CLASSIFICATION OF PROBLEMS 
219 
higher curves. 1 * Another classification of loci divides them 
into loci on lines (tottol Trpos ypappais) and loci on surfaces 
{tottol rrpos emfaveiaLs)? The former term is found in 
Proclus, and seems to he used in the sense both of loci which 
are lines (including of course curves) and of loci which are 
spaces bounded by lines; e.g. Proclus speaks of ‘the whole 
space between the parallels’ in Fuel. I. 35 as being the locus 
of the (equal) parallelograms ‘on the same base and in the 
same parallels’. 3 Similarly loci on surfaces in Proclus may 
be loci which are surfaces; but Pappus, who gives lemmas 
to the two books of Euclid under that title, seems to imply 
that they were curves drawn on surfaces, e.g. the cylindrical 
helix. 4 
It is evident that the Greek geometers came very early 
to the conclusion that the three problems in question were not 
'plane, but required for their solution either higher curves 
than circles or constructions more mechanical in character 
than the mere use of the ruler and compasses in the sense of 
Euclid’s Postulates 1-3. It was probably about 420 b. c. that 
Hippias of Elis invented the curve known as the quadratrix 
for the purpose of trisecting any angle, and it was in the first 
half of the fourth century that Archytas used for the dupli 
cation of the cube a solid construction involving the revolution 
of plane figures in space, one of which made a tore or anchor 
ring with internal diameter nil. There are very few records 
of illusory attempts to do the impossible in these cases. It is 
practically only in the case of the squaring of the circle that 
we read of abortive efforts made by ‘ plane ’ methods, and none 
of these (with the possible exception of Bryson’s, if the 
accounts of his argument are correct) involved any real 
fallacy. On the other hand, the bold pronouncement of 
Antiphon the Sophist that by inscribing in a circle a series 
of regular polygons each of which has twice as many sides 
as the preceding one, we shall use up or exhaust the area of 
the circle, though it was in advance of his time and was 
condemned as a fallacy on the technical ground that a straight 
line cannot coincide with an arc of a circle however short 
its length, contained an idea destined to be fruitful in the 
1 Cf. Pappus, vii, p. 662, 10-15. 
3 lb., p. 395. 5. 
2 Proclus on Euch I, p. 894. 19. 
4 Pappus, iv, p. 258 sq.