The Geometry of Piero della Francesca
Vasari, writing the Lives of the Painters some 60 years after Piero's death, calls him "the greatest geometer of his time, or perhaps of any time" , and claims that he had written many books ("molti scritti"). Indeed, we do know of three books authored by Piero, rediscovered relatively recently. All were manuscript books, but all are now available in modem printed editions , ,  with introductions by their editors. They tend to confir-m Vasari, at least in suggesting that Piero was the greatest geometer of his time.
Of course one does not expect very much from Piero's time. Marshall Clagett, in Archimedes in the Middle Ages , documents in detail the very imperfect transmission of classical knowledge that was still the principal occupation of mathematicians as late as 1500. Piero's time was still the age of the manuscript. The first printed Euclid did not appear until 1482 , when Piero was past 60, and the first printed Archimedes was not until 1544  Before that the paucity of manuscript books makes it possible sometimes to know not only which edition was the source of a writer's knowledge, but exactly which volume. It is nearly certain, for example, that Piero studied Archimedes in the volume in the library at Urbino, now in the Vatican Library .
In addition to classical mathematics Piero knew another kind of mathematics, commercial arithmetic, taught in the so-called abacus schools . By the indirect evidence of his own writings, Piero must have attended such a school, because his books resemble, in form, the abacus school texts: they are, like those texts, long series of worked examples, not really meant to be read, but to be worked through. (The texts for the abacus schools derive from the works of Leonardo Pisano, called Fibonacci, who introduced computation with Hindu-Arabic numerals into Europe around 1200 after himself learning the method in North Africa.) Piero was very proficient in arithmetic and very pedantic about it, another hint that he had attended an abacus school. However, the abacus school and the occasional look at a classical mathematical work in manuscript copy hardly form a promising basis for doing original mathematics, especially at a time when original mathematics was hardly thought of. This is what makes Piero's books so surprising.
Piero was a painter, and the great revolution in painting in the 1400's was the introduction of linear perspective. The basic idea was clear, as stated by Leon Battista Alberti : light rays travel in straight lines from points in the observed scene to the eye, forming a kind of pyramid with the eye as vertex. The painting should represent a section of that pyramid by a plane. But this conception by itself does not immediately tell the painter how to paint.
The first really practical treatise on perspective painting was Piero's De Pmspectiva Pingendi . The series of perspective problems posed and solved builds from the simple to the complex in a very methodical way. In Book I, after some elementary constructions to introduce the idea of the apparent size of an object being actually its angle subtended at the eye, and referring to Euclid's Elements Books I and VI, and Euclid's Optics, he turns, in Proposition 13, to the representation of a square lying flat on the ground in front of the viewer. What should the artist actually draw? After this, objects are constructed in the square (tilings, for example, to represent a tiled floor), and corresponding objects are constructed in perspective; in Book II prisms are erected over these planar objects, to represent houses, columns, etc.; but the basis of the method is the original square, from which everything else follows.
Piero's exposition has been much criticized by his successors  and by art historians , . What no one seems to say is that his construction is remarkably simple, efficient, and original.
Piero's Basic Construction in Perspective
A horizontal square with side BC is to be viewed from point A, which is above the ground plane and in front of the square, over the point D. The construction is shown in Figure 1. Although the square is supposed to be horizontal, it is shown, rather surprisingly, as if it had been raised up and were standing vertically, as the square BCGF. The construction lines AC and AG cut the vertical side BF in points E and H, respectively. These points give the crucial dimensions. BE, subtending the same angle at A as the horizontal side BC, represents the height occupied by the square in the drawing.
EH, subtending the same angle at A as the far side of the square, CG, represents the length of that side of the square in the drawing. Now, Piero says, draw parallels to BC through A and E, and locate a point A' on the first of these to represent the viewer's position horizontally with respect to the edge of the square BC. Draw A'B and A'C, cutting the parallel through E at D' and E'. The construction is done! The square in perspective is BCE'D'.(1) It is clear that BCE'D' occupies the height BE in the drawing, but why is E'D' = EH?
Theorem: E'D' = EH.
Piero gives the following proof, but leaves it partly to the reader to notice which similar triangles justify each equality
Thus, Piero says, E'D' and EH are either similar or equal: but they are equal, because BC = CG
This theorem is the first new European theorem in geometry after Fibonacci. It is a significant one, arguably the beginning of projective geometry. And it is beautiful, as the careful reader will surely agree. The result is not obvious, coming as it does through a series of similarity relations which move cleverly through the diagram. The way the perspective square suddenly appears, with a 90 degree twist in viewpoint, is unexpected and delightful. Even with the proof before us, are we sure it is correct? Can we really determine EH using a vertical side and then apply it to a horizontal side? Yes, but the reason is subtle: we see the painting foreshortened too. Because the result depends entirely on similarity, a powerful projective invariance is at work. Piero discusses none of this, but I have no doubt he thought of it. His laconic proof is aloof and cool: he could say more, but he stops. He has brought the reader to an intellectual result, but it seems more like the threshold of a deeper mystery. This is just the effect his paintings have on many viewers.
The Mathematical Treatises
Piero's next book, known now as the Trattato d'Abaco , claims to be an abacus text, or rather to present methods useful for merchants. In fact, though, it is extremely ambitious. Some of it is problematic, notably the algebraic part, which unsuccessfully tackles polynomial equations of 3rd, 4th, and 5th degree. Even this has interest, just as an attempt
, , and it certainly belies the modest language of the introduction.
The geometric section, on the other hand, is both ambitious and successful.
The problems here were reworked by Piero in his old age, together with
new ones, as his third book, dedicated to the young Duke Guidobaldo da
Montefeltro at Urbino (2) to be shelved together with
Prospectiva Pingendi in the great library, under the title Libellus De Quinque Corporibus Regularibus, also called, for short, the Libellus [ 15]. Piero says in his dedication that he intends the Libellus to be his memorial, (3) a notion that will strike some as incomprehensible. The Libellus survives in only this one manuscript copy
Its program is to complete the arithmetization of geometry which was begun in the work of Flbonacci, as Piero implicitly says in the dedication. (4) To understand his claim one must know that the geometry of the abacus schools was very arithmetical. It dealt to a great extent with finding lengths, areas, and volumes. Geometrical figures invariably came with lengths given numerically, and the problems posed had numerical answers. The lack of a decent notation prevented these relationships from being expressed algebraically, but it was understood that the rules for doing the arithmetic of a given problem, which would be given as a numerical example, fell into patterns which amounted to algebraic formulae. In what follows I will sometimes represent Piero's statements algebraically, for this is their clear meaning, and lets us see that he is working toward an analytic geometry.
For Piero, the natural thing to do with a triangle with sides a, b, and c, is to find an altitude h (see Fig. 2). He would do this in two steps, first finding the length x which this altitude cuts from the side c (say), and then finding the altitude itself from the Pythagoras theorem. The length x is given by
recognizable as a version of the Law of Cosines stated entirely in terms of lengths, or, after some algebra, as an equivalent of Heron's formula He got this from Fibonacci.
Book 1, Problem 1 of the Libellus says you do this with any triangle. It amounts to saying you should routinely take orthogonal projections of line segments onto other lines (as one does in using Cartesian coordinates).
From this starting point one can translate Euclidean constructions into arithmetic, going, for example, from the construction of a regular pentagon to a formula for its area in terms of its side. The plane geometry of Euclid is transformed this way in Fibonacci, and the introductory Book I of Libellus reviews this. What is not in Fibonacci is some of the three-dimensional geometry in the later books of Euclid, notably the regular polyhedra of Book XIII. This was the gap Piero set out to fill in Book 11 of Libellus. He is never grandiose about it-indeed, in the exaggeratedly self-deprecating language of dedications he calls his work .poor, empty fruit"5 and himself "a rude and clumsy fanner bringing rustic apples to an opulent and splendid table."6 But he also says "in respect to its novelty, at any rate, it cannot displease."' This is the closest Piero ever comes to saying he has done something new.