Ghiyāth al-Dīn Jamshīd Kāshānī | |
|---|---|
%3b_Miftah_al-Hisab_signed_ibn_Muhammad_Mu%2527min_Taj_al-Din_al-Shirazi%2c_Iran%2c_dated_31_May_1656.jpg) Opening bifolio of a manuscript of al-Kashi'sMiftah al-Hisab. Copy created inSafavid Iran, dated 1656 | |
| Title | al-Kashi |
| Personal life | |
| Born | c. 1380 |
| Died | 22 June 1429 (1429-06-23) (aged 48) |
| Era | Islamic Golden Age-Timurid Renaissance |
| Region | Iran |
| Main interest(s) | Astronomy,Mathematics |
| Notable idea(s) | Pi decimal determination to the 16th place Law of cosines |
| Notable work(s) | Sullam al-sama' |
| Occupation | PersianMuslimscholar |
| Religious life | |
| Religion | Islam |
| Muslim leader | |
Influenced | |
Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (oral-Kāshānī)[2] (Persian:غیاثالدین جمشید کاشانیGhiyās-ud-dīn Jamshīd Kāshānī; c. 1380 – 22 June 1429) was a Persianastronomer andmathematician during the reign ofTamerlane.
Much of al-Kāshī's work was not brought toEurope and still, even the extant work, remains unpublished in any form.[3]
%3b_al-Risala_al-Kamaliya%2c_Safavid_Iran%2c_dated_26_June_1520.jpg)
Al-Kashi was born in 1380, inKashan, in central Iran, to aPersian family.[4][5] This region was controlled byTamerlane, better known as Timur.
The situation changed for the better when Timur died in 1405, and his son,Shah Rokh, ascended into power. Shah Rokh and his wife,Goharshad, a Turkish princess, were very interested in thesciences, and they encouraged their court to study the various fields in great depth. Consequently, the period of their power became one of many scholarly accomplishments. This was the perfect environment for al-Kashi to begin his career as one of the world's greatest mathematicians.
Eight years after he came into power in 1409, their son,Ulugh Beg, founded an institute inSamarkand which soon became a prominent university. Students from all over theMiddle East and beyond, flocked to this academy in the capital city of Ulugh Beg's empire. Consequently, Ulugh Beg gathered many great mathematicians and scientists of theMiddle East. In 1414, al-Kashi took this opportunity to contribute vast amounts of knowledge to his people. His best work was done in the court of Ulugh Beg.
Al-Kashi was still working on his book, called “Risala al-watar wa’l-jaib” meaning “The Treatise on the Chord and Sine”, when he died, in 1429. Some state that he was murdered and say that Ulugh Beg probably ordered this, whereas others suggest he died a natural death.[6][7] Regardless, after his death, Ulugh Beg described him as "a remarkable scientist" who "could solve the most difficult problems".[1][8]
Al-Kashi produced aZij entitled theKhaqani Zij, which was based onNasir al-Din al-Tusi's earlierZij-i Ilkhani. In hisKhaqani Zij, al-Kashi thanks theTimurid sultan and mathematician-astronomerUlugh Beg, who invited al-Kashi to work at hisobservatory (seeIslamic astronomy) and hisuniversity (seeMadrasah) which taughttheology. Al-Kashi producedsine tables to foursexagesimal digits (equivalent to eightdecimal places) of accuracy for each degree and includes differences for each minute. He also produced tables dealing with transformations betweencoordinate systems on thecelestial sphere, such as the transformation from theecliptic coordinate system to theequatorial coordinate system.[9]
He wrote the bookSullam al-sama' (Stairway to the Heavens, 1407) on the resolution of difficulties met by predecessors in the determination of sizes and distances ofheavenly bodies such as theEarth, theMoon, theSun, and thestars.
In 1416, al-Kashi wrote theTreatise on Astronomical Observational Instruments, which described a variety of different instruments, including thetriquetrum andarmillary sphere, theequinoctial armillary andsolsticial armillary ofMo'ayyeduddin Urdi, thesine andversine instrument of Urdi, thesextant ofal-Khujandi, the Fakhri sextant at theSamarqand observatory, a double quadrantAzimuth-altitude instrument he invented, and a small armillary sphere incorporating analhidade which he invented.[10]
Al-Kashi invented the plate of conjunctions, ananalog computing instrument used to determine the time of day at whichplanetary conjunctions will occur,[11] and for performinglinear interpolation.[12]
Al-Kashi also invented a mechanical planetarycomputer which he called the Plate of Zones, which could graphically solve a number of planetary problems, including the prediction of the true positions inlongitude of theSun andMoon,[12] and theplanets in terms ofelliptical orbits;[13] thelatitudes of the Sun, Moon, and planets; and theecliptic of the Sun. The instrument also incorporated analhidade andruler.[14]
Al-Kashi made the most accurateapproximation ofπ to date in hisal-Risāla al-muhītīyya (Treatise on the Circumference).[15] He correctly computed2π to 9sexagesimal digits[16] in 1424,[9] and he converted this estimate of 2π to 16decimal places of accuracy.[17] This was far more accurate than the estimates earlier given inGreek mathematics (3 decimal places byPtolemy, AD 150),Chinese mathematics (7 decimal places byZu Chongzhi, AD 480) orIndian mathematics (11 decimal places byMadhava ofKerala School,c. 14th Century). The accuracy of al-Kashi's estimate was not surpassed untilLudolph van Ceulen computed 20 decimal places ofπ 180 years later.[9] Al-Kashi's goal was to compute the circle constant so precisely that the circumference of the largest possible circle (ecliptica) could be computed with the highest desirable precision (the diameter of a hair).
In Al-Kashi'sRisālah al-watar waʾl-jaib (Treatise on the Chord and Sine), he computed sin 1° to nearly as much accuracy as his value forπ, which was the most accurate approximation of sin 1° in his time and was not surpassed untilTaqi al-Din in the sixteenth century. Inalgebra andnumerical analysis, he developed aniterative method for solvingcubic equations, which was not discovered in Europe until centuries later.[9]
A method algebraically equivalent toNewton's method was known to his predecessorSharaf al-Din al-Tusi. Al-Kāshī improved on this by using a form of Newton's method to solve to find roots ofN. Inwestern Europe, a similar method was later described byHenry Briggs in hisTrigonometria Britannica, published in 1633.[18]
In order to determine sin 1°, al-Kashi discovered the following formula, often attributed toFrançois Viète in the sixteenth century:[19]
Al-Kashi'sMiftāḥ al-ḥisāb (Key of Arithmetic, 1427) explained how tosolve triangles from various combinations of given data. The method used when two sides and their included angle were given was essentially the same method used by 13th century Persian mathematicianNaṣīr al-Dīn al-Ṭūsī in hisKitāb al-Shakl al-qattāʴ (Book on the Complete Quadrilateral, c. 1250),[20] but Al-Kashi presented all of the steps instead of leaving details to the reader:
Another case is when two sides and the angle between them are known and the rest are unknown. We multiply one of the sides by the sine of the [known] angle one time and by the sine of its complement the other time converted and we subtract the second result from the other side if the angle is acute and add it if the angle is obtuse. We then square the result and add to it the square of the first result. We take the square root of the sum to get the remaining side....
— Al-Kāshī'sMiftāḥ al-ḥisāb,
translation by Nuh Aydin, Lakhdar Hammoudi, and Ghada Bakbouk[21]
Using modern algebraic notation and conventions this might be written
After applying the Pythagorean trigonometric identity this is algebraically equivalent to the modernlaw of cosines:
InFrance, the law of cosines is sometimes referred to as thethéorème d'Al-Kashi.[22][23]
In discussingdecimal fractions,Struik states that (p. 7):[24]
"The introduction of decimal fractions as a common computational practice can be dated back to theFlemish pamphletDe Thiende, published atLeyden in 1585, together with a French translation,La Disme, by the Flemish mathematicianSimon Stevin (1548-1620), then settled in the NorthernNetherlands. It is true that decimal fractions were used by theChinese many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal andsexagesimal fractions with great ease in hisKey to arithmetic (Samarkand, early fifteenth century).[25]"
In consideringPascal's triangle, known in Persia as "Khayyam's triangle" (named afterOmar Khayyám), Struik notes that (p. 21):[24]
"The Pascal triangle appears for the first time (so far as we know at present) in a book of 1261 written byYang Hui, one of the mathematicians of theSong dynasty inChina.[26] The properties ofbinomial coefficients were discussed by the Persian mathematician Jamshid Al-Kāshī in hisKey to arithmetic of c. 1425.[27] Both in China and Persia the knowledge of these properties may be much older. This knowledge was shared by some of theRenaissance mathematicians, and we seePascal's triangle on the title page ofPeter Apian'sGerman arithmetic of 1527. After this, we find the triangle and the properties of binomial coefficients in several other authors.[28]"
In 2009,IRIB produced and broadcast (through Channel 1 of IRIB) a biographical-historical film series on the life and times of Jamshid Al-Kāshi, with the titleThe Ladder of the Sky[29][30] (Nardebām-e Āsmān[31]). The series, which consists of 15 parts, with each part being 45 minutes long, is directed byMohammad Hossein Latifi and produced by Mohsen Ali-Akbari. In this production, the role of the adult Jamshid Al-Kāshi is played by Vahid Jalilvand.[32][33][34]
AL-KASHl Or AL-KASHANI, GHIYATH AL-DIN DjAMSHlD B. MASCUD B. MAHMUD, Persian mathematician and astronomer who wrote in his mother tongue and in Arabic.
Al-Kāshī, or al-Kāshānī (Ghiyāth al-Dīn Jamshīd ibn Mas˓ūd al-Kāshī (al-Kāshānī)), was a Persian mathematician and astronomer.
On donne deux côtés et un angle. [...] Que si l'angle donné est compris entre les deux côtés donnés, comme l'angle A est compris entre les deux côtés AB AC, abaissez de B sur AC la perpendiculaire BE. Vous aurez ainsi le triangle rectangle [BEA] dont nous connaissons le côté AB et l'angle A; on en tirera BE, EA, et l'on retombera ainsi dans un des cas précédents; c. à. d. dans le cas où BE, CE sont connus; on connaîtra dès lors BC et l'angle C, comme nous l'avons expliqué[Given [...] the angle A is included between the two sides AB AC, drop from B to AC the perpendicular BE. You will thus have the right triangle [BEA] of which we know the side AB and the angle A; in that triangle compute BE, EA, and the problem is reduced to one of the preceding cases; that is, to the case where BE, CE are known; we will thus know BC and the angle C, as we have explained.]
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