440 
EUCLID 
such loci as were cones, cylinders and spheres. The two 
lemmas given by Pappus lend some colour to this view. The 
first of these 1 and the figure attached to it are unsatisfactory 
as they stand, but Tannery indicated a possible restoration. 2 
If this is right, it suggests that one of the loci contained all 
the points on the elliptical parallel sections of a cylinder, and 
was therefore an oblique circular cylinder. Other assump 
tions with regard to the conditions to which the lines in the 
figure may be subject would suggest that other loci dealt with 
were cones regarded as containing all points on particular 
parallel elliptical sections of the cones. In the second lemma 
Pappus states and gives a complete proof of the focus-and- 
directrix property of a conic, viz. that the locus of a point 
the distance of which from a given point is in a given ratio 
to its distance from a fixed straight line is a conic section, 
which is an ellipse, a parabola or a hyperbola according as the 
given ratio is less than, equal to, or greater than unity? Two 
conjectures are possible as to the application of this theorem in 
Euclid’s Surface-Loci, (a) It may have been used to prove that 
the locus of a point the distance of which from a given straight 
line is in a given ratio to its distance from a given plane 
is a certain cone. Or (6) it may have been used to prove 
that the locus of a point the distance of which from a given 
point is in a given ratio to its distance from a given plane is 
the surface formed by the revolution of a conic about its major 
or conjugate axis. 4 
We come now to Euclid’s works under the head of 
Applied mathematics. 
(a) The Phaenomena. 
The book on sphaeric intended for use in astronomy and 
entitled Phaenomena has already been noticed (pp. 349, 351-2). 
It is extant in Greek and was included in Gregory’s edition of 
Euclid. The text of Gregory, however, represents the later 
of two recensions which differ considerably (especially in 
Propositions 9 to 16). The best manuscript of this later 
recension (b) is the famous Vat. gr. 204 of the tenth century ^ 
1 Pappus, vii, p. 1004. 17 ; Euclid, ed. Heiberg-Menge, vol. viii, p. 274. 
2 Tannery in Bulletin des sciences mathématiques, 2 e série, YI, p. 149. 
3 Pappus, vii, pp. 1004. 28-1014 ; Euclid, vol. viii, pp. 275-81. 
* For further details, see The Works of Archimedes, pp. Ixii-lxv.