The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge (Sources and Studies in the History of Mathematics and Physical Sciences)

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).

This is a very detailed treatment of perspective in painting and the history of geometry that follows. It is a very thoroughly researched book. I know Kirsti Andersen, and I highly recommend her book as a reliable source on the history. On the art side it has some weaknesses, but on the whole it is a fine book.

The best history of geometry in art one could hope for.