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 Sir Isaac Newton, in Latin, first published 5 July 1687. After annotating and correcting his personal copy of the first edition, Newton also published two further editions, in 1713 and 1726. The Principia states Newton’s laws of motion, forming the foundation of classical mechanics, also Newton’s law of universal gravitation, and a derivation of Kepler’s laws of planetary motion (which Kepler first obtained empirically). The Principia is “justly regarded as 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 a 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, Hypotheses non fingo (“I contrive no hypotheses”).
Expressed aim and topics covered
In the preface of the Principia, Newton wrote
[…] 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 phenomena […]
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.
The opening sections of the Principia contain, in revised and extended form, nearly all of the content of Newton’s 1684 tract De motu corporum in gyrum.
The Principia begins with ‘Definitions’ and ‘Axioms or Laws of Motion’ and continues in three books:
Book 1, De motu corporum
Book 1, subtitled De 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.
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 11–31 establish 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 include Newton’s theorem about ovals (lemma 28).
Propositions 43-45 are demonstration that in an eccentric orbit under centripetal force where the apse may 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 57–69 deal 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.
Propositions 70–84 deal 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 enables the inverse square law of gravitation to be applied to the real solar system to a very close degree of approximation. The conclusion made by Isaac Newton, stating that the change of movement of some bodies is caused by the impact of other bodies – is wrong and does not correspond to the facts.
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; thus Section 1 discusses resistance in direct proportion to velocity, and Section 2 goes on to examine the implications of resistance in proportion to the square of velocity. Book 2 also discusses (in Section 5) hydrostatics and the properties of compressible fluids. The effects of air resistance on pendulums are studied in Section 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 bodies of different shape, attempts to derive the speed of sound, and gives 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 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
Book 3, subtitled De 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 developed several of the features and irregularities of the orbital motion of the Moon, especially the variation. 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 Jupiter and going on by stages to show that the law is of universal application. He also gives starting at Lemma 4 and Proposition 40) the theory of the motions of comets, for which much data came from John Flamsteed and Edmond Halley, and accounts for the tides, attempting quantitative estimates of the contributions of the Sun and Moon to the tidal motions; and offers the first theory of the precession of the equinoxes. Book 3 also considers the harmonic oscillator in 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 recognized 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”. Newton rejected the second alternative after adopting the position that “the centre of the system of the world is immoveable”, which “is acknowledg’d by all, while some contend that the Earth, others, that the Sun is fix’d in that centre”. Newton estimated the mass ratios Sun:Jupiter and Sun:Saturn, and pointed out that these put the centre of the Sun usually a little way off the common center of gravity, but only a little, the distance at most “would scarcely amount to one diameter of the Sun”.
Commentary on the Principia
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 mass
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.
To some modern readers it can appear that some dynamical quantities recognized today were used in the Principia but not named. The mathematical aspects of the first two books were so clearly consistent that they were easily 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.
Rules of Reasoning in Philosophy
Perhaps to reduce the risk of public misunderstanding, 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.
The General Scholium is a concluding essay added to the second edition, 1713 (and amended in the third edition, 1726). It is not to be confused with the General Scholium at the end of Book 2, Section 6, which discusses his pendulum experiments and resistance due to air, water, and other fluids.
Here Newton used what became his famous expression Hypotheses non fingo, “I frame no hypotheses”, in response to criticisms of the first edition of the Principia. (‘Fingo’ is sometimes nowadays translated ‘feign’ rather than the traditional ‘frame’.) Newton’s gravitational attraction, an invisible force able to act over vast distances, had led to criticism that he had introduced “occult agencies” into science. Newton firmly rejected such criticisms and wrote that it was enough that the phenomena implied gravitational attraction, as they did; but the phenomena did not so far indicate the cause of this gravity, and it was both unnecessary and improper to frame hypotheses of things not implied by the phenomena: such hypotheses “have no place in experimental philosophy”, in contrast to the proper way in which “particular propositions are inferr’d from the phenomena and afterwards rendered general by induction”.
Newton also underlined his criticism of the vortex theory of planetary motions, of Descartes, pointing to its incompatibility with the highly eccentric orbits of comets, which carry them “through all parts of the heavens indifferently”.
Newton also gave theological argument. From the system of the world, he inferred the existence of a Lord God, along lines similar to what is sometimes called the argument from intelligent or purposive design. It has been suggested that Newton gave “an oblique argument for a unitarian conception of God and an implicit attack on the doctrine of the Trinity”, but the General Scholium appears to say nothing specifically about these matters.
WRITING AND PUBLICATION
Halley and Newton’s initial stimulus
In January 1684, Halley, Wren and Hooke had a conversation in which Hooke claimed to not only have derived the inverse-square law, but also all the laws of planetary motion. Wren was unconvinced, Hooke did not produce the claimed derivation although the others gave him time to do it, and Halley, who could derive the inverse-square law for the restricted circular case (by substituting Kepler’s relation into Huygens’ formula for the centrifugal force) but failed to derive the relation generally, resolved to ask Newton.
Halley’s visits to Newton in 1684 thus resulted from Halley’s debates about planetary motion with Wren and Hooke, and they seem to have provided Newton with the incentive and spur to develop and write what became Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). Halley was at that time a Fellow and Council member of the Royal Society in London, (positions that in 1686 he resigned in order to become the Society’s paid Clerk). Halley’s visit to Newton in Cambridge in 1684 probably occurred in August. When Halley asked Newton’s opinion on the problem of planetary motions discussed earlier that year between Halley, Hooke and Wren, Newton surprised Halley by saying that he had already made the derivations some time ago; but that he could not find the papers. (Matching accounts of this meeting come from Halley and Abraham De Moivre to whom Newton confided.) Halley then had to wait for Newton to ‘find’ the results, but in November 1684 Newton sent Halley an amplified version of whatever previous work Newton had done on the subject. This took the form of a 9-page manuscript, De motu corporum in gyrum (Of the motion of bodies in an orbit): the title is shown on some surviving copies, although the (lost) original may have been without title.
Newton’s tract De motu corporum in gyrum, which he sent to Halley in late 1684, derived what are now known as the three laws of Kepler, assuming an inverse square law of force, and generalized the result to conic sections. It also extended the methodology by adding the solution of a problem on the motion of a body through a resisting medium. The contents of De motu so excited Halley by their mathematical and physical originality and far-reaching implications for astronomical theory, that he immediately went to visit Newton again, in November 1684, to ask Newton to let the Royal Society have more of such work. The results of their meetings clearly helped to stimulate Newton with the enthusiasm needed to take his investigations of mathematical problems much further in this area of physical science, and he did so in a period of highly concentrated work that lasted at least until mid-1686.
Newton’s single-minded attention to his work generally, and to his project during this time, is shown by later reminiscences from his secretary and copyist of the period, Humphrey Newton. His account tells of Isaac Newton’s absorption in his studies, how he sometimes forgot his food, or his sleep, or the state of his clothes, and how when he took a walk in his garden he would sometimes rush back to his room with some new thought, not even waiting to sit before beginning to write it down. Other evidence also shows Newton’s absorption in the Principia: Newton for years kept up a regular programme of chemical or alchemical experiments, and he normally kept dated notes of them, but for a period from May 1684 to April 1686, Newton’s chemical notebooks have no entries at all. So it seems that Newton abandoned pursuits to which he was normally dedicated, and did very little else for well over a year and a half, but concentrated on developing and writing what became his great work.
The first of the three constituent books was sent to Halley for the printer in spring 1686, and the other two books somewhat later. The complete work, published by Halley at his own financial risk, appeared in July 1687. Newton had also communicated De motu to Flamsteed, and during the period of composition he exchanged a few letters with Flamsteed about observational data on the planets, eventually acknowledging Flamsteed’s contributions in the published version of the Principia of 1687.
The process of writing that first edition of the Principia went through several stages and drafts: some parts of the preliminary materials still survive, others are lost except for fragments and cross-references in other documents.
Surviving preliminary materials show that Newton (up to some time in 1685) conceived his book as a two-volume work: The first volume was to be De motu corporum, Liber primus, with contents that later appeared in extended form as Book 1 of the Principia.
A fair-copy draft of Newton’s planned second volume De motu corporum, Liber secundus still survives, and its completion has been dated to about the summer of 1685. What it covers is the application of the results of Liber primus to the Earth, the Moon, the tides, the solar system, and the universe: in this respect it has much the same purpose as the final Book 3 of the Principia, but it is written much less formally and is more easily read.
It is not known just why Newton changed his mind so radically about the final form of what had been a readable narrative in De motu corporum, Liber secundus of 1685, but he largely started afresh in a new, tighter, and less accessible mathematical style, eventually to produce Book 3 of the Principia as we know it. Newton frankly admitted that this change of style was deliberate when he wrote that he had (first) composed this book “in a popular method, that it might be read by many”, but to “prevent the disputes” by readers who could not “lay aside the[ir] prejudices”, he had “reduced” it “into the form of propositions (in the mathematical way) which should be read by those only, who had first made themselves masters of the principles established in the preceding books”. The final Book 3 also contained in addition some further important quantitative results arrived at by Newton in the meantime, especially about the theory of the motions of comets, and some of the perturbations of the motions of the Moon.
The result was numbered Book 3 of the Principia rather than Book 2, because in the meantime, drafts of Liber primus had expanded and Newton had divided it into two books. The new and final Book 2 was concerned largely with the motions of bodies through resisting mediums.
But the Liber secundus of 1685 can still be read today. Even after it was superseded by Book 3 of the Principia, it survived complete, in more than one manuscript. After Newton’s death in 1727, the relatively accessible character of its writing encouraged the publication of an English translation in 1728 (by persons still unknown, not authorised by Newton’s heirs). It appeared under the English title A Treatise of the System of the World. This had some amendments relative to Newton’s manuscript of 1685, mostly to remove cross-references that used obsolete numbering to cite the propositions of an early draft of Book 1 of the Principia. Newton’s heirs shortly afterwards published the Latin version in their possession, also in 1728, under the (new) title De Mundi Systemate, amended to update cross-references, citations and diagrams to those of the later editions of the Principia, making it look superficially as if it had been written by Newton after the Principia, rather than before. The System of the World was sufficiently popular to stimulate two revisions (with similar changes as in the Latin printing), a second edition (1731), and a ‘corrected’ reprint of the second edition (1740).
Halley’s role as publisher
The text of the first of the three books of the Principia was presented to the Royal Society at the close of April, 1686. Hooke made some priority claims (but failed to substantiate them), causing some delay. When Hooke’s claim was made known to Newton, who hated disputes, Newton threatened to withdraw and suppress Book 3 altogether, but Halley, showing considerable diplomatic skills, tactfully persuaded Newton to withdraw his threat and let it go forward to publication. Samuel Pepys, as President, gave his imprimatur on 30 June 1686, licensing the book for publication. The Society had just spent its book budget on aHistory of Fishes, and the cost of publication was borne by Edmund Halley (who was also then acting as publisher of the Philosophical Transactions of the Royal Society): the book appeared in summer 1687.
LOCATION OF COLLECTIONS
Several national rare-book collections contain original copies of Newton’s Principia Mathematica, including:
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.