tensor analysis history

Revued’histoiredesmath´ ematiques, 2 (1996), p. 215–264. ON THE GENESIS OFTHE CONCEPT OFCOVARIANT DIFFERENTIATION Luca D ELL’AGLIO(*) ABSTRACT . — The purpose of this paper is to reconsider the genesis of the concept of covariant differentiation, which is interpreted as arising out of two traditions runningthrough 19th-century research work. While the first tradition, of an algebraic nature,was responsible for the “algorithmic” emergence of the concept, the second, analyti-cal in character, was essentially concerned with the import of covariant differentiationas a broader kind of differentiation. The methodological contrast that these two tra-ditions exhibit, concerning the use of algebraic and variational methods, was mainlyevidenced in Ricci-Curbastro’s work, and was a significant factor in the genesis of tensor analysis. The emergence of the notion of covariant differentiation in his research work may, indeed, be interpreted as the resolution of that methodological contrast into thedefinitive form of a conceptual synthesis. R´ESUM´E.—SUR L’ORIGINE DU CONCEPT DE D ´ERIVATION COVARIANTE . Cet article se propose d’interpr´ eter l’origine du concept de d´ erivation covariante comme cons´equence de deux traditions de recherche au XIXesi`ecle. Alors que la premi` ere tra- dition, de nature alg´ ebrique, est ` a l’origine de l’´ emergence /angbracketleft/angbracketleftalgorithmique /angbracketright/angbracketrightdu con- cept, la seconde, de caract` ere analytique, se rapporte essentiellement ` a la significa- tion de la d´ erivation covariante comme extension ou g´ en´eralisation de la d´ erivation usuelle. L’opposition m´ ethodologique que manifestent ces deux traditions, ` ap r o p o s de l’utilisation de m´ ethodes alg´ ebriques ou variationnelles, apparaˆ ıt principalement dans l’œuvre de Ricci-Curbastro, et fut un facteur fondamental dans la gen` ese de l’analyse tensorielle. L’´ emergence de la notion de d´ erivation covariante dans son travail de recherche peut, de fait, ˆ etre interpr´ et´ee comme la r´ esolution de cette opposition m´ethodologique sous la forme d´ ecisive d’une synth` ese conceptuelle. 1. INTRODUCTION Emerging at the end of the 19th century with the work of the Italian (*) Texte re¸ cu le 3 f´ evrier 1995, r´ evis´e le 24 juillet 1996. Luca DELL’AGLIO,U n i v e r s i t ` a degli studi della Calabria, Dipartimento di matematica, 87036 Arcavacata di Rende (Cs), Italia. /circlecopyrtCS O C I´ET´EM A T H´EMATIQUEDE FRANCE, 1996 216 L. DELL’AGLIO mathematicianG.Ricci-Curbastro,absolutedifferentialcalculus(andsub- sequently tensor analysis) appears historically as one of the most impor- tant links between Riemann’s concept of space and the relativistic theoryof gravitation. As an extension of the usual calculus to general geomet-rical contexts, this theory indeed represents one of the most important developments of Riemann’s geometrical conceptions in the latter part of the 19th century. On the other hand, with reference to such notions asthat of covariant differentiation and of the tensor, the theory set out bythe Italian mathematician pointed to the possibility of an invariant for- mulation of analytical problems, possibly of a physical nature: a technical possibilitywhichwasto playaleadingroleinthe mathematicalexpressionof Einstein’s ideas, some decades later. Bearing that in mind, the aim of this paper is to provide a recon- struction of the emergence of the first fundamental concept of absolute differential calculus — that of covariantdifferentiation — as marking the convergence of various research traditions in mathematical thought, pre-vailing in the 19th century. More specifically, this reconstruction is basedon a number of historiographical tenets, concerning different instances of the impact of the idea of invariance, which I shall now detail. First of all, one may hold that there was an “algorithmic” genesis of the concept of covariant differentiation, arising out of a purely algebraicresearchtradition. As already suggested by other authors, 1prior to Ricci- Curbastro’s work, this concept had originated in Christoffel’s approach, as the result of a research tradition, consisting in the application of themethods of the theory of algebraicinvariantsto analyticalmatters.In thiscontext, the algorithm of covariantdifferentiation was used by Christoffelasawell-definedtechniqueinaparticularfieldofresearch,thatofdifferen- tial quadratic forms: in particular, it had the specific function of allowing a general programme to be carried out, that of the “reduction” of thetheory of differential invariantsto that of algebraic forms. As we shall see,this researchtradition had clearlyexerted an influence on Ricci-Curbastro in a period before his work was directly concerned with the creation of the absolute differential calculus. Such a methodological influence is nochancefeature:asweshallsee,anembryonicformofthealgebraicresearch 1In particular, see the recent book [Reich 1994]. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 217 tradition on differential invariants was already at work in the mathemat- ical community of post-Unification Italy, with the geometrical work of Casorati. Ontheotherhand,despitethefundamentalsignificanceofthealgebraic tradition with respect to differential invariants, it is tenable that theconceptual origin of covariant differentiation — as a generalisation of theusualdifferentiation—wasindependentofthattradition.Theappearanceof a Riemannian differentiation, indeed, finds its true justification onlywhen one takes into accountthe emergence of a second researchtradition,which to some extent ran counter to the former from a methodologicalpoint of view. More specifically, this second tradition was concerned withacloseinvestigationof“differentialparameters”,asarisingoutofthework of the French mathematician G.Lam´ e and developed mainly through the research work of E.Beltrami. This new tradition made its presence feltin the process leading to the emergence of absolute differential calculus,most recognisably when Christoffel’s research programme was extendedby Ricci-Curbastro to the study of differential parameters. The point that needs to be emphasised is the contrast inherent in such a switch in topics of investigation. At that time, indeed, the researchtradition concerned with differential parameters was grounded, methodo-logically speaking, on the use of the calculus of variations and only partlyon algebraic methods. This was no chance feature, since this second tradi-tion was closely connected to the thrust of classical mathematical physics, and hence to the study of partial differential equations. As we shall see, Ricci-Curbastro effected the introduction of the concept of covari-ant differentiation precisely for the purposes of furthering the study ofdifferential equations, his aim being to arrive at an invariant expressionof these equations in order to simplify their investigation. It is this verycross-over of the contexts of interpretation and methods — i.e., to usemodern terminology, the analytical interpretation of an algebraic tech-nique introduced to tackle some analytical problems — that warrantedthe emergence of the concept of covariant differentiation. Thus, the emergence and the very genesis of the concept of covariant differentiation appears as a specific synthesis of many research traditions concerning the idea of invariance, running through the 19th century: dif- ferential invariants, differential parameters and algebraic invariants. This 218 L. DELL’AGLIO fact — which, indeed, means that absolute differential calculus, together with Klein’s “Erlangen programme”, represented one of the most signif-icant products of the idea of invariance in 19th-century mathematicalthought — was especially significant with regard to the physical aspects of invariance that were to emerge with general relativity. And, as a final point, reconstruction of the genesis of the concept of covariant differentiation makes it possible, post factum ,t oe x a m i n et h e specificfeaturesofRicci-Curbastro’sscientificworkand,moregenerally,oftheItalianmathematicalcommunity’scontributionasspecificcontextsfor the appearance of absolute differential calculus. In effect, from a strictly historical point of view, one may view the present paper as a comparativestudy of some aspects of Ricci-Curbastro’s work in differential geometry. 2. RESEARCHTRENDSIN THE19TH-CENTURYTHEORYOF DIFFERENTIALINVARIANTS As is well known, the context of research in which Ricci-Curbastro’s analytical methods originated was provided by the theory of differentialinvariants, i.e. the study of differential quantities that are invariant with respect to any particular transformation of coordinates. 2In this man- ner, the Italian mathematician’s work may be considered as an aspect ofa more general phenomenon — the pervasiveness of the idea of invari- ance — which was a characteristic feature of a large part of mathematics throughout the 19th century [Bell 1945, chap. 20]. In this general context — where concepts of geometrical and algebraic invariance were coming to the fore — the study of differential invari-ants reflected various analytical requirements associated with the idea of invariance.Indeed, the modern theory of differentialinvariantsreachedits unified form only at the beginning of the 20th century, 3as the outcome of many research traditions at work in the course of the 19th century. 2According to M. Kline, tensor analysis “ is actually no more than a variation on anoldtheme,namely,thestudyofdifferentialinvariantsassociatedprimarilywithaRiemannian geometry ” [Kline 1972, p. 1122]. On this subject see also [Reich 1994, 4.1.2.1], [Tonolo 1954, pp. 2–6]. 3This may be considered to be a result of Klein’s thought. See [Veblen 1927, p. 15]. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 219 Apart from the approach of G.Halphen [1878] and S.Lie [1884] — which emerged much later in the century — there are essentially twotheoretical thrusts which were of major importance in this field. 4 The first direction — which will be referred to here as the “restricted [or special] theory of differential invariants” — arose from the context of 19th-century differential geometry. In effect, this was a direct sequelof Gauss’s geometrical opinions: according to this tradition, a differential invariant is the analytical reflection of the intrinsic properties of surfaces (such as the line element, curvature, and the angle between two directionson a surface). At the same time, more general invariants— the so-called “differential parameters” — were being studied by another line of research,arising out of the workof Lam´ eon the equations of classical mathematical physics. In this context, differential parameters are quantities — such as the Lapla- cianofafunction—bymeansofwhichitispossibletoshowtheinvariance of specific differential equations, in a well-defined geometrical situation. For quite some time, these research traditions developed, to a large extent, independently. They pursued similar aims but in different fields of research: intrinsic geometry, on the one hand, and the theory of partialdifferential equations, on the other. They actually converged only in the post-Riemannian period. Although exhibiting different concerns and activities, the two thrusts of research into differential invariantsshared one common methodologicalelement. Both traditions, indeed, were characterised by the implementa- tionoftwodistinct technicalmethodologies:the theoryofalgebraicforms, on the one hand, and the calculus of variations, on the other. The func-tion of these theoretical methods was operational, involved as they both were in the demonstration of the invariance (with respect to particular transformations of coordinates) of known differential quantities and thesearch for new, analogous, quantities. From an operational standpoint, this methodological duality was of no particular significance for the development of the theory of differential invariants: as we shall see later, apart from some particular cases, the 4On this subject, see [Reich 1973], [Struik 1933], [Veblen 1927], [Vincensini 1972], [Weitzenb¨ ock 1921]. 220 L. DELL’AGLIO two methodological approaches coexisted without any significant prob- lems. On the other hand, the use of well-defined theoretical methods in a new domain of research may not be considered in an abstract fashion,irrespective of their contexts of origin. In other words, the problem tobe analysed is to what extent the use of certain theoretical tools impliedan actual carrying over of their original conceptual features into the newresearch context. Specifically, in the case of the history of the theory of differential invariants,the twotheoretical methods (the algebraicand the variational)clearly exhibited different conceptual backgrounds. 2.1. The programme of “algebraic reduction” of the theory of differential invariants Owingtotheanalogywiththepropertiesofalgebraicinvariants,theuse ofalgebraicinstrumentswasofparticularsignificanceforthehistoryoftherestricted theory of differential invariants. More generally, this approachwas a natural development in 19th-century mathematical thought, andfoundareadyplace,onthebasisoftheleadingroledevolvingtothetheory of algebraic forms during this century and of the consequent tendency, evinced by quite a few mathematicians of the time, to extol the centralityof algebraic methods. Casoratiand“eliminationtheory” A highly significant example of the tendency to use algebraic meth- ods in the theory of differential invariants in pre-Riemannian times isto be found in the work of F.Casorati, one of the most important fig- ures of post-Unification Italian mathematics. Indeed, although they are known essentially for their analytical concerns, his works included a longpaper of a geometricalnatureaboutdifferentialinvariants[Casorati1860–1861], 5which may be viewed as one of the first systematic studies on the topic. This in particular may be argued from the fact that, in this con-text, there was an explicit and general definition of differential invariants. More specifically, a differential invariant — funzione inalterabile —w a s introduced by Casorati [ Ibid., p. 136] as any function fsuch that (1) f/parenleftbig a 11,a12,a22,∂ a11/∂x1,∂ a11/∂x2,etc./parenrightbig 5About this paper see [Bertini 1892, pp. 1221–1222], [Vivanti 1935, p. 135]. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 221 =f/parenleftbig A11,A12,A22,∂ A11/∂X1,∂ A11/∂X2,etc./parenrightbig where the arsandArsare the coefficients of the differential quadratic forms representing the metric element of a surface for the two different coordinate systems, xrandXr. In his pursuit of a systematic investigation, Casorati insisted on two methodological elements whose significance was to be realised later. On the one hand, from a structural point of view, his study was based on a preliminary classification of differential invariants by ordini,w h e r e the order of this invariant was defined in terms of that of the derivatives of the coefficients arsincluded in it. Consequently, from a methodologi- cal point of view, Casorati’s analysis of differential invariants exhibits a “vertical” approach in which the study of the first orders is given pride of place in a natural way. On the other hand, as we have mentioned, Casorati’s investigation of differential invariants was essentially algebraic in character, since it was based solely on the “elimination theory”. This second element — induced by the development of algebraic studies in Italy in the period around Unification [Bottazzini 1980] — may be considered as stemming from a definite methodological choice: “The purpose of this short paper is to show a method of finding the fundamental equations for the investigation of absolute properties; among such equations, the main one is that which expresses Gauss’s famous the- orem on the measure of curvature ....It is easy to point out that the way chosen by Gauss and other distinguished geometers to prove this theorem ..., does not make apparent as fully as may be wished the principal source of the significance of that theorem, that is the fact of its being the simplest [equation]of such a class; nor [does it make apparent ]those that actually follow it in order of simplicity .”6 6“Scopo di questo breve lavoro ` e l’esposizione di un modo di trovare le equazioni fondamentaliperlostudiodellepropriet` aassolute;fralequaliequazionilaprincipale` e quellaesprimenteilcelebreteoremadiGausssullamisuradellacurvatura ....`Efacile rilevare che la via tenuta da Gauss nel dimostrare il detto teorema, e quelle seguiteda altri insigni Geometri ..., non mettono in tutta l’evidenza desiderabile n` ec i `o dondevieneprincipalmentelaimportanzadelmedesimo,cio` ediessereilpi` usemplice possibiledisiffattacategoria;n` equaliindubbiamentesienoquellichetenganglidietro perordinedisemplicit` a” [Casorati 1860, p. 134]. 222 L. DELL’AGLIO This critical attitude of Casorati towards earlier investigations of dif- ferential invariantsarose out of his opinions concerning the significance of algebraic methods for the treatment of differential questions: “On these grounds especially, I believe the method to be worthy of some attention, that I am propounding here, which, consisting in a process ofelimination without exceptions, necessarily leads us to find the equationssought for one after the other and exactly in the order of their impor-tance.” 7 From a technical point of view, Casorati’s method was based on the elimination of the Xh-derivatives of the functions xrfrom the laws of transformation for the coefficients ars (2)/summationdisplay r/summationdisplay sars∂xr ∂Xh∂xs ∂Xk=Ahk(h,k=1,2), andfromtheirsuccessive Xr-derivatives.This,indeed,makesitpossibleto obtain certain expressions that are independent of the coordinate systemchosen, as containing only the quantities a rsand their derivatives. Through this algebraic method of resolution, Casorati was able to determine a large number of differential invariants of surfaces. In par-ticular, he discussed the search for differential invariants up to the fourthorder, first proving the non-existence of such invariantsfor the first order;going on to construct an invariant for the second (Gaussian curvature)and third orders and, finally, providing a well-defined procedure to obtainthose of the fourth order. It must be pointed out that this simple method of elimination was to be a characteristic instrument in much of the subsequent research onthis subject, where algebraic methods were broughtto bear on differentialquestions. ThemethodologicalcontrastbetweenLipschitzandChristoffel As is well known, in the 19th century, the main impulse leading to the development of the theory of differential invariants came from Riemann’sconceptions. More specifically, the lead came from the publication of the 7“`Especialmenteperquestoriguardoch’iocredopossameritarequalcheattenzioneil metodocheespongo,ilquale,consistendoinunprocessodieliminazionenonsoggettoadeccezioni,conducepernecessit` aatrovareleequazioniindiscorsol’unadopol’altra precisamente in quell’ordine con cui si succedono nella importanza ” [Casorati 1860, p. 134]. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 223 Germanmathematician’s Habilitationsschrift of1868[Riemann1854].The main technical topic of this paper — namely, the equivalence of differ- ential quadratic forms ( Aequivalenzproblem ) — was in fact taken up by E.B.Christoffel and R.Lipschitz in papers published in the same issueof the “Journal de Crelle” one year later [Christoffel 1869 a,b;L i p s c h i t z 1869].8 Although bearing on the same technical argument,9Christoffel’s and Lipschitz’s papers were methodologically quite different. Indeed, while Christoffel followed a purely algebraic approach to the Aequivalenz- problem, Lipschitz solved it by means of a “mixed” approach, where, in addition to algebraic methods, the calculus of variations played a signifi- cant role. This is actually the first occurrence of a methodological differ- ence that was to characterisethe later developments of the general theory of differential invariants. More specifically, his constant reference to the methods of the theory of algebraicforms notwithstanding, it was essentially by means of the cal- culus of variations that Lipschitz arrived at the conditions for the trans- formability of a differential quadratic form into another with constant coefficients. As is well known, such conditions are given by the vanish- ing of the following 4-index symbol, already considered independently by Riemann (3) ( ghki)=∂ ∂xiΓgh,k−∂ ∂xhΓgi,k+/summationdisplay p(Γp giΓhk,p−Γp ghΓik,p), where Γ gh,kand Γp giare the Christoffel symbols of the first and second kind.10In addition, after 1869, Lipschitz extensively used the calculus of variations in a number of papers directly concerned with the search 8Christoffel’s works are discussed from different points of view in [Ehlers 1981], [Leichtweiss 1981], [Pinl 1981] and [Reich 1994]. 9As is well known, the topics considered by these papers were actually very similar: while Christoffel faced up to the problem in the general case, Lipschitz, in a wayanalogous to Riemann’s [1861], but quite independently, investigated the conditionsfor the transformability of one differential quadratic form into another with constantcoefficients. 10The symbol (3) was introduced by Riemann in his Commentatio [1861] and inde- pendently by Christoffel and Lipschitz: see on this matter [Farwell, Knee 1990]. Theexpression given here is that of Christoffel, as it is relevant to the subsequent discussionof his work on differential quadratic forms. 224 L. DELL’AGLIO for the differential invariants of a differential quadratic form [Lipschitz 1870, 1871]. In these papers, the original problem would be reduced to the theoretical domain of the calculus of variations, as a rule by means ofa mechanical interpretation. 11For instance, in order to determine the dif- ferentialinvariantsofadifferentialquadraticform,Lipschitzwouldreduce the problem to that of obtaining the maximum and minimum values of the pressure exerted bya mechanicalpointmovingon a surface defined bymequations in an n-dimensional space and having for its line element the differential quadratic form under consideration. This procedure involved an algebraic equation of degree ( n−m), of the form D(ω)=0 ,t h ec o e f - ficients from which may be used to construct the differential invariants. It is worthy of note that, despite his intensive use of the calculus of variations,Lipschitzattachedmoreimportanceto algebraicmethods from a heuristic point of view. This emerges from the clear-cut distinction he madebetweenthealgebraicandthevariationalmethods,wheretheformerwere denoted as “direct”: “The point of view from which the forms of ndifferentials are here being considered makes the bilinear form, which points to the conditions of integrability, and the quadrilinear form, that denotes the measure of curvature, appear as the first elements of a chain; there remains the needto continue this sequence of direct methods, in order to solve the givenproblem.In what follows, this problem will, on the contrary, be generally dealt with through an indirect method, whereby, for any forms, an associ- ated problem of the calculus of variations shall be considered as solved. ” 12 This methodological duality — which is similar to what, at the same time, was happening in the theory of differential parameters, as we shall see — was totally absent from Christoffel’s work. His investigation of the equivalence of differential quadratic forms was based solely on algebraic 11For the link between mechanics and geometry in Lipschitz’s work, see [L¨ utzen 1995, pp. 34–45]. 12“DerGesichtspunkt,vonwelchemausdieFormenvon nDifferentialenhierbetra- chtet sind, l¨asst die bilineare Form, welche auf die Bedingungen der Integrabilit¨ at hinweist,unddiequadrilineareForm,welcheaufdasKr¨ ummungsmassdeutet,alsdie ersten Glieder einer Kette erscheinen; es bleibt das Bed¨ urfniss, diese Reihe directer MethodenzurL¨ osungdergestelltenAufgabefortzusetzen.DieseAufgabewirddagegen imFolgendendurcheineindirecteMethode,beiderf¨ urjedeFormeinentsprechendes Problem der Variationsrechnung als gel¨ ost vorausgesetzt ist, allgemein erledigt wer- den” [Lipschitz 1869, p. 74]. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 225 arguments; a fact which is hardly surprising, from a historical point of view, since the German mathematician approached the Aequivalenzprob- lemfrom the angle of earlier studies on algebraic invariants [Christoffel 1868a,b]. The algebraic character of Christoffel’s work on differential quadratic forms stems essentially from one theorem — the so-called Reduktionssatz [Klein 1927, p. 198] — which established a close link between the studyof differential forms and the theory of algebraic invariants. The German mathematician arrived at this theorem by way of technical steps of fun- damentalsignificance for the emergence of tensor analysis,since they con-tained the first consideration of the technical expression of covariant dif-ferentiation: this justifies recounting these steps in outline. After reducing the original problem — that of the transformabil- ity of one given quadratic differential form F=/summationtext ikωikdxidxkinto another F/prime=/summationtext ikω/prime ikdx/prime idx/prime k, involving a locally reversible transforma- tion:xi=xi(x/prime 1,...,x/primen) — to the investigationof the following system of partial differential equations (4)∂2xλ ∂x/primeα∂x/primeα s=/summationdisplay r(Γr ααs)/primeuλr−/summationdisplay iisΓλ iisuiαuis αs, where ui α=∂xi/∂x/prime α, Christoffel showed that the conditions of integra- bility of this Pfaffian system may be put in the form (5) ( αβγδ)/prime=/summationdisplay gkhi(gkhi)ug αuhβukγuiδ, where ( gkhi) is the Riemann 4-index symbol (3). In this analytical context, Christoffel interpreted the expressions (5) algebraically, considering them as the Transformationsrelationen13of a quadrilinear form G4having for coefficients the symbols ( gkhi); i.e., as the conditions required of the coefficients for the transformability of thismultilinear form into another, G /prime 4. It was this very view of expressions (5) that led Christoffel to his wholly algebraic treatment of the Aequivalen- zproblem . That view, indeed, corresponded to his switching his attention 13This term was taken up by Christoffel in the context of the theory of algebraic forms and, specifically, from Aronhold’s work: [1863, p. 283]. This notion is historicallysignificant, in that it constitutes an “algorithmic” definition of the tensor as given byChristoffel: see [Reich 1994, pp. 59–60]. 226 L. DELL’AGLIO to multilinear forms with the construction, starting from G4,o fap a r t i c - ular sequence of multilinear differential forms G5,G6,...:w es h a l ls e ei n the following section how this construction was effected. More specifically, Christoffel’s solution of the Aequivalenzproblem was closely related to the consideration of a well-defined property of the sequence G4,G5,G6,...and of the sequence G/prime 4,G/prime 5,G/prime 6,..., obtained in like manner with G/prime 4as starting point; i.e., this was linked to the real- isation of the fact that equation Gµ=G/prime µentails the consequent one, Gµ+1=G/prime µ+1. Consequently, the Reduktionssatz asserts that the equiva- lence between the quadratic differential forms FandF/primedepends only on the algebraic compatibility of the system of equations: F=F/prime,G4=G/prime 4, G5=G/prime 5,....14 Thus, the differential forms F,G4,G5,G6,...may be considered as purely algebraic forms of their differentials and the Aequivalenzproblem may thus be reduced to a problem of equivalence of algebraic forms; and,ultimately — as a result of the theory of algebraic invariants — this wasreduced by Christoffel to the coincidence betweenthe sets of simultaneousinvariants of the forms F,G 4,G5,G6,...and those of forms F/prime,G/prime 4,G/prime 5, G/prime 6,....15 To sum up, it may be said that Christoffel’s investigation of the Rie- mannian Aequivalenzproblem exhibited a programme of “reduction” of the theory of differential forms to the theory of algebraic forms. From an operational point of view, this approachwas the opposite of that adopted by Lipschitz, which was characterised by its essential reliance on varia-tional methods. In analogous fashion, following from this, the search fordifferential invariants brings out the fact that different approaches were used by the two German mathematicians. Indeed, Christoffel’s method also implicitly entailed an algebraic reduction of the search for differen-tial invariants: the common algebraic invariants of the forms F,G 4,G5, G6,...may also be considered as differential invariants,even though this resultwasnotexplicitly stated bythe Germanmathematicianfor the gen- eral case. Such a viewpointis similar to that ofCasorati,but is technically more powerful owing to the use of stronger algebraic methods. 14See [Christoffel 1869 a, p. 369]. 15For a modern version of this theorem on algebraic invariants, see [Weitzenb¨ ock 1923, pp. 199–203]. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 227 The“algorithmic”genesisofcovariantdifferentiation There are a number of reasons accounting for the central significance, asregardstheemergenceoftensoranalysis,ofthemethodologicalcontrastwe have just oulined, between algebraic and variational methods. As weshall see, indeed, the same contrast — on a different level — was toreappear in Ricci-Curbastro’s work and characterises the genesis of thattheory. There is, however, one specific aspect in Christoffel’s solution of the Aequivalenzproblem that is directly connected to the rise of absolute dif- ferential calculus. As was already mentioned, in effect, the German math-ematician’s algebraic treatment of the Aequivalenzproblem contained the first occurrence of what was later to be referred to as covariant differen- tiation by Ricci-Curbastro. 16It is necessary, however, to understand how it is possible to make this claim. The introduction by Christoffel of the algorithms of “covariant differ- entiation” occurred in the middle of his 1869 paper, as the method toconstruct the sequence of differential multilinear forms G 4,G5,G6,.... It is interesting to observe how, bringing out a clear methodological anal-ogy,Christoffel’sprocessmadeuseofthe“eliminationtheory”inafashionsimilar to that which had characterised Casorati’s geometrical work. Given a µ-linear form G µwith coefficients ( i1i2...i µ), thex/prime α-differen- tiation of their Transformationsrelationen (6) ( α1α2...α µ)/prime=/summationdisplay i1i2...iµ(i1i2...i µ)ui1 α1···uiµ αµ leads to the equations (7)∂(α1α2...α µ)/prime ∂x/primeα =/summationdisplay ii1i2...iµ∂(i1i2...i µ) ∂x/prime iui αui1 α1···uiµ αµ +/summationdisplay λi2...iµ∂2xλ ∂x/primeα∂x/primeα 1(λi2···iµ)ui2 α2···uiµ αµ +/summationdisplay i1λ...iµ∂2xλ ∂x/primeα∂x/primeα 2(i1λ...i µ)ui1 α1···uiµ αµ+···. 16The algorithms of covariant differentiation are also present in Lipschitz’s work, but only as a consequence of their use by Christoffel: see [Lipschitz 1871, p. 17]. 228 L. DELL’AGLIO LikeCasorati,Christoffel used the expressions(4) to replace the second derivatives in equations (7), thus obtaining the following (8) ( αα1...α µ)/prime=/summationdisplay ii1...iµ(ii1...i µ)ui αui1 α1···uiµ αµ, where (9) ( ii1...i µ)=∂(i1i2...i µ) ∂x/prime i−/summationdisplay λ/bracketleftbig Γλ ii1(λi2...i µ) +Γλ ii2(i1λ...i µ)+···/bracketrightbig . As usual, the expressions (8) were considered by Christoffel as the Transformationsrelationen ofa (µ+1)-linearform Gµ+1whosecoefficients are given by the quantities (9). Thus, the previously-considered process yields the definition, taking Gµas starting point, of a multilinear differ- ential form Gµ+1,f o rw h i c ht h e Transformationsrelationen (8) still hold; i.e., in such a way that the equation Gµ=G/prime µentails as its consequent Gµ+1=G/prime µ+1, as required. One may, therefore, claim that the introduction of the algorithms (9) of covariantdifferentiation constituted the corner-stone of the entire alge-braic project followed by Christoffel. Evidence for such a role is that, inthe German mathematician’s work, the expressions (9) were never con-sidered from an analytical point of view but only from an algebraic one;actually,theyweremerelyalgorithmsfortheiterativeproductionofdiffer- ential forms. This may be underscored by pointing out that, in Christof- fel’s work, the expressions (9) were only special techniques introduced toreduce the original problem to another context, which was presumed tohave a broader heuristic power. One may refer to such a functional deviceas alinking technique . 17 2.2.The researchtraditionofthetheory ofdifferentialparameters As we shall see, the actual origin of the concept of covariant differ- entiation — understood as an extension of ordinary differentiation — 17This notion may be seen to be closely connected with that of “transplantation”, as proposed by E. Koppelman [1975, p. 459]. More specifically, one may view a linking technique as a well-defined technique in a field of research which — when interpreted in another field of research — allows the application of the latter to the former. Thereason for introducing this notion stems from the requirement to make a clear-cutdifference between a technical and a conceptual stage in the history of the idea ofcovariant differentiation. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 229 may be viewed as the result of a particular interpretation given by Ricci- Curbastro of Christoffel’s algorithms (9). Analytical in nature, this inter- pretation arose in connection with a context of research — the theory ofdifferential parameters — different from that in which the expressions (9)first emerged — the theory of differential forms. This fact tends to givethe theory of differential parameters great significance for the history of absolute differential calculus. Theresearchprogrammeof theearlytheoryofdifferentialparameters Like the investigations of algebraic and differential invariants, the the- ory of differential parameters presented an embryonic stage of develop-ment characterised by the study of the invariant properties of certainwell-defined quantities. On the other hand, unlike the investigations intothe other types of invariants,which mostly arose in geometrical contexts, the study of differential parameters came from a typically analytical field, i.e., fromclassicalFrenchmathematical physics.It wasindeed while work-ing on elasticity theory in 1834 that Lam´ e demonstrated the invariance of the following expressions ∆ 1f=/parenleftBig∂f ∂x/parenrightBig2 +/parenleftBig∂f ∂y/parenrightBig2 +/parenleftBig∂f ∂z/parenrightBig2 , (10) ∆2f=∂2f ∂x2+∂2f ∂y2+∂2f ∂z2, (11) which he respectively called “ param` etres diff´ erentiels du premier et du second ordre ”[ L a m ´e 1834, p. 215]. After initially considering rectangular Cartesian coordinates, he generalised these results to the case of orthog- onal curvilinear coordinates two decades later [Lam´ e 1859]. Lam´e’s attention to differential parameters was neither due to chance nor did it lack significance. Actually, it was connected with a generalresearch programme in the field of partial differential equations. As is well known [Kline 1972, 28.6], this programme was based on the theory of systems of triply orthogonal families of surfaces [Reich 1973, VI.5], sinceit consisted in the search for a particular system of curvilinear coordi-nates, to reduce differential equations to a form resolvable by means ofa separation of variables. This involved consideration of how one mightexpress the differential equations of a given problem in a general form, i.e. with respect to a general system of curvilinear coordinates. This technical 230 L. DELL’AGLIO programme,in particular,was already presentin Lam´ e’spaper of 1834,in which the differential equations for the propagation of light in the etherwere expressed with respect to the system of curvilinear coordinates ρ,ρ 1, ρ2, with the only condition that ρbe the parameter “ des surfaces d’´ egale densit´ ed el ’ ´ ether”. More specifically, using some properties of orthogonal surfacesstudied in the secondpartofhis paperandessentiallybasinghim-self on consideration of the differential parameters of the first and secondorder, Lam´ e put these equations in the following form (12)  P 1=hh2∂ϕ ∂ρ2,P2=−hh1∂ϕ ∂ρ1, h2∂ ∂ρ/bracketleftBig h2 1∂ ∂ρ1(ρ3ϕ)/bracketrightBig =h2 1∂F ∂ρ1, h2∂ ∂ρ/bracketleftBig h2 2∂ ∂ρ2(ρ3ϕ)/bracketrightBig =h2 2∂F ∂ρ2, ρh1h2 h/bracketleftBig∂ ∂ρ1/parenleftBighh1 h2∂ϕ ∂ρ1/parenrightBig +∂ ∂ρ2/parenleftBighh2 h1∂ϕ ∂ρ2/parenrightBig/bracketrightBig +ρ∂ ∂ρ/parenleftBigF ρ3/parenrightBig =1 A∂2ϕ ∂t2, where h,h1,h2are the differential parameters of the first order of ρ,ρ1, ρ2with respect to the rectilinear variables x,y,z;fandFare functions ofρ,ρ1,ρ2,t;a n d Ais a numerical coefficient. In this fashion, in the final part of the paper, Lam´ e was able to solve the aforegoing system of equations for the particular case of a single spherical and homogeneousparticle acting “ sur l’´ether environnant ”. In this technical programme,which wasalso the fundamental matter of Lam´e’smain work— his Le¸cons sur les coordonn´ ees curvilignes [1859]—, differential parameters played an essential role; indeed, they represented the invariantquantities through which it waspossible to express the givenequations in an invariant form: “Cette constance de forme et de valeur explique, en quelque sorte, com- ment il se fait que presque toutes les ´ equations aux diff´ erences partielles, qui concentrent les lois des ph´ enom` enes physiques, peuvent s’exprimer ` a l’aide de certaines fonctions-de-point et de leurs param` etres diff´ erentiels du second ordre, sans qu’il soit n´ ecessaire de sp´ ecifier le syst` eme de coor- donn´ees que l’on adopte ”[ L a m ´e 1859, p. 24]. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 231 As a result, differential parameters — along with curvilinear coordi- nates — were to be extensively taken up after Lam´ e’s research work.18It is worthy of note that this reception was marked by the deep involvementof the Italian mathematical community, from the middle of the 19th cen-tury on. This development— which may be explained in terms of the riseof differential geometry, already manifest in pre-Unification Italy 19—i s evidenced first and foremost by the work of Beltrami. Beltramiandthe“mixed” approachtothetheoryofdifferentialparameters The study of differential parameters was indeed one of the main topics of Beltrami’s scientific production. It constituted, in fact, the principalcommon technical element linking his earlier research work in geometry and his later work on mathematical physics. 20 On the one hand, in effect, Beltrami extended the quantities defined by Lam´e to the case of surfaces and manifolds in his “Ricerche di analisi applicataallageometria”[1864–1865].21It isworthnotingthat this exten- sion was concomitant with a direct generalisation of Casorati’s notion of funzione inalterabile ; in particular, this was tantamount to a first general definition of differential parameters as quantities analytically dependingon a set of arbitrary functions: “But the idea of these functions, which may be termed absolute like the geometric properties which they represent, is susceptible of a usefulextension.With the functions E,F,G, let us consider the other functions ϕ,ψ,...ofu,v, and suppose that the same change of variables that transforms expression (34)into(34 /prime)22also transforms ϕ,ψ,... i n t o ϕ/prime, ψ/prime, ... . An expression formed with E,F,G,ϕ,ψ,...and their partial 18Among the principal studies concerning differential parameters around 1850–1860, one may mention: [Brioschi 1854], [Chelini 1853], [Codazzi 1868], [Jacobi 1847], [Neu-mann 1860, 1867], [Somov 1865]. 19“In my opinion Liouvillewas among the mathematicians who bestunderstoodthe ideaofintrinsicgeometryaround1850,onlyrivaledperhapsbyafewItalianssuchasBrioschiandChelini ”[ L ¨utzen 1989, p. 86]. 20For this central aspect of Beltrami’s work, see: [Loria 1901], [Pascal 1901, pp. 71, 73, 77], [Struik 1981, p. 600], [Tazzioli 1993, pp. 3–4]. 21Differential parameters are also discussed in [Beltrami 1867 a,b]. 22E,F,G andE/prime,F/prime,G/primeare the coefficients of the differential quadratic form in the two systems u, vandu/prime,v/primeand equations (34) and (34/prime) are the expressions of that form with respect to the two systems of coordinates. 232 L. DELL’AGLIO derivatives with respect to u,vwill be termed invariable if ...it changes into another expression analogously formed with E/prime,F/prime,G/prime,ϕ/prime,ψ/prime,... and their partial derivatives with respect to u/prime,v/prime.”23 On the other hand, Beltrami extensively employed differential param- eters in his later work in mathematical physics, concerning the geomet-rical nature of physical space. In particular, he used such parameters tosearch for a general expression of the equations of classical mathematical physics — and, above all, in potential theory — in non-Euclidean con- texts: a thrust of research which would lead to extensive developments inthe German and Italian contexts of researchshortly thereafter. 24In addi- tion to a clear Riemannian influence, these studies of Beltrami showed a renewal of Lam´ e’s scientific orientation, now taken up in more gen- eral geometrical contexts.25In this manner, emphasis was again placed on the major significance of the mathematical tool of curvilinear coordi-nates (oblique, as opposed to the orthogonal ones considered by Lam´ e) in analytical research: “Now as regards the general usefulness of oblique curvilinear coordi- nates in physico-mathematical matters ..., it is not out of place to note that, considering the frequency of the cases in which the very nature ofthe problem suggests a priori a certain system of surfaces as an essentialinstrument for resolution ..., it is a natural assumption that discussion may often be made easier by use of formulae not bound to the hypothesis of threefold orthogonality. ” 26 23“Ma il concettodi questefunzioni, chesipossono chiamare assolutecomelepro- priet`ageometrichecherappresentano,` esuscettibilediun’utileestensione.Consideri- amo,oltrele E,F,G,altrefunzioni ϕ,ψ,...diu,v,esupponiamochequellostesso cambiamento di variabili il quale trasforma l’espressione (34)nella (34/prime)trasformi parimente le ϕ,ψ,...nelle ϕ/prime,ψ/prime,....Un’espressioneformatacolle E,F,G,ϕ,ψ, ...ecolleloro derivateparziali rispetto alle u,vsi dir`ainvariabilequando ...essa sitrasformer`ainun’espressioneformataanalogamentecolle E/prime,F/prime,G/prime,ϕ/prime,ψ/prime,...e colleloroderivateparzialirispettoalle u/prime,v/prime” [Beltrami 1864, p. 142]. 24[Lipschitz 1870], [Schering 1870], [Lipschitz 1872], [Schering 1873], [Tonelli 1882], [Killing 1885]. On this matter, see [Tazzioli 1993, pp. 5–7]. 25Unlike the French physical mathematician, however, Beltrami would appear to have followed more typically physical aims, in particular in connection with Maxwell’selectromagnetic theory. 26“Quantopoi all’utilit` adi massima che pu` oaverel’uso delle coordinatecurvilinee obliquenellequestionidifisicamatematica ...non`efuordiluogoilnotarechestante lafrequenzadeicasiincuilanaturastessadelproblemasuggerisce ap r i o r iuncerto ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 233 Besides playing a major part in much of his scientific work, the topic of differential parameters was also developed by Beltrami in the guise of a purely analytical study. His paper, “Sulla teorica generale dei parametridifferenziali” [Beltrami 1868 b], may, indeed, be considered as the first autonomous analysis of such quantities. In particular, in this context, heproved the Riemannian invariance of Lam´ e’s quantities (10), (11), now expressed as follows ∆ 1U=/summationdisplay rsArs∂U ∂xr∂U ∂xs, (13) ∆2U=1 √ a/summationdisplay r∂ ∂xr/parenleftBig√ a/summationdisplay sArs∂U ∂xs/parenrightBig , (14) where ais the determinant of the fundamental differential form and theArsare the coefficients of its reciprocal form. Moreover, he also demonstrated the invariance of the following expression — the so-called parametro differenziale misto (15) ∆ 1(UV)=/summationdisplay rsArs∂U ∂xr∂V ∂xs· This last work of Beltrami is of the greatest importance for a historical reconstruction of the genesis of tensor analysis. Indeed, it clearly showedthe state of the theory of differential parametersa short time before Ricci- Curbastro’s work on the matter. This remark is of particular significance from a methodological point of view. Like the studies of Lipschitz on the special theory of differential invari- ants, Beltrami’s research methodology was characterised by its “mixed”nature, as it was based on both algebraic and variational methods. Morespecifically, the invariance of the differential parameters (13), (15) wasestablished only on the basis of certain properties of algebraic forms, forwhich the essentials of the theory were extensively discussed in the firstpart of Beltrami’s paper. Conversely, in his treatment of the expressions (14), Beltrami made use of the variation of an integral, with explicit ref- erence to a paper by Jacobi on potential theory [Jacobi 1847]. The actualform of the differential parameters (14) derived from the application of sistemadisuperficiecomestrumentoessenzialedisoluzione ...`enaturalepresumere chelatrattazionepossaesserenondiradoagevolatadall’usodiformolenonvincolateall’ipotesidellatripliceortogonalit` a” [Beltrami 1884, p. 138]. 234 L. DELL’AGLIO the calculus of variations to the subject topic of differential invariants. Indeed, by extending the procedure employed by Jacobi, Beltrami arrived at these expressions by applying variational methods to the equation (16)(n)/integraldisplay ∆1U·√ a·dx1···dxn=(n)/integraldisplay ∆1U·√ b·dy1···dyn, where bis the determinant of the fundamental form with regard to the variables yi.27Thus, using the results of the calculus of variations forn-integrals yields (17)(n)/integraldisplay δU/summationdisplay r∂ ∂xr/parenleftbig Ur√ a/parenrightbig dx1···dxn =(n)/integraldisplay δU/summationdisplay r∂ ∂yr/parenleftbig U/prime r√ b/parenrightbig dy1···dyn, where (18) Ur=1 2∂ ∂∂U ∂xr(∆1U)=/summationdisplay sArs∂U ∂xs; and by way of (17), Beltrami was able to demonstrate the invariance of (14). It is worth noting, however,that, notwithstanding the analogy of their methodological duality, Lipschitz’s and Beltrami’s opinions on the respec-tive significance of the algebraicand variationalmethods did not coincide.Unlike that of the German mathematician, indeed, Beltrami’s treatment of differentialparametersdid not explicitly indicate a definitly greatersig- nificance for algebraic methods over and above variational ones. In fact,the latter were often designated by Beltrami as more direct methods of research,thus evincing a different stance from that of the German mathe- matician. Thus,in the introductionofhis essayon differential parameters,he wrote: “I hope that the simplicity of the method considered, which essentially does not differ from that of Jacobi ..., may be conducive to considering it as the most natural and direct way to achieve the aim ”. 28 27In particular, b=ap2,w h e r e pis the Jacobian of the functions xi=xi(y1,...,y n). 28“Spero che la semplicit` a del metodo usato, il quale nei suoi principali lineamenti ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 235 To sum up, it may be said that, around 1870, the theory of differential parameters exhibited a “mixed” character with algebraic methods coex-isting with variational ones. Moreover, this “mixed” character appears more marked than what was happening at the same time in the special theory of differential invariants, as a result of the impact, in this latter context of research, of Christoffel’s work. 3. THERISEOF TENSORANALYSISINTHECONTEXTOF THE WORK OF RICCI-CURBASTRO The work of Ricci-Curbastro on the introduction of absolute differ- ential calculus goes back to the final two decades of the 19th century, including gradual definitions of the main concepts and the first systematicaccounts of the theory. As previously mentioned, the theoretical context from which the basic concepts of absolute differential calculus — and in particular that of covariant differentiation — emerged was provided by the theory of differential invariants. Starting from 1884, indeed, a major part of the Italian mathematician’s research work was directed to thisfield of research, being concerned with the topics of differential quadratic forms [Ricci-Curbastro 1884]and differential parameters [Ricci-Curbastro 1886 a]. This part of Ricci-Curbastro’s work may be seen as taking place in a precise historical and methodological context. On the one hand, it represented the stage of initial maturity of Ricci- Curbastro’s production. His previous contributions, indeed, may all be seen as arising out of his scientific background, which — apart from a year of specialisation in Munich, where he attended Klein’s and vonBrill’s lectures — had unfolded entirely at the Scuola Normale Supe- riore of Pisa. Virtually all this early work of the Italian mathematician nondifferiscedaquellodiJacobi, ...,inducalapersuasionechelaviadaessoaperta `elapi`unaturaleelapi` udirettapergiungerealloscopo ” [Beltrami 1868 b,p .7 5 ] . This attitude of Beltrami was altogether consistent with the extensive use of the cal- culus of variations in his research work on the non-Euclidean expression of the equationsof classical mathematical physics. At first glance, then, one may view the divergencebetween Beltrami and Lipschitz as to the use of the term “direct” as resulting from thefact that this use occurred in connection with different fields of research: on the onehand, differential quadratic forms, with a potential connection with algebraic forms,and, on the other, differential parameters, in connection with the use of the calculusof variations in investigations in the field of mathematical physics. 236 L. DELL’AGLIO was concerned with physico-mathematicalmatters, especially electromag- netism, as a result of the influence exerted on him by E.Betti [Ricci-Curbastro 1877 b,c].29Not surprisingly, in 1880 Ricci-Curbastro started teaching mathematical physics at the University of Padua. It is in thiscontext of research that his interest for differential invariants arose. Inparticular,in his only workto be published between1880and 1883[Ricci-Curbastro 1882], his treatment of galvanic currents intersected with thetopic of differential parameters. On the other hand, from a methodological point of view, Ricci- Curbastro’s papers on differential invariants exhibit a common element,namely, one and the same research programme. More specifically, thesepapers evidence the same systematic reduction of the study of differen-tial invariants to that of algebraic invariants,which, as we have seen, hadcharacterised Casorati’s and, above all, Christoffel’s approach. It is thisfact, essentially, that enabled Ricci-Curbastro to bring a new twist toChristoffel’s algorithms (9). 3.1.Thefirst occurrenceofthe programmeof“algebraicreduction”in the workofRicci-Curbastro There are various points in the course of Ricci-Curbastro’s research work where the continuing impact may be recognised, of Casorati’s andChristoffel’s algebraic approach to the theory of differential invariants.The first instance was the previously-mentioned paper [Ricci-Curbastro1884] on the theory of differential quadratic forms. The first clue is proffered by the theoretical assumptions of the paper, showingclearlythe influenceofCasorati’swork.Indeed, Ricci-Curbastro’sinvestigation features a “vertical” aspect, being based on an initial classi-fication of differential quadratic forms into classi, closely recalling Caso- rati’s analysis of differential invariants into ordini. More specifically, a differential quadratic form ϕ=/summationtext rsarsdxrdxswas termed by Ricci- Curbastro of classe h,i fhbe the smallest positive number for which it is possible to express the form as follows (19) d s2=n+h/summationdisplay r=1dy2 r. 29In Pisa, U. Dini also exerted an influence on Ricci-Curbastro, this being specifically vouched for by the paper [Ricci-Curbastro 1877 a]. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 237 Wholly taken up by the study of forms of classes 0 and 1, the paper included, in its final part, a short analysis of the search for invariants of quadratic differential forms of the first class. And it is in this very context that the tradition may be recognised at work, of the algebraic approachto the theory of differential invariants. Let us see how this occurs. After setting out the usual laws of transformation for the coefficients a rsof a differential quadratic form (of class 1) (20) ars=/summationdisplay pqbpq∂up ∂xr∂uq ∂xs, Ricci-Curbastro considered the quantities ( pq), defined as follows (21) ( pq)=1 √ a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ 2y1 ∂xp∂xq∂2y2 ∂xp∂xq···∂2yn+1 ∂xp∂xq ∂y1 ∂x1∂y2 ∂x1···∂yn+1 ∂x1... ... ... ... ∂y1 ∂xn∂y2 ∂xn···∂yn+1 ∂xn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, where ais the determinant of the a rsand the yrare the variables appear- ing in (19). On this basis, one can see that the quantities ( pq)v a r yi n similar fashion to the coefficients arsof the fundamental form, i.e. (22) ( rs)=/summationdisplay pq(pq)/prime∂up ∂xr∂uq ∂xs· AsinChristoffel’sresearchwork,thisremarkimpliedawhollyalgebraic view of the topic of differential invariants by Ricci-Curbastro. Indeed, thedifferential quadratic form having the quantities ( pq)a sc o e ffi c i e n t s— and which the Italian mathematician called forma derivata di ϕ—s h o w s itselfalgebraicallysimilartothefundamentalone,duetothesimultaneousvalidity of (20) and (22). In analogous fashion to Christoffel, this factled Ricci-Curbastro to consider the two differential forms from a purelyalgebraic standpoint, considering their simultaneous algebraic invariants.However, owing to the special nature of the forma derivata di ϕ,t h e s e algebraic invariants will also be differential invariants. In this way, Ricci- Curbastro arrived at the following result — where Casorati’s notion offunzione inalterabile reappearsas invariante differenziale di ordine m ofϕ, i.e., as an expression depending only on the a rsand their derivatives to them-th order: 238 L. DELL’AGLIO “Forn>2, every differential quadratic form of class 1 with n variables has n differential invariants of the 2nd order, which are obtained through construction, by the methods [already]known, of the set of absolute alge- braic invariants that are common to ϕand its derivative form .”30 It is obvious that Ricci-Curbastro’s method of seeking differential invariants — unlike Lipschitz’s use of indirecte Methoden — is analogous to Christoffel’s solution of the Riemannian Aequivalenzproblem ;i nb o t h instances, indeed, the method of resolution is based on the introductionof differential forms which are covarianti with respect to the fundamental form and, ultimately, on the searchfortheir simultaneousalgebraicinvari- ants. In this respect, it is no mere chance that Ricci-Curbastro shoulddirectly refer to Christoffel’s work of 1869 at this point and, in particular, to the final part where the German mathematician had solved the three- dimensional case of the Aequivalenzproblem by proving the existence of three differential invariants of two differential quadratic forms. Consequently, one may claim that there was a clear influence, of a purely methodological nature, of Christoffel’s work on Ricci-Curbastro’s.In particular,it is importantto laystress onthe analogybetweenthe Ger- manmathematician’salgorithms(9)andRicci-Curbastro’s forma derivata diϕ, whose coefficientsare givenby the expressions(21). Both constructs, in fact, carried out the essential role of making it feasible to establish a connection betweenproblems involvingdifferential invariantsand an alge- braicmode ofresolution.In otherwords,Ricci-Curbastro’s forma derivata diϕhad the same function, i.e. that of a linking technique , as the algo- rithms (9) in Christoffel’s work. 31 3.2. Ricci-Curbastro’s algebraic approach to the theory of differential parameters The second occurrence of the programme of “algebraic reduction” in 30“Ogni forma differenziale quadratica ϕdi 1aclasse ad nvariabili, per n>2, ammette ninvarianti differenziali di 2oordine, i quali si ottengono costruendo coi metodi noti, il sistema di invarianti algebrici assoluti comuni a ϕed alla sua forma derivata ” [Ricci-Curbastro 1884, p. 170]. 31In effect, the expressions (21) — that in modern terms are the second fundamental coefficients of the hypersurface of line element ϕin the ( n+ 1)-dimensional Euclidean space — are implicitly connected to the algorithms (9) by Gauss equations [Ricci-Curbastro 1884, p. 159, (14)]. This however does not represent a real consideration ofsuch algorithms. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 239 Ricci-Curbastro’sworkconcernsthetheoryofdifferentialparameters.The significance of this development appears principally as a reflection on the “mixed”naturewhich,aswehavealreadynoted,characterisedthattheoryat the time. Thus, Ricci-Curbastro’s paper of 1886 on differential param-eters may also be viewed as a breaking point as well as a breakthrough inthe history of this theory. Ricci-Curbastro’s approach to the theory of differential parameters involved essentially two innovations, which mirrored precisely Casorati’sapproach to the special theory of differential invariants. The first new departure was an element of systematicity. This merely implied the possibility of a wider theory, i.e., of an investigation no longerrestrictedtoconsideringonlyspecialquantitiesofanintrinsicnature.ThisorientationbyRicci-CurbastroentailedproducinganexplicitexpressionofBeltrami’s general definition of differential parameters. Moreover, in thiscontext, Casorati’s concept of the “order” of a differential invariantagainappeared, in the guise of the followingdefinition of differential parametersby Ricci-Curbastro: “We shall term differential parameters of the form itself [of the fun- damental form ]all expressions that include the coefficients of ϕ,o n eo r more arbitrary functions and the derivatives of all these quantities, andthat do not change their form when, for the variables x 1,x2,...,x n,n e w ones are substitued, u1,u2,...,u n.Their order [of these parameters ]is given by that of the derivatives of highest order included in them .”32 The second element of innovation concerns the methodological hetero- geneity of the earlier treatments of differential parameters. Like Casorati, Ricci-Curbastro viewed as “indirect” some of the methods initially imple- mented for these treatments: “From this remark there naturally arises the doubt that, even restricting oneself to expressions of the second order, not solely those generally known...be worthy of that name; and the more so since, to date, that very property has been demonstrated in their case by artificial and indirectmethods. 32“Chiameremo parametri differenziali dellaformastessa [dellaformafondamentale ], tutte le espressioni, che contengono i coefficientidi ϕ,u n aop i`u funzioni arbitrarie e le derivate di tutte queste quantit` a, e non cambiano forma quando alle variabili x1,x2,...,x nsenesostituisconodellenuove u1,u2,...,u n.Illor oor dinesidesume daquellodellederivatepi` ualteinessicontenute ” [Ricci-Curbastro 1886 a, p. 180]. 240 L. DELL’AGLIO As a direct method for these investigations a natural candidate is that proposed by Prof.Casorati ..., which consists in eliminating, as between two systems of quantities corresponding to two different systems of vari- ables, the derivatives of the former with respect to the latter, in order to arrive at the equations that express precisely the essential property under consideration .”33 The true object of Ricci-Curbastro’s criticism was the use of the cal- culus of variations in the theory of differential parameters. Although Lip-schitz was not mentioned at this juncture, the aforegoing passage was clearlyreferringto hisqualitativedistinctionbetween directeandindirecte Methoden . Unlike the German mathematician, however, Ricci-Curbastro was not contentwith a theoretical distinction, subject to specific practical requirements. On the contrary, Ricci-Curbastro’s approach called for the actual exclusion of the indirecte Methoden from the theory of differential parameters, which may only be studied in algebraic terms. One way ofaccounting for this stance of the Italian mathematician, again showing Christoffel’s influence, is to claim that it was connected with his general, systematic aims, as in the case of the search for invariants of differential quadratic forms. In any event,one might inquire why Ricci-Curbastro — with his strong background in mathematical physics — set himself against one of themost important technical instruments of this scientific tradition, and at a time when the calculus of variations was undergoing considerable devel- opments. To answer this historical question, one particular aspect of Ricci- Curbastro’sscientificbackgroundmustbe examined. Thisaspectis specif- ically linked to the role played by von Brill, whose course of lectures the Italian mathematician had followed in Munich in 1877, as mentioned 33“Daquestaosservazioneemananaturalmenteildubbioche,anchelimitandosialle espressionidi2oordine,nonsoltantoquellecomunementeconosciute ...meritinotal nome;eci`otantopi`ucheperessilapropriet` amedesima`estatafinoadoradimostrata conmetodiindirettiedartificiosi. Comemetododirettopertaliindaginisioffrenaturalmentequelloseguitodalprof. Casorati ...che consiste nell’eliminare tra due sistemi di quantit` a corrispondenti a due diversi sistemi di variabili, le derivate delle une rispetto alle altre per giungerealleequazioniesprimentiappuntolaricordatapropriet` aessenziale ”[Ibid., p. 177]. The property of the differential parameters which Ricci-Curbastro was referring to here isnaturally their intrinsic nature. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 241 above. Indeed, Ricci-Curbastro’s stance, laying greater store by algebraic methods, maybe linkedtoBrill’sapproachtothe theoryofalgebraicfunc- tions; his approach, indeed, had been characterised by a purely algebraic treatment of the subject, as opposed to the use of transcendental methods that had previously prevailed [Pogrebyssky 1981]. Thus,hereisanotherpossibleinfluence,actingonRicci-Curbastrofrom a purely methodological point of view, originating in the German research tradition on algebraic functions, as propounded by Clebsch, Gordan and thereafter by M.Noether and Brill. It is this further influence, in par- ticular — together with that of Casorati —, which may well have been the main reason for Ricci-Curbastro’s embracing Christoffel’s algebraic approach to the theory of differential invariants.Such an influence, more- over, is hardly surprising from a historiographic point of view: in this respect, we need only recall the impact of that self-same research tradi- tion on another segment of the Italian mathematical community at the time, i.e. the famous school of geometry founded by Segre and Enriques. 3.3.The emergenceofthe conceptofcovariantdifferentiation Ricci-Curbastro’s radical approach to the theory of differential param- eters tended to emphasise the contrast between algebraic and variational methods which, as we have seen, characterised — to varying degrees — many lines of research into differential invariants. Indeed, one may claim thatthe Italianmathematician’soppositiontotheuseofvariationalmeth-ods turned this divergence from a coexistence of methods showing differ- ences in emphasis into an actual methodological conflict. It is precisely this changed context that may be viewed as one of the principal reasonsfor the emergence ofthe idea ofcovariantdifferentiation. One consequence, indeed, was a clear-cut separation of various aspects thathadgoneintotheresearchtraditionsondifferentialinvariants.Specif- ically, this entailed the emergence of a break between the algebraic and analytical aspects of the question, to use these terms in the modern sense. Broadlyspeaking,thealgebraicaspectislinkedtoChristoffel’sapproach to the theory of differential quadratic forms and points to the tensor substratum of the notion of covariant differentiation — i.e., the valid- ity of (8) — with its connection with the theory of multilinear forms. Conversely, the analytical aspect concerns the essential nature of differ- ential parameters viewed as objects of the theory of partial differential 242 L. DELL’AGLIO equations. As we shall see, this had already involvedthe emergence of the requirement for a broader concept of differentiation in the work of Lam´ e and of Beltrami. These different aspects are clearly reflected in the process of genesis of the concept of covariant differentiation in Ricci-Curbastro’s work, whichmay be divided into three basic stages. The first stage — characterisedby the resurgence of Christoffel’s algorithms (9) in the context of the the- ory of differential parameters — was essentially algebraic in nature, on the lines of the German mathematician’s program of “algebraic reduc-tion”. Conversely, the second stage was strictly analytical in character,as it involved interpreting the expressions (9) as differential operators, inaccordance with the researchtradition on differential parameters. Finally, the last stage was characterised by the true autonomy of the concept of covariant differentiation — as evidenced by the total independence ofthis notion from a linguistic point of view. Of course, this separation intostages should not be considered in absolute terms, but only by way of a reconstruction of a specific rational process. The algorithmic emergenceof covariant differentiationin the work of Ricci- Curbastro The particular approach adopted by Ricci-Curbastro for the theory of differential parameters — calling for a systematic treatment of thesubject using a methodologically coherent procedure — was what led tohis reappraisal of Christoffel’s algorithms (9). Indeed, as we have already mentioned, the claim of the centrality of the directe Methoden in the theory of differential parameters coincided with a second instance of theimplementation of Christoffel’s programme of algebraic reduction in theItalian mathematician’s work. Indeed,Ricci-Curbastro’sanalysisofdifferentialparameters[1886 a]was groundedentirelyonalgebraicarguments.The investigationofdifferentialparameters of order 0 and 1 actually hinged, in the first case (in order to prove the non-existence of such parameters), on the absence of absolute algebraic invariants of the fundamental form; and, in the second case, oncertain points made by Beltrami about algebraic forms. 34 34These are the arguments which Beltrami had adduced to show the invariance of differential parameters (13), (15). ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 243 However, the algebraic character of Ricci-Curbastro’s approach was peculiarly well suited for the investigation of the differential parameters of the higher orders.This investigation,indeed, wascast once againto the self-same methodological plan that had characterised Christoffel’s workon differential quadratic forms and which the Italian mathematician hadpreviously implemented to search for differential invariants of forms ofclass 1, as we have seen. In other words, in order to arrive at the differ-ential parameters, Ricci-Curbastro brought in a differential form whosecoefficients change in like manner to those of the fundamental form —i.e., a form covariante a ϕ. Thus, he could apply the results of the the- ory of algebraic invariants. 35This is precisely why Ricci-Curbastro looked again to Christoffel’s algorithms (9): it is interesting to see in more tech-nical terms how this was achieved. First, Ricci-Curbastro considered the laws of transformation for the coefficients a rsof the fundamental form and for their derivatives a(g) rs under a given change of coordinates (apq)=/summationdisplay rsarsx(p) rx(q) s, (23) (a(i) pq)=/summationdisplay rsga(g) rsx(i) gx(p) rx(q) s+/summationdisplay rsars/bracketleftbig x(pi) rx(q) s+x(p) sx(qi) r/bracketrightbig , (24) where the x(p) rhave the same meaning as Christoffel’s up r. After introduc- ing the Christoffel symbols of the first kind ars,i, and using the laws of transformation for them (25) ( ars,i)=/summationdisplay gx(i) g/bracketleftBig/summationdisplay hkahk,gx(r) hx(s) k+/summationdisplay hahgx(rs) h/bracketrightBig , Ricci-Curbastro arrived at the following expressions for the x(rs) h (26) x(rs) h=/summationdisplay pq(cpq)(ars,q)x(p) h−/summationdisplay pqtchpaqt,px(r) qx(s) t, 35As a result of this paper of Ricci-Curbastro’s, the method of studying differential parameters by means of algebraic covariant forms widely characterized the subsequentdevelopment of the theory of differential parameters [Somigliana 1890; Frobenius 1892;Knoblauch 1893, 1895]. On the other hand, while he noted the significance of Ricci-Curbastro’s method, E. Padova — one of his colleagues at the University of Padua —insisted on using the calculus of variations [Padova 1887]. Further, Ricci-Curbastro’smethod was incorporated by Levi-Civita into his general treatment of differential invari-ants [Levi-Civita 1893-1894], which was also based on Lie’s approach to the topic. 244 L. DELL’AGLIO wherethe crsarethe coefficientsofthe form reciprocalto the fundamental one. Thusfar,everythinghadproceededalongthe linesofChristoffel’sargu- ment. Now, however, Ricci-Curbastro, in order to examine differentialparameters, went on to consider an arbitrary function Uand the laws of transformation for its derivatives U (h)andU(hk) /parenleftbig U(r)/parenrightbig =/summationdisplay hU(h)x(r) h, (27) /parenleftbig U(rs)/parenrightbig =/summationdisplay hkU(hk)x(r) hx(s) k+/summationdisplay hU(h)x(rs) h. (28) With respect to U, Ricci-Curbastro also brought in expression (18) as envisaged by Beltrami, and now written as follows (29) Ur=/summationdisplay scrsU(s). UnlikeBeltrami,whohadusedthese quantitiesin avariationalcontext, Ricci-Curbastrointroducedthem asthe basisof his algebraictreatmentofthe search for differential parameters. In particular, by substituting (26),the expressions (28) may be written as follows (30) ( U rs)=/summationdisplay hkUhkx(r) hx(s) k, wherethe quantities Uhkareobtainedfromthe Urin thefollowingmanner (31) Uhk=U(hk)−/summationdisplay iahk,iUi. These last expressions may be seen as the coefficients of a differential quadratic form that is “covariant”with respect to the fundamental form,by virtue of (30); i.e., such that the coefficients of both forms change in like manner. As ever, this view of the matter led Ricci-Curbastro to reduce the search for differential parameters to that for certain algebraicinvariants: “If one constructs the system of absolute algebraic invariants, common to the form ϕand to other forms having for coefficients respectively U rs, Vrs,Wrs,...all formed , like the Urs, with the coefficients of the arbitrary ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 245 functions U,V,W,..., this yields differential parameters of the second order with any number of arbitrary functions. ”36 In the 1886 paper itself, this method of systematic generation of dif- ferential parameters was extended by Ricci-Curbastro to the higher-ordercases.In particular,when outlining the analysisforthe differentialparam-eters of the third order, he also considered the following quantities (32) U hkj=U(hkj)−/summationdisplay pqcpq/bracketleftbig ahk,pUqj+ajk,pUqh+ahj,pUqk/bracketrightbig −/summationdisplay g/bracketleftbig a(j) kh,g−/summationdisplay pqcpqahk,qagj,q/bracketrightbig Ug having the same function as the terms Uhk. In hindsight, these quanti- ties, together with (31), may be seen as the first occurrences in Ricci-Curbastro’s work of what he was to call subsequently covariant differen-tiation. A minor formal difference concerning the application to the “con-travariant” quantities (29) notwithstanding, they were wholly analogousto the expressions (9) of Christoffel. This analogy essentially concerns thefunction devolvingto quantities (31), (32), for Ricci-Curbastro,and quan-tities (9) for Christoffel, in their programmes of “algebraic reduction” ofthe theory of differential invariants. That is to say, the expressions (31),(32) were once again cast in the self-same role, i.e. as linking techniques , as Christoffel’s quantities (9). Thus,theresurrectionofChristoffel’salgorithms(9)byRicci-Curbastro does not appear surprising. In effect, the Italian mathematician’s intro-duction of quantities (31), (32) was no more than a variation, to suit thecase of differential parameters, of the suggestion made by the Germanmathematician for the expressions (9): the basic proof of (30), indeed, isentirely analogous to that of Christoffel for (8), since in both instancesthe expressions were obtained through substitution, respectively in equa-tion (27) — or its equivalent (7). Actually, this method had originated inelimination theory, once more manifesting the close connection betweenRicci-Curbastro’s work and that of Casorati. 36“Sesicostruisceilsistemadiinvariantialgebriciassoluticomuniallaforma ϕeda pi`uformerispettivamentedicoefficienti Urs,Vrs,Wrs,...formatituttianalogamente alleUrscoi coefficienti delle funzioni arbitrarie U,V,W,..., si ottengono dei parametridifferenzialidi2oordineconunnumeroqualsivogliadifunzioniarbitrarie ” [Ricci-Curbastro 1886 a, p. 183]. 246 L. DELL’AGLIO Tosum up: itcan be claimedthat Ricci-Curbastrowasheir to Christof- fel with regardto his “algorithmic” genesis of the concept of covariantdif-ferentiation,since this emergencewascloselyconnected to the programmeof “algebraic reduction” that had characterised the German mathemati-cian’s approach to the topic of differential invariants. 37 Differentialparametersand the emergenceof a broader concept of differen- tiation In spite of their methodological similarities, the lines of research pur- sued by Ricci-Curbastro and Christoffel already diverged quite notablyat this initial stage of the Italian mathematician’s work. In this context, indeed, the quantities (31), (32) were seen by Ricci-Curbastro as more general expressions of the customary second and third derivativesof func-tionU. This fundamental shift in emphasis occurred in singular fashion, in the introduction of Ricci-Curbastro’s paper on differential parameters; it was effected en passant , independently of the technical aims pursued in the paper. While pointing to a possible extension of the results arrived at,Ricci-Curbastro wrote: “Just as ...the second derivatives of Uwere expressed by the U rs,f o r the third derivatives of Uone substitutes, by way of analogous methods and results, the coefficients of a cubic form covariant with respect to ϕ, and, in general, for all the derivatives of Uof the m-th order one is to substitute the coefficients of a form of degree m covariant with respect tothe one under consideration. ” 38 There is an obvious difference between this passage and what had been involved in Christoffel’s work; there, indeed, as we have already pointedout, the expressions (9) were only given an algebraic meaning, as specific 37Oddly enough, Ricci-Curbastro [1886 a] made no direct reference to Christoffel’s work. Nevertheless, acknowledgement of the decisive significance of German mathe-matician’s research work is to be found in many other places in Ricci-Curbastro’sresearch. For example: “ L’algorithmeduCalculdiff´ erentielabsolu ...setrouveentier dansuneremarquedue` aM.Christoffel ” [Ricci-Curbastro, Levi-Civita 1901, p. 127]. 38“Come ...lederivatesecondedi Usonostateespresseperle Urs,cos`ıallederivate terzedi U,sisostituisconoconmetodierisultatianaloghiicoefficientidiunaforma cubicacovariantea ϕ,eingeneraleallederivatedell’ordine m-esimodi Usisostituis- conoicoefficientidiunaformadigradomcovarianteallaproposta ” [Ricci-Curbastro 1886a, p. 179]. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 247 techniques for the progressive generation of differential forms. How is one to account, then, for the conceptual step taken by Ricci- Curbastro? One reason, of course, seems to be predominant in this context. In the research work pursued by Christoffel and Ricci-Curbastro one finds aclear shift in topics, from the Riemannian Aequivalenzproblem to the sys- tematic investigation of differential parameters. From a technical point ofview, this shift entailed a change in the way Christoffel’s algorithms (9)were to be viewed: they now could be considered by Ricci-Curbastro inclose connection with the analyticalcharacterof the arbitrary function U. The aforementioned minor formal difference between Christoffel’s expres-sions(9)andRicci-Curbastro’s(31),(32)representedanobvioustechnical manifestation of this changed situation. However, there is another, and more important, reason why this shift in subject matter, from the investigation of differential quadratic formsto that of differential parameters, was of fundamental significance as faras the emergence of a broader concept of differentiation was concerned. In effect, one can show that this idea had already been at work in the research on the theory of differential parameters. An embryonicmodel of a generalisationof the usual differentiation had actually already been present in Lam´ e’s work. In that context, the gener- alisation had focused essentially on the differential parameters of the sec- ond order, in order to emphasise their symbolic significance for analytical research. Thus, after setting out the principal equations of mathematicalphysics in terms of these parameters, Lam´ e claimed: “En r´esum´e, lorsqu’une classe de ph´ enom` enes physiques d´ epend des variations d’une certaine fonction-de-point, c’est presque uniquement par son param` etre diff´ erentiel du second ordre que cette fonction intervient. Comme si ce param` etre ´etait une d´eriv´ee naturelle , plus essentielle, plus simple, et en mˆ eme temps plus compl` ete, que toutes les d´ eriv´ees partielles, choisies plus ou moins arbitrairement, que l’on a l’habitude de consid´ erer” [Lam´e 1859, p. 25]. Lam´e’s call for something plus essentiel andplus simple than the usual differentiation had clearly stemmed from the intrinsic nature of the differ-ential parameters. On the other hand, the notion of “intrinsic” propertiescharacterised the entire tradition of the theory of differential parameters, 248 L. DELL’AGLIO as indeed of other forms involvedin the investigationof invariance; more- over,ithadbeen closelyconnectedwith thedevelopmentofmathematical, and, in particular, geometricalthought in the 19th century.Thus, it is not surprising that Beltrami — who was working in a Riemannian context — should havetaken up and expanded Lam´ e’sidea of a possible extension of the usual differentiation. Once again, attention was being focused on dif- ferential parameters, although now also including those of the first order. This involved a very abstract view of such quantities, as expounded by Beltrami, which may be recognised as an embryonic kind of the modern differential operators: “The functions ∆1,∆2relative to a certain system of curves drawn on this surface can take an infinity of different values.The system of curves ϕ=c o n s t ., in fact, is not geometrically different from the system f(ϕ)=c o n s t .; but the parameters ∆1and∆2are different according to whether one uses one form rather than the other.Indeed, there is no difficulty in establishing that ∆1f(ϕ)=f/prime(ϕ)∆1ϕ,∆2f(ϕ)=f/prime(ϕ)∆2ϕ+f/prime/prime(ϕ)(∆1ϕ)2, formulae which bear indeed a strong analogy with those which, in the calculus, are used for the differentiation of composite functions .”39 It is worth noting that this passage ends on a direct quotation of the words of Lam´ e we have just seen: “This fact, which is most noteworthy for function ∆2, accounts to some extent for the reason why this function is spontaneously introduced in many studies ‘as if, as M.Lam´ es a y s ...,i tw e r ea natural derivative , that is more essential, more simple, and also more complete than all the partial derivatives that are usually considered and which are more or less arbitrarily chosen’ .”40 39“Lefunzioni ∆1,∆2relativeaduncertosistemadicurvetracciatesullasuperficie possono avere infiniti valori differenti.Infatti il sistema di curve ϕ=c o s t .non dif- ferisce,geometricamente,dalsistema f(ϕ)=c o s t .;maiparametri ∆1e∆2sonodiffe- rentisecondochesiadottal’unaol’altraforma.Edinverositrovafacilmente ∆1f(ϕ)=f/prime(ϕ)∆1ϕ,∆2f(ϕ)=f/prime(ϕ)∆2ϕ+f/prime/prime(ϕ)(∆1ϕ)2, formole che presentano una grande analogia con quelle che nel calcolo differenziale, servonoalladifferenziazionedellefunzionicomposte ” [Beltrami 1864, pp. 151–152]. 40“Questa circostanza, che ` e massimamente notevole per la funzione ∆2,r e n d ei n ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 249 In this context,while takingup this questfor a broaderidea ofdifferen- tiation, Ricci-Curbastro’s research work involved the essential innovation of transferring attention from differentialparameters to Christoffel’s algo-rithms (9). This development was clearly signposted by Ricci-Curbastro’sdirect mention of Lam´ e, while laying the same emphasis on Christoffel’s algorithms that the French mathematician had put on differential param- eters of the second order: “When the line element has the form/summationtext rdx2 r, they coincide with the derivatives of Uof the m-th order and it is possible to consider them, rather than differential parameters perhaps, as Lam´ e said of the latter, as something more essential, more simple, and also more complete than allpartial derivatives .” 41 The recurrence of this quotation clearly shows the emergence of a concept over the evolution of a research tradition, in connection with its specific operational circumstances. Partialdifferentialequationsandthe conceptof covariantdifferentiation The point, however,that must decisively corroborate the historical sig- nificance of the work of Lam´ e and Beltrami for the emergence of tensor analysis has to do with the reasons put forward by Ricci-Curbastro for consideringtheexpressions(31),(32)asderivativesofamoregeneralkind.These reasons, indeed, showed a direct connection with Lam´ e’s research programme into the theory of partial differential equations. To wit, while switching attention to the quantities (31), (32), Ricci-Curbastro put for- ward the same requirements as for the theory of differential parameters.Once again, indeed, emphasis was placed on the usefulness of choosinga particular system of curvilinear coordinates in the study of differen-tial equations: certomodoragionedelperch´ equestafunzionesiintroducaspontaneamenteinungran numerodiricerche‘comeseessa,diceilsig.Lam´ e...,fosseuna derivata naturale ,pi`u essenziale,pi`usempliceedinparitempopi` ucompletadituttelederivateparzialiche sisoglionoconsiderareechesiscelgonopi` uomenoarbitrariamente’ ”[Ibid., p. 152]. A similar purpose, of extending the usual meaning of differentiation is also presentelsewhere in Beltrami’s work [1867 b]. 41“Nel caso che l’elemento lineare abbia la forma/summationtext rdx2 r, essi coincidono colle derivate di ordine mdiUe ad esse forse meglio che ai parametri differenziali, si addiceilconsiderarle,comedissediquestiilLam´ e,comequalchecosa plus essentielle, plus simple et en mˆ eme temps plus compl` ete que toutes les d´ eriv´ees partielles” [Ricci- Curbastro 1886 a, p. 179] (in French in the original text). 250 L. DELL’AGLIO “It seems to me that this very substitution should often turn out to be useful in analytical studies since, in the change of variables, the coefficientsthemselves explicitly introduce nothing but the first derivatives of the old[variables ]with respect to the new ones and, depending on the form of the line element, they will indicate naturally, and in every case, the coordi-nates that are to be preferred, to give the greatest possible simplicity to theequations of the problem itself .” 42 The fundamental import of these remarks lay in the fact that they effectively switched attention from an algebraic view of the quantities (31), (32) to an analytical one. In particular, this was concomitant with a shift in the significance of the implementation of Christoffel’s algorithms:i.e., a shift from their use as linking techniques within the framework of the theory of differential parameters to their employment as techniquesfor the investigation of partial differential equations. If the initial charac-terisation,as afunctional device,oftheexpressions(31),(32)hadbroughtabout the “algorithmic” emergence of the concept of covariant differenti-ation, it was only their use for the study of partial differential equationsthat truly underpinned the introduction of this concept, as an extensionof the usual differentiation. The close link between the introduction, by Ricci-Curbastro, of the concept of covariant differentiation and Lam´ e’s research programme was further demonstrated in the Italian mathematician’s subsequent work.From this point of view, it is most important to note how the concept ofcovariant differentiation emerged. While the paper of 1886 on differentialparameters had already made clear the essential import of covariant dif-ferentiation, it actually carried only general considerations, rather thanthe actual introduction of this concept, which, however, made its appear-ance in a paper of 1887 [Ricci-Curbastro 1887 a]. It is highly significant, at any rate, that this formal introduction came only after prior employmentof the quantities (31), (32) in the context of the theory of partial differ- 42“Questa medesima sostituzione parmi debba spesso tornare utile nelle ricerche analitiche, perch` e i coefficienti stessi non introducono nei cambiamenti di variabili, esplicitamente, se non le derivate prime delle antiche rispetto alle nuove e, dipen-dendodallaformadell’elementolineare,indicherannoinogniproblemanaturalmentelecoordinatedapreferireperdarealleequazionidelproblemastessolamaggiorepos-sibilesemplicit` a”[Ibid]. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 251 ential equations, along the lines of the research tradition on differential parameters. This first form of the analytical use of covariant differentiation arose in the treatment of a particular extension of Lam´ e’s research, namely the extension of the problem of systems of triply orthogonal families of surfaces — as developed by Darboux in his early research work [Dar- boux 1878] — to the case of nvariables. Ricci-Curbastro discussed this matter in generalised form in a paper [Ricci-Curbastro 1886 b] that imme- diately followed the one on differential parameters.43From an analytical point of view, the problem addressed by the Italian mathematician cor- responded to the search for the existence conditions of ( n−1) integrals ρ1,ρ2,...,ρ n−1of the equation (33)/summationdisplay rYr∂ϕ ∂xr=0, such as to be mutually orthogonal on a manifold with metric ds2=/summationtext rsarsdxrdxs.44Ricci-Curbastro reduced the resolution of this problem to the study of a particular system of partial differential equa- tions. In the course of this investigation, at one point and under well- defined conditions, Ricci-Curbastro introduced the following quantities,which are totally analogous to (29), (31) and (32) ρ r=/summationdisplay scrs∂ρ ∂xs, (34) ρrs=∂2ρ ∂xr∂xs−/summationdisplay iars,iρi, (35) ρrsq=∂3ρ ∂xr∂xs∂xq−/summationdisplay hkchk/bracketleftbig arq,kρhs+asq,kρhr+asr,kρhq/bracketrightbig (36) −/summationdisplay k/bracketleftBig∂ ∂xqars,k−/summationdisplay hichiakq,hars,i/bracketrightBig ρk, obtaining the equations (37)  /summationdisplay rsqρrsqρqHrhHsk=2/summationdisplay pqrscpqρprρqsHrhHsk, /summationdisplay rsqρrsqHqiHrhHsk=0. 43A technical analysis of this work — from a modern point of view — is to be found in [Tonolo 1961]. 44I.e., such that/summationtext rscrs(∂ρh/∂xr)(∂ρk/∂xs)=0 ,w i t h h/negationslash=k. 252 L. DELL’AGLIO In this process, the expressions (34), (35), (36) played the same func- tional role as the differential parameters in the work of Lam´ ea n dB e l - trami, i.e. that of a method to produce the invariant expression of a given equation; the difference, however,lay in the fact that the quantities ρr,ρrs,ρrsq,asagainstdifferentialparameters,weretruegeneralisationsof the usual derivatives. Thus, the equations (37) recast the initial problem in a generalised and invariant form and may indeed be considered as the first “tensor” formulation of a specific analytical problem. Moreover,they were very similar to the general — and, specifically, the non-Euclidean — expressions of the laws of mathematical physics that were being con- sidered in the research work on differential parameters. Compared to the latter, however, the equations (37) were now involved in a programme which was more strictly mathematical in character. In Ricci-Curbastro’scontext of research, indeed, the “tensor” formulation of the problem of systems of orthogonal families of surfaces played a fundamental technical role essentially in that it enabled him to obtain more information about the resolution of the given equations. The equations (37) indeed were of remarkable heuristic value for Ricci-Curbastro since they showed: “how the degree of difficulty as to the existence of the orthogonal sys- tems in a manifold does not depend solely on the number of dimensions, but also on the nature of the manifold.Given the number n, the difficulty is smallest when the manifold is flat or Euclidean .” 45 The same purposes were to be expressed by Ricci-Curbastro more clearly in an extended version of his paper on systems of orthogonal sur- faces, published the following year: “I believe that the problem as stated and the results arrived at have their own intrinsic value as manifesting a new aspect of the theory of linear and homogeneous partial differential equations of the first order and as constituting, in highly general cases, a notable reduction of the problem of their integration .”46 45“come il grado di difficolt` a per la esistenza dei sistemi ortogonali in una variet` a nondipendasoltantodalnumerodelledimensioni,maanchedallanaturadellavariet` a stessa. Fermo il numero n, la difficolt`a`e minima se la variet` a`e piana od euclidea ” [Ricci-Curbastro 1886 b, p. 197]. 46“Parmi che il problema enunciato e i risultati ottenuti abbiano un interesse loro proprio come quelli, che mettono in evidenza un nuovo aspetto della teoria delleequazioni lineari ed omogenee a derivate parziali di 1 oordine e costituiscono in casi ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 253 Itisnotsurprising,therefore,thatsimilarremarkswerealsotobefound in the note of 1887 which included the formal introduction of covariantdifferentiation.Once again,the purposes ofthat introductionreflectedtheoperational goals concerning the expression of an analytical problem —againin conformitywith theresearchtraditionofthe theoryofdifferential parameters: “The usefulness of this substitution is manifest, in particular in the investigations that are essentially independent of the nature of the man-ifold or of the choice of coordinates in any given manifold.Thus, for instance, those expressions ...necessarily yield a simpler and clearer form for all expressions endowed with the characteristic property of differentialparameters; and they have enabled me to put the equations, which theparameter of a family of (n−1)-dimensional loci in any n-dimensional manifold must satisfy ..., in a form as simple as that given by Darboux for the case of a flat or Euclidean manifold with orthogonal Cartesiancoordinates .” 47 Thestageof autonomyofthe conceptof covariantdifferentiation In this overall context of continuity, the formal introduction of the notion of covariant differentiation was essentially conceptual in nature.Indeed, this introduction occurred in the briefest of forms, in an accountsolely concerned with the new kind of differentiation. Moreover,the intro- duction was presented in a highly schematic and technical fashion, being basedonacompletequotationfromthecentralpartofChristoffel’s Reduk- tionssatz . More specifically, after introducing the usual quantities (38)U r1r2...rprp+1=∂Ur1r2...rp ∂xrp+1−/summationdisplay qscqs/summationdisplay harhrp+1,sUr1r2...rh−1qrh+1...rp, molto generali una riduzione notevole del problema della loro integrazione ” [Ricci- Curbastro 1887 b, p. 205]. 47“La utilit`a della sostituzione stessa in ispecie nelle ricerche, che sono per loro essenza indipendentidalla natura della variet` ao dalla sceltadelle coordinatein una variet`adata,`eevidente.Cos` ı,peresempio, questeespressioni ...danno necessaria- menteformapi` usempliceeperspicuaatutteleespressioni,chegodonodellapropriet` a caratteristicadeiparametridifferenziali,emihannopermessodidarealleequazioni,cuidevesoddisfareilparametrodiunafamigliadiluoghiad (n−1)dimensioniinuna variet`aqualsivogliaad ndimensioni ...,unaformatantosemplicequantoquelladata dal Darboux nel caso in cuila variet` aproposta sia piana od euclidea e le coordinate sianocartesianeortogonali ” [Ricci-Curbastro 1887 a, p. 199]. 254 L. DELL’AGLIO Ricci-Curbastro reported Christoffel’s statement in the following terms: “If the expressions Ur1r2...rpare the coefficients with pindices of a form which is covariant with respect to ϕ2,t h eUr1r2...rprp+1given by (4) [i.e.(38)] are the coefficients with (p+1)indices of a form which is also covariant with respect to ϕ2.”48 The explicit definition of covariant differentiation was no more than a consequenceofthis statement,fromwhich,ofcourse,theterm “covariant”derived: “By means of the theorem demonstrated above, we can thus construct successively expressions with 2,3,...,p indices, such that those with p indices are the coefficients of forms covariant with respect to ϕ 2and include the derivatives of Uup to the order p.Further it will be easily seen that these [expressions ]will all be linear relative to the derivatives themselves and that each one will include just one derivative of order p. We shall call them covariant derivatives of order pin the manifold which is intrinsically defined by the expression ϕ2of the square of its line ele- ment.”49 In spite of the algebraic character of its name, the new concept was highly analytical in nature, and was to be viewed as a differential oper-ator. This is clearly vouched for by the fact that, in connection with Beltrami’s work concerning differential parameters, a large part of Ricci- Curbastro’spresentation[1887 a]isdevotedtoaparticularfunctionalprop- erty of covariant differentiation, namely its partial commutativity. Further, the essentially analytical nature of covariant differentiation was attested by a significant peculiarity of Ricci-Curbastro’s aforegoing formal definition. This definition, indeed, as in the case of the usual scalar differentiation,onlyconcernsthe single-functioncase.Thatisthemeaning 48“Seleespressioni Ur1r2...rpsonocoefficientia pindicidiunaformacovariantea ϕ2,leUr1r2...rprp+1datedalle (4) [(38)]sonocoefficientia p+1indicidiunaforma purecovariantea ϕ2”[Ibid., p. 201]. 49“Medianteilteoremasopradimostratopossiamodunquecostruiresuccessivamente delleespressionicon 2,3,...,pindici,perguisachequelleconpindicisianocoeffici- entidiforme covariantia ϕ2econtenganolederivatedi Ufinoall’ordine p.Sive de dipi`ufacilmentecheessesarannotuttelinearirispettoallederivatestesse,echecon- tengonociascunaunasoladerivatadiordine p.Noilechiameremo derivate covarianti di ordine pnellavariet`a,che`edefinitains` edallaespressione ϕ2delquadratodelsuo elementolineare ”[Ibid., p. 202]. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 255 of the expression derivazione di ordine pused by the Italian mathemati- cian. In other words, in this context, covariantdifferentiation was consid- ered as a differentiation iteratively defined on function U.O fc o u r s e ,t h i s also entailed an operation of differentiation on the quantities Ur1r2...rp— even though this consideration had not as yet been explicitly stated by Ricci-Curbastro.Inotherwords,inthisinitialcontext,hehadnointention of introducing a differentiation on “tensor fields”, i.e., on specific systems of functions. The simplest way of accounting for this fact is to claim that, notwith- standing the extensive use of the algebraic substrate provided by the the- ory of multilinear forms, an autonomous concept of the “tensor” had not yet emerged. In effect, Ricci-Curbastro’s sole aim here was to define an analytical tool generalising customary differentiation and, in so doing, he disregarded the nature of what that tool was intended to be applied to. Hence, one may claim that, in Ricci-Curbastro’s work, the introduc- tion ofthe notionofcovariantdifferentiationconceptually anticipatedand implied that of the “tensor field”. Consequently,acompletetechnicalformulationoftheconceptofcovari- ant differentiation had to await the formalisation of the notion of tensor, i.e., until 1888. From then on, in every systematic treatment of his meth- ods, Ricci-Curbastro would always term as covariantdifferentiation a dif- ferentiation defined on a particular system of functions [Ricci-Curbastro 1889, 1892; Ricci-Curbastro, Levi-Civita 1901]. For instance, in his 1888 essay “Delle derivazioni covarianti e controvarianti e del loro uso nella analisi applicata”, Ricci-Curbastro introduced covariant (and contravari- ant) differentiation after calling the expressions Ur1r2...rmsistema m-plo covariante : “The operation by which, in accordance with (8) [i.e., (38) ], one goes over from the m-ply system Ur1r2...rmto the(m+1)-ply system Ur1r2...rm+1 is what I call differentiation covariant with respect to the differential form ϕ2of the latter system from the former one .”50 Thus, the objects of covariant differentiation were the systems of func- tionsUr1r2...rm, the single-function case merely being a particular case. 50“Laoperazione,percuisecondole (8) [(38)]sipassadalsistema mploUr1r2...rmal sistema (m+1)ploUr1r2...rm+1`equellachechiamoderivazionecovarianteallaforma differenziale ϕ2delsecondosistemadalprimo ” [Ricci-Curbastro 1888, p. 251]. 256 L. DELL’AGLIO Hence, after also introducing the contravariant case, Ricci-Curbastro claimed: “I may thus state that, by means of the repeated application of covariant or contravariant differentiation, from one m-ply covariant or contravari- ant primitive system, others of the same nature can be obtained in indef- inite numbers, i.e. one (m+1)-ply system, one (m+2)-ply system, etc. A single function Umay be regarded as the most elementary [instance]of both the covariant and the contravariant systems .”51 Froma conceptual point of view, this may be seen as the starting point of the thrust towards a logical reconstruction of the theory, in which the concept of tensor would assume the dominant role. 4. CONCLUSION :WHY WASTENSORANALYSISBORN INITALY? The aforegoing considerations on the genesis of the concept of covari- ant differentiation also tell us something about the historical reason why tensor analysis was born in Italy, and indeed why it should have sprung from the work of Ricci-Curbastro. As a matter of fact, the situation ofItalian differential geometry in the second half of the 19th century, as wellas Ricci-Curbastro’s position in this context appear highly specific. Let us spell this out, and adduce some reasons. The crucial element of our subject is the relationship between Ricci- Curbastro’s work and the mathematical idea of invariance: a relationship which appears more developed than in other fields of research in the post-Riemannian period. In particular, there was a difference betweenRicci-Curbastro and the other mathematicians who continued Riemann’s work on differential geometry. The works of these latter, indeed, show the presence of certain aspects of the mathematical idea of invariance — andof their respective research programmes — in the 19th century, but neverall of these aspects actually together. Indeed, Christoffel’s work on differential quadratic forms developed on lines that were essentially concerned with two types of occurrence of the 51“Potr`ocos`idirechemediantelaapplicazioneripetutadelladerivazionecovariante o controvariante da un sistema m.plo primitivo covariante o controvariante, se nepossono ottenerealtridella stessanaturainnumeroindefinito ecio` euno (m+1 ) plo uno(m+2)ploecc.Unafunzioneunica Upu`origuardarsicomeilpi` uelementaretanto traisistemicovariantiquantotraicontrovarianti ”[Ibid., p. 252]. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 257 idea of mathematical invariance. On the basis of the Reduktionssatz ,h i s procedure,in fact, consistedin the globalapplication ofthe theoryofalge- braic invariants to the theory of Riemannian differential invariants,as wehave seen. On the other hand, Lipschitz’s work on differential invariantsand that of Beltrami on differential parameters showed no such exclu-sive reliance on algebraic methods: their analyses, indeed, were essen-tially grounded on the calculus of variations,an instrument that remainedalien to Christoffel’s methodology. This was no mere chance event. Theabsence of the calculus of variationsfrom Christoffel’s work on differential quadratic forms may be seen as a specific reflection of the way he stood apart from the tradition of research on differential parameters, which, onthe contrary, characterised Beltrami’s (and, to some extent, Lipschitz’s)work. Partly linked to this fact, the latter’s work on differential invari-ants manifested an element of an analytical nature that was not to befound in Christoffel’s work. That element was the considerable signifi-cance attributed to certain differential quantities (the differential param-eters), owing to the central role they played in much analytical research, an importance which allowedBeltrami — as, before him, Lam´ e—toview them as expressions generalising the operation of customary differentia-tion. In this context the role played by the calculus of variations — whichh a dn op a r ti nL a m ´ e’s work — was essentially that of amplifying the ana- lytical import of the topic of differential invariants in Beltrami’s researchwork. Thus, the position occupied by Ricci-Curbastro is highly distinctive. His work,in effect, representedthe intersectionpointofvariousinfluences,showing the simultaneous presence of elements associated with partly divergent research traditions. Actually, the investigation that led Ricci-Curbastro to bring in the algorithm of covariant differentiation, as we have seen, was characterisedby the global application of the methods of the theory of algebraic invari-ants to the theories of differential invariants and differential parameters.In this process, the natural premise and starting point was represented byChristoffel’s work, consolidated by the algebraic nature of Casorati’s geo-metrical research. This character of Ricci-Curbastro’s work represented a true break with the earlier methods implemented in the investigation of differential invariants and parameters, essentially based as they were 258 L. DELL’AGLIO on the calculus of variations. On the other hand, his analytical inter- pretation of the algorithm of covariant differentiation as a more generalform of differentiation occurred as a result of the reappraisal of certaintopics emanating from the research tradition on differential parameters.Thus, the original algorithm of covariant differentiation, introduced asalinking technique to reduce the investigation of an analytical problem (the equivalence of differential quadratic forms, the systematic search fordifferential parameters) to an algebraic perspective, was reconsidered byRicci-Curbastro from an analytical point of view, on an independent levelof interpretation. Thus, compared with Christoffel, Ricci-Curbastro had recourse to a methodologically similar reliance on the theory of algebraic invariants foranalytical matters, while generalising the German mathematician’s inter- pretative viewpoint. At the same time, Ricci-Curbastro eschewed Bel- trami’s methodological approach to differential parameters, while takingup the conceptual import of the Italian mathematician’s research work. One mayclaim, therefore,that Ricci-Curbastro’sworkinvolveda pecu- liar“mix”ofdivergentresearchtraditions.Thus,thegenesisoftheconceptof covariant differentiation appears as the point of convergence of differ-ent contexts of developmentof the idea of invariance:algebraic invariants,differential invariants and parameters. And this is a fact that truly char-acterises the emergence of tensor analysis as one of the most importantsyntheses of the idea of invariance to have occurred in the second half ofthe 19th century. 52 On the other hand, one should also note that the conflict between research traditions, manifest in such an explicit form in Ricci-Curbastro’swork, equally characterised the development of Italian differential geom- etry at large. It is essentially this element that may be of help in under- standing the historical reason why tensor analysis should haveemerged inthat mathematical community. One may claim, indeed, that this conflictwas also inherent in the virtual opposition between Casorati’s algebraicapproach to geometrical research and that of Beltrami, which was of a 52One should note that, in addition, Klein also exerted an influence on Ricci- Curbastro, who had attended his courses at the University of Munich in 1877. Althoughit is not possible to consider this as a truly direct influence [Levi-Civita 1925, p. 393], itis clear that Ricci-Curbastro inherited from the German mathematician the awarenessof the centrality of invariance in geometrical research. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 259 physico-mathematical nature. Despite its virtual character, rooted as it was in a kind of operational coexistence,53this contrast in fact exhibited an intrinsic potentiality:the possibilityof entertaining simultaneouslydif- ferent points of view concerning the use of the idea of invariance in con- texts of research concerned with the investigation of differential matters. BIBLIOGRAPHY ARONHOLD (S.) [1863] Ueber eine fundamentale Begr¨ undung der Invariantentheorie, Journal f¨ur diereineundangewandteMathematik , 62 (1863), pp. 281–345. BELL(E.T.) [1945]Thedevelopmentofmathematics , New York: McGraw-Hill, 1945. BELTRAMI (E.) [Opere ]Operematematiche , 4 vols., Milano: Ulrico Hoepli, 1902–1920. [1864-1865] Ricerche di analisi applicata alla geometria, Giornaledimatematiche ,2 (1864), pp. 267–282, 297–306, 331–339, 355–375, and 3 (1865), pp. 15–22,33–41, 82–91, 228–240, 311–314; Opere 1, pp. 107–198. [1867 a] Intorno ad una trasformazione di variabili, Ibid., 5 (1867), pp. 24–27; Opere 1, pp. 306–309. 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[1884] Principi di una teoria delle forme differenziali quadratiche, Ann.mat.pura appli., (II) 12 (1884), pp. 135–167; Opere I, pp. 138–171. [1886a] Sui parametri e gli invarianti delle forme quadratiche differenziali, Ibid.,( I I ) 14 (1886), pp. 1–11; Opere I, pp. 177–188. [1886b] Sui sistemi di integrali indipendenti di una equazione lineare ed omogenea a derivate parziali del 1oordine,Attidell’AccademiadeiLincei.Rendiconti , (IV) 2 (1886), pp. 119–122, 190–194; Opere I, pp. 189–198. [1887a] Sulla derivazione covariante ad una forma quadratica differenziale, Ibid., (IV) 3 (1887), pp. 15–18; Opere I, pp. 199–203. [1887b] Sui sistemi di integrali indipendenti di una equazione lineare ed omogenea a derivate parziali del 1oordine,Ann. mat. pura appli. , (II) 15 (1887), pp. 127–159; Opere I, pp. 204–238. [1888] Delle derivazioni covarianti e controvarianti e del loro uso nella analisi appli- cata, inStudieditidall’Universit` adiPadovaacommemorarel’ottavocen- tenario della origine della Universit` ad iB o l o g n a , vol. III, Padova, 1888, pp. 3–23;Opere I, pp. 245–267. [1889] Sopra certi sistemi di funzioni, Atti Accad. Lincei Rc. , (IV) 5 (1889), pp. 112–118; Opere I, p. 268–275. ON THE GENESIS OF THE CONCEPT OF COVARIANT DIFFERENTIATION 263 [1892] R´ esum´e de quelques travaux sur les syst` emes variables de fonctions associ´ es `a une forme diff´ erentielle quadratique, Bulletindessciencesmath´ ematiques , (II) 16 (1892), pp. 167–189; Opere I, pp. 288-310. [1893] Di alcune applicazioni del calcolo differenziale assoluto alla teoria delle forme differenziali quadratiche binarie e dei sistemi a due variabili, AttiIst.veneto sci., (VII) 4 (1893), pp. 1336–1364; Opere , I, pp. 311–335. RICCI-CURBASTRO (G.), LEVI-CIVITA (T.) 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