144 HISTORY OF SCIENCE. 



Franciscan friar, called usually LUCAS DE BORGO, He travelled in the 

 East, and afterwards taught mathematics at Naples, Venice, and Milan. 

 He published, about the middle of the sixteenth century, a book on 

 arithmetic, algebra, and geometry ; another on regular figures, poly- 

 gons, etc. ; and-a third on the division of a line into sections in extreme 

 and mean ratio. BENEDETTO, another Italian mathematician, pub- 

 lished at Turin, in 1585, a treatise on geometrical analysis. Many 

 translations of the works of the ancient geometers, and commentaries 

 on those works, appeared in the sixteenth century. The editors and 

 the commentators contributed greatly to the progress of their science, 

 forming, as they did, a body of pioneers who undertook much useful 

 and necessary work, though it was not of a kind to require any detailed 

 notice in these pages, 



A great problem for algebraists has been the discovery of some 

 general methods of solving equations of the higher degrees. Equa- 

 tions of first and second degrees that is, those in which the unknown 

 quantities enter simply, or under the second power, which are other- 

 wise called respectively simple and quadratic equations were plainly 

 capable of solution in every case. But, when attempts were made to 

 solve equations of the third degree (cubic equations], it was found that, 

 except in certain obvious cases, no general method could be assigned. 

 The most persevering efforts have been made by mathematicians in 

 vain, to discover a general method of solving equations of every de- 

 gree ; they have hitherto had to content themselves with approximative 

 methods, which however are sufficient in practice. The theory and 

 solution of equations received attention at the hands of the early alge- 

 braists. A method of solving a large class of cubic equations was first 

 given to the world by CARDAN in 1545, and this circumstance will per- 

 petuate his name in connection with mathematics, although the dis- 

 covery was not his own. It is related that Cardan first obtained his 

 knowledge of the method from NICOLO TARTAGLIA, a teacher of mathe- 

 matics at Venice, who, having revealed it to Cardan under an oath of 

 secrecy, was greatly provoked when Cardan published it as his own dis- 

 covery. The rule, as explained by Tartaglia, was, however, extended 

 by Cardan, who also supplied a demonstration of its truth, and pointed 

 out a method of approximation for those cases to which the rule could 

 not be applied. The problem of the exact solution of cubic equations 

 has not been carried further than Garden brought it. BOMBELLI, an 

 Italian who also wrote on algebra, pointed out that those problems 

 which involved cubic equations not reducible by Cardan's rule, admit 

 of a geometrical construction by the trisection of a circular arc. The 



signs for plus and minus, H , were first used in a work published at 



Nuremberg in 1544, and in the first treatise in the English language 

 on algebra, published by Robert Recorde in 1557, the sign of equality 

 ~ appears for the first time. Another great improvement in algebra 

 was made by VIETA, a native of Fontenoy, in France. He held an 



