THE COmCS AND SURFACE-LOCI 
439 
in the ‘ elements of conics ’, he clearly refers to these two 
treatises, and the other propositions to which he refers as well 
known and not needing proof were doubtless taken from the 
same sources. Euclid still used the old names for the conic 
sections (sections of a right-angled, acute-angled, and obtuse- 
angled cone respectively), but he was aware that an ellipse 
could be obtained by cutting (through) a cone in any manner 
by a plane not parallel to the base, and also by cutting a 
cylinder; this is clear from a sentence in his Phoenornena to 
the effect that, ‘If a cone or a cylinder be cut by a plane not 
parallel to the base, tins section is a section of an acute-angled 
cone, which is like a shield {Ovpeos)-’ 
(8) The Surface-Loci (touch upas lirLcjxxveia). 
Like the Data and the Porisms, this treatise in two Books 
is mentioned by Pappus as belonging to the Treasury of 
Analysis. What is meant by surface-loci, literally ‘ loci on a 
surface’ is not entirely clear, but we are able to form a con 
jecture on the subject by means of remarks in Proclus and 
Pappus. The former says (l) that a locus is ‘ a position of a 
line or of a surface which has (throughout it) one and the 
same property V and (2) that ‘ of locus-theorems some are 
constructed on lines and others on surfaces ’ 2 ; the effect of 
these statements together seems to be that ‘ loci on lines ’ are 
loci which are lines, and ‘ loci on surfaces ’ loci which are 
surfaces. On the other hand, the possibility does not seem to 
be excluded that loci on surfaces may be loci traced on sur 
faces ; for Pappus says in one place that the equivalent of the 
quadratrix can be got geometrically ‘ by means of loci on 
surfaces as follows ’and then proceeds to use a spiral de 
scribed on a cylinder (the cylindrical helix), and it is consis 
tent with this that in another passage 4 (bracketed, however, by 
Hultsch) ‘linear’ loci are said to be exhibited (SeiKvvvraL) or 
realized from loci on surfaces, for the quadratrix is a ‘ linear ’ 
locus, i.e. a locus of an order higher than a plane locus 
(a straight line or circle) and a ‘ solid ’ locus (a conic). How 
ever this may be, Euclid’s Surface-Loci probably included 
1 Proclus on Eucl. I, p. 894. 17. 
s Pappus, iv, p. 258. 20-25. 
2 lb., p. 394. 19. 
4 lb. vii. 662. 9.