| Title | Simplicius’s Proof of Euclid’s Parallels Postulate |
| Type | Article |
| Language | English |
| Date | 1969 |
| Journal | Journal of the Warburg and Courtauld Institutes |
| Volume | 32 |
| Pages | 1-24 |
| Categories | no categories |
| Author(s) | Sabra, A. I. |
| Editor(s) | |
| Translator(s) |
| A commentary by Simplicius on the premises to Book I of Euclid’s Elements survives in an Arabic translation, of which the author and the exact date of execution are unknown. The translation is reproduced by the ninth-century mathematician al-Fadl ibn Hâtim al-Nayrîzî in the course of his own commentary on the Elements. Of Nayrîzî’s commentary, which is based on the earlier translation of the Elements by al-Hajjâj ibn Yûsuf ibn Matar, we have only one manuscript copy at Leiden and Gerard of Cremona’s Latin translation, both of which have been published. The passages quoted by Nayrîzî, owing to their extensiveness and consecutive order, would strongly lead one to assume that they together make up the whole of Simplicius’s text. In what follows, however, I shall argue that they suffer from at least one important omission: a proof by Simplicius himself of Euclid’s parallels postulate. Since the omission occurs both in the Leiden manuscript and in Gerard’s translation, it cannot simply be an accidental feature of the former. My argument will consist in (i) citing evidence (Document I) to the effect that such a proof was known to some Arabic mathematicians, and (ii) producing a hitherto unnoticed text (Document II), which, in the light of the evidence cited, may well be taken to be the missing proof. In addition, I shall show how Simplicius’s proof entered Arabic discussions on parallels, first, by being made subject to criticism (Document I), and then by being incorporated into a new proof, which was designed to take that criticism into account (Document III). The title of Simplicius’s work in question appears in the Arabic sources in slightly different forms. Nayrîzî concludes the last citation from that work with the following words: “There end the matters which Simplicius has put forward in the commentary to the musädara of Euclid for the first part of the book of Elements.” The word musädara has here something a little unexpected about it. Usually, as in translations of Euclid and Aristotle, it corresponds to the Greek αἴτημα (aitêma), and it is used in this sense in the body of Simplicius’s commentary itself. (The Arabic verb sädara appropriately means “to demand.” Musädara: demanding, or that [proposition] which is demanded.) But the commentary is not restricted to the αἰτήματα (postulates) at the beginning of the Elements, but also treats of the common notions (κοιναί ἔννοιαι: 'ulüm muta‘ärafa) and the definitions (ὅροι: hudüd). Could musädara be used here in a general sense that covers all three groups of Euclid’s premises? Such a hypothesis would derive at least partial support from a statement in Proclus that some ancient writers applied the term αἴτημα to axioms (or common notions) as well as to postulates. Proclus quotes Archimedes as an example. In agreement with this usage, the titles of at least two Arabic works on geometry employ the plural musädarät as a collective term for the axioms, definitions, and postulates. It was probably this sense that the eleventh-century scholar Abü cAbd Allah al-Khwarizmï had in mind when he gave the following explanation in his Keys of the Sciences: “al-musädara are those premises of the question which are put at the beginning of a book or chapter of geometry.” The tenth-century bibliographer Ibn al-Nadïm gives a somewhat different version of the title of Simplicius’s book: “A commentary on the sadr of the book of Euclid, which is the introduction to geometry.” Sadr means fore-part or front and is frequently used to refer to the introductory part of a book; it might have rendered the Greek προοίμιον (prooimion). The latter part in this version, “which is the introduction to geometry,” looks like a description of the book supplied, perhaps, by Ibn al-Nadïm himself, but it may also have been an alternative title of the book. Nayrîzî’s version of the title agrees with Khwarizmï’s definition in applying the singular musädara to a multitude of premises, but we shall see that the thirteenth-century author of Document I cites the same title with musädarät in the plural. Simplicius prefaces his comments on the individual postulates of Euclid with a long passage on the meaning and function of postulates in general. It will be useful to quote this passage here in full, since it is one of the channels through which Greek discussions of mathematical methodology were transmitted to the Islamic world—particularly discussions connected with the question of parallels. [introduction p. 1-2] |
| Online Resources | https://uni-koeln.sciebo.de/s/DNibNx7ADIjjT3W |
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The translation is reproduced by the ninth-century mathematician al-Fadl ibn H\u00e2tim al-Nayr\u00eez\u00ee in the course of his own commentary on the Elements. Of Nayr\u00eez\u00ee\u2019s commentary, which is based on the earlier translation of the Elements by al-Hajj\u00e2j ibn Y\u00fbsuf ibn Matar, we have only one manuscript copy at Leiden and Gerard of Cremona\u2019s Latin translation, both of which have been published.\r\n\r\nThe passages quoted by Nayr\u00eez\u00ee, owing to their extensiveness and consecutive order, would strongly lead one to assume that they together make up the whole of Simplicius\u2019s text. In what follows, however, I shall argue that they suffer from at least one important omission: a proof by Simplicius himself of Euclid\u2019s parallels postulate. Since the omission occurs both in the Leiden manuscript and in Gerard\u2019s translation, it cannot simply be an accidental feature of the former. My argument will consist in (i) citing evidence (Document I) to the effect that such a proof was known to some Arabic mathematicians, and (ii) producing a hitherto unnoticed text (Document II), which, in the light of the evidence cited, may well be taken to be the missing proof. In addition, I shall show how Simplicius\u2019s proof entered Arabic discussions on parallels, first, by being made subject to criticism (Document I), and then by being incorporated into a new proof, which was designed to take that criticism into account (Document III).\r\n\r\nThe title of Simplicius\u2019s work in question appears in the Arabic sources in slightly different forms. Nayr\u00eez\u00ee concludes the last citation from that work with the following words: \u201cThere end the matters which Simplicius has put forward in the commentary to the mus\u00e4dara of Euclid for the first part of the book of Elements.\u201d The word mus\u00e4dara has here something a little unexpected about it. Usually, as in translations of Euclid and Aristotle, it corresponds to the Greek \u03b1\u1f34\u03c4\u03b7\u03bc\u03b1 (ait\u00eama), and it is used in this sense in the body of Simplicius\u2019s commentary itself. (The Arabic verb s\u00e4dara appropriately means \u201cto demand.\u201d Mus\u00e4dara: demanding, or that [proposition] which is demanded.) But the commentary is not restricted to the \u03b1\u1f30\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 (postulates) at the beginning of the Elements, but also treats of the common notions (\u03ba\u03bf\u03b9\u03bd\u03b1\u03af \u1f14\u03bd\u03bd\u03bf\u03b9\u03b1\u03b9: 'ul\u00fcm muta\u2018\u00e4rafa) and the definitions (\u1f45\u03c1\u03bf\u03b9: hud\u00fcd). Could mus\u00e4dara be used here in a general sense that covers all three groups of Euclid\u2019s premises?\r\n\r\nSuch a hypothesis would derive at least partial support from a statement in Proclus that some ancient writers applied the term \u03b1\u1f34\u03c4\u03b7\u03bc\u03b1 to axioms (or common notions) as well as to postulates. Proclus quotes Archimedes as an example. In agreement with this usage, the titles of at least two Arabic works on geometry employ the plural mus\u00e4dar\u00e4t as a collective term for the axioms, definitions, and postulates. It was probably this sense that the eleventh-century scholar Ab\u00fc cAbd Allah al-Khwarizm\u00ef had in mind when he gave the following explanation in his Keys of the Sciences: \u201cal-mus\u00e4dara are those premises of the question which are put at the beginning of a book or chapter of geometry.\u201d\r\n\r\nThe tenth-century bibliographer Ibn al-Nad\u00efm gives a somewhat different version of the title of Simplicius\u2019s book: \u201cA commentary on the sadr of the book of Euclid, which is the introduction to geometry.\u201d Sadr means fore-part or front and is frequently used to refer to the introductory part of a book; it might have rendered the Greek \u03c0\u03c1\u03bf\u03bf\u03af\u03bc\u03b9\u03bf\u03bd (prooimion). The latter part in this version, \u201cwhich is the introduction to geometry,\u201d looks like a description of the book supplied, perhaps, by Ibn al-Nad\u00efm himself, but it may also have been an alternative title of the book. Nayr\u00eez\u00ee\u2019s version of the title agrees with Khwarizm\u00ef\u2019s definition in applying the singular mus\u00e4dara to a multitude of premises, but we shall see that the thirteenth-century author of Document I cites the same title with mus\u00e4dar\u00e4t in the plural.\r\n\r\nSimplicius prefaces his comments on the individual postulates of Euclid with a long passage on the meaning and function of postulates in general. It will be useful to quote this passage here in full, since it is one of the channels through which Greek discussions of mathematical methodology were transmitted to the Islamic world\u2014particularly discussions connected with the question of parallels. [introduction p. 1-2]","btype":3,"date":"1969","language":"English","online_url":"","online_resources":"https:\/\/uni-koeln.sciebo.de\/s\/DNibNx7ADIjjT3W","doi_url":null,"categories":[],"authors":[{"id":396,"full_name":"Sabra, A. I.","role":{"id":1,"role_name":"author"}}],"book":null,"booksection":null,"article":{"id":1055,"journal_id":null,"journal_name":"Journal of the Warburg and Courtauld Institutes","volume":"32","issue":"","pages":"1-24"}},"sort":[1969]}
| Title | Simplicius’s Proof of Euclid’s Parallels Postulate |
| Type | Article |
| Language | English |
| Date | 1969 |
| Journal | Journal of the Warburg and Courtauld Institutes |
| Volume | 32 |
| Pages | 1-24 |
| Categories | no categories |
| Author(s) | Sabra, A. I. |
| Editor(s) | |
| Translator(s) |
A commentary by Simplicius on the premises to Book I of Euclid’s Elements survives in an Arabic translation, of which the author and the exact date of execution are unknown. The translation is reproduced by the ninth-century mathematician al-Fadl ibn Hâtim al-Nayrîzî in the course of his own commentary on the Elements. Of Nayrîzî’s commentary, which is based on the earlier translation of the Elements by al-Hajjâj ibn Yûsuf ibn Matar, we have only one manuscript copy at Leiden and Gerard of Cremona’s Latin translation, both of which have been published. The passages quoted by Nayrîzî, owing to their extensiveness and consecutive order, would strongly lead one to assume that they together make up the whole of Simplicius’s text. In what follows, however, I shall argue that they suffer from at least one important omission: a proof by Simplicius himself of Euclid’s parallels postulate. Since the omission occurs both in the Leiden manuscript and in Gerard’s translation, it cannot simply be an accidental feature of the former. My argument will consist in (i) citing evidence (Document I) to the effect that such a proof was known to some Arabic mathematicians, and (ii) producing a hitherto unnoticed text (Document II), which, in the light of the evidence cited, may well be taken to be the missing proof. In addition, I shall show how Simplicius’s proof entered Arabic discussions on parallels, first, by being made subject to criticism (Document I), and then by being incorporated into a new proof, which was designed to take that criticism into account (Document III). The title of Simplicius’s work in question appears in the Arabic sources in slightly different forms. Nayrîzî concludes the last citation from that work with the following words: “There end the matters which Simplicius has put forward in the commentary to the musädara of Euclid for the first part of the book of Elements.” The word musädara has here something a little unexpected about it. Usually, as in translations of Euclid and Aristotle, it corresponds to the Greek αἴτημα (aitêma), and it is used in this sense in the body of Simplicius’s commentary itself. (The Arabic verb sädara appropriately means “to demand.” Musädara: demanding, or that [proposition] which is demanded.) But the commentary is not restricted to the αἰτήματα (postulates) at the beginning of the Elements, but also treats of the common notions (κοιναί ἔννοιαι: 'ulüm muta‘ärafa) and the definitions (ὅροι: hudüd). Could musädara be used here in a general sense that covers all three groups of Euclid’s premises? Such a hypothesis would derive at least partial support from a statement in Proclus that some ancient writers applied the term αἴτημα to axioms (or common notions) as well as to postulates. Proclus quotes Archimedes as an example. In agreement with this usage, the titles of at least two Arabic works on geometry employ the plural musädarät as a collective term for the axioms, definitions, and postulates. It was probably this sense that the eleventh-century scholar Abü cAbd Allah al-Khwarizmï had in mind when he gave the following explanation in his Keys of the Sciences: “al-musädara are those premises of the question which are put at the beginning of a book or chapter of geometry.” The tenth-century bibliographer Ibn al-Nadïm gives a somewhat different version of the title of Simplicius’s book: “A commentary on the sadr of the book of Euclid, which is the introduction to geometry.” Sadr means fore-part or front and is frequently used to refer to the introductory part of a book; it might have rendered the Greek προοίμιον (prooimion). The latter part in this version, “which is the introduction to geometry,” looks like a description of the book supplied, perhaps, by Ibn al-Nadïm himself, but it may also have been an alternative title of the book. Nayrîzî’s version of the title agrees with Khwarizmï’s definition in applying the singular musädara to a multitude of premises, but we shall see that the thirteenth-century author of Document I cites the same title with musädarät in the plural. Simplicius prefaces his comments on the individual postulates of Euclid with a long passage on the meaning and function of postulates in general. It will be useful to quote this passage here in full, since it is one of the channels through which Greek discussions of mathematical methodology were transmitted to the Islamic world—particularly discussions connected with the question of parallels. [introduction p. 1-2] |
| Online Resources | https://uni-koeln.sciebo.de/s/DNibNx7ADIjjT3W |
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The translation is reproduced by the ninth-century mathematician al-Fadl ibn H\u00e2tim al-Nayr\u00eez\u00ee in the course of his own commentary on the Elements. Of Nayr\u00eez\u00ee\u2019s commentary, which is based on the earlier translation of the Elements by al-Hajj\u00e2j ibn Y\u00fbsuf ibn Matar, we have only one manuscript copy at Leiden and Gerard of Cremona\u2019s Latin translation, both of which have been published.\r\n\r\nThe passages quoted by Nayr\u00eez\u00ee, owing to their extensiveness and consecutive order, would strongly lead one to assume that they together make up the whole of Simplicius\u2019s text. In what follows, however, I shall argue that they suffer from at least one important omission: a proof by Simplicius himself of Euclid\u2019s parallels postulate. Since the omission occurs both in the Leiden manuscript and in Gerard\u2019s translation, it cannot simply be an accidental feature of the former. My argument will consist in (i) citing evidence (Document I) to the effect that such a proof was known to some Arabic mathematicians, and (ii) producing a hitherto unnoticed text (Document II), which, in the light of the evidence cited, may well be taken to be the missing proof. In addition, I shall show how Simplicius\u2019s proof entered Arabic discussions on parallels, first, by being made subject to criticism (Document I), and then by being incorporated into a new proof, which was designed to take that criticism into account (Document III).\r\n\r\nThe title of Simplicius\u2019s work in question appears in the Arabic sources in slightly different forms. Nayr\u00eez\u00ee concludes the last citation from that work with the following words: \u201cThere end the matters which Simplicius has put forward in the commentary to the mus\u00e4dara of Euclid for the first part of the book of Elements.\u201d The word mus\u00e4dara has here something a little unexpected about it. Usually, as in translations of Euclid and Aristotle, it corresponds to the Greek \u03b1\u1f34\u03c4\u03b7\u03bc\u03b1 (ait\u00eama), and it is used in this sense in the body of Simplicius\u2019s commentary itself. (The Arabic verb s\u00e4dara appropriately means \u201cto demand.\u201d Mus\u00e4dara: demanding, or that [proposition] which is demanded.) But the commentary is not restricted to the \u03b1\u1f30\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 (postulates) at the beginning of the Elements, but also treats of the common notions (\u03ba\u03bf\u03b9\u03bd\u03b1\u03af \u1f14\u03bd\u03bd\u03bf\u03b9\u03b1\u03b9: 'ul\u00fcm muta\u2018\u00e4rafa) and the definitions (\u1f45\u03c1\u03bf\u03b9: hud\u00fcd). Could mus\u00e4dara be used here in a general sense that covers all three groups of Euclid\u2019s premises?\r\n\r\nSuch a hypothesis would derive at least partial support from a statement in Proclus that some ancient writers applied the term \u03b1\u1f34\u03c4\u03b7\u03bc\u03b1 to axioms (or common notions) as well as to postulates. Proclus quotes Archimedes as an example. In agreement with this usage, the titles of at least two Arabic works on geometry employ the plural mus\u00e4dar\u00e4t as a collective term for the axioms, definitions, and postulates. It was probably this sense that the eleventh-century scholar Ab\u00fc cAbd Allah al-Khwarizm\u00ef had in mind when he gave the following explanation in his Keys of the Sciences: \u201cal-mus\u00e4dara are those premises of the question which are put at the beginning of a book or chapter of geometry.\u201d\r\n\r\nThe tenth-century bibliographer Ibn al-Nad\u00efm gives a somewhat different version of the title of Simplicius\u2019s book: \u201cA commentary on the sadr of the book of Euclid, which is the introduction to geometry.\u201d Sadr means fore-part or front and is frequently used to refer to the introductory part of a book; it might have rendered the Greek \u03c0\u03c1\u03bf\u03bf\u03af\u03bc\u03b9\u03bf\u03bd (prooimion). The latter part in this version, \u201cwhich is the introduction to geometry,\u201d looks like a description of the book supplied, perhaps, by Ibn al-Nad\u00efm himself, but it may also have been an alternative title of the book. Nayr\u00eez\u00ee\u2019s version of the title agrees with Khwarizm\u00ef\u2019s definition in applying the singular mus\u00e4dara to a multitude of premises, but we shall see that the thirteenth-century author of Document I cites the same title with mus\u00e4dar\u00e4t in the plural.\r\n\r\nSimplicius prefaces his comments on the individual postulates of Euclid with a long passage on the meaning and function of postulates in general. It will be useful to quote this passage here in full, since it is one of the channels through which Greek discussions of mathematical methodology were transmitted to the Islamic world\u2014particularly discussions connected with the question of parallels. [introduction p. 1-2]","btype":3,"date":"1969","language":"English","online_url":"","online_resources":"https:\/\/uni-koeln.sciebo.de\/s\/DNibNx7ADIjjT3W","doi_url":null,"categories":[],"authors":[{"id":396,"full_name":"Sabra, A. I.","role":{"id":1,"role_name":"author"}}],"book":null,"booksection":null,"article":{"id":1055,"journal_id":null,"journal_name":"Journal of the Warburg and Courtauld Institutes","volume":"32","issue":"","pages":"1-24"}},"sort":["Simplicius\u2019s Proof of Euclid\u2019s Parallels Postulate"]}