Torricelli’s Indivisibles

Bascelli, Tiziana

doi:10.1007/978-3-319-00131-9_6

Tiziana Bascelli⁵

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

Abstract

Evangelista Torricelli made a significant contribution to the theory of indivisibles. He studied and challenged Cavalieri’s method of indivisibles. In this paper I shall examine the essays in which Torricelli considers indivisibles and their applications even when these essays are fragmentary or not more than working notes. I shall provide a logical reconstruction of how he arrived at his definition of indivisible also by analysing the theoretical framework in which he applied notions like magnitude, quantity, and continuity in his analysis of a wide range of geometric objects. Eventually I shall show how he applied indivisibles to other fields.

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Notes

1.
[L]’oblique prove con doppie false posizioni, e costruzzioni possonsi alla semplice, e diretta maniera ridurre […] i principij nostri, che pure dalle viscere di Euclide sono tratti non solo a cio fare saranno bastevoli, ma anco stabiliranno i principij ingegnosissimi de gli indivisibili ch’a giorni nostri incominciarono a sorgere con meravigliosa felicità. Antonio Nardi, Scene, Scena 7, Veduta 8, f. 1010, quoted in Belloni (1987, p. 34).
2.
Torricelli, Opere, 1919–1944.
3.
Torricelli’s starting points to the theory of indivisibles are found in his essays “Against Infinite Aggregates”, Contro gl’infiniti in Torricelli, Opere, 1919–1944, I part 2, pp. 45–48, and “On the Doctrine of Indivisibles Wrongly Applied”, De indivisibilium doctrina perperam usurpata in Opere, 1919–1944, I part 2, pp. 417–432, which contains a chapter, entitled “Examples of Application of Curved Indivisibles”, Exempla pro us[u] curvorum indivisibilium, pp. 426–432. Examples of application can be found in the “Quadrature of the Parabola Resolved in Several Ways by Means of the New Geometry of Indivisibles”, Quadratura parabolae per novam indivisibilium geometriam pluribus modis absoluta in Opere, 1919–1944, I part 1, pp. 139–162, which is inserted in the essay “On the Dimensions of the Parabola”, De dimensione parabolae, pp. 89–172; “Field of Truffles”, Campo di tartufi in Opere, 1919–1944, I part 2, pp. 1–43; “On Vase-like Solids”, De solidis vasiformis in Opere, 1919–1944, I part 2, pp. 101–123; “On the Infinite Parabolas”, De infinitis parabolis in Opere, 1919–1944, I part 2, pp. 275–328, particularly the section “On the Tangents of Infinite Parabolas Drawn by Means of Additional Lines”, Delle tangenti delle parabole infinite per lineas supplementares [sic], pp. 320–324.
4.
For Italian translations, see Bortolotti (1925a, 1928, 1939).
5.
About François De Gandt’s most important contributions to the understanding of Torricelli’s works see De Gandt (1987 and 1995).
6.
Poichè questa contiene tante cilindriche quanti sono i punti del semiquadrante AB e l’altra figura residua ne contiene tanti quanti sono i punti del semiquadrante GA e ciascuna eguale a ciascuna etc., Torricelli, Opere, I part 2, p. 20. See, also, next footnote 13.
7.
Aristotle, Physics, III, 200b17–20. In this chapter, Aristotle is quoted in Aristotle, Physics, translated by Robin Waterfield with introduction and notes by David Bostock, Oxford University Press, 2008.
8.
Aristotle, Physics, VI, 231a21–24. See Chap. 2.
9.
Euclid, Elements, III, Heath, p. 371.
10.
Euclid, Elements, III, Heath, p. 14.
11.
Euclid, Elements, II, Heath, p. 420.
12.
This is Proposition 1: “The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.” Archimedes (1953, p. 90).
13.
See in particular Proposition 19 and 20. Archimedes (1953, pp. 129–130).
14.
[…] una lunghezza, cioè una estensione di punti continuati, Torricelli, Opere, 1919–1944, II, p. 247.
15.
The original subtitle of “On the Dimensions of the Parabola” reads that Torricelli is going to prove the quadrature of a parabola in 20 different ways, in quo quadratura parabolae xx modis absolvitur, Torricelli, Opere, 1919–1944, I, part 1, p. 89. Actually, the Propositions he proved are 21.
16.
Esto parabola ABC, cuius diameter[,] BD, triangulum inscriptum ABC; Dico parabolam esse sesquitertiam trianguli ABC[.] [s]ibi inscripti, Torricelli, Opere, 1919–1944, I, part 1, pp. 120–121.
17.
Torricelli, Opere, 1919–1944, I, part 1, p. 112. The property is proved by means of Lemma 7, which comes from Archimedes’ Quadrature of the Parabola Proposition 21: “If Qq be the base, and P the vertex, of any parabolic segment, and if R be the vertex of the segment cut off by PQ, then ΔPQq = 8ΔPRQ”, Archimedes (1953, p. 248). Referring to Fig. 6.5, Qq is AC, P is B, and R is E.
18.
Torricelli, Opere, 1919–1944, I, part 1, p. 121.
19.
Ibid.
20.
Ibid.
21.
Torricelli, Opere, 1919–1944, I, part 1, pp. 102–138.
22.
Torricelli, Opere, 1919–1944, I, part 1, pp. 139–162.
23.
Cavalieri (1635), see Chap. 3, pp. nnn.
24.
Par cette disposition, le lecteur est d’abord invité à constater combien les démonstrations sont pénibles et peu naturelles sans les indivisibles. Puis vient la géométrie des indivisibles, cette voie nouvelle et admirable (I, I, p. 139), qui démontre d’innombrables théorèmes “par de démonstrations brèves, directes et affirmatives” (I, I, p. 140), et auprès de laquelle la géométrie antique fait figure “pitoyable”, Miseret me veteris Geometriae, I, I, p. 173, De Gandt (1987, pp. 152–153).
25.
Parabola igitur nihil aliud est quàm aggregatum quoddam infinitarum numerum magnitudinum in propotione quadrupla, quarum prima est triangulum ABC, secunda verò constat ex duobus triangulis ADB, BEC. Propterea prima magnitudo ABC media proportionalis erit inter primam differentiam, et aggregatum omnium, nempe parabolam, Torricelli, Opere, 1919–1944, I, part 1, pp. 150–151.
26.
“[Cavalieri’s] method, thus, relies upon a special type of isomorphism: the proportionality between two ‘aggregates’ of either lines or planes may be mapped over those figures from which those lines and planes were cut without the necessity of deciding whether the lines ‘compound’ the figure, or the planes ‘compound’ the solid”, La méthode [de Cavalieri] repose ainsi sur une sorte d’isomorphisme : la proportionnalité entre deux ‘agrégats’ de lignes ou de plans peut être transférée aux figures sur lesquelles on a découpé ces lignes ou ces plans, sans que l’on ait à décider si les lignes ‘composent’ la figure, si les plans ‘composent’ le solide, De Gandt (1987, p. 162).
27.
Omnes autem parabolae, atque ipse conus idem sunt, Torricelli, Opere, 1919–1944, I, Part 1, p. 156.
28.
The identity is due to Lemma 7, already quoted (see Footnote 16, p. 4), which shows why triangle ABC is equal to eight times triangle ADB.
29.
Torricelli, Opere, 1919–1944, I, part 1, p. 151.
30.
Suppositis infinitis rectis lineis continua proportione maioris inaequalitatis, rectam lineam, quae praedictis omnibus sit aequalis reperire, Torricelli, Opere, 1919–1944, I, part 1, pp. 148–149.
31.
Euclid, Elements, Definition 5, Book 5.
32.
Torricelli’s theory of proportions is described in his “Book on Proportions”, De proportionibus liber, Torricelli, Opere, 1919–1944, I, part 1, pp. 293–327.
33.
“Indeed, we must congratulate inventors of logarithm computations for having removed all the obstacles to the difference between quantities and proportions”, Nam sedulo gratulandum esset Logarithmorum computatoribus, si nihil discriminis inter quantitates, et proportiones intercederes, Torricelli, Opere, 1919–1944, I, part 1, p. 295; “Moreover, for the sake of discussion we claim to embrace not several, but every experiments done with numbers. Therefore, what should we say about incommensurable magnitudes which, doubtless, cannot have the same ratio as numbers have?”, Porro concedamus disputationis gratia, non plurima, sed omnia numerorum experimenta arridere, quidnam denique censendum erit de magnitudinibus incommensurabilibus, quae quidem inter se non possunt habere rationem illam, quam habet numerus ad numerum?, Torricelli, Opere, 1919–1944, I, part 1, p. 299; “In this and in the four next theorems we are going to prove a statement, which is true only for lines, until in the Appendix at the end of this short treatise we will show it is true also for the other types of quantities”, Demonstrabimus in hoc, et in quatuor seguentibus theorematibus propositionem in lineis tantum, donec in Appendice ad finem libelli ostendamus veram esse etiam in aliis quantitatis generibus, Propositio 6, Torricelli, Opere, 1919–1944, I, part 1, p. 311.
34.
Aequales magnitudines quotcumque sint eamdem habent rationem, quam habent numeri a quibus numerantur. Exempli gratia: Septem lineae palmares ad quatuor lineas palmares eamdem habent rationem quam habet numerus 7 ad numerum 4. Vel quinque quadrata palmaria ad novem quadrata palmaria eamdem habent rationem, quam habet numerus quinque ad numerum novem, Torricelli, Opere, 1919–1944, I, part 1, p. 306.
35.
Demonstravimus […] eas omnes, quas scitu necessarias judicavimus. […] Sed jam tempus exigit ut ostendamus in aliquibus tantum exemplis quomodo ea, quae de solis lineis demonstravimus ad superficies etiam et ad solida propagari possint: et hoc in gratiam eorum, qui […] dubitare adhuc poterunt, an ea vera sint etiam in superficiebus, in solidis, in temporibus, et in omni alio genere quantitatis, Torricelli, Opere, 1919–1944, I, part 1, p. 319.
36.
Torricelli, Opere, 1919–1944, I, part 1, p. 149.
37.
Torricelli, Opere, 1919–1944, I, part 1, pp. 91–172.
38.
… omnes primae simul, nempe parallelogrammum AE, ad omnes secundas simul, nempe ad trilineum ABCD, ut sunt omnes tertiae simul, nempe cylindrus AE, ad omnes quartas simul hoc est ad conum ACD, Torricelli, Opere, 1919–1944, I, part 1, p. 140.
39.
Dato trilineo mixto, sub lineà parabolica, eiusque tangente, et alià rectà diametro parallela compraehenso; possibile est in dato trilineo figuram inscribere constantem ex parallelogrammis aequealtis, quae figura deficiat à trilineo mixto minori differentià quàm sit quaecumque data magnitudo, Torricelli, Opere, 1919–1944, I, part 1, pp. 128–129.
40.
Torricelli, Opere, 1919–1944, I, part 2, pp. 1–43. A set of notes tests the method of indivisibles in simple geometrical situations, and are collected in a section entitled “Against Infinite Aggregates”, Contro gl’infiniti. These are Propositions 73–80, Torricelli, Opere, 1919–1944, I, part 2, pp. 20–23. There is another chapter in Torricelli’s collected works with the same title, where we can find almost identical notes, Torricelli, Opere, 1919–1944, I, part 2, pp. 46–48.
41.
Parmi tous les exemples, il en est un, très simple, qui peut servir de paradigme et de fil conducteur, parce que son étude et sa généralisation ont conduit Torricelli lui-même à des développements très féconds, De Gandt (1987, pp. 147–206; p. 164).
42.
See Exemplum I, in De indivisibilium doctrina perperam usurpata, Torricelli, Opere, 1919–1944, I, Part 2, p. 417.
43.
See Exemplum II, pp. 417–418.
44.
See Exemplum III, p. 418.
45.
Exemplum IV, pp. 418–419.
46.
Exemplum XI, pp. 418–419. Torricelli examined this geometric circumstance in another page in Contro gl’infiniti, Torricelli, Opere, 1919–1944, I, Part 2, p. 47.
47.
Exemplum V, p. 419.
48.
Exemplum VI, p. 420.
49.
Torricelli, Opere, 1919–1944, I, Part 1, p. 174.
50.
Exemplum I, in Exempla pro usu curvorum indivisibilium, Torricelli, Opere, 1919–1944, I, Part 2, pp. 426–427.
51.
Exemplum II, pp. 427–428.
52.
The bowl is a cylinder that has a hole of cylindrical or hemispherical shape, see Exemplum III, p. 427.
53.
They are, in order, Exempla IV, V, VIII, IX, X, and XIII pp. 427–428, p. 428, p. 429, pp. 429–430, p. 430, p. 431.
54.
… les paradoxes … ne sont pas conformes aux canons requis par La Geometria. C’est flagrant dans certains cas : la comparaison entre la surface de la sphère et celle du cylindre (exemple n ^o 5 …) est impossible dans le cadre de la méthode de Cavalieri, parce que la sphère n’est pas développable, et qu’il est impossible de la balayer à la manière d’une figure plane.
Dans la plupart des exemples, c’est la ‘regula’ qui est absent. Cavalieri imposait que le plan mobile qui découpe les figures ou les solides restât toujours parallèle à une droite fixée (la ‘regula’), De Gandt (1987, pp. 147–206; p. 171).
55.
De indivisibilium doctrina perperam usurpata is the title of Vincenzo Viviani’s Latin copy of Torricelli’s notes. In that short treatise Torricelli did not provide a systematic account or a complete mathematical theory of indivisibles but rewrote the geometrical examples worked out especially in the pages of Contro gl’infiniti, adding note that underlined where the crucial problem is, Torricelli, Opere, 1919–1944, I, part 2, pp. 47–48.
56.
Che gli indivisibili tutti sieno eguali fra loro, cioè i punti alli punti, le linee in larghezza alle linee e le superficie in profondità alle superficie, è opinione a giudizio mio non solo difficile da provarsi, ma anco falsa., Torricelli, Opere, 1919–1944, I, part 2, p. 320, De Gandt 1995, p. 188). I shall quote this translation as Wilson followed by the number of the page.
57.
This is what results from Torricelli’s analysis, which is described in detail in the section “On the Tangents of Infinite Parabolas Drawn by Means of Additional Lines”, Delle tangenti delle parabole infinite per lineas supplementares in Torricelli, Opere, 1919–1944, I, Part 2, pp. 320–324.
58.
Se nel triangolo ABC che abbia il lato AB maggiore del BC ci immagineremo tirate tutte le infinite linee parallele alla base AC, tanti saranno i punti stampati dal segmento su la retta AB, quanti su la BC; dunque un punto di quella ad un punto di questa sta come tutta la linea a tutta la linea, Delle tangenti delle parabole infinite per lineas supplementares, Ibid, pp. 320–321.
59.
When Torricelli uses the word quantum he refers to Galileo. Galileo’s quantum-like point is a finite, continuous, and minuscule magnitude, that is to say an infinitesimal entity of an order above the order of indivisibles. Therefore, a sum of an infinite amount of quanta is always infinite, while a sum of an infinite amount of indivisibles is finite. See Chap. 5.
60.
Wilson (1995, pp. 188–189). Se siano due circoli concentrici, e dal centro s’intendano tirate tutte le linee a tutti i punti della periferia maggiore, non è dubbio che altrettanti punti faranno i transiti delle linee sulla periferia minore, e ciascuno di questi sarà tanto minore di ciascuno di quelli, quanto il diametro è minore del diametro. [Nota a margine:] Perché le linee sono accuminate, ma tirando una sola linea i punti saranno eguali. Delle tangenti delle parabole infinite per lineas supplementares, Torricelli, Opere, 1919–1944, I, Part 2, p. 321.
61.
Torricelli, Opere, 1919–1944, I, Part 2, p. 321; Wilson p. 189. See Chatton’s position in Chap. 2.
62.
Euclid, Elements, Heath, I, p. 326.
63.
“… in triangle ABC … a point on the former [straight line AB] is to a point on the latter [straight line BC] as the whole straight line to the whole straight line”, … nel triangolo ABC … un punto di quella [retta AB] ad un punto di questa [retta BC] sta come tutta la linea a tutta la linea, Torricelli, Opere, 1919–1944, I, Part 2, pp. 320–321. “… hence one [of the lines of one set] is equal to one [of the lines of the other set]. But they are of unequal length; therefore, although indivisibles, they are of unequal widths, reciprocally proportional to their lengths”, … dunque una [delle rette dell’insieme] è uguale ad una [delle rette dell’altro insieme], ma sono disugualmente lunghe, adunque benché indivisibili sono di larghezza ineguale, e reciproca alle lunghezze. Ibid, p. 321; Wilson, p. 189. “Moreover, if points and lines are unequal, then unequal are the lines and the surfaces passing through those given points and lines”, E se i punti e le linee sono disuguali così saranno le linee e le superficie le quali passeranno per detti punti, e per dette line. Ibid, p. 321.
64.
Torricelli, Opere, 1919–1944, I, Part 2, p. 322; Wilson p. 190.
65.
L’istesso si dice nelle parabole. Sia la parabola ABC, e preso qualunque punto B in essa (per le cose provate altrove) sarà la figura BCE alla BC[G] come l’esponente delle app[licate] a quello delle diametrali. Ma se, divisa per mezzo la FE in I, applicheremo la ID, sarà la figura DCE alla DCF nel medesimo modo come l’esponente all’esponente; et se faremo, o supporremo fatta questa div[ision]e infinite volte, resteranno in cambio di figure due linee CE, et CF, le quali, non secondo la longitudine, ma secondo la quantità, saranno nel medesimo modo come l’esponente all’esponente, Torricelli, Opere, 1919–1944, I, Part 2, p. 322.
66.
See Chap. 3.
67.
Torricelli, Opere, 1919–1944, I, part 2, p. 322.
68.
See De solido acuto hyperbolico problema alterum, Torricelli, Opere, 1919–1944, I, Part 1, pp. 173–190.
69.
Ibid, pp. 191–221.
70.
Problema secundum: “solidum acutum hyperbolicum infinitè longum, sectum plano as axem erecto, unà cum cylindro suae basis, aequale est cylindro cuidam recto, cuius basis sit latus versum, sive axis hyperbolae, altitudo verò sit aequalis semidiametro basis ipsius acuti solidi”. Ibid, pp. 193–194.
71.
Prima io dimostrai che infinite quantità in proporzione Geometrica majoris inaequalitatis sono misurabili. Poi mostrai che un solido infinitamente lungo era misurabile; e sapevo che descrivendosi una figura piana le cui linee andassero continuamente decrescendo come i circoli del predetto solido, anche quella era misurabile onde l’invenzione del piano infinitamente lungo perde assai dopo pubblicata quella del solido in Torricelli’s letter to Michelangelo Ricci, Florence 17 March 1646, Torricelli, Opere, 1919–1944, III, p. 361. See also Bortolotti (1925a, pp. 49–58, 1925b, pp. 139–152).
72.
Bortolotti (1987, pp. 118–121).
73.
A mixed quadrilateral is a plane figure with four sides with at least one of them being curvilinear.
74.
The dignitas of a point on a hyperbole is x-coordinate or abscissa and y-coordinate or ordinate of the point in a perpendicular system of reference in which the axes are the asymptotes of the hyperbole.
75.
Esto hyperbola quaelibet, cujus asymptoti EA, AB, sumaturque quodvis ejus frustum DCBG, ductis [C]B, DG et DE, CF ad asymptotos parallelis. Dico quadrilineum EDCF ad frustrum DCBG esse ut dignitas BA ad AE dignitatem in De infinitis hyperbolis, Torricelli, Opere, 1919–1944, I, Part 2, p. 256.
76.
Two lines, BD and BF, are supplementary lines when they are equal in area because they are semi gnomons.
77.
… sia una delle inf[inite] parabole ABC et al punto B devasi dare la tangente. Sia tangente BE, applicata BD et finiscasi la figura DF; et intendasi che per quel medesimo punto B per il quale passa la tangente passino anco adeguatamente la BD et la BF. Saranno dunque BD et BF linee supplementari; e la lunghezza EB alla BG, overo come la quantità FB alla BG, overo come la quantità DB alla BG (per esser FB, et BD supplementari) cioè come l’esponente all’esponente. Torricelli, Opere, 1919–1944, I, Part 2, pp. 322–323.
78.
Pro confirmanda prima Galilei et sequentem nostram, Torricelli, Opere, 1919–1944, I, Part 2, p. 259.
79.
Galileo (1638, VIII, pp. 208–209).
80.
… ut AVDB ad ACDB, ita quantitas omnium punctorum GF ad quantitatem omnium nempe totidem punctorum OH, sive ita GF ad OH, in Pro confirmanda prima Galilei et sequentem nostram, Torricelli, Opere, 1919–1944, I, Part 2, p. 259, Wilson, p. 113.
81.
Torricelli, Opere, 1919–1944, I, Part 2, pp. 277–328.
82.
… spatia vero temporum aequalium erunt ut omnes gradus velocitatis ad omnes gradus velocitatis, nempe ut [parallelogrammum] BD ad trilineum AB[C] in pro tangentibus infinitarum parabolarum, Torricelli, Opere, 1919–1944, I, Part 2, p. 313.
83.
Torricelli, Opere, 1919–1944, I, Part 2, p. 309.
84.
“Let us assume, thus, that a body in motion is falling in such a way that speed increases like the square of the time, for instance; … the space traversed in the same time interval is three times the time of fall … I claim that spaces traversed are the compound ratio of the ratio of speeds with the ratio of times. … Definition of linear, square, cubic acceleration. … another falling body having a square acceleration. I claim that it draws a cubic parabola. … What we have just exemplified in the cubic example for sake of concision can be said of any parabola”, Supponamus iam mobile aliquod descendente ita ut velocitates crescant ut quadrata temporum, exempli gratia; […] spatium conficiet tempore aequali tempori descensus […] triplum. […] Dico sic spatia peracta habent rationem compositam ex ratione velocitatum et ex ratione temporum. […] Definitio lineariter, quadratice, cubicae etc. accelerat[ionis]. […] alterum descendentem acceleratum quadratice. Dico parabolam cubicam fieri. […] Quae vero braevitatis causa exemplificavimus in cubica dici possent de quacunque parabola, Torricelli, Opere, 1919–1944, I, Part 2, pp. 309–310.
85.
We are referring to statement [49]: “Let AB be the time […], and during this time AB let a body traverse the lines GF and OH–on the one hand, GF with a uniform motion having a constant degree of velocity AV, and on the other hand, OH with a non-uniform motion having degrees of velocity homologous to the lines AC or ME. I say that the spaces traversed GF and OH are between them as the figures ACDB and AVDB”, Esto tempus AB, moveaturque mobile, et tempore AB percurrat rectas GF, OH, sed rectam GF currat motu aequabili cum gradu velocitatis semper eodem AV, rectam vero OH currat motu non aequabili cum gradibus velocitatis homologis ad lineas AC, sive ME. Dico spatium GF ad OH esse ut figura ACDB ad figuram AVDB, Torricelli, Opere, 1919–1944, I, Part 2, p. 259; Wilson, p. 113.
86.
See Chap. 5.

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University of Padua, Padua, Italy
Tiziana Bascelli

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Tiziana Bascelli
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Département de Philosophie, Université de Nantes, Nantes, France
Vincent Jullien

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Bascelli, T. (2015). Torricelli’s Indivisibles. 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_6

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