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.
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 Rudolphinae in 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!
Our daily landscape is full of images like this one. Every day we see chains that bar our way and we assume them naturally. However, which shape do these chains take? Can we identify them with known curves? These questions have thrilled some of the greatest mathematicians of all times and have led to the development of elaborated techniques valid today.
The main characters of this post are the Bernoulli brothers, Jakob (1655-1705) and Johann (1667-1748). They studied together Gottfried W. Leibniz’s works until they mastered it and widened the basis of what is nowadays known as calculus. One of the typical problems was the study of new curves which were created in the 17th century to prove the new differential methods. René Descartes and Pierre de Fermat invented new algebraic and geometrical methods that allowed the study of algebraic curves (those which the coordinates x and y have a polynomial relation). Descartes didn’t consider in his Géométrie curves that were not of this kind, and he called “mechanical curves” to all those ones that are not “algebraic”. In order to study them, he developed non algebraic techniques that allowed the analysis of any kind of curves, either algebraic or mechanical. These mechanical curves had already been introduced a lot of time ago, and were used to solve the three classical problems: squaring the circle, angle trisection and doubling the cube.
When developing calculus, Leibniz’s objective was to develop this general method that Descartes asked for. When Bernoulli brothers started to study curves and its mechanical problems associated, calculus would become its principal solving tool. For instance, in 1690 Jakob solved in Acta Eruditorum, a new problem proposed by Leibniz. In this document, he proved that the problem was equivalent to solve a differential equation and the power of the new technique was also shown.
During the 17th century, mathematicians often proposed problems to the scientific community. The first challenge that Jakob exposed was to find the shape that takes a perfectly flexible and homogeneous chain under the exclusive action of its weight and it is fixed by its ends. This was an old problem that hadn’t been solved yet. As we can see in the picture, the shape taken by the chain is very similar to a parabola and, owing to the fact that it is a well-known curve for centuries, it comes easy to think that it is indeed a parabola (for example, Galileo Galilei thought that he had solved the problem with the parabola). However, in 1646 Christiaan Huygens (1629-1695) was capable to refute it using physical arguments, despite not being able to determine the correct solution.
When Jakob aunched the challenge, Huygens was already 60 years old and successed in finding the curve geometrically, while Johann Bernoulli and Leibniz used the new differential calculus. All of them reached the same result, and Huygens named the curve “catenary”, derived from the latin word “catena” for chain.
Hence the first picture shows a catenary which we observe without paying much attention although all its history. Nowadays, we know it can be described using the hyperbolic cosine, although any of all these great mathematicians couldn’t notice it, as the exponential function had not been introduced yet. Then… How did they do it? Using the natural geometric propreties of the curve so Huygens, Johann Bernoulli and Leibniz could construct it with high precision only usiny geometry. Jakob didn’t know the answer when he proposed the problem and neither found the solution by himself later. So Johann felt proud of himself for surpassing his brother, who had been his tutor (Jakob, who was autodidactic, introduced his younger brother to the world of mathematics while he was studying medicine). What is more, the catenary problem was one of the focusses of their rivalry.
Jakob and Johann were also interested in the resolution of another fmous problem: the braquistochrone. Now, Johann proposed the challenge of finding the trajectory of a particle which travels from one given point to another in the less possible time under the exclusive effect of gravity. In the deadline, only Leibniz had come up with a solution which was sent by letter to Johann with a request of giving more time in order to receive more answers. Johann himself had a solution and in this additional period Jakob, l’Hôpital and an anonymous english author. The answer was a cycloid, a well-known curve since the 1st half of the 17th century. Jakob’s solution was general, developing a tool that was the start of the variational calculus. Johann gave a more imaginative solution based on Fermat’s principle of maximums ans minimums and Snell’s law of refraction of light: he considered a light beam across a medium that changed its refraction index continuously. Given this diference between their minds, Johann enforced his believe that he was better due to his originality and brightness, in contrast with his brother, who was less creative and worked more generally.
A third controversial curve was the tautochrone which is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point. This property was studied by Huygens and he applied it to the construccion of pendulum clocks.
While we can see catenaries in a park to prevent children from danger or in the entrance to this Catalan beach, we luckily do not see cycloid slides. Thus, children will not descend in the minimum time but will reach the floor safe and sound!
But… who was the misterious english author? For Johann Bernoulli the answer to this question did not involve any mistery: the solution carried inside Isaac Newton’s genius signature -to whom we apologise for having mentioned Leibniz as the calculus inventor-, and he expressed that in the famous sentence:
I recognize the lion by his paw.
Here we see one more example of a catenary in our daily life:
You have more information in this older post.
This post has been written by Bernat Plandolit and Víctor de la Torre in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).
Location: Ocata beach (map)
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)
In my last post about the Hewelanium Centre of Gdansk, I must show you the caricatures of the famous mathematicians and astronomers which you find on the walls (and you also can buy as a puzzle in the shop of the museum). You have pictures of Archimedes, Pascal, Copernicus:
Halley and Hevelius:
Sir Isaac Newton:
and Albert Einstein:
These aren’t good pictures but the posters are in 3D and my camera is not the best camera in the World!
This doodle was published in August 25, 2009, to celebrate the 400th anniversary of Galileo’s telescope.
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:
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 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.
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:
On both sides we have some of the most important men in the history of astronomy like Hipparcus of Rhodas,
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”:
In another house there also are the allegorical Astronomia rounded by Cellarius’ heliocentric systems:
And finally we find other allegories like the Architecture, the Geometry or the Geography in the opposite side of the square:
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:
Location: Długi Targ in Gdansk (map)
One of the most beautiful buildings which can be visited in Zaragoza is the hall of the former Faculty of Sciences. Itwas constructed by Ricardo Magdalena in 1893 and is decorated with 72 statues and roundels designed by Dionisio Lasuén (1850-1916). These allegorical sculptures are dedicated to Medicine and Science and we find some very important mathematicians among all the scientifics represented on them. For example:
We also fins a representation of the Theorem of Pythagoras next to these two great names:
Other important mathematicians are:René Descartes…
…the great Euclid…
…Hipparchus of Rhodes…
We also find Spanish scientific and mathematicians as the Andalusi Abû al-Qâsim al-Zahrawî (Al-Zahra, Cordova,936-Cordoba,1013), also known as Abulcasis. He was an important physician, surgeon and doctor who wrote the Kitab at-Tasrif (Arabic,كتاب التصريف لمن عجز عن التأليف) or The Method of Medicine (compiled in 1000 AD) which had an enormous impact in all Medieval Europe and the Islamic World.
Pedro Sanchez Ciruelo (Daroca,1470 – Salamanca, 1550) was an important Spanish mathematician of the 16th century who wrote some mathematical treatiseslike the Cursus quattuor mathematicarum artium liberalium (1516) thorugh which Bradwardine’s Arithmetic and Geometric work was taught in Spain.
Jorge Juan (1713-1773) and Antonio Ulloa (1716-1795) were two Spanish scientifics who participated in the measurement of the Terrestrial Meridian organized by the Academy of Sciences of Paris:
Gabriel Ciscar (1759-1829) wrote the Curso de Estudios Elementales de la Marina, divided in a volume dedicated to Arithmetics and another dedicated to Geometry.
Finally, José Rodríguez González (1770-1824) and José Chaix (1765-1811) participated in the triangulations of the meridian arc from Dunkerque to Barcelona.Furthermore,Chaix wrote the Instituciones de Cálculo Diferencial e Integral and publicó the Memoria sobre un nuevo método general para transformar en serie las funciones trascendentes which were so popular in Spain because of the explanations of the differential calculus.
So, the building is so beautiful and you can learn History of Mathematics while walking around it. Do you want anything else?
Location: Hallof the Faculty of Science in Zaragoza (map)
Last Wednesday I went to MMACA (Museum of Mathematics of Catalonia) with some of my students. This museum is located in Mercader Palace in Cornellà de Llobregat (near Barcelona) since February and we enjoyed a very interesting “mathematical experience”.
The museum is not so big but you can “touch” and discover Mathematics in all its rooms. I think that there are enough experiences to enjoy arithemtical and geometrical properties, simmetries, mirrors, impossible tessellations, Stadistics,…
For example, students could check the validity of theorem of Pythagoras in two ways. First of all, they coud weigh wooden squares and check that the square constructed on the hypotenuse of a right triangle weighs the same as the two squares constructed on the other two sides of the triabgle. Later, they discovered that the first square could be divided in some pieces of Tangram with which they could construct the other two squares. So the visitors demonstrated the theorem in a very didactic way: playing with balances and playing with tangram.
Students also learnt some properties of the cycloid and they could check its brachistochronic characteristic. I imagine Galileo or some of Bernoulli brothers in the 17th century doing the same experiments with a similar instrument. What a wonderful curve! The ball always reaches the central point in the same time and its initial position doesn’t matter!
Another of the studied curves is the catenary which is one of the emblematic mathematical symbols of Antoni Gaudi’s architecture in Barcelona.
Of course, polyhedra are very important in the exhibition and visitors can play with them so they discover some of their most important properties. For example, which is the dual polyhedron of the dodecahedron? Playing with it the students could see that the hidden polyhedron is a… You must visit MMACA and discover it!
Another example: look at these three wooden pieces…
The dodecahedron has an ortonormal symmetry and we can check it with an ortonormal set of mirrors:
There are more mirrors and more wooden pieces to play and construct other different Platonic and Archimedian polyhedra.
And… did you know that it’s possible to draw a right line playing with two circles? If the red circle rotates within the black one… what figure is described by the yellow point?
In the 13th century, the great Nasîr al-Dîn al-Tûsî had to build one similar instrument to improve the astronomical geometrical systems with his “Al-Tûsî’s pair”:
Rotating a circle within another one, he could move a point in a right line without denying Aristotelian philosophy. This dual system was used by al-Tûsî in his Zîj-i Ilkhanî (finished in 1272) and Nicolas Copernicus probably read this innovation together with other Arabic astronomical models. Thinking about them, he began to improve the astronomical system of his De Revolutionibus (1543). Al-Tûsî’s pair was very famous until the 15th century.
In Erathostenes Room there are some Sam lloyd’s puzzles, games about tesselations, Stadistics, Probablility and this quadric:
I didn’t know that it could be described only with a multiplication table! Is its equation z = xy? Yes, of course! My students also played to build the famous Leonardo’s bridge and they could see that there isn’t necessary any nail to hold a bridge.
Ah! And I can’t forget to say that if you visit MMACA with a person that don’t like Maths, he/she can always admire this beautiful XIX century Mercader Palace:
Furthermore, one of the rooms of the palace is decoratd by a chess lover!
So… you must go to MMACA and enjoy Mathematics in a way ever done!