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).