it

which they principally excelled. A few names, will be perceived, occur in both lists, because they may, with equal propriety, be classed with the scientific and intellectual philosophers of their age.

CHAPTER II.

THE PROGRESS OF THE ABSTRACT SCIENCES.

SECTION I.

ON THE PRINCIPAL MATHEMATICIANS OF THE SIXTEENTH CENTURY.

245. THE history of mathematical science has been sketched, both in its earliest period and its intermediate stages: we now proceed to view it in its full vigour and maturity of growth. The retrospect of the scientific history of the middle ages, it will be remembered, terminated with two illustrious individuals, Purbach and Regiomontanus, the latter of whom died towards the close of the fifteenth century. But the impulse of his genius was felt long after his death, and its effects were quickly apparent in the rapid improvement of those sciences, to the cultivation of which his distinguished talents had been devoted. Among those who seem to have been stimulated by his example, and encouraged by his brilliant success, was WERNER, who is stated to have been "the first among the moderns acquainted with the method of geometrical analysis." In an earlier part of this elementary history, some account was given of the invention of this beautiful method of demonstration. (§ 72.) It seems, however, to have been long forgotten, if not wholly

lost in subsequent ages: nor was any book known to exist which contained a development of this system, by the help of which the ancient mathematicians were enabled to demonstrate so many simple and elegant propositions. "Euclid's Data" was the only work then discovered, bearing, however remotely, on the subject. With this slender assistance, Werner re-discovered the "Method of Analysis," and applied it to the solution of some of the most difficult problems. His mathematical works are now little known, having been long since superseded by others of far greater value.

246. CARDAN and TARTAGLIA were contemporary algebraists, whose united labours contributed in no ordinary degree to the advancement of that science. The former was born at Milan, A. D. 1501; the latter at Brescia, a few years after. Cardan is best known to posterity on account of the celebrated rule for solving equations of the third order, which still bears his name; but to Tartaglia seems rather to have belonged the honour of the invention. A hateful rivalry subsisted between these two distinguished mathematicians, which only terminated with their death, in consequence of Cardan having taken to himself the merit of this scientific discovery, which Tartaglia asserted to have been communicated to him in confidence, and under a promise of secrecy. Cardan, on the contrary, affirmed, that though the formula was communicated to him by Tartaglia, the method of demonstrating it was discovered jointly by himself and his pupil Ferrari, the latter of whom carried the process farther, by solving equations of