434 
PAPPUS OF ALEXANDRIA 
about FL as if L were the fulcrum of a lever. Now the 
weight A acts vertically downwards along a straight line 
through E. To balance it, Pappus supposes a weight B 
attached with its centre of gravity at G. 
Then A: B = GF: EF 
= {EL — EF): EF 
[= (1 — sin 0): sin 6, 
where Z KMN = d\; 
and, since ¿KMN is given, the ratio EF: EL, 
and therefore the ratio {EIj — EF) : EF. is 
given ; thus B is found. 
Now, says Pappus, if I) is the force which will move B 
along a horizontal plane, as C is the force which will move 
A along a horizontal plane, the sum of C and D will be the 
force required to move the sphere upwards on the inclined 
plane. He takes the particular case where 6 = 60°. Then 
sin $ is approximately (he evidently uses \. ff for ^ ^3), 
and A :B= 16:104. 
Suppose, for example, that A = 200 talents; then B is 1300 
talents. Suppose further that G is 40 man-power; then, since 
D :C = B:A, i) = 260 man-power; and it will take D + C, or 
300 man-power, to move the weight up the plane! 
Prop. 10 gives, from Hei'on’s Bar ulcus, the machine con 
sisting of a pulley, interacting toothed wheels, and a spiral 
screw working on the last wheel and turned by a handle; 
Pappus merely alters the proportions of the weight to the 
force, and of the diameter of the wheels. At the end of 
the chapter (pp. 1070-2) he repeats his construction for the 
finding of two mean proportionals. 
Construction of a conic through five 'points. 
Chaps. 13-17 are more interesting, for they contain the 
solution of the problem of constructing a conic through five 
given points. The problem arises in this way. Suppose we 
are given a broken piece of the surface of a cylindrical column 
such that no portion of the circumference of either of its base