1 Pappus, iv, p. 270. 5-17. 
WORKS BY ARISTAEUS AND EUCLID 117 
lyperbola, 
ate, draw 
ely. Let 
angle; 
iree conic 
350 B.C., 
ar by the 
vorks on 
Pappus, 
Ionics o£ 
lourished 
but his 
Aristaeus 
scribes it 
, (rvve^fj) 
,’s Conics 
assage is 
his dis- 
3 him or 
articular, 
le three- 
i as was 
claiming 
om these 
arrange- 
n in his 
time, whereas the work of Aristaeus Was more specialized and 
more original. 
‘Solid loci’ and ‘solid 'problems'. 
‘ Solid loci ’ are of course simply conics, but the use of the 
title ‘ Solid loci ’ instead of ‘ conics ’ seems to indicate that 
the work was in the main devoted to conics regarded as loci. 
As we have seen,‘ solid loci ’ which are conics 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 conics. 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 pf solid figures (but so do some higher 
curves, e.g. the spiric curves which are sections of the andpa 
or tore). But some light is thrown on the subject by the corre 
sponding 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 pro 
perly 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 pro 
blem, 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.’ 1 
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 th^t the problems in question can 
be solved by the ordinary plane methods of transformation of