Review From the reviews: "The book in question is a survey of the history of this subject … . I do not know the author, but I came to trust her voice in the book, to trust her honesty and her judgments. I appreciated the clarity of her illustrations and her concern for the reader’s understanding. I found the book to be carefully written, and I was impressed with the immense amount of work that must have gone into writing it." (Greg St. George, Zentralblatt für Didaktik der Mathematik, Vol. 39, 2007) "I was very pleased to find Kristi Andersen’s book on the history of the geometrical evolution of perspective. … Reading it from the point of view of someone who is interested in geometry as well as art, it is a fascinating book, but it also has much to offer the historian of mathematics. … it is extremely well produced and researched and makes an invaluable contribution to the literature on perspective as well as the history of geometry." (John Sharp, Journal of Mathematics and the Arts, Vol. 1 (4), 2007) Read more From the Back Cover This monograph describes how the understanding of the geometry behind perspective evolved between the years 1435 and 1800 and how new insights within the mathematical theory of perspective influenced the way the discipline was presented in textbooks. In order to throw light on these issues, the author has chosen to focus on a number of key questions, including:• What were the essential innovations in the mathematical theory of perspective?• Was there any interplay between the developments of the mathematical theory of perspective and other branches of geometry?• What were the driving forces behind working out an advanced mathematical theory of perspective? • Were there regional differences in the mathematical approach to perspective? And if so, how did they relate to local applications of perspective?• How did mathematicians and practitioners of perspective interact?In fact, the last issue is touched upon so often that a considerable part of this book could be seen as a case study of the difficulties in bridging the gap between those with mathematical knowledge and the mathematically untrained practitioners who wish to use this knowledge.The author has based her work on more than 200 books, booklets, and pamphlets on perspective. She starts with the first treatise known to deal with geometrical perspective, Leon Alberti Battista’s De pictura, and ends around 1800, when the theory of mathematical perspective as an independent discipline was absorbed first into descriptive geometry and later into projective geometry.The prominent protagonists are Guidobaldo del Monte, Simon Stevin, Willem ’sGravesande, Brook Taylor, and Johann Heinrich Lambert. As far as data were available, the author has provided brief biographies of all the writers on perspective whose work she studied. The book also contains an extensive bibliography divided into two parts, one for primary sources on perspective, and the second for all other literature.Kirsti Andersen is Associate Professor of History of Science at the University of Aarhus, Denmark. She is the author of Brook Taylor’s Work on Linear Perspective, also published by Springer.  Read more
V**O
Few but beautiful ideas
This is a dry and thorough study of the history of perspective. As one would expect, historiographical considerations have priority, making ideas of perspective rare and sometimes opaque. Still, even for those of us who don't care very much about nitpickery, there are some absolutely wonderful ideas in here that mathematicians should never have forgotten. I wish to point to my favourite, the visual ray construction of 'sGravesande (1711). We shall draw the perspective image of a ground plane. To do this we rotate both the eye point and the ground plane into the picture plane: the ground plane is rotated down about its intersection with the picture plane (the "ground line") and the eye is rotated up about the horizon. Consider a line AB in the ground plane. The intersection of AB with the ground line is of course known. The image of AB intersects the horizon where the parallel to AB through the eye point meets the picture plane, and parallelity is clearly preserved by the turning-in process. So to construct the image of AB we turn it into the picture plane and mark its intersection with the ground line and then draw the parallel through the eye point and mark its intersection with the horizon; the image of AB is the line connecting these two points. This enables us to construct the image of any point A by constructing the images of two line going through it. This process is simplified by letting one of the lines be the line connecting the turned-in point and the turned-in eye point. The image point will be on this line because if we turn things back out the eye point-to-horizon part of the line will be parallel to the A-to-ground line part of the line, so that the image part of this line is indeed the image of the A-to-ground-line line. In other words: in the turned-in situation, a point in the ground plane, its image and the the eye point will be collinear. This makes it particularly interesting to construct the perspective image A'B'C' of a triangle ABC. By the image-of-a-line construction, intersections of extensions of corresponding sides are all on a line, namely the ground line, and by the collinearity property A'B'C' and ABC are in perspective from the eye point, so we have Desargues's theorem. These ideas are also great for doing ruler-only geometry. For instance, Lambert (1774) constructed the parallel m to a given line l trough a given point P, given a parallelogram ABCD, by considering l as the ground line, m as the horizon and P vanishing point of the images of AD and BC. "It would have been a grand finale to the story on the development of the matematical theory of perspective to say that it helped give rise to [projective geometry], but I'm afraid the conclusion is that neither Lambert nor perspective contributed in any essential way to the birth of projective geometry. It was only after creating this subject that Poncelet realized that a few of the problems he took up had also been treated by Lambert---and by Desargues." (p. 703).
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