A history of Greek mathematics - Wilbourhall.org

THE COLLECTION. BOOK VIII 433
with its extremities on AC, AB and so that AC :
ratio,
BD is a given
then the centre of gravity of the triangle ADC will lie
on a straight line.
Take E, the middle point of AC, and Fa, point on BE such
that DF = 2 FE. Also let H be a point on BA such that
BH=2HA. Draw FG parallel to AC.
Then AG = J AD, and AH=^AB;
#G = § 5Z).
Also .TO = § ,4# = § ,4C.
therefore
Therefore,
since the ratio AC:BD is given, the
ratio GH: GF is given.
And the angle FGH (= A) is given ;
therefore the triangle FGH is given in
species, and consequently the angle GHF
is given. And if is a given point. *
Therefore HF is a given straight line, and it contains the
centre of gravity of the triangle A DC.
The inclined plane,.
Prop. 8 is on the construction of a plane at a given inclination
to another plane parallel to the horizon, and with this
Pappus leaves theory and proceeds to the practical part.
Prop. 9 (p. 1054. 4 sq.) investigates the problem 'Given
a weight which can be drawn along a plane parallel to the
horizon by a given force; and a plane inclined to the horizon
at a given angle, to find the force required to draw the weight
upwards on the. inclined plane'. This seems to be the first
or only attempt in ancient times to investigate motion on
an inclined plane, and as such it is curious, though of no
value.
Let A be the weight which can be moved by a force C along
a horizontal plane.
Conceive a sphere with weight equal to A
placed in contact at L with the given inclined plane ;
the circle
OGL represents a section of the sphere by a vertical plane
passing through E its centre and LK the line of greatest slope
drawn through the point L.
Draw EGH horizontal and therefore
parallel to MN in the plane of section, and draw LF
perpendicular to EH. Pappus seems to regard the plane
as rough, since
he proceeds to make a system in equilibrium
1523.2 Y f