Cavalieri’s reply on the bowl and cone, 2 October 1634, with the 12 September 1634 letter and Aproino 1635 and Santini 1636 on indivisibles (Opere XVI)
Source: Internet Archive
le-opere-di-galileo-galilei-edizione-nazionale-volume-16; local OCR excerpt.txt, archive_djvu.txtlines 8833–8907 (letter 2988), 9110–9259 (letter 2992), 15093–16039 (Aproino) and 25801–26984 (Santini), retrieved 2026-09-08. Metadata: Le Opere di Galileo Galilei, Edizione Nazionale, vol. XVI, Carteggio 1634–1636 (Florence, 1905): no. 2988, Bonaventura Cavalieri to Galileo in Florence, Bologna, 12 September 1634; no. 2992, Cavalieri to Galileo in Arcetri, Bologna, 2 October 1634, printed pp. 136–138, Bibl. Naz. Fir. Mss. Gal. P. VI, T. XII, car. 85–86, autograph; Paolo Aproino to Galileo, Venice, 3 March 1635; Antonio Santini to Galileo, Milan, 16 January 1636. Extraction: archive.org OCR; Italian prose is clean. Visual check: printed pp. 135–138 (PDF pages 138–141 of the archive PDF) were rendered and read on 2026-09-08; the header, date, manuscript reference, the bowl-and-cone paragraph, the semicircle example, the concentric-circles paragraph and the closing sentence on composition were confirmed against the page images. The archive’s full-text index places “scodella” on its pages 138–139 and “componere” on 140. An earlier version of this companion dated the reply 12 September because a generic header pattern missed the bracketed header of no. 2992; the date is corrected here. Rights: public domain.
Source digest
Letter 2992 answers Galileo’s objection from the bowl and the cone. Cavalieri opens by regretting that Galileo’s ill health prevents a close examination of the Geometria, acknowledges that Galileo finds “molte difficoltà” in a method that passes to the infinite, and says he added Book VII to prove the same results by another route free of that infinity. He then restates the objection: if all the lines of two equal surfaces are equal, and they diminish equally, their last vanishings should be equal, which fails in the bowl and cone, where a circumference remains in one and a point in the other. His answer: the indivisibles in that example are planes; removing equal parts from cone and bowl leaves equal remainders until the null plane is reached in both, so the last vanishings are equal, “negatively rather than positively”, and it does not matter that a point is left in one and a line in the other, since neither is a plane. A semicircle example follows, in which the rectangle under the diameter and the point d equals the square on d, both null. On the concentric circles he separates points of right transit from points of oblique transit and takes only the former as measures. Finally: “assolutamente io non mi dichiaro di componere il continuo d’indivisibili, ma solo mostro che i continui hanno la proportione delli aggregati di questi indivisibili”. Letter 2988 precedes it; Aproino writes on the indivisible and the infinite as Galileo teaches them; Santini discusses indivisibles at length. For this project, letter 2992 is the direct exchange on Democritus’s dilemma between the two founders of the modern method, and its closing sentence is the position the project’s cut estimator formalizes: compare aggregates, never compose.
Passage anchors
- Printed p. 136, lines 18–20: Galileo’s objection restated with the bowl and the cone.
- Printed p. 137, lines 26–33: equal parts removed until the null plane, “essendo noi arrivati al nullo piano tanto nel cono quanto nella scodella”.
- Printed p. 137, lines 51–60: the concentric circles and right versus oblique transit.
- Printed p. 138, lines 69–72: “non mi dichiaro di componere il continuo d’indivisibili, ma solo mostro che i continui hanno la proportione delli aggregati di questi indivisibili”.
Coverage and limits
Reading level passage with visual verification for
letter 2992; metadata for letter 2988, Aproino and Santini
beyond their keyword lines.