A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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 ;
the curve in itself
we only know from Pappus that he proved two
properties, (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.
200 SUCCESSORS OF THE GREAT GEOMETERS
The second name is that of Diocles. We have already
(vol. i, pp. 264-6) seen him as the discoverer of the curve
known as the cissoid, which he used to solve the problem
of the two mean proportionals, and also (pp. 47-9 above)
as the author of a method of solving the equivalent of
a certain cubic equation by means of the intersection
of an ellipse and a hyperbola. We are indebted for our
information on both these subjects to Eutocius, 1 who tells
us that the fragments which he quotes came from Diocles's
work rrepl nuptiais, On burning-mirrors. The connexion of
the two things with the subject of this treatise is not obvious,
and we may perhaps infer that it was a work of considerable
scope. What exactly were the forms of the burning-mirrors
discussed in the treatise it is not possible to say, but it is
probably safe to assume that among them were concave
mirrors in the forms (1) of a sphere, (2) of a paraboloid, and
(3) of the surface described by the revolution of an ellipse
about its major axis. The author of the Fragmentum mathematicum
Bobiense says that Apollonius in his book On the
burning-mirror discussed the case of the concave spherical
mirror, showing about what point ignition would take place
and it is certain that Apollonius was aware that an ellipse has
the property of reflecting all rays through one focus to the
other focus. Nor is it likely that the corresponding property
of a parabola with reference to rays parallel to the axis was
unknown to Apollonius. Diocles therefore, writing a century
or more later than Apollonius, could hardly have failed to
deal with all three cases. True, Anthemius (died about
A. D. 534) in his fragment on burning-mirrors says that the
ancients, while mentioning the usual burning-mirrors and
saying that such figures are conic sections, omitted to specify
which conic sections, and how produced, and gave no geometrical
proofs of their properties. But if the properties
were commonly known and quoted, it is obvious that they
must have been proved by the ancients, and the explanation
of Anthemius's remark is presumably that the original works
in which they were proved (e.g. those of Apollonius and
Diocles) were already lost when he wrote. There appears to
be no trace of Diocles's work left either in Greek or Arabic,
1
Eutocius, loc. cit., p. 66. 8 sq., p. 160. 3 sq.