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The combinations of numbers constituted one of the principal objects of his researches: and all anti< quity testifies, that he had carried them to the highest degree. He clothed his philosophy in emblems, which, necessarily differing from the ideas they were intended to represent, became still more obscure in process of time, and occasioned whimsical systems to be attributed to him, which we can hardly suppose to have been the productions of so great a genius.

According to some authors, Pythagoras is at the head of the inventors of the ancient kabbala: he attached several mysterious virtues to numbers, and swore by nothing but the number four, which was to him the number of numbers. In the number three likewise he discovered various marvellous properties, and said, that a man perfectly skilled in arithmetic possessed the sovereign good, &c. But if he did advance such propositions, were they to be taken strictly according to the letter? Is it not more probable, either that his words were erroneously reported, or that they included allegories, with the meaning of which we are unacquainted?

This conjecture appears to be the better founded, as, according to other authors, Pythagoras never having written any thing on the different subjects of philosophy, his doctrines were preserved for a long time solely in his own family and among his scholars; and afterward Plato, with other philosophers, committed them to writing, and corrupted them from vague and confused tradition. On this obscure question, however, which is but little interesting in the present day, I shall say no more. Of all the real or supposed dis

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coveries of Pythagoras in arithmetic Time has respected only his multiplication table: but the taste for investigating numbers and their properties, with which he inspired his school, gave rise to some very ingenious theories; such for instance as that of figurate numbers, which was unfolded by degrees, and of which many useful applications have been made.

It is not possible to follow the progress of arithmetic among the ancients step by step, during the night of time. We can only judge, from their works that have come down to us, that it must have advanced with rapidity, as the first of all the sciences, and the key to the rest. Beside addition, subtraction, multiplication, and division, which are it's principal objects, the ancients possessed methods of extracting the square and cube roots; and they were acquainted with the theory of proportions and arithmetical and geometrical progression. Generally speaking, the combinations of numbers and the reduction of ratios to the most simple form of which they are susceptible, was well known to them: for instance the famous sieve of Eratosthenes, librarian of the Alexandrian museum, affords an easy and commodious method of finding prime numbers, the investigation of which is curious in itself, independently of it's use in the theory of fractions.

By prime numbers, every one knows, those are understood, which have no other divisors but themselves and unity. In the series of even numbers two is the only prime: all the rest, therefore, must be sought among uneven numbers. With this view Eratosthenes wrote on a thin board, or on a sheet of paper stretched

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tight, the series of uneven numbers; and then under every third, fifth, seventh, &c. of these numbers he made a hole in the board or paper. Thus he formed a kind of sieve, through the holes of which he supposed the number above each to fall, and then the remaining numbers would be primes*.

Diophantus, one of the most celebrated mathematicians of the Alexandrian school, made one remark able step in arithmetic. About 350 A. C. he invented the analysis of indeterminate problems, of which so many curious and useful applications have been made, both in pure arithmetic, and in algebra, as well as in the higher geometry.

When a problem, translated into the language of arithmetic or algebra, leads to an equation which contains but one unknown quantity, it is called a determinate problem; and the roots of the equation give all the solutions it admits. Problems of this kind have ultimately no other difficulty, than what arises from the resolution of the equations. But if a problem contain more unknown quantities, than there. are conditions to be expressed,, it is indeterminate; and in this case we cannot find all the unknown quantities, but by affixing to some of them determinate values, taken arbitrarily, or subjected to particular restrictions. Hence two very different cases arise. In the first, that is, when the values.care assumed arbitrarily, the solution is easy, and requires no other precaution than that of avoiding such values

For an explanation and abstract of a similar method I may refer to my Traité d'Arithmétique.

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as would lead to absurd results: but in the second, the choice of some unknown values constitutes of itself an indeterminate problem, which is to be resolved only by a particular art. It was in this art that Diophantus displayed a sagacity truly original. If, for example, the following questions were proposed: to divide a square number into two other square numbers; to find two numbers, the sum of which should be in a given ratio to the sum of their squares; to find two square numbers, the difference of which should be a square: nothing could be more easy than to resolve them, if we were allowed to employ any kind of numbers. But if it were made a condition, that the numbers sought should be rational, and fractions be excluded, the solution would require some address. Diophantus found the method of subjecting all questions of this nature to certain rules, exempt from every kind of conjectural proceeding. His methods bear an evident analogy to those we now employ for the resolution of equations of the first and second order, and hence some authors have taken occasion, to ascribe to him the invention of algebra. He wrote thirteen books of arithmetic, the first six of which have reached us: the rest are lost, if a seventh, which is found in some editions of Diophantus, be not his work. This seventh book contains some learned investigations of the properties of figurate numbers.

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This writer had a number of interpreters among the ancients, but most of their works are loft. these we regret the commentary of the learned Hypatia. A. D. 410. The talents, virtues, and misfor

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tunes of this illustrious victim of fanaticism have a claim to the homage of posterity, and we cannot dispense with paying her this tribute.

The philosopher Theon, her father, had taken such pains to instruct her, and she made so considerable a progress in a short space of time, that she was chosen, when very young, to teach mathematics in the school of Alexandria. All historians agree in saying, that in Hypatia personal beauty was united with uncommon modesty, purity of morals, and consummate prudence. These advantages procured her great respect at Alexandria, particularly from Orestes, the governor of that city. Some wretched theological disputes having excited a bitter dissension between Orestes and St. Cyril, the monks of St. Cyril's faction stirred up the people to massacre Hypatia, as the author of the troubles, in consequence of the advice she gave the governor. This action,' says the historian Socrates, brought great reproach upon Cyril, and the church of Alexandria; for such acts of violence are totally inconsistent with christianity.' Fleury, a man of justice and moderation, but perhaps too much attached to the dogmas of an intolerant religion, does not express with sufficient energy the horrour, with which such an abominable crime should have inspired him.

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