THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES !^ r-=H he'- UNIVERSITY of CALIFORNl AT LOS ANGELES LIBRARY asiter^iDe €Ducational jmonograpljjs EDITED BY HENRY SUZZALLO PRESIDENT OF THE UNIVERSITY OF WASHINGTON SEATTLE, WASHINGTON THE TEACHING OF HIGH SCHOOL MATHEMATICS BY GEORGE W. EVANS HEADMASTER OF THE CHARLESTOWN HIGH SCHOOL HOUGHTON MIFFLIN COMPANY BOSTON NEW YORK CHICAGO SAN FRANCISCO COPYRIGHT, 59II, BY HOUGHTON MIFFLIN COMPAWV ALL RIGHTS RESERVED QTijt Bibtrsitit JOnii CAMBRIDGE . MASSACHUSETTS U . S . A r ■ ^- ^ CONTENTS to Editor's Introduction ... v I. The Modern Point of View . . i 11. The Order of Topics ... 13 III. Equations and their Use . . • 17 r4 IV. Some Rules of Thumb ... 23 '^j,,^^^ V. Geometry as Algebraic Material . 32 *\ VI. The Graphical Method ... 41 VII. The Bases of Proof in Geometry . 49 VIII. The Method of Limits ... 60 ^ IX. Simpson's Rule and the Curve of Sections 74 X. The Teacher 85 Outline 91 EDITOR'S INTRODUCTION The ideal of practicality has now entered the schools with telling force. It has been manifested in its demand for vocational training, and it is reconstructing the older cultural training by elim- inations and additions. Its effects on the curri- cula of liberal schools are quite obvious. Materials once accepted without question, when schools had a margin of energy, are now displaced by the pressure of new demands. These new demands force the teachers of a crowded curriculum to re- consider every traditionally taught fact or process from the standpoint of its relative rather than its absolute worth. Without scorn for the value of any truth, old disciplines disappear and new ones enter. The ends of life are the ends of the school, and what is social is becoming academic, thus freeing the scholastic from the contempt of men who work in the world at large. What the practical ideal is doing for the pur- poses of the school, the ideal of psychological EDITOR'S INTRODUCTION efficiency is doing for its methods. It is perceived that truths that stand in vital relation to human need are more readily mastered and retained than those that do not ; that a series of facts is better presented in an order that pays heed to the stu- dent's mental development rather than to the logical steps in a perfected system of knowledge. Hence, old approaches to the mastery of studies are giving way to more objective, more reasonable, and more circuitous, but more efficient means of progression. It is interesting to note how often the demands for more practical results and more efficient psy- chological methods cooperate toward a common reformation of school work. The practicalists tell us that the schools must equip men with what they will most use as citizens and as workers. The psychologists say it is folly to master those minor knowledges and disciplines, which life will not utilize, for, neglected, the memory of them soon fades, and the school will have had only its pains for its trouble. The world asks for live men who know the conditions of real life in the com- bat as they find it, men who can think through vi EDITOR'S INTRODUCTION a new situation, when old knowledge fails to point the way. And, for reasons quite his own, the skilled teacher, who knows human nature, prefers to let his pupil think slowly through his difficulties, far more solicitous that the child get a confident command over his own mental resources than that he accumulate many facts quickly by the direct method of memorization. Psychology and practicality are the twin reform- ers of the schools, and like true twins they man- ifest a deal of similarity in their manner of ex- pressing themselves in actual practice. The modern demands of a sound educational sociology and a rational educational psychology do not appear so revolutionary in the elementary school as in the secondary schools ; for element- ary teachers have become more used to revolu- tion, and the violations of professional habit do not seem so monstrous where, as in the element- ary school, the sanctuaries of tradition are not so ancient. The secondary school teacher is also far more of a specialist and less open to the con- sideration of the total effects of a youth's edu- cation; he is a little nearer to the university vii EDITOR'S INTRODUCTION where truth for its own sake is a dominant ideal. In consequence, he suffers a considerable wrench when the logical arrangements of systematic knowledge arc disturbed for the purposes of more vital presentation. The modern programme of reform therefore finds a less easy acceptance in the secondary school. But high school teachers, just because they are specialized in interest and responsibility, are not all of one pattern. They are likely to resist change in varying degrees. One can readily see that in- structors in history and in science could be more easily led to reconstruct their courses of study so as to interpret present problems than could teachers of the classics or of mathematics. These last named subjects have had a very old and hon- orable position in the curriculum. The classics, by virtue of their antiquity, could not stand much modernization without losing their essence; and advanced mathematics, the most abstract and formal of all the high school disciplines, lends itself to concreteness and practical application only with great effort. Modify the logical se- quence of mathematics at a single point, and viii EDITOR'S INTRODUCTION you disturb the whole structure. A single mino? change that has a far-reaching effect is therefore likely to be resisted. But traditional resistance of an heroic sort may finally be futile. Sound sanctions at last decide every controversy. What is right must prevail in the teaching of mathematics, as in morals or politics. With such faith in mind, this volume on the teaching of high school mathematics is given to the teaching profession. A reflection of modern demands upon the school and an expres- sion of recent pedagogical ideals, it summarizes the modern reform programme as applied to mathematical teaching. It aims in considerable degree to make mathematics yield practical ef- ficiency, and it derives from the experiences of the students themselves the impulse and power to use high school mathematics as an instrument for the solution of real needs. The chaotic condition in which the discussions of the past decade have left the subject of math- ematical teaching suggests the desirability of pre- senting, in small compass, a systematic restate- ment, not merely in terms of a general theory, ix EDITOR'S INTRODUCTION but also in the more useful form of a series of concrete suggestions as to the material and meth- ods to be used. This volume is offered with the assurance that it serves this definite purpose. THE TEACHING OF HIGH SCHOOL MATHEMATICS I THE MODERN POINT OF VIEW The purpose of most of the recent innovations in the teaching of high school mathematics is to provide a more immediate application of the knowledge acquired; and to make the success- ive steps of the pupil's progress available even if his education is interrupted before the com- pletion of the high school course. It is to be remembered at the start that the pu- pils of high schools, the country over, do not for the most part go to any other educational insti- tution upon graduation ; that only a minority graduate at all ; and that a very large part of the entering class does not complete the first year. Let us grant, then, that it is worth while for these schools to furnish preparation for college or technical schools to the very small fraction of their membership that can make use of this priv- I TEACHING MATHEMATICS ilege ; but we cannot avoid the conclusion that it is no less important to give to those pupils who cannot stay through, or cannot go farther, some definite advantage even from the curtailed study that they can give to the subject. This is the justification of the innovations to which I have referred. In some schools a six-year secondary school course is given, beginning with the seventh year of the pupil's school life ; and it is altogether probable that this plan will be largely extended. The obstacle to it is the insufficient training obtainable for the teachers of the seventh and eighth years of the elementary schools, and the lack of departmental organization for those years. For this last reason teachers who are excellent in one line of work must give a part of their time to the teaching of subjects for which they have less taste and aptitude — and less skill. Mathe- matics suffers more than any other subject in this respect. In spite of this obstacle the plan is being urgently advocated, especially in New York, and devices of reorganization have been proposed that seem feasible. 2 THE MODERN POINT OF VIEW • Most secondary schools, however, receive pu- pils at the end of the eighth year, practically ignorant of algebra and geometry, and with no confidence whatever in their own ability to per- form the computations which they have been studying and practicing for years. The high school teacher's problem is to furnish to these pupils instruction that will immediately improve their efficiency and at the same time so contri- bute to their progress in mathematical know- ledge that at the end of four years they can take up college work. The problem is not yet com- pletely solved, but much progress has been made. When it is completely solved a new problem may be hoped for. On the new programme of high school mathe- matics the first thing is the treatment of compu- tation — of the four operations in arithmetic — with emphasis on self-reliance and on accuracy. Commercial computations may be left to the commercial courses, but accuracy and self-reli- ance can be cultivated in such computations as more intelligent artisans use. By this I mean only that in the successive development of the 3 TEACHING MATHEMATICS different subjects of instruction the problems be related to things that are actually done in the world, — let us say in play, as well as in work, not by any means that the mathematical teacher should give an exhaustive discussion of trade problems for their own sake. It is out of the question that the multitudinous activities of life can be so prepared for that every pupil on leaving school shall find his perplexities all solv- able as corollaries of his school problems — but it is not out of the question that his school pro- blems should be such as men in the world about him have solved and are solving for the daily needs of civilization. The most obvious points of contact between mathematical science and practical matters are the graphical method and the use of formulae. The graphical method can be used from the start, even in the elementary school, and through- out the high school course with increasing ad- vantage ; and the use of formulae can serve to enliven the practice in arithmetic on the one hand, and on the other to introduce the subject of algebra. In these two things the pupil has his 4 THE MODERN POINT OF VIEW introduction to algebra and geometry, as rein- forcements of his old enemy, arithmetic. To his father's question, "What were you doing in school to-day?" he will reply, "Mathematics. Something like arithmetic. " Let us hope he will add that it is more interesting. It is doubtful if there is any advantage at this stage in definitions of algebra, geometry, or mathematics — or even of arithmetic — if they can be avoided. The pupil's own definition of the study that he has known in the elementary school as arithmetic would be interesting if he would talk sense, instead of trying to say what he thinks is expected. His own definition of al- gebra and geometry on such slight acquaintance would be as futile as Dickens's impressions of this country, recorded in the "American Notes." Clearness and definiteness of aim in teaching requires, however, that the teacher should have in mind definitions of his subjects from the high school point of view. For this reason the follow- ing definitions are selected. Mathematics is the science that draws neces- sary conclusions. 5 TEACHING MATHEMATICS Algebra is a systematic method of abbreviat- ing the words used in discussing numbers. Geometry is a method of investigating the shape and size of material things by means of diagrams, and of expressing the relations of shape and size among these diagrams by means of numbers. The word mathematics comes from a Greek word meaning to know, and was originally used for science in general, becoming ls>.ter restricted to numbers, geometry, and kindred subjects; and at one time even to astronomy. The com- prehensive definition here given is quite mod- ern ; the fact that it includes logic gives a valu- able point of view to the teacher of geometry. The word geometry indicates its origin in the barter of land. Abraham's purchase of the field of Machpelah, notwithstanding the friendly dis- claimer of Ephron the Hittite ("What is that betwixt me and thee .-' " ) shows us a conviction, even among the kindly children of Heth, that equitable dealing was better than kind hearts. At any rate their neighbors the Egyptians, cultivating lands on which the landmarks were periodically washed away, found it necessary 6 THE MODERN POINT OF VIEW to locate and measure their holdings. From this utilitarian practice the Greeks derived their bases for a science which they felt to be abstract and ideal, and which has come down to our own day, only slightly changed in ma- terial and method, as a subject for high school teaching. With the Greeks it was a study of the relative shapes and sizes of ideal diagrams, mostly plane ; but it was also a study of certain relations of num- bers such as need not be obtained by counting ; these numbers being represented by lines, and their relations by the relations of the lines in dia- grams. With the notation of modern algebra, the numerical part of the Greek geometry has been superseded by the study of formulas ; and there has arisen a superstitious belief in the necessity for the absolute separation of geometry and alge- bra, which has materially impeded, for more than a century, the natural development of teaching in mathematics. For our purposes, geometry is not only a numerical investigation of the ideas of space, but a diagrammatic representation of number. A chief purpose must be to correlate 7 TEACHING MATHEMATICS the ideas of number and space so as to enlarge the idea of number by that association. To be sure there are important properties and relations of our diagrams that are not numerical, and the investigation of them is by no means an unimportant part of our study. Again, we use geometrical arguments as a type of valid in- ference ; by them we teach our pupils to avoid fallacies and to examine demonstrations with confidence in their own judgment. Finally, from the beginning of his study of geometry, the pupil is learning to comprehend the precise meaning of definite statements about simple things, and to separate from his knowledge of a particular object the connotations that are not postulated as a part of the basis of his argument. In all these respects the subject which we call geometry differs notably from land-measurement, which the term originally implied ; but the measurement of land and of plane areas gener- ally furnishes for our childhood, as it did for the childhood of science, an excellent introduc- tion and point of departure. The word algebra is the Arabic name of a 8 THE MODERN POINT OF VIEW single but important detail in the rule that the Arabian algebraists used for the reduction of simple equations. Their rule was first to add to each member of the equation what would get rid of all negative terms ; then to subtract from each side what would leave the unknown term standing alone. To the first of these two opera- tions was applied the Arabic word from which algebra is derived. We may conclude then that so far as the derivation of its name may in- dicate, algebra was originally concerned mainly with equations ; and that is not a bad notion to have in mind in planning the first year of high school mathematics. To the pupil beginning, however, it is neces- sary to show why equations should exist. It is unwise to take ready-made equations to show him, giving him successively greater degrees of com- plication in their structure and manipulation, bidding him to have faith in the overruling wisdom of those who plan his studies for bene- ficent ends, and reserving the problems which may give rise to the equations until the equation itself is a perfected and more or less mechanical 9 TEACHING MATHEMATICS instrument in his hands. On the contrary, the equation must appear to him at the begin- ning as a convenience, if not a necessity, in the discussion of things that the pupil may see a reason for discussing ; that is, as a statement of one item in a scheduled explanation of a problem. Now the explanations of such problems as are traditional in the teaching of algebra were originally written out in words, — what Nessel- man calls rhetorical algebra ; then a good many systematic abbreviations were used for the num- bers involved and for their relations, though the form of exposition was still that of ordinary speech ; finally a system of notation was devised which had no necessary connection with the particular words, and which did not follow the rules of speech. This is the modern, or sym- bolic, form. It is capable of translation into words, at least for the simple equations that are used in first year work ; and the words of the problems that present themselves for solution by algebraic means must always be capable of being so recast that the necessary data of the problem lo THE MODERN POINT OF VIEW can be " translated into algebra, " that is, ex- pressed as an algebraic equation. It is perfectly consistent for the teacher, then, and reasonably clear for the pupil, to define algebra as a system- atic method of abbreviating the words used in discussing numbers ; though the teacher knows that in order to utilize this method it is necessary to rearrange and schematize the suc- cessive steps of the discussion. This definition has the further advantage that it includes the use of algebra for formulae, as a substitute for verbal rules of computation. The words describing the numerical data, and the operations which the rule directs to be performed on them, are all represented by a systematic symbolism. There is no need to guard this definition of algebra on account of any scruples the teacher may have as to the existence of purely symbolic equations and expressions. As the pupil pro- gresses in his work he will gradually introduce himself to symbolic manipulation which he does not care to translate into words, — at least, until he arrives at a rather simple result ; and there II TEACHING MATHEMATICS will be no awkwardness in his progress towards a state of mind in which he will complacently put a problem in at one end of an algebraic solu- tion, and machine out his answer at the other, with the comfortable assurance that his algebra has served him rather as a substitute for thought than as a means for briefly writing his thought down. This is no mere figure of speech, for machines have been constructed for performing highly complicated algebraic work, and even for deduc- ing, by rigid logical rules, simple and necessary conclusions from data so complex that an un- trained thinker might easily fail to make a com- plete deduction. All this the pupil would not understand at the start, and it is better that he should not ; but as his progress continues, these things shall be added unto him. n THE ORDER OF TOPICS It is a bad habit of teachers of all kinds of sub- jects that they hunger and thirst after thorough- ness ; not, alas ! thoroughness on the pupil's part, but thoroughness of exposition on the teacher's part. Thus makers of text-books, find- ing that square root may be of use in solving quadratic equations, exploit not only the subject of square root of numbers and of literal express- ions, but also the subject of radicals, and com- pile sets of examples under these heads until not only the pupils, but probably the teachers also, forget how little the material serves for handling equations. For the sake of unity, the first year student should be spared all such elaborate treatment; his attention should be confined to two things : first, the association of number with magnitude, and the consequent representation of number by diagrams ; and second, the use of the equation 13 TEACHING MATHEMATICS as a means of solving problems. These two things can be closely related by using the formulae of geometry as a source of some of the equations upon which the student practises his growing skill. The manner in which these desiderata are attained will probably be worked out by each teacher for himself. The following order of topics is, consequently, merely suggestive. I. Problems giving rise to simple equations of gradually increasing complication. II. The use of formulae illustrated by the men- suration of plane figures. III. Computation, with economy in the number of figures used ; square root of numbers. IV. Simple transformations, including frac- tions ; long division and multiplication with binomial factors only ; the parenthesis and the radical sign as symbols of convenience, not as sources of problems. V. Algebraic theorems, illustrated by the dif- ference of two squares, the square of the sum, and the square of the difference, with illustrations from arithmetic and from ge- ometry. Factoring of expressions like x^- 5;r-i4 by inspection, and of expressions like jir2-ioo.:r -|- 2491 by "completing the square." 14 THE ORDER OF TOPICS VI. Quadratic equations. VII. Similar triangles and the Pythagorean theorem as a source of quadratic equations. VIII. The Graphical Method ; the straight line as the locus of a two-letter equation of the first degree. IX. Elimination "by addition or subtraction " for two-letter equations of the first degree ; rough check by intersection of loci. X. Elimination " by substitution " for linear- quadratic pairs. The equation of a circle with given centre and radius. The stand- ard parabola j = ;tr2. Graphic checking of certain equation-pairs and of a few one- letter quadratics. This will complete the first year work. With the second year the pupil will have two things to do : to investigate the methods of manipulating algebraic expressions, and to make a beginning of the study of geometry as a logical structure. Some of the topics under these heads, such as the binomial theorem, or the method of limits, might with great advantage be postponed in favor of the simpler topics in solid geometry and in trigonometry. Those that are taken will be the IS TEACHING MATHEMATICS more welcome as the pupil now realizes that algebraic expressions have been of use to him, and as he has had experience in the intelligent discussion of geometrical inferences. Ill EQUATIONS AND THEIR USE For the points of beginning in high school math- ematics we have problems that give rise to alge- braic equations, and the numerical investiga- tion of geometrical diagrams. The ends to be sought are, in the first place, a conscientious logic and a sense of responsibility for numer- ical results ; eventually, the power to repre- sent numerical results by diagrams, and to inter- pret the relations of diagrams numerically; and incidentally, facility in the manipulation of sym- bolic statements that represent or replace argu- ment. " Real applied problems " in algebra for first- year high school pupils are scarce. Arithmetic in the elementary school has been mainly commer- cial, and reflection will show that every day affairs do not present many problems in which algebraic treatment is imperative. It is not expe- dient to give explanations of trades or of science 17 TEACHING MATHEMATICS hitherto unknown to the pupil, for the sake of new problem material. The bulk of the subject-matter, then, in the problems of the first part of algebra, must be money, percentage, distances, lapse of time, and weights. The idea of ratio must be used from the first ; it is usually wrapped in confusing words, like the mist in which yEneas walked in the sunny square at Carthage. If ratio is defined plainly as a multiplier (or as a quotient) and is not followed by the usual generalized treatment of proportion, experience shows that pupils have no difificulty with it. The equations to which these problems are to give rise must be at first extremely simple ; so simple in fact that it cannot be claimed that their use makes the problem one whit easier to solve. The motive to be presented for using alge- bra is that by its use the explanation of the pro- blem can be systematized and briefly recorded. This system and this brevity can afterwards be applied in the study of problems so much more complicated that the pupil trying to solve them without algebra would surely become confused, i8 EQUATIONS AND THEIR USE These introductory problems, again, must be brief in statement, and unmistakable in meaning. New words, when introduced, must come one at a time, with full explanation and illustration, but not necessarily with formal definition. The data, clad in terms of money, distance, time, etc., should be as follows: — First, one of two numbers, and its ratio to an- other. Second, the ratio of two numbers, and their sum. Third, the difference of two numbers, and their sum. Thus we introduce simple equations in which only positive terms appear. After the pupil has learned how to deal with small numbers, attention can be focused on the processes of reduction by using large numbers. The range of material can be enlarged by introducing the measurement of angles, with a protractor, in degrees and decimals of a degree ; the words complement and supplement can be used ; and the number of degrees in an arc distinguished from the length of the arc in inches or feet. In the next type of problem negative terms 19 TEACHING MATHEMATICS are introduced. These cannot be avoided, for example, when the sum of two numbers is given and an equation formed with multiples of them. Thus, in the problem : — Two adjacent building lots have a total front- age of 104 feet; one is 50 feet deep, the other 80, and their areas are equal. Find the frontage of each lot. SOX = 80 (104— Jf) For the purpose of dealing with this negative term it is not desirable to go into a general treat- ment of negative number as a new feature of algebra. The pupil as yet has no need of the idea of negativ^e number, except as the subtrahend which he has always considered it ; he will not have need of the idea until he gets to the nega- tive solution of a quadratic equation. The method of dealing with the negative term in the equation ^ox = 8320— 8o;r is indicated by the original meaning of the word "algebra," referred to in the first chapter; that is, "the making up of shortages." The second member of the equation, if it were not for the 20 EQUATIONS AND THEIR USE term — 8o;tr, would be 8320; it is now short of 8320, and the shortage is Sox. We can make up that shortage by adding Sox to each side of the equation, and the rest of the work is clear to the pupil. If this plan is adopted, the teacher must get rid of his foolish partiality for the left side of the equation, and be ready to have his ;r-terms on the right if they come that way. Thus, 65— -f = 4 X becomes 65 = 5^ by adding x to each side ; and 13 = X is then just as good as X = IS Another advance in manipulation is necessary for the multiplication 80 (104— ;r) . and this should be dealt with here, explained, illustrated, emphasized, and practiced upon, as if that was the only multiplication difficulty in algebra. 21 TEACHING MATHEMATICS The geometric material can be increased by the idea of a stripe (the figure formed by two parallel lines) and by the sum of the angles of a triangle. Single capital letters can be used for the number of degrees in an angle, and the va- rious theorems about the angles made by a transversal of two parallels should be condensed into a single sentence, such as: "All the acute angles formed by any transversal of a stripe are equal." SOME RULES OF THUMB The first year of high school work is the place to introduce certain features of computation which are important to practical men as well as to mathematicians. The chief of these is the measure of precision in a measurement-number, as indicated by the number of significant figures in it. This may be illustrated by the mean distance of the earth from the sun, which is about 92 J4 million miles. This may be a hundred thousand miles or more out of the way. If written 92,500,- 000 the first three figures would be significant, the others merely fix the position of the decimal point. Again, the United States Government defines the inch by saying that 39.370 inches make a meter. Here the o is significant, because it indicates that this specification is accurate to five figures. Still another illustration is the length of a micron, a unit that may be used to 23 TEACHING MATHEMATICS measure in the field of a microscope. A micron is .000039370 inches long. Here the four zeros are not significant, because they serve merely to fix the position of the decimal point; the last zero is significant. A change in the unit of measurement will change the position of the decimal point, but will not change the degree of accuracy to which a measurement is made. Thus a distance of 1.3420 miles would really be measured in feet, and the measurement number 7085.8 for the measure- ment in feet is no less accurate, though having only one decimal place, than the number 1.3420 which has four decimal places. The precision of a measurement, then, is in- dicated by the number of significant figures. Measurement is actually much less accurate than is generally supposed. A carpenter's care- ful measurements are usually made to three fig- ures, an ordinary surveyor's to four figures, a civil engineer's measurements of a city lot to five figures. The most careful measurements pos- sible are those of the international standards of length, like the "prototype meters" which serve 24 SOME RULES OF THUMB as the legal standards of length in the United States. These are bars made of a durable alloy, kept in a sealed room at a constant temperature. On the bars the length of a meter is supposed to be indicated by five scratches. The scratchef are actually from 6 to 8 microns wide, and th< length indicated by them cannot be said to be really ascertainable within .2 of a micron. That is, the limit of human accuracy in the most im- portant measurements ever made is seven signi- ficant figures; while even five-figure measure- ment requires expensive instruments and trained skill. Bearing these things in mind, one may easily see waste in ordinary computation. This is no place to present the argument for " contracted " multiplication and division, which good teachers of mathematics are now using for the prevention of this waste. The greatest obstacle to teaching these common-sense improvements is the unwill- ingness of elementary teachers to have their pupils begin learning to multiply from the left of the multiplier instead of from the right. In the sixteenth century both methods were taught ; 25 TEACHING MATHEMATICS chance seems to have favored the right-hand end, unluckily, and though the custom now turns out to be foolish it is intrenched in the proverbial conservatism of school-teachers. Responsibility and self-reliance are promoted by the practice of " checking " results of com- putation. The well-known check by " casting out nines" is valueless except where all the figures used are kept to the end. Multiplication can be effectively checked by interchanging multiplier and multiplicand; division, by multi- plying the quotient by the divisor. The solu- tions of algebraic problems should always be checked by substituting the answers /';/ tJie words of the problem and verifying that its con- ditions are fulfilled. The square root of a number is needed al- most from the beginning, but the traditional method for finding it, besides being difficult for the pupil to understand and remember, seems to him somewhat artificial also, and remote from the definition of a square as the product of equal factors. If it were not for the necessity of emphasizing this definition, the use of a table 26 SOME RULES OF THUMB of square roots would be just as clearly ad- visable as the use of a table of sines or tangents is in trigonometry. The best method to teach under the circumstances is the so-called " guess and try" method, although, as will be readily seen, it is no more guess-work than the method ordinarily used. Thus, let it be required to find the square root of 830.0. Since the square of 25 is 625 and the square of 30 is 900, let us try 27.00 as the square root required. 27.00)830.0(30.74 810.0 Here we have two unequal factors of 830.0, namely, 27.00 and 30.74 ; the square root must be between them. Try the number half way bettveen: 27 TEACHING MATHEMATICS 28.72)830.0(28.90 574-4 2556 2298 258 258 And again: 28.81)830.0(28.81 5762 ^ This method is based on the ( erroneous ) assumption that if the number given is repre- sented by d^, and if our guess has an error b, then the quotient obtained by using it as a divisor will have the same error b in the opposite direction ; in other words, that the factors obtained for the * The argument here given should certainly not form a part of the first-year work. 28 SOME RULES OF THUMB given number would be a + b and a — b, the quotient and the divisor. The product of these two factors, however, is not a^, but a^ — b"^. For example, when we had the two factors a — b = 28.72 and a-\-b = 28.90 the value of b was .09, and 32 = .008 1, which would not affect the last figure of the given number. If we had realized that, there would have been no need of the last division. It would be convenient if we could be sure that when, as above, the value of b comes in the last place (e.g. within the fourth figure in a four-figure number) the true value of the root can be found by averaging the divisor and quotient, as above. This is always the case for four-figure numbers, for in that case b is less than I in the third figure of a and consequently b"^ is less than i in the fifth figure of a^. So much the more is it true for a greater number of figures. Three-figure numbers present little dif- ficulty. This method possesses the advantage that it is easily remembered, and the further advantage that it is a constant reminder of the definition 29 TEACHING MATHEMATICS of square root. It can be applied to cube root, by using the trial root as a divisor twice suc- cessively, and then averaging the second quotient with the two divisors. There is a greater error, but on the whole the process is not more cumbrous than the traditional algorithm for cube root. Cube root, however, does not have to be used much in high school mathematics, and not at all in the first part ; when it is needed cube root tables or logarithms would ordinarily be available. It is regrettable that college examinations still insist on the square root, and sometimes on the cube root, of algebraic polynomials. The require- ment is not, however, so significant as it might seem, for the following reasons : First, because no self-respecting examiner can give an example in algebraic square or cube root that does not "come out even" ; unless he speci- fies restrictions (on the relative size of the num- bers represented by the letters) that are out of the question in elementary algebra. Second, because such examples, if they do come out even, can easily be solved by inspec- 30 SOME RULES OF THUMB tion, as far as those that have four terms in the root. There is a little difficulty about the signs when there are four terms, but nothing that a pupil will not learn to do with glee, to avoid remembering either of the rules that vexed our younger days. Third, because neither of the rules is of any use except for emphasizing two simple cases of the binomial theorem ; and if the schools stop teaching them with the motive of saving the pupil's effort for things more worth while, the colleges will stop examining for them. Whatever is done with the " evolution " of polynomials should be reserved for the mora formal discussions of the second or third year. V GEOMETRY AS ALGEBRAIC MATERIAL The formulae for the measurement of triangles and quadrilaterals, the principal facts about sim- ilar figures, the Pythagorean theorem, and the use of similar right triangles for the indirect meas- urement of distances are all of great value in fur- nishing material for the algebraic work of the first year. They not only illustrate the rational derivation and the numerical application of for- mulae, but they serve for the construction of equa- tions many of which might actually occur in practical application, and all of which strengthen the mental association between numbers on the one hand and lengths and areas on the other. The high school pupil begins his work with at least one mathematical item among his assets : he knows that the area of a rectangle is obtained by multiplying its length and its breadth. All the geometrical theorems referred to above can be founded upon that ; without too great insist- 32 GEOMETRY AS ALGEBRAIC MATERIAL ence on the proof of reasonably obvious things, ji good idea can be given of successive logical dependence ; and for the rectangle-rule itself a proof can be given that on the one hand relates to his previous knowledge, and on the other hand prepares him for a rigid " limit " proof in his later work. This program makes necessary a considerable departure from the order of topics and from the methods of presentation that seem appropriate for geometry taught by itself in the second stage of high school mathematics. Only a sketch can be given here, but one innovation in terminology may be mentioned. This is the word "stripe," used in German school-books for the figure formed by two parallel lines. It gives a name to a con- figuration that every geometry student learns to imagine, for example when he thinks of two par- allelograms or triangles with the same altitude ; and of course it is known in popular use with its exact (or shall we say scientific .-') meaning. Again, a "strip" is a limited portion of a stripe ; for example, an inch-wide strip along the bottom of a triangle. 33 TEACHING MATHEMATICS The proof for the area of a rectangle is briefly summarized as follows. The teacher will readily adapt it to the more concrete point of view that the pupil regards it from. If the length of a rectangle can be expressed by a decimal number, the area of the unit-wide strip along the side of the rectangle can be ex- pressed by the same number. For on each unit of length stands the unit square, which is the unit of area ; on each tenth of a unit stands a portion of the unit square congruent with every other such portion, that is, a tenth of the unit area ; and so on. If also the width of the rectangle can be ex- pressed by a decimal number, the area of the rectangle can be expressed by the product of that number by the length-number. For each unit in the width is one end of a unit-wide strip whose area-number is the length-number ; each tenth of a unit is one end of a tenth of the unit-wide strip ; and so on. Rhomboids, triangles and trapezoids can then be treated as usual. Similar triangles are covered by the following succession of theorems. [A fig- 34 GEOMETRY AS ALGEBRAIC MATERIAL ure is said to be inscribed in a stripe when its vertices lie in the sides of the stripe.] Two triangles inscribed in the same stripe have the same ratio as their bases. If two triangles have one angle the same, their ratio is equal to the ratio of the products of the sides that include the equal angles. If two triangles have the angles of one respect- ively equal to the angles of the other, the ratio of any two corresponding sides is equal to the ratio of any other two corresponding sides ; and the ratio of their areas is the square of the ratio of two corresponding sides. In all these proofs the cumbrous algebra of the geometry text-book is replaced by a notation which always uses a single letter for one number, and which not only represents by that letter the measurement-number, but also uses it for the name of the line, or the angle, or the area measured. Thus, in the accom- fig. i panying figure, vS represents the area of the upper triangle S, X represents the area of the quad- 35 TEACHING MATHEMATICS rilateral below S, and vS + A" is the name of the triangle composed of the quadrilateral X and the triangle 6'. Angles, and the number of degrees in an angle or in an arc, are represented by capital letters ; areas are represented by capital letters ; lines are named by small letters, and length-numbers are represented by small letters (lower-case let- ters). The context always prevents confusion of signification, and the great advantage is gained of similarity of notation in geometry and algebra. It is only recently that text-books of geometry have ventured to express even purely algebraic proofs in the ordinary notation of algebra. Equa- tions like the following are still found even in the best of recent text-books : aABC^AB AC KDEF~ DE^ DF This is objectionable not only because the hasty pupil maybe tempted to write his product as - ^ , but also because the attempt is made to create the erroneous impression that — J^lr 36 GEOMETRY AS ALGEBRAIC MATERIAL is a ratio between the geometrical figures, and is some mysterious thing different from the ratio of their area-numbers. This is the place to comment on the vicious custom of lettering angles or lines with Arabic numerals. Imagine the confusion in stating val- ues, e.g., the line 12 = 13 ( !) Of course a bright pupil can be made to skate over these places, but why introduce unneces- sary elements of confusion when there are plenty of difficulties with the clearest notation ? The practice of draughtsmen should not mislead us ; they are not teaching ; and when our pupils get to the draughting-table they can learn this custom if they care to be so injudicious. The single-letter notation should be adhered to, if for no other reason, because all the equa- tions in geometry are really equations between numbers or combinations of numbers. There is, however, the additional reason that by its use it is easier to indicate the relations of duality that appear here and there in elementary geometry ." not so many in plane as in solid. 37 94259 TEACHING MATHEMATICS These considerations will be my excuse for introducing here one of the many modifications of Euclid's proof that the square on the hypote- nuse of a right triangle is the sum of the squares on the other two sides. The proof of the Pythagorean theorem can be based on similar triangles or on the rule for the area of a trapezoid. The proof given below is selected because it brings into view the advan- tages of a good notation, and at the same time illustrates the omission of essential steps in a proof for the sake of a unified impres- sion on the be- ginner's mind. It will be noticed that proofs by sup- erposition are passed over as obvious at this stage. 38 B / / ^^•^ ^^-^^ ^>. C 6 Fig. 2 GEOMETRY AS ALGEBRAIC MATERIAL Here the square c^ is inscribed in the same stripe as a rhomboid which we can call R ; and they have the same base a. Consequently Q) R = a^ Again the rectangle cr is inscribed in the same stripe as a rhomboid which we can call S ; and they have the same base c. Consequently (2^ S = ex A tracing of the rhomboid R can be made to fit exactly on the rhomboid S, by keeping the corner B fixed and turning the tracing of R upon that as a pivot. That is, 0^ = 5 Substituting equations (T^ and (2) in (j^ we have r^S a^ = ex In the same way, on the other side of the tri- angle, we obtain and by adding these ^6^ cP' -^^ b"^ = ex -V cy or ^2 + ^2 == ^2 39 TEACHING MATHEMATICS It is common enough to illustrate the applica- tion of similarity by estimating the height of an object by the length of its shadow. Some recent books even introduce the sine, cosine, and tangent into plane geometry. Certainly it seems wise to make use of measurable angles, and to show that the measurement-numbers of inaccessible lines can be computed. For the first-year work, however, a clearer im- pression will be made if we confine ourselves to one of the six ratios of the sides of the right tri- angle. The most immediately applicable is the tangent. A three-figure table of tangents can go on a single page, without the confusion attend- ant upon reading backwards for angles over 45°. Solving triangles in which angles are included among the data will then give practice in appli- cations of algebra and geometry which have an aspect of reality, not only because they are ob- viously such as human beings have to work at, but also because the arithmetic, though compli- cated, does not seem to have been purposel;' complicated for the good of the pupil. 40 VI THE GRAPHICAL METHOD The graphical method of comparing numerical data and of exhibiting statistics is of wide pop- ular use. We compare crudely the numerical strength of armies and navies by the pictures of giants and dwarfs in the uniforms appropriate to the service, or by sketches of battleships white for our side and black for the other. Census re- ports, magazine articles, and even schoolboy com- positions compare the cotton crop or the popu- lation of different states by black lines in length proportional to the numbers. Beet-sugar and cane-sugar production are exhibited in curves that sweep upward and cross at a significant date. The political muck-raker sets out the expense accounts that he is hunting in a curve, and delights to locate his adversary's extravagance by its undu- lation. It has become, in the course of the last generation, a mode of public expression ; it is addressed to the intelligent ; it should be a part, 41 TEACHING MATHEMATICS therefore, of the education of youth ; and its place is in the mathematics course. Probably the first point at which the advan- tage of this method is obvious to the pupil is in the discussion of elimination between two-letter equations. For example, the equation 5^-37=20 has a list of number-pairs that will satisfy it, in other words, "answers"; and there is no limit to the number of such answers. Yet there are num- ber-pairs that will not satisfy this equation. The equation implies a restriction upon number-pairs. What sort of restriction .? We begin by defining an algebraic scale as a straight line upon which one point is marked for zero, and every other point represents some num- ber, positive or negative, the integers succeeding each other at convenient equal intervals. Two of these scales, set perpendicular to each other with the zero-points together, we call axes. Every point in the plane is then opposite to two num- ber-points in the axes, one in each, and repre- sents that number-pair. It is a highly artificial device, but so is algebra; and, in fact, arithmetic. 42 THE GRAPHICAL METHOD Now the points that represent the number* pairs serving as answers to the equation we started with all lie on a straight line. The pupil will infer that himself, when he has plotted a few. The proof that all the points satisfying (i. e., representing number-pairs that satisfy ) a two- letter equation of the first degree lie on a straight line is one of the simplest " exercises " in similar triangles ; instead of being reserved for college, it is well within first-year high school work. It is probably the easiest and clearest example of a locus, much clearer than the bisector proposi- tions, which foster, temporarily at least, or among the laggards, the notion that a " locus-of-points " is some kind of a bisecting line. It is also a valuable argument to refer to later, when the time comes to speak of " necessary and sufficient conditions," or propositions and their converses. This representation of equations is not merely for illustration, but is also a means of checking the solution of a pair of two-letter equations. The pupil can easily estimate the intercepts of each equation on the axes, and thus locate the two lines, and their intersection, which sat- 43 TEACHING MATHEMATICS isfies both equations. He will balk at this if he is expected to draw the lines for every one of a dozen number-pairs ; a much easier plan is to rule a line on a sheet of tracing paper, lay that line through one pair of intercepts and a straightedge through the other. The point thus located will roughly check his answer. The objection will be raised that this is teach- ing analytic geometry in the high school ; those to whom that objection seems reasonable can- not be argued with. Others will see that the method furnishes a good point of contact between the pupil's algebra and his geometry, and increases his command over both. Even more so the next suggestion, perhaps for second- year work — not later. Why should not the pupil be taught that every circle has an equation in the form x^ +JJ/2 -\- ax ^^ by -\- c ^^ o and conversely that one and only one circle can be constructed for every such equation } No knowledge of geometry beyond the Pythagorean theorem is required, and a great addition is made to the pupil's capacity. 44 THE GRAPHICAL METHOD Be it remembered that these two things — the circle-equation as well as the straight-line equa- tion — are proposed as additions to the work of the first two years. There is one advantage to this which every teacher of experience will ap- preciate ; that is, that " real problems " in elimin- ation, which have been scarce, are at hand here in plenty. For example: — Find the point of intersection of AB and PQ, where the number-pairs of the four points are respectively as follows : A, 5,6 ; B, 1,2 ; P, 5,1 ; Q, 2,7. We can take for AB the equation ax+by= i, and for /"(J the equation /;ir+^j=i ; substituting the values for A, that is, x=^ andjj/ = 6, we get 5 « + 6^=i ; 2indior B,a-\-2 b-=i, whence a = — i and b=i ] in the same way 5 / + ^ = i and 2p + yq = 1, whence p = -^^ and q = -^^. Then for AB the equation is— ;f+ J = i and for P^, 2x+y=\i ; whence x=T)^ and j = 4, as the number-pair lo- cating the required intersection. This answer can of course be readily checked by the tracing-paper and straightedge, as before. The circle through three points would in 45 TEACHING MATHEMATICS the same way give three-letter equations of the first degree to determine the coefficients in the equation thus for the circle through the three points x = 2, y = 3; x=4, jj/=5 ; x = 3, y=i; we have the three equations 2^ + 3 b + c= — 13 3 a + b + c= — 10 There are other exercises which will at the same time throw side lights on elimination and on the fundamental theorems of geometry. These will readily suggest themselves to an enterpris- ing teacher. A very important exercise is the careful plot- ting of a curve of squares. The squares of the numbers from i to 10 being plotted in class, each section between successive integers can be assigned to a small group of pupils to locate the intermediate points for tenths [(2.1)^, (2.2)^, (2.3)^, and so on] ; this will serve to illustrate the con- venience of arithmetical interpolation in the flat regions of the curve. 46 THE GRAPHICAL METHOD It is very instructive to plot the areas of fig- ures that vary under some restriction (say squares, similar rectangles, similar triangles, circles, par- allelograms with the sides given but not the an- gles, and so on), so that one number will serve to determine the area of one figure in the series. The mere fact that an area-number is repre- sented by a line is significant. The use of a " standard parabola " (curve for y=x^) has been suggested for checking the solution of one-letter quadratics. Thus for the quadratic we take a carefully plotted curve ior y=x'^, find the intercepts on the same axes for y=-px-q and lay a straightedge through them. The in- tersections of this line with the curve ior y^x^ will have x^^ —px—q, that is, x'^-\-px+q=o and the values of x will be the roots of the equa- tion. It is not advisable to go into the discussion of ellipses, parabolas, or hyperbolas in high school 47 TEACHING MATHEMATICS work, or to have the pupil expected to plot every two-letter quadratic that he deals with. Many of the curves that he would have to deal with, es- pecially very broad or very narrow hyperbolas, would be difficult for him to recognize from the few roughly-plotted points he can get ; and when we realize that plotting an algebraic curve by points is a stupid and profitless job anyway, one that a college student will be taught to dodge wherever possible, it is just as well for high schools to confine themselves, for the most part, to simple curves that can be utilized for collat- eral information ; for help on the algebra or on the geometry that forms the main part of his present burden. Nor, on the whole, is it advisable to resort to the plotting of parabolas often used to "explain" the double or the imaginary roots of a one-letter quadratic. These can be exhibited much more easily and simply by the straightedge and stand- ard parabola, as above. VII THE BASES OF PROOF IN GEOMETRY Euclid (1,4) uses the following phraseology: "If we fit the triangle ABF upon the triangle AEZ, and if we put not only the point A upon the point A, but also the straight line AB upon the straight line AE ...*'; so much of motion, then, has always been regarded as orthodox, though some recent purists object even to that. The motions implied by Euclid in this quotation constitute no insignificant part of the definition of a plane ; it is implied also that a geometrical figure can be moved about without distortion. Again, it is im- plied, though Euclid does not mention it or use the inference otherwise, that a plane figure can be overturned upon its plane, so that the order of the parts of one of the triangles may be re- versed; for he makes no exception to his theo- rem. Although a first treatment of geometry should 49 TEACHING MATHEMATICS not include proofs of theorems that do not cry for proofs, and although it is quite out of the ques- tion to point out all the assumptions that are logically necessary and no others, it is desirable to take up at some time in the high school course a system of propositions based upon assumptions that will seem to the pupil of that age necessary and sufficient, so that he can ar- rive at an appreciation of the "monument of human reason " that systematic elementary geo- metry has for centuries been to the world. He should have had a previous experience in geo- metry demonstrations, so as to have an appetite for a cogent argument ; he should have had an experience in the perception of successive logical dependence, so as to have a general no- tion of what he is driving at when he starts in. As, in algebra, we cannot justify to the beginner the use of an equation as a means of simplifying his work, so in systematic geome- try we cannot justify to the beginner the early proofs as necessary to his acceptance of their conclusions. At least one of our basic as- sumptions — the axiom of parallels — is in mod- 50 THE BASES OF PROOF IN GEOMETRY ern times known to be only an inference from experience. Many of the propositions that we carefully prove can also be so regarded. The only justification for the proofs is, that we want to reduce to as small a compass as we conveniently can the facts for which we must depend upon observation, so that experience can be foretold. In this way a comparatively small number of elementary truths, with standard methods of inference, will serve to connect and unify a large number of facts that might to hasty observation seem independent. For this reason the first group of theorems should be confined to simple figures, and should serve to illustrate standard methods of inference ; first, by utilizing the characteristic qualities of a plane surface, and next, by obtaining numbers called measurement-numbers, by which a geo- metric magnitude can be reconstructed out of standard or unit magnitudes of the same kind. The characteristic qualities of a plane can be appreciated by comparing it with other surfaces which do not possess all of these qualities. They are : — 51 TEACHING MATHEMATICS I. Any figure in a plane can be moved about upon it until any specified point of the figure coincides with any specified point of the plane. This motion will be called sliding. II. Any figure in a plane can be rotated about any specified point of the figure, until any specified line of the figure through that point coincides with any specified line of the plane through that point. This motion will be called rotation. III. Any figure in a plane can be taken out of the plane, turned over, and laid back upon the plane, so that the order of its parts is thereby reversed. This motion is called overturning. For all of these motions the figure must be thought of as a separate thing from the plane, though the figure lies in it, and may be named as if a part of it. Tracing paper furnishes a con- venient means of illustration. On a cylindrical surface (cylinder of revolu- tion) the first motion, sliding, is the only one of the three that is possible without distortion ; on a spherical surface the first two are possible, but not the third ; on the plane only are the three motions possible, 52 THE BASES OF PROOF IN GEOMETRY The proof, therefore, that circles are congruent if they have equal radii is valid for a cylinder (if one can succeed in defining what a circle is on a cylinder) ; but the proof that the diameter of a cir- cle divides it into two congruent parts is not valid for a cylinder, though it is, if properly chosen, valid fora sphere. Symmetry propositions, being proved on a plane by overturning, cannot be regarded as true for a sphere without additional argument. The question of order of parts requires care- ful teaching. The pupil must be reminded that the figure L cannot be got into the position J by any amount of sliding and rotating. He must be shown that there is not only a right and left order, but a clockwise and a counter-clockwise order ; and should be taught to name the parts of a figure with attention to this distinction. Tracing paper again is a ready means of experi- ment for this purpose ; it is well to letter the first diagrams with such letters as T, U, V, to confine the pupil's attention to the figures them- selves, and not bother him with the strange ap- pearance of reversed unsymmetrical letters, like B, F, G, and so on. 53 TEACIHNG MATHEMATICS All this matter of reversal is within the experience of every one who has seen for example the lettering on a glass door; nevertheless it will be interesting to the pupil as a scientific point of view for his vague experience. And though the proof of congruence propositions such as the fol- lowing will be made more detailed than under the loose treatment hitherto customary, the ar- gument will be more interesting to him, and more satisfactory. Let us take the proposition Euclid I, 4, to which I have previously referred : — If two triangles have two sides and the in- cluded angle of one equal respectively to the corres- ponding parts of the other, the triangles are congruent. Suppose a=x, b=y, and the angle Z= C, all in the same order in the two fig- ures. To prove ABC=XYZ. Slide ^FZ'untilthe point Z comes to C. Rotate XVZ about 6" as a pivot 54 THE BASES OF PROOF IN GEOMETRY until y fits exactly on b. Then the point X will fit exactly over A, because j/ = <^. Since the parts are in the same order, Cand Z will be on the same side of b, and x will fit on a because the angle C=Z\ and the point Fwill then fit exactly over B because x=a. Then all the vertices of XYZ exactly cover the vertices of ABC, and the triangles are con- gruent. Again, suppose a=x, b=y, and the angle C=Z, all the parts in ABC being in reverse order to XYZ. To prove, as before, ABC ^ XYZ. Overturn XYZ, so that the order of parts will be the same as ABC. Then proceed with the same -^' proof as before. The motion of sliding could be exhibited as a pure translation by drawing B Y, to cut ZX as at Q, and Z' then sliding XYZ along B Y until Y reaches B ; then we could rotate 55 TEACHING MATHEMATICS about B until ;r falls on a and so on. The objec- tion is of course the artificial character of the device ; and for the sake of a theoretical ad- vantage, remote at best, it seems hardly worth while. Figures are symmetrical about an axis when one can be made to fit exactly upon the other by overturning its half of the plane about that line, as a door upon its hinge. By analogy with a mir- ror, one of the figures may be called the image of the other. Two symmetrical lines intersect the axis at the same point and make equal angles with it ; two symmetrical points are in a straight line per- pendicular to the axis, and are equally distant from it. A very important theorem, and one that should be placed early, is that the figure formed by two intersecting circles is symmetrical about its line of centres. This theorem enables us to prove immediately the congruence of mutually equilat- eral triangles. When we have proved the three propositions about congruent triangles, and the two converse 56 THE BASES OF PROOF IN GEOMETRY propositions about the angles of a stripe,^ the whole subject of congruence and symmetry in a plane, and the whole subject of mensuration for plane rectilinear figures is within our reach. In selecting an order of theorems the teacher should avoid successive dependence where it is not necessary ; for example, neither of the two propositions about the angles of a stripe should be made to depend upon the other. Neglect of this precaution will lead to distrust of perfectly good proofs. The theorems about the angles at the base of an isosceles triangle, and the converse theorem, can be proved by overturning the triangle and showing it to be congruent with its old position. The theorem that only one perpendicular can be drawn from a point to a straight line has an amusing proof. Overturn the figure about one of the two lines ; if both were perpendicular this could be repeated indefinitely, the foot of each falhng in the straight line, until the straight line returned into itself. To tell the truth, I have never 1 If the two lines are parallel, certain angles are equal; if certain angles are equal, the two lines are parallel. 57 TEACHING MATHEMATICS found a pupil who considered this as amusing as it seemed to me. The congruence of figures cannot be proved without overturning except when the order of parts is immaterial ; that is, when the figures to be proved congruent are themselves symmet- rical. Among the very first theorems then, for sim- plicity of proof are : — If two central angles are equal their arcs are. If two arcs are equal, their central angles are. If two arcs are equal, their chords are. If two chords are equal, their arcs are. The first two of these are at the foundation of measurement; their position at the beginning of the subject is therefore a strategic advantage. The execution of problems of construction in geometry has no logical connection with the de- velopment of theorems, except where the con- structions show the existence of the figures re- ferred to in the theorems. The question of the existence of such figures is settled by showing that their properties are not contradictory, either of each other, or of the general postulates of space 58 THE BASES OF PROOF IN GEOMETRY in which they are assumed to exist. In most cases, therefore, since this consistency is shown by the extensive development of systems of the- orems based upon the properties of the figures in question, their existence may safely be as- sumed and this particular requirement for the execution of constructions may safely be ignored. On the other hand, they do furnish excellent practice in the application of theorems, and should be freely used for this purpose. When, however, the methods of construction, instead of conforming to the practice of draughtsmen, confine themselves to the two Euclidean instru- ments, — compass and unmarked straightedge, — the historical and the logical reasons for that limitation should be pointed out. None of the inequality theorems has any nec- essary application in the early part of geometry, though they are generally placed early. They should follow the measurement theorems rather than precede them. They are necessary to the theorems on the value of tt, and to those only. VIII THE METHOD OF LIMITS The direct measurement of a quantity is ac- complished by dividing it up into parts that are congruent with the unit or with submultiples of the unit. Now there are some quantities which cannot be thus decomposed. The diagonal of a square whose side is the unit of length is such a quantity. Its length, in terms of the unit, can- not be exactly expressed in figures, for if it could, these figures would represent parts of the line that would be multiples either of the unit itself or of one of its aliquot parts. We say that such a quantity is not commensurable (has not a common measure) with the unit ; and we call the number that exactly expresses its length an incommensurable number. Algebraic symbols may be used for the number ( like V2, or, in the case of a circumference, a number like 10 tt), but it cannot be expressed in figures. The proofs of geometry for which the method 60 THE METHOD OF LIMITS of limits is used all refer to incommensurable numbers. The task attempted in every case is to show that computations from certain direct measurements will give the measurement-num- ber of the quantity in question. In general the method of attack serves to ob- scure the problem. That method is an inheritance from Euclid, in whose time units were in such a confused state that they could hardly be spoken of as standard even for a particular time and in a particular country ; and for whom the number system was cumbrous in the extreme. According to that time-honored method, quantities are com- pared with each other as if there were no unit. From that comparison we moderns deduce, as an incidental consequence, of great practical but of small theoretical import, the case where one of the quantities has fallen from its high estate and become a mere unit. Since one of our important aims in teaching geometry must be to foster in the pupil's mind the concept of a series of numbers exactly corre- sponding to every series of quantities, we should be disappointed if he did not almost instinct« 6i TEACHING MATHEMATICS ively think, for example, of an angle as a cer- tain number of degrees. He does, in fact. The teacher, moreover, will give numerical illustra- tions of his theorem, tacitly assuming the very truth he is engaged in proving. The result is that instead of investigating one measurement- ratio, that of a measurable quantity to the unit of that kind of quantity — rather an abstract problem at best, involving the fundamental ide? of number — the pupil is really dealing in a somewhat vague way, wholly divorced from such experience as he may have had with actual things, with the ratio of two numbers each of which is it- self a ratio ( as of course all measurement num- bers are); though the formal words of the proof do not mention those numbers at all, and the teacher ( of whom the text-book is a part ) will deny, in spite of his numerical illustrations, that there are any such numbers in sight until he has planted his foot on the last step of the proof. No loss in rigor, and much gain in clearness, would result if we started in every case with the unit as one of the two quantities compared. For ex' 62 THE METHOD OF LIMITS ample, the theorem that a central angle is pro- portional to its intercepted arc can be stated thus : The measurement-number of any central angle is the same as that of its intercepted arc ; or perhaps, to avoid confusion with the length- number of an arc, thus : — The number of degrees in an angle is the same as the number of degrees in its intercepted arc. The proof consists first in defining a degree of arc, and showing, by a previous theorem, that, while it may be of different lengths on dif- ferent circles, on the same or equal circles a de- gree of arc is congruent with every other degree of arc. Consequently an angle of an integral number of degrees intercepts an arc of the same number of degrees. Again a tenth of a degree of angle is one of ten mutually congruent parts of a degree ; each will intercept, according to the same theorem before referred to, one of ten mutually con- gruent parts of a degree of arc ; and likewise for other aliquot parts of a degree. Thus we have established the theorem for 63 TEACHING MATHEMATICS all cases where the number of degrees in the angle can be expressed in figures. If we have an angle in which the number of degrees can- not be so expressed, in other words, where it is incommensurable, we must still base our proof upon degrees and fractions of a degree. We can show the existence of a series of central angles, each greater than the one before it, and each less than the angle we have to measure ; and we can also show the existence of a series of arcs each greater than the one before it, and each less than the arc we have to measure ; moreover each arc is intercepted by the angle corresponding to it in the other series, and the arc and its correspond- ing angle have the same measurement-number. These series can be continued until the last angle differs from the angle to be measured by an angle less than any specified subdivision of a degree; the last arc will then differ from the arc to be measured by an arc less than the same specified subdivision of a degree ; and the mea- surement-number, the number of degrees for the last angle, being the same, as we have seen, for the last arc also, differs from either of the 64 THE METHOD OF LIMITS measurement-numbers sought by less than the specified subdivision of unity ; the measurement- number of the arc, then, cannot differ from the measurement-number of the angle, because either is, by a definition to be given later, the limit of the same sequence of numbers. This proof is so important, and so typical of proofs for the measurement of quantities incom- mensurable with the unit, that it should be fully illustrated. Again, it must be remembered that the pupil's conception of incommensurable num- bers must be developed by just such proofs ; it follows then that he will be perplexed not only by the proof itself but by the question of the necessity for it. For the first instance of this kind of proof, therefore, it is advisable to select, for illustrative purposes only, a quantity that actually has a com- mensurable measurement-number, but one that cannot be expressed in decimal notation. Let us say, the arc intercepted by a central angle of io|°. Here we can first show that the measure- ment-number, io|, is the same for the arc and for the angle, as stated in the theorem. Then we 65 TEACHING MATHEMATICS can describe the construction of the series of angles, and the series of arcs, with the corre- sponding series of measurement-numbers as fol- lows : — 10.3°, 10.33°, 10.333°, 10-3333°. and so on. We can show that the successive angles, arcs, and measurement-numbers differ, respectively, from those we are after by less than .1°, .01°, .001°, .0001°, and so on ; and consequently, even if we did not know that the required measure- ment number was 10^, we could prove that the number for the arc could not differ from the number for the angle by any decimal fraction that might be specified in advance, however small that fraction might be. This kind of proof is more welcome to the pu- pil, for one reason at least : namely, that a standard system of subdivision for the unit is less vague than "any convenient fraction," and recommends itself to a healthy practical sense. There is also, as I have said, the postponement of this new idea of incommensurable numbers until the ar- gument that establishes their reason for existence is more familiar. Perhaps I am wrong in saying 66 THE METHOD OF LIMITS that the idea should be postponed ; at least it should not, while new and abstruse itself, be made a necessary part of an entirely new kind of ar- gument, itself sufficiently abstruse. Precisely the same problem presents itself when we begin to deal with the measurement of the rectangle. Considering first the strip of unit width along one base of the rectangle, we have a unit square standing on each length-unit of the base, a tenth of a unit square on each tenth of a length-unit, and so on. The theorem upon which our argument is founded is that two rectangles with the bases and altitudes respectively equal are congruent. When the length-number of the base is incommensurable we have the same kind of correspondence as before, this time of a series of lines, a series of rectangular unit-wide strips on those lines as bases, and a series of measure- ment numbers. The argument is identical with that just given for the central angle and its inter- cepted arc. When we come to deal with the entire area of the rectangle, the different series of correspond- ing quantities and measurement-numbers is a 67 TEACHING MATHEMATICS little more complicated. Let us represent the length-numbers of the sides of the rectangle (whether commensurable or not ) by a and b, and its area-number by 5". We can form a series of pairs of sides, whose commensurable length-num- bers are represented by x and y, each pair form- ing a rectangle whose area Q consists of ;r unit- wide strips each of area y. We have then for each rectangle in the series the equation Q=xy. Now when for example we measure the sides of the rectangle to tenths of the length-unit, the rectangle Q cannot differ from the rectangle 5 by an amount so great as the area of a strip -J^ of a unit wide running around two sides of the rectangle Q. That is and, since ;r ■< « and y <^b, and also ^o "^C ^ S-Q<-h{a + b + I) In the same way when we measure to hun- dredths, thousandths, or any other decimal sub- division. For any such subdivision, say to ;/ths, we have S-Q As) /% • • • • each of which is greater than the one preceding. Then if we start with the circumscribed hex- agon, and repeatedly double the number of sides, we can obtain another series of length-numbers which we may represent by ^6. $^12, q-ih ^48. $^96 • • • • each of which is less than the one preceding. We have two things to prove before going on; first, that any circumscribed polygon has a 70 THE METHOD OF LIMITS greater perimeter than any inscribed polygon; and second, that we can find enough correspond- ing pairs of terms in the two series so that the difference between the last / and the last q shall be less than any number specified in advance. If we now imagine the two series of numbers to be represented by two series of points on a line, that is, in the case of a circle with one-inch radius, so that OPq= just 6 inches, (9^e = 6.928 inches ; P12, P^x, Pis ^^^ so on would be points between Pq and Q^, and ^12, Q2i, Qa and so on would also be points between P^ and Q^. In this diagram the points /'12, /*24> etc., re- presenting the numbers /i2» ^24) etc., would begin Tbis-T?ay for -^ P04 Q„. ^« ' I I I" I II ! I'll Fig. 5 at Pe and succeed each other towards the right, while the Q's begin at Q^ and succeed each other towards the left. No P can appear on the right of any point Q, and no Q can appear on the left of any point P. We can make a P-Q pair of points as close to each other as we please by 71 TEACHING MATHEMATICS continuing the process of computing polygons of double the number of sides, but there is always the inexorable law that no /*- point can appear in any part of the region in which a Q has yet appeared, or in which, by continuation of our work, a Q may hereafter appear. If we assume then, that there is a point L, to the right of all the Fs and to the left of all the Qs ; and that there is a corresponding number /, greater than every one of the/'j and less than every one of the q's ; then we can call that point and its corresponding number the limits to which the points and numbers that we have been con- sidering approach. It satisfies the definition of a limit, as follows: — A variable is said to approach a constant as a limit when, no matter what small number is specified in advance, some one of the regular sequence of values assumed by the variable, as well as every value thereafter, differs from the constant by an amount less than the small number arbitrarily specified in advance. It has been proposed to define the length of the circumference of a circle as the limit of the length of the perimeter of an inscribed polygon, 72 THE METHOD OF LIMITS the number of sides being doubled again and again indefinitely. This has at least the advant- age of being the only definition possible. If it is objected to on the ground that it is too abstruse for this stage of education, that is a reason for omitting the argument by limits from the study of the circle in high-school geometry; it is not a good reason for passing over the definition of the length of a curved line as if there were no diffi- culty there. The objection that we get a different series of numbers if we begin, say, with a square or a pentagon is seen to be of no moment when we remember that each of the numbers of any such series as A» /sj /i6. /32> A4 • • • • must still be less than any q whatever, whether of the series corresponding to this series of /'j, or of any other ; and consequently the limit / must be the limit of every such series. IX y^ SIMPSON'S RULE AND THE CURVE OF SECTIONS Simpson's Rule for plane areas is a formula for obtaining the area of a figure bounded by a curv^ed line ; for exam- ple, the water-line plan of a ship. It depends on the area of a double strip such as is shown in this diagram. Here the figure bounded by yiy 2 k, j'3, and the curve is approximated to by a rectangle and two trapezoids, each having the breadth § k. These figures have areas as follows : — First trapezoid : |(ji + J2) I ^ = I (ji + J'2) Rectangle: (§/^)j2 = 1(272) Second trapezoid : h (72 -^y^) l^ = i ( J2 + Ja) The entire polygon, then, which is intended to 74 /T 'T^ 1 1 1 1 1 1 1 \ 1 1 — •( Fig. 6 Fig. 7 SIMPSON'S RULE be an approximation to the area of the given fig- ure, has an area S = | (ji + 4/2 + Js) Generally there are sev- eral double strips, as in the diagram here given. The end-ordinate may be zero, but takes its place in the formula just the same. Thus in this diagram the areas of the five double strips are : — First double strip : | (jo + 4/i + J2) Second " " | (/2 + 4/3 + n) Third " " I (j/4 + 4/5 +/6) and so on ; the entire area being given by the formula 5=1 (j'o +4/1 + 2/2 + 4J'3 + 2j4 + 475 + 2/ + 4f- + 2/8 + 4J9 +/10) The numbers i, 4, 2, 4, 2 .... are called " Simpson's multipliers." For a curve that is not too steep the results from this formula are very accurate. Without it the pupil has no means of handling any curve 75 TEACHING MATHEMATICS except the circle. With it he has a practical ap. plication of his knowledge of geometry and a very satisfactory command over any kind of area. As an example of its use consider the follow- ing computation for the area of a circle 20 inches in radius, by which the value of tt to 6 figures is obtained with great simplicity. The equation of the circle with radius 20 and centre at O is x^ -V y^ = 400 From this, by the use of a ta- ble of square roots, we obtain the following values of J, for values of x for every inch from Fig. 8 o to lo inches. We use them to find the ordinates of half a 60° 76 SIMPSON'S RULE segment. From this we can get the six seg- ments to be added to the area of the inscribed hexagon. Subtracting the length OP =/ from each value of y, we obtain the length of the cor- responding ordinate of the segment, which is to be used in finding its area by Simpson's Rule. X r y y-p Simps o 400 20.0000 2.6795 X I 399 19.9750 2.6545 4 2 396 19.8997 2.5792 2 3 391 ^9-im 2.4532 4 4 384 19-5959 2.2754 2 S 375 19.3649 2.0444 4 6 364 19.0788 1.7583 2 7 351 18.7350 1-4145 4 8 336 18.3303 1.0098 2 9 319 17.8606 0.5401 4 lO 300 17-3205 0,0000 X Then, using Simpson's multipliers as indicated we have the following products : — 77 TEACHING MATHEMATICS 2 g^ge The sum of these products gives the 10.6180 value of the parenthesis, in the for- _ „ mula, which we have to multiply by -. 9.8128 ^ -^ -'3 4.5508 In this case /^= I. 8-1776 The area of this half-segment, then, 3.5166 6q8o ^^ ^ (54-3517); the six segments that 2.0196 lie around the inscribed hexagon will 2.1604 , have an area 6 Q) (54.3517) which 0.0000 -^ ^ Jj /y reduces to 4(54.3517) square inches. 54-3517 'Yhe area of the hexagon, ^, is 4 6(40o)(i. 73205) ,, "^ — — ^ = 6(173.205) square mches. 4 The entire area of the circle, then, will be 1256.637 square inches; then, since 40077=1256.637 77 = 3.14159 Suppose the area of the base of a solid is 3.1 1 sq. in. ; the area of a section parallel to the base and one inch above it, 3.02 sq. in. ; of another section 2 inches above the base, 2.86 sq. in. ; 3 78 SIMPSON'S RULE inches above, 2.63 sq. in. ; 4 inches, 2.36 sq. in. ; 5 inches, 2.01 sq. in. ; and so on. These numbers could be laid off as ordinates, 3.11 2.01 Q li Fig. 9 on any convenient scale, as in the diagram. If now we suppose that a great many other sec- tions are measured, say at intervals of j\ inch, or even ^^^ inch or j^qq inch, the ordinates laid off for them between those here shown, and a curve drawn through the tops of all these ordi- nates, the curve will be a graph of the areas of horizontal sections, the distances OP, PQ, etc., showing the distances between two sections. Such a curve is called a curve of sectional areas (or, briefly, a curve of sections ) for the solid. I shall presently prove that the area-number of the curve of sections is the same as the vol- 79 TEACHING MATHEMATICS ume-number of the solid for which it is drawn. This theorem is of use to prove the " Principle of Cavalieri," namely : — If two solids have equivalent bases, and if sections parallel to the bases and equally distant from them are equivalent, then the solids are equivalent. Two solids such as those here described would have the same curve of sections. The volume-number of a right prism of unit thickness (altitude) is equal to the area-number of its base. For upon every square unit in its base can be laid a cubic unit; on every tenth of a square unit, a tenth of a cubic unit, and so on. All the propositions about congruent or equiva- lent plane figures can be shown to be true of the right prisms of unit thickness standing upon them. For the sake of generahty we may substi- tute cylinder for prism in this theorem, under- standing by a right cylindrical surface that gen- erated by a line tracing out the perimeter of the base and remaining perpendicular to its plane. The area-number of the base of this unit-thick cylinder being laid off as an ordinate, and the 80 SIMPSON'S RULE unit thickness being measured off along OX, the volume of the cylinder will be represented by the area of the rectangle thus constructed. The volume of a cylinder -^^ of a unit thick would be represented by a rectangle ^^ of a unit wide, and so on. If now we take any solid standing on a hor- izontal base, and divide it by means of horizontal sections into slices, the areas of the sections will appear as ordinates of the curve of sections at the appropriate points on OX. A right cylinder standing upon one of those sections, and having a thickness equal to the thickness of the slice, would have its volume represented by the area of a rectangle formed upon the corresponding ordinate in the curve of sections. Let us now suppose that the solid is divided up into very thin slices of the same thickness, and that upon the base of each slice we con- struct a right cylinder of the same thickness. The total volume of the whole series of cylinders would differ from the volume of the solid by less than a layer of a certain definite thickness ex- 8i TEACHING MATHEMATICS tending over the whole lateral surface of the solid. If we now halve the thickness of the slices, reconstructing the cylinders to correspond, the necessary thickness of this layer would become less ; and by repeating the halving process we could finally arrive at a point where the differ- ence between the volume-number of the solid and that of the series of right cylinders would be less than any small number that may have been specified in advance. At the same time the narrow rectangles, con- structed as above on the ordinates of the curve of sections, have a total area that differs from the area under the curve by an amount less than the area of a band of a certain definite width extend- ing along the upper boundary. By repeating the process of halving slices, reconstructing cylinders, and reconstructing the rectangles that represent the volumes of the cylinders, we could finally arrive at a point where the difference between the area-number of the curve of sections and that of the series of rectangles would be less than any small number that may have been specified in advance. 82 SIMPSON'S RULE We shall thus have a sequence of numbers, each of which expresses not only the volume of the series of right cylinders but also the area of the series of rectangles ; this sequence approach- ing a definite limit, which may or may not be commensurable ( i.e., expressible in figures) ; and this limit being the expression not only of the volume of the solid but also of the area of its curve of sections. In other words, the volume of any solid having a curve of sections is equal to the area under that curve. The Principle of Cavalieri follows immediately from this theorem. It was proved originally by supposing the two solids to be composed of very thin but uniform layers, as of sheets of paper ; since each layer in one was equivalent to ( con- tained as much paper as) the corresponding layer in the other, the total amount in one was the same as in the other. The use of the curve of sections leads at once to the so-called "Prismatoid Formula" for the area of a solid. This can be directly proved for a prismatoid, that is, for a solid bounded by planes, having all of its vertices in two parallel 83 TEACHING MATHEMATICS planes. It is used also, as a formula of approxi- mation, for other solids, such as those computed for excavations and embankments. It consists merely in the application of Simpson's Rule to the section-areas, on the hypothesis that they are the ordinates of the curve of sections ; for the prismatoid only the mid-section is used in addition to the two bases. There is need only to mention the great eco- nomy in the demonstration of the mensuration theorems of solid geometry from the early proof of this " principle " ; an economy not merely of actual effort in demonstration, but also of the interest of the pupil, in the presentation of one comprehensive method of attack instead of a number of widely different methods. X THE TEACHER Text-books may be written embodying these or other reforms in method or subject-matter ; but the success of such projects must depend upon the teacher. However well convinced the teacher is of the value of the changes that he wishes to effect, he cannot ignore the value that exists in the old ways ; nor can he avoid the high duties to which the new times will call him, and from which his propositions for reform will not excuse him. They ascribe to Euclid a fable about a mule, which met a donkey at a ford, and entertained him with astute remarks about the sizes of their respective burdens. From these remarks the learned, of ancient times, were wont to infer the load each of these gossipy animals bore. Ever since that time problems of a like "unpractical" character have occupied the attention of stu- dents. Even within the memory of men now 85 TEACHING MATHEMATICS living, mathematical earwigs have disported themselves upon perfectly rigid and unbeliev- ably slender rods, policemen having no dimen- sions have chased infinitesimal culprits up math- ematical alleyways, and we have rearranged the soldiers of Napoleon in phalanxes that even Xerxes could have seen were useless. Yet per- haps these problems were not entirely useless. What did Apollo want to double the size of his altar for ? The very futility of the task, as in the quest of the alchemists, filled the world with results of great value. Let us remember the civilization of China, for which confident philanthropists would substitute the ways of Europe or the United States. It has maintained itself for three thousand years : it cannot be worthless. Though we have, to our own satisfaction, demolished the position of those who would defend the established customs of teaching, let us be gentle with the poor old world. There is something eternal even in the mistakes that we condemn. It is good to be enthusiastic ; but to be intolerant is bad strategy. The proposition to unite the different branches 86 THE TEACHER of school mathematics into one progressive sub- ject has a long history of defeat, as the present customs of teaching show. The reason for that defeat lies in the desire of the community, as well as of the teachers, for distinctly marked stages of advancement, recognizable successive tasks, for which teacher and pupils can severally be held responsible, and for which books can be ordered without too much scrutiny. The hope for success in the present wide- spread attempt rests upon new conditions. For one thing, teachers are better informed, less dis- tracted by demands to teach from all parts of the cyclopaedia, more wide awake not only to progress in the art of teaching but also, let us hope, to the widening scope and the beauty of the science of mathematics. Again, the modern aim, while not forgetting that instruction is to be sound and in line with later study, seeks also immediate effi- ciency wherever possible ; so that the credentials of progress are the increased powers of the pupil, rather than documents to show that he " is in " or " has been through " the fields of knowledge that bear the orthodox text-book names. 87 TEACHING MATHEMATICS These considerations point to new demands upon the teacher. The most obvious is that he can no longer — any more than can the teacher of history or French — rest content with the programme of the text-book to which his year's work is married. The worded problems particu- larly he must study out in detail, with especial reference to two things : first, the degree of diffi- culty in constructing the equation ; and second, the type of solution required by that equation. In geometry the teacher must not only master with great minuteness the logical relations and the information presented in the book his pupils use, but he must himself be able to vary funda- mentally those logical relations, to classify and reclassify that body of knowledge, so as to build parts of it from time to time coherently about the topics on which he succeeds in arousing the interest of the class. More than all else, and in mathematics more than in any other subject, the teacher should have the enthusiasm of the achieving student. I wish I could reproduce here the eloquent words in which I once heard a distinguished scholar 88 THE TEACHER urge, upon prospective teachers of mathematics in an Eastern university, the need and the sure reward of hard study in lines not too remote from the field of teaching. Certainly no teacher of mathematics should rest content with the bare knowledge of the matters he is at work upon with his class ; much of the material to which it is directly preparatory should also be within his grasp ; he should have some knowledge of such obvious applications as are known to the machin- ist, the engineer, the mariner — as well as the salesman and the usurer; and formal logic — not necessarily the time-honored gabble of technical terms, but the meat of the subject, with some idea of its diagrams and its algebra — is almost indispensable. Not only these things that he must have as the weapons of his daily war, but other treasures that lie in great abundance, waiting only for re- solute endeavor to seize them — treasures that genius and incredible industry have heaped for centuries upon the altars of wisdom : these must the true teacher search for his own adornment. Lest his mind become dulled in repeating stale 89 TEACHING MATHEMATICS arguments to the feebler minds of children, let him whittle upon matters hard enough to test its edge. Let him feel for himself the triumph, the glow of discovery, that he sees shining in the eyes of those to whom he is a. sage. For this purpose some will choose one study, some an-' other ; there is certainly variety enough to keep conversation sweet. THE END OUTLINE I. THE MODERN POINT OF VIEW 1. The aim for immediate efficiency I 2. Computation and self-reliance 3 3. Practical aspects 4 4. Definitions for the teacher 5 5. Greek geometry and modern beginners .... 7 6. Algebra named for an old Arabic rule 8 7. Purely symbolic manipulation 1 1 II. THE ORDER OF TOPICS 1. Pedantry in teaching 13 2. Equations and numberecl diagrams 13 3. Outline for the first year 14 4. Subjects for the second year 15 III. EQUATIONS AND THEIR USE 1. The ends to be sought 17 2. The subject-matter of problems 18 3. Positive terms only 19 4. Negative terms, and the word " algebra" .... 20 5. Geometric material 22 IV. SOME RULES OF THUMB 1. Measure of precision 23 2. What a significant figure is 23 3. The degree of precision attainable 24 4. Reasonable economy in computation 25 91 OUTLINE 5. The cultivation of self-responsibility 26 6. Square root of numbers 26 7. The "guess and try " method 27 8. Its theory 28 9. Its advantages 28 10. Possible extension to cube root 30 11. Algebraic evolution to be omitted 30 V. GEOMETRY AS ALGEBRAIC MATERIAL 1. The facts available 32 2. The rectangle rule fundamental 32 3. Rearrangement necessary 33 4. The stripe idea 33 5. Proof for the rectangle rule 34 6. The logical scheme of theorems 34 7. Appropriate notation 35 8. Illustration by the Pythagorean Theorem ... 38 9. The use of measurable angles 40 10. Three-figure table of tangents 40 11. '• Practical problems" again 40 VI. THE GRAPHICAL METHOD 1. Importance and popular use 41 2. Application to elimination 42 3. The locus idea 43 4. The graphical check 43 5. The circle-equation 44 6. A source of elimination problems 45 7. The standard parabola 47 8. Check for one-letter quadratics 47 9. Things to be avoided 47 92 OUTLINE VII. THE BASES OF PROOF IN GEOMETRY 1. The idea of motion in Euclid 49 2. Logical system desirable 50 3. Warrant for the early proofs 50 4. Characteristic motions of plane figures . . . .51 5. Comparison with cylinders and spheres .... 52 6. Cyclic order of parts S3 7. Illustration by Euclid I, 4 54 8. Congruence, symmetry, and mensuration ... 57 9. The early theorems 57 ID. Problems of construction 58 II. Inequality theorems to be postponed 59 VIII. THE METHOD OF LIMITS 1. Incommensurable numbers 60 2. Computations from direct measurement .... 61 3. Our cumbersome methods inherited 61 4. The measurement of an angle 63 5. Type of the "limit" proof 65 6. Division of difficulties for the pupil . . . , . 65 7. The measurement of the rectangle 67 8. The argument in regard to tt 69 9. Graphical illustration of a limit 7' 10. Definition of a limit 72 1 1. Definition of the length of the circumference . . 72 IX. SIMPSON'S RULE AND THE CURVE OF SECTIONS 1. Simpson's rule for plane areas 72 2. The double strip polygon 74 3. Simpson's multipliers 75 93 OUTLINE 4. Application to the evaluation of w 76 5. Curve of sectional areas 78 6. The principle of Cavalieri 80 7. Volume of a right cylinder 80 8. Generalization of volume-mensuration .... 84 X. THE TEACHER 1. The need of tolerance 85 2. New hopes of progress 87 3. New tasks for the teacher 88 The teacher a student ... 88 4- This book is DUE on the last date stamped below "IL 1 5 19311 MAY ^ 5 1934 'H' ' ^ 1934 ^^y 2 1936 ' DEC ' -^ JIAY I 195^ «>'5' AUG ^ V A^3« jUAy i ^s^*» MAY 2 5 1939 SEP 1 8194? Form L-9-3om-8,'28 mi SEPzzim \ Iiliinii.i I MS An QAll E vans - E92t The teaching of high school mathematics. L 007 062 039 8 TT^' Mite UC SOUTHERN REGIONAL LIBRARY FACILITY AA 000 792 281 Q^ll ' cf ':ALIFORNrA E92t ANGELES BRARY