“Comme EX UNGUE LEONEM”
An important disclaimer: there are several versions of how the facts really happened mainly with how long it took Newton to answer this problem. Thus, I’ll base the narration on historical facts but include fictitious or not proven parts and by the power of narration make it real. However, I’ll leave some references for those purists at the end so everyone gets his version of the story.
In June 1696 the Swiss mathematician Johann Bernoulli published in the journal Acta Eruditorum (the first scientific journal in German-speaking Europe) the following challenge to European mathematicians:
During the XVI century, it was a common practice to post unsolved mathematical problems in journals and challenge the most brilliant minds in Europe to solve them as a way to progress in science. J. Bernoulli wasn’t the first to do it. Previously, the French mathematicians Blaise Pascal, and Pierre de Fermat challenged the scientific community of the time in the same way. However, Bernoulli’s publication was a trap instead of a challenge. He wanted to test the man who proposed the idea of gravity as well as (one of) the inverter(s) of calculus, the British mathematician Issac Newton, who was in his 50s when that article was published. Recognizing the plot in Bernoulli’s challenge Newton solved the problem, but instead of sending it to Acta Eruditorum he published it anonymously in Philosophical transactions (an English journal) before the deadline given by Bernoulli in a clear violation of the terms and spirit of the contest. And as if that were not enough, Newton omitted the details of his solution, only saying what the correct answer is!
How did Bernoulli take Newton's answer to his challenge? Why on the earth J. Bernoulli wanted to confront Newton? Why is the shortest time curve between two points subject only to the action of gravity the arc of a cycloid? How is this related to the brachistochrone problem? To answer these questions and to know why Newton became the biggest troll in history, we must have some context of what Europe was like at the end of the 17th century (scientifically speaking of course).
First a map of Europe at the beginning of the 17th century. Portugal, Spain, the south of Italy, Belgium, and a part of the Netherlands were a unique nation called the Iberian Union, The first Reich, the holy roman empire, which includes the current Germany, Switzerland, and the north of Italy was a kind of European Union of the time. Finland still was a Swedish duchy, Norway and Denmark were a single nation, the Ottoman Empire annexed the Balkan peninsula and further territories in the north, Poland lived its golden age with the biggest territory in its history with the Republic of the two nations, Russia was ruled by the first Romanov, the Ksar Michael I, and Great Britain and Ireland Islands was living the period of the Union of the crowns under the sovereignty of the king James VI (Scotland) and I (England and Ireland). It is in this Europe where the events took place.
Brachistochrone is a combination of two greek words: βράχιστος (brákhistos) which means shortest and χρόνος (khrónos) which means time. Thus, the brachistochrone curve is the shortest time curve, which despite what an inattentive reader might think, isn’t a straight line. The reason for this is easy to grasp, the straight line is the shortest distance. However, the shortest time is a competition between the distance and how fast an object can go from point A to point B when gravity is acting on it. The brachistochrone is, therefore, a compromise between distance and depth. The curve should be curved enough to take advantage of the gravity force, but not so much depth to the distance, and gravity plays against it. The problem, I think, is intuitively easy to understand, but hard to solve, at least it was in the late 17th century when Johann Bernoulli published the challenge.
The shape of the brachistochrone was the interest of mathematicians during the whole 17th century. The first one to say something about this problem was Galileo Galilei in his scientific book “Discorsi e dimostrazioni matematiche intorno a due nuove scienze” (Discourses and Mathematical Demonstrations Relating to Two New Sciences), commonly just called the two sciences:
The Galileo problem is similar to but different from the one proposed by Bernoulli, and I won’t enter into these details. but, here is a good reference for those interested. The point here is that Galileo already said that the shortest time curve couldn't be the shortest distance, instead, it should be an arc of a circle. However, In footnote 48 of Galileo’s statement of the Scholium (in the two sciences), Drake wrote:
“All that could properly be deduced was that the shortest descent is along some kind of curve. The curve is in fact only approximately circular and was later shown to be cycloidal”
Later in 1648, Blaise Pascal challenged the mathematicians of Europe to find the Tautochrone curve i.e. 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 on the curve. The solution to this problem was given by the Dutch mathematician Christiaan Huygens in the next year, proving that the Tautochrone curve is in fact an arc of a cycloid! Then, the cycloid is both the shortest time curve and also the curve of equal time (tautochrone is also a combination of two Greek words: tauto meaning same and khrónos meaning time). Moving now in time to 1684. the year when all the demons were unleashed, an ex-Ph. D. student of Huygens published the “Nova Methodus pro Maximis et Minimis.” Considered the first publication on the calculus of infinitesimals, commonly called simply calculus. The author, a Saxonian mathematician raised in the heart of the Holy Roman Empire, Gottfried Wilhelm Leibniz, claimed publicly to be the inventor of a method to determine maxima or minima of varying quantities, a.k.a. differential calculus. Among several achievements, Leibniz had the merit to be the Ph.D. advisor of Jakob Bernoulli, the older brother of Johannes who also was the one who taught him mathematics. In 1695 Leibniz and Jakob were in charge of developing calculus into a magnificent tool for solving a variety of problems. Therefore, the Bernoullis had a close relationship with Leibniz and spread the ideas of calculus around Europe. Thus, between 1684 and 1695 calculus showed a powerful tool to solve a generality of problems which before should be attacked on the premise that every case is a case. What led to the demons being unleashed? Well, between this period, specifically in 1687 (three years after the first Leibniz publication about calculus) one of the most famous books in the history of physics was published, the “Philosophiæ Naturalis Principia Mathematica” (The Mathematical Principles of Natural Philosophy) called simply Principia. The author of the book Issac Newton dedicates the section I of chapter I of his Principia to a method he invented that could be useful to determine maxima and minima of varying quantities, the method of fluxions (a.k.a. differential calculus, and yes the same one Leibniz (also) invented). He used “fluent” to designate independent variables, “relata quantitas” to indicate dependent variables, and “genita” to refer to quantities obtained from others using the four fundamental arithmetical operations. The reason for these weird names of “fluent”, “relata qunatitas”, and “genita” is because the concept of function hadn't been invented yet then the idea of a varying quantity as y = f(x) as we used now was unknown at that time.
After the publication of his Principia, Newton realized that the mathematicians of continental Europe already knew how to determine maxima and minima by means of infinitesimal calculus. The methodology employed for the continental Europeans had a different notation, his fluxions were written in other terms, and his operations had other symbols, but the soul of the theory was the same, using infinitesimal quantities to determine maxima and minima. Newton was more than angry, he was furious. –Who dares to steal my work and declare it as yours– thought Newton while reading his fluxions prostituted in the European scientific journal as a Leibniz invention. That thieving swine has plagiarized my ideas, and has claimed a merit that belongs to me alone, but how? –Newton wondered himself–. Since when did the Holy Roman Empire propagate English inventions as its own? And that Leibniz, he is a burglar, he came to my country, several years ago, and surely, he read my work on fluxions. Certainly, someone let him read my manuscripts on fluxions and this worthless rodent published them as his own!
In fact, before publishing his work, Leibniz was traveling around Europe and spent some time in England. Although there is no proof that Leibniz had stolen any of Newton's ideas, probably, his mind was influenced by Newton’s fluxion theory and later developed independently when Leibniz came back to the Holy Roman Empire. Despite this, there’s no doubt about Leibniz being the first one to make public the ideas of calculus to the scientific community. Newton, on the other hand, was a renewed scientist when this dispute started, and for good or for bad, his prestige was higher than Leibniz's. I think even nowadays, everyone knows or at least has heard about Newton, but it will be hard to find a scientist who knows who Leibniz was and his contributions to science. Thus, you can get an idea of how unequal this controversy was! Currently, the scientific community agrees that both mathematicians were the inventors of calculus and even when Leibniz published first, they arrived at the same concept independently. So, let's come back to the XVII century. Since Newton was a renowned scientist, who, in principle, did not have reason to appropriate the idea of others, the scientific community split into Team Newton, which use the fluxions and Newton's notation of calculus, and Team Leibniz which use the Leibniz version of calculus, which is, by the way, the one is taught in our days. As you can imagine, Johann Bernoulli was on the Leibniz side and supported him as the inventor of calculus.
He is lying, the method to determine maxima and minima is my creation! – yells Leibniz to his friend Johann – How dare this skinny English worm to accuse me of plagiarism – Leibniz continues – To me, to meeeeeeee – Leibniz spittles flew from his lips as he shouted those words – To me, to meeeee, Is this what it has come to? These mediocre English mathematicians have been using my methods for years, they are no more than a bunch of cowards at Newton's service – yells Leibniz when walking around the room. Our scientists and most of continental Europe are with you Gottfried, please take a seat, I have an idea to solve this situation – says J. Bernoulli trying to chill Leibniz –. The English mathematicians are the cancer of knowledge, the scum of the world mathematicians, I will never steal ideas from them. But if he thinks I let him be with the credit of my invention, he’s wrong, I’ll write to the Royal Society in London for them to decide who the real thief is – Says Leibniz to Bernoulli, now calmer. – To the royal society, did you lose your mind? the royal society is full of Newton adepts, you certainly will lose – said Bernoulli trying to persuade Leibniz. – No, No, No – shouted Leibniz, losing his composure once again – It should be the royal society, even though if this bunch of cowards is Newton adepts, they cannot simply ignore the evidence, my articles were published before his Principia, in honor to science they will be obliged to recognize me as the inventor of “fluxions” – Leibniz said this last word imitating a British accent. This will be a bad idea, Britons won’t recognize you, Gottfried, listen to me, we should destroy him with science, I have a better idea to prove that Newton is the biggest liar in the history of science – Says Bernoulli advising Leibniz once more. Do whatever you want Johann, I’ve already decided to send a letter to the royal society, this skinny worm has no idea who he's messed with – replied Leibniz to Bernoulli's advice.
Leibniz, as planned, sent a request to the royal society to decide who was the real inventor of calculus. However, Leibniz was unaware of important detail about the Royal Society. Newton had recently been chosen as its president, and despite Newton not taking part in the decision, he chose his adepts to judge the authorship of Calculus. I think you can guess the result. Yes, the royal society chose Newton as the inventor of calculus instead of Leibniz.
Bernoulli, on the other hand, was even more ambitious than Leibniz and tried to destroy Newton with his weapons. Thus, considering Newton was both the inventor of calculus and also the master of gravity, he decided to challenge the most brilliant mathematicians in the world to solve a minimization problem in which only gravity was involved. Making his publication in Acta Eruditorum a clear and public contest for Newton's ``inventions”. Since the context cannot be forever Bernoulli established the end of the year as a deadline. Indeed, the media of the time was not as advanced as today's media. However, no later than September the challenge has already arrived in the United Kingdom, and some of Newton's colleagues started to work on a solution to the problem. Newton, feeling himself in a superior position and with no need to receive the approval or recognition of anyone, simply ignored the contest and kept himself far away from any possible allusion to the challenge published in Acta Eruditorum. When December of 1696 arrives Bernoulli had only received two solutions to his problem, one given by his older brother Jakob Bernoulli and the other one for Leibniz, which also requested Bernoulli an extension of the deadline arguing that probably the information of the challenge arrived later than expected and they will be in disadvantage with the mathematician of the Holy Roman empire. Bernoulli immediately understood the hidden lines in the Leibniz message and extended the deadline until Easter in 1697. But this time to be sure Newton could not refuse his challenge, he composed a leaflet on the contest with an extra provocation:
“Only the celebrated Leibniz, who is so justly famed in the higher geometry has written to me that he has by good fortune solved this, as he himself expresses it, very beautiful and hitherto unheard of problem. . .”
The leaflet was sent around Europe, and Bernoulli himself made a point to have one delivered directly to Newton. Thus, there was no way that Newton, or any other reader, could miss Bernoulli’s explicit targeting of Newton. When Newton read the leaflet immediately realized that the challenge couldn’t be avoided, this time, Leibniz was announced as a master of calculus, and being able to solve a minimization problem involving gravity. In the middle of his resignation for not being able this time to answer Bernoulli's challenge with silence, Newton said to himself: “I do not love to be printed upon every occasion much less to be dunned and teased by foreigners about Mathematical things.” Immediately after reading it, Newton planned his revenge. I’ll solve this problem today before sleeping, and I’ll send the solution tomorrow morning, my fluxions are more than enough to deal with this problem – said Newton to himself –. But, I’ll do it better, I won’t show my procedure, I simply will give the answer, and I’ll do it anonymously, and not in Acta I’ll publish in an English journal, before the deadline. Thus my solution will be the first one to be known, and there will no longer be any doubt that Leibniz plagiarized my fluxions, and this time I will have published my results before him.
And that's exactly what he did, his answer was published in Philosophical Transactions before Easter, in a clear violation of the spirit of the contest. And to be sure the contest won’t be canceled he only answered without an explanation or a new method to solve the problem, just the answer:
“The shortest time is an arc of a cycloid”
When the deadline arrived, Bernoulli had received solutions from Jakob Bernoulli, Gottfried Leibniz, Guillaume François Antoine, Marquis de l'Hôpital a Ph. D. student of Johann Bernoulli, and an anonymous English solution. Bernoulli published all these solutions as the correct answers to his challenge and even commented on the anonymous one: “He has taken us for idiots, he thinks we don't recognize his style, his way of acting, this answer, because it cannot be called a solution was certainly given by the egocentric skinny worm, I immediately recognized it comme EX UNGUE LEONEM (just as a lion by its claw marks). ”
Thus, in the end, Newton trolls both, Bernoulli and Leibniz. The problem of the shortest curve of the brachistochrone is not a problem of infinitesimal calculus, instead belongs to a different branch of mathematics called the calculus of variations. However, a formal theory of calculus of variation will be born in 1756 by a Ph.D. student of Johan Bernoulli, his name was Leonhard Euler, and he could be considered one of the most brilliant mathematicians who has ever exists. Sadly, the theory arrives quite late because Leibniz dead in 1716, Newton in 1727, and Johann Bernoulli in 1748. Thus, they never see the calculus of variations, a completely new branch of mathematics that was born from an ego dispute between two mathematicians. Then sometimes being an egocentric scientist pay off :0).
I’d like to finish this post with some references that were very useful when writing this post:
The Wikipedia link to this topic includes the solution given by Jakob Bernoulli, Johann Bernoulli, and Isaac Newton.
This one is an article I found while doing my research, and I think is very interesting, it tells a dark side of this story, really worth it.
These two youtube videos about the problem (including its solution) are video_1 and video_2.
A video of a very mature post-doc who went to the Polish science museum in Warsaw and move away from all the children in the museum to show them how the experience should be done.
Comments
Post a Comment