eath's Mathematical Monographs |\ Issued under the general editorship of Webster Wells, S. B. f?rofc88or of Mathematics in the Massachusetts Institute of Technology ON TEACHING GEOMETRY BV FLORENCE MILNER L)ei«oii Univkmity School Dk'iHOIT, MlCHlOAN QPi^Ql Heath & Co., Publishers ^ C^O New York Chicago Price, Ten Cents The Ideal Geometry Must have Clear and concise types of formal dem- onstration. Many safeguards against illogical and inaccurate proof. Numerous carefully graded, original problems. Must Afford ample opportunity for origin- ality of statement and phraseology without permitting inaccuracy. Call into play the inventive powers without opening the way for loose demonstration. Wells's Essentials of Geometry meets all of these demands. Half leather y Plane and Solid, J^Q pages. Price $1.2^. Plane, 75 cents. Solid, 75 cents. . C. H EATH & CO., Publishers BOSTON NEW YORK CHICAGO ^^\^^ .^^STNUT HILL, Ma..; ^ THE SKIFULL GRADATION OF ORIGINAL WORK IN 800 EXERCISES Is A Marked Feature of Wells's Essentials of Geometry As soon as the art of rigorous logic is acquired, the simpkr and more axiomatic steps are omitted from the given proof, and the student is required to supply them for himself. Ample opportunity for originality of statement is afforded, but no inaccuracy permitted. The inventive powers are called into play without opening the way for loose demonstration. As power of independent thought and reasoning grows, the student is given fewer helps, and finally, is entirely dependent on himself to make his constructions and prove his propositions. sturdy self-reliance, resourcefulness, and ingenuity, are the results. I commend particularly that feature of the Geometry in which the details of proof are left grad- ually to the pupil. GEO. BUCK. Steele High School, Dayton, O. I like the original exercises which are not at first too difficult, but by their gradation encourage and stimulate. G. K. BARTHOLEMEW. English School, Cincinnati, O. Half Leather, Plane and Solid, $1.25. Plane, 75 Cents. Solid, 75 Cents. D. C. HEATH & CO. BOSTON NEW YORK CHICAGO ESSENTIALS OF ALGEBRA By WEBSTER WELLS, S.B., Professor of Mathematics in the Massachusetts Institute of Technology. This book fully meets the most rigid requirements now made in secondary schools. Like the author's other Algebras, it has met with marked success and is in extensive use in schools of the highest rank in all parts of the country. The method of presenting the fundamental topics differs at several points from that usually followed. It is simpler, more logical and more philosophical, yet by reason of its admirable grading and superior clearness The Essentials of Algebra is not a difficult book. The examples and problems number over three thousand and are very carefully graded. They are especially numerous in the important chapters on Factoring, Fractions, and Radicals. All of them are new, not one being a duplicate of a problem in the author's Academic Algebra. In accurate definitions, clear and logical demonstrations, well selected and abundant problems, in systematic arrangement and completeness, this Algebra is unequalled. Half leather. ^58 pages. Introduction price, $1.10. D. C. HEATH & CO., Publishers, Boston, New York, Chicago FACTORING AS PRESENTED IN WELLS' ESSENTIALS ^/ALGEBRA I. Advanced processes are not presented too early. II. Large amount of practice work. No other algebra has so complete and well-graded development of this important subject, presenting the more difficult principles at those stages of the student's progress when his past work has fully prepared him for their perfect comprehension. The Chapter on Factoring contains these simple processes : CASE L When the terms of the expression have a common monomial factor thirteen examples, n. When the expression is the sum of two binomials which have a common binomial factor — twenty examples. IIL When the expression is a perfect trinomial square — twenty-six examples, IV. When the expression is the difference of two perfect square? fifty-five examples. V. When the expression is a trinomial of the form x^-\- ax-\-b — sixty-six examples. VL When the expression is the sum or difference of two perfect cubes twenty examples. Vn. When the expression is the sum or difference of two equal odd powers of two quantities — thirteen examples. Ninety-three Miscellaneous and Review Examples. Further practice in the application of these principles is given in the two following chapters — Highest Common Factor and Lowest Common Multiple. In the discussion of Quadratic Equations, Solution of Equations by Factoring is made a special feature. Equations of the forms Ar^—j';\r — 24 = 0, 2x'^ — x=-0, x'^ -\- ^x^ — x — a. = 0, and .AT?— I =0 are discussed and illustrated by thirty examples. The factoring of trinomials of the form ^x~-^6x-\-c and ax'^-^l>x'^-{-c, which involves so large a use of radicals, is reserved until Chapter XXV, where it receives full and lucid treatment. The treatment of factoring is but one of the many features ol superiority in Wells' Essentials of Algebra. Half Leather, JjS pages. Frice $1.10. D. C. HEATH & CO., Publishers BOSTON RnOTl/^ll WyORK CHICAGO CHESTNUT Hitr. M/^t (i) in general excellence (2) in special fitness for use Central High School, Philadelphia, Pa. I have examined, with interest and appreciation, Wells's Essen- tials of Geometry both with reference to its improvements over his former edition and also with reference to its excellence when com- pared with others of the present date. I have no hesitation in saying that it seems to me superior both in general excel- lence and special fitness for use in the schoolroom, to any which I have seen. George W. Schock, Prof ess 07' of Mathematics. (i) in the order of theorems (2) in the proof of corroUaries (3) in the grading of the original exercises (4) in the opportunity for original work Boys' High School, Brooklyn, N. Y. In the order of theorems, the proof of corroUaries, the grading of the original exercises, in the diagrams for the exercises, and in the opportunity for original work, Wells's Essentials of Geometry is notably superior. C. A. Hamilton, Instructor in Mathematics. Decidedly the Best Central High School, Cleveland, Ohio. Wells's Essentials is decidedly the best geometry for general school work so far published. Walter D. Mapes, Instructor in Mathematics. ESSENTIALS OF GEOMETRY PLANE AND SOLID By WEBSTER WELLS, S.B. Professor of Mathe77iatics in the Massachusetts Institute of Technology In this new work, issued in 1899, the ideal of modern teach- ing of Geometry is made practical by a method which neither discourages the pupil nor helps him to his hurt. The author recognizes the needs of the beginner, and meets them in such a way as to arouse his interest and enthusiasm. The college re- quirements are heeded, both in letter and spirit, without sacrifice of organic unity. The exercises are about 800 in number, and are carefully graded. An important feature consists in giving figures and sug- gestions in connection with the exercises. In Books I and VI, a figure is given for nearly every non-numerical exercise ; in the other books they are given less frequently. It is believed that with this aid the exercises are brought within the capacity of the aver- age pupil, and that his interest in the solution of the original exercises will be stimulated. Every definition, demonstration and discussion has been sub- jected to rigorous criticism in order to secure clearness, brevity and absolute accuracy. It is believed that no other text in Geometry is so free from ambiguous and loosely constructed statements. The Appendix contains rigorous proofs of the limit-statements used in connection with the demonstrations of Book IX. Half leather. Pages, viii -\-jgi. Introduction price, $1.2^. Plane Geometry, separately, y^ cents. Solid Geojuetry, separately, 75 cents. D. C. HEATH & CO., Publishers, Boston, New York, Chicago Good Advice from Harvard GEOMETRY " As soon as the pupil has begun to acquire the art of rigorous demonstration, his work should cease to be merely receptive, he should be trained to devise constructions and demonstrations for himself, and this training should be carried through the whole of the work of Plane Geometry. Teachers are advised, in their selection of a text-book, to choose one having a clear tendency to call out the pupil's own powers of thought, prevent the formation of mechanical habits of study, and encourage the concentration of mind which it is a part of the discipline of mathematical study to foster." — Extract from the Harvat'd University Catalogue^ igoo, page 304. Wells's Essentials of Geometry fully meets these broad requirements, and meets them more successfully than any other book. Half Leather, Plane and Solid, ^1.25. Plane, 75 cents. Solid, 75 cents. D. C. HEATH ^ CO., Publishers BOSTON NEW YORK CHICAGO ^Br ON TEACHING GEOMETRY BY FLORENCE MILNER DETROIT UNIVERSITY SCHOOL DETROIT, MICHIGAN BOSTON COLLEGE LI. UAUif CHESTNUT HILL, MA6S. MATH, DEPT. BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS 1900 Copyright, 1900, By D. C. Heath & Co. 150013 ON TEACHING GEOMETRY. Of writing many geometries there is no end. With any of them, or without them all, the good teacher will get good results; with the best of them, the poor teacher cannot rise above medioc- rity. Under both conditions, however, there is wisdom in a careful choice, for a strong book not only lessens the labors of a good teacher, buf makes it possible for a class to get some value out of the work in spite of poor teaching. Yet we as teachers are inclined to ask too much of text- books, and we expect them not only to do their own work, but also to become responsible for a large share of ours. It is the province of the text- book to present clearly, according to its established sequence, the subject matter of geometry, not to teach how to teach it, and that book is best which has in it least of anybody's method, even of thei present writer's, and most of clearly expressed] geometry. So, however good the book, there always remains, and wisely too, much for the teacher to do. It is not the present purpose to outline any par- ticular method of teaching, but to call attention to a few general lines which, in harmony with the 2 On Teaching Geometry. subject, should be followed out, and to touch upon some points in a wider training which a correct handling of the subject should give. Geometry differs markedly from the preceding studies, and success or failure depends largely upon getting at the beginning the right point of view. To play a game of chess one must learn the moves of individual pieces. These, in infinite combination, but always under the same fixed rules, make up the most intricate of intellectual games. Geometry is analogous. We expect no one to play a game of chess until he learns the moves; we should not expect a pupil to work intelligently in geometry until he is helped to the mental attitude demanded by the subject, and knows something of the few simple truths that are the guiding thread through the seemingly intricate labyrinth. In geometry the pupil encounters for the first time formal logic, and we are too prone to plunge a class into it without adequate preparation. Be- fore opening the book at all, the pupil should be I taught something of the nature of logic, something \ of its requisites, something of its method of work. Certain texts adhere to set, formal demonstrations, others give nothing but original work, while be- tween these two extremes are all grades of com- promise, with more or less practical application of estabhshed truths. Whether we are working out original theorems or are following the demonstra- On Teaching Geonietry. 3 tion of a conventional proposition or are using our knowledge in the solution of a practical problem, the process is identical. Axioms and definitions are the foundation upon which the whole superstructure of geometry is builded, and in the beginning a class should be thoroughly grounded in an understanding of their nature and province. Definitions should receive first attention. If you and I are to carry on a discussion about an apple, we must agree as to its characteristics. If you define it as " the fleshy pome or fruit of a rosaceous tree," while I insist that it is a hard, black mineral, we shall never get far in our argu- ment. If you say that " a triangle is a plane fig- ure bounded by three right lines," while I call a solid bounded by four curved surfaces a triangle, our discussion will again come to grief. It be- comes necessary, then, that you and I agree upon a common conception of the object under consider- ation, and that we shall so describe it that ambi- guity is impossible. This is the province of the definition. A definition is such a description as will bring up to all minds the same conception. The definitions in geometry have long been agreed upon by mathematicians, and the importance of knowing, and knowing with verbal accuracy, the universally accepted description of geometric con- cepts cannot be too strenuously insisted upon, and the teacher who in place of accurate technical Ian- 4 On Teaching Geometry. guage accepts careless and verbose statements, does a class a great wrong. A lover of literature will have little patience with one who, pretending to give Hamlet's soliloquy or Milton's sonnet On his Blmd?tess would dare substitute his own meagre words for the matchless language of the great masters. In certain directions the phraseology of mathematics has crystallized equally with other literature, and the giving of familiar definitions is not the place for original work, — that comes later. The attitude toward axioms should next be made right. Every sane mind finds itself unconsciously in possession of certain knowledge. We know that certain things are true. No one can remem- ber when he learned that the whole of anything is greater than any one of its parts, but the babe creeping along the floor has a practical knowledge of the truth, and he will contest it with you to the limit of his physical strength. Later he will for- mulate the idea, and possibly when he gets into the high school, some teacher will set him to learning this axiom as though it were something new. On the contrary, the youth should be taught that cer- tain facts are so evidently true that everybody must needs accept them or be counted of unsound mind. When he begins the study of geometry, many of these truths should be put into compact form for future use. Here again comes the value of systematic and even stereotyped phraseology. He should learn now to state in the technical Ian- On Teaching Geometry. 5 guage of mathematics what he and everybody else have long known. When the pupil has been put in the right atti- tude toward axioms and definitions, he is ready to open his geometry and learn how to use them. Certain general definitions should be studied. He should know what a proposition is ; that there are various kinds, differing in purpose and in form of expression. As he comes to them, he should discriminate carefully between theorem, problem, corollary, and lemma. Most geometries give some preHminary work consisting largely of definitions and other discus- sion of geometrical concepts. This should be studied with more or less thoroughness, as the preparation of the class demands. Entering upon the work peculiar to geometry, we come to the opening section, which may treat of perpendicular straight Hnes, of triangles, or of whatever the sequence of the particular text in use demands. After reading the first theorem, the hypothesis should be isolated that the pupil may know exactly what is to be his by gift of this same hypothesis, and he should be taught to enumerate these gifts. Perhaps the enumeration will run like this : " A Une, a point without that line, and a per- pendicular from the given point to the given line." Show the class that it is not sufficient to say that a line and a point are given; this allows one to locate the point within the line. Whenever they 6 On Teaching Geometry. fail to be thus accurate in statement, draw figures, taking such latitude as they leave you by inaccu- rate statement. A few illustrations will show them the necessity of talking sharply to the facts, and the pupils will soon learn to hold each other to statements that admit of no ambiguity. A certain power of reconstructive imagination should also be cultivated. Besides stating accu- rately the conception that is in one's own mind, there must be the ability to construct, rapidly and clearly, mental pictures of all statements made in class. Every sentence uttered should add some- thing to the picture or present some new phase of it. Every recitation should be a series of con- stantly changing views in which everybody, in his mind's eye, sees the same things. This result is not easy to obtain, but it is possible, and the degree of excellence reached by a class becomes a crucial test of the teacher. Impress upon a class that the formal statement of a proposition is always a genei'al truth, true of all figures falling under the given conditions. These statements are conventional and should be as accurately stated as definitions and axioms. In proving a proposition, our human limitations require us to fix our minds upon a particular case, and accordingly a special statement follows. Given i. Line AB. 2. Point P without the line. 3. Perpendicular PD upon the line On Teaching Geometry. 7 To prove, that PD is the shortest distance from P to the Hne AB. If there is more than one conclusion, bring out the facts in clear, mathematical language. Do not be satisfied to let a class say, " To prove that from the point P one perpendicular and only one can be let fall to the line ABr It should be stated, " To prove : first, that one perpendicular can be let fall upon the line AB\ and second, that only one can be let fall from the same point upon the line." This may seem to some like splitting hairs, but mathematics is an exact science, and in the begin- ning too much cannot be done in training toward accurate thought and exact expression. Now comes important work in preparing for a formal demonstration. Here the class must learn that the regular, logical form of every argument is a syllogism. Pupils will not be frightened at the new word, and they will like to use it when they comprehend its meaning. Discuss it with them. Explain the major premise, the minor premise, and show how the conclusion inevitably follows. Give them an example and set them to hunting for others. You will wonder where they find so many. Next teach them that every step in a well-con- structed demonstration is taken by means of a syl- logism. As the major premise must always enun- ciate a general truth, the major premise in their first demonstration must be either an axiom or a definition, for these are the only general geometric 8 On Teaching Geometry. . truths in their possession. The minor premise isolates and states formally the special case now present and in harmony with the major premise. The conclusion is the new knowledge revealed by bringing these two premises together. This con- clusion in turn may become the minor premise, and so on to the end of the argument. Show a class how a syllogism may be worked out from some of the early definitions and axioms, letting the minor premise be a fact given. Tell them, for instance, to draw the line CD to the point D in the line AB, making the angle CDB a right angle. This is the fact given, and if dem- onstration were to follow, might constitute the hypothesis. Now if the fact is of any value in constructing a syllogism it must fall under some general definition or axiom. Some one will dis- cover that the definition for a perpendicular fits the case. Then you can show how the inevitable conclusion that CD is perpendicular to AB must follow. Finally show the formal structure of a syllogism and write out the one developed. 1. The line CD meets the line AB, making the angle CDB a right angle. 2. A perpendicular to a line is a line which makes right angles with a given line. 3. CD is perpendicular to AB. Repeat the process until you make them see that when the first two terms are rightly selected and agreed upon, there is no escape from the On Teaching Geometry. 9 conclusion. It follows like a decree of inexorable fate. Every demonstration is a chain of syllogisms, in all of which the major premise must be some gen- eral conclusion already established, either axiom, corollary, theorem, definition, or algebraic truth. Of course as the work goes on, the formal state- ment that involves frequent repetition can be con- tracted, but the habit of looking at every demon- stration from this side is invaluable. Text-books do not follow this form, for they rarely give complete demonstrations. The text is an outline of the subject, a note-book, keeping the general trend of the argument but leaving the pupil to fill in the suggested discussion. Teachers rarely insist upon so closely logical a demonstra- tion as is here outlined, but experience and careful comparison have convinced me that the syllogistic method closely adhered to in the beginning pro- duces results that come from no other kind of work. In the illustration just used, a pupil might say and with truth, " CD is perpendicular to AB because CDB is a right angle," and "When a line meets," etc., but that is making a statement and then going back to try to make the statement good. The other method never leaves a chance to question a point, for one follows another in the logical order that carries conviction at every step. If you announce to an opponent the thing of lo On Teaching Geometry. which you expect to convince him, the effect is usually to arouse his antagonism, and he will then and there make up his mind, and no argument of yours, no matter how convincing, is likely to change his opinion. He may listen to you politely, but when you are through he will probably say, "Yes, but as I said in the first place." Don't give him a chance to say anything in the first place, but make him grant your premises, one after another, then when your conclusion is reached, there is nothing left for him to do but accept it. If this method is used in conventional and formal demonstration, it becomes of greatest value in original work. A pupil thus trained will, when given original work, have a definite and efficient method of attack. He will first study his hypoth- esis, isolating each individual part of it. Taking one fact for a minor premise, he will next examine his stock in trade and see what definition, theorem, corollary, or other authority he can bring to bear on the case in hand. He will draw his conclusion and see if it advances his argument or brings him nearer the desired end. If it does not, then he decides that his minor premise is wrong and re- turns to his hypothesis for another fact, and con- structs the syllogism. Sooner or later he is bound to reach the right conclusion. Sometimes teachers allow pupils to say that such or such a thing is true "by a previous proposi- tion." This is a much abused and greatly over- On Teaching Geometry. ii worked expression. It becomes to a poor student like a cloak of charity, and if permitted, will be used to cover a multitude of hazy, illogical, and vague ideas. Do you think that a judge in court would admit as evidence the statement that somewhere, some- time, somebody had made a decision in a case somewhat similar to the one in hand ? Indeed not. The lawyer would have to produce his authority, giving title, volume, and page of report, with date of decision. In all demonstrations equal accuracy should be demanded. Not that chapter and verse be given, but pupils should be made to quote geo- metric scripture in proof of every conclusion. And "quote" here means exactly what the word indi- cates, not a slipshod attempt at giving the idea. There are certain definite and wide-reaching re- sults that should come from the right teaching of geometry. Mention has already been made of the importance of accurate and exact expression, but this must be preceded by equally exact and accu- rate thinking. Better than any other subject, geometry will train the youth to keep close watch and ward over the action of his mind and accustom him to express clearly and honestly the result of his own mental happenings. The ability to make wise selection under varying circumstances, is repeatedly demanded. After a figure is drawn, the pupil should examine it carefully. He may discover in it many things which he knows, from 12 On Teaching Geometry. construction, hypothesis, or other conditions, are all true, but only one of them has any concern with the business in hand. He should be trained to see that one fact, and be able to make proper use of it. Teach him to go straight after the one that he needs, and make it serve him. He should also be taught that certain things lie within his power to do, while others are as abso- lutely beyond his control as are the turbulent waves, the floating clouds, or the sweeping course of the planets. He may push a stone from the edge of an over- 1 hanging cliff, but after that he must let it go i crashing down the mountain side. He may elect I to let fall a perpendicular from a certain point to I a given line, but he cannot dictate where it shall strike the line. He may know certain things about I two figures, and it may be wise to try the applica- f tion of one to the other. He may lift one figure and place a line of it or an angle upon its equal, : but there his control over the matter ceases. From : that moment he is under law, and face to face with \ the eternal. Let him stand there a humble spec- tator, knowing that till heaven and earth shall pass, ''one jot nor one tittle shall in nowise pass from the law " over which his finite mind has no control. He was responsible for placing the figures together, but before the resulting consequences he is help- less. He may watch to see if in the finality there is anything that concerns him. Here again he will On Teaching Geometry. 13 discover various things accomplished ; but again he must recognize the one conclusion for which he has been striving, must isolate it from the rest, and hold it up to view with the strength of conviction. The young mind is naturally unreasoning, and often utters words without consciousness of a defi- nite idea back of them. The pupil will watch the ^ teacher rather than himself, to determine whether I he is travelling the right road. It is very easy for • a teacher unconsciously to take this responsibility, j and by various gentle leading-strings keep the pupil '■■ in the path ; but such work is not teaching geome- try. If a class leans upon you, or has the habit of watching you, rest assured that your teaching is \ not right. If the development is correctly carried on, the pupil will be aroused to watch his own mind, his own statements, and the work in hand ; the more nearly he can forget j/oii, the better. He \ must learn to discover what is actually happening in his own mind as the process of reasoning goes { on ; he must be trained to faith in his own con- j victions, and to fearlessness in expressing them. • Nothing is more pitiful than a mind that can be shifted from any position by an incredulous ques- tion. At first your pupils will not think inde- pendently. Rouse them from this condition, and | force them to a conclusion of some sort. A wrong \ opinion is better than no opinion at all, for activity ^ is better than stagnation. In a demonstration the teacher must be con- 14 On Teaching Geometry. stantly in the attitude of the doubting Thomas, who took nothing on faith, but always demanded evidence. In this way the burden of proof rests with the pupil, and his aim should be to anticipate every possible question. An exceptionally good teacher of geometry says that the right results I have not been attained until a pupil not only ceases to lean upon a teacher, but is also able to stand up against a teacher. Do not rest until you see your pupils thinking independently, and at the I same time clearly, fearlessly talking out their con- I elusions. For a complete setting in order of one's mental house, the High School course offers nothing better than georaetry. The facts learned have some value, but the greatest good is the mental poise, the clear- I ness of vision, and the honesty of expression that it develops. In addition to all this, a pupil rightly trained will learn to measure his own strength, will recognize his limitations, and will bow in reverent respect before some things greater than himself. On Teaching Geometry. 15 Note. — Some teachers may be interested to see how the preceding discussion would tend to elaborate in reci- tation the demonstrations as they appear in our best text- books. It has seemed wise, accordingly, to add such a detailed demonstration. The particular theorem is selected because it offers illustrations of more points than does any other in the early part of the work. The syllogistic method has been strictly adhered to until the last few steps of the concluding argument are reached. At this point the mind works so rapidly as to become impatient of talking out in slow words what it grasps instantly, if the preceding argument has been properly built up. You have seen children stand domi- noes in a row at regular intervals, working with pains- taking care to get them rightly placed. A single touch upon the last one overthrows it, and the others inevitably fall in rapid succession. So in this argument. When the conclusion that CEC cannot be a straight line is reached, all the rest of the structure comes tumbling down so rapidly that no time is left to do more than to watch each domino as it falls. See to it that the arguments are rightly placed as the demonstration is built up, and the rest will take care of itself. Of course the syllogisms can be easily repeated in this last part as well, and it is often a good exercise to allow a class to supply them. i6 On Teaching Geometry. Book I. Proposition VI. Theorem.* From a given point witJioitt a straight line a perpen- dicular can be drawn to the line, and but one. Given the line AB and the point C without the Hne. To prove (i) that from the point C one J_ can be drawn to the Hne AB ; (2) that only one J_ can be drawn from the point C to the line AB. Proof. I. Draw the auxiliary line FG, and let UK be drawn from H 1^ to AB. 2. At a given point in a straight line a per- pendicular to the line can be drawn, and but one. (25) .-. 3. BKisl.ioFG2XB. K F- H ■a E\ D He' Apply the line FG to the line AB, and move it along until HX passes through C. Let V be the point where H falls. Draw the hne CB>. I . CD has two points, C and D, which coincide with points in HX by construction. * From Wells's Essentials of Plane and Solid Geometry. On Teaching Geometry. 17 2. But one straight line can be drawn between two points. (Ax. 3.) .'. 3. CD coincides with HK, and CDB coincides with and is equal to KHG. 1. Xi7(S^ is a rt. Z. 2. Z CDB = AKHG. .'. 3. CDB is a rt. Z. I. Ci^^isart. Z. 2. WTien a line makes a right angle with another line, it is said to be perpendicular to it. (24) .-. 3. CD is ± to AB. Hence one perpendicular can be drawn from the point C to the Hne AB. If there can be another _L from C to AB, let it be C£. Produce CD, making CD = CD. Draw C'£, 1. CD = CD by construction. £D is _L to CC by construction. 2. If lines be drawn to the extremities of a straight line from any point in the perpendicular erected at its middle point, they make equal angles with the perpen- dicular. (44) /. 3. Z C'£D = Z C£D. 1. ACED = ACED, 2. Z CED is a rt. Z by hypothesis. .-. 3. Z CED is also a rt. Z. Add Z C^/) and Z C'^Z>. 1 8 On Teaching Geometry. " 1. Z. CED is a rt Z by hypothesis. Z CED is a rt. Z by proof. 2. If the sum of two adjacent angles is equal to two right angles, their exterior sides lie in the same straight line. (37) .-. 3. CEC is a straight line. 1. CEC is a straight line by proof. CDC is a straight line by construction. 2. But one straight line can be drawn between two points. (Ax. 3.) .'. 3. As we know that CDC is a straight line by con- struction, CEC cannot be a straight hne. If CEC is not a straight line, then CED + CED is not = to two rt. A. If CED + CED is not equal to two rt. A, then CED^ which is half of this sum, is not a rt. Z. If CED is not a rt. Z, then CE is not _L to AB. As CjS" is any other possible _L from C to AB, then CZ^ is the only _L from C to the line AB. Hence there can be only one perpendicular from C to the line AB, THREE (3) POINTS FROM MANY IN WHICH WELL5'5 ESSENTIALS OF GEOMETRY EXCELS : Accuracy No other Geometry is so free from ambiguous and loosely constructed statements. Ever}- definition and demonsti-a- tion has been subjected to rigorous criticism in order to secure clearness, brevity and absolute accuracy. Adaptation to the needs of beginners. The difiiculties that confront the pupil are recognized and met in such a way as to arouse his interest and enthusiasm. Propositions and original exercises are presented in a manner at once more teachable and more educative than ever before attempted. Adequacy to the demands of the colleges and technical schools. The entrance requirements are heeded, both m letter and spirit, without sacrifice of organic unity. {I. Accurate for Everybody 2. Interesting to Pupils 3. Satisfactory to Teachers No teacher in search of the best and most practical text on Geometry can afford to disregard the merits of Wells's Essentials. D. C. HEATH & CO., Publishers BOSTON NEW YORK CHICAGO EXERCISE BOOK IN ALGEBRA ^Designed for supplementary or review work in connection with any text-hook in ^Igehra. By MATTHEW S. McCURDY, M.A., Itistructor in Mathematics in the Phillips Academy, Andover, Mass. This book is designed to furnish a collection of exercises similar in character to those in the ordinary text-books, of medium grade as to difficulty, and selected with special reference to giving an opportunity for drill upon those subjects which experience has shown to be difficult for students to master. Though intended primarily to be supplementary to some regu- lar text-book, a number of definitions and a few rules have been added, in the hope that it may also be found useful as an inde- pendent review and drill book. With or without answers. Cloth. Pages, vi -f- 220. Introduction price, 60 cents. ALGEBRA LESSONS By J. H. GILBERT. This series is intended for supplementary or review work, and contains three numbers: No. i — To Fractional Equations, No. 2 — Through Quadratic Equations, No. 3 — Higher Algebra. Paper. Tablet form. Price, $1.44 per do^en. REVIEW AND TEST PROBLEMS IN ALGEBRA By S. J. PETERSON and L. F. BALDWIN. The problems in this manual are original — none have been copied from any other author. They illustrate points of special importance, and are sufficiently varied and difficult for written drills for those preparing for college entrance examinations. Paper. 8 j pages. Introduction price, ^o cents. D. C. HEATH & CO., Publishers, Boston, New York, Chicago NEW HIGHER ALGEBRA By WEBSTER WELLS, S.B. Professor of Mathe?fiatics in the Massachusetts Institute of Technology The first 358 pages of this book are identical with the author's Essentials of Algebra, in which the method of presenting the fundamental topics differs at several points from that usually fol- lowed. It is simpler and more logical. The latter chapters present such advanced topics as compound interest and annuities, permutations and combinations, continued fractions, summation of series, the general theory of equations, solution of higher equations, etc. Great care has been taken to state the various definitions and rules with accuracy, and every principle has been demonstrated with strict regard to the logical principles involved. The examples and problems are nearly 4,000 in number, and thoroughly graded. They are especially numerous in the impor- tant chapters on factoring, fractions and radicals. The New Higher Algebra is adequate in scope and difficulty to prepare students to meet the maximum requirements in ele- mentary algebra for admission to colleges and technical schools. The work is also well suited to the needs of the entering classes in many higher institutions. Half leather. Pages, viii -\- 4g6. Introduction price, $1.32. Wells's Academic Algebra. For secondary schools. $1.08. Wells's Essentials of Algebra. For secondary schools. $1.10. Wells's Higher Algebra, $1.32. Wells's University Algebra. Octavo. $1.50. Wells's College Algebra, $1.50. D. C. HEATH & CO., Publishers, Boston, New York, Chicago COLLEGE ALGEB By WEBSTER WELLS, S.B., Professor of Mathematics in the Massachusetts Institute of Technology. The first eighteen chapters have been arranged with reference to the needs of those who wish to make a review of that portion of Algebra preceding Quadratics. While complete as regards the theoretical parts of the subject, only enough examples are given to furnish a rapid review in the classroom. Attention is invited to the following particulars on account of which the book may justly claim superior merit : — The proofs of the five fundamental laws of Algebra — the Com- mutative and Associative Laws for Addition and Multiphcation, and the Distributive Law for Multiplication — for positive or negative integers, and positive or negative fractions ; the proofs of the fundamental laws of Algebra for irrational numbers ; the proof of the Binomial Theorem for positive integral exponents and for fractional and negative exponents ; the proof of Descartes's Rule of Signs for Positive Roots, for incomplete as well as complete equations ; the Graphical Representation of Functions ; the so- lution of Cubic and Biquadratic Equations. In Appendix I will be found graphical demonstrations of the fundamental laws of Algebra for pure imaginary and complex numbers ; and in Appendix II, Cauchy's proof that every equa- tion has a root. Half leather. Pages, vi + ^78. Introduction price, $1. po- part I I, he ginning with Quadratics. ^41 pages. Introduction price, $1 .^2. D. C. HEATH & CO., Publishers, Boston, New York, Chicago Demonstrations Clear and logical Problems Well graded and abundant Order of Topics Suited to the learner Scope Adequate for the best schools THESE are a few of the characteristics of a successful Algebra, The one book that has them all, and in the right combination is the ESSENTIALS OF ALGEBRA By Webster Wells, S.B. Professor of Mathematics in the Massachusetts Institute of Technology. IT fully meets the most rigid requirements now made in secondary schools. Like the author's other Algebras, it has met with marked success and is in extensive use in schools of the highest rank in all parts of the country. The method of presenting the fundamental topics differs at several points from that usually followed. It is simpler, more logical and more philosophical, yet by reason of its admirable grading and superior clearness The Essentials of Algebra is not a difficult book. Half leather. 3,$^ pages. Introduction Price, $i.io. D. C. HEATH & CO., Publishers, Boston, New York, Chicago Wells's Mathematical Series. ALGEBRA. Wells's Essentials of Algebra ..... $i.io A new Algebra for secondary schools. The method of presenting the fundamen- tal topics is more logical than that usually followed. The superiority of the book also appears in its definitions, in the demonstrations and proofs of gen- eral laws, in the arrangement of topics, and in its abundance of examples. Wells's New Higher Algebra ..... 1.32 The first part of this book is identical with the author's Essentials of Algebra. To this there are added chapters upon advanced topics adequate in scope and difficulty to meet the maximum requirement in elementary algebra. Wells's Academic Algebra ..... 1.08 This popular Algebra contains an abundance of carefully selected problems. Wells's Higher Algebra ...... 1.32 » The first half of this book is identical with the corresponding pages of the Aca- *% demic Algebra. The latter half treats more advanced topics. Wells's College Algebra ...... 1.50 A modern text-book for colleges and scientific schools. The latter half of this book, beginning with the discussion of Quadratic Equations, is also bound sep- arately, and is known as Wells's College Algebra, Part II. $1.32. W^ells's University Algebra ..... 1.32 GEOMETRY. Wells's Essentials of Geometry — Plane, 75 cts.; Solid, 75 cts.; Plane and Solid ....... 1.25 This new text offers a practical combination of more desirable qualities than any other Geometry ever published. Wells's Stereoscopic Views of Solid Geometry Figures . .60 Ninety-six cards in manila case. Wells's Elements of Geometry — Revised 1894. — Plane, 75 cts.; Solid, 75 cts.; Plane and Solid ..... 1.25 TRIGONOMETRY. Wells's New Plane and Spherical Trigonometry (1896) . $1.00 For colleges and technical schools. With Wells's New Six-Place Tables, $1.25. Wells's Plane Trigonometry . . . . • -75 An elementary work for secondary schools. Contains Four-Place Tables. Wells's Essentials of Plane and Spherical Trigonometry . .go For secondary schools. The chapters on Plane Trigonometry are identical with those of the book described above. With Tables, $i.o8. Wells's New Six-Place Logarithmic Tables . . . .60 The handsomest tables in print. Large Page. Wells's Four-Place Tables . . . . . .25 ARITHMETIC. Wells's Academic Arithmetic . . . . . $1.00 Correspondence regarding ter77is for introduction and exchange is cordially invited. D. C. Heath & Co., Publishers, Boston, New York, Chicago BOSTON COLLEGE 3 9031 01550115 8 Mathematifsl Barton's Theory of Equations. A treatise for college classes. $1.50. Bowser's Academic Algebra. For secondary schools. $1.12. Bowser's College Algebra. A full treatment of elementary and advanced topics. $1.50. Bowser's Plane and Solid Geometry. $1.25. Plane, bound separately. 75 cts. Bowser's Elements of Plane and Spherical Trigonometry. 90 cts.; with tables, $1.40. Bowser's Treatise on Plane and Spherical Trigonometry. An advanced work for col- leges and technical schools. $1.50. Bowser's Five-Place Logarithmic Tables. 50 cts. Fine'sNumber System in Algebra. Theoretical and historical. $1.00. ^.ilbcrfs Algebra Lessons. Thr "" "' "1 riimations: No. through Quadratic Equations; .opkins's Plane Geometry, 'idyland's Elements of the Co ■fevre's Number and its Alg i>yman's Geometry Exerci^ McCurdy's Exercise Bool/ '? Plane and Spherj iih six-place tables, | Nichol's Analytic Geoip« Nichols's Calculus. Osborne's Differential oterson and Baldwin obbins's Surveying a chwatt's Geometric^ Waldo's Descriptive G with suggestions 'ells's Academic i ' ells' s Essentials Veils' s Academic l vells's New Highe fells's Higher Algftw" yells' 8 University 4y v/ells's College Algel Wells's Essentials of Wells's Elements of ( Wells's New Plane ai $1.00. With six pi v/elis's Complete T Plane, bound saps Wells's New Six-Pla( Wells's Four-Place T For Arithn. MATH. DEFT c>^^4 Jo i D.C. HEATH BOSTON COLLEGE LIBRARY UNIVERSITY HEIGHTS CHESTNUT HILL. MASS. Books may be kept for two weeks and may be renewed for the same period, unless reserved. Two cents a day is charged for each book kept overtime. If you cannot find what you want, ask the Librarian who will be glad to help you. The borrower is responsible for books drawn on his card and for all fines accruing on the same. /5^ HEATH'S MATHEMATICAL MONOGRAPH! ISSUED UNDER THE GENEIL\L EDITORSHIP OF WEBSTER WELLS, S.B. Professor of Mathematics in the Massachusetts Institute of 7'echnology It is the purpose of this series to make direct contrihu- tion to the resources of teachers of mathematics, by pre-| senting freshly written and interesting monographs upon] the history, theory, subject-matter, and methods of teachq ing both elementary and advanced topics. The first five] numbers are as follows : — 1. FAMOUS GEOMETRICAL THEOREMS AND PROBLEMS AND] THEIR HISTORY. By William W. Rupert, C.£. i. The Greek Geometers, ii. The Pythagorean Proposition. 2. FAMOUS GEOMETRICAL THEOREMS. By William W. Rupert^ ii. The Pythagorean Proposition (concluded), iii. Squaring the Circle 8. FAMOUS GEOMETRICAL THEOREMS. By William W. Rupert.^ iv. Trisection of an Angle, v. The Area of a Triangle in Terms of|1 its Sides. I' 4. FAMOUS GEOMETRICAL THEOREMS. By William W. Rltert. / vi. The Duplication of the Cube. vii. Mathematical Inscription upon the Tombstone of Ludolph van Ceulen. 6. ON TEACHING GEOMETRY. By Florence Milner. Others in preparation. PRICE, 10 CENTS EACH D. C. HEATH & CO., Publishers BOSTON NEW YORK CHICAGO