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.
II
The Progress of Self-conscious Theory
A characteristic of Greek social life is responsible both for the next phase of the development of mathematical thought and for the misapprehension of its nature by so many moderns. "When Archytas and Menaechmus employed mechanical instruments for solving certain geometrical problems, 'Plato,' says Plutarch, 'inveighed against them with great indignation and persistence as destroying and perverting all the good that there is in geometry; for the method absconds from incorporeal and intellectual or sensible things, and besides employs again such bodies as require much vulgar handicraft: in this way mechanics was dissimilated and expelled from geometry, and being for a long time looked down upon by philosophy, became one of the arts of war.' In fact, manual labor was looked down upon by the Greeks, and a sharp distinction was drawn between the slaves who performed bodily work and really observed nature, and the leisured upper classes who speculated, and often only knew nature by hearsay. This explains much of the naïve dreamy and hazy character of ancient natural science. Only seldom did the impulse to make experiments for oneself break through; but when it did, a great progress resulted, as was the case of Archytas and Archimedes. Archimedes, like Plato, held that it was undesirable for a philosopher to seek to apply the results of science to any practical use; but, whatever might have been his view of what ought to be in the case, he did actually introduce a large number of new inventions" (Jourdain, The Nature of Mathematics, pp. 18-19). Following the Greek lead, certain empirically minded modern thinkers construe geometry wholly from an intellectual point of view. History is read by them as establishing indubitably the proposition that mathematics is a matter of purely intellectual operations. But by so construing it, they have, in geometry, remembered solely the measuring and forgotten the land, and, in arithmetic, remembered the counting and forgotten the things counted.
Arithmetic experienced little immediate gain from its new association with geometry, which was destined to be of momentous import in its latter history, beyond the discovery of irrationals (which, however, were for centuries not accepted as numbers), and the establishment of the problem of root-taking by its association with the square, and interest in negative numbers.
The Greeks had only subtracted smaller numbers from larger, but the Arabs began to generalize the process and had some acquaintance with negative results, but it was difficult for them to see that these results might really have significance. N. Chuquet, in the fifteenth century, seems to have been the first to interpret the negative numbers, but he remained a long time without imitators. Michael Stifel, in the sixteenth century, still calls them "Numeri absurdi" as over against the "Numeri veri." However, their geometrical interpretation was not difficult, and they soon won their way into good standing. But the case of the imaginary is more striking. The need for it was first felt when it was seen that negative numbers have no square roots. Chuquet had dealt with second-degree equations involving the roots of negative numbers in 1484, but says these numbers are "impossible," and Descartes (Geom., 1637) first uses the word "imaginary" to denote them. Their introduction is due to the Italian algebrists of the sixteenth century. They knew that the real roots of certain algebraic equations of the third degree are represented as results of operations effected upon "impossible" numbers of the form a + b√-1 (where a and b are real numbers) without it being possible in general to find an algebraic expression for the roots containing only real numbers. Cardan calculated with these "impossibles," using them to get real results [(5 + √-15) (5 - √-15) = 25 - (-15) = 40], but adds that it is a "quantitas quae vere est sophistica" and that the calculus itself "adeo est subtilis ut est inutilis." In 1629, Girard announced the theorem that every complete algebraic equation admits of as many roots, real or imaginary, as there are units in its degree, but Gauss first proved this in 1799, and finally, in his Theory of Complex Quantity, in 1831.
Geometry, however, among the Greeks passed into a stage of abstraction in which lines, planes, etc., in the sense in which they are understood in our elementary texts, took the place of actually measured surfaces, and also took on the deductive form of presentation that has served as a model for all mathematical presentation since Euclid. Mensuration smacked too much of the exchange, and before the time of Archimedes is practically wholly absent. Even such theorems as "that the area of a triangle equals half the product of its base and its altitude" is foreign to Euclid (cf. Cajori, p. 39). Lines were merely directions, and points limitations from which one worked. But there was still dependence upon the things that one measures. Euclid's elements, "when examined in the light of strict mathematical logic, ... has been pronounced by C. S. Peirce to be 'Riddled with fallacies'" (Cajori, p. 37). Not logic, but observation of the figures drawn, that is, concrete symbolization of the processes indicated, saves Euclid from error.