Let a : b :: c : d : then a − b : b :: c − d : d;
| Dem.—Since | a : b | :: c : d, | |||||||||
| = ; | |||||||||||
| therefore | − 1 | = − 1, | |||||||||
| or | = ; | ||||||||||
| therefore | a − b : b | :: c − d : d. |
PROP. XVIII.—Theorem.
If four magnitudes be proportionals, the sum of the first and second : the second :: the sum of the third and fourth : the fourth (componendo).
Let a : b :: c : d; then a + b : b :: c + d : d.
| Dem.—Since | a : b :: | c : d, | |||||||||
| = | ; | ||||||||||
| therefore | + 1 = | + 1, | |||||||||
| or | = | ; | |||||||||
| therefore | a + b : b :: | c + d : d. |
PROP. XIX.—Theorem.
If a whole magnitude be to another whole at a magnitude taken from the first it to a magnitude taken from the second, the first remainder : the second remainder :: the first whole : the second whole.
Let a : b :: c : d, c and d being less than a and b;
then a − c : b − d :: a : b.
| Dem.—Since | a : b | :: c : d, | |||||||||
| then | a : c | :: b : d [alternando], | |||||||||
| and | c : a | :: d : b [invertendo]; | |||||||||
| therefore | = , | ||||||||||
| and | 1 − | = 1 −, | |||||||||
| or | = . | ||||||||||
| Hence | a − c : | b − d :: a : b. |
Prop. E.—Theorem (Simson).
If four magnitudes be proportional, the first : its excess above the second :: the third : its excess above the fourth (convertendo).