galileo galilei`s location,shape and size of dante`s inferno

GALILEO GALILEI’S LOCATION,SHAPE AND SIZE
OF DANTE’S INFERNO:
AN ARTISTIC AND EDUCATIONAL PROJECT
MAGNAGHI-DELFINO Paola - NORANDO Tullia
Paola Magnaghi and Tullia Norando are members of the School of Engineering of the Politecnico di
Milano and, in particular, of the FDS Laboratory.
The FDS laboratory (Formation, Science Communication, Didactics and Experimental Teaching)
operates in Mathematics' teaching and in science communication.
Teaching activities are developed in co-operation with the world of primary and high school, for the
training of teachers and students and the experimentation of non-traditional forms of learning. In
this context, FDS promotes large-scale initiatives for the dissemination and the "demystification" of
the difficulties of mathematics .
If you are interested in more information about our activities, we are at your disposal. You can visit
this website.
Now we start our presentation. We can see the text using this QR code.
Introduction
There is a close relation between Mathematics and Fine Arts during the Renaissance: mathematical
knowledge is applied in drawings and paintings with the use of symmetry, producing ratios and
proportions. Within the study of such a context arises our artistic and educational project as a
collaboration between the FDS Laboratory and Accademia di Belle Arti di Brera.
The project is inspired by the first of two lectures held by Galileo Galilei at the Accademia
Fiorentina in 1588. These lectures were commissioned by the Accademia to solve a literary
controversy concerning the interpretation of Dante’s Inferno. In these lessons Galileo took the
opportunity to show his mathematical abilities combined with his strong background in Humanities.
When giving his lectures Galileo probably used drawings to explain how to map Dante’s Inferno,
because the difficulty of the subject which does not admit of easy explication in writing.
Galileo’s manuscript survives and is catalogued in the Filza Rinucciniana 21 of the Biblioteca
Nazionale di Firenze, but the drawings are lost.
The artistic and educational project
The artistic and educational project “Galileo Galilei’s location, shape and size of Dante’s Inferno”
was proposed to a group of students of Graphic Art course in the Accademia di Belle Arti di Brera.
Students were the actors in the project, the first addressees of all communication; each of them was
the creator of his artwork.
The project here presented was meant as an opportunity for the students of Graphic Art to
investigate the relationship between geometric representation and artistic interpretation.
The work plan was divided in two parts: the mathematical laboratory and the artistic work. In the
second part the students were followed by their teacher and artist Alessandra Angelini.
The mathematical laboratory
The students followed lessons about the cultural environment and the mathematical aspects of the
topic and then they went into the concept of mathematical perspective and the use of proportion and
similarity in order to render mathematically the precise positions of Inferno’s rings.
They studied the Inferno’s architecture, the Manetti’s plan and estimated the sizes, the widths, the
lengths of the eight levels and finally the height of Lucifer.
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The artistic work: technical drawing and free creative artwork
First, each student created a “technical drawing” that is a scaled drawing of Dante’s Inferno, based
on Galileo’s calculations, using different types of paper and free-chosen drawing techniques.
The choice of different colouring techniques and papers made possible that every drawing could
give emotions strongly different, despite being equal ratio and proportion.
Therefore there was the same Inferno, in shape and measure, but much different in impressions and
feelings.
After, each student created a “creative artwork” aroused from his artistic vision and inspired to the
Commedia’s verses. They were free to choose the artistic techniques, supports and dimensions of
them works, so all students created very different works.
They realized drawings, paintings, original engravings and various dimensions woodcuts, rich in
colour and sign and all tightly related to the author’s reflections.
At the end of our presentation, you will see a selection of the technical drawings and the free
creative artworks and can help to understand the thinking and the creative force that characterized
this project and the students’ different solutions.
The students’ works were displayed on the exhibition that was held at Politecnico di Milano (May
2012). After the works were exhibited at the Museo Dantesco of Ravenna (September 2013) and at
the Bergamo Science Festival (XI Edition, October 2013).
Some sketch from Mathematical Laboratory
Galileo Galilei.
In 1586, the young Galileo Galilei wrote his first scientific book La Bilancetta (The little balance).
He sent his work to many Italian mathematicians and he got a favorable reply from Guidobaldo Del
Monte, Inspector of Fortifications of the Granduca of Tuscany, Ferdinando I de’ Medici.
When the chair of Mathematics at Pisa became open, Guidobaldo arranged an invitation for Galileo
to address to the Accademia Fiorentina two lectures on mathematical topics.
The Accademia Fiorentina was founded in Florence in 1540 in accordance with the wishes
of Cosimo I de' Medici. Cosimo instituted the Fiorentina’s public lectures to supplement its private
meetings, allotted stipends to members and encouraged the Accademia to render “every science
from every other language into our own”. The principal topic of discussion of the Accademia was
the question of what should constitute the basis for Italian language. Indeed at those times it was
referred to as volgare, roughly "the common tongue", and not yet organized into a framework of
rules. The Accademia Fiorentina believed it should be based on contemporary Florentine usage and
on the language of Dante.
Galileo’s audience at the Accademia Fiorentina was not a mathematical one: it was a lecture on
Literature that would turn Galileo's fortunes. In 1589 indeed, he was appointed to the chair of
Mathematics in Pisa.
For a long time the manuscript of the two lectures was forgotten, perhaps hidden by Galileo
himself, because it contains a mistake about the question of the scale invariance.
The last book of Galileo “Two New Sciences” (1633) begins with the subject of scaling and the
observations on scaling in general are ingenious and deserving of the prominent place he gives
them. It is clear that Galileo assign enormous importance to the problem. The beginning of his
interest in this issue is certainly to be found in these two lectures.
In the lectures, Galileo examines the opposing views concerning the structure of the Inferno
proposed by Antonio di Tuccio Manetti and Alessandro Vellutello.
The two arguments are identical as regards the general appearance of the Inferno, but are
considerably different regarding the shape and the size.
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Antonio Manetti
Antonio di Tuccio Manetti (1423–1497) was a Florentine mathematician and architect, member of
Accademia Fiorentina, and biographer of the architect Filippo Brunelleschi.
Manetti thus wanted to show that Architecture is not only a useful technique, but has a role
comparable to that of the Humanities.
For this reason Manetti attempted to map out the realm Dante described, based on the textual
evidence and using scientific methods. Although Manetti never himself published his research
regarding Dante's Inferno, the earliest Renaissance Florentine editors of the poem, Cristoforo
Landino and Girolamo Benivieni, reported the results of his research in their respective editions of
the Divina Commedia.
Dante’s geography
Manetti’s plan is based on Dante’s geographical knowledge and the religious beliefs. The sphericity
of the Earth was of course well known in the Middle Ages: the Earth was represented in maps
called Mappae Orbis Terrae, or T-O maps, where a T-shaped Mediterranean sea divided the three
continents known at the time (Asia, Africa and Europe), surrounded by the Great Ocean. The
reason is that during the Middle Ages, it was believed that land occupied half of the Earth’s surface,
while the other hemisphere was occupied by the Ocean; maps were thus related only to land. Maps
also had a symbolic meaning: Jerusalem was always at the center and Asia was always in the upper
section. In the maps the regions North of the Arctic Circle and the part that lies South of the Tropic
of Cancer (Hic sunt leones) were missing, what remains was an area called Spherical Trapezium
and had a shape similar to a cloak (cape of Ptolemy). The land, called the Oecumene, ranged from
the Pillars of Hercules (Cadiz) to the mouth of Ganges, whose distance for Dante was 180°, while in
reality it is 120°, so it seems that the Earth was “stretched” from East to West.
Jerusalem, the center of Mankind, was exactly at the center of Oecumene.
In the Middle Ages, people believed that the terrestrial radius was about 3250 Florentine miles (1
Florentine mile is about 1,74 Km), so the measure is about the 88% of the terrestrial radius.
Manetti’s Inferno
Manetti’s Inferno is a cone-shaped region in the Earth, with the vertex in the center of the Earth and
the base on the surface, centered on Jerusalem.
The cone is generated by rotation of the circular sector which has radius identical to the terrestrial
radius. Because the distance from Jerusalem to Cuma was believed to be 1700 miles, the arc which
is drawn from Jerusalem to the edge of the mouth of the Inferno is of 1700 miles. Therefore the
circular sector has the angle at the vertex of 60°.
The Inferno does not occupy the whole spherical sector but only the part of the cone which is, under
Jerusalem, at the depth of 1/8 of the terrestrial radius.
The funnel is made of nine circles. The first circle is the widest; progressively, the ninth circle is the
smallest. This ninth circle surrounds Lucifer.
The First Six Levels
The first six levels of Manetti’s Inferno are regularly spaced, in fact the they are equidistant with
1/8 the radius of the Earth between each level and the next.
Level
Limbus
level 2
level 3
level 4
level 5
level 6
Distance from the Earth’s
center
2839 17/22
2434 1/11
2028 9/22
1622 8/11
1217 1/22
811 4/11
3
In order to deduce the widths of the first six levels, Manetti divided the length of the arc on the
surface from Cuma to Jerusalem into two parts: 1000 miles + 700 miles.
We can sketch the Manetti’s plan as in the following drawing
In the first 1000 miles he marked 10 spaces, each one of 100 miles, beginning from the mouth; from
these he deduced the widths of the first six levels. The reason of this partition into two parts is that
in the Middle Ages geography the distance from Cuma to the island of Crete was considered exactly
1000 miles. When Dante arrived to the sixth level of Hell, he is located exactly below the Mount
Ida, where was the statue of Veglio di Creta (Grand Old Man) which is the mythical origin of the
infernal rivers (Inferno, XIV, 103-120). Galileo did not care about these details, but in the Girolamo
Benivieni’s book we read this explanation about the Dante’s path: Dante covers only a tenth of each
ring and so completes the circle after ten rings (Inferno XIV, 121 – 129).
Manetti supposed that this spiral drawing correspond to the Dante’s path
In order to assign the widths to the first six levels the method used by Manetti is based on the
Thales similarity theorem, also known as intercept theorem. Using similarity we can derive that the
length of the arc intercepted by an angle is proportional to the radius. Since the angle at the center is
the same, the circular arcs lengths are proportional to the corresponding radii.
Galileo calculated this table
4
Limbus
level 2
level 3
level 4
level 5
level 6
width
87 1/2
75
62 1/2
50
112 1/2
75
on the Earth’s surface
100
100
100
100
300
ring 1
ring 2
ring 3
37 1/2
37 1/2
37 1/2
300
ring 1
ring 2
ring 3
25
25
25
Malebolge
The seventh level contains the whole of Malebolge, which is depth of the Geryon’s ravine, and the
eighth and last level embraces the four spheres of ice including Lucifer. The first six distances from
one level to the other are equal, but it is not possible for the distances from the seventh and the eight
levels, because of some points of Dante’s text noticed by Manetti. Indeed Dante says that the ninth
bolgia turns through 22 miles, and, in consequence, the diameter must be 7 miles:
Tu non hai fatto sì a l’altre bolge;
pensa, se tu annoverar le credi,
che miglia ventidue la valle volge.
(Thou hast not done so at the other Bolge;
consider, if to count them thou believes,
that two-and-twenty miles the valley winds)
(Inferno, XXIX, 7-9)
Then Dante also says (Inferno, XXX, 82-87) that the tenth bolgia turns through 11 miles, and, in
consequence, the diameter must be 3 1/2 miles:
cercando lui tra questa gente sconcia,
con tutto ch’ella volge undici miglia,
e men d’un mezzo di traverso non ci ha.
(Seeking him out among this squalid folk,
although the circuit be eleven miles,
and be not less than half a mile across)
(Inferno, XXX, 85-87)
Manetti thus supposed that the radii of the bolge were in aritmetic progression and obtained
Bolgia
10
9
8
7
6
5
4
3
2
1
Arc length
11
22
33
44
55
66
77
88
99
110
5
diameter
3 1/2
7
10 1/2
14
17 1/2
21
24 1/2
28
31 1/2
35
radius
1 3/4
3 1/2
5 1/4
7
8 3/4
10 1/2
12 1/4
14
15 3/4
17 1/2
Galileo concluded that the distance of Malebolge from the Earth’s center is 81 3/22 miles, via
Thales similarity theorem, and the Geryon’s ravine is 730 5/22 depth.
The Well of Giants
After the bolge but still within the seventh level there is an empty land which leads on down into
the tomb of Lucifer. On the far side of this land enormous Giants are buried in the ground. Its top is
flat and the inner side slopes in such a way that one can climb up the Giants and then slide down
from the wall into the eighth level.
As Galileo says, that he learned from Dante, the width of the well is 1 mile in radius, the width of
that space which remains between the last bolgia and the well is 1/4 mile, that of the last bolgia 1/2.
on the Earth’s
width
surface
bolgia 1
1 3/4
70
bolgia 2
1 3/4
70
bolgia 3
1 3/4
70
bolgia 4
1 3/4
70
bolgia 5
1 3/4
70
bolgia 6
1 3/4
70
bolgia 7
1 3/4
70
bolgia 8
1 3/4
70
bolgia 9
1 3/4
70
bolgia 10
1/2
20
land Malebolge-Well
1/4
10
Well
1
40
In the Divina Commedia however we were not able to find that the width of the well is 1 mile in
radius, so on which basis did Galileo claim this? From these verses
Facemmo adunque più lungo viaggio,
volti a sinistra; e al trar d’un balestro
trovammo l’altro assai più fiero e maggio.
(Therefore a longer journey did we make,
Turned to the left, and a crossbow-shot oft
We found another far more fierce and large.)
(Inferno, XXXI, 82 -84
we can argue that “Dante and Virgilius turn around the well” and so the well must have a circular or
polygonal shape, and that the distance from one Giant to the other is about 300 braccia (a
crossbow-shot). We recall that 1 Florentine braccio is about 0,58 m.
The size of Lucifer and the spheres of ice
Manetti calculated the size of Lucifer from the verses
Lo ‘mperador del doloroso regno
da mezzo ‘l petto uscia fuor de la ghiaccia;
e più con un gigante io mi convegno,
che i giganti non fan con le sue braccia
(The Emperor of the kingdom dolorous
From his mid-breast forth issued from the ice,
And better with a giant I compare
Than do the giants with those arms of his)
(Inferno, XXXIV, 28 – 31)
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So Dante makes a greater comparison with a Giant than a Giant makes with one arm of Lucifer. If
therefore Manetti knew the size of Dante and that of the Giant he would be able from that to find
the size of Lucifer. But Dante was a man of average stature, which is 3 braccia. To investigate the
size of a Giant, Manetti used the following verses:
La faccia sua mi parea lunga e grossa
come la pina di San Pietro a Roma,
e a sua proporzione eran l’altre ossa
(His face appeared to me as long and large
As is at Rome the pine-cone of Saint Peter's,
And in proportion were the other bones)
(Inferno, XXXI, 58 – 60)
Manetti thus found the length of the face of the Giant, basing on that of the sculpture called
Pinecone (5 1/2 braccia)
Pinecone is bronze artefact of Roman origin, which is now in the Belvedere’s Garden (Città del
Vaticano, Rome).
Because men are ordinarily 8 heads high, Manetti and Galileo proposed that the Giant was up 8
times the height of his head, so that a Giant was 44 braccia in height.
Because a man to a Giant has greater comparison than a Giant to an arm of Lucifer, Manetti
founded that one arm of Lucifer was more than 645 braccia. Because the length of an arm is 1/3 of
the total height, the height of Lucifer is 1935 braccia. But because the comparison is greater
between a man and a Giant than between a Giant and an arm of Lucifer, Manetti concluded that
Lucifer was 2000 braccia height.
Dante says that Lucifer has his navel at the Earth’s center and protrudes out of the lowest ice sphere
up from the middle of the breast. The distance from the navel of Lucifer to the middle of the breast
is 1/4 of the total height of Lucifer, so the distance is 500 braccia. Consequently the radius of the
lowest sphere of ice is 500 braccia. Manetti judged that the other radii are in arithmetic progression:
1000, 1500, and 2000.
Pinecone
Nembrot
Dante
Arm of Lucifer
Lucifer
navel- middle of the breast
7
braccia
5 ½
44
3
645 1/3
1936
484

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