Tag Archives: Compass

Kepler’s last home in Regensburg

Photography by Carlos Dorce

Photography by Carlos Dorce

Kepler’s last home is this orange house located in Keplerstrasse 5 in Regensburg. Reading the famous Kepler’s biography written by Max Caspar:

[…] On November 2 [1630] he rode, tired, on a skinny nag, over The Stone Bridge into Regensburg. He took up quarters in Hillebrand Billj’s house in the street now named after him. This acquaintance was a tradesman and later an innkeeper.

Only a few days after his arrival Kepler came down with an acute illness. His body was weakened by much night study, by constant worry, and also by the long journey at a bad time of year. In the beginning he attributed no significance to his being taken ill. He had often before suffered from attacks of fever. He believed that his fever originated from “sacer ignis”, fire-pustules. As the illness became worse, an attempt was made to help him by bleeding. But soon he began to lose consciousness and became delirious. Several pastors visited him and “refreshed him with the vitalizing water of consolation”. It is not said anywhere that holy communion was afforded him. In the throes of death Pastor Christoph Sigmund Donauer rendered him aid. When, almost in the last moment of his life, he was asked on what he pinned his hope of salvation, he answered full of confidence: only and alone on the services of Jesus Christ; in Him is based, as he wanted to testify firmly and resolutely, all refuge, all his solace and welfare. At noon on November 15 this pious man breathed his last. […].

A plaque on the facade says that this is the house which I was looking for when I have arrived at Regensburg:

Photography by Carlos Dorce

Photography by Carlos Dorce

Bad luck! This small museum is only open in the weekends and it’s possible to rent a guided visit only for groups! I’ve not arrived here to give up! Finally, I’ve been able to visit it and the first thing that I’ve seen… the magnificent bust of the last owner of the house…

Photography by Carlos Dorce

Photography by Carlos Dorce

… over a plaque in German language where it’s possible to read a little part of this story:

Photography by Carlos Dorce

Photography by Carlos Dorce

The museum located in the house is very small and explains Kepler’s life and works focussing the interest in his astronomical discoveries and his three laws.

Photography by Carlos Dorce

Photography by Carlos Dorce

There is also a representation of the barrels whose volume was calculated precisely by Kepler in 1615:

Photography by Carlos Dorce

Photography by Carlos Dorce

Another bust representing the great mathematician is in the room of the first floor next to some information about his commemorative monument also in Regensburg.

Photography by Carlos Dorce

Photography by Carlos Dorce

There are a lot of Kepler’s works (which seem to be original) and this wonderful German edition of Napier’s logarithms (1631) which couldn’t be used by Kepler but exemplifies the great impact that this powerful calculator had in the beginning of the 17th century.

Photography by Carlos Dorce

Photography by Carlos Dorce

Of course, his Astronomia Nova, his Harmonices mundi,… and his Tabula Rudolphinae are also exhibited.

Photography by Carlos Dorce

Photography by Carlos Dorce

There also are explanation about his relation with Tycho Brahe and the Copernican system and a lot of astronomical instruments like sextants, globes, compasses,…

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally, I want to say goodbye looking at this famous portrait. This man discovered the elliptical orbits of the palnets and his obsession with numbers let him find the second and the third law. Copernicus was right and Newton will be confirm all this theories. The World was explained (Wait Einstein, wait!).

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Kepler’s museum in Regensburg (map)

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Another Kepler in Weil der Stadt

Photography by Carlos Dorce

Photography by Carlos Dorce

This statue of Johannes Kepler is located in a garden next to Kepler’s museum in Weil der Stadt. It seems to be a private monument in a garden but it’s possible to go inside.

Location: Monument in Weil der Stadt (map)

A masonic geometric symbol in Barcelona

Photography by Paula Arrebola and Abel Hernadez

Photography by Paula Arrebola and Abel Hernadez

There are a lot of hidden enigmas and misteries in the modern buildings of Barcelona and some of these details are unknown by almost everyone. For example, on the facade at 11 Portaferrissa Street in the Old Town (Barri Gòtic), we can see the sculpture of two little boys above the main door on the top of a pile of bricks between them and the boy on the right holds some kind of square, whereas the one on the left holds a compass and a paddle. At first sight, it may seem another figure of the architectural style of the 19th Century. Nevertheless, it is a masonic sign indeed. The square and the compass were some of the most remarkable symbols of Freemasonry since both appear in the Masonry emblem.

Source. Wikimedia Commons

Source: Wikimedia Commons

In ancient times, the compass symbolized the Heavens inasmuch as it was used to study the starry Heavens, while the square represented the Earth because it was used to measure it. Nowadays both might have some philosophical and ethic connotations such as boundaries, so as to keep the equilibrium, and morality respectively. So, according to the Barcelona Historical Archive, the construction was made in the late 19th Century since there is a file with a works license of 1867, projected by Domingo Sitjas. It is interesting to notice that, in the original plans of the project, there is no sign of this sculpture. Due to Masonry persecution, everything related to it had to be kept secret. So, logically, the sculpture doesn’t appear in the plans!

Freemasonry describes itself as a beautiful system of morality, veiled in allegory and illustrated by symbols. Traditionally, masons are fond of architecture and are dedicated to the seven Liberal Arts: Grammar, Rhetoric, Arithmetic, Logic, Music, Geometry and Astronomy. Therefore, since Arithmetic, Geometry and Logic belong to mathematics, there is a link between Masonry and mathematics somehow. For instance, number 1 is represented by a point, which has no dimensions and turns to be the generator of any imaginable figure. According to the Masons, one is the arithmetic symbol of the Unity as well as the point is the geometric image of the Being. On top of that, if there are two points, they can be joined with a compass considering the straight segment connecting them like the one-dimensional projection of the geometrical link. It is important to note that the symbol of the Unit is the generator of duality, ternary, etc. using the compass. In addition to that, Geometry is the basis on which the Masonic superstructure stands and is considered by them as the mother of Science. However, Arithmetic has an important role in Masonry philosophybecause under Freemasons’ point of view, each of the four fundamental operations corresponds to a present value in their lives. For example, the sum is related to adding knowledge to our cognizance. In regard to such importance of the Unity as the genesis of the rest, it should be noted that the quaternary arises just like the ternary does. The last one would be represented in the form of a square. Mathematically, this generating method from the Unit would be considered as N = 1 + n (0<n<9 integer number), where N is associated with geometric figures representing values ​​or facts of their daily life. It must be kept in mind that this is a cycle whose culmination is the following expression: 9+1=10=1+0=1. Let’s look at some examples of the corresponding symbolism to certain geometrical figures identified, as we have seen, with numbers. Remember that the “construction” of these is given by using a compass and a square like the ones in the image:

  1. The triangle, which represents number 3, is the geometric figure par excellence. The rest of figures can be represented as a set of triangles. It is known among Masons as Radiant Delta and each of its vertices represents space, time and energy. The union of these vertices turns out to be the force that gives structure to the universe, which is the Great Architect of the Universe (G), i.e., God.
  2. The square, which represents number 4, symbolizes at the same time two squares or the union of two triangles, that is, harmony and balance. If these two squares are within a circle, then they represent the harmony between the Earth and the Sacred World.
  3. The 5-pointed star (or pentagrammon) represents number 5 and symbolizes the man and life, what is masculine and what is feminine, and the union represents androgyny.

It’s imperative to comment that, although there may be no direct link between Pythagoras and Freemasonry, the teachings of Pythagoras have greatly influenced its structure and its teachings: numbers (especially numbers from 1 to 10) are symbols representing the philosophical universal organization and the only way to reach God. Furthermore, every degree of the Masonic initiation corresponds to one number.

Finally, it is necessary to remark that this is just a selection of the whole mathematics they studied, since the 47th Euclid problem and the golden proportion should be also mentioned.

In conclusion, the sculpture emblems a fraternal community full of mysteries which has always been passionate about mathematics. One last mathematical fact: if you come in the building, you will see 7 steps followed by 7 steps and 14 steps more!

Photography by Paula Arrebola and Abel Hernandez

Photography by Paula Arrebola and Abel Hernandez

This post has been written by Paula Arrebola and Abel Hernandez in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).

Location Casa Domingo Sitjas (map)

A plaque in the Cloth Hall in Krakow

Photography by Carlos Dorce

Photography by Carlos Dorce

In the ground floor of the Cloth Hall of Krakow there is this modern plaque (2007) designed by Czeslaw Dźwigaj and the selection of the text was made by Jerzy Wyrozumski and Alexander Kravchuk:

1257 Krakow city rights advocates by the German tradition and the situation of the market and the houses and affected courts

It was Bolesław V, “the Chaste One”, (1226–1279), Duke of Sandomierz in Lesser Poland from 1232 and High Duke of Poland from 1243 until his death, who introduced in 1257 the city rights modelled on the Magdeburg law allowing for tax benefits and new trade privileges for the citizens and this plaque was designed to conmemmorate the 750th anniversary of this moment.

The Cloth Hall is one of the most emblematic symbols of Krakow because is probably the oldest shopping mall in the World:

Source: Wikimedia Commons

It’s in the middle of the main square of Krakow and everybody visit it and buy something in the shops of the ground floor.

Photography by Carlos Dorce

Photography by Carlos Dorce

So next time that you visit Krakow look at the floor to this conmemorative plaque:

Location: The Cloth Hall in Krakow (map)

Bernard Bolzano’s birthhouse

Photography by Carlos Dorce

Photography by Carlos Dorce

The Municipal Library of Prague (Marianské Square No.101) is in the same place where was Bernard Bolzano’s birthhouse. The house was detroyed but we have a photography of the main door from 1906:

The grey building is not beautiful although is full of allegories and symbols and among them we can see a ruler and a compass:

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Bolzano’s birthhouse in Prague (map)

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)

Copernicus’ House in Torun

Photography by Carlos Dorce

Photography by Carlos Dorce

Copernicus’ house was built in the 14th/15th century in the Gothic style of the Hanseatic towns. It has a high entrance hall with an open kitchen on the ground floor and the merchant’s office. The rooms are upstairs.

Photography by Carlos Dorce

Photography by Carlos Dorce

The building was the property of Copernicus’ family in the years 1463-1480 and it may well be here that he was born in 1473.  In an old picture it’s possible to see Napoleon Bonaparte walking in front of the present Copernicus Street nr 40. Till the end of the 19th century there was the wrong opinion that Copernicus had been born there although it seems to be the present numbers 15 and 17 the right houses. In the end of the 15th century, Copernicus’ father owned these three houses and another one in 36 and 37 Old Market Square.

Nowadays, there is a restaurant in 40 Copernicus Street…

Photography by Carlos Dorce

Photography by Carlos Dorce

…and a new building is a department store (“Dom Towarowy”):

Photography by Carlos Dorce

Photography by Carlos Dorce

The interior of the house (17 Copernicus Street) is reconstructed and the imagination is the only way to see Copernicus familiy in it.

Photography by Carlos Dorce

Photography by Carlos Dorce

The museum is full of portraits (I’m going to write another post about them!), the replicas of the instruments, scultures about the heliocentric system… and information about Copernicus’ life and works. For example, here you have a very modern design for Copernicus’ office:

Photography by Carlos Dorce

Photography by Carlos Dorce

And the room dedicated to his De Revolutionibus:

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally, hthe replicas of Copernicus’ instruments:

Photography by Carlos Dorce

Photography by Carlos Dorce

A few days ago I went to Frombork where Copernicus died and today I’ve been in the house where he was born. I’ve been in Olsztyn too so… what about going to Krakow where he studied? Thus… next step: Krakow!

Location: Copernicus’ house in Torun (map)

The Long Market in Gdansk

Source: Wikimedia Commons

The Long Market (Długi Targ) is one of the most important touristic attractions of Gdansk. It was a merchant road in the 13th century. After the massacre of Gdansk citizens on 13 November 1308 by Teutonic Knights, the place became the main street of the city and is name “Longa Platea” was first written in 1331. Nowadays it’s a very beautiful long square full of typical shops and restaurants which are the soul of this cosmopolutan city. One of its most representative houses is the town hall from the 16th century and Neptune’s Fountain, the main symbol of the city, is also there. This fountain was constructed in 1617 from Abraham van den Blocke’s designs.

Photography by Carlos Dorce

Photography by Carlos Dorce

Thus, if you visit Gdansk, you must have time to take a beer or a coffee in one of the cafes or have a typican Polish dinner in one of the restaurants which fill all the beautiful houses which can be admire in the square.

Among all these houses we also find a lot of mathematical symbols which allow me to talk of them in this new post. For example, Radisson Blue hotel is located in number 19 and the allegorical paintings of the facade are a joy for the mathematical freak:

Photography by Carlos Dorce

Photography by Carlos Dorce

On both sides we have some of the most important men in the history of astronomy like Hipparcus of Rhodas,

Photography by Carlos Dorce

Photography by Carlos Dorce

Ptolemy:

Photography by Carlos Dorce

Photography by Carlos Dorce

Copernicus:

Photography by Carlos Dorce

Photography by Carlos Dorce

Galileo Galilei:

Photography by Carlos Dorce

Photography by Carlos Dorce

and Hevelius:

Photography by Carlos Dorce

Photography by Carlos Dorce

Approaching the town hall, there is another red house which is full of artists ans it’s coronated by a replica of Aristotle and Plato from Raffaello’s “School of Athens”:

Photography by Carlos Dorce

Photography by Carlos Dorce

In another house there also are the allegorical Astronomia rounded by Cellarius’ heliocentric systems:

Photography by Carlos Dorce

Photography by Carlos Dorce

And finally we find other allegories like the Architecture, the Geometry or the Geography in the opposite side of the square:

Photography by Carlos Dorce

Photography by Carlos Dorce

For example…

Photography by Carlos Dorce

Photography by Carlos Dorce

…or:

Photography by Carlos Dorce

Photography by Carlos Dorce

As you can see, this is an excuse to admire the beautiful facades of the houses in this square which I never tire of walking through it.

By the way, there is a beautiful sundial in the town hall:

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Długi Targ in Gdansk (map)

Copernicus in Warsaw

Photography by Carlos Dorce

Photography by Carlos Dorce

Today is our first day in Warsaw! My wife, my son, my daughter and me are going to visit Poland for twelve days and I am sure that the mathematical tourist is going to see a lot of things!

This morning we have been in the city centre and we have walked through Krakowskie Przedmiescie and Nicolas Copernicus was there! His natural position is the middle of Copernicus square which is called with his name and he is rounded by his heliocentric system:

Photography by Carlos Dorce

We can read in wikipedia:

Nicolaus Copernicus (Polish: Mikołaj Kopernik ; German: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance mathematician and astronomer who formulated a heliocentric model of the universe which placed the Sun, rather than the Earth, at the center.

Portrait, 1580, Toruń Old Town City Hall Source: Wikimedia Commons

The publication of Copernicus’ book, De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), just before his death in 1543, is considered a major event in the history of science. It began the Copernican Revolution and contributed importantly to the scientific revolution.

Copernicus was born and died in Royal Prussia, a region of the Kingdom of Poland since 1466. Copernicus had a doctorate in canon law and, though without degrees, was a physician, polyglot, classics scholar, translator, governor, diplomat, and economist who in 1517 set down a quantity theory of money, a principal concept in economics to the present day, and formulated a version of Gresham’s law in 1519, before Gresham.

The bronze statue was designed by the Danish sculptor Bertel Thorvaldsen (c.1770-1844) in 1822 and was completed eight years later. The phiolosopher Stanislaw Staszic (1755-1826) made important donations for erecting it in Torun because he knew that Napoleon was very surprised for not finding a monument dedicated to the Polish astronomer in his birthplace.

The inscription says “To Nicolas Copernicus [from a] grateful nation” in Latin and Polish although during the German occupation of Poland in 1944, the inscription was translated to German saying “To Nicolaus Copernicus [from] the German Nation”.

copernic02

Photography by Carlos Dorce

In 2007, Copernicus’ solar system was represented rounding him representing an image from his revolutionary De Revolutionibus Orbium Coelestium (1543).

Photography by Carlos Dorce

Photography by Carlos Dorce

The statue is in front of Staszic Palace (1620), the seat of the Polish Academy of Sciences. In 1967, a bronze replica was erected in Chaboillez Square in Montreal during the World’s Fair:

Source: Wikimedia Commons

In 1973, on the 500th anniversary of Copernicus’ birth, another bronze replica was installed in front of Adler Planetarium in Chicago:

Source: Wikimedia Commons

Location in Warsaw: map

Location in Montreal: map

Location in Chicago: map

Pythagoras in Temple Bar Moument

Photography by Carlos Dorce

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

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

Photography by Carlos Dorce

Location: Temple Bar (map)