Euclid, Elements V 23©
translated by Henry Mendell (Cal. State U., L.A.)

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English translation of V 23 from the Latin translation of the Arabic version of Euclid's Elements, commonly ascribed to Gerard of Cremona (ed., H.L.L. Busard (Leiden: Brill, 1984)

Notes on the use of alternando in the Greek text and its absence in the Arabo-Latin version.

Prop. 23: If three magnitudes and others equal to them in plêthos, taken by two’s, are in the same ratio, but their proportion is perturbed, they will also be through an equal (ex aequali) in the same ratio.

(general diagram)
(diagram based on manuscript Vatican gr. 190 (P), 9th cent.)

Diagrams for book 5 proposition 23, following the display, construction, and demonstration(diagram 1) Let there be three magnitudes, A, B, G, and others equal to them in plêthos being taken by two’s are in the same ratio, but their proportion is perturbed, they will be through an equal (ex aequali) in the same ratio: as A to B, so E to Z, and as B to G, so D to E. I say that it is: as A to G, so D to Z.

(diagram 2) Let there be taken equal-times multitples of A, B, D, i.e., H, Q, K, (diagram 3 = gen. diag.) and others, whatever happens, equal-times multiples of G, E, Z, i.e., L, M, N. (diagram 4) And since H, Q are equal-times multiples of A, B, but the parts have the same ratio as the multiples in the same way (v 15), therefore, it is: as A to B, so H to Q. (diagram 5) For the same reasons, in fact, it is: as E to Z, so too M to N. (diagram 6) (diagram 7) Therefore, as H to Q, so to M to N. (diagram 8) And since it is: as B to G, so too D to E, alternando as B to D, so too G to E.1 (v 16) (diagram 9) And since Q, K are equal-times multiples of B, D, but the parts have the same ratio as the equal-times multiples, therefore, it is: as B to D, so Q to K. (v 15) (diagram 10) But as B to D, so G to E. (alternando above, diagram 11) Therefore as Q to K, so too G to E. (v 11) (diagram 12) Again, since L, M are equal-times multiples of G, E, therefore, as G to E, to L to M. (v 15) But as G to E, so Q to K. (diagram 13) Therefore, as Q to K, so too L to M (v 11), and alternando as Q to L, K to M. (v 16) (diagram 14) But it was shown that as H to Q, so to M to N. And so, since three magnitudes H, Q, L, and others equal to them in plêthos, K, M, N, taken by twos are in the same ratio, and their proportion is perturbed, therefore, through an equal (ex aequali), if H exceeds L, K also exceeds N, and if equal, equal, and if smaller, smaller. (v 21) (diagram 15) And H, K are equal-times multiples of A, D, while L, N are of G, Z. Therefore, it is: as A to G, so D to Z. Therefore, if three magnitudes and others equal to them in plêthos, taken by two’s, are in the same ratio, but their proportion is perturbed, they will also be through an equal (ex aequali) in the same ratio, just what it was required to show.

1. (Manuscripts B p V (added)) equal-times multiples of B, D have also been taken, Q, K and equal-times multiples of G, E, whatever happens, L, M, therefore, it is: as Q to L, so K to M.



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The Arabic translation(s) of this theorem is different in one important way. Here is an English translation of V 23 from the Latin translation of the Arabic version of Euclid's Elements, commonly ascribed to Gerard of Cremona (ed., H.L.L. Busard (Leiden: Brill, 1984). The Arabic text on which the Latin translations are based has yet to be edited and published. For simplicity, I have kept the lines vertical, which appear as horizonal in Busard's edition. Vertical lines are typical in Greek and Arabic manuscripts in any case. Note that there are two differences in the lettering. One difference is merely from the translation, where the letter 'Q' and 't are the same', 'Q' being my substitution for theta, and 't' standing in Latin texts for Arabic ط, which also stands for theta. The other change is more significant. The Greek text keeps the letter ordering as the multiples are introduced. The Arabo-Latin text keeps the ordering of the multiples as they relate to what they are multiples of, that is the multiples of a, b, g are the three letters after d, e, z, namely h, t, k. Thus, the letters k, l are switched with L, K.

Prop. 23: If there are as many quantities as you please and others with just so many of their number, such that two of the very first are in proportion from the second two, and the proportion differs in the antecedent and consequent, then they will be in proportion of them following the proportion of equality (ex aequali).

(general diagram)

(diagram 1) For instance: Let there be quantities, a and b and g, and let there be other quantities d and e and z in a single enumeration and let two of the very first be in proportion with two of the second, and let the proportion differ in quantities according to antecedent and consequent. Let the proportion of a to b be as the proportion of e to z, and the proportion of b to g be as the proportion of d to e. I say, therefore, that following equality (ex aequali) the proportion of a to g is just as the proportion of d to z.

The demonstration of it: (diagram 2) For let me assume for quantities a and b and d equal multiples h and t and l, (diagram 3 = gen. diag.) and for quantities e and z and g equal multiples, that is, m and n and k. (diagram 4) And because h is thus a multiple of quantity a just as t is a multiple of quantity b, and the proportion of the parts whose multiples are equal is so between them as the proportion of the multiples is between them, therefore, the proportion of a to b is as the proportion of h to t. (The order of the argument here is slightly different, but I have kept the numbering of the diagrams the same.) (diagram 6) But the proportion of a to b is as the proportion of e to z. Therefore, the proportion of e to z is as the proportion of h to t. (diagram 5) And besides, since m is so a multiple of quantity e as n is a multiple of quantity a, (diagram 7) therefore, the proportion of e to z is just as the proportion of m to n. But the proportion of e to z is as the proportion of h to t. Therefore, the proportion of h to t is just as the proportion of m to n. (Here the Greek puts in alternando) (diagram 8) And besides, since the proportion of b to g is as the proportion of d to e, (diagram 9) and we just assumed for quantities b and d equal multiples, that is, t and l (Here the arguments diverge) and for quantities g and e equal multiples, that is, k and m, (Here the arguments converge)(diagram 13) therefore the proportion of t to k is as the proportion of l to m. (diagram 14) But it was just shown that the proportion of h to t is as the proportion of m to n. (diagram 15) Therefore, two quantities, h and l either are together larger quantities than k and n, or together equal them, or are together smaller than them. But h and l are equal multiples of quantities a and d, and k and n are equal multiples of g and z. Therefore, the proportion a to g is as the proportion of d to z. And that is what we aimed to demonstrate.

All the other proofs in Elements V pay close attention to the principle of homogeneity specified in V def 2 and explained in V def. 3. Let us suppose that a : b = c : d is part of the statement of the theorem. Unless a ratio of a : c is in the statement of the theorem, it should not appear in the proof, and so too for theorems about equal multiples. Theorem V 23 is such a theorem and, according to the principle of homogeneity, should not allow taking a ratio a : c. Of course, it is possible that Euclid is not as rigorous as readers might expect (so I. Mueller, Philosophy of Mathematics and Deductive Structure in Euclid's Elements (Cambridge: MIT Press, 1981), 133, who is inclined to think of the violation as an “inexplicable exception.”).

It is, therefore, important that the Heiberg's Greek text and the Arabo-Latin texts diverge significantly in two places, where the Greek text uses alternando (V 16). In the second, the argument of the Arabo-Latin text is unclear. Three manuscripts of the Greek text, which Heiberg treated as less important, preserve exactly the reading of the Arabo-Latin text (see the footnote above), which, as Vitrac observed in his French translation of Euclid (ii 121 n. 124), indicates a different line of argument intruding into the Greek text.

However, since alternando just is: a : b = c : da : c = b : c, its use in a theorem whose statement does not likewise allow a : c as a ratio would be illegitimate, especially if the theorem is to be used in cases where, for example, one pair of magnitudes is lines and the other figures. Any appllication of perturbando to the two collections would become incoherent, since the application would presuppose that there is a ratio of a line to a figure. It is but small solace that perturbando is not a common rule. It does not occur in Euclid again (cf. Mueller, Ibid., 156 for VI, but Mueller does not cite its use anywhere else). It occurs in two proofs of Archimedes, Sphere and Cylinder I 4 and Plane Equilibria II 9, and in Pappus, Collectio VII 932.9-13, 988.19-23. Saito does not identify its being used in Apollonius, Conica I-IV.

Hence, from a strictly deductive point of view, the Greek text is notoriously defective.

The Arabo-Latin text does not fill in the argument, but we could do it in two ways, one matching the Greek text and one that preserves the integrity of the argument. I have aligned the lettering of the Arabo-Latin argument to that of the Greek

Greek Rule Arabo-Latin (Gregory) and fragment in three Greek mss. for lines 9-13 Rule
1. A : B = E : Z given 1. A : B = E : Z given
2. B : G = D : E given 2. B : G = D : E given
3. H, Q, K are equimultiples of A, B, D const 3. H, Q, K are equimultiples of A, B, D const
4. L, M, N are equimultiples of G, E, Z const 4. L, M, N are equimultiples of G, E, Z const
5. A : B = H : Q 3 V 15 5. A : B = H : Q 3 V 15
6. E : Z = M : N 4 V 15 6. E : Z = M : N 4 V 15
7. H : Q = M : N 1, 5, 6 V 11 7. H : Q = M : N 1, 5, 6 V 11
8. B : D = G : E 2 V 16 (alt) 8. B : G = D : E 2 Repitition
9. B : D = Q : K 3 V 11 9. Q, K are equimultiples of B, D 3 some veriosn of simplification
10. Q : K = G : E 8, 9 V 11 10. L, M are equimultiples of G, E 4 some veriosn of simplification
11. G : E = L : M 4 V 11 and G : E is a ratio in 8    
12. Q : K = L : M 10, 11 V 11    
13. Q : L = K : M 12 V 16 (alt) 13. Q : L = K : M ?

14. H > L iff K > N &
H = L iff K = N
H < L iff K < N

7, 13 V 21

14. H > L iff K > N &
H = L iff K = N
H < L iff K < N

7, 13 V 21

A : G = D : Z

2, 3, 14 V def. 4

A : G = D : Z

2, 3, 14 V def. 4

The reasoning, lines 8-13 of the Greek text, inter alia, uses lines 2, 3, and 4, the lines cited in Gregory's translation. It is also somewhat more complex than might be neceessary. Given that we have no account of Gregory's argument, we are free to fill it in however we choose. Here is one possibility:

Arabo-Latin (Gregory) Rule
11. G : L = E : M 10 V 4
12. Q : B = K : D 9 V 4
13. Q : L = K : M 12, 8, 11 V 22 (ex aequali) twice

Three points:

1. The application of V 4 requires that a : a = b : b, which is obvious from the definition of same ratio, V def. 4 (or at least would require no more to prove than this), and then if c, d are equi-multiples of a, b, V 4 allows that a : c = b : d.

2. In his copy of Busard's edition of Gregory's translation, Knorr wrote next to this part of the proof, "cf. alt Heib 39n," that is Heiberg's introduction to his edition of the scholia, Elements vol. V.1 (p. 52 in the original edition, p. 39 in that of Stamatis), where, citing his note to V 3, Heiberg observes that Euclid sometimes uses δι᾽ ἴσου (ex aequali) differently from how he defines it. Perhaps, Knorr is referring to its use with τεταραγμένη (perturbed), here and in V 21, as well as at V 22 (vulgo) in the Greek text.. However, it is notable if the Arabo-Latin text does not use any form of 'ex aequali' in this part of the argument but supposes that the reader will supply some argumentl. So it is possible that Knorr is referring to this absence. If so, he suspected the ex aequali argument.

3. The alternate reading in the Bodleian 301, Paris Gr. 2466, and the margin of Vind. 103, concludes by ex aequali from the Arabo-Latin, namely as Q to L, so K to M. These three manuscripts lie in a group that Heiberg labeled Theonine, as opposed to Vat. gr. 190 that Heiberg preferred. Knorr (“Wrong Text of Euclid,” Centaurus 38 (1996): 208-76), followed by S. Rommevaux, A. Djebbar, B. Vitrac, “Remarques sur l'Histoire du Texte des Élément d'Euclide,” Arch. for Hist. of Exact Sciences 55 (2001): 221-295, argued that these manuscripts better represented the original text.

Let's suppose that the Arabo-Latin text is closer to Euclid. Then the original argument had a gap, which Euclid assumed one could fill in trivially, e.g.., with an ex aequali argument such as that given above. An editor of the Elements filled in the argument with an argument unnecessarily using alternando (V 16) twice and the multiple theorem (V 15). This becomes the ancestor of the Greek manuscripts, with the exception of the three manuscripts that collated the problematic step. However, V 4, 22 (ex aequali) better represent Euclid's intention.

If this is the case, the originial logical structure of book V improves significantly.

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