¹ Euclid, VII. Def. 20. All quotations of Euclid are from T.L. Heath's translation of the Elements [Sir Heath, Thomas L., The Thirteen Books of Euclid's Elements, 2nd edn, 3 vols. (New York, 1956)].Google Scholar In modern notation, this definition is: Four numbers A, B, C, D are proportional, i.e. A:B:: C:D, if A = mB and C = mB or A = (1/n)B and C = (1/n)D; or A = (m/n)B and C = (m/n)D; where m, n are positive integers, m < n.

² Eculid, V. Def. 5. In modern notation: A:B:: C:D if, whatever be the positive integers m, n, mA > nB implies mC > nD; mA = nB implies mC = nD; and mA < nB implies mC < nD.+nB+implies+mC+>+nD;+mA+=+nB+implies+mC+=+nD;+and+mA+<+nB+implies+mC+<+nD.>Google Scholar

³ For more details on these matters, see Plooij, E.B., Euclid's Conception of Ratio and His Definition of Proportional Magnitudes as Criticized by Arabian Commentators (Rotterdam, 1950); andGoogle ScholarYouschkevitch, A.P., Les mathématiques arabes (Paris, 1976), pp. 80–90.Google Scholar Regarding the mathematicians which vindicated Euclid's definition, see Plooij, Euclid's Conception of Ratio, ch. II; Isaac Barrow's Mathematical Lectures (published under the title The Usefulness of Mathematical Learning by Frank Cass and Co. Ltd., 1970), Lectures XVIII–XXIII; andGoogle ScholarSaunderson, N., The Elements of Algebra (Cambridge, 1740), pp. 439–46.Google Scholar

⁴ For a detailed account of al-Khayyām's life and writings, see Youschkevitch, A.P. & Rosenfeld, B.A., “Al-Khayyāmī,” in Dictionary of Scientific Biography (New York, 1973), VII, pp. 323–34.Google Scholar

⁵ Abū al-Fath ‘Umar Ibn Ibrāhīm al-Khayyāmī, Risāla fi sharh mā ashkala min musādarāt kitāb Uqlīdis. We have made our translation from the following manuscripts: Leiden, Bibliotheek der Rijksuniversiteit, MS Or. 199/8 (hereafter: Leiden), fols. 75^r–100^v; and Paris, Bibliothèque nationale, MS Ar. 4946/4 (hereafter: Paris), fols. 38^v–73^v. Sezgin, F. [Geschichte des Arabischen Schriftums (Leiden, 1974), V, p. 109] mentions a third manuscript: Saray, Ahmet III, 1584 (98b–110b); but, as we have been told, these folios contain in reality two philosophical treatises of al-Khayyām, and not his commentary on Euclid.Google Scholar – This work has already been translated into English by Amir-Moez, A.R. in “Discussion of difficulties in Euclid by Omar Ibn Abrahim al-Khayyami (Omar Khayyam),” Scripta Mathematica, 24 (1959): 275–303; but this translation, which contains a certain amount of errors and omissions, is unreliable and therefore unusable.Google Scholar

⁶ We recall here Euclid's definition of greater ratio, viz. Euclid, Def. V. 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.” In modern notation: A:B > C:D if positive integers m, n can be found such that mA > nB and mC ≦ nD.

⁷ Al-Khayyām, , Risāla fi sharh m¯ ashkala min musādarāt kitāb Uqlīdis, LeidenGoogle Scholar, fol. 86^r; Paris, fol. 53^r. The original text is:

This statement refers to Euclid, V. Def. 3, viz. “A ratio is a sort of ralation in respect of size between two magnitudes of the same kind.”

⁸ Ibid., Leiden, fol. 86^v; Paris, fol. 53^r–53^v. The original text is:

⁹ It must be mentioned that for al-Khayyām, both numbers and magnitudes belong to the genus of quantity.

¹⁰ We will, given four magnitudes A, B, C and D, always indicate hereafter the common proportionality by the symbolic notation A:B:: C:D, and the true proportionality by A/B = C/D.

¹¹ Al-Khayyām, , LeidenGoogle Scholar, fol. 89^r; Paris, fols. 56^v–57^r. The original text is:

The process involved in this definition, namely the anthyphairetic process or Euclidean algorithm, is, in algebraic notation, the following: Given two magnitudes A and B, one considers the sequence of non-negative integers a;_i and the sequence of real numbers r _i, i = 0, 1, 2, etc., defined by the relations 0 ≦ A – a₀B = r ₀ < B, 0 ≦ B – a₁r₀ = r ₁ < r ₀, 0 ≦ r ₀ – a ₂r₁ = r ₂ < r _1,, etc.; where the a_i's are called the quotients, and the r_i's the remainders. All the a_i's will necessarily be > 0, except perhaps a ₀ which may be = 0 in case A < B. If the magnitudes A and B are commensurable, one will finally arrive at a remainder that will measure exactly the preceding one leaving a remainder = 0; so that the process will terminate. Otherwise, the sequences a _i and r _iwill be infinite, and r _i > 0 for i = 1, 2 &c. It follows, from the previous set of relations, that the ratio A/B can be expressed as a finite or an infinite simple continued fraction with partial quotients a _i, We will hereafter represent an infinte simple continued fraction with partial quotients a ⁱ by the usual notation [a ₀, a ₁, a ₂, …]. But if a simple continued fraction be finite, for example [a ₀, a ₁, a ₂], we will, for convenience, still represent it by the notation [a ₀, a ₁, a ₂, …] and suppose conventionnally that a _i = ∞ for i = 3. According to this convention, the following rules will hold: r _i = 0 is equivalent to a _i+1 = ∞; and r _i > 0 is equivalent to a _i+1 < ∞. Also, in case the anthyphairetic process be applied to two ratios, for example to A/B and C/D, we will denote the remainders obtained during the anthyphairetic process by r(A/B)_i respectively. – Now al-Khayyām's definition of proportional magnitudes is, in modern notation: Given four magnitudes A, B, C and D, where A < B and C < D, if both the ratios A/B and C/D are expanded into the same infinite simple coninued fraction, viz. if A/B = [0, a ₁, a_2, …] and also C/D = [0, a₁, a₂, …, then A/B = C/D.

¹² In modern notation: Four magnitudes A, B, C, D being given, if A = B and C < D, or A > B and C ≦ D; or if A = (m/n)B and C = (r/s)D where m, n, r, s are positive integers such, that (m/n) > (r/s); then A/B > C/D.

¹³ Al-Khayyām, Leiden, fols. 89^v–90^r; Paris, fols. 57^v–58^r. The original text is:

In modern notation: If A/B = [0, a₁, a₂, …] and C/D = [0, b₁, b₂, …], and if a₁ < b₁; or if a₁ = b₁, but a₂ > b₂; or if a₁ = b₁ and a₂ = b₂, but a₃ < b₃; or if r(A/B)₃ or r(A/B)₁ = 0, but r(C/D)₃ or r(C/D)₁ > 0 (which is equivalent to: or if a₄ or a₂ = ∞, but b₄ or b₂ < ∞); then A /B > C/D. On the whole: If r(A/B)₁; or a(A/B)_2i+1 = 0, but r(C/D)₁ or r(C/D)_2i+1 > 0; or if r(A/B)_2i+1 < r(C/D)_2i+1; or if r(A/B)₀ or r(A/B)_2i+1 > 0, but r(C/D)₁ or r(C/D)_2i = 0; or if r(A/B)_2i > r(C/D)_2i; then A/B > C/D (This last statement is equivalent to: If a₂ or a_2i = ∞, but b₂ or b_2i < ∞ or if r(A/B)_2i+1 < r(C/D)_2i+1; or if a₁ or a_2i+1 < ∞, but b₁ or b_2i+1, = ∞; or if r(A/B)_2i > r(C/D)_2i; then A/B > C/D).

¹⁴ In modern notation: Four magnitudes A, B, C and D being given, if A:B:: C:D, then A/B = C/D; and if A/B = C/D, then A:B:: C:D. Also, if A:B > C:D, then A/B > C/D; and if A/B > C/D, then A:B > C:D.

¹⁵ The reader may think that the arguments considered in this paragraph are obvious or unnecessary. But this would be a consequence of his having in mind the modern theory of continued fractions. And to make use of that theory, or of its notation, would indeed constitute a reasoning a posteriori; whereas we are trying to reason a priori. Consequently, we will make no use, in this paragraph, of any modern notation, this notation being unknown to the mathematicians of the period under consideration.

¹⁶ Cf. Plooij, Euclid's Conception of Ratio, ch. III & IV, wherein the reader will find some extracts from the commentaries on Book V of Euclid's Elements by al-Māhānī (d. ca. 880), al-Nayrīzī (d. ca. 922), Ibn al-Haytham (965–1039), as well as from al- Khayyām's.

¹⁷ It should be mentioned that Euclid, VII. Prop. 3 is used in Prop. 33 of the same Book to reduce a ratio to its least terms.

¹⁸ It appears that this technique most probably dates back to Greek mathematics. In fact, J. Itard has shown that certain approximations to numerical ratios found in Aristarchus of Samos and in Archimedes could be obtained straightforwardly by the use of the Euclidean algorithm; and that this algorithm might have been used to approximate certain non-numerical ratios, as the ratio of the diagonal and side of a square [Cf. Itard, J., Les livres arithmétiques d'Euclide (Paris, 1961), pp. 26–32 & 41–7].Google Scholar W.R. Knorr is also of the same opinion [Cf. Knorr, W.R., The Evolution of the Euclidean Elements (Dordrecht, 1975), pp. 33, 36, 125 & 256–7].CrossRefGoogle Scholar And according to B. Vitrac: “il est quasiment certain que l'anthyphérèse a été utilisée dans les manipulations des rapports, au moins des rapports numériques, en particulier pour trouver des approximations de rapports entre grands nombres; il est également vraisemblable qu'elle a servi à déterminer des approximations de rapports de grandeurs incommensurables.” [d'Alexandrie, Euclide, Les Eléments, traduction et commentaires par Bernard Vitrac, 2 vols. (Paris, 1990 & 1994), II, p. 515].Google Scholar – As to the possibility of a Greek pre-Eudoxean definition of equal ratios based on the Euclidean algorithm and the problems involved therein, we refer the reader to Knorr, The Evolution of the Euclidian Elements; Fowler, D.H., The Mathematics of Plato's Academy, A New Reconstruction (Oxford, 1987); and to the comments ofGoogle ScholarVitrac, B. in d'Alexandrie, Euclide, Les Éléments, II, pp. 507–38.Google Scholar

¹⁹ In fact, this definition was precisely the one admitted by the 11th century mathematician al-Jayyānī (see Plooij, Euclid's Conception of Ratio, ch. II, wherein an English translation of al-Jayyānī's Commentary on Ratio can be found).

²⁰ Now considered as an interpolated definition, but which is mentioned by al-Khayyām after Euclid, V. Def. 3.

²¹ It appear that E.B. Plooij alludes to this when he says, refering to the view of those who criticized Euclid, namely the advocates of the anthyphairetic definition (Euclid's Conception of Ratio, p. 63): “The subtle difference is that this non

²² Cf. Plooij, Euclid's Conception of Ratio, p. 58.

²³ which is a necessary condition in order to consider any ratio qua number.

²⁴ Al-Khayyām, , Leiden, fol. 97^r; Paris, fol. 68^v.Google Scholar The original text is:

In modern notation: If A, B, C are three given magnitudes, then (A/C) = (A/B).(B/C). This proposition is not found in T.L. Heath's translation of the Elements; but al-Khayyām considers it as a postulate which is found at the beginning of the fifth Book of the Elements.

²⁵ Notably in Euclid, VI. Prop. 23, where it is said without proof: “But the ratio of K to M is compounded of the ratio of K to L and of that of L to M.”

²⁶ Al-Khayyām, Leiden, fol. 97^r; Paris, fol. 68^r. The original text is:

The translation of the passage “nothing but themselves is associated therein” presents some substantial difficulties because of the presence of two pronouns which could refer to magnitudes, to relation and to ratio; which prevents us from providing a littéral translation. We can only refer the reader to two different translations which have been made of a passage of al-Khayyām's Algebra which corresponds word for word to the one we ace considering now (As we had to quote the whole sentence to avoid any misinterpretation, we have italicized the passage under consideration). The first translation is to be found in L'œuvre algèbrique d'al-Khayyām, établie, traduite et analysée par Roshdi Rashed est Ahmad Djebbar, Sources and Studies in the History of Arabie Mathematics 3 (Alep, 1981), p. 13: “L'art de l'algèbre et d'al-muqàbala est un art scientifique dont l'objet est le nombre absolu et les grandeurs mesurables, en tant qu'inconnus mais rapportés à une chose connue par laquelle on peut les déterminer; et cette chose est soit une quantité, soit un rapport, de manière que rien d'autre qu'elle ne leur soit commensurable […].” The second translation is to be found in L'Algèbre d'Omar Alkhayyāmī, publiée, traduite et accompagnée d'extraits de manuscrits inédits, par Woepcke, F. (Paris, 1851), p. 5:Google Scholar “L'algèbre est un art scientifique. Son objet, ce sont le nombre abssolu et les grandeurs mesurables, étant inconnus, mais rapportés à quelque chose de connu de maniàre à pouvoir être déterminés; cette chose connue est une quantité ou un rapport individuellement déterminé […].” - The reader which understands either the Persian or the Russian language may consult the following translations of al-Khayyām's commentary on Euclid: a Russian translation made by Rosenfeld, B.A. and Youschkevitch, A.P. in Omar Khayyām, Treatises (Moscow, 1961);Google Scholarand a Persian translation made by Homāi, J. in Homāi, J., Khayyāmī Nāmeh (Tehran, 1967).Google Scholar

²⁷ In modern notation: A ratio A/B being given, one fixes in advance a magnitude U as unit, that is, one supposes U = 1, and then takes the magnitude G such, that 1/G = A/B. – We think this to be a mistake, and that al-Khayyām should have taken G such, that G/1 = A/B. For he says further on that G is the ratio A/B, in other words, that G = A/B; so that this would lead to the equation 1/G = G, which is absurd (unless G = ± 1). But as the same thing occurs with all the following ratios where a unit is involved, and that, apart from what we have just mentioned, the text is coherent, we will not amend here al-Khayyām's text, and only indicate in the footnotes what we consider as necessary amendments.

²⁸ Al-Khayyām, Leiden, fol. 97^v; Paris, fol. 69^r–69^v. The original text is:

²⁹ Ibid., Leiden, fol. 99^r–99^v; Paris, fol. 71^v. The original text is:

³⁰ Ibid., Leiden, fol. 99^v; Paris, fols. 71^v–72^v The original text is:

³¹ Ibid., Leiden, fols. 99^v–100^r; Paris, fol. 72^v. The original text is:

In modern notation: Let A, B, C be given. We fix a magnitude U as 1. Suppose 1/G = A/B, 1/D = A/C, and E/1 = C/B. As A/C = 1/D and C/B = E/1, we have ex æquali A/B = E/D (Euclid, V. Prop. 23). But A/B = 1/G. So that E/D = 1/G. Consequently 1.D = E.G. But G = A/B, E = B/C. and D = A/C. So that (A/B).(B/C) = 1.(A/C) = A/C. Q. E. D. - As we have said, we think that this proof is wrong, and that the correct proof should run thus: Let A, B, C be given. We fix a magnitude U as 1. Suppose G/l = A/B, D/l = A/C (so that C/A = l/D), and E/l = B/C. As B/C = E/l and C/A = 1/D, we have ex sequali B/A = E/D (Euclid, V. Prop. 22). But B/A = 1/G. So that E/D = 1/G. Consequently 1.D = E.G. But G = A/B, E = B/C and D = A/C. So that (A/B)XB/0 = l.(A/C) = A/C. Q. E. D.

³² viz. “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. 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.” In modern notation: Given the magnitudes X₀, X_t, X₂, X… X_n, if X₀/X₁ = X₁/X₂ = … = X_n-1/X_n, then X₀/X_n, = (X₀/X₁)ⁿ.

³³ Cf. Rashed, R., Entre arithmétique et algébre (Paris, 1984), pp. 27, 32, 34–6, 48, 250–1 & 309–12; andGoogle Scholar Youschkevitch, Les mathématiques arabes, pp. 80–1 & 83.