# A catenary welcomes you to the beach in Ocata

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)

# 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)

# Some caricatures of famous mathematicians

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:

Galileo:

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!

**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)

# Hall of the former Faculty of Sciences in Zaragoza

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…

…Galileo Galilei…

…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)

# Immortal Books, essential instruments (II)

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 workDe Revolutionibus Orbium Coelestiuminthe 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):

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 RudolphinaeandAstronomia Nova(1609). InAstronomia Novahe explained two of the three fundamental laws describing the movement of the planets; the third one was explained inHarmonices 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!

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:

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*):

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)

# Isaac Newton in the Simpsons

Isaac Newton was born in January 4, 1643 (in the Gregorian calendar) so today it’s his birthday although he was really born in December 25, 1642!

I’ve been looking for something which could celebrate this day and I’ve thought in his appearance in the chapter “The Last Temptation of Homer” of the Simpsons. Homer is visited by an angel disguised as Newton to show him what his life would be without Marge.

# Isaac Newton’s doodle

This is the doodle published for Sir Isaac Newton’s 367th birthday in January 4, 2010.

Will we see another doodle dedicated to the great genius in two days?

# The National Portrait Gallery in London

This was one of the great moment in my last holidays in England! Newton and me together in the same picture! (I must thank the guard because he allowed me to take this picture) Today is 25 December and this is the reason because I am publishing today this picture: Newton was born on December 25, 1642 (Julian Calendar) so… Happy Birthday Great Mind!

Sir Isaac Newton (1642-1727) [by Sir Godfrey Kneller (1646-1723)]

An immensely influential mathematical scientist, in one year (1665-6), when driven from Cambridge by plague, Newton formulated a series of important theories concerning light, colour, calculus and the ‘universal law of gravitation’. According to tradition, he developed the latter theory after seeing an apple fall from a tree. He published

Principia(1687) and theOptiks(1704), and was knighted in 1705. Newton was President of the Royal Society from 1703 until his death.

Newton is not alone and he is accompanied by other great English scientist like Edmund Halley. The portrait of Halley is attributed to Isaac Whood (1688-1752) from 1720. Halley has a chart showing his predicted path accross Southern England of the total solar eclipse of 22 April 1715.

Edmond Halley (1656-1742)

Astronomer. At the age of twenty-two in 1678 he published his catalogue of the stars of the southern hemisphere, and in 1705 his celebrated work on comets. Halley published Newton’s Principia at his own expense, 1687; he was appointed Astronomer Royal in 1721. He successfully predicted the reappearance of the great comet in 1758 (‘Halley’s Comet’).

Sir Christopher Wren (showing a plan of St. Paul’s Cathedral) is also in the Gallery:

Wren was an architect and scientist. After the Great Fire of 1666, he rebuilt St. Paul’s Cathedral and many of the London City Churches; his work includes the Sheldonian Theatre in Oxford (1664-9), Trinity College Library in Cambridge (1674-84), Chelsea Hospital and Greenwich Hospital (from 1696). He was professor of Astronomy at Oxford and later President of the Royal Society.

Herschel and Boyle are also exhibited in the Gallery but it was almost impossible to take a picture of them so it’s better if you go to the National Portrait Gallery web and you’ll see better pictures of them.

Before ending this post, we must look at this anamorphic picture of King Edward VI:

Edward VI 1537-53 by William Scrots (active 1537-53). Oil on panel, 1546.

This unusual portrait of Edward was painted in 1546 the year before he became king. He is shown in distorted perspective (anamorphosis), a technique to display the virtuosity of the painter and amaze the spectator. Anamorphic portraits were relatively popular in mainland Europe at this time, but this painting was considered particularly remarkable […].

The anamorphosis is a very interesting mathematical technique which must be explained in detail but I am not going to do it now.

MERRY CHRISTMAS… or ….

HAPPY NEWTON’S BIRTHDAY!

**Location**: National Portrait Gallery in London (map)

# The King’s School in Grantham

My last visit in Grantham was the King’s School. Why? Read it:

A second plaque remebers us the value of this old building:

**Location**: The King’s School (map)