Talk:Philosophiae Naturalis Principia Mathematica

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Contents 1 Location of Copies 2 Physics Today 3 Euclid, Descartes and Newton 4 More on Euclid, Descartes, and Newton 5 Additions and removal 6 On "I feign no hypotheses" 7 What is it 8 Absence of Proof 9 The Converse Theorem 10 Newton's Geometry 11 Newton's Apple 12 The Mathematical Language 13 Principia Discordia

[edit] 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 [1]. Indeed, the good online version of the exhibition includes a photo of a better page [2] 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 [3]. 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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 —
  • 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)

[edit] 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)

[edit] 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 — now reverted. -- Solipsist 09:02, 13 December 2005 (UTC)

is this confusing, or should i read this book?

[edit] 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:

• 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.

Lestrade 01:52, 6 February 2006 (UTC)Lestrade

• 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.

Lestrade 12:53, 6 February 2006 (UTC)Lestrade

[edit] 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

[edit] 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

[edit] 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)

[edit] 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.

[edit] 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)