Proof that pi is transcendental
Escantidu
(Mathematische Annalen Band 43, a reprint of the original),
actually the first paper in his collected works,
titled "On the transcendence of e and pi".
On the same journal there appeared a variant of the proof
that e is transcendental, by Hurwitz.
Searching the net I have seen another variant, used by Niven
(but perhaps not originated by him). All the variants are pretty similar:
Hilbert estimates integrals, Hurwitz estimates derivatives, and Niven
also does something very similar.
(1) Does anyone know what is the origin of the third variant?
(2) Does anyone know of any other variants to the Hilbert proof?
(3) Does anyone know of another proof (Hermite-Lindemann theorem
does not count!)?
If anyone is interested, I made a transcript of the Hilbert and Hurwitz
papers in LaTeX... they are in German though (how unfortunate it is not to
know German!)
In a quite different vain, does anyone know of an interesting, elementary
paper by some famous (Italian) mathematician, in Italian? Oddly enough,
the paper where Peano describes his axioms, dating from the beginning of
the 20th century, is in Latin!
Bennett Standeven
> A month ago I have read a paper of Hilbert dating 1893
> (Mathematische Annalen Band 43, a reprint of the original),
> actually the first paper in his collected works,
> titled "On the transcendence of e and pi".
> On the same journal there appeared a variant of the proof
> that e is transcendental, by Hurwitz.
> Searching the net I have seen another variant, used by Niven
> (but perhaps not originated by him). All the variants are pretty similar:
> Hilbert estimates integrals, Hurwitz estimates derivatives, and Niven
> also does something very similar.
>
> (1) Does anyone know what is the origin of the third variant?
> (2) Does anyone know of any other variants to the Hilbert proof?
> (3) Does anyone know of another proof (Hermite-Lindemann theorem
> does not count!)?
>
> If anyone is interested, I made a transcript of the Hilbert and Hurwitz
> papers in LaTeX... they are in German though (how unfortunate it is not to
> know German!)
>
> In a quite different vain, does anyone know of an interesting, elementary
> paper by some famous (Italian) mathematician, in Italian?
Does Galileo count? I'm pretty sure he was writing in Italian.
David C. Ullrich
Standeven) wrote:
[...]
>>
>> In a quite different vain, does anyone know of an interesting, elementary
>> paper by some famous (Italian) mathematician, in Italian?
>
>Does Galileo count? I'm pretty sure he was writing in Italian.
Do you have any reason to be sure of this? My guess would have
been Latin.
David C. Ullrich
*********************
"Sometimes you can have access violations all the
time and the program still works." (Michael Caracena,
comp.lang.pascal.delphi.misc 5/1/01)
Herman Rubin
David C. Ullrich <ull...@math.okstate.edu> wrote:
>On 16 Jun 2001 14:08:38 -0700, be...@pop.networkusa.net (Bennett
>Standeven) wrote:
>[...]
>>> In a quite different vain, does anyone know of an interesting, elementary
>>> paper by some famous (Italian) mathematician, in Italian?
>>Does Galileo count? I'm pretty sure he was writing in Italian.
>Do you have any reason to be sure of this? My guess would have
>been Latin.
Galileo wrote much in Italian; he DELIBERATELY did this
whenever possible.
Italian was considered a major mathematical language for
quite some time. Latin pretty much died out in the early
19th century.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Robert J. Kolker
"David C. Ullrich" wrote:
>
> Do you have any reason to be sure of this? My guess would have
> been Latin.
Read Gallileo's two Dialogues translated from the Italian. They
are a good read in English as well. Gallileo was a very witty
writer and he had good fun making fun of the Pope.
One of the reasons why G. got in trouble with the Church is
that he did write in the vernacular.
Strangely though, Newton wrote in Latin.
Bob Kolker
David C. Ullrich
Rubin) wrote:
>In article <3b2cab00...@nntp.sprynet.com>,
>David C. Ullrich <ull...@math.okstate.edu> wrote:
>>On 16 Jun 2001 14:08:38 -0700, be...@pop.networkusa.net (Bennett
>>Standeven) wrote:
>
>>[...]
>
>>>> In a quite different vain, does anyone know of an interesting, elementary
>>>> paper by some famous (Italian) mathematician, in Italian?
>
>>>Does Galileo count? I'm pretty sure he was writing in Italian.
>
>>Do you have any reason to be sure of this? My guess would have
>>been Latin.
>
>Galileo wrote much in Italian; he DELIBERATELY did this
>whenever possible.
Oh.
>Italian was considered a major mathematical language for
>quite some time. Latin pretty much died out in the early
>19th century.
>
>
>--
>This address is for information only. I do not claim that these views
>are those of the Statistics Department or of Purdue University.
>Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
>hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
David C. Ullrich
Dave L. Renfro
[sci.math 15 Jun 2001 13:39:54 -0700]
<http://forum.swarthmore.edu/epigone/sci.math/fawharbin>
wrote (in part):
<< Questions about different proofs that Pi is transcendental. >>
> In a quite different vain, does anyone know of an interesting,
> elementary paper by some famous (Italian) mathematician, in
> Italian? Oddly enough, the paper where Peano describes his
> axioms, dating from the beginning of the 20th century, is in Latin!
Regarding your questions about various ways of proving that Pi is
transcendental, you might want to look at the references I give
in these posts --->>>
IRRATIONALITY AND TRANSCENDENCE OF e AND Pi [May 21, 2000; Updated
into Parts A and B on March 25, 2001]
<http://forum.swarthmore.edu/epigone/sci.math/zhingpheldkoi>
Regarding Italian papers, you might want to try looking up some
famous Italian mathematicians at
<http://www.emis.de/MATH/JFM/JFM.html>.
Here are some names that you can try: Peano, Volterra, Dini,
Betti, Ascoli, Ricci, Bianchi, Arzela, Tonelli. The reviews are
in German, which you said you can't read. Neither can I, but
this will help --->>> <http://translation.langenberg.com/>.
Off-hand, one paper I know that's fairly elementary and interesting
(the latter to me, at least) that's in Italian is
Giulio Vivanti, "Sugli aggregati perfetti", Rend. del Circ. Mat.
di Palermo 13 (1899), 86-88.
Vivanti proves that the Cantor-Bendixson decompostion of a closed
set into a perfect set and a countable set is unique. Cantor and
Bendixson only proved (in 1882) that such a decomposition exists.
Incidentally, not every countable set arises in such a
decomposition. The sets that do arise are the scattered sets.
Among the various ways to characterize scattered sets (one of
which is being a countable G_delta set), here's an interesting
one, due to Denjoy I think -- A set is scattered if and only if it
is nowhere dense in every nonempty perfect set. Equivalently, it
fails to be dense in every portion of every perfect set. ["Portion"
means the intersection of the set with an open interval.] This
means that scattered sets are extremely sparse. When a set fails
to be dense in every portion of every interval (yes, I know the
"every portion" part here is redundant; I'm using this to help
show a parallel between two conditions), it's nowhere dense.
Nowhere dense sets are usually considered to be fairly sparse.
However, scattered sets not only fail to be dense in every portion
of every interval, they also fail to be dense in every portion of
sets such as the Cantor middle thirds set, Cantor sets with Hausdorff
dimension zero, etc. In other words, they're so sparse that they're
like spider webs even when compared to the thinnest Cantor sets
that you can come up with.
Here are two Italian papers where Volterra independently (of Cantor,
Smith, and a few others) came up with constructions of Cantor sets
having positive measure. Volterra was an undergraduate (age 20 or
so) when he wrote these papers, by the way.
Vito Volterra, "Alcune osservazioni sulle funzioni punteggiate
discontinue", Giornale di Matematiche 19 (1881), 76-86. [Reprinted
on pp. 7-15 of Volterra's "Opere Matematiche" 1 (1954).]
Vito Volterra, "Sui principii del calcolo integrale", Giornale di
Matematiche 19 (1881), 333-372. [Reprinted on pp. 16-48 of
Volterra's "Opere Matematiche" 1 (1954).]
Dave L. Renfro
James Hunter
Herman Rubin wrote:
> In article <3b2cab00...@nntp.sprynet.com>,
> David C. Ullrich <ull...@math.okstate.edu> wrote:
> >On 16 Jun 2001 14:08:38 -0700, be...@pop.networkusa.net (Bennett
> >Standeven) wrote:
>
> >[...]
>
> >>> In a quite different vain, does anyone know of an interesting, elementary
> >>> paper by some famous (Italian) mathematician, in Italian?
>
> >>Does Galileo count? I'm pretty sure he was writing in Italian.
>
> >Do you have any reason to be sure of this? My guess would have
> >been Latin.
>
> Galileo wrote much in Italian; he DELIBERATELY did this
> whenever possible.
>
> Italian was considered a major mathematical language for
> quite some time. Latin pretty much died out in the early
> 19th century.
Those are days are gone too though.
If you can't speak computers and robots , you can't speak the lingo.
You're just praying to some nonexistent mongoloid for 3.14 .... to be
really, really, really, really, really ..... real.
James Hunter
"Robert J. Kolker" wrote:
He didn't have much choice at that time, they wouldn't
have been accepted almost anywhere, if he wrote in Olde English.
The Scarlet Manuka
news:3B2D437C...@Jhuapl.edu...
> "Robert J. Kolker" wrote:
> > Strangely though, Newton wrote in Latin.
> He didn't have much choice at that time, they wouldn't
> have been accepted almost anywhere, if he wrote in Olde English.
In fact, Newton wrote in both Latin and English.
IIRC he published some of his later work in English,
notably _Opticks_, so that it would not have much
circulation on the Continent; by this means he
hoped to avoid the type of controversy that had
accompanied some of his earlier publications.
--
The Scarlet Manuka