CHAPTER II 
MATHEMATICAL INTRODUCTION 
Mathematics is thought moving in the sphere of complete abstrac 
tion from any particular instance of what it is talking about. . . . 
The certainty of mathematics depends upon its complete abstract 
generality. But we can have no a priori certainty that we are right 
in believing that the observed entities in the concrete universe form 
a particular instance of what falls under general reasoning. 
A. N. Whitehead, “ Science and the Modern World,” 1926 
Mathematics is a system of symbolic logic constituting a machinery 
for the evolution of results to which the finite unaided human intellect 
could not otherwise attain. 
W. Peddie 
1. The Linear Harmonic Oscillator 
I N his treatment of the radiation problem Planck selected a 
system composed of ideal linear oscillators or resonators. 
This was done with a view to simplifying as far as possible the 
nature and arrangement of the systems emitting and absorbing 
radiation. Each resonator may be thought of as consisting of 
two poles, charged with equal quantities of electricity of opposite 
sign, which may move relatively to one another on the fixed 
axis of the resonator. The centre of mass of each resonator is 
regarded as stationary. The vibration of the resonator entails 
one degree of freedom only. Each resonator is supposed to 
possess a definite natural frequency of vibration v. The equation 
of motion of such a resonator when it is in its steady state, that 
is, when it neither emits nor absorbs energy, may be written 
in the familiar form typical of simple harmonic motion 
T 
2 : i 
where « is a mass, or an inertia coefficient, ju corresponds to 
the restoring force per unit displacement, and q is called the 
positional coordinate, in this case the distance between the electric 
I 
poles of the resonator. The momentum will be denoted 
by p, which is called in Hamiltonian dynamics the impulse 
coordinate. These two coordinates p and q may be taken to 
16 
dt