Al-K?sh

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Al-K?sh? (or Al-K?sh?n?), Ghiy?th Al-D?n Jamsh?d Mas’?d

(b. K?sh?n, Iran; d. Samarkand [now in Uzbek, U.S.S.R], 22 June 1429)

astronomy, mathematics.

The biographical data on al-K?sh? are scattered and sometimes contradictory. His birthplace was a part of the vast empire of the conqueror Tamerlane and then of his son Sh?h Rukh. The first known date concerning al-K?sh? is 2 June 1406 (12 Dh??l-Hijja, A.H. 808), when, as we know from his Khaq?n? z?j, he observed a lunar eclipse in his native town. ¹ According to Suter, al-K?sh? died about 1436; but Kennedy, on the basis of a note made on the title page of the India Office copy of the Khaq?n?i z?j, gives 19 Ramad??n A.H. 832, or 22 June 1429². The chronological order of al-K?sh?’s works written in Persian or in Arabic is not known completely, but sometimes he gives the exact date and place of their completion. For instance, the Sullam al-sam?? (“The Stairway of Heaven”), a treatise on the distances and sizes of heavenly bodies, dedicated to a vizier designated only as Kam?l al-D?n Mahm?d, was completed in K?sh?n on 1 March 1407.³ In 1410–1411 al-K?sh? wrote the Mukhtas?ar dar ilm-ihay?at (“Compendium of the Science of Astronomy”) for Sultan Iskandar, as is indicated in the British Museum copy of this work. D. G. Voronovski identifies Iskandar with a member of the T?m?rid dynasty and cousin of Ulugh B?g, who ruled Fars and Is?fah?n and was executed in 1414.⁴ In 1413–1414 al-K?sh? finished the Khaq?n? z?j. Bartold assumes that the prince to whom this z?j is dedicated was Sh?h Rukh, who patronized the sciences in his capital, Herat;⁵ but Kennedy established that it was Sh?h Rukh’s son and ruler of Samarkand, Ulugh B?g. According to Kennedy, in the introduction to this work al-K?sh? complains that he had been working on astronomical problems for a long time, living in poverty in the towns of Iraq (doubtless Persian Iraq) and mostly in K?sh?n. Having undertaken the composition of a z?j, he would not be able to finish it without the support of Ulugh B?g, to whom he dedicated the completed work.⁶ In January 1416 al-K?sh? composed the short Ris?la dar shar?-i ?l?t-i ras?d (“Treatise on . . . Observational Instuments”), dedicated to Sultan Iskandar, whom Bartold and Kennedy identify with a member of the K?r? Koyunl?, or Turkoman dynasty of the Black Sheep.⁷ Shishkin mistakenly identifies him with the above-mentioned cousin of Ulugh B?g.⁸ At almost the same time, on 10 February 1416, al-K?sh? completed in K?sh?n Nuzha al-?ad?iq (“The Garden Excursion”), in which he described the “Plate of Heavens,” an astronomical instrument he invented. In June 1426, at Samarkand, he made some additions to this work.

Dedicating his scientific treatises to sovereigns or magnates, al-K?sh?, like many scientists of the Middle Ages, tried to provide himself with financial protection. Although al-K?sh? had a second profession—that of a physician—he longed to work in astronomy and mathematics. After a long period of penury and wandering, al-K?sh? finally obtained a secure and honorable position at Samarkand, the residence of the learned and generous protector of science and art, Sultan Ulugh B?g, himself a great scientist.

In 1417–1420 Ulugh B?g founded in Samarkand a madrasa—a school for advanced study in theology and science—which is still one of the most beautiful buildings in Central Asia. According to a nineteenth century author, Ab? T?hir Khw?ja, “four years after the foundation of the madrasa,” Ulugh B?g commenced construction of an observatory; its remains were excavated from 1908 to 1948.⁹ For work in the madrasa and observatory Ulugh B?g took many scientists, including al-K?sh?, into his service. During the quarter century until the assassination of Ulugh B?g in 1449 and the beginning of the political and ideological reaction, Samarkand was the most important scientific center in the East. The exact time of al-K?sh?’s move to Samarkand is unknown. Ab? T??hir Khaw?ja states that in 1424 Ulugh B?g discussed with al-K?sh?, Q?d?? Z?de al-R?m?, and another scientist from K?sh?n, Mu?in al-D?n, the project of the observatory.¹⁰

In Samarkand, al-K?sh? actively continued his mathematical and astronomical studies and took a great part in the organization of the observatory, its provision with the best equipment, and in the preparation of Ulugh B?g’s Zij, which was completed after his (al-K?sh?’s) death. Al-K?sh? occupied the most prominent place on the scientific staff of Ulugh B?g. In his account of the erection of the Samarkand observatory the fifteenth-century historian Mirkhw?nd mentions, besides Ulugh B?g, only al-K?shi, calling him “the support of astronomical science”and “the second Ptolemy.”¹¹ The eighteenth-century historian Sayyid Raq?m, enumerating the main founders of the observatory and calling each of them maulan? (“our master,” a usual title of scientists in Arabic), calls al-K?sh? maulan?-i ?lam (maulan? of the world).¹²

Al-K?sh? himself gives a vivid record of Samarkand scientific life in an undated letter to his father, which was written while the observatory was being built. Al-K?sh? highly prized the erudition and mathematical capacity of Ulugh B?g, particularly his ability to perform very difficult mental computations; he described the prince’s scientific activity and once called him a director of the observatory.¹³ Therefore Suter’s opinion that the first director of the Samarkand observatory was al-K?sh?, who was succeeded by Q?d?? Z?de, must be considered very dibious.¹⁴ On the other hand, al-K?sh? spoke with disdain of Ulugh B?g’s nearly sixty scientific collaborators, although he qualified Q?di Z?de as “the most learned of all.”¹⁵Telling of frequent scientific meetings directed by the sultan, al-K?sh?gave several examples of astronomical problems propounded there. These problems, too difficult for others, were solved easily by al-K?sh?. In two cases he surpassed Q?d?? Z?de, who misinterpreted one proof in al-B?r?n?’s al-Q?n?n al-Mas??d? and who was unable to solve one difficulty connected with the problem of determining whether a given surface is truly plane or not. Nevertheless his relations with Q?d?i Z?de were amicable. With great satisfaction al-K?sh? told his father of Ulugh B?g’spraise, related to him by some of his friends. He emphasized the atmosphere of free scientific discussion in the presence of the sovereign. the letter included interesting information on the construction of the observatory building and the instruments. This letter and other sources characterize al-K?sh?as the closest collaborator and consultant of Ulugh B?g, who tolerated al-K?sh?’s ignorance of court etiquette and lack of good manners.¹⁶ In the introduction to his own Zij Ulugh B?g mentions the death of al-K?sh? and calls him “a remarkable scientist, one of the most famous in the world, who had a perfect command of he science of the ancients, who contributed to its development, and who could solve the most difficult problems.”¹⁷

Al-K?sh? wrote his most important works in Samarkand. In July 1424 he completed Ris?la al-mu?it?iyya (“The Treatise on the Circumference”), masterpiece of computational technique resulting in a the determination of 2? to sixteen decimal places. On 2 March 1427 he finished the textbook Mift?? al-?is?b (“The Key of Arithmetic”), dedicated to Ulugh B?g. It is not known when he completed his third chef d’oeuvre, Ris?la al-water wa?l-jaib (“The Treatise on the Chord and Sine”), in which he calculated the sine of 1° with the same precision as he had calculated ?.Apparently he worked on this shortly before his death; some sources indicate that the manuscript was incomplete when he died and that it was finished by Q?d?? Z?de.¹⁸ Apparently al-K?sh? had developed his method of calculation of the sine of 1° before he completed Mift?? al-?is?b, for in the introduction to this book, listing his previous works, he mentions Ris?la al-watar wa?l-jaib.

As was mentioned above, al-K?sh? took part in the composition of Ulugh B?g’s Z?j. We cannot say exactly what he did, but doubtless his participation was considerable. The introductory theoretical part of the Z?j was completed during al-K?sh?’s lifetime, and he translated it from Persian into Arabic.¹⁹

Mathematics. Al-K?sh?’s best-known work is Mift?? al-?is?b (1427), a veritable encyclopedia of elementary mathematics intended for an extensive range of students; it also considers the requirements of calculators—astronomers. land surveyors, architects, clerks, and merchants. In the richness of its contents and in the application of arithmetical and algebraic methods to the solution of various problems, including several geometric ones, and in the clarity and elegance of exposition, this voluminous textbook is one of the best in the whole of medieval literature; it attests to both the author’s erudition and his pedagogic ability.²⁰ Because of its high quality the Mift?? al-?is?b was often recopied and served as a manual for hundreds of years; a compendium of it was also used. The book’s title indicates that arithmetic was viewed as the key to the solution of every kind of problem which can be reduced to calculation, and al-K?sh? defined arithmetic as the “science of rules of finding numerical unknowns with the aid of corresponding known quantities.”²¹ The Mift?? al-?is?b is divided into five books preceded by an introduction: “On the Arithmetic of Integers,” “On the Arithmetic of Fractions,” “On the ‘Computation of the Astronomers’”(on sexagesimal arithmetic), “On the Measurement of Plane Figures and Bodies,” and “On the Solution of Problems by Means of Algebra [linear and quadratic equations] and of the Rule of Two False Assumptions, etc.” The work comprises many interesting problems and carefully analyzed numerical examples.

In the first book of the Mift??, al-K?sh? describes in detail a general method of extracting roots of integers. The integer part of the root is obtained by means of what is now called the Ruffini—Horner method. If the root is irrational, (a and r are integers), the fractional part of the root is calculated according to the approximate formula ²²Al-K?sh? himself expressed all rules of computation in words, and his algebra is always purely “rhetorical.” In this connection he gives the general rule for raising a binomial to any natural power and the additive rule for the successive determination of binomial coefficients; and he constructs the so-called Pascal’s triangle (for n = 9). The same methods were presented earlier in the J?mi?al-?is?b bi?l takht wa?l-tuz?b (“Arithmetic by Means of Board and Dust”) of Nas??r al-Din al-T??s? (1265). The origin of these methods is unknown. It is possible that they were at least partly developed by al-Khayy?m? the influence of Chinese algebra is also quite plausible.²³

Noteworthy in the second and the third book is the doctrine of decimal fractions, used previously by al-K?sh? in his Ris?la al-muh?t?yya. It was not the first time that decimal fractions appeared in an Arabic mathematical work; they are in the Kit?b al-fus?l fi?l-his?b al-Hindi (“Treatise of Arithmetic”) of al Ul?dis? (mid-tenth century) and were used occasionally also by Chinese scientists.²⁴ But only al-K?sh? introduced the decimal fractions methodically, with a view to establishing a system of fractions in which (as in the sexagesimal system) all operations would be carried out in the same manner as with integers. It was based on the commonly used decimal numeration, however, and therefore accessible to those who were not familiar with the sexagesimal arithmetic of the astronomers. Operations with finite decimal fractions are explained in detail, but al-K?sh? does not mention the phenomenon of periodicity. To denote decimal fractions, written on the same line with the integer, he sometimes separated the integer by a vertical line or wrote in the orders above the figures; but generally he named only the lowest power that determined all the others. In the second half of the fifteenth century and in the sixteenth century al-K?sh?’s decimal fractions found a certain circulation in Turkey, possibly through ?Al? Q?shj?, who had worked with him at Samarkand and who sometime after the assassination of Ulugh B?g and the fall of the Byzantine empire settled in Constantinople. They also appear occasionally in an anonymous Byzantine collection of problems from the fifteenth century which was brought to Vienna in 1562.²⁵ It is also possible that al-K?sh?’s ideas had some influence on the propagation of decimal fractions in Europe.

In the fifth book al-K?sh? mentions in passing that for the fourth-degree equations he had discovered “the method for the determination of unknowns in. . . seventy problems which had not been touched upon by either ancients or contemporaries.”²⁶ He also expressed his intention to devote a separate work to this subject, but it seems that he did not complete this research. Al-K?sh?’s theory should be analogous to the geometrical theory of cubic equations developed much earlier by Abu’l-J?d Muhammad ibn Laith, al-Khayy?m? (eleventh century), and their followers: the positive roots of fourth-degree equations were constructed and investigated as coordinates of points of intersection of the suitable pairs of conics. It must be added that actually there are only sixty-five (not seventy) types of fourth-degree equations reducible to the forms considered by Muslim mathematicians, that is, the forms having terms with positive coefficients on both sides of the equation. Only a few cases of fourth-degree equations were studied before al-K?sh?.

Al-K?sh?’s greatest mathematical achievements are Ris?la al-muhitiyya and Ris?la al-watar wa?l-jaib, both written in direct connection with astronomical researches and especially in connection with the increased demands for more precise trigonometrical tables.

At the beginning of the Ris?la al-mu??t??yya al-K?sh? points out that all approximate values of the ratio of the circumference of a circle to its diameter, that is, of ?, calculated by his predecessors gave a very great (absolute) error in the circumference and even greater errors in the computation of the areas of large circles, Al-K?sh? tackled the problem of a more accurate computation of this ratio, which he considered to be irrational, with an accuracy surpassing the practical needs of astronomy, in terms of the then-usual standard of the size of the visible universe or of the “sphere of fixed stars.”²⁷ For that purpose he assumed, as had the Iranian astronomer Qutb al-Din al-Sh?r?z? (thirteenth-fourteenth centuries), that the radius of this sphere is 70,073.5 times the diameter of the earth. Concretely, al-K?sh? posed the problem of calculating the said ratio with such precision that the error in the circumference whose diameter is equal to 600,000 diameters of the earth will be smaller than the thickness of a horse’s hair. Al-K?sh? used the following old Iranian units of measurement: I parasang (about 6 kilometers) = 12,000 cubits, 1 cubit = 24 inches (or fingers), 1 inch = 6 widths of a medium-size grain of barley, and I width of a barley grain = 6 thicknesses of a horse’s hair. The great-circle circumference of the earth is considered to be about 8,000 parasangs, so al-K?sh?’s requirement is equivalent to the computation of ? with an error no greater than 0.5 ·10^-17. This computation was accomplished by means of elementary operations, including the extraction of square roots, and the technique of reckoning is elaborated with the greatest care.

Al-K?sh?’s measurement of the circumference is based on a computation of the perimeters of regular inscribed and circumscribed polygons, as had been done by Archimedes, but it follows a somewhat different procedure. All calculations are performed in sexagesimal numeration for a circle with a radius of 60. Al-K?sh?’s fundamental theorem—in modern notation—is as follows: In a circle with radius r,

where crd ?° is the chord of the arc ?° and ?° < 180°. Thus al-K?sh? applied here the “trigonometry of chords” and not the trigonometric lines themselves. If ? = 2?° and d = 2, then al-K?sh?’s theorem may be written trigonometrically as

which is found in the work of J. H. Lambert (1770). The chord of 60° is equal to r, and so it is possible by means of this theorem to calculate successively the chords c₁, c₂, c₃. . . . of the arcs 120°, 150°, 165°, in general the value of the chord c_n of the arc will be . The chord c_n being known, we may, according to Pythagorean theorem, find the side of the regular inscribed 3 · 2ⁿ-sided polygon, for this side a_n is also the chord of the supplement of the arc ?_n° up to 180°. The side b_n of a similar circumscribed polygon is determined by the proportion b_n: a_n = r: h, where h is the apothem of the inscribed polygon. In the third section of his treatise al-K?sh? ascertains that the required accuracy will be attained in the case of the regular polygon with 3·2²⁸ = 805, 306, 368 sides.

He resumes the computation of the chords in twenty-eight extensive tables; he verifies the extraction of the roots by squaring and also by checking by 59 (analogous to the checking by 9 in decimal numeration); and he establishes the number of sexagesimal places to which the values used must be taken. We can concisely express the chords c_n and the sides a_n by formulas

and

where the number of radicals is equal to the index n. In the sixth section, by multiplying a₂₈ by 3·2²⁸, one obtains the perimeter p₂₈ of the inscribed 3·2²⁸-sided polygon and then calculates the perimeter p₂₈ of the corresponding similar circumscribed polygon. Finally the best approximation for 2? r is accepted as the arithmetic mean whose sexagesimal value for r = 1 is 6 16^I 59^II 28^III 1^IV 34^V 51^VI 46^VIII 50^IX, where all places are correct. In the eighth section al-K?sh? translates this value into the decimal fraction 2?= 6.2831853071795865, correct to sixteen decimal places. This superb result far surpassed all previous determinations of ?. The decimal approximation ? ? 3.14 corresponds to the famous boundary values found by Archimedes, Ptolemy used the sexagesimal value 3 8^I 30^II (? 3.14166), and the results of al-K?sh?’s predecessors in the Islamic countries were not much better. The most accurate value of ? obtained before al-K?sh? by the Chinese scholar Tsu Ch?ung-chih (fifth century) was correct to six decimal places. In Europe in 1597 A. van Roomen approached al-K?sh?’s result by calculating ? to fifteen decimal places; later Ludolf van Ceulen calculated ? to twenty and then to thirty-two places (published 1615).

In his Ris?la al-walar wa?l-jaib al-K?sh? again calculates the value of sin 1° to ten correct sexagesimal places; the best previous approximations, correct to four places, were obtained in the tenth century by Abu?l-Waf?? and Ibn Y?nus. Al-K?sh? derived the equation for the trisection of an angle, which is a cubic equation of the type px = q + x³—or, as the Arabic mathematicians would say, “Things are equal to the cube and the number.” The trisection equation had been known in the Islamic countries since the eleventh century; one equation of this type was solved approximately by al-B??n? to determine the side of a regular nonagon, but this method remains unknown to us. Al-K?sh? proposed an original iterative method of approximate solution, which can be summed up as follows: Assume that the equation

possesses a very small positive root x; for the first approximation, take ; for the second approximation, ; for the third, , and generally x₀ = 0.

It may be proved that this process is convergent in the neighborhood of values of . Al-K?sh? used a somewhat different procedure: he obtained x₁ by dividing q by p as the first sexagesimal place of the desired root, then calculated not the approximations x₂, x₃, . . . themselves but the corresponding corrections, that is, the successive sexagesimal places of x. The starting point of al-K?sh?’s computation was the value of sin 3°, which can be calculated by elementary operations from the chord of 72° (the side of a regular inscribed pentagon) and the chord of 60°. The sin 1° for a radius of 60 is obtained as a root of the equation

The sexagesimal value of sin 1° for a radius of 60 is 1 2^I 49^II 43^III 11^IV 14^V 44^VI 16^VII 26^VIII 17^IX; and the corresponding decimal fraction for a radius of 1 is 0.017452406437283571. All figures in both cases are correct.

Al-K?sh?’s method of numerical solution of the trisection equation, whose variants were also presented by Ulugh B?g, Q?d?i Z?de, and his grandson Ma?m?d ibn Mu?ammad M?r?m Cheleb? (who worked in Turkey),²⁸ requires a relatively small number of operations and shows the exactness of the approximation at each stage of the computation. Doubtless it was one of the best achievements in medieval algebra. H. Hankel has written that this method “concedes nothing in subtlety or elegance to any of the methods of approximation discovered in the West after Viéte.”²⁹ But all these discoveries of al-K?sh?s’s were long unknown in Europe and were studied only in the nineteenth and twentieth centuries by such historians of science as Sédillot, Hankel, Luckey, Kary-Niyazov, and Kennedy.

Astronomy. Until now only three astronomical works by al-K?sh? have been studied. His Kh?q?n? Zij, as its title shows, was the revision of the ?lkh?n? Zij of Nas??r al-D?n al-T??s?. In the introduction to al-K?sh?’s Zij there is a detailed description of the method of determining the mean and anomalistic motion of the moon based on al-K?sh?’s three observations of lunar eclipses made in K?sh?n and on Ptolemy’s three observations of lunar eclipses described in the Almagest. In the chronological section of these tables there are detailed descriptions of the lunar Muslim (Hijra) calendar, of the Persian solar (Yazdegerd) and Greek-Syrian (Seleucid) calendars, of al-Khayy?m?’s calendar reform (Malik?) of the Chinese-Uigur calendar, and of the calendar used in the II-Khan empire, where Nas??r al-D?n al-T??s? had been working. In the mathematical section there are tables of sines and tangents to four sexagesimal places for each minute of arc. In the spherical astronomy section there are tables of transformations of ecliptic coordinates of points of the celestial sphere to equatorial coordinates and tables of other spherical astronomical functions.

There are also detailed tables of the longitudinal motion of the sun, the moon, and the planets, and of the latitudinal motion of the moon and the the planets. Al-K?sh? also gives the tables of the longitudinal and latitudinal parallaxes for certain geographic latitudes, tables of eclipses, and tables of the visibility of the moon. In the geographical section there are tables of geographical latitudes and longitudes of 516 points. There are also tables of the fixed stars, the ecliptic latitudes and longitudes, the magnitudes and “temperaments” of the 84 brightest fixed stars, the relative distances of the planets from the center of the earth, and certain astrological tables. In comparing the tables with Ulugh B?g’s Zij, it will be noted that the last tables in the geographical section contain coordinates of 240 points, but the star catalog contains coordinates of 1,018 fixed stars.

In his Mift?? al-?is?b al-K?shi mentions his Zij al-tashil?t (“Zij of Simplifications”) and says that the also composed some other tables.³⁰ His Sullam alsam??, scarcely studied as yet, deals with the determination of the distances and sizes of the planets.

In his Ris?la dar shar?-i ?l?t-i ras?d (“Treatise on the Explanation of Observational Instruments”) al-K?shi briefly describes the construction of eight astronomical instruments: triquetrum, armillary sphere, equinoctial ring, double ring, Fakhr? sextant, an instrument “having azimuth and altitude,” an instrument “having the sine and arrow,” and a small armillary sphere. Triquetra and armillary spheres were used by Ptolemy; the latter is a model of the celestial sphere, the fixed and mobile great circles of which are represented, respectively, by fixed and mobile rings. Therefore the armillary sphere can represent positions of these circles for any moment; one ring has diopters for measurement of the altitude of a star, and the direction of the plane of the ring determines the azimuth. The third and seventh instruments consist of several rings of armillary spheres. The equinoctial ring (the circle in the plane of the celestial equator), used for observation of the transit of the sun through the equinoctial points, was invented by astronomers who worked in the tenth century in Sh?r?z, at the court of the Buyid sultan ?Ad??d al-Dawla. The Fakhr? sextant, one-sixth of a circle in the plane of the celestial meridian, used for measuring the altitudes of stars in this plane, was invented about 1000 by al-Khujand? in Rayy, at the court of the Buyid sultan Fakhr al-Dawla. The fifth instrument was used in the Mr?gha observatory directed by Nas? al-Din al-T??s?. The sixth instrument, al-K?sh? say, did not exist in earlier observatories; it is used for determination of sines and “arrows” (versed sines) of arcs.

In Nuzha al-?ad?iq al-K?sh? describes two instruments he had invented: the “plate of heavens” and the “plate of conjunctions.” The first is a planetary equatorium and is used for the determination of the ecliptic latitudes and longitudes of planets, their distances from the earth, and their stations and retro-gradations; like the astrolabe, which it resembles in shape, it was used for measurements and for graphical solutions of problems of planetary motion by means of a kind of nomograms. The second instrument is a simple device for performing a linear interpolation.

NOTES

1. See E. S. Kennedy, The Planetary Equatorium . . ., p. 1.

2. H. Suter, Die Mathematiker und Astronomen . . ., pp. 173–174; Kennedy, op. cit., p. 7.

3. See M. Krause, “Stambuler Handschriften . . .,” p. 50; M. T?ab?t?ab??i,” “Jamsh?d Ghiy?th al-D?n K?sh?n?,” p. 23.

4. D. G. Voronovski, “Astronomuy Sredney Sredney Azii ot Muhammeda al-Havarazmi do Ulugbeka i ego shkoly (IX-XVI vv.),” pp. 127, 164.

5. See V. V. Bartold, Ulugbek i ego uremya, p. 108.

6. Kennedy, op. cit., pp. 1-2.

7. Bartold, op. cit., p. 108; Kennedy, op. cit., p. 2.

8. V. A. Shishkin, “Observatoriya Ulugbeka i ee issledovanie,”p. 10.

9. See T. N. Kary-Niyazov, Astronomicheskaya shkola Ulugbeka, 2nd ed., p. 107; see also Shishkin, op. cit.

10. See Kary-Niyazov, loc. cit.

11. See Bartold, op. cit. p. 88.

12. Ibid., pp. 88–89.

13. E. S. Kenedy, “A Letter of Jamsh? al-K?sh? to His Father,” p. 200.

14. Suter, op, cit., pp. 173, 175; E. S. Kennedy, “A Survery of Islamic Astronomical Tables,” p. 127.

15. Kennedy, “A Letter...,” p. 194.

16. See Bartold, op, cit., p. 108.

17. See Zij-i Ulughbeg, French trans., p. 5.

18. See Kennedy, The Planetary Equatorium..., p. 6.

19. See Ta?r?b al-zij; Kary-Niyazov, op. cit., 2nd ed., ppl. 141–142.

20. See P. Luckey, Die Rechenkunst... A. P. Youschkevitch; Geschichte der Mathematik im Mittelalter p. 237 ff.

21. al-K?sh?, Klyuch arifinetiki..., p. 13.

22. See P. Luckey, “Die Auszichung des n-ten Wurzel...”

23. P. Luckey, “Die Ausziehung des n-ten Wurzel...”; Juschkewitsch, op. cit., pp., 240–248.

24. See A. Saiden, “The Earliest Extant Arabic Arithmetic...”; Juschkewitsch, op. cit., pp. 21–23.

25. H. Hunger and K. Vogel, Ein byzantinisches Rechenbuch des 15. Jahrhunderts, p. 104.

26. al-K?shi, Klyuch arifmetiki..., p. 192.

27. Ibid., p. 126.

28. Kary-Niyazov, op. cit., 2nd ed, p. 199; Q?d?? Z?de, Ris?la f? istikhra?j jaib daraja w?hida; M?r?m Cheleb? Dast?r al-?amal wa tas??? al-jadwal.

29. H. Hankel, Zur Geschichte der Mathematik..., p. 292.

30. al-K?h?Klyuch arifmetiki..., p. 9.

BIBLIOGRAPHY

I. Original Works. Al-K?sh?’s writings were collected as Majm?? (“Collection”; Teheran, 1888), an ed. of the matematicheskie issledoveniya, 7 (1954), 9–439, Russian trans. by B. A. Rosenfeld and commentaries by Rosenfeld and A. P. Youschkevitch; and Klyuch arifmeti. Traktat of okruzhnosti ( “The Key of Arithmetic. A Treatise on Circumference”), trans. bty B. A. Rosenfeld, ed. by V. S. Segal and A. P. Youschkevitch, commentaries by Rosenfeld and Youschkevitch, with photorepros. of Arabic MSS.

His individual works are the following:

1. Sullam al-sam?? fi ?all ishk?l waqa?a li?l-muqaddim? fi?l-ab??d wa?l-ajr?m ( “The Stairway of Heven, on Resolution of Difficulties Met by Predecessors in the Determination of Distances and Sizes”; 1407). Arabic MSS in London, India Office 755; and Oxford, Bodlye 888/4.

2. Mukhtas?ar dar ?lim-i hay? at (“Compendium on the Science of Astronomy”) or Ris?la dar hay? at ( “Treatise on Astronomy”; 1410–1411). Persian MSS in London and Yezd.

3. Zij-i Khaq?ni f? takm?l-i Zij-i ?lkh?n? (“Khaq?ni Zij— perfection of ?lkh?n? Zij” 1413–1414). Persian MSS in London, Istanbul, Teharan, Yezd, Meshed, and Hyderabad-Deccan, the most important being London, India Office 2232, which is described in E. S. Kennedy, “A Survey of Islamic Astronomical Tables,” pp. 164–166.

4. Ris?la dar shar?-i ?l?t-i ras?d (”Treatise on the Explanation of Observational Instruments”; 1416). Persian MSS in Leiden and Teharan, the more important being Leiden, Univ. 327/12, which has been pub. as a supp. to V. V. Bartold, Ulugbek i ego uremya; and E. S. Kennedy,” Al-K?shi’s Treatise on Astronomical Observation Instruments,” pp. 99, 101, 103. There isd an English trans. in Kennedy, “Al-K?sh?’s Treatise...,” pp. 98–104; and a Russian trans. in V. A. Shishkin, “Observatoriya Ulugebeka i ee issledovanie,” pp. 91–94.

5. Nuzha al-?ad?iq fi kayfiyya s?an?sa al-?la al-musamm? bi t?abaq al-man?t?iq (“The Garden Excursion,; on the Method of Construction of the Instrument Called Plate of Heavens”; 1416). Arabic MSS are in London, Dublin, and Bombay, the moist important being London, India Office Ross 210. There is a litho. ed. of another MS as a Supp. to the Teheran ed. of Mift?? al-?is? see also Ris?la fi?l-?amal bi ashal ?la min qabl al-nuj?m; G. D. Jala-lov, “Otlichie ’Zij Guragani’ ot drugikh podobnykh zijey” and “K voprosu o sostavelnii planetnykh tablits samarKandskoy observatorii”; T. N. Kary-Niyazov, Astronomicheskaya shkola Ulugbeka; and E. S. Kennedy, “Al-K?sh?’s ‘Plate of Conjunctions.‘”

6. Ris?al-mu??it??yya (“Treatise on the Circumference”; 1424). Arabic MSS are in Istanbul, Teheran, and Meshed, the most important being Istanbul, Ask. müze. 756. There is an ed. of another MS in Majm?? and one of the Istanbul MS with German trans. in P. Luckey, Der Lehrbrief über den Kreisumfang von Gamš?d b. Mas??d al-K?ši. Russian trans. are in “Matematicheskie trakaty,” pp.327–379; and in Klyuch arifmetiki, pp. 263–308, with photorepro . of Istanbul MS pp. 338–426.

7. Ilka??t an-Nuzha (“Supplement to the Excursion” 1427). There is an ed. of a MS in Majm??.

8. Mift?? al-?is?b (”The Key of Arithmetic”) or Mift?? al-?uss?b fi ’ilm al-?is?b (“The Key of Reckoners in the Science of Arithmetic”). Arabic MSS in Leningrad, Berlin, Paris, Leiden, London, Istanbul, Teheran, Meshed, Patna, Peshawar, and Rampur, the most important being Leningrad, Publ. Bibl. 131; Leiden, Univ. 185; Berlin, Preuss. Bibl. 5992 and 2992a, and Inst. Gesch. Med. Natur. 1.2; Paris, BN 5020; and London, BM 419 and India Office 756. There is a litho. ed. of another MS (Teheran, 1889). Russian trans. are in “Matematicheskie traktaty,” pp. 13–326; and Klyuch arifmetiki, pp. 7–262, with photorepro. of Leiden MS on pp. 428–568, There is an ed. of the Leiden MS with commentaries (Cairo, 1968). See also P. Luckey, “Die Ausziehung dos n-ten Wurzel...” and “Die Rechenkunst bei ?amšid b. Mas?ud al-K?ašs?...”

9. Talkh?is al-Mift?ah (“Compendium of the Key”). Arabic MSS in London, Tashkent, Istanbul, Baghdad, Mosul, Teheran, Tabriz, and Patna, the most important being London, India Office 75; and Tashkent, Inst. vost. 2245.

10. Ris?la al-watar wa?l-jaib (“Treatise on the Chord and Sine”). There is an ed. of a MS in Majm??.

11. Ta?rib al-zij (“The Arabization of the Z?j”), an Arabic trans. of the intro. to Ulugh B?g’s Z?j. MSS are in Leiden and Tashkent.

12. Wuj?uh al-?amal al-d?arb fi?l-takht wa?l-tur?b (“Ways of Multiplying by Means of Board and Dust”). There is an ed. of an Arabic MS in Majm??.

13. Nat??ij al-?aq??iq (“Results of Verities”). There is an ed. of an Arabic MS in Majm??.

14. Mift?? al-asb?b fi?ilm al-zij ( “The Key of Causes in the Science of Astronomical Tables” ). There is an Arabic MS in Mosul.

15. Ris?la dar sakht-i ast?url?b (“Treatise on the Construction of the Astrolabe”). There is a Persian MS in Meshed.

16. Ris?la fi ma?rifa samt al-qibla min d?ira hindiyya ma?r?fa (“Treatise on the Determination of Azimuth of the Qibla by Means of a Circle Known as Indian”). There is an Arabic MS at Meshed.

17. Al-K?sh?’s letter to his father exists in 2 Persian MSS in Teheran. There is an ed. of them in M. T?ab?t?ab???, “N?ma-yi pisar bi pidar,” in Am?zish wa parwarish,10 , no. 3 (1940), 9–16, 59–62. An English trans. is E S. Kennedy, “A Letter of Jamsh?d al-K?sh? to His Father” English and Turkish trans. are in A. Sayili, “Ghiy?th al-D?n al-K?sh?’s Letter on Ulugh B?g and the Scientific Activity in Samarkand,” in Türk tarih kurumu yayinlarinden, 7th ser., no. 39 (1960).

II. Secondary Literature. See the following: V. V. Bartold, Ulugbek i ego uremya (“Ulugh B?eg and His Time”; Petrograd, 1918), 2nd ed. in his Sochinenia (“Works”), II, pt. 2 (Moscow, 1964), 23–196, trans into German as “Ulug Beg und Seine Zeit,” in Abhandlungen für die Kunde des Morgenlandes,21 no, 1 (1935); L. S. Bretanitzki and B. A. Rosenfeld, “Arkhitekturnaya glava traktata ‘Klyuch arifmetiki’ Giyas ad-Dina Kashi” (“An Architectural Chapter of the Treatise ‘ The Key of Arithmetic’ by Ghiy?th al-D?n K?sh?”), in Iskusstvo Azerbayjana, 5 (1956), 87–130; C. Brockelmann, Geschichte der arabischen literature 2nd ed., II (Leiden, 1944), 273 and supp. II (Leiden, 1942), 295; M?r?m Cheleb?, Dast?r al-?amal was tas???h al-jadwal (“Rules of the Operation and Correction of the Tables”; 1498), Arabic commentaries to Ulugh B?g’s Z?j, contains an exposition of al-K?sh?’s. Ris?la al-watar wa? l-jaib—Arabic MSS are in Paris, Berlin, Istanbul, and Cairo, the most important being Pairs, BN 163 (a French trans. of the exposition is in L. A. Sédillot, “De l?algèbre chez les Arabes,” in Journal asiatique, 5th ser., 2 [1853], 323–350; a Russian trans. is in Klyuch arifmetiki, pp. 311–319); A. Dakhel, The Extraction of the n-th Root in the Sexagesimal Notation. A Study of Chapter 5, Treatise 3 of Mift?? al H?is?b, W. A. Hijab and E. S. Kennedy, eds.(Beirut, 1960); H. Hankel, Zur Geschichte der Mathematik im Altertum und Mittelalter (Leipzig, 1874); and H. Hunger and K. Vogel, Ein byzantinisches Rechenbuch des 15. Jahrhunderts (Vienna, 1963), text, trans., and commentary.

See also G. D. Jalalov,“Otlichie ‘Zij Guragani’ ot drugikh podobnykh zijey” (“The Difference of ‘Gurgani Zij’ from Other Z?jes”), in Istoriko-astronomicheskie issledovaniya, 1 (1955), 85–100; “K voprosu o sostavlenii planetnykh tablits samarkandskoy observatorii” (“On the Question of the Composition of the Planetary Tables of the Samarkand Observatory”), ibid., 101–118; and “Giyas ad-Din Chusti (Kashi)—krupneyshy astronom i matematik XV veka” (“Ghiy?th al-D?n Ch?st? [K?sh?]— the Greatest Astronomer and Mathematician of the XV Century”), in Uchenye zapiski Tashkentskogo gosudarstvennogo pedagogicheskogo instituta, 7 (1957), 141–157; T. N. KaryNiyazov, Astronomicheskaya shkola Ulugeka (Moscow-Leningrad, 1950), 2nd ed. in his Izbrannye trudy (“Selected Works”), VI (Tashkent, 1967); and “Ulugbek i Savoy Jay Singh, “in Fiziko-matematicheskie nauki v stranah Vostoka, 1 (1966), 247–256; E. S. Kennedy, “Al-K?sh?’s Plate of Conjunctions,” on Isis, 38 , no. 2 (1947), 56–59; “A Fifteenth-Century Lunar Eclipse Computer,” in Scripta mathematica, 17 , no. 1–2 (1951), 91–97; “An Islamic Computer for Planetary Latitudes,” in Journal of the American Oriental Society, 71 (1951), 13–21; “A Survey of Islamic Astronomical Tables,” in Transactions of the American philosophical Society, n.s. 46 no. 2 (1956), 123–177; “Parallax Theory in Islamic Astronomy,” in Isis, 47 , no. 1 (1956), 33–53; The Planetary Equatorium of Jamshid Ghiy?th al-Din al-K?shi (Princeton, 1960); “A Letter of Jamshid al-K?shi to His Father. Scientific Research and Personalities of a Fifteenth Century Court,” in Commentarii periodici pontifici Instituti biblici, Orientalia, n.s. 29 , fasc. 29 (1960), 191–213; “Al-K?shi’s Treatise on Astronomical Observation Instruments,” in Journal of Near Eastern Studies, 20 , no. 2 (1961), 98–108; “A Medieval Interpolation Scheme Using Second-Order Differences,” in A Locust’s Leg. Studies in Honour of S. H. Tegi-zadeh (London, 1962), pp. 117–120; and “The Chinese-Uighur Calendar as Described in the Islamic Sources,” in Isis, 55 , no. 4 (1964), 435–443; M. Krause, “Stambuler Handschriften islamischer Mathematiker,” in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abt. B, 3 (1936), 437–532; P. Luckey, “Die Ausziehung des n-ten Wurzel and der binomische Lehrsatz in der islamischen Mathematik,” in Mathematische Annalen, 120 (1948), 244–254; “Die Rechenkunst bei ?amšid b. Mas’?d al-K?ši mit Rückblicken auf die ältere Geschichte des Rechnens,” in Abhandlungen für die Kunde des Morgenlandes, 31 (Wiesbaden, 1951); and Der Lehrbrief uber den Kreisumfang von ?amšid b. Mas’?d al-K?si, A. Siggel, ed. (Berlin, 1953); Ris?la fi?l-?amal bi ashal ?la min qabl alnuj?m (“Treatise on the Operation With the Easiest Instrument for the Planets”), a Persian exposition of al-K?shi’s Nuzha—available in MS as Princeton, Univ. 75; and in English trans. with photorepro. in E. S. Kennedy, The Planetary Equatorium; B. A. Rosenfeld and A. P. Youschkevitch, “O traktate Q?d??-Z?de ar-R?mi ob opredelenii sinusa odnogo gradusa” (“On Q?d?i-Z?de al-R?mi’s Treatise on the Determination of the Sine of One Degree”), in Istoriko-matematicheskie issledovaniya, 13 (1960), 533–556; and M?s? Q?d?i Z?de al-?mi, Ris?la f? istikhr?j jaib daraja w?hida (“Treatise on Determination of the Sine of One Degree”), an Arabic revision of al-K?shi’s Ris?la al-watar wa’l-jaib—MSS are Cairo, Nat. Bibl. 210 (ascribed by Suter, p. 174, to al-K?shi himself) and Berlin, Inst. Gesch. Med. Naturw. 1.1; Russian trans. in B. A. Rosenfeld and A. P. Youschkevitch, “O traktate Q?d??-Z?de...” and descriptions in G. D. Jalalov, “Giyas ad-D?n Chusti (Kashi)...” and in S??lih Zak? Effend?, Ath?r b?qiyya, I.

Also of value are A. Saidan, “The Earliest Extant Arabic Arithmetic. Kit?b al-fus??l fi al-?is?b al-Hind? of... al-Uql?dis?,” in Isis, 57 , no. 4 (1966), 475–490; S??lih Zak? Effend?, Ath?r b?qiyya, I (Istanbul, 1911); V. A. Shishkin, “Observatoriya Ulugbeka i ee issledovanie” (“Ulugh B?g’s Observatory and Its Investigations”), in Trudy Instituta istorii i arkheologii Akademii Nauk Uzbekskoy SSR, V, Observatoriya Ulugbeka (Tashkent, 1953), 3–100; S. H. Sirazhdinov and G. P. Matviyevskaya, “O matematicheskikh rabotakh shkoly Ulugbeka” (“On the Mathematical Works of Ulugh B?g’s School”), in Iz istorii epokhi Ulugbeka (“From the History of Ulugh B?g’s Age”; Tashkent, 1965), pp. 173–199; H. Suter, Die Mathematiker und Astronomen der Araber und ihre Werke (Leipzig, 1900); M. Tab?tab??i, “Jamshid Ghiy?th al-Din K?sh?ni,” in Amuzish wa Parwarish, 10 , no. 3 (1940), 1–8 and no. 4 (1940), 17–24; M. J. Tichenor, “Late Medieval Two-Argument Tables for Planetary Longitudes,” in Journal of Near Eastern Studies, 26 , no. 2 (1967), 126–128; D. G. Voronovski, “Astronomy Sredney Azii ot Muhammeda al-Havarazmi do Ulugbeka i ego shkoly (IX-XVI vv.)” (“Astronomers of Central Asia from Muhammad al-Khw?rizmi to Ulugh B?g and His School, IX-XVI Centuries”), in Iz istorii epokhi Ulugbeka (Tashkent, 1965), pp. 100–172; A. P. Youschkevitch, Istoria matematiki v srednie veka (“History of Mathematics in the Middle Ages”; Moscow, 1961); trans. into German as A. P. Juschkewitsch, Geschichte der Mathematik in Mittelalter (Leipzig, 1964); and Zij-i Ulughb?g (“Ulugh B?g’s Z?j”), or Zij-i Sult??n? or Zij-i jad?d-? Gurag?n? (“New Gurag?n? Zij”), in persian, the most important MSS being Paris, BN 758/8 and Tashkent, Inst. Vost. 2214 (a total of 82 MSS are known)—an ed. of the intro. according to the Paris MS and a French trans. are in L.A. Sédillot, Prolegomènes des tables astronomiques d’Oloug-Beg (Paris, 1847; 2nd ed., 1853), and a description of the Tashkent MS is in T. N. Kary-Niyazov, Astronomicheskaya shkola Ulugbeka (2nd ed., Tashkent, 1967), pp. 148–325.