MAGIC SQUARES• M. Bachet endeavoured to discover a general me thod of constructing those squares, and succeeded where the number of cells is the square of the odd numbers, but he failed in those cases when they are the squares of the even numbers. M. Frenicle advanced considerably be yond the method of Bachet ; he not only gave methods for the squares of the odd numbers, but also new me thods for those of the even numbers, and sheaved that both were susceptiole of the greatest variety. He did not even rest satisfied with the natural difficulties of this kind of research, but created others for himself, to have the pleasure of surmounting them. In these in vestigations, it must be confessed, he has shewn an un common degree of address ; we only wish that it had been devoted to subjects of more useful inquiry. Poig uard, a canon of Brussels, made an additional improve ment to these learned trifles, by extending his method to a series of numbers in geometrical or harmonic pro gression, with this difference in the result, that, in the one case, the product of the numbers composing the respective columns were equal ; in the other, the pro gression was still harmonic. M. de la Loubere found traces of magic squares in India ; and even mentions the manlier in which the mathematicians of that country con struct them.
About the same time that magic squares were invent ed, Lucas Paccioli, a native of Tuscany, called also Lucas de Burgo, published a treatise, entitled, Sunzma de Aritlimeticli, Geonzetrid, &c. This work, besides the common rules of arithmetic, contains several inventions of the Arabs, particularly the rules of False Position, which are there given for the first time, and called the Rules of Elcataim. About this tirne,lso, an important change took place in the new method of notation, by which the decimal scale was extended to fractions, as well as integers. The first approach to this improve ment was made by Purbach, a native of a village of the same name, which is situated on the frontiers of Austria and Bavaria. Instead of 3600, (60 x60.) he divided the circle into 600,000 parts, and computed the sines of the arcs for every 10 minutes in terms of this new division.
His pupil, John Muller, afterwards made tables of the sines for every minute ; but finding, in the course of his attention to this subject, that the division of Purbach was, in some respects, inconvenient, he resolved to sub stitute another in its place, and accordingly divided the radius into 1,000,000 parts. The sines were then ob tained in millionth parts of the radius ; but it was soon observed, that the division of the radius was not neces sarily restricted to millionth parts, and that the value of the sines might be ascertained to any degree of exactness by extending the division decimally. In short time, the advantages of this method of expressing fractions over the scxagesimal, brought it into general use in other divisions. Ramus employs decimal fractions in his Az itninetic, which was published in 1550 ; and they had even been employed some time before by Buckley and Recorde, in the extraction of roots. Stevinus, however, is the first who wrote an express treatise on them, which he published in 1582, under the title of La Pra ague d'Arithmetique. Since that period, decimal frac tions have been considered essential to every system of arithmetic.
The light of discovery now began to diffuse its salu tary beams over Europe ; invention quickly succeeded invention, and improvements were introduced with un exampled rapidity into every department of science. About the beginning of the seventeenth century, Baron Napier of Merchiston immortalized his name by the invention of logarithms ; an invention which has not only abbreviated the operations of arithmetic, but conferred the greatest benefit on the higher geometry. By means of it multiplication and division have been reduced to addition and subtraction ; while those operations, in their turn, have taken the place of involution and evolution. To trigonometry, in particular, logarithms have render ed the highest services, and given a degree of simpli city to its calculations, which was scarcely to be ex pected.