AL-T??S?, SHARAF AL-D?N AL-MUZ?AFFAR IBN MU?AMMAD IBN AL-MUZ?AFFAR

(b. T??s [?]. Iran; d. Iran, ca. 1213/1214)

astronomy, mathematics.

The name of Sharaf al-D?n’s birthplace, T?s, refers both to a city and to its surrounding region, which with Mashhad and N?sh?pur formed a very prosperous area in the twelfth century.¹ A century earlier, T?s had given Islam one of its most profound thinkers, al-Ghaz?l? (d. 1111); and it was soon to produce a great astronomer and theologian, Nas??r al-D?n (d. 1274). Nothing is known about the first years of al-T??s?’s life; but it is reported that, faithful to the tradition of medieval scholars, he went on a long journey to some of the major cities of the time. His itinerary can be reconstructed from undated information preserved in biographies of his contemporaries.

Al-T??s? taught at Damascus, probably about 1165.² His most distinguished student there was Abu’l-Fadl (b. ca. 1135). an excellent carpenter who helped make the wood paneling of the B?m?rist?n al-N?r? (1154-1159) before discovering the joys of Euclid and Ptolemy.³ Al-T??s? most probably then stayed at Aleppo, where one of his pupils was a respected member of the city’s Jewish community, Abu’l-Fadl Biny?m?n (d. 1207/1208), whom he instructed in the science of numbers, the use of astronomical table, and astrology, and, at a less advanced level, in the other rational sciences.⁴ From the nature of these courses, it is reasonable to suppose that they lasted about three years.

Al-T??s?’s most outstanding pupil, however, was Kam?l al-D?n Ibn Y?nus (d. 1243) of Mosul, through whom al-T??s?’s teachings passed to Nas?r al-D?n and Ath?r al-D?n al-Abhar? (d. 1263/1265).⁵ Al-T??s? was apparently in Mosul in the years preceding 1175, ⁶ for around this date two physicians from Damascus went there to study with him, but he had already left. ⁷ One of them then went to the neighboring city of irbil, where he became a pupil of Ibn al-Dahh?n.⁸ About this time, however, the latter left Irbil to join Saladin, who had just seized Damascus (1174). ⁹ Al-T??s? returned to Iran, where he died around 1213, at an advanced age.

Al-T??s? is known for his linear astrolabe (al-T??s?’s staff), a simple wooden rod with graduated markings but without sights. It was furnished with a plumb line and a double cord for making angular measurements and bore a perforated pointer. This staff reproduced, in concrete form, the meridian line of the plane astrolabe-that is, the line upon which the engraved markings of that instrument are projected. (These markings are of stars, circles of declination, and heights.) Supplementary scales indicate the right ascensions of the sun at its entry into the signs of the zodiac as well as the hourly shadows. Al-T??s? described the construction and use of the linear astrolabe in several treatises, praising its simplicity and claiming that an amateur could build it in about an hour. His staff made it possible to carry out the observations used to determine the height of the stars, the time, the direction of the Ka’ba, and the ascendants. The instrument, although inexpensive to construct, was less accurate than the ordinary astrolabe. It also was less decorative, and perhaps for this reason it was of little interest to collectors. In any case, not a single linear astrolabe has survived.¹⁰

Al-T??s?’s greatest achievement is recorded in a work that has not yet been analyzed by historians, the manuscript Loth III, 767, in the collection of the India Office, London. This manuscript is actually a reworking of the original by an unknown author who proudly states that he has eliminated the mathematical tables and shortened some of the long passages. He makes no further claims; and even if he had wished to make more substantial changes, the great difficulty of the work would have discouraged him. The entire contents of the work may, therefore, confidently be attributed to al-T??s?. The treatise, which may have been mentioned by al-Sinj?r?,¹¹ is not the first of its kind by an Arab author. A cross check of citations from Jamsh?d al-K?sh? and T?sh Kopru Z?deh reveals that al-Mas’?d?, a disciple of al-Khayy?m?, wrote on the numerical solution of third-degree equations.¹² The existence of an earlier author is not explicity indicated, but, about 1350, Yahy? al-K?sh? noted several similar writings, without specifiying dates or names.¹³ In the following paragraphs we shall present the most remarkable results in al-T?s?’s treatise, but we cannot state the degree of originality for each.

The treatise divides the twenty-five equations of degree n ? 3 into three groups. The first includes twelve equations: those of degree n ?2 or that reduce to that degree, plus the equation x³=a. The second contains the eight equations of the third degree that always admit one (positive) solutions.¹⁴ The third group is composed of the five equations that can give rise to impossible soluitons:¹⁵

x³+c=ax²

x³+c=b²x

x³+3ax²+c=3b²x

x³+b²x+c=3ax²

x³+c=3ax²+3b²x

We shall not give details of the geometric solutions, since they do not differ from those presented by al-Khayy?m?. (The care that al-T??s? bestows on the study of the problem of the relative position of two conics is, however, worth noting.) On the other hand, the outstanding discussion of the existence of the roots of the group of equations that can give rise to impossible solutions merits the closest examination. Accordingly, we shall outline, by way of example, al-T??s?’s treatment of the fourth equation of this group, which, like the others, is based on the calculation of a maximum. Given that x³ < 3 ax²; therefore x< (3 a. Then b²x < x²(3 a - x), so that b² < x(3 a - x. The maximum of x(3 a – x) is (3 a/2) ² · ¹⁶ Therefore b < 3 a /;2. We consider x² + b² / 3=2 ax and take its root . A discussion of its existence does not arise, since b < 3 a/2. We form f(x ₁)= x₁² (3 a - x₁) - b²x₁. If f (x₁)= c, the equation x³ + b³x - c =3 ax² has a solution x = x₁. If f (x₁) < c, there is no soluion. If f (x ₁) > c, the equation has two roots separated by x₁. Turining to an evaluation of al-T??s?’s treatment in the light of the diffrential calculus, we set f(x) =3 ax² –x³–b² x; then f ’(x)=6 ax -3 x²-b². Thus f ’ (x) reduces to zero when x²-2 ax + b²/3=0. Accordingly. the roots x ₀ and x ₁ are equal to . Finally, f(x₁) > 0 implies b < 3a/2.

The text does not say what led al-T?s? to such profound and beautiful results. The idea of determining the maximum of x₂(a-x), x(b²-x₂), ... might have been suggested by the solution of x (a-x)=b₂. The value of the maximum of x₂ (a-x) might have been borrowed from Archimedes, who, unlike al-T?s?, established it geometrically, ¹⁷ Yet, even if al-T?s? started from this point, he still had far to go. Pursuing his solution of the equation x³+bx²+c=3ax², he shows that the two solutions are, respectively, x¹1 + X, where X is the root of X³+3(x₁-a) X= f(x₁)-c, and x₁-x, where X is the root of X³ + f(x₁)-c=3(X₁- a) X. This method contains the genesis of a genuine change of variables, and one must admire the author’s intention of interrelating the various equations—an approach quite different from traditional Arab thinking on this topic, which emphasized independent solutions of problems (as in the classic solution of the second degree equations).

We shall conclude with a very schematic presentation of al-T??s?’s soluion of the equaltion x³ + 3ax=N, using the example x³+36x=91,750,087.· ¹⁸ Let x₁ be the number in the hunnreds’ place of the root; then x₁³ will represent millions and 3ax₁will represent hundreds· Therefore, we place x₁ in the millions’ box (the upper line in Table I) and a=12 in the hundreds’ box (on the lower line; actually, since a is grater then nine, it is carried over

into that of the thousands). We then calculate the greatest x₁ such that x³₁ ? 91; this yields x₁=4. We remove x³₁ +36x₁from N, obtaining N 27,735,687. We next place x²₁=16 under x₁ in the line containing a and decrease the lower line by one rank and x₁ by two. The result is Table II.

We now calculate the figure in the tens’ place. It will be the greatest x₂ such that 3x₂ multiplied by 16 can be subtracted from 277. Accordingly, x₂=5, and we place it to the right of 4 in the upper line. In the lower line we put x₁x₂ in the position under x₁ =4. We then subtract from N₁ the total of x³₂ and the product of 3x₂ times the lower line — that is, 3x₂(x²₁ + x₁x₂ + a) = 15(180,012). This yields N₂. We add x₁x₂ (that is, 20) to the lower line in the position under x₁=4 and x²₂ = 25 in the position under x₂ = 5. the line becomes 202, 512. We decrease it by one rank and decrease the upper line by two. The result is Table III.

Finally, we calculate x₃ such that the product of 3x₃ times 20?;60. Thus x₃ = 1. We place it to the right of 5. To the lower line we add 45 and subtract from N₂ the total of x³₃ and the product of 3x₃ times the lower line (202,962). The remainder is 0. The root of the equation is therefore 451. The method is independent of the system of numeration and permits as close an approximation of the root as desired; it suffices to add a row of three to the last remainder and to continue operating in the same manner. The treatise also gives analogous methods of numerical resolution for the other equations, even for those of the second degree.

NOTES

1. Guy Le Strange, The lands of the Eastern Caliphate (Cambridge, 1909). See the chapter on Khur?s?n (with references to the Arab geographers).

2. See Ibn Ab? Usaybi’a. ‘Uy?n al-anb?, II, 190–191.

3. This was a hospital built by Sultan N?r al-D?n ibn Zenki, famous for his wars against the Crusaders. See Ibn al-Athir, al-T?r?kh al-B?hir fi’l dawl l-at?bikiyya, A.A. Tualym?;t. ed. (Cairo. 1963). 170; and Shawkat al-Shatt?;. M?ujaz t?r?kh al-tibb’ind al-’Arab (Damascus. 1959). 22. See also Ibn Ab? Usaybi’a. loc. cit

4. Ibnl al-Qift?. T?rikh (Caito. 1948). 278.

5. See Ibn Khallih?n. Wafay?t al-a’yan. IV. no. 718: and G. Sarton. Introduction to the History of Science. II. 600. and II. pt. 2. 100–1013.

6. In 1193 Ibn Y?nus went to Baghdad to continue his religious studies; see Ibn Khallik?n, loc. cit. See also T?sh Kopru Z?deh, Mift?h al-sa’?da. 11 , 214–215.

7. They were Ibn al-H?jib and Muwaffaq al-D?;n. See Ibn Ab? Usaybi’a. II. 181–182. 191–192.

8.Ibid.

9. See Ibn Khallik?n. IV. no. 655.

10. See Henri Michel. Traité de l’astrolabe, 22. the same point is also made in L.A. Mayer, Islamic Astrolabists and Their Works (Geneva. 1956).

11. Al-Sinj?r?. Irsh? al-q?sid (Beirut. 1904). 124. Although probably valid, the citation raises some doubt. In fact, the title, Kit?b al-Muzaffar al-T?s?, becomes, in certain editions of T?sh Kopru Z?deh’s Mift?h al-sa’?da (for instance, 1,327) which, however, derive from al-Sinj?r?: Kit?b al-Zafar of al-T?s? (nas?r al-D?n).

12. Jamsh?d al-K?sh?. Miftah al-his?b, MS Paris Ar. 5020. fol. 98; and T?sh Korpy Z?deh, Miftah al-sa’ada, I. 327. Sharafal-D?n Muhammad ibn Mas’?d ibn MUhammad al-Mas’?d? is cited in the article on Muhammad ibn Ahmad al-Shur-w?ni in Safad?, al-W?f?, Ritter. ed. (Istanbul), II, 497, as having taught the Ish?r?r of Ibn S?n? to Fakhr al-D?n al-Z?zi (1164–1238) after having stidied under al-Khay-y?m. He is the author of al-Kif?ya fi’l-hid?ya; see H?jj? Khal?fa. Kashf al-Zun?n, II. col. 1500. Khal?fa also cites his algebra (1, col, 857).

13. yahya al-K?shi, al-Lub?b fi’l-His?b. Aya Sofya MS 2757. See fol. 65r, 1. 21’ fol, 65v. 1.3: and fol. 67r. 1. 25. The MS, written in 1373, bears notes in the authors hand. See the article on al-K?sh? in Sarton, Introduction to the History of Science, III, pt. 1, 698.

14. Only x³ + a x = b x² + c can admit up to three positive solutions.

15.Kit?b fi’l-jabr wa’l-muq?bala. India office (London), Loth 767. the equations are found on pp. 101r– 112r; 112r– 121r; 121r– 130r; 130r– 142v; and 142v– 179r.

16. This is an immediate cpnsequence of Eucid’s Elements, ii , 5.

17. T.L. Heath, The Works of Archimedes (New York, 1953), 67– 72.

18. In the treatise (folw. 54v– 55v) the equation actually solved is x³+ 36x = 33,087,717. the root of which is 321.

BIBLIOGRAPHY

I. Original Works. Al-T??s?’s works include the following.

1. Kit?b fi’l jabr wa’l muq?bala, Indian Office (London), Loth 767.

2. Ris?la fi’l-asturl?b al-khatt?. British Museum, Or. 5479.

3. Ma’rifat al-asturl?b al-musattah wa’l-’amal bihi. Leiden 1082. The MS does not bear this title, which was erroneously given to it by some bibliographers, and discusses the linear astrolabe, not the plane astrolabe. The third part, containing demonstrations, is missing from the MS.

4. Kit?b fi ma’rifat al astrul?b al-musattah wa’l mal bihi, Seray 3505, 2nd. If Max Krause’s identifications of this MS with Leiden 1082 is correct, it would be necessary to conclude that we do not have al-T?s?’s treatise on the plane astrolabe.

5. Ris?la fi’l-astrul?b al-Khatt?, Seray 3342, 7.

6. Ris?la fi’l-asturl?b al-Khatt?, Seray 3464, 1.

7. Jaw?b ’al? su’?l li’am?r al-umar?’ Shams al-D?n.

Leiden 1027; Columbia University, Smith, Or. 45, 2. This work concerns the division of a square into three trapezoids and a rectangle, with the relationships preassigned.

8. Fi’l-Khattayn alladhayn yaqrub?n wa la yaltaqiy?n, Aya Sofya 2646, 2, 71r–v, deals with the existence of an asymptote to the (equilateral) hyperbola and contains the same demonstration as in Kit?b fi’l-jabr wa’l-muq?bala, (1), fols. 38r–40r.

II. Secondary Literature. See the following:

9. Ibn Khallik?n, Wafay?;t al-a’y?;n (Cairo, 1948).

10. Ibn Ab? Usaybi’a, ’Uy?n al-anb?;’ (Cairo, 1882).

11. T?;sh Kopru Z?;deh, Mift?;h al-sa’?;da (Hyderabad, 1910-1911).

12. H??jj? Khal?fa, Kashf al-zun?n (Istanbul, 1941-1943).

13. H. Suter, Die Mathematiker und Astronomen der Araber (Leipzig, 1900), 134 (no. 333).

14. Max Krause,“Stambuler Handschriften islamischer Mathematiker,”in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abt. B, Studien, 3 (1936), 437–432, see 490.

15. C. Brockelmann, Geschichte der arabischen Literatur, I, 2nd ed. (Leiden, 1943), 472, and supp. I (Leiden, 1937), 858.

16. G. Sarton, Introduction to the History of Science, II, pt. 2 (Baltimore, 1950), 622–623.

17. Carlo Nallino, article on the astrolabe (asturl?;b) in Encyclopaedia of Islam, 1st ed., I (1913); and by Willy Hartner, ibid., 2nd ed., I, 722-728.

18. Henri Michel, Traité de l’astrolabe (Paris, 1947), 115-122; and“L’astrolabe linéaire d’al-T??s?,”in Ciel et terre (1943), nos. 3-4. A description, sketch, and note on the use of al-T?s?’s linear astrolabe can be found on p. 21.

19. R. Carra de Vaux,“L’astrolabe linéaire ou bâton d’al-Tousi,”in Journal asiatique, 11th ser., 5 (1895), 464-516. This article reproduces the text of al-Hasan al Marr?;kush? with a French translation.

Adel Anbouba