# Kepler’s last home in Regensburg

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:

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…

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

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.

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

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.

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.

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

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

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!).

**Location**: Kepler’s museum in Regensburg (map)

# The mathematicians in the Walhalla

The Walhalla is a neo-classical hall of fame which honours the most important people in German history. It was conceived in 1807 by Ludwig I of Bavaria (king from 1825 to 1848) and its construction took place between 1830 and 1842 designed by Leo von Klenze.

The Walhalla was inaugurated on October 18, 1842 with 96 busts and 64 commemorative plaques for people with no available portrait and everything was presided by the great King Ludwig:

Among all these very famous people related with the German history there are some… of course… mathematicians who share this space with Bach, Göethe, Beethoven, Guttemberg, Luther, Otto von Bismarck,… First of all, Dürervis the great German painter from the Renaissance who applied a lot of perspective new techniques to his paintings:

The great astronomers are also here. Regiomontanus,…

Herschel,…

Copernicus,…

and Kepler:

The great Leibniz…

and the greatest Gauss (added in 2007), also have their busts in this hall of fame:

Finally, Albert Einstein’s bust was added in 1990:

I must say that the commemorative plaques also mention Alcuin of York, Albertus Magnus and the Venerable Bede, all ot them related with the wonderful Arithmetics!

Come to Regensburg to see this beautiful (and strange) place!

**Location**: Walhava in Donaustauf (map)

# Another Kepler in Weil der Stadt

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)

# Kepler-Museum in Weil der Stadt

This building is the theorical Kepler’s bithplace in Weil der Stadt which hosts a very small museum about Kepler’s life and work:

At the age of six Kepler attends the German school. Continuing with Latin school he has to interrupt his attendance several times to help his parents with their work in the fields and at their inn. As a result he requires five years to complete the usual three school years.

The sickly child shows more enthusiasm at school than for hard work in the fields. His parents decide to send him to monastery school: First to the Adelberg monastery school (lower seminary) and then to Maulbronn (higher seminary).

His school comrades and teachers give him a hard time: At an early stage he starts to have his own ideas of church doctrine. His main struggle is with the meaning of Predestination and Communion.

Two celestial phenomena arouse his interest in astronomy: His mother shows him a comet, his father a lunar eclipse. Both phenomena remain in his mind for a long time. On the other hand, he never mentions his astronomy lessons in his written work.

Furthermore…

During the Age of Reformation the University of Tübingen, founded in 1477, forms the intellectual centre for Southern German Lutheran and for the Duchy of Württemberg. In 1536, Duke Ulrich orders the accomodation of poor students in Tübingen’s Stift. His aim is to ensure more graduates for loyal service in church and administration.

Coming from a humble background, Kepler wins a scholarship at the Stift. In 1589, he takes up his studies at the Faculty of Arts providing a general education, where the talented student receives many important stimuli. In particular, he studies the works of the Neoplatonists, whose ideas of a harmonically built creation make a deep impact on him.

However, his Professor of astronomy, Mästlin, influences him the most. Like a fatherly friend he familiarizes him with the ideas of Copernicus. Kepler sees an analogy in the central position of the sun to God’s omnipotence and consequently becomes a convinced advocate of the heliocentric view.

Kepler passes the baccalaureat exam at the Faculty of Arts as the second best in his class. […]. Before graduating, he accepts the position as provincial mathematician in Graz.

These were the first steps in Kepler’s life and the first thing that you see after entering the museum is the bust of this great mind:

Since 1594, as a provincial mathematician in Graz, Kepler…

[…] has to teach at the Lutheran seminary and write astrological calendars. His enthusiasm for astronomy inspires him to do his own research, and in 1596 he publishes his first work on astronomy

Mysterium Cosmographicum.He attempts to prove that a harmonic creation allows for only six planets. He regards the five regular Platonic polyhedra as elements of the planetary system, which, nested in the proper order, should determine the planet’s distance to the sun. As this approximately corresponds with the Copernican planetary distances, the work catches the attention of such important astronomers as Galileo Galieli and Tycho Brahe.

In spite of his fame, Kepler has to worry about his position in Graz. The Counter-Reformation creates great tension between the Lutheran inhabitants and the Catholic authorities. To recommend himself to the archduke Kepler dedicates a treatise to him on the solar eclipse of July 10th, 1600.

However, this does not prevent his expulsion from Graz one month later.

So these years in Graz were the period in which Kepler dreamt of his *Mysterium cosmographicum* and the possibility of the God’s design for the universe based in the regular polyhedra:

After Graz, Kepler became Tycho Brahe’s assistnat in Prague. After Tycho’s death, he assumed his position as imperial mathematician for Emperor Rudolph II:

[…] The quality of the [astronomical] data depends on the exactness of the particular orbit theory. Since all tables used around 1600 are inaccurate, Emperor Rudolph commissions Brahe and Kepler with the creation of the

Tabulae Rudolphinaein 1601. When Brahe dies in the same year Kepler has to continue the work on his own.It takes 22 years to complete the final version of the tables. Alone, the development of the elliptical orbits takes Kepler eight years. When he hears about the development of Napier’s logarithm, he integrates this into his tables and manages to simplify the calculation of orbital positions […]

Kepler discovered his first law and published it in his *Astronomia nova *(1609) and ten years later, he publishes his *Harmonice mundi* with the second and the third laws. Furthermore, Kepler had also time to wpork on infinitesimal calculus to compute the volume of some tonnels of wine:

I could follow explaining more things about Kepler’s life and works but this museum is very small so you must visit it. And Weil der Stadt is a very beautiful town!

**Location**: Kepler-Museum in Weil der Stadt (map)

# Johannes Kepler Monument in Weil der Stadt

Weil der Stadt is located near Stuttgart. Johannes Kepler was born in this very veautiful town on December 27, 1571 and his memory is still there: this big statue is in the middle of the Market Square…

… and Copernicus, Mästlin, Tycho Brahe and Jobst Bürgi are with him in this monumental sculpture.

The four scientists are in the corners of the base of the statue and the words “Astronomia”, “Optica”, “Mathematica” and “Physica” are graved on each of the four sides.

I must say that here we have the first (imaginary) bust of Bürgi that I know. Bürgi was one of the originators of the logarithms because Kepler said that he had seen Bürgi using logarithms in astronomical calculus (*Rudolphine Tables *(1627)) before their “official” first occurrence in Napier’s *Mirifici Logarithmorum Canonis Descriptio *(1614). Furthermore, Bürgi published his logarithms in his *Aritmetische und Geometrische Progreß tabulen* (1620) but his “red numbers” and “black numbers” couldn’t never win the “logarithms” which were the first calculator in all history.

Notice that this statue is not very similar to this other portrait from 1620:

The base of the statue also have four graved images representing moments in Kepler’s life like thispicture with Kepler in the middle explaining the Copernican system…

… with Hipparchus and Ptolemy watching how a central Sun brights in the middle of the universe.

Can you imagine Kepler investigating about his elliptical orbits?

Next to Market Square there is his bithplace which hosts… no, no, no! Tomowwor will be another day!

**Location**: Weil der Stadt (map)

# The golden number in Cosmocaixa

In the wonderfull wall full of formulas (already mentioned in this blog) that you can see in the Cosmocaixa in Barcelona, there also is the sacred equation which solution is the famous golden ratio:

Of course, one of the solutions of *x*^{2} = *x* + 1 is the number *x* = 1.6180339887498948482… (the other is -0.6180339887498948482…). At first sight it may seem a regular solution for a regular equation, but this number has revealed to the world of mathematics a whole new conception of nature and proportionality and this is the reason why it is interesting to know the history of this number and who dared to study its wonderful properties.

Since the golden ratio is a proportion between two segments, some mathematicians have assigned its origin to the ancient civilizations who created great artworks such as the Egyptian pyramids or Babylonian and Assyrian steles, even though it is thought that the presence of the ratio was not done on purpose. We can go forward on history and find the paintings and sculptures in the Greek Parthenon made by Phidias, whose name was taken by Mark Barr in 1900 in order to assign the ratio the Greek letter phi. So we can associate the first conscious appearance of the golden ratio with the Ancient Greece because of its multiple presence in geometry. Although it is usually thought that Plato worked with some theorems involving the golden ratio as Proclus said in his *Commentary* on Euclid’s *Elements*, Euclid was the first known person who studied formally such ratio, defining it as the division of a line into extreme and mean ratio. Euclid’s claim of the ratio is the third definition on his sixth book of *Elements*, which follows: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser”. He also described that the ratio could not be obtained as the division between two integers, referring to the golden ratio as an irrational number.

In the 13th century, Leonardo de Pisa (also known as Fibonacci) defined his famous serie in the *Liber abaci* (1202) although he wasn’t aware that phi is asymptotically obtained by dividing each number in the serie by its antecedent, thus, lots of natural phenomena which follows the Fibonacci sequence in any way, are related to the golden proportion.

Another important work from the 16th century is *De Divina Proportione* (1509) by Luca Pacioli, where the mathematician and theologian explains why the golden ratio should be considered as “divine”, comparing properties of our number like its unicity, immeasurability, self-similarity and the fact its obtained by three segments of a line, with divine qualities as the unicity and omnipresence of God and the Holy Trinity.

In the Renaissance, the golden ratio was chosen as the beauty proportion in the human body and all the painters and artists used it for his great masterpieces, like Leonardo da Vinci in his Mona Lisa or his famous Vitruvian Man.

The golden ratio was known in the world of mathematics as the Euclidean ratio between two lines and it wasn’t until 1597 that Michael Maestlin considered it as a number and approximated the inverse number of phi, describing it as “about 0.6180340”, written in a letter sent to his pupil Johannes Kepler. Kepler, famous by his astronomical theory about planetary orbits, also talked about the golden ratio and claimed that the division of each number in the Fibonacci sequence by its precursor, will result asymptotically the phi number. He called it a “precious jewel” and compared its importance to the Pythagoras theorem.

About one century later, the Swiss naturalist and philosopher Charles Bonnet (1720-1793) found the relation between the Fibonacci sequence and the spiral phyllotaxy of plants andthe German mathematician Martin Ohm (1792-1872) gave the ratio its famous “golden” adjective. If we want to talk about artists who introduced the ratio in their paintings in the modern times, a good example would be Salvador Dalí, whose artwork is plenty of masterpieces structured by the golden ratio.

This is just a brief summary of the history behind the golden ratio, which suffices to show that the interest induced by this number over the minds of the greatest mathematicians hasn’t ceased since the Ancient Greece, and even people non-related with mathematics have used it in their own work, which shows the importance and the multiple presence of mathematics and this special number in places that one could not imagine

*This post has been written by Pol Casellas and Eric Sandín in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).*

**Location**: Cosmocaixa in Barcelona (map)

# The Collegius Maius of the Jagiellonian University

Copernicus studied in the Collegius maius between 1491 and 1495. On the list of 69 students matriculated in 1491 at the Cracok Academy were “Nicolaus Nicolai de Thuronia” and aslso his brother “Andreas Nicolai”. The Jagiellonian University consisted offour faculties at the time (the Theological Faculty, the Canonical La Faculty, the Medical Faculty and the Liberal Arts Faculty). Copernicus began his studies learning the grammar of Latin, poetry and rhetoric but he early started to attend lectures on Euclidean geometry and astronomy. During the 15th and early 16th centuries, the University gained importance in Central Europe as a scientific center due to the high level of astronomical and mathematical sciences: the distinguished professors of the time included Marcin Hrol (c.1422-c.1453), Wojciech of Brudzewo (1445-1495), Jan of Glogow (c.1445-1507) and Maciej of Miechow (1453-1523). In the second semester of 1493 he attended lectures of Jerzy Peürbach, with the comments of Wojciech of Brudzewo, and the lectures about Aristotle’s *De Caelo*. It’s unknown when Copernicus brothers finished their studies n Cracow but they surely didn’t receive their degrees. Perhaps their mother’s death in 1495 caused their return to Prussia.

Thus one of the required mathematical visits that must be done in Cracow is this College:

The building hosts an interesting museum with a lot of old objects which are not directly related to the College but I must recognize that it’s possible to imagine how the academical life was in the 15th century. The first room is a big hall full of shelves with books, statutes, quadrants, portraits, maps and spheres:

Everything takes you back to a ‘kitsch’ Renaissance:

There is space for our Copernicus, of course,…:

…and also for Galileo:

There is a special small room dedicated exclusively to Copernicus with astrolabes, charts, books and copies of some interesting documents:

For example, look at this interesting torquetum made by Hans Dorn in 1480 (the astrolabe was also made by Dorn in 1486)…:

…or this portrait of Kepler from the 18th century:

Furthermore, a bust of Isaac Newton…

… is on the top of the door through which you enter a room full of astronomical and mathematical instruments:

Can you see this little Aechimedes screw?

Before ending the visit, Newton (again!) says goodgye to the visitors in a very modern picture:

And Kepler too!

One thing more… Go to the ticket office and you will see some mathematical objects more like these English Napier Rods from the 17th century:

**Location**: Collegius Maius (map)

# Playing with Maths in the Hewelianum Centre

The “puzzle” exhibition isn’t the only place in the Hewelanium Centre where you can discover mathematical facts. For example, in the exhibition about the History of the Centre there are cannons in a defensive fortress with which you can learn a lot about parabolic shots…

…or how many cannonballs you have in a pyramid… Is Kepler’s theorem right? Do you think about a better way of stacking cannonballs?

There also is space for optical illusions, technology,… and a very modern Archimedes screw:

You can also play with the Galilean experiments about movement and see how a piece of wood climbs a path down:

In a hidden corner of the museum, a sextant tells you goodbye:

**Location**: Hewelianum Centre in Gdansk (map)

# Museum of Nicolaus Copernicus, Frombork

In a previous post I began to talk about this museum located inside Frombork castle. You can learn almost everything about him, his life and his works on medicine, economies and, of course, astronomy, including the replicas of his instruments (we saw them also in Warsaw). For example, it’s possible to see some facsmile editions of his works and also a recreation of his desk:

Among the references about his publication of his works, we can find this engraving showing Copernicus in a lecture for the Cracovian scientists in 1509:

Or this other wonderful one (1873) with Copernicus in he middle of the picture talking about his heliocentric system:

How proud he is of his heliocentric theory!

And who are his guests? First of all, Hipparcus (with the armillar spher) and Ptolemy (with his geocentric system) are listening the theory which will finish theirs. Ptolemy looks askance at Tycho Brahe meanwhile Newton is looking at Laplace:

Galileo Galilei is behind Copernicus looking at him with great reverence:

And Hevelius, the other great Polish astronomer, agrees Copernicus’ theories although he never had the telescope to check them.

Finally, Johannes Kepler seems to be bored of listening this obvious theory although his ellipses will be the curves which will change the astronomy.

A beautiful picture for a beautiful museum. Next step: the cathedral!

**Location**: Frombork castle (map)

# The National Technical Museum in Prague

This is one of the Top 10 Museums in Prague! The museum was founded in 1908 and has been in its current location since 1941. It’s a very big building and the collection exhibited is so big although the exhibition about transports is its main attraction:

But for me, the exhibition about Astronomy has been the interesting part of the museum and I have been able to visit it on my own meanwhile my children were playing in another room with some technical toys. The astronomical rooms are very dark so it has been very difficult to take good pictures although I’ve tried to do my best. The collections has sundials, armilar spheres, quadrants, astrolabes,… and a lot of other astronomical instruments:

For example, the polyhedrical sundials are so beautiful like this constructed on a cube by German David Beringer around 1750:

Or… what about this other constructed by Mathias Karl Krausler in 1691?

The oldest exhibited astrolabe is this unsigned one from around 1450:

And there also is an unsigned torquetum from the late 16th century:

One of the instruments which have surprised me has been Joost Bürgi’s sextant for measuring the angles of celestial bodies (I knew that Bürgi, one of the inventor of logarithms, had constructed a lot of clocks and astronomical instruments but I didn’t expect to find one here!). Kepler used it to measure two consecutive oppositions of the planet Mars in 1602 and 1604.

There also is Habermel’s sextant, built by Erasmus Habermel (1538 – 15th of November of 1606 in Prag) who was mechanic at the court of Emperor Rudolph II:

The prevailing opinion for a long time was that the instrument belonged to Brahe and so it was called the “Tychonian sextant”.

Habermel was specialised in small devices and portable sundials and one example is this sundial in the form of a book (c.1600)…

… and another is this equinoctial sundial (1585):

Finally, look at this armilar sphere from the second half of the 16th century! It’s a piece of art!

**Location**: National Technical Museum in Prague (map)