Geometrical
Proportion

Like
the material in arithmetical proportions, to put in a general form what
Macdonald has described in his introduction “Ratios and Proportions”, an geometrical
proportion is of the form . Two other ways of
expressing this symbolically are and . The latter symbols
may be read as is to as is to . As in arithmetical
proportions, the terms and are called
antecedents and the terms and are called consequents. Likewise, the terms and are called the extremes and the terms and are called the means.

Macdonald
provides four rules for geometrical proportions. Once again, he illustrates the
rules with numbers; I will do it with symbols.

1. Multiplying each geometrical
ratio by a constant maintains a geometrical proportion (for any constant , ).

2. The product of the
means is equal to the product of the extremes ().

3. In Macdonald’s words,
“If the terms of two geometrical proportions be multiplied together, term by
term, that is, antecedent [by antecedent] and
consequent by consequent will constitute a new proportion.” What appears in the
square brackets [ ] is missing from the text in the notebook which makes it a
little confusing until a numerical example is given. From Macdonald’s example,
the rule is: multiply the appropriate terms in the geometrical proportion by the appropriate terms in to obtain the geometrical proportion
.

4. Given three of the
numbers in a geometrical proportion the fourth can be determined, in
Macdonald’s words, by multiplying the given means together and dividing by the
given extreme (if or then ).

As
in arithmetical progressions, Macdonald
introduces the idea of a geometrical
progression which is treated later in the notebook. For a geometrical
proportion in which the consequent in the first ratio is the same as the
antecedent in the second ratio, the geometrical proportion is obtained. The term is called the geometrical mean. When there is
a series of continued geometrical proportions () the distinct terms
in the geometrical ratios form a geometrical progression (, , , , , , are in geometrical progression).