Newton’s *Principia* is one of the most original and epoch making works in the history of science. There is absolutely nothing original in Newton’s *Principia*. These two seemingly contradictory judgements of Isaac Newton’s *Philosophiæ Naturalis Principia Mathematica* are slightly exaggerated versions of real judgements that have been made at various points in the past. The first was the general hagiographical view that was prevalent for much of the eighteenth, nineteenth and twentieth centuries. The second began to appear in the later part of the twentieth century as some historian of science thought that Newton, or better his reputation, needed to be cut down a bit in size. So, which, if either of them, is correct? The surprising answer is, in a way, both of them.

The *Principia* is a work of synthesis; it synthesises all of the developments in astronomy and physics that had taken place since the beginning of the fifteenth century. All of the elements that make up Newton’s work were, so to speak, laid out for him to integrate into the book. This is what is meant when we say that there is nothing original in the *Principia*, however the way that Newton integrated them and what he succeeded in creating was at the time unique and totally original. The *Principia* was truly a case of the whole being greater than the parts. Before we take a brief look at the contents of the *Principia* there are a couple of anomalies in its construction that need to be addressed.

The first concerns the general methodological structure of the book. Medieval science was dominated, not exclusively, by the theories of Aristotle and Aristotelian methodology. The developments in astronomy, physics and mathematics that we have covered up to now in this series have seen a gradual but steady deconstruction of the Aristotelian structures and theories. In this situation it comes as a bit of surprise that the methodology of the *Principia* is classically Aristotelian. Aristotle stated that true episteme (Greek) or scientia (Latin), what we would term scientific knowledge, is achieved by setting out a set of first principles or axioms that are perceived as being true and not in need proof and then logically deducing new knowledge from them. Ironically the most famous example of this methodology is the Elements of Euclid, ironically because Aristotle regarded mathematics as not being real knowledge because it doesn’t deal with objects in the real world. This is the methodology that Newton uses in the *Principia*, setting out his three laws of motion as his basic principles, which we will come back to later, and not the modern methodologies of Francis Bacon or René Descartes, which were developed in the seventeenth century to replace Aristotle.

The second anomaly concerns the mathematics that Newton uses throughout the *Principia*. Ancient Greek mathematics in astronomy consisted of Euclidian geometry and trigonometry and this was also the mathematics used in the discipline in both the Islamic and European Middle Ages. The sixteenth and seventeenth centuries in Europe saw the development of analytical mathematics, first algebra and then infinitesimal calculus. In fact, Newton made major contributions to this development, in particular he, together with but independently of Gottfried William Leibniz, pulled together the developments in the infinitesimal calculus extended and codified them into a coherent system, although Newton unlike Leibniz had at this point not published his version of the calculus. The infinitesimal calculus was the perfect tool for doing the type of mathematics required in the *Principia*, which makes it all the more strange that Newton didn’t use it, using the much less suitable Euclidian geometry instead. This raises a very big question, why?

In the past numerous people have suggested, or even claimed as fact, that Newton first worked through the entire content of the *Principia* using the calculus and then to make it more acceptable to a traditional readership translated all of his results into the more conventional Euclidian geometry. There is only one problem with this theory. With have a vast convolute of Newton’s papers and whilst we have numerous drafts of various section of the Principia there is absolutely no evidence that he ever wrote it in any other mathematical form than the one it was published in. In reality, since developing his own work on the calculus Newton had lost faith in the philosophical underpinnings of the new analytical methods and turned back to what he saw as the preferable synthetic approach of the Greek Euclidian geometry. Interestingly, however, the mark of the great mathematician can be found in this retrograde step in that he translated some of the new analytical methods into a geometrical form for use in the *Principia*. Newton’s use of the seemingly archaic Euclidian geometry throughout the Principia makes it difficult to read for the modern reader educated in modern physics based on analysis.

When referencing Newton’s infamous, “If I have seen further it is by standing on the sholders [*sic*] of Giants”, originally written to Robert Hooke in a letter in 1676, with respect to the *Principia* people today tend to automatically think of Copernicus and Galileo but this is a misconception. You can often read that Newton completed the Copernican Revolution by describing the mechanism of Copernicus’ heliocentric system, however, neither Copernicus nor his system are mentioned anywhere in the *Principia*. Newton was a Keplerian, but that as we will see with reservations, and we should remember that in the first third of the seventeenth century the Copernican system and the Keplerian system were viewed as different, competing heliocentric models. Galileo gets just five very brief, all identical, references to the fact that he proved the parabola law of motion, otherwise he and his work doesn’t feature at all in the book. The real giants on whose shoulders the Principia was built are Kepler, obviously, Descartes, whose role we will discuss below, Huygens, who gets far to little credit in most accounts, John Flamsteed, Astronomer Royal, who supplied much of the empirical data for Book III, and possibly/probably Robert Hooke (see episode XXXIX).

We now turn to the contents of the book; I am, however, not going to give a detailed account of the contents. I Bernard Cohen’s *A Guide to Newton’s Principia*, which I recommend runs to 370-large-format-pages in the paperback edition and they is a whole library of literature covering aspects that Cohen doesn’t. What follows is merely an outline sketch with some comments.

As already stated the book consists of three books or volumes. In Book I Newton creates the mathematical science of dynamics that he requires for the rest of the book. Although elements of a science of dynamics existed before Newton a complete systematic treatment didn’t. This is the first of Newton’s achievement, effectively the creation of a new branch of physics. Having created his toolbox he then goes on to apply it in Book II to the motion of objects in fluids, at first glance a strange diversion in a book about astronomy, and in Book III to the cosmos. Book III is what people who have never actually read *Principia* assume it is about, Newton’s heliocentric model of the then known cosmos.

Mirroring The Elements of Euclid, following Edmond Halley’s dedicatory ode and Newton’s preface, Book I opens with a list of definitions of terms used. In his scholium to the definitions Newton states that he only defines those terms that are less familiar to the reader. He gives quantity of matter and quantity of motion as his first two definitions. His third and fourth definitions are rather puzzling as they are a slightly different formulation of his first law the principle of inertia. This is puzzling because his laws are dependent on the definitions. His fifth definition introduces the concept of centripetal force, a term coined by Newton in analogy to Huygens’ centrifugal force. In circular motion centrifugal is the tendency to fly outwards and centripetal in the force drawing to the centre. As examples of centripetal force Newton names magnetism and gravity. The last three definitions are the three different quantities of centripetal force: absolute, accelerative and motive. These are followed by a long scholium explicating in greater detail his definitions.

We now arrive at the Axioms, or The Laws of Motions:

1) *Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.*

This is the principle of inertia that Newton had taken from Descartes, who in turn had taken it from Isaac Beeckman.

2) *A change of motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.*

Somewhat different from the modern formulation of F=ma, this principle has its origin in the work of Huygens although there is not a one to one correspondence.

3) *To any action there is always an opposite and equal reaction, in other words, the actions of two bodies upon each other are always equal and always opposite in direction.*

This law originates with Newton and its source is not absolutely clear. It seems to have been inspired by Newton’s examination of Descartes laws of inelastic collision but it might also have been inspired by a similar principle in alchemy of which Newton was an ardent disciple.

Most people are aware of the three laws of motion, the bedrock of Newton’s system, in their modern formulations and having learnt them, think that they are so simple and obvious that Newton just pulled them out of his hat, so to speak. This is far from being the case. Newton actually struggled for months to find the axioms that eventually found their way into the Principia. He tried numerous different combinations of different laws before finally distilling the three that he settled on.

Having set up his definitions and laws Newton now goes on to produce a systematic analysis of forces on and motion of objects in Book I. It is this tour de force that established Newton’s reputation as one of the greatest physicist of all time. However, what interests us is of course the law of gravity and its relationship to Kepler’s laws of planetary motion. The following is ‘plagiarised’ from my blog post on the 400^{th} anniversary of Kepler’s third law.

In Book I of Principia, the mathematics and physics section, Newton first shows, in Proposition 11[1], that for a body revolving on an ellipse the law of the centripetal force tending towards a focus of the ellipse is inversely as the square of the distance: i.e. the law of gravity but Newton is not calling it that at this point. In Proposition 14[2] he then shows that, *If several bodies revolve about a commo***[3]***n center and the centripetal force is inversely as the square of the distance of places from the center, I say that the principal latera recta of the orbits are as the squares of the areas which bodies describe in the same time by radii drawn to the center.* And Proposition 15: *Under the same supposition as in prop. 14, I say the square of the periodic times in ellipses are as the cubes of the major axes*. Thus Newton shows that his law of gravity and Kepler’s third law are equivalent, although in this whole section where he deals mathematically with Kepler’s three laws of planetary motion he never once mentions Kepler by name. Newton would go one to claim the rights to laws one and two as he had, in his opinion, provided their first real proof. He acknowledges, however, Kepler’s claim to the third law.

Book II as already mentioned appears to go off a tangent in that it deals with motion in a fluid medium, as a result it tends to get ignored, although it is as much a tour de force as Book I. Why this detour? The answer can be found in the theories of René Descartes and Newton’s personal relationship to Descartes and his works in general. As a young man Newton undertook an extensive programme of self-study in mathematics and physics and there is no doubt that amongst the numerous sources that he consulted Descartes stand out as his initial primary influence. At the time Descartes was highly fashionable and Cambridge University was a centre for interest in Descartes philosophy. At some point in the future he then turned totally against Descartes in what could almost be describe as a sort of religious conversion and it is here that we can find the explanation for Book II.

Descartes was a strong supporter of the mechanical philosophy that he had learnt from Isaac Beeckman, something that he would later deny. Strangely, rather like Aristotle, objects could only be moved by some form of direct contact. Descartes also rejected the existence of a vacuum despite Torricelli’s and Pascal’s proof of its existence. In his *Le Monde*, written between 1629 and 1633 but only published posthumously in 1664 and later in his *Principia philosophiae*, published in 1644, Descartes suggested that the cosmos was filled with very, very fine particles or corpuscles and that the planets were swept around their orbits on vortexes in the corpuscles. Like any ‘religious’ convert, Newton set about demolishing Descartes theories. Firstly, the title of his volume is a play upon Descartes title, whereas Descartes work is purely philosophical speculation, Newton’s work is proved mathematically. The whole of Book II exists to show that Descartes’ vortex model, his cosmos full of corpuscles is a fluid, can’t and doesn’t work.

Book III, entitled *The System of the World*, is as already said that which people who haven’t actually read it think that the *Principia* is actually about, a description of the cosmos. In this book Newton applies the mathematical physics that he has developed in Book I to the available empirical data of the planets and satellites much of it supplied by the Astronomer Royal, John Flamsteed, who probably suffered doing this phase of the writing as Newton tended to be more than somewhat irascible when he needed something from somebody else for his work. We now get the astronomical crowning glory of Newton’ endeavours, an empirical proof of the law of gravity.

Having, in Book I, established the equivalence of the law of gravity and Kepler’s third law, in Book III of *The Principia*: *The System of the World* Newton now uses the empirical proof of Kepler’s third law to establish the empirical truth of the law of gravity[4] Phenomena 1: *The circumjovial planets, by radii drawn to the center of Jupiter, describe areas proportional to the times, and their periodic times—the fixed stars being et rest—are as 3/2 powers of their distances from that center*. Phenomena 2: *The circumsaturnian planets, by radii drawn to the center of Saturn, describe areas proportional to the times, and their periodic times—the fixed stars being et rest—are as 3/2 powers of their distances from that center*. Phenomena 3: *The orbits of the five primary planets—Mercury, Venus, Mars, Jupiter, and Saturn—encircle the sun*. Phenomena 4: *The periodic times of the five primary planets and of either the sun about the earth or the earth about the sun—the fixed stars being at rest—are as the 3/2 powers of their mean distances from the sun*. “This proportion, which was found by Kepler, is accepted by everyone.”

Proposition 1: *The forces by which the circumjovial planets are continually drawn away from rectilinear motions and are maintained in their respective orbits are directed to the center of Jupiter and are inversely as the squares of the distances of their places from that center*. “The same is to be understood for the planets that are Saturn’s companions.” As proof he references the respective phenomena from Book I. Proposition 2: *The forces by which the primary planets are continually drawn away from rectilinear motions and are maintained in their respective orbits are directed to the sun and are inversely as the squares of the distances of their places from its center*. As proof he references the respective phenomenon from Book I.

In the 1^{st} edition of Principia Newton referenced the solar system itself and the moons of Jupiter as system that could be shown empirically to Kepler’s third law and added the moons of Saturn in the 3^{rd} edition.

Book III in the first edition closes with Newton’s study of the comet of 1680/81 and his proof that its flight path was also determined by the inverse square law of gravity showing that this law was truly a law of universal gravity.

I have gone into far more detain describing Newton’s *Principia* than any other work I have looked out in this series because all the various streams run together. Here we have Copernicus’s initial concept of a heliocentric cosmos, Kepler’s improved elliptical version of a heliocentric cosmos with it three laws of planetary motion and all of the physics that was developed over a period of more than one hundred and fifty years woven together in one complete synthesis. Newton had produced the driving force of the heliocentric cosmos and shown that it resulted in Kepler’s elliptical system. One might consider that the story we have been telling was now complete and that we have reached an endpoint. In fact, in many popular version of the emergence of modern astronomy, usually termed the astronomical revolution, they do just that. It starts with Copernicus’ *De revolutionibus* and end with Newton’s *Principia* but as we shall see this was not the case. There still remained many problems to solve and we will begin to look at them in the next segment of our story.

[1] Isaac Newton, *The Principia*: *Mathematical Principles of Natural Philosophy*, A New Translation by I: Bernard Cohen and Anne Whitman assisted by Julia Budenz, Preceded by *A Guide to Newton’s Principia*, by I. Bernard Cohen, University of California Press, Berkley, Los Angeles, London, 1999 p. 462

[2] Newton, *Principia*, 1999 p. 467

[3] Newton, *Principia*, 1999 p. 468

[4] Newton, *Principia*, 1999 pp. 797–802

Thank you for this fascinating post and for all the “Emergence of Modern Astronomy…” series. My knowledge of the “Principia” is very spotty and comes from secondary sources (I am a mathematician, not a historian). So please take what I am going to write as good faith question from someone without expertise. Still, I have the impression that Newton went beyond synthetising and proving previous knowledge. For instance, I understand that he started the approximate analysis of the three body problem in order to understand why the orbit of the moon deviates from a Keplerian ellipse around the Earth. Did he have precursors in this, or was this a wholly original contribution?

The old myth was that Newton first worked out his results using calculus, and then translated them into classical Euclidean geometry for the

Principia. The new myth one often encounters is that thePrincipiais written entirely in the classical Euclidean style. I think Thony hints at the problem with this formulation when he says,In fact, much of the

Principiais suffused with the modes of thought that also gave birth to calculus. A perfect example is Newton’s proof of Kepler’s 2nd law, Bk I Sec 2 Prop 1 Thm 1. This concludes:You will find nothing like this passage to the limit in Euclid, or (to give a canonical instance) Archimedes’

On the Sphere and Cylinder. In contrast, the classical Greek treatment relies on the Eudoxus method of exhaustion, where one proves an equality A=B viareductio ad absurdum: assuming either that A>B or A<B leads to a contradiction.I think that Newton’s loss of faith in the “new analytical methods” could be traced to his unease about the foundations. (Recall George Berkeley’s 1734 remark in his book “The Analyst” about fluxions being “the ghosts of departed quantities”.) Newton was undoubtedly aware of the issue.

As for Hooke, I suspect that Newton would have been very displeased at Thony’s suggestion that the Principia owed anything to him at all.

As for Hooke, you are quite right. For example, this is from a letter he wrote to Halley:

As for analytical methods, I have to disagree. First, the dates are backwards: the three editions of Newton’s

Principiawere published in 1687, 1714, and 1726, while The Analyst was published in 1734.Second, the

Principiais replete with “ghosts of departed quantities”. For example, this passage:Or this:

And Berkeley attacks the

Principiadirectly in one passage, concerning Newton’s derivation of what we’d call the product rule for derivatives.Viewed generously, Newton had premonitions of the modern notion of limit. But

Principiacomes nowhere close to either ancient Greek or modern standards of rigor.By “he wrote to Halley”, I meant of course “Newton wrote to Halley”.

I agree with your remark about the dates not lining up regarding Berkeley’s book (Newton having died in 1727).

My point is that Newton was aware that using expressions such as “nascent principles of finite magnitudes” is really just arm waving. That’s what Berkeley recognised and criticised.

However, as you’ve shown with your quotations, he could not avoid using the new tools that he’d created. So my guess is that he adopted the style of Euclid’s Elements to show that his deductions were completely rigorous.

Is this just speculation about Newton’s motives, or do you have some evidence for it? To me, this looks like attributing a 19th century mathematical sensibility to Newton. Newton did

a lotof arm waving in thePrincipia, and it doesn’t seem to have bothered him, as far as I can tell. His “lemma on ovals” is a good example; even after Huygens and Leibniz offered counterexamples, Newton just threw in a one-sentence qualification in the 2nd edition, without making any attempt at all to figure out what was wrong with his original proof, nor why the qualification would fix the problem. (It doesn’t, it just rules out a few counterexamples.) Not the attitude of someone who takes rigor all that seriously!As I indicated, Newton’s

Principia, and his other works, contain inklings of all sorts of later developments. But just inklings. Here’s how Westfall put it, in his criticism of Chandrasekhar’sNewton’s Principia for the Common Reader:Thony writes,

It’s certainly a work of synthesis, but it wasn’t

onlya work of synthesis, and I’d argue strongly not evenprimarilya work of synthesis. Historians, naturally focused on broad trends, often tend to minimize more purely technical achievements. But these can have a profound impact over time.Chandrasekhar’s

Newton’s Principia for the Common Readeris an excellent guide tothis aspectof thePrincipia. Chandrasekhar was not a historian, but rather one of the foremost astrophysicists of the 20th century. To quote from Westfall’s (on the whole hostile) review:Three examples: (1) Newton showed that the inverse square law implies Kepler’s 1st law. None of his contemporaries achieved this, and even today this stands out as a “hard” result, unlike the derivation of Kepler’s 2nd law (almost a trivial consequence of conservation of angular momentum), or of Kepler’s 3rd law for circular orbits (easy algebra from Huygens’ centrifugal force formula plus the inverse square law).

(2) Newton showed that the attraction of a spherically symmetrical mass on any exterior point is the same as if all the mass were concentrated at the central point. This is

crucialfor his famous “apple/moon” calculation. Nowadays we have a conceptual proof using Gauss’s law, but at the time it was fairly difficult.(3) As another commenter pointed out, Newton inaugurated the study of perturbations: Kepler’s laws aren’t 100% correct, because planetary orbits are not truly a two-body problem.

But the

Principiais chock-full of other stuff as well. Of Newton’s “theorem on ovals”, V.I. Arnol’d, another famous physicist said that his argument was “an astonishingly modern topological proof of a remarkable theorem on the transcendence of Abelian integrals”. Wigner (again with the famous physicists!) wrote about Newton’s treatment of (as we’d now call it) the conservation of energy, “The surprising discovery of Newton’s is just this, the clear separation of laws of nature on the one hand and initial conditions on the other.” Final example: the Principia also treats the inversecubelaw, with some remarkable results.The comments by Arnol’d and Wigner are the sort of things that Westfall rightly complains about: reading back subsequent developments into Newton’s text. We can be sure that Newton didn’t anticipate Abel’s integral theorem, let alone 20th century algebraic topology. On the other hand, it is true to say that many rivulets, originating in the

Principia, grew into major rivers of mathematics in the centuries to follow.On the other hand, it is true to say that many rivulets, originating in the Principia, grew into major rivers of mathematics in the centuries to follow.Absolutely true. There is a small book by David L and Judith R Goodstein from Caltech titled “Feynman’s Lost Lecture: The motion of the planets around the Sun” which reconstructs one of the few lectures in the series by Feynman that didn’t make it into “The Feynman Lectures on Physics”;. It is remarkable that even though the proof requires nothing beyond high school geometry it is still not easy to follow. Feynman himself had tried to reproduce Newton’s proof and had got stuck at one point so, quite typically, had created his own proof (which later was found to have been discovered originally by the physicist William Rowan Hamilton in the 19th Century).

That even Feynman could have had difficulty with Newton’s proof illustrates that in Newton’s time there was a much better understanding of some of the more subtle geometrical properties of conics (this is what Goodstein says in the book). The rise of algebra and calculus from Newton’s time, which made solving many problems easier that were difficult or impossible to solve through geometry meant that the study and teaching of geometry atrophied until the 19th Century discovery of non-Euclidean geometries.

The youtube channel minutephysics has a video explaining the Hamilton/Feynman proof. It’s a guest video by Grant Sanderson, author of the marvelous youtube channel 3blue1brown.

Most people may think that Newton’s three laws are simple and obvious, but most people don’t actually understand them. (I do not include readers of this blog, a highly atypical sample.) David Hestenes demonstrated this in the 80s and 90s with his Force Concept Inventory. For example, he says that “nearly 80% of the [students completing introductory college physics courses] could state Newton’s Third Law at the beginning of the course. FCI data showed that less than 15% of them fully understood it at the end”. The Third Law isn’t the only issue. Hestenes’ work uncovered a mental substratum of intuitive physics, far more akin to Aristotle than to Newton; this is the starting point for nearly everyone, and not many in the population at large ever get past it.

Aristotle’s physics is largely how we experience the everyday world ourselves as is geocentric astronomy. We have to be taught the alternative, ‘correct’ but counter intuitive physics and astronomy.

Everyday experience isn’t a complete explanation, and probably not even the main reason. For example, “heavier things fall faster” seems intuitively correct. But it is totally at odds with even the simplest direct observation. I have a crumpled piece of paper, 4 grams, and a paperweight, 467 grams. I drop them from as high up as I can reach, and there is no perceptible difference in when they hit the ground. Try it yourself!

(There is a modern myth that viscosity is supposed to account for the Aristotelian belief. This is rubbish–the viscosity of air is negligible for everyday experience. On the other hand, if you don’t crumple the paper, it flutters to the ground, rather than falling.)

Hestenes and his group spent years researching these issues. Our deeply imbedded intuitions about physics do not derive only from everyday experience with inanimate objects; other psychological factors weigh in just as much, if not more.

OTOH, when it come to geocentric astronomy (or basically, just the notion that the earth isn’t moving), I agree.