A history of Greek mathematics - Wilbourhall.org

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
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