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\u2019s 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 [fols 93r-94r<\/a>], tables of fixed stars [fols 38r-44v<\/a>], and the laws of motion [fols 81r-83v<\/a>, 85r-86v<\/a>, 87r-87v<\/a>], as one would expect given Newton\u2019s best-known discoveries of this period. Less obvious are annotations on inheritance laws [fol. 32v<\/a>], anatomical drawings of the optical and auditory systems of (possibly) a small rodent and the eye of a bird [fols 35r, 37v<\/a>], and the subdivision of the intervals in the musical scale [fol. 31r<\/a>]. On Newton\u2019s study of the musical scale see also Add. 3970, fols 544r-545v<\/a>; Add. 4000, fols 138-143r<\/a>.<\/p>

The \u2018October 1666 Tract on Fluxions\u2019 [fols 48v-63v<\/a>] owes its name to a date squeezed onto the page (in a different ink?) and added apparently in retrospect 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 Add. 3960.1. fols 1-50<\/a> and a copy (probably in William Jones\u2019s 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\u2019s version of the rules of the differential calculus can be seen applied to a cubic curve on fol. 52v<\/a>. The same result, dated 13 November 1665 (possibly, again, in retrospect) can be found in the Waste Book [Add. 4004, fol. 57r<\/a>]. The logarithm calculations based on the binomial theorem [fols 78r-80v<\/a>] are closely related to Newton\u2019s discovery of the calculus. Similar calculations can be found in the Waste Book [Add. 4004, fols. 80r-81v<\/a>]. Newton calculates logarithms up to more than 50 decimal places! Newton reworked and further developed the \u2018October 1666 Tract on Fluxions\u2019 in *De Analysi per aequationes numero terminorum infinitas<\/i> (MS 81/4, Royal Society Library, London) and in the so-called De methodis serierum et fluxionum<\/i> [*

*The \u2018Vellum manuscript\u2019 [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 [Add. 4004, fol.1r<\/a>], and to the study of uniform circular motion that can be found in 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\u2019s surface upwards. Newton shows that on the Earth\u2019s surface the \u2018endeavour from the centre\u2019 ( conatus recedendi a centro<\/i>) due to daily rotation is much smaller than the \u2018endeavour of approaching to the centre [of the Earth] in virtue of gravity\u2019 (conatus accedendi ad centrum virtute gravitatis<\/i>) [fol. 87r<\/a>]. Newton compares the acceleration of bodies on the Earth\u2019s surface (this is a measurement of g<\/i> [the acceleration of gravity at the earth\u2019s surface] obtained from experiments on the motion of pendulums) to the \u2018fall\u2019 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 Philosophiae naturalis principia mathematica<\/i> in which the theory of universal gravitation was clearly stated and set out fully in mathematical terms (Newton\u2019s own annotated copies are Adv.b.31<\/a> and Adv.b.39.2<\/a>). <\/p>*