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Bernardino Baldi's in Mechanica Aristotelis Problemata Exercitationes
Bernardino Baldi’s In mechanica Aristotelis problemata exercitationes ii Max Planck Research Library for the History and Development of Knowledge Series Editors Jürgen Renn, Robert Schlögl, Bernard F. Schutz. Edition Open Access Development Team Lindy Divarci, Beatrice Gabriel, Jörg Kantel, Matthias Schemmel, and Kai Surendorf, headed by Peter Damerow. Scientific Board Markus Antonietti, Ian Baldwin, Antonio Becchi, Fabio Bevilacqua, William G. Boltz, Jens Braarvik, Horst Bredekamp, Jed Z. Buchwald, Olivier Darrigol, Thomas Duve, Mike Edmunds, Yehuda Elkana, Fynn Ole Engler, Robert K. Englund, Mordechai Feingold, Rivka Feldhay, Gideon Freudenthal, Paolo Galluzzi, Kostas Gavroglu, Mark Geller, Domenico Giulini, Günther Görz, Gerd Graßhoff, James Hough, Manfred Laubich- ler, Glenn Most, Pier Daniele Napolitani, Alessandro Nova, Hermann Parzinger, Dan Potts, Circe Silva da Silva, Ana Simões, Richard Stephen- son, Mark Stitt, Noel M. Swerdlow, Liba Taub, Martin Vingron, Scott Walter, Norton Wise, Gerhard Wolf, Rüdiger Wolfrum, Zhang Baichun. Sources 3 Edition Open Access 2011 Bernardino Baldi’s In mechanica Aristotelis problemata exercitationes Elio Nenci Communicated by Jürgen Renn and Antonio Becchi Edition Open Access 2011 Max Planck Research Library for the History and Development of Knowledge Sources 3 Communicated by Jürgen Renn and Antonio Becchi Translated from Italian into English by Adriano Carugo Copyedited by Lindy Divarci ISBN 978-3-86931-961-2 First published 2011 Printed in Germany by epubli, Oranienstraße 183, 10999 Berlin http://www.epubli.de Edition Open Access http://www.edition-open-access.de Published under Creative Commons by-nc-sa 3.0 Germany Licence http://creativecommons.org/licenses/by-nc-sa/3.0/de/ The Deutsche Nationalbibliothek lists this publication in the Deutsche Na- tionalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. -
Geometry and Arithmetic in the Medieval Traditions of Euclid's
Geometry and Arithmetic in the Medieval Traditions of Euclid's Elements: a View from Book II Leo Corry – Tel Aviv University Dedicated to my dear friend Sabetai Unguru on his 82th birthday Contents 1. Abstract .......................................................................................................... 1 2. Introduction .................................................................................................... 2 3. Euclid’s Elements - Book II and Geometric Algebra ............................................... 7 3.1. Elements Book II – An Overview ........................................................................................... 7 3.2. Geometric Algebra – An Overview ..................................................................................... 14 3.3. Visible and Invisible Figures ............................................................................................... 18 4. Book II in Late Antiquity and in Islam Mathematics ............................................. 22 4.1. Heron’s Commentary of the Elements ................................................................................ 22 4.2. Al-Khwārizmī and Abū Kāmil .............................................................................................. 27 4.3. Thābit ibn Qurra ............................................................................................................... 35 4.4. Al-Nayrīzī ........................................................................................................................ -
The Development of Euclidean Axiomatics the Systems of Principles and the Foundations of Mathematics in Editions of the Elements in the Early Modern Age
Arch. Hist. Exact Sci. (2016) 70:591–676 DOI 10.1007/s00407-015-0173-9 The development of Euclidean axiomatics The systems of principles and the foundations of mathematics in editions of the Elements in the Early Modern Age Vincenzo De Risi1 Received: 4 August 2015 / Published online: 24 February 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract The paper lists several editions of Euclid’s Elements in the Early Modern Age, giving for each of them the axioms and postulates employed to ground elementary mathematics. Contents 1 Introduction ..................................593 1.1 The Euclidean tradition .........................593 1.2 Axioms and postulates .........................602 2 The principles .................................618 2.1 The principles of Euclid .........................618 2.2 Further principles in the Greek tradition ................619 2.3 Additional common notions and the foundations of the theory of magnitudes ...............................619 2.4 Principles on mereological composition and multiples of magnitudes . 620 2.5 The theory of ratios and proportions ..................622 2.6 The Axiom of Archimedes and the theory of indivisibles .......626 2.7 Principles of arithmetic and number theory ...............626 2.8 General principles of space, situation, and givenness ..........628 2.9 Principles of intersection, incidence and connection ..........629 2.10 Principles of continuity .........................631 2.11 Principles of congruence, motion, and transportation -
Euclid (In the Renaissance)
EUCLID (IN THE RENAISSANCE) Jens Høyrup Section for Philosophy and Science Studeis Roskilde University, Denmark Article written for Marco Sgarbi (ed.), Encyclopedia of Renaissance Philosophy Preprint 20 March 2017 Abstract Although the Latin Middle Ages received a number of versions of Euclid’s Elements and several other Euclidean works, by the 14th century only the Campanus redaction from c. 1259 was in circulation. In the 14th and 15th century, this redaction was encountered by students of Arts or Medical university faculties, even though we have scant evidence that Euclid impressed their minds. In the fifteenth century, other circles discovered him: Alberti took over the idea of elements, Regiomontanus used Euclid alongside Archimedes as an argument for the superiority of mathematics over philosophy, and one Florentine abbacus school tradition was able to give correct references to the Elements. A turn arrived with book printing. In 1482, the Campanus Elements were printed, and in 1498 and 1501 Giorgio Valla inserted pseudo-Euclidean and Euclidean material in two bulky volumes. A new though somewhat problematic Latin translation from the Greek (including also some minor works) was published by Zamberti in 1505, and until 1540 a number of reprints or re-editions of Campanus’s and Zamberti’s texts were published – at times in combination. From the 1540s onward, revisions, selections and vernacular translations began to appear, all based on the same two texts. In 1572, however, Commandino made a new Latin translation from Zamberti’s text and a sounder manuscript, and in 1574 Clavius produced a didactically oriented redaction. These two set the scene for the next two centuries. -
PAPPUS of ALEXANDRIA (About 320 AD) by HEINZ KLAUS STRICK, Germany
PAPPUS OF ALEXANDRIA (about 320 AD) by HEINZ KLAUS STRICK, Germany PAPPUS OF ALEXANDRIA is considered the last of the great Greek geometers. Almost nothing is known about his life - not even exactly when he lived. The only historical link is a commentary he wrote on a solar eclipse that he himself observed in Alexandria, which can be dated to October 320 by a modern calculation. It is known that he lived in Alexandria and led a "school" (or Academy) there. His main work was entitled Synagoge (Collection) and consisted of eight books. It represented a successful attempt to revive the classical geometry of the Greeks. PAPPUS was obviously not interested in replacing the books of the "ancients", but in bringing the meaning of these books (which probably all still existed at the time) back to consciousness and supplementing them with insights that had been added subsequently by other scholars. The collection also contained some references to writings by authors whose existence we might otherwise not have known about. The first translation of the Synagoge into Latin was by FEDERICO COMMANDINO in 1589, but it was then another few decades before RENÉ DESCARTES, PIERRE DE FERMAT and ISAAC NEWTON recognised the importance of the work and made it the basis of their own research. Book I on arithmetic has been completely lost, of Book II only a part exists (the fragment was discovered in 1688 by JOHN WALLIS in the Savillian Library in Oxford). It dealt with a problem of recreational mathematics: In ancient Greece, numerals were represented by letters, among others in the Milesian notation. -
DEFENDING HYPATIA Archimedes NEW STUDIES in the HISTORY and PHILOSOPHY of SCIENCE and TECHNOLOGY
DEFENDING HYPATIA Archimedes NEW STUDIES IN THE HISTORY AND PHILOSOPHY OF SCIENCE AND TECHNOLOGY VOLUME 25 EDITOR Jed Z. Buchwald, Dreyfuss Professor of History, California Institute of Technology, Pasadena, CA, USA. ASSOCIATE EDITORS Jeremy Gray, The Faculty of Mathematics and Computing, The Open University, Buckinghamshire, UK. Sharon Kingsland, Department of History of Science and Technology, Johns Hopkins University, Baltimore, MD, USA. ADVISORY BOARD Henk Bos, University of Utrecht Mordechai Feingold, California Institute of Technology Allan D. Franklin, University of Colorado at Boulder Kostas Gavroglu, National Technical University of Athens Anthony Grafton, Princeton University Paul Hoyningen-Huene, Leibniz UniversitätHannover Trevor Levere, University of Toronto Jesper Lützen, Copenhagen University William Newman, Indian University, Bloomington Lawrence Principe, The Johns Hopkins University Jürgen Renn, Max-Planck-Institut für Wissenschaftsgeschichte Alex Roland, Duke University Alan Shapir , oUniversity of Minnesota Nancy Siraisi, Hunter College of the City University of New York Michael R. DietrichDartmouth, College Michel Morange , IHPST, Paris Hans-Jörg RheinbergerMax-Planck-Institut, für Wissenschaftsgeschichte Noel Swerdlow, University of Chicago Archimedes has three fundamental goals; to furtherintegration the of the histories of science and technology with one another: to investigate the technical, social and practical histories of specific developments in science andnology; tech and finally, where possible and desirable, to bring the histories of science and technology intocontact closer with the philosophy of science. To these ends, each volume will have its own themetitle and and will be planned by one or more members of the Advisory Board in consultation with the editor. Although the volumes have specific themes, the series itself will not be limited to one or even to a few particular areas. -
Max Planck Institute for the History of Science Archimedes
MAX-PLANCK-INSTITUT FÜR WISSENSCHAFTSGESCHICHTE Max Planck Institute for the History of Science 2017 PREPRINT 487 Jens Hµyrup Archimedes: Knowledge and Lore from Latin Antiquity to the Outgoing European Renaissance Abstract With only Apuleius and Augustine as partial exceptions, Latin Antiquity did not know Archimedes as a mathematician but only as an ingenious engineer and astronomer, serving his city and killed by fatal distraction when in the end it was taken by ruse. The Latin Middle Ages forgot even much of that, and when Archimedean mathematics was translated in the 12th and 13th centuries, almost no integration with the traditional image of the person took place. With the exception of Petrarca, who knew the civically useful engineer and the astrologer (!), fourteenth-century Humanists show no interest in Archimedes. In the 15th century, however, “higher artisans” with Humanist connections or education took interest in Archimedes the technician and started identifying with him. In mid-century, a new translation of most works from the Greek was made by Jacopo Cremonensis, and Regiomontanus and a few other mathematicians began resurrecting the image of the geometer, yet without emulating him in their own work. Giorgio Valla’s posthumous De expetendis et fugiendis rebus from 1501 marks a watershed. Valla drew knowledge of the person as well as his works from Proclus and Pappus, thus integrating the two. Over the century, a number of editions also appeared, the editio princeps in 1544, and mathematical work following in the footsteps of Archimedes was made by Maurolico, Commandino and others. The Northern Renaissance only discovered Archimedes in the 1530s, and for long only superficially. -
HISTORIA MATHESEOS. EARLY DEVELOPMENT STAGE 4Th
HISTORIA MATHESEOS. EARLY DEVELOPMENT STAGE HISTORY OF MATHEMATICS. HISTORIOGRAPHY Sinkevich G.I. Saint Petersburg State University of Architecture and Civil Engineering Vtoraja Krasnoarmejskaja ul. 4, St. Petersburg, 190005, Russia Summary. This article focuses on evolvement of the history of mathematics as a science and development of its methodology from the 4th century B.C. to the age of Enlightenment. Key words: methodology of the history of mathematics. The history of mathematics was evolving together with the mathematics itself. Its methodology gradually established. Having started with short excursuses in the history of certain issues and biographical details of scientists, the history of mathematics ended up with research using historical, textual, and mathematical methods, and achieved significant results. One would find wealth of diverse literature devoted to the history of mathematics – from popular literature to rigorous research. People in all countries studied the history of mathematics; it is included in education courses; it is interesting for all lovers of mathematics. 4th century B.C., Eudemus of Rhodes. Mathematics was laid down as a science in Ancient Greece, and the first famous work devoted to the history of mathematics was the History of Geometry (Γεωμετρικὴ ἱστορία) by Eudemus of Rhodes1, Aristotle’s student. This work was repeated in Comments to Euclid by Proclus (The Catalogue of Geometers) [1, p. 82-86]. Eudemus is known to have also written The History of Arithmetic and The History of Astronomy, none of which has survived to our days and are only mentioned by other authors. He concluded that arithmetic resulted from trade activities of the Phoenicians, and geometry, from geodetic activities of Egyptians. -
Architectural Criticism, Science, and Visual Eloquence: Teofilo Gallaccini in Seventeenth- Century Siena Author(S): Alina A
Architectural Criticism, Science, and Visual Eloquence: Teofilo Gallaccini in Seventeenth- Century Siena Author(s): Alina A. Payne Source: The Journal of the Society of Architectural Historians, Vol. 58, No. 2 (Jun., 1999), pp. 146-169 Published by: Society of Architectural Historians Stable URL: http://www.jstor.org/stable/991482 Accessed: 13/11/2009 15:09 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=sah. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Society of Architectural Historians is collaborating with JSTOR to digitize, preserve and extend access to The Journal of the Society of Architectural Historians. -
Aristarchus's Book on the Sizes and Distances of The
ARISTARCHUS’S BOOK ON THE SIZES AND DISTANCES OF THE SUN AND OF THE MOON WITH SOME EXPLANATIONS OF PAPPUS ALEXANDRINUS. IN LATIN TRANSLATED AND ILLUSTRATED WITH COMMENTARY FROM FEDERICO COMMANDINO OF URBINO P E S A R O, by Camillus Francischinus M D L X X I I With privilege granted by PONT.MAX. for then years Introduction, translation and notes by Antonio Mancini Latin text in front 3 ‘Consider ye the seed from which ye sprang: ye were not made to live like unto brutes, but for pursuit of virtute and knowledge. So eager did I render my companions, with this brief exortation, for the voyage, that then hardly could have held them back. Dante, The Inferno XXVI 5 5 In memory of Emma Castelnuovo 6 Acknowledgments I are grateful to Livia Borghetti, former director of the Central National Library Vittorio Emanuele II of Rome, for facilitating my access to ancient texts, and to Letizia Jengo, professor of mathematics, for helping me in difficulty moments. I thank also specially my wife Paola, my daughter Chiara and my granddaughter Flaminia for their support in the revision of the text. 7 INTRODUCTION I FEDERICO COMMANDINO AND THE DISCLOSURE OF THE ANTIQUE WRITTEN OF MATHEMATICS The evolutionary impulsion imposed on the present society by the spread of computer technology reminds us of the great acceleration that the Gutenberg’s movable types printing invention, in the mid- fifteenth century, gave to the knowledges diffusion, which at the end of the century, facilitated the so-called scientific revolution. In the last years of the fifteenth century hundreds of thousands of books, writin with this new technique, were circulating, and many literates, artisans, soldiers, architects and students could easily access texts unavailable in the past for their rarity, as handwritten or products with primitive print shapes. -
ARCHIMEDES (IN the RENAISSANCE) Jens Høyrup Section for Philosophy and Science Studeis Roskilde University, Denmark
ARCHIMEDES (IN THE RENAISSANCE) Jens Høyrup Section for Philosophy and Science Studeis Roskilde University, Denmark Article written for Marco Sgarbi (ed.), Encyclopedia of Renaissance Philosophy Preprint 20 March 2017 Abstract With only Apuleius and Augustine as partial exceptions, Latin Antiquity did not know Archimedes as a mathematician but only as an ingenious engineer and astronomer, serving his city and killed by fatal distraction when in the end it was taken by ruse. The Latin Middle Ages forgot even much of that, and when Archimedean mathematics was translated in the twelfth and thirteenth centuries, almost no integration with the traditional image of the person took place. With the exception of Petrarca, who knew the civically useful engineer and the astrologer, fourteenth-century Humanists show no interest in Archimedes. In the fifteenth century, “higher artisans” with Humanist connections or education took interest in Archimedes the technician and started identifying with him. In mid-century, a new translation of most works from the Greek was made by Jacopo Cremonensis, and Regiomontanus and a few other mathematicians began resurrecting the image of the geometer, yet without emulating him. Giorgio Valla’s posthumous De expetendis et fugiendis rebus from 1501 marks a watershed. Valla drew knowledge of the person as well as his works from Proclus and Pappus, thus integrating the two. Over the century, a number of editions also appeared, the Editio princeps in 1544, and mathematical work following in the footsteps of Archimedes was made by Maurolico, Commandino and others. The Northern Renaissance only discovered Archimedes in the 1530s, and for long only superficially. -
Apollonius De Pergé, Coniques. Tome 4: Livres VI Et VII. Commentaire Historique Et Mathématique
Apollonius de Pergé, Coniques. Tome 4: Livres VI et VII. Commentaire historique et mathématique. Édition et traduction du texte arabe by Roshdi Rashed Berlin/New York: De Gruyter, 2009. Pp. .ISBN 978–3–11– 019940–6. Cloth € 137.00 xii + 572 Reviewed by Aldo Brigaglia Università di Palermo [email protected] In memoriam Hélène Bellosta With the publication of the fourth volume of the new edition of the Conics by Apollonius of Perga (ca 262–180 bc), Roshdi Rashed has completed his very important work on the edition of the Arabic text, its translation into French, and a vast mathematical commentary. Apollonius’ treatise itself may well be considered one of the highest achievements of Greek mathematics at its most brilliant. In fact, together with the corpus of the mathematical work of Archimedes (287–216 bc), the Conics constitute the greater part of Greek higher mathematics. Rashed’s edition of the text of the Conics is the latest episode in the long and intriguing history of the transmission of this major mathematical work to us.The first four books arrived to Western mathematical culture through the edition by Eutocius (fifth century ad), which was translated into Latin in the 16th century by Johannes Baptista Memus, Francesco Maurolico, and Federico Commandino. Books 5, 6, and 7 of the Conics arrived in Europe only through the Arabic translations of the Greek text: the first text of the lost Greek books was contained in an Arabic compendium of the Conics written by Abu’l-Fath Mahmud al-Isfahani (second half of the 10th century).