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The Invention of Infinity: Mathematics and Art in the Renaissance

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Renaissance artists learned how to draw perspective images. Perhaps the realisation of how to draw checkered floors sparked the development. Then one learned to draw perspective views of cubes and prisms and such. By approximation, these techniques could be of use when drawing more complex bodies and well, but apparently many artists preferred mechanical tricks instead (eye line strings, reference grids, etc.). Indeed, the mathematical sophistication of the Renaissance artists should not be over stated; a detailed 15-page analysis of one of the most celebrated perspective frescos, Masaccio's Holy Trinity, fails to reveal much mathematical depth ("we are dealing with a painting not a theorem"; p. 59). A notable exception is Piero della Francesca, who was a competent mathematician and wrote his time's most sophisticated treatise on perspective, earning him 50 pages here, but even he compromised mathematical precision in some of his paintings. The convergence of mathematics and art was at least as fruitful in the other direction. Desargues saw the importance of the idea of perspectivity in geometry, where it unifies the theory of conics and paves the way for essentially projective theorems such as "Desargues' Theorem".

This is basically a scholarly book, but at times one gets the impression that Field is more interested in showing off pretty pictures and telling amusing side stories rather than explaining the development of ideas on perspective in a clear and structured manner. One of the largest images is a full page reproduction of Titian's portrait of Ranuccio Farnese (p. 153), which has nothing to do with perspective except that the subject once had a book dedicated to him. Why not use the space for more relevant paintings instead? For example, Piero della Francesca's An Ideal Town would go beautifully with the discussion of his perspective treatise. Discussing Taylor's work on perspective, Field remarks that Taylor introduced the term "vanishing point" but then says "Taylor does not quite explain what is supposed to vanish at the vanishing point ... The readers he was addressing were presumably not expected to be so literal minded as to ask that question" (p. 229). I would say that more probably the readers were not expected to be so stupid as to fail to grasp this very simple concept by themselves; but that aside: if Field had wanted to explain ideas rather than to poke fun at people, she could simply have quoted from Taylor's 1719 edition, where, on page 15, he explains: "the further any object is off, upon any line, the smaller is its projection ... and when it comes to this point, its magnitude vanishes, because the original object is at an infinite distance. This is easily conceived by imagining a man to be going from you in a long walk, who appears to be smaller and smaller, the further he goes." (Incidentally, this would also have made clear that, to Taylor at least, a vanishing point is not the same as a centric point, as Field mistakenly implies.)

This is the ideal book for anybody who wants to really understand how painting lead to the birth of projective geometry. It's a sholarly book, and at times it requires some effort on your part, but you will be highly rewarded in the end!