Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

ESSENTIAL DISCRIMINATIONS 105
Euclidean plane. We saw that, a pair of axes and a
distance unit being chosen, to any point P of the plane
there belongs a pair (x, y) of real numbers, and conversely; and that to each line there belongs an equation
zx+By +C=0, and conversely. Now such a pair and
such an equation are respectively what we called in
Lecture VI a dyad and a system of dyads. The question
is: Is not the dyad doctrine D4 simply ordinary Euclidean
geometry Di in disguise? I might answer, quite justly,
that D4 is no more and no less D1 in disguise than Di is
D4 in disguise. You may now wish to say: very well,
are not D1 and D4 identical? The answer is no, for D1
is a doctrine about spatial things-points and lineswhile D4 is a doctrine about non-spatial things-dyads
and systems of dyads of pure real numbers. Perhaps
you would rejoin, saying: Is not D4 simply the analytic,
or algebraic, geometry of the Euclidean plane? It is
evidently just to answer: D4 is that, no more and no less
than D1 is the geometric algebra of N, which is the field of
number dyads and systems thereof just as the plane is the
field of points and lines. And you know that Di-the
ordinary plane geometry of Euclid-is not an algebra.
The fact is that, unless we are content to confound things
that are essentially different, we must here distinguish
four different things: namely, D1, D4, and two converse
aspects of what is in usage somewhat indiscriminately
called analytic, or algebraic, geometry of the Euclidean
plane. One of these aspects ought to be called analytic,
or algebraic, geometry; and the other, geometric analysis
or geometric algebra. "Ought," I mean, for the sake of
philosophic clarity, not necessarily in common every-day
parlance or practice. Let us be quite clear in this business. What is commonly called the analytic, or algebraic,