Problem 3

If it be
required to place a number of men in rank and file or to plant a number of
trees or other objects so that the rank may be to the file or in other words
the length may be to the breadth in a given ratio and so that there may be a given
distance between man and man, tree and tree, or object and object you will find
how many must be placed in rank or length or in file or breadth likewise how
many square feet or yards they stand upon by the following rule.

Comment: The
problem is an exercise in geometrical
proportions and square roots.

Rule

As the
ratio in length is to the ratio in breadth so is the number of men, trees,
&c to a fourth number whose square root is the number in breadth; and as
the ratio in breadth is to the ratio in length so is the number of things to a
fourth number whose square root is the number in length. As unity is to the
distance so is the number in length to a fourth number; as unit [is] to the
distance so it the number in breadth to a fourth number. These two numbers last
found multiplied together will give the square feet or yards upon which they
stand.

Example

I would set
out and orchard of 600 trees so that the length shall be to the breadth as 3 to
2 and the distance of each tree from the other 7 yards. How many trees must
there be in length? And how many square yards do they stand upon?

Solution

Here is
Macdonald’s solution with a bit more detail. Let , , and be the unknown fourth numbers

or so that is the length

or so that is the breadth

What is not
explicitly stated is that since is the number of trees in length then is the number of units of distance in length.
Likewise is the number of units of distance in breadth.

Therefore,

so that and

so that .

Consequently,
the whole plot is square yards. Macdonald writes the answer as
26999 __square feet__, again showing his lack of attention to the problem.