Corollary If any magnitudes are proportional, then they are also proportional inversely.
Corollary. If magnitudes are proportional taken jointly, then they are also proportional in conversion.
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.)
V.1. Multiplication by numbers distributes over addition of magnitudes.
V.2. Multiplication by magnitudes distributes over addition of numbers.
V.3. An associativity of multiplication.
V.5. Multiplication by numbers distributes over subtraction of magnitudes.
V.6. Uses multiplication by magnitudes distributes over subtraction of numbers.
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 x_{1}:y_{1} = x_{2}:y_{2} = ... = x_{n} : y_{n}, then each of these ratios also equals the ratio (x_{1} + x_{2} + ... + x_{n}) : (y_{1} + y_{2} + ... + 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 x_{1}:x_{2} = y_{1}:y_{2}, x_{2}:x_{3} = y_{2}:y_{3}, ... , and x_{n1}:x_{n} = y_{n1}:y_{n}, then x_{1}:x_{n} = y_{1}: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.
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. 
