¹ The dangers of neglecting these medieval treatises have been pointed out by many since Pierre Duhem, but grievous sins of omission are still committed by scholars of high repute. See. for example, Sambursky, S. on time as a geometrical dimension, The Physical World of the Greeks (London, 1956), 152Google Scholar, and cf. Hoskin, M. A., “Commentary on Zoubov's paper”, Actes du X^e Congrès International d'Histoire des Sciences (Paris, 1964), i, 78–80.Google Scholar

² Madison, , Wis., 1959.Google Scholar

³ On Swineshead see especially Clagett, M., The Science of Mechanics, 201–204Google Scholar and “Richard Swineshead and late medieval physics”, Osiris, ix (1950), 131–161Google Scholar; Thorndike, Lynn, A History of Magic and Experimental Science, iii (New York, 1934), ch. 23Google Scholar; and Wilson, Curtis, William Heytesbury (Madison, Wis., 1956), passim.Google Scholar

⁴ We are very grateful to Fr. J. A. Weisheipl, O.P., who made available to us his important doctoral thesis (unpublished), “Early fourteenth century physics of the Merton ‘School’”.

⁵ For example, the gist of one of his arguments (1520 edn, f. 16r) may be rendered in modern terminology as follows. Consider a body divided along its length into parts so that the nth part is of length proportional to 1/2ⁿ, and is cold proportional to 2ⁿ if n is a multiple of 3, but is hot proportional to 2ⁿ otherwise. If we denote hotness by positive numbers and coldness by negative numbers, his argument reduces to the following: (a) Since the body as a whole is hot. (b) Since the body as a whole is not hot. Note that (b) involves there being as many positive as negative numbers in the sequence This of course antedates by some three centuries Galileo's proof that there are as many perfect squares as positive integers. For a discussion of Swineshead's use of infinite series, see Boyer, Carl B., The Concepts of the Calculus (New York, 1949), 74–79Google Scholar; and also Maier, A., Die Vorläufer Galileis im 14 Jahrhundert (Rome, 1949), 127–128.Google Scholar

⁶ Clagett, , The Science of Mechanics, 202.Google Scholar

⁷ Thorndike, , op. cit., 373.Google Scholar

⁸ Gualteri Burlei in Physicam Aristotelis … (Venice, 1501), f. 106v: Dico quod pars non movetur sursum nisi per accidens, et dico quod motus illius partis sursum ad motum totius deorsum ad locum suum naturalem non est violentus parti, sed magis naturalis, quia illa pars non movetur per se sed ad motum totius et motus totius est naturalis.

⁹ Clagett, , The Science of Mechanics, 570Google Scholar. See also Oresme's, Le Livre du Ciel et du MondeGoogle Scholar, ed. by Menut, A. D. and Denomy, A. L., in Mediaeval Studies, iv (1942), 293.Google Scholar

¹⁰ Alberti de Saxonia questiones super quatuor libros Aristotelis de celo et mundo, in Lokert, G. (ed.), Questiones et decisiones physicales insignium virorum (Paris, 1518), Lib. ii, qu. XXIV, f. cxvii^vGoogle Scholar: Iuxta istam conclusionem dubitatur posito quod aliquod grave uniforme in gravitate habeat (habent ed.) medium sue gravitatis extra medium mundi omni impedimento extrinseco amoto, nihil sibi addendo nee aliquid ab eo removendo, an sic moveretur quod tandem medium sue gravitatis esset medium mundi. Dico quod non, quia prius quod medium sue gravitatis fieret medium mundi descenderet aliqua tarditate et dupla ad illam et quadrupla ad illam et sic in infinitum, et ideo, si perpetuo maneret, perpetuo descenderet, et nunquam medium sue gravitatis fieret medium mundi. (But see also Clagett, , The Science of Mechanics, Document 9.2.)Google Scholar

¹¹ For a discussion of this, see Clagett, , The Science of Mechanics, ch. 7.Google Scholar

¹² See Clagett, ibid.; Maier, A., Die Vorläufer Galileis im 14 Jahrhundert, ch. 4Google Scholar; Crosby, H. Lamar Jr., Thomas of Bradwardine's Tractatus de proportionibus (Madison, Wis., 1954)Google Scholar; and Miss Maier's review of Crosby's volume in Isis, xlviii (1957), 85–87.Google Scholar

¹³ See Murdoch, John E., “The medieval language of proportions” in Crombie, A. C. (ed.), Scientific Change (London, 1963), 237–271.Google Scholar

¹⁴ Clagett, , The Science of Mechanics, 440–444.Google Scholar

^{page 168 note 1} ad descensum: ad ascensum 56.

^{page 168 note 2} et … inequalitatis: om. 3.

^{page 168 note 3} circa: citra ed. 1347.

^{page 168 note 4} ante non: omnino 12569.

^{page 168 note 5} ita … moveretur: terra motione 3; factum (?) motione 4; quod illud quando (?) moveretur 6; sive motione 7; quod illud moveretur 8; quod illud quod moveretur 9.

^{page 168 note 6} proportionem: motionem 3; a proportione 2.

^{page 169 note 7} post duo: notabilia ed.

^{page 169 note 8} quam inter 6 … maior proportio: om. ed.

^{page 169 note 9} 7 et 6: 6 et 7 ed.

^{page 169 note 10} primo tangitur: tangitur prius 3.

^{page 169 note 11} locali: om. 6789.

^{page 169 note 12} excessus: est excessus ed. 12; excessus est maior 347; maior est excessus 9.

^{page 169 note 13} c eo … b ad a: ita excessus e supra b est maior excessu b supra a 2.

^{page 170 note 14} ergo … d ad c: om. 5689.

^{page 170 note 15} tertium … excedetur: terminum minus equaliter ista duo illud excedit ut constat 569; tertiem minus equaliter excedunt 8.

^{page 170 note 16} secundum: om. 2; secundo 67.

^{page 170 note 17} portionalitate: proportione ed. 4; om. 12.

^{page 170 note 18} sunt equalia: est equale ed. 123457.

^{page 170 note 19} sicut: ad medium ed. 12; sicut medium 3.

^{page 170 note 20} d: idem 367; id 9.

^{page 170 note 21} d: om. ed. 1249.

^{page 170 note 22} secundum: tertium ed. 12.

^{page 171 note 23} quorum primus … excessus inter tertium et quartum: qui 5689.

^{page 171 note 24} scilicet … erit: scilicet primum et secundum ed. 1; om. 347.

^{page 171 note 25} … b minus: explicit 2.

^{page 171 note 26} continue: om. ed. 1467.

^{page 171 note 27} quartam … incipiens ultimam: et quartam 568; quartam 9.

^{page 171 note 28} et gradua … incipiens secundam quartam: om. 4.

^{page 171 note 29} accipiendos bis exceptis: recipiendos vel ex 56; recipiendo vel ex 8; recipiendo 9.

^{page 171 note 30} vocando … illorum: om. 568.

^{page 172 note 31} maiorem proportionem: plus 5689.

^{page 172 note 32} et ista: quarum proportio maxima ed. 1347.

^{page 172 note 33} excessus: latitudinis 589.

^{page 172 note 34} per … notabilis: om. 58; per antecedens 9.

^{page 172 note 35} illas … excessus: illos duos excessus intermedios 5689.

^{page 172 note 36} totius … quatuor: totius 347; aggregati ex illis quatuor 568.

^{page 172 note 37} ita si … in infinitum: om. 1589.

^{page 172 note 38} quartam totius … constituunt: om. 4.

^{page 172 note 39} quartam totius … octavam partem totius: octavam partem totius 37; octavam partem ipsius totius sive aggregati 6.

^{page 172 note 40} item quotcumque … sit par: ita quod 167.

^{page 172 note 41} item: arguitur quod 9.

^{page 172 note 42} hoc modo … inter eos: om. 89.

^{page 173 note 43} constituunt: constituent partem aliquotam totius denominatatim 1.

^{page 173 note 44} post patet: per intuenti 6; intuanti 9.

^{page 173 note 45} constituunt: continent 569.

^{page 173 note 46} octo: 16 347.

^{page 173 note 47} proportionalitate: proportione ed.

^{page 173 note 48} composito: compositum ed. 58.

^{page 173 note 49} quintum: quartum 37.

^{page 173 note 50} quartum: quintum 37.

^{page 173 note 51} primum … quartum: om. 67.

^{page 173 note 52} medietatis: medietatem 6.

^{page 173 note 53} medium: idem medio ed. 1; idem cum medio 347.

^{page 173 note 54} post duobus: sive equalibus sive ed. 1.

^{page 173 note 55} post accipiendo: ly ed. 1; li 3.

^{page 173 note 56} illorum … ergo quidlibet: om. 37.

^{page 174 note 57} in numero par: om. ed. 1; pares 347; numero pares 5.

^{page 174 note 57a} proportionalitate: proportione ed.

^{page 174 note 58} pares: partes ed. 189.

^{page 174 note 59} sic: tune sit sic quod ed.; tunc 347.

^{page 174 note 60} semper … terminos maiores: excessus ultimus scilicet inter terminos minores sit minimus 1.

^{page 174 note 61} denominatam: denominata vero 3; diminutis 5; denominatis 9.

^{page 174 note 62} primo notabili dicto: primo dicto 37; notabili predicto 689.

^{page 174 note 63} equevelociter: equaliter ed. 15.

^{page 174 note 64} equevelociter maioris: equevelocitatem maioris 3; maioris equalem 569; equale maioris 8.

^{page 174 note 65} proportionata: composita 5.

^{page 175 note 66} deperditi continetur: deperdita invenitur ed. 1347.

^{page 175 note 67} post deperditi: etc. ed. 137.

^{page 175 note 68} 16 partes: 15 partes ed.; 15 partes proportionales 5689.

^{page 175 note 69} minor … erit: om. 347.

^{page 175 note 70} post maiores: et inter terminos maiores maiores ed. 1.

^{page 175 note 71} quam ipsa: contra illa proportio ed. 1.

^{page 175 note 72} post quod: terra 3.

^{page 175 note 73} ex se ipso: centrum ipsum ed. 137; se ipso 4.

^{page 175 note 74} figure: om. 137.

^{page 175 note 75} d: om. ed. 1347.

^{page 175 note 76} post et: sic sequitur quod medietas a et tantum 4.

^{page 175 note 77} post et: si 689.

^{page 175 note 78} post et: continens ed. 1.

^{page 175 note 79} quandocumque: si ed. 1.

^{page 175 note 80} simplicis: terre 5689.

^{page 176 note 81} d: c 17.

^{page 176 note 82} ergo … et quantum: deperditam et que ed. 138.

^{page 176 note 83} post ergo: terre 6; terre erit et 4.

^{page 176 note 84} et quantum: e quantitatem 4.

^{page 176 note 85} post proportionali: illius divisionis ed. 1.

^{page 176 note 86} erit motus: est pars 5; est 9.

^{page 176 note 87} post gradu: subduplo ad gradum ed. 1.

^{page 176 note 88} erit motus: remittetur motus et erit 569.

^{page 176 note 89} quia: om. 5689.

^{page 176 note 90} remitteretur … subduplum: om. 5689.

^{page 176 note 91} per idem argumentum: quia tertia pars proportionalis est in duplo minor secunda et motus in tertia erit minor quam subduplus ad motum in secunda 5679.

^{page 177 note 92} ut patet. Sed: cum ergo 347.

^{page 177 note 93} ut patet. Ergo: patet quia 347.

^{page 177 note 94} antecedens: minor 1.

^{page 177 note 95} quia: qui 37; et 1.

^{page 177 note 96} quia: et 3479.

^{page 177 note 97} probationem: portionem ed. 357; proportionem 8.

^{page 177 note 98} hic: sic ed. 13.

^{page 177 note 99} conclusio: om. 5.

^{page 177 note 100} quod: ut 347.

^{page 177 note 101} ergo … sequitur: om. 589.

^{page 177 note 102} conclusione: positione 59.

^{page 177 note 103} impossibile: irrationale 9.

^{page 177 note 104} moveri: om. 3456789.

^{page 177 note 105} velocius: necessario ed. 1.

^{page 178 note 106} post moveretur: cuius scilicet una pars minor quam medietas esset ultra centrum ed. 1.

^{page 178 note 107} nititur: intendatur 347.

^{page 178 note 108} deorsum … versus om. 4.

^{page 178 note 109} nitetur: intendetur 37.

^{page 178 note 110} sed … etiam: iuvabit totum ad descensum quia 589.

^{page 178 note 111} levitatem: gravitatem ed. 37; om. 1.

^{page 178 note 112} ultra: om. 568.

^{page 178 note 113} post proportione: geometrica ed. 1.

^{page 178 note 114} simpliciter: simul 6789.

^{page 178 note 115} maiorabitur … alia: om. 3479.

^{page 178 note 116} finita … minorationem: sic tardius et tardius proportionaliter minorabitur 5689.

^{page 178 note 117} excessus: om. 5689.

^{page 178 note 118} sit d: om. 347.

^{page 179 note 119} sic capiatur: sit ed. 1; sic 3

^{page 179 note 120} terra: gravitas ed.

^{page 179 note 121} terre: gravitati ed.

^{page 179 note 122} ascensum: descensum ed. 13478.

^{page 179 note 123} nec resistit: om. 69.

^{page 179 note 124} elementi: illa 1; obiecti 4.

^{page 179 note 125} appeteret: haberet 347.

^{page 179 note 126} naturaliter … scilicet: appeteret ut 5689.

^{page 179 note 127} eiusdem … ilio: om. 345678.

^{page 179 note 128} si … moveretur: om. 8.

^{page 179 note 129} moveretur: tendit 56.

^{page 179 note 130} consequens … eius: om. 569.

^{page 179 note 131} ubi: ut ibi 569; ut 4.

^{page 180 note 132} sit … que: om. 59; que 8.

^{page 180 note 133} et: quia 346789.

^{page 180 note 134} ymmo: ideo ed. 1347; imago 5.

^{page 180 note 135} remitteretur: debet sequi 3.

^{page 180 note 136} resistentiam: distantiam 4.

^{page 180 note 137} eundem: dimidium (?) 6; dividum (?) 5.

^{page 180 note 138} apparet: arguitur 56; arguetur 89.

^{page 180 note 139} nec non … centro: om. ed. 15.

^{page 180 note 140} quid: quia 689.

^{page 180 note 141} cum puncto medio: per punctum medium 45678.

^{page 180 note 142} secundum extremum: per exemplum 345; per extremum 789; per punctum extremum 6.

^{page 181 note 143} eius: sicut 56.

^{page 181 note 144} partis: totius 347; om. 1.

^{page 181 note 145} post appetitus: ideo iuvat ed. 9.

^{page 181 note 146} et… appetitus: om. 568.

^{page 181 note 147} ut: et ed. 1.

^{page 181 note 148} ideo ut: et quia 5689.

^{page 181 note 149} post totius: descentis 589; descendentis 6.

^{page 181 note 150} rationabile: possibile 568.

^{page 181 note 151} post iste: appetitus ille parte 56; appetitus iste partes 89.

^{page 181 note 152} rationale: possibile 5689.

^{page 181 note 153} et ubi: ubi enim 56.

^{page 181 note 154} fieret… partis: om. 5689.

^{page 181 note 155} corpori: om. ed. 1.

^{page 181 note 156} ultra: om. 5689.

^{page 181 note 157} sed … appetitu: om. 14.

^{page 182 note 158} quia dicitur: et sicut 5689.

^{page 182 note 159} ita quod: om. 45689.

^{page 182 note 160} quod: quia ibi 1347; quod ibi 5689.

^{page 182 note 161} ideo … motum: om. 3.

^{page 182 note 162} impossibile: irrationabile 9.