Isaac Newton

Philosophiae naturalis principia mathematica
Authorised by the Royal Society, 5th July 1686; published, London, 1687.

Isaac Newton, Philosophiae naturalis principia mathematica

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  Sir Isaac Newton's Principia is rightly regarded as one of the fundamental texts of Western science. Its foundational concept of one universal mathematical law binding all matter together - from the smallest particles to astronomical bodies - lies at the heart of physical science. On the other hand, Newton's masterpiece has to be understood in both its scientific and its historical context, for while a work of undoubted genius, it came as the crowning achievement of a tradition in European physics that went back a century before 1687. And we must be careful not to see Principia as a flash of truth in murky darkness, as was stated in Alexander Pope's Westminster Abbey epitaph on Newton:

        Nature, and Nature's Laws lay hid in night,
       God said 'Let Newton be', and all was light!

For ideas about 'gravitas' were being discussed by scientific writers at least back to the time of Aristotle in the fourth century BC, where it was associated with weight, heaviness, and a tendency to fall. Objects with less 'gravitas', such as sponges, fell more slowly than those with much 'gravitas', such as lumps of iron; although this idea was first contradicted by medieval scientists working in Oxford, and given its classic articulation by Galileo in the 1620s.

Central to the problem with which Newton was to wrestle for over 20 years, from c. 1665 to 1683, was what force made the planets rotate around the sun, and the moon around the earth. For after Tycho Brahe, Johannes Kepler, and others had advanced serious observational and physical objections to the ancient idea that the planets were attached to a set of concentric 'crystalline' spheres (doubts raised largely from the shape of cometary orbits after c.1580), the question was: How are the moon and planets suspended, and what force makes them move so predictably through seemingly empty space? And why, moreover, does this force appear to diminish with distance, so that Mercury orbits the sun much faster than Saturn (the outermost known solar system body in 1687)? By 1619 Kepler had published all three of his 'Laws of Planetary Motion', and had defined a series of geometrical relationships that existed between the sun and an orbiting planet, based on the unprecedentedly accurate observations of Tycho Brahe, who had died in 1601. In these 'Laws', Kepler had elucidated the elliptical shape of a planet's orbit, its acceleration and retardation as it came near to and then receded from the sun, along with the Third Law, which defined an exact proportionate geometry in the motion from which the sun-planet distance could be computed. But what was the nature of the invisible force that acted with an exactitude no less rigid than a rod or taut line connecting the two bodies? Various possibilities were considered.

Drawing analogies from William Gilbert's De Magnete (1600), Kepler suggested a kind of magnetic force which emanated from the sun, and swept the planets along in its flux, rather like a whirlpool. This view gained strength after 1612, when Galileo, Johannes Fabricius, Thomas Harriot, and others discovered from sunspots that the sun rotated on its axis in 28 days. It rotated, moreover, in the same direction as the planets.

Further experimental discoveries, however, undermined this 'magnetical hypothesis' of the solar influence, as Galileo, Pierre Gassendi, Jeremiah Horrocks, Robert Hooke, Christiaan Huygens, and others, 1620-1670, showed the 'gravitating force' to be quite different from that of magnetism. But were there different types of 'gravity' that acted, for instance, between terrestrial bodies, such as falling projectiles? And how did these terrestrial forces relate to the astronomical force? And did all objects possess gravity in differing ways and strengths? How did gravity relate to inertia, and to the fact that long pendulums swing more slowly than short ones? And was the gravity force stronger at the bottom of a deep mine than at the top of a mountain? And how could the moon produce two oceanic tides per day, rather than one? In 1664-5, Robert Hooke even suggested (Micrographia, 1665, p. 246) that the moon must possess a gravitating power, for otherwise how could it have such a regular spherical shape, compacting its matter around a centre, and be capable of retaining objects on its surface without their flying off into space! Indeed, in 1674 Hooke drew most of what was then known about gravity into three 'Laws', namely (1) all bodies possess gravity and all influence each other; (2) an object will move in a straight line until another object deflects it into an elliptical or other curve; (3) the force of gravity diminishes with the distance between objects (though Hooke admitted, 1674, that he had not yet worked out in what mathematical proportion). Hooke and Newton corresponded on the subject of gravitational attraction over November and December 1679: a correspondence which Newton broke off after Hooke had supplied a correct solution to Newton's erroneous analysis of the path which would be described by a body rotating towards or rotating around a gravitational centre.

And while in no way diminishing the genius and monumental standing of Newton and his Principia, the reality was rather different from Alexander Pope's heroic language about night and light, or the 'Newton worship' that became almost official in the eighteenth century.

Where Newton displayed undoubted genius, however, was in the production of what was not only a brilliant, inspired treatise on mathematical physics, but also an inspired and wide-ranging work of intellectual synthesis. A work that drew disparate observations and conjectures together, to demonstrate how the geometrical shape of planetary orbits, the production of the tides, and the velocities of pendulums, and how the earth is an 'oblate sphere', wider across the equator than between the poles, were all the product of one Law: Universal Gravitation. 'Universal' in so far as it applied to all matter across the universe, from specks of dust to the sun and stars, although in 1687 there was really no evidence for its action beyond the solar system. That would come later.

Newton had wrestled with the problems of motion since around 1665, when he was 23 years old. He was fully aware of the observations and measurements of his predecessors mentioned above, though one could suggest that what caused things to fall into place and crystallise into the intellectual unity of the future Principia was reading the works of and conversing and corresponding with Robert Hooke, Christiaan Huygens, Sir Christopher Wren, Edmond Halley, and others. For Principia seemed to crystallise rapidly at the end, driven in no small degree by Newton's disciple, the 29-year-old Edmond Halley, who also put up most of the mone' to meet the publication costs.

Content and Structure of Principia

The Latin text of Principia is at heart a geometrical treatise, with parallels of structure and argument going back to Euclid and the classical Greek geometers whose works were seen as laying the foundations of mathematics. For like them, Principia is based on axioms, definitions, and logical sequences, each building upon its predecessors.

Principia opens (pp. 12-25 in the original Latin text) with a preliminary statement of 'Axiomata sive leges Motus' (Axioms or Laws of Motion), which would become the classic Newtonian definitions of mass, momentum, acceleration, and inertia. This includes his famous three laws of motion, which are: (1) that in theoretical isolation, all stationary bodies will remain at rest, and moving bodies will continue to move in a straight line at a perfectly uniform velocity (this is generally regarded as a rule or principle of inertia); (2) that the change in velocity of a moving body relates directly to the 'force' acting upon it; (3) that to every action of force there must be an equal reaction.

Principia then divides into three 'Books', or sections.

Book I (Liber Primus, 'De Motu Corporum' ('On the Movement of Bodies'), pp. 26-235) deals with fundamental dynamic properties, for example those of bodies moving through a non-resisting medium and those in a resisting medium: such as those moving in space or those moving in air. It examines the geometrical motions described by bodies moving in curved orbits and under centripetal (attracting to a centre) forces. Book I, Section 13 looks at attraction between irregular and non-spherical bodies.

Book II is really an extension of Book I, 'On Motion', further developing ideas of motion and attraction. It deals, amongst other things, with the action of gravity upon swinging pendulums, while 'Lemma II' deals with an early form of calculus or 'Fluxions', whereby the constantly changing attractive force between elliptically orbiting bodies could be computed. And this calculus was to become one of the fundamental aspects of subsequent mathematics. Book II also contains Newton's refutation of René Descartes' theory of mechanical 'vortices', in which Descartes (died 1650) had argued that planets orbited the sun as a result of a sort of whirlpool of particles which carried them around.

Book III (Liber Tertius, 'De Mundi Systemate' ('On the System of the World'), pp. 401-510) sees the application of the previous principles and axioms to the heavens and to the movement of astronomical bodies. At its heart is the concept of the Inverse Square Law of Universal Gravitational Attraction, and its application to a variety of phenomena. This includes an analysis of why, when pendulums of exactly the same length are swung on the equator and in the north polar circle, they do so at different velocities (slower at the equator, faster near the poles), in 'Proposition XIX'. The reason is that the earth is an oblate sphere, with an equatorial diameter slightly greater than the polar - now known to be about 26 miles more - so that a pendulum on the equator is slightly further away from the earth's point mass centre than a polar pendulum, and hence, the pull of gravity is slightly weaker.

Book III also contains Newton's analyses and explanations of the velocities of Jupiter's satellites (p. 428), the shape of the lunar orbit around the earth and the causes of the tides (429), the Precession of the Equinoxes (470), and a great deal of material on comets orbiting the sun and the shape of their orbits, based largely on exact observations, especially by the Astronomer Royal, the Reverend John Flamsteed, of the brilliant comet of 1680-1681. (Flamsteed came to be increasingly angry with Newton in so far as he did not pay proper acknowledgement to the thoroughness of Flamsteed's cometary and later lunar orbital observations - as did Robert Hooke, whom Newton treated in similar fashion.) Principia Book III also laid out in logical mathematical form a gravitational analysis of what would come to be immortalised as the 'Three Bodies Problem': namely, the complex elliptical orbits described by the earth and the moon as they rotated around the sun.

Principia, however, while conceptually and mathematically exacting, does not use the 'new' algebraic mathematical notation, but the older geometrical demonstration notation. This makes it, in many ways, a remarkably visual book; and it would be Newton's great German rival, Gottfried Leibniz, who would be the pioneer of the algebraic calculus.


It is impossible to calculate the continuing impact of Principia from 1687 onwards. For not only did Newton supply an integrated and elegant solution to the whole domain of terrestrial and celestial physics, but his work was to mould the whole of subsequent physical thinking, and to provide methods and techniques of analysis that later generations would apply to the study of electromagnetism, optics, engineering, thermodynamics, and even atomic physics. For it was the limitations imposed by the small, essentially solar-system-based Newtonian universe that were to stimulate Sir William Herschel's deep-space cosmology by 1790, and then the mathematical physics of Albert Einstein's relativity theory, and the electron-proton, quantum and nano-physics of twentieth-century science. For without Principia the essential bedrock of first an establishment of a fundamental mathematical physics, and then a going beyond it to realms that were imaginable in 1687, could never have been laid.

By 1800 four major branches of physical study had developed in the immediate wake of Principia. These were the following. (1) Edmond Halley's fundamental study of comets, and their possible orbital predictability. (2) The Reverend Dr James Bradley's investigations into the earth-moon-sun elliptical orbital system, using instruments of the highest accuracy, leading to the discovery of the Aberration of Light in 1728 (which provided the first direct physical proof of the earth's motion around the sun), and then in 1748, the 'Nutation', or systematic orbital 'nodding', of the earth in relation to the changing gravitational positions of the moon and sun. (3) By 1735 teams of French Académiciens, from observing the positions of zenith stars in Lapland and Peru, were able to physically quantify the extent of the earth's oblate spherical shape, which was found to be in accordance with Newtonian criteria. (4) Firstly in 1775 the Reverend Dr Nevil Maskelyne (the Astronomer Royal), then in 1798 the Hon. Henry Cavendish, one using an astronomical method and the other a laboratory physics technique, were able to determine the Gravitational Constant, or 'Big G', with relation to the density of water. This made it possible, using Newtonian criteria, to calculate the respective densities of the moon and other planets, to establish a unified mathematical physics for the solar system.

After its first publication in 1687, there were two further editions of Principia in Newton's lifetime, in 1713 and 1726, the latter only a few months before his death. The later editions contained ideas and information not present in the first. One very important idea concerned the very nature of gravity itself. For Cartesian physicists in continental Europe had criticised Newton for bringing occult, mystical, and invisible forces back into physics which Descartes' swirling vortices had supposedly banished, and which it had been one of the key purposes of Book II to refute. But towards the end of the 1713 edition, in the newly-added concluding 'General Scholium' essay, Newton came to grips with some of the more philosophical implications of his gravitation theory. He plainly admitted that he had no idea what gravity was as a physical entity; what he was doing was elucidating the exact mathematical laws in accordance with which gravity acted. As Newton phrased it in the Latin, hypotheses non fingo, or 'I frame no hypotheses' about what gravity is. Gravity simply exists and acts. There is evidence that Newton, who was deeply learned theologically as well as mathematically, saw gravity as an aspect of God's power, the action of which the Almighty had revealed to the human intellect through mathematics. But that was all. As a concept, however, Newton's recognition of the ultimate un-knowableness of the nature of gravity was to have an immensely liberating effect upon subsequent science, as future generations of scientists no longer saw themselves as having to explain the powers of nature (such as light, electromagnetism, or by the 1840s, energy itself), but rather as free to explore their physical properties and to establish exact mathematical expression s for the same. For it came to be accepted that it was the scientist's job to describe nature and to elucidate nature's laws, and that it was the task of the theologian or the philosopher to explore why these laws were so.

So while we need to be cautious in our reading of Pope's epitaph on Newton, and remember that his great achievement came not as simple flash of illumination into a world of darkness, we must not forget that perhaps no other single book has exerted such a profound and sustained impact upon scientific thinking as Sir Isaac Newton's Principia.

Text kindly provided, especially for this exhibition, by Allan Chapman