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.