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 eif and e-if, respectively. He uses these to find cosmf, sinmf, and their product cosmf sinmf. 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.

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