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.