Category Archives: France

The Pascaline

Pascaline (1652). Musée des arts et métiers (París).

When Blaise Pascal (1623 – 1662) was only 17 y.o., his father Etienne Pascal (another great mathematician) was appointed chief of the taxmen in Normandy, France. What did this clever teenager think when saw his poor father calculating everytime? Well, as good son as he was, he decided to give him the best present which Etienne could receive: a calculating machine (1652) which was recalled with the name Pascaline a lot of years later. This analogic machine was only able to add numbers but Etienne’s work became easier than before. Pascal introduced several improvements to his invention and the Pascaline was patented. Unfortunately, the manufacturing of the machine was too expenseive and Pascal’s small business could only make fifty calculators and one of them still survives in the Musée des arts et métiers in Paris.

In addition , we also report the visit to see some of the instruments that Pascal used to study the atmospheric pressure.

Location: Musée des arts et métiers in Paris (map)

A Trigonometric lesson


Teacher drawing a geometrical figure (1658). Painter: Ferdinand Bol (1616-1680).  Source: Wikimedia Commons

This is one of the mathematical pictures which can be seen in Louvre Museum! Dutch Ferdinand Bol painted a maths teacher explaining a lesson about Trigonometry with the aid of the famous trigonometric circle and their geometrical representation. I think that there will be a good idea travelling to Paris with all my students and explain the sinus, cosinus, tangent,… in the second floor of this wonderful museum. What do you think?

Location: Louvre Museum in Paris (map)

And finally… the Astronomy!

Allegory of the Astronomy
Laurent de la Hire
Musée des Beaux-Arts (Orleans)

The third mathematical chapter of the De la Hire’s Liberal Arts is the Astronomy. We can see in the painting a lot of astronomical instruments and books!

Location: Musée des Beaux-Arts (map)

The Labyrinth of Amiens

Cathedral of Amiens
Source: Wikimedia Commons

The cathedral of Amiens is tha tallest complete cathedral in France. According to Wikipedia, its stone-vaulted nave reaching a height of 42.30 metres (138.8 ft). It also has the greatest interior volume of any French cathedral, estimated at 200,000 cubic metres (260,000 cu yd). The cathedral was built between 1220 and c.1270 and has been listed as a UNESCO World Heritage Site since 1981. Although it has lost most of its original stained glass, Amiens Cathedral is renowned for the quality and quantity of early 13th century Gothic sculpture in the main west facade and the south transept portal, and a large quantity of polychrome sculpture from later periods inside the building.

The reason why it is referred now is its maze (yesterday I began to talk about it) which represented the symbolic journey of salvation. The original maze is dated around 1288 and the modern path is a very good copy from it.

I’ve found this very beautiful picture of the maze in the net:

Maze in the cathedral of Amiens

I am sure that I am going to try to travel to Amiens to make a good picture of the maze!

Location: Cathedral of Amiens (map)

Labyrinths in Cathedrals

Figure on page 67 from The Book of Fun, Mirth and Mystery (1924)

Figure on page 67 from The Book of Fun, Mirth and Mystery (1924)

I am so lucky because I’ve been able to buy R. A. Hummerston’s The Book of Fun, Mirth & Mystery (1924). I was writing an article for a Catalan magazine about Mathematics and I needed to read some paragraphs of this interesting book so I decided to buy it in a special book shop from London. Yesterday, I was reading quietly some of the very funny articles of the book and… I discovered page 67 entitled “Labyrinths in Cathedrals”:

It was customary during the Middle Ages to insert in the floor of the nave of certain Cathedrals a maze or labyrinth of black and white stones or coloured tiles. These labyrinths were also known as “Roads of Jerusalem”, owing, it is supposed, to the fact that workshippers were accustomed to traverse the sharp stones of the maze upon their knees.

Fig. 1, in the Cathedral of Sens, is on circular form, and was encrusted with lead. It measured 66 feet across, and the length of the circuit, with took an hour to traverse, was over a mile and a quarter. The maze at Saint Owen (fig. 2) was formed of blue and yellow tiles, and measured about three-quarters of a mile. The labyrinth of Saint Quentin (fig. 3) was taken away in 1792, because of children playing the game of, “Who can get into it quickest?” disturbed the workshippers.

The maze of Bayeux (fig. 4) is formed of black squares, bearing yellow riffings, roses, and armorial bearings.

I’ve looked for real images of these mazes and I’ve found some of them in the net as the next one:

Source: Not all those who wander are lost (blog)

And this picture from the Cathedral of Bayeux:

Source: Not all those who wander are lost (blog)

I think that this is a very interesting subject which must be studied better. I’m going to spent some time looking for information about some mazes inside European Medieval Cathedrals. Maybe tomorrow I’ll have some interesting information!

A Roman abacus in London

Roman abacus
Science Museum (London)
Photography by Carlos Dorce

This is a replica of an ancient Roman abacus which we can find in the great Science Museum of London (there are some abacus like this in other museums as the British Museum or the National Library of Paris). It consists in a small metallic plate with nine parallel slits: the right slot is related with ounces and next to it, from right to left, the other slots are for units, tens, hundreds, thousands units, thousand tens, thousand hundreds and millions, with its corresponding figures:

Roman figures

The seven left slots are divided in two different sections: the upper one have one moving pieces and the lower have four moving pieces. The Roman represented the units of each power of 10 putting the lower pieces near the centre of the abacus if they were less than 5. When they need to represent 5 units, they only moved one upper piece to the centre. Thus, number 6 was one upper piece and one lower piece in the centre, 7 was one upper piece and two lower ones,… For example, the abacus located in the British Museum represents number 7.656.877 (or a similar number):

Roman abacus
British Museum (London)
Font: BM website

The two first right slots corresponded to the ounces: the Roman unit (= as) were divided in 12 equal ounces. The slot marked with the symbol O had five moving pieces in the lower slot and it was used to count the multiples of 11 and 12 ounces. The first slot was divided in three parts with four moving pieces:

If the upper piece was next to the “pound” symbol (in the left), the piece value was 1/2 ounce or 1/24 of as (= semuncia); if it was next to the symbol in the middle of the slot (= sicilius), the piece value was 1/4 ounce or 1/48 of as; and if it was next to the “number 2” (in the right), the piece value was 1/72 of as (= duella).

This kind of abacus were very popular as small fast calculators among the Romans as we can see in a marble sarcophagus in the Capitoline Museums of Rome: there is a young boy standing at his feet holding an abacus (he is probably counting the money which the deceased is holding in a money purse):

A Roman abacus represented in a sarcophagus (IInd c. AD)
Capitoline Museums (Rome)

 Location: Science Museum at London (map)

A Parisian Arithmetic

The Arithmetic (XVIth century)
Musée de Cluny (Paris)

This medieval Flemish tapestry is entitled L’Arithmétique and it is part of an anonymous series on the theme of the seven liberal arts. The composition is organized around a young woman standing behind a table. This woman is teaching some traders and bankers how to manage with the Arabic figures. The inscription says:

Monstrat ars numeri que virtus possit habere. / Explico per que sit proportio rerum.

I should look for the Geometry and the Astronomy to complete this allegorical series, don’t I?

Location: Musée of Cluny at Paris (map)

Two tax-gatherers by Van Reymerswaele

Two tax-gatherers (c.1540)
Marinus van Reymerswaele (c.1490–c.1546)
National Gallery (London). Font: Wikimedia Commons

The Two-gatherers is a painting attributed to Marinus van Reymerswaele which we can see in the National Gallery of London and also in the Louvre Museum of Paris (the French copy was executed before the English one) although we have simplified versions in Belgium and Poland. A very good article written by Paul Ackroyd, Rachel Billinge, Lorne Campbell and Jo Kirby gives the details of the painting:

Both the London and the Paris pictures show, behind the two men, a wooden cupboard on top of which are piled documents, an oval deed-box, a ledger, a turned wooden sand-box and a brass candlestick; across its base lies a pair of snuffers. The folded document above the head of the man on our right is a deed issued, according to the inscription, in 1515 by two aldermen of Reymerswale. The name of the first is concealed by the folding of the document; the second, Cornelis Danielsz, was indeed an alderman in 1514-15. The man on the left is writing in his ledger an account of the income of  a town over a period of 7 months -from the excise duties on wine and beer, the ‘first-bridge’, the weight-house the ‘hall’, the ferries, fees for deeds, charges raised for specific expenses, loans and the civic mills.

I still continue counting money!

Location 1: National Gallery in London (map)

Location 2: Louvre Museum in Paris (map)

More bankers counting money

The Banker and His Wife (XVIth c)
Marinus van Reymerswaele (c.1490–c.1546)
Musée des Beaux-Arts (Valenciennes). Font: Wikimedia Commons

I didn’t know that Marynus van Reymerswaele had painted a similar situation to the last post: a banker or a moneylender counting money. I’m surprised! If I had to say something about this painting I’d notice that here there is the son of the banker and his wife behind her. And… have you seen that the banker is weighting the coins?

I am sure that this painting is not going to be the last financial situation represented by Van Reymerswaele!

Location: Musée des Beaux-Arts in Valenciennes (map)

Van Gent’s Claudius Ptolemy

Portrait of Claudius Ptolemy (c. 1475)
Justus van Gent (c.1410 – c.1480)
Louvre Museum. Font: Wikimedia Commons

This is not the first time in which I talk about Justus van Gent’s paintings. Indeed, the first post of this blog was dedicated to the portrait of Euclid of Alexandria made by this Dutch painter. This portrait is in Louvre Museum and it’s part of the serie of portraits of some important people made by Van Gent.

Claudius Ptolemy lived and worked in the IInd century AD in Alexandria. We know little about him and we can place his life from his own astronomical observations recorded in his great work entitled Mathematical Collection. His first observation was an eclipse of the Moon made in Alexandria in the 5 April of 125 AD and the last one was the observation of the maximum elongation of Mercury made in the 2 February of 141 AD. This recorded data mean that Ptolemy worked in his astronomical book in the period 125-141 AD. Furthermore, we know that in the year 147/148 AD he erected a stele in the town of Canopus about 25 Km East of Alexandria. we can also observe that his name Claudius Ptolemy is a good definition of his life: Claudios is a Greek name whereas Ptolemaios could indicate that he came from one of the various Egyptian towns named after the Ptolemaic kings.

Ptolemy’s Mathematical Collection was the most important Greek astronomical work. Later the Arabs called it with the superlative Al-Majistî (Almagest)  and with this name the Latin Europe adopted as the referencial astronomical handbook. In the IVth century, Pappus of Alexandria (c.290-c.350) made a Commentary on it and part of the commentary on Book V (the Almagest is divided in 13 Books) as well as his commentary on Book VI are actually extant in the original. Theon of Alexandria (c.335–c. 405) wrote another commentary on the Mathematical Collection in 11 books incorporating as much as was available of Pappus’ work. Theon was assisted by his daughter Hypatia of Alexandria (c.360–March 415) and the whole text was published at Basel by Joachim Camerarius (April 12, 1500 – April 17, 1574) in 1538.

Joachim Camerarius (1500-1574)
Font: Wikimedia Commons

The Mathematical Collection arrived at the Muslim World and it was translated into Arabic, first by translators unnamed at the instance of Yahyâ b. Khâlid b. Barmak, then by al-Hajjâj, the translator of Euclid (c.786-835), and again by Ishaq b. Hunain (d.910) whose translation was improved by Thâbit b. Qurra (c.826–February 18, 901). The first edition to be published (Venice, 1515) was the Latin translation made by Gherard of Cremona from the Arabic, which was finished in 1175. Although there was a previous Latin translation from the Greek, the first Latin translation from the Greek to be published was that made by Georgius of Trebizond in 1451 and the editio princeps of the Greek text was brought out by Grynaeus at Basel in 1538.

According to Sir Thomas Heath in his A History of Greek Mathematics (II, 275), the Almagest is most valuable for the reason  that it contains very full particulars of observations and investigations by Hipparchus, as well as of the earlier observations recorded by him. The indispensable preliminaries to the study of the Ptolemaic system, general explanations of the different motions of the heavenly bodies in relation to the Earth as centre, propositions required for the preparation of Tables of Chords, the Table itself, some propositions in spherical trigonometry,… are in Books I and II; Book II deals with the length of the year and the motion of the Sun on the eccentric and epicycle hypotheses; Book IV is about the length of the months and the theory of the Moon; in Book V we find the construction of an astrolabe and the theory of the Moon continued, the diameters of the Sun, the Moon and the Earth’s shadow, the distances between them and their dimensions; the conjunctions and oppositions of Sun and Moon, the solar and lunar eclipses and their periods are studied in Book VI; Books VII and VIII are about fixed stars and the precession of the equinoxes and Books IX-XIII are devoted to the movements of the planets.

Illustration of the Ptolemic system
Harmonia Macrocosmica by Andreas Cellarius (1661). Font: Wikimedia Commons.

Location: Louvre Museum in Paris (map)