Abstract
From the very beginnings of statics we were able to discern two types of minds grappling with the difficulties of this science: the intuitive versus the deductive mind. We have seen Aristotle, or rather whoever the author of the Mechanical Problems might be, look deeply into the principle governing the science of equilibrium, without, however, consolidating his views into perfect order. On the contrary, we have seen Archimedes striving not to set forth any proposition which could not be rigorously deduced from clear and explicitly stated postulates.
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References
Other authors spell his name differently. Favaro, for example, writes, Guidobaldo dal Monte. We follow the spelling adopted by Pigafetta in the translation of the Mechani-corum liber (T. N.: The Latin title reads, The Book on Mechanics) which Pigafetta published in Italian in 1581, during the author’s lifetime.
Hieronymi Cardani Mediolanensis civisque Bononiensis Opus novum de proportioni-bus numerorum… Propositio CLXXVI, p. 197; Basileae, MDLLXX. (T. N.: The Latin title reads, A New Work on the Ratios of Numbers by the Milanese Jerome Cardan, a citizen of Bologna… Proposition 176, p. 197; Basel, 1570).
Admiranda Archimedis monumenta omnia quae exstant, ex traditione D. Francisci Maurolyci. Panormi, ap. Cyllenium Hispanicum, DMCLXXXV. (T. N.: The Latin title reads, All the Extant and Admirable Monuments of Archimedes,expounded by Francesco Maurolico. Palermo, Cyllenius Hispanicus, 1685).
Federici Commandini Liber de centro gravitatis solidorum, Bononiae, MDLXV. (T. N.: The Latin title reads, A Book on the Center of Gravity of Solids, by Frederico Commandino, Bologna, 1565).
Lucae Valerii De centro gravitatis solidorum libri III, MDCIV. (T. N.: the Latin title reads, Three Books on the Center of Gravity of Solids, by Luca Valerio, 1604)
Galileo Galilei, Dialoghi delle Scienze nuove… Giornata seconda. (T. N.: The Italian reads, Dialogues on New Science… Second Day. Duhem must be referring to the Dialogues Concerning Two New Sciences.)
Guido Ubaldi e Marchinonibus Montis De cochlea libri quatuor, superiorum permissu et privilegio. Venetiis, apud Evangelistam Deuchinum, MDCXV. (T. N.: The Latin title reads, Four Books on the Screw by Guido Ubaldo, Marquis del Monte, with the permission and privilege of his Superiors. Venice, Evangelista Deuchinus, 1615)
Guidi Ubaldi e Marchionibus Montis In duos Archimedis aequiponderantium libros paraphrasis, scholiis illustrata. Pisauri, apud Hieronymum Concordiam, MDLXXXVIIL (T. N.: The Latin title reads, A Paraphrase of Two Books of Archimedes on Equilibrium illustrated with scholia. Pesaro, Jerome Concordia, 1583.)
Guidi Ubaldi e Marchionibus Montis Mechanicorum liber,in quo haec continentur: De libra, de vecte, de trochlea, de axe in peritrochio, de cuneo, de cochlea. Superiorum permissu et privilegio. Pisauri, apud Hieronymum Concordiam, MDLXXVII. (T. N.: The Latin title reads, The Book on Mechanics of Guido Ubaldo, Marquis del Monte, which contains the following: On the Balance. On the Lever, On the Block and Tackle, On the Windlass, On the Wedge, On the Screw, With the permission of his Superiors. Persaro; Jerome Concordia, 1577.) — The same work has been reprinted in: Venetiis, apud Evangelistam Deuchinum, MDCXV. — It has also been translated into Italian under a title of rather curious grandiloquence: Le mechaniche dell’illustriss. Sig. Guido Ubaldo de Marchesi del Monte, tradotte in volgare dal Sig. Filippo Pigafetta. Nelle quali si contiene la vera dottrina di tutti gli istrumenti principali di mover pesi grandissimi con piccola forza. A beneficio di chi si diletta di questa nobilissima scienza; et massimamente di capitani di guerra, ingegnieri, architetti, et d’ogni artefice, che intenda per via di machine far opre maravigliose, e quasi sopranaturali. Et si dichiarano i vocabili, et luoghi più difficili. In Veneria, appresso Francesco di Franceschi Sanese, MDLXXXI. — A second edition of this translation was published in Venice in 1615. (T. N.: The Italian reads, The Mechanics of the very illustrious Guido Ubaldo, Marquis del Monte, translated into Italian by Filippo Pigafetta. In which is included the true doctrine of all the principal instruments for moving very large weights with a small force. For the benefit of those who take pleasure in this extremely noble science and most of all for captains of war, engineers, architects and every artisan who intends to do marvelous and almost supernatural things with the help of a machine. With an explantion of the most difficult words and passages. In Venice, by Francesco di Franceschi Sanes, 1581.
Guidi Ubaldi Mecanicorum liber, ad Franciscum Mariam II, Urbinatum ducem, praefatio. (T. N.: The Latin title reads, The Book on Mechanics by Guido Ubaldo, to Francesco Maria II, Duke of Urbino, Preface.)
Guidi Ubaldi Mecanicorum liber, de libra, Propositio IV. (T. N.: The Latin title reads, The Book on Mechanics by Guido Ubaldo, “On the Balance, Proposition IV.”)
We purposely omit, while presenting Guido Ubaldo’s argumentation, the convergence of verticals, which troubles the author.
T. N.: The lever arm is pivoting about point D.
Les Méchaniques de Galilée, mathématicien et ingénieur du duc de Florence, avec plusieurs additions rares et nouvelles, utiles aux architectes, ingénieurs, fonteniers, philosophes et artisans; traduites de l’Italien par L. P. M. M. (le P. Mersenne, Minime). A Paris, chez Henry Guenon, MDCXXXIV; 2e addition, p. 23. (T. N.: The French titles read, The Mechanics of Galileo, mathematician and engineer to the Duke of Florence, with several rare and new additions useful to architects, engineers, excavators, philosophers and craftsmen; translated from Italian By L. P. M. M.: Father Mersenne, of the Order of the Minims. Paris, Henry Guenon, 1634, 2nd edition, p. 24.)
The Book on Mechanics by Guido Ubaldo, “On the Lever,” Proposition X.
(a) Pierre Hérigone, Cours de Mathématique, Vol. III: les Méchaniques, Proposition VI, Paris, 1634. (T. N.:The French title reads, Pierre Hérigone, Course on Mathematics,Vol. III.; Mechanics, Proposition VI, Paris 1634).
(b) Oeuvres posthumes de J. Rohault (publiées par Clerselier). Traité des Méchaniques. Proposition XI Paris, 1682. (T. N.: The French title reads, Posthumous Works of J. Rohault (published by Clerselier). Treatise on Mechanics, Proposition XL Paris, 1682.)
Lagrange, loc. cit. Art. 16.
Guidi Ubaldi Mecanicorum liber, de vecte. Propositio III. (T. N.: The Latin title reads, The Book on Mechanics by Guido Ubaldo, “On the Lever,” Proposition III.)
Guidi Ubaldi Mecanicorum liber, de trochlea, Propositiones X ad XXVIII. (T. N.: The Latin title reads, The Book on Mechanics by Guido Ubaldo, “On the Block and Tackle,” Propositions X to XXVIII.)
Guidi Ubaldi Mecanicorum liber, de trochlea, Propositio XXVI, Corollarium. (T. N.: The Latin title reads, The Book on Mechanics by Guido Ubaldo. “On the Block and Tackle.” Proposition XXVI, Corollary.)
Id., Guidi Ubaldi Mecanicorum liber, de trochlea, de trochlea, Propositio XXVIII, Corollarium II.
Guidi Ubaldi Mecanicorum liber, de trochlea, de axe in peritrochio. (T. N.: The Latin title reads, The Book on Mechanics by Guido Ubaldo, “On the block and tackle, On the Windlass.”)
Id., Guidi Ubaldi Mecanicorum liber, de trochlea, de axe in peritrochio, de cuneo (T. N.: The Latin title reads, “On the Wedge,”)
Id., Guidi Ubaldi Mecanicorum liber, de trochlea, de axe in peritrochio, de cochlea. (T. N.: The Latin title reads, “On the Screw.”)
Jo. Baptistae Benedicti, patritii Veneti, philosophi, Diversarum speculationum mathematicarum et physicarum liber, quarum seriem sequens pagina indicabit. Ad serenissimum Carolimi Emmanuelem Allobrogum et Subalpinorum ducem invictis-simum. Taurini, apud haeredem Nicolai Beveliquae, MDLXXXV. (T. N.: The Latin title reads, A book of Diverse Speculations on Mathematics and Physics, with the sequence indicated on the following page, by Giovanbattista Benedetti, citizen of Venice and philosopher. To his most Serene Charles Emmanuel, the invincible Duke of Savoy and the Sub-Alpines. Turin, the Heir of Nicolò Bevilaqua, 1585.)
J. B. Benedicti Diversarum speculationum… p. 141. De mechanicis.
T. N.: The Latin quote reads, “And by this means alone, I would have left clear proof that I had lived among mortals.”
De resolutione omnium Euclidis problematum aliorumque ad hoc necessario inven-torum una tantummodo circuii data apertura, per Joannem Baptistam de Benedictis inventa. On the last page: Venetiis, apud Bartholomaeum Caesanum. MDLIII. (T. N.: The Latin title reads, On the Resolution of all the Problems of Euclid and of others by a single setting of the Compass, invented by Giovanbattista Benedetti, Venice, Bartho-lomaeus Caesanus, 1553.)
This passage was reproduced by Libri, Histoire des Sciences mathématiques en Italie, note XXV. vol. III. p. 258.
J.B. Benedicti Diversarum speculationum… Disputationes de quibusdam placitis Aristotelis, Caput, X, p. 174. — This passage is also reproduced by Libri, loc. cit., p. 264. (T. N.: The Latin title reads, Giovanbattista Benedetti’s Diverse speculations… Disputations on certain views of Aristotle, Chapter X, p. 174.)
Hieronymi Cardani Mediolanensis, civisque Bononiensis, philosophi, medici et mathematici Opus novum de proportionibus; Basileae, ex officina Henricpetrina, Anno Salutis MDLXX, Mense Martio. Liber V, Propositio CX, p. 104.
Simonis Stevini Mathematicorum Hypomnematum Statica. Appendix Statices. Caput II: Res motas impedimentis suis non esse proportionales, p. 151. Lugodini Batavorum, MDCV. (T. N.: The Latin reads, Simon Stevin, Mathematical Mémoires on Statics. Appendix on statics. Chapter II: Moving Bodies are not proportional to their resistances, p. 151. Leyden, 1605.)
J. B. Benedicti Diversarum speculationum… De mechanicis, Caput XI, p. 153.
Id., ibid., Caput VII et Caput VIII.
J. B. Benedicti Diversarum speculationum… De mechanicis, Caput I.
J. B. Benedicti Diversarum speculationum… De mechanicis, Caput II.
Id., J. B. Benedicti Diversarum speculationum… De mechanicis, Caput III.
Les Manuscrits de Léonard de Vinci,published by Ch. Ravaisson-Mollien, Ms. I of the Bibliothèque de l’Institut, folio 30, recto. Cf. chapter II.
T. N.: This quote is not clear as it appears in the French.
J. B. Benedicti Diversarum speculationum…. De mechanicis, Caput XII.
T. N.: Although Benedetti uses the term “gravity,” the text implies “positional gravity.”
Les Manuscrits de Léonard de Vinci,published by Ch. Ravaisson-Mollien, Ms. E. of the Bibliothèque de l’Institut, folio 57, verso; folio 58, recto; folio 59, recto. — Cf. Chapter VIII, Section 1.
Cf. Chapter IX, Section 2.
J. B. Benedicti Diversarum speculationum… De mechanicis, Caput XXI.
Les Manuscrits de Léonard de Vinci,published by Ch. Ravaisson-Mollien, Ms. E. of the Bibliothèque de l’Institut, folio 55, recto.
J. B. Benedicti Diversarum speculationum… De mechanicis, Caput V.
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Duhem, P. (1991). The Reaction Against Jordanus Guido Ubaldo — G. B. Benedetti. In: The Origins of Statics. Boston Studies in the Philosophy of Science, vol 123. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3730-0_10
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