Prosthaphaeresis before logarithms

Today the sum-to-product formulae look like a minor detail of trigonometry, but between the late sixteenth and early seventeenth centuries they were a genuine computational tool, used by astronomers to sidestep an enormous practical difficulty: multiplying many-digit numbers by hand is slow and error-prone, whereas adding is easy.

The idea is contained in the Werner formula $\cos\alpha\cos\beta = \tfrac{1}{2}\bigl[\cos(\alpha-\beta) + \cos(\alpha+\beta)\bigr].$ Suppose we must multiply two numbers $x$ and $y$ lying between $-1$ and $1$. From a table of cosines we look up the angles $\alpha$ and $\beta$ with $\cos\alpha = x$ and $\cos\beta = y$; we compute the two combinations $\alpha-\beta$ and $\alpha+\beta$ (easy operations); we read off $\cos(\alpha-\beta)$ and $\cos(\alpha+\beta)$ from the same table; finally we add them and halve the result. The product $xy$ is thus obtained without ever carrying out a multiplication, using only additions, subtractions and table look-ups.

The method was given the Greek name prosthaphaeresis — from prósthesis (addition) and aphaíresis (subtraction) — precisely because it reduced everything to additions and subtractions. The underlying identities go back to Johann Werner, a mathematician of Nuremberg, at the start of the sixteenth century; their systematic use in astronomical computation, however, is due to Paul Wittich, who around 1580 introduced it at Tycho Brahe’s famous observatory on the island of Hven, where reducing the planetary observations demanded endless multiplications (Boyer; Katz).

Prosthaphaeresis had a short life: in 1614 Napier published logarithms, which turn every product directly into a sum ($\log(xy) = \log x + \log y$) with no restriction to numbers between $-1$ and $1$. From then on the sum-to-product formulae went back to being what they are for us: trigonometric identities useful for factorising a sum of sines or cosines (Dunham).

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Topics: Prostaferesi werner
Concepts: Formule di prostaferesi · Formule di werner
People: Tycho Brahe · Paul Wittich · Johann Werner · Nepero