# Al-T?usi, Sharaf Al-din Al-muz?affar Ibn Mu?ammad Ibn Al-Muz?affar

# AL-ṬūSī, SHARAF AL-DīN AL-MUẒAFFAR IBN MUḥAMMAD IBN AL-MUẒAFFAR

(*b.* Ṭūs [?]. Iran; *d.* Iran, *ca.* 1213/1214)

*astronomy, mathematics*.

The name of Sharaf al-Dīn’s birthplace, Tūs, refers both to a city and to its surrounding region, which with Mashhad and Nīshāpur formed a very prosperous area in the twelfth century.^{1} A century earlier, Tūs had given Islam one of its most profound thinkers, al-Ghazālī (*d.* 1111); and it was soon to produce a great astronomer and theologian, Naṣīr al-Dīn (*d.* 1274). Nothing is known about the first years of al-Ṭūsī’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-Ṭūsī 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 Bīmāristān al-Nūrī (1154-1159) before discovering the joys of Euclid and Ptolemy.^{3} Al-Ṭūsī most probably then stayed at Aleppo, where one of his pupils was a respected member of the city’s Jewish community, Abu’l-Fadl Binyāmīn (*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-Ṭūsī’s most outstanding pupil, however, was Kamāl al-Dīn Ibn Yūnus (*d.* 1243) of Mosul, through whom al-Ṭūsī’s teachings passed to Nasīr al-Dīn and Athīr al-Dīn al-Abharī (*d.* 1263/1265).^{5} Al-Ṭūsī 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-Dahhān.^{8} About this time, however, the latter left Irbil to join Saladin, who had just seized Damascus (1174). ^{9} Al-Ṭūsī returned to Iran, where he died around 1213, at an advanced age.

Al-Ṭūsī is known for his linear astrolabe (al-Ṭūsī’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-Ṭūsī 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-Ṭūsī’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-Ṭūsī. The treatise, which may have been mentioned by al-Sinjārī,^{11} is not the first of its kind by an Arab author. A cross check of citations from Jamshīd al-Kāshī and Tāsh Kopru Zādeh reveals that al-Mas’ūdī, a disciple of al-Khayyāmī, wrote on the
numerical solution of third-degree equations.^{12} The existence of an earlier author is not explicity indicated, but, about 1350, Yahyā al-Kāshī noted several similar writings, without specifiying dates or names.^{13} In the following paragraphs we shall present the most remarkable results in al-Tūsī’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 *x ^{3}=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}

*x*^{3}+*c*=*ax*^{2}

*x*^{3}+*c*=*b*^{2}*x*

*x*^{3}+3*ax*^{2}+*c*=3*b*^{2}*x*

*x*^{3}+*b*^{2}*x*+*c*=3*ax*^{2}

*x*^{3}+*c*=3*ax*^{2}+3*b*^{2}*x*

We shall not give details of the geometric solutions, since they do not differ from those presented by al-Khayyāmī. (The care that al-Ṭūsī 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-Ṭūsī’s treatment of the fourth equation of this group, which, like the others, is based on the calculation of a maximum. Given that *x*^{3} < 3 *ax*^{2}; therefore *x*< (3 *a*. Then *b*^{2}*x* < *x*^{2}(3 *a* - *x*), so that *b*^{2} < *x*(3 *a* - *x*. The maximum of *x*(3 *a* – *x*) is (3 *a*/2) ^{2} · ^{16} Therefore *b* < 3 *a* /;2. We consider *x*^{2} + *b*^{2} / 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})= *x*_{1}^{2} (3 *a* - *x*_{1}) - *b*^{2}*x*_{1}. If *f* (*x*_{1})= *c*, the equation *x*^{3} + *b*^{3}*x* - *c* =3 *ax*^{2} has a solution *x* = *x*_{1}. If *f* (*x*_{1}) < *c*, there is no soluion. If *f* (x _{1}) > *c*, the equation has two roots separated by x_{1}. Turining to an evaluation of al-Ṭūsī’s treatment in the light of the diffrential calculus, we set *f(x)* =3 *ax*^{2} –x^{3}–b^{2} x; then *f* ’(x)=6 *ax* -3 *x*^{2}-b^{2}. Thus *f* ’ (x) reduces to zero when x^{2}-2 *ax* + b^{2}/3=0. Accordingly. the roots x _{0} and x _{1} are equal to
. Finally, *f*(x_{1}) > 0 implies b < 3*a*/2.

The text does not say what led al-TŪsĪ to such profound and beautiful results. The idea of determining the maximum of *x*_{2}(a-x), *x(b ^{2}-x_{2}),* ... might have been suggested by the solution of

*x*(a-x)=b

_{2}. The value of the maximum of

*x*

_{2}(a-x) might have been borrowed from Archimedes, who, unlike al-TŪsĪ, established it geometrically,

^{17}Yet, even if al-TŪsĪ started from this point, he still had far to go. Pursuing his solution of the equation x

^{3}+bx

^{2}+c=3ax

^{2}, he shows that the two solutions are, respectively, x

^{1}1 + X, where X is the root of X

^{3}+3(x

_{1}-a) X=

*f*(x

_{1})-c, and x

_{1}-x, where X is the root of X

^{3}+ f(x

_{1})-c=3(

*X*

_{1}-

*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-Ṭūsī’s soluion of the equaltion x^{3} + 3*ax*=*N*, using the example *x*^{3}+36*x*=91,750,087.· ^{18} Let x_{1} be the number in the hunnreds’ place of the root; then x_{1}^{3} will represent millions and 3ax_{1}will represent hundreds· Therefore, we place x_{1} 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

Table I | ||||||||

x_{1} | 4 | |||||||

N | 9 | 1 | 7 | 5 | 0 | 0 | 8 | 7 |

N_{1} | 6 | 4 | 1 | 4 | 4 | |||

2 | 7 | 7 | 3 | 5 | 6 | 8 | 7 | |

a | 1 | 2 | ||||||

Table II | ||||||||

4 | 5 | |||||||

N_{1} | 2 | 7 | 7 | 3 | 5 | 6 | 8 | 7 |

1 | 2 | 5 | ||||||

2 | 7 | 0 | 0 | 1 | 8 | 0 | ||

N_{2} | 6 | 0 | 8 | 8 | 8 | 7 | ||

1 | 6 | 1 | 2 | |||||

2 | 0 |

Table III | ||||||||

N_{2} | 6 | 0 | 8 | 8 | 8 | 7 | ||

1 | ||||||||

6 | 0 | 8 | 8 | 8 | 6 | |||

2 | 0 | 2 | 5 | 1 | 2 | |||

4 | 5 |

into that of the thousands). We then calculate the greatest x_{1} such that *x*^{3}_{1} ≤ 91; this yields *x*_{1}=4. We remove *x*^{3}_{1} +36*x*_{1}from *N*, obtaining *N* 27,735,687. We next place *x*^{2}_{1}=16 under *x*_{1} in the line containing *a* and decrease the lower line by one rank and *x*_{1} by two. The result is Table II.

We now calculate the figure in the tens’ place. It will be the greatest *x*_{2} such that 3*x*_{2} multiplied by 16 can be subtracted from 277. Accordingly, *x*_{2}=5, and we place it to the right of 4 in the upper line. In the lower line we put *x*_{1}*x*_{2} in the position under *x*_{1} =4. We then subtract from *N*_{1} the total of *x*^{3}_{2} and the product of 3*x*_{2} times the lower line — that is, 3*x*_{2}(*x*^{2}_{1} + *x*_{1}*x*_{2} + *a*) = 15(180,012). This yields *N*_{2}. We add *x*_{1}*x*_{2} (that is, 20) to the lower line in the position under *x*_{1}=4 and *x*^{2}_{2} = 25 in the position under *x*_{2} = 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 *x*_{3} such that the product of 3*x*_{3} times 20≤;60. Thus *x*_{3} = 1. We place it to the right of 5. To the lower line we add 45 and subtract from *N*_{2} the total of *x*^{3}_{3} and the product of 3*x*_{3} 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 Khurāsān (with references to the Arab geographers).

2. See Ibn AbĨ Usaybi’a. ‘*Uyūn al-anbā*, II, 190–191.

3. This was a hospital built by Sultan Nūr al-Dīn ibn Zenki, famous for his wars against the Crusaders. See Ibn al-Athir, *al-Tārīkh al-Bāhir fi’l dawl l-atābikiyya*, A.A. Tualymā;t. ed. (Cairo. 1963). 170; and Shawkat al-Shattā;. *Mūujaz tārīkh al-tibb’ind al-’Arab* (Damascus. 1959). 22. See also Ibn Abī Usaybi’a. *loc. cit*

4. Ibnl al-Qiftī. *Tārikh* (Caito. 1948). 278.

5. See Ibn Khallihān. *Wafayāt 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 Yānus went to Baghdad to continue his religious studies; see Ibn Khallikān, *loc. cit*. See also Tāsh Kopru Zādeh, Miftāh al-sa’āda. **11** , 214–215.

7. They were Ibn al-Hājib and Muwaffaq al-Dā;n. See Ibn Abī Usaybi’a. II. 181–182. 191–192.

8.*Ibid*.

9. See Ibn Khallikān. 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-Sinjārī. *Irshā al-qāsid* (Beirut. 1904). 124. Although probably valid, the citation raises some doubt. In fact, the title, Kitāb al-Muzaffar al-Tāsī, becomes, in certain editions of Tāsh Kopru Zādeh’s Miftāh al-sa’āda (for instance, 1,327) which, however, derive from al-Sinjārī: *Kitāb al-Zafar of al-Tūsī* (nasīr al-Dīn).

12. Jamshīd al-Kāshī. *Miftah al-hisāb*, MS Paris Ar. 5020. fol. 98; and Tāsh Korpy Zādeh, *Miftah al-sa’ada*, I. 327. Sharafal-Dīn Muhammad ibn Mas’ūd ibn MUhammad al-Mas’ūdī is cited in the article on Muhammad ibn Ahmad al-Shur-wāni in Safadī, *al-Wāfī*, Ritter. ed. (Istanbul), II, 497, as having taught the Ishārār of Ibn Sīnā to Fakhr al-Dīn al-Zāzi (1164–1238) after having stidied under al-Khay-yām. He is the author of al-Kifāya fi’l-hidāya; see Hājjī Khalīfa. *Kashf al-Zunūn*, II. col. 1500. Khalīfa also cites his algebra (1, col, 857).

13. yahya al-Kāshi, al-Lubāb fi’l-Hisāb. 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-Kāshī in Sarton, *Introduction to the History of Science*, III, pt. 1, 698.

14. Only *x*^{3} + *a x* = *b x*^{2} + *c* can admit up to three positive solutions.

15.*Kitāb fi’l-jabr wa’l-muqābala*. 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 *x*^{3}+ 36*x* = 33,087,717. the root of which is 321.

## BIBLIOGRAPHY

I. Original Works. Al-Ṭūsī’s works include the following.

1. *Kitāb fi’l jabr wa’l muqābala*, Indian Office (London), Loth 767.

2. *Risāla fi’l-asturlāb al-khattī*. British Museum, Or. 5479.

3. *Ma’rifat al-asturlāb 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. *Kitāb fi ma’rifat al astrulāb 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-Tūsī’s treatise on the plane astrolabe.

5. *Risāla fi’l-astrulāb al-Khattī*, Seray 3342, 7.

6. *Risāla fi’l-asturlāb al-Khattī*, Seray 3464, 1.

7. *Jawāb ’alā su’āl li’amīr al-umarā’ Shams al-Dīn*.

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 yaqrubān wa la yaltaqiyān*, Aya Sofya 2646, 2, 71r–v, deals with the existence of an asymptote to the (equilateral) hyperbola and contains the same demonstration as in *Kitāb fi’l-jabr wa’l-muqābala*, (1), fols. 38r–40r.

II. Secondary Literature. See the following:

9. Ibn Khallikān, *Wafayā;t al-a’yā;n* (Cairo, 1948).

10. Ibn Abī Usaybi’a, ’*Uyūn al-anbā*;’ (Cairo, 1882).

11. Tā;sh Kopru Zā;deh, *Miftā;h al-sa’ā;da* (Hyderabad, 1910-1911).

12. Ḥājjī Khalīfa, *Kashf al-zunūn* (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 (*asturlā;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-Ṭūsī,”in *Ciel et terre* (1943), nos. 3-4. A description, sketch, and note on the use of al-Tūsī’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 Marrā;kushī with a French translation.

Adel Anbouba

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