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

NON-GEOMETRIC INTERPRETATION 97
whence
x-xi = p(b-bi) - (I -pm),
-X2- p(b-b)2) (- pm),
where p is a proportionality factor. You see that x-xi
and x-X2 are unlike or like in sign according as b-bi
and b-b2 are unlike or like in sign; that is, d is between
or not between d1 and d2 according as b is between or not
between bi and b2; hehce the theorem. We may agree to
say that the dyad dî(xl, yi) is on the positive or the
negative side of s according as b > or <b.
DEFINITIONS.-If d be a dyad of a system s, the dyads
of s on a same side of d constitute a half-system emanating
from d. [So it is seen that any dyad of an s separates the
remaining dyads of s into two opposite half-systemns.l
A pair of half-systeits, h and k, emanating from a dyad d
and not belonging to a same system, is an angle (h, k);
d is the angle's vertex, and h and k its sides; its interior is
the class of dyads such that, if di and d2 be anytwo of them,
the segment dd2 contains no dyad of h or k; its exterior
consists of all othet dyads of N except d and the dyads
of h and k. Let d and d' be the vettices of two angles
(h, k) and (h', k'); let ddi and dd2 be two segments of
h and k respectively, and let d'di' and d'd2' be segments of
h' and k' respectively; suppose ddi is congruent with
d'di' and dd2 with d'd2'; then, if did2 is congruent with
di'd2', we shall say that the given angles are congruent.
[Note that we have defined congruence of angles in terms
of congruence of segments. Note also and note carefully
that, though we have for the sake of convenience used
such terms as angle, vertex, side, and so on, which smell
of geometry and suggest space, there is in such use no
logically implicit geometric or spatial reference whatever.