<p style='text-align: justify;'>Ms. Add. 3958 is a gathering of notes and short essays that Newton composed from the mid 1660s until the early 1670s. These writings shed light on Newton’s earliest discoveries. The breadth of subjects in which he was interested is impressive. Most of the notes concern mathematics. We find also some notes on optics [<a href='' onclick='store.loadPage(181);return false;'>fols 93r-94r</a>], tables of fixed stars [<a href='' onclick='store.loadPage(71);return false;'>fols 38r-44v</a>], and the laws of motion [<a href='' onclick='store.loadPage(157);return false;'>fols 81r-83v</a>, <a href='' onclick='store.loadPage(165);return false;'>85r-86v</a>, <a href='' onclick='store.loadPage(169);return false;'>87r-87v</a>], as one would expect given Newton’s best-known discoveries of this period. Less obvious are annotations on inheritance laws [<a href='' onclick='store.loadPage(60);return false;'>fol. 32v</a>], anatomical drawings of the optical and auditory systems of (possibly) a small rodent and the eye of a bird [<a href='' onclick='store.loadPage(65);return false;'>fols 35r, 37v</a>], and the subdivision of the intervals in the musical scale [<a href='' onclick='store.loadPage(57);return false;'>fol. 31r</a>]. On Newton’s study of the musical scale see also <a href='/view/MS-ADD-03970/1107'>Add. 3970, fols 544r-545v</a>; <a href='/view/MS-ADD-04000/288'>Add. 4000, fols 138-143r</a>.</p><p style='text-align: justify;'>The most important items in Add. 3968 constitute the so-called ‘October 1666 Tract on Fluxions’, Newton’s first attempt to systematize his discoveries on calculus, and the calculations on the Moon’s motion that might also be related to the famous story of the falling apple. </p><p style='text-align: justify;'>The ‘October 1666 Tract on Fluxions’ [<a href='' onclick='store.loadPage(92);return false;'>fols 48v-63v</a>] owes its name to a date squeezed onto the page (in a different ink?) and added apparently in retrospect <a href='' onclick='store.loadPage(93);return false;'>on top of fol. 49r</a>. At some point, this essay must have circulated in manuscript, since there are extant copies of it. The University Library owns a later version in <a href='/view/MS-ADD-03960/1'>Add. 3960.1. fols 1-50</a> and a copy (probably in William Jones’s hand) in the Macclesfield Collection [Add. 9597.9.1]. In both versions the original notation for velocities, terminology and some methods of proof have been altered. In this essay Newton attempted to systematize results in the calculus (the method of series and fluxions, as he named it) that he had achieved in the previous two years. Newton’s version of the rules of the differential calculus can be seen applied to a cubic curve on <a href='' onclick='store.loadPage(100);return false;'>fol. 52v</a>. The same result, dated 13 November 1665 (possibly, again, in retrospect) can be found in the Waste Book [<a href='/view/MS-ADD-04004/117'>Add. 4004, fol. 57r</a>]. The logarithm calculations based on the binomial theorem [<a href='' onclick='store.loadPage(149);return false;'>fols 78r-80v</a>] are closely related to Newton’s discovery of the calculus. Similar calculations can be found in the Waste Book [<a href='/view/MS-ADD-04004/163'>Add. 4004, fols. 80r-81v</a>]. Newton calculates logarithms up to more than 50 decimal places! Newton reworked and further developed the ‘October 1666 Tract on Fluxions’ in <i>De Analysi per aequationes numero terminorum infinitas</i> (MS 81/4, Royal Society Library, London) and in the so-called <i>De methodis serierum et fluxionum</i> [<a href='/view/MS-ADD-03960/325'>Add. 3960.14</a>]. </p><p style='text-align: justify;'>The ‘Vellum manuscript’ [<a href='' onclick='store.loadPage(85);return false;'>fol. 45r + scraps 46r-46v</a>] was so christened by John Herivel in the 1960s because Newton wrote some calculations on the back of a legal parchment. These far from easy-to-interpret numbers are related to early discoveries on the mathematics of plane and conical pendula and on uniform circular motion [<a href='/view/MS-ADD-04004/13'>Add. 4004, fol.1r</a>], and to the study of uniform circular motion that can be found in <a href='' onclick='store.loadPage(169);return false;'>fol. 87r-87v</a>. Newton appears to be responding to an old anti-Copernican objection that if the Earth rotated around its own axis, then centrifugal force would fling bodies placed on the Earth’s surface upwards. Newton shows that on the Earth’s surface the ‘endeavour from the centre’ (<i>conatus recedendi a centro</i>) due to daily rotation is much smaller than the ‘endeavour of approaching to the centre [of the Earth] in virtue of gravity’ (<i>conatus accedendi ad centrum virtute gravitatis</i>) [<a href='' onclick='store.loadPage(169);return false;'>fol. 87r</a>]. Newton compares the acceleration of bodies on the Earth’s surface (this is a measurement of <i>g</i> [the acceleration of gravity at the earth’s surface] obtained from experiments on the motion of pendulums) to the ‘fall’ of the Moon, in uniform circular motion, towards the Earth. The claim that Newton discovered universal gravitation in his youth is based on these few pages, probably written in the middle and the late 1660s. Scholars are much divided about how to interpret them. Newton resumed researches on the motion of planets and gravitation after corresponding with Robert Hooke in 1679-80. A few years later, in 1684-7, he was busy in writing his masterpiece, the <i>Philosophiae naturalis principia mathematica</i> in which the theory of universal gravitation was clearly stated and set out fully in mathematical terms (Newton’s own annotated copies are <a href='/view/PR-ADV-B-00039-00001/1'>Adv.b.31</a> and <a href='/view/PR-ADV-B-00039-00002/1'>Adv.b.39.2</a>). </p><p style='text-align: justify;'>Niccolò Guicciardini, Università degli Studi di Milano, and Scott Mandelbrote, Peterhouse, Cambridge. <br /><br /></p><p style='text-align: justify;'><iframe width="560" height="315" src="//www.youtube.com/embed/3tDlPmX3tgw?rel=0" frameborder="0" allowfullscreen></iframe></p><p style='text-align: justify;'>Fluxions and the direct algorithm Oct 1666 tract. <a href='' onclick='store.loadPage(100);return false;'>Folio 52v</a></p><p style='text-align: justify;'><iframe width="560" height="315" src="//www.youtube.com/embed/uQVrB4_5Lh0?rel=0" frameborder="0" allowfullscreen></iframe></p><p style='text-align: justify;'>Newton, Wallis and the binomial theorem. <a href='' onclick='store.loadPage(139);return false;'>Folio 72r</a></p>