Tartaglia’s Italian Euclid of 1543 is geometry in the narrow sense. But the big two books of 1543, Copernicus’ De revolutionibus and Vesalius’ De humani corporis fabrica are also geometry, if a slightly wider sense of the term is allowed. Though Copernicus writes on physics, he does not speak of forces, energies, masses or the like: there are only the appearances of the heavens from certain points of view. Though Vesalius is biology, there is little physiology, or mechanical analogy, or discussion of causes: the emphasis is on how parts of the body look from suitable points of view. But the three books share more than just pictures, and it is this extra element that is the focus of this article. Euclid’s Elements is not a picture book of shapes. The point of Euclid is to reason about the diagrams, and expose the necessary interrelations of the spatial parts. So it is with Copernicus and Vesalius. The text of Copernicus is an exercise in reasoning about which geometrical scheme will best fit the sequences of spatial points recorded in the astronomical tables. Vesalius uses the best of the discoveries of artists to make easy for the reader inference about how the systems of the body look in isolation, and in relation to one another. The difference between a Vesalian diagram and a photograph is exactly that the former allows one to work out structural facts which are almost invisible in the photograph. The plates also allow muscles in the partially dissected cadaver to be drawn with the natural tensions of the living body—obviously impossible with a photograph. An exclusively geometrical focus persisted in science in the work of Kepler, in both his ellipse theory and his Platonic solids theory of the planetsl (as well as in his optical theory). Galileo’s first success, the Sidereus nuncius, is, amongst other things, about inferring the shape of the moon’s surface from optical considerations of light and shadow.2 Galileo’s famous saying that the universe is written in the language of mathematics, which when quoted in 53 0...