Chapter 1: Introduction

Glasner, Ruth; Baraness, Avinoam

doi:10.1007/978-3-319-77303-2_1

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

The text of SMA is preserved in a single manuscript in the British library (henceforth BL manuscript). It is the sixth work in a codex of eight scientific books. The focal subject of this codex is the astrolabe. The first group of texts (1–4) are copied in a single hand, Sephardic script. Texts 5–7, by Abraham Ibn Ezra, Alfonso, and Emanuel ben Ya‛aqov were copied in another Sephardic-Italian hand in the last third of the fifteenth century. Text 8 by Ya‛aqov ben Makhir is copied in a third hand. The corpus belonged to an erudite owner who wrote commentaries and added diagrams in the margins. Nurit Pasternak suggested and Malachi Beit-Arié confirmed that this owner was Mordekhai Finzi. Finzi was at the center of the mathematical and astronomical activity in the Mantua community, in the third quarter of the fifteenth century, and had Jewish and Christian scientific colleagues. It seems that texts 5–7 were copied in Finzi’s milieu and perhaps were ordered by Finzi himself.

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Notes

1.
BL Add. ms. 26984 fols. 93b–128a; Margalioth catalog, vol. 3, p. 320, 1002/6; institute of microfilmed Hebrew manuscripts in Jerusalem F 5656. We rely on the work of Nurit Pasternak who examined the manuscript at the British library.
2.
It is made of paper and parchment. The parchment is Italian.
3.
Edna Engel suggested that the copyist was a pupil of Abraham Fariṣol, perhaps the one who copied the Parma manuscript that is photocopied in Engel (1992, 158).
4.
On the basis of comparing the marginal notes (not in SMA, but in the astronomical parts of the codex) to the known autographs of Finzi. Interestingly the line in the letter in Finzi’s autographs is usually straight and in the marginal comments in the BL manuscript it is sometimes straight and sometimes curved, even in the same comment.
5.
Langermann (1988, 8); Wagner (2013, 195–6).
6.
On fol. 122b. See picture in III.24 below.
7.
On fol. 109a.
8.
We are again indebted to Prof. Beit Arié for confirming this.
9.
There are a few corrections above the lines, e.g., in fol. 115a, but these could have been made by the copyist.
10.
Langermann (1988, 37–9; 1996, 34–5). See Oxford Bodl. ms. Mich 350, Neubauer 2052, IMHM 19337, fol. 92b. There are minor differences between Alfonso’s text and Finzi’s copy such as replacing the suffix by (e.g., by ) or omitting a few words. Another text in Finzi’s notebook is dated 1473.
11.
In 1473 the scholar and copyist Abraham Farissol who lived in Mantua at the time left for Ferara in 1473; Mordekhai Finzi died in 1475.
12.
Freudenthal (2005, Chap. IX, 1).
13.
Gluskina (1974, 1978).
14.
Lévy (1992); Langermann (1996a).
15.
Baraness (2016, 2018).
16.
Gluskina (1978).
17.
E.g., the use of the Talmudic, but uncommon phrase by both Abner and Alfonso. Gluskina 1978, 70–1.
18.
Freudenthal (2005, Chap. IX, 2–4).
19.
Freudenthal (2005, Chap. IX, 2–3). See Section I.3.1 below.
20.
An Italian origin could have brought us somewhat closer to understanding how Alfonso learned about the construction of the conchoid by Nicomedes (SMA III.29). This construction is known to us from Pappus’ Collection, vol. 4. Sabetai Unguru (1974, 319–321) very cautiously, but reasonably offers that an archetype of Pappus’ collection circulated in Italy before 1533. He argues that Witelo was familiar with parts of the collection and suggests that William of Moerbeke translated for him parts of it. The single manuscript of SMA and the only quotation from it known to us are from north Italy in the 1470s. The one explicit reference to a Christian scholar is to the Italian mathematician Campanus. Thus, prima facie, an Italian origin seemed to be likely.
21.
See Section 6, vii, example [i] below. See also commentary to Section II.5.5.
22.
See for instance Gershenzon (1984, 14).
23.
Ryan Szpiech (2006, Appendix 4, 645–657) compiled a complete list of the sources of Alfonso’s polemical book Mostrador de justicia (extant only in Castilian) and has shown that the vast majority of the citations are from Jewish traditional texts.
24.
Several important dissertations have been writes: Gershenzon (1984); Sainz de la Maza Vicioso (1990); Hecht (1993); Szpiech (2006); Sadik (2011).
25.
See Sadik (2008).
26.
Szpiech (2006, 524–8). Szcpiech further points out that Abner’s “attitude to his sources was ambivalent, split between a real objective interest in the ideas that he proffers, and an overriding concern with his messianic polemical arguments” (2006, 528).
27.
Most notably in I.4.2, I.3.1 and the poetical preface.
28.
Gershenzon (1984, 12–4).
29.
Szpiech (2010, 194–195); see also 14–5.
30.
He probably would not have made a good chess player.
31.
It is possible that “Alfonso” was Abner’s Spanish name already before his conversion, but Shalom Sadik, for several reasons, considers it as unlikely. We are grateful to Sadik for his advice.
32.
Elements V definitions 3, 4.
33.
Aristotle, Meta. V.13, 1020a11–13. In the definitions of Part III, Alfonso Replaces the strict Greek language of homogeneous ratios by a more flexible language of geometrical operators. “I call an expansion [] of a straight line by a straight line the rectilinear right-angled triangle enclosed by the two lines”. He by no means shifts to a symbolic algebraic language, but allows more flexibility within geometry.
34.
On the translation see Section I.1.1, note on the term .
35.
The notions of Euclidean and mental superposition are discussed in more detail in the introductory note to Section I.2.
36.
See citation in the commentary to IV.1.
37.
Sadik (2007, 136–7).
38.
Sadik( 2011, 71–82) (Section 1.6.2.2). Allusions to this peculiar notion of dimensionality may be found in SMA as well: “He [God] determines the actual specific limits of distances that are potentially unlimited” (II.4.2); “first quantity is not an attribute of the first body” (II.1.7).
39.
Gershenzon 137. Baer (1958, 287) finds in the concept of hitlabbshut a mixture of Kabbalistic and Christian ideas.
40.
Gershenzon, Chap. V; Sadik (2011, Chap. 1.6.2).
41.
The metaphor of the mirror as an entity in which an object can be in many places without being anywhere (Szpiech 2006, 157) may help us understand Abner’s imaginative and rather unique way of associating geometrical and theological conceptualization.
42.
See Szpiech ibid.
43.
This is notable in the mathematical texts that were written by authors who were brought up in Christian Spain: Yehuda ben Shlomo’s Midrash ha-Ḥokhma, IsḤāq Isra’eli’s Yesod ‘Olam and IsḤāq al-AḤhadab’ Epistle of Number (on the latter see Wartenberg 2007, 33).
44.
Abraham ben David of Posquières (Rabad), David QimḤi and Yosef QimḤi. Rashi (eleventh century France) is also quoted. See Szpiech (2006, 657, 659).
45.
Lévy (1997a, 185–6; 2005, 107–8).
46.
Szpiech (2006, 593–642).
47.
Szpiech (2006, 445).
48.
Vajda (1959).
49.
Harvey (2003, 66–7).
50.
E.g., by al-Nayarizi and Ibn al-Haytham.
51.
Langermann (1996b); Koningsveld (1992); Assis (2012); Hacker (2015).
52.
Sainz (1990, 153) has convincingly argued that Alfonso was a physician.
53.
Sirat and Geoffroy (2005).
54.
Szpiech (2006, 444–5); Sadik (2016a, 99).
55.
Sadik (2016a, 98, 102).
56.
Sainz (1990, 167).
57.
See Poetical Preface and Section I.3.1.
58.
Our attempt to find some of Alfonso’s “sources” in the catalog of the Monasterio did not yield any results.
59.
Sainz (1990, 171).
60.
Sainz (1990, 190, note 86).
61.
To our disappointment the catalog includes no books in Arabic. This does not necessarily imply, however, that Arabic books were not accessible in the monastery.
62.
Sainz (1990, 166–7).
63.
Geoffroy and Steel (2001, 87–91); Steel and Guldentops (1997, 87–9); Meirinhos (2007, 58–63).
64.
Meirinhos (2007).
65.
Steel and Guldentops (1997, 89).
66.
Sections I.3.5, I.4.3–4, II.5.5.
67.
He studied at the university of Paris medicine (1329 and 1332) and theology (1342–5). See Meirinhos (2007, 52).
68.
Meirinhos (2007, 50–51).
69.
Meirinhos (2007, 51).
70.
E.g., Merinhos (2007, 54).
71.
As already noted, in Part III he fairly consistently modifies the arguments that he uses by shifting from the classical Greek language of proportion to a language of expansion (). On this term and its importance for Alfonso see the discussion in the introductory note to Part III.
72.
Section I.3.1.
73.
See commentary on Section I.2.3.
74.
Langermann (1996a). See commentary on Section III.23.
75.
See commentary on Sections I.1.3 and I.3.5.
76.
It has been suggested that in his other writings, when he mentions Plato by name Alfonso sometimes refers to Plotinus’ Enneads, most frequently to the fifth Ennead. See Szpiech (2006, 662); Sadik (2011, notes 208, 257, 287, 331). This text was available in Arabic but not in Hebrew. See Zimmermann (1986, 112–13, 128–31). It does not seem that Alfonso relies on Plotinus in SMA.
77.
See commentary on Sections I.3.2 and I.3.4.
78.
Gluskina (1978, 69).
79.
Sadik (2007).
80.
Physics V.1. Alfonso refutes Aristotle’s tenet that oscillatory motion must come to a rest at the endpoints (III.33).
81.
Section 4 above.
82.
Glasner (2009, Chap. 8).
83.
See commentary on I.1.1, I.1.3.
84.
See <InternalRef RefID="Sec19" >Appendix</Internal Ref> below.
85.
There are no allusions at all to the added 14th and 15th book.
86.
Djebbar (1996, 96–8); Brentjes (2001b, 39; 2006, 171–4; 2001a, 24).
87.
On the relation between Hajjaj’s original translation and al-Nairizi’s commentary see Engroff (1980, 19); Brentjes (2001, 24–8). See also the section on al-Nayrizi below.
88.
Brentjes (2018).
89.
See Lévy (1997b); Elior (forthcoming).
90.
Sections III.0, III.11.
91.
Explicit references: to I.11; to I.13 = Heiberg I.17; to I.12 = Heiberg I.13 (all in II.3.2); to I.34 (I.2.2), to I.16, I.26 (II.3.7), to I.33 (II.4.1), to I.47 (III.11).
92.
To V.13 = Heiberg V.12 (in I.5.4).
93.
The propositions used are: V.8, 9, 10, 12, 14, 16, 19, 24; VI. 1, 2, 4, 8, 12, 16, 20, 22, 23, 25, 31.
94.
It is not clear whether these are author’s or copyist’s errors, or whether they carry some information on the version that Alfonso used. On the numbering of the proposition in the Arabic versions of the Elements see Jaouiche (1986, 138–144); Lévy (1997a, 223, n. 59).
95.
See II.5.5.
96.
In I.1.2 he ascribes to Euclid the three circles example, perhaps assuming that it is included in Elements XII. In I.3.4 he ascribes to Euclid the statement that any solid is divisible to pyramids, perhaps assuming that it is included in Elements XI.
97.
I.1.3–4.
98.
On the secondary transmission of Elements in Arabic see Brentjes (2001, 39–40).
99.
Three of the four manuscripts of Qalonymos’ Hebrew translation of The Sphere and the Cylinder offer an unusual spelling of Archimedes’ name: or In SMA as well as in one of the two Hebrew translation of The Measurement of the Circle, the name is spelled , which is the correct transcription of the common medieval Arabic spelling
100.
On the Arabic translations see Rashed (1997, 3–4). See also Rashed (1993).
101.
Rashed (1997, 4–5).
102.
Lévy (2011, 129).
103.
Lévy (2011, 134–7).
104.
Glasner (2013).
105.
See Pedersen and Jones (2010, 49, 79–81).
106.
“In a quadrilateral inscribed in a circle, the sum of the products of opposite sides with one another is equal to product of its two diagonals with one another” (Toomer 50–1).
107.
The quotation is neither a literal translation from the Arabic nor a verbatim quotation from the Hebrew translation by Ya‘aqov Anatoly (Cf. Wien ms. 194/40, IMHM 1317, fol. 20b20–12a11). Alfonso “translates” the theorem to his own language of ribui. See Part III, introduction and commentary on Property III.12.
108.
Sabra (1969); Jaouiche (1986, 18–19).
109.
Simplicius In Phys. 61–69; Langermann (1995, 34).
110.
Two extant copies of the commentary are preserved: Codex Leidenesis 399 (edited by Besthorn and Heiberg 1897–1905) and Qom ms. See Brentjes (1996, 203, note 13, 206); Lo Bello (2003, pp. xv–xvi, xxviii).
111.
See Lévy (2000, 302, 309).
112.
See Section II.2.3.
113.
On Ibn al-Haytham’s contribution to mathematics see Rashed (2013a, b).
114.
This text was translated into Hebrew by Moshe Ibn Tibbon.
115.
Rashed (2013a, Chap. 1).
116.
Szpiech (2010).
117.
A manuscript which is partly in Hebrew letters testifies that Ibn Ridwan’s commentary on Ptolemy’s Tetrabiblos circulated among Jews in the Iberian Peninsula. See Koningsveld, p. 102, item 90.
118.
See commentary to sections I.1.1, I.1.3, I.2.3, I.3.2–4.
119.
In his dissertation Murdoch (1957) argues that Duns Scotus, who was the main Latin source for geometrical arguments against indivisibilism, drew on the Latin translation of al-Ghazali. Podkonski (2006) contends that this thesis should not be accepted without reservation, and in a later presentation Murdoch put the thesis more cautiously: one source of the geometrical argument against indivisibilism was al-Ghazali’s Metaphysics (Murdoch 2009, 19).
120.
See <InternalRef RefID="Sec19" >Appendix</Internal Ref>.
121.
Sadik (2016b, 35) writes that Alfonso refers to Maimonides more than seventy times in his writings, more than to any other post-Talmudic Jewish author.
122.
Maimonides, Guide I.73; see Freudenthal (1998). Alfonso does not refer to this passage, and offers a different, more complex solution using a conchoid (which apparently interested him more). Also, Maimonides asserts in his commentary on the Mishna (‘Eruvin 1.5) that the ratio of the diameter diagonal to its circumference cannot be known, but only approximated. Alfonso does not mention this statement either.
123.
Freudenthal (2005, IX, 4); Hebrew (1991, 986). See also Lévy (1992, 40); Encyclopedia Judaica, vol. 14, pp. 551–2.
124.
Freudenthal and Zonta (2012, 232).
125.
Hughes (2008, xxi). Seville fell into the hands of the Christians in 1248.
126.
Verlinden (1953, 200–2).
127.
Busard (2005). Books XIV and XV are late additions to the Elements.
128.
Grant (1974, 172. Col. 1).
129.
Clagett (1964, I, 581–605).
130.
Clagett (1864, I, 581).
131.
Section II.5.6.
132.
Quoted from Szpiech (2006, 524–5). Szpiech’ translation.
133.
Grant (1971, 107). We could not locate the passage in Ibn Rushd.
134.
Sefer ha-‘Olam I #24.1–7. See Sela, forthcoming.
135.
Sela, forthcoming; see also Sela (2017).
136.
We are very grateful to Shlomo Sela for clarifying this point. Grant (1971, 111–6) led us to assume that Ibn Ezra could have been Alfonso’s source, but Shlomo Sela ruled out this possibility.
137.
Grant (1994, 499).
138.
Szpiech (2006, 524–5).
139.
Grant (1971, 120, note 97).
140.
In Elements X 115 Euclid proves that “from medial straight line there arises irrational straight lines infinite in number, and none of them is the same as any of the preceding”; in the second part of X.111 he proves that “the apotome and the irrational straight lines following it are neither the same with the medial straight line nor with one another”.
141.
Section II.5.5.
142.
Grant (1966, 247).
143.
Grant (1966; 1971).
144.
See commentary to III.33.
145.
Murdoch (1957, 15–16); Grellard and Robert (2009, 2).
146.
I.2.1, II.5.1–4.
147.
Murdoch (2009, 23 esp. note 26). Alfonso adopts this view in I.2.1 and II.5.4.
148.
I.4.4.
149.
Section II.1.3.
150.
Berggren in Katz et al. (2016, 383–4); Rashed (2012, 727); Hogendijk (1994); de Young (2002).
151.
The closest to verbatim quotations from first-hand sources ae al-Nayrizi’s lemmas in Part II and Ptolemy’s theorem in Part III.
152.
HGC is a long text, preserved in the library of the Jewish community of Mantua and carries the title The book of Euclid Langermann (1984, 74; 2004); Glasner.
153.
Two of the theorems common to SMA and HGC appear also in Ibn Hud’s Istikmal: Ptolemy’s theorem and Archimedes’ Sphere and the Cylinder I.21.
154.
Langermann (1996a, 41).
155.
Langermann (1996a, 35) suggests that in addition to his Recension of Euclid’s Elements, al-Maghribī wrote also a geometrical compendium titled The Book of Euclid.
156.
Alfonso’s attempt to prove the parallels postulate.
157.
A few examples: the word (appropriate) appears 25 times in the general discussion and is absent in the mathematical parts; the root (to be necessary or obligatory) occurs 15 times in the general discussion and not once in the mathematical discussion. The root (to limit, also to construct) appears 26 times in Parts I–II (24 times as verb twice as noun), not once in Part III. The verb (first person plural of ) is used in Part II (II.1.5, II.2.1, II.4.1) in the sense of “imagine”, usually in the context of imagining moving of lines; In III.24 and III.25 it is used in the simpler sense of “assume”.
158.
For instance, the word occurs 91 times in Parts I–II, only once in Part III, whereas the combination is very frequent in Part III and in mathematical passages of part II (86 times) and occurs only once in the philosophical Part I. The word seven more times in Parts I–II, and once in the philosophical paragraph of Part III, i.e., the end of Property 33.
159.
In one context (II.2.3), when citing two lemmas of al-Nayrizi, Alfonso translates shakl as haqdama rather than tavnit.
160.
The Provençal translators usually translate shakl (in both senses) as temuna (Sarfatti 1968, 193–4). Ofer Elior remarks that in manuscripts Firenze 137 and Mantua 1 (that is based on Firenze 137) of the Hebrew translation of the Elements the term “tavnit” is used.
161.
Aristotle Post. An. I.13 79a6–9. Langermann (1992, 273–4) shows that this term was used in medicine to refer to the special properties of medicines that cannot be accounted for in terms of the basic qualities.
162.
E.g., Katz et al. (2007, 667).
163.
E.g., Hojendick (1991, 232, 235, 241, 245, 247, 252, 254, 256).
164.
This text was later translated into Hebrew, probably by Don Benveniste ben Lavi in Saragossa. See Aradi (2015, 11). We are grateful to Naomi Aradi for letting us consult this translation.
165.
He uses the expressions and (Sha‛ar C, Sha‛ar D). Isra’eli uses the word Segula to refer to a mathematical proposition that characterizes a mathematical object or concept, e.g., the segula of parallels (Limud 18), of triangles (Limud 20), of similar triangles or of ratios.
166.
Yesodei ha-Tevuna, 51.
167.
Sha‛ar D is systematically divided into propositions. The term for proposition is limud. In Sha‛ar B and Sha‛ar C several terms are used to divide the text: yesod musad, ‛iqar efshar, or simply ‛iqar. The text is from 1310.
168.
See Commentary on the Premises of Euclid’s Elements, Hooper-Sude’s edition, Arabic text pp. 46, 50, 55. Al-Tūsī also uses Shakl in his suggested proof.
169.
See commentary on the definitions of Part III.
170.
Shmuel Ibn Tibbbon in his Hebrew lexicon used the spelling (Sarfati 177), and this form was adopted by the other Tibbonides in their translations of Euclid, by the compiler of the Hebrew Geometrical Compendium, by Qalonimus in his translation of Archimedes and by other Provencal translators and writers. The word is found also in Provencal philosophical translations, e.g., in the anonymous translation of Ibn Rushd’s long commentary on the Physics, Paris ms. 154b.
171.
See Sarfatti (1968, 195).
172.
The Mishnaic word does not seem to be relevant.
173.
III.25. See note to the text there.
174.
Teshuvot, see Hecht (1993, 367–8).
175.
Teshuvot, see Hecht (1993, 372); Gershenzon (1984, 12); Szpisch (2006, 634).
176.
I.3.4.
177.
Ibn Rushd MC De gen. et corr., Kurlad edition, 59. The underlying text is: “The account given in Timaeus… says that it is a sort of substratum prior to the so-called elements as gold to the artefacts made of gold… but in the case of generation and corruption it is not possible to call something by the name of the thing from which it has come to be” (De gen. II.1329a17–24, Joachim’s translation).
178.
Parma ms. 69a5–6.
179.
Much has been written on the study of al-Ghazali among the Jews. See, for instance, Harvey (2001); Freudenthal and Zonta (2012, 244, 247–248).
180.
Szpiech (2006, 620).
181.
See Harvey (2001, 364).
182.
II.1.1.
183.
Paris ms. 901, IMHM 26857 fol 60b24–61a6.
184.
Paris ms. 904, IMHM 26860 fol 42b9–20.
185.
Oxford Bodl. ms. 1346, Add. 4048, IMHM 22372 fol 31b31.
186.
In the translation of R. Ya‘aqov and in Firenze manuscript 137 the book is called without any additional title. I am greatful to Ofer Elior for this comment.
187.
SMA I.5.3.
188.
Paris, Private collection Ms. IMHM 39116, fol. 1b18–20.
189.
Oxford Ms. Hunt. 16, IMHM 19290, fol. 2a5–6.
190.
Mantua com. 2, Ms. IMHM 783 fol. 2.b13–15.
191.
Section III.12.
192.
Wien Ms. 194/40, IMHM 1317 fol. 11b–12a. Also Munich Ms. 182, IMHM 1638.
193.
See discussion of this term in the introductory note to Part III.

Author information

Authors and Affiliations

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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Appendices

Since most of the manuscripts mentioned here are relevant only to the <InternalRef RefID="Sec19" >Appendix</Internal Ref> they are not listed in the bibliography, but instead are referred to directly here.

8 Appendix: Translation Comparisons

Since most of the manuscripts mentioned here are relevant only to the <InternalRef RefID="Sec19" >Appendix</Internal Ref> they are not listed in the bibliography, but instead are referred to directly here.

Ibn Rushd

Tahafut al-Tahafut

This book was translated into Hebrew by Qalonimos ben David under the title Alfonso refers to it as ^{Footnote 173}

Epistle on the Possibility of Conjunction

An anonymous Hebrew translation is titled Alfonso refers to this text as ^{Footnote 174} He quotes one sentence from the book.

Alfonso	Anonymous Hebrew translation

Middle Commentary on De Generatione et Corruptione

Alfonso	Qalonimos ben Qalonimos’ translation	Falaquera, De‛ot ha-Filosofim
^{Footnote 177}	^{Footnote 178}	^{Footnote 179}

Al-Ghazali

Alfonso was acquainted with both Maqāṣid al-falāsifah and Tahāfut al-Falāsifa.^{Footnote 178}

The Tahāfut was translated into Hebrew by ZeraḤya ben IṣḤāq ha-Levi under the title but referred to by Alfonso as ^{Footnote 179}

The Maqāṣid was translated into Hebrew perhaps three times, the first translation was made by IṣḤāq Albalag at the end of the thirteenth century and was completed by IṣḤāq Pulgar (who was Alfonso’s student and opponent after his conversion). The second was made by Yehuda ben Shlomo Nathan, in Provence in the fourteenth century. The third anonymous translation was the one which Moshe Narboni used when writing his commentary on this text.^{Footnote 180} Steven Harvey notes (note 18) the similarity between Yehuda’s translation and the anonymous translation. The comparison below supports Harvey’s suggestion that these two translations are not independent.

Maqāṣid al-falāsifah

Alfonso	Albalag’s translation	Yehuda ben Shlomo Natan’s translation	Anonymous translation
^{Footnote 183}	^{Footnote 184}	^{Footnote 185}	^{Footnote 186}

Euclid

Euclid’s Elements (στοιχεία, ) was translated into Hebrew under the title Sefer ha-shorashim both by Moshe Ibn Tibbon and by his nephew Ya‛aqov ben Makhir.^{Footnote 185} Alfonso referred to it as Sefer ha-‛Iqarim.

Elements Postulate 5

Alfonso	Moshe Ibn Tibbon’s translation	Ya‛aqov ben Makhir’s translation	Geometrical compendium
^{Footnote 188}	^{Footnote 189}	^{Footnote 190}	^{Footnote 191}

Ptolemy

Ptolemy’s Theorem, Almagest I.10

Alfonso	Ya‛aqov Anatoli’s translation
^{Footnote 192}	^{Footnote 193}

Anatoli uses the preposition (which in this context means “by”) to denote product. Alfonso’s uses his favorite term which means half product.^{Footnote 192} The factor 1/2 introduced by the use of this term is irrelevant because in this context it appears in an equation and the factor can be cancelled.

Edition and Translation Conventions

The Edition

The text of SMA was edited by Gita Gluskina and published with a Russian translation and comments in 1983. We consulted her edition as well as her introduction, notes, and two articles (that were translated for us from the Russian). In order that our edition be compatible with hers, we followed Gluskina’s Latin transliteration of the letters of the diagrams:

The editing methods are somewhat different. Gluskina’s method was to provide the reader with a precise transcription of the manuscript, adding her own corrections in the apparatus. Our policy was to offer the reader a readable text, so where we believed the text to be mistaken we put what we took to be the correct reading in the text, citing the manuscript reading in the apparatus. The justification for this policy is that the manuscript is often misleading in the rendering of the letters of the diagrams, and must be corrected in order to make the text comprehensible. Most of the corrections were already made by Gluskina, and we usually adopted her corrections. In a few places our interpretation of the text and consequently our corrections were different from hers.

We did not comment in the apparatus when we corrected the readings of graphically similar letters as or or sometimes We did not reproduce some single letters scattered in the text and apparently irrelevant, and we also skipped repeated words. We completed truncated words without noting it, and only in cases that it was not obvious we added brackets.

The division of the text into four parts (she‘arim) is Alfonso’s; the division into sections and the numbering of the paragraphs are editorial additions. Added numbers and titles are in square brackets.

The Translation

Our translation policy was to offer a literal translation if possible in order to preserve some of the flavor of Alfonso’s style. In this vein we tried to translate words consistently. For instance, we rendered always as “superpose” even if in some cases “be congruent” sounds better; we use the more literal translation “we consider triangle ABC…” of the Hebrew rather than the more elegant expression “let ABC be a triangle….” In Part III, however, we preferred mathematical clarity, so the translation is somewhat freer. For instance, we use “rectangle” instead of “four-sided right-angled figure.”

In those cases where more than one translation of a word was used, the different translations are listed in the glossary.

We do not insist on strictly parallel punctuation of the Hebrew and English versions, and a few times divide in English a long sentence that could not be divided in the Hebrew text.

Notations

[] Editors’ additions. What appears within brackets is not a part of the text.

[…] Missing text.

[?] Unreadable text (no. of question marks = no. of unreadable words).

<> Parts of the text that seem to be irrelevant or out of place.

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Glasner, R., Baraness, A. (2021). Chapter 1: Introduction. 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_1

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DOI: https://doi.org/10.1007/978-3-319-77303-2_1
Published: 27 November 2020
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-77302-5
Online ISBN: 978-3-319-77303-2
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