A history of Greek mathematics - Wilbourhall.org

ARISTAEUS'S SOLID LOCI 119
the same way the point will lie on a conic section given in
position.' x
The reason why Apollonius referred in this connexion to
Euclid and not to Aristaeus was probably that it was Euclid's
work that was on the same lines as his own.
A very large proportion of the standard properties of conies
admit of being stated in the form of locus-theorems ; if a
certain property holds with regard to a certain point, then
that point lies on a conic section. But it may be assumed
that Aristaeus's work was not merely a collection of the
ordinary propositions transformed in this way ; it would deal
with new locus-theorems not implied in the fundamental
definitions and properties of the conies, such as those just
mentioned, the theorems of the three- and four-line locus.
But one (to us) ordinary property, the focus-directrix property,
was, as it seems to me, in all probability included.
Focus-directrix property known to
Euclid.
It is remarkable that the directrix does not appear at all in
Apollonius's great treatise on conies. The focal properties of
the central conies are given by Apollonius, but the foci are
obtained in a different way, without any reference to the
directrix; the focus of the parabola does not appear at all.
We may perhaps conclude that neither did Euclid's Conies
contain the focus-directrix property ;
for, according to Pappus,
Apollonius based his first four books on Euclid's four books,
while filling them out and adding to them. Yet Pappus gives
the proposition as a lemma to Euclid's Surface-Loci, from
which we cannot but infer that it was assumed in that
treatise without proof. If, then, Euclid did not take it from
his own Conies, what more likely than that it was contained
in Aristaeus's Solid Loci ?
Pappus's enunciation of the theorem is to the effect that the
locus of a point such that its distance from a given point is in
a given ratio to its distance from a fixed straight line is a conic
section, and is an ellipse, a parabola, or a hyperbola according
as the given ratio is less than, equal to, or greater than unity.
1
Pappus, vii, p. 678. 15-24.