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)
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)
Descartes (1596-1650) and his Discours de la méthode (1637) is exhibited here although there is no word about his Géométrie. I could cry for it!
D’Alembert and Diderot are also here but the important D’Alembert’s mathematical works are not mentioned either. They compiled the first Encyclopèdie ou Dictionnaire raisonné des sciences, des arts et des métiers (1751-1772) and we have to settle this:
Finally, a little mention to Albert Einstein (1879-1955) and his Über die spezielle und die allgemeine Relativitätstheorie: Gemeinverständlich (1917):
As you can see it’s a very good opportuniry to learn some things about all these great scientists and their works! The other scientist are Lavoisier, Lyell, Darwin, Bernard, Maxwell, Ramón y Cajal, Curie, Dirac and Morgan.