The Geometry of Pierodella Francesca
The Side of an Icosahedron
An example of a typical problem is Problem 37 of Book II which asks, given a regular icosahedron with volume 400 brachiae, what is the side x? Piero finds
x = (806400 - V 597196800000)1/6
Perhaps the first thing to say is that this is not something that any artist or merchant has the remotest need to know. Strangely, Piero adheres to a convention which requires all factors to be brought inside the radical, often leading to very large integers and other awkwardness. The computation seems to be at the limit of what quattrocento arithmetic is capable of handling. For this, as for all the other surface areas, volumes, containing spheres, etc., Piero describes three-dimensional constructions to guide the computation. He does this with great patience and precision; the very few errors appear to be copyist errors in the arithmetic-transposition of digits, for example. One is tempted again to make comparisons with his paintings, which are painstaking in the extreme, even to the details in wall panels and ceilings which are too high and far away for any observer to be expected to see them. As Martin Kemp has said, for Piero " 'Getting it right' was an ethical imperative.  It seems like a good description of these computations as well, the first to treat the Platonic solids comprehensively.
Doubts about Arithmetic?
In addition to results of this kind there are oddities. Book III is an arithmetization of Euclid's Book XV (no longer recognized as genuinely belonging to the Elements), which deals with the regular polyhedra in relation to each other. That one can neatly inscribe a regular tetrahedron in a cube, for example, by taking alternate vertices, is Proposition 1 of Book XV, and problem 2 of Piero's Book III. Problem 1 of Book III, however, is to find the side of the regular octahedron inscribed in a regular tetrahedron of side 12, corresponding to Euclid XV Proposition 2. The situation is illustrated in Figure 3: the midpoints of the 6 sides of the tetrahedron are the vertices of the regular octahedron. If the tetrahedron has side 12, the octahedron has side 6. We would not expect Piero to spend many words on this. Amazingly, though, he gives it by far the most complicated solution he presents to any problem. He drops altitudes in the faces and through the interior, including one in a very asymmetric position. He notes non-obvious similar triangles, and at one point must recognize *108 *108 = *48 (his cumbersome algorithm for knowing when such simplification is possible is given in the Trattato). In the end he finds the side of the octahedron as *36 and he leaves it that way.
Is this some kind of sophisticated joke? We cannot know for sure, but I believe Piero was concerned whether arithmetic was really adequate to represent geometry. Such doubts, going back to the discovery of the irrationality of *2 were not really resolved until the construction of the real numbers. In Book III Problem 1 he seems deliberately to use every single arithmetic method in his entire repertoire in a roundabout solution to a problem whose correct solution is obvious, perhaps as a test of consistency. If this is what he was doing, arithmetic passed the test In his usual cool manner, he does not comment on what he has done. He never does it again.
Another oddity may be related, however. In Book IV Problem 17 Piero suddenly departs from the methods he has been following to find volumes of increasingly complicated solids. For a very complicated solid, like a statue of a human figure, he suggests lowering it into a carefully constructed rectangular vat of water and measuring the change in the water level in order to find its volume. This suggestion jars. It seems out of place just because it is practical and approximate, hardly mathematical at all, unlike any of the foregoing problems. I suggest that this observation may serve a more mathematical purpose than at first appears. Piero might have wondered how far this process of finding volumes could go. If a shape is so complicated that the utmost ingenuity could still not decompose it into shapes with computable volumes, would it still have a volume? What seems a merely practical method may really be a "thought experiment" to prove that even in the most complicated case arithmetic can still produce a number which represents the geometric quantity "volume." It goes to the relationship between geometry and arithmetic. In more modem language it suggests the existence of the integral, with the water doing the integrating.
A Proof by Computation
In Book III Problem 9, considering an octahedron symmetrically inscribed in a dodecahedron (Euclid XV, 9), Piero proves a new and purely geometrical result (i.e., one that Euclid could have discovered), but he proves it with a computation, perhaps the first time a proof of this kind was ever done (in Europe at least). The theorem is that a certain diameter of a regular dodecahedron is equal to the side plus the chord of the pentagonal face. The diameter in question joins the midpoint of a side to the midpoint of a side, and is the diameter of the containing sphere of the inscribed octahedron. His proof is to compute both quantities in case the dodecahedron has side 4, and show that they are the same number, namely 6 + V20 He must have found this number as the diameter of the containing sphere of the inscribed octahedron and then, because of his thorough familiarity with the arithmetic of these situations, noticed the numerical coincidence with the side plus the chord. The theorem is illustrated in Figure 4.
The Volume of a General Tetrahedron In Terms of Its Sides
An unexpected problem of a different kind occurs in Book II, Problem 11: find the altitude of a general tetrahedron given its sides. Piero's result here was surely new. His superb intuition for three dimensions guides the computation, and is worth seeing as an example of his method, which uses only Eq. (2) and the Pythagoras theorem. It is illustrated in Figure 5.
Construct altitudes of triangles, as usual, in this case the two altitudes to BC, which are DE and Al. Now take III parallel to DE and equal to it. Join DH. DIRE is a rectangle in the base plane of the tetrahedron, and AHI is in a plane perpendicular to the base plane, because it is perpendicular to the line BC. The altitude AG of AM is thus the altitude of the original tetrahedron. This implies the following algorithm for finding the altitude h:
This is Piero's result (which he gives as a numerical example). He does not give the volume of the tetrahedron, although that is the unspoken motivation.
It is truly unfortunate that the algebra of Piero's time was so primitive. His formula for the altitude together with Heron's formula for the area of the base implies a fascinating formula for the volume V of a general tetrahedron which manifestly has the symmetry of the tetrahedron, a three-dimensional analogue of Heron's formula, (8) which manifestly has the symmetry of the tetrahedron.
Piero's result was never lost. It was printed, without the proof, by Luca Pacioli [151 in his Summa Aritkmetica  in 1494, plagiarized from the Trattato d Abaco immediately after Piero's death, and again in Pacioli's De Divina Proportione [181 in 1509, plagiarized from the Libellus, where Piero had added the proof. It seems not to have been very well known, though. I believe the complete volume formula does not appear in print until the 19th century, when it assumes determinantal form. The first modem proof is by J.J. Sylvester in 1852 [191, who credits Staudt (1843) and Cayley (1841), and adds in a note at the bottom of his first page, "Query, Is not this expression for the volume of a pyramid in terms of its sides to be found in some previous writer? It can hardly have escaped inquiry."
The Cross Vault
Piero's most sophisticated geometrical result is probably in Libellus Book IV, Problems 10 and 11: the volume and surface area of a cross vault, the chamber formed by the intersection of two equal right circular cylinders whose axes intersect at right angles. This volume was also found by Archimedes, but Piero could not have known that, because it only came to light on a 10th-century palimpsest discovered by Heiberg in Constantinople in 1906 . The Archimedes manuscript contains no proof of the volume result. It is called On Method, and describes, in a letter to Eratosthenes, something perhaps more interesting, Archimedes's intuitive way of thinking of problems of this type. As Thomas Heath has said, "The method will be seen to be not integration, as certain geometrical proofs in the great treatises actually are, but a clever device for avoiding ... integration ... and making the solution depend, instead, upon another integration, the result of which is already known."  That is also a description of what Piero does.
In brief, Piero relates the cross vault and the sphere.(9)
To visualize it, think of the two cylinders forming the cross vault
to be lying horizontally, as they would be in an architectual realization.
Take the center of the roof of the vault to correspond to the north
pole of the sphere. Piero introduces a right circular cone inscribed
in the hemisphere with its base on the equator and vertex at the north
pole, and the analogous pyramid on the "equatorial" section
of the cross vault (which is square) having vertex at the center of
the roof, There is no difficulty in finding the volume of this pyramid
over the square. But then, Piero says, the upper half of the cross vault
must have exactly twice that volume, because the ratio of the volume
of the hemisphere to that of the cone is 2: 1. His proof of this non-obvious
Piero had already described the perspective rendering of a cross vault
in De Prospectiva Pingendi, and it is clear that his geometric intuition
for this situation is very good-- perhaps too good. As in many other
places, one wishes he had said more. His result for the surface area
of the inner concave surface of the vault is even more laconic. This
area has the same relation to the volume of the cross vault as the surface
area of a sphere has to its volume: just multiply the volume by 3/R
(where R is the radius of the cylinder in the case of the vault). This
seems sufficiently obvious to him that he does not even give an argument.
It is correct, of course.
The Archimedean Solids
It has long been recognized that Piero rediscovered 6 of the 13 Archimedean solids [91, . He did not think in terms of a complete classification, but simply obtained new semiregular polyhedra by truncating each regular polyhedron at the vertices. The reason he found 6 Archimede an at the solids instead of 5 is that there is more than one way to truncate, and in going from the Trattato to the Libellus he substituted one new one. He draws no attention to the construction itself, treating it as obvious, but turns immediately to the computation of the sides, surface areas, and volumes of these figures.
Piero's books are a mass of detail--detailed arithmetic, detailed instructions. In the case of De Prospectiva Pingendi, though, we have another medium, the paintings, to reveal what it is really about. We see that a simplistic reading would completely miss the point. With the mathematical treatises we are not so fortunate-there is no other medium. If we want to know the real meaning, we have to construct it from the treatises alone by getting behind the superficial details and discovering the mathematical thought. Beneath the surface, the thought is surprisingly deep. Piero was a real mathematician--one can say it without apology.
Piero worked ambitiously in mathematics, proved new theorems, attempted difficult problems, and even formulated and completed a coherent research program, the arithmetization of the later books of Euclid. In the context of the fifteenth century this is impressive. In fact it is at odds with conventional histories of mathematics, which state that the fifteenth century produced no original mathematics at all (in spite of Vasari's well-known characterization of Piero). What happened? Was Piero forgotten so soon after Vasari? The truth is stranger than that: he was forgotten even before Vasari.
In his Lives of the Painters Vasari is primarily concerned with art and artists, but the book's charm is enhanced by stories on many topics woven into the biographical sketches. In Piero's case, the story is of a lone mathematical researcher whose reputation has come to nothing because his work has been stolen by another and printed under his own name: Luca Pacioli. Vasari records not Piero's mathematical fame, as one might have thought, but rather his mathematical obscurity.
This obscurity is borne out by contemporary documents. In 1559, for example, after Vasari's first edition and before his second, both of which tell the Piero story, Nicolo Tartaglia published an encyclopedic mathematical work called Trattato Generale di Numeri et Misure . It cites Pacioli freely, and many other contemporary mathematicians, and is, on the whole, very gossipy, but it seems never to have heard of Piero. Tartaglia, is very impressed by some of Piero's results, though. For example, he describes the construction of the altitude of a general tetrahedron twice, in two different contexts, giving the entire argument both times, which he calls "ingenious" (23]. The numbers and symbols he uses are the same as those of Piero, as printed by Pacioli, so there is little doubt where Tartaglia learned it, but he cites no one. A little further on he discusses what was then called the fifteenth book of Euclid, on inscriptions of one regular polyhedron in another. Although many believe the twelve inscriptions given there are the only ones possible, he says, he has found two more ("due altre ne ho ritrovate"), but does not say that he found the first one, the icosahedron inscribed in the cube, which is yet another nice construction due to Piero (Libellus Book III Problem 4), in Pacioli's De Divina Proportione. He should have cited Pacioli, but the irascible Tartaglia never concealed his contempt for Pacioli, whose mediocrity was clear to him, and it appears he could not bring himself to give him credit. Vasari, with reference to Pacioli's plagiarisms from Piero, says Pacioli "covered his ass's hide with the glorious skin of a lion." Tartaglia undoubtedly knew that characterization: polemics on mathematical plagiarism were right in his line. If he saw that the lion's skin didn't fit Pacioli, though, he had no certain evidence that the lion was Piero either, whatever he might have suspected. After all, Pacioli plagiarized from others as well: Tartaglia specifically points out a possible plagiarism (although a very trivial one) from Piero Borgi da Venetia [24). Tartaglia was far readier to cite others for their errors than for their accomplishments, and in these cases he cited no one. In effect, Piero had dropped out of the mathematical record.
Far from being the expression of a widely held opinion, Vasari's allegation that Piero had been defrauded of his rightful mathematical reputation was completely unsubstantiated and widely doubted for the next 350 years. Naturally it was much debated, but the question was always whether an injustice had been done, not to Piero, but to Pacioli, and the usual verdict was that Pacioli had been maligned [251. Pacioli's brother Franciscans were inclined to defend him. Vasari had said Piero's writings could still be found in Borgo San Sepolcro, or (a few pages later) in the ducal library at Urbino, which makes the story sound plausible, but as centuries passed, and no one produced these books, it seemed to suggest the opposite: that they had never existed. Vasari's inconsistency in saying where to look was derided, and his many other errors in matters of fact were called to witness how unreliable he was. What did an artist like Vasari know about mathematics anyway? Someone in Piero's family must have planted this exaggerated story, which the credulous Vasari accepted. Defenders of Pacioli pointed out that Pacioli, so far from being an enemy, lauds Piero highly, calling him Monarca della Pittura, an epithet which has stuck. (In retrospect Pacioli's praise looks carefully calculated to avoid mentioning Piero's mathematics.) As late as 1911 the Encyclopedia Britannica dismisses Vasari's allegation on grounds like these.
By this time, however, the Libellus had already been found, and it confirmed Vasari in every essential respect. It was found by G. Pitarelli in 1903 in the Vatican Library. Any doubt that it was the work of Piero vanished when the marginal corrections were found by G. Mancini to be in Piero's own hand. It is identical in content with a section of Pacioli's De Divina Proportione. Piero's Trattato d'Abaco came to light soon after, and the geometrical section was found to be incorporated into Pacioli's Summa Arithmetica. Despite the vague consensus that has always existed that Piero is "mathematical," it was only with the rediscovery of these books that it became possible to think of Piero as a mathematician who had actually done something. Thus it is no wonder that histories of mathematics written before 1920 do not mention him.
The reassessment of Piero in light of these materials is still going on. Although he usually rates mention now, in standard histories of mathematics Piero is still a footnote to Pacioli. More space is given, as a rule, to explaining that Leonardo da Vinci did no mathematics than is given to anything that Piero did. It may seem that no reassessment is necessary of the period generally, since these works were never lost, only mislabeled, but this is to underestimate the baleful influence of Pacioli, who is considered emblematic of the fifteenth century. There is a quality to his work which makes it difficult to believe it could contain anything of merit. When Piero's work is segregated from it and correctly identified as the work of a different author, it takes on an integrity and intensity which appears altogether different.
Most of the work on these materials has been done by art historians, but their opinions are also still evolving. The only book-length study of Piero's mathematical treatises maintains that the geometry of the Libellus derives from his art and intends to be useful to artists, a view which seriously misunderstands what is going on mathematically, and which is not representative of current thinking in the field. More recently there has been exploration of the subtler interplay of Piero's art and his mathematics [131. This endeavor raises truly interesting questions of aesthetics and mathematics, in the context of a person and a period of great fascination.
This work was supported by NEH-NSF award EW-20327 for "Science and Humanities: Integrating Undergraduate Education," under the title "Mathematics Across the Curriculum." Thanks also to the Mount Holyoke College Archives and Special Collections.
 M.A. Lavin, "The Piero Project," in Piero della Francesca and His Legacy, ed. by M.A. Lavin, University Press of New England, Hanover, NH (1995), 315-323.
 G. Vasari, Le Opere, ed. G. Milanesi, vol. 2, Florence (1878), 490.
 Piero della Francesca, De Prospectiva Pingendi, ed. G. Nicco
 Piero della Francesca, Trattato d'Abaco, ed. G. Arrighi, Pisa (1970).
 Piero della Francesca, L'opera "De corporibus regularibus'' di Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli, ed. G. Mancini, Rome, (1916).
 M. Clagett, Archimedes in the Middle Ages, University of Wisconsin Press, 1971.
 T.L. Heath, The Thirteen Books of Euclid's Elements, Cambridge University Press (1908), 97.
 T.L. Heath, The Works of Archimedes, Cambridge University Press (1897), xxix.
 M. Clagett, op. cit. vol 3, pp. 383-415.
 P. Grendler, "What Piero Learned in School: Fifteenth-Century Vernacular Education," in Piero della Francesca and His Legacy, ed. M.A. Lavin, University Press of New England (1995), 161-176.
 Leon Battista Alberti, On Painting, ed. Martin Kemp, trans. Cecil
 M. Kemp, "Piero and the Idiots: The Early Fortuna of His Theories of Perspective," in Piero della Francesca and His Legacy, ed. M.A. Lavin, University Press of New England (1995), 199-212.
 J.V. Field, "A Mathematician's Art," in Piero della Francesca and His Legacy, ed. M.A. Lavin, University Press of New England (1995), 177-198.
 J. Elkins, "Piero della Francesca and the Renaissance Proof of Linear Perspective," Art Bulletin 69 (1987), 220-230.
 M.D. Davis, Piero della Francesca's Mathematical Treatises, Longo Editore, Ravenna (1971).
 S.A. Jayawardene, 'The Trattato d'abaco' of Piero della Francesca," in Cultural Aspects of the Italian Renaissance, Essays in Honour of Paul Oskar Kristeller, ed C. Clough Manchester (1976).
 L. Pacioli, Summa Arithmetica, Venice (1494) Book 11, fol. 725
 L. Pacioli, De Divina Proportione, Venice (1509).
 J.J. Sylvester, "On Staudt's Theorems Concerning the Contents of Polygons and Polyhedrons, with a Note on a New and Resembling Class of Theorerns, 'Philosopftal Magazine IV (1852), 335-345.
 T.L. Heath, The Method of Archimedes, Recently Discovered by Heiberg, Cambridge University Press (19 12).
 P.R. Cromwell, "Kepler's Work on Polyhedra", The Mathematical Inteffigencer Vol. 17, No. 3, New York (1995), 23-33.
 Nicolo Tartaglia, Trattato Generale di Numeri e Misure, Venice
 Nicolo Tartaglia, op. cit., Part IV Book 2, and Part V Book 2.
Note added in proof: Proceedings of a 1992 conference held in Arezzo and Sansepolcro appeared after this work was done. Piero della Francesca tra arte e scienza, ed. by Marisa Dalai Emiliani and Valter Curzi, Venice, 1996. Cecil Grayson reports on a project to bring out new editions of all of Piero's treatises; J. V. Meld's analysis of Piero's perspective square construction is virtually identical to mine Another recent source is the World Wide Web: search on "de quinque corporibus regularibus" to find a page from Libellus showing Piero's construction of the icosahedron in the cube.