establishing indubitably the proposition that mathematics is a matter of purely intellectual operations. But by so construing it, they have, in geometry, remembered solely the measuring and forgotten the land, and, in arithmetic, remembered the counting and forgotten the things counted. Arithmetic experienced little immediate gain from its new association with geometry, which was destined to be of momentous import in its latter history, beyond the discovery of irrationals (which, however, were for centuries not accepted as numbers), and the establishment of the problem of root-taking by its association with the square, and interest in negative numbers. The Greeks had only subtracted smaller numbers from larger, but the Arabs began to generalize the process and had some acquaintance with negative results, but it was difficult for them to see that these results might really have significance. N. Chuquet, in the fifteenth century, seems to have been the first to interpret the negative numbers, but he remained a long time without imitators. Michael Stifel, in the sixteenth century, still calls them "Numeri absurdi" as over against the "Numeri veri." However, their geometrical interpretation was not difficult, and they soon won their way into good standing. But the case of the imaginary is more striking. The need for it was first felt when it was seen that negative numbers have no square roots. Chuquet had dealt with second-degree equations involving the roots of negative numbers in 1484, but says these numbers are "impossible," and Descartes (_Geom._, 1637) first uses the word "imaginary" to denote them. Their introduction is due to the Italian algebrists of the sixteenth century. They knew that the real roots of certain algebraic equations of the third degree are represented as results of operations effected upon "impossible" numbers of the form _a_ + _b_ sqrt{-1} (where _a_ and _b_ are real numbers) without it being possible in general to find an algebraic expression for the roots containing only real numbers. Cardan calculated with these "impossibles," using them to get real results [(5 + sqrt{-15}) (5 - sqrt{-15}) = 25 - (-15) = 40], but adds that it is a "quantitas quae vere est sophistica" and that the calculus itself "adeo est subtilis ut est inutilis." In 1629, Girard announced the theorem that every complete algebraic equation admits of as many roots, real or imaginary, as there are units in its degree, but Gauss first proved this in 1799, and finally, in his _Theory of Complex Quantity_, in 1831.