A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

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A SEQUEL TO EUCLID.

to 0, the centre of perspective; then each of the
pencils (A. PCA'B), (A'. PC'AB') is equal to the pencil (0. PCAB); hence they are equal to one another,
and they have the ray AA' common. Hence the intersections of the three corresponding pairs of rays AC, A'C',
AP, A'P, AB, A'B', are collinear.
Cor. 3.-If two vertices of a variable A ABC move
on fixed right lines LMT, LN, and if the three sides
pass through three fixed collinear points 0, P, Q, the
locus of the third vertex is a right line.
Let the side AB pass through O, BC through P, CA
through Q, and let A'B'C' be another position of the A;
then the two As AA'Q, BB'Q, have the lines joining
their corresponding vertices concurrent; hence the intersections of the corresponding sides are collinear.
Hence the Proposition is proved.
Prop. 6.-If on a right line OX three pairs of points
A, A'; B, B'; C, C' be taken, such that the three rectangles
OA. OA', OB. OB', OC. OC', are each equal to a constant,
say k2, then the anharmonic ratio of any four of the six
points is equal to the anharmonic ratio of their four conjugates.
Dem.-Erect OY at right Z s to OX, and make OY
- k; join AY, A'Y, BY, B'Y, CY, C'Y. Now, by
hypothesis, OA. OA'= OY2;.. the 0 described about
the A AA'Y touches OY at Y;.'. the Z OYA = OA'Y.
In like manner, the Z OYB = OB'Y; hence the Z AYB
o) 0 A B C D C' B A X
= A'TYB': similarly the Z BYC = B'YC', &c.;.'. the
- of ilic pencil (Y. ABCC')= the Zs of the pencil