A history of Greek mathematics - Wilbourhall.org

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430 PAPPUS OF ALEXANDRIA
in better logical sequence than his predecessors had attained.
The sort of questions to be dealt with are (1) a comparison
between the force required to move a given weight along
a horizontal plane and that required to move the same weight
upwards on an inclined plane, (2) the finding of two mean
proportionals between two unequal straight lines, (3) given
wheel with a certain number of teeth, to find the
a toothed
diameter of, and to construct, another wheel with a given number
of teeth to work on the former. Each of these things, he says,
will be clearly understood in its proper place if
the principles
'
on which the centrobaric doctrine ' is built up are first set out.
'
It is not necessary, he adds, to define what is meant by heavy
'
'
and light ' or upward, and downward motion, since these
matters are discussed by Ptolemy in his Mathematical but
the notion of the centre of gravity is so fundamental in the
whole theory of mechanics that it is essential in the first
place to explain what is meant by the ' centre of gravity
of any body.
On the centre of gravity.
Pappus then defines the centre of gravity as ' the point
within a body which is such that, if the weight be conceived
to be suspended from the point, it will remain at rest in any
position in which it is put '} The method of determining the
point by means of the intersection, first of planes, and then of
straight lines, is next explained (chaps.
1,2), and Pappus then
proves (Prop. 2) a proposition of some difficulty, namely that,
if D, E, F be points on the sides BG, GA, AB of a triangle ABG
such that
BD:DC= GE:EA = AF:FB,
then the centre of gravity of the triangle ABG is also the
centre of gravity of the triangle DEF.
Let H, K be the middle points of BG, GA respectively;
join AH, BK. Join EK meeting DE in L.
Then AH, BK meet in G, the centre of gravity of the
triangle ABG, and AG = 2 GH, BG = 2 BK, so that
.
GA :AK = AB:HK = BG: GK = AG :
GH.
1
Pappus, viii, p. 1030. 11-13.