eli5: How were square roots calculated in the first place, before calculators?

Question

eli5: How were square roots calculated in the first place, before calculators?

In: Mathematics

Answer 1

I was at school before calculators were widely available, and we used [log tables](https://en.wikipedia.org/wiki/Mathematical_table) – books of logarithm values to do large multiplications and divisions, and to calculate roots.

To find a square root of a number: look up its logarithm in the table, divide it by 2, then look up the related exponential value.

Here’s a trivial example to illustrate the method…
To get the square root of 25:

log(10) of 25 is 1.39794 (looked up in the base-10 log table)

1.39794 / 2 = 0.69897

10^0.69897 = 5 (looked up by working backwards in the same table, or by using an “anti-log” table)

There are tricks to how to scale and combine values to make it possible to apply the table to various large and small numbers, not just the values that can be directly looked up in the table.