A history of Greek mathematics - Wilbourhall.org

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420 PAPPUS OF ALEXANDRIA
opposite sides and the two diagonals respectively, Pappus's
resiilt is equivalent to
AB^B'C^
GA
TW7M ~ C 7 A'
Props. 127, 128 are particular cases in
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which the transversal
is parallel to a side; in Prop. 131 the transversal passes
through the points of concourse of opposite sides, and the
result is equivalent to the fact that the two diagonals divide
into proportional parts the straight line joining the points of
concourse of opposite sides; Prop. 132 is the particular case
of Prop. 131 in which the line joining the points of concourse
of opposite sides is parallel to a diagonal; in Prop. 133 the
transversal passes through one only of the points of concourse
of opposite sides and is parallel to a diagonal, the result being
CA = 2 GB . GB\
Props. 129, 136, 137, 140, 142, 145 (Lemmas 3, 10, 11, 14, 16,
19) establish the equality of the anharmonic ratios which
four straight lines issuing from a point determine on two
transversals ; but both transversals are supposed to be drawn
from the same point on one of the four straight lines.
Let
AB, AC, AD be cut by transversals HBGD, HEFG. It is
required to prove that
HE.FG HB.GD
EG.EF" HD.BC'
Pappus gives (Prop. 129) two methods of proof which are
practically equivalent. The following is the proof 'by compound
ratios '.
Draw HK parallel to AF meeting DA and AE produced