Geometry and analysis in Euler’s integral calculus

Euler developed a program which aimed to transform analysis into an autonomous discipline and reorganize the whole of mathematics around it. The implementation of this program presented many difficulties, and the result was not entirely satisfactory. Many of these difficulties concerned the integral calculus. In this paper, we deal with some topics relevant to understand Euler’s conception of analysis and how he developed and implemented his program. In particular, we examine Euler’s contribution to the construction of differential equations and his notion of indefinite integrals and general integrals. We also deal with two remarkable difficulties of Euler’s program. The first concerns singular integrals, which were considered as paradoxical by Euler since they seemed to violate the generality of certain results. The second regards the explicitly use of the geometric representation and meaning of definite integrals, which was gone against his program. We clarify the nature of these difficulties and show that Euler never thought that they undermined his conception of mathematics and that a different foundation was necessary for analysis.

1. By this term, we refer to both the theory of integration of functions and that of the integration of differential equations.

2. On Euler’s analytical program, see Ferraro (2010).

3. In his (1727b, 408) Euler stated that Johan Bernoulli was “the most renowned of masters ... who not only was my teacher, greatly fostering my inquiries into such matters, but also looked after me as a patron.”

4. On the role of the construction of equations in the seventeenth-century mathematics, see Bos (1984, 1993b).

5. See Bos (1993a, 35).

6. See, for example, Bernoulli (1964c).

7. An example is the construction of the elastica with equation \(\mathrm{d}y=\frac{x^{2}\hbox {d}x}{\sqrt{a^{4}-x^{4}}}\) in Bernoulli (1694a). In this paper, after having constructed the algebraic curve \(\frac{x^{2}\hbox {d}x}{\sqrt{a^{4}-x^{4}}}\), Bernoulli assumed that one could determine a rectangle ay equal to \(\int _{0}^{x}\frac{ax^{2}}{\sqrt{a^{4}-x^{4}}}\hbox {d}x\). In this way, given the value of the abscissa x, one could find the point (x, y) on the elastica (Bos 1993b, 31–32).

8. An example is the construction of the paracentricisochrone of equation \(\frac{d(ay)}{2\sqrt{ay}}=\frac{a^{2}\hbox {d}x}{\sqrt{a^{4}-x^{4}}}\) in Bernoulli (1694b). In that case, Bernoulli found \(\sqrt{ay}=\int _{0}^{x}\frac{a^{2}\hbox {d}x}{\sqrt{a^{4}-x^{4}}}\) and interpreted the integral \(\int _{0}^{x}\frac{a^{2}\hbox {d}x}{\sqrt{a^{4}-x^{4}}}\) as the arc length of the elastica. By assuming the rectification of elastica as known, one could determine the paracentric isochrones point by point (Bos 1993b, 32–33).

9. We note that, given the meaning of this term in Euler [see below and Ferraro (2000)], the function Z could always be constructed by quadrature, rectification, and ruler and compass.

10. A tractrix is a curve such that the tangent segments between the curve and the axis have constant length.

11. On the use of tractorial motion in the constructions, see Bos (1988), Tournès (2003).

12. For a discussion of this construction, see Tournès (2009, 97–108).

13. In Cartesian coordinates, the equation of the curve is \(u=\frac{nb}{n+1}\hbox {log}t\) and \(t\hbox {d}u=\frac{nb\mathrm{d}t}{n+1}\), namely BN is a logarithmic curve and its constant subtangent is \(\frac{nb}{n+1}\).

14. On Riccati’s book, see Tournès (2009).

15. Dieser Grundtheil der Mathematic wird die Analytic oder Algebra genennet.

    6. In der Analytic werden also blos allein Zahlen betrachtet, wodurch die Größen angezeiget werden, ohne sich um die besondere Art der Größen zu bekümmern, als welches in den übrigen Theilen der Mathematic geschiehet (Euler 1770, \(\S \S \)5–6).

16. See Tournès (2009).

17. “Of course, if we take x as the abscissa and denote the corresponding orthogonal ordinate by P, the formula \(\int P\hbox {d}x\) expresses the area of the curve above the abscissa x, and, on putting \(x=a\), the area is considered to be equal to the value \(y=\int P\hbox {d}x\)—just as we defined it [namely an integral of the type \(\int _{a}^{b} P(x,u)\hbox {d}x\), where u is regarded as a constant]. Therefore, this integral can—as usually said—be assigned by the quadratures of curves, by which this way of integrating is conveniently called construction by quadratures” (Scilicet dum sumta x pro abscissa, si P denotet applicatam orthogonalem ei convenientem, formula \(\int P\hbox {d}x\) exprimet aream eiusdem curvae abscissae x insistentem, ac posito \(x =a\), area habetur determinata valori \(y=\int P\hbox {d}x\) , prouti eum modo definivimus, aequalis, quae ergo, uti loqui solent, per quadraturas curvarum assignari potest, ex quo haec integrandi ratio commode appellatur constructio per quadraturas) (1768–1770, 2: \(\S \)1016).

18. Quanquam autem formula \(\int P\hbox {d}x\) spectata quantitate u ut constante actu integrari nequit, tamen eius integrale in hoc negotio pro cognito accipi potest, quia eius valor saltem per approximationes assignari potest (Euler 1768–1770, 2: \(\S \) 1016).

19. On this question, see also Fraser (1989), Panza (1996).

20. As seen in Ferraro (2000, 117–188), in the strict sense of the term, a function was precisely what one could to exhibited as a solution to a problem and, therefore, a known object.

21. See Ferraro (1998, 2000, 2001).

22. These new instruments were later named special functions; Euler sometimes named them as functions, more often, he merely termed them as quantities.

23. Euler spoke of a special calculus (peculiaris calculus) concerning them.

24. For instance, by mean of a table of values.

25. Euler wrote \(y=\int e^{KQ} P\hbox {d}x\) and specified that the integral vanishes for \(x=b\) and gives the value of y for \(x=a\).

26. Other instances are in Euler (1768–1770, 2: \({\S }\)1026, \({\S }\)1027, and \({\S }\)1033), where Euler used the transformations \(y=\int \limits _{0}^c {e^{ux}x^{n}(c-x)^{\nu } \hbox {d}x}\), \(y=\int \limits _{0}^c {x^{n}\sqrt{\frac{u^{2}+x^{2}}{c^{2}-x^{2}}}\hbox {d}x}\) and \(y=\int \limits _{0}^c {x^{n-1}(u^{2}+x^{2})^{\mu } (c^{2}-x^{2})^{\nu } \hbox {d}x}\), and showed that they solved the equations \(\frac{u\mathrm{d}^{2}y}{\mathrm{d}x^{2}}+(n+\nu +2-au)\frac{\mathrm{d}y}{\mathrm{d}u}-(n+1)ay=0\), \(u(c^{2}+u^{2})d^{2}y-(n+1)(c^{2}+u^{2})\mathrm{d}u\mathrm{d}y+(n+1)uy\mathrm{d}u^{2}=0\), and \(u(c^{2}+u^{2})d^{2}y-(n+2\mu -1)(c^{2}+u^{2})\mathrm{d}u\mathrm{d}y-2(\mu +\nu )u^{2}\mathrm{d}u\mathrm{d}y +2\mu (n+2\mu +2\nu )uy\mathrm{d}u^{2}=0\).

27. In Institutionum calculi integralis, Euler made some reference to the relation between a definite integral and the area under the curve, but in those cases he assumes that this was a fact known (see, for instance, quotation in footnote n. 17).

28. Euler had intended this work, which was published posthumously in 1862 with the title Institutionum calculi differentialis sectio III, to form the third part of his treatise on differential calculus. It is probable that this third part should have played a role similar to the second part of the Introductio in analysis infinitorum, where Euler applied the analytical notions investigated in the first part to geometry.

29. “quantitas, cujus differentiale = ydx exhibebit aream inter lineam curvam et coordinatas x et y contentam” (Euler 1862, \({\S }\)13).

30. “Integral calculus is the method for finding, from a given relation of differentials, the relation of the quantities themselves; The operation, which is employed to this end, is usually called integration (...) Therefore, since differential calculus teaches how to investigate the relation of differentials from the given relation of variable quantities, integral calculus is the inverse method (...) Let X be any function of x, the function, the differential of which is equal Xdx, is called its integral” (Calculus integralis est methodus ex data differentialium relatione inveniendi relationem ipsarum quantitatum, et operatio, qua hoc praestatur, integration vocari solet... Cum igitur calculus differentialis ex data relatione quantitatum variabilium relationem differentialium investigare doceat, calculus integralis methodum inversam suppeditat ...X sit functio quaecunque ipsius x, illa functio, cuius differentiale est = Xdx , huius vocatur integrale) (Euler 1768–1770, 1: \({\S }{\S }\)1, 2, and 7). The term ‘anti-differential’ is not found in Euler’s papers, we use it for brevity.

31. Quemadmodum in calculo differentiali cuiusvis quantitatis differentiale investigatur, ita vicissim calculi species constituitur quoque in inventione eius quantitatis, cuius differentiale proponitur, qui calculus integralis vocatur. Si enim propositum fuerit differentiale quodcunque, eius respectu ea quantitas, cuius est differentiale, vocari solet integrale (Euler 1755, \({\S }\)139).

32. In this paper, for short, we will use the modern expressions “definite integral” and “indefinite integral,” which are not found in Euler. We also observe that initially Euler expressed the definite integral by means of circumlocutions: “if, after integration, a determinate value is given to a variable quantity” [see, for instance, Euler (1743, 1766)] or “integration extended from the value \(x=a\) to the value \(x=b\)” [see, e.g., Euler (1771)]. Only later, he introduced the symbol \(\int f(x)\hbox {d}x\left[ {{\begin{array}{l} \hbox {ab}\quad x=a \\ \hbox {ad}\quad x=b \\ \end{array}}} \right] \) similar to the modern notation \(\int \limits _{a}^{b} {f(x)\hbox {d}x}\) [see, e.g., Euler (1777, 1785, 1789)]. We will employ the modern symbol.

33. Omnes functiones per calculum integralem inventae sunt indeterminatae ac requirunt determinationem ex natura quaestionis, cuius solutionem suppeditant (Euler 1768–1770, 1: \({\S }\)31).

34. Integrale completum exhiberi dicitur, quando functio quaesita omni extensione cum constante arbitraria repraesentatur. Quando autem ista constans iam certo modo est determinata, integrale vocari solet particulare (Euler 1768–1770, 1: \({\S }\)36). In his (1768–1770, 1: \({\S }\)540) Euler gave the name particular integral to a relation between the variables that it satisfied the given equation and that did not contain any new constant quantity.

35. Si functiones, quae in calculo integrali ex relatione differentialium quaeruntur, algebraice exhiberi nequeant, tum eae vocantur transcendentes, quandoquidem earum ratio vires Analyseos communis transcendit.... Ita si formula differentialis Xdx integrationem non admittit, eius integrale, quod ita indicari solet \(\int X \hbox {d}x\), est functio transcendens ipsius x (Euler 1768–1770, 1: \({\S }{\S }\)24–25).

36. Cum calculus integralis ex inversione calculi differentialis oriatur, perinde ac reliquae methodi inversae ad notitiam novi generis quantitatum nos perducit. Ita si a tirone primorum elementorum nihil praeter notitiam numerorum integrorum positivorum postulemus, apprehensa additione, statim atque ad operationem inversam, subtractionem scilicet, ducitur, notionem numerorum negativorum assequetur. Deinde multiplicatione tradita, cum ad divisionem progreditur, ibi notionem fractionum accipiet. Porro postquam evectionem ad potestates didicerit, si per operationem inversam extractionem radicum suscipiat, quoties negotium non succedit, ideam numerorum irrationalium adipiscetur haecque cognitio per totam Analysin communem sufficiens censetur. Simili ergo modo calculus integralis, quatenus integratio non succcedit, novum nobis genus quantitatum transeendentium aperit. Non enim, uti omnium differentialia exhiberi possunt, ita vicissim omnium differentialium integralia exhibere licet (Euler 1768–1770, 1: \({\S }\)29).

37. See Ferraro (2000).

38. In Sect. 5, we will see that sometimes Euler explicitly used geometric principles.

39. (1) Neque vero, statim ac primi conatus in integratione expedienda fuerint initi, functiones quaesitae pro transcendentibus sunt habendae; fieri enim saepe solet, ut integrale etiam algebraicum nonnisi per operationes artificiosas obtineri queat. (2) Deinde quando functio quaesita fuerit transcendens, sollicite videndum est, num forte ad species illas simplicissimas logarithmorum vel angulorum revocari possit, quo casu solutio algebraicae esset aequiparanda. (3) Quod si minus successerit, formam tamen simplicissimam functionum transcendentium, ad quam quaesitam reducere liceat, indagari conveniet. (4) Ad usum autem longe commodissimum est, ut valores functionum transcendentium vero proxime exhibeantur, quem in finem insignis pars calculi integralis in investigationem serierum infinitarum impenditur, quae valores earum functionum contineant (Euler 1768–1770, 1: \({\S }\)30).

40. Point (4) suggests the use of series for calculating the approximate values of the integral.

41. Euler did not ask for a proof that a certain integral could not be expressed in elementary form or in terms of some other transcendental. It is difficult to state whether he did not think of this impossibility proof as something one ought to find or he just did not know how to do it.

42. In Euler (1768–1770, 1: \({\S }\)213), integrating by parts Euler obtained:

    $$\begin{aligned} \int \frac{x^{m-1}}{\mathrm{log}^{n}x}\hbox {d}x= & {} -\frac{x^{m}}{(n-1)\mathrm{log}^{n-1}x} -\frac{mx^{m}}{(n-1)(n-2)\mathrm{log}^{n-2}x}-\frac{m^{2}x^{m}}{(n-1)(n-2)(n-3)\mathrm{log}^{n-3}x}\\&-\cdots -\frac{m^{n-2}}{(n-1)(n-2)\ldots 1}\frac{x^{m}}{\mathrm{log}\,x} +\frac{m^{n-1}}{(n-1)(n-2)\ldots 1}\int \frac{x^{m-1}}{\mathrm{log}\,x}\hbox {d}x. \end{aligned}$$

43. Hae ergo integrationes pendent a formula \(\int \frac{x^{m-1}}{lx}\hbox {d}x\), quae posito \(x^{m}=z\ldots \) reducitur ad hanc simplicissimam formam \(\int \frac{\mathrm{d}z}{lz}\); cuius integrale si assignari posset, amplissimum usum in Analysi esset allaturum, verum nullis adhuc artificiis neque per logarithmos neque angulos exhiberi potuit; quomodo autem per seriem exprimi possit, infra ostendemus (\({\S }\)228). Videtur ergo haec formula \(\int \frac{\mathrm{d}z}{lz}\) singularem speciem functionum transcendentium suppeditare, quae utique accuratiorem evolutionem meretur. Eadem autem quantitas transcendens in integrationibus formularum exponentialium frequenter occurrit, quas in hoc capite tractare instituimus, propterea quod cum logarithmicis tam arcte cohaerent, ut alterum genus facile in alterum converti possit: veluti ipsa formula modo considerata \(\frac{\mathrm{d}z}{lz}\) posito \(lz=x\), ut sit \(z=e^{x}\) et \(\mathrm{d}z=e^{x}\mathrm{d}x\), transformatur in hanc exponentialem \(e^{x}\frac{\mathrm{d}x}{x}\), cuius ergo integratio aeque est abscondita (Euler 1768–1770, 1: \({\S }\)219).

44. This has been Leibniz’s definition of the integral (summatrix). Starting from this definition, he had deduced the relationship between integration and differentiation. As early as the 1690s, Johann Bernoulli [see Engelsman (1984, 46)] had preferred to define integration as anti-differentiation and, during the eighteenth century, this definition was the prevailing one [see, for instance, Hermann (1726), d’Alembert (1765), Lagrange (1797)].

45. Neque etiam calculus differentialis in quantitate differentialum, quae nulla est, indaganda occupatur, sed in eorum ratione mutual definienda, quae ratio utique certam obtinet quantitatem (Euler 1765, \({\S }\)9).

46. Euler did not distinguish between these notions.

47. On this question, see Ferraro and Panza (2012).

48. See Ferraro (2004).

49. This notion of continuous quantity was substantially a primitive notion in Euler’s calculus.

50. The expression “singular integrals” is not found in Euler’s writings.

51. Problème II. Sur l’axe AB trouver la courbe AMB telle, qu’ayant de son point quelconque M la tangente TMV, elle coupe en sorte les deux droites AE et BF, tirées perpendiculairement sur l’axe AB, en deux points donnés A et B, que le rectangle formé par les lignes AT et BV soit partout de la même grandeur (Euler 1756, \({\S }\)18).

52. Pour l’exemple que je viens d’alléguer ici, comme il est formé à fantaisie, on pourroit aussi douter, si ce cas se rencontre jamais dans la solution d’un probleme réel (Euler 1756, \({\S }\)37).

53. on pourroit croire que c’est quelque bizarrerie dans les problemes mecaniques, qui n’auroit plus lie sans les problemes de Geometrie; ou que ce ne seroit pas un reproche, qu’on pourroit faire directement à l’Analyses même (Euler 1756, \({\S }\)36).

54. Already in the 1730s, in his first studies on gamma function, Euler had used the beta integral and had showed that the calculation of \(\int \limits _{0}^{1}{(-\log x)^{n}\hbox {d}x}\) could be reduced to the calculation of the beta integral, namely \(\int \limits _{0}^{1}{(-\log x)^{n}\hbox {d}x} =\frac{(f+g)(f+2g)\ldots (f+(n+1)g)}{g^{\mathrm{n+1}}}\int \limits _{0}^{1} {x^{\frac{f}{g}}} (1-x)^{n}\hbox {d}x\) (in a more modern notation \(\Gamma (n+1)=B\left( {\frac{f}{g}+1,n+1} \right) \frac{(f+g)(f+2g)...(f+(n+1)g)}{g^{\mathrm{n+1}}}\), where \(\Gamma (x)\) and B(x, y) are the gamma and beta functions) (Euler 1730–1731, 13–14). Among other articles on the beta integral, see, e.g., Euler (1739a, b).

55. See above Sect. 3.

56. Euler applied the formula for the change of the order of integration that is found in his (1775b).

57. Euler used the symbol \(\int P\hbox {d}x\left[ {{\begin{array}{l} \hbox {ab}\quad x=a \\ \hbox {ad}\quad x=b\\ \end{array}}}\right] \). In Euler (1785, \({\S }\)1), he explained that \(\int P\hbox {d}x\left[ {{\begin{array}{l} \hbox {ab}\quad x=a \\ \hbox {ad}\quad x=b\\ \end{array}}}\right] \)denoted the integral \(\int P\hbox {d}x\) so that it vanished posing \(x=a\) and one stated \(x=b\) (Hac signandi ratione \(\int P\hbox {d}x \left[ {{\begin{array}{l} \hbox {ab} \quad x=a\\ \hbox {ad}\quad x=b\\ \end{array}}}\right] \) declaratur, integrale \(\int P\hbox {d}x\) ita esse affirmatum, ut evanescat posito \(x=a\), tum vero statui \(x=b\)).

58. In Euler’s original words: Lemma 1. \(\int P\hbox {d}x\left[ {{\begin{array}{l} {\hbox {ab}\quad x=a} \\ {\hbox {ad}\quad x=b} \\ \end{array}}}\right] =-\int P\hbox {d}x\left[ {{\begin{array}{l} {\hbox {ab}\quad x=b} \\ {\hbox {ad}\quad x=a} \\ \end{array}} } \right] \)

    Quoniam enim, sit b ut maius spectetur quam a, formula posterior \(\int P\hbox {d}x\left[ {{\begin{array}{l} {\hbox {ab}\quad x=b} \\ {\hbox {ad}\quad x=a} \\ \end{array}}}\right] \) eandem aream AaBb refert quam prior, sed ordine retrogrado, ista expressio pro negativa erit habenda, sicque erit quoque

    $$\begin{aligned} \int P\hbox {d}x\left[ {{\begin{array}{l} {\hbox {ab}\quad x=a} \\ {\hbox {ad}\quad x=b} \\ \end{array}}}\right] +\int P\hbox {d}x\left[ {{\begin{array}{l} {\hbox {ab}\quad x=b} \\ {\hbox {ad}\quad x=a} \\ \end{array}}}\right] =0. \end{aligned}$$

59. On Euler’s treatment of discontinuous functions, see Ferraro (2000). A different approach to the question can be found in Spalt (2011).

60. In opposition functions composed by one only analytical formula were termed continuous functions.

61. ... calculus integralis ad functiones duarum variabilium accommodatus plurimum differt a calculo integrali communi, ubi non nisi functiones unius varibilis occurunt, et praecepta omnino singularia postulat, praeterquam quod in eo omnia quoque artificia prioris partis sint in usum vocanda. Verum haud diu est, ex quo haec pars Analyseos coli est coepta, ita ut vix adhunc prima eius elementa satis sint evoluta (Euler 1765, \({\S }\)17).

62. Quis autem in curva quacunque libero manus ductu descripta applicatas abscissis

    $$\begin{aligned} x+a\sqrt{-1y} \hbox { et } x-a\sqrt{-1y} \end{aligned}$$

    respondentes animo saltem imaginari ac summam earum realem assignare valuerit aut differentiam, quae per \(\sqrt{-1}\) divisa etiam erit realis? Hic ergo haud exiguus defectus calculi cernitur, quem nullo adhuc modo supplere licet (Euler 1768–1770, \({\S }\)301).

63. Euler referred to functions of one variable that could be expressed by only one analytical formula (in other terms he referred to the theory of elementary functions, which included their development into power series).

64. See Ferraro (2007).

We thank Jesper Lützen and Jeremy Gray for useful suggestions and improving our English.

Authors and Affiliations

1. Università del Molise, Campobasso, Italy

   Giovanni Capobianco & Giovanni Ferraro

2. Università della Basilicata, Potenza, Italy

   Maria Rosaria Enea

Corresponding author

Correspondence to Giovanni Capobianco.

Communicated by Jesper Lützen.

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