Arithmetical
Proportion

To
put in a general form what Macdonald has described in his introduction “Ratios and Proportions”, an
arithmetical proportion is of the form . The terms and are called antecedents and the terms and are called consequents. Further, the terms and are called the extremes and the terms and are called the means.

Macdonald
provides six rules for arithmetical proportions. He illustrates the rules with
numbers; I will do it with symbols.

1. Adding a constant to
each arithmetical ratio maintains an arithmetical proportion (for any constant , ).

2. Multiplying each
arithmetical ratio by a constant maintains an arithmetical proportion (for any
constant , )).

3. The sum of the means
is equal to the sum of the extremes ().

4. Transferring the two
means or transferring the two extremes does not destroy the equality, i.e. it
maintains an arithmetical proportion (transferring the means:
; transferring the
extremes ).

5. Several arithmetical
proportions added together is also an arithmetical
proportion.

(If

then

)

6. Given three of the
numbers in an arithmetical proportion the fourth can be determined, in Macdonald’s
words, by adding the means and subtracting and extracting the one extreme from
the result (if then ).

Then
Macdonald introduces the idea of an arithmetical
progression which is treated later in his notebook. When in an arithmetical
proportion, the same number which has been the consequent in the first ratio is
the antecedent in the second ratio, the proportion is called a continued
arithmetical proportion (). When there is a
series of continued arithmetical proportions () the distinct terms
in the arithmetical ratios form an arithmetical progression (, , , , , , are an arithmetical progression).