420 
PAPPUS OF ALEXANDRIA 
opposite sides and the two diagonals respectively, Pappus’s 
result is equivalent to 
AB.B'C _ GA 
A'B'. BG' ~ G'A' ' 
Props. 127, 128 are particular cases in 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 
G A 1 = CB . CB\ 
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, AG, AD be cut by transversals 11 BCD, HEFO. It is 
required to prove that 
HE.FG HB.CD 
HG . EF ~~ HD. BG 
Pappus gives (Prop. 129) two methods of proof which are 
practically equivalent. The following is the proof ‘by com 
pound ratios ’. 
Draw HK parallel to AF meeting DA and AE produced