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182. C. 1.





THE Differential and the Integral Calculus have been established upon entirely different axioms and definitions by the several founders of those sciences. The primary ideas of infinitesimals, fluxions, and exhaustions, though their results coincide, for the simple reason that all pure truth is consistent with itself, are widely diverse in their abstract nature. In writing, therefore, on the principles of either Calculus, a difficulty presents itself in the necessity of electing between. systems, each of which has the sanction of high authority and peculiar intrinsic merits.

This consideration is of especial importance in a "Rudimentary Treatise," which cannot, of course, fulfil the profession of its title without singleness and simplicity of its fundamental ideas, and an exactness of thought and language often very difficult of attainment. The choice of methods in the present work has been determined partly by historical considerations. The discoverers of new truths usually search after them by the simplest and most familiar considerations; and it seems natural to presume that, as far at least as abstract principles are concerned, the way of discovery is the easiest way of instruction.

The original idea upon which Newton based the system of fluxions, regarded a differential coefficient as the rate of increase of a function. The idea upon which Leibnitz and the Bernouillis established the Integral Calculus, regarded an integral as the limit of the summation of an indefinite number of indefinitely diminishing quantities. The facility

with which the idea of "rate" may be conceived and applied to the science of which Newton was the great founder, and the similar advantages of the idea of summation in the Integral Calculus, determined the selection of the first idea as the basis of the "Manual of the Differential Calculus" by the present writer, and the second as the basis of the present treatise.

The value and importance of what is termed by Professor De Morgan the "summatory" definition of integration, has been insisted upon by him and others of the most eminent modern mathematicians; but the present is probably an almost solitary attempt to establish the Integral Calculus on that definition exclusively. Throughout the entire range of the practical applications of the Integral Calculus-to Geometry, Mechanics, &c.-the idea of summation is solely and universally applied. The rival definition of the Integral Calculus-as the inverse of the Differential Calculus-has a merely relative signification, and is, therefore, essential only in analytical investigations of the relations of the two sciences.

But whatever system be adopted for establishing either calculus must of necessity involve the idea of limits and limiting values. An unreasonable reluctance has been sometimes exhibited in adopting this idea in elementary treatises, whereas that it is one by no means difficult to be conceived is shewn by its adoption in the first ages of mathematics. By far greater difficulties have arisen from the shifts to which resort has been had to evade it in theorems of which demonstrations without it are necessarily illogical.

The idea of limits occurs, or ought to occur, much earlier in the study of exact science than is generally allowed. This idea is essentially involved in Arithmetic, Euclid, and Algebra. The laws of operation with recurring decimals and surds cannot be accurately established without limits— for in what sense is the fraction equal to 3333...., or ✔ equal to another interminable decimal, except as the limits of the two infinite convergent series represented by the decimals? Euclid's definition of equality of ratios

(Book V., Def. V.), is made to include incommensurable ratios by considerations dependent on the method of limits, which also occurs repeatedly in Book XII. In Algebra, as the present writer has endeavoured to shew elsewhere (Cambridge Mathematical Journal, Feb., 1852), an exact demonstration of the Binomial Theorem must involve the method of limits. The same remark applies to the operation of equating indeterminate coefficients and the theorem a° = 1. Neglect of these considerations involves the writers of some treatises in obscurities, errors, and inconsistencies, which bring to remembrance the supposed common origin of the words "gibberish" and "algebra."*

Throughout the present work, the language of infinites and infinitely small quantities has been carefully avoided, partly because they cannot, except by an inaccuracy of language, be spoken of as really existing magnitudes which may be subjected to analytical operations, partly because the language of the method of limits is equally concise, and is, moreover, exact.

That infinity has a real existence must be admitted; for let us conceive any distance, however great, such that the remotest known star is comparatively near; we cannot say that space terminates at that distance. What is beyond the boundary? A void, perhaps, but still space; so that unless we can conceive the existence of a boundary which includes all space within it, and to which no space is external, we are forced to admit the existence of infinite space. But this admission is altogether different from that which subjects infinity to mathematical operations. How is the infinity thus operated upon to be defined? As a magnitude than which none other is greater? But by hypothesis it is the subject of analytical

* Algebra." Some, however, derive it from various other Arabic words, as from Geber, a celebrated philosopher, chemist, and mathematician, to whom they ascribe the invention of this science."-Hutton's Mathematical Dictionary. Gibberish.-"It is probably derived from the chemical cant, and originally implied the jargon of Geber and his tribe."-Johnson's Dictionary.

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