Let a : b :: c : d; then a : a − b :: c : c − d.
| Dem.—Since | a : b | :: c : d, | |||||||||
| = ; | |||||||||||
| therefore | = [Dem. of xvii.], | ||||||||||
| therefore | ÷ | = ÷, | |||||||||
| or | = , | ||||||||||
| therefore | a : a − b | :: c : c − d. |
PROP. XX.—Theorem.
If there be two sets of three magnitudes, which taken two by two in direct order have equal ratios, then if the first of either set be greater than the third, the first of the other set is greater than the third; if equal, equal; and if less, less.
Let a, b, c; a′, b′, c′ be the two sets of magnitudes, and let the ratio a : b = a′ : b′, and b : c = b′ : c′; then, if a be greater than, equal to, or less than c, a′ will be greater than, equal to, or less than c′.
| Dem.—Since | a | : b :: a′ : b′, | |||||||||
| we have | = , | ||||||||||
| In like manner, | = , | ||||||||||
| Hence | × | = ×, | |||||||||
| or | = . |
Therefore if a be greater than c, a′ is greater than c′; if equal, equal; and if less, less.
PROP. XXI.—Theorem.
If there be two sets of three magnitudes, which taken two by two in transverse order have equal ratios; then, if the first of either set be greater than the third, the first of the other set is greater than the third; if equal, equal; and if less, less.
Let a, b, c; a′, b′, c′ be the two sets of magnitudes, and let the ratio a : b = b′ : c′, and b : c = a′ : b′. Then, if a be greater than, equal to, or less than c, a′ will be greater than, equal to, or less than c′.