Application and Use of the Square Root

Macdonald lists seven problems all requiring the calculation of a square root in the solution. For the most part, each problem is described generally, and then a general rule is given to solve it. In each case at least one numerical example is given to illustrate the problem and the rule. The first six problems are numbered. In this group, some of these problems involve the concept of geometrical proportion. Others involve the area of a circle or the area of an ellipse. The rules are given such that the student did not need to know the formula, for example, for the area of a circle. The problems are set to provide the student practice in calculation not in understanding the geometrical concepts and relations. All the problems in this part of the notebook are taken from Nicholas Pike’s A New and Complete System of Arithmetick, pp. 171 – 175.

The final problems are applications of what is known as Pythagoras’ Theorem in geometry. Macdonald has copied in the notebook, “The following questions may be all solved by Euclid 1st Book Prop 47”. This proposition reads, “In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.” This part of the notebook contains the only set of diagrams related to the arithmetical problems that Macdonald is solving. Here is one page from this section. 