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# Autograph manuscript, [Cambridge, c. May-July 1694], revisions to three sections of the first edition of the Philosophiae naturalis principia mathematica, a heavily corrected draft with three additional notes by the Scottish mathematician and astronomer David Gregory.

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ISAAC NEWTON (1642-1727)

Autograph manuscript

In Latin, on paper, 1½ pages, 220 x 189mm, the recto comprising 27 lines in Newton’s hand and 9 by Gregory, the verso (rotated 90 degrees in relation to the recto) comprising 12 lines by Newton and 5 by Gregory, as well as two diagrams labelled by Gregory and likely in his hand. Numbered ‘39’ in Gregory’s hand on the recto; three passages on the recto lightly cancelled by Newton.

RECTO:

The recto comprises revisions by Newton of two passages from the first edition of the

The first, of 17 lines, relates to page 302, line 28, ‘vires comprimentes erunt ut latera cubica dignitatis E

The second revision, of 10 lines, relates to page 490, ‘quae sit ad 4/3

These are followed by two notes in Gregory’s hand: the first, dated Oxford, 31 May 1694, refers to the experiments on air resistance in Book II of the

VERSO:

The text in Newton’s hand comprises two paragraphs, closely relating to passages in

The first paragraph of four lines, beginning ‘For the perpendicular dropped from the focus of the figure to its tangent is the mean proportional between the distance from whose furthest end the tangent is drawn and the minimum distance if the figure is a parabola …’ is almost a quotation of Lemma XIV (‘A perpendicular dropped from the focus of a parabola to its tangent is a mean proportional between the distance of the focus from the point of contact and its distance from the principal vertex of the figure’). The second paragraph, which opens ‘For if, the focus and the principal vertex being common, those conic sections are described as intersecting mutually anywhere…’ and concludes ‘… for that reason the perpendicular to the tangent of the hyperbola will be smaller and the perpendicular to the tangent of the ellipses will be bigger than that mean proportional’ links this argument to Proposition XVI, Corollary 6 (‘in an ellipse the velocity varies in a ratio that is greater than this, and in a hyperbola in a ratio that is less’).

The text in Gregory’s hand at the top of the page appears to be a prompt for this exposition, and comprises two diagrams, the second of which is virtually identical to that published in Lemma XIV of the

The publication of Newton's

One such relationship was with the young Scottish mathematician, David Gregory (1659-1708). Born in Aberdeen, Gregory's interest in mathematics was first prompted by the study of the papers of his uncle James Gregory (1638-1675), whom he succeeded in the chair of mathematics at the University of Edinburgh in 1683. He had written to Newton in 1684, and again in 1687 when he composed a fulsome letter of praise after reading the

Newton had first contemplated a second edition of the

It was out of this visit and the correspondence which ensued between the two scientists over the following two months that the present manuscript emerged: its format, on a scrap of paper annotated by both Newton and Gregory, is closely comparable to a number of other leaves among Gregory's surviving papers. While Newton's drafts on the recto are clearly formal revisions to passages in books 2 and 3 of

We are grateful to Scott Mandelbrote for his assistance in the preparation of this catalogue note.

Autograph manuscript

*,**[Cambridge, c. May-July 1694], revisions to three sections of the first edition of the Philosophiae naturalis principia mathematica, a heavily corrected draft with three additional notes by the Scottish mathematician and astronomer David Gregory.*In Latin, on paper, 1½ pages, 220 x 189mm, the recto comprising 27 lines in Newton’s hand and 9 by Gregory, the verso (rotated 90 degrees in relation to the recto) comprising 12 lines by Newton and 5 by Gregory, as well as two diagrams labelled by Gregory and likely in his hand. Numbered ‘39’ in Gregory’s hand on the recto; three passages on the recto lightly cancelled by Newton.

**Newton’s authorial revisions to the****Principia***,***‘the greatest work in the history of science’ (PMM), ‘perhaps the greatest intellectual stride that it has ever been granted to any man to make’ (Einstein). This is a working manuscript for Newton’s projected second edition**.**Apparently unpublished. Autograph scientific manuscripts by Newton are of the greatest rarity on the market, and no other autograph manuscript relating to the****Principia****has been publicly offered since 1999.**RECTO:

The recto comprises revisions by Newton of two passages from the first edition of the

*Principia,*in each case located by their page reference (the first by page and line number), and a brief quotation in square brackets:The first, of 17 lines, relates to page 302, line 28, ‘vires comprimentes erunt ut latera cubica dignitatis E

*n*+2’ – i.e. Book II, Proposition 23, Theorem 17, on the calculation of the forces acting in the compression of liquids: Newton’s revision (itself heavily redrafted), which begins ‘For if fluids enclosed in similar spaces ACE, ace consist of particles equal in number, and supposing the densities of the fluids are E and e…’, elaborates a calculation of the force exerted on the enclosing surface by the particles of a liquid which has undergone compression.The second revision, of 10 lines, relates to page 490, ‘quae sit ad 4/3

*Tt*in ratione subduplicata*St*ad*SQ*’, i.e. Book III, Proposition 41, on how to calculate the orbit of a comet moving in a parabola on the basis of three observations. Newton’s revision refines his explanation of the method for calculating the orbit, in this instance involving a ratio compounded of two terms, one of which is ‘the ratio of the radius to the secant of the angle which the orbit of the comet makes with the plane of the ecliptic’.These are followed by two notes in Gregory’s hand: the first, dated Oxford, 31 May 1694, refers to the experiments on air resistance in Book II of the

*Principia*, suggesting that ‘What Newton should have written is that the proportion and law of the resistance of the air can be found from the difference of the height to which the water rises (dans les jets d’eau) and the height of the reservoir’, with a cross-reference to the*Traité du movement des eaux*(Paris, 1686) of Edme Mariotte. Above this, Gregory has inserted a second brief note dated 13 July 1694 on the scholium to section VI of Book I of the*Principia*, discussing the mathematics of planetary trajectories, and making a specific reference to Gregory’s own*Exercitatio geometrica de dimensione figurarum*(Edinburgh, 1684).VERSO:

The text in Newton’s hand comprises two paragraphs, closely relating to passages in

*Principia*Book I, section III (‘The motion of bodies in eccentric conic sections’), specifically propositions X-XII which propose the laws governing the centripetal forces for bodies moving in an ellipse or a hyperbola.The first paragraph of four lines, beginning ‘For the perpendicular dropped from the focus of the figure to its tangent is the mean proportional between the distance from whose furthest end the tangent is drawn and the minimum distance if the figure is a parabola …’ is almost a quotation of Lemma XIV (‘A perpendicular dropped from the focus of a parabola to its tangent is a mean proportional between the distance of the focus from the point of contact and its distance from the principal vertex of the figure’). The second paragraph, which opens ‘For if, the focus and the principal vertex being common, those conic sections are described as intersecting mutually anywhere…’ and concludes ‘… for that reason the perpendicular to the tangent of the hyperbola will be smaller and the perpendicular to the tangent of the ellipses will be bigger than that mean proportional’ links this argument to Proposition XVI, Corollary 6 (‘in an ellipse the velocity varies in a ratio that is greater than this, and in a hyperbola in a ratio that is less’).

The text in Gregory’s hand at the top of the page appears to be a prompt for this exposition, and comprises two diagrams, the second of which is virtually identical to that published in Lemma XIV of the

*Principia*, and a proposition: ‘Leaving everything else unchanged, AT will be greater in the ellipse than in the parabola, and smaller in the hyperbola…’.**ISAAC NEWTON AND DAVID GREGORY**The publication of Newton's

*Principia*on 5 July 1687 was almost at once recognised, both in Britain and on the Continent, as an event of epochal significance, and transformed Newton's position from one of voluntarily obscure scholarly isolation in Cambridge to a place amongst the foremost natural philosophers in Europe. The following years saw Newton adopting an increasingly public, London-centred existence, with a vastly expanded social circle including connections with a number of leading intellectuals, notably Christiaan Huygens and John Locke.One such relationship was with the young Scottish mathematician, David Gregory (1659-1708). Born in Aberdeen, Gregory's interest in mathematics was first prompted by the study of the papers of his uncle James Gregory (1638-1675), whom he succeeded in the chair of mathematics at the University of Edinburgh in 1683. He had written to Newton in 1684, and again in 1687 when he composed a fulsome letter of praise after reading the

*Principia*. Gregory first met Newton in person during a journey to England in the summer of 1691, and impressed him enough to gain his recommendation (ahead of Edmond Halley) for the Savilian professorship of astronomy at Oxford, to which he was elected later that year. Newton's continuing patronage was to be responsible for Gregory's later appointments as mathematics tutor to the Duke of Gloucester (son of the future Queen Anne) in 1699 and as overseer of the Scottish mint in 1707.Newton had first contemplated a second edition of the

*Principia*in 1691, when he discussed it with the brilliant young Swiss mathematician, Nicolas Fatio de Duillier. The project was however suspended by a number of factors, not least the nervous breakdown, characterised by outbreaks of paranoia and insomnia, which Newton suffered in the autumn of 1693; there is also evidence for a fire which damaged or destroyed a number of his papers at this period. The plan for a second edition was revived early in 1694 by Gregory, who had been conducting a close study of the*Principia*in the years since its publication*(his notes and commentary survive in the library of the Royal Society (Ms. 210)), and it was the subject of a bout of intense activity during a visit by Gregory to Newton in Cambridge on 4-10 May 1694. Gregory's surviving memoranda from the visit bear witness to such a blaze of intellectual energy on Newton's behalf that Gregory 'could hardly write fast enough to take down notes of projects on which Newton was at work or at least pretended plausibly to be' (Richard S. Westfall,**The Life of Isaac Newton*, Cambridge: University Press, 1993, 217).It was out of this visit and the correspondence which ensued between the two scientists over the following two months that the present manuscript emerged: its format, on a scrap of paper annotated by both Newton and Gregory, is closely comparable to a number of other leaves among Gregory's surviving papers. While Newton's drafts on the recto are clearly formal revisions to passages in books 2 and 3 of

*Principia*, and*intended for the planned second edition, those on the verso were evidently not written in the same context (as indicated by the 90-degree rotation of the paper): although these relate closely to passages in book 1, they appear to be a looser redrafting in response to a proposition by Gregory at the head of the page. Relating as they do to sections in each of the three books of the**Principia*, Newton's drafts in the present manuscript almost sketch in miniature the argument of the work, beginning in Book 1 with the laws of motion in the absence of a resisting medium, applying the same laws to resisting mediums in book 2, and in book 3 deriving the law of universal gravitation, and demonstrating its application. The section on the recto relating to the calculation of the orbit of a comet is of particular interest: the successful application of the law of gravitation to comets was a major triumph for Newton's new system, and his calculation of their orbits is considered to be one of his most brilliant achievements, based as it is on a small number of limited observations made on earth as it moves in an elliptical orbit on a different plane to the conical orbit of the comet. Ultimately, the projected new edition of the*Principia*was abandoned amongst Newton's multitude of projects in the mid-1690s, including his definitive move from Cambridge to London to take up a post as warden of the Royal Mint in 1696, and no second edition was produced until Roger Cotes’s entirely independent edition of 1713. Newton’s exchanges with Gregory remained unknown until after the deposit of Gregory's papers at the Royal Institution in the 1860s: the present manuscript, which evidently became separated from the main group of papers before that date, remains unpublished.We are grateful to Scott Mandelbrote for his assistance in the preparation of this catalogue note.

Provenance

The archive of David Gregory (his numbering '39'): the present leaf evidently escaped before the 1860s, when Gregory’s Newtonian papers were presented to the Royal Society (Ms. 247), with the residue going to the library of the University of Edinburgh (Ms. Dc. 1.61). The leaf was most recently in the autograph collection of Maurice Car (1908-1968).

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