then
a + b : b = c + d : d.
Prop. 19. If a, b, c, d are quantities of the same kind, and if
a : b = c : d,
then
a − c : b − d = a : b.
§ 54. Prop. 20. If there be three magnitudes, and another three, which have the same ratio, taken two and two, then if the first be greater than the third, the fourth shall be greater than the sixth: and if equal, equal; and if less, less.
If we understand by
a : b : c : d : e : ... = a′ : b′ : c′ : d′ : e′ : ...
that the ratio of any two consecutive magnitudes on the first side equals that of the corresponding magnitudes on the second side, we may write this theorem in symbols, thus:—