Square Root

The square root of a given number is the value that, when multiplied by itself, yields the given number. Square roots are easy to find using the proper button on a calculator, but not so easy when it comes to finding it by hand. In the notebook, John A has written down an algorithm to calculate the square root of a number by hand. The algorithm is copied directly from Nicholas Pike’s A New and Complete System of Arithmetick, Composed for the Use of the Citizens of the United States. Over several numerical examples, John A puts into play the algorithm to extract the square root. Pike has some worked examples and other examples where only the answer is given. Macdonald worked on finding the square roots of a few of the numbers where only the answer was given and a few where presumably his teacher had set the problem and no answer was given.

Here are some of Macdonald’s square root problems for whole numbers and decimal fractions. One cannot be solved using a typical pocket calculator because there are too many digits. Another cannot be found exactly; Macdonald obtained it to three decimal places.

Number	Square Root
10342656
5499025
964.5192360241
0.000729
0.04209

Other square root problems involved fractions. Once again, one cannot be found exactly; Macdonald obtained it to four decimal places.

Number	Square Root
144/15129
2793/8208
42 1/4
17 3/8