THE COLLECTION. BOOK VII 
419 
i.e, (if DA . AG be subtracted from each side) 
that AD . DC + FD. DB = AC. DB + AF. CD, 
i.e. (if AF. CD be subtracted from each side) 
that FD . DG+ FD . DB = AC. DB, 
or FD.GB = AG.DB: 
which is true, since, by (1) above, FD : DB = AC: GB. 
(£) Lemmas io the ‘ Porisms ’ of Euclid. 
The 38 Lemmas to the Porisms 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 1 
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 Gutting-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, G' be the points in which the transversal meets the pairs of 
1 Chasles, Les trois livres de Ponsmes d'Euclide, Paris, 1860, pp. 74 sq. 
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