Introducing differential calculus in Spain: The fluxion of the product and the quadrature of curves by Tomàs Cerdà

Abstract

Differential calculus was introduced into eighteenth-century Spain through the teaching of several authors in different scientific institutions. One of the more noteworthy of these Spanish authors was the Jesuit Tomàs Cerdà (1715–1791), who taught mathematics at the College of Cordelles in Barcelona and at the Imperial College in Madrid. This mathematician introduced differential calculus through the manuscript entitled ‘Tratado de Fluxiones’ (1757–1759), which had as a main source The Doctrine and Application of Fluxions (1750) by Thomas Simpson (1710–1761). Our aim in this paper is to analyse Cerdà’s special contribution to the introduction into Spain of the Newtonian theory of fluxions based on Simpson’s definition of a fluxion. Specifically, the paper shows that Cerdà deduced the fluxion of the product of two variables and the area under a curve by previously establishing the fluxion of a curvilinear surface, a particular and different approach to that employed by other contemporaneous mathematicians in Spain.

Acknowledgments

I am especially indebted to M Rosa Massa-Esteve for her encouragement throughout the period of research and for stimulating discussions on the subject of my study. This paper was written with the support of the project HAR2016-75871-R: ‘Matemáticas e Ingeniería: Nuevas perspectivas críticas (siglos XVI-XX)’.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 It is worth mentioning the contributions by Mahoney (1980) and Mancosu (1996).

2 Among the long list of historians devoted to the history of the differential calculus, we may mention some authors who in recent years have contributed to the analysis of different aspects of its development such as Feingold (1993), Grabiner (1997), Schubring (2005), Malet (2008), Serfati (2014), Jullien (2015), Massa-Esteve (2018) and Probst (2018).

3 The main goal in our thesis (Berenguer 2016) is to analyse Cerdà’s role in the introduction of differential and integral calculus into Spain, based on the study of his treatise on fluxions. Our research has been included in the projects HAR2013-44643-R and HAR2016-75871-R: ‘Matemáticas e Ingeniería: Nuevas perspectivas críticas (siglos XVI–XX)’ guided by Antoni Roca-Rosell and Maria Rosa Massa-Esteve, Centre de Recerca per a la Història de la Tècnica, Universitat Politècnica de Catalunya.

4 See Berenguer (2015).

5 See Clarke (1929) for more information about Simpson and his life. See Blanco (2013) for Simpson’s contribution to the theory of fluxions.

6 As regards the introduction of fluxional calculus into Spain, in addition to Cerdà’s treatise, we must at the very least take into account Padilla’s work (1753–1756) and the manuscript attributed to Rieger (1761–1765). Padilla was indeed the first to publish a treatise in which the notion of fluxion appears but, unlike in Newtonian fluxions, in Padilla’s treatise the fluxion is not related to the velocity but is rather an infinitely small increment, like the Leibnizian difference. Rieger also adopted the same concept of fluxion as Simpson and Cerdà, although in a much shorter text (Berenguer 2016, annex 9). For this reason it can be said that Cerdà was definitely the main introducer into Spain of the Newtonian fluxion, related to velocity.

7 In the introduction to his book Doctrine of Fluxions (1736), James Hodgson (1672–1755) states that a basic point for further calculation in the method of fluxions is to find the fluxion of the rectangle, that is, the product of two variables: ‘As the main or fundamental Point in the Method of Fluxions upon which all the future Operations depend, is to find the Fluxion of the Rectangle, or Product of two indeterminate or flowing Quantities … ’ (Hodgson 1736, V–VI).

8 Bos (1974, §2.22) explains about this principle of homogeneity: ‘(…) there is a second kind of homogeneity, which requires that all the terms of an equation should be of the same order of infinity. A quantity which is infinitely small in relation to another quantity can be disregarded in comparison with that quantity’.

9 According to Gassiot (1996), who states that the copy of the baptismal registry booklet was provided to him by Xavier Peralta Huguet.

10 Data provided by Josep M Benítez i Riera, extracted from the annual catalogues of the Society of Jesus.

11 Pézenas was the translator of several scientific treatises from English into French, one of which is The Elements of the Method of Fluxions (1742) of Colin Maclaurin (1692–1756). See Boistel (2005).

12 Cerdà published three treatises: Liciones de Mathemática o Elementos Generales de Arithmética y Algebra para el uso de la clase (1758), Lecciones de mathematica o Elementos generales de Geometria para el uso de la clase (1760) and Lección de Artilleria para el uso de la clase (1764).

13 All these treatises are kept in manuscript form at the Real Academia de Historia (Royal Academy of History) in Madrid (RAH): ‘Tratado de Fluxiones’, ‘Tratado de Astronomia’, ‘Elementos Generales de Mechánica’, ‘Secciones cónicas’, ‘Óptica’, ‘Hidrostática’, ‘Hidráulica’, ‘Neumática’, ‘Navegación’.

14 In some lists of books belonging to Cerdà there appear the names of some Central European authors such as Ruggiero Giuseppe Boscovich (1711–1787), Johannes Kepler (1571–1630), Gaspar Schott (1608–1666), Christoph Sturm (1635–1703) and Christian Wolff (1679–1754) and some Italians such as Giovanni Battista Beccaria (1716–1781), Giovanni Cassini (1625–1712), Bonaventura Cavalieri (1598–1647), Galileo Galilei (1564–1642) and Vincenzo Riccati (1707–1775).

15 Cerdà (1758): ‘Tractatum de fluxionibus nunc dispono hic te ducem et Magistrum sequor, et ut ingenue fatear, quamquam alios et Italos et Germanos de Calculo diferentiali et integrali disserentes Authores legam, tamen tota mihi mens est te in perspecuitate et methodo tua emulanda et me felicem existimarem si tractatulum meum de Fluxionibus non meus sed tuus […]’.

16 The Catalan mathematician also held the position of Chief Cosmographer of the Indies, associated with the Chair of Mathematics at the Imperial College of Madrid. He held these positions until 1767 when the Jesuits were expelled from Spain. Cerdà moved to Forlí in Italy and died in 1791.

17 The first historian to discover Cerdà’s manuscripts on fluxions was Cuesta Dutari with his disciple Eulogio Hernández Alonso (1922–1997) in 1973. Some initial studies on Cerdà's contribution to infinitesimal calculus are collected in Ausejo et al. (2010). Finally, Berenguer (2015) published the analysis and transcription of part of Cerdà’s ‘Tratado de Fluxiones’ and submitted his doctoral thesis in 2016. See Berenguer (2016, 154–165).

18 The first version would be a draft of the second. On the other hand, the second version of these 14 chapters would be what Cerdà had ready to print. There are also differences between the two versions. Although the ‘Tratado de Fluxiones’ is a treatise written from the Newtonian perspective, some expressions of the Leibnizian terminology appear in the first version that do not appear in the second one. See Berenguer (2016, 154–187).

19 Newton’s Tractatus de Methodis Serierum et Fluxionum was written in 1671 but was first published in an English translation in 1736.

20 One of the first articles Leibniz published on differential calculus was ‘Nova methodus pro maximis et minimis, itemque tangentibus’ in the journal Acta Eruditorum in (Leibniz 1684).

21 The main mathematicians who influenced the development of differential and integral calculus in Spain in the mid-eighteenth century were Maclaurin and Simpson, on the one hand, and L’Hôpital and Wolff on the other.

22 The original Latin text in Newton’s Tractatus de Methodis Serierum et Fluxionum was: ‘l. Spatij longitudine continuo (sive ad omne tempus) data, celeritatem motus ad tempus propositum invenire. 2. Celeritate motus continuo data longitudinem descripti spatij ad tempus propositum invenire’ (Whiteside 1967–1981, v. 3, 70).

23 The original Latin text in Newton’s Tractatus de Methodis Serierum et Fluxionum was: ‘Quantitates autem quas ut sensim crescentes indefinite considero […], posthac denominaba fluentes, […]. Et celeritates quibus singulre a motu generante fluunt et augentur (quas possim fluxiones vel simpliciter celeritates vocitare)’ (Whiteside 1967–1981, v. 3, 72).

24 Leibniz (1710, 159): ‘Hic dx significat, elementum, id est, incrementum vel decrementum (momentaneum) ipsius quantitatis x (continue) crescentis. Vocatur & differentia, nempe inter duas proximas x, elementaliter (seu inassignabiliter) differentes; dum una sit ex altera (momentanée) crescente vel decrescente.’ A draft of ‘Monitum’ was writen in 1696. The translation of this quoted text into English comes from Bos (1974, 18).

25 Leibniz (1846, 36): ‘Fundamentum calculi: Differentiae et summae sibi reciprocae sunt, hoc est summa differentiarum seriei est seriei terminus, et differentia summarum seriei est ipse seriei terminus, quorum illud ita enuntio: $\int dxaequ.x$; hoc ita: $d\int xaequ.x$.’ The translation of this quoted text into English comes from Child (1920, 142).

26 Many historians have studied the role played by infinitesimals in Leibnizian calculus, namely the role of the laws of homogeneity and continuity. Even mathematicians working under the recently developed non-standard analysis claim their connection with Leibnizian infinitesimal calculus. See Bos (1974), Knobloch (2002, 2013), Bascelli et al. (2014) and Bair et al. (2017).

27 Stone published The Method of Fluxions, both Direct and Inverse (1730) which contains a translation of l’Hôpital’s treatise as well as an original part on the inverse method of fluxions.

28 In his Methodus Incrementorum Directa et Inversa (1715), Taylor develops the calculus of fluxions as a particular case of the calculus of finite differences.

29 Several authors have analysed the role of indivisibles and infinitesimals in the beginning of the differential calculus, such as Malet (1996), Jullien (2015), Malet and Panza (2015) and Massa-Esteve (2015).

30 Colin Maclaurin (1698–1746), in his treatise The Elements of the Method of Fluxions (1742), also avoided infinitesimal increments: see Bruneau (2011).

31 Several historians have analysed the texts of Newton’s followers concerning the concept of fluxion and the rules for deducing the fluxion of algebraic expressions. In this sense, in addition to Cajori’s analysis (1919), it is important to mention Guicciardini (1989), who deals with the diffusion of differential calculus in Great Britain from John Craig (?–1731) to Robert Woodhouse (1773–1827). It is also worth noting Blanco (2013), who makes a comparative study between the two books by Simpson (1737 and 1750) in which the texts of Maclaurin, Stone and Blake are also analyzed in regard to the concept of fluxion and the rules of fluxional calculus.

32 ‘It has been hinted above, that, though the Increments of Quantities are not, strictly, like Fluxions, yet from them the Ratio of the Fluxions may be deduced; and it appears that the smaller those Increments are taken, the nearer their Ratio will approach that of the Fluxions. Therefore, if we can, by any Means, find the Ratio to which the said Increments, by conceiving them less and less, do perpetually converge, and which they may approach, before they vanish, nearer than any assignable Difference, that Ratio (called here after for Distinction Sake, the Ratio limiting that of Increments) will be strictly that of the Fluxions.’ (Simpson 1750, 152).

33 ‘The foregoing Principles of the Doctrine of Fluxions being chiefly abstracted and Analytical, I shall here endeavour, after a general manner, to shew something analogous to them in Geometry and Mechanicks; by which they may become not only the object of the Understanding, and of the Imagination, (which will only prove their possible existence) but even of Sense too, by making them actually exist in a visible and sensible form’. (Newton 1736, 266).

34 ‘The word Fluxion properly apply’d always supposes the generation of some quantity (term’d fluent of flowing quantity) with an equable, accelerated, or retarded velocity, and is itself the quantity which might be uniformly generated, in a constant portion of Time, with the amount or remainder of that velocity, at the instant of finding such Fluxion’. (Blake 1741, 2).

35 ‘Likewise his definition [i.e. Simpson’s] (in which he pretends to exclude velocity) cannot be made intelligible without introducing velocity into it. Again, he mistakes the effect for the cause; for the thing generated must owe its existence to something, and this can only be the velocity of its motion; but it can never be the cause of itself, as his definition would erroneously suggest’ (Monthly Review 4, London 1750, 129–130).

36 Cerdà (1757–1759), ‘Capitulo 1. Explícase la Naturaleza de las Fluxiones’. Cortes 9/2812 f. 85r: ‘Para comprender perfectamente el Método de las Fluxiones, téngase presente que toda Magnitud Geométrica se reduce a Línea, Superficie o Plano y Sólido. La Línea se concibe formada por el Movimiento continuo de un Punto que la describe, la Superficie por el movimiento continuo de una Línea y el Sólido o Cuerpo por el movimiento continuo de una Superficie o Plano; y aquella parte de Línea, Superficie o Sólido que describiría el Punto, Línea o Figura generatriz en un tiempo dado, si perseverase constante e invariable en la velocidad, que en algún punto o posición determinada tiene, es la que llamamos Fluxión en aquel punto de la tal cantidad que así se forma, llamada por esto Fluente’.

37 Cerdà (1757–1759), ‘Capitulo 1. Explícase la Naturaleza de las Fluxiones’. Cortes 9/2812 f. 85r–85v.: ‘De aquí es que si el movimiento del Punto, Línea, Figura generatriz es uniforme la Fluxión será precisamente la misma, o igual a la cantidad que actualmente describe el Punto o Figura generatriz; pero si el Movimiento en realidad es o acelerado o retardado, la Fluxión será menor o mayor que la cantidad con que actualmente se aumenta la Fluente porque la variación de celeridad que entretanto sobreviene ha de causar también una diferencia de espacio actualmente descrito respecto del que describiera si la velocidad perseverase constantemente la misma que en un Punto o posición tiene y así, si la velocidad es acelerada, la Fluxión será menor, si retardada, la Fluxión será mayor que el Espacio actualmente descrito por el Punto, Línea o Figura generatriz’.

38 Cerdà (1757–1759), ‘Cap. [2,3] De las Fluxiones de Cantidades Algebraicas’. Cortes 9/2812 f. 97r. : ‘La expresión de la fluxión de una variable expresada por sola una letra es la misma letra precedida de la letra ‘d’, así la fluxión o diferencia de x es dx, que quiere decir la diferencia o fluxión elemento de la variable x […]. Los ingleses lo expresan de otra suerte, esto es por la misma letra con un punto arriba así $\dot{x}$ es la diferencia de x, pero semejante modo de expresar tiene el inconveniente [de que está] expuesto a errores de impresión […]’.

39 Cerdà (1757–1759), ‘Cap 4. De las Fluxiones Superiores’. Cortes 9/2792/28 f. 2.: ‘Pero, antes de entrar en este punto es menester dar alguna noticia del Cálculo Diferencial, que algunos confunden con el Método directo de las Fluxiones, […]. En este Cálculo Diferencial se hacen las operaciones como en el Método Directo de las Fluxiones y salen los mismos Resultados, aunque es forzoso confesar haber alguna diferencia entre estas Infinitésimas y las que nosotros llamamos Fluxiones’.

40 The word ‘space’ used by Cerdà is equivalent to the increment made by the fluent, that is to say, the variable that can be either a line, a surface or a solid.

41 Cerdà (1757–1759), ‘Cap 4. De las Fluxiones Superiores’. Cortes 9/2792/28 f. 2.: ‘Hemos inculcado bastante que Fluxión es el Espacio o partes que describiera la Figura generatriz si prosiguiese constante con el movimiento que en una posición dada tiene. Los que siguen el Cálculo Diferencial llaman Diferencia no lo que la Figura generatriz describiera sino lo que actualmente describe […]’.

42 A highly relevant study on this subject can be found in Guicciardini (1999, §2.2.6.2) and in Guicciardini (2009, §8.2.6).

43 Cerdà (1757–1759), ‘Cap 1: Explícase la Naturaleza de las Fluxiones’. Cortes 9/2812 f. 86r. : ‘Sea la línea mn que con su movimiento describe la área continuamente variante en longitud como sucede en la formación de las Áreas de las curvas, aun la Fluxión de la Área Amn será siempre el Rectángulo de la misma Línea mn y de la Fluxión Dd de la base AD. Porque la Fluxión de la Área Amn es el espacio que se describiría si la línea nm perseverase invariable ya en longitud ya en el movimiento que tiene en la posición DC; pero si desde la posición DC quedase la Línea generatriz invariable y prosiguiese con la misma velocidad que tiene en DC, el Rectángulo Dc seria la Fluxión de la Área Amn; luego también por más que varíe la línea mn ya en longitud ya en movimiento, será dicho Rectángulo la Fluxión de el área’.

44 See in Bos (1974, §2.22) and Guicciardini (1999, §2.2.5).

45 L’Hôpital (1696), proposition II, p.4: ‘1° La différence de xy est ydx+xdy. Car y devient y + dy lors x devient x + dx, & partant xy devient alors xy + ydx + xdy + dxdy qui est le produit de x + dx par y + dy, & la différence sera ydx + xdy + dxdy, c’est-à-dire ydx + xdy: puisque dxdy est une quantité infiniment petite par rapport aux autres termes ydx, & xdy’.

46 Blanco (2013) has analysed in great detail the fluxion of the product that appears in Simpson (1750) compared with the one that the same author presents in 1737. According to Blanco (2013, 58), the definition of fluxion adopted by Simpson is based on that given by Blake (1741), and, specifically, the fluxion of the rectangle is also influenced by Blake, as can be seen from the correspondence exchanged between the two mathematicians during the summer and autumn of 1740 (Blanco, 2013, 62).

47 Cerdà (1757–1759), ‘Cap 2. Algunos problemas para encontrar las Fluxiones de las cantidades algebraicas’. Cortes 9/2812 f. 89v.: ‘Concíbanse las líneas RC, BF que cortándose entre sí en ángulos Rectos varían entrambas a un tiempo separándose paralelamente de las líneas AL y AE de manera que formando con su movimiento el Rectángulo BC, el Punto de Intersección D describa la línea ADG. Sea Cc (dx) Fluxión de AC (x) y Bb (dz) Fluxión de AB (z). Completando pues los Rectángulos Cm, Bn, será Cm (Fluxión del Área ACD) = CD (AB) × Cc = zdx y Bn (Fluxión del Área ADB) = BD (AC) × Bb = xdz, por consiguiente la Fluxión de ACD + ABD = BC = xz será = zdx + xdz’.

48 Some years later, Newton, in the Principia (Newton 1687, book II, lemma II, 261), avoiding infinitesimals, tried to deduce the moment of the product of two quantities where the moment does not appear as an infinitesimal increment.

49 Cerdà (1757–1759), ‘Cap. 11 Aplicación del Método Inverso de las Fluxiones para la Cuadratura de las Curvas’. Cortes 9/2792/29 f. 11v: ‘Concíbase la línea bRg moviéndose paralelamente a sí misma a lo largo de la línea o Eje AB, a quien [= al cual] le es perpendicular, y que su velocidad en esta dirección o lo que es lo mismo la Fluxión de la Abscisa Ab, sea bd. Según lo que dijimos, explicando las Fluxiones, será el Rectángulo de esta Fluxión bd de la Abscisa por la ordenada Rb la Fluxión o Elemento del Área ARb. Por lo tanto suponiendo la Abscisa Ab = x, la ordenada Rb = y, será este Rectángulo o Fluxión del Área ydx y así, substituyendo según las Curvas en esta Expresión el valor particular de y o de dx, tomando su respectiva Fluente y corrigiéndola, si es menester, será esta Fluente corregida el Área de la Curva, cuyas ordenadas sean perpendiculares al Eje’.

50 Cerdà (1757–1759), ‘Cap. [11] Aplicación del Método inverso de las Fluxiones para la Cuadratura de las curvas’. Cortes 9/2792/46 f. 13r.: ‘Cuadrar una curva es encontrar el área de su plano y cómo este plano no es otra cosa que la suma de todas las fluxiones que son rectángulos, según vimos al principio, todo el arte de esta cuadratura se reducirá, dada su Ecuación de la curva, buscar la fluxión de su plano y encontrada ésta, integrando dicha fluxión se tendrá la fluente que es la suma de todas las fluxiones que la componen’.

51 See Newton (1736, preface, XI–XII).