|
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 prothesi and afairo, meaning addition and subtraction, two steps in the process. Flowcharts are often used to represent algorithms. ...
(15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ...
(16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ...
In its simplest form, multiplication is a quick way of adding identical numbers. ...
Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine and cosine. ...
In mathematics, if two variables of bn = x are known, the third can be found. ...
Events April 5 - In Virginia, Native American Pocahontas marries English colonist John Rolfe. ...
The Greek language (Greek Ελληνικά, IPA – Hellenic) is an Indo-European language with a documented history of some 3,000 years. ...
Although the originator of the technique is not known for certain, its contributors included Paul Wittich, Ibn Yunis, Joost Bürgi, Johannes Werner, Christopher Clavius, and François Viète. In fact, Wittich, Yunis, and Clavius have all been credited with its discovery by various sources. Its most well-known proponent was Tycho Brahe, who used it extensively for astronomical calculations (in fact, Yunis, Wittich, and Clavius were also astronomers). It was also used by John Napier, who is credited with inventing the logarithms that would supplant it. Christopher Clavius, born Christoph Clau, (1538 – February 12, 1612) was a German mathematician and astronomer who was the main architect of the modern Gregorian calendar. ...
François Viète. ...
Tycho Brahe (December 14, 1546 Knudstrup, Denmark – October 24, 1601 Prague, Bohemia (now Czech Republic)) was a Danish nobleman, well known as an astronomer/astrologer (the two were not yet distinct) and alchemist. ...
John Napier (1550–April 4, 1617) was a Scottish mathematician and astrologer. ...
Just as ordinary slide rules are based on logarithms, researchers have conceived a little-known slide rule based on prosthaphaeresis. [1] (http://www.findarticles.com/p/articles/mi_qa3950/is_200401/ai_n9372466) The slide rule is a portable, mechanical, analog computer usually consisting of three interlocking calibrated strips and a sliding cursor used to record intermediate results. ...
The identities
The trigonometric identities exploited by prosthaphaeresis relate products of trigonometric functions to sums. They include the following: In mathematics, trigonometric identities (or trig identities for short) are equations involving trigonometric functions that are true for all values of the occurring variables. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
- sin a sin b = ½[cos(a - b) - cos(a + b)]
- cos a cos b = ½[cos(a - b) + cos(a + b)]
- sin a cos b = ½[sin(a - b) + sin(a + b)]
- cos a sin b = ½[sin(a - b) - sin(a + b)]
The first two of these are believed to have been derived by Bürgi, who related them to 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 works as follows: - Scale down: By shifting the decimal point to the left, scale both numbers to a value between -1 and 1.
- Inverse cosine: Using an inverse cosine table, find two angles 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.
- Scale up: Shift the decimal place back to the right as many places as you shifted either decimal place to the left in the first step.
For example, say we want to multiply 105 and 720. Following the steps: - Scale down: Shift the decimal 3 to the left in each. We get: 0.105, 0.720
- Inverse cosine: cos(84°) is about 0.105, cos(44°) is about 0.720
- Sum and difference: 84 + 44 = 128, 84 - 44 = 40
- Average the cosines: ½[cos(128°) + cos(40°)] is about ½[-0.616 + 0.766], or 0.075
- Scale up: We shifted 105 and 720 each 3 to the left, so shift our answer 6 to the right. The result is 75,000. This is very close to the actual product, 75,600.
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.
Notice how similar the above algorithm is to the process for multiplying using logarithms, which follows the steps: scale down, take logarithms, add, take inverse logarithm, scale up. It's no surprise that the originators of logarithms had used prosthaphaeresis.
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: - sin a + sin b = 2sin[½(a + b)]cos[½(a - b)]
- sin a - sin b = 2cos[½(a + b)]sin[½(a - b)]
- cos a + sin b = 2sin[½(a + b)]cos[½(a - b)]
- cos a - cos b = -2sin[½(a + b)]sin[½(a - b)]
Sources |
Feeds |
Contact