<p style='text-align: justify;'>Add. 4000 is a small notebook that Newton appears to have begun using in the winter of 1663-4. He continued to write in it extensively over the next two or three years, mostly jotting down personal annotation of his mathematical discoveries. It contains notes on his mathematical and geometrical reading, drawn from Frans van Schooten’s <i>Exercitationum</i>mathematicarum (Leiden, 1657) and from his edition of the works of François Viète (Leiden, 1646), as well as from the writings of William Oughtred and of the Oxford mathematician, John Wallis. Newton was particularly interested in Wallis’ <i>Arithmetica infinitorum</i> (included in his <i>Operum Mathematicorum Pars Altera </i>(1656)), in which problems of mathematical methods of reasoning about the quadrature of areas and volumes through consideration of their indivisible elements of line and surface were set out. This notebook also contains evidence of the development of Newton’s own mathematical skill in this period, especially his study of infinite series and development of the binomial theorem, the evolution of the differential calculus, and its application to the problem of quadratures and integration. Although Newton had been inspired in his mathematical work on curved lines and surfaces by reading Wallis and most likely by listening to the lectures that Barrow delivered in 1664, his knowledge of Descartes’ geometry was also critical in these discoveries. This notebook also contains Newton’s investigation of the musical scale in ‘Of Musick’ (c. 1665) [<a href='' onclick='store.loadPage(288);return false;'>fols 138r-143r</a>]. Musical theory was at the time considered as part of the mathematical sciences. </p><p style='text-align: justify;'> One of the later entries in this notebook derives from Newton’s knowledge of Descartes’ <i>La dioptrique</i> (1637), which Newton probably read in a Latin edition published at Paris in 1656. By September 1664, Newton had learned about the sine law of refraction through his reading of Descartes’ works, and, in the winter of 1665-6, he investigated Descartes’ findings about refraction at curved surfaces. This is probably the date of the composition of the essay ‘Of Refractions’ [<a href='' onclick='store.loadPage(55);return false;'>fols 26r-33r</a>]. Newton later told Oldenburg that he had first applied himself to grinding lenses that were not spherical in the winter of 1666 and this essay includes the detailed description of a machine for shaping a wheel to a hyperbolical profile, which could later be used for grinding lenses. Such a machine had been discussed in chapter ten of <i>La dioptrique</i>, but Newton improved the design and gave a geometrical demonstration of its operation. </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/eTVWQwwkkrU?rel=0" frameborder="0" allowfullscreen></iframe></p><p style='text-align: justify;'>The Key Developments in Newton's Thinking</p><p style='text-align: justify;'><iframe width="560" height="315" src="//www.youtube.com/embed/Chy8eCoT9sw?rel=0" frameborder="0" allowfullscreen></iframe></p><p style='text-align: justify;'>Newton and the fundamental theorem. <a href='' onclick='store.loadPage(253);return false;'>Folio 120v</a></p>