Where does one start regarding Leonhard Euler (1707-1783).? We all no doubt know that he was blind for the last 17 years of his life but that his Mathematical output during these years did not diminish or that he is the most prolific Mathematician in history (followed closely by J.Jacobi (1804-1851) of the Jacobian fame) or that he became a Mathematical superstar overnight by solving the Basel problem and its corollaries and advanced analysis beyond recognition in a single swoop.
Euler’s Solution and its Corollaries
I recall that when I first saw Euler’s absolutely beautiful proof of the sum of the square of the reciprocals of the natural numbers, I applied his technique to the sum of the reciprocals of fourth (and sixth…) powers and then commenced with the cosine (instead of the sine as Euler did) and to determine wonderful analytic expressions for sums of reciprocal of other odd and even exponents only to find that Euler had already done this. But I was happy to have seen that what I had done was correct and followed the footsteps of the “Giant himself”.
Number Theory & Basel Problem
Euler made advances in number theory (consider the Zeta function), applied Mathematics (his equation of motion), complex numbers (Euler’s theorem), differential equations (the integrating factor), Euler’s differential equation, the list is almost endless.
Euler was from Basel Switzerland he spent much of his academic life in St Petersburg establishing a school of Mathematics there.
“Read Euler, read Euler, he is the master of us all”. P. S. de Laplace (1749-1827) considered to be the Newton of France.
Staying with quotes of one giant praising another the last quote reminds me of another by J. Lagrange (1736-1813) regarding the beheading of A. Lavoisier (1743-1794) the father of Analytical Chemistry during the French revolution.
“It has cost them but a moment to cut off that head; but a hundred years will not be sufficient to produce another like it”.
Resolution of the Königsberg Bridge Problem
Remaining with quotes I will return to Euler:
Thus you see, most noble Sir, how this type of solution to the Königsberg bridge problem bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle.
Leonhard Euler.
Legacy and Recognition
Here he solved negatively the problem of traversing all the bridges in Königsberg and visiting all the surrounding land masses, creating graph theory and topology. My favourite equation of Euler is his polyhedral characteristic formula i.e. X = V – E + F, where V=number of vertices, E=number of edges, F=number of faces. Many parts of the world call the number e = 2.7182818… the Euler number, even though Chinese scholars knew it earlier due to their study of compound interest.
I could go on but it is difficult to know when to stop. Until next time, Happy Birthday “old friend”
