# Celebrating the scientific, mathematical, and technological accomplishments of the Banu Musa Brothers

## Celebrating the scientific, mathematical, and technological accomplishments of the Banu Musa Brothers

*The Bana Musa brothers were three brothers: Jafar Muhammad ibn Musa ibn Shakir, Ahmad ibn Musa ibn Shakir and al-Hasan ibn Musa ibn Shakir. They are almost indistinguishable but we do know that although they often worked together, they did have their own areas of expertise.*

The three links above give details specific to each of the brothers but most of the information about them is on this page.

Jafar Muhammad worked mainly on geometry and astronomy while Ahmad worked mainly on mechanics and al-Hasan worked mainly on geometry. It is quite impossible to write separate biographies of the three brother, who are usually known as the Banu Musa, and we shall not attempt to do so.

The Banu Musa brothers were among the first group of mathematicians to begin to carry forward the mathematical developments begun by the ancient Greeks. It is therefore worth looking at the background to how Arabic mathematics came to fill this role....

We now turn to the important mathematical contributions made by the Banu Musa brothers. As al-Dabbagh writes in [1]:-

The most studied treatise written by the Banu Musa is Kitab marifat masakhat al-ashkal Ⓣ. This work became well known through the translation into Latin by Gherard of Cremona entitled Liber trium fratum de geometria Ⓣ. The treatise considers problems similar to those considered in the two texts by Archimedes, namely On the measurement of the circle and On the sphere and the cylinder.

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 2k 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]:-

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.

The three links above give details specific to each of the brothers but most of the information about them is on this page.

Jafar Muhammad worked mainly on geometry and astronomy while Ahmad worked mainly on mechanics and al-Hasan worked mainly on geometry. It is quite impossible to write separate biographies of the three brother, who are usually known as the Banu Musa, and we shall not attempt to do so.

The Banu Musa brothers were among the first group of mathematicians to begin to carry forward the mathematical developments begun by the ancient Greeks. It is therefore worth looking at the background to how Arabic mathematics came to fill this role....

We now turn to the important mathematical contributions made by the Banu Musa brothers. As al-Dabbagh writes in [1]:-

*The Banu Musa were among the first Arabic scientists to study the Greek mathematical works and to lay the foundation of the Arabic school of mathematics. They may be called disciples of Greek mathematics, yet they deviated from classical Greek mathematics in ways that were very important to the development of some mathematical concepts.*The most studied treatise written by the Banu Musa is Kitab marifat masakhat al-ashkal Ⓣ. This work became well known through the translation into Latin by Gherard of Cremona entitled Liber trium fratum de geometria Ⓣ. The treatise considers problems similar to those considered in the two texts by Archimedes, namely On the measurement of the circle and On the sphere and the cylinder.

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 2k 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.

http://www-history.mcs.st-andrews.ac.uk/Biographies/Banu_Musa.html

**Rashmun**- Posts : 4036

Join date : 2011-08-18

## Re: Celebrating the scientific, mathematical, and technological accomplishments of the Banu Musa Brothers

Another reference for these remarkable brothers:

http://islamsci.mcgill.ca/RASI/BEA/Banu_Musa_BEA.htm

http://islamsci.mcgill.ca/RASI/BEA/Banu_Musa_BEA.htm

**Rashmun**- Posts : 4036

Join date : 2011-08-18

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