Prop. 22. If four straight lines be proportionals, the similar rectilineal figures similarly described on them shall also be proportionals; and if the similar rectilineal figures similarly described on four straight lines be proportionals, those straight lines shall be proportionals.
This is essentially the same as the following:—
If
a : b = c : d,
then
a² : b² = c² : d².
§ 66. Now follows a proposition which has been much discussed with regard to Euclid’s exact meaning in saying that a ratio is compounded of two other ratios, viz.:
Prop. 23. Parallelograms which are equiangular to one another, have to one another the ratio which is compounded of the ratios of their sides.
The proof of the proposition makes its meaning clear. In symbols the ratio a : c is compounded of the two ratios a : b and b : c, and if a : b = a′ : b′, b : c = b″ : c″, then a : c is compounded of a′ : b′ and b″ : c″.
If we consider the ratios as numbers, we may say that the one ratio is the product of those of which it is compounded, or in symbols,