¹ “La Géométrie au XIe Siècle”, in Tannery, Paul, Mémoires Scientifiques, v (Paris and Toulouse, 1922), 79–102Google Scholar. The article was originally published in 1897.

² Tannery, P., Mémoires Scientifiques, v, 101–102.Google Scholar

³ Clagett, M., Archimedes in the Middle Ages, i (Madison, 1964).Google Scholar

⁴ Clagett, M., “The Medieval Latin Translations from the Arabic of the Elements of Euclid, with Special Emphasis on the Versions of Adelard of Bath”, Isis, xliv (1953), 16–42.CrossRefGoogle Scholar

⁵ In the Middle Ages geometry was divided into two parts: theoretical and practical (geometria speculativa and geometria practica). Practical geometry dealt with practical problems of mensuration, and numerous treatises on it are extant, but I do not consider this branch of the subject in this article.

⁶ Baur, L. (ed.), Die Philosophischen Werke des Robert Grosseteste (Münster, 1912), 59–60Google Scholar. (This work forms vol. ix of Beiträge zur Geschichte der Philosophie des Mittelalters). Cf. Crombie, A. G., Robert Grosseteste (2nd imp., Oxford, 1961), 110.Google Scholar

⁷ Bridges, J. H. (ed.), The “Opus Maius” of Roger Bacon (Oxford, 1897), i, 97Google Scholar. Cf. Little, A. G. (ed.), Roger Bacon: Essays (Oxford, 1914), 167.Google Scholar

⁸ Murdoch, J. E., Geometry and the Continuum in the Fourteenth Century: A Philosophical Analysis of Thomas Bradwardine's Tractatus de Continuo (microfilm, Ann Arbor, 1957), 401Google Scholar. Cf. Weisheipl, J. A., The Development of Physical Theory in the Middle Ages (London and New York, 1959), 73.Google Scholar

⁹ MS., Oxford, Bodleian, Digby 76, f. 78ar: Geometer igitur considerat vias possibiles nature, quia propter opera nature certificanda constituta est (ex in MS) geometria primo et per se, deinde propter opera humana. Auctores enim perspective nobis estendunt quod linee et figure declarant nobis totam Operationem nature, et principia et efieclus. Et simililer patet per celestia, de quibus naturalis et astronomia communicant.

¹⁰ Several early printed editions of this work exist, all dependent on the first one, edited by Pedro Sanchez Cirvelo and published at Paris in 1495. I have prepared a new text in my Cambridge doctoral dissertation of 1967 which is entitled: The Geometria Speculativa of Thomas Bradwardine: Text with Critical Discussion. It is not easy to assign a precise date to this work. Comparison with the Tractatus de proportionibus suggests that at times in this latter work Bradwardine was improving upon certain material in the Geometria. Further, Bradwardine's spell of teaching in the Faculty of Arts would seem to have been a suitable time for writing a geometrical textbook, and I am grateful to Dr. J. A. Weisheipl, o.p., for giving me his opinion on the question of dating. See pp. 16, 350, 358, 363–364 of my dissertation.

¹¹ Cantor, M., Vorlesungen über Geschichte der Mathematik, ii (2nd edn., Leipzig, 1900), 116Google Scholar. See my discussion of this question in The Geometria Speculativa of Thomas Bradwardine, (10) 321–328.Google Scholar

¹² MS. Munich, Bayerische Staatsbibliothek, Clm. 14908, f. 224r: Hoc opus geometricum continet fere omnes demonstraciones geometricas quas adducit philosophus gratia exempli vel in loyca vel phylosophia.

¹³ From the title page of the Paris, 1495 edition: Geometria speculatiua Thome brauardini recoligens omnes conclusiones geometricas studentibus artium & philosophie aristotelis valde necessarias.

¹⁴ On the place of the imaginatio in late medieval natural philosophy see especially Wilson, C., William Heytesbury (Madison, 1960), 24–25, 174Google Scholar. Cf. also McLuhan, M., The Gutenberg Galaxy (London, Routledge and Kegan Paul edn., 1962), 80–81et passim.Google Scholar

¹⁵ Neumann, S., “Gegenstande und Methode. Der Theoretischen Wissenschaften nach Thomas von Aquin aufgrund der Expositio super Librum Boethii de Trinitate”, Beiträge zur Geschichte der Philosophie des Mittelalters; XLI, Heft 2 (1965), 132–135.Google Scholar

¹⁶ Quodlibet II, questio 9: Quodlibeta Magistri Henrici Goethals a Gandavo doctoris Solemnis (Paris, 1518), f. 36rGoogle Scholar. Cf. Klibansky, R., Panofsky, E. and Saxl, F., Saturn and Melancholy (London, 1964), 338–339.Google Scholar

¹⁷ Clagett, M., The Science of Mechanics in the Middle Ages (Madison and London, 1959), 368; cf. p. 347.Google Scholar

¹⁸ To marshal the evidence concerning the extent to which Oresme was acquainted with Mertonian works would be outside the scope of this article. Clagett, M. (op. cit. in n. 17, p. 331)Google Scholar says that the “distinctive vocabulary and principal theorems” of Mertonian kinematics passed to Italy and France about 1350. For our purposes it is only necessary to assume that Oresme had a certain familiarity with the spirit of the Mertonian methods, and this could easily have been obtained merely by hearsay.

¹⁹ For the distinction between Roger Swineshead and Richard Swineshead see Weisheipl, J. A., “Roger Swyneshead, o.s.b., Logician, Natural Philosopher, and Theologian” in Oxford Studies presented to Daniel Callus (Oxford, 1964), 231–252.Google Scholar

²⁰ de Bury, Richard, Philobiblon, the Text and Translation of E. C. Thomas (Oxford, 1960), 104–107.Google Scholar

²¹ Book V, definition 5.

²² Murdoch, J. E., “The Medieval Language of Proportions” in Crombie, A. C. (ed.), Scientific Change (London, 1963), 251 sqq.Google Scholar

²³ Comment on Book V, definition 5.

²⁴ Boethius, , De Institutions Arithmetica Libri Duo…, ed. Friedlein, G. (Leipzig, 1867), 45–72.Google Scholar

²⁵ Thus 2:1 is called dupla proportio; 3:2, sesquialtera; 17:4, quadrupla sesquiquarta; 19:5, triplex superquadripartiens quintas (or, in the elliptical form actually given by Boethius, , triplex super-quadripartiensGoogle Scholar).

²⁶ It is of course no part of the aim of Book V of the Elements to provide numerical specifications of ratios.

²⁷ Bacon, Roger, Opera hactenus inedita, xvi (Oxford, 1940), 80.Google Scholar

²⁸ Campanus, comment on Book V, definition 3. Cf. Bacon, Roger, op. cit. in n. 27, 79–80Google Scholar. following the reading of MS. siglum D.

²⁹ Molland, A. G., The Geometria Speculativa of Thomas Bradwardine (see note 10), 121Google Scholar: Proportio autem irrationalis non sic immediate denominatur ab aliquo numero licet ab aliqua proportione numerali, quoniam non est ibi possibile ut secundum aliquem numerum pars aliqua minoris maiorem numeret. Contingit tarnen mediate denominari proportionem irrationalem a numero, ut proportio dyametri ad costam est medietas duple proportionis, et ita capiunt alie species huius proportionis denominationes a numero.

³⁰ Tannery, P., Mémoires Scientifiques, iii (Paris and Toulouse, 1915), 70–73Google Scholar; Szabó, Á., “Die Frühgriechische Proportionenlehre im Spiegel ihrer Terminologie”, Archive for History of Exact Sciences, ii (1962–1966), 251–255.Google Scholar

³¹ Book V. definitions 9, 10; Book VI, proposition 23.

³² It may be helpful for the reader at this point to turn to the appendix to this article, for the terminology that I am adopting is at variance with that used in most of the modern literature on the subject. The justification for my convention will become more apparent in the course of the article, but it should be remarked here that if we, like the Schoolmen, do not regard a ratio as a number, then we are free to define addition of ratios in any way we please, providing of course that the operation which we select satisfies the associative and commutative laws. By speaking of the compounding of ratios as addition we get closer to word-for-word correspondence with medieval language, and we no longer have to accuse medieval mathematicians of using such terms as dupla in two senses in the same sentence. (On the subject of this choice of convention see my review of Grant, E.'s edition of Oresme's De proportionibus proportionum and Ad pauca respicientes in Annals of Science, xxii (1966), 296–297Google Scholar). In the symbolism that I have adopted I have tried to ease the path of the modern reader by placing the signs for operations upon ratios in Clarendon (or bold-face) type.

³³ Grant, E., “Part I of Nicole Oresme's Algorismus proportionum”, Isis, lvi (1965), 340–341.Google Scholar

³⁴ See, for example, Crosby, H. L. (ed.), Thomas of Bradwardine: His Tractatus de Proportionibus (Madison, 1955), p. 78, II. 293–310Google Scholar. There Bradwardine makes it quite clear that he is going to use the terms duplus, triplus, quadruplus in such a way that doubling, tripling or quadrupling a ratio will for us correspond to squaring, cubing or raising to the fourth power the fraction that we form from the ratio.

³⁵ Oresme, Nicole, De proportionibus proportionum and Ad pauca respicientes, ed. Grant, E. (Madison, Milwaukee, and London, 1966), 158.Google Scholar

³⁶ We may think of it as scalar multiplication.

³⁷ Oresme, Nicole, op. cit. in n. 35, 160.Google Scholar

³⁸ See e.g. Wieleitner, H., “Zur Geschichte der gebrochen Exponenten”. Isis, vi (1924), 509–520CrossRefGoogle Scholar, and Grant, E., op. cit. in n. 33, 340–341.Google Scholar

³⁹ See note 35.

⁴⁰ Oresme, Nicole, op. cit. in n. 35, 150–155, 321–322Google Scholar; Maier, A., Die Vorläufer Galileis im 14. Jahrhundert (Rome, 1949), 90, no. 20Google Scholar. A detailed examination of Oresme's difficulties would be out of place here, but basically he is trying to deal with ratios of lesser inequality by analogy with the procedures adopted for ratios of greater inequality. The obvious analogy would lead to such results as 1:9 being double 1:3, but Oresme regards it as too much of a verborum abusio that the lesser should be double the greater. He therefore adopts a different procedure (which involves 1:3 being double 1:9), but the operation of addition which he defines (or at least indicates) on this basis is non-commutative: he does not explore the consequences. If he had been familiar with negative numbers he would not have regarded it as unreasonable that the lesser should be double the greater, for twice –4 is –8. In fact, if we extend the precise mathematical analogy between ratios of greater inequality and positive real numbers, then the ratio of equality corresponds to zero and ratios of lesser inequality correspond to negative numbers.

⁴¹ Physica IV. 8, 215b1–216a7Google Scholar; VII. 5, 249b27–250b10.

⁴² Crosby, H. L., op. cit. in n. 34, 110.Google Scholar

⁴³ Oresme, Nicole, op. cit. in n. 35, 262Google Scholar: Velocitas sequitur proportionein potentie motoris ad mobile seu ad resistentiam eius. Cf. Bradwardine, in Crosby, H. L., op. cit. in n. 34, 110Google Scholar: Proportio velocitatum in motibus sequitur proportionem potentiae motoris ad potentiam rei motae. Bradwardine's writing of proportio velocitatum instead of velocitas obscures his statement.

⁴⁴ See nn. 32, 34, and the appendix to this article.

⁴⁵ Thus, if F ₁ is as 9, F ₂ as 3, R ₁ as 4, R ₂ as 2, we have (F ₁:R ₁):(F ₂:R ₂)=(9:4):(3:2)=2:1=V ₁:V ₂.

⁴⁶ Maier, A., Die Vorläufer Galileis im 14 Jahrhundert (Rome, 1949), 86–95Google Scholar. For an earlier discussion of Bradwardine's law see Clagett, M., Giovanni Marliani and Late Medieval Physics (New York, 1941), 130–137Google Scholar. Although Clagett does not in this work emphasize the novelty of Bradwardine's view, I am unable to agree with Maier's opinion that “Clagett hat Bradwardines Lehre völlig missverstanden” (op. cit., 108, n. 55).Google Scholar

⁴⁷ Galilei, Galileo, Dialogues concerning Two New Sciences, tr. H. Crew and A. de Salvio (Dover reissue of 1914 edition), 161.Google Scholar

⁴⁸ Maier, A., op. cit. in n. 46, 104–107.Google Scholar

⁴⁹ Physica IV. 11, 220a24–25.Google Scholar

⁵⁰ A new text of this tract may be found in Hoskin, M. A. and Molland, A. G., “Swineshead on Falling Bodies”, British Journal for the History of Science, iii (1966–1967), 150–182CrossRefGoogle Scholar. In the early printed editions of the Liber calculationum it appears as Tractatus XI.

⁵¹ Hoskin, M. A. and Molland, A. G., loc. cit., 161–163, 176–177.Google Scholar

⁵² Quodl. I, art. 8.

⁵³ de Cervantes Saavedra, Miguel, The Adventures of Don Quixote, tr. Cohen, J. M. (Penguin edn., 1950), 539.Google Scholar