Isaac Newton

How to use this course

Isaac Newton gave physics its first complete mathematical system and gave mathematics the calculus, the general binomial series and the most used root-finding algorithm in numerical analysis. He also built the first working reflecting telescope, showed that white light is a mixture of colours, ran the English coinage for three decades and wrote more on alchemy and theology than on physics. This course follows the life in Parts I and III and the science in Part II.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Newton wrote in Latin and, in the Principia, in the language of Euclidean geometry with limits. This course states his results in modern notation and says where his own route differed.

Dates. England used the Julian calendar (Old Style, O.S.) until 1752, ten days behind the Gregorian calendar in the seventeenth century and eleven days behind from 1700. The course gives Old Style dates, as English sources of the time do, unless marked N.S.

Timeline

Woolsthorpe and Grantham, 1642–1661

Isaac Newton was born at Woolsthorpe Manor, near Grantham in Lincolnshire, on Christmas Day 1642 by the calendar then used in England. He was premature and small. His father, also Isaac, was a yeoman farmer with land and livestock who could not sign his own name; he had died in October, three months before the birth.

When the boy was about three, his mother Hannah Ayscough married Barnabas Smith, the elderly rector of North Witham, and moved to his house. Isaac stayed at Woolsthorpe with his grandmother Margery Ayscough. The separation marked him. In a list of sins he wrote at nineteen he recorded threatening to burn his mother and stepfather and the house over them. Smith died in 1653, and Hannah returned to Woolsthorpe with three young half-siblings.

Grantham

Around 1653 Newton entered the Free Grammar School in Grantham, where he learned Latin and some Greek and lodged with the apothecary William Clark. Early reports called him idle and inattentive. Later stories, many collected after his death, describe the model windmill he built with a mouse inside, sundials cut into walls, water clocks and kites carrying lanterns at night. His mother then withdrew him to run the farm. He was a poor farmer: he read under hedges while animals strayed into neighbours’ fields.

Two men changed his course. The schoolmaster Henry Stokes and Hannah’s brother William Ayscough, a Cambridge graduate, persuaded her to send him back to school to prepare for the university. In 1660 he returned to Grantham and lodged with Stokes. In June 1661 he went up to Cambridge.

Trinity and the plague years, 1661–1667

Newton was admitted to Trinity College on 5 June 1661 as a sizar, a student who paid his way by serving fellows and wealthier students, although his mother could have afforded more. The official curriculum was still built on Aristotle. Newton read it, then left it. Around 1664 he began a notebook with the heading Quaestiones quaedam philosophicae, in which he worked through René Descartes, Pierre Gassendi, Thomas Hobbes, Robert Boyle and Galileo Galilei, and recorded experiments on light, colour and motion.

His serious mathematics started in 1663–64. He read Euclid in Barrow’s edition, William Oughtred, Descartes’s Géométrie in Frans van Schooten’s Latin edition with its commentaries, François Viète, and John Wallis’s Arithmetica infinitorum. Within about a year he had moved from reader to discoverer. He was elected a scholar on 28 April 1664 and took his BA in 1665.

The plague years

In the summer of 1665 plague closed the university. Newton spent most of the next two years at Woolsthorpe. In this period he found the general binomial series, developed the method of fluxions as far as the fundamental theorem of calculus, carried out his prism experiments, and compared the force holding the Moon in its orbit with gravity at the Earth’s surface. Half a century later he wrote that in those years he was in the prime of his age for invention.

Historians now treat the phrase annus mirabilis with care. The work spread over about 1664–1667 and continued for years after; the theory of colours was not complete until about 1670, and the dynamics of the Principia came only in 1684–86. The famous apple appears in a conversation Newton had with William Stukeley in 1726, sixty years after the event. It is a story Newton told, and no more can be said.

The university reopened in 1667. In October Newton was elected a minor fellow of Trinity, and in 1668, after his MA, a major fellow.

The Lucasian professor, 1667–1684

Isaac Barrow, the first Lucasian Professor of Mathematics, discovered Newton late. In July 1669 he sent Newton’s manuscript De analysi per aequationes numero terminorum infinitas to the mathematical intelligencer John Collins in London, who copied it and showed it to others. That autumn Barrow resigned the chair to turn to divinity, and Newton, aged 26, succeeded him. He lectured first on optics, from January 1670. The lectures were poorly attended; his assistant later recalled that he sometimes lectured to the walls.

The telescope and the Royal Society

Late in 1668 Newton built a small reflecting telescope with a mirror of speculum metal he cast and ground himself. A second instrument reached the Royal Society at the end of 1671, and in January 1672 he was elected a Fellow. In February he sent the Society his theory of light and colours (Lesson 07). Robert Hooke, the Society’s curator of experiments, criticised it sharply; Christiaan Huygens raised objections; a group of English Jesuits at Liège disputed his experiments for years. Newton found controversy unbearable. In 1675 Hooke implied that Newton had taken ideas from him. Newton’s reply of February 1676 contains his line about standing on the shoulders of giants. He withdrew from public debate for most of a decade.

The 1670s

In these years Newton turned to alchemy and theology with the same intensity he had given to mathematics (Lesson 12). He wrote the treatise on fluxions and series of 1671, answered Leibniz’s enquiries in 1676 (Lesson 05) and composed the algebra lectures later printed as Arithmetica universalis (1707). His mother died in 1679; he nursed her through her final illness.

In late 1679 Hooke, now secretary of the Royal Society, wrote to ask Newton’s view of his idea that planetary motion is compounded of straight-line motion along the tangent and an attraction toward the Sun. In the exchange Hooke stated that the attraction varied inversely as the square of the distance. Newton privately worked out that an ellipse with the Sun at a focus required such a force, and told no one.

Halley’s visit

In August 1684 the astronomer Edmond Halley came to Cambridge. He, Hooke and Christopher Wren had discussed in London what curve a planet would follow under an inverse-square attraction, and none could prove the answer. Newton said at once that it was an ellipse; he had calculated it. He could not find the papers and promised to send a proof. In November Halley received a nine-page tract, De motu corporum in gyrum. Over the next eighteen months it grew into the Principia.

The binomial series and infinite series

For the general reader. Everyone learns that \((1+x)^2=1+2x+x^2\). Around 1665 Newton saw that the same pattern works for fractional and negative powers, provided you are willing to write down an infinite sum. With this single tool he could compute areas, roots and logarithms to as many digits as he liked, and it became the engine of his calculus.

The theorem

For any real number \(\alpha\) and integer \(k\ge0\) define \(\binom{\alpha}{k}=\dfrac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k!}\), with \(\binom{\alpha}{0}=1\).

Newton never proved the theorem. He checked it by squaring the series for \((1+x)^{1/2}\) and recovering \(1+x\), and by comparing with long division and root extraction. He announced it to Leibniz in his letter of 13 June 1676. A rigorous proof for real and complex exponents was given by Niels Henrik Abel in 1826.

How he found it

Wallis had computed the areas \(\int_0^x(1-t^2)^n\,dt\) for integer \(n\) and tried to interpolate between them to reach \(n=\tfrac12\), the quarter circle. Newton changed the problem: keep the upper limit \(x\) variable and look at the coefficients of the powers of \(x\). For \(n=0,1,2,3\) they are the rows of Pascal’s triangle with alternating signs, divided by odd numbers. He saw that each coefficient follows the rule \(\binom{n}{k}=\binom{n}{k-1}\cdot\frac{n-k+1}{k}\), and that this rule makes sense for \(n=\tfrac12\). Applied to \(n=\tfrac12\) it gives

Integrating term by term then gives the area of a circular segment as a series. This is the essence of De analysi: expand into a power series, integrate each power by the rule \(\int x^{n}=\frac{x^{n+1}}{n+1}\), and sum.

Computing \(\pi\)

Consider the circle of diameter 1, \(y=\sqrt{x-x^2}\), and the area \(A\) under it from \(x=0\) to \(x=\tfrac14\). Elementary geometry gives \(A=\frac{\pi}{24}-\frac{\sqrt3}{32}\): a sector of angle \(60^\circ\) minus a triangle. On the other hand \(\sqrt{x-x^2}=x^{1/2}(1-x)^{1/2}\), so

The terms shrink by roughly a factor of four each time. Newton carried this computation to about sixteen decimal places around 1666, and later confessed in a letter that he was ashamed of how far he had taken it.

Fluxions and the priority dispute

For the general reader. Newton thought of a curve as traced by a moving point, and of every quantity as flowing in time. The speed of the flow he called a fluxion. Finding fluxions (differentiation) and recovering flowing quantities from their fluxions (integration) turned out to be inverse operations. Leibniz reached the same calculus independently a decade later, published first, and the question of credit became the most bitter priority dispute in the history of science.

Fluents, fluxions, moments

In the treatise of 1671 a flowing quantity, or fluent, \(x\), has a fluxion \(\dot x\), its rate of change. In an indefinitely small interval \(o\) of time, \(x\) increases by the moment \(\dot x\,o\). To find the relation between fluxions implied by an equation \(f(x,y)=0\), substitute \(x+\dot x o\) and \(y+\dot y o\), subtract the original equation, divide by \(o\), and discard terms still containing \(o\). Newton set two problems as the core of the method:

1. given the relation between fluents, find the relation between their fluxions;
2. given a relation involving fluxions, find the relation between the fluents.

In modern terms these are differentiation and the solution of differential equations. In the Principia he replaced the infinitesimal \(o\) by his method of first and last ratios, a geometric doctrine of limits.

Newton proved Rule I by reversing it. Let the area be \(z=\frac{n}{m+n}a x^{(m+n)/n}\). Increase \(x\) by a moment \(o\); the area increases by approximately \(o\,y\), the strip of height \(y\). Expanding \(z(x+o)\) with the binomial series and comparing first-order terms gives \(y=a x^{m/n}\). That is, the rate of growth of the area is the ordinate. This is the fundamental theorem of calculus, used as a working method:

Combined with the binomial series, this let Newton integrate any algebraic function he could expand, and to solve curves “by infinite equations”, which was his name for power series.

The priority dispute

The documented sequence is as follows. Newton’s calculus dates from 1665–66 (an October 1666 tract survives), De analysi circulated in 1669, and the 1671 treatise stayed in manuscript until John Colson’s English edition of 1736. In 1673 and again in 1676 Leibniz visited London; on the second visit Collins showed him papers including De analysi. Leibniz developed his own calculus in Paris from 1675, with the notation \(dx\) and \(\int\). Newton’s two letters of 1676, sent through Henry Oldenburg, described the binomial series and concealed the method of fluxions in an anagram. Leibniz published the differential calculus in Nova methodus pro maximis et minimis in the Acta Eruditorum in 1684, and the integral calculus in 1686.

The quarrel began with insinuations. In 1699 Nicolas Fatio de Duillier suggested in print that Leibniz had borrowed from Newton. An anonymous review of Newton’s De quadratura in the Acta Eruditorum, written by Leibniz, implied the reverse. John Keill then accused Leibniz of plagiarism in the Philosophical Transactions. Leibniz appealed to the Royal Society, whose president was Newton. In 1712 the Society appointed a committee, and its report, published with supporting letters as the Commercium epistolicum (dated 1712, distributed early 1713), found for Newton. Newton drafted the report himself, and in 1715 wrote an anonymous review of it in the Philosophical Transactions. Johann Bernoulli sided with Leibniz. The dispute continued until Leibniz’s death in 1716, and Newton pursued it afterwards.

Historians today agree that the two men invented the calculus independently: Newton first, Leibniz first in print. Leibniz’s notation was better suited to calculation, and British mathematicians, keeping to Newton’s dots and geometric style, fell behind the Continent for much of the eighteenth century.

Newton’s method, interpolation and cubic curves

For the general reader. Most equations cannot be solved by a formula. Newton gave a way to improve a rough guess for a root step by step, each step roughly doubling the number of correct digits. He also showed how to fit a polynomial through tabulated values, the basis of numerical tables for three centuries, and classified the curves of degree three.

Newton’s method

In De analysi Newton solved \(y^3-2y-5=0\). Starting from \(y=2\), he set \(y=2+p\), obtaining \(p^3+6p^2+10p-1=0\); neglecting the higher powers gives \(p\approx0.1\). He then set \(p=0.1+q\), found a new polynomial in \(q\), kept its linear part, and continued. Each step rewrites the polynomial. The method appeared in print in Wallis’s Algebra of 1685. Joseph Raphson published in 1690 the simpler form that works directly with the original polynomial, and the general formula with the derivative was given by Thomas Simpson in 1740:

Proof sketch. By Taylor’s theorem, \(0=f(r)=f(x_k)+f'(x_k)(r-x_k)+\tfrac12f''(\xi_k)(r-x_k)^2\) for some \(\xi_k\) between \(r\) and \(x_k\). Divide by \(f'(x_k)\) and rearrange: \(x_{k+1}-r=\dfrac{f''(\xi_k)}{2f'(x_k)}(x_k-r)^2\). Near \(r\) the factor is bounded by some \(M\), so \(|x_{k+1}-r|\le M|x_k-r|^2\), and if \(M\delta<1\) the errors shrink to zero, with the stated limit.

Interpolation

Newton’s interpolation formulas, developed from the 1670s and printed as Methodus differentialis in 1711, give the polynomial of degree \(n\) through \(n+1\) points. With divided differences \(f[x_0]=f(x_0)\), \(f[x_0,\dots,x_k]=\dfrac{f[x_1,\dots,x_k]-f[x_0,\dots,x_{k-1}]}{x_k-x_0}\):

For equally spaced nodes this becomes the Gregory–Newton forward-difference formula, \(P(x_0+sh)=\sum_k\binom{s}{k}\Delta^kf(x_0)\), which is the binomial series again, now in the shift operator. Newton used it to interpolate astronomical tables and comet positions.

Cubic curves

The Enumeratio linearum tertii ordinis, written in the 1690s and appended to the Opticks in 1704, sorted plane cubic curves into 72 species; six more were found later. Newton reduced every cubic to one of four forms and stated that every cubic is a projection of one of the five “divergent parabolas” \(y^2=ax^3+bx^2+cx+d\), a claim proved later in the eighteenth century. It was the first serious work on the algebraic geometry of curves beyond conics.

Optics and the theory of colours

For the general reader. Before Newton, most natural philosophers thought that white light was pure and that a prism added colour to it. Newton showed the opposite. White light is a mixture of rays, each bent by a definite amount and each with its own colour, and a prism only sorts them. From this he explained the rainbow, the coloured fringes of telescope lenses and the rings of colour in thin films.

The experiments

Newton let sunlight through a small hole in a window shutter into a darkened room and through a prism. If refraction acted equally on all light, Snell’s law predicted a round image on the far wall. He saw an oblong spectrum about five times as long as it was wide. In the experimentum crucis he isolated one colour with a board and a second hole and passed it through a second prism: it was bent by a fixed amount and did not change colour. Rays that were bent more by the first prism were bent more by the second.

Newton argued that light consisted of particles, but he insisted that his theory of colours did not depend on that hypothesis. Hooke and Huygens attacked the particle view, and the debate blurred the experimental result. Newton named seven colours in the spectrum, adding orange and indigo partly by analogy with the seven notes of the musical scale.

The telescope

Because each colour has its own \(n\), a lens focuses each colour at a different distance: chromatic aberration. Newton concluded, wrongly as it turned out, that this could not be corrected, and built a telescope that formed its image with a curved mirror, which reflects all colours alike. Achromatic doublets combining two kinds of glass came in the 1730s and 1750s.

The rainbow

A ray entering a spherical drop at incidence angle \(i\), refracted at angle \(r\) with \(\sin i=n\sin r\), and reflected once inside, is deviated by \(D(i)=\pi+2i-4r\). Rays bunch where \(D\) is stationary. Descartes had found the angle of the bow; Newton explained its colours by the variation of \(n\) with colour.

Derivation. \(dD/di=2-4\,dr/di=0\) gives \(dr/di=\tfrac12\). From \(\cos i=n\cos r\,(dr/di)\) we get \(\cos i=\tfrac n2\cos r\). Squaring and using \(\sin^2 r=\sin^2 i/n^2\): \(4\cos^2 i=n^2-\sin^2 i=n^2-1+\cos^2 i\), so \(\cos^2 i=(n^2-1)/3\).

Newton’s rings

Pressing a convex lens of radius \(R\) on a flat glass plate leaves an air gap of thickness \(t\approx\rho^2/(2R)\) at distance \(\rho\) from the contact point. Newton measured the rings of colour that appear and found that the squares of their radii go up as the integers, so that the thicknesses go as \(0,1,2,3,\dots\) for dark rings and \(1,3,5,\dots\) half-units for bright ones. He explained the periodicity with “fits of easy reflection and easy transmission”, a periodic property of the rays. Thomas Young reinterpreted his measurements as interference in 1801–02 and used them to calculate wavelengths.

The laws of motion and the Principia

For the general reader. The Principia begins by defining mass and momentum and stating three laws of motion. From these, and from one law of force, it derives the motion of the planets, the Moon, comets and tides. It was the first time a single mathematical theory covered both the heavens and the Earth.

The book

Halley handled the edition and paid for it. Samuel Pepys, as President of the Royal Society, gave the imprimatur in 1686, and the book appeared in July 1687. It opens with Definitions (quantity of matter, quantity of motion, inertia, impressed and centripetal force), a Scholium on absolute space and time, and the Axioms or Laws of Motion. Book I treats motion under central forces in empty space. Book II treats motion in resisting media and fluids, and argues against Descartes’s vortex theory of the planets. Book III, De mundi systemate, applies the theory to the solar system. The second edition (1713), edited by Roger Cotes, added the General Scholium with the phrase hypotheses non fingo: Newton declined to frame hypotheses about the cause of gravity. The third edition (1726) was edited by Henry Pemberton.

“Motion” here means momentum, mass times velocity. Law II in modern form is \(\mathbf F=\dfrac{d}{dt}(m\mathbf v)\). Newton often used it for impulses, as a change of momentum produced by a force acting briefly, which is the form in the next proof. The form \(\mathbf F=m\mathbf a\) as a differential equation was written down by Leonhard Euler in 1750.

Newton’s proof. Divide time into equal intervals. In the first the body moves from \(A\) to \(B\). With no force, by Law I it would move to \(c\) with \(Bc=AB\), and triangles \(SAB\) and \(SBc\) have equal area (equal bases, same height). At \(B\) an impulse toward \(S\) acts, so by Corollary 1 the body actually reaches \(C\), where \(cC\) is parallel to \(BS\). Triangles \(SBc\) and \(SBC\) share base \(SB\) and have equal heights, so they have equal areas. Hence \(SAB=SBC\), and so on. Let the number of triangles grow and their width shrink: the polygon tends to a curve and the impulses to a continuous central force. In modern terms, \(\frac{d}{dt}(\mathbf r\times m\mathbf v)=\mathbf r\times\mathbf F=0\), so angular momentum \(\mathbf L\) is constant and \(\frac{dA}{dt}=\frac{|\mathbf L|}{2m}\).

Proposition 2 proves the converse: equal areas in equal times imply a central force. Proposition 6 then gives Newton’s general measure of a central force from the geometry of an orbit, which Lesson 09 uses.

Universal gravitation and Kepler’s laws

For the general reader. Johannes Kepler found from observations that planets move in ellipses, sweep equal areas in equal times, and have periods related to their distances by a fixed rule. Newton showed all three laws follow from one force: every particle attracts every other with a force proportional to their masses and inversely proportional to the square of the distance. The same force makes apples fall and holds the Moon.

Kepler’s laws

In Astronomia nova (1609) Kepler stated the first two laws for Mars: (1) the orbit is an ellipse with the Sun at a focus; (2) the radius sweeps equal areas in equal times. In Harmonices mundi (1619) he gave the third: (3) the square of the period is proportional to the cube of the mean distance. The second law is Proposition 1 of Lesson 08 and holds for every central force. The first law determines the force.

Modern derivation. In polar coordinates with \(u=1/r\) and \(h=r^2\dot\theta\) constant (law of areas), the radial equation of motion for a central force \(F(r)\) per unit mass becomes Binet’s equation

The ellipse \(r=\dfrac{p}{1+e\cos\theta}\) has \(u=\dfrac{1+e\cos\theta}{p}\), so \(u''+u=1/p\) and \(F=h^2u^2/p\propto1/r^2\). Conversely, if \(F=GM/r^2\) the equation is \(u''+u=GM/h^2\), whose general solution \(u=\dfrac{GM}{h^2}\big(1+e\cos(\theta-\theta_0)\big)\) is a conic with focus at the origin. Newton’s own proof of Proposition 11 is geometric, using Proposition 6 and properties of conjugate diameters.

The Moon test

Book III, Proposition 4: the Moon is about 60 Earth radii away and orbits in 27.32 days. If gravity falls as \(1/r^2\), its acceleration toward the Earth should be \(g/60^2\).

Newton made a first, rougher comparison in the 1660s. In the Principia he concluded that the force holding the Moon in orbit is terrestrial gravity.

This theorem justifies the Moon test, which treats the Earth as a point. With it, Newton stated universal gravitation in Book III, Propositions 7–8: \(F=\dfrac{Gm_1m_2}{r^2}\) between any two bodies. The constant \(G\) was not measured in his lifetime; Henry Cavendish’s torsion experiment of 1798 determined the Earth’s density, equivalent to fixing \(G\).

Newton saw that Kepler’s third law is exact only when \(m\ll M\), and used the orbits of satellites to weigh the planets that have moons: Jupiter, Saturn and the Earth, relative to the Sun.

Tides, comets and the shape of the Earth

For the general reader. Book III of the Principia showed what the law of gravity could explain. The Moon and Sun raise two tides a day because they pull the near side of the Earth more strongly than the far side. The spinning Earth bulges at the equator, and the pull of the Sun and Moon on the bulge makes its axis turn slowly. Comets follow the same law as planets, on long orbits.

Tides

A body of mass \(M\) at distance \(d\) produces acceleration \(GM/d^2\) at the Earth’s centre. At a point displaced by \(r\ll d\) along the line to the body, the difference is the tidal acceleration.

Hence there are two bulges, one facing the body and one opposite, and two high tides a day. Because the effect goes as \(M/d^3\), the Moon’s tide exceeds the Sun’s although the Sun’s pull on the Earth is far greater. Spring tides occur when Sun and Moon align, neap tides at the quarters. Newton treated these points in Book III, Propositions 24 and 36–37, and used the observed size of tides to estimate the mass of the Moon, which he overestimated.

The shape of the Earth

Newton argued that a rotating fluid Earth must be an oblate spheroid. He imagined two canals of water reaching the centre, one from a pole and one from the equator, and required that the two columns weigh the same, allowing for the centrifugal effect in the equatorial canal and for the change of gravity with shape. For a homogeneous Earth he obtained (Book III, Propositions 18–20)

which gives \(\varepsilon\approx1/230\). The modern value is about \(1/298\); the real Earth is denser toward the centre. He explained why Jean Richer’s pendulum clock had run slow at Cayenne in 1672, and predicted that a degree of latitude is longer near the poles. The French expedition to Lapland under Pierre-Louis Moreau de Maupertuis (1736–37) confirmed the oblate shape against the prolate figure favoured by the Cassini school.

Precession

The Sun and Moon pull on the equatorial bulge and exert a torque that turns the Earth’s axis slowly around the pole of the ecliptic, the precession of the equinoxes, known since Hipparchus. Newton was the first to give a physical cause (Book III, Proposition 39). His numerical agreement with the observed rate of about \(50''\) per year rested partly on compensating errors; Jean le Rond d’Alembert gave the first correct treatment in 1749.

Comets

For the great comet of 1680 Newton fitted a parabola with the Sun at its focus to observations, including those of John Flamsteed, and showed that a single orbit accounted for both the approaching and the receding comet, which many had taken for two. Near the Sun a long ellipse is indistinguishable from a parabola. Using Newton’s methods, Halley computed orbits for 24 comets in 1705, found that those of 1531, 1607 and 1682 were nearly identical, and predicted a return around 1758. The comet was recovered on 25 December 1758, after Alexis Clairaut and colleagues had refined the date by computing the perturbations of Jupiter and Saturn.

London: the Mint and the Royal Society, 1687–1727

The Principia made Newton famous. In 1687 he joined the Cambridge resistance to James II’s order that a Benedictine monk be admitted to a degree without the oaths, and in January 1689 the university elected him to the Convention Parliament, which offered the crown to William and Mary. In London he met John Locke and the young Swiss mathematician Nicolas Fatio de Duillier, with whom he formed an intense friendship. In 1693 that friendship ended and Newton suffered a breakdown. He slept little, accused Locke and Samuel Pepys in strange letters, then apologised. Causes proposed include exhaustion, depression and mercury poisoning from his alchemical work; none is established.

The Mint

In 1696 Charles Montagu, Chancellor of the Exchequer and a former fellow of Trinity, made Newton Warden of the Royal Mint. England’s silver coinage was clipped and counterfeited, and the government had begun the Great Recoinage, calling in the old coin and reminting it. Newton took the post far more seriously than his predecessors. He studied the assaying and minting processes, and he pursued counterfeiters as Warden, gathering evidence, interrogating witnesses and securing prosecutions. The most notorious, William Chaloner, was hanged in 1699. At the end of that year Newton became Master of the Mint, a better paid office he held until his death. In a report of 1717 he recommended valuing the gold guinea at 21 shillings; the overvaluation of gold relative to silver drew silver out of circulation and set Britain on the path to a gold standard.

Newton resigned his Cambridge posts in 1701 and sat again as MP for the university in 1701–02. His half-niece Catherine Barton kept his London house and later married John Conduitt, who succeeded him at the Mint and collected material for a biography. Newton never married.

President of the Royal Society

Hooke died in March 1703. In November Newton was elected President of the Royal Society and was re-elected every year for the rest of his life. He revived the Society’s meetings and ruled it firmly. In 1704 he published the Opticks, in English, with the treatises on quadrature and cubic curves appended. Its closing Queries, 16 in 1704 and 31 in later editions, speculated about the ether, the forces between particles and chemistry, and shaped eighteenth-century experimental science. In April 1705 Queen Anne knighted him at Trinity College.

His presidency was also marked by conflict. He pressed the Astronomer Royal John Flamsteed for his lunar and stellar observations, and in 1712 had them published, edited by Halley, without Flamsteed’s consent. He directed the Royal Society’s case against Leibniz (Lesson 05). In 1715–16 his friend Samuel Clarke conducted the famous correspondence with Leibniz on space, time and God. Newton prepared the second and third editions of the Principia with Roger Cotes and Henry Pemberton. He invested heavily in the South Sea Company and lost a large sum when the bubble burst in 1720.

Alchemy, theology, death and legacy

Alchemy

From the late 1660s to the mid-1690s Newton studied alchemy intensively. He bought furnaces and chemicals, copied and annotated a large body of alchemical texts, and kept laboratory notebooks with careful records of experiments, many on the chemistry of metals and antimony. He published almost nothing of this. Historians such as Betty Jo Teeter Dobbs and William Newman have studied the manuscripts closely and read them as serious, experimental chemistry within the alchemical tradition, connected to his ideas about active principles and forces between particles that surface in the Queries of the Opticks.

Theology

Newton read the Bible and the Church Fathers with the same rigour as Kepler’s astronomy. He concluded privately that the doctrine of the Trinity was a corruption introduced in the fourth century, and he regarded worship of Christ as God as idolatry. These views were illegal for a fellow of Trinity, and he kept them secret. He avoided ordination, which fellows were normally required to take, through a royal dispensation obtained in 1675 for the Lucasian chair. He wrote at length on prophecy, on the history of the early Church and on the chronology of ancient kingdoms. The Chronology of Ancient Kingdoms Amended appeared in 1728 and Observations upon the Prophecies of Daniel, and the Apocalypse of St. John in 1733, both posthumously. In one unpublished manuscript he reasoned that the end of the present age could not come before 2060.

Death

In his last years Newton suffered from a bladder stone. He died in Kensington on 20 March 1727 (O.S.), aged 84. He lay in state in the Jerusalem Chamber and was buried in the nave of Westminster Abbey on 28 March, with the Lord Chancellor among his pallbearers. His monument by Michael Rysbrack to a design by William Kent was unveiled in 1731.

The papers

Newton’s manuscripts passed through Catherine Barton’s family to the Earls of Portsmouth. The scientific papers went to Cambridge University in the late nineteenth century. The rest were sold at Sotheby’s in London on 13–14 July 1936. The economist John Maynard Keynes bought most of the alchemical papers and left them to King’s College, Cambridge, in 1946; the scholar Abraham Yahuda bought much of the theology, now in the National Library of Israel. In an essay published after his death, Keynes called Newton “the last of the magicians”. The Newton Project now publishes the manuscripts online.

Legacy

• Mechanics: the laws of motion and universal gravitation, reformulated by Euler, Joseph-Louis Lagrange, Laplace and William Rowan Hamilton, governed physics until relativity and quantum theory, and still govern engineering and spaceflight.
• Mathematics: the calculus (shared with Leibniz), power series as a general tool, Newton’s method, interpolation, and the first classification of cubic curves.
• Optics: the composition of white light, spectroscopy in embryo, and the reflecting telescope, the design of almost every large modern telescope.
• Method: the Principia’s Rules of Reasoning and the Opticks’ Queries set the model of mathematical, experimental natural philosophy for the Enlightenment.
• Honours in name: the SI unit of force, the newton; Newton’s rings; Newton polygons; Newton–Cotes integration; Newton’s identities for power sums.

Connections

People linked to Isaac Newton across this collection. A name without a link has no page here yet. See Isaac Newton in the network of all 337 lives →

Teachers and advisors

• Isaac Barrow 1664-1669

Influenced by

• Apollonius of Perga
• François Viète
• Galileo Galilei
• Johannes Kepler
• John Wallis
• Madhava of Sangamagrama
• Ole Rømer
• Pappus of Alexandria
• Pierre de Fermat
• René Descartes
• Tycho Brahe

Influenced

• Alexis Clairaut
• Benjamin Franklin
• Brook Taylor
• Colin Maclaurin 1719-1727
• Daniel Bernoulli
• David Brewster
• Ernst Mach
• Henry Cavendish
• James Bradley
• Jean le Rond d'Alembert
• John Dalton
• Joseph-Louis Lagrange
• Leonhard Euler
• Pierre-Louis Moreau de Maupertuis
• Pierre-Simon Laplace
• Subrahmanyan Chandrasekhar
• Thomas Bayes
• Thomas Young
• William Rowan Hamilton
• Émilie du Châtelet
• Étienne-Louis Malus

Colleagues and collaborators

• Christiaan Huygens
• Christopher Wren
• Edmond Halley 1684-1727
• Robert Boyle
• Henry Pemberton
• John Keill
• Roger Cotes 1709-1713

Rivals

• Giovanni Domenico Cassini
• Gottfried Wilhelm Leibniz 1676-1716
• Johann Bernoulli
• John Flamsteed 1680-1719
• Robert Hooke 1672-1703

Friends

• Abraham de Moivre 1692-1727
• Nicolas Fatio de Duillier 1689-1693
• Samuel Clarke

Also mentioned in this life

• Niels Henrik Abel
• Aristotle
• Euclid
• Hipparchus
• Blaise Pascal
• George Gabriel Stokes

Fields

• mechanics 27 lives
• calculus 9 lives
• optics 38 lives
• infinite series 3 lives
• numerical analysis 6 lives

On CoursesHub

• Newton & Leibniz · History of Mathematics
• Classical Mechanics
• Optics & Photonics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Isaac Newton, MacTutor History of Mathematics.
2. Isaac Newton, Wikipedia (with extensive references).
3. R. S. Westfall, Never at Rest: A Biography of Isaac Newton, Cambridge University Press, 1980.
4. The Newton Project, University of Oxford: Newton’s manuscripts online; The Sotheby sale.
5. Newton and the Mint, University of Oxford: Newton at the Mint.
6. Westminster Abbey, Sir Isaac Newton.
7. B. J. T. Dobbs, The Foundations of Newton’s Alchemy, Cambridge University Press, 1975; W. R. Newman, Newton the Alchemist, Princeton University Press, 2019.
8. J. M. Keynes, Newton, the man, in Newton Tercentenary Celebrations, Royal Society, 1947.

Mathematics and physics

1. I. Newton, The Principia: Mathematical Principles of Natural Philosophy, trans. I. B. Cohen and A. Whitman, University of California Press, 1999.
2. Philosophiæ Naturalis Principia Mathematica, Wikipedia.
3. I. Newton, Opticks, London, 1704; overview.
4. D. T. Whiteside (ed.), The Mathematical Papers of Isaac Newton, 8 vols., Cambridge University Press, 1967–1981.
5. A. R. Hall, Philosophers at War: The Quarrel between Newton and Leibniz, Cambridge University Press, 1980; Leibniz–Newton calculus controversy, Wikipedia.
6. S. Chandrasekhar, Newton’s Principia for the Common Reader, Oxford University Press, 1995.
7. N. Guicciardini, Isaac Newton on Mathematical Certainty and Method, MIT Press, 2009.
8. T. J. Ypma, Historical development of the Newton–Raphson method, SIAM Review 37 (1995), 531–551.
9. G. E. Smith, Newton’s Philosophiae Naturalis Principia Mathematica, Stanford Encyclopedia of Philosophy.