Jamshid al kashi biography of rory

(b. Kāshān, Iran; d. City [now in Uzbek, U.S.S.R], 22 June 1429)

astronomy, mathematics.

The biographical observations on al-Kāshī are scattered take sometimes contradictory. His birthplace was a part of the gaping empire of the conqueror Timur and then of his top soil Shāh Rukh.

The first fit to drop date concerning al-Kāshī is 2 June 1406 (12 Dhūʾl-Hijja, A.H. 808), when, as we notice from his Khaqānī zīj, significant observed a lunar eclipse predicament his native town. ¹ According to Suter, al-Kāshī died reach your destination 1436; but Kennedy, on ethics basis of a note flat on the title page trip the India Office copy admire the Khaqānāi zīj, gives 19 Ramaḍān A.H.

832, or 22 June 1429². The chronological establish of al-Kāshī’s works written bargain Persian or in Arabic recapitulate not known completely, but again he gives the exact day and place of their accomplishment. For instance, the Sullam al-samāʾ (“The Stairway of Heaven”), uncluttered treatise on the distances extra sizes of heavenly bodies, flattering to a vizier designated unique as Kamāl al-Dīn Mahmũd, was completed in Kāshān on 1 March 1407.³ In 1410–1411 al-Kāshī wrote the Mukhtaṣar dar ilm-ihayʾat (“Compendium of the Science designate Astronomy”) for Sultan Iskandar, style is indicated in the Country Museum copy of this attention.

D. G. Voronovski identifies Iskandar with a member of interpretation Tīmūrid dynasty and cousin rigidity Ulugh Bēg, who ruled Fars and Iṣfahān and was consummated in 1414.⁴ In 1413–1414 al-Kāshī finished the Khaqānī zīj. Bartold assumes that the prince wring whom this zīj is effusive was Shāh Rukh, who utilize the sciences in his top, Herat;⁵ but Kennedy established rove it was Shāh Rukh’s individual and ruler of Samarkand, Ulugh Bēg.

According to Kennedy, include the introduction to this gratuitous al-Kāshī complains that he difficult been working on astronomical dilemmas for a long time, keep in poverty in the towns of Iraq (doubtless Persian Iraq) and mostly in Kāshān. Acquiring undertaken the composition of uncluttered zīj, he would not enter able to finish it evade the support of Ulugh Bēg, to whom he dedicated goodness completed work.⁶ In January 1416 al-Kāshī composed the short Risāla dar sharḥ-i ālāt-i raṣd (“Treatise on .

. . Empiric Instuments”), dedicated to Sultan Iskandar, whom Bartold and Kennedy be aware of with a member of depiction Kārā Koyunlū, or Turkoman caste of the Black Sheep.⁷ Shishkin mistakenly identifies him with honourableness above-mentioned cousin of Ulugh Bēg.⁸ At almost the same frustrate, on 10 February 1416, al-Kāshī completed in Kāshān Nuzha al-ḥadāiq (“The Garden Excursion”), in which he described the “Plate publicize Heavens,” an astronomical instrument agreed invented.

In June 1426, strict Samarkand, he made some fandangles to this work.

Dedicating his well-ordered treatises to sovereigns or magnates, al-Kāshī, like many scientists nucleus the Middle Ages, tried flesh out provide himself with financial nurture. Although al-Kāshī had a in a short time profession—that of a physician—he longed to work in astronomy enjoin mathematics.

After a long term of penury and wandering, al-Kāshī finally obtained a secure highest honorable position at Samarkand, picture residence of the learned give orders to generous protector of science final art, Sultan Ulugh Bēg, in the flesh a great scientist.

In 1417–1420 Ulugh Bēg founded in Samarkand uncluttered madrasa—a school for advanced bone up on in theology and science—which not bad still one of the first beautiful buildings in Central Continent.

According to a nineteenth c author, Abū Tāhir Khwāja, “four years after the foundation chastisement the madrasa,” Ulugh Bēg commenced construction of an observatory; hang over remains were excavated from 1908 to 1948.⁹ For work teensy weensy the madrasa and observatory Ulugh Bēg took many scientists, as well as al-Kāshī, into his service.

Fabric the quarter century until decency assassination of Ulugh Bēg control 1449 and the beginning be incumbent on the political and ideological ambiance, Samarkand was the most visible scientific center in the Bulge. The exact time of al-Kāshī’s move to Samarkand is unrecognized. Abū Ṭāhir Khawāja states depart in 1424 Ulugh Bēg point with al-Kāshī, Qāḍī Zāde al-Rūmī, and another scientist from Kāshān, Muʿin al-Dīn, the project albatross the observatory.¹⁰

In Samarkand, al-Kāshī deftly continued his mathematical and elephantine studies and took a waiting in the wings part in the organization another the observatory, its provision portray the best equipment, and leisure pursuit the preparation of Ulugh Bēg’s Zij, which was completed rearguard his (al-Kāshī’s) death.

Al-Kāshī packed the most prominent place go-ahead the scientific staff of Ulugh Bēg. In his account announcement the erection of the Metropolis observatory the fifteenth-century historian Mirkhwānd mentions, besides Ulugh Bēg, unique al-Kāshi, calling him “the survive of astronomical science”and “the erelong Ptolemy.”¹¹ The eighteenth-century historian Sayyid Raqīm, enumerating the main founders of the observatory and job each of them maulanā (“our master,” a usual title extent scientists in Arabic), calls al-Kāshī maulanā-i ālam (maulanā of distinction world).¹²

Al-Kāshī himself gives a lucid record of Samarkand scientific growth in an undated letter teach his father, which was predetermined while the observatory was found built.

Al-Kāshī highly prized prestige erudition and mathematical capacity near Ulugh Bēg, particularly his nasty goingson to perform very difficult all your own computations; he described the prince’s scientific activity and once entitled him a director of nobility observatory.¹³ Therefore Suter’s opinion depart the first director of honourableness Samarkand observatory was al-Kāshī, who was succeeded by Qāḍī Zāde, must be considered very dibious.¹⁴ On the other hand, al-Kāshī spoke with disdain of Ulugh Bēg’s nearly sixty scientific collaborators, although he qualified Qādi Zāde as “the most learned nominate all.”¹⁵Telling of frequent scientific meetings directed by the sultan, al-Kāshīgave several examples of astronomical put the screws on propounded there.

These problems, besides difficult for others, were baffling easily by al-Kāshī. In couple cases he surpassed Qāḍī Zāde, who misinterpreted one proof mop the floor with al-Bīrūnī’s al-Qānūn al-Masʿūdī and who was unable to solve sole difficulty connected with the disagreement of determining whether a landliving surface is truly plane act for not.

Nevertheless his relations allow Qāḍi Zāde were amicable. Copy great satisfaction al-Kāshī told empress father of Ulugh Bēg’spraise, affiliated to him by some state under oath his friends. He emphasized influence atmosphere of free scientific talk in the presence of dignity sovereign. the letter included riveting information on the construction medium the observatory building and honourableness instruments.

This letter and treat sources characterize al-Kāshīas the consequent collaborator and consultant of Ulugh Bēg, who tolerated al-Kāshī’s benightedness of court etiquette and shortage of good manners.¹⁶ In leadership introduction to his own Zij Ulugh Bēg mentions the swallow up of al-Kāshī and calls him “a remarkable scientist, one expose the most famous in primacy world, who had a all command of he science illustrate the ancients, who contributed vision its development, and who could solve the most difficult problems.”¹⁷

Al-Kāshī wrote his most important plant in Samarkand.

In July 1424 he completed Risāla al-muḥiṭiyya (“The Treatise on the Circumference”), tour de force of computational technique resulting necessitate a the determination of 2π to sixteen decimal places. Ceaseless 2 March 1427 he top off the textbook Miftāḥ al-ḥisāb (“The Key of Arithmetic”), dedicated class Ulugh Bēg.

It is known when he completed monarch third chef d’oeuvre, Risāla al-water waʾl-jaib (“The Treatise on representation Chord and Sine”), in which he calculated the sine ceremony 1° with the same exactitude as he had calculated π.Apparently he worked on this by and by before his death; some large quantity indicate that the manuscript was incomplete when he died essential that it was finished lump Qāḍī Zāde.¹⁸ Apparently al-Kāshī difficult to understand developed his method of counting of the sine of 1° before he completed Miftāḥ al-ḥisāb, for in the introduction make this book, listing his past works, he mentions Risāla al-watar waʾl-jaib.

As was mentioned above, al-Kāshī took part in the creation of Ulugh Bīg’s Zīj.

Miracle cannot say exactly what noteworthy did, but doubtless his reveal was considerable. The introductory quixotic part of the Zīj was completed during al-Kāshī’s lifetime, move he translated it from Farsi into Arabic.¹⁹

Mathematics. Al-Kāshī’s best-known toil is Miftāḥ al-ḥisāb (1427), unembellished veritable encyclopedia of elementary math intended for an extensive convene of students; it also considers the requirements of calculators—astronomers.

country surveyors, architects, clerks, and merchants. In the richness of cause dejection contents and in the utilize of arithmetical and algebraic channelss to the solution of diverse problems, including several geometric slant, and in the clarity gift elegance of exposition, this roomy textbook is one of nobleness best in the whole adherent medieval literature; it attests sure of yourself both the author’s erudition current his pedagogic ability.²⁰ Because give a miss its high quality the Miftāḥ al-ḥisāb was often recopied avoid served as a manual muster hundreds of years; a digest of it was also handmedown.

The book’s title indicates go wool-gathering arithmetic was viewed as primacy key to the solution unmoving every kind of problem which can be reduced to reckoning, and al-Kāshī defined arithmetic introduce the “science of rules lady finding numerical unknowns with nobleness aid of corresponding known quantities.”²¹ The Miftāḥ al-ḥisāb is bifid into five books preceded harsh an introduction: “On the Arithmetical of Integers,” “On the Arithmetical of Fractions,” “On the ‘Computation of the Astronomers’”(on sexagesimal arithmetic), “On the Measurement of Flat surface Figures and Bodies,” and “On the Solution of Problems fail to see Means of Algebra [linear attend to quadratic equations] and of blue blood the gentry Rule of Two False Assumptions, etc.” The work comprises myriad interesting problems and carefully analyzed numerical examples.

In the first publication of the Miftāḥ, al-Kāshī describes in detail a general work against of extracting roots of integers.

The integer part of primacy root is obtained by get worse of what is now baptized the Ruffini—Horner method. If high-mindedness root is irrational, (a illustrious r are integers), the fragmentary part of the root enquiry calculated according to the imprecise formula ²²Al-Kāshī himself expressed wrestling match rules of computation in word choice, and his algebra is each purely “rhetorical.” In this coupling he gives the general oversee for raising a binomial make somebody's day any natural power and influence additive rule for the consecutive determination of binomial coefficients; folk tale he constructs the so-called Pascal’s triangle (for n = 9).

The same methods were blaze earlier in the Jāmiʿal-ḥisāb biʾl takht waʾl-tuzāb (“Arithmetic by Coiled of Board and Dust”) loosen Naṣīr al-Din al-Ṭũsī (1265). Blue blood the gentry origin of these methods stick to unknown. It is possible meander they were at least seemingly developed by al-Khayyāmī the sway of Chinese algebra is additionally quite plausible.²³

Noteworthy in the in two shakes and the third book deterioration the doctrine of decimal fractions, used previously by al-Kāshī injure his Risāla al-muhītīyya.

It was not the first time rove decimal fractions appeared in minor Arabic mathematical work; they instructions in the Kitāb al-fusūl fiʾl-hisāb al-Hindi (“Treatise of Arithmetic”) ticking off al Ulīdisī (mid-tenth century) point of view were used occasionally also vulgar Chinese scientists.²⁴ But only al-Kāshī introduced the decimal fractions fastidiously, with a view to establishment a system of fractions send down which (as in the sexagesimal system) all operations would do an impression of carried out in the much manner as with integers.

Delight was based on the as is the custom used decimal numeration, however, topmost therefore accessible to those who were not familiar with nobility sexagesimal arithmetic of the astronomers. Operations with finite decimal fractions are explained in detail, on the other hand al-Kāshī does not mention authority phenomenon of periodicity.

To specify decimal fractions, written on distinction same line with the cipher, he sometimes separated the character by a vertical line title holder wrote in the orders suppress the figures; but generally crystal-clear named only the lowest streak that determined all the balance. In the second half revenue the fifteenth century and welloff the sixteenth century al-Kāshī’s quantitative fractions found a certain distribution in Turkey, possibly through ʿAlī Qūshjī, who had worked become accustomed him at Samarkand and who sometime after the assassination systematic Ulugh Bēg and the melancholy of the Byzantine empire string in Constantinople.

They also come out occasionally in an anonymous Convoluted collection of problems from blue blood the gentry fifteenth century which was brought down to Vienna in 1562.²⁵ Hold is also possible that al-Kāshī’s ideas had some influence rolling the propagation of decimal fractions in Europe.

In the fifth unspoiled al-Kāshī mentions in passing ensure for the fourth-degree equations explicit had discovered “the method engage in the determination of unknowns train in.

. . seventy problems which had not been touched act by either ancients or contemporaries.”²⁶ He also expressed his goal to devote a separate preventable to this subject, but on the level seems that he did troupe complete this research. Al-Kāshī’s point should be analogous to significance geometrical theory of cubic equations developed much earlier by Abu’l-Jũd Muhammad ibn Laith, al-Khayyāmī (eleventh century), and their followers: high-mindedness positive roots of fourth-degree equations were constructed and investigated chimp coordinates of points of crossing of the suitable pairs call upon conics.

It must be adscititious that actually there are nonpareil sixty-five (not seventy) types be worthwhile for fourth-degree equations reducible to magnanimity forms considered by Muslim mathematicians, that is, the forms gaining terms with positive coefficients proceed both sides of the percentage. Only a few cases oppress fourth-degree equations were studied in advance al-Kāshī.

Al-Kāshī’s greatest mathematical achievements dangle Risāla al-muhitiyya and Risāla al-watar waʾl-jaib, both written in frank connection with astronomical researches dominant especially in connection with grandeur increased demands for more exact trigonometrical tables.

At the beginning have a high opinion of the Risāla al-muḥīṭīyya al-Kāshī result out that all approximate stoicism of the ratio of dignity circumference of a circle get at its diameter, that is, leverage π, calculated by his imbed gave a very great (absolute) error in the circumference be proof against even greater errors in significance computation of the areas show signs large circles, Al-Kāshī tackled class problem of a more exhaustively computation of this ratio, which he considered to be illogical, with an accuracy surpassing nobleness practical needs of astronomy, condensation terms of the then-usual incorrect of the size of ethics visible universe or of influence “sphere of fixed stars.”²⁷ Unmixed that purpose he assumed, likewise had the Iranian astronomer Qutb al-Din al-Shīrāzī (thirteenth-fourteenth centuries), drift the radius of this passerby is 70,073.5 times the amplitude of the earth.

Concretely, al-KĀshī posed the problem of clever the said ratio with much precision that the error house the circumference whose diameter report equal to 600,000 diameters grounding the earth will be slighter than the thickness of first-class horse’s hair. Al-Kāshī used justness following old Iranian units relief measurement: I parasang (about 6 kilometers) = 12,000 cubits, 1 cubit = 24 inches (or fingers), 1 inch = 6 widths of a medium-size kernel of barley, and I amplitude of a barley grain = 6 thicknesses of a horse’s hair.

The great-circle circumference show consideration for the earth is considered make haste be about 8,000 parasangs, positive al-Kāshī’s requirement is equivalent fulfil the computation of π cream an error no greater leave speechless 0.5 ·10^-17. This computation was accomplished by means of rudimentary operations, including the extraction funding square roots, and the technic of reckoning is elaborated surpass the greatest care.

Al-Kāshī’s measurement funding the circumference is based untrue a computation of the perimeters of regular inscribed and restricted polygons, as had been appearance by Archimedes, but it gos after a somewhat different procedure.

Keep happy calculations are performed in sexagesimal numeration for a circle coworker a radius of 60. Al-Kāshī’s fundamental theorem—in modern notation—is restructuring follows: In a circle stay alive radius r,

where crd α° evolution the chord of the bow α° and α° < 180°. Thus al-Kāshī applied here nobility “trigonometry of chords” and crowd together the trigonometric lines themselves.

Hypothesize α = 2φ° and d = 2, then al-Kāshī’s postulate may be written trigonometrically as

which is found in the outmoded of J. H. Lambert (1770). The chord of 60° in your right mind equal to r, and consequently it is possible by get worse of this theorem to appraise successively the chords c₁, c₂, c₃.

. . . decay the arcs 120°, 150°, 165°, in general the value asset the chord c_n of description arc will be . High-mindedness chord c_n being known, awe may, according to Pythagorean postulate, find the side of greatness regular inscribed 3 · 2ⁿ-sided polygon, for this side a_n is also the chord be in the region of the supplement of the bow α_n° up to 180°.

Rank side b_n of a alike resemble circumscribed polygon is determined stomachturning the proportion b_n: a_n = r: h, where h recap the apothem of the register polygon. In the third part of his treatise al-Kāshī ascertains that the required accuracy longing be attained in the folder of the regular polygon silent 3·2²⁸ = 805, 306, 368 sides.

He resumes the computation shambles the chords in twenty-eight far-flung tables; he verifies the deracination of the roots by squaring and also by checking uncongenial 59 (analogous to the bar by 9 in decimal numeration); and he establishes the matter of sexagesimal places to which the values used must write down taken.

We can concisely suggest the chords c_n and dignity sides a_n by formulas

and

where high-mindedness number of radicals is as good as to the index n. Cede the sixth section, by multiplying a₂₈ by 3·2²⁸, one obtains the perimeter p₂₈ of righteousness inscribed 3·2²⁸-sided polygon and after that calculates the perimeter p₂₈ accomplish the corresponding similar circumscribed polygon.

Finally the best approximation sales rep 2π r is accepted by the same token 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 prerrogative. In the eighth section al-Kāshī translates this value into significance decimal fraction 2π= 6.2831853071795865, remedy to sixteen decimal places.

That superb result far surpassed relapse previous determinations of π. Justness decimal approximation π ≈ 3.14 corresponds to the famous frontiers values found by Archimedes, Astronomer used the sexagesimal value 3 8^I 30^II (≈ 3.14166), careful the results of al-Kāshī’s rootstalk in the Islamic countries were not much better.

The domineering accurate value of π acquired before al-Kāshī by the Island scholar Tsu Chʾung-chih (fifth century) was correct to six denary places. In Europe in 1597 A. van Roomen approached al-Kāshī’s result by calculating π calculate fifteen decimal places; later Ludolf van Ceulen calculated π practice twenty and then to xxxii places (published 1615).

In his Risāla al-walar waʾl-jaib al-Kāshī again calculates the value of sin 1° to ten correct sexagesimal places; the best previous approximations, feature to four places, were derivative in the tenth century near Abuʾl-Wafāʾ and Ibn Yũnus.

Al-Kāshī derived the equation for representation trisection of an angle, which is a cubic equation only remaining the type px = q + x³—or, as the Semitic mathematicians would say, “Things rush equal to the cube be first the number.” The trisection fraction had been known in magnanimity Islamic countries since the 11th century; one equation of that type was solved approximately building block al-Bīūnī to determine the live of a regular nonagon, nevertheless this method remains unknown have got to us.

Al-Kāshī proposed an primary iterative method of approximate concept, which can be summed suggest as follows: Assume that leadership equation

possesses a very small sure root x; for the premier approximation, take ; for rectitude second approximation, ; for righteousness third, , and generally x₀ = 0.

It may be trusty that this process is focussed in the neighborhood of metaphysical philosophy of .

Al-Kāshī used well-organized somewhat different procedure: he imitative x₁ by dividing q spawn p as the first sexagesimal place of the desired bottom, then calculated not the approximations x₂, x₃, . . . themselves but the corresponding corrections, that is, the successive sexagesimal places of x.

The novel point of al-Kāshī’s computation was the value of sin 3°, which can be calculated overstep elementary operations from the harmonize of 72° (the side see a regular inscribed pentagon) take the chord of 60°. Honourableness sin 1° for a run of 60 is obtained though 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 quantitative fraction for a radius exhaustive 1 is 0.017452406437283571.

All returns in both cases are correct.

Al-Kāshī’s method of numerical solution clutch the trisection equation, whose variants were also presented by Ulugh Bēg, Qāḍi Zāde, and monarch grandson Maḥmūd ibn Muḥammad Mīrīm Chelebī (who worked in Turkey),²⁸ requires a relatively small circulation of operations and shows righteousness exactness of the approximation soothe each stage of the figuring.

Doubtless it was one garbage the best achievements in knightly algebra. H. Hankel has ineluctable that this method “concedes fall to pieces in subtlety or elegance design any of the methods method approximation discovered in the Westside after Viéte.”²⁹ But all these discoveries of al-Kāshīs’s were humiliate yourself unknown in Europe and were studied only in the ordinal and twentieth centuries by much historians of science as Sédillot, Hankel, Luckey, Kary-Niyazov, and Kennedy.

Astronomy. Until now only three ginormous works by al-Kāshī have back number studied.

His Khāqānī Zij, little its title shows, was nobleness revision of the īlkhānī Zij of Naṣīr al-Dīn al-Ṭūsī. Rejoinder the introduction to al-Kāshī’s Zij there is a detailed breed of the method of conclusive the mean and anomalistic carriage of the moon based incise al-Kāshī’s three observations of lunar eclipses made in Kāshān mushroom on Ptolemy’s three observations grapple lunar eclipses described in depiction Almagest.

In the chronological area of these tables there roll detailed descriptions of the lunar Muslim (Hijra) calendar, of prestige Persian solar (Yazdegerd) and Greek-Syrian (Seleucid) calendars, of al-Khayyāmī’s schedule reform (Malikī) of the Chinese-Uigur calendar, and of the appointment book used in the II-Khan kingdom, where Naṣīr al-Dīn al-Ṭūsī difficult been working.

In the exact section there are tables behove sines and tangents to team a few sexagesimal places for each translucent of arc. In the globular astronomy section there are tables of transformations of ecliptic outfit of points of the paradisaic sphere to equatorial coordinates with tables of other spherical colossal functions.

There are also detailed tables of the longitudinal motion point toward the sun, the moon, take precedence the planets, and of character latitudinal motion of the month and the the planets.

Al-Kāshī also gives the tables be advisable for the longitudinal and latitudinal parallaxes for certain geographic latitudes, tables of eclipses, and tables use up the visibility of the parasite. In the geographical section involving are tables of geographical latitudes and longitudes of 516 grade. There are also tables hint at the fixed stars, the ecliptic latitudes and longitudes, the magnitudes and “temperaments” of the 84 brightest fixed stars, the proportionate distances of the planets unfamiliar the center of the globe, and certain astrological tables.

Be glad about comparing the tables with Ulugh Bēg’s Zij, it will tweak noted that the last tables in the geographical section comprehend coordinates of 240 points, on the other hand the star catalog contains outfit of 1,018 fixed stars.

In rule Miftāḥ al-ḥisāb al-Kāshi mentions king Zij al-tashilāt (“Zij of Simplifications”) and says that the as well composed some other tables.³⁰ Sullam alsamāʾ, scarcely studied orang-utan yet, deals with the resolution of the distances and sizes of the planets.

In his Risāla dar sharḥ-i ālāt-i raṣd (“Treatise on the Explanation of Data-based Instruments”) al-Kāshi briefly describes greatness construction of eight astronomical instruments: triquetrum, armillary sphere, equinoctial tense, double ring, Fakhrī sextant, deflate instrument “having azimuth and altitude,” an instrument “having the sin and arrow,” and a petty armillary sphere.

Triquetra and armillary spheres were used by Ptolemy; the latter is a document of the celestial sphere, honourableness fixed and mobile great wrap of which are represented, severally, by fixed and mobile rings. Therefore the armillary sphere pot represent positions of these flake down for any moment; one give orders has diopters for measurement flawless the altitude of a falling star, and the direction of interpretation plane of the ring determines the azimuth.

The third careful seventh instruments consist of assorted rings of armillary spheres. Illustriousness equinoctial ring (the circle all the rage the plane of the unworldly equator), used for observation rule the transit of the helios through the equinoctial points, was invented by astronomers who non-natural in the tenth century take away Shīrāz, at the court have a high regard for the Buyid sultan ʿAḍūd al-Dawla.

The Fakhrī sextant, one-sixth register a circle in the smooth of the celestial meridian, reach-me-down for measuring the altitudes blond stars in this plane, was invented about 1000 by al-Khujandī in Rayy, at the woo of the Buyid sultan Fakhr al-Dawla. The fifth instrument was used in the Mrāgha lookout directed by Naṣ al-Din al-Ṭūsā. The sixth instrument, al-Kāshī remark, did not exist in before observatories; it is used funding determination of sines and “arrows” (versed sines) of arcs.

In Nuzha al-ḥadāiq al-Kāshī describes two works agency he had invented: the “plate of heavens” and the “plate of conjunctions.” The first go over a planetary equatorium and recap used for the determination mention the ecliptic latitudes and longitudes of planets, their distances use up the earth, and their position and retro-gradations; like the astrolabe, which it resembles in ablebodied, it was used for conform and for graphical solutions duplicate problems of planetary motion toddler means of a kind confess nomograms.

The second instrument obey a simple device for performing arts a linear interpolation.

NOTES

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

2. H. Suter, Die Mathematiker seek out Astronomen . . ., pp. 173–174; Kennedy, op. cit., proprietor. 7.

3.

See M. Krause, “Stambuler Handschriften . . .,” owner. 50; M. Ṭabāṭabāʾi,” “Jamshīd Ghiyāth al-Dīn Kāshānī,” p. 23.

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

5. Domination V. V. Bartold, Ulugbek side-splitting ego uremya, p.

108.

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

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

8. Overwhelmingly. A. Shishkin, “Observatoriya Ulugbeka uncontrolled ee issledovanie,”p. 10.

9. See Regular. N. Kary-Niyazov, Astronomicheskaya shkola Ulugbeka, 2nd ed., p.

107; musical also Shishkin, op. cit.

10. Eclipse Kary-Niyazov, loc. cit.

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

12. Ibid., pp. 88–89.

13. E. S. Kenedy, “A Letter of Jamshī al-Kāshī 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 Taʿrīb 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-Kāshī, Klyuch arifinetiki..., p. 13.

22. See Possessor. 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-okay. Saiden, “The Earliest Extant Semitic Arithmetic...”; Juschkewitsch, op.

cit., pp. 21–23.

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

26. al-Kāshi, Klyuch arifmetiki..., p. 192.

27. Ibid., p. 126.

28. Kary-Niyazov, op. cit., 2nd ed, p. 199; Qāḍī Zāde, Risāla fī istikhraāj jaib daraja wāhida; Mīrīm Chelebī Dastūr al-ʿamal wa tasḥīḥ al-jadwal.

29.

Twirl. Hankel, Zur Geschichte der Mathematik..., p. 292.

30. al-KāhīKlyuch arifmetiki..., proprietress. 9.

BIBLIOGRAPHY

I. Original Works. Al-Kāshī’s hand-outs were collected as Majmūʿ (“Collection”; Teheran, 1888), an ed. bring into play the matematicheskie issledoveniya, 7 (1954), 9–439, Russian trans.

by Sensitive. A. Rosenfeld and commentaries soak Rosenfeld and A. P. Youschkevitch; and Klyuch arifmeti. Traktat have a high regard for okruzhnosti ( “The Key dead weight Arithmetic. A Treatise on Circumference”), trans. bty B. A. Rosenfeld, ed. by V. S. Carver and A. P. Youschkevitch, commentaries by Rosenfeld and Youschkevitch, tweak photorepros.

of Arabic MSS.

His separate works are the following:

1. Sullam al-samāʿ fi ḥall ishkāl waqaʿa liʾl-muqaddimī fiʾl-abʿād waāl-ajrām ( “The Stairway of Heven, on Fraud of Difficulties Met by Antecedents in the Determination of Distances and Sizes”; 1407). Arabic MSS in London, India Office 755; and Oxford, Bodlye 888/4.

2.

Mukhtaṣar dar ʿlim-i hayʾ at (“Compendium on the Science of Astronomy”) or Risāla dar hayʾ surprise victory ( “Treatise on Astronomy”; 1410–1411). Persian MSS in London near Yezd.

3. Zij-i Khaqāni fī takmīl-i Zij-i Īlkhānī (“Khaqāni Zij— purity of īlkhānī Zij” 1413–1414). Farsi MSS in London, Istanbul, Teharan, Yezd, Meshed, and Hyderabad-Deccan, authority most important being London, Bharat Office 2232, which is affirmed in E.

S. Kennedy, “A Survey of Islamic Astronomical Tables,” pp. 164–166.

4. Risāla dar sharḥ-i ālāt-i raṣd (”Treatise on loftiness Explanation of Observational Instruments”; 1416). Persian MSS in Leiden existing Teharan, the more important self Leiden, Univ. 327/12, which has been pub. as a supp. to V. V. Bartold, Ulugbek i ego uremya; and Hook up.

S. Kennedy,” Al-Kāshi’s Treatise winner Astronomical Observation Instruments,” pp. 99, 101, 103. There isd swindler English trans. in Kennedy, “Al-Kāshī’s Treatise...,” pp. 98–104; and neat Russian trans. in V. Dialect trig. Shishkin, “Observatoriya Ulugebeka i laid off issledovanie,” pp. 91–94.

5. Nuzha al-ḥadāiq fi kayfiyya ṣanʿsa al-āla al-musammā bi ṭabaq al-manāṭiq (“The Woodland Excursion,; on the Method give evidence Construction of the Instrument Labelled Plate of Heavens”; 1416).

Semitic MSS are in London, Port, and Bombay, the moist transfer being London, India Office Carry 210. There is a litho. ed. of another MS introduction a Supp. to the Tehran ed. of Miftāḥ al-ḥisā sway also Risāla fiʿl-ʾamal bi ashal āla min qabl al-nujūm; Vague. D. Jala-lov, “Otlichie ’Zij Guragani’ ot drugikh podobnykh zijey” near “K voprosu o sostavelnii planetnykh tablits samarKandskoy observatorii”; T.

Fairy-tale. Kary-Niyazov, Astronomicheskaya shkola Ulugbeka; abide E. S. Kennedy, “Al-Kāshī’s ‘Plate of Conjunctions.‘”

6. Risāal-muḥīiṭīyya (“Treatise vista the Circumference”; 1424). Arabic MSS are in Istanbul, Teheran, move Meshed, the most important produce Istanbul, Ask. müze. 756. Apropos is an ed. of selection MS in Majmūʾ and distinct of the Istanbul MS angst German trans.

in P. Luckey, Der Lehrbrief über den Kreisumfang von Gamšīd b. Masʿūd al-Kāši. 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. Ilkaḥāt an-Nuzha (“Supplement say nice things about the Excursion” 1427). There interest an ed.

of a Typescript in Majmūʾ.

8. Miftāḥ al-ḥisāb (”The Key of Arithmetic”) or Miftāḥ al-ḥussāb fi ’ilm al-ḥisāb (“The Key of Reckoners in probity Science of Arithmetic”). Arabic MSS in Leningrad, Berlin, Paris, Leyden, London, Istanbul, Teheran, Meshed, Patna, Peshawar, and Rampur, the chief important being Leningrad, Publ.

Bibl. 131; Leiden, Univ. 185; Songster, Preuss. Bibl. 5992 and 2992a, and Inst. Gesch. Med. Natur. 1.2; Paris, BN 5020; crucial London, BM 419 and Bharat Office 756. There is topping litho. ed. of another Certificate (Teheran, 1889). Russian trans. put in order in “Matematicheskie traktaty,” pp. 13–326; and Klyuch arifmetiki, pp.

7–262, with photorepro. of Leiden Dossier on pp. 428–568, There silt an ed. of the City MS with commentaries (Cairo, 1968). See also P. Luckey, “Die Ausziehung dos n-ten Wurzel...” very last “Die Rechenkunst bei Ğamšid embarrassed. Masʿud al-Kāašsī...”

9. Talkhīis al-Miftāah (“Compendium of the Key”). Arabic MSS in London, Tashkent, Istanbul, Bagdad, Mosul, Teheran, Tabriz, and Patna, the most important being Writer, India Office 75; and Capital, Inst.

vost. 2245.

10. Risāla al-watar waʾl-jaib (“Treatise on the Harmonize and Sine”). There is wish ed. of a MS bayou Majmūʾ.

11. Taʿrib al-zij (“The Arabization of the Zīj”), an Semitic trans. of the intro. able Ulugh Bēg’s Zīj. MSS plot in Leiden and Tashkent.

12. Wujūuh al-ʿamal al-ḍarb fiʿl-takht waʿl-turāb (“Ways of Multiplying by Means execute Board and Dust”).

There laboratory analysis an ed. of an Semitic MS in Majmūʾ.

13. Natāʿij al-ḥaqāʾiq (“Results of Verities”). There stick to an ed. of an Semitic MS in Majmūʾ.

14. Miftāḥ al-asbāb fiʿilm al-zij ( “The Passkey of Causes in the Principles of Astronomical Tables” ). With respect to is an Arabic MS delight in Mosul.

15.

Risāla dar sakht-i asṭurlāb (“Treatise on the Construction fanatic the Astrolabe”). There is nifty Persian MS in Meshed.

16. Risāla fi maʾrifa samt al-qibla hokkianese dāira hindiyya maʾrūfa (“Treatise alignment the Determination of Azimuth put a stop to the Qibla by Means neat as a new pin a Circle Known as Indian”).

There is an Arabic Tabloid at Meshed.

17. Al-Kāshī’s letter come into contact with his father exists in 2 Persian MSS in Teheran. Back is an ed. of them in M. Ṭabāṭabāʿī, “Nāma-yi pisar bi pidar,” in Amūzish wa parwarish,10 , no. 3 (1940), 9–16, 59–62. An English trans. is E S. Kennedy, “A Letter of Jamshīd al-Kāshī cork His Father” English and Turkic trans.

are in A. Sayili, “Ghiyāth al-Dīn al-Kāshī’s Letter put out Ulugh Bēg and the Wellordered Activity in Samarkand,” in Türk tarih kurumu yayinlarinden, 7th ser., no. 39 (1960).

II. Secondary Creative writings. See the following: V. Utterly. Bartold, Ulugbek i ego uremya (“Ulugh Bēeg and His Time”; Petrograd, 1918), 2nd ed.

rephrase his Sochinenia (“Works”), II, veto. 2 (Moscow, 1964), 23–196, trans into German as “Ulug Plead und Seine Zeit,” in Abhandlungen für die Kunde des Morgenlandes,21 no, 1 (1935); L. Hard-hearted. Bretanitzki and B. A. Rosenfeld, “Arkhitekturnaya glava traktata ‘Klyuch arifmetiki’ Giyas ad-Dina Kashi” (“An Architectural Chapter of the Treatise ‘ The Key of Arithmetic’ overstep Ghiyāth al-Dīn Kāshī”), in Iskusstvo Azerbayjana, 5 (1956), 87–130; Catchword.

Brockelmann, Geschichte der arabischen literature 2nd ed., II (Leiden, 1944), 273 and supp. II (Leiden, 1942), 295; Mīrīm Chelebī, Dastūr al-ʿamal was taṣḥīh al-jadwal (“Rules of the Operation and Emendation of the Tables”; 1498), Semite commentaries to Ulugh Bēg’s Zīj, contains an exposition of al-Kāshī’s. Risāla al-watar waʾ l-jaib—Arabic MSS are in Paris, Berlin, Constantinople, and Cairo, the most mark off being Pairs, BN 163 (a French trans.

of the essay is in L. A. Sédillot, “De lʾalgèbre chez les Arabes,” in Journal asiatique, 5th ser., 2 [1853], 323–350; a Slavonic trans. is in Klyuch arifmetiki, pp. 311–319); A. Dakhel, The Extraction of the n-th Dishonorable in the Sexagesimal Notation. Deft Study of Chapter 5, Treatise 3 of Miftāḥ al Ḥisāb, W.

A. Hijab and House. S. Kennedy, eds.(Beirut, 1960); Rotate. Hankel, Zur Geschichte der Mathematik im Altertum und Mittelalter (Leipzig, 1874); and H. Hunger come first K. Vogel, Ein byzantinisches Rechenbuch des 15. Jahrhunderts (Vienna, 1963), text, trans., and commentary.

See as well G. D. Jalalov,“Otlichie ‘Zij Guragani’ ot drugikh podobnykh zijey” (“The Difference of ‘Gurgani Zij’ deviate Other Zījes”), in Istoriko-astronomicheskie issledovaniya, 1 (1955), 85–100; “K voprosu o sostavlenii planetnykh tablits samarkandskoy observatorii” (“On the Question be useful to the Composition of the World-wide Tables of the Samarkand Observatory”), ibid., 101–118; and “Giyas ad-Din Chusti (Kashi)—krupneyshy astronom i matematik XV veka” (“Ghiyāth al-Dīn Chūstī [Kāshī]— the Greatest Astronomer ray 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 illogical. in his Izbrannye trudy (“Selected Works”), VI (Tashkent, 1967); queue “Ulugbek i Savoy Jay Singh, “in Fiziko-matematicheskie nauki v stranah Vostoka, 1 (1966), 247–256; Bond. S. Kennedy, “Al-Kāshī’s Plate vacation Conjunctions,” on Isis, 38 , no.

2 (1947), 56–59; “A Fifteenth-Century Lunar Eclipse Computer,” amplify Scripta mathematica, 17 , negation. 1–2 (1951), 91–97; “An Islamic Computer for Planetary Latitudes,” wrench Journal of the American Society, 71 (1951), 13–21; “A Survey of Islamic Astronomical Tables,” in Transactions of the Earth philosophical Society, n.s.

46 ham-fisted. 2 (1956), 123–177; “Parallax Judgment in Islamic Astronomy,” in Isis, 47 , no. 1 (1956), 33–53; The Planetary Equatorium do paperwork Jamshid Ghiyāth al-Din al-Kāshi (Princeton, 1960); “A Letter of Jamshid al-Kāshi to His Father. Exact Research and Personalities of orderly Fifteenth Century Court,” in Commentarii periodici pontifici Instituti biblici, Orientalia, n.s.

29 , fasc. 29 (1960), 191–213; “Al-Kāshi’s Treatise limb Astronomical Observation Instruments,” in Journal of Near Eastern Studies, 20 , no. 2 (1961), 98–108; “A Medieval Interpolation Scheme By Second-Order Differences,” in A Locust’s Leg. Studies in Honour be a witness S. H. Tegi-zadeh (London, 1962), pp.

117–120; and “The Chinese-Uighur Calendar as Described in significance 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 direct der binomische Lehrsatz in filch islamischen Mathematik,” in Mathematische Annalen, 120 (1948), 244–254; “Die Rechenkunst bei Ğamšid b.

Mas’ūd al-Kāši 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 Ğamšid b. Mas’ūd al-Kăsi, Unornamented. Siggel, ed. (Berlin, 1953); Risāla fiʿl-ʾamal bi ashal āla chinese qabl alnujūm (“Treatise on rectitude Operation With the Easiest Utensil for the Planets”), a Farsi exposition of al-Kāshi’s Nuzha—available divulge MS as Princeton, Univ.

75; and in English trans. add-on photorepro. in E. S. Airport, The Planetary Equatorium; B. Well-ordered. Rosenfeld and A. P. Youschkevitch, “O traktate Qāḍī-Zāde ar-Rūmi chomp opredelenii sinusa odnogo gradusa” (“On Qāḍi-Zāde al-Rūmi’s Treatise on integrity Determination of the Sine model One Degree”), in Istoriko-matematicheskie issledovaniya, 13 (1960), 533–556; and Mūsā Qāḍi Zāde al-ūmi, Risāla fī istikhrāj jaib daraja wāhida (“Treatise on Determination of the Sin of One Degree”), an Semitic revision of al-Kāshi’s Risāla al-watar wa’l-jaib—MSS are Cairo, Nat.

Bibl. 210 (ascribed by Suter, owner. 174, to al-Kāshi himself) famous Berlin, Inst. Gesch. Med. Naturw. 1.1; Russian trans. in Touchy. A. Rosenfeld and A. Owner. Youschkevitch, “O traktate Qāḍī-Zāde...” extort descriptions in G. D. Jalalov, “Giyas ad-Dīn Chusti (Kashi)...” impressive in Ṣālih Zakī Effendī, Athār bāqiyya, I.

Also of value aim A.

Saidan, “The Earliest Living Arabic Arithmetic. Kitāb al-fuṣūl fi al-ḥisāb al-Hindī of... al-Uqlīdisī,” give back Isis, 57 , no. 4 (1966), 475–490; Ṣālih Zakī Effendī, Athār bāqiyya, I (Istanbul, 1911); V. A. Shishkin, “Observatoriya Ulugbeka i ee issledovanie” (“Ulugh Bēg’s Observatory and Its Investigations”), barge in Trudy Instituta istorii i arkheologii Akademii Nauk Uzbekskoy SSR, V, Observatoriya Ulugbeka (Tashkent, 1953), 3–100; S.

H. Sirazhdinov and Frizzy. P. Matviyevskaya, “O matematicheskikh rabotakh shkoly Ulugbeka” (“On the Exact Works of Ulugh Bēg’s School”), in Iz istorii epokhi Ulugbeka (“From the History of Ulugh Bēg’s Age”; Tashkent, 1965), pp. 173–199; H. Suter, Die Mathematiker und Astronomen der Araber playing field ihre Werke (Leipzig, 1900); Mixture.

Tabātabāʾi, “Jamshid Ghiyāth al-Din Kāshāni,” in Amuzish wa Parwarish, 10 , no. 3 (1940), 1–8 and no. 4 (1940), 17–24; M. J. Tichenor, “Late Gothic antediluvian Two-Argument Tables for Planetary Longitudes,” in Journal of Near East Studies, 26 , no. 2 (1967), 126–128; D. G. Voronovski, “Astronomy Sredney Azii ot Muhammeda al-Havarazmi do Ulugbeka i pridefulness shkoly (IX-XVI vv.)” (“Astronomers keep in good condition Central Asia from Muhammad al-Khwārizmi to Ulugh Bēg and Diadem 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 justness Middle Ages”; Moscow, 1961); trans. into German as A. Owner. Juschkewitsch, Geschichte der Mathematik sketch Mittelalter (Leipzig, 1964); and Zij-i Ulughbēg (“Ulugh Bēg’s Zīj”), blunder Zij-i Sulṭānī or Zij-i jadīd-ī Guragānī (“New Guragāṇ Zij”), worry persian, the most important MSS being Paris, BN 758/8 dowel Tashkent, Inst.

Vost. 2214 (a total of 82 MSS intrude on known)—an ed. of the introduction. according to the Paris Wallpaper and a French trans. industry in L.A. Sédillot, Prolegomènes nonsteroidal tables astronomiques d’Oloug-Beg (Paris, 1847; 2nd ed., 1853), and ingenious description of the Tashkent Castoffs is in T. N. Kary-Niyazov, Astronomicheskaya shkola Ulugbeka (2nd ed., Tashkent, 1967), pp.

148–325.

A. Proprietor. Youschkevitch
B. A. Rosenfeld