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From the INTRODUCTION. The branch of Mathematics called the Differential Calculus was originally invented by Sir Isaac Newton, and later independently by Leibnitz, for the purpose of dealing with variable quantities; that is to say, with any quantity which does not remain constant, of which kind of quantity the whole of Physics and Engineering and daily life is full. For in so far as things remain constant they are stagnant; every kind of activity involves variable quantities, and therefore to every kind of activity the differential calculus is applicable. It is called the differential calculus because it attends not so much to the quantities themselves as to their variations DEGREES their increases or differentials: it is the calculus which deals with differentials, especially with the ratio of two differentials to each other. For by a differential is understood an infinitesimal difference or increase or increment (or, for the matter of that, decrease or decrement), and whenever one quantity changes there is always some other quantity which changes too, the two quantities being called connected variables; or either of them is called a function of the other. Although it is customary to attend to very minute increments, and to consider each differential as infinitesimal, yet the ratio of two such microscopic differentials will in general be finite, and may be large. Thus, if a thing advances 1/100 part of an inch in the millionth part of a second, it is then going at the rate of 10,000 inches per second, much faster than an express train, and nearly as fast as a bullet. Or if a slope descends the thousandth of a millimeter for each hundredth of a millimetre along, it has a gradient of 1 in 10, and is too steep for a railway without cogs. Or, if a rod expands the millionth part of its length for one-tenth of a degree rise in temperature, it has about the expansibility of iron. In this last example the two connected variables are the temperature and the length of the rod. It may be asked, why deal with infinitesimal quantities at all? Why not attend to appreciable changes of magnitude and take their ratio 1 If we could depend on quantities varying uniformly, or if they always bore to one another the relation of simple proportion, this would be the natural and sufficient thing to do. But in practice it is only a few quantities which are thus simply connected, and if we were constrained to attend always to finite differences their ratio would in general give us a mere average result, not an actual result at any instant. To know that a bullet has travelled a mile in ten seconds does not tell us with what speed it left the muzzle; and instruments adapted to ascertain this or any other actual velocity must be chronographic instruments able to record extremely small increments of time and the corresponding moderately small distance travelled. In the laboratory it is to be observed that we are bound to deal with finite changes, and thus are limited to a kind of average result: we may make the observed intervals small, but we cannot make them infinitesimal. But in theory we are not so limited, and the theoretical treatment of infinitesimal changes is decidedly simpler and easier than the treatment of finite changes; except when the observed quantities are varying at a steady or a proportional rate. In that case the finite difference becomes as easy to deal with as the differential.....

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