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Ex aequali Ratios in the Greek and Arabic Euclidean Traditions

Published online by Cambridge University Press:  24 October 2008

Gregg De Young
Affiliation:
The American University in Cairo, 113 Sharia Kasr el Aini, P.O. Box 2511, Cairo, Egypt

Abstract

Euclid discusses the ex aequali relationship twice in the Elements. The first is in Book V (based on definitions 17 and 18, propositions 22 and 23), during his discussion of arithmetical relations between mathematical magnitudes in general. The second is in Books VII–IX (developed using proposition VII,14), where he focuses on arithmetical relations in the case of numbers only. Although the distinction between mathematical magnitudes in general and numbers in particular often seems somewhat forced to contemporary philosophers, it was apparently very real to Euclid. Because Euclid seemed so conscious of the differences between the subject matter of Book V (magnitudes) and Books VII–IX (numbers), he was not much troubled by the differences between his treatment of ex aequali ratios in these two contexts. Later generations of mathematicians, however, found these differences less acceptable and tried to minimize them in various ways. This paper summarizes Euclid's use of the ex aequali relation in developing his mathematics. The paper then outlines the fate of the post-Theonine Greek attempts to “improve” the Euclidean discussion when the Elements entered the Arabic/Islamic intellectual tradition. The study concludes with the attempts by Ibn al-Hayṯam and Ibn al-Sarī to improve the parallelism between the discussions of ex aequali ratios in Book V and Book VII.

Euclide discute de la relation ex aequali deux fois dans les Éléments: la première fois dans le Livre V (fondé sur les définitions 17 et 18, les propositions 22 et 23), au cours de sa discussion des relations arithmétiques entre les grandeurs mathématiques en général; la seconde fois dans les Livres VII–IX (développement qui utilise la proposition VII, 14), où il se concentre sur les relations arithmétiques dans le cas des nombres seulement. Bien que la distinction entre grandeurs mathématiques en général et nombres en particulier paraisse souvent quelque peu forcée aux yeux de philosophes contemporains, elle fut apparemment très réelle pour Euclide. Parce qu'Euclide semblait être aussi conscient des differences entre le sujet du Livre V (grandeurs) et les Livres VII–IX (nombres), il ne fut pas très gêné par les différences entre son traitement des rapports ex aequali en ces deux contextes. Cependant des générations ultérieures de mathématiciens trouvèrent ces différences moins acceptables et essayèrent de les minimiser de diverses manières. Cet article présente en résumé l'utilisation par Euclide du rapport ex aequali dans le développement de ses mathématiques. Ensuite l'article met en évidence le destin des tentatives grecques post-Théonines visant à “améliorer” la discussion euclidienne quand les Éléments firent leur entrée dans la tradition intellectuelle arabo-musulmane. L'étude conclut avec les tentatives d'Ibn al-Haytham et d'Ibn al-Sarī visant à améliorer le parallélisme entre les discussions des rapports ex aequali dans le Livre V et le Livre VII.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1996

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References

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“A ratio ex aequali is the ratio of the extremes to one another when the number of magnitudes is greater than two magnitudes [and] in addition there are other magnitudes according to their number and in the same ratio and the number of the intermediate <magnitudes> is equal.”

15 This phrase is omitted by Gerard of Cremona in his Latin translation of al-Nayrīzī's commentary. Cf. Curtze, M., Anaritii in decem libros priores Elementorum Euclidis Commentarii ex interpretatione Gherardi Cremonensis in codice Cracoviens 569 servata (Leipzig, 1899), p. 168.Google Scholar

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The “Pseudo-Ṭūsī” printed Taḥrīr has a very similar formulation of the definition, suggesting a close connection to the work of al-Ṭūsī, . Kitāb Taḥrīr Uṣūl li-Uqlīdis (Roma, 1594), p. 111:Google Scholar

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Two marginal notes offer two alternative formulations of this definition:

“And the ex aequali <ratio> is that the ratio of the first <from the initial magnitudes> to the last of them is as the ratio of the first among the other <magnitudes> to the last of them.”

“And the ex aequali <ratio> is that the ratio of the first extremity of the initial magnitudes to the last of them is as the ratio of the first extremity of the other <magnitudes> to the last of them.”

34 Ibn al-Hayṯam, , Kitāb fi ḥall šukūk Kitāb Uqlīdis, Istanbul, MS Fātiḥ 3439/2, fol. 111a. Cf. note 24, above.Google Scholar

35 Al-Nayrīzī, , Codex, III, 26:Google Scholar

“The ordered ratio is when the antecedent to the consequent is as the antecedent to the consequent and the consequent to something else is as the consequent to something else.” Cf. note 11.

36 Ibid.:

37 Engroff, , Elements, Book V, pp. 62, 167:Google Scholar

This definition is omitted from Oxford, Bod. Lib., Huntington 435; from the Andalusian/Maṭribī family of manuscripts: Escurial, or. 907; Rabat, al-Mālik 1101; Rabat, al-Mālik 53; and from the manuscript sub-family comprised of Istanbul, Fātih 3439/1 and Copenhagen LXXXI. Most manuscripts containing this interpolated definition attribute it to Ṯābit ibn Qurra. Those omitting the attribution are: Rampur, Riḍā Lib. 3656; Uppsala, University Lib., Tornberg 321; Oxford, Bod. Lib., Thurston 11; Leningrad, Akad. Nauk, or. C 2145; and Teheran, Malik 3586. Cf. ibid., pp. 66, 171.

38 Ibn Sīnā, , al-Šifā', p. 155:Google Scholar

“And the ratio of the consequent, if it becomes the antecedent to another consequent, is as the ratio of the consequent from the others to another consequent.”

Cf. al-Abharī, Iṣlāḥ, fol. 31a:

“The ordered <ratio> is that which has magnitudes and other magnitudes according to their number and the ratio of the antecedent to the consequent is as the ratio of the corresponding antecedent to the corresponding consequent and the ratio of the consequent to another consequent is as the ratio of the corresponding consequent to another consequent.”

39 Engroff, , Elements, Book V, pp. 62, 167.Google Scholar This definition is omitted from Oxford, Huntington 435 and given only in the margin in Oxford, Bod. Lib., Thurston 11. In two manuscripts (Cambridge, University Lib., add. 1075 and Uppsala, University Lib., Tornberg 321), it is referred to another manuscript of Ṯābit ibn Qurra. In Leningrad, Akad. Nauk, or. C 2145, it is said to be found in other manuscripts but these are not specified to be those used by Ṯābit in his editing. Cf. ibid., pp. 66, 171.

40 Al-Maģribī, , Taḥrīr, fol. 35a:Google Scholar

41 Heath, , Elements, II, 176.Google Scholar

42 Ibid., II, 175.

43 Ibid., II, 179.

44 Al-Nayrīzī, , Codex, III, 76:Google Scholar

“If there be magnitudes and other magnitudes according to their number, each pair from the first <magnitudes> according to the ratio of a pair from the others, and the ratio is ordered, then the initial magnitude from the first magnitudes is in an ex aequali ratio when, if it is greater than the last, then the first <magnitude> from the other magnitudes is greater than the last magnitude; and if it is equal to it, then it is equal to it; and if it is less than it, then it is less than it.”

Ibid., p. 78:

“If there be some magnitudes and other magnitudes according to their number, each pair from the first <magnitudes> according to the ratio of a pair from the others, and the ratio is perturbed, the initial magnitude from the first magnitudes is in an ex aequali ratio when, if it is greater than the last, the first <magnitude> from the others is greater than the last; and if it is equal to it, then it is equal to it; and if it is less than it, then it is less than it.”

45 Leningrad, , Akad. Nauk, MS or. C 2145, fol. 88a-b:Google Scholar

MSS Cambridge, University Lib., add. 1075; Uppsala, University Lib., Tornberg 321; Oxford, Bod. Lib., Thurston 11 and Huntington 435 contain the same formulation. Cf. Engroff, , Elements, Book V, p. 133.Google Scholar

46 MSS Chester Beatty 3035; Teheran, Maǧlis Šūrā 200; and Rampur, Riḍā Lib. 3656 omit this clause.

47 Engroff, , Elements, Book V, p. 244:Google Scholar

In addition to the manuscripts in Engroff's “Family of the Six”, this phraseology is found in Teheran, Malik 3586; Rabat, al-Mālik 1101; Rabat, al-Mālik 53.

48 Ibn Sīnā, , al-Šifā', p. 174:Google Scholar

“The ratio of A to B is as <the ratio of> D to E and <the ratio of> B to G is as <the ratio of> E to Z. Thus, by the ex aequali relation, if A is equal to or greater than or less than G, in the same way is D <in relation> to Z.”

49 Al-Abharī, , Iṣlāḥ, fols. 34b–35a:Google Scholar

“The ratio of A to B is as the ratio of D to E and the ratio of B to G is as the ratio of E to Z so, because of the ordered <ratios>, I say by the ex aequali relation that if A is greater than G, D is greater than Z; and if it is equal, then it is equal; and if less, then it is less.”

“The ratio of A to B is as the ratio of E to Z and the ratio of B to G is as the ratio of D to E so, because of the perturbed <ratios>, I say by the ex aequali relation that if A is greater than G, D is greater than Z; and if it is equal, then it is equal; and if less, then it is less.”

50 The manuscript has D here.

51 The manuscript has D here.

52 Al-Ṭūsī, Taḥrīr, fol. 87a:

53 “Pseudo-Ṭūsī,” Taḥrīr, pp. 125–7:Google Scholar

“Any two groups of magnitudes equal in number, whatever the number may be, each pair from a group according to the ratio of a pair from the other group, with the ratio ordered, are in an ex aequali relation when, if the first from the first group is greater than the last from it, the first from the second group is greater than the last from it; and if equal, it is equal; and if less, it is less.

Let A, B, G, D, E, Z be two groups of the same number, and the ratio of A to B be as the ratio of D to E and the ratio of B to G as the ratio of E to Z. I say that if A is greater than G, D is greater than Z; and if A is equal to G, D is equal to Z; and if A is less than G, D is less than Z.

The proof is to let A be greater than G. Then, because the ratio of D to E is as the ratio of A to B, and the ratio of A to B is greater than the ratio of G to B according to proposition eight, then by proposition twelve, the ratio of D to E is greater than the ratio of G to B. But the ratio of G to B is as the ratio of Z to E. Thus the ratio of D to E is greater than the ratio of Z to E by using proposition twelve. Therefore, by proposition ten, D is greater than Z.

And if A is equal to G, then, because the ratio of D to E is as the ratio of A to B, and A is equal to G, the ratio of G to B is as the ratio of A to B, according to proposition nine. Thus, by proposition eleven, the ratio of D to E is as the ratio of G to B. But the ratio of Z to E is as the ratio of G to B by inversion. Thus, the ratio of D to E is as the ratio of Z to E, according to proposition eleven. Therefore, D is equal to Z by proposition nine.

And if A is less than G, then, by inversion, the ratio of E to D is as the ratio of B to A. But the ratio of B to A is greater than the ratio of B to G, according to proposition eight. Thus, by proposition twelve, the ratio of E to D is greater than the ratio of B to G. But the ratio of B to G is as the ratio of E to Z. Thus the ratio of E to D is greater than the ratio of E to Z, by using proposition twelve. Therefore, by proposition ten, D is less than Z. The proposition is established, and that is what we wanted to show.”

“<Any two groups of magnitudes equal in number, whatever the number may be,> each pair from a group according to the ratio of a pair from the other group, and the ratio being perturbed, are in an ex aequali relation when, if the first from the first group is greater than the last from it, the first from the other group is greater than the last from it; and if it is equal, it is equal; and if it is less, it is less.

Let A, B, G, D, E, Z be two groups of magnitudes of the same number, and the ratio of A to B be as the ratio of E to Z and the ratio of B to G as the ratio of D to E. I say that if A is greater than G, D is greater than Z; and if it is equal, then it is equal; and if it is less, it is less.

The proof is that, because the ratio of E to Z is as the ratio of A to B, and A is greater than G, the ratio of A to B is greater than the ratio of G to B by proposition eight. Thus, the ratio of E to Z is greater than the ratio of G to B according to proposition twelve. But, by inversion, the ratio of G to B is as the ratio of E to D. Thus, the ratio of E to Z is greater than the ratio of E to D by using proposition twelve. Therefore, D is greater than Z by proposition ten.

But if A is equal to G, then, because the ratio of E to Z is as the ratio of A to B and A is equal to G, the ratio of G to B is as the ratio of A to B by proposition seven. By proposition eleven, the ratio of E to Z is as the ratio of G to B. But by inversion, the ratio of E to D is as the ratio of G to B. Thus, by proposition eleven, the ratio of E to Z is as the ratio of E to D. Therefore, D <and> Z are equal by proposition nine.

And if A is smaller than G, then, by inversion, the ratio of E to D is as the ratio of G to B. But the ratio of G to B is greater than the ratio of A to B by proposition eight. Thus, by proposition twelve, the ratio of E to D is greater than the ratio of A to B. But the ratio of A to B is as the ratio of E to Z. Thus the ratio of E to D is greater than the ratio of E to Z by using proposition twelve. Therefore, D is less than Z by proposition ten. And that is what we wanted to show.”

54 Heath, , Elements, II, 177–8.Google Scholar

55 Ibid., II, 179.

56 Al-Maģribī, Taḥrīr, fol. 40a:

“<There being> three magnitudes and three other magnitudes, each pair of succeeding magnitudes from the first according to the ratio of a pair of succeeding magnitudes from the other and the ratios between the two of them being ordered, the first is in an ex aequali relation when, if it is greater than the third, the fourth is greater than the sixth; and if equal to it, it is equal to it; and if less than it, it is less than it.”

“<There being> three magnitudes and three other magnitudes, each pair of magnitudes from the first according to the ratio of a pair of magnitudes from the other and the ratios between them being perturbed in respect to antecedence and consequence, the first is in an ex aequali relation when, if it is greater than the third, the fourth is greater than the sixth; and if equal to it, it is equal to it; and if less than it, it is less than it.”

57 Heath, , Elements, II, 182–3.Google Scholar

58 Al-Nayrīzī, , Codex, III, 82, 84, 86:Google Scholar

“If there be some magnitudes and other magnitudes according to their number, each pair from the first according to the ratio of a pair from the other and the ratio is perturbed, when they are in an ex aequali relation, they are proportional. An example is that magnitudes A, B, G are the first and magnitudes D, E, Z are the others and every two magnitudes from A, B, G are according to the ratio of two magnitudes from D, E, Z and the ratio is perturbed – I mean that the ratio of A to B is as the ratio of E to Z and the ratio of B to G is as the ratio of D to E. I say that in an ex aequali relation, the ratio of A to G is as the ratio of D to Z.

The proof is that we take equimultiples for magnitudes A, B, D, namely H, T, L, and we take equimultiples for magnitudes E, Z, G, namely M, N, K. Then, because whatever is in H of the multiples of A is equal to that which is in T of the multiples of B and <because> when there are magnitudes which are equimultiples of them, the ratio of one magnitude to another is as the ratio of its equimultiple to another, which was shown in proposition 15, the ratio of A to B is as the ratio of H to T. But we specified that the ratio of A to B was as the ratio of E to Z. So if we drop out the intermediate terms, as was shown in proposition 11, the ratio of E to Z is as the ratio of H to T.

Likewise, because whatever is in M of the multiples of E is equal to whatever is in N of the multiples of Z, it is evident also from proposition 15 that the ratio of E to Z is as the ratio of M to N. But it was shown that the ratio of E to Z is as the ratio of H to T. So if we drop out the intermediate terms, as was shown in proposition 11, the ratio of H to T is as the ratio of M to N.

Likewise, because the ratio of B to G is as the ratio of D to E and equimultiples were taken for magnitudes B <and> D, namely T <and> L, and for magnitudes G <and> E, namely K <and> M, it is evident from proposition 15 with proposition 11 that the ratio of T to K is as the ratio of L to M. But it was shown that the ratio of H to T is as the ratio of M to N. Therefore, magnitudes H, T, K <are> the first and magnitudes L, M, N <are> the others, and each pair from magnitudes H, T, K are according to a pair from magnitudes L, M, N and the ratio is perturbed – I mean that the ratio of H to T is as the ratio of M to N and the ratio of T to K is as the ratio of L to M. It is evident, therefore, from proposition 21 that H <and> L either together exceed K <and> N or together equal them, or together as less than them. But H <and> L are equimultiples of magnitudes A <and> D and K <and> N are equimultiples of magnitudes G <and> Z. Thus it is evident from proposition 15 that the ratio of A to G is as the ratio of D to Z and that is what we wanted to show.”

59 Ibn Sīnā, , al-Šifā', p. 175:Google Scholar

A, B, G, D, E, Z <being> according to a single ratio, by the ex aequali <relation> A, G <are> as D, Z.”

60 Al-Maģribī, Taḥrīr, fols 40b–41a:

“<There being> three magnitudes and three others with them, each two contiguous magnitudes from the first according to the ratio of two contiguous magnitudes from the other and the ratios are perturbed in antecedence and consequence, when they are in an ex aequali ratio they are proportional. An example is that the ratio of A to B is as the ratio of D to E and the ratio of B to G is as the ratio of Z to D, the ratio being perturbed, when they are in an ex aequali relation, the ratio of A to G is as the ratio of Z to E.

We take for magnitudes A, B, Z multiples T, K, L of the same number and for G, D, E multiples M, N, S of the same number. Thus the ratio of T to K is as the ratio of A to B, that is to say, D to E – I mean N to S – and the ratio of K to M is as the ratio of L to N. Therefore, T and L either together exceed or together equal or together are less than M and S. Therefore, the ratio of A to G is as the ratio of Z to E.”

61 Al-Abharī, Iṣlāḥ, fol. 35b:

“The ratio of A to B is as the ratio of E to Z and the ratio of B to G is as the ratio of D to E by perturbation. I say that, by the ex aequali relation, the ratio of A to G is as the ratio of D to Z.

Let H <and> T be multiples equal in number to A <and> D and K <and> L to B <and> E and M <and> N to G <and> Z. Then the ratio of H to K is as the ratio of L to N and the ratio of K to M is as the ratio of T to L. Thus, if H is greater than M, T is greater than N; and if equal, then it is equal; and if less, then it is less. Therefore, the ratio of A to G is as the ratio of D to Z and that is what we wanted to show.”

62 “Pseudo-Ṭūsī,” Taḥrīr, p. 129:Google Scholar

63 Al-Ṭūsī, Taḥrīr, fols 87a–88b:

64 Ibid., fol. 88b:

65 Heath, , Elements, II, 180–3.Google Scholar

66 Al-Nayrīzī, , Codex, III, 86, 88, 90:Google Scholar

“These two propositions – I mean the twenty second and the twenty third – may be generalized to all magnitudes that are in a relation to other magnitudes, however many they may be. But mathematics only gives demonstrations for the least magnitudes (numerically) which are in the smallest ratios. Thus, since the least proportion is in three magnitudes <definition eight>, these have been demonstrated for three magnitudes because the method of the Elements is to set out the proof from an enunciation least in number and likewise according to a specification whose elements are least in number even when the demonstration is completely generalized for that category <of magnitudes>. Now as for the previous two propositions – I mean the twentieth and the twenty first – it is not possible to demonstrate them for more than three magnitudes, and even if it were possible, one would not be able to use the demonstration since the initial assumptions were that there are three magnitudes. These two propositions – I mean the twenty second and the twenty third – may, however, be generalized to any <number of> magnitudes.

We specify that four magnitudes <are in the> first, namely A, B, G, D, and four magnitudes <are in the> second, namely E, Z, H, T, each pair from the first according to a pair from the second in ordered ratios – I mean that the ratio of A to B is as the ratio of E to Z and the ratio of B to G is as the ratio of Z to H and the ratio of G to D is as the ratio of H to T. I say that the ratio of A to D is as the ratio of E to T.

The proof is that the ratio of A to B is as the ratio of E to Z and the ratio of B to G is as the ratio of Z to H. Thus, in an ex aequali relation, the ratio of A to G is as the ratio of E to H, just as was shown in proposition 22. Likewise, because the ratio of A to G is as the ratio of E to H and the ratio of G to D is as the ratio of H to T, then, in an ex aequali relation, the ratio of A to D is as the ratio of E to T. In the same way one may demonstrate the case of more than four magnitudes, however many may be specified in terms of number, and that is what we wanted to show.”

“If the ratio of A to B is as the ratio of H to T and the ratio of B to G is as the ratio of Z to H and the ratio of G to D is as the ratio of E to Z, I say that the ratio of A to D is as the ratio of E to T.

The proof is that the ratio of A to B is as the ratio of H to T and the ratio of B to G is as the ratio of Z to H. Then it is obvious, from proposition 23 that, when they are in an ex aequali relation, they are proportional, the ratio of A to G is as the ratio of Z to T. Then, because the ratio of A to G is as the ratio of Z to T and the ratio of G to D was specified as the ratio of E to Z, then, when they are in an ex aequali relation, as was demonstrated in proposition 23, the ratio of A to D is as the ratio of E to T. In the same way it may shown in the case of other magnitudes greater in number than A, B, G, D, and E, Z, H, T, whatever the number may be, and that is what we wanted to show.”

67 Heath, , Elements, II, 200.Google Scholar

68 Ibid., II, 204.

69 Ibid., II, 206.

70 Ibid., I, 306–7.

71 Ibid., II, 209.

72 Ibid., II, 235.

73 Ibid., II, 114.

74 Ibid., II, 132–3.

75 Ibid., II, 240.

76 Ibid., II, 247.

77 Ibid., II, 189–90.

78 Al-Nayrīzī, , Codex, III, 96:Google Scholar

79 MSS Leningrad, Akad. Nauk, or. C 2145; Oxford, Bod. Lib., Huntington 435; Oxford, Bod. Lib., Thurston 11; Cambridge, University Lib., add. 1035; Teheran, Malik 3586.

80 MSS Uppsala, University Lib., Tornberg 321; Copenhagen LXXXI; Istanbul, Fātiḥ 3439/1; Dublin, Chester Beatty Lib. 3035; Teheran, Maǧlis Šūrā 200; Rampur, Riḍā Lib. 3656; Escurial, or. 907; Rabat, al-Mālik 1101; Rabat, al-Mālik 53.

81 Heath, , Elements, II, 251.Google Scholar

82 Ibid., I, 343–5 and 372–4.

83 Unguru, S. and Rowe, D., “Does the quadratic equation have Greek roots? A study of ‘geometric algebra’, ‘application of areas’, and related problems,” Libertas Mathematica, 1 (1981): 149; 2 (1982): 1–62.Google Scholar

84 Herz-Fischler, R., A Mathematical History of Division in Extreme and Mean Ratio (Waterloo, 1987), pp. 3748.Google Scholar

85 De Young, G., “Abu Sahl's additions to Book II of Euclid's Elements,” Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften, VII (1992): 73135.Google Scholar

86 Heath, , Elements, II, 313.Google Scholar

88 Ibid., II, 345.

89 Ibid., II, 131.

90 Aristotle, Eth. Nic. 1131 a31, as quoted in Heath, Elements, II, 131.

91 Heath, , Elements, II, 114.Google Scholar

92 Ibid., II, 354.

93 Ibid., II, 411.

94 Ibid., II, 421.

95 Ibid., III, 24.

96 Ibid., III, 26.

97 Ibid., III, 25.

98 Ibid., III, 34.

99 Ibid., II, 314.

100 Al-Ṭūsī, Taḥrīr, fol. 112a:

101 Heath, , Elements, II, 158.Google Scholar

102 Ibid., II, 314.

103 Al-Ṭūsī, Taḥrīr, fol. 112a:

104 Ibn Sīnā, , Šifā', p. 224:Google Scholar

105 Heath, , Elements, II, 313.Google Scholar

106 Ibid., III, 329.

107 Ibid., III, 392.

108 Ibid., III, 410.

109 Ibn al-Hayṯam, Šukūk, fol. 114a–b:

110 Ibid., fol. 114a:

“Euclid founds this proposition on the twenty second, and there is no necessity for that in it.”

111 Ibid., fol. 111a:

“Between a line and a plane figure, and likewise between a plane figure and a volume, and between a straight line and a curved line no ratio exists. For even if the geometers make a ratio between the circumference of a circle and its diameter, that is an approximation in that they take for the curved line a large number of small straight lines, joining them together and putting these into a ratio with the diameter…”

Ibid., fol. 111b:

“Now, his statement <was> that magnitudes, some of which have a ratio to others, are those which are able, when multiplied, to exceed one another. It must be added that magnitudes between which a ratio may exist are those between whose two species an equalivalence occurs. Now perhaps the equalivalence might not appear between them until after one multiplies one of the two of them, for in lunulate figures, each of which is surrounded by two arcs, there exists that which is equivalent to a straight-sided triangle. Thus these <lunulate figures> are proportionate to straight-sided figures. And perhaps it might not be possible to subtract the equal of one from the other. But if a triangle equal to a lunulate figure be multiplied several times, it is possible that there occurs in the interior of its multiple a lunulate figure equal to one of its multiples. And we have already shown in the case of lunulate figures how there is found a lunulate figure equal to a straight-sided triangle. Indeed, it is permissible that <a ratio> exists between the two of them and it is possible to multiply one of the two of them by the multiple of the other because each one of the two of them is a plane figure, so that the two of them are of a single kind….”

112 As Heath remarked, if one accepts that Euclid, by his parallel discussions of magnitudes (in Book V) and numbers (in Books VII–IX) seems to differentiate numbers from magnitudes in species, such mixed ratios will lead to an untenable position within the logic of Euclidean geometry. Cf. Heath, , Elements, III, 25Google Scholar. One might have expected a mathematician of the stature of Ibn al-Hayṯam to have objected to Euclid's use of mixed ratios in propositions X, 5–8, where the ratio of one magnitude to another is compared to a ratio of one number to another. In these propositions, if the alternation of ratios is used, the result will be ratios containing a number and a magnitude. But alternation can be used only when all four magnitudes are of the same species. Ibn al-Hayṯam, however, ignores this problem, contenting himself with offering alternative proofs for propositions X, 5 and 6. Cf. Ibn al-Hayṯham, , Kitāb fi ḥall shukūk kitāb Uqlīdis fi'l-uṣūl wa-sharḥ ma'ānīhi, Institut für Geschichte der arabisch-islamischen Wissenschaften, Series C, Facsimile editions, vol. 11 (1985), pp. 339–40:Google Scholar

“Proposition Five: For any pair of <commensurable> magnitudes, the ratio of one of the two of them to the other is as the ratio of a number to a number. There is no doubt concerning this proposition, but it is possible to demonstrate it by a proof other than the proof of Euclid.

Let there be two commensurable magnitudes, AB <and> GD. I say that the ratio of AB to GD is as the ratio of a number to a number.

The proof of that is that the two magnitudes, AB <and> GD are commensurable by specification. Thus for each of them there is a magnitude which measures the two of them. Let the magnitude which measures the two of them be E and let AB be divided by the equals of E and let its divisions be AH <and> HB. <Let> GD be divided by the equals of E and let its divisions be GZ, ZT <and> TD. Thus the divisions of AB are equal, each one of them, to each one of the divisions of GD. Therefore, one of the divisions of AB is a part of GD and all of AB is parts of GD so the ratio of AB to GD is as the ratio of a number to a number. That is what we wanted to prove, and the basis of this proposition is that the two of them are commensurable by specification.”

“Proposition Six: Any two magnitudes, the ratio of one of the two of them to the other <being> as the ratio of a number to a number, the two of them are commensurable. This proposition is the opposite of the previous one. There is no doubt concerning it, but it is possible to demonstrate it by a proof other than the proof of Euclid.

Let there be two magnitudes, A <and> B and <let> the ratio of A to B be as the ratio of a number to a number. I say that A <and> B are commensurable.

The proof of that is that the ratio of A to B is as the ratio of a number to a number. But every number is parts of every number. Thus A is parts of B and each part from the parts of A called by the name of the number corresponding to it is a part of B and the one part of A measures B. But it <also> measures A. Thus, A <and> B are commensurable and that is what we wanted to show. The basis for this proposition is that the numerical ratio is the ratio of parts one to another.”

113 De Young, G., “Ibn al-Sarī on ex aequali ratios: His critique of Ibn al-Haytham and his attempt to improve the parallelism between Books V and VII of Euclid's Elements, Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften, 9 (1994): 99152.Google Scholar

114 Heath, , Elements, II, 164.Google Scholar

115 Ibid., II, 313.

116 Sezgin, , GAS, V, 310.Google Scholar

117 De Young, , “Ibn al-Sarī,” pp. 104, 112–13, 134–5.Google Scholar

118 Heath, , Elements, II, 247.Google Scholar

119 Ibid., II, 354.

120 Cf. Ibid., III, 24–6.

121 De Young, , “Ibn al-Sarī,” pp. 151–2 and 128:Google Scholar

122 Heath, , Elements, II, 318.Google Scholar

123 Al-Ṭūsī, Taḥrīr, fol. 113b:

124 Heath, , Elements, II, 320Google Scholar. On this latter point, see also Gerard of Cremona's Latin translation of al-Nayrīzī's Commentary. Cf. Curtze, , Anaritii, pp. 191–2.Google Scholar

125 Heath, , Elements, II, 320.Google Scholar