How Did Astronomers Calculate Multiplication?
We take multiplication for granted. Need to calculate 420 × 69.9? Just type it into a calculator or search bar, and boom — there’s your answer.
We know multiplication is just repeated addition. If you add 5 to itself 8 times, you’re really doing:
5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 5 × 8 = 40
We can perform repeated addition in our mechanical or scientific calculator pretty easily. But in the 16th century, astronomers trying to calculate planetary positions or navigate using the stars had no access to this kind of computational power. They relied on long multiplication, which is tedious and error-prone, especially with large numbers.
Long Multiplication Example
Until a clever algorithm was invented that turned multiplication into addition and subtraction using trigonometry.
Prosthaphaeresis — The Algorithm
Prosthaphaeresis is a mathematical trick that appeared in the 1580s. It converts multiplication into addition and subtraction using trigonometric identities.
The name comes from Greek:
- Prosthesis means addition
- Aphaeresis means subtraction
Together, they describe the steps of the method: add and subtract angles to simplify a multiplication problem.
This method was especially useful in the pre-logarithm era. It let mathematicians and astronomers perform faster computations with the help of trigonometric tables.
Product-to-Sum Formula
At the heart of Prosthaphaeresis is the product-to-sum formula.
For example, the formula for the product of two cosines:
This formula lets us treat a multiplication operation as an addition and subtraction of angles — making the whole process faster using precomputed cosine values.
Calculating 420 × 69.9 using Prosthaphaeresis
So, wait — people in the 1580s couldn’t multiply large numbers easily, but they could calculate cosine values?
Not quite.
People built lookup tables with precomputed values for sine and cosine. These trigonometric tables were essential tools that allowed them to use techniques like Prosthaphaeresis effectively.
The result may have had a significant margin of error compared to exact multiplication, but it was still much faster and more practical than performing long multiplication by hand — especially when dealing with very large numbers.
This algorithm was widely used for about 25 years, until the introduction and formalization of logarithms, which made multiplication and division even easier and more accurate.
Final Thoughts
Prosthaphaeresis may be a forgotten relic of math history, but it’s a brilliant reminder of how creative people have been in solving everyday problems — even something as “simple” as multiplication.
Before calculators, cleverness was the calculator.
❤️ SUPRIYA