By C.H.Jr. Edwards

The calculus has served for 3 centuries because the crucial quantitative language of Western technological know-how. during its genesis and evolution the most basic difficulties of arithmetic have been first con fronted and, throughout the chronic labors of successive generations, eventually resolved. as a result, the historic improvement of the calculus holds a distinct curiosity for somebody who appreciates the worth of a ancient point of view in instructing, studying, and having fun with arithmetic and its ap plications. My objective in scripting this booklet used to be to give an account of this improvement that's available, now not exclusively to scholars of the heritage of arithmetic, yet to the broader mathematical group for which my exposition is extra particularly meant, together with those that research, train, and use calculus. The scope of this account will be delineated in part by means of comparability with earlier works within the similar normal region. M. E. Baron's The Origins of the Infinitesimal Calculus (1969) offers an informative and trustworthy deal with ment of the precalculus interval as much as, yet no longer together with (in any detail), the time of Newton and Leibniz, simply whilst the curiosity and speed of the tale start to quicken and accentuate. C. B. Boyer's famous booklet (1949, 1959 reprint) met good the targets its writer set for it, however it used to be extra ap propriately titled in its unique edition-The techniques of the Calculus than in its reprinting.

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Quadrature of the Parabola On the Sphere and Cylinder On Spirals On Conoids and Spheroids The Method The first five of these develop the method of exhaustion into a technique of remarkable power which Archimedes applied to a wide range of problems that today are typical applications of the integral calculus, and which provided the starting point for the modem development of the calculus. Treatise (6), which was unknown until its rediscovery in 1906, describes the heuristic infinitesimal method by which Archimedes first discovered many of his results.

2 7 It is generally believed that the extant Measurement of a Circle is only a fragment of Archimedes' original and more comprehensive treatment of the circle. In a recent article W. R. Knorr [10] argues persuasively that, to obtain a more accurate approximation to 'IT, Archimedes started with inscribed and circumscribed decagons (regular 100sided polygons) and successively doubled sides six times to obtain inscribed and circumscribed regular polygons with 640 sides. 35 The Quadrature of the Parabola ExERCISE 6.

At this point Archimedes derives the elementary identity I I I I I 4 1 + '4 + 42 + . . + 4" + 3 4" = 3 . (7) This follows from the observation that 1 1 1 4 4k +3 . 4k = =3 . +1. ) 4 3 4 4 =3' It is tempting to simply sum the geometric series by letting n ~ 00 in (7) to obtain I I 1 4 1 + '4 + 42 + . . + 4" + . . + ... + ... -+ ... ) 4 4 a(APB) = -a 3 4" 4 = -a(,6,APB) 3 o as desired. No doubt Archimedes intuitively obtained the answer 4/3 in similar fashion but, rather than taking limits explicitly, he concluded the proof with a typical double reductio ad absurdum argument which we leave to the reader.