English Title

A contribution to the calculations of sines


Leonhard Euler

Enestrom Number


Fuss Index


Original Language


Archive Notes

According to Jacobi, a treatise with the title "Subsidium doctrinae sinuum" was read to the Berlin Academy on March 9, 1752. According to St. Petersburg Academy records, it was presented to that Academy on March 12, 1753.

Content Summary

Let u = cos(φ†) + i sin(φ) and v = cos(φ†) – i sin(φ†). Euler shows these are eiφ† and e-iφ†, respectively. He uses these to find cosm(φ†), sinm(φ†), and their product cosm(φ†)∙sinm(φ†). He sums series of complex numbers to get more things that look like Fourier series, including the series φ†/2 = sin(φ†) – sin(2φ†)/2 + sin(3φ†)/3 – sin(4φ†)/4 ± .... Apparently, we may also find this theorem in E447 and E655, as well as in works by Cauchy and Abel.

Published as

Journal article

Published Date


Written Date


Original Source Citation

Novi Commentarii academiae scientiarum Petropolitanae, Volume 5, pp. 164-204.

Opera Omnia Citation

Series 1, Volume 14, pp.542-584.

Record Created


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Mathematics Commons