THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES I This book is DUE on the last date stamped below 192- 27 MAR 8 1929 2 1 1929 7 1331 r ROV 2 7 193S \7 1933T JL / JAN 22 1942 2 1954 APR 2 2 RECfl On the Foundation and Technic of Arithmetic By George Bruce Halsted A. B. and A. M., Princeton; Ph. D., Johns Hopkins; F. R. A. S. 2- S S O O Chicago The Open Court Publishing Company 1912 COPYRIGHT BY THE OPEN COURT PUBLISHING CO. 1912 J-ihra CONTENTS. CHAPTER PACK Introduction I I. The Prehuman Contributions to Arithmetic 3 The natural individual, 3. The artificial individual, 4. Primary number, 5. Our base ten, 7. II. The Genesis of Number 8 III. Counting and Numerals 10 Correlation, 10. To count, n. The primitive standard sets, n. The abacus, 12. The word-numeral system, 12. Periodicity, 13. A partitioned unit, 14. Number without counting, 14. Decimal word-numerals, 14. Invariance of cardinal, 15. IV. Genesis of our Number Notation 17 Positional counting, 17. The abacus, 17. Recorded symbols, 18. The Hindu numerals, 19. The zero, 20. Our present notation, 22. V. The Two Direct Operations, Addition and Multiplication. 26 Notation, 26. The symbol =, 26. Inequality, 27. Parentheses, 28. Expressions, 28. Substitution, 29. Addition, 29. Formulas, 32. Ordinal addition, 33. Properties of addition, 33. Multiplica- tion, 35. VI. The Two Inverse Operations, Subtraction and Division. . 39 Inversion, 39. Subtraction, 39. Division, 41. VII. Technic 44 Addition, 44. Subtraction, 44. Multiplication, 45. Verify multi- plication, 46. Shorter forms, 47. Division, 47. Verify division, 48. VIII. Decimals 49 Decimals, 49. Product, 52. Quotient, 53. IX. Fractions 55 Generalizations' of number, 55. Principle of permanence, 56. Fractions, 56. Fractions ordered, 59. Division of fractions, 61. Multiplication of fractions, 61. X. Relation of Decimals to Fractions 63 ist Decimals into fractions, 63. 2d. Fractions into decimals, 63. Base, 65. Change of base, 67. IV FOUNDATION AND TECHNIC OF ARITHMETIC. CHAPTER PAGE XI. Measurement 68 Why count ? 68. The measure device, 70. Counting prior to meas- uring, 70. New assumptions, 72. XII. Mensuration 75 Geometry, 75. Length of a sect. 76. Length of the circle, 76. Area, 77. Volume, 78. XIII. Order 81 Depiction, 82. Infinite, 82. Sense, 82. Analysis of order, 83. Ordered set, 84. Finite ordinal types, 85. Number series, type of order, 85. Well-ordered sets, 86. XIV. Ordinaf Number 88 Ordinal number, 88. Children's counting, 88. Uses of ordinals, 89. Nominal number, 92. XV. The Psychology of Reading a Number 94 XVI. Arithmetic as Formal Calculus 101 XVII. On the Presentation of Arithmetic no ist Grade: Previous blunders, no. Begin with ordinals, no. Cardinal from ordinal, in. Ordinal counting, in. Cardinal counting, 112. Number precedes measure, 112. Cardinal number, 113. How to begin, 113. Ordinal games, 114. The call, 114. Ordinal operations, 116. The simplest cardinal, 116. Triplets and quartets, 117. The "how many" idea, 117. Symbols, 117. Car- dinal counting, 117. Recognition of the cardinal, 117. Cardinal addition, 118. Summary, 118. 2d Grade: Measurement, 119. The decimal, 120. Carrying, 121. Subtraction, 121. Fractions, 122. Multiplication, 122. Division, 123. Summary, 124. 3d Grade, 125. 4th Grade, 127. sth Grade, 127. 6th Grade, 128. 7th Grade, 129. Index 131 INTRODUCTION. In the French Revolution, when called before the tribunal and asked what useful thing he could do to deserve life, Lagrange answered: "I will teach arith- metic." Almost invariably now arithmetic is taught by those whose knowledge of mathematics is most meager. No wonder it and the children suffer. In this day of the arithmetization of mathematics and later its logiciza- tion, are the beauty, the elegance of arithmetical proce- dures to remain still unexplained? Is the singular, the lonely precision of this science and art to remain un- heralded, unexpounded? In arithmetic a child may taste the joy of the genius, the joy of creative activity. Arithmetic is for man an integrant part of his world construction. Thus do his fellows make their world, and so must he. Now this is not by passive apprehension of something presenting itself, but by permeating vitali- zation spreading life and its substance through what the ignorant teacher would present as the dead mechanism of mechanical computation. More than in any other science, there has been in mathematics an outburst of most unexpected, most deep- reaching progress. Its results, if made available for the 2 FOUNDATION AND TECHNIC OF ARITHMETIC. teacher, will revivify this first, most precious of edu- cational organisms; the more so since mathematics is seen to possess of all things the most essential, most fundamental objective reality. CHAPTER I. THE PREHUMAN CONTRIBUTIONS TO ARITH- METIC. Properly to understand or to teach arithmetic, one should have a glimpse of its origin, foundation, meaning, aim. Arithmetic is the science of number, but for the ordi- nary school-teacher it is to be chiefly t|ie doctrine of pri- mary natural number, the decimal and later the fraction, and the art of reckoning with them. Numbers are of human make, creations of man's mind; but they are first created upon and influenced by a basis which comes from the prehuman. Before our ancestors were men, they represented to themselves, as do some animals now, the world as con- The natural sisting of or containing individuals, definite individual. objects of thought, things. They exercised an individuating creative power. In now understanding by thing a. definite object of thought, conceived as indi- vidual, we are using a method of world presentation which served animals before there were any men to serve. The child's consciousness certainly begins with a sense-blur into which specification is only gradually in- troduced. At what stage of animal development the vague and fluctuating fusion, which was the world, be- gins to be broken up into persistently separate entities 4 FOUNDATION AND TECHNIC OF ARITHMETIC. would be an interesting comparative biologi co-psycho- logic investigation. However that might turn out, yet things, separate objective things, are a gift to man from the prehuman. Yet simple multiplicity of objects present to perception or even to consciousness does not give number. The duck does not count its young. The crow, wise old bird, has no real counting power to help its cunning. The animals' senses may be keener than ours, yet they never give number. A babe sees nothing numeric. Even an older child may attend to diverse objects with no suggestion from them of number. Sense-perception may be said to have to do with natural individuals, but never, unaided by other mind-act, does it give number. To the animal habit of postulating entities as separate must be added, before cardinal number comes, the human The artificial unification of certain of them into one whole, individual. one totality, one assemblage or group or set, one discrete aggregate or artificial individual man-made. This artificial whole, this discrete aggregate it is to which cardinal number pertains. Thus number rests upon a prehuman basis, yet is not number itself pre- human. Cardinal number involves more than the animal or natural individuals or things. It comes only with a human creation, the creation of artificial individuals, dis- crete aggregates taken each as an individual, an indi- vidual of human make, fleeting perhaps as our thought, transient, yet the necessary substratum for cardinal num- ber. Unification is necessary. The mind must make of the distinct things a whole, a totality. Else no cardinal number. Now to an educated man a number concept is sug- gested when a specific simple aggregate of objects is at- PREHUMAN CONTRIBUTIONS TO ARITHMETIC. 5 tended to. Not so to any animal, though just the same individual objects be recognized and attended to. The animal has the unity of the natural object or individual, but that unity is not enough. There is needed the new, the artificial, the man-made individuality of the total aggregate. To this artificial individual it is that the cardinal number pertains. There is thus a unity, man- made, of the aggregate of natural individuals, of the set of constituent units. To this unity made of units car- dinal number belongs. Going for quite different articles, or to accomplish entirely different things, may we not help and check memory by fixing in our mind that we are to get three things, or that we are to do three things? How man- made, arbitrary, and artificial, this conjoining of acts most diverse into a fleeting unified whole! Each finger of the left hand is different. A dog might be taught to recognize each as a separate and dis- tinct individual. Only a man can make of all at once an individual which, conceived as a whole, is yet multiple, multiplex, a manifold, fivefold, a five of fingers, a prod- uct of rational creation beyond the dog. A primary cardinal number is a character or attribute of an artificial unit made of natural units. It needs this Primary single individuality and this multiplicity of number. individuals. The fingered hand has fiveness only if taken as an individual made of individuals. Number is a quality of a construct. If three things are completely amalgamated, emulsified, like the com- ponents of bronze or the ingredients of a cake, there re- mains no threeness. If some things are in no way taken together the number concept is still inapplicable, we do not see them as a trio. 6 FOUNDATION AND TECHNIC OF ARITHMETIC. The animally originated primitive individuals, how- ever complete in their distinctness, have no numeric sug- gestion. The creative synthesis of a manifold must pre- cede the conscious perception of its numeric quality. The set must be conceived as a whole before discriminated as a dozen. It is only to man-made conceptual unities that the numeric quality pertains. This "number of natural individuals" in an artificial individual is called its cardinal number or cardinal. The cardinal n of a set j is the class of all sets similar to s. Primary number would seem in some sense a normal creation of man's mind. No primitive language has ever been investigated without therein finding records of the number idea, unmistakable though perhaps slight, limited, meager, it may be not going beyond our baby stage, one, two, many. There is a baby stage when no many is specialized but two. One, two, many, then baby waits how long be- fore that many called three is specialized ? Numeric one as cardinal only comes into existence in contrast with many. It involves a distinction between the class whose only member is x, and the thing x itself. The Stoic Chrysippos (282-209 B. C.) spoke of the "aggregate or assemblage one." Number comes when we make a vague many specific. The world-mind rose from the animal to the human when it grouped, aggregated, made wholes of, made arti- ficial individuals of the distinct individual objects pre- viously created by the animal mind. We may see babies recapitulating the race in this. The number of a particular totality represents the par- ticular multiplicity of its individual elements and nothing more. So far as represented in a number, each natural PREHUMAN CONTRIBUTIONS TO ARITHMETIC. 7 individual loses everything but its distinctness; all are alike, indistinguishably equivalent. The idea of unity is doubly involved in number, which applies to a unity of a plurality of units. The units are arithmetically identical; not so the complex unities man-made out of collections of the units. To these pertain the differing cardinal numbers. In our developed number systems certain manys take Our base on a peculiar prominence, are of basal char- acter. Of these ten has now permanently the upper hand. What is the origin of this preeminence? Its origin is prehuman. Our system is decimal, not because ten is scientifically, arithmetically a good base, a superior number, but solely because our prehuman an- cestors gave us five fingers on each of two hands. CHAPTER II. THE GENESIS OF NUMBER. In nature, distinct things are made and perceived as individual. Each distinct thing is a whole by itself, - .. . a qualitative whole. The individual thing Cardinals. . , ... , . is the only whole or distinct object in na- ture. But the human mind takes individuals together and makes of them a single whole of a new kind, and names it. Thus we have made the concept a flock, a herd, a bevy, a covey, a genus, a species, a bunch, a gang, a host, a class, a family, a group, an array, a crowd, a party, an assemblage, an aggregate, a manifold, a throw, a set, etc. These are artificial units, discrete magnitudes ; the unity is wholly in the concept, not in nature; it is artificial. We constitute of certain things an artificial individual when we distinguish them collectively from the rest of the world, making out of subsidiary individ- uals a single thing, a system, of which each component is recognizable as distinct from all others. From the contemplation of the natural individual or element in relation to the artificial individual, the group, spring the related ideas "many" and "one." We must have numeric many before we can have cardinal one. A natural quali- tative unit thought of in contrast to a "many" as not- many gives the idea "one" as cardinal. An aggregate may contain only a single element. Thus we have a set containing an element with which every element is iden- THE GENESIS OF NUMBER. ^ tical. So we get "one." A unity, a "many" composed of a "one" and another "one" is characterized as two. The unity, the "many" composed of "one" and the special many "two" is characterized as three. Among the primitive ideas of cardinal number, the idea of "two" is the first to be formed definitely. There are ever present doublets, things which can be grasped in pairs. This two is the very simplest many, the simplest recognized form of plurality. It is incalculably simpler than three, as witness whole savage tribes whose spoken number system is "one, two, many" ; as witness the mind- wasting primitive stupidities of the dual number in Greek grammar. The special many, a one made of three, a trinity, a trio, triplets, here is an advance. When to the grasp of the pair, the dominance over the trio is added, when the three is created, then after-progress is rapid. With a couple of pairs goes four; with a couple of threes, six. A hand represents five coming in between four and six. A pair of hands says ten. A pair of tens is twenty, a score. A pair of fours is eight. A trio of threes is nine. A pair of sixes or a trio of fours is twelve, a dozen. Arithmetic flowers like a rocket. That seven is left out, is missed, makes it the sacred, the mystic number of superstition. To numbers, however complicated their genesis, is finally ascribed a certain objective reality. In our mind the number concepts finally become simple things, objectively real. CHAPTER III. COUNTING AND NUMERALS. The ability of mind to relate things to things, to correlate, to represent something by some- Correlation. thing else, to make or perceive a correspon- dence between things or thought creations is funda- mental, essential, necessary. The operation of establishing such a correspondence between two sets that every thing or element of each set is mated with, paired with, just one particular thing or element of the other, is called establishing a one-to-one correspondence between the sets. Two sets which can be so mated are said to be equivalent as regards plural- ity, or to have the same potency. Two sets equivalent to the same are equivalent to each other, their elements correlated to the same element being thereby mated. Two sets between which a one-to-one relation exists have the same cardinal number and are said to be cardinally similar. A set's cardinal number is what is common to the set and every equivalent set. Thus a set's cardinal is independent of every characteristic or quality of any element beyond its distinctness. To find the cardinal of a set, we count the set. Counting is the establishing of a one-to-one corres- pondence of aggregates, one of which belongs to a well- known series of aggregates. If a group of things have COUNTING AND NUMERALS. 11 this correspondence with this standard group, then those properties of this standard group which are carried over by the correspondence will belong to the new group. They are properties of the group's cardinal number. To count an aggregate, an artificial individual, is to identify it as to numeric quality with a familiar assem- blage by setting up a one-to-one correspon- To count. , , Al _ dence between the elements of the two groups. Thus counting consists in assigning to each natural individual of an aggregate one distinct individual in a familiar set, originally a group of fingers, now usu- ally a set of words or marks. So counting is essentially the numeric identification, by setting up a one-to-one cor- respondence, of an unfamiliar with a familiar group. Thus it ascertains, it fixes the nature of the less familiar through the preceding knowledge of the more familiar. Primitively the known groups were the groups of fingers. The fingers gave the first set of standard groups The primitive an d formed the original apparatus for count- standard sets. m g ) anc j serve( j f or the symbolic transmis- sion of the concepts, the number ideas generated. More than that, this finger counting gave the names of the numbers, the numeric words so helpful in the further development of numeric creation. The name of a number, when referring to an artificial unit, as of sheep, denoted that a certain group of fingers would touch successively the natural units in the discrete magnitude indicated, or a certain finger would stand as a symbol for the numerical characteristic of that group of natural units. Our word "five" is cognate with the Latin quinque, Greek pente, Sanskrit pancha, Persian pendji; now in Persian penjeh or pentcha means an outspread hand. 12 FOUNDATION AND TECHNIC OF ARITHMETIC. In Eskimo "hand me" is tamuche; "shake hands" is tallalue; "bracelet" is talegowruk; "five" is talema. In the language of the Tamanocs of the Orinoco, five means "whole hand" ; six is "one of the other hand" ; and so up to ten or "both hands." Philology confirms that the original counting series or outfit was the series of sets of fingers, and this primi- tive method preceded the formation of numeral words. The use of visible signs to represent numbers and aid reckoning is not only older than writing, but older than the development of numerical language. In very many languages the counting words come directly and recog- nizably from the finger procedure. But of the fingers there are only a few distinct ag- gregates, only ten. Developing man needs more, needs to enlarge and extend his standards. The Chinese, even at the present day, extend the series of primary groups, the finger-groups, by substi- tuting groups of counters movably strung The abacus. & s F . r w*i on rods fixed in an oblong frame. With this abacus, which they call shwanpan, reckoning board, and the Japanese call soroban, they count and perform their arithmetical calculations. In many languages there are not even words for the first ten groups. Higher races have not only named The word- * nese g rou P s > but have extended indefinitely numeral this system of names. They no longer count directly with their fingers, but use a series of names, so that the operation of counting an assemblage of things consists in assigning to each of them one of these numeral words, the words being always taken in order, and none skipped, each word being thus capable of representing not merely the individual with which it COUNTING AND NUMERALS. 13 is associated, but the entire named group of which this individual is the last named. In making this series of word-numerals, there is evidently need for a system of periodic repetition. The prehuman fixes five, ten, or twenty as the Periodicity. r , . , . . , ^ r number after which repetition begins. Ut these, ten has become predominant. Thus come our word-numerals, each applicable to just one of a counted set and to the aggregate ending with this one. This dekadic word-system makes easy, with a simple, a light numerational equipment, the perfectly definite expression of any number, however advanced. So for us to count is to assign the numerals one, two, three, etc., successively and in order, to all the individual objects of a collection, one to each. The collection is said to be given in number, the number of things in it, by the cardinal number signified by the numeral as- signed to the last natural unit or component of the col- lection in the operation of counting it. Numerals are also called numbers. The numeral and a word specify- ing the kind of objects counted make what is called a concrete number. In distinction from this, a number is called an abstract number. When children are to count, the things should be sufficiently distinct to be clearly and easily recognizable as individual, yet not so disparate as to hinder the human power to make from them an artificial individual. The objects should not be such as to individually distract the attention from the assemblage of them. With little children use a binary system. Build with twos. Then go on, as did the Romans, to a quinary- binary system, which suits counting on the fingers. 14 FOUNDATION AND TECHNIC OF ARITHMETIC. In counting, an artificial individual may take the place of a natural individual. Children enjoy counting A partitioned Dv fives. Inversely, a unit may be thought unit> of as an artificial individual, composed of subsidiary individuals, as a dollar of 100 cents. An interesting exercise is the instantaneous recog- nition of the cardinal, the particular numeric quality of Number with- the collection, its specification without count- out counting. j ng g ut ^jg power to picture all the sep- arate individuals and to recognize the specific given pic- ture is very limited. If it be attempted to facilitate this recognition by arrangement, the recognition may easily become that of form instead of number. It is then simply recognizing a shape which we know should have just so many elements. Every teacher should remember when using blocks in developing the number-concept that only if very few can their number be perceived without the help of counting or addition. If 4 blocks lie close their number may be perceived immediately, but seven are dealt with as two groups. It is believed that the limit, even for adults and under favorable conditions, is about 4. We know that even IIII was replaced by IV. Try the children to see if their primitive number perception, that of II, has grown, and how far. In the making of numeral words it is necessary to fix upon one after which repetition is to begin. Other- Decimal w * se tnere would be no end to the number word- of different words required. We have noted numerals. , , , ,. , , that the prehuman has narrowed the choice, by the fiveness of the extremities of mammalian limbs, to five, ten or twenty. The majority of races, especially the higher, in prehistoric time chose ten, the number of our fingers. Then was developed a system to express COUNTING AND NUMERALS. 15 by a few number-names a vast series of numbers. If we interpret eleven as "one and ten" and twelve as "two and ten," teen as "and ten," ty as "tens," then English, until it took "million," ("great thousand," Latin mille, a thou- sand,) bodily from the French and Italian, used only a dozen words in naming numbers, in making a series of word-numerals with fixed order. The systematic formation of numerical words is called numeration. The cardinal number of any finite set of things is the same in whatever order we count them. Invariance This is so fundamental a theorem of of cardinal, arithmetic, it may be well to make its reali- zation more intuitive. That the number of any finite group of distinct things is independent of the order in which they are taken, that beginning with the little finger of the left hand and going from left to right, a group of distinct things comes ulti- mately to the same finger in whatever order they are counted, follows simply from the hypothesis that they are distinct things. If a group of distinct things comes to, say, five when counted in a certain order, it will come to five when counted in any other order. For a general proof of this, take as objects the letters in the word "triangle," and assign to each a finger, be- ginning with the little finger of the left hand and ending with the middle finger of the right hand. Each of these fingers has its own letter, and the group of fingers thus exactly adequate is always necessary and sufficient for counting this group of letters in this order. That the same fingers are exactly adequate to touch this same group of letters in any other order, say the alphabetical, follows because, being distinct, any pair 16 FOUNDATION AND TECHNIC OF ARITHMETIC. attached to two of my fingers in a certain order can also be attached to the same two fingers in the other order. In the new order I want a to be first. Now the letters t and a are by hypothesis distinct. I can therefore inter- change the fingers to which they are assigned, so that each finger goes to the object previously touched by the other, without using any new fingers or setting free any previously employed. The same is true of r and e, of i and g, etc. As I go to each one, I can substitute by this process the new one which is wanted in its stead in such a way that the required new order shall hold good behind me, and since the group is finite, I can go on in this way until I come to the end, without changing the group of fingers used in counting, that is without altering the cardinal number, in this case 8. The group of fingers exactly adequate to touch a group of objects in any one definite order is thus exactly adequate for every order. But when touching in one definite order each finger has its own particular object and each object its own particular finger, so that the group of fingers exactly adequate for one peculiar order is always necessary and sufficient for that one order. But we have shown it then exactly adequate for every order; therefore it is necessary and sufficient for every order. CHAPTER IV. GENESIS OF OUR NUMBER NOTATION. The systematic decimal system in accordance with which, even in the times of our prehistoric ancestors, a Positional f w number names were used to build all counting. numeral words, is paralleled by the proce- dure, even at the present day, of those Africans who in counting use a row of men as follows rjihe first begins with the little finger of the left hand, and indicates, by raising it and pointing or touching, the assignment of this finger as representative of a certain individual from the group to be counted; his next finger he assigns to another individual; and so on until all his fingers are raised. And now the second man raises the little finger of his left hand as representative of this whole ten, and the first man, thus relieved, closes his fingers and begins over again. When this has been repeated ten times, the second man has all his fingers up, and is then relieved by one finger of the third man, which finger therefore represents a hundred ; and so on to a finger of the fourth man, which represents a thousand, and to a finger of the fifth man, which represents a myriad (ten thousand). An advance on this actual use of fingers with a posi- tional value depending only on the man's place in the row, is seen in the widely occurring abacus, a rough instance of which is just a row of grooves in which pebbles can slide. With most races, as 18 FOUNDATION AND TECHNIC OF ARITHMETIC. with the Egyptians, Greeks, Japanese, the grooves or col- umns are vertical, like a row of men. The counters in the right-most column correspond to the fingers of the man who actually touches or checks off the individuals counted; it is the units column. But in the abacus a simplification occurs. One finger of the second man is raised to picture the whole ten fingers of the first man, so that he may lower them and begin again to use them in representing individuals. Thus there are two designations for ten, either all the fingers of the first man or one finger of the second man. The abacus omits the first of these equivalents, and so each column contains only nine counters. For purposes of counting, a group of objects can be represented by a graphic picture so simple that it can be Recorded produced whenever wanted by just making symbols. a mar k f or Gac fo distinct object. Thus the marks I, II, III, IIII, picture the simplest groups with a permanence beyond gesture or word; and for many im- portant purposes, one of these stroke-diagrams, though composed of individuals all alike, is an absolutely per- fect picture, as accurate as the latest photograph, of any group of real things no matter how unlike. The ancient Egyptians denoted all numbers under ten by the corresponding number of strokes; but with ten a new symbol was introduced. The Romans regularly used strokes for numbers under five, using V for five. The ancient Greeks and Romans both however indicated numbers by simple strokes as high as ten. The Aztecs carried this system as high as twenty, but they used a small circle in place of the straight stroke. I have seen the same thing done in Japan. Each stroke of such a picture-group may be called a GENESIS OF OUR NUMBER NOTATION. 19 unit. Each group of such units will correspond always to the same group of fingers, to the same numeral word. Though to this primitive graphic system of number- pictures there is no limit, yet it soon becomes cumbrous. The Hindu Abbreviations naturally arise. Those the numerals. world now uses, the Hindu numerals, have been traced back to inscriptions in India probably dating from the early part of the second century B. C. The oldest inscription using them positionally with local value and developed form is of 595 A. D. The Egyptians had no positional notation for number, though they had a hieroglyph for nothing, which they substituted for one side when applying their formula for a quadri- lateral to a triangle. The Babylonians had a sign of this kind, not used in calculation, consisting of two angu- lar marks, one above the other. About A. D. 130, Ptol- emy in Alexandria used, in his Almagest, the Babylonian sexagesimal fractions, and designated voids by the first letter of the word ovSe'v, nothing. This letter was not used as a zero. M. F. Nau gives in French translation in Journal asiatique, Vol. 16 (10th series), 1910, pp. 225-227, a quotation from Severus Sebokt, of Quennesra, on the Euphrates, near Diarbekr, written in 662 A. D., more than two centuries before the earliest known appearance of the numerals in Europe: "I refrain from speaking of the science of the Hin- dus, who are not Syrians, of their subtile discoveries in this science of astronomy more ingenious than those of the Greeks and even of the Babylonians and of their facile method of calculating and computing, which sur- passes words. I mean that made with nine symbols." 20 FOUNDATION AND TECHNIC OF ARITHMETIC. But probably a long time was yet to pass before the creation of the most useful symbol in the world, the naught, the zero, not merely a sign for noth- ing, but a mark for the absence of quantity, the cipher, whose first known use in ring form in a document is in 738 A. D.* This little ellipse, picture for airy nothing, is an indis- pensable corner-stone of modern civilization. It is an Ariel lending magic powers of computation, promoting our kindergarten babies at once to an equality with Cae- sar, Plato or Paul in matters arithmetical. The user of an abacus might instead rule columns on paper and write in them the number of pebbles or coun- ters. But zero, 0, shows an empty column and so at once relieves us of the need of ruling the columns, or using the abacus. Modern arithmetic comes from ancient counting on the columns of the abacus, immeasurably improved by the creation of a symbol for an empty col- umn. The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing not merely a local habitation and a name, a picture, a symbol, but helpful power, is characteristic of the Hindu race whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power. From the second half of the eighth century Hindu writ- ings were current at Bagdad. After that the Arabs knew positional notation. They called the zero gifr. The Arab word, a substantive use of the adjective gifr ("empty"), was simply a translation of the Sanskrit name sunya, * E. C. Bayley, 1882 Doubted by G. F. Hill, 1910, who substi- tuted an inscription of 876 A. D. GENESIS OF OUR NUMBER NOTATION. 21 literally "empty." It gave birth to the low-Latin zephi- rum or zefirum (used by Leonard of Pisa, 1202), whence the Italian form zefiro, contracted to zefro, and (1307) zeuero, then zero, whose introduction in print goes back to the 15th century (1491). In the oldest known French treatise on algorithm (author unknown, of the thirteenth century) we read, "iusca le darraine ki est appellee cifre 0." In the thir- teenth century in Latin the word cifra for "naught" is met in Jordan Nemorarius and in Sacrabosco who wrote at Paris about 1240. In MS. Egerton 2622, one of the earliest arithmetics in our language, on leaf \2>7b, we read: "Nil cifra significat sed dat signare sequent!. "Expone this verse. A cifre tokens noyt, bot he makes the figure to betoken that comes aftur hym more than he schuld & he were away, as thus 10. here the figure of one tokens ten. it may happe aftur a cifre schuld come a nothur cifre, as thus 200." Maximus Planudes (1330) uses tziphra. Euler used (1783) in Latin the word cyphra. We still say "cipher" or "cypher." In German Ziffer has taken a more gen- eral meaning, as has the equivalent French word chiffre, the most important numeral coming to mean any. The oldest coin positionally dated is of 1458. Zero, originally the sign of a blank or nil or vacant column, may be looked upon as indicating that a class is void, containing no object whatever, that it is the null class. Thus it is one of the answers to the question, "How many?", and so is a cardinal. It is also given a place in the ordinal series of natural numbers, and is chief in the series of algebraic numbers. Only in the 22 FOUNDATION AND TECHNIC OF ARITHMETIC. sixteenth century does naught appear as common sym- bol for all differences in which minuend and subtrahend are equal, and thus show itself as ready for its second great application, to standardize algebraic forms. By the first meaning of cipher, "empty," we have 20 = twain ten, but 2 + = 2. Hankel, 1867, calls modu- lus of an operation that which combined by the operation with something leaves this unchanged. So to-day we use nine digits and have no digit corresponding to the Roman X, for X is all the fingers of the first man, while we, like the abacus, use 10, which is one finger of the second man. Thus the ten, hundred, thousand are only expressed by the position of the number which multiplies them. In the written numeral IIII, we still see in the symbol the units of which the fourfold unit four is composed. Later abbreviation veils the constituent units, but their independence and all-alike-ness remain fundamental, giv- ing to cardinal number its independence of the order in which the things are enumerated. The use of the digits (Latin, digitus, a "finger"), the substitution of a single symbol for each of the first nine Our present picture-groups, and that splendid creation of notation. the Hindus, the zero, 0, naught, cipher, made possible our present notation for number. This still has a base, ten, in which the sins of our fathers, the mammals, are visited on their children. Its perfection is in its use of position with digits and zero, a positional notation for number, which the decimal point (or unital point) empowers to run down below the units, giving the indispensable decimals. This positional notation for number consists in the very refined artifice of representing every number as a GENESIS OF OUR NUMBER NOTATION. 23 sum of terms expressed by a row of digits each standing for a product of two factors, one factor the intrinsic, the face factor, indicated by the digit itself, the other factor, the local, the place factor, indicated by the place of this digit in the row, the local factor being a power of the base, for units' place, or column b or one, for the next place to the left b 1 or b (the base), to the right b~ l or \/b, etc. The summation of these binary products is indicated by the juxtaposition in the row of the digits representing them by their form and their place in the row with reference to units' place. Calculus, (Latin, "a pebble"), ciphering, which thus by the aid of zero attains an ease and facility which would have astounded the antique world, consists in com- bining given numbers according to fixed laws to find certain resulting numbers. Teaching is to enable the ordinary child to do what the genius has done untaught. A Hindu genius created the zero. The common, even the stupid, child is now to be taught to understand and use this wonderful creation just as it is taught to use the telephone. So the teacher incites, provokes the self- activity of the child's mind and guides it and confirms it, stopping this kaleidoscope at a certain turn, when the evershifting picture is near enough for life to the picture in the teacher's mind. Without theory, no practice, yet need not the theory be conscious. There is a logic of it, yet the child need not necessarily know, had perhaps better not know, that logic. The teacher should know, the child practise. It is striking to realize the centuries that passed after the present system of number-naming, numeration, had 24 FOUNDATION AND TECHNIC OF ARITHMETIC. been developed, before it had analogous, adequate sym- bolization, adequate written notation. As compared with their number-names, how bungling the Greek and Roman numerals, how arithmetically help- less the men of classic antiquity for lack of just one writ- ten symbol, the Hindu naught, giving us a written system which, except for its base ten, seems to be final and for all time, a world sign-language more perspicuous and compendious than any word-language. That prehuman parasite, the ten, is fixed on us like an Old Man of the Sea, else we could take the easily superior base twelve. The number of digit figures required is one less than the base; since 10 represents the base, whatever it be. In each case the prebasal figures, by help of the zero, always express as written in succession to left or to right of the units place (fixed by the unital point) multiples of ascending and descending powers of the base. But while the two and six of twelve are like the two and five of ten, yet twelve has three and four besides as divisors, as submultiples, for which tremendous advantage ten offers no equivalent whatsoever. The prehuman imposi- tion of ten as base, disbarring twelve, is thus a permanent clog on human arithmetic. The mere numerals, 1, 2, 3,.... or the numeral words, "one," "two," "three," .... are signs for what are called "natural numbers," or positive integers. In- teger with us shall always mean positive integer. If pure numbers, integers, have an intrinsic order, so do these, their symbols. The unending series, 1, 2, 3, 4, 5,. ... or one, two, three, four, five,.. ..is called the "natural scale," or the scale of the natural numbers, or the number series. Each symbol in it, besides its ordinal, positional sig- GENESIS OF OUR NUMBER NOTATION. 25 nificance in the sequence of symbols, is used also to in- dicate the cardinal number of the symbols in the piece of the scale it ends, and so of any group correlated to that piece. Thus the ordinal system is the original from which the cardinal system is derived. In the primary ordinal system the symbols refer to the individual objects, while in the derived cardinal sys- tem these same symbols refer to the successively larger sets whose names are determined as the name of the last individual counted ordinally. CHAPTER V. THE TWO DIRECT OPERATIONS, ADDITION AND MULTIPLICATION. The symbolic representation of numbers and ways of combining numbers comes under the head Notation. . " . of what is called notation. The natural numbers, as shown in the primitive nu- meral pictures, I, II, III, IIII, begin with a single unit, and, cardinally considered, are changed to the next al- ways by taking another single unit. A number, an integer, is said to be equal to, or the same as, a number otherwise expressed, when their units The symbol being counted come to the same finger, the same numeral word. The symbol =, read equals, is called the sign of equality, and takes the part of verb in this symbolic language. It was invented by an Englishman, Robert Recorde, replacing in his algebra, The Whetstone of Witte* the sign z used for equality in his arithmetic, The Grounde of Artes, 1540. Equality is a relation reflexive, symmetric, invertible. Equality is a mutual relation of its two members. If x=y, then y-x. Equality is a transitive relation. If x-y and yz, then xz. A symbolic sentence using this verb is called an equality. Ordinally, x=y means that x and y denote the same * London (no date, preface 1557). ADDITION AND MULTIPLICATION. 27 number in the natural scale. Formally, x-y means that either can at will be substituted for the other anywhere. When the process of counting the units of one num- ber simultaneously one-to-one with units of a second number ends because no unit of the second number remains uncounted, but the units of the first number are not all counted, then the first number is said to contain more units than the second number, and the second number is said to contain less units than the first. If a number contains more units than a second, it is called greater than this second, which is called the lesser. By successively incorporating single units with the lesser of two primitive numbers we can make the greater. Thomas Harriot* (1560-1621), tutor to Sir Walter Raleigh and one of "the three magi of the Earl of North- umberland," devised the symbol >, published 1631, read "is greater than," and called the sign of inequality. In- equality is a sensed relation. Turned thus < its symbol is read "is less than." Inequality in the same sense is transitive. If x > y and y > 2, then x > z. Since the result of counting is independent of the order of the individuals counted, therefore of two un- equal natural numbers the one once found greater is always the greater. Without knowing the number n, we can write "either n>5, or n=5, or n < 5." Any number which succeeds another in the natural scale is greater than this other. Ordinally, x < y means that x precedes y in the scale. * Harriot was sent to America by Raleigh in the year 1585. He made the first survey of Virginia and North Carolina, the maps of these being subsequently presented to Queen Elizabeth. He started the standardizing of algebraic forms and the theory of functions by writing every equation as a function equal to zero. 28 FOUNDATION AND TECHNIC OF ARITHMETIC. When by any definite process we select one or more elements of any aggregate A, these form another aggre- gate B, called a part of A. If any element of A remains unselected, B is called a proper part of A. It is possible for an aggregate to be equivalent to a proper part of it- self ; the aggregate is then called infinite. For example: for every number there is an even number; again, for every point on a foot there is a point on an inch. When we can get a third number from two given numbers by a definite operation, the two given numbers joined by the sign for the operation and Parentheses. J , '. enclosed m parentheses may be taken to mean the result of that combination. The result can now be again combined with another given number, and so we may get combinations of several numbers though each operation is performed only with two. Parentheses indicate that neither of the two numbers enclosed, but only the number produced by their combina- tion, is related to anything outside the parentheses. Parentheses (first used by the Flemish geometer Al- bert Girard in 1629) may without ambiguity be omitted: First, When of two operations of like rank the pre- ceding (going from left to right) is to be first carried out; Second, When of two operations of unlike rank the higher is the first to be carried out. The representation of one number by others with symbols of combination and operation is called an ex- pression. By enclosing it in parentheses, Expressions. r . J any expression however complex in any way representing a number, may be operated upon as if it were a single symbol of that number. If an expression already involving parentheses is enclosed in parentheses, ADDITION AND MULTIPLICATION. 29 each pair, to distinguish it, can be made different in siae or shape. The three most usual forms are the parenthesis (, the bracket [, and the brace {. In translating the ex- pression into English, ( should be called first parenthesis, and ) second parenthesis ; [ first bracket, ] second bracket ; { first brace, } second brace. No change of resulting value is made in any expres- sion by substituting for any number its equal however expressed. From this it follows that two Substitution. ... t numbers each equal to a third are equal to one another. This process, putting one expression for another, substitution, is a primitive yet most important proceeding. A single symbol may be substituted for any expression whatever. Permutation consists in a simultaneous carrying out of mutual substitution, interchange. Thus a and b in an expression, as abc, are permuted when they are inter- changed, giving bac. More than two symbols are per- muted when each is replaced by one of the others, as in abc giving bca or cab. Suppose we have two natural numbers written in their primitive form, as III and IIII; if we write all these units in one row we indicate another Addition. natural number; and the process of getting from two numbers the number belonging to the group formed by putting together their groups to make a single group is called addition. This operation of incorporating other units into the preceding diagram is indicated by a symbol first met in print in the arithmetic by John Widman, (Leipsic, 1489), a little Greek cross, +, read plus. If one artificial individual be combined with another to give a new artificial individual in which each unit of 30 FOUNDATION AND TECHNIC OF ARITHMETIC. the components appears retaining its natural indepen- dence and natural individuality, while the artificial indi- viduality of the two components vanishes, the number of the new artificial individual is called the sum of the numbers of the two components, and is said to be ob- tained by adding these two numbers (the terms or sum- mands). The first of two summands may be called the augment; the second, the increment. The sum of two numbers, two terms, is the numeric attribute of the total system constituted of two partial systems to which the two terms respectively pertain. In the child as in the savage, the number idea is not dissociated from the group it characterizes. But educa- tion should help on the stage where the number exists as an independent concept, say the number five with its own characteristics, its own life. Therefore we have number-science, pure arithmetic. So though it might per- haps be argued that there is only one number 5, yet we may properly speak of combining 5 with 5 so as to retain the units unaffected while the fiveness vanishes in the compound, the sum, 10. Addition is a taking together of the units of two num- bers to constitute the units of a third, their sum. This may be obtained by a repetition of the operation of form- ing a new number from an old by taking with it one more unit; thus 3 + 2 = 3 + 1 + 1. If given numbers are written as groups of units, e. g. (exempli gratia) , 2=1 + 1, 3 = 1 + 1 + 1, the result of adding is obtained by writing together these rows of units, e.g., 2 + 3=(l + !) + (! + 1 + 1) = 1 + 1 + 1 + 1 +1=5. Since cardinal number is independent of the order of counting, therefore in any natural number expressed ADDITION AND MULTIPLICATION. 31 in its primitive form, as IIII, the permutation of any pair of units produces neither apparent nor real change. The units of numeration are completely interchange- able. Therefore we may say adding numbers is finding one number which contains in itself as many units as the given numbers taken together. In defining addition, we need make no mention of the order in which the given numbers are taken to make the sum. A sum is independent of the order of its parts or terms. This is an immediate consequence of the theo- rem of the invariance of the number of a set. For a change in the order of the parts added is only a change in the order of the units, which change is without in- fluence when all are counted together. To write in symbols, in the universal language of mathematics, that addition is an operation unaffected by permutation of the order of the parts added, though applied to any numbers whatsoever, we cannot use nu- merals, since numerals are always absolutely definite, particular. If, following Vieta's book of 1591, we use letters as general symbols to denote numbers left other- wise indefinite, we may write a to represent the first number not only in the sum 2 + 3, but in the sum 4 + 1 and in the sum of any two numbers. Taking b for a second number, the symbolic sentence a + b = b + a is a statement about all numbers whatsoever. It says, addi- tion is a commutative operation. The words commutative and distributive were used for the first time by F. J. Servois in 1813. The previous grouping of the parts added has no effect upon the sum. Brackets occurring in an indicated sum may be omitted as not affecting the result. The general statement or formula (a + b) +c = a+ (b + c) says, 32 FOUNDATION AND TECHNIC OF ARITHMETIC. addition is an associative operation, an operation having associative freedom. Rowan Hamilton in 1844 first explicitly stated and named the associative law. For addition it follows from the theorem of the invariance of the number of a group. Equalities having to do only with the very nature of the operations involved, and not at all Formulas. . , . . , , with the particular numbers used are called formulas. A formula is characterized by the fact that for any letter in it any number whatsoever may be substituted without destroying the equality or restricting the values of any other letter. In a formula a letter as symbol for any number may be replaced not only by any digital num- ber, but also by any other symbol for a number whether simple or compound, in the last case bracketed. Thus a + b = b + a gives (a+c) +b = b+ (a+c). So from a formula we can get an indefinite number of formulas and special numerical equations. Each side or member of a formula expresses a method of reckoning a number, and the formula says that both reckonings produce the same result. A formula trans- lated from symbols into words gives a rule. As equality is a mutual relation always invertible, a formula will usually give two rules, since its second member may be read first. By definition, from the inequality a > b we know that a could be obtained by adding units to b. Calling this unknown group of units n, we have a = b + n. Inversely, if a=b + n then a > b, that is a sum of finite natural numbers is always greater than one of its parts. A sum increases if either of its parts increases. ADDITION AND MULTIPLICATION. 33 Addition may also be defined and its properties es- Ordinal tablished from the ordinal view-point. addition. start from the natural scale. To add 1 to the number x is to replace x by the next following ordinal. So if we know x, we know x+\. When we have defined adding some particular num- ber a to x, when we have defined the operation x + a, the operation x+(a+\} shall be defined by the formula (1) ..... .r+(a+ l) = (* + a) + l. We shall know then what ;r+(a + l) is when we know what x + a is, and as we have, to start with, defined what x+\ is, we thus have successively and "by recur- rence" the operations x + 2, x + 3, etc. The sum a + b is thus defined ordinally as the bth term after the ath. It serves to represent conventionally a new number univocally deduced by a definite given procedure from the numbers summed or added together. Associativity : a+ (b + c) = (a + b) +c. This theorem is by definition true for c=l, since, by Properties formula (1), a+ (b + 1) = (a + b) + 1. Now of addition, supposing the theorem true for c = y, it will be true for c = y + l. For supposing it follows that (2) . . . . [ (fl + fe) + y] + 1 = which is only adding one to the same number, to equal numbers. Now by definition ( 1 ) the first member of this equa- tion (2) .... (3), 34 FOUNDATION AND TECHNIC OF ARITHMETIC. as we recognize that it should be, since y is the number preceding y+l. But by the same formula (1), read backward, the second member of equation (2) as we see it should be, since b + y is the number preceding b + y + l. But again by (1), the second member of (4), Therefore [by (5), (4) and (3)], (2) may be writ- ten, Hence the theorem is true for c = y+l. Being true for c = l, we thus see successively that so it is for c = 2, for c = 3, etc. This is a proof by mathematical induction or demon- stration by recurrence, a procedure first explicitly used, although without a general enunciation, by Maurolycus in his work, Arithmetic orum libri duo (Venice, 1575). Commutativity : 1.... a + l = l+a. This theorem is identically true for a=l. Now we can verify that if it is true for a y it will be true for a = y + l ; for then by associativity. But it is true for a = 1, therefore it will be true for a = 2, for a = 3, etc. 2 ---- This has just been demonstrated for & = 1; it can be verified that if it is true for b-x, it will be true for For, if true for b = x, then we have by hypoth- ADDITION AND MULTIPLICATION. 35 esis a + x = x + a; whence, by formula (1), by 1 and asso- ciativity, o+ (-*+!) = (a + .*) + ! = (.* + a) + l=#+(a+l) =*+(! + a) (*+!)+ a. The proposition is therefore established by recurrence. Sums in which all the parts are equal frequently occur. Such additions are often laborious and liable to error. Multiplica- But such a sum is determined if we know tion. one t h e e q ua i parts and the number of parts. The operation of combining these two numbers to get the result is called multiplication; the result is then called the product. The part repeated is called the multipli- cand, and the number which indicates how often it occurs is called the multiplier. Multiplicand and multiplier are each factors of the product. Such a product is a multiple of each of its factors. In forming such a product, the multiplicand is taken once as summand for each unit in the multiplier. More generally, a product is the number related to the multiplicand as the multiplier to the unit. Following Wm. Oughtred (1631), we use the sign x to denote multiplication, writing it before the multiplier but after the multiplicand. Thus 1 xlO, read one multi- plied by ten, or simply one by ten, stands for the product of the multiplication of 1 by 10, which by definition equals 10. The multiplication sign may be omitted when the product cannot reasonably be confounded with any- thing else, thus la means 1 xa, read one by a, which by definition equals a. From our definition also axl, that is a multiplied by 1, must equal a. Commutativity. Multiplication of a number by a num- ber is commutative. Multiplier and multiplicand may be interchanged with- out altering the product. 36 FOUNDATION AND TECHNIC OF ARITHMETIC. 11111 For if we have a rectangular array of 11111 a rows each containing b units, it is also b 11111 columns each containing a units. Therefore bxa = axb. Taking apposition to mean successive multiplication, for example, abode = {[(ab}c]d}e, calling the numbers involved factors, and the result their product, we may prove that commutative freedom extends to any or all factors in any product. For changing the order of a pair of factors which are next one another does not alter the product. abcd- acbd. For c rows of a's, each row containing a a a a a b of them, is b columns of a's, each con- a a a a a taining c of them. So c groups of ab units a a a a a comes to the same number as b groups of ac units. This reasoning holds no matter how many factors come before or after the interchanged pair. For example abcdefg=abc ed fg, since in this case the product abc simply takes the place which the number a had before. And e rows with d times abc in each row come to the same number as d columns with e times abc in each column. It remains only to multiply this number successively by whatever factors stand to the right of the interchanged pair. It follows therefore that no matter how many num- bers are multiplied together, we may interchange the places of any two of them which are adjacent without altering the product. But by repeated interchanges of adjacent pairs we may produce any alteration we choose in the order of the factors. ADDITION AND MULTIPLICATION. 37 This extends the commutative law of freedom to all the factors in any product. Associativity. To show with equal generality that multiplication is associative, we have only to prove that in any product any group of the successive factors may be replaced by their product. abcdefgh = abc(def}gh. By the commutative law we may arrange the factors so that this group comes first. Thus abcdefgh = def abc gh. But now the product of this group is made in carry- ing out the multiplication according to definition. There- fore abcdefgh = def abc gh= (def) abc gh. Considering this bracketed product now as a single factor of the whole product, it can, by the commutative law, be brought into any position among the other fac- tors, for example, back into the old place; so abcdefgh = def abc gh- (def} abc gh = abc (def) gh. Distributivity. Multiplication combines with addition according to what is called the distributive law. Instead of multiplying a sum and a number we may multiply each part of the sum with the number and add these partial products. 4x5 = 4(2+ 3) = (2 + 3)4=2x4+3x4=5x4. Four by five equals five by four, and four rows of (2 + 3) units may be counted as four rows of two units together with 4 rows of 3 units. As the sum of two numbers is a num- 38 FOUNDATION AND TECHNIC OF ARITHMETIC. ber, we may substitute (a + b) for b in the formula , which thus gives So the distributive law extends to the sum of how- ever many numbers or terms. Since a(b+c)> ab and (a+b)b>ab, there fore a prod- uct changes if either of its factors changes. A product increases if either of its factors increases. Notwithstanding the historical origin of addition from counting and of multiplication from the addition of equal terms, it is now advantageous to consider multi- plication, not as repeated addition, but as a separate operation, only connected with addition by the distribu- tive law, an operation for finding from two elements, x, y, an element univocally determined, xy, called "the product, x by y" which by commutativity equals x times y. CHAPTER VI. THE TWO INVERSE OPERATIONS, SUBTRAC- TION AND DIVISION. In the preceding direct operations, in addition and multiplication, the simplest problem is, from Inversion. . , . ,, . , two given numbers to make a third. If a and b are the given numbers, and x the unknown number resulting, then , or x = ax, according to the operation. An inverse of such a problem is where the result of a direct operation is given and one of the components, to find the other component. The operation by which such a problem is solved is called an inverse operation. Since by the commutative law we are free to inter- change the two parts or terms of a given sum, as also the two factors of a given product, therefore here the inverse operation does not depend upon which of the two components is also given, but only upon the direct operation by which they were combined. Suppose we are given a sum which we designate by s, and one part of it, say, p, to find the corresponding other part, which, yet unknown, we repre- Subtraction. _. * sent by x. Since the sum of the numbers p and x is what p + x expresses, we have the equality 40 FOUNDATION AND TECHNIC OF ARITHMETIC. But this equation differs in kind from the literal equal- ities heretofore used. It is not a formula, for any digital number substituted for one of these letters restricts the simultaneous values permissible for the others. Such an equality is called a conditional equality or a synthetic equation, or simply an equation. The inverse problem for addition now consists just in this, to solve the synthetic equation when a and b are given ; in other words, to find a definite number which placed as value for x will satisfy the equa- tion, that is which added to b will give a, and thus verify the equation. The number found, which satisfies the equation is called a root of the equation. If the operation by which from a given sum a and a given part of it b we find a value for the corresponding other part x is called from a subtracting b, then, using the minus sign (-) to denote subtraction, we may write the result a-b, read a minus b. We may get this result, remembering that a number is a sum of units, by pairing off every unit in b with a unit in a, and then counting the unpaired units. This gives a number which added to b makes a. The expression or result a- b is called a difference. The term preceded by the minus sign is called the subtrahend', the other the minuend. Thus (a-b)+b = a-b + b = a; also Ordinally, to subtract y from x is to find the number occupying the ;yth place before x. Postulating the "rule of signs," that a-(b-c) = a-b + c, subtraction is associative and commutative. SUBTRACTION AND DIVISION. 41 The term division has two distinct meanings in ele- mentary mathematics. There are two ope- Division. . ... . . . rations called division: 1, Remainder divi- sion ; 2, Multiplication's inverse. 1, Given two numbers, a > b, a the dividend, and b the divisor, the aim of remainder division may be con- sidered the putting of a under the form bq + r, where r < b, and b not 0. We call q the quotient, and r the re- mainder. Both are integral. There is a definite proba- bility that r will not be 0. The remainder division of a by & answers the two questions: 1, What multiple of b if subtracted from a gives a difference or remainder less than 6? 2, What is this remainder? Remainder division will regroup a given set, the divi- dend, into smaller sets each with the same cardinal as a given set, the divisor, and a remaining set whose cardinal is less than that of the divisor. The number of the equivalent subsets is here the quotient. There is no implication that the original units are equal in size. So it would be a blunder to call this process measuring. Again remainder division will regroup a given set, the dividend, into equivalent subsets and a less remainder, when the number of subsets, the divisor, is given. The cardinal of each subset is here the quotient. This has sometimes been called partitive division. But these two applications of remainder division are not two kinds of division, and should not be emphasized. In arithmetical division, dividend and divisor are two given numbers fixing a third, the quotient. So the division of 15 by 4 tells how often 15 eggs contain 4 eggs and equally well 42 FOUNDATION AND TECHNIC OF ARITHMETIC. how many dollars in each of the 4 equivalent pieces of 15 dollars. When r is 0, then a is a multiple of b, and a is exactly divisible by b. The case 6 = is excluded. In this excluded case the problem would be impossible if a were not 0. But if a-0 and b = 0, every number, q, would satisfy the equal- ity a = bq. So this case must be excluded to make the operation of division unequivocal, that is, in order that the problem of division shall have always one and only one solution. A second solution q', / would give a = bq + r = bq'+r* t b(q-q')=r'-r. But r / -r , such that ten of them make the unit. But 1st. Decimals into just this same thing is meant by 1/10. Therefore any decimal may be instantly written as a fraction; e. g., 0.234 = 2/10 + 3/100 + 4/1000 = 234/1000. First Method. Any fraction equals the quotient of its numerator divided by its denominator. Consider the 2d. Fractions into fraction, then, simply as indicating an ex- ample in division of decimals, and proceed to find the quotient. Thus for 1/2 we have : . 5 2)1.0 So 1/2 = 0.5. For 3/4 we have . 75 4)3.00 So 3/4=0.75. For 7/8 we have: .875 8)7.000 So 7/8 = 0.875. 64 FOUNDATION AND TECHNIC OF ARITHMETIC. Second Method. Apply the principle : The value of a fraction is un- altered by multiplying both numerator and denominator by the same number. Thus 7/8-7/(2x2x2) = (7x5x5x5)/(2x5x2x5x2x5) = 875/1000-0.875. Considering the application of this second method to 1/3, we see there is no multiplier which will convert 3 into a power of 10, since 10 contains no factors but 2 and 5. Ten does not contain 3 as a factor, so we cannot convert 1/3 into an ordinary decimal. We cannot, as an example in division of decimals, divide 1 by 3 without remainder. But we can freely apply remainder-division, at any length. Thus 333 3)1. .1 .01 .001 The procedure is recurrent, and if continued the 3 would simply recur. . 142857 In division by n, not more than n 1 7) 1 . different remainders can occur. But as .3 soon as a preceding dividend thus re- 2 curs, the procedure begins to repeat it- 6 self. Here then this division by 7 must 4 begin to repeat, and the figures in the 5 quotient must begin to recur. 1 If the recurring cycle begins at once, immediately after the decimal point, the decimal is called a pure re- curring decimal. As notation for a pure recurring deci- RELATION OF DECIMALS TO FRACTIONS. 65 mal, we write the recurring period, the repetend, dotting its first and last figures thus 1/11 =-6$; 1/9 =!. Every fraction is a product of a decimal by a pure recurring decimal. Thus l/6=(l/2)(l/3)=0.5X-3. To convert recurring decimals into fractions : 12x 100 = 12-12 12x 1= -12 Therefore subtracting, 12x 99 = 12 .12 = 12/99=4/33 Rule : Any pure recurring decimal equals the fraction with the repeating period for a numerator, and that many nines for denominator. The base of a number system is the number which indicates how many units are to be taken together into a composite unit, to be named, and then to Base. , . . ... be used in the count instead of the units composing it, this first composite unit to be counted until, upon reaching as many of them as units in the base, this set of composite units is taken together to make a com- plex unit, to be named, and in turn to be used in the count, and enumerated until again the basal number of these complex units be reached, which manifold is again to be made a new unit, named, etc. Thus twenty-five, twain ten + five, uses ten as base. Using twelve as base, it would be two dozen and one. Using twenty, it would be a score and five. In positional notation for number, a digit in the units' place means so many units, but in the first place to the left of units' place it means so many times the base, while in the first 66 FOUNDATION AND TECHNIC OF ARITHMETIC. place to the right of the units' place it means so many subtmits each of which multiplied by the base gives the unit. And so on, for the second, etc., place to the left of the units' column, and for the second, etc., place to the right of the units' column. It is the systematic use of a base in connection with the significant use of position, which constitutes the for- mal perfection of our Hindu notation for number. The actual base itself, ten, is a concession to our fingers. The complete formula for a number in the Hindu positional notation is db n . . . . + db* + db 3 + db 2 + db 1 + db () +db- 1 +db- 2 + db~ n where juxtaposition of the d (digit) and b (base) means multiplication. This we condense to d. . .ddd.ddd. . .d, where the omitted ^-factor is indicated by the position of the d with reference to the units column, fixed by the unital point written to its right in the ordered row. Juxtaposition here means addition. If no base be speci- fied, ten is understood. Compare these subunital expressions for the funda- mental fractions, to base ten, to base twelve, to base two. DECIMALLY. DUODECIMALLY. DUALLY. [IN THE DENARY [DUODENARY [DYADIC SCALE.] SCALE.] SCALE.] 1/2=0-5 1/2 = 0-6 1/2 = 1/10 =0-1 1/3= .3 1/3=0-4 1/3 = 1/11 = -61 2/3= -6 2/3= -8 1/4 = 1/100 = -01 1/4=0-25 1/4=0-3 1/6=1/110 = -OOl 3/4= -75 3/4= .9 1/8 = 1/1000= -001 1/5= .2 1/5 =-249+ 1/9 = 1/1001= -OOOlli 1/6=0-16 1/6=0-2 1/8=0-125 1/8=0-16 3/8= .375 3/8= -46 1/9= -1 1/9=0-14 RELATION OF DECIMALS TO FRACTIONS. 67 To express a given number to a new base, divide it Change of an d the successive quotients by the new base base. until a quotient is reached less than the new base; this quotient and the successive remainders will be digits. Express 1594 to base twelve. 1 1 - Using x for ten and s for eleven, the 12)132-10 answer is sOx. 12)1594 Express sxs (base twelve) to base ten. 1-7 Answer 1715. 1)15-1 x)123-5 Express 98 to base two. 1-1 Answer 1100010. 2)3-0 2)6-0 2)12-0 2)24-1 2)49-0 2)98 Express 1111 (base two) to base ten. 1-101 Answer 15. 1010)1111 CHAPTER XI. MEASUREMENT. Says Dr. E. W. Hobson: "It is a very significant fact that the operation of counting, in connection with which numbers, integral and fractional, have their origin, is the one and only absolutely exact operation of a mathe- matical character which we are able to undertake upon the objects which we perceive. On the other hand, all operations of the nature of measurement which we can perform in connection with the objects of perception contain an essential element of inexactness. The theory of exact measurement in the domain of the ideal objects of abstract geometry is not immediately derivable from intuition." Arithmetic is a fundamental engine for our creative construction of the world in the interests of our dom- inance over it. The world so conceived bends to our will and purpose most completely. No rival construct now exists. There is no rival way of looking at the world's discrete constituents. One of the most far-reaching achievements of constructive human thinking is the arith- metization of that world handed down to us by the think- ing of our animal predecessors. In regard to an aggregate of things, why do we care .. . to inquire "how many" ? Why do we count Why count? . , / _ ,_., J an assemblage of things? Why not be satis- fied to look upon it as an animal would? How does the cardinal number of it help? MEASUREMENT. 69 First of all it serves the various uses of identification. Then the inexhaustible wealth of properties individual and conjoined of exact science is through number assimi- lated and attached to the studied set, and its numeric potential revealed. Mathematical knowledge is made ap- plicable and its transmission possible. Thus the number is basal for effective domination of the world social as well as natural, Number arises from a creative act whose aim and purpose is to differentiate and dominate more perfectly than do animals the perceived material, primarily when perceived as made of individuals. Not merely must the material be made of individuals, but primarily it must be made of individuals in a way amenable to treatment of this particular kind by our finite powers. Powers which suffice to make specific a clutch of eggs, say a dozen, may be transcended by the stars in the sky. Number is the outcome of an aggressive operation of mind in making and distinguishing certain multiplex objects, certain manifolds. We substitute for the things of nature the things born of man's mind and more obe- dient, more docile. They, responsive to our needs, give us the result we are after, while economizing our output of effort, our life. The number series, the ordered de- numerable discrete infinity is the prolific source of arith- metic progress. Who attempts to visualize 90 as a group of objects? It is nine tens. Then the fingers tell us what it is, no graphic group visualization. First comes the creation of artificial individuals having numeric qual- ity. The cardinal number of a group is a selective rep- resentation of it which takes or pictures only one quality of the group but takes that all at once. This selective picture process only applies primarily to those particular 70 FOUNDATION AND TECHNIC OF ARITHMETIC. artificial wholes which may be called discrete aggregates. But these are of inestimable importance for human life. The overwhelming advantages of the number picture led, after centuries, to a human invention as clearly a The measure device of man for himself as the telephone, device. This was a device for making a primitive individual thinkable as a recognizable and recoverable artificial individual of the kind having the numeric qual- ity, having a number picture. This is the recondite de- vice called measurement. Measurement is an artifice for making a primitive individual conceivable as an artificial individual of the group kind with previously known elements, conven- tionally fixed elements, and so having a significant num- ber-picture by which knowledge of it may be transmitted, to any one knowing the conventionally chosen standard unit, in terms of this previously known standard unit and an ascertained number. From the number and the standard unit for measure the measured thing can be approximately reproduced and so known and recovered. No knowledge of the thing measured must be requisite for knowledge of the stand- ard unit for the measurement. This standard unit of measure must have been familiar from previous direct perception. So the picturing of an individual as three- thirds of itself is not measurement. All measurement is essentially inexact. No exact measurement is ever possible. Counting is essentially prior to measuring. The sav- Counting a & e ' makin g the first faltering steps, fur- prior to nished number, an indispensable prerequisite for measurement, long ages before measure- ment was ever thought of. The primitive function of MEASUREMENT. . 71 number was to serve the purposes of identification. Count- ing, consisting in associating with each primitive indi- vidual in an artificial individual a distinct primitive indi- vidual in a familiar artificial individual, is thus itself essen- tially the identification, by a one-to-one correspondence, of an unfamiliar with a familiar thing. Thus primitive count- ing decides which of the familiar groups of fingers is to have its numeric quality attached to the group counted. To attempt to found the notion of number upon measure- ment is a complete blunder. No measurement can be made exact, while number is perfectly exact. Counting implies first a known ordinal series or a known series of groups; secondly an unfamiliar group; thirdly the identification of the unfamiliar group by its one-to-one correspondence with a familiar group of the known series. Absolutely no idea of measurement, of standard unit of measure, of value is necessarily involved or indeed ordinarily used in counting. We count when we wish to find out whether the same group of horses has been driven back at night that was taken out in the morning. Here counting is a process of identification, not connected fundamentally with any idea of a standard measurement-unit-of-reference, or any idea of some value to be ascertained. We may say with perfect certainty that there is no implicit presence of the measurement idea in primitive number. The number system is not in any way based upon geometric congruence or measure- ment of any sort or kind. The numerical measurement of an extensive quantity consists in approximately making of it, by use of a well- known extensive quantity used as a standard unit, a col- lection of approximately equal, quantitatively equal, quan- tities, and then counting these approximately equal quan- 72 FOUNDATION AND TECHNIC OF ARITHMETIC. tities. The single extensive quantity is said to be numer- ically measured in terms of the convened standard quan- titative extensive unit. Any continuous magnitude is measured by discreting it into a standardized set and a negligible residue, and counting the standard units in this set. For measurement, assumptions are necessary which are not needed for counting or number. Spatial measure- New ment depends upon the assumption that assumptions, there is available a standard body which may be transferred from place to place without under- going any other change. Therein lies not only an as- sumption about the nature of space but also about the nature of space-occupying bodies. Kindred assumptions are necessary for the measuring of time and of mass. Now in reality none of these assumptions requisite for measurement are exactly fulfilled. How fortunate then that number involves no measurement idea! But still other assumptions are made in measurement. After this device for making counting apply to some- thing all in one piece has marked off the parts which are to be assumed as each equal to the standard, their order is unessential to their cardinal number. But it is also assumed that such pieces may be marked out be- ginning anywhere, then again anywhere in what remains, without affecting the final remainder or the whole count. Moreover measurement, even the very simplest, must face at once incommensurability. Whatever you take as standard for length, neither it nor any part of it is exactly contained in the diagonal of the square on it. This is proven. But the great probabilities are that your stand- ard is not exactly contained in anything you may wish MEASUREMENT. 73 to measure. There is a remainder large or small, per- ceptible or imperceptible. Measurement then can only be a way of pretending that a thing is a discrete aggregate of parts equal to the standard, or an aliquot part of it. We must neglect the remainder. If we do it uncon- sciously, so much the worse for us. No way has been discovered of describing an object exactly by counting and words and a standard. Any man can count exactly. No man can measure exactly. Arithmetic applies to our representation of the world, to the constructed phenomena the mind has created to help, to explain, its own perceptions. This representa- tion of things lends itself to the application of arithmetic. Arithmetic is a most powerful instrument for that order- ing and simplification of perception which is fundamental for dominance over so-called nature. Measurement may be analyzed into three primary procedures: 1. The conventional acceptance or determi- nation of a standard object, the unit of measure. 2, The breaking up of the object to be measured into pieces each congruent to the standard object. 3. The counting of these pieces. The standard unit for any particular sort of magni- tude might have been any magnitude of the same kind. Race, locality, convenience, chance, have contributed to establish and maintain diverse units for magnitudes of the same kind, some wholly bad, stupid, indefensible, like the acre (160 times 30% square yards). A magnitude is often measured indirectly, perhaps by substituting for it and its standard unit two other magnitudes know to have the same quantuplicity rela- tion; thus an angle may be measured indirectly by using 74 FOUNDATION AND TECHNIC OF ARITHMETIC. two arcs ; the thermometer serves for the indirect meas- urement of a temperature by use of two volumes ; a mass is usually measured indirectly by use of two weights at the same station. CHAPTER XII. MENSURATION. Never forget that no exact measurement is ever pos- sible, that no theorem of arithmetic, algebra, or geometry could ever be proved by measurement, that measure could never have been the basis or foundation or origin of number. But the approximate measurements of life are im- portant, and the best current arithmetics give great space to mensuration. Geometry is an ideal construct. Of course the point and the straight are to be assumed as elements, without definition. They are Geometry. equally immeasurable, the straight in Eu- clidean geometry being infinite. What we first measure and the standard with which we measure it are both sects. A sect is a piece of a straight between two points, the end points of the sect. The sides of a triangle are sects. A ray is one of the parts into which a straight is di- vided by a point on it. An angle is the figure consisting of two coinitial rays. Their common origin is its vertex. The rays are its sides. When two straights cross so that the four angles made are congruent, each is called a right angle. One ninetieth of a right angle is a degree (1). A circle is a line on a plane, equidistant from a point 76 FOUNDATION AND TECHNIC OF ARITHMETIC. of the plane (the center). A sect from center to circle is its radius. An arc is a piece of a circle. If less than a semicircle it is a minor arc. One quarter of a circle is a quadrant. One ninetieth of a quadrant is called a degree of arc. A sect joining the end points of an arc is its chord. A straight with one, and only one, point in common with the circle is a tangent. To measure a sect is to find the number L (its length) Length of when the sect is conceived as ~Lu + r, where a sect. u j s f^ standard sect and r a sect less than u. In science, u is the centimeter. Thus the length, L, of the diagonal of a square centi- meter, true to three places of decimals, is 1 .414. Since there are different standard sects in use, it is customary to name u with the L. Here 1 .414 cm. Knowing the length of a sect, from our knowledge of the number and the standard sect it multiplies we get knowledge of the measured sect, and can always approxi- mately construct it. We assume that with every arc is connected one, and only one, sect not less than the chord, and if the arc Length of De minor, not greater than the sum of the the circle. sects on the tangents from the extremities of the arc to their intersection, and such that if the arc be cut into two arcs, this sect is the sum of their sects. The length of this sect we call the length of the arc. If r be the length of its radius, the length of the semicircle is nr. Archimedes expressed IT approximately as 3 + 1/7. True to two places of decimals, 7r = 3.14 or 3.1416 true to four places. MENSURATION. The approximation 7r = 3 + l/7 is true to three signifi- cant figures. But since TT = 3. 1416 = 3 + 1/7- 1/800, a second approximation, true to five significant figures, can be obtained by a correction of the first. Again ir = 3. 1416= (3 + 1/7) (1-. 0004), which gives the advantage that in a product of factors including ir, the value 3 + 1/7 can be used and the product corrected by subtracting four ten-thousandths of itself. The circle with the standard sect for radius is called the unit circle. The length of the arc of unit circle inter- cepted by an angle with vertex at center is called the size of the angle. The angle whose size is 1, the length of the standard sect, is called a radian. A radian intercepts on any circle an arc whose length is the length of that circle's radius. The number of radians in an angle at the center intercepting an arc of length L on circle of radius length r, is L/r. 180=7r/>. An arc with the radii to its endpoints is called a sector. The area of a triangle is half the product of the length of either of its sides (the base) by the length of the corresponding altitude, the perpendicular upon the straight of that side from the opposite vertex. A figure which can be cut into triangles is a polygon, whose area is the sum of theirs. Its perimeter is the sum of its sides. Area of Circle. In area, an inscribed regular polygon (one whose sides are equal chords) of 2n sides equals a triangle with altitude the circle's radius r and base the perimeter of an inscribed regular polygon of n sides. 78 FOUNDATION AND TECHNIC OF ARITHMETIC. A circumscribed regular polygon (one with sides on tangents) of n sides equals a triangle with altitude r and base the polygon's perimeter. There is one, and only one, triangle intermediate be- tween the series of inscribed regular polygons and the series of circumscribed regular polygons, namely that with altitude r and base equal in length to the circle. This triangle's area, rc/2 = r 2 7r, is the area of the circle, r 2 *. From analogous considerations, the area of a sector is the product of the length of its arc by the length of half its radius. A tetrahedron is the figure constituted by four non- coplanar points, their sects and triangles. Volume. _ * -L n j v -j. The four points are called its summits, the six sects its edges, the four triangles its faces. Every summit is said to be opposite to the face made by the other three; every edge opposite to that made by the two remaining summits. A polyhedron is the figure formed by n plane polygons such that each side is common to two. The polygons are called its faces; their sects its edges; their vertices its summits. One-third the product of the area of a face by the length of the perpendicular to it from the opposite vertex is the volume of the tetrahedron. The volume of a polyhedron is the sum of the vol- umes of any set of tetrahedra into which it is cut. A prismatoid is a polyhedron with no summits other than the vertices of two parallel faces. The altitude of a prismatoid is the perpendicular from top to base. MENSURATION. 79 A number of different prismatoids thus have the same base, top, and altitude. If both base and top of a prismatoid are sects, it is a tetrahedron. A section or cross-cut of a prismatoid is the polygon determined by a plane perpendicular to the altitude. To find the volume of any prismatoid. Rule : Multi- ply one-fourth its altitude by the sum of the base and three times the cut, at two-thirds the altitude from the base. Halsted's Formula: V= (o/4)(B + 3C). All the solids of ordinary mensuration, and very many others heretofore treated only by the higher mathematics, are nothing but prismatoids or covered by Halsted's For- mula. A pyramid is a prismatoid with a point as top. Hence its volume is aB/3. A circular cone is a pyramid with circular base. A prism is a prismatoid with all lateral faces paral- lelograms. Hence the volume of any prism =aB. A circular cylinder is a prism with circular base. A right prism is one whose lateral edges are per- pendicular to its base. A parallelepiped is a prism whose base and top are parallelograms. A cuboid is a parallelepiped whose six faces are rect- angles. A cube is a cuboid whose six faces are squares. Hence the volume of any cuboid is the product of its length, breadth and thickness. The cube whose edge is the standard sect has for volume 1. 80 FOUNDATION AND TECHNIC OF ARITHMETIC. Therefore the volume of any polyhedron tells how oft it contains the cube on the standard sect, called the unit cube. Such units, like the unit square, though traditional, are unnecessary. A sphere is a surface equidistant from a point (the center). A sect from the center to sphere is its radius. A spherical segment is the piece of a sphere between two parallel planes. If a sphere be tangent to the parallel planes containing opposite edges of a tetrahedron, and sections made in the sphere and tetrahedron by one plane parallel to these are of equal area, so are sections made by any parallel plane. Hence the volume of a sphere is given by Halsted's For- mula. V= (a/4) (B + 3C) = (3/4)aC. But a = 2r and C= (2/3)nr(4/3)r. So Vol. sphere = (4/3) Trr 3 . Hence also the volume of a spherical segment is given by Halsted's Formula. Area of sphere = 4?rr 2 . The area of a sphere is quadruple the area of its great circle. As examples of solids which might now be introduced into elementary arithmetic, since they are covered by Hal- sted's Formula,- may be mentioned : oblate spheroid, pro- late spheroid, ellipsoid, paraboloid of revolution, hyper- boloid of revolution, elliptic hyperboloid, and their seg- ments or frustums made by planes perpendicular to their axes, all solids uniformly twisted, like the squarethreaded screw, etc. CHAPTER XIII. ORDER. In the counting of a primitive group, any element is considered equivalent to any other. But in the use even of the primitive counting apparatus, the fingers, appeared another and extraordinarily important character, order. If always when any two elements a, & of a set are taken, a definite criterion fixes one or other of two alter- native relations, symbolized by a generalized use of > and <, such that if a < b then b> a, while if a > b then b < a, and such that if a < b and b < c, then a < c, we say the criterion arranges the set in order. So arranged, it becomes an ordered set. The savage in counting systematically begins his count with the little finger of the left hand, thence proceeding toward the thumb, which is fifth in the count. When number-words or number-symbols come to serve as ex- tended counting apparatus, order is a salient character- istic. Each is associated with a definite next succeeding number. The set possesses intrinsic order. By one-to-one adjunction of these numerals the in- dividuals of a collection are given a factitious order, the familiar order of the number-set. When the order is emphasized the number-names are modified, becoming first, second, third, fourth, etc., and are called ordinal numbers or ordinals, but this designa- tion is now applied also to the ordinary forms, one, two, 82 FOUNDATION AND TECHNIC OF ARITHMETIC. three, etc., when order is made their fundamental char- acteristic. If we can so correlate each element of the set A with a definite element of the set B that two different elements of A are never correlated with the same Depiction. . _. .. . . . , element of B, the element of A is consid- ered as depicted or pictured or imaged by the correlated element of B, its picture or image. Such a correlation we call a depiction of the set A upon the set B. The elements of A are called the originals. An assemblage contained entirely in another is called a component of the latter. A proper component or proper part of an assemblage is an aggregate made by omitting some element of the assemblage. An assemblage is called infinite if it can be depicted _ . upon some proper part of itself, or distinctly Infinite. -11 < < 1 imaged, element for element, by a constit- uent portion, a proper component of itself. Otherwise it is finite. Stand between two mirrors and face one of them. Your image in the one faced will be repeated by the other. If this replica could be separately reflected in the first, this reflection imaged by itself in the second, this image pictured as distinct in the first, this in turn depicted in the second, and so on forever, this set would be in- finite, for it is depicted upon the proper part of it made by omitting you. It is ordered. You may be called 1, your image 2, its image 3, and so on. A relation has what mathematicians call sense, if, when A has it to B, then B has to A a relation different, but only in being correlatively op- posite. Thus "greater than" is a sensed relation. "Greater ORDER. 83 than" and "less than" are different relations, but differ only in sense. Any number of numbers, all individually given, form a finite set. If numbers be potentially given through a given operand and a given operation, law, of successive eduction, they are still said to form a set. If the law educes the numbers one by one in definite succession, they have an order, taking on the order inherent in time or in logical or causal succession. A set in order is a series. Intrinsic order depends fundamentally upon relations Analysis of having sense, and, for three terms, upon a order. relation and its opposite in sense attaching to a given term. The unsymmetrical sensed relation which determines the fixed order of sequence may be thought of as a logic- relation, that an element shall involve a logically sequen- tial element creatively or as representative. An individual or element 1 has its shadow 2, which in turn has its shadow 3, and so on. Linear order is established by an unsymmetrical re- lation for one sense of which we may use the word "pre- cede," for the opposite sense "follow." The ordering relation may be envisaged as an opera- tion, a transformation, which performed upon a preced- ing gives the one next succeeding it; turns 1 into 2, and 2 into 3, and so on. If we have applicable to a given individual an opera- tion which turns it into a new individual to which in turn the operation is applicable with like result, and so on without cease, we have a recurrent operation which recreates the condition for its ongoing. If in such a set we have one and only one term not so created from 84 FOUNDATION AND TECHNIC OF ARITHMETIC. any other, a first term, and if every term is different from all others, we have a commencing but unending ordered series. The number series, 1, 2, 3, and so on, may be thought of as the outcome of a recurrent opera- tion, that of the ever repeated adjunction of one more unit. It is a system such that for every element of it there is always one and only one next following. This successor may be thought of as the depiction of its pred- ecessor. Every element is different from all others. Every element is imaged. There is an element which though imaged is itself no image. Thus the series is depicted without diminution upon a proper part of itself; is infinite, and by constitution endless. It has a first element, but no element following all others, no "last" element. Any set which can be brought into one-to-one cor- respondence with some or all of the natural numbers is said to be countable, and if not finite, is called countably infinite. An order of a set is constituted by a relation between the elements of the set. The same set may Ordered set. , . ,. have at the same time many different orders. The particular order is defined by the particular serial or arranging relation. A set of elements is said to be in simple order if it has two characteristics: 1. Every two distinct arbitrarily selected elements, A and B, are always connected by the same unsym- metrical relation, in which relation we know what role one plays, so that always one, and only one, say A, comes before B, is source of B, precedes B, is less than B; while B comes after A, is derived from A, follows A, is greater than A. ORDER. 85 2. Of three elements ABC, if A precedes B, and B precedes C, then A precedes C. So an arranging relation implies diversity of the ele- ments, is transitive, and connects any two different ele- ments related by it to a third. Thus the moments of time between twelve and one o'clock, and the points on the sect AB as passed in going from A to B are simply ordered sets. Two ordered sets A, B are called similar when a one- to-one correspondence can be established between their elements such that if a. From mb = an and m'b'=a'n' it follows that (w&)(m / & / ) = (aw)(aV). Hence, by II 2 and II 3, (mm'} (bb'} = (aa') (nn r ). Hence, by Theorem 3, mm'/nn'=aa'/bb'. Theorem 5: In any case, m/n + m'/n' = (mn' + nm')/ nn', and (m/nxm'/n')=mm'/nn'. Proof : See Theorem 4 and Definition 3. 106 FOUNDATION AND TECHNIC OF ARITHMETIC. Definition 4: If m and n are any two integers (the same or different) then m/n is called a positive fraction, for which fraction is only an abbreviation. Conversely, every fraction is m/n, where m and n are integers (the same or different). Theorem 6: Every integer is a fraction. Proof'. If m is an integer, then, by Definition 1, mm/m=m. Hence, by II 2 and Definition 4, m is a fraction. Capital letters are used here to designate fractions only. Theorem 7 : If A, B, C are fractions, then the follow- ing statements are true : F 1. A + B and AxB are fractions. F 2. A+B = B + Aand AxB = BxA. F 3. (A + B)+C = A+(B + C), and (AB)C = A(BC). F 4. There is not more than one fraction, D, such that A + D = B, and there is not more than one fraction E such that AxE-B. F 5. There is a fraction F such that AxF = B. F 6. There is a fraction G such that, if H is any fraction whatsoever, then, GH = H. F 7. A(B + C)=AB + AC Proof of F 1 : See Theorem 5, Definition 4, I 1 and II 1. Proof of F 2 : a. By Theorem 5, m/n + m'/n' = (mn' + nm' )/'. ARITHMETIC AS FORMAL CALCULUS. 107 Hence, by I 2 and II 2, m/n + m'/n'= (m'n + n'm)/ nn'. Hence, by Theorem 5, m/n + m'/n'=m' /n' + m/n. b. By Theorem 5, (w/wxw'/V) =mm'/nn f , Hence, by II 2 and Theorem 5, (ra/nxw'/n') = Proof of F 3 : a. By Theorem 5, m/n+ (m'/n' + m"/n") = m/n + ( m'n" + n'm" ) /n'n" = O(Vtt") + ttO'n" + w'm")]/(w'"), which by HI, = [w(w'w") + (ro'n") + ('w")]/('n"), which by II 3, = [(mw')w" + (nw')" + (nn')m"]/(nn')n", which by II 2 and III, = ( mw' + nm') /nn' + m" / n" = ( m/n + m'/n' ) + m"/n". b. By Theorem 5 and II, 3, m/nx (W'/M'XW"/W") =m/nx (m'm" '/n'n") = m(m'm")/n(b'n") = (mm'}m" / (nn')n" = (W/MXW'/W') xm r/ / w// - Proo/ of F 4 : a. If a/b + x/y = c/d and a/b+x'/y' = c/d, then, by Theorem 5 , ( m'/n f means there exist x, y such that m/n- m'/n'+x/y. m/,n < m'/ri means m f /n' > m/n. Assumption IV: A necessary and sufficient condition that integer a should be different from integer b is the ARITHMETIC AS FORMAL CALCULUS. 109 existence of an integer x such that either a + x=b or If this assumption IV is added to the others, then the following additional statements may be added in The- orem 7: F 8. Either A < B, A = B, or A > B. But no two of these three statements are simultaneously true. F 9.* If A > B, and B > C, then A > C F 10. If A>B, then A + OB + C, and AC> BC * F 9 and F 10 may be proved without use of IV. CHAPTER XVII. ON THE PRESENTATION OF ARITHMETIC FIRST GRADE. All schools heretofore have commenced the study of number by asking and considering the answers to the Previous questions, "how many?" "how much?" "how blunders. f ar? " how i on g?" They have thus begun with the cardinal, and with it alone have continued. Thus all teaching of the beginnings of arithmetic has uncon- sciously overlooked and missed the more fundamental and prerequisite question, "which one ?", and so remained unconscious of, and blind to the infinitely precious and in fact indispensable succor and aid of order, of the ordinal. Had study of the child been fructified by foreknowl- edge of the modern higher mathematics, it could not have Begin with overlooked in the spontaneous creative ac- ordinals. tivities of the child, the prominence and absolutely basal character of the ordinal, non-cardinal ideas, the serial, arranging and identifying ideas, histor- ically and developmentally preceding and prerequisite for the very apparatus subsequently used for the ascertain- ment of the "how many." In the counting of a primitive group, any element is considered equivalent to any other. But in the use even of the primitive counting apparatus, the fingers, appeared another and extraordinarily important character, order. ON THE PRESENTATION OF ARITHMETIC. Ill The savage, in counting, systematically begins his count with the little finger of the left hand, thence pro- ceeding toward the thumb, which is fifth in the count. When number-words come to serve as extended counting apparatus, order is not only a salient but an absolutely essential and indispensable characteristic of the apparatus. The number series, 1, 2, 3, and so on, is a system such that for every element of it there is always one and only one next following. Numbers are ordinal as individuals in a well-ordered set or series, and used ordinally when taken to give to any one object its position in an arrangement and thus individually to identify and place it. The ordinal process has also as outcome knowledge of the cardinal. When we have in order ticketed the _ ,. . ninth, we have ticketed nine. Thus the last Cardinal from ordinal used tells the result of the count. ma ' But this very ordering process precedes all cardinal ideas, as is shown by that use of count which occurs in the spontaneous games of little children, in their counting out or counting to fix who shall be it. This counting is characterized by order pure and simple. There is successive designation with no attempt Ordinal a * simultaneous apprehension, simply the as- countmg. signment of order to a collection and the ascertainment of place in the series made by this putting in order. Our instrument for this is the number series, and it is upon the order in the system that we ourselves rely to get a working hold of the individual number, especially when beyond the point where we can have any complete appreciation of the simultaneous multiplicity of the units involved in the corresponding cardinal. 112 FOUNDATION AND TECHNIC OF ARITHMETIC. It is fortunate then, and natural, that the modern child, despite the blindness of its teachers hitherto, gets the words of the ordinal series before it gets the cardinal concepts we attach to them. The ordinal coherence of the number series and its independence of cardinal concepts is shown by the child. Each name depicts a natural individual, not the so-far group of natural individuals, not a new kind of unity composed of units. Our apparatus for the ascertainment of cardinal num- ber involves, is based upon and uses order, ordinal num- Cardinal her. The child should be counting up to counting. a hundred before it can recognize a group of seven objects. When the symbols of the number series, the natural scale, are mated in sequence with the elements of an aggregate, the last symbol used is also taken as designation of the particular whole set so far used, and this identification of the unknown set with a known set it is which gives the cardinal property or quality of the hitherto unknown set. In this sense we say the last symbol used gives the outcome of the count, tells the cardinal number of the counted aggregate. First of all then let the teacher put out of her mind the blunder, pedagogic as well as scientific, that number N . was in any way dependent upon measure- precedes ment for origin. Number was created and used for individual ordering and identifica- tion and for group identification centuries before any measurement. There are tribes now using number that never have used measurement. All natural children use number long before measurement can even be explained to them. Measurement is a recondite device. Number is enormously more simple and primitive. Its uses in ON THE PRESENTATION OF ARITHMETIC. 113 identification both of individuals and groups are vastly important and quite independent of measurement. They long precede any thought of measurement. The number concepts are wholly apart from measure- ment, from length, from size, from the late-coming con- ventional standards for measurement, from the yard, the mile, the grain, the liter or any other standard for measurement. Valuation is a false associate for primi- Cardinal tive number. Number implies no exact size number. image. Cardinal number is a quality of a group. Two eyes and an ear-ache is a less dangerous trio than three yards, lest the teacher make the mistake of supposing number in any way dependent upon meas- urement. It is the acme of stupidity to attempt to found the number concepts upon "how much"; for example, "my desk is greater in length than in width." Begin by letting the child sing the number names as far as it enjoys the singing. Follow this up by exercises How to m designating or tagging objects with these begin. number-names as identifying tags. Paper horses may be used, named one, two, three, etc. Paper automobiles may be named, as the real ones are tagged, one, two, three, etc. Objects so tagged may be jumbled up and then arranged in the order of their names. Then differing objects, say the various differing animals in animal crackers, may be named, each with a number. Then the qualities of No. 2 may be contrasted with those of No. 4. The children may each be given a number as name. The teacher and the children may invent games using the ordinal properties, carefully avoiding as yet any "how many." 1 14 FOUNDATION AND TECHNIC OF ARITHMETIC. (1) Thus an instructive ordinal game is using a set of ordinals to count out the class. Choose a set Ordinal f ordinals, say the first nine. Distribute games. them in order and let the child to whom the nine comes be out. Then begin again with the re- maining children and again distribute the ordinals in order, dropping as out the child upon whom the nine now falls. When there are only eight children remaining, the count will more than go around, and the child tagged with one will also be tagged with nine and so be out, etc. (2) Give each child the same set of disarranged num- bers. See who can arrange quickest. (3) One, two; Buckle my shoe. Three, four; Open the door. Five, six; Pick up sticks. Seven, eight; Lay them straight. Nine, ten; A big, fat hen. Eleven, twelve; Dig and delve. (4) One, two, three, four, five; I caught a bird alive. Six, seven, eight, nine, ten; I let it go again. The call. (5) One, two; Glad to see you. Three, four; ON THE PRESENTATION OF ARITHMETIC. 115 Open the door. Five, six; My dog does tricks. Seven, eight; Walk to the gate. Nine, ten ; Please come again. (6) Mix up nine blocks numbered from one to nine. Let the child draw them out of the heap and put down each in its relative place when drawn until all are arranged in their proper order. (7) Hang about the neck of each of nine children a numbered tag. Let the children arrange themselves in order in line. Bend the line into a closed curve. Call out one number. The child so numbered goes within the en- closure. The others march about him. At a signal he calls a number. The child so designated takes his stand within the encircling line, and the caller finds his proper place in the line. (8) Give a number to each animal in a Noah's ark. This so far is only a nominal number, a name for a natural individual. Then introduce the ordinal by let- ting the child arrange the numbered animals in accord- ance with their number-names. Animal crackers may be substituted for a Noah's ark. (9) Have colored strips of paper numbered con- secutively in correspondence with the colors in the pri- mary rain-bow. Let the children arrange them in order to make a rain-bow. (10) Shuffle a pack of numbered cards. Give the pack to the child to arrange in the order of the numbers. 116 FOUNDATION AND TECHNIC OF ARITHMETIC. (11) Let the aisles in the school-room be numbered streets, and the broad cross passage-ways numbered ave- nues, and each desk a numbered house. Let the children write and address notes giving the house address, and let a messenger-child carry and deliver the letters. Addition : Ordinal I n the ordered row of children ask: operations. Which is the third after the second? An- swer: the fifth. Subtraction : Which is the third before the fifth? Answer: the second. Multiplication : Which is the third second? Answer: the sixth. Which is the second third? Answer: the sixth. When the child is thoroughly familiar with the or- dered names as applied to natural individuals, we are The simplest ready for their first application to artificial cardinal. individuals of the group kind, and first the application of the ordinal two to a pair. Make couples, partners, pairs, mates, and call each pair two. The cardinal two, the simplest cardinal, is that prop- erty of a set whereby it can be mated, one to one, with a child's thumbs, or it is the class of such sets. The idea of a cardinal, belonging as it does to a set of things as a whole, is a comparatively late concept. It must follow the concept of a whole composed of parts, constituents permanently distinguishable. Later comes the attribution of the geometirc quality of relative size, big and little, to numbers. For the next step make trios. The cardinal three is ON THE PRESENTATION OF ARITHMETIC. 117 the class of all triplets, or that quality of a set whereby Triplets and ^ can be mated, one to one, with a child's quartets. eves an( j nO se, also with the ordinal set one, two, three; the last of which is used as a tag or name for the group, the trio. Quartets are groups mateable, individual to individual, with the fingers of the left hand, or the words one, two, three, four; the last of which is to be used as a name for every such set ; and so on. There may follow in rich variety the construction, the identification, the tagging, of small The "how groups. This is at last the "how many" many" idea. jd ea L et j t fi rst be the natural and useful question of simple identification of groups, recognition of like or unlike cardinal. Herein lies abundant opportunity for constructive work. Give the child the first five ordinals. Let him then construct groups whose name shall be five, conse- quently whose "how many" shall be five, the cardinal. Explain how simple groups were used as symbols for the numeric quality of all like groups. Thus, II, III, IIII, are symbols for their own cardinal quality two, three, four. Then may come the Hindu symbols 2, 3, 4, primarily as ordinals, then Cardinal secondarily as cardinals. Now is the time counting. f or cardinal counting, counting as group- identification, using first the ten different groups of fin- gers as known groups with one of which the unknown group is to be identified by setting up a one-to-one correspondence between the individuals of the unknown R niti rou P an d the individuals of a finger group, of the car- Then we go to cardinal counting using the first dozen groups of ordinal words as known groups. All in good time, a test that the idea of 1 18 FOUNDATION AND TECHNIC OF ARITHMETIC. the cardinal has taken root and germinated, is practice in the instantaneous recognition of the cardinal of a small group suddenly exhibited, then veiled. The question "how many" is to be answered without conscious count- ing. Then larger groups may be used recognizable by use of symmetry in the arrangement or grouping, as on playing cards. Then we may begin to train for the instantaneous recognition from two components of the cardinal of their Cardinal compound, for example, the thinking of addition. seven upon seeing three and four. Here we should stop to train until every pair from one plus one up to nine plus nine arouses the image of its sum instantly and automatically. Coins, cents, nickels, dimes, dollars, are admirably adapted at this stage as anchors for the ideas created, while at the same time bringing home to the child the precious aid of number (anterior to any measurement) in the child's social relations, in the interest growing out of and attaching to the very life of the child itself. Games of buying, and perhaps actual buying, with the consequent paying and change- making, are here in place. Constructive processes familiarize and endear to the child the ideal numeric creations. Summary (First Grade). A. Ordinal counting. Utilize the spontaneously child- Ordinal arith- create( ^ ordinal systems. Also rhymes and metic, then jingles, cardinal. R The number symbolS) 1, 2, 3, 4, 5, 6 etc. as ordinals. C. Ordinal applications, identification, arrangement, factitious order. ON THE PRESENTATION OF ARITHMETIC. 119 D. Ordinal tagging. E. Ordinal games. F. Group-making, group distinction, group familiari- zation. G. Group identification, cardinal counting. H. Cardinal applications. I. Cardinal games. J. The number symbols as cardinals. K. Positional notation for number. L. Addition tables. M. Coins and their applications and games. N. Exercises in making conscious the number-needs of the child's own life, individual and social. O. Problems oral and motor ; ordinal ; to be solved by ordinal identification and arrangement. Cardinal; to be solved by cardinal identification, by addition, by corre- lation. SECOND GRADE. The number work of the second grade, as in all grades, is to be related as closely as may be to the actually existing interests and immediate needs of the child. Do not bend for a moment to the false and exploded idea that number was originated or created by measure- Measure- ment, a palpable absurdity, since we must ment. already be able to count before we can meas- ure, and since the preexistent counting is absolutely exact while no measuring ever can be exact. But now that the child has the prerequisite number-equipment, we may envisage measurement. The "muchness" of a quantity is not determined by the "how many" parts in it, unless these be all of a fixed, a preestablished size. Hence in addition to, and 120 FOUNDATION AND TECHNIC OF ARITHMETIC. outside of the number-ideas, the child must now be con- fronted with the new and difficult idea of definite con- ventional standards, the so-called units-for-measure or units of measure, the inch, the quart, the pound, the second, the degree. To measure is to break the thing up into pieces each equal to one of these standards, or a like standard, and to count the pieces. The child must combine his old knowledge of the number obtained with his new knowledge of the standard now used. Measurement then can only come after much prac- tice in counting. Finally begin measuring by measur- ing a length. Show that nothing would be gained here by actually breaking off the pieces, as we do in measuring milk. We need only see where they could be broken off. Now we are ready for the consideration of the actual problems presented to the child by its own occupations. It may be called upon to use so much rope or board or food. The outcome of the measure- ment is a graphic description in known terms, a num- ber and a unit; and now inversely a metric description should evoke a graphic image, a picture. Since mensuration is combined with arithmetic, there may be training to familiarize the various units and their subunits, yard, foot, inch, gallon, quart, pint, hour, min- ute, second, pound, ounce, gram, etc. Now should be given a thorough-going presentation of our positional notation for number, and as the neces- sary extension of it, the decimal. Decimals The decimal. , .. , ... . , , , are made up of the subunits inevitably des- ignated by the extension of our positional notation to the right of the units' column. As the self-interpreting extension of this positional notation for number to the right of the units' column, ON THE PRESENTATION OF ARITHMETIC. 121 we have decimals. We need no new elements, nothing but the already mastered digits, base, column. The deci- mal is not a fraction; it has no denominator. Decimals are significant figures to the right of the units' column; to indicate units' column, we henceforth use the decimal point. One thousand (1000) means ten of such units as stand in the adjacent column to the right ; and one of these, one hundred (100), means ten of such as stand in the next column ; and one of these, ten ( 10) , means ten of our primal units, such as stand in our units' column ; and one of these, One (1 ), means ten of such as stand in the next column to the right, that is in the first column to the right of our units' column ; and one of these, one-tenth, . 1 , has the same relation to one in the next column. We have an excellent available illustration in our coins. Taking the dollar as the primal unit, one-tenth, .1, is one dime or ten cents; .01 is one cent, or ten mills. These columns are to be named so that units' column be axis of sym- metry; twenty (20) gives tens; so 0.2 gives tenths; three hundred (300) gives hundreds; so 0.03 gives hun- dredths ; then 4000 gives thousands ; so . 004 gives thou- sandths. As no new elements come with decimals, nothing but our old digits, base, column, so no new principle is in- volved in their addition, subtraction, multiplication and division. The child who has the equipment for inter- preting 23 has that for interpreting 3.14159265. Our _ . explanation of positional notation contains the explanation of "carrying" in addition. Whenever the digit X is reached in any column, it is carried, appearing as one in the next column to the left. So we have this word already available when Subtraction. . * . we reach subtraction, which is always to be 122 FOUNDATION AND TECHNIC OF ARITHMETIC. worked by addition. Look upon difference as the num- ber which if added to the subtrahend gives the minuend. Thus to subtract, 9004 5126 3878 Think six and eight make fourteen ; carry 1 ; three and seven make ten ; carry 1 ; two and eight make ten ; carry 1 ; six and three make nine. We carry one to balance a one put in to facilitate our procedure. Thus in sub- tracting, 8% say two-fifths and four- fifths make six-fifths; 6% carry 1 ; seven and one make eight. ~I%~ A fraction is an ordered number-pair where the sec- ond number, the denominator, tells what Fractions. ... 11.10 sort of units are represented by the nrst number, the numerator. Thus 2/3 means two of such units (subunits) that three of them make the primal unit. When we come to multiplication, the idea of column Multiplied- i s to dominate. The fundamental admoni- tion. tj on j s . Always keep your columns. Always begin to multiply with left-most figure of the multiplier. Thus we get the most important partial product first. Rule : The figure put down stands as many places to the right or left of the digit multiplied as the multiplier is from units' column. 21 .354 Another form of the rule is : Mul- 200 . 003 tiplying shifts as many places right 4270.8 or left as the multiplier is from 64062 units' column. Note as an important 4270.864062 special case of our rule: // of two ON THE PRESENTATION OF ARITHMETIC. 123 figures multiplied one is in units' column, the figure put down stands under the other. There are two interpretations of division, namely . . . Remainder Division and Multiplication's In verse. Remainder division may be taught before the multiplication of fractions. It is to find how many times one number, the divisor, is contained in an- other, the dividend; and what then remains. For ex- ample, if eggs are four cents apiece, how many can be bought for three nickels? Answer three Or in count- ing with a compound unit, the divisor, how many times is it taken before overstepping the dividend? Historically it was in connection with measurement that fractions had their origin. By way of review and advance combined, we may now introduce subtraction, of course never to be worked by anything but addition, the "making change" method. Again multiplication may now be introduced, with the tables for doubling, tripling, quadrupling. Here may be given the symbols, +, -, x, /, -T-, =. Pairs of numbers may now be exhibited for the child to give their difference; then pairs of numbers, the second number a 1, 2, 3, 4, or 5, for the child to give the product. For games we have dominoes, bean matching, and the like. Use the savage device of a row of men for counting, to make easy our positional nota- tion for number. Thus familiarize digits of different orders. Sticks and stick-bundles can be correlated with cents, nickels, dollars, halves, quarters. If the sticks be marked off in tenths, decimals may be illustrated. Thus numbers of two and three orders are familiar- ized, as also the shifting of the decimal point. 9876 mills 124 FOUNDATION AND TECHNIC OF ARITHMETIC. are 987.6 cents, or 98.76 dimes, or 9.876 dollars. Deci- mals and fractions are made simple by the idea of a prin- cipal unit and subunits. Give each child a cheap foot rule; here inches are subunits. Actual familiarity with standards for measure is essential, the more so as these are no part of pure arith- metic or number, but only extraneous components of a device for the application of number, namely measure- ment. Practice in simple multiplication, envisaged first as condensed addition, may go up through doubling, tripling, quadrupling, quintupling. Multiplication by ten is equiv- alent to shifting the decimal point to the right. Quin- tupling is shifting the point and halving. Measurements for the application of number knowledge to the attain- ment of ends desired by the child are in place, but the so-called "formal work" and "mechanical drill" may give more joy and interest to the child than any measurement. From Teachers College Record we quote : "Upon be- ing given their choice one morning between going to the new gymnasium and remaining in the room to learn a new multiplication table, all but three of a class of thirty chose the mental gymnastics. This is cited to show that much of the so-called 'formal work,' 'systematic me- chanical drill,' which sounds so formidable to an out- sider, may bring much delight to one of our eight year old children, and that the mechanism of number may be secured with no sacrifice of interest." Summary (Second Grade) A. The extra-arithmetical idea of a standard for measurement. B. The usual standards for measure. ON THE PRESENTATION OF ARITHMETIC. 125 C. Explanation of "to measure." D. Knowledge obtained by measuring is a combine of number-knowledge and knowledge of the standard. E. Length, area, volume, capacity, weight, tempera- ture; with their standards, foot, square, cube, quart, pound, degree. F. Metric description evoking visual image. G. Positional notation for number. H. Decimals. Basal subunits. Significant figures to right of units column. I. Fractions. Any subunits. J. Subtraction. Difference. K. Multiplication. L. Symbols. M. Games. O. Change of unit. Shifting the decimal point. P. Problems; written work. Q. Multiplication tables through quintupling. THIRD GRADE. We are more than ever to aim at helping the develop- ment of the child in mental power, accuracy, and pre- cision, mind-mastery, ability to direct and fix the atten- tion, and withal to a distinct growth in technically arith- metical equipment for efficiency and life. There very often seems here to bloom out spontane- ously in the child a love for what has sometimes been called the abstract formal part of arithmetic. It is seen to give delight. The play-joy, which is perhaps a greater ingredient in pure science than has been suspected, now shows forth to illumine the work, and beautify the seem- ingly mechanical. 126 FOUNDATION AND TECHNIC OF ARITHMETIC. Review Work. A. Counting with a compound unit, by 2's, by 3's, by 4's, by 5's, by 10's. Beginning with zero or any number. B. Addition with "carrying." C. Subtraction, with "carrying." (Never use any but the addition method.) Advance Work. D. Multiplication. Complete the tables through 8 constructively. Explain the nine, ten, and eleven tables, so that they need not be memorized. For example, to "nine-times" a digit, write the preceding digit and adjoin what it lacks of being nine : e. g., 9x8 = 72. Connect the eights with the fours. Written multiplication; begin always with the left-most figure of the multiplier. E. Division. Two kinds, but teach first Remainder Division. First utilize the multiplication work. Teach to divide by one digit, then by two. Contrast remainder division and multiplication's inverse. F. Decimals. The point in addition and subtraction. Shifting the point in multiplication and division. G. Fractions. 1/2, 1/4, 1/8, 1/3, 1/6; change the subunit. Addition; subtraction. H. Measurement. Square measure. When objects are used, it should be remembered that after they have once served their purpose they only ham- per children and teacher. But buying, selling, making change may often be used. Let the children, where pos- sible, make their own problems. Groups of objects may be used to introduce division. Let a child realize what he is working to accomplish. ON THE PRESENTATION OF ARITHMETIC. 127 FOURTH GRADE. A. A review of addition, subtraction, and multiplica- tion; but a very extensive presentation and mastery of remainder division. B. Verifications. Verify addition and subtraction by the commutative principle. Verify multiplication and division by the simplest method of casting out nines. C. Multiplication's inverse. No remainder. Fraction in quotient. D. Invention of problems. E. Tests of accuracy and speed. F. Measurements. Include decimals and fractions in the problems apart and together. Cubic. G. Plotting on squared paper. Graphic representa- tion. H. Illustrations of the life-value of facility and ac- curacy in the four operations. I. Divisibility. Factors. Multiples. K. Emphasize the form of arrangement of written work. FIFTH GRADE. A. Decimals. The identity of decimal notation with the ordinary positional notation used throughout the first four grades. B. Reading of all decimals in the new method. C. Addition and subtraction shown to involve nothing new. D. Illustrations from our money. E. Multiplication of decimals; (all multiplications begin with the left-most figure of the multiplier). Shift- ing the point. 128 FOUNDATION AND TECHNIC OF ARITHMETIC. F. Division of decimals. Shifting the point. G. Problems. H. Checking results. I. Percentage. Applications to discount, commission, simple interest. The one hundred months method for interest. 1. Find a percent of a number, (given number and rate). 2. Find what percent one number is of another. 3. To find a number from a given percent of it. J. Geometric forms. Denominate numbers. K. Prime numbers. Prime factors. L. Business problems. SIXTH GRADE. Fractions. Meaning of fractions. Gain by the notation. A. Reduction, that is, change of the subunit. B. Addition and subtraction. Meaning of these ope- rations for fractions. C. Least common multiple. Common denominator. Simplest form. D. Extension of the idea of multiplication. E. Multiplication by a fraction. F. "Of" not multiplication symbol, yet %xQ or % times Q equals % of Q. G. Cancellation. H. Division by a fraction. I. The so-called business fractions and their percent equivalents. J. Expression of decimals as fractions and fractions ON THE PRESENTATION OF ARITHMETIC. 129 as decimals. Show by squared paper and diagrams the identity of different expressions for the same fraction. K. Scale drawing. SEVENTH GRADE. Review. A. Symbols: Row of savages. Zero. Decimal point. Fracjonal notation. Parentheses. Units added counted together are thereby taken as equivalent. Illustrations. Adding with time limit. B. Business forms and operations. Banks. Interest. Deposit slips. Checks. Drafts. Notes. Discount. Stocks. Bonds. Coupons. C. Meaning of per cent and percentage. Decimals and fractions in percentage. D. Percentage equivalents of 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 1/6, 1/8, 3/8, 5/8, 7/8, when considered as ope- rators; and vice versa. Percentage equivalents of deci- mals when considered as operators. E. Problems on percentage. Commission, taxes tariff, insurance. F. Longitude and time. G. Hundred Months Method. Interest for one hundred months at twelve percent equals principal. Interest for one month at twelve per- cent equals .01 of principal. Interest for a number of months, an aliquot part of one hundred, is just that part of the principal. Interest for 3 days is . 001 of the prin- cipal. Thus to get interest at twelve percent for eight months, shift point two places to left in principal and multiply by eight. 130 FOUNDATION AND TECHNIC OF ARITHMETIC. Interest at 8, 6, 4, 3, 2 % is 2/3, 1/2, 1/3, 1/4, 1/6 of that at 12. H. Mensuration ; rectangle ; parallelogram ; trapezoid ; regular polygon; circle; prismatoid; Halsted's Formula: V= (a/4) (B + 3C) ; prism; cylinder; pyramid; cone; sphere. I. Evolution; use of tables. Logarithms. Negative and positive numbers. The equation. The unknown. The variable. The constant. The parameter. Coordi- nates. The graph. The function. J. General review. INDEX. abacus 12, 17 addition 29, 33, 44, 60, 1 16 Ahmes 55 angle 75 Archimedes 76 area 77 arithmetic 68, 73, 101 artificial 4 associative 32 assumptions 72, 102 base 7, 24, 65 Bayley 20 begin 113 binary 13 Birch $5 blunders no Bosworth 94 Britannica 98 Brown 97 Byrhtferth 97 calculus 22, 1 01 cardinal 5, in, 113 cardinals 8 carrying 121 Cassini 99 Century 94 Chambers 96 child 4, no Chrysippos 6 cipher 20 columns 122 commutative 31 correlation 10 count n, 68, m countable 84 counting 10, 71, 88, 117 Cowper 97 cross-cut 79 cuboid 79 Cursor 97 decimal 14, 51, 120 decimals 22, 49, 63 degree 75, 76 difference 40, 122 digits 22 distributive 3t division 41, 47, 58, 61, 123 dozen 97 Egerton 21 equivalent 10 Eskimo 12 fingers n five n formulas 32 fraction 106 fractions 56, 63, 121 games 113 geometry 75 Gerbier 97 Girard 28 grades no, 118 Hale 97 Halsted's 79, 80, 130 Hamilton 32 Hankel 22, 56 Harriot 27 Hickes 96 Hill 20 Hindu 19 Holmes 97 Hooke 97 hundred 94, 129 hundredths 50 132 FOUNDATION AND TECHNIC OF ARITHMETIC. individual 3 induction 34 inequality 27 infinite 82 integer 24 interest 129 intrinsic 23 invariance 15 inverse 39 Lagrange i, 99 Langland 98 length 76 Leonard 21 local 23 Maurolycus 34 Mazarin 99 measurement 68, 73, 119 million 98 modulus 22 Moore 101 multiple 42 multiplication 35, 38, 45, 61 Murray 95, 97 Napier 49 Napoleon 99 natural 24 Nau 19 Nemorarius 21 nines 46 nine-times 126 nominal 92 notation 26 number 5, 14, 69, 88, 92 numbers 3 numeral 12 numeration 15, 23 one 6 order 81, 83 ordered 59 ordinal 25, 33, 88, in ordinals Si, 89, no Oughtred 35 parentheses 28 part 28 partitioned 14 Peacock 56, 96 periodicity 13 permanence 56 Planudes 21 pi ay- joy 125 plus 29, 53 point 51 position 17 positional 22 prehuman 3 presentation no prism 79 prismatoid 78 product 35, 52 Ptolemy 19 quartets 117 quotient 41, 53 radian 77 Raleigh 27 ray 75 read 51, 94 reciprocal 58 Recorde 26 recur 64 remainder 41 roundheads 93 Sacrabosco 21 scale 59 schoolmaster's 99 Sebokt 19 sect 75 sense 82 Servois 31 seven 9 Shakespeare 98 shift 50 solidus 43 sphere 80 standard n, 73, 124 Stevinus 49 straight 75 substitution 29 subtraction 39, 44, 57, 116 sum 30 summits 78 symbol 26 symbols 18 symmetrical 43 teaching 23 technic 44 telephone 92 ten 7 tenths 50 INDEX. 133 terms 30 verify 46, 48 thousand 96, 98 Vieta 31 thousandths 51 volume 78, 80 three 9, 116 twenty 22 well-ordered 86 two, 6, 9, 116 Whitney 99 Widman 29 unification 4 "nit, 8, 73 zero 20 THE OPEN COURT MATHEMATICAL SERIES A Brief History of Mathematics. By the late DR. KARL FINK, Tubingen, Germany. Trans- lated by Wooster Woodruff Beman, Professor of Math- ematics in the University of Michigan, and David Eugene Smith, Professor of Mathematics in Teachers' College, Columbia University, New York City. With ^biographical notes and full index. Second revised edition. Pages, xii, 333. Cloth, $1.50 net. (5s. 6d. net.) "Dr. Fink's work Is the most systematic attempt yet made to present a compendious history of mathematics." The Outlook. "This book Is the best that has appeared In English. It should find a place In the library of every teacher of mathematics." The Inland Educator. Lectures on Elementary Mathematics. By JOSEPH Louis LAGRANGE. With portrait and biography of Lagrange. Translated from the French by T. J. Mc- Cormack. Pages, 172. Cloth, $1.00 net (4s. 6d. net.) "Historical and methodological remarks abound, and are so woven to- gether with the mathematical material proper, and the whole is so vivified by the clear and almost chatty style of the author as to give the lectures a charm for the readers not often to be found in mathe- matical works." Bulletin American Mathematical Society. A Scrapbook of Elementary Mathematics. By WM. F. WHITE, State Normal School, New Paltz, N. Y. Cloth. Pages, 248. $1.00 net. (5s. net) A collection of Accounts, Essays, Recreations and Notes, selected for their conspicuous interest from the domain of mathematics, and calculated to reveal that domain as a world in which invention and imagination are prodigiously enabled, and in which the practice of generalization is car- ried to extents undreamed of by the ordinary thinker, who has at his command only the resources of ordinary lan- guage. A few of the seventy sections of this attractive book have the following suggestive titles : Familiar Tricks, Algebraic Fallacies, Geometric Puzzles, Linkages, A Few Surprising Facts, Labyrinths, The Nature of Mathematical Reasoning, Alice in the Wonderland of Mathematics. The book is supplied with Bibliographic Notes, Bibliographic Index and a copious General Index. "The book Is interesting, valuable and suggestive. It Is a book that really fills a long-felt want. It is a book that should be In the library of every high school and on the desk of every teacher of mathematics." >T/ie Educator-Journal, THE OPEN COURT MATHEMATICAL SERIES Essays on Mathematics. Articles by HENRI POINCARE. Published in the Monist. Price, 60 cents each. On the Foundations of Geometry Oct. , 1898 The Principles of Mathematical Physics Jan., 1905 Relations Between Experimental Physics and Mathematical Physics July, 1902 The Choice of Facts April, 1909 The Future of Mathematics Jan., 1910 Mathematical Creations July, 1910 Chance Jan. , 1912 The New Logics April, 1912 Portraits of Eminent Mathematicians. Three portfolios edited by DAVID EUGENE SMITH, Ph. D., Professor of Mathematics in Teachers' College, Columbia University, New York City. Accompanying each portrait is a brief biographical sketch, with occasional notes of interest concerning the artist represented. The pictures are of a size that allows for framing 11x14. Portfolio No. 1. Twelve great mathematicians down to 1700 A. D.: Thales, Pythagoras, Euclid, Archi- medes, Leonardo of Pisa, Cardan, Vieta, Napier, Descartes, Fermat, Newton, Leibnitz. Price, per set, $3.00. Japanese paper edition, $5.00. Portfolio No. 2. 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With many new additions still unpublished in German. Translated by E. J. TOWN- SEND, Ph. D., Associate Professor of Mathematics in the University of Illinois. Pages, viii, 132. Cloth, $1.00 net (4s. 6d net.) "Professor Hilbert has become so well known to the mathematical world by his writings that the treatment of any topic by him commands the attention of mathematicians everywhere. The teachers of elemen- tary geometry in this country are to be congratulated that it is possible for them to obtain in English such an important discussion of these points by such an authority." Journal of Pedagogy. Euclid's Parallel Postulate : Its Nature, Val- idity and Place in Geometrical Systems. By JOHN WILLIAM WITHERS, Ph. D. Pages vii, 192. Cloth, net $1.25. (4s. 6d. net.) "This is a philosophical thesis, by a writer who Is really familiar with the subject on non-Euclidean geometry, and as such it is well worth reading. 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A practical presentation of arithmetic for the use of teachers. There has been in mathematics an outburst of unexpected deep reaching progress and properly to understand or to teach arithmetic, one should have a glimpse of its origin, foundation, meaning and aim. Non-Euclidean Geometry, a Critical and Historical Study of its Development. By ROBERTO BONOLA. With an Introduction by FEDERIGO ENRIQUES. Translated by H, S. CARSLAW. Cloth, $2.00. Pages, 268. Illus- trated. A clear exposition of the principles of elementary geometry especially of that hypothesis on which rests Euclid's theory of parallels, and of the long discussion to which that theory was subjected; and of the final discovery of the logical possibility of the different Non-Euclidean Geometries. In Preparation: Bibliography of 1OO selected books on the History and Phi- losophy of Mathematics. Price, $1.00. THE OPEN COURT MATHEMATICAL SERIES Geometric Exercises in Paper-Folding. By T. SUNDARA Row. Edited and revised by W. W. BE- MAN and D. E. SMITH. With half-tone engravings from photographs of actual exercises, and a package of papers for folding. Pages, x, 148. Price, cloth, $1.00 net. (4s. 6d. net.) "The book is simply a revelation In paper folding. All sorts of things are done with the paper squares, and a large number of geometric figures are constructed and explained in the simplest way." Teachers' Institute. Magic Squares and Cubes. By W. S. ANDREWS. With chapters by PAUL CARUS, L. S. FRIERSON and C. A. BROWNE, JR., and Introduction by PAUL CARUS. Price, $1.50 net. (7s. 6d. net.) The first two chapters consist of a general discussion of the general qualities and characteristics of odd and even magic squares and cubes, and notes on their construction. The third describes the squares of Benjamin Franklin and their characteristics, while Dr. Carus adds a further analysis of these squares. 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The volume is attractive in appearance, and what is of the greatest importance in such a work, the proof-reading has been careful." The Nation. The Foundations oi Mathematics. A Contribution to The Philosophy of Geometry. BY DR. PAUL CARUS. 140 pages. Cloth. Gilt top. 75 cents net. (3s. 6d. net.) The Open Court Publishing Co. 623-633 Walmsh Avenue Chicago Engineerings Mathematical Sciences Library M72 UC SOUTHERN 'REGIONAL ,||mr|im|ljimilll A 000210473 5 STATE NOR; ;HQOL LOS AKC