Mengoli's mathematical ideas in Leibniz's excerpts

Abstract

Dedicated to the memory of Jacqueline Stedall

In the seventeenth century many changes occurred in the practice of mathematics. An essential change was the establishment of a symbolic language, so that the new language of symbols and techniques could be used to obtain new results. Pietro Mengoli (1626/7–86), a pupil of Cavalieri, considered the use of symbolic language and algebraic procedures essential for solving all kinds of problems. Following the algebraic research of Viète, Mengoli constructed a geometry of species, Geometriae Speciosae Elementa (1659), which allowed him to use algebra in geometry in complementary ways to solve quadrature problems, and later to compute the quadrature of the circle in his Circolo (1672). In a letter to Oldenburg as early as 1673, Gottfried Wilhelm Leibniz (1646–1716) expressed an interest in Mengoli's works, and again later in 1676, when he wrote some excerpts from Mengoli's Circolo. The aim of this paper is to show how in these excerpts Leibniz dealt with Mengoli's ideas as well as to provide new insights into Leibniz's mathematical interpretations and comments.

Acknowledgements

I am grateful to Eberhard Knobloch, Siegmund Probst, Antoni Roca-Rosell and Monica Blanco each of whom read earlier version of this article and made some remarks concerning content and language. This research is included in the project HAR2013-44643-R of the Ministerio de Economía y Competitividad.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ If we observe Viète's equation, we can appreciate the rhetorical form. We provide one example to show how Viète writes an equation: ‘B in A − Aquad. Aequatur Zquad’, which in modern notation would be written Bx − x² = Z².

² Viete's algebra was spread by some authors, thereby facilitating the algebraicization of mathematics. One example is Thomas Harriot (c. 1560–1621) in his mathematical writings. On Harriot's algebraic works we can see Stedall's seminal researches in Stedall (2003, 2007). Another example is the encyclopaedic work by Pierre Hérigone (1580–1643), Cursus mathematicus, Paris (1634, 1637, 1642), which consists of six volumes, one of which is on algebra. On Hérigone's work we can see a comparative analysis between Viète's and Hérigone's algebra in Massa-Esteve (2008), the treatment of Euclid's Elements in Hérigone's work in Massa-Esteve (2010) and the influence of Viète's work on Hérigone's work and from that to Mengoli's work in Massa-Esteve (2012).

³ Pierre de Fermat (1607–65) was among the mathematicians who used algebraic analysis to solve geometric problems. He did not publish any of his work during his lifetime, although it circulated in the form of letters and manuscripts and was referred to in other publications. On Fermat's works see Fermat (1891–1922, 65–71 and 286–292) and Mahoney (1973, 229–232). The most prominent figure in this process of algebraicization was René Descartes (1596–1650), who published La géométrie in 1637. There are many excellent useful studies on Descartes, including Giusti (1987, 409–432), Mancosu (1996, 62–84), and Bos (1981 and 2001).

⁴ For more biographical information on Mengoli, see Natucci 1970–91; Massa-Esteve 1998, 2006b; and Baroncini and Cavazza 1986.

⁵ Although he published nothing between 1660 and 1670, the latter year saw the appearance of three works: Refrattioni e parallase solare (1670), Speculationi di musica (1670), and Circolo (1672). These reflected Mengoli's new aim of pursuing research not on pure but on mixed mathematics, such as astronomy, chronology, and music. Furthermore, his research was clearly in defence of the Catholic faith. Mengoli went on writing in this line, publishing Anno (1675) and Mese (1681) on the subject of cosmology and Biblical chronology, and Arithmetica rationalis (1674) and Arithmetica realis (1675) on logic and metaphysics.

⁶ ‘Si tamen idem et Mengolus praestitit, non miror; saepe enim concurrere solent diverse’ (Oldenburg 1986, vol IX, 648). According to Probst (2015), Leibniz probably did not get access to the Novae quadraturae of Mengoli during his stay in Paris. However when he visited London a second time in October 1676 he made excerpts from the correspondence between James Gregory and John Collins. In the sections copied by Leibniz, there is also a passage on Mengoli's proof of the divergence of the harmonic series, characterized by Leibniz in a marginal note as ‘ingeniose’ (A III, 1 N 88_2 p 486f).

⁷ The Geometriae Speciosae Elementa (1659) consists of an introduction entitled Lectori elementario, which provides an overview of the six individually titled chapters or Elementa that follow. In the first Elementum, De potestatibus, à radice binomia, et residua (1659, 1–19), Mengoli shows the first 10 powers of a binomial given with letters for both addition and subtraction, and explains that it is possible to extend his result to higher powers. The second, De innumerabilibus numerosis progressionibus (1659, 20–94), contains calculations of numerous summations of powers and products of powers in Mengoli's own notation, as well as demonstrations of some identities. In the third, De quasi proportionibus (1659, 95–147), he defines the ratios ‘quasi zero’, ‘quasi infinity’, ‘quasi equality’ and ‘quasi a number’. With these definitions, he constructs a theory of quasi proportions on the basis of the theory of proportions found in the fifth book of Euclid's Elements. The fourth Elementum, De rationibus logarithmicis (1659, 148–200), provides a complete theory of logarithmical proportions. He constructed a theory of proportions between the ratios in the same manner as Euclid with the magnitudes in the fifth book of Elements. From this new theory in the fifth Elementum, De propriis rationum logarithmis (1659, 201–347) he found a method of calculation of the logarithm of a ratio and deduced many useful properties of the ratios and their powers. Finally, in the sixth Elementum, De innumerabilibus quadraturis (1659, 348–392) he calculates the quadrature of figures determined by the ordinates now represented by y = K · x^m · (t − x)ⁿ. A detailed analysis of this work can be found in Massa-Esteve (1998).

⁸ ‘Cercai, fino da giovinetto, il Problema Della quadratura del Circolo, il più desiderato di tutti nella Geometria’ (Mengoli 1672, 1).

⁹ In the opening pages of the Circolo, Mengoli explains that he had found this result, the quadrature of the circle, in 1660, but had not published it because, according to him, he only wanted to publish the mathematics he needed to explain natural events (Mengoli 1672, 1).

¹⁰ The word abscissa had appeared in 1646 in Fermat's works (1891–1922, 195), in Torricelli's work (1919, III, 366), in Cavalieri's work in 1647 (1966, 858–859), and in Degli Angeli (1659, 175–179). Other authors also used the word ‘diameter’ with the same meaning.

¹¹ Mengoli used the word ‘ordinata’ instead of the word ‘applicata’, which was commonly in use at that time. Descartes defined the ordinates as ‘celles qui s'appliquent par ordre’ (Descartes 1954, 67). Smith in this note explains: ‘The equivalent of “ordinate application” was used in the fifteenth-century by translating Apollonius’. The note also cites that the Mathematical dictionnary of Hutton (1796) gave ‘applicata’ as the word corresponding to the ordinate, and explained that the expression ‘ordinata applicata’ was also used. In fact, Fermat and Cavalieri used ‘applicata’. Mengoli in Circolo (1672) named them ‘ordinatamente applicate’ (Mengoli 1672, 5).

¹² ‘10. Super basi describatur quadratum: & ab uno quolibet puncto in basi sumpto, recta ducatur, usque ad oppositum latus, reliquis lateribus quadrati parallela: quae dicetur, Ordinata in quadrato.’ (Mengoli 1659, 368)

¹³ ‘Unaquaelibet ordinata, est abscissa secunda’ (Mengoli 1659, 369).

¹⁴ The word forma dates from the previous century and was identified by measuring the intensity of a given quality. (Clagett 1968, 91–92; Crombie 1980, 82–95).

¹⁵ ‘23. Et generaliter, si super basi concipiatur figura, extensa non nisi per ordinatas in quadrato: & in qua, unaquaelibet ordinata, est assumpta quaedam in tabula proportionalium: dicetur, Forma omnes tales proportionales aptoque significabitur charactere. vt Forma omnes abscissae tertiae, FO.a³: Forma omnes biprimae, FO.a²r: Forma omnes unisecundae, FO.ar²: Forma omnes residuae tertiae, FO.r³. & sic deinceps’ (Mengoli 1659, 369).

¹⁶ ‘Formae propositae, in data basi, per datum punctum, ordinatam invenire’ (Mengoli 1659, 377).

¹⁷ ‘Esto proposita FO.10 a²r³, super data basi AR, in qua datum punctum B. Oportet per B ordinatam invenire’ (Mengoli 1659, 377).

¹⁸ Throughout the book, Mengoli presented Theorems and Problems. In this case, he wrote the word Construction before the demonstration and explained the construction used in it, as Euclid did in Mengoli's source Hérigone (Massa-Esteve 2010).

¹⁹ ‘Data AR, datisque AB, BR, inveniatur recta BC, ad quàm AR, rationem habet compositam ex datis rationibus, AR ad AB duplicata, AR ad BR triplicata, & ex ratione subdecupla:& collocetur BC perpendiculariter ad AR. Dico BC, esse ordinatam per B, in FO.10 a²r³’ (Mengoli 1659, 377).

²⁰ ‘Ratio AR ad BC, componitur ex rationibus AR ad AB duplicata, AR ad BR triplicata, & ex subdecupla: sed AR, est u; AB est a; BR est r: Ergo AR ad BC ratio, componitur ex rationibus u ad a duplicata, u ad r triplicata, & ex subdecupla: sed ex ijsdem componitur u ad 10 a²r³: ergo AR ad BC est ut u ad 10 a²r³: sed AR est u: ergo BC est 10 a²r³: ergo BC est ordinata per B, in FO. 10 a²r³.Quod&c’ (Mengoli 1659, 378).

²¹ The combinatorial triangle is known in history as Pascal's triangle, because Blaise Pascal (1623–62) explained and proved the properties in a very clear style (Bosmans 1924, 25–36; Pascal 1954, 91–107; Edwards 2002, 57–86). Mengoli may not have known about Pascal's treatise since it was published in 1665, but he knew its source very well, which was Hérigone's work. On Mengoli's triangular tables see Massa (1997).

²² The harmonic triangle, also called Leibniz's triangle, is formed by the reciprocal of the elements (binomial coefficients) of the binomial triangle times their own numbers. Its definition is related to the successive differences of the harmonic sequence. See Edwards 2002, 106.

²³ ‘15. Le quali tutte hò dimostrato, che sonoproportionali, come le quantità disposte nella terza tavola triangulare; ed è il quadrato Della Rationale FO.u, homologa all'unità; e i triangoli FO.a, eFO.r, homologhi allà metà; e le FO.a², FO.ar, FO.r², homologhe alle parti quarta, duodecima, duodecima, e quarta, dell'unità, e dello stesso quadrato; e cosí tutte le altre forme per ordine: como ivi en el sesto elemento si può dedurrè per corollario dalla prop. 10’ (Mengoli 1672, 6).

²⁴ ‘Hoc schediasma collocavi apud Excerpta ex Mengoli Circulo’ (Leibniz A VII, 3 No 57_1). This was later removed to another place within the collection of Leibniz's manuscripts (at an unknown date before the end of the nineteenth century).

²⁵ Leibniz may not have read Mengoli's system of coordinates, because in Circolo Mengoli did not include any drawing of geometric figures.

²⁶ Sit trilineum ARDCA duabus rectis AR.RD et curva ACD. Inclusum. Recta AR tota sit t.Abscissa AB sit a. Residua BR sit r. Ordinata BC sit y (Leibniz 1676, AVII 3, 57_ 2, 2003, 737).

²⁷ Si ponamus iam esse y Π t. ut si pro figura ARDCA erit quadratum ARDE. Si y sit Π a vel r, seu si ordinata BF sit Π AB. pro figura ARDCA, erit triangulum a semiquadratum ARD. Sin y Πa²/t vel posita tΠ 1, si y Πa² Π BG, pro figura ARDCA, erit ARDGA trilineum parabolicum, et ita de caeteris altioribus, ut si y sit a³, vel a⁴ (Leibniz 1676, AVII 3, 57_2, 2003, 737).

²⁸ Obviously the expression was originated with Cavalieri.

²⁹ Porro quoniam harum figurarum omnium quarum ordinatae tab. I exhibentur, datae sunt quadraturae (sunt enim omnes ex genere paraboloeidum, quippe rationales integrae) ideo area figurarum completarum seu summas omnium ordinatarum ab A ad R. expressimus tab. II. Nempe omnes a. seu area ARD est quadrati ARDE. Omnes a², seu area ARDGA, est 1/3 eiusdem; et ita de caeteris (Leibniz 1676, AVII 3, 57_2, 2003, 738).

³⁰ Series Tabulae ⊙ horizontales multiplicentur, prima seu vertex 1. per 1. seu 2/2. secunda seu 1. 1. Per 3/2. tertia seu 1. b. 1. Per 4/2 seu 2. quarta 1. 3/2. 3/2. 1. per 5/2. Et ita porro. Termini singuli producti invertantur, unitato numeratore in nominatorem et contra, fiet Tabula  (Leibniz 1676, AVII 6 No 13₁, 2012, 118).

³¹ Tabula ♀ ex figura adjecta explicatur AF Π a. abscissa, FC Π r residua, AC tota. FG ordinata erit aliquis terminus tabulae, ♀. ut si AGD sit quadrans circuli FG erit √ar, si parabola sit cujus axis AB, erit FG, √a vel √r (Leibniz 1676, AVII 6 No 13₁, 2012, 119).

³² Termini autem seriei explicant aream figurae AGDCLA, posito AC, quadrato totius AC, 1 (Leibniz 1676, AVII 6 No 13₁, 2012, 119).