It. is certainly a very curious circumstance, that Fohi is accounted by the Chinese the author of the first of these systems ; and Vou-vang, who reigned more than 1100 years before the commencement of the Christian wra, the author of the second. That two philosophers, living at different periods, and in countries remote from each other, should coincide in the discovery of some great physical truth ; or that two poets, in similar circum stances, should agree in some general sentiment, with out the charge of plagiarism being justly applicable to either, is not only possible, but very likely to happen. The phenomena of nature are always the same, and de. pend on the immutable laws which the Author of all things has established ; and the embellishments of poe tical imagery, though less definite, are subject to the unvarying operations of human thought ; but nothing short of actual communication can explain a coincidence of opinion between the inhabitants of two distant na tions, when the object of that opinion is not only im probable, but fanciful in the extreme. In short, we may infer, almost with the confidence of demonstration, that Pythagoras, or some of his disciples, borrowed those opinions from the East, more especially as we . have good grounds for believing, that the system of notation, which is now used in Europe, came originally from the same quarter.
But though the objects of research which the Py thagoreans held in view, were sometimes imaginary and often absurd, their labours were not altogether unpro ductive. They discovered many useful properties of numbers ; and the common table of multiplication is com monly ascribed to Pythagoras himself. The truth is, the first efforts of science must be stimulated by the prospect of advantages that are perhaps altogether fanci ful ; time, sooner or later, winnows the produce, and se parates truth from error. The alchemists did not find out the philosophers' stone, or an universal elixir, but their attempts to discover these ideal substances laid the foundation of chemistry, and made several valuable additions to medicine. One of the greatest benefits which the Pythagoreans conferred on arithmetic, was their rectangular triangles of numbers, or triangles whose sides are so related to one another, that the square of the greatest is equal to the sum of the squares of the other two. Problems of this sort are not without their celebrity in modern times, and have afforded subjects of challenge between mathematicians of the first eminence. They are well calculated to sharpen the powers of in vention, as they generally require new and peculiar me thods of solution.
We regret that we are unable to follow the progress of the arithmetic of the ancients with a satisfactory minuteness of detail. All the information we can glean
with respect to it is exceedingly scanty ; though, con sidering the great and indispensible utility of arithmetic to the other sciences, as well as to the common business of life, we may presume that it always continued to ad vance with more or less rapidity. It appears, by the mutilated remains of Grecian arithmetic, which have descended to our times, that, besides the fundamental operations of addition, multiplication, subtraction, and division, the ancients were acquainted with the methods of extracting the square and cube roots ; and that they understood the theory of arithmetical and geometrical progressions. The methods by which they conducted their calculations were, without doubt, tedious and com plicated, and very unlike those of the present day, but, on the whole, their knowledge of the combinations of numbers, and of the methods of reducing ratios to their simplest forms, was extremely accurate, as well as ex tensive.
Lagus, one of the successors of Alexander the Great, had no sooner obtained the peaceable possession of Egypt, than he began to turn his attention to the encourage ment of science. His munificence and generosity attracted the sages of Greece to his capital, and in a short time rendered it the seat of learning and philosophy. He established the famous school of Alexandria, which con tinued to flourish there during ten centuries, and did more honour to his memory than all the military atchieve ments of his predecessor. This seminary produced many distinguished mathematicians, and among others Euclid, so justly celebrated for his Elements of Geometry ; the seventh, eighth, and ninth books of which, form the oldest treatise on arithmetic extant.
The sieve of Eratosthenes deserves to be noticed in a hsitory of arithmetic, both as an object of curiosity, and as an invention of some importance in the theory of fractions. This sieve, as it is whimsically called, con sists in an easy, though somewhat tedious, method of finding the prime numbers, or those numbers which are divisible only by themselves and unity. For this pur pose, Eratosthenes wrote in succession all the odd num bers (the number 2 being the only even prime. number) as far as he wished his table to extend. He then began at unity, and excluded all the multiples of 3 ; next those of 5 ; next those of 7, &c. cutting them out as he ad vanced. When he had exhausted the whole of the multiples of the odd numbers by these successive inter cisions, the remaining numbers were all prime ; and the table itself, on account of the holes which had been made in it, received the name of sieve, from its resemblance to that utensil.