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

110 INTRODUCTION TO MATHEMATICS
any equations like x+3 = 2 or (x+3)2 = -2
can now always be interpreted into terms of
vectors, and solutions found for them. In
seeking for such interpretations it is well to
note that 3 becomes (3,0) and -2 becomes
(-2,0), and x becomes the "unknown"
couple (u, v): so the two equations become
respectively (u, v) + (3,0) = (2,0), and { (u,v)
+(3,0)2 = ( -,0).
We have now completely solved the initial
difficulties which caught our eye as soon as
we considered even the elements of algebra.
The science as it emerges from the solution is
much more complex in ideas than that with
which we started. We have, in fact, created
a new and entirely different science, which
will serve all the purposes for which the old
science was invented and many more in addition. But, before we can congratulate ourselves on this result to our labours, we must
allay a suspicion which ought by this time to
have arisen in the mind of the student. The
question which the reader ought to be asking
himself is: Where is all this invention of new
interpretations going to end? It is true that
we have succeeded in interpreting algebra so
as always to be able to solve a quadratic
equation like x2-2x+4=0; but there are
an endless number of other equations, for
example, x3-2x +4 =0, x4+x3 + = 0, and so
on without limit. Have we got to make a