Guldin, Centrobaryca, Book IV (1641): the preface to the reader on Cavalieri and Galileo, and chapter V against the Geometria indivisibilibus
Source: Max Planck Institute for the History of Science, Digital Libraries Connected, record 868389986, DOI 10.48644/868389986, IIIF image API; fourteen page images stored in
guldin/: preface pages 1–7 at canvases 0397–0403 (p1, p2, p3, p4, p5, p6, p7) and printed pages 339–345 at canvases 0735–0741 (p339, p340, p341, p342, p343, p344, p345), retrieved 2026-09-08 at 1400 pixels on the long side. Metadata: Paul Guldin, De centro gravitatis trium specierum quantitatis continuae (Centrobaryca), Liber quartus (Vienna: Matthäus Cosmerovius, 1641); the record holds all four books (1635–1641) in 811 canvases, extent [ca. 20], 227, 99, [ca. 50], 401 pages. Book IV opens at canvas 0397 with “Lectori S.”; chapter V, “Perpenduntur quaedam ex Nova Geometria Bonaventurae Cavalerij desumpta”, begins on printed p. 339. Extraction: page images only; no OCR exists for this record. The preface passages below were read from the images by the session and normalised (long s to s, ligatures resolved); they are a target for verification by a second reader. Chapter V pages were identified from a thumbnail montage by their headings. Rights: public domain original; the MPIWG serves the images under its digital-library terms with a DOI, cited here.
Source digest
The preface of Book IV names Cavalieri as “Geometra insignis”, praised by Galileo in his Dialogues on local motion, and says that Guldin, who knew Galileo personally, never thought Cavalieri deserved harsh judgement. It then states that Cavalieri took his cue from Kepler to bring the method of indivisibles into the light, that Guldin will not say how students of purer geometry approve it, and that Galileo himself, in the same Dialogue on local motion, disputing about the infinite and about properties of finite things that may not be applied to infinites, concludes against Cavalieri. Cavalieri, following Sover and Kepler, uses lines and parallel indivisible and infinite planes and concludes from them what is asked about finites; Guldin sees a grave dispute among the learned and no Aeacus or Minos to settle it. Chapter V then examines Book II of the Geometria proposition by proposition: Propositio I on the preliminaries of Cavalieri’s Book II, Propositio II on its first proposition and scholium, Propositio III on its second and third propositions with corollaries, a Scholium, and Propositio IV on the first proposition of Book VII. For this project, this is the objection Cavalieri answers in the Exercitatio III of 1647, and the preface’s sentence on Galileo ties Guldin’s critique to the First Day paradoxes stored in this folder.
Passage anchors
- Preface p. 3, canvas 0399: “Geometriam Promotam Bonaventurae Cavalerij Ordinis Iesuatorum, Geometrae insignis, à Galileo … in suis Dialogis de Motu Locali, &c. laudati”.
- Preface p. 4, canvas 0400: “Galileus profecto in eodem Dialogo de Motu locali, disputans de infinito, de proprietatibus finitorum, quas infinitis applicare minime liceat, contra ipsum concludit.”
- Printed p. 339, canvas 0735: “CAPUT V. Perpenduntur quaedam ex Nova Geometria Bonaventurae Cavalerij desumpta.”
- Printed pp. 340–345: Propositiones I–IV and the Scholium on Cavalieri’s Book II and Book VII.
Coverage and limits
Reading level passage for preface pp. 3–4 from the
images; metadata with page identification for chapter V,
whose text remains to be transcribed. The chapter continues beyond
p. 345; later pages were not stored.
Transcription
Preface p. 3 (canvas 0399), from the middle of the page:
… Geometriam Promotam Bonaventurae Cavalerij Ordinis Iesuatorum, Geometrae insignis, à Galileo (quem eodem tempore licuit videre) in suis Dialogis de Motu Locali, &c. laudati, cui palmariam olim famam inter excellentes aevi nostri Mathematicos pollicebatur. Et ut verum fatear quem Galileus, mihi ex facie & indole notus laudavit, nunquam mordaciori iudicio distringendum censui; qui si tot habuisset laudis fautores, quot habuit obtrectatores, per ipsum floridior augustiorque staret honos disciplinarum Mathematicarum. Reperi autem …
Preface p. 4 (canvas 0400), whole page:
… autem in his Libris Geometriae Promotae, Authorem ansam arripuisse ex Keplero, ut suam non tam inveniret, quam ab umbris in lucem publicam vindicaret, Methodum Indivisibilium. Quae tamen methodus quomodo purioris Geometriae studiosis probetur non edicam; eam tamen, propter rationes hic minimè importuno silentio supprimendas, respuendam non censeo. Galileus profecto in eodem Dialogo de Motu locali, disputans de infinito, de proprietatibus finitorum, quas infinitis applicare minime liceat, contra ipsum concludit. Hic ergo Cavalerius secutus tam Bartholomaeum Soverum, qui Libro quinto de Curvi ac Recti proportione promota, parallelarum ac figurarum analogarum virtutes ac proprietates tradit, quam Keplerum, qui per infinita plana & corpora minutissima, maiora componit (si quidem componit) ut videbimus Libro 4. Cap. 4. putabat se sensum attigisse Archimedis, quo ille sua facile quidem proponat, demonstret autem & spinose & laboriosissime, per novam istam resolutionem & compositionem facillimo modo demonstrare se, ipso etiam Archimede consentiente, iudicat. Hos igitur Cavalerius non solum imitari, sed & ea, quae ipsi in paucis duntaxat & sigillatim tantum persecuti sunt, ipse generatim in pluribus, immo in infinitis & universe demonstrare conatus est. Sed hoc refellit Alexander Andersonius Scotus, ut in loco patebit. Plana autem illa Kepleriana cum in punctum omnia, & corpora in axem coirent, minus ad rem facere visa sunt Cavalerio; utitur ergo ille lineis, & parallelis planis indivisibilibus & infinitis, deque illis concludit id quod de finitis quaeritur. Verum circa hanc rem gravi litigio doctissimorum virorum animi iure merito dividi possunt; nec satis video quis tot dissidia tamque diversa, aut Aeacus, aut Minos componat. Agnovit Cavalerius solidorum Rotundorum, quemadmodum …