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The Geometrical Lectures of Isaac Barrow, Translated, with Notes And

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The Geometrical Lectures of Isaac Barrow, Translated, with Notes And Open Court Classics of Science and Philosophy, iOMETRICAL LECTURES OP ISAAC BARROW J. M, CHILD CO GEOMETRICAL LECTURES ISAAC BARROW Court Series of Classics of Science and Thilosophy, 3\(o. 3 THE GEOMETRICAL LECTURES OF ISAAC BARROW . HI TRANSLATED, WITH NOTES AND PROOFS, AND A DISCUSSION ON THE ADVANCE MADE THEREIN ON THE WORK OF HIS PREDECESSORS IN THE INFINITESIMAL CALCULUS BY B.A. (CANTAB.), B.Sc. (LOND.) 546664 CHICAGO AND LONDON. THE OPEN COURT PUBLISHING COMPANY 1916 Copyright in Great Britain under the Ad of 191 1 33 LECTI ONES Geometric^; In quibus (praffertim) GEHERALJA Cttr varum L iacaruw STMPTOUATA D E. C L A II 7^ A T^ T !(, Audore IsAACoBARRow Collegii S S. Trhiitatn in Acad. Cant ah. SociO, & Socictatif lie- ti< Sodale. Oi H< TO. <fvftt Aoj/r/ito; iruiv]<t nxSHfufla, at tr& wVi^C) o%*( OITTI a* i r>riu- GfetJfft , rttrv -mufAttli <iow it, yupi mt , *'Ao &.'> n$uw, O 6)f TO fi>1/Si' H tr;t ^urt^i oTi a'uT&i/yi'jrf **ir. Plato de Repub. VII. L ON D IN I, Cttlielmi Godkd & venales Typis , proftant apud OR.ivi.innm Jh-tnitf>t Tt-wmsre, & Pullejn Juniorcm. UW. D C L X X. " Note the absence of the usual words Habitae Cantabrigioe," which on the title-pages of his other works indicate that the latter were delivered as Lucasian Lectures. J. M. C. PREFACE ISAAC BARROW was Ike first inventor of the Infinitesimal Calculus ; Newton got the main idea of it from Barrow by personal communication ; and Leibniz also was in some measure indebted to Barrow's work, obtaining confirmation of his oivn original ideas, and suggestions for their further development, from the copy of JJarrow's book (hat he purchased in 1673. The above is the ultimate conclusion that I have arrived at, as the result of six months' close study of a single book, my first essay in historical research. By the "Infinitesimal Calculus," I intend "a complete set of standard forms for both the differential and integral sections of the subject, together with rules for their combination, such as for a a or a of a function also a product, quotient, power ; and recognition and demonstration of the fact that differentiation and integration are inverse operations." The case of Newton is to my mind clear enough. Barrow was familiar with the paraboliforms, and tangents and areas connected with them, in from 1655 to 1660 at the very latest; hence he could at this time differentiate and inte- grate by his own method any rational positive power of a variable, and thus also a sum of such powers. He further developed it in the years 1662-3-4, and in the latter year probably had it fairly complete. In this year he com- municated to Newton the great secret of his geometrical constructions, as far as it is humanly possible to judge from a collection of of circumstantial tiny scraps evidence ; and it was probably this that set Newton to work on an attempt to express everything as a sum of powers of the variable. During the next year Newton began to "reflect on his method of fluxions," and actually did produce his Analysis per sfLquationes. This, though composed in 1666, was not published until 1711. viii BARROW'S GEOMETRICAL The case of Leibniz wants more argument that I am in a position at present to give, nor is this the place to give it. 1 hope to be able to submit this in another place at some future time. The striking points to my mind are that Leibniz " bought a copy of Barrow's work in 1673, anc^ was a^^e to " communicate a candid account of his calculus to Newton in 1677. In this connection, in the face of Leibniz' per- sistent denial that he received any assistance whatever from Barrow's book, we must bear well in mind Leibniz' twofold " " idea of the calculus : (i) the freeing of the matter from geometry, (ii) the adoption of a convenient notation. Hence, be his denial a mere quibble or a candid statement without any thought of the idea of what the "calculus" really is, it is perfectly certain that on these two points at any rate he derived not the slightest assistance from Barrow's for the first of be work ; them would dead against Barrow's practice and instinct, and of the second Barrow had no knowledge whatever. These points have made the calculus the powerful instrument that it is, and for this the world has to thank Leibniz; but their inception does not mean the invention of the infinitesimal calculus. This, the epitome of the work of his predecessors, and its completion by his own discoveries until it formed a perfected method of dealing with the problems of tangents and areas for any curve in general, i.e. in modern phraseology, the differentiation and integration of any function whatever (such as were known in Barrow's time), must be ascribed to Barrow. Lest the matter that follows may be considered rambling, and marred by repetitions and other defects, I give first some account of the circumstances that gave rise to this volume. First of all, I was asked by Mr P. E. B. Jourdain to write a short account of Barrow for the Monist ; the request being accompanied by a first edition copy of Barrow's Lectiones Opticce. et Geometries. At this time, I do not mind confessing, my only knowledge of Barrow's claim to fame was that he had been "Newton's tutor": a " notoriety as unenviable as being known as Mrs So-and-So's husband." For this article I read, as if for a review, the book that had been sent to me. My attention was arrested PREFACE ix by a theorem in which Barrow had rectified the cycloid, which I happened to know has usually been ascribed to Sir C. Wren. My interest thus aroused impelled me to make a laborious (for I am no classical scholar) translation of the whole of the geometrical lectures, to see what else I could find. The conclusions I arrived at were sent to the Monist for publica- tion ; but those who will read the article and this volume will find that in the article I had by no means reached the stage represented by this volume. Later, as I began to still further appreciate what these lectures really meant, I con- ceived the idea of publishing a full translation of the lectures together with a summary of the work of Barrow's more immediate predecessors, written in the same way from a personal translation of the originals, or at least of all those that I could obtain. On applying to the University Press, Cambridge, through my friend, the Rev. J. B. Lock, I was referred by Professor Hobson to the recent work of Professor Zeuthen. On communicating with Mr Jourdain, I was invited to elaborate my article for the Monist into a 2oo-page volume for the Open Court Series of Classics. I can lay no claim to any great perspicacity in this dis- of if I call it so all that follows is due covery mine, may ; rather to the lack of it, and to the lucky accident that made me (when I could not follow the demonstration) turn one of Barrow's theorems into algebraical geometry. What I found induced me to treat a number of the theorems in the same way. As a result I came to the conclusion that Barrow had got the calculus; but I queried even then whether Barrow himself recognized the fact. Only on com- pleting my annotation of the last chapter of this volume, Lect. XII, App. Ill, did I come to the conclusion that is as the of this Preface for I then given opening sentence ; found that a batch of theorems (which I had on first reading noted as very interesting, but not of much service), on careful revision, turned out to be the few missing standard forms, the set that necessary for completing for integration ; and one of his problems was a practical rule for finding the area under any curve, such as would not yield to the theoretical rules he had given, under the guise of an "inverse-tangent" problem. The reader will then understand that the conclusion is x HARROW'S GEOMETRICAL LECTURES the effect of a gradual accumulation of evidence (much a^ a detective picks up clues) on a mind previously blank as regards this matter, and therefore perfectly unbiased. This he will see reflected in the gradual transformation from tentative and imaginative suggestions in the Introduction to direct statements in the notes, which are inset in the text of the latter part of the translation. I have purposely refrained from altering the Introduction, which preserves the form of my article in the Monist, to accord with my final ideas, because I feel that with the gradual developrhent thus indicated I shall have a greater chance of carrying my readers with me to my own ultimate conclusion. The order of writing has been (after the first full trans- lation had been made) : Introduction, Sections I to VIII, excepting III; then the text with notes; then Sections III of the Introduction altera- and IX ; and lastly some slight tions in the whole and Section X. In Section I, I have given a wholly inadequate account of the work of Barrow's immediate but I felt predecessors ; that this could be enlarged at any reader's pleasure, by to it reference the standard historical authorities ; and that was hardly any of my business, so long as I slightly expanded my Monist article to a sufficiency for the purpose of showing that the time was now ripe for the work of Barrow, Newton, and Leibniz.
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