Abstract
VJ. Cavalieri was born in Milan most probably on 1598. His family was not wealthy; he entered into Jesuats of St. Jerome order and he staid in Pisa, where he met Benedetto Castelli, lecturer in Mathematics at the University. He arrived in Pisa, completely lacking in mathematical preparation. Cavalieri went to Florence and remained in this town for more than 1 year (1617–1618) until he was asked to come back to Pisa to substitute for Castelli in his lectureship. By that time he had mastered Euclid and Archimedes, and probably Apollonius, and had begun to study Ptolemy. In 1619—very young—he applied for the vacant chair of Magini, professor in mathematics in Bologna and did not obtain it. Ten years later, probably through Galileo’s influence, Cavalieri eventually obtained a professorship in mathematics at the University of Bologna. Cavalieri taught mathematics in this town until his death in 1647. He had systematic scientific relations with Galilei. His main works are Lo specchio ustorio overo trattato delle settioni coniche (1632), Geometria indivisibilibus continuorum nova quadam ratione promota (1635, quoted as Geometria) and Exercitationes geometricae sex (1647, quoted as Exercitationes). He published other books on mathematics, mathematical sciences and a table of logarithms. One of the books treating astrology was published under the pseudonym Silvio Filomantio.
This chapter has been organized by Vincent Jullien, with material taken from two classical works on Cavalieri’s Indivisibles: Giusti (1980a) and Andersen (1985). We have noted “E.G.” at the beginnings of paragraphs due to Enrico Giusti, “K.A.” at those due to Kirsti Andersen. Some paragraphs has been translated by Sam Brightbart.
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Notes
- 1.
References to the Geometria in this chapter are to the second edition, Cavalieri, 1653. According to Urbano D’Aviso, brother in religion to Cavalieri.
- 2.
Marie (1883–1888, vol. 4, p. 90).
- 3.
Boyer (1941, 8, p. 85).
- 4.
See Chap. 4.
- 5.
Published and commented by Giusti (1980a, pp. 85–90).
- 6.
Here, we use a modern terminology where Cavalieri uses plane figures and solid bodies.
- 7.
Cavalieri to Galileo, 15 December 1621, Galilei Galileo (Ed. Naz.), vol. 13, pp. 81–82, n° 1515.
- 8.
A discussion of the contributions to geometry during this early period would be beyond the scope of this introduction; it will suffice to quote F. Commandino, whose translations of the classics have long been a model, and L. Valerio, called “nuovo Archimede dell’eta nostra”, Galileo (1638, vol. 8, p. 76).
- 9.
Il est vray qu’ils [les anciens] n’ont pas aussy entièrement receu les sections coniques en leur geometrie, 1637, Descartes, La géometrie, Livre 2, A.T. VI, p. 389.
- 10.
Si per oppositas tangentes cuiuscumque datae planae figurae ducantur duo planas invicem parallela, recta, sive inclinata ad planum datas figurae, hinc inde indefinite producta; quorum alterum moveatur versus reliquum eidem semper aequidistans donec illi congruerit: singulae rectae lineae, quae in toto motu fiunt communes sectiones plani moti, & datae figurae, simul collectae vocentur : Omnes lineae talis figurae, sumptae regula una earumdem. Geometria, Book II, def. 1, p. 99 and Exercitatio I, Definitio I.
The following figure is a reconstruction not to be found in Cavalieri’s Geometry; see Andersen (1985, p. 301).
- 11.
For a fuller discussion, see Chap. 2, p. nnn.
- 12.
Si, proposito quocumque solido, eiusdem opposita plana tangentia regula quacumque ducta fuerint, hinc inde indefinite producta, quorum alterum versus reliquum moveatur semper eiusdem aequidistans, donec illi congruerint; singula plana, quae in toto motu concipiuntur in proposito solido, simul collecta, vocentur : Omnia plana propositi solidi, sumpta, regula eorundem uno; Geometria, Book II, def. II, p. 100, and Exercitatio I, Prop. III; stress added.
- 13.
Figurae planae habent inter se eandem rationem, quam eorum lineae iuxta quamvis regulam assumptae; et figurae solidae, quam eorum omnia plana iuxta quamvis regulam assumpta, Geometria, Book II, th. III, prop. III, p. 113, and Exercitatio I, Prop. III.
The first proposition of this second book is: All the lines of several plane figures and all the planes of several solid figures are magnitudes which have a ratio to each other.
Quarumlibet planarum figurarum omnes lineae recti transitus; & quarumlibet solidarum omnia plana sunt magnitidines inter se rationem habentes. (id. p. 108).
- 14.
With this key the paradox of two figures having different areas, and whose indivisibles can be set in a one-to-one correspondence, might possibly be read. See for instance Exercitatio III, p. 180.
- 15.
See for instance, Exercitatio III, p. 180.
- 16.
See Chap. 2.
- 17.
See Chap. 4.
- 18.
Exercitatio III, p. 203.
- 19.
Besides the references see Geometria II, Scholium p. 111, Th. II, Prop. II, Corollarium, pp. 113–114 and Exercitatio III, p. 199.
- 20.
Occasionally I apply the term collection of lines in stead of “all the lines” or alternatively collection of lines should be changed to “all the lines”.
- 21.
aliquid aliud, Geometria II, Scholium, p. 111.
- 22.
This point of view is also expressed in Lombardo-Radice 1966, e.g. p. 206, and in Cellini 1966, p. 9.
- 23.
Artificio autem tali usus sum, quale ad propositas quaestiones ab solvendas Algebratici adhibere solent; qui quidem numerorum radices, quamvis ineffabiles, surdas, ac ignotas, nihilominus simul aggregantes, subtrahentes, multiplicantes, ac dividentes, dunimodo proposite rei exeptatum sibi notitia enucleare valeant, sua satis ob sse munera (ibi persuadent. Non aliter ipse ergo indivisibilium sive linearum, sive planarum congerie licet quoad corundem numerum innomabilis, surda, ac ignora, quoad magnitudinem tamen conspicuis limitibus clausa, ad continuorum investigandam mensuram usus sum, ut legenti in processu operis apparebit. Geometria, Praefatio, p. 2.
- 24.
This is not the Giusti’s, neither the Andersen’s point of view.
- 25.
Andersen (1985, p. 308).
- 26.
Exercitatio I, post. II, XIII, pp. 15–16.
- 27.
Geometria, p. 108.
- 28.
See more commentaries, p. 49.
- 29.
Parallelogramma in eadem altitudine existentia inter se sunt, ut bases… Geometria II, Th. V, prop. V, p. 117.
- 30.
Kepler, Nova Stereometria, Gesammelte Werke, Band IX. See Chap. 4.
- 31.
Solidum, cuius omnes descriptae figurae silmiles sunt omnia plana, dicetur, solidum similare genitum ex proposita … Geometria, II, def. VIII, p. 103, and Exercitatio I, def. VIII, p. 10.
- 32.
Geometria, Book 2, and Exercitatio I, Propositio XXXIII.
- 33.
Geometria, Book 2, Propositio XV. See Exercitatio I, Propositio XXIII.
- 34.
After a first version based on curved indivisibles a version whose remnants might be found in the proof of Proposition IX, as well as in Definitions I and II of the sixth book. That is to say the booklet of 1622 on spirals, see Giuntini (1985).
- 35.
Geometria, Book 2, and Exercitatio I, Corollarium IV Generale, Section IX.
- 36.
Quoniam vero solida ad invicem similaria genita ex duabus figuris planis, iuxta datas regulas, totuplicia sint, quotuplices sunt figurae similes, quae dicuntur, omnes figurae similes duarorn genitricium figurarum, cum eisdem regulis assumpta, iuxta quas dicta solida similaria genita dicuntur, figurarum autem variationes nullo dato numero clauduntur, ideo nec horum simiIarium solidorum variationes ullo dato coartantur numero, unde evidentissime apparet hanc demonstrandi methodum, ipsamque demonstrationem, infinite (ut ita dicam) locupletem esse… Geometria III, Th. XXXIII. Prop. XXXIV, Scholium, p. 257.
- 37.
So is the abovementioned Proposition XV of the second book, and the corresponding Proposition XVII for solids, two among the most hard won results achieved by Cavalieri.
- 38.
Definitions X and XI; propositions XXVII & fol.
- 39.
Geometria III, def. VIII, p. 103.
- 40.
Geometria II, Appendix, p. 107.
- 41.
See what this notion has in common with the ductus plani ad planum in Gregory of St. Vincent’s works or in Roberval’s or Pascal’s concepts in relevant chapters of this book.
- 42.
Exposito parallelogrammo quocumque in eoque ducta diametro, omnia quadrata parallelogrammi ad omnia quadrata cuiusuis triangulorum per dictam diametrum constitutorum erunt in ratione tripla, uno laterum parallelogrammi communi regula existente. Geometria II, Th. XXIV, prop. XXIV, p. 159.
- 43.
Figurae planae quaecumque in eisdem parallelis constitutae, in quibus, ductis quibuscumque eisdem parallelis aequidistantibus rectis lineis, conceptae cuiuscumque recta lineae proportiones sunt inter se, ut cuiuslibet alterius in eisde figuris conceptae proportiones (homologis tamen in eadem figura simper existentibus) eandem inter se proportionem habebunt, quam dictae proportiones. Geometria, Book 7, Proposition II (which is a generalization of Prop. I) p. 497 and Exercitatio II, Th. II, prop. II, p. 103.
- 44.
Euclid, Elements, book V, definition IV.
- 45.
Euclid, Elements, Book I. Proposition XXXVIII.
- 46.
Cupimus ostendere modo Archimedeo aequalitatem duorum corporum; nobis inscribenda erent, in praefatis corporibus, alia corpora, nimirum, vel cylindri, vel prismata, &c. ostendendo unicuique inscriptum in uno corpore, aequari aliud in alio corpore inscriptum; unde tandem colligemus, omnia corpora inscripta in uno corpore, aequari omnibus in aIio corpore inscriptis. Verum, quoniam isthaec corpora in illis corporibus inscripta, Iicet sint partes illorum corporum, attamen, nequaquam sunt omnes partes, minimeque sunt partes aliquotae, et aliquantae, ideo ad colligendam aequalitatem inter ipsa corpora, deductio ad impossibilem omnino necessaria conspicitur. Secus accideret si partes illae & aliquotae, & omnes essent. Statim enim probata aequalitate omnium unius corporis, cum omnibus alterius corporis partibus, clarissima, directaque consequentia, aequaIitas corporum innotesceret. Cur ergo ratiocinatio per indivisibilia semper est regia, semper directa? Non alia sane videtur assignabiIis germana causa, nisi quia indivisibiIibus utendo, utimur omnibus magnitudinum partibus.
De Angelis (1659).
- 47.
Congruentium planarum figurarum omnes lineae .. sunt congruentes. Geometria II, post. I, p. 108.
- 48.
Aequalium planarum figurarum omnes lineae sunt aequales. Geometria II, Th. II, Prop. II, p. 112. To understand the difference between the two statements we recall that equal means always equivalent, or of equal areas.
- 49.
Cavalieri started in the first theorem of Geometria, Book VII, Th. I, prop. I, p. 485, by imagining that situation.
- 50.
See the letter to Torricelli, Bologna 10 March 1643.
- 51.
Geometria II, Th. IV, prop. IV, p. 115. See supra, p. 40.
- 52.
See supra, pp. 39–42, about the Principle, ut unum ad unum sic omnia ad omnia.
- 53.
Ut igitur…unum antecedentium ad unum consequentium, ita erunt omnia antecedentia, nempe omnes lineae figurae CAM,…, ad omnia consequentia, scilicet ad omnes lineas figurae CME. Geometria, II, p. 116.
- 54.
See Chap. 4.
- 55.
Analysis Tetragonistica ex Centrobarycis, pars secunda. G.W. Leibniz to Oldenburg, 29 October 1675. Leibniz, Math. Schr., I, p. 154.
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Andersen, K., Giusti, E., Jullien, V. (2015). Cavalieri’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_3
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