Isaac Newton: Birth of a Mathematician

Isaac Newton: Birth of a MathematicianAuthor(s): D. T. WhitesideSource: Notes and Records of the Royal Society of London, Vol. 19, No. 1 (Jun., 1964), pp. 53-62Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/3519861 .

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53

ISAAC NEWTON: BIRTH OF A MATHEMATICIAN

By D. T. WHITESIDE

Research Assistant, University of Cambridge

N the development of a human being to intellectual maturity it is often difficult to pinpoint the crucial changes of phase: those, for example, when

a child first recognizes significance in a spoken jumble of sound, or first perceives the logical compulsion of a train of argument. Of such a kind is the glimmering recognition of the structured orderings of conventional symbols which we loosely call mathematics. Not least that we may know ourselves better we find it tempting to analyze this intellectual growth in the giants of the past in all fields of human endeavour, artistic and scientific-where we can, that is, for all too often through lack of historical fact we can know almost nothing of the growth to manhood of a Shakespeare or an Archimedes. A recently renewed interest in the documentary basis which supports the conventional picture of the life and thought of Isaac Newton has led to a vigorous and variegated study of the wealth of existing manuscript material, still too little known, which both cleans and illuminates it and sometimes demands that we repaint it wholly afresh (I). I myself for many years past have been absorbed in the minute study of Newton's unpublished mathematical papers and find myself drawn back again and again to the question: Where and when did Newton's mathematical inspiration begin? Let me present a personal viewpoint on the matter which while urging new facts re-interprets the old.

Newton, born on Christmas Day 1642 (old style) to the already widowed Hannah Newton (nde Ayscough) at Woolsthorpe in the parish of Colsterworth in Lincolnshire, was parted from his mother within two years when she married the Reverend Barnabas Smith of nearby North Witham (2). For a time he was left in the care of his grandmother Ayscough but a little later, perhaps about the age of five, he was entered successively in local day schools at Skillington and Stoke [Rochford?] (3). Later still, some time in the early I650o's, he was placed in the grammar school at Grantham and there he stayed (apart from a prolonged absence about 1656 when he was tem- porarily recalled by his once more widowed mother to manage the Woolsthorpe farm) till in the early summer of 1661 he left the security of Lincolnshire for the immediately strange but ultimately vitalizing undergraduate life of Cambridge.

What, mathematically, Newton learnt in his schooldays we do not



54 know but we can hazard a shrewd guess. The dame schools would teach him little more than the elements of reading, writing and 'rithmetic learned by rote: having learnt his numbers he would chant a few tables in unison with his fellow pupils and might possibly be introduced to simple forms of addition and subtraction but little more. His formal education must certainly have begun with his entry to Grantham grammar school and could have contained little that was mathematical (4). As with other late medieval foundations the patents of Edward VI given to Grantham school required that its pupils be soundly grounded in Latin and Greek grammar, literature and history, and indeed one of Newton's early notebooks (5) (dated March 1659/60 and introducing his signature 'Isaac Newton' by the charming couplet 'Quisquis in hunc librum teneros conjecit ocellos,/Nomen subscriptum perlegat ipse meum') reveals his struggle to learn Latin prosody, while a second (6) indicates that he had acquired by 1662 a passing acquaintance with the Hebrew script, some sign of the depth of his early biblical studies (7). How- ever, in the mid-seventeenth century a reform movement was slowly spreading through the grammar schools, supporting the study of such new subjects as English, modern history and arithmetic, none of which indeed were taught in the universities of Oxford and Cambridge at the time. In the study of arithmetic, in particular, Robert Recorde's Grounde ofArtes (i 540), strongly promoted in the late sixteenth century by Richard Mulcaster, had been limitedly successfully in promoting 'cyphering' as a worthy school subject and a few school texts of arithmetic began to appear, notably that of Billingsley (8). We may tentatively suppose that this reform movement had spread to Lincolnshire and that Newton's schoolmaster John Stokes, by Stukeley's account both kindly disposed and dedicated, drummed into his pupils a basic familiarity with (if not understanding of) standard methods of addition, subtraction, multiplication and division, reduction of fractions and the rule of proportion and their application to elementary weight and money problems, perhaps even simple techniques of casting accounts. But little else could have been taught to Newton: nothing of such true mathematical structures as algebra, trigonometry or geometry, all then adult studies and the last almost exclusively the domain of university scholars. Whatever else in mathematics he learnt he must have taught himself, and I do not think it can have been much.

Newton of course, alone in his garret lodging over Clark's apothecary's shop in Grantham's High Street, did read a great deal, and later, taken away from school by his mother, he had and took the opportunity in hours snatched away from his farm of being allowed to ransack the parcels of books, mainly



55

relating to scientific subjects (9), which Clark had stored away there. Back, too, in his Woolsthorpe manor house, perhaps already in the little study he blocked off in a sunny corner of the south bedroom, he had books to read, including many theological ones from his late step-father Barnabas Smith. In particular, thanks to the researches of Andrade and Huxley (Io) we now know that John Bate's Mysteries of Nature and Art (which reached a third edition in 1654) was the source of many of Newton's mechanical contrivances, mills, water-clocks and 'lanthorns' so much admired by Stukeley sixty years later, and of the rules for drawing and making colours which he copied into an early notebook (11). Unfortunately in later life Newton gave away many of his boyhood books and we can only guess at the titles of the mathematical texts in Clark's garret. What is perhaps more important, nothing in all the early Newtoniana we now have allows us to assume that he taught himself any- thing of significance from them: most, indeed, must have been way above his head. The little there was of mathematical precocity in the youth reveals itself, if at all, in the mechanical facility so strongly urged by Stukeley (12) as a seed of his future greatness, and possibly in a few rough diagrams scratched on the walls of his Woolsthorpe manor house.

We need not go very far into the former. Stukeley, almost our only informant on this point, has given (13) an admirable if largely second-hand description of his interest in carpentry, his mouse-mills and fiery kites and the many other aspects well described in the conventional account. Perhaps most revealing of a possible future dedication to exact science was his early interest in 'making dyals of divers forms and constructions everywhere about the house, in his own chamber, in the entrys and rooms where ever the sun came'. This first sign of a keen observational interest which continued throughout his life (14) we can happily assess at first hand for ourselves. One of Newton's dials, lacking a gnomon and taken from the south wall of the house, was at the beginning of last century re-erected-apparently wrong side up!-by Edmund Turnor in the north wall of Colsterworth Church: it is a small, roughly cut semicircular dial of no great accuracy. A second of Newton's dials (which I have not seen), likewise taken down by Charles Turnor from the south wall and similarly lacking its gnomon, 'was marked on a large stone at the angle of the building, and about six feet from the ground' (I5). Stukeley also informs us (16) that the walls of Newton's Grantham garret 'were full of the drawings he had made upon them, with charcole: There were birds, beasts, men, ships, plants and mathematical figures as he took them, being circles and triangles . . .'. This matter, too, we can now judge for ourselves to some extent, for in 1947 when a window



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in the north bedroom of Woolsthorpe manor house, blocked two and a half centuries before to evade tax, was reopened several crude geometrical diagrams were found carved in the stone and on subsequent search other diagrams, notably a large sketch of Grantham Church in the downstairs passage, showed up elsewhere scratched in the wall plaster (17). If these are youthful examples of Newton's work-and we may gladly accept that with some reservation (18)-then we must discard once for all the view that the young Newton was bursting with hidden talent. It would need the blindness of maternal love to read into these sets of intersecting circles and scrawled line-figures either burgeoning artistic prowess or mathematical precocity: indeed, the shakily drawn mathematical figures which came from Newton's adult pen are proof enough that his artistic deftness remained well within the ordinary.

The conclusion so far is clear. Whatever lay deep-seated in Newton's mind, by the time (June 1661) he first travelled down the old North Road to Cambridge his creative mathematical talent had not exploded into fire- nor should we really have expected it to in the sleepy countryside of Woolsthorpe or the slow rural life of a country market town like Grantham, where he was alike deprived of access to mathematical ideas, expert guidance and any real stimulus to throw himself into theoretical pursuits. The fuse for Newton's intellectual powder lay slow-burning in the still inexperienced, intellectually disciplined pan of academic research.

With Newton's entry halfway through his nineteenth year into undergraduate life at Trinity College we pass from loose hearsay knowledge tentatively backed by a little direct information to a factual world firmly controlled by the mass of Newton's extant personal papers. Thanks to West- fall's recent deciphering of shorthand notes made by Newton around the early summer of 1662 (19) we can now know exactly how immature socially if not intellectuallyhe then was-'a wholly unsophisticated provincial puritan', in fact, already stern in his religious views but with a hidden streak of temper that could flash out on occasion. This impression of Newton's undergraduate diffidence is confirmed by Nicholas Wickin's account half a century later (20) of his father John meeting him for the first time in their undergraduate days in 'ye Walks [College backs?], where he found Mr Newton solitary and dejected', unable to withstand his boisterous room-mate. We may accept, then, that he found it difficult to make friends with his fellow undergraduates,Wickens excepted, and presume that he would not be forward in seeking the companionship of the seniors in his college. Intellectually, I think, during those years he must have thrived on a lonely diet of what he



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could pick up at his university and college lectures and what with little or no outside help, he could read for himself in the books available to him in the Cambridge libraries. With regard to the former we must remember that the undergraduate curriculum was still severely scholastic in structure, with an excessive stress on the teachings of Aristotle, heavily systematized in commentary and minutely inculcated through the media of the lecture and the disputation, each laid out in their formal subdivisions of argument and counter-argument point by point (21). From one of the most precious of Newton's early notebooks (22) with its uncompleted Greek notes on Aristotle's Organon and Ethics succeeded by his annotations of such seventeenth century commentaries as those of David Stahl, Eustachius and Gerard Vossius, we can appreciate that Newton went without favour through the mental grinding of the three base stones, logic, ethics and rhetoric, of the scholastic mill and could then pass on to a fuller understanding of Johannes Magirus' commentary on Aristotelian physics. The benefits of this disciplined training in the orderly syllogistic presentation of argument on an intellect such as Newton's, well able to withstand the dangerous side-effects of monotony, rigidity and sterility, we need not stress. Undoubtedly it helped to shape his adult powers of thought and independent judgement as he grew physically to maturity during his four undergraduate years. A sure sign of that developing maturity was the decision, made about the beginning of 1664, to change his early, rather ornate handwriting for the simpler, less pretentious form which was to remain his throughout the rest of his life. Much more significantly, that year 1664 (his last as an undergraduate) was to see Newton's creative intellect burst forth in a scarcely controlled blaze, not least in its mathematical aspects.

About the beginning of the year we find him beginning to read widely for himself, taking full advantage of the growing Cambridge practice of allowing last-year undergraduates a degree of freedom from the rigid scholastic curriculum. Quickly he studied not only Charleton's English epitome of Gas- sendi's work (his 1654 Physiologia) and philosophical works of Hobbes, Glanvill, Kenelm Digby and Henry More but soon, provoked perhaps by a new-found interest in Descartes which seems suddenly to have excited the 'brisk part of the university' (23), was tackling that philosopher's Principia Philosophic and works of Boyle and Gassendi himself. On all those, in the very same pocket book in which he had entered his Aristotelian notes, he began to draft an elaborate set of Qucestiones qucerdam Philosophicae whose systematic ordering and very name bear out the fruitful effect of Newton's scholastic training (24). Once it was caught there was no holding his fever

5



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for scientific knowledge: by the year's end he was browsing through such contemporary scientific works as those ofDescartes and Kepler on optics and of Gassendi and Streete on astronomy. And suddenly in the middle of that magic year 1664 his mathematical spirit too took fire.

It was inevitable that, sooner or later, Newton would have had to acquire a degree of mastery of existing mathematical techniques if he was to pursue his scientific reading with profit: without that mastery whole stretches of the work of Descartes, Kepler, Galileo and even Streete's modest Astronomia Carolina (1661) were denied to him. We should, I think, expect Newton's introduction to the subject to have been at the hands of some college senior, and it is tempting, if a little unwise, to think of the Lucasian Professor and Fellow ofTrinity, Isaac Barrow (25). However, ifwe are to believe De Moivre's later account (26), one well worthy of quotation, Newton's first baptism was somewhat unusual:

'In 63 [Newton] being at Sturbridge fair bought a book of Astrology, out of a curiosity to see what there was in it. Read in it till he came to a figure of the heavens which he could not understand for want of being acquainted with Trigonometry.

'Bought a book of Trigonometry, but was not able to understand the Demonstrations.

'Got Euclid to fit himself for understanding the ground of Trigonometry. 'Read only the titles of the propositions, which he found so easy to

understand that he wondered how any body would amuse themselves to write any demonstrations of them. Began to change his mind when he read that Parallelograms upon the same base & between the same Parallels are equal, & that other proposition that in a right angled Triangle the square of the Hypothenuse is equal to the squares of the two other sides.

'Began again to read Euclid with more attention than he had done before & went through it.

'Read Oughtreds [Clavis] which he understood tho not entirely, he having some difficulties about what the Author called Scala secundi and tertii gradus, relating to the solution of quadratick [&] Cubick Equations. Took Descartes's Geometry in hand, tho he had been told it would be very difficult, read some ten pages in it, then stopt, began again, went a little farther than the first time, stopt again, went back again to the beginning, read on till by degrees he made himself master of the whole, to that degree that he understood Descartes's Geometry better than he had done Euclid.

'Read Euclid again & then Descartes's Geometry for a second time. Read next Dr Wallis's Arithmetica Infinitorum, & on the occasion of a certain



59

interpolation for the quadrature of the circle, found that admirable Theorem for raising a Binomial to a power given. But before that time, a little after reading Descartes Geometry, wrote many things concerning the vertices Axes [&] diameters of curves, which afterwards gave rise to that excellent tract de Curvis secundi generis ...'

Whatever the truth of Newton's first readings in Euclid and Descartes' Geometrie, we cannot doubt that he applied his mind minutely to both: his well-thumbed and marginally annotated copy of the Elements (in Barrow's 1655 edition) (27) shows that from about mid-i664 he made repeated study of its content (particularly the 'arithmetical' books II, V, VII and X) while it is scarcely possible to overemphasize the role of the Geometrie, read in its copiously commentated second Latin edition of I659/166I, in the development of Newton's first mathematical researches in algebra, analytical geometry and especially calculus. As for the works of Oughtred and Wallis -and also Vikte, Schooten and others not here mentioned-we have, in con- firmation of De Moivre's account, not only the direct evidence of Newton's extant annotations of them (28) but also his own corroborating testimony given out on several occasions in little varying form. In particular, having had opportunity to consult his early notes on the mathematical work of John Wallis, he entered on a facing page:

'July 4th 1699. By consulting an accompt of my expenses at Cambridge in the years 1663 & 1664 I find that in ye year I664 a little before Christmas I being then senior Sophister [undergraduate], I bought Schooten's Mis- cellanies & Cartes's Geometry (having read this Geometry & Oughtreds Clavis above half a year before) & borrowed Wallis's works and by conse- quence made these Annotations out of Schooten & Wallis in winter between the years I664 & 1665. At wch time I found the method of Infinite series. And in summer 1665 being forced from Cambridge by the Plague I computed ye area of ye Hyperbola at Boothby in Lincolnshire to two & fifty figures by the same method.

Is. Newton' (29).

We need say little more. From those beginnings in 1664 Newton over the next two years (and not only of course in mathematics) was to develop a remarkable series of researches formidable in technical content and effer- vescent with still untested creative thoughts, and their detailed systematization, carried through by a typically stubborn perseverance and massive power of mental concentration, was to take most of the rest of his life. To our guiding question: When and where did Newton's mathematical inspiration begin?



6o

we have found no easy answer in youthful prodigiousness or the exploitation of a freak chance. The truth is both simple and complex. Newton was self- taught in mathematics, deriving his factual knowledge from books he bought or borrowed and with little or no outside assistance, but it was the vastness of his innate intellect and the sweat of his unremitting toil which bore him so quickly to the frontiers of existing knowledge. Let us throw away once and for all any remaining vestiges of the demi-god and accept him for what he was, the most intelligent of men who used his native genius to the full.

NOTES

(I) For an outline of current Newtonian research consult I. B. Cohen's 'Newton in the Light of Recent Scholarship', Isis, 51 (1960), 489-514 and my 'Expanding World of Newtonian research', Hist. Sci., I (1962), 16-29.

(2) Smith died in late 1653. For the general details of Newton's family background C. W. Foster's Sir Isaac Newton's Family (Associated Architectural Societies' Reports and Papers, 39 (1928): Reports and Papers of the Architectural and Archaeological Society of the County of Lincoln, 1-62) is indispensable.

(3) According to Conduitt in a draft (King's College, Keynes MS. 130.2) for his unfinished Lfe.

(4) I know of no extended account which deals with the 'petty' schools of the period, while for the country grammar schools Ben Jonson's comment that they taught 'small Latine and lesse Greeke' was not outdated. See Foster Watson's The Beginnings of the Teaching of Modern Subjects in England (London, 1909), especially 288-331, Ch. VIII. 'The teaching of arithmetic.' Further details on the organization and curricula of the contemporary grammar school may be had from Watson's The English Grammar Schools to 166o (Cambridge, 1908) and from T. W. Baldwin's William Shakspere's Small Latine & Lesse Greeke, Urbana, Illinois, 1944).

(5) Now in the library of Trinity College, Cambridge. (6) That which is now in the Fitzwilliam Museum, Cambridge. (7) Newton's near-contemporary William Stukeley informs us (Memoirs of Sir Isaac Newton's

Life [I752], ed. A. Hastings White, 1936, 44) that Newton learned to write his fine youthful hand 'of one old Barley, ... writing master to the school'.

(8) R. Billingsley, An Idea ofArithmetick (London, 1655). (9) Comprising, according to Stukeley's Memoirs [note 7], 50, 'physic, botany, anatomy,

philosophy, mathematics, astronomy and the like'. (lo) See E. N. da C. Andrade, 'Newton's Early Notebook', Nature, Lond., 135 (I935), 360,

and G. L. Huxley, 'Two Newtonian Studies. I. Newton's Boyhood Interests', Harvard Libr. Bull., 13 (I959), 348-54.

(11ii) That now in the Pierpont Morgan Library, New York. See D. E. Smith, 'Two Un- published Documents of Sir Isaac Newton' in Isaac Newton 1642-1727, ed. W. J. Greenstreet, London, 1927, 16 fif.

(I2) Memoirs [note 7], 54/55. (13) See his letter to Richard Mead of 26 June 1727 (King's College, Keynes MS. 136, partially

printed by Edmund Turnmor in his Collections for the History of the Town and Soke of Grantham, London, 1806, 174-80), which was later incorporated in his Memoirs [note 7], especially 39-44.



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(i4) In unpublished drafts for his Life (King's College, Keynes MS. I30.4) Conduitt notes Newton's ability in adult life tojudge the time of day accurately from window shadows on the floor of a room.

(15) Proc. Roy. Soc., 5 (Abstracts of the Papers communicated... from 1843 to 1850 ...), London, 185I, 513: June 13, 1844, ? 2. 'An Account of the Newtonian Dial presented to the Royal Society'... By the Rev. Charles Turnmor, F.R.S.

(I6) Memoirs [note 7], 42. (17) H. W. Robinson, 'Note on Some Recently Discovered Geometrical Drawings in the

stonework of Woolsthorpe Manor House', Notes & Records Roy. Soc. Lond., 5 (I947/48), 35-6.

(18) One figure, carved in the east window of the north bedroom (and reproduced in Robinson's plate I), is reminiscent of a diagram noted by Newton from Viete in the winter of 1664/5 (ULC. Add 4000, iIV). Further, a set of scratchings in the opposite wall, a long series of short near-vertical lines converging upwards to a centre, may be the tally marks of some sort of upright pendulum: if so they can hardly have been made earlier than the autumn of 1665 when (cf. ULC. Add 3958.2, 29v) Newton's interest in the topic was roused.

(19) See R. S. Westfall's 'Short-writing and the state of Newton's conscience, 1662', Notes & Records Roy. Soc. Lond., 18 (I963) o10-16. Newton's system was slightly adapted from that laid out by Thomas Shelton in his Zeiglographia, A New art of Short-writing never before published (which went through many editions during the seventeenth century, and in particular one at London in I654).

(20o) In King's College, Keynes MS. 137, printed by Brewster in his Life, 2 (1855), 88. (21) To my mind the best account of the prevailing educational structure is that presented

by W. T. Costello in his The Scholastic Curriculum at Early Seventeenth Century Cambridge, Harvard, 1958. A more general and not entirely consistent viewpoint is expressed in M. H. Curtis' Oxford and Cambridge in Transition, 1558-1642, Oxford, I959. Rouse Ball's brief Notes on the history of Trinity College, Cambridge, London, I899, gives a helpful picture of contemporary conditions in Newton's own college.

(22) ULC. Add 3996. Compare A. R. Hall's 'Sir Isaac Newton's Note-book, I66I-I665', Camb. Hist. ]., 9 (1948), 239-50.

(23) ULC. MS. Baker 37 [Roger North's autobiography], I63r/1i63v, quoted from Curtis [note 21], 257.

(24) See R. S. Westfall's 'The Foundations of Newton's Philosophy of Nature', Brit. J. Hist. Sci., I (1962), 171-82.

(25) Let me lay a ghost which has haunted the conventional account of Newton's early life for more than two centuries. Newton was never Barrow's pupil, for Charles II's letter of 18 January 1663/4. (ULC. Baker MS. 29, 403; cf. J. Edleston, Correspondence of... Newton and ... Cotes, London, 1850, xlv) which confirmed the Lucasian statutes forbade the Professor to take any but a Fellow-commoner as his pupil, and Newton was never that. In fact when he took his B.A. inJanuary 1664/5 his tutor was Benjamin Pulleyn, later to become Regius Professor of Greek (Edleston, xli). Newton did later admit (ULC. Add 3968. 41, 84v) that he attended Barrow's lectures in 1665 but would not allow that they were helpful to him: a plausible statement when we look at the non-technical nature of those lectures (Lectiones XXIII ... habitae Cantabrigia 1664- 1666, London, 1685). The tradition of Barrow's early influence on Newton begins only at the time of the latter's death with its acceptance by Stukeley (letter to Mead of 26 June 1727 [note 13]), onto whose account Conduitt grafted the anecdote, not in



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itself implausible, of Barrow's examining Newton in Euclid as an undergraduate and finding him wanting (King's College, Keynes MS. 130.4; cf. Brewster's Life, I, 21 f.).

(26) The extract is taken from his memorandum given to Conduitt in November 1727. The original (in Conduitt's hand), sold in 1936 at Sothebys', is now rumoured to be in private possession in New York and the text here transcribed is taken from a late nineteenth century copy in Luard's hand (ULC. Add 4007, 706r-707r).

(27) Now in Trinity College, Cambridge (NQ. 16.201). (28) Especially in the little undergraduate pocket book ULC. Add 4000. Complete repro-

duction of these annotations is made in Part I of the first volume of my edition of Newton's mathematical papers, now in the press. (I may note that volume I covers the years 1664-1666. Volumes 2-5, dealing with the years 1667-1684, are in a high state of preparation, and the edition will be completed in a further three volumes.)

(29) ULC. Add 400ooo, I4v.

