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

THE GROUP CONCEPT 209
ence of the other shapes. In speaking of the dog and the
table, I have been using the word " combination " in a
very general sense. Can it, in this connection, be made
precise enough for our use? Is it possible to find or frame
a rule by which, any two visible shapes being given, these
can be combined? If so, is the result of the combination
a visible shape? If so, the system consisting of the rule
and the total class of shapes has the group property.
Does the system satisfy the remaining three conditions
for a group?
And what of sounds-sensations of sound? Are sounds
combinable? Is the result always a sound or is it sometimes silence? If we agree to regard silence as a species
of sound,-as the zero of sound,-has the system of sounds
the property of a group? There is the question of thresholds: sound is a vibrational phenomenon; if the rate of
vibration be too slow or too great,-say, Ioo,ooo per
second,-no sound is heard. If you disregard the thresholds, has the sound system the group property? Is it a
group? If so, what is the identical element? And what
would you say is the reciprocal of a given sound or tone?
Consider other vibrational phenomena-as those of
light or electricity. Can you so conceive them as to get
group systems? Sharpen your questions and then carry
them to physicists. You need have no hesitance-the
service is apt to be mutual.
The Infinite lbelian Group of Angel Flights.-We are
accustomed to think of ourselves as being in a boundless
universe of space filled with what we call points any two
of which are joined by what we call a straight line,
Imagine one of those curious creatures which are to-day
for most of us hardly more than figures of speech but
which for many hundreds of years were very real and very