DOCSLIB.ORG

Albrecht Dürer's Personal Underweysung Der Messung

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Albrecht Dürer's Personal Underweysung Der Messung View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Ghent University Academic Bibliography Word & Image A Journal of Verbal/Visual Enquiry ISSN: 0266-6286 (Print) 1943-2178 (Online) Journal homepage: http://www.tandfonline.com/loi/twim20 Albrecht Dürer’s personal Underweysung der Messung Noam Andrews To cite this article: Noam Andrews (2016) Albrecht Dürer’s personal Underweysung der Messung, Word & Image, 32:4, 409-429, DOI: 10.1080/02666286.2016.1216821 To link to this article: http://dx.doi.org/10.1080/02666286.2016.1216821 Published online: 15 Nov 2016. Submit your article to this journal Article views: 10 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=twim20 Download by: [Thomas J. Watson Library, The Metropolitan Museum of Art] Date: 02 December 2016, At: 11:34 Albrecht Dürer’s personal Underweysung der Messung NOAM ANDREWS Abstract The article discusses the handwritten revisions and drawn additions by Albrecht Dürer in his own copy of the treatise on geometry, Underweysung der Messung (1525). Situating Dürer’s interest in mathematics within the scholarly milieu of Renaissance Nuremberg, the article addresses the shifts in style and content that Dürer proscribes and offers new perspectives on the artist’s relation to his own late work. The article concludes by displaying several drawings bound into the edition by Dürer that illustrate variations on perspectival apparatuses. These lesser known drawings illuminate the evolution of Dürer’s conceptualization of the dynamics between artist and subject, as well as the tools used to facilitate this interaction. Keywords Albrecht Dürer, editing, geometry, measurement, perspective, Platonic solids Albrecht Dürer’s(1471–1528) most important contribution to six- ratios as a means of determining the human body—asetof teenth-century geometry remains the first of his two major treatises theoretical concerns that potentially tracks back more easily to on measurement and proportion, Underweysung der Messung his artistic œuvre. Many of Dürer’s extant working drawings of (1525), a text which synthesized the internal structure, principles, figures are designs for prints in Vier Bücher von menschlicher and precision of Euclid’s Elements with various classical and con- Proportion, which show Dürer grappling with generating geome- temporary geometrical texts available to him in his home city of trical ratios of body proportions using a compass, though he 1 Nuremberg. Beginning with Euclidean principles and spreading eventually moves towards systems of ratios that balanced the into a diverse set of “mixed-mathematical,” or applied-geometrical dimensions of body parts to each other, sometimes even survey- 4 applications, such as column design, sundials, calligraphy, the ing the ratios of the body’s parts together in elaborate detail. modeling of polyhedra, and perspective, Underweysung systematically Perhaps the relative lack of interest in divining the permu- organizes a wealth of knowledge accumulated over a lifetime of tations of Dürer’s first geometrical work explains the scarcity artistic practice, and was intended, as Dürer declared in of scholarship on Dürer’s extraordinary personal copy of Underweysung’s introduction, to be “not only for painters, but also Underweysung der Messung, which has been located at the for goldsmiths, sculptors, stonemasons, carpenters, and all those for Bayerische Staatsbibliothek in Munich since 1839 and can 2 5 whom using measurement is useful.” Envisioned more as a peda- be consulted under the call number 4° L.impr.c.n.mss. 119. gogical provocation than a personal dissection of his own creative Still, this fascinating volume deserves another look. The book process, Underweysung would go on to form the theoretical base upon is a complete rendition of Dürer’s own edits and additions to which the next generations of German artists built their approach Underweysung, made sometime after its 1525 printing and to geometrical research and experimentation. Artists like Augustin before his death on April 6, 1528. The volume includes Hirschvogel (1503–53), Wenzel Jamnitzer (1507/8–85), Heinrich numerous unpublished geometrical diagrams (some of which Lautensack (1522–90), Lorentz Stöer (c.1530–after 1621), and others have been reproduced in this article) as well as two very developed proficiencies in geometricality by adopting the physical- significant, yet widely unknown, drawings of figures that 6 ity and tactility of Dürer’s geometry, in the process altering the have nearly escaped attention. Dürer’s notes, marked directly nature of geometrical knowledge from the abstract and theoretical in the margins in his own hand, bound together as separate discussions of Euclid into an intimate and approachable field of pages, and contained on small scraps of paper glued onto the investigation. This in turn generated new geometrical objects pages they reference, attest both to Dürer’s prolonged and intended to be touched, handled, and manipulated, such as the passionate engagement with the treatise even after its initial many three-dimensional polyhedral solids that appear in sixteenth- publication and to the authority he commanded as an 7 century German print culture and decorative arts. author. There is no “range of interpretive possibilities” for Nevertheless, Underweysung never became as popular with the workshop of Hieronymous Andreae (d. 1556)—the printer historians as Dürer’s second major treatise, the posthumously of the first and second posthumous 1538 edition of published Vier Bücher von menschlicher Proportion (1528), which had Underweysung as well as Dürer’s monumental Triumphal Arch been conceptualized as a more advanced treatise on drawing to (first printed 1526) for Emperor Maximilian I (1459–1519); no be studied after Underweysung and deployed geometry as a leeway to “schreibt hrein, was Ir wollt,” as has been discussed 3 stereometric aid in the representation of the human body. in relation to the blank sign in Dürer’s drawing The Recording 8 After all, Vier Bücher von menschlicher Proportion reveals Dürer’s of the Thoughts of the Pious and the Wicked (c.1500–15). All Dürer’s 9 developing and experimental interest in the use of proportional edits were carried over and faithfully reproduced. WORD & IMAGE, VOL. 32, NO. 4, OCTOBER–DECEMBER 2016 409 http://dx.doi.org/10.1080/02666286.2016.1216821 # 2016 Taylor & Francis Tracing the contours of Dürer’s acute investment in his During this time, the “Regiomontan-Waltersche Bücherei” geometrical treatise provides a yet untapped access into the became an intellectual resource and reference library for a new mind of the artist in the last years of his life, and, moreover, generation of Nuremberg-based scholars such as Johannes indicates a route towards extrapolating more general conclu- Werner (1468–1522), who worked on spherical trigonometry sions about how Dürer’s revisions and edits are at once and conic sections; Dürer’s close friend the humanist reflective, and in excess, of the cultural expectations and Willibald Pirckheimer (1470–1530); Joachim Camerarius material practices of the Renaissance print workshop. By (1500–74), the classical scholar, Dürer’s biographer, and the triangulating between all three versions of Underweysung—the first translator of Dürer’s Underweysung into Latin; the influential 1525 original, Dürer’scopy,andthe1538 second edition—it is globe-maker and cosmographer Johannes Schöner (1477–1547), 13 possible to reconstruct the evolution of the treatise and and others. It was only in 1519, after a failed attempt to sell Dürer’s thoughts on it; to confirm the extent to which the books to Elector Friedrich the Wise of Saxony, that a Dürer’s edits and proscriptions were posthumously followed portion of the books was sold for the sum of 150 gulden, most by the printing house; and their relation to the numerous of which went to Pirckheimer, who recorded the purchase in 14 fragments of preparatory material for Underweysung scattered his personal manuscripts. throughout the world, as well as direct attention to the Dürer’s status as Nuremberg’s most celebrated artist would 10 numerousdrawingsin4°L.impr.c.n.mss.119. Beginning have been sufficient to provide him with access to the by contextualizing Dürer’s lifelong interest in geometry, and Regiomontanus-Walther library, irrespective of his close asso- Euclid in particular, the article centers on the corrections ciation with Pirckheimer and the other Nuremberg scholars. and insertions that Dürer made to his personal copy of But given his direct connections to Walther, he may even have Underweysung der Messung, commencing with the fastidious edit- had access to the library while Walther was alive. His parents ing of the text he undertook and concluding with his first had known Walther personally and Walther’s wife had been drafts for his famous perspectival scenes printed in the 1538 the godmother to Dürer’s sister Christina, who was born in edition. As Dürer himself notes in a handwritten inscription 1488. That Dürer was intimately familiar with the house itself is on the title page of 4°L.impr.c.n.mss.119: attested to by the fact that in 1509, five years after Walther’s death, he purchased the house from Walther’s heirs and moved I have properly corrected this book. If one wants to print a into it with his wife Agnes (1475–1539), living there until his further new copy then close attention should be paid to the 1528 corrections and care taken throughout to properly arrange death in when he was said to have been found with books 11 15 the old and new diagrams relative to the appropriate text. from the library scattered near him. Perhaps some of these books were those he had bought in 1523 from the library ’ Let us begin by accepting Durer s invitation. because of their “usefulness to painters,” when he had paid 16 ten florins for a selection of ten.
Load more

Recommended publications
  • Symmetry As an Aesthetic Factor
    Comp. & Maths. with Appls, Vol. 12B, Nos. I/2, pp. 77-82. 1986 0886-9561/86 $3,1)0+ .00 Printed in Great Britain. © 1986 Pergamon Press Ltd. SYMMETRY AS AN AESTHETIC FACTOR HAROLD OSBORNEt Kreutzstrasse 12, 8640 Rappersvill SG, Switzerland Abstract--In classical antiquity symmetry meant commensurability and was believed to constitute a canon of beauty in nature as in art. This intellectualist conception of beauty persisted through the Middle Ages with the addition doctrine that the phenomenal world manifests an imperfect replica of the ideal symmetry of divine Creation. The concept of the Golden Section came to the fore at the Renaissance and has continued as a minority interest both for organic nature and for fine art. The modern idea of symmetry is based more loosely upon the balance of shapes or magnitudes and corresponds to a change from an intellectual to a perceptual attitude towards aesthetic experience. None of these theories of symmetry has turned out to be a principle by following which aesthetically satisfying works of art can be mechanically constructed. In contemporary theory the vaguer notion of organic unity has usurped the prominence formerly enjoyed by that of balanced symmetry. From classical antiquity the idea of symmetry in close conjunction with that of proportion dominated the studio practice of artists and the thinking of theorists. Symmetry was asserted to be the key to perfection in nature as in art. But the traditional concept was radically different from what we understand by symmetry today--so different that "symmetry" can no longer be regarded as a correct translation of the Greek word symmetria from which it derives--and some acquaintance with the historical background of these ideas is essential in order to escape from the imbroglio of confusion which has resulted from the widespread conflation of the two.
    [Show full text]
  • De Divino Errore ‘De Divina Proportione’ Was Written by Luca Pacioli and Illustrated by Leonardo Da Vinci
    De Divino Errore ‘De Divina Proportione’ was written by Luca Pacioli and illustrated by Leonardo da Vinci. It was one of the most widely read mathematical books. Unfortunately, a strongly emphasized statement in the book claims six summits of pyramids of the stellated icosidodecahedron lay in one plane. This is not so, and yet even extensively annotated editions of this book never noticed this error. Dutchmen Jos Janssens and Rinus Roelofs did so, 500 years later. Fig. 1: About this illustration of Leonardo da Vinci for the Milanese version of the ‘De Divina Proportione’, Pacioli erroneously wrote that the red and green dots lay in a plane. The book ‘De Divina Proportione’, or ‘On the Divine Ratio’, was written by the Franciscan Fra Luca Bartolomeo de Pacioli (1445-1517). His name is sometimes written Paciolo or Paccioli because Italian was not a uniform language in his days, when, moreover, Italy was not a country yet. Labeling Pacioli as a Tuscan, because of his birthplace of Borgo San Sepolcro, may be more correct, but he also studied in Venice and Rome, and spent much of his life in Perugia and Milan. In service of Duke and patron Ludovico Sforza, he would write his masterpiece, in 1497 (although it is more correct to say the work was written between 1496 and 1498, because it contains several parts). It was not his first opus, because in 1494 his ‘Summa de arithmetic, geometrica, proportioni et proportionalita’ had appeared; the ‘Summa’ and ‘Divina’ were not his only books, but surely the most famous ones. For hundreds of years the books were among the most widely read mathematical bestsellers, their fame being only surpassed by the ‘Elements’ of Euclid.
    [Show full text]
  • CARTOGRAPHIC IMAGES of SLOVENIA THROUGH TIME KARTOGRAFSKE UPODOBITVE SLOVENIJE SKOZI ^AS Primo` Ga{Peri~
    acta47-2.qxd 17.1.2008 7:26 Page 245 Acta geographica Slovenica, 47-2, 2007, 245–273 CARTOGRAPHIC IMAGES OF SLOVENIA THROUGH TIME KARTOGRAFSKE UPODOBITVE SLOVENIJE SKOZI ^AS Primo` Ga{peri~ PRIMO@ GA[PERI^ Cartographic material is a part of national cultural heritage. Kartografsko gradivo je del kulturne dedi{~ine vsakega naroda. acta47-2.qxd 17.1.2008 7:26 Page 246 Primo` Ga{peri~, Cartographic images of Slovenia through time Cartographic images of Slovenia through time UDC: 912.43(497.4)(091) 528.9(497.4)(091) COBISS: 1.02 ABSTRACT: The territory of Slovenia appears on Europe's oldest maps. Slovenia's location at the junction of differ- ent natural and geographical units and various political formations influenced its traffic and border situation. As a result, for many centuries the territory of Slovenia appeared inaccurately on maps and as part of for- eign national units. With the development of cartographical skills and the recognition of cartography in our area in the 16th century and especially from the 17th century on, Slovene territory was depicted equal- ly on maps of the times through the efforts of Slovene and foreign individuals. At first, maps of individual Slovene regions dominated, but from the middle of the 19th century there are more frequent depictions of the entire Slovene territory. Cartographical work thus also became a means of expressing national demands and hopes. KEY WORDS: geography, historical geography, maps, Slovenia The article was submitted for publication on October 5, 2006. ADDRESS: Primo` Ga{peri~, B.
    [Show full text]
  • Michael Kühn Detlev Auvermann RARE BOOKS
    ANTIQUARIAT 55Michael Kühn Detlev Auvermann RARE BOOKS 1 Rolfinck’s copy ALESSANDRINI, Giulio. De medicina et medico dialogus, libris quinque distinctus. Zurich, Andreas Gessner, 1557. 4to, ff. [6], pp. AUTOLYKOS (AUTOLYCUS OF PYTANE). 356, ff. [8], with printer’s device on title and 7 woodcut initials; a few annotations in ink to the text; a very good copy in a strictly contemporary binding of blind-stamped pigskin, the upper cover stamped ‘1557’, red Autolyci De vario ortu et occasu astrorum inerrantium libri dvo nunc primum de graeca lingua in latinam edges, ties lacking; front-fly almost detached; contemporary ownership inscription of Werner Rolfinck on conuersi … de Vaticana Bibliotheca deprompti. Josepho Avria, neapolitano, interprete. Rome, Vincenzo title (see above), as well as a stamp and duplicate stamp of Breslau University library. Accolti, 1588. 4to, ff. [6], pp. 70, [2]; with large woodcut device on title, and several woodcut diagrams in the text; title a little browned, else a fine copy in 19th-century vellum-backed boards, new endpapers. EUR 3.800.- EUR 4.200.- First edition of Alessandrini’s medical dialogues, his most famous publication and a work of rare erudition. Very rare Latin edition, translated from a Greek manuscript at the Autolycus was a Greek mathematician and astronomer, who probably Giulio Alessandrini (or Julius Alexandrinus de Neustein) (1506–1590) was an Italian physician and author Vatican library, of Autolycus’ work on the rising and setting of the fixed flourished in the second half of the 4th century B.C., since he is said to of Trento who studied philosophy and medicine at the University of Padua, then mathematical science, stars.
    [Show full text]
  • Syddansk Universitet Review of "Henning, Herbert: La Divina
    Syddansk Universitet Review of "Henning, Herbert: La divina proportione und die Faszination des Schönen oder das Schöne in der Mathematik" Robering, Klaus Published in: Mathematical Reviews Publication date: 2014 Document version Accepted author manuscript Citation for pulished version (APA): Robering, K. (2014). Review of "Henning, Herbert: La divina proportione und die Faszination des Schönen oder das Schöne in der Mathematik". Mathematical Reviews. General rights 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 ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 14. Feb. 2017 MR3059375 Henning, Herbert La divina proportione und die Faszination des Sch¨onenoder das Sch¨onein der Mathematik. (German) [The divine proportion and the fascination of beauty or beauty in mathematics] Mitt. Math. Ges. Hamburg 32 (2012), 49{62 00A66 (11B39 51M04) The author points out that there is a special relationship between mathematics and the arts in that both aim at the discovery and presentation of truth and beauty.
    [Show full text]
  • Leonardo Universal
    Leonardo Universal DE DIVINA PROPORTIONE Pacioli, legendary mathematician, introduced the linear perspective and the mixture of colors, representing the human body and its proportions and extrapolating this knowledge to architecture. Luca Pacioli demonstrating one of Euclid’s theorems (Jacobo de’Barbari, 1495) D e Divina Proportione is a holy expression commonly outstanding work and icon of the Italian Renaissance. used in the past to refer to what we nowadays call Leonardo, who was deeply interested in nature and art the golden section, which is the mathematic module mathematics, worked with Pacioli, the author of the through which any amount can be divided in two text, and was a determined spreader of perspectives uneven parts, so that the ratio between the smallest and proportions, including Phi in many of his works, part and the largest one is the same as that between such as The Last Supper, created at the same time as the largest and the full amount. It is divine for its the illustrations of the present manuscript, the Mona being unique, and triune, as it links three elements. Lisa, whose face hides a perfect golden rectangle and The fusion of art and science, and the completion of the Uomo Vitruviano, a deep study on the human 60 full-page illustrations by the preeminent genius figure where da Vinci proves that all the main body of the time, Leonardo da Vinci, make it the most parts were related to the golden ratio. Luca Pacioli credits that Leonardo da Vinci made the illustrations of the geometric bodies with quill, ink and watercolor.
    [Show full text]
  • Leonardo's Vitruvian Man Drawing: a New Interpretation Looking at Leonardo's Geometric Constructions
    Nexus Netw J (2015) 17:507–524 DOI 10.1007/s00004-015-0247-7 RESEARCH Leonardo’s Vitruvian Man Drawing: A New Interpretation Looking at Leonardo’s Geometric Constructions Vitor Murtinho1 Published online: 9 April 2015 Ó Kim Williams Books, Turin 2015 Abstract Generally speaking, today’s scientific community considers that the famous figure drawn by Leonardo da Vinci at the end of the fifteenth century was made using the Golden ratio. More specifically, the relationship established between the circle’s diameter and the side of the square is a consequence of the geometric relationship probably discovered by Euclid, but made famous by Luca Pacioli in his De Divina proportione. Aware of the close working relationship between Leonardo and Pacioli, namely in the writing of this last book, the theory that establishes a close relationship between these two figures, making use of this remarkable mathematical relationship, has gained credibility. In fact, the use of the Divina proporzione, despite being a very stimulating construction on an intellectual level, presents too great a margin of error, especially for such a competent geometrician as Leonardo da Vinci was. For that reason, the relationship between these two figures (square and circle) is grounded on a much simpler geometric relationship than the one found at the base of the definition, for instance, of Le Corbusier’s Modulor. Keywords Leonardo da Vinci Á Vitruvian man Á Vesica piscis Á Golden number Á Proportion Introduction No human investigation may claim to be a true science if it has not passed through mathematical demonstrations, and if you say that the sciences that begin and end in the mind exhibit truth, this cannot be allowed, but must be & Vitor Murtinho [email protected] 1 CES-Department of Architecture, University of Coimbra, Colegio das Artes, Largo D.
    [Show full text]
  • Reposs #11: the Mathematics of Niels Henrik Abel: Continuation and New Approaches in Mathematics During the 1820S
    RePoSS: Research Publications on Science Studies RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation and New Approaches in Mathematics During the 1820s Henrik Kragh Sørensen October 2010 Centre for Science Studies, University of Aarhus, Denmark Research group: History and philosophy of science Please cite this work as: Henrik Kragh Sørensen (Oct. 2010). The Mathematics of Niels Henrik Abel: Continuation and New Approaches in Mathemat- ics During the 1820s. RePoSS: Research Publications on Science Studies 11. Aarhus: Centre for Science Studies, University of Aarhus. url: http://www.css.au.dk/reposs. Copyright c Henrik Kragh Sørensen, 2010 The Mathematics of NIELS HENRIK ABEL Continuation and New Approaches in Mathematics During the 1820s HENRIK KRAGH SØRENSEN For Mom and Dad who were always there for me when I abandoned all good manners, good friends, and common sense to pursue my dreams. The Mathematics of NIELS HENRIK ABEL Continuation and New Approaches in Mathematics During the 1820s HENRIK KRAGH SØRENSEN PhD dissertation March 2002 Electronic edition, October 2010 History of Science Department The Faculty of Science University of Aarhus, Denmark This dissertation was submitted to the Faculty of Science, University of Aarhus in March 2002 for the purpose of ob- taining the scientific PhD degree. It was defended in a public PhD defense on May 3, 2002. A second, only slightly revised edition was printed October, 2004. The PhD program was supervised by associate professor KIRSTI ANDERSEN, History of Science Department, Univer- sity of Aarhus. Professors UMBERTO BOTTAZZINI (University of Palermo, Italy), JEREMY J. GRAY (Open University, UK), and OLE KNUDSEN (History of Science Department, Aarhus) served on the committee for the defense.
    [Show full text]
  • Reflections on the Pragmatics of the Illustrated Perspective Treatise Performative Failures and (Pre-) Romantic Innovations Eduardo Ralickas
    Document generated on 09/25/2021 8:06 a.m. Intermédialités Histoire et théorie des arts, des lettres et des techniques Intermediality History and Theory of the Arts, Literature and Technologies Reflections on the Pragmatics of the Illustrated Perspective Treatise Performative Failures and (Pre-) Romantic Innovations Eduardo Ralickas exposer Article abstract displaying This essay reconsiders visual demonstrations contained in a selection of Number 15, printemps 2010 illustrated perspective treatises. Based on a fundamental distinction made in the field of pragmatics, the author argues that the images designed to URI: https://id.erudit.org/iderudit/044680ar demonstrate/teach/instantiate the perspectival system are plagued by a DOI: https://doi.org/10.7202/044680ar contradiction between the conceptual “content” of perspective and the figurative means deployed to display such content. In all cases, this aporia, which defines the teaching of perspective by means of images, arises when the See table of contents figurative discourse of perspective attempts to integrate a representation of its user within the system itself. In closing, the author suggests that the perspective treatise’s “pragmatic unconscious” allows one to shed new light on Publisher(s) the pictorial innovations of German romanticism, particularly in the work of Caspar David Friedrich, which is tied to fundamental (and hitherto unforeseen) Revue Intermédialités (Presses de l’Université de Montréal) ways to the “failures” of the classical age. ISSN 1705-8546 (print) 1920-3136 (digital) Explore this journal Cite this article Ralickas, E. (2010). Reflections on the Pragmatics of the Illustrated Perspective Treatise: Performative Failures and (Pre-) Romantic Innovations. Intermédialités / Intermediality, (15), 163–185.
    [Show full text]
  • On a Human Scale. Drawing and Proportion of the Vitruvian Figure Veronica Riavis
    7 / 2020 On a Human Scale. Drawing and Proportion of the Vitruvian Figure Veronica Riavis Abstract Among the images that describe the proportions of the human body, Leonardo da Vinci’s one is certainly the most effective, despite the fact that the iconic drawing does not faithfully follow the measurements indicated by Vitruvius. This research concerned the geometric analysis of the interpretations of the Vitruvian man proposed in the Renaissance editions of De Architectura, carried out after the aniconic editio princeps by Sulpicio da Veroli. Giovanni Battista da Sangallo drew the Vitruvian figure directly on his Sulpician copy, very similar to the images by Albrecht Dürer in The Symmetry of the Human Bodies [Dürer 1591]. Fra Giocondo proposes in 1511 two engravings of homo ad quadratum and ad circulum in the first Latin illustrated edition of De Architectura, while the man by Cesare Cesariano, author of the first version in vernacular of 1521, has a deformed body extension to adapt a geometric grid. Francesco di Giorgio Martini and Giacomo Andrea da Ferrara also propose significant versions believed to be the origin of Leonardo’s figuration due to the friendship that bound them. The man inscribed in the circle and square in the partial translation of Francesco di Giorgio’s De Architectura anticipates the da Vinci’s solution although it does not have explicit metric references, while the drawing by Giacomo Andrea da Ferrara reproduces a figure similar to Leonardo’s one. The comparison between the measures expressed by Vitruvius to proportion the man and the various graphic descriptions allows us to understand the complex story of the exegesis of the Roman treatise.
    [Show full text]
  • Sources and Studies in the History of Mathematics and Physical Sciences
    Sources and Studies in the History of Mathematics and Physical Sciences Editorial Board J.Z. Buchwald J. Lützen G.J. Toomer Advisory Board P.J. Davis T. Hawkins A.E. Shapiro D. Whiteside Springer Science+Business Media, LLC Sources and Studies in the History of Mathematics and Physical Seiences K. Andersen Brook Taylor's Work on Linear Perspective H.l.M. Bos Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction 1. Cannon/S. Dostrovsky The Evolution of Dynamics: Vibration Theory from 1687 to 1742 B. ChandIerlW. Magnus The History of Combinatorial Group Theory A.I. Dale AHistory of Inverse Probability: From Thomas Bayes to Karl Pearson, Second Edition A.I. Dale Most Honourable Remembrance: The Life and Work of Thomas Bayes A.I. Dale Pierre-Simon Laplace, Philosophical Essay on Probabilities, Translated from the fifth French edition of 1825, with Notes by the Translator P. Damerow/G. FreudenthallP. McLaugWin/l. Renn Exploring the Limits of Preclassical Mechanics: A Study of Conceptual Development in Early Modem Science: Free Fall and Compounded Motion in the Work of Descartes, Galileo, and Beeckman, Second Edition P.l. Federico Descartes on Polyhedra: A Study of the De Solworum Elementis B.R. Goldstein The Astronomy of Levi ben Gerson (1288-1344) H.H. Goldstine A History of Numerical Analysis from the 16th Through the 19th Century H.H. Goldstine A History of the Calculus of Variations from the 17th Through the 19th Century G. Graßhoff The History of Ptolemy's Star Catalogue A.W. Grootendorst Jan de Witt's Eiementa Curvarum Linearum, über Primus Continued after Index The Arithmetic of Infinitesimals John Wallis 1656 Translated from Latin to English with an Introduction by Jacqueline A.
    [Show full text]
  • Bridges Conference 2009
    Non-flat tilings with flat tiles Rinus Roelofs Sculptor Lansinkweg 28 7553AL Hengelo The Netherlands E-mail: [email protected] www.rinusroelofs.nl Abstract In general a tiling is considered to be a set of tiles placed next to each other in a flat plane. The tiles are placed in the plane in such a way that there are no gaps and no overlaps. But what if we leave out the condition that the plane has to be flat? For when there are no gaps and no overlaps between the tiles we still can call it a tiling. The consequences for the possible shapes of the tiles in non-flat tilings as well as the possible symmetrical structures that can be used are discussed in this paper. 1. Tilings 1.1. Definition. A tiling, or tessellation, is a covering of a plane without gaps or overlaps by polygons, all of which are the same size and shape. That is one of the definitions of a tiling. Another definition is the following: Tiling: a pattern made of identical shapes; the shapes must fit together without any gaps and the shapes should not overlap. Although the second definition doesn’t speak about a plane, it is mostly assumed that the tiles do cover a plane. But we can take the definition literal, and then the only conditions are that the tiles do not overlap and do not leave gaps. Tilings in which all the tiles have the same shape are mostly called monohedral tilings [1]. In this paper all the tilings will be monohedral.
    [Show full text]