Subject: [HM] Bolzano, Cauchy, Epsilon, Delta (3/5)
From: Walter Felscher (walter.felscher@uni-tuebingen.de)
Date: Wed Feb 02 2000 - 06:01:44 EST
4C. d'Alembert versus Cauchy
The definitions of a limit quoted from d'Alembert
On dit qu'une grandeur est la limite d'une autre grandeur, quand
la seconde peut approcher de la premi\ere plus pr\es que d'une
grandeur donn/ee, si petite qu'on la puisse supposer, ...
and from Cauchy
Lorsque les valeurs successivement attribu/ees \a une m^eme
variable s'approchent ind/efiniment d'une valuer fixe, de mani\ere
\a finir par en diff/erer aussi petit que l'on voudre, cette
derni\ere est appel/ee la limite de toutes les autres.
have the same content. They both contain the "for every epsilon" in
the form of
approcher .. plus pr\es que d'une grandeur donn/ee, si petite
qu'on la puisse supposer
and
diff/erer aussi petit que l'on voudre ;
yet neither cares to mention the "there exists delta" - presumably as
their authors considered it as obvious that an approximation, once
achieved, would finally progress to the better. The difference between
the two authors is the use they make of their definitions: Cauchy
wanted to exhibit more than d'Alembert, wanted to prove consequences
of that definition, and to this end he disvovered the delta and its
role. J.V.Grabiner in her two articles
The origins of Cauchy's theory of the derivative.
Historia Math. 5 (1978) 379-409
Who gave you the Epsilon ? Cauchy and the origins of rigorous calculus.
American Math.Monthly (1983) 185-194
has pointed out the location where this took place: the Th/eorem\e in 1823 ,
Oevres 4 , p.44 ; Cauchy there even employed both the letters epsilon and
delta as they are used in today's definition. [While these articles list
Bolzano's booklet in their references, their appreciation expressed of Cauchy
"It is a commonplace that Augustin-Louis Cauchy gave the first generally
acceptable account of the basic concepts of the calculus" (1978 , p.380)
... "The epsilon notation was introduced into analysis by Cauchy" (1978
, p.382) ... "Cauchy gave the calculus a rigorous basis" (1983 , p.188 )
appears as rather one-sided. ]
4D. Continuity
Cauchy's definition of continuity in 1821 pp.34-35 (Oevres p.43) reads
Cela pos/e, la fonction f(x) sera, entre les deux limites assign/ees
\a la variable x , fonction continue de cette variable, si, pour
chaque valeur de x interm/ediaire entre ces limites, la valeur
num/erique de la diff/erence f(x+a)-f(x) d/ecro^it ind/efiniment
avec celle de a . En d'autres termes, la fonction f(x) restera
continue par rapport \a x entre les limites donn/ees, si, entre ces
limites, un accroissement infiniment petit de la variable produit
toujours un accroissement infiniment petit de la fonction elle-m^eme.
This are actually two formulations. In 1823 only the second one
appears in the definition of continuity.
It should be noted that in the first formulation x actually is kept
fixed and that here the letter a now denotes a variable. The first
formulation says that |f(x+a)-f(x)| decreases indefinitely "together"
with a decreasing indefinitely. Here the "avec celle de a " may
appear symmetrical; if it were understood as a one-sided conditional
"si celle de a d/ecroit ind/efiniment", then it would express C- or
B-continuity (depending on whether the variable a is viewed to decrease
in a sequence or not). There is no reason to assume that Cauchy,
would he have been pressed to the point, would not have chosen this
second phrasing.
The formulation "en d'autres termes" expresses the meaning: if the
assignment with values a is a quantit/e infiniment petite then so is
the assignment with values |f(x+a)-f(x)|. But if x is fixed and a is
a variable, then a converges to zero if, and only if, x+a converges
to x , and as f(x) then is fixed as well, also |f(x+a)-f(x)| converges
to zero if, and only if, f(x+a) converges to f(x) . So here Cauchy
again defines C- or B-continuity.
As far as I can see, Cauchy nowhere uses the first formulation in his
examples. His second formulation he puts to use, among other things,
in order to characterize the continuous solution of his functional
equation (1821 , p.103 ff), and later to prove the intermediate-value
theorem (1821 , p.462). A particularly striking application of
C-continuity occurs during the latter proof in the form of the
observation that, if two variables converge towards the same value a ,
then "puisque la fonction f(x) reste continue" also the associated
values of f both converge towards f(a).
5. Bolzano versus Cauchy
The works of Bolzano and Cauchy, from which the information discussed
here was taken, were written for different purposes. Cauchy wrote
textbooks, destined to support his introductory lectures on analysis.
The purpose of lectures is to teach new things, and it is a didactical
technique to also use motivations from known things and from intuition.
What Bolzano wrote was not a textbook, but an essay laying the conceptual
foundation for a particular, basic notion of analysis: continuity.
What for Cauchy were tools for things to come, for Bolzano was the
object of the investigation itself.
Cauchy in his textbooks covered many more theorems than Bolzano in
his foundational essay. [There are extensive manuscripts on analysis
of Bolzano's published from his estate, but they were dated from 1830.]
Cauchy was read by contemporaries and the following generations, Bolzano
was not. Bolzano's actual influence then was negligeable compared to
Cauchy's.
Both Bolzano and Cauchy have given definitions of continuity which
express today's B- respectively C-continuity. Both made their definitions
precise, using them in the sense of today; both employed them by comparing
numbers and their differences with help of inequalities in order to prove
important theorems in analysis. More generally, Cauchy defined and used
the notion of limit, Bolzano did not.
Cauchy once used the letters epsilon and delta in connection with limits;
Bolzano instead used D and omega in connection with continuity [though
he did use both letters epsilon and delta, albeit in a different
connection, in the proof of his pages 47-48]. Were we to write a
history of letters used as mathematical notations, Cauchy might be
mentioned in a comparition with Bolzano.
But the history of mathematics is not one of denotations, it is a
history of inventions and concepts. And there 1817 counts four years
before 1821 . (Of course, there seems to be no reason to assume that
Cauchy ever was aware of Bolzano's work.)
- o -
Outside of mathematics, the fates of both men were decisively
determined by the political developments of their times.
An extensive biographical notice on Bolzano can be found in the
"Biographisches Bibel-Lexikon", accessible online at www.bautz.de/bbkl .
Having had his mind formed at the time of Josephine enlightenment,
Bolzano was called to a chair established with the purpose to counteract
the liberalist mindset flowing in from France. With his lectures to
the university community, Bolzano was most successful in spreading
awareness of the social components of Catholic ethics. Yet until 1811
he had been under orders to use a textbook written by a Viennese
theologian. Against these he protested - finally with success, but
making himself influential personal enemies by that. When in 1819 a
friend of Bolzano's became involved with a local student unrest, his
enemies used the occasion to denounce him as a dangerous radical, and
in 1820 he was dismissed from his chair and forbidden to teach; an
ensuing Church-internal process, aiming to have his missio canonica
withdrawn, failed in 1825 through the support given to him from his
bishops. After his dismissal, Bolzano lived from occasional work as
private instructor, as sometimes substituting priest, and from the
support of personal friends.
Lazare Carnot in 1792 had been one of the conventionels regicides;
later he had been Napol/eon's last minister of the interior. In 1815
he fled France never to return, and in 1816 was struck from the list
of members of the Acad/emie. Cauchy was appointed his successor, not
yet 27 years old. At the Polytechnique, Cauchy held his chair until
1830 when in July the French liberals, mobilizing the Paris mob,
forced king Charles X to abdicate, ending the house of Bourbon,
and set up an Orl/eans as their kinglet, the son of the infamous
Philippe /Egalit/e. Cauchy refused to swear allegiance to the
usurpator and, consequenly, lost his position, remaining, though, a
member of the Acad/emie. He left France, taught in Torino and Prag
and returned to Paris only in 1838 . In 1839 the ministry vetoed the
vote of his colleagues who had elected him to a seat at the Coll\ege
de France. At the same year, the members of the Bureau des Longitudes
elected him to their body, but after he had served there for four
years, the ministry had the election annulled. In 1848 the republic
appointed Cauchy to a chair at the Facult/e des Sciences; in June
1852 he had to give it up when an oath of allegiance was required
again by Louis Napol/eon ; in 1854 finally he was granted the chair
by special privilege and sans condition.
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