Lectures on Elementary Mathematics

"Diophantus... does not proceed beyond equations of the second degree, and we do not know if he or any of his successors... ever pushed... beyond this point."

"Although the work of Diophantus contains indeterminate problems almost exclusively, the solution of which he seeks in rational numbers,— problems which have been designated after him Diophantine problems, —we nevertheless find in his work the solution of a number of determinate problems of the first degree, and even of such as involve several unknown quantities. In the latter case, however, the author invariably has recourse to... reducing the problem to a single unknown quantity, —which is not difficult."

"It was geometry really that suggested to us the use of negative quantities, and herein consists one of the greatest advantages that have resulted from the application of algebra to geometry, —a step which we owe to Descartes."

"Algebra is a science almost entirely due to the moderns... for we have one treatise from the Greeks, that of Diophantus... the only one which we owe to the ancients in this branch of mathematics. ...I speak of the Greeks only, for the Romans have left nothing in the sciences, and to all appearances did nothing."

"Diophantus was not known in Europe until the end of the sixteenth century, the first translation having been a wretched one by Xylander made in 1575. Bachet de Méziriac... a tolerably good mathematician for his time, subsequently published (1621) a new translation... accompanied by lengthy commentaries, now superfluous. Bachets translation was afterwards reprinted with observations and notes by Fermat [1670]."

"Prior to the discovery and publication of Diophantus... algebra had already found its way into Europe. Towards the end of the fifteenth century [1494] there appeared in Venice a work by... Lucas Paciolus on arithmetic and geometry in which the elementary rules of algebra were stated."

"At this period, Italy, which was the cradle of algebra in Europe, was still almost the sole cultivator of the science, and it was not until about the middle of the sixteenth century that treatises on algebra began to appear in France, Germany, and other countries."

"Tartaglia expounded his solution in bad Italian verses in a work treating of divers questions and inventions printed in 1546, a work which enjoys the distinction of being one of the first to treat of modern fortifications by s."

"The works of Peletier [1554 ] and Buteo [i.e., Jean Borrel, who published the algebraic text, Logistica (1559)] were the first which France produced in this science..."

"In the subsequent period the resolution of equations of the third degree was investigated and the discovery for a particular case ultimately made by... Scipio Ferreus (1515). ...Tartaglia and Cardan subsequently perfected the solution of Ferreus and rendered it general for all equations of the third degree."

"[T]he Europeans, having received algebra from the Arabs, were in possession of it one hundred years before the work of Diophantus was known to them. They made, however, no progress beyond equations of the first and second degree."

"He [Diophantus] gives, also, the solution of equations of the second degree, but is careful so to arrange them that they never assume the affected form containing the square and the first power of the unknown quantity. ...he always arrives at an equation in which he has only to extract a square root to reach the solution..."