Chapter 2: English Translation and Commentary

Glasner, Ruth; Baraness, Avinoam

doi:10.1007/978-3-319-77303-2_2

Ruth Glasner¹⁰ &
Avinoam Baraness¹¹

Part of the book series: Sources and Studies in the History of Mathematics and Physical Sciences ((SHMP))

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Abstract

[93b] This book includes five parts. The first [serves as] an introduction to the book and inquires whether the imagining [] of equally divided motion [] can qualify as one of the first principles of geometry. The second is on the accidents associated with motion, which prevent its imagining from being a first principle of geometry, and how several earlier scholars became misled by them. The third is about the propria of rectilinear magnitudes and areas that are useful in this science. The fourth explains how a body is divided into surfaces, the surface into lines, and the line into points, as Plato taught, and the way of evaluating and measuring them by one another. The fifth is about some of the propria of divisions of bodies into surfaces, of surfaces into lines, and of lines into points, which are needed for the rectifying of the circle, and for the measurement of the sphere and the cylinder and the cone and the spherical annulus and the square annulus and the circular annulus and their sections, and their enclosing lines and surfaces, and on transforming them, one into the likeness of another, and on equating them to rectilinear [figures], and on the division of angles into any number of equal parts.

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Notes

1.
Bar-Ḥiyya’s vocabulary is notable throughout SMA. Here we find for pyramid, and the unusual spelling side by side with the common spelling for a sphere. See Sarfatti (1968, 86, 122). See also Section I.1.4 below.
2.
The word (proprium, property) was discussed in the introduction. The key terms to the understanding of Alfonso’s agenda, and (evaluate, measure, imagine, equally divided) are discussed in Sections I.3.4, I.4.2 and I.5.3 below; (form) in I.5.5, (sections) in I.5.4 and (square) in III.0. The word (spherical, square or circular), translated here as “annulus”, appears only in the outline and we have no information on its use. It may be a translation of the Arabic ḥalaqa.
3.
We understand the word in this sentence as a verb.
4.
Allusion to Bab. Talmud, Megila ch. 1 6b, tr. M.L. Rodkinson.
5.
Ecc. 12:12. Most English translations use “weariness.” We used “exertion” to be consistent with the translation of the previous sentence from the Talmud.
6.
Isa. 40:17.
7.
Ecc. 12:10, following here the New International Version.
8.
The Biblical source is Numbers 11:17 or 11:25, . The expression appears in Yehuda ha-Levi’s poem Barkhi Atzula, and the exact expression in Yehuda al-Ḥarizi’s Taḥkemoni, p. 24.
9.
Dt. 5:24.
10.
Ps. 27:4.
11.
Ps. 27:4.
12.
Allusion to Isa 43:7.
13.
Ps. 96:6.
14.
Isa. 6:3.
15.
Sections I.1.3–4, I.3.3, II.1.6, I.4.1, I.5.5, II.1.6–7, II.5.6.
16.
Sections III.26–7.
17.
The term “superposition” is discussed in the introductory note to Section I.2 below.
18.
Asserting the preservation of equality under addition and subtraction.
19.
“To construct a square equal to a given rectilinear figure.”
20.
Hilbert Grundlage der Geometrie # 18. In the first 34 propositions of Elements I equality is used in the simple sense of congruence; from Proposition I.35 the term is used in a wider sense of equality of rectilinear areas. Proclus comments: “it means that the areas, not the sides, are equal, for the statement is about included spaces, the areas” (Commentary 398 Heiberg; Morrow’s translation 314).
21.
Elements III, definition 1. Euclid defines also similarity between sections of a circle.
22.
Hooper-Sude 41, 43.
23.
Rashed (2015, 91, 440).
24.
On equalities between ratios see introductory note and commentary to III.24–27.
25.
See Heath (1908/1956, II, 116–120); Mueller (1971, 2; 1981, 132); Lachterman (1989, 33–4). Mueller suggests that the requirement of homogeneity was “reworked” by Euclid and that it cannot be ascribed to Eudoxus the way Euclid presents it.
26.
Lachterman (1989, 35).
27.
Lachterman (1989, 37).
28.
We are grateful to Nathan Sidoli for suggesting this term.
29.
Elements XII.2, XII.10, XI.11.
30.
Archimedes, Measurement of Circle 1; Sphere and Cylinder I.33, 34.
31.
Literally: the thought of those who have doubts.
32.
We are grateful to Donna Shalev for reading this paragraph and for her comments.
33.
In the discussion of atomism (Section I.3) and are used in the sense of “composed of.”
34.
Guide I.70.
35.
See Sections II.1.7; II.4.2; Introduction Section 3 argument ii.
36.
Common Notion no. 4 in Heath’ list (1908/1956 I, 224); no. 7 in Heiberg’s list (1883, I, 10).
37.
Elements Heath’ translation I, 224–5, 249.
38.
Elements Heath translation I, 247, 249.
39.
According to Heath’ numbering.
40.
Besthorn’s edition 28.12, also in the manuscripts of Yisḥāq-Ṭhābit’s version. I am grateful to Ofer Elior for this information. Elior also noted that Ibn Sina uses in his rendering of common notion 4 and Proposition I.4.
41.
Wehr’s dictionary, 645.
42.
Besthorn’s edition 52.19. Also in the manuscripts of version.
43.
Besthorn’s edition, Introduction, p. 28, lines 12–3, p. 29, footnote 5.
44.
Moshe Ibn Tibbon (Paris ms. fol. 2b24–5; Oxford ms. fol. 2b9), Yaaqov ben Makhir (Oxford ms. fol. 2b6) and the compiler of the Hebrew Geometrical Compendium (Mantova ms. 2, fol. 3b13, 14), all use .
45.
Oxford ms. fol. 1b20; Paris ms. fol. 2a1. In Proposition I.4 where both the active and the passive senses are used Moshe Ibn Tibbon and the compiler of the Hebrew Geometrical Compendium use both and (Paris ms. 3a2. Mantova ms. 3b16, 17).
46.
For example, in the Hebrew translations of Ibn Rushd by Qalonimus ben Qalonimus. See MC Phys. Hamburg ms. fol. 3b6; MC Soph. Ref. Paris ms. fol. 111a5.
47.
Oxford ms. fol. 2a10.
48.
Mantova ms. fol. 2b5–6.
49.
See Section I.2 below.
50.
See discussion of terms in I.2.3 below.
51.
occurs in II.1.1, and is also the term used by Shmuel Ibn Tibbon in his translation of the Guide.
52.
For example, in I.1.4 he refers twice to Archimedes’ proofs, which should be , according to his standard, as .
53.
Klatzkin’s Thesaurus, vol. I, 257.
54.
Aristotle’s book Sophistical Refutations, named in Arabic Kitāb al Safsaṭa was translated into Hebrew by Qalonimos ben Qalonimos as . This term was used earlier in Shmuel Ibn Tibbon’s translation of the Guide of the perplexed and in other Hebrew texts. See Klatzkin, Thesaurus I.168.
55.
Soph. Ref. 11, 171b8. On this distinction and the notion of peirastic argument in this context see Schreiber (2003, 191, note 6).
56.
Ibn Rushd MC Soph. Ref. Arabic Jéhamy, vol. 2, 694; Hebrew 110b11–13.
57.
See Harvey’s edition, text 186.4, translation 250, Hamburg ms. 3b16. The Arabic text of the middle commentary is not extant. Zeraḥya ben Shealtiel uses the word .
58.
Wardy (1990, 263).
59.
Phys. VII.4, 248a18–b7.
60.
Wardy (1990, 266–73).
61.
Rashed (2015, 441–9).
62.
See references in I.1.1 and I.1.3.
63.
Shifā’, Physics IV.5, McGinnis’ edition and translation, vol. 2, 425–6. See discussion in Rashed (2015, 446–9). Also in IV.3. “So we say that in truth neither the straight line nor the circular line undergoes alteration as to become the other” (McGinnis’ edition and translation, vol. 2, 414). It is also possible that Alfonso found a similar statement in Ibn Sīnā’s Oriental Philosophy, a text with which he was familiar and that is no longer extant. See Szpiech (2010).
64.
See MC Phys. VII.5; MC. Hamburg ms. fol. 89b16–90a15; Post. An, Arabic Jéhamy, vol. 2, 396; Hebrew 10a; MC Soph. Ref. Arabic Jéhamy, vol. 2, 694; Hebrew Paris ms. fols. 110b-111a. The issue is not raised in the Epitome on Logic.
65.
LC Post. An. I comment 67, Badawi edition, p. 293. The complete passage is quoted in I.1.3 below.
66.
MC Soph. Ref. Arabic Jéhamy, vol. 2, 694; Hebrew Paris ms. fol. 111a3–5.
67.
Ibn Rushd, MC De cael. Arabic 92, Herbrew 21b8.
68.
Physics I.2185a18. In his long commentary on this passage Ibn Rushd uses the terms (doubts, natural doubts, eristic doubts). See LC Phys. Not extant in Arabic; Anon. Hebrew translation Paris ms. fol. 9b.
69.
For example, Pr. An. II.26, 69a27–36; Post. An I.9, 75b37–76a3; Soph. Ref. 11171b16–18, 171b34–172a8; Physics I.1185a14–17. More information about these arguments comes from the Greek commentators.
70.
Rashed (2013a, 99).
71.
Alfonso’s two proofs of the quadrature of the lunes (III.23) do not follow Ibn al-Hytham’s and nowhere does he allude to the last step in his predecessor’s argument about the squaring of the complete circle.
72.
Rashed (2015), text and translation 480–1; analysis 450–51.
73.
See Introduction Section 6 and I.1.2, I.2.1, II.5, III.33.
74.
Archimedes, Sphere and Cylinder I.33, 34.
75.
Rashed (2015, 484 (Arabic 485); 522 (Arabic 523)).
76.
See commentary to previous section and note with references there.
77.
Heath (1921, vol. I, 185).
78.
See commentary to I.2.3. In I.2.3 and I.2.5 he mentions Hippocrates by name.
79.
See Introduction, Section 6.
80.
The construction of the squaring of the lunes, for instance, is found in a few Hebrew manuscripts It is appended to the Paris Private collection manuscript of the Elements fol. 169b; it is also included in the Hebrew Geometrical Compendium Mantova ms. fol. 70b. On this text see Introduction Section 6.
81.
Besthorn edition part II fasciculus 1, p. 8 in the online file, fol. 51r of the Arabic text.
82.
Rabat ms. fol. 142–143; Paris ms. fol. 41a. Euclid uses consistently the verb eggraphesthai.
83.
The compiler of the Hebrew Geometrical Compendium used both expressions and .
84.
For example, Moshe Ibn Tibbon’s translation Paris ms. 95b; Compendium, Mantova ms. 29b.
85.
He uses this verb in other contexts in the sense of “describe” or “mark.” See glossary.
86.
See also Moshe Ibn Tibbon’s translation of Elements I.46.
87.
Post. An I.9, 75b39–76a2; Soph. Ref. 11, 172a2–7; on the commentators see Heath (1921, I, 223).
88.
MC. Soph. Ref. Arabic Jéhamy, vol. 2, 695; Hebrew Paris ms. fol. 111a, 20–24; MC Post. An. Jéhamy, vol. 2, 396.5; The passage from the LC is sited below. Bryson is not mentioned in Alfarabi’s commentary on the Sophistical Refutations.
89.
In Arabic (right). Among the Hebrew manuscripts, Munich and Parma manuscript read (right), Wien manuscript reads (not right). Comparison with the middle commentary confirms the first reading: Bryson’s argument “is not demonstrative although he made right premises” (MC. Post. An. Paris ms. fol. 10a. 15–6).
90.
In the middle commentary “because they are general and comprehensive.”
91.
LC Post. An. I, Badawi’s Arabic edition, pp. 292–3 (there is no numbering of the texts-comments), text-comment 67 according to the numbering in the margin of the Hebrew translation: Munich ms. fol. 116b, Oxford ms. fol. 44a–b, Parma Palatina ms. 3022/2 De Rosi 285, IMHM 13751 (there is no page numbering, but texts-comments are numbered in the margin).
92.
Badawi edition, p. 69.
93.
MC. Soph. Ref. Paris ms. fol. 111a8.
94.
The rule of inference: if P (minor premise) and P → Q (major premise), then Q.
95.
See for instance Maimonides, Introduction to Logic, p. 37. See also Bobzien (2002, 360–1).
96.
Heath (1921, vol. I, 224).
97.
Rashed (2015, 454) refers to the two types of superposition in al-Ṭūsī as “superposition géométrique” and “superposition cinématique.”
98.
Bertrand Russel commented (1903, 161 # 152 β) that defining equality in terms of superposition is circular because superposition assumes the rigidity of the superposed body, but the notion of rigidity depends on this of equality.
99.
Plato Tim. 37c–38c. Aristotle Meta. E.1.
100.
Euclid deals with the notion of “position” in his Data.
101.
“It seems clear that common notion 4, as here formulated, is intended to assert that superposition is a legitimate way of proving the equality of two figures… At the same time it is clear that Euclid disliked the method and avoided it wherever he could.” Heath (1908/1956 I, 225).
102.
Sabra (1968, 19–20), Sabra’s translation with a minor modification; Jaouiche (1986, 151); Rashed and Houzel 43–46.
103.
Jaouiche (1986, 162–4).
104.
Sabra (1968, 21), Sabra’s translation.
105.
Commentary on the Premises of Euclid’s Elements, see Jaouiche (1986, 162).
106.
Rashed (2015, 454).
107.
See introductory note to Section I.1 above.
108.
Namely not relying on the parallels postulate.
109.
In II.3.2, II.3.4, II.3.5.
110.
For example, in III.33 when he proves that the planet always moves on the diameter of the deferent.
111.
Alfonso’s notion has little or nothing to do with the methods of the Oxford calculators and with the controversial notion of “Bradwardine’s function.” See Maier (1982).
112.
Kleiner (1989, 283).
113.
Ponte (1992, 2–5).
114.
Pedersen Jones (2010, 49, 80–81), See also North (1987).
115.
II.5.1.
116.
Almagest I.15, Toomer’s translation, p. 72. This value is somewhat larger than the modern.
117.
Elements I, Common Notion 4.
118.
Literally “similarly.”
119.
The sentence seems to be interrupted.
120.
Rashed (2015, 454). See introductory note to Section I.2.
121.
II.3.2, II.3.4, II.3.5, III.33.
122.
Reading , rather than .
123.
Heath (1921 I, 184–200).
124.
See comment to I.1.2 above.
125.
The question of Hippocrates’ “fallacy” is studied in detail in Lloyd (1987).
126.
Notably Soph. ref. 11, 171b13–15; Phys. I.2, 185a14–17, Pr. An. II.25, 69a30–35. In the Physics Hippocrates’ name is not mentioned and there is an argument among modern commentators about it. See Lloyd (1987, 104, note 3); Heath (1921 I, 184–200).
127.
See commentary to III.23 below.
128.
See Langermann (1996, 36–7).
129.
Two of the four quadratures that Simplicius quotes from Eudemus. See Heath (1921, 187, 191–195); Netz (2004, 249–252).
130.
Here and in paragraph I.2.5.
131.
For example, Ibn Rushd, LC Phys. Comment I.10, Hebrew Paris ms. 9a20–26.
132.
MC Soph. ref. Arabic Jéhamy, vol. 2, 694; Hebrew Paris ms. fol. 110b20–24.
133.
Heath (1921, 186); Lloyd (1987, 107). It is mentioned by Simplicius and Proclus who praised Hippocrates as “a man of genius, when it came to constructions, if there ever was one.” See, Commentary 213 Friedline, Morrow’s translation 167.
134.
Heath (1921, 185–6).
135.
Eudoxus’ proof is presented in Euclid, Elements XI.2. Archimedes was the most sophisticated user of the method. For example, Measurement of the circle Proposition 1, Sphere and Cylinder I. 13, I.14, I.33, I.34.
136.
In the text “geometry” (). Should probably be “physics” ().
137.
Elements I.2.
138.
Gitin 7. The word was used also by Abraham Ibn Ezra (Sefer ha-‘Ibur 4, 90).
139.
See for instance Brentjes (1996, 127).
140.
See Sarfatti 193 # 247.
141.
I.2.5, II.1.6, II.3.2, II.4.1.
142.
Metaphysics V.13, 1020a12–14.
143.
Aristotle, Meta. V.13.
144.
Huppert-Sude 21.12–14.
145.
Huppert-Sude 22.13–14. For surface see 24.25.
146.
I.3.5 below.
147.
Euclid Elements I.10; Aristotle, Meta. V.13.
148.
In the Metaphysics, referring to Plato, Aristotle says: “He gave the name of principle to the line—and this he often posited—to the indivisible lines” (Meta. I.9, 992a20–22).
149.
The translation of this section is taken from Freudenthal (2005, IX, 2–3).
150.
Freudenthal (2005, IX, 3).
151.
The hiuli is referred to sometimes as singular, sometimes as plural.
152.
Gerem in the singular appears once in I.4.3.
153.
Tim. 48E–51B.
154.
Plato had a difficulty to explain the receptacle’s mode of existence (Tim. 48E–52C), and used several metaphors to explain it. Gold persists, whereas the things that come to be in it are temporary. In Aristotle’s presentation: “the truest thing to say that each of them is gold.” De gen. et corr. II.1329a21. An associated concept is chora, i.e., place (Tim. 52A, 52D).
155.
The word gerem was common in astronomy but less in geometry. Sarfatti (1968, 202–3) considers it as a synonym of geshem, i.e., body.
156.
In I.3.3 below the word geramim is not mentioned, and Alfonso says that the cause of heaviness is “bodies triangles.” The two words appear one after the other with a bigger than usual space between them. Apparently, the copyist had a confused or corrected text before him and could not decide between the two words. Perhaps this sentence belongs to an earlier stratum of the text, before Alfonso settled on the term geramim.
157.
De caelo III.6, 305a18–19.
158.
See Sadik (2007, 5).
159.
De gen. et corr. I.2315b26–31.
160.
De gen. et corr. I.2316a11–12. MC de gen. et corr. I.3.2 Kurland 8.35–6, 46–7. Ibn Rushd writes “Plato” where Aristotle “Platonists.”
161.
Ibn Rushd (in both his short and middle commentaries on De caelo) explicitly says that Plato associated weight with the planes. Ibn Rushd, however, does not use the word , but the word . Epitome on De caelo, Horesh edition 67. 3,5; MC De caelo III.3.3.2.1, Arabic, p. 286; Hebrew Berlin ms. fol. 61a.
162.
Lane’s Arabic Lexicon, vol. I, 412–4.
163.
Ben Yehuda dictionary, vol. 2, 844.
164.
See commentary on I.4.3 below.
165.
Sarfatti (1968, 86–7, 124).
166.
Gershenzon (1984, 10).
167.
Szpiech (2006, 662); Hecht, note 240; Sadik, notes 93, 262, 287, 391.
168.
Sadik refers once in his dissertation (note 353) to Ennead II. The reference is not sufficient to confirm direct familiarity with Plotinus’ text.
169.
MC De gen. et corr II.1.1, (http://dare.uni-koeln.de/sourceviewer?type=text&textid=26, Chunk 37), translated by Moshe Ibn Tibbon into Hebrew also , Kurland edition 59.52.
170.
Tim. 48E–49A; 52A–B.
171.
Plato cautiously remarks (Tim. 53D–E): “the principles which are prior to these God only knows, and he of men who is the friend of God.”
172.
The Pythagorean agenda to build up solids of points (or numbers) collapsed after the discovery of the incommensurability of the side and the diameter of the square (or the pentagon). Plato, apparently wished to “save” the Pythagorean idea of mathematization of natural bodies, but to avoid the pitfall of transition from surfaces to lines (that was equivalent to the extraction of square root).
173.
Aristotle, De gen. et corr. II.1329a23. This was also well mentioned by Falaquera: “it was stated by Plato in his book Timaeus that bodies are made of planes” (Deʽot ha-filosofim, Parma ms. 65b5–6. It does not seem, however, that Falaquera was Alfonso’s source.
174.
De gen. et corr 316a12.
175.
Ibn Rushd, MC De gen. et corr. Kurland, 8.43–9.49; 60.1. According to Ibn Rushd Plato bases his theory of indivisible areas on the triangle, because all areas are divisible into triangles, and the triangles are further divisible into smaller triangles but not into other areas. The triangle is, thus the beginning of area.
176.
The Timaeus is referred to explicitly three times, but Aristotle mentions also some close variants on the theme, referring to “those who construct bodies of planes” or to “Some who subject all bodies to generation by means of composition and separation of planes” (e.g., De caelo III.1, 299a1–4).
177.
“Those who start from fire as a single element…. some of them give fire a particular shape, like those who make it a pyramid…” (De caelo III.3, 304a9–10; III.5, 304a20, 304b4).
178.
Aristotle, De caelo III.1299a7–9; Ibn Rushd’s middle commentary on De caelo III.3.1 Arabic 284.7–14; Hebrew Berlin ms. fol. 60b3–10. According to Aristotle’s testimony “they [the Pythagoreans] say that a moving line generated a surface and a moving point a line, the movements of unites will also be lines, for the point is a unit having position” (De an. 409a3). Much has been written on this subject. See for instance Guthrie (1962, vol. I, 256–265); Philip (1966).
179.
O’Brien (1984, Chap. 5) tries to reconstruct Plato’s “geometrical theory of weight,” mainly from Timaeus 55D–56C. See also Cornford 222–3.
180.
Timaeus 62c–64a. See also Cornford (1937, 264–6).
181.
O’Brien (1984, 79).
182.
Timaeus 56B. Fire is the lightest “being composed of the smallest number of similar parts.”
183.
De caelo III.1299b24–300a4; IV.2308b15–19; see comment on I.3.4 below. This argument is mentioned by Ibn Rushd in his epitome on De caelo (Horesh edition 66. 3–8), and in detail in his middle commentary (III.3.3.2.1, Arabic, p. 286; Hebrew Berlin ms. fol. 61a).
184.
The relation between them is not clear in Plato either. Plato introduces the receptacle after announcing a “new beginning” (Tim. 48E) and does not explain how the new analysis accords with the previous one.
185.
De gen. et corr. II.1329a16–24, Quotation from II.1329a22–3, Williams’ translation; Ibn Rushd, MC De gen. et corr. II.1.1, Kurland edition, pp. 58–60.
186.
De caelo III.5, 304b3–6; III.7, 305b34–306a1, 306a30–34: Ibn Rushd, MC De caelo III.5.3, Arabic 319.6–12, Hebrew Berlin ms. fol. 70a21–27).
187.
De caelo III.5, 304a11.
188.
De caelo III.1, IV.2; Ibn Rushd MC De cael. Chapters III.3.2.1; [III.5. 3]; IV.1.3.
189.
In I.3.2 above we do not find this statement of thesis [ii] but the revised one. Perhaps I.3.2 was corrected after the reconsideration reported in I.3.3.
190.
The two words gufim meshulashim appear one after the other with a large space between them. Perhaps the original manuscript was confused or corrected and the copyist hesitated which word to choose. See commentary on the word geramim after I.3.2 above.
191.
The tradition about this script goes back to late antiquity. See www.math.ucdenver.edu/~wcherowi/courses/m4010/historians.pdf.
192.
Gluskina’s edition, p. 93, note 53.
193.
MC De caelo III.5.3. The Arabic text is defective; Hebrew Berlin ms. fol. 70a8.
194.
See commentary on I.3.2.
195.
On this reference see commentary on thesis [i] below.
196.
The opposition of vs. was used by Abraham Bar-Ḥiyya (Klatzkin, Thesaurus, vol. IV, 191).
197.
Translating as “truest” follows Joachim’s and Williams’ English translations of Aristotle.
198.
The order of the discussion of the three theses is different from the order in I.3.2 above.
199.
Literally: limited to.
200.
This description calls to mind Aristotle’s descriptions of body as form outside, matter inside. De cael. II.13293b13–16; Phys. III.7207a35–b1.
201.
See Sadik (2007, 2); see also Introduction Section 3, argument ii. Perhaps Alfonso concept of matter is somewhat similar to that of Duns Scotus. See Gordon (2016, Section 1).
202.
Tim. 56A–B.
203.
De caelo III.5, 304a23–34.
204.
De caelo Loeb Classic edition 300–301.
205.
“The element of water will be smaller than that of air, but the least quantity is contained in the greater. Therefore, the air element is divisible” (De caelo III.5, 304a33–b1).
206.
MC De caelo III.5.3, Arabic 318, Hebrew Berlin ms. 70a9–18.
207.
Tomas Bradwardine, Alfonso’s English contemporary in his Tractatus de Proportionibus Written in 1328, suggests that “The four elements are linked in continuous proportionality.” It seems that Alfonso was not familiar with his work.
208.
Aristotle De gen. I.7. 323b11–16. Ibn Rushd MC. De gen. et corr. Is available online via the DARE project. Both Aristotle and Ibn Rushd mention by name only Democritus in this context.
209.
Tim. 53C.
210.
De caelo I.4, 303b1.
211.
Physics VI.1231a24–26, 231b15–16.
212.
I.2.2 above.
213.
Gluskina corrects to .
214.
Soph. ref. 11, 172a7; Phys. I.2185a17.
215.
Heath (1921, 221–3).
216.
For example, MC Soph. Ref. Arabic Jéhamy, vol. 2, 694; Hebrew Paris ms. fol. 110b26–111a4; MC Phys. Hamburg ms. fol. 3b3–6. LC Phys. I.11 Paris ms. fol. 9a26–b6. In the long commentary Antiphon’s name is mentioned but the spelling is different—.
217.
Vlastos (1995, 294–5).
218.
In I.5.3 below Alfonso will try to solve the problem of “natural composition” by identifying the Platonic geramim with the Aristotelian minimal parts.
219.
Quoted above. See introductory note to Section I.3.
220.
The Hebrew sentence is not syntactically correct, so the translation is not certain,
221.
I.2.1 above.
222.
For example, I.4.1–2, 4.4, I.5.4 and in III.33.
223.
For example, II.2.3–4, II.4.1, and III.29. The term was used by Bar-Ḥiyya in this sense, e.g., Ha-Meshiha Part I, #12, p. 9.
224.
The term was used by Bar-Ḥiyya in this sense, e.g., Ha-Meshiha Part I, #33, p. 17; #53, p. 34.
225.
See Klatzkin, Thesaurus.
226.
Phys. III.2, 201b30–32.
227.
Ibn Rushd, MC. Phys. Hamburg ms. fol. 23b22–23 Hebrew translation by Qalonimus ben Qalonimus (this commentary is extant only in two Hebrew translations). A similar expression was used by Alfonso’s contemporary Moshe ben Yehuda with respect to first matter. See Eisenmann (2009, 411).
228.
Or “in the same ratio.” See Terms below on the translation.
229.
Physics VI.1231b18, VI.4.235a15.
230.
Alfonso alludes to Aristotle’s definition “time is the number of motion with respect to before and after” (Phys. IV.11219b1).
231.
The sentence is difficult. Presumably its meaning is that distance and time are divisible into indivisible parts in the same way by motion.
232.
Wolfson (1973, 484–5).
233.
Wolfson (1973, 490). Shmuel Ibn Tibbon defines it in his glossary as “the actuality of the mental capacity in the human soul.” The word taṣawwur appears in Maimonides Guide and is translated by Munk into French as “conception,” and by Pines into English as “spontaneous perception.”
234.
Proclus, Commentary 55 Friedline, Morrow’s translation 44.
235.
Lachterman (1989, 88–9).
236.
Netz (1999, 52).
237.
For example, with respect to dimensions, in pp. 3–5 of the Arabic text, see Hooper-Sude edition.
238.
See commentary on II.2.1.
239.
The fairly literal translation homeomerous belongs to the vocabulary of natural science rather than of geometry. and sometimes also are the Hebrew terms for homeomerous (a term that was used several times in Aristotle’s De generatione et corruptione to denote a substance in which the structure of the parts and of the whole are identical). This meaning is not relevant in the present context.
240.
See Introduction, Section 7, example iii.
241.
See Klatzkin, Thesaurus, III.172.
242.
“Distance is analyzable into indivisible parts.”
243.
Cf. Klatzkin, Thesaurus.
244.
Phys. VI.1, 231b18; see also VI.4235a15.
245.
According to the Jewish tradition a man should not profit from the disgrace of another (). Gluskina noted that Abner of Burgos uses this expression in Sefer Tshuvot la-Meharef. The use of this somewhat unusual expression was one of the linguistic arguments that supported her identification of Alfonso with Abner. We are indebted to Shalom Sadik for letting us use his copy of Sefer Tshuvot la-Meharef.
246.
There is a minimal amount of water that can be considered water. This theory was briefly alluded to by Aristotle, and developed in the Arabic Aristotelian tradition by Ibn Rushd. For a summary of the subject see Glasner (2009, 143–6).
247.
This charge could be based on Aristotle’s complaint that “Plato in the Timaeus says that matter (hule) and place (chora) are the same.” Physics IV.2, 209b11. See also Keyt (1961).
248.
The text is confused and we tried to clarify it by adding brackets and parentheses.
249.
There were several discussions of the continuity of air that took place in Alfonso’s lifetime. For instance, Adam Woodland’s argument that when a sphere is placed on a plane “some continuum would have been interposed, and consequently those objects would not touch.” A similar argument was made by Buridan (Grellard 2011, 72–3).
250.
Grant (1981, 91–2).
251.
Grant (1981, 86–7).
252.
See Sadik (2007, 138–9).
253.
Gluskina (1974, 70); Sadiq (2007, 134).
254.
The first principles of a science, according to Aristotle are posits, suppositions, and definitions (An. post I.272a15–24). The use of these terms, remarks Barnes, was not consistent and “this terminology remained in a state of flux” An. Post. Barnes’ edition (1993, 99).
255.
See Mueller (1981, 15–17).
256.
Elements I.4.
257.
Literally “by forcing.”
258.
By not admitting that it is an axiom. The Hebrew word is not clear and the translation is uncertain.
259.
That is, geometry that allows translation, rotation, and reflection of geometrical magnitudes.
260.
Rashed (2015, 440).
261.
Sabra (1968, 20); Jauiche (1986, 151–2).
262.
Sabra (1968, 20).
263.
ʽUmar al-Khayyām criticized Ibn al-Haytham for his “permissive” interpreting of Euclid’s definitions of the circle and the sphere. Jaouiche (1986, 186–7); Rosenfeld and Youschkevitch (1997, 143).
264.
Proclus, commentary 249–50 Friedline, Morrow’s translation; 194–5; Heath (1908/1956 I, 254).
265.
The proof is known to us via Proclus’ commentary. No Arabic translation of Proclus’ commentary is mentioned in the bibliographical literature but, comments Rashed (2015, 8, note 3), his ideas were not unknown to Arabic mathematicians. Rashed names al-Sijzī as an example. There is evidence that the author of the Hebrew Geometrical Compendiun (preserved in manuscript Mantova com. 2) was familiar with Proclus commentary (Glasner, 2019).
266.
On the question of the incompatibility of mathematical techniques and Aristotle’s notion of demonstration as explanatory see Mancosu (1996, Chap. 1). Harary (2008), Harari (2012).
267.
I.5.2 below.
268.
For example, II.3.2, II.3.4, III.33.
269.
Euclid, Elements I, Postulate 5, known as the parallels postulate. Alfonso’s translation differs from the three main Hebrew translations by Moshe Ibn Tibbon, Ya‛aqov ben Makhir and R. Ya‛aqov.
270.
For a brief summary and bibliography see https://en.wikipedia.org/wiki/Parallel_postulate.
271.
For a study of the transmission of this example see Freudenthal (1988).
272.
Proclus, Commentary 192, Friedline, Morrow’s translation 151.
273.
Guide I.73 tenth premise. Pines’ English translation I, 210. For a discussion of this example see Freudenthal (1988).
274.
Or “ratios of magnitudes.” See Terms on I.4.2.
275.
Or “ratio of magnitudes.” See Terms on I.4.2.
276.
Literally: “and of equal parts.”
277.
Aristotle defines time as the number of motion. Physics IV.11, 219b1.
278.
The word is not certain; the translation follows Gluskina’s reading.
279.
Or “ratio of magnitudes.” See Terms on I.4.2.
280.
According to Heiberg’s numbering should be the twelfth.
281.
The sentence is difficult, and the translation is not certain.
282.
The term tishboret was used by Bar-Ḥiyya to denote area, and shevarim was used by Ibn Ezra with the same denotation. Sarfati (1968, 137, 142).
283.
It was used by Ibn al-Haytham in this sense. See for instance Hooper-Sude, Arabic text 2, 6, 37.
284.
Ibn al-Haytham mentions the term for practical measurement. See Hooper-Sude Arabic text 37. This word has a Hebrew phonetic translation , that was used by Bar-Ḥiyya and by Alfonso himself for measurement and for area. Alfonso uses it in I.1.0 above and in the translation of Archimedes’ Measurement of the circle that was made supposedly by him (Glasner 2013). For Bar-Ḥiyya see Sarfatti (1968, 124).
285.
Like in II.1.1 below.
286.
I.1.3; I.2.3.
287.
See also Introduction, Section 4.
288.
“It follows therefore that if to one revolution of the sun are added twelve revolutions of the moon for an infinite time, there would now be a greater number of revolutions of the moon than of the sun, and similarly that there are potentially more numbers exceeding two than exceeding 100. This argument holds even if one proceeds to infinity” (Dales 1984, 299).
289.
Murdoch (2009, 22–3); Wood (2009, 40, note 8).
290.
Following Isa. 40:4.
291.
https://en.wikipedia.org/wiki/Cavalieri‘s_principle.
292.
See introductory note to Section I.3 above.
293.
Mcginnis (2010, 5, 23) remarks that Arabic natural philosophers preferred “mathematical-style proofs” against the possibility of actual infinity.
294.
Meta VI.1; XI.2, 1060a13–15. Planes and lines can be moved accidentally as parts of a moving three-dimensional body.
295.
For example, Pines (1936/1997, 21–31); Wolfson (1976, 132–182), and, arguing with Wolfson, Frank (1967, 249–253).
296.
Yessod ‘Olam Sha‘ar Alef, p. 3 in the online edition, p. 3b in Goldberg-Rosenkranz edition.
297.
See note @ above (reference to McGinnis).
298.
De caelo I.5272b25–9. Aristotle’s argument is presented in detail in Ibn Rushd’s Middle Commentary on De caelo Arabic 98.10–100.3; Hebrew Berlin ms. fol. 24a21–b2.
299.
The Physics of the Healing II.8, argument 8, McGinnis translation, vol. 1, 184–186. See also McGinnis (2006).
300.
McGinnis (2006, 14) testifies that “a cursory examination” of early Arabic sources “has produced nothing even vaguely similar to Avicenna’s argument.” He also is of the opinion that “nothing like Avicenna’s argument ever explicitly appeared in Greek commentaries on the Physics” (ibid, 8).
301.
According to McGinnis (2006, 15) there are different versions of this argument: “every text has at least some variant reading of the lettering of the points, such that there are multiple ways to construct the diagram, each different and sometimes significantly so.”
302.
Prima facie in the argument quoted by Alfonso only a finite line segment AB rotates, but since the argument involves the straight extension of the line segment, the assumption of this argument is not truly weaker than that of Aristotle’s original argument.
303.
Ibn Sīnā’, Treatise on the Now, Mcginnis (1999, 76–8); Physics VI.5236a11–16.
304.
Mcginnis (2006, 6–7).
305.
Wolfson (1929, 346–7). The example was used late also by Christian scholars. For example, Murdoch (1982, 571).
306.
Mcginnis (2006, 18–9).
307.
Eran, forthcoming.
308.
I.5, Yehuda Ibn Tibbon’s translation, Zifroni’s edition (1949, 115).
309.
Cf. for instance Yehuda ben Shlomo Natan’s translation. Paris ms. fol. 42b20–43a5.
310.
Szpiech (2010).
311.
See Appendix to the introduction.
312.
Literally “to.”
313.
Phys. V.2.
314.
Phys. IV, 4, 212a5, 212a28.
315.
De gen. et corr. I.6, 322b32–323a3.
316.
Phys.III.5, 205b32; IV.1, 208b28.
317.
Meta. XIV.5, 1092a18–20, Annas’ translation.
318.
Phys. IV.1, 208b23–5.
319.
See Introduction, Section 6.
320.
Hogendijk (2003, 22–23).
321.
Sabra (1968, 15).
322.
London Beth Din ms. 138/16, IMHM 4799, fols. 89a–b. After this passage follows a construction for finding two mean proportionals, also using motion, that may be by the same author. Alfonso offers a different construction of two mean proportional using conchoid in III.31.
323.
II.2.1–2.
324.
See I.2, introductory note.
325.
Aristotle, Phys. II.2, 193b31–194a5; Meta. M 3.
326.
Aristotle, Meta. XI.21060 b12–16; XIII.31078a10.
327.
Since common notion 4 is a basic principle of geometry.
328.
The reference to Aristotle at the end may be to Physics IV. A possible example may be Ibn Rushd’s presentation of Physics IV.4212a23–8: “When a radius of a circle rotates about the center, every point on the radius describes a circle and this is true also of the center point itself. The circle generated by the center point must be either in a void or in a spherical body. The first alternative is ruled out. The circle is therefore in a spherical body [i.e., corporeal]. This spherical body must be either in motion or at rest. In the first case the argument can be repeated. We can thus assume that it is at rest. Furthermore, the center cannot be a point because a point can be in motion or at rest only accidentally.” MC Phys. IV.1.9, Hamburg MS, 39b14–25.
329.
II.4.1 below.
330.
Two unreadable words in the text.
331.
The sentence is difficult and cannot be translated literally.
332.
Elements I Postulate 2.
333.
Elements I Postulate 1.
334.
See Klatzkin, Thesaurus, vol. 3, 294.
335.
Hooper-Sude 23.13–15.
336.
Hooper-Sude 26.24.
337.
Elements I, postulate 2, Proposition 10.
338.
See Introduction Section 3, argument iv, and Section II.4.2 below.
339.
There is a problem in the syntax: it is the imagining of motion, rather than the accidents, that is said to be a principle of geometry.
340.
I.4.2, II.1.7 above.
341.
See Jauiche (1986), commentary 57–64, translation of the text 161–175; Huper-Sude (1974), 93.12–106.19; Commentary 109–119.
342.
Lévy (1992, 47).
343.
Hooper-Sude 50.26–51.1.
344.
Hooper-Sude 53, 1–3. It should be noted that Ibn al-Haytham’s premise is not valid in hyperbolic geometry.
345.
Hooper-Sude 53.6–14.
346.
He repeats again and again that points on the moved lines produce by their motion “similar lines” (e.g., Hooper-Sude 51.24–52.7, 52.11–16, 52.20–25).
347.
Thābit first proof, fifth proposition: Sabra (1968, 25); Jaouiche (1986, 148–9).
348.
Ibn al-Haytham’s proof of the theorem relying on this premise is summarized below.
349.
See Section II.3.6 below.
350.
Jauiche (1986, 163); Hooper-Sude 94–5.
351.
Jaouiche, p. 167; Huper-Sude, p. 95.
352.
The postulate is not included in Heiberg’s edition but is mentioned by Heath (1908/1956 I, 232).
353.
Ibn al Haytham relies on Euclid’s Proposition 34 (“In parallelogramic areas the opposite sides and angles are equal to one another”) which relies on Proposition 29, in which Postulate 5 is used.
354.
Hooper-Sude 96.1–2.
355.
Hooper-Sude 53.1–3.
356.
I.2.1 above, II.5 below.
357.
Rashed and Vahabzadeh (1999, 273–4).
358.
Two sentences of Nayrizi’s text are missing here.
359.
The reference should be to the first lemma that, in turn, relies on Elements I.19.
360.
Al-Nayrizi mentiones the angle DGE, adjacent to EGC.
361.
II.2.1; II.2.2 above.
362.
Al-Nayrizi Commentary, Besthorn 120.13.
363.
Al-Nayrizi Commentary, Lo Bello 159–60 (English). Besthorn 120–123 (Arabic and Latin).
364.
Al-Nayrizi Commentary, Lo Bello 158.
365.
Al Nayrizi Commentary, Lo Bello 158–166. See Lo Bello’s introduction, p. 17 and note L5, pp. 224–9.
366.
In the text CD. It should probably be HI, because the parallelism of CD to AB follows from Elements I.28, but HI is a parallel to AB through G but it does not necessarily follow that it is perpendicular to EG.
367.
Elements I, def. 23.
368.
Proclus, Commentary 176 Friedline, 138 Morrow’s translation.
369.
Namely a parallelogram. See commentary on II.3.1.
370.
The fact that Euclid does not use postulate 5 until Proposition I.29 was noted by Proclus (Commentary 364 Freidline, 284 Morrow). Al-Nayrizi ascribed this comment to Simplicius (Lo Bello 158).
371.
Baraness (2018).
372.
Part of the passage that claims to belong to case II seems actually to deal with case III.
373.
See comment on the term in Section I.2.5 above.
374.
Euclid, Elements I.11.
375.
Book I Postulate 2.
376.
Ibn al-Haytham applied twice his technique of drawing two perpendiculars on a line and then dropping a perpendicular from a point on one of them to the other.
377.
Presumably a point on line BD.
378.
Euclid, Elements I.17. See introduction about proposition numbering.
379.
For another interpretation see Baraness (2018).
380.
In the text CHK. Accepting Gluskina’s correction.
381.
A possible understanding is that a–d are subcases of II. This interpretation is discussed in Avinoam Baraness (2018).
382.
Probably referring to a diagram in the original manuscript.
383.
Cf. Klatzkin, III, pp. 238–9. In Sefer Yessod ‘Olam (by Alfonso’s Castilian contemporary Iṣḥaq Israeli) the word is used to refer to a proposition.
384.
Seems that a sentence is missing here.
385.
Presumably the 13th proposition according to Heiberg. See introduction about Alfonso’s numbering of the proposition in Euclid’s Elements.
386.
The correct order, here and in the sequel is ADBC.
387.
Redundant text in the manuscript: then point B of area ABCD.
388.
In the text “right.”
389.
In sentences 3–4 the letter was corrected to and the letter was corrected to . The corrections are listed in the notes to the Hebrew text.
390.
On the distinction between Euclidean and mental superposition see introductory note to Section I.2.
391.
A, B in quadrilaterals ABDC; G, H in quadrilateral AGBH respectively.
392.
The first part of the sentence follows from the first figure, the second part from the second figure.
393.
In the text AD. Gluskina’s correction.</Emphasis>
394.
In the text CBD. Gluskina’s correction.</Emphasis>
395.
Sabra (1968, 26–8); Jaouiche 159–160.
396.
In Thābit’s construction the use of letter is different: the given line is AB, the perpendiculars AC and BD from a point E on AC a perpendicular is drawn that meets BD. See Jaouice (1986, 157–8), Sabra (1968, 25).
397.
After the Persian mathematician Omar Khayyam (1048–1131) and the Italian mathematician Giovanni Girolamo Saccheri (1667–1733).
398.
In the text GAH.
399.
In the text AB. Gluskina’s correction.
400.
In the text AK. Gluskina’s correction.
401.
Elements V definition 4.
402.
See note to II.3.4 above.
403.
Assuming that D is far enough.
404.
Elements I.31. As to the term see I.2.5 above.
405.
II.1.1 above.
406.
Cf. Aristotle Phys. IV.11219a30–b2.
407.
That is, the time interval in which the point crosses over the distance between the two parallel lines.
408.
Job 38, 4–5, NIV translation.
409.
Job 38, 11, NIV translation.
410.
In medieval Aristotelian philosophy the origin of dimensionality is discussed in terms of “prime matter” and “corporeal form.” See Wolfson (1929, 101, 579–590); Elior (2012).
411.
Gershenzon (1984, 140–155). See above, Introduction, Section 3 argument iv.
412.
Sadik (2007, 136–7; 2011, Chap. 1.6.2.2).
413.
Gershenzon (1984, 137).
414.
Gershenzon (1984, 137, and Chap. V); Sadik (2011, Chap. 1.6.2).
415.
The style of the following passage from Minhat Qena’ot illustrates this point: Quoted from Baer (1959, 201); See also Sadik (2011, 76).
416.
Pseudo Aristotle, Mechanica, Chap. 24, Forster’s translation. On the history of this problem see Drabkin (1950); Palermino (2001).
417.
Abattouy (2001, 103).
418.
Chapter I. 7.
419.
See editors’ introduction to Hero’s Opera, vol. II, p. xviii.
420.
Drabkin (1950); Palermino (2001).
421.
Drabkin (1950, 172).
422.
Abattouy (2001, 103, notes 18, 19). See Cardano 221–2 (240–1 in online version).
423.
Rashed (2015, 470) has no doubt that Hero’s text was available to the anonymous mathematician who criticized al-Shīrāzī. On Alfonso’s acquaintance with al-Ṭūsī’s schools see Introduction Section 6.
424.
Palermino (2001, 383).
425.
Since Murdoch’s (1957) dissertation and subsequent articles much has been written on this controversy. For a recent brief presentation of the subject see Grellard and Aurélien (2009, 1–14); Murdoch (2009, 16). Although the controversy had strong underlying theological motives (Sylla 1997), a major part of the arguments adduced against indivisibilism by Aristotle, al-Ghazali, Duns Scotus, Bradwardine and by other divisiblists were mathematical to different degrees. We are grateful to Liran Gordon for sharing with us his digitalized version of Duns Scotus’ Commentaria Oxeniensia, vol. II.
426.
See Introduction, Section 6.
427.
Rashed (2015, 469).
428.
We ignored the word (equal) that appears in the text here.
429.
Rashed (2015, 578–582).
430.
When speaking of the rolling circle Alfonso adds “for instance the base of a cylinder,” thus guaranteeing the rigidity of the motion. According to Aristotle one or two-dimensional entities have no independent existence and can be said to be moved only qua sections of three-dimensional bodies (e.g., Meta. III.5, 1002a15–b10). The Arabic anonymous mathematician adduces another argument against al-Shīrāzī by comparing the rolling of a cylinder on the floor to that of rolling a cone with the same base on the floor (Rashed 2015, 588–592).
431.
For a dynamic illustration see: http://mathworld.wolfram.com/AristotlesWheelParadox.html.
432.
Word not certain. Corrected after the first appearance in this sentence.
433.
G is on the line CK as in the previous section.
434.
Phys. VI.1231b18, VI.4.235a15.
435.
Zupko (1993, 160).
436.
Harclay was criticized by several of his scholastic contemporaries, most notably by Thomas Bradwardine in his De continuo written between 1328 to 1335. See Murdoch (1957, 200; 1987, 114–5).
437.
Cf. Rashed (2015, 592–3). See also Section II.5.1 above.
438.
Referring to the views of Ibn Rushd and Ibn Sīnā, I.1.1 above.
439.
Alfonso does not return to this point in the extant part of SMA.
440.
See Szpiech (2006, 525); Smith (1925/1953, vol. II, 252).
441.
Translating the definitions of Elements X, Moshe Ibn Tibbon rendered the passive expression yuntaq bi-hi as . See Paris ms. fol. 83a9–10. Cf. Judeo-Arabic manuscript fol. 94a.10–11.
442.
Pappus, Commentary on book X, p. 191, #1, line 2.
See http://www.wilbourhall.org/pdfs/pappus/PappusBookX.pdf.
443.
Sarfatti (1968, 82, 128).
444.
Heath (1921 I, 220–221).
445.
Hooper-Sude 85.
446.
Knorr (1986, 44 n. 46; 284 n. 88).
447.
See more on this example in the Introduction Section 6, (vii), example [i].
448.
I.2.5 above.
449.
Gluskina identifies him as Johannes Campanus. See Introduction, Section 6.
450.
See comment on terms after I.5.3.
451.
Sarfatti (1968, 69, 91).
452.
Sarfatti (1968, 68, 88, 171).
453.
The text Quadranta circuli is published in Clagett (1964, 591–605).
454.
Grant (1974, 172. Col. 1).
455.
Alfonso’s several terms for geometrical propositions are discussed in the Introduction, Section 7.
456.
Property18 (following Archimedes), 23 (following Hippocrates), 33 (following al-Ṭūsī), and 29–32 which deal with asymptotic lines and mechanical constructions.
457.
Definitions [d] and [e] add two variants to this basic concept.
458.
See introductory comment to Part III, Section II.
459.
For example, in Properties III.11–13.
460.
Heath’ translation.
461.
Smith II, 488.
462.
See Saito (1985) on the use of invisible diagrams.
463.
See commentary on Property 13 below.
464.
This terms appears many times in III.10–23, 31.
465.
III.10.
466.
In I.1.4; I.2.4.
467.
III.17, III.23 and III.29.
468.
Three-dimensional expansion according to Definition [e].
469.
Three-dimensional expansion as used in Property 32.
470.
The Measurement of the Circle Proposition 1, quoted by Alfonso in I.2.4 above.
471.
Literally: “the expansion of a straight line by the hypotenuse of a given angle.” The word “hypotenuse” seems to be out of place.
472.
See Terms below.
473.
On the meaning of see I.3.2 comment on terms.
474.
So it is used in Properties 11–13. In Property 1 DH and CG are half chords of central angles, which equal chords of inscribes angles.
475.
II.3.5; III.1, III.20. and without reference to an angle are used many times.
476.
Perhaps Alfonso intended to use it in the following parts that are lost.
477.
See Appendix B.
478.
See Rashed (2012, 25–9).
479.
Sarfatti (1968, 107–8).
480.
We are grateful to Akiva Kadary, who gave a lecture on the Hebrew terms for sine.
481.
Goldstein Chabas (2000, 152); Goldstein (2019, 134).
482.
Sarfatti (1968, 82).
483.
Naḥmias, A.II.8, Hebrew, p. 196, Arabic, p. 60.
484.
See Sarfatti (1968, 195).
485.
The text is resumed here after a long lacuna. The title “Proposition 10” is missing but it seems that the text of Proposition 10 is complete.
486.
On visible vs. invisible diagrams see Saito (1985).
487.
See introductory note to Part III above.
488.
The correct order of the letters should be BAD.
489.
The wording “intersecting one another and cutting the circumference” and the presentation of the circle as “ABCD” accord with the situation that E falls inside the circle.
490.
Literally “a figure with four straight sides.”
491.
I.2.1. See introductory note to I.2. There is a Hebrew translation (from the Arabic) by Ya‘aqov Anatoly (cf. Wien ms. 194/40, IMHM 1317, fol. 20b20–12a11).
492.
See introductory note to Part III.
493.
Ptolemy (1816, 29).
494.
Ptolemy, Hebrew text, Paris ms. fol. 7a. On this term see Bar-Ḥiyya Hibbur ha-Meshiḥa, 10. For a summary of Alfonso’s terms see table in the commentary on III.0.
495.
Mantova MS fol. 66a.
496.
See introductory comment to Part III.
497.
See Rashed (2012, 46 and 767).
498.
For example, in Fibbonaci. See Barnabas (2008, 80–81).
499.
Accepting Gluskina correction of (complete) to (discard).
500.
Sarfatti (1968, 85); e.g., Sefer ha-Meshiha we-ha-Tishboret 3, 4, 18.
501.
Sarfatti (1968, 178).
502.
The expression appears in the version of Euclid’s Elements in Firenze 137 ms. in the proof of Proposition I.26. We are grateful to Ofer Elior for this comment.
503.
Not accepting Gluskina’s correction AD. Her correction, as well as her diagram, indicate that she considered only the case that points C and D coincide.
504.
An alternative proof using VI.22: from the proportion of the sides of the triangles (AB : AC = DE : DG) it can be directly concluded that the ratio of the square on AB to the square on DE is as the ratio of the ratio of the expansion of AB by DE to the expansion of AC by DG. It isn’t clear why Alfonso preferred to use the mediation of the squares on the sides.
505.
AB should not be listed because the expansion of AB by itself was mentioned before.
506.
Netz (2004, 113–4). The proof is analogous for any even number of points of division, but is different in the case of an odd number of points of division.
507.
Proposition 313.12 in Hogendijk analytical table (1991, 239). Also Rashed (2012, 756–7).
508.
Mantova ms. 73b–74a. See Glasner (2019).
509.
Being angles of a polygon the angles are less than 360° and their halves less than 180°. Consequently the sines are positive. The equality of the corresponding chords follows from the equation sin α = sin (π − α).
510.
In the text DB. Gluskina’s correction.
511.
Theorem I.47 pertains to squares on the sides of the triangles, Alfonso uses it for expansions, i.e., half squares.
512.
Heath (1908/1956 I, 373, 388–9).
513.
In the text DE. Gluskina’s correction.
514.
In the text DA. Gluskina’s correction.
515.
In the text ACD. Gluskina’s correction.
516.
The Hebrew word is not certain. Gluskina’s reading is . We suggest that it may be .
517.
Missing in the text. Gluskina’s correction.
518.
Alfonso skips the explanation. He presumably alludes here to his source, but unfortunately does not provide the reference.
519.
In the text “triangle.” Gluskina’s correction.
520.
In the text AD. Gluskina’s correction.
521.
In the text AB. Gluskina’s correction.
522.
In the text AEG. Gluskina’s correction.
523.
In the text C. Gluskina’s correction.</Emphasis>
524.
Literally to the “triangle of segments.”
525.
In the text DB. Gluskina’s correction.</Emphasis>
526.
Simplicius Commentary on Aristotle’s Physics. Book I. 60.22–68.32 Diels; Heath (1921 I, 184–200); Lloyd (1987); Netz (2004). Lloyd (1987, 110) remarks that the testimonies of Themistius and Philoponus contribute little to the reconstruction of Hippocrates’ argument.
527.
Sorabji (1990, 301).
528.
Ibn Rushd used Alexamder’s commentary intensively when he wrote his long commentary on the Physics. See Glasner (2009, 24–5, note 23).
529.
Heath (1921, 191–3); Netz (2004, 249–50).
530.
London BL MS 1013 Or. 2806, IMHM 6385 at the beginning; Paris Private collection MS of Euclid’s Elements (IMHM 39116), fol. 169b.
531.
See Introduction, Section 6.
532.
Langermann (1996a, 33–35).
533.
Langermann suggests that Alfonso drew on the Hebrew Geometrical Compendium and abridged the argument. The relation between these two texts is not easy to understand. See Introduction, Section 6 above.
534.
Langermann (1996, 41–2).
535.
III.0 above.
536.
Elements V definition 3.
537.
In the definitions of Book VII a general concept of ratio is not introduced, but three types of ratios between numbers are defined: part, parts and multiple. The word “ratio,” however, is used in the propositions.
538.
Heath (1908/1956 II.132); Mueller (1981, 118, 154). Euclid recognized only compound ratios that can be reduced to a single ratio (i.e., a:b∙c:d where b = c). See Netz (2004, 203, note 102).
539.
Vitrac (2000); Rashed and Vahabzadeh (1999, 298).
540.
These propositions were not used later in the Elements (Heath 1908/1956 II, 132–3, 176–8, 189–90).
541.
Vitrac (2000, 6). Heiberg and Heath agree that this definition is an interpolation (Heath 1908/1956 II, 189).
542.
For example, Judeo-Arabic ms. fol. 57a8–11; Hebrew Paris private collection ms. fol. 48a18–9. There are some differences between the mss. In the Hebrew Geometrical Compendium (Mantova ms. 2, fol. 46b) the two definitions are more neatly stated, applying multiplication and division to the ratios themselves, not mentioning “sizes of ratios.”
543.
Euclid, Elementa, vol. II, 72, Heiberg.
544.
Rashed and Vahabzade (1999, 278); Vitrac (2000, 23–4).
545.
In Part V of this treatise. See Glasner, forthcoming.
546.
For example, Heath @@, II. 120–129.
547.
Vitrac (2002, 4–8); Rashed and Vahabzade (1999, 275); Vahabzade (2013, 225); De Risi (2014, 18).
548.
Vahabzadeh (2012, 11).
549.
Much has been written on this early theory of proportions. For example, Becker (1933); Szabo (1978, Part 2); Fowler (1981, 1982); Thorup (1992).
550.
Vitrac (2002, 11).
551.
Rashed and Vahabzadeh (1999, 274–9); Vahabzadeh (2013, 225–7); Vitrac (2002, 13–17, 30). The need to correct Euclid’ theory of proportion was still an issue in the modern era from the early Renaissance to the beginning of the eighteenth century (De Risi 2014, 17–24). On the persistence of the early theory of proportions in post-Euclidean Greek geometry see Knorr (1978).
552.
Ibn Hud in his Istikmāl follows Euclid’s definition (Hognedijk 1991, 231). Gersonides, in his commentary on the fifth book of the Elements, provides the rigorous axiomatic basis that should underlie Euclid’s definition (Carlebach 1910, 168–172).
553.
Plooij (1950, 63–4).
554.
De Risi (2014, 195–201).
555.
Alfonso had a copy of al-Nayrizi’s commentary on Elements 1 (II.2.3 above) but we cannot be sure that his copy of al-Nayrizi included book V, in which al-Nayrizi presents the antyphairetic theory.
556.
See note to the Hebrew text of Property 24.
557.
On the “errors” in Part III see Introduction, Section 7, example [iii].
558.
Sarfatti (1968, 82, 135).
559.
Naḥmias, e.g., Section A.II.10, Hebrew text, p. 196, Arabic text, p. 60.
560.
Paris private collection ms. fol. 55a7.
561.
See Introduction, Section 1.
562.
See below Appendix B.
563.
If the condition were symmetrical, the proof would have been valid regardless of whether strong or weak inequalities were used in the proof. See Appendix B below.
564.
The finding of the fourth proportional here is not trivial because the magnitudes are not of the same kind (see Bos 2001, 105). Alfonso uses a construction of a fourth proportional in the non-homogeneous case also in III.32. Perhaps he was not aware of the difficulty or ignored it.
565.
See Appendix B below on the use of weak vs. strong inequalities.
566.
The requirement of homogeneity is made in the exposition of the theorem but is not mentioned in the proof.
567.
II.3 above.
568.
There are examples of the use of mixed inequalities in the sixteenth century. See, e.g., Wagner (2010, 506).
569.
The assumptions (1) ∑ a_i > AB ⇒ ∑ b_i > CD, (2) ∑ a_i < AB ⇒ ∑ b_i < CD are not necessarily equivalent to ∑a_i ≤ AB ⇒ ∑ b_i ≤ CD, ∑ a_i ≥ AB ⇒ ∑ b_i ≥ CD, so it is possible that ∑a_i ≤ AB but ∑b_i > CD; In that case EB: CD ≥ EB: ∑ b_i. But this inequality cannot lead (together with the inequality EB: ∑ b_i ≤ ∑ a_i: ∑ b_i ) to the conclusion EB: CD < p, so the desired contradiction cannot be achieved, and the theorem cannot be proved.
570.
The theorem can be proved also under these assumptions, but in a slightly different way.
571.
In this case ∑a_i = AB ⇔ ∑ b_i = CD, and the conclusion EB : CD < p is valid.
572.
Van Brummelen (2009, 177).
573.
Mantova ms fol. 212 B. We are grateful to Akiva Kadari who drew our attention to this marginal note.
574.
Avinoam Baraness had already published his interpretation of these four properties in Katz (2016).
575.
See Heath (1921, vol. I, 225–232, 264–266).
576.
It can also be viewed as the trace of a fixed point on a straight-line through the pole, moving along the ruler. See commentary on Property 29 below.
577.
Its Cartesian equation is little more complicated: (y − a)² (x² + y²) = d²y².
578.
Nothing is known of his life; the dates were estimated by references to his work. See Heath (1921, 238); Toomer (2008).
579.
Pappus’ Collection, Eutocius’ Commentary on Archimedes’ Sphere and Cylinder, Proclus’ Commentary on the First Book of Euclid’s Elements. See Toomer (2008).
580.
Proclus, Commentary 138–9 Morrow translation (176–7 Friedline).
581.
Heath (1908/1956, vol. I, 160–1).
582.
Sefrin-Weis (2010, 144, see also pp. 144–6, 271–3); Heath (1921, vol. ii, 362).
583.
http://xahlee.info/SpecialPlaneCurves_dir/ConchoidOfNicomedes_dir/conchoidOfNicomedes.html.
584.
In case the greater line is double of the less. The reduction was probably established in the fifth century BCE by Hippocrates of Chios (Heath 1981, 244–246).
585.
Toomer (2008).
586.
Toomer (2008).
587.
Rashed (2012, 66–9); see also Heath, The Works of Archimedes, c-cii and p. 310 note.
588.
In the text KHC. Gluskina’s correction.</Emphasis>
589.
The curve was called “conchoid” by Proclus, but Pappus has called it “cochloid” (Heath 1921, 238).
590.
Ms. Parma, fol. 45b, Following Nehemiah 2:13.
591.
This result can be justified by simple geometric (or trigonometric) considerations. Alfonso—as he often does—skips the details. It is possible that the subject was discussed in the beginning of chapter three that is missing in the manuscript.
592.
Alfonso mentions en passant that the constructions can be done by means of the conchoid, not explaining the neusis property of the conchoid on which the demonstrations of Propositions 30 and 31 rely.
593.
See Knorr (1987, 303); Sefrin-Weis (2010, 243–4 and 244, note 3).
594.
In the text DHB. Gluskina’s correction.</Emphasis>
595.
In the text BDE. Gluskina’s correction.</Emphasis>
596.
As noted in the previous property, Alfonso most probably took for granted the use of a special instrument to draw the conchoid.
597.
Relying on Elements I.1 an angle of 60° is constructed, which can then be bisected.
598.
In Pappus. See Sefrin-Weis (2010, 149).
599.
Heath (1921, 235); Sefrin-Weis (2010, 148–149).
600.
In the text GM. Gluskina’s correction.
601.
In the text NE. Gluskina’s correction.
602.
In the text BL. Gluskina’s correction.
603.
In the text “with.” Gluskina’s correction.
604.
In the text GN. Gluskina’s correction.
605.
In the text AL. Disagreeing with Gluskina here.
606.
In the text F instead of G and the order is reversed: “MB to LF.”
607.
If AB = CE the proposition becomes trivial.
608.
See Knorr (1989, pp. 251–319); Heath (1981, pp. 244–268); Clagett (1964–1984, I pp. 335–345, 658–665, III pp. 27–30, 849–854, 1163–1179); Rashed (2011–2013, pp. 60–69, 103–107).
609.
See Heath, pp. 260–262.
610.
In the text EG, our correction.
611.
In the text HG, our correction.
612.
In the text MI, Gluskina’s correction.</Emphasis>
613.
This is not the simple classic problem of finding a fourth proportional, because not all the magnitudes are of the same kind. In such a case neither the existence nor the constructability of the fourth proportional is obvious. See Bos (2001, 125). Alfonso does not give any details about the way this construction should be carried out (unless he did it in one of the missing Properties 2–9).
614.
See Bos (2001, 79–81).
615.
Heath (1921, 246); Knorr (1986, 23).
616.
Langermann (1996a, 34).
617.
See Introduction, Section 6.
618.
In the text: E.
619.
In the text: E. Several readings are possible. Seems that “D” agrees better with the rest of Property 33.
620.
In the text: the motion of E is twice that of D.
621.
In the text DH, Gluskina’s corrects to AG.
622.
Gluskina reads AC.
623.
Gluskina correct to CI.
624.
In the text: C.
625.
The word is not certain.
626.
Barker and Heidarzadeh (2016, 21–2).
627.
Questiones de spera, Droppers edition English 285–9, Latin 286–290; Kren (1971, 490–1), Kren’s translation.
628.
Goddu (2010, 478–484).
629.
Kren (1971, 496).
630.
For example, Goddu (2010, 476). Blåsjö (2014) are skeptical as to eastern influence on Copernicus. Barker and Heidarzadeh (2016) adduce strong arguments to support the possibility of such influence.
631.
Langermann (1996, 34–5). See also Ragep (2017, 190).
632.
Oresme’s proof, as Kren (1971, 492) and others noted, is confused.
633.
Nice animations are available in the internet. For example, http://en.wikipedia.org/wiki/Tusi_couple.
634.
The angular velocity of the inner circle is twice of that of the outer circle but the tangential velocities are equal and therefore the velocity of the planet at A is zero.
635.
Physics V.1, mainly 224b1–225a12.
636.
Physics VIII.8, 262a14.
637.
Rashed (2005, Appendix Section II). I am grateful to Prof. Rashed for the reference and for sending me the appendix.
638.
Langermann (2009).
639.
The natural places as causes of the natural movements play a major role in Aristotle’s natural philosophy, yet, Aristotle himself is ambiguous on the question whether place can be considered a cause. Physics IV.1209a 18–22, 208b8–25. See Glasner (2015, 23, especially note 21).
640.
Pedersen Jones (2010, 295–8).
641.
Copied from Ragep (1993, vol. I, English translation 198, Arabic text 199).
642.
Outline, Section 0 above.
643.
On the significance of this correction see commentary to I.3.2, thesis ii.
644.
Phys. VI.1, 231a21–b18; Meta. V.131020a11–14.
645.
Elements I, definitions 1–6, XI, definitions 1–2.
646.
In the text: HG.
647.
Alexander (2014, 138).

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The Faculty of Humanities, Hebrew University of Jerusalem, Jerusalem, Jerusalem, Israel
Ruth Glasner
The Faculty of the Humanities, Hebrew University of Jerusalem, Jerusalem, Jerusalem, Israel
Avinoam Baraness

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Glasner, R., Baraness, A. (2021). Chapter 2: English Translation and Commentary. In: Alfonso's Rectifying the Curved. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-77303-2_2

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