Review: Menso Folkerts: Die älteste lateinische Schrift über das indische Rechnen nach al-Hwarizmi. Verlag d. bayerischen Akademie der Wissenschaften, 1997
The publications of the Studies series are dedicated to key subjects in the history and developme... more The publications of the Studies series are dedicated to key subjects in the history and development of know ledge, bringing together perspectives from different fields and combining sourcebased empirical research with theoretically guided approaches. The Proceedings series presents the results of scientific meetings on current issues and supports, at the same time, further cooperation on these issues by offering an electronic platform with further resources and the possibility for comments and interactions. The Textbooks volumes are prepared by leading experts in the relevant fields.
Copyright and moral rights for the publications made accessible in the public portal are retained... more Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain. • You may freely distribute the URL identifying the publication in the public portal.
In an extensive paper published due to the kindness of the editorial staff of the present journal... more In an extensive paper published due to the kindness of the editorial staff of the present journal, I have suggested a geometrical reinterpretation of so-called Old Babylonian "algebra". 1 Among the texts analyzed in the paper were the mathematical texts No. XVI and No. IX from Susa, which turned out to contain highly illuminating didactical commentaries of a kind not known from the Babylonian core area -be it because teaching in a peripheral area felt a need to make explicit what could be left to a stable oral tradition in the core, or simply because the Susa teachers had a bent for loquacity. Text No. XVI turned out not to contain solutions of problems but only a didactical discussion of transformations of linear "equations" of two unknowns (as usually, the us ("length") and ("width") of a rectangle, with the usual values 30 ' and 20' [nindan] 2 ). No. IX contained initial didactical discussions of the transformations of complex into simpler second-degree "equations" followed by use of the technique just taught for the solution of a sophisticated set of equations. These texts are not the only Susa texts to contain illuminating didactical commentaries. In the present paper I shall analyze two further texts, of which one contains an explicit explanatory part, while the other employs some of the concepts introduced in the former. An extra reason for reanalyzing the two texts is that the treatment given by
As stated in the title, the book tells the mathematical history of division in extreme and mean r... more As stated in the title, the book tells the mathematical history of division in extreme and mean ratio (DEMR) from the beginnings through the 18th century. The nonmathematical history of the same famous ratio, under which aspect it is most often called by its alias "the Golden Number," is in reserve for another book. The mathematical history, on the other hand, includes both the theory and the construction of the ratio itself and constructions which make use (or might make use) of the division. The book is thus also a history of constructions and discussions of the pentagon, the decagon, the icosahedron, and the dodecahedron. The earliest extant source mentioning DEMR is Euclid's Elements. It is therefore fully appropriate that the book starts with a detailed exposition of all propositions and definitions from the Elements somehow related to the DEMR or its applications. After a chapter on mathematical topics essential to the discussion (concepts relating to the question of "geometrical algebra," "side and diagonal numbers," incommensurability, and the "Euclidean algorithm"), the author goes back in time to see how much can be inferred about the pre-Euclidean history of the concept. Next comes an (apparently exhaustive) presentation of constructions, discussions, and calculations related to DEMR from Archimedes to Mascheroni. Finally, a first appendix arrays the variety of names given to DEMR, while a second appendix quotes the opinions of Campanus, Pacioli, Ramus, Kepler, and others concerning its marvellous power. Up to the time of the Elements (ca. 300 B.C.), the history of DEMR has to be based on indirect evidence. Since the mid-19th century, a wealth of theories, hypotheses, and speculations has grown from this absence of constraints on scholarly imagination. Herz-Fischler offers a broad and liberal survey that includes all serious attempts to deal with the subject (and a few less serious), and then discusses these various interpretations in the light of available evidence. Many properties of DEMR are easily demonstrated by means of the pentagram, which sources from later Antiquity connect with the Pythagorean order. Therefore, Herz-Fischler goes through the early occurrences of the pentagram, in ostraca, graffiti, and coins. He argues convincingly that prehistoric and Bronze Age pentagrams and an Old Babylonian calculation of the area of a pentagon have nothing to do with the matter, and, what is more important, that a figure in such widespread use (be it for decoration or with some hypothetical symbolic meaning) could hardly have been utilized as a specific symbol of recognition by the early Pythagoreans. He demonstrates, moreover, that there is no particular reason to believe brought to you by CORE View metadata, citation and similar papers at core.ac.uk
Written mathematical traditions in Ancient Mesopotamia
Ugarit-Verlag eBooks, Mar 25, 2015
... found) could be our first hint of the social setting of Old Babylonian sophisticated mathemat... more ... found) could be our first hint of the social setting of Old Babylonian sophisticated mathematics which, though written in school format, appears not to have been part of the normal scribal curriculum as we know it from Nippur and as analyzed by Christine Proust [2008] 22 This ...
From a sociological point of view, pre-Modern "non-theoretical geometry" is not adequately descri... more From a sociological point of view, pre-Modern "non-theoretical geometry" is not adequately described as merely "practical". The "practical geometry" we find in written treatises is mostly that of "scribal" environments, and aims at calculating lengths, areas or volumes from already performed measurements. As a rule it is not interested in geometrical construction, nor in the making of measurements -the fields, broadly speaking, of master builders/architects and surveyors. The paper discusses two cases -one fairly well-established, another more conjectural -where none the less "scribal" practical geometry does reveal traces of (very simple) geometrical construction. Both of these concern the "long run", connecting Old Babylonian, classical ancient and late medieval material. A final instance of weak communication between "scribal" and "surveying" geometry is located in thirteenth-fourteenth33-century France.
Jacobus de Florentia,<i>Tractatus algorismi</i>(1307), the chapter on algebra (Vat. Lat. 4826, fols 36<sup>v</sup>-45<sup>v</sup>)
Centaurus, 2000
Ce texte retranscrit une traduction anglaise accompagnee d'un bref commentaire mathematique e... more Ce texte retranscrit une traduction anglaise accompagnee d'un bref commentaire mathematique et historique de ce qui represente le plus ancien texte traitant de l'algebre en Europe. Il provient d'un manuscrit incluant une copie du Tractatus algorismi de Jacopo da Firenze
How to transfer the conceptual structure of Old Babylonian Mathematics: solutions and inherent problems. With an Italian Parallel.: Beitrag zur Tagung “Zur Übersetzbarkeit von Wissenschaftssprachen des Altertums”, Johannes Gutenberg-Universität Mainz, 27.–29 Juli 2009. Preprint
Copyright and moral rights for the publications made accessible in the public portal are retained... more Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain. • You may freely distribute the URL identifying the publication in the public portal.
Review: Menso Folkerts: Die älteste lateinische Schrift über das indische Rechnen nach al-Hwarizmi. Verlag d. bayerischen Akademie der Wissenschaften, 1997
The publications of the Studies series are dedicated to key subjects in the history and developme... more The publications of the Studies series are dedicated to key subjects in the history and development of know ledge, bringing together perspectives from different fields and combining sourcebased empirical research with theoretically guided approaches. The Proceedings series presents the results of scientific meetings on current issues and supports, at the same time, further cooperation on these issues by offering an electronic platform with further resources and the possibility for comments and interactions. The Textbooks volumes are prepared by leading experts in the relevant fields.
Copyright and moral rights for the publications made accessible in the public portal are retained... more Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain. • You may freely distribute the URL identifying the publication in the public portal.
In an extensive paper published due to the kindness of the editorial staff of the present journal... more In an extensive paper published due to the kindness of the editorial staff of the present journal, I have suggested a geometrical reinterpretation of so-called Old Babylonian "algebra". 1 Among the texts analyzed in the paper were the mathematical texts No. XVI and No. IX from Susa, which turned out to contain highly illuminating didactical commentaries of a kind not known from the Babylonian core area -be it because teaching in a peripheral area felt a need to make explicit what could be left to a stable oral tradition in the core, or simply because the Susa teachers had a bent for loquacity. Text No. XVI turned out not to contain solutions of problems but only a didactical discussion of transformations of linear "equations" of two unknowns (as usually, the us ("length") and ("width") of a rectangle, with the usual values 30 ' and 20' [nindan] 2 ). No. IX contained initial didactical discussions of the transformations of complex into simpler second-degree "equations" followed by use of the technique just taught for the solution of a sophisticated set of equations. These texts are not the only Susa texts to contain illuminating didactical commentaries. In the present paper I shall analyze two further texts, of which one contains an explicit explanatory part, while the other employs some of the concepts introduced in the former. An extra reason for reanalyzing the two texts is that the treatment given by
As stated in the title, the book tells the mathematical history of division in extreme and mean r... more As stated in the title, the book tells the mathematical history of division in extreme and mean ratio (DEMR) from the beginnings through the 18th century. The nonmathematical history of the same famous ratio, under which aspect it is most often called by its alias "the Golden Number," is in reserve for another book. The mathematical history, on the other hand, includes both the theory and the construction of the ratio itself and constructions which make use (or might make use) of the division. The book is thus also a history of constructions and discussions of the pentagon, the decagon, the icosahedron, and the dodecahedron. The earliest extant source mentioning DEMR is Euclid's Elements. It is therefore fully appropriate that the book starts with a detailed exposition of all propositions and definitions from the Elements somehow related to the DEMR or its applications. After a chapter on mathematical topics essential to the discussion (concepts relating to the question of "geometrical algebra," "side and diagonal numbers," incommensurability, and the "Euclidean algorithm"), the author goes back in time to see how much can be inferred about the pre-Euclidean history of the concept. Next comes an (apparently exhaustive) presentation of constructions, discussions, and calculations related to DEMR from Archimedes to Mascheroni. Finally, a first appendix arrays the variety of names given to DEMR, while a second appendix quotes the opinions of Campanus, Pacioli, Ramus, Kepler, and others concerning its marvellous power. Up to the time of the Elements (ca. 300 B.C.), the history of DEMR has to be based on indirect evidence. Since the mid-19th century, a wealth of theories, hypotheses, and speculations has grown from this absence of constraints on scholarly imagination. Herz-Fischler offers a broad and liberal survey that includes all serious attempts to deal with the subject (and a few less serious), and then discusses these various interpretations in the light of available evidence. Many properties of DEMR are easily demonstrated by means of the pentagram, which sources from later Antiquity connect with the Pythagorean order. Therefore, Herz-Fischler goes through the early occurrences of the pentagram, in ostraca, graffiti, and coins. He argues convincingly that prehistoric and Bronze Age pentagrams and an Old Babylonian calculation of the area of a pentagon have nothing to do with the matter, and, what is more important, that a figure in such widespread use (be it for decoration or with some hypothetical symbolic meaning) could hardly have been utilized as a specific symbol of recognition by the early Pythagoreans. He demonstrates, moreover, that there is no particular reason to believe brought to you by CORE View metadata, citation and similar papers at core.ac.uk
Written mathematical traditions in Ancient Mesopotamia
Ugarit-Verlag eBooks, Mar 25, 2015
... found) could be our first hint of the social setting of Old Babylonian sophisticated mathemat... more ... found) could be our first hint of the social setting of Old Babylonian sophisticated mathematics which, though written in school format, appears not to have been part of the normal scribal curriculum as we know it from Nippur and as analyzed by Christine Proust [2008] 22 This ...
From a sociological point of view, pre-Modern "non-theoretical geometry" is not adequately descri... more From a sociological point of view, pre-Modern "non-theoretical geometry" is not adequately described as merely "practical". The "practical geometry" we find in written treatises is mostly that of "scribal" environments, and aims at calculating lengths, areas or volumes from already performed measurements. As a rule it is not interested in geometrical construction, nor in the making of measurements -the fields, broadly speaking, of master builders/architects and surveyors. The paper discusses two cases -one fairly well-established, another more conjectural -where none the less "scribal" practical geometry does reveal traces of (very simple) geometrical construction. Both of these concern the "long run", connecting Old Babylonian, classical ancient and late medieval material. A final instance of weak communication between "scribal" and "surveying" geometry is located in thirteenth-fourteenth33-century France.
Jacobus de Florentia,<i>Tractatus algorismi</i>(1307), the chapter on algebra (Vat. Lat. 4826, fols 36<sup>v</sup>-45<sup>v</sup>)
Centaurus, 2000
Ce texte retranscrit une traduction anglaise accompagnee d'un bref commentaire mathematique e... more Ce texte retranscrit une traduction anglaise accompagnee d'un bref commentaire mathematique et historique de ce qui represente le plus ancien texte traitant de l'algebre en Europe. Il provient d'un manuscrit incluant une copie du Tractatus algorismi de Jacopo da Firenze
How to transfer the conceptual structure of Old Babylonian Mathematics: solutions and inherent problems. With an Italian Parallel.: Beitrag zur Tagung “Zur Übersetzbarkeit von Wissenschaftssprachen des Altertums”, Johannes Gutenberg-Universität Mainz, 27.–29 Juli 2009. Preprint
Copyright and moral rights for the publications made accessible in the public portal are retained... more Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain. • You may freely distribute the URL identifying the publication in the public portal.
Following the lead of Hans Georg Zeuthen (not that of Tannery, as often claimed), many historians... more Following the lead of Hans Georg Zeuthen (not that of Tannery, as often claimed), many historians of mathematics since the outgoing nineteenth century have referred to " geometric algebra " as a constituent of ancient Greek theoretical geometry. In the 1930s, Otto Neugebauer added that this geometric algebra was indeed a translation of a Babylonian numerical algebra, necessitated by the discovery of irrational ratios. Then, beginning in 1969, the notion was attacked, first by Árpád Szabó, Michael Mahoney and Sabetai Unguru. In particular the views of the latter have since been accepted as the " new orthodoxy ". As analysis of the writings of the actors involved shows, these have rarely read each other's works with much care. That already holds for many of those who have claimed inspiration from Zeuthen, but those who have criticized the idea have felt even less obliged to show that they knew what they spoke about. The paper deals with this relationship between the participants in the debate. It is not intended to discuss who is right when it comes to analysis of the historical material.
As Ahmed Djebbar has pointed out, 11th-century and earlier al-Andalus produced a “solid
research ... more As Ahmed Djebbar has pointed out, 11th-century and earlier al-Andalus produced a “solid research tradition in arithmetic”. So far, no continuation of this tradition has been known, but analysis of three sections of two Latin works suggest that they borrow material that can hardly come from elsewhere: 1. The Liber mahameleth, likely to be a more or less free translation made by Gundisalvi or somebody close to him of an Arabic original presenting “mu a¯mala¯t mathematics vom höheren Standpunkt aus” contains systematic variations, for instance of proportions :: p g P G (claimed to deal with prices and goods), where the givens may be sums, differences, products, sums or differences of square roots, etc., solved sometimes by means of algebra, sometimes with appeals to Elements II.5–6, often after reduction by means of proportion techniques. 2. A passage in Chapter 12 of the Liber abbaci first presents the simple version of the recreational problem about the “unknown heritage” (likely to be of late Ancient or Byzantine origin): a father leaves to his first son 1 monetary unit and 1/n of what remains, to the second 2 units and 1/n of what remains, etc.; in the end, all get the same, and nothing remains. Next it goes on with complicated cases where the arithmetical series is not proportional to 1 – 2 – 3 ..., and the fraction is not an aliquot part. Fibonacci gives an algebraic solution to one variant and also general formulae for all variants – but these do not come from his algebra, and he thus cannot have derived them himself. A complete survey of occurrences once again points to al-Andalus. 3. Chapter 15 Section 1 of Fibonacci’s Liber abbaci mainly deals with the ancient theory of means though not telling so. If M is one such mean between A and B, it is shown systematically how each of these three numbers can be found if the other two are given – once more by means of algebra, Elements II.5–6, and proportion techniques. The lettering shows that Fibonacci uses an Arabic or Greek source, but no known Arabic or Greek work contains anything similar. However, the structural affinity suggests inspiration from the same environment as produced the Liber mahameleth. So, this seems to be a non-narrative, a story that was not revealed by the participants, and was not discovered by historians so far.
As stated in the title, the book tells the mathematical history of division in extreme and mean r... more As stated in the title, the book tells the mathematical history of division in extreme and mean ratio (DEMR) from the beginnings through the 18th century. The nonmathematical history of the same famous ratio, under which aspect it is most often called by its alias "the Golden Number", is in reserve for another book. The mathematical history, on the other hand, includes both the theory and construction of the ratio itself and constructions which make use (or might make use) of the division. The book is thus also a history of constructions and discussions of the pentagon, the decagon, the icosahedron and the dodecahedron.
Review of:
Chambon, Grégory: Normes et pratiques: L’homme, la
mesure et l’écriture en Mésopotamie... more Review of: Chambon, Grégory: Normes et pratiques: L’homme, la mesure et l’écriture en Mésopotamie. I. Les mesures de capacité et de poids en Syrie Ancienne, d’Ebla à Émar. Gladbeck: PeWeVerlag 2011. 200 S. 8° = Berliner Beiträge zum Vorderen Orient 21. Hartbd. € 29,80. ISBN 978-3-935012-08-9.
Review of:
Anne-Marie Vlasschaert, Le Liber mahameleth: Édition critique et commentaires. (Boethi... more Review of: Anne-Marie Vlasschaert, Le Liber mahameleth: Édition critique et commentaires. (Boethius: Texte und Abhandlungen zur Geschichte der Mathematik und der Naturwissenschaften 60.) Stuttgart: Franz Steiner, 2010. Pp. 184 (commentary), 429 (edition). €74. ISBN: 978-3-515-09238-2.
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research tradition in arithmetic”. So far, no continuation of this tradition has been known,
but analysis of three sections of two Latin works suggest that they borrow material that
can hardly come from elsewhere:
1. The Liber mahameleth, likely to be a more or less free translation made by Gundisalvi
or somebody close to him of an Arabic original presenting “mu a¯mala¯t mathematics vom
höheren Standpunkt aus” contains systematic variations, for instance of proportions :: p
g
P
G
(claimed to deal with prices and goods), where the givens may be sums, differences,
products, sums or differences of square roots, etc., solved sometimes by means of algebra,
sometimes with appeals to Elements II.5–6, often after reduction by means of proportion
techniques.
2. A passage in Chapter 12 of the Liber abbaci first presents the simple version of the
recreational problem about the “unknown heritage” (likely to be of late Ancient or
Byzantine origin): a father leaves to his first son 1 monetary unit and 1/n of what remains,
to the second 2 units and 1/n of what remains, etc.; in the end, all get the same, and nothing
remains. Next it goes on with complicated cases where the arithmetical series is not
proportional to 1 – 2 – 3 ..., and the fraction is not an aliquot part. Fibonacci gives an
algebraic solution to one variant and also general formulae for all variants – but these
do not come from his algebra, and he thus cannot have derived them himself. A complete
survey of occurrences once again points to al-Andalus.
3. Chapter 15 Section 1 of Fibonacci’s Liber abbaci mainly deals with the ancient theory
of means though not telling so. If M is one such mean between A and B, it is shown
systematically how each of these three numbers can be found if the other two are given –
once more by means of algebra, Elements II.5–6, and proportion techniques. The lettering
shows that Fibonacci uses an Arabic or Greek source, but no known Arabic or Greek work
contains anything similar. However, the structural affinity suggests inspiration from the
same environment as produced the Liber mahameleth.
So, this seems to be a non-narrative, a story that was not revealed by the participants,
and was not discovered by historians so far.
Chambon, Grégory: Normes et pratiques: L’homme, la
mesure et l’écriture en Mésopotamie. I. Les mesures
de capacité et de poids en Syrie Ancienne, d’Ebla à
Émar. Gladbeck: PeWeVerlag 2011. 200 S. 8° = Berliner
Beiträge zum Vorderen Orient 21. Hartbd. € 29,80. ISBN
978-3-935012-08-9.
Anne-Marie Vlasschaert, Le Liber mahameleth: Édition critique et commentaires. (Boethius: Texte und Abhandlungen zur Geschichte der Mathematik und der Naturwissenschaften 60.) Stuttgart: Franz Steiner, 2010. Pp. 184 (commentary), 429 (edition).
€74. ISBN: 978-3-515-09238-2.