Prosthaphaeresis

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 prosthesi and afairo, meaning addition and subtraction, two steps in the process.

The algorithm

Using the second formula above, the technique works as follows:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Scale down: By shifting the decimal point to the left or right, 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 in the answer to the right (or left) as many places as you shifted the decimal place to the left (or right) in the first step, for each input.

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 is about ½, 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.

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.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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 the steps: scale down, take logarithms, add, take inverse logarithm, scale up. It's no surprise that the originators of logarithms had used prosthaphaeresis.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~