E246  Subsidium calculi sinuum
(A contribution to the calculations of sines)
Summary:
Let u = cos f
+ i sin f and
v = cos f 
i sin f.
Euler shows these are
e^{if} and
e^{if},
respectively. He uses these to find
cos^{m}f,
sin^{m}f,
and their product
cos^{m}f
sin^{m}f.
He starts summing series of complex numbers and gets more things that look
like Fourier series, as well as the series:
f/2 =
sinf 
sin(2f)/2 +
sin(3f)/3 
sin(4f)/4 + ....
We can apparently find this theorem in
E447 and
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.
Publication:

Originally published in Novi Commentarii academiae scientiarum Petropolitanae 5, 1760, pp. 164204

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.
Documents Available:
 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 settheory, 18701985.” Archive for History of Exact Sciences, 45 (4), pp. 281334 (1993).
 Ferraro G., “Functions, functional equations, and the laws of continuity in Euler.” Historia Mathematica, 27 (2), pp. 107132 (May 2000).
 Katz VJ., “The calculus of the trigonometric functions.” Historia Mathematica, 14 (4), pp. 311324 (Nov 1987).
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