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Napier's Analogies

Let a spherical triangle have sides a, b, and c with A, B, and C the corresponding opposite angles. Then

(sin[1/2(A-B)])/(sin[1/2(A+B)])=(tan[1/2(a-b)])/(tan(1/2c))
(1)
(cos[1/2(A-B)])/(cos[1/2(A+B)])=(tan[1/2(a+b)])/(tan(1/2c))
(2)
(sin[1/2(a-b)])/(sin[1/2(a+b)])=(tan[1/2(A-B)])/(cot(1/2C))
(3)
(cos[1/2(a-b)])/(cos[1/2(a+b)])=(tan[1/2(A+B)])/(cot(1/2C))
(4)

(Smart 1960, p. 23).

See also

Spherical Triangle, Spherical Trigonometry

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 131 and 147-150, 1987.Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 109-110, 1998.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 16, 2003.Smart, W. M. Text-Book on Spherical Astronomy, 6th ed. Cambridge, England: Cambridge University Press, 1960.Zwillinger, D. (Ed.). "Spherical Geometry and Trigonometry." §6.4 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 468-471, 1995.

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Napier's Analogies

Cite this as:

Weisstein, Eric W. "Napier's Analogies." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NapiersAnalogies.html

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