Beginnings of Arithmetic and Geometry
The most primitive mathematical activity of man is counting, but here his first efforts are lost in the obscurity of the past. The lower races, however, yield us evidence that is not without value. Although the savage mind is not identical with the mind of primitive man, there is much in the activities of undeveloped races that can throw light upon the behavior of peoples more advanced. We must be careful in our inferences, however. Among the Australians and South Americans there are peoples whose numerical systems go little, or not at all, beyond the first two or three numbers. "It has been inferred from this," writes Professor Boas (Mind of Primitive Man, pp. 152-53), "that the people speaking these languages are not capable of forming the concept of higher numbers.... People like the South American Indians, ... or like the Esquimo ... are presumably not in need of higher numerical expressions, because there are not many objects that they have to count. On the other hand, just as soon as these same people find themselves in contact with civilization, and when they acquire standards of value that have to be counted, they adopt with perfect ease higher numerals from other languages, and develop a more or less perfect system of counting.... It must be borne in mind that counting does not become necessary until objects are considered in such generalized form that their individualities are entirely lost sight of. For this reason it is possible that even a person who owns a herd of domesticated animals may know them by name and by their characteristics, without even desiring to count them."
And there is one other false interpretation to be avoided. Man does not feel the need of counting and then develop a system of numerals to meet the need. Such an assumption is as ridiculous as to assume prehistoric man thinking to himself: "I must speak," and then inventing voice culture and grammar to make speaking pleasant and possible. Rather, when powers of communication are once attained, presumably in their beginnings also without forethought, man being still more animal than man, there were gradually dissociated communications of a kind approaching what numbers mean to us. But the number is not yet a symbol apart from that of the things numbered. Picture writing, re-representing the things meant, preceded developmentally any kind of symbolization representing the number by mere one-one correspondence with non-particularized symbols. It is plausible, although I have no anthropological authority for the statement, that the prevalence of finger words as number symbols (cf. infra) is originally a consequence of the fact that our organization makes the hand the natural instrument of pointing.
The difficulty of passing from concrete representations to abstract symbols has been keenly stated by Conant (The Number Concept, pp. 72-73), although his terminology is that of an old psychology and the limitations implied for the primitive mind are limitations of practice rather than of capacity as Mr. Conant seems to believe. "An abstract conception is something quite foreign to the essentially primitive mind, as missionaries and explorers have found to their chagrin. The savage can form no mental concept of what civilized man means by such a word as soul; nor would his idea of the abstract number 5 be much clearer. When he says five, he uses, in many cases at least, the same word that serves him when he wishes to say hand; and his mental concept when he says five is a hand. The concrete idea of a closed fist, of an open hand with outstretched fingers, is what is uppermost in his mind. He knows no more and cares no more about the pure number 5 than he does about the law of conservation of energy. He sees in his mental picture only the real, material image, and his only comprehension of the number is, "these objects are as many as the fingers on my hand." Then, in the lapse of the long interval of centuries which intervene between lowest barbarism and highest civilization, the abstract and concrete become slowly dissociated, the one from the other. First the actual hand picture fades away, and the number is recognized without the original assistance furnished by the derivation of the word. But the number is still for a long time a certain number of objects, and not an independent concept."
An excellent fur trader's story, reported to me by Mr. Dewey, suggests a further impulse to count besides that given by the need of keeping a tally, namely, the need of making one thing correspond to another in a business transaction. The Indian laid down one skin and the trader two dollars; if he proposed to count several skins at once and pay for all together, the former replied "too much cheatem." The result, however, demanded a tally either by the fingers, a pebble, or a mark made in the sand, and as the magnitude of such transactions grows the need of a specific number symbol becomes ever more acute.
The first obstacle, then, to overcome—and it has already been successfully passed by many primitive peoples—is the need of fortuitous attainment of a numerical symbol, which is not the mere repeated symbol of the things numbered. Significantly, this symbol is usually derived from the hand, suggesting gestures of tallying, and not from the words of already developed language. Consequently, number words relate themselves for the most part to the hand, and written number symbols, which are among the earliest writings of most peoples, tend to depict it as soon as they have passed beyond the stage mentioned above of merely repeating the symbol of the things numbered. W. C. Eells, in writing of the Number Systems of the North American Indians (Am. Math. Mo., Nov., 1913; pp. 263-72), finds clear linguistic evidence for a digital origin in about 40% of the languages examined. Of the non-digital instances, 1 was sometimes connected with the first personal pronoun, 2 with roots meaning separation, 3, rarely, meaning more, or plural as distinguished from the dual, just as the Greek uses a plural as well as a dual in nouns and verbs, 4 is often the perfect, complete right. It is often a sacred number and the base of a quarternary system. Conant (loc. cit. p. 98) also gives a classification of the meanings of simple number words for more advanced languages; and even in them the hand is constantly in evidence, as in 5, the hand; 10, two hands, half a man, when fingers and toes are both considered, or a man, when the hands alone are considered; 20, one man, two feet. The other meanings hang upon the ideas of existence, piece, group, beginning, for 1; and repetition, division, and collection for higher numerals.
A peculiar difficulty lies in the fact that when once numbering has become a self-conscious effort, the collection of things to be numbered frequently tends to exceed the number of names that have become available. Sometimes the difficulty is met by using a second man when the fingers and toes of the first are used up, sometimes by a method of repetition with the record of the number of the repetition itself added to the numerical significance of the whole process. Hence arise the various systems of bases that occur in developed mathematics. But the inertia to be overcome in the recognition of the base idea is nowhere more obvious than in the retention by the comparatively developed Babylonian system of a second base of 60 to supplement the decimal one for smaller numbers. Among the American Indians (Eells, loc. cit.) the system of bases used varies from the cumbersome binary scale, that exercised such a fascination over Leibniz (Opera, III, p. 346), through the rare ternary, and the more common quarternary to the "natural" quinary, decimal, and vigesimal systems derived from the use of the fingers and toes in counting. The achievement of a number base and number words, however, does not always open the way to further mathematical development. Only too often a complexity of expression is involved that almost immediately cuts off further progress. Thus the Youcos of the Amazon cannot get beyond the number three, for the simplest expression for the idea in their language is "pzettarrarorincoaroac" (Conant, loc. cit., pp. 145, 83, 53). Such names as "99, tongo solo manani nun solo manani" (i.e., 10, understood, 5 plus 4 times, and 5 plus 4) of the Soussous of Sierra Leone; "399, caxtolli onnauh poalli ipan caxtolli onnaui" (15 plus 4 times 20 plus 15 plus 4) of the Aztec; "29, wick a chimen ne nompah sam pah nep e chu wink a" (Sioux), make it easy to understand the proverb of the Yorubas of Abeokuta, "You may be very clever, but you can't tell 9 times 9."
Almost contemporaneously with the beginnings of counting various auxiliary devices were introduced to help out the difficult task. In place of many men, notched sticks, knotted strings, pebbles, or finger pantomime were used. In the best form, these devices resulted in the abacus; indeed, it was not until after the introduction of arabic numerals and well into the Renaissance period that instrumental arithmetic gave way to graphical in Europe (D. E. Smith, Rara Arithmetica, under "Counters"). "In eastern Europe," say Smith and Mikami (Japanese Mathematics, pp. 18-19), "it"—the abacus—"has never been replaced, for the tschotü is used everywhere in Russia to-day, and when one passes over into Persia the same type of abacus is common in all the bazaars. In China the swan-pan is universally used for the purposes of computation, and in Japan the soroban is as strongly entrenched as it was before the invasion of western ideas."
Given, then, the idea of counting, and a mechanical device to aid computation, it still remains necessary to obtain some notation in which to record results. At the early dawn of history the Egyptians seem to have been already possessed of number signs (cf. Cantor, Gesch. de. Math., p. 44) and the Phœnicians either wrote out their number words or used a few simple signs, vertical, horizontal, and oblique lines, a process which the Arabians perpetuated up to the beginning of the eleventh century (Fink, p. 15); the Greeks, as early as 600 B. C., used the initial letters of words for numbers. But speaking generally, historical beginnings of European number signs are too obscure to furnish us good material.
Our Indians have few number symbols other than words, but when they occur (cf. Eells, loc. cit.) they usually take the form of pictorial presentation of some counting device such as strokes, lines dotted to suggest a knotted cord, etc. Indeed, the smaller Roman numerals were probably but a pictorial representation of finger symbols. However, a beautiful concrete instance is furnished us in the Japanese mathematics (cf. Smith and Mikami, Ch. III). The earliest instrument of reckoning in Japan seems to have been the rod, Ch'eou, adapted from the Chinese under the name of Chikusaku (bamboo rods) about 600 A. D. At first relatively large (measuring rods?), they became reduced to about 12 cm., but from their tendency to roll were quickly replaced by the sangi (square prisms, about 7 mm. thick and 5 cm. long) and the number symbols were evidently derived from the use of these rods:
For the sake of clearness, tens, hundreds, etc., were expressed in the even place by horizontal instead of vertical lines and vice versa; thus 1267 would be formed
The rods were arranged on a sort of chessboard called the swan-pan. Much later the lines were transferred to paper, and a circle used to denote the vacant square. The use of squares, however, rendered it unnecessary to arrange the even places differently from the odd, so numbers like 38057 came to be written
instead of
as in the earlier notation.
Somewhere in the course of these early mathematical activities the process has changed from the more or less spontaneous operating that led primitive man to the first enunciation of arithmetical ideas, and has become a self-conscious striving for the solution of problems. This change had already taken place before the historical origins of arithmetic are met. Thus, the treatise of Ahmes (2000 B. C.) contains the curious problem: 7 persons each have 7 cats; each cat eats 7 mice; each mouse eats 7 ears of barley; from each ear 7 measures of corn may grow; how much grain has been saved? Such problems are, however, half play, as appears in a Leonardo of Pisa version some 3000 years later: 7 old women go to Rome; each woman has 7 mules; each mule, 7 sacks; each sack contains 7 loaves; with each loaf are 7 knives; each knife is in 7 sheaths. Similarly in Diophantus' epitaph (330 A. D.): "Diophantus passed 1/6 of his life in childhood, 1/12 in youth, and 1/7 more as a bachelor; 5 years after his marriage, was born a son who died 4 years before his father at 1/2 his age." Often among peoples such puzzles were a favorite social amusement. Thus Braymagupta (628 A. D.) reads, "These problems are proposed simply for pleasure; the wise man can invent a thousand others, or he can solve the problems of others by the rules given here. As the sun eclipses the stars by its brilliancy, so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems, and still more if he solves them" (Cajori, Hist. of Math., p. 92).
The limitation of these early methods is that the notation merely records and does not aid computation. And this is true even of such a highly developed system as was in use among the Romans. If the reader is unconvinced, let him attempt some such problem as the multiplication of CCCXVI by CCCCLXVIII, expressing it and carrying it through in Roman numerals, and he will long for the abacus to assist his labors. It was the positional arithmetic of the Arabians, of which the origins are obscure, that made possible the development of modern technique. Of this discovery, or rediscovery from the Hindoos, together with the zero symbol, Cajori (Hist. of Math., p. 11) has said "of all mathematical discoveries, no one has contributed more to the general progress of intelligence than this." The notation no longer merely records results, but now assists in performing operations.
The origins of geometry are even more obscure than those of arithmetic. Not only is geometry as highly developed as arithmetic when it first appears in occidental civilization, but, in addition, the problems of primitive peoples seem to have been such that they have developed no geometrical formulæ striking enough to be recorded by investigators, so far as I have been able to discover. But just as the commercial life of the Phœnicians early forced them self-consciously to develop arithmetical calculation, so environmental conditions seem to have forced upon the Egyptians a need for geometrical considerations.
It is almost platitudinous to quote Herodotus' remark that the invention of geometry was necessary because of the floods of the Nile, which washed away the boundaries and changed the contours of the fields. And as Proclus Diadochus adds (Procli Diadochi, in primum Euclidis elementorum librum commentarii—quoted Cantor, I, p. 125): "It is not surprising that the discovery of this as well as other sciences has sprung from need, because everything in the process of beginning proceeds from the incomplete to the complete. There takes place a suitable transition from sensible perception to thoughtful consideration and rational knowledge. Just as with the Phœnicians, for the sake of business and commerce, an exact knowledge of numbers had its beginning, so with the Egyptians, for the above-mentioned reasons, was geometry contrived."
The earliest Egyptian mathematical writing that we know is that of Ahmes (2000 B. C.), but long before this the mural decorations of the temple wall involved many figures, the construction of which involved a certain amount of working knowledge of such operations as may be performed with the aid of a ruler and compass. The fact that these operations did not earlier lead to geometry, as ruler and compass work seems to have done in Japan in the nineteenth century (Smith and Mikami, index, "Geometry"), is probably due to the stage at which the development of Egyptian intelligence had arrived, feebly advanced on the road to higher abstract thinking. It is everywhere characteristic of Egyptian genius that little purely intellectual curiosity is shown. Even astronomical knowledge was limited to those determinations which had religious or magically practical significance, and its arithmetic and geometry never escaped these bounds as with the more imaginative Pythagoreans, where mystical interpretation seems to have been a consequence of rather than a stimulus to investigation. An old Egyptian treatise reads (Cantor, p. 63): "I hold the wooden pin (Nebi) and the handle of the mallet (semes), I hold the line in concurrence with the Goddess Sạfech. My glance follows the course of the stars. When my eye comes to the constellation of the great bear and the time of the number of the hour determined by me is fulfilled, I place the corner of the temple." This incantation method could hardly advance intelligence; but the methods of practical measuring were more effective. Here the rather happy device of using knotted cords, carried about by the Harpedonapts, or cord stretchers, was of some moment. Especially, the fact that the lengths 3, 4, and 5, brought into triangular form, served for an interesting connection between arithmetic and the right triangle, was not a little gain, later making possible the discovery of the Pythagorean theorem, although in Egypt the theoretical properties of the triangle were never developed. The triangle obviously must have been practically considered by the decorators of the temple and its builders, but the cord stretchers rendered clear its arithmetical significance. However, Ahmes' "Rules for attaining the knowledge of all dark things ... all secrets that are contained in objects" (Cantor, loc. cit., p. 22) contains merely a mixture of all sorts of mathematical information of a practical nature,—"rules for making a round fruit house," "rules for measuring fields," "rules for making an ornament," etc., but hardly a word of arithmetical and geometrical processes in themselves, unless it be certain devices for writing fractions and the like.