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 fragments on geometry. He gives the theorem of the right triangle, but proves it after Hindoo fashion and only for the simplest case, when the right triangle is isosceles. He then calculates the areas of the triangle, parallelogram, and circle. For $$\scriptstyle{\pi}$$ he uses the value $$\scriptstyle{3\frac{1}{7}}$$, and also the two Indian, $$\scriptstyle{\pi=\sqrt{10}}$$ and $$\scriptstyle{\pi=\frac{62882}{20000}}$$. Strange to say, the last value was afterwards forgotten by the Arabs, and replaced by others less accurate. This bit of geometry doubtless came from India. Later Arabic writers got their geometry almost entirely from Greece.

Next to be noticed are the three sons of Musa ben Sakir, who lived in BagdadBaghdad [sic] at the court of the Caliph Al Mamun. They wrote several works, of which we mention a geometry in which is also contained the well-known formula for the area of a triangle expressed in terms of its sides. We are told that one of the sons travelled to Greece, probably to collect astronomical and mathematical manuscripts, and that on his way back he made acquaintance with Tabit ben Korra. Recognising in him a talented and learned astronomer, Mohammed procured for him a place among the astronomers at the court in BagdadBaghdad [sic]. Tabit ben Korra (836-901) was born at Harran in Mesopotamia. He was proficient not only in astronomy and mathematics, but also in the Greek, Arabic, and Syrian languages. His translations of Apollonius, Archimedes, Euclid, Ptolemy, Theodosius, rank among the best. His dissertation on amicable numbers (of which each is the sum of the factors of the other) is the first known specimen of original work in mathematics on Arabic soil. It shows that he was familiar with the Pythagorean theory of numbers. Tabit invented the following rule for finding amicable numbers: If $$\scriptstyle{p=3 \cdot 2^n-1}$$, $$\scriptstyle{q=3 \cdot 2^{n-1}-1}$$, $$\scriptstyle{r=9 \cdot 2^{2n-1}-1}$$ (n being a whole number) are three primes, then $$\scriptstyle{a=2^n pq,~b=2^n r}$$ are a pair of amicable numbers. Thus, if $$\scriptstyle{n=2}$$, then $$\scriptstyle{p=11}$$,