A scientific or technical study always consists of the following three steps:
1. One decides the objective.
2. One considers the method.
3. One evaluates the method in relation to the objective.

He who seeks for methods without having a definite problem in mind seeks for the most part in vain.

I have no fault to find with those who teach geometry. That science is the only one which has not produced sects; it is founded on analysis and on synthesis and on the calculus; it does not occupy itself with the probable truth; moreover it has the same method in every country.

I see with much pleasure that you are working on a large work on the integral Calculus [ ... ] The reconciliation of the methods which you are planning to make, serves to clarify them mutually, and what they have in common contains very often their true metaphysics; this is why that metaphysics is almost the last thing that one discovers. The spirit arrives at the results as if by instinct; it is only on reflecting upon the route that it and others have followed that it succeeds in generalising the methods and in discovering its metaphysics.

In our search after the Knowledge of Substances, our want of Ideas, that are suitable to such a way of proceeding, obliges us to a quite different method. We advance not here, as in the other (where our abstract Ideas are real as well as nominal Essences) by contemplating our Ideas, and considering their Relations and Correspondencies; that helps us very little, for the Reasons, and in another place we have at large set down. By which, I think it is evident, that Substances afford Matter of very little general Knowledge; and the bare Contemplation of their abstract Ideas, will carry us but a very little way in the search of Truth and Certainty. What then are we to do for the improvement of our Knowledge in Substantial beings? Here we are to take a quite contrary Course, the want of Ideas of their real essences sends us from our own Thoughts, to the Things themselves, as they exist.

It seems perfectly clear that Economy, if it is to be a science at all, must be a mathematical science. There exists much prejudice against attempts to introduce the methods and language of mathematics into any branch of the moral sciences. Most persons appear to hold that the physical sciences form the proper sphere of mathematical method, and that the moral sciences demand some other method—I know not what.

Since religion intrinsically rejects empirical methods, there should never be any attempt to reconcile scientific theories with religion. [An infinitely old universe, always evolving may not be compatible with the Book of Genesis. However, religions such as Buddhism get along without having any explicit creation mythology and are in no way contradicted by a universe without a beginning or end.] Creatio ex nihilo, even as religious doctrine, only dates to around AD 200. The key is not to confuse myth and empirical results, or religion and science.

The idea of making a fault a subject of study and not an object to be merely determined has been the most important step in the course of my methods of observation. If I have obtained some new results it is to this that I owe it.

The second [argument about motion] is the so-called Achilles, and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.
Statement of the Achilles and the Tortoise paradox in the relation of the discrete to the continuous.; perhaps the earliest example of the reductio ad absurdum method of proof.

The transition from a paradigm in crisis to a new one from which a new tradition of normal science can emerge is far from a cumulative process, one achieved by an articulation or extension of the old paradigm. Rather it is a reconstruction of the field from new fundamentals, a reconstruction that changes some of the field's most elementary theoretical generalizations as well as many of its paradigm methods and applications. During the transition period there will be a large but never complete overlap between the problems that can be solved by the old and by the new paradigm. But there will also be a decisive difference in the modes of solution. When the transition is complete, the profession will have changed its view of the field, its methods, and its goals.

The true method of discovery is like the flight of an aeroplane. It starts from the ground of particular observation; it makes a flight in the thin air of imaginative generalization; and it again lands for renewed observation rendered acute by rational interpretation.

This example illustrates the differences in the effects which may be produced by research in pure or applied science. A research on the lines of applied science would doubtless have led to improvement and development of the older methods—the research in pure science has given us an entirely new and much more powerful method. In fact, research in applied science leads to reforms, research in pure science leads to revolutions, and revolutions, whether political or industrial, are exceedingly profitable things if you are on the winning side.