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Home , Isaac Newton, John Conduitt, Woolsthorpe Manor

8 Isaac 8.1 Potted biography

Rural background

• 1642: born in , of . • He was a small & weak baby. • Father died before birth. Mother remarried when he was 2, and moved to live with new husband, leaving N. with his grandparents on the farm until he was 12. • Indifferent performance at school until there was an altercation with another student, after which N. applied himself and came top of the class. • Major interest was making models and mechanical toys, but also was always absorbed in reading, and developed a life-long habit of keeping a diary.

School years University • Left school at 16 and was meant to take over running the • 1661: went to College Cambridge as a sizar. farm, but showed no aptitude or interest and was a At this time Cambridge was rather backward intellectually, disaster. • and new of Copernicus, Galileo, Kepler, Descartes • His mother realized that Newton should go to largely hadn’t penetrated: scholarship was focused on university--with strong encouragement of his school and on voluminous theological studies. headmaster, and so arranged for him to finish school. • However, Trinity was a hotbed of Cartesian . • This required boarding with a family in nearby town of . • The University (and the country) were also just recovering from the Puritanism of the Protectorate & Charles II was • Husband of this family was a pharmacist and Newton proclaimed monarch in 1661. developed a lifelong interest in chemistry, as well as Newton would have studied arithmetic, Euclid, and doctoring himself with various chemical remedies. • trigonometry, classical & medieval Latin, & ethics. • Also fell in love with a Miss Storey, a stepdaughter of his But he also studied with his tutor, read Kepler’s Optics host, and became engaged to marry her ca. 1661. • and struggled through Descartes’ Analytical Geometry. • 1665 graduated with B.A. degree. Back in Woolsthorpe Newton’s

• In 1665 the university was closed due to a resurgence of the Plague. • Newton spent much of the next two years back at the family manor, quietly reading and thinking. • During these “golden years” he made three classic discoveries that revolutionized the future of : ! the mathematical method of “fluxions,” i.e. the , ! understanding the composition of , & ! the law of gravitation. • The chromatic of led to his invention of a reflecting telescope.

Cambridge professor Scientific disputes • His publication on light also began a long dispute with • In 1667 Newton returned to Cambridge as a Fellow of , who claimed priority in Newton’s Trinity. discoveries in optics. (There were more to follow.) • Engagement to Miss Storey ended (or faded away). • A result of these disputes was to strengthen Newton’s • In 1669 he was appointed (2nd) Lucasian Professor. opposition to publishing his , believing that others would draw him into further disputes which he • First lecture course was on optics. considered a waste of time. • His invention of the reflecting telescope brought • A major difference in outlook between Newton, fellowship in the Royal Society in 1671. , & Hooke was Newton’s insistence on • A year later he published a letter in the Transactions of the quantitative , rather than qualitative Royal Society on his “new theory of light and colours.” speculation. • Huygens appears not to have appreciated the significance • He did not send any further papers to the Royal Society of this work, nor completely understood it. on optics, and in 1675 offered his resignation as as Fellow, in part because he believed the Society had not This was quite a disappointment to Newton. • supported his ideas.

Problems with orthodoxy Life in Cambridge • Newton had always done much theological reading and thinking, and as a result became convinced that the doctrine of • At the request of one of his friends, in 1676 Newton the Trinity was not Biblical. exchanged two letters with Gottfried Leibniz describing some of his mathematical discoveries. • This (Arian ) was a major problem for him: if this were to become known he would have lost his college fellowship. • These were courteous and open, but later would become part of a long controversy over priority in discovery of • The law required that the Lucasian Professor become ordained the calculus. in the , which would mean affirming the 39 Articles of Religion, including the doctrine of the Trinity, which • During this period (from ca. 1669) Newton did much he refused to do. experimentation in chemistry & (i.e. transmutation of metals and finding an elixir for • By 1675 he had given up hope, but at the last minute the King immortality), having studied all the ancient alchemical issued a royal dispensation, lifting the requirement that the . Lucasian Professor must take holy orders. • Newton’s lifestyle was largely solitary. His concentration • Thus Charles II saved from oblivion one of the greatest was extraordinary: he would become so engaged on a of all time. problem he would forget to eat and sleep. Publication of the Principia Parliamentary service

In 1684 Edmund Halley visited Newton and asked what • In 1687 James II became King. the shape of a planetary would be if the • between it and the varied as the inverse square of • He was a devout Roman Catholic and eventually the distance between them. declared his intent to overthrow the English Church. • Newton replied that it would be an , and that he • As an example the monarch sent a letter to Cambridge had worked it out years before. commanding that a Benedictine monk should be given an M.A. degree without requiring him to sign the 39 He later produced a proof for Halley, in the course of • Articles of Religion of the Anglican Church. which he began developing a book-length discussion on the of of bodies in orbit. • Newton encouraged the University to resist, and in the end the King backed down. • This was written out during 1685 & 1686 into his . • In 1688 the University chose Newton to represent them in the House of Commons for two years. • In 1687 Newton published the Philosophiae Naturalis Principia Mathematica (Mathematical of • He seems to have especially enjoyed this period in ). London and participation in the halls of government.

Newton’s breakdown Newton: portrait of • In 1689 Newton’s mother died. • In 1692 a young Swiss , Fatio de Duillier 1689 who was an admirer and had proposed to edit a new edition of the Principia, abruptly ended his friendship. (aged 46) • In this period Newton also became obsessed with obtaining a government position, and felt let down by his friends when a position was not forthcoming. • In 1692 he suffered a major mental breakdown. • By 1693 he seems to have recovered, but had no interest in creative scientific work, despite the that the Principia was out of print and others were following up his ideas.

Life in London The Great Man • In 1699 the dispute between Leibniz and Newton became • In 1694 one of Newton’s former Trinity students, Charles a major international conflict. Montague (later Lord Halifax), became Chancellor of the Yet another major quarrel erupted between Newton & Exchequer. • , the Royal. One of his decisions was the recoinage of English • In 1703 Newton was elected President of the Royal currency. • Society, and continued so until he died. In 1696 Newton was appointed , and • In 1704 Newton published his (in English), & the the recoinage project was executed with great success. • Latin translation in 1706. It was wildly popular. He moved to London and set up house with his niece, • In 1705 he was knighted. Catharine Barton. • By this time Newton had become one of the most In 1699 Newton was appointed , and • • celebrated persons of his time. held this position until his death. In 1727 Newton died and was buried in a spectacular There is no doubt he carried out his duties as a civil • • tomb at . servant with distinction. Newton’s tomb, Newton, 1726 Westminster Abbey

8.2 The Principia

Newton’s title page changes for the 2nd edition of the Principia Principia summary

• Book I ! Laws of motion • Book II ! and hydrodynamics, in which Newton demolishes Descartes’ vortex model for the . • Book III ! Law of universal gravitation ! ! Accounts of oblate shapes of the Earth & other planets ! of celestial objects in terms of of the Earth.

The Principia The Principia, cont’d

• Using his laws of motion and gravitation, Newton • In short, Newton’s laws of motion and the law of showed that he could deduce mathematically all three of universal gravitation not only described precisely the Kepler’s laws of planetary motion (which Kepler had of of planets orbiting the Sun, they enabled people course worked out painstakingly by years of of calculate things they had not previously were decades of ). possible. • He also showed how his laws of motion and gravitation • Comparison of predictions of Newton’s theory with could explain new phenomena: observations confirmed the validity of Newton’s ideas. ! the tides • Newton could predict from his theory all the observational work of centuries. ! the oblate shapes of Earth and other planets. • This was a spectacular achievement, something no one ! He calculated the mass of a in terms of the had ever done before. Earth’s mass. • And so it is not surprising that Newton was regarded as ! And he showed that the Great of 1680 was in one of the great intellectual giants of all time. a Keplerian orbit about the Sun. The First Law:

• Newton’s First Law of Motion (Inertia): “Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by impressed upon it.” 8.2.1 Newton’s • As we have seen, both Descartes and Galileo were working toward an of inertia. • Descartes argued that , once in motion, continued Laws of Motion to move in a straight line until it collided with another bit of matter, although this assumption is based on purely metaphysical reasoning. • Galileo’s ideas on inertia were derived from his experiments with balls rolling on inclined planes, and on projectile motion.

Newton’s Law of Inertia Astronomical application

• Newton’s First Law effectively says that the natural • If we accept Newton’s First Law, what would we expect motion of an at rest is to remain at rest, or, if it is the of the Moon or a planet would be, if there moving, to move in a straight line at constant . were no forces on it? • In other words, if there are no forces on an object, it will • A straight line! either remain at rest or move with constant speed in a • Since the Moon and planets orbit about the Sun, a straight line: it takes an external force to change either consequence of the First Law is that there must be a the speed or direction of motion. force which pulls them toward the Sun, as Kepler • Note that Newton is adopting the idea of natural motion argued. that goes back to Aristotle, but with a rather different formulation. • Note also that the same natural motion applies to objects on Earth and in the heavens.

The Second Law Newton’s Second Law Newton’s Second Law of Motion: “The change of motion is • Repeating: Force ! change in of motion per unit time. proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.” If mass is constant, then this becomes: Note that Newton defines “the quantity of motion” as “the • Force (mass) x (change in per unit time), velocity and quantity of matter conjointly,” so that “quantity of ! motion” is not velocity but what we would call = and since = (change in velocity)/time, mass x velocity. Newton’s Second Law becomes F (mass) x (acceleration). • We see that, whatever force is, it can be measured ! quantitatively by its effect on the momentum of an object. With appropriate of units this is the famous Note also that “quantity of motion” has a direction, and the • F = ma. change in quantity of motion is in the same direction as the direction of the force. This shows that the mass of an object is its resistance to • This can be written: Force is proportional to change in accelerated, i.e. a 5 lb force in a slingshot will accelerate a BB quantity of motion in a given time. to high speed, but will hardly move a bowling ball. The idea of gravitation

• During the two years back at Woolsthorpe Newton was puzzling just what force could pull planets into orbit round the Sun. • The story goes that he was in the garden at Woolsthorpe 8.2.2. The path to and he saw an fall from the apple tree. • He realized that the same force that pulled the apple to earth should extend much further could be the force that gravitation pulled the circling Moon toward the Earth. • Although this story is sometimes said to be apocryphal, it was written down by , who married Newton’s niece Catharine Barton, and who was his assistant at the Mint for many years, and is stated by several independent sources.

Drawing from 1820 of Woolsthorpe Manor and the (only) A photograph from 1998 taken with the same view as the apple tree in the Manor garden. 1820 drawing shown previously. The present tree is rooted off the fallen trunk of the original tree.

Discovery of Gravitation & the planets

• Newton had worked out that is proportional to v2/r. • Gravity is the force • He used Kepler’s third law of planetary motion to find an between a planet and expression for v in terms of r: the Sun that continually accelerates the planet 2 3 (period) = (radius of orbit) . both in speed (Kepler’s The period = 2!r/v & therefore (2!r/v)2 = r3, second law) and in direction (Kepler’s first so v2 = 4!2/r. law). Substituting this into the expression for the force required to keep a planet (or the Moon) in its orbit: Force of gravity: mv2/r ∝ m/r2. • This is how Newton convinced himself that the force required to keep a planet in orbit is proportional to 1/r2. Newton’s theory of gravity Newton’s gravitational force

Newton realized that to make a planet take an elliptical • • In Newton’s gravitational force equation: (or circular) orbit required an attractive force between 2 the planet and the Sun. F = Gm1m2/r • He formulated this idea in his law of universal G is the proportionality constant. gravitation: the attractive force between any two bodies • In MKS units (mass in kg, distance r in meters, force in is proportional to the product of their masses (kg) divided Newtons) by the square of the distance between them: G = 6.67x10-11. m m F = G 1 2 • This proportionality constant, and experimental • 2 to confirm Newton’s theory, were made r by Cavendish in 1797 & 1798. • r • (This enabled Cavendish to then calculate the mass of Mm1 1 m2 the Earth.)

Problems with Newtonian Gravity

• How is the gravitational force transmitted between the Sun and the planet? • Most people in this period could only imagine forces being applied by contact (e.g. Descartes and Huygens). 8.2.3 Applications • Newton himself was reluctant to accept the idea that gravity was an innate of matter and could be transmitted without the mediation of some intervening medium: of Newton’s "...that one body may act upon another at a distance through a without the mediation of anything else, by and through which their action and force may be conveyed theories from one to another, is to me so great an absurdity that, I believe no man, who has in philosophic a competent faculty of thinking, could ever fall into it."

Overview

• We should not let these comments on loose ends in • Newton adopted the helio-centric model for the solar Newtonian gravity distract from the fact that Newton . used his and theory of gravitation to solve • He was able to show purely from calculation Kepler’s problems or provide quantitative explanations no one three laws of planetary motion--which, remember, else had every been able to do before. Kepler had discovered empirically from decades of • In Book III, “The System of the World,” Newton applies precise observations made by Tycho. this theories to a variety of astronomical situations, and • This marks the absolute demise of the old geo-centric is able to make quantitative predictions which were picture of the heavens. revolutionary. • It is also a striking confirmation of the validity of • He checks them against measurements when possible to Newton’s theories of dynamics and gravitation. see how well his theory does. • We have now, for the first time, a theory of dynamics • For the many people who could not understand the which provides quantitative predictions of phenomena. Principia itself, confirmation of these quantitative predictions was simply astounding. Tides

• More generally, Newton showed that the orbits of • Newton’s calculations showed that gravitational force objects moving under the influence of gravity may be from the Moon is the major cause of the tides. , , or hyperbolas (these are all conic • (The Sun contributes about 25% of the effect of the sections). Moon.) • Newton showed that the Great Comet of 1680 was in an • He calculated the expected heights of tides. elliptical orbit, and that two apparitions of a comet were in fact the same comet. • He also showed that one gets the strong spring tides when the Sun and Moon are lined up and the weaker That all this worked proved that comets are “, • neap tides when the Sun and Moon are perpendicular. compact, fixed, and durable” rather than vapors emitted by the Earth, the Sun, or the planets.

Centrifugal bulges of planets Masses of planets

• Newton argued that, since planets are rotating about an • To calculate the masses of planets Newton could use his axis, self-gravity in a direction perpendicular to the spin form of Kepler’s third law: axis is somewhat offset by centrifugal force, compared to P2 = 4!2 R3/GM, the direction along the spin axis. but he did not know the for G. • This makes the planets into oblate (pumpkins) with the equatorial diameter larger than the polar • So he used the periods, P, and orbit radii, R, for the diameter. of a planet and the Earth to get the product GM for each, and thereby the ratio of the planet’s mass to • He compared his theoretical predictions with that of the Earth. measurements of the slightly slower movement of clocks at the Earth’s equator compared to the • Example: Jupiter’s moon Callisto. polar regions, and found the difference implied that the Period ~ 16 days and R ~ 4.9 x Moon’s orbit radius. equatorial diameter is 27 km (the modern value is about 40 km). These give MJup/MEarth ~ 330. Modern value is 318.

In summary Ode to Newton

• What Newton showed in Book III is that there are Here is the last stanza of Halley’s Ode to Newton which prefixed universal principles or “laws” that apply to a wide range the Principia: of physical phenomena. Then ye who now on heavenly nectar fare, • These laws can be expressed in mathematical form, and Come celebrate with me in song the name can make quantitative predictions of the outcomes of Of Newton, to the Muses dear; for he experiments. Unlocked the hidden treasuries of : • This set the for physical science which remains So richly through his had Phoebus cast with us to today. The radiance of his own divinity. Nearer the no mortal may approach. • This also gave people the realization that there are likely other universal laws or principles which can similarly be else can say so clearly the profound effect Newton’s found by creative thinking of a similar sort to Newton’s achievements had on intellectual life in the 17th C. use of for his discoveries.