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# Newton's Lemma 28

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### Dan Piponi

Mar 14, 2006, 6:42:20 PM3/14/06
to
According to "Abel's Proof" by P Pesic, Newton stated a lemma, lemma
28, in which he demonstrates that pi is a transcendental number, though
not using that terminology of course. The argument *seems* like
complete nonsense to me, but it may be that there is a reading of it
that can be turned into a valid proof. Pesic has published a paper on
the validity of this proof, but it appears I have to pay money to get
access to this paper. Does anyone know what Pesic's conclusions are in
that paper? Does anyone know a more rigorous statement of what Newton
was doing? Did Newton have a valid idea about how to prove the
transcendence of pi before Lindemann?

### Andrew Usher

Mar 14, 2006, 9:47:07 PM3/14/06
to

I remember reading this - was it in the 'Principia'? I will look.
I believe it sounded like it could be turned into a proof.

Andrew Usher

### Gerry Myerson

Mar 14, 2006, 10:51:08 PM3/14/06
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"Dan Piponi" <goog...@sigfpe.com> wrote:

> According to "Abel's Proof" by P Pesic, Newton stated a lemma, lemma
> 28, in which he demonstrates that pi is a transcendental number, though
> not using that terminology of course. The argument *seems* like
> complete nonsense to me, but it may be that there is a reading of it
> that can be turned into a valid proof. Pesic has published a paper on
> the validity of this proof, but it appears I have to pay money to get
> access to this paper. Does anyone know what Pesic's conclusions are in
> that paper?

You might have made it a bit easier for us by telling us where
and when Pesic published this paper. I believe you're referring to

Pesic, Peter
The validity of Newton's Lemma 28.
Historia Math. 28 (2001), no. 3, 215--219.

The review in Math Reviews (2002g:01006) goes like this:

In Lemma 28 in Book I of his Principia Newton gives a simple proof that
the areas of oval figures are not expressible in algebraic equations
with a finite number of terms.

This proof has been the subject of controversy since Newton's time. The
distinguished Newton scholar D. T. Whiteside offered a counterexample
which he considered decisive evidence of flaws in the lemma. But the
author observes that this counterexample is not infinitely smooth and
thus would not have been admitted by Newton.

After a careful analysis, the author concludes that "Lemma 28 shows the
deeper grounds of Newton's conscious reliance on the outward use of
geometry, not so much out of reverence for the ancients but in order to
encompass the infinite transcendence that Descartes could not grasp"
(p. 218), a point that would be interesting to develop in greater
extent.

Reviewed by Massimo Galuzzi

I tried to download the article from the web (the library has
a subscription) but got a message that the file was damaged
and could not be repaored.

Also of interest might be this:

(2002b:01023)
Pourciau, Bruce
The integrability of ovals: Newton's Lemma 28 and its counterexamples.
Arch. Hist. Exact Sci. 55 (2001), no. 5, 479--499.

Lemma 28 occurs in Section 6 of Book 1 of Newton's Philosophiae
naturalis principia mathematica (1687). Section 6 is devoted to the
so-called Kepler problem: the aim is to find a method for calculating
the area of a focal sector of the ellipse. This would enable one to
determine the position of a planet, provided that the orbit is assumed
to be an unperturbed ellipse. Newton found ways for approximating the
position, but could one solve the problem in finite terms? Lemma 28
gives very general reasons for believing that the planet's position
(even in the two-body problem) cannot be an algebraic function of time.
The lemma reads as follows: "No oval figure exists whose area, cut off
by straight lines at will, can in general be found by means of
equations finite in their number of terms and dimensions". Nowadays
this might perhaps read as: "no oval is algebraically integrable". The
statement is ambiguous, since it is not clear what Newton meant by
"oval figure". Lemma 28 aroused the attention and the criticisms of
Newton's contemporaries (most notably Christiaan Huygens and Gottfried
Wilhelm Leibniz) and several views have been held until recently.

In this delightful and clear paper Pourciau paves the way for an
understanding of Newton's lemma. Pourciau considers various readings of
it in dependence on what meaning one gives to the terms "oval" and
"integrable". He shows that the disagreements between the commentators
depend on the fact that they read Lemma 28 differently. The conclusion
reached is that, while Newton lacked rigor in his enunciation and
sketchy proof, his insight was profound and far ahead of his times.

Reviewed by Niccolò Guicciardini

--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

### Andrew Usher

Mar 15, 2006, 12:28:44 AM3/15/06
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Gerry Myerson wrote:

OK, this is good. I see by:

> Pesic, Peter
> The validity of Newton's Lemma 28.
> Historia Math. 28 (2001), no. 3, 215--219.

that it is from the Principia and read Newton's argument. I admit his
introducing this spiral is confusing, and his reference to the length
of the spiral is plain wrong (he must have meant distance from the
origin).

But in this source:

> (2002b:01023)
> Pourciau, Bruce
> The integrability of ovals: Newton's Lemma 28 and its counterexamples.
> Arch. Hist. Exact Sci. 55 (2001), no. 5, 479--499.

I read a more intelligible restatement of the theorem. It seems that
Newton's argument really does show that no 'isolated' algebraic curve
can be algebraically integrable. In particular this proves that pi is
transcendental.

Andrew Usher

### Andrew Usher

Mar 15, 2006, 12:30:32 AM3/15/06
to
Andrew Usher wrote:

> that it is from the Principia and read Newton's argument. I admit his
> introducing this spiral is confusing, and his reference to the length
> of the spiral is plain wrong (he must have meant distance from the
> origin).

Damn it, posted too soon. Seems Newton did not make this mistake,
don't know how I read it wrong ...

Andrew Usher

### Robert Israel

Mar 15, 2006, 1:30:38 AM3/15/06
to
Andrew Usher <k_over...@yahoo.com> wrote:

>> (2002b:01023)
>> Pourciau, Bruce
>> The integrability of ovals: Newton's Lemma 28 and its counterexamples.
>> Arch. Hist. Exact Sci. 55 (2001), no. 5, 479--499.
>
>I read a more intelligible restatement of the theorem. It seems that
>Newton's argument really does show that no 'isolated' algebraic curve
>can be algebraically integrable. In particular this proves that pi is
>transcendental.

I don't think so (although I haven't read that article).
As I understand it (from reading Arnol'd, "Huygens &
Barrow, Newton & Hooke"), Newton proves e.g. that the area cut off
from an algebraic oval by a vertical line is not an algebraic
function of the x coordinate of the line. But transcendental
functions can have algebraic values at particular points.
Arnol'd says:

The connection between the transcendency of functions and the
transcendency of numbers, to which Leibniz alluded in the last
cited letter to Huygens, is deeper than appears at first sight.
In modern times Leibniz's conjecture reads: an Abelian integral
along an algebraic curve with rational (algebraic) coefficients
taken between limits which are rational (algebraic) numbers is
generally a transcendental number. Unlike Hilbert's conjecture
on transcendental numbers, which has been proved by Gelfond,
this conjecture of Leibniz seems to be still unproved.

Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

### Dan Piponi

Mar 15, 2006, 5:08:15 PM3/15/06
to
Gerry Myerson said:

> The review in Math Reviews (2002g:01006) goes like this:

> In Lemma 28 in Book I of his Principia Newton gives a simple proof that...

Thanks for the detailed response!
--
Dan

### Gerry Myerson

Mar 15, 2006, 5:33:21 PM3/15/06
to
In article <dv8cae\$7vc\$1...@nntp.itservices.ubc.ca>,
isr...@math.ubc.ca (Robert Israel) wrote:

I read the Pourciau article (but not the Pesic article also
mentioned in this thread), hoping to see him discuss (or at
least mention) any relation to the transcendence of pi, but
that topic does not arise in the paper. If there were a close
relation, I think Pourciau would mention it, so I take its
absence as evidence for Robert Israel's position.

I recommend the Pourciau article - it's quite nice.

### Andrew Usher

Mar 16, 2006, 10:21:13 PM3/16/06
to
Robert Israel wrote:

> >I read a more intelligible restatement of the theorem. It seems that
> >Newton's argument really does show that no 'isolated' algebraic curve
> >can be algebraically integrable. In particular this proves that pi is
> >transcendental.
>
> I don't think so (although I haven't read that article).
> As I understand it (from reading Arnol'd, "Huygens &
> Barrow, Newton & Hooke"), Newton proves e.g. that the area cut off
> from an algebraic oval by a vertical line is not an algebraic
> function of the x coordinate of the line. But transcendental
> functions can have algebraic values at particular points.

Yes, right. Without Leibniz's conjecture, one can't prove that any
particular area is transcendental. The fact that the conjecture still
is open, though, indicates its truth.

Andrew Usher

### Robert Israel

Mar 17, 2006, 2:46:59 AM3/17/06
to

Indicates? Suggests, perhaps. There have been plenty of conjectures
that have turned out to be wrong. Not many that are this old,
though.

### Andrew Usher

Mar 17, 2006, 8:12:52 PM3/17/06
to
Robert Israel wrote:

> >Yes, right. Without Leibniz's conjecture, one can't prove that any
> >particular area is transcendental. The fact that the conjecture still
> >is open, though, indicates its truth.
>
> Indicates? Suggests, perhaps. There have been plenty of conjectures
> that have turned out to be wrong. Not many that are this old,
> though.

OK, 'very strongly suggests' (i.e. no one seriously believes it's
false).

Andrew Usher

### Andrew Usher

Mar 17, 2006, 8:32:48 PM3/17/06
to
Robert Israel wrote:

> >I read a more intelligible restatement of the theorem. It seems that
> >Newton's argument really does show that no 'isolated' algebraic curve
> >can be algebraically integrable. In particular this proves that pi is
> >transcendental.
>
> I don't think so (although I haven't read that article).
> As I understand it (from reading Arnol'd, "Huygens &
> Barrow, Newton & Hooke"), Newton proves e.g. that the area cut off
> from an algebraic oval by a vertical line is not an algebraic
> function of the x coordinate of the line. But transcendental
> functions can have algebraic values at particular points.

Actually, I don't think even that much is proved. Newton, by exhibiting
a particular case, shows that the curve is not in general algebraically
integrable. That itself does not imply that there is no family of lines
giving an algebraic function; in the nomenclature of the Pourciau
paper, there being no P such that P(a,b,c,S) = 0 does not imply that
there is not, for some fixed b and c, P such that P(a,S) = 0, right?

Andrew Usher

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