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

210 MATHEMATICAL PHILOSOPHY
lovely or very terrible things for millions of men, women
and children and were studied and discoursed about
seriously by men of genius: I mean angels. Angels can
fly. Let us confine their flights to straight lines but
impose no other restrictions. I am going to ask you to
understand a flight as having nothing but length, direction
and sense; if it is parallel to a given straight line, it has
that line's direction; if it goes from A towards B, it has
that sense; if from B toward A, the opposite sense; a
flight from A to B and one from C to D are the same if
they agree in length, direction and sense. Consider a
flight a from point A to point B followed by a flight b
from B to C; you readily see that a and b are two adjacent
sides of a parallelogram, one of whose diagonals is the
direct flight d from A to C; d is called the resultant or
flight-sum of a and b because d tells us how far the angel
has. finally got from the starting place; and so we write
aob =d. If flight b' goes from P to Q but agrees with b
in length, direction and sense, we write aob' =d as before
for, as already said, b and b' are one and the same. Now
let S12 denote the system whose C is the class of all
possible angel flights including rest, or zero flight, and
whose rule of combination is flight summation as above
explained; you see at once that S12 is a closed system, has
the group property, for the combination of any two
flights is a flight; if a, b and c be three flights, we may suppose them to go respectively from A to B, from B to C,
and from C to D; consider (aob)oc; aob=d, the flight
from A to C; doc =e, the flight from A to D; now consider ao(boc); boc=d', flight from B to D; aod'=e',
flight from A to D; so e=e' and (aob)oc =ao(boc); hence
summation of flights is associative-condition (b) is satisfied. Condition (c) is satisfied with zero (0) flight for i;