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HISTORIA MATEMATICA Sunday, April 9 2000 Volume 02 : Number 012
~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE of SUBJECTS ~~~~~~~~~~~~~~~~~~~~~~~~~~
Re: [HM] Bezier curves
Re: [HM] Bourbaki, theory, and problems
Re: [HM] Bourbaki, theory, and problems
[HM] Geometry and Algebra in ancient civilizations
Re: [HM] The Unfortunate Wallace!
Re: [HM] Fundamental Theorem of Algebra
Re: [HM] nabla origin
Re: [HM] Fundamental Theorem of Algebra
Re: [HM] Inductive proof of unique factorization
Re: [HM] Fundamental Theorem of Algebra
Please see the end of this digest.
----------------------------------------------------------------------
Date: Sat, 08 Apr 2000 10:50:54 +0200
From: Roger CUCULIERE
Subject: Re: [HM] Bezier curves
Voici deux documents au sujet de Pierre B/ezier.
Cordialement,
Roger CUCULIERE
35, Rue d'Alsace
92110 CLICHY
FRANCE
Tel. (33) 01 47 39 23 14
cuculier@imaginet.fr
****************************
Date : mardi 7 decembre 1999 08:27
Objet : Pierre BEZIER
Pierre BEZIER died on November 25, 1999.
The following short biography has been published in volume 22, number 9
(november 1990) of "Computer Aided Design", a special issue devoted to
Bezier techniques.
Pierre Etienne Bezier was born on September 1, 1910 in Paris. Son and
grandson of engineers, he chose this profession too and enrolled to study
mechanical engineering at the Ecole des Arts et Metiers and received his
degree in 1930. In the same year he entered the Ecole Superieure
d'Electricite and earnt a second degree in electrical engineering in 1931.
In 1977, 46 years later, he received his DSc degree in mathematics from the
University of Paris.
In 1933, aged 23, Bezier entered Renault and worked for this company for 42
years. He started as Tool Setter, became Tool Designer in 1934 and Head of
the Tool Design Office in 1945. In 1948, as Director of Production
Engineering he was responsable for the design of the transfer lines
producing most of the 4 CV mechanical parts. In 1957, he became Director of
Machine Tool Division and was responsable for the automatic assembly of
mechanical components, and for the design and production of an NC drilling
and milling machine, most probably one of the first machines in Europe.
Bezier become managing staff member for technical development in 1960 and
held this position until 1975 when he retired.
Bezier started his research in CADCAM in 1960 when he devoted a substantial
amount of his time working on his UNISURF system. From 1960, his research
interest focused on drawing machines, computer control, interactive
free-form curve and surface design and 3D milling for manufactoring clay
models and masters. His system was launched in 1968 and has been in full
use since 1975 supporting about 1500 staff members today.
Bezier's academic career began in 1968 when he became Professor of
Production Engineering at the Conservatoire National des Arts et Metiers.
He held this position until 1979. He wrote four books, numerous papers and
received several distinctions including the "Steven Anson Coons" of the
Association for Computing Machinery and the "Doctor Honoris Causa" of the
Technical University Berlin. He is an honorary member of the American
Society of Mechanical Engineers and of the Societe Belge des Mecaniciens,
ex-president of the Societe des Ingenieurs et Scientifiques de France,
Societe des Ingenieurs Arts et Metiers, and he was one of the first
Advisory Editors of "Computer-Aided Design".
+-----------------------------------------------------------+
* Pierre-Jean LAURENT *
* LMC-IMAG, Universite Joseph Fourier *
* F-38041 GRENOBLE cedex 09 (France) *
* Phone: 33 4 76 51 46 11, Fax: 33 4 76 63 12 63 *
* Email: pjl@imag.fr or Pierre-Jean.Laurent@imag.fr *
* http://www-lmc.imag.fr/MGA/ *
+-----------------------------------------------------------+
****************************
Du quotidien franc,ais "Le Monde", 10 decembre 1999
DISPARITIONS
Pierre Bezier
Le concepteur de la representation numerique des formes complexes.
Ingenieur des Arts et Metiers, Pierre Bezier est mort, jeudi 25 novembre a
Gallardon (Eure-et-Loir). Ne le 11 septembre 1910 a Paris, Pierre Bezier
est celebre dans le monde entier pour les courbes mathematiques qui portent
son nom. Diplome des Arts et Metiers en 1930, il passe par l'Ecole
superieure d'electricite avant d'obtenir un doctorat en mathematiques.
En 1933, il entre chez Renault comme ajusteur-outilleur. Il restera
quarante-trois ans chez le constructeur automobile et y fera une carriere
remarquable qui le conduira, au debut des annees 60, a concevoir une
representation numerique des formes complexes (courbes et arrondis) qui
donnent aux voitures leur style et leur aerodynamisme.
Avant lui, la realisation de l'outillage des presses qui emboutissent
les toles de carrosserie restait entierement manuelle. Maquettes, gabarits,
modeles et moulages conduisaient a l'usinage des outillages sur des
machines a copier. Un processus laborieux qui introduisait une serie
d'erreurs rendant souvent delicat l'assemblage final des pieces de
carrosserie. Il manquait une reference, un modele mathematique
indiscutable auquel se referer. Bezier a apporte ce precieux outil. Ces
courbes ont marque le point de depart d'une nouvelle technique, la
conception et fabrication assistee par ordinateur (CFAO) qui allait
revolutionner l'industrie manufacturiere, de l'automobile a l'aeronautique
(logiciel Catia) en passant par l'ensemble des entreprises de mecanique
et de tolerie. Aujourd'hui, la plupart des logiciels de dessin utilisent
les courbes de Bezier qui definissent egalement les Polices de caracteres
vectorielles (Postcript) des imprimantes.
Mathematiquement, "ce ne fut pas particulierement genial", confiait
Pierre Bezier a Philippe Grange dans le numero special sur "40 ans
d'innovation", publie par la revue Industries et Techniques en 1998.
Cette modestie cache une remarquable demarche d'ingenieur qui a
parfaitement concilie l'analyse d'un probleme de fabrication industrielle,
l'exploitation de connaissances theoriques et le recours a un tout nouvel
outil pour l'epoque : l'ordinateur. Un modele de processus d'innovation.
Il faut dire que Pierre Bezier n'en etait pas a son premier succes. Des
1945, il avait introduit dans les ateliers les machines transferts,
monstres de mecanique de precision et d'automatisation. De quoi produire
quotidiennement trois cents 4 CV en 1948 et mille Dauphine en 1956.
L'ingenieur fit ensuite partie des pionniers des machines a commande
numerique.
En 1970, raconte Pierre Bezier a Philippe Grange, une reunion de trente
directeurs devait prendre une decision sur l'utilisation de la CFAO chez
Renault. Lors du vote, le "Oui" l'emporta par deux voix contre
vingt-huit abstentions. C'est dire la fragilite des inventions, quelle
que soit la dose de genie quelles contiennent.
Michel Alberganti
------------------------------
Date: Sat, 08 Apr 2000 12:34:46 -0300
From: Julio Gonzalez Cabillon
Subject: Re: [HM] Bourbaki, theory, and problems
At 07:23 p.m. 07/04/00, Colin McLarty wrote:
|
| Christian Houzel (as quoted by David Aubin) found that mathematics
| in the the 1970s was "characterized by a tendency to revive an old interest
| in more concrete problems" rather than the more abstract large scale
| theoretical apparatus of the 50s and 60s.
|
| Volker Eisermann wrote:
| >
| > As I have a similar impression as Houzel (maybe I would date the tendency
| > back to the more concrete a little bit later), I am very interested in
| > the question, how to make a statement like this more precise and how
| > to collect evidence for (or against) it.
|
| I think that actually no such tendency has ever existed (at least not
| in the 20th century). Individual mathematicians shift their attention from
| one thing to another. Yet the pace of abstraction has never lessened on any
| large scale, and concrete problems have always gotten plenty of attention.
|
First of all, many thanks to Volker and Colin for their thoughtful posts. In
short, I find myself in tune with most of their points and concerns. Despite
"the pace of abstraction has never lessened on any large scale, and concrete
problems have always gotten plenty of attention", I would like to point out
that this has not been as such outside the _math factory_. J.Dieudonne et al
had a tremendous impact on the transmission and reception of mathematics for
a long time, which changed curricula at schools - at all levels - during the
60s and 70s (and also 80s here in UY). This minor (?) aspect should be taken
into consideration (at least tangentially) in this discussion.
Kind regards,
Julio Gonzalez Cabillon
------------------------------
Date: Sat, 08 Apr 2000 08:52:20 -0700
From: David Stump
Subject: Re: [HM] Bourbaki, theory, and problems
It seems to me that the following discussion raises a general and very
important question, namely how do we as historians determine the "major
trends" of mathematics.
Do we look at research, at teaching in graduate schools, undergraduate
schools, mathematical associations, journals, all of the above? What
is mathematics, in the phrase 'history of mathematics'?
David Stump
At 12:34 PM 4/8/00 -0300, Julio Gonzalez Cabillon wrote:
>
snip
>
> First of all, many thanks to Volker and Colin for their thoughtful
> posts. In short, I find myself in tune with most of their points
> and concerns. Despite "the pace of abstraction has never lessened
> on any large scale, and concrete problems have always gotten plenty
> of attention", I would like to point out that this has not been as
> such outside the _math factory_. J.Dieudonne et al had a tremendous
> impact on the transmission and reception of mathematics for a long
> time, which changed curricula at schools -- at all levels -- during
> the 60s and 70s (and also 80s here in UY). This minor (?) aspect
> should be taken into consideration ( at least tangentially) in this
> discussion.
David J. Stump voice: (415) 422-6153
Department of Philosophy FAX: (415) 422-5356
University of San Francisco email: stumpd@usfca.edu
2130 Fulton Street
San Francisco, CA 94117-1080 USA
------------------------------
Date: Sat, 08 Apr 2000 13:04:05 -0300
From: Julio Gonzalez Cabillon
Subject: [HM] Geometry and Algebra in ancient civilizations
At 05:30 p.m. 07/04/00 +0000, M.Carmen Hernandez-Martin typed:
| ...
| Somebody (I can no remember who) wrote a paper in "Archives for
| History of Exact Sciences" defending the need of rewriting the history
| of Greek mathematics ...
Carmen, you're referring to the controversial paper "On the Need to Rewrite
the History of Greek Mathematics" [_Archive for History of Exact Sciences_
vol 15 (1975/76), no. 1, 67-114] written by listmember Sabetai Unguru. He
may speak up for himself.
Afectuosamente, Julio GC
------------------------------
Date: Sat, 08 Apr 2000 13:22:44 -0300
From: Julio Gonzalez Cabillon
Subject: Re: [HM] The Unfortunate Wallace!
|
| > Wallace's work was on geometry and Simson's line (which is definitely
| > not due to Simson!) appears first in a paper of Wallace in 1799.
|
| I'd appreciate very much a precise reference to this paper.
|
See Leybourn's _Mathematical Repository_ [ first series, vol II (1799)
page 111 ]. I just found this reference in one of my notes, but I cannot
locate right now my Xerox copy of the referenced paper.
Julio Gonzalez Cabillon
------------------------------
Date: Sat, 8 Apr 2000 14:05:31 -0400 (EDT)
From: John Conway
Subject: Re: [HM] Fundamental Theorem of Algebra
The simplest proof of the "fundamental theorem" has always
been this:
1) prove that for any real C we can find an R such that
|f(z)| > C for |z| > R.
{This is easy - if the polynomial is
z^n + az^n-1 + ... + k
then if |z| > C( 1 + |1| + ... + |k|) I think we're OK.
2) So the infimum of |f(z)| over the whole plane equals
that over some closed disc |z| =< R.
3) Prove or quote that the infimum of a continuous function
over a closed and bounded set is attained. [The most
easily-comprehended proof is by the bisection method
of the Bolzano-Weierstrass theorem.]
4) So we can suppose that there's a particular value Z
of z at which |f(z)| is minimal. Shift to make Z = 0.
We now want to prove f(0) = 0.
5) If not, we can suppose f(0) = 1, so f looks like
1 + az + bz^2 + ... .
Now if only |z| is sufficiently small we have
|bz^2 + ... | < |az|/2.
6) Now by chose of arg(z) we can make az be negative real,
and then if it's small enough, it's easy using 5) to
see that | 1 + az + bz^2 + ... | < 1, contradicting
our assumption.
John Conway
------------------------------
Date: Sat, 8 Apr 2000 14:17:02 -0400
From: "Randy K. Schwartz"
Subject: Re: [HM] nabla origin
Steven Schwartzman, on p. 142 of his book _The Words of Mathematics_
(MAA, 1994), states that the Irish mathematician William Rowan Hamilton
called the inverted uppercase delta symbol (the symbol he chose for that
operator which yields the vector gradient of a real-valued multivariable
function) the "nabla" because the symbol resembled an ancient Hebrew
stringed instrument called nabla. The instrument is or was believed to have
been a type of harp. Schwartzman adds that "nabla" was the Greek form of a
Hebrew word that was likely of Phoenician origin, and that the word is
sometimes rendered in English as "nable" or "nebel."
Jeff Miller, in his website "Earliest Uses of Various Mathematical
Symbols" (http://members.aol.com/jeff570/mathsym.html), states the following
(as slightly edited by me):
Nabla (the Hamiltonian operator), which is also called a "del" or "atled"
(delta spelled backwards), was introduced by William Rowan Hamilton
(1805-1865) in 1853 in _Lectures on Quaternions_, according to Florian
Cajori in his _A History of Mathematical Notations_ (vol. 2, page 135).
David Wilkins has found the symbol used earlier by Hamilton in the
_Proceedings of the Royal Irish Academy_ of the meeting held on July 20,
1846. The volume appeared in 1847. However the symbol is rotated 90 degrees.
According to Sherman K. Stein and Anthony Barcellos on page 836 of their
_Calculus and Analytic Geometry_ (McGraw Hill, Inc., 1992), Hamilton denoted
the gradient with an ordinary capital delta in 1846.
=============================================
Prof. Randy K. Schwartz
Department of Mathematics
Liberal Arts Building
Schoolcraft College
18600 Haggerty Road
Livonia, MI 48152-2696 USA
email rschwart@schoolcraft.cc.mi.us
voice 734/462-4400 extn. 5290
fax 734/462-4558
"In the Inn of the world there is room for
_everyone_. To turn your back on even one
person, for whatever reason, is to run
the risk of losing the central piece of
your jigsaw puzzle." - Leo F. Buscaglia
=============================================
------------------------------
Date: Sat, 08 Apr 2000 15:54:36 -0300
From: Julio Gonzalez Cabillon
Subject: Re: [HM] Fundamental Theorem of Algebra
At 11:50 a.m. 07/04/00 -0700, John Dawson typed:
|
| Is there a proof of the fundamental theorem of algebra that can be
| understood by students with only a calculus background? ...
For a handy and good reference see Michael Spivak's "Calculus".
Kind regards, Julio GC
------------------------------
Date: Sat, 8 Apr 2000 22:03:44 +0200 (MEST)
From: Walter Felscher
Subject: Re: [HM] Inductive proof of unique factorization
There are at least two indirect proofs for the uniqueness of the prime
factor decomposition. They both can be found in Arnold Scholz:
Einf"uhrung in die Zahlentheorie, De Gruyter, reprinted 1945 .
The first is said to be by Zermelo und runs as follows. Let m be the
smallest number for which the decomposition is not unique. Let p be the
smallest prime factor of m ; then (1) m/p = h has a unique decomposition
as it is smaller than m , and m = ph . By assumption there must be
a second, different decomposition of m , and among the prime factors of
that there is a smallest one which I write q . Hence m = qk , and q is
different from p , for otherwise also h=k which by (1) has a unique
decomposition. Hence q>p by minimality of p , and n = m - pk = qk - pk
= (q-p)k is positive and smaller than m . Thus n has a unique
decomposition, and as p divides n = ph - pk = p(h-k) = (q-p)k , now p
occurs among the prime factors of (q-p) or of k . But by choice of q ,
the prime factors in k are larger than q , hence larger than p , and
so p cannot divide k but must divide q-p . So q-p = pr , hence q =
(p+1)r whence q is not prime !
The second is said to be due to G.Klappauf and runs as follows. Let m
the smallest number with two different prime factor decompositions
m = pp' ... p" = qq' ... q" . It follows from this minimality that none
of the p is one of the q and vice versa. There is a smallest of the
numbers p, ... and q, ... , and I may assume that it is q . Now I
divide with remainder each of p by that one of the q : p = aq + r ,
p' = a'q + r' , ... , p" = a" + a"q + r" . As q now is smaller than
every p , the a, a', ... a" all are positive, and as q does not divide
any p also the r, r', ... r" are positive. For the product m of the q ,
the multiplication of these expressions gives a result m = Aq + rr'... r"
where A and R = rr' ... r" are positive, hence also R < m . Now q
divides R while R = rr' ... r" is a product of factors each of which
is below r . Hence R , although smaller than m , does not have a
unique decomposition !
In this second proof, the reduction step from m to R reaches farther
down than in the first proof the step from m to n .
W.F.
------------------------------
Date: Sun, 9 Apr 2000 03:41:04 +0200 (IST)
From: Avinoam Mann
Subject: Re: [HM] Fundamental Theorem of Algebra
However, the proof to which Hans refers below needs the fact that any
(non-constant) polynomial has roots in some extension field of the
rationals. To prove that one needs to develop a few properties of
polynomial rings, so it is not as elementary as was asked for.
Avinoam Mann
On Sat, 8 Apr 2000, Hans Lausch wrote:
>
> One of Gauss's proofs, the "most algebraic" one, uses induction on the
> exponent m of 2, where 2^m is the largest 2-power dividing the degree of
> the polynomial (which can be assumed to have no multiple roots. For m=0,
> the start of the induction, one invokes the intermediate value theorem,
> which remains the only "calculus" tool used in the proof.
------------------------------
End of HISTORIA MATEMATICA V2 #12
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HISTORIA MATEMATICA Monday, April 10 2000 Volume 02 : Number 013
~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE of SUBJECTS ~~~~~~~~~~~~~~~~~~~~~~~~~~
Re: [HM] Fundamental Theorem of Algebra
Re: [HM] Matematica precolombina
Re: [HM] Inductive proof of unique factorization
Re: [HM] Mathematics during the Roman Empire
Re: [HM] "maths" in lieu of "math"
[HM] Mathematics and Music
[HM] HPM Gathering in Chicago
Re: [HM] mystery-woman
Re: [HM] Fundamental Theorem of Algebra
Please see the end of this digest.
----------------------------------------------------------------------
Date: Sun, 9 Apr 2000 14:12:45 +0200 (GMT)
From: Lambrou Michael
Subject: Re: [HM] Fundamental Theorem of Algebra
The following question appeared in HM.
>
> Date: Fri, 07 Apr 2000 11:50:53 -0700
> From: John Dawson
> Subject: [HM] Fundamental Theorem of Algebra
>
> Is there a proof of the fundamental theorem of algebra that can be
> understood by students with only a calculus background?
> ...
Here is a summary of a proof I give my students, many years now. It is
a trivial modification of one of Gauss' proofs.
Let me start with Gauss' proof refered too:
If the polynomial p (of a complex variable z = r*e^(iq), where I denote
theta by q) did not have a root then the two quantities
A=the real part of p and B=the imaginary part of p
would not be simultaneously zero. Express A and B in terms of r and q.
Note that the (rational function in r, q) C = 1/(A^2 + B^2) would be
well defined. Then Gauss considers the double integral
/ /
| | C drdq
/ /
for appropriate range of values of r and q (I don't remember them now
but I think they are 0 < r < 1, 0 < q < 2(pi) (where pi = 3,14159...).
He evaluates this integral in two different ways (first way is by
doing doing the r integration first, and the second is by reversing the
order of integration and doing q first). He finds two different values
for the integral (namely 0 and 1 respectively) from where he infers the
contradiction.
All this is, of course, way out of beginner calculus students.
Here is how to trivially modify this and make it student level:
The idea is that integrals are approximated by double sums.
So for my students, then, I write an easy enough double sum in terms of r
and theta. This sum is simply along a judiciously chosen fine subdivision
of the intervals that r and q range. Then I estimate the sum in two
different ways (the second is by reversing the order of summation). The
first method gives a value strictly less that 1/2 (recall the exact value
is 0) and the second strictly larger that 1/2 ( the exact value being
1). Hey presto.
You should be able to fill in the details without any difficulties.
Believe me you can really make it easy and approachable to students.
Let me also comment on the following
> Date: Fri, 07 Apr 2000 17:44:36 -0700
> From: "Peter Ross"
> Subject: Re: [HM] Fundamental Theorem of Algebra
>
> Volume 28 #1 (January 1997) of The College Mathematics Journal has
> a one-page proof on p.58, entitled "A Proof That Polynomials Have
> Roots". The author Uwe F. Mayer near the start of the article says,
> "Your readers may enjoy a proof whose ingedients are known to any
> student who has taken a year of calculus." This statement must be
> taken with a grain of salt; for instance, the student needs to know
> that a continuous function on a closed disk achieves a minimum.
> However it is a nice proof and is accessible to students with a
> good calculus background.
I have not seen the proof, but you must bare in mind that you CANNOT avoid
some use of the supremum axiom of real numbers. (Otherwise the proof would
be valid for the rational field too.) In the proof you sight, it is in the
proof of the property of continuous functions you quote.
I hope the above answer your question.
Michael Lambrou
University of Crete
Greece.
------------------------------
Date: 9 Apr 00 11:15:35 -0400 (EDT)
From: Milo Gardner
Subject: Re: [HM] Matematica precolombina
Laura's request for Mayan and Incan numerology is not of interest to
me. What is of interest is the astronomies and mathematics of both
cultures. The March/April edition of the magazine Archeology suggests
that Incan astronomers understood lunar sidereal time, and many other
linked calendar relationships. Incan math was base 10.
George G. Joseph suggested in 1991, Crest of the Peacock, as debated
by Marcia Ascher in the journal Historia Mathematica, that quipu
arithmetic included the use of abacus registers. I tend to agree
with Joseph since Mayan base 20 was clearly written up by George I.
Sanchez, 1961, Arithmetic in Maya, also as using a 'mental' abacus
registger, where 0-3, 0-4 described base 20.
Mayan astronomy, as the Dresden Codex describes does include a form
of numerology, using base 13 for the underworld, an aspect that did
not enter into the Mayan daily numeration system. Interestingly, Mayan
arithmetic did not use fractions, but rather modular arithmetic in
the form of remainders. Loundsbury writing in The Sky in Mayan
Literature, offers an approach to solving one key diophantine equation
found in the Dresden Codex.
Generally, the issue of Mayan division and its non-use of fractions
has tended to be ignored by the Mayanist community. Thompson's 1930's
views of Mayan numeration will end one day. The suggestion that Mayan
numeration was historically based on 'finger and toes' thinking is
of the the saddest myths that I hear from Mayanists.
Mayan base 4 was often seen as linked to the four directions, a very
positive spiritual aspect that found its way to the majority of North
American tribes. Kroeber wrting in Handbook of California
Indians found base 20 used in conjunction with local numeration
systems (in two southern California tribes), a point that suggests
trade of obsidian and other products like turquoise, had a long
standing. Kroeber also found that the non- military dominated
California tribes freely used base 4, 8, 10, 12, 16, 20 and
combinations thereof, depending upon their unique view of cosmology.
Milo Gardner
------------------------------
Date: 9 Apr 00 12:20:43 -0400 (EDT)
From: John Stillwell
Subject: Re: [HM] Inductive proof of unique factorization
The inductive proof of unique prime factorization attributed to
Zermelo is mentioned by Hasse in his paper Ueber eindeutige
Zerlegung ... in Crelle 159 (1928) pp. 3--12. I don't know
whether Zermelo himself ever published the proof.
It should also be pointed out that Gauss, in the Disquisitiones,
gives a proof by descent of Euclid's result that if prime p|ab
then p|a or p|b. Since unique prime factorization follows easily
from this, Gauss's proof might also be considered to be an inductive
proof of it.
John Stillwell
------------------------------
Date: Sun, 9 Apr 2000 20:40:08 +0100
From: John Bibby
Subject: Re: [HM] Mathematics during the Roman Empire
I awaited further responses on Roman mathematics with interest, because I
was recently asked to produce a book on the subject for Channel 4
television. It was aimed at 7-11 year-olds and arguably dealt with a lot of
things that are not "maths" (or "math") _sensu structu_ (measurement,
symmetry, surveying etc etc).
There is also a video on the subject (even emptier in mathematical content,
I regret.)
Further details available from qed@talk21.com
JOHN BIBBY
-------------------------- Original Message --------------------------
>
> From: Ubiratan D'Ambrosio
> Sent: 22 February 2000 13:40
> To: historia-matematica@chasque.apc.org
> Subject: Re: [HM] Mathematics during the Roman Empire
>
>
>
> The best source I know is Vitruvius' Ten Books of Architecture.
>
> A new edition [20 pages Introduction + 113 pages translation + 182
> commentaries] by Ingrid D. Rowland (Art History, University of
> Chicago) and Thomas Noble Howe (Architecture History, Southwestern
> University}, has been published by Cambridge University Press, 1999.
>
> Yours, Ubi
>
>
>
> > From: Dinesh Maheshwari
> >
> > My wife's [she is from a Roman town in the foothill of Alps]
> > remarks on pragmatism during the Roman empire have made me
> > curious about mathematics and the basic sciences under the
> > Roman empire.
> >
> > Could some one please shed some light on the contributions to
> > mathematics under the Roman Empire.
------------------------------
Date: Sun, 9 Apr 2000 20:40:02 +0100
From: John Bibby
Subject: Re: [HM] "maths" in lieu of "math"
While mathematics/math/maths is undeniably singular, the same cannot be said
of statistics/stats/stat?
I never heard of "stats" as a student (in the 60's). When did it/they first
arise?
JOHN BIBBY
PS: I find the "Germanic" explanation of math quite convincing. My germanic
friends were immediately amused at the fact that my second lad is called
"Matt" (and it took me a long time to get the joke!)
------------------------------
Date: Sun, 9 Apr 2000 16:12:42 -0500 ()
From: Clark Kimberling
Subject: [HM] Mathematics and Music
Long ago, I read a strong sentence something like this: The science of
modern mathematics can claim to be the most original creation of mankind;
another claimant to this position is music.
Can someone identify the source and give the correct wording?
The sentence raises two questions in the history of mathematics:
(1) Are there earlier publications asserting that mathematics is unique -
or extreme - with regard to originality (and/or creativity)?
(2) Are there earlier publications in which mathematics and music, to the
exclusion of other subjects, are placed so high?
Regards,
Clark Kimberling
------------------------------
Date: Sun, 9 Apr 2000 21:34:38 EDT
From: Karen Dee Michalowicz
Subject: [HM] HPM Gathering in Chicago
This years HPM Meeting in Chicago will be in conjunction with the
presentation by the authors of the six secondary mathematics history modules
developed by way of Victor Katz's and Karen Michalowicz's NSF Grant. We
invite any and all HPM members to attend the presentation on Saturday morning
8:30am, Session 1001, Gold Coast Room Hyatt. It is ticketed. However, don't
worry if you are unable to get a ticket. We would like to go out to lunch
after the presentation, about noon, with anyone who is around.
We are pleased to announce that Bob Stein, Cal State, will be taking over the
leadership of the Americas Section of HPM beginning in May. We thank Victor
Katz for his leadership during these last years.
We have begun planning for the NCTM meeting in Orlando. If you are
interested in making a 30min-45minute presentation, please contact Bob Stein
at bstein@csusb.edu.
Karen Dee Michalowicz
HPM, Assistant to the President
HPM Representative to the National Council of Teachers of Mathematics
------------------------------
Date: Sun, 9 Apr 2000 23:28:22 -0500 (CDT)
From: James Propp
Subject: Re: [HM] mystery-woman
A few months ago, I asked:
> Andre Weil, in his autobiography, mentions having received a copy of
> Mordell's thesis in 1925 from a woman mathematician visiting from the
> U.S. who gave a lecture on Diophantine equations. Weil does not mention
> the woman's name, but it seems likely to me that there were comparatively
> few female American mathematicians with an interest in number theory at
> that time, so that it might be possible to deduce who this woman was.
> Does anyone know about this (or know who'd be likely to know)?
>
> Jim Propp
> Department of Mathematics
> University of Wisconsin
Franz wrote:
> Yep, it's M.I. Logsdon, one of Dickson's students. I'm pretty sure that
> I know this from
>
> N. Schappacher,
> D\'eveloppement de la loi de groupe sur une cubique,
> S\'em. Th\'eor. Nombres Paris 1988-89, 159--184
>
> There's a recent article by Della Fenster on Dickson's female students
> that you should find in the online version of the MR.
Here's a reply from Judy Green that supports this and (I think) definitively
settles the matter:
From jgreen@phoenix.marymount.edu Fri Jan 14 16:51 CST 2000
To: James Propp
cc: Jeanne LaDuke
Subject: Re: Andre Weil and M.I. Logsdon (fwd)
Dear Prof. Propp
It is certainly Logsdon. Weil describes her in more detail on page 524 of
volume 1 of his collected works. From that description, a visitor to Rome
from Chicago who had authored a paper on cubics in Trans. Am. Math. Soc. in
1925, it is very clear that the woman in question is Mayme I. Logsdon.
Logsdon served as instructor at Chicago from 1921 to 1925 at which time she
was promoted to assistant professor. During the year 1925-1926 she was an
International Education Board Fellow studying in Rome. Her paper "Complete
groups of points on a plane cubic curve of genus one" appeared in TAMS 27
(1925): 474-490.
You can find a short description of her life and work in the paper I wrote
with Jeanne LaDuke (DePaul University) - "Contributors to American
Mathematics: An Overview and Selection" in Women of Science: Righting the
Record, edited by G. Kass-Simon and Patricia Farnes, 117-146.
Bloomington: Indiana University Press, 1990.
I hope this is of some help.
Judy Green
------------------------------
Date: Mon, 10 Apr 2000 10:22:49 +0200 (MESZ)
From: luene@mathematik.uni-kl.de
Subject: Re: [HM] Fundamental Theorem of Algebra
On Sunday, 9 Apr 2000, Avinoam Mann wrote:
>
>
> However, the proof to which Hans refers below needs the fact that any
> (non-constant) polynomial has roots in some extension field of the
> rationals. To prove that one needs to develop a few properties of
> polynomial rings, so it is not as elementary as was asked for.
>
> Avinoam Mann
>
>
> On Sat, 8 Apr 2000, Hans Lausch wrote:
> >
> > One of Gauss's proofs, the "most algebraic" one, uses induction on the
> > exponent m of 2, where 2^m is the largest 2-power dividing the degree of
> > the polynomial (which can be assumed to have no multiple roots. For m=0,
> > the start of the induction, one invokes the intermediate value theorem,
> > which remains the only "calculus" tool used in the proof.
>
This is exactly what Gauss critized in the first paper he wrote on the
fundamental theorem of algebra that Euler and Lagrange assume --speaking
in modern language -- the existence of a splitting field of the polynomial
under consideration. Later he also critized Laplace for the same token.
The first proof by Gauss is of a topological nature. The second one is
the most algebraic one and I believe that Hans Lausch is referring to
this proof. Now, Gauss does not assume in this proof the existence of a
splitting field, as Avinoam Mann suggests one has to. This makes the
proof very long and tedious. I once reproduced it in an algebra course
and it took me six hours to 45 minutes each (the university hours). The
proof, in fact, is a master piece. Gauss uses Waring's theorem on
symmetric polynomials in great virtuosity over and over again in the
following way. Given a polynomial identity for the elementary symmetric
functions one can replace the symmetric functions by indeterminates
according to this theorem and viceversa. The second tool he uses is the
evaluation homomorphism, i. e., the fact that inserting ring elements
for the indeterminates of a polynomial ring yields a homomorphism of
the polynomial ring into the ring.
The most beautiful proofs of the fundamental theorem of algebra for me
are the proof given by Laplace using symmetric polynomials and the
Artin-Schreier variant of it, where the argument using the symmetric
polynomials is replaced by an argument using the fundamental theorem
of Galois theory.
All the best, Heinz Lueneburg
------------------------------
End of HISTORIA MATEMATICA V2 #13
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HISTORIA MATEMATICA Tuesday, April 11 2000 Volume 02 : Number 014
~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE of SUBJECTS ~~~~~~~~~~~~~~~~~~~~~~~~~~
Re: [HM] Early Algebraic Proofs of FTA
Re: [HM] Fundamental Theorem of Algebra
Re: [HM] Mathematics and Time
Re: [HM] Mathematics and Music
Re: [HM] Mathematics and Music
Re: [HM] Fundamental Theorem of Algebra
Re: [HM] nabla origin
[HM] Were Euclid's numbers geometrical entities?
Re: [HM] Fundamental Theorem of Algebra
[HM] Euler: To Negative Quantities Through Infinity
[HM] HISTORY and PEDAGOGY of MATHEMATICS
[HM] Mersenne primes and Russian mathematicians
Re: [HM] Early Algebraic Proofs of FTA
Re: [HM] Euler: To Negative Quantities Through Infinity
Please see the end of this digest.
----------------------------------------------------------------------
Date: Mon, 10 Apr 2000 10:31:20 -0400 (EDT)
From: Christopher Baltus
Subject: Re: [HM] Early Algebraic Proofs of FTA
Dear HM list members,
While early proofs of the Fundamental Theorem of Algebra [FTA] are under
discussion, I hope members can help me with several questions concerning
early algebraic proofs.
First, I offer a brief outline of the history, as I understand it.
The first clear enunciation and claim for the FTA was by Euler, in a
letter to J. Bernoulli in 1739, that every real polynomial is the product
of real factors of 1st and 2nd degree. Euler read a proof to the Berlin
Academy in Nov 1746, later developed into a published version for the 1749
Memoire of the Academy. d*Alembert sent a proof in Dec 1746, based on
algebraic equations, which appeared in the Memoire for 1746 of that same
academy.
Euler claimed that any real polynomial of degree 2**n, n > 1, was the
product of two real polynomials of equal degree. Daviet de Foncenex, in
Reflexions sur les quantites imaginaires, Miscellanea Taurinensis 1759,
offered a variation, claiming that any real polynomial with degree P*
2**m, P odd, had a real quadratic factor; a coefficient, u, of the
proposed factor satisfied an equation of degree Q*2**(m-1), Q odd. Thus
one proceeded by what we now call induction to an odd degree equation
which, as was obvious even to Gauss, has a real root. Laplace, in his
Lecons de Mathematiques donnees a lEcole Normale en 1795, followed the
plan of Foncenex, but in a cleaner form which evaded one of the
difficulties of the 1759 proof. (Gauss followed this plan in his 2nd
proof, 1815.)
The proofs of Euler and Foncenex shared the same deficiencies, most of
which were cleared up by Lagrange in papers of 1771 and 1772. Except for
the one Gauss considered fatal in his paper of 1799, that the existence of
roots of some form was assumed.
Question 1. As reported by E. Netto, in Vol IV, p 308, of M. Cantor*s
Vorlesungen uber Geschichte der Mathematik, J. B. J. Delambre disclosed in
his *Eloge de Lagrange* that Foncenex was a schoolmate and friend of
Lagrange and that, for a work on the complex logarithm, Lagrange had given
his own results to Foncenex who developed them and published them under
his own name. The work on the FTA also delved into the logarithm and
included a long footnote by Lagrange himself.
Does anyone know more about the role of Lagrange in the 1759 proof of
Foncenex?
Question 2. The concept of Symmetric Functions played a central role in
the early algebraic proofs of the FTA. The earliest use of *symmetric
functions,* as a defined term, of which I am aware, is in S. F. Lacroix*s
Complements des elements dAlgebre, An VIII - 1799. This seems an unlikely
place for the introduction of so crucial a definition. Does anyone know
of an earlier use?
Question 3. The 1795 proof by Laplace was the clearest of the early
proofs; it still appears (e.g., M. Mignotte, Mathematics for Computer
Algebra) and it was the one proof of the FTA given by Lacroix, 1799. When
was it first published? [The journal publication was in 1812.]
It is described in Lagrange*s Traite . . . Equations Numeriques
. . . of An VI - 1798, but E Netto, in the article Le Theoreme
Fondamental, in Encyclop. des Sciences Mathemat., 1907 (French trans. of
the German work), refers to *Lecons de Mathematiques donnees a lEcole
Normale en l*an III,* in Seances des ecoles Normales recueillies par des
stenographes et revues par les professeurs, An III. What is this
publication? The recent biography *Pierre-Simon Laplace,* by Gillispie,
1997, says (p 168) that Laplace*s lectures were originally published from
stenographic transcripts in 1800. Did Netto err in his date? Does it
matter?
[Note that Gauss devoted two articles of his 1799 thesis on the FTA to
Foncenex, but does not mention Laplace.]
Christopher Baltus
Oswego, NY
------------------------------
Date: Mon, 10 Apr 2000 14:13:25 -0400
From: "Daniel E. Otero"
Subject: Re: [HM] Fundamental Theorem of Algebra
John Conway wrote:
> 5) If not, we can suppose f(0) = 1, so f looks like
>
> 1 + az + bz^2 + ... .
>
> Now if only |z| is sufficiently small we have
>
> |bz^2 + ... | < |az|/2.
>
> 6) Now by chose of arg(z) we can make az be negative real,
>
> and then if it's small enough, it's easy using 5) to
> see that | 1 + az + bz^2 + ... | < 1, contradicting
> our assumption.
>
> John Conway
A minor lacuna: if a = 0, we're in trouble here. Of course, just let
az^k = lowest nonzero nonconstant term, and we can make |z| so small
that |(all terms higher than the kth, if any)| < |az^k|/2. Step (6)
works the same way with az^k for az. Yes?
Danny Otero
------------------------------
Date: Mon, 10 Apr 2000 15:21:58 -0400 (EDT)
From: William C Waterhouse
Subject: Re: [HM] Mathematics and Time
On Apr 7, 2000, "M.Carmen Hernandez-Martin" wrote:
> That the thirteen books of Euclid's "Elements" are geometry is
> something I have not invented. It have been said by lots of authors
> through the centuries. The number-segments are even drawn in every
> proposition of books VII, VIII, and IX. ...
If we take this literally, we will certainly be forced to some
unlikely conclusions. For instance, Definition 4 in Book VII
tells us that an even number is one that can be split in two
("dicha"). But I.10 shows how to cut a segment in two ("dicha").
Thus if numbers are segments, all numbers are even.
...
> I have just found Boyer: "Throughout these books (VII, VIII, IX)
> each number is represented by a line segment, so that Euclid will
> speak of a number as A B...."
This should be considered an abbreviated notation. If you like,
you can represent a number by a line segment TOGETHER WITH a
specified or understood "unit" segment. But there are many
such representations, and "the number" is not identical with
any one of them. As Proclus said (In Eucl. 95, 23), "It is clear
to everyone that numbers are purer and more abstract than
magnitudes."
William C. Waterhouse
Penn State
------------------------------
Date: Mon, 10 Apr 2000 16:45:51 -0500
From: Steve Kennedy
Subject: Re: [HM] Mathematics and Music
Dear Professor Kimberling,
You wrote:
> Long ago, I read a strong sentence something like this: The science
> of modern mathematics can claim to be the most original creation of
> mankind; another claimant to this position is music.
> Can someone identify the source and give the correct wording?
"The Science of pure mathematics, in its modern developments, may claim to
be the most original creation of the human spirit. Another claimant for
this position is music. But we will put aside all rivals, and consider the
ground on which such a claim can be made for mathematics. The originality
of mathematics consists in the fact that in mathematical science
connections between things are exhibited which, apart from the agency of
human reason, are extremely unobvious. Thus the ideas, now in the minds of
contemporary mathematicians, lie very remote from any notions which can be
immediately derived by perception through the senses; unless indeed it be
perception stimulated and guided by antecedent mathematical knowledge.
This is the thesis which I proceed to exemplify."
This is the opening paragraph of Alfred North Whitehead's "Mathematics as
an Element in the History of Thought," which is included in Newman's "The
World of Mathematics. This is the essay which also contains the entertaining:
"I will not go so far as to say that to construct a history of thought
without profound study of the mathematical ideas of successive epochs is
like omitting Hamlet from the play which is named after him. That would be
claiming too much. But it is certainly analogous to cutting out the part
of Ophelia. This simile is singularly exact. For Ophelia is quite
essential to the play, she is very charming---and a little mad."
Steve Kennedy
skennedy@carleton.edu
------------------------------
Date: Mon, 10 Apr 2000 10:03:26 GMT
From: "Tony Mann"
Subject: Re: [HM] Mathematics and Music
Clark Kimberling wrote:
> Long ago, I read a strong sentence something like this: The science of
> modern mathematics can claim to be the most original creation of mankind;
> another claimant to this position is music.
>
> (2) Are there earlier publications in which mathematics and music, to the
> exclusion of other subjects, are placed so high?
>
Doesn't answer your questions, but "The Music of the Spheres" by
Jamie James (Penguin paperback in UK) argues that for most of its
history music, like maths, was an attempt to understand the structure
of the universe. I found it very stimulating!
Tony
Tony Mann
School of Computing and Mathematical Sciences,
Maritime Greenwich University Campus, 30 Park Row,
Greenwich, London SE10 9LS
Tel: +(44) (0)20 8331 8709
Fax: +(44) (0)20 8331 8665
Email: A.Mann@gre.ac.uk
Web site http://www.gre.ac.uk/~A.Mann
------------------------------
Date: Mon, 10 Apr 2000 04:24:33 -0400 (EDT)
From: Abe Shenitzer
Subject: Re: [HM] Fundamental Theorem of Algebra
Apropos of the proofs of the FTA. It seems to me that a beginner who can
follow any of the proofs mentioned on HM should be encouraged to read the
wonderful little book "First Concepts of Topology" by Chinn and Steenrod
(NML 18). Among other things, this book provides the remarkable context
for the theorem and ends with its proof.
Abe Shenitzer
------------------------------
Date: Tue, 11 Apr 2000 00:07:43 +0100
From: David Wilkins
Subject: Re: [HM] nabla origin
Randy K. Schwartz wrote
>
> Steven Schwartzman, on p. 142 of his book _The Words of Mathematics_
> (MAA, 1994), states that the Irish mathematician William Rowan Hamilton
> called the inverted uppercase delta symbol (the symbol he chose for that
> operator which yields the vector gradient of a real-valued multivariable
> function) the "nabla" because the symbol resembled an ancient Hebrew
> stringed instrument called nabla. The instrument is or was believed to have
> been a type of harp. Schwartzman adds that "nabla" was the Greek form of a
> Hebrew word that was likely of Phoenician origin, and that the word is
> sometimes rendered in English as "nable" or "nebel."
>
> Jeff Miller, in his website "Earliest Uses of Various Mathematical
> Symbols" (http://members.aol.com/jeff570/mathsym.html), states the following
> (as slightly edited by me):
>
> Nabla (the Hamiltonian operator), which is also called a "del" or "atled"
> (delta spelled backwards), was introduced by William Rowan Hamilton
> (1805-1865) in 1853 in _Lectures on Quaternions_, according to Florian
> Cajori in his _A History of Mathematical Notations_ (vol. 2, page 135).
>
> David Wilkins has found the symbol used earlier by Hamilton in the
> _Proceedings of the Royal Irish Academy_ of the meeting held on July 20,
> 1846. The volume appeared in 1847. However the symbol is rotated 90 degrees.
>
>
> According to Sherman K. Stein and Anthony Barcellos on page 836 of their
> _Calculus and Analytic Geometry_ (McGraw Hill, Inc., 1992), Hamilton denoted
> the gradient with an ordinary capital delta in 1846.
>
>
>
I can add some further details, and I have made the texts
themselves (apart from the `Lectures on Quaternions') available
on the Web.
First then the `Proceedings of the Royal Irish Academy', for the
meeting of July 20, 1846. This paper appeared in volume 3 (1847),
pp. 273--292. You will find the relevant occurrences on
page 12 of my online edition of this paper, which is located at
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Pascal/
(available in Plain TeX, DVI, PostScript and PDF). The nabla
symbol is rotated through ninety degrees.
Hamilton also introduced this symbol in a long paper, or series,
`On Quaternions; or on a new System of Imaginaries in Algebra';
which he published in installments in the Philosophical Magazine
between 1844 and 1850. The relevant portion of the paper consists
of articles 49-50, in the installment which appeared in October 1847
in volume 31 (3rd series, 1847) of the Philosophical Magazine,
pp. 278-283. This is reproduced on pp. 45-46 of my online edition
of this paper, which is located at
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/
There is an interesting footnote, which reads
`In that paper itself, the characteristic was written
$\nabla$; but this more common sign has been so often
used with other meanings, that it seems desirable to
abstain from appropriating it to the new signification
here proposed.'
(Here `that paper' refers to an unpublished paper that Hamilton
had prepared for a meeting of the British Association
for the Advancement of Science, but which had been forwarded
by mistake to Sir John Herschel's home address, not to the
meeting itself in Southampton, and which therefore was
not communicated at that meeting.)
In other words, according to the above footnote, Hamilton had
originally intended to use the same symbol, $\nabla$, that we
use today, but then decided to rotate it through 90 degrees
to avoid confusion with other uses of $\nabla$.
(I have used $\triangleleft$ in my online edition of these
papers, as being the nearest equivalent in TeX to what Hamilton
used. But the upper and right sides should be thickened in order
to reproduce more accurately the appearance of Hamilton's
symbol.)
I am puzzled by the following:
> According to Sherman K. Stein and Anthony Barcellos on page 836 of
> their _Calculus and Analytic Geometry_ (McGraw Hill, Inc., 1992),
> Hamilton denoted the gradient with an ordinary capital delta in 1846.
I am sure that I have never seen the gradient denoted by an
`ordinary capital delta' in any paper of Hamilton published
in his lifetime.
With reference to the following:
> Nabla (the Hamiltonian operator), which is also called a "del" or "atled"
> (delta spelled backwards), was introduced by William Rowan Hamilton
> (1805-1865) in 1853 in _Lectures on Quaternions_, according to Florian
> Cajori in his _A History of Mathematical Notations_ (vol. 2, page 135).
>
I would observe that the symbol is rotated through 90 degrees in
the `Lectures', exactly as in the 1847 occurences in the
`Proceedings of the Royal Irish Academy' and the
`Philosophical Magazine'. It is introduced on pages 610-611
of the `Lectures on Quaternions' (1853). (I have those pages
open in front of me as I write this.)
A quick search has not located any instance of the nabla operator
in the posthumously published `Lectures on Quaternions'. According
to the Preface to the Second Edition, Hamilton intended to introduce
it in the final section that he was working on at the time of his
death.
I am not aware of any published writing of Hamilton in which he
calls this operator `nabla', or anything else for that matter.
Nor is there any mention in his writings of any Hebrew stringed
harp, so far as I am aware.
I should add that in the past year I have several times consulted
the original published text of all those papers of Hamilton cited
above.
------
In the Hamilton section of my History of Mathematics Website at
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/
I have made available on the Web my online edition of the texts
of all mathematical papers published by Hamilton in his lifetime
that are listed in the bibliography in the third and final volume
of the biography by Robert Perceval Graves. I have not included
the two books on quaternions. Also I have deemed some items in
that bibliography not to be `mathematical papers' (e.g., mere
titles of lectures, speeches, presidential addresses, reviews),
and these are not included in my collection. Also I have not
included material from Hamilton's Nachlass that is still
unpublished, or which first appeared in print in the three
volumes of the Royal Irish Academy edition of Hamilton's papers
that have so far been published. (I have seen an advertisement
for the fourth and final volume of that edition, which should
appear in the autumn of this year, and which will apparently
include a substantial amount of further material from the
Nachlass.)
___
David Wilkins
School of Mathematics,
Trinity College, Dublin.
------------------------------
Date: Mon, 10 Apr 2000 19:51:28 -0300
From: Romulo Lins
Subject: [HM] Were Euclid's numbers geometrical entities?
"M.Carmen Hernandez-Martin" wrote:
>
> Dear all:
>
> Romulo Lins wrote:
>
> > Given the sureness with which M.Carmen Hernandez-Martin asserts
> > the above, I would ask her or other list members to let me know
> > the argument that makes possible to dismiss so completely Jacob
> > Klein's 'Greek Mathematical Thought and the Origins of Algebra,'
> > to mention just one book.
>
> That the thirteen books of Euclid's "Elements" are geometry is
> something I have not invented. It have been said by lots of authors
> through the centuries. [...]
Certainly. But that is not to say that such view is necessarily to be
taken as correct; that's why I asked for the arguments. Particularly on
this subject there has been a lot of debate.
> The number-segments are even drawn in every
> proposition of books VII, VIII, and IX.
>From J. Klein's 'Greek Mathematical Thought..." (p111)
"The geometric form of their [numbers] presentation is suggested not
only by a consideration of the great problem of 'incommensurability'
which forces a thorough 'geometrization' on Greek mathematics (cf p43).
The 'pure' units of which the numbers to be studied are composed are
here understood precisely only as _'units of measurement'_ such as can
be represented most simply by straight lines which are _directly
measurable_ [...] quite independently of whether they form a 'linear'
(prime), 'plane' or 'solid' number. [...]"
That is, 'measuring' is the operation that is intended by the
representation by lines, not the association between number and lines
_within arithmetic_ (and one cannot simply deny that Euclid does use the
word 'number'). On the other hand, the interpretation you suggest for
the arithmetical books would be quite natural for the "algebra" in
Vieta's 'Introduction to the Analytical Art,' where plane numbers _must_
be given in its two 'dimensions' and cannot be directly added to a
'linear' number. It also applies to the 'geometrical algebra' of Omar
Khayyam.
I see Euclid's use of lines there as similar to when we want to
represent R^2xR^2 and place two axis calling each of them R^2.
> The denomination of geometric
> algebra for the second book is classical also. You can see van der
> Waerden for example: "Geometry and algebra in Ancient civilizations".
> As a matter of fact the term "geometric algebra" was coined by
> Zeuthen.
As I understand it, this goes directly against your argument that
everything was geometry, as they were tryng to a great extent say that
everything was algebra!
> I can remember now Abel Rey (and may be also Tannery) using
> the term "geometric arithmetic", and stating that in Euclid, number
> is supplanted by geometric magnitude. You can see "La science dans
> l'Antiquite" volume II (La jeunesse de la science grecque). And of
> course you can see also Boyer's "History of Mathematics".
I am almost afraid of disagreeing with such great writers, but I cannot
help. As to van der Waerden it seems he was unable to move an inch from
his own conceptualisation, a major obstacle to anyone trying to make
sense of what people do in other cultures.
> I cannot see
> the contradiction with Klein.
>From J. Klein's "Greek..." (p122)
"[...] Zeuthen was not the first to understand the ancient mode of
_presenting_ [my emphasis] mathematical facts as a 'geometrical
algebra,' although he was the first to use this concept consistently. To
anticipate, this interpretaion can arise only on the basis of an
insufficient distinction between the _generality of the method_ and the
_generality of the object_ of investigation. [...]"
and p123:
"Although in Euclid the linear presentation of the 'arithmetical' books
is tailored to the requirements of the tenth book, with with the earlier
books have a 'systematic' connection, yet the objects of these books
are, in spite of the _sameness_ of method, _different_: books VII-IX
deal with numbers which are so defined as to be always commensurable
(cf. VII, def. 2, 14-16), while book X, on the other hand, deals with
manitudes whose ratios cannot be reduced to number ratios (cf X, 5-8)
and which are incommensurable precisely on this account."
I can hardly imagine a more direct disagreement.
> The only way
> of saving this crisis is considering number not as a construction of
> points but as something represented by a segment of right line. This
> "ambiguity" of number, which becomes a kind of variable, allows the
> irrationals to enter the system in book X.
ibid., pp123-4:
"[...] And again, when in the arithmetical books an arithmetical, or
more exactly, a logistical proposition is demonstrated _generally_ with
the aid of lines, this does not in the least mean that there exists
either a general number or the concept of a 'general,' i.e.,
indeterminate, number corresponding to this general proof [...]"
And this is in direct disagreement with what you said, I guess.
To say that two lines A and B are commensurable if there are _numbers_ a
and b such that a:b=A:B is _not_ to say that A and B are of the same
nature as a and b, as becomes clear when we think of a ratio of plane
figures being equal to a ratio of lines.
>
> I have just found Boyer: "Throughout these books (VII, VIII, IX)
> each number is represented by a line segment, so that Euclid will
> speak of a number as A B. The discovery of the incommensurable had
> shown that not all line segments could be associated with whole
> numbers; but the converse statement - that numbers can always be
> represented by line segments - obviously remains true. Hence Euclid
> does not use the phrases 'is a multiple of' or 'is a factor of', for
> he replaces these by 'is measured by' and 'measures' respectively"
> (A History of Mathematics pag 126).
I do not think that the notion of 'measure of a set' would take us very
far if understood in a literal way, but because there are _aspects_ of
the notion of 'measure' that are useful there, the use is justified. It
seems to me that Euclid's use of the line presentation has to be
understood in this sense: useful and not literal (oops!).
[...]
> Somebody (I can not remember who) wrote a paper in "Archives for
> History of Exact Sciences" defending the need of rewriting the history
> of Greek mathematics in its exact terminology as the translation into
> modern terminology becomes often misleading.
If I remember well, Unguru's paper argued against 'geometrical algebra'.
I wonder what he has to say about 'arithmetical geometry.' Or
'geometrical arithmetic."
Christian Marinus Taisbak wrote:
>
> Euclid's number theory is not geometry, he has asked me to tell you.
>
> One should not be misguided by the fact that the Elements books
> vii-ix illustrate arbitrary numbers by line segments. That was one
> way to denote their arbitrariness, since the alphabet would convey
> distinctly determined numbers. THERE IS NOT A SINGLE GEOMETRICAL
> ARGUMENT in those books, not even when he is dealing with square
> numbers and cube numbers.
I think we can move beyond that justification for the use of lines. It
has been said that Diophantus did not solve his problems using 'literal'
notation for the same reason as above, but the fact is that he invented
a number of symbols and had a very ingenious use of the number 1 to
stand for a second unknown; I prefer to follow (again...) Jacob klein
and think that he would not write general/generic expressions simply
because they did not make sense within his conceptualisation of number:
number had to be a _determinate_ number and to solve a problem by giving
an expression like (-b+-sqr(b^2_4ac))/2 as an answer was precisely _not_
to solve the problem. Also it is hard to think that a man (or woman, or
women, or men, or people) like Euclid had no knowledge of other
alphabets he could use as we use Greek letters. The Hindus used colour
names, the Chinese used coloured rods. Euclid coulde have used Greek
letters with a special mark over it (as Newton did). That he did not do
it suggests that the 'line notation' was adequate and useful: number is
the result of measuring a collection with a unit, and the idea of
measurement can hardly be more easily grasped than with lines.
all the best,
Romulo
------------------------------
Date: 10 Apr 00 09:58:33 -0400 (EDT)
From: Peter Flor
Subject: Re: [HM] Fundamental Theorem of Algebra
I have preferred this proof (without Conway´s normalizations, as I
have to admit) to all others I know since I first learned it. It is
frequently called "Argand´s proof", and I stuck to this usage in some
lectures. Is this attribution historically correct?
Best wishes to all, Peter.
------------------------------
Date: Tue, 11 Apr 2000 01:55:06 +0200
From: "Jose' Correia"
Subject: [HM] Euler: To Negative Quantities Through Infinity
In "De seriebus divergentibus", to defend sums for divergent series,
Euler writes:
"(...) it seems in accord with the truth if we say that the same
quantities which are less than zero can be considered to be greater than
infinity. For not only from algebra but also from geometry, we learn
that there are two jumps from positive quantities to negative ones, one
through nought or zero, the other through infinity (...)."
My question is: what does Euler mean when he says "not only from
algebra but also from geometry"?
We found a part of the answer in Kline's "Mathematical Thought from
Ancient to Modern Times":
"An interesting argument against negative numbers was given by Antoine
Arnauld (1612-94) (...). Arnauld questioned that -1:1=1:-1 because, he
said, -1 is less than +1; hence, How could a smaller be to a greater as
a greater is to a smaller?" (p. 252)
"Thought Wallis was advanced for his times and accepted negative
numbers, he thought they were larger than infinity but not less than
zero. In his 'Arithmetica Infinitorum' (1665), he argued that since the
ratio a/0, when a is positive, is infinite, then, when the denominator
is changed to a negative number, as in a/b with b negative, the ratio
must be greater than infinity." (p. 253)
What about the remaining part, i.e., "from geometry"?
Jose Correia
------------------------------
Date: Mon, 10 Apr 2000 22:45:27 EDT
From: Karen Dee Michalowicz
Subject: [HM] HISTORY and PEDAGOGY of MATHEMATICS
My humble apologies. The Open Study Group on the Relations Between HISTORY
and PEDAGOGY of MATHEMATICS (HPM), Americas Section (which includes both
North America, Central America, and South America) meets each year at the
National Council of Teachers of Mathematics Annual Meeting. NCTM will begin
this Thursday. Next year NCTM will be the first week of April. We are
trying to make sure we have a presence at NCTM and the joint Mathematical
Association of America/American Mathematical Association meetings.
I know that many on this list have received the HPM Newsletter in the past.
We hope to have it up and going again very soon.
Sincerely,
Karen Dee Michalowicz
------------------------------
Date: Tue, 11 Apr 2000 09:57:43 +-200
From: Mariano Mataix
Subject: [HM] Mersenne primes and Russian mathematicians
Dear friends:
Could any of you answer the following questions:
1.- Which are the Mersenne primes 31 to 38th?
2.- Have you some information on the life and work of
Russian mathematicians A. T. Fromenko and Vladimir Voevodsky?
Thank you
Mariano Mataix. Madrid. Spain
------------------------------
Date: Tue, 11 Apr 2000 10:14:24 +0200 (MESZ)
From: Heinz Lueneburg
Subject: Re: [HM] Early Algebraic Proofs of FTA
Christopher Baltus wrote:
> [Note that Gauss devoted two articles of his 1799 thesis on the FTA
> to Foncenex, but does not mention Laplace.]
>
Gauss mentions Laplace later in the announcement of his second proof
Goettingische gelehrte Anzeigen. 1815, Dezember 23.
Collected Papers 3, 105-106
There he also criticizes that Laplace assumes the existence of roots.
Best regards, Heinz Lueneburg
------------------------------
Date: Tue, 11 Apr 2000 15:31:45 +1200 (NZST)
From: John Harper
Subject: Re: [HM] Euler: To Negative Quantities Through Infinity
On Tue, 11 Apr 2000, Jose' Correia wrote:
>
> In "De seriebus divergentibus", to defend sums for divergent series,
> Euler writes:
> "(...) it seems in accord with the truth if we say that the same
> quantities which are less than zero can be considered to be greater than
> infinity. For not only from algebra but also from geometry, we learn
> that there are two jumps from positive quantities to negative ones, one
> through nought or zero, the other through infinity (...)."
>
> My question is: what does Euler mean when he says "not only from
> algebra but also from geometry"?
I don't know what Euler did mean but here is one thing he could have
meant.
Imagine a pointer with one end at a fixed point P and free to turn in the
plane of P and a fixed line, with the origin O on the line at its nearest
point to P:
P
- --------------------------------O-------------------------------
If the pointer starts by pointing at O and turns anticlockwise, it will
pointing at all the + numbers to +infinity then through the - numbers
from -infinity to 0, then through the + numbers again, ... One could of
course turn the pointer the other way. In this context the jumps from
+ to - or - to + through O or infinity are indeed rather similar. Many
of us first met this when contemplating how tan(x) varies with x.
John Harper, School of Mathematical and Computing Sciences,
Victoria University, Wellington, New Zealand
e-mail john.harper@vuw.ac.nz phone (+64)(4)463 5341 fax (+64)(4)463 5045
------------------------------
End of HISTORIA MATEMATICA V2 #14
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HISTORIA MATEMATICA Wednesday, April 12 2000 Volume 02 : Number 015
~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE of SUBJECTS ~~~~~~~~~~~~~~~~~~~~~~~~~~
[HM] Info about conferences on the History/Philosophy of Mathematics
Re: [HM] Fundamental Theorem of Algebra
[HM] Tartaglia
Re: [HM] nabla origin
Re: [HM] Mersenne primes and Russian mathematicians
Re: [HM] Mersenne primes and Russian mathematicians
Re: [HM] Mathematics and Music
Re: [HM] Tartaglia
Re: [HM] Tartaglia
Re: [HM] Matematica precolombina
Re: [HM] Tartaglia
Re: [HM] Tartaglia
Please see the end of this digest.
----------------------------------------------------------------------
Date: Tue, 11 Apr 2000 13:52:24 +0200 (GMT+0200)
From: Ilana Wartenberg
Subject: [HM] Info about conferences on the History/Philosophy of Mathematics
Hi everyone,
Could someone give me a good source of information about conferences on
the History/Philosophy of Mathematics?
Thanks,
Ilana Wartenberg
Tel Aviv University
------------------------------
Date: Tue, 11 Apr 2000 15:01:06 +0200 (MESZ)
From: Heinz Lueneburg
Subject: Re: [HM] Fundamental Theorem of Algebra
Peter Flor wrote:
>
> I have preferred this proof (without Conway´s normalizations, as I
> have to admit) to all others I know since I first learned it. It is
> frequently called "Argand´s proof", and I stuck to this usage in some
> lectures. Is this attribution historically correct?
>
I and others call this proof the Cauchy-Argand proof of the FTA. I got from
Doerrie the quotation
Argand, Annales de Gergonne 1815.
I have never seen this paper. Cauchy has published his proof in his
Cours d'analyse. Paris 1821. Oeuvres, Ser. 2, Vol 3. Paris 1897
and in
Exercises de mathe/matiques. Quatrie\me anne/e. Paris 1829
Cauchy writes in his Cours d'analyse that his proof is a variant of the proof
Legendre gives in his The/orie des Nombres, .I.re Partie, Sect. XIV. I looked
it up this afternoon. In this section, Legendre developes reel roots of
polynomials in continued fractions the way Lagrange did. Then he showes how to
improve a given approximation to an imaginary root of a polynomial. The step of
improving the approximation must be the step Cauchy used to get the
contradiction that the minimum |f(k)| of the polynomial f inside a certain
circle around 0 can in fact be made smaller if it is not 0. I have not checked
the details.
Legendre, The/orie des Nombres. Paris, an VI. Pages 161 ff.
Sect. XIV starts on page 133.
Best regards, Heinz Lueneburg
------------------------------
Date: Tue, 11 Apr 2000 13:24:31 +0100
From: "Victor E. Hill IV"
Subject: [HM] Tartaglia
The entry on Tartaglia found at
http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Tartaglia.html
states, in part, "So, in March 1539, Tartaglia left Venice and travelled to
Milan.
. . . Tartaglia, after much persuasion, agreed to tell Cardan his
method, if
Cardan would swear never to reveal it and furthermore, to only ever write it
down in code so that on his death, nobody would discover the secret from his
papers. This Cardan readily agreed to, and Tartaglia divulged his formula in a
poem, to help protect the secret, should the paper fall into the wrong hands."
Do any correspondents know whether the poem is extant, whether in Italian, in
Latin, or in translation? Can anyone provide a reference?
=========================================================
Victor E. Hill IV
Thomas T. Read Professor of Mathematics, Williams College
Mail: P.O. Box 11, Williamstown MA 01267 USA
Office: (413) 597-2428 -- fax: (413) 597-4061
http://www.williams.edu/Mathematics/vhill/
=========================================================
------------------------------
Date: Tue, 11 Apr 2000 09:59:26 EDT
From: "James A. Landau"
Subject: Re: [HM] nabla origin
In a message dated 4/10/00 8:13:25 PM Eastern Daylight Time,
dwilkins@maths.tcd.ie writes:
> In the Hamilton section of my History of Mathematics Website at
>
> http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/
>
> I have made available on the Web my online edition of the texts
> of all mathematical papers published by Hamilton in his lifetime
In the paper
On the Argument of Abel, respecting the Impossibility of expressing a Root of
any General Equation above the Fourth Degree, by any finite Combination of
Radicals and Rational Functions.
published in the Transactions of the Royal Irish Academy, volume 18 (1839),
pp. 171-259, and available at URL
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quintic/
the upside-down delta appears on pages 39 and 40. Here it is used NOT as the
gradient operator but to signify a type of permutation. There is no mention
of why the upside-down delta is chosen, or whether it is a standard symbol,
or where it came from. In Hamilton's own words, as rendered into ASCII
We have therefore four and only four distinct sorts of changes of
arrangement, which...may be denoted by the four characteristics
nabla-sub-1, nabla-sub-2, nabla-sub-3, nabla-sub-4,
or more fully by the following,
nabla-super-a,b-sub-1, nabla-super-a,b-sub-2,nabla-super-a,b,c-sub-3,
nabla-super-a,b,c-sub-4,
nabla-super-a,b-sub-1, implying, when prefixed to any functions (alpha, beta,
gamma, delta), that we are to interchange the a-th and b-th of the roots on
which it depends...
nabla-super-1,2-sub-1 (alpha, beta, gamma, delta) =
(beta, alpha, gamma, delta)
<\quote>
>Hamilton also introduced this symbol ["nabla"] in a long paper, or series,
>`On Quaternions; or on a new System of Imaginaries in Algebra';
>which he published in installments in the Philosophical Magazine
>between 1844 and 1850. The relevant portion of the paper consists
>of articles 49-50, in the installment which appeared in October 1847
i>n volume 31 (3rd series, 1847) of the Philosophical Magazine,
>pp. 278-283. This is reproduced on pp. 45-46 of my online edition
>of this paper, which is located at
>
> http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/
>
>There is an interesting footnote, which reads
>
> `In that paper itself, the characteristic was written
> $\nabla$; but this more common sign has been so often
> used with other meanings, that it seems desirable to
> abstain from appropriating it to the new signification
> here proposed.'
>
I suppose the 1839 quote is an example of why Hamilton refers to the nabla as
"this more common sign" that "has been so often used with other meanings".
Did people other than Hamilton also use the upside-down delta?
>...the `Proceedings of the Royal Irish Academy', for the
>meeting of July 20, 1846. This paper appeared in volume 3 (1847),
>pp. 273--292. You will find the relevant occurrences on
>page 12 of my online edition of this paper, which is located at
>
> http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Pascal/
>
>(available in Plain TeX, DVI, PostScript and PDF). The nabla
>symbol is rotated through ninety degrees.
One last note: during Hamilton's lifetime the only methods of preparing
material for printing were engraving and lithography (which as far as I know
were used only for illustrations and graphics) and hand-set raised type,
which is probably what was used for Hamilton's papers. I don't know if the
individual type elements use by the typesetter were square or rectangular,
but it is possible they were rectangular. Now if the capital Greek letter
delta were on a rectangular piece of type, then it would be quite easy for
the typesetter to insert the piece of type upside down to create the nabla
symbol, but difficult to insert the piece of type sideways to create the
delta-rotated-ninety-degrees.
James A. Landau
systems engineer
FAA Technical Center (ACT-350V/BCI)
Atlantic City Airport NJ 08405 USA
P.S. Hamilton's _Lectures on Quaternions_ is available on the Web from the
Cornell University Digital Library at URL
http://moa.cit.cornell.edu/dienst-data/cdl-math-browse.html
James A. Landau
------------------------------
Date: Tue, 11 Apr 2000 14:50:14 -0700
From: Peter Ross
Subject: Re: [HM] Mersenne primes and Russian mathematicians
In answer to the first question below, the website
http://www.utm.edu/research/primes/largest.html
includes, among other interesting lists, a list of the ten
largest known of the 38 known Mersenne primes. It notes
that the largest listed may not be the 38th according to
size since "the region between the largest two and the
previous primes has not been completely searched".
The abbreviated website
http://www.utm.edu/research/primes/
has an astonishing amount of information on primes, including
ways to see them graphically and even hear them.
Peter Ross
Santa Clara University
> 1. Which are the Mersenne primes 31 to 38th?
>
> 2. Have you some information on the life and work of
> Russian mathematicians A. T. Fromenko and Vladimir Voevodsky?
------------------------------
Date: Tue, 11 Apr 2000 20:39:55 +0200
From: Ignacio Larrosa Can~estro
Subject: Re: [HM] Mersenne primes and Russian mathematicians
> Which are the Mersenne primes 31 to 38th?
That is immediate history! The Mersenne numbers are 2^N-1. The exponents
of Mersenne prime numbers 31 to 38 are:
31 216091 Slowinski 1985
32 756839 Slowinski & Gage 1992
33 859433 Slowinski & Gage 1994
34 1257787 Slowinski & Gage 1996
35 1398269 GIMPS 1996
36* 2976221 GIMPS 1997
37* 3021377 GIMPS 1998
38* 6972593 GIMPS 1999
* Its not full proved that there are not other mersenne prime less than
those
The last four were found for GIMPS, in 1996, 1997, 1998 and 1999. GIMPS
is a project of distributed computation on Internet.
At this page http://www.ctv.es/USERS/gbv/GIMPS/prime.htm , in spanish,
you can find links to mersenne primes, their search and history.
In particular you could be interested in
http://www.scruznet.com/~luke/mersenne.htm
and
http://www.utm.edu/research/primes/mersenne.shtml
Un saludo,
Ignacio Larrosa Can~estro
A Corun~a (Espan~a)
ilarrosa@linuxfan.com
------------------------------
Date: Tue, 11 Apr 2000 07:36:18 EDT
From: Bill Everdell
Subject: Re: [HM] Mathematics and Music
<>
is from Alfred North Whitehead, "Science and the Modern World," Lowell
Lectures, Harvard University, 1925
Bill Everdell, Brooklyn
------------------------------
Date: Wed, 12 Apr 2000 09:16:15 +0200
From: "Heinrich C. Kuhn"
Subject: Re: [HM] Tartaglia
> . . . Tartaglia, after much persuasion,
[...]
> and Tartaglia divulged his formula in a
> poem,
> Do any correspondents know whether the poem is extant, whether
> in Italian, in Latin, or in translation?
Italian and German translation in:
http://www.mathematik.uni-kl.de/~luene/miszellen/Tartaglia.html
HTH
Heinrich C. Kuhn
+-------------------------------------------------------------------
| Dr. Heinrich C. Kuhn
| Seminar fuer Geistesgeschichte der Renaissance
| Ludwig-Maximilians-Universitaet Muenchen
| D-80539 Muenchen / Ludwigstr. 31/IV
| T.: +49-89-2180 2018, F.: +49-89-2180 2907
| inst. URL: http://www.phil-hum-ren.uni-muenchen.de/
| priv. URL: http://www.phil-hum-ren.uni-muenchen.de/php/Kuhn/
+-------------------------------------------------------------------
------------------------------
Date: Wed, 12 Apr 2000 10:25:45 +0200 (MESZ)
From: Heinz Lueneburg
Subject: Re: [HM] Tartaglia
Dear all,
Victor E. Hill IV wrote:
>
>
> The entry on Tartaglia found at
>
> http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Tartaglia.html
>
> states, in part, "So, in March 1539, Tartaglia left Venice and travelled
> to Milan.
> . . . Tartaglia, after much persuasion, agreed to tell Cardan his
> method, if
> Cardan would swear never to reveal it and furthermore, to only ever
> write it down in code so that on his death, nobody would discover the
> secret from his papers. This Cardan readily agreed to, and Tartaglia
> divulged his formula in a poem, to help protect the secret, should the
> paper fall into the wrong hands."
>
> Do any correspondents know whether the poem is extant, whether in
> Italian, in Latin, or in translation? Can anyone provide a reference?
The poem is in Quesito XXXIII in
Tartaglia, Quesiti et inventioni diverse. Riproduzione in facsimile
dell'edizione del 1554. Edita con parti introduttorie da Arnaldo Masotti.
Brescia 1959
The poem is in fact plain text for mathematicians of the 16th century. The
text here is copied from a TeX-file. I think that you can interprete the
accents and Umlaute.
Quando chel cubo con le cose appresso
Se agguaglia \`a qualche numero discreto
Trouan dui altri differenti in esso.
Da poi terrai questo per consueto
Che'llor produtto sempre sia eguale
Al terzo cubo delle cose neto,
El residuo poi suo generale
Delli lor lati cubi ben sottratti
Varra la tua cosa principale.
In el secondo de cotesti atti
Quando che'l cubo restasse lui solo
Tu osseruarai quest'altri contratti,
Del numer farai due tal part'\`a uolo
Che l'una in l'altra si produca schietto
El terzo cubo delle cose in stolo
Delle qual poi, per commun precetto
Torrai li lati cubi insieme gionti
Et cotal somma sara il tuo concetto.
El terzo poi de questi nostri conti
Se solue col secondo se ben guardi
Che per natura son quasi congionti.
Questi trouai, \& non con passi tardi
Nel mille cinquecent\'e, quatroe trenta
Con fondamenti ben sald'\`e gagliardi
Nella Citta dal mar'intorno centa.
Here my translation of the poem into German.
Wenn der Kubus mit den Cossen daneben x^3 + px = q
gleich ist einer diskreten Zahl,
finden sich als Differenz zwei andere in dieser.
Dann halte es wie gew\"ohnlich,
dass n\"amlich ihr Produkt gleich sei
dem Kubus des Drittels der Cossen,
Und der Rest dann, so die Regel,
ihrer Kubusseiten wohl subtrahiert
wird sein deine Hauptcoss.
In dem zweiten von diesen F\"allen, x^3 = px + q
wenn der Kubus allein steht
und du betrachtest die anderen zusammengezogen,
Von der Zahl mache wieder zwei solche Teile,
dass der eine in den anderen multipliziert
den Kubus des Drittels der Coss en ergibt.
Von jenen dann, so die gemeine Vorschrift,
nimm die Kubusseiten zusammen vereint
und diese Summe wird dein Konzept sein.
Die dritte nun von diesen unseren Rechnungen x^3 + q = px
l\"ost sich wie die zweite, wenn du wohl beachtest,
dass sie von Natur aus gleichsam verwandt sind.
Dieses fand ich, nicht schwerf\"alligen Schritts,
im Jahre tausendf\"unfhundertvierunddreissig
mit Begr\"undungen triftig und fest
In der Stadt vom Meer rings umg\"urtet. Venice
Observe that Tartaglia could not solve the general cubic equation. He did
not know that one can transform each cubic equation into one without a
square term. Cardano knew at least two such transformations.
Concerning the oath Cardano should have sworne, here is what Tartaglia
wrote about it, also in Quesiti XXXIII.
Io ui giuro, ad sacra Dei euangelia, \& da real gentil'huomo, non solamente
da non publicar giamai tale uostre inventioni, se me le insignate. Ma
anchora ui prometto, at impegno la fede mia da real Christiano, da notarmele
in zifera, accio che dapoi la mia morte alcuno non le possa intendere, se me
il uoleti mo credere credetilo, se non lassatilo stare.
notarmele in zifera = enciphered
Note that Ludovico Ferrari says somewhere in the Cartelli di Sfide that
Tartaglia is lying here and that he - Tartaglia - should not forget that
he - Ferrari - was present when Cardano and Tartaglia were talking to each
other. Ferrari, of course, is Cardano's partisan. So it is not clear whether
Cardano ever swore an oath in this context. Cardano is very vague in his
autobiography on this issue. He barely mentions it.
The St. Andrew entry "Tartaglia" is not reliable.
Best regards, Heinz Lueneburg
------------------------------
Date: Tue, 11 Apr 2000 21:16:45 +0200
From: "Prof.ssa Laura Laurencich"
Subject: Re: [HM] Matematica precolombina
Dear list members:
I agree with Ubiratan D'Ambrosio's review of Gary Urton's "The Social
Life of Numbers": this book is a very important contribution to Quechua
philosophy of numbers!
I agree also with Milo Gardner that a linguistic research on Aymaran
language may open new perspectives on Andean ethnophilosophy of numbers.
However I regret that Gary Urton, when presenting his manuscript of "The
Social Life of Numbers" to his editor, did not know yet the early 17th
century Jesuitic document "Historia et Rudimenta Linguae Piruanorum" which,
for instance, let us read the meaning of the numbers and of the geometrical
figures of the Inca's shirts pictured on Guaman Poma's Nueva Coronica y Buen
Gobierno. If he could have read it, his interpretation of Quechua math
beeing a mathematics of rectification might be more puctual.
As a matter of fact, this document and moreover the jesuitic manuscript
Exsul Immeritus Blas Valera Populo Suo (from now on EI), written by the
mestizo Father Blas Valera in 1618 (who was still alive, as far as the two
documents claim, as well as other archival documents recently found), opens
new perspectives on the Inca mathematics. I presented EI on the
international symposium "Guaman Poma y Blas Valera" (Instituto Italo-Latino
Americano, Roma, Sept.29th-30th 1999) and on HISTORIA MATEMATICA V1, number
274 (of December 16th 1999, subject: An Italian archive) I wrote a very
short review of the many contributions that EI and Historia et Rudimenta
give to the early colonial Peruvian history, to the Peruvian cronicles, to
the Inca system of writing and to the Inca mathematics; on the same review I
wrote also a very short bibliography just to open the discussion on the two
documents and on what there is written on the Inca's mathematics.
According to the two Jesuitic documents, the Inca's math was a mathematics
which organized the whole universe into numbers. In other words they do not
reject Urton's interpretation of Quechua math beeing a math of rectification
but they answer why: the aim of Inca mathematics was a continuous
rectification for organizing the universe, i.e. it was a trascendental math
where the numbers were either gods or cosmic forces. According to EI, the
Inca and his wise men (i.e. the astronomer-astrologists, the historians,
the poets) worked for connecting the whole Tahuantinsuyu to the Upper and to
the Inner worlds by the literary quipu, by the numerical quipu, by the
yupana (i.e. by numbers and by geometry).
This ethnophilosophy of mathematics fits into the conception of the Inca
theocratic empire where the Inca was the sun-god in the earth, the pivot
between men and gods, and is similar to the Maya conception of astrononomy
and of mathematics: as a matter of fact both cultures belonge to the ancient
times of the history of the world and of the Americas but the Inca's
mathematics conception seems more monolithic to me than the Maya's, i.e. the
Inca reduced into numbers not only the gods and the supernatural forces (as
the stars and the planets) but also all the tributary sources as well as
the inhabitants which were all counted, that is organized into numbers and
they all were means, in the hands of the Inca, for keeping the Universe
organized.
Anyhow further researches are needed on the Inca mathematic and on the Inca
phylosophy of arithmetic according to Historia et Rudimenta Linguae
Piruanorum and to EI.
Regards to you all!
Laura Laurencich Minelli
Dip. di Paleografia e Medievistica
Universita' dei Bologna
Bologna, Italy
------------------------------
Date: Wed, 12 Apr 2000 12:12:57 +0200
From: Ivo Schneider
Subject: Re: [HM] Tartaglia
Moritz Cantor, Vorlesungen ueber Geschichte der Mathematik, vol. II, second
edition, Teubner 1900 and Reprint Johnson 1965, quotes the poem on p. 488 f.
According to Cantor Tartaglia published the poem in his Quesiti et
inventioni diverse, Venice 1554, p. 266.
Ivo Schneider
------------------------------
Date: Wed, 12 Apr 2000 12:37:52 +0200
From: Milan Bozic
Subject: Re: [HM] Tartaglia
"Victor E. Hill IV" wrote:
> The entry on Tartaglia found at
>
> http://www-groups.dcs.st-and.ac.uk/history/Mathematicians/Tartaglia.html
>
> states, in part, "So, in March 1539, Tartaglia left Venice and travelled
> to Milan. ... Tartaglia, after much persuasion, agreed to tell Cardan
> his method, if Cardan would swear never to reveal it and furthermore, to
> only ever write it down in code so that on his death, nobody would
> discover the secret from his papers. This Cardan readily agreed to, and
> Tartaglia divulged his formula in a poem, to help protect the secret,
> should the paper fall into the wrong hands."
But, has anyone explanation for such "eternaly extended secrecy". Or, if
Tartaglia's idea was to keep the algorithm secret even after death, what
was the point behind it? To keep it secret forever? Why? On whose benefit?
Milan Bozic
------------------------------
End of HISTORIA MATEMATICA V2 #15
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HISTORIA MATEMATICA Thursday, April 13 2000 Volume 02 : Number 016
~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE of SUBJECTS ~~~~~~~~~~~~~~~~~~~~~~~~~~
Re: [HM] Fundamental Theorem of Algebra
Re: [HM] Tartaglia
Re: [HM] mystery-woman
Re: [HM] Tartaglia
Re: [HM] Tartaglia
[HM] Fundamental theorem of calculus
Re: [HM] Fundamental theorem of calculus
Please see the end of this digest.
----------------------------------------------------------------------
Date: Wed, 12 Apr 2000 09:37:19 -0400 (EDT)
From: Christopher Baltus
Subject: Re: [HM] Fundamental Theorem of Algebra
On Tue, 11 Apr 2000, Heinz Lueneburg wrote:
> I and others call this proof the Cauchy-Argand proof of the FTA. I
> got from Doerrie the quotation
>
> Argand, Annales de Gergonne 1815.
>
> I have never seen this paper. Cauchy has published his proof in his
>
> Cours d'analyse. Paris 1821. Oeuvres, Ser. 2, Vol 3. Paris 1897
>
> and in
>
> Exercises de mathe/matiques. Quatrie\me anne/e. Paris 1829
> . . . .
Cauchy*s proof, essentially the same as the 1820 version, is presented
in two Notes of 1817, found in Oeuvres Series 2 Tome 2, p 210-216 and p
217-222. Argand's proof is particularly clear and simple (it assumes that
a continuous function takes on its max and min on a closed disk.) The
1814/1815 version was reprinted in an 1874 book with the name of the
original article, Essai sur une Maniere de Representer les Quantites
Imaginaire, editor Houell, reissued in 1971 by Librairie Scientifique et
Technique, Albert Blanchard, Paris.
Christopher Baltus
Oswego, NY
------------------------------
Date: Wed, 12 Apr 2000 10:26:54 -0500
From: "David Kullman"
Subject: Re: [HM] Tartaglia
Dear List Members:
An English translation, by F. R. Smith, of (at least a portion of)
Tartaglia's poem will be found in Fauvel and Gray, _The History of
Mathematics - A Reader_, MacMillan Press, 1987, pp. 255-256.
Best wishes,
David Kullman
Miami University
Oxford, Ohio
> ------------------------------
>
> Date: Tue, 11 Apr 2000 13:24:31 +0100
> From: "Victor E. Hill IV"
> Subject: [HM] Tartaglia
>
> Do any correspondents know whether the poem is extant, whether in
> Italian, in Latin, or in translation? Can anyone provide a reference?
>
------------------------------
Date: Wed, 12 Apr 2000 11:08:03 -0300
From: Julio Gonzalez Cabillon
Subject: Re: [HM] mystery-woman
Dear Jim,
On page 383 of the article _Women in American Mathematics: A Century
of Contribution_ [1] there is a lovely picture [2] of members of the
Chicago Mathematics Department Faculty (mid-20s). There appears your
"mystery-woman", Mayme F. Irwin Logsdon [3], along with E.H. Moore,
G.A. Bliss, L.E. Dickson, W.D. MacMillan, F.R. Moulton and H.E. Slaught.
Notes:
[1] written by listmember Judy Green and Jeanne LaDuke. See pp 379-398
in "A Century of Mathematics in America, Part II" (edited by P. Duren,
R.A. Askey & U.C. Merzbach), AMS, vol 2, 1989.
[2] Date unknown (mid-20s).
[3] See also http://www.agnesscott.edu/lriddle/women/logsdon.htm
Kind regards,
Julio Gonzalez Cabillon
At 11:28 p.m. 09/04/00 -0500, James Propp typed:
|
| A few months ago, I asked:
|
| : Andre Weil, in his autobiography, mentions having received a copy
| : of Mordell's thesis in 1925 from a woman mathematician visiting
| : from the U.S. who gave a lecture on Diophantine equations. Weil
| : does not mention the woman's name, but it seems likely to me that
| : there were comparatively few female American mathematicians with
| : an interest in number theory at that time, so that it might be
| : possible to deduce who this woman was. Does anyone know about
| : this (or know who'd be likely to know)?
| :
| : Jim Propp
| : Department of Mathematics
| : University of Wisconsin
|
| Franz wrote:
|
| > Yep, it's M.I. Logsdon, one of Dickson's students. I'm pretty sure that
| > I know this from
| >
| > N. Schappacher,
| > D\'eveloppement de la loi de groupe sur une cubique,
| > S\'em. Th\'eor. Nombres Paris 1988-89, 159--184
| >
| > There's a recent article by Della Fenster on Dickson's female students
| > that you should find in the online version of the MR.
|
|
| Here's a reply from Judy Green that supports this and (I think) definitively
| settles the matter:
|
| From jgreen@phoenix.marymount.edu Fri Jan 14 16:51 CST 2000
| To: James Propp
| cc: Jeanne LaDuke
| Subject: Re: Andre Weil and M.I. Logsdon (fwd)
|
| Dear Prof. Propp
|
| It is certainly Logsdon. Weil describes her in more detail on page 524 of
| volume 1 of his collected works. From that description, a visitor to Rome
| from Chicago who had authored a paper on cubics in Trans. Am. Math. Soc. in
| 1925, it is very clear that the woman in question is Mayme I. Logsdon.
| Logsdon served as instructor at Chicago from 1921 to 1925 at which time she
| was promoted to assistant professor. During the year 1925-1926 she was an
| International Education Board Fellow studying in Rome. Her paper "Complete
| groups of points on a plane cubic curve of genus one" appeared in TAMS 27
| (1925): 474-490.
|
| You can find a short description of her life and work in the paper I wrote
| with Jeanne LaDuke (DePaul University) - "Contributors to American
| Mathematics: An Overview and Selection" in Women of Science: Righting the
| Record, edited by G. Kass-Simon and Patricia Farnes, 117-146.
| Bloomington: Indiana University Press, 1990.
|
| I hope this is of some help.
|
| Judy Green
------------------------------
Date: Wed, 12 Apr 2000 14:37:11 +0200 (MESZ)
From: Heinz Lueneburg
Subject: Re: [HM] Tartaglia
Milan Bozic asks:
>
> But, has anyone explanation for such "eternaly extended secrecy".
> Or, if Tartaglia's idea was to keep the algorithm secret even after
> death, what was the point behind it? To keep it secret forever?
> Why? On whose benefit?
>
According to Tartaglias own testimony, he did not want that someone
else publishes his discovery. Before publishing the results he wanted
to do further research, but he did not have the time for that at this
particular moment, because he was busy translating Euclid's elements
into Italian.
Heinz Lueneburg
------------------------------
Date: Wed, 12 Apr 2000 15:05:19 +0200 (MESZ)
From: Heinz Lueneburg
Subject: Re: [HM] Tartaglia
This is a postscriptum to my message containing Tartaglia's poem.
According to Tartaglia, he composed the poem as an aide-me\moire
for himself, i. e., he did not compose it for Cardano, as it is
told so often. This strengthens my saying in the first message that
the poem is plain-text for mathematicians of the 16. century.
Best regards, Heinz Lueneburg
------------------------------
Date: Wed, 12 Apr 2000 09:27:15 -0700
From: Charles Craig
Subject: [HM] Fundamental theorem of calculus
Who provided the first proof the Fundamental Theorem of Calculus
and when. I have found many references to Newton first stating it,
but nothing stating he was the first to prove it.
Charles Craig
------------------------------
Date: Wed, 12 Apr 2000 12:55:10 -0700 (PDT)
From: Barnabas Hughes
Subject: Re: [HM] Fundamental theorem of calculus
Perhaps if you were to define (i.e., state) what you understand as the
FTC, it would be easier to answer your question. You see, Newton and
Leibniz had different ideas about the FTC to which there is more than the
common idea that differentiation and integration are inverse operations.
Barnabas Hughes
On Wed, 12 Apr 2000, Charles Craig wrote:
> Who provided the first proof the Fundamental Theorem of Calculus
> and when. I have found many references to Newton first stating it,
> but nothing stating he was the first to prove it.
>
> Charles Craig
------------------------------
End of HISTORIA MATEMATICA V2 #16
*********************************
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