The First Six Books of the Elements of Euclid

by Euclid; John Casey

Exercises on Book V.

Def. I.—A ratio whose antecedent is greater than its consequent is called a ratio of greater inequality; and a ratio whose antecedent is less than its consequent, a ratio of lesser inequality.

Def. II.—A right line is said to be cut harmonically when it is divided internally and externally in any ratios that are equal in magnitude.

1. A ratio of greater inequality is increased by diminishing its terms by the same quantity, and diminished by increasing its terms by the same quantity.

2. A ratio of lesser inequality is diminished by diminishing its terms by the same quantity, and increased by increasing its terms by the same quantity.

3. If four magnitudes be proportionals, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference (componendo et dividendo).

4. If two sets of four magnitudes be proportionals, and if we multiply corresponding terms together, the products are proportionals.

5. If two sets of four magnitudes be proportionals, and if we divide corresponding terms, the quotients are proportionals.

6. If four magnitudes be proportionals, their squares, cubes, &c., are proportionals.

7. It two proportions have three terms of one respectively equal to three corresponding terms of the other, the remaining term of the first is equal to the remaining term of the second.