iiewpoint

The Geometry of Paradise MARK A . PETERSON

The Viewpoint column offers

mathematicians the opportunity to write about any issue o f interest to the international mathematical

community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts

responsibility for them. Viewpoint should be submitted to the editor-inchief, Chandler Davis.

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athematics before 1700 presents a peculiar picture. It is difficult to avoid the impression that there was a g o l d e n age of mathematics in the Hellenistic p e r i o d of Euclid a n d Archimedes, that a new mathematical golden age b e g a n in the 17th century, the golden age in which w e are n o w living, and that in the long p e r i o d in between, mathematics for s o m e reason languished. Medieval Arab and European cultures inherited classical mathematics, and g o o d mathematical minds were u n d o u b t e d l y at work, but the circumstances put t h e m someh o w at a disadvantage. Lucio Russo's remarkable b o o k The Forgotten Revolution [1] argues that the d a m a g e to mathematics as a collective enterprise was d o n e already in the Rom a n period. Thus w h a t later civilizations inherited was already s o m e h o w m a i m e d , cut off from the p r o b l e m s that gave rise to it. Medieval cultures were in the peculiar condition of b e i n g unmathematical cultures in p o s s e s s i o n of sophisticated mathematics. T h e y poss e s s e d it in the sense of having the b o o k s , studying them and translating them, a n d even doing s o m e mathematics, but they had no clear indication w h e r e this rich subject h a d come from or w h a t it w o u l d b e g o o d for. T h e y d i d not know, in o u r terms at least, w h a t it was. The sto W is c o m p l i c a t e d b y exceptions to this s w e e p i n g characterization. Certain constructions called geometria practica, useful in building a n d commerce, h a d a continuous existence right t h r o u g h the period, changing hardly at all, as if they w e r e already a d e q u a t e to their problems, with no n e e d for innovation. Methods a n d notations for doing arithmetic certainly changed, but ancient civilizations had already b e e n g o o d at arithmetic, so the m a i n innovation here was p e r h a p s the diffusion of arithmetical c o m p e t e n c e to a large commercial and professional class. A b o v e all, algebra d e v e l o p e d , with algorithms for the solution of p o l y n o m i a l

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equations a n d systems of equations, in r e s p o n s e to p r o b l e m s that arose first in commerce, a n d then t o o k on an abstract life of their own. This d e v e l o p ment looks like normal mathematics, a n d it is n o t e w o r t h y that algebra d i d not have a n y essential connection to the G r e e k mathematics of the classical past. Geometry, on the other hand, was moribund. It is as if its high sophistication, precisely its roots in the classical past, s o m e h o w d i s a d v a n t a g e d it, and m a d e it almost a d e a d subject, in spite of its high status. I have stated this view of m e d i e v a l g e o m e t r y m o r e starkly than I could justify. It is not given as the topic of this article (I a m not so ambitious as to undertake a p r o p e r evaluation of it), but as a b a c k g r o u n d against w h i c h one exa m p l e stands out dramatically: Dante's g e o m e t r y in The Divine Comedy, specifically in Paradise. Dante seems to have an unusual mathematical gift, but in an u n m a t h e m a t i c a l age this gift finds a peculiar outlet. What Dante d o e s with mathematics m a y bear out the previous suggestion that late medieval E u r o p e a n culture p o s s e s s e d mathematics, but without k n o w i n g what it was. W h e r e w e e x p e c t m a t h e m a t i c s to find application in practical, earthly p r o b l e m s , m e d i e v a l mathematics, apart from geometria practica a n d c o m m e r c i a l arithmetic, d i d not c o m e with that exp e c t a t i o n at all. If anything, its associations w e r e with a s t r o n o m y a n d celestial things [2]. It must have s e e m e d natural to D a n t e to find, as w e will see, a p p l i c a t i o n s of mathematics in theology! A n d p e r h a p s it is w e w h o are not sufficiently imaginative. Mathematics, as the ultimate in abstraction, d o e s not c o m e with any p r e s c r i p t i o n for w h a t it might mean, s o - - t h e o l o g y , w h y not?

Dante's Universe is S3 It has b e e n noticed by m a n y readers that Dante's universe is topologically S 3 [3, 4, 5, 6]. Still, the occurrence of a compact 3-manifold without b o u n d a r y

in a late m e d i e v a l p o e m is so unexpected, and the suggestion s e e m s so implausible, that it might b e g o o d to go o v e r the evidence here, lines of the p o e m that cumulatively leave little doubt. Dante invented c o n c e p t s that w e r e reinvented long afterward. P a r a d i s e represents the ascent of Dante and Beatrice through the spheres of the Aristotelian heaven, concentric with the Earth, beginning with the s p h e r e of the Moon, then Mercury, Venus, the Sun, etc. [7]. In Canto 27, Dante and Beatrice m a k e the ascent from the s p h e r e of the fixed stars to the Primum Mobile, the outermost s p h e r e of the universe, the o n e that turns all the others. B e y o n d that is the Empyrean, which has no conventional geometric description, but w h i c h Dante must n o w describe. The exposition begins as early as the e n d o f Canto 22, in lines that are add r e s s e d to Dante, but are also preparing the reader, "Tu se' si p r e s s o a l'ultima salute," cominci6 Beatrice, "che tu d e i aver le luci tue chiare e acute; e per6, prima che tu pit? t'inlei, rimira in giO, e vedi q u a n t o m o n d o sotto li piedi girl esser ti fei . . . " Beatrice began: "Before long thou wilt raise Thine eyes and the S u p r e m e G o o d thou wilt see; H e n c e thou must s h a r p e n a n d make clear thy gaze, Before thou nearer to that Presence be, Cast thy look d o w n w a r d a n d consider there

H o w vast a w o r l d I have set u n d e r thee... " [8] Par. 22:124-129 Dante d o e s look d o w n , seeing "this little threshing floor" the Earth below, s u r r o u n d e d b y the heavenly spheres through w h i c h he has ascended, e tutti e sette mi si dimostraro quanto s o n grandi e q u a n t o son veloci e c o m e s o n o in distante riparo. All seven being displayed, I c o u l d admire H o w vast they are, h o w swiftly they are spun, And h o w r e m o t e they dwell . . . Par. 22:148-150 This calling attention in Canto 22 to the sizes a n d velocities of the h e a v e n l y spheres before ascending to the s p h e r e of the fixed stars is a kind of foreshadowing, to be recalled in Canto 28. In Canto 27 Beatrice asks Dante once again to l o o k down, admiring the spheres b e l o w , as they ascend from the sphere of the fixed stars to the Primum Mobile, the ninth sphere. Dante is careful to say of the Primum Mobile Le parti sue vivissime ed eccelse si uniforme son, c h T n o n so dire qual Beatrice per loco mi scelse. This heaven, the liveliest and loftiest, So equal is, which part I cannot say My Lady for m y sojourn there d e e m e d best. Par. 27:100-102

That is, this s p h e r e has full rotational symmetry, and the ascent they have chosen is in no w a y distinguished from any o t h e r w a y t h e y might have come. That SO(3) s y m m e t r y is a crucial ingredient of Dante's image, a n d he d o e s not w a n t it to b e missed. At the b e g i n n i n g of Canto 28, Dante sees reflected in Beatrice's eyes a bright Point. Turning, he sees the Point in reality, s u r r o u n d e d b y nine whirling circles, m o v i n g the m o r e slowly as t h e y are larger. The seventh of these is already larger than the rainbow's circle. It e m e r g e s in the next lines, w h e r e this w h o l e structure is discussed, that these angelic circles are a kind of mirror image of the h e a v e n l y circles below. Mirror symmetry is already suggested in the w a y that Dante first sees them, as a reflection. The Point, representing God, the center o f the angelic circles, is the mirror i m a g e of the center of the material universe, the center of the Earth d o w n below, w h e r e Satan is fixed in ice. La d o n n a mia, che mi v e d e a in cura forte sospeso, disse: "Da quel p u n t o d e p e n d e il cielo e tutta la natura." Observing w o n d e r in m y every feature, My Lady told m e w h a t I set below: "From this Point hang the heavens a n d all nature." Par. 28:40-42 The w o r d " d e p e n d e " u s e d in this w a y expresses essentially w h a t w e m e a n w h e n w e say S 3 is s 2, the s u s p e n s i o n of the 2-sphere. Dante even describes himself as "sospeso," s u s p e n d e d , per-

MARK A. PETERSON is Professor of Physics and Mathematics on the Alumnae Foundation at Mount Holyoke College. His interest in Renaissance mathematics goes back to h~s graduate student days in physics at Stanford. He has just completed a book from which the material of this article is taken, Gahleo'sMuse: The Renaissance Re-invention of Science by the Arts. He also works on geometrical methods for continuum mechanics. Department of Mathematics and Statistics Mount Holyoke College South Hadley, MA 01075-6420 USA e-mail: [email protected]

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haps suggesting the precariousness of the soul b e t w e e n t h e s e theological poles, in addition to the more literal meaning of being "in suspense," although w h a t is literal a n d what is figurative here is hard to pin down. In any case, the picture is the suspension construction, with a sense of "higher" and "lower" i m p o s e d o n it. Dante is b o t h e r e d b y something that seems w r o n g to him. The heavenly spheres turn faster the larger they are, but the angelic o n e s turn slower the larger they are. We h a v e already b e e n alerted to these velocities, but n o w the matter is brought u p explicitly. As Dante puts it, he w o u l d like to k n o w w h y ' T e s s e m p l o e l'essemplare non v a n n o d ' u n modo," that is, w h y the pattern and the c o p y d o n ' t move in the same way. Beatrice laughs "Se li tuoi diti n o n s o n o a tal n o d o sufficienti, non e maraviglia: tanto, p e r non tentare, ~ fatto sodo!" "There's naught to marvel at, if to untie This tangled knot thy fingers are unfit, So tight 'tis g r o w n for lack of will to try. Par. 28:58-60 This is an assertion that w e are looking at n e w mathematics! Beatrice explains that the s p h e r e s are o r d e r e d b y velocity, a n d that t h e y turn faster the higher they are (in the sense of higher knowledge, higher love, that is, proximity to God), not the larger they are, another nice w a y to think of S 3. The Primum Mobile m a y b e the largest, but it is only the e q u a t o r o f the universe as a whole, being only m i d w a y in the ordering. The smallest circles, closest to the Point, turn really fast, as Beatrice points out. It is clear that Dante invents the notion of manifold here, in building the universe out of two balls, glued along their c o m m o n b o u n d a r y . That is the meaning of ' T e s s e m p l o e l'essemplare." The reader may b e b o t h e r e d by the frequent use of the w o r d "circle" where the right w o r d w o u l d seem to be "sphere," but Dante explicitly says, in another place that w e will see later, that he uses the w o r d circle for both, and in general for anything round. In any case he makes clear m o r e than once that

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the circles about the Point are actually spheres. For example, "l'essemplo" is the spheres of the conventional universe, and it is the m o d e l for "l'essemplate," the angelic "circles." Also, the hom o g e n e o u s uniformity of the Primum Mobile, in which Dante can't distinguish any location, implies that the angelic "circles" are not just off to o n e side of the Primum Mobile but surround it. A skeptic might reluctantly agree that this looks very m u c h like S 3, described in several different ways, in fact, b u t that it is simply impossible: Dante could not so easily h a v e overc o m e the normal t e n d e n c y of the hum a n m i n d to regard space as infinite ( a n d Euclidean). I w o n d e r , though. It might have b e e n easier for D a n t e to invent S 3 than for, say, I m m a n u e l Kant. Aristotle's universe, which w a s also Dante's universe, was explicitly declared to be finite, although in Aristotle's version it w e n t only u p to the Prim u m Mobile a n d no resolution was offered for the puzzle of w h a t lay beyond. From the beginning, therefore, Dante was describing a finite universe. G o i n g on, it a p p e a r s that he d i d not have the Euclidean prejudice in favor of infinite straight lines. This objection might not even have o c c u r r e d to him. His preferred geometrical object was the circle, and a space built o u t of circles might well b e S 3. The internal evidence of Dante's writing suggests that although he k n e w Euclid's geometry, and m a d e casual, e a s y use of it, he d o e s not necessarily r e g a r d it as a m o d e l for space, especially globally. Rather he regards it as a b r a n c h of p h i l o s o p h y w h o s e p r o p o sitions are true with peculiar certainty (as Aristotle also regards geometry). There are two Euclidean t h e o r e m s in Paradise, but neither of t h e m carries a m e a n i n g that has anything to d o with space. The t h e o r e m in Par. 13:101-102 is Elements III.31, a triangle inscribed in a semicircle is a right triangle. This theo r e m occurs as just o n e o f several l e a r n e d propositions in a list, the other p r o p o s i t i o n s not being from geometry. The list consists, tellingly p e r h a p s , of things that King S o l o m o n d i d not ask to k n o w w h e n he was g r a n t e d wisdom. This rather b a c k h a n d e d reference might even be read as slightly dismissive of geometry.

In Par. 17:13-18 Dante says, addressing Beatrice, that she sees the future as clearly as m e n see that no triangle can h a v e two obtuse angles. Neither this occurrence nor the previous one uses Euclid to describe s o m e thing in space. W h e n Dante asks a b o u t his future, he d o e s not m e a n triangles. Rather, these t h e o r e m s are cited as examples of things that are k n o w n with certainty to b e true. In short, g e o m e t r y seems to b e less geometrical for Dante than it is for us.

DaRle's Geometer Dante discusses the seven liberal arts of the trivium a n d quadrivium, g e o m etry b e i n g o n e of them, in his earlier unfinished b o o k of classical learning, The Banquet. From The Banquet II.13, G e o m e t r y moves b e t w e e n t w o things antithetical to it, namely the point a n d the c i r c l e - - a n d I m e a n "circle" in the b r o a d sense of anything round, w h e t h e r a solid b o d y or a surface; for, as Euclid says, the point is its beginning, a n d as h e says, the circle is its most perfect figure, w h i c h must therefore be conceived as its end,Therefore G e o m e try m o v e s b e t w e e n the point a n d the circle as b e t w e e n its beginning a n d end, a n d these two are antithetical to its certainty; for the point c a n n o t b e m e a s u r e d b e c a u s e of its indivisibility, a n d it is impossible to square the circle perfectly b e c a u s e of its arc, and so it cannot be m e a s u r e d exactly. G e o m e t r y is furthermore most white insofar as it is without taint of error a n d most certain both in itself a n d in its handmaid, which is called Perspective. [9] This p a s s a g e cites Euclid, but the sentiments attributed to Euclid are virtually unrecognizable. The implied m e a n i n g of g e o m e t r y in this passage is precise m e a s u r e m e n t , a n d the point a n d the circle are "antithetical" to the certainty of g e o m e t r y b e c a u s e they can't b e m e a sured, not at all a Euclidean idea. Nor d o e s Euclid call the circle "most perfect." The enthusiasm for the circle exp r e s s e d h e r e must be Dante's own. The p r o b l e m of measuring the circle, given such p r o m i n e n c e here, is of course not a p r o b l e m o'f Euclid. O n e is left with the impression, con-

sistent with the two t h e o r e m s in Paradise, that although Dante k n o w s and respects Euclid, he d o e s not find him very interesting. The p a s s a g e in The Banquet summarizing g e o m e t r y essentially ignores Euclid, even as it cites him. The certainty of g e o m e t r y seems less interesting to Dante than its opposite, the antithetical point a n d circle, for he devotes most of this little statem e n t to them. The unmeasurability of the circle definitely interests him.

Dante and Archimedes Dante returned to the p r o b l e m of m e a s u r i n g the circle in o n e of the most astonishing passages he ever wrote, the final image of Paradiso. He is looking at an image of the Trinity, as three circles, a n d staring especially at the seco n d of these, representing the Son: d e n t r o da se, del suo c o l o r e stesso mi parve pinta de la nostra effige: p e r che '1 mio viso in lei tutto era messo. Qual e '1 geometra che tutto s'affige p e r misurar lo cerchio, e n o n ritrova, p e n s a n d o , quel principio o n d ' elli indige, tal era io a quella vista nova: v e d e r voleva come si c o n v e n n e l'imago al cerchio e c o m e vi s'indova; m a non eran da ci6 le p r o p r i e penne: se n o n c h e l a mia mente fu percossa da un fulgore in che sua voglia venne. A l'alta fantasia qui m a n c 6 possa; m a gift volgeva il mio disio e '1 velle, si c o m e rota ch'igualmente e mossa, l ' a m o r che m o v e il sole e l'altre stelle. Par. 33:130-145 Within itself, of its o w n coloration I saw it p a i n t e d with o u r o w n hum a n form: So that I gave it all m y attention. Like the geometer, w h o exerts himself completely To m e a s u r e the circle, a n d doesn't succeed, Thinking w h a t principle he needs for it, Just so was I, at this n e w sight. I w a n t e d to see h o w the h u m a n image

Conforms itself to the circle, a n d finds its place there; But there were not the m e a n s for that, Except that my m i n d was struck By a flash of lightning, b y w h i c h its will was accomplished. Here strength for the high imagining failed me, But a l r e a d y the love that m o v e s the Sun a n d the other stars Turned m y desire a n d m y will Like a w h e e l that is turned evenly. I have preferred the unpoetic translation here in o r d e r to be as literal as possible, for the p u r p o s e of a close reading. Notes to this passage always p o i n t out the futility of trying to square the circle. T h e y suggest that squaring the circle functions here as a m e t a p h o r for the impossibility of understanding the mystery of salvation b y Christ's crucifixion. In the last century or so, notes on this p a s s a g e even cite Lindemann's 1882 p r o o f that ~r is transcendental! That result cannot b e relevant to Dante's intention in this image, but w e already have Dante's o w n o p i n i o n in The Banquet that the circle cannot b e squared. The m e s s a g e of futility might a p p e a r to be u n a v o i d a b l e , in view of the general a g r e e m e n t that w h a t the g e o m e t e r is trying to d o is impossible, but there are subtle p r o b l e m s with this reading. In the first place, it just d o e s n ' t s o u n d like Dante to give up. W h y w o u l d he c o m e to the e n d of his amazing epic p o e m a n d t h e n admit d e f e a t b y introd u c i n g an i m p o s s i b l e p r o b l e m in t h e very last lines? It isn't e v e n so clear that he is d e f e a t e d . That flash o f lightning might indicate the o p p o s i t e . Typical n o t e s s u g g e s t that the flash is a m e t a p h o r for the a c c e p t a n c e of G o d ' s grace, as if the struggle with g e o m e try w e r e over, b u t the g e o m e t r i c a l m e t a p h o r s e e m s to c o n t i n u e e v e n after the l i g h t n i n g flash, in the i m a g e of the turning w h e e l . Scholarship has ret u r n e d to this enigmatic p a s s a g e a g a i n a n d again, w i t h o u t a w h o l l y satisfactory conclusion. Dante n e v e r mentions the n a m e Archimedes, but Archimedes's little treatise On the Measure of the Circle had b e e n translated several times before Dante wrote, from Arabic b y b o t h

Plato of Tivoli a n d Gerard of Cremona, and from G r e e k a r o u n d the time of Dante's birth b y William Moerbeke. Given Dante's interest in the question of measuring the circle, he w o u l d naturally have sought out this treatise a n d studied it. It s e e m s impossible that he w o u l d not have. It is short and easily copied, especially its Proposition I, which is the relevant one. According to Marshall Clagett [10], versions of the Gerard translation were widely circulated, a n d we will notice evidence below that this is the version that Dante knew. I believe that Dante used the Archimedes p r o o f as an e x t e n d e d m e t a p h o r in the last lines of Paradise for the drama of salvation, as I will n o w explain. It will follow that Dante und e r s t o o d the p r o o f perfectly, and u s e d it with precision. Let m e recall the familiar magisterial argument of A r c h i m e d e s in On the Measure of the Circle. Archimedes shows b y a m e t h o d of exhaustion that the circle of radius R and circumference C is, in area, neither larger nor smaller than 89 Thus it is exactly 89 For assume that the circle is larger than 89 We construct a s e q u e n c e of regular polygons inside the circle, b e g i n n i n g with the inscribed square, a n d doubling the n u m b e r of sides at e a c h step, as in Figure 1. More than half the remaining area outside the p o l y g o n is incorporated at each step into the next polygon in the sequence. Hence, by our ass u m p t i o n that the circle is larger than 89 s o m e p o l y g o n in the s e q u e n c e will also have area larger than 89 But this is impossible, b e c a u s e the area of the p o l y g o n is the sum of the triangular w e d g e s in Figure 2, n a m e l y 89 in the notation defined there, a n d h < R, a n d Nb < C. Thus the circle is not greater than 89 A similar argument using a s e q u e n c e of p o l y g o n s outside the circle s h o w s that the circle is also not less than 89 Figure 1 s h o w s the s e q u e n c e of drawings that a n y o n e w o u l d m a k e w h o actually carried out the constructions of the proof. Only the rightmost figure of those four is f o u n d in the manuscripts, and only the Gerard translation manuscripts s h o w the w h o l e p o l y g o n [10]. The M o e r b e k e translations confine the construction lines to the u p p e r left quadrant, so that the visual impression is quite different, although s o m e o n e re-

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Figure I. A sequence of regular polygons is constructed in the circle.

capitulating the process might still d r a w it as in Figure 1. I b e l i e v e that Dante w o r k e d with the G e r a r d translation, a n d m a d e the figures in the s e q u e n c e s h o w n in Figure 1, b e c a u s e the s e c o n d figure in that sequence, the cross in the circle, must have struck him as crucially significant (pun intended). The app e a r a n c e o f the cross in an argument that already s e e m e d to have a transcendent m e a n i n g m u s t have b e e n irresistible to him. If w e e x a m i n e again the last lines of The Divine Comedy, w e see that they follow the Archimedes p r o o f thought for thought. The s e c o n d circle, painted with man's image, is the Son, a n d "la nostra effige," our o w n h u m a n form, is the cross, that is, a m a n stretched out (crucified), exactly the s e c o n d figure in Figure 1. The strange w o r d "painted" depicts the geometer's literally adding the lines of the cross to the circle with a drawing instrument. The cross gives rise to the square, then to the octagon, a n d so forth. The g e o m e t e r wants to k n o w "how the h u m a n image/Conforms itself to the circle, and finds its place there." That is just the question, geometrically, h o w the s e q u e n c e of p o l y g o n s a p p r o a c h e s the circle, and theologically h o w the h u m a n and measurable b e c o m e s the divine and immeasurable. The g e o m e t e r knows that n o matter h o w m a n y sides it has, a p o l y g o n still has straight sides and so cannot b e c o m e the circle, which is curved. It is just Dante's point in The Banquet, that the circle cannot b e measured "because of its arc." He vainly seeks the principle, until s u d d e n l y there is "a flash of lightning," which resolves the problem. This is the argum e n t of A r c h i m e d e s that shows h o w the p o l y g o n s do b e c o m e the circle in the limit. That result is finally asserted in Dante's saying that the w h e e l turns evenly, as only a circular w h e e l can do, not a polygon.

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THE MATHEMATICALINTELLIGENCER

The Archimedes p r o o f m a k e s precise s e n s e of so m a n y o d d details in these last lines of the p o e m that it s e e m s quite believable that D a n t e had this p r o o f in m i n d as an e x t e n d e d m e t a p h o r for the union of h u m a n (straight) and divine (curved). If so, he u n d e r s t o o d that the limit of a s e q u e n c e can h a v e a p r o p e r t y that no m e m b e r of the s e q u e n c e had, in this case the p r o p e r t y of being curved, since that is the p o i n t of the m e t a p h o r a n d the mystery to w h i c h he is leading us. Like the construction of S 3 in Canto 28, it is possible that this final image was u n d e r s t o o d in mathematical terms b y no c o n t e m p o r a r y of Dante, but there is an o d d hint in this latter case that s o m e p e o p l e did. The first c o m m e n t a r y that actually mentions the g e o m e t e r is B e n v e n u t o da Imola (1375), w h o says [11]: Et explicat s u m m u m conatum suum p e r u n a m comparationem elegantissimam de geometra, qui volens mensurare circulum colit se totum sibi; et quamvis autor videatur loqui com-

muniter d e geometria, tamen iste actus et casus q u e m ponit maxime verificatur d e Archimede philosopho; ad q u o d est p r a e n o t a n d u m q u o d sicut scribit Titus Livius etc. And he e x p l a i n s his highest effort b y a most elegant c o m p a r i s o n with a geometer, who, wanting to measure the circle, gives himself c o m p l e t e l y to it; a n d h o w e v e r m u c h the author is seen to s p e a k generally a b o u t geometry, this particular case, w h i c h he places most highly, is p r o v e d b y the p h i l o s o p h e r Archimedes, a b o u t w h o m it is well known, as Livy writes, etc. Benvenuto seems to k n o w that there is a proof of Archimedes b e h i n d this image! He d o e s not, however, refer to Archimedes's treatise, but to Roman histories that tell the story of Archimedes. That suggests that he is not one of the p e o p l e w h o has actually seen or understood the proof. The next commentators s e e m to have m i s u n d e r s t o o d this idea in a

Area of polygon = (1/2) h.circumference Figure 2. The area of the regular polygon is lhNb, where N is the number of sides.

rather hilarious way. Chiose Vernon (1390) [11], p r o b a b l y from r e a d i n g Benvenuto, thinks that the g e o m e t e r staring at the circles is A r c h i m e d e s at the m o m e n t of his death at the siege of Syracuse, and inserts that w h o l e story into his commentary. Since this makes a ridiculous ending for The Divine Comedy, and is clearly impossible, the idea was d r o p p e d in the n e x t generation of commentaries, and all connection to the Archimedes p r o o f seems thereafter to have b e e n forgotten. If w e restore the idea of the Archimedes proof as metaphor, w e see that Dante's image might well represent not the failure of h u m a n intellect to comp r e h e n d the divine, as it is usually understood, but rather something more positive, more like a triumph of the hum a n intellect, and more characteristic of Dante himself. Understanding the mathematics behind the image potentially changes its meaning.

Geometry as Philosophy It is n o t e w o r t h y that although Dante refers to geometry, and e v e n does geometry, in w a y s that w e can recognize (with s o m e difficulty), the meaning of mathematics for him is philosophical. Euclid and Archimedes are philosophers. What w e call mathematics is, for him, and p r e s u m a b l y for his

c o n t e m p o r a r i e s and for his culture, a corner of p h i l o s o p h y having to do with the celestial part of creation, exemplifying a particular kind of truth. In particular, mathematics does not deal with messy, earthly problems, or with terrestrial space. Is it credible that a philosophical stance of this kind, even if it is a c c e p t e d b y a w h o l e culture, could change the nature of mathematics so drastically in practice? Restricting the p r o b l e m s that mathematics could address a p p e a r s to have restricted mathematics itself. In principle, mathematics could be an abstraction that feeds on its o w n abstract problems. The Romans believed, a n d therefore later civilizations also believed, that the origin of G r e e k mathematics was a love of abstraction. If the long m e d i e v a l p e r i o d a t t e m p t e d to see w h e t h e r mathematics could flourish without earthly applications, the answer s e e m s to be a r e s o u n d i n g no. The 17th-century revolution in mathematics came w h e n it b e g a n addressing questions that w e n o w call physics, concrete p r o b l e m s with experiments and data. Even if w e formally share the mathematics inherited from the ancients, w h a t w e m a k e of it d e p e n d s on o u r culture, not simply on the contents of mathematics books. To a surprising degree, the m e a n i n g of mathematics is

what w e think it is, and w h a t w e w a n t it to be. REFERENCES

[1] Russo, L., The Forgotten Revolution, Springer-Verlag, Berlin, Heidelberg, New York, 2003. [2] Ptolemy, Almagest, Book I. [3] Speiser, Andreas, Klassische StOcke der Mathemat& Verlag Orell F0selli, ZQrich, 1925. [4] Callahan, James, "The curvature of space in a finite universe," Scientific American 235, August, 1976, 90-100. [5] Peterson, Mark, "Dante and the 3-sphere," American Journal of Physics, 47, 1979, 1031-1035. [6] Osserman, Robert, Poetry of the Universe, Garden City, NY: Doubleday, 1995. [7] An amusing rift on these spheres is Osmo Pekonen, "The Heavenly Spheres Regained," The Mathematical Intelligencer 15, No. 4, 1992, 22-26. [8] Verse translations are those of Barbara Reynolds from Dante's Paradise, Penguin Books, 1962. [9] Dante's II Convivio (The Banquet), tr. R. H. Lansing, Garland Publishing, New York, 1990, 72. [10] Clagett, Marshall, Archimedes in the Middle Ages, Vol. 1, University of Wisconsin Press, Madison, 1964. [11] Over 70 Dante commentaries can be searched online at dante.dartmouth.edu.

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