Tag Archives: Abacus

The Museum of Mathematics in the Collegius Maius

Photography by Carlos Dorce

Photography by Carlos Dorce

Another interesting museum located in the Collegius Maius of the Jagiellonian University is an exhibition about mathematics where children can play and learn a lot! There are old calculators from the 20th century…

Photography by Carlos Dorce

Photography by Carlos Dorce

…abacus and slide rulers:

Photography by Carlos Dorce

Photography by Carlos Dorce

Children can play with Geography and learn that straight lines in a map are not the shortest ways for the planes:

Photography by Carlos Dorce

Photography by Carlos Dorce

They can also learn the theorem of Pythagoras scrolling this interesting figure:

Photography by Carlos Dorce

Photography by Carlos Dorce

There are polyhedra and a lot of geometrical and topological games:

Photography by Carlos Dorce

Photography by Carlos Dorce

The museum is very small but all the tourist are inside Collegius Maius so you can be very quiet watching all the exhibited objects and toys, like the Rodin’s Thinker:

Photography by Carlos Dorce

Photography by Cristina Martínez

Finally… here you have my two children playing with Eulerian graphs! They are lovely! Aren’t they?

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Collegius Maius (map)

Advertisements

Kircher’s Organum Mathematicum

Photography by Carlos Dorce

Photography by Carlos Dorce

Yesterday I didn’t remember to show Kircher’s Organum Mathematicum:

Organum Mathematicum was invented in 1661 by the Jesuit astronomer and mathematician Athanasius Kircher. This device is a comprehensive portable encyclopedia and is designed for the following disciplines: arithmetic, geometry, fortifications, chronology, gnomonics (sundials), astronomy, astrology, steganography (encoding) and music. The case contains tables for calculations without ‘tiring the mind’. Each of the nine disciplines contains 24 flat boards of different colours, with definitions and information.

This is Athanasius Kicher:

Athanasius Kircher (1602-1680) pictured in the book Mundus Subterraneus (1664) by Cornelius Bloemart (1603-1680). Source: Wikimedia Commons

Of course, in the exhibition you can also find compasses, rules, abacus, slide rules, the Napier bones,…

Photography by Carlos Dorce

Photography by Carlos Dorce

…and calculators from the 20th century:

Photography by Carlos Dorce

Photography by Carlos Dorce

LocationNational Technical Museum in Prague (map)

Whipple Museum of the History of Science (and V)

Photography by Carlos Dorce

Photography by Carlos Dorce

The last post dedicated to Whipple Museum is for the calculators and their predecessors. All these objects are located in the next room which contains a lot of things in shelves and drawers as if they were in a store. There are calculators and a drawer dedicated to the Napier’s rods or bones:

Photography by Carlos Dorce

Photography by Carlos Dorce

There also are some interesting abacus like this one:

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally, different slide rules fill some drawers. You must be very patient and it’s a pity that this museum isn’t located in a larger building. 

WM27

Photography by Carlos Dorce

I have more pictures but you must go there if you want to have a real idea of the exhibition. It’s impossible to summarize it in some photos!

LocationWhipple Museum of the History of Science in Cambridge (map)

The Science Museum in London (III)

Photography by Carlos Dorce

Photography by Carlos Dorce

A lot of calculating machine from different times are on display and John Napier and his arithmatic inventions are part of this trasure. There is his Rabdologia (1617) where he described his famous Napier’s bones or rods and we have also some examples of them.

Photography by Carlos Dorce

Photography by Carlos Dorce

The box located in the bottom of the picture is Napier’s own Napier’s bones. There other sexagessimal bones are also very curious:

Photography by Carlos Dorce

Photography by Carlos Dorce

Napier’s bones were very popular and they were used until the 19th century as we can see them in this wooden box:

Photography by Carlos Dorce

Photography by Carlos Dorce

The exhibiion continues with “The Art of reckoning”:

As the level of trade increased throughout the Renaissance, the European counting boards and abacusses were gradually replaced by the use of pen and paper. Merchants andgentlemen taught themselves and their sons the new method.

In England, during the 16th and 17th centuries, many books were written encouraging people to learn arithmetic, and many gadgets invented to aid the beginner. By the 18th century, ready reckoners, devices to simplify calculation, were available to many tradesmen.

These words introduces all the calculating machines world but it’s also the moment of the former counting methods. For example, what do you think about this replica of a 16th century counting cloth?

Photography by Carlos Dorce

Photography by Carlos Dorce

LocationScience Museum in London (map)

The Museum of the History of Science (III)

Photography by Carlos Dorce

Photography by Carlos Dorce

The mathematicalinstruments are also part of the collection. For example, there is a 17th-century box with some wooden polyhedra and some models for the study of Spherical Trigonometry:

Photography by Carlos Dorce

Photography by Carlos Dorce

And more wooden models in this mathematical box:

Photography by Carlos Dorce

Photography by Carlos Dorce

John Rowley was one of the leading London instrument makers in the late 17th and early 18th centuries and there are some mathematical compasses and instruments made by him in the collection:

Photography by Carlos Dorce

Photography by Carlos Dorce

Object number 1 is a proportional compass meanwhile number 5 is a ruler with pencil and dividers and number 6 is a slide rule.

Of course, if we are in a museum where the History of the Mathematics is exhibited, Napier’s rods must be here:

Photography by Carlos Dorce

Photography by Carlos Dorce

Unsigned, English, c. 1679

Photography by Carlos Dorce

Photography by Carlos Dorce

Unsigned, English, 17th century?

Photography by Carlos Dorce

Photography by Carlos Dorce

Charles Cotterel’s Arithmetical Compendium, Unsigned, English, c.1670

As in the Pitt Rivers Museum, the abacus also have their space in the showcases:

Oriental abacuses use beads on rods to represent numbers. Addition and substraction can be quickly performed by flicking the beads to and fro. Rather than ten beads in each column, the Chinese abacus uses five ‘unit’ beads and two ‘five’ beads (1 and 2). The Japanese abacus has just four ‘unit’ beads and one ‘five’ in each column (3).

Photography by Carlos Dorce

Photography by Carlos Dorce

The next Arithmetical instrument was made in the 18th century for counting. Addition was performed by turning the brass discs but since there isn’t no mechanism it was up to the user to carry tens:

Photography by Carlos Dorce

Photography by Carlos Dorce

I am going to finish this post with this reproduction of the Measurers by the Baroque painter Van Balen (1575 – 17 July 1632) which can be seen upstairs:

The Measurers. Hendrick van Balen

Location: The Museum of the History of Science (map)

The Pitt Rivers Museum

The Pitt Rivers Museum Photography by Carlos Dorce

The Pitt Rivers Museum
Photography by Carlos Dorce

The Pitt Rivers Museum cares for the University of Oxford’s collection of anthropology and world archaeology. It is next to the Oxford University Museum of Natural History which was closed in August and it was very surprising for me and also for my kids (I think it’s an idela museum for children!).

Photography by Carlos Dorce

Photography by Carlos Dorce

There are some interesting mathematical objects in the collection and I am going to list some of them. First of all, we must focus our interest in the showcase dedicated to “counting”:

Photography by Carlos Dorce

Photography by Carlos Dorce

There are some old counting strings:

Photography by Carlos Dorce

Photography by Carlos Dorce

and this “swampan”:

Photography by Carlos Dorce

Photography by Carlos Dorce

“Swampan” or calculating board with sliding beads, used in casting accounts. The two upper balls on each bar = 5 each, the lower balls = units, similar to the roman abacus. China.

There also is the typical “soroban” which is next to a icture of a Roman abacus and in the upper right corner of the next picture:

Photography by Carlos Dorce

Photography by Carlos Dorce

“Soroban” or calculating board for casting accounts, similar to and derived from the Chinese “swampan”. Japan.

There is also a picture of a “quipu”.

There also are astrolabes and clocks. For example, there are a brass astrolabe dated in 1673 and sme interesting portable sundials:

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally there is some showcases dedicated to games, dice, chess,… in the upper floor:

Photography by Carlos Dorce

Photography by Carlos Dorce

Before finishing this post, look at the next picture and try to guess who is this great man:

Photography by Carlos Dorce

Photography by Carlos Dorce

The Oxford University Museum of Natural History was closed but it was possible to walk around the inner yard and it was possible to see one of the famous statues dedicated to the great scientific men. So it was possible to take a photography of Gottfried Wilhelm Leibniz!

Location: Pitt Rivers Museum (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)

Darius vase

Darius' vaseSource: Museo Archeologico Nazionale di Napoli

Darius vase
Source: Museo Archeologico Nazionale di Napoli

According to Wikipedia, The Darius Painter was an Apulian vase painter and the most eminent representative at the end of the “Ornate Style” in South Italian red-figure vase painting. His works were produced between 340 and 320 BC. The Darius Painter’s conventional name is derived from his name vase, the “Darius Vase”, which was discovered in 1851 near Canosa di Puglia and now on display at the Museo Archaeologico Nazionale, Naples (H3253). Many of his works, mostly volute kraters, amphorae and loutrophoroi, are of large dimensions. He most frequently depicted theatrical scenes, especially ones from the Classical tragedies by Euripides, and mythological themes. A number of mythological motifs not represented in surviving literary texts are known exclusively from his vases. On other shapes, especially pelikes, he also painted as wedding scenes, erotes, women, and dionysiac motifs. In contrast to other contemporary painters, sepulchral scenes (naiskos vases) by him are rare; where such motifs occur, they are virtually always on the back of the vessel. Some of his paintings, like those on the Darius Vase itself, show historical subjects.

The vase conserved in Naples is important because of its representation of a man counting on a board:

Pg.072_imatge 01 (original)

Nevertheless, the description of this Persian vase in the Museo Archeologio web page doesn’t mention the mathematical interest:

This volute krater with grotesque masks is undoubtedly one of the most famous vases in the collection. On the neck of side A there is a painted scene of an Amazonomachia, with Amazons wearing oriental costumes and armed with battle axes engaged in duels – which take place on two levels – against naked Greek warriors wearing crested Corinthian helmets who are equipped with a circular shield and long spear. The figurative decoration of the body is organised into three registers, in each of which there is a seated figure in a central position. In the upper register is Zeus, with a wingedNike kneeling down; to the left are Aphrodite with a swan on her lap and Artemis on a deer, while on the other side are Athena, Hellas, Achates with two torches and Asia, seated on an altar with the image of a deity. The central band shows Darius on his throne, behind whom stands figures who are presumably members of his bodyguard, carefully listening to a messenger standing erect in the king’s presence on a circular podium, surrounded by seated dignitaries and, it would seem, his pedagogue, who can be identified as the old man leaning on a stick. The last frieze shows five Orientals around a seated man, presumably the treasurer; three of them are kneeling, pleading for mercy. Side B, which has a similar structure, shows the myth of Bellerophon: in the upper part, Bellerophon rides Pegasus while a winged Nikecrowns him with a laurel wreath; to the left, a naked young man clasps a laurel branch in his hands while in front of him Poseidon, holding his trident in his left hand, sits on a rocky spur. To the right Pan, holding a pyxis and laurel branch, stands opposite Athena, seated on a rock, with a long spear in his left hand. In the middle of the central frieze is Chimera, depicted as a two-headed monster with a leonine body, the head of a lion and a goat, and the tail of a snake, while on the right two Amazons are fleeing; on the left there are two more Amazons, one of whom is attacking. The lowest register shows two fallen Amazons, armed with a spear and an axe respectively, and a marsh bird. On the neck of this side of the picture is a Dionysian scene with a group featuring a Maenad and Silenus on the left, a man and a woman on the sides of the fountain and lastly a second Maenad. The main scene has been interpreted in various ways: the identification of the characters is certain since beside each figure appears the name. What has proven more difficult is contextualising it. Some scholars have argued that it shows a scene from Phrynicos’ tragedy in which Persia is about to declare war on Hellas; more recently, an analysis of the compositional structure has led to the conclusion that the space is used symbolically to allude to the actual space of the theatre with the chorus in the lower register, theproscenium in the centre and the tribune of the gods above. Alternatively, the entire decorative layout could refer to the revolt of the Greek cities of Asia and may re-echo the troubled period of the wars against the Lucanians and the Messapians in Magna Graecia, specifically during the period in which the Darius painter was working.

The man of the picture is a tax collector counting on a special board in which we can read the letters M (= 10.000), Ψ (= 1.000), H (= 100) and Δ (= 10) and the former symbols used to represent the Greek coins (drachma, obol, half an obol and a quarter of obol). The collector has an opened book in which we can read the letters T A Λ and N. These letters correspond to another Greek coin named talent so we can suppose that this counting boards were used to make calculus with different kinds of coins.

Location: Museo Archeologico Nazionale of Naples (map)

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)

Estonian History Museum

Estonian History Museum (Tallinn)
Photography by Carlos Dorce

The Estonian History Museum is one of the most interesting attractions of Tallinn. We can read in the official web site that the story of the Museum began in 1802, when Tallinn’s town hall pharmacist, Johann Burchard (1776–1838), started a collection called Mon Faible (“My weakness”).  The first exhibit was a Chinese opium pipe and Burchard put on the exhibition “Antiquities and rarities” at the House of the Brotherhood of the Blackheads in 1822. Twenty years later, the Estonian Literature Society was founded in Tallinn and one of its aims was to establish a museum “to broaden our knowledge of this country by studying its history, art, manufacturing, technology and nature”. Extensive collections were compiled over the following twenty years, which formed the basis of the Provincial Museum of the Estonian Literature Society, founded in 1864 at the house of St. Canute’s Guild.

In 1911, the Estonian Literature Society purchased premises on Toompea at 6 Kohtu Street, where the museum’s innovative activities could flourish. As the only museum in the city, it became an important focal point in Tallinn’s cultural life with educational lectures and exhibitions. The Museum retained its important position through its valuable collections although the Estonian National Museum in Tartu (founded in 1909) became the most important museum in the Republic of Estonia. In 1940 Estonia was incorporated to the U.S.S.R. and  The museum was nationalized and the History Museum of the Estonian Soviet Socialist Republic was established in its place. Overbearing ideological pressure ruined the museum in the following years. In addition to subjugating the museum employees, items that were deemed harmful were eliminated, which meant destroying everything that reminded people of the republic of Estonia. The Museum preserved the greater part of the main collection and it moved to its current location at the Great Guild Hall in 1952. Finally, in 1989 the Museum was renamed the Estonian History Museum and many important exhibitions that introduced the contemporary history of Estonia were held in the late 1980s and early 1990s.

Estonia is a very young country and in the main exhibition we find a lot of references to its former background and to the Soviet occupation. Among all these interesting object there also are some references to mathematical objects. For example, we can see different scholar material as two geometrical figures…

Two geometrical figures (1920-1930)
Photography by Carlos Dorce

 and a mathematical ruler:

Mathematical ruler (c.1950)
Photography by Carlos Dorce

The this section of the exhibition about the schools we can read that the first educated Estonians were said to be monk Nicolaus, who was appointed by the Pope to carry out missionary work in Estonia in 1170, and the parish priest Johannes, who worked with the Livonians at the beginning of the 13th century. In the mid-13th century cathedral schools were established in Tartu, Pärnu, Tallinn and Haapsalu. Next to the monasteries, there were monastic schools, and in the 15th and 16th centuries, town schools were founded. The 17th century was of great importance to Estonian cultural history because it was when that high schools, along with print shops, were first founded in the country. The University of Tartu was founded in 1632. In the 1680s a network of village schools was created but regular school education in rural areas only took hold in the early 19th century. Peasant schools were mandatory, and students were taught in the native language; however, most of the children were allowed to study at home. Writing and arithmetic were taught starting in the mid-19th century. The 1897 census revealed that 94% of Estonians were able to read, which was the highest percentage among the nations of the Russian Empire. The Imperial University of Tartu reopened in 1802 and became a centre of science and intellectuals in the Baltic provinces. In 1803 the university employed an Estonian and a Latvian-language lecturer. On December 1, 1919, the University of Tartu of the Republic of Estonia was opened, marking the beginning of regular higher education in the Estonian language.

Another interesting object is in the lower lever: it’s an abacus which was taken from Ivan Mazepa’s (1639-1709) tent after he Battle of Poltava in 1709. The wooden box has a framed mirror and an abacus with beads made of white and red bone. Mazepa wanted to unite the Ukranian terriories into a single state. During the Great Northern War he initially supported Russia but later changed sides and lost the Battle of Poltava alongside Swedish troops.

Mazepa’s abacus
Photography by Carlos Dorce

Finally, there is a three-dimensional wooden puzzle made in 1920s or 1930s. It consists in 16 pieces. Different wooden puzzles were made in Estonia already in the 19th century:

Estonian Puzzle (c.1925)
Photography by Carlos Dorce

As we can see, the Estonian History Museum is very interesting for a Mathematical lover. Furthermore, there is another object which must be mentioned but I’m going to talk about it in the next post.

Location: Estonian History Museum at Tallinn (map)