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 1^{st} 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.