A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

i.e. (if DA .
THE COLLECTION. BOOK VII 419
be subtracted from each side)
AC
that • AD.DC + FD.DB = AC.DB + AF.CD,
CD be subtracted from each side)
that FD . DC+
i.e. (if AF .
FD.DB = AC. DB,
or
*
FD.CB = AC.DB:
which is true, since, by (1) above, FD : DB = AC :
(£) Lemmas fo the ' Porisms
'
of Euclid.
CB.
The 38 Lemmas to the For isms of Euclid form an important
collection which, of course, has been included in one form or
other in the ' restorations ' of the original treatise. Chasles x
in particular gives a classification of them, and we cannot
do better than use it in this place :
'23 of the Lemmas relate
to rectilineal figures, 7 refer to the harmonic ratio of four
points, and 8 have reference to the circle.
'
Of the 23 relating to rectilineal figures, 6 deal with the
quadrilateral cut by a transversal ; 6 with the equality of
the anharmonic ratios of two systems of four points arising
from the intersections of four straight lines issuing from
one point with two other straight lines ;
4 may be regarded as
expressing a property of the hexagon inscribed in two straight
lines ; 2 give the relation between the areas of two triangles
which have two angles equal or supplementary ; 4 others refer
to certain systems of straight lines; and the last is a case
of the problem of the Cutting-off of an area.'
The lemmas relating to the quadrilateral and the transversal
are 1, 2, 4, 5, 6 and 7 (Props. 127, 128, 130, 131, 132, 133).
Prop. 130 is a general proposition about any transversal
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
•
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
whatever, and is equivalent to one of the equations by which
we express the involution of six points. If A, A'; B, B' ;
C, C be the points in which the transversal meets the pairs of
1
Chasles, Les trois livres de Porismes d'Euclide, Paris, 1860, pp. 74 sq.
E e 2
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