An introduction to mathematics, by A. N. Whitehead.

102 INTRODUCTION TO MATIIEMATICS
(e) Furthermore the law involving both
addition and multiplication, called the distributive law, must be satisfied, namely
(x, y)x{(a, b)+(c, d)}
-i(x, y) X(a, b)i +i(x, y) X(c, d)l.
All these conditions (a), (/3), (y), (8), (e)
can be satisfied by an interpretation which,
though it looks complicated at first, is capable
of a simple geometrical interpretation.
By definition we put
(x, y) X (x', y') = {(xx' - yy'), (xy' + x'y) l (A)
This is the definition of the meaning of the
symbol X when it is written between two
ordered couples. It follows evidently from
this definition that the result of multiplication is another ordered couple, and that the
value of the right-hand side of equation (A)
is not altered by simultaneously interchanging x with x', and y with y'. Hence conditions (a) and (/3) are evidently satisfied. The
proof of the satisfaction of (y), ($), (e) is
equally easy when we have given the geometrical interpretation, which we will proceed to do in a moment. But before doing
this it will be interesting to pause and see
whether we have attained the object for
which all this elaboration was initiated.
We came across equations of the form
sx2 -3, to which no solutions could be