The Euclid's Elements of Geometry

Geometry is the study of figures. Euclid's Elements provides themost fundamental way of learning geometry geometrically.

Book V: Theory of abstract proportions

Definitions

1. A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.
2. The greater is a multiple of the less when it is measured by the less.
3. A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
4. Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.
5. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
6. Let magnitudes which have the same ratio be called proportional.
7. When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.
8. A proportion in three terms is the least possible.
9. When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.
10. When four magnitudes are continuously proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion.
11. Antecedents are said to correspond to antecedents, and consequents to consequents.
12. Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.
13. Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.
14. A ratio taken jointly means taking the antecedent together with the consequent as one in relation to the consequent by itself.
15. A ratio taken separately means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.
16. Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.
17. A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, the first is to the last among the first magnitudes as the first is to the last among the second magnitudes. Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.
18. A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, antecedent is to consequent among the first magnitudes as antecedent is to consequent among the second magnitudes, while, the consequent is to a third among the first magnitudes as a third is to the antecedent among the second magnitudes.

Propositions

1.  If any number of magnitudes are each the same multiple of the same number of other magnitudes, then the sum is that multiple of the sum.
2. If a first magnitude is the same multiple of a second that a third is of a fourth, and a fifth also is the same multiple of the second that a sixth is of the fourth, then the sum of the first and fifth also is the same multiple of the second that the sum of the third and sixth is of the fourth.
3. If a first magnitude is the same multiple of a second that a third is of a fourth, and if equimultiples are taken of the first and third, then the magnitudes taken also are equimultiples respectively, the one of the second and the other of the fourth.
4. If a first magnitude has to a second the same ratio as a third to a fourth, then any equimultiples whatever of the first and third also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.
5. If a magnitude is the same multiple of a magnitude that a subtracted part is of a subtracted part, then the remainder also is the same multiple of the remainder that the whole is of the whole.
6. If two magnitudes are equimultiples of two magnitudes, and any magnitudes subtracted from them are equimultiples of the same, then the remainders either equal the same or are equimultiples of them.
7. Equal magnitudes have to the same the same ratio; and the same has to equal magnitudes the same ratio.
   Corollary: If any magnitudes are proportional, then they are also proportional inversely.
8. Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.
9. Magnitudes which have the same ratio to the same equal one another; and magnitudes to which the same has the same ratio are equal.
10. Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.
11. Ratios which are the same with the same ratio are also the same with one another.
12. If any number of magnitudes are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents.
13. If a first magnitude has to a second the same ratio as a third to a fourth, and the third has to the fourth a greater ratio than a fifth has to a sixth, then the first also has to the second a greater ratio than the fifth to the sixth.
14. If a first magnitude has to a second the same ratio as a third has to a fourth, and the first is greater than the third, then the second is also greater than the fourth; if equal, equal; and if less, less.
15. Parts have the same ratio as their equimultiples.
16. If four magnitudes are proportional, then they are also proportional alternately.
17. If magnitudes are proportional taken jointly, then they are also proportional taken separately.
18. If magnitudes are proportional taken separately, then they are also proportional taken jointly.
19. If a whole is to a whole as a part subtracted is to a part subtracted, then the remainder is also to the remainder as the whole is to the whole.
    Corollary. If magnitudes are proportional taken jointly, then they are alsoproportional in conversion.
20. If there are three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first is greater than the third, then the fourth is also greater than the sixth; if equal, equal, and; if less, less.
21. If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them is perturbed, then, if ex aequali the first magnitude is greater than the third, then the fourth is also greater than the sixth; if equal, equal; and if less, less.
22. If there are any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali.
23. If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then they are also in the same ratio ex aequali.
24. If a first magnitude has to a second the same ratio as a third has to a fourth, and also a fifth has to the second the same ratio as a sixth to the fourth, then the sum of the first and fifth has to the second the same ratio as the sum of the third and sixth has to the fourth.
25. If four magnitudes are proportional, then the sum of the greatest and the least is greater than the sum of the remaining two.