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

206 MATHEMATICAL PHILOSOPHY
foregoing illustrations of the group concept I am going
to add a few further ones,-some of them very simple,
some of them more complex,-trusting that the former
may not seem to you too trivial nor the latter too hard.
Every one has seen the pretty phenomenon of a grey
squirrel rapidly rotating a cylindrical wire cage enclosing
it. It may rotate the cage in either of two opposite ways,
senses or directions. Let us think of rotation in only one
of the ways, and let us call any rotation, whether it be
much or little, a turn. Each turn carries a point of the
cage along a circle-arc of some length, short or long.
Denote by R the special turn (through 360~) that brings
each point of the cage back to its starting place. Let
Sio be the system whose C is the class of al possible
turns and whose o is addition of turns so that aob shall
be the whole turn got by following turn a by turn b.
You see at once that S has the group property for the sum
of any two turns is a turn; it is equally evident that the
associative law-condition (b)-is satisfied. Note that R
is equivalent to no turn,-equivalent to rest,-equivalent
to a zero turn, if you please; note that, if a be a turn
greater than R and less than 2R, then a is equivalent to
a's excess over R; that, if a be greater than 2R and less
than 3R, then a is equivalent to a's excess over 2R; and
so on; thus any turn greater than R and not equal to a
multiple of R is equivalent to a turn less than R; let us
regard any turn that is thus greater than R as identical
with its equivalent less than R; we have, then, to consider no turns except R and those less than R-of which
there are infinitely many; you see immediately that, if a
be any turn, aoR =Roa =a, which means that condition
(c) is satisfied with R for identical element. Next notice
that for any turn a there is a turn a' such that aoa' a