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State of the Sciences among the Chinese and Hindoos.
we were to discuss the high opinion, which has been hitherto entertained of the acquirements of the chinese in all branches of science, we should find no very solid foundation for it in the period under our consideration. The arithmetic and geometry of that nation still remain very imperfect; and we find among them no new theory, no interesting application, of the principles of mechanics. It is true the chinese have been assiduous observers of the stars: but all their observations are confined to the most common objects in astronomy, such as eclipses, the positions of the planets, the solstitial altitudes of the Sun, and occultations of stars by the Moon: while no deduction of importance to the progress of the science stands forward to meet the eye. I shall only remark, that the emperor Hupilay, the fifth successor of Genghis khan in China, and the founder of the dynasty of Ywen in 1271, was a great patron of astronomy. He was brother to Houlagou, already mentioned, and resembled him in disposition. He appointed Co-CheouKing chief of the tribunal of mathematics; an industrious observer, who carried the chinese astronomy to a degree of precision, which it had never reached before. But this lustre was transient: the chinese astronomy fell back into it's former languor, and did
not again raise it's head, till about a century after, under the emperors of a new dynasty, who gave the direction of the mathematical tribunal into the hands of mohammedan astronomers.
On the history of the sciences among the hindoos at the same time we may be still more brief. Their knowledge never extended beyond the elements of mathematics; and their astronomy had nearly the same fate as that of the persians after the death of Ulugh Beg.
State of the Sciences among the modern Greeks,
On the destruction of the alexandrian school, the men of science, who were dispersed over all parts of Greece, contributed at first to keep up a taste for the mathematics in that country; but in the state of neglect, to which they were reduced, they could not fail to decline. Indeed many ages elapsed, before any modern greek exhibited the least spark of that genius, which had animated Euclid, Archimedes, Apollonius, &c. Zonaras and Tzetzes, who have been quoted on occasion of the burning glasses of Archimedes, were mere compilers, and in many cases but little acquainted with the subjects on which they treated. At length, in the beginning of the fifteenth century [A. D. 1420], Emanuel Moschopulos, a greek monk, made the very ingenious discovery of magic squares, It is true this was of no practical utility; but it ranks among those theoretical and subtle speculations, which exercise the mind while they amuse it: and as I cannot dispense with mentioning it here, I shall give at once a general sketch of the labours of modern geometricians on this subject, that I may not have to return more than once to a matter of mere curiosity.
Let a geometrical square, each side of which is represented by a given number, as for example the
number 5, be traced on a plane; and let every side, both vertical and horizontal, be divided into five equal parts, and the points that mark the divisions be joined by vertical and horizontal lines. Thus the square will be divided into twenty-five equal cells; and if, beginning from one of the angular cells, and proceeding successively through all the horizontal or vertical rows, the series of numbers, 1, 2, 3, 4, 5, 6, &c. be written, the last cell will contain the number 25, which is the square of 5. This dis position of the figures in their natural order forms in consequence a natural square; the numbers in each row compose an arithmetical progression; and the sums of all these progressions are different. But if the order of the numbers be changed, and they be arranged in such a manner, that all the rows, including even the two diagonal rows, produce the same sum, the square obtains the name of magical. This epithet may have been derived from the singular property of these squares, at a time when the mathe matics were considered as a sort of magic: but it is not improbable, that it was taken from the superstitious application of these squares to the construction of talismans in an age of ignorance. For instance, Cornelius Agrippa, who lived in the fifteenth century, has given in his treatise on Occult Philosophy the magic squares of all the numbers from three to nine: and these squares, according to Agrippa and the followers of the same doctrine, are planetary: the square of three belongs to Saturn; of four, to Jupiter; of five to Mars; of fix, to the Sun; of seven,
to Venus; of eight, to Mercury; and lastly, that: of nine to the Moon. Hist. de l'Acad. 1705, p. 71.
The methods of Moschopulos for the formation of magic squares extend only to certain particular cases, and required to be generalized. Bachet de Meziriac, a very learned analyst about the beginning of the seventeenth century, found a method for all squares with uneven roots; such as 25, 49, 81, &c., the roots of which are 5, 7, 9, &c. In these cases there is a central cell, which facilitates the solution of the problem; but Bachet could not solve it com pletely for squares, the root of which is an even number.
Frenicle de Bessi, one of the oldest members of the academy of sciences, a profound arithmetician, considerably increased the numbers of cases and combinations that produce magical squares, both for even and uneven numbers. For instance, a skilful algebraist had supposed, that the sixteen numbers, which fill the cells of the natural square of four, could produce no more than sixteen magical squares; but Frenicle showed, that they were capable of forming 880. To this research he added a new difficulty. Having formed, for example, one of the magic squares of the number 7; if the two extreme horizontal ranks, and the two extreme vertical ones, be taken from the 49 cells which composed it, a square will remain, which in general will not be a magic square, but may be so, if the primitive magic square be properly chosen for the purpose. Frenicle taught