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Philosophiae Naturalis Principia Mathematica

2007 Schools Wikipedia Selection. Related subjects: British History 1500-1750

   Newton's own copy of his Principia, with handwritten corrections for
   the second edition.
   Enlarge
   Newton's own copy of his Principia, with handwritten corrections for
   the second edition.

   The Philosophiae Naturalis Principia Mathematica (Latin: "mathematical
   principles of natural philosophy", often Principia or Principia
   Mathematica for short) is a three-volume work by Isaac Newton published
   on July 5, 1687. It contains the statement of Newton's laws of motion
   forming the foundation of classical mechanics as well as his law of
   universal gravitation. He derives Kepler's laws for the motion of the
   planets (which were first obtained empirically).

   In formulating his physical theories, Newton had developed a field of
   mathematics known as calculus. However, the language of calculus was
   largely left out of the Principia. Instead, Newton recast the majority
   of his proofs as geometric arguments.

   It is in the Principia that Newton expressed his famous Hypotheses non
   fingo ("I feign no hypotheses").

The historical context

The beginnings of the scientific revolution

   Nicolaus Copernicus had firmly moved the Earth away from the centre 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 centers 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 foundations 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.

Newton's role

   Newton had studied these books, or, in some cases, secondary sources
   based on them, and taken notes entitled Quaestiones quadem
   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, the central
   notion of inertia still remained outside his understanding. He
   summarized this work in a note that he called "The lawes of Motion"
   (preserved in the Cambridge University Library as the Additional MS
   3958).

   Over the following years, he published his experiments on light and the
   resulting theory of colours, to overwhelmingly favourable response, and
   a few inevitable scientific 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. 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 experienced the famous
   revelation under an apple tree in Woolesthorpe, which led him to
   conclude 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).

   Hooke, in 1674, wrote Newton a letter (later published in 1679 in his
   book Lectiones Cutlerianes) 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.

Writing and publication

   In November 1684, Halley received a treatise of nine pages from Newton
   called De motu corporum in gyrum (On 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 December
   10, 1684 (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, was
   authorised on 30 June to licence 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 third book
   was finally completed a year later in April, 1687, and published that
   summer.

The contents of the book

   In the preface of the Principia, Newton wrote^6

     ... rational mechanics will be the science of motion resulting from
     any forces whatsoever, and of the forces required to produce any
     motion ... and therefore I offer this work as the mathematical
     principles of philosophy, for the whole burden 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 ...

   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 practised
   almost three and a half centuries after his beginning. Today the two
   aspects that Newton outlined would be called analysis and synthesis.

   The Principia consists of three books
    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 also
       contains propositions and proofs that have little to do with
       dynamics but demonstrate the kinds of problems that can be solved
       using calculus.
    2. The first book was divided into a second volume because of its
       length. It contains sundry applications such as motion through a
       resistive medium, a derivation of the shape of least resistance, a
       derivation of the speed of sound and accounts of experimental tests
       of the result.
    3. De mundi systemate (On the system of the world) is an essay on
       universal gravitation that builds upon the propositions of the
       previous books and applies them to the motions observed in the
       solar system — the regularities and the irregularities of the orbit
       of the moon, the derivations of Kepler's laws, applications to the
       motion of Jupiter's moons, to comets and tides (much of the data
       came from John Flamsteed). It 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 exactly the same as in all textbooks today. Newton first
   set out the definition of mass^6

     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 mass of phenomena that could be
   organised by the theory was so impressive that younger "philosophers"
   soon adopted the methods and language of the Principia.

The mathematical language

   The reason for Newton's use of Euclidean geometry as the mathematical
   language of choice in Principia is puzzling in two respects. The first
   is the trouble that today's physicists, trained in modern analytical
   methods, face in following the arguments. This mathematical language
   reportedly baffled Richard Feynman to the extent that he tried to work
   out alternative Euclidean proofs to his own satisfaction. S.
   Chandrasekhar, in one of his last major efforts, translated the
   Principia into modern mathematical language so that physicists of today
   can read and appreciate the book that founded modern physics.

   The second puzzle is historical. Why did Newton revert to Euclidean
   methods, when seventeenth century mathematics increasingly used
   Descartes' analytical geometry for its transactions? Newton himself had
   written earlier tracts using this language. Even his earlier
   communications on the calculus of differentials referred to a new
   language of fluxions that he had invented. In fact, his early notebooks
   suggest strongly that he learnt Cartesian geometry long before he came
   to Euclid. Some commentators have suggested that Newton used the
   mathematical language of Euclid in order to make a rhetorical point
   about how his methods followed easily from the Greek tradition.
   However, this piece of rhetoric was unnecessary for his contemporaries,
   who knew very well the general nature of Newton's mathematical
   discoveries.

Location of copies

   A page from the Principia
   Enlarge
   A page from the Principia

   Several national rare-book collections contain original copies of
   Newton's Principia Mathematica, including:
     * The Wren Library in Trinity College, Cambridge, has Newton's own
       copy of the first edition, with handwritten notes for the second
       edition.
     * 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 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 invented
       calculus (although he published it later, Newton argued that he
       developed it earlier). As an interesting side note, the copy shows
       clear signs that Leibniz was an avid smoker.
     * 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 was published in 1972 by Alexandre Koyré and I.
   Bernard Cohen, Cambridge University Press, 1972, ISBN 0-674-66475-2.

   Two more editions were published during Newton's lifetime:

Second edition

   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 1709^1. Under the weight of Cotes' efforts, but
   impeded by priority disputes between Newton and Leibniz^2, and by
   troubles at the Mint^3, Cotes was able to announce publication to
   Newton on 30 June 1713^4. 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
     * David Gregory

   However, Newton omitted acknowledgements to some because of the
   priority disputes. John Flamsteed, the Astronomer Royal, suffered this
   especially.

Third edition

   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.^5

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