¹ Euclid, VII. Def. 20 (the preceding notation means: the twentieth definition of the Seventh Book of Euclid's Elements). All quotations of Euclid are taken from Heath's translation of the Elements [Sir Thomas Heath, L., The Thirteen Books of Euclid's Elements, 2nd edn, 3 vols. (Dover, 1956)]. This definition is, in modern notation: Four numbers A, B, C and D are proportional, that is A:B = C:D (or: A/B = C/D), if A = mB and C = mD; 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.Google Scholar

² Euclid, V. Def. 5. In modern notation: Four magnitudes A, B, C and D have the same ratio, that is 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.

³ If there was any.

⁴ However it must be mentioned that this algorithm is mainly used in the Elements to find the greatest common measure of two numbers (see Euclid, VII. Prop. 2, X. Prop. 3, and the demonstration of VII. Prop. 4), to reduce a ratio to its least terms (see Euclid, VII. Prop. 33), and as a criterion for the incommensurability of two magnitudes (see Euclid, X. Prop. 2); but never as a definition of equal ratios.Google Scholar

⁵ In modern notation: A:B = C:D if both the ratios A:B and C:D are represented by the same finite or infinite continued fraction.

⁶ For more details on these subjects in Arabic Mathematics, see Plooij, E. B., Euclid's Conception of Ratio, and His Definition of Proportional Magnitudes as Criticized by Arabian Commentators (Rotterdam, 1950), ch. I, III, and IV;Google ScholarYouschkevitch, A. P., Les mathématiques arabes (Paris, 1976), pp. 80–90;Google Scholarand regarding the necessity for Arabian algebraists of an abstract concept of (irrational) number, Rashed, R., Entre arithmétique et algèbre(Paris, 1984), pp. 27, 32, 34, 35, 48, 250, 251.Google ScholarFor more details in Latin Mathematics, see SirHeath, Thomas L., The Thirteen Books of Euclid's Elements, II, 121;Google ScholarMurdoch, J. E., “The Medieval Language of Proportions,” in Crombie, A. C. (ed.), Scientific Change (Oxford, 1961), pp. 237–71; and Isaac Barrow's Mathematical Lectures (published under the title The Usefulness of Mathematical Learning by Frank Cass & Co. Ltd., 1970), Lectures XXI, XXII, and XXIII.Google Scholar

⁷ Not much is known of his life. Apparently, he lived in the 11th century. For biographical details, see Dold-Samplonius, Y. and Hermelink, H., “Al-Jayyānī”, in Dictionary of Scientific Biography (New-York, 1973), vol. VII, pp. 82–3;Google ScholarSabra, A. I., “The authorship of the Liber de crepusculis, an eleventh-century work on atmospheric refraction,” Isis, 58, 1 (1967): 77–85, see on pp. 84–5;Google Scholarand Smith, A.Mark, “The Latin version of Ibn Mu'ādh's treatise: “On Twilight and the Rising of Clouds”, ” Arabic Sciences and Philosophy, 2, 1 (1992): 83–116, see on pp. 83–4.CrossRefGoogle Scholar

⁸ Al-Jayyānī, Maqāla fī sharḥ al-nisba, MS Algiers 1446, fols. 74–82 (A facsimile of the manuscript is found in Plooij, Euclid's Conception of Ratio, ch. II, with an English translation on the opposite page).Google Scholar

⁹ See Introduction above.

¹⁰ That is, Euclid, V. Def. 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.

¹¹ Al-Jayyānī, Maqāla fī sharḥ al-nisba, p. 1 of the Arab text.Google Scholar

¹² Corresponding to Euclid, V. Def. 1: “A magnitude is a part of a magnitude, the less of the greater, when it measures the greater”; and to Euclid, V. Def. 2: “The greater is a multiple of the less when it is measured by the less.”

¹³ Corresponding to Euclid, X. Def. 1: “Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.”

¹⁴ A1-Jayyānī,Maqāla fī sharḥ al-nisba, p. 2 of the Arab text. The latter statement replaces Euclid, V. Def. 4: “Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.”Google Scholar

¹⁵ Corresponding to Euclid, V. Def. 3: “A ratio is a sort of relation in respect of size between two magnitudes of the same kind”; and to Euclid, V. Def. 6: “Let magnitudes which have the same ratio be called proportional.”

¹⁶ A1-Jayyānī, Maqāla fī sharḥ al-nisba, p.2 of the Arab text.Google Scholar

¹⁷ Ibid. In modern notation, by parts of a magnitude A, al-Jayyānī means the magnitude (m/n)A, where m, n are any positive integers.Google Scholar

¹⁸ A1-Jayyānī, Maqāla fī sharḥ al-nisba, pp. 2, 3 of the Arab text.Google Scholar

¹⁹ In modern notation: A:B ≠ C:D if there be two positive integers m, n such that A > (m/n)B and C < (m/n)D, or A = (m/n)B and C < (m/n)D; or such that A < (m/n)B and C gt; (m/n)D, or A = (m/n)B and C > (msol;n)D.+(m/n)B+and+C+<+(m/n)D,+or+A+=+(m/n)B+and+C+<+(m/n)D;+or+such+that+A+<+(m/n)B+and+C+gt;+(m/n)D,+or+A+=+(m/n)B+and+C+>+(msol;n)D.>Google Scholar

²⁰ A1-Jayyānī, Maqāla fī sharḥ al-nisba, p.3 of the Arab text.Google Scholar

²¹ Ibid.

²² For the meaning of the word ‘denomination” (or “name”), see Euclid, VII. Prop. 37–39.

²³ Al-Jayyānī, Maqāla fī sharḥ al-nisba, pp.4,5 of the Arab text. In modern notation: If A:B = C:D, then, whatever be the positive integers m, n, A < (m/n)B implies C < (m/n)D; A = (m/n)B implies C = (m/n)D; and A > (m/n)B implies C > (m/n)D.+(m/n)B+implies+C+>+(m/n)D.>Google Scholar

²⁴ Ibid., pp.5,6 of the Arab text. This converse is, in modern notation: Let A, B, C, D be four magnitudes such that whatever be the positive integers m, n, A < (m/n)B implies C < (m/n)D; A = (m/n)B implies C = (m/n)D; and A > (m/n)B implies C > (m/n)D. Then A:B = C:D. For if not, let E be such that A:B =E:D. Suppose for example E > C. Take two positive integers m, n such that C lt; (m/n)D < E. Then (M/N)B < A, because A:B = E:D (according to the previous proposition). So that m, n hace been found such that A > (m/n)B and C > (m/n)D, which contradicts the hupothesis.+(m/n)B+implies+C+>+(m/n)D.+Then+A:B+=+C:D.+For+if+not,+let+E+be+such+that+A:B+=E:D.+Suppose+for+example+E+>+C.+Take+two+positive+integers+m,+n+such+that+C+lt;+(m/n)D+<+E.+Then+(M/N)B+<+A,+because+A:B+=+E:D+(according+to+the+previous+proposition).+So+that+m,+n+hace+been+found+such+that+A+>+(m/n)B+and+C+>+(m/n)D,+which+contradicts+the+hupothesis.>Google Scholar

²⁵ Ibid., p. 6 of the Arab text.

²⁶ Ibid.

²⁷ Ibid.

²⁸ In modern notation: If A, B be two magnitudes, and m, n, positive integers, then A < (m/n)B is equivalent to nA < mB; A = (m/n)B to nA = mB; and A > (m/n)B to nA >mB.

²⁹ Al-Jayyānī, Maqāla fī sharḥ al-nisba, pp. 6–8 of the Arab text.Google Scholar

³⁰ Ibid., pp. 8–10 of the Arab text. In modern notation, this equivalence is an obvious consequence of the fact mentioned in footnote 28.

³¹ Ibid., pp. 10, 11 of the Arab text.

³² Ibid., p. 11 of the Arab text.

³³ Ibid. In modern notation: If A, B, C, and D are such that A > (m/n)B and C ≤ (m/n)D for two positive integers m, n, then A:B > C:D.

³⁴ In modern notation: If A:B > C:D, then positive integers m, n can be found such that A > (m/n)B and C ≤ (m/n)D.

³⁵ These are, in modern notation: 1. If A, B, C and D are such that whatever be the positive integers m, n, A > (m/n)B implies C > (m/n)D, and A < (m/n)B implies C < (m/n)D; then A:B = C:D. 2. If AB > C:D, then A < (m/n)B implies C < (m/n)D, whatever be the positive integers m, n. The demonstration of the above-mentioned converse is based on the fact that if A > (m/n)B implies C > (m/n)D whatever be the positive integers m, n, then A:B = C:D, which contradicts the hypothesis; so that some positive integers m, n, must be found such that A > (m/n)B and C > (m/n)D.

³⁶ Al-Jayyānī, Maqāla fī sharḥ al-nisba, pp. 11–14 of the Arab text.Google Scholar

³⁷ Saunderson, N., The Elements of Algebra, 2 vols. (Cambridge). The first volume is dated 1741, and the second 1740 (the two volumes are numbered consecutively).Google Scholar

³⁸ Ibid., pp. 439–46.

³⁹ Ibid., p. 439.

⁴⁰ Ibid., pp. 439, 440.

⁴¹ Of which he had previously demonstrated the incommensurability.

⁴² As a consequence of Euclid, I. Prop. 47.

⁴³ Saunderson, Algebra, p. 440.Google Scholar

⁴⁴ Ibid., pp. 440, 441.

⁴⁵ Ibid., p. 441.

⁴⁶ Ibid.

⁴⁷ Ibid., p. 442.

⁴⁸ Ibid., pp. 442, 443. This converse is demonstrated by supposing E to be a fourth proportional to A, B and D, so that A:B = E:D, and showing that the difference between C and E is less than (1/10)D, (1/100)D, (1/1000)D, (1/10000)D, and so on ad infinitum, so that C must be equal to E.

⁴⁹ Saunderson, Algebra, p. 443.Google Scholar

⁵⁰ Ibid.

⁵¹ Ibid., p. 444.

⁵² Ibid., pp. 443–5.

⁵³ In modern notation: If, for example, positive integers m, n can be found such that mA > nB but mC ≤ nD, then A:B > C:D; or such that mA < nB but mC ≥ nD, then A:B <C:D.

⁵⁴ In modern notation: If A:B < C:D, then there exists positive integers m, n such that mA < nB and mC ≤ nD. For if such positive integers did not exist, then necessarily either A:B = C:D, or A:B < C:D.

⁵⁵ Saunderson, , Algebra, p. 446.Google Scholar

⁵⁶ Which is, in modern notation: If A:B = C:D, then, for any positive integers m, n, it is not possible that A > = < (m/n)B, unless C > = < (m/n)D accordingly.+=+<+(m/n)B,+unless+C+>+=+<+(m/n)D+accordingly.>Google Scholar

⁵⁷ Which is, in modern notation: A:B ≠ C:D if there be two positive integers m, n such that A> (m/n)B and C ≤ (m/n)D, or A = (m/n)B and C ≥ (m/n)D; or such that A < (m/n)B and C ≥ (m/n)D, or A = (m/n)B and C > (m/n)D.+(m/n)B+and+C+≤+(m/n)D,+or+A+=+(m/n)B+and+C+≥+(m/n)D;+or+such+that+A+<+(m/n)B+and+C+≥+(m/n)D,+or+A+=+(m/n)B+and+C+>+(m/n)D.>Google Scholar

⁵⁸ See his Algebra, article 180, pp. 288–90.Google Scholar

⁵⁹ We will not, however, try to settle this matter, as the difficulty we have found in interpreting what Saunderson had exactly in mind when stating his principle could be due to a lack of understanding on our behalf.

⁶⁰ The following extract from Saunderson's comment on this definition makes it clear: “3 hath a greater ratio to 4 than 1 hath to 2, because 3 hath more magnitude in comparison of 4 than 1 hath in comparison of 2; for 3 is more than the half of 4, whereas 1 is but just the half of 2.” (Saunderson, Algebra, p. 448).Google Scholar

⁶¹ A1-Jayyānī, Maqāla fī sharḥ al-nisba, pp. 4–5 of the Arab text. The original text is:

⁶² Saunderson, Algebra, p. 442.Google Scholar

⁶³ A1-Jayyānī, Maqāla fī sharḥ al-nisba, p.5 of the Arab text. The original text is:

⁶⁴ Saunderson, Algebra, p. 442.Google Scholar

⁶⁵ Al-Jayyānī, Maqāla fī sharḥ al-nisba, p. 3 of the Arab text. The original text is: As to the translation we have given, the following remark should be made: the word is the nomen verbi (maṣdar) of the verb which means “to clarify” but also “to explain, to demonstrate,” so that the word means “clarification” but also “explanation, demonstration.” Moreover, is what grammarians call a specification (tamyīz). So that the words of the sentence which we have translated by “does not become clearer by adding more words to it,” mean literally: “it does not increase”; “by adding more words to it” “in regard to clarification” or “in regard to explanation” or “in regard to demonstration,” according to the translation we chose for the word. Consequently, a very literal translation of this passage could be: “it does not increase in explanation by adding more words to it” or “it does not increase in demonstration by adding more words to it.”

⁶⁶ Saunderson, Algebra, p. 442.Google Scholar

⁶⁷ Al-Jayyānī, Maqāla fī sharḥ al-nisba, p. 10 of the Arab text. The original text is: .

The word , which we have translated by “better,” could also be translated by “more elegant.”

⁶⁸ Saunderson, Algebra, p. 444.Google Scholar