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PERIOD THE SECOND.

STATE OF THE MATHEMATICS, FROM THEIR REVIVAL AMONG THE ARABS TO THE END OF THE FIFTEENTH CENTURY.

THE mathematics still flourished in Greece, and more particularly in the school of Alexandria, when a little before the middle of the seventh century a tremendous storm arose, which threatened their total destruction in those climes. A. D. 638. Filled with all the enthusiasm a militant religion inspired, the successors of Mohammed ravaged that vast extent of country, which stretches from the east to the southern confines of Europe. All the cultivators of the arts and sciences, who from every nation had assembled in Alexandria, were driven away with ignominy. Some fell beneath the swords of the conqueror : others fled into remote countries, to drag out the remainder of their lives in want. The places and the instruments, which had served for making an immense number of astronomical observations, were involved in one common destruction. In fine, that precious depository of human knowledge, the library of the Ptolemics, which had already suffered by fire under Julius Caesar, was entirely delivered to the

flames

Eames by the arabs. The khalif Omar directed all It's books to be burnt; because, observed he, if they, agree with the Koran, they are useless; if they differ from it, they ought to be held in detestation, and destroyed an argument worthy of a fanatic robber.

The fate of the sciences, thus attacked and annihilated in the heart of their empire, must seem absolutely desperate. But that same vicissitude, which produces so many misfortunes and so many crimes, sometimes brings about revolutions, by which mankind is benefitted. Such was the change, that speedily took place in the manners of the arabs. This people, like all those of the east, had formerly possessed some notions of the sciences, and in particular of astronomy. If the fanaticism of a sanguinary religion at first overwhelmed these seeds, it did not totally dry up their germes. When the different nations were weary of mutually exterminating each other, their ferocity was softened, and the leisure of peace recalled the active minds of the arabs to less empty and more agreeable occupations, than disputing on the dogmas of the Koran. Scarcely had a hundred and twenty years elapsed after the death of Mohammed, when they began themselves to cultivate those arts and sciences, which they had endeavoured to proscribe. Soon they had their poets, orators, mathematicians, &c. In this number are reckoned several of the khalifs of the arabs; and afterward several of the emperours of the persians, when these people became a separate empire.

The arabs derived the principles of all parts of the mathematics from an assiduous study of the greck writers.

writers. Furnished with these acquisitions they became emulous of their masters, and rendered themselves capable of translating, commenting, and sometimes adding to their discoveries. Hence the substance of several works of the greeks has reached us only through the medium of arabic versions. By the arabs other nations were taught, and the sciences were revived with such success, as posterity ought never to forget. Let us proceed to particulars.

СПАР.

CHAP. I.

The Arithmetic and Algebra of the Arabs.

For the ingenious system of arithmetical numeration, which all modern nations employ, they are indebted to the arabs. It has the advantages of clearness and simplicity over all those of the ancients. With ten characters, made to occupy different places, we are able to express in the most commodious manner a number immense in multitude. Some writers assert, that the arabs derived this idea from the hindoos but the arguments they adduce do not appear to me very convincing. Without entering into this idle disquisition, I shall content myself with observing, that we derive our arithmetic, as we practise it at present, immediately from the arabs. The celebrated Gerbert, afterward pope Silvester 11, went to Spain, then under the dominion of the arabs, to study this science, and thence diffused it over the rest of Europe, about the year 960.

The first notions of algebra, which are found in Diophantus, were unfolded by the arabs. Cardan even considers these people as the real inventors of algebra. The celebrated analyst Wallis, adopting this opinion, assigns as a reason for it, that the arabs in the denominations of the powers employed a system different from that of Diophantus; whence he concludes, that their principles were likewise different.

In the greek author the second, third, fourth, fifth, and sixth power, and so on, are called the square, cube, quadrato-quadratum, quadrato-cubus, cubo-cubus, &c.; so that each power took it's denomination from the two inferiour powers, by which it was produced. With the arabs they are the square, cube, quadrato-quadratum, sursolid, quadrato-cubus, second sursolid, &c.; where we find, that those powers, which are not the product of two powers of the same kind, are called sursolids. For instance, in Diophantus the quadrato-cubus, or cube multiplied by the square, forms the fifth power: the arabs mean by the same expression the square of the cube, or the cube of the square, which with them forms the sixth power. The reader will weigh the force of this conjecture of Wallis.

We do not accurately know the extent of the progress, which the arabs made in algebra; but we have some indications, that they had advanced so far as to resolve equations of the third order, and even some particular cases in the fourth; in which they went farther than Diophantus, who did not get beyond the second. As a proof of this it is asserted, that in the Leyden library there is an arabic manuscript, entitled, The Algebra of cubic Equations, or the Solution of solid Problems.

СНАР.

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