Abstract:
The history of Euler's formula $ln(cos\phi +isin\phi)=i\phi$ or $cos\phi + isin\phi = e^{i\phi}$ and the history of the most beautiful formula in mathematics $e^{i\pi}=-1$ or $e^{i\pi}+1=0$ are mostly known, except for one small detail: who first wrote them in the form we have given? We will try to shed light on this issue.
It all started when L. Euler was not yet seven years old: the English astronomer and mathematician R. Cotes developed the J. Bernoulli’s idea of a connection between logarithms and circular functions and received the first of the formulas we have given. Then I. Bernoulli and G. Leibniz discussed the value of a negative number logarithm, getting conflicting conclusions. Close to resolving the issue was Giulio Fagnano. 34-year-old Euler received the first and second formulas connecting the exponential and trigonometric functions, and then the expression for the negative number logarithm. Euler's work contains the values of the logarithm for various arcs, including for $\pi$, but there is no explicit expression $e^{i\pi}=-1$. This equality appeared more than half a century later in the work of the French engineer and mathematician Jacques Français among several special cases of Euler's formula. Theoretical mathematicians, in particular, A. Cauchy, did not single out this formula among others. The origin of these formulas from geometric and mechanical problems has already been forgotten. Among specialists from adjacent sciences – physics, astronomy, geodesy, cartography, logic, philosophy – admiration for the beauty and mystery of Euler identity grew. A mystical halo began to appear around this formula (B. Pierce). At the end of the 20th century, according to a survey of readers of the journal Mathematical Intelligenñer, Euler's identity was recognized as the most beautiful mathematical result out of 24 proposed. In our story, the chronology of previous mathematical events will be presented.
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