{"id":1560,"date":"2011-10-03T09:00:28","date_gmt":"2011-10-02T21:00:28","guid":{"rendered":"https:\/\/blogs.otago.ac.nz\/emxphi\/?p=1560"},"modified":"2012-09-25T02:11:39","modified_gmt":"2012-09-24T14:11:39","slug":"newton-and-the-case-of-the-missing-calculus","status":"publish","type":"post","link":"https:\/\/blogs.otago.ac.nz\/emxphi\/newton-and-the-case-of-the-missing-calculus\/","title":{"rendered":"Newton and the Case of the Missing Calculus"},"content":{"rendered":"

Kirsten Walsh writes…<\/strong><\/p>\n

The case of the missing calculus is well-known.\u00a0 Newton (co-)invented calculus<\/a> in the late 1660s, and he wrote Principia<\/em><\/a> <\/em>in the late 1680s.\u00a0 It would be natural to expect that Newton used the calculus in Principia<\/em>.\u00a0 But it seems that he didn\u2019t.\u00a0 Instead, Newton wrote Principia<\/em> in the style of Euclid\u2019s Elements<\/em><\/a>, that is, using Classical Greek geometry. \u00a0This is surprising indeed, given the powerful new tool he had at his disposal.\u00a0 What should we make of this?<\/p>\n

Almost thirty years after the publication of Principia<\/em>, Newton explained that he had used algebraic calculus to discover the propositions of Principia<\/em>, but used classical geometry to demonstrate them:<\/p>\n

    \u201cBy the help of the new Analysis<\/em> [i.e. algebraic calculus] Mr. Newton found out most of the Propositions in his Principia Philosophiae<\/em>: but because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically, that the System of the Heavens might be founded upon good Geometry.\u00a0 And this makes it now difficult for unskilful men to see the Analysis by which those Propositions were found out.\u201d<\/ol>\n

    But Newton was lying.\u00a0 Scholars<\/a> have found no evidence that he wrote or developed Principia<\/em> in any other way than the published form.\u00a0 Moreover, few, if any, of the propositions in Principia<\/em> can even be presented in the form of algebraic calculus.<\/p>\n

    This raises two questions:<\/p>\n

      \n
    1. Why did Newton lie?<\/li>\n
    2. Why did Newton eschew modern algebraic calculus in favour of classical geometry?<\/li>\n<\/ol>\n

      These questions have been discussed by numerous scholars including A. Rupert Hall<\/a> and I. Bernard Cohen<\/a>.\u00a0 The answer to (1) can be found in Newton\u2019s priority dispute with Leibniz.\u00a0 The answer to (2) was summarised neatly by Thony Christie<\/a> last year:<\/p>\n

        \u201cPut simply Newton had serious doubts about the reliability of the new analytical mathematics and that is why he didn\u2019t use it for his magnum opus.\u201d<\/ol>\n

        But what caused these doubts?<\/p>\n

        In 1714, Newton wrote that the algebraic calculus is \u201carithmetic applied to geometrical matters… Its operations are complicated and excessively susceptible to errors, and can be understood by the learned in algebra alone\u201d.\u00a0 Whereas geometry \u201cmay be appreciated by the great majority and thus most impress the mind with [its] clarity\u201d.\u00a0 One might wonder why Newton bothered to invent algebraic calculus at all!<\/p>\n

        Well it seems that Newton wasn\u2019t always so anti-algebra, nor was he always so interested in classical geometry.\u00a0 In fact, as an undergraduate, Newton didn\u2019t read the ancients.\u00a0 Rather, he read a few modern summaries of the ancient texts, building his own mathematics on the algebraic work of mathematicians such as Descartes, Wallis and Barrow.<\/p>\n

        Newton seems to have become interested in classical geometry in the late 1670s, after re-reading Descartes\u2019 La G<\/em>\u00e9om<\/em>\u00e9trie<\/em><\/a>.\u00a0 La G<\/em>\u00e9om<\/em>\u00e9trie<\/em> was an attempt to unite algebra and geometry \u2013 Descartes aimed to show how symbolic algebra could be applied to the study of plane curves.\u00a0 Guiccardini<\/a> writes:<\/p>\n

          \u201c[Descartes\u2019] tract could be read as a deliberate proof of the superiority of the new analytical method, uniting symbolic algebra and geometry, over the purely geometrical ones of the ancients.\u201d<\/ol>\n

          Newton was very critical of Descartes\u2019 text, writing comments such as \u201cerror\u201d and \u201cI hardly approve\u201d in the margins.\u00a0 He even drafted a paper entitled \u2018Errors in Descartes\u2019 Geometry\u2019.<\/em> To find support for his position, Newton began to read the ancient texts, including Pappus<\/a>.<\/p>\n

          Newton wrote:<\/p>\n

            \u201cTo be sure, [the ancients\u2019] method is more elegant by far than the Cartesian one.\u00a0 For [Descartes] achieved the result by an algebraic calculus which, when transposed into words (following the practice of the Ancients in their writings), would prove to be so tedious and entangled as to provoke nausea, nor might it be understood.\u00a0 But they accomplished it by certain simple propositions, judging that nothing written in a different style was worthy to be read, and in consequence concealing the analysis by which they found their constructions.\u201d<\/ol>\n

            Newton was neither the first, nor the only, philosopher to equate algebra and geometry with the ancient methods of analysis and synthesis respectively.\u00a0 But he was<\/em> the first to reject modern algebraic calculus in favour of ancient geometry.\u00a0 (If only because he was the first to invent<\/em> it!)\u00a0 Does Newton\u2019s rejection of algebraic calculus stem from his anti-Cartesian stance?\u00a0 What if Newton had never re-read Descartes\u2019 G<\/em>\u00e9om<\/em>\u00e9trie<\/em>?\u00a0 Could his priority dispute with Leibniz have been avoided?<\/p>\n","protected":false},"excerpt":{"rendered":"

            Kirsten Walsh writes… The case of the missing calculus is well-known.\u00a0 Newton (co-)invented calculus in the late 1660s, and he wrote Principia in the late 1680s.\u00a0 It would be natural to expect that Newton used the calculus in Principia.\u00a0 But […]<\/p>\n","protected":false},"author":4582,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[113],"tags":[275,224],"class_list":["post-1560","post","type-post","status-publish","format-standard","hentry","category-ideas","tag-mathematics","tag-newton"],"_links":{"self":[{"href":"https:\/\/blogs.otago.ac.nz\/emxphi\/wp-json\/wp\/v2\/posts\/1560","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.otago.ac.nz\/emxphi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.otago.ac.nz\/emxphi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.otago.ac.nz\/emxphi\/wp-json\/wp\/v2\/users\/4582"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.otago.ac.nz\/emxphi\/wp-json\/wp\/v2\/comments?post=1560"}],"version-history":[{"count":0,"href":"https:\/\/blogs.otago.ac.nz\/emxphi\/wp-json\/wp\/v2\/posts\/1560\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.otago.ac.nz\/emxphi\/wp-json\/wp\/v2\/media?parent=1560"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.otago.ac.nz\/emxphi\/wp-json\/wp\/v2\/categories?post=1560"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.otago.ac.nz\/emxphi\/wp-json\/wp\/v2\/tags?post=1560"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}