A history of Greek mathematics - Wilbourhall.org

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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.