{"id":629,"date":"2022-07-27T12:02:50","date_gmt":"2022-07-27T16:02:50","guid":{"rendered":"http:\/\/people.clas.ufl.edu\/ufhatch\/?p=629"},"modified":"2026-03-19T09:03:24","modified_gmt":"2026-03-19T13:03:24","slug":"newtons-demonstration-of-the-law-of-areas","status":"publish","type":"post","link":"https:\/\/people.clas.ufl.edu\/ufhatch\/2022\/07\/27\/newtons-demonstration-of-the-law-of-areas\/","title":{"rendered":"Newton&#8217;s Demonstration of the Area Law"},"content":{"rendered":"\r\n<section class=\"fullwidth-text-block\">\r\n\t<div class=\"container px-0 pt-5\">\r\n\t\t<div class=\"row align-items-start\">\r\n\t\t\t<div class=\"col-12\">\r\n\t\t\t\t\n<h1 class=\"wp-block-heading\">Newton&#8217;s Demonstration of the Area Law<\/h1>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"size-full aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"450\" height=\"340\" src=\"http:\/\/people.clas.ufl.edu\/ufhatch\/files\/newtonsDemonstrationLawsOfArea.jpg\" alt=\"\" class=\"wp-image-630\" style=\"width:450px;height:auto\" srcset=\"https:\/\/people.clas.ufl.edu\/ufhatch\/files\/newtonsDemonstrationLawsOfArea.jpg 450w, https:\/\/people.clas.ufl.edu\/ufhatch\/files\/newtonsDemonstrationLawsOfArea-300x227.jpg 300w, https:\/\/people.clas.ufl.edu\/ufhatch\/files\/newtonsDemonstrationLawsOfArea-414x313.jpg 414w\" sizes=\"auto, (max-width: 450px) 100vw, 450px\" \/><\/figure>\n<\/div>\n\n\n\n\n<p class=\"wp-block-paragraph\">In Proposition I of the <em>Principia<\/em> Newton derives Kepler&#8217;s Law of Areas on the assumption that a revolving body is subject to a centripetal force directed to a fixed point.  Newton&#8217;s demonstration employs equal finite time intervals, after each of whlch the force gives a finite impulse to the body towards center S. During one interval, from A to B, the motion remains the same; then at B it is suddenly changed by the additional motion BV. In the second interval, instead of continuing the motion along Bc=AB the body follows the resulting path BC. Since the areas SAB and SBc are equal, and because impulse BV is directed toward S, areas SBc and SBC are equal, and area SBC is equal to SAB, which holds for each succeeding interval: (Every next triangular area is equal to the preceding one). Hence all triangular areas described in equal intervals are equal, and will lie all in the same place. This holds also when the time intervals are taken ever smaller and their number ever greater in the same rate until, finally, we have a continuously acting force and a curved orbit, for which the areas described are proportional to the time used.<\/p>\n\n\n\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t<\/div>\r\n<\/section>\r\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1262,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"featured_post":"off","footnotes":"","_links_to":"","_links_to_target":""},"categories":[4,5],"tags":[],"class_list":["post-629","post","type-post","status-publish","format-standard","hentry","category-history-of-science","category-study-guide"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/ufhatch\/wp-json\/wp\/v2\/posts\/629","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/people.clas.ufl.edu\/ufhatch\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/people.clas.ufl.edu\/ufhatch\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/ufhatch\/wp-json\/wp\/v2\/users\/1262"}],"replies":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/ufhatch\/wp-json\/wp\/v2\/comments?post=629"}],"version-history":[{"count":3,"href":"https:\/\/people.clas.ufl.edu\/ufhatch\/wp-json\/wp\/v2\/posts\/629\/revisions"}],"predecessor-version":[{"id":1405,"href":"https:\/\/people.clas.ufl.edu\/ufhatch\/wp-json\/wp\/v2\/posts\/629\/revisions\/1405"}],"wp:attachment":[{"href":"https:\/\/people.clas.ufl.edu\/ufhatch\/wp-json\/wp\/v2\/media?parent=629"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/ufhatch\/wp-json\/wp\/v2\/categories?post=629"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/ufhatch\/wp-json\/wp\/v2\/tags?post=629"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}