Principia E-Book

Isaac Newton

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Beschreibung

Philosophiæ Naturalis Principia Mathematica (Latin for Mathematical Principles of Natural Philosophy), often referred to as simply the Principia, is a work in three books by Isaac Newton, in Latin, first published 5 July 1687. After annotating and correcting his personal copy of the first edition, Newton published two further editions, in 1713 and 1726. The Principia states Newton's laws of motion, forming the foundation of classical mechanics; Newton's law of universal gravitation; and a derivation of Kepler's laws of planetary motion (which Kepler first obtained empirically). The Principia is considered one of the most important works in the history of science. The French mathematical physicist Alexis Clairaut assessed it in 1747: "The famous book of Mathematical Principles of Natural Philosophy marked the epoch of a great revolution in physics. The method followed by its illustrious author Sir Newton ... spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses." A more recent assessment has been that while acceptance of Newton's theories was not immediate, by the end of the century after publication in 1687, "no one could deny that" (out of the Principia) "a science had emerged that, at least in certain respects, so far exceeded anything that had ever gone before that it stood alone as the ultimate exemplar of science generally". In formulating his physical theories, Newton developed and used mathematical methods now included in the field of Calculus. But the language of calculus as we know it was largely absent from the Principia; Newton gave many of his proofs in a geometric form of infinitesimal calculus, based on limits of ratios of vanishing small geometric quantities. In a revised conclusion to the Principia (see General Scholium), Newton used his expression that became famous. The Principia deals primarily with massive bodies in motion, initially under a variety of conditions and hypothetical laws of force in both non-resisting and resisting media, thus offering criteria to decide, by observations, which laws of force are operating in phenomena that may be observed. It attempts to cover hypothetical or possible motions both of celestial bodies and of terrestrial projectiles. It explores difficult problems of motions perturbed by multiple attractive forces. Its third and final book deals with the interpretation of observations about the movements of planets and their satellites. It shows: • How astronomical observations prove the inverse square law of gravitation (to an accuracy that was high by the standards of Newton's time); • Offers estimates of relative masses for the known giant planets and for the Earth and the Sun; • Defines the very slow motion of the Sun relative to the solar-system barycenter; • Shows how the theory of gravity can account for irregularities in the motion of the Moon; • Identifies the oblateness of the figure of the Earth; • Accounts approximately for marine tides including phenomena of spring and neap tides by the perturbing (and varying) gravitational attractions of the Sun and Moon on the Earth's waters; • Explains the precession of the equinoxes as an effect of the gravitational attraction of the Moon on the Earth's equatorial bulge; and • Gives theoretical basis for numerous phenomena about comets and their elongated, near-parabolic orbits.

Leseprobe

PRINCIPIA

TheMathematical PrinciplesofNatural Philosophy

Isaac Newton

(1643-1727)

Translated

By

Andrew Motte

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ABOUT THE BOOK & AUTHOR

Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, theologian, and author (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time and as a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

In Principia, Newton formulated the laws of motion and universal gravitation that formed the dominant scientific viewpoint until it was superseded by the theory of relativity. Newton used his mathematical description of gravity to prove Kepler's laws of planetary motion, account for tides, the trajectories of comets, the precession of the equinoxes and other phenomena, eradicating doubt about the Solar System's heliocentricity. He demonstrated that the motion of objects on Earth and celestial bodies could be accounted for by the same principles. Newton's inference that the Earth is an oblate spheroid was later confirmed by the geodetic measurements of Maupertuis, La Condamine, and others, convincing most European scientists of the superiority of Newtonian mechanics over earlier systems.Newton built thefirst practical reflecting telescopeand developed a sophisticated theory of colour based on the observation that aprismseparateswhite lightinto the colours of thevisible spectrum. His work on light was collected in his highly influential bookOpticks, published in 1704. He also formulated anempirical law of cooling, made the first theoretical calculation of thespeed of sound and introduced the notion of aNewtonian fluid. In addition to his work on calculus, as a mathematician Newton contributed to the study of power series, generalised thebinomial theoremto non-integer exponents, developeda methodfor approximating theroots of a function, and classified most of thecubic plane curves.

Newton was a fellow ofTrinity Collegeand the secondLucasian Professor of Mathematicsat theUniversity of Cambridge. He was a devout but unorthodox Christian who privately rejected the doctrine of theTrinity. Unusually for a member of the Cambridge faculty of the day, he refused to takeholy ordersin theChurch of England. Beyond his work on the mathematical sciences, Newton dedicated much of his time to the study ofalchemyandbiblical chronology, but most of his work in those areas remained unpublished until long after his death. Politically and personally tied to theWhig party, Newton served two brief terms asMember of Parliament for the University of Cambridge, in 1689–90 and 1701–02. He wasknightedbyQueen Annein 1705 and spent the last three decades of his life in London, serving asWarden(1696–1700) andMaster(1700–1727) of theRoyal Mint, as well as president of theRoyal Society(1703–1727).

Early life and education

Isaac Newton was born on Christmas Day, 1642, at Woolsthorpe, a village in southwestern Lincolnshire, England. His father died two months before he was born. When he was three years old, his mother remarried and moved away, leaving Isaac in the care of his grandmother. After a basic education in local schools, at the age of twelve he was sent to the King's School in Grantham, England, where he lived in the home of a pharmacist (one who prepares and distributes medication) named Clark. Newton was interested in Clark's chemical library and laboratory and built mechanical devices to amuse Clark's daughter, including a windmill run by a live mouse, floating lanterns, and sun dials.

After Newton's stepfather died, his mother returned to Woolsthorpe, and she pulled him out of school to help run the family farm. He preferred reading to working, though, and it became apparent that farming was not his destiny. At the age of nineteen he entered Trinity College, Cambridge, England. After receiving his bachelor's degree in 1665, Newton stayed on for his master's, but an outbreak of the plague (a highly infectious and deadly disease often carried by rats)

Caused the university to close. Newton returned to Woolsthorpe for eighteen months, from 1666 to 1667, during which time he performed the basic experiments and did the thinking for his later work on gravitation (the attraction the mass of the Earth has for bodies near its surface) and optics (the study of light and the changes it experiences and produces). The story that a falling apple suggested the idea of gravitation to him seems to be true. Newton also developed his own system of calculus (a form of mathematics used to solve problems in physics).

Returning to Cambridge in 1667, Newton quickly completed the requirements for his master's degree and then began a period of expanding on the work he had started at Woolsthorpe. His mathematics professor, Isaac Barrow, was the first to recognize Newton's unusual ability. When Barrow resigned to take another job in 1669, he recommended that Newton take his place. Newton became a professor of mathematics at age twenty-seven and stayed at Trinity in that capacity for twenty-seven years.

Experiments in optics

Newton's main interest at the time was optics, and for several years his lectures were devoted to the subject. His experiments in this area had grown out of his interest in improving the effectiveness of telescopes (instruments that enable the user to view distant objects through the bending of light rays through a lens). His discoveries about the nature and properties of light had led him to turn to suggestions for a reflecting telescope rather than current ones based on the refractive (bending) principle. Newton built several reflecting models in which the image was viewed in a concave (rounded like the inside of a bowl) mirror through an eyepiece in the side of the tube. In 1672 he sent one of these to the Royal Society (Great Britain's oldest organization of scientists).

Newton was honored when the members of the Royal Society were impressed by his reflecting telescope and when they elected him to their membership. But when he decided to send the society a paper describing his experiments on light and the conclusions he had drawn from them, the results almost changed history for the worst. The paper was published in the society's Philosophical Transactions. Many scientists refused to accept the findings, and others were strongly opposed to conclusions that seemed to show that popular theories of light were false. At first Newton patiently answered his critics with further explanations, but when these produced more criticism, he became angry. He vowed he would never publish again, even threatening to give up science altogether. Several years later, at the urging of the astronomer Edmund Halley (c. 1656–1743), Newton put together the results of his work on the laws of motion, which became the great Principia.

His major work

Newton's greatest work, Philosophiae naturalis principia mathematica, was completed in eighteen months. It was first published in Latin in 1687, when Newton was forty-five. Its appearance established him as the leading scientist of his time, not only in England but in the entire Western world. In the Principia Newton, with the law of universal gravitation, gave mathematical solutions to most of the problems relating to motion with which earlier scientists had struggled.

In the years after Newton's election to the Royal Society, the thinking of his peers and of scholars had been slowly developing along lines similar to those which his had taken, and they were more open to his explanations of the behavior of bodies moving according to the laws of motion than they had been to his theories about the nature of light. Yet the Principia 's mathematical form made it difficult for even the sharpest minds to follow. Those who did understand it saw that it needed to be made easier to read. As a result, in the years from 1687 to Newton's death, the Principia was the subject of many books and articles attempting to better explain Newton's ideas.

London years

After the publication of the Principia, Newton became depressed and lost interest in scientific matters. He became interested in university politics and was elected a representative of the university in Parliament. Later he asked friends in London to help him obtain a government appointment. The result was that in 1696, at the age of fifty-four, he left Cambridge to become warden and then master of the Mint (place where money is printed or manufactured). Newton took the job just as seriously as he had his scientific pursuits and made changes in the English money system that were effective for over one hundred years.

Newton's London life lasted as long as his professorship. He received many honors, including the first knighthood given for scientific achievement and election to life presidency of the Royal Society. In 1704 he published the Opticks, mainly a collection of earlier research, which he revised (changed) three times. In later years he supervised two updated versions of the Principia, he carried on a correspondence with scientists all over Great Britain and Europe, he continued his study and investigation in various fields, and, until his very last years, he performed his duties at the Mint.

His Opticks

The Opticks was written and originally published in English rather than Latin, and as a result it reached a wide range of readers in England. The reputation the Principia had prepared the way for the success of Newton's second published work. Also, its content and manner of presentation made the Opticks more approachable. It contained an account of experiments performed by Newton himself and his conclusions drawn from them, and it had greater appeal for the experimentally minded public of the time than the more mathematical Principia.

Of great interest for scientists were the questions with which Newton concluded the text of the Opticks —for example, "Do not Bodies act upon Light at a distance, and by their action bend its rays?" These make up a unique expression of Newton's ideas; posing them as negative (incorrect) questions made it possible for him to suggest ideas that he could not support by experimental evidence or mathematical proof, paving the way for further research by future scientists.

Later years

Two other areas to which Newton devoted much attention were chronology (the science of assigning to events their proper dates) and theology (the study of religion). His Chronology of Ancient Kingdoms, published in full after his death, attempts to link Egyptian, Greek, and Hebrew history and myths and to establish dates of historical events. In his Observations upon the Prophecies of Daniel and the Apocalypse of St. John, his aim was to show that the predictions of the Old and New Testaments had so far come true.

Newton died on March 20, 1727. His surviving writings and letters reveal a person with tremendous powers of concentration, the ability to stand long periods of intense mental strain, and the ability to remain free of distractions. The many portraits of Newton show him as a man with natural dignity, a serious expression, and large searching eyes. He had developed a mathematical explanation of the universe and opened the door for further study. In changing from pursuit of answers to the question "Why?" to focus upon "What?" and "How?," he prepared the way for the age of technology (a scientific way of achieving a practical purpose).

* * *

About the Principia

Philosophiæ Naturalis Principia Mathematica (Latin for Mathematical Principles of Natural Philosophy), often referred to as simply the Principia, is a work in three books by Isaac Newton, in Latin, first published 5 July 1687. After annotating and correcting his personal copy of the first edition, Newton published two further editions, in 1713 and 1726. The Principia states Newton's laws of motion, forming the foundation of classical mechanics; Newton's law of universal gravitation; and a derivation of Kepler's laws of planetary motion (which Kepler first obtained empirically).

The Principia is considered one of the most important works in the history of science. The French mathematical physicist Alexis Clairaut assessed it in 1747: "The famous book of Mathematical Principles of Natural Philosophy marked the epoch of a great revolution in physics. The method followed by its illustrious author Sir Newton ... spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses."

A more recent assessment has been that while acceptance of Newton's theories was not immediate, by the end of the century after publication in 1687, "no one could deny that" (out of thePrincipia) "a science had emerged that, at least in certain respects, so far exceeded anything that had ever gone before that it stood alone as the ultimate exemplar of science generally".

In formulating his physical theories, Newton developed and used mathematical methods now included in the field ofCalculus. But the language of calculus as we know it was largely absent from thePrincipia; Newton gave many of his proofs in ageometricform ofinfinitesimal calculus, based on limits of ratios of vanishing small geometric quantities.In a revised conclusion to thePrincipia(seeGeneral Scholium), Newton used his expression that became famous.

In the preface of thePrincipia, Newton wrote:

... Rational Mechanics will be the sciences of motion resulting from any forces whatsoever, and of the forces required to produce any motion, accurately proposed and demonstrated ... And therefore we offer this work as mathematical principles of his philosophy. For all the difficulty of philosophy seems to consist in this—from the phenomenas of motions to investigate the forces of Nature, and then from these forces to demonstrate the other phenomena ...

ThePrincipiadeals primarily with massive bodies in motion, initially under a variety of conditions and hypothetical laws of force in both non-resisting and resisting media, thus offering criteria to decide, by observations, which laws of force are operating in phenomena that may be observed. It attempts to cover hypothetical or possible motions both of celestial bodies and of terrestrial projectiles. It explores difficult problems of motions perturbed by multiple attractive forces. Its third and final book deals with the interpretation of observations about the movements of planets and their satellites.

It shows:

·how astronomical observations prove the inverse square law of gravitation (to an accuracy that was high by the standards of Newton's time);

·offers estimates of relative masses for the known giant planets and for the Earth and the Sun;

·defines the very slow motion of the Sun relative to the solar-system barycenter;

·shows how the theory of gravity can account for irregularities in the motion of the Moon;

·identifies the oblateness of the figure of the Earth;

·accounts approximately for marine tides including phenomena of spring and neap tides by the perturbing (and varying) gravitational attractions of the Sun and Moon on the Earth's waters;

·explains theprecession of the equinoxesas an effect of the gravitational attraction of the Moon on the Earth's equatorial bulge; and

·gives theoretical basis for numerous phenomena about comets and their elongated, near-parabolic orbits.

The opening sections of thePrincipiacontain, in revised and extended form, nearlyall of the content of Newton's 1684 tractDe motu corporum in gyrum.

ThePrincipiabegin with "Definitions"and "Axioms or Laws of Motion",and continues in three books:

Book 1,“De motu corporum”:

Book 1, subtitledDe motu corporum(On the motion of bodies) concerns motion in the absence of any resisting medium. It opens with a mathematical exposition of "the method of first and last ratios",a geometrical form of infinitesimal calculus.

Newton's proof of Kepler's second law, as described in the book. If a continuous centripetal force (red arrow) is considered on the planet during its orbit, the area of the triangles defined by the path of the planet will be the same. This is true for any fixed time interval. When the interval tends to zero, the force can be considered instantaneous. (Click image for a detailed description).

The second section establishes relationships between centripetal forces and the law of areas now known as Kepler's second law (Propositions 1–3),and relates circular velocity and radius of path-curvature to radial force (Proposition 4), and relationships between centripetal forces varying as the inverse-square of the distance to the center and orbits of conic-section form (Propositions 5–10).

Propositions 11–31establish properties of motion in paths of eccentric conic-section form including ellipses, and their relation with inverse-square central forces directed to a focus, and includeNewton's theorem about ovals(lemma 28).

Propositions 43–45are demonstration that in an eccentric orbit under centripetal force where theapsemay move, a steady non-moving orientation of the line of apses is an indicator of an inverse-square law of force.

Book 1 contains some proofs with little connection to real-world dynamics. But there are also sections with far-reaching application to the solar system and universe:

Propositions 57–69deal with the "motion of bodies drawn to one another by centripetal forces". This section is of primary interest for its application to theSolar System, and includes Proposition 66along with its 22 corollaries:here Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions, a problem which later gained name and fame (among other reasons, for its great difficulty) as thethree-body problem.

Propositions 70–84deal with the attractive forces of spherical bodies. The section contains Newton's proof that a massive spherically symmetrical body attracts other bodies outside itself as if all its mass were concentrated at its centre. This fundamental result, called theShell theorem, enables the inverse square law of gravitation to be applied to the real solar system to a very close degree of approximation.

Book 2, part 2 of “De motu corporum”:

Part of the contents originally planned for the first book was divided out into a second book, which largely concerns motion through resisting mediums. Just as Newton examined consequences of different conceivable laws of attraction in Book 1, here he examines different conceivable laws of resistance; thusSection 1discusses resistance in direct proportion to velocity, andSection 2goes on to examine the implications of resistance in proportion to the square of velocity. Book 2 also discusses (inSection 5) hydrostatics and the properties of compressible fluids; Newton also derivesBoyle's law.The effects of air resistance on pendulums are studied inSection 6, along with Newton's account of experiments that he carried out, to try to find out some characteristics of air resistance in reality by observing the motions of pendulums under different conditions. Newton compares the resistance offered by a medium against motions of globes with different properties (material, weight, size). In Section 8, he derives rules to determine the speed of waves in fluids and relates them to the density and condensation (Proposition 48;this would become very important in acoustics). He assumes that these rules apply equally to light and sound and estimates that the speed of sound is around 1088 feet per second and can increase depending on the amount of water in air.

Less of Book 2 has stood the test of time than of Books 1 and 3, and it has been said that Book 2 was largely written on purpose to refute a theory ofDescarteswhich had some wide acceptance before Newton's work (and for some time after). According to this Cartesian theory of vortices, planetary motions were produced by the whirling of fluid vortices that filled interplanetary space and carried the planets along with them.Newton wrote at the end of Book 2his conclusion that the hypothesis of vortices was completely at odds with the astronomical phenomena, and served not so much to explain as to confuse them.

Book 3,“De mundi systemate”:

Book 3, subtitledDe mundi systemate(On the system of the world), is an exposition of many consequences of universal gravitation, especially its consequences for astronomy. It builds upon the propositions of the previous books, and applies them with further specificity than in Book 1 to the motions observed in the Solar System. Here (introduced by Proposition 22,and continuing in Propositions 25–35) are developedseveral of the features and irregularitiesof the orbital motion of the Moon, especially thevariation. Newton lists the astronomical observations on which he relies, and establishes in a stepwise manner that the inverse square law of mutual gravitation applies to Solar System bodies, starting with the satellites of Jupiterand going on by stages to show that the law is of universal application.He also gives starting at Lemma 4and Proposition 40the theory of the motions of comets, for which much data came fromJohn FlamsteedandEdmond Halley, and accounts for the tides,attempting quantitative estimates of the contributions of the Sunand Moonto the tidal motions; and offers the first theory of the precession of the equinoxes. Book 3 also considers theharmonic oscillatorin three dimensions, and motion in arbitrary force laws.

In Book 3 Newton also made clear his heliocentric view of the Solar System, modified in a somewhat modern way, since already in the mid-1680s he recognised the "deviation of the Sun" from the centre of gravity of the Solar System.

For Newton, "the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World",and that this centre "either is at rest, or moves uniformly forward in a right line".

Table of Contents

The Mathematical Principles of Natural Philosophy

ABOUT THE BOOK & AUTHOR

THE AUTHOR’S PREFACE

DEFINITIONS.

Definition i.

Definition ii.

Definition iii.

Definition iv.

Definition v.

Definition vi.

Definition vii.

Definition viii.

Scholium.

AXIOMS, OR LAWS OF MOTION.

Law I.

Law ii.

Law iii.

Corollary I.

Corollary ii.

Corollary iii.

Corollary iv.

Corollary V.

Corollary vi.

Scholium.

BOOK I.

OF THE MOTION OF BODIES

Section I.

Section II.

Section iii.

Section iv.

Section V.

Section vi.

Section vii.

Section viii.

Section ix.

Section X.

Section xi.

Section xii.

Section xiii.

Section xiv.

BOOK II.

OF THE MOTION OF BODIES

Section I.

Section ii.

Section iii.

Section iv.

Section V.

Section vi.

Section vii.

Section viii.

Section ix.

BOOK III.

RULES OF REASONING IN PHILOSOPHY.

PHAENOMENA, OR APPEARANCES.

PROPOSITIONS

OF THE MOTION OF THE MOON’S NODES.

GENERAL SCHOLIUM.

THE AUTHOR’S PREFACE

§

SINCE the ancients (as we are told by Pappus), made great account of the science of mechanics in the investigation of natural things: and the moderns, laying aside substantial forms and occult qualities, have endeavoured to subject the phenomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it regards philosophy. The ancients considered mechanics in a twofold respect; as rational, which proceeds accurately by demonstration ; and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accurate is called geometrical , what is less so, is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic; and if any could work with perfect accuracy, he would be the most perfect mechanic of all ; for the description if right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn; for it requires that the learner should first be taught to describe these accurately, before he enters upon geometry ; then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics ; and by geometry the use of them, when so solved, is shown ; and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things. Therefore geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that geometry is commonly referred to their magnitudes, and mechanics to their motion. In this sense rational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any motions, accurately proposed and demonstrated. This part of mechanics wascultivated by the ancients in the five powers which relate to manual arts, who considered gravity (it not being a manual power), ho Otherwise than as it moved weights by those powers. Our design not respecting arts, but philosophy, and our subject not manual but natural powers, we consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or impulsive; and therefore we offer this work as the mathematical principles of philosophy; for all the difficulty of philosophy seems to consist in this from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena ; and to this end the general propositions in the first and second book are directed. In the third book we give an example of this in the explication of the System of the World: for by the propositions mathematically demonstrated in the former books, we in the third derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, we deduce the motions of the planets, the comets, the moon, and the sea. I wish we could derive the rest of the phenomena of nature by the same kind of reasoning from mechanical principles; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto at tempted the search of nature in vain; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy. In the publication of this work the most acute and universally learned Mr. Edmund Halley not only assisted me with his pains in correcting the press and taking care of the schemes, but it was to his solicitations that its becoming public is owing; for when he had obtained of me my demonstrations of the figure of the celestial orbits, he continually pressed me to communicate the same to the Royal Society, who afterwards, by their kind encouragement and entreaties, engaged me to think of publishing them. But after I had begun to consider the inequalities of the lunar motions, and had entered upon some other things relating to the laws and measures of gravity, and other forces; and the figures that would be described by bodies attracted according to given laws ; and the motion of several bodies moving among themselves; the motion of bodies in resisting mediums; the forces, densities, and motions, of mediums ; the orbits of the comets, and such like ;deferred that publication till I had made a search into those matters, and could put forth the whole together. What relates to the lunar motions (being imperfect), I have put all together in the corollaries of Prop. 66, to avoid being obliged to propose and distinctly demonstrate the several things there contained in a method more prolix than the subject deserved, and interrupt the series of the several propositions. Some things, found out after the rest, I chose to insert in places less suitable, rather than change the number of the propositions and the citations. I heartily beg that what I have here done may be read with candour; and that the defects in a subject so difficult be not so much reprehended as kindly supplied, and investigated by new endeavours of my readers.

Isaac Newton.

Cambridge, Trinity College May 8, 1688.

In the second edition the second section of the first book was enlarged. In the seventh section of the second book the theory of the resistances of fluids was more accurately investigated, and confirmed by new experiments. In the third book the moon’s theory and the praecession of the equinoxes were more fully deduced from their principles; and the theory of the comets was confirmed by more examples of the calculation of their orbits, done also with greater accuracy.

In this third edition the resistance of mediums is somewhat more largely handled than before; and new experiments of the resistance of heavy bodies falling in air are added. In the third book, the argument to prove that the moon is retained in its orbit by the force of gravity is enlarged on; and there are added new observations of Mr. Pound’s of the proportion of the diameters of Jupiter to each other: there are, besides, added Mr. Kirk’s observations of the comet in 1680; the orbit of that comet computed in an ellipsis by Dr. Halley; and the orbit of the comet in 1723 computed by Mr. Bradley.

* * *

DEFINITIONS.

§

Definition i.

The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.

THUS air of a double density, in a double space, is quadruple in quantity ; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body ; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.

Definition ii.

The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly.

The motion of the whole is the sum of the motions of all the parts ; and therefore in a body double in quantity, with equal velocity, the motion is double ; with twice the velocity, it is quadruple.

Definition iii.

Force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present stale, whether it be of rest, or of moving uniformly forward in a right line.

This force is ever proportional to the body whose force it is ; and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name, be called vis inertia, or force of inactivity. But a body exerts this force only, when another force, impressed upon it, endeavours to change its condition ; and the exercise of this force may be considered both as resistance and impulse ; it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished ; nor are those bodies always truly at rest, which commonly are taken to be so.

Definition iv.

An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.

This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its vis inertiae only. Impressed forces are of different origins as from percussion, from pressure, from centripetal force.

Definition v.

A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre.

Of this sort is gravity, by which bodies tend to the centre of the earth magnetism, by which iron tends to the loadstone ; and that force, what ever it is, by which the planets are perpetually drawn aside from the rectilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits. A stone, whirled about in a sling, endeavours to recede from the hand that turns it ; and by that endeavour, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity, and as soon as ever it is let go, flies away. That force which opposes itself to this endeavour, and by which the sling perpetually draws back the stone towards the hand, and retains it in its orbit, because it is directed to the hand as the centre of the orbit, I call the centripetal force. And the same thing is to be understood of all bodies, revolved in any orbits. They all endeavour to recede from the centres of their orbits ; and wore it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal, would fly off in right lines, with an uniform motion. A projectile, if it was not for the force of gravity, would not deviate towards the earth, but would go off from it in a right line, and that with an uniform motion, if the resistance of the air was taken away. It is by its gravity that it is drawn aside perpetually from its rectilinear course, and made to deviate towards the earth, more or less, according to the force of its gravity, and the velocity of its motion. The less its gravity is, for the quantity of its matter, or the greater the velocity with which it is projected, the less will it deviate from a rectilinear course, and the farther it will go. If a leaden ball projected from the top of a mountain by the force of gunpowder with a given velocity, and in a direction parallel to the horizon, is carried in a curve line to the distance of two miles before it falls to the ground ; the same, if the resistance of the air were taken away, with a double or decuple velocity, would fly twice or ten times as far. And by increasing the velocity, we may at pleasure increase the distance to which it might be projected, and diminish the curvature of the line, which it might describe, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole earth before it falls ; or lastly, so that it might never fall to the earth, but go forward into the celestial spaces, and proceed in its motion in infinitum. And after the same manner that a projectile, by the force of gravity, may be made to revolve in an orbit, and go round the whole earth, the moon also, either by the force of gravity, if it is endued with gravity, or by any other force, that impels it towards the earth, may be perpetually drawn aside towards the earth, out of the rectilinear way, which by its innate force it would pursue; and would be made to revolve in the orbit which it now describes ; nor could the moon with out some such force, be retained in its orbit. If this force was too small, it would not sufficiently turn the moon out of a rectilinear course : if it was too great, it would turn it too much, and draw down the moon from its orbit towards the earth. It is necessary, that the force be of a just quantity, and it belongs to the mathematicians to find the force, that may serve exactly to retain a body in a given orbit, with a given velocity ; and vice versa, to determine the curvilinear way, into which a body projected from a given place, with a given velocity, may be made to deviate from its natural rectilinear way, by means of a given force.

The quantity of any centripetal force may be considered as of three kinds; absolute, accelerative, and motive.

Definition vi.

The absolute quantity of a centripetal force is the measure of the same proportional to the efficacy of the cause that propagates it from the centre, through the spaces round about.

Thus the magnetic force is greater in one load-stone and less in another according to their sizes and strength of intensity.

Definition vii.

The accelerative quantity of a centripetal force is the measure, of the same, proportional to the velocity which it generates in a given time.

Thus the force of the same load-stone is greater at a less distance, and less at a greater : also the force of gravity is greater in valleys, less on tops of exceeding high mountains ; and yet less (as shall hereafter be shown), at greater distances from the body of the earth ; but at equal distances, it is the same everywhere ; because (taking away, or allowing for, the resistance of the air), it equally accelerates all falling bodies, whether heavy or light, great or small.

Definition viii.

The motive quantity of a centripetal force, is the measure of the same proportional to the motion which it generates in a given time.

Thus the weight is greater in a greater body, less in a less body ; and in the same body, it is greater near to the earth, and less at remoter distances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or, as I may say, its weight ; and it is always known by the quantity of an equal and contrary force just sufficient to hinder the descent of the body.

These quantities of forces, we may, for brevity’s sake, call by the names of motive, accelerative, and absolute forces ; and, for distinction’s sake, con sider them, with respect to the bodies that tend to the centre ; to the places of those bodies ; and to the centre of force towards which they tend ; that is to say, I refer the motive force to the body as an endeavour and propensity of the whole towards a centre, arising from the propensities of the several parts taken together ; the accelerative force to the place of the body, as a certain power or energy diffused from the centre to all places around to move the bodies that are in them : and the absolute force to the centre, as endued with some cause, without which those motive forces would not be propagated through the spaces round about ; whether that cause be some central body (such as is the load-stone, in the centre of the magnetic force, or the earth in the centre of the gravitating force), or anything else that does not yet appear. For I here design only to give a mathematical notion of those forces, without considering their physical causes and seats.

Wherefore the accelerative force will stand in the same relation to the motive, as celerity does to motion. For the quantity of motion arises from the celerity drawn into the quantity of matter : and the motive force arises from the accelerative force drawn into the same quantity of matter. For the sum of the actions of the accelerative force, upon the several ; articles of the body, is the motive force of the whole. Hence it is, that near the surface of the earth, where the accelerative gravity, or force productive of gravity, in all bodies is the same, the motive gravity or the weight is as the body : but if we should ascend to higher regions, where the accelerative gravity is less, the weight would be equally diminished, and would always be as the product of the body, by the accelerative gravity. So in those regions, where the accelerative gravity is diminished into one half, the weight of a body two or three times less, will be four or six times less.

I likewise call attractions and impulses, in the same sense, accelerative, and motive; and use the words attraction, impulse or propensity of any sort towards a centre, promiscuously, and indifferently, one for another ; considering those forces not physically, but mathematically : wherefore, the reader is not to imagine, that by those words, I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points); when at any time I happen to speak of centres as attracting, or as endued with attractive powers.

Scholium.

Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be under stood in the following discourse. I do not define time, space, place and motion, as being well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common.

I. Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ; such as an hour, a day, a month, a year.

II. Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces ; which our senses determine by its position to bodies ; and which is vulgarly taken for immovable space ; such is the dimension of a subterraneous, an aereal, or celestial space, determined by its position in respect of the earth. Absolute and relative space, are the same in figure and magnitude ; but they do not remain always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into which the air passes; at another time it will be another part of the same, and so, absolutely understood, it will be perpetually mutable.

III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative. I say, a part of space; not the situation, nor the external surface of the body. For the places of equal solids are always equal; but their superfices, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same thing with the sum of the motions of the parts; that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places ; and therefore the place of the whole is the same thing with the sum of the places of the parts, and for that reason, it is internal, and in the whole body.

IV. Absolute motion is the translation of a body from one absolute place into another ; and relative motion, the translation from one relative place into another. Thus in a ship under sail, the relative place of a body is that part of the ship which the body possesses; or that part of its cavity which the body fills, and which therefore moves together with the ship : and relative rest is the continuance of the body in the same part of the ship, or of its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space, in which the ship itself, its cavity, and all that it contains, is moved. Wherefore, if the earth is really at rest, the body, which relatively rests in the ship, will really and absolutely move with the same velocity which the ship has on the earth. But if the earth also moves, the true and absolute motion of the body will arise, partly from the true motion of the earth, in immovable space; partly from the relative motion of the ship on the earth ; and if the body moves also relatively in the ship ; its true motion will arise, partly from the true motion of the earth, in immovable space, and partly from the relative motions as well of the ship on the earth, as of the body in the ship ; and from these relative motions will arise the relative motion of the body on the earth. As if that part of the earth, where the ship is, was truly moved toward the east, with a velocity of 10010 parts; while the ship itself, with a fresh gale, and full sails, is carried towards the west, with a velocity expressed by 10 of those parts ; but a sailor walks in the ship towards the east, with 1 part of the said velocity ; then the sailor will be moved truly in immovable space towards the east, with a velocity of 10001 parts, and relatively on the earth towards the west, with a velocity of 9 of those parts.

Absolute time, in astronomy, is distinguished from relative, by the equation or correction of the vulgar time. For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time ; astronomers correct this inequality for their more accurate deducing of the celestial motions. It may be, that there is no such thing as an equable motion, whereby time may H accurately measured. All motions may be accelerated and retarded; but the true, or equable, progress of absolute time is liable to no change. The duration or perseverance of the existence of things remains the same, whether the motions are swift or slow, or none at all : and therefore it ought to be distinguished from what are only sensible measures thereof ; and out of which we collect it, by means of the astronomical equation. The necessity of which equation, for deter mining the times of a phaenomenon, is evinced as well from the experiments of the pendulum clock, as by eclipses of the satellites of Jupiter.

As the order of the parts of time is immutable, so also is the order of the parts of space. Suppose those parts to be moved out of their places, and they will be moved (if the expression may be allowed) out of themselves. For times and spaces are, as it were, the places as well of themselves as of all other things. All things are placed in time as to order of succession ; and in space as to order of situation. It is from their essence or nature that they are places ; and that the primary places of things should be moveable, is absurd. These are therefore the absolute places ; and translations out of those places, are the only absolute motions.

But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body considered as immovable, we define all places ; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs ; but in philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred.

But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest ; but impossible to know, from the position of bodies to one another in our regions whether any of these do keep the same position to that remote body; it follows that absolute rest cannot be determined from the position of bodies in our regions.

It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. For all the parts of revolving bodies endeavour to recede from the axis of motion ; and the impetus of bodies moving forward, arises from the joint impetus of all the parts. Therefore, if surrounding bodies are moved, those that are relatively at rest within them, will partake of their motion. Upon which account, the true and absolute motion of a body cannot be determined by the translation of it from those which only seem to rest ; for the external bodies ought not only to appear at rest, but to be really at rest. For otherwise, all included bodies, beside their translation from near the surrounding ones, partake likewise of their true motions ; and though that translation were not made they would not be really at rest, but only seem to be so. For the surrounding bodies stand in the like relation to the surrounded as the exterior part of a whole does to the interior, or as the shell does to the kernel ; but, if the shell moves, the kernel will also move, as being part of the whole, without any removal from near the shell.

A property, near akin to the preceding, is this, that if a place is moved, whatever is placed therein moves along with it ; and therefore a body, which is moved from a place in motion, partakes also of the motion of its place. Upon which account, all motions, from places in motion, are no other than parts of entire and absolute motions ; and every entire motion is composed of the motion of the body out of its first place, and the motion of this place out of its place ; and so on, until we come to some immovable place, as in the before-mentioned example of the sailor. Where fore, entire and absolute motions can be no otherwise determined than by immovable places : and for that reason I did before refer those absolute motions to immovable places, but relative ones to movable places. Now no other places are immovable but those that, from infinity to infinity, do all retain the same given position one to another ; and upon this account must ever remain unmoved ; and do thereby constitute immovable space.

The causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved : but relative motion may be generated or altered without any force impressed upon the body. For it is sufficient only to impress some force on other bodies with which the former is compared, that by their giving way, that relation may be changed, in which the relative rest or motion of this other body did consist. Again, true motion suffers always some change from any force impressed upon the moving body ; but relative motion docs not necessarily undergo any change by such forces. For if the same forces are likewise impressed on those other bodies, with which the comparison is made, that the relative position may be pre served, then that condition will be preserved in which the relative motion consists. And therefore any relative motion may be changed when the true motion remains unaltered, and the relative may be preserved when the true suffers some change. Upon which accounts; true motion does by no means consist in such relations.

The effects which distinguish absolute from relative motion arc, the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute circular motion., they are greater or less, according t the quantity of the motion. If a vessel, hung: by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water ; after, by the sudden action of another force, it is whirled about the contrary way, and while the cord is untwisting itself, the vessel continues for some time in this motion ; the surface of the water will at first be plain, as before the vessel began to move : but the vessel; by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave figure (as I have experienced), and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shows its endeavour to recede from the axis of its motion ; and the true and absolute circular motion of the water, which is here directly contrary to the relative, discovers itself, and may be measured by this endeavour. At first, when the relative motion of the water in the vessel was greatest, it produced no endeavour to recede from the axis ; the water showed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its endeavour to recede from the axis ; and this endeavour showed the real circular motion of the water perpetually increasing, till it had acquired its greatest quantity, when the water rested relatively in the vessel. And therefore this endeavour does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defined by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axis of motion, as its proper and adequate effect ; but relative motions, in one and the same body, are innumerable, according to the various relations it bears to external bodies, and like other relations, are altogether destitute of any real effect, any otherwise than they may perhaps partake of that one only true motion. And therefore in their system who suppose that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them ; the several parts of those heavens, and the planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another (which never happens to bodies truly at rest), and being carried together with their heavens, partake of their motions, and as parts of revolving wholes, endeavour to recede from the axis of their motions.

Wherefore relative quantities are not the quantities themselves, whose names they bear, but those sensible measures of them (either accurate or inaccurate), which are commonly used instead of the measured quantities themselves. And if the meaning of words is to he determined by their use, then by the names time, space, place and motion, their measures are properly to be understood ; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. Upon which account, they do strain the sacred writings, who there interpret those words for the measured quantities. Nor do those less defile the purity of mathematical and philosophical truths, who confound real quantities themselves with their relations and vulgar measures.