The main part of the third volume of Dr Whiteside's annotated and critical edition of all the known mathematical papers of Isaac Newton reproduces, from the original autograph, Newton's elaborate tract on infinite series and fluxions (the so-called Methodus Fluxionum), including a formerly unpublished appendix on geometrical fluxions. Ancillary documents include, in Part 1, papers on the integration of algebraic functions and, in Part 2, short texts dealing with geometry and simple harmonic motion in a cycloidal arc. Part 3 reproduces, from both manuscript versions of Newton's Lectiones Opticae and from his Waste Book, mathematical excerpts from his researches into light and the theory of lenses at this period. An appendix summarizes mathematical highlights in his contemporary correspondence.

This volume reproduces the texts of a number of important, yet relatively minor papers, many written during a period of Newton's life (1677-84) which has been regarded as mathematically barren except for his Lucasian lectures on algebra (which appear in Volume V). Part 1 concerns itself with his growing mastery of interpolation by finite differences, culminating in his rule for divided differences. Part 2 deals with his contemporary advances in the pure and analytical geometry of curves. Part 3 contains the extant text of two intended treatises on fluxions and infinite series: the Geometria Curvilinea (c. 1680), and his Matheseos Universalis Specimina (1684). A general introduction summarizes the sparse details of Newton's personal life during the period, one - from 1677 onwards - of almost total isolation from his contemporaries. A concluding appendix surveys highlights in his mathematical correspondence during 1674-6 with Collins, Dary, John Smith and above all Leibniz.

The fifth volume of this definitive edition centres around Newton's Lucasian lectures on algebra, purportedly delivered during 1673-83, and subsequently prepared for publication under the title Arithmetica Universalis many years later. Dr Whiteside first reproduces the text of the lectures deposited by Newton in the Cambridge University Library about 1684. In these much reworked, not quite finished, professional lectiones, Newton builds upon his earlier studies of the fundamentals of algebra and its application to the theory and construction of equations, developing new techniques for the factorizing of algebraic quantities and the delimitation of bounds to the number and location of roots, with a wealth of worked arithmetical, geometrical, mechanical and astronomical problems. An historical introduction traces what is known of the background to the parent manuscript and assesses the subsequent impact of the edition prepared by Whiston about 1705 and the revised version published by Newton himself in 1722. A number of minor worksheets, preliminary drafts and later augmentations buttress this primary text, throwing light upon its development and the essential untrustworthiness of its imposed marginal chronology.

This volume reproduces mathematically significant extracts from the extant manuscript record of Newton's researches during 1684-5 into the dynamical motion of bodies under the deviating action of a central force, and his subsequent struggles thereby to explain the observed motions of solar comets and of the moon. The short tract De motu Corporum, which Newton initially composed on this topic in the early autumn of 1684, was primarily built around his earlier proof that in the absence of external perturbation a planetary eclipse may be traversed under an inverse-square force pull to its solar focus, but also discussed the simplest case of resisted ballistic motion. In epilogue, excerpts from his abandoned grand scheme for revising the Principia in the early 1690s detail Newton's planned refinements to his printed exposition of central force, both simplifying and extending it, introducing therein a novel general fluxional measure of such force - but failing adequately to apply it to the primary case of conic motion.

Newton's mathematical researches during the last five years of his stay in Cambridge before leaving in April 1696 to take up his duties at the Mint in London have three main centres of interest: methods of fluxions and series, classical pure geometry, and Cartesian analytical geometry. Part 1 reproduces Newton's advances at this time in further extending the techniques of his combined calculus of fluxions and fluent, and of expansion into infinite series. Part 2 gives publication of Newton's lengthy excursions in the early 1690s into the modes of geometrical analysis used by the 'ancient' geometers, based - by way of Commandino's Latin translation - on the account of this little understood field of the Greek 'topos analuomenos' which was given by Pappus in the prolegomenon to the seventh book of his Mathematical Collection. Part 3 gives prominence to the final text of the Enumeratio Linearum Tertii Ordinis which Newton put together in June 1695.

When Newton left Cambridge in April 1696 to take up, at the age of 53, a new career at the London Mint, he did not entirely 'leave off Mathematicks' as he so often publicly declared. This last volume of his mathematical papers presents the extant record of the investigations which for one reason and another he pursued during the last quarter of his life. In January 1697 Newton was tempted to respond to two challenges issued by Johann Bernoulli to the international community of mathematicians, one the celebrated problem of identifying the brachistochrone; both he resolved within the space of an evening, producing an elegant construction of the cycloid which he identified to be the curve of fall in least time. In the autumn of 1703, the appearance of work on 'inverse fluxions' by George Cheyne similarly provoked him to prepare his own ten-year-old treatise De Quadratura Curvarum for publication, and more importantly to write a long introduction to it where he set down what became his best-known statement of the nature and purpose of his fluxional calculus.

The bringing together, in an annotated and critical edition, of all the known mathematical papers of Isaac Newton marks a step forward in the publication of the works of this great natural philosopher. In all, there are eight volumes in this present edition. Translations of papers in Latin face the original text and notes are printed on the page-openings to which they refer, so far as possible. Each volume contains a short index of names only and an analytical table of contents; a comprehensive index to the complete work is included in Volume VIII. Volume I covers three exceptionally productive years: Newton's final year as an undergraduate at Trinity College, Cambridge, and the two following years, part of which were spent at his home in Lincolnshire on account of the closure of the university during an outbreak of bubonic plague.

The second volume of Dr Whiteside's annotated edition of all the known mathematical papers of Isaac Newton covers the period 1667-70. It is divided into three parts: Part 1 contains the first drafts of an attempted classification of cubics, together with more general studies on the properties of higher algebraic curves and researches into the 'organic' construction of curves. Part 2 comprises papers on miscellaneous researches in calculus, including the important De Analysi which introduced Newton to John Collins and others outside Cambridge; Newton's original text is here accompanied by Leibniz's excerpts and review, and by Newton's counter review. Part 3 contains Mercator's Latin translation of Kinckhuysen's introduction to algebra, with Newton's corrections and 'observations' upon it, and an account of researches into algebraic equations and their geometrical construction.