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The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Andersen, Kirsti. The mathematical treatment of anamorphoses from Piero della Francesca to Niceron. History of mathematics: states of the art, 3--28, Academic Press, San Diego, CA, 1996.
Discusses the mathematics of anamorphoses and the history of the subject from the mathematical point of view. Begins with a short discussion of problems stemming from the well-known fact that cylindrical columns seem smaller towards the top. Dürer discussed how one can use letters of different size on such a column so that rows of print will all appear the same size. His student Erhard Schön did some work using anamorphoses proper. (This was about the same time as Hans Holbein's Ambassadors.) Piero della Francesca's De Prospectiva Pingendi includes a discussion of how to construct a particular anamorphic drawing, but little further progress was made until the 1600s. The author notes that artists didn't seem to use the same mathematical techniques when using more extreme perspectives as they used with more normal perspectives. In fact, written works from the time suggest that orthogonal projections were used. The author gives examples from the work of of Daniele Barbaro [Italy 1500s], Paolo Giovanni Lomazzo [Italy 1500s], Egnazio Danti [Italy 1500s], Guidobaldo del Monte [France 1600s], Samuel Marolois [France 1600s], and Salomon de Caus [France 1600s]. (The case of Lomazzo is unclear: he suggested using threads for the construction, but didn't state clearly how they were to be used.) After Niceron, more mathematically accurate techniques were used; the author gives an example of a work by Emmanuel Maignan [France 1600s], who was influenced by Niceron. The problems of mirror anamorphoses apparently originated in China by about 1600. Artists apparently either worked intuitively (as in China), or by using approximate constructions. Approximate constructions still appear today in the work of the 20th century Swedish artist Hans Hamngren. A mathematically precise treatment of the problem (and of a problem using a conical mirror) was given by Jean-Louis Vaulezard in the 1600s, but even Niceron gave only an approximate method. The author suggests that Vaulezard's students were perhaps the only ones who constructed curved-mirror anamorphoses using mathematically accurate methods. (Computer analyses might be useful to verify this.) Using a computer algebra system, the author has derived the equations for the curves which will project to a coordinate grid. The curve is not given in the text, but the author tells us that it is not one of the familiar curves, has degree 6, and has rather complicated coefficients. Closely related topics: Anamorphoses, Albrecht Dürer, Erhard Schön, Piero della Francesca, China, Jean-Louis Vaulezard, and Jean-François Niceron.
Cox, Steven J. The shape of the ideal column. Math. Intelligencer 14 (1992), no. 1, 16--24. (Reviewer: Peeter Müürsepp.) SC: 01A99 (00A69), MR: 93a:01072.
Discusses the shape of the "ideal" column. Shows how the aesthetic and perceptual ideals of Greek and Roman times were relayed by Vitruvius and later by Alberti and others. Then shows how later scientists considered the problem from the point of view of structural strength instead. A key player in this new point of view was Lagrange. The author discusses mistakes in Lagrange's work and in the work of some later scientists and mathematicians. It is interesting that the author himself has made investigations in this area (together with M. L. Overton). The article Kirmser, Philip G. and Hu, Kuo-Kuang, The shape of the ideal column reconsidered is critical of these investigations, and includes a response by Cox. Closely related topics: Vitruvius, Leone Battista Alberti (1404?--1472), Statics, and Joseph Louis Lagrange.
Kirmser, Philip G. and Hu, Kuo-Kuang. The shape of the ideal column reconsidered. With a reply by Steven Cox. Math. Intelligencer 15 (1993), no. 3, 62--68. (Reviewer: Peeter Müürsepp.) SC: 73K05 (00A69 01A99 49N55 73H05), MR: 94e:73039.
This article criticizes some of the conclusions of Cox, Steven J., The shape of the ideal column, and contains a new derivation of the shape of the "ideal" column. In Cox's view the problem of the ideal column remains far from solved. Cox acknowledges some of the criticism, but in turn objects to the way Kirmser and Hu have had tacitly assumed the existence of a strongest column in order, which he considers far from clear. He says "Faced with their outright contempt for the question of existence of a strongest column, I find solace in L. C. Young's invocation of Perron's paradox." (This paradox starts "Let N be the largest positive integer", and then shows that there exists a larger number.) The mathematics involved is somewhat technical. Closely related topics: Statics and Paradox.