By David Perkins
Calculus & Its Origins is an outline of calculus as an highbrow pursuit having a 2,000-year history.
Author David Perkins examines the level to which mathematicians and students from Egypt, Persia, and India absorbed and nourished Greek geometry, and info how the students wove their inquiries right into a unified theory.
Chapters conceal the tale of Archimedes discovery of the world of a parabolic section; ibn Al-Haytham s calculation of the quantity of a revolved quarter; Jyesthadeva s rationalization of the countless sequence for sine and cosine; Wallis s deduction of the hyperlink among hyperbolas and logarithms; Newton s generalization of the binomial theorem; Leibniz s discovery of integration by means of parts--and a lot more.
Each bankruptcy additionally includes routines by way of such mathematical luminaries as Pascal, Maclaurin, Barrow, Cauchy, and Euler. Requiring just a easy wisdom of geometry and algebra--similar triangles, polynomials, factoring--and a willingness to regard the endless as metaphor--Calculus & Its Origins is a treasure of the human mind, pearls strung jointly by way of mathematicians throughout cultures and centuries.
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Although the rates differ, then, the quantity cannot escape its fate; it is consumed either way. 4. 3 Descartes corrects Fermat’s method. In taking issue with Fermat’s methods described in this chapter, Descartes wrote a letter (to a third party) in which he hoped to repair the methods. 8. Today we would describe curve ABD using x D y 3 . 14) (b) Line HBD is not tangent to the curve, but if we allow DF to slide left until it is superimposed on BC (which, it is worth noting, would slide H closer to A), then the new line HB would be tangent.
1 Region ACD is revolved around side CD to create a volume much like a solid upside-down bowl. Ibn al-Haytham sliced the area within region ACD into thin rectangular strips and asked what volume each strip would create when revolved around CD to create a thin cylinder. Totaling the volumes of these stacked cylinders, he estimated the volume within the upside-down bowl. Thinner slices give better estimates. 3, so that the height h of each slice is 1=n. In the figure, only the kth slice from the bottom is drawn.
1. The heights of the rectangles climbing up the left-hand side are all 1 unit, but the widths of those growing across the bottom are the cubes of successive integers (and thus, for convenience, the scale of the widths is only suggestive). The area of each rectangle appears within it; note the fourth powers appearing as areas. 1. Ibn al-Haytham equated the entire area of the figure with the sum of its rectan- gular parts. Consider the total area of the figure in two ways: as a simple height-times-width, and as the sum of all the areas of the rectangular pieces.