“All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers.”

I think it ought to occupy a prominent place in our investigations, and that we ought to endeavour to construct a mental representation of all the details of its action, and this has been my constant aim in this treatise.

The mathematical difficulties of the theory of rotation arise chiefly from the want of geometrical illustrations and sensible images, by which we might fix the results of analysis in our minds.

Wherein they fancy intersperses Long avenues of universes, While Klein and Clifford fill the void With one finite, unbounded homoloid, And think the Infinite is now at last destroyed.

When I reappear it will be in the dusky ring, which is something like the state of the air supposing the siege of Sebastopol conducted from a forest of guns 100 miles one way, and 30,000 miles the other, and the shot never to stop, but go spinning away round a circle, radius 170,000 miles.

In every branch of knowledge the progress is proportional to the amount of facts on which to build, and therefore to the facility of obtaining data.

The vast interplanetary and vast interstellar regions will no longer be regarded as waste places in the universe. We shall find them to be already full of this wonderful medium; so full that no human power can remove it from the smallest portion of space or produce the slightest flaw in its infinite continuity.

The popularisation of scientific doctrines is producing as great an alteration in the mental state of society as the material applications of science are effecting in its outward life.

The experimental verification of the mathematical results therefore is no evidence for or against the peculiar doctrines of this theory.

The true Logic for this world is the Calculus of Probabilities, which takes account of the magnitude of the probability.

But though the professed aim of all scientific work is to unravel the secrets of nature, it has another effect, not less valuable, on the mind of the worker. It leaves him in possession of methods which nothing but scientific work could have led him to invent.

It is perfect in form, and unassailable in accuracy, and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electro-dynamics.

Such indeed is the respect paid to science, that the most absurd opinions may become current, provided they are expressed in language, the sound of which recals [sic] some well-known scientific phrase.

Very few of us can now place ourselves in the mental condition in which even such philosophers as the great Descartes were involved in the days before Newton had announced the true laws of the motion of bodies.

Every existence above a certain rank has its singular points; the higher the rank the more of them. At these points, influences whose physical magnitude is too small to be taken account of by a finite being may produce results of the greatest importance.

The dimmed outlines of phenomenal things all merge into one another unless we put on the focusing-glass of theory, and screw it up sometimes to one pitch of definition and sometimes to another, so as to see down into different depths through the great millstone of the world.

It is of great advantage to the student of any subject to read the original memoirs on that subject, for science is always most completely assimilated when it is in the nascent state.