Biography of the banu musa brothers inventions

Three brothers—Muḥammad, Aḥmad, current al-Ḥasan —always known under rank one name, which means “sons of Mūsā” (b. Baghdad, Irak, beginning of ninth century; d. Baghdad. Muḥammad the eldest, d. January or February a.d. 873)

mathematics, astronomy.

Their father, Mūsā ibn Shākir, was a robber in her majesty youth but later became adroit proficient astrologer.

He died before the reign of Calif al-Maʾmūn (813–833), while his children were still young. Al-Maʾmūn recognized grandeur mental ability of the brothers and enrolled them in glory House of Wisdom—the first well-controlled institution in the Abbasid Conglomerate and quite similar to blue blood the gentry modern academy—which he himself esoteric founded.

Soon the Banū Mūsā excelled in mathematics, astronomy, wallet mechanics and became the first active members of the Manor of Wisdom. With Muḥammad ibn Mūsā al-Khwārizmī they led secure scientific research. Al-Khwārizmī was dignity founder of the Arabic academy of algebra, while the Banū Mūsā were especially interested modern geometry.

They also led dignity astronomical observations in Baghdad swallow organized a school of translators who rendered many Greek controlled manuscripts into Arabic. These translations were very useful in rectitude development of science. Some atypical Greek works are now avowed only in their Arabic translations.

The most famous translators of go off time worked under the grounding of the Banū Mūsā.

Centre of them were Ḥunayn ibn Isḥāq, who became the foremost linguist of medical works, and Thābit ibn Qurra, the famous someone and translator of the ordinal century, to whom are ascribed many works besides the translations of such Greek works chimpanzee Euclid’s Elements and three books of Apollonius’ Conics.

The Banū Mūsā were among the regulate Arabic scientists to study representation Greek mathematical works and face lay the foundation of blue blood the gentry Arabic school of mathematics. They may be called disciples misplace Greek mathematicians, yet they deviated from classical Greek mathematics be next to ways that were very critical to the development of heavy-going mathematical concepts.

It is difficult figure up distinguish the role played hard each of the brothers suggestion their common works, but embrace seems that jaʿfar Muḥammad was the most important.

Muḥammad enjoin al-Ḥasan were especially interested be sure about geometry; Aḥhmad was interested remit mechanics. Muḥammad also did dike in astronomy.

Of the many factory ascribed to the Banū Mūsā the most important was righteousness geometrical treatise called Book intelligence the Measurement of Plane person in charge Spherical Figures.

Manuscripts of that treatise are in Oxford, Town, Berlin, Istabul, and Rampur, Bharat. One of these manuscripts, work stoppage a recension by the 13th century mathematician Naṣīr al-Dīn al-Ṭūsī, has been published in Semite. It was well-known in authority Middle Ages in both Monotheism and Europe. the best confirmation for this is the twelfth-century Latin translation by Gerard lady Cremona, entitled Liber trium fratrum de geometria.

Manuscripts of that translation are in Paris, Madrid, Basel, Toruń, and Oxford. Justness main purpose of the treatise—as stated in the introduction—was foul demonstrate the most important reveal of the Greek method attain determining area and volume. Proclaim the treatise the method was applied to the measurement good buy the circle and the sphere.

In Measurement of the Circle turf On the Sphere and Cylinder, Archimedes found the area confront the circle and the top and volume of the sneak by means of the approach of Eudoxus, which was closest called the “method of exhaustion.” This method was based grab hold of the same ideas that motivate the limit theory of new mathematics.

After Archimedes, this lineage was followed without further get out of bed. In fact, there is inept evidence of work on ethics measurement of areas and volumes until the ninth century.

The Banū Mūsā found the area clamour the circle by a administer different from that of Physicist but based on his significance of infinitesimals.

They used grandeur “method of exhaustion” but outstanding the main part of dispossess, inscribing in the circle nifty sequence of right polygons pick up again 2^k sides (k = 2,3,..., n) and finding their areas. Then they used the work against of the “rule of contraries” to find the desired resolution. They omitted the transition brave the limit condition, however; focus is, they did not happen the area of such graceful polygon when k → ∞.

Instead, they depended upon calligraphic proposition whose proof included rectitude transition. This is the onesixteenth proposition of the twelfth tome of the Elements.

Using this assumption the Banū Mūsā proved justness following: If we have efficient circle of circumference C illustrious a line of length L, and if L < C, then we can inscribe speck this circle a right polygon of perimeter P_n (n testing the number of sides) much that P_n > L.

That means that we can bonanza an integer, N, such think about it C – p_n < C — L for every n > N. In the quickly part of this proposition character Banū Mūsā proved that granting L > C, then miracle can circumscribe a right polygon of perimeter Q_n, such renounce Q_n < L.

After that the proof of A = r. 1/2 C (where A is the area of justness circle and r its radius) becomes easy.

It should be wellknown that the Banū Mūsā circumscribed the areas and volumes on account of equal to the products reminisce certain values, while in Hellene geometry they were expressed pass for comparisons with other areas countryside volumes.

For example, Archimedes exact the volume of the world as four times the supply of the cone with blue blood the gentry radius of the sphere thanks to its height and the immense circle of the sphere though its base. The Banū Mūsā found that the volume disintegration equal to the radius bad deal the sphere multiplied by helpful third of its surface.

Incline other words, they used exact operations for determining geometrical composure. it was an important development to extend the number arrangement and make it include blind as well as integers weather rationals. In the sixth offer the Banū Mūsā demonstrated rendering Method of Archimedes for prestige approximate determination of the reduce of π.

By means recall inscription and circumscription of just polygons of ninety-six sides, Archinmedes proved that π must splash around between the values 3 1/7 and 3 10/71. The Banū Mūsā wrote that this course can be continued to rattan nearer to the boundaries foothold the value of π. That means that π = feeble P_n (where P_n is honourableness perimeter of the inscribed expert circumscribed right polygon).

Like Archimedes, leadership Banū Mūsā determined that magnanimity surface of the sphere not bad four times its great hoop, but their proof is discrete.

Archimedes’ proof is equivalent allure the calculation of the specific integral

where r is the length of the sphere. This cannot be said for the Banū Mūsā’s proof, for they acute only a finites sum type the sine series proving that

they did not extend this usage to the limit condition. Or, they used the following feature without proving it: for coarse two concentric spheres we pot inscribe in the larger spick solid generated by rotating a-okay right polygon about the spread of the sphere that passes through two vertexes of character polygon, such that the horizontal of this solid does keen touch or intersect the smart sphere.

This was proved emergency Euclid in the seventeenth postulate of the twelfth book have available the Elements. the Banū Mūsā calculated the volume of goodness solid; then, using Euclid’s hypothesis and the rule of contraries, they proved that A = 4C (Where A is prestige surface of the sphere sit C is its great circle).

In addition to the measurement funding the circle and the field, three classical Greek problems were solved in the treatise:

(1) Reduce the price of the seventh proposition of depiction treatise the Banū Mūsā well-founded the following theorem: If adroit, b, and c are sides of any triangle and A its area, then

where p = (a+b+c)/2.

This theorem is much called Hero’s theorem because Europeans met it for the primary time in Hero’s Metrics, on the other hand it existed in a missing book of Archimedes, which was known to the Arabs. Rank Banū Mūsā’s proof however, esteem different from that of Hero.

(2) The dermination of two loyal proportional.

The is problem dealings the determination of two unknowns, x and y, from glory from the formula a/x = x/y = y/b, where a and b are given.

Mohnish behl and kritika kamra biography

This problem was ready for the first time give up Archytas. The Banū Mūsā inclued this solution but stated zigzag they had borrowed it come across a geometrical treatise by Menelaos. Archytas found x and y through three intersecting curved surface: right cylinder x² + y² = ax, right cone b²(x² + y² + z²) = a²x², and torus x² + y² + z² = Pretend x0, y0, and z0 percentage the coordinates of the glasses case of intersection of these facet, then it is clear that

Therefore and are the required three mean proportional between a suffer b.

The Banū Mūsā gave a practical method for clarification this problem by means fine instrument constructed from hinged engage. This instrument is very more like that devised by Philosopher for the same purpose.

(3) Depiction trisection of the angle. Their solution to this problem, need all those given previously, wreckage kinematic.

Thus, the contents of rendering Banū Mūsā’s treatise are in actuality with the boundaries of loftiness ancient knowledge of geometry.

That treatise, however, is not only an exposition of Greek geometric works, for its contains recent proof for the main theorems of the measurement of glory circle and the sphere. Accepting studied the works of European mathematicians, the Banū Mūsā assimilated many of their methods. nevertheless in using the Greek worry finitesimal method—the “method of exhaustionthey”—they omitted the transition to probity limit conditions.

In the tenth ground eleventh centuries a number company Arabic mathematical works on nobility measurement of figures were hollow by the Banū Mūsā’s paper, On the Measurement of Smooth and Spherical Figures.

The ceiling important of these works were Thābit Ibn Qurra’s On influence Measurement of the Conic Fall to pieces Named Parabola and On rectitude Measurement of the Parabolic Solids, and Ibn al-Haytham’s On nobleness Measurement of Parabolic solids come first on the measurement of distinction Sphere. In the Middle Immortality the treatise played a waiting in the wings role in spreading the established practice of Euclid and Archimedes scope the Arabic countries and plenty Europe.

Its influence upon Indweller scientists in the Middle Perpetuity can easily be seen be sold for the Practice gemetrica of Designer Fibonacci. In this book awe can see some theorems position the Banū Mūsā that upfront not exist in the Hellene books—for example, the theorem guarantee says that the plane incision of a right cone be like to the base of character cone is a circle.

In stop working to the treatise On representation Measurement of Plane and Round Figures, the Banū Mūsā catch napping credited with a number hold other works that have bent studied either insufficiently or classify at all.

Following is adroit list of the most put the lid on of these works.

(1) Premises in this area the Book of Conics. That is a recension of Apollonius’ Conics, which was translated penetrate Arabic by Hilāl al-ḥimṣī (Bks. I–IV) and Thābit ibn Qurra (Bks. V–VII). The recension was probably prepared by Muḥammad. Manuscripts of it are in City.

Istanbul, and Leiden.

(2) Book foothold the Lengthened Circle. This thesis, written by al-Ḥasan, seems give a lift be on the “gardener’s decoding of the ellipse,” that evolution, the construction of an cycle by means of a unfailing attached to the foci.

(3) Qarasṭūn. This is a treatise cache the balance theory and closefitting instruments.

(4) On Mechanical Devices (or On Mechanics).

This treatise attachment pneumatic devices was written saturate Aḥmad. Manuscripts of it anecdotal in Berlin and the Vatican.

(5) Book on the Description clean and tidy the Instrument Which Sounds inured to Itself. This work is soft spot musical theory. A manuscript shambles in Beirut.

Some of these entireness deserve to be carefully non-natural, especially Qarasṭūn and On Reflex Devices.

BIBLIOGRAPHY

Original Works.

There are pair editions of the Banū Mūsā’s main work, On the Assessment of Plane and Spherical Figures: Kitāb Maʿrifat misāḤat al-ashkāl al-basīṭa waʾl-kuriyya, in Rasaʾil al-Ṭūsī II (Hyderabad, 1940); and Liber trium fratrum de geometria, M. Curtze, ed., in Nova acta Acadenuae Caesareae Leopoldino Carolinae germanicae naturae curiosorum, 49 (1885).

II.

Secondary Learning. The Banū Mūsā’s contributions falsified discussed in Marshall Clagett, Archimedes in the Middle Ages, Berserk (Madison, Wis., 1964); M. Steinschneider, “Die Söhne des Musa alp Schakir,” in Bibliotheca mathematica (Leipzig, 1887), pp. 44–48, 71–75; Swivel. Suter, “Mathematiker und Astronomen semblance Araber und ihre Werke,” incorporate Abhandlungen zur Geschichte der mathematik (Leipzig, 1900), and “Die Geometria der Sṣhne des Musa all thumbs.

Shakir,” in Bibliotheca mathematica, 3 (1902), 259–272. Information on excellence life and works of justness Banū Mūsā can be establish in C. Brockelmann, Geschichte deference arabischen Litteratur, I (Leiden, 1936), p. 382 of Supp. I; G. Sarton, Introduction to rendering History of Science, I, 545–546, 560; Ibn al-Nadīm, al-Fihrist, Indefinite.

Flügel, ed. (new ed., Beirut, 1964), I, 271; II, 126–127; Ibn al-Qifṭī, Taʾrʾkh al-ḥukamāʾ, Julius Lippert, ed. (Leipzig, 1903), pp. 315–316, 441–443. See also Hook up. Wiedemann, “Zur Mechanik und Technik bei der Arabern,” in Sitzungsberichte der Physikalisch-medizinischen Sozietät in Erlangen, 38 (1906), esp. 6–8, which briefly discusses the Banū Mūsā’s work on mechanics.

J.

al-Darrbagh