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INFINITESIMAL CALCULUS

a satisfactory account of them. While new and varied

of Brook Taylor and Colon Maclaurin were absorbed into the rapidly growing continental analysis, and the more precise conceptions reached through a critical scrutiny of the true nature of Newton's fiuxions and moments stimulated a like scrutiny of the basis of the method of differentials. 30. This method had met with opposition from the first. Christiaan Huygens, whose opinion carried more weight than opposh that of any other scientific man of the day, declared tion that the employment of differentials was unnecessary, ° "Ie and that Leibnitz's second differential was meaningless “'°"'"s' (1691). A Dutch physician named Bernhard Nieuwentijt attacked the method on account of the use of quantities which are at one stage of the process treated as somethings and at a later stage as nothings, and he was especially severe in commenting upon the second and higher differentials (1694, 169 5). Other attacks were made by Michel Rolle (1701), but they were directed rather against matters of detail than against the general principles. The fact is that, although Leibnitz in his answers to Nieuwentijt (1695), and to Rolle (1702), indicated that the processes of the calculus could be justified by the methods of the ancient geometry, he never expressed himself very clearly on the subject of differentials, and he conveyed, probably without intending it, the impression that the calculus leads to correct results by compensation of errors. In England the method of fiuxions had to face similar attacks. George Berkeley, bishop and philosopher, wrote in 1734 a tract entitled The Analyst; or a Discourse addressed to an I njidel 'M mathematician, in which he proposed to destroy the presumption that the rhc"AM opinions of mathematicians in matters of faith are ys, ,, mm likely to be more trustworthy than those of divines, rmversy. by contending that in the much vaunted fluxional calculus there are mysteries which are accepted unquestioningly by the mathematicians, but are incapable of logical demonstration. Berkeley's criticism was levelled against all intinitesirnals, that is to say, all quantities vaguely conceived as in some intermediate state between nullity and finiteness, as he took Newton's moments to be conceived. The tract occasioned a controversy which had the important consequence of making it plain that all arguments about infinitesimals must be given up, and the calculus must be founded on the method of limits. During the controversy Benjamin Robins gave an exceedingly clear explanation of Newton's theories of fluxions and of prime and ultimate ratios regarded as theories of limits. In this explanation he pointed out that Newton's moment (Leibnitz's “ differential ”) is to be regarded as so much of the actual difference between two neighbouring values of a variable as is needful for the formation of the iluxion (or differential coefficient) (see G. A. Gibson, “The Analyst Controversy, ” Proc. Math. Soc., Edinburgh, xvii., 1899). Colin Maclaurin published in 1742 a Treatise of Fluxions, in which he reduced the whole theory to a theory of limits, and demonstrated it by the method of Archimedes. This notion was gradually transferred to the continental mathematicians. Leonhard Euler in his I nstilutioues Calculi differeutialis (17 5 5) was reduced to the position of one who asserts that all differentials are zero, but, as the product of zero and any finite quantity is zero, the ratio of two zeros can be a finite quantity which it is the business of the calculus to determine. Jean le Rond d'Ale1nbert in the Encyclopédie métlwdique (1755, 2nd ed. 1784) declared that differentials were unnecessary, and that Leibnitz's calculus was a calculus of mutually compensating errors, while Newton's method was entirely rigorous. D'Alembert's opinion of Leibnitz's calculus was expressed also by Lazare N. M. Carnot in his Réjlexions sur la métaphysique du calcul infinitesimal (1799) and by Joseph Louis de la Grange (generally called Lagrange) in writings from 1760 onwards. Lagrange proposed in his Théorie des frmctians analytiques (1797) to found the whole of the calculus on the theory of series. It was not until 1823 that a treatise on the differential calculus founded upon the method of limits was published. The treatise was the Résumé des legons sur le calcul infinilésimal of Augustin Louis Cauchy. Since that time it has been understood that the use of the phrase “infinitely small” in any mathematical argument is a figurative mode of expression pointing to a limiting process. In the opinion of many eminent c“”"'-V" method of

mathematicians such modes of expression a.re ”, ,, ,, , confusing to students, but in treatises on the calculus the traditional modes of expression are still largely adopted.

31. Defective modes of expression did not hinder constructive work. It was the great merit of Leibnitz's symbolism that a mathematician who used it knew what was to be Arith-

done in order to formulate any problem analytically, metlcal even though he might not be absolutely clear as to the b="'~= vf proper interpretation of the symbols or able to render "'°"°" analysis.

results were promptly obtained by using them, a long time elapsed before the theory of them was placed on a sound basis. Even after Cauchy had formulated his theory much remained to done, both in the rapidly growing department of complex variables, and in the regions opened up by the theory of expansions in trigonometric series. In both directions it was seen that rigorous demonstration demanded greater precision in regard to fundamental notions, and the requirement of precision led to a gradual shifting of the basis of analysis from geometrical intuition to arithmetical law. A sketch of the outcome of this movement-the “ arithmetization of analysis, ” as it has been called-will be found in FUNCTION. Its general tendency has been to show that many theories and processes, at first accepted as of general validity, are liable to exceptions, and much of the work of the analysts of the latter half of the 19th century was directed to discovering the most general conditions in which particular processes, frequently but not universally applicable, can be used without scruple.

III. Outlines of the Injnitesimal Calculus. 32. The general notions of functionality, limits and continuity are explained in the article FUNCTION. Illustrations of the more immediate ways in which these notions present themselves in the development of the differential and integral calculus will be useful in what follows.

33. Let y be given as a function of x, or, more generally, let x and y be given as functions of a variable t. The first of these cases is included in the second by putting x=t. If certain conditions are satisfied the aggregate of the points de- am' I. termined by the functional relations form a curve. The Fez, ” first condition is that the aggregate of the values of t to m S which values of x and y correspond must be continuous, or, in other words, that these values must consist of all real numbers, or of all those real numbers which lie between assigned extreme numbers. When this condition is satisfied the oints are “ ordered, " and their order is determined b the order ofp the numbers t, supposed to be arranged in order oiy increasing or decreasing magnitude; also there are two senses of description of the curve, according as t ls taken to increase or to diminish. The second condition is that the aggregate of the points which are determined by theifunctional relations must be “ continuous.” This condition means that, if any point P determined by a value of t is taken, and any distance 6, however small, is chosen, it is possible to find two points Q, Q' of the aggregate which are such that (i.) P is between Q and Q', (ii.) if R, R are any points between Q and Q' the distance RR' is less than 6. The, meaning of the word “between " in this statement is fixed by the ordering of the points. Sometimes additional conditions are imposed upon the functional relations before they are regarded as dehning a curve. An aggregate of points which satisfies the two conditions stated

above is sometimes called a

“ lordan curve." It by no

means follows that every

curve of this kind has a tangent..h

In order that the curve

may ave a tangent »

at P it is necessary Tangents

that, if any angle ut, however R,

small, is specified, a distance 5

can be found such that when

P is between Q and Q', and

PQ and PQ' are less than 6,

the angle RPR' is less than .

a for all pairs of points R, R' which are between and Q. Ol between P and Q' (fig. 8). When this condition is satisfied y is a. function of x which has a differential coefficient. The only way of RR

P Q.

R

QI

Fro. 8.