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And here I must say a few words on the exactness of our processes. It is supposed that the results cannot be truly accurate, because certain quantities are neglected; thus, a finite number of infinitesimals is neglected if it is added to or subtracted from a finite quantity; and similarly a finite quantity must be and is neglected when it is added to or subtracted from an infinity. And it is supposed that an error is hereby introduced, which vitiates and destroys the accuracy of the whole work, and that the results are at last true only because a compensation is made. Now let the nature of our Calculus be clearly understood; it is of itself a part of the science of number; its subject matter is continuous number; and according as its results and theorems in number are exact or not, so is its exactness to be tested. It is capable of application to other sciences; those of Geometry, Motion, &c.; but its truth is not to be tried by the results of these subjects, because their matter may not be so conformable to that of our Calculus, but that discrepancy may be between them. In Infinitesimal Calculus properly so called, our symbols are symbols of number only: we make our own materials; our infinitesimals and infinities are created by us; and are subject to certain conditions which we choose to impose on them; we make them subject to certain laws; and so long as they are employed consistently with these conditions and these laws, and in accordance with the rules of correct inference, the conclusions which they lead to are strictly correct. There is no error: the neglecting of infinitesimals is a necessary stop in our process, and therefore it is that I have used the language of necessity: I have said in Theorem VI that a finite sum


of infinitesimals must be neglected, not may be neglected, when it is added to or subtracted from a finite quantity: were this not done, infinitesimals would not be what they are, and our rules for the discovery of them would be other than they are. When however we apply to other subject matter, say of space or of motion, the conception of infinitesimals, it may be that this particular subject-matter does not admit of the continuous infinitesimal change with that exactness which the conception of numerical infinitesimal growth is capable of; it may be that there is a discrepancy, and consequently an error; for which afterwards compensation has to be made. Thus, for instance, we may in our conception of the infinitesimal Calculus as applied to Geometry assume the line joining two consecutive points in a circle to be straight, and represent it by a symbol which denotes a straight line; whereas from the geometrical definition of a circle we know that the curvature of a circle is continuous, and that the line joining two points of it, however near together they are, cannot be straight; and thus our symbols, though representatives of such straight lines, only approximately represent them. In this case doubtless there may be an error; an error not in the work of the calculus; that is true and exact; but because the geometrical quantities are not adequately expressed by the symbols; but when by means of integration we pass from the infinitesimal element to the finite function, then the finite function becomes the exact and adequate representative of the geometrical quantity, and a compensation has taken place in the act of passing from the infinitesimal element to the finite function. On investigation it will, I venture to think, be found that the exactness of the Calculus has been impugned on these and similar grounds; and therefore that it has been unfairly impugned: let it be tried on its own principles; on them I venture to say it will stand the attack. It creates its own materials, and is subject to its own laws; let it not be condemned because other materials, which you try to bring within its grasp, refuse to submit to these laws.

The Volume consists of two Parts: in the former are investigated the Theorems of the Differential Calculus so called, and in the latter the applications to Geometry of two and three dimensions are discussed.

The Chapters mark the salient divisions of the matter: and these again are subdivided into Articles, which for the sake of reference are numbered continuously through the Volume, and have their numerals placed in the corners on the top of the pages. The bracketed numerals attached to the more important equations are separate for each Chapter; and the references are usually made to the numbers of the equation and of the Article. The Analytical Table of Contents exhibits a re'sume' of the matter of the Treatise.

Pembroke College, Oxford,
Junk 6, 1857.

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