Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
used in the late 16th century and early 17th century for approximate
multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
and
division using formulas from
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
. For the 25 years preceding the invention of the
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
in 1614, it was the only known generally applicable way of approximating products quickly. Its name comes from the
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
''prosthesis'' (πρόσθεσις) and ''aphaeresis'' (ἀφαίρεσις), meaning addition and subtraction, two steps in the process.
Prosthaphaeresis
by Brian Borchers
History and motivation
In 16th-century Europe, celestial navigation
Celestial navigation, also known as astronavigation, is the practice of position fixing using stars and other celestial bodies that enables a navigator to accurately determine their actual current physical position in space (or on the surface o ...
of ships on long voyages relied heavily on ephemerides
In astronomy and celestial navigation, an ephemeris (pl. ephemerides; ) is a book with tables that gives the trajectory of naturally occurring astronomical objects as well as artificial satellites in the sky, i.e., the position (and possibly ...
to determine their position and course. These voluminous charts prepared by astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ...
s detailed the position of stars and planets at various points in time. The models used to compute these were based on spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...
, which relates the angles and arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
s of spherical triangles (see diagram, right) using formulas such as
:
and
:
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 taxingly 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
Abu al-Hasan 'Ali ibn 'Abd al-Rahman ibn Ahmad ibn Yunus al-Sadafi al-Misri (Arabic: ابن يونس; c. 950 – 1009) was an important Egyptian astronomer and mathematician, whose works are noted for being ahead of their time, having been base ...
, Johannes Werner
Johann(es) Werner ( la, Ioannes Vernerus; February 14, 1468 – May 1522) was a German mathematician. He was born in Nuremberg, Germany, where he became a parish priest. His primary work was in astronomy, mathematics, and geography, although he ...
, Paul Wittich, Joost Bürgi, Christopher Clavius
Christopher Clavius, SJ (25 March 1538 – 6 February 1612) was a Jesuit German mathematician, head of mathematicians at the Collegio Romano, and astronomer who was a member of the Vatican commission that accepted the proposed calendar inve ...
, and François Viète
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
. 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
Tycho Brahe ( ; born Tyge Ottesen Brahe; generally called Tycho (14 December 154624 October 1601) was a Danish astronomer, known for his comprehensive astronomical observations, generally considered to be the most accurate of his time. He was ...
, who used it extensively for astronomical calculations such as those described above. It was also used by John Napier
John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston. His Latinized name was Ioan ...
, who is credited with inventing the logarithms that would supplant it.
Nicholas Copernicus mentions "prosthaphaeresis" several times in his 1543 work , meaning the "great parallax" caused by the displacement of the observer due to the Earth's annual motion.
The identities
The trigonometric identities exploited by prosthaphaeresis relate products of trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s to sums. They include the following:
:
The first two of these are believed to have been derived by Jost Bürgi
Jost Bürgi (also ''Joost, Jobst''; Latinized surname ''Burgius'' or ''Byrgius''; 28 February 1552 – 31 January 1632), active primarily at the courts in Kassel and Prague, was a Swiss clockmaker, a maker of astronomical instruments and a ma ...
, who related them to ycho?Brahe; the others follow easily from these two. If both sides are multiplied by 2, these formulas are also called the Werner formulas.
The algorithm
Using the second formula above, the technique for multiplication of two numbers works as follows:
# Scale down: By shifting the decimal point to the left or right, scale both numbers to values between and , to be referred to as and .
# Inverse cosine: Using an inverse cosine table, find two angles and whose cosines are our two values.
# Sum and difference: Find the sum and difference of the two angles.
# Average the cosines: Find the cosines of the sum and difference angles using a cosine table and average them, giving (according to the second formula above) the product .
# Scale up: Shift the decimal place in the answer the combined number of places we have shifted the decimal in the first step for each input, but in the opposite direction.
For example, say we want to multiply and . Following the steps:
# Scale down: Shift the decimal point three places to the left in each. We get and .
# Inverse cosine: is about 0.105, and is about .
# Sum and difference: , and .
# Average the cosines: is about .
# Scale up: For each of and we shifted the decimal point three places to the left, so in the answer we shift six places to the right. The result is . This is very close to the actual product, (a percent error In any quantitative science, the terms relative change and relative difference are used to compare two quantities while taking into account the "sizes" of the things being compared, i.e. dividing by a ''standard'' or ''reference'' or ''starting'' va ...
of ≈0.8%).
If we want the product of the cosines of the two initial values, which is useful in some of the astronomical calculations mentioned above, this is surprisingly even easier: only steps 3 and 4 above are necessary.
To divide, we exploit the definition of the secant as the reciprocal of the cosine. To divide by , we scale the numbers to and . The cosine of is . Then use a table of secants to find out that is the secant of . This means that is the cosine of , and so we can multiply by using the above procedure. Average the cosine of the sum of the angles, , with the cosine of their difference, ,
:
Scaling up to locate the decimal point gives the approximate answer, .
Algorithms using the other formulas are similar, but each using different tables (sine, inverse sine, cosine, and inverse cosine) in different places. The first two are the easiest because they each only require two tables. Using the second formula, however, has the unique advantage that if only a cosine table is available, it can be used to estimate inverse cosines by searching for the angle with the nearest cosine value.
Notice how similar the above algorithm is to the process for multiplying using logarithms, which follows these steps: scale down, take logarithms, add, take inverse logarithm, scale up. It is no surprise that the originators of logarithms had used prosthaphaeresis. Indeed the two are closely related mathematically. In modern terms, prosthaphaeresis can be viewed as relying on the logarithm of complex numbers, in particular on Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
:
Decreasing the error
If all the operations are performed with high precision, the product can be as accurate as desired. Although sums, differences, and averages are easy to compute with high precision, even by hand, trigonometric functions and especially inverse trigonometric functions are not. For this reason, the accuracy of the method depends to a large extent on the accuracy and detail of the trigonometric tables used.
For example, a sine table with an entry for each degree can be off by as much as 0.0087 if we just round an angle off to the nearest degree; each time we double the size of the table (for example, by giving entries for every half-degree instead of every degree) we halve this error. Tables were painstakingly constructed for prosthaphaeresis with values for every second, or 3600th of a degree.
Inverse sine and cosine functions are particularly troublesome, because they become steep near −1 and 1. One solution is to include more table values in this area. Another is to scale the inputs to numbers between −0.9 and 0.9. For example, 950 would become 0.095 instead of 0.950.
Another effective approach to enhancing the accuracy is linear interpolation
In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
Linear interpolation between two known points
If the two known poi ...
, which chooses a value between two adjacent table values. For example, if we know that the sine of 45° is about 0.707 and the sine of 46° is about 0.719, we can estimate the sine of 45.7° as 0.707 × (1 − 0.7) + 0.719 × 0.7 = 0.7154. The actual sine is 0.7157. A table of cosines with only 180 entries combined with linear interpolation is as accurate as a table with about entries without it. Even a quick estimate of the interpolated value is often much closer than the nearest table value. See lookup table
In computer science, a lookup table (LUT) is an array that replaces runtime computation with a simpler array indexing operation. The process is termed as "direct addressing" and LUTs differ from hash tables in a way that, to retrieve a value v w ...
for more details.
Reverse identities
The product formulas can also be manipulated to obtain formulas that express addition in terms of multiplication. Although less useful for computing products, these are still useful for deriving trigonometric results:
:
See also
* Slide rule
The slide rule is a mechanical analog computer which is used primarily for multiplication and division, and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which ...
References
{{reflist
External links
Prosthaphaeresis formulas
* Daniel E. Otero
Introduction: the need for speed in calculation.
* Adam Mosley
University of Cambridge.
* IEEE Computer Society
* ttps://web.archive.org/web/20180520205524/http://www.pballew.net/arithm18.html#Prostha Math Words: Prosthaphaeresis* Beatrice Lumpkin.
African and African-American Contributions to Mathematics
'. Discusses Ibn Yunis's contribution to prosthaphaeresis.
Prosthaphaeresis
and beat phenomenon in the theory of vibrations, by Nicholas J. Rose
Trigonometry
Arithmetic