<p style='text-align: justify;'>The most cherished legacy that Newton received from his stepfather, Barnabas Smith (1582-1653), seems to have been this vast manuscript commonplace book Add. 4004. Smith was rector of North Witham, a wealthy clergyman who married Newton’s mother on 27 January 1646. The immediate consequence of this union was that the three-year old Isaac Newton had to be sent to live with his grandmother. On Smith’s death, Newton appears to have inherited his library, most of which he gave away much later in life to a kinsman in Grantham. Smith himself had made extensive use of these books, in compiling a volume of theological commonplaces. This consisted of hundreds of folios bound in pasteboard, ruled at the top and in the margin of each folio to allow space for a heading and references to each entry. Newton was not interested by the very pedestrian efforts in divinity, largely the culling of quotations, with which Smith had begun to fill the book since its inception on 12 May 1612. He wanted its paper, as the title that he wrote on its original cover in February 1664 (‘Waste Book’) suggested.</p><p style='text-align: justify;'><iframe width="560" height="315" src="https://www.youtube.com/embed/-MObQA7X2MY" frameborder="0" allowfullscreen></iframe></p><p style='text-align: justify;'> By September 1664, Newton had started to use some of the pages for the optical and mathematical calculations, inspired by Descartes and van Schooten, that were beginning to occupy him (see <a href='/view/MS-ADD-03996'>Add. 3996</a> and Fitzwilliam Museum, MS. 1-1936). Over the next two years, Newton broadened his reading only slightly. Nevertheless, through the study of Wallis’ works and of the other authors (Johann Hudde, Hendrick van Heuraet, and Jan de Witt) whose writings were presented by van Schooten in his edition of Descartes’ <i>Geometria</i> (1639-41), Newton gradually mastered the analysis of curved lines, surfaces, and solids. He learned how to use the method of infinite series and extended it by discovering how to expand binomials with fractional indices. Most significantly, he developed an approach to the measurement of curved lines that mapped the motion that produced them. This arose out of dissatisfaction with the method of infinitesimals and the advances towards describing curves through their tangents that Newton had made with it. By autumn 1665, Newton had worked out a method for replacing the use of infinitesimal increments of space in his calculations with instantaneous changes in the velocity of a moving point by which curved lines were described. Stimulated entirely by his reading, Newton had invented the method of fluxions, or calculus, through the working in his ‘Waste Book’ [<a href='' onclick='store.loadPage(117);return false;'>fols 57r-57v</a>].</p><p style='text-align: justify;'> Newton was at this stage completely unknown. Others were groping for the solutions that he had found, and, encouraged by Barrow and Collins, Newton both worked up his own methods and began to think of publishing them. By 1672, he started to have doubts about the wisdom of doing so. Later, the dates given to work recorded in the ‘Waste Book’, some of which must have been added retrospectively [<a href='' onclick='store.loadPage(103);return false;'>fol. 47r</a>], provided a timeline for Newton’s activity that was useful for the arguments that he presented to assert his priority in discovering the calculus. Newton recorded in the ‘Waste Book’ the anagrams that concealed his methods for dealing with fluxions and infinite series [<a href='' onclick='store.loadPage(166);return false;'>fol. 81v</a>]. These had been used in letters that he sent in 1676 to Leibniz about his discoveries. Judging from copies in the Macclesfield Collection, it seems likely that at least one of Newton’s champions in the controversy that later broke out with Leibniz, William Jones, had the opportunity to check the chronology of the calculus against the manuscript itself. The ‘Waste Book’ was not retired by Newton after his initial mathematical labours. He continued to use it extensively for calculations and rough working on the topics that concerned him most. Thus, in the 1680s or perhaps even the 1690s, he set down information about the motion of comets in this manuscript.</p><p style='text-align: justify;'>D.T. Whiteside (ed.), <i>The Mathematical Papers of Isaac Newton</i>, 8 vols (Cambridge, 1967-81), vol. 1, pp. 145-54</p><p style='text-align: justify;'>Richard S. Westfall, <i>Never at Rest. A Biography of Isaac Newton</i> (Cambridge, 1980), pp. 105-39.</p><p style='text-align: justify;'>Niccolò Guicciardini, Università degli Studi di Milano, and Scott Mandelbrote, Peterhouse, Cambridge.</p>