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In formulating his physical theories, Newton developed and used mathematical methods now included in the field of calculus, expressing them in the form of geometric propositions about "vanishingly small" shapes.

In a revised conclusion to the , Newton emphasized the empirical nature of the work with the expression Hypotheses non fingo ("I frame/feign no hypotheses").

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.

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.

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.

Book 2 also discusses (in Section 5) hydrostatics and the properties of compressible fluids; Newton also derives Boyle's law.

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 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 to refute a theory of Descartes which had some wide acceptance before Newton's work (and for some time after).

According to Descartes's 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.

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.

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 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".

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".

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.

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.

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

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.

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.

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.

An extensive explanation is given of the third rule, concerning the qualities of bodies, and Newton discusses here the generalisation 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 revolutionised 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.

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 the expression hypotheses non fingo, "I formulate no hypotheses", in response to criticisms of the first edition of the Principia.

From the system of the world, he inferred the existence of a 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".

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.

Halley was at that time a Fellow and Council member of the Royal Society in London (positions that in 1686 he resigned 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.

Halley then had to wait for Newton to "find" the results, and 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 a title.

Newton's tract De motu corporum in gyrum, which he sent to Halley in late 1684, derived what is now known as the three laws of Kepler, assuming an inverse square law of force, and generalised 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.

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 formally 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.

Surviving materials show that Newton (up to some time in 1685) conceived his book as a two-volume work.

The first volume was to be titled 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 survives, its completion dated to about the summer of 1685.

It covers 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.

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.

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 De Historia piscium, 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.

After Halley had personally financed the publication of Principia, he was informed that the society could no longer afford to provide him the promised annual salary of £50.

Instead, Halley was paid with leftover copies of De Historia piscium.

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 1619 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.

Newton was criticized for apparently introducing forces that acted at distance without any medium.

Work on calculus is shown 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.

Surviving manuscripts of the 1660s also show Newton's interest in planetary motion and that by 1669 he had shown, for a circular case of planetary motion, that the force he called "endeavour to recede" (now called centrifugal force) had an inverse-square relation with distance from the center.

After his 1679–1680 correspondence with Hooke, described below, Newton adopted the language of inward or centripetal force.

According to Newton scholar J. Bruce Brackenridge, although much has been made of the change in language and difference of point of view, as between centrifugal or centripetal forces, the actual computations and proofs remained the same either way.

They also involved the combination of tangential and radial displacements, which Newton was making in the 1660s.

The difference between the centrifugal and centripetal points of view, though a significant change of perspective, did not change the analysis.

Newton also clearly expressed the concept of linear inertia in the 1660s: for this Newton was indebted to Descartes' work published 1644.

Hooke clearly postulated mutual attractions between the Sun and planets, in a way that increased with nearness to the attracting body, along with a principle of linear inertia.

Hooke's statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions.

Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses.

Hooke also did not provide accompanying evidence or mathematical demonstration.

In 1686, when the first book of Newton's Principia was presented to the Royal Society, Hooke claimed that Newton had obtained from him the "notion" of "the rule of the decrease of Gravity, being reciprocally as the squares of the distances from the Center".

At the same time (according to Edmond Halley's contemporary report) Hooke agreed that "the Demonstration of the Curves generated therby" was wholly Newton's.

A recent assessment about the early history of the inverse square law is that "by the late 1660s", the assumption of an "inverse proportion between gravity and the square of distance was rather common and had been advanced by a number of different people for different reasons".

Newton himself had shown in the 1660s that for planetary motion under a circular assumption, force in the radial direction had an inverse-square relation with distance from the center.

The background described above shows there was basis for Newton to deny deriving the inverse square law from Hooke.

On the other hand, Newton did accept and acknowledge, in all editions of the Principia, that Hooke (but not exclusively Hooke) had separately appreciated the inverse square law in the Solar System.

Newton's reawakening interest in astronomy received further stimulus by the appearance of a comet in the winter of 1680/1681, on which he corresponded with John Flamsteed.

In 1759, decades after the deaths of both Newton and Hooke, Alexis Clairaut, mathematical astronomer eminent in his own right in the field of gravitational studies, made his assessment after reviewing what Hooke had published on gravitation.

A survey published in 1953 located 189 surviving copies with nearly 200 further copies located by the most recent survey published in 2020, suggesting that the initial print run was larger than previously thought.

Cambridge University Library 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.

Throughout are Latin annotations written by Thomas S. Savage.

These handwritten notes are currently being researched at The College.

The Frederick E. Brasch Collection of Newton and Newtoniana in Stanford University also has a first edition of the Principia.

A first edition forms part of the Crawford Collection, housed at the Royal Observatory, Edinburgh.

The Uppsala University Library owns a first edition copy, which was stolen in the 1960s and returned to the library in 2009.

The Folger Shakespeare Library in Washington, D.C. owns a first edition, as well as a 1713 second edition.

The Huntington Library in San Marino, California owns Isaac Newton's personal copy, with annotations in Newton's own hand.

The Martin Bodmer Library keeps a copy of the original edition that was owned by Leibniz.

It contains handwritten notes by Leibniz, in particular concerning the controversy of who first formulated calculus (although he published it later, Newton argued that he developed it earlier).

The University of St Andrews Library holds both variants of the first edition, as well as copies of the 1713 and 1726 editions.

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 Linda Hall Library holds the first edition, as well as a copy of the 1713 and 1726 editions.

The Teleki-Bolyai Library of Târgu-Mureș holds a 2-line imprint first edition.

One book is also located at Vasaskolan, Gävle, in Sweden.

Dalhousie University has a copy as part of the William I. Morse collection.

McGill University in Montreal has the copy once owned by Sir William Osler.

The University of Toronto has a copy in the Thomas Fisher Rare Book Collection.

Newton referred to his plans for a second edition in correspondence with Flamsteed in November 1694.

Newton also maintained annotated copies of the first edition specially bound up with interleaves on which he could note his revisions; two of these copies still survive, but he had not completed the revisions by 1708.

Newton had almost severed connections with one would-be editor, Nicolas Fatio de Duillier, and another, David Gregory seems not to have met with his approval and was also terminally ill, dying in 1708.

Nevertheless, reasons were accumulating not to put off the new edition any longer.

The correspondence of 1709–1713 shows Cotes reporting to two masters, Bentley and Newton, and managing (and often correcting) a large and important set of revisions to which Newton sometimes could not give his full attention.

Under the weight of Cotes' efforts, but impeded by priority disputes between Newton and Leibniz, and by troubles at the Mint, Cotes was able to announce publication to Newton on 30 June 1713.

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.

In 1739–1742, two French priests, Pères 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.

Unlike LeSeur and Jacquier's edition, hers was a complete translation of Newton's three books and their prefaces.

She also included a Commentary section where she fused the three books into a much clearer and easier to understand summary.

She included an analytical section where she applied the new mathematics of calculus to Newton's most controversial theories.

Previously, geometry was the standard mathematics used to analyse theories.

The first, from 1729, by Andrew Motte, was described by Newton scholar I. Bernard Cohen (in 1968) as "still of enormous value in conveying to us the sense of Newton's words in their own time, and it is generally faithful to the original: clear, and well written".

The 1729 version was the basis for several republications, often incorporating revisions, among them a widely used modernised English version of 1934, which appeared under the editorial name of Florian Cajori (though completed and published only some years after his death).

The second full English translation, into modern English, is the work that resulted from this decision by collaborating translators I. Bernard Cohen, Anne Whitman, and Julia Budenz; it was published in 1999 with a guide by way of introduction.

The third such translation is due to Ian Bruce, and appears, with many other translations of mathematical works of the 17th and 18th centuries, on his website.

Dana Densmore and William H. Donahue have published a translation of the work's central argument, published in 1996, along with expansion of included proofs and ample commentary.

The book was developed as a textbook for classes at St. John's College and the aim of this translation is to be faithful to the Latin text.

In 2014, British astronaut Tim Peake named his upcoming mission to the International Space Station Principia after the book, in "honour of Britain's greatest scientist".

Tim Peake's Principia launched on December 15, 2015 aboard Soyuz TMA-19M.

I. Bernard Cohen, Introduction to Newton's Principia (Harvard University Press, 1971).

Richard S. Westfall, Force in Newton's physics; the science of dynamics in the seventeenth century (New York: American Elsevier, 1971).

S. Chandrasekhar, Newton's Principia for the common reader (New York: Oxford University Press, 1995).

Guicciardini, N., 2005, "Philosophia Naturalis..." in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics.

Andrew Janiak, Newton as Philosopher (Cambridge University Press, 2008).

François De Gandt, Force and geometry in Newton's Principia trans.

Curtis Wilson (Princeton, NJ: Princeton University Press, c1995).

Steffen Ducheyne, The main Business of Natural Philosophy: Isaac Newton's Natural-Philosophical Methodology (Dordrecht e.a.

John Herivel, The background to Newton's Principia; a study of Newton's dynamical researches in the years 1664–84 (Oxford, Clarendon Press, 1965).

Brian Ellis, "The Origin and Nature of Newton's Laws of Motion" in Beyond the Edge of Certainty, ed.

Burtt, Metaphysical Foundations of Modern Science (Garden City, NY: Doubleday and Company, 1954).

Colin Pask, Magnificent Principia: Exploring Isaac Newton's Masterpiece'' (New York: Prometheus Books, 2013).

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