The Forgotten Method of Prosthaphaeresis


Image Credit: Angela Yuriko Smith

Around the fifteenth century, technological advances such as the magnetic compass, lateen sail, and stern-post rudder enabled Europe to enter what is now known as the Age of Discovery. Oceanic routes, at first restricted to coastal areas, spread into open ocean with the adoption and subsequent improvements to the astrolabe and quadrant.

Navigation became increasingly computational without visual clues from the coastline. In the sixteenth century, navigators relied on ephemerides (journals of positions of astronomical bodies at known times) to determine position and heading. It’s no stretch to say that navigators became astronomers, who used spherical trigonometry to create models of the earth and the sky above.

The fundamental identity in spherical trigonometry is the spherical law of cosines, which – like its Euclidean cousin – relates the angles and side lengths inside a (spherical) triangle. With reference to the figure below, this formula may be written

\displaystyle \cos a = \cos b \cos c + \sin b \sin c \cos \alpha.

If all but one quantity is known in the formula above, the last can be obtained through multiplication, division, and application of a trigonometric table. In this process, hand multiplication became an algorithmic bottleneck, since it was infeasible to create large multiplication tables. (Unlike trigonometric functions, multiplication is a binary operation.)


Today, we know that the algorithmic problem of multiplication can be solved using the logarithm (and tables of its values), since the identity

\displaystyle \log (a \cdot b) = \log a + \log b

relates a product (inside a logarithm) to a sum (of logarithms). To multiply a \cdot b, one simply finds \log a and \log b, adds them, and looks in the table to find an x for which \log x = \log a + \log b. The process of constructing logarithmic tables was still time-consuming, but it was a process that only had to be done once (and could be done on dry land). Once made, application of these tables was easy. In the (perhaps apocryphal) words of Pierre-Simon Laplace, logarithms were

…an admirable artifice which, by reducing to a few days the labor of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations.

While logarithms are, in hindsight, the simplest way to perform multiplication with a single variable look-up table, they were not the first. That honor goes to the Babylonian Quarter Square Algorithm, which applies the identity

\displaystyle (x+y)^2 - (x-y)^2 = x^2 + 2xy+y^2 - x^2 +2xy-y^2 = 4xy

to reduce the multiplication of xy to two squaring operations. The algorithm could be made efficient through the use of a table of squares, but it appears that this was not implemented until Antoine Voisin’s table of 1000 integral squares in 1817.

There was another technique, invented in the late sixteenth century, that saw widespread use in navigation for about twenty-five years before the introduction of Napier’s logarithm in 1614 and the first logarithmic table in 1617. This technique was called prosthaphaeresis, after the Greek prosthesis (addition) and aphaeresis (subtraction), and used trigonometric angle addition formulas to reduce multiplication to addition/subtraction with the assistance of trigonometric tables. For example, one has

\displaystyle \sin \alpha \sin \beta = \tfrac{1}{2} \left( \cos (\alpha-\beta) - \cos(\alpha+\beta) \right).

To use prosthaphaeresis to multiply x \cdot y, one scales down x and y to lie in [-1,1], then represents each as a value of sine (through application of a sine table) to find \alpha and \beta. The values are added (prosthesis) and subtracted (aphaeresis), then \cos(\alpha-\beta) and \cos(\alpha+\beta) are computed using another table look-up. Their average gives \sin \alpha \sin \beta, which is rescaled to give x \cdot y.

The angle addition laws, as relayed within European mathematicians, are believed to originate with Jost Bürgi (1552-1632). Globally, they predate Bürgi by several centuries, having been derived by the Persian mathematician and astronomer Abū al-Wafā Būzhjānī sometime in the late tenth century.

As a historical anecdote, the method of prosthaphaeresis highlights the role that technological advancement (in particular, the newfound ability to traverse open ocean) plays in the development of mathematics. Prosthaphaeresis also highlights the way in which trigonometric functions paved the way for other transcendental functions like the logarithm (1614), exponential function (1748, although the idea of exponentiation predates this), and the solutions to differential equations.

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