through the principle of homogeneity hitherto regarded as in-
violable on geometrical grounds, he was enabled to express the
sum, difference, product and quotient of any two line segments as
another line segment. As a result, he could also express line seg-
ments obtained by a combination of these four fundamental
operations in a finite number of single steps. The number con-
cept, originally restricted to natural numbers and extended to
fractions, negative numbers and irrationals by the most labor-
ious step after step, henceforth embraced the entire domain of
algebraic numbers. Accordingly, Descartes considered problems
solvable by purely algebraic methods (so-called geometrical) as
belonging to exact mathematics, and all others (so-called me-
chanical) as belonging to the mathematics of approximation,
which he no longer included in pure mathematics.

Descartes knew that all the geometrical problems of the linear
and quadratic types could be constructed by straight edge and
compasses and he classified such problems as plane problems
(designation after the manner of Apollonius). In agreement
with the cossists, problems of the 3rd and 4th degrees were
called solid problems. Descartes solved the solid problems
graphically ( 1628-29) by drawing a single parabola and
cutting it with a circle which could be determined by plane
constructions. Ferrari's method (see I, p. 84) for the solution of
the 4th degree equation x⁴ + px² + qx + r = 0 was transformed
by means of 2t - p = y². After the solution of the auxiliary equa-
tion y⁶ + 2py⁴ + (p² - 4^r) y² - q² = 0, the value of x was found
from x² ∓ xy + y² + p/2 ± q/2y = 0. In accordance with his
usual practice, Descartes gave only the prescribed formula for
the calculation. His contemporaries verified it at the expenditure
of a great deal of effort (Debeaune, printed 1649; van Schooten,
printed 1659).

Descartes conceived of the graphical solution of equations of

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