Philosophiæ Naturalis Principia Mathematica

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The Philosophiæ Naturalis Principia Mathematica, which is Latin for "mathematical principles of natural philosophy", often Principia or Principia Mathematica for short, is a three-volume work by Isaac Newton first published on 5 July 1687.^[1][2] Newton also published two further editions, the second in 1713, and the third in 1726.^[3] The Principia contains the statement of Newton's laws of motion forming the foundation of classical mechanics, as well as his law of universal gravitation and a derivation of Kepler's laws for the motion of the planets (which were first obtained empirically). The Principia is "justly regarded as one of the most important works in the history of science".^[4]

In formulating his physical theories, Newton had developed a field of mathematics now known as calculus. However, the language of calculus as we know it was largely absent from the Principia. Instead, Newton cast the majority of his proofs in geometric form, although with many calculus-like arguments based on limits of vanishing small geometric quantities.

In a supplement to the Principia, entitled General Scholium, Newton expressed his famous Hypotheses non fingo ("I feign no hypotheses" or "I make no guesses").

Contents 1 The historical context 1.1 The beginnings of the scientific revolution 1.2 Newton's role 2 Writing and publication 3 The contents of the book 4 Rules of Reasoning in Philosophy 5 Location of copies 5.1 Second edition, 1713 5.2 Third edition, 1726 5.3 Annotated editions 5.4 English translations 6 General Scholium 7 See also 8 References 9 Further reading 10 External links

[edit] The historical context

[edit] The beginnings of the scientific revolution

Nicolaus Copernicus had firmly moved the Earth away from the center of the universe with the heliocentric theory for which he presented evidence in his book De revolutionibus orbium coelestium (On the revolutions of the heavenly spheres) published in 1543. The structure was completed when Johannes Kepler wrote the book Astronomia nova (A new astronomy) in 1609, setting out the evidence that planets move in elliptical orbits with the sun at one focus, and that planets do not move with constant speed along this orbit. Rather, their speed varies so that the line joining the centres of the sun and a planet sweeps out equal areas in equal times. To these two laws he added a third a decade later, in his otherwise forgettable book Harmonices Mundi (Harmonies of the world). This law sets out a proportionality between the third power of the characteristic distance of a planet from the sun and the square of the length of its year.

The foundation of modern dynamics was set out in Galileo's book Dialogo sopra i due massimi sistemi del mondo (Dialogue on the two main world systems) where the notion of inertia was implicit and used. In addition, Galileo's experiments with inclined planes had yielded precise mathematical relations between elapsed time and acceleration, velocity or distance for uniform and uniformly accelerated motion of bodies.

Descartes' book of 1644 Principia philosophiae (Principles of philosophy) stated that bodies can act on each other only through contact: a principle that induced people, among them himself, to hypothesize a universal medium as the carrier of interactions such as light and gravity—the aether. Another mistake was his treatment of circular motion, but this was more fruitful in that it led others to identify circular motion as a problem raised by the principle of inertia. Christiaan Huygens solved this problem in the 1650s and published it much later.

[edit] Newton's role

Newton had studied these books, or, in some cases, secondary sources based on them, and taken notes entitled Quaestiones quaedam philosophicae (Questions about philosophy) during his days as an undergraduate. During this period (1664–1666) he created the basis of calculus, and performed the first experiments in the optics of colour. In addition he took two crucial steps in dynamics: first, in the course of an analysis of the impact between two bodies, he deduced correctly that the centre of mass remains in uniform motion; second, he made his first, but mistaken, analysis of circular motion assuming that there must exist a (repulsive) centrifugal force. At this time, his proof that white light was a combination of primary colours (found via prismatics) replaced the prevailing theory of colours and received an overwhelmingly favourable response, and occasioned bitter disputes with Robert Hooke and others, which forced him to sharpen his ideas to the point where he composed sections of his later book Opticks already by the 1670s in response. He wrote up bits and pieces of the calculus in various papers and letters, including two to Leibniz. He became a fellow of the Royal Society and the second Lucasian Professor of Mathematics (succeeding Isaac Barrow) at Trinity College, Cambridge.

In the plague year of 1665, Newton had already concluded that the strength of gravity falls off as the inverse square of the distance, by substituting Kepler's third law into his derivation of the centrifugal force (muddled as it was through his misunderstanding of the nature of circular motion in The lawes of motion). This conclusion is apocryphally purported to be the result of seeing an apple fall while in an orchard at Woolesthorpe.

Hooke, in 1674, wrote Newton a letter (later published in 1679 in his book Lectiones Cutlerianae) through which Newton first understood of the role of inertia in the problem of circular motion—that the tendency of a body is to fly off in a straight line, and that an attractive force must keep it moving in a circle. In reply Newton drew (and described) a fancied trajectory of a body, initially with only tangential velocity, falling towards a centre of attraction in a spiral. Hooke noted this error and corrected it, saying that with an inverse square force law the correct path would be an ellipse, and made the exchange public by reading both Newton's letter and his correction to the Royal Society in 1676. Newton tried a rearguard action by giving the orbits in various other kinds of central potentials in another letter to Hooke, thus showing his mastery over the method. In 1677, in a conversation with Christopher Wren, Newton realized that Wren had also arrived at the inverse square law by exactly the same method as he.

Reflections on what can be deduced from common sense about aspects of circular motion brought him to his concept of "absolute space". In the Principia Newton presents the example of a rotating bucket to show that in everyday life it can readily be discerned that in a rotating motion another factor besides the motion relative to other objects is involved.

Newton had still not completed all the steps in the construction of the Principia by 1681, when a comet was observed to turn around the sun. The astronomer royal, John Flamsteed, recognised the motion as such, whereas most scientists believed that there were two comets, one that disappeared behind the sun, and another that appeared later from the same direction. The correspondence between Flamsteed and Newton showed that the latter had not appreciated the universality of the law of gravity.

This was the state of affairs when Edmund Halley visited Newton in Cambridge in August 1684, having rediscovered the inverse-square law by substituting Kepler's law into Huygens' formula for the centrifugal force. In January of that year, Halley, Wren and Hooke had a conversation where Hooke claimed to not only have derived the inverse-square law, but also all the laws of planetary motion. Wren was unconvinced, and Halley, having failed in the derivation himself, resolved to ask Newton. Newton said that he had already made the derivations but could not find the papers. Matching accounts of this meeting come from Halley and Abraham De Moivre to whom Newton confided.

[edit] Writing and publication

In November 1685, Halley received a treatise of nine pages from Newton called De motu corporum in gyrum (Of the motion of bodies in an orbit). It derived the three laws of Kepler assuming an inverse square law of force, and generalized the answer to conic sections. It extended the methodology of dynamics by adding the solution of a problem on the motion of a body through a resisting medium. After another visit to Newton, Halley reported these results to the Royal Society on 10 December 1685 (Julian calendar). Newton also communicated the results to Flamsteed, but insisted on revising the manuscript. These crucial revisions, especially concerning the notion of inertia, slowly developed over the next year-and-a-half into the Principia. Flamsteed's collaboration in supplying regular observational data on the planets was very helpful during this period.

The text of the first of the three books was presented to the Royal Society at the close of April, 1686. Hooke's priority claims caused some delay in acceptance, but Samuel Pepys, as President, gave his imprimatur on 30 June 1686, licensing it for publication. Unfortunately the Society had just spent their book budget on a history of fish, so the initial cost of publication was borne by Edmund Halley: the book appeared in summer 1687.^[5]

[edit] The contents of the book

In the preface of the Principia, Newton wrote^[6]

[...] 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 [...] And therefore we offer this work as 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 phaenomena [...]

It was perhaps the force of the Principia, which explained so many different things about the natural world with such economy, that caused this method to become synonymous with physics, even as it is practiced almost three and a half centuries after its beginning. Today the two aspects that Newton outlined would be called analysis and synthesis.

The Principia consists of three books:

Book 1: De motu corporum (On the motion of bodies) is a mathematical exposition of calculus followed by statements of basic dynamical definitions and the primary deductions based on these. It contains some propositions and proofs that have little to do with dynamics but demonstrate the kinds of problems that can be solved using calculus. But Book 1 also contains sections with far-reaching application to the solar system and universe:-

• Section XI (Propositions 57-69), which deals with the "motion of bodies drawn to one another by centripetal forces". This section is of primary interest for its application to the solar system, and includes Proposition 66 along 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 the three-body problem.
• Section XII (Propositions 70-84), which deals with the attractive forces of spherical bodies. This 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 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 of the contents originally planned for the first book was divided out into a second book, largely concerning motion through resisting mediums, hydrostatics and the properties of compressible fluids; it also includes derivations of the shape of least resistance, of the speed of sound, and accounts of experimental tests of the result. 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 of Descartes which 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 2 (in the Scholium to proposition 53) his 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 (On the system of the world) is an exposition of many consequences of universal gravitation, and builds upon the propositions of the previous books, applying them with further specificity than in Book 1 to the motions observed in the solar system. Here are developed several of the features and irregularities of the orbital motion of the Moon (see Lunar theory -- Newton); the derivations of Kepler's laws, and applications of the theory of gravity to the motion of Jupiter's moons, to comets (for which much data came from John Flamsteed and from Edmond Halley), and to the tides, and to the precession of the equinoxes. Book 3 also considers the harmonic oscillator in three dimensions, and motion in arbitrary force laws.

The sequence of definitions used in setting up dynamics in the Principia is recognisable in many textbooks today. Newton first set out the definition of mass⁶

The quantity of matter is that which arises conjointly from its density and magnitude. A body twice as dense in double the space is quadruple in quantity. This quantity I designate by the name of body or of mass.

This was then used to define the "quantity of motion" (today called momentum), and the principle of inertia in which mass replaces the previous Cartesian notion of intrinsic force. This then set the stage for the introduction of forces through the change in momentum of a body. Curiously, for today's readers, the exposition looks dimensionally incorrect, since Newton does not introduce the dimension of time in rates of changes of quantities.

He defined space and time "not as they are well known to all". Instead, he defined "true" time and space as "absolute" and explained:

Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to perceptible objects. And it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common. [...] instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs; but in philosophical discussions, we ought to step back from our senses, and consider things themselves, distinct from what are only perceptible measures of them.

It is interesting that several dynamical quantities that were used in the book (such as angular momentum) were not given names. The dynamics of the first two books was so self-evidently consistent that it was immediately accepted; for example, Locke asked Huygens whether he could trust the mathematical proofs, and was assured about their correctness.

However, the concept of an attractive force acting at a distance received a cooler response. In his notes, Newton wrote that the inverse square law arose naturally due to the structure of matter. However, he retracted this sentence in the published version, where he stated that the motion of planets is consistent with an inverse square law, but refused to speculate on the origin of the law. Huygens and Leibniz noted that the law was incompatible with the notion of the aether. From a Cartesian point of view, therefore, this was a faulty theory. Newton's defence has been adopted since by many famous physicists—he pointed out that the mathematical form of the theory had to be correct since it explained the data, and he refused to speculate further on the basic nature of gravity. The sheer number of phenomena that could be organised by the theory was so impressive that younger "philosophers" soon adopted the methods and language of the Principia.

[edit] Rules of Reasoning in Philosophy

Perhaps to reduce the risk of public misunderstanding how the "Principia" was intended by its author to be understood, Newton included at the beginning of Book 3 (in the second (1713) and third (1726) editions) a section entitled "Rules of Reasoning in Philosophy". In the four rules, as they came finally to stand in the 1726 edition, Newton effectively offers a methodology for handling unknown phenomena in nature and reaching towards explanations for them. The four Rules of the 1726 edition run as follows (omitting some explanatory comments that follow each):

Rule 1: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.

Rule 2: Therefore to the same natural effects we must, as far as possible, assign the same causes.

Rule 3: The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.

Rule 4: In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, not withstanding any contrary hypothesis that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.

This section of Rules for philosophy is followed by a listing of 'Phenomena', in which are listed a number of mainly astronomical observations, that Newton used as the basis for inferences later on, as if adopting a consensus set of facts from the astronomers of his time.

Both the 'Rules' and the 'Phenomena' evolved from one edition of the 'Principia' to the next. Rule 4 made its appearance in the third (1726) edition; Rules 1-3 were present as 'Rules' in the second (1713) edition, and predecessors of them were also present in the first edition of 1687, but there they had a different heading: they were not given as 'Rules', but rather in the first (1687) edition the predecessors of the three later 'Rules', and of most of the later 'Phenomena', were all lumped together under a single heading 'Hypotheses' (in which the third item was the predecessor of a heavy revision that gave the later Rule 3).

From this textual evolution, it appears that Newton wanted by the later headings 'Rules' and 'Phenomena' to clarify for his readers his view of the roles to be played by these various statements.

In the third (1726) edition of the Principia, Newton explains each rule in an alternative way and/or gives an example to back up what the rule is claiming. The first rule is explained as a philosophers' principle of economy. The second rule states that if one cause is assigned to a natural effect, then the same cause so far as possible must be assigned to natural effects of the same kind: for example respiration in humans and in animals, fires in the home and in the Sun, or the reflection of light whether it occurs terrestrially or from the planets. An extensive explanation is given of the third rule, concerning the qualities of bodies, and Newton discusses here the generalization of observational results, with a caution against making up fancies contrary to experiments, and use of the rules to illustrate the observation of gravity and space.

Isaac Newton’s statement of the four rules revolutionized the investigation of phenomena. With these rules, Newton could in principle begin to address all of the world’s present unsolved mysteries. He was able to use his new analytical method to replace that of Aristotle, and he was able to use his method to tweak and update Galileo’s experimental method. The re-creation of Galileo’s method has never been significantly changed and in its substance scientists use it today.

[edit] Location of copies

Several national rare-book collections contain original copies of Newton's Principia Mathematica, including:

• The library of Trinity College, Cambridge, has Newton's own copy of the first edition, with handwritten notes for the second edition.
• The Earl Gregg Swem Library at the College of William & Mary has a first edition copy of the Principia [1]
• The Frederick E. Brasch Collection of Newton and Newtoniana in Stanford University also has a first edition of the Principia.[2]
• The Whipple Museum of the History of Science in Cambridge has a first-edition copy which had belonged to Robert Hooke.
• Fisher Library in the University of Sydney has a first-edition copy, annotated by a mathematician of uncertain identity and corresponding notes from Newton himself.
• The Pepys Library in Magdalene College, Cambridge, has Samuel Pepys' copy of the third edition.
• The Martin Bodmer Library[3] keeps a copy of the original edition that was owned by Leibniz. In it, we can see handwritten notes by Leibniz, in particular concerning the controversy of who discovered calculus (although he published it later, Newton argued that he developed it earlier).
• A first edition is also located in the archives of the library at the Georgia Institute of Technology. The Georgia Tech library is also home to a second and third edition.
• A facsimile edition (based on the 3rd edition of 1726 but with variant readings from earlier editions and important annotations) was published in 1972 by Alexandre Koyré and I. Bernard Cohen.^[3]
• A first edition forms part of the Crawford Collection, housed at the Royal Observatory, Edinburgh. The collection also holds a third edition copy.
• The Uppsala University Library owns a first edition copy, which was stolen in the 1960s and returned to the library in 2009. [4]
• The University of Michigan Special Collections Library owns several early printings, including the first (1687), second (1713), second revised (1714), unnumbered (1723), and third (1726) editions of the Principia.
• The Burns Library at Boston College contains a 1723 copy published between the second and third editions.
• The George C. Gordon Library at the Worcester Polytechnic Institute hold a third edition. [5]

Two later editions were published by Newton:

[edit] Second edition, 1713

Richard Bentley, master of Trinity College, influenced Roger Cotes, Plumian professor of astronomy at Trinity, to undertake the editorship of the second edition. Newton did not intend to start any re-write of the Principia until 1709.^[7] Under the weight of Cotes' efforts, but impeded by priority disputes between Newton and Leibniz,^[8] and by troubles at the Mint,^[9] Cotes was able to announce publication to Newton on 30 June 1713.^[10] Bentley sent Newton only six presentation copies; Cotes was unpaid; Newton omitted any acknowledgement to Cotes.

Among those who gave Newton corrections for the Second Edition were: Firmin Abauzit, Roger Cotes and David Gregory. However, Newton omitted acknowledgements to some because of the priority disputes. John Flamsteed, the Astronomer Royal, suffered this especially.

[edit] Third edition, 1726

The third edition was published 25 March 1726, under the stewardship of Henry Pemberton, M.D., a man of the greatest skill in these matters ...; Pemberton later said that this recognition was worth more to him than the two hundred guinea award from Newton.^[11]

[edit] Annotated editions

In 1740-42 two French priests, Peres Thomas LeSeur and François Jacquier (of the 'Minim' order, but sometimes erroneously identified as Jesuits) produced with the assistance of J-L Calandrini an extensively annotated version of the 'Principia' in the 3rd edition of 1726. Sometimes this is referred to as the 'Jesuit edition': it was much used, and reprinted more than once in Scotland during the 19th century.^[12]

[edit] English translations

Two English translations of Newton's 'Principia' have appeared, both based on the 3rd edition of 1726: The first, from 1729, by Andrew Motte,^[2] was the basis for several republications and a linguistically revised version of 1934 which appeared under the editorial name of Florian Cajori (but was completed after his death). A recent translation is from 1999, by I Bernard Cohen and Anne Whitman, and is accompanied by an extensive guide by way of introduction.^[13]

[edit] General Scholium

The second edition of 1713 had an essay attached, titled General Scholium (which received some amendments and additions in the third edition of 1726), which was to become one of Newton's most notable writings. Newton criticizes Descartes and Leibniz, and famously states Hypotheses non fingo "I feign no hypotheses", besides obliquely attacking the doctrine of Trinity.^[14][15]

• trans. Motte (1729)

[edit] See also

• Galileo, Descartes, Robert Hooke and Christian Huygens
• Previous writings by Newton, including Quaestiones quadem philosophicae, De motu corporum in gyrum
• Elements of the Philosophy of Newton

[edit] References

[edit] Further reading

• Guicciardini, N., 2005, "Philosophia Naturalis..." in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 59-87.

[edit] External links

• First edition at Google books
• Newton's Principia - read and search

• Principia Text in Latin
• Principia in Text and PDF