If two magnitudes of the same kind (a, b) have to a third magnitude (c) ratios equal to those which two other magnitudes (a′, b′) have to a third (c′), then the sum (a + b) of the first two has the same ratio to their third (c) which the sum (a′ + b′) of the other two magnitudes has to their third (c′).
| Dem.—Since | a : c | :: a′ : c′, | |||||||||
| we have | = . | ||||||||||
| In like manner, | = ; | ||||||||||
| therefore, adding, | = . | ||||||||||
| Hence | a + b : c | :: a′ + b′ : c′. |
PROP. XXV.—Theorem.
If four magnitudes of the same kind be proportionals, the sum of the greatest and least is greater than the sum of the other two.
Let a : b :: c : d; then, if a be the greatest, d will be the least [xiv. and a]. It is required to prove that a + d is greater than b + c.
| Dem.—Since | a : b :: c : d, | ||||||||||
| a : c :: b : d [alternando]; | |||||||||||
| therefore | a : a − c :: b : b − d [E].; | ||||||||||
| but | a is greater than b (hyp.), | ||||||||||
| therefore | a − c is greater than b − d [xiv.]. | ||||||||||
| Hence | a + d is greater than b + c. |
Questions for Examination on Book V.
1. What is the subject-matter of this book?
2. When is one magnitude said to be a multiple of another?
3. What is a submultiple or measure?