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Arch. Hist. Exact Sci. 54 (1999) 101–135.c© Springer-Verlag 1999

The First Modern Definition of the Sumof a Divergent Series:

An Aspect of the Rise of 20th Century Mathematics

Giovanni Ferraro

Communicated byJ. Gray

Introduction

In this paper my aim is to clarify the conceptual aspect of the rise of the modernconcept of summability. I shall limit myself to the origin of the first modern definitionof the sum of divergent series which was formulated by Ernesto Cesaro1 in 1890 aftervarious attempts starting from 1882. The historical circumstances of these attempts,closely related to the rise of 20th century axiomatics, are a very interesting example ofthe transition to modern mathematics. The developments following Cesaro’s definition,above allEmile Borel’s contributions, which actually gave rise to summability theory,are beyond the scope of this article.

In the first part, I outline the conceptions underlying the different solutions of theproblem of the sum of divergent series and point out that the modern use of the term‘definition’ differs from that of 18th and 19th century. I thus highlight that Cesaro’sreal novelty was precisely to give the firstmoderndefinition of the sum andnot simplythe first definition, which is actually due to Euler at the middle of 18th century.

1 Cesaro was born in Naples in 1859. From 1870 his originally affluent family came moreand more in economical difficulties and supported his studies only unsatisfactorily. To shortenthe long course of study necessary to be admitted to a university in Italy, he went to study inLi ege. In Belgium he devoted himself to the scientific research under the influence of Eugene-Charles Catalan (1814–1894) and his circle. When he went back to Italy in 1883, Cesaroenrolled at the University of Rome, where he met Giuseppe Battaglini (1826–1894). He didnot graduate, but wrote almost 100 papers from 1880 and 1886. Thanks to the good offices ofCatalan, Charles Hermite (1822–1901), Luigi Cremona (1830–1903), and Luigi FedericoMenabrea (1809–1896), he was appointed full professor in Palermo in 1886. In 1891 he moved toNaples; however, he was on bad terms with Neapolitan Academicians, who traditionally attachedmore importance to career developments than scientific contributions. In 1906, a short time beforehe was to be transferred to the University of Bologna, he died by drowning. Cesaro worked onumbral methods, number theory, divergent series, differential geometry, probability theory andmathematical physics.

102 G. Ferraro

In the second part, I briefly illustrate the two sources of Cesaro’s definition: thepermanence of the formal method during the 19th century and the consequences of theinterpretation of Cauchy’s rigourous style by epsilontics in the 1880s (inter alia I pointout that one of Holder’s theorems was later incorrectly conceived as defining the sumof divergent series).

In the third part, I present in specific and selected detail the evolution of Cesaro’sthought from 1880 to 1890. The acknowledgement of the usefulness of divergent series,the attempts to “establish the fundamental principles of an asymptotic theory of num-bers” [1888a, 292], and the developments of the foundations of mathematics led him todifferent approaches to the problem of their sum.

Part one

1. Mathematical definitions

In his [1949, 5–6], G. H. Hardy stated: “[18th century mathematicians] had not thehabit of definition: it was not natural to them to say, in so many words, ‘byX wemeanY ’ . . . it is broadly true to say that mathematicians before Cauchy asked not ‘How shallwedefine1 − 1 + 1 − · · ·?’ but ‘What is 1 − 1 + 1 − · · ·?”’

I view the matter in a partially different way because the meaning and use of theword definitionchanged from 18th century to today. I prefer to say that 18th centurymathematicians lacked the habit ofmoderndefinition rather than the habit of defini-tion. Definitions were not lacking in 18th century; they were, however, definitions inEuclid’s sense and, as such, differed from definitions as conceived nowadays, after thedevelopment of axiomatic theories. Certainly, 18th century mathematicians did not ask‘How shall wedefine X?’ but ‘What is X?’; nevertheless the latter question is preciselythe hardcore of classical Euclidean definitions, which were used by mathematicians(Cauchy included) till the end of 19th century.

Today,2 a termX of a given theory is either a primitive term, implicitly definedby means of the axiomsX1, X2, X3, . . . , or it is explicitly defined by means of termsY1, Y2, Y3, . . . defined beforeX or by the primitive termsP1, P2, P3, . . .. The implicitdefinitions of primitive terms are the only starting point of modern mathematical theories,which have more than one model or interpretation according to the different interpre-tations of primitive terms. An objectX of the theory has a property if and only if thisproperty can syntactically be derived from the axioms of the theory by means of givenrules of derivation.

This is precisely the case of the definition of summability. Nowadays, the “sum”, ormore precisely the P-sum, of a series

∑an is a numberSassociated with

∑an according

2 Here I merely stress some aspects of the standard use of the term ‘definition’ in 20th centurymathematics, after Hilbert and Bourbaki. An epistemological inquiry of the concepts of thedefinitions is beyond the scope of this paper. In the same manner I do not intend to face anyquestions about Greek mathematics, and by speaking of Euclid and hisElements, I mainly referto the interpretation of Euclid in the 18th century.

The Definition of the Sum Divergent Series 103

to a given procedure P. If no numbers can be associated with∑an by the procedure P we

say that the series is not P-summable. The ordinary procedure is:S = limn→∞∑ni=1 ai .

Instead if we takeS0n = ∑n

i=1 ai ,

Skn =n∑i=1

Sk−1i =

n−1∑r=0

(r + k

k

)an−r , for k = 1, 2, 3 . . . ,

and consider

limn→∞

Skn(n− 1 + k

k

) = limn→∞

k!Sknnk

,

we have Cesaro’s procedure: if

limn→∞

Skn(n− 1 + k

k

)

exists, then the series is said to be C-summable and its (C,k) sum is

S = limn→∞

Skn(n− 1 + k

k

) .

Our choice of the procedure P is, in principle, arbitrary. Of course, when a mathemati-cian gives a definition of the sum, it is usual and appropriate that some properties arerequired for ausefulprocedure P of summation. One of these requirements is regularityor consistency (a procedure is said to be regular if it sums every convergent series toits ordinary sum). Nevertheless these requirements are useful but not essential: if wewish, we might imagine a procedure that assigns a sum different from its ordinary sumto a convergent series. Generally speaking, series have different sums or have no sumaccording to the chosen procedure, i.e., the sum is not a notion intrinsic to the conceptof series. The freedom of the choice of P is related to the conception according to whichtheories are free creative acts of our mind: we are free to make any assumptions or giveany definitions we please according to our taste or inclination, or the particular ends inview.

Contrary to modern ones, Euclidean definitions are ‘descriptive’ definitions, as theygive appropriate descriptions of certain objects which are taken as given. They limit themeaning of terms and fix the objects of the research precisely. In a sense, Euclidean def-initions discover the nature of mathematical objects, whereas modern definitions createmathematical objects, i.e., they themselves are tools for generating mathematical objectswhich exist in virtue of implicit or explicit definitions. Before the rise of axiomatics,mathematical objects were nota priori constructs; instead they were considered as mir-rors of reality, idealisations derived from the physical world, and having an intrinsicexistence before and independently from their definition. For this reason descriptivedefinitions always included semantic aspects which were closely related to syntacticalaspects. According to the different degrees of semantic and syntactical content, descrip-tive definitions can be subdivided into real and nominal definitions.

104 G. Ferraro

A real definition is of the kind: ‘byX we meanY’, whereY is an object that doesnot belong to the given theory. For example, the first two definitions of Euclid3 (‘Apoint is that which has no part’; ‘A line is breadthless length’) refer to pre-mathematicalpractices or experiences, i.e., to the pre-mathematical notions of part, breadth, length.Real definitions play an entirely semantic role and provide the interpretation of the theory.They intuitively, but unequivocally, link the mathematical theory with objects outside it(that is, ultimately, with empirical reality). And reality being unique, two mathematicaltheories, the first of which contradicts the second, cannot exist simultaneously. In modernwords, a theory has only one model; more, it is constructed starting from that modeland solely serves to investigate it. Of course, real definitions are lacking in modernmathematics and are replaced by undefined terms, which can be interpreted in differentsemantic ways.

Nominal definitions are of the kind: ‘byX we meanY’, whereY is a combination ofterms denoting known mathematical objects. Different from modern explicit definitions,which are merely syntactical and inside a given theory, having no semantic reference,nominal definitions are neither simply syntactical nor merely inside the theory. Theyalways refer to some real object and have a certain degree of semantic content, as amathematical object is not merely characterised by its definition but also by the refer-ence to non-mathematical objects or concepts. For instance, definition 23, Book I, anddefinition 2, Book III, of Euclid’s Elements(‘Parallel straight lines are straight lineswhich, being in the same plane and being produced indefinitely in both directions, do notmeet one another in either direction’; ‘a straight line is said to touch a circle which, meet-ing the circle and being produced, does not cut the circle’) refer to non-mathematical,empirical concepts of ‘meeting’, ‘cutting’, and ‘continuation’.4 In a sense, they are ‘in-complete’, not exhaustive definitions as they do not characterise mathematical objectsentirely.

For this reason, today Euclid’s Elementsseems to have many ‘holes’. Before ax-iomatics, mathematicians, however, viewed the matter differently. Euclidean geometrywas not a merely syntactical theory, the concept of which was lacking entirely till thelast century; we may indeed understand it only if we refer to the semantic content ofits notions. Today we state that two equal circles of radiusr intersect each other if theseparation of their centers is less than 2r, if and only if an appropriate axiom (or a theo-rem based upon appropriate axioms) guarantees their intersection. We can not refer thesemantic meaning of the term circle, the idealisation of material circles, to derive thisproperty. But that is precisely what Euclid did in the proof of his very first proposition,where he constructed an equilateral triangle.

3 It has recently been argued by Russo [1998] that these definitions may not be by Euclidhimself, but that does not affect the philosophical point being made here.

4 There are also definitions very similar to modern ones, for example, definition 21, BookI: ‘Of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angledtriangle that which has an obtuse angle, and an acute-angled triangle that which has its three anglesacute’ (Heath’s translation [1908]). Here Euclid defined right-angled, obtuse-angled, and acute-angled triangles by means of terms defined earlier; he, however, classified triangles that are takenas given in reality (even if this can be conceived as an immaterial or idealised reality).


Such a concept of definition permeated 18th and 19th century analysis at least tillthe rise of the new conceptions of Dedekind, Weierstrass, etc. In the 18th centurymathematics was considered as a ‘science of nature’ (see, e.g., the preliminary discourseto theEncyclopedieof d’Alembert) and analysis “was an extension of ordinary Algebraused to investigate the geometry of curves. The problems and theorems of the subjectarose in the course of these investigations; they did not appear as arbitrarily formulatedpropositions” (Fraser [1989, 330]). D’Alembert considered analysis5 as “the methodof making calculations with all kinds of quantities in general, representing them by en-tirely universal signs”.6 The combinations of such signs, i.e. analytical forms, were notmerely syntactical expressions, but the abstract representation of geometric quantities.The principles of analysis “were based upon merely intellectual notions, upon ideas thatwe ourselves shaped by abstraction, by simplifying and generalising the ‘first’ ideas;thus these principles, properly, include only what we have put in them, and namely thesimplest our perceptions” [1767, 5:154]. In operating upon analytical expressions, 18thcentury mathematicians, basically, had curves in mind and used pre-analytical proper-ties of geometric quantities (e.g. continuity). Their aim was not merely the syntacticalcorrectness of analytical forms; rather they started from forms expressing truths aboutquantities and arrived at forms expressing truths about quantities.

New combinations of symbols, such as log(−1), 1− 1+ 1− 1+ · · ·, 0/0,a0, werenot introduced arbitrarily, but were the results of a transformation of signs abstractlyexpressing objects given in our perceptions. It seemed obvious to them that 0/0,a0,1 − 1 + 1 − 1 + · · · had a ‘natural’ meaning and that mathematicians were to discoverit. If the needs of the calculus involved in a new symbol6, they actually asked ‘Whatis the value of the symbol6?’ (quaeratur valor) and not ‘How shall we define6?’ Forinstance, if one met with the undefined symbolic notationa0, one did not takea0 = 1by a useful but arbitrary ‘definition’; rather one ‘discovered’ that thetrue meaning ofa0 was 1 sincea0 = an−n = an : an. Similarly, in 18th century treatises, we seesuch problems asfind the value of the expressions(b − √

b2 − x2)/x2 for x = 0, and1−x+x2−x3+x4−· · · for x = 1; whereas, today, the same problems are formulated:define(b − √

b2 − x2)/x2 for x = 0, and 1− x + x2 − x3 + x4 − · · · for x = 1.

2. The question of the sum of divergent series

When mathematicians ran up against the question of the sum of divergent seriesat the beginning of the 18th century, they agreed that the matter consisted in seekingthe true value of divergent series and that more than one value could not exist. They

5 I use the term ‘Analysis’ whereas d’Alembert used the word ‘Algebra’, because his distinc-tion between Algebra and Analysis is of no importance to my argument. According to him, Analysis“is properly the method for solving mathematical problems by reducing them to equations. . .. Inorder to solve problems, Analysis resorts to Algebra, or the calculus of magnitudes in general:thus, these two words, Analysis and Algebra, are often regarded as synonyms” [Anal, 400b].

6 la methode de faire en general le calcul de toutes sortes de quantites, en les representant pardes signes tres universels” [Anal, 400b].

106 G. Ferraro

disagreed, however, on the possibility that the search would meet with success, i.e. onthe existence of the unique true value of the sum. Some thought that

f (x) =∑

fn(x) (1)

denoted an equality among numbers: they based their argument upon the practical ob-servation that certain series approach closer and closer to a single fixed value, whichwas considered as its unique, ultimate and true value (later I shall call this interpretationof (1) numerical). Of course, in this sense, divergent series have no sum. Most 18thcentury mathematicians, however, realised that the intrinsic principles of their analysisled necessarily to admitting divergent series. They sought the true meaning of (1) not ina mere relation among numbers but in a more general relation among analytical formsconceived as ‘abstract’ or ‘general’ quantities.7 The equation (1) meant that

∑fn(x)

was derived from an analytical formf (x) (called the sum of the series) by means ofquasi-algebraic procedures, i.e., procedures that were born of an infinitary extension ofalgebraic laws (later I shall call this conceptionformal). The formal conception wasde factoused in the first half of 18th century before Euler formalised it in 1750s bydefining the sum of an infinite series as “the finite expression, the expansion of whichgenerates the given series” (see [1755, 2:§.110–111] and [1754–55, 593–594]). Accord-ing to Woodhouse’s words, in the equation (1) the sign = denoted the “expansion orthe result of any operation. . . and assuming this signification of the sign =, when anarithmetical equality results between the function and its expansion, such an equality re-sults not necessarily but contingently. . . it is the law of expansion, the connection of thecoefficients of its terms, which is useful to consider and the convergency or divergencyof the series is then a useless consideration” [1803, 15].

Eighteenth century mathematicians were fully aware of the difference between theformal and the numerical conception. For instance, Daniel Bernoulli [1771] spokeof the sums which were falsein concreto(i.e., numerically) but truein abstracto(i.e.,formally), and proposed to designate the abstractly true sums by means of a ‘hybrid’term: incongruenter veraesums. Other crucial characteristics of the formal conceptionwere:

– A numerical series∑anwas considered as a special case of a function series

∑fn(x)

with fn(c) = an.– The numerical validity of formal results was relevant and was verified only if se-

ries were applied to numerical, geometrical or physical problems. In this context,approximations, evaluations of errors, and even convergence criteria were used.

– Since computational and combinatorial relationships among analytical forms or ex-pressions were studied, results had to be explicitly exhibited. There were no indirectexistence theorems.

At the beginning of the 19th century the formal conception of the sum was rejected.In [1821a], Cauchy considered the validity of the mathematical formulas dependent

7 I discuss in detail elsewhere ([1998] and [A]) the 18th century concept of the sum. On 18thcentury analysis, see the very interesting works of Marco Panza (in particular [1992]) and CraigG. Fraser (especially [1989]).


on appropriate conditions: if an equalityf (x) = g(x) was verified for some values ofthe variablex it could not be generalised to all the values of the variable. Consequently,according to Cauchy, one had to admit propositions that, at a first sight, seem “a littlehard”, such as “a divergent series does not have a sum”, and “before summing anyseries, I had to examine when the series can be summed, or, in other terms, what arethe conditions for their convergence” [1821a, v].8 The core of Cauchy’s conception onseries can be summarised as follows:9

– Relations among mathematical objects, not forms by themselves, were of impor-tance: formulas were to be considered insofar as they represent relations among realquantities.10

– The sum of a numerical series was defined as the limit of its par tial sums, when itexisted.11

– Numerical series were preliminary to functions series: the equality∑fn(x) = f (x)

was considered valid in an interval only if it was satisfied for allx in the interval.– The existence of the result was determineda priori by means of the study of condi-

tions of convergence.– In no case could a result proved on a determinate interval be extended by analogy

and no symbol could be introduced without an appropriate definition which referredit to real quantities.

Cauchy stressed the importance of definitions in mathematics; he, however, neverconceived mathematics in an axiomatic way. Thus, his definition of the sum is a mereclarification of the intuitive observation that ultimately the partial sums of a given seriesdiffer from a fixed value as little as desired. Cauchy’s definition described a conceptthat existed before it and was not ‘created’ by it; as such, it could not be varied and oneseries might not have different sums according to different definitions.

Euler’s and Cauchy’s conceptions shared the crucial idea of the unique, true sum;and such an idea was based on the conviction that mathematics was an expression of the

8 “une serie divergente n’a pas de somme. . . avant d’effectuer la sommation d’aucune series,j’ai du examiner dans quels cas les series peuventetre sommees, ou, en d’autres termes, quellessont les conditions de leur convergence” [1821a, v].

9 The short summary is mainly derived from Laugwitz’s works on Cauchy. I, above all,refer to his [1987] and [1989].

10 “ Determining these conditions and these values, and fixing in a precise way the sense ofthe notations I use, I make any uncertainty vanish; and then the different formulas involve nothingmore than relations among real quantities, relations which are always easy to verify on substitutingnumbers for the quantities themselves.” (“En determinant ces conditions et ces valeurs, et fixantd’une maniere precise le sens des notations dont je me sers, je fais disparaitre toute incertitude;et alors les differentes formules ne presentent plus que des relations entre les quantites reelles,relations qu’il est toujours facile de verifier par la substitution de nombres aux quantites elles-memes” [1821a, iii–iv].”) Cauchy usually studied functional relations expressed by formulas.However, unlike 18th century mathematicians, Cauchy mainly saw functions as relations, laws,and not as formulas.

11 Cauchy actually used what we today call topological concepts, and expressed them bymeans of the intuitive ideas of limit and neighbourhood.

108 G. Ferraro

unique reality. To use Hardy’s words, both Euler and Cauchy asked ‘What is 1−1+1− · · ·?’, namely ‘What is the (unique) mathematical object equal to 1− 1+ 1− · · ·?’Of course, their answers were different. Euler answered ‘1/2’; Cauchy instead statedno mathematical object is equal to 1− 1+ 1− · · ·’ and, for this reason, he thought thatone could never define the sum of 1− 1 + 1 − · · ·.

From Cauchy to 1890, the numerical concept, grounded upon a basic intuition,seemed to be theunique, true, and rigorous concept of the sum. However, on the onehand, it was felt necessary to use divergent series; on the other hand, the idea of limits washandled by theε-δ method. Epsilontics broke with the naturalist and intuitive aspectsof Cauchy’s conception, of which it was not a mere improvement but a substantialmodification.

This fact enabled mathematicians to make the numerical conception more rigorousand, at the same time, to recuperate the use of divergent series. The result was a Hegeliansynthesis of the numerical conception (rethought according toε-δ method) and formalviews (which lasted throughout the last century) based on overcoming the uniquenessof the notion of a sum and related to the new conception of mathematical theories asstructures developing the logical consequences of appropriate axioms and definitions.In the 19th century, the latter conception began to develop starting from non-Euclideangeometries, whose validity does not depend on the ‘true’ geometry of the physical world.Summability was the first occurrence of this conception in analysis and arose onlywhen theconventionalaspect of Cauchy’s definition was highlighted by Cesaro andBorel.12 For instance, the former wrote: “After all, if one regards the indefinite form ofconvergent series, is it not in virtue of a convention that they intervene in calculations?”13

The latter stated: “Besides, one can be led. . . to attribute several different sums to adivergent series” and, in a footnote, added “And similarly to a convergent series.”14

Part two

1. Formal algorithms: the permanence of 18th century methodsduring the 19th century

Cauchy’s program was felt to be excessively restrictive by some mathematicians.Cauchy himself examined the possibility of legitimating the use of Stirling’s series:

Nowadays geometers generally recognise the dangers deriving from the introduction ofdivergent series into Analysis and rightly assume that these series do not have sums.Nevertheless the series employed by Stirling for the approximate determination of thelogarithm of a product whose factors increase in an arithmetic progression, and of otherdivergent series of the same kind, furnish approximate values of the function whose

12 It is, however, appropriate to note that neither Cesaro’s papers on series nor Borel’s onescan be considered as axiomatic works in Hilbert’s sense.

13 “Apr es tout, n’est-ce pas en vertu d’une convention que les series convergentes, prises sousleur forme indefinie, interviennent dans les calculs?” [1890, 360].

14 “On pourrait d’ailleursetre amene . . . a attribuer plusieurs sommes differentesa une seriedivergente”, “Et aussia une serie convergente” [1901, 15].


development they represent effectively, provided one stops them at a certain number ofterms. It should be important to examine if it is possible to legitimate the use of similarseries.15

Cauchy, however, always reaffirmed the primary principles of his program16 and re-duced the study of divergent series to the evaluation of errors in order to approximatecertain functions.17Although Stirling’s series is of importance in the history of summa-bility, research pursuing Cauchy’s point of view analysed the possible use of asymptoticseries to approximate functions and led to Stieltjes’s and Poincare’s ideas on the rep-resentation and calculation of functions by means of divergent series rather than to thequestion of assigning a sum to a divergent series.

Another field of research is more relevant to my argument: the calculus of oper-ations,18 the main expression of the formalist tradition that developed starting fromthe analogy between the law of exponents (anam = an+m) and repeated operations,such as (dn/dxn)(dmf (x)/dxm), and that lasted throughout the 19th century in spite ofCauchy’s emphasis on the numerical calculability. The calculus of operations was ananalogical calculus based on the idea that symbols of operations could be handled sep-arately from the subjects on which they operate, and treated as if they were symbols ofquantities. This technique, also known as the method of the separation of the symbols ofoperation from quantity, allowed the use of algebraic laws out of their domains. Typicalexamples are the rules:

(*) Af = Bf ⇒ A = B,(**) Af + Bf = (A+ B)f , and, more generally,

∑∞n=1Anf = (∑∞

n=1An)f ,

(***) A(Bf ) = (A · B)f ,

whereA,An, andB are operations andf a function (or, as they said, a quantity).In the calculus of operations, (*) and (**) were merely the result of the application of

algebraic laws to the symbolsA,B, f regardless their nature, and (***) was based uponthe idea of replacing iteration by multiplication. Today, we view the matter differently.Given, for instance, a vector spaceU , and the operatorsA,B, andC fromU toU ,

(§) {Au = Bu (for eachu ∈ U) ⇔ A = B} is the definition of the equality ofA andB;(§§) (A + B)u = Au + Bu (for eachu ∈ U ) is simply the definition of the sum ofAandB.

15 “Les geometres reconnaissent generalement aujourd’hui les dangers que peut offrirl’introduction des series divergentes dans l’Analyse, et ils admettent avec raison que ces seriesn’ont pas de sommes. Toutefois la serie employee par Stirling, pour la determination approx-imative du logarithme d’un produit dont les factuers croissent en progression arithmetique, etd’autres series divergentes du meme genre fournissent effectivement, quand on les arrete apresun certain nombre de termes, des valeurs approchees des fonctions dont elles representent lesdeveloppements. Iletait important d’examiner s’il possible de rendre legitime l’emploi de sem-blables series” [1843, 18].

16 On Cauchy’s use of divergent series, see Laugwitz [1989, 197].17 Around 1840 Cauchy’s view on Stirling’s series was sufficiently spread: see, for instance,

Jacobi [1835] and Malmsten [1847].18 On the calculus of operations, see Koppelman [1971].

110 G. Ferraro

To state(A + B)u = Cu ⇒ A + B = C we therefore define, first, the equalityof two operators, and, second, a new operation over the setS of the operators fromUto U , which is usually denoted by +, but differs from the old operation + acting onU . Infinite sums of operators

∑∞n=1An are a more complicated question because they

involve topological concepts.In the calculus of operations there was a confusion between different algebraic struc-

tures (albeit related to each other): only quantities formed a structured set; while thecollections of other objects, such as the operations acting upon quantities, borrowed thealgebraic structure of quantities. The sumA+B was not defineda priori, ratherA+Bwas viewed as a mere expression on which one operated as ifA andB were quantitiesand + the usual sum of quantities. In the same way, albeit that the composition of oper-atorsA · B was a known operation, it was transformed into multiplication of algebraicsymbols, namely· was treated as the usual product of quantities.

As an illustration, consider the following two examples. The first concerns the so-called Lagrange’s theorem1n = (

ehD − 1)n

. Since the forms of the expansions

ehDf (x) − 1 =∞∑n=1

hn

n![Df (x)]n and1f = f (x + h)− f (x) =

∞∑n=1

hn

n!Dnf (x)

are similar, it was deduced that1f = (ehD − 1

)f (or, if you wish, (***) and (**) were

applied). By (*), the latter equation was also expressed in the form1 = ehD − 1, andit was inferred that1n = (

ehD − 1)n

or, more generally,F(1) = F(ehD − 1

)for any

functionF(x).The second example is the theorem: To legitimate the equivalenceeA+B · u =

eA · eB · u (i.e., eA+B(u) = eA(eB(u))) it is necessary and sufficient thatA andBare commutative. The proof (see Murphy [1837, 188] and Carmichael [1855, 13])consisted in expanding both sides and observing that one should have(A + B)2 =A2 + 2AB + B2 = A2 + AB + BA+ B2 in order that

∞∑n=0

(A+ B)n

n!u =

( ∞∑n=0

An

n!

)( ∞∑n=0

Bn

n!

)u ,

and consequently the latter equation could not hold unless the symbolsA andB werecommutative.

By 1850 the hard core of analogical calculi (the symbols of operations and symbolsof quantities obey the same laws of combination, or, as we may say if we possess theconcept of mathematical structure, the set of operations and that of quantities havethe same algebraic structures) was entirely clear, mainly thanks to works of DuncanGregory, Robert Murphy and George Boole. Scholars characterised the three lawsof combination of symbols upon which the calculus of operations was based as:

– distributive(A+ B)u = Au+ Bu,– commutativeA(B(u)) = B(A(u)), and– index lawAnAmu = An+mu.

They observed that any theorem that was true for any one symbol which satisfied theselaws was true for every symbol which satisfied them. “Now the symbols ofnumberssatisfied them. . . Again, the symbols ofdifferentiationsatisfy those laws. . . Hence we


deduce the important consequence, that every theorem in Algebra, which depends onthose laws, has an analogue in the Differential Calculus.” (Carmichael [1855, 9]).

Symbolists, however, did not extend their analysis to cover what later mathemati-cians would call topological aspects of the problems raised.19 They generally ignoredproblems of convergence and had no hesitation about considering infinite series of ar-bitrary operations, on the bases of the mere analogy of forms with power series. Thus,in the above example, one can consider eA = ∑∞

n=0An/n! for any operationA, as one

knows the equality ex = ∑∞n=0 x

n/n! for a variable quantityx.Obviously the formalist point of view contrasted with Cauchy’s conception. Cauchy

wrote some papers on the calculus of operations: he however stated that this calculuswas, in general, an inductive and heuristic tool to arrive at many results with ease,but these were rigorous only if one limited oneself to consider polynomial functionsF(d/dx,1, . . .) of symbols. If the functionsF(d/dx,1, . . .) were expressed by powerseries, the results were valid under certain determined conditions, ordinarily concerningthe convergence of series one dealt with.20

Nevertheless mathematicians who did research in symbolic methods went on withtheir unrestricted ‘symbolical reasonings’: the accepted point of view was that the veri-fication of the numerical validity of formal results was put into effecta posteriori. Theirconception was still essentially that of the 18th century and was based upon a distinctionsimilar to Bernoulli’s difference between falsein concretoproperties and truein ab-stractoproperties. At the end of the 19th century Pincherle still distinguished “betweenformal and effective properties. Formally, one can always say that the equation

ϕ(x) =∫(c)

e−xzf (z)dz

is solved by

F(z) =∫(c′)

e+xzϕ(x)dx .

The difficulty is in giving the limits within which this solution has a meaning and indetermining the path of integration.”21

19 This crucial character of all the formal calculi of the 19th century was transmitted to math-ematicians such as Salvatore Pincherle (1853–1936) who is counted among the initiators ofmodern functional calculus (for instance see Amaldi [1937, 12]). This is, perhaps, one of thereasons why Pincherle’s work on the functional calculus was not very popular and his contri-butions were part of the background noise of the rise of modern analysis functional (see Gray[1984, 141]).

20 The influence of Cauchy is likely one of the reasons why, after the 1830s, the calculus ofoperation was developed mainly in Great Britain, where it was linked with the rise of abstractAlgebra (see Koppelman [1971]).

21 “. . . fra le proprieta formali edeffettive. Formalmente, si puo sempre dire che l’equazione

ϕ(x) =∫(c)

e−xzf (z)dz

112 G. Ferraro

Although the calculus of operations lost importance in the second half of the 19thcentury, the basic idea of switching from powers to indices of operations or of sequencesremained and gave rise to the umbral calculus. Today, the umbral calculus is synonymouswith the theory of binomial enumeration and, after the contribution of Rota and hisschool (for instance see Roman-Rota [1978] and Roman [1984]), it has a firm logicalfoundation based upon operator methods. In the 19th century, it was instead a merelyanalogical calculus, and was considered as a matter of notation, subject to rules ofmanipulation and interpretation. It was related to an ancient tradition of Enlightenmentorigin, particularly spread in the circle of theIdeologues, which considered analysis asa well made language (une langue bien faite)22 and, as such, a powerful tool whichenabled mathematicians to arrive at new results and to extend their investigations. Forinstance, Laplace stated:

The language of analysis, the most perfect of all the languages, in itself being a powerfulinstrument of discovery, if its notations are necessary and are happily imagined, they arethe germs of new calculi, of a well-made language, the simplest notions of which haveoften become the source of the deepest theories.23

The calculus of operations and the umbral calculus directly influenced the rise ofC-summability as they retained the non-numerical sense of the equality (1), and, moregenerally, the use of divergent series. The 19th century umbral calculus (which was oftenalso termed the ‘symbolic calculus’) substantially concerned the operatorEun = un+1working on the sequenceun and was based upon the similarity ofunum = un+m andEnu0E

mu0 = En+mu0. There are so many parallels between certain sequencesun andthe powers sequenceun that the indexn in un seemed to be theumbra(i.e., the shadow)of the exponentn in un. One could thus discover identities involving a sequenceun byoperating on un by means of the usual rules of Algebra and replacing the superscriptby the subscript. For instance, Bernoulli numbersBr and Euler numbersEr are definedby the relationst/(et − 1) = ∑∞

r=0 (Br/r!)tr (|t | < 2π ) and 2/(et + e−t ) = secht =∑∞

r=0 (Er/r!)tr (|t | < (1/2)π) , respectively. Formally, these series can be obtained

from the expansions∑∞r=0 (B

r tr )/r! of eBt and∑∞r=0 (E

r tr )/r! of eEt by replacing

si risolve mediante

F(z) =∫(c′)

e+xzϕ(x)dx

la difficolta sta nel dare i limiti entro i quali tale soluzione ha un significato e nella determinazionedelle linee d’integrazione.” This quotation is in a letter which is found in Cesaro’s Nachlass, keptin Department of Mathematics and Applications of University ‘Federico II’, Naples). See alsoPincherle [1887]. Remarkably, according to Struppa and Turrini [1991, 508], Pincherle’smethod “is essentially what is done in modern algebraic analysis, as it is clearly shown in”Kashiwara, Kawai and Kimura [1986].

22 See d’Alembert [1767, 5:164] and Condillac [1797].23 “La langue de l’analyse, la plus parfaite de toutes,etant par elle-meme un puissant instrument

de decouverte; ses notations, lorsqu’elles sont necessaires et heureusement imaginees, sont lesgermes des nouveaux calculs, d’une langue bien faite, que ses notions les plus simples sontdevenues souvent la source des theories les plus profondes” [1812, 7].


the powersBr andEr byBr andEr . In the 19th century umbral method, one assumedthe symbolic equalities

∞∑r=0

Br

r!t r =

∞∑r=0

Brtr

r!and

∞∑r=0

Er

r!t r =

∞∑r=0

Ertr

r!(2)

and manipulated the umbrae B and E instead of the more complicated sequencesBrandEr .24 In fact, the relations defining the Bernoulli and Euler numbers can be writtensymbolically as eBt = t/(et − 1) and eEt = 2/(et + e−t ) by (2). If one operates uponthis umbral forms algebraically, one obtains

e(B+1)t − eBt = t and e(E+1)t − e(E−1)t = 2 . (3)

Since the expansion of e(B+1)t−eBt is∑∞r=0 [(B + 1)r − Br ]t r/r! and e(E+1)t−e(E−1)t

is equal to∑∞r=0 [(E + 1)r − (E − 1)r ]t r/r!, the symbolic equalities (3) can be written

as∞∑r=0

[(B + 1)r − Br ]t r

r!= t and

∞∑r=0

[(E + 1)r − (E − 1)r ]t r

r!= 2

Equating coefficients oft r/r! of each of these equalities, one deduces

(B + 1)n − Bn ={

1, n = 10 n = 0, 2, 3, 4 . . . .

and

(4)(E + 1)n + (E − 1)n =

{2, n = 00 n = 1, 2, 3, 4, . . . .

which are the umbral version of the recurrence formulas ofBr andEr . From (4), it iseasy to derive the Euler-Maclaurin summation formulaf ′(1)+f ′(2)+f ′(3)+· · ·+f ′(n− 1) = f (n+ B)− f (B) of a polynomialf (x), and its analogue for alternatingseries

f (1)− f (3)+ f (5)− · · · − f (4n− 1) = 12 [f (E)− f (E + 4n)] . (5)

Indeed, expandingf [a + (1+ B)], f [a + B], f [a + (E + 1)], andf [a + (E − 1)] inTaylor series, wherea is a real number, we have

24 By interchanging exponents with suffixes, we can verify that many rules of manipulationfor usual algebraic symbols are also valid for umbrae. For example,(d/dt)eBt = BeBt . In fact:

d

dteBt = d

dt

( ∞∑r=0

Brtr

r!

)= d

dt

( ∞∑r=0

Brtr

r!

)=

∞∑r=1

Brtr−1

(r − 1)!

=∞∑r=0

Br+1t r

r!=

∞∑r=0

Br+1 tr

r!= B

∞∑r=0

Brtr

r!= BeBt .

114 G. Ferraro

f [a + (1 + B)] − f [a + B]

= [(B + 1)0 − B0]f (a)+ [(B + 1)1 − B1]f ′(a)+ [(B + 1)2 − B2]f ′′(a)1 · 2

+ · · ·and

f [a + (E + 1)] − f [a + (E − 1)]

= [(E + 1)0 − (E − 1)0]f (a)+ [(E + 1)1 − (E − 1)1]f ′(a)

+ [(E + 1)2 − (E − 1)2]f ′′(a)1 · 2

+ · · ·

Applying the recurrence formulas (4), these infinite expressions are simply reduced to

f [a + (1 + B)] − f [a + B] = f ′(a) and

f [a + (E + 1)] + f [a + (E − 1)] = 2f (a) .

Takinga = 0, 1, 2, . . . , n − 1 in the former equation, anda = 1, 3, . . . ,4n − 1 (andchanging the sign appropriately) in the latter, we obtain

f (1 + B)− f (B) = f ′(0) and f (E + 2)+ f (E) = 2f (1)

f (2 + B)− f (1 + B) = f ′(1) and − f (E + 4)− f (E + 2) = −2f (3)· · ·

f (n+ B)− f (n− 1 + B) = f ′(n− 1) and

−f (E + 4n)− f (E + 4n− 2) = −2f (4n− 1) .

and by adding we derive the Euler-Maclaurin summation formula and (5). Thisprocedure can formally be applied to functions which are analytic at the origin.

In conclusion, although Cauchy’s numerical notion of the sum triumphed widelyin the 19th century, the formal conception was still remembered and yielded symbolicmethods. These were at times used with caution and with attention paid to convergence,but often series were handled formally. Precisely the justification of the results derivedby umbral procedures was the starting point of the research that led to C-summability.

2. On some research concerning series in the 1880s

In the 1870s and 1880s, epsilontics provided the numerical conception with newand more solid bases. Pre-mathematical and naturalistic aspects of the limit conceptwere definitively abandoned in favour of theε-δ formulation and new, subtler notions(supremum, infimum, accumulation point, etc.) were introduced. Within the generalreformulation of the whole of analysis, much interesting research dealt with convergencecriteria, analyticity, mean theorems, multiplication theorems, and touched on the topicof divergent series. Such research mainly aimed at generalising or making old theoremsrigorous; the results sometimes overlapped and gave rise to polemics. Looking back froma modern prospective, some mathematicians have rethought several theorems formulated


in that time as theorems about summability theory and have even considered certainpropositions as implicit definitions of the sum of divergent series.

An example is appropriate to illustrate the state of the art and how certain the-orems were later reinterpreted differently on the basis of 20th century views. In hisAnalyse alg´ebrique, Cauchy [1821a, 133] proved that limn→∞ f (n)/n = limn→∞[f (n)− f (n− 1)

], provided the latter limit exists. This theorem was generalised by

Johan L.W.V. Jensen (1859–1925), who stated:

Theorem (J).Given the sequencesϕn andan, if limn→∞ ϕn = ϕ, limn→∞ an = ∞,and (1/|an|)

∑ni=1 |ai − ai−1| < M (where M is a fixed real number), then

limn→∞

(a1 − a0)ϕ1 + (a2 − a1)ϕ2 + · · · + (an − an−1)ϕn

an= ϕ .

Theorem (J) was first published in 1884, in a Danish journal,Tidsskrift for Math-ematik [1884], and then in 1888, inComptes rendus de l’Academie des SciencesofParis [1888a]. In 1888, Cesaro, who did not know the first of Jensen’s papers, claimedhis priority [1888h], since he had published the same result inNouvelles Annales deMathematiques, formulating it in these terms [1888g, 318]:

Theorem (C1). Given the sequencesan, and pn, if pn > 0,∑pn = ∞, and

limn→∞ an = a, then

limn→∞

∑ni=0piai∑ni=0pi

= a ,

and inRendiconti dell’Accademia dei Lincei, where he used it to prove:

Theorem (C2).Given the sequencesan, bn, εn, and the real number H, if (b1 + b2 +· · · + bn)εn+1 − (b1ε1 + b2ε2 + · · · + bnεn) is increasing, ((b1 + b2 + · · · + bn)εn+1)/

(b1ε1 + b2ε2 + · · · + bnεn) < H , and limn→∞ [(b1 + b2 + · · · + bn)εn+1− (b1ε1 +b2ε2+ · · · + bnεn)] = ∞, then

limn→∞

a1ε1 + a2ε2 + · · · + anεn

b1ε1 + b2ε2 + · · · + bnεn= limn→∞

a1 + a2 + · · · + an

b1 + b2 + · · · + bn,

provided the right-hand side exists[1888a, 292].

Theorems (J) and (C1) suggest a method of summation, similar to Norlund means,usually denoted by (N,pn)25 and Hardy even interpreted theorem (C2) as a theoremrelating the (N,pn) and (N, qn) sums. In his [1949], Hardy proved:

Theorem (H). If pn > 0, qn > 0,∑pn = ∞,

∑qn = ∞, H is a real number,

and eitherqn+1/qn <= pn+1/pn, or pn+1/pn <= qn+1/qn and alsoPn/pn <= HQn/qn,wherePn = ∑n

i=1pn andQn = ∑ni=1 qn, then

25 If pn >= 0, p0 > 0,∑pn = ∞, Sn = ∑n

0 ai , and limn→∞∑n

i=0 pisi/∑n

i=0 pi = S, thenwe say thatS is the (N, pn) sum of

∑∞0 an (notation

∑∞0 an = S(N, pn)). On this method, see

Hardy [1949, 57].

116 G. Ferraro

∑an = s(N, pn) implies

∑an = s(N, qn) .

He said [1949, 63] that theorem (H) “is due to Cesaro, Atti d. R. Accad., d. Lincei[Rendiconti (4), 4 (1888), 452–7]” (namely, [1888a]), whereas Cesaro proved nothingbut (C2), and used it to determine some asymptotic laws of numbers without referenceto summability.

This attitude gave rise to a historiographic tradition according to which the first mod-ern definition of the sum dates back to 1882 when Otto Holder (1859–1937) wouldhave defined H-summability26 in hisGrenzwerte von Reihen an der Convergenzgrenze.For instance, Tucciarone said that Georg Frobenius (1849–1917), Holder,Cesaro and Borel “were striving for an extension of the traditional convergence, in or-der to ‘sum’ divergent series. . . they posed new definitions of summability which wouldsum ordinarily convergent series, and they then tried to frame the definitions to includein their scope as many divergent series as possible. It was clear to these men that to solvethe problem by starting with a definition applicable only to divergent series could leadto difficulties” [1973, 3–4].

This statement is correct only in so far as it refers to Cesaro and Borel: indeedno definition of the sum of divergent series is found in Frobenius [1880] and Holder[1882], where the authors remained within the bounds of the ordinary conception of thesum. In [1880], Frobenius merely limited himself to prove the following theorem (F)by a classic application ofε-δ method:

Theorem (F). If Sn=∑n−1i=0 ai andlimn→∞ 1

n

∑n−10 Si=M, then the series

∑∞0 anx

n

converges for−1< x < 1 and limx→1−∑∞

0 anxn = M.

Theorem (F) is a generalisation of Abel’s well-known theorem (if the seriesa0 +a1x + · · · + anx

n + · · · = F(x) converges whenx = b, then it also converges for|x| < b and limx→b− F(x) = F(b)), and provides a continuation of the function∑anx

n, defined for|x| < 1, at the point 1. There is nothing else in Frobenius’s shortpaper. It is true that Frobenius began his note by mentioning Leibniz’s solution ofthe paradox 1− 1 + 1 − 1 + · · · = 1/2: the sum of 1− 1 + 1 − 1 + · · · is 1/2because(s2n + s2n+1)/2 = 1/2 with sn = ∑n−1

i=0 an. Frobenius, however, thought thatLeibniz had stated the generalisation (F) of Abel’s theorem (sic), although withoutdemonstration.27 In other words, for Frobenius, Leibniz merely formulated, but didnot prove, a first, intuitive version of (F). On the contrary, Leibniz actually intended

26 Given the divergent series∑an and settingS0

n = ∑n

i=1 ai, Skn = 1

n

∑n

i=1 Sk−1i , k =

1, 2, 3, . . . , if lim n→∞∑n

i=1 Skn = S, then we say that

∑an is summable(H, k) with sumS.

27 “Albeit without proof (this kind of reasoning, even if it seems to be metaphysical rather thanmathematical, is however sound), Leibnitz thus states the following proposition, a generalisationof a well-known theorem of Abel (in this journal [i.e., Crelle’s Journal] vol. 1, p. 314): Letsn = a0 + a1 + · · · + an be and let(S0 + S1 + · · · + Sn−1)/n approach a certain finite limitM forincreasingn, then the seriesa0+a1x+a2x

2+· · · is convergent for values ofx between−1 and +1and the function represented by it approaches the valueM as a limit, ifx converges towards 1 bycontinuously increasing” (“Leibnitz behauptet also, allerdings ohne Beweis (hoc argumentandigenus, etsi Metaphysicum magis quam Mathematicum videatur, tamen firmum est) den folgendenSatz, eine Verallgemeinerung eines bekannten Abelschen Satzes (dieses Journal Bd. 1, S. 314):


to associate 1/2 with 1− 1 + 1 − 1 · · · and to provide a justification of Grandi’sparadox.28 Frobenius was so far from the idea of summing a divergent series that heeven interpreted Leibniz in a restrictive and incorrect way. (F) is really only one ofthe attempts which aimed at improving and generalising series theory; and, as such, itwas noticed by Holder and Thomas Jan Stieltjes (1856–1894) who provided twodifferent generalisations of (F), both published in 1882.

In his [1882, 84–90], Stieltjes proved that

Theorem (S1). Given a sequenceSn, one has:

1) if limn→∞ 1n

∑n−10 Si exists, then limx→1−(1 − x)u[S0 + (u/1)S1x + (u(u+ 1)/

2!)× S2x2 +(u(u+ 1)(u+ 2)/3!)S3x

3 + · · ·] = limn→∞ 1n

∑n−10 Si for u > 0;29

2) if limn→∞ 1n

∑n−10 Si exists, then limx→1−(S0 + 1

2S1x + 13S2x

2 + 14S3x

3

+ · · ·)/log(1/(1 − x)) = limn→∞ 1n

∑n−10 Si ;

3) if pn >= 0,∑∞

0 pnxnis convergent for0 < x < 1 and divergent forx = 1, and

limn→∞Sn = M, thenlimx→1−∑∞

0 pnSnxn/∑∞

0 pnxn = M.

Although Stieltjes [1882], like Frobenius [1880] and Holder [1882], did notintend to define the sum of divergent series but only to develop some consequences ofthe theory of convergent series, (S1) suggest another definition of the sum to us, modernreaders, and, if we wish, we could call it the Stieltjes sum of

∑∞0 an:

Theorem (S2). If the power series∑∞

0 pnxn has radius of convergence equal to 1,∑∞

0 pn = ∞, Sn = ∑n−10 ai , andlimx→1−

∑∞0 pnSnx

n/∑∞

0 pnxn exists and is equal

to the number A, we say that A is the Stieltjes sum of∑∞

0 an with respect to∑∞

0 pnxn.30

Starting from the observation that (F) fails if limn→∞ 1n

∑n−10 Si does not exist,

Holder proved

Theorem (Ho). If S1n = 1

n

∑ni=1 Si , S

kn = 1

n

∑ni=1 S

ki and limn→∞Skn = C exists for

some k, thenlimx→1−∑∞

0 anxn−1 = C.

Ist sn = a0 + a1 + · · · + an und nahert sich(S0 + S1 + · · · + Sn−1)/n bei wachsendemn einerbestimmten endlichen GrenzeM, so ist die Reihea0 + a1x + a2x

2 + · · · fur die Werthe vonxzwischen−1 und+1 convergent, und die durch sie dargestellte Function nahert sich, wennxbestandig zunehmend gegen 1 convergirt, dem WertheM als Grenze” [1880, 262].

28 Eandem lineam infinities positam, et infinities subtractam relinquere sui medietatem, i.e., byadding and subtracting a same segment DV infinitely, one obtains a segment equal to the half of DV.This was the first formulation, geometrically expressed, of the idea that 1−1+1−1+· · · = 1/2.

29 I recall that (F) may also be formulated as follows: if limn→∞ 1n

∑n−10 Si=M, then

limx→1−(1 − x)∑∞

0 Snxn = M.

30 On this method of summation, see Hardy [1949, 79–81]. It is a variant of the class ofmethods, of which the best known is Borel’s, defined thus: IfJ (x) = ∑∞

0 pnxn is an integralfunction, not a polynomial, with non-negative coefficientspn, the J-sum of the series

∑∞0 is

defined as limx→∞∑∞

0 pnSnxn/∑∞

0 pnxn, if this limit exists (Sn are the partial sums of

∑∞0 an).

If we setJ (x) = ex we have the Borel sum.

118 G. Ferraro

The proof takes up his [1882] almost wholly (it is summarised by Tucciarone[1973]): only in the last two paragraphs Holder did give two examples. The first con-cerns the series

∑∞1 (−1)nnxn−1 which is equal to−1/(1 + x)2 for |x| < 1 and di-

verges whenx = 1. Frobenius’s theorem can not be applied becauseS12n = 0 and

S12n+1 = −(n+ 1)/(2n+ 1) with regard to

∑∞1 (−1)nn. Instead limn→∞ S2

n = −1/4,and, by theorem (Ho), limx→1−

(∑∞1 (−1)nnxn−1

) = −1/4 as one can also verifydirectly because limx→1− (−1/(1 + x)2) = −1/4.

At p. 10 of his [1973], Tucciarone observed: “On the basis of this outcome, itwould seem quite natural to assign the value−1/4 as the ‘sum’ of the original series,and characterise it as theS(2)-Sum, or Second Holder Sum”. Early, at p. 7, referringto the fact that the sequences of repeated meansS1

n, S2n, S3

n,. . . can provide a convergentsequence, Tucciarone had also written: “Once this convergent sequence was reached,Holder suggested that its limit be the value assigned as the sum of the original se-ries.” Certainly Frobenius’s and Holder’s theorems suggest that limn→∞ Skn may beassumed as a new definition of the sum; neither Frobenius nor Holder however tookthis opportunity, and restricted themselves to theorems (F) and (Ho), conceived as twonew improvements and generalisations of the ordinary theory of convergent series.

Hardy gave a slightly different interpretation of Frobenius’s and Holder’s papers:“In modern times it [the (C,1) or (H,1) definition] was used implicitly by Frobeniusand Holder in 1880 and 1882; but it does not seem to have been stated formally as adefinition until 1890, when Cesaro published a paper on the multiplication of series inwhich, for the first time, a ‘theory of divergent series is formulated explicitly” [1949, 8].

In this quotation, ‘implicit use of the (H,1) definition’ means that divergent seriesare used as such and associated with real numbers (for instance, 1−1+1−· · · with 1/2)so that operations on divergent series correspond to operations on associated numbers(for instance, the sum of two series correspond to the sum of their associated number);however such a fact, which is the hardcore of the concept of the sum, is not defined,formalised, or recognised explicitly (1/2 is not termed the sum of 1−1+1−· · ·, but it iseffectively used as the sum of 1−1+1−· · ·). The concept is used but the formalisationis lacking.

An ‘implicit use’ of the sum (in this sense) occurred in the first half of 18th century,when divergent series and their sums often appeared in analytical works, but only in his[1754–55] Euler did publish the famous definition which formalised and justified theearlier use. Instead, Frobenius [1880] and Holder [1882] did not use divergent seriesas such; these scholars did not connect 1− 1 + 1 − · · · with 1/2, were not aware thata sum could be associated with a divergent series in a modern and rigorous sense, andremained thus within the limits of convergent series theory.

In [1949], Hardy actually re-interpreted theorems (F) and (Ho) from the point ofview of the new theory of summability which developed in 20th century. In the samemanner, many other ancient theorems can, however, be viewed as theorems of summableseries theory (and, as mentioned above, Hardy himself rethought theorem (C2) as histheorem (H) on summable series). This attitude with respect to previous mathematicsis perfectly reasonable for a mathematician like Hardy, whose aim is to construct anew general theory including older results. It is, however, historically unsatisfactory,because a historian, as far as possible, hasabove allto enquire into how 19th century


mathematicians effectively thought and what their ideas really were, and not to attributethem later concepts alien to their mentality.

The importance of (F) and (Ho) is merely to form part of the ‘background noise’of summability, together with many other theorems, such as (J), (C1), (C2), (S1); wemay interpret them as suggestions of some definitions of the sum or as propositions onsummability onlya posteriori, looking back.

Part three

In this third part I examine the birth of Cesaro’s definition of the sum. Until 1882he based the use of divergent series upon symbolic methods. A short time afterwards(1883), Cesaro derived the first idea of the sum by arithmetic means from the study ofasymptotic laws of numbers: however, this formulation was still related to the intuitiveconcepts of limit, existence,. . .. After 1886 Cesaro rejected the intuitive aspects ofthe infinitesimal analysis and tried to provide a more solid base to asymptotic numbertheory: within such research he formulated the C-summability.

1. Absurd equalities

One of the first fields of research of Cesaro was the umbral calculus. His interestin this method was raised by the reading ofNouvelle Correspondance Mathematique.In answering some questions which appeared in this journal, edited by Catalan, herediscovered several umbral formulas; most of them had already been given by FrancoisEdouard Anatole Lucas (1842–1891), one of the first who studied umbral methods,of whose numerous contributions Cesaro however had no knowledge. Only later, after acommunication from Catalan, did Cesaro come to know and study Lucas’s writings,with whom he shared the ancient idea of analysis as a well made language (Lucaswas explicitly to refer to Laplace in chapter XII of his [1891]). However, Lucas’sinterpretation of the umbral calculus (“a rapid method for writing certain formulas”, “ashorthand of formulas of Arithmetic and Algebra” [1891, 211]) was too restrictive fromCesaro’s point of view. Following Cauchy, Lucas limited himself to applying symbolicmethods to polynomials, while Cesaro applied them to power series. This was mainlydue to the influence of the Neapolitan Academic circles, which were very backward asconcerns analysis.31 In Naples, not only was the newε-δ approach neglected, but someof the most influential mathematicians were even indifferent to Cauchy’s methodology.For instance, Nicola Trudi (1811–1884) explicitly stated that he intended to use serieswithout regard to their convergence or divergence [1862, 135]. In particular, Cesaro

31 Cesaro learned his mathematics in places that were backward in the field of analysis, not onlyNaples, but also Belgium which had not progressed beyond Cauchy’s teaching. Cremona com-municated his opinion of the state of mathematics in Belgium to Cesaro in a letter of 13/2/1882:“keep in your mind that the best mathematics is not made in Belgium. Be silent so that the excellentCatalan does not hear” (“tenete presente che lamigliore matematicanon si fa nel Belgio. Zitti chenon ci oda l’ottimo Catalan” [Cesaro’s Nachlass]).

120 G. Ferraro

studied Trudi [1865], [1867], [1879], Fergola [1856], and Torelli [1867] carefully.These works still followed the methods of algebraic analysis, which consisted of thealgebraic treatment of functions and infinitesimal calculus born of Euler’s Introductioin analysin infinitorum, and had an approach analogous to that of the ‘CombinatorialSchool’ of Carl Friedrich Hindeburg (1741–1808) and German mathematicians ofthe first decades of the 19th century (cf. Jahnke [1993]).

In his earlier papers, Cesaro had an ambiguous attitude to divergence. InPrincipesdu calcul symbolique, he indeed stated that if one expands a functionf (x) in a powerseries and replaces the powersxi by a sequenceNi , these sequences “must be chosenin such a way that the series are convergent” [1883a, 2]. In practice, however, he lim-ited himself to such observations as: “by means of certain conditions of convergence,that are difficult to determine in a general way”, Euler’s transform

∑ni=0 (−1)iui =∑∞

i=0 (−1/(−2)i)1iu0 is derived from the expansion of 1/(1 + E) = 1/(2 +1)

(whereEup = up+1, 1up = up+1 − up, and, therefore,1 = E − 1) [1883a, 8];or he merely postponed the analysis of the convergence to further studies which henever made. Thus, although the umbral proof of the Euler-Maclaurin sum formulaimplicitly assumed analyticity off (x) at the origin (see part two), Cesaro wrote: “Inorder to simplify, I delay the question of convergence, which is specific in every par-ticular case, to a subsequent examination”32 and applied the Euler-Maclaurin for-mula not only to functions such asxp+1/(p + 1) (obtaining the widely known result1p+2p+3p+· · ·+np = ((n+ B)p+1 − Bp+1)/(p + 1)),33 but also to 1/x and logx.Forf (x) = 1/x, he derived

1 + 1

2+ 1

3+ 1

4+ · · · + 1

n= logn+ 1

2n− 1

12n2+ 1

120n4− 1

252n6+ · · · + C

(C being Euler’s constant); forf (x) = logx, he obtained the Stirling’s formulan! = √

2πnnne−n+g(n) with g(x) = ∑∞n=1B2n/(2n(2n− 1)x2n−1) [1883a, 4–5].

Actually such references to convergence merely served to avoid the accusation (easyto make of a 23 year old student) of using techniques most mathematicians believed def-initely banished from analysis as lacking in rigor, and, above all, they were a concessionto his teacher Catalan who helped him to publish his first papers and repeatedly advisedhim to be cautious in the use of divergent series. The reverent mention of the “conditionsof convergence” disappeared entirely in his later articles on the umbral calculus, writtenunder the more massive influence of Neapolitan analysts.34 For instance, it is missing in

32 “ Pour simplifier, la question de la convergence, specialea chaque cas particulier, est reserveea un examen ulterieur” [1883a, 4].

33 Cesaro defined Bernoulli numbers by the umbral relation(B + 1)n−Bn = n. This generatesa sequenceBn that is equal to the so-called even-index Bernoulli numbers (i.e., Bernoulli numbersdefined by the modern definitiont/(et − 1) = ∑∞

n=0 (Bn/n!)tn) for n = 0, 2, 3, 4, . . ., whileB1 = 1/2 (in the modern even suffix notation it isB1 = −1/2). In this third part I maintainCesaro’s notation.

34 From the combinatorial techniques of Neapolitan analysts Cesaro derived an inter-esting extension of the umbral calculus to symmetric functions, which he termed isobariccalculus. He did not hesitate to apply this calculus and other similar calculi to infinite se-ries. He, inter alia, proved and generalised Lagrange’s inversion formula:f (y)= f (z) +


[1886c], where Cesaro derived one of Euler’s theorems [1755, 2:240–242], which hemust have learned by reading Fergola [1862]: Given a functionf (x) = ∑∞

n=0 anxn,

and a sequenceun, the sumF(x) of the series∑∞n=0 unanx

n is

F(x) = f (xu) = f (xEnu0) = f (x + x1)

=∞∑n=0

xn

n!f (n)(x)(x1)n =

∞∑n=0

xn

n!f (n)(x)1nu0 [1886c, 195].

Forf (x) = 1/(1 − x), andun = np(p > 0), we haveF(x) = 1px + 2px2 + 3px3 +4px4+· · · = ∑p

n=11n(0p)(xn/(1 − x)n+1), and, forx = −1, 1p−2p+3p−4p+· · · =∑p

n=11n(0p)((−1)n+1/2n+1), which we may write

1p − 2p + 3p − 4p + · · · = 2p+1 − 1

p + 1Bp+1 (6)

as the Bernoulli numbersBp+1 are equal to(p + 1)/(2p+1 − 1)∑p

n=11n(0p)((−1)n+1/

2n+1) [1886b, 316–317].Equation (6) expresses nothing but the(C, p+ 1)-sum of 1p − 2p + 3p − 4p + · · ·

and is one of the ‘conventional formulas’, to which Cesaro paid special attention duringhis study of the umbral calculus. In a letter to Catalan of 1/6/1882 published inSurdiverses questions d’arithmetique, he already stated that his aim was to give a rigorousmeaning to them: “However, what I propose to do above all is to bring together in onebody of theory the conditions by means of which one can to use the equality (6) and theother conventional equalities that are derived from it, in an entirely rigorous way”.35

He called equalities of the type (6) absurd equalities, an ambiguous and cautiousname, due to Catalan, who had referred to divergent series disparagingly (for instancesee [1870, ch. 4] and [1885, 283–285]). For Cesaro the equation (6) “is nothing but aformula of convention: it is a mere algorithm, nothing but a tool, which I use successfullyin the study of certain series.”36 Equation (6) and other similar formulas can be used,provided they are well interpreted (“with circumspection and observing certain rulesestablished in advance”37). Besides: “I say that, although these formulas are false, theymay be used as a base of a theory, which shall not be more absurd than the theory ofImaginaries” [1883c, 248].38 Cesaro indeed thought that one could use equation (6) as

∑∞p=0 (x

p+1/(p + 1)!)dp/dzp[f ′(z)ϕp+1(z)

], which expresses the expansion of every (analytic)

functionf (y), wheny satisfies the equationy = z+xϕ(y)with arbitraryϕ (x being a parameter)[1885, 506].

35 “Mais ce que je me propose surtout de faire, c’est reunir, en un corps de theorie, les conditionsmoyennant lesquelles on peut se servir, en toute rigueur, de l’egalite [(6)], et des autresegalitesconventionnelles qui s’en deduisent” [1883c, 249].

36 Equation (6) “n’est qu’uneformule de convention: c’est un pur algorithme, elle n’est qu’unoutil, dont je me sers, avec assez de succes, dans l’etude de certains series” [1883c, 248].

37 “[A]vec circonspection, et respectant certaines regles, prealablementetablies” [1883c, 248].38 “[C]es formules, dis-je, quoique fausses, peuvent servir de la basea une theorie, qui ne serait

pas plus absurde que la theorie des Imaginaires” [1883c, 248].

122 G. Ferraro

a definition (analogously to√−1 = i) to derive symbolic equalities such as

f (x)− f (2x)+ f (3x)− f (4x)+ · · · = 12f (Ax) ,

f (x) being an appropriate function andA the umbra ofAp = (2(2p+1 − 1)/(p + 1))Bp+1. He was still far from modern conceptions and stressed the conventional aspect ofabsurd equalities, a technical instrument that made thinking easier39 and enabled us toderive exact formulas. They were conceived as formal equalities, substantially differentfrom exact (i.e., numerical) equalities: he thought that 1− 1+ 1− 1+ · · · was equal to1/2 by convention, whereas 1+ 1/2 + 1/4 + · · · was intrinsically equal to 2.

2. An asymptotic definition of the sum

Cesaro quickly realised that the objective of furnishing a justification of the use ofdivergent series cannot be achieved within the umbral calculus. He therefore endeav-oured to arrive at his aim differently, and drew his inspiration on asymptotic laws fornumber-theoretic functions. In his [1838, 351], Dirichlet had proposed to call a func-tion f (n), simpler than another functiong(n), an asymptotic law for the complicatedfunctiong(n) if lim n→∞ f (n)/g(n) = 1. He also noted that certain functionsg(n) varyirregularly and, for this reason, it is better to study the asymptotic behaviour of theirmean value(g(1)+ g(2)+ · · · + g(n))/n. For instance ifτ(n) is the function whichgives the total number of divisors of a natural numbern, andD(n) = ∑n

i=1 τ(i) itssummatory function, Dirichlet proved limn→∞

[(D(n)/n)− logn

] = 2C−1, whereCis Euler’s constant. Following Dirichlet, Cesaro used infinite series to obtain variousnumber-theoretic laws; he employed the notion of mean equality:

Definition (A). A functionψ(n) is said to be equal in mean to the functionψ ′(n) withregard to the‘ function of reference’ 8(n), andψ ′(n) is said to be the mean expressionor value ofψ(n) if

39 He referred to the tradition of analysis as a language of discovering: “It is quite remarkablethat the most fruitful theories are precisely those where to shorten the path of thought one uses theseconventional ideas, provided, during the research, one takes care to eliminate everything that cangive rise to a misinterpretation of the same ideas. The theory of imaginary and the symbolic calculusis of this kind, and it can lead to false results and, sometimes, to the most bewildering paradoxes, ifthey are misinterpreted. Instead, if one always pays attention to the merely conventional characterof the starting point and does not take a step beyond what initial conventions allow, one arrives atthe most unexpected results by means admirable for their simplicity and elegance” (“Il est bienremarquable que les theories les plus fecondes sont precisement celles ou, pour abreger le cheminde la pensee, on fait usage de ces idees de convention, en ayant soin d’ecarter, dans le cours desrecherches, tout ce qui peut donner lieua une fausse interpretation des memes idees. Il en est ainside la theorie des imaginaires et du calcul symbolique, lesquels mal interpretes, peuvent conduirea des resultats faux, et, quelquefois, aux paradoxes les plusetonnants; tandis que, si l’on a soin detenir toujours present le caractere purement conventionnel du point de depart, en ne faisant pointun pas de plus, qui ne soit consenti par les conventions initiales, on arrive aux resultats par lesplus inattendus, par des moyens admirables de simplicite et d’elegance” [1883c, 248–249]).


limn→∞

ψ(1)+ ψ(2)+ · · · + ψ(n)

8(n)= limn→∞

ψ ′(1)+ ψ ′(2)+ · · · + ψ ′(n)8(n)

,

provided these limits are finite[1883c, 123].40

A short time afterwards he introduced thecomplete mean equalityand used this no-tion to sum divergent series.41 Given the functionsf (x), g1(x), g2(x), g3(x),. . . 81(x),82(x), 83(x), . . ., the complete mean equality of the functionf (x) with regard tothe scale of reference81(x), 82(x), 83(x), . . . is denoted by the symbolf (x) =g1(x)+ g2(x)+ g3(x)+ · · ·, which meansg1(x) is the mean expression off (x) withregard to the function of reference81(x); g2(x) is the mean expression off (x)−g1(x)

with regard to82(x); g3(x) is the mean expression off (x)−g1(x)−g2(x)with regardto83(x); etc.

In Medie ed assintotiche espressioni in aritmetica, Cesaro said that “a function hasalways a mean expression but can fail to have an asymptotic expression”42 and thatthe calculus of mean expressions differs from the asymptotic calculus. Basing himselfon this distinction he stated that the question of mean expressions is not different fromthat of divergent series; the term “sum” indeed denotes the asymptotic expression ofthe partial sums of a series when it is constant. When this constant does not exist it ispossible to consider the mean expression. Cesaro called

Definition (B). “The sum of a series the mean expression of the sum of its x first terms,the function of reference being x”.43

Indeed the sum of∑anis limn→∞ 1

n

∑ni=1A1(i), whereA1(n) = ∑n

i=1 ai = S0n.

If the limit does not exist, the mean value is to be determined by repeated calculations.Cesaro does not formalise the definition in the most general case but restricts himselfto illustrate it by some examples. In order to sum 1− 2 + 3 − 4 + · · ·, he calculatesA1(n) = (−1)n+1 [(n+ 1)/2], where [x] is the integral part ofx, and

40 Here I correct a trivial misprint which is found in the original work and also in the carelessedition of Cesaro’s Opere scelteby Carlo Miranda. In [1881, 100] Cesaro had given a differentdefinition: a functionψ ′(n) is the mean value of the functionψ(n) if lim n→∞ |(ψ(1)+· · ·+ψ(n))−(ψ ′(1)+ · · · + ψ ′(n)| = 0.

41 The definitions of the mean expression and sum were published in [1887a] inGiornaledi Matematiche, a little known journal printed in Naples, in Italian. Originally these definitionsformed part of a long memoir of number theory, written in French and entitledNouvelles Notesd’Arithmetique Asymptotique, which was, unfortunately, already lost in 1884. At the present day,some pieces of the first version of this memoir are kept in Cesaro’s Nachlass. Among them there isa short note, entitledCalcul Asymptotique et Calcul Moyen, which is given over to the definitionsof the sum and mean. It did not differ substantially from the first part of [1887a] and dates back1883. Albeit the Italian version ofCalcul Asymptotique et Calcul Moyenwas published whenCesaro had already accepted epsilontics, it shows traces of more backward concepts.

42 “Una funzione ha sempre un’ espressione media, mentre puo mancarle una espressioneasintotica” [1887a, 253]. Cesaro took the existence of means for granted for an appropriatechoice of the scale of reference. He tacitly assumed that one, sooner or later, would be arrived atone result by the iteration of the procedure.

43 “dicesi somma d’una serie l’espressione media della somma dei primi x termini, essendo xla funzione di riferimento” [1887a, 254–255].

124 G. Ferraro

S1n = A2(n) =

n∑i=1

A1(i) = 1 − (−1)n

2· n+ 1

2.

SinceA1(n), 1nA1(n), and 1

nA2(n) do not converge, he computes

S2n = A3(n) =

n∑i=1

A2(i) = 1

2

[n+ 1

2

]2

+ 1

2

[n+ 1

2

]

and finds thatA3(n) is equal in mean ton2/8 with regard to the function of referencen3.FromA3(n) = n2/8, Cesaro deducesA2(n) = n/4 andA1(n) = 1/4; consequently hetakes 1− 2+ 3− 4+ · · · = 1/4.44 To sum 1− 4+ 9− 16+ · · ·, he similarly computes

A5(n) = 1

6

[n+ 1

2

]3

+ 1

2

[n+ 1

2

]2

+ 1

3

[n+ 1

2

]

and derives

A5(n) = n3

48, A4(n) = n

16

2, A3(n) = n

8, A2(n) = 1

8, A1(n) = 0 .

He, therefore, assigns the sum 0 to 1− 4+ 9− 16+ · · · (the step fromA2(n) toA1(n)

is really not homogeneous with respect to others).We may re-phrase Cesaro’s procedure thus: given a divergent series

∑an, if j is

the first index of the sequence

Ak(n) =n∑i=1

Ak−1(i) = Sk−1n =

n−1∑r=0

(r + k − 1k − 1

)an−r , k = 2, 3, · · ·

for whichAj(n) is asymptotic tor · nh, with h < j , then the number{0 if h < j − 1r · (j − 1)! if h = j − 1

is said to be the sum of the series. In other terms, if j is the first index such thatlimn→∞Aj(n)/n

j−1 exists and is equal toR, then the sum isR · (j − 1)!

44 Although these steps are clearly based upon the fact that∑n

k=1 k/4 = n(n+ 1)/8 and∑n

k=1 1/4 = n/4, they cannot be entirely justified by(A) and(B) and imply an implicit general-isation of(B). In fact, the mean expression ofA2(n) is preciselyn4 with regard to the function ofreferencen2, because

limn→∞

n∑i=1A2(i)

n2= lim

n→∞A3(n)

n2= lim

n→∞

12

[n+1

2

]2 + 12

[n+1

2

]n2

= limn→∞

n(n+1)8

n2= lim

n→∞

n∑k=4

k

4

n2.

Instead limn→∞(∑n

i=1A1(i))/n = limn→∞ (A2(n))/n = limn→∞ (1 − (−1)n)/2 · (n+ 1)/2)/n

does not exist and it is not possible to deduce that the mean expression of the partial sumsA1(n)

is 1/4 with regard to the function of referencen.


This seems to be the definition ofC-summability and actually leads to the sameresults. However, providing a definition of the sum of divergent series does not guar-antee a modernity of thought by itself: as above mentioned, Euler also gave a famousdefinition of the sum. In my opinion, Cesaro’s procedure did not involve an entirelymodern notion of the sum. Cesaro [1887a, 254–255] indeed stated that “it will not beallowed” to assign a sum different from 1/2 to the series 1−1+1−1+· · ·: he was stillseeking thetruesum of a series and was unaware of the plurality of possible definitions.In particular, he was far from considering Cauchy’s definition of the sum conventional,as he did in [1890]. Besides, inMedie ed assintotiche espressioni, in aritmetica, Cesaroused the notion of limit in an intuitive, naturalistic way and never faced the problem ofthe existence of the limita priori as he thought that the existence was guaranteed by theexhibition of a result. This is due to the permanence of the formal conception, which paidattention to algorithmic and combinatorial aspects, and not to existence theorems. Onlyafter 1866, when he accepted theε-δ method, Cesaro attached the proper importanceof existence theorems.45

3. A probabilistic definition of the sum

After 1886 Cesaro took part in the research stream on series theory which aimedto rigorise and generalise earlier results from the epsilontic point of view. He, however,had an original stance in this stream as he deepened the notions of continuity, derivativesand limits with concepts of a probabilistic nature.

Stimulated by the reading ofRemarque sur la th´eorie des series [1886], in whichMathias Lerch (1860–1922) had determined a convergent series

∑an such that the ra-

tio an+1/an is not bounded, Cesaro analysed the convergence of series by the followingtheorem:

Theorem (C3). If the subsequencesa(i)nk , i = 1, 2, . . . , r are a set partition of a givensequencean, if limk→∞ a

(i)nk = li , i = 1, 2, . . . , r, and ifωi = limn→∞ ti (n)/n, where

ti (n) denotes the number of the terms ofa(i)nk not greater thann, then one has

limn→∞

1

n

n∑1

an =r∑1

ωili [1887b, 272−273] . (7)

By applying (7) toan = log(un/un−1) with a0 = 1, Cesaro derived that if the sub-sequences ofun/un−1 go to l1, l2, . . . , lr , then limn→∞ n

√un = l

ω11 l

ω22 · · · lωrr . He also

observed that (7) enables us to give a meaning to the expression “the sum of a divergentseries” and that, for example, 1− 1 + 1 − 1 + · · · is equal to 1/2 [1887b, 274].

The analysis of the convergence tests persuaded Cesaro that the limit concept andother fundamental notions dependent on it (continuity, derivation, integration) had tobe generalised. He, however, used probability theory, a tool unsuitable for his purpose,

45 In spite of the statements of principle, Cesaro often transformed the calculation of a meanexpression into the calculation of an asymptotic expression. This fact, at times, gave rise to confu-sions that were definitely solved only in 1888 after a controversy with Jensen (see footnote 48).

126 G. Ferraro

because it itself needs the limit concept. In [1888k], Cesaro, first, introduced the degreeof discontinuityω(x) of a functionf (x) at the pointx:

For everyh > 0, andε > 0, if y is chosen at random from (x, x + h)46, letprh,x {|1f | > ε} be the probability that|1f | = |f (y)−f (x)| > ε. Let furtherωε(x) =limh→0prh,x {|1f | > ε}. Then the degree of discontinuityω(x) is limε→0ωε(x).

He then considered|dωε/dε|, which he termed the intensity of aspiration (intensitad’aspirazione) after the degree of discontinuity, and stated that if limε→0 (ω − ωε)/ε =dωε/dε does not exist, one is obliged to evaluate this limit by probabilistic criteria. “It is,therefore, necessary to extend the limit concept” (Occorre dunque estendere il concettodi limite) [1888k, 283].

Finally, Cesaro defined the mean limit off (x). Given a real numberr, letpε,h(x, r)be the probability that|f (y)− r| < ε, wheny ∈ (x, x+h) is chosen at random, then, iflimh→0pε,h(x, r) = pε(x, r) and limε→0pε(x, r) = p(x, r),47 themean limitof f (x)at the pointx is the numberλ = ∑

r∈< rp(x, r). For instance 1/2 is the mean limit of

f (x) ={

1x

−[

1x

]for x 6= 0

0 for x = 0

whenx → 0.Cesaro was, however, pessimistic with regard to the use of mean limits in the calculus

because it was not possible to extend all the usual properties of limits to mean limits. He,nevertheless, thought that this notion was relevant when one restricted its use to the studyof mathematical events and numerical distributions [1888k, 286], and therefore whenf (x) is a discrete function. In this simpler case he defined themean limitof a sequencean asλ = ∑

r∈< rp(r), where, for any real numberr, p(r) = limε→0pε(r), andpε(r)is the probability that, given a numbern, chosen at random,|r−an| < ε. If the sequencean is subject to the same constraints as those of theorem (C3), then the mean limitλ isprecisely

∑r1ωili , and, therefore,λ = limn→∞ 1

n

∑ni=1 ai . This fact persuaded Cesaro

‘to define the limit of a sequence otherwise’ and to classify sequences into genera:

Definition (M). A convergent sequencean is of genus 0; iflimn→∞ an does not exist,we consider the‘derived’ sequence1

n

∑ni=1 ai , and if limn→∞ 1

n

∑ni=1 ai exists and is

equal toλ, we assumeλ to be the mean limit or, simply, the limit ofan and say thatanis of genus 1; if the ‘derived’ sequence1

n

∑ni=1 ai has not limit, ‘we derive a third one,

and so on’ [1888k, 285].

This definition differs from that of [1887a] in two respects. First, if we apply (M)to the partial sumssn of a series, it provides H-summability, and not C-summability.Second, in [1888k] there is no longer the idea that the iteration of the procedure, sooneror later, regularises the sequence. In fact, Cesaro stated that although these ‘derived’sequences tend to becomea1, a1, a1, a1, . . . , there exist sequencesan whose ‘derived’sequences have no determinate limit [1888k, 285].

46 A similar definition may be given fory ∈ (x − h, x).47 Cesaro stated thatp(x, r) is the aspiration off (x) after the valuer; in fact if f (x) is

continuous inx0 one hasp(x0, r) = 1, for r = f (x0), andp(x0, r) = 0, for r 6= f (x0).


4. C-summability

After 1886 Cesaro, inter alia, worked on the foundation of the asymptotic the-ory of numbers, obtaining several results on means which form part of a progressiveenlargement of Cauchy’s theorems. For instance, in [1888d], to prove rigorously thatlimn→∞ (ϕ(1)+ ϕ(2)+ · · · + ϕ(n))/n2 = 3/π2, whereϕ(n) is the totient function(which gives the number of positive integers less or equal ton and relatively prime ton), he showed48 that if a functionF(n) is equal in mean toσ , then

limn→∞

1rF (1)+ 2rF (2)+ · · · + nrF (n)

nr+1= σ

r + 1

and generalised this result demonstrating:

Theorem (C4). If limn→∞ 1n

∑ni=1 ci exists andlimn→∞ n

(1 − vn+1

vn

)6= 0, then

limn→∞

n∑i=1

civi/ n∑i=1

vi = limn→∞

1n

n∑i=1

ci [1888d, 342− 343] .

Now Cesaro used epsilontics and the most modern analytic concepts going farbeyond the teaching of Catalan, who considered the new conceptions as ‘metaphysi-cal’ [letter to Cesaro of 14/2/1892, Cesaro’s Nachlass], and the even more backwardNeapolitan analysts. After 1886, Cesaro looked at his old ideas on divergent seriesfrom a new perspective: he, however, did not reject them, but rather strove to legitimatethem by new and more valid reasonings. Thus he often underlined the vagueness ofthe bounds between convergence and divergence: “It is impossible to separate cleanlythe class of convergent series from that of divergent series. The necessary imperfectionof all the particular characters of convergence is due to this impossibility.”49 Indeed if∑vn is divergent, one can construct another slower divergent series50∑ un by taking

un = f (∑ni=1 vi)−f (

∑n−1i=1 vi), wheref (x) is a function such that limx→∞ f (x) = ∞

and limx→∞f ′(x) = 0.51

The synthesis between Cesaro’s early view, related to the 18th century formal con-ception, and theε-δ method, which had reinterpreted the numerical conception, was

48 Cesaro’s paper was the object of a controversy with Jensen, who, in his [1888b], noticeda confusion between mean and asymptotic behaviour. Cesaro ([1888m] and [1889a]) answeredby distinguishing between asymptotism (in the strict sense of word) and mean asymptotism andreferring to the definition of limit given in [1888k].

49 “Il est impossible de separer nettement la classe des series convergentes de celle des seriesdivergentes. C’esta cette impossibilite que nous devons l’imperfection necessaire de tous lescaracteres speciaux de convergence” [1888c, 288].

50 Given the series∑un and

∑vn, the former is said slower divergent than the latter if

limn→∞∑n

i=1 ui/∑n

i=1 vi = 0.51 Analogously, in [1888g, 314–317], Cesaro constructed a slower convergent series than a

given convergent series∑un, by takingvn = f

(∑∞i=n vi

) − f(∑∞

i=n+1 vi), wheref (x) is a

function such that limn→∞ f (x) = 0 and limn→∞ f ′(x) = ∞. The series∑vn is said slower

convergent than∑un if lim n→∞βn/αn = ∞, whereαn is the remainder of

∑un andβn is the

remainder of∑vn.

128 G. Ferraro

formulated, in a mature and conscious way, inSur la multiplication des s´eries[1890].The first part of this paper is devoted to a generalisation of Cauchy’s theorem con-cerning the product of series, whose hypothesis had already been weakened by FranzMertens (1840–1927) in his [1875]. Cesaro first proved this lemma: If limn→∞ an = a

and limn→∞ bn = b then limn→∞ (a1bn + a2bn−1 + · · · + anb1)/n = ab.Second, he demonstrated the proposition: If

∑un and

∑vn converge toU andV ,

respectively, and∑wn is their Cauchy product, then

limn→∞

W1 +W2 + · · · +Wn

n= limn→∞

U1Vn + U2Vn−1 + · · · + UnV1

n= UV ,

whereUn, Vn andWn are the partial sums of∑un,

∑vn, and

∑wn.

Third, he generalised these two results and proved that: if limn→∞ an/nr−1 = a and

limn→∞ bn/ns−1 = b, for some integersr > 0, s > 0, then

limn→∞

a1bn + a2bn−1 + · · · + anb1

nr+s−1= 0(r)0(s)

0(r + s)ab ,

where0(x) is the gamma function; and ifW,U, V are the mean values of∑un,

∑vn

and∑wn(where

∑wn is the Cauchy product of

∑un and

∑vn), thenW = UV .

At this juncture, Cesaro, finally, defined the sum of divergent series. Given a di-vergent series

∑un, he termed it asimply indeterminate series, if the sequenceUn of

the partial sums of∑un converge towards a mean valueU , which is called thesum

of the series. WhenUn does not converge in mean, Cesaro considered the sequencesU(1)n , U

(2)n , U

(3)n , . . . whereU(r)n is defined by the relation

(r + n− 1n− 1

)U(r)n =

n−1∑i=0

(i + r

i

)un−i ,

and called the series∑un k-fold indeterminateif k is the minimum integer such that

U(r)n converges to a numberU , which is thesumof

∑un by definition.

After proving that limn→∞ U(r+1)n = limn→∞ U

(r)n , provided the right-hand side

exists, by using (C1) and the relation(r + n

n− 1

)U(r+1)n =

n∑i=1

(i + r − 1i − 1

)U(r)i ,

Cesaro gave two examples ofr-fold indeterminate series,

1r−1 − 2r−1 + 3r−1 − 4r−1 + · · · = 2r−1rBr , and

1r−1 − 3r−1 + 5r−1 − 7r−1 + · · · = 12Er−1 ,

which underline the link of C-summability with his first research on absurd equalities. Henoted: “This yields a classification of indeterminate series, which is doubtless incompleteand insufficiently natural but, for the time being, it suffices to show that one can use


indeterminate series in calculations quite rightly, whatever most geometricians think. Itis rash to assert that non-convergent series will never be useful.”52

This idea had already been expressed by Cesaro in his early writings and the 1890definition actually provides the same results as the definition (B). Nevertheless Cesaro’slanguage was changed and this modification presupposes the transition towards a moremodern conception of mathematics; he had overcome his original conception basedupon the distinction between symbolic and real expressions, to use Pincherle’s words(see part two). Now Cesaro clearly stated that even the usual definition of the sumS of the series

∑∞n=1 an is a conventionalway to denote limn→∞ Sn = S. Cauchy’s

definition is no longer considered necessarily true from a conceptual view: it is moreuseful and simpler than other conventions, but it is merely a convention; the existenceof a mathematical object always depends on an appropriate definition.

Cesaro, probably, derived this frame of mind from geometry. In fact in a text onpolyhedra inn-dimensional spaces, he stated:

The preceding research was written more than eight years ago, in complete ignoranceof what had been done before. . . When I had occasion to talk about this research tothe renowned Prof. Gabriele Torelli, I received the very polite communication of asingular and interesting work, due to W.I. Stringham, whose title is: ‘Regular figures inn-dimensional space’ (American Journal of mathematics, vol. III, 1). This work, whichis rather different from mine. . . [because] the authorwould not be surprised if the tallbuilding collapsed, since he believed to have used methods extremely subject to errors.I instead have taken care of highlighting what was the merely hypothetical content ofmy methods, – which coincide with Stringham’s, all things considered –. Given certaindefinitions, the conclusions at which one arrives can be nothing but rigorously exact, andwe would be amazed only if they were not exact.53

These words include the crucial idea of modern axiomatics and [1890], as far as Iknow, is the first clear and aware application of such ideas to analysis.

52 “Il r esulte de la une classification des series indeterminees, qui est sans doute incomplete etpas assez naturelle, mais qui nous suffit, pour le moment, pour montrer qu’on peut parfaitement biense servir des series indeterminees dans les calculs, quoi qu’en pensent la plupart des geometres. Ilest temeraire d’affirmer que les series non convergentes n’auront jaimais d’utilite” [1890, 360]. In[1890], Cesaro gave only one theorem illustrating the usefulness of divergent series: The Cauchyproduct series of an r-fold indeterminate series with sumU and of an s-fold indeterminate serieswith sumV is an(r + s + 1)-fold indeterminate series with sumUV . This is the first theorem ofthe theory of summability.

53 “Le ricerche precedenti sono state redatte, or son piu di otto anni, in una completa ignoranzadi cio che si era fatto prima di noi. . . Avendo avuto occasione di parlare di queste ricercheal ch. prof. Gabriele Torelli, ne abbiamo ricevuto la gentilissima comunicazione di un lavorocurioso ed interessante, dovuto a W.I.Stringham, ed intitolato: ‘Regular figures inn-dimensionalspace’ (American Journal of mathematics, vol. III, 1). Questo lavoro, che differisce alquanto dalnostro,. . . [in quanto] l’autore, credendo di aver posto in opera metodi estremamente soggetti aderrore,non sarebbe sorpreso se l’edificio innalzato cadesse. Noi, invece, abbiamo avuto cura dimettere in evidenza quello che i nostri metodi, – che coincidono, in fondo, con quelli di Stringham–, contengono di puramente ipotetico. Poste certe definizioni, le conclusioni alle quali siamo giuntinon possono essere che rigorosamente esatte, e saremmo sorpresi solamente nel caso che tali nonfossero” [1886a, 73].

130 G. Ferraro

In the following years, Cesaro was concerned with other branches of mathematicsand his references to C-summability were very rare. The only two brief hints are in[1894, 432–433], an article very suitable for applying the new definition, and in [1891,4], a work on probability where he stated that divergent series would be brought to thecalculus in an entirely rigorous way and noted: “One already writes

(1 − 1 + 1 − 1 + · · ·)2 = 1 − 2 + 3 − 4 + · · ·and asserts that both the sides are equal to 1/4”. There are also only a few referencesto divergent series in Cesaro’s correspondence. He, probably, discussed of them withGiuseppe Peano54 and claimed his priority in a letter to Pincherle.55 After 1890 itwas Borel who actually transformed Cesaro’s definition into the starting point of atheory of divergent series.56

Acknowledgements.I thank Jeremy Gray for the useful suggestions and for improving myEnglish.

54 On 14/1/1892 Peano wrote to Cesaro: “As concerns to the liberal point of view fromwhich I consider Taylor’s formula (cf. Peano [1891]), it is old and is implicit in many passages,e.g. Euler’s [. . . ] What I believe to be new is the enunciation of which are the legitimate stepsand which the illegitimate ones. I’ll examine Mr Stieltjes’s note (Peano refers to [1886]) thatyou point out”. (“Riguardo al punto di vistaliberale da cui considero la formula di Taylor,essoe vecchio; ede implicito in tanti passi p.e. di Eulero. . . Quello che io ritenevo nuovo, siel’enunciazione dei passaggi leciti, e di quelli non leciti. Esaminero la nota del sig. Stieltjes cheElla mi indica” Palladino [1991, 261]). The reciprocal influences between Cesaro and Peanohave not yet been studied. In particular, Peano [1894] pursued Cesaro’s mean theorems and partVIII of Peano’s Formularioshows traces of Cesaro’s results. Conversely Cesaro [1897], [1905]and [1906] follow Peano [1890].

55 On 4/5/1906, Pincherle wrote to Cesaro: “In the footnote at p. 137, I touched on Borel,not to ascribe to him a priority which is not his due, but only because he has written most widelyon this topic. For this reason I used the locution ‘in a particular way’. However, I shall not forgetto rectify the omission of your name in the next edition: on the other hand, Borel himself,several times, has acknowledged that the germ of his study is in your research.” (“Nella nota ap. 137 ho accennato al Borel, non per attribuirgli la priorita che non gli spetta, ma solo percheha scritto piu diffusamente sull’argomento, e a cio la locuzione ‘in particolar modo’ che housata. Ma non manchero, in una prossima edizione di riparare all’omissione del Suo nome: delresto il Borel stesso, a piu riprese, ha riconosciuto che il germe del suo studio si trova nelledi Lei ricerche.” [Cesaro’s Nachlass]). Pincherle refers to his [1906], where he stated that itis “possible, under appropriate conventions, to give a meaning and a practical application evento divergent or indeterminate series, as various authors, and in a particular way Borel, haveshown recently” (“[e] possibile, sotto opportune convenzioni, di dare un significato ed una praticaapplicazione anche a serie divergenti od indeterminate, come hanno mostrato di recente vari autori,ed in particolar modo il Borel” [1906, 137]).

56 I do not know when Borel first learned of Cesaro’s notion. I merely note that in the sametome of the Bulletin of Darboux whereSur la multiplication des seriesappeared, about fifteenpages before Cesaro’s paper, there is the noteSur le changement de l’ordre des termes d’uneserie semi-convergenteby Emile Borel, “eleve de l’ecole Normale superieure”.


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(Received February 9, 1999)

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