itators. 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.

Geometry, however, among the Greeks passed into a stage of abstraction
in which lines, planes, etc., in the sense in which they are understood
in our elementary texts, took the place of actually measured surfaces,
and also took on the deductive form of presentation that has served as a
model for all mathematical presentation since Euclid. Mensuration
smacked too much of the exchange, and before the time of Archimedes is
practically wholly absent. Even such theorems as "that the area of a
triangle equals half the product of its base and its altitude" is
foreign to Euclid (cf. Cajori, p. 39). Lines were merely directions, and
points limitations from which one worked. But there was still dependence
upon the things that one measures. Euclid's elements, "when examined in
the light of strict mathematical logic, ... has been pronounced by C. S.
Peirce to be 'Riddled with fallacies'" (Cajori, p. 37). Not logic, but
observation of the figures drawn, that