This article is taken from the last number of ENEL’s magazine, Oxygen

In 1587, a promising young scholar by the name of Galileo Galilei held Two lectures to the Florentine Academy on the Shape, Location and Size of Dante’s Inferno (1). This very serious work is actually a literary essay of explanation, commissioned of Galileo in light of his growing notoriety. The young man leapt at the chance to have his mathematical abilities recognized: an excellent promotional opportunity for an ambitious twenty four year old who would soon be appointed to the chair of mathematics at the University of Pisa. After having published his first scientific works (De Motu and La bilancetta (The Little Balance)), Galileo wished to demonstrate that mathematical physics need not consist of only technically efficient calculations, but can also contribute to upper level cultural debates and discussions, thus acquiring an intellectual status comparable to classical and artistic studies (2). In fact, the Florentine academicians asked Galileo to resolve a literary controversy regarding the interpretation of Dante’s Inferno.

In 1506 the Florentine Antonio Manetti had published a description of the geography and geometry of the Inferno just as it had been described by Dante (often quite obscurely). To be given particular attention was the evaluation of the reliability of the figurative representations given by Botticelli in the Nineties of the Fifteenth century in a luxurious illustrated edition which followed the first sketches of Giuliano da Sangallo. The illustrations were drawn based on measurements established through complicated calculations taken from references in Dante’s text and which, for the intellectual circles of an era for which The Divine Comedy was a fundamental reference point, needed to be accurately established. But in 1544 Alessandro Velutello from Lucca, Florence’s rival city, published a severe critique of Manetti’s work and proposed a very different description of Inferno. Galileo was called on the resolve the debated question which he did, predictably, in favor of the Florentine, Manetti. Obviously, neither at the beginning of the 15th century for Manetti and Velutello and their readers, nor at the end of that same century for Galileo and his listeners, was Dante’s description to be taken seriously from a theological point of view. But the importance of The Divine Comedy in Tuscan culture made evident the necessity of commenting on it and of completely understanding its every aspect – including the topography – so as to make this complex literature easier to read and comprehend (3).

Galileo carefully chose and commented on the verses of Dante that were called into dispute, beginning with the confirmation of Manetti’s description: the Inferno is a concave cavity of which the vertex is found at the center of the Earth, and the base, which sits at the level of the Earth’s surface, is placed at Jerusalem (obviously…); the circle at the base has a diameter equal to the arc of the Earth (4), which is to say that in a central section of the Earth which passes through the cone’s base, the infernal section occupies one sixth of the disc’s total area. Those not well versed in three-dimensional geometry could thus think that the same proportions apply to the volumes. At this point Galileo brought his personal knowledge into play:

But wanting to know its size in respect to the whole volume of earth and water, we should not just follow the opinion of some who have written about the Inferno, who believe it to occupy the sixth part of the volume, because making the computation according to the methods proved by Archimedes in his book On the Sphere and the Cylinder, we will find that the space of the Inferno occupies a little less than 1/14 part of the whole volume; I say this if that space should extend all the way to the surface of the earth, which it doesn’t: on the contrary, the mouth remains covered by a great vault of earth, whose summit is Jerusalem and whose thickness is the eighth part of the radius.

Archimedes’ treatises were still little known at the time, and were included in scholastic mathematics papers that previous commentators of Dante, themselves literati, could not have been familiar with. It is interesting to quickly verify Galileo’s estimations with the help of the algebraic equations now available (though these remain university level calculations (5)). In his comments, Galileo did not present himself as only a mathematician, but also called his knowledge of physics into play. In this role he offered a crippling critique of Velutello’s comments in which the different levels of Inferno were conceived as portions of a cylinder, the walls of which were parallel to a common axis. Galileo rightly brought attention to the fact that these walls are not at all vertical, because to be so they must be generated by an arc that passes through the Earth’s center: in two distant points, the directions of the vertical locales are not parallel, but convergent. Thus the cliffs that bordered the cylindrical levels must, by definition, be oblique to vertical and with a pronouncedly sheer face, as a result absolutely instable.

For if we assume [like Velutello] that the defile rises up with its sides equidistant from each other, the upper parts will find themselves without support to hold them up, and that being the case, they will undoubtedly collapse. For heavy objects, in falling, go along a line which conducts them straight to the center, if in this line they do not find anything that impedes and supports them.

This means that Galileo was already thinking about the problem of falling bodies while he was reading Dante! In reality Galileo’s criticisms were ironically invalidated by successive developments in physics. Newton’s theory of gravity demonstrated, effectively, that the acceleration of gravity is directed towards the center of the Earth only in the case that it be a solid sphere (or, in any case, that it presents spherical symmetry). The existence of an outlandish pit of such notable dimensions as Dante’s Inferno would modify the gravitational field surrounding it considerably. Galileo’s calculations, obviously, could not be carried out, so showing us that Velutello’s naïve hypothesis was closer to reality then Galileo’s theory was…(6) In closing, Galileo takes issue with the resistance of the materials, as an experimenter, to respond to objections raised against the existence of the vaulted cover of Inferno:

Here one might oppose that the Inferno cannot be so large as Manetti makes it, since, as some have suspected, it doesn’t seem possible that the vault that covers the Inferno could support itself and not fall into the hole […], beyond being no thicker than the eighth part of the radius of the earth […]. To this one answers easily that such a thickness is more than sufficient; for taking a little vault which will have an arch of 30 braccias, it will need a thickness of about 4 braccias, which not only is enough, but even if you used just 1 braccia to make an arch of 30 braccias, and perhaps just 1/2, and not 4, it would be enough to support itself.

This comparison between the infernal cap and a stonemason’s vault offered by Galileo erases any doubts regarding the relationship of the Inferno’s structure and the architecture of the celebrated cupola of Florence’s famed Cathedral, designed by Brunelleschi, which had an emblematic role in the Renascence (7). But although this analogy has profound cultural meaning, its scientific value is zero: a gigantic domed vault such as the Inferno’s, if it had the same geometric proportions of a small stonework vault, would certainly not be as solid. In light of modern concessions to gravitational acceleration and the strength of materials, the cover of the Inferno would undoubtedly be destined to collapse. In fact, the strength of a dome increases by the area of its sections, while its weight varies according to its volume. If all the dimensions are multiplied by the same factor, by 10 for example, the weight would be multiplied by 1,000, but the resistance against collapse only by 100; it would thus be, proportionately, 10 times more fragile. Therefore we can see that there is a limit to the stability of a structure obtained by simply changing its scale, going from a smaller structure to a larger (8). And in the case if Inferno’s vaulted ceiling, when compared to the small, domed, stone ceiling considered by Galileo, where the scale factor is increased by more than hundreds of thousands, the limit is obviously surpassed by quite a lot. But this argument has been demonstrated and developed… by Galileo himself, in Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Dialogue Concerning Two New Sciences) (9).

Here we come to a crucial point regarding the Lectures and their role in the development of Galileo’s ponderings. In point of fact, it would seem that the scientist quickly understood his mistaken reasoning, which was derived from a strictly geometrical point of view and did non take into account the laws of scale for the physical properties of the materials. It seems reasonable to suppose that the awareness of this error is what brought about his work on the resistance of materials, found in Dialogue. This theory, proffered by Mark Paterson (10), is based on serious arguments. The fact that Galileo understood soon after the faulty character of his ideas on the change of scale presented in Lectures on Inferno would first of all explain the discretion, not to say the reticence, he showed almost immediately afterwards regarding the work. The scientist surely dedicated much intense reflection to the question between 1590 and 1600. One could even surmise that Galileo, in understanding his error, suffered a very real psychological shock, echoes of which are heard in Dialogue:

Sagredo: My brain already reels. My mind, like a cloud momentarily illuminated by a lightning-flash, is for an instant filled with an unusual light, which now beckons to me and which now suddenly mingles and obscures strange, crude ideas. From what you have said it appears to me impossible to build two similar structures of the same material, but of different sizes and have them proportionately strong.

These lines constitute an explicit confutation to the preceding argument presented in Lectures in which he compares the cupola of Inferno to a small stonemason’s vault. The same argument invalidates the picturesque passage in Lectures in which Galileo attempts to estimate the height of Lucifer by homothetic transformation (the result being 2,000 braccia). There is therefore ample reason to believe that Galileo quickly grasped his error. In fact, though Dialogue was only published in 1638, the material that makes up the work was ready before 1610, the time in which Galileo dedicated himself to astronomy and published his first important work, Sidereus Nuncius (Sidereal Messenger). It is then legitimate to consider Lectures on Inferno as a crucible for the path taken by Galileo in his essential works, developed in Dialogue.

Jean-Marc Lévy-Leblond is physics and philosophy professor at Nice’s University

Translation by Mia Adelman

Notes

1) G. Galilei, Due lezioni all’Accademia Fiorentina circa la figura, sito e grandezza dell’Inferno di Dante, in Le opere di Galileo Galilei, A. Favaro, vol. IX, 29-57, Barbera, Florence 1968. This work was recently translated into French: Galilée, Leçons sur l’Enfer de Dante, Fayard, Paris 2008, translation and preface by Lucette Degryse, afterward by Jean-Marc Lévy-Leblond. An English translation; Two Lectures to the Florentine Academy On the Shape, Location and Size of Dante’s Inferno, translation by Mark A. Peterson, Mount Holyoke College, is available at www.mtholyoke.edu/courses/mpeterso/galileo/inferno.html
2) The context of Galileo’s work is outlined in Lucette Degryse’s introduction to the French translation of op. cit. As well as in: Th.B. Settle, Experimental Sense in Galileo’s Early Work and its Likely Sources, in Largo campo di filosofare, Proceedings Eurosymposium Galileo 2001, Fundación Canaria Orotava de Historia de la Ciencia, La Orotava, Tenerife 2002; Th.B. Settle, Dante, the Inferno and Galileo, soon to be published.
3) The maps and measurements of Inferno and their iconography constitute an argument in the literary exegesis of Dante’s work and can be found in: G. Agnelli, Topo-cronografia del Viaggio dantesco, Hoepli, Milano 1891; S. Orlando, Geografia dell’Oltretomba dantesco, in Guida alla Commedia, Bompiani, Milano 1993.
4) This position assures that the best access points to inferno are found on this circle (at the point in which the width of the dome is reduced to nothing). We can see (cheating a little on the numbers of Earth’s radius), that this circle passes close to the entrances to hell already well known in ancient times (in Greece, Sicily and Campania). See A. Nadaud, Aux portes des Enfers, Actes Sud, Arles 2004.
5) See J.-M. Lévy-Leblond, Appendice I in the afterward to the French edition, op. cit., pp. 167-168.
6) See J.-M. Lévy-Leblond, Appendice I in the afterward to the French edition, op. cit., pp. 170-173.
7) See S. Toussaint, De l’Enfer à la Coupole. Dante, Brunelleschi et Ficin A propos des “Codici Caetani di Dante”, L’Erma di Bretschneider, Rome 1997. Here can be seen how Manetti, whom Galileo defended in Lectures, was also the biographer of Brunelleschi. 
8) See J.-M. Lévy-Leblond, Appendice III in the afterward to the French edition, op. cit., pp. 174-175.
9) G. Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze, in Le opere di Galileo Galilei, by A. Favaro, vol. VIII, Barbera, Firenze 1968. In English, G. Galilei, Dialogue Concerning Two New Sciences, translated by Henry Crew and Alfonso de Salvio, MacMillan, 1914.
10) M.A. Peterson, Galileo’s Discovery of Scaling Laws, in “American Journal of Physics”, 70, 575, 2002.