## An Elementary Treatise on the Differential and Integral Calculus, Volume 25As part of their campaign to introduce the continental notation into Great Britain, Babbage and his friends Herschel and Peacock translated and published S. F. Lacroix's Sur le calcul differentiel et integral (1802). Babbage had begun the task of translation while still at Cambridge, but for one reason or another set it aside uncompleted. "A few years later Peacock called on me in Devonshire Street, and stated that both Herschel and himself were convinced that the change from the dots to the d's would not be accomplished until some foreign work of eminence should be translated into English. Peacock then proposed that I should either finish the translation which I had commenced, or that Herschel and himself should complete the remainder of my translation. I suggested that we should toss up which alternative to take. It was determined by lot that we should make a joint translation." Some months after, the translation of the small work of Lacroix was published (Babbage 1864, 39). Part I of Lacroix's work, on differential calculus, was translated by Babbage; Part II, on integral calculus, was translated jointly by Peacock and Herschel. The "Appendix" was written by Herschel, and he and Peacock collaborated on the notes. Acceptance of the new notation was slow at first, but by 1820 the d-notation had triumphed at Cambridge, largely due to the support and influence of William Whewell, the future dean of Trinity College. |

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abscissae algebraical applied arbitrary constant arbitrary function axis becomes circle co-ordinates complete integral consequently considered const corresponding cosz cycloid deduce denominator denote determined developement differen Differential Calculus differential coefficients differential equation dº y dºu dºy eliminate equa evanescent example exponent expression factor ferential fluxions follows formula fraction func given gives increment Integral Calculus limits logarithms manner maxima and minima method multiply obtain ordinate osculating circle parabola partial fraction particular solution plane preceding primitive equation proposed curve proposed equation proposed function radius ratio reduced represent result second order shew sine ſº substituting suppose tangent Taylor's theorem theorem tion unity vanish variables whence whole number

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Page 583 - Proposition 14. The surface of any isosceles cone excluding the base is equal to a circle whose radius is a mean proportional between the side of the cone [a generator] and the radius of the circle which is the base of the cone.

Page 35 - The part 2az, which is independent of h, is therefore the limit of the ratio of the increment of the function to that of the variable.

Page 124 - It is the curve described by a point in the circumference of a circle, while the circle itself rolls in a straight line along a plane.

Page v - D'Alembert, in the place of the more correct and natural method of Lagrange, which was adopted in the former. The first part of this Treatise, which is devoted to the exposition of the principles of the Differential Calculus, was translated by Mr. Babbage. The translation of the second 'part, which treats of the Integral Calculus, was executed by Mr. G. Peacock, of Trinity College, and by Mr. Herschel, of St. John's College, in nearly equal proportions.

Page 582 - DO, do, of the inscribed circles. The surfaces of these polygons are to each other as the squares of the homologous sides AB, ab (B.

Page 497 - It was also shown in the same article, that the differential of the sum of any number of functions is equal to the sum of their...

Page 309 - x". It is not "dy" divided by "dx" or "d" multiplied by "y" divided by "d" multiplied by "x." In precise mathematical terms a derivative of a function is the limit of the ratio of the increment of the function to the increment of the independent variable when the latter increment varies and approaches zero as a limit.

Page 588 - Geométrica censetur; malui demonstrationes rerum sequentium ad ultimas quantitatum evanesceutium summas et rationes, primasque nascentium, id est ad limites summarum et rationum deducere; et propterea limitum illorum demonstrationes qua potui brevitate praemittere. His enim idem praestatur quod per methodum Indivisibilium ; et principiis demonstratis jam tutius utemur.

Page 581 - ... less than any assigned area, however small ; and since the circle is necessarily less than the first, and greater than the second, it must differ from either of them by a quantity less than that by which they differ from each other...

Page 251 - B sin A cos B = £{sin (A + B) + sin (A - B)}. cos A sin B = £{sin (A + B) - sin (A - B)}. cos A cos B = i{cos (A + B) + cos (A - B)}.