Talk:Philosophiæ Naturalis Principia Mathematica

Location of Copies
I think I added most of the original information in this section, but I have come across a possible confusion concerning Newton's own copy of Principia.

Having added the photograph of the Principia on the Wren Library, Cambridge, I came across a news item saying that Newton's Principia was currently in a touring exhibition at the New York City Library between October 8, 2004 through February 5, 2005. Indeed, the good online version of the exhibition includes a photo of a better page with copious notes for the second edition (it looks like the page shown is on gravity and orbits).

Initially I worried about how the book could be in two places at once, but this can easily be satisfied since the book is actually three volumes. However, the book in NYC is identified as coming from the Portsmouth Collection in Cambridge University Library. And this reminded me that indeed most of Newton's best scientific papers were found in the Portsmouth library and are now securely kept at the University Library. The edges of the bindings of the books in the two photos look quite different and they both seem to have very different catalogue numbers, which would imply that the one in the Wren Library isn't on loan from the Portsmouth collection.

Does anyone know if there is more than one set of Newton's 'own' copy of Principia around? -- Solipsist 07:49, 13 Nov 2004 (UTC)

When I was a student as the University of Sheffield, there was indeed a copy of the Principia in their rare books collection. Was this simply on some sort of tour/loan, or is this simply not listed in this article? 63.162.143.5Gadee —Preceding signed but undated comment was added at 11:59, 28 September 2007 (UTC)

There is a first edition copy of the book in the Crawford collection, housed at the Royal Observatory Edinburgh. Saw it today with my own eyes. Also in this collection is a third edition copy. Verification available at []. Will I add the entry, or does someone else want to? Marmite disaster 23:44, 25 October 2007 (UTC)

On December 29, 2015 I saw Principia at the Huntington Library in San Marino, California. The nearby annotation said,

Isaac Newton (1642 - 1727) Philosophiae naturalis principia mathematica  (Mathematical principles of natural philosophy)  London, 1687. Isaac Newton's and later, Edmond Haley's copy.

I saw it with my own eyes and have photographs. Furthermore the docent spoke proudly of it. Of course, if I edit this based on my own observation, I could be accused of original research. And certainly if the Huntington Library staff edited the Wikepedia article, it would be original research. What do the Wikipedia experts recommend here? DMJ001 (talk) 06:24, 3 January 2016 (UTC)

Addendum to my comments above. I found a reference to the Huntington Library owning Newton's own copy of Principia at http://www.huntington.org/webassets/templates/general.aspx?id=16993. I'll go ahead a modify the article. DMJ001 (talk) 06:34, 3 January 2016 (UTC)

Physics Today
"This is all mentioned to point out how difficult it is to deal with the motion of bodies in pure geometry. Modern day physicists have a much more powerful and elegant toolkit to deal with such problems."

Can someone please provide a link or explanation of how modern physics has circumvented the problems that pure geometry in proofs incurred? -- 212.159.91.204 5 July 2005 09:07 (UTC)
 * You can start with history of physics. See also history of mathematics and history of science. When Newton invented calculus, he was describing the mathematics of changes of the subject of study with time. In his time geometry was a time-static field of study. Now our concepts in geometry can handle changes in time as well, of course. Ancheta Wis 5 July 2005 09:54 (UTC)
 * These concepts (and the toolkit) were developed by Newton in Principia (although the modern notation is due to Leibnitz). While modern day physicists do have a much more powerful toolkit, where Newton's problems are concerned, the tools in the kit are exactly the same as Newton's. --Bambaiah 10:53, July 19, 2005 (UTC)

Euclid, Descartes and Newton
Something decidedly wrong with this article. I'd expected a discussion of the contents of the book and the cirrcumstances surrounding its writing and publication. But this heart of the article is missing!

Also, it seems something of an overstatement to say (in the section called readabiility) that Euclid's geometry was a "hot topic" in the 17th century. It was certainly part of every educated person's background, just as classical (Newtonian) mechanics is part of every science student's background today. The "hot topic" was really analytical (Cartesian) geometry. Westfall's book, in fact, treats at length the question of why Newton used Euclidean geometry in his book, to the exclusion of the more obvious analytical methods (note that many independent sources, and Newton's notebooks, testify that Newton learnt Descartes before Euclid). It would also be appropriate in this section to reduce the unnecessary bits about Feynman and Principia and instead say something about Chandrasekhar's book on Newton's principia. Or better, delete this section altogether. Bambaiah 14:30, July 11, 2005 (UTC)
 * Moved the link to the in-depth article to the top of the page. Hope this helps. It may be helpful to note that Newton found Euclid "obvious", on his first reading. I would support a link to the Feynman's lost lecture article, from which the readability section was copied. Feel feel to replace with a link to the lost lecture, if you have the time. For example, "Feynman spent a sabbatical studying the copy of Principia at the Huntington Library", if you like. (I believe that Caltech does not have an original copy of Principia.) Ancheta Wis 06:53, 12 July 2005 (UTC)
 * Recent exhibit on Newton and Principa at the Huntington Library
 * If you have the time and inclination, how about replacing the "readability" section with a reference to Chandrasekhar's book under "See also"? Ancheta Wis 07:04, 12 July 2005 (UTC)
 * Hi. I took the phrase "hot topic" from the introduction of the rewrite of the lost lecture.  I took it at face value from that book. JohnFlux 09:16, 20 July 2005 (UTC)

Good to have the link to the longer article at the beginning, but a section like the one I'm putting in is more like what I had in mind. This is after all a book which founded modern physics: it needs a heart and a circulatory system.

On the other matter: There may even be a case for an article on "Chandra on Newton". Let me take a while to go through the book before deciding. Bambaiah 10:35, July 12, 2005 (UTC)

Circumstances Surrounding Writing
Various Web sites claim that Cambridge was closed in 1665 consequent to a plague (eg. google "cambridge university 1665 plague"). Further that during that time Newton retired to the family home where he was exceptionally creative and productive. What substantiates the claims? Likely there are records about Cambridge being closed. Is there any tangible evidence that he spent the time at home? Does anyone else consider the topic worthy of mention? Regards, ... PeterEasthope (talk) 23:18, 16 March 2020 (UTC)

More on Euclid, Descartes, and Newton
I'm confused about the following two sentences:


 * A second point that has been made is that Newton had to reject Descartes' views on inertia in order to understand and generalise Galileo's new ideas.

Who made the point? How is Galileo's ideas new to the 17thc, when Galileo was writing a generation or two before? What is it about Descartes' views on inertia that Newton had to reject?


 * He simultaneously rejected Descartes' new language of geometry: since he recognised that it was equivalent to Euclid's, and it lay within Newton's powers to recast the calculus in these terms.

"He" is now back to Newton. What evidence did he have to say it was "equivalent" to Euclid's? That's a bold statement I think. "Equivalent" isn't defined in this article. Equivalent how? In terms of its proof power?

Thanks! --M a s 20:15, 9 May 2006 (UTC)


 * THE NEWTONIAN CREED: We are all pretending that Newton used geometric proofs to show that an inverse-square force law must result in an elliptical orbit. Several books contain deep, sometimes long, explanations of this proof. However, no one can simply, briefly, and clearly give this proof. Most authors bypass it. Richard Feynman, John Locke, and Thomas Edison were baffled by it. It is one of mankind's hidden, sacrosact mysteries. Only the high priests, they say, really understand it, and we take them at their word. The mass of humans, including Robert Hooke, were not meant to know how that inverse-square force law necessarily results in an elliptical orbit. Humanity must believe it. We affirm our faith.Lestrade 23:32, 12 May 2006 (UTC)Lestrade

Thanks Lestrade. I think you are demanding that the simplicity of a theorem or law of nature imply the simplicity of a proof or derivation. I understand Fermat's Last Theorem as stated by Fermat, but I, along with a lot more people smarter than me, know nothing about Andrew Wiles' proof. Should I then say that Wiles' failed in his goal? Or should I accuse those who have taken the time to fully devour it as being but high priests? Thanks! --M a s 17:34, 13 May 2006 (UTC)


 * In Prop. 11, Prob. 6, Newton supposedly proved that elliptical orbits occur with inverse-square attractions. Read it yourself and see if it convinces you. In subsequent propositions, he then supposedly proved that circular, parabolic,and hyperbolic orbits also occur with inverse-square attractions. But, he never proved that inverse-square attractions necessarily result in elliptical orbits. So, he never really solved Halley's problem. Why can't minds like Feynman and Chandrasekhar show how Newton gave this assumed proof? I consider this to be an important question. The situation almost seems to resemble the famous "Emperor's New Clothes" fable in that it is generally agreed that Newton proved that an inverse-square force necessarily results in an elliptical orbit.Lestrade 15:36, 14 May 2006 (UTC)Lestrade

Thanks Lestrade. I'm sorry; the language of this article, (and I admit as well as the Principa itself,) is a little confusing for me to make sense of what you are asking. When you say, "But, he never proved that inverse-square attractions necessarily result in elliptical orbits," the statement that inverse-square attractions necessarily result in elliptical orbits is as you've already stated patently false- because inverse-square also leads to any conic (as long as there's only two bodies and one has negiligable mass relative to the other, etc...) So you fault the history for claiming that Newton had proof to something that he could not have really proven?

Or am I misunderstanding your "necessarily?" In which case maybe you meant "But, he never proved that inverse-square attractions are sufficient to result in elliptical orbits." Thanks! --M a s 01:38, 15 May 2006 (UTC)


 * Halley asked Newton, "If there is an inverse-square attractive force, what shape is the orbit?" Newton replied, "An ellipse, and I have proved it." He looked for his proof but unfortunately couldn't find it. Instead of simply providing the proof of the solution to this problem, he wrote On the Motion of Bodies, in which he tried to show that elliptical orbits could be related to such a force. Then he wrote the Principia in which he included this demonstration and added many more topics. But, he never really provided the proof that he mentioned to Halley. Most people attribute the obscurity of Newton's so-called proof to the brilliance of his eccentric and peculiar mathematics, which are supposed to be beyond the comprehension of ordinary minds. Dana Densmore wrote a 525 page book in order to explain Newton's claim. It is titled Newton's Principia: The Central Argument, Green Lion Press, ISBN 1-888009-23-3. It took that many pages for a very intelligent author to try to clarify Proposition 11, Problem 6, in Newton's Principia. In spite of this, many books seem to consider it an easy, settled matter and quickly move on to other issues.Lestrade 12:27, 15 May 2006 (UTC)Lestrade

Fair enough Lestrade. So, you are taking the statement:


 * IF bodies obey an inverse square law of attraction (and also standard assumptions about one body of negligible mass to the other, etc.) THEN one body will move as an ellipse relative to the other,

and you are arguing that Newton provided a questionable proof of this, or in a sense failed to prove it succinctly. But it's not true- inverse square implies any conic. Perhaps Newton meant "a conic, and I have proved it."

Regardless, you then seem to imply Newton, for want of priority? or for other motives? fudged his answer and response to Halley... and because of this, Newton's current reputation should be tarnished? should be reduced? And I think I would argue that Newton was a nut-job egomaniac who had a strong sense of Schadenfreude, and in addition studying the historiography of Principia is an interesting exercise in 17thC British academic culture, but we can't take the leap from being a despicable man to being a despicable scientist. Thanks! --M a s 16:12, 15 May 2006 (UTC)

I don't see the point of Lestrade's objection. According to OED, in Newton's time an orbit was "the way or course of the Sun, particularly called the Ecliptick, as also of any other Planet moving on according to the Circle of its Latitude." The word is Latin for "wheel-track", i.e., something round. While it is true that for the two-body system all possible trajectories are conic sections, actual orbits, as Newton and Hooke understood the term, are ellipses (and the circle is simply one type of ellipse, so Lestrade's objection there says more about him than Newton). Bodies in hyperbolic and parabolic motion do not orbit. They come in and go away again, never to return. I'm afraid Lestrade is trying to be pedantic and not succeeding.134.121.40.106 (talk) 09:18, 24 March 2009 (UTC)

Additions and removal
I put in three new sections: one on the (physics and mathematics) context of the Principia, one a brief description of the book and a third on the "mathematical language". In this I tried to correct the errors in the section on "readability" that I'd mentioned above. I removved the section called "readability" and merged some of the information into the new sections. A para-by-para justification:
 * Para 1: the core information is merged into the section on "mathematical language". The material which was dropped was of two types &mdash;
 * information which was more about Feynman than about the Principia
 * some of the other material was in error (see my note dated July 11 above).
 * Para 2: all the points are included in the section on the book
 * Para 3: the difficulty with geometry is (for the problems that Newton tackled) a matter of training, as demonstrated by Chandra's book. Therefore this sentence was pov, and I decided to remove it.

--Bambaiah 10:31, July 19, 2005 (UTC)


 * BTW, the published version of Feyman's lost lecture is not even due to Feynman; it is a reconstruction of what Feynman may have said by someone who did not understand Feyman's way of redoing Newton! --Bambaiah 10:42, July 19, 2005 (UTC)

On "I feign no hypotheses"
The terse paraphrase, "I do not assert that any hypotheses are true", does not seem supported by the block quote following the statement. If I understand Newton's remark, it is not that he is offering hypotheses but refusing to defend them, as I think the paraphrase suggests, but that he does not pretend that what he exposits is a hypothesis, an explanation, for what he observes. To whatever exactly the quote refers is offered as description of phenomena. IHendry 19:16, 16 November 2005 (UTC)

"Feign" is a mistranslation of "fingo", which has the meaning 'devise', contrive' or 'fabricate'. 'Feign' means 'to pretend to be affected by (a feeling, state, or injury' [ ]   (It's OK to cite a copyright page in the discussion, isn't it?). Markcymru     19:27, 5 July 2010 (UTC)

Actually, Newton was saying that he does not make hypotheses. He said as much in his words following "hypotheses non fingo", which were translated as: "...for whatever is not deduced from the phenomena, is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy." The translation of "hypotheses non fingo" that seems most consistent with Newton's intent was that made in the first English translation of Newton's work, published a few years after he died "I don't frame hypotheses." In modern English, one might simply translate the phrase as "I don't make hypotheses." If you follow the revisions of the Principia from the first Edition, before the philosophy section was in there, to the last, you see a reorganization of the material - in the first edition the term "hypothesis" is used quite frequently. And then in the revisions its culled out... so clearly he went through this transition, which is consistent with his explicit rejection of the terminology in his experimental philosophy writings in both the Optiks and in the Principia. Gacggt (talk) 12:13, 24 August 2012 (UTC)

What is it
In excerpts from book, one paragraph begins with "hi, i am karthik". Replace it please
 * Just some vandalism introduced by User:203.174.79.131 an hour ago &mdash; now reverted. -- Solipsist 09:02, 13 December 2005 (UTC)

is this confusing, or should i read this book?

Absence of Proof
Newton showed how an elliptical (as well as a circular, hyperbolic, and parabolic) orbit can be associated with an inverse-square force. However, he never supplied Halley with the proof that he had promised. That is, Newton never proved that an inverse-square force necessarily results in an elliptical orbit. Speaking logically, x may possibly result in y is not the same as y necessarily results in x. Elliptical orbits may possibly have inverse-square forces. But, inverse-square forces do not necessarily result in elliptical orbits. This has been noted by such scholars as Johann Bernoulli, Ferdinand Rosenberger, Aurel Wintner, Johannes A. Lohne, François De Gandt, and Robert Weinstock.Lestrade 01:52, 6 February 2006 (UTC)Lestrade
 * References:

Lestrade 01:52, 6 February 2006 (UTC)Lestrade Lestrade 12:53, 6 February 2006 (UTC)Lestrade
 * Johann Bernoulli's Letter in The Correspondence of Isaac Newton, edited by A. Rupert Hall and Laura Tilling, Cambridge University Press, New York, 1975, Volume V (1709-1713), pp. 5-6.
 * Rosenberger, Ferdinand, Isaac Newton und seine physikalischen Principien, Barth, Leipsig, 1895, pp. 183-184.
 * Wintner, Aurel, Analytical Foundations of Celestial Mechanics, Princeton University Press, Princeton, 1941, pp. 421-422.
 * Lohne, Johannes A., "Hooke versus Newton", Centaurus 7, pp. 6-52.
 * De Gandt, François, Force and Geometry in Newton's Principia, Princeton University Press, Princeton, 1995.
 * Weinstock, Robert, "Dismantling a Centuries-Old Myth: Newton's Principia and Inverse-Square Orbits", American Journal of Physics 50, 1982, pp. 610-617.

In Book I, Propositions 11, 12, and 13, Newton shows that motion along the three conic sections (hyperbola, parabola, ellipse/circle) about a fixed force center requires an inverse square force. Lestrade is partly right and partly wrong. When he says Elliptical orbits may possibly have inverse-square forces he is wrong; elliptical orbits must be caused by an inverse square law force, according to Newton. Lestrade is right when he says the converse does not follow; inverse square laws might, from these propositions, take other types of paths, and conic sections would be one subset. However, in Proposition 17 Newton shows that given an initial position and velocity with respect to an inverse square force center, only one conic path is specified.

Restating the argument:

A conic section is one possible path for a body attracted by an inverse square law. (Proposition 17)

A body attracted by an inverse square law has only one possible path, given its position and velocity at one time. (Proposition 42)

In Corollary I to Proposition 13 he states the conclusion:

A body attracted by an inverse square law must move along a conic section, given its position and velocity at one time.

To sum up, Newton proved that if you find an object moving along a conic section, an inverse square force is the only force that can be responsible. He ALSO proved that if you have an inverse square force, ONLY paths along conic sections are possible. This answers one of Lestrade's objections. See "Kepler's Laws and universal gravitation in Newton's Principia, James T. Cushing, American Journal of Physics 50 617, (1982).

The other is answered by noting that an "orbit" has to be an ellipse, because hyperbolas and parabolas are not closed paths, and that a circle is also an ellipse.

Showing that Newton was wrong about something, or left something undone, is a bee in the bonnets of a certain class of people. But there have been 400 years of work on the mechanics that Newton founded. If Newton had said everything there was to say, physicists would be out of a job.Shrikeangel (talk) 22:47, 24 March 2009 (UTC)


 * The problem is even stronger. The force on a body doesn't depend only on the geometrical shape of the path but also on the time schedule this path is absolved. Therefore the second law of Kepler is crucial to determine the forces. So take a look at my intervention concerning the second law of Kepler. It can't be proved with Newtons laws.


 * Another problem is the quantification of forces. How does Newton come to a quantification of forces? How does he know that forces are linear proportional to his geometric data and not logarithmic e.g.? The quantification of his centripetal forces is done by the quantification of corresponding centrifugal forces, well quantified by his predecessors (e.g. Huygens). The point is that the centrifugal forces are a condition for Newtons arguments.

ThvAq (talk) 10:05, 18 August 2015 (UTC)


 * ... and to be pedantic: Lestrade is completely right when he says Elliptical orbits may possibly have inverse-square forces. Only if the attractor is in the focus of the ellipse the inverse square law applies. If elliptical orbits of planets would have the sun in the center, the law of force would be 1/r, following to Newton. There are even other places of the attractor thinkable.

ThvAq (talk) 10:36, 18 August 2015 (UTC)


 * However, in Proposition 17 Newton shows that given an initial position and velocity with respect to an inverse square force center, only one conic path is specified.

That's exactly right. But the problem is that the inverse square law is not a law that's only founded on Newtons laws. Its main source is the centrifugal force F=m*v*v/r. So Newtons laws alone don't even discribe a conic path in this case. Only together with the law of the centrifugal force they can produce the inverse square law. And don't tell me the centrifugal force is a consequence of Newtons laws. It's the inverse. With the concrete experience of the realistic centrifugal force Newton found the concrete inverse square law from his abstract laws. The centrifugal law is a condition for the inverse square "law". ThvAq (talk) 11:59, 18 August 2015 (UTC)

The Converse Theorem
Robert Hooke, Christopher Wren, Edmund Halley, and Christiaan Huygens believed that there was a relationship between the inverse-square force law and elliptical orbits. Newton told Halley that he had proved that the inverse-square force resulted in an elliptical orbit. In his subsequent publications, however, Newton tried to prove the converse. That is, Newton tried to prove that if an orbit is an ellipse (or any other conic section), then the force is inverse-square. I find that I can't fit this fact into the article as it now exists.Lestrade 02:02, 18 February 2006 (UTC)Lestrade

There's no need, as Newton proved both the theorem and converse in Principia. There is no way to go back in time and find out what Newton proved and when exactly he proved it; i.e. if he told people he'd proved something and he hadn't, and rushed home to prove it that day and said he did it twenty years before, so I can't imagine this is a very important issue. I suspect that he had proved it using calculus to his satisfaction, and then when other people wanted to see his work, he had to figure out how to do it geometrically, because not everyone would have accepted a calculus-based proof. It doesn't matter. If he was lying about when he did it, no one now can prove it; but what he can prove is that he did prove both theorems in Principia, in Propositions 11,12,13,17 and 42.Shrikeangel (talk) 22:55, 24 March 2009 (UTC)

Newton's Geometry
Newton did not use Euclidean geometry. He used his own geometry. This was a geometry that investigated triangles that have curved, instead of straight, sides.Lestrade 02:07, 18 February 2006 (UTC)Lestrade

On what do you base this statement? In Principia I see diagrams of triangles with straight lines. I also see arcs. Perhaps you are confused by Newton's lemmas about the small angle approximation? There he shows what looks like triangles with one curved side, but he is just showing that x, sin x, and tan x are approximately equal for small angles.Shrikeangel (talk) 23:01, 24 March 2009 (UTC)


 * Very few diagrams in the Principia show rectilinear figures. Most figures are curvilinear and many show a triangle that has one side as a curve. This is because Newton was investigating curved orbits. In so doing, he was concerned with curved conics such as circles, ellipses, parabolas, hyperbolas, and spirals. Euclid's rectilinear plane geometry was inadequate for this task. Newton tried to use traditional geometry, but had to invent his own version. Dana Densmore, in her Newton's Principia: The Central Argument (ISBN 1-888009-24-1) stated that "Newton chose to express his proofs in an idiosyncratic modification of the classic geometry of the ancient Greeks" (p. xlvi). In his study of orbits, Newton's own geometry was used to study triangles that had one side as a curve or arc. In order to accomplish this, he used his mysterious concept of "ultimate limits." At the ultimate limit, a curve starts to become a straight line and can be considered as such. The lines retain their curved existence but can be considered as straight lines for purposes of calculation. Lemma 3, Corollary 4 shows how "these ultimate figures &hellip; are not rectilinear, but curvilinear limits of rectilinear figures." By reducing curved orbital lines to very small straight lines, Newton tried to use a kind of Euclidean geometry to solve his problems. This procedure is very awkward and difficult. Many intelligent people confess to being unable to follow Newton's mathematical proofs and, instead, merely accept the results of his propositions on his authority. During his investigations, Newton stopped using his idiosyncratic version of geometry and invented calculus. With calculus, a curve can be calculated as though it is a sum of very small straight tangential lines. That is the very essence and purpose of differential calculus and is the main reason why Newton developed his fluxions or calculus. When he wrote the Principia, however, he did not express his proofs by using his new calculus, Instead, he used his idiosyncratic version of traditional geometry, with its curved triangles and its ultimate limits.Lestrade (talk) 15:02, 25 March 2009 (UTC)Lestrade

Newton is using rectilinear geometry to approximate curved paths in the limit of small displacements. This was ancient and respectable; one of the first to use this approach was Archimedes. There is no physicist or mathematician working today who would call what Newton did "his own geometry". They would call it "first-order approximation". Since Newton also worked on infinite series this wasn't strange to him, or anyone. As far as limits go, they've been mathematically respectable for for several centuries now and are not a "mysterious concept" to physicists or mathematicians.

Many intelligent people confess to being unable to follow Newton's mathematical proofs and, instead, merely accept the results of his propositions on his authority. But where did Newton's "authority" come from? It came from being right. Mathematicians of his day had no trouble following the proofs. Modern physicists find them unfamiliar, but they are not difficult to understand if one is willing to put in the effort.Shrikeangel (talk) 07:36, 27 March 2009 (UTC)
 * "Idiosyncratic" means "peculiar to one individual." As Dana Densmore asserted, this is true of Newton's geometry. Regarding ultimate limits, Berkeley described their mystery. They are fictions that Newton used as a device for his purpose. I am not convinced that mathematicians of his day or of our day can follow his proofs with "no trouble." Regarding physicists, two Nobelists, Feynman and Chandrasekhar, struggled with Newton's proofs. Feynman admitted that he could not follow them, and Chandrasekhar tried to convert them to calculus in order to explain them because he couldn't clearly render them lucid. Of course, the old defense can be used that Newton was just smarter, so if people cannot follow his proofs, it is because they are just not intelligent enough to understand. That ruse never fails, but it also never leads to clear explanations. Feynman and Chandrasekhar were intelligent men, but they were not able to follow the proofs. In the end, we just accept or reject everything and simply move on to other topics. The problem is resolved by its disappearance, but no clarity is ever attained.Lestrade (talk) 13:47, 27 March 2009 (UTC)Lestrade

Nonsense, Lestrade. Archimedes and other ancient mathematicians used the concepts of limits and straight lines to approximate curves as you can discover here(Method of Exhaustion), so they are not "peculiar to Newton". Berkeley's objections to the concept of limits are three hundred years out of date--I might as well cite Galen against germs being caused by disease.

I am not convinced that mathematicians of his day or of our day can follow his proofs with "no trouble. I'm sure you're not, but neither was the Tortoise convinced by Achilles.  All of your objections to, and misunderstandings of, Newton have been answered.  I'm sure you can cite 50 more people who say they don't understand Newton's proofs, but in the sixteenth century people understood them perfectly well, which is why Newton gained the renown he did.  Lots of people don't understand things like evolution, or relativity, or quantum mechanics, but that doesn't mean they are not proven.Shrikeangel (talk) 00:32, 28 March 2009 (UTC)

Being unable to understand this is no big thing. It requires a deeper understanding of geometry than I have. Also, not everyone's brains work in the same way. Inability to understand someone else's complex non-standard math and logic doesn't mean much. Noghiri (talk) 18:22, 20 April 2010 (UTC)

Newton's Apple
"In the plague year of 1665, Newton had already experienced the famous revelation under an apple tree in Woolesthorpe, which led him to conclude that the strength of gravity falls off as the inverse square of the distance..."

This seems to refer to the apocryphal story of the apple falling from a tree. Although Newton told this in his old age (and by that time he has undergone a transformation - moving to London, becoming the master of the mint, MP and president of the RS which was an extraordinary change considering his earlier life). He spent time lying in the apple orchard in Woolesthorpe but I sincerely doubt that there was any falling apple that created a revelation in his mind. I don't believe he ever wrote anything in his notes to this effect (and they are copious) although many minor details are in those notes. To my mind, to say that an apple falling from a tree created the revelation of gravitational attraction holding the moon in orbit would be as amusingly laughable as claiming that had Newton seen a banana he would have been prompted to use the calculus to prove planetery orbits were eliptical! I think that Newton probably told this story to help others to understand some of the basic concepts of his works - remember that very few scientists could understand the Principia at all and so some assistance from the mundane would have been useful. By trivialising the actual thought behind his work he could assist others in making the leave to understand that there was no physical mechanism to hold the planets in orbit.

I think that perhaps there should be a page devoted to the story in both fact and fiction as it is of considerable interest. In this text the reference to the apple tree revelation needs to be struck. I also think that revelation may also need to be removed. Any comments?

Candy 03:23, 9 September 2006 (UTC)

The Mathematical Language
I'm putting this is a seperate section becasue I am trying to draw attention to the language used in "The reason for Newton's use of Euclidean geometry as the mathematical language of choice in Principia is puzzling in two respects. The first is the trouble that today's physicists, trained in modern analytical methods, face in following the arguments."

Does that sound very odd to anyone else. The reason for Newton's use ..... is puzzling to me implies that we do not understand why Newton did what he did clearly. However, then to mention that the trouble is how modern physicists find it difficult to follow the arguments is non sequitur. If one said, "Two puzzles are created from Newton's use of Euclidean geometry as the mathematical langauge of choice in Principia. The first is that today's .... I would be happier .. or am I missing a point here?

If I can also be so bold as to suggest why Newton used Euclidean geometry I would suspect that it is because like many "new fangled" developments sometimes we all have the potential to be intellectual luddites (just look at the number of scholars using pre-Rechtschreibung reform German regardless of the new rules) but more importantly Newton's work with Alchemy involved his belief that the Greek myths held coded secrets of the fundamental truths behind the structure of matter. With this in mind I can see why he used Euclidean geometry as it was closer to the "truth".

Candy 03:44, 9 September 2006 (UTC)

I don't that passage either. To me it feels derogatory, which is very presumptuous.

Clearly the author wishes to contrast Newton's geometric approach with more modern algebraic methods. This, I feel, is a valuable contribution. But the passage seems to suggest that algebra is in some way better than geometry and that Newton’s mathematics could do with some editing. Algebra is not superior to geometry. As Descartes’ demonstrates they are isomorphic. Further, Newton does use algebra in some parts of the principia. E.g. in book 2, prop 7, theorem 5, lemma 2.

Also, there is criticism of Newton’s using an approach similar to Euclid’s. But Euclid’s Elements is one of the greatest achievements of mankind. In the Element's Euclid uses as few and as simple definitions as possible to derive some magnificent theorems. e.g. Euclid 9.20 proves that there is no highest prime using only geometry (i.e. without using numbers).

Anyone familiar with The Elements reading the principia will recognise the same style used in the principia. Newton uses a few simple definitions and laws to derive such things as the forces with which the sun disturbs the motion of the moon (book 3 prop 25 problem 4)

Yes the principia is difficult to read. But its presentation is deliberate. Further, in my opinion Newton's proofs are as elegant as those of Archimedes. Newton was truly as great a mathematician as Mozart was a musician. To change the principia's geometric proofs to algebra would be like painting a moustache on the Mona Lisa.

Principia Discordia
Principia directs here? I was hoping for at least a disambiguation page, if it didn't go straight to the Principia Discordia. Mathiastck 13:21, 14 September 2006 (UTC)

Could someone experienced make a disambiguation? Noghiri (talk) 18:26, 20 April 2010 (UTC)

General Scholium
This article indicates General Scholium was attached to the second edition of Principia. However, the article on Hypotheses non fingo, which is linked to from the section on Hypotheses non fingo in this article, indicates it was the third edition. I have no information as to which is correct, I only noticed the disagreement. Djkeng65 21:49, 5 July 2007 (UTC)


 * This page is right: second edition. But why does it say the General Scholium attacks the Trinity? It certainly does not do so explicitly. I'm not a theologian but I can't see that any of the comments there would count as such an attack. And I would have thought that even a fairly subtle attack would have been very risky for a public official like Newton (this was published when he was Master of the Mint.) PaddyLeahy 02:17, 9 August 2007 (UTC)

I got this from here:. how about
 * Every soul that has perception is, though in different times and in different organs of sense and motion, still the same indivisible person. There are given successive parts in duration, co-existent parts in space, but neither the one nor the other in the person of a man, or his thinking principle; and much less can they be found in the thinking substance of God.

he is denying that the "substance of God" can be divided in "co-existent parts in space", which is tantamount to denying the possibility of God incarnate. dab (𒁳) 20:53, 17 August 2007 (UTC)


 * The natural reading of this quote is that Newton is defending himself from those who would claim that Newton's very literal interpretation of omnipresence implicity allows God to be separated into physical parts. A great deal of the Scholium (also Cotes' preface to the 2nd edition) appears to be targeted at Leibniz, and this question of the presence of God at each point in space was certainly an issue in the Leibniz-Clarke correspondence. Of course you may be right that this is a very subtle dig at the Trinity as well (your blog reference doesn't explicitly highlight this passage). But I'm sure this was not perceived as an attack on the Trinity at the time, except possibly by Newton's very close friends. Newton had enough enemies on both sides of the channel to cause a huge fuss if they had even a suspicion of this. So I think including the comment on the article page without qualification is very misleading. PaddyLeahy 21:30, 18 August 2007 (UTC)

1686 or 1687
On the first page of the book it says MDCLXXXVII (1687) but the article states the book is from 1686. Which on is it? Is the date in the book something else or is the article wrong? --131.174.17.112 (talk) 10:13, 3 March 2008 (UTC)
 * I. Bernard Cohen wrote that Principia was first published in 1687, but he gave no month or day.Lestrade (talk) 19:51, 3 March 2008 (UTC)Lestrade

Lestrade (talk) 20:03, 3 March 2008 (UTC)Lestrade The image of the book in Wikipedia says both "MDCLXXXVII" (1687) in the bottom and "Julii 5, 1686" in "IMPRIMATUR". Does the "July 5" comes from there? If so, should it be 1686? 189.5.141.84 (talk) 21:35, 29 October 2009 (UTC) Jaglion (talk) 19:15, 25 April 2015 (UTC) from what I could understand 5 july 1686 was the date that he got imprimatur: here . and probably 5 july 1687 was the date it was published.

WikiProject class rating
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 10:01, 10 November 2007 (UTC)

≈ —Preceding unsigned comment added by 71.111.147.232 (talk) 03:36, 23 January 2008 (UTC)

Basis for the refimprove?
Wouldn't specific fact checks be more appropriate here? 74.78.162.229 (talk) 00:24, 5 July 2008 (UTC)

Screwed up sentence
As of July 5 2008 there's an incomprehensible sentence in the section "Newton's Role"; looks like a cut-and-paste problem. Can't fix it, because I don't know what it's supposed to mean. Any ideas? The sentence is: "At this time, the central not the resulting theory of colours, to overwhelmingly favourable response, and a few inevitable scientific disputes with Robert Hooke and others, which forced him to sharpen his ideas to the point where he composed sections of his later book Opticks already by the 1670s." —Preceding unsigned comment added by 89.176.31.200 (talk) 06:27, 5 July 2008 (UTC)


 * I cleaned it based on commonly known facts. 74.78.162.229 (talk) 11:57, 5 July 2008 (UTC)

"Chronology of Ancient Kingdoms" as part of Principia
This might (or might not) seem a bit odd, but according to some sources The Chronology of Ancient Kingdoms was "... originally intended to be the last book of the Principia, The System of the World" according to these guys. Shouldn't this info be included somewhere in this article? The two works to me seem entirely disconnected, but perhaps not to Newton. --Trippz (talk) 08:50, 4 August 2008 (UTC)
 * [From Terry0051]Forgive me (and this response looks pretty late, but better late than never), but if you look at the semicolons and commas in your cited source, I think you'll see that what it means is that the originally-intended last book of (what became) the 'Principia' is the next, not the last, item in their list, i.e. they mean the book eventually published as 'System of the World'. This is now dealt with in the main article here, under the subsection "A preliminary version". Terry0051 (talk) 19:24, 1 October 2009 (UTC)
 * Ah, now makes more sense. I believe you are correct. Thanks for checking that out. --Trippz (talk) 20:09, 1 October 2009 (UTC)

Intro/Volumes
There were 3 books, but the article's intro says there were 2 volumes. Were the three books published as 2 volumes, or is it wrong? Have all three books also been subsequently published in a single volume? Jackd88 (talk) 23:14, 25 September 2009 (UTC)
 * [From Terry0051] There were lots of different editions in different numbers of 'n' physical volumes -- where 'n' ranges from 1 to 4, I think! (I've made an edit that looks to me consistent with the facts, E & OE.) Terry0051 (talk) 19:15, 1 October 2009 (UTC)

Ellipse theorem

 * (The following discussion, copied here in relevant part, took place on Reference desk/Mathematics starting 29 Sept 2009. It's copied here (in part) because it may indicate the content of a suitable explanatory comment that may be used to improve the main article. Terry0051 (talk) 19:13, 1 October 2009 (UTC))

In Proposition XI, Problem VI of the Principia (the inverse square law for ellipse focus), Newton refers to the 'writers on Conics' for proof of this property of the ellipse : all 'tangentially circumscribing' parallelograms have the same area.

I've tried Appolonius and Archimedes for this theorem but cannot find it there. Geometric proof anyone?

Dunloskinbeg (talk) 06:48, 29 September 2009 (UTC)


 * That doesn't sound right at all to me, are you sure you're quoting him right? A circle is an ellipse and you can get as large an area as you like with a rhombus round it. Dmcq (talk) 09:55, 29 September 2009 (UTC)
 * [ Comment agreeing that it doesn't sound right ][ ... ] Think of a unit circle $$x^2 + y^2 = 1$$ and a rhombus with vertices $$(\pm 1/\sin\alpha, 0) $$ and $$(0,\pm 1/\cos\alpha)$$ for some small alpha. --CiaPan (talk) 11:45, 29 September 2009 (UTC)

Hah! The explanation is in De motu corporum in gyrum article, first section 'Contents of "De Motu"', part 2 Lemmas:
 * All parallelograms touching a given ellipse (to be understood: at the end-points of conjugate diameters) are equal in area.

Conjugate diameters do not have their own article on Wikipedia nor in http://mathworld.wolfram.com/ but they are two such chords of an ellipse that each of them halves all chords parallel to the other. --CiaPan (talk) 12:41, 29 September 2009 (UTC)
 * Now they have, thanks to Jim.belk — see conjugate diameters --CiaPan (talk) 06:31, 1 October 2009 (UTC)
 * Now the proof is quite simple: an ellipse is an image of a circle in some affine transformation; a pair of conjugate diams is an image of a pair of perpendicular diams of the circle, and the parallelogram is an image of a square tangent to the circle. All such squares have equal area, and area ratios are preserved by affine transformation (as long as it's not degenerate, i.e. detA≠0), so all parallelograms considered have equal areas, too. --CiaPan (talk) 12:52, 29 September 2009 (UTC)

I've seen the phrase "Writers on X say..." or "Books on X say..." before, I think always in books written before 1900. Nowadays of course we use the much more self explanatory "It is well known that...".--RDBury (talk) 14:43, 29 September 2009 (UTC)
 * [ End of copied discussion ]

Copies
You can find the Latin version here: http://www.gutenberg.org/etext/28233  The English version is available via Bittorrent.--Noghiri (talk) 18:03, 16 April 2010 (UTC)

Latin, right?
I assume the book was written in Latin. The article does not explicitly say this. BigJim707 (talk) 13:33, 14 September 2011 (UTC)

Reference
Hoe about a reference to "On the Shoulders of Giants", Ed. Stephen Hawking, Running Press, ISBN 0-7624-1348-4 - Writings of Copernicus, Galileo, Kepler, Newton, Einstein. I've lost ready access myself, but it should be reasonably accessible and is in English. 94.30.84.71 (talk) 22:48, 12 December 2011 (UTC)

Linking to WikiSource instead of Google Books?
I love that this article links to the specific propositions and sections while describing the layout of the book. WikiSource has an excellent 1846 translation of the Principia, including scans of the diagrams. I think it'd be a great idea to replace the links to Google Books with links to this editions: as internal links they'd look cleaner on the page, and would provide support to WikiSource as well! -- Gaurav (talk) 09:04, 12 January 2012 (UTC)
 * Nobody has moved on this reasonable suggestion, presumably considering it too much work or not having enough knowledge of WikiSource citation. Instead, I added a book citation of the Google book English translation version of 1729 using Wikipedia citation tool for Google Books. The many plain links to this Google book in the article could be converted to page range references using this citation, for the first volume and equivalent ones for the later volumes, to be consistent with Wikipedia style and avoid linkrot. I tried User:Dispenser/Reflinks and this got rid of the bare links but I was unable to change the plain links possibly due to my lack of knowledge of this tool. I only wanted the basic book citation for my editing elsewhere and don't know how to do the page range references, but others are welcome to jump in. Puzl bustr (talk) 18:43, 16 March 2012 (UTC)
 * You manually edit the ‘page=X-Y’ field of the cite book citation to indicate page numbers. Or a shorter way is to just use to briefly indicate the page range referring to the main citation. Puzl bustr (talk) 19:12, 16 March 2012 (UTC)

External links banner
In July 2011 Yworo added the link warning banner external links with this edit summary: "There should not be text-linked external links in text, they should be moved to footnotes".

This might be persuasive if we were linking to external reference works, but here we are just trying to bring up the text section from Principia Mathematica that this article is commenting on at the time. If we require people to go through the reference section every time, then they need to make two clicks to bring up the text for review rather than one. In case a part of the objection is seeing the external link syntax scattered through the text, please see my proposed remedy in the section called Book 2. I edited that section to make the external links display differently. I used ... to make the external links display in the same style as internal ones.

I agree with User:Gaurav above that linking to WikiSource would be better, just because WikiSource provides the same translation with modern spelling, while keeping Newton's original diagrams. If anyone has another proposal for fixing the complaint about the external links, you can go ahead and revert my changes to the 'Book 2' section. Here is a link to WikiSource for Book 2 Section 1. Here is a link to the same section using Google Books. To appreciate the improved presentation in WikiSource, locate the sentence 'If a body is resisted in the ratio of its velocity..' in both editions.

If nobody disagrees with this, I'll eventually complete my change to the link syntax, switch to WikiSource, and then remove the external links banner. EdJohnston (talk) 02:27, 22 May 2012 (UTC)

New photos on Commons from the Royal Society Library
As part of WikiProject Royal Society a special photo session in the Royal Society Library in London has resulted in Commons:Category:Royal Society Library, with over 50 photos of their treasures, mostly 17th century manuscripts, including several of one of the early minute books, Boyle's notebooks etc, the manuscript fair copy of Newton's Principia etc. Please add these as appropriate. Thanks! Wiki at Royal Society John (talk) 22:05, 25 June 2014 (UTC)

The proof of the second law of Kepler is wrong
I can't hide that a proof of the second law of Kepler is not possible from Newtons laws. Conservation of angular momentum is typical for a central force field. Newtons laws don't demand a central force field. They are only about linear momentum.

The beautiful animated "proof" you give is not from Newton. Only the ideas of this proof are found in its works.

The proof is false. The linear movement in the animation is not perpendicular to the attraction of the sun. An acceleration in direction of the sun will take place. The secant of the second segment will not have the same length as the first one. — Preceding unsigned comment added by ThvAq (talk • contribs) 08:31, 18 August 2015 (UTC)

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Blunder?
Is this WP:original research (currently "citation" [81])

"Several commentators have followed Hooke in calling Newton's spiral path mistaken, or even a 'blunder', but there are also the following facts: (a) that Hooke left out of account Newton's specific statement that the motion resulted from dropping 'a heavy body suspended in the Air' (i.e. a resisting medium), see Newton to Hooke, 28 November 1679, document #236 at page 301, 'Correspondence', vol. 2 cited above, and compare Hooke's report to the Royal Society on 11 December 1679, where Hooke reported the matter 'supposing no resistance', see D Gjertsen, 'Newton Handbook' (1986), at page 259); and (b) that Hooke's reply of 9 December 1679 to Newton considered the cases of motion both with and without air resistance: The resistance-free path was what Hooke called an 'elliptueid'; but a line in Hooke's diagram showing the path for his case of air resistance was, though elongated, also another inward-spiralling path ending at the Earth's centre: Hooke wrote 'where the Medium ... has a power of impeding and destroying its motion the curve in wch it would move would be some what like the Line AIKLMNOP &c and ... would terminate in the center C'. Hooke's path including air resistance was therefore to this extent like Newton's (see 'Correspondence', vol. 2, cited above, at pages 304–306, document #237, with accompanying figure). The diagrams are also available online: see Curtis Wilson, chapter 13 in 'Planetary Astronomy from the Renaissance to the Rise of Astrophysics, Part A, Tycho Brahe to Newton', (Cambridge UP 1989), at page 241 showing Newton's 1679 diagram with spiral, and extract of his letter; also at page 242 showing Hooke's 1679 diagram including two paths, closed curve and spiral. Newton pointed out in his later correspondence over the priority claim that the descent in a spiral 'is true in a resisting medium such as our air is', see 'Correspondence', vol. 2 cited above, at page 433, document #286."

I can't find any commentators that use the term "blunder", let alone "several"? --𝕁𝕄𝔽 (talk) 20:56, 12 January 2024 (UTC)