Essays On Mathematical Education

ESSAYS ON M A T H E M A T I C A L EDUCATION

B Y

G. S T . L . C A R S O N

W I T H AN I N T R O D U C T I O N BY

DAVID EUGENE

L O N D O N

A N D

G I N N A N D COMPANY, l

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SMITH

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PUBLISHERS

C O P Y R I G H T , 1913, B Y G I N N A N D ALL RIGHTS RESERVED

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COMPANY

INTRODUCTION I t has always been hard for people to judge with any accuracy the work of their own age, and it is hard for us to do so to-day. I n spite of our optimism and of our certainty that we are pro­ gressing, what we conceive to be an era of great educational awakening may appear to the historian of the future as one in which noble ideals were sacrificed to the democratizing of the school, and the twentieth century may not rank with the sixteenth when the toll is finally taken. I t is, therefore, with some hesitancy that we should assert that we live in a period of remarkable achievement in all that pertains to education. That the period is one of advance is in harmony with the general principle of evolution, but that all that we do is uniformly progressive is not at all in accord with general experience. Certain it is that the present time is one of agita­ tion, of the shattering of idols, and of the setting up of strange gods in their places. Nothing is sacred to the iconoclast, and he is found in the school as he is found in the church, in govern­ ment, and in the social world. Among the objects of attack in this generation is " the science venerable" that has come down to us from Pythagoras and Euclid, from Mohammed ben Musa and Bhaskara, and from Cardan, Descartes, and Newton. A n d yet it does not seem to be mathematics itself that is challenged so much as the way in which it has been presented to the youth in our schools, and to most of us the challenge seems justified. With all the excel­ lence of Euclid, his work is not for the child ; and with all the value of formal algebra, the science needs some other introduction than the arid one until recently accorded to it. iii

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I t is on this account that M r . Carson's work in the English schools and before bodies of English teachers has great value. H e is thoroughly trained as a mathematician, is a product of the college where Newton studied and taught, is a lover of the science in its purest form, and has had an unusual amount of experience in the technical applications of the subject; but he is a teacher by instinct and by profession, and is imbued with the feeling that mathematics can be saved to the school only through an improvement in our methods of teaching and in our selection of material. H e stands for the principle that mathe­ matics must be made to appeal to the learner as interesting and valuable, and he has shown in his own classes that, after this appeal has been successful, pupils need to be held back rather than driven forward in this branch of learning. I t is because of this feeling on the part of M r . Carson that his essays on the teaching of mathematics have peculiar value at this time. They will encourage teachers to continue their advocacy of a worthy form of mathematics, at the same time seeking better lines of approach and endeavouring to relate the subject in a reasonable manner to the various other interests of the pupil. The problem is much the same everywhere, but the ties of a common language, a common spirit of freedom, and a common ancestry make it practically identical in English-speak­ ing lands. On this account we, in the United States, feel that M r . Carson's message is quite as much to us as to his own countrymen, and we shall appreciate it as we have appreciated the noteworthy work that he has already achieved in the teaching of mathematics in England. D A V I D TEACHERS COLUMBIA NEW

COLLEGE UNIVERSITY

YORK

CITY

E U G E N E

S M I T H

CONTENTS PAGE SOME

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INTUITION T H E

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SOME P R I N C I P L E S OF MATHEMATICAL EDUCATION ( R e p r i n t e d from The Mathematical

Gazette,

January, 1 9 1 3 )

SOME

PRINCIPLES OF M A T H E M A T I C A L EDUCATION

O f all the problems w h i c h have perplexed teachers o f mathematics i n this generation, probably none has been more i r r i t a t i n g and insistent than the choice of assumptions w h i c h must be made i n each branch of the science. I n geometry, i n analysis, i n mechanics, one and the same difficulty arises. A r e we to prove that any two sides of a triangle are greater than the t h i r d ? T h a t the l i m i t of the sum of a finite number o f functions is equal to the sum of their limits ? T h a t the total m o m e n t u m of two bodies is uninfluenced by their mutual action ? A n d i n every such case, on what is the proof to depend ? A clear under­ standing of the answers to such questions, or, better still, a clear understanding of principles by w h i c h answers may be found, would go far to co-ordinate and simplify elemen­ tary t e a c h i n g ; the object of this paper is to state such principles and indicate their application. A X I O M , POSTULATE, PROOF

I t is first necessary to lay down definitions, as precise as may be possible, of the terms " a x i o m , " " p o s t u l a t e , " " proof." I t is not i m p l i e d that these definitions should be insisted on, or the terms used, i n elementary t e a c h i n g ; n o t h i n g could be more likely to lead to failure. B u t a full comprehension of each is essential to every teacher of mathematics, and is too often l a c k i n g i n current usage. 3

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A n axiom, or " c o m m o n n o t i o n " i n E u c l i d ' s language, is a statement which is true of all processes of thought, whatever be the subject matter under discussion. T h u s the f o l l o w i n g are a x i o m s : " I f A is identical w i t h B, and C is different f r o m B, then C is different from A . " " I f B is a necessary consequence of A , and also C of B, then C is a necessary consequence of A . " B u t " T w o and two make four " and " T h e straight line is the shortest distance between two points " are not axioms, although they may be considered no less obvious. A statement is not an axiom because i t is obvious, but because i t concerns u n i ­ versal forms of thought, and not a particular subject matter such as arithmetic, geometry, and the l i k e . A postulate is a statement which is assumed concerning a particular subject m a t t e r ; for example, " T h e whole is greater than a p a r t " (subject matter, finite aggregates); " A l l r i g h t angles are equal " (subject matter, Euclidean space). I t is essential to observe that, whereas an axiom is an axiom once for all, a postulate i n one treatment of a science may not be a postulate i n another. I n E u c l i d ' s development o f geometry, the statement that any two sides of a triangle are together greater than the t h i r d side is not a postulate, because i t is deduced f r o m other statements (postulates) w h i c h are avowedly assumed; but i n many current developments i t is adopted at once, without refer­ ence to other statements, and is therefore a postulate i n such cases. T o use an unconventional but expressive t e r m , postulates are " jumping-off places " for the logical explo­ ration of a subject. T h e i r number and nature are i m m a t e r i a l ; they may be readily acceptable, or difficult of credence. T h e i r one function is to supply a basis for reasoning,

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w h i c h is conducted i n accordance w i t h the axioms. Postu­ lates are thus doubly relative : they relate to one particular subject matter (number, space, and so on) and to one par­ ticular method of v i e w i n g that subject matter. A statement w h i c h is deduced, by use of the axioms, f r o m two or more postulates is said to be proved. T h e r e is thus no such t h i n g as absolute proof. Proofs are related to the postulates o n w h i c h they are based, and a demand for a proof must inevitably be met by a counter demand for a place to start f r o m , that is, for some postulates. W h e n a statement is said to have been proved, what is meant is that i t has been shown to be a logical consequence of some other statements w h i c h have been accepted; i f these statements are found to be incorrect, the statement w h i c h is said to be proved can no longer be accepted, t h o u g h the logical character of the proof is i n no way i m p u g n e d . T h u s the type of a proof is, " I f A , then B" ; relentless and final certainty surrounds " then " ; but A , w h i c h is assumed i n the " i f , " may nevertheless be utterly fantastic as viewed i n the l i g h t of experience. T H E T H R E E FUNCTIONS OF M A T H E M A T I C S

T h e first application of mathematics to any domain of knowledge can now be explained. S t a r t i n g f r o m postulates, the t r u t h of w h i c h is no concern of mathematics, sets of deductions are evolved by use of the a x i o m s ; agreement of the results w i t h experience strengthens the evidence i n favour of these postulates. I f this evidence be deemed sufficient, as, for example, i n geometry and mechanics, then deduction yields acceptable results w h i c h could not other­ wise have been predicted or ascertained.

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I t is here that the prevailing concept of the power of mathematics ends ; but such a concept presents a view of the subject so l i m i t e d and distorted as to be almost gro­ tesque. T h e process just described may be regarded as an upward development; a downward research is also possible, and no less valuable. I t consists of a logical review of the set of postulates w h i c h have been adopted ; i n the result, either it is shown that some must be rejected, or the evidence i n favour of all may be considerably enhanced. T h i s review consists of two processes, w h i c h w i l l be described i n t u r n . I t is first necessary to ascertain whether the set of pos­ tulates is consistent; that is, whether some a m o n g t h e m may not be logically contradictory of others. F o r example, E u c l i d defines parallel straight lines as coplanar lines w h i c h do not intersect, and proves i n his twenty-seventh proposi­ t i o n that such lines can be drawn ; for this purpose he uses his fourth postulate, w h i c h makes no allusion to parallels. I f he had included a m o n g his postulates another, stating that every pair of coplanar lines intersect i f produced suffi­ ciently far, and had o m i t t e d his definition of parallel lines, his postulates would not have been consistent; for the twenty-seventh proposition proves that i f the fourth postu­ late be granted, then the existence of non-intersecting co­ planar lines must be admitted also. I t is essential to realize that the contradiction i m p l i e d i n the t e r m " i n c o n s i s t e n t " is based on logic, not on experience ; assumptions w h i c h are contrary to all experience are not thereby inconsistent. T h e r e is n o t h i n g i n logic to veto the assumption that, for certain types of matter, weight and mass are inversely propor­ tional ; or that life may exist where there is no atmosphere, as on the m o o n . Such assumptions are not inconsistent

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w i t h the other postulates of mechanics or b i o l o g y ; they are merely contrary to all experience gained up to the present time. H e r e , then, is the second function of the mathematician — the investigation of the consistence of a set of postulates. A n d the task is not superfluous. Physical measurements are perforce inaccurate, and a set of inconsistent assump­ tions m i g h t well appear to be consistent w i t h actual obser­ vations. M o r e accurate measurements must, of course, expose the discrepancy, but these may for ever remain beyond our powers ; logic renders t h e m superfluous by demonstrating the consistence or otherwise of each set considered. T h e next investigation concerns the redundance of a set of postulates. Such a set is said to be redundant i f some of its members are logical consequences of others. F o r example, any ordinary adult w i l l accept without difficulty the properties of congruent figures, the angle properties of parallel lines, and the properties of similar figures, as i n maps or plans, regarding t h e m as " i n the nature of t h i n g s . " A n d electricians may, by experiment, convince themselves first, that Coulomb's law of attraction is very approximately t r u e ; and secondly, that w i t h i n the limits of observation there is no electric force i n the interior of a closed con­ ductor. I n neither the one case nor the other need there be the least suspicion that the statements are logically connected, so that they must stand or fall together. Y e t so i t is, and the fact is expressed by the statement that the assumptions are redundant. T h e investigation of redundance, and the demonstration that sets of postulates are free therefrom, forms the t h i r d

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function of the mathematician. I t s value, i n connection w i t h any subject matter to w h i c h it may be applied, may not at once be evident. I t is based on the fact that all experiments, necessary and inevitable t h o u g h they be, are nevertheless sources of uncertainty; i t reduces this uncer­ tainty to a m i n i m u m by r e m o v i n g the redundant assump­ tions into the category of propositions, and exposing the science i n question as based on a m i n i m u m of assumption. A n d m o r e ; it can offer several alternative sets of assump­ tions for choice, that one being taken w h i c h is most nearly capable of verification. T h e labours of Faraday resulted i n the offer of such a choice to electricians ; either, they were told, you can base electrostatics {inter alia) on Cou­ lomb's experiment, or on the absence of electric force i n ­ side a closed conductor; i t is logically immaterial w h i c h course you adopt. T h e latter experiment, being far more capable of accurate demonstration i n the laboratory, is chosen as the p r i m a r y basis for faith i n the deductions of electrostatics — a faith w h i c h is, of course, very m u c h strengthened as such deductions are found to accord w i t h our experience. B u t these considerations are for the physi­ cist ; the task of the mathematician is ended when he has put forward, for choice by the physicist, alternative sets of assumptions w h i c h are at once consistent w i t h each other and free f r o m redundance. I n this way does he free the physicist, so far as may be, from the uncertainties of assumption, and assure h i m that no further increase of such freedom can be attained. 1

1

T h e antithesis between mathematician and p h y s i c i s t does not imply

that the functions are of n e c e s s i t y p e r f o r m e d by different i n d i v i d u a l s ; it is u s e d m e r e l y to enforce the argument.

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Such is the range of application of mathematics to other sciences. W h e n complete i t reveals each science as a firmly k n i t structure of logical reasoning, based on assump­ tions whose number and nature are clearly exposed; of these assumptions i t can be asserted that no one is incon­ sistent w i t h the others, and that each is independent of the others. T h e r e is thus no fear that contradictions may i n t i m e emerge, and no false hope that one assumption may i n t i m e be shown to be a logical consequence of the others. F i n a l i t y has been reached. T h e acute critic may, of course, ask the mathematician whether his o w n house is i n order. W h a t is the precise statement of the axioms w h i c h are the basis of his science, and can they be shown to depend on a set of consistent mental postulates, free f r o m redundance ? H e r e i t need only be said that the labours of the last generation have done m u c h to answer these questions, and that their com­ plete solution is certainly possible, i f not actually achieved; to go further would be beyond the limits of this paper. THE

D I D A C T I C PROBLEM

T h e complete application of mathematics to any branch of knowledge b e i n g thus exhibited, the didactic problem can now be stated i n explicit terms. I n any g i v e n science— geometry, mechanics, and so o n — what is the r i g h t point of entry to the structure, and i n what order should its exploration be made ? W h a t results should be regarded as postulates, and should their consistence and possible inter­ dependence one o n another be investigated before upward deduction is undertaken ? Should the m i n i m u m number be chosen o n the g r o u n d that the p u p i l should at once be

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placed i n possession of the ultimate point of view ? O r should some larger number be taken, and i f so, on what principles should they be chosen ? B e a r i n g i n m i n d that the pupils concerned are not presumed to be adults, i t is easy to indicate principles f r o m w h i c h answers to such questions may be deduced. One of the few really certain facts 'about the juvenile m i n d is that i t revels i n exploration of the u n k n o w n , but loathes analysis of the k n o w n . I t is often said that boys and girls are indifferent to, and cannot appreciate, exact l o g i c ; that i t is unwise to force detailed reasoning upon t h e m . Few statements are farther f r o m the t r u t h . L o g i c , provided that i t leads to a comprehensible goal, is not only appreciated, but demanded, by pupils whose i n ­ stincts are n o r m a l . B u t the goal must be comprehensible ; it must not be a result as easily perceived as the assump­ tions o n w h i c h the proof is based. L e t any one w i t h expe­ rience i n e x a m i n i n g consider the types of answer given to two p r o b l e m s ; one, an " obvious " rider on congruence, i n v o l v i n g possibly the pitfall of the ambiguous case; the other, some simple but not obvious construction or rider concerning areas or circles. I n the former, paper after paper exhibits f u m b l i n g uncertainty or bad l o g i c ; i n the latter, there is usually success or silence, and more usually success; bad logic is hardly ever f o u n d . T h e phenomenon is too universal to be comfortably accounted for by abuse of the teachers; the abuse must be transferred to the crass methods w h i c h enforce the premature application of logic to analysis of the k n o w n , rather than to exploration of the u n k n o w n . T h e natural order of exploration should now be evident. L e t the leading results of the science under consideration

PRINCIPLES OF E D U C A T I O N

I I

be divided into two groups : one, those w h i c h are accept­ able, or can be rendered acceptable by simple illustration, to the pupils under consideration ; the other, those which would never be suspected and whose verification by exper­ iment would at once produce an unreal and artificial atmos­ phere. L e t the former group — w h i c h i n geometry would include many o f Euclid's propositions — be adopted as postulates, and let deductions be made f r o m t h e m w i t h full rigour. W h e r e v e r possible, let the results of such deduc­ tions be tested by experiment, so as to give the utmost feeling of confidence i n the whole structure. Later, when speculation becomes more natural, let i t be suggested that gratuitous assumption is perhaps inadvisable, and let the meanings of the consistence and redundance of the set of postulates be explained. Finally, i f i t prove possible, let the postulates be analysed, their consistence and independ­ ence be demonstrated, and the science exposed i n its ultimate f o r m . These second and t h i r d stages are even more essential to a " liberal education " than the first, for they exhibit scientific method and human knowledge i n t h e i r true aspect. I t is not suggested that they can be dealt w i t h i n schools, except perhaps tentatively i n the last year of a l o n g course. B u t i t is definitely asserted that the general ideas involved should f o r m part of the compulsory element of every U n i ­ versity course, even t h o u g h details be excluded, for they are of the very essence of the spirit of mathematics. T h e method of developing such ideas remains to be considered. I t may be presumed that the pupils concerned have some knowledge of arithmetic, geometry, the calculus, and mechanics, each subject h a v i n g been developed f r o m a

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redundant set of postulates. I n which, then, of these four branches is i t most natural to suggest the analysis of these assumptions ? Since analysis of the k n o w n may still be presumed to have its dangers, the branch chosen must be that one i n which the investigation bears this aspect least p r o m i n e n t l y . N o w the m a i n ideas of arithmetic, geometry, and the cal­ culus are so firmly held by boys and girls, that any attempt to discuss t h e m i n detail produces revolt or boredom. Such attempts account for m u c h ; the writer can well remember his feelings o n first seeing a formal proof that the sum of a definite number of continuous functions is itself a con­ tinuous f u n c t i o n ; and at the same t i m e he realized to the full that the proposition m i g h t well be untrue i f the n u m ­ ber of functions were not finite. G r o u n d such as this is unfavourable for the development of this new analysis. T h e same is by no means true of mechanics. H e r e the postulates, acceptable t h o u g h they be, have been elucidated w i t h i n the memory of the pupils, and they may reasonably be asked to examine the facility w i t h w h i c h these assump­ tions were made, and to consider whether the evidence can i n any way be strengthened. T h i s being done, the ideas of consistence and redundance can be developed, and some idea of the structure of a science imparted. E v e n then i t may probably be wise to lay little stress o n analysis of the geometrical postulates ; i f the ideas are realized i n connec­ tion w i t h mechanics, we may w e l l leave the seed to mature i n minds to which i t is congenial. I n the view o f mathematics here taken, its various branches are regarded as structures w i t h many possible entrances, and the discussion has been concerned w i t h the

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choice of entrance and the route to be taken t h r o u g h the edifice. W e cannot hope that our pupils w i l l ever k n o w more than the outline of each structure. E v e n we w h o are the guides cannot k n o w each detail of any one ; the laby­ r i n t h is too vast. B u t the best guide to a structure is he who knows its m a i n outlines most completely, and a teacher who has clear ideas of mathematical principles can do much, i n leading his pupils t h r o u g h such avenues o f the structure as they can attain, to give t h e m a view of the whole. O f the i m p o r t and beauty of this view more need not be said.

INTUITION address delivered to the Mathematical A s s o c i a t i o n , and reprinted from The Mathematical

Gazette,

March, 1 9 1 3 )

INTUITION I f there be one duty more incumbent than any other upon mathematicians, i t is to have a clear and c o m m o n understanding of every t e r m w h i c h they use. I do not say a formal definition, though that is most advisable i f and when i t can be obtained ; but a class of entities must be k n o w n and recognised before i t can be defined, and no t e r m should be used unless i t at least gives rise to definite, recognisable, and identical images i n the minds of the speaker and listener. I t cannot fairly be said that mathe­ maticians are at fault i n this respect, when dealing w i t h their own special subjects ; but I fear they cannot so easily be acquitted when discussing the didactic side of their w o r k . Concrete and utilitarian, axiom and postulate, intu­ ition and assumption; how many of us have definite meanings for these terms, and can feel certain that they represent the same meanings to others ? T h e t e r m w h i c h I have chosen as the title of this paper is one of the most commonly used and, as i t seems to me, most often misun­ derstood ; at the same time, the ideas and processes for which i t stands lie at the root of all elementary teaching. I have therefore thought it w o r t h while to discuss its meaning and to show the bearing of the process on math­ ematical education. T h e r e is, I t h i n k , little doubt that to most of those w h o use the t e r m " i n t u i t i o n , " i t connotes some peculiar quality of material certainty. T a k e , for example, the equality of 17

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all r i g h t angles, or the angle properties of parallel lines, and ask one who understands these statements w i t h what degree of certainty he asserts their t r u t h . I t w i l l be found almost invariably that he regards t h e m as far more certain than statements such as " the sun w i l l rise to-morrow m o r n i n g " or " a l l m e n are mortal " ; these, he admits, m i g h t be upset by some perversion of the order which he has regarded as customary, but the geometrical statements appear to be of the essential nature of things, eternal and invariable verities. So m u c h , indeed, is this the case that the very idea of practical tests is grotesque. W h o has ever experimented to ascertain whether, i f two pieces of paper are folded, and the folds doubled again on themselves, the corners so f o r m e d are superposable ? I f the individual under examination be questioned as to the basis for this faith, he can only reply that i t is the nature of things, or that he knows i t i n t u i t i v e l y ; o f the degree o f his faith there is no doubt. I t is to statements asserted i n this manner that the t e r m " i n t u i t i o n " is c o m m o n l y applied ; other facts, such as the mortality of all men, w h i c h are justified by the fact that all h u m a n experience points to them, are not classified under this heading nor, as I have said, are they accepted w i t h the same faith. These alleged certainties can of course be dissipated by purely philosophical considerations concerning the relations and differences between concepts and percepts; but " an ounce of practice is w o r t h a ton of theory," and I propose here to show, mainly f r o m historical considerations, that there is no g r o u n d for absolute faith i n certain intuitions, however tenaciously they may be held. T a k e first the idea, still held by many, that a body i n m o t i o n must be urged

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on by some external agent i f its velocity is to be main­ tained. U n t i l the time of Galileo this belief was held uni­ versally, even men o f eminence who had considered the subject being convinced of its t r u t h . N o w this faith was of just such a k i n d , and just as strongly held, as the faith i n geometrical statements w h i c h I have m e n t i o n e d ; i t was, and still is by many, regarded as i n the nature of things that a body should stop m o v i n g unless i t is propelled by some external agency. A n d yet others, of w h o m Galileo was the forerunner, see the nature of things i n a l i g h t wholly different. T h e y regard i t as utterly certain that a body can o f itself neither increase nor retard its own mo­ t i o n . A s k a clever boy who has learnt some mechanics, or even a graduate who has not thought overmuch on the foundations of the subject, w h i c h he regards as more un­ likely : that an isolated body should, contrary to N e w t o n ' s first law, set itself i n m o t i o n , or that the secret of i m m o r ­ tality should be discovered. H e w i l l tell you that the second m i g h t happen, t h o u g h personally he does not be­ lieve that i t ever w i l l ; but that a body can never begin to move unless i t has some other body " to lever against." W e thus see two contradictory intuitions i n existence, each held w i t h equal strength. C o m i n g to more recent history, let me r e m i n d you of the development of the theory of parallels, and the rise of non-Euclidean geometries. U n t i l the last century i t may fairly be said that no one had ventured to doubt the so-called t r u t h of the parallel postulate, t h o u g h many emi­ nent mathematicians had endeavoured to deduce i t f r o m the other postulates of geometry. T h e genius of Bolyai and Lobachewsky, however, put the matter i n quite another

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light. T h e y showed that a completely different theory of parallels was just as m u c h i n accord w i t h the nature of things as that hitherto h e l d ; and that, to beings w i t h more extended experience or finer perceptions than ours, this different theory m i g h t appear to correspond with ob­ servation while the current belief failed to do so. I n other words, they showed that there are several ways of account­ i n g for such space observations as we can make w i t h our restricted o p p o r t u n i t i e s ; just as i t was then well k n o w n that there were two theories w h i c h fitted the observations of astronomers, of w h i c h Newton's was the more simple and self-consistent. I t thus becomes clear that intuitions are no more than w o r k i n g hypotheses or assumptions ; they are on the same footing as the p r i m a r y assumptions concerning gravitation, electrostatics, or any other branch of knowledge based on sensation. T h e y differ f r o m these i n that they are formed unconsciously, as a result of universal experience rather than conscious e x p e r i m e n t ; and they are so formed i n regard to those experiences — space and motion — w h i c h are forced on all of us i n virtue of our existence. I t is not i m p l i e d that their possessor is even fully conscious of t h e m ; ask some comparatively untrained adult how to test rulers for straightness, and he may be at a loss or give some i n ­ effective r e p l y ; but suggest placing t h e m back to back and then reversing one, and he at once assents. H e re­ gards this not as new information, but as something so simple and obvious that i t had not occurred to h i m . I t is to h i m the essential nature of t h i n g s ; he has held this view f r o m so early an age, and it has remained so entirely free f r o m challenge, that he revolts at the suggestion that

INTUITION

21

things, viewed from another standpoint, may appear to have a different nature. T h e formation of such w o r k i n g hypotheses is the normal method by w h i c h the m i n d investigates natural phenom­ ena. A f t e r observation of a certain set of events, a theory is formed to fit them, the simplest being chosen i f more than one be found to fit the facts equally well. T h i s theory is developed, and its consequences compared w i t h the results of further observations; so l o n g as these are i n accord, and so l o n g as no simpler theory is found to ac­ commodate the fact, the first theory holds the field. B u t , should either of these events occur, i t is abandoned r u t h ­ lessly i n favour of some better description of the recorded observations. T h e r e are famous historical cases of each e v e n t ; Newton's corpuscular theory of light yielded de­ ductions i n actual disaccord w i t h observation, and was therefore abandoned. T h e ancient theory of astronomy, wherein the stars were imagined to be fixed on a crystal sphere on w h i c h the planets travelled i n epicycles, was abandoned i n favour of the modern theory, not because i t could not be modified to accord w i t h observation, but be­ cause of its greater complexity. I n every such case the question of absolute t r u t h is irrelevant and beyond our reach ; the problem is to find the simplest theory i n accord w i t h all the facts, abandoning i n the quest each theory as a successor is found w h i c h better fulfils these requirements. Shortly, then, we may say that intuitions are merely a particular class of assumptions or postulates, such as f o r m the basis of every science. T h e y are distinguished f r o m other postulates first, i n that they, w i t h their subject matter — for example, space or m o t i o n — are c o m m o n

22

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EDUCATION

f r o m an early age to every h u m a n being endowed w i t h the ordinary senses; and secondly, i n that no other assump­ tions fitting the sensations concerned ever occur to those who make t h e m . T h e i r formation is forgotten, and they are therefore regarded as e t e r n a l ; they h o l d the field un­ challenged, and are therefore regarded as inviolable. Before passing to the consideration o f the bearing of intuition o n the teaching of mathematics, i t may be well to illustrate what has been said by the consideration of a few particular cases. First, suppose that one sees a jar o n a shelf, and puts his hand up to find out whether it is empty. I s the act based on an i n t u i t i o n from the appearance of the jar ? T h i s is not the case ; i f asked before the act, one would not express any final certainty that the hand could enter the j a r ; i t m i g h t have a l i d or be a d u m m y . T h e i n ­ dividual can make more than one assumption which corre­ sponds to the sight sensation; the first assumption made — that the hand can enter the jar — is merely the most likely as j u d g e d by experience. N e x t suppose that a knock is heard i n a r o o m . T h e natural exclamation " W h a t is that ? " is based on i n t u i t i o n , for i t expresses the now universal conviction that such a noise is an invariable accompaniment of some happening which, given opportunity, w i l l also appeal to the other senses. A c c u m u l a t i o n of h u m a n experience has led to the belief that such is invariably the case ; but belief i t is, and not certainty. I f the reply were, " I t is n o t h i n g ; under no circumstances could you have correlated any sight or feel­ i n g w i t h that sound," i t would be received w i t h complete incredulity.

INTUITION

23

Consider again the statement that, given a sufficient number of weights, no matter how small, one can w i t h t h e m balance a single weight, however large. N o one would doubt this or treat i t as a n y t h i n g but the most obvi­ ous of truisms, and yet i t is a pure assumption, formed unconsciously as the result of general experience ; i t an­ swers i n every respect to our definition of an i n t u i t i o n . I t may be thought by some that the statement can be proved arithmetically, but i n every such alleged proof the assump­ tion itself w i l l be found somewhere concealed. W e have, i n fact, no warrant for assuming that the phenomenon called weight retains the same character, or even exists, for portions of matter w h i c h are so small as to be beyond our powers of subdivision. Finally, consider the statement, " I knew intuitively that you would come to-day." I n what respect do those who use i t regard i t as differing from, " I thought it was almost certain that you would come t o - d a y " ? I t may fairly be said that the former expresses less basis of knowledge but more feeling of certainty than the l a t t e r ; it means, " I don't i n the least k n o w how I knew i t , but I d i d k n o w beyond all doubt that you would come." Such ideas, w i t h or without the use of the actual term " i n t u i t i o n , " are com­ m o n enough. T h e y are here quoted to justify the statement, made above, that the t e r m connotes to many of those w h o use it some peculiar degree of certainty. Such statements are not intuitions ; they are mere superstitions, and those who are subject to t h e m fail to realise how often they are unjustified by the event. Belief i n the absolute t r u t h of the angle properties of parallels or of the Laws of M o t i o n is equally a superstition, though these are, u n t i l now, justified

24

MATHEMATICAL

EDUCATION

by the event. T h e t r u t h is that they can never receive this absolute justification, for no material observation is beyond the possibility of error, nor can i t be certain that some simpler theory w i l l not be formed, accounting equally well for the observations; i t is the belief i n this impossible finality w h i c h constitutes the superstition. T u r n i n g now to the more educational 'aspect of the subject, the first problem w h i c h confronts us is t h i s : children, when they commence mathematics, have formed many intuitions concerning space and m o t i o n ; are they to be adopted and used as postulates w i t h o u t question, to be tacitly ignored, or to be attacked ? H i t h e r t o teaching methods have tended to ignore or attack such i n t u i t i o n s ; instances of their adoption are almost non-existent. T h i s statement may cause surprise, but I propose to justify i t by classifying methods w h i c h have been used under one or other of the two first heads, and I shall urge that com­ plete adoption is the only method proper to a first course in mathematics. Consider first the treatment of formal geometry, either that of E u c l i d or of almost any of his modern r i v a l s ; i n every case i n t u i t i o n is ignored to a greater or less extent. E u c l i d , of set purpose, pushes this policy to an e x t r e m e ; but all his competitors have adopted i t i n some degree at least. Deductions of certain statements still persist, al­ though they at once command acceptance when expressed in non-technical f o r m . F o r example, i t is still shown i n elementary text-books that every chord of a circle perpen­ dicular to a diameter is bisected by that diameter. D r a w a circle on a wall, then draw the horizontal diameter, mark a point on it, and ask any one you please whether he w i l l

INTUITION

25

get to the circle more quickly by g o i n g straight up or straight down from this point. Is there any doubt as to the a n s w e r ? A n d are not those who deduce the propo­ sition just quoted, f r o m statements no more acceptable, i g n o r i n g the intuition w h i c h is exposed i n the immediate answer to the question ? A l l that we do i n u s i n g such methods is to make a chary use of i n t u i t i o n i n order to reduce the detailed reasoning of E u c l i d ' s scheme; our attitude is that statements w h i c h are accepted intuitively should nevertheless be deduced f r o m others of the same class, unless the proofs are too involved for the juvenile m i n d . W e oscillate to and fro between the Scylla of ac­ ceptance and the Charybdis of proof, according as the one is more revolting t o ourselves or the other to our pupils. 1

A t this point I wish to suggest that a distinction should be drawn between the terms " d e d u c t i o n " and " p r o o f . " T h e r e is no doubt that proof implies access of material conviction, while deduction implies a purely logical process i n w h i c h premisses and conclusion may be possible or impossible of acceptance. A proof is thus a particular k i n d of deduction, wherein the premisses are acceptable (intuitions, for exam­ ple), and the conclusion is not acceptable u n t i l the proof carries conviction, i n virtue of the premisses on w h i c h i t is based. F o r example, E u c l i d deduces the already accept­ able statement that any two sides of a triangle are together greater than the t h i r d side f r o m the premiss {inter alia) that all r i g h t angles are equal to one another ; but he proves that triangles o n the same base and between the same 1

T h e r e is often apparent doubt;

but it will usually be found that

this is due to an attempt to estimate the want of truth of the c i r c l e as drawn.

26

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EDUCATION

parallels are equal i n area, starting from acceptable prem­ isses concerning congruent figures and converging lines. T h e distinction has didactic importance, because pupils can appreciate and obtain proofs l o n g before they can understand the value of deductions; and it has scientific importance, because the functions of proof and deduction are entirely different. Proofs are used i n the erection of the superstructure of a science, deductions i n an analysis of its foundations, undertaken i n order to ascertain the number and nature of independent assumptions involved therein. I f two intuitions or assumptions, A and B, have been adopted, and i f we find that B can be deduced from A , and A f r o m B, then only one assumption is involved, and we have so m u c h the more faith i n the bases of the science. H e r e i n lies the value of deducing one accepted statement from another ; the element of doubt involved i n each acceptance is thereby reduced. N e x t , to justify the statement that intuition has been attacked. B o t h E u c l i d and his modern rivals knew well enough that their schemes must be based on some set of assumptions; they differed only i n the choice. Each agrees that i n t u i t i v e assumptions are undesirable, but the m o d e r n school regards the extreme logic entailed by E u c l i d ' s principle of the m i n i m u m of assumption as impossible for y o u n g pupils. T h e r e is, however, a t h i r d school w h i c h pursues a different course; i t professes to replace i n t u i t i o n by experimental demonstration. Pupils are directed to draw pairs of intersecting lines,- measure the vertically opposite angles, and state what they observe; to perform similar processes for isosceles triangles, parallel lines, and so o n . Instead of being asked, " D o you t h i n k that, i f these lines

INTUITION

27

were really straight, and you cut out the shaded pieces, the corners would fit ? " they are told to find out, by a clumsy method, a belief w h i c h they had previously held, t h o u g h i t had never, perhaps, entered definitely into t h e i r consciousness. T h e question suggested is, i n these homely terms, just sufficient to b r i n g the idea before t h e m , and i t is at once recognised as according w i t h the child's previous notions; he does not regard i t as new, but merely as some­ t h i n g of w h i c h he had not before t h o u g h t so definitely. I t is this type of exercise i n d r a w i n g and measurement w h i c h I regard as an attack u p o n i n t u i t i o n . I t replaces this natural and inevitable process by hasty generalisation f r o m experiments of the crudest type. Some advocates o f these exercises defend t h e m o n the g r o u n d that they lead to the formation of intuitions, and that the pupils were not previously cognisant of the facts involved. B u t i n the first place, a conscious induction f r o m deliberate experiments is not an i n t u i t i o n ; i t lacks each o f the special elements connoted by the t e r m . A n d as to the alleged ignorance o f the elementary idea o f space, i t appears to me to be a mis­ taken impression, based o n undoubted ignorance of mathe­ matical terminology. I f you say to a c h i l d o f twelve, " A r e these angles equal?" he has to stop to t h i n k first, what an angle is, and next, when angles are e q u a l ; by the t i m e he has done this his m i n d is incapable o f grasping the pecul­ iar relations of the angles i n question, and he is labelled as ignorant o f the answer. T h e real difficulty, and i t is not a small one, is to lead the child to express familiar facts i n precise mathematical t e r m i n o l o g y ; to say "angles e q u a l " rather than " c o r n e r s fit." U n t i l this terminology is thoroughly familiar, the effort of using i t must absorb

28

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EDUCATION

a large part of the child's attention, leaving little available for the matter i n hand. T h i s paper is not concerned w i t h the methods or practice of teaching, but I would strongly urge all those who are concerned w i t h y o u n g children to guard against this danger, by constant transition to and fro between c o m m o n and technical phraseology, appealing at once to the former at the least sign o f doubt or hesita­ t i o n . T h e l e a r n i n g of technical terms should not appear as part of the definite w o r k , or it w i l l inevitably be regarded as the major p a r t ; i t should come incidentally and by gradual transition, as I have suggested. 1

T h e only alternative to this evasion or suppression of i n t u i t i o n is to accept i t f r o m the commencement as the natural basis for p r i m a r y education. B u t to be of any avail, the acceptance must be unquestioned and complete ; every i n t u i t i o n w h i c h can be f o r m e d by the pupils must, without suggestion of doubt, be adopted as a postulate, none being deduced f r o m others w h i c h are themselves no more easy of acceptance. Such a course leads, i t need hardly be said, to considerable simplification i n the early treatment of any subject. F o r example, i n geometry the angle properties of parallel lines, properties of figures evident f r o m symmetry, and the theory o f similar figures (excluding areas) appear as postulates; i n the calculus i t is not proved that the differential coefficient of the sum of a finite number of functions is equal to the sum of their differential coeffi­ cients ; the statement is illustrated by, say, consideration 1

I t is no good to say, " C o m e now, what is an angle ? " A p p e a l first

to the tangible fact i n the child's m i n d by s a y i n g , " C a n n o t you see that those c o r n e r s must

fit ? " a n d t h e n r e m i n d h i m that " equal a n g l e s "

merely m e a n s the same thing.

INTUITION

29

of some expanding rods placed end to end, and at once commands acceptance. H e r e the question of terminology again arises ; I have often been struck, i n teaching school­ boys and students, by their slowness to accept this and similar results i n the calculus; the clue was given to me by a boy who remarked that i t was t a k i n g h i m all his t i m e to remember what a rate of increase was, and he could not manage any more at the m o m e n t . Since that t i m e I have avoided many seeming difficulties w i t h elementary and advanced pupils by appeal f r o m technical to familiar terms, always of course rephrasing the result i n the proper f o r m before leaving the matter i n hand. I t w i l l , I know, be t h o u g h t by many that this adoption of all natural intuitions involves an appalling lack of rigour. B u t I would ask those who are of this opinion to do one t h i n g before passing j u d g m e n t , and that is, to define and ex­ emplify w i t h some care the meaning of the t e r m " r i g o u r . " W h e n they have done this, I t h i n k they may be disposed to agree w i t h the answer to their accusation w h i c h I am now g o i n g to put forward. I t is that the scheme suggested is perfectly rigorous, provided that every deduction made f r o m the postulates adopted is logically sound; on the other hand, i t is admitted that the mathematical t r a i n i n g thus imparted is not complete, because no attempt has been made to analyse these i n t u i t i v e postulates into their com­ ponent parts, s h o w i n g how many must perforce be adopted i n the most complete system of deduction. I n other words, we may be rigorous i n regard to logical reasoning, or i n regard to lessening the number of assumptions w h i c h f o r m the basis of a science. T h e view for w h i c h I contend is, that i n a l l stages of mathematical education, deductions

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EDUCATION

f r o m the assumptions made should be r i g o r o u s ; but that i n the earlier stages every acceptable statement or i n t u i t i o n should be taken as an assumption, the analysis of these, to show on how small an amount of assumption the science can be based, being deferred. T o avert misapprehension, let me say again that I pro­ pose that, w h e n all intuitions are accepted as postulates, this should be done w i t h o u t question or discussion other than that necessary to give t h e m some precision. T o em­ bark o n a discussion of t h e i r nature, or to appear to cast doubt u p o n t h e m , would be fatal, as fatal as has been the apparently futile process of deducing one accepted state­ m e n t f r o m another. T h e p u p i l is already i n possession of a body o f accepted t r u t h ; let us b u i l d o n that and defer its analysis, or a n y t h i n g that pertains thereto, u n t i l he is sufficiently mature to appreciate the motive. T h e first course of mathematics would, then, range from arithmetic and analysis t h r o u g h geometry to mechan­ ics. I n this last subject there is little scope for i n t u i t i o n . Most of the mechanical intuitions formed by the race as a whole have been mistaken, and i t is just this fact w h i c h gives some indication of the proper commencement for the second course, i n w h i c h the intuitive postulates are to be analysed and reduced as far as possible. L e t the student learn s o m e t h i n g of the history of mechanics, realising that ideas w h i c h he regards as impossible and absurd were held, by m e n o f great eminence, w i t h faith just as strong as that w h i c h he places i n his geometrical postulates. T h e n let i t be suggested to h i m that this renders care i n regard to assumption o f vital importance, and so commence an analysis of the mechanical postulates, h i t h e r t o redundant,

INTUITION

31

obtaining deductions of one f r o m another to show their inter-connection. T h i s completed, and the task is not a large one, i t is natural to suggest that the postulates of geometry deserve some examination, and so, according to the t i m e available and the ability of the p u p i l , we may pass backward t h r o u g h a review of the foundations of geometry to an examination of the foundations of analysis and arith­ metic. I t is not, o f course, i m p l i e d that every student of mathematics can reach this g o a l ; few can ever get beyond some consideration of the foundations of geometry, w i t h a clear understanding of the end to be attained i n its general application to all sciences. B u t I do wish to put forward, w i t h such emphasis as I can, this general scheme o f math­ ematical education; namely, an upward progress, based o n i n t u i t i o n , f r o m arithmetic t h r o u g h geometry to mechanics, followed by consideration i n the reverse order o f the founda­ tions of each branch, the upward progress constituting the first course, and the downward review the second course. I t would, I believe, give an intelligible unity to the whole subject, a n d would do s o m e t h i n g to restore that purely intellectual appreciation w h i c h has so largely declined d u r i n g the past generation. Mathematics is a useful tool, but i t is also something far greater, for i t presents i n unsullied outline that model after w h i c h all scientific thought must be cast. I have endeavoured to show how this outline may be developed, starting f r o m those intuitions w h i c h are c o m m o n to us all, and e n d i n g i n an analysis demonstrating their true nature. T h e concrete illustrations, so necessary and i l l u m i n a t i n g i n elementary teaching, are so many draperies, fashioned to render this outline visible to those who cannot otherwise

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appreciate i t . E v e n the several branches—analysis, geom­ etry, mechanics—serve

the same e n d ; behind t h e m all

is the one pure structure of mathematical thought. T h e y who most appreciate the structure w i l l best fashion the draperies, and so render i t most clearly visible to those w h o m they instruct.

T H E USEFUL A N D T H E REAL

THE

USEFUL AND THE

REAL

A m o n g the many changes i n mathematical education d u r i n g the last twenty years, and a m o n g the many and often conflicting ideals w h i c h have directed these changes, one element at least appears t h r o u g h o u t ; a desire to relate the subject to reality, to exhibit i t as a l i v i n g body of thought w h i c h can and does influence h u m a n life at a m u l t i t u d e of points. T h e old scholastic ideal of development i n the most abstract way, the realities b e i n g allowed to take care of themselves, is exploded for this as for most branches of education ; i t is recognised that the separated mediaeval worlds of t h o u g h t and action must be replaced by a single w o r l d wherein each exerts profound influences on the other. O u r c h i l d r e n must learn to t h i n k , and to t h i n k about the w o r l d as i t now is and the manner of its evolution. Some few there may be w h o can w i t h profit to us all devote themselves to one or other side of this w o r l d of t h o u g h t and action, but the mass of m e n must be fitted to play their part between the two. So far all are agreed, but c o m m u n i t y of pious opinion has before now been k n o w n to result i n discord, and dis­ cord none the less acute because due to diversity of policy alone. Such has been the case w i t h mathematical educa­ tion ; the c o m m u n i t y of ideals just described has not resulted i n c o m m u n i t y of action ; i t is more nearly true that each man is a law unto himself i n his method of for­ warding t h e m . L i k e most disorganised armies we have 35

36

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EDUCATION

our shibboleths, and a m o n g the most p r o m i n e n t are " r e a l , " " u s e f u l , " " c o n c r e t e . " A n examination of what these do and should represent may not be without profit. Starting f r o m agreement that the w o r l d of thought is to be related to the w o r l d of action or reality (not thereby dependent u p o n or l i m i t e d by that world), the natural course is to attempt to f o r m some concept of the particular w o r l d of reality w i t h w h i c h we are concerned. Suppose that one desires to explain the principles o f the calculus to an assemblage of doctors; tables of population, mortality, and the l i k e f o r m one obvious w o r l d of reality f r o m w h i c h thought can be developed; i f to an assemblage of mer­ chants, statistics of trade and finance w o u l d f o r m such a world, and so o n for other avocations. B u t suppose that the assembly consisted o f m e n engaged i n no one p u r s u i t ; the difficulty would be greatly enhanced, for there would be no obvious w o r l d o f reality c o m m o n and familiar to them a l l . So also i n his dealings w i t h y o u n g c h i l d r e n must the teacher of mathematics determine fitting worlds of reality and develop his instruction for t h e m . B u t what is reality ? W h a t considerations determine the entities w h i c h have this attribute ? F o r me, m y hands, m y furniture, this town, E n g l a n d , Cromwell, Macbeth, the binomial theorem, are all real ; but Cromwell's hands, the furniture i n a strange house, Hepscott ( I take the name at random f r o m a gazetteer), F a n n i n g Island, B e n Jonson, H e d d a Gabler, n i t r a t e s — a l l lack reality. Each one o f the first is related to some definite recognisable sensation or concept of m y own, but each of the second is (for me) a mere name w h i c h bears no relation to any such sensation or concept; I k n o w n o t h i n g of F a n n i n g Island, nor sufficient

THE USEFUL AND T H E REAL

37

of B e n Jonson to distinguish h i m f r o m other writers of his t i m e ; I have not read Ibsen, and I k n o w little chemistry. T h e essence o f reality is thus found i n definite recognisable percepts or concepts, and is therefore a func­ t i o n of the individual and the t i m e ; what is real to me is not necessarily real to another, and m u c h that was real to me i n childhood is no longer so. I t is for the teacher to determine the realities of his pupils and exemplify mathe­ matical principles by as many as are suitable for the purpose. H e w i l l also find i t necessary to enlarge their spheres of reality, but he must avoid confusion between a name and a t h i n g ; he must, for example, make sure that his pupils k n o w what a parallelogram is before they use the name. I t is at this p o i n t that various policies have arisen, des­ pite general agreement o n ideals. O n e of these confuses the many worlds o f reality, different for each individual, w i t h some absolute w o r l d of reality supposed to be com­ m o n to a l l . T h i s absolute w o r l d is usually based o n those applications of mathematics which have some commercial or scientific utility, such utility being considered to involve reality for the p u p i l . T h e result of this confusion o f the useful w i t h the real is seen i n problems w h i c h deal w i t h such mysteries as resistance i n pounds per t o n weight, the extension of helical springs, efficiency and load, ton-inches of t w i s t i n g moment. T o all children (and many adults) these phrases are as meaningless as the symbols of the purely scholastic algebra of t h i r t y years ago ; they are merely a cumbrous way of w r i t i n g the x and y of that alge­ bra and i m p l y as little to those for w h o m they are intended. B u t their use may t e n d to impart an idea that realities are being dealt w i t h — an idea thoroughly vicious i n that i t

38

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replaces entities by words. W e may name entities w h i c h are direct sensations, and we may name entities w h i c h are pure creations of the i m a g i n a t i o n ; but to imagine that a name w h i c h is co-related to neither sensation nor imagina­ t i o n possesses any sort of reality is the grossest of errors. T o o many teachers are content to use words for w h i c h they have no definite meanings, and to allow their pupils to imagine that they have acquired something i n learning such words ; but we need not go out of our w ay to spread this error, the more so as we are concerned w i t h the one subject w h i c h should suppress i t most completely. 7

I t is not, of course, suggested that any existing courses of mathematics are l i m i t e d to such applications alone. B u t there is an obvious tendency t o judge applications by such standards, a t t r i b u t i n g more and more importance to those w h i c h accord w i t h t h e m . A n excellent instance of such judgments is the condemnation of the traditional problems dealing w i t h tanks w h i c h are emptied and filled simultane­ ously by different pipes. I t is argued that no adult ever deals w i t h a cistern i n this way, and that the problems should therefore be replaced by others h a v i n g more reality. T h e t e r m " r e a l i t y " begs the whole question, for i t has no absolute m e a n i n g for all people at all ages and must be defined by those who use i t . A n d i t is here confused w i t h utility, a very different attribute. T h e essence of the con­ tention is that no application should be used i n education unless i t is of actual use i n some branch of science or walk of life. T h i s is a far cry f r o m the pupil's w o r l d of reality ; the formalist attempted to transport h i m to a w o r l d of ab­ stract thought wherein the entities are typified by letters x and but our utilitarian proposes to l i m i t the play of

T H E USEFUL AND T H E REAL

39

his imagination to matters used by adults, no matter how far these may be removed from his cognisance or interest. T h e r e is at bottom little difference between the two, but the formalist is the more open i n that he does not cloak his meaning under a mass of words w h i c h are f u l l of sound and signify n o t h i n g . A variant o f this school may reply that they are being unjustly accused; that they are i n entire accord w i t h the rejection of matters such as voltage and twisting m o m e n t on the g r o u n d that they have no reality for the pupils con­ cerned, but that there are plenty of applications w h i c h are real and also useful. T h i s may be so, t h o u g h examination of modern text-books hardly supports the c l a i m ; but i n any case we cannot o n such lines develop mathematical t h o u g h t f r o m any large portion of the pupil's w o r l d of reality ; i t is related to those parts only of that w o r l d w h i c h coincide w i t h the worlds o f various adults, and these may well be neither the most interesting nor the most familiar portions of his own w o r l d . A second policy, exemplified for the most part i n con­ nection w i t h geometry, interprets the child's world o f reality as the w o r l d of his senses, and more particularly the senses of sight and touch, and so is allied w i t h the concrete rather than the useful. I t endeavours to develop t h o u g h t f r o m manipulations and measurements performed by the p u p i l himself, and is thus l i m i t e d to the perceptory or concrete portion of his realities. I n itself and so far as i t goes this is an entire and most valuable gain as compared w i t h the practice of t h i r t y years ago ; the pupils feel that they are dealing w i t h matters w i t h i n their own personal cognisance instead of abstractions w h i c h are evidently familiar only

4°

MATHEMATICAL

EDUCATION

to m e n w i t h w h o m , intellectually, they have and w i l l have little i n c o m m o n . Unfortunately, however, there has been a strong tend­ ency to l i m i t the w o r k to this concrete domain, refusing any part to that world of imagination which is, especially i n children, just as real and a great deal more v i v i d . I n t r o ­ ductory courses of geometry consist of the construction and measurement by the p u p i l of figures whose dimensions are prescribed. T h e y develop a detailed knowledge of perceptory space but make no use of that m u c h larger and more i m p o r t a n t conceptual space wherein the creations of his imagination move and have their being. Travels, ad­ ventures, romances, history, and the hundred and one utterly useless but apparently practical things which interest a boy are situated i n this space, and here, as well as i n the smaller concrete space of t h e senses, should the w o r l d of thought be exemplified, for these t h i n g s also are realities for h i m . T h r e e distinct policies have now been discussed : the first rejects all applications and insists throughout o n devel­ opment i n abstract t e r m s ; the second insists that illustra­ tions must be drawn f r o m applications w h i c h are relevant to some branch of science, industry, or commerce; and the t h i r d insists that development must originate i n the i m m e ­ diate evidence of the senses. O f course, no m a n or body of m e n holds one of these views to the entire exclusion of the other two, nor is the w o r l d of imagination entirely ignored i n current practice. B u t most text-books and writ­ ings on mathematical education are influenced mainly by some one of them, and may be placed i n a class w h i c h holds that particular policy as paramount. T h e r e is most i n

T H E USEFUL AND T H E REAL

41

common between those who hold the second or t h i r d view, for they give a c o m m o n allegiance to the use o f reality and differ only i n the scope of the t e r m . M a n y teachers are, indeed, influenced by considerations of utility i n algebra and considerations of concrete reality i n geometry, their utterances on one subject often contradicting those on the other. N o w among the many uncertainties and conflicts w h i c h surround these (as all) questions of education, two state­ ments at least stand out as certain beyond dispute. T h e first is that the operations and processes o f mathematics are i n practice concerned at least as m u c h w i t h creations of the imagination as w i t h the evidences of the senses; i t is enough to m e n t i o n points, complex numbers, ether, electric charges, to make this plain. T h e second is that the purpose of mathematical education is to put the p u p i l i n a " mathematical way " ; to permeate his whole being w i t h the elementary principles of the science so that he w i l l apply t h e m spontaneously i n considering any matter to w h i c h they may be relevant. T h e formalists held that i f principles were imparted i n their utmost generality, each individual could and would make such applications as he m i g h t require, a statement not justified by experience and not i n accord w i t h such knowledge of the m i n d as we possess; the moderns believe that principles can only be seen by their exemplification throughout the w o r l d of reality of the p u p i l . T h e formalists thus seek u n i t y of treatment for a class i n generality of presentation, the moderns seek it among the experiences and concepts of the various pupils. Fortunately for education i n general, this modern search is certain to prove successful as regards children, because

42

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their experiences and imaginations r u n i n grooves more or less alike ; they are interested i n puzzles, hidden treasures, travels, railways, ships, and the like, and problems concerned w i t h these entities are real to t h e m no matter how absurd they appear from the standpoint of practical life. T h e i r educational utility is not to be measured by their commercial or scientific value, but by their degree o f reality for the pupils under instruction. P u t t i n g the matter i n more or less mathematical phrase­ ology, we may say that the mathematical instruction of a beginner must be exemplified by a m a x i m u m number of his realities i n order that the principles may permeate his whole b e i n g ; i n dealing w i t h a class we must therefore find the greatest c o m m o n measure of their realities and w o r k f r o m that. I f the class is composed of adults h a v i n g v a r y i n g antecedents, this c o m m o n measure may be small compared w i t h the realities of any one member; but i f the pupils are children, i t is large i n comparison w i t h their individual realities, and the task of the teacher is corre­ spondingly simplified. L e a v i n g generalities w h i c h may have appeared somewhat vague, we may now consider a few problems w h i c h are real for children but not directly useful to t h e m or any one else. F i r s t take the type already mentioned, which deals w i t h the e m p t y i n g and filling o f a tank. T h e r e is no doubt that this is sufficiently real for any c h i l d ; he can visualise the whole process, and its value is increased because the entities are imagined and not perceived t h r o u g h the senses. T h e purpose served is the exemplification of the m e t h o d o f adding or comparing several rates by reducing all to a common unit, an idea sufficiently important i n after life.

THE

USEFUL AND T H E REAL

43

Those who attack such an illustration must find others w h i c h w i l l serve the same purpose and satisfy their test of utility, and i n d o i n g this they w i l l i n all probability pass beyond the limits of reality. T h e r e is no doubt that the e m p t y i n g of cisterns, the coincidence of clock hands, and other seeming trivialities do exemplify the h a n d l i n g of rates i n ways w h i c h are more real to y o u n g children than others w h i c h have more actual utility, and they are therefore to be welcomed rather than condemned. T h e mistake i n their treatment, and as gross a mistake as could well be made, has been their g r o u p i n g by subject matter instead of p r i n ­ ciple. A l l questions w h i c h deal w i t h one principle should be grouped together and the subject matter varied continually. N e x t consider the Progressions, which have of late been attacked on the score that they are comparatively useless i n mathematics or anywhere else. T h i s is true, and ad­ vocates of their retention have done their cause no good by saying vaguely that they have their uses and t h e n fail­ i n g to give specific instances. T h e y do, however, provide a number of problems w h i c h have reality for children, and they exemplify three most important matters : the concept of a series, the value of w h i c h extends far beyond math­ ematics ; the insight w h i c h can be gained by a proper g r o u p i n g of various entities; and the construction of a formula or law to cover any number of discrete cases. Consider again the well-known problem i n the calculus of a man who is on a common and wishes to reach a point on a straight road, along which he can walk more quickly than on open country, as soon as possible. I f such a problem ever has practical utility, i t is not for one man i n ten thou­ sand, and to regard it as i n any way generally useful is

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obviously grotesque ; but again i t is real for those who study it, and i t exemplifies the comparison of different modes of transition f r o m one state to another, and the selection of the most suitable. A n o t h e r illustration is provided by the use of statistical graphs i n the introduction of the calculus. Such graphs are of service i n exemplifying the meaning of a differential coefficient and a definite integral by means w h i c h possess reality for the students, and their whole function is described i n this statement. N o w i t may well happen that a set of statistics w h i c h have no practical use of any k i n d , or are even i n actual disaccord w i t h the results o f some branch of science or industry, may serve these purposes better than others which have some direct use or are i n accord w i t h experience. F o r example, excellent problems can be made concerning the consumption of coal by locomotives, but they would never occur i n the practice of any engineer, nor would the numbers w h i c h happen to give good graphs occur i n the w o r k i n g of any conceivable locomotive. B u t this is i n no way to the detriment of the problem for the purposes of instruction. T h e inaccuracy of the information contained i n the figures is surely immaterial i f students are told that actual numbers can be found i n any handbook for engineers should they ever chance to need them, and no other objection seems relevant to the purpose of the problem. I t exemplifies principles t h r o u g h illustrations w h i c h are real for the particular students, and thus fulfils its a i m . 1

These examples exhibit the tests by w h i c h applications should be judged. T h e y must exemplify those leading ideas 1

T h e y would not be real for a class of locomotive engineers, a n d the

example would not be u s e d for s u c h a class.

T H E USEFUL AND T H E REAL

45

which it is desired to impart, and they must do so t h r o u g h media w h i c h are real to those under instruction. T h e reality is f o u n d i n the students, the utility i n their acquisition of principles. T h e outcome of our discussion is, then, that illustrations must above all be r e a l ; they must be useful as well, i f that be possible, and particularly w i t h reference to other branches of study such as physics; but reality is the crucial test. A n d reality is a function of the individual and the time, so that no absolute schedule of the more and the less real can be devised ; but there is sufficient c o m m u n i t y between children o f the same age to handle t h e m i n groups, while adults m i g h t , o n the other hand, require classification i n regard to their realities before they could receive efficient instruction i n groups. M a n y problems w h i c h interest and even excite children are to t h e m hope­ lessly banal, and others must be used more i n touch w i t h their particular spheres of reality. Finally, each principle must be exemplified i n as many ways as possible so that unity may be perceived i n principle rather than subject matter. W e have travelled far from the useful applications of mathematics i n our quest for fitting illustrations ; we have been led to consider reality as the proper criterion, and to recognise that the t e r m is essentially relative. B u t so also is "useful " a relative t e r m ; what is useful for one purpose is useless for another, and i t may well be said that many applications of mathematics which are grotesquely useless in any branch of science or commerce are of the utmost use i n education for their v i v i d illustration of ideas so abstract as to be otherwise vague or invisible.

SOME U N R E A L I S E D P O S S I B I L I T I E S OF M A T H E M A T I C A L E D U C A T I O N ( A n address delivered to the Mathematical A s s o c i a t i o n , and reprinted from The Mathematical

Gazette,

M a r c h , 1912)

SOME UNREALISED POSSIBILITIES OF MATHEMATICAL EDUCATION T h e last half-century has seen a great and significant change i n the popular estimation of mathematics. F o r m e r l y the subject was regarded as utterly unpractical and there­ fore useless i n the narrow sense of this term, t h o u g h i t was recognised as p r o v i d i n g a t r a i n i n g unique i n its char­ acter, i n logical thought and i n accurate expression. N o w it is regarded, and correctly regarded, as having enormous practical importance i n science and engineering. Most, i f not all, of those discoveries and inventions w h i c h are so profoundly m o d i f y i n g civic and national life have found their origin, or development, or both, i n the labours of mathematicians, and this fact is widely k n o w n . T h e mathematician is no longer regarded as a dreamer of dreams; he is classed w i t h the doctor, the engineer, the chemist, and all those whose specialised labours have had immense i m p o r t for the human race. B u t simultaneously a change of no less magnitude has taken place i n the mathematical w o r l d . T h e type of i n ­ vestigation which bore such fruit i n the hands of Faraday, Clerk M a x w e l l , K e l v i n , and many others no longer occupies the attention of those who are i n the forefront of mathe­ matical investigation. T h e theories of pure number, of space, of functions, and such names as D e d e k i n d , Cantor, Grassmann, K l e i n , and, i n our own country, Hobson, W h i t e h e a d , and Russell, have little or no connotation for 49

SO

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EDUCATION

the outer w o r l d . I n so far as this outer w o r l d is cognisant of their existence, these theories, and the men to w h o m they are due, appear as chimerical and unpractical as would the labours of Clerk M a x w e l l have appeared to the L a n ­ cashire cotton spinner of 1850. A n d I fear that this view is too often shared, consciously or unconsciously, by mathe­ maticians themselves, and especially by those who teach the subject. Here, they say or t h i n k , is a type of thought or investigation of great interest to those who can appre­ ciate it, but it is utterly and permanently out of touch w i t h the w o r l d at large. I t can have no relevance or i m p o r t for the ordinary boys and girls who learn mathematics at school, and can i n no way assist t h e m to become efficient citizens. B u t is this really the case ? H e would be a bold man who would say w i t h certainty that any branch of scientific investigation must be regarded, once for all, as h a v i n g no bearing on the development of the individual or race. Is not the better answer that the practical i m p o r t of these investigations has not yet been perceived ; that i t behoves all mathematicians, but especially those w h o are engaged i n teaching, and therefore have some knowledge of the youthful m i n d , to do what they can to correlate this w o r k w i t h the outer world, and to examine to what extent i t can now influence the manner or matter of teaching i n our schools ? T h e question w i l l probably receive an affirmative answer f r o m each one of you, but you may perhaps add that I am w a l k i n g i n the mists w h i c h hide from us the development of future centuries; that sufficient unto the day is the vision thereof; and that the ground to which I invite you is a morass w h i c h may conceivably be made firm by our great-grandchildren.

SOME U N R E A L I S E D POSSIBILITIES

51

Nevertheless, I am g o i n g to ask you to bear w i t h me while I endeavour to convince you that we can now com­ mence to bridge the morass. I admit that i t is one. H e s i t a t i n g and imperfect our endeavours may be, but I am honestly convinced that the t i m e is ripe for a com­ mencement, and that the future of mathematics as a universal subject i n the curricula of schools depends, i n some part at least, on this commencement being made at once. M y ground for this conviction is best stated tersely. I believe that the modern theories of pure mathematics are destined to i l l u m i n e our understanding of the human m i n d and of cities and nations, just as the pure mathe­ matics of fifty years ago has already i l l u m i n e d the previ­ ously dark and chaotic field of physical science; that modern mathematics is or w i l l be to psychology, history, sociology, and economics as has been the older mathe­ matics to electricity, heat, light, and other branches of physical science. F o r example, i t may well be that the theory of sets of points or the theory of groups w i l l find fruitful application i n economics. Y o u w i l l see that I am suggesting that the range of applied mathematics may be widened far beyond its present scope. I t was asserted recently at a meeting of head-masters that the reign of pure mathematics was closed. W o u l d i t not be more accurate to say that pure mathematics has of late extended and co-ordinated its dominions to an amazing extent, and that corresponding extensions of applied mathematics have yet to be found ? I f I am right, then our subject has an irresistible claim. W e may trust our lives to engineers and scientists, just as we entrust our bodies to doctors and surgeons ; but each member of a human society should,

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EDUCATION

so far as he may, be competent to analyse and estimate for himself the w o r k i n g s of his own m i n d and the devel­ opment of the society of w h i c h he is a unit. I n the more detailed remarks w h i c h I am about to make, I w i l l ask you to bear i n m i n d that their m a i n inspiration and justi­ fication lies i n what I have just said — that they represent an individual attempt to relate mathematical education to h u m a n thought and social development. Mathematics has been defined by Russell as the class of propositions, " I f A , then and is applied to classes of entities concerning which certain propositions A are assumed; the t r u t h of these is no concern of the subject. T h e entities f o r m the universe of discourse. T h e y can be ordered i n respect of each of the attributes w h i c h charac­ terise their class. T h i s universe of discourse may be of any number of dimensions from one u p w a r d s ; i n arith­ metic i t is one-dimensional, and i n geometry i t should be three-dimensional, but is more often two-dimensional. I may remark i n passing that some attempt to estimate the number of dimensions, that is, of quantities required for exact specification, of the entities discussed i n such subjects as economics would often t h r o w considerable l i g h t o n these subjects. T h e abstract idea of entities and their dimen­ sions is too often w a n t i n g . Hence arithmetic forms the basis of mathematics, since i t explores the properties o f onedimensional fields. A n y treatment of arithmetic which fails to explore the whole domain of such fields is ipso facto incomplete, and its v i c t i m is i n possession of an imper­ fect instrument which cripples h i m alike i n concrete and abstract applications. M y first plea is, therefore, for a mathe­ matical treatment of arithmetic from the earliest stages.

SOME U N R E A L I S E D POSSIBILITIES

53

T h e r e is m u c h w h i c h m i g h t be said concerning integers and fractions, and i n particular scales of notation. M y omission of these subjects is only to be interpreted as an admission that decimals and the theory of exact measure­ m e n t are of more immediate importance, and must occupy such t i m e as I can devote to arithmetic. T o m y t h i n k i n g , young children are h u r r i e d on to fractions far too soon. T h e r e are many unexplored fields of concrete problems, possessing real interest for y o u n g pupils, the study of w h i c h would give a m u c h firmer basis for future develop­ ments than is now obtained. A n d the proofs of such simple rules as " casting out the nines " may provide easy exercises i n deduction, not w i t h o u t value. T o commence, then, w i t h measurement. W h e n , i n actual practice, one measures a length, there are three distinct objects, any one of w h i c h may be i n view. T h e purpose may be either ( i ) to state a l e n g t h greater than that of the given object, but as little greater as may be, or (2) to state a l e n g t h less than that of the given object, but as little less as may be, or (3) to state two lengths as close together as may be, between which the given object lies. I venture to suggest that t r a i n i n g i n measurement can only become of any value (other than manipulative) if i t proceeds on these lines, phrases such as " nearly " and " e x a c t l y " b e i n g abolished as inexact, and therefore unscientific. " Nearly " is useless u n t i l we are told how near or w i t h i n what nearness, and " exactly " only means " as nearly as I can see." B y the use of a vernier — the theory of w h i c h should be included i n every course of arithmetic — children should learn how nearly they can see, and then say, for example, 13-40111. w i t h i n 0-2 m m .

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W e should thus sweep away all the loose statements w h i c h are, I honestly believe, responsible for m u c h of that lack of accurate thought which is the subject of present com­ plaint, and replace t h e m by a t r a i n i n g i n the exact expres­ sion of practical measurements, the final f o r m being of the type " between 7-38 and 7-39 c m . " T h e g r o u n d is now prepared for the extension of the idea of number, this being done, probably, i n connection w i t h mensuration, that is, by questions such as " F i n d the length of the side of a square whose area is 2 sq. i n . " Few trials are necessary i n order to ensure conviction of the fact that the number of inches is not a fraction, and systematic approximation f r o m above and below is at­ tempted. B y actual trial, u s i n g multiplication only, i t is found that the following pairs of numbers are respectively smaller and greater than the number r e q u i r e d : ( 1 , 2), (1-4, 1-5), (1-41, 1-42), and so on. So far n o t h i n g more appears than can be realised by measurement, but i t is at once seen that (1) this process can be continued indefi­ nitely, given t i m e and energy; and (2) that there is no l i m i t to the closeness of the approximation. T h e human m i n d , by this systematic approach, has thus ridden rough­ shod over the imperfections of physical measurement. T h e latter leaves, and must always leave, an unexplored gap w h i c h cannot be diminished, but the method of suc­ cessive approximation enables us to d i m i n i s h the gap below any l i m i t , however small. N o w this process, i f carefully developed, is not beyond the comprehension of y o u n g pupils, and i t may fairly be said to contain the germ of any proper study of functions and the calculus, whether this be undertaken on a graphical

SOME U N R E A L I S E D POSSIBILITIES

55

or analytical basis. I n either case this method of inclu­ sion between converging pairs is essential to any exact comprehension of the subject. A n d beyond this i t develops the theory of pure number so far as to give the pupils — however unconsciously — an early example of a perfect mental structure, fashioned by extension from concrete experience, and i t gives t h e m the only true ideal for the exact estimation of any set of phenomena. Shortly, then, I suggest the continuous development of the idea of a cut or ScJudtt o f the rational numbers, commencing i t at an early age i n connection w i t h a scientific treatment of simple measurements, the purpose being to give a true concept of number i n its relation to measurement. I next make some reference to algebra, stating first that I am not to be taken as i m p l y i n g that the subject should be taught before geometry. O n the contrary, I am convinced f r o m actual experience that geometry should have been studied for two years at least before algebra is commenced. A t the risk of appearing to raise needlessly large issues, I must ask the question, W h a t is an algebra for our present purpose, and what educational purpose may be served by its study ? T o m y m i n d there are two essential steps i n the development of an algebra: the first is the development of a symbolism w h i c h is usually suggested by certain combinations of entities, for example, a + b = b + a, ab—ba; and the second is the extension of this symbol­ ism to cases w h i c h bear no interpretation i n terms of these entities, and its subsequent application to other classes of entities. B y this I mean the interpretation of symbols such

56

MATHEMATICAL

a s

m

e

a

c

n

EDUCATION w

m

3 —• S> 3 + ^~ 7> ° f c h the entities origin­ ally considered are found to f o r m part of a larger class. I propose to allude shortly to each of these steps. A s regards the first I have little to say, for the unrealised possibilities w i t h which I am concerned are here not con­ spicuous. B u t I do feel that the laws of algebra have re­ ceived far too little attention i n current and past teaching, i n that their interpretation is so exclusively confined to the domain of pure number. A n y ordinary boy or g i r l of 1 5 is able to realise that a + b = b + a and a + (b + c) = a + b + c are true w h e n a, b, c are vectors, and to make simple deductions therefrom, as, for example, the proof of the median properties of a triangle. Such work, even i f only a little t i m e be devoted to it, gives a larger and truer view of algebra as a language w i t h more than one inter­ pretation. A n d i t gives the idea of an algebra relevant to any field of human thought, an idea far more s t i m u l a t i n g and fruitful for the ordinary man or woman than the nar­ rower view of one absolute algebra, w h i c h is too often the only result o f our teaching. B u t , w h e n all is said and done, this first part of the subject only presents itself as the formation on methodical lines of a shorthand language; every step i n the solution of equations, factorisation, or what you w i l l , can be expressed i n words whether the entities be numbers or vectors, and no new methods are involved. B u t now take the second step, the interpretation of algeP q

braic symbols such as x or V — y w h i c h have at first no meaning. T h e process involved is, or should be, purely logical. W e assume that such laws of combination as x x x =x and (x ) = x must also hold i n cases f

m

n

m + n

m n

mn

SOME UNREALISED POSSIBILITIES

57

which already bear interpretation, and then find that the p 9

p

one interpretation x = ~^lx is consistent with each of these laws. I t is too often assumed without proof that, because the one law x x x" = x leads to this interpretation, the other laws, such as (x ) = x , must also be true in this case. I do not believe that complex exercises in the manipulation of fractional and negative indices can be of any profit, but I am convinced that a complete and logi­ cal interpretation of these indices, if only in particular numerical cases, can and should form part of every course in algebra. I t is one of the best examples of constructive logic to be found in elementary mathematics, and it gives a sense of new methods for the discovery of hidden fields of entities which is hardly to be found elsewhere. Passing now to imaginary expressions, I would suggest that the geometrical interpretation of these is not beyond the capacity of pupils of seventeen or eighteen years of age, and, further, that it provides a valuable link between the symbolism thus far developed and geometry of two dimensions. Not much knowledge of trigonometry is re­ quired in order to understand the expressions a + bi, (a + bi) (c + di), nor is it necessary to plunge into useless elaborations. The pupils have ample scope for exercise in written descriptions of the processes ; for example, in show­ ing that this interpretation satisfies the laws r , =
m + n

m

n

mn

zJ l

2

g

x

z

z

z

l

1

z

1

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MATHEMATICAL

EDUCATION

A t this point the pupil may well review his experience of algebra. O n e after another apparent impossibilities of interpretation have been surmounted. I s there an end to the process, or can we go on i n this manner indefinitely ? T h e answer is, of course, that the performance of any algebraic operation on a quantity of the type a + bi pro­ duces another quantity o f the same type, and the process is closed. I would suggest that there is no inherent diffi­ culty i n the proof of this, granted a knowledge of elemen­ tary trigonometry, and that the view of algebra so gained is of real value as showing that the exploration of the field of entities under discussion has been completed. I f the boys and girls of the future can reach this point, they may, I admit, forget, and r i g h t l y forget, many of the details of their education, but this idea of the exploration of a field of entities, and the demonstration that this exploration is complete, may remain w i t h t h e m . I f this be so, I do not t h i n k that you w i l l question its value i n dealing w i t h the problems which present t h e m s e l v e s — o r should present themselves — to every citizen of a modern state. 1

Finally, I must make some reference to geometry. T h e primary value of the subject is, i n m y opinion at least, that it develops a power of dealing logically w i t h manifolds of two and three dimensions. W h e n we prove that, i f A and B are fixed points, and the p o i n t P moves so that the angle ABB is constant, then P must lie on one of two arcs of circles, we are selecting f r o m all the points of the plane 1

T h e entities are typified by the points of a plane, denoted by sym­

bols s u c h as 2, — \, V3, 7 + 4 /, a n d the exploration is not c a r r i e d to three dimensions, as might have b e e n expected after the extension from a line to a plane.

SOME U N R E A L I S E D POSSIBILITIES

59

those w h i c h enjoy a certain property, and are showing that a certain other property is a necessary consequence of this principle of selection. A n d we develop the consequences i n order to encourage the dormant faculty of selecting some set o f a class of entities (the points i n the plane) and ex­ a m i n i n g their properties, not by the imperfect method of measurement, but w i t h the relentless certainty of logical reasoning. B u t I w i l l not dilate further o n this aspect of geometry, as i t can hardly be called an unrealised possi­ bility of education. M y first suggestion i n regard to geometry is that some simple idea of methods of transformation, such as projec­ tion and inversion, should f o r m part of every course, at any rate for pupils w h o continue the subject u n t i l they are eighteen or nineteen years of age. Such transformations contain the idea, not illustrated so completely elsewhere i n elementary mathematics, of a correlation between two sets of entities, such that to each entity of one set corre­ sponds a definite entity of the other set; and f r o m the k n o w n properties of one set we derive properties of the other set. I t may, I know, be said that the study of graphs involves this idea, but graphs deal w i t h one-dimensional sets only, and a general idea cannot be gained by one i l ­ lustration. T h e problems w h i c h concern both the ordinary citizen and the w o r k m a n i n the trades must often involve sets of entities of several dimensions, and i f he has at­ tained to some idea of the correlation of such sets, and the examination of a new set i n the l i g h t of k n o w n proper­ ties of an older set, he must thereby have more likelihood of f o r m i n g some definite conclusions instead of floundering i n vague uncertainties.

6o

MATHEMATICAL

EDUCATION

M y last suggestion, and perhaps the most startling at first sight, is that older pupils should be given some idea of the nature of non-Euclidean geometry. One of the most vicious fallacies w i t h w h i c h we are encumbered is the idea that our postulates of space, and i n particular the parallel postulate, possess an absolute certainty w h i c h is denied to every other statement that is the result of experience. M o s t of us regard the parallel postulate as more obvious and certain than, say, the statement that all men must die some day, and we are utterly w r o n g i n so doing. A n outline of the idea and history of non-Euclidean geometries — I would refer especially to Poincar^'s illus­ tration and the recent paper by C a r s l a w — is sufficient to dispel the idea, and to exhibit our space postulates as mere assumptions w h i c h fit our experience more simply and nearly than any others which can be made. I a m not speaking at random ; I have aroused keen interest i n a f o r m o f classical specialists whose knowledge of geometry was distinctly l i m i t e d . T o what end, y o u may ask. I n showing the true relation between t h o u g h t and experi­ ence, the manner i n w h i c h the m i n d deals w i t h the sensa­ tions w h i c h reach it f r o m the outer w o r l d . Far as we have progressed, the saying " M a n , k n o w t h y s e l f " still has force. N o experience w i t h which I am acquainted shows so conclusively the relation of each of us to the universe as the discovery that the supposed certainties of space are pure assumptions ; as much so as Newton's laws of gravitation and motion, or D a r w i n ' s theory of evolution. 1

1

S e e J . W . Y o u n g , Fundamental Concepts of Algebra and Geometry; also the article b y P r o f e s s o r C a r s l a w in Proceedings of the Edinburgh Mathe­ matical Society, V o l . X X V I I I ; W . B . F r a n k l a n d , Theories of Parallelism.

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61

Y o u may, I fear, regard me as an unpractical visionary who has put before you a host of ludicrously impossible suggestions. B u t I would ask you to stop and consider whether they really are impossible, and I w o u l d r e m i n d you that I have suggested n o t h i n g that has not been at­ tempted, i n outline at any rate, w i t h ordinary pupils, and w i t h some measure of success. A n d I would ask you to remember one t h i n g more. T h e whole w o r l d is g o i n g t h r o u g h a transformation, due i n part to scientific and mechanical i n v e n t i o n and i n part to the g r o w t h of sepa­ rate nations, each w i t h its own methods and ideals, o f w h i c h no m a n can see the outcome. O u r function, the function of all teachers, is to produce m e n and w o m e n competent to appreciate these changes and to take their part i n guid­ i n g t h e m so far as may be possible. Mathematical t h o u g h t is one fundamental equipment for this purpose, but mathe­ matical teaching has not hitherto been devoted to i t , be­ cause the need has but recently arisen. B u t now that i t has arisen and is appreciated, we must meet i t or sink, and sink deservedly. N e i t h e r the arid f o r m a l i s m of older days nor — I say i t i n no spirit of disrespect — the workshop r e c k o n i n g introduced of late w i l l save us. T h e only hope lies i n grasping that inner spirit of mathematics w h i c h has i n recent years simplified and co-ordinated the whole structure of mathematical thought, and i n relating this spirit to the complex entities and laws of modern civil­ isation. E v e n t h o u g h every suggestion that I have made be fallacious and impossible, this one statement remains, and the future lies w i t h those who first achieve success i n directing mathematical education to this end.

T H E T E A C H I N G OF E L E M E N T A R Y ARITHMETIC ( A n address delivered to the S o u t h e a s t e r n A s s o c i a t i o n of T e a c h e r s of Mathematics,

and

reprinted

from March,

the Journal 1912)

of

the

Association,

THE

TEACHING

OF

ELEMENTARY

ARITHMETIC I propose to commence m y discussion of this subject by raising the question, a question joyous of sound to many a boy and g i r l wearied w i t h obvious futilities — W h y should we teach arithmetic at all ? I raise i t i n no whimsical or revolutionary spirit, but i n order that we may, if possible, agree upon the motives w h i c h determine the appropriation of so many valuable hours i n the life of a c h i l d to this one subject. O u r mission is not merely to occupy our pupils' time, nor to make t h e m efficient but unintelligent beasts of burden ; i t is to educate t h e m to take their places as efficient citizens of a free c o m m u n i t y . I t is i n the interpretation of this mission that subjects should be included i n the curriculum, and i n its furtherance should guidance as to matter and method of teaching be found. B u t , some may say, what has this to do w i t h teachers themselves ? A r e they not i n the hands of those who draw up schemes and syllabuses, and can they w i t h profit do more than carry out such instructions as they receive ? T h e question is not unnatural, but i t is based on grave misconception of the duties and privileges of each and every teacher. Schemes and syllabuses there must be, and by them all must be bound w i t h i n reasonable limits, or anarchy w i l l r e s u l t ; but their interpretation is i n the hands of the teachers themselves. T h i s interpretation can be 6s

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performed either as a mechanical duty, or i n free and w i l l ­ i n g co-operation, and those w h o ask the question I have suggested i m p l y that they regard their duties as mechani­ cal rather than co-operative ; that they attend to the letter rather than the spirit. T h e better method is, surely, to endeavour to appreciate the motives of the schemes under which we work, and to shape t h e m to the best advantage for our pupils. I n so far as this is done, i n so far w i l l our profession acquire its proper influence i n the general con­ duct of education, and associations such as this may do m u c h to that end. F r a n k l y , I am one of those who t h i n k that the body — I wish I could say corporate body — of teachers should have more voice i n educational affairs than is now the case, and the remedy is largely i n our own hands. I t is by consideration o f the why, as well as the how, of teaching that we shall best utilise our unique experience a m o n g the c h i l d r e n themselves, and so gain our true position. W h y , then, do we teach arithmetic ? First, of course, because a certain m i n i m u m knowledge is essential to the conduct of life. W e must all be able to use money, keep our o w n simple accounts, and so o n ; but to how m u c h does this amount ? A t most to simple operations i n sums of money, lengths, and so on ; certainly not to the cum­ brous barbarisms w h i c h disfigure the pages of so many text-books. " F i n d the cost of 17 tons 11 cwt. 7 qr. 14 l b . at £9 16s. 4 ^ d . per t o n , " or " F i n d the compound interest on ^ 2 7 3 16s. 7 d . for four years at 3 | per cent, per a n n u m paid half-yearly." W h o on earth wants to do these sums i n everyday life but a merchant's clerk, and what conceivable

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mental value can they have ? I f the need for such results does arise, may we not like the clerk use a ready reckoner as we use other time-saving devices made for us by the specialised labour of others ? E v e r y one should k n o w what compound interest is, and w h y more frequent pay­ ments increase the amount, but simple examples and few of t h e m suffice for this. I f , then, utility of the narrow personal k i n d is the only reason for teaching arithmetic, let us ensure full proficiency i n the operations of everyday life, show the use of ready reckoners when needed, and utilise the t i m e so gained i n teaching something more likely to assist i n the production of capable citizens. Provided that c h i l d r e n can perform simple calculations w i t h fair speed and accuracy, they should learn the proper use o f ready reckoners before they leave school, i n accordance w i t h the modern tendency to use labour-saving devices and so obtain greater efficiency. T h e individual is thus freed for the performance of other functions, and so increases his power of production. Those who deprecate this sug­ gestion m i g h t as well deprecate the use of sewing-machines, and their i n t r o d u c t i o n into girls' schools. I n neither the one case nor the other is there a loss of independence ; on the contrary, there is a gain. B u t can arithmetic fulfil no other function i n our schools ? I am probably preaching to the converted when I say that there are two other aspects of the subject w h i c h not merely recommend but enforce its study i n schools of all types. T h e y are its application to the social life of cities and states (for example, to the intelligent consideration of schemes of insurance and pensions), and the concept of orderly and precise methods of thought w h i c h i t may convey, i n

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hardly less degree than the study of geometry. T h e two are indeed l i n k e d together, for these methods of thought find some of their best applications i n the study of con­ crete problems which have some touch of reality for the children concerned. T o sum up, then, we base our teaching of arithmetic o n three foundations : practical use, furtherance of the proper understanding of social and political problems, and develop­ ment of power of independent t h o u g h t ; and we accept the use of labour-saving devices wherever possible, even though we are now compelled to waste t i m e which w i l l be better employed when our examinations are more rationally conducted. First, then, for practical utility. W e have to ensure the ready and accurate use of figures i n concrete problems, and their combination by addition, subtraction, m u l t i p l i ­ cation, and division ; and the idea of a fraction must be gained — f o r its utility alone i t is a necessary part of the equipment of every civilised being. A t the base of all these things lies our scale of notation. N o w I do not propose to enlarge upon the way i n which this should be explained. I have never, unfortunately, taught i t to y o u n g c h i l d r e n . I would only commend the use of the abacus to those not familiar w i t h it, as h a v i n g historical sanction and being justified by modern experience. B u t I do wish to enlarge upon the importance of a correct understanding of the method of the scale, not only for its own sake, but for its applications also. I t is the first example of orderly classi­ fication reached by the child, and as such deserves full elucidation, for i f he once acquires the idea that things are taught to h i m which he need not and cannot understand,

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the impression w i l l dog h i m and his teachers as an evil spectre for many a weary year. T h e difficulty, such as i t is, lies i n the fact that only one scale of notation is presented as such, and the u n d e r l y i n g principles cannot be grasped from a study of this or any one case. I t is too little realised that our E n g l i s h systems of money, weights, etc., are also scales of notation. T h e notations hundreds tens units 7 3 5 that is,

7 x 1 0 x 1 0 + 3 x 1 0 + 5 units ; and pounds

7 that is,

shillings 3

pence 5

7 x 2 0 x 1 2 + 3 x 1 2 + 5 pence

are exactly similar i n method, though not i n detail, for they each f o r m groups of g r o u p s : tens of tens of units i n the one case, twenties of twelves of pence i n the o t h e r ; and this is done merely to save t i m e . S h i l l i n g s and pounds are not necessary, but they are convenient as saving t i m e and labour i n speaking, i n w r i t i n g , and i n carrying money. T h e r e is even a somewhat vague scale of notation i n geog­ raphy — hamlet, village, town, county, country, continent. I believe that such general considerations can and should be brought before children d u r i n g their education, for they enlarge the m i n d and lead to the formation of general concepts f r o m particular cases. T h e y may invent examples for themselves, finding, for example, what 235 would mean for a race of beings who, h a v i n g only eight fingers, counted in eights.

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Beyond this we have to deal w i t h money, weights, and measures, and simple sums concerned w i t h t h e m . H e r e again there is little of practical value to be said; methods of teaching and w o r k i n g have been thrashed out ad nauseam. T h e only plea one can make is for the utmost speed and accuracy i n simple mental calculations, such as the cost of 2.\ l b . of tea at i s . 7 d . Frequent practice for short periods is the only way to ensure this, and such practice has its reward i n an increase o f general alertness and vigour. I wish, however, to suggest a type of easy problem which has hitherto been neglected i n elementary teaching. Take, first, an illustration. " F o u r boys A , B, C, D are to sit on one bench, and the teacher knows that A and C, i f placed together, w i l l talk. Show all the ways i n w h i c h he can seat t h e m . " T h e solution should be systematic; A and C may be placed i n six ways, t h u s : A

—

C

—

C

—

A

—

—

A

—

C

—

C

—

A

A

—

—

C

C

—

—

A

and then B and D may be placed i n two ways i n each case, g i v i n g twelve ways altogether. T h e question can then be narrowed ; A and B may have to share a book, and so o n . O t h e r problems are easily devised. " I n how many ways can four people sit r o u n d a table ? " " T h r e e men are to be chosen out of five to perform a piece of w o r k ; A and B refuse to w o r k w i t h C. H o w many teams can be chosen?" Such questions not only give t r a i n i n g i n classification,

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they develop the idea that some things can be . done i n several ways, and that i t may be w o r t h while to reckon up all the ways and choose the best. W e now consider fractions, r e m a r k i n g first that they may and should be taught before l o n g multiplication and division. T h e whole theory can be taught i n concrete ap­ plications without the use of large numbers, and is only obscured by their introduction. H a r d e r examples are ac­ cessible without further theory, once the fundamental proc­ esses are fully assimilated. T h e first p o i n t is to develop the idea of a fraction, and the last way to do this is to commence w i t h the notation | . T h i s should come late — m u c h later than is usually the case. T h e symbol means three-fifths of something, say of a pound or a line o n the blackboard, and i t should be re­ garded as three units of a new size. T h e r e is m u c h diffi­ culty i n g e t t i n g as far as this. A few concrete examples taken orally may suffice, w r i t t e n work, i f there be any, being expressed w i t h the denominator i n w o r d s ; thus, 3-fifths. I t is essential that some u n i t be stated, 3-fifths of a pound or an a p p l e ; the abstract 3-fifths is far too general a concept at this stage. A l t h o u g h there is not m u c h difficulty i n i m p a r t i n g the idea of a fraction, i t is vital that this, as any other mathe­ matical concept, should acquire l i v i n g reality for the pupil, and not remain an arid tract of schoolroom f o r m a l i s m . T h e best safeguard against this danger is considerable practice i n estimating one magnitude as a fraction of an­ o t h e r — two lines drawn o n the board, the areas of two pages, the sizes of two pieces of wood (to be tested by weighing), and so o n . A little practice i n this, the pupils

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being told w h i c h estimates were most nearly accurate, w i l l soon induce that sense of proportion w h i c h is of the es­ sence of fractions, and is so essential i n practical life. A n d here we come to a method, w h i c h is two thousand years old, for the exact specification of one line or other mag­ nitude as a fraction of another. Suppose that two lines AB, PQ are to be compared, and let AB be the shorter. L a y off lengths equal to AB along PQ as far as possible, and let the remainder, i f any, be BQ, T h e n lay off lengths RQ along AB and let the remainder, i f any, be CB. T h e n lay off lengths CB along BQ, w i t h remainder, i f any, SQ, and so on. T h e remainders w i l l soon become indistinguishably small, so that AB and PQ are expressed, w i t h such accuracy as our instruments allow, as exact multiples of the smallest remainder visible, whence the fraction is at once obtained. T h e r e is much more to be said of this process, but i t pertains to the education of older pupils. A n y one who can understand the process can, however, realise also that he has i n this method a logical process w h i c h w i l l continue u n t i l his instrument fails h i m ; i n other words, i t beats the instrument every t i m e and so illustrates the superiority o f mental process over empirical measurement. N e x t comes the addition and subtraction of fractions, still i n concrete problems. F o r example: " A farmer wishes to sow two-thirds of his land w i t h barley and one-quarter w i t h w h e a t ; what fraction is left for other purposes ? " T h e best way to surmount the very considerable difficulty of this question for y o u n g children is to lay stress on the idea of change of unit. W e may commence by saying, " Can y o u add 7 dollars to 4 francs and call i t 11 ? N o ! W h a t do you

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73

do ? Convert t h e m both to pence : 7 x 50 pence + 4 x 1 0 pence = 390 pence = 39 francs, or 7 dollars and 8 0 cents. I n the same way we must convert 2-thirds and 1-quarter to the same k i n d of t h i n g before we can add t h e m . " N o w draw a line and demonstrate o n i t that 2-thirds = 8-twelfths and 1-quarter = 3-twelfths ; we can then say 2-thirds + 1-quarter = 8-twelfths -f- 3-twelfths = 11-twelfths. I t is unnecessary to enlarge upon this process; its nature is evident, and text-books contain many suitable examples i n the chapters on ratio and proportion and i n other parts.

T h e essentials

are constant verbal expression and continual illustration by division o f a line or area u n t i l real comprehension is attained. T h e r e is no need to be particular about the lowest c o m m o n denominator ; we may well allow our pupils to say : 1 -quarter + 1 -sixth = 6-twenty-f ourths + 4-twenty-f ourths = 1 o-twenty-f ourths = 5-twelfths, for at this stage the a i m is clearness and accuracy, not brevity gained at their expense. A s soon as the pupils have become fluent i n such state­ ments as 2-thirds + 3-quarters = 8-twelfths + 9-twelfths = 17-twelfths, the fraction notation may be introduced on the g r o u n d that it saves t i m e . Some stress should be laid upon this point, and i t should be illustrated by analogies.

T h u s we save

t i m e by h a v i n g one w o r d " s c h o o l , " instead of saying " a place where c h i l d r e n are t a u g h t " ; " chair " instead of " a t h i n g to sit u p o n . " T h e fractional symbol thus assumes

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its proper aspect as a short expression o f an idea already comprehended, and the c h i l d is receiving a valuable lesson on the m e a n i n g and use of language. E v e n w h e n the notation is i n use, frequent practice should be given i n the verification, by division of lines or areas, o f such statements as § = y f . Unless they are understood they w i l l inevitably be misapplied. N e x t comes the multiplication of fractions. H e r e I wish to make a strong protest against the usual premature use of the t e r m " m u l t i p l y " i n such statements as " § m u l t i p l i e d by | . " W h a t does the c h i l d understand by multiplication ? Surely n o t h i n g but repeated addition. I f , then, we say to h i m " | m u l t i p l i e d by 5," he can see that this means five times two of a certain t h i n g (thirds), and is therefore ten of these things. B u t to tell h i m that " to m u l t i p l y two fractions you m u l t i p l y their numerators and denominators " confuses the t e r m hopelessly. I t has had one meaning, clearly com­ prehended, and now acquires a second w h i c h is apparently a mere j u g g l e w i t h figures. A l l sense of logic and exact use o f language must depart w i t h this step. I t is well to recognise that there is no obvious or easily apparent justification for the use of the same name for these two processes. T h i n g s receive the same name be­ cause they have something i n c o m m o n . W e are all called h u m a n beings because, amidst m u c h diversity, we have certain c o m m o n attributes. N o w the c o m m o n element is not at all manifest, at first sight, i n such statements as the following: 7 X 3 = 7 + 7 + 7 = 14 + 7 = 2 1 , 3

2 _ 3 X 2 _

8

5~8 x 5 "40*

6

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75

Unless we can see the c o m m o n element we have no r i g h t to name t h e m alike, and u n t i l the child perceives a c o m m o n element i t is absolutely pernicious to suggest a common name, for i n so d o i n g we debase that most wonderful creation of the h u m a n race—language as a clear expression of thought. So soon as we do this we may say farewell to clear t h i n k i n g or exact expression o n the part of the pupils. I honestly believe that this one step is responsible for most, i f not all, of the doubt and haze which hang like a nightmare over many children i n their dealings w i t h fractions. A good introduction is to consider questions such as, " A man left § of his land to his children, and | of this to his eldest son. W h a t fraction of D E G K the land d i d the eldest son g e t ? " Representing the land by ABCD, we divide i t into thirds by PQ, RS, and then p | | p0 | | |Q into fifths by EF, GH, IJ, KL. T h e n \ of | of the land F I I is seen to be POGJD, and this is seen to be -pj of the land. W i t h many such questions the general idea that f of -| of a t h i n g is the same as of that t h i n g is imparted, and the formal rule is seen to hold i n all cases. T h e drawings can be discarded, except for revision, w h e n this formal rule is grasped and not be­ fore. A t no stage need they, or should they, be made w i t h great accuracy. Freehand sketches are better than drawings w i t h instruments, for they enforce the lesson that the process is essentially one of reasoning and not one of measurement.

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Finally, we come to the division of one fraction by another. H e r e again, and for precisely the same reason, this t e r m should be discarded u n t i l its application can be comprehended. B y concrete questions the problem is raised, " T w o - t h i r d s of a t h i n g is taken, and three-quarters of the same t h i n g . W h a t fraction is the former of the latter ? " and the idea is evolved from such discussions. I t is not pretended that this w o r k is easy, or that i t can be learnt by r o t e ; but experience shows that i t is w i t h i n the comprehension of ordinary boys and girls of twelve years of age or even less, and i t has far more value, as a practical mental t r a i n i n g , than purposeless j u g g l i n g w i t h numbers. A child who can perform these processes feels that he is u s i n g his o w n m i n d to answer definite questions w i t h logical certainty. T h e mere appreciation of this fact raises h i m f r o m drudgery and gives h i m an ideal of men­ tal independence w h i c h he may, perchance, i n some part retain i n after years. Before leaving the subject of fractions some further reference should be made to the premature use of the terms " multiplication " and " division " as applied to such n u m ­ bers. F i r s t take the statement, " T o m u l t i p l y | by | , do to -| what is done to u n i t y to obtain | . " T h i s definition is hopelessly defective i n that i t omits to state exactly what is done to unity. Is \ subtracted f r o m i t , or is i t increased by 2 and the result divided by 4, or w h i c h other of the innumerable ways of obtaining | f r o m i t is meant ? T h e upholders of this definition w i l l reply that this is s p l i t t i n g hairs ; that every one knows i t to mean that unity is to be divided into four equal parts and three of these parts taken. Certainly this is so, and the statement

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as thus amended loses its gross ambiguity. B u t what analogy is there between this process and the original view of multiplication as repeated addition to justify the same name for both processes ? T h i s question must still be answered before the definition can be accepted. T h e justification is twofold. First, i t can be shown that the latter definition includes the former as a particular case; that multiplication by according to the definition for fractions amounts to the same t h i n g as multiplication by 7 according to the first definition. A n d secondly i t can be shown that the formal laws such as ba — ab, a(b + c) = ab + ac, w h i c h are so easily seen to be consequences of the first definition, are consequences of the second also; that is, they are true when a, b, and c are fractions as well as when they are integers. T h e coincidence being complete, the use of the same t e r m is justified, but i t is evident that considerations of this nature can hardly be included i n a first course of arithmetic. F o r pupils who are revising the subject for the second or t h i r d t i m e they may be interest­ i n g and profitable, but before then the use of the t e r m " o f , " as i n \ of f , is preferable to the apparently ambiguous " multiply by." T h e considerations above may serve to show the spirit w h i c h i t is suggested should i n f o r m the treatment of the theory of arithmetic ; we now pass to a discussion of some elementary applications w h i c h may go towards fitting the pupils to be capable citizens as well as efficient clerks. O f all the various applications w h i c h appear i n very ele­ mentary text-books of arithmetic, the theory of averages suffers perhaps more than any other f r o m the banality of its treatment. A n d yet no other application possible i n

MATHEMATICAL

78

EDUCATION

such books is possessed of equal ease and interest; nor are there many, i f indeed any, others w h i c h have so direct a bearing o n almost every question o f national or civic i m ­ portance. T h e r e are few such questions into which n u m ­ bers or statistics do not enter i n some shape or f o r m , and their correct treatment by averaging is almost invariably essential to a proper view of the facts. T h e one, and usually the only, t h i n g w h i c h is taught i n connection w i t h averages is the rule for obtaining the average of a set of numbers ; it is t h e n applied without intelligence to problems fit and unfit for the purpose. Consider the two sets of numbers :

and

10,

7,

12,

8,

9,

9,

8,

n,

14,

12,

o,

23,

2,

45,

6,

15,

o,

7,

1,

1.

Each has 10 for its average, but i t can at once be seen that there is no real significance i n this statement. I n the first set the numbers are grouped closely r o u n d this average, but i n the second they bear no special relation to i t ; i t is n o t h i n g more than a levelling up of things widely diverse one f r o m another, and has little or no other i m p o r t . Considering the first set a little more closely, we may make a table s h o w i n g how many o f the numbers come w i t h i n different percentages of the average, thus : Percentage of average 10 Number within 4 Percentage w i t h i n 40

20 8 80

30 9 90

40 10 100

Such a table shows w i t h what nearness the average repre­ sents the group of numbers, and enables us to compare the relative values of different averages. F o r example, the

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cricket averages of different players may be treated i n this manner, when m u c h information is gained as to their steadiness of play. O r the average age of each f o r m and of the whole school can be so c o m p a r e d ; the contrast be­ tween the irregularity of the distribution i n many small forms and its u n i f o r m i t y for the whole school w i l l convey its o w n lesson. T h e results can be exhibited graphically, laying off horizontally differences of i , 2, 3, . . . p e r c e n t , f r o m the average, and vertically the percentage of the whole number of observations w h i c h fall w i t h i n each l i m i t . A l l such graphs should be drawn to one u n i f o r m scale, so that a glance w i l l indicate the relation of the average to the set of numbers, and oral or w r i t t e n statements of what­ ever can be seen f r o m the w o r k should be insisted on i n every case. A second application of averages concerns the " smooth­ i n g o u t " of a series of statistics which, though liable to large irregular variations, obey on the whole some definite law of change. Suppose, for example, that the shade tem­ perature is observed each day at noon for a period of six months ; the results w i l l be very irregular but w i l l show o n the whole a steady increase, and the object is to eliminate the irregularities so far as may be possible and thus exhibit the general law of increase. T h i s is done by t a k i n g the average for a number of consecutive days (say five) and assigning i t to the m i d d l e of the period, this being done for every such period i n the six months. T h e accidental irregularities, due mainly to the direction of the w i n d and the amount of cloud, are thus spread out and the increase corresponding to the change of season becomes apparent. I n practice, periods of five days would be too short, as the

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w i n d often holds i n one direction for a longer time, but they commence the smoothing process and longer intervals may be considered afterwards. T h e comparison of results for different periods is of interest as showing how the effect of a l o n g period of h i g h or low temperature is gradually eliminated. A n o t h e r method of obtaining the same final result, and one w h i c h is simpler i n itself, is to take the average tem­ perature over a series of years for each particular date and so smooth out the irregularities i n a different way. B u t this m e t h o d would be impossible i n other cases, such as the study of the mortality f r o m consumption or the price of corn i n L o n d o n , for these phenomena are not recurrent like the seasons and we cannot, therefore, eliminate acci­ dental irregularities by r e v i e w i n g several cycles of change. Moreover, the temperature averages obtained f r o m a period of years assume that every season is the same apart f r o m irregularities, and so conceal any possible change i n the seasons f r o m year to year; but a comparison of these aver­ ages w i t h those obtained by the m e t h o d of consecutive days w i l l reveal such changes i f they exist. O t h e r materials for the application of this method of smoothing out w i l l be found i n any book of reference and i n most text-books of arithmetic. I t can be applied to any set of statistics, but consideration of weather records is of special use, partly f r o m their interest, but more f r o m the repetition f r o m year to year w h i c h has just been discussed. Y e t another important application of averages is the method for obtaining the mean value of a continuously v a r y i n g quantity, such as the height of the barometer or the depth of a tidal river. I n a certain t o w n (there may

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be many such) the m a x i m u m and m i n i m u m temperatures for each day are recorded and the mean is taken to be their half-sum, w h i c h is solemnly written down as the mean temperature for the day. N o w i t is obvious that this method w i l l give results which are too low i n summer and too h i g h i n w i n t e r ; for i n summer the temperature stays i n the neighbourhood of its m a x i m u m for the greater part of the day, and i n winter i t lingers near its m i n i m u m . T h i s application w i l l show how a proper estimate of the mean may be obtained. I t is clear that the t r u t h would be shown more nearly by averaging readings taken every three hours, and that still better results would be gained i f the readings were still more frequent. Such averages should be taken and the results compared w i t h each other and w i t h the mean of the m a x i m u m and m i n i m u m readings ; statistics are easily obtained from the charts given i n many newspapers or f r o m an instrument dealer who has recording instruments. W i t h such charts we can, however, obtain an even better estimate of the mean by f i n d i n g the height of a rectangle whose base is the horizontal w i d t h of the graph and whose area is equal to the area under the graph, for it is obvious that this height is the true average of the heights of a l l points on the curve. T h e area of the curve can be esti­ mated i n the usual way by counting squares, and the aver­ age height is then found by d i v i d i n g by the base. T h i s method of estimating mean values is of m u c h importance i n theory and practice, and examples of its use are not lacking i n interest. 1

1

T h e London

Daily

Telegraph,

for example.

I n the U n i t e d States,

the reports distributed freely by the W e a t h e r B u r e a u m a y be used.

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I t has seemed worth while to discuss averages i n some detail, even to the exclusion of other applications o f arith­ metic, for the work conflicts little, i f at all, with the syllabi to w h i c h most schools are subject, and combines ease, interest, and value i n exceptional degree. B u t I would mention also the understanding of insurance tables (not the formulae from w h i c h they are constructed) and the mean­ ing of the value of money and its variations i n t i m e and place as matters w h i c h should be considered by teachers of arithmetic ; the fallacy of the thirty-shilling wage would find no wide acceptance i f education were all that i t should be. I have endeavoured to suggest some simple and perhaps novel considerations concerned w i t h the teaching of ele­ mentary arithmetic. I t may perhaps be felt that they have some slight interest and value, but that i t is hopeless t o attempt the application o f some at least, i n view o f prevail­ i n g custom and requirements. T h i s frame of m i n d , excel­ lent as a balancing factor, is nevertheless to be regarded w i t h m u c h caution, for salvation f r o m our present difficul­ ties can come only f r o m the efforts and experiments o f teachers themselves. Educational matters are i n a ferment. M e n are asking more and more insistently why this and that are done, and they are r i g h t i n their insistence. U n ­ less f i t t i n g answers are ready, our w o r k w i l l stand con­ demned ; the degradation of our subject to the domain o f purely immediate utility w i l l surely follow, as also the loss of that higher mental t r a i n i n g w h i c h is so essential to the for­ mation of an efficient citizen. A m a n who has no power of intelligent numerical t h o u g h t is to this extent a serf intel­ lectually, and i t is hard to believe that teachers as a whole w i l l fail to p o i n t out the evil and insist upon its avoidance.

T H E E D U C A T I O N A L V A L U E OF GEOMETRY ( R e p r i n t e d by p e r m i s s i o n of the C o n t r o l l e r of H i s Majesty's Stationery Office f r o m the Special Reports of the Board of Education on The Teaching of Mathematics, N o . 15* 1912)

THE

EDUCATIONAL VALUE

OF

GEOMETRY

" E v e r y great study is not only an e n d in itself, but also a m e a n s of c r e a t i n g a n d sustaining a lofty habit of m i n d ; a n d this purpose should be k e p t always i n view throughout the t e a c h i n g a n d l e a r n i n g of mathe­ matics." — B E R T R A N D

RUSSELL

T h e title o f this paper has been chosen to indicate that the discussion w i l l not be concerned w i t h the value of geometry as applied to other sciences or to practical ends, nor even w i t h its place and importance i n schemes of mathematical education. T h e purpose is to state the rea­ sons w h i c h appear to have led to the universal acceptance of the subject as a necessary element i n education, to as­ certain to what extent geometrical teaching i n this country can find justification i n t h e m , and to give some slight ac­ count of experiments i n teaching made on this basis by the writer and his colleagues at T o n b r i d g e School. L e s t it should be thought, however, that this avoidance of the practical importance of the subject and its relation to other branches of knowledge imposes unreasonable limitations, it may be well to state the reasons for i t . T h e danger of g i v i n g undue importance to considera­ tions of practical utility need hardly be enlarged upon, since it is not proposed to consider geometry from this p o i n t of view. I a m more concerned to point out that i f the advo­ cates of the subject rest any part of t h e i r case on such considerations, they at once enter into competition w i t h a host of other interests, many of w h i c h have, on such §5

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grounds, m u c h higher claims. T h e parents of a boy w h o is to adopt a business career w i l l r i g h t l y prefer, i f his edu­ cation is guided by his future requirements, that he should spend his t i m e on geography or economics, arguing that surveying and bridge b u i l d i n g can have no relevance to his future interests; while those who take a wider but still utilitarian view w i l l insist that subjects such as civics and the chemistry of food have stronger claims to a place in the education of every c h i l d . S t i l l more dangerous is the plea that every educated m a n should have some idea of a subject of such wide u t i l ­ ity. A p a r t from the claims of many other branches of knowledge, i t has a further demerit i n that the object of teaching the subject is implied to be the acquisition o f encyclopaedic knowledge, rather t h a n the development of the mental faculties. T h e o l d conception of education as the acquisition of i n f o r m a t i o n is dead, and i t least becomes mathematicians to do a n y t h i n g to revive i t . T h e use o f justifications o f this type, even t h o u g h i t be only i n sec­ ondary positions, is likely to defeat the aims of those w h o advance t h e m and to do m u c h h a r m to educational ideals. A discussion of the value of geometry i n relation to other branches of science would be appropriate i n a paper dealing w i t h the co-ordination and relative importance of these branches. M y object here is, however, to show that the subject has for its o w n sake a claim to a place i n the education of every h u m a n being. Such a discussion could, therefore, give only a secondary and relatively weak sup­ port to this claim, a support w h i c h only becomes valid w h e n the claims of these other subjects to a universal place i n education have been admitted. I f the view here taken

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is unjustified, geometry must then make its own place i n such volume of scientific knowledge as may be found necessary to a liberal education. T h i s place would be an important one, especially i n view of the now almost uni­ versal teaching of natural science, but i t is hoped that the considerations to be stated i n support of the stronger view are such as w i l l meet w i t h agreement among mathemati­ cians and convince laymen of its t r u t h . T h i s is, of course, no new claim. Plato inscribed over the entrance to his Academy, " L e t no one enter who is ignorant of geometry," and almost every university now imposes a similar condition. Such recognition of the sub­ ject by educationalists w h o are not mathematicians implies an inherent value w h i c h must be expressible i n non-tech­ nical terms, and i t behoves all those who teach i t to assure themselves that they appreciate this value and that the education i n schools is such as to realise i t as fully as may be. I n this country at the present t i m e the duty is espe­ cially urgent. Educational systems and ideals are changing w i t h some rapidity, and almost every subject i n school curricula has been challenged to justify its place, geometry being one of the few exceptions, i f indeed i t still be one. I t is almost certain that the motives for this forbearance are of utilitarian type, combined, perhaps, w i t h some vague idea that the subject may train a boy to chop logic and hold his own i n argument. T h u s the lay advocates o f the subject, o n w h o m its continuance must depend, base their support on reasons w h i c h are open to successful attack from those who take a too material view of education, and are almost beneath attack f r o m those who have higher ideals. O f this weakness they must become conscious;

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signs are not w a n t i n g that this is i n process, and unless mathematicians themselves take the initiative i n defence they may find the attack developed w i t h some sudden­ ness. I f the boy w h o specialises i n science obtains exemp­ t i o n from the study of Greek, his fellow who specialises in classics or history w i l l almost certainly claim exemp­ tion from mathematics, as also w i l l those who intend to devote themselves to subjects such as law, medicine, or music. T h e question for mathematicians is, then, whether they can convince others that the appropriation to the study of geometry of a portion of the school t i m e of every boy and g i r l is really expedient. T o do this i t will almost certainly be necessary, even t h o u g h those w h o are to be convinced have themselves had some portion of their school time so appropriated, to explain i n some detail what geometry really is. T h e first element i n the explanation must be that the subject is based on agreement as to a certain n u m ­ ber of cardinal facts, this agreement resting on foundations of general experience c o m m o n to every civilised h u m a n being. T h e equality o f vertically opposite angles, the angle properties of parallels, and those properties of a circle w h i c h can be perceived from considerations of symmetry are instances. I t is essential to understand that these facts should not depend, even for their elucidation, on numerical experiments made i n class-rooms or laboratories. R o u g h descriptive illustrations there may be, b u t their only pur­ pose is to recall or intensify conceptions previously formed 1

1

T h e latter part of this s e n t e n c e defines the subject as taught in

schools.

I t is, of course, an essentially vicious limitation from a more

general point of view.

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subconsciously, for i t is this universal acceptance of postu­ lates without conscious experiment w h i c h differentiates geometry f r o m physics or chemistry. T h e necessary ex­ periments and inductions are made i n infancy w i t h o u t the aid of ruler, compasses, and protractor; the dog who fol­ lows his master by a " c u r v e of p u r s u i t " fails to describe a curve w h i c h is grasped by y o u n g children, although i t is the one actually taken by infants who can only just crawl. T h e y , i n their apparently aimless wanderings, are i n the true geometrical laboratory, p e r f o r m i n g experiments and m a k i n g inductions. T r i t e as this statement may appear to many, i t is of some importance here. T h e appreciation of c o m m o n conviction should, when possible, be the first aim of teacher and pupil, as of all those who would make a concerted effort, and here there is found a number of facts of which all pupils can say, " Yes, I know that these things are t r u e . " T h e y feel that they are not using arbitrary rules, as i n the study of languages; nor dealing w i t h facts asserted by others to be true, as i n geography or h i s t o r y ; nor dealing w i t h experiments put before t h e m by some one who knows what w i l l happen, as i n experimental science : here they rest on their own convictions. I need not enlarge upon the value of this consideration, but I would point out that its importance can be realised by the layman and should influence h i m considerably. A m i d the many arbitrary rules and asserted facts which, perforce, find place i n education, the presence of schemes of deduction based o n statements

1

1

T h e basis for a n d m e a n i n g of this assertion do not enter into the

question. portance.

T h e fact that it is made so universally is the point of im­

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w h i c h find universal acceptance as descriptions of our space impressions must make for good i n the child's development. T h e peculiar nature of the premisses on w h i c h geom­ etry is based h a v i n g been explained, i t m i g h t appear suffi­ cient to complete the description of the subject by stating that i t consists of a series of deductions f r o m these prem­ isses and therefore supplies a useful t r a i n i n g i n the art of deduction. B u t this statement would, i f not amplified, be so bald as to mislead. T h e full sequence of processes involved i s : 1. T h e separation of essential f r o m irrelevant consider­ ations involved i n the appreciation of points, lines, and planes and their mutual relations. 2. T h e erection on this appreciation of continuous chains of reasoning, one result leading to another i n such a way that each chain can be comprehended as an ordered whole and its construction realised as fully as that of each sepa­ rate l i n k . 3. A discussion of the interdependence of the various •premisses and their precise statement. Since some such sequence is c o m m o n to every h u m a n construction, the educational value of a subject w h i c h pro­ vides a t r a i n i n g i n these processes is indisputable ; for this purpose geometry stands alone i n that its bases can be appreciated, and the deductions performed, at an age earlier than is possible i n the study of any other subject h a v i n g the same purpose. T h e r e is yet one more consideration, and that not the least important, to be urged i n favour of the subject. T h e appreciation of literary and artistic beauty has of late re­ ceived increasing recognition as a necessary element i n the

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t r a i n i n g of the y o u n g . T h e study of the E n g l i s h language now includes l i t e r a t u r e ; d r a w i n g and music advance i n importance i n schools of every type. B u t the element of intellectual beauty has not yet received general recognition, despite the supreme position ascribed to i t by almost all schools of thought from the Greeks downward, and the loss is a great one. T h e contemplation of unassailable mental structures such as are found i n mathematics cannot but raise ideals of perfection different i n nature f r o m those found i n the more emotional creations of literature and art. I t must induce an appreciation of intellectual unity and beauty w h i c h w i l l play for the m i n d that part w h i c h the appreciation of schemes of shape and colour plays for the artistic faculties; or again, that part w h i c h the appre­ ciation of a body of religious doctrine plays for the ethical aspirations. Fanciful t h o u g h this may appear to many, I believe that i t may be an i m p o r t a n t factor i n d e t e r m i n i n g the retention of geometry i n schools. T h e conception of a body of t r u t h invulnerable o n a l l sides, a conception w h i c h finds one of its best and most common expressions i n the quotation, " F o u r square to all the winds that b l o w , " appeals to most men, and they w i l l welcome an effort to b r i n g i t to their children i n one of its purest forms. 1

Such being the basis on w h i c h i t is thought that the universal teaching of geometry may be justified, i t remains to develop the leading principles of a scheme of education, and to examine to what extent teaching practice i n this country accords w i t h t h e m . 1

E u c l i d e a n geometry is still invulnerable i n that it is based on the

simplest k n o w n d e s c r i p t i o n of our space p e r c e p t i o n s , a n d s u c h de­ scriptions form the foundation of most " t r u t h s . "

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EDUCATION

T h e division of the processes of geometry into three classes implies a corresponding division of the period of education into three epochs. I n the first, the imagination is stimulated and developed and some general power of reasoning should be acquired without any formal presenta­ t i o n ; i n the second, ordered systems of reasoning are developed f r o m facts w h i c h are now w i t h i n the scope of the i m a g i n a t i o n ; and i n the t h i r d , the true basis for the assertion of these facts is discussed and their interconnec­ t i o n investigated. I t is at least doubtful whether this t h i r d stage would be a subject proper to a school course i f i t were w i t h i n the pupils' grasp; and i t is almost certain that its nature and the problem involved are as much as can be brought home to t h e m . I t must therefore be assumed that a school course should end w i t h the second of these epochs, and the first two only need be considered i n detail. T h e increase i n power of imagination, w h i c h is the m a i n object i n the first epoch, can only be effected by extension from those impressions i n which this power is already developed to some extent. A child may k n o w what is meant by a p o i n t and a line, and be able to recognise t h e m at sight, and yet not be able to t h i n k of points and lines, just as an adult may recognise the whole of a tune when he hears i t and yet not be able to reproduce a single note. I t is therefore inadvisable to commence by d r a w i n g points, lines, and angles, and p e r f o r m i n g constructions and measurements, because this opens up a new field of un­ familiar ideas h a v i n g no connection w i t h any of that knowl­ edge w h i c h has been so far assimilated as to be a possible subject for imagination. Houses, roads, mountains, islands, and the like can all be imagined by a child of ten years of

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age, and his geometrical imagination is developed by stat­ i n g problems i n such terms, but the construction of t r i ­ angles h a v i n g given sides leads nowhere at this age and is a mere gymnastic. I t is not hard to devise problems on these lines which stimulate the imagination, excite the spirit of research, and provide exercise i n the simpler forms of geometrical reason­ i n g . T h e y may be divided into groups, i n each of w h i c h one of the following methods is introduced : 1. Construction of triangles and polygons when lengths only are given. 2. Simple constructions for heights of buildings, ships' courses, and the like, depending on compass bearings and angles of elevation. 3. Construction of triangles and polygons when lengths and angles are g i v e n . 4. E x t e n s i o n of all the above to problems i n more than one plane. 5. D e t e r m i n a t i o n of a point as the intersection of two loci, or its limitation to lie inside or outside two or more loci. These loci need not be lines or circles and are con­ structed by actual plotting. I t may be w o r t h while to give a few specimen problems to show what is intended, and i t should be stated that these and subsequent suggestions are the outcome o f experience and have been tested i n T o n b r i d g e School. E a c h of these problems is well w i t h i n the scope of a boy of eleven or twelve years of age, after he has received a reasonable amount of teaching i n the shape of questions i n v o l v i n g the same p r i n c i p l e s ; the concrete terms should, of course, be varied constantly.

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a {Type i ) . A straight road runs east and west; a house is 95 yards n o r t h of the road and a well is 4 0 yards n o rt h ­ east of the house. D r a w an accurate plan and locate upon it the position of a cow-shed w h i c h is to be built 65 yards from the well, w i t h i n 70 yards of the road, and as far as possible f r o m the house. b {Types 2, 3, 4 ) . T h e height o f a tree or chimney may be f o u n d from measurements i n one or i n two planes, the measurements being made by the pupils themselves. c {Types 2, 3, 4 ) . A h i l l rises due n o r t h at a gradient of 1 i n 10. F i n d the direction of a road w h i c h rises to the eastward o n the h i l l at a gradient of 1 i n 30. d {Type 5). A l i g h t s h i p is 2 miles from a straight shore. A submarine, w h i c h is cruising i n the neighbour­ hood, explodes and sinks, the only survivor being the man in the c o n n i n g t u r r e t ; he can only say that the wreck is equidistant from the l i g h t s h i p and shore, and the watch on the l i g h t s h i p says i t was 3 miles away when the explosion happened. Show on an accurate plan where the wreck must lie. c (Type 5). T w o towns, A , B, are 5 miles apart, and a man who lives w i t h his parents at a place 6 miles f r o m A and 7 miles f r o m B, bicycles to A , then to B, and then home every day. A f t e r a time he wishes to reduce his daily ride to 14 miles. Show on an accurate plan all the places where he can live, and indicate those places w h i c h are nearest to and farthest from his parents' home. T h e first three types require no further explanation, but since the f o u r t h and fifth involve departures from current practice, discussion of these may be of some value. T h e early introduction of solid geometry (examples b, c) may

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cause some surprise, especially i n problems of such appar­ ent difficulty. B u t I am convinced, f r o m the experience of myself and m y colleagues at T o n b r i d g e , that boys of twelve or thirteen grasp these ideas w i t h m u c h more ease and rapid­ ity than those who have deferred the w o r k for two or three years, and that their whole outlook is i m p r o v e d i n conse­ quence. T h e first problems should concern things w h i c h can be seen ; for example, the height of one corner of the class-room can be determined by measurements taken f r o m the ends of a desk not i n line w i t h i t , or the diagonal o f a block of wood may be found by construction of two rightangled triangles. A problem such as c above is s i m p l y illustrated by u s i n g one corner of a book half open to represent the hill-side. These aids to imagination are soon found to be unnecessary, fairly complex construc­ tions being undertaken without their assistance, and there is no lack of material f r o m w h i c h examples may be constructed. Since the m e t h o d described i n the fifth type and exam­ ples d and e is, except for the introduction of other loci, used i n the preceding sections, its separate m e n t i o n may appear redundant, but i t is easily seen to involve principles w h i c h can hardly receive too m u c h attention. T h e statement of the laws of selection w h i c h define two or more manifolds i n such f o r m as to exhibit the common elements ( i f any) of t h e m all is one of the most familiar forms of mental activity, and experience has shown that c h i l d r e n can under­ stand and p e r f o r m the process i n cases such as those above. I n so d o i n g they apply i t , not to new and unfamiliar ab­ stractions, but to classes of objects so familiar as to be possible subjects for mental operations.

g6

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EDUCATION

Such applications f o r m the second o f four stages i n t o which education i n logical processes may be divided. I n the first they are applied to definite objects, as i n the methodical selection f r o m a number of tiles of all those w h i c h have given shape, size, and thickness ; i n the second, to mental images of classes of objects, as when " a house " is not t h o u g h t of as any particular house ; i n the t h i r d , to abstractions, such as point, line, colour, sense, i n s t i n c t ; while i n the last the processes themselves are considered i n their utmost purity. F o r our purpose the first stage is dealt w i t h i n the kindergarten and i n courses of practical meas­ urement, and the t h i r d i n formal g e o m e t r y ; the second does not, i n my o p i n i o n , receive sufficient recognition i n the teaching of geometry at the present time, and the exercises above are intended to remedy this deficiency. T h e course may also be regarded as an introduction to the ideas of a manifold and a function. T h e i r importance has of late been recognised and need not occupy us here, but their difficulty, especially i n regard to manifolds, is hardly realised. I f a c h i l d is directed to m a r k on paper a number of points such that the sum of their distances f r o m two given points has a given value, he w i l l do so without any idea that all such points f o r m a continuous curve, and i f he is told this he will not grasp the fact. B u t allow h i m to go on m a r k i n g points and he w i l l ultimately attain to a conception of the whole and their assemblage along a continuous curve. T h i s cannot be t a u g h t ; the teacher must wait for the child to reach i t , and the value of the w o r k is lost i f mechanical means of describing the curve continuously are adopted. Such a process has its own value, but i n this connection i t obscures the idea of

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the curve as a manifold of points selected f r o m the m a n i ­ fold w h i c h forms the plane. T h e early formation of such habits of thought, even though subconscious and unsystematised, has a technical importance w h i c h deserves m e n t i o n despite our general limitation to educational considerations. Recent research has exhibited the whole structure of mathematics as founded on the processes and conceptions w h i c h underlie t h e m , and has gone far, w i t h their aid, to perfect a co-ordination be­ tween the different branches of the subject. These funda­ mental ideas are now seen to be essential to the under­ standing of even elementary mathematics, and their absence is responsible for most of the difficulties w h i c h occur i n the introduction to more advanced w o r k . T h e difficulty of the child who i n c o m m e n c i n g algebra says, " Y e s , but what is x ?" of the boy who cannot solve a simple rider, and of the student who finds the calculus puzzling, are now all traced to this c o m m o n o r i g i n . T h e moral for the teacher is obvious. U n t i l the y o u n g child can i n some way attain to such ideas i n relation to matters w i t h i n his o w n experi­ ence he has not grasped the g r o u n d w o r k of mathematical thought and so cannot erect the structure on proper foun­ dations, and the present problem for teachers is to b r i n g t h e m to h i m i n as many forms as possible. F o r this pur­ pose some understanding of the philosophy of arithmetic, geometry, and algebra is essential, and I believe that the l i g h t w h i c h this philosophy throws on problems of ele­ mentary education constitutes by no means its least value. T h e transition f r o m the preliminary stage just described to a first course of formal geometry is marked by the i n ­ troduction of connected reasoning leading from one result

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to another, and a gradual cessation o f the aid to imagina­ tion involved i n the use of houses, roads, and the l i k e . T h e assumption of postulates as a basis for this reason­ i n g should not be stated f o r m a l l y ; unless the preliminary course has failed i n its object the pupils w i l l accept t h e m without difficulty as they are required. B u t i t is essential that the choice of these postulates should not be left to chance or dictated by convenience of presentation; i t must be guided by the considerations w h i c h determine the adoption of the subject as a mental training. I f these considerations be such as are indicated above, there is no difficulty i n specifying the nature and extent of the assumptions w h i c h should be adopted as facts i n a first course of geometry. I t must be possible to induce the c h i l d to accept t h e m without the aid of numerical measurement of any k i n d , and every statement for which this is possible should be regarded as a postulate. T h i s definition includes the f o l l o w i n g : ( i ) the equality of verti­ cally opposite angles; (2) the angle properties of parallel l i n e s ; (3) properties of figures w h i c h are evident f r o m s y m m e t r y ; (4) properties of figures w h i c h can be dem­ onstrated by superposition, a method w h i c h should be once for all discarded as a proof. 1

T h e m e a n i n g and intention of this definition is best illustrated by reference to recent developments i n the teaching of geometry. T h e r e has been a tendency i n preliminary courses to associate such ideas w i t h numerical measurement, i f not, indeed, to profess demonstration by 1

D e m o n s t r a t i o n s b y folding are not proofs as the term is here used.

T h e y l a c k the essential element of deduction from two or m o r e state­ ments already admitted.

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its aid. F o r example, children may be instructed to draw two straight lines, measure the vertically opposite angles, and state what they observe. I t is hard to see what good can be derived from such exercises, and they may do m u c h h a r m . I f i t be true that the education of the child should follow the development of the race, they are condemned at once, for i t is inconceivable that these ideas were suggested by such processes, or that there was a geometrical Faraday who announced t h e m to the ancient w o r l d . T h e y can only be regarded as intuitions f r o m rough experience; a person i n w h o m they are w a n t i n g is ignorant of space, and a knowledge of (Euclidean) space is neither more nor less than their full comprehension — a comprehension resting on foundations wider by far than any schoolroom experi­ ments. A n attempt to aid its formation by numerical exercises may not unfitly be compared to an attempt to teach music by e x p l a i n i n g the mechanism of a piano, and the relation between notation and keyboard, before the p u p i l has heard a single tune. I t is a c r i p p l i n g of subjec­ tive growth at its most sensitive stage by the crudest f o r m of materialism. A m i n d w h i c h has ranged over all its ex­ perience and has made these intuitions has gained a sense of power and accepted t r u t h w h i c h cannot be induced to an equal extent by any substitute for the process. T h e statement that all possible intuitions must be taken as postulates is hardly less important than the definition (for our purpose) of a postulate as any statement of com­ m o n acceptance. T h e object of teaching geometry has been stated as twofold : the sense of logical proof is to be developed, and the conception of a chain of proofs is to be formed. T h e r e are many ways of d o i n g this ; they vary

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w i t h the selection of the theorems used and the order of their sequence, but each should be subject to the condition that every theorem should prove some new fact which could not have been perceived by direct i n t u i t i o n , sym­ metry, or superposition. I f this condition be violated the pupil's sense of a proof is confused; he appears to arrive w i t h m u c h painful labour at a result w h i c h he and every one else knew before to be true, and the impression of inevitable but unforeseen t r u t h is not imparted. T a k e , for example, the theorems concerning bisected chords of a circle : i f a boy cannot perceive the t r u t h of these w i t h , possibly, a little stimulus, his knowledge of space is so meagre that he ought not to have commenced formal geometry. I f , on the other hand, he can see their t r u t h , there can be few processes more destructive than the elaboration of deductions f r o m other intuitions w i t h the intention of i m p a r t i n g a sense of proof. T h e details of the propositions and the order of their sequence is hardly germane to this paper, but the neces­ sity for presenting t h e m i n sequences each of w h i c h can be grasped as a whole deserves further m e n t i o n . I f the Euclidean tradition be ignored, such sequences can still be found. F o r example, experience has shown that the angle properties of polygons, arcs of circles, and tangents to circles can be arranged i n one sequence, so that the unity of the group can be appreciated by pupils to w h o m it is the first example of formal geometry. A n o t h e r sub­ ject is found i n the theory of the regular p o l y h e d r a ; the limitation of their number, the investigation of their shapes and dimensions, and their construction, can be grasped by boys of ordinary ability at an early age. I t w i l l probably

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be agreed that a considerable element of solid geometry is essential; the tetrahedron, cone, and sphere are other themes w h i c h have been found possible and valuable. I t is, of course, true that E u c l i d presented geometry i n a series of such sequences, and despite the logical defi­ ciencies i n his scheme his genius is probably still unrivalled. B u t his theme was the reduction of the number of postu­ lates to a m i n i m u m and the introduction of each as late as possible. A s soon as i t is granted that all possible intui­ tions should, i n a first course, be accepted as postulates, his sequence fails to have value for this purpose, and has, indeed, demerits. I t is therefore a necessary consequence of the theory of geometrical education here developed that a deliberate effort should be made to replace this scheme by one more suitable. T h e need is not met by allowing propositions to be treated as intuitions when possible and retaining his sequence, for this destroys the whole meaning of that sequence ; his motive enforces the introduction of solid geometry as late as possible, while ours demands i t as early as possible, and numberless other contradictions arise. T h i s conclusion induces doubts whether even this resid­ u u m of Euclid's propositions, taken i n any order, forms the best material for geometrical t r a i n i n g of an educative value. Some of i t undoubtedly deserves its place, both for its own value and its necessity i n other parts of mathe­ matics ; but stripped of unessentials this is small i n content and less than can be acquired i n an ordinary school course. Before i n c l u d i n g more than this bare m i n i m u m i t is advis­ able to ascertain whether, i n the whole domain of geomet­ rical knowledge, there may not be other matter of more

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educative value w i t h i n the grasp of ordinary boys and girls. H e r e I a m passing beyond the range of m y own experi­ ence, but I have, as the outcome of a m a t u r i n g conviction, made tentative experiments w i t h individual pupils and small classes and deem i t worth while to state the result. I f , without regard to the age of the pupils, one asks where the methods and ideals of geometry are presented in the greatest unity, simplicity, and beauty, the answer must be that geometry of position and projective geometry have no rivals. Estimated o n such standards Euclid's w o r k is dwarfed by these modern creations, and not least so i n respect of the ease and generality of their conceptions. M u s t i t be said, i n spite of this, that they are to remain the property of professed mathematicians since they are beyond the grasp of the n o r m a l adolescent, or may i t be that their power and beauty can be appreciated, even i f only i n some comparatively crude f o r m ? E v e n those, i f there be any, who at once deny the possibility w i l l not dispute the i m ­ portance of the question and the value of success i f i t can be attained. M y own conviction, fortified by such l i m i t e d experience as I have indicated, is that the elementary con­ cepts and methods of projective geometry can be grasped by ordinary p u p i l s ; that they would excite a greater inter­ est and fuller spirit o f enquiry than any f o r m of Euclidean geometry, and that their educational value would be far greater. T h e amount of knowledge required is not so great as m i g h t be imagined. A p u p i l who has had some expe­ rience of three-dimensional w o r k can grasp the relations between a figure i n one plane and its projection i n another plane, i n c l u d i n g the particular cases wherein a p o i n t or line i n either plane have their corresponding element i n

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the other at infinity. T h e theorem of Desargues, w i t h its simpler consequences, is then w i t h i n his grasp, and he may so gain some idea of the extent and variety of the results w h i c h can be deduced from the axioms of position only, and the manner i n w h i c h they unite i n one statement results w h i c h he had regarded as disconnected. I t is then natural to enquire whether there is any simple relation between corresponding segments i n the two planes. T h e position ratio AP : PJ5, w h i c h defines the position of a p o i n t P on a l i n e w h e n t w o of its points A , B are given, should have become familiar i n connection w i t h earlier work, and i t is now easy to prove that corresponding position ratios remain i n a constant ratio as the points P, P' move along their respective lines. H e n c e the ratio PB

: ^ = , formed from two such ratios, is unchanged by QB

'

projection. T h e metric properties of quadrangles and quadrilaterals (deduced by projection f r o m a parallelogram), and the simple properties of the conic regarded as the projection of a circle, can then be investigated to such extent as may appear possible or desirable. I have myself, w i t h ordinary boys of eighteen, reached Carnot's theorem without great difficulty. I t may be suggested that this w o r k is unnecessary, and that when the m i n i m u m of geometry has been acquired the p u p i l should proceed to trigonometry or other branches of mathematics.

A s to this I can only say that this paper

is i n part an attempt to justify the teaching i n schools of an amount of geometry m u c h larger than has hitherto been thought possible, and this without increase of the t i m e devoted to the subject.

M y own experience is that

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an early commencement of trigonometry can and should be made by some sacrifice of the large amount of time now devoted to algebra by pupils who are too y o u n g to understand the subject. T r i g o n o m e t r y is a valuable stimulus to geometrical thought, but is no substitute for i t . I t only remains, i n considering the scheme of geometri­ cal education, to refer to the interdependence of the pos­ tulates w h i c h have been adopted. I do not believe that any detailed or systematic discussion of this is possible or advisable at the school age, but i f towards the end of this period examples of deduction of some postulates f r o m others were shown, i t m i g h t be possible to lead the pupils to realise the ideal of a geometry based on a m i n i m u m number of assumptions concerning the nature of space. Such considerations would then acquire more reality for them i n that they would have some acquaintance w i t h mechanics and physical science and could therefore con­ ceive the general ideal of a m i n i m u m of induction and a m a x i m u m of deduction. A n d I feel bound to state m y conviction that every student of whatever subject, who pro­ ceeds to a university education w o r t h y of the name, should gain some slight idea of the nature of non-Euclidean geom­ etry. T h e simpler portions of the paper by Carslaw i n the Proceedings

of the Edinburgh

Mathematical

Society

for

ip/O, or the description of Poincare's well-known illustra­ tion given i n Y o u n g ' s " Elementary Concepts of Geometry and A l g e b r a , " already mentioned on page 60, are w i t h i n the grasp of any one who has even a slight acquaintance w i t h the geometry of the c i r c l e ; an appreciation of these ideas throws a l i g h t on the space-concept i n particular and our so-called knowledge i n general which can be

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gained i n no other way. I t may fairly be said that there are few portions of mathematical knowledge w h i c h have more educational value. T h e nature of the current teaching of geometry i n E n g ­ land is best understood by reference to recent history. U n t i l a few years ago the use of Euclid's text i n matter and sequence was universal o w i n g to the regulations of e x a m i n i n g bodies. A t t e m p t s to secure more freedom had not been w a n t i n g . T h e Association for the I m p r o v e m e n t of Geometrical T e a c h i n g was formed w i t h this object as early as 1S71, and undoubtedly succeeded i n awakening interest i n the presentation of elementary geometry, though no tangible result appeared. I n 1 9 0 1 , however, a paper read by Professor Perry before the B r i t i s h Associa­ t i o n at Glasgow aroused fresh interest, and a committee was formed by the association to consider and report upon the teaching of elementary mathematics. I n their report, issued a year later, they advocated a p r e l i m i n a r y course of practical geometry, and stated that i t was i n their o p i n i o n unnecessary and undesirable that one text-book or one order of development should be placed i n a position of authority. Simultaneously the Mathematical Association, a develop­ ment of the association founded i n 1 8 7 1 , had formed a committee to consider the subject, and this body issued a report i n M a y , 1902. I n principle i t proceeds on lines similar to the report of the B r i t i s h Association Committee, but i t contains a definite statement that " i t is not proposed to interfere w i t h the logical order of E u c l i d ' s series of theorems." I n effect i t simplifies the development by i n ­ troducing hypothetical constructions, changes the order of

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certain groups of propositions, and introduces an algebraic treatment (for commensurables only) of ratio and propor­ tion. A s a consequence of the w o r k of these committees and their supporters the e x a m i n i n g bodies yielded one by one, announcing that E u c l i d ' s order and methods would not be enforced, and setting questions i n v o l v i n g numerical construction and calculation. Finally, the U n i v e r s i t y of Cambridge issued i n 1903 a syllabus i n d i c a t i n g the amount of geometrical knowledge required for its entrance exam­ ination, and stated that the examiners would accept any proof w h i c h appeared to f o r m part of a systematic treat­ ment of the subject. T h i s syllabus, w h i c h is still i n force, omits portions of E u c l i d ' s text and introduces no addi­ tional matter. T a k i n g these changes i n detail, we are first concerned w i t h the postulates. T h e intention appears to have been the increase of their number i n order to simplify certain proofs w h i c h had presented difficulty; proofs of facts w h i c h can be perceived by intuition are still retained, notably i n regard to congruent figures and properties of the circle. T h e present scheme may therefore be not unfairly described as E u c l i d ' s ideal of the m i n i m u m n u m ­ ber of assumptions, tempered by consideration for the age of the pupils — a consideration w h i c h renders the whole scheme meaningless f r o m his p o i n t of view. I t may be doubted whether such a compromise is likely to be suc­ cessful. T h e broad lines of educational advance should be based on fixed principles and not on expediency. I f the pupils can understand the development of a scheme based on a m i n i m u m of assumption, and can also understand this motive, let i t be adopted as the highest ideal. B u t i f

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not, as experience has shown w i t h some certainty, the only logical alternative is to allow all possible intuitions as pos­ tulates. T h e educational advantages of this course have already been described, and i t has the further advantage of lessening the amount of t i m e and effort required, and so clearing the way for some study of more advanced geometry, w h i c h is not yet represented i n any of the ex­ aminations referred to. Should any doubt be felt as to the persistence of Euclidean methods despite the abandonment of his ideal, consideration of the fact that even the elements of solid geometry are not included i n school courses, ex­ cept i n p r e l i m i n a r y w o r k (to w h i c h reference w i l l be made), may carry conviction. A l l e x a m i n i n g "bodies (the C i v i l Service Commission excepted) appear to i m p l y by their schedules that sufficient t r a i n i n g i n schools can be obtained w i t h o u t i t , the fact being, of course, that its omission is a survival of E u c l i d ' s order. T h u s i t may perhaps be said that what is often called the abolition of E u c l i d must involve the complete abolition of his order and methods and the construction of an­ other theory o f development, and that i n both respects the changes are incomplete. T h e structure, always of course fallacious i n its foundations, has now been shattered and we are g r o p i n g among the fragments. T u r n i n g to educational methods, there are two important developments: the introduction of preliminary courses, and the use of numerical examples. T h e latter only requires brief reference; i t must give greater reality and preci­ sion to the results w h i c h i t is intended to illustrate, and there is c o m m o n agreement that i t has done this. T h e m e a n i n g and proofs of propositions are admittedly better

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comprehended, but the comprehension is too often of isolated results rather than structures of reasoning. T h e object of the preliminary course is to enable the pupils to acquire some familiarity w i t h the leading facts and concepts of the subject. For the most part such courses consist of exercises i n measurement and construction, coupled ( i n some cases at any rate) w i t h numerical intro­ ductions to or illustrations of the axioms. So l o n g as such exercises are confined to the performance of constructions of k n o w n type there is little to be said for or against t h e m i n logic or philosophy, t h o u g h they are, i n the opinion of some, as deadening to the intellect as the excessive per­ formance of algebraic simplifications. B u t when the angle properties of parallels and triangles are introduced by measurement w i t h a protractor, instead of by t u r n i n g a rod, and w h e n we find an example such as: " D r a w a triangle whose sides are 2, 3, and 4 inches and then draw the perpendiculars f r o m each vertex to the opposite side. These lines should meet i n a p o i n t ; see that they do so," the matter becomes more serious, for numerical measurement has been substituted for intuition or demon­ stration and the impression is hard to eradicate. T h e r e is also some general introduction dealing w i t h space-concepts, and here there is usually some allusion to objects i n three d i m e n s i o n s ; beyond this, solid geometry finds as a rule no place i n such courses. I t s persistent neglect by teachers, e x a m i n i n g bodies, and writers of text-books is one of the most m a r k e d and regrettable features i n the developments of recent years. A p a r t f r o m this, the m a i n criticisms, f r o m the p o i n t of view of this paper, to be directed against such courses are

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that they are not based on a gradual extension from previ­ ous experience and imagination, and that there is a distinct tendency, as has been said, to relate the postulates to nu­ merical processes. O f the second I need not speak f u r t h e r ; of the first i t may be said that i t violates the principle that the development of the powers of imagination, abstraction, and reasoning should be made continuously from experi­ ence and knowledge gained i n daily life. T o construct a preliminary course w h i c h consists of w o r k concerning angles, lines, triangles, and circles, w i t h perhaps a pass­ i n g reference to a few surveying problems, is to place the child suddenly i n a new w o r l d where things are replaced by abstractions, and to give h i m an occasional glimpse of his o w n sphere as from behind bars — bars which are not made thinner by assigning numerical measures to the lengths and angles w i t h w h i c h he deals. I n the alterna­ tive w h i c h I have suggested the endeavour is to lead h i m gradually to this new w o r l d of abstract t h o u g h t and ideal t r u t h , or, perhaps, to present i t as an outcome of and one w i t h that more l i m i t e d w o r l d of which alone he is at first cognisant. T h e result of this period of freedom has been summed up i n a circular published by the Board of Education ( N o . 7 1 1 , 1909), w h i c h contains luminous and practical suggestions to teachers, based on the experience of the Board's inspectors. For our present purpose the most s t r i k i n g statement made therein is that the t i m e taken to acquire the matter of the first three books of E u c l i d varies from one to three years i n different schools, and that i t is where the work proceeds quickly that i t is best, and nearly always where i t proceeds slowly i t is poor. T h e difference

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is ascribed to the manner of treating the earlier part of the work, w i t h w h i c h the circular is mainly concerned. A s to this, I w i l l only say that, while its suggestions are far i n advance of the matter contained i n most text-books, i t perhaps hardly lays sufficient stress on the need of devel­ opment f r o m the child's previous experience, nor does i t suggest concrete problems r e q u i r i n g a considerable amount of imagination and reasoning. T h e importance of solid geometry is pointed out w i t h some force, but no very definite suggestion is made as to the t i m e or manner of its introduction i n a deductive course, a p o i n t on w h i c h most teachers are i n need of guidance, and especially those who now succeed i n covering a matriculation course i n two or three years. T h e circular also deals w i t h the general effect of the changes, stating that i t has been beneficial. " U n i n t e l l i g e n t learning by rote has practically disappeared, and classes, for the most part, understand what they are doing, t h o u g h they often lack power of insight and have but a narrow extent of knowledge." H a d the changes already described been the only edu­ cational changes d u r i n g this period, a fairly conclusive inference m i g h t be drawn. B u t i t must not be forgotten that the modern secondary school, w i t h graduates of teach­ i n g universities for its teachers (often trained) and a cur­ r i c u l u m designed to develop all the pupils rather than to benefit the few of exceptional ability, has, d u r i n g the same time, come into being, and the issue is thus confused. Greater comprehension, as due to improved teaching, would have been likely even though E u c l i d had not been dethroned. T h e older schools have for the most part been

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comparatively unaffected by such developments d u r i n g this time. Speaking as one who has some experience, both as teacher and examiner, of these and other schools, I can but state m y opinion that the improvement i n the modern secondary schools is far greater than i n the schools of other types, i n c l u d i n g those which have adopted the changes i n geometry most fully. D e v o t i n g roughly an equal amount of t i m e to the subject, they obtain better results at an earlier age. I f I am r i g h t i n this — and I believe that most m e n who have acquaintance w i t h the various types of school i n this country would confirm the statement — i t follows that most of the recent improve­ ment i n modern secondary schools is due, not to any recent changes i n the syllabus of geometry, but to the ac­ quisition of teachers who not only understand the subject but also k n o w how to teach i t . Some confirmation of this view is given by an enquiry made some three years ago a m o n g the professors and examiners i n the modern universities. T h e y declared themselves as dissatisfied alike w i t h the results of the older and more modern methods, the major­ ity against the modern methods being the larger. T h e i m p r o v e m e n t i n quality of knowledge was admitted i n many cases; i t was rather the material and its co-ordina­ t i o n that were criticised. Finally, then, i t may be said that improvements i n teaching methods and i n personnel of the teachers have produced their natural results, and to the teachers must be ascribed m u c h of the admitted improvement. Schemes of geometrical education i n this country are l a c k i n g i n foun­ dation, method, and extent, and this arises from the fact that Euclid's scheme — itself utterly unsuitable as an

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introduction to the subject—has been so far tampered w i t h that hardly any scheme remains. So l o n g as no attempt is made to devise a connected development based o n the many intuitions w h i c h are c o m m o n to all civilised beings before they reach maturity, so l o n g w i l l the subject realise a painfully small proportion of its potential value. I have endeavoured i n this paper to interpret the quo­ tation at its head i n its reference to m y subject. I do not forget that children w i l l have to do the w o r k of the w o r l d and must be fitted for i t , and I believe that a t r a i n i n g such as is here described w i l l assist t h e m i n this. B u t they w i l l not be less fitted i f their education provides t h e m w i t h w i d e n i n g and i n s p i r i n g subjects for contemplation w h e n they reach maturity, nor indeed is such fitness the sole end of life.

T H E P L A C E OF D E D U C T I O N I N ELEMENTARY MECHANICS ( R e p r i n t e d by permission from the Proceedings of the Congress of Mathematicians, August, 1912)

International

THE

PLACE

OF

DEDUCTION

TARY

IN

ELEMEN­

MECHANICS

A science consists of a definite class or of definite classes of entities, a set or sets of postulates relating them, and a series of deductions w h i c h are logical consequences of these postulates. I n the earlier stages of its evolution i t may be — i t usually has been — that the sets of postulates contain redundancies. O n l y when the logical consistence of these sets has been investigated, and the number o f independent postulates established, can the science be termed complete, for t h e n only is i t certain that the number o f assumptions has been reduced to a m i n i m u m , and that no one o f t h e m conflicts w i t h any other. A l t h o u g h the concept of a perfect science was attained by E u c l i d i n connection w i t h geometry, the first approximately successful presentation of a science i n this f o r m came as late as N e w t o n , and then i n connec­ t i o n w i t h mechanics; geometry, o n the other hand, has only been completed w i t h i n the last generation, and this after struggles e x t e n d i n g over t w o thousand years. T h i s historic distinction between geometry and mechanics implies a corresponding didactic distinction. I t is the pecul­ iarity of geometry, as opposed to other physical sciences, that its postulates, and many of the deductions w h i c h can be made f r o m t h e m , are and have l o n g been the c o m m o n property of civilised m a n k i n d . W h o doubts, whether he has learnt geometry or not, that all r i g h t angles are equal, that only one parallel can be drawn t h r o u g h a p o i n t to a "5

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given line, or that any diameter divides a circle into two identical parts ? Most, i f not all, of the postulates of mechanics are, o n the other hand, k n o w n only to those who have consciously adopted t h e m . Indeed, many persons, otherwise educated, w i l l dispute their t r u t h . T h e problems before the teacher are therefore entirely different i n the two cases. I n geometry he steps into an inheritance o f preacquired space concepts, crude perhaps, but formed beyond d o u b t ; he can develop deductions w i t h little trouble con­ cerning postulates. B u t i n mechanics he must set up the entities and develop the postulates f r o m the commence­ ment ; he steps into no such inheritance as does the teacher of geometry. W i t h methods of e x h i b i t i n g mechanical entities, and the choice of postulates for use i n a first treatment of the subject, this paper is not concerned. I t is assumed, how­ ever, i n accordance w i t h m o d e r n practice i n most schools, that the number of postulates is more than the m i n i m u m . M y first concern is to p o i n t out that, treat i t how we may, the direct evidence w h i c h can be brought before a boy i n support of any of these postulates is singularly narrow and unconvincing. A n d i t is an essential part of the argument that this weakness should be exposed at the outset. T a k e , for instance, the triangle of forces ; the p u p i l may fairly ask W i t h i n what degree of accuracy has i t been demonstrated ? I s i t true whatever be the body on w h i c h the forces act ? I s i t true at all places and at a l l times ? I s i t true under any circumstances o f m o t i o n ? I s i t true for forces of all k i n d s — electrical, magnetic, or any other ? O r again, take the proportionality between force and acceleration, i f the subject be developed so that this is a

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postulate. T h e p u p i l may be convinced that, i n his own locality and for some substances, the statement is approx­ imately true. B u t does the functional relation between force and acceleration involve no other variables, for ex­ ample, temperature ? A n d is its f o r m the same for all kinds of matter ? M a y not the force be proportional to the square of the acceleration for some substances other than those used i n the experiments ? Pupils should be trained, and trained from the outset, to question i n this manner the degree of accuracy of every measurement, and the gener­ ality of the circumstances under w h i c h each experiment is performed. I t is scientific method, and education of the most practical and valuable character. B u t , then, the pupils may say, what is the use of g o i n g further ? M u s t not some better evidence be obtained ? T h e r e are, of course, many reasons which can and should be given for g o i n g o n i n faith, but one of the most i l l u m i ­ n a t i n g and interesting illustrations is contained i n Faraday's verification of Coulomb's law of electrostatic attraction. Few boys fail to find interest i n the picture o f Faraday, basing h i g h l y complex calculations on Coulomb's crude experi­ ments, testing t h e m inside the h i g h l y charged i r o n box, and e m e r g i n g from i t w i t h a demonstration that the law corresponds really closely w i t h observed facts. T h e p u p i l thus realises the true importance of deduction as an aid to his very imperfect powers of observation. I n place of b u i l d i n g complacently on a foundation whose imperfections have been glossed over, too often w i t h pulleys mounted on ball bearings and other viciously misleading trivialities, he has a sane idea of what he is doing. H e sees that, o n these meagre foundations, he is to erect a structure w h i c h w i l l

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come into contact w i t h practical experience at many points, and must be judged by its degree of accordance w i t h such experience at a l l these points. T h e structure h a v i n g been erected on a number of these very dubious supports, i t becomes necessary to examine their possible interconnection, to ascertain whether the t r u t h or falsity of any one of these assumptions involves of logical necessity the t r u t h or falsity of any others. T o illustrate the process suggested, I have dealt w i t h the postulates of statics, but the method is equally applicable to dynamics, or to any combination of mechanical postulates. T h e customary assumptions i n a first treatment of statics — suggested, and r i g h t l y so, by crude experiment — are three i n number, namely, the triangle of forces, the principle of the lever, and the principle of moments for two forces acting along intersecting lines. A l l three have been adopted, but any or all may be true or false. T h u s the possibilities are eight i n number, as shown i n the f o l l o w i n g table, and among t h e m must the t r u t h be s o u g h t :

I 2 3 4 5 6 7 8

TRIANGLE OF FORCES

PRINCIPLE OF LEVER

P R I N C I P L E OF MOMENTS

true false true true true false false false

true true false true false true false false

true true true false false false true true

N o w i t is possible to show, by logical deduction, that any two of these assumptions are necessary consequences of the

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1

t h i r d . W h e n this is done (and the proofs are well w i t h i n the comprehension of a boy of seventeen), the alternatives 2 to 7 disappear, and the three assumptions have become one. T h e only possibility now being the simultaneous t r u t h or falsity of all, the rough experimental results acquire greater i m p o r t . I f all three were false, i t is trebly unlikely that they should every one accord fairly well w i t h experi­ ment, and the only alternative to the falsity of all is the t r u t h of a l l . T h e probability of this t r u t h has thus been strongly reinforced by processes purely logical i n nature. I t is worthy of notice that the conventional deductive method does not give an equal amount o f strength to the hypothesis. Postulating the triangle of forces, the principle of the lever and the principle of moments are obtained by logical deduction, the possible alternatives, five i n number, being left t h u s : TRIANGLE OF FORCES

PRINCIPLE OF LEVER

PRINCIPLE OF MOMENTS

true

true

true

false

true

true

false

true

false

false

false

true

false

false

false

T h e proofs that any two of these postulates can be de­ duced f r o m the t h i r d are not given, and the f u l l power of deduction to reinforce assumption is not exhibited. T r e a t i n g other groups of postulates i n the same manner, the structure is seen to be based not o n weak, isolated 1

I f this is done, selecting a n y one assumption only, the consistence

of the three assumptions follows at once, and the demonstration of con­ sistence should not be o v e r l o o k e d .

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supports, but o n i n t e r l i n k e d groups of such supports, the strength of the foundation being greatly increased by these interconnections. A n d finality is reached w h e n the sup­ ports have been i n t e r l i n k e d into groups, between which i t can be demonstrated that no such logical interconnections are possible. T h i s , I conceive, is the true aspect of New­ ton's achievement i n the statement of his three laws of motion ; he stated a number of consistent and independent hypotheses, and developed the whole subject from these by purely logical processes. I do not imagine that N e w t o n had any such general concept of a science as is set out at the b e g i n n i n g of this paper, but he perceived, intuitively or sub-consciously, that mechanics could be based on three sets of postulates, each set referring to a different class of entities. I n their statement his logic was defective, but this is t r i f l i n g compared w i t h the greatness of his achieve­ ment ; he formed an ideal for mechanics similar to that of E u c l i d for geometry, and he attained practical success i n its elaboration, i n contrast w i t h E u c l i d ' s decided failure. W e have now before us three distinct didactic treatments of mechanics — the o l d method, the m e t h o d now current, and a development such as has here been suggested. T h e old method presents the science i n its complete f o r m , w i t h no indication of its evolution. T h r e e postulates are laid down, to be accepted i n b l i n d faith, and f r o m t h e m the subject is developed by logical processes; i t is a course i n applied deduction. T h e current method presents a number of mechanical assumptions, based on foundations whose strength is hardly discussed, and uses t h e m i n application to various problems. T h e r e is little or no attempt to discuss their logical interconnection, and certainly no suggestion

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of its scientific m e a n i n g ; i t is a course i n applied com­ putation. B u t i f this current method were followed by a logical discussion, e x h i b i t i n g mechanics as based on inde­ pendent supports, each consisting of i n t e r l i n k e d assump­ tions as has been described, I venture to suggest that i t m i g h t fairly be called a course i n applied mathematics. T h e tendency of modern education, as i t seems to me, is to lay undue stress on direct sensation as the one and only basis for faith. Undoubtedly education must find its origins, and these as widespread as possible, i n direct sen­ sation ; but a false and very dangerous ideal is left, unless finally these origins are l i n k e d together by logical process, so as to give t h e m their m a x i m u m strength and expose their ultimate weakness. F i n a l contentment w i t h a set of postulates w h i c h may or may not be inconsistent or re­ dundant, and for w h i c h there has appeared little real justi­ fication, is vicious ; vicious also is the attempt i n a first course to develop any science f r o m a m i n i m u m of hypoth­ esis. T h e one method is an undue suppression of per­ ception, the other an undue glorification. T h e first step of the teacher should be to develop a wide spirit of en­ q u i r y ; the second should be to breed a " d i v i n e discontent" w i t h the imperfections of perceptual evidence. Recognis­ i n g its essential nature, our inevitable bondage to i t , we may yet liberate ourselves, so far as may be, i n each branch of knowledge. I t is the peculiar function of mathematics to point the way to this freedom i n each science, and it is here that m o d e r n developments of mathematical thought may yet find application i n other sciences.

A C O M P A R I S O N OF G E O M E T R Y W I T H MECHANICS ( A paper r e a d before the L i v e r p o o l A s s o c i a t i o n of T e a c h e r s of Mathematics and P h y s i c s )

A

COMPARISON

OF GEOMETRY

WITH

MECHANICS T h e subject of this paper may at first sight seem unlikely to provide m u c h opportunity for profitable discussion. W h e n i t has been said that the bodies which are the con­ cern of mechanics move i n the space w h i c h is the concern of geometry, and are therefore subject to the laws of that space, and when i t is added that the same processes of arithmetic are applied to measurement i n each case, i t may be thought that the title is exhausted. T h e comparison to w h i c h I invite you is not, however, directly concerned w i t h the matter or the domain of either subject. M y object is to compare t h e m i n their relation to mathematical education; to examine i n how far each may fulfil the ideals of that education, and i n how far each may supplement the deficiencies of the other. T o do this, the basis of comparison must first be assured; that is, i t is necessary that I should explain what are the ends which I conceive to be furthered by the teaching of mathematics. I n the conflict of educational interests which has i n the last fifteen years become so acute, mathematicians have borne their part, but they cannot be said to have spoken w i t h one voice, whether i n advocacy or defence. I do not k n o w that this is to be regretted, for no progressive devel­ opment is likely, except as the outcome of such differ­ ences ; but the fact enforces, o n any who would discuss mathematical education, a clear statement of their creed. 125

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E v e n yet, however, the preliminary enquiry is unfinished. I have just said that differing views on the aims of mathe­ matical teaching are held by those who are concerned w i t h the subject. F o r the most part they are held consciously and can be explained at w i l l . B u t an enquiry as to the nature of mathematics itself — the short question, " W h a t is mathematics ?" — is apt to produce no immediate or definite response. T o o often, I fear, the only reply w i l l be that mathematics consists of algebra, geometry, trigonom­ etry, the calculus, and so on, arithmetic being, for some unaccountable reason, o m i t t e d from the category. N o w this is a t r i f l i n g w i t h logic by just those people who ought, above all others, to be i n this respect beyond suspicion. I f they consider heat, light, sound, they can see w h y these are grouped under the c o m m o n t e r m " physics " ; i f they study atoms, decomposition, elements, they can defend the one term " chemistry " for these. B u t w h y the one t e r m " mathe­ matics " for algebra, geometry, calculus, and all the other branches ? W h a t are the common elements i n these sub­ jects w h i c h entitle t h e m to a generic t e r m ? Failure to answer is a failure i n logic, for to group entities i n one class w i t h o u t cognisance of common elements among t h e m is to offend the cardinal principles o f classification and definition. I t may, I know, be said w i t h t r u t h that mathematics is concerned w i t h reasoning — pure reasoning, i f you w i l l . B u t this is logic, and even those who are most uncertain as to the definition of mathematics are equally certain that there is some distinction between the study of mathematics and the study of logic. A necessary preliminary to the discussion w h i c h I have undertaken is, then, to define this

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distinction, and this must be m y first task. A f t e r the explanation of the nature of mathematics, its relation to education must be discussed ; our comparison of geometry and mechanics can then be developed. T h e nature of mathematics may best be explained by showing its relation to the physical sciences. T h e study of any one of these commences by the assertion of a number of statements, on bases more or less uncertain. T h e assump­ tion of laws of m o t i o n here on the earth, from observation of the planets, is a daring one, however exactly these laws describe the motions of the planets themselves; the direct evidence for the law of conservation of energy, or that for most other laws of physics and chemistry, is weak to a degree. T o see general possibilities i n a maze of results apparently unco-ordinated, to fashion various hypotheses to fit the facts as they are observed, is the function of the natural philosopher ; i t is no concern of the mathematician. T h e process is exemplified i n the lives of any of the great natural philosophers — above all, perhaps, i n the life of N e w t o n , i n that the development of his m i n d is k n o w n i n such detail. T h e natural philosopher, h a v i n g thus fulfilled his first task, presents the hypotheses so formed to the mathemati­ cian, who accepts t h e m for investigation without regard to the evidence for or against t h e m . L e t us call these hypoth­ eses A , B , C, . . . . T h e mathematician does three things : First, he makes deductions from the hypotheses; that is, he says to the natural philosopher: " I f what you say is true (and that is no concern o f mine) then must certain other statements P, Q, R, . . . also be true ; further, i f cer­ tain of these be tnie, then must certain of A , B , C, . . . also

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be t r u e . " T h e natural philosopher then tests the t r u t h of P, Q , R, . . . and finds his hypotheses strengthened or destroyed as the case may be. N e x t , the mathematician informs the natural philosopher that he has examined the hypotheses A , B , C, . . . and finds that they are consistent; that is, that there is n o t h i n g in any one of t h e m to negative any other by the force of logic. I t should here be borne i n m i n d that all measure­ ments are more or less inexact, and i t is therefore possible, from actual observations, u n k n o w i n g l y to frame hypotheses which are logically contradictory one of another. Finally, the mathematician informs the natural philos­ opher that some of his assumptions were unnecessary; that is, that some are logical consequences of others, and so need not have been assumed. O r he tells h i m that all were necessary, no one being deducible f r o m some or all of the others. T h a t is, he investigates the possible redun­ dance of the hypotheses, and tells the natural philosopher to how many distinct assumptions he is really committed. Shortly, then, the mathematician receives sets of hypoth­ eses from the natural philosopher, tests their consistence, examines how many assumptions are i n fact involved, and develops logical consequences from t h e m . H i s function is entirely impartial — material t r u t h is not his concern, but is that o f the natural philosopher; but I would p o i n t out to you that the really great natural philosophers, from A r c h i m e d e s downwards, have been m e n who i n themselves combined both these functions. I n no one have they been exemplified more nearly i n their due proportions than i n N e w t o n ; there have been greater mathematicians and there have been m e n whose power of speculation was more rapid

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and prolific, but no other man has so balanced the one w i t h the other, and therein lay the secret o f his pre-eminence. I must not, however, leave an impression that the mathe­ matician has no use for his imagination and performs no creative functions, though the slightest consideration of any branch of the subject suffices to show the fatuity of such an idea. T h e straight line at infinity, the circular points at infinity, complex numbers, lines of force, the ether, at once occur to the m i n d as commonplace instances of creations i n w h i c h the natural philosopher has had no direct share. H i s entities are groups of sensations, and relations between t h e m are suggested by further sensa­ tions ; but the mathematician creates other entities w h i c h would forever remain beyond the vision of the natural philosopher, and has often, by their means, revealed unsus­ pected unities i n his w o r k . T h e creations of the mathe­ matician w h i c h are opposed to the suggestions o f the senses must forever rank among the most s t r i k i n g o f all h u m a n creations, and the shortness o f this allusion to t h e m is only excused by their irrelevance to the matter under discussion. I t is now easy to explain the function of mathematics i n a well-balanced education. T h e purpose of teaching natural science is to develop i n combination the powers of observation and speculation; to train the p u p i l to use his senses and, from the material which they afford h i m , to frame hypotheses w h i c h accord w i t h that material as nearly as may be possible. T h e purpose of teaching mathematics is to enable h i m to develop the consequences o f these hypotheses, to test their consistence, and to reduce t h e m to the m i n i m u m of pure assumption. I say to train h i m i n these things, but i t were perhaps better to speak of setting

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them before h i m as ideals for w h i c h he must strive i n his dealings w i t h things as he finds t h e m . A m a n who has i n his m i n d t h i s chain of processes, observation, speculation, proof of consistence i n speculation, rejection of redundant speculation, and finally the erection of deductions on this foundation, is i n possession o f an intellectual creation w h i c h , i n beauty alone, is worthy to rank w i t h the creations of poetry, music, or a r t ; and beyond this, i t is a possession which, i n so far as i t guides his life, w i l l make of h i m a more efficient labourer and a better citizen. W e must now examine the relations of geometry and mechanics to the description of mathematics which has been given, and so ascertain w h i c h of the three processes w h i c h have been said to pertain to the mathematician are most clearly exemplified i n each subject. First, however, I must p o i n t out that this description o f mathematics con­ signs geometry, and even arithmetic, to the domain of applied mathematics. T h e one shows the application of mathematics to number, the other to space; i n neither is the u n d e r l y i n g essence seen, except t h r o u g h illustrations of one k i n d or the other. B u t the first t h i n g to consider w i t h respect to a machine is to see what i t does, rather than to find out how i t performs its functions, and we need not, therefore, cavil at the idea that our so-called branches of mathematics are really applications of mathematics, by examination and contrast o f w h i c h we may perceive t h e underlying unity. W h e n a c h i l d commences geometry, what is his personal position i n regard to space ? Certainly i t is far different from his position i n regard to matter when he commences mechanics, or f r o m his position i n regard to l i g h t or heat

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when he commences physics. Personal experience of space — his space — has been forced on h i m from his earliest days i n his every movement, and f r o m this experi­ ence he has formed ideas or hypotheses concerning a space beyond, w h i c h he cannot reach w i t h his own limbs, and is therefore not his space. Show h i m a triangle ABC cut out i n cardboard, and make another by t a k i n g a tracing of the corner A and p r o d u c i n g its sides u n t i l they are equal to AB and A C, and t h e n ask h i m i f these triangles are an exact fit. H e w i l l not have much doubt about this, nor w i l l he question the equality of all r i g h t angles, i f i t be similarly suggested to h i m . N e x t draw a circle, rule a diameter, and ask h i m i f one part of the curve so divided w i l l fit the other. A b o u t this also he w i l l not have m u c h doubt. H e w i l l assert the t r u t h of these things wherever and whenever the acts are performed, not merely i n his own personal space; that is, he asserts t h e m for the imagined space beyond, concerning w h i c h he has formed ideas fashioned f r o m experience i n his own space. N o w as a matter of fact, as is well k n o w n to all of us here, the t h i r d statement concerning the circle is a necessary a n d inevitable consequence of the first two. These three asser­ tions are i n t r u t h redundant, to use our technical phrase, as also are many others which the child w i l l make w i t h equal certainty concerning this imagined space beyond his personal space. T h u s , i n c o m m e n c i n g the study of geometry, we apply mathematics to a subject — space — concerning w h i c h the p u p i l is already i n possession of a set of beliefs w h i c h are, as a matter of fact, interdependent one o n another. A n d these beliefs are held w i t h great tenacity; n o t h i n g w i l l

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induce the pupil to doubt any of them, for they have, i n t r u t h , become a part of his very being. I t is further to be remarked, for purposes of comparison, that we elders and experts have not recanted any of these beliefs; we h o l d them as does the child, though we k n o w t h e m to be but beliefs ; our imagined space has the same properties as his. N e x t let us consider the position of a p u p i l i n regard to matter, when he commences the study of mechanics. I t is true that here also experience has been forced o n h i m from his earliest days, but i t is of the narrowest k i n d , for it concerns little more than the sensation of l i f t i n g , and few assumptions are made. H e w i l l say, i f two boxes are k n o w n to be exactly alike and one feels heavier than the other, that this one has something inside i t ; and, w h i c h is a con­ sideration fully as important, i f neither feels to h i m heavier than the other, he w i l l refuse to say that each is empty, p o i n t i n g out that the possible contents of one may be so l i g h t that he does not notice t h e m . B u t he is unconscious of the concept of mass (or nearly so), of the triangle of forces, and of any of the mechanical concepts and laws w h i c h are to us so familiar. M o r e , he only receives t h e m w i t h difficulty; explanation, illustration, and distinct effort are required before the concepts are grasped and the state­ ments accepted as more or less nearly corresponding w i t h the results of experience. Some of these statements, as, for example, Newton's first and t h i r d laws, are indeed received w i t h i n c r e d u l i t y ; who has not heard i t argued that the horse pulls the cart more than the cart pulls the horse ? A t the outset, then, there is a sharp distinction between geometry and mechanics. I n geometry the p u p i l com­ mences w i t h a number of ideas and beliefs concerning

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space—beliefs held so tenaciously that to question t h e m i n any way produces bewilderment. I n mechanics he has but a few crude concepts and few or no beliefs, and he greets w i t h disbelief some of those held by his elders. I n the terms of our earlier discussion, he has, i n regard to space, been his own natural philosopher — though all uncon­ s c i o u s l y — and has produced a set of beliefs w h i c h are ready for examination by the mathematician w i t h i n h i m ; but i n regard to the matter the natural philosopher w i t h i n h i m has yet to play his part before the mathematician can receive the materials for his task. R e t u r n i n g to geometry, to w h i c h of the three processes performed by the mathematician should the p u p i l first be led i n his study of this subject ? Shall he find whether his beliefs concerning space are consistent w i t h one an­ other, or enquire whether some of t h e m may not be logical consequences of others ; or shall he, without any such analyses, pass on to make deductions f r o m his spatial creed, deferring its analysis for the t i m e at least ? T h e r e can, I t h i n k , be little doubt as to the answer. T o analyse the number of his beliefs, to question and examine their foundations, is repugnant to any normal child, t h o u g h welcome to most educated adults; we should then avoid these processes and allow the study of geometry to centre round deductions from the pupils' spatial beliefs. I n p r i ­ mary education this subject can only illustrate the deductive side of mathematics ; i t can do little or n o t h i n g to show the analytical processes which are concerned w i t h consist­ ence and redundance. W e have said that, i n commencing mechanics, the c h i l d must play the natural philosopher before the mathematician

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can find scope for his efforts. T h i s subject thus provides the first example of the methods of natural philosophy, and the part of the teacher is one of supreme importance. Shall he allow the child's fancy to roam whither i t w i l l , or set h i m down to prescribed tasks w i t h definite ends, or endeavour to lead h i m to the examination of natural phe­ nomena by those methods w h i c h history has shown to be most productive ? T h e r e is, I know, a strong movement i n favour of rely­ i n g upon the " interest " o r " play " motive i n primary edu­ cation ; and the vocal organs of this movement are h i g h l y developed. T h e r e is also, I believe, a perhaps stronger movement, whose vocal organs are as yet rudimentary, i n favour of severe l i m i t a t i o n of the use of these methods. T o some extent each party misjudges the other. T h e up­ holders of interest accuse their opponents of enforcing meaningless drudgery, while these i n their t u r n accuse their opponents of a l l o w i n g education to degenerate into disconnected fripperies, and each is more or less unjust in so d o i n g . F o r myself, and speaking only i n regard to this present subject, I would say that the mere perform­ ance of prescribed tasks, w h i c h arise apparently f r o m the brain o f the teacher or the designer of apparatus for use i n schools, can have n o t h i n g to do w i t h the development of the spirit of natural philosophy. O n the other hand, to allow education to be dominated by interest is to cast aside all hope of discipline — not discipline of class by teacher, but discipline of p u p i l by himself. T o show the exact m e a n i n g of this statement, let me recall to you the behaviour of y o u n g boys learning to play football or cricket. T h e i r interest is to toss the ball aimlessly f r o m one to another,

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and to scamper about as irregularly as y o u n g horses; as much discipline is required to induce t h e m to play the game as is used i n many a class-room to ensure the due perform­ ance of work. T h e effect of this discipline is to replace casual interest i n passing sensations by continued interest in definite achievements ; i n like manner, the effect of our teaching should be to leave the pupils w i t h this desire for achievement, rather than w i t h a mere craving for a t i c k l i n g of the fancy. I do not mean to i m p l y that interest is not to be considered; on the contrary, I w i l l assert most em­ phatically that teaching w h i c h is met by a continued lack of interest must of necessity be at fault. B u t I assert also that the mere presence of interest is insufficient as a testimony to the value of education ; and, further, that courses which are chosen on the basis of m a x i m u m imme­ diate interest are, i n all probability, thereby grievously at fault. W e must then, i n c o m m e n c i n g the study of mechanics, guide our pupils to the attitude of the natural philosopher towards the phenomena w h i c h he studies. B u t what is this attitude ? H o w far is i t concerned w i t h matters of common experience, and how far w i t h the more artificial happen­ ings of the laboratory ? A n d is there any order of priority as between c o m m o n experience and laboratory investiga­ tion ? H e r e there can be little doubt, whether the question be viewed i n the l i g h t of history or of common sense. T h e foundation of this w o r k must be c o m m o n experience; i t must be reviewed, questions must be asked to be an­ swered i n the laboratory, and hypotheses must be made to be tested there. T w o illustrations may show m y meaning more precisely.

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T h e customary experimental introduction to the triangle of forces starts from n o t h i n g ; bodies are suspended and balanced on pulleys w i t h o u t suggestion o f a why or a where­ fore. T h e attitude w h i c h seems more natural is to give to or draw from the pupils everyday illustrations of the balanc­ i n g of two forces by a t h i r d , and to obtain from t h e m as m u c h information as possible about this t h i r d force; namely, that i t always exists, that i t lies between the two original forces, and is nearer to the larger of these. T h e n the question of the law of determination arises ; this is a ques­ tion to be answered i n the laboratory, and the answer found must receive every possible verification. N e x t , let us consider some hypothesis w h i c h may first be made and then be tested i n the laboratory. Several such hypotheses are possible, but perhaps the best is found i n the m o t i o n of a falling body. T o surmise a rule for the composition of two forces is beyond any c h i l d ; but specu­ lation on the actual law of acceleration of a falling body, its dependence on distance or time, is natural f r o m the outset, and should be encouraged before investigation is undertaken. I t is not difficult to divide mechanical inves­ tigations into two classes; namely, those i n which initial speculation is possible, and those i n w h i c h the pupil can only ask a question w i t h no power of suggesting a possible answer, and this division does m u c h to simplify the early treatment of mechanics. So m u c h , then, for the part o f the natural philosopher i n mechanics; the p u p i l has now a body of statements w h i c h he is w i l l i n g to accept, for the t i m e at any rate, though he recognises them as assumptions w h i c h may be unwarranted. Observe that I call them statements, not

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beliefs, and do not refer to t h e m as his mechanical creed; this because they are accepted i n no such b l i n d and u n ­ reasoning faith as the assumptions i n geometry to w h i c h I gave such terms. T h e mathematician i n the c h i l d is now presented w i t h these accepted statements; what is he to do w i t h t h e m ? Can any or all of his three functions here be exercised w i t h profit to the p u p i l i n that he w i l l so gain some idea of their nature and i m p o r t ? I n particular, can we now develop the two questions of consistence and redundance i n speculation, w h i c h were put on one side i n the consideration of geometry? F i r s t consider deduction. T h i s process is possible and necessary, but its area of application is l i m i t e d i n compari­ son w i t h g e o m e t r y ; the l i n k polygon, the centre of mass and its motion, the path of a body under gravity, are i n ­ stances, and others w i l l occur to you. B u t there is no such wealth of propositions and riders as is found i n g e o m e t r y ; most of the problems i n mechanics are applications of principles rather than logical deductions. N e x t take analysis. I t must first be said that the appli­ cation of such processes to mechanics is at least legitimate for pupils i n schools. T o question and investigate their geometrical creed at this age must lead to perplexity and boredom, as has already been said ; but the accepted state­ ments of mechanics lie under no such ban. T h e y are assumptions made consciously o n the best evidence obtain­ able, and investigations t e n d i n g to their confirmation are at least acceptable ; this i n contrast to geometry, of w h i c h the reverse has just been said. T o discuss i n detail the possibility of u n d e r t a k i n g this analysis for mechanics w o u l d lead me too far. H e r e I can

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only say that for certain parts of mechanics, at any rate, i t is not only possible but easy and interesting. F o r example, any boy can understand proofs that f r o m any one of the three assumptions k n o w n as the triangle of forces, the principle of the lever, and the principle of moments, the other two can be deduced; he can thus see that only one of t h e m need be assumed, of course that one for w h i c h the evidence is strongest; and he can see further that there need be no fear that, at some future time, one of these assumptions w i l l be found i n logical conflict w i t h another. F o r the m e a n i n g of such conflict there is ample illustration i n mechanics and physics. T h i s concludes our comparison of geometry and me­ chanics. W h a t , i n short, is the outcome ? First, we have seen that there are three leading processes i n mathematics,— deduction, analysis of consistence, and analysis of redun­ dance, — these being exercised by the mathematician on material presented to h i m by the natural philosopher. N e x t , geometry was seen to be well suited for exercise i n deduc­ tion, but not suited for illustration of the other processes; this is because the natural philosopher acted too early i n regard to space, and w i l l not now brook criticism of his results. Finally, this deficiency was seen to be remedied by mechanics, w h i c h should provide the first deliberate exercise i n natural philosophy, and so present material better suited for illustration of these r e m a i n i n g processes of mathematics. I t may be asked whether i t can be hoped that a p u p i l may end his school career w i t h these ideas fully de­ veloped. F r a n k l y , I do not t h i n k that he c a n ; indeed, I am sure that he cannot. W e may regard such ideas as

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mountain peaks, standing far above the mists of the partic­ ular applications, two of which we have i n particular been discussing to-night. H e who has scrambled longest among the mists sees these peaks most clearly; some indeed have pierced the clouds and seen them i n their full beauty — have even scaled t h e m and viewed one f r o m another. T h e function of the teacher is to lead the c h i l d t h r o u g h the mists by such ways as w i l l give h i m glimpses, even though they are but shadowy, of the higher ground beyond. These w i l l remain and develop i n minds to w h i c h they are suited — minds, I am convinced, far more common than is generally supposed.