THE LOGIC OF 
 
 ARITHMETiv 
 BY M. E,
 
 THE LIBRARY 
 OF 
 
 THE UNIVERSITY 
 
 OF CALIFORNIA 
 
 LOS ANGELES
 
 LECTURES ON THE 
 LOGIC OF AEITHMETIC 
 
 BY M. E. BOOLE 
 
 ' Heaven lies about us in our infancy.' 
 'That true Heaven, the recovered Past.'
 
 HENRY FROWDE, M.A. 
 
 PUBLISHER TO THE UNIVERSITY OF OXFORD 
 
 LONDON, EDINBURGH 
 
 NEW YORK
 
 fif 
 
 TO LADY LOW 
 
 MY DEAR FRIEND, 
 
 But for your resolute energy I think I 
 should never have communicated to the educa- 
 tional world the method which I have long used 
 for reviving the faculties of children suffering 
 from mathematical rickets and logical paralysis. 
 These diseases are common ; they are induced 
 by the practice of teaching mathematical pro- 
 cesses on a hypothesis about the nature of 
 Mathematics directly opposed to that which 
 underlies the original invention and formulation 
 of these processes. 
 
 I have, as you know, taken no part in recent 
 attempts to improve methods of teaching, be- 
 cause, as James Hinton said, I can see no use 
 in trying to invent good methods for teaching 
 truth on a basis of falsehood ; the only remedy 
 in which I have any faith is to tell the pupil 
 one or two simple laws of the relation of the 
 human mind to Scientific Truth, and then to 
 see that he forms the habit of working in 
 accordance with those laws. 
 
 For nearly forty years, authorities of various 
 kinds have been assuring me that it would be 
 impossible to do this except in individual cases ;
 
 4 DEDICATION 
 
 that the public will not tolerate being shown 
 the glowing heart of the mathematical dis- 
 coverer, will not dare to let itself know to what 
 virgin inspiration it is paying homage when it 
 confers medals and honours on an original 
 mathematician. 
 
 Your Ladyship has intervened in the situa- 
 tion, refusing to see objections and difficulties, 
 or to take ' no ' for an answer. You insist that 
 the public shall at least have the option of 
 deciding for itself whether its children shall 
 know what lies at the heart of genuine Mathe- 
 matical Science. I must therefore ask you to 
 accept your share of the responsibility of offer- 
 ing to parents and teachers what I fear is after 
 all only a very feeble presentation of Arithmetic 
 treated as a branch of the Art of Thinking, 
 founded on the general Science of the Laws of 
 Thought. 
 
 Yours affectionately, 
 
 M. E. BOOLE.
 
 CONTENTS 
 
 PAGE 
 
 PREFACE . . . . . . 7 
 
 INTRODUCTORY ADDRESS TO YOUNG 
 
 TEACHERS 29 
 
 I. How MEN LEARNED IN OLD TIMES . 31 
 
 II. ON COUNTING BY TENS ... 37 
 
 III. WHY WE DO NOT ALWAYS DO SUMS 
 
 THE WAY THAT COMES NATURAL . 45 
 
 IV. ARITHMETICAL SHORTHAND ... 52 
 V. KEEPING ACCOUNTS .... 56 
 
 VI. DlS-MEMBERING AND RE-COLLECTING . 58 
 
 Vll. WEIGHTS AND MEASURES ... 65 
 
 VIII. MULTIPLYING BY MINUS ... 69 
 
 IX. ADDING MINUS ..... 75 
 
 X. DIVIDING AND SHARING ... 78 
 
 XI. IN WHAT CONSISTS ECONOMY . . 81 
 
 XII. ECONOMY OF MIND-FORCE ... 87 
 
 XIII. EXERCISE IN KELEVANCE; INTRODUCING 
 
 THE IDEA OF PROBABILITY . . 94 
 
 XIV. EXERCISE ON ZERO .... 97 
 XV. = oo 98 
 
 XVI. EQUIVALENT FRACTIONS . . . 101 
 
 XVII. GREATEST COMMON MEASURE . . 106 
 
 XVIII. STANDARD WEIGHTS AND MEASURES 117
 
 6 CONTENTS 
 
 PAGE 
 
 XIX. WHAT CAN BE SETTLED BY HUMAN LAW 122 
 
 XX. PAPER MONEY 127 
 
 XXI. THE DOG'S PATH . . . .132 
 
 XXII. THE BALL'S PATH . . . .136 
 
 XXIII. EXERCISE TO PREPARE FOR GENERAL 
 
 FORMULAE 139 
 
 APPENDIX: ON PROPORTION. 142
 
 PREFACE 
 
 THE motto chosen for the title-page of this 
 little work may seem unsuitable as an Intro- 
 duction to a course of lessons on Arithmetic ; 
 a subject which to many persons seems so emi- 
 nently un-heavenly and dull. But then, the main 
 reason why it seems so is that their teachers 
 failed to put them in possession of that ' Re- 
 covered Past/ the bearing of which on the 
 Present forms the great clue to that knowledge 
 of subtle forces which gives its possessor the 
 key of the Future. 
 
 The present little volume is not intended 
 to interfere with ordinary methods of teaching 
 Arithmetic, or to supersede the books already 
 in use. It can be used under any School system 
 and in conjunction with any text-book. Not 
 more than one chapter is intended for use in 
 any one term ; the earlier chapters are suited 
 to little children, the later ones for children of 
 fourteen or fifteen. The lesson for the term 
 may be repeated each week with varying illus- 
 trations. 
 
 Teachers of such subjects as Electricity com- 
 plain of the difficulty of getting pupils to apply 
 what they know of Mathematics (at whatever
 
 8 PREFACE 
 
 level) to the analysis and manipulation of real 
 forces. It is not that the pupil does not know 
 enough (of Arithmetic, or Algebra, or the Calcu- 
 lus, as the case may be), but he too often does not 
 see, and cannot be got to see, how to apply what 
 he knows. Some faculty has been paralysed 
 during his school-life ; he lacks something of 
 w r hat should constitute a living mathematical 
 intelligence. In truth he usually lacks several 
 things. 
 
 In the first place, though he knows a good 
 deal about antithesis of operations (e.g. he 
 knows that subtraction is the opposite of addi- 
 tion, division of multiplication, movement in 
 the direction minus x of movement in the 
 direction x, and so on), he has not the habit of 
 observing in what respects antithetic operations 
 neutralize each other, and in what respects 
 they are cumulative ; and surely no habit is 
 more needed than this as preparation for 
 making calculations in electricity or mechanics. 
 
 In the next place, he too often knows, 
 about the idea of relevance, only enough to be 
 foggy about it. The reason for this is that his 
 study of the idea of relevance itself began where 
 it ought to have ended ; his attention was never 
 called to it till the stage was reached when it 
 would have been right that he should direct his 
 action in regard to it sub-consciously from long
 
 PKEFACE 9 
 
 habit, leaving conscious attention free for dealing 
 with the actual elements of some question which 
 is difficult enough to need thinking about. He 
 was not made to grasp the fundamental idea 
 that a statement may be relevant to one question 
 and irrelevant to another, till some knotty prob- 
 lem occurred involving consideration of which 
 statements are relevant to the special question 
 in hand. So he had to try to grasp at once the 
 idea of relevance and the question, what is 
 relevant to what, in a special problem. Such 
 thrusting on the young brain of two difficulties 
 of different kinds at once, is contrary to all 
 accepted canons of Psychology. Examples are 
 here suggested (Lesson XIII) in which there can 
 be no doubt as to what is relevant to the ques- 
 tion at issue ; the child's attention is therefore 
 free to focus itself on the idea that there can 
 be facts concerning a thing which in no way 
 concern the particular question which is just 
 now being asked about the thing. 
 Then again, whatever skill he may have 
 acquired in the manipulation of those notations 
 and formulae which he has been taught to use, 
 he knows hardly anything about the manner 
 in which such things come into being. Now 
 an applier of Mathematics to real forces should 
 be able, when occasion requires, to modify his 
 notation, or invent a new formula, for himself.
 
 10 PREFACE 
 
 He cannot begin to learn how to do this, straight 
 away, while his mind is struggling with problems 
 of electricity or mechanics ; he should have had, 
 from the first, the habit of seeing through for- 
 mulae and notations ; of watching them coming 
 into being, of helping to construct them. 
 
 The special difficulties found in interesting 
 pupils in Arithmetic are in no way inherent in 
 the nature of the subject ; they belong rather 
 to the domain of medical psychology ; their 
 cause can be fitly described only in the language 
 of that science. However logical may be the 
 course of a teacher's demonstration, given his 
 premises, those premises are usually introduced 
 to the pupil under conditions extraordinarily 
 unfavourable to concentration of the attention- 
 power. One can only describe them by saying 
 that if analogous conditions were set up in the 
 stomach-nerves at the beginning of dinner, the 
 result would be lessened appetite and digestion- 
 power; if in the muscles at the outset of a 
 cricket-match, the running would in consequence 
 be slower, the batting, bowling, and fielding less 
 accurate, and the heart-action more irregular ; if 
 in the eye- and finger-nerves before a drawing- 
 lesson, the result would be dazzled sight and 
 trembling fingers. 
 
 It may be objected that if this temporary 
 partial paralysis is set up, in the Arithmetic
 
 PEEFACE 11 
 
 class, only on the occasion of the first presenta- 
 tion of some new idea or principle, i.e. perhaps 
 once in a term or two, it is not frequent enough 
 to induce disease or permanent weakness of the 
 attention-power. But if any condition of nerve 
 disturbance or lowered vitality accompanies the 
 Jirst presentation of an idea or object, there is 
 a strong probability that each subsequent con- 
 tact with that object or idea will produce, by 
 association, a recurrence of similar disturbed 
 action. Thus is set up a habit of scattered 
 attention which retards the development of the 
 faculty of concentration, and makes any real 
 grasp of principles difficult, especially in regard 
 to Mathematics. One mam object of the present 
 volume is to present the premises involved- in 
 various arithmetical operations in a manner 
 which concentrates instead of scattering the 
 power of attention, and thus leaves the pupils 
 fit to attend to the teacher's reasoning. 
 
 My own experience, however, is that a child 
 who has been helped to grasp the premises of 
 a mathematical argument, in any hygienic man- 
 ner, seldom needs much assistance in making 
 out for himself the logical consequences. 
 
 The Preface to a recently issued text-book of 
 Arithmetic l asserts that ' the young are always 
 eager for a rule or formula which will save 
 1 Arithmetic, Kirkman and Field.
 
 12 PREFACE 
 
 them the trouble of thinking for themselves.' 
 The young who are reeling under a blow on the 
 head from a bludgeon may not unnaturally be 
 eager to be taken home to bed in the easiest 
 vehicle at hand ; and there is no telling what 
 habits of lethargy might not be generated by 
 a systematic course of such treatment ; but the 
 young who have been left in possession of their 
 normal faculties prefer using those faculties in 
 active personal exercise. 
 
 Another teacher says T : 
 
 ' In stocks and shares the pupil is usually 
 dealing with things the nature of which he 
 .does not understand. A little political economy 
 must be drilled into him. Bring the meaning 
 of the abstractions of the subject home to 
 him. . . . The great difficulty is in problems. 
 Nothing but stern practice can avail here. Dis- 
 courage all formulae. Anybody who knows any- 
 thing about boys knows that if, by any means 
 under heaven, they can avoid using their brains 
 and attain the same result by a mechanical 
 application of a formula they will do so. They 
 cling to a formula as to a straw, in the ocean 
 of new ideas in which they are drowning.' 
 Exactly so ; children's minds are too often left 
 to clutch at straws ' in an ocean of new ideas.' 
 
 1 Workman, ' Oxford Junior Local,' in Educational 
 Review.
 
 PREFACE 13 
 
 If, two or three terms before a pupil has to do 
 with fluctuating values, he had any clear idea 
 why a Bank of England note has a fixed value, 
 it would be easier than it usually is found to 
 make him grasp the meaning of stocks and 
 shares. 
 
 The mechanism by means of which man 
 learns is intended to play among the concep- 
 tions Unity, Negation, and Fraction; Number 
 is less cognate to it. The kind of fractions dealt 
 with in Arithmetic books is, it is true, too 
 recondite to be thrust on a brain at an early 
 stage. But if mathematical faculty is to de- 
 velop normally in the youth, the child should 
 have early practice in dealing with the idea of 
 a unit broken into bits, which Jit together to 
 make up a whole. 
 
 Nature provides savages with opportunities 
 for dealing with this idea, to an extent of which 
 it is difficult for us to form any adequate con- 
 ception. Something has or has not been caught 
 for dinner ; and when it has been caught it 
 has to be shared and is entirely eaten. Next 
 day the child again sees an animal of the same 
 kind whole. The child's mind plays con- 
 stantly between the idea of rabbit (or whatever 
 animal may be used for food) whole, rabbit 
 shared, no rabbit; and again, whole rabbit. 
 This has a very different effect from what is
 
 14 PKEFACE 
 
 induced by the sight of a loaf or joint appear- 
 ing as a matter of course ; a loaf from which 
 every one is helped, the rest being taken away 
 when no longer interesting. This is one of the 
 reasons why an existence uniformly, uninter- 
 ruptedly prosperous is not always the most 
 favourable to intellectual development. 
 
 In all ranks of life, it is true, infants provide 
 themselves with opportunities for exercising 
 the faculty of reconstructing a unit from its 
 fractions, by tearing or breaking things and 
 fitting the bits together again. Our Kinder- 
 garten systems to some extent provide for 
 systematic exercise of the same kind by means 
 of various toys. Even there, however, though 
 much practice is given in constructing, too little 
 is given in re-constructing a unit out of its 
 fractions. But in schools the whole provision 
 for this practice, which is the essential fibre of 
 all mental growth, seems left to chance, and 
 too much crowded out. 
 
 In Arithmetic, where it is specially important, 
 hardly any opportunity is afforded for practice 
 in swinging the mind between the conceptions 
 Unity, Negation, and Fraction. The mass 
 material of Arithmetic itself (i.e. the art of 
 dealing with numbers) is packed into the mind 
 artificially ; little or no possibility is provided 
 for it to build itself up by natural accretion
 
 PEEFACE 15 
 
 round its organic supporting fibre. This tends 
 to induce a condition of what I have called 
 mental rickets. 
 
 It is desirable to fix children's minds on the 
 act of Negation as a positive act of mind, to 
 a far larger extent than is usually done. Oppor- 
 tunities for this might be taken in connexion 
 with sums in multiplication and long division ; 
 the zeros should be written in, as statements of 
 fact, for some time after the pupil has grasped 
 the idea that their omission does not actually 
 affect the ultimate answer. 
 
 Use might also be made of such exercises as 
 those of Chapter XIII. 
 
 Again, the teaching of Arithmetic is much 
 facilitated, if, besides actual exercises on o and i 
 and very simple fractions, logical exercises 
 parallel to the operations of Arithmetic and 
 constructed on the same models, are carried on 
 in that borderland region where mental opera- 
 tions parallel to those involved in Arithmetic 
 deal with questions of Art, or such simple 
 portions of History, Ethics, or social relations, 
 as come fairly within the scope of a child's 
 intelligence. Each operation of Arithmetic may 
 find its analogue in one of these borderland 
 exercises in Logic. 
 
 It is of primary importance that children 
 should clearly distinguish between what they
 
 16 PREFACE 
 
 do or recite because they are told to do so, and 
 what they themselves grasp or see. They must 
 do certain things because they are told, and 
 commit to memory facts which they will need 
 to know ; but they should always be taught to 
 distinguish between : 
 
 'I repeat this because I was told/ and 'I see 
 that this is so.' 
 
 Some of the worst mental habits are induced 
 by the practice of teachers making a statement 
 as if ex cathedra, and then proceeding to bring 
 forward proofs of its truth ; this is Euclid's 
 method, but Euclid apparently wrote for grown- 
 up men, perfectly well able to take care of their 
 own minds ; he wrote probably for the best 
 intellects of his time. His method is unsuited 
 to children. If a teacher has anything to say 
 to children as a statement, he should say it, 
 not exactly as dogma which they are bound to 
 believe, but as working hypothesis which they 
 are to assume as a basis for the present. Any- 
 thing which he intends to prove should never 
 be stated ; children should be led up to find it 
 out for themselves by successive questions. No 
 pains should be spared to keep the two sets of 
 statements apart. 
 
 To return to Mr. Workman's complaint of 
 the abuse of formulae, surely the remedy lies 
 not in discouraging the use of them, but in
 
 PREFACE 17 
 
 encouraging children to use them intelligently 
 and to make them for themselves. They should 
 begin early the practice of entering certain 
 kinds of results in a book. This book may be 
 divided into two parts : anything specially 
 needful to remember, if found out by the 
 children, should be entered in one part ; if 
 told them by the teacher, in the other. 
 
 The sentimental people who assert that every- 
 thing in Arithmetic can be ' proved ' to children 
 have, usually, no idea of what rigid proof means ; 
 it is not necessary that the child should see 
 the evidence for every hypothesis on which he 
 works ; what is necessary to mental health is 
 a clear understanding of what constitutes evi- 
 dence, and the power to distinguish between 
 what is, and what is not, proved. 
 
 The children should be encouraged to use 
 freely, for reference, any tables or formulae 
 constructed by themselves. This should be 
 permitted as an indulgence, a labour-saving 
 luxury, the reward for accuracy in recording 
 the results of previous investigation. In private 
 teaching and in very small classes, no child 
 should be allowed to enter any formula into 
 the form-book till he is individually able to 
 work it out unaided. In ordinary class-teach- 
 ing this may be impossible. In that case the 
 process should be gone over at intervals, till
 
 18 PREFACE 
 
 the class as a whole can, amongst them, evolve 
 every step. The formula should then be entered 
 by each child in his own form-book ; and the 
 impression should be conveyed that it is the 
 communal work of the class, the expression of 
 its conjoint logical investigation. 
 
 The formula-book should begin with a blank 
 form of multiplication-table, which is to be 
 filled in by the child as practice in addition. 
 The operation of multiplying should be intro- 
 duced to the child's mind in some such way as 
 this : ' Add together four and four and four 
 and four and four.' When the addition sum 
 has been done correctly, say : ' You will have 
 to add five fours together very often. You 
 may enter the result in your table-book ; and 
 next time I give you five fours to add together, 
 I will allow you to look for the answer in your 
 book instead of adding.' 
 
 All the modern higher Mathematics is based 
 on a Calculus of Operations, on Laws of 
 Thought. All Mathematics, from the first, 
 was so in reality ; but the evolvers of the 
 modern higher Calculus have known that it 
 is so. Therefore elementary teachers who, 
 at the present day, persist in thinking about 
 Algebra and Arithmetic as dealing with 
 Laws of Number, and about Geometry as 
 dealing with Laws of Surface and Solid Con-
 
 PREFACE 19 
 
 tent, are doing the best that in them lies to 
 put their pupils on the wrong track for 
 reaching in future any true understanding of 
 the higher Algebras. Algebra deals not with 
 Laws of Number, but with such laws of the 
 human thinking machinery as have been dis- 
 covered in the course of investigations on 
 number. Plane Geometry deals with such 
 Laws of Thought as have been discovered by 
 men intent on finding out how to measure 
 surface ; and solid Geometry with such 
 additional Laws of Thought as were discovered 
 when men began to extend Geometry into 
 three dimensions. The branch of Mathematics 
 called Quaternions deals with such Laws of 
 Thought as reveal themselves during the pro- 
 cess of investigating the relations between n 
 and n + 1 dimensions. The sooner pupils are 
 made to see all Laws of Arithmetic as Laws of 
 Thought, not of things, .the simpler and more 
 satisfactory will their future course be. 
 
 This is especially the case with regard to the 
 sign ; it should never be interpreted in any 
 such way as to convey the impression that it 
 indicates negative quantity, subtraction, or 
 diminution of numbers or of things ; but 
 always as indicating something about the point 
 of view from which the things or numbers are 
 considered. Neglect of this caution lands one 
 
 B 2
 
 20 PREFACE 
 
 in endless metaphysical contradictions and 
 absurdities. Such for instance as that a rect- 
 angular surface becomes = o if we happen to 
 place the origin of co-ordinates in the middle of 
 it ; for it then divides naturally into the four 
 rectangles X x Y, X x (- Y), (- X) x (- Y), 
 ( X) x Y. The amount of confusion and 
 hindrance to progress thus induced is sickening 
 to think of. 
 
 Teachers seldom observe how early many 
 children are affected by such anomalies as 
 these, or how profoundly they are affected by 
 them. If the same boys who are thus pain- 
 fully impressed became teachers in their turn, 
 they would no doubt remember their own past 
 experience, and a better mode of presenting 
 the primary conceptions of Arithmetic and 
 Geometry might soon be evolved. But, un- 
 fortunately, these logical thinkers are usually 
 repelled by the impossibility of extricating 
 themselves from the network of insincerity, or 
 what seems to them such ; give up the study 
 of Mathematics, and betake themselves to some 
 other pursuit. Very much of the waste and 
 confusion which gather round the sign might 
 be obviated by making children go through such 
 exercises as those given in Chapters VIII and IX. 
 
 Much needless difficulty is caused by the 
 attitude of Mathematical teachers towards the
 
 PKEFACE 21 
 
 symbol oc , which they often interpret as some- 
 how connected with endless length or exhaust- 
 less quantity. The effort to conceive of either 
 endless length or exhaustless quantity is a 
 metaphysical gymnastic very unhealthy for 
 a young mind. It sets up also a set of ideas 
 and connotations quite irrelevant to any use of 
 the symbol oo in the higher Mathematics. 
 GO has reference not to length or quantity, 
 whether great or small ; but to release from 
 certain restrictions to which the values specified 
 as finite have been subject. (See Lesson XV.) 
 
 The whole terminology connected with the 
 operations called in Arithmetic books 'Multipli- 
 cation' and 'Division' is excessively misleading. 
 The words ' multiply' and ' divide ' might have 
 been invented on purpose to create confusion in 
 passing from integral to fractional Arithmetic. 
 
 The continued use of the word ' dividend ' 
 especially, in Arithmetic, is a symptom of the 
 carelessness of teachers in the matter of avoiding 
 causes of confusion. In old days, when most 
 people who had money hoarded it in the house or 
 deposited it in a bank, till they could use it in 
 some business of their own, the word 'dividend' 
 had no connotation for a child's ear, till it was 
 explained to him as ' an amount to be divided.' 
 But in these days of limited liability companies, 
 the word ' dividend ' is incorrectly, but very
 
 22 PKEFACE 
 
 commonly used, at home in the hearing of 
 children, to indicate a sum of money which is 
 the result, or quotient, of the division and 
 distribution of the year's profits among the 
 shareholders : ' I will do so-and-so when my 
 dividends come in,' &c. Even if the teacher 
 is careful to explain that the word in the text- 
 book does not mean what it means in common 
 parlance, which not all teachers remember to 
 do, that explanation in itself causes distraction ; 
 some word should be chosen which has no mis- 
 leading associations. Of course children ' out- 
 grow ' any misapprehension which may be thus 
 caused, and learn that ' dividend ' has two 
 meanings. But one wonders what teachers 
 suppose is the effect on the mental habits, on 
 the brain tissue, of struggling, for even a few 
 days, to understand division before this dis- 
 covery has been made ! The present writer 
 has not authority to devise a nomenclature ; 
 and therefore in this introduction employs that 
 in common use, though under protest. It 
 should be avoided in speaking to the class. 
 
 The teacher should, in dealing with any 
 branch of the operation called ' multiplication,' 
 have in his mind this idea : ' Multiplication ' 
 means doing to the operand (multiplicand) what, 
 if done to unity (1), would produce the operator 
 (multiplier).
 
 PREFACE 23 
 
 In dealing with ' division ' he should keep in 
 his mind the idea : ' division ' means doing to 
 the operand (dividend) what, if done to the 
 operator (divisor), would produce unity (1). 
 
 The operation of division should be introduced 
 at first, not as the inverse of multiplication but 
 as a succession of subtractions, thus : 
 
 If five boys are to share twenty apples among 
 them, how many can each boy have ? Let us 
 give each boy one apple first ; that takes off? 
 Five apples. And leaves ? Fifteen. Now we 
 give each boy another apple : that takes off? 
 Five more. And leaves ? Ten. Another to 
 each boy leaves? Five. Another to each 
 leaves ? None. So now we have finished. 
 How many times did we go the round and 
 give each boy an apple ? Then each boy has ? 
 Four apples. 
 
 A little practice in such successive sub 
 tractions should be gained, before the idea is 
 introduced of doing long division straight off. 
 Division should not be treated as the mere 
 inverse of multiplication, but as a way of cutting 
 short a laborious series of subtractions. The 
 child should be led to see that the multiplica- 
 tion table can be used to cut short not only 
 series of similar additions, but also series of 
 similar subtractions. A great deal of the diffi- 
 culty afterwards found in dealing with such
 
 24 PREFACE 
 
 operations as G.C.M. is traceable to neglect 
 of this simple precaution. 
 
 The manner in which that most logical of all 
 text-books, Euclid, has been misused to induce 
 illogical habits of mind, repeats itself in most 
 departments of elementary Mathematics. A 
 good instance of such misuse can be shown in 
 connexion with the subject of equivalent frac- 
 tions (Lesson XVI). Let us suppose that the 
 teacher wishes to prove to the class that three- 
 quarters of one is equal to one-quarter of three. 
 He takes as his unit of thought the concept 
 apple. He too often begins by stating a thing 
 which is not true in itself: viz. that three- 
 quarters of an apple is the same thing as a 
 quarter of three apples. For the purpose with 
 which the teacher's mind is occupied at the 
 moment (the equivalence of fractions), the state- 
 ment is true ; it is as true as that, from the 
 shop-keeper's point of view, twelve pence are 
 a shilling. But the child, who has not yet 
 been introduced to the conception of equivalence 
 of fractions, is at a point of view quite different. 
 To the child's imagination ' great lots of grub ' 
 have a fascination quite independent of the 
 size of his own share. To suggest (for instance) 
 that a bun for one child is the same as a hundred 
 buns for a hundred children would be to insult 
 the pupil's understanding and his feelings
 
 PEEFACE 25 
 
 feelings which affect him all the more keenly 
 that as yet he does not know how to express 
 them. If he could believe you, what would be 
 the meaning of school treats and social picnics ? 
 In proportion as you ultimately induce him to 
 feel that one bun for a child is the same as 
 a hundred buns for a hundred children, he will 
 be, in future, the worse man, citizen, political 
 economist ; and the less fit to apply mathe- 
 matics to problems in real forces. But we are 
 speaking now of the effect of such statements 
 on his intellectual processes, at the age when 
 his sensations are as yet unwarped. The state- 
 ment that a quarter of three apples is 'the 
 same as ' or ' equal to ' three-quarters of an 
 apple has brought prominently into the field of 
 his imagination two pictures : an apple, and 
 a group of three apples. The natural way of 
 making an apple equal to three apples is to 
 bring forward two more apples. It is the way 
 things used to be made equal when he was 
 learning addition ; why not now ? The teacher 
 has therefore started him looking out for the 
 two more apples ; and as what is said next does 
 not seem to be tending in that direction, his 
 attention is distracted ; he gives only half his 
 mind to what is said. Thus he fails to get hold 
 of one or more links in the chain of argument. 
 Some children soon recover themselves, and
 
 26 PKEFACE 
 
 attend to the rest of what is said. And, as 
 the teacher seems to be satisfied with the 
 reasoning and to expect them to be so, they 
 imagine they are so. This helps to form a habit 
 of intellectual dishonesty, of confusing sham 
 proof with real proof. The children who are 
 either less sharp or more logical and thorough, 
 look longer for the non-appearing two apples, 
 and miss more links of the chain of proof. 
 These children feel that the whole demonstration 
 has passed in that region where grown-ups 
 conduct a self-satisfied mental life in which 
 children cannot share. Thus are formed habits 
 of 'hopeless non-comprehension/ perhaps even 
 of ' self-protecting and contemptuous non-atten- 
 tion.' The chapter on equivalent fractions 
 (Lesson XVI) suggests ways of attacking the 
 problem not open to this objection. 
 
 It may occur to some that too much use is 
 made of examples relating to food. But we 
 cannot make mathematicians by insisting upon 
 a non-existing superiority to physical facts. 
 Apple or bun forms the natural unit for a child ; 
 the sharing of a cake or fruit is the natural 
 fraction as well as the true introduction to the 
 higher ethical life. As a matter of fact the 
 Arithmetic of grown people is largely occupied 
 over questions of food supply and of personal or 
 family interest. To place children's ideas about
 
 PREFACE 27 
 
 such subjects on an honest basis and give them 
 a social direction would surely tend more to- 
 wards ethicalizing the community than the 
 setting up of topics of fictitious interest, and 
 then teaching children to be satisfied with 
 reasoning which is either imperfect in itself or 
 but partially understood by them. To let the 
 children be thoroughly logical about their own 
 sphere of action among things which they do care 
 about, and deal practically with duties which they 
 must perform, is far better training than to en- 
 courage them to express opinions about subjects 
 the data of which lie beyond their personal ken. 
 The pauses of sitting with slack muscles and 
 taking slow quiet breaths, to make mind-pictures, 
 is not intended to supersede the usual permitted 
 'intervals' between lessons; they are an integral 
 part of the Logic lesson itself. They, as well as 
 the rest of the scheme here suggested, form, it is 
 believed, the first attempt to adapt to elementary 
 education here the magnificent method of study 
 described in Gratry 's Logique. A certain element 
 of normal mental life, essential to sound intel- 
 lectual development, and which can only be 
 received during suspension of the active per- 
 ceptions of intellect, is being driven out by 
 modern educational grind. This tends to 
 produce a sort of mental and moral rickets, 
 analogous to the physical disease induced by
 
 28 PEEFACE 
 
 a diet deficient in the bone-forming element. 
 Teachers who have studied any sort of medical 
 psychology bitterly lament that the require- 
 ments of existing systems force them to inflict 
 this grievous wrong upon the rising generation. 
 In the present volume, an attempt is made 
 to re-introduce at least a small share of the 
 skeleton-forming elements of mental life, in a 
 manner which does not necessitate disturbing 
 existing arrangements. The author has been 
 kindly assisted in this matter by Mrs. Archer 
 and her pupil Miss Cross, of Coombe Hill School. 
 
 English children have now to be taught to 
 relax, to breathe, to see. It is probable that 
 the cultivated art of relaxation will ultimately 
 prove more satisfactory than the mere following 
 of natural impulse to relax. 
 
 But it is certain that the teachers who have 
 weeded out the wild growths must see about 
 planting the cultivated ones. Such exercises as 
 those here recommended tend to promote sane 
 and healthy mental action, and to make the 
 discipline of school more organic and harmonious. 
 They have a great fascination for children, as 
 my young friends at Coombe Hill School know. 
 I am sure they will be glad that I am attempt- 
 ing to give to other boys and girls a share in 
 the fun that we had over our little sums. 
 
 MARY EVEREST BOOLE.
 
 INTRODUCTORY ADDRESS TO YOUNG 
 TEACHERS 
 
 SOME of you may perhaps have seen what are called 
 Japanese Flowers. You buy a penny packet of things 
 which look like rather clumsy wafers, each of which 
 has a fine line round its outer edge. You put one of 
 the wafers into warm water, and (as the printing on the 
 envelope says) 'Watch the result.' The line round 
 the edge melts and slips off. Then perhaps, from the 
 wafer come out first one little coloured flower, then 
 another, and another. Or there may come out leaves, 
 or sea- weed. The change seems wonderful, but the 
 explanation of it is simple enough : some one in Japan, 
 by great care and skill, made each leaf and flower ; then 
 they were linked together by stalks made of thread, and 
 then the whole set was rolled up into a tight form 
 convenient for carrying about, and fastened round with 
 a little strip of paper. And we have to put them into 
 warm water which acts as a solvent (that is to say, 
 a melter), before we can see how pretty they are. 
 
 Arithmetic seems to some people dry and un-beautiful ; 
 but that is because they have not soaked it in that 
 solvent which is called sympathy. If we had sympathy 
 with the struggles and labours of others, Arithmetic 
 would be easier to understand and pleasanjber to learn 
 than many children find it. 
 
 It must, however, be distinctly understood that any 
 of the following lessons are to be omitted or modified, 
 if, in the judgment of the teacher, it is unsuited to the 
 condition of the class. 
 
 This remark applies especially to Lesson I. If the 
 discipline of the class is in such condition that the
 
 30 INTRODUCTORY ADDRESS 
 
 teacher would not find it easy to restore order after 
 a burst of laughter, he may suggest to the children, in 
 some quieter way than that here given, that men held 
 up all their fingers, as a way of saying ' Many,' before 
 they had learned to count how many. But my own 
 experience and that of several of my friends is that the 
 dramatic introduction of the savage element, where this 
 can be safely permitted, has a wonderful effect in 
 conquering the apathy of which so many teachers 
 complain, and making Arithmetic real and living. This 
 remark applies to children of all ages, and at all social 
 levels. 
 
 In this book you will find no long sums. The 
 following caution may be found of use in dealing with 
 those which you set from the ordinary text-books. 
 Children should be carefully taught to suspend attention 
 at proper pausing-places ; as, for instance, at the ends 
 of lines, in multiplication, long division, practice, or 
 equations. Just as, in reading aloud, one must take 
 breath at suitable stopping-places, or one will be driven 
 to stop for breath involuntarily at the wrong place, so 
 it is with the attention-power in Mathematical work : 
 if it is not trained to take a moment's rest at the right 
 place, it rests itself (or, as we say, ' the mind wanders ') 
 in the middle of a line of figures ; a wrong figure is 
 written down during the lapse, and sets the whole sum 
 wrong. This acts very unfavourably on those children 
 who have specially short-breathed attention-power ; who 
 are often, otherwise, the cleverest, and might become, if 
 properly trained, the most intelligent mathematicians. 
 On the other hand, a training in the proper care of the 
 attention-power is good for all children alike.
 
 HOW MEN LEAKNED IN OLD TIMES 
 
 You have heard, my dears, that, a long time 
 ago, there were no people in this country except 
 savages. Perhaps some of you have read that 
 delightful book, The Story of Ab ; or you may 
 have seen the picture of a savage with his wife 
 and baby, at the beginning of Clodd's book, 
 Primitive Man. 
 
 How do you think children learned lessons in 
 those days ? Do you think Ab's wife ever told 
 little Mok to wash his face and hands 1 , and come 
 to lessons ? Do you think that the hairy woman 
 in Clodd's book ever called out to her boy : 
 '-The school bell is ringing ; make haste, or you 
 will be late ' ? Oh ! no ; nothing of that sort 
 
 1 [In old Miracle Plays the first woman is represented 
 as calling her boys in to wash their hands and brush 
 their hair, because to-morrow will be Sunday and they 
 will be examined on the Catechism. The future villain 
 of the piece initiates his evil career by some such breach 
 of good manners as taking off his cap with his left 
 hand ! These and countless other literary phenomena 
 of the same kind go to show how far astray the unin- 
 structed imagination goes in its presentation of historic 
 data ; and how necessary it is to take frequent oppor- 
 tunities of pulling it into line with the reality of things.]
 
 32 LOGIC OF AKITHMETIC 
 
 ever happened at that time. Yet the children 
 did not grow up quite ignorant. We are going 
 to have a little talk to-day about how people 
 learned in those days, and what sort of teachers 
 they had to make them learn. 
 
 Why do the monkeys in the Zoological Gar- 
 dens like climbing on their ropes ? Why is it 
 found good for the health of children to climb 
 the gymnasium ladder and cling to the bar by 
 their hands ? Why do most of you enjoy doing 
 so as soon as you have got a little practice at 
 it ? Why do dogs not climb ladders or ropes, 
 and why is it not found good for their health to 
 teach them to do so ? You and the monkeys 
 had ancestors who climbed trees in order to be 
 out of the way of wild beasts of the dog kind. 
 These ancestors of yours forced themselves to 
 learn to climb, by great labour and effort, driven 
 by great fear and desperate need. But they 
 went on, generation after generation, trying and 
 trying, till it came quite easy and pleasant, till 
 they could climb for pleasure and fun. And 
 that has worked into the very muscles of your 
 flesh and the blood of your veins, the power and 
 the wish to climb. But the dogs seem to have 
 had no ancestors who learned to climb ; when 
 their ancestors were afraid of enemies, they 
 protected themselves in some different way. 
 Primitive Man climbed trees quite easily, and
 
 HOW MEN DID LEARN 33 
 
 so did Ab, and Ab's wife, and Mrs. Primitive 
 Man ; they carried their babies up trees till the 
 babies were old enough to learn how to go up ; 
 and then the mothers 'made the children learn. 
 And all that has helped to make climbing 
 natural and good for children now. 
 
 But another thing was happening at the same 
 time. Have you noticed what sort of words 
 a child learns first ? Names of things, or names 
 of numbers ? For instance, if a child of about 
 a year and a half old saw six birds on a wall, 
 would he say ' Six,' or would he say ' Dickie- 
 birds ' ? You know that he would say 
 ' Dickie-birds ' much earlier in his life than he 
 could count ' six.' Well, savages of very long 
 ago, such as Primitive Man, could not talk as 
 we do ; indeed, one name for them is ' Speech- 
 less Man.' But they must have had words 
 or grunts or growls or signs of some sort for 
 some few things ; even a hen can tell her 
 chickens that she- has found food, or that she 
 sees a hawk. We may be sure that Primitive 
 Man could tell his wife that there was a wolf 
 near when he could not say ' Six wolves.' Per- 
 haps he said 'Woo-oo-oo-oo' by way of a sort of 
 imitation of a wolf's howl. 
 
 And presently he began to find out that 
 it would be convenient if he could somehow 
 tell his wife whether there was one wolf or
 
 34 LOGIC OF ARITHMETIC 
 
 many. He had no words, as yet, to tell her 
 with. 
 
 Now what does an omnibus conductor do, 
 when you have heard, or guessed, that he said 
 ' Fares, please ! ' ; but, because of the rattling 
 noise, he cannot make you understand whether 
 your particular fare is two pence or three ? He 
 holds up fingers. Can we not fancy Primitive 
 Man doing that ? If he said his word or grunt 
 for wolf, and held up one finger, that might 
 mean : ' Just stand behind me with baby, my 
 dear, while I kill the wolf with a stone.' But 
 if he spread out all his fingers, that might 
 mean : ' Let us get up the tree quick ; for there 
 are more wolves coming along than you and 
 I can manage/ 
 
 We see now that the savages of long ago had 
 teachers ; though they had neither desks nor 
 books nor school bells. Outside of themselves 
 there were wild beasts which taught them to 
 be sharp and try to escape danger. Inside of 
 them they had feelings which taught them that 
 it is not always enough to climb up a tree by 
 oneself, that they must help each other and 
 take care of the baby. And, between the two 
 sets of teachers, they somehow learned to say 
 ' One ' and ' Many ' on the fingers of their hands. 
 Neither set of teachers could have taught them 
 alone ; neither the wolves outside them nor the
 
 HOW MEN DID LEAKN 35 
 
 feelings inside. But between the two sets, the 
 savages learned their lesson so well that, to 
 this very day, we have never forgotten it. 
 Children count on fingers still, unless there is 
 some reason for using counters instead. 
 
 Poor Mrs. Primitive Man, and all the rest of 
 our wild ancestors ! I often think of them when 
 I am puzzled and worried myself. They little 
 guessed what delightful gymnastics they were 
 preparing for you children, or what a convenient 
 way of doing sums they were helping to pre- 
 pare for us all, by their care of their babies and 
 by their brave struggles against laziness and 
 cowardice. Perhaps, thousands of years hence, 
 people will be enjoying some wonderful science 
 or delightful art, of which we now have no 
 notion at all ; and will be looking back with 
 thankfulness to the struggles and troubles of 
 us now, which are preparing for their enjoy- 
 ment. 
 
 Now before you go to the next lesson and 
 forget this one, I want you to make a picture in 
 your minds that you will remember in after life. 
 
 Shut your eyes ; lean back comfortably in 
 your seats. Let your hands lie quite slack on 
 your laps. Take a few long easy breaths. Now 
 then. 
 
 (The Teacher should read the following with a pause 
 between each sentence.) 
 
 C 2
 
 36 LOGIC OF ARITHMETIC 
 
 Think of the funny hairy man in Clodd's 
 book. See him start suddenly. He calls out : 
 ' Woo-oo-oo-oo.' The little ones may call out 
 ' Woo-oo-oo-oo ' if they like. Now be quiet again. 
 He calls out ' Woo-oo-oo,' and puts up his hands 
 with all his fingers spread out. His wife comes 
 out of her dreamy mood. Jumps up and carries 
 her baby to the foot of the tree. Clasps the 
 baby between her long hind feet. Climbs the 
 tree like a monkey by her hands only. The 
 man climbs after her. When they have reached 
 a high branch, she takes baby in her arms : she 
 and the man sit side by side, and wait till the 
 wolves have gone by. 
 
 (Two or three minutes should now be spent in silence, 
 sitting at ease, before the class breaks up. Attention 
 should be paid to relaxation of that muscular tension 
 which accompanies all active attention to external facts. 
 
 Modern children are fast forgetting the secret of 
 slow, deep natural breathing, as well as that relaxed 
 attitude and meditative picture-soakage which is one of 
 Nature's most powerful educators. The loss of these 
 things is probably one great factor in the failure of 
 modern educational schemes. 
 
 As the sudden rush from active attention to one topic 
 to active attention to another is in itself bad for the 
 young brain, the last few minutes of each class seems 
 a good opportunity for practising concentration. 
 
 The above method for impressing mind-pictures on 
 children is an adaptation, suited to ordinary class teach- 
 ing, of a more elaborate method carried out at Coombe 
 Hill School.)
 
 II 
 
 ON COUNTING BY TENS 
 BLACK-BOARD. 
 
 WHAT is this, 1 ? And this, 2 ? And this 
 3 ? (and so on), and this, 9 ? 
 
 Now I want to write ten : how shall I do it ? 
 
 Put 1 and 0. 
 
 But what has ten done to be different from 
 the rest ? 
 
 Why should it have two signs instead of one 
 like its neighbours ? and why does it take signs 
 belonging to its neighbours, instead of having 
 one of its own ? 
 
 Did ten ask to have two signs ? Did it wish 
 to have two ? No ; then why do we give it 
 two ? 
 
 I once asked a young friend of mine why he 
 did something in his sum ; and he answered : 
 ' My reason is that I was told to do it at school ; 
 but I know I ought to have another reason, and 
 I know I haven't.' I thought that was a sensi- 
 ble answer. It applies to most things in 
 arithmetic : your reason for writing ten with 
 two signs is that your teachers told you to do
 
 38 LOGIC OF AEITHMETIC 
 
 so ; their reason for telling you to do so is that 
 it has been found a convenient and suitable 
 way for people to write ten. But there is 
 a reason why it is suitable and convenient, and 
 you ought to know that reason. All your sums 
 will seem to you more interesting, and, I think, 
 easier, when you know it. 
 
 We have seen under what sort of pressure 
 people must have found out how to tell each 
 other that there were one or many wild beasts. 
 But when once they had learned that much, 
 they could apply it further. By-and-by they 
 began to keep flocks of sheep or goats, and they 
 made stone walls to fence in the sheep by night, 
 so that wolves could not get at them. Perhaps 
 they might wish to count their flocks, to see if 
 all the animals were safe in for the night ; or 
 one man might wish to sell his sheep to another. 
 Well, at first the owner might count by putting 
 up a finger for each sheep, as it passed in 
 through the gap in the fence. That did very 
 well up to ten. But what next ? How could 
 he show anybody that there were sixteen sheep 
 or twenty-nine sheep ? He might put up all his 
 fingers for ten, and then put them down ; and 
 then put up six fingers for the other six sheep, 
 that would mean sixteen. He might put up 
 all his fingers twice for two tens, and then nine 
 fingers ; that would mean twenty-nine.
 
 ON COUNTING BY TENS 39 
 
 But if he had many tens of sheep, he might 
 lose count of the number of times he put his 
 hands up and down. What was to be done 
 about that ? 
 
 The man who had very many sheep to count 
 must contrive somehow to make a mark for 
 each time he put his hands up. 
 
 There was no paper in those days, no pens or 
 pencils, not even balls strung on wires such as 
 we see in some infant schools. I believe people 
 had not even names for more than a very few 
 numbers. There are still people in the Malay 
 Islands who have no names for any numbers 
 except one, two, and three. People had to 
 think about keeping flocks in order to be sure 
 of having food and clothes for their families, 
 before they began thinking about what was the 
 best way of doing sums ! What they wanted 
 was to make sure somehow that all the sheep 
 they sent out in the morning were safe inside 
 the fence at night. There was a quicker way 
 to manage that than inventing figures, and 
 finding out how to make pencils. The man 
 could call up his little daughter, who had no 
 school, to go to, and say to her : ' Would you 
 like to help Daddy count the sheep ? Just 
 stand nice and quiet beside me, and I'll tell you 
 presently what to do/ Then each time he 
 finished putting up all his fingers, he would
 
 40 LOGIC OF AEITHMETIC 
 
 make her put up one finger, till all the sheep 
 had passed out. Then he would look well at 
 his own fingers and the little girl's, and make 
 a picture on his mind of how they looked. 
 Then he would perhaps say : ' Now, dear, 
 when the sheep come home we must get our 
 hands looking just like this ; and we must not 
 be satisfied till they do.' In the evening, if his 
 hands and hers looked as they did in the morn- 
 ing, he could take her in to supper, and have 
 a game of play with her, and then go to bed 
 quite happy, because all his sheep were safe in 
 for the night. But if either he or she had too 
 few fingers up, and no more sheep seemed 
 coming in, he knew that he must send her in 
 to her mother, and call his big son, and go out 
 on the moor to look for the rest of the sheep. 
 
 Now I think we had better spell a number 
 in the way that the shepherd and his child did ; 
 then you will know better what sort of way it 
 was. Tom, come here please ; stand at the 
 right hand of the class ; you shall play at being 
 the shepherd. Jane, you stand at the left side 
 of the class ; you shall play at being the 
 daughter. I cannot let the rest of you rush 
 about pretending to be sheep, in school ; you 
 can do that in the play-ground if you like ; but, 
 in school, I will say ' sheep,' and you children 
 must try to make a mind-picture of sheep going
 
 ON COUNTING BY TENS 41 
 
 out through a gateway, one by one, as fast as 
 I count. Tom, every time I say ' sheep/ put 
 up a thumb or finger. Jane, wait till I tell 
 you what to do. Now, Tom : 
 
 ^ One sheep ; two sheep ; three sheep ; four 
 sheep ; five sheep ; six sheep ; seven sheep ; 
 eight sheep ; nine sheep ; ten sheep. 
 
 Now you have no more fingers to put up. 
 Hold your hands up high, and show the class 
 that you have no more fingers left to count 
 with. Put your hands down. Jane, put up 
 one finger ; that means that Tom has put up 
 all his fingers once. Hold it high up for 
 a minute, for the class to see that you have one 
 finger up. Now, you need not keep your hand 
 so high any longer, it would make your 
 shoulder ache ; but keep that finger out 
 steadily all the time till I tell you to leave off. 
 Now Tom : One sheep ; (repeat ). 
 
 Hands down, Tom. Jane, put up one more 
 finger to show that Tom has gone over all his 
 fingers once more. Tom, one sheep; (repeat >*(). 
 
 Hands down, Tom. Jane, one finger more. 
 Tom, one sheep ; two sheep ; three sheep ; 
 four sheep ; five sheep ; six sheep ; seven 
 sheep. 
 
 Now, both of you, hold your hands high, 
 so that the class can see them. How many 
 fingers has Jane got up .? Three. And each of
 
 42 LOGIC OF AKITHMETIC 
 
 the fingers means? That Tom has counted 
 ten on his fingers. So Tom has counted ten 
 three times ; how many does that make ? Ten 
 three times ? Thirty. That is right. But 
 Tom has done something else besides putting 
 up ten fingers three times ; he has also put up 
 seven fingers besides. How many sheep has 
 he counted then ? Thirty-seven. We will 
 pretend that this is early morning and the 
 sheep have just gone out to feed. Look well 
 at the hands. When the sheep come home at 
 night, the hands must look just so, or else the 
 shepherd cannot go happily to bed ; he must 
 go and see where the missing sheep are.' 
 
 If you play that game out of school, you can 
 play at counting other numbers ; forty, fifty, 
 seventy, ninety sheep. But that would take 
 too long now. 
 
 For many, many generations people had only 
 fingers to count with ; and, all that time, one 
 finger of some other person may have stood for 
 ten fingers of the person counting. So, at last, 
 when people found out how to write, it came 
 natural to them to make one in the left-hand 
 place of the paper stand for ten in the right-hand 
 place, just as it comes natural to monkeys and 
 boys to climb things, because their forefathers 
 had done it for so long that they did it without 
 much thinking about it. It seemed easier to
 
 ON COUNTING BY TENS 43 
 
 do it that way than to think about any other 
 way ; the inclination to do it so had got worked 
 into the very marrow of their brains. And so, 
 when they came to want to reckon more than ten 
 tens, they put the figure for ten tens in a fresh 
 place ; and that is what we call the hundreds' 
 place of a row of figures. And so they went 
 on and on, making fresh places for the different 
 powers of ten (as some people call it) because 
 their great-grandfathers had ten fingers to 
 count with. 
 
 Now we come back to the questions we 
 started with just now. Why does ten have 
 two signs belonging to other numbers instead 
 of one sign of its own ? It does not ; it has 
 one sign. Its sign is a ' 1,' ivritten in the tens' 
 place, and that means : once the hands are put 
 down, after counting ten, in order to begin 
 counting over again. When we write twelve, 
 which means ten-two, we write it so ' 12 ' ; 
 that is, one ten or one double-hand, and two in 
 the place of units as we call them. But if 
 there are no things over, no units, we say so ; 
 we put in the units' place. That helps to 
 remind the writer, and to show to other people, 
 that the 1 stands for one ten, not one thing. 
 
 If ever you are teaching a baby brother or 
 sister to count, there are three things I should 
 like you to be careful about. Never teach
 
 44 
 
 a baby to say ' one, two, three,' like a parrot ; 
 always teach him to count things. If you have 
 no counters, use some sort of blocks, or bricks, 
 or collect little pebbles, or buttons. (Only 
 then you must be careful not to let him swallow 
 them.) Or he might count your fingers and 
 toes, or the railings. But it is best to have 
 things he can shift about, such as buttons ; and 
 let him push each, as he counts it, from the 
 uncounted heap to the counted heap. Never 
 make a little child say ' eleven, twelve ' ; 
 always tell him to say ' ten-one ; ten-two ; 
 ten-three ; ten-four ' ; grown-up people say 
 ' eleven, twelve,' and now you are old enough 
 to go to school, you must do the same ; but 
 a little child should always say ' ten-one ; ten- 
 two ' ; like ' twenty-one, twenty- two.' And 
 as soon as he has counted ten things, make 
 him put them in a little box or in a heap by 
 themselves ; that will make him understand 
 what he is about much better than most little 
 children do. And then you can tell him tKe 
 stories about Mr. and Mrs. Primitive Man, 
 and the baby, and the wolves ; and about the 
 shepherd and his little daughter.
 
 Ill 
 
 WHY WE DO NOT ALWAYS DO SUMS 
 THE WAY THAT COMES NATURAL 
 
 IT is often best to do things the way that 
 ' comes natural,' as long as there is no reason 
 for altering it. Very many things have to be 
 altered as we become civilized and have to live 
 in towns. It comes natural to children to count 
 on their fingers, but they have to be made 
 to use counters or blocks instead, because 
 counters can be arranged in convenient pat- 
 terns : 
 
 BLACK-BOARD. 
 
 Whereas fingers are much less convenient. 
 
 It came natural to bees to make round cells, 
 because their forefathers had made little holes 
 by rubbing their bodies round ; but when the 
 bees took to living a great many of them in one
 
 46 LOGIC OF ARITHMETIC 
 
 hole, they had to be stopped doing what was 
 natural, and made to learn to fit their cells into 
 each other, because it saved room and was more 
 convenient. Poor little bees ! They didn't at 
 first like learning the new way of making cells, 
 I dare say ; but they had to. It comes natural 
 to you to do some things that have to be stopped 
 because they are inconvenient. It comes natural 
 to a baby to grab with its hands at everything 
 it wants ; but it has to be taught to eat pro- 
 perly with a spoon. There are such quantities 
 of things that we must alter, that it is generally 
 best to go on doing things in the old way until 
 a reason comes for altering them. It would 
 take a good deal of trouble now to learn to 
 count ordinary things otherwise than in tens. 
 There is no particular reason why we should 
 do so, so we all go on counting by tens ; and 
 you were made to learn to do the same. Half 
 the use of school is that children should learn 
 which things they had better do the way that 
 their forefathers did them, and which things 
 we must now learn to do in some other 
 way. 
 
 Once upon a time a dear little girl about a 
 year old was sitting on her mother's lap, and 
 a bright teapot full of hot water was on the 
 table close by. It comes natural to babies to 
 try to get hold of anything bright that they
 
 SIMPLE ADDITION 47 
 
 see. So this baby put out her hand towards 
 the teapot. Mother said ' No/ and pushed 
 the little hand away. But baby thought she 
 would do what was natural ; she screamed and 
 put out her hand again. Mother began to 
 think that if baby was so anxious to get at 
 the bright things, she might make a grab at the 
 candle or the lamp some evening, or crawl to 
 the fire, and have a bad accident. So she 
 thought she would let baby have her way at 
 once, when it could not do any real harm ; she 
 pretended to give in, and let baby touch the 
 teapot. It was very hot. It did not actually 
 injure the child. But somehow, after that, baby 
 understood why it isn't always nice and com- 
 fortable to do just what comes natural ! 
 
 You know that we must do sums in certain 
 ways because we are told to do them so ; but 
 also we ought, when we can, to have another 
 reason. It is a good thing to play sometimes, 
 when it does not matter, at doing a sum the 
 ' wrong ' way, the upside-down way to what 
 we have been taught ; then we find out why 
 we were taught a way that does not seem 
 natural. We are going to do that to-day. 
 
 Suppose I have in my purse a shilling and 
 two pence ; and some one pays me two shillings 
 and three pence, how shall I write the account 
 of what it ought to come to ?
 
 48 LOGIC OF ARITHMETIC 
 
 BLACK-BOARD. 
 s. d. 
 
 1 2 
 
 2 3 
 
 1 Three pence and two pence are . . . ' 
 Stop a minute. Which does one care most 
 about, the pence or the shillings ? The 
 shillings. Well, why don't we reckon the 
 shillings first ? ' It is the wrong way.' Why 
 is it the wrong way ? Would it make the sum 
 come wrong ? 
 
 (Usually some children in a class think it would 
 make the sum come wrong.) 
 
 Well, we are only playing to-day, it does not 
 matter if the sum does come wrong. Let us 
 try. 
 
 One shilling and two shillings make three 
 shillings. Two pence and three pence make 
 five pence. 
 
 s. d. 
 
 3 5 
 
 That is the answer we get by beginning at the 
 shillings end. Now let us try beginning at the 
 pence end, as we are told is the right way at 
 lessons. Two pence and three pence make five 
 pence ; one shilling and two shillings make 
 three shillings. So we have five pence and 
 three shillings. Does that come to the same as
 
 SIMPLE ADDITION 49 
 
 three shillings and five pence? Yes. And 
 ' three and five pence ' is more natural to say 
 than five pence and three shillings. Then why 
 are we taught to do the sums in the way 
 that is not natural, since both ways come to 
 the same answer? 
 
 We must find out if we can. We must know 
 the real reasons for things when we can. Let 
 us try another sum : 
 
 BLACK-BOARD. 
 s. d. 
 
 2 6 
 
 3 8 
 
 Three shillings and two shillings make five 
 shillings. Eight pence and six pence make 
 fourteen pence. How do we write fourteen 
 pence? Two pence and one shilling. Where 
 must we put the one shilling ? Along with the 
 five shillings. So now our sum looks like 
 
 this : 
 
 s. d. (a) 
 
 2 6 
 
 3 8 
 
 6
 
 50 LOGIC OF ARITHMETIC 
 
 or else like this 
 
 s. d. (6) 
 
 2 6 
 
 3 8 
 
 5 
 
 1 2 
 
 Now let us do the sum the way we are shown 
 at lessons. Eight and six are fourteen. Put 
 down two pence and carry one shilling ; one 
 shilling and three make four ; four and two 
 make six. We have our sum looking so : 
 s. d. (c) 
 
 2 6 
 _3 8_ 
 
 6 2 
 
 ^ Now do you see why you were told to do 
 addition sums in a way that is not the natural 
 way ? Fancy if a shopman gave us our bills 
 all crowded up with scratchings out and extra 
 figures at every turn, like (a) or (6)! It is 
 necessary that you should learn, while you are 
 young, to do some things in the way that you 
 would not think natural. Your teachers pick out 
 for you which way you must do them in class ; 
 they have experience and know ; but in holidays 
 you may always try doing easy sums the upside- 
 down way to the way you are told in school,
 
 SIMPLE ADDITION 51 
 
 so as to gain experience for yourselves, and find 
 out the reasons of the school rules. 
 
 It is not safe to experiment, while you are 
 young, with real things with fire or food or 
 your own bodies, because you might do harm 
 that could not be undone. But sums can be 
 wiped off or scratched out, when you get them in 
 a muddle ; so they are capital things to experi- 
 ment with, and great fun too ; and you will 
 understand your real work at school much 
 better if you experiment in play. Always do 
 your school-work, in term-time, exactly as your 
 teacher tells you ; that is the way to grow 
 smart and handy and useful. In holidays, do 
 easy sums the upside-down way to the school 
 way. That is the way to understand your 
 work, and grow clever enough to find out things 
 for yourself. 
 
 At ^ the teacher may interpolate other examples of 
 beginning sums at the wrong end, taking an easy 
 example of subtraction, multiplication, and short 
 division. 
 
 Look well at the black-board. Now shut 
 your eyes and settle at ease. 
 
 Can you see in your minds the sums (a) and 
 
 (6) ' (and any others that may have been done in round- 
 about ways). If not, open your eyes and look. 
 Shut your eyes again. What do you see ? 
 
 D 2
 
 IV 
 ARITHMETICAL SHORTHAND 
 
 BLACK-BOARD. 
 
 12 Buttons are a dozen. 
 12 Pence are a shilling. 
 
 Are those statements true ? 
 
 Are they both true ? 
 
 Are they always true ? 
 
 Are they true in the same sense ? 
 
 Let us see. 
 
 Suppose you go into a shop and say : f I want 
 a dozen of those buttons, please,' and a friend 
 says : ' And I will take twelve of the same 
 buttons, please/ do your two purchases look 
 alike ? Would the two lots weigh the same ? 
 Would the owner of one of the lots be any the 
 worse or the better off, if the parcels were 
 changed by accident ? If the twelve buttons 
 were sewn on to your coat, would any one know 
 that they were riot the dozen ? If we wanted 
 to play a game with twelve counters and had no 
 proper counters, we might use the twelve but- 
 tons for counters ; would the dozen do instead ? 
 Yes, just as well. The dozen is twelve, and 
 twelve is a dozen ; and for every purpose for
 
 AEITHMETICAL SHOETHAND 53 
 
 which one could be used, the other would do 
 just as well. 
 
 You say twelve pence are a shilling. Do 
 they look like a shilling ? Are they the same 
 colour, size, weight ? If I wanted things to 
 use instead of counters, I might use twelve 
 pennies : would a shilling do instead ? No. 
 Sometimes in cooking, if we have not small 
 weights, we use a coin as a weight ; we might 
 be told in a cookery book to take the weight of 
 a sixpence or shilling of carbonate of soda or 
 of some spice. How would it be if we used 
 twelve pence instead ? 
 
 Twelve pence are not a shilling, not in any 
 way like a shilling. Why do you say they are 
 a shilling ? They are of the same value. Value 
 for what ? Not for counters, nor for weighing 
 things. 
 
 Twelve pence are of the same value as a shil- 
 ling when we want to buy things. Yes, now we 
 have it right ; twelve pence are not a shilling, 
 and cannot be used instead of a shilling for any 
 real use of the things themselves. But the 
 chief purpose of coins is to exchange for other 
 things ; and, for that purpose, twelve pence are 
 of the same value as a shilling. 
 
 Then if that is what we mean, why don't we 
 say sol Why do we say 'Twelve pence are 
 a shilling,' when we don't mean it ?
 
 54: LOGIC OF ARITHMETIC 
 
 Ah ! why ? Now we have come to something 
 which it is very important you children should 
 understand and remember. 
 
 Think how often, in your arithmetic, you 
 have to make the change from pence to a shil- 
 ling or from a shilling to pence. How would it 
 be if the teacher had to stop eveiy time and 
 say, ' It is arranged by law that, for purposes 
 of buying and selling, twelve pence shall be 
 considered as of the same value as one shilling ' ? 
 The teacher would be tired of saying it ; you 
 would be tired of hearing it ; it would waste 
 time, and disturb you from attending to your 
 lessons ; so it has been arranged that people 
 may always say, ' Twelve pence are a shilling/ 
 for shortness, whenever the business on hand is 
 using coins as money, or talking of them as 
 money, as things to buy other things with ; but 
 not when any other business is on hand. 
 
 ' Twelve buttons are a dozen ' is true always, 
 everywhere where people are speaking English, 
 whatever use they mean to make of the buttons. 
 But ' twelve pence are a shilling ' is not really 
 true in itself; it is true as far as the business 
 on hand is concerned, whenever people are 
 talking about using coins to buy things with. 
 
 You might go some day to a class in what 
 is called physical science, where the business on 
 hand is learning about the weights of things
 
 ARITHMETICAL SHORTHAND 55 
 
 or the qualities of things ; or to a class in 
 cookery. If the teacher told you to go and 
 fetch a dozen bottles, and you came back and 
 said, ' Here are the twelve bottles you sent me 
 for,' nothing would be said about it ; you would 
 be all right. But if the teacher told you to 
 take the weight of a shilling in some powder, 
 or to take as much as would lie on a shilling, 
 or to see what effect some liquid had when it 
 touched a shilling, and he found you using 
 twelve pennies instead, there would be some- 
 thing said to you then! You would perhaps 
 be told that you never could learn anything 
 while you were so stupid and clumsy and 
 inaccurate. 
 
 I don't suppose you would ever make just 
 that particular mistake. But I have known 
 many children hopelessly puzzled over sums 
 and other things ; and, when I came to see 
 what puzzled them, I found they had been 
 taking something some teacher had said, as 
 meant to be true in itself, when it was only 
 a shorthand sort of sentence which meant some- 
 thing else : a shorthand which was true for 
 one purpose but not for another. 
 
 Try to understand always whether your 
 teacher means what he says to be true always 
 and everywhere ; or whether he means it as 
 a bit of shorthand talk fit for that particular
 
 56 LOGIC OF ARITHMETIC 
 
 class. If you cannot find out for yourself, ask. 
 Never go on using a sentence till you are sure 
 whether it is meant for literal truth or con- 
 venient shorthand. Think of the 1, the stroke 
 which the old people cut on their tallies, and 
 which meant, not one thing, but once putting 
 all the fingers up and then down so as to begin 
 counting again. Arithmetic is full, from begin- 
 ning to end, of just such nice, clever, convenient 
 bits of shorthand as that. Sums are difficult 
 and puzzling chiefly because children forget 
 this. 
 
 V 
 KEEPING ACCOUNTS 
 
 SUPPOSE you have a shilling and spend three- 
 pence, what have you left ? Ninepence. Is that 
 all that is left ? 
 
 Suppose that you go out with a shilling in 
 your purse and spend three-pence , on buying 
 flower-roots, what do you bring back ? Nine- 
 pence in your purse. Anything else ? Any- 
 thing not in the purse ? The roots. Well, is 
 that an end of the matter ? What is to happen 
 next ? The roots have to be planted, and then 
 watered ; or else they will die, and you will 
 have wasted your money.
 
 KEEPING ACCOUNTS 57 
 
 You might have bought something else, not 
 roots. But whatever you did with- part of your 
 money, something of some sort is left besides 
 the change. If you bought food, it would be 
 there ; and must be seen to that it might not be 
 wasted. If you ate the food while you were 
 out, you would bring back the strength you got 
 out of it ; or if it was unwholesome food, you 
 might bring 'home ninepence and some very 
 uncomfortable feelings. Whenever money has 
 been spent, something remains besides the 
 mere balance of cash. 
 
 Yet if you have been taught to keep accounts, 
 or if any grown-up person allows you to see 
 how she keeps accounts, you know that when 
 the money spent one week has been subtracted 
 from the cash in hand, the page on which the 
 things bought during the week have been 
 written down is turned over, and only the 
 remaining cash is entered on the new page for 
 next week. 
 
 Now shut your eyes for a minute or two, and 
 think. See an account-book open. On one 
 page is written : 
 
 ' Mother gave me Is.' 
 
 On the other page is written : 
 
 ' Plants, 3d.' 
 
 You subtract the 3c. from the shilling, and 
 write 9d. Then you turn over the leaf, arid
 
 58 LOGIC OF ARITHMETIC 
 
 on the next page you write, ' Brought over : 9dL' 
 The writing about the plants is hidden out of 
 sight ; nothing is on the open page except the 
 account of the money still left in your purse. 
 The roots are not in sight ; they are in another 
 room, waiting till you can go to them. But they 
 have not gone out of the world : they are there, 
 waiting ; and you will have to see to them 
 presently. 
 
 Then why is nothing written about them in 
 the new page of the account-book? Think 
 about that a little, and we will have a lesson 
 about it next time. 
 
 VI 
 DIS-MEMBERING AND RE-COLLECTING 
 
 THE question you were to think over was : 
 Why, when we turn over a page in an account- 
 book, we enter on the next page nothing that 
 was on the last page, except the balance in 
 cash ; we make no mention of the things bought 
 last week, though those things may still be left 
 for us to deal with. The plants we bought last 
 week still need watering ; part of the food we 
 bought may be there still to eat ; the strength 
 we got out of what we ate may remain to 
 be used ; the illness we got if the food was
 
 DIS-MEMBERING, RE-COLLECTING 59 
 
 unwholesome may be still uncured. Yet we 
 make no mention of these things in this week's 
 accounts. 
 
 Why not? 
 
 I think you feel why not, though perhaps .you 
 cannot quite explain it. To-day we are going 
 to have a little talk about that question. 
 
 Human beings, you and I for instance, are 
 finite creatures ; that is to say, we cannot be 
 everywhere at once or do many things at once. 
 Our bodies are made so that we can see only 
 a short way across our big world ; and our minds 
 are made so that we can only attend to a. small 
 part of any big business at once. All the rules 
 of Arithmetic are made to help us to do big 
 sums by attending to little bits at a time. If 
 our minds were bigger we could do big sums, 
 straight off, without any rules ; but, as it is, 
 we have to attend to a bit at a time ; and the 
 rules are made in order to fit the bits together 
 properly. It is not only sums that have to be 
 attended to a bit at a time, in proper order 
 according to rule. Suppose you are going to 
 a school-treat or picnic. If you were fairies, 
 you could start off straight away; and the 
 proper clothes and boots would grow on you, 
 if you needed them, as you flew along. But 
 you are not fairies, so the clothes and boots 
 have to be put on first, and carefully too, or
 
 60 LOGIC OF ARITHMETIC 
 
 you cannot go. Suppose you were thinking 
 about what you would do out of doors, and not 
 tying up your boot-laces properly, mother might 
 say : ' Come, attend to your boot-laces now ; 
 and when they are tied up safely, you can think 
 of out of doors.' But ' out of doors ' has not 
 gone away because you have to put it out of 
 your mind for a little while and attend to your 
 boot-laces. It is waiting, waiting, till you are 
 at leisure to attend to it. So it is always. 
 When we put anything out of sight and out of 
 mind, in order to attend to something which 
 for the moment we must attend to, the thing 
 we put out of mind is not gone out of the 
 world ; in some shape or form it is waiting, 
 waiting, waiting, and will have to be reckoned 
 with some time or other. 
 
 Now we will do a little sum. 
 
 How many days has a man lived who has 
 lived forty-seven years ? 
 
 If I asked you how many days a child has 
 lived who has lived two weeks, you could tell 
 me straight off, without any need to write on 
 the black-board. There are how many days in 
 a week? (7) and twice 7 are? (14). If your 
 minds were giant minds, you could answer the 
 other question, about the days in forty-seven 
 years, just as easily as that one. But we, you 
 and I, have such tiny, helpless little minds that
 
 DIS-MEMBERING, RE-COLLECTING 61 
 
 we cannot manage a question like that. If we 
 were giants we could pick pears off the top of 
 a very big tree ; but human people have to 
 climb up on ladders to reach high-growing 
 pears. And in the same way we have to make 
 a sort of mind-ladder before we can reach such 
 big numbers as the days of forty-seven years. 
 What are we going to do ? Write on the board. 
 Yes ; tell me what I am to write. 
 
 A 3 and a 6 and a 5. And then, under- 
 neath, a 4 and a 7. 
 
 What does the 3 stand for ? Three hundreds 
 of days. And the 6 ? Six tens of days. The 5 ? 
 Five single days. The 4 ? Four tens of years. 
 And the 7 ? Seven years. 
 
 Now tell me what I am to write. Seven times 
 five are thirty-five , . . Why, what are you 
 about ? I asked you to find the number of 
 days in forty-seven whole years ; and here you 
 are, telling me how many days seven times five 
 days are. 
 
 Had you forgotten the hundreds of days and 
 the tens of days and the forty years ? Yes, for 
 the moment ; we had to forget them, to push 
 them out of our minds, so that we could attend 
 properly to just the bit we were doing at a time. 
 Go on then, seven times five are 35. There, 
 I have written that down. Have we done now ? 
 Is the sum right ? Will you go home and tell
 
 62 LOGIC OF ARITHMETIC 
 
 your mothers that you have found out at school 
 how many days there are in forty-seven years, 
 and there are just thirty-five days in all those 
 years ? Much good there would be in going to 
 a school where that sort of thing was allowed ! 
 Thirty-five is only the first rung of the ladder ; 
 we haven't reached our answer yet. The three 
 hundreds of days and the six tens of days and 
 the forty years are waiting quietly, till we have 
 finished attending to the seven times five ; next 
 it will be the turn of seven times six tens to be 
 attended to, while the hundreds are waiting, 
 waiting. And now we multiply the three 
 hundreds by seven ; and all this time the forty 
 years are waiting, waiting, waiting, still. 
 
 Well, now we have attended to each bit, and 
 here are all our bits written down, one under 
 the other \ But which of all the bits is the 
 answer ? Which is the true right answer to 
 the question we began about ? 
 
 Alice, you say, 7x6 = 42. And, Mary, you 
 think that 4x5 = 20. And, Julia, your opinion 
 is that 4 x 6 = 24. 
 
 Which is the true view of the matter ? Are 
 you going to quarrel about which is right ? Or 
 would you rather have a nice long solemn argu- 
 ment, each trying to prove that she has the true, 
 the only true, answer to the question I asked ? 
 1 See note at end of chapter.
 
 DIS-MEMBERING, KE-COLLECTING 63 
 
 You won't do either * Well then, what will 
 you do ? We hiust fit them all together to see 
 what they all come to, before we have a right 
 to tell people how many days there are in forty- 
 seven years ; because our minds are so tiny, and 
 the number of the days of the years are so very, 
 very big. We have to make ourselves forget 
 some things while we attend to other things ; 
 but before we dare tell any one that we have 
 found the actual truth, we must call back all 
 that we made ourselves forget, and try to 
 re-collect. 
 
 Well then, 2 and 4 are 6 ; and 1 are seven. 
 Is that right ? No ? Why not ? Some of the 
 figures mean hundreds and some tens and some 
 only single days, and we must sort them pro- 
 perly before we fit them together, and arrange 
 them so that each sort of figure shall stand for 
 its true value. 
 
 Now we understand better the question about 
 the account-book. The use of the book is to 
 keep account of money. Our minds are so small 
 that we cannot think of cash accounts and other 
 things at the same time. We wrote down 
 once : ' Plants, 3d.,' because we may some day 
 wish to look back and see what different things 
 cost us ; but once writing it was enough : we do 
 not wish to be constantly reminded of plants and 
 other nice things, while we are busy adding up
 
 64 LOGIC OF ARITHMETIC 
 
 accounts ; we try to forget the things and attend 
 to the cash accounts, just while we are using 
 that book. But when we have done the accounts 
 and shut up the books, we should recollect the 
 things ; for everything that we have spent 
 money on is still somewhere, in some shape or 
 form, waiting, waiting, waiting. 
 
 And when we go to recollect the things, 
 we must try to arrange them in their proper 
 order so as to give to each its -true value and 
 meaning. 
 
 That is enough for to-day. Sit slack, shut 
 your eyes, and rest before you go to the next 
 class. I am going to give you two mind- 
 pictures : 
 
 A little boy was so excited about going to 
 a treat that he would keep talking about it, 
 and would not tie his shoe-laces properly. On 
 his way downstairs his lace came untied ; he 
 stepped on it and fell and sprained his ankle, 
 so he could not go to the treat after all. 
 
 A servant was asked to get some children 
 dressed to go out. She got them tidy, and their 
 boots nicely cleaned and tied on ; the children 
 felt impatient because they were in a hurry to 
 go out ; but they knew they could not go till 
 they were dressed, so they were good and quiet. 
 When they were ready, they thought they were 
 to go. But the nurse grumbled and said :
 
 WEIGHTS AND MEASURES 65 
 
 * Now you are dressed and all tidy and clean, 
 I don't want you to go out, for fear you should 
 tumble your hair or get your boots dirty. I was 
 told to dress you, and I have dressed you : that 
 was the important thing ; that was my real duty. 
 Going out is all nonsense. Sit still all the after- 
 no* n, and look at your nice clean boots.' 
 
 Try to fancy how those children would feel, 
 and what they would think. Try for a minute 
 or two to fancy yourselves in the place of those 
 children, and to think what you would feel like. 
 
 Now open eyes and stand up. Go and get 
 ready for the History Class. 
 
 NOTE. 
 
 The answers had better be written on the board at 
 first in this form : 
 
 7x5 = 35. 
 
 7x6 = 42. 
 
 7x3 = 21. 
 
 4x5 = 2O. 
 
 4x6 = 24. 
 
 4x3 = 12. 
 
 VII 
 WEIGHTS AND MEASURES 
 
 On the Table : A set of weights, one-ounce, two- 
 ounce, four-ounce, half-a-pound, one-pound. 
 
 WHAT is this ? A weight. What weight ?
 
 66 LOGIC OF AKITHMETIC 
 
 An ounce. What is it for ? To weigh things 
 with. Yes, this ounce weight is one of the 
 things that has no proper use of its own ; its 
 use is to make other things more useful by 
 keeping them exact and orderly. 
 
 Let us think of the uses of weights. They 
 are used in shops, for the shopman to know 
 exactly how much sugar or tea or meat he is to 
 give for his customer's money. 
 
 But they have another use as well. 
 
 Very long ago people had no notion how to 
 make cakes or puddings. When they caught 
 an animal, they roasted it by a fire ; and when 
 they found roots fit to eat, they roasted them 
 in hot ashes ; and that was all they knew 
 about cooking. Then they found out how to 
 make pots of some kind, and then they found 
 they could boil things in water. Gradually 
 they found that some things- improve the 
 taste of other things, when they are boiled 
 together : salt improves soup, and sugar im- 
 proves puddings. But there must not be too 
 much of one sort of flavouring. People began 
 to try to find out how much of each kind of 
 thing made the nicest mixture. They could 
 find out only by trying. That is what we call 
 experiment : a woman finds the soup too salt 
 or not salt enough one day, she thinks it nicer 
 when she puts in less salt or more salt ; then
 
 WEIGHTS AND MEASURES 67 
 
 she tries to remember for the future what is 
 the right quantity. 
 
 But it is a pity for each one to have to make 
 all the experiments for herself. It saves time 
 and trouble, if one person's experiments can be 
 made useful to other people. Suppose a woman 
 is a very clever cook and finds out how to make 
 things just right, it is well for her to be able to 
 tell other women just what she found answered 
 best. The other women's husbands and children 
 may not have quite the same taste as the first 
 woman's ; they may like a little more salt or 
 sugar or spice than the first woman said ; but 
 still, it is useful to know what other people find 
 answer, and then we can alter it to suit our- 
 selves and our families. 
 
 Now how is the woman, who has made a nice 
 cake, to tell others how much she put in ? She 
 has to use weights or else measures. She says: 
 ' I took a pound of flour, and a quarter of a 
 pound of sugar, and half a tea-spoonful of baking 
 powder,' and so on ; weights and measures are 
 a language in which one person can tell others 
 how much of anything to use. 
 
 Well, a w r oman read in a book a receipt, as it 
 is called, for making a nice small cake. One of 
 the ingredients was an ounce of lard. But she 
 was preparing for a tea for a great many people, 
 and she wanted to mix all her dough at once 
 
 E 2
 
 68 LOGIC OF AEITHMETIC 
 
 for several cakes. She wanted to mix sixteen 
 times as much dough as the receipt told about. 
 So she had to use more lard, sixteen times as 
 much as the receipt told about. She had only 
 one ounce weight. Look at my set of weights 
 here ; there is no other of this size. Most sets 
 of weights have only one of each size. The 
 woman I am telling you of had only one weight 
 of this size, and she wanted to weigh sixteen 
 times as much. What did she do? Did she 
 weigh out each ounce separately? If you think, 
 you will see she could do something handier 
 and quicker than that : she could weigh the 
 whole sixteen at once with the pound weight. 
 
 Now I should like you to go over the weights 
 and see how they are arranged for use. 
 
 If we want to weigh one ounce we use ? The 
 ounce weight. If we want to weigh two ounces ? 
 The two-ounce weight. 
 
 If we want to weigh three ounces, what shall 
 we do ? There is no three-ounce weight. Put 
 in the scales the two-ounce and the one- 
 ounce. 
 
 For four ounces ? The quarter of a pound 
 weight. 
 
 For five ounces? The quarter of a pound 
 and the ounce. 
 
 For six ounces ? The quarter of a pound and 
 the two-ounce.
 
 MULTIPLYING BY MINUS 69 
 
 For seven ounces ? The quarter and the two- 
 ounce and one-ounce. 
 
 For eight ounces ? The half-pound. 
 
 (The teacher goes on thus up to thirty-one ounces.) 
 
 So, you see, we can weigh any number of 
 ounces we please, up to thirty-one ounces, by 
 having only these five weights, one ounce, two 
 ounces, quarter of a pound, half a pound, and 
 one pound. 
 
 If we want to weigh more than thirty-one 
 ounces, we shall have to get other weights; 
 such as two pounds, four pounds, &c. 
 
 But we can weigh thirty-one different quan- 
 tities with only five different weights. 
 
 VIII 
 MULTIPLYING BY MINUS 
 
 Many persons who have learned a certain amount of 
 Algebra are confused between what are called negative 
 quantities, (movements, or actions), and the mental act of 
 negation of quantity, (movement, or action) ; i. e. between 
 the sign which indicates active fact in the direction 
 contrary to the one (arbitrarily) chosen as positive, and 
 the symbol 0, which states denial of existence of fact. 
 Practice should be given in exercises which involve these 
 signs and show the distinction between them. 
 
 It is also desirable, for many reasons, that children 
 should be accustomed to use letters for unknown
 
 70 LOGIC OF AEITHMETIC 
 
 quantities at an early age. It assists the imagination 
 in keeping a clear distinction between a quantity in 
 itself and the effect on the quantity of a certain opera- 
 tion or group of operations. Whatever a number may 
 be, the group of operations represented by 7 + 4 9 + 2 
 diminishes it by 10. The statement x 7 + 4 9 + 2 
 = #10 has a meaning which can be made in- 
 telligible at an early age ; whereas the mere statement 
 7 + 4 9 + 2 = 10 is confusing to a child's imagina- 
 tion. Even for those who are never to learn formal 
 Algebra, the practice of using letters, for numbers which 
 are to be the subjects of operation, imparts to Arithmetic 
 something of Algebraic clearness of thought, which will 
 be of use in 'itself ; while those who are to learn Algebra 
 should not be exposed to the mental violence of being 
 introduced to Algebraic notation for the first time, just 
 when they will have to begin to learn the actual 
 manipulation of unknown quantities by processes 
 properly Algebraic. The following is a specimen of 
 the kind of form in which elementary exercises in 
 ordinary sums may with advantage be given. 
 
 BLACK-BOARD. 
 T 
 
 THE figure on the board is supposed to be 
 a counter in a shop. We are going to do a sum 
 about buying something over the counter. What
 
 MULTIPLYING BY MINUS 71 
 
 does the shopman call the thing in which he 
 keeps his money ? A till. Well then, we will 
 call the amount of money in it T. What do we 
 call the thing a customer carries money about 
 in ? A purse. We will call the money in this 
 customer's purse P. We do not know how 
 much T is, nor how much P is ; and we do not 
 want to know just now. Provided that the 
 customer has brought enough to pay for what 
 she buys and the shopman has plenty of change, 
 that is all we shall need to know at present. 
 When we do not know a number, we often call 
 it by the name of some letter. 
 
 The customer sees boxes of sweets marked 
 lid. She takes a box and pushes a shilling 
 towards the shopman. What does he do ? Does 
 he push anything towards her ? A penny. 
 
 Does the shilling which she gives to him add 
 to what is in the till or does it make it less ? 
 Adds to it. When we add two things together, 
 we put this mark between them +. So now 
 we write what is in the till T + Is. T means 
 what was in the till before the customer came 
 in. T has not altered. What is now in the 
 till is T + Is., which we read : T plus Is. But 
 what has the shopman to give to the customer? 
 \d. Is that added to T + Is., or taken from it ? 
 Taken from it. We write that this way : IcZ., 
 and we call it minus Id.
 
 72 LOGIC OF ARITHMETIC 
 
 So now this is how we write what is in the 
 till: T + ls.-ld. 
 
 Just for to-day we are thinking about things 
 as the shopman thinks about them ; we are 
 agreeing with him. It is his till to which 
 a shilling has been added, and his till from 
 which a penny was taken. He thinks about 
 the shilling as plus and the penny as minus. 
 Some other day we will see what the customer 
 thinks, but the shopman's thinkings are enough 
 for to-day. He thinks pushing something from 
 P to T is a plus action ; and pushing from T 
 towards P is a minus action. As long as we 
 are thinking like the shopman, everything that 
 goes from P towards T is marked plus ; and 
 everything that goes from T towards P is 
 marked minus. 
 
 For to-day, P to T is the plus direction. T to 
 P is the minus direction. 
 
 By such a lesson as the above, we lay a skeleton, 
 round which may be grouped a great variety of examples 
 in ordinary addition and subtraction of money. The 
 customer may ask for several articles in succession, and 
 a bill be made out. Several customers may come in, 
 and require amounts of change. All the money which 
 passes should at first be registered as added to or taken 
 from T. Sometimes the customer has the exact coins 
 needed. In such case no change will pass ; this fact 
 should be carefully recorded by writing 0. 
 
 When the teacher wishes to extend the scope of the 
 operations, he should go back to the one shilling given 
 and one penny change ; so as to leave the children's
 
 MULTIPLYING BY MINUS 73 
 
 thoughts free to understand the operation involved, 
 undistracted by questions of number. 
 
 We are going now to look at the business of 
 paying for things, from the customer's point 
 of view, to see how she thinks about it. When 
 she pushes a shilling towards the shopman, is it 
 added to or taken from P ? Taken from P. 
 And the penny of change is ? Added to P. Then 
 the customer's account now is ? P Is. + \d. ; 
 The shopman's is ? T + Is. \d. 
 
 After this, money sums should be done sometimes 
 from the shopman's, sometimes from the customer's 
 point of view. Occasionally, a sum should be written 
 out from both points of view. Every shop transaction 
 should be dealt with either as an addition to or a 
 diminution of T or P ; or, preferably, as both. 
 
 At a still later stage we suggest : 
 
 We are going to see whether the money has 
 been paid and the change given all right. For 
 that purpose we must know what was in the till 
 before the customer came in. T was 3 9s. Qd. 
 So on the shopman's side we ought to have 
 3 9s. Qd. + ls.-ld. P was 4 75. 8d. 
 Therefore the customer ought now to have 
 4 Is. Qd.-ls. 
 
 Then more sums are grouped round that conception. 
 There is a further stage of the same scheme. 
 
 A customer asks for twelve articles value 
 19s. each. She lays down 12 sovereigns. The
 
 74 LOGIC OF ARITHMETIC 
 
 shopman hands back a shilling change for each 
 sovereign : 12s. 
 
 One of the articles is then found to be faulty. 
 There are no more in stock. The customer says : 
 ' Never mind ; eleven will do for the present.' 
 The shopman hands her back 1. Who has 
 too much now ? Who must give a Is. in change? 
 The customer. 
 
 The account then stands thus : 
 
 On the shopman's side, 
 
 T + l2-12s.-L + ls., 
 and on the customer's side, 
 
 P- 12 + 12s. + l-ls. 
 
 In order to emphasize this lesson, it should also be 
 written out in another form : 
 
 1-1*. 
 x(ldoz.-l) 
 
 12-12*. 
 -1 
 
 Now then, what does 1 x 1 come to ? In 
 which direction must the last shilling be pushed? 
 From P towards T. But that is, from the shop- 
 man's point of view, the plus direction. So minus 
 multiplied by minus comes io plus. 
 
 Now from the customer's point of view : 
 
 x(l doz.-l) 
 
 -12 + 12*. 
 
 + 1-1*.
 
 ADDING MINUS 75 
 
 This lesson carefully gone over, with the simple 
 numbers given above, and afterwards repeated with 
 more complex sums, would save all future trouble about 
 the much-vexed problem of 'minus multiplied by 
 minus.' 
 
 IX 
 
 BLACK-BOAHD. 
 
 S H _C ^ 
 
 M M M 
 
 S is a shop where a boy works. H is the 
 house where he lives and sleeps. C is a house 
 where a customer of his master lives. Each M 
 is a mile-stone on the road. The distances 
 from S to H and from M to M are, each, a 
 small picture of a mile along the road. 
 
 Now, do you remember what those letters 
 stand for ? Because I don't want you to have 
 to think about that presently when I am ex- 
 plaining something else. So say those letters 
 over : S, shop ; H, boy's home ; C, customer's 
 house ; M, mile-stone. Again : S, shop ; H, 
 boy's home ; C, customer's house ; M, mile- 
 stone. 
 
 You see which way the arrow points. Do 
 you remember that we had a lesson about
 
 76 
 
 money pushed across a counter ? When it was 
 pushed across one way, it added to the shop- 
 man's money ; so, when we were doing his 
 accounts, we called that direction plus ; and 
 when he pushed change back, we called that 
 minus. Miles on a road are sometimes counted 
 in the same way : we call one direction plus 
 and the other minus. Well, the boy who works 
 at the shop lives one mile from the shop in the 
 direction that we are going to call plus ; what 
 then shall we call a mile in the opposite 
 direction ? A mile minus ; he walks a mile 
 minus in going to work ; every evening he 
 walks a mile plus in going home. C is two 
 miles further on than H. So all the distance 
 from S to C is how many miles ? In what 
 direction ? S to C is three miles plus. 
 
 One night the master says : ' Bill, there is 
 a parcel to be taken to Mr. Smith's house (C). 
 You live in that direction ; you must take it.' 
 
 What then has Bill got to do ? How many 
 miles plus has he to walk ? Three ; and what 
 has he to do next ? He must get home some- 
 how. What must he do ? Walk two miles 
 minus. So three miles plus and two miles minus 
 brings him to the same place, H, as he gets to 
 on other evenings by walking one mile plus. 
 
 We said that the boy's task, set him by his 
 master, was to walk three miles plus. He
 
 ADDING MINUS 77 
 
 neutralized or counteracted part of that walk 
 by adding to it a walk of two miles minus. 
 
 But now suppose that before Bill starts with 
 his parcel a workman says to the master : ' I 
 am going to see a friend who lives close to 
 Mr. Smith's house ; may I take the parcel, sir?' 
 And suppose the master gives leave. What 
 has the workman done to Bill's task of three 
 miles plus ? He has taken off from it two 
 miles plus. Bill still has to walk one mile plus, 
 because he must get home. 
 
 If the workman takes the parcel, he subtracts 
 or takes off from Bill's task, two miles plus ; if 
 Bill takes the parcel he neutralizes two of the 
 plus miles by adding to his three miles plus 
 another walk of two miles minus. His walk 
 in either case ends at the same point H. But 
 now I want to know : are the two ways of 
 getting home the same in any other respect ? 
 For instance, will he get home to supper at the 
 same time ? No, he will be more than an hour 
 later for supper if he takes the parcel himself. 
 And what about exercise and health and all 
 that ? If Bill is cashier, and has been sitting 
 still all day in a close shop, the five miles walk, 
 three plus and two minus, would do him a great 
 deal of good. But if Bill is a delicate little 
 chap and has been on his feet all day, that five 
 mile walk instead of one mile might make him
 
 78 LOGIC OF ARITHMETIC 
 
 quite ill. It would make a difference to wear 
 and tear of shoe-leather, and to the length of 
 time he would have for playing with his sister 
 before going to bed. The only thing about 
 which adding two miles minus comes to the same 
 as subtracting two miles plus, is the point, one 
 mile from the shop, where he gets to at the end 
 of his walk. 
 
 So you see, if you are asked a question about 
 the point which Bill will reach at last, it will 
 be quite right to say, ' Adding minus two comes 
 to the same as subtracting plus two.' If you 
 are asked about Bill's health, or his shoes, or 
 the time he gets his supper, you must not say 
 anything of the kind ; because, in relation to 
 those questions, it would not be true. 
 
 MlND-PlCTURE. 
 
 Bill trudging home tired ; near the middle M ; 
 with his back to C and his face towards H. He 
 is looking tired, and is glad to have got as far 
 as M on his way back. 
 
 X 
 
 DIVIDING AND SHAKING 
 
 WHAT is the half of five ? Two and a half. 
 If two children are to share five ounces of any- 
 thing, what is the share of each ? Two and
 
 DIVIDING AND SHAKING 79 
 
 a half ounces. Is that so always ? Are you 
 sure? Well now, let us see. 
 
 Suppose two children are to share a cake 
 weighing five ounces, the share of each is ? 
 Two and a half ounces of cake. How do we 
 divide it ? Do we give one all the top and the 
 other all the bottom ? No, that would not be 
 a fair division ; we cut it so that each gets half 
 the currants, half the sugar on the top, and 
 half the bit of candied peel in the middle, as 
 well as half the dough. 
 
 Suppose five marbles, weighing an ounce each, 
 are given to two boys to play with, what is the 
 share of each 1 Two marbles ; and there will 
 be one over (as you say in division sums). What 
 would the boys do with the one over ? I think 
 they would put it on the ground between them, 
 and each shoot at it in turns, with the marbles 
 that were his own share. 
 
 Suppose I have a doll weighing five ounces, 
 and give it to two little girls. What is the 
 share of each ? Will they cut it in two ; and 
 each have half a dolly ? Shall each have one 
 leg and one arm and half a head ; half the 
 calico, half the plaster, and half the saw-dust ? 
 No. Well, what would be the share of each ? 
 If it was a naked doll, there would be its clothes 
 to make ; they would do the work between 
 them. When they had time to nurse it for an
 
 80 LOGIC OF ARITHMETIC 
 
 hour, they would each nurse it for half an hour. 
 When they had it to tea, they would sit one 
 each side of it. That would be the fair sharing 
 of a five-ounce dolly. 
 
 Now suppose a flower- bulb, weighing five 
 ounces, is given to two children, how would 
 they share it \ Would they cut it in two down 
 the middle ? No ; they would plant it, and 
 take turns at watering and tending it. 
 
 Children who have become mechanicalized, here add : 
 ' And share the flowers when it blossomed.' 
 
 Do you mean what you are saying ? Just 
 think a minute. Babies might cut a hyacinth 
 up for the sake of sharing the flower-bells 
 between them, and each being able to say, 
 ' These are mine ' ; but I never saw two children 
 old enough to tend plants who would do such a 
 stupid thing. What would you really do if the 
 plant were yours ? Keep it growing, to look 
 at, and show to friends ; and how about the 
 sharing? If one child looks at it, does that 
 prevent the other seeing it ? If one smells it, 
 does that prevent the other smelling it ? If one 
 child brings in her special school-friend to see 
 it, does that prevent the other from bringing 
 in her special friend to see it too ? No. But 
 neither child can say : ' This is my plant.' 
 
 So now this is what we have come to. 
 
 If two people are to share five ounces of
 
 IN WHAT CONSISTS ECONOMY 81 
 
 cake, each is to have two and a half ounces 
 of cake, each share to contain half of each kind 
 of ingredients that the cake is made of. 
 
 If they are to share five marbles weighing 
 an ounce each, the share of each is two marbles 
 and the fun of shooting them at the fifth marble. 
 
 If they are to share a five-ounce doll, the 
 share of each is half the work of dressing it, 
 the fun of nursing it for half the time they 
 have to spare ; and the fact of sitting beside it 
 at tea just as if it were all her own. 
 
 If they are to share a flower-bulb, the share 
 of each is, half the work of tending it, all the 
 pleasure of seeing it, all the pleasure of smelling 
 it, all the pleasure of letting friends see it ; and 
 none at all of the pleasure of saying : ' This is 
 mine, my very own, and no one may look at 
 it or smell it without my leave.' 
 
 XI 
 IN WHAT CONSISTS ECONOMY 
 
 BLACK-BOARD. 
 A. B. 
 
 Things such that I Things such that 
 no one else can other people can 
 have the part I have the part 
 thereof that I thereof that I 
 have. I have.
 
 82 LOGIC OF AKITHMETIC 
 
 You remember what we said about sharing 
 cake and sharing a lily-bulb. If two children 
 share five ounces of cake equally, the share 
 of each is ? Tw and a half ounces. If one 
 has three ounces, the other can only have ? 
 Two. If one has four ounces, the other can only 
 have ? One ounce. If one has five ounces, the 
 other gets ? None. Now look at the black- 
 board. On which side shall we write cake ? 
 Under A. 
 
 Now about the lily-bulb. If two children 
 share a bulb, the share of each is ? 
 
 Half the labour of growing it. 
 
 All the pleasure of seeing it grow. 
 
 All the pleasure of smelling the flowers when 
 they come out. 
 
 Alt the pleasure of seeing friends enjoy the 
 flowers. 
 
 None at all of the pleasure of saying : ' This 
 is my very own, and no one else can share it.' 
 So we shall write the bulb under ? B. 
 
 Suppose two people grow bulbs for sale, they 
 get ? Money. What happens to the money ? 
 It is shared between them. If they get six- 
 pence for a bulb, the share of each is ? Three- 
 pence. If one has four-pence, the other can 
 only get ? Two-pence. We will write money 
 
 got by selling bulbs under ? A. 
 
 Now I want you to think about a little row
 
 IN WHAT CONSISTS ECONOMY 83 
 
 of houses, six comfortable houses, all about the 
 same size, built near together on a common, 
 away from shops. The gardens are as yet 
 scarcely planted at all. Suppose one of you 
 is the mistress of the end house of the row. 
 
 One morning, two hawkers come up the road 
 and come past your house first. One has on 
 his barrow a lot of nice fresh fruit, about as 
 much as he thinks the people in your row will 
 want to buy. The other has young plants. 
 
 Now remember, we are not here talking of 
 shipwrecked sailors, or desert islands. I am 
 not asking what people should do who are shut 
 up together, with not enough food. I am not 
 asking what is the heroic thing to do in ex- 
 ceptional circumstances ; but of what is the 
 common -sense thing to do in commonplace 
 circumstances. There is enough fruit on the 
 barrow for every one in the row; and every one 
 has enough money to buy some. We will 
 admit at starting that the first duty of a house- 
 keeper, in ordinary civilized society, is to see 
 that there is proper food in the house for all 
 the household ; so the first thing you do, when 
 the hawkers call, is to buy as much fruit as 
 your household will need while it keeps good. 
 
 The people from one of the other houses are 
 coming in to tea ; so you buy enough for them 
 to have plenty as well as your own family. 
 
 F 2
 
 84 LOGIC OF AKITHMETIC 
 
 We will suppose that all the rest of the 
 necessary provisions are in the house already. 
 You have just bought fruit enough ; and you 
 have a few shillings which you decide to spend 
 on 'extras.' There are two things you can do 
 with them. 
 
 It comes into your head that it would look 
 rather grand if your neighbour were to see 
 on the table a huge heap of strawberries : 
 two or three times as many as could be eaten 
 that day. Will you buy that extra fruit ? 
 Or will you buy some plants for the garden ? 
 
 Let us see what would come of each of these 
 proceedings. 
 
 Suppose you spend your spare shillings on 
 buying more fruit than you need, some other 
 family will not be able to get any. So we 
 must enter Strawberries on the A side of 
 the black-board. 
 
 If you buy up that fruit, what happens to 
 it ? A good deal is left for next day ; not 
 quite as nice as it was the first day. No one 
 can enjoy it now quite as much as the people 
 who got none would have done if they had 
 got it while it was fresh. Everybody in the 
 house is perhaps tempted to eat a little more 
 than they otherwise would do, because it is 
 there and must not be wasted. Yet in spite 
 of that some of it is wasted ; goes really bad
 
 IN WHAT CONSISTS ECONOMY 85 
 
 and has to be destroyed at last. Meantime 
 there has been about the house a faint smell 
 of not quite fresh fruit, which takes away 
 everybody's appetite and makes every one un- 
 comfortable. For the next few days, no one 
 cares as much about fruit as they did before. 
 You will often find this the case about things 
 
 in the A class of this division. (Point to black- 
 board.) It is right that you should have enough 
 of each of them ; but the least bit more than 
 enough is too much, too much even for your 
 own good. 
 
 Well now, suppose that instead of buying 
 too much fruit you buy plants for your garden, 
 what happens ? All the people in the house 
 enjoy .them ; visitors enjoy them ; and the 
 people who pass by on the road enjoy them 
 too. 
 
 We will write the plants down in Column ? 
 B. 
 
 I have known many servants who liked 
 living in houses where the mistress bought 
 what is called in Ireland * lashings and leav- 
 ings ' of all sorts of food-things ; they called 
 a mistress stingy who bought only as much as 
 was of real use. You might see, in such houses, 
 ends of joints going bad, bits of loaves on the 
 floor among dirty boots, milk going sour, fish 
 going stale, candles guttering away in waste ;
 
 86 LOGIC OF ARITHMETIC 
 
 and the garden quite neglected because the 
 family were too poor to keep it up and had no 
 money to spend on plants. And the servants 
 thought such masters 'rale gintry.' You all 
 know that that was because these servants 
 were ignorant and uneducated. I think you 
 will find that this is a good test of true 4 
 education. 
 
 (The teacher now rubs out the instances, leaving only 
 the headings of the two columns ; and writes under 
 them this sentence : ) 
 
 True education tends to make 
 people satisfied with just enough 
 of the things in Column A, and 
 leads them to spend spare time, 
 money, and energy on things in 
 Column B. 
 
 Read what is on the black-board. Copy it 
 in your books. Read it out. Shut your eyes 
 and try to say it. Learn it by heart (in 
 preparation time). 
 
 Now I am going to tell you something rather 
 difficult. You will not quite understand it yet, 
 but if you think about it and fix it in your 
 minds you will understand it some day, and 
 it will help you to understand many other 
 things. 
 
 You might, if you could afford it, buy more 
 kinds of plants than you could attend to
 
 ECONOMY OF MIND-FOKCE 87 
 
 properly, or than the garden could well hold. 
 Some people do that kind of thing ; they can- 
 not make up their minds at once to do without 
 what they cannot use ; so they crowd up their 
 houses and premises with things that are in 
 the way. 
 
 Shut your eyes and think of this for a 
 minute or two : Plants that are crowding 
 each other up, and preventing each other grow- 
 ing, or that are not planted and are littering 
 up the house, are not in Class B ; I am not 
 sure that they get as far as to be properly in 
 Class A. 
 
 XII 
 ECONOMY OF MIND-FOKCE 
 
 TRUE education tends to make people satisfied 
 with just enough of the things in Column A, 
 and leads them to spend spare time, money, and 
 energy on things in Column B. 
 
 BLACK-BOARD. 
 A. B. 
 
 Things such that Things such that 
 no one else can I other people can 
 have the part that have the part that 
 I have. * I have.
 
 88 LOGIC OF AKITHMETIC 
 
 You know that when I want to make you 
 remember a sentence which I have written on 
 the board, I let you all say it at once. But 
 when I want to help you to understand some- 
 thing, or to see whether you understand, I allow 
 only one child to answer at a time. I will 
 
 write : repeating a thing to help me 
 to remember, in Column ? B. 
 
 I will write: showing how much I 
 know and how far I understand, in 
 
 Column ? A. 
 
 There are several things you have to learn in 
 school, besides the actual lessons in the books, 
 such as sums and geography. One of the most 
 important is to answer questions properly. You 
 must learn to think what was the exact question 
 asked ; to think what the question means ; to 
 answer it quickly, quietly, and politely ; to tell 
 the truth, the whole truth so far as you know 
 it, and nothing but the truth, about that 
 question ; to put it in plain words, and not to 
 use unnecessary and roundabout phrases. All 
 that needs practice. It is the teacher's duty to 
 give each child a share of such practice just 
 as it is the housewife's business to give every 
 child a proper amount of food. And as we said, 
 while Mary is answering, Claire cannot answer. 
 
 Therefore we put : practice in answer- 
 ing questions, in Column ? A,
 
 ECONOMY OF MIND-FORCE 89 
 
 Another thing that is necessary to learn is to 
 listen to other people's conversation, without 
 interrupting till your turn comes. If Alice is 
 answering me, and Mary is sitting quiet, can 
 Claire sit quiet too ? Yes. Then we will 
 
 write : practice in listening to con- 
 versation without interrupting till 
 my turn conies, in Column? B. 
 
 But we must not sit idle while other people 
 are discussing interesting or useful things, we 
 must learn to profit by what is said. If I am 
 teaching Alice and making her answer, and 
 Mary is listening and learning by what Alice 
 and I say, can Claire listen and learn too ? Yes. 
 
 Then we will write : learning from what 
 
 is going On, in Column ? B. 
 
 It is necessary to collect a good stock of mind- 
 pictures for our future use. While one child is 
 sitting nice and quiet making a mind-picture, 
 does that hinder the others from doing so too ? 
 No ; the quieter each one sits, and the more 
 steadily she makes her mind-picture, the better 
 every one else can do the same. We will write : 
 
 making mind-pictures, in Column ? B. 
 
 It is necessary that a child should not pass 
 up into a higher class (form, or standard) till 
 he is quite fit for it. Therefore, there must be 
 pass-examinations, to see who is fit. If one 
 child passes an examination creditably, does
 
 90 LOGIC OF ARITHMETIC 
 
 that hinder another from doing so ? No. We 
 
 will write : passing school examina- 
 tions creditably, in Column ? B. 
 
 It is necessary that Governments should know 
 who is fit to become a doctor, or a lawyer, or 
 a school-teacher, or a Civil Service clerk, or 
 a postman, because they ought not to allow 
 unfit persons to undertake responsible posts. 
 If one man shows he knows his work properly, 
 does that hinder another from showing that he 
 knows it too ? No. We will write : passing 
 
 professional pass-examinations, under 
 
 Column ? B. 
 
 But for some purposes it is right that the very 
 fittest persons who can be got should be chosen 
 for a post. It is sometimes right therefore that 
 the Government should know, not only who is 
 fit, but who is most fit. If one man is at the 
 top, can another be at the top ? No. We will 
 write : being at the top, in Column ? A. 
 
 Now let us read over carefully our two 
 columns. 
 
 (Kead out the columns.) 
 
 Now I am going to speak about something 
 which puzzles many children. You are always 
 being told that it is kind to help other people 
 and share with them, yet if you help another 
 child at an examination you are told it is 
 naughty ; and if you let another child help you
 
 ECONOMY OF MIND-FORCE 91 
 
 you are punished. Even at class, if I ask one 
 child a question, and another whispers the 
 answer to her, or writes it for her to see, I 
 stop you at once. 
 
 Let us see what all that means. 
 
 If I say to one child, ' I do not know whether 
 any dogwood grows in this neighbourhood ; will 
 you keep a look-out when you are out walking ? ' 
 Would it be right for another child to help her 
 to look for it ? Yes ; because the thing I want 
 to know is whether there is dogwood ; and the 
 thing the child wants to do for me is to find 
 the dogwood if it is there. Two pair of eyes 
 are better than one ; two children, by helping 
 each other, are more likely to succeed in doing 
 what is wanted than one alone. 
 
 But if I ask one of you a question at class for 
 instance, what are nine times eight 1 I do not 
 want to know what nine times eight are ; I know 
 that already. What I want is, first, to know 
 whether that child remembers nine times eight 
 and can say it without help ; and, secondly, to 
 give that child practice in grasping for herself 
 the question she was asked and answering it 
 for herself. If another child interferes or tries 
 to help, she does not help, she hinders the very 
 purpose for which the question was asked. 
 
 So it is at examinations. The examiner does 
 not want to know the answer to the questions
 
 92 LOGIC OF ARITHMETIC 
 
 he has asked ; he wants to know which children 
 are fit to begin the work of a higher class. If 
 you ' help ' a friend (as you call it) to answer 
 questions, you are really hindering her progress, 
 by helping to get her into a class she is not fit 
 for. You help her to be set to lessons too 
 difficult for her. You help her to have duties 
 which she cannot properly do. You help her 
 to brain-muddle and overstrain and many sorts 
 of bad things. 
 
 BLACK-BOARD. 
 
 When the object of doing a thing 
 is to get the thing done, it is right to 
 accept the help of other people. 
 
 When the object of doing a thing 
 is either to get practice in doing it, 
 or to show that I can do it, I must 
 not accept help from other people. 
 
 There is something that I should like to tell 
 the elder children just here. Little ones must 
 please sit still, and practise trying to learn 
 what they can from conversation between their 
 elders. 
 
 There are, in every subject, some parts which 
 can be learned on purpose to show that you 
 know them ; for instance, on purpose to pass 
 examinations in them. There are other parts 
 which cannot be learned so, which must be
 
 ECONOMY OF MIND-FORCE 93 
 
 learned while you are not thinking of showing 
 what you know. Now you can never really 
 thoroughly understand either of these parts 
 unless you also learn something about the 
 other. You want to know why this is so ? 
 Well, I will answer that question after you 
 have answered me a few questions. Which 
 makes a man strongest for exercise, walking 
 all day, or leaving off to take food sometimes ? 
 Which has the better nourished body, the boy 
 who sits trying to stuff down food all day, 
 or the boy who does a fair amount of running 
 about between times ? Which will put forth 
 most leaves and fruit to show, the plant that 
 has been cut from its root and now has only 
 branches, or the plant that is growing from 
 a root hidden under ground \ 
 
 I once knew a selfish, greedy little boy, who 
 would rather make himself ill than let the 
 servant (who made the pudding, and brought 
 it to him, and was going to wash up his plate) 
 have any herself. Do you think that boy 
 could learn the true use of food, and the proper 
 care of his digestion, while he thought of his 
 food in that sort of way \ You know that 
 he could not learn so. Why ? You cannot 
 explain, but you know he could not. 
 
 Well, tell me why all those things are so, and 
 then I will tell you why you understand your
 
 94 LOGIC OF ARITHMETIC 
 
 examination work best on the whole, if you do 
 some portions of work which do not tell at 
 examinations at all. It is our ' nature to/ as 
 the old rhyme says. We are made so, and we 
 cannot help ourselves. 
 
 All the great men whose names live for 
 centuries after their death, did a great deal 
 of work which the world never heard of. Men 
 who will do nothing except that part which 
 they can show, never do anything which lasts. 
 
 A minute or two of silence. 
 
 Now all of you read aloud what is on the 
 black-board. Copy it into your books. Read 
 it again. Shut your eyes. Try if you can say 
 it. Learn it by heart (in preparation), and say it 
 to me next time. 
 
 I am not going to give you a mind-picture 
 to-day. Shut your eyes and sit at ease, and 
 try to put together in your minds the different 
 things we have been talking about. 
 
 Five minutes' silence. 
 
 XIII 
 
 EXERCISE IN RELEVANCE; INTRO- 
 DUCING IDEA OF PROBABILITY 
 
 SUPPOSE a pony is shut up in a field alone. 
 How many heads will be in the field ? How
 
 EXEKCISE IN RELEVANCE 
 
 95 
 
 many legs ? Eyes ? Hoofs ? Tails ? Hands ? 
 How many pieces of mischief do you think will 
 be done in an hour ? Suppose another pony is 
 turned in. (Repeat same questions.) Twenty ponies 
 come in. (Repeat.) A hen comes in. (Repeat.) A boy 
 comes in. (Repeat.) Another boy comes in, the 
 second boy has a monkey. The master comes 
 
 in. (Repeat the questions each time.) Now let US put 
 
 all that on the black-board. 
 
 
 
 03 
 
 a 
 
 <3 
 
 
 
 03 
 
 bD 
 
 03 
 
 
 
 o 
 o 
 
 03 
 
 i I 
 
 "3 
 
 
 
 B 
 
 il 
 
 H 
 
 B 
 
 EH 
 
 1st 
 
 Pony 
 
 1 
 
 4 
 
 2 
 
 4 
 
 1 
 
 2nd 
 
 Pony 
 
 1 
 
 4 
 
 2 
 
 4 
 
 1 
 
 20 
 
 Ponies 
 
 20 
 
 80 
 
 40 
 
 80 
 
 20 
 
 
 Hen 
 
 1 
 
 2 
 
 2 
 
 
 
 1 
 
 
 Boy 
 
 1 
 
 2 
 
 2 
 
 
 
 
 
 2nd 
 
 Boy 
 
 1 
 
 2 
 
 2 
 
 
 
 
 
 Monkey 14201 
 Master 12200 
 
 wS^ 1 
 
 ^S 
 
 *2* 
 
 .~j M iH 
 
 HPHffl 
 
 
 
 
 
 2 
 2 
 4 
 2 
 
 (It is important that all zeros in this exercise be 
 filled in. For though zero is not anything, negation is 
 a mental act, the cognition of a fact. It is important 
 that children should learn to cognize the fact and deal 
 rightly w th it, at an early age. 
 
 The seventh column is to be filled in, in each case, 
 according to the decision of the class after discussion of 
 the probabilities of that case. 
 
 Exercises such as the above can be varied indefinitely.
 
 96 LOGIC OF ARITHMETIC 
 
 They are of great importance ; they fix attention on the 
 questions of relevance, and of cumulation or neutraliza- 
 tion (e. g. the advent of the master adds to the number 
 of legs, is irrelevant to questions about hoofs or tails, 
 and diminishes the amount of mischief). They call 
 attention to the line of demarcation between certain, 
 calculable knowledge (number of heads, &c.), and pro- 
 bable knowledge, or that which is contingent on facts 
 not yet known, or not ascertainable (the character of 
 the boys). They also call attention to the fact that an 
 additional element may either add its own bulk to that 
 previously existing (heads, tails), or may raise that 
 previously existing to a higher power. A boy once 
 suggested to me that, if two boys were in a field, they 
 would not only do mischief themselves but would 
 start the ponies trying to break fences ; and added : 
 ' One boy alone might not think of it, but two together 
 would be bound to.' I affirm that the boy's mathe- 
 matical insight must have been increased by his having 
 made such a suggestion. All such ideas are better 
 suggested by examples of the above kind, where no 
 intellectual work is involved in the question itself and 
 the mind is free for new conceptions to surge up of 
 themselves from the ' abysmal depths of consciousness,' 
 than by explanations thrust in ab extra while the child's 
 mind is occupied in struggling with the difficulties of 
 a problem in Arithmetic or Algebra. 
 
 For town children, playground may be substituted 
 for field, dogs and cats for ponies). 
 
 MlND-PlCTURE. 
 
 Shut your eyes, &c. See a field ; see ponies, two 
 boys, and monkey. Monkey jumps on the back of 
 a pony and frightens it. Boys shout and scamper. 
 Ponies become wild and break fences. Hen runs 
 away scared and falls into pond, &c., &c., &c. 
 
 Master comes in and restores order.
 
 XIV 
 EXERCISE ON ZERO 
 
 SHUT eyes, &c. Make a mmd-picture : Me 
 lifting the chalk to the black-board. I make 
 one stroke and then put my hand down. I do 
 this action three times ; how many strokes will 
 be on the board ? 
 
 If, instead of making one stroke on the board, 
 I made two and put my hand down ; how many 
 strokes would be on the board when I had done 
 the action once ? Twice ? Three times ? Four 
 times ? Before I had done it at all ? 
 
 Open eyes, sit up. Shut eyes, &c. Make a 
 mind-picture : A clean black-board, me holding 
 the chalk and then putting it down, without 
 touching the board ; what would be on the 
 board ? Nothing. Now make another picture : 
 Me making a stroke. Now I rub the stroke 
 out. What is on the board that you now see 
 in your mind ? Nothing. So if I do nothing, 
 or if I make a stroke and rub it out, the result 
 is the same as far as the board is concerned. 
 Were the two ways of getting it the same ? No. 
 
 That was a mind-picture black-board. Now 
 open eyes and look at the real one. Here is 
 a quite clean board; I make a stroke ; I rub it 
 out. Is it really quite as clean as when it has
 
 98 LOGIC OF ARITHMETIC 
 
 been cleaned on purpose for class ? No. What 
 do you see on it ? A smudge. 
 
 Suppose I lifted the chalk to the board, and 
 made a dot and no stroke at all ; how many 
 strokes would tie on the board when I had 
 done the action three times ? Four times ? Six 
 times ? Nine times ? Once ? Before I had 
 done it ? 
 
 One stroke no time ? 
 
 Two strokes no time ? 
 
 No stroke three times ? 
 
 No stroke one time ? 
 
 No stroke no time ? 
 
 XV 
 
 The idea of fraction is readily introduced by accus- 
 toming children, when the concept ' child ' is taken as 
 arithmetical unit, to think of that unit as divided into 
 halves, each side being a half; or the unit may be 
 a monkey, each hand representing a quarter. For 
 instance, a common exercise in Arithmetic is the 
 following : 
 
 If each boy is to have two apples, how many 
 shall I want for two boys, three boys, four boys ? 
 &c. 
 
 The problem might be stated thus : 
 
 If each boy is to have an apple in each hand, 
 how many will be given to two boys ? Three
 
 i = oo 99 
 
 boys ? &c. One boy ? Half a boy ? If each 
 monkey is to have two nuts in each hand, how 
 many will three monkeys have ? Two monkeys ? 
 One monkey ? Half a monkey ? A quarter of 
 a monkey ? Three quarters of a monkey ? 
 
 Such questions as the above may sound foolish, 
 because teachers are not yet accustomed to see their 
 importance in the development of arithmetical faculty. 
 Let us once learn to think of the human mind as 
 intended to build up the material sciences round an 
 organic skeleton made of acts of positing the unit, 
 negation, fraction and reconstruction of the broken unit, 
 and our estimate of the relative values of many things 
 in education will undergo rapid change. Every ele- 
 mentary exercise in number should be applied to the 
 concepts unity, zero, one-half, one-quarter, &c. In con- 
 structing his tables, the child should be asked 'Twice 
 one ? ' ' Twice nought ? ' and ' Twice a half ? ' as often 
 as he is asked ' Twice three ? ' 
 
 Another useful exercise is the following : 
 
 I have twenty apples on a plate. Boys pass 
 through the room, each coming up for his 
 allowance. If I give one to each boy, how 
 many boys can J serve before my stock is 
 exhausted? If I give one to each half-boy? 
 If I give two to each boy ? If I give two to 
 each half-boy ? If I give four, five, ten, twenty, 
 to each boy ? 
 
 Then come down the series. 
 
 If I give ten, five, four, two, one, to each boy ? 
 If I give half an apple to each boy ? Half an apple 
 to each half-boy, a quarter-apple to each boy ? 
 
 G 2
 
 100 LOGIC OF AEITHMETIC 
 
 The questions should be kept playing round the 
 fraction of apple and fraction of child, till the children 
 see quite clearly through the whole process, and are 
 familiar with the conception that, at the point where 
 the share of each child is one apple, i. e. where the 
 unitary concept of the class 'children' coincides with the 
 unitary concept of the class ' apples/ a crisis is reached ; 
 something happens ; at that point the number of boys 
 is equal to the number of apples ; on one side of it the 
 number of boys is less than that of apples ; on the 
 other side it is greater. 
 
 The series of questions should be repeated with ten 
 apples in the plate ; then with forty apples, then the 
 three series (ten, twenty, forty apples) should be inter- 
 mixed. And as soon as it can be done without 
 confusion, this question should be asked : 
 
 If I give none to each boy, how many can 
 pass through the room before my stock is 
 exhausted ? 
 
 We thus introduce the true mathematical conception 
 of l infinity,' free from all that is hazy or doubtful, or 
 which makes a strain on the nerves or imagination ; we 
 call attention to the simple fact that no number of boys 
 passing through the room affects the remaining stock of 
 apples in any way ; that, when the share of each boy 
 passes from fraction of apple to negation of apple, the 
 relevance of the number of apples on the plate to the 
 number of boys is suddenly broken. 
 
 Then ask : 
 
 If the share of each half-boy is none, what is 
 the share of each boy ? And how many boys 
 can pass through before the supply is exhausted? 
 
 Exercises of this kind should be strictly limited to 
 such simple fractions as are indicated by the two hands 
 of the child and the four hands of the monkey. No
 
 EQUIVALENT FRACTIONS 101 
 
 attempt should be made to use them to impart any 
 premature information about the branch of Arithmetic 
 commonly called ' fractions.' 
 
 The object of these examples is not to teach 'fractions,' 
 but to supply elementary Arithmetic with that normal 
 thinking-fibre which is, for a human creature, essential 
 to clear ideas about anything, and which is generated, 
 as above stated, by the mutual inter-action of the ideas 
 of unit, negation and fraction. 
 
 xvr 
 
 EQUIVALENT FRACTIONS 
 
 SIT at ease, shut eyes. Make a mind- 
 picture. 
 
 Each one of you is to think of some grown- 
 up person that he likes. I am going to call 
 the person ' Mother ' ; but you may think of 
 Father or Auntie or any one else you like. 
 Now then. 
 
 There is a plate on the table, and a knife, 
 and an apple. Mother cuts the apple into two 
 pieces the same size. We call those pieces ? 
 Halves. Now Mother cuts each of those pieces 
 into two. We call those pieces halves of 
 balves, or ? Quarters. Mother eats one of the 
 quarters, and says you may have the rest. 
 How many quarters are left for you ? Three. 
 Two of them are the two halves of one half-
 
 102 LOGIC OF AKITHMETIC 
 
 apple ; the other quarter is one of the halves 
 of the other half-apple. 
 
 Now, what is left for you when some one 
 takes one quarter of an apple and leaves you 
 the rest ? Three quarters. 
 Open eyes, sit up. 
 
 
 BLACK-BOARD. 
 
 If a person takes one quarter of 
 anything and leaves me the rest, 
 what I get is three quarters. 
 
 Is that right ? Do you quite understand it ? 
 Now write it in your table-books. Bead it 
 out. 
 
 Now sit at ease again. Shut eyes. Make a 
 picture. Me with four of you children. Each 
 of you is to think of himself and three others ; 
 any three others you like. 
 
 There is a table in front of me, and three 
 apples on a plate. I am going to divide those 
 three apples among the four children. If I give 
 one of the apples to each of your friends, what 
 will be left for you ? Nothing. Well, that 
 won't do, will it ? 
 
 We must try a better way than that. I cut 
 an apple into quarters. How many pieces will 
 there be ? Four. I give one piece to each of 
 you ; what sized piece will each child have ? 
 A quarter. What will be left of that apple on
 
 the plate ? Nothing. Then I take the second 
 apple and give each child a quarter of it. So 
 now each child will have ? Two quarters of 
 apple ; and there will be left on the plate one 
 apple and no piece. Now 1 cut up the third 
 apple and give each child a quarter. And now 
 there is left on the plate ? Nothing. So we 
 have shared up the three apples among four 
 children. And the children have all got just 
 the same quantity. What is each child's 
 share? Three quarters of apple. Are you 
 sure you understand ? Shall we go over it all 
 again ? 
 
 (Repeat the paragraph if necessary.) 
 
 Open eyes. Sit up. 
 
 BLACK-BOARD. 
 
 If three apples are shared between 
 four children, the share of each is 
 three quarters of an apple. 
 
 Copy into your table-books under the 
 sentence you wrote last. Read both sentences 
 over. You see that if three things are shared 
 between four children, each one's share is the 
 same size as it would have been if there were 
 only one thing, and some one took away one 
 quarter and left him all the rest. What is 
 alike in the two cases ? Your share. Only 
 that. Everything else is different. When we
 
 104 LOGIC OF ARITHMETIC 
 
 are talking short, we say that ' A quarter of 
 three is equal to, or the same as, three quarters 
 of one/ And we write it this way : 
 
 BLACK-BOARD. 
 |ofl = iof3. 
 
 Copy that in your books. 
 
 When this exercise is gone over; another day, the 
 following should be given as a variation : 
 
 I cut up the three apples into quarters. 
 How many pieces are now on the plate ? 
 Twelve. And there are four children. How 
 many pieces can I give to each child ? Three. 
 And each piece is ? A quarter of apple. So 
 each child's share is ? Three quarters of apple. 
 
 We now have this : 
 
 BLACK-BOARD. 
 
 Three quarters of one apple. 
 A quarter of each of three apples. 
 A fourth part of twelve quarter- 
 apples. 
 All are the same size. 
 
 Copy that into your books. 
 
 If four children are to share one apple, how 
 many quarter-apples does each child get ? 
 
 If two children are to share two apples, how 
 many quarter-apples does each get ?
 
 EQUIVALENT FRACTIONS 105 
 
 If two children are to share three apples ? 
 four apples ? 
 
 five apples? 
 six apples ? 
 
 no apple. 
 
 Before children are introduced to the practice of 
 1 cancelling ' out, in proportion or fraction sums, it 
 would be desirable to introduce them to the idea under- 
 lying such processes in some such way as this : 
 
 Suppose I have four apples to share among 
 eight boys, there are two ways in which I can 
 do it. I can cut up all the apples into halves, 
 and give one piece to each boy, or I can give 
 a whole apple to each two boys, and let one of 
 each pair divide the apple belonging to that 
 pair. I haver to-- begin with, eight boys and 
 four whole apples?, on as. plate, I shall have at 
 the end eight boys each holding a half-apple. 
 One of the ways of doing, this is to make first, 
 four groups, each consisting of two boys and an 
 apple. The other way is to cut up all the 
 apples into halves, and give one to each boy. 
 Whichever way I set about it, I get at the last 
 eight boys each holding half an apple. 
 
 It is desirable to avoid suggesting to the child's 
 imagination, at any stage of the process, that two boys 
 and an apple is supposed to be in any sense the equiva- 
 lent of either eight boys and four apples, or one boy 
 and half an apple. Therefore make a clear mind-
 
 106 LOGIC OF ARITHMETIC 
 
 picture of the four groups, each consisting of two boys, 
 one of whom is engaged in cutting an apple into 
 halves. 
 
 Enter in books (pupil's own part) : 
 
 ' If four apples are to be shared among eight 
 boys, the share of each boy is the same, whether 
 one man cuts up all the apples, or whether four 
 people each cut one apple in halves/ 
 
 All lessons concerning equivalence of fractions and 
 the dividing of numerator and denominator by the same 
 factor should be linked in the children's minds with 
 this fundamental conception of a set of molecules or 
 groups, each consisting of so many boy-units and so 
 many apple-units ; it should be shown that what is can- 
 celled out is the number of such groups ; and that the 
 number of the groups is rejected for the present, not as 
 being unimportant in itself, but as being irrelevant to 
 the immediate question on hand. 
 
 XVII 
 GREATEST COMMON MEASURE 
 
 THIS is the picture of a ceiling of a room. 
 Suppose a painter wishes to ornament the edge 
 where the ceiling meets the walls, with a little 
 simple pattern that will fit exactly into the 
 lengths of all the sides, without needing any 
 extra corner-ornaments. Suppose he asks us 
 to find out for him the length of the longest
 
 GREATEST COMMON MEASURE 107 
 
 pattern that will fit in exactly. How shall we 
 set about it ? 
 
 The pattern is to be repeated over and over 
 again. One repeat must end and another begin 
 at A ; one must end and another begin at B ; 
 the same at C ; and also at D. 
 
 A-i ,-B 
 
 I- 
 
 L- 
 H- 
 
 E- 
 
 C- 
 
 -D 
 
 Let us measure off" the length A B from A 
 towards C. We will call the end of that 
 measure E. Some number of repeats will go 
 exactly into A B. We do not yet know what
 
 108 LOGIC OF AKITHMETIC 
 
 number of repeats ; but some number is to go 
 into A B exactly. The same number will go 
 exactly into A E. E therefore will be one of 
 the points where one pattern ends and another 
 begins. Some number of patterns will fit 
 exactly into C E. We must find a length that 
 will fit some number of times into A E, and some 
 other number of times into E C. For the present 
 we can leave A B alone ; any pattern that will 
 fit exactly into A E can be ' copied exactly into 
 A B, because A B is the same length as A E. So 
 now we are going to attend to finding a pattern 
 that will fit exactly into A E and into E C. 
 
 Let us measure off the length of E C from E 
 towards A ; we will call the end of that 
 measurement F. So F is another point where 
 a pattern begins and ends. Now let us measure 
 off the length C E again, from F towards A ; 
 and call the end of that measurement G. Then 
 G is a point where one pattern ends and another 
 begins. We measure off the same length again 
 and call the end H ; and again, and call the 
 end I ; H and I are points where one pattern 
 ends and another begins. 
 
 If we try to measure the length C E from I 
 towards A, we cannot do it ; there is not room, 
 I A is not long enough. 
 
 Any pattern that will fit into I H will fit also 
 into H G, and G F and F E, and E C ; because
 
 GEEATEST COMMON MEASURE 109 
 
 all those bits are the same length. So, for the 
 present, we will attend only to finding a length 
 which can repeat some number of times in I A, 
 and some other number of times in I H. Let 
 us measure off the length A I from I towards 
 H. We will call the end of that measurement 
 L. L, then, is one of the points where a pattern 
 ends and another begins. Now we will measure 
 off the same length, A I from L towards H. 
 Ah ! now we find that we have come exactly 
 back to H. So that last measurement gives us 
 no new points. 
 
 Now let us see what we have got. 
 
 A I, I L, and L H, are all the same length. 
 The length of A I has gone twice into I H, 
 therefore it will go into H G twice ; and into 
 G F, and F E, and E C twice each. So it fits 
 exactly into A C, and repeats in A C ? 1 1 times, 
 Will it fit exactly into A B ? Yes, it fits exactly 
 into A E, and A B is the same length as A E. 
 
 Now we will work this off as a sum on another 
 black-board. We will keep this one up, so that 
 we can see what we have been doing. 
 
 (If there is no second black-board, the children should 
 be told to work the sum off on paper or slates. On no 
 account must the diagram be rubbed out till the sum 
 has been worked.) 
 
 We ticked off the length A B from A to- 
 wards C ; that left E C over. If A B is 18
 
 110 LOGIC OF ARITHMETIC 
 
 units and A C is 22 units, how many units is 
 EC? Four. How do you know that ? Because 
 18 from 22 = 4. We will put that down as 
 a subtraction sum. 
 
 22 = AC 
 
 18 = A E = A B 
 
 ~4 = EC 
 
 Then we ticked off 4 from the 18. 
 18 = AE 
 4 =EP 
 
 14 = AF 
 4 = FG 
 
 10 = AG 
 4 = GH 
 
 6 = AH 
 4 = HI 
 
 ~2 = AI 
 
 Then we ticked off 
 
 2 = A I = I L from I H 
 
 and this left L H without any remainder. 
 
 Then we saw that the length A I would 
 measure off exactly, without leaving a remainder, 
 into all the lengths H G, G F, F E, E C, because 
 they are all equal to I H. 
 
 Now we will go over that in a different way.
 
 GKEATEST COMMON MEASURE 111 
 
 How often will 1 8 units go into 22 units ? 
 Once, leaving a remainder 4. 
 
 18) 22 (1 
 18 
 
 Remainder, used as ) 4) 18(4 
 
 new divisor i 16 
 
 Ditto 2) 4 (2 
 
 4 
 
 Remainder 
 
 Now, will A I fit exactly into A B ? What 
 is A B equal to ? A E. Well, if the pattern 
 fits into A E, will it fit into A B ? And into 
 CD? Yes, because C D is the same length as 
 AB. 
 
 And will it fit into B D ? Yes. Because ? 
 B D is the same length as A C. 
 
 You remember that the question we started 
 with was : What is the length of the longest 
 pattern that will fit exactly into all the lengths 
 of the ceiling edge ? We have done that. We 
 have done it in three different ways, and we 
 know the longest pattern is 2 units long. We 
 do not know yet how often the pattern will 
 repeat in the room. 
 
 How many times will the length A I fit into 
 the length A C ?
 
 112 LOGIC OF ARITHMETIC 
 
 In A I ? 1 
 
 In I L ? 1 
 
 InLH? 1 
 
 In H G? 2 
 
 InGF? 2 
 
 InFE? 2 
 
 In EC? 2 
 
 11 
 
 Eleven times altogether. 
 
 How many times in A B ? 9 
 
 InBD = AC? 11 
 
 InCD=AB? 9 
 
 Answer 40 
 
 The above exercise repeated occasionally prepares 
 children for understanding G.C.M. ; especially if they 
 have learned Division as a shortened method for getting 
 the result of a series of similar subtractions. It may 
 also be given with great advantage to older children 
 who have already been badly taught G.C.M. and become 
 foggy. In their case it should be followed up by 
 a further explanation. Thus : 
 
 ' Why could you not understand the rule 
 before ? I think it was partly for this reason : 
 
 You have been accustomed to think that the 
 most important part of the answer to a Division 
 sum is the quotient ; the remainder matters 
 much less. And when you began to do G.C.M. 
 you were told to do a Division sum ; and no
 
 GREATEST COMMON MEASURE 113 
 
 notice was taken of the quotient ; you were 
 made to go on as if the remainder were the 
 important thing, and the quotient of no con- 
 sequence. And that seemed to you somehow 
 dishonest and not quite true ; and it puzzled 
 you and prevented your taking in what was 
 really going on. But now you can see. When 
 the painter comes to paint the ceiling border, 
 he must provide paint enough to go all round 
 the room. It would not be honest if he only 
 painted from A to H. If he sends his assistant 
 to block out the points where the patterns must 
 begin and end, the assistant must take the 
 trouble to go all round and divide up the whole 
 edge. If we wanted to find out how many 
 points he will have to mark off, we should have 
 to take account of the quotients and the re- 
 mainders. Missing a quotient would put us 
 wrong, even more than missing its remainder. 
 
 But we did not start to paint the room, nor 
 to block out the edge, nor to find out how many 
 times the pattern would be repeated. We were 
 set to find out, first, what length of pattern 
 would fit in. For that purpose, you see, the 
 remainders matter, and the quotients do not. 
 
 Now do you see what mistake you made when 
 G.C.M. was being explained to you before ? 
 
 Mistakes of just that kind are made about 
 many things in life. I am going to tell you 
 
 BOOLE H
 
 114 LOGIC OF AEITHMETIC 
 
 something that may help you not to make 
 them in future. 
 
 Sit at ease. Shut eyes. Here is a mind- 
 picture. 
 
 There is a village. It has water-works and 
 a reservoir. At the other end of the village 
 there are a few houses which have a deep well 
 of their own. A doctor notices that the people 
 in those houses do not have a kind of illness 
 which is common in the rest of the village. He 
 tells the corporation that he suspects the water 
 of the reservoir is not wholesome somehow, but 
 he does not know exactly what is the matter 
 with it. There is in the village a man who is 
 fond of science, and knows a good deal about it. 
 The corporation ask this man to try to help 
 them to get the water right. He goes to the 
 reservoir. What do you see him doing ? Does 
 he slash in pounds and pounds of stuff at 
 random, to try to cure the water before he 
 knows what is wrong with it ? No. He has 
 a bottle with him. The bottle holds about a 
 pint. It is perfectly clean, and has a nice clean 
 stopper. He takes a good long look round the 
 reservoir. He then dips the bottle in, fills it, 
 and stoppers it up. He carries it home. He 
 sits down in his study. He puts a drop of the 
 water out of his bottle onto a glass slide, puts 
 the slide under his microscope, and takes a long,
 
 GKEATEST COMMON MEASURE 115 
 
 long, careful look at it. He turns a little wheel 
 round, and looks again and again. 
 
 Then he- puts some water put of his bottle 
 into a small glass, and adds a little of some 
 stuff, and looks, and looks. Then he puts some 
 more in another little glass, and adds a bit of 
 some other stuff, and looks again. 
 
 In the middle of all this, two neighbours 
 come in. One says : ' How can you waste 
 your time amusing yourself with your micro- 
 scope and all these little glasses that are more 
 fit for a doll's house than for a grown man, 
 when your neighbours are suffering and dying 
 because the water is wrong ? Why don't you 
 do something to help us ? ' 
 
 The other neighbour says : ' I heard you 
 were employed by the corporation to help the 
 town to get the water pure. Why aren't you 
 at the reservoir doing it ? Do you think it is 
 honest to use only a grain of your stuff and 
 purify only a pint- bottle full of water, when you 
 are engaged to cure the reservoir ? You will 
 want a hundredweight of disinfectant, not that 
 tiny parcel of it.' 
 
 The scientific man answers : ' When we know 
 what is the matter with the water, the corpora- 
 tion will send the right people to cure the evil, 
 and they must take the right amount of stuff, 
 and do the right kind of cleansing. If they 
 
 H 2
 
 116 LOGIC OF ARITHMETIC 
 
 attempt to do with less, that will be dishonest. 
 But at present we do not know what is the 
 matter, nor what treatment the water needs. 
 I have been asked to find out these things, and 
 am doing my best to find out. When you are 
 ignorant, experiment on a small scale at first ; 
 that is the way to learn.' 
 
 Have you taken in that picture ? Remember 
 it sometimes, when you are tempted to think 
 that settling something big, in a hurry, before 
 you understand what you are about, will do 
 more good than careful, patient study on a 
 small scale. 
 
 (All that is on the two black-boards, to be ultimately 
 entered in form-books.)
 
 XVIII 
 STANDARD WEIGHTS AND MEASURES 
 
 You remember I told you that in teaching 
 a baby to count, it is best to teach him to pack 
 up each ten of his counters or pebbles in a box 
 or heap to itself. You will find many things 
 come easier in your sums, if you take notice 
 that a great deal of what you are learning is to 
 pack things up in your minds, so that they 
 shall be handy and convenient. It would have 
 been inconvenient and troublesome for the 
 shepherds of long ago to make a notch for each 
 sheep that passed through the gateway into 
 the fence ; they packed up the number of sheep 
 into tens, and made one stroke put in the 
 tens' place stand for ten sheep. Later on, 
 people packed up tens, and made one stroke in 
 the hundreds' place do for ten tens. 
 
 In the same way our pennies are packed up 
 into shillings. We need not carry about as 
 many pennies as we shall want to spend ; if we 
 want to buy twelve penny eggs, we can pay 
 one little shilling for the lot ; because the value 
 of twelve pennies is packed up into a shilling. 
 We do not need to weigh out sixteen 
 separate ounces of lard to make the dough for
 
 118 LOGIC OF ARITHMETIC 
 
 sixteen small cakes ; sixteen ounce weights are 
 packed into one pound weight. 
 
 You will understand sums better if you will 
 remember that much of what you hear about 
 in the Arithmetic-class is a sort of packing of 
 things in your mind or memory. 
 
 Things have to be packed in parcels or 
 bundles of different shapes and sizes, according 
 to the nature and use of the things themselves. 
 Let us think of a few different kinds of mind- 
 packing that we know of. 
 
 Ordinary numbers are packed in ? Tens. And 
 then into? Tens of tens. That plan was started 
 because ? Savages found they had ten fingers to 
 count on, before they had names for numbers or 
 knew how to write signs for numbers. 
 
 Some things are counted in dozens, such as ? 
 Eggs, buttons. Why that is done I really do 
 not know ; I have heard that there was once 
 a set of people who had a thumb and five 
 fingers on each hand ; that gave them ? Twelve 
 counters on their two hands. Perhaps these 
 men invented counting by dozens, but that 
 I cannot say. 
 
 Weights go quite differently at least in 
 England. They are arranged so that each one 
 is twice the weight of the one next below, or, 
 as it is called, they are arranged in powers of 2. 
 For some purposes this is a convenient arrange-
 
 WEIGHTS AND MEASURES 119 
 
 ment, and it is a very interesting one to learn 
 about. 
 
 Things to drink are arranged in something of 
 the same kind of way ; in pints and quarts and 
 gallons. A pint, I believe, is the size that 
 people used to like for their tumblers to drink 
 out of. People in Bavaria still use pint tum- 
 blers ; it looks very funny to see the children 
 lifting great pint pots to their mouths. We in 
 England have taken to have our tumblers, only 
 half that size ; we call a tumbler, half a pint ; 
 and we call as much as two tumblers hold, 
 a pint ; and what four will hold, a quart ; and 
 so on. 
 
 Can you tell me how inches are arranged? 
 In twelves. Twelve inches are called a ? 
 Foot ; though very few men have feet quite 
 twelve inches long. Three twelve-inch feet are 
 called a ? Yard. Four inches and a half are 
 the length of an average-sized woman's middle 
 finger; some of you have seen your mothers 
 measuring on their fingers. How many ' fingers ' 
 make a yard ? Eight. 
 
 Some women measure tape or cloth by hold- 
 ing one end of the tape to their noses and then 
 stretching their hands back ; that measures off 
 a yard. In old books we read of a cubit (that 
 is, the length from the tip of the fingers to 
 the elbow) ; and a hands-breadth. All these
 
 120 LOGIC OF ARITHMETIC 
 
 measurements are rough and not exact, till 
 some Government gives what are called standard 
 weights and measures. This saves a good deal 
 of disagreement and quarrelling. People in 
 shops are made to use weights and measures 
 that have been standardized, that is to say, 
 stamped by people appointed by the Govern- 
 ment ; so that customers can know for certain 
 whether they are getting a full pint, or pound, 
 or yard. 
 
 Now when a Government is going to stan- 
 dardize weights or measures, a good deal of 
 thinking has to be done about which are the 
 exact sizes that will pack up into each other 
 most conveniently. You know what I mean by 
 packing, in sums ? I do not mean packing 
 actual things in trunks ; I mean packing ten 
 ones into a one in the tens' place ; and the 
 value of twelve pence in one shilling. Napo- 
 leon, more than a hundred years ago, made the 
 French people pack all weights and measures 
 in tens. The way they measure cloth and 
 tapes and roads is this : Ten centimetres make 
 a decimetre ; ten decimetres make a metre ; 
 ten metres make a dekametre, and so on. 
 Weights go : Ten centigrammes make a deci- 
 gramme ; ten decigrammes a gramme, and so 
 on. That is what is called the decimal arrange- 
 ment. "We in England have arrangements that
 
 WEIGHTS AND MEASUKES 121 
 
 are not so regular ; sixteen ounces make a 
 pound ; fourteen pounds make a stone ; twelve 
 inches make a foot ; three feet make a yard ; and 
 so on. I do not think that either you or I know 
 enough about the matter to be able to judge 
 which plan is best on the whole. Everything 
 costs something in this world ; whatever is ar- 
 ranged, something good and nice and interesting 
 will have to be sacrificed to carry it out. The 
 thing of most consequence in the matter is that 
 we should learn to understand other people, 
 and to speak so that they can understand us. 
 We ought to know how they weigh and 
 measure things, and what they mean when 
 they say a pound or a yard ; what we are to 
 expect them to give us when we ask for a yard 
 or pound. 
 
 If we keep shops by-and-by, we ought to 
 know what we are to give each customer. 
 This is partly why children go to school : to 
 make them understand other people. People 
 used to measure things by ' rule of thumb,' as 
 it is called ; every household taking its measures 
 from the length of the father's thumb or the 
 mother's fingers ; or the tribe took its measures 
 from the length of the chiefs foot. Then when 
 they sold to each other, there were quarrels ; 
 because people's feet or fingers did not measure 
 the same length. In civilized countries, people
 
 122 LOGIC OF AEITHMETIC 
 
 must have standard weights, and all learn to 
 use the same ; so whatever the English Govern- 
 ment arrange about weights and measures by- 
 and-by, I hope you will be ready to learn it 
 quickly and use it good-temperedly ; and remem- 
 ber that understanding what your neighbours 
 mean and do is of far more consequence than 
 anything you may happen to think, one way or 
 other, about decimal coinage or decimal weights 
 and measures. 
 
 And, in order that you may be ready to learn 
 new measures easily and use them cheerfully, 
 you ought to understand a good deal more than 
 most of you do about the different ways in 
 which such things pack up, as we called it. So 
 one day soon we wiU have a lesson on packing 
 quantities in our sums. 
 
 XIX 
 
 WHAT CAN BE SETTLED BY 
 HUMAN LAW 
 
 On Table : An ounce weight, a pound weight ; 
 a yard tape marked off in inches ; a metre ruler marked 
 off in centimetres. 
 
 WHEN you see a dozen eggs, or buttons, or 
 stamps, you see the whole twelve in the dozen, 
 and can count them separately.
 
 HUMAN LAW 123 
 
 But when you see a shilling, do you see the 
 twelve pennies ? No ; the pennies are not there 
 at all, only the cash value of them is in the 
 shilling. 
 
 Here is a pound weight. A pound is? 
 Sixteen ounces. Can you see the ounces sepa- 
 rate ? No. If I took this to an ironworker's, 
 is there anything he could do to make it show 
 its sixteen ounce weights ? Yes, he could melt it 
 down, divide it into ounces, and stamp each ' one 
 ounce'; then each would be a proper ounce 
 weight, just like this one. If I had sixteen like 
 this one, could they be made into a pound 
 weight ? Yes ; the ironworker could melt them 
 all down together, and make them into a one 
 pound weight like this one. 
 
 Is there anything that any one could do to 
 a shilling to make it into twelve pennies ? No. 
 Is there any way that any man could make 
 a shilling out of twelve pennies ? No. 
 
 So you see there are different sorts of packing 
 in our sums. A dozen buttons is twelve buttons ; 
 you can see them and count them separately as 
 they lie on their card. 
 
 A pound weight can be made into sixteen 
 ounces by melting down and re-stamping ; and 
 meantime it already weighs the same as sixteen 
 ounces, and contains the same amount of the 
 $ame metal.
 
 124 LOGIC OF ARITHMETIC 
 
 A shilling could not be made into twelve 
 pennies, and does not weigh them down, and is 
 not the same metal as pennies. But the value 
 of twelve pennies is packed into a little bit of 
 a dearer kind of metal. 
 
 What is packed into this yard measure ? 
 Three feet. What is packed into a foot ? 
 Twelve inches. So thirty-six inches are packed 
 in a yard measure. Here is a French metre 
 measure ; what is packed into that ? A hundred 
 centi-metres. 
 
 What are these yard and metre measures 
 most like : dozens, shillings, or weights ? They 
 are like dozens, in so far that we can see the 
 inches as they lie along the yard. 
 
 Governments can alter the length of the 
 divisions of a yard or metre, in a way that no 
 Government could alter the size of an egg. Eggs 
 grow, inches are only marked off by men. 
 
 Men might agree to sell eggs in tens instead 
 of dozens ; but they could not alter the size that 
 the eggs grow by merely marking off differently. 
 
 What are packed into a year ? Months, weeks, 
 days. How many days are packed into a year ? 
 Three hundred and sixty-five. Who arranged 
 that ? Are there the same number of days in 
 a year in France as here ? Could the Govern- 
 ment alter that ? What makes the length of 
 a day ? The sun rising and setting, as we call
 
 HUMAN LAW 125 
 
 it ; which means, you know, that the earth 
 turns us to the sun and then away from the 
 sun. What makes the year ? Something else 
 that happens between the sun and the earth. 
 The change from day to night and back to day 
 happens three hundred and sixty-five times for 
 once that the other changes happen, the ones 
 that make a year. 
 
 Who arranged that ? Was it the king ? Was 
 it the Parliament 1 Was it Napoleon or the 
 French Government ? Do you think that if all 
 these great people agree together and decided 
 to alter the number of days in a year, they could 
 do it "? No. Long ago a number of learned 
 men spent time on finding out how many days 
 there are in a year. It took them a very long 
 time to do, but they found out at last. But as 
 for altering the number, oh, no ! no one has 
 ever been able to do that. 
 
 You see how many different things had to be 
 thought about in arranging the tables in your 
 sum-books. The number of fingers we have; 
 the most convenient way to divide lengths ; the 
 most comfortable-sized tumbler to hold ; the 
 most convenient coins to carry about ; the length 
 of days and years ; all sorts of things have been 
 taken into account at different times. So the 
 least we can do is to take things as they come, 
 make the best of them, and learn to understand
 
 126 LOGIC OF ARITHMETIC 
 
 Nature, and each other's needs and wishes ; and 
 to fit in amongst it all as best we can. 
 
 Now sit at ease, shut eyes, and make mind- 
 pictures. 
 
 A king sitting in a Parliament House. He 
 says : ' The length of my thumb-joint shall be 
 called an inch ; and thirty-six inches shall be 
 called a yard.' London shopkeepers now go to 
 an office at the Guildhall, and take tapes to be 
 marked in inches. 
 
 A French general in a cocked hat, with a 
 uniform and sword. He says : ' A metre shall 
 be divided into a hundred centimetres.' French 
 shopkeepers go to an office in Paris, and get their 
 metre tapes measured. 
 
 Another picture. Now the king and the 
 general sit waiting. Learned men come and 
 say : ' There are three hundred and sixty-five 
 days in a year.' The king and the general bow 
 their heads. The almanack-makers write down 
 what the learned men said 1 . 
 
 1 There are 365 days and a few hours in a year. But 
 it is not advisable to enter on details in the present 
 lesson.
 
 XX 
 PAPEK MONEY 
 
 On Table : A 5 note and a 10 note. 
 
 I HOPE you have begun to see that Arithmetic 
 is very much concerned with the question of 
 packing things in convenient shapes and sizes, 
 to carry them in our understanding and our 
 memories : sometimes even with convenient 
 packing of things to carry in our hands or 
 pockets. Buttons are sewn in dozens on cards 
 to avoid the need of counting them at busy 
 times, just as grocers put up sugar in pounds 
 ready for customers. Eggs are ready packed 
 in boxes so that a whole dozen can be handled 
 as one boxful. We pack up enough iron or 
 brass to make sixteen ounce weights into one 
 lump, which we call ? A pound weight. And 
 so on. If you think, you can see that, if an 
 ancient shepherd made a notch for every ten 
 sheep that had been counted, he was really 
 doing something of the same kind. In the 
 same way we put a mark for a foot at the end 
 of every twelve inches, or a mark for a yard at 
 the end of thirty-six inches. 
 
 When we take a shilling in our purse instead 
 of twelve pennies, the packing is of a different
 
 128 LOGIC OF ARITHMETIC 
 
 kind. We do not pack the twelve pennies into 
 one lump ; we do not take the pennies themselves 
 in any shape. Do you remember what it is that 
 is packed into the little shilling which we use 
 instead of the twelve pennies ? The value of 
 twelve big clumsy copper pennies is packed into 
 one little bit of the more valuable metal, silver. 
 
 In multiplication, when we say 6 times 8 are 
 48, we pack into a short form the business of 
 adding six eights, one after the other. When 
 we divide 37 by 5, what we are really doing is 
 to see how often we can subtract 5 from 37; 
 instead of doing the subtractions one after 
 another, we pack the whole lot together and do 
 it all in one sentence. If you think about it 
 sometimes, as you go on learning, you will find 
 that a great deal of your Arithmetic is simply 
 a sort of packing up to make short cuts. We 
 get the value, so to speak, of a long roundabout 
 proceeding, packed into a convenient bit of 
 work. 
 
 Well now, let us go back to our twelve 
 pence and our shilling. You perfectly under- 
 stand that the reason why this shilling is worth 
 as much as two sixpences is because there is 
 twice as much silver in it as in one sixpence. 
 It weighs twice as much. And the reason why 
 the silver is worth as much as twelve pennies is 
 because it is a more costly material. Can you
 
 PAPER MONEY 129 
 
 tell me any other instance of one coin being 
 worth a great many others because it is of 
 a more costly material ? A sovereign is worth 
 twenty shillings or eight big half-crowns, be- 
 cause gold is much more costly than silver. 
 How many pennies is a sovereign worth ? 240. 
 Do you think you quite clearly understand why ? 
 
 Well now, look at these two pieces of paper. 
 What are they worth ? One is worth 5, the 
 other 10. Why? Is paper a more costly 
 material than gold ? No. You know that it is 
 very much cheaper by weight. And it would 
 take a great many of such papers as these to 
 weigh down a sovereign. I am sure that these 
 papers are not worth in themselves a penny 
 apiece as paper. And they are exactly alike ; 
 that is, the paper part of them is alike ; one is 
 not bigger or thicker than the other. Yet 
 I could get five sovereigns for one and ten for 
 the other. Why is that ? What makes them 
 so valuable ? And why is one worth twice as 
 much as the other ? What does it all mean ? 
 Something must be packed into a little piece of 
 paper, to make it worth so many sovereigns. 
 Let us see if we can find out what is 
 packed into these bits of paper to make them 
 valuable. 
 
 It is never worth while guessing at the 
 meaning of a thing until after you have looked
 
 130 LOGIC OF AKITHMETIO 
 
 at all that there is to be seen in it, and found 
 out all that there is to learn about it. So we 
 will begin by reading all that is on these papers. 
 And then we shall see if we can make out what 
 it is that has been packed into them. 
 Bank of England. 
 
 L promise to pay the Bearer on Demand the 
 sum of Five Pounds. 
 
 London 5 Sept. 1902 
 
 For the Gov rs and Comp a of the 
 Bank of England. 
 J. G. Nairne, Chief Cashier. 
 
 The 'Bank of England promises to pay to 
 whoever brings this piece of paper to the Bank, 
 .5. And people feel so sure that the Bank will 
 pay the five pounds that anybody will give five 
 pounds for the paper. They feel they will always 
 get their money back. 
 
 How did people come to feel so sure that the 
 Bank would keep its promises ? Because it has 
 gone on so many generations always, keeping 
 them. 
 
 In some countries you may see written on 
 a Bank note a promise to pay five dollars, 
 and you may hear people bargaining as to 
 whether that promise (or that bit of paper) is 
 worth two dollars or three. Which means that 
 taking paper money in these countries is more 
 or less a sort of gambling or speculation. People
 
 PAPER MONEY 131 
 
 do not feel sure enough that the promise will 
 be kept to give the full value for the piece of 
 paper that it is written on. 
 
 What, then, is it that is packed into these 
 pieces of paper that we call Bank notes ? A 
 habit of trust in the honour of the Bank of 
 England. 
 
 We have been hearing a good deal of late 
 years about the honour of England and the 
 glory of England, and there have been pro- 
 cessions and shows and military displays, which 
 are supposed to be for the honour of the country. 
 I should like you to notice, my dears, that any 
 country can wave flags and bang drums and ring 
 bells and walk about in processions and shout 
 about its honour and glory, if it pleases. The 
 real glory and honour of England are much better 
 expressed, it seems to me, by the fact that, if 
 its National Bank promises to pay 5 or 500, 
 the people of the country take that promise for 
 its full value. They believe that their Govern- 
 ment will keep its promises. 
 
 J 2
 
 XXI 
 THE DOG'S PATH 
 
 If children are to learn Curve-Tracing, or any sort 
 of Analytical Geometry, the following exercise forms 
 a good introduction to the subject, as it puts their 
 minds into line with the alternation of tendencies, the 
 apparent conflict of forces, by which physical Nature 
 and human evolution are alike worked out. It obviates 
 any sense of mystery about what the whole thing means. 
 Once a child has grown accustomed to pay attention 
 alternately to the aspirations of the dog and to that of 
 the rabbit, and to see a curve growing up under the 
 impulsion of his own alternating sympathy, it is then 
 legitimate to tell him that there are other forces, besides 
 the wishes of animals, at work making curves, and that 
 he can learn to follow the action of many of these, even 
 when he can trace no conscious motive which is setting 
 them to work. 
 
 TRY to fancy that this black-board is a field. 
 It has a brick wall all round it, with no opening 
 except at A, where there is a gate. 
 
 At Z there is a rabbit hole. A rabbit came 
 out into the field at Z and wandered about till 
 he came to here (write the figure i ), where he found 
 something he liked to eat. After a little while, 
 a dog came in at the gate A ; the rabbit caught 
 sight of him, and directly afterwards the dog 
 caught sight of the rabbit. Now let us try if 
 we can make out what happened. First we 
 must try to think what the animals would each
 
 THE DOG'S PATH 133
 
 134 LOGIC OF AKITHMETIC 
 
 like to have happen ; the rabbit saw the dog first ; 
 what do you think he wished ? To get away. 
 Perhaps his first idea is to run straight away 
 from the dog. But he can't ; the wall prevents 
 him. What will he do next ? If he was a very 
 foolish rabbit, he might stand trying to get 
 through the wall, and fussing, and saying : ' Oh ! 
 dear ! I must get out here, there is no other 
 way. Oh ! this cruel wicked wall ! it is pre- 
 venting me from getting out ! Oh ! what shall 
 I do ? This is the straight way out ; and I 
 must knock a hole in this wall ! Oh ! what 
 shall I do ? ' 
 
 If the rabbit was a foolish rabbit, he might 
 go on in that way, till the dog caught him and 
 ate him up. 
 
 But we will suppose he is a sensible rabbit 
 who is not fond of knocking his head against 
 brick walls, and who has learned to use his 
 brains properly about his own business. What 
 do you think he will think of next ? He will 
 think of trying to get to his hole. He would 
 like to jump straight to his hole ; but he cannot 
 go right across a field in one jump. However, 
 we will draw a line to show what he would like 
 
 to do. (Draw the line 1 Z.) 
 
 But just when the rabbit gave up the notion 
 of getting out through the wall at 1, the dog 
 saw him. The dog had his little wishes about
 
 THE DOG'S PATH 135 
 
 the rabbit. What do you think he wished ? 
 What would the dog like to do ? Jump on the 
 rabbit and kill him. The dog can no more get 
 across a big field in one jump than the rabbit 
 could; but he would like to do so. We will 
 draw a line to show what the dog thinks he 
 
 would like to do. (Draw the line A 1.) As the 
 
 dog cannot jump from A to 1 he jumps as far 
 as he can ; his first jump takes him to B. But 
 by the time he has got to B, he sees the rabbit 
 has got to 2. Do you think he will go on 
 scampering down the line A 1 now he sees the 
 rabbit is not at 1 , but at 2 ? Of course not. He 
 will wish now to jump from B to 2. But that 
 again is too far for him to go in one jump ; he 
 jumps to C, but by the time he gets there the 
 rabbit is at 3. 
 
 The teacher should go on step by step, drawing the 
 successive wishes of the dog, and marking off the jumps 
 of both animals, taking care to keep up, all the time, the 
 children's consciousness of what the animals are each 
 thinking and doing ; till the diagram has evolved itself 
 on the black-board. Then : 
 
 Now tell me, what do all these lines repre- 
 sent ? The line 1 to Z represents the path 
 which the rabbit would like to -go along in one 
 jump, and does take in eighteen jumps. The 
 lines A 1, B 2, C 3, and all the other straight 
 lines, each represent a line that the dog at 
 some moment wished to jump along ; he jumped
 
 136 LOGIC OF ARITHMETIC 
 
 along a bit of one, and then changed his mind 
 and jumped a bit of the next, and so on. We 
 drew all those straight lines ; you saw me draw 
 them by the ruler, did you not ? 
 
 But here is a curved line A to Z. Who drew 
 that ? I drew no line except straight ones by 
 the ruler. Look at it well. Make sure that 
 you see it and all the lines on the black-board. 
 
 Now sit slack, shut eyes, and think what the 
 curved line is and how it came. 
 
 Open eyes and sit up. What is the curved 
 line ? The path which the dog really ran, when 
 at each step he meant only to go down some 
 straight line. 
 
 XXII 
 THE BALL'S PATH 
 
 You remember that we drew the path of a 
 dog in a field, by drawing straight lines to show 
 what he wished to do but could not. The dog 
 was dragged several ways, first one way and 
 then another. The dog was dragged only by 
 his own wishes and thoughts. 
 
 But a thing which has no wishes or thoughts 
 of its own, may be dragged by other forces. We 
 call what makes things move, force. Well, as
 
 THE BALL'S PATH 137 
 
 I said, things may be moved by other forces 
 besides their own thoughts and wishes. If you 
 throw a ball, you make it go along. We do not 
 know that it has any wishes of its own ; your 
 wish is a force which moves it. But your wish 
 is not the only force which moves it ; for if you 
 aim exactly at the top corner of the house, the 
 ball will not hit exactly the top corner; some 
 other force, what we call its weight, pulls it a 
 little downwards as it goes along ; so, between 
 the two forces, the ball makes a curve ! When 
 a thing moves in a curve, it is usually because 
 two forces, or more than two, are pulling or 
 pushing it in different directions. 
 
 You are going to begin learning about curves. 
 It will help you to keep out of many muddles, 
 if you will try to remember that, when you see 
 a curve in a book or on paper, it represents 
 some real form or movement ; or something more 
 or less like a real form or movement. It may 
 be simpler than the real path or shape ; but it 
 is only simpler ; it is really more or less like 
 something meant to be real. 
 
 But when you see straight lines, they are 
 seldom meant for anything real. A straight 
 line represents either a path that one force alone 
 would have taken something along if no other 
 force had interfered ; or else it is just put in 
 for convenience, to measure by. You have per-
 
 138 LOGIC OF ARITHMETIC 
 
 haps seen tailors' fashion books, with directions 
 for taking measures. You see a picture of a 
 man with a coat on, and straight lines drawn 
 across the shoulders or bust. You would be 
 dreadfully puzzled if you thought of those lines 
 as parts of the picture ; because no coat has 
 lines across the shoulders or bust ; but you 
 know they are meant, not for seams in the coat, 
 but for a measuring-tape supposed to be stretched 
 across the man in order to measure the width 
 of his coat. I have known children puzzled out 
 of their wits, and never able to understand their 
 Geometry for years of their school-time, because 
 they mistook straight lines for real parts of 
 some curved thing. 
 
 And if you ever feel worried and puzzled over 
 your Geometry lessons, shut your eyes for a 
 minute and think about the rabbit and the dog. 
 Then open eyes, and look at your book ; and say 
 to yourself, 'Which parts of this picture are 
 meant to be real like the real path of the dog ; 
 and which parts are only like the straight lines 
 that teacher drew on the black-board on purpose 
 to make us children understand how the dog 
 came to run in a path in which at first he did 
 not mean to go ? '
 
 xxin 
 
 EXERCISE TO PREPARE FOR 
 GENERAL FORMULAE 
 
 1 and 3 are? 4. And 5? 9. And 7? 16. 
 And 9 ? 25. 
 
 And so on, adding successively the odd numbers. 
 The results should be entered in a column on the right 
 hand of the black-board, and ultimately on the right 
 hand of the page in the formula-books. On the left 
 hand should be entered successively : 
 
 1x1 = 1 
 
 2x2=4, &c., thus : 
 
 1x1 = 1 
 
 2x2 = 4 
 3x3 = 9 
 
 1 
 3 
 
 4 
 5 
 
 9 
 
 7 
 
 4 x 4 = 16 16 
 9 
 
 5x5 = 25 25 
 
 &c. &c. 
 
 This exercise should be repeated occasionally, as 
 mere practice in adding and multiplying, till the fact 
 of the identity of results of the two processes has been
 
 140 
 
 LOGIC OF ARITHMETIC 
 
 worked into the children's consciousness, simply as 
 a fact, empirically observed, and, as yet, unexplained. 
 
 Then, they should be told to build up the successive 
 odd numbers, either with cubes, or on squared paper (A). 
 
 They should be led to see that, by building 3 round 
 1, they get a square block of the same size and shape as 
 by building 2 on 2 ; 5 round the block of 4 gives a block 
 the same size and shape as three rows of three, and 
 so on. 
 
 These things should be done simply as exercises in 
 adding, multiplying, and neat pencil-shading or chalking. 
 They should each be repeated many times. Great 
 
 accuracy should be insisted on ; but no explanation 
 should be attempted. The good mathematician will see 
 that none is necessary. An ordinary teacher who 
 attempted one would be almost sure to confuse and 
 mislead. The important thing to secure is that the 
 children's eyes, fingers, and attention should be occupied 
 about the diagrams, thus lodging clear images of them 
 on the memory, ready to crop up on some future 
 occasion ; as explained in my Address On the Prepara- 
 tion of the Unconscious Mind for Science ] . 
 
 In case the pupil afterwards takes up Algebra or the 
 Calculus, the theory of General (or Sliding) Formulae 
 
 1 Parent's Eeview, 1899,
 
 FORMULAE 141 
 
 will, when it has to be explained to him, find ready 
 a substratum of observed and familiar fact, illustrating 
 one such Formula : 
 
 ( + 1) 2 = n z +2 n + 1 
 
 The lack of any such basis, the consequent difficulty of 
 making the pupil grasp the idea of a General Formula, 
 forms one of the great obstructions to sound progress in 
 Mathematics. 
 
 But, even if he is never to go beyond ordinary 
 Arithmetic, no harm will have been done. He will 
 have gained a little practice, useful in itself, in the 
 comparison of numerical results arrived at by different 
 methods. The time spent on gaining familiarity with 
 the Genesis of Tables of Squares will not have been 
 wasted. Arithmetic itself is always best taught, when 
 it is taught on methods which constitute a sound prepara- 
 tion for the study of the higher Algebra.
 
 APPENDIX 
 
 I HAVE not succeeded in drawing out any 
 quite satisfactory scheme for suggesting the 
 feeling of proportion to the imagination of 
 children who have never used any optical 
 instrument. Perhaps some reader may devise 
 one. For children accustomed to use a magni- 
 fying-glass, the following mode of conveying 
 the idea is simple. Draw on paper two or three 
 short but unequal lines, and mark each with 
 a letter. Let the children look at each, first 
 with the naked eye, then with a magnifier. 
 Tell them to draw (or to suppose drawn) the 
 lines as seen through the glass. Then say that 
 the little line marked (a) is to the big line 
 marked (a) in the same proportion as the little 
 line marked (b) is to the big line marked (6), 
 and as the little line marked (c) is to the big 
 line marked (c). Draw also a small triangle 
 and quadrilateral, and a copy of each as seen 
 through the magnifier ; say that the little 
 triangle is to the big triangle in the same 
 proportion as the little quadrilateral to the big 
 one. Then show (a) through a weak magnifier
 
 APPENDIX 143 
 
 and (6) through a stronger one ; and say that, 
 now, the pairs of lines are not in proportion to 
 each other. 
 
 I would again emphasize the caution that 
 the lessons in this book are mere specimen 
 types, intended, not to be slavishly read aloud 
 to a class, but to suggest methods, such as 
 I have found useful, of dealing with various 
 arithmetical difficulties. On p. Ill a sum is 
 put in the form kindly shown me by an 
 elementary-school teacher, as the one familiar 
 to the pupils to which she is accustomed. 
 There is a different form more suitable for 
 advanced pupils doing long sums. Each teacher 
 should use the form to which his pupils are 
 accustomed. 
 
 On p. 61, the lesson is given as suited to 
 children accustomed to begin multiplying by 
 the unit figure of the multiplier ; any teacher 
 who habitually begins at the other end of the 
 multiplier should modify the lesson to suit his 
 class. 
 
 When I am giving the lesson on G. C. M., 
 I usually connect it with a story of a painter in 
 Rome, who knew nothing about our system of 
 notation, and who, having to decorate a ceiling, 
 asked his children to spare his labour by mark- 
 ing off, on a plan, as many as they could of the 
 points where he would have to make a pattern
 
 144 APPENDIX 
 
 end. This particular story would confuse chil- 
 dren not yet accustomed to realize that there 
 have been civilized and artistic people ignorant 
 of our notation ; such an idea, suggested for 
 the first time, would distract attention from 
 the arithmetical fact. I have therefore omitted 
 it in the chapter on G. C. M. But I would 
 take this opportunity of reminding teachers 
 that the thought, the imaginative idea, of 
 being engaged in helping a real toiler, has 
 a great effect in checking the tendency to 
 showy guessing, and in fixing attention on the 
 exact nature of the question under considera- 
 tion. It strengthens the feeling that the 
 object of study is real knowledge and not the 
 mere power to answer examination questions. 
 
 Oxford : Printed at the Clarendon Press, by HORACE HART, M. A.
 
 University of California 
 
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