Problem 6

To find the diameter of a circle equal in area to an ellipsis (or oval) whose transverse and conjugate diameters are given.

Rule

Multiply the two diameters of the ellipsis together and the square root of that product will be the diameter of the circle equal to the ellipsis.

Comment: No results in geometry are covered in the notebook including the area of a circle which is where is the radius, or the relation between the area of a circle and the area of an ellipse. The area of the ellipse is , where is the length of the semi-major axis and is the length of the semi-minor axis. The circle is the special case of the ellipse in which . Given the statement of the rule no knowledge of any of this material is necessary.

Example

The transverse diameter of a circle [sic] is 48 yds and the conjugate is 36. What is the diameter of an equal circle?

Comment: Once again Macdonald is not paying attention to what he is doing. The transverse diameter should be of an ellipse.

Solution

Multiply the two diameters together to get . Then , which can be found longhand by the algorithm given earlier in Macdonald’s notes, which was taken from Nicholas Pike’s A New and Complete System of Arithmetick. For some unexplained reason Macdonald has tacked on to the end of his answer.