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It is well known that over the eighteenth century the calculus moved away from its geometric origins; Euler, and later Lagrange, aspired to transform it into a “purely analytical” discipline. In the 1780 s, the Portuguese mathematician José Anastácio da Cunha developed an original version of the calculus whose interpretation in view of that process presents challenges. Cunha was a strong admirer of Newton (who famously favoured geometry over algebra) and criticized Euler’s faith in analysis. However, the fundamental propositions of his calculus follow the analytical trend. This appears to have been possible due to a nominalistic conception of variable that allowed him to deal with expressions as names, rather than abstract quantities. Still, Cunha tried to keep the deﬁnition of ﬂuxion directly applicable to geometrical magnitudes. According to a friend of Cunha’s, his calculus had an algebraic (analytical) branch and a geometrical branch, and it was because of this that his deﬁnition of ﬂuxion appeared too complex to some contemporaries. 1 Geometry and analysis in eighteenth-century calculus When the calculus appeared at the end of the seventeenth century, it concerned variable geometrical quantities associated with curves: abscissa, ordinate, arc-length, and son on (Bos 1974,5). As the eighteenth century progressed, algebraic, or analytical, expressions, which at ﬁrst were tools for studying geometrical objects, gained ascendance. Between 1748 and 1770, Leonhard Euler published a set of treatises on the calculus where, for (Bos 1974), already cited, covers this process from Leibniz to Euler; (Fraser 1987) addresses Lagrange’s later, more radically algebraic, version of the calculus; (Fraser 1989) identiﬁes the common algebraic characteristics of Euler’s and Lagrange’s versions (which were different in other aspects). (Jahnke 2003) gives a general view of the calculus in the eighteenth century; (Domingues 2008, 53–59) presents a picture of the question of the foundations of the calculus at about the same period as Cunha’s work. Communicated by Craig Fraser. B João Caramalho Domingues jcd@math.uminho.pt Centro de Matemática, Universidade do Minho, Braga, Portugal 123 J. C. Domingues the ﬁrst time, this was presented as being primarily about functions – “function of a variable quantity” being deﬁned as an “analytical expression composed in any way from that variable quantity and numbers or constant quantities”; as examples, “a + 3z; z 2 az − 4zz; az + b (aa − zz); c ; &c. are functions of z”. In the preface to his treatise on differential calculus Euler states that in it “all is contained within the boundaries of pure Analysis, so that no ﬁgure is necessary to explain all the rules of this calculus”. This move away from geometry and towards analysis was not immediately followed by every author. In particular, most textbook authors resisted or ignored it. A clear example of survival of a geometrical version of the calculus can be found in the section on the calculus in (Bézout 1767). This was an extremely successful text, reprinted several times up to the end of the eighteenth century; it is quite relevant to us that this section on the calculus was translated into Portuguese (Bézout 1774) and adopted as a textbook in the newly founded Faculty of Mathematics of the University of Coimbra. In that textbook, the word “function” is ﬁrst deﬁned nearly 70 pages after “differential”, as a mere detail in a section on multiple points of curves and for a second time at the beginning of the integral calculus: “We will call function of a quantity, any expression for calculation where that quantity enters, whatever way it enters” . Geometrical applications occupy a major portion of Bézout’s calculus (about two thirds of the differential calculus); and several results are based on geometrical reasonings—for instance, the determination of maxima and minima comes from the study of tangents that are parallel to the axis of abscissas (Bézout 1774, 51, 55), while dy in (Euler 1755, 580–581) the condition = 0 comes from the Taylor series expansion dx of y as function of x. What has been said above applies directly to the Leibnizian calculus, which was dominant in continental Europe. In Britain, Newton’s method of ﬂuxions prevailed. Although equivalent for many purposes, these two calculi were conceptually distinct and followed different paths. Overall, it may be said that the method of ﬂuxions was more consistently geometric, lacking an analytic version such as Euler’s. It is true that formal manipulation of series was a fundamental component—so much so that Newton called it “method of ﬂuxions and of series”; however, after an analytical youth, Newton came to see geometry as epistemologically superior to analysis. The objects of the method were geometrical quantities generated by motion (ﬂuents), a ﬂuxion being the velocity of a ﬂuent’s generation, or ﬂow. Moreover, after the famous “Functio quantitatis variabilis, est expressio analytica quomodocunque composita ex illa quantitate vari- abili, & numeris seu quantitatibus constantibus. […] Sic a + 3z; az − 4zz; az + b (aa − zz); c ; &c. sunt Functiones ipsius z.” (Euler 1748,4). “Hic autem omnia ita intra Analyseos purae limites continentur, ut ne ulla quidem ﬁgura opus fuerit, ad omnia huius calculi praecepta explicanda.” (Euler 1755, xx). On Bézout, see (Alfonsi 2011); on the adoption of several of Bézout’s textbooks in Portugal, see (Saraiva 2015); and on Bézout’s calculus see, for example, (Blanco 2013; Lamandé 1988). “[…] F, F ,&c., T denoting quantities composed as one may wish of x, y and constants, which, to abbreviate, are usually called functions of x, y and constants” (“[…] F, F ,&c., T marquant des quantités composées, comme on le voudra, de x, y & de constantes, ce que, pour abréger, on appelle des fonctions de x, y & de constantes” (Bézout 1767, 78); cf. (Bézout 1774, 74)); that is, the word “function” appeared only as an “abbreviation”, it did not correspond to a fundamental concept. “Nous appellerons fonction d’une quantité, toute expression de calcul dans laquelle cette quantité entrera, de quelque maniere qu’elle y entre d’ailleurs.” (Bézout 1767, 95); cf. (Bézout 1774, 98–99). 123 Geometry and analysis in Anastácio da Cunha’s calculus attack by Berkeley on the use of inﬁnitely small quantities in 1734, most British mathematicians adopted the stance that the method of ﬂuxions was a generalization of the ancient Greek geometers’ method of exhaustion (Guicciardini 1989, 47–51). The analytic perspective only gained ground in Britain in the 19th century. It is a well-known fact that the word “analysis” has multiple meanings. It should be made explicit that the important distinction in this text is that between analysis and geometry, rather than that between analysis and synthesis. It is, in a sense, an ontological distinction, rather than a methodological distinction: we are interested in the nature of the fundamental objects of the calculus, not in how the presentation of this subject is organized. We will see arguments that are ontologically analytical, because they consist of manipulations of analytical expressions and do not appeal to geometrical properties, but are methodologically synthetical, because they do not show how a result can be obtained, only that it is true. However, we should keep in mind that since the seventeenth century there was a traditional association between the synthetic method and classical geometry, as the paradigm of the synthetic method was Euclid’s Elements; while the word “analysis” was often given as synonymous of “algebra”. 2 José Anastácio da Cunha, a heterodox mathematician José Anastácio da Cunha (1744–1787) was certainly the most original Portuguese mathematician of the 18th century. Cunha was initially educated at the Oratorian college of his hometown, Lisbon, where he studied elementary mathematics reading works by Andreas Tacquet, Tomás Vicente Tosca, and Alexis Claude Clairaut (Rodrigues et al. 2013, 55–56). In 1764, he joined the army and was stationed in the northern border town of Valença. There he befriended several foreign ofﬁcers who worked for the Portuguese army. Among these were captain Richard Muller, son of John Muller, the ﬁrst director of the Royal Mil- itary Academy at Woolwich, and colonel James Ferrier, a Scotsman. These two gave Cunha access to British scientiﬁc books, including Simpson’s Algebra and Newton’s Arithmetica Universalis and Principia Mathematica (Rodrigues et al. 2013, 56–57). In 1773, Cunha was appointed professor at the newly founded Faculty of Mathemat- ics of the University of Coimbra—part of a major reformation of the university ordered by the Marquis of Pombal, the all-powerful prime minister who was a reformer aligned with the European enlightenment, but also a ruthless autocrat. In Coimbra Cunha had access to advanced works of continental European mathematics; his friend and biog- rapher José Maria de Sousa told a story of how Cunha borrowed Euler’s Integral Calculus from his colleague José Monteiro da Rocha (1734–1819) to study it, and Guicciardini addresses the rare exceptions to this in a chapter called “The analytic art (1755–85)” (1989, 82–91). (Queiró 1988) is a very good introduction to José Anastácio da Cunha in English, but it is outdated in some aspects, particularly because several manuscripts by or about Cunha have since been discovered. In English, see also (Oliveira 1988) and (Domingues 2014). (Ferraz et al. 1990) and (Ralha et al. 2006,I) contain important studies about Cunha, mostly in Portuguese but also, in the former case, a few in French or English. 123 J. C. Domingues later had to explain a particular passage in it to Monteiro da Rocha (Rodrigues et al. 2013, 62–63). There is an inventory of Cunha’s personal library in 1778, when it was conﬁscated by the Inquisition, and it is possible to say that its mathematical section was modest, when compared with the literary section: apart from some elementary books, we ﬁnd several works by Newton (eight volumes in total), d’Alembert’s Traité de l’équilibre, Bossut’s Traité de hydrodynamique, and three unidentiﬁed works by Euler bound in one volume (Giusti 1990, 35–37). But of course his readings were not limited to the books he owned: besides borrowing books from Monteiro da Rocha, he could use the University’s library. Cunha’s position in the university only lasted 5 years, because a political turn in the country (including the dismissal of Pombal) led to a persecution of free thinkers by the Inquisition (which, although much weakened, still existed). Since Valença, Cunha had had opinions and behaviours that were not in keeping with Roman Catholic orthodoxy of the time. He read, and translated, authors such as Alexander Pope and Voltaire (besides writing his own poetry, which was often also heterodox), and at least neglected religious observance. He was arrested in July 1778 and found guilty of heresy and apostasy. Cunha was detained in Lisbon, at the Oratorian house of Necessidades. This was in the same building where the new Science Academy of Lisbon (Academia Real das Sciencias de Lisboa, founded in 1779) was based. Although he was never admitted into the Academy, the Oratorian priest Teodoro de Almeida, his friend and spiritual director, was a founding member; thus, Cunha had close, albeit indirect contact with the Academy in its early years (Estrada et al. 2006). During this time, he kept working in mathematics: he wrote a text entitled “Principios do Calculo Fluxionario” (“Principles of ﬂuxionary calculus”), which survives only in a fragmentary state, with the date March 1780 (Cunha 2006b; Domingues et al. 2006). Cunha was released in 1781, but forbidden from returning to Coimbra. He was appointed director of studies of a school for poor boys in Lisbon, but apparently by 1785 he had lost that position too. In his ﬁnal 2 or 3 years, he depended on friends, as he was jobless and his health was frail (Rodrigues et al. 2013, 71–72). He died on the 1st January 1787. In 1785–86, he was involved in two polemics with other mathematicians. The most important one was with his former university colleague José Monteiro da Rocha. What is left of it are three letters, two by Cunha and one by Monteiro da Rocha, which were published in the 1890s in the journal O Instituto and reprinted in (Ferraz et al. 1990). In particular, the ﬁrst one (Cunha 1785), addressed to his friend João Manuel de Abreu, but which appears to have circulated in manuscript copies, is an important source for Cunha’s opinions on several issues in his ﬁnal years. He was highly critical of how mathematics was usually taught in Portugal and of the scientiﬁc level of the Science Academy of Lisbon (where Monteiro da Rocha was the foremost mathematician). He repeatedly praised Newton and d’Alembert, while presenting Monteiro da Rocha as His ﬁle at the Inquisition has been published as (Ferro 1987). This happens not only in (Cunha 1785) but also in other texts: for example, in an undated essay on the principles of mechanics (Cunha 1807) and in one of the fragments comprising (Cunha 2006b). 123 Geometry and analysis in Anastácio da Cunha’s calculus as an ardent follower of Euler, as if projecting d’Alembert’s rivalry with Euler on to his own disagreements with Monteiro da Rocha. Around 1782, a book by Cunha, entitled Principios Mathematicos (Mathematical Principles), began being printed; according to João Manuel de Abreu, who later trans- lated the book into French, as each section of the book was printed, it was used in the college where he worked at the time (Cunha 1790, French transl., iii). But the printing of the book was interrupted when Cunha lost his position. Only 3 years after his death was it published (Cunha 1790). (Cunha 1790) is a relatively short book (little over 300 pages) that tries to present in a logical order the main branches of pure mathematics, from elementary geometry to some calculus of variations. To cover so much ground, it is naturally an extremely concise text. It also has a few peculiarities, both in the organization of the subjects and in several deﬁnitions. As Grattan-Guinness (1990, 59) put it: “Impressive but odd, powerful but cryptic, this book […] ‘interesting’, but too off-beat to gain the attention that he deserved”. A French translation of (Cunha 1790), by his friend João Manuel de Abreu, was published in 1811 (and reissued in 1816) but it did not have much impact (Duarte and Silva 1990; Domingues 2014). In the late twentieth century Cunha’s book received some attention from historians of mathematics, particularly for three originalities: • in book 9 he deﬁned “convergent series” as one that satisﬁes what would later be called the Cauchy criterion, proceeding to actually prove the convergence of some series using this deﬁnition ; b bbcc bbbccc • also in book 9 he deﬁned the power a as 1 + bc + + + &c., where c 2 2×3 cc ccc b is such that a = 1 + c + + + &c. (i. e., a is deﬁned via the power series 2 2×3 b log(a) for e ) covering rational, real and even complex exponents in the deﬁnition; • in book 15 he deﬁned “ﬂuxion” in a way that has been described as corresponding to the modern deﬁnition of differential. Only the third will be directly relevant here. Although Cunha, naturally, used power series in his calculus, he did not address their convergence in that context. As far as I can tell, the word “convergent” does not appear after book 9. Notice that all these originalities are related to the issue of how to (properly) deﬁne particular concepts. Notice also that they are not merely descriptive deﬁnitions (as often happened in the eighteenth century): they are actually used in proofs and in the development of theories (albeit short theories, because of the concise nature of the book). Another posthumous publication (Cunha 1807), about the principles of mechanics, should be mentioned. According to Cunha, the ﬁrst principles of mechanics cannot be proven mathematically (unlike what many authors tried to do in the eighteenth By the end of his life, Monteiro da Rocha owned most of Euler’s books, and drew on his work about the orbits of comets; but there is no evidence, apart from Cunha’s “accusation”, that Monteiro da Rocha was more Eulerian than the average mathematician of his time (Domingues 2007, 97–100). Unfortunately, the French translation is faulty in these passages, and the French version of this deﬁnition contains a fallacy. On Cunha’s convergence of series, see (Queiró 1988, 40–41), (Oliveira 1988), and (Giusti 1990, 42–45). 123 J. C. Domingues century). There are then two possibilities: in a physico-mathematical work these ﬁrst principles must be proven experimentally or come from observation of nature; in a purely mathematical work they must be taken as axioms. In the latter case, the author is, in theory, free to assume the laws of mechanics at will, even that light propagates in a circular, rather than straight, line: “mathematical truth consists solely in the legitimacy with which theorems and solutions of problems are derived from deﬁnitions, postulates, and axioms” . It is true that, to avoid being criticized for lack of usefulness, the mathematician should take as axioms factual truths taught by nature. But that theoretical freedom was very unusual, to say the least, in the eighteenth century. 3 Cunha’s ﬂuxionary calculus: geometry or analysis? It is possible to glimpse the evolution of José Anastácio da Cunha’s personal views on the foundations of the calculus, but it is not possible to have a full picture. In his essay on the principles of mechanics, whose date of composition is not known, he spoke of ultimate ratios (Cunha 1807, 344–345) and used the dot notation for ﬂuxions. Thus, it seems that at some point Cunha was a canonical follower of the Newtonian calculus of ﬂuxions. A manuscript discovered in 2005 and published in (Ralha et al. 2006, II) bears the title “Principles of the ﬂuxionary calculus” and the date March 1780 (Cunha 2006b). But it is only a copy, by someone else, of very incomplete fragments from at least two different versions of Cunha’s work (Domingues et al. 2006, 265–266). In the ﬁrst part (the one actually dated 1780), Cunha gives a deﬁnition of ﬂuxion very close to the one that later appeared in (Cunha 1790), and uses the d notation; in another part, on higher-order ﬂuxions, he uses the dot notation; near the end, he refers to a deﬁnition of 2 3 limit (which is not extant) and says that “ A is the limit of A + By + Cy + Dy + &c. in regard to inﬁnitesimal y”. The word “inﬁnitesimal” should be understood in the non-Leibnizian sense of a variable (not a magnitude) capable of assuming arbitrarily small (but ﬁnite) values; this is the sense in which Cunha deﬁned it in the ﬁrst part of the manuscript and later in (1790). Finally, it must be mentioned that João Manuel de Abreu reported that among the manuscripts that Cunha had left, one had the title “Against the doctrine of prime and “A verdade mathematica não consiste senão na legitimidade com que os theoremas, e as soluções dos problemas se derivam das deﬁnições, postulados e axiomas” (Ferraz et al. 1990, 340). The editor of the 1807 edition added footnotes with the Leibnizian d notation, and the editor of the 1856 edition used only the d notation; the editors of the 1990 edition copied the 1856 edition, but acknowledged this issue (Ferraz et al. 1990, 315). On Newtonian calculus, see (Guicciardini 1989, 2003, 74–85). But with all verbs in the indicative, rather than subjunctive mood (see Sect. 5.4). 17 2 3 “ A he o limite de A + By + Cy + Dy + &c. a respeito de y inﬁnitessimo” (Cunha 2006b, 54–55). On Cunha’s non-Leibnizian concept of inﬁnitesimal, see (Domingues 2004, 23–26) and page 17 below. 123 Geometry and analysis in Anastácio da Cunha’s calculus ultimate ratios of nascent and evanescent quantities” . Neither the date nor the content of this text are known. But in (Cunha 2006b, 50–51) he distanced himself from the idea, used by Newton, of quantities being generated by motion, which would entail the consideration of time in geometry. All this suggests that Cunha’s opinions moved from a canonical Newtonian calculus to a somewhat original take on d’Alembert’s proposal of using limits (not surpris- ing, given that d’Alembert, according to himself, was following Newton), and later developed into a more original version of the calculus, using a peculiar deﬁnition of inﬁnitesimal. The last one, the version that appeared in (Cunha 1790), is the only one that survives in a form that we may call complete, and the only that will be considered henceforth. We will see that Cunha used the Leibnizian notation dx , d x (and also dx x) in (1790), but he kept the Newtonian word “ﬂuxion” (as well as “ﬂuent”). In (2006b, 52–53), he had commented that those names might appear improper, but added that “it matters little: in the deﬁnitions lies everything” . Cunha’s deﬁnition of ﬂuxion is the following: “Some magnitude having been chosen, homogeneous to an argument x,tobe called ﬂuxion of that argument, and denoted by dx; we will call ﬂuxion of d x x, and will denote by d x, the magnitude that would make constant and dx (x +dx )− x d x − inﬁnitesimal or zero, if dx were inﬁnitesimal and all that dx dx does not depend on dx constant.” Speaking of this deﬁnition, Youschkevitch (1973, 19) said that “it was Cunha who, for the ﬁrst time, formulated a rigorous analytical deﬁnition of the differential, taken up again and used later by the mathematicians of the nineteenth century” .Mawhin (1990, 100) was more speciﬁc, saying that it “corresponds to the modern deﬁnition of differential of f at x as a linear function h → Ah such that f (x + h) − f (x ) − Ah = hB(h) where B(h) → 0 when h → 0” ; that is, d x is a linear function of dx (x +dx )−(x )−d x d x (since is constant) such that lim = 0. Of course, this dx →0 dx dx “correspondence” must be taken with a grain of salt. Even apart from some linguistic or conceptual differences (for instance, Cunha does not explicitly say that d x is a function of dx, even though he spoke of functions), his deﬁnition is not strictly equivalent, in the mathematical sense, to the modern one, nor could it be without a modern theory of real functions; among other details, and like all his contemporaries, “Contra a doutrina das razoens primeiras e ultimas das quantidades nascentes e fenescentes” (Ferraz et al. 1990, 355); also “Contre la méthode des premiers et derniers rapports des quantités naissantes et évanouissantes de Newton” (Cunha 1790, Fr. transl., ii). “isso pouco importa: nas deﬁnições está tudo”. “Escolhida qualquer grandeza, homogénea a uma raiz x, para se chamar ﬂuxão dessa raiz e denotada d x assim dx; chamar-se-á ﬂuxão de x, e se denotará assim d x, a grandeza que faria constante e dx (x +dx )− x d x − inﬁnitésimo ou cifra, se dx fosse inﬁnitésimo e constante tudo o que não depende de dx dx dx” (Cunha 1790, 194). “C’est da Cunha qui a, pour la première fois, formulé une déﬁnition analytique rigoureuse de la dif- férentielle, reprise et utilisée plus tard par les mathématiciens du XIX siécle.” “correspond […] á la déﬁnition moderne de différentielle de f en x comme fonction linéaire h → Ah telle que f (x + h) − f (x ) − Ah = hB(h) oú B(h) → 0 lorsque h → 0”. 123 J. C. Domingues he assumed that all functions were differentiable, or considered only differentiable functions. A question that naturally arises is whether Cunha’s version of the calculus was more geometrical or followed the analytical trend of the late eighteenth century. Historians or mathematicians who have studied Cunha’s calculus have focused mostly on how rigourous it was, and have not really addressed this question. We will take a brief look at a couple of passing remarks, by Youschkevitch and Gomes Teixeira, that apparently point in opposite directions, simply to show that the classiﬁcation of Cunha’s calculus as geometrical or analytical is not immediate. On one hand, Youschkevitch, as quoted above, explicitly stated that “Cunha […] formulated a rigorous analytical deﬁnition of the differential”. It is far from straight- forward that in this sentence the word “analytical” is particularly meaningful or used in a sense similar to the one described in section 1; however, it should be remarked that Youschkevitch immediately pointed out that “a precise deﬁnition of the differential had already been given, under a geometrical form, by Leibniz” (but also that this precise geometrical deﬁnition by Leibniz, dependent on subtangents, was useless for calculations). On the other hand, Gomes Teixeira , although praising the rigour of Cunha’s ﬂuxionary calculus, included too large a role for geometrical intuition as one of its few ﬂaws: “It would sufﬁce to introduce in the exposition the word limit, which Anastácio da Cunha, bound to the Greek tradition, did not want to employ, to make explicit some conditions included in proofs, and to give a less intense role to geometrical intuition, in order to reduce our geometer’s doctrine to the modern form.” Cunha’s personal opinions about Newton and Euler seem to suggest that he favoured geometry over analysis (speaking of general approaches to mathematics, not limited to the calculus). Cunha repeatedly expressed his admiration for Newton, while he disliked Euler, and in particular Euler’s faith in analysis. In a letter included in the polemic against Monteiro da Rocha (see page 5 above), he wrote, right after praising d’Alembert: “But in Coimbra c’est tout une autre chose [it is completely different] Newton, d’Alembert, ne sont que de petits génies [are only little geniuses]. Euler is the only god of mathematics, and Monteiro [da Rocha] his prophet. And which author could our masters, nos sages maîtres [our wise masters], ﬁnd more suitable to the characters and interests but the one who established implicit faith in matters of mathematics? I do not know if I have ever told you that this author, when “une déﬁnition exacte de la différentielle avait déjá été donnée, sous une forme géométrique, par Leibniz” (Youschkevitch 1973, 19). Francisco Gomes Teixeira (1851–1933) was, by far, the foremost Portuguese mathematician of his time. An analyst at ﬁrst, he then turned his attention to geometry and, in his later years, to the history of mathematics in Portugal. His History of Mathematics in Portugal (1934) is, regrettably, still the most recent general account of the subject; it is, naturally, quite dated. “Bastaria introduzir na exposição a palavra limite, que Anastácio da Cunha, prêso á tradição grega, não quis empregar, tornar explícitas algumas condições incluídas nas demonstrações e dar á intuïção geométrica um papel menos intenso, para reduzir a doutrina do nosso geómetra á forma moderna.” (Teixeira 1934, 257). 123 Geometry and analysis in Anastácio da Cunha’s calculus perplexed between manifest truths and Algebra, which contradicts them, would close his eyes and cry out as a faithful algebraist: Quidquid sit, calculo potius, quam judicio nostro, est ﬁdendum! [Whatever the question, we should rely on calculation, better than on our judgement!]” Some of Cunha’s philosophical opinions, which will be the subject of the next section, also suggest, at ﬁrst sight, a preference for geometry. But, as we will see in later sections, things are not so simple. In a sense, both Youschkevitch and Gomes Teixeira were right: Cunha’s calculus had an analytical part and a geometrical part. And Euler was probably a bigger inﬂuence than Cunha himself would like to admit. 4 Cunha’s mathematical ontology One of the most marked characteristics in José Anastácio da Cunha’s Principios Math- ematicos is the near absence of commentaries or explanatory notes. A consequence is that no motivation is presented there for the frequently unusual and sometimes truly original paths that the text follows. However, Cunha also left several shorter manuscripts on particular mathematical topics, and in those texts he did include several methodological and philosophical reﬂexions, often very critical of the ways in which several mathematical topics were usually developed in the eighteenth century. Based on one of the few of those texts then known (his essay on the principles of mechanics, already mentioned), Norberto Ferreira da Cunha noted in (2001) Anastácio da Cunha’s nominalistic, or anti-essentialist, stance: he rejected the real existence of universals (abstract ideas). But one of the most clear passages in this respect can be “Mas em Coimbra c’est tout une autre chose Newton, d’Alembert, ne sont que de petits génies. Euler é o único Deus da Mathematica, e Monteiro o seu propheta. E que auctor podiam os nossos mestres, nos sages maîtres, achar mais acommodado aos characteres e interesses senão o que instituiu a fé implícita em pontos de Mathematica? Não sei se algum dia lhe contei, que este auctor, quando se via perplexo entre verdades manifestas, e a Algebra, que as contradiz, fechava os olhos, e exclamava como ﬁel algebrista: Quidquid sit, calculo potius, quam judicio nostro, est ﬁdendum!” (Cunha 1785, 367) A sentence very close to the last one (“Quicquid autem sit hic calculo potius, quam nostro iudicio est ﬁdendum”) occurs in (Euler 1736, I, 108), in a discussion of a body under a force of attraction inversely proportional to the distance: this body reaches the centre of attraction with inﬁnite speed but, contrary to what common judgement would imagine, does not go beyond it, because if it did its speed would become imaginary. However, the possibility should not be excluded that Cunha knew this sentence from a satyrical pamphlet by Voltaire, part of a polemic against Maupertuis, who was supported by Euler: a supposed “peace treatise” where Euler begged forgiveness to all logicians for having written such a sentence (Voltaire 1877–1885, XXIII, 578). A similar conclusion was drawn already by Giusti (1990, 39), not about the calculus but about (Cunha 1790) at large. The word “essentialism” was proposed by Popper (1944, 94) to refer to the belief in the actual existence of universals (or essences); the traditional term is “realism”, but “essentialism” has gained some ground in the last decades. “Nominalism” is the traditional name for the view that universals are merely names. Popper’s example is the following: “The universal term ‘white’, for instance, seemed to [the nominalists] to be nothing but a label attached to a set of many different things, snowﬂakes, table-cloths, and swans, for instance. […] Essentialists deny that we ﬁrst collect a group of single things and then label them ‘white’; rather, they say, we call each single white thing ‘white’ on account of a certain intrinsic property that they have in common—their ‘whiteness’” (Popper 1944, 94). 123 J. C. Domingues found in another text, discovered only in 2005, a prologue for a presentation of the principles of geometry : “[there is no reason to] seriously consider, analyse and combine beings of rea- son,mere Aristotelian substantial forms, such as would be, in the literal sense of almost every author, point, line, surface, angle, ratio between two magnitudes, incomparable indivisibles, inﬁnitely large and inﬁnitely small [quantities], ﬂux- ions, prime and ultimate ratios, velocity, momentum, force, action, reaction, collision, attraction, repulsion. It is not usually noticed that such words are but descriptions of phenomena, abbreviations of phrases, of arguments, sometimes intricate and even unfeasible; and this negligence together with the unproﬁtable mistake or imprudence of taking them for names of substances, such as are e. g. the words man, tree, ﬂower, Sun, stars, etc. has been an extremely plentiful source of logomachies and relevant errors”. Cunha’s concern with proper deﬁnitions (pages 6-7 above) was, at least in part, a consequence of his nominalism. For most mathematicians of the 18th century, deﬁni- tions were merely descriptions of mathematical objects that were assumed to exist a priori; they were intended to convey the general meaning of a word, but did not need to exhaust that meaning [Ferraro 1999, 103–104; Petrie 2012, 282–285]. Not so for the nominalist Cunha: for instance, in a manuscript (in English) on logarithms and powers, he complained of authors who “employ sophistry to prove what the narrowness of their deﬁnition renders not only incapable of demonstration, but even unintelligible. They deﬁne the power of a number to be what is form’d by its continual multiplication. Admit this, and 1 1 2 3 then I will ask you what does a or a signify?” (Cunha 1778, 58). For Cunha, the word “power”, or the symbol a , could mean only what its deﬁnition said it meant; hence he sought to deﬁne power, in (Cunha 1778) and in (Cunha 1790, 108–109), in ways not limited to integer exponents. We have seen in page 9 that Those principles of geometry were probably an early version of the ﬁrst few “books” of (Cunha 1790). “[Não há razão para] seriamente contemplar, analysar e combinar entes de razão, meras formas substan- ciaes, aristotelicas, quaes seriam no sentido literal de quasi todos os Autores o ponto, a linha, a superﬁcie, o angulo, a razão de duas grandezas, os indivisiveis incomparaveis, inﬁnitamente grandes,e inﬁnita- mente pequenos, ﬂuxões, razões, primeiras e ultimas, velocidade, quantidade de movimento, força, acção, reacção, percussão, attracção, repulsão. Geralmente n[ão] se custuma reparar que semelhantes palavras não são senão [des]cripções de phenomenos, abreviaçoens de frases, de discursos, as vezes, entrincados, e athe impraticaveis: e esta incuria junta com a mal succedida equivocação ou temeridade de as tomar por nomes de substancias, como o são v. g. as palavras homem, arvore, ﬂor, Sol, Estrellas, &c. tem sido um manancial copiosissimo he Logomachias, e de relevantes erros” (Cunha 2006a, II, 6–7) Although not using the word “deﬁnition”, Euler opens the chapter on powers of his Elements of Algebra stating that “When a number is multiplied several times by itself, the product is called a power” (“Wann eine Zahl mehrmalen mit sich selbsten multiplicirt wird, so wird das Product eine […] Potenz […] genennet” 0 −1 1 (Euler 1770, I, 99)). Later, he concludes that a = 1, a = , a = a, and so on (Euler 1770,I, 104–105, 116–117). There is more to these attempts than a philosophical standpoint on deﬁnitions: Cunha also wished to present a proper proof of the binomial theorem. 123 Geometry and analysis in Anastácio da Cunha’s calculus when discussing the appropriateness of the names “ﬂuent and ﬂuxion”, he concluded that “it matters little: in the deﬁnitions lies everything”. Another, but related, aspect of Cunha’s ontology is his physicalism. In the same prologue to the principles of geometry, following a quotation from Newton, Cunha concludes that “thus, in the opinion of Sir Isaac, geometry is properly a part of physics. And in truth I do not know what else it might be” . Accordingly, Cunha deﬁned the simpler objects of geometry (points, lines, surfaces) as “bodies”: for instance, the ﬁrst deﬁnition in Cunha (1790) reads “The Body, whose length is such that no remarkable error comes from disre- garding it, is called Point” . More complex objects (for instance, ﬂuxion or velocity) are just names, words that abbreviate more intricate phrases. Anti-essentialism, nominalism, or physicalism are not, of course, originalities of Cunha (although we will see that he drew some original consequences from his nom- inalism). David Sepkoski (2005) identiﬁed nominalist and physicalist conceptions in Barrow and Newton (and, as had been said, Newton was one of Cunha’s mathematical heroes). It is important to notice that the anti-essentialism of Barrow and Newton is asso- ciated to their preference for geometry over algebra. Analytic/algebraic methods and concepts, being more abstract, would be more palatable to mathematicians with essen- tialist stances. Giovanni Ferraro, in a paper on “analytical symbols and geometrical ﬁgures in eighteenth-century calculus” (2001), used Aristotelian references to inter- pret the analytical deﬁnitions of variable, in particular those of Euler and Lagrange. While for authors of geometrical versions of the calculus, a variable was literally a (geometrical) quantity that varied, increasing or decreasing, for the great analysts Euler and Lagrange, a variable was “an indeterminate or universal quantity” (Euler) or an “abstract quantity” (Lagrange); being “generated from particular geometrical quantities by means of a process of abstraction […] the notion of a variable concerned the essence of quantity” (Ferraro 2001, 541) (emphasis in the original). Actually, Cunha’s deﬁnition of variable may have have been inspired by Euler’s, but with a crucial nominalistic twist: for Euler, to quote in full, “a variable quantity is an indeterminate or universal quantity, which comprises in itself absolutely all determinate values” ; for Cunha, “if an expression can assume more than one value, while another can assume only one, the latter will be called constant, and the former “Hé pois propriamente a geometria na opinião de Sir Isaac huma parte da physica. E na verdade, não sei que outra cousa possa ser […]” (Cunha 2006a, 6–7). “O Corpo, cujo comprimento he tal, que de se naõ attender a elle, naõ resulta erro notavel, chama-se Ponto” (Cunha 1790,1). This distinction made here between simpler and more complex objects is somewhat artiﬁcial: “point” is also an abbreviation, for the more intricate phrase “body whose length is such, that no remarkable error comes from disregarding it”. “Quantitas variabilis est quantitas indeterminata seu universalis, quae omnes omnino valores determi- natos in se complectitur” (Euler 1748,I,4). 123 J. C. Domingues variable” — that is, Cunha’s variable is an expression (rather than a quantity) that can assume, if not all values like Euler’s, at least several. Cunha’s deﬁnition seems more distant from the traditional geometrically-inspired deﬁnitions (quantities that vary), of which he was very critical (notice the opposition between the explanation by “common authors”, which allegedly results in a contra- diction, and the understanding of “the geometer”, i. e. a proper mathematician): “Common authors [say] that, e. g. in a given circle, the diameter is constant and the chord is variable; and [the reader] understands that the same magnitude is now the chord of 10° then of 11° etc.; that is, one magnitude is and is not the same. — The geometer understands by variability only what consists in the possibility of denoting several magnitudes by a single expression.” 5The algebraic and the geometrical branches in Cunha’s calculus The question of whether Cunha’s calculus was more geometrical or more analyti- cal received an answer over 200 years ago, from João Manuel de Abreu, a friend of Cunha’s and the translator of (Cunha 1790) into French. He was not trying to answer this question. A review of the French edition of (Cunha 1790) had appeared in the Edinburgh Review (anonymously but almost certainly by the Scottish mathematician John Playfair) (Domingues 2014, 37–38). This review was globally positive, but crit- icised several aspects of Cunha’s book. Abreu published a reply (but in Portuguese, in a Portuguese periodical published in London) (Abreu 1813–1814). In that reply, addressing Playfair’s criticism that Cunha’s deﬁnition of ﬂuxion was “very difﬁcult to be understood”, Abreu stated that “[Anastácio da Cunha] divided his theory of ﬂuxions into two branches, an algebraic one, composed of proposition 1 of book 15, and of all propositions that depend on it; and a geometrical one, whose ﬁrst proposition is Archimedes’ axiom, and which is composed of propositions 13, 14, 15, 17, and 18, of book 15, and 39, 40, 41, of book 16, &c. In the ﬁrst, algebraic, branch he followed his ordinary method, always resorting to the fundamental deﬁnition, or to theorems deduced from it; in the second, geometrical, branch, he adopted the ancients’ method of proof, commonly called of exhaustion. Now, deﬁnition 4 of book 15 is common to both; thus, it must be more complex, and consequently less intelligible than any deﬁnition of ﬂuxion that comprehends but one of the two branches.” “Se huma expressaõ admittir mais de hum valor, quando outra expressaõ admitte hum só, chamarse-ha esta constante, e aquella variavel” (Cunha 1790, 193). “Os autores vulgares [dizem] que, v. g. em hum circulo dado, o diametro hé constante eacorda variavel; e ﬁca entendendo que huma mesma grandeza hé ora corda de 10° ora de 11° &c. ; isto hé, que huma mesma grandeza hé e não hé a mesma. — O geometra não entende por variabilidade se não o que consiste na possibilidade de notar com huma so expressão grandezas diversas.” (Cunha 2006b, 54–55). Playfair’s review and Abreu’s reply were reprinted as appendices in (Ferraz et al. 1990). “[Anastácio da Cunha] dividio a sua theorica das ﬂuxoens em dous ramos, hum algebraico, que se compoem da proposiçaõ 1 do livro 15, e de todas as que della dependem; outro geometrico, cuja proposiçaõ 123 Geometry and analysis in Anastácio da Cunha’s calculus These “branches” do not reﬂect the formal organization of (Cunha 1790), nor are they ever mentioned in Cunha’s known writings. Rather, they reﬂect Abreu’s classiﬁ- cation of those propositions, a classiﬁcation made about 25 years after Cunha’s death. But it is a classiﬁcation that makes sense: as we will see, the “algebraic” branch is composed of purely analytical propositions (and will often be called in the following “analytical”, rather than “algebraic”), while geometrical objects and arguments appear in the geometrical branch. This classiﬁcation even seems to reﬂect, if we restrict our- selves to book 15, a subtle difference in language, namely in some verb moods (see Sect. 5.4). 5.1 The algebraic/analytical branch in book 15 (Cunha 1790) is organized in chapters called “books”, following the Euclidean model. Book 15 is dedicated to the calculus, starting with fundamental deﬁnitions. We have seen that, according to João Manuel de Abreu, deﬁnition 4, of ﬂuxion, is common to both the algebraic (analytical) and the geometrical branches. How can we classify the other deﬁnitions in book 15? We have already seen deﬁnition 1, of constant and variable (pages 14–15 above). Its classiﬁcation is not straightforward. It is not a typical analytical deﬁnition (vari- able as an universal or abstract quantity), but it is even more distant from traditional geometrical traditions (quantities that vary). It may be a nominalistic adaptation of Euler’s analytical deﬁnition. A similar difﬁculty occurs with deﬁnition 2: “A variable always capable of assuming a value greater than any proposed mag- nitude will be called inﬁnite; and a variable always capable of assuming a value smaller than any proposed magnitude will be called inﬁnitesimal.” Throughout the eighteenth century there were plenty of discussions about the nature of inﬁnite and inﬁnitesimal quantities, and whether they actually existed or instead some- thing like limits was more advisable. This deﬁnition by Cunha, which must be read in conjunction with his deﬁnition 1, does not correspond to any of the common solutions of the period: it introduces inﬁnitesimals, but as variables and hence expressions, not quantities. Footnote 41 continued primeira he o axioma de Archimedes, e que se compoem das proposiçoens 13, 14, 15, 17, e 18, liv. 15, e 39, 40, 41, liv. 16, &c. No primeiro ramo algebraico seguio o seu methodo ordinario, recorrendo sempre á deﬁniçaõ fundamental, ou á theoremas deduzidos della; no segundo ramo Geometrico adoptou o methodo de demonstraçaõ dos antigos, chamado vulgarmente d’exhaustaõ. Ora a deﬁniçaõ 4, liv. 15, he comum a ambos; logo deve ser mais complicada, e por consequencia menos intelligivel que qualquer deﬁniçaõ de ﬂuxaõ, que naõ comprehenda senaõ hum dos dous ramos.” (Abreu 1813–1814, 451–452) On deﬁnitions 1 and 2, in their eighteenth-century context, see also (Domingues 2004). “A variavel que podér sempre admittir valor maior que qualquer grandeza que se proponha chamarse- ha inﬁnita;eavariavelque podér sempre admittir valor menor que qualquer grandeza que se proponha, chamarse-ha inﬁnitessima.” (Cunha 1790, 193). It is obvious that Cunha’s inﬁnites and inﬁnitesimals are potential, rather than actual. This is consistent with all that we know about Cunha (“the attitude of considering only potential inﬁnites and inﬁnitesimals permeates all the Principios” (Queiró 1988, 41)). But there is more here than the classical opposition 123 J. C. Domingues Like deﬁnition 1, deﬁnition 2 may best be classiﬁed as nominalistic. Notice that neither of them introduces new ontological categories, but only names for certain types of expressions. However, being about expressions, they are in a sense (albeit not the traditional one) in an analytical domain. Deﬁnition 3 reinforces the analytical course: “If the value of an expression A depends on another expression B, A will be called function of B” . It is signiﬁ- cant that “function” is deﬁned so early in Cunha’s calculus, suggesting that this is a central object here, as it was in Euler’s; and indeed it is. This is followed by deﬁnition 4, of ﬂuxion, two deﬁnitions (of ﬂuent as antideriva- tive, and of higher order ﬂuxions) that are not important for our purpose, some remarks on notation, and then propositions. According to Abreu, propositions 1 to 12 are part of the algebraic branch. Indeed, in these propositions, Cunha presents fundamental results of the differential calculus in a purely analytical context, without any geometrical concepts or arguments. A simple example of the typical format of these propositions is proposition 2, to the effect that n n−1 d(x ) = nx dx. Proposition 1 had established that a polynomial in an inﬁnitesimal n−1 variable is itself inﬁnitesimal; this is now used to verify that nx dx satisﬁes the conditions in the deﬁnition of ﬂuxion (page 9 above): n−1 nx dx n−1 “dx inﬁnitesimal and what does not depend on dx constant make = nx dx n n n−1 (x +dx ) −x nx dx n−1 n−2 n−1 n−2 n−3 2 constant and − = n x dx + n × x dx + dx dx 2 2 3 &c. inﬁnitesimal.” Another example, whose analytical character is very obvious, is proposition 8, where the ﬂuxion of the logarithm is obtained differentiating term by term the series of the exponential: “Let x stand for any number and l indicate hyperbolic logarithms: then dx = xdl x. 1 1 1 1 2 3 4 5 For dx = d 1 + lx + (lx ) + (lx ) + (lx ) + (lx ) + &c. = dl x + 2 6 24 120 2 3 2 4 3 5 4 1 2 (lx )dl x + (lx ) dl x + (lx ) dl x + (lx ) dl x + &c. = 1 + lx + (lx ) 2 6 24 120 2 1 3 1 4 47 + (lx ) + (lx ) + &c. dl x = xdl x.” 6 24 In this case Bézout (1774, 23–26) is not explicitly geometrical, but neither really analytical: the result equivalent to this is obtained going back to the deﬁnition of loga- Footnote 44 continued between potential and actual inﬁnities: in Cunha’s time, the usual route for those who rejected the actual inﬁnite was to follow the method of limits, as proposed by d’Alembert in the Encyclopédie (Domingues 2008, 57–59); Cunha apparently at some point used limits (see page 8), but later changed paths. “Se o valor de huma expressaõ A depender de outra expressaõ B, chamarse-ha A funcçaõ de B” (Cunha 1790, 193). n−1 nx dx 46 n−1 “dx inﬁnitessimo, e o que de dx naõ depende, constante, fazem = nx constante e dx n n n−1 (x +dx ) −x nx dx n−2 n−1 n−2 n−2 n−3 2 − [= n x dx + n × x dx + &c.] inﬁnitessimo.” (Cunha 1790, 2 2 3 dx dx 195). “Represente x qualquer numero e indique l logarithmos hyperbolicos: será dx = xdl x.Poishe dx = 1 2 1 3 1 4 1 5 2 3 2 4 3 d 1 +lx + (lx ) + (lx ) + (lx ) + (lx ) +&c. = dl x + (lx )dl x + (lx ) dl x + (lx ) dl x + 2 6 24 120 2 6 24 5 4 1 2 1 3 1 4 (lx ) dl x + &c. = 1 + lx + (lx ) + (lx ) + (lx ) + &c. dl x = xdl x.” (Cunha 1790, 196). 120 2 6 24 123 Geometry and analysis in Anastácio da Cunha’s calculus Fig. 1 Diagram for propositions 13 and 14 of book 15 of (Cunha 1790) rithms as terms in an arithmetical progression in a correspondence with a geometrical ma( y − y) progression, establishing the relation = x − x between consecutive terms in these progressions and then imagining the differences y − y and x − x inﬁnitely small. Other authors from this period are more directly geometrical: (Cousin 1777, 29–30) uses the logarithmic curve (deﬁned by the property that, if the abscissas are in arithmetical progression, then the ordinates are in geometrical progression); while (Saladini 1775, II, 44–47) uses the characterization of the logarithm as the area under a hyperbola. 5.2 The geometrical branch in book 15 According to João Manuel de Abreu, the geometric branch starts in proposition 13, whose enunciation reads: “Let AB stand for the abscissa and BC for the ordinate corresponding to an arbitrary arc AC of a regular curve AD (that is, of a curve whose ordinate is a function of the abscissa); let any other ordinate DE be drawn and the parallelogram BF be completed: if BE is ﬂuxion of AB, BF will be ﬂuxion of the area AC B.” There is here a surprisingly analytical detail: the explicit condition that the ordinate be a function of the abscissa. However, the proof is geometrical: Cunha assumes that the ordinate function is monotonic and trusts the diagram (Fig. 1; notice oblique coordinates) to convince the reader that area CD F is contained in the parallelogram with diagonal CD (not drawn): a is the ﬁrst term in the geometrical progression and m is the quotient between the difference between the two ﬁrst terms in the arithmetical progression and the difference between the two ﬁrst terms in the geometrical progression. “Represente AB a abscissa, e BC a ordenada correspondentes ao arco qualquer AC de huma curva regular AD [isto he, de huma curva, cuja ordenada he funcçaõ da abscissa]; tire-se outra qualquer ordenada DE e complete-se o parallelogramo BF:se BE for ﬂuxaõ de AB,será BF ﬂuxaõ da area AC B.” (Cunha 1790, 200). Or, at least, piecewise monotonic. In a proof included in a letter to João Manuel de Abreu, Cunha wrote: “Let the ordinates always increase or always decrease from 0to x (for all cases may be reduced to this one) […]” (“Cresçam sempre ou diminuam sempre as ordenadas desde 0 até x (pois a este caso de podem reduzir todos) […]” (Ferraz et al. 1990, 363)). 123 J. C. Domingues BF “ will be the perpendicular drawn from point C to line BE, produced if need BE BF be; let + π be the perpendicular drawn from point D to the same line AE; BE CD F then <π. AB constant and BE inﬁnitesimal would make BC constant, BE BF BF constant, π inﬁnitesimal and the area CD F inﬁnitesimal; and therefore BE BE AD E − AC B BF BC D E BF CD F constant and − (= − = <π) inﬁnitesimal. BE BE BE BE BE Therefore if BE is ﬂuxion of AB, BF will be ﬂuxion of the area AC B.” BF ( , that is, the area of parallelogram BF divided by the length of base BE,isthe BE height of parallelogram BF; Cunha assumes that the curved region CD F is contained in the parallelogram CD, so that the area of that region divided by the length of base BE = CF is less than the height π of parallelogram CD; it remains only to verify the conditions of the deﬁnition of ﬂuxion — page 9 above) A modern reader might be tempted to see in this proposition a version of the (ﬁrst) fundamental theorem of the calculus. However, as was usual in the eighteenth century, for Cunha the deﬁnite integral was not a central concept: it has already been observed that “ﬂuent” is deﬁned as an antiderivative (“every magnitude is called ﬂuent of its ﬂuxion” ). Therefore, proposition 13 is only the ﬁrst geometrical application of the ﬂuxionary calculus, equivalent to deriving the area under the graph of a function. The remaining propositions in the geometrical branch of book 15 are yet geometrical applications of the calculus: the characteristic triangle, with the tangent to the curve and the ﬂuxion of the arc; and the ﬂuxion of the volume of a solid. 5.3 The following books of Principios Mathematicos Book 16 is dedicated to trigonometry. The initial approach is geometrical, sine, tangent, etc. being deﬁned as lines. Cunha even waits thirteen pages (and 28 propositions) until he assumes the radius of the circle to be 1; thus, his versions of the basic trigonometric sin ζ cos z+cos ζ sin z formulas must take the radius in account (for instance, sin(ζ +z) = ). This is partly due to some peculiarities in the organization of the subject. Euler had givenin[1748, I, 93–107; 1755, 164–177] a purely analytical calculus of trigonometric functions, but he had done so assuming the reader to already know the basic formu- las of trigonometry (for example, si n.( y + z) = si n. y.cos.z + cos. y.si n.z (Euler 1748, I, 94)), presumably from more elementary books, that certainly used geomet- rical arguments. Cunha could have done something similar, presenting a geometrical version of elementary trigonometry in an earlier book of (1790) and then, after book BF BF “ será a perpendicular conduzida do ponto C á recta BE, produzida se necessario for; seja + π a BE BE CD F perpendicular conduzida do ponto D á mesma recta AE;será <π. AB constante e BE inﬁnitessima BE BF BF fariam BC constante, constante, π inﬁnitessima e a área CD F inﬁnitessima; e logo constante, e BE BE AD E − AC B BF BC D E BF CD F − [= − = <π] inﬁnitessimo. Logo se for BE ﬂuxaõ de AB,será BF BE BE BE BE BE ﬂuxaõ da area AC B.” (Cunha 1790, 200–201). This theorem was fundamental in the creation of the calculus, but by the middle of the eighteenth century, with the integral seen almost always as an antiderivative, it had become only a geometrical application of the calculus—for instance, in (Bézout 1767, 111–113). It is absent from (Euler 1768–1770), because this treatise does not include geometrical applications. It became fundamental again in the 19th century, when the deﬁnite integral became a fundamental concept. “Toda a grandeza se chama ﬂuente da sua ﬂuxaõ […]” (Cunha 1790, 194). 123 Geometry and analysis in Anastácio da Cunha’s calculus 15, developing the ﬂuxionary calculus of trigonometric functions in an analytical way. Instead, in his very economical style, he concentrated all of trigonometry in book 16, organizing it in a peculiar order: it practically starts with the ﬂuxion of the sine (rd sen z = dz cos z), demonstrated with a geometrical argument and invoking propo- sition 14 of book 15 (part of the geometrical branch); from there, he derives the power series for the sine and cosine, and it is from these that comes the formula for the sine of the sum of two arcs. In spite of frequent use of analytical arguments such as this, book 16 must be classiﬁed as geometrical, because the basic deﬁnitions and some fundamental arguments are geometrical. In book 17, we ﬁnd topics of elementary differential geometry of curves: multiple points, asymptotes, radius of curvature. Naturally, it is geometrical. The next three books, however, are purely analytical. In book 18, we ﬁnd several techniques of integration (such as partial fraction decomposition), and L’Hôpital’s rule, which is proven using Taylor series expansions of the numerator and of the denominator. Book 19 deals with differential equations, in a purely analytical way, including Euler’s solution for linear differential equations with constant coefﬁcients (Baroni 2001, 34–35). Book 20 deals with the calculus of ﬁnite differences. Book 21, the last one, is a case apart. It is a miscellany, probably compiled from several short manuscripts left by Cunha on diverse topics, by whoever arranged for the ﬁnal publication of (Cunha 1790). It was almost certainly not revised by Cunha. It is here that we see what is possibly the only case in (Cunha 1790) of a fundamental proposition of the calculus that, not being about a geometrical object, resorts to a geometrical reasoning; but the argument is so vague that it is not clear whether it is geometrical. The proposition in question is proposition 12: “To ﬁnd the maximum value of a given function x” . Cunha just: 1 - states that “the ﬂuxions of any two values of x that are each on each side of the maximum, will be opposite [i. e., will have opposite signs]”, , but does not explain why (the relationship between the sign of the ﬂuxion and the increasing or decreasing property of the function is not explained before), and 2 - invokes a scholium from book 17 according to which “experience has shown geometers that any variable whose values have inﬁnitesimal differences, when 1 56 passing from positive to negative becomes equal either to 0 or to ” —a version of the intermediate value property, grounded on experience, for lack of a proof. 5.4 A linguistic distinction between the two branches Let us recall Cunha’s deﬁnition of ﬂuxion: “Some magnitude having been chosen, homogeneous to an argument x,tobe called ﬂuxion of that argument, and denoted by dx; we will call ﬂuxion of d x x, and will denote by d x, the magnitude that would make constant and dx “Achar o maximo valor de huma funcçaõ proposta x” (Cunha 1790, 294). “As ﬂuxoens de quaesquer dois valores de x, que estaõ hum de huma parte, outro de outra do maximo, seraõ contrarias”. “A experiencia tem mostrado aos Geometras, que toda a variavel, entre cujos valores ha differenças inﬁnitessimas, ao passar de positiva para negativa, se acha igual, ou a 0, ou a .” (Cunha 1790, 247). 123 J. C. Domingues (x +dx )− x d x − inﬁnitesimal or zero, if dx were inﬁnitesimal and all that dx dx does not depend on dx constant.” Are Cunha’s ﬂuxions inﬁnitesimal, using this word in the sense of deﬁnition 2 (p. 17 above)? The counterfactual clause “if dx were inﬁnitesimal” suggests that dx is not. Furthemore, if ﬂuxions were inﬁnitesimal, according to deﬁnition 2 they would be variables, that is, expressions that can assume multiple values; while the deﬁnition of ﬂuxion says explicitly that a ﬂuxion is a magnitude—thus, presumably, having only one value. See the quotation in pages 14–15 above, which shows his reservations about talking of variable geometrical magnitudes. Yet, going beyond the deﬁnition and looking at the language used in propositions of book 15, we see a clear distinction in this respect between the analytical branch and the geometrical branch, and in the former ﬂuxions seem to actually be inﬁnitesimal. In fact, in the analytical branch, Cunha systematically uses phrases like “dx n−1 inﬁnitesimal and what does not depend on dx constant make […] nx constant n n n−1 (x +dx ) −x nx dx and − […] inﬁnitesimal” (prop. 2, quoted above; my emphasis), dx dx using the indicative mood, which indicates that ﬂuxions are inﬁnitesimal. On the other hand, in the geometrical branch, at least in book 15, we ﬁnd phrases such as “ AB constant and BE inﬁnitesimal would make […] area CD F inﬁnitesimal” (prop. 13, quoted above; my emphasis). In the geometrical branch of book 15, ﬂuxions are never said to actually be inﬁnitesimal. At ﬁrst hand, this may seem inconsistent. However, in the analytical branch x, dx, x, d x,…are expressions that stand for multiple magnitudes, so that they can be inﬁnitesimal in Cunha’s sense; apparently, this did not happen in the geometrical branch, perhaps because he did not see the phrase “line BE” as representing several segments. An actual inconsistency in (Cunha 1790) is that from book 16 onwards these sub- tleties of language disappear, and we ﬁnd phrases stating that geometrical magnitudes are inﬁnitesimal. For instance, in book 16: “let AD be constant and DF inﬁnitesimal; BEm 58 will be inﬁnitesimal” . Maybe all this concern with language might be difﬁcult DF to maintain, and it was enforced only in book 15. Or, perhaps, Cunha’s death in 1787 prevented him from revising the text from book 16 onwards in order to introduce subjunctives when speaking of magnitudes. Still, it seems clear that in book 15 Cunha made an effort to use language consistent with the following scheme: • Speaking of geometrical magnitudes, their ﬂuxions are magnitudes, homogeneous to them, therefore not inﬁnitesimal (although, in calculating them, one operates as if they were inﬁnitesimal); • Speaking of expressions that may represent several magnitudes (that is, variables), their ﬂuxions are naturally also variables, and are indeed inﬁnitesimal. The French translation appears to be faithful in this respect. The passages quoted in these two paragraphs n−1 are rendered as “dx inﬁnitiéme, et ce qui ne dépend pas de dx constant, rendent […] nx constante et n n n−1 (x +dx ) −x nx dx − […] inﬁnitiéme” and “ AB constante et BE inﬁnitiéme rendroient […] l’aire dx dx CD F inﬁnitiéme” (Cunha 1790, Fr. tr., 198, 203), my emphases. Of course, the translator was the same João Manuel de Abreu who wrote about the two branches. 58 BEm “Seja AD constante, e DF inﬁnitessima; será […] inﬁnitessimo” (Cunha 1790, 222). DF 123 Geometry and analysis in Anastácio da Cunha’s calculus In practical terms, this linguistic distinction is inconsequential, but it suggests that Cunha actually distinguished in his mind between the two branches, and indicates that, at a theoretical level, the geometrical branch took precedence—the deﬁnition of ﬂuxion is worded in the manner of the geometrical branch. 6 Final remarks José Anastácio da Cunha’s personal and philosophical ideas, such as his dislike of Euler and of Euler’s faith in algebra, his admiration for Newton, his preference for a geometry grounded on physical bodies, his anti-essentialism, all suggest at ﬁrst sight that his version of the calculus should be geometrical, not in line with the analytical trend of the late eighteenth century. Also, an attentive reading of Cunha’s deﬁnition of ﬂuxion brings out geometrically inclined characteristics: ﬂuxions are deﬁned as magnitudes, and dx must be homoge- neous to x (concern with homogeneity is a hallmark of geometrical thinking). And yet, we see that he developed the fundamental propositions of his version of the calculus in an analytical way, thanks to an original, nominalistic, conception of variable that allowed him to talk of functions and inﬁnitesimals as mere expressions, not, as he would put it, “beings of reason”. What Abreu called the “geometrical branch” and what has been classiﬁed as geometrical in Sect. 5.3 are really geometrical applications, not results that might instead be derived analytically. It is signiﬁcant that Cunha felt that his deﬁnition of ﬂuxion had to speak of mag- nitudes and conform with homogeneity to accommodate geometrical objects. Unlike Euler, he did not see variables as a more general kind of magnitude. But in practice, apart from a convoluted deﬁnition, his calculus was mainly analytical; certainly more analytical than the one adopted in the University of Coimbra (Bézout 1774). Actually, the conclusion must be drawn (or reinforced) that, despite his dislike of Euler, Cunha was heavily inﬂuenced by him. He was always very critical, but he managed to reconcile his anti-essentialism with the analytical ways that were gaining ground in his time. Acknowledgements The author would like to thank Antoni Malet and an anonymous reviewer for their comments on an earlier version of this paper that helped to clarify several of its passages. Funding Open access funding provided by FCT|FCCN (b-on). This research was partially ﬁnanced by Por- tuguese Funds through FCT (Fundação para a CiênciaeaTecnologia) within theProjects UIDB/00013/2020 and UIDP/00013/2020. Declarations Conﬂict of interest The author has no competing interests to declare that are relevant to the content of this article. 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