384 
PAPPUS OF ALEXANDRIA 
Now (sector HPN on sphere): (sector HKL on sphere) 
= (chord HP) 2 : (chord HL) 2 
(a consequence of Archimedes, On Sphere and Cylinder, I. 42). 
And HP 2 : HL 2 = CB 2 : CA 2 
Therefore 
= CB 2 : CE 2 . 
(sector HPN) : (sector HKL) = (sector CBG): (sector CEF). 
Similarly, if the arc LL' be taken equal to the arc KIj and 
the great circle through H, 1/ cuts the spiral in P', and a small 
circle described about H and through P' meets the arc HPL 
in p; and if likewise the arc BB' is made equal to the arc BC, 
and CB' is produced to meet AF in E', while again a circular 
arc with G as centre and CB' as radius meets GE in h, 
(sector HP'p on sphere): (sector HLL' on sphere) 
= (sector CB'h): (sector GE'E). 
And so on. 
Ultimately then we shall get a figure consisting of sectors 
on the sphere circumscribed about the area S of the spiral and 
a figure consisting of sectors of circles circumscribed about the 
segment GBA; and in like manner we shall have inscribed 
figures in each case similarly made up. 
The method of exhaustion will then give 
S: (surface of hemisphere) = (segmt, ABC): (sector GAF) 
= (segmt. ABC): (sector DAG). 
[We may, as an illustration, give the analytical equivalent 
of this proposition. If p, co be the spherical coordinates of P 
with reference to H as pole and the arc HNK as polar axis, 
the equation of Pappus’s curve is obviously co = 4 p. 
If now the radius of the sphere is taken as unity, we have as 
the element of area 
dA = dco (1 —cosp) = 4dp (1 —cosp). 
Therefore 
A = 
'i* 
Hip (1 — cosp) — 2tt — 4. 
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