ATHEMMICAL SERIES GF ALGEBRA ING TRIGONOMETRY) .PERCY NLIMN GIFT OF Miss Emily Palmer ^> / -p Cy C> CALIFORNIA . CALIFORNIAI LONGMANS MODERN MATHEMATICAL SERIES General Editors P. Abbott, B.A., C. S. Jackson, *M.A. F. S. Macaulav, M.A., D.Sc. THE TEACHING OF ALGEBRA BY THE SAME AUTHOR Uniform with this Volume. EXERCISES IN ALGEBRA (INCLUDING TRIG- ONOMETRY). Parti. Without Answers, 3s.6d.; with Answers, 4s. Contents. — Section I, "Non-directed Numbers"; Section II, "Directed Numbers"; Section III, "Log- arithms " ; Supplementary Exercises. EXERCISES IN ALGEBRA (INCLUDING TRIG- ONOMETRY). Part II. Without Answers, 6s. ; with Answers, 6s. 6d. Contents. — Section IV, " Mainly Revision " ; Sec- tion V, " The Trigonometry of the Sphere"; Section VI, "Complex Numbers"; Section VII, "Periodic Functions"; Section VIII, "Limits"; Section IX, " Statistics ". LONGMANS, GREEN AND CO. LONDON, NEW YORK, BOMBAY, CALCUTTA, AND MADRAS Xonomang' /B^o^ern /IDatbemattcal Series THE TEACHING OF ALGEBRA (INCLUDING TRIGONOMETRY) T. PERCY _^^UNN, M.A., D.Sc PROFESSOR OF EDUCATION IN THE UNIVERSITY OF LONDON VICE-PRINCIPAL OF THE L.C.C. LONDON DAY TRAINING COLLEGE (UNIVERSITY OF LONDON) ; FORMERLY SENIOR MATHEMATICAL AND SCIENCE MASTER WILLIAM ELLIS SCHOOL WITH DIAGRAMS LONGMANS, GREEN AND CO. 39 PATERNOSTER ROW, LONDON FOURTH AVENUE & 30th STREET, NEW YORK BOMBAY, CALCUTTA, AND MADRAS I9I4 ,\^ / fs PREFACE. In 1909 and 1910 the author had the honour of giving courses of lectures on the teaching of Algebra addressed respectively to masters and mistresses in Secondary- Schools. The present volume is a practical handbook based upon those lectures, and containing what seem to the author the most useful things he has learnt during the fifteen years of his work as a mathematical master and the ten years in which it has been his duty to discuss with teachers, actual and prospective, the problems of their craft. It is accompanied by two col- lections of examples — ''Exercises in Algebra, Part I," and " Exercises in Algebra, Part II " — which are in- tended together to cover all stages of school instruction in the subject. Thus the three volumes constitute a single work. In the view of the author the term "Algebra" should include in its reference all the Trigonometry, plane and spherical, which it is desirable to teach in schools, together with an exposition of the fundamentals of the Calculus. He has sought, there- fore, to present these subjects, both in this book and in the " Exercises," as a unified whole. " Exercises, Part I," is intended to supply materials 6c22r>7 vi PREFACE for a course which every boy or girl who remains at a secondary school until the age of sixteen or seventeen may reasonably be expected to cover. Thus it meets and in some directions exceeds the present requirements of University Entrance and similar examinations. An important note upon this point will be found on page 60 of the present book. " Exercises, Part II," presents a continuation of this universal course, to be taken, as a whole or in part, by boys and girls who remain at school until the age of eighteen or nineteen, and by those who, at an earlier age, forge ahead of their fel- lows. It is hoped that it may also prove useful to students preparing to enter the scientific professions and to students in Training Colleges for Teachers. In '* Exercises, Part I," explanatory matter has been limited to a few brief notes — on the ground that in the earlier stages of the pupil's progress exposition is of little use unless given verbally by the teacher. Thus the discussions and arguments presupposed in the exercises have been relegated to the present volume, where they could be given in a form which, it is hoped, the younger teacher will find more helpful, and his experienced colleague more suggestive and provocative. In " Exer- cises, Part II," each set of examples is accompanied by exposition intended for the student's reading. To make it suitable for this purpose it has seemed neces- sary to abandon the traditional reticence of the mathe- matical textbook — a fact to be borne in mind in con- nexion with the otherwise misleading length of the book. The corresponding sections of the present book consist of critical introductions, commentaries upon the exposition and exercises, suggestions for illustrations PREFACE vii and other teaching devices, and solutions of typical examples. There is considerable evidence that teachers of the present generation are no longer satisfied with either the curriculum or the methods of instruction in Algebra which they have inherited from their predecessors. Moreover, there are signs that their dissatisfaction is reflected in the public examining bodies which exercise such immense influence upon mathematical teaching in this country. These circumstances may be held to justify an addition to the few works written expressly for the teacher who wishes, while maintaining his hold upon all that is sound in the traditional methods, to orient himself in accordance with the present drift of well-considered opinion. The author has worked out in " Exercises, Part I," a curriculum congruent on the whole with the recent Eeport of the Com- mittee of the Mathematical Association, though in some respects more radical and in some more conserva- tive. He has noted with special satisfaction that, taking the whole work into consideration, the programme of studies which he has proposed is in close agreement with the one outlined by Dr. A. N. Whitehead in the very valuable address printed in the Mathematical Gazette for March, 1913. With regard to method, the author has sought to take due account both of the pedagogical pragmatism of which Professor Perry has long been our most influential advocate and also of the modern critical movement represented so brilliantly in this country by the authors of Principia Mathematica. In addition, recognizing that mathematical ideas are viii PREFACE apt to become " flat " through long confinement in textbooks, he has sought, by drawing them afresh from their historical sources, to present them with as much as possible of their original vigour. Acknowledgment has been made in the text wherever the author has consciously taken advantage of the work of writers whose ideas have not yet been absorbed into the common stock. He has the pleasure of recording here many other obhgations, general and specific. From his three Editors he has constantly received criticism and advice of the most helpful kind. Mr. C. 0. Tuckey of Charterhouse read a good deal of the earlier part of the work in typoscript and made several important sug- gestions, gratefully accepted. Professor C. Spearman of University College very kindly read and criticised the section on Statistics — a subject to which he has himself made such important contributions. Similar service was rendered, in connexion with other sections, by Messrs. G. B. Jeffery of University College, B. M. Neville of the William Ellis School, and E. Wasser- man of Owen's School. Dr. L. Silberstein, from his inexhaustible store of fertile mathematical ideas, has kindly contributed those embodied in several important examples. Intercourse with Mr. Benchara Branford and Mr. David Mair has had an influence upon the book which is not the less substantial because it is not located in any particular pages. The author's thanks are due also to Miss Doris Brookes, Miss Elsa Nunn and Miss Clotilde von Wyss for assistance given in the preparation of the diagrams, and to a number of his present and former students for undertaking the un- grateful task of providing the answers to the examples. PREFACE IX Lastly, he has to acknowledge the help received at every stage of the work from his colleague, Miss Margaret Punnett, who not only drew all the diagrams which required special care or laborious calculations, corrected all the proofs, and compiled the index of the present volume, but also gave constant criticism and encourage- ment of the greatest value. It should be unnecessary to add that the author does not seek by these acknowledgments to make his friends share the responsibility for anything in the work that may be judged unsatisfactory, or to suggest that they accept his views upon all the disputed or disputable questions to which he has offered answers. London Day Training College (University op London), May, 1914. CONTENTS. General Introduction, chaptbb page I. The Nature op Algebra 1 II. Method and Curriculum 16 III. The Formula 26 IV. The Graph . 31 PAET I. Alternative Schemes op Study .... 60 V. Introduction to Part I 61 SECTION I. NON-DIRECTED NUMBERS. The Exercises op Section I 62 VI. The Programme op Section I (Exercises I-XVI) . 63 VII. Factorization 82 A. Factorization ot ac + be (p. 82) ; B. Factoriza- tion of a2 - 62 (p^ 87). VIII. Square Root. Surds 90 A. The Calculation of Square Roots (p. 90) ; B. The Radical Form (p. 93). IX. Fractions 96 A. Fractions with Monomial Denominators (p. 96) ; B. Fractions with Binomial Denominators (p. 98). X. Changing the Subject op a Formula . . . 104 XI. Programme op Section I (Exercises XVEI-XXVI) . 109 X CONTENTS xi CHAPTBR PAOE XII. Direct Proportion 117 XIII. Trigonometrical Ratios (I) 121 A. The Tangent of an Angle (p. 121) ; B. The Sine and Cosine, Vectors (p. 124). XIV. Trigonometrical Ratios (II) 129 A. Circles of Latitude, Middle Latitude Sailing (p. 129) ; B. Relations between the Sine, Cosine, and Tangent (p. 132). XV. The Combining op Formula 136 A. The Determination of Constants in a Formula (p. 136) ; B. Common Values of Two Relations (p. 139) ; C. Elimination (p. 142). XVI. Further Types op Proportionality . . . 146 A. Inverse Proportion (p. 145) ; B. Direct Propor- tion to the Square or Square Root (p. 149) ; C. Inverse Proportion to the Square or Square Root (p. 152) ; D. Combinations of Types of Proportion (p. 154). SECTION II. DIRECTED NUMBERS. The Exercises op Section II 157 XVII. The Programme op Section II (Exercises XXVII- XXXVIII) 159 XVIII. Directed Numbers 181 A. The Uses of Directed Numbers (p. 181); B. Algebraic Addition and Subtraction (p. 184); C. The Multiplication and Division of Directed Numbers (p. 193). XIX. CONSTANT-DlPPERBNCE SbRIES 199 A. The Summation of Constant-Difference Series (p. 199) ; B. The Calculation of Certain Areas and Volumes (p. 203). XX. Algebraic Multiplication 207 A. Algebraic Multiplication (p. 207); B. The Binomial Expansion (p. 211). rii CONTENTS CHAPTER PAGE XXI. Positive and Negative Indices .... 214 A. The Uses and Laws of Positive Indices (p. 214) ; B. Negative Indices (p. 217). XXII. Algebraic Division 222 A. Algebraic Division (p. 222) ; B. Geometric Series (p. 224). XXIII. The Complete Number-Scale 228 XXIV. The Programme op Section II (Exercises XXXIX-L) 235 XXV. Linear Functions. Extended Use of Sine, Cosine, AND Tangent 258 A. Linear Functions (p. 258) ; B. Extension of meaning of Sine and Cosine (p. 261). XXVI. The Hyperbolic and Parabolic Functions . . 264 A. Hyperbolic Functions (p. 264) ; B. Parabolic Functions (p. 266) ; C. Quadratic Equations (p. 270) ; D. Inverse Functions (p. 274). XXVII. Wallis's Law 279 A. Area Functions (p. 279) ; B. DiSerential For- mulae (p. 282). XXVIII. The Calculation op it and the Sine-Table . . 292 A. The Calculation of tt (p. 294) ; B. The Calcula tion of the Sine-Table (p. 295). SECTION III. LOGARITHMS. The Exercises op Section III .... 298 XXIX. The Programme op Section III ... . 299 XXX. The Graphic Solution op Growth-Problems . 312 XXXI. The Gunter Scale 319 XXXII. Logarithms 325 A. The Slide Rule (p. 325) ; B. Logarithms (p. 329). XXXIII. Common Logarithms 333 A. Gunter's Scale and Logarithms Obtained by Calculation (p. 333) ; B. Common Logarithms (p. 335) ; C. The Use of Tables (p. 337). CONTENTS xiii CHAPTER PAGB; XXXIV. The Logarithmic and Antilogarithmic Functions 341 XXXV. Nominal and Effective Growth-Factors . . 346 EXERCISES SUPPLEMENTARY TO SECTIONS II AND III. Supplementary Exercises 354 XXXVI. The Programme op Exercises LX-LXV . . 355 XXXVII. The Programme op Exercises LXVI-LXIX . . 366 PAET II. Alternative Schemes op Study .... 380 XXXVIII. Introduction to Part II 381 SECTION IV. MAINLY REVISION. The Exercises of Section IV 402 XXXIX. Number-Systems and Numerical Operations . 403 XL. Functions 422 XLI. The Exponential Function and Curve . . . 428 SECTION V. THE TRIGONOMETRY OF THE SPHERE. The Exercises op Section V 440 XLII. Projections 441 XLIII. The Trigonometry of Spherical Triangles . . 454 SECTION VI. COMPLEX NUMBERS. The Exercises op Section VI 468 XLIV. The Nature op Complex Numbers .... 469 XLV. Relations between a Real and a Complex Variable 478 XLVI, Relations between two Complex Variables . 486 xiv CONTENTS SECTION VII. PERIODIC FUNCTIONS. CHAPTER PAQB The Exercises of Section VII .... 498 XLVII. The Circular Functions . . . . . .499 XLVIII, Wave-Motion 515 XLIX. Differential Formula of the Circular Functions 528 L. The Hyperbolic Functions 531 SECTION VIII. LIMITS. The Exercises of Section VIII LI. Differentiation and Integration . LII. Expansions. Supplementary Examples . 540 541 557 SECTION IX. STATISTICS. The Exercises of Section IX . . ... 568 Lin. Frequency-Distribution 569 LIV. The Calculation of Frequencies. Probability . 583 LV. Correlation 602 Index 611 COLOUEED DIAGRAMS. Figs. 48, 49, 50 Fig. 51 Fig. 58 Fig. 59 Figs. 60, 61, 62 To face page 200 M .. 201 „ „ 208 „ 209 .. 210 GENEEAL INTEODUCTION. CHAPTER I. THE NATUEE OF ALGEBEA. § 1. A discussion of the proper definition of " Algebra " may easily be carried to the point where its interest becomes academic rather than practical. It will be limited here to a review of the general nature of the topics traditionally studied and taught under a name of ancient and obscure origin.^ In- spection of these topics shows the presence of at least four important elements : Analysis, the direct use of symbolism, the extended use of symbolism, and the manipulation of symbolism. § 2. Analysis. — The most fundamental is analysis. The diffi- culty of finding a precise boundary between arithmetic and alge- bra (as these terms are commonly understood) is well known. It is due to the fact that the distinction between them con- 'sists not so much in a difference of subject-matter as in a difference of attitude towards the same subject-matter. A simple example may make the difference clear. A small boy has learnt that the " area " of a figure is the number of unit squares (say, square inches) which would entirely cover it. With this definition before him he is asked to determine the area of a rectangle measuring 7 inches by 5. He soon ob- serves that the unit squares into which the figure is to be mapped out can be regarded as forming five rows each con- 1 It is unfortunate that a writer is compelled by convention to place his introductory chapter at the threshold of his work. The disadvantages of the arrangement are obvious. An introduction often represents the author's attempt to meet difficulties and to answer objections which the reader has not yet felt. It is therefore liable to be misunderstood or misused. It is to be hoped that the reader who finds in the present chapter little relevance to the urgent problems of algebra teaching will by judicious skipping reach the end without serious waste of time. He will find at many points of the sequel references which will send him back, perhaps to a more prohtable reading of these preliminary discussions. T. 1 2 ALGEBRA taining seven squares. This observation enables him to shorten the process of finding the area ; for it is obvious that the rectangle nxust contain 7 x 5 = 35 square inches. So ^ar arithmetic. But now let the boy's attention shift from the actual manipulation of the figures to the process which ilie maaipiila:ion follows ; and let him observe that the essence of that process is the multiplication of the length of the rectangle by its breadth. At this moment he has crossed the frontier which separates arithmetic from algebra ; for it is an important part of the business of algebra to disengage the essential features of an arithmetical process of given type from the numerical setting which a particular case presents. The result of the boy's analysis of the process of area-cal- culation will be expressed in a statement or "rule" that makes no reference to the particular numbers, 7 and 5, and would therefore hold good in any other area-calculation of the same type. For this reason it is usually called a " generalization ". Strictly speaking, however, the terms an- alysis and generalization refer to two distinct mental move- ments. In the former I bring to light the essential process concealed in a particular or accidental numerical garb. In the latter I recognize that this process may be followed identically in solving all problems of the same type. The distinction is not a trivial one, but demands the teacher's serious attention. The neglect of it is largely responsible for the common belief that the process of generalization must of necessity build upon numerous examples of the truth to be generalized, and that the security of the result depends upon the number of instances upon which it is based. This belief makes the schoolmaster regard "generalizing from a single instance" as one of the most dangerous manifestations of original intellectual sin. The truth is that in some cases the certainty of a generali- zation does depend upon the multiplicity of its data, while in other cases the number of data is logically irrelevant ; a single instance will establish the rule as securely as a hundred. Thus, when John Wallis (1655) had shown that what we now call the " integrals " of x, x^, x^, x^ are ^x^, ^x^, {x^, ix^, he felt entitled to generalize the result and to assume that the integral of re" "^ would be - x" for all integral values of n. This was a generalization of the former kind and had probable THE NATURE OF ALGEBRA 3 truth only. The degree of probability of such a generaliza- tion clearly depends in the first instance upon the number of data upon which it is based. Its credit will subsequently rise if results deduced from it are found to agree with known truths, or will collapse if a " negative instance " can be pro- duced. Thus the numbers 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, etc., are all prime and are all included in the formula n^ + n + 4:1. But the hypothesis that this formula always yields primes is at once discredited when it is seen that it fails for n = 40.^ On the other hand there is no possibility that the rule for the area of a rectangle should break down.^ Its certainty is due, not to verification in a large number of instances, nor to its simplicity, but merely to the fact that it is founded on analysis. It may be objected that the two types of generalization cannot be distinguished in this way ; that, for example, Wallis had to analyse his individual results before he could include them all in a single algebraic statement. This is, of course, true, but it should be noted that in Wallis's case the analysis was limited to the discovery that, in several instances, expres- sions of the form re" ~ ^ actually have integrals of the form ^ x". Analysis in the sense intended here would deal essentially with the circumstances on which this relation depends. An analysis in this sense of any one of Wallis's instances of integration would establish for ever the certainty of all instances covered by his formula.^ Thus the question whether a generalization requires the evidence of a number of examples is the question whether or not it is founded on analysis. If it is not so founded the generalization has at best only probability in its favour, a probability which increases with the number of instances in which it is verified. If it is so founded, though in practice several examples may be needed to point the way to the generalization, yet in theory a single instance is sufficient to render it certain. "^ 1 Quoted from Jevons, Principles of Science. ^ It is assumed for the present that the dimensions are integral. See § 3. ^ I.e. in which n is integral, * Logicians (e.g. Bradley) have called the first kind of generaliza- tion an "empirical universal," the second kind an "uncondi- tional universal ". The philosophically minded reader may follow 1* 4 ALGEBRA Each form of generalization has played an important part in the history of mathematical discovery. Bach has its place in school instruction. They have, however, very unequal value. While a mathematical truth may for a time be ac- cepted and used — as Newton accepted and used the binomial theorem for a fractional index — on the guarantee of a number of instances in which it is known to hold good, yet its position is not felt to be satisfactory until it has been placed on the firmer foundations of analysis. On the other hand, it must not be forgotten that the power to reach by analysis a com- plete guarantee of a mathematical truth is subject to the weakness which limits all human powers. Thus Pythagoras — probably as the result of the examination of a number of special instances — first enunciated the proposition that the sum of the squares on the sides of a right-angled triangle is equal to the square on the hypothenuse. Euclid's " de- monstration " was an attempt to guarantee the truth of the proposition by analysis. Apparently the analysis seemed to Euclid complete, and satisfied geometers down to modern times. But it involves the proposition about congruent tri- angles known to our youth as Prop. 4, and this truth is not really guaranteed by Euclid's analysis of it. A recent brilliant critic, generalizing illegitimately, like Macaulay, from the single instance of his own amazing precocity, says that Euclid's proof of Prop. 4 " strikes every intelligent child as a juggle ". Euclid's proof of the theorem of Pythagoras thus offers an instance of a piece of mathematical analysis, long thought to be adequate, which the modern critical sense requires to be corrected and made more complete. Similar incidents are constantly occurring in various parts of mathe- matical theory. They show us that mathematical reasonings do not necessarily issue from the brains of mathematicians in full-blown perfection, like Athene from the head of Zeus. For them, as for all other human productions, perfection is a goal to be reached, if at all, only after a long process of puri- fication by criticism and reconstruction. § 3. Direct Use of Symbolism. — For the task of algebra as described in the foregoing section only two tools are, in up the question raised above in James's Principles of Psychology, i., ch. XII., and Prof. Bosanquet's paper in the Proceedings of the Aristotelian Society, 1910-11. THE NATURE OF ALGEBRA 5 principle, necessary : the power (which every mind possesses in some measure) of discerning the abstract essential process in the concrete arithmetical case, and a sufficient command of language to express it when discerned. But in practice something more is needed. Neither the analysis of arith- metical procedure nor the expression of the results of such analysis can proceed very far without the help of symbolism. Thus it is difficult to suppose that the generalization commonly expressed in the form {x + af = X'' + nx''-^ a + -^^^^ — -^■"~'^ot?' + ... + a" could ever have been reached without the aid of symbols. Moreover its expression in language would not only be in- tolerably prolix but almost useless for practical purposes ; only the conciseness of the symbolic expression makes it possible for the student either to grasp the generalization or to apply it. The development of a symbolism with the pro- perties of making analysis easier and the expression of its results more concise and available is, then, the second funda- mental element in algebra. This function of symbolism has an importance that reaches much beyond the field of school mathematics. " The ideal of mathematics," writes Dr. Whitehead, ^ " should be to erect a calculus to facilitate reasoning in connexion with every pro- vince of thought, or of external experience, in which the succession of thoughts or of events can be definitely ascer- tained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a cal- culus." The definition of algebra suggested by this passage is, of course, much too wide for our present purpose, but it brings out a point of great importance. The algebra with which we are all familiar is only one of an indefinite number of possible algebras. Wherever there is a field for inquiry of a certain type an algebra may be invented to facilitate that inquiry. The field of common algebra is that of numbers and their relations, but the school curriculum itself yields an example of an algebra concerned with an entirely different region. For in chemical formulae and " equations " we have a system ^ Universal Algebra, p. viii. 6 ALGEBRA of symbolism expressly designed to facilitate the expression and investigation of truths in the field of chemical composition and reaction. Optimistic philosophers have even hoped to devise algebras that should impart to the treacherous ground of theological and political controversy the solidity and cer- tainty of mathematical inquiry. An invitation from one theological algebraist to another to "sit down and calculate " would then take the place of the denunciations of rival champions inspired with odium theologicum.^ It is important to note that the aim of all these algebras is the same : namely to correct the weaknesses and supple- ment the deficiencies of language as an instrument of abstract investigation and exact statement. Words and phrases as the vehicles of ideas are replaced by symbols — with a con- sequent gain in clearness and conciseness. A formula, consisting of an arrangement of symbols, is free from the ambiguity which often besets the arrangement of verbal units into a sentence, and is, besides, a more effective vehicle of a complicated meaning. It is easier to move forward in an argument when the steps can be expressed in symbolic form, and it is easier to check the correctness of the movements afterwards. The practical deduction from this general theory of algebra is that in ordinary algebra the symbols are to be thought of as substitutes, not immediately for numbers, but for words as the vehicles of general ideas. The res, as and 6s of a common algebraic statement refer to numbers only in the same way as the verbal forms which they replace ; they do not stand for numbers. To speak of a symbol as a " general- ized number "is to employ a phrase which (with all deference to the great algebraists who have used it) has no clear mean- ing and is incompatible with modern logical ideas. In this book, then, it will be assumed that such symbolisms as (a + h)'^ = a^ + 2ab + b^, V = 7rr% (for the volume of a cylinder), y = ax^ + bx + c, may always be regarded as verbal statements about numbers expressed for a special purpose in a conventional form, the letters and the graphic symbols being immediately substitutes for words and " stand- ing " for numbers only in the same sense as the verbal units 1 See the account of Leibniz' Universal Language in Venn's Empirical Logic, ch. xxii. THE NATURE OF ALGEBRA 7 to which they correspond stand for them. This practice will be found to remove much of the difficulty which the beginner has in understanding what algebra is " all about ". He can- not easily see how a can mean a particular number without meaning either this particular number or that one ; but he at once appreciates the sport of expressing mathematical state- ments in a new kind of " shorthand ". It is quite true that the miracle of " ambiguous reference " is present in the words " add any two numbers together " in exactly the same way as in the symbolism "a + b". But it is so familiar a thing in the former case that nobody but a philosopher finds anything mysterious in it ; in the latter case the novelty of the ex- pression brings the strangeness of the fact to our notice. This point is so important as to be worth repeating in another form. Every one knows that mathematics is essenti- ally concerned with "variables". For instance, in the formula V = Ah, which gives the rule for the volume of a cylindrical solid, or in the " equation of the ellipse," x'^/a^ + y'^jb'^ = 1, the symbols all represent variables. That is, while the numerical connexion expressed by the symbolism V = Ah holds good only between particular volumes, areas and heights, the formula refers ambiguously to any set of the volumes, areas and heights which could exist in combina- tion. What is not generally noticed is that variables are almost as common outside mathematics as within.^ Thus in the statement " The King of England is a constitutional monarch " the element " the King of England " is a variable in exactly the same sense as V in the formula Y = Ah. The sole difference is that while V refers ambiguously to one of an indefinite collection of number?, " the King of England " refers ambiguously to one of an indefinite collection of per- sons. " Edward VII " and " George V " are particular values which may be " substituted " for the latter variable, just as the number of cubic centimetres of an actual cylinder may be substituted for the former. The invention of variables was, perhaps, the most import- ant event in human evolution. The command of their use remains the most significant achievement in the history of the individual human being. Ordinary algebra simply carries to a higher stage of usefulness in a special field the ^ See Russell, Principles of Mathematics (references in index). 8 ALGEBRA device which common language employs over the w^hole range of discourse. The prudent teacher will, therefore, in the interests of clear understanding and economy of effort, present the technical use of variables in mathematics not as a new thing but as merely a modification of linguistic uses which the pupil mastered, in principle, at his mother's knee. The use of the sign " = "is sometimes thought to be an obstacle to the view here expounded. How can symbols be anything but numbers, it is argued, when they are connected by a sign of equality? The objection would have more weight if there were not other algebras in which the sign " = " is used to connect symbols which are certainly not numbers : for example, M^ + H2O = M^O + H2. Con- sideration shows two common elements in the meaning of the sign " = "in all algebras, nB,me\y identity and equivalence. For example, in the chemical equation just quoted, the sign '* = " implies that the " matter " referred to by the symbolism M^ + H2O and M^O + H^ is identically the same matter manifesting itself in two different forms. The two sides of the equation are therefore equivalent in the sense that one gives us in a different shape all the matter that is contained in the other. Similarly in {a + b)'^ = a'^ + 2ab + 6^ or in y = ax'^ + bx + c Y7e have the ideas, first, that the two sides of the equation have reference to identically the same number and, second, that this number can be regarded in two equivalent ways. Thus a chain of symbolical expressions each linked to its predecessor by the sign " = " represents a series of mental occurrences that can be likened to the " trans- formations of energy " that take place when (say) the ex- plosion of gas in an engine cylinder makes the fly-wheel turn round and so generates in a dynamo an electric current which in turn makes a lamp filament become incandescent. In both cases we have the thought of something which remains fundamentally identical in amount but is made to assume a number of different forms. This is the reason why such a chain of statements as 3-1-5 = 84-3 = 11 + 7 = 18 . . . is inadmissible : it implies the introduction of new matter where there ought to be identity of matter throughout. The practical conclusion from this discussion is that, in ordinary algebra, while the sign " = " always implies the presence of numerical identity beneath changes of form, yet THE NATURE OF ALGEBRA 9 it may appear in different contexts with varying force. Thus in the chain of statements d , ' V T . sin (x + h) - sin x — (sma?) = Lit ^^ ^ ax . h sm^ = L^ cos (x + o) * = cos X sin| h 2 the sign has an obviously different meaning each time it occurs. In the first line it may be regarded as heralding either a definition or a practical rule, in the second it asserts the equivalence of two symbolic expressions, in the third it claims validity for a certain regrouping of the symbols, in the last its force is best given by a simple " is ". We may read it ''equals" in each case, but it is important, especially in the earlier stages of the subject, to make the pupil realize the varying colour of that convenient word. The history of algebra throws interesting light on the origin and functions of symbolism. After 230 years it may still be read in the pages of John Wallis as profitably as any- where. Wallis points out that Vieta (about 1590) greatly improved -algebra by first denoting known numbers as well as unknown by "Marks or Notes," and by exercising "all the Operations of Arithmetick in such Notes and Marks as were before exercised in the common Numerical Figures " {Algebra, ch. xiv.). Suppose a problem to concern "any three numbers ". Then the various numbers in so far as they may be chosen to be the first, the second or the third of the " any three numbers " may be regarded as forming three inde- finitely numerous classes. In denoting any member of the first, second or third class by the symbol A, B or C, Vieta followed (says WaUis) the custom of lawyers who "put cases in the name of John an-Oaks and John a-Stiles or John a-Down, and the like, (by which names they mean any person indefinitely, who may be so concerned;) and of later times (for brevity 10 ALGEBRA sake) of J. O. and J. S. or J. D. ; (or yet more shortly) of A, B, C, etc." This practice is (Uke Euclid's use of letters in referring to lines and angles) to save the labour of describing the members of each class ** by long periphrases or tedious De- scriptions ". In chapter xv. Wallis shows how the English Oughtred {Clavis Mathematicae, 1631) " who affected brevity, and to deliver what he taught as briefly as might be, and reduce all to a short view," carried Vieta's " improvement " still farther. " Thus what Vieta would have written A Quadrate, into B Cube _ , _,^ , OPE s^ii^ Equal to FG plane, would with him be thus expressed ODE ~ A2B3 From this symbolism to the modern pryps = EG is but a short though a very important step.^ § 4. Extended Use of Symbolism. — A third distinct element appears in algebra when a piece of symbolism originally in- vented to express a simple arithmetical operation is found so. convenient that our definition of the operation is deliberately changed in order to bring other less simple operations within the scope of the same symbolism. Here again is a feature of such importance that some writers have regarded it as the distinctive characteristic of the science. It is, perhaps, more illuminating to think of it as simply a further development of the power of symbolism to bring effici- ency and the " short view ". It may be illustrated by continu- ing a little further our observation of the small boy who was studying areas in § 2. We left him at the point where he had found that the area of a rectangle with integral dimensions is obtained by multiplying the length by the breadth. Let us follow him in his investigation of rectangles whose dimensions are fractional. There is, to begin with, no difficulty in a rectangle of frac- tional length but integral breadth. For if the length is, say, 6 J ^Prof. U. G. Mitchell has contributed a useful review of the growth of algebraic symbolism to Young's Fundamental Concepts of Algebra and Geometry, p. 22b. THE NATURE OF ALGEBRA 11 inches and the breadth 4 inches the rectangle can be divided into 4 inch-strips, each containing 6 J square inches. The area will, therefore, be found by taking four times 6J — an operation which comes under the formula " multiply length by breadth ". But if the breadth is also fractional — say 4:^ inches — the matter is not so simple. The area cannot be divided into an exact number of inch-strips, and cannot, therefore, be calculated by multiplication — as multiplication has hitherto been understood. We are driven to the more complicated calculation : — area = 6J x 4 + | of 6^ Now if the breadth were 5 inches, the calculation of the area would again come under the simple formula " multiply length and breadth together ". The formula holds good then, for integral breadths, 3 inches, 4 inches, 5 inches, etc., but not for intermediate breadths, 3 J inches, 4^ inches, 5f inches, etc. But this is a very unsatisfactory state of affairs. If some of these areas are determined by multiplication it seems reason- able to regard them all as determined by that process. This view is possible if we agree so to extend our notion of multi- plication that an operation of the type 6;^ x 4 4- -^^ ^^ ^i shall be defined as " multiplying 6 J by 4^ ". By this device the useful formula, area = length x breadth, is made to include every case in which the lengths of the sides of a rectangle can be expressed in integral or fractional numbers. The introduction of negative and fractional indices is another typical instance of this influence of a useful symbolism upon the original definition of an arithmetical process. Thus, if a sum of money, say £250, increases at 3 per cent com- pound interest for 5 years its arnount is given by the cal- culation £250 X (1-03)^ — the symbolism implying (by defini- tion) 5 successive multiplications by 1*03. Similarly, the amount at the end of 6 years is obtained by 6 successive multiplications, symbolized by £250 x (1*03)''. Now we must not say that the amount after 5-^ years is found by 6^ successive multiplications, for the statement would be mean- ingless. On the other hand it seems unreasonable that the operation needed in the last case should not be regarded as the same operation and be denoted by the same symbolism as in the other cases. If we yield to this argument we shall agree so to change our original definition of the symbolism 12 ALGEBRA £250 X (l-03)-^ that the operation (whatever it may be) which determines the amount after 5 "5 years shall be capable of being described by the symbolism £250 x (l-03)^*^ In this way all problems of this type can be brought under the single formula Amount = Pr" where P is the principal, r the amount of £1 after 1 year and n the time in years, integral or fractional. The use of negative numbers is, of course, the capital in- stance in elementary algebra of the characteristic considered in this article. § 5. Manipulation of Symbols. — Wallis signalized two great merits in Vieta's use of " species " or symbols. First, that they made general arithmetical statements much more con- cise ; second, that the symbols could be "exercised" like numerals, in all the ordinary operations of arithmetic. The second of these properties is the fourth fundamental character- istic of algebra. It has two chief uses. The first is exemplified by " identities" such as (a + b) {a - b) = a^ - b^. It is profit- able to examine carefully the meaning of this piece of symbolism. It is, we have seen (§ 3), to be regarded as a "shorthand" transcription of a verbal statement: "If any two numbers be selected the product of their sum and differ- ence will be the same as the difference between their squares ". It can easily be seen how this result comes about in any specific instance. Consider, for example, the product 15 x 3, expressed as (9 + 6) (9 - 6). Simple arithmetical considera- tions show that the product can be written in the form (9 4- 6) (9 - 6) = 92 + 6 X 9 - 9 X 6 - 62 = 92-62 Now this analysis of the way in which the result comes about shows clearly that it is quite independent of the fact that we selected the numbers 9 and 6 for our experiment. The process of multiplication would have taken the same typical course if we had chosen any other numbers. We can, that is, describe the process in words without using the names of any particular numbers, and we can reduce our statement THE NATURE OF ALGEBRA 13 to a '' short view " by expressing it in " species ". We then have (a + b) (a - b) = a^ + ba - ab - b^ = a^ - ¥ But when we have gone through this process of analysing a particular case and expressing its permanent or essential features by the aid of a symbolic statement we may notice that our work might have been much abbreviated. The symbolism chosen is of such a character that, line by line, it imitates the arrangement of the figures in the arithmetical process. It follows that there was really no necessity to go through the arithmetical process first and afterwards to de- scribe its essential features in symbols. By manipulating the letters as if they were figures we could with perfect certainty have predicted the typical form of the arithmetical result. The " simplification of an algebraic fraction " may be taken as another example of the same property. Suppose that a quantity t is such that 7 ^ Z "^ 7- Then we have t 4 7 7 + 4 whence 4x7 4x7 7 + 4 There are certain features of this process which are obviously independent of the particular numbers which here enter into it. It is possible, therefore, by the employment of " species '' to give a generalized account of its steps. Putting a and b for the verbal units ** the first denominator," " the second de- nominator," the three stages of the process can be analysed as follows : — 1 = 1 + 1 tab b + a ~~ ab ab t = , 14 ALGEBRA But it is obvious that here, as in the case previously examined, the consideration of an arithmetical example is really unnecessary. By treating the letters as if they were figures we might have passed at once and with confidence from the first symbolic statement to the third. That is, we might be sure that if the typical form of an expression for the reciprocal of a number is 1/a + 1/6, the typical form of the number itself is ahjih + a). It should now be easy to understand why "species" are so often taken to be numbers. For on the one hand the symbolic statement about a numerical relation imitates exactly the arrangement of a particular numerical instance of that relation — as, for example, — \- - imitates - + ~ ah 4 7 and, on the other hand, the general features of the result of transforming a given arithmetical expression can always be predicted by imitating the steps of the transformation with the letters of the corresponding symbolic expression. ^ But while it would be pedantic never to speak of "adding" or " multiplying" symbolic expressions, yet it should be made clear to the pupil that these are merely convenient forms of speech; that, strictly speaking, we can add and multiply nothing except numbers ; and that the operations which we carry out upon symbols are merely a means of determining, without considering a specific example, the general or typical features of the result of manipulating an arithmetical ex- pression in a given way. We may summarize this discussion by saying that from one statement expressed in algebraic symbols it is possible to deduce, merely by manipulating the symbols in accordance with the laws of arithmetic, an indefinite number of state- ments each of which will be true if the original statement is true. Thus if a number can truly be expressed in the form (a + h) (a - h) it is certain that it can be expressed with equal truth in the form a^ - 6^. But by far the most striking instances of this property consist in the transformations which are used in the solutions of equations. Suppose, for example, that I am confronted with the formula ^ These remarks also make it clear why the Greeks, in the absence of a convenient arithmetical notation, were unable to develop algebra to any considerable extent. THE NATURE OF ALGEBRA 15 C = nr + R taken from an electrician's note-book. Merely by treating the letters as if they were figures I can obtain from this statement a number of other symbolic statements, such as r = E/C - B/n R = w (E/C - r) Now the important thing here is that I may be so grossly ignorant of electricity that the original formula is meaning- less to me. Yet I shall be quite certain that if the original formula was valid the formulae I have derived from it are equally valid. In this way it is possible for me to discover electrical facts of which my friend the electrician (who though an excellent practical man is, perhaps, but an indifferent algebraist) was actually unaware.^ In such a way, too, given a symbolic statement of one property of a curve and a few geometrical ideas, it is possible to deduce statements of an endless series of other properties of the curve — properties which may in this way become known for the first time. Point is given to earlier arguments of the chapter by the observation that the property of yielding new truths as a result of a merely mechanical rearrangement of symbols is not confined to the algebra of number. It is to be found in some form wherever a system of symbolism has been developed " to facilitate reasoning " in a particular province of thought. The late Prof. Jevons actually invented a " logical machine " in which the exploration of the field of truth could be carried out by pulling levers and turning handles. It would probably not be impossible, if only it were worth while, to construct an " algebra machine " which could in a similar way be made to yield from a given formula other formulae which follow from it. ^ Of course it will require his practical knowledge to give mean- ing to my discoveries. CHAPTEE II. METHOD AND CURRICULUM. § 1. Our Aim in Teaching Mathematics. — Clear notions about the nature of algebra are essential to a profitable dis- cussion of teaching problems but do not themselves constitute solutions of such problems. The purposes to be held in view in teaching algebra to boys and girls, the proper selection of topics, the best methods of presenting them : these questions constitute a distinct subject of inquiry and are to be settled by considerations of v^hich those adduced in the previous chapter form only a part. For a full discussion of such matters the reader must look elsewhere than in a practical handbook ; but he may be asked to accept the following brief statement of general principles as the author's confession of faith. Mathematical truths always have two sides or aspects. With the one they face and have contact with the world of outer realities lying in time and space. With the other they face and have relations with one another. Thus the fact that equiangular triangles have proportional sides enables me to determine by drawing or by calculation the height of an unscaleable mountain peak twenty miles away. This is the first or outer aspect of that particular mathematical truth. On the other hand I can deduce the truth itself with complete certainty from the assumed properties of congruent triangles. This is its second or inner aspect. The history of mathe- matics is a tale of ever-widening development on both these sides. From its dim beginnings by the Euphrates and the Nile mathematics has been on the one hand a means by which man has constantly increased his understanding of his environment and his power of manipulating it, and on the other hand a body of pure ideas, slowly growing and con- solidating into a noble rational structure. Progress has brought about, and, indeed, has required, division of labour. 16 METHOD AND CURRICULUM 17 A Lagrange or a Clerk Maxwell is chiefly concerned to enlarge the outer dominion of mathematics over matter ; a Gauss or a Cantor seeks rather to perfect and extend the inner realm of order among mathematical ideas themselves. But these different currents of progress must not be thought of as independent streams. One never has existed and probably never will exist apart from the other. The view that they represent wholly distinct forms of intellectual activity is partial, unhistorical, and unphilosophical. A more serious charge against it is that it has produced an infinite amount of harm in the teaching of mathematics. Our purpose in teaching mathematics in school should be to enable the pupil to realize, at least in an elementary way, this two-fold significance of mathematical progress. A person, to be really "_educated," should have been taught the import- ance of mathematics as an instrument of material conquests and of social organization, and should be able to appreciate the" value and significance of an ordered system of mathe- matical ideas. There is no need to add that mathematical instruction should also aim ai ^Ijdis ciplini ng his mind" or giving him "mental training ". So far as t^e idealFintended by these phrases are sound they are comprehended in the wider purpose already stated. Nor should we add a clause to safeguard the interests of those who are to enter the mathematical professions. The treatment of the subject prescribed by our principle is precisely the one which best supplies their special needs. § 2. The Choice of Curriculum. — This principle adopted, we have the practical problem of selecting for our curriculum in algebra topics which shall illustrate adequately the two aspects of mathematics. To the question whether the two must be provided for separately we may reply with a confi- dent negative. The theoretical questions which are of most importance in an elementary course are just those which arise naturally out of attempts to apply mathematical ideaa and methods to practical purposes. We shall, therefore, choose those subjects in which the practical value of algebra is- most clearly exhibited, confident that this is the best means of securing opportunities for fruitful theoretical discussions. In accordance with this view algebra should be introduced to the young pupil as a symbolic language specially adapted for making concise statements of a numerical kind about T, 2 18 ALGEBRA matters with which he is already more or less familiar. In other words, the earliest lessons in the subject should teach the use of the formula, illustrations being drawn largely from the "Pocket Book" of the engineer and similar formularies. In these lessons the various algebraic notations should be in- troduced as symbolic idioms needed for the transcription of pieces of important or interesting fact. A little later comes the study of those manipulations of a formula by which it may be made to yield truths unknown or unperceived before (ch. I., § 5). It is of the first importance that from the out- set of his work the pupil should be made to perceive clearly and feel constantly that both formulae and manipulations always refer to realities beyond themselves. It is scarcely too much to say that incompetence in, and distaste for, mathe- matics nearly always spring from the neglect of this funda- mental teaching principle. Even in the case of those who have a natural fondness for the technique of mathematics the same neglect often leads to an astonishing blindness to the real significance of mathematical ideas and operations. The gradual elaboration of the formula as an instrument of description and investigation is, then, the first business of the course in algebra. The pursuit of this task leads naturally to two kinds of inquiries both of which belong to the inner aspect of mathematical thought. In the first place, the attempt to build up an effective system of symbolic ex- pression leads to a number of problems of great importance from the theoretical point of view. These necessarily include questions about the meaning and use of negative numbers, the interpretation of fractional indices, etc., and may range as far as the fascinating inquiries into the ultimate nature of num- bers and numerical operations which are so characteristic of modern mathematics. In the second place, in trying to give an account from the numerical standpoint of the concrete things with which his formulae deal, the young algebraist can hardly fail to notice and to become interested in the fact that "variables" of widely differing character are yet often bound to one another by identical quantitative laws. From that moment onwards it is natural to give an increasing amount of attention to these general forms of connexion be- tween variables. Eventually — under the rather forbidding name of " functions " — they may become the main object of study. METHOD AND CURRICULUM 19 § 3. The Position of Trigonometry and the Calculus. — The foregoing principles point directly to two important modifica- tions in present teaching practice. The first concerns the position of trigonometry, the second that of the differential and integral calculus. At present it is usual to teach both of these as " subjects," distinct from algebra and distinct from one another. The reason is easily found. The professional mathematician, interested chiefly in perfecting the technique of his subject, finds it natural as well as most effective to take a special group of allied methods or allied problems and to develop them as far as he can without concerning himself too greatly about the practical value of his work. Now this systematic exploration of special parts of mathematics is, no doubt, of vital importance for the continued growth of the science. It does not, however, by any means follow that the branches of mathematics should be presented to beginners with the formal elaboration which is the inevitable mark of their treatment as separate subjects. The absurdity to which this practice may lead is sufficiently illustrated by the custom of one very important public examination in which, until re- cently, a paper was set requiring considerable technical know- ledge of the differential calculus while assuming no knowledge at all of integration. The mischief which follows from it is still more evident from the fact — surely little less than a scandal — that so far as the operation of University Leaving Certificates is concerned a boy or a girl may at present [1913] pass through a secondary school without making the smallest acquaintance with the fascinating and powerful methods of elementary trigonometry. If we admit that the custom of teaching trigonometry and the calculus as distinct subjects has no defensible founda- tion and is responsible for very unsatisfactory consequences, we shall be prepared to inquire what position is assigned to them by the principles laid down in this chapter. Taking first the case of trigonometry, it becomes at once evident that the pupil's acquaintance with the tangent, sine and cosine should begin in the region where arithmetic marches with elementary geometry. As soon as the symbols of the trigono- metrical ratios are recognized as capable of entering into formulae and of being manipulated they should be regarded as belonging to the vocabulary of algebra. There is, indeed, no principle, except the invalid principle of formal segrega- 2* 20 ALGEBRA tion, upon which we can include the study of a;" or of a^ in the algebra course and exclude sin x or tan x. All alike are pieces of symbolism invented for the description and interpretation of facts of the external world. Each re- presents a typical kind of " function ". To each corresponds a specific form of curve which may be regarded as the graphic symbol of the function. Both algebra and trigono- metry would gain by fusion : the former through an added variety and richness in the illustrations of its main themes ; the latter by the removal of the excessive formalism which at present obscures its value and interest for the begin- ner. Fusion upon these lines is attempted in this book. The formal work which constitutes so large a part of the ordinary treatment of trigonometry is greatly reduced and much more stress is laid in Part I upon the practical, and in Part II upon the functional, aspect of the subject. Among other advantages this plan permits the inclusion of a brief section upon the trigonometry of the spherical surface. The customary neglect of a branch of inquiry which is of such importance and interest, in which also the essential results are to be obtained with such ease, is, of course, only another glaring instance of the mischief brought about by the over- formal treatment of mathematics in school. There is ab- solutely no reason why the main elements of the trigonometry of the sphere should not be taught except the ridiculous one that spherical trigonometry is a separate " subject " which can be studied only after " plane trigonometry " has been disposed of. In arguing for the assimilation of trigonometry in the algebra course we are arguing against the artificial separation between problems which must in any case be attacked in a similar spirit and by similar weapons. When we consider the position of the differential and integral calculus we have to protest against a tradition which forbids all but exceptional pupils to become acquainted with the most powerful and attractive of mathematical methods. As in the former case, the mischief is the result of a technical elaboration which, though essential to the historical development of the calculus, has had the effect of making the really simple ideas upon which it is built inaccessible to the ordinary boy or girl at school. In this instance the history of the subject suggests a remedy for a state of things which is generally regarded METHOD AND CURRICULUM 21 as unsatisfactory. The calculus began, in the writings of John Wallis and others, merely as a special kind of alge- braic argument which might be introduced at any appropriate point and without the apparatus of a technical notation. The remedy suggested by this observation is adopted in the present work. Notions which form part of the doctrine of the calculus are introduced at an early stage and are developed side by side with other algebraic ideas ; but only towards the end of the work are the technical symbols introduced which have been known to so many students only as hostile standards floating above an impregnable citadel. § 4. Some Practical Suggestions. — It is probable that enough has been said to indicate the point of view adopted in this book, and there would be nothing to gain by anticipating here what will be found in detail in later chapters. The whole course treated in these chapters is divided into two main stages upon a principle already explained in the Preface. These stages are represented respectively by Parts I and II of the Exercises in Algebra. The ground assigned to the two stages, their subdivisions, and the order of treatment recom- mended are dealt with in ch. v. as far as concerns Part I, and in ch. xxxviii. as far as concerns Part II. The reader who wishes to gain a complete view of the whole course may with advantage turn immediately from this chapter to those. Meanwhile it may be convenient to give some account of the plan of the book and to suggest how it should be used. The present or Introductory Section contains two more chapters. These deal in some detail with the two instruments of expression which the student of algebra has to use at every stage of his progress — namely, the formula and the graph. The conclusions reached are applicable all through the course, but they have special reference to the first be- ginnings of algebra. Thus they are illustrated largely by re- ferences to Exercises I and II, which are intended to be regarded as preliminary work to be covered before the systematic study of the subject is begun. The rest of the book — setting aside the general introduc- tions to Part I and Part II — is divided into sections which correspond to the ten sections of the exercises. In Part I each of these divisions contains a chapter or chapters in which the general ideas underlying the exposition of the section are explained, and hints are given for the illustration 22 ALGEBRA of the teacher's lessons, etc. These chapters also offer remarks upon the examples of each exercise. Matters which require emphasis are pointed out and the solutions recom- mended for typical examples are given. The remaining chapters indicate the substance of the lessons which the teacher should give before setting his class to work upon the various exercises. He should understand that these notes do not always contain a complete discussion of the topic with which they deal. They aim merely at carrying the discussion up to the point at which the pupil can profitably attack the exercise. In many cases the subject is developed farther in the course of the exercise itself. For this reason alone it is important that the teacher should take occasion, after an exercise has been disposed of, to summarize afresh what has been learnt both from the preliminary lesson and from the subsequent examples. It may be added that sometimes the examples, instead of developing farther a topic which has re- ceived preliminary discussion in class, prepare the way for a subsequent lesson. In such cases warning is always given that certain examples are specially important, and what is to be learnt from them is summarized in the chapter devoted to the following lesson. Part II is written upon a similar plan, with the important difference that there are no systematic suggestions for lessons preliminary to the exercises. These are unnecessary in view of the detailed discussions which accompany the exercises. With regard to the use of the exercises many points must be left to the teacher's own discretion and experience. For instance, it is hardly ever desirable that the whole class shall at the first attack work through all the examples of an exercise. Certain examples must be taken because, as we have said, the theory of the subject is developed from them, while others are essential as a preparation for instruction to come later. The teacher must use his judgment as to the number of examples, outside these, to be exacted from every pupil. Some may be left for subsequent homework, some for re- vision in class, some (especially in the harder sections of the exercises) for the cleverer and quicker boys or girls. The teacher should bear in mind that it is often a good thing to run away from a difficulty in order to fight it another day ; that of many a difficulty, especially in mathe- matics, it may be said, solvitur ambulando ; and that a class METHOD AND CURRICULUM 23 bored by long-continued study of a single topic is making its minimum rate of progress and gaining the minimum profit from its labours. To these remarks the author may, perhaps, be permitted to add for the benefit of his younger colleagues that it pays always to exact from a boy or girl the best work of which he or she is capable. The beginner, recognizing that his real business is to keep his pupil's mind active, is apt to underestimate the closeness of the connexion between orderly ways of thinking and writing, and to be imposed upon by specious little rogues who have developed a precocious talent for concealing lazi- ness. He should be on his guard against this weakness, and should constantly check any tendency to accept careless or untidy work merely because it shows intelligence. When he has acquired the diagnostic powers which only experience can give he may trust his ability to determine the cases in which he may safely relax. Even then these cases should be rare. References for Reading. The present author has dealt more fully with some of the points raised in chs. i. and ii. in the following papers : — "On the Method of School Algebra." *' School," Sept. 1905. (John Murray, 6d.) "The Arithmetic of Infinites." "Mathematical Gazette," Dec. 1910 and Jan. 1911. (Bell & Co., Is. 6d. each.) "The Aim and Methods of School Algebra." "Mathematical Gazette," Dec. 1911 and Jan. 1912. The following are among the most important and acces- sible books upon the logic, pedagogy and history of mathe- matics. Some of them contain bibliographies : — A. N. Whitehead, "Introduction to Mathematics". (Home Uni- versity Library, Williams & Norgate, Is.) P. E. B. Jourdain, "The Nature of Mathematics ". (The People's Books, T. C. & E. C. Jack, 6d.) J. W. A. Young, "Lectures on Fundamental Concepts of Algebra and Geometry". (The Macmillan Co., 7s.) An admirable review of the subject, intended specifically for teachers. Bertrand Russell, " The Principles of Mathematics ". (Cambridge Univ. Press, 25s.) A book of the highest originality, impor- tance and authority. 24 ALGEBRA B. Branford, *' A Study of Mathematical Education". (Clarendon Press, 4s. 6d.) The most important and original of recent English contributions to the pedagogy of mathematics. D. E. Smith, ''Teaching of Elementary Mathematics". (Mac- millan Co., 4s. 6d. net.) J. W. A. Young, "Teaching of Mathematics in the Elementary and Secondary School ". (Longmans^ 6s.) The last two are excellent American textbooks. G. St. L. Carson, "Essays on Mathematical Education". (Ginn & Co., 3s. net.) J. Perry, "Report of a Discussion on the Teaching of Mathe- matics," British Association, 1901. (Macmillan & Co., Is.) A Report which has had a great influence upon the reform of mathematical teaching in England. A. Holier, "Didaktik und Methodik des Rechnen und der Mathe- matik". (Leipzig, Teubner, 12s.) W. W. Rouse Ball, "Short Account of the History of Mathe- matics ". (Macmillan & Co., 10s.) F. Cajori, "A History of Elementary Mathematics". (The Macmillan Co., 6s. 6d. net.) J. Tropfke, " Geschichte der Elementar-Mathematik, " 2 vols. (Leipzig, Veit & Co., 9s. each.) CHAPTEE III. THE FORMULA. -Algebra regarded as " general- ized arithmetic " should have no formal beginning. As soon as the child who sees the teacher write upon the blackboard area = length x breadth can translate this into the words: "To find the area of the floor I must multiply its length by its breadth," ^ he has, without knowing it, already begun his study of the subject. What the teacher has set before him has the two character- istics of a "formula": {a) it is a statement of a general " rule " applicable to any one of a definite class of problems ; and (6) the statement is expressed in a conventional form chosen for its properties of conciseness and ready compre- hensibility (ch. i., §§ 2, 3). By his twelfth year lessons in arithmetic and science should have afforded the pupil abundant opportunity of learning to write down and use simple formulae of this kind. Formulae must, of course, never be used in arithmetic un- less the pupil clearly understands the processes which they prescribe. In other words he is entitled to use a formula only if it represents genuine results of his own thinking. He may then with advantage write it at the head of his calcula- tion as a memorandum of the process which he intends to employ. Used in this way, the formula makes for greater clearness both of the pupil's thinking and of his written state- ments. These preliminary exercises will give occasion for the use ^ The purist may make him say : " the number which measures its length by the number which measures its breadth ". This greater scrupulosity of diction need not affect the formula. 25 26 ALGEBRA in verbal formulae of all the ordinary symbols of arithmetic : area e.g. , ,. , or area/length, (length)^, (length)^. The teacher must also seek in them his opportunity for introducing his pupils to the extremely important device of replacing the constituent phrases of the verbal formula by single letters. The use of words in an abbreviated form supplies a natural transition to the stage in which this practice is definitely adopted. § 2. Use of Literal Symbols. — The use of single letters can best be explained (and made attractive) by teaching the class to regard formulae as "shorthand" memoranda of the rules which they have established in the course of their work and are constantly needing. The principles of this " short- hand" are {a) to represent certain constantly recurring words (such as "multiply," "divide," "square") by conven- tional symbols or " grammalogues " ; and (6) to reduce other words or verbal expressions in the full statement of the rule to single letters, chosen so as to suggest those words or ex- pressions as readily as possible to the reader of the memo- randum. Thus " circumference of the circle " may be reduced to C, " rate of interest per cent per annum," to r, "the number of passengers " to either n or p, according to con- venience or the choice of the writer. Moreover, it will be in accordance with the notion that we are developing a " short- hand " to replace always by the same letters any words or verbal expressions which frequently occur in the problems in which our formulae are employed. Thus A can generally be taken without special explanation to mean " area," the parti- cular figure whose area is in question being known from the context. This method of procedure follows so obviously from the function of the formula as a labour-saving device, that it is, perhaps, unnecessary to point out how it accords with the practice of all persons who employ formulae for serious pur- poses. To the electrical engineer or the actuary it is a matter of no small importance that his text-books and formularies should employ consistently the same symbols, and that these should readily suggest the verbal units for which they stand. ^ 1 The Institute of Actuaries have published an official set of symbols which are used, without explanation, in all the papers and discussions of their members. THE FORMULA 27 When the verbal statement of the rule contains a numerical constant the practice must be taught of placing it before the literal symbols. Thus the rule that the volume of a pyra- midal solid is obtained by multiplying the area of the base by one-third of the height is, in accordance with this convention, to be written neither in the form V = AJ/i, nor in the form V = Ah^, but in the form V = ^Ah. The use of the symbol -n- appears to contradict the statement that symbols are to be taught as representing not numbers but words or verbal expressions. This is not really the case. There would be no point in using the symbol tt in a formula, rather than a concrete number, if it was not understood by the class that the ratio signified is one that in different calcula- tions is taken to have different values according to the degree of approximation required. Thus tt does not immediately represent a number but is the " shorthand " rendering of the phrase : " the ratio of the circumference of a circle to its diameter, taken to the degree of approximation which the problem requires ". § 3. Example of Method. — It is a matter of little import- ance at what precise point the class should be taught to adopt definitely the device of representing by a single letter a word or larger verbal unit. Whenever the step is taken the teacher will find it convenient to proceed much as in the following example. The topic under discussion is supposed to be the mode of calculating the volume of a solid of uniform cross- section, e.g. a cylinder. The teacher is provided with a cylindrical tin over the bottom of which he has pasted a piece of paper divided into centimetre squares. He proposes to the class the problem of finding the number of cubic centimetres in the content of the tin and conducts with them a colloquy to the following effect : — On counting up the number of square centimetres that cover the base of the tin (making due allowance for incom- plete squares) I find that there are (we will say) exactly 32. If now I placed in the tin a layer of clay 1 cm. thick and just large enough to cover the base I should evidently have a centi- metre cube of clay standing on each of the squares. There would be, therefore, 32 c.cm. in the layer altogether. If the tin is (say) 15 cm. high, I can pack 15 of such layers of clay on top of one another, so that the total amount of clay 28 ALGEBRA that the tin would hold must be 32 x 15 c.cm. Now if there had been on the base of the tin 17 or 148 or any other number of square centimetres and 82 or 2003 or any other number of linear centimetres in its height it is evident that I could calculate the amount of clay the tin would hold in exactly the same way. Thus we have the rule that the number of cubic centimetres which the tin would hold (the "volume" of the tin) is found by multiplying the number of square centimetres in the area of the base by the number of linear centimetres in the height — or, expressed more shortly, by multiplying the area of the base by the height. If the area of the base had been fractional — say 32-7 sq. cm. — each of the slabs of clay would have contained 32*7 c.cm. instead of 32 c.cm. If, moreover, the height of the tin had been 15-3 cm., instead of exactly 15 cm., then it is clear that I could have packed into the tin 15 slabs and a thinner slice of clay 3/10 of a centimetre thick, and therefore containing 3/10 of 32*7 c.cm. Thus the total volume of the clay would be given by the product 32-7 x 15*3. The rule that the volume of the tin is obtained by multiplying the area of the base by the height evidently holds good, then, when the area or the height is measured by a fractional number as well as when the measures are whole numbers. We may now proceed to write this rule upon the black- board. It is unnecessary to write every word in full, for you will have no difficulty in knowing what I mean if I shorten it down to the following : — vol. of cyl. = base x height or if I make it briefer still : — vol. of cyl. = base x ht. Now there are a great many persons who have constantly to make use of notes or memoranda of this kind. They are such people as engineers, who have to keep notes of all sorts of rules in regard to the weights which their materials will bear, etc., ship-builders, electricians, architects, military officers, sailors, etc., etc. Some of their rules are so compli- cated that their notes would be very cumbersome even if they shortened the words down as we have, and employed symbols such as " = " and " x ". They find it necessary to use, there- fore, a kind of shorthand in which they can express their memoranda much more briefly even than we have expressed THE FORMULA 29 the rule for finding the volume of the cylinder. The principle is to use one letter only to represent a word such as " height " or a phrase such as "area of the base". As far as possible letters are chosen which readily suggest the words for which the letters stand. Thus if we were to write our formula in this '' shorthand " way we could choose the letter V to stand for the words " volume of a cylinder," the letter B to stand for the words " area of the base " and the letter h to stand for the word "height". The rule would then take the ex- tremely short form : — V = B X /i But if we make up our minds never to use more than one letter to represent a word or group of words, the formula may be shorter still. We can agree to indicate that two numbers are to be multiplied together simply by writing the letters which are the shorthand descriptions of them side by side. Upon this plan our formula becomes : — V = B/t Such a formula is, remember, merely a shorthand way of writing down the sentence " The volume of the cylinder is obtained by multiplying the area of the base by the height ". The sign " = " can be read " is obtained by " (or equivalent words), while the word " multiply " is supplied by the fact that the letters are side by side. If it is necessary to make clear in your note-book what words the various letters stand for it is best to write as follows : — V ^ " volume of a cylinder " Thus the symbol " = " is to be read " is the symbol for the words ". From this point onwards the teacher should take occasion whenever a rule in connexion with mensuration, arithmetic, or elementary science has been formulated in words to dis- cuss with the class how to express it in " shorthand " form. In this way the class will, without receiving special lessons in algebra, acquire facility in handling the simpler forms of symbolical expression. Exercise I is intended to indicate the range of algebraic expression which should be covered in these incidental dis- cussions. It will, therefore, be most conveniently used as a 30 ALGEBRA means of revising and assuring the pupil's preliminary know- ledge before he begins, in Ex. Ill, the systematic study of the art of formulation. If the pupil has no knowledge of the uses of the formula, Ex. I must be worked through with considerable care. In either case it is of much importance that the method recommended in this chapter should be followed ; that is, that the beginner should be taught to regard a formula as nothing more than a " shorthand " transcription of a verbal rule or other statement. § 4. Setting Down of Work. — By writing his formulae at the head of arithmetical calculations the young student will also learn the rudiments of the art of "substitution". In connexion with this topic it is hardly possible to lay too much stress upon the importance of cultivating a neat and orderly way of setting down the steps in an arithmetical or algebraical argument. A piece of algebraic symbolism should be as capable of straightforward and continuous reading as a passage from a newspaper. To achieve this end the teacher will find it a sound rule never to permit a line to contain more than two expressions connected by the sign of equality, and to insist upon the pupil's setting the signs of equality, in successive lines of the argument, directly underneath one another. Thus such expressions as Y = BA, = 32-7 X 12-4 = 405-48 c.cm. should be excluded both from the exercise book and the black- board in favour of the arrangement : — ^ Y = Bh = 32-7 X 12-4 = 405-48 c.cm. § 5. No Manipulation of Symbols. — It should be noted that in this preliminary work no manipulations of the symbols will be taught and no question of the " sign " of the numbers that are substituted in the formulae will be raised. Both these matters belong distinctly to the formal study of algebra. ^ The need of economizing space unfortunately compels the author to break this rule in his book. The reader should attend to the precept and ignore the examples. CHAPTER IV. THE GRAPH. § 1. The Graph and the Formula. — Throughout algebra the graph, alone or in conjunction with the formula, plays an important part as an instrument of analysis and generaliza- tion. It is obvious that it shares many of the properties of the formula. Like the formula it can be used to bring out and express the *' law " or identity which underlies the diver- sity of a number of concrete numerical facts. Like the formula it delivers its message in a form readily taken in by the eye, and so ministers to the " short view ". It may often be regarded as a general statement from which, as by sub- stitution in a formula, an endless number of new particulars can be deduced. Lastly, it can in some cases be manipulated like a formula so as to yield new and unsuspected generaliza- tions. On the other hand, it is inferior to the symbolic formula in many important respects. Its accuracy depends largely upon mechanical or non-intellectual conditions, such as the skill of the draughtsman and the exactness of the squared paper. It is less compact and less easily reproduced. Its message is frequently inarticulate and obscure. For these and similar reasons it should be regarded as a subsidiary algebraic instrument which fulfils its best office when it either leads up to a formula by which it may itself be superseded, or serves to unfold more fully the implications of a formula whose properties have been only partially explored. As contrasted with the symbolic formula the usefulness and the limitations of the graph both rest upon the same circumstance — its relatively concrete character. Thus the curve which represents the relations between the values of two variables is abstract enough to be a means of concentrat- ing attention upon the law of connexion as distinguished from the variables connected, and is at the same time concrete 31 32 ALGEBRA enough to make a vivid appeal to intuition. On the other hand, though its concreteness makes it a more impressive form of statement than the formula, it also makes it a much less flexible instrument of investigation. In view of these characteristics it is not surprising to find that graphic methods were used as an effective instrument of mathematical thinking before algebraic symbolism had de- veloped beyond the rudiments. Thus the Greeks, who never succeeded in producing a satisfactory algebraic method, yet performed analytic feats of high importance with the aid of graphic forms. The propositions of Euclid's second, fifth and most of the later books exemplify this statement ; a simpler, but very striking illustration, is afforded by the argument in proof of the rule for extracting a square root which is given in ch. viii. This method (as the present author found after publishing it as his own !) was used, in practically the same form, by the Alexandrian astronomer, Ptolemy, about a.d. 120.1 The superior vividness and intuitability of graphic modes of expression suggests the conclusion that the young pupil should be taught their simpler uses before he makes acquaint- ance with the abstruser though more powerful instrument, the formula. The historical circumstances just adduced will seem to many to add support to this conclusion. It is true that it is opposed to the prevailing practice, but no one is likely to contest it who has observed the readiness with which a child will express himself in pictures long before he has learnt to command the more abstract medium of written words. Thus the gradual penetration of graphic methods into elementary instruction in mathematics and science may be welcomed as one of the most significant features of present pedagogical tendencies. § 2. First Lessons in Graphic Bepresentation (Ex. II, A.). — In accordance with the observations of the last article the second of the two preliminary exercises (Ex^ II) consists of a number of examples illustrating those uses of the graph which foreshadow the more effective use of the formula to be learned at a later stage. Like those of Ex. I these examples are best used as a means of revising and extending a know- ^ See the School World, Feb. 1911 ; also Heath, Archimedes^ p. Ixxvi. THE GRAPH 33 ledge of graphic representation which the pupil acquired before he began the systematic study of algebra. They sum- marize, therefore, a course of instruction which will be all the more useful if it has been spread over two or three years of school life. If the pupil has not been practised in graphic representation in his earlier lessons in arithmetic and geometry this exercise will require special attention. It is,|however, neither necessary nor desirable to work through all the examples before going on to Ex. III. A few of the examples in divisions A and B should be worked in class. The rest can be set from week to week as homework, or worked in class alternately with the first few examples of Ex. III. The only thing essential is that the main principles of graphic representation (i.e. those exemplified in divisions A and B) should be well understood before the class attacks Ex. IV. The best way to teach the graph is to let it grow out of the use of the picture. This principle is illustrated in the examples of Ex. II. Thus division A begins with graphs which are very little removed from pictures, and goes on to others in which the pictorial element is constantly less prominent, until, in the examples of division B, the pupil is prepared to dis- pense with it almost entirely. Nos. 1, 2 and 4 are exercises which form part of a course of lessons in Nature study given to children of about nine years old. (The author owes them to the kindness of his colleague, Miss C. von Wyss.) In No. I a base line is divided into equal parts and graduated to represent minutes. In the middle of each interval a perpendicular line is drawn of the same length as the path traced out by the snail during the corresponding minute. This length is to be determined by laying a piece of cotton thread along the line in the diagram. Eig. 1 1 shows the solution of No. 2. Vertical lines are drawn through the minute graduations of the base, and the perpen- diculars drawn in No. 1 are placed end to end across the spaces between them. The firm sloping lines represent in this way the movements of the snail which traced the path AB ; the broken lines represent those of his competitor. The variations in the average speed of the snails are, of course, ^ I.e. in this book. Figures in Exercises in Algebra, Part I and Part II, are numbered consecutively among themselves. A refer- ence to one of them will in this book always be prefaced by " Exercises " for Part I, and by " Exercises II" for Part II. T. 3 34 ALGEBRA Fig. 1. expressed by the variations in the slope of the lines. Thus the pupil acquires in his first graphical exercise the germ of the idea of measuring a rate of change by the " gradient " of a graph. On the whole, however, these two examples illustrate the lowest grade of usefulness of a graph. The diagrams add practically nothing to the information contained in the original figures ; they merely present it in a more effective and easily assimilable shape. From this point of view No. 4 repre- sents an important step forward, for it introduces the process of interpolation. Having graduated his base-line as before, the pupil sets up at the end of each seg- ment a vertical equal to the length of the tulip as shown in the corresponding drawing in Exercises, fig. 2. A vacant place must, of course, be left at the end of the third segment. The pupil is then to judge how long the tulip would have been found to be if this particular measure- ment had not been omitted. After (rather than before) he has expressed his judgment the curve through the tops of the ordinates is to be drawn, and it is to be made clear that, consciously or unconsciously, the course of this curve really determines the judgment. The significance of the process of interpolation consists in the draughtsman's analysis, out of the given lengths of the tulip, of the law of succession which they suggest. He may be quite unable to give a precise account of this law — he may even not understand what is meant by calling it a law — but the graphic presentation of the data forces it upon his atten- tion as an actual fact. Nos. 5 and 6, taken together, are meant to fortify the incipient notion of a law by contrast- ing a case where law exists with one from which it is absent. In No. 5 the " lawfulness " in the growth of the tulip leads the pupil to expect a corresponding " lawfulness " in its weekly increase. We may permit him at this stage to act upon the expectation, even though he cannot give an adequate defence of it. Thus when the growth during the fourth week is determined by interpolation he will expect it to be the same THE GRAPH 36 as the ditference between the length of the plant at the end of the third week and the length inserted by interpolation in No. 4. On the other hand it is evident that there is no law in No. 3 and, therefore, that nothing can be deduced by inter- polation. An important technical detail must not be forgotten. The vertical ^ lines in No. 4 represent the height of the plant at the end of each week ; they should be drawn, therefore, at the end of the corresponding segments of the base line. On the other hand the verticals in No. 5 represent growth during the interval and should, accordingly, be erected at the mid-points of the time-intervals. Of the remaining examples in division A nothing need be said except that No. 7 describes an elementary type of astronomical observations which may with great advantage be carried out practically. Such observations lead to the recognition (i) that the sun reaches its highest point in the sky at a time which varies from about a quarter to twelve to about a quarter past by local time, and (ii) that when at its highest point it is always exactly in the same direction — the direction called " south ". § 3. Ex. II, B. — In division B the pictorial element present in all the examples of division A disappears. Instead of thinking chiefly of his representations of the data the pupil is now to attend directly to the graphic expression of the law which governs their variation. The **note " before No. II is intended to prepare him for this shifting of the centre of interest. The teacher will see that there is an important gradation in the abstractness of the examples. In the earlier ones the ordinates, if they were drawn, would represent the data pictorially, for the data are all lengths. In the later ones the ordinates do not represent lengths, and, therefore, cease to be even implicitly pictorial. Much of the difficulty which beginners experience in understanding the representa- tive character of a graph is met with at this point. The teacher does not always make allowance for the gap which the child feels between the case in which a length represents ^ The terms " horizontal" and " vertical " will be used through- out this book to denote the directions upon a sheet of paper in which the axes of x and y are respectively drawn according to the common usaw-e. They may be taken to refer originally to lines drawn upon a blackboard arranged with its plane vertical. 3* 36 ALGEBRA a length and the case in which it represents something totally different from itself. The note prefacing No. 15 suggests a way of bridging the gap. In this example the ordinates are to represent sums of money, and the pupil is told to think of each of them as showing the height of a certain pile of shillings or pence. Such devices facilitate the transition to the stage in which the abscissae and ordinates become purely re- presentative and the graph an expression of an entirely abstract law of connexion between the numerical values of variables. When a graph ceases to be pictorial we need a principle to decide which of the two variables shall be represented by horizontal measurements and which by vertical. There is a perfectly clear rule upon this point which the pupil should be taught to apply unaided. In every case in which a graph is drawn we can regard one set of measurements as having been chosen, and the question is how the other set of measure- ments depends upon these. Thus in No. 15 we select from the dealer's catalogue statements about the monthly payment demanded for furniture of a certain total value, and the question is how does the former sum depend upon the latter. The former may be called the dependent variable, the latter by contrast the independent variable. Then the rule in question states that the independent variable is to be represented by horizontal, and the dependent variable by vertical measurements. The teacher may choose to add the information that the former measurements are called abscissce and the latter ordinates, but these technical terms are probably better reserved until a later point. § 4. The Column-graph (Ex. II, C.).— So far the variables whose connexion is the object of inquiry have been in each case represented in the graph by measured lengths. There have been, however, certain cases in which this mode of re- presentation is not completely satisfactory. No. 5 offers a typical instance. Here the thing to be represented is certainly a length, but it is a length acquired by the growing tulip, not at any particular moment of time, but gradually during the course of the week. There is therefore something arbitrary in connecting the representative line with the middle or any other particular point of the base-segment. It is obvious that a more satisfactory mode of representation would con- nect the weekly growth with the whole of the segment repre- senting the time in which it accrued. THE GRAPH 37 t Zmi f - IIIIIIIIIIIII?! IIIII IIIIIIIIIIIIIIZ ::::;:::::fi?f:::::::E:::;:::;::::: J .._ ::::::::::::::::::::!:?"::::::::::: :::::|:::::::::::::::::::|::::::: t ^ "!""!::i:i::i::::::::::::::5"ii mil mil IIIII IIIII mil IIIII ii_ii a 3 4 Fig. 2. The best way to meet this objection is to represent each weekly increase by an area erected upon the corresponding segment of the base. But the representation of a length by an area is more difficult to understand even than the repre- sentation of a sum of money by a length. It is important, therefore, to introduce this new form of gra- phic symbolism by examples in which it is readily intelligible. This condition is ful- filled by Nos. 21 and 22, the solution of which is shown in fig. '2. The horizontal axis is here divided into equal segments to re- present intervals of one day. Upon each segment is set a rect- angle whose area measures the amount of fresh ground swallowed up by the flood during the corresponding day. A larger square of the paper is taken to represent an acre. When completed the series of rectangles constitutes a " histo- graph " or (as we shall prefer to call it) a " column-graph " representing pictorially what we are told about the history of the flood. It is obvious that the total area flooded at the end of the first, second, third, . . . days is represented by the combined area of the first, the first two, the first three, . . . rectangles. The data must in the first instance be represented by rect- angles, for the reason that they consist merely of state- ments of the total increase of the area of the flood during each day. We are supplied with no direct information about the way in which the flood grew from hour to hour. Yet as contemplation of the pictures given in Exercises, fig. 2, in- evitably suggests dimensions for the missing member of the series, so contemplation of the gross results of the successive days' flooding inevitably suggests probable details of its history. We may assume, to begin with, that the growth of the flood was continuous, and we know how much was added to its surface during the course of each day. The assumption com- 38 ALGEBRA bined with the knowledge suffice to suggest the interpolations represented by the broken curve which is added in fig. 2 to the original column-graph. The principle followed is to draw a continuous curve across the rectangles in such a way that the total area above each of the original segments of the base remains unchanged. It is possible, of course, to draw many curves which fulfil this condition, but if we qualify it by the further condition that the curve shall be as smooth as possible — that is, if we assume the changes in the rate of flooding to have been as little violent as the data permit us to suppose — the results obtained by different draughtsmen will show a satisfactory agreement. To answer such a question as No. 22 (i) we shall, of course, refer to the column-graph as cor- rected by the added curve. The total area under water at the end of 1^ days is represented with much probability by the area under the curve from the ordinate at to the ordinate at 1'5. The column-graph is the means of representation which should be used whenever the data state, not the values of one variable corresponding to definite values of the other, but the gross amount of the change in the first variable corresponding to a given change in the second. As in the case of the linear graph, the representation varies with the nature of the variables from a pictorial to a purely symbolic stage. The earlier stages in the gradation are illustrated by the examples of division C. It is unnecessary to delay the progress of the class in order to work through these examples. It will be sufficient, and probably best, to take them from time to time side by side with Exs. III-XV.* They will be found of great importance in later stages of the course, for they are not only constantly used by modern statisticians but are also made in this book to play an essential part in the exposition of the main ideas of the differential and integral calculus. Thus the method is one which the pupil will often be required to use. § 5. Ex. II, D. — The last division of Ex. II is made up of examples of varying interest and importance. They are given here chiefly in order to prevent the pupil from acquiring an unduly narrow conception of the scope of graphic methods. The chief value of Nos. 26 and 30 is that they introduce the pupil to curves, very different from the circle, yet having just as definite an individuality. Nos. 27 and 28 show how the THE GRAPH 39 course of a curve may be defined, either partially or entirely, by angular " co-ordinates ". The solution of No. 27 is given in fig. 3, that of No. 28 in fig. 4. No. 29 (solved in fig. 5) is an example of a type which always proves very interesting to beginners and has an obvious value as foreshadowing some ideas of importance in the calculus. It is suggested in A Bhythmic Approach to Mathematics, the little book in which Miss Somervell has illustrated some of Mrs. Mary Boole's interesting ideas. Let AB be the path of the cyclist and the points on it marked 1, 2, 3, . . . the positions which he occupies at the end of the first, second, third, . . . inter- vals of time. Let D be the point from which the dog starts. 40 ALGEBRA Then we may suppose that during the first interval the dog runs in the direction DA and covers the distance Dl which is, by hypothesis, half as long again as Al. At the end of the first interval we may suppose the dog to observe his master's change of position and to amend his own direction accordingly. Thus during the second interval he will run along 11 and will cover the distance 12. x\gain he changes his direction and during the third interval runs along the line 22, reaching the point 3 at the end of it. The process is continually repeated until the dog is found to have caught the man up. Lastly we argue that our solution of this artificial problem needs only slight amendment to fit the given case. The polygonal figure D123 . . . indicates a con- tinuous curve which may easily be accepted as representing the actual path of the dog if he adjusted his direction at every moment to the varying position of his master. Such a curve is a " Curve of Pursuit ". § 6. Certain Principles of Method. — The examples of Ex. II all conform to at least one principle which should never be contravened. It is that a graph should not be drawn unless there is a clear purpose to be served by it — a purpose that the pupil can understand and accept. " A train is going at 30 miles an hour. Determine by a graph how far it will go in 3 hours." This is a type of " graphical work " which offends the common- sense of the pupil and prejudices him against its legitimate uses. He gets to regard it as merely a capricious, inconvenient, and uncertain way of treating problems which can be solved with ease and certainty by ordin- ary arithmetic. He entertains the same sound objection to solving by graphs equations which yield without difi&culty to algebraic methods. If there is nothing to be gained by it he sees no reason why the thing should be done. The ends to be achieved by drawing a graph will, of course, vary greatly from case to case. From the logical point of view the simplest cases are those in which it serves as a " ready reckoner " — either by presenting a number of numeri- cal facts in a form convenient for inspection or by solving troublesome calculations by a mechanical device. In most other cases the special service it renders is to disengage from a set of numerical data knowledge which is not obtainable from the numbers considered separately. The extent and value of this knowledge also vary considerably. The simplest THE GRAPH 41 case is typified by the records of the snails' movements in Ex. II, No. 1, or the column-graph of No. 21 before the ad- dition of the continuous curve. Such records are useful be- cause the facts can be " taken in " from them as a whole more readily than from a column of figures. They give a clearer impression of how the snails' speed varied or the area of the flood increased — and one more easily retained in memory. The graph performs a much higher function when, as in No. 4, No. 22, and in most of the other examples of the exercise, it is used to bring to light the mathematical law underlying a set of data. It is important that clear thinking — graduated in " rigour " to the age and experience of the pupil — should accompany this use of the graph, otherwise there is some danger that a fine heuristic instrument may be perverted to the encouragement of slip-shod intellectual habits and the blunting of the logical sensibility. We have seen that graphs can in these cases be regarded as generalizations (§ 1). It must be noted, however, that they are not generalizations based upon analysis, but belong to the inferior type which requires a number of instances (ch. i., § 2) ; it is impossible to construct a graph upon one result. The trustworthiness of a graph de- pends, therefore, in the first instance upon the number and variety of the data originally plotted. Thus in Ex. II, No. 7, if the sun had been hidden by clouds from 10.20 to 12.30 the drawing of the middle part of the graph would have been at- tended with much uncertainty. It would have been unsafe to adopt any very definite conclusion about either the time of noon or the minimum length of the shadow. But while the risk attending these graphic generalizations should always be clearly presented it should not be exagger- ated. After all it is of precisely the same character and de- gree as the risk run by a physicist or a chemist who publishes a new law after carefully examining a number of well-chosen instances. He predicts that all other instances will be found to follow the law which he detects in his data. The confidence of the mathematician in a generalization not based upon analysis rests on the same foundation as that of the man of science — a belief in the prevalence of simplicity and continuity among natural phenomena of all kinds. " Simplex sigillum veri " ; " natura nihil per saltumfacit ". Return in this con- nexion to Ex. II, No. 4, and let a, 6, c, e, f be the terminal points of the lines which represent the recorded lengths of 42 ALGEBRA the tulip. Then the problem before the pupil is not merely to join these points by a smooth curve — a problem soluble, of course, in an infinite number of ways — but to find the con- tinuous curve which in addition to passing through a, b, c, etc., satisfies certain other definite conditions. Some of these conditions represent actual knowledge. For example, it may be taken as certain that the height of the tulip will not rise and fall rhythmically, so the graph must rise continuously from left to right. Other conditions are the expressions of more or less reasonable assumptions. Thus, although a plant may show rhythmic variations in its rate of growth, yet in the case of observations of a plant which cover a short period at the beginning of its career it may safely be assumed that the acceleration which is a characteristic of the successive weeks' growth when recorded was also a feature of the growth during the vacant fortnight. This assumption, translated into graphical terms, implies that the curve must show no " waves ". Thus in the end ambiguity in the solution of the problem is reduced within very moderate limits. It has already been shown how similar assumptions limit the solu- tion of problems hke that of No. 22. From the point of view of the logical training of the pupil the important thing is that he should recognize that he is making certain assumptions and should understand how they work out in graphical terms. Fortunately the investigation of such matters adds not difii- culty but interest to the lesson. The same principle applies in a modified form in instances of which Ex. IV, No. 9, may be taken as typical. Here the pupil is to draw a graph in order to find by interpolation the square roots of numbers which cannot be determined by in- spection. He plots the square roots of 1, 4, 16, etc., against the numbers themselves and draws a smooth curve through the points thus defined. This curve he uses to find the square roots of intermediate numbers. It should be noted that in this case there is no knowledge or plausible assumption from which we can deduce that the curve must be " smooth ". The choice of the smooth curve is determined merely by its simplicity and uniqueness. Of all possible continuous curves through the given points this is the one which it is reasonable to try first. But in this case — as distinguished from those hitherto considered — we already know the law of which the graph is to be the expression. We can therefore use it to THE GRAPH 43 test the success of our venture. Each pupil chooses at random one or two numbers within the range of the graph, and reads off the numbers which the graph asserts to be their square roots. The truth of this pretension is verified by multiplica- tion. It is now impossible to doubt that though our initial choice of a smooth curve might have proved erroneous yet as a matter of fact it has been justified at least to the extent that square roots deduced from it may be expected as a rule to be accurate within the degree obtainable by inspection of the graph. § 7. Graphs in Practical Work. — When the points which ^re to determine the course of a graph represent the results of practical measurement the problem of drawing the appro- priate curve is complicated by other considerations. Meas- urement is always subject to error, and the uncertainty about the correct position of the graphic points may, for this reason, be great enough to make it doubtful how the curve should be drawn even when its general form is already known. When the general form of the graph is itself unknown the draughtsman will attach much weight to considerations of simplicity. Thus if the assumption that the graph is " meant to be " a straight line is not grossly discordant with the actual position of the points, he will adopt it in preference to the as- sumption of a more complicated curve. But when he has learnt or has assumed that the graph has a certain general form he has still to determine the particular instance. In advanced work it is customary to apply in this connexion the " method of least squares " which is illustrated in the last section of this volume. In elementary work no systematic method of dealing with the difficulty is generally used. The pupil is told to select the straight line or curve which his eye judges to fit the points best. In Ex. XXVI, D, a simple method is discussed for which there is a good deal to be said on statistical grounds, while it offers at least a definite principle for dealing with the kind of situation which is now in view. It may be illustrated by No. 17 {Exercises, p. 144). The solution is shown in fig. 6 in this book. Since it is known that P is proportional to Q^ the eleven given values of P are plotted (in the left-hand portion of the figure) against numbers which represent the squares of the values of Q. Each of the graphic points corresponds, of course, to a definite possible value of the ratio P/Q^. This value would be the tangent of 44 ALGEBRA 1 — — — rrr r^ [j:::^ •-«« "■"•- rrt ^ ■:^L i-^ -2i ^ ^ i^ ^i ^ ^ S V^ ~~1 I ( 5 I t 1 i r*~ ) < ) ' t c 1 _^ ,\ \ \ \ \ A \ V \ « V v^\ \ V ^^^ \ \ \ \ \ v \ ^\ ''\ \ \ '^ \\> V ^^ ^ I* *k \ \ , ^ Oeo«D;;t^OO