how to make this choice. By his method, on taking away the outer circumference from a magic square, or any circumference when the square has enough to admit this, or even several circumferences together, the remaining square is still magic. He likewise inverted this condition, requiring that any circumference taken at pleasure, or several circumferences, should be inseparable from the square; that is, that It should cease to be a magic square, if these were taken away, and not if others were. In 1703 Mr. Puignard, a canon of Brussels, published a book on magic squares, in which he made two innovations, that extend and embellish the problem. 1st: Instead of taking all the numbers that fill a square, for instance the thirty six consecutive numbers that would fill all the cells of the natural square of six, he takes only so many of these numbers as there are units in the side of the square, that is in this case six; and these six numbers alone he disposes in such a manner, in the thirty six cells, that no one is repeated twice in the same rank, either horizontal, vertical, or diagonal: whence it necessarily follows, that all the ranks, taken in any direction whatever, uniformly produce the same sum. 2dly: Instead of taking these numbers only in the natural series, or in arithmetical progression, he takes them likewise in geometrical and harmonical progression; but with the last two progressions the magical contrivance of the square necessarily changes. In squares filled by numbers in geometrical progression, the products arising from the multiplication of the 2 numbers numbers in each rank must be equal; and in the harmonic progression the numbers of each rank in like manner follow this progression. Poignard likewise makes squares of these three progressions repeated. La Hire, a geometrician of the academy of sciences, having his attention called to the subject by these researches, in which mere conjecture had frequently been employed, investigated and demonstrated it's principles in two very curious papers in the Memoirs of the Academy for 1705. In these he adds several new problems, which, extending and generalizing the theory still farther, render it the more interesting to those, who are fond of the combinations of numbers. The demonstrations of all these learned men appearing to Sauveur, another geometrician of the academy of sciences, too complex, and too little connected, he undertook to subject this theory to analytical calculation, and uniform methods; whence he might afterward deduce as corollaries simple and easy means of constructing magic squares in all cases. Pajot Osembrai considered the question in the same point of view; and to him we are indebted for a new analytical method for magical squares of even numbers, those for uneven numbers having been sufficiently investigated. Lastly Rallier des Ourmes still farther improved and extended all these methods in an excellent memoir presented to the academy of sciences. We have every reason to suppose, that the subject is now exhausted. NA This This discovery of magic squares by Moschopulos may be called the last breath of the greek mathematicians. After the taking of Constantinople by Mohammed 11, we hear of them in these climes no longer. CHAP. State of the Sciences among the Christians in the West, to the End of the thirteenth Century. THE christians in general for a long time displayed a great aversion to the sciences. Subjected from the origin of christianity to a multitude of superstitious opinions, which tended to convert man into a contemplative automaton, they looked with indifference or disdain on all occupations foreign to religious worship, or to the labours absolutely necessary to procure them subsistence. However, when they had begun to drive the arabs out of some parts of Spain, in the beginning of the tenth century, the voluntary or compulsive intercourse which they had with these people excited the electric fire of genius among the christians, and many of them were eager to acquire knowledge from those moors, whose religion they held in abhorrence. We have already said, that pope Silvester 11 had learned arithmetic from the arabs of Spain. Alphonsus 11, king of Castile, founded in his capital a sort of college or lyceum for the advancement of astronomy, and entrusted it's principal direction to some arabs. He himself made observations and calculations with them. These mutual labours produced the celebrated Alphonsine Tables, more accurate and complete than any that had preceded them. And the study of astronomy was pursued in Castile long after the death of Alphonsus. But the interests of ambi tion, which nothing can withstand, perpetually cherished the seeds of hatred and discord between the arabs and the christians. The latter, never losing sight of their project of retaking all Spain, gained ground every day in proportion as their victories multiplied, the sciences were neglected: and these at length received their mortal blow, when the moors were completely expelled from Spain by the loss of Granada, a. d. 1492; an event to be regretted in the annals of the human mind, and advantageous to nothing but the catholic religion, the empire of which it extended on the ruins of islamism. In the other christian countries of Europe we find. many men distinguished for the extent of their know ledge, considering the time in which they lived; or by the proofs of genius, which they exhibited, and from which society might have derived the most striking benefits, had not ecclesiastical authority, ever intole rant, and ever clothed in thunder, too frequently checked or totally stopped their career. The italians are the first, that here present themselves to our view; and at the outset their attention was drawn to algebra by an accidental circumstance. One Leonard, a rich merchant of Pisa, made several voyages to the East in pursuing his commerce; and his intercourse with the arabs affording him an opportunity of attaining a knowledge of algebra, which was then considered as the sublime part of arithmetic; and he imparted his acquirements to his countrymen about the beginning of the thirteenth century. It was supposed till very lately, on the authority of Vossius and some modern italian authors, that Leonard of |