Prosthaphaeresis (from the Greek προσθαφαίρεσις) was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the invention of the logarithm in 1614, it was the only known generally applicable way of approximating products quickly. Its name comes from the Greek πρόσθεν, prosthen, 'before', and ἀφαίρεσις, aphaeresis, 'taking away, subtraction'. In ancient times the term was used to mean a reduction to bring the apparent place of a moving point or planet to the mean place (see Equation of the center). Nicholas Copernicus mentions "prosthaphaeresis" several times in his 1543 work De Revolutionibus Orbium Coelestium 'great parallax', caused by the displacement of the observer due to the Earth's annual motion.
History and motivation
In 16th-century Europe, celestial navigation of ships on long voyages relied heavily on ephemerides to determine their position and course. These voluminous charts prepared by astronomers detailed the position of stars and planets at various points in time. The models used to compute these were based on spherical trigonometry, which relates the angles and arc lengths of spherical triangles (see diagram, right) using formulas such as
cos a = cos b cos c + sin b sin c cos α {\displaystyle \cos a=\cos b\cos c+\sin b\sin c\cos \alpha }
and
sin b sin α = sin a sin β , {\displaystyle \sin b\sin \alpha =\sin a\sin \beta ,}
where a, b and c are the angles subtended at the centre of the sphere by the corresponding arcs. When one quantity in such a formula is unknown but the others are known, the unknown quantity can be computed using a series of multiplications, divisions, and trigonometric table lookups. Astronomers had to make thousands of such calculations, and because the best method of multiplication available was long multiplication, most of this time was spent taxingally multiplying out products. Mathematicians, particularly those who were also astronomers, were looking for an easier way, and trigonometry was one of the most advanced and familiar fields to these people. Prosthaphaeresis appeared in the 1580s, but its originator is not known for certain; its contributors included the mathematicians Ibn Yunis, Johannes Werner, Paul Wittich, Joost Bürgi, Christopher Clavius, and François Viète. Wittich, Ibn Yunis, and Clavius were all astronomers and have all been credited by various sources with discovering the method. Its most well-known proponent was Tycho Brahe, who used it extensively for astronomical calculations such as those described above. It was also used by John Napier, who is credited with inventing the logarithms that would supplant it.
The identities
The trigonometric identities exploited by prosthaphaeresis relate products of trigonometric functions to sums. They include the following:
sin a sin b = cos ( a − b ) − cos ( a + b ) 2 cos a cos b = cos ( a − b ) + cos ( a + b ) 2 sin a cos b = sin ( a + b ) + sin ( a − b ) 2 cos a sin b = sin ( a + b ) − sin ( a − b ) 2 {\displaystyle {\begin{aligned}\sin a\sin b&={\frac {\cos(a-b)-\cos(a+b)}{2}}\\[6pt]\cos a\cos b&={\frac {\cos(a-b)+\cos(a+b)}{2}}\\[6pt]\sin a\cos b&={\frac {\sin(a+b)+\sin(a-b)}{2}}\\[6pt]\cos a\sin b&={\frac {\sin(a+b)-\sin(a-b)}{2}}\end{aligned}}}
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