Euclid has shown through Books 1-4 that he has a specific method of organizing his work; each definition seems to be carefully placed in relation to each proposition, each Book put together thoughtfully. Euclid seems to abandon his previous mode of organization and coordination between books, however, when he begins Book 5. Suddenly, none of the propositions from previous Books are used to prove the propositions in Book 5, and Euclid begins working with magnitudes rather than geometric shapes. While Book 5 does seem to stand alone from Euclid’s first four Books, it maintains a specific organization within itself. I have discovered an alternate proof for proposition 22, and while it proves the same enunciation as Euclid’s version, it does so differently, changing the the way in which the proposition contributes to Book 5 overall. A Euclidean proof must be understandable while also contributing to the Book as a whole; while my alternate proof makes more logical sense to me, Euclid’s proof of Proposition V.22 does a better job of contributing to Book 5 as a whole. The question becomes, what is the point of a proof? to contribute to the book as a whole or to make logical sense?
Proposition 22’s enunciation, that is, Euclid’s original and my alternate, states that “If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio ex aequali”(V.22). Just as the original and alternate propositions share the same enunciation, they also share a given,“any number of magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two together are in the same ratio, so that, as A is to B, so is D to E, and, as B is to C, so is E to F”(V.22), and what needs to be proved, “that they will also be in the same ratio ex aequali”(V.22). Euclid’s proof goes on to take equimultiples of the given magnitudes, G of A, H of D, K of B, L of E, M of C, and N of F, and prove, based on Proposition V.4, that since the original first magnitude has the same ratio to the second as the third has to the fourth the equimultiples of the original first and third magnitudes will have the same ratio to any equimultiples of the original second and fourth magnitudes. In other words, with regard to Proposition 22, Proposition 4 proves that G:K::H:L and K:M::L:N. Euclid then goes on to prove by Proposition V.20 that since “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 be greater than the third, the fourth will also be greater than sixth; if equal, equal; and if less, less”(V.20), or, that since there are six multiples of the original six magnitudes and they are in the same ratio when taken two and two together, by ex aequali, if G is greater than, equal to, or less than M, the same relationship holds true for M and H. Finally, Euclid proves his enunciation. By Definition V.5, which states that magnitudes are in the same ratio if equimultiples taken of the original first and third and the original second and fourth are greater, equal, or less than their equimultiples respectively, Euclid proves that the original magnitudes are in ratio ex aequali.
The alternate proof, however, avoids taking equimultiples; instead it takes the original given ratio of magnitudes,
A:B::D:E and B:C::E:F
and by Proposition V.16, takes the alternates of these ratios:
A:D::B:E and B:E::C:F.
Next, by Proposition V.11, which states that “ratios which are the same with the same ratio are also the same with one another”(V.11), the alternate proof shows that
Next, the alternate proof proves the enunciation by Proposition V.16, which again allows the alternate to be taken, so
an ex aequali relationship as set out in Definition V.17.
While both Euclid’s Proposition 22 and the alternate of Proposition 22 prove the same enunciation, they do so differently. Euclid’s version uses Propositions V.4 and V.20 in conjunction with Definition V.5 while the alternate version of Proposition 22 uses Propositions V.11 and V.16 in conjunction with Definition V.17. The obvious loss in the alternate proof is the use of multiples, though the extensions of Propositions V.4 and V.20 are also lost in the alternate version. Euclid’s Proposition 22 extends Proposition V.4, which states that “if a first magnitude have to a second the same ratio as a third to a fourth, and equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order” by showing that the same relationship that exists in Proposition V.4 between four given magnitudes and their four equimultiples also exists ex aequali between four of six given magnitudes and four of their six equimultiples. While Euclid’s Proposition 22 seems like a more universal Proposition V.20, the alternate proof eliminates the elaboration on Proposition V.20. Proposition V.20 states that “if there be 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 be greater than the third, the fourth will also be greater than the sixth; if equal, equal; if less, less” while Proposition 22 states nearly the same thing, but, instead of specifically three magnitudes and three other magnitudes equal to them in multitude, Proposition 22 allows for “any number of magnitudes whatever, and others equal to them in multitude” to be in the same ratio ex aequali if they are in the same ratio when taken two and two together.
Another difference between the enunciations of Propositions V.20 and V.22 is that the former provides the ex aeqauli relationship as a given which is what has to be proved in Proposition 22. This brings up the question, then, of why Euclid decided to use three magnitudes in his proof of Proposition 22 if Proposition V.20 also uses three magnitudes. Since Proposition 22 extends Proposition V.20 to hold true for any number of magnitudes it would seem that it might be more useful for Euclid to use a different number of magnitudes than are used in the Proposition it is built off of, but Euclid has a point. By using the same number of magnitudes in each proposition but by taking vertical multiples, that is, multiples of A and D, B and E, and C and F, Euclid shows a relationship between the original magnitudes that may not have been evident from the outset of the proposition. By relating the original magnitudes(A,B,C) to the others equal to them in multitude(D,E,F) through multiples which he can manipulate, Euclid is illustrating Proposition V.16, that alternate relationships exist, without explicitly stating so. Therefore, though the reasoning may seem convoluted, multiples do indeed have a purpose. Not only do Euclid’s vertical multiples create a visualization of alternate ratios, without taking multiples in Proposition 22, Definition 5 would have no place in the Proposition and it would contribute less to the Book.
The fact that both versions of the Proposition prove the enunciation does not render them equally valuable. While the alternate proof may be logically sound and easier to understand, it is ultimately not as useful since it does not contribute as much to Book 5. Not only does Euclid’s version of Proposition 22 use the cited Propositions V.4 and V.20, it also uses Proposition V.16 without stating it while the alternate proof of Proposition 22 only uses Propositions V.11 and V. 16. While both versions of the Proposition share in their use Definition 5, ultimately Euclid’s version connects back to the most information from Book 5, therefore contributing more to the book as a whole. Euclid’s organization is purposeful and intricate, as can be seen through the comparison of these two different methods of proving Proposition 22. This comparison convincingly shows also that the point of a proof is not simply to prove the enunciation, but to go further and contribute to the overall Book, which Euclid’s Proposition 22 does while the alternate proof falls short.
Euclid. Elements. Translated by Thomas L. Heath. Santa Fey: Green Lion Press: 2002. Print