¹ Other papers in which I have reported the results of my researches include: “The Concept of Motion in the Sixteenth Century”, Proceedings of the American Catholic Philosophical Association, xli (1967), 184–195Google Scholar; and “The Enigma of Domingo de Soto: Uniformiter Difformis and Falling Bodies in Late Medieval Physics”, in Isis (in press).

² Etudes sur Léonard de Vinci (Paris, 1906–1913). The principal writers among the “Calculatores” to whom I refer were associated with Merton College, Oxford; they include Thomas Bradwardine, William Heytesbury (Hentisberus), and Richard Swineshead (Suiseth), and, to a lesser extent, Walter Burley.

³ For a chronology of the details of Soto's life, see Vicente Beltrán de Heredia, O.P., Domingo de Soto: Estudio biográfico documentado. Biblioteca de Teologos Españoles, vol. 20 (Salamanca, 1960); for a history of the Spanish universities, see Rashdall, Hastings, The Universities of Europe in the Middle Ages, 3 vols., ed. Powicke, F. M. and Emden, A. B. (Oxford, 1936), ii, 63–114.Google Scholar

⁴ One of the important sources for sixteenth-century studies in mechanics are the various expositions of Heytesbury written by Italian commentators and published at Venice in 1494; this work is described in Wilson, Curtis, William Heytesbury: Medieval Logic and the Rise of Mathematical Physics, University of Wisconsin Publications in Medieval Science, No. 3 (Madison, 1960), 4, fn.Google Scholar, and passim. Another source is a compilation of treatises on ratios (proportiones) published at Venice in 1505, including the Tractatus proportionum introductorius ad calculationes Suisset by Bassanus Politus and the commentary on Albert of Saxony's Tractatus proportionum by Benedictus Victorius Faventinus, as well as the better-known treatises by Bradwardine and Nicole Oresme. Writers of Soto's own Order who discussed methods of calculating ratios include Isidorus de Isolanis, O.P., whose De velocitate motuum was printed at Pavia in 1522, and Chryso-stomus Javellus, O.P., whose Quaestiones in libros Physicorum, completed before 1532, appears in printed editions of Venice, 1564 and Lyons, 1568. Other significant “calculatory” works by authors who were known to Galileo and his teachers are the opusculum In quaestione de motuum proportionibus of Alessandro Achillini, in the Omnia opera printed at Venice in 1545, and the Opus novum de proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum mensurandarum of Girolamo Cardano, printed at Basle in 1570. Critical studies of these works are yet to be made, but undoubtedly a comparison of their contents with those treatises with similar titles appearing in France and Spain during the same period will shed considerable light on the evolution of thought that led to Galileo's contribution.

⁵ de Heredia, Beltrán, op. cit. (3), 16–17.Google Scholar

⁶ Ibid., 26, 34; on Encinas (Enzinas), see Vicente Muñoz Delgado, O. de M., La Logica Nominalista en la Universidad de Salamanca (1510–1530). Publicaciones del Monasterio de Poyo, n. 11 (Madrid, 1964), 130–131.Google Scholar

⁷ See Muñoz, , op. cit. (6), especially pp. 77–88.Google Scholar

⁸ The best account of Celaya is to be found in Villoslada, R. G., S.J., La Universidad de Paris durante los estudios de Francisco de Vitoria, O.P. (1507–1522). Analecta Gregoriana (Rome, 1938), xiv, 180–215Google Scholar; on Jean Mair (John Major), see Elie, Hubert, “Quelques Maitres de l'université de Paris vers l'an 1500”, Archives d'histoire doctrinale et littéraire du moyen âge, xviii (1950–1951), 193–243.Google Scholar

⁹ Apart from the works already cited by Duhem (2), Villoslada (8), and Elie (8), one should consult Renaudet, Augustin, Préréforme et humanisme à Paris pendant les premières guerres d'Italie, Bibliothèque de l'Institut Français de Florence, 1^re Série, Tome vi (Paris, 1916; new edn. Paris, 1953)Google Scholar, for further details of this period.

¹⁰ Many of these textbooks are described in Smith, D. E., Rara ArithmeticaGoogle Scholar: A Catalogue of the Arithmetics written before the year MDCI with a description of those in the library of George Arthur Plimpton of New York (Boston/New York, 1908). Apart from a series of works on logic, Lax produced several mathematical treatises that were quite influential, including his Arithmetica speculativa (Paris, 1515) and Proportiones (Paris, 1515); he wrote also a series of Questiones phisicales (Saragossa, 1527) that appear to have been lost. For details concerning Lax, see Solana, Marcial, Historia de la Filosofía Española: Epoca del Rinacimiento (Siglo XVI). 3 vols. (Madrid, 1941), iii, 19–33.Google Scholar

¹¹ Ribeyro, in particular, was a devoted disciple of Celaya, as attested by Villoslada (8), 209–211. Soto wrote a laudatory preface to Celaya's exposition of the Posterior Analytics, where he identifies himself as a disciple. On Celaya's close association with Alvaro Thomaz, see Villoslada, , p. 190.Google Scholar

¹² Villoslada, , op. cit. (8), 191.Google Scholar

¹³ The full title is Liber de triplici motu proportionibus annexis magistri Alvari Thome Ulixbonensis philosophicas Suiseth calculations ex parte declarans (Paris, 1509).

¹⁴ See fn. 4 above; on Politus, consult Duhem, , op. cit. (2), iii, 533.Google Scholar

¹⁵ Among others, Alvaro cites by name Burley, Bradwardine, Heytesbury, the “Calculator” (Swineshead), Albert of Saxony, Oresme, Paul of Venice, Gaetano da Thiene, James of Forli, John of Casali, Peter of Mantua—referring to his treatise De primo et ultimo instanti—and Bassanus Politus.

¹⁶ For particulars on Alvaro's disagreement with Oresme, see Grant, Edward, Nicole Oresme: De proportionibus proportionum and Ad pauca respicientes (Madison, 1966), fn., 56–58.Google Scholar

¹⁷ Some details are given in Pastor, J. Rey, Los Matemáticos Españoles del Siglo XVI (Toledo, 1926), 82–89.Google Scholar

¹⁸ This work is essentially a commentary on the eight books of Aristotle's Physics, interspersed with questions on the more controversial topics being discussed in the schools of the period.

¹⁹ The fuller title is Expositio … in octo libros Phisicorum Aristótelis cum questionibus ejusdem, secundum triplicem viam beati Thome, realium et nominalium; the “three ways” are those of Thomism, Scotism, and Ockhamism respectively.

²⁰ Coronel states: “…Ferme omnia que dicta sunt de difformibus qualitatibus possunt applicari difformi motui, quapropter in istis non insisto. Videantur regulae Hentisberi in tractatu de motu locali, que sunt satis bone et faciles, et qui in vacuum vellet tempus terere videat regulas Suyset: quia ego inutile reputo peramplius in his insistere.”—Lib. 3, pars 4, fol. 86r.

²¹ In Celaya's words: “…Conclusiones non solum ad medicinam, verum ad sacram theo-logiam applicari valent mutando ilium terminum ‘moveri’ vel ‘motus’ in aliquem istorum terminorum, seil., ‘febris’ vel ‘meritum’ vel ‘mereri’.”—Lib. 3, fol. 88rb.

²² On this point, see Duhem, , op. cit. (2), iii, 548–549.Google Scholar

²³ This movement is well described by Muñoz, , op. cit. (6), passim.Google Scholar

²⁴ Soto falls approximately in the middle of this development. His predecessors and contemporaries are sketched in what follows, while his students and disciples, such as Francisco Toledo, Pedro de Oña, Domingo Bañez, and Diego Mas were still influential at the end of the century.

²⁵ Tractatus arithmetice practice qui dicitur Algorismus (Paris, 1495). Rey Pastor cites further editions of 1502, 1505, 1509, 1513, and 1514—op. cit. (17), 155; see also 54–61.

²⁶ The title of this work is Cursus quattuor mathematicarum artium liberalium. It appeared first at Alcalá in 1516, and subsequent editions followed in 1523, 1526, and 1528.

²⁷ The Paris edition of 1498–1499 is described by Thorndike, Lynn, The Sphere of Sacrobosco and Its Commentators (Chicago, 1949), 39 and fn. 78Google Scholar; the title of the Alcalá edition of 1526 reads Opusculum de Sphera mundi Joannis de Sacro Busto: cum additionibus et familiarissimo commentario Petri Cirueli Darocensis, nunc recenter correctis a suo autore, insertis etiam egregiis questionibus Petti de Aliaco.

²⁸ See the prefatory letter to these Paradoxae quaestiones (Salamanca, 1538), addressed by Ciruelo to his students at the University of Salamanca, where he explains the title: “Quia fere omnia erunt preter communem doctorum opinionem, ea vocabulo greco ‘paradoxa’ censui noncupanda.”

²⁹ The arithmetic is entitled: Liber arithmetice practice astrologis, phisicis, calculatoribus admodum utilis (Paris, 1513); other editions appeared in 1514, 1519, and 1526 under slightly different titles. For the contacts between Silíceo and Soto, consult the index to Beltrán de Heredia's volume cited above.

³⁰ The title reads in part: Calculatoris Suiset anglici sublime et prope divinum opus … cura atque diligentia philosophi Silicei (Salamanca, 1520), Villoslada notes another edition of 1524, op. cit. (8), 191, fn. 19.

³¹ Diest is mentioned by Villoslada, but otherwise has been unnoticed by those working in this area. He is important for having transmitted some elements of Oresme's and Albert of Saxony's teaching to Spain in the early sixteenth century. See my forthcoming article on “The Enigma of Domingo de Soto …” (fn. 1, above).

³² The full title reads: Magistri Didaci Diest questiones phisicales super Aristotelis textum sigillatim omnes materias tangentes in quibus difficultates que in theologia et aliis scientiis ex physica pendent discusse suis locis inseruntur (Saragossa, 1511).

³³ As he notes in his introduction, Diest taught the arts course “in collegio fratrum minorum de observantia” at Saragossa; his intention is to cover “omnia quae dirficultatem penes Aquinatem Thon am, Joannem Scotum, Guillermum Okam, Gregorium Ariminensem, ceterosque moderniores horum sequaces in artibus facere possunt” (fol. 1v).

³⁴ On Margallo, see Muñoz, , op. cit. (6), 122–126Google Scholar; also Villoslada, , op. cit. (8), 397.Google Scholar

³⁵ The work contains a very brief summary of the Aristotelian corpus on natural philosophy, a synopsis of the Sphere of Sacrobosco, a treatise on ratios, and a somewhat disorganized discussion of selected topics relating to matter, form, privation and alternative qualities, including the intension and remission of forms.

³⁶ References to the “Calculator” occur on folios 24r, 28r, 29v, 30r, 30v, and 32r; the reference to “Neotericus Albarus” is on fol. 29r.

³⁷ Apart from Juan de Ortega, however, one should note the Dominican Tomas Durán, who edited Bradwardine's arithmetic and geometry at Valencia in 1503, and Diego Deza, a Dominican who later became Archbishop of Seville and is usually referred to in the literature of the day as “Hispalensis”, who composed a commentary on the Sentences (Seville, 1517) that discussed topics of nominalist and “calculatory” interest.

³⁸ On Astudillo, see Ramirez, S. M., O.P., “Hacia una renovación de nuestros estudios filosóficos: Un indice de la producción filosófica de los Dominicos españoles”, Estudios Filosóficos, i (1952), 8–9.Google Scholar

³⁹ The title reads Quaestiones super acto libros physicorum et super duos libros de generalione Aristotelis, una cum legitima textus expositione eorundem librorum ( Valladolid, 1532).

⁴⁰ Peter (Crokaert) of Brussels composed a series of Questiones phisicales, published at Paris in 1521; a convert to Thomism from nominalism, Peter seems responsible for the dialogue between the Dominicans and the disciples of Jean Mair at Paris.

⁴¹ References to “Sysset Calculator” are to be found on folios 14va, 22vb, 33va, 35vb, and 39va; those to “Alvarus Thomas” are on folios 14va, 22vb, 26rb, and 39rb.

⁴² A copy of Meygret, 's Questiones Fratris Amadei Meygreti Lugdunensis Ordinis Predicatorum in libros De celo et mundo Aristotelis (Paris, 1514)Google Scholar is in the university library at Salamanca; its presence there, and the use indicated by Astudillo's references, attests to the fact that the work of the Paris Dominicans was known in Spain during the early sixteenth century.

⁴³ The Latin reads: “Alia fieri soient argumenta contra alias regulas, sed ex istorum solutione solvi poterunt. Calculatorias autem disputationes reliqui, ne confunderem incipientium iudicia, qui communiter mathematicam nesciunt. Potissimum autem ne scriptura mea causa sit legentibus ut in talibus inutilibus questionibus tempus vane consumant, specialiter theologi, qui saluti debent animarum consulere. Ad quod melius faciendum, illa phisicalia scripsi, dumtaxat que ad theologalia utilia esse videri possunt”—fol. 133rb. Later, in Question 11, article 2, on the first book of De generatione, he remarks similarly: “Quantum ad secundum articulum, principales secundi articuli sunt due opiniones que communiter deffenduntur a doctoribus calculantibus, quas breviter recitabo: turn quia mihi inutile videtur in calculationibus tempus consumere; tum quia false mihi apparent”—fol. 24vb.

⁴⁴ One reference of Soto, in Question 4 on Book 7 of the Physics, where he states, “Miror tarnen quosdam schole nostre, qui aiunt regulam primam, contrariam huius conclusionis, veram esse in Universum, posila constantia potentie motive”, can only be directed at Astudillo and his students.

⁴⁵ Delgado, Vicente Muñoz, “La Logica en Salamanca durante la primera mitad del siglo XVI”, Salmanticensis, xiv (1967), 171–207, especially pp. 192–195.Google Scholar

⁴⁶ Ibid., p. 195.

⁴⁷ See Pastor, Rey, op. cit. (17), 156Google Scholar; Solana, , op. cit. (10), iii, 612.Google Scholar

⁴⁸ The title page reads: “Philosophia naturalis Petri a Spinosa artium magistri: opus inquam tripartitum quod continet tres partes. Prima pars erit emporium refertissimum bone philosophie, currens per omnes textus Philosophi cum aptis questionibus ibidemque propriis. Secunda pars erit Calculatoria: quam appello Roseam. Tertia pars erit Flos campi, Lilium agri, continens omnes naturales questiones ordine alphabetico. Nil optabis quod hec philosophia non clare tibi ostendat. Si textum ibidem habes expositionem lucidissimam. Si questiones ad idem. Si calculationes habes cas in secunda parte. Si denique problemata habes omnia ordine alphabetico: quo sit tibi minor labor inveniendi quod vellis.”—Unfortunately only the first part of this work is preserved in the copy I have used at the Biblioteca Nacional in Madrid.

⁴⁹ In the Biblioteca de Santa Cruz at Valladolid there is a collection of treatises on the Sphere of Sacrobosco entitled Spherae tractatus (Venice, 1531 ) which contains all of the works cited by Espinosa; he may have used this particular edition.

⁵⁰ This appeared in two volumes: the first, Super octo libros physicorum commentarii, provided the exposition of Aristotle's doctrine; the second, Super octo libros physicorum quaestiones, took up special problems and applications in the context of sixteenth-century thought.

⁵¹ See Duhem, , op. cit. (2), iii, 555–562Google Scholar; the relevant passages from Soto are given in Clagett, Marshall, The Science of Mechanics in the Middle Ages (Madison, 1959), pp. 555–556.Google Scholar For the genesis of this teaching, see my paper, “The Enigma of Domingo de Soto …” (fn. 1, above).

⁵² Apart from his own writings, Soto's teachings were promulgated by his Dominican students and supporters at Salamanca, among whom should be noted the following seventeenth-century writers: Diego Ortiz, Cosme de Lerma, Froilán Díaz, Jacinto de la Parra, and Domingo Lince. Practically all of these, unfortunately, paid no attention to the more mathematical portions of Soto's works.

⁵³ The work is entitled Commentaria et quaestiones in duos Aristotelis Stagfritae de generatione et corruptione libros (Salamanca, 1585). In the preface, Bañez points out that the work was composed some thirty years previously and dictated to his students at that time; this would coincide roughly with the completed edition of Soto's Physics, and may have been prepared for use in conjunction with it.

⁵⁴ The first edition appeared in Mexico in 1557; the second, at Salamanca in 1562. On Veracruz, see the article by Reinhardt, K. F. in the New Catholic Encyclopedia (New York, 1967), xiv, 607.Google Scholar

⁵⁵ An edition of this work was published at Lyons in 1551; it is a very brief summary of all of Aristotle's natural philosophy, including treatises on minerals, plants, and animals.

⁵⁶ Veracruz's prologue, in fact, is derogatory of the “calculatory” tradition. He writes: “Quis enim non ex animo doleat, quanta iactura temporis (quo nihil pretiosius) adolescentumque olei, et operis amissio sit in tractandis quae de maximo et minimo naturali multiplicantur argumentis, in illis voluendis, quae a Calculatore diffuse valde tractantur, atque de motuum et mobilium proportione, et ad invicem comparatione sophystice proponuntur. Atque (ut unico verbo multa dicam) quae de triplici motu ab Alvaro Thoma sunt excogitata ? Hoc unum vere tales asserere posse affirmo: ‘Per totam noctam laborenles nihil cepimus’.”

⁵⁷ With the exception, that is, of the adumbration of the “law of falling bodies”, which Veracruz apparently did not understand!

⁵⁸ The Venice 1582 edition of Soto's Quaestiones antedated Galileo's studies at both Pisa and Padua, begun ca. 1584.

⁵⁹ Commentaria una cum questionibus in octo libros Aristotelis de physica auscultatione (Venice, 1580).

⁶⁰ De communibus omnium rerum naturalium principiis et affectionibus libri guindecim (Rome, 1562).

⁶¹ Consult the index to Galileo's Juvenilia in Volume I of Le Opere di Galileo Galilei, Edizione Nazionale, ed. Favaro, Antonio (Florence, 1890).Google Scholar