E246 -- Subsidium calculi sinuum
(A contribution to the calculations of sines)
Let u = cos f
+ i sin f and
v = cos f -
i sin f.
Euler shows these are
respectively. He uses these to find
and their product
He starts summing series of complex numbers and gets more things that look
like Fourier series, as well as the series:
sin(4f)/4 + ....
We can apparently find this theorem in
E655, as well as
in Cauchy and Abel.
According to C. G. J. Jacobi, a treatise with the title: “Subsidium doctrinae sinuum” was read to
the Berlin Academy on March 9, 1752.
According to the records, it was presented to the St.
Petersburg Academy on March 12, 1753.
Originally published in Novi Commentarii academiae scientiarum Petropolitanae 5, 1760, pp. 164-204
Opera Omnia: Series 1, Volume 14, pp. 542 - 584
- According to Jacobi, the manuscript of the printed treatise can be found in
the archive of the Berlin Academy.
- Original publication: E246
- The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E246 include:
- Cooke R., “Uniqueness of trigonometric series and descriptive set-theory, 1870-1985.” Archive for History of Exact Sciences, 45 (4), pp. 281-334 (1993).
- Ferraro G., “Functions, functional equations, and the laws of continuity in Euler.” Historia Mathematica, 27 (2), pp. 107-132 (May 2000).
- Katz VJ., “The calculus of the trigonometric functions.” Historia Mathematica, 14 (4), pp. 311-324 (Nov 1987).
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