Projective geometry, by Oswald Veblen and John Wesley Young.

146

ALGEBRA OF POINTS

[CHAP. VI

COROLLARY 3. The operationt of multiplication is associative; i.e. we
have (Ip' Jy). P = P. (P P) for every three points P, P,, for which
these products are defined. (A, E)
Proof (fig. 76). The proof is entirely analogous to the proof for
the associative law for addition. Let the point P1. be constructed
O Pi PZ y Ky Pyz.Py)-PeF) P7
FIG. 76
as in the definition by means of three fundamental lines lo, 11, lo,
the point Py being determined by the line XY. Denote the line PY
by l', and construct the point I. - P = (T. * ) * P, using the lines lo, l, lo
as fundamental. Further, let the point P,. = P be constructed by
means of the lines 10, l, ln, and then let P. Pz, = P (Py ) be constructed by means of 10, 11, l,. It is then seen that the points P Pz,
and PI y - are determined by the same line.
By analogy with Theorem 1, Cor. 4, we should now prove that multiplication is also commutative. It will, however, appear presently
that the commutativity of multiplication cannot be proved without
the use of Assumption P (or its equivalent). It must indeed be clearly
noted at this point that the definition of multiplication requires the
first factor PJ in a product to form with P and 1P a point triple of
the quadrangular set on I (cf. p. 49); the construction of the product
is therefore not independent of the order of the factors. Moreover,
the fact that in Theorem 3, Chap. II, the quadrangles giving the points
of the set are similarly placed, was essential in the proof of that