Infinite analytical procedures for the computation of logarithms in works by Benito Bails (1731–1797)

Abstract

During the Spanish eighteenth century, a process of modernization took place in scientific knowledge, partly driven by the circulation and appropriation of new scientific ideas.

In this context, the Spanish mathematician Benito Bails (1731–1797) published his course Elementos de Matemática (Elements of Mathematics) consisting of ten volumes (1779–1799), in which, among other subjects, he presented one of the most complete mathematical developments of logarithmic calculation methods of his time, by using the infinity through infinite series.

The aim of our article is to demonstrate how algebraic analytical reasoning enabled Bails to obtain new and more efficient infinite algorithms that converge more quickly in the computation of logarithms in any system. We show how Euler's number ‘e’ is calculated, probably for the first time in Spanish teaching, in an eighteenth century mathematical text.

Our analysis concludes that Bails’ course constituted an innovation and provides evidence of its creativity, originality and ingenuity.

Acknowledgements

An earlier version of this article was presented on-line in the workshop: Research in Progress (RiP), Oxford, 27 February 2021, organized by British Society for the History of Mathematics. This paper was written with the support of the project PID2020-113702RB-I00: ‘Matemáticas, Ingeniería y Patrimonio: Nuevos Retos y Prácticas, XVI-XIX’ of the Ministerio de Ciencia e Innovación of Spain. We are very grateful to Benjamin Wardhaugh, Isobel Falconer, Snezana Lawrence and the reviewer for their valuable comments for improving our article.

Disclosure statement

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

Notes

1 This was a process that Bos (2001, 10) calls the ‘degeometrization of analysis’.

2 For instance, the idea that a function could be represented as a series of powers had some influence on the work Théorie des fonctions analytiques by Joseph-Louis Lagrange (1736–1813), published in 1797. An extensive bibliography exists on the development of the concept of function and its relevance for the foundations of calculus, among which we may mention Youschkevitch (1976), Edwards (1979), Schubring (2005), Ferraro (2008), Sonar (2021).

3 For an approach to the different ways in which Bails introduced the concept of function, see Martínez-Verdú and Massa-Esteve (2021).

4 Consult Martínez-Verdú (2017) for an overview of the course.

5 The San Fernando Real Academy of Fine Arts in Madrid was created in 1752 and played an important role in the development of various arts during the second half of the 18th century. More information about the history of the Royal Academy of San Fernando, about the types of studies and the students and architects can be found in Quintana (1983), Bédat (1989), León and Sanz (1994).

6 There are historical studies describing science and technology in Spain during the Enlightenment: see Garma (2002), Peset (2002), Lafuente and Valverde (2003). On science policy and the process of militarization of Spanish science in the 18th century, see Lafuente and Peset (1985), Capel et al. (1988).

7 Translations by the authors of (Bails 1787, Prólogo, iv): ‘¿Podemos temer que se nos pregunte, á que viene la aplicación de las series, de que sirven tantos cálculos para buscar los logaritmos de los números, y las lineas trigonométricas, así naturales como artificiales, una vez que mas de cien años há tenemos calculadas de unos y otras tablas tan socorridas para quanto pueda ofrecerse? Al que nos hiciere esta pregunta le diríamos […] nada está por demas, como se ciña dentro de los límites regulares su autor, en obra alguna cuyo asunto sea la instrucción de nuestros hombres’.

8 Bails (1788, Principios de Matemática, vol 1, prólogo, ii): ‘[…] de las mejoras hechas á la Aritmética, la principal es la declaración de la Teórica y Práctica de los logaritmos. Es esta una doctrina de mucha importancia por transcendental á todos los ramos de la Matemática; y sus fundamentos bien declarados facilitan no poco la inteligencia de remontadas investigaciones en que se han exercitado Escritores muy afamados de este siglo’. Note: The Principios de Matemática (Principles of Mathematics) is Bails' other important mathematical work, a compendium of the Elements, as will be seen in the following section.

9 An extensive literature exists on the life and work of Bails, among which we highlight the Obituary Notice (Noticia necrológica) issued by the Real Academia de San Fernando, (RABASF 1799), and the studies by Bédat (1968), Garma (1983), León and Sanz (1994), Arias de Saavedra (2002), as well as the review published by García Camarero in the Electronic Biographical Dictionary of the Royal Academy of History (Diccionario Biográfico Electrónico de la Real Academia de la Historia: https://dbe.rah.es/biografias/12643/benito-bails).

10 Plan del Curso de Arquitectura civil para la instrucción de los discípulos de la Real Academia de San Fernando. RABASF Archive (1759, sig. 3–121: 60r). Minutes of the Private Meeting of February 25, 1759). In the first paragraph, the Course Plan is mentioned: «[…] propose the Architectural Course Plan to be worked on / […] proponen el Plan del Curso de Arquitectura que han de trabajar».

11 RABASF Archive (1768, sig. 3–121: 344r-344v). Minutes of the Private Meeting of September 19, 1768): ‘And in attention to the urgent need to form the course of Architecture having reiterated Mr. Dn. Jorge Juan the good reports that he has given about the sufficiency of Dn. Benito Bails, to write it, […] by the plan, that the Academy approved on February 25, 1759’ / ‘Y en atención à la urgente necesidad de formar el curso de Arquitectura haviendo reiterado el Sr. Dn. Jorge Juan los buenos informes que tiene dados acerca de la suficiencia de Dn. Benito Bails, para escrivirlo, […] por el plan, que aprobó la Academia en 25 de febrero de 1759’.

12 Jorge Juan Santacilia (1713–1773) was marine, scientific, diplomat, and academician, member of the Royal Society of London, of the Academies of Sciences of Paris, Berlin and Stockholm, and an academician of the Royal Academy of San Fernando.

13 For a detailed account of Bails’ difficulties with the Inquisition, consult Bédat (1968, 32–50).

14 A more comprehensive bibliographical account of Bails’writings and their editions can be found in Aguilar Piñal (1981, 491–495), detailing the libraries, public and religious centers, seminaries and economic societies where copies of Bails’ work can be found.

15 RABASF Archive (1770, sig. 3–122: 2v-5r). Minutes of the Private Meeting of January 14, 1770.

16 This text was followed by others that were widely distributed and had a significant influence in Europe, such as: Cursus seu mundus mathematicus (1674) by Claude François Milliet Dechales (1621–1678); Cours de mathématique, qui comprend toutes les parties de cette science les plus utiles et les plus nécessaires à un homme de guerre, et a tous ceux qui se veulent perfectionner dans les mathématiques (1693) by Jacques Ozanam (1640–1718); Elementa Matheseos Universae (1713–1715) composed by Christian Wolff (1679–1754); or Nouveau Cours de Mathématique, à l’usage de l’artillerie et du génie où l’on applique les Parties les plus utiles de cette science à la Théorie & à la Pratique des différents sujets qui peuvent avoir rapport à la Guerre (1725) by Bernard Forest de Bélidor (1698–1761). In Spain, we have the Compendio Mathematico (1707–1715) by Tomás Vicente Tosca (1651–1723) and the Curso Mathematico para la Instrucción de los Militares (1739–1744) by Pedro de Lucuce (1692–1779). For more information about mathematical courses as encyclopaedic texts that collected the knowledge of the new science, as well as about the meaning of pure, mixed or Physical-mathematical in these courses and Lucuce’s course, the reader is referred to Massa-Esteve et al. (2011, 235–238). An extensive study of Hérigone and his work Cursus mathematicus (Paris, 1634/1637/1642) can be found in (Mellado Romero 2022).

17 Minutes of the Private Meeting of January 17, 1759 (Archivo de la Real Academia de San Fernando (hereinafter RABASF Archive), sig. 3–121: 52r–52v).

18 In the Foreword (Prólogo General) to his work, included in the first volume of the Elements, Bails criticizes different European mathematical courses: The French, German, Dutch, Italian, English, and the Spanish (Bails 1779 [1772], i-xvi). When in the article we refer to a volume, in addition to the date of publication, the date of printing will be added in square brackets.

19 (Bails 1779 [1772], vol 1, ii): ‘para el tiempo presente, por razón de los muchos adelantamientos que ha hecho la Matemática […] es sin duda incompleta y diminuta […] porque no trata ni del cálculo diferencial ni del integral, y así debía ser, una vez que es tan poco lo que trahe de Algebra, y omite la teórica de las curvas, doctrina muy necesaria para las investigaciones peculiares á la analisis superior […]’

20 (Bails 1779 [1772], vol 1, iii-iv): ‘[…] es tambien notorio que despues de publicada su obra [Elementa Matheseos Universæ] se han dado á luz otras muchas, que han perfeccionado inmensamente todos los ramos de la Matemática, ya se atienda á los inventos de sus autores, ya á su singular destreza en proponer con mejor método los asuntos. Euler, Cramer, Ricati, Stirling, &c. han promovido muchísimo, despues que Wolfio escribió, la teórica de las series; todos los tratados que tenemos hoy dia de cálculo integral son posteriores á la publicacion de su Curso; y la Dinámica, Hidrodinámica, Óptica y Astronomía han adelantado infinito con las investigaciones de Juan y Daniel Bernouli, de M. d'Alembert, Euler, Bouguer, Clairaut, Micheloti, La Lande, Bossut, Lambert, Halley, La Caille, &c. siendo cierto que desde que salió á luz la obra de Wolfio han mudado enteramente de semblante las Matemáticas. Estas obras modernas, de donde hemos sacado todo lo que incluye la nuestra, la grangean un grado de estimacion superior al que merece la del Escritor Aleman, por la mucha ventaja que los materiales de que nos hemos valido llevan á los que él tuvo á mano’.

21 For more information about the Academy's recommendations for the mathematics course, see Martínez-Verdú (2022).

22 Where it says: ‘Volume VI. on the 15th January 74’, should read: ‘Volume VI. on the 15th January 75’. This error was corrected by Bails in the second edition (Bails, 1793, vol 1, Prólogo, xviii, footnote (1)).

23 Volume X was printed in 1776 and published in 1787, but Bails explained Cagnoli's new and more convergent methods (Cagnoli 1786) in 1787, so our reference to volume X will always be to 1787: (Bails 1787).

24 Volumes seven and eight were printed in 1775, but were not published during Bails' lifetime; they are therefore posthumous in 1799. However, the Royal Academy decided that the title page should bear the printing year of 1775 and not that of their publication in 1799, see acta de la junta particular de 7 de octubre de 1798, sig. 3–125: 116r) and Figure 2.

25 At the same time, Bails extracted the printed volumes resulting in the Principles, or Curso pequeño, as Bails confirms: 'Because this work, as we have just insinuated, is an extract from a larger one' / (Bails 1776, Principles, vol 1, Prologue, 2): ‘Por ser esta obra, según acabamos de insinuar, estracto de otra mayor’. Principles was published in 1776; in this way, Bails' ideas and the content of the course had another means of dissemination, not only within the Academy, but also outside it, as the bound volumes of the Principles could be purchased by individuals, educational institutions or private academies at the Royal Academy headquarters where they were sold.

26 In this direction, for more information, see Sambricio (1986), León and Sanz (1994).

27 The Elements had different editions in Spain, but never as a complete work, but as separate volumes. Volume I was republished in 1793 in a second edition, corrected and enlarged. Later, in 1801, the Royal Academy commissioned the Academy's director of mathematics, Antonio Varas (?-1847), to prepare a book that would bring together Arithmetic and Geometry, with the purpose of teaching it to the disciples of the Practical Geometry class. In order to create this volume, Varas used the first volume of Bails' Elements and chose the subjects that would compose the new volume that the Academy wanted, [Real Academia de San Fernando (Antonio Varas), 1801, Prólogo, 1–2]. The volume compiled by Varas from volume I of the Elements was published under the title Aritmética y Geometría Práctica de la Real Academia de San Fernando, Madrid: En la imprenta de la Viuda de Ibarra, 1801. Let us look at other reprints of the Elements. With regard to Volume IX, Part I, De la Arquitectura Civil, published in 1783, the Royal Academy produced a second edition, corrected by Bails, in 1796. And Volume X, Tabla de Logaritmos, saw a second reprint in 1804.

28 The priest and scientist José Celestino Bruno Mutis y Bosio (1732–1808) was appointed 'Perpetual Professor of Mathematics' at the Colegio de Nuestra Señora del Rosario in Bogotá. Mutis argued that although Wolff’s course had until then been the most complete one and the model for those that followed, with subsequent scientific and technical advances it had become outdated and it was necessary to replace it with those of Bails, which according to Mutis were 'the most excellent works' in Europe (Gredilla 2009, 36). The introduction of Bails in Colombia is extensively discussed in Arboleda (2020).

29 Del Cero, del Infinito, y de las Cantidades Imaginarias, Bails (1779 [1772], vol 2, 182–197). The part dealing with the notions and operations of zero and infinity goes from page 182 to page 191, and consists of 12 articles or paragraphs enumerated from 177 to 188; the remaining of pages contain the presentation and explanation of Imaginary Quantities.

30 Bails (1779 [1772], vol 2, viii): ‘ … no es este, según se hará patente, una cantidad que no puede ser mayor; pues ninguna hay, por grande que la supongamos, que no admita incremento … una cantidad que de puro grande no admitiese aumento alguno, sería un infinito metafísico, un ente de razón en las cosas criadas, un absurdo que argüiría de limitada la Omnipotencia del Criador’.

31 Bails (1779 [1772], vol 2, 187): ‘ … se echa de ver que la idea del infinito no es mas que una noción abstracta’.

32 ‘The more terms we take from this series, the greater sum they will compose; and that we can take so many terms, that their sum will be equal to as great a number as we wish. This is the idea that we must form of the infinite’. Bails (1779 [1772], vol 2, 189): ‘quantos mas términos tomásemos de dicha serie, tanto mayor suma compondrá; y que podremos tomar tantos términos, que su suma podrá ser igual á un número tan grande como se quisiere. Esta es la idea que nos hemos de formar del infinito’.

33 'the infinity, as Mathematics considers it, is really the limit of the finite'. Bails (1779 [1772], vol 2, 188): ‘ … el infinito, conforme le considera la Matemática, es en la realidad el límite del finito’.

34 Bails (1779 [1772], vol 2, xvi): ‘Hemos alargado la doctrina de las series con manifestar su aplicacion al cálculo de los logaritmos, y de las lineas trigonométricas’.

35 Bails (1779 [1772], vol 2, xvi).

36 Tabla de Logaritmos de todos los números naturales desde 1 hasta 20000; y de los logaritmos de los senos, tangentes de todos los grados y minutos del quadrante del círculo (Bails 1787).

37 Orígenes de los logaritmos (Bails 1787, 37–188).

38 Doctrina de los logaritmos por Aritmética; Origen y doctrina de los logaritmos por la curva logarítmica; Doctrina de los logaritmos por la logarítmica y la hipérbola (Bails 1787, 37–84; 84–119; 119–124).

39 Aplicación del Análisis a la doctrina de los logaritmos (Bails 1787, 124–147).

40 Étienne Bézout (1730–1783), French mathematician, was elected as a member of the Académie des Sciences and appointed instructor of the Marine Royale as well as examiner for the Corps de l’Artillerie. He wrote several textbooks designed for teaching mathematics to his students, such as the Cours de Mathématiques à l'usage des Gardes du Pavillon et de la Marine, published in Paris between 1764 and 1769.

41 The names of the sections are the following (in parenthesis we show the corresponding Bézout titles and after each slash we translate them into English): De los Logaritmos [including the table of logarithms from 1 to 200] (Des Logarithmes, Table des Logarithmes des Nombres natureles depuis 1 jusqu’a 200) / On Logarithms; Propiedades de los logaritmos (Propriétés des Logarithmes) / Properties of logarithms; Usos de los Logaritmos (Usages des Logarithmes) / Usages of logarithms; De los Números cuyos Logaritmos no se hallan en las tablas (Des Nombres dont les Logarithmes ne se trouvent point dans les Tables) / About numbers whose logarithms do not appear in the tables; De los Logaritmos cuyos números no se hallan en la tablas (Des Logarithmes dont les Nombres ne se trouvent point dans les Tables) / About logarithms whose number do not appear in the tables; Del Complemento aritmético (Remarque [it includes Complément artihmétique]) / About the arithmetic complement. (Bails, vol 1, 1779 [1772], 166–195; Bézout, vol 1, 1764, 212–247).

42 Odoardo Gherli (1730–1780), Italian mathematician and Dominican religious, was a professor of theology at the University of Modena, and of mathematics at the University of Pisa. The doctrine of logarithms by arithmetics according to Gherli can be found in Gherli (1770, 195–234).

43 (Bails 1779 [1772], vol 1, General Prologue, xi-xii): ‘[El Curso Matemático es obra] del P. Gherli y la mejor sin duda alguna que hemos visto hasta el dia de hoy entre tantas como hemos registrado. Todos los asuntos que incluye los toma el P. Gherli desde sus primeros fundamentos, y los propone con tan feliz explicacion, que dudamos se hayan publicado hasta ahora Elementos de Matemática que tanto puedan honrar á un Escritor. […] no hay Curso alguno donde los que desearen hacer progresos sólidos y rápidos en la Matemática, puedan aprovechar tanto como en los Elementos del P. Gherli’.

44 Bails, contrary to Gherli, does not label his columns with (A, B, C, D, H). Notice also that in the last cell of Bails’ table there is a mistake, the fraction $9{\displaystyle \frac{2}{3}}$ must be replaces by $10{\displaystyle \frac{2}{3}}$, as in Gherli’s table.

45 John Keill, Scottish mathematician, studied at the University of Edinburgh and was a student of David Gregory (1659–1708).

46 The text of De Natura et Arithmetica Logarithmorum was inserted in the final pages (551–581) of another Keill’s work, Introductiones ad veram Physicam et veram Astronomiam (Keill 1725).

47 (Bails 1787, Prólogo, iii): ‘La doctrina de los logaritmos por la curva logarítmica es un tratado apreciable de Keil, dignamente celebrado de todos los Matemáticos, que, por incidencia, ó de propósito han tratado esta materia’.

48 Bails refers to Euler’s work Introductio in analysin infinitorum (1748), see his footnote in Bails (1779 [1772], vol 2, prólogo, xv).

49 (Bails 1779 [1772], vol 2, prólogo, xv): ‘nos engolfamos en la doctrina de las series, dándolas á conocer con las mismas expresiones que usa Leonardo Euler en una obra muy profunda y original, porque aun los asuntos que no son nuevos los trata con maravillosa novedad, y su estudio tendrá mucha conveniencia á los que desearen adelantar en la ciencia del Analisis’.

50 An excellent literature exists for a better understanding of the importance of the role of power series in the development of the analysis, among which we highlight Knobloch (2006), Ferraro (2008), Sonar (2021).

51 L’Abbé Marie studied theology at the Sorbonne and in 1762 succeeded L’Abbé Nicolas Louis de La Caille (1713–1762) as professor of mathematics at the Mazarin college. For more biographical information about Marie, consult Michaud (1860, 648–649).

52 Antonio Cagnoli was an Italian astronomer and mathematician. He published Trigonometria piana e sferica in Paris in 1776. In the same year, and under Cagnoli's supervision, his Trigonometria was translated from Italian into French by Nicolas Maurice Chompré (1750–1825) as the Traité de Trigonométrie Rectiligne et Sphérique. This French version was the one used by Bails (Bails 1787, Prólogo, iii) to compose the analytical part of the logarithms in volume ten of the Elements. Cagnoli promoted the introduction of scientific ideas of the Enlightenment into the Italian society. The asteroid 11112 discovered in 1995 bears his name. For more information about the Cagnoli’s biography and work, the reader is referred to Labus (1818), Tipaldo (1840).

53 Among the mathematical works by L’Abbé Marie, the main one is the enlargement in successive editions of the de La Caille’s work Leçons élémentaires de Mathématiques, in such a way that they are now referenced as de La Caille-Marie. In 1768, L'Abbé Marie reprinted (third edition) the logarithmic tables by de La Caille and Joseph Jérôme Le Français de Lalande (1732–1807) (Tables de logarithmes: pour les sinus & tangentes de toutes les minutes du quart de cercle, & pour all natural names after 1 jusqu'à 20000; avec une exposition abrégée de l'usage de ces tables, Paris: Chez Desaint, 1768). In turn, La Caille and Lalande used those published in 1740 by the French mathematician Antoine Deparcieux (1703–1768), which he included in his work Nouveau traité de trigonométrie. According to Bails (1787, Prólogo, v), the tables of logarithms edited by L'Abbé Marie were finally revised and corrected by Cagnoli, see (Cagnoli 1786). Cagnoli's revision served Bails for the correction of the errors in the Table of Logarithms in volume ten of the Elements, since it was a copy of that by L'Abbé Marie.

54 See de La Caille-Marie (1770, 201–204).

55 Bails schematizes it in the same way as L'Abbé Marie (La Caille 1770, 201).

85 It should be noted that Bails, unlike L'Abbé Marie, uses the horizontal bar or vinculum [see, $m.\overline{m-1}$] as a sign of aggregation of terms in addition to the parenthesis. Among the signs used to express aggregation of terms, 'parentheses prevailed for typographical reasons' and have ‘won their place in competition with the horizontal bar or vinculum’ and other signs used (Cajori, 1993, 384–385).

56 We follow L’Abbé Marie-Bails’ notation and extend the series until the fourth term in order to include the coefficient D.

57 (Bails 1779 [1772], vol 2, 380): ‘un mismo número (1+x) puede tener una infinidad de logaritmos diferentes’. In L'Abbé Marie’s words: ‘que par consequent le même nombre 1+x peut avoir une infinité de Logarithmes différens' (La Caille 1770, 201).

58 (Bails 1779 [1772], vol 2, 381). At this point, L’Abbé Marie comments on the Napier’s work Mirifici Logarithmorum canonis Descriptio (1614) [Johannes Neper (1550–1617)], which Bails omits.

59 Note that $L\left(a+z\right)=L\left(a\left(1+{\displaystyle \frac{z}{a}}\right)\right)=\mathrm{La}+L\left(1+{\displaystyle \frac{z}{a}}\right)$ and $L\left(a-z\right)=\mathrm{La}+L\left(1-{\displaystyle \frac{z}{a}}\right).$

60 At this point, L’Abbé Marie recognizes that ‘it is easy, therefore, to find by this method, (the same in essence as that of Mr. Halley, but demonstrated in a much clearer way) the Logarithms of the first numbers'. / ‘Il est donc aisé de trouver par cette méthode, (la même quant au fonds que celle de M. Halley, mais démontrée d’une maniere bien plus claire) les Logarithmes des nombres premiers’ (La Caille 1770, 203). It should be emphasized that Edmund Halley (1656–1742) also made an important contribution to the subject with his article A most compendious and facile Method for Constructing of Logarithms, exemplified and demonstrated from the Nature of Numbers, without any regard to the Hyperbola, with a speedy Method for finding the Number from the Logarithm given (Halley, 1695, 58–67), where, in addition, Halley mentions the mathematicians James Gregory (1638–1675), Nicolaus Mercator (1620–1687) and John Wallis (1616–1703).

61 About the inverse methods of series, consult Edwards (1979), Jahnke (2003), Stillwell (2010), Sonar (2021). In 1698, Abraham de Moivre (1667–1754) gave an explicit recursive formula for the reversion of the series (Jahnke 2003, 120). In the article by De Moivre (1698, 191), we can read that ‘This Theorem may be applied to what is called the Reversion of Series, such as finding the Number from its Logarithm given’. According to Stillwell, ‘De Moivre (1698) gave a formula for inverting series’ (Stillwell 2010, 169).

62 Del Método inverso de las Series (Bails 1779 [1772], vol 2, 329–332).

63 L’Abbé Marie (La Caille 1770, 196) refers to the text by William Emerson (1701–1782), A Treatise of Algebra in Two Books, London: J. Nourse, 1764, in which the inversion of the series and its application to logarithms is dealt with. For example, Problem LXXXV. A logarithm being given; to find the quantity belonging to it, or its number, by a series. Or to turn logarithms into numbers (Emerson 1764, 247–250).

64 (Bails 1779 [1772], vol 2, 384).

65 The reader interested in the calculations required to obtain the coefficients can proceed in a similar way to that we have shown in the question of, given a number, finding its logarithm.

66 For more information on how the e number was introduced in eighteenth-century Spain, see Navarro-Loidi (2008, 154).

67 (La Caille 1770, 204).

68 (La Caille 1770, 204): ‘Ce nombre sert très-souvent dans le Calcul intégral. Le voilà calculé d’avance’.

69 (Bails 1779 [1772], vol 2, 204): ‘Este número es de muchísimo uso en el ramo mas dificultoso del cálculo infinitesimal’.

70 Bibliographic cite of Bails: 'c) Traité de Trígonométrie rectiligne et sphérique. Contenant des métodes et des formules nouvelles, avec des applications à la plupart des problémes de l'Astronomie. Par Mr. Cagnoli, Citoyen de Vérone, membre de la Sóciété Italienne. Traduit de l’Italien par Mr. Chompré. Paris 1786, un tomo en 4'. (Bails 1787, prólogo, v).

71 (Bails 1787, Prólogo, iii-iv): ‘Todo lo demas que le añadimos está sacado de la Trigonometría de Cañoli, del diestro Cañoli, tan justamente ponderada de todos los Diaristas, que la han dado á conocer, y que por mas que la elogien nunca podrán excederse en alabarla’.

72 (Cagnoli 1786, 84): ‘Étant donné un nombre, trouver son logarithme’ (Cagnoli 1786, 79); and ‘Étant donné un logarithme, trouver le nombre auquel il correspond’.

73 Cagnoli asserts: ‘La solution qui suit est en partie tirée des Éléments de Mathématiques de M. l’Abbé Marie’ (Cagnoli 1786, 79) / ‘The solution that I follow is partially taken from M. l’Abbé Marie’s Éléments de Mathématiques’.

74 See Cagnoli (1786, 82): ‘Mais nous allons donner des formules nouvelles, beaucoup plus convergentes, et qui pourront avec avantage remplacer toutes les précédentes’. / ‘But we are going to give new formulas, much more convergent, and which can advantageously replace all the previous ones’.

75 (Bails 1787, 133): ‘vamos a dar fórmulas nuevas, mucho mas convergentes, las quales con mucha ventaja pueden suplir por todas las que hemos sacado hasta aquí’.

76 (Bails 1787, 132): ‘el punto está en señalar el aumento que entonces compete á su logaritmo’.

77 Cagnoli (1786, 82) poses the question as follows: ‘Si un nombre quelconque n reçoit un accroissement ϑn, on demande quelle est l'augmentation correspondante du log. de n’/ ‘If an arbitrary number n receives an increment θn, it is asked what is the corresponding increase in log.n’. Here, Δlog. n is meant the difference $\log \left(n+\mathrm{\Delta n}\right)-\mathrm{logn}$.

78 (V.) in Bails’ text (Bails 1787, 133), (E) for Cagnoli (1786, 82).

79 It suffices to recall the formulas for $\log \left(1+x\right)$ and $\log \left(1-x\right)$ already mentioned in our study, and to notice that $\log \left({\displaystyle \frac{1+x}{1-x}}\right)=\log \left(1+x\right)-\log \left(1-x\right).$

80 See Cagnoli (1786, 83): ‘Cette série est, sans comparaison, plus convergente que la série (D) pour calculer la difference d'un logarithme connu à un autre plus grand’ / ‘This series is, without comparison, more convergent than series (D) to calculate the difference of a known logarithm to another larger’.

81 ‘Esta serie por la qual se calcula la diferencia que va de un logaritmo conocido a otro mayor, es mas convergente que la serie (IV)’ / This series, from which the difference that goes from a known logarithm to another greater is calculated, is more convergent than the series (IV) (Bails 1787, 133).

82 This Table 2 does not belong to Bails, but has been drawn up by the authors in order to show the effectiveness of the new algorithm (VI).

83 (Bails 1787, 135): ‘Con calcular solos quatro términos de esta serie se halla la cantidad que se ha de añadir al logaritmo de 4 para sacar el logaritmo hiperbólico de 5 con siete decimales cabales’.

84 (Bails 1787, 138).

Additional information

Funding

This work was supported by the project ‘Matemáticas, Ingeniería y Patrimonio: Nuevos Retos y Prácticas, XVI-XIX' of the Ministerio de Ciencia e Innovación of Spain: [Grant Number PID2020-113702RB-I00].