Do not imagine that mathematics is harsh and crabbed, and repulsive to common sense. It is merely the etherealisation of common sense.

I have no satisfaction in formulas unless I feel their arithmetical magnitude.

In physical science the first essential step in the direction of learning any subject is to find principles of numerical reckoning and practicable methods for measuring some quality connected with it. I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of Science, whatever the matter may be.
Often seen quoted in a condensed form:If you cannot measure it, then it is not science.

Let nobody be afraid of true freedom of thought. Let us be free in thought and criticism; but, with freedom, we are bound to come to the conclusion that science is not antagonistic to religion, but a help to it.

Mathematics is the only good metaphysics.

Mathematics is the only true metaphysics.

The life and soul of science is its practical application, and just as the great advances in mathematics have been made through the desire of discovering the solution of problems which were of a highly practical kind in mathematical science, so in physical science many of the greatest advances that have been made from the beginning of the world to the present time have been made in the earnest desire to turn the knowledge of the properties of matter to some purpose useful to mankind.

The vortex theory [of the atom] is only a dream. Itself unproven, it can prove nothing, and any speculations founded upon it are mere dreams about dreams.

There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.

You, in this country [the USA], are subjected to the British insularity in weights and measures; you use the foot, inch and yard. I am obliged to use that system, but must apologize to you for doing so, because it is so inconvenient, and I hope Americans will do everything in their power to introduce the French metrical system. ... I look upon our English system as a wickedly, brain-destroying system of bondage under which we suffer. The reason why we continue to use it, is the imaginary difficulty of making a change, and nothing else; but I do not think in America that any such difficulty should stand in the way of adopting so splendidly useful a reform.

[Referring to Fourier's mathematical theory of the conduction of heat] … Fourier's great mathematical poem ...

The saying often quoted from Lord Kelvin… that 'where you cannot measure your knowledge is meagre and unsatisfactory,' as applied in mental and social science, is misleading and pernicious. This is another way of saying that these sciences are not science in the sense of physical science and cannot attempt to be such without forfeiting their proper nature and function. Insistence on a concretely quantitative economics means the use of statistics of physical magnitudes, whose economic meaning and significance is uncertain and dubious. (Even wheat is approximately homogeneous only if measured in economic terms.) And a similar statement would even apply more to other social sciences. In this field, the Kelvin dictum very largely means in practice, 'if you cannot measure, measure anyhow!'

When Lord Kelvin was in this country [U.S.], he said that nothing interested him so much as Mr. Hewitt's work and his vacuum lamp.
Referring to the mercury lamp invention.

Geologists have not been slow to admit that they were in error in assuming that they had an eternity of past time for the evolution of the earth's history. They have frankly acknowledged the validity of the physical arguments which go to place more or less definite limits to the antiquity of the earth. They were, on the whole, disposed to acquiesce in the allowance of 100 millions of years granted to them by Lord Kelvin, for the transaction of the whole of the long cycles of geological history. But the physicists have been insatiable and inexorable. As remorseless as Lear's daughters, they have cut down their grant of years by successive slices, until some of them have brought the number to something less than ten millions. In vain have the geologists protested that there must somewhere be a flaw in a line of argument which tends to results so entirely at variance with the strong evidence for a higher antiquity, furnished not only by the geological record, but by the existing races of plants and animals. They have insisted that this evidence is not mere theory or imagination, but is drawn from a multitude of facts which become hopelessly unintelligible unless sufficient time is admitted for the evolution of geological history. They have not been able to disapprove the arguments of the physicists, but they have contended that the physicists have simply ignored the geological arguments as of no account in the discussion.

I came into the room, which was half dark, and presently spotted Lord Kelvin in the audience and realised that I was in for trouble at the last part of my speech dealing with the age of the earth, where my views conflicted with his. To my relief, Kelvin fell fast asleep, but as I came to the important point, I saw the old bird sit up, open an eye and cock a baleful glance at me! Then a sudden inspiration came, and I said Lord Kelvin had limited the age of the earth, provided no new source (of energy) was discovered. That prophetic utterance refers to what we are now considering tonight, radium! Behold! the old boy beamed upon me.