A history of Greek mathematics - Wilbourhall.org

NICOMEDES 199
tators, and especially by Pappus and Eutocius. Some of
these are very interesting, and it is evident from the
extracts from the works of such writers as Diodes and
Dionysodorus that, for some time after Archimedes and
Apollonius, mathematicians had a thorough grasp of the
contents of the works of the great geometers, and were able
to use the principles and methods laid down therein with
ease and skill.
Two geometers properly belonging to this chapter have
already been dealt with. The first is Nicomedes, the inventor
of the conchoid, who was about intermediate in date between
Eratosthenes and Apollonius. The conchoid has already been
described above (vol. i, pp. 238-40). It gave a general method
of solving any vevcris where one of the lines which cut off an
intercept of given length on the line verging to a given point
is a straight line ; and it was used both for the finding of two
mean proportionals and for the trisection of any angle, these
problems being alike reducible to a vevo-is of this kind. How
far Nicomedes discussed the properties of
is uncertain ;
properties,
the curve in itself
we only know from Pappus that he proved two
(1) that the so-called 'ruler' in the instrument for
constructing the curve is an asymptote, (2) that any straight
line drawn in the space between the ruler ' ' or asymptote and
the conchoid must, if produced, be cut by the conchoid. 1 The
equation of the curve referred to polar coordinates is, as we
have seen, r = a + b sec 6. According to Eutocius, Nicomedes
prided himself inordinately on his discovery of this curve,
contrasting it with Eratosthenes's mechanism for finding any
number of mean proportionals, to which he objected formally
and at length on the ground that it was impracticable and
entirely outside the spirit of geometry. 2
Nicomedes is associated by Pappus with Dinostratus, the
brother of Menaechmus, and others as having applied to the
squaring of the circle the curve invented by Hippias and
known as the quadratrix, z which was originally intended for
the purpose of trisecting any angle. These facts are all that
we know of Nicomedes's achievements.
1
Pappus, iv, p. 244. 21-8.
2
Eutoc. on Archimedes, On the Sphere and Cylinder, Archimedes,
vol. iii, p. 98.
3
Pappus, iv, pp. 250. 33-252. 4. Cf. vol. i, p. 225 sq.