rture from them.

But in the present state of science we can no longer overlook the
important questions which arise when we deal with bodies not rigid in
the mathematical sense of the word. Let us, for instance, take the
simplest of the laws to which we have referred, the great law of Kepler,
which asserts that a planet will revolve for ever in an elliptic path of
which the sun is one focus. This is seen to be verified by actual
observation; indeed, it was established by observation before any
theoretical explanation of that movement was propounded. If, however, we
state the matter with a little more precision, we shall find that what
Newton really demonstrated was, that if two _rigid_ particles attract
each other by a law of force which varies with the inverse square of the
distance between the particles, then each of the particles will describe
an ellipse with the common centre of gravity in the focus. The earth is,
to some extent, rigid, and hence it was natural to suppose that the
relative behaviour of the earth and the sun would, to a corresponding
extent, observe the simple elliptic law of Kepler; as a matter of fact,
they do observe it with such fidelity that, if we make allowance for
other causes of disturbance, we cannot, even by most careful
observation, detect the slightest variation in the motion of the earth
arising from its want of rigidity.

There is, however, a subtlety in the investigations of mathematics
which, in this instance at all events, transcends the most delicate
observations which our instruments enable us to make. The principles of
mathematics tell us that though Kepler's laws may be true for bodies
which are absolutely and mathematically rigid, yet that if the sun or
the planets be either wholly, or even in their minutest part, devoid of
perfect rigidity, then Kepler's laws can be no longer true. Do we not
seem here to be in the presence of a contradiction? Observation tells us
that Kepler's laws are true in the planetary system; theory tells us
that these laws cannot be true in the planetary system, because the
bodies in that system are not perfectly rigid. How is this discrepancy
to be removed? Or is there really a discrepancy at all? There is not.
When we say that Kepler's laws have been proved to be true by
observation, we must reflect on the nature of the proofs which are
attainable. We observe the places of the planets with the instruments in
our observatories; these places are measur