Quas nos quantitates variabiles vocamus, eas Angli nomine magis idoneo quantitates fluentes vocant, et earum incrementa infinitè parva seu evanescentia Auxiones nominant, ita ut fluxiones ipsis idem sint, quod nobis differentialia. Hæc diversitas loquendi ita jam usu invaluit ut conciliatio vix unquam sit expectanda; equidem Anglos in formulis loquendi lubenter imitarer, sed signa quibus nos utimur illorum signis longe anteferenda videntur. Euler, Cal. Int. vol. i. p. 3. LONDON: Acer William it butto 10.04.1935 PREFACE. THIS Work will contain the substance of a course of lectures read to pupils some few years ago. Part of the leisure which I have had since that period has been devoted to such alterations and corrections as may make it more generally useful. It is with this view that I have not confined myself to the Newtonian doctrine of limits, but have introduced the principles of La Grange's Theory of Functions. The two systems meet in Taylor's Theorem, and that being once established, the difference is merely nominal. I have endeavoured to render this work as independent of all others as possible, and have required as little previous knowledge as the nature of the subject will admit. This consists of the elements of Geometry, Trigonometry, Algebra, and Conick Sections. The works on these subjects which I have always used, and to which I of course refer, are Simson's Euclid, Professor Woodhouse's Trigonometry, third edition, Dr. Wood's Algebra, sixth edition, and Mr. Peacock's Conick Sections, second edition. To complete the work, it should be extended to a discussion of certain curves—to a treatise on the calculus of fluents—the integration of fluxional equations—the calculus of variations—the application of the calculus to curved surfaces, including those of double curvature. This, together with its application to such parts of Natural Philosophy as usually form an academical course of lectures, may appear before the publick in a second volume. CONTENTS. 14_20 . 52 Page CHAP. I. On the first fluxions of a function con- Rules for the differentiation of algebraical quantities 6 Integration of algebraical forms 36-40 Series for the computation of logarithms Series for the sine and cosine of the multiple arc 61 Imaginary logarithmick and circular expressions 62-65 by particular artifices of calculation By the method of indeterminate coefficients 105-114 Rules for the differentiation of functions 114-120 . . CHAP. IV. On the different orders of fluxions ; the developement of functions into series; and Developement of a function of two independent va- riables Maclaurin's theorem for two variables Developement of a function of three independent Developement of dependent variables Certain advantages derivable from differentiation com- Developement of functions continued J. Bernoulli's theorem CHAP. V. The cases in which the developement of CHAP. VI. Maxima and minima functions of one Functions of two independent variables . . |