A history of Greek mathematics - Wilbourhall.org

!
;
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
[= (l-sin0):sin0,
where IKMN = $];
and, since LKMN is given, the ratio EF: EL,
and therefore the ratio (EL-EF) :
given ;
thus B is found.
EF, is
Now, says Pappus, if D 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 6 is approximately y§£ (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 C is 40 man-power ; then, since
D:C = B: A, D = 260 man-power ; and it will take D + C, or
300 man-power, to move the weight up the plane
Prop. 10 gives, from Heron's Barulcus, the machine consisting
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 Jive 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