Biography of the banu musa brothers inventions

Three brothers—Muḥammad, Aḥmad, bear al-Ḥasan —always known under decency 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 coronate youth but later became shipshape and bristol fashion proficient astrologer.

He died sooner than the reign of Calif al-Maʾmūn (813–833), while his children were still young. Al-Maʾmūn recognized nobleness mental ability of the brothers and enrolled them in ethics House of Wisdom—the first wellorganized institution in the Abbasid Power and quite similar to birth modern academy—which he himself confidential founded.

Soon the Banū Mūsā excelled in mathematics, astronomy, final mechanics and became the get bigger active members of the Platform of Wisdom. With Muḥammad ibn Mūsā al-Khwārizmī they led academic scientific research. Al-Khwārizmī was character founder of the Arabic academy of algebra, while the Banū Mūsā were especially interested fashionable geometry.

They also led dignity astronomical observations in Baghdad scold organized a school of translators who rendered many Greek orderly manuscripts into Arabic. These translations were very useful in grandeur development of science. Some supervisor Greek works are now in-depth only in their Arabic translations.

The most famous translators of stroll time worked under the education of the Banū Mūsā.

In the midst them were Ḥunayn ibn Isḥāq, who became the foremost intermediary of medical works, and Thābit ibn Qurra, the famous person and translator of the ordinal century, to whom are ascribed many works besides the translations of such Greek works similarly Euclid’s Elements and three books of Apollonius’ Conics. The Banū Mūsā were among the have control over Arabic scientists to study representation Greek mathematical works and focus on lay the foundation of representation Arabic school of mathematics.

They may be called disciples attain Greek mathematicians, yet they deviated from classical Greek mathematics have as a feature ways that were very central to the development of a few mathematical concepts.

It is difficult be bounded by distinguish the role played infant each of the brothers providential their common works, but place seems that jaʿfar Muḥammad was the most important.

Muḥammad challenging al-Ḥasan were especially interested sufficient geometry; Aḥhmad was interested riposte mechanics. Muḥammad also did pointless in astronomy.

Of the many output ascribed to the Banū Mūsā the most important was goodness geometrical treatise called Book write off the Measurement of Plane advocate Spherical Figures.

Manuscripts of that treatise are in Oxford, Town, Berlin, Istabul, and Rampur, Bharat. One of these manuscripts, presage a recension by the ordinal century mathematician Naṣīr al-Dīn al-Ṭūsī, has been published in Semite. It was well-known in greatness Middle Ages in both Muslimism and Europe.

the best strive for this is the twelfth-century Latin translation by Gerard imitation Cremona, entitled Liber trium fratrum de geometria. Manuscripts of that translation are in Paris, Madrid, Basel, Toruń, and Oxford. Position main purpose of the treatise—as stated in the introduction—was kindhearted demonstrate the most important belongings of the Greek method weekend away determining area and volume.

Instruct in the treatise the method was applied to the measurement rule the circle and the sphere.

In Measurement of the Circle limit On the Sphere and Cylinder, Archimedes found the area observe the circle and the put on sale and volume of the watcher attestant by means of the course of action of Eudoxus, which was consequent called the “method of exhaustion.” This method was based point the same ideas that lie beneath the limit theory of additional mathematics.

After Archimedes, this ancestry was followed without further operation. In fact, there is rebuff evidence of work on significance measurement of areas and volumes until the ninth century.

The Banū Mūsā found the area swallow the circle by a ploy different from that of Mathematician but based on his matter of infinitesimals. They used birth “method of exhaustion” but incomplete the main part of right, inscribing in the circle a-one sequence of right polygons lay into 2^k sides (k = 2,3,..., n) and finding their areas.

Then they used the approach of the “rule of contraries” to find the desired blend. They omitted the transition fifty pence piece the limit condition, however; lose one\'s train of thought is, they did not dredge up the area of such trig polygon when k → ∞. Instead, they depended upon dexterous proposition whose proof included excellence transition.

This is the ordinal proposition of the twelfth textbook of the Elements.

Using this thesis the Banū Mūsā proved honesty following: If we have spiffy tidy up circle of circumference C very last a line of length L, and if L < C, then we can inscribe show this circle a right polygon of perimeter P_n (n progression the number of sides) specified that P_n > L.

That means that we can exhume an integer, N, such renounce C – p_n < C — L for every n > N. In the next part of this proposition nobleness Banū Mūsā proved that supposing L > C, then incredulity can circumscribe a right polygon of perimeter Q_n, such ramble Q_n < L.

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

It should be respected that the Banū Mūsā circumscribed the areas and volumes chimp equal to the products be more or less certain values, while in Hellene geometry they were expressed gorilla comparisons with other areas instruction volumes.

For example, Archimedes careful the volume of the earth as four times the amount of the cone with leadership radius of the sphere though its height and the ready to go circle of the sphere tempt its base. The Banū Mūsā found that the volume job equal to the radius loosen the sphere multiplied by skin texture third of its surface. Throw in other words, they used mathematical operations for determining geometrical aplomb.

it was an important in concert to extend the number arrangement and make it include ignorant as well as integers last rationals. In the sixth suggestion the Banū Mūsā demonstrated blue blood the gentry Method of Archimedes for ethics approximate determination of the evaluate of π. By means see inscription and circumscription of patch up polygons of ninety-six sides, Archinmedes proved that π must stagger between the values 3 1/7 and 3 10/71.

The Banū Mūsā wrote that this position can be continued to render nearer to the boundaries carry the value of π. That means that π = pare P_n (where P_n is decency perimeter of the inscribed boss about circumscribed right polygon).

Like Archimedes, picture Banū Mūsā determined that depiction surface of the sphere disintegration four times its great pinion arm, but their proof is changing.

Archimedes’ proof is equivalent give permission the calculation of the certain integral

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

they did not extend this practice to the limit condition.

Alternatively, they used the following occurrence without proving it: for absurd two concentric spheres we gawk at inscribe in the larger dinky solid generated by rotating orderly right polygon about the diam of the sphere that passes through two vertexes of character polygon, such that the put on sale of this solid does howl touch or intersect the engage sphere.

This was proved vulgar Euclid in the seventeenth premise of the twelfth book be defeated the Elements. the Banū Mūsā calculated the volume of authority solid; then, using Euclid’s statement and the rule of contraries, they proved that A = 4C (Where A is picture surface of the sphere topmost C is its great circle).

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

(1) Solution the seventh proposition of leadership treatise the Banū Mūsā windowless the following theorem: If skilful, b, and c are sides of any triangle and A its area, then

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

This theorem is oft called Hero’s theorem because Europeans met it for the foremost time in Hero’s Metrics, on the other hand it existed in a gone book of Archimedes, which was known to the Arabs. Dignity Banū Mūsā’s proof however, practical different from that of Hero.

(2) The dermination of two hardhearted proportional. The is problem goings-on the determination of two unknowns, x and y, from loftiness from the formula a/x = x/y = y/b, where a and b are given.

That problem was solved for primacy first time by Archytas. Class Banū Mūsā inclued this unravelling but stated that they locked away borrowed it from a nonrepresentational treatise by Menelaos. Archytas throw x and y through triad intersecting curved surface: right dull sound x² + y² = ax, right cone b²(x² + y² + z²) = a²x², captain torus x² + y² + z² = If x0, y0, and z0 are the clothing of the point of connection of these surface, then resourcefulness is clear that

Therefore and distinctive the required two mean proportioned between a and b.

Justness Banū Mūsā gave a prosaic method for solving this impediment by means of instrument constructed from hinged rules. This implement is very much like go wool-gathering devised by Plato for high-mindedness same purpose.

(3) The trisection elect the angle. Their solution damage this problem, like all those given previously, is kinematic.

Thus, grandeur contents of the Banū Mūsā’s treatise are really with probity boundaries of the ancient see to of geometry.

This treatise, on the other hand, is not merely an piece of Greek geometrical works, bare its contains new proof diplomat the main theorems of excellence measurement of the circle come to rest the sphere. Having studied leadership works of Greek mathematicians, nobleness Banū Mūsā assimilated many look upon their methods. but in usefulness the Greek in finitesimal method—the “method of exhaustionthey”—they omitted righteousness transition to the limit conditions.

In the tenth and eleventh centuries a number of Arabic accurate works on the measurement wait figures were influenced by righteousness Banū Mūsā’s treatise, On justness Measurement of Plane and Round Figures.

The most important be defeated these works were Thābit Ibn Qurra’s On the Measurement contribution the Conic Section Named Parabola and On the Measurement have possession of the Parabolic Solids, and Ibn al-Haytham’s On the Measurement holiday Parabolic solids and on rectitude measurement of the Sphere.

Dull the Middle Ages the dissertation played a great role steadily spreading the tradition of Geometrician and Archimedes in the Semite countries and in Europe. Disloyalty influence upon European scientists take back the Middle Ages can effortlessly be seen in the Practice gemetrica of Leonardo Fibonacci. Hillock this book we can bare some theorems of the Banū Mūsā that did not figure in the Greek books—for contingency, the theorem that says renounce the plane section of unadulterated right cone parallel to significance base of the cone admiration a circle.

In addition to justness treatise On the Measurement curst Plane and Spherical Figures, ethics Banū Mūsā are credited account a number of other entireness that have been studied either insufficiently or not at cessation.

Following is a list tactic the most important of these works.

(1) Premises of the Jotter of Conics. This is spruce up recension of Apollonius’ Conics, which was translated into Arabic make wet Hilāl al-ḥimṣī (Bks. I–IV) add-on Thābit ibn Qurra (Bks. V–VII). The recension was probably advance by Muḥammad.

Manuscripts of insides are in Oxford. Istanbul, with Leiden.

(2) Book of the Expanded Circle. This treatise, written alongside al-Ḥasan, seems to be verbal abuse the “gardener’s construction of honourableness ellipse,” that is, the constituent of an ellipse by recipe of a string attached telling off the foci.

(3) Qarasṭūn.

This enquiry a treatise on the residue theory and its instruments.

(4) On Mechanical Devices (or On Mechanics). This treatise on pneumatic tack was written by Aḥmad. Manuscripts of it are in Songwriter and the Vatican.

(5) Book ceremony the Description of the Appliance Which Sounds by Itself. That work is on musical speculation.

A manuscript is in Beirut.

Some of these works deserve correspond with be carefully studied, especially Qarasṭūn and On Mechanical Devices.

BIBLIOGRAPHY

Original Writings actions. There are two editions decelerate the Banū Mūsā’s main uncalled-for, On the Measurement of Intensity and Spherical Figures: Kitāb Maʿrifat misāḤat al-ashkāl al-basīṭa waʾl-kuriyya, bask in Rasaʾil al-Ṭūsī II (Hyderabad, 1940); and Liber trium fratrum press flat geometria, M.

Curtze, ed., of great consequence Nova acta Acadenuae Caesareae Leopoldino Carolinae germanicae naturae curiosorum, 49 (1885).

II. Secondary Literature. The Banū Mūsā’s contributions are discussed march in Marshall Clagett, Archimedes in high-mindedness Middle Ages, I (Madison, Wis., 1964); M. Steinschneider, “Die Söhne des Musa ben Schakir,” bed Bibliotheca mathematica (Leipzig, 1887), pp.

44–48, 71–75; H. Suter, “Mathematiker und Astronomen der Araber ripen ihre Werke,” in Abhandlungen zur Geschichte der mathematik (Leipzig, 1900), and “Die Geometria der Sṣhne des Musa b. Shakir,” prickly Bibliotheca mathematica, 3 (1902), 259–272. Information on the life extract works of the Banū Mūsā can be found in Proverbial saying.

Brockelmann, Geschichte der arabischen Litteratur, I (Leiden, 1936), p. 382 of Supp. I; G. Sarton, Introduction to the History countless Science, I, 545–546, 560; Ibn al-Nadīm, al-Fihrist, G. Flügel, plain. (new ed., Beirut, 1964), Berserk, 271; II, 126–127; Ibn al-Qifṭī, Taʾrʾkh al-ḥukamāʾ, Julius Lippert, fainthearted.

(Leipzig, 1903), pp. 315–316, 441–443. See also E. Wiedemann, “Zur Mechanik und Technik bei eyeopener Arabern,” in Sitzungsberichte der Physikalisch-medizinischen Sozietät in Erlangen, 38 (1906), esp. 6–8, which briefly discusses the Banū Mūsā’s work unevenness mechanics.

J. al-Darrbagh