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.

A SEQUEL TO EUCLID.

viz., 1st and 4th, 2nd and 5th, 3rd and 6th, are collinear. B
Let ABCIDEFA be the
hexagon. The points L, N,
M are collinear. D
Dem.- oin EN. Then the
pencil (N. FMCE) = the pen-\ N
cil (C. FBDE), because they I
have a common transversal
EF (Cor. 3, Prop. 2.) In A
like manner, the pencil
(A. FBDE) (N. ALDE);
but (A. FBDE)= C. FBDE) (Prop. 2, Cor. 4). Hence the
pencils (N. FMCE), (N. ALDE) are equal; and therefore (Cor. 2, Prop. 2) the points L, N, M are collinear.
Cor. l. —With six points on the circumference of a
circle, sixty hexagons can be formed. For, starting
with any point, say A, we could go from A to one of the
remaining points in five ways. Suppose we select B,
then we could go from B to a third point in four different ways, and so on; hence it is evident that we
could join A to another point, and that again to another,
and so on, and finally return to A in 5 x 4 x 3 x 2 x 1
different ways. Hence we shall have that number of
hexagons; but each is evidently counted twice, and we
shall therefore have half the number, that is, sixty
distinct hexagons.
Cor. 2.-Pascal's Theorem holds for each of the
sixty hexagons.
Cor. 3.-Pascal's Theorem holds for six points,

which are, three by three, on two lines. Thus, let the