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

WORKS BY ARISTAEUS AND EUCLID 117
time, whereas the work of Aristaeus was more specialized and
more original.
'
Solid loci ' and
'
solid problems \
'
Solid loci ' are of course simply conies, but the use of the
title ' Solid loci ' instead of ' conies ' seems to indicate that
the work was in the main devoted to conies regarded as loci.
As we have seen, ' solid loci ' which are conies are distinguished
from ' plane loci ', on the one hand, which are straight lines
and circles, and from ' linear loci ' on the other, which are
curves higher than conies. There is some doubt as to the
real reason why the term ' solid loci ' was applied to the conic
sections. We are told that ' plane ' loci are so called because
they are generated in
a plane (but so are some of the higher
curves, such as the quadratrix and the spiral of Archimedes),
and that ' solid loci ' derived their name from the fact that
they arise as sections of solid figures (but so do some higher
curves, e.g. the spiric curves which are sections of the a-irelpa
or tore). But some light is thrown on the subject by the corresponding
distinction which Pappus draws between plane ' ',
'
solid ' and ' linear ' problems.
'Those problems', he says, 'which can be solved by means
of a straight line and a circumference of a circle may properly
be called plane ;
for the lines by means of which such
problems are solved have their origin in a plane. Those,
however, which are solved by using for their discovery one or
more of the sections of the cone have been called solid ;
for
their construction requires the use of surfaces of solid figures,
namely those of cones. There remains a third kind of problem,
that which is called linear ; for other lines (curves)
besides those mentioned are assumed for the construction, the
origin of which is more complicated and less natural, as they
are generated from more irregular surfaces and intricate
movements.'
'
The true significance of the word plane ' as applied to
problems is evidently, not that straight lines and circles have
their origin in a plane, but that the problems in question can
be solved by the ordinary plane methods of transformation of
1
Pappus, iv, p. 270. 5-17.
118 CONIC SECTIONS
areas, manipulation of simple equations between areas and, in
particular, the application of areas ;
in other words, plane
problems were those which, if expressed algebraically, depend
on equations of a degree not higher than the second.
Problems, however, soon arose which did not yield to plane
'
methods. One of the first was that of the duplication of the
cube, which was a problem of geometry in three dimensions or
solid geometry. Consequently, when it was found that this
problem could be solved by means of conies, and that no
higher curves were necessary, it would be natural to speak of
them as 'solid' loci, especially as they were in fact produced
from sections of a solid figure, the cone. The propriety of the
term would be only confirmed when it was found that, just as
the duplication of the cube depended on the solution of a pure
cubic equation, other problems such as the trisection of an
angle, or the cutting of a sphere into two segments bearing
a given ratio to one another, led to an equation between
volumes in one form or another, i. e. a mixed cubic equation,
and that this equation, which was also a solid problem, could
likewise be solved by means of conies.
Aristaeus's
Solid Loci.
The Solid Loci of Aristaeus, then, presumably dealt with
loci which proved to be conic sections. In particular, he must
have discussed, however imperfectly, the locus with respect to
three or four lines the synthesis of which Apollonius says that
he found inadequately worked out in Euclid's Conies. The
theorems relating to this locus are enunciated by Pappus in
this way
If three straight lines be given in position and from one and
'
the same point straight lines be drawn to meet the three
straight lines at given angles, and if the ratio of the rectangle
contained by two of the straight lines so drawn to the square
on the remaining one be given, then the point will lie on a
solid locus given in position, that is, on one of the three conic
sections. And if straight lines be so drawn to meet, at given
angles, four straight lines given in position, and the ratio of
the rectangle contained by two of the lines so drawn to the
rectangle contained by the remaining two be given, then in