Prosthaphaeresis

Prosthaphaeresis was an algorithm used in the late 16th century and early 17th century for approximating products 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 prosthesi and afairo, meaning addition and subtraction, two steps in the process.

History and motivation

In the sixteenth century, celestial navigation of European ships on long voyages at sea relied heavily on charts prepared by astronomers called ephemeral charts to determine their position and course. These voluminous documents detailed the position of various celestial objects in the sky 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 such as the one in the diagram to the right using formulas such as the following:

Related Topics:
Celestial navigation - Astronomer - Ephemeral chart - Spherical trigonometry

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• cos a = cos b cos c + sin b sin c cos α
• sin b sin α = sin a sin β

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 only method of multiplication available was the grade-school method, most of this time was spent taxingly multiplying out products.

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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 Paul Wittich, Ibn Yunis, Joost Bürgi, Johannes Werner, Christopher Clavius, and François Viète. Wittich, 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.

Related Topics:
Paul Wittich - Ibn Yunis - Joost Bürgi - Johannes Werner - Christopher Clavius - François Viète - Tycho Brahe - John Napier

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Just as ordinary slide rules are based on logarithms, recently researchers have conceived a little-known slide rule based on prosthaphaeresis. http://www.findarticles.com/p/articles/mi_qa3950/is_200401/ai_n9372466

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