of the Renaissance, including Vieta. Yet even they mentioned these so-called false or fictitious roots only in passing, and without grasping their real significance and importance. On this subject Cardan and Bombelli had advanced to about the same point as had the Hindoo Bhaskara, who saw negative roots,
but did not approve of them. The generalisation of the conception of quantity so as to include the negative, was an exceedingly slow and difficult process in the development of algebra.
We shall now consider the history of geometry during the
Renaissance. Unlike algebra, it made hardly any progress. The greatest gain was a more intimate knowledge of Greek geometry. No essential progress was made before the time of Descartes. Regiomontanus, Xylander of Augsburg, Tartaglia,
Commandinus of Urbino in Italy, Maurolycus, and others,
made translations of geometrical works from the Greek. John Werner of Nürnberg published in 1522 the first work on
conics which appeared in Christian Europe. Unlike the geometers of old, he studied the sections in relation with the cone, and derived their properties directly from it. This mode of studying the conics was followed by Maurolycus of Messina (1494–1575). The latter is, doubtless, the greatest geometer of the sixteenth century. From the notes of Pappus, he
attempted to restore the missing fifth book of Apollonius
on maxima and minima. His chief work is his masterly and original treatment of the conic sections, wherein he discusses
tangents and asymptotes more fully than Apollonius had done, and applies them to various physical and astronomical problems.
The foremost geometrician of Portugal was Nonius; of
France, before Vieta, was Peter Ramus, who perished in the
massacre of St. Bartholomew. Vieta possessed great familiarity with ancient geometry. The new form which he gave to algebra, by representing general quantities by letters, enabled him to point out more easily how the construction of the roots of cubics depended upon the celebrated ancient problems of the duplication of the cube and the trisection of an
angle. He reached the interesting conclusion that the former problem includes the solutions of all cubics in which the radical in Tartaglia's formula is real, but that the latter problem includes only those leading to the irreducible case.
The problem of the quadrature of the circle was revived in
this age, and was zealously studied even by men of eminence and mathematical ability. The army of circle-squarers became