![]() ![]() Taking differences gives $3xxdx 3yydy = axdy aydx$, and $dy = \frac=0$, whence one derives $2cx^3-2ccxx-aaxx aacc=0$, and dividing by $x-c$ yields $2cxx-aax-aac=0$, of which one of the roots gives for CE a value such that the perpendicular ED passes through the pulley F and the plumb bob D, when they are at rest.This image and those below appear courtesy of the Boston Public Library. Suppose that $x^3 y^3=axy$ (AP $= x$, PM $= y$, AB $= a$) gives the nature of the curve MDM. Historical note: the text of the Infiniment petits I worked with (a 1768 edition available online, with plates here), once belonged to John Quincy Adams, and bears copious annotations in his hand. He analyzes a curve from its equation by calculating the relation between the differences $dx$ and $dy$, and locates maxima and minima by looking for points where $dy=0$ (it is understood that $dx\neq 0$). The concepts of function and derivative were still in the future. ![]() Notation: Much of L'Hospital's notation, derived from Leibnitz, has remained standard, although following Descartes he writes $xx$ instead of $x^2$. TI 89 Calculus: Step by Step The Tautochrone Problem / Brachistrone Problem The tautochrone problem addresses finding a curve down which a mass placed anywhere on the curve will reach the bottom in the same amount time, assuming uniform gravity. " Unfortunately De ScientiĆ¢ infiniti, if written, was never published. I am only making this matter public because he asked me to do so in his Letters. Leibnitz having written me that he was working on the subject in a Treatise which he entitles De ScientiĆ¢ infiniti, I did not want to risk depriving the Public of such a beautiful Work which must include all the most intriguing facts about the inverse tangent Method, about rectifications of curves, about quadrature of the spaces they enclose, about those of the surfaces of the bodies they describe, about the dimensions of those bodies, about the location of centers of gravity,
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