382 
PAPPUS OF ALEXANDRIA 
curve, describes a certain plectoid, which therefore contains the 
point I. 
Also IE = EF, IF is perpendicular to BG, and hence IF, and 
therefore I, lies on a fixed plane through BG inclined to ABC 
at an angle of 
Therefore I, lying on the intersection of the plectoid and the 
said plane, lies on a certain curve. So therefore does the 
projection of I on ABC, i.e. the point E. 
The locus of E is clearly the quadratrix. 
[This result can also be verified analytically.] 
(5) Digression: a spiral on a sphere. 
Prop. 30 (chap. 35) is a digression on the subject of a certain 
spiral described on a sphere, suggested by the discussion of 
a spiral in a plane. 
Take a hemisphere bounded by the great 'circle KLM, 
with H as pole. Suppose that the quadrant of a great circle 
HNK revolves uniformly about the radius HO so that K 
describes the circle KLM and returns to its original position 
at K, and suppose that a point moves uniformly at the same 
time from H to K at such speed that the point arrives at K 
at the same time that HK resumes its original position. The 
point will thus describe a spiral on the surface of the sphere 
between the points H and K as shown in the figure. 
Pappus then sets himself to prove that the portion of the 
surface of the sphere cut off towards the pole between the 
spiral and the arc HNK is to the surface of the hemisphere in