Monthly Archives: September, 2013

A sundial in Dean Street

Photography by Carlos Dorce

Photography by Carlos Dorce

There is a very big sundial over the pub called Golden Lion in Dean Street. It’s a nice place to drink a beer and you can check in the sundial if it’s too late to continue drinking more beers!

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: The Golden Lion (map)

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Brook Taylor’s tomb

Church of St. Anne Photography by Carlos Dorce

Church of St. Anne
Photography by Carlos Dorce

I tried to find Brook Taylor’s tomb when I was in London. Taylor (1685-1731) is well known from his famous Taylor series which we can find in all the mathematical texts of our students. He contributed to the development of calculus (he also made some experiments in magnetism) and became one of the most important English mathematicians of the 17th century.

Portrait of Brook Taylor attributed to the studio of Hans Hysing.                            Source: BBC’s web

When I read that Taylor was buried in St. Anne’s churchyard I thought that I had to find a grave in a cemetery but… the churchyard is a public garden nowadays! So the tombs have been replaced by tables, chairs, cold beverages and music. For the moment I was content with the information I read on a bulletin board:

The Church of St. Anne, Soho, built 1677-1685 and probably designed by Sir Christopher Wren or William Talman, possibly both, was consecrated by Henry Compton, Bishop of London, on 21st March 1686. […] The churchyard was included in the original proposals for the establishment of a new parish and church, and for some one hundred and sixty years it served as the last resting place for most of Soho’s cosmopolitan population. It has been stimated that there were over 100.000 interments in this plot of just three-quarters of an acre, with about 14.000 of them in the period 1830-1850.

By the middle of the 19th century the insanitary conditions caused by overcrowded burial grounds had given rise to so much public concern that Parliament was obliged to pass a series of Burials Acts. In 1853 churchyards within the Metropolis were closed to further interments which thereafter took place in the new suburban cemeteries. […]

So, Taylor’s tomb is somewhere in this plot of three-quarters of an acre together with 99.999 people more.

Location: St. Anne’s Churchyard (map)

Egyptian Mathematics in the BM

Photography by Carlos Dorce

Photography by Carlos Dorce

I am going to begin my Egyptian visit to the British Museum with the limestone game-board in the form of a coiled snake used for the game called “mehen” (2890-2686 BC). The body of the coiled snake is divided into rectangular spaces but the number of these spaces is not important for the game.

Game-boards in the form of coiled snakes are known from the Early Dynastic period whengames became a regular item of tomb equipment. Several examples were discovered in the excavation of the Second-Dinasty tomb of King Khasekhemwy at Abydos. The game for which the snake-board was used was called mehen and although the exact method of play is not known, later representations show that it involved two players. The game-pieces consisted of spherical stone marbles and small figures of lions and lionesses usually made of bone or ivory.

The other popular game in Ancient Egypt was the ‘senet’ and there is one ivory sene board with a drawer for storing the gaming pieces with the glazed gaming pieces. This second board is in the special exhibition dedicated to the tomb-chapel of Nebamun (1350 BC):

Photography by Carlos Dorce

Photography by Carlos Dorce

The secnond mathematicl object found in the British Museum is this sandstone stela of the Egyptian Viceroy of Kush, Merymose, who served under Pharaoh Amenhotep III (c.1400 BC). A hieroglyphic text describes his campaign against the Nubians of Ibhet:

Photography by Carlos Dorce

Photography by Carlos Dorce

The hieroglyphic text is full of numbers and figures:

Photography by Carlos Dorce

Photography by Carlos Dorce

Other hieroglyphic numbers are found in the limestone relief of Rahotep (c.2600 BC) which was fixed in the offering-chapel of a brick mastaba tomb.

Photography by Carlos Dorce

Photography by Carlos Dorce

The relief shows Rahotep seated before offerings which are detailed in a formal list on the right of the slab and all these offerings are accompanied of the number of them. We can see the ‘lotus’ for the thousands…

Photography by Carlos Dorce

Photography by Carlos Dorce

and a lot of examples of units, tens and hundreds:

Photography by Carlos Dorce

Photography by Carlos Dorce

It’s also interesting this bone identifying label from an item of funerary equipment (3100 BC). The front of the label bears the name of Queen Neithhotep and on the back is the numeral 135:

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally, the limestone false door stela of Niankhre (2450 BC) from Saqqara which comes from the mastaba-tomb of the superintendant of the hairdressers of the Palace Niankhre. You can see the number 4.000 in the top of the stela:

Photography by Carlos Dorce

Photography by Carlos Dorce

LocationBritish Museum in London (map)

Mesopotamian Mathematics in the BM

Photography by Carlos Dorce

Photography by Carlos Dorce

This is the famous Royal Game of Ur (2600-2300 BC). This wooden game board was in at least six graves in the Royal Cemetery so it’s an early example of a game that was played all over the ancient Near East for about 3.000 years.

The game is a race for two players using dice with seven identical pieces each. All playing squares are decorated, but on later boards only the five ‘rosette’ squares are marked. […] Pieces are ‘at war’ along the central path but turn off to their own side to exit.

Playing pieces were discs of shell or lapis lazuli. The tetrahedrical dice of the game are also exhibited.

Apart of the Royal Game of Ur, the only exhibited objects which are related with Mespotamian mathematics are the Archaic and Cuneiform tablets. For example, look at this tablet containing a five day ration list (Jemdet Nasr, 3000-2900 BC):

Photography by Carlos Dorce

Photography by Carlos Dorce

Each line contains rations for one day and the sign for ‘day’ and numbers 1, 2, 3, 4 and 5 are easily identificable (at the beginning of the line!).

Photography by Carlos Dorce

Photography by Carlos Dorce

This Gypsum tablet with Archaic numbers (Uruk, 3300 BC) has 3 units (round impressions) and 3 ‘tens’ (elongated impressions).

Photography by Carlos Dorce

Photography by Carlos Dorce

This tablet above contains the daily barley beer ration for the workers (3300-3100 BC). Here there are also identificable all the marks representing units and tens and it’s the same in the next tablet containig food rations (3300-3100 BC):

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally, there is another tablet from the Late Uruk Period (3300-3100 BC):

Photography by Carlos Dorce

Photography by Carlos Dorce

However, mathematical tables are not only clay tablets with figures and numbers. For example, the next tablet contains a set of problems relating to the calculation of volume, together with the solutions.

Photography by Carlos Dorce

Photography by Carlos Dorce

You can see the details of the tablet in the next two pictures:

Photography by Carlos Dorce

Photography by Carlos Dorce

Photography by Carlos Dorce

Photography by Carlos Dorce

There is also a tablet recording observations of the planet Venus from c.1700 BC:

Photography by Carlos Dorce

Photography by Carlos Dorce

Astronomical tablets were so common in Mesopotamia and here we have a representation of the heavens in eight segments which include drawings of the constellations.

Photography by Carlos Dorce

Photography by Carlos Dorce

The next piece of cuneiform tablet contains a star chart which was found in Ashurbanipal’s library:

Photography by Carlos Dorce

Photography by Carlos Dorce

According the British Museum’s web…

Ashurbanipal, whose name (Ashur-bani-apli) means, ‘the god Ashur is the creator of the heir’, came to the Assyrian throne in 668 BC. He continued to live in the Southwest Palace of his grandfather, Sennacherib, in Nineveh, which he decorated with wall reliefs depicting his military activity in Elam. He also had a new residence built at Nineveh, known today as the North Palace. The famous lion hunt reliefs, some of which are now in The British Museum, formed part of the new palace’s decorative scheme.

Throughout his reign, Ashurbanipal had military problems, mainly at the borders of the empire. He also continued his father’s policy of attacking Egypt. Campaigns in 667 and 664 BC led to the defeat of the Egyptian Twenty-fifth Dynasty and the appointment of a pro-Assyrian ruler in the Nile Delta. Assyria also attacked Elam, possibly in 658-57 BC, following the receipt of insulting letters from the Elamite king. In 652 BC Shamash-shum-ukin, Ashurbanipal’s brother, and ruler of Babylonia, revolted against Assyria with the support of the Elamites. The Assyrian army invaded Elam and Babylonia. Babylon was captured in 648 BC and the following year the Elamite city of Susa was destroyed. There is little surviving evidence that can help us to reconstruct the last years of Ashurbanipal’s reign. Ashurbanipal boasted of his ability to read the cuneiform script, and was responsible for the collection and copying of a major library of contemporary literary and religious texts

LocationBritish Museum in London (map)

Babbage in the Science Museum

Photography by Carlos Dorce

Photography by Carlos Dorce

My last post about the Science Museum is about Charles Babbage. We’ve seen that there is a difference machine in the groundfloor of the museum but in the mathematical section there is a special space dedicated to him and his machines:

Charles Babbage (1791-1871) is widely regarded as the first computer pioneer and the great ancestral figure in the history of computing. Babbage excelled in a variety of scientific and philosophical subjects though his present-day reputation rests largely on the invention and design of his vast mechanical calculating engines. His Analytical Engine conceived in 1834 is one of the startling intellectual feats of the nineteenth century. The design of this machine possesses all the essential logical features of the modern general purpose computer. However, there is no direct line of descent from Babbage’s work to the modern electronic computer invented by the pioneers of the electronic age in the late 1930s and early 1940s largely in ignorance of the detail of Babbage’s work.

Apart of his portrait from 1860, we can see in the exhibition the right sagittal section with cerebellum of his brain. Babbage’s son donated it for research to the Hunterian Museum at the Royal College of Surgeons in England. There also are extracts from his diary (1844):

Photography by Carlos Dorce

Photography by Carlos Dorce

Charles Babbage invented the Difference Engine in 1821 but never built a full example. The only complete Difference Engine built during Babbage’s lifetime was made by Swedish engineers George and Edvard Scheutz. Inspired in Babbage’s ideas, and encouraged by Babbage himself, they printed the first ever mathematical tables calculated by machine. The Scheutz brothers went on to sell two further Difference Engines of which this is the second:

Photography by Carlos Dorce

Photography by Carlos Dorce

Practical and finantial problems meant that Babbage and his engineer Joseph Clement completed only about a seventh of Babbage’s original mechanism, which is on display in the ‘Making the Modern World’ gallery on the ground floor. Known as Difference Engine No. 1, it is one of the finest examples of precision egineering from 19th-century England.

The world’s first mechanical computer was also invented by Babbage in 1834. He never saw his Analytical Machine finished and this small section was under construction when he died:

Photography by Carlos Dorce

Photography by Carlos Dorce

There is also a model of the Difference Engine No. 2:

Photography by Carlos Dorce

Photography by Carlos Dorce

Charles Babbage designed this mechanical calculating machine, called Difference Engine No. 2, between 1847 and 1849. He aimed to print mathematical tables that were much more accurate than the hand-produced versions available to Victorian engineers, scientists and navigators.

Babbage called his machine a Difference Engine because it calculated tables of sums automatically using ‘the method of finite differences’. This mathematical method involves only addition and subtraction, and avoids multiplication and division, which are more difficult to mechanise.

I have a picture in front of this machine with my students from our visit to the museum in February 2012:

Who took this picture for us?

Who took this picture for us?

Nowadays, the machine is part of this section dedicated exclusively to Babbage.

I am going to visit the Science Museum again next February and I am sure that another post will be written because there are a lot of pictures and things that I don’t have time now to share with you!

LocationScience Museum in London (map)

The Science Museum in London (VI)

Photography by Carlos Dorce

Photography by Carlos Dorce

Let’s play with Topology! The exhibition is full of polyhedra, curious surfaces,… and Möbius strips:

Photography by Carlos Dorce

Photography by Carlos Dorce

August Möbius (1790-1868)

Professor Möbius discovered the surface now known as the Möbius strip in the course of an investigation of the properties of polyhedra. The discovery was made in 1858 but was not published until 1865.

There is certainly also the Klein’s bottle:

Photography by Carlos Dorce

Photography by Carlos Dorce

We can imagine what a wonderful surface we can get if a plane is deformed!

Photography by Carlos Dorce

Photography by Carlos Dorce

Sometimes, it depends on our point of view and there are surfaces which are homeomorphic to other ones that seems very different to them! If we look at the example of the conics, the circumference, the ellipse, the hyperbola and the parabola are different points of view of the same reality, aren’t they?

Photography by Carlos Dorce

Photography by Carlos Dorce

LocationScience Museum in London (map)

The Science Museum in London (V)

Photography by Carlos Dorce

Photography by Carlos Dorce

I must finished this wonderful walk for the calculating machines going downstairs to the groundfloor again and showing some instruments more. Number 41 is a calculating machine from c.1955 made by Bell Punch Company Limited and number 42 is the ‘Eckel dial rule’ from the same year. The calculator number 51 is from 1955-1965.

Among the instruments which made our life more confortable there is also space for the sundials, clocks and quadrants. For example, look at this Gunter’s quadrant from the beginning of the 19th century:

Photography by Carlos Dorce

Photography by Carlos Dorce

One example of sundial is this inclining sundiel from 1800:

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally, we notice this universal ring dial from the mid-eighteenth century. Sundials were still needed to set the clocks and watches that had superceded them as timekeepers:

Photography by Carlos Dorce

Photography by Carlos Dorce

LocationScience Museum in London (map)

The Science Museum in London (IV)

Photography by Carlos Dorce

Photography by Carlos Dorce

John Napier invented published his Mirifici Logarithmorum Canonis Descriptio in 1614 and the invention of the logarithms was the beginning of a new method of computing. Henry Briggs met Napier in Edimburg in the summer of 1615 and 1616 and these two men together decided to improve the invention creating the decimal logarithms which were published by Briggs some years later. In 1620 Edmund Gunter published his Canon Triangulorum where he described one of the first attempts to create a slide rule:

Photography by Carlos Dorce

Photography by Carlos Dorce

After the Gunter’s scale was invented, some other descriptions of the rule appeared like the one made by Wingate in paris in 1624. Gunter’s scale was very popular because all the trigonometrical resolutions of the triangles were reduced to additions and substractions on the rule:

Photography by Carlos Dorce

Photography by Carlos Dorce

The slide rule was invented by William Oughtred who designed both a circular and a straight form of slide rule in about 1621 but did not publish his work until much later. Richard Delamain, one of his former pupils, published a description of a circular slide rule in 1630, and claimed priority of invention although he copied Oughtred’s ideas. In 1660’s Thomas Browne invented the spiral slide rule consisting in fixed scales and moveable index arms similar to Oughtred’s circles of proportion:

Photography by Carlos Dorce

Photography by Carlos Dorce

LocationScience Museum in London (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 Science Museum in London (II)

Photography by Carlos Dorce

Photography by Carlos Dorce

There is asection dedicated to Mathematics and it’s a paradise for the mathematical freaks! It’s full of calculators, Klein’s bottles, polyhedra,… and it’s possible to learn a lot of things only reading the information next to them. Shall we begin?

After the Second World War, the teaching of arithmetic to children in Britain became less focussed on repeated sums and tables, and more orientated towards understanding through experience. The use ofcalculators was always controversial.
In 1971, the rapid introduction of silicon chips ushered in the age of the pocket electronic calculator. However, in Japan, use ofthe ‘soraban’ or Japanese abacus was so instilled that, even a generation later, older people relied on them.

Educational toys became increasingly popular during the 20th century as parents realised they could improve their children’s performance and manufacturers realised there was a large potential market. For example, we can see the ‘Tell Bell’ educational toy from c.1930 in the first picture of this post.

All these first calculators also have their space in the museum. Here we can see some examples. The first one is the ‘Addiator’ mechanical adder from 1924. Until the 1970s mechanical adders remained essentially the same as those made in the 19th century but used new materials and designs:

Photography by Carlos Dorce

Photography by Carlos Dorce

The ‘Alpina’ mechanical adder was produced in Germany in c.1955:

Photography by Carlos Dorce

Photography by Carlos Dorce

From c.1950 we find the ‘Baby’ and the ‘Exactus’ mechanical adders and the ‘ Magical Brain’ mechanical adder is from c.1960:

Photography by Carlos Dorce

Photography by Carlos Dorce

But… when did everything begin? The first known attempt to make a caclulating machine was by Wilhelm Schickard (1592-1635), Professor at the University of Tübingen. Sketches of the machine appear in two of his letters written to Johannes Kepler in 1623 and 1624. Unfortunately the machine was destroyed by fire and there is no record of a replacement. Two decades later, Blaise Pascal completed a calculating machine in 1642 when he was 19 y.o. It was designed for addition and subtraction, using a stylus to move the number wheels:

Photography by Carlos Dorce

Photography by Carlos Dorce

The crude constructional methods of the time resulted in unreliable operation. Although several machines were later offered for sale, the venture was not a commercial success.

Photography by Carlos Dorce

Photography by Carlos Dorce

Sir Samuel Morland was the first Englishman to venture into the field of calculating machines designing this one (1666) on Pascal’s invention.

Photography by Carlos Dorce

Photography by Carlos Dorce

Gottfried W. Leibniz continued the construction of new calculating machines with this new one which wasn’t on display when I was in the museum. The mechanism was based on the stepped reckoner which eventually became the foundation of the Arithmometer:

Photography by Carlos Dorce

Photography by Carlos Dorce

It is unworthy of excellent men to lose hours like slaves in the labour of calculation, which could be safely relegated to anyone else if machines were used.

Leibniz, 1671

Finally, we must take a look to the Facit:

Photography by Carlos Dorce

Photography by Carlos Dorce

Was it really the World Champion in its class?

Photography by Carlos Dorce

Photography by Carlos Dorce

LocationScience Museum in London (map)