Abû al-Wafâ’ al-Buzjanî’s doodle
This beautiful doodle was published by google in the Persian and Arabic countries last 10th June because in 10th June 940 the great Abû al-Wafâ’ al-Buzjanî was born in Persia. Since 959, he worked in the Caliph’s court in Baghdad among other distinguished mathematicians and scientists who remained in the city after Sharâf al-Dawlâh became the new caliph in 983. He continued to support mathematics and astronomy and built a new observatory in the gardens of his palace in Baghdad (June 988) which included a quadrant over 6 metres long and a sextant of 18 metres.
Abû al-Wafâ’ wrote commentaries on works of Euclid, Ptolemy, Diophantus and al-Khwârizmî, and his works were very important in the developement of Trigonometry and Astronomy.
A solar “eight” in the Cathedral of Palma
Source: Wikimedia Commons
Palma’s cathedral, also known as La Seu, like almost every building of this kind hide curiosities and mathematical facts that surprise due to the construction’s age. We can find trigonometry relations in a common cathedral or church, some of them even with a religious meaning behind. But in La Seu we can find two particular incidents that catch our eye. The first one happens every 2nd of February (2/2) and 11th of November (11/11), when the sunbeams go throw the east-faced rose window (known as Oculus Maior) and they are projected under the west-faced rose window, tangentially, in such a way that their centers lie on a line that is perpendicular to the ground.
Source: XEIX webpage
It is not a coincidence the days in which the event takes place, as they are both in a similar position regarding the winter solstice. Thanks to this light effect and the data currently available on the internet, we can easily calculate La Seu’s orientation. When this phenomenon takes place, the direction of the sunbeams coincide with the nave’s orientation. Then, we only have to set the time and figure out the exact azimuth considering this time and geographic situation. With some of the available programs, we can find out that it has a value of 122,4º and the angle of solar elevation is 10,2º (both calculus with an error smaller than 0,5º!).
Source: XEIX webpage
The other incident takes place during the days near to the winter solstice. We can stare at the sunrise going through both rose windows causing a rather impressive light effect. Then again, we can find the value of the azimuth which results in 120,3º. Curiously, the bell tower, which has a square base, doesn’t have an axis of symmetry parallel to the central nave, they are out of place about 10º from each other.
Source: XEIX webpage
Like many others cathedrals and churches, this one is built on an ancient mosque. In this culture, it was very important to have the ”qibla” oriented towards Mecca, specifically, towards the Ka’ba, following the precepts of the Koran. The muslim domination in Mallorca happened between 903 and 1231. In this period of time, the solution of the “qibla problem” was already known, solved by al-Khwârizmî in the 9th century. In effect, if one traces the line segment bisector of the east-faced bell tower and lengths it over the terrestrial sphere it matches the Ka’ba with an astonishing precision.
You can read more information about these two facts in this interesting article.
This post has been written by Aina Ferra in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)
Location: Cathedral of Palma (map)
The “Taula” in Torretrencada
Photography by Laura Barber
The name of the Talaiotic culture comes from conic towers built with stones, probably used as a dwelling, watch towers and defense towers. These tables (“Taules” in Catalan) consist in a vertical rectangular stone and another one placed horizontally on its top, so the name of the table is given by the form of “T”. But… why these old monuments are mathematic? The front of most of them is oriented to the south! This orientation is related to the possible use as calendar in this former culture. The construction of the first monuments in Balearic islands dates from the end of the 2nd millenium BC to the beginnings of the 1st millenium BC. At this moment, these monuments began to proliferate on Mallorca and Menorca (there are 31 only in this small Mediterranean island!) appearing in isolated fashion as a territorial boundary stone.
The tables served as sanctuaries next to other monuments and all of them were built in almost the same latitude (and longitude?). For example, Sa naveta des Turons (latitude = 39.99º and longitude = 3.93º), Torretrencada (latitude = 40.003º and longitude = 3.89º) and Torre d’en Gaumès (latitude = 39.93º and longitude = 4.12º) seems to be aligned!
Naveta des Tudons
Source: Wikimedia Commons
In 1996, Vicente Ibáñez Orts published his hypothesis on the Table explaining that their design was very well computed and not the result of chance. Regarding Torretrencada, it seems that the monument was built from some mathematical computation indicating that Talaiotic men had a system of writing numbers and a deep knowledge of arithmetic and geometry)
This post has been written by Laura Barber and Anabel Luís in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).
Location: Torretrencada (map)
Kukulkan’s temple in Chichen Itza
Kukunkan’s temple
Photography by Roberto Lara
Today, the ancient Mayans are particularly famous by their incredible calendar. In fact, Mayans made a really powerful calendar inspired by astronomical events, as they really were essentially farmers and very superstitious. This is the reason why they didn’t have an unique counting system in their calendar, that is, they had ‘sub-calendars’ which different periods as reference. For example, they had a holy calendar (called Tzolkin), which had 260 days, and also a civil solar calendar (called Haab) with 365 days (it’s not clear what was the motivation for the Tzolkin). Tzolkin means “division of days” was probably based on the 224-day Venus sidereal period although there are some hypothesis which defend that it is related with the human gestation period. The Haab calendar consisted in 18 months of 20 days each plus an additional period of five days at the end of the year. It was first used around 500 BC. Mayans were so religious and these astronomical calendars were exposed in their most important buildings like the World-wide famous Temple of Kukulkan (“Feathered serpent”) in the archeological site of Chichen Itza. The temple was founded around 525 AD although the current building was completed between the 9th and the 12th centuries. The pyramid has four sides, each one with 91 steps, which adds up to 364 steps. If we count the last platform as a step we get 365 steps, which is equal to the days we find in the Haab calendar.
Photography by Roberto Lara
But the most famous thing about the Kukulkan’s temple is the descent of Kukulkan: during the autumn and spring equinoxes the late afternoon Sun strikes off the northwest corner of the pyramid and casts a series of triangular shadows against the northwest balustrade, creating the illusion of a feathered serpent ‘crawling’ down the pyramid. We should remark that the balustrade and corners of the pyramid are perfectly aligned, which makes us admire even more the work that Mayans had on the building:
The Feathered Serpent in the Spring Equinox
Source: Wikimedia Commons
The pyramid also shows us that Mayans had some knowledge about acoustics. If you stand in front of any of the four stairway and clap your hands, the pyramid reflects the sound in such way that you hear the sing of a quetzal, a bird from the jungle. It’s fascinating! Isn’t it? Moreover, the shaman was known as ‘the man with the great voice’, because when people met for a ritual, he didn’t have to speak loudly, as everybody could hear him perfectly.
From all these facts, we can easily conclude that mathematics in the ancient Mayan world wasn’t only a help for agriculture but a tool through which the leaders could control the population. In fact, in the picture below we can see the ruins of a Mayan school. Only those from the upper class had access to the education, and we can see from the building they truly wanted to keep it as a privilege!
Photography by Roberto Lara
The hole at the right of the picture was made by an adventurer who thought gold was hiding inside it and used dynamite to enter the building.
This post has been written by Roberto Lara Martín in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).
Location: Chichen Itzá (map)
Al-Bîrûnî’s birthday
Abû al-Rayhân Muhammad ibn Ahmad al-Bîrûnî was born near Kath in the region of Khwârazm (now Kara-Kalpakskaya) in September 4, 973, and was died in Gbazna(?) after 1050. He lived in Kath and in Jurjanîyya and we know that he began his studies under Abû Nasr Mansûr (970-1036). He became a good mathematician and astronomer very fast and he measured the latitude of Kath observing the maximum altitude of the Sun when he was only 17 y.o. He also wrote some astronomical and mathematical works before 995 as we can check in his Cartography (a book about map projections).
About 995, al-Bîrûnî left the civil war in Khwârazm and moved to Rayy (now near Tehran) where he lived in poverty. We know that he worked with al-Khujandî who had a large sextant with which he had determined the obliquity of the ecliptic.
Through the observational data registered by him we know that he spent some days in Rayy and that he was back in his birthplace in 1004. That year he became protected by the rulers of the region and he got enough money to build an instrument at Jurjanîyya to observe solar meridian transits. He made observations with it in 1016 and one year later he and Abû Nasr Mansûr were made prisoners by Mahmûd, the new ruler of Khwârazm. Al-Bîrûnî continued working as astronomer but he had a lot of problems with victorious Mahmûd. However, between 1018 and 1020 he made observations from Ghazna supported by Mahmûd and this work allowed him to determinate the latitude of the place.
Al-Bîrûnî travelled to India together with Mahmûd’s military expedition and he spent at least five years working on his India, in which he computed latitudes of cities and explained calendars, geography, literature…
After Mahmud’s death, the next rulers allow al-Bîrûnî to be free to travel and work in his interests and he became the most prolific Arabic mathematician in the World. He produced 146 works with more than 13.000 pages!
Russian stamp (1973)
So, al-Bîrûnî’s doodle published two years ago in the Arabic countries is a very good example of a great scientific contribution!
Pythagoras in Temple Bar Moument
Photography by Carlos Dorce
The Temple Bar Memorial (1880) stands in the middle of the road opposite Street’s Law Courts marking the place where Wren’s Temple Bar used to stand as the entrance to London from Westminster.
The monument has two standing statues dedicated to Queen Victoria and the Prince of Wales because both were the last royals to pass through the old gate in 1872.
Photography by Carlos Dorce
The reliefs round Queen Victoria contains some allegories which includes the first picture about the Euclidean demonstration of the theorem of Pythagoras. We also find a ruler and a globe with the ecliptic.
Photography by Carlos Dorce
Location: Temple Bar (map)
Immortal Books, essential instruments (II)
Photography by Carlos Dorce
The Astronomical Revolution is visited after the Greek books and Copernicus (1473-1543) and his De Revolutionibus orbium coelestium are the next couple to study:
He was born in Poland in a very rich family. His parents died and his uncle (bishop of Warmia) took care of him. He went to the University of Krakow and he studied Canonic Law in Bologna some years later. He was under the Italian Humanism there and he began to have interest for Astronomy. He completed his studies and also Mechanics in Padova and read his doctoral dissertation in Canonic Law in the University of Ferrara. After this, he came back to his country and entered the Bishop’s court. In 1513 he wrote the Commentariolus – manuscript which circulated anonymously- where astronomers could read his new astronomical system. He was invited to reform the Julian calendar. He wrote his great work De Revolutionibus Orbium Coelestium inthe last days of his life and he defended the heliocentrical hypothesis in it. His disciple Rheticus brought a copy of the manuscript to the printing in 1542 and it was published in 1543. Copernicus died in Frombork and his theory was condemned by the Church in 1616 and was in the List of Prohibited Books until 1748.
I think that I’m going to go to Poland next holidays!
One of the most important followers of the heliocentrism was Johannes Kepler (1571-1630):
Photography by Carlos Dorce
The scientist who opened the way to the modern astronomy was born in Weil der Stadt, Germany. He suffered from myopia and double vision caused from smallpox and this wasn’t a problem for him to discover the laws which explain the movements of the planets around the Sun. He studied Theology in the University of Tubingen under his teacher Michael Mastlin and he soon noticed his unusual skills reading Ciopernicus’ heliocentrism. He mainly lived in Graz, Prague and Linz. He met Tycho Brahe in Prague and some years later he became Imperial Mathematician under Rudolph II’s protection. It wa sin this period when he developed his great works: Tabulae Rudolphinae and Astronomia Nova (1609). In Astronomia Nova he explained two of the three fundamental laws describing the movement of the planets; the third one was explained in Harmonices Mundi Libri V (1619). Kepler was the first scientific in needing phisician demonstrations to the celestial phenomena.
Who is the next? Galileo (1564-1642), of course!
Photography by Carlos Dorce
His book is the Dialogo sopra i due massimi sistemi del mondo Tolemaico, e Copernicano (1632). In this book he defended the Copernicanism against the Ptolemaic system although the book was prohibited by the Inquisition and he was condemned to house arrest.
Galileo died in 1642 and Newton (1642-1727) was born some months after his death. His Philosophiae Naturalis Principia Mathematica was one of the most important scientific books of all the History of Science. I am not going to talk about Newton and his book after my visit to Englang last holidays but here you have his portrait:
Photography by Carlos Dorce
The other scientists of this epoch are Vesalius (De humani corporis fabrica), Harvey (Exercitatio anatomica de motu cordis et sanguini), Linneo (Systema naturae) and Hooke (Macrographia):
Photography by Carlos Dorce
There is another important mathematician from the 17th century but… it will be presented tomorrow!
Location: MUNCYT in Madrid (map) and MUNCYT in A Coruña (map)
Immortal Books, essential instruments (I)
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
[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
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
Ptolemy (2nd century) is the next and his Almagest was the most important astronomical book since the 16th century.
Photography by Carlos Dorce
There is also an interesting wooden astrolabe from 1630 (“Claudii Ricchardi”):
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)
Whipple Museum of the History of Science (III)
Photography by Carlos Dorce
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Indian astrologer’s celestial globe from the 19th century:
Photography by Carlos Dorce
This celestial globe is extremely unusual in being inscribed in Arabic, Persian and Urdu, representing a blending of traditions. It was probably designed for astrological use.
English Ptolemaic armillary sphere, by Richard Glynne (c. 1715)
Photography by Carlos Dorce
Ptolemy’s cosmology placed the moon along with Mercury, Venus, the sun, Jupiter and Saturn in orbit around the Eart, which stood at rest at the centre of the universe. Although a sun-centred universe was more widely accepted among astronomers in the 18th century, Ptolemaic armillary spheres such as this one continued to be made and sold.
Upstairs there is another exhibition related to globes and armilar spheres and we can find some terrestial globes and Copernican and more Ptolemaic spheres. Here you have one Ptolemaic one dated in 1790:
Photography by Carlos Dorce
Finally, I must point to two delicatessen. The first one is this little ivory plaque showing astronomers working with their instruments:
Photography by Carlos Dorce
The other is this picture showing a figure pointing to a globe from a section of the Roman mosaic from mid-2nd century (Museo Nacional de Arte Romano):
Location: Whipple Museum of the History of Science in Cambridge (map)
Whipple Museum of the History of Science (II)
Photography by Carlos Dorce
The collection of astrolabes is very interesting and there are pieces from the 14th to the 19th century.
Photography by Carlos Dorce
For example, look at this English astrolabe from the 14th century:
Photography by Carlos Dorce
One of the most interesting objects of the exhibition is this Egyptian sundial from the 1st to 3rd century AD (Roman Imperial date):
Photography by Carlos Dorce
Sundial construction requires expert knowledge of the apparent movement of the Sun, both in its daily and annual motion. Hence the construction of sundials has often been considered part of astronomy.
The diagram that represents annual motion is called the ‘analemma’ and was known in classical antiquity. Many different designs of sundial can be derived from the analemma. This dial has two scales, one for the morning, one for the afternoon; the shadow is cast by the prominence between them.
Of course there are more sundials like this pillar dial made in 1542 and this ring dial made in 1588…
Photography by Carlos Dorce
… or all these different dials of different kinds, epochs and countries:
Photography by Carlos Dorce
Upstairs there is a replica of an antique globe fragment which is on display in the Neues Museum:
Location: Whipple Museum of the History of Science in Cambridge (map)
THE MATHEMATICAL TOURIST 