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The Role of Indivisibles in Mengoli’s Quadratures

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Seventeenth-Century Indivisibles Revisited

Part of the book series: Science Networks. Historical Studies ((SNHS,volume 49))

Abstract

The name of Pietro Mengoli (1626–1686) appears in the University of Bologna registry for the period 1648–1686. He studied with Bonaventura Cavalieri and ultimately succeeded him in the chair of mechanics. He graduated in philosophy in 1650 and 3 years later in canon and civil law. In his first period, he wrote three mathematical books, Novae quadraturae arithmeticae seu de additione fractionum (Bologna, 1650), Via Regia ad Mathematicas per Arithmeticam, Algebram Speciosam, & Planimetriam, ornata, maiestati Serenissimae D. Christinae Reginae Suecorum (Bologna, 1655) and Geometriae Speciosae Elementa (Bologna, 1659). He took holy orders in 1660 and was prior at the church of Santa Maria Maddalena in Bologna until his death.

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Notes

  1. 1.

    Although Mengoli published nothing between 1660 and 1670, the latter year saw the appearance of two works: Refrattioni e parallase solare (Bologna, 1670), Speculationi di musica (Bologna, 1670), and later Circolo (Bologna, 1672). These reflected Mengoli’s new aim of pursuing research not on pure but on mixed mathematics like astronomy, chronology and music. Furthermore, his research was clearly in defence of the Catholic faith. Mengoli went on writing in this line, publishing Anno (Bologna, 1675) and Mese (Bologna, 1681) on the subject of cosmology and Biblical chronology and Arithmetica rationalis (Bologna, 1674) and Arithmetica realis (Bologna, 1675) on logic and metaphysics. For more biographical information on Mengoli, see Natucci (1970) and Massa Esteve (2006b).

  2. 2.

    Information on these subject may be found in the following sources: on Fermat, see Mahoney (1973); on Roberval see Auger (1962), Walker (1986), and Jullien (1996b); on Pascal see Bosmans (1924) and Boyer (1943) and on Wallis see Stedall (2001).

  3. 3.

    We may also cite Roberval, who in 1636, in a letter written to Fermat, enunciated the rule for finding the infinite sum of powers, and explained how he employed it for calculating quadratures. Fermat, for his part, stated in a letter to Cavalieri, written before 1644, that he had squared the parabolas, giving both the rule and an example. Ten years later, Pascal arrived, apparently independently, at a similar conclusion in the work Potestatum numericarum summa, see Pascal, Oeuvres de Blaise Pascal (1954). In 1657, Fermat himself proved the quadratures for a positive rational number n, see Fermat, Oeuvres, (1891–1922). Furthermore, Wallis also proved these same quadratures in his Arithmetica Infinitorum (1655) using the sum of powers, see Wallis, Opera, (1972).

  4. 4.

    In fact Mengoli was influenced by Viète’s algebra through Hérigone’s algebra in his Cursus Mathematicus (1634/1637/1642). On a comparative analysis between Viète’s specious algebra and Hérigone’s algebra see Massa Esteve (2008) and on Hérigone’s influence in Mengoli’s works see Massa Esteve (2012). On Viète’s specious algebra, see Viète, Opera, (1970).

  5. 5.

    Ipsae satis amabiles litterarum cultoribus visae sunt utraque Geometria, Archimedis antiqua, & Indivisibilium nova Bonaventura Cavallerij Praeceptoris mei, necnon & Vietae Algebra: quarum non ex confusione, aut mixtione, sed coniuntis perfectionibus, nova quaedam, & propria laboris nostri species, nemini poterit displicere (Mengoli 1659, pp. 2–3).

  6. 6.

    This sixth Elementum, with the title De innumerabilibus quadraturis contains (besides a letter to Cassini), three triangular tables, 36 definitions, 11 propositions (4 of them he named problems) and lastly, two pages on barycentre.

  7. 7.

    Giandomenico Cassini (1625–1712) was a professor of astronomy at the University of Bologna from 1650 to 1669, before moving in the latter year to Paris. Baroncini, Gabriele and Cavazza, Marta (eds) (1986) La corrispondenza di Pietro Mengoli see Baroncini and Cavazza (1986).

  8. 8.

    Ante annos duodecim, occasione cuiusdam problematis mihi propositi à D. Io. Antonio Rocca Regiensi, de figura unilinea describenda, quae secaret ellipsim in duobus punctis innumerabiles huiusmodi figuras excogitavi, quas tunc per Geometriam indivisibilium quadrabam, adhibito tamen priùs hoc lemmate (Mengoli 1659, p. 348). I am unable to identify the question posed by Antonio Rocca (1607–1656), who was a friend and correspondent Cavalieri’s and of many other scientists at this time. For more information see Favaro (1983) and Rocca (1785).

  9. 9.

    Jean de Beaugrand (1595–1640) was also a mathematician; in 1635 he spent an entire year in Italy and visited Cavalieri in Bologna. He published a version of In Artem Analyticen Isagoge, which was in fact Viète’s work extended with some “scolies” and a mathematical compendium.

  10. 10.

    Mengoli states: “Furthermore, in order to obtain this in a shorter way, we will proceed using Speciosa Algebra”/“Ut autem breviori via id obtineamus, procedemus per Algebram Speciosam”, Mengoli, Geometria, (1659, p. 349). Furthermore, Cavalieri in the Exercitatio after the lemma claims: “But in order to obtain this, the reader who does not ignore these algebraic products will understand that this way is much easier than the Euclidian approach. We have used its longer structure for Propositions 17 and 18.” Ex his ergo Lector harum multiplicationum Algebraicarum non ignarus, intelliget hanc viam multò faciliorem esse quàm Euclidianam, cuius longiorem texturam in Propos 17. & 18. prosecuti sumus (Cavalieri 1647, p. 286).

  11. 11.

    Ibid, 243–245. On Cavalieri’s chapter see Bosmans (1922).

  12. 12.

    Est autem hoc lemma affine illi, quod recitat Bonaventura Cavallerius b. m. Praeceptor meus ex Io. Beaugrand: quod idcircò in expositione placet imitari (Mengoli 1659, p. 349).

  13. 13.

    Esto parallelogrammum AB, cuius diameter CD: dividaturque CD bifariam in E: ducanturque per E, rectae FG, IH, parallelogrammi AB lateribus parallele: ducanturque hinc inde ab E distantes quantumlibet, sed aequaliter, & intra quadratum, duae KLMN, & OPQR. Dico sub triangulis ACD, BCD, omnes sextuplas uniprimas, aequales esse, omnibus secundis potestatibus parallelogrammi AB. (Mengoli 1659, p. 358).

  14. 14.

    Proposition XX of Cavalieri states that: \( 3\;{\displaystyle {\int}_0^t{x}^2}dx={\displaystyle {\int}_0^t{t}^2}dx \), (Cavalieri 1647).

  15. 15.

    His demonstratis, cogitabam si possent aliae quadraturae inveniri ex inventis compositae, in quas insignis aliqua resolvatur, quemadmodum in triangula, parabolam Archimedes resolvit (Mengoli 1659, p. 363). Indeed, Mengoli says that he knew these quadratures by indivisibles in 1647, and in 1650 he published the Nova, in which he proves infinite sums. A reading of the preface to the Nova makes the relation between these works clear. Mengoli explains the relation between these sums and the calculation of a universal quadrature. See Mengoli (1650) and Giusti (1991).

  16. 16.

    Mengoli had already published this work, in which he employed infinite series, adding them together and giving them suitable properties. On this subject see Giusti (1991).

  17. 17.

    Et generaliter inveni, figuram, in qua ordinatae sunt omnes potestates abscissarum, & deinceps omnes figuras, in quibus ordinatae sunt productae sub ijsdem potestatibus abscissarum, & sub residuarum potestatibus omnifariam, simul aggregatas, aequales esse figurae, in qua ordinatae, sunt omnes potestates abscissarum ordinis proximè inferioris. (Mengoli 1659, pp. 363–364).

  18. 18.

    We can suppose that this “insignis” quadrature he was looking for was the quadrature of the circle. In fact, at the beginning of his Circolo (1672), Mengoli stated that he had found the quadrature of the circle in 1660, but he had not published it because, according to him, he only wanted to publish the mathematics he needed to explain natural events.

  19. 19.

    Ipsam interim accessionem, quam Geometriae Indivisibilium feceram, praeterivi: veritus eorum authoritatem, qui falsum putant suppositum, omnes rectas figurae planae infinitas, ipsam esse figuram planam: non quasi hanc sequens partem; sed illam quasi non prorsus indubiam devitans: tentandi animo, si possem demum eamdem indivisibilium methodum, aut aliam equivalentem novis, & indubijs prorsus constituere fundamentis (Mengoli 1659, p. 364).

  20. 20.

    A knowledge of algebraic language enabled Mengoli to extend the Euclidean theory of proportions and create new theories. On the importance of Mengoli’s work on the Euclidean theory of proportions, see Massa (2003).

  21. 21.

    On these explanations see Massa (1997).

  22. 22.

    In his Circolo of 1672, Mengoli again uses quasi ratios and explains: Dissi quasi, e volsi dire, che vadino accostandosi ad essere precisamente tali (Mengoli 1672, p. 49). On Mengoli’s Circolo see Massa Esteve-Delshams (2009).

  23. 23.

    To clarify the notion of “ratio quasi infinite” Mengoli in his Geometria considered values up to 10 in the ratio O.a to t; for instance, if t = 4, then the ratio is 6 to 4; if t=7 then the ratio is 21 to 7; if t = 10 then the ratio is 45 to 10. He argued that the ratio takes increasingly greater values as the value of t increases, so the ratio is quasi infinite. For the ratio quasi null, he considered values up to 10 in the ratio O.a to t3.

  24. 24.

    1. Ratio indeterminata determinabilis, quae in determinari, potest esse maior, quam data, quaelibet, quatenus ita determinabilis, dicetur, Quasi infinita. 2. Et quae potest esse minor, quàm data quaelibet, quatenus ita determinabilis, dicetur, Quasi nulla. 3. Et quae potest esse minor, quàm data quaelibet minor inaequalitas; & maior, quàm data quaelibet minor inaequalitas, quatenus ita determinabilis, dicetur, Quasi aequalitas. Vel aliter, quae potest esse propior aequalitati, quàm data quaelibet non aequalitas, quatenus talis, dicetur, Quasi aequalitas. 4. Et quae potest esse minor, quàm data quaelibet non maior, proposita quadam ratione; & maior, quàm data quaelibet minor, propositâ eâdem ratione, quatenus ita determinabilis, dicetur, Quasi eadem ratio. Vel aliter, quae potest esse propior cuidam propositae rationi, quàm data quaelibet alia non eadem, quatenus talis, dicetur, Quasi eadem. 5. Et rationum quasi earundem inter se, termini dicentur, Quasi proportionales. 6. Et quasi aequalitatum, dicentur, Quasi aequales (Mengoli 1659, p. 97).

  25. 25.

    The inaequalitas of a ratio denotes a number other than unity, and so ratios minor inaequalitas and maior inaequalitas correspond to numbers smaller and larger than unity, respectively.

  26. 26.

    On this subject, see Massa (1997).

  27. 27.

    He defined the abscissa as our x, but in a segment measuring the unit u or t. Mengoli always worked within a finite base in which the abscissa was represented by the letter “a” and the remainder was represented by the letter “r = t-a” or “1-a”, depending on whether the base was a given value t or the unit u, see Massa (2006).

  28. 28.

    The word figure or forma, which dates from the previous century, was identified by measuring the intensity of a given quality; see Massa (2006).

  29. 29.

    For these demonstrations Mengoli used the definitions from the Elementum tertium of quasi equality.

  30. 30.

    33. Figura vero ex tot parallelogrammis, quot sunt ordinatae per puncta divisionum, & ad ipsas ordinatas iacentibus composita, dicetur, Adscripta formae (Mengoli 1659, p. 371).

  31. 31.

    He used Proposition 67 of Elementum quintum, which established ratios of quasi equality between two magnitudes situated between two quasi equals.

  32. 32.

    “Theor. 6. Prop. 10. Omnes quadraturae super eadem basi constitutae, sunt inter se aequales (Mengoli 1659, p. 389).

  33. 33.

    Also according to Malet’s interpretation of Gregorie’s work, see Malet (1996).

  34. 34.

    On the construction of Mengoli’s harmonic triangle and interpolated harmonic triangle see Massa Esteve-Delshams (2009).

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    M. Rosa Massa Esteve

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Esteve, M.R.M. (2015). The Role of Indivisibles in Mengoli’s Quadratures. In: Jullien, V. (eds) Seventeenth-Century Indivisibles Revisited. Science Networks. Historical Studies, vol 49. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-00131-9_13

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