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Letter from Descartes to Desargues 1 (19 June 1639)
Appendix 1 Letter from Descartes to Desargues 1 (19 June 1639) Sir, The openness I have observed in your temperament, and my obligations to you, invite me to write to you freely what I can conjecture of the Treatise on Conic Sections, of which the R[everend] F[ather] M[ersenne] sent me the Draft. 2 You may have two designs, which are very good and very praiseworthy, but which do not both require the same course of action. One is to write for the learned, and to instruct them about some new properties of conics with which they are not yet familiar; the other is to write for people who are interested but not learned, and make this subject, which until now has been understood by very few people, but which is nevertheless very useful for Perspective, Architecture etc., accessible to the common people and easily understood by anyone who studies it from your book. If you have the first of these designs, it does not seem to me that you have any need to use new terms: for the learned, being already accustomed to the terms used by Apollonius, will not easily exchange them for others, even better ones, and thus your terms will only have the effect of making your proofs more difficult for them and discourage them from reading them. If you have the second design, your terms, being French, and showing wit and elegance in their invention, will certainly be better received than those of the Ancients by people who have no preconceived ideas; and they might even serve to attract some people to read your work, as they read works on Heraldry, Hunting, Architecture etc., without any wish to become hunters or architects but only to learn to talk about them correctly. -
Desargues, Girard
GIRARD DESARGUES (February 21, 1591 – October 1661) by HEINZ KLAUS STRICK, Germany GIRARD DESARGUES came from very wealthy families of lawyers and judges who worked at the Parlement, the highest appellate courts of France in Paris and Lyon. Nothing is known about GIRARD's youth, but it is safe to assume that he and his five siblings received the best possible education. While his two older brothers were admitted to the Parisian Parlement, he was involved in the silk trade in Lyon, as can be seen from a document dating from 1621. In 1626, after a journey through Flanders, he applied to the Paris city council for a licence to drill a well and use the water from the well. His idea was to construct an effective hydraulic pump to supply water to entire districts, but this project did not seem to be successful. After the death of his two older brothers in 1628, he took over the family inheritance and settled in Paris. There he met MARIN MERSENNE and soon became a member of his Academia Parisiensis, a discussion group of scientists including RENÉ DESCARTES, GILLES PERSONNE DE ROBERVAL, ÉTIENNE PASCAL and his son BLAISE. (drawing: © Andreas Strick) The first publication by DESARGUES to attract attention was Une méthode aisée pour apprendre et enseigner à lire et escrire la musique (An easy way to learn and teach to read and write music). In 1634 MERSENNE mentioned in a letter to his acquaintances that DESARGUES was working on a paper on perspective (projection from a point). But it was not until two years later that the work was published: only 12 pages long and in a small edition. -
1 H-France Forum Volume 14 (2019), Issue 4, #3 Jeffrey N. Peters, The
1 H-France Forum Volume 14 (2019), Issue 4, #3 Jeffrey N. Peters, The Written World. Space, Literature, and the Chorological Imagination in Early Modern France. Evanston: Northwestern University Press, 2018. vii + 272 pp. Figures, notes, and Copyright index. $34.95 (pb). ISBN 978-0-8101-3697-7; $99.95 (cl). ISBN 978-0- 8101-3698-4; $34.95 (Kindle). ISBN 978-0-8101-3699-1. Review Essay by David L. Sedley, Haverford College The Written World has on its cover an image from La Manière universelle de M. Desargues, pour pratiquer la perspective (1648). This book, written and illustrated by Abraham Bosse and based on the projective geometry of Girard Desargues, extends the theories of perspective codified by Leon Battista Alberti and his followers. [1] Alberti directed painters to pose a central point at the apparent conjunction of parallel lines in order to lend depth and coherence to their compositions. Desargues reinterpreted and renamed Alberti’s central point (and other points like it) as a point at infinity. Consequently, the convergent lines of a visual representation could be taken to indicate the infinite more emphatically than before. As an illustration of the art of putting objects in a perspective that emphasizes their connection to infinity, Bosse’s image suits Peters’ book to a T. Peters represents his objects of study—mainly a series of works of seventeenth-century French literature—with an eye to showing their affinity with the infinite. He frequently discusses infinity through chora, the Ancient Greek term used by Plato and adopted by Jacques Derrida to denote the space underlying all finite places and place- based thought. -
Mathematics and Arts
Mathematics and Arts All´egoriede la G´eom´etrie A mathematical interpretation Alda Carvalho, Carlos Pereira dos Santos, Jorge Nuno Silva ISEL & Cemapre-ISEG, CEAFEL-UL, University of Lisbon [email protected] [email protected] [email protected] Abstract: In this work, we present a mathematical interpretation for the mas- terpiece All´egoriede la G´eom´etrie (1649), painted by the French baroque artist Laurent de La Hyre (1606{1656). Keywords: Laurent de La Hyre, \All´egoriede la G´eom´etrie",baroque art, mathematical interpretation, perspective. Introduction The main purpose of this text is to present a mathematical interpretation for the masterpiece All´egoriede la G´eom´etrie, from a well-known series of paint- ings, Les 7 arts lib´eraux, by the French baroque artist Laurent de La Hyre (1606{1656). Figure 1: All´egoriede la G´eom´etrie (1649), oil on canvas. Recreational Mathematics Magazine, Number 5, pp. 33{45 DOI 10.1515/rmm{2016{0003 34 allegorie´ de la geom´ etrie´ Laurent de La Hyre painted the series Les 7 arts lib´eraux between 1649 and 1650 to decorate G´ed´eonTallemant's residence. Tallemant was an adviser of Louis xiv (1638{1715). The king was 10 years old at the time of the commission. According to the artist's son, Philippe de La Hire (1640{1718), writing around 1690 [5], (. .) une maison qui appartenoit autrefois a M. Tallemant, maistre des requestes, sept tableaux repres´entant les sept arts liberaux qui font l'ornement d'une chambre. Also, Guillet de Saint-Georges, a historiographer of the Acad´emieRoyale de Peinture et de Sculpture, mentioned that it was Laurent's work for the Ca- puchin church in the Marais which led to the commission for the \Seven Liberal Arts" in a house [2]. -
PAINTING and PERSPECTIVE 127 the Real World, That the Universe Is Ordered and Explicable Rationally in Terms of Geometry
PAINTING AND PERSPECTIVE 127 the real world, that the universe is ordered and explicable rationally in terms of geometry. Hence, like the Greek philosopher, he be lieved that to penetrate to the underlying significance, that is, the reality of the theme that he sought to display on canvas, he must reduce it to its mathematical content. Very interesting evidence of x the artist's attempt to discover the mathematical essence of his sub ject is found in one of Leonardo's studies in proportion. In it he Painting and Perspective tried to fit the structure of the ideal man to the ideal figures, the flquare and circle (plate VI). The sheer utility of mathematics for accurate description and the The world's the book where the eternal sense philosophy that mathematics is the essence of reality are only two Wrote his own thoughts; the living temple where, Painting his very self, with figures fair of the reasons why the Renaissance artist sought to use mathematics. He filled the whole immense circumference. There was another reason. The artist of the late medieval period and T. CAMPANELLA the Renaissance was, also, the architect and engineer of his day and so was necessarily mathematically inclined. Businessmen, secular princes, and ecclesiastical officials assigned all construction problems During the Middle Ages painting, serving somewhat as the hand to the artist. He designed and built churches, hospitals, palaces, maiden of the Church, concentrated on embellishing the thoughts cloisters, bridges, fortresses, dams, canals, town walls, and doctrines of Christianity. Toward the end of this period, the and instru ments of warfare. -
The Encounter of the Emblematic Tradition with Optics the Anamorphic Elephant of Simon Vouet
Nuncius 31 (2016) 288–331 brill.com/nun The Encounter of the Emblematic Tradition with Optics The Anamorphic Elephant of Simon Vouet Susana Gómez López* Faculty of Philosophy, Complutense University, Madrid, Spain [email protected] Abstract In his excellent work Anamorphoses ou perspectives curieuses (1955), Baltrusaitis con- cluded the chapter on catoptric anamorphosis with an allusion to the small engraving by Hans Tröschel (1585–1628) after Simon Vouet’s drawing Eight satyrs observing an ele- phantreflectedonacylinder, the first known representation of a cylindrical anamorpho- sis made in Europe. This paper explores the Baroque intellectual and artistic context in which Vouet made his drawing, attempting to answer two central sets of questions. Firstly, why did Vouet make this image? For what purpose did he ideate such a curi- ous image? Was it commissioned or did Vouet intend to offer it to someone? And if so, to whom? A reconstruction of this story leads me to conclude that the cylindrical anamorphosis was conceived as an emblem for Prince Maurice of Savoy. Secondly, how did what was originally the project for a sophisticated emblem give rise in Paris, after the return of Vouet from Italy in 1627, to the geometrical study of catoptrical anamor- phosis? Through the study of this case, I hope to show that in early modern science the emblematic tradition was not only linked to natural history, but that insofar as it was a central feature of Baroque culture, it seeped into other branches of scientific inquiry, in this case the development of catoptrical anamorphosis. Vouet’s image is also a good example of how the visual and artistic poetics of the baroque were closely linked – to the point of being inseparable – with the scientific developments of the period. -
A Tale of the Cycloid in Four Acts
A Tale of the Cycloid In Four Acts Carlo Margio Figure 1: A point on a wheel tracing a cycloid, from a work by Pascal in 16589. Introduction In the words of Mersenne, a cycloid is “the curve traced in space by a point on a carriage wheel as it revolves, moving forward on the street surface.” 1 This deceptively simple curve has a large number of remarkable and unique properties from an integral ratio of its length to the radius of the generating circle, and an integral ratio of its enclosed area to the area of the generating circle, as can be proven using geometry or basic calculus, to the advanced and unique tautochrone and brachistochrone properties, that are best shown using the calculus of variations. Thrown in to this assortment, a cycloid is the only curve that is its own involute. Study of the cycloid can reinforce the curriculum concepts of curve parameterisation, length of a curve, and the area under a parametric curve. Being mechanically generated, the cycloid also lends itself to practical demonstrations that help visualise these abstract concepts. The history of the curve is as enthralling as the mathematics, and involves many of the great European mathematicians of the seventeenth century (See Appendix I “Mathematicians and Timeline”). Introducing the cycloid through the persons involved in its discovery, and the struggles they underwent to get credit for their insights, not only gives sequence and order to the cycloid’s properties and shows which properties required advances in mathematics, but it also gives a human face to the mathematicians involved and makes them seem less remote, despite their, at times, seemingly superhuman discoveries. -
Projective Geometry Seen in Renaissance Art
Projective Geometry Seen in Renaissance Art Word Count: 2655 Emily Markowsky ID: 112235642 December 7th, 2020 Abstract: For the West, the Renaissance was a revival of long-forgotten knowledge from classical antiquity. The use of geometric techniques from Greek mathematics led to the development of linear perspective as an artistic technique, a method for projecting three dimensional figures onto a two dimensional plane, i.e. the painting itself. An “image plane” intersects the observer’s line of sight to an object, projecting this point onto the intersection point in the image plane. This method is more mathematically interesting than it seems; we can formalize the notion of a vanishing point as a “point at infinity,” from which the subject of projective geometry was born. This paper outlines the historical process of this development and discusses its mathematical implications, including a proof of Desargues’ Theorem of projective geometry. Outline: I. Introduction A. Historical context; the Middle Ages and medieval art B. How linear perspective contributes to the realism of Renaissance art, incl. a comparison between similar pieces from each period. C. Influence of Euclid II. Geometry of Vision and Projection A. Euclid’s Optics B. Cone of vision and the use of conic sections to model projection of an object onto a picture. Projection of a circle explained intuitively. C. Brunelleschi’s experiments and the invention of his technique for linear perspective D. Notion of points at infinity as a vanishing point III. Projective geometry and Desargues’ Theorem A. Definition of a projective plane and relation to earlier concepts discussed B. -
Girard Desargues, Maître De Pascal Eishi KUKITA
Girard Desargues, maître de Pascal Eishi KUKITA La vie et l’œuvre de Girard Desargues (1591-1661), savant et ingénieur du XVIIe siècle, sont aujourd’hui presque inconnues1. Pourtant les plus lucides de ses contemporains, dont Descartes et Pascal, tous deux grands géomètres qui ont marqué chacun à sa manière l’aube des sciences naturelles modernes, lui trouvaient une intelligence hors du commun. Une lecture attentive des correspondances de Descartes nous permettra de constater que le philosophe du cogito n’exagérait point en écrivant à Mersenne, au sujet de ses Méditations (1641) : Je serai bien aise que M. Desargues soit aussi un de mes juges, s’il lui plaît d’en prendre la peine, et je me fie plus en lui qu’en trois théologiens2. Quant à Pascal, publiant à l’âge de seize ans son premier article Essai pour les Coniques en 1640, il rendit un hommage vibrant à celui qu’il considérait comme son maître : Nous démontrerons aussi cette propriété, dont le premier inventeur est M. Desargues Lyonnais, un des grands esprits de ce temps et des 1 La littérature sur Desargues est franchement pauvre. Autant que nous le sachions, seuls quatre livres en français ont été publiés depuis XIXe siècle. N.-G. Poudra, Œuvres de Desargues, 2 vol., Léiber, 1864, rééd. fac-similé, Cambridge Univ. Press, 2011 (sigle : Pou). R. Taton, Œuvres mathématiques de G. Desargues, PUF, 1951, 2e éd., Vrin, 1988 (sigle : Tat). J. Dhombres et J. Sakarovitch (dir.), Desargues en son Temps, Blanchard, 1994 (sigle : DS). M. Chaboud, Girard Desargues, Bourgeois de Lyon, Mathématicien, Architecte, Lyon, Aléas, 1996 (sigle : Cha). -
A New Perspective in Perspective New Technique Debating Old Hand-Drawing Perspective Methods Taught in Schools of Architecture
American Journal of Engineering, Science and Technology (AJEST) Volume 7, 2021 A New Perspective in Perspective New Technique Debating Old Hand-Drawing Perspective Methods Taught in Schools of Architecture Yasser O. El-Gammal, Saudi Arabia Abstract This paper introduces a new technique in hand-drawing perspective that is more convenient for students, instructors, and architectural design professionals - than the difficulties associated with old hand drawing methods taught in schools of architecture. The paper starts with an overview exploring origins and fundamental concepts of perspective drawing in an attempt to find a better approach in perspective drawing by investigating the works of ancient artists and painters, Afterwards; The paper proceeds with debating older perspective drawing methods taught in schools of architecture through exploring both students and instructors opinions about disadvantages of the old methods, and the recent trends in teaching perspective drawing. At the end, the paper introduces the concept of the suggested new method, its relationship with the magnification factor option in the AutoCAD universe, and its advantages, together with some early experimentation Keywords: Perspective Sketching, Perspective Drawing Method, Perspective Drawing Techniques PROBLEM Students learning perspective hand drawing in schools of architecture find old drawing methods very tedious, time consuming, and end up with unsatisfactory results, not to mention the difficulty they experience when setting the building perspective scene measurements into scale, and from within too many drawing construction lines. Although both undergraduates and architects nowadays are heavily depending on 3Dimensional computer software for developing accurate perspective scenes; There will always still the need for using perspective hand-drawing; Since every architectural concept starts with a hand-drawn sketch. -
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics Editors S. Axler K.A. Ribet Undergraduate Texts in Mathematics Abbott: Understanding Analysis. Chambert-Loir: A Field Guide to Algebra Anglin: Mathematics: A Concise History Childs: A Concrete Introduction to and Philosophy. Higher Algebra. Second edition. Readings in Mathematics. Chung/AitSahlia: Elementary Probability Anglin/Lambek: The Heritage of Theory: With Stochastic Processes and Thales. an Introduction to Mathematical Readings in Mathematics. Finance. Fourth edition. Apostol: Introduction to Analytic Cox/Little/O’Shea: Ideals, Varieties, Number Theory. Second edition. and Algorithms. Second edition. Armstrong: Basic Topology. Croom: Basic Concepts of Algebraic Armstrong: Groups and Symmetry. Topology. Axler: Linear Algebra Done Right. Cull/Flahive/Robson: Difference Second edition. Equations: From Rabbits to Chaos Beardon: Limits: A New Approach to Curtis: Linear Algebra: An Introductory Real Analysis. Approach. Fourth edition. Bak/Newman: Complex Analysis. Daepp/Gorkin: Reading, Writing, and Second edition. Proving: A Closer Look at Banchoff/Wermer: Linear Algebra Mathematics. Through Geometry. Second edition. Devlin: The Joy of Sets: Fundamentals Berberian: A First Course in Real of Contemporary Set Theory. Second Analysis. edition. Bix: Conics and Cubics: A Dixmier: General Topology. Concrete Introduction to Algebraic Driver: Why Math? Curves. Ebbinghaus/Flum/Thomas: Bre´maud: An Introduction to Mathematical Logic. Second edition. Probabilistic Modeling. Edgar: Measure, Topology, and Fractal Bressoud: Factorization and Primality Geometry. Testing. Elaydi: An Introduction to Difference Bressoud: Second Year Calculus. Equations. Third edition. Readings in Mathematics. Erdo˜s/Sura´nyi: Topics in the Theory of Brickman: Mathematical Introduction Numbers. to Linear Programming and Game Estep: Practical Analysis in One Variable. Theory. Exner: An Accompaniment to Higher Browder: Mathematical Analysis: Mathematics. -
The Historiography of Perspective and Reflexy-Const in Netherlandish
The Historiography of Perspective and Reflexy-Const in Netherlandish Art Sven Dupré Introduction In The Heritage of Apelles Ernst Gombrich famously drew a distinction between art North and South of the Alps along optical lines. Painters in fifteenth-century Florence, such as Domenico Veneziano, used different optical features than their contemporaries in Bruges, the likes of Jan Van Eyck, to create the illusion of space. Gombrich commented: „We all associate Florentine art with the development of central perspective, and thus with the mathematical method of revealing form in ambient light. The other aspect of optical theory, the reaction of light to various surfaces, was first explored in modern times by painters North of the Alps. It was there that the mastery of lustre, sparkle and glitter was first achieved, permitting the artist to convey the peculiar character of materials. Indeed, for a time, during the first decades of the fifteenth century, the two schools of painting appeared thus to have divided the kingdom of appearances between them.‟1 While Italian painters of the fifteenth century used perspective, Netherlandish artists studied and painted the reflection of light from surfaces of different textures and materials to create the illusion of space. For Gombrich, the point of contact between North and South was Leonardo da Vinci, „the greatest explorer of natural appearances‟, who must have been „a keen student of Northern painting‟.2 Here I am less interested in Gombrich‟s geography of optics, but more in how his juxtaposition of perspective and painterly light reflects and complicates the historiography of perspective. For reasons which fall outside the scope of this paper, since the seminal work of 1 Erwin Panofsky linear perspective has become a locus classicus of the study of artistic practice and science, and the history of perspective, which Panofsky in Die Perspektive als symbolische Form (1927) disconnected from optics, a field of research of its own, independent from the history of science and the history of art.