Dante · Galileo · the square-cube law

The Roof of Hell

In the late 1580s a young Galileo measured Dante's Hell for the Florentine Academy: its depth, its circles, the height of Lucifer, and the thickness of the rock over it. His arithmetic is redone here, in your browser, from the page he wrote. It holds. His one piece of engineering does not, and he spent the rest of his life finding out why.

Some time before the end of 1592, and by the usual dating in 1587 or 1588, long before the telescope, Galileo stood up in the Florentine Academy "to obey the command of one who can command us" and explained the size and shape of Dante's Hell. He was in his twenties. The two lectures survive in a fair copy in his own hand, and Antonio Favaro printed them from it in 1899, in volume IX of the national edition of his works. They are mostly arithmetic, and the arithmetic can be done again.

The quarrel was between two readings of the Inferno. Antonio Manetti, a Florentine, had made Hell a cone reaching from under Jerusalem down to the centre of the Earth. Alessandro Vellutello, from Lucca, had published a much smaller one and mocked the Academy for Manetti's. Galileo was defending the Academy. Everything below is recomputed in your browser from the numbers he printed, by engine.mjs, which names the page each one came from.

A cone as deep as the Earth is wide

Draw a line from the centre of the Earth up to Jerusalem, Galileo says, and an arc along the surface from Jerusalem a twelfth of the way round the globe. Close the sector and spin it about the Jerusalem line. The solid it sweeps is Hell. Its point is at the centre of the world, where Lucifer is fixed in the ice, and its mouth is a circle whose diameter equals the Earth's radius, because the chord under a sixth of a circle is equal to the radius.

Every measurement in the lecture comes from this cone and from four things the poem says. Two are in consecutive cantos. In Inferno XXIX Virgil rebukes Dante for staring into the ninth ditch of Malebolge:

Pensa, se tu annoverar le credi, / che miglia ventidue la valle volge. If you mean to count them, consider that the valley runs round twenty-two miles. Inferno XXIX 8-9 (Petrocchi); our translation

and in XXX a damned forger says of the tenth that it turns eleven miles. Galileo takes the ditches as circles and divides by 22/7: diameters of 7 and 3½ miles, so each ditch is 1¾ miles wide. Then comes his own step, which he says his predecessors had left out. The 1,700-mile arc had been cut into hundreds, a hundred for each of the first four circles, three for the fifth, three for the sixth, and 700 remain for Malebolge and the well. Widths shrink in proportion to distance from the centre, so if Malebolge's 17½-mile radius is what 700 surface miles become, it must lie 40 times closer to the centre than the surface does.

With his inputs every row comes out to his fraction exactly: 3245 5/11 is 20,400 divided by twice 22/7, so his Earth is a globe of 20,400 miles measured with Archimedes' value of π. Galileo was delighted by the last step. Nine ditches of 1¾, a tenth of ½, a quarter-mile gap and a one-mile well, each multiplied by 40, add to 700, a punto, the exact number left over on the surface. Mirabilmente, he says: Manetti had found the mind of the poet.

Two of those inputs are not in the poem. The one-mile radius of the well and the quarter-mile gap are introduced as drawn from Dante come di sopra dicemmo, "as we said above", but nothing earlier in the lecture derives them, and they are exactly what is needed to make the sum come out. With the tenth ditch at 3½ miles across, the well and the gap together must take 1¼ of its 1¾-mile radius for the tenth ditch to get the half-mile that XXX gives it.

Two Hells, one scale

Vellutello's Hell was a stack of wells and terraces around the centre, reaching up about a tenth of the radius. Lesson II describes it segment by segment: a giants' well a mile across, Malebolge sloping fourteen miles, a sheer cliff of 140, the circle of the violent, a wall of seventy up to the city of Dis, then five scarps each rising fourteen miles while widening seventeen, to a mouth 280 miles across. Galileo makes three claims about its size, and all three can be checked by turning that profile into a solid.

All three of Galileo's claims hold, and with room. Manetti's solid, cone and cap together, is a spherical sector of half-angle 30°, whose share of the sphere Archimedes' theorems give as (1 − cos 30°)/2: 1 part in 14.93, "something less than" a fourteenth. The opinion he rejected, that Hell fills a sixth of the Earth, was too big by a factor of two and a half. For Vellutello's the reading of his profile involves small choices (where the floor of Lucifer's pit sits, whether the plain around the mouth counts); none of them moves the answer anywhere near one part in thirty thousand.

How tall is Lucifer

Galileo wanted the size of the four spheres of ice at the bottom, and the poem gives them only through the body frozen in them. Lucifer stands out of the ice from mid-chest, and Dante says of him:

e più con un gigante io mi convegno, / che i giganti non fan con le sue braccia and I compare better with a giant than the giants do with his arms. Inferno XXXIV 30-31 (Petrocchi); our translation

So Dante is to a giant as a giant is to Lucifer's arm, or a little more. The giant comes from canto XXXI, where Nimrod's face is "as long and as broad as the pine cone of Saint Peter's in Rome", the great bronze cone that stood in the forecourt of the old basilica. Galileo gives it as 5½ braccia, makes a man eight heads tall, and makes Dante an ordinary three braccia.

Galileo's arm is 44 × 44 ÷ 3 = 645 braccia (he drops the third), Lucifer three arms, 1935, and he rounds up to 2000 for Dante's "more". Navel to mid-chest is a quarter of a body, 500 braccia, and that is the radius of the smallest sphere of ice; the other three he makes 1000, 1500 and 2000. In Lesson II he reports that Vellutello got 3000 by using nine heads and by measuring the cone "before it was broken"; with nine heads the cone would need to be about six braccia for that. He answers that Dante, if he measured it at all, measured it as it was in his own time.

The roof that could not stand

Near the end of Lesson II Galileo raises an objection to Manetti and answers it. If Hell is that large, the dome of rock over it, with Jerusalem on its crown, is only an eighth of the radius thick, about 405 miles, less whatever the seas and the grotto of the neutrals take out. Will it not fall in?

tal grossezza è suffizientissima: perciò che, presa una volta piccola, fabricata con quella ragione, se arà di arco 30 braccia, gli rimarranno per la grosseza braccia 4 in circa, la quale non solo è bastante, ma quando a 30 braccia di arco se gli desse un sol braccio, e forse ½, non che 4, basteria a sostenersi that thickness is more than sufficient: for take a small vault built in that proportion; if it spans 30 braccia it will have about 4 braccia of thickness, which is not only enough, but if a 30-braccia span were given a single braccio, perhaps a half, never mind four, it would hold itself up. Galileo, Lesson II, Favaro IX p. 55; our translation

The argument is that shape is what matters: a vault that stands at thirty braccia stands at any size if its proportions are kept. Hell's roof and his small vault are nearly the same shape (four braccia in thirty is 0.13; 405 miles over a mouth one Earth-radius wide is 0.125). The instrument below keeps the shape fixed and changes only the size.

At thirty braccia (about 17.5 m) the vault carries roughly 0.22 megapascals of compression from its own weight, hundreds of times less than sound stone can bear. Galileo was right that it would stand, and right that it would stand much thinner. What he took for granted is that a vault which stands at one size stands at every size, and that is false. Keep the shape and the material, multiply every length by a factor, and the weight multiplies by the cube of the factor while the section of stone that carries it multiplies only by the square. The compression goes up in proportion to size. Thickness cancels out of it entirely: in the model the page uses (a thin dome carrying only its own weight, whose compression at the springing is density × gravity × radius), a fatter dome is no stronger.

With stone of 150 MPa and a density of 2,600 kg/m³ the crossing comes at a span of about 12 km. The mouth of Hell is one Earth-radius across, 3245 5/11 of Galileo's miles, which with the Tuscan mile of 2833⅓ braccia is about 5,370 km. There the same dome carries about 68 gigapascals, some four hundred and fifty times what the stone can take. The honest figure for the actual roof, a shell concentric with the Earth so that gravity presses square on it everywhere, is half the density times gravity times the radius of curvature, about 64 GPa: the same verdict. Measured as a model, his thirty-braccia vault is a copy of Hell's roof at roughly one braccio to a hundred miles, a factor of about 300,000.

Fifty years later, blind and under house arrest, Galileo published Two New Sciences (1638). It opens in the Venetian arsenal. An old shipwright has told the company that a big ship needs heavier cradles and bracing at its launch than a small one, or it may part under its own weight, and Sagredo dismisses this as a saying current among the ignorant, on exactly the ground of the Inferno lecture:

“If, therefore, a large machine be constructed in such a way that its parts bear to one another the same ratio as in a smaller one, and if the smaller is sufficiently strong for the purpose for which it was designed, I do not see why the larger also should not be able to withstand any severe and destructive tests to which it may be subjected.” Two New Sciences, First Day, Ed. Naz. VIII p. 50; tr. Crew and de Salvio (1914)

Salviati takes the shipwright's side: the larger the machine, the greater its weakness, and for every structure there is set a necessary limit beyond which neither art nor nature can pass. In the Second Day he draws a bone lengthened three times and thickened until it can do its work, and concludes that a giant built like an ordinary man will fall and be crushed under his own weight; just before it he names the impossibility of building ships, palaces, or temples of enormous size.

The physicist Mark Peterson argued in 2002 that the lectures are where this began: that Galileo's vault is a scale model of Hell's roof, that it is a convincing argument that Manetti’s model can support itself, but only until you realize that the argument assumes scale invariance!, and that the scaled-up version would immediately collapse of its own weight. He reads the giants of Two New Sciences the same way, against the eight-heads-tall giants of the lecture, and calls part of his case speculation, of course. It is a reconstruction; no document has Galileo admitting the error. What can be checked is the pair of texts, and the instrument above: the vault he offered as proof is the exact case his later book says cannot be scaled.

What the check found

Reading a sixteenth-century calculation against its source turns up small things, and they are part of what the lecture is. The recomputation used Favaro's printed text, re-read against the page images, because the online transcription is marked unproofread.

• The arithmetic holds. Every figure in the ledger, the widths of all the circles, the 700-mile sum, and his radius itself come out to his printed fraction exactly when π is 22/7. With the true π the Earth would be 3246.76 miles in radius, and the ditches 7.003 and 3.501 miles across.
• One slip. The giants' well is derived twice as 81 3/22 miles (pp. 37 and 39); the closing tour of Lesson I gives its depth as miglia 81 1/2 (p. 46). Favaro's apparatus notes no correction there.
• A misquotation that does not matter. Galileo quotes Inferno XXX 87 as E men d’un terzo di traversa non ci ha, "not less than a third across". The critical text (Petrocchi) reads e men d’un mezzo di traverso, "a half", and a half is the width his own sum gives the tenth ditch.
• Another. His first lecture opens by quoting the gate of Hell as Uscite di speranza, voi ch’entrate (p. 31). The line is Lasciate ogne speranza, voi ch’intrate (Inferno III 9). We found no edition with his wording and assume he quoted from memory; The Gate Built by Love follows what later English did to the same line.
• Two numbers the poem does not give. The well's one-mile radius and the quarter-mile gap (see above).
• Three size claims, all true. Manetti's Hell is 1 part in 14.93 of the Earth (under 1/14); Vellutello's, built from Lesson II's own dimensions, 1 part in about 55,000 (under 1/30,000) and about 3,700 times smaller than Manetti's (more than the 1,000 he claims). Only an absurdly generous reading, a solid cylinder 315 miles wide and a tenth of the radius tall, would break the 1/30,000 bound.
• The pine cone. The Vatican Museums give the Pigna as almost four metres high today, with a later base and a replacement for its lost tip. Galileo's 5½ braccia is 3.2 m. Put four metres into the Lucifer instrument with eight heads and he comes out near 3,000 braccia, Vellutello's figure; Galileo says Vellutello measured the cone prima che fosse rotta, "before it was broken". Which base and which tip anyone measured, we cannot say.
• A detail in the secondary literature. Peterson sets the model beside Brunelleschi's dome of the Florentine cathedral, which, as Galileo says, is less than 4 braccias thick. Neither lecture mentions Brunelleschi, the cathedral or its dome; the only vault in Lesson II is the unnamed thirty-braccia one.
• The date. Favaro himself fixes the lectures only as before December 1592, from a letter of 1594 and a later testimony; 1587 to 1588 is Ottavio Gigli's inference, which Favaro reports.