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If then x and y are connected by some fixed law, as must be the case if the extremity of y traces out some regular curve, the relation between them may be expressed by an equation, which will remain one however often it may be differentiated or again integrated, and whatever modifications or transformations it may receive by mathematical processes which do not alter the essential equality of the two sides connected by the symbol of equality =. Thus by differentiating and casting off as evanescent all differentials of a lower order than that which we are working with, we may arrive at forms of which we know the integrals, and by integrating get back to the results in ordinary numbers, which we were in search of but could not attain directly.

The same thing will apply if our symbols are more numerous, and if they express relations of motion as well as of space, or, in fact, any relations which are governed by fixed laws expressible by equations. If I have succeeded in conveying to the readers any idea of this celebrated calculus, they will perceive what an analogy it presents to the idea of modern physical and chemical science, that of molecules, atoms, and ether, forming differentials of successive orders of the infinitely small. It is certainly most remarkable that while the former was a purely intellectual idea based on mathematical abstractions, and which was invented and worked as an instrument for solving the most intricate astronomical problems for nearly two centuries, without a suspicion that it represented any objective reality: the latter idea, based on actual experiment, seems to show that differentials and integrals have their real counterpart in nature and represent fundamental facts in the constitution of the universe.