Banu Musa Brothers – III

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continued…

There are many similarities in the methods employed by the Banu Musa and those employed by Archimedes. More significant, however, is the fact that there are also many differences which, although at first sight may not seem of major importance, yet were providing the first steps towards a new approach to mathematics. The Banu Musa apply the method of exhaustion invented by Eudoxus and used so effectively by Archimedes. However, they omitted that part of the method which involves considering polygons with 2^k sides as k tends to infinity. Rather they chose to use a proposition which itself required this passage to infinity in its proof. This in itself may not have been a step forward for, as the author of [2] suggests, this may have been due to a lack of understanding of the finer points of Greek geometric thinking. As used by the Banu Musa the “method of exhaustion” loses most of its subtlety and power.

In another aspect, however, the Banu Musa made a definite step forward. The Greeks had not thought of areas and volumes as numbers, but had only compared ratios of areas etc. The Banu Musa’s concept of number is broader than that of the Greeks. For example they describe π as [2]:-

… the magnitude which, when multiplied by the diameter of a circle, yields the circumference.

In the text areas as described as products of linear magnitudes, so the terminology of arithmetic is perhaps for the first time applied to the operations of geometry. The Banu Musa also introduce geometrical proofs which involve thinking of the geometric objects as moving. In particular they used kinematic methods to solve the classical problem of trisecting an angle.

In astronomy the brothers made many contributions. They were instructed by al-Ma’mun to measure a degree of latitude and they made their measurements in the desert in northern Mesopotamia. They also made many observations of the sun and the moon from Baghdad. Muhammad and Ahmad measured the length of the year, obtaining the value of 365 days and 6 hours. Observations of the star Regulus were made by the three brothers from their house on a bridge in Baghdad in 840-41, 847-48, and 850-51.

(Source: http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Banu_Musa.html)