# 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 ** a:b = c:d** and

**. The latter symbols may be read as**

*a:b :: c:d**is to*

**a****as**

*b**is to*

**c****. As in arithmetical proportions, the terms**

*d***and**

*a***are called antecedents and the terms**

*c***and**

*b**are called consequents. Likewise, the terms*

**d****and**

*a***are called the extremes and the terms**

*d***and**

*b**are called the means.*

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

- Multiplying each geometrical ratio by a constant maintains a geometrical proportion (for any constant
, ).*k* - The product of the means is equal to the product of the extremes (
x*a*=**d**x*b*).*c* - 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 .
- 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*a:b :: c:x*then ).*a:b = c:x*

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 ** b** 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 (

*,*

**a****,**

*b**,*

**c****,**

*d**,*

**e***,*

**f****are in geometrical progression).**

*g*