THE COLLECTION. BOOK VIII 
with its extremities on AC, AB and so that AC : BD is a given 
ratio, then the centre of gravity of the triangle ADC will lie 
on a straight line. 
Take E, the middle point of AC, and F a point on DE such 
that DF = 2 FE. Also let H be a point on BA such that 
BH = 2HA. Draw FQ parallel to AG. 
Then AG = | AD, and AH = ^AB; A 
therefore HG = BD. /\ 
Also FG = §AE = ¿¡AC. Therefore, 
since the ratio AG:BD is given, the * 
ratio GH ; G F is given. /s' \ 
And the angle FGH ( =.- A) is given ; r~ q 
therefore the triangle FGH is given in / 
species, and consequently the angle GHF b/ 
is given. And H is a given point. 
Therefore HF is a given straight line, and it contains the 
centre of gravity of the triangle ADC. 
The inclined plane. 
Prop. 8 is on the construction of a plane at a given inclina 
tion 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 G 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 there 
fore 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