Euclid's Elements Book V : Theory of abstract proportions.

### Definitions

- Definition 1
- A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.
- Definition 2
- The greater is a multiple of the less when it is measured by the less.
- Definition 3
- A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
- Definition 4
- Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.
- Definition 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.
- Definition 6
- Let magnitudes which have the same ratio be called proportional.
- Definition 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.
- Definition 8
- A proportion in three terms is the least possible.
- Definition 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.
- Definition 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.
- Definition 11
- Antecedents are said to correspond to antecedents, and consequents to consequents.
- Definition 12
- Alternate ratio means taking the antecedent in relation to the antecedent and the consequent
in relation to the consequent.
- Definition 13
- Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.
- Definition 14
- A ratio taken jointly means taking the antecedent together with the consequent as one
in relation to the consequent by itself.
- Definition 15
- A ratio taken separately means taking the excess by which the antecedent exceeds
the consequent in relation to the consequent by itself.
- Definition 16
- Conversion of a ratio means taking the antecedent in relation to the excess by
which the antecedent exceeds the consequent.
- Definition 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.
- Definition 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

- Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 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.

- Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 11
- Ratios which are the same with the same ratio are also the same with one another.
- Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 15
- Parts have the same ratio as their equimultiples.
- Proposition 16
- If four magnitudes are proportional, then they are also proportional alternately.
- Proposition 17
- If magnitudes are proportional taken jointly, then they are also proportional taken separately.
- Proposition 18
- If magnitudes are proportional taken separately, then they are also proportional taken jointly.
- Proposition 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 also proportional in conversion.

- Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 25
- If four magnitudes are proportional, then the sum of the greatest and the least is greater than the sum of
the remaining two.

Euclid's Elements: Book IV <--- Book V --->Euclid's Elements: Book VI