By Natalia Kokoromyti
Niels Abel's yearning to produce as much mathematics as he could, combined with his eagerness to get to know the most famous mathematicians of his time, would unfortunately prove to be inversely proportional to the time he had to live.
September 1824 found him in a carriage destined to Denmark. He had already obtained a math degree from the University of Kristiansand, the region where he was born, and a grant from the Norwegian government to visit Europe. Although his reputation as a mathematician had already spread, this grant was approved with great difficulty, due to the financial hardships that post-war Norway was facing.
Niels got in the carriage carrying his meager possessions and sat down, waiting impatiently for it to start. There were no other passengers in the carriage. Soon he took out of his bag his work on "Solving the general quintic equation in radicals" and looked at it beaming with pride. He had given an answer to a problem that had been posed for a long time and had brought the greatest mathematicians to a standstill. Since algebraic formulas were found to solve equations up to the fourth degree, the challenge of finding a corresponding formula for the fifth degree equation was great. Abel had given the final answer: There is no such formula.
He caressed the cheap paper and smiled.
Abel discovered mathematics at the age of 15, under Bernt Michael Holmboe’s guidance. Having studied the groundbreaking approaches of Newton, Euler, Lagrange and Gauss, it seemed quite unbelievable that he had managed to give an answer, where these great sages had failed to. He had spent a lot of money, relative to what he had at his disposal, to print a few copies of his work: one of these was intended for Carl Friedrich Gauss. In describing Gauss’s mathematical prowess one thing was certain to Abel; that he had to get his feedback.
“Mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not succeeded yet. I therefore dare to hope that they will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations," he thought to himself.
He started coughing uncontrollably yet the command of the wagon driver to the horses drowned the sound; but Abel almost drowned himself, not that much from his cough that had been troubling him since his childhood, but because a well-dressed and beautiful woman came in and sat opposite to him.
Abel politely stood up to greet and help her with her belongings, but she nodded and smiled at him. As she took off her hat, she stared at him, impressed. Indeed Abel was a very handsome man. He lowered his own eyes, under her penetrating gaze, and when he looked at her again, she, as if in surprise, closed hers. This was repeated several times. Whenever Abel had his eyes lowered, she would look at him and when he looked at her, she would hastily close them as if she was asleep. Their conspiratorial “game” gave him an idea.
He took a notebook out of his bag and wrote:
I will write 1 every time her eyes are open and -1 every time she closes her eyes. Then he thought: I will add 1 and -1 and if the result is 1, then I will talk to her, if it is -1, then I will dedicate myself to studying my book for the rest of the journey and if it is 0, I will continue jotting down numbers.
So he wrote: 1-1+1-1+1-1+1-1+1 -...
While looking at this sum, with unwavering focus, he forgot about the woman for a while. As he leaned over the notebook, the curls of his hair hid his face, but little did he care about his appearance right now. He started wondering, with his childlike enthusiasm, how much this sum was, if it continued indefinitely.
If we write this sum like this: Σ = (1-1) + (1-1) + (1-1) + ..., then it is equal to 0.
But if we write it Σ = 1 + (- 1 + 1) + (- 1 + 1) + (- 1 + 1) + ..., then it is equal to 1.
If we write Σ = 1- (1-1 + 1-1 + 1-1 + ...), then we have Σ = 1-Σ, 2Σ = 1, therefore Σ = 1/2.
So what is the real value of the sum?
These infinite series are wicked, he thought and justified Euler, who, years ago, when he studied some of them, had made serious mistakes. Somehow though, this little game offered him glimpses to a world he never knew existed! This gave him the resolve to find the conditions under which a series converges or diverges.
Relieved, he raised his head, pushed his curls with his hand and looked at the woman in front of him again. She now had her eyes closed and seemed to be sleeping lightly. He looked out of the window, which gave him ample opportunity to enjoy the surroundings. Abel had a sincere love for nature.
His surprise from the abrupt stop of the carriage brought him a wave of intense cough. The woman opened her eyes, sighed and stood up. As his cough continued, she took a handkerchief from her bag and offered it to him with a smile. He took it awkwardly and the woman hurried out of the carriage. He didn't even manage to say "thank you" and when he looked at her handkerchief he saw a drop of blood that had come out of his mouth.
He stayed a few days in Copenhagen, where he met local mathematicians and then he continued to Germany. He was really looking forward to visiting Gauss and seeing first hand the evaluation of his work by one of the greatest mathematicians of his time. But when he heard that Gauss had not even looked at his dissertation, considering that it was yet another, unworthy work, of the many he was receiving, Abel was greatly distressed, but he was not disappointed. He was very confident in himself, but he developed a dislike for Gauss. So he decided to go to Berlin, instead of Göttingen where Gauss was living.
There at last Abel’s optimistic nature justified him. He met an engineer by profession, but a lover of mathematics, who was planning to publish a journal with purely research mathematical content of the time. His name was Leopold Crelle and he was to become Abel’s "agent" in some way. Crelle had published a mathematical article, which by coincidence Abel had read. He decided to pay him a visit in his office.
"Please," said Crelle, watching the young man enter hesitantly. “What would you like?”
"I want to talk about math”, Abel said, in broken German.
"What exactly do you mean by that, sir? Please sit down."
"I'm from Norway," Abel said humbly. “I have read and dare say understood, the classics: Newton, Euler, Lagrange, Gauss. My individual research at the moment concerns the proof of the binomial theorem, the proof that the fifth degree equation is not solved algebraically and…”
"Oh, you are Mr. Abel! I'm glad to know you. I have seen your work, but I have to confess that I did not understand many things. You see I love mathematics, but I haven’t fully mastered it."
"I have also read your published work and I have to tell you that it contains some errors," Abel said with disarming directness.
Crelle was stunned, but not offended. Although this young man looked kind of arrogant when he talked about mathematics, he was kind and timid.
"What could I do for you?" Crelle said finally.
"I would like to be included in the authors of your scientific journal, the one that you intend to publish. I would also like to contribute to the publishing costs, but my financial situation does not allow me to do so."
"Thank you very much, Niels. I feel that you will publish a lot of interesting work. As for the money, don’t worry, I can cover the cost."
This is how a long-term collaboration between the two men started. Abel had a platform to get his work known and Crelle's magazine gained prestige exactly because of Abel's brain-fuddling approaches. In the very first issues, great papers were published and Crelle not only did not ask money from Abel, but on the contrary, he tried to support him financially.
Abel stayed in Germany for over a year. There he laid the foundation for his most important work, which he completed later on in Paris. French mathematicians treated him with courtesy, but also with some disdain. During his stay, he polished his dissertation on transcendental functions and paved the way for a new branch in analysis.
More confident than ever before, he decided to submit his work to the “Académie des sciences.” Mathematicians Legendre and Cauchy were appointed judges. However his living conditions and the climate of Paris worsened Abel’s state of health. He was diagnosed with tuberculosis. Frightened by this development, he left for Vienna, entrusting the presentation of his work to a colleague. His dissertation was not understood and therefore its value was not recognized. In fact, legend has it that Cauchy lost the copy of the paper and never had the chance to read it. Exhausted physically and psychologically, Abel decided to return to his homeland with no money for his family.
His work eventually reached mathematician Jacoby, who promoted it to be later expanded by Weierstrass and Riemann. The French Academy tried to rectify matters by awarding him, Krele succeeded in securing him a professorship in Berlin, and Holmboe published some of his works. But all this happened too late. Tuberculosis had defeated Abel at the age of 27.
The "Abelian" groups, the "Abelian" functions, the "Abel Theorem," the "Abel Crater" on the moon and the "Abel International Prize" were named after him.
One thing is for sure though: we are still waiting for the apologies of Gauss and Cauchy, for underestimating such a great mind.