Definition:Prosthaphaeresis
Definition
Prosthaphaeresis is the process whereby multiplication and division are performed by:
- $(1): \quad$ converting the numbers into a form where they can be added or subtracted
- $(2): \quad$ performing the addition or subtraction
- $(3): \quad$ converting the result back again, using the inverse process to step $(1)$.
Trigonometrical Technique
The trigonometric technique of prosthaphaeresis is based on the formula:
- $\dfrac {\sin \alpha + \sin \beta} 2 = \map \sin {\dfrac {\alpha + \beta} 2} \, \map \cos {\dfrac {\alpha - \beta} 2}$
Logarithmic Technique
The logarithmic technique of prosthaphaeresis is based on the rule of the Sum of Logarithms:
- $\log_b x y = \log_b x + \log_b y$
Any base $b$ can be used.
The original tecnhique used Napierian logarithms where the base was $0.9999999$
In due course the common (base $10$) logarithm was used, as developed by Henry Briggs, called Briggsian logarithms.
Also known as
Some sources spell the word prosthapheiresis.
$\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the prosthaphaeresis spelling, purely because it appears to be more common.
Also see
Linguistic Note
The word prosthaphaeresis or prosthapheiresis is a neologism coined some time in the $16$th century from the two Greek words:
- prosthesis, meaning addition
- aphaeresis or apheiresis, meaning subtraction.
With the advent of machines to aid the process of arithmetic, this word now has only historical significance.
Ian Stewart, in his Taming the Infinite from $2008$, accurately and somewhat diplomatically describes the word as "ungainly".
Sources
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $5$: Eternal Triangles: Napierian logarithms