ÜiÜâif 
V» 
VII 
SPECIAL PROBLEMS 
Simultaneously with the gradual evolution of the Elements, 
the Greeks were occupying themselves with problems in 
higher geometry; three problems in particular, the squaring 
of the circle, the doubling of the cube, and the trisection of 
any given angle, were rallying-points for mathematicians 
during three centuries at least, and the whole course of Greek 
geometry was profoundly influenced by the character of the 
specialized investigations which had their origin in the attempts 
to solve these problems. In illustration we need only refer 
to the subject of conic sections which began with the use 
made of two of the curves for the finding of two mean pro 
portionals. 
The Greeks classified problems according to the means by 
which they were solved. The ancients, says Pappus, divided 
them into three classes, which they called plane, solid, and 
linear respectively. Problems were plane if they could be 
solved by means of the straight line and circle only, solid 
if they could be solved by means of one or more conic sections, 
and linear if their solution required the use of other curves 
still more complicated and difficult to construct, such as spirals, 
quadratrices, cochloids (conchoids) and cissoids, or again the 
various curves included in the class of ‘ loci on surfaces ’ (tottol 
7Tpos kTTLcpavdcus), as they were called. 1 There was a corre 
sponding distinction between loci : plane loci are straight 
lines or circles; solid loci are, according to the most strict 
classification, conics only, which arise from the sections of 
certain solids, namely cones ; while linear loci include all 
1 Pappus, iii, pp. 54-6, iv, pp. 270-2.