Newton Papers : On the Quadrature of Curves

Newton, Isaac, Sir, 1642-1727

Newton Papers

In Add. 3962 one finds some of Newton’s papers related to the ‘quadrature of curves’. In Newton’s time, providing the ‘quadrature’ of a curve meant calculating the area enclosed or subtended to it. Thus, ‘squaring the circle’ meant calculating the circle area: literally, determining a square the area of which is equal to the area of the circle. As Newton and his contemporaries realised, in most cases it is possible to attain only an approximate value. Most of the papers in Add. 3962 were written in the early 1690s, and in some cases in the early eighteenth century when Newton prepared these results for publication with the title Tractatus de Quadratura Curvarum as an appendix to the Opticks (1704), pp. 165-211. An interesting item is 3962.1, which is the final text for its introduction. Other manuscripts related to quadratures, both drafts preceding the publication and later intended revisions, can be found elsewhere in the Portsmouth and Macclesfield collections, most notably in <a href='/view/MS-ADD-03960'>Add. 3960.6-11, 3960.13</a> and Add. 9597.2.8-10. One reason why these manuscripts are interesting is that we find preliminary drafts, texts prepared for the printer, and later revisions of the printed text. It is rare to have the opportunity of following so closely Newton’s practices of writing: from annotations jotted down on working sheets to texts deemed ready for publication.The last item [<a href='' onclick='store.loadPage(139);return false;'>Add. 3962.6</a>] on The quadrature of all curves whose æquations consist of but three terms is tentatively dated 1671 by Whiteside, mostly on the basis of the handwriting and the choice of English. In 1685 Newton drew on this paper in planning a scholium for the Principia, and he reused it again in writing his early versions of De quadratura in the early 1690s. This manuscript and its later use by Newton shows very clearly how the geometrical language of the Principia often hides the use of rather advanced techniques in integration, which Newton was happy to publish in the second volume of Wallis’s Opera mathematica (1693). In this case, Newton – continuing researches on quadrature in Problem 8 of the so-called De methodis serierum et fluxionum (Add. 3960.14) – is interested in what we would call the integration of irrational functions by ‘comparison with the area of the hyperbola or ellipsis’. That is, by variable substitution Newton performs these integrations (to use modern language) in terms of logarithms and trigonometric functions.Niccolò Guicciardini, Università degli Studi di Milano, and Scott Mandelbrote, Peterhouse, Cambridge.