ARISTAEUS’S SOLID LOCI 
119 
is and, in 
ds, plane 
j, depend 
* second, 
o ‘ plane ’ 
on of the 
aisions or 
that this 
that no 
speak of 
produced 
ity of the 
rt, just as 
of a pure 
ion of an 
s bearing 
between 
equation, 
ern, could 
ealt with 
he must 
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says that 
lies. The 
3 appus in 
n one and 
the three 
rectangle 
Re square 
1 lie on a 
hree conic 
b, at given 
le ratio of 
wn to the 
n, then in 
the same way the point will lie on a conic section given in 
position.’ 1 
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 conics 
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 conics, 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 conics. The focal properties of 
the central conics 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 Conics 
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 Conics, 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.