The Project Gutenberg eBook #36640: Lectures on Elementary Mathematics

LECTURE III.

ON ALGEBRA, PARTICULARLY THE RESOLUTION OF EQUATIONS OF THE THIRD AND FOURTH DEGREE.

Algebra is a science almost entirely due to the moderns. I say almost entirely, for we have one treatise from the Greeks, that of Diophantus, who flourished in the third^∗ century of the Christian era. This work is the only one which we owe to the

ancients in this branch of mathematics. When I speak of the ancients I speak of the Greeks only, for the Romans have left nothing in the sciences, and to all appearances did nothing.

Diophantus may be regarded as the inventor of algebra.^† From a word in his preface, or rather in his letter of dedica-

tion, (for the ancient geometers were wont to address their pro- ductions to certain of their friends, a practice exemplified in the prefaces of Apollonius and Archimedes), from a word in his pref- ace, I say, we learn that he was the first to occupy himself with that branch of arithmetic which has since been called algebra.

His work contains the first elements of this science. He em- ployed to express the unknown quantity a Greek letter which corresponds to our st^‡ and which has been replaced in the trans-

lations by N . To express the known quantities he employed num-

bers solely, for algebra was long destined to be restricted entirely to the solution of numerical problems. We find, however, that in


^∗The period is uncertain. Some say in the fourth century. See Cantor,

Geschichte der Mathematik, 2nd. ed., Vol. I., p. 434.—Trans.

^†On this point, see Appendix, p. 136.—Trans.

^‡According to a recent conjecture, the character in question is an abbre- viation of αρ the first letters of ¤ριθµός, number, the appellation technically applied by Diophantus to the unknown quantity.—Trans.
setting up his equations consonantly with the conditions of the problem he 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. Although the work of Diophantus contains in- determinate problems almost exclusively, the solution of which he seeks in rational numbers,—problems which have been des- ignated 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 particular artifices for reducing the problem to a sin- gle unknown quantity,—which is not difficult. He gives, also, the solution of equations of the second degree, but is careful so to ar- range them that they never assume the affected form containing the square and the first power of the unknown quantity.

He proposed, for example, the following question which in- volves the general theory of equations of the second degree: