Bolzano, Cauchy, Epsilon, Delta

In this article, I describe how the notions of limit and continuity were explained and used between 1817 and 1823 by Bolzano and Cauchy.
In a closing section I discuss the extent to which the technique of epsilon and delta serves a tool to write finite proofs of statements which, involving limits and continuity, refer to infinite processes.
 
 

1. Today's terminology

Let f be a function defined in a neighbourhood 
V = {x: |a-x| < q} of a number a .

A number b is defined to be the " B-limit of f at a " if

  for every e > 0
  there exists d > 0 (with d < q)
  for every y : if |a-y| < d  then  |b-f(y)| < e  .

Let s be a sequence <x(i): i < w>   of numbers x(i) indexed by the set w  of natural numbers. The sequence s is defined to " converge
to a " , and a is then the " limit of s ", if

  for every e > 0
  there exists n in w
  for every i in w : if i > n then |a - x(i)| < e  .

A number b is defined to be the " C-limit of f at a " if

  for every sequence s = <x(i): i < w> with values in    V
  if s converges to a   then   fs = <f(x(i)): i < w> converges to b .
 

Lemma: b is B-limit of f at a if  and only if b is C-limit of f at a .
 

Let % be one of B or C . The function f is defined to be %-continuous at a  if  f(a) is the %-limit of f at a 
 

Lemma: f is B-continuous at a if and only if f is C-continuous at a .
 
 

2. D'Alembert's program.

  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, sans pourtant que la grandeut qui approche, puisse jamais surpasser la grandeur  dont elle approche; en sorte que la diff/erence d'un pareille
quantit/e \a sa limite est absolument inassignable ...

  A proprement parler, la limite ne conincide jamais, ou ne devient  jamais /egale, a la quantit/e dont elle est la limite; mais celle-ci s'en approche toujours deplus en plus et peut en diff/erer aussi
 peut qu'on voudra ...

In its 4th volume 1754 of he wrote in the article "Diff/erentiel"

  Celui-ci nous paro^it suffire pour faire entendre aux commen,cans la vraie m/etaphysique du calcul diff/erentiel. Quand une fois on l'aura bien comprise, on sentira que la supposition que l'on y
fait de quantit/es infiniment petites, n'est que pour abr\eger &  simplifier les raisonnemens; mais que dans le fond le calcul  diff/erentiel ne suppose point n/ecessairement l'existence de ces  quantit/es; que ce calcul ne consiste qu'a d/eterminer alg/ebriquement la limite d'un rapport de laquelle on \a d/eja l'expression en lignes , & \a /egaler ces deux limites, ce qui
fait trouver une des lignes que l'on cherche.

  Il ne s'agit point, comme on le dit encore ordinairement, de quantit/es infiniment petites dans le calcul diff/erentiel; il s'agit uniquement de limites de quantit/es finites. Ainsi la m/etaphysique & des quantit/es infiniment petites plus grandes ou  plus petites les unes que les autres, est totalement inutile au  calcul diff/erentiel. On ne se sert du terme d'infiniment petit,  que pour abr\eger les expressions. Nous ne dirons donc pas avec bien des g/eometres qu'une quantit/e est infiniment petite, non avant qu'elle s'/evano"uisse, non apr\es qu'elle est /evano"uie,
mais dans l'instant m^eme o\u elle s'/evano"uit; car que veut dire une d/efinition si fausse, cent fois plus obscure que ce  qu'on veut d/efinir ? Nous dirons qu'il n'y a point dans le calcul diff/erentiel de quantit/es infiniment petites.
 

Reading these words today, we may receive the impression that they might as well have been written at the time of Weierstrass, of Cantor,
or even by a contemporary mathematician: all that matters in analysis is the notion of limit, and there is no place at all to conceive infinitesimals. Yet when they were written, the details of how to mathematically work with limits were hardly worked out, and what d'Alembert wrote here was less a description of the actual state of affairs, but rather a program to be carried out in the
future. It _was_ carried out, with the decisive steps performed by Bolzano and Cauchy.

3. Bolzano.

In this position, in 1817 Bolzano published a book of 60 pages with the title

  Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewaehren, wenigstens
 eine reelle Wurzel der Gleichung liege

Here  he gave a definition of continuity of a function f at an argument x :

  if x is such argument then the difference f(x+w)-f(x)  can be made smaller than any given quantity if only w is assumed as small as wanted

I have shown that whenever Bolzano uses his definition in his proofs, this use consists precisely in the verification of

  given e > 0 , there is d > 0 such that w < d implies |f(x+w)-f(x)| < e .

For example, in the case of  a particular f Bolzano determines, on p.58 of his book, for a given e (he uses the letter D ) the number
d as the smaller of two numbers w_1  and D/S .

Thus Bolzano's notion of continuity is intended precisely as that of B-continuity.

At first sight, Bolzano's concepts, presented with unambiguous perspicacity, appear to have sprung from his head, as Pallas did spring from that
of Zeus.

Bolzano does not appear to have had contact with mathematicians apart from his acquaintances in Prague. In consequence, his mathematical work remained completely unknown and came to the notice of the mathematical community only thirty years after his death. In particular, Bolzano's writings had no influence upon the re-discovery
of B-continuity by Weierstrass and others.
 
 

4. Cauchy

4A. Cauchy on variables and their limits

Augustin Louis Cauchy (21.8.1789 - 23.5.1857) set out to fulfill d'Alembert's program in his two textbooks of analysis :

  Cours d'Analyse de l'/Ecole Royale Polytechnique , Paris 1821 (reprinted in Oevres Compl\etes, 
s/er.2 , vol.3 )

  R/esum/e des Le,cons donn/ees a l'/Ecole Royale Polytechnique sur l'Calcul Infinit/esimal , Paris 1823 (reprinted in Oevres Compl\etes, s/er.2 , vol.4 )

In both books, Cauchy begins by attempting to give a mathematical description of what is  conceived as a limiting process. He explains on page 4 of 1821 (Oevres 3 , p.19) the term "variable quantity"

  On nomme quantit/e variable celle que l'on consid\ere comme devant recevoir successivement plusieurs valeurs diff/erentes les une  apr\es des autres.  On d/esigne une semblable quantit/e par une lettre prise parmi les derni\eres de l'alphabet.  ...

He proceeds to explain limits of such assignments:

  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.  ...

In particular, there are variables with assignments which have the limit zero:

  Lorsque les valeurs num/eriques successives d'une m^eme variable  d/ecroissant ind/efiniment, de mani\ere \a s'abaisser au-dessous  de tout nombre donn/e, cette variable devient ce qu'on nomme un  infiniment petit ou une quantit/e infiniment petite. Une variable de cette esp\ece a z/ero pour limite.

The same definitions appear in 1823 (Oevres 4 , p.16). In Cauchy's further text then, when these definitions are referred to, the phrase "aussi petit que l'on voudre" usually is expressed by saying that the difference, between the values of an assignment to the variable, and the fixed value can be made smaller than any given positive number.
 

4B. Comments

A variable for Cauchy then _is not_ a letter, but a concept: the concept of a form to be filled by _assignments_ A of values. The domains L of such assignments A remain unspecified; apparently
they shall at least carry an order (such that one can speak of "valeurs les unes apr\es des autres"). In particular, assignments _may_ be conceived as sequences defined on w , but they may also be
more general assignments, e.g. the identity map on a whole interval.

The ranges of assignments are not specified either, but it appears clear from Cauchy's words that they shall be num/eriques, i.e. consist of real numbers.

 Note  : Cauchy says that values are assigned to a variable, he does not use the defineed notion of an assignment. Nor does he distinguish notationally between 

(1) a variable, for which he
  may write x , 

(2) an assignment, and 

(3) the values of that assignment:  quite often x then also stands for a value assigned.]

 
A variable with an assigment converging to zero is said "to become" a "quantit/e infiniment petite". Since already a variable was introduced as a "quantit/e variable" by Cauchy, the word "quantit/e" here appears to have a more abstract meaning than that of number or of magnitude in
the geometric continuum. [Of course, it is left open whether a quantit/e variable, with an assigment converging to zero, actually _is_ or only _becomes_ a quantit/e infiniment petite.] Thus a quantit/e infiniment petite is of quite a different species than are numbers (the valeurs num/eriques of variables) or, equivalently, geometric magnitudes.

Cauchy's purpose when introducing the quantit/ees infiniment petitesis expressed in 
1823 , p.9 :

  Mon but pricipal a /et/e de concilier la rigeur, dont je m'/etais fait une loi dans mon Cours d'analyse, avec le simplicit/e qui r/esulte de la cond/eration directe des quantit/es infiniment petites.

It rests on the observation that a number c is the limit of the assignment of values j to the variable x if and only if 0 is the limit of the assigment of j-c to the variable x-c , i.e. this variable under that
assignment becomes infiniment petite. Based on it, statements about limits can be translated into statements about quantit/ees infiniment petites.

As an informative example, let me quote from 1823 , Oevres 4 , p.18 :

    Cela pos/e, si la variable y est exprim/ee en fonction de la    variable x par l' /equation

    (1)  y = f(x) ,

    D_y , ou l'accroissement de y correspondent \a
    l'accroissement D_x de la variable x , sera    d/etermin/e par la  formule

    (3)  y + D_y = f(x + D_x)   .

    [p.19]  ... Il est bon d'observer que, des /equations (1) et (2)   r/eunies, on conclut

    (5)  D_y = f(x + D_x) - f(x) .

    Soient maintenant h et i deux quantit/es distinctes, la premi\ere   finie, la seconde infiniment petite, et a = i/h  le  rapport infiniment petit de ces deux quantit/es. Si l'on attribue
    \a D_x la valeur finie h , la valeur de D_y , donn/ee par l'equation (5), deviendra ce qu'on appelle la diff/erence de la   fonction f(x), et sera ordinairement une quantit/e finie. Si, au  contraire, l'on attribue \a D_x une valeur infiniment petite, si
l'on fait par example

    D_x = i = a.h  ,

    la valeur de D_y , savoir

    f(x+i) - f(x)   ou   f(x + a.h) - f(x)

    sera ordinairement une quantit/e infiniment petite. C'est ce que    l'on verfiera ais/ement \a l'/egard des fonctions

    A^x , ...

    auxquelles correspondent les diff/erences

    A^x+i - A^x = (A^i - 1).A^x  , ...

    dont chacune renferme un facteur A^i - 1 ou ... qui converge   ind/efiniment avec i vers zero.

So Cauchy here considers a quantit/e infiniment petite i and a quantit/e finite h (of which we are not certain whether it shall be understood as a positive number or as a variable with an assignment converging to something different from zero). Cauchy then writes a = i/h which is (a variable with) an assigment having as values the quotients of the values of i and the value(s) of h (there is the obvious notational confusion which comes from denoting both the variable i and its
values by the same letter). If now D_x is i = a.h then D_y is f(x+i)-f(x) . This he illustrates by the exponential function f(x)=A^x where

   D_y : A^x+i - A^x = (A^i - 1).A^x  .

Thus here D_y is a quantit/e infiniment petite, presented by the quantit/e infiniment petite B with values A^i - 1 and the constant number A^x .

In particular, the above example shows that the three quantit/ees infiniment petites i , D_y and B have as values ordinary real numbers(and all converge to zero). There are no 'infinitesimal', non-Archimedean numbers ever used by Cauchy for his quantit/ees infiniment petites.
 

4C. d'Alembert versus Cauchy

  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 , Ouevres 4 , p.44 ; Cauchy there even employed both the letters e and
d 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

  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

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- and 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 e and d in connection with limits; Bolzano instead used D and omega in connection with continuity [though he did use both letters e and d, 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 comparison 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 of counteracting
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 until1830 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 usurper and, consequenly, lost his position, remaining, though, a member of the Acad/emie. He left France, taught in Torino and Prague 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.
 
 

6. Bestiarium infinitesimale

Abraham Robinson (1918-1974) embedded the real numbers R into a non-Archimedean field S in which all L-sentences true in R hold as well. [There L is a language with names for all real numbers, predicate symbols for all sets and relations of and between real numbers, and with function symbols for all real functions. It then becomes possible to extend the notions of 'standard' analysis from R to a 'non-standard' analysis on S where now the presence of infinitesimal numbers, i.e. numbers less than 
1/n for all  natural n, permits to phrase certain arguments in a new and rigorous way which Leibniz and his contemporaries had had to leave
in vagueness.

In his book "Nonstandard Analysis" of 1966 Robinson quoted Cauchy's definitions on pp.269-270 and then continued:

  We gather from the above passages that infinitely small quantities are fundamental in Cauchy's approach to Analysis. However, these
quantities are not numbers but variables, or rather, states of  variables whose limit is zero. ...

  Whatever the precise picture of an infinitely small quantity may have been in Cauchy's mind, we may examine his subsequent definitions and see what they amount to if we interpret the infinitely small and infinitely large quantities mentioned in them in the sense of  Non-standard Analysis. For the notion of continuity, Cauchy's
definition may thus be interpreted as stating that for f(x) defined in the interval [a,b] , f(x) is continuous in that interval if, for  infinitesimal a  the difference f(x+a)-f(x) is always   infinitesimal. If now we interpret 'always' as meaning 'for all standard  x ' then we obtain ordinary continuity in the interval, but if by  'always' we mean 'for all x ' then we obtain uniform continuity.

With this last paragraph Robinson introduces what I choose to call the Robinson interpretation:

  (a) where Cauchy writes about quantit/ees infiniment petites   assume them as infinitesimal numbers in the sense of Non-standard  Analysis and

  (b) read Cauchy's following developments under this assumption,   together with assuming today's (or Weierstrass') standard knowledge   about distinctions such as usual versus uniform continuity etc.

This interpretation resulted in beautiful mathematical insights. In particular, certain erroneous statements of Cauchy's (on series of
continuous functions, on continuity in several variables ...) in this way can be read as correct statements in Non-standard Analysis. In this connection there are two informative articles by John P.Cleave:

  Cauchy, Convergence and Continuity. British J.Phil.Sci. 22 (1971) 27-37

  The concept of 'variable' in nineteenth century analysis. British J. Phil.Sci. 30 (1979) 266-278
 

But it must be emphasized that Robinsons's interpretation is in no way an explanation of the historical content of Cauchy's writings, and it is clear from Robinson's word's above that it does not purport to be one. Cauchy wrote explicitly that his quantit/ees infiniment petites have valeurs num/eriques, real numbers :

  "Lorsque les valeurs num/eriques successives d'une m^eme variable ... ".

Of course, the non-Archimedean, infinitesimal numbers of Pascal, say, were still in the back of Cauchy's and his contemporaries' minds. But
for his quantit/ees infiniment petites Cauchy purposely stipulated that they were assignments to variables the values of which were real
numbers. There is no mention of infinitesimals anywhere in Cauchy's writings.

And so we have had a glance at the bestiarium infinitesimale.
 
 

7. The workings of finitarization

It is not surprising that one can prove insights about triangles or about numbers, for instance the decomposition into prime factors and its uniqueness. But analysis deals with infinite processes, or at least what we imagine to be such - and yet we manage to prove our statements by writing only a finite number of lines.

We speak of a function or a variable approaching some value indefinitely; we imagine a limiting process. Bolzano's and Cauchy's analysis of the notions of limit and of continuity started from e, as a measure of approximation, and proceeded to the d on which e depends. Expressed with e and
d, the conception of a limiting processes appears to have been described in a finitary manner - at least as far as the mathematical use is concerned.

So far, the e and d (and in case of sequences also the n and N ) appear as handles, as grips, affixed to the stages of those infinite processes. If appropriately handled in our mental exercises,
that is, with the correct use of the quantifiers 'for all' and 'there exists'  they permit us to get away with finitely many arguments in order to prove statements which, in the end, will speak about all the stages of the infinite process. 

And so the quantifier rules are some of the few places in the mathematical teddy bear's fur where the stitches become visible.  The seams below
which the linguistical sawdust and splinters are hidden, the letters serving as variables say, which make up its body.  It is the use of these
linguistical tools, the recourse to the plain facts of language, which enables us to finitarize our handling of infinite processes.
 

W.F.
 

 Last update February 4, 2000

Author: Phill Schultz, schultz@maths.uwa.edu.au