Lectures on Elementary Mathematics Quotes

“Mathematics”

“History of algebra”

“Joseph Louis Lagrange”

“Lectures on Elementary Mathematics 2nd ed. (1901)*oogleBooks”

“We can even determine in this manner, from the shape of the curve itself, whether the problem has possible solutions satisfying the proposed conditions.”

“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.”

“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.”

“[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.”

“The solution of equations of the third and fourth degree was quickly accomplished. But the successive efforts of mathematicians for over two centuries have not succeeded in surmounting the difficulties of the equation of the fifth degree.”

“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...”

“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.”

“[S]ince most frequently it is not necessary to know all possible values of the unknown quantity but only such as solve the problem in hand, it will be sufficient to describe that portion of the curve which corresponds to these roots, thus saving much unnecessary calculation.”

“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.”

“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.”

“[H]e uses the known and the unknown quantities alike. And herein consists virtually the essence of algebra, which is to employ unknown quantities, to calculate with them as we do with known quantities, and to form from them one or several equations from which the value of the unknown quantities can be determined.”

“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...”

“Diophantus regarded the rule of the signs [the principle that in multiplication, $+$ and $-$, give $-$; and $-$ and $-$, give $+$] as a self-evident principle not in need of demonstration. ...[H]e is... likely to have considered it as an axiom, as did Euclid some of the principles of geometry.”

“Vieta and Descartes... Harriot... and Hudde... were the first after the Italians... to perfect the theory of equations, and since their time there is scarcely a mathematician of note that has not applied himself...”

“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 bastions.”

“His [Diophantus'] work contains the first elements of this science [algebra]. He employed to express the unknown quantity a Greek letter which corresponds to our st and which has been replaced in the translations by N. To express the known quantities he employed numbers solely, for algebra was long destined to be restricted entirely to the solution of numerical problems.”