Prosthaphaeresis (Calculating a Product from a Sum) and Logarithm Tables
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As a method for quickly calculating products, it is said that before the 16th century, a technique called “Prosthaphaeresis”—which involves converting the problem into a sum using the formula cos(a)cos(b) = (cos(a+b) + cos(a-b))/2—was used.Later, the process became even easier by using the property log(ab) = log(a) + log(b) from logarithm tables. Let’s take a look at that.
| |1|: A number normalized to 1 (Part 1) | |1|: Numbers Normalized to 1, Part 2 |
| a in cos(a) | °b in cos(b) ° |
| (cos(a+b)+cos(a-b))/2 | ° |
| Product after reversing normalization | |
| The logarithm of Number 1 | |
| The Logarithm of Number 2 | |
| Sum of Logarithms | |
| Product Calculated from Logarithms | |
| The actual product value |
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