At the time the book of Marquis de l'Hôpital had appeared, and almost all mathematicians began to turn to the new geometry of the infinite [that is, the new infinitesimal calculus], until then little known. The surprising universality of the methods, the elegant brevity of the proofs, the neatness and speed of the most difficult solutions, a singular and unexpected novelty, all attracted the mind and there was in the mathematical world a well marked revolution [une révolution bien marquée.

The physicist needs a facility in looking at problems from several points of view. The exact analysis of real physical problems is usually quite complicated, and any particular physical situation may be too complicated to analyze directly by solving the differential equation. But one can still get a very good idea of the behavior of a system if one has some feel for the character of the solution in different circumstances. Ideas such as the field lines, capacitance, resistance, and inductance are, for such purposes, very useful. ... On the other hand, none of the heuristic models, such as field lines, is really adequate and accurate for all situations. There is only one precise way of presenting the laws, and that is by means of differential equations. They have the advantage of being fundamental and, so far as we know, precise. If you have learned the differential equations you can always go back to them. There is nothing to unlearn.

The application of mathematics to the solution of the problems presented by the motion of the heavenly bodies has had a larger degree of success than the same application in the case of the other departments of physics. This is probably due to two causes. The principal objects to be treated in the former case are visible every clear night, consequently the questions connected with them received earlier attention; while, in the latter case, the phenomena to be discussed must ofttimes be produced by artificial means in the laboratory; and the discovery of certain classes of them, as, for instance, the property of magnetism, may justly be attributed to accident. A second cause is undoubtedly to be found in the fact that the application of quantitative reasoning to what is usually denominated as physics generally leads to a more difficult department of mathematics than in the case of the motion of the heavenly bodies. In the latter we have but one independent variable, the time; while in the former generally several are present, which makes the difference of having to integrate ordinary differential equations or those which are partial.

So intimate is the union between Mathematics and Physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create for each successive class phenomena a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature. Sometimes the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armory of the mathematician the weapons which he needed ready made to his hand. But much oftener, the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma.

In the case of partial differential equations employed in connection with physical problems, their use must be given up in most circumstances, for two reasons: first, it is in general impossible to get the general solution or general integral, and second, it is in general of no use even when it is obtained.

As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation (can be compared with experience).

The theory of s in approximately one century old, although its origin may be traced back much further. As originally formulated by and subsequently used throughout his work, the theory was intended as a tool to be used in the study of geometric problems. After two periods of theoretical development, one in the 1930s and the other in the 1960s, there has recently been a renewed interest in exterior differential systems as providing a systematic framework for the study of geometric problems. It is my opinion that this development is just beginning, and that exterior differential systems should become a standard tool for geometers, especially for questions where the differential equations expressing the problem are overdetermined systems, and for global questions. When used properly, the theory has a marvelous ability to reveal the underlying geometry in a complicated problem.

Any progress in the theory of partial differential equations must also bring about a progress in Mechanics.

As professor in the Polytechnic School [autumn of 1858] in Zurich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt, more keenly than ever before, the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis.

Why was the discovery of the ... so important for the physics of the 1920s and 30s? The answer is manifold. The Dirac equation provided a relativistic description of spin ½ particles and in particular of the electron. In doing so, it gave a relativistic description of spin and opened the way for the application of group theory to the description of particles of arbitrary spin. The reinterpretation of the Dirac equation as a field equation that followed from Dirac's theory of holes was decisive in the conceptual transformation of single particle theory to many particle (quantum field) theory. The resulting quantum electrodynamics of spin ½ particles, refined by two generations of theoretical work, is the best theory we have. Although it is an approximation since it does not include the effects of weak and strong interactions it has survived many stringent experimental tests when applied to electrons and s.

At the Stourbridge Fair in 1663, at age twenty, he purchased a book on astrology, “out of a curiosity to see what there was in it.” He read it until he came to an illustration which he could not understand, because he was ignorant of trigonometry. So he purchased a book on trigonometry but soon found himself unable to follow the geometrical arguments. So he found a copy of Euclid’s Elements of Geometry, and began to read. Two years later he invented the differential calculus.

[F]or the further improvement of natural philosophy a more advanced geometry must be found. ...[T]he reason why physical science has here been brought to a level that is the envy of foreigners is the knowledge... of some more universal geometry. Of what part of this the learned owe to this renowned university and in it to the prince of geometers I shall not speak lest I appear to be fawning, which in a mathematician would be unseemly.

Great physics does not automatically imply complicated mathematics!

In fact, the science of thermodynamics began with an analysis, by the great engineer Sadi Carnot, of the problem of how to build the best and most efficient engine, and this constitutes one of the few famous cases in which engineering has contributed to fundamental physical theory. Another example that comes to mind is the more recent analysis of information theory by Claude Shannon. These two analyses, incidentally, turn out to be closely related.