Euclid's Elements, Book V

Definitions (18)
Propositions (25)
Guide to Book V
Logical structure of Book V

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

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.

Guide for Book V

Background on ratio and proportion

Book V covers the abstract theory of ratio and proportion. A ratio is an indication of the relative size of two magnitudes. The propositions in the following book, Book VI, are all geometric and depend on ratios, so the theory of ratios needs to be developed first. To get a better understanding of what ratios are in geometry, consider the first proposition VI.1. It states that triangles of the same height are proportional to their bases, that is to say, one triangle is to another as one base is to the other. (A proportion is simply an equality of two ratios.) A simple example is when one base is twice the other, therefore the triangle on that base is also twice the triangle on the other base. This ratio of 2:1 is fairly easy to comprehend. Indeed, any ratio equal to a ratio of two numbers is easy to comprehend. Given a proportion that says a ratio of lines equals a ratio of numbers, for instance, A : B = 8:5, we have two interpretations. One is that there is a shorter line CA = 8C while B = 5C. This interpretation is the definition of proportion that appears in Book VII. A second interpretation is that 5 A = 8 B. Either interpretation will do if one of the ratios is a ratio of numbers, and if A : B equals a ratio of numbers that A and B are commensurable, that is, both are measured by a common measure.

Many straight lines, however, are not commensurable. If A is the side of a square and B its diagonal, then A and B are not commensurable; the ratio A : B is not the ratio of numbers. This fact seems to have been discovered by the Pythagoreans, perhaps Hippasus of Metapontum, some time before 400 B.C.E., a hundred years before Euclid’s Elements.

The difficulty is one of foundations: what is an adequate definition of proportion that includes the incommensurable case? The solution is that in V.Def.5. That definition, and the whole theory of ratio and proportion in Book V, are attributed to Eudoxus of Cnidus (died. ca. 355 B.C.E.)

Summary of the propositions

The first group of propositions, 1, 2, 3, 5, and 6 only mention multitudes of magnitudes, not ratios. They each either state, or depend strongly on, a distributivity or an associativity. In the following identities, m and n refer to numbers (that is, multitudes) while letters near the end of the alphabet refer to magnitudes.

V.1. Multiplication by numbers distributes over addition of magnitudes.

m(x₁ + x₂ + ... + x_n) = m x₁ + m x₂ + ... + m x_n.

V.2. Multiplication by magnitudes distributes over addition of numbers.

(m + n)x = mx + nx.

V.3. An associativity of multiplication.

m(nx) = (mn)x.

V.5. Multiplication by numbers distributes over subtraction of magnitudes.

m(x – y) = mx – my.

V.6. Uses multiplication by magnitudes distributes over subtraction of numbers.

(m – n)x = mx – nx.

The rest of the propositions develop the theory of ratios and proportions starting with basic properties and progressively becoming more advanced.

V.4. If w : x = y : z, then for any numbers m and n, mw : mx = ny : nz.

V.7. Substitution of equals in ratios. If x = y, then x : z = y : z and z : x = z : y.

V.7.Cor. Inverse proportions. If w : x = y : z, then x : w = z : y.

V.8. If x < y, then x : z < y : z but z : x > z : y.

V.9. (A converse to V.7.) If x : z = y : z, then x = y. Also, if z : x = z : y, then x = y.

V.10. (A converse to V.8.) If x : z < y : z, then x < y. But if z : x < z : y, then x > y

V.11. Transitivity of equal ratios. If u : v = w : x and w : x = y : z, then u : v = y : z.

V.12. If x₁:y₁ = x₂:y₂ = ... = x_n : y_n, then each of these ratios also equals the ratio (x₁ + x₂ + ... + x_n) : (y₁ + y₂ + ... + y_n).

V.13. Substitution of equal ratios in inequalities of ratios. If u : v = w : x and w : x > y : z, then u : v > y : z.

V.14. If w : x = y : z and w > y, then x > z.

V.15. x : y = nx : ny.

V.16. Alternate proportions. If w : x = y : z, then w : y = x : z.

V.17. Proportional taken jointly implies proportional taken separately. If (w + x):x = (y + z):z, then w : x = y : z.

V.18. Proportional taken separately implies proportional taken jointly. (A converse to V.17.) If w : x = y : z, then (w + x):x = (y + z):z.

V.19. If (w + x) : (y + z) = w : y, then (w + x) : (y + z) = x : z, too.

V.19.Cor. Proportions in conversion. If (u + v) : (x + y) = v : y, then (u + v) : (x + y) = u : x.

V.20 is just a preliminary proposition to V.22, and V.21 is just a preliminary proposition to V.23.

V.22. Ratios ex aequali. If x₁:x₂ = y₁:y₂, x₂:x₃ = y₂:y₃, ... , and x_n-1:x_n = y_n-1:y_n, then x₁:x_n = y₁:y_n.

V.23. Perturbed ratios ex aequali. If u : v = y : z and v : w = x : y, then u : w = x : z.

V.24. If u : v = w : x and y : v = z : x, then (u + y):v = (w + z):x.

V.25. If w : x = y : z and w is the greatest of the four magnitudes while z is the least, then w + z > x + y.

Logical structure of Book V

Book V is on the foundations of ratios and proportions and in no way depends on any of the previous Books. Book VI contains the propositions on plane geometry that depend on ratios, and the proofs there frequently depend on the results in Book V. Also Book X on irrational lines and the books on solid geometry, XI through XIII, discuss ratios and depend on Book V. The books on number theory, VII through IX, do not directly depend on Book V since there is a different definition for ratios of numbers.

Although Euclid is fairly careful to prove the results on ratios that he uses later, there are some that he didn’t notice he used, for instance, the law of trichotomy for ratios. These are described in the Guides to definitions V.Def.4 through V.Def.7.

* Some of the propositions in Book V require treating definition V.Def.4 as an axiom of comparison. One side of the law of trichotomy for ratios depends on it as well as propositions 8, 9, 14, 16, 21, 23, and 25. Some of Euclid’s proofs of the remaining propositions rely on these propositions, but alternate proofs that don’t depend on an axiom of comparison can be given for them.

Propositions 1, 2, 7, 11, and 13 are proved without invoking other propositions. There are moderately long chains of deductions, but not so long as those in Book I.

The first six propositions excepting 4 have to do with arithmetic of magnitudes and build on the Common Notions. The next group of propositions, 4 and 7 through 15, use the earlier propositions and definitions 4 through 7 to develop the more basic properties of ratios. And the last 10 propositions depend on most of the preceding ones to develop advanced properties.