Tag Archives: Euclid

Immortal Books, essential instruments (I)

Photography by Carlos Dorce

Photography by Carlos Dorce

This post is about a very interesting exhibition about 26 selected scientific books which I visited in Madrid in August and it can be visited now in A Coruña (from the 17th October). There are explanation of the 26 books and their authors and I am going to talk about the mathematical ones (of course!). Furthermore, there are Eulogia Merle‘s drawings of every scientist exhibited here so this is another interesting attraction to visit it.

The first great mathematician is Euclid (c.295 BC).

Photography by Carlos Dorce

Photography by Carlos Dorce

[In Spanish:] Es difícil precisar datos de la biografía del más destacado matemático de la antigüedad grecolatina, considerado el Padre de la Geometría. Solo se conocen con certeza dos hechos indiscutibles: vivió en una época intermedia entre los discípulos de Platón y los de Arquímedes, y formó una gran escuela de matemáticas en Alejandría. Según el filósofo bizantino Proclo, Euclides enseñó en esta ciudad del delta del Nilo durante el mandato de Ptolomeo I Sóter, es decir, entre los años 323 y 285 a.C. Murió en torno al año 270 a.C. Su fama radica en ser el autor de los Elementos, un tratado de geometría que ha servido de libro de tecto en la materia hasta comienzo del siglo XX. Está compuesto por trece libros que tratatn de geometría en dos y tres dimensiones, proporciones y teoría de números. Presenta toda la geometría basándose en teoremas que pueden derivarse a partir de cinco axiomas o postulados muy simples que se aceptan como verdaderos.

There are two different digital editions of the Elements and a compass from the 16th or 17th century with all this information:

Photography by Carlos Dorce

Photography by Carlos Dorce

The next Greek mathematician is Archimedes (287-212 BC) although his book here is On the floating bodies which is less mathematical than phisician.

Photography by Carlos Dorce

Photography by Carlos Dorce

Ptolemy (2nd century) is the next and his Almagest was the most important astronomical book since the 16th century.

Photography by Carlos Dorce

Photography by Carlos Dorce

There is also an interesting wooden astrolabe from 1630 (“Claudii Ricchardi”):

Photography by Carlos Dorce

Photography by Carlos Dorce

Arsitotle, Hippocrates and Pliny the Younger are the other three Greek scientists represented in the exhibition.

Location: MUNCYT in Madrid (map) and MUNCYT in A Coruña (map)

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Works in the street

Photography by Carlos Dorce

Photography by Carlos Dorce

A new building for the Bodleian Library is being built opposite the old building. It will be finished in 2015 but until then we can see  long fence that obscures the works. Why I am saying this? Because there are some pictures on it refering different books of the Library qnd the E is for Euclid and the H for Hooke!

Photography by Carlos Dorce

Photography by Carlos Dorce

LocationBodleian Library (map)

A mathematical tomb in Bath Abbey

Photography by Carlos Dorce

Photography by Carlos Dorce

This musical tomb or burial monument is located in one of the inner walls of Bath Abbey. Unfortunately, there are some needless pictures which cover one part of it but we can notice our mathematical interest in the carved drawing. It consist in an organon with a curious rectangular triangle in its base:

Photography by Carlos Dorce

Photography by Carlos Dorce

Harmony represented by the theorem of Pythagoras! The triangle is the classical (3,4,5) in its version (60, 80, 100) so it’s easy to identify the longitudes of the rest of the segments. The grave has the name of the composer Henry Harington who also studied medicine and  founded the Bath Harmonic Society. We can read:

Memoriae sacrum HENRICI HARINGTON. M:D: verè nobili HARINGTONORUM stirpe de Kelston. In agro Somerset: oriundi: Qui natus Septembris 29 A.D. 1727, obiit Januarii 15. A.D. 1816. Per sexaginta annos suae Bathoniae saluti. Omnibus officiis afsidue studebat. Optimas artes ad municipum suorum. Nelectacionnem et militatem excolens…

One of the books in the right lower corner of the picture is from Euclid:

Photography by Carlos Dorce

Photography by Carlos Dorce

Bath Abbey is a wonderful construction and it’s possible that almost everybody don’t notice this particular monument. So you have a very good oportunity to admire a mathematical objecti inside a church. If not, you can always follow the angels climbing on its beautiful facade:

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Bath Abbey in Bath (map)

A capital in Doge’s Palace in Venice

Doge's palace Photography by Carlos Dorce

Doge’s palace
Photography by Carlos Dorce

Doge’s Palace in Piazza San Marco is one of the most touristic attractions of Venice. The palace (XIVth c.) is very beaytiful and there is a hidden mathematical secret in one of the capitals of its columns. The capitals of the columns of the palace are dedicated to some biblical passages, quotidian Medieval scenes and… there is one capital dedicated to the Liberal Arts!  So we can find here our famous three most representative figures of the Arithmetic, the Geometry and the Astronomy:

Pythagoras Photography by Carlos Dorce

Pythagoras
Photography by Carlos Dorce

Euclid Photography by Carlos Dorce

Euclid
Photography by Carlos Dorce

Venice08

Ptolemy
Photography by Carlos Dorce

Pythagoras is counting money and next to his coins we can read the number 1399 and Euclid has a compass in one of his hands. The column is the first one next to the corner in front of the sea:

Photography by Carlos Dorce

Photography by Carlos Dorce

You must see this column in Venice!

Location: Piazza San Marco (map)

The wonderful vault of a Royal Library

Library of the monastery
Source: Wikimedia Commons

The King Philip II of Spain decided in 1550’s that he wanted to have a great library near his court in Madrid and he chose the new Monastery of San Lorenzo de El Escorial to place it in spite of other bigger villages. He didn’t want that the new library was a regular room inside a monastery so it had to be a very important place. Therefore, the library was placed on the second floor of the monastery just above his royal chambers but never above the basilica. Between 1565 and 1576, the king bought almost 5.000 books and manuscripts and the library became one of the most important libraries in all Europe.

Imaginary portrait of Juan de Herrera (1791) from
the book “Retratos de Españoles ilustres
Source: Wikimedia Commons

The mathematician and architect Juan de Herrera (1530-1597) designed a large room (54 m. long x 9 m. wide x 10 m. high) with big windows in both sides under a great barrel vault. This vault had to be decorated by an important painter and Philip II decided that Peregrino Tibaldi (1527–1596) had to be the right artist to do the work. Philip II was advised by Juan de Herrera and other humanists and he decided that the main subject of the paintings of the vault had to be the Liberal Arts. Furthermore, the seven arts would be together with the Philosophy and the Theology on both ends of the room. The Philosophy represented the compendium of the Human knowledge and she is accompanied by Aristotle, Plato, Seneca and Socrates:

The Philosophy

The Philosophy

The Theology is on the side next to the convent and she represented the Divine knowledge. Therefore the vault represented the way from the Human Philosophy to the Divine knowledge through the seven Liberal Arts: the Arithmetic, the Geometry, the Astronomy (Astrology), the Music, the Rhetoric, the Grammar and the Dialectic. We can see a mathematical detail on the fresco below the Philosophy: it represents the School of Athens and there is a discussion between the Academics leaded by Socrates and the Stoics leaded by Zeno of Elea.

The School of Athens

The School of Athens

The scholars aren’t listening to the speakers because each of them is “playing” with something different. We can see at the lower left corner a man measuring something with a compass and two books, a sphere and an armilar sphere, a dodecahedron and a compass in the middle of the picture:

Detail of the School of Athens

Detail of the School of Athens

Going from the Philosophy to the Theology, we arrive at the Arithmetic after admiring the Grammar, the Rhetoric and the Dialectic. The Arithmetic is a woman turned to a table with simple mathematical operations rounded by muscled young men with tablets with arithmetical operations ans counting with their fingers:

The Arithmetic

The Arithmetic

There is also a representation of the Queen of Saba talking with King Solomon  According to the Book of the Kings (I,10,1), the Queen of Saba went to meet Solomon to ask some enigmas to him so we can see a ruler, a balance and a tablet with some numbers written on it. In the red tablecloth we can read “Everything has number, weight and measure” in Hebrew:

King Salomon and the Queen of Saba talking about numbers

King Salomon and the Queen of Saba talking about numbers

The other panel next to the Arithmetic represents the school of the Gymnosophists who lived near the Nile and thought their philosophical theories from the numerical computations. In the middle of the picture we can see one of the gymnosophist with a compass looking at a triangle with the word “Anima” and the arithmetic progression 1, 2, 3 and 4 and the geometric 1, 3, 9 and 27 written on it. The other gymnosophists are computing with numbers written on the sand:

The Gymnosophists

The Gymnosophists

Finally, at both sides of the Arithmetic on the roof we find four people related with this subject: Archytas of Tarentum (c.428–c.347 BC) and Boethius (c.480-c.525) in one side and the Platonic Xenocrates (c.396/5 – 314/3 BC) and Jordan in the other. They are writing numbers in their tablets.

There is the Music after the Arithmetic and we find the Geometry after it:

The Geometry

The Geometry

She has a compass in one of her hands and the young men around her have different geometrical instruments. The two scenes which are on the corresponding walls next to her are dedicated to some Egyptian monks drawing geometrical figures on the sand…

Egyptian monks measuring the lands

Egyptian monks measuring the lands

and Archimedes’ death:

Archimedes' death

Archimedes’ death

Notice that Archimedes is drawing the demonstration of the Theorem of Pythagoras made by Euclid!

Finally, the four chosen figures are the Astronomer Aristarchus of Samos (IIIrd c. BC) and the Persian astrologer Abd del Aziz also known as Alcabitius (Xth century) in one side and Archimedes (c.287-212 BC) and Regiomontanus (1436-1476)  in the other. Aristarchus is measuring angles and has a dodecahedron at his feet, Alcabitius has a carpenter’s square, Archimedes has a compass and a sphere to measure the Earth and Regiomontanus is pointing at a dodecahedron.

The last Liberal Art is the Astrology. She is backed on a terrestrial globe and her eyes are looking at the sky. She has a compass in one of her hands and the little boys around her have an armilar sphere and some astronomical books:

The Astrology

The Astrology

In one of the two panels on the walls we can see Dionysius the Areopagite observing a solar eclipse the day of Jesuschrist’s death in Athens (Luke, 23,45) We can notice a quadrant and an astrolabe in the hands of the amazed men!

Dionysius the Areopagite observing a solar eclipse

Dionysius the Areopagite observing a solar eclipse

The other fresco represents King Ezekiel resting in bed and looking how time is delayed 15 years by God because of the repentance of his sins:

astronomia2

Ezekiel resting in bed

The four famous men are Euclid, Ptolemy, Alfonso X and Johannes of Sacrobosco. Euclid is represented here meaning the relationship between Astrology and Geometry. He has drawn three geometrical schemes. One is a triangle and a square inscribed in a circle and another square. Another scheme seems to be two overlaid squares partially hidden by Euclid’s name. In the middle of both pictures there is a man measuring the stars. Johannes of Sacrobosco has a quadrant in his right hand.

Euclid and Johannes of Sacrobosco

Euclid and Johannes of Sacrobosco

King Alfonso X of Castile (XIIIth. c) is the author of the Libros del Saber de Astronomía (“Books of the Astronomical knowledge”) and on the tablet which he has in his hands we notice a compass and the Ursa Maior (the compass is anachronistic!). His left hand has an open book with a horoscope

Alfonso X

Alfonso X

So you can see that this wonderful vault is an open mathematical book designed by Tibaldi and Juan de Herrera. I’ve been twice in the library and now I am waiting for the next time that I could enjoy this artistic part of the monastery of San Lorenzo de El Escorial.

Location: San Lorenzo de El Escorial (map)

The Liberal Arts in El Prado

The Seven Liberal Arts (c.1435)
Giovanni dal Ponte
Source: Museo del Prado

In room 56B of the Museo del Prado we can admire Giovanni dal Ponte’s Seven Liberal Arts. There are some masterpieces in the same room 56B as Fra Angelico’s Annuntiation and therefore people don’t use to stop in front of this mathematical panel. In the web of the museum we can read:

This decoration of the front of a chest depicts the seven Liberal Arts, accompanied by an equal number of figures that represent the most relevant personages in each discipline. All are being crowned with laurel wreaths by small angels.

Astronomy presides over the composition, carrying the heavenly sphere, with Ptolemy (first and second centuries A.D.) sitting at his feet and reading one of the thirteen volumes in which he surveyed the history of Greek astronomy. To the right, Geometry holds an angle iron and a compass, walking hand-in-hand with Euclid (fourth and third centuries B.C.). He is followed by Arithmetic, who carries a counting tablet and is accompanied by Pythagoras (sixth century B.C.). At the right end of the composition, Music bears an organ, followed by its inventor, Tubalcain. To the left of Astronomy, Rhetoric carries a scroll and is accompanied by Cicero (first century B.C.), who carries one of his texts. Then comes Dialectics, who carries an olive branch as a symbol of agreement among the Arts, and a scorpion, whose pincers represent the opposing positions of dialectical thought. He is accompanied by Aristotle. At the left end of the composition is Grammar, with its disciplines, preceeded by two children and accompanied by Donatus (fourth century A.D.) or Priscian (fifth and sixth centuries A.D.).

This work exemplifies the coexistence in the arts of that period between the late Gothic heritage —visible in the use of gold and lineal calligraphy— and the new Renaissance style, which is clear in the solid and monumental definition of the figures, recalling works by Masaccio (1401-1428)

Our Ptolemy, Pythagoras and Euclid are the guest stars again and we have here a mathematical reason to visit room 56B. For example, Euclid is following the Geometry who is wearing a ruler and a compass:

Detail of the painting: Euclid and the Arithmetic

Detail of the painting: Euclid and the Geometry

After Euclid, Pythagoras is following the Arithmetic who holds a counting tablet:

Detail of the painting: Pythagoras and the Arithmetic

Detail of the painting: Pythagoras and the Arithmetic

Finally, Ptolemy is below the Astronomy:

Detail of the painting: Ptolemy

Detail of the painting: Ptolemy

Location: Museo del Prado (map)

De la Hire’s Geometry

Allegory of the Geometry
Laurent de la Hire

De la Hire also painted the allegories of the Geometry and the Astronomy. The Geometry is  a young woman with a paper in her left hand in which we can see some geometrical constructions as the famous Euclid’s demonstration of the theorem of Pythagoras:

Detail of the painting

In her left hand, she also holds a right angle edge and a compass. We can also see that there is a sphinx and an Egyptian background in the right which represents the Egyptian origin of the Geometry. Proclus stated that:

Since, then, we have to consider the beginnings of the arts and sciences with reference to the particular cycle [of the series postulated by Aristotle] through which the universe is at present passing, we say that, according to most accounts, geometry was first discovered in Egypt, having had its origin in the measurement of areas. For this was a necessity for the Egyptians owing to the rising of the Nile which effaced the proper boundaries of everybody’s lands.

Herodotus says that Ramses II distributed the land among the Egyptians in equal rectangular plots on which he levied an annual tax. When therefore the river swept away a portion of a plot and the owner applied for a corresponding reduction in the tax, surveyors had to sent down to certify what the reduction of the area had been.

The Geometry is next to a globe which nods to the science devoted o measuring the Earth (Geo + metry = Earth + measurement). There is also a snake representing the ancient goddess of the Earth.

Location: The Toledo Art Museum (map)

The Spanish Chapel in Santa Maria Novella

PPhotography by Carlos Dorce

The Triumph of St. Thomas Aquinas
Photography by Carlos Dorce

The Spanish Chapel is one of the most wonderful chapels which can be enjoyed in Santa maria Novella. My students didn’t want to visit it but the teacher could convince most of them to enter the church and they weren’t disappointed.

The fresco entitled The Triumph of St. Thomas Aquinas was painted by Andrea di Bonaiuto (1365-1367) and it was dedicated to…

the great Dominican Doctor of the Church who, illuminated by the spirit of Wisdom, as described in the book lying open in his hands, and supported by the Theological and Cardinal Virtues and the study of the biblical writers of both the Old and New Testaments, defeats heresy, personified by Nestor, Arius and Averroes, and dominates the sciences. These are represented by fourteen allegorical female figures, alluding in part to the Sacred Sciences (left) and in part to the Liberal Arts (right). Each of these is accompanied by a historical personage, famous for having distinguished himself in that articular discipline.

The Theological and Cardinal Virtues are the Charity (over St. Thomas), the Faith and the Hope (at her respectively left and right sides), the Prudence (below the Faith), the Temperance (at the left side of the Prudence), The Justice (below the Hope) and the Fortitude (at her right). On the left of St. Thomas, there are (from left to right) the Biblical authors Job, David, St. Paul, St. Mark and St. Matthew, and on his right (from left to right), St. John the Evangelist, St. Luke, Moses, Isaiah and Solomon. Below St. Thomas, we find Nestor, Arius and Averroes:

AverroesPhotography by Carlos Dorce

Averroes
Photography by Carlos Dorce

The fourteen allegorical women and the corresponding eminent men are (from left to right): the Civil Law with Justinian, the Canonical Law with Clement V, the Philosophy with Aristotle, the Holy Scriptures with St. Jerome, the Theology with St. John of Damascus, the Contemplation with St. Dionysius the Areopagite, the Preaching with St. Augustine, the Arithmetic with Pythagoras, the Geometry with Euclid, the Astronomy with Ptolemy, the Music with Tubalcain, the Dialectics with Pietro Ispano (?), the Rhetoric with Cicero and the Grammar with Priscian (?):

Pythagoras, Euclid and PtolemyPhotography by Carlos Dorce

Pythagoras, Euclid and Ptolemy
Photography by Carlos Dorce

Finally, here you are my privileged students who enjoyed the wonderful Spanish Chapel:

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Santa Maria Novella (map)

Euclid in the Campanile of Florence

Nino Pisano's EuclidPhotography by Carlos Dorce

Nino Pisano’s Euclid
Photography by Carlos Dorce

In the XIVth century, the Italian sculptor Nino Pisano collaborated in the panels for the Campanile (the bell tower) of Santa Maria del Fiore of Firenze. One of his panels has a lot of interest here due to the picture represented in it: it’s Euclid with a compass!

You can find this panel next to the exit door of the church and this isn`t the only mathematical picture that you can find in it! If you follow the frieze you’ll find Ptolemy looking at the night sky using a quadrant in the south side of Giotto’s bell tower:

Ptolemy in Giotto's CampanilePhotography by Carlos Dorce

Ptolemy in Giotto’s Campanile
Photography by Carlos Dorce

Location: Giotto’s Campanile (map)

The Liberal Arts in the Spanish Chapel

Pythagoras, Euclid and Ptolemy
The Spanish Chapel at Santa Maria Novella (Florence)

This wonderful picture shows Pythagoras (left), Euclid (middle) and Ptolemy (right) sitting in front of the Arithmetic, the Geometry and the Astronomy respectively. We can see the Arithmetic holding a tablet, the Geometry holding a compass and the Astronomy holding an armillar sphere and the three men are holding their books. Of course, Euclid has his Elements and Ptolemy may be writing the Almagest. This section is part of the painting which we find in the Spanish Chapel at Santa Maria Novella. According to Wikipedia, the Spanish Chapel is the former chapter house of the monastery. It is situated at the north side of the Chiostro Verde and it was commissioned by Buonamico (Mico) Guidalotti as his funerary chapel. Construction started c. 1343 and was finished in 1355. The Guidalotti chapel was later called “Spanish Chapel”, because Cosimo I assigned it to Eleonora of Toledo and her Spanish retinue. The Spanish Chapel was decorated from 1365 to 1367 by Andrea di Bonaiuto and the large fresco on the right wall depicts the Allegory of the Active and Triumphant Church and of the Dominican order. It is especially interesting for us the fresco called The Triumph of Saint Thomas Aquinas:

The Triumph of Saint Thomas Aquinas

We can see St. Thomas Aquinas holding the Book of Wisdom with the words:

And so I prayed, and understanding was given me; I entreated, and the spirit of Wisdom came to me. I esteemed her more than scepters and thrones; compared with her, I held riches as nothing.

Book of Wisdom 7:7, 8

There are seven figures over him which are the Seven Virtues: the three figures on the top from left to right are the Faith (holding a cross), the Charity (with her arms open and the Hope (holding an olive branch); the four figures on the bottom from left to right are: the Temperance (holding a upright branch of peace), the Prudence (holding a book to educate people in the correct way), the Justice (holding a scepter and the crown of the power) and the Fortitude (wearing an armor and holding a sword and a tower).

Next to St. Thomas sitting in the same row as him there are ten Biblical figures (from left to right): Job, David, Saint Paul, Matthew, John, Luke, Moses (holding the two sheets of the Law), Isaiah and Solomon. Under St. Thomas there are three heretic figures: Nestor, Arius and Ibn Rushd (Averroes).

Averroes

Finally, the bottom row is full of allegorical figures. On the image’s left (from left to right): the Civil Law(and the Emperor Justinian whose code was the law of the Roman Empire, sitting at her feet), the Canonical Law (and Pope Clement V), the Philosophy (and Aristotle), the Holy Scripture (and Jerome), the Theology (and John of Damascus), the Contemplation (and Dionysius the Areopagite) and the Preaching (and St. Agustine). On the right, the Arithmetic (and Pythagoras), the Geometry (and Euclid), the Astronomy (and Ptolemy) and the Music (and Tubal Cain) represent the Quadrivium. Finally, the Dialectics (and Pietro Ispana), the Rhetoric (and Cicero) and Grammar (and Priscian) represent the Trivium.

Location: Santa Maria Novella at Florence (map)