9-928420' 
 
 • 32 
 
 9.795789 
 
 4 
 
 68 
 
 10-204211 
 
 60 
 
 I 
 
 724412 
 
 3 
 
 ll 
 36 
 
 928342 
 
 ■32 
 
 796070 
 
 4 
 
 68 
 
 203930 
 
 
 2 
 
 3 
 
 724614 
 724816 
 
 3 
 3 
 
 9 28263 ; 
 928183 
 
 •32 
 •32 
 
 796351 
 796632 
 
 4 
 4 
 
 68 
 68 
 
 203649 
 203368 
 
 4 
 
 725017 
 
 3 
 
 36 
 
 928104 
 
 •32 
 
 796913 
 
 4 
 
 68 
 
 2o3o8t 
 202806 
 
 5 
 
 723219 
 
 3 
 
 36 
 
 928025 
 
 •32 
 
 797194 
 
 4 
 
 68 
 
 55 
 
 6 
 
 725420 
 
 3 
 
 35 
 
 927946 
 
 •32 
 
 797475 
 797 7 55 
 
 4 
 
 68 
 
 202525 
 
 54 
 
 I 
 
 725622 
 
 3 
 
 35 
 
 927867 
 
 •32 
 
 4 
 
 68 
 
 202245 
 
 53 
 
 •725823 
 
 3 
 
 35 
 
 927781 
 
 •32 
 
 7g8g36 
 
 4 
 
 67 
 
 201964 
 
 02 
 
 9 
 
 10 
 
 726024 
 
 3 
 
 35 
 
 927708 
 927629 
 
 •32 
 
 798316 
 
 4 
 
 67 
 
 201684 
 
 5i 
 
 726225 
 
 3 
 
 35 
 
 •32 
 
 798596 
 
 4 
 
 67 
 
 201404 
 
 5o 
 
 11 
 
 9.726426 
 
 3 
 
 34 
 
 9-927549 
 
 •32 
 
 9.798877 
 
 4 
 
 67 
 
 10-201123 
 
 % 
 
 12 
 
 726626 
 
 3 
 
 34 
 
 927470 
 
 ■ 33 
 
 799157 
 
 4 
 
 67 
 
 200843 
 
 i3 
 
 726827 
 
 3 
 
 34 
 
 927390 
 
 •33 
 
 799437 
 
 4 
 
 67 
 
 200563 
 
 % 
 
 14 
 
 727027 
 727228 
 
 3 
 
 34 
 
 927310 
 
 • 33 
 
 799717 
 
 4 
 
 67 
 
 200283 
 
 15 
 
 3 
 
 34 
 
 927231 
 
 • 33 
 
 799997 
 800277 
 
 4 
 
 66 
 
 200003 
 
 45 
 
 16 
 
 727428 
 
 3 
 
 33 
 
 927151 
 
 ■33 
 
 4 
 
 66 
 
 199723 
 
 44 
 
 \l 
 
 727628 
 
 3 
 
 33 
 
 927071 
 
 ■ 33 
 
 8oo557 
 
 4 
 
 66 
 
 199443 
 
 43 
 
 727828 
 728027 
 
 3 
 
 33 
 
 926991 
 
 • 33 
 
 ,8oo836 
 
 4 
 
 66 
 
 199164 
 • 198884 
 
 42 
 
 19 
 
 3 
 
 33 
 
 92691 1 
 
 ■ 33 
 
 801 116 
 
 4 
 
 66 
 
 41 
 
 20 
 
 728227 
 
 3 
 
 33 
 
 926831 
 
 ■ 33 
 
 801396 
 
 4 
 
 66 
 
 198604 
 
 40 
 
 21 
 
 9-728427 
 
 3 
 
 3a 
 
 9-926751 
 926671 
 
 .33 
 
 9-801675 
 
 4 
 
 66 
 
 10-198335 
 
 38 
 
 22 
 
 728626 
 
 3 
 
 32 
 
 ■ 33 
 
 - 8oi 9 55 
 
 4 
 
 66 
 
 198045 
 
 23 
 
 728825 
 
 3 
 
 32 
 
 926591 
 
 1-33 
 
 802234 
 
 4 
 
 65 
 
 197766 
 
 IT 
 
 24 
 
 729024 
 
 3 
 
 32 
 
 9265i 1 
 
 1-34 
 
 8o25i3 
 
 4 
 
 65 
 
 197487 
 
 25 
 
 729223 
 
 3 
 
 3i 
 
 926431 
 
 1-34 
 
 802792 
 
 4 
 
 65 
 
 197208 
 
 35 
 
 26 
 
 729422 
 
 3 
 
 3.1 
 
 92635i 
 
 [-34 
 
 803072 
 
 4 
 
 65 
 
 196928 
 
 34 
 
 ll 
 
 729621 
 
 3 
 
 3i 
 
 926270 
 
 1-34 
 
 8o335i 
 
 4 
 
 65 
 
 196649 
 
 33 
 
 729820 
 
 3 
 
 3i 
 
 926190 
 
 1-34 
 
 8o363o 
 
 4 
 
 65 
 
 196370 
 
 32 
 
 20 
 
 730018 
 
 3 
 
 3o 
 
 926110 
 
 1-34 
 
 803908 
 
 4 
 
 65 
 
 196092 
 
 3i 
 
 3o 
 
 730216 
 
 3 
 
 3o 
 
 / 926029 
 
 [■34 
 
 804187 
 9-804466 
 
 4 
 
 65 
 
 195813 
 
 3o 
 
 3i 
 
 9-73o4i5 
 
 3 
 
 3o 
 
 9-925949 
 925868 
 
 [-34 
 
 4 
 
 64 
 
 10.195534 
 
 •3 
 
 32 
 
 73061 3 
 
 3 
 
 3o 
 
 r-34 
 
 8o4745 
 
 4 
 
 64 
 
 195255 
 
 33 
 
 73o8i 1 
 
 3 
 
 3o 
 
 925788 
 
 [-34 
 
 8o5o23 
 
 4 
 
 64 
 
 194077 
 
 194698 
 
 27 
 
 34 
 
 731009 
 731206 
 
 3 
 
 29 
 
 925707 
 
 [■34 
 
 8o53o2 
 
 4 
 
 64 
 
 26 
 
 35 
 
 3 
 
 29 
 
 925626 
 
 1-34 
 
 8o558o 
 
 4 
 
 64 
 
 194420 
 
 25 
 
 36 
 
 73i4o4 
 
 3 
 
 29 
 
 925545 
 
 1-35 
 
 8o585 9 
 
 4 
 
 64 
 
 194141 
 
 24 
 
 ll 
 
 731662 
 
 3 
 
 29 
 
 925465 
 
 1-35 
 
 806137 
 
 4 
 
 64 
 
 193853 
 
 23 
 
 731799 
 
 3 
 
 29 
 
 925384 
 
 1-35 
 
 80641 5 
 
 4 
 
 63 
 
 193585 
 
 22 
 
 3 9 
 
 731996 
 
 3 
 
 28 
 
 9253o3 
 
 [-35 
 
 806693 
 
 4 
 
 63 
 
 193307 
 
 21 
 
 4o 
 
 732193 
 
 3 
 
 28 
 
 925222 
 
 1-35 
 
 806971 
 
 4 
 
 63 
 
 193029 
 
 20 
 
 41 
 
 9-732390 
 732587 
 
 3 
 
 28 
 
 9-925141 
 
 [■35 
 
 9-807249 
 
 4 
 
 63 
 
 10-192751 
 
 II 
 
 42 
 
 3 
 
 28 
 
 925o6o 
 
 1-35 
 
 807527 
 807805 
 
 4 
 
 63 
 
 192473 
 
 43 
 
 732784 
 
 3 
 
 28 
 
 924979 
 924897 
 
 [•35 
 
 4 
 
 63 
 
 192195 
 
 \l 
 
 44 
 
 732980 
 
 3 
 
 27 
 
 [-35 
 
 8o8o83 
 
 4 
 
 63 
 
 191917 
 
 45 
 
 733177 
 
 3 
 
 27 
 
 924816 
 
 i-35 
 
 8o836i 
 
 4 
 
 63 
 
 191639 
 
 i5 
 
 46 
 
 7 333 7 3 
 
 3 
 
 27 
 
 924735 
 
 [-36 
 
 8o8638 
 
 4 
 
 62 
 
 191362 
 
 14 
 
 % 
 
 733569 
 733765 
 
 3 
 
 27 
 
 924654 
 
 i-36 
 
 808916 
 
 4 
 
 62 
 
 191084 
 
 i3 
 
 3 
 
 27 
 
 924572 
 
 [-36 
 
 809193 
 
 4 
 
 62 
 
 190807 
 
 12 
 
 £ 
 
 733961 
 
 3 
 
 26 
 
 924491 
 
 1-36 
 
 809471 
 
 4 
 
 62 
 
 190529 
 
 11 
 
 734r57 
 
 3 
 
 26 
 
 924409 
 
 1-36 
 
 809748 
 
 4 
 
 62 
 
 IO0252 
 
 10-189975 
 
 10 
 
 h 
 
 9-734353 
 
 3 
 
 26 
 
 9-924328 
 
 [-36 
 
 9-810025 
 
 4 
 
 62 
 
 I 
 
 §2 
 
 734549 
 
 3 
 
 ?6 
 
 924246 
 
 [-36 
 
 8io3o2 
 
 4 
 
 62 
 
 189698 
 
 5-3 
 
 734744 
 
 3 
 
 25 
 
 924164 
 
 [-36 
 
 8io58o 
 
 4 
 
 62 
 
 189420 
 
 I 
 
 54 
 
 73493 2 
 
 3 
 
 25 
 
 924083 
 
 [-36 
 
 810857 
 
 -4 
 
 62 
 
 189143 
 188866 
 
 55 
 
 735i3d 
 
 3 
 
 25 
 
 924001 
 
 [.36 
 
 811 i34 
 
 4 
 
 61 
 
 5 
 
 56 
 
 73533o 
 
 3 
 
 25 
 
 923919 
 
 1-36 
 
 811410 
 
 4 
 
 61 
 
 188590 
 
 4 
 
 8 
 
 735525 
 
 3 
 
 •25 
 
 923837 1 
 
 [-36 
 
 81 1687 
 
 4 
 
 61 
 
 i883i3 
 
 3 
 
 735719 
 
 3 
 
 •24 
 
 923755| 
 
 .-3 7 
 
 811964 
 
 4 
 
 61 
 
 i88o36 
 
 1 
 
 5o 
 
 735914 
 
 3 
 
 •24 
 
 923673 
 
 1.37 
 
 812241 
 
 4 
 
 61 
 
 187759 i 
 
 $P 
 
 736iog 
 
 3 
 
 •24 
 
 923591 
 
 [•3 7 
 
 812517 
 
 4-61 
 
 18748J 
 
 
 
 
 Cpiino 
 
 D, 
 
 Sine |5T° 
 
 Cotang, 
 
 r>. 
 
 Tan K . 
 
 ST 
 
 fc ■r-r- 
 
 
 
 
 
 
 
 
 
 
 
SINES A ND V TAH'GENTS. (33 DEGREES.) 
 
 51 
 
 M. 
 
 Sine | D. 
 
 Cosine 
 
 D. | Tang. 
 
 D. 
 
 Cotang. 
 
 
 
 
 9-736109 
 
 3-24 
 
 9-923591 
 
 i-3 7 
 
 9-812517 
 
 4-6i 
 
 10-187482 
 
 60 
 
 I 
 
 7363o3 
 
 3-24 
 
 923509 
 
 1 -3 7 
 
 812794 
 
 4-6i 
 
 187206 
 
 is 
 
 3 
 
 736498 
 
 3-24 
 
 923427 
 
 i-3 7 
 
 813070 
 
 4-6i 
 
 186930 
 186653 
 
 3 
 
 736692 
 
 3-23 ■ 
 
 923345 
 
 i-3 7 
 
 8i3347 
 
 4.60 
 
 57 
 
 4 
 
 736886 
 
 3-23 
 
 923263 
 
 i-3 7 
 
 8i3623 
 
 4.60 
 
 186377 
 
 56 
 
 5 
 6 
 
 737080 
 737274 
 
 ,3-23 
 3-23 
 
 923181 
 
 923098 
 
 ,.3 7 
 1.37 
 
 813899 
 814170 
 
 4.60 
 4.60 
 
 186101 
 185825 
 
 55 
 54 
 
 I 
 
 737467 
 
 3-23 
 
 923016 
 
 i-37 
 
 8i4452 
 
 4.60 
 
 185548 
 
 53 
 
 737661 
 
 3-22 
 
 922933 
 922851 
 
 !-3 7 
 
 814728 
 
 4-6o 
 
 185272 
 
 52 
 
 9 
 
 10 
 
 737855 
 
 3-22 
 
 i-3 7 
 
 8i5oo4 
 
 4.60 
 
 184906 
 
 5i 
 
 738048 
 
 3-22 
 
 922768 
 
 i-38 
 
 815279 
 9-8i5555 
 
 4-6o 
 
 •184721 
 
 5o 
 
 ii 
 
 9-738241 
 
 3-22 
 
 9-922686 
 
 i-38 
 
 4-59 
 
 io-i84445 
 
 49- 
 
 12 
 
 738434 
 
 -3-22 
 
 922603 
 
 i-38 
 
 8i583i 
 
 4-5 9 
 
 184169 
 i838 9 3 
 
 48 
 
 i3 
 
 738627 
 
 8-21 
 
 922520 
 
 1-38 
 
 816107 
 
 4- 5 9 
 
 47 
 
 14 
 
 738820 
 
 3-21 
 
 922438 
 
 1-38 
 
 8:1 6382 
 
 4, 5 9 
 
 i836i8 
 
 46 
 
 i5 
 
 739013 
 
 3-2! 
 
 922355 
 
 1-38 
 
 8i6658 
 
 4.59 
 
 183342 
 
 45 
 
 16 
 
 739206 
 
 3-21 
 
 922272 
 922189 
 
 1-38 
 
 . 816933 
 
 4-5 9 
 
 183067 
 
 44 
 
 \l 
 
 739398 
 
 3-21 
 
 1-38 
 
 817209 
 
 4-59 
 
 182791 
 
 43 
 
 736590 
 
 3-20 
 
 922106. 
 
 1-38 
 
 817484 
 
 4-59 
 
 182516 
 
 42 
 
 <9 
 
 20 
 
 739783 
 
 3-20 
 
 922023 
 
 1-38 
 
 817759 
 8i8o35 
 
 4-5o 
 
 182241 
 
 -41' 
 
 739975 
 
 3-20 
 
 921940 
 9-921857 
 
 1-38 
 
 4-58 
 
 181965 
 
 40 
 
 31 
 
 9-740I67 
 
 3-20 
 
 i-3 9 
 
 g-8i83io 
 
 4-58 
 
 10-181690 
 
 18 
 
 22 
 
 740359 
 
 3-20 
 
 921774 
 
 1-39 
 
 8 1 8585 
 
 4-58 
 
 181415 
 
 23 
 
 74o55o. 
 
 3-19 
 
 921691 
 
 1 -3q 
 
 ■ Si 8860 
 
 4-58 
 
 181 140 
 
 37 
 
 24 
 
 740742 
 
 3-19 
 
 921607 
 
 1.39 
 
 819135 
 
 4-58 
 
 i8o865 
 
 36 
 35 
 
 25 
 
 740934 
 
 3-19 
 
 921524 
 
 i-3 9 
 
 819410 
 
 4-58 
 
 180590 
 
 26 
 
 741 1 25 
 
 3-io 
 
 921441 
 
 i-3o 
 
 819684 
 
 '4-58 
 
 i8o3i6 
 
 34 
 
 27 
 
 74i3i6 
 
 3-io 
 
 921357 
 
 1-39 
 
 819969 
 
 4-58, 
 
 1 8004 1 
 
 33 
 
 28 
 
 741 5o8 
 
 3-i8 
 
 921274 
 
 i-3 9 
 
 820234 
 
 4-58 
 
 179766 
 
 32 
 
 29 
 
 741699 
 
 3-i8 
 
 921 190 
 
 1-39 
 
 82o5o8 
 
 4-67 
 
 179492 
 
 3i | 
 
 3o 
 
 741889 
 
 3-i8 
 
 921107 
 
 1 -3g 
 
 820783 
 
 4-57 
 
 179217 
 10-178943 
 
 3o 
 
 3i 
 
 9-742080 
 742271 
 742462 
 
 3-i8 
 
 9-921023 
 
 1-39 
 
 9-821057 
 
 4-57 
 
 20 \ 
 
 32 
 
 3-i8 
 
 920939 
 920856 
 
 i-4o 
 
 82i332 
 
 4-57 
 
 178668 
 
 28 
 
 33 
 
 3-17 
 
 i-4o 
 
 821606 
 
 4-5 7 
 
 178394 
 
 11 
 
 34 
 35 
 
 742652 
 742842 
 
 3-i 7 
 3-i 7 
 
 920772 
 920688 
 
 i-4o 
 1-40 
 
 821880 
 822154 
 
 4-57 
 4- 5 7 
 
 178120 
 177846 
 
 26 
 
 25 
 
 36 
 
 743o33 
 
 3-i 7 
 
 920604 
 
 i-4o 
 
 822429 
 822700 
 
 4-57 
 
 177571 
 
 24 
 23 
 
 -37 
 
 743223 
 
 3-i 7 
 
 920520 
 
 i-4o 
 
 4.57 
 
 177297 
 
 38 
 
 7434i3 
 
 3-i6 
 
 920436 
 
 i-4o 
 
 822977 
 
 4-56 
 
 177023 
 176760 
 
 22 
 
 3 9 
 40 
 
 7436o2 
 743792 
 
 3-i6 
 
 920352 
 
 i-4o 
 
 823250 
 
 4-56 
 
 21 
 
 3-i6 
 
 „ 92026^ 
 
 i-4o 
 
 823524 
 
 4-56 
 
 176476 
 
 20 
 
 41 
 
 9.743982 
 
 " 3-i6 
 
 9-920184 
 
 i-4o 
 
 9-823798 
 
 4-56 
 
 10-176202 
 178928 
 175655 
 
 '7 
 
 42 
 43 
 44 
 45 
 46 
 
 7,44i7i 
 74436i 
 744550 
 
 744739 
 744928 
 
 74 5l V7 
 7453o6 
 
 74549* 
 
 3-i6 
 
 3-i5 
 
 92009c 
 92001' 
 
 i-4o 
 i-4o 
 
 824072 
 824345 
 
 4-56 
 4-56 
 
 3-i5 
 3- 15 
 3-i5 
 
 9i993i!i -41 
 919846;! -41 
 9197621-41 
 
 824619 
 
 | 824893 
 
 825i66 
 
 4-56 
 4-56 
 4-56 
 
 175381 
 175107 
 i 7 4834 
 
 16 
 i5 
 14 
 
 il 
 
 3- 1 5 
 3-i4 
 
 919677 1 -41 
 919593 r -4^ 
 
 82543a 
 82571; 
 
 4-55 
 4-55 
 
 1 74661 
 174287 
 
 i3 
 12 
 
 it 
 
 3-i4 
 
 919508 1 -41 
 
 825986 
 
 4-55 
 
 1 740 1 4 
 
 11 
 
 745683 
 
 3-14 
 
 9i9424|i-4i 
 
 826255 
 
 4-55 
 
 173741 
 
 10 
 
 5i 
 
 52 
 
 9-745871 
 746o5c 
 
 3-14 
 3-U 
 
 9-9193391 -4i 
 91925411 -41 
 
 9-826532 
 82680E 
 
 4-55 
 4-55 
 
 io- 17346E 
 
 173195 
 
 I 
 
 53 
 
 746248 
 746436 
 746624 
 7468 r 2 
 -746999 
 747187 
 747374 
 7475M 
 
 3-i3 
 
 9I9i6q!i-4i 
 
 827076 
 
 4-55 
 
 172922 
 
 I 
 5 
 
 54 
 55 
 
 4-i3 
 3-i3 
 
 91908* 
 9 1 900c 
 
 i-4> 
 1-41 
 
 827351 
 827624 
 
 4-55 
 4-55 
 
 17264c 
 172376 
 
 56 
 
 57 
 58 
 
 iS 
 60 
 
 3i3 
 
 3-i3 
 
 91891! 
 9 1 883c 
 
 1-43 
 1-45 
 
 82789- 
 82817c 
 
 4-54 
 4-54 
 
 172 1 o3 
 17183c 
 
 3 
 
 3-12 
 
 3-ia 
 
 3-12 
 
 91874' 
 
 9 1 865c 
 
 I 91857* 
 
 I-4S 
 
 I -45 
 
 J-i-41 
 
 82844! 
 82871; 
 82898- 
 
 4-54 
 4-54 
 4-54 
 
 I7I55S 
 
 171285 
 17101; 
 
 2 
 
 I 
 
 ° 
 
 Cosine 
 
 D. 
 
 I Sine |58< 
 
 Cotang. 
 
 D 
 
 Tang. 1 M.. 
 
 24* 
 
52 
 
 (^ 
 
 DEGREES ) A TABLE OF LOGARITHMIC 
 
 
 M. 
 
 . Sine 
 
 D. 
 
 Cosmo j D. 
 
 Tang. 
 
 D. 
 
 Cotnng. 
 
 
 
 
 9-747562 
 
 3-12 
 
 9-918574! 
 918489' 
 
 t-42 
 
 9-828987 
 
 4 
 
 54 
 
 10-171013 60 
 
 1 
 
 747749 
 
 3 
 
 12 
 
 1-42 
 
 829260 
 
 4 
 
 54 
 
 170740 
 
 38 
 
 a 
 
 747936 
 748123 
 
 3 
 
 12 
 
 918404, 
 
 1-42 
 
 829532 
 
 4 
 
 54 
 
 170468 
 
 3 
 
 3 
 
 n 
 
 9 i83i8 
 
 1-42 
 
 829805 
 
 4 
 
 54 
 
 170195 
 
 57 
 
 4 
 
 7483io 
 
 3 
 
 n 
 
 918233 
 
 1-42 
 
 830077 
 
 4 
 
 54 
 
 169923 
 
 56 
 
 5 
 
 748497 
 748683 
 
 3 
 
 11 
 
 918147 
 
 1-42 
 
 83o349 
 
 4 
 
 53 
 
 1 6965 1 
 
 55 
 
 6 
 
 3 
 
 11 
 
 918062 
 
 1-42 
 
 83o62i 
 
 4 
 
 53 
 
 169379 
 
 54 
 
 o 
 
 748870 
 
 3 
 
 n 
 
 917976 
 
 1-43 
 
 83o8 9 3 
 83n65 
 
 4 
 
 53 
 
 169107 
 
 53 
 
 749056 
 
 3 
 
 10 
 
 917891 
 
 [-43 
 
 4 
 
 53 
 
 168835 
 
 5] 
 
 9 
 
 749243 
 
 3 
 
 10 
 
 917805 
 
 •43 
 
 83U37 
 
 4 
 
 53 
 
 ■68563 
 
 5i 
 
 10 
 
 749429 
 9-749615 
 
 3 
 
 10 
 
 917719 
 
 •43 
 
 831709 
 
 4 
 
 53 
 
 168291 
 
 5o 
 
 n 
 
 3 
 
 10 
 
 9-917634 
 
 •43 
 
 9.831981 
 
 4 
 
 53 
 
 10-168019 
 
 4o 
 
 12 
 
 749801 
 
 3 
 
 10 
 
 917548 
 
 •43 
 
 832253 
 
 4 
 
 53 
 
 '67747 
 
 i3 
 
 749987 
 
 3 
 
 09 
 
 917462 
 
 -43 
 
 832525 
 
 4 
 
 53 
 
 167475 
 
 47 
 
 14 
 
 750172 
 
 3 
 
 09 
 
 917376 
 
 1-43 
 
 832796 
 
 4 
 
 53 
 
 167204 
 
 46 
 
 15 
 16 
 
 75o358 
 75o543 
 
 3 
 3 
 
 09 
 09 
 
 917290 
 917204 
 
 [-43 
 ■43 
 
 833o68 
 833339 
 
 4 
 4 
 
 52 
 
 02 
 
 166932 
 166661 
 
 45 
 44 
 
 \l 
 
 750729 
 
 3 
 
 3 
 
 917118 
 
 •44 
 
 8336n 
 
 4 
 
 52 
 
 i6638 9 
 166118 
 
 43 
 
 750914 
 
 3 
 
 917032 
 
 1-44 
 
 833882 
 
 4 
 
 52 
 
 42 
 
 19 
 
 751099 
 
 3 
 
 08 
 
 916946 
 
 •44 
 
 834i 54 
 
 4 
 
 52 
 
 J65846 
 
 41 
 
 20 
 
 751284 
 
 3 
 
 08 
 
 916859 
 
 •44 
 
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 DEGREES.) A 
 
 TABLE OF LOGARITHMIC 
 
 
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 Sine 
 
 
 0. 
 
 Cosine j D. 
 
 >Tang. 
 
 D. 
 
 Cotang 
 
 60 
 
 
 
 9.769219 
 76939J 
 
 2-90 
 
 9 -9079581 1 -53 
 
 9-861261 
 
 4-43 
 
 10-138739 
 138473 
 
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 89 
 
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 53 
 
 861 527 
 
 4 
 
 43 
 
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 53 
 
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 53 
 
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 4 
 
 42 
 
 137942 
 
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 4 
 
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 89 
 
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 53 
 
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 42 
 
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 56 
 
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 53 
 
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 4 
 
 42 
 
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 55 
 
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 88 
 
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 53 
 
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 42 
 
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 54 
 
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 88 
 
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 54 
 
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 42 
 
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 53 
 
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 88 
 
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 54 
 
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 9 
 
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 54 
 
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 10 
 
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 88 
 
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 54 
 
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 4 
 
 42 
 
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 54 
 
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 87 
 
 54 
 
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 42 
 
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 54 
 
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 4 
 
 42 
 
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 2 
 
 87 
 
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 54 
 
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 41 
 
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 2 
 
 87 
 
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 54 
 
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 4 
 
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 134760 
 
 45 
 
 16 
 
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 2 
 
 87 
 
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 54 
 
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 4 
 
 41 
 
 1 344o5 
 1 34230 
 
 44 
 
 13 
 
 772159 
 
 2 
 
 87 
 
 906389 1 
 
 55 
 
 865770 
 
 4 
 
 41 
 
 43 
 
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 2 
 
 86 
 
 906296 1 
 
 55 
 
 866o35 
 
 4 
 
 41 
 
 i33 9 65 
 
 42 
 
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 86 
 
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 55 
 
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 4 
 
 41 
 
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 20 
 
 772675 
 
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 86 
 
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 55 
 
 866564 
 
 4 
 
 41 
 
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 40 
 
 21 
 
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 86 
 
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 55 
 
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 10-133171 
 
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 22 
 
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 86 
 
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 55 
 
 867094 
 86 7 358 
 
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 41 
 
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 23 
 
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 86 
 
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 4 
 
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 132642 
 
 37 
 
 24 
 
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 2 
 
 85 
 
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 55 
 
 867623 
 
 4 
 
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 132377 
 
 36 
 
 25 
 
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 2 
 
 85 
 
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 41 
 
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 35 
 
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 85 
 
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 55 
 
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 4 
 
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 131848 
 
 34 
 
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 7 7 38 7 5 
 
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 85 
 
 905459' 1 
 
 55 
 
 868416 
 
 4 
 
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 33 
 
 774046 
 
 2 
 
 85 
 
 9o5366ji 
 
 56 
 
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 4 
 
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 32 
 
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 774217 
 
 2 
 
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 56 
 
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 56 
 
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 56 
 
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 32 
 
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 84 
 
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 56 
 
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 28 
 
 33 
 
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 84 
 
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 56 
 
 870001 
 
 4 
 
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 129735 
 
 27 
 
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 84 
 
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 56 
 
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 4 
 
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 26 
 
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 56 
 
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 129207 
 
 24 
 
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 83 
 
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 56 
 
 4 
 
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 23 
 
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 57 
 
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 4 
 
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 83 
 
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 57 
 
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 83 
 
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 57 
 
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 42 
 
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 57 
 
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 17 
 
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 82 
 
 
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 4 
 
 39 
 
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 16 
 
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 82 
 
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 57 
 
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 57 
 
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 39 
 
 126570 
 
 14 
 
 3 
 
 777275 
 
 2 
 
 81 
 
 9o358i 
 
 
 57 
 
 873694 
 873957 
 
 4 
 
 3 9 
 
 1 263o6 
 
 ■ 3 
 
 777444 
 
 2 
 
 81 
 
 903487 
 
 
 57 
 
 4 
 
 3 9 
 
 1 26043 
 
 12 
 
 P 
 
 777613 
 
 2 
 
 81 
 
 903392 
 
 
 58 
 
 874220 
 
 4 
 
 3 9 
 
 125780 
 
 u 
 
 5o 
 
 77778i 
 
 2 
 
 81 
 
 903298 
 
 
 58 
 
 874484 
 
 4 
 
 3 9 
 
 I255i6 
 
 10 
 
 5i 
 
 9-777950 
 
 2 
 
 81 
 
 9-9o32o3 
 
 
 58 
 
 9-874747 
 
 4 
 
 3 9 
 
 io- 125253 
 
 8 
 
 i2 
 
 778119 
 
 2 
 
 81 
 
 903108 
 
 
 58 
 
 875010 
 
 4 
 
 3 
 
 1 24990 
 
 '.3 
 
 778287 
 
 2 
 
 80 
 
 9o3oi4 
 
 
 58 
 
 875273 
 
 4 
 
 124727 
 
 I 
 
 '>i 
 
 778455 
 
 2 
 
 80 
 
 902919 
 902824 
 
 
 58 
 
 875536 
 
 4 
 
 38 
 
 1 24464 
 
 55 
 
 778624 
 
 2 
 
 80 
 
 
 58 
 
 875800 
 
 4 
 
 38 
 
 124200 
 
 5 
 
 ;* 
 
 778792 
 
 2 
 
 80 
 
 902729 
 
 
 58 
 
 876063 
 
 4 
 
 38 
 
 123937 
 
 4 
 
 ll 
 
 778960 
 
 2 
 
 80 
 
 902634 
 
 
 58 
 
 876326 
 
 4 
 
 38 
 
 1 23674 
 
 3 
 
 77912b 
 
 2 
 
 80 
 
 902539 
 
 
 5 9 
 
 876589 
 
 4 
 
 38 
 
 1 23411 
 
 2 
 
 5 9 
 
 779295 
 
 2 
 
 79 
 
 902444 
 
 
 5 9 
 
 876851 
 
 4 
 
 38 
 
 I 23 1 49 
 
 ■ 
 
 6o 
 
 779463 
 
 2-79 
 
 902349 
 
 1^9 
 
 877114 
 
 4-38 
 
 122886 
 
 _M. 
 
 r~ 
 
 Cosine 
 
 J). 
 
 Sine 
 
 53° 
 
 Cotang. 
 
 D. 
 
 Tang. 
 

 SINKS AND TANGENTS. 
 
 (37 DEGREES.) 
 
 
 55 
 
 M. 
 o 
 
 Bins 
 
 D. 
 
 Cosine j D. 
 
 Tang. 
 
 D. | 
 
 Cottng. 
 
 60 
 
 9-779463 
 
 a-79 
 
 c ■ 902349 1 - 59 
 902253 1 -69 
 
 9-877114 
 
 4-38 
 
 10-122886 
 
 i 
 
 779631 
 
 2-79 
 
 877377 
 
 4-38 
 
 122623 
 
 it 
 
 a 
 
 779798 
 
 •3.- 79 
 
 902i58,i -59 
 
 877640 
 
 4-38 
 
 122360 
 
 3 
 
 779966 
 
 2-79 
 
 902063,1 -59 
 
 877903 
 
 4-38 
 
 122097 
 1 21 835 
 
 s 
 
 4 
 
 7Soi33 
 
 2-79 
 
 901967 J 1 -59 
 901872 1-59 
 
 8 7 8i65 
 
 4-38 
 
 56 
 
 5 
 
 780300 
 
 2-78 
 
 878428 
 
 /■38 
 
 121572 
 
 55 
 
 6 
 
 780467 
 
 2-78 
 
 901776 
 901681 
 
 1-59 
 
 878691 
 878953 
 
 4-38 
 
 121309 
 
 54 
 
 7 
 
 780634 
 
 2-78 
 
 1 -59 
 
 4-37 
 
 121047 
 
 53 
 
 8 
 
 780801 
 
 2-78 
 
 90 1 585 
 
 1-59 
 
 879216 
 
 4-37 
 
 120784 
 
 5a 
 
 9 
 
 780968 
 
 2-78 
 
 901490 
 
 1 -5g 
 
 879478 
 
 4-37 
 
 120522 
 
 5i 
 
 10 
 
 78i'i34 
 
 2-78 
 
 901 3 9 4 
 
 i-6o 
 
 879-41 
 
 4-37 
 
 120259 
 
 5o 
 
 ii 
 
 q-78i3oi 
 
 2-77 
 
 9-901298:1-60 
 
 9-88ojo3 
 
 4-37 
 
 10-119997 
 
 49 
 
 12 
 
 781468 
 
 a- 77 
 
 901202 
 
 i-6o 
 
 880265 
 
 4-3T 
 
 119735 
 
 48 
 
 i3 
 
 ■78-634 
 
 2-77 
 
 901 106 
 
 I- 60 
 
 8So528 
 
 4-37 
 
 119472 
 
 47 
 
 14. 
 
 781800 
 
 2-77 
 
 901010 
 
 i-6o 
 
 (480790 
 8Sio52 
 
 4-37 
 
 119210 
 
 46 ' 
 
 15 
 
 781966 
 
 2-77 
 
 900914 
 
 I- 60 
 
 4-37 
 
 118948 
 
 45 
 
 16 
 
 782132 
 
 2-77 
 
 9O0818 
 
 I- 60 
 
 88i3i4 
 
 4-37 
 
 1 1 8686 
 
 44 
 
 "7 
 
 ,- 782298 
 
 2-76 
 
 900722 
 
 i-6o 
 
 88 1 5 7 6 
 
 4-37 
 
 1 1 8424 
 
 43 
 
 18 
 
 782464 
 
 2-76 
 
 900626 
 
 i-6o 
 
 881839 
 
 4-37 
 
 118161 
 
 42 
 
 19 
 
 •j82630 
 
 2-76 
 
 90o52o 1 -6c 
 
 882101 
 
 4- 3 7 
 
 1 17899 
 117637 
 
 41 
 
 20 
 
 782796 
 
 2-76 
 
 900433 
 
 i-6i 
 
 882363 
 
 4-36 
 
 40 
 
 21 
 
 9'78296l 
 
 2-76 
 
 9-900337 
 
 i-6i 
 
 9-882626 
 
 4-36 
 
 10-117375 
 
 ll 
 
 22 
 
 783:27 
 
 2-76 
 
 900240 
 
 1-61 
 
 882887 
 883i48 
 
 4-36 
 
 1 171 13 
 
 23 
 
 783292 
 
 733458 
 
 2-75 
 
 900144 
 
 f -61 
 
 4-36 
 
 1 1 6852 
 
 37 
 
 24 
 
 2-75 
 
 900047 
 899951 
 899854 
 
 i-6i 
 
 883410 
 
 4-36 
 
 1 16590 
 
 36 
 
 25 
 
 7 83623 
 
 2-75 
 
 1-61 
 
 883672 
 
 4-36 
 
 n6328 
 
 35 
 
 26 
 
 783 7 8g 
 
 2-75 
 
 1-61 
 
 883934 
 
 4-36 
 
 1 16066 
 
 34 
 
 3 
 
 783953 
 
 2-75 
 
 8997J7 
 
 1 -61 
 
 884196 
 884457 
 
 4-36 
 
 n58o4 
 
 33 
 
 7841 ft 
 
 2-75 
 
 899660 
 
 i-6i 
 
 4-36 
 
 115543 
 
 32 
 
 29 
 
 784282 
 
 2-74 
 
 899564 
 
 1-61 
 
 884719 
 
 4-36 
 
 , Ii528i 
 
 3i 
 
 3o 
 
 784447 
 
 2-74 
 
 899467 
 
 1-62 
 
 884980 
 
 4-36 
 
 Il5o20 
 
 3o 
 
 3i 
 
 9-784612 
 
 2-74 
 
 9-899370 
 
 1-62 
 
 9-885242 
 
 4-36 
 
 I0-H4758 
 
 ■29 
 
 32 
 
 784776 
 
 2-74 
 
 899273 
 
 1-62 
 
 8855o3 
 
 4-36 
 
 1 14497 
 
 28' 
 
 33 
 
 784941 
 
 2-74 
 
 899176 
 
 1-62 
 
 885 7 65 
 
 4-36 
 
 II4235 
 
 2 Z 
 
 34 
 
 .785 1 o5 
 
 2-74 
 
 899078 
 898981 
 
 1-62 
 
 886026 
 
 4-36 
 
 1 13974 
 
 26 
 
 35 
 
 785?6g 
 
 2 = 73 
 
 1 -62 
 
 886288 
 
 4-36 
 
 113712 
 
 25 
 
 36 
 
 785433 
 
 2-73 
 
 898884 
 
 1-62 
 
 886549 
 
 4-35 
 
 1 1 3454 
 
 24 
 
 Si 
 
 785597 
 
 2- 7 3 
 
 898787 
 
 1-62 
 
 886810 
 
 4-35 
 
 113190 
 
 23 
 
 785761 
 
 * 2-73 
 
 898689 
 
 1 -62 
 
 887072 
 
 4-35 
 
 1 1 2928 
 
 22 
 
 3 9 
 
 .785925 
 
 i- 7 3 
 
 898592 
 
 1-62 
 
 887333 
 
 4 r 35 
 
 1 1 2667 
 
 21 
 
 40 
 
 786089 
 
 2-73 
 
 898494 
 
 I-.63 
 
 887594 
 9.887855 
 
 4-35 
 
 1 1 2406 
 
 ao 
 
 41 
 
 9-786252 
 
 2-72 
 
 9-898397 
 
 1-63 
 
 4-35 
 
 io-H2i45 
 
 \l 
 
 42 
 
 7864:6 
 
 2-72 
 
 898299 
 
 1-63 
 
 8881 16 
 
 4-35 
 
 .111884 
 
 43 
 
 786579 
 
 2-72 
 
 898202 
 
 1-63 
 
 888377 
 
 4-35 
 
 1 1 1623 
 
 \l 
 
 44 
 
 786742 
 
 2-72 
 
 898104 
 
 1-63 
 
 88863 9 
 
 4-35 
 
 1 1 i36i 
 
 45 
 
 786906 
 
 2-72 
 
 898006 
 
 1-63 
 
 £88900 
 
 4-35 
 
 IIIIOO 
 
 i5 
 
 46 
 
 787069 
 
 2-72 
 
 897908 
 
 1-63 
 
 889160 
 
 4-35 
 
 1 1 0840 
 
 1 4. 
 
 47 
 
 787232 
 
 2-71 
 
 897810 
 
 1-63 
 
 ' 889421 
 
 4-35 
 
 1 1057c 
 
 i3 
 
 48 
 
 787395 
 
 2-71 
 
 8977 1 2 
 
 1-63 
 
 889682 
 
 4-35 
 
 Iio3ic 
 
 12 
 
 g 
 
 787567 
 
 2-71 
 
 897614 
 
 1-63 
 
 889943 
 
 4-35, 
 
 Iioo5- 
 
 11 
 
 787720 
 
 2-71 
 
 897516 
 
 1-63 
 
 89020^ 
 
 4-34 
 
 10979^ 
 -10- 10953^ 
 
 10 
 
 5i 
 
 9-787883 
 788045 
 
 2-71 
 
 9-897418 
 
 1-64 
 
 9-89046' 
 
 4-34 
 
 9 
 
 52 
 
 2-71 
 
 897320 
 
 1-64 
 
 890725 
 
 - 4-34 
 
 109275 
 
 
 
 53 
 
 788208 
 
 2-71 
 
 897222 
 
 1-64 
 
 890986 
 
 4-34 
 
 109014 
 
 I. 
 
 54 
 
 788370 
 
 2-70 
 
 89712; 
 
 1-64 
 
 891247 
 
 4-34 
 
 10875; 
 
 55 
 
 7 88532 
 
 ' 2-70 
 
 897025 
 
 1-64 
 
 891 5o- 
 
 4-34 
 
 108490 
 io8a32 
 
 5 
 
 56 
 
 78869/ 
 788856 
 
 2-70 
 
 896926 
 
 1-64 
 
 891768 
 
 4-34 
 
 4 
 
 % 
 
 2-70 
 
 896828 
 
 1-64 
 
 89202S 
 
 '4-34 
 
 107972 
 
 3 
 
 789018 
 
 2-70 
 
 896729 
 
 1-64 
 
 892289 
 
 4-34 
 
 107711 
 
 3 
 
 H 
 
 789180 
 
 2-70 
 
 896631 
 
 1-64 
 
 892549 
 
 4-34 
 
 lo-J45i 
 
 I 
 
 789342 
 
 2-69 
 
 896532 
 
 1-64 
 52° 
 
 89281c 
 
 4.34 
 
 10719c 
 
 
 
 Cosire 
 
 D. 
 
 Sine 
 
 1 Cotang. 
 
 D. 
 
 Tang. 
 
 M. 
 
66 
 
 (38 DEGREES.) A TABLE OF LOGARITHMIC 
 
 M. 
 
 Sine 
 
 19 
 20 
 21 
 22 
 23 
 
 24 
 25 
 
 26 
 
 u 
 ll 
 
 3i 
 
 32 
 
 33 
 34 
 35 
 36 
 
 ll 
 3 9 
 4o 
 41 
 4S 
 43 
 44 
 45 
 46 
 
 % 
 
 % 
 5i 
 52 
 53 
 54 
 55 
 56 
 
 % 
 
 9-789342 
 789504 
 789665 
 789827 
 
 D. 
 
 790149 
 7903 10 
 790471 
 790632 
 790793 
 790954 
 9-791 1 i5 
 791275 
 791436 
 79i5o6 
 791767 
 791917 
 792077 
 792237 
 792397 
 792557 
 9-792716 
 792876 
 793o35 
 793195 
 793354 
 79?5i4 
 793673 
 7g3832 
 793991 
 794130 
 9-794308 
 794467 
 794626 
 794784 
 794942 
 795101 
 795259 
 795417 
 795575 
 7 9 5733 
 9-7 9 58 9 i 
 796049 
 796206 
 796364 
 796521 
 796679 
 796836 
 796993 
 797i5o 
 797307 
 9-797464 
 797621 
 
 191111 
 797934 
 798091 
 798247 
 7 9 84o3 
 7g856o 
 798716 
 798872 
 
 2-69 
 
 2-69 
 
 2-69 
 
 2-69 
 
 2-69 
 
 2-69 
 
 2-68 
 
 2-68 
 
 2-68 
 
 2-68 
 
 2-68 
 
 2-68 
 
 2-67 
 
 2-67 
 
 2-67 
 
 2-67 
 
 2-67 
 
 2-67 
 
 2-66 
 
 2-66 
 
 2-66 
 
 2-66 
 
 2-66 
 
 2-66 
 
 2-65 
 
 2-65 
 
 2-65 
 
 2-65 
 
 2-65 
 
 2-65 
 
 2-64 
 
 2-64 
 
 2-64 
 
 2-64 
 
 2-64 
 
 2-64 
 
 2-64 
 
 2-63 
 
 2-63 
 
 2-63 
 
 2-63 
 
 2-63 
 
 2-63 
 
 Cosine 
 
 63 
 2-62 
 2-62 
 2-62 
 
 2-62 
 2-62 
 2-6l 
 2-6l 
 2-6l 
 2-6l 
 2-61 
 2-6l 
 2-6l 
 2-6l 
 2- 60 
 
 2-60 
 S-6o 
 2-60 
 
 9-896532 
 896433 
 896335 
 896236 
 896137 
 896038 
 895939 
 895840 
 895741 
 895641 
 895542 
 
 9-895443 
 895343 
 895244 
 895145 
 895045 
 894945 
 894846 
 894746 
 894646 
 894546 
 
 9-894446 
 894346 
 894246 
 894146 
 894046 
 893946 
 893846 
 893745 
 893645 
 893544 
 •893444 
 893343 
 893243 
 893142 
 893041 
 
 892940 
 
 892839 
 892739 
 8 9 2638 
 892536 
 • 892430 
 892334 
 892233 
 892132 
 892030 
 891929 
 891827 
 891726 
 891624 
 891523 
 •891421 
 891319 
 891217 
 891115 
 891013 
 89091 1 
 890809 
 
 Tang. 
 
 1-64 
 1-65 
 1-65 
 -65 
 1-65 
 1-65 
 
 • 65 
 
 • 65 
 1-65 
 1-65 
 1-65 
 
 66 
 1-66 
 1-66 
 1-66 
 1-66 
 1-66 
 1-66 
 1-66 
 1-66 
 1-66 
 ,.67 
 ,.67 
 1-67 
 1.67 
 1.67 
 
 67 
 1. 67 
 1-67 
 
 67 
 
 67 
 
 1-68 
 
 1-68 
 
 • 798 1 
 Cosine | D. 
 
 1-68 
 !-68 
 1-68 
 1-68 
 1-68 
 1-68 
 1-68 
 1-69 
 1-69 
 1.69 
 1 -69 
 1 -69 
 1-69 
 1-69 
 1-69 
 1-69 
 70 
 1-70 
 •70 
 ■70 
 • 70 
 
 ■7° 
 ■70 
 
 .y --1° 
 890707 1-70 
 890605 1-70 
 89o5o3 1-70 
 Sine 51° 
 
 1-892810 
 893070 
 8 9 333 1 
 893591 
 8 9 385i 
 894111 
 894371 
 894632 
 
 895152 
 895412 
 
 1-895672 
 8 9 5 9 32 
 8961. 
 8964 
 896712 
 89697 1 
 89723 
 89749 
 897751 
 898010 
 
 ;■ 898270 
 898530 
 
 D. 
 
 89904' 
 89930' 
 899568 
 '899827 
 
 900346 
 goo6o5 
 
 9-900864 
 901 1 24 
 90i383 
 901642 
 901901 
 902160 
 902419 
 902679 
 902938 
 903197 
 
 9-903455 
 903714 
 903973 
 904232 
 904491 
 904750 
 900008 
 905267 
 905526 
 905784 
 
 9 • 906043 
 906302 
 906560 
 906819 
 907077 
 907336 
 907594 
 907802 
 9081 1 1 
 908369 
 
 4-34 
 
 4-34 
 
 4 -.3-4 
 
 4-34 
 
 4-34 
 
 4-34 
 
 4-34 
 
 4-33 
 
 4-33 
 
 4-33 
 
 4-33 
 
 4-33 
 
 4-33 
 
 4-33 
 
 4-33 
 
 4-33 
 
 4-33 
 
 4-33 
 
 4^33 
 
 4-33 
 
 4-33 
 
 4-33 
 
 4-33 
 
 4-33 
 
 4-32 
 
 4-32 
 
 4-32 
 
 4-32 
 
 4-32 
 
 4-32 
 
 4-32 
 
 4-32 
 
 4-32 
 
 4-32 
 
 4-32 
 
 4-32 
 
 4-32 
 
 4-32 
 
 4-32' 
 
 4-32 
 
 4-3i 
 
 4-3i 
 
 Cotanpr. 
 
 Cctung. 
 
 •3i 
 ■3i 
 4-3i 
 4-3i 
 4-3i 
 4-3i 
 4-3i 
 4-3i 
 4-3i 
 4-3i 
 4-3i 
 4-3i 
 4-3i 
 4-3i 
 4-3i 
 4-3i 
 4-3i 
 4-3o 
 4-3o 
 
 0-107190 
 106930 
 106669 
 106409 
 106149 
 105889 
 105629 
 105368 
 io5ro8 
 104848 
 104588 
 
 10-104328 
 104068 
 io38o8 
 io3548 
 103288 
 103029 
 102769 
 102509 
 102249 
 101990 
 0-101730 
 101470 
 101211 
 100951 
 100692 
 100442 
 100173 
 099014 
 099654 
 ~ i5 
 
 - -A ib 
 
 098876 
 
 098617 
 o 9 8358 
 098099 
 097840 
 097581 
 097321 
 097062 
 096803 
 
 10-096545 
 096286 
 096027 
 095768 
 095509 
 095250 
 094992 
 094733 
 094474 
 094216 
 
 10-09395' 
 09369! 
 093440 
 093 181 
 092023 
 092664 
 092406 
 092148 
 091889 
 09163 
 
 09939! 
 io-o99i3< 
 
 60 
 
 u 
 
 55 
 54 
 53 
 
 52 
 
 5i 
 
 So 
 
 t 
 
 % 
 45 
 44 
 43 
 42 
 41 
 40 
 
 ll 
 
 ll 
 35 
 34 
 33 
 32 
 3i 
 3o 
 
 It 
 
 25 
 
 24 
 
 23 
 22 
 21 
 20 
 
 18 
 
 '7 
 16 
 i5 
 14 
 i3 
 12 
 
 D. 
 
 Tang. M. 
 

 8INES 
 
 AND TANGENTS. 
 
 (39 BEOREES.] 
 
 
 57 
 
 M. 
 
 Sine 
 
 D. 
 
 Cosine 
 
 D. 
 
 Tang. 
 
 D. 
 
 Cotimg. .' 
 
 o 
 
 0-798872 
 
 2-6o 
 
 9 -890503 
 
 • 70 
 
 9'9o83t9 
 908628 
 
 4-3o 
 
 10-091631 
 
 60 
 
 i 
 
 799028 
 
 2- 
 
 60 
 
 890400 
 
 •71 
 
 4-3o 
 
 091372 
 
 .S 
 
 2 
 
 799184 
 
 2 
 
 00 
 
 890298 
 
 •7 1 
 
 908886 
 
 4-3o 
 
 091114 
 
 3 
 
 799339 
 
 2 
 
 5 9 
 
 890195 
 
 •T 
 
 909144 
 
 4-3o 
 
 090856 
 
 5 7 
 
 4 
 
 799493 
 7996D1 
 
 2 
 
 5 9 
 
 890093 
 
 •7< 
 
 909402 
 
 4-3o 
 
 090598 
 
 56 
 
 5 
 
 .2 
 
 5 9 
 
 889990 
 889888! 
 
 ■7' 
 
 909660 
 
 4-3o 
 
 090340 
 
 55 
 
 6 
 
 799806 
 
 2 
 
 5 9 
 
 ■7' 
 
 909918 
 
 4-3o 
 
 090082 
 
 54" 
 
 •2 
 
 799962 
 8001 17 
 
 2 
 
 5 9 
 
 889785 
 
 ■7" 
 
 9101-77 
 
 4-3o 
 
 089823 
 
 53 
 
 2 
 
 5 9 . 
 
 889682 
 
 [•71 
 
 910435 
 
 4-3o 
 
 089565 
 
 52 
 
 9 
 
 800272 
 
 2 
 
 58 
 
 889579 
 
 •71 
 
 9io6o3 
 ' 910961 
 
 4-3o 
 
 089307 
 
 5i 
 
 10 
 
 800427 
 
 2 
 
 58 
 
 889477. 
 
 [•71 
 
 4-3o 
 
 , 089049 
 
 10-088791 
 
 088533 
 
 5o , 
 
 ii 
 
 9 -800582 
 
 2 
 
 58 
 
 9-889374 
 
 [■72 
 
 9-911209 
 
 4-3o 
 
 49 
 
 12 
 
 800737 
 
 2 
 
 58 
 
 889271! 
 
 [•72 
 
 911467 
 
 4-3o 
 
 48 . 
 
 i3 
 
 800892 
 
 2 
 
 58 
 
 889168 
 
 [-72 
 
 911724 
 
 4-3o 
 
 088276 
 
 % 
 
 14 
 
 801047 
 
 2 
 
 58 
 
 889064 
 
 1-7? 
 
 911982 
 
 4-3o 
 
 088018 
 
 i5 
 
 801201 
 
 2 
 
 58 
 
 888961 
 
 1-72 
 
 912240 
 
 4-3o 
 
 087760 
 
 45 
 
 16 
 
 80 1 356 
 
 2 
 
 57 
 
 888858 
 
 '•72. 
 
 9 1 2498 
 912706 
 
 4-3o 
 
 087502 
 
 44 
 
 '7 
 
 8oi5ii 
 
 2 
 
 57 
 
 888755 
 
 1-72 
 
 4-3o 
 
 087244 
 
 43 
 
 IS 
 
 80 1 665 
 
 2 
 
 57 
 
 888651 
 
 1-72 
 
 913014 
 
 4-29 
 
 086986 
 
 42 
 
 >9 
 
 801819 
 801973 
 
 2 
 
 57 
 
 888548 
 
 1-72 
 
 913271 
 
 4-29 
 
 086729 
 
 41 
 
 20 
 
 2 
 
 57 
 
 888444 
 
 i- 7 3 
 
 913529 
 
 4-29 
 
 086471 
 
 40 
 
 21 
 
 9-802128 
 
 2 
 
 57 
 
 9-888341 
 
 i- 7 3 
 
 9-9U787 
 
 4-29 
 
 10-086213 
 
 1 
 
 22 
 
 802282 
 
 2 
 
 56 
 
 888237 
 
 i- 7 3 
 
 914044 
 
 4-29 
 
 085956 
 
 23 
 
 802436 
 
 2 
 
 56 
 
 888 1 34 
 
 .. 7 3 
 
 914302 
 
 4-29 
 
 ,085698 
 
 37 
 
 24 
 
 8o258o 
 802743 
 
 2 
 
 56 
 
 888o3o 
 
 1-73 
 
 914060 
 
 4-29 
 
 '085440 
 
 36 
 
 25 
 
 2 
 
 56 
 
 887926 
 887822 
 
 1-73 
 
 914817 
 
 4-29 
 
 o85i83 
 
 35 
 
 26 
 
 802807 
 8o3o5o 
 
 2 
 
 56 
 
 i- 7 3 
 
 91 5o75 
 
 4-29 
 
 084925 
 
 34 
 
 =7 
 
 2 
 
 56 
 
 887718 
 
 1.73 
 
 9i5332 
 
 4-29 
 
 084668 
 
 33 
 
 28 
 
 8o32o4 
 
 2 
 
 56 
 
 887614 
 
 !- 7 3 
 
 915590 
 
 4-29 
 
 084410 
 
 32 
 
 29 
 
 8o3357 
 
 2 
 
 55 
 
 88 7 5io 
 
 1.73 
 
 915847 
 
 ' 4-29 
 
 o84i53 
 
 3l 
 
 3o 
 
 8o35u 
 
 2 
 
 55 
 
 . 887406 
 
 1-74 
 
 .916104 
 
 . 4-29 
 
 083896 
 10 •o83638 
 
 3o 
 
 3i 
 
 g-8o3664 
 
 2 
 
 55 
 
 9-887302 
 
 1-74 
 
 9-916362 
 
 4-29 
 
 11 
 
 32 
 
 8o38i7 
 
 2 
 
 55 
 
 887198 
 
 1-74 
 
 916619 
 
 4-29 
 
 o8338i 
 
 33 
 
 803970 
 
 2 
 
 55 
 
 887093 
 
 1-74 
 
 916877 
 
 4-29 
 
 o83i23 
 
 =7 
 
 3 4 
 
 804123 
 
 2 
 
 55 
 
 886989 
 886885 
 
 1-74 
 
 9'7'34 
 
 4-29 
 
 082866 
 
 26 
 
 35 
 
 804276 
 
 2 
 
 •54 
 
 1-74 
 
 917391 
 
 4-29 
 
 082609 
 
 25 
 
 36 
 
 804428 
 
 2 
 
 •54 
 
 886780 
 
 1-74 
 
 917648 
 
 4-29 
 
 o82352 
 
 24 
 
 ii 
 
 8o458i 
 
 2 
 
 •54 
 
 886676 
 
 i-74 
 
 917905 
 
 4- 29 
 4-28 
 
 082095 
 081837 
 
 23 
 
 804734 
 804886 
 
 2 
 
 .54 
 
 886571 
 
 1-74 
 
 918163 
 
 22 
 
 3 9 
 
 2 
 
 •54 
 
 886466 
 
 1-74 
 
 918420 
 
 4-28 
 
 o8i58o 
 
 21 
 
 4o 
 
 8o5o39 
 
 2 
 
 •54 
 
 886362 
 
 1 -75| 918677 
 
 4-28 
 
 o8i323 
 
 20 
 
 4i 
 
 9-805191 
 
 2 
 
 •54 
 
 9.886257 
 
 1-75 9-918934 
 
 4-28 
 
 10-081066 
 
 !S 
 
 42 
 
 8o5343 
 
 2 
 
 -53 
 
 886i52 
 
 i- 7 5 
 
 919191 
 
 4-28 
 
 080809 
 
 43 
 
 8o5495 
 
 2 
 
 • 53 
 
 886047 
 
 ,- 7 5 
 
 .919446 
 
 4-28 
 
 o8o552 
 
 '7 
 
 44 
 
 8o5647 
 
 2 
 
 ■ 53 
 
 88 5 9 42 
 
 i- 7 5 
 
 919705 
 
 4-28 
 
 080295 
 o8oo38 
 
 16 
 
 45 
 
 805799 
 
 2 
 
 -53 
 
 885837 
 
 i. 7 !j 
 
 919962 
 
 2-28 
 
 i5 
 
 46 
 
 805961 
 
 2 
 
 ■53 
 
 885732 
 
 L75 
 
 920219 
 
 4-28 
 
 079781 
 
 r4 
 
 4? 
 
 806 1 o3 
 
 2 
 
 • 53 
 
 885627 
 
 !. 7 5 
 
 920476 
 
 4-28 
 
 079S24 
 
 i3 
 
 48 
 
 806254 
 
 2 
 
 • 53 
 
 885522 
 
 i- 7 5 
 
 920733 
 
 4-28 
 
 079267 
 
 12 
 
 49 
 
 806406 
 
 2 
 
 •52 
 
 8854i6 
 
 i- 7 5 
 
 920990 
 
 4-28 
 
 079010 
 078753 
 
 11 
 
 5o 
 
 806557 
 
 2 
 
 •52 
 
 8853 1 1 
 
 1.76 
 
 92124-; 
 
 4-28 
 
 10 
 
 5i 
 
 0-806709 
 806860 
 
 2 
 
 •52 
 
 9-885205 
 
 1.76 
 
 9-92i5oc 
 
 4-28 
 
 10-078497 
 
 I 
 
 '52 
 
 2 
 
 •52 
 
 885 1 00 
 
 1-76 
 
 921760 
 
 4-28 
 
 078240 
 
 53 
 
 80701 1 
 
 2 
 
 •52 
 
 884994 
 884889 
 884783 
 
 i. 7 6 
 
 922017 
 
 4-28 
 
 077983 
 
 7 
 
 54 
 
 807163 
 
 2 
 
 •52 
 
 1.76 
 
 922274 
 
 4-28 
 
 077726 
 
 6 
 
 55 
 
 807314 
 
 2 
 
 •52 
 
 1.76 
 
 92253o 
 
 4-28 
 
 077470 
 
 5 
 
 56 
 
 807465 
 
 2 
 
 • 5i 
 
 884677 
 
 1.76 
 
 922187 
 
 4-28 
 
 077213 
 
 4 
 3 
 
 tl 
 
 807615 
 
 2 
 
 • 5i 
 
 884572 
 
 1.76 
 
 923o4i 
 
 4-28 
 
 076956 
 
 807766 
 
 2 
 
 • 5i 
 
 884466 
 
 1.76 
 
 9233oo 
 
 4-28 
 
 076700 
 
 2 
 
 5o 
 
 807917 
 808067 
 
 2 
 
 •5! 
 
 88436o 
 
 1.76 
 
 923557 
 
 4-27 
 
 07644c 
 
 1 
 
 66 
 
 2 
 
 •5i 
 
 884254 
 
 i;77 
 
 9238i3 
 
 4-27 
 D. 
 
 076187 
 
 
 
 
 Cosine 
 
 D. 
 
 Sine 
 
 50° 
 
 Cotang. 
 
 Tang. 
 
 M.j 
 
58 
 
 (40 DEGREES.) A TABLE OF LOGARITHMIC 
 
 M. Sine 
 
 D. 
 
 •9 
 
 S3 
 U 
 25 
 26 
 27 
 
 28 
 
 ?9 
 3o 
 3i 
 32 
 33 
 34 
 35 
 36 
 
 ll 
 3 9 
 40 
 4i 
 42 
 43 
 44 
 45 
 46 
 
 % 
 
 £ 
 
 Si 
 52 
 53 
 54 
 55 
 56 
 
 u 
 
 0-808067 
 
 808218 
 
 8o8368 
 
 ' 8o85i 9 
 
 803669 
 
 808819 
 
 809119 
 809269 
 809419 
 809561 
 9-80971 
 
 810017 
 810167 
 8io3i6 
 8io465 
 810614 
 810763 
 810912 
 811061 
 9-811210 
 8u358 
 8u5o7 
 8u655 
 81 1 804 
 811952 
 81 2 100 
 812248 
 812396 
 812544 
 9-812692 
 812840 
 812988 
 8i3i35 
 8i3283 
 8i343o 
 8i35 7 8 
 813725 
 8i38 7 2 
 814019 
 9-814166 
 8i43i3 
 814460 
 814607 
 8i47.53 
 814900 
 8 1 5o46 
 815193 
 8i533( 
 81 548 
 9-8i563 
 815778 
 815924 
 816069 
 81621$ 
 8i636i 
 •8i65o7 
 8i6652 
 816798 
 816943 
 Cosine 
 
 2-5l 
 2-5l 
 2-5l 
 
 2-5o 
 2-5o 
 2-5o 
 2-5o 
 
 2-30 
 
 2-5o 
 2-49 
 2-49 
 2-49 
 2-49 
 2-49 
 2-4g 
 2-48 
 2-48 
 
 2-47 
 
 2-47 
 
 2-47 
 
 2-47 
 
 2-47 
 
 2-47 
 
 2-47 
 
 2-46 
 
 2.46 
 
 2-46 
 
 2-46 
 
 2-46 
 
 2-46 
 
 2-46 
 
 2-45 
 
 2-45 
 
 2-45 
 
 2-45 
 
 2-45 
 
 2-45 
 
 2-45 
 
 2-44 
 
 2-44 
 
 2-44 
 
 2-44 
 
 2-44 
 
 2-44 
 
 2-44 
 
 2-43 
 
 2-43 
 
 Cosine 
 
 D. 
 
 • 43 
 •43 
 ■ 43 
 ■43 
 •43 
 •42 
 •42 
 •42 
 2-42 
 
 9-884254 
 884148 
 884042 
 
 883o36 
 
 883829 
 
 883 7 23 
 
 8836n 
 
 8835io 
 
 883404 
 
 883297 
 
 883 191 
 
 9 -883084 
 
 882977 
 
 882871 
 
 882764 
 
 882657 
 
 88255o 
 
 882443 
 
 882336 
 
 882229 
 
 882121 
 
 9-882014 
 
 881907 
 
 881799 
 
 8816 
 
 881 5 . 
 
 881477 
 
 881369 
 
 881261 
 
 88u53 
 
 881046 
 
 9 -880938 
 
 88o83o 
 
 880722 
 
 880613 
 
 88o5o5 
 
 880397 
 
 880289 
 
 880180 
 
 880072 
 
 87996.3 
 
 9-879855 
 
 879746 
 
 879637 
 
 879529 
 
 879420 
 
 8793 1 1 
 
 879202 
 
 879093 
 
 878984 
 
 878875 
 
 9-878766 
 
 878656 
 
 878547 
 
 878438 
 
 878328 
 
 878219 
 
 878109 
 
 877999 
 
 877890 
 
 877780 
 
 1-77 
 
 1-77 
 1.77 
 1.77 
 1-77 
 '■77 
 '•77 
 '•77 
 
 7 2 
 78 
 
 t 
 1.78 
 
 1.78 
 
 78 
 
 1-78 
 
 78 
 
 '•79 
 1-79 
 
 '•79 
 '■79 
 1.79 
 
 '•79 
 '■79 
 '•79 
 '■79 
 
 79 
 1.80 
 I-.8o 
 i-8o 
 I-8o 
 i-8o 
 
 80 
 i-8o 
 
 80 
 
 80 
 1-8 
 1-8 
 
 1-8 
 
 i-8 
 
 i-8 
 
 i-8 
 
 1-8 
 
 1-8 
 
 1-82 
 
 1.82 
 
 1-82 
 
 Tang. 
 
 9-9238i3 
 924070 
 924327 
 924583 
 924840 
 925096 
 925352 
 925609 
 925865 
 926122 
 926378 
 
 9.926634 
 
 Sine 49° 
 
 D. Ootang. 
 
 1-82 
 
 82 
 
 1-82 
 
 ■82 
 
 1-82 
 1-82 
 
 1-83 
 1-83 
 1-83 
 1-83 
 1-83 
 
 927U7 
 927403 
 927639 
 927915 
 928171 
 928427 
 928683 
 928940 
 9-929196 
 929452 
 929708 
 929964 
 930220 
 93o475 
 930731 
 930987 
 93i243 
 931499 
 9-931755 
 932010 
 932266 
 932522 
 932778 
 -933033 
 933289 
 933540 
 9338oo 
 934066 
 9-934311 
 , 934567 
 934823 
 935078 
 935333 
 935589 
 935844 
 g36l .10 
 g 36355 
 936610 
 936866 
 937121 
 937376 
 937632 
 937887 
 938142 
 9383 9 8 
 9 38653 
 938908 
 939163 
 
 Cotang. 
 
 10-076187 
 
 075930 
 
 075673 
 
 075417 
 
 075160 
 
 074904 
 
 074648 
 
 074391 
 
 074i35 
 
 073878 
 
 073622 
 
 10-073366 
 
 073110 
 
 072853 
 
 072597 
 
 072341 
 
 072085 
 
 071829 
 
 071573 
 
 071317 
 
 071060 
 
 10 070804 
 
 070548 
 
 070292 
 
 070036 
 
 069780 
 
 069025 
 
 069269 
 
 0690 1 3 
 
 068757 
 
 068001 
 
 o-.o68245 
 
 067990 
 
 067734 
 
 067478 
 
 067222 
 
 066967 
 
 066711 
 
 066455 
 
 066200 
 
 065944 
 
 io-o65689 
 
 065433 
 
 065177 
 
 064922 
 
 064667 
 
 06441 1 
 
 0641 56 
 
 063900 
 
 063645 
 
 o633oo 
 
 !0'o63i34 
 
 062879 
 
 062624 
 
 062368 
 
 062113 
 
 06 1 858 
 
 061602 
 
 061347 
 
 061092 
 
 0608J7 
 
 60 
 
 3 
 
 50 
 55 
 54 
 53 
 52 
 ■5 1 
 5o 
 
 % 
 
 t 
 45 
 44 
 43 
 42 
 41 
 40 
 
 18 
 
 37 
 
 36 
 35 
 34 
 33 
 32 
 3i 
 3o 
 
 % 
 27 
 26 
 
 25 
 
 24 
 
 23 
 22 
 21 
 20 
 
 lo 
 
 !o 
 
 i5 
 14 
 i3 
 12 
 n 
 10 
 
 _T2££_ 
 
 M. 
 

 67NES 
 
 AND TANGENTS. 
 
 (41 DKGKEKS. 
 
 k 
 
 r 
 
 5 
 
 [W. 
 
 Sine 1 T>. 
 
 Cosine [ D. 
 
 Tang. 
 
 D. 
 
 Cotang. 
 
 
 c 
 
 0-816943 2-42 
 
 9. 877780' 1 -83 
 
 9-939163 
 
 4-25 
 
 10-060837 
 
 60 
 
 1 
 
 817088 
 
 2 
 
 42 
 
 877670 1 
 
 83 
 
 939418 
 
 4 
 
 25 
 
 060682 
 
 59 
 
 2 
 
 817233 
 
 2 
 
 42 
 
 877560 1 
 
 83 
 
 939673 
 
 4 
 
 25 
 
 060327 
 
 58 
 
 3 
 
 817379 
 
 2 
 
 42 
 
 877450,1 
 
 83 
 
 9.39928 
 
 4 
 
 25 
 
 060072 
 
 -57 
 
 4 
 
 817524 
 
 2- 
 
 41 
 
 8773401 
 
 83 
 
 940183 
 
 4 
 
 25 
 
 059817 
 
 06 
 
 5 
 
 817668 
 
 2 
 
 41 
 
 8772301 
 
 84 
 
 940438 
 
 4 
 
 25 
 
 059562 
 
 55 
 
 6 
 
 817813 
 
 2 
 
 41 
 
 877120' 1 
 
 84 
 
 940694 
 
 4 
 
 25 
 
 069306 
 
 54 
 
 I 
 
 817958 
 
 2 
 
 41 
 
 877010! 1 
 
 84 
 
 940949 
 
 4 
 
 25 
 
 o5oo5i 
 
 53 
 
 8i8io3 
 
 2 
 
 41 
 
 8768991 
 
 84 
 
 941 204 
 
 . 4 
 
 25 
 
 058796 
 
 52 
 
 9 
 
 818247 
 
 2 
 
 41 
 
 8767891 
 8766781 
 
 84 
 
 94S458 
 
 4 
 
 25 
 
 058542 
 
 5i 
 
 10 
 
 8i83o2 
 o.8i8536 
 
 2 
 
 41 
 
 84 
 
 94m4 
 
 4 
 
 25 
 
 058286 
 
 5o 
 
 ii 
 
 2 
 
 40 
 
 9. 876568' 1 
 
 84 
 
 9-041968 
 
 4 
 
 25 
 
 I0-o58o32 
 
 % 
 
 13 
 
 818681 
 
 2 
 
 40 
 
 876457 1 
 
 84 
 
 9422'i3 
 
 4 
 
 25 
 
 057777 
 
 i3 
 
 818825 
 
 2 
 
 40 
 
 876347.1 
 
 84 
 
 942478 
 
 4 
 
 25 
 
 057622 
 
 47 
 
 i4 
 
 818969 
 819113 
 
 2 
 
 40 
 
 876236.1 
 
 85 
 
 942733 
 
 4 
 
 25 
 
 067267 
 
 46 
 
 •5 
 
 2 
 
 40 
 
 876125,1 
 
 85 
 
 94^988 
 
 4 
 
 25 
 
 057012 
 
 45 
 
 16 
 
 819257 
 
 2 
 
 40 
 
 876014 1 
 
 85 
 
 943243 
 
 4 
 
 25 
 
 066757 
 
 44 
 
 \l 
 
 819401 
 
 2 
 
 40 
 
 875904 1 
 
 85 
 
 943498 
 943722 
 
 4 
 
 25 
 
 056502 
 
 43 
 
 819645 
 
 2 
 
 39 
 
 875793,1 
 
 85 
 
 4 
 
 25 
 
 056248 
 
 42 
 
 19 
 
 819689 
 
 2 
 
 39 
 
 875682|i 
 
 85 
 
 044007 
 
 4 
 
 25 
 
 055993 
 055738 
 
 4i 
 
 20 
 
 819832 
 
 2 
 
 39 
 
 8 7 55 7 ii 
 
 85 
 
 944262 
 
 4 
 
 25 
 
 40 
 
 21 
 
 9-819976 
 
 2 
 
 3 9 
 
 9. 875459 [ 1 
 
 85 
 
 9.944517 
 
 4 
 
 25 
 
 10-055483 
 
 ll 
 
 22 
 
 820120 2 
 
 3 9 
 
 875348,1 
 
 85 
 
 944771 
 
 4 
 
 24 
 
 055229 
 
 23 
 
 820263 
 
 2 
 
 3 9 
 
 875237' 1 
 
 85 
 
 945026 
 
 4 
 
 24 
 
 054974 
 
 ?"» 
 
 24 
 
 820406 
 
 2 
 
 ll 
 
 875126 1 
 
 86 
 
 945281 
 
 4 
 
 24 
 
 054719 
 05446S 
 
 35 
 
 25 
 
 82o55o 
 
 2 
 
 875oi4'i 
 
 86 
 
 945535 
 
 4 
 
 24 
 
 35 
 
 26 
 
 820693 
 820836 
 
 2 
 
 38 
 
 874903 ' 1 
 
 86 
 
 945790 
 
 4 
 
 24 
 
 ■054210 
 
 34 
 
 3 
 29 
 
 • 2 
 
 38 
 
 87479' !i 
 
 86 
 
 946045 
 
 4 
 
 24 
 
 053955 
 
 33 
 
 820979 
 821122 
 
 2 
 2 
 
 38 
 38 
 
 874680;! 
 874568 1 
 
 86 
 86 
 
 946299 
 946554 
 
 4 
 4 
 
 24 
 
 24 
 
 053701 
 053446 
 
 32 
 
 3i 
 
 3o 
 
 821265 
 
 2 
 
 38 
 
 8744561 
 
 86 
 
 946808 
 
 4 
 
 24 
 
 053192 
 
 3o 
 
 3i 
 
 9-821407 
 
 2 
 
 38 ' 
 
 9 -874344' 1 
 
 86 
 
 9-947063 
 
 4 
 
 24 
 
 10-052937 
 
 29 
 
 32 
 
 82i55o 
 
 2 
 
 38 
 
 874232J1 
 
 87 
 
 9473 18 
 
 4 
 
 24 
 
 052682 
 
 28 
 
 33 
 
 821693 
 
 2 
 
 37 
 
 874121 1 
 
 87 
 
 947572 
 
 4 
 
 24 
 
 062428 
 
 27 
 
 34 
 
 82i835 
 
 2 
 
 37 
 
 874009 1 
 
 87 
 
 947826 
 
 4 
 
 24 
 
 052174 
 
 26 
 
 35 
 
 821977 
 
 2 
 
 37 
 
 8738 9 6ji 
 
 87 
 
 948081 
 
 4 
 
 24 
 
 051919 
 
 25 
 
 36 
 
 822120 
 
 2 
 
 37 
 
 873784 1 
 
 87 
 
 948336 
 
 4 
 
 24 
 
 o5i664 
 
 24 
 
 37. 
 
 822262 
 
 2 
 
 37 
 
 873672 1 
 
 87 
 
 948590 
 
 4 
 
 24 
 
 o5i4io 
 
 23 
 
 38 
 
 822404 
 
 2 
 
 37 
 
 873560 1 
 
 87 
 
 948844 
 
 4 
 
 24 
 
 05n56 
 
 22 
 
 3 9 
 
 822546 
 
 2 
 
 ll 
 
 873448 ; i 
 
 87 
 
 949099 
 949353 
 
 4 
 
 24 
 
 000901 
 
 21 
 
 40 
 
 822688 
 
 2 
 
 8 7 3335;i 
 
 87 
 
 4 
 
 24 
 
 050647 
 
 20 
 
 41 
 
 9-822830 
 
 2 
 
 36 
 
 9-873223,1 
 
 87 
 
 9-949607 
 
 4 
 
 24 
 
 io-o5o393 
 o5oi38 
 
 :g 
 
 42 
 
 822972 
 
 2 
 
 36 
 
 8731 10 1 
 
 88 
 
 949862 
 
 4 
 
 24 
 
 43 
 
 823114 
 
 2 
 
 36 
 
 872998 
 
 
 88 
 
 9501 16 
 
 4 
 
 24 
 
 049884 
 
 '7 
 
 44 
 
 823255 
 
 2 
 
 36 
 
 872885 
 
 
 88 
 
 950370 
 
 4 
 
 24 
 
 049630 
 
 16 
 
 45 
 
 823397 
 
 2 
 
 36 
 
 872772 1 
 
 88 
 
 950625 
 
 4 
 
 24 
 
 049375 
 
 i5 
 
 46 
 
 823539 
 
 2 
 
 36 
 
 872659' 1 
 
 88 
 
 950879 
 q5i i33 
 
 4 
 
 24 
 
 049121 
 048867 
 
 14, 
 
 5 
 
 82368o 
 
 2 
 
 35T 
 
 872547:1 
 
 88 
 
 4 
 
 24 
 
 i3 
 
 823821 
 
 2 
 
 35 
 
 87243411 
 
 88 
 
 , 9 5 1 388 
 
 4 
 
 24 
 
 048612 
 
 12 
 
 49 
 
 823963 
 
 2 
 
 35 
 
 872321 1 ! 
 
 88 
 
 951642 
 
 4 
 
 24 
 
 048358 
 
 11 
 
 5o 
 
 824104 
 
 2 
 
 35 
 
 87220811 
 
 88 
 
 961896 
 
 4 
 
 24 
 
 048104 
 
 10 
 
 5i 
 
 9-824245 
 
 2 
 
 35 
 
 9-872005 I 
 
 89 
 
 9-952160 
 
 4 
 
 24 
 
 10-047850 
 
 I 
 
 52 
 
 824386 
 
 2 
 
 35 
 
 871981 I 
 
 89 
 
 9524o5 
 
 4 
 
 24 
 
 047595 
 
 53 
 
 824527 
 S24668 
 
 2 
 
 35 
 
 871868 I 
 
 89 
 
 902659 
 952913 
 
 4 
 
 24 
 
 047341 
 
 7 
 
 54 
 
 2 
 
 34 
 
 871755 I 
 
 89 
 
 4 
 
 24 
 
 047087 
 
 6 
 
 55 
 
 824808 
 
 2 
 
 34 
 
 871641'! 
 
 89 
 
 953167 
 
 4 
 
 23 
 
 046833 
 
 5 
 
 56 
 
 - 824949 
 
 2 
 
 34 
 
 871528 I 
 
 89 
 
 953421 
 
 4 
 
 23 
 
 •046679 
 
 4 
 
 u 
 
 825090 
 
 2 
 
 34 
 
 871414 I 
 
 89 
 
 953675 
 
 4 
 
 23 
 
 046323 
 
 3 
 
 825230 
 
 2 
 
 34 
 
 87'i3oi!i 
 
 89 
 
 953929 
 954183 
 
 4 
 
 23 
 
 046071 
 
 2 
 
 5 9 
 
 825371 
 
 2 
 
 34 
 
 8711871 
 
 89 
 
 4 
 
 23 
 
 045817 
 
 I 
 
 60 
 
 8255i 1 
 
 2 
 
 34 
 
 871073 1 
 
 90 
 
 954437 
 
 4-23 
 
 045563 
 
 
 
 Cosine 
 
 IX 
 
 Sine |4 
 
 8° 
 
 Ootong. 
 
 
 0. 
 
 Tang. 
 
 .«L 
 
 25 
 
80 
 
 (42 
 
 DEUREES.) A 
 
 rABLE OF "LOGARITHMIC 
 
 
 M. 
 
 
 
 Sine 
 
 D. 
 
 Cosine | D. | Tang. 
 
 D. 
 
 Cotang. 
 
 
 9-8255II 
 
 2-34 
 
 9-871073 1-90! 9-964437 
 
 4 
 
 23 
 
 10-045563 
 
 60 
 
 i 
 
 825651 
 
 2 
 
 33 
 
 870960' 1 
 870846'! 
 
 90 
 
 964691 
 
 4 
 
 23 
 
 o453oo 
 o45o55 
 
 ol 
 
 2 
 
 825791 
 825931 
 
 2 
 
 33 
 
 90 
 
 954945 
 
 4 
 
 23 
 
 3 
 
 2 
 
 33 
 
 870732 1 
 
 90 
 
 955200 
 
 4 
 
 23 
 
 044800 
 
 n 
 
 4 
 
 826071 
 
 2 
 
 33 
 
 870618 1 
 
 90 
 
 955454 
 
 4 
 
 23 
 
 044546 
 
 5 
 
 826211 
 
 2 
 
 33 
 
 8-jo5o4 
 
 
 90 
 
 955707 
 
 4 
 
 23 
 
 o442o3 
 o44o3o 
 043785 
 
 55 
 
 6 
 
 82635i 
 
 2 
 
 33 
 
 870390 
 
 
 90 
 
 955961 
 
 4 
 
 23 
 
 54 
 
 I 
 
 826491 
 
 2 
 
 33- 
 
 870276 
 
 
 90 
 
 9562i5 
 
 4 
 
 23 
 
 53 
 
 82663i 
 
 2 
 
 33 
 
 870161 
 
 
 90 
 
 956469 
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 97 
 
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 52 
 
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 37 
 
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 28 
 
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 29 
 
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 35 
 
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 37 
 
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 38 
 
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 22 
 
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 58 
 
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 4 
 
 21 
 
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 2 
 
 18 
 
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 4 
 
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 66 
 
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 4-21 
 
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 57 
 
 4 
 
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 842424 
 
 2-17 
 
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 4-21 
 
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 55 
 
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 2-17 
 
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 4-21 
 
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 54 
 
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 842685 
 
 2-17 
 
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 2-04 
 
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 4-21 
 
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 53 
 
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 2-17 
 
 855 9 56 
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 4-21 
 
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 52 
 
 9 
 
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 2-04 
 
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 4-21 
 
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 10 
 
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 9-8432o6| 2-16 
 
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 2-oJ 
 
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 4-21 
 
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 9-855588 
 
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 9-987618 
 
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 12 
 
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 4-21 
 
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 4-21 
 
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 47 
 
 14 
 
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 855219 
 
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 4-21 
 
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 46 
 
 15 
 
 S43t25 2' 16 
 
 843855 2- 16 
 
 855096 
 
 2-o5 
 
 988629 
 
 4-21 
 
 011371 
 
 -45 
 
 16 
 
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 2-oS 
 
 988882 
 
 4-21 
 
 011118 
 
 44 
 
 \l 
 
 843984 
 
 2-l6 
 
 85485o 
 
 2-o5 
 
 9S9134 
 
 4-21 
 
 010866 
 
 43 
 
 8441 14 
 
 2-l5 
 
 854727 
 
 2-o6 
 
 989387 
 
 4-21 
 
 oio6i3 
 
 42 
 
 '9 
 
 844243 
 
 2-l5 
 
 8546o3 
 
 2-o6 
 
 989640 
 
 4-21 
 
 oio36o 
 
 41 
 
 20 
 
 844372 
 
 2-l5 
 
 854480 
 
 2- 06 
 
 989893 
 
 4-21 
 
 010107 
 
 4o 
 
 21 
 
 9-844502 
 
 2- 15 
 
 9-854356 
 
 2-06 
 
 9-990145 
 
 4-21 
 
 10-009855 
 
 ll 
 
 22 
 
 844631 
 
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 2-o6 
 
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 4-21 
 
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 23 
 
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 37 
 
 24 
 
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 4-21 
 
 009097 
 
 36 
 
 25 
 
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 853862 
 
 2-o6 
 
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 4-21 
 
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 35 
 
 26 
 
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 4-21 
 
 008591 
 
 34 
 
 3 
 
 845276 
 
 2-14 
 
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 2-07 
 
 991662 
 
 4-21 
 
 oo8338 
 
 33 
 
 845405 
 
 214 
 
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 2-07 
 
 991914 
 
 4-21 
 
 008086 
 
 32 
 
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 2-14 
 
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 4-21 
 
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 2-14 
 
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 2-07 
 
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 4-21 
 
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 9-845790 
 
 214 
 
 9-853u8 
 
 2-07 
 
 9-992672 
 
 4-21 
 
 10-007328 
 
 8 
 
 32 
 
 845919 
 
 2-14 
 
 852994 
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 2-07 
 
 992925 
 
 4-21 
 
 007075 
 
 33 
 
 846047 
 
 2-14 
 
 2-07 
 
 993178 
 
 4-21 
 
 006822 
 
 27 
 
 34 
 
 846175 
 
 2-14 
 
 852 7 45 
 
 2-07 
 
 99343o 
 
 4-21 
 
 006570 
 
 26 
 
 35 
 
 8463o4 
 
 2-14 
 
 852620 
 
 2-07 
 
 993683 
 
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 006317 
 
 25 
 
 36 
 
 846432 
 
 2-l3 
 
 8524 9 6 
 
 2-08 
 
 993936 
 
 4-21 
 
 006064 
 
 24 
 
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