v = gt). There was a mention of indivisibles here, but they were
not accepted generally. In 1637, Descartes took up the Aristotel-
ian dogma of the lack of comparability between the straight
line and the curved line (see I, p. 20) stated in the form, no al-
gebraic curve can be rectified algebraically. As evidence to the
contrary, he offered the rectification of a transcendental curve,
which could be carried out algebraically, namely, the logarithmi-
cal spiral. He defined this curve as the isogonal trajectory of
rays through a point ( 1638). The period to which the post-
humous "rounding out" of a given square belongs is uncertain.
This was an isoperimetric transformation of the square into a
regular polygon of 8, 16, . . . sides. Nor is it clear to what
extent this work could have been instigated by similar undertak-
ings by Nicolaus Cusanus (see I, p. 79).

Descartes also knew the construction for normals to the
cycloid from the momentary pole and suggested how one could
make the rolling of one curve upon another clear to himself
through the use of systems of equal chords. Mention was made
on this occasion, of the quadrature of the cycloid on the basis
of the equality of sections of figures having equal altitudes
( Cavalieri's principle: see I, p. 120). Shortly before this, Des-
cartes gave, without proof, some quadratures, cubatures and
determinations of centroids of plane parabolic surfaces and their
solids of revolution "elucidated by the very statement of the
results." He did not say how he had arrived at his results. Des-
cartes was able ( 1638) to properly evaluate the quadrature of
curves having the tangent property

The Geometrie, the one mathematical work by the great phi-

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