Strabo mentions, that, in his time, the invention of arithmetic was ascribed to the Phoenicians ; and Cedre nus affirms, that the first system of this science was written in the Phoenician language, by Agenor, the son of Phccnix but this is by no means probable. It will readily be granted, that the operations of arithmetic might have been improved by the Phoenicians, who were a commercial and enlightened people ; but there can be no question of their having borrowed the first notions of number from their neighbours the Egyptians.
On the other hand, Josephus maintains, perhaps with a degree of national partiality, that Abraham was the inventor of arithmetic, and that the descendants of that patriarch communicated the knowledge of numbers to the Egyptians. Whatever credit we may be disposed to give to this assertion, it must be admitted, that tho Greeks, at least, copied both their alphabet and method of notation from the Hebrews. The latter employed the first nine letters of their alphabet, aleph, beth, ghi nzel, daleth, he vau, &c. to represent the nine digits ; and we cannot obtain a more convincing proof, that the Greeks, in this respect, followed their example, than the circumstance of their introducing a new character pat) to denote six, because they had no vau in their own alphabet, and this letter, according to the Hebrew notation, corresponds to that number ; so that if we had to determine the merit of the invention of numbers between the Hebrews and the Greeks alone, we should not hesitate to ascribe it to the former ; though the Greeks were undoubtedly the first European nation among whom arithmetic arrived at any degree of perfection.
The mathematics had already begun to be cultivated in Greece when Thales appeared ; but from that period we may date the commencement of a more rapid pro gress. This eminent philosopher resolved to travel in search of knowledge among the Indians and Egyptians, at a time of life when, in general, the ardour of new pursuits begins to subside, and the desire of informa tion yields to the more selfish cares of approaching old age. During his residence in Egypt, he measured the height of the pyramids, by means of their shadows pro jected on the plain below ; and thus gave the first in stance, upon record, of the measurement of an inac cessible object by the relations of the sides of similar triangles, and, probably too, one of the first applications of arithmetic to geometry. When he was fully instruct ed in all the learning of the East, Thales returned to his own country to compare and arrange the materials of his knowledge, and soon diffused a taste for science throughout Greece.
The Hebrews, Greeks, and Romans, seem to have employed nearly the same kind of notation, but the man ner in which they performed their calculations is very imperfectly ascertained. Goguet is of opinion that, first of all, they used pebbles or small stones for that pur pose ; and 1,1,4(pzezv, calculare to calculate, derived from calculus a small stone, certainly gives much plausibility to the supposition. To this day, in many provinces of Russia, as well as China, arithmetical ope rations are performed by means of stringed beads, with great accuracy, and considerable expedition.
Boethius, whom we shall afterwards mention more particularly, informs us, that certain disciples of Py thagoras invented, or at least employed, peculiar cha racters called apices for respresenting numbers, which enabled them greatly to simplify the operations of arith metic. These apices were nine in number, and appear to have had no small resemblance to the arithmetical characters which were employed at a later 'period by the Arabs. There is some reason, however, for suspect ing, that they have been inserted by some copyist, as they are only found in manuscripts of the fifteenth cen tury, and resemble the Arabian figures used about that time.
Pythagoras was a disciple of Thales, who was soon led to form the highest expectations of the future celebrity of his pupil. Nor was he disappointed ; for to no man, perhaps, has science been more indebted. Like his mas ter, lie travelled among the Indians and Egyptians, and scrupled not to comply with the customs peculiar to the eastern countries, in order to obtain freer accesss to the learned men. As he spent two-and-twenty years in those parts, and to great original talents joined the most persevering industry, and the most ardent love of knowledge, he collected much important and valuable information. Among other objects of attention, he did not overlook the science of arithmetic ; and if his fol lowers really made use of the characters mentioned by Boethius, he himself probably transplanted them from India to Greece. It does not appear, however, that the method of computing by them was ever brought into general use among the Greeks. The Pythagorean philosophers either wished to conceal them from the vulgar, or were not aware themselves of all the advan tages they possessed ; at least they must have fallen again into disuse before the time of Pappus, who lived towards the close of the fourth century, as is evident from a fragment which has descended to us of that dis tinguished mathematician. The complicated contri vances which he has there employed, to simplify the ordinary methods of multiplication by large numbers, render it highly probable, that at the time in which he lived, the Greeks performed their calculations, as the common people do at present, by mere dint of thought, and without the aid of visible signs. The very circum stance of the Pythagoreans devoting their exclusive at tention to discover the abstract properties .of numbers, instead of trying to simplify the methods of calculation, gives additional force to this supposition. We must here remark, that numbers have two kinds of properties, one of which is essential and inseparable from their very nature ; and another, which is accidental, and derived entirely from the manner of representing them. It is an essential property of number, for instance, that the successive sums of the odd numbers are squares 1, 1+3=4, 1+3+5=9, 1+3+5+7=16, &c.; but it is an accidental property of 9, that the sum of the digits which represent its products, is always either 9 itself, or a multiple of 9. Now the Pythagoreans confined their attention solely to the former of these properties, dis tributing numbers into classes of perfect and imperfect, abundant and defective, &c. and applying to them dis tinctions which are totally independent of the scale of notation. They even pretended to discover occult quali ties, and mysterious relations of numbers, and were absurd enough to believe, that the world itself was created with a reference to their abstract properties. The first four odd numbers 1, 3, 5, 7, according to their wild speculations, represented the pure and celestial parts of the universe ; while the first four even num bers being less exalted in dignity, represented the same elements combined with terrestrial matter. The sum of all these numbers is 36, which, of course, was con ceived to possess high and wonderful virtues ; according to Plutarch, it was held in such veneration by the Py thagoreans, that to swear by it was to contract the most solemn of all obligations. Plato was supposed to have carried the Tetrachys, for so this number was called, to a still higher degree of perfection, by advancing it to 40. Instead of the odd numbers 1, 3, 5, 7, he substitu ted I, 3, 7, 9, and made 5, which holds invisibly the mid dle place in this series, to denote the supreme intelli gence, or Ness. The even numbers 2, 4, 6, 8, were left to perform the same functions as before.