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Mengoli on ``Quasi Proportions''* HISTORIA MATHEMATICA 24 (1997), 257±280 ARTICLE NO. HM962147 Mengoli on ``Quasi Proportions''* MA.ROSA MASSA Centre d'Estudis d'HistoÁria de les CieÁncies, Universitat AutoÁnoma de Barcelona, 08193 Bellaterra, Barcelona, Spain This paper aims to analyze the ®rst three elementa of the Geometriae speciosae elementa (Bologna, 1659) of Pietro Mengoli (1625±1686), probably the most original pupil of Bonaven- tura Cavalieri (1598±1647). In this work, Mengoli develops a new method for the calculation of quadratures using a numerical theory called ``quasi proportions.'' He grounds quasi propor- tions in the theory of proportions as presented in the ®fth book of Euclid's Elements, to which he adds some original ideas: the ratio ``quasi zero,'' the ratio ``quasi in®nite,'' and the ratio ``quasi a number.'' A detailed analysis of this theory demonstrates the originality of Pietro Mengoli's work as regards both its content and his method of exposition. 1997 Academic Press Le but de ce papier est l'analyse des trois premiers elementa de la Geometriae speciosae elementa (Bologna, 1659) de Pietro Mengoli (1625±1686), probablement le disciple le plus original de Bonaventura Cavalieri (1598±1647). Dans cette oeuvre Mengoli deÂveloppe une nouvelle meÂthode pour calculer des quadratures en utilisant une theÂorie numeÂrique nommeÂe ``theÂorie des quasi-proportions.'' Il fonde cette theÂorie des quasi-proportions sur la theÂorie des proportions du cinquieÁme livre des EleÂments d'Euclid aÁ laquelle il ajoute quelques ideÂes originales: proportions ``quasi-nulles,'' ``quasi-in®nies,'' et ``quasi un nombre.'' Une analyse deÂtailleÂe de cette theÂorie deÂmontre l'originalite du travail de Pietro Mengoli. 1997 Aca- demic Press L'objectiu d'aquest article eÂs analizar els tres primers elementa de la Geometriae speciosae elementa (Bolonya, 1659) de Pietro Mengoli (1625±1686), que fou possiblement el deixeble meÂs original de Bonaventura Cavalieri (1598±1647). En aquesta obra Mengoli desenvolupa un nou meÁtode per calcular quadratures utilitzant una teoria numeÁrica anomenada de ``quasi proporcions.'' Mengoli fonamenta les quasi proporcions en la teoria de proporcions del llibre cinqueÁ dels Elements d'Euclides, a la qual hi afegeix unes nocions originals: rao ``quasi nulla,'' ``quasi in®nita,'' i ``quasi un nombre.'' Una exhaustiva anaÁlisi d'aquesta teoria demostra l'originalitat de l'obra de Pietro Mengoli tant pel que fa a la seva forma d'exposicio com pel que fa al seu contingut. 1997 Academic Press MSC 1991 subject classi®cations: 01A45, 40-03, 40A25. KEY WORDS: Mengoli, 17th century, proportion, limit. THE BACKGROUND TO MENGOLI'S WORK Most 17th-century mathematicians worked on problems of quadrature. From 1600 to 1680, the tools they used gave way to varied versions of in®nitesimals and indivisibles.1 Bonaventura Cavalieri (1598±1647) was one of the ®rst to develop a new method involving indivisibles, at a moment when there were two clear prece- * A ®rst version of this work was presented at the University AutoÁ noma of Barcelona on October 1, 1993 for the Master's degree in the history of science. 1 Among the many studies on this subject, the following are particularly useful: [5±7; 20; 25; 26; 28±30; 43]. 257 0315-0860/97 $25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved. 258 MA. ROSA MASSA HM 24 dents: the technique of ancient times, today called the method of exhaustion (due to Eudoxus and Archimedes), and the work of Johannes Kepler (1571±1630).2 Pietro Mengoli's Geometriae speciosae elementa (Bologna, 1659) also contains a new method for the calculation of quadratures. At the beginning of this work, in a letter dedicated to D. Fernando Riario, Mengoli outlines the relationship between his method of quadratures and the methods known up to that point: Both geometries, the old one of Archimedes and the new one of Indivisibles of my tutor, Bonaventura Cavalieri, as well as VieÁte's algebra, are regarded as pleasurable by the learned. Neither through the confusion nor the mixture of these, but through their perfect conjunction, a somewhat new form [of geometry will arise]Ðour ownÐwhich cannot displease anyone. [32, 2]3 As this makes clear, Mengoli, who knew the work of Archimedes and Cavalieri well, introduces a new element into his geometry, namely, VieÁte's algebra speciosa, which he quotes constantly.4 Another of Mengoli's sources is Euclid's Elements, which he mentions throughout the book in passages such as ``. and I believe that I am not taking anything from others, except from the ®rst nine [books] of Euclid's Elements'' [32, 9]5 and ``I am not taking anything from others, except for certain aspects of Euclid, in the ®fth and sixth, which I quote in the margin of the passages where I use it'' [32, 2].6 Indeed, throughout the Geometriae speciosae elementa, Mengoli repeatedly uses Euclid's de®nitions and theorems, but it is in the elaboration of the new theory of ``quasi proportions'' in the Elementum tertium of his Geometria that his use of the Elements is clearest. He ®rst de®nes the notion of quasi proportion and then proves that quasi proportions satisfy all of the properties of standard proportions as found in the ®fth book of the Elements. Mengoli wants to establish his new theory by grounding it in the ®rst principles of Euclid's Elements. To what extent, however, does Mengoli's new method of quadratures follow from Cavalieri's theory of indivisibles? After all, Mengoli was Cavalieri's pupil, so some commonality would seem reasonable. Surprisingly, a study of Mengoli's work reveals that the basis of his method was the theory of quasi proportions, a numerical theory of summations of powers and limits of these summations which has nothing to do with Cavalieri's Omnes lineae.7 It is not clear why Mengoli did not follow in the path of his master. Cavalieri's 2 Cavalieri's method is set forth in [16; 17]. Kepler's work on quadratures and cubatures is found in [24]. 3 ``Ipsae satis amabiles litterarum cultoribus visae sunt, utraque Geometria, Archimedis antiqua, & Indivisibilium nova Bonaventura Cavallerij Praeceptoris mei, necnon & Viettae Algebra: quarum, non ex confusione, aut mixtione, sed coniunctis perfectionibus, nova quaedam, & propria laboris nostri species, nemini poterit displicere.'' 4 Mengoli may have known VieÁte's algebra speciosa through the second volume of Herigone's textbook [23], as he cited Herigone as a source for his notation. See [32, 12]. 5 ``ideoque nihil alienum sumpsi, praeterquam ex prioribus novem Elementis Euclidis.'' 6 ``Nihil alienum sumo; praeter quaedam, ex Euclide, in quinto, & sexto: quae suis locis allego, in margine.'' 7 Mengoli had already published a work, Novae quadraturae arithmeticae (Bologna, 1650), in which he worked with in®nite series, adding them together and giving them properties. See [3; 21]. HM 24 MENGOLI ON ``QUASI PROPORTIONS'' 259 method did receive considerable criticism, and Mengoli might have been affected by this. In the letter dedicated to Giandomenico Cassini8 in the Elementum sextum of the Geometriae, Mengoli explains that 11 years before he had found many quadratures of plane ®gures using Cavalieri's method. He goes on to acknowledge that he did not make them known on account of the attacks leveled against that method: Meanwhile I left aside this addition that I had made to the Geometry of indivisibles, because I was afraid of the authority of those who think false the hypothesis that the in®nity of all the lines of a plane ®gure is the same as the plane ®gure; I did not publish it not because I agreed with them, but because I was doubtful of it, and I tried . to establish new and secure foundations for the same method of indivisibles or for other methods which were equivalent. [32, 364]9 In this letter, Mengoli recognizes that the basis of Cavalieri's method of indivisibles was not sound enough, and, as he wanted to ground this method of indivisibles solidly, he started on a new path, that of in®nite series. In fact, after 1650 through the in¯uence of VieÁte and, above all, Descartes, algebraic methods became ever more accepted in the ®eld of geometry, and interest in numerical work, such as interpolation and approximation, also increased. Other mathematicians of the periodÐsuch as Pierre de Fermat (1601±1665), Gilles Personne de Roberval (1602±1675), John Wallis (1616±1703), and Blaise Pascal (1623±1662) also used these methods.10 They aimed, among other things, to calculate the result which today would look like 1p 1 ...1tp 1 lim 5 , F tp11 G p11 for t tending to in®nity. This would have allowed them to square the parabolas y 5 xp, for p any positive integer. Mengoli also calculates this using his theory of quasi proportions, and he does so in an original and general way in Theorem 42 of the Elementum tertium (see below). PIETRO MENGOLI'S GEOMETRIAE SPECIOSAE ELEMENTA The name of Pietro Mengoli (1625±1686) appears in the register of the University of Bologna in the period 1648±1686. He studied with Bonaventura Cavalieri and 8 Giandomenico Cassini was a professor of astronomy at the University of Bologna from 1650 to 1669, before moving to Paris [9, 37]. 9 ``Ipsam interim accessionem, quam Geometriae Indivisibilium feceram, praeterivi: veritus eorum authoritatem, qui falsum putant suppositum, omnes rectas ®gurae planae in®nitas, ipsam esse ®guram planam: non quasi hanc sequens partem; sed illam quasi non prorsus indubiam debitans: tentandi animo, si possem demum eandem indivisibilium methodum, aut aliam equivalentem novis, & indubijs prorsus constituere fundamentis.'' 10 Information on these mathematicians may be found in the following sources: on Fermat, [26, 230]; Roberval, [5, 18±21; 43, 41±44]; Wallis, [44, 365±392; 38, 34]; and Pascal, [13, 240; 37, 171].
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