## Al-Sabi Thabit ibn Qurra al-Harrani (826 - 901)

Born in Turkey, Thabit Ibn Qurra was a member of a sect that based its fundamental beliefs on the worship of stars.

The sect had the name of ‘Sabians’ and hence the title - Al Sabi (The Sabian) - in his name.

The Sabians had strong historic roots in Greece and this brought with it all their knowledge and accumulated wisdom in the form of books.

Their own beliefs in the stars gave them additional motivation to study mathematics and astronomy in particular.

The net result is that much of Greek mathematics and astronomy was preserved, translated and transmitted. Without this much would have been lost.

This background was useful to Thabit Ibn Qurra when he moved to Baghdad, the capital city of the vast Abbasid Empire. His knowledge of Greek, Arabic and mathematics meant that he joined the other men involved in translating the Greek classics into Arabic.

He translated and amended the work of many important Greeks thinkers. Without him, the writings of Archimedes, Euclid, Apollonius and Ptolemy would be unknown to us today as the Greek copies have been lost.

The traditional view of scholars such as Thabit Ibn Qurra was that they were passive translators - content merely to set down Greek works in another language, word for word. The evidence does not support such a view. Their understanding of mathematics and the ancient works, led them to make criticism and improvements to the works they dealt with.

## Thabit Ibn Qurr’s mathematics

A flavour of the way that Thabit Ibn Qurra improved on the works he was translating is given by the contributions he made to the work of Euclid and Pythagoras. Each of them were concerned with number theory. A number is perfect if it is the sum of its divisors; so

6 = 1+2+3

is perfect, while 9 is not since

9\neq 1+3

The next perfect number, after 6, is 28, but then the sequence becomes complicated. Euclid, had discovered that an even number is perfect if it has the form,

2^{p-1}(2^p-1)

in which both p and 2^p-1 are prime numbers. Using this we find that the first few even perfect numbers given by the formula are,

6, 28, 496, 8128 and 33,550,336.

Euler would prove the converse of this result some two thousand years later - that all even perfect numbers are of this form. But what about odd perfect numbers? We still do not know if any exist at all - but we do know that if they do, they are very large, greater in fact than 1050 and that’s a very, very large number.

But back to Thabit Ibn Qurra. In another of his works he remarks that Euclid and Pythagoras looked at perfect numbers but ignored an obvious generalisation of them. He then goes on to show what this is - if a number wasn’t perfect then either it was abundant or deficient, depending on the sum of its divisors. Then he gives the numbers 220 and 284 as an amicable pair since,

284 = 2.2.71

and so the sum of its divisors is,

1+2+71+2.2+2.71 = 1+2+71+4+142 = 220;

similarly the sum of the divisors of 220 is,

1+ 2+ 4+ 5+ 10+ 11+ 20+22+ 44+ 55+110 = 284.

Not content with this, he gives a general rule: for n > 1 let,

p_n = 3.2^n - 1 and q_n = 9.2^{2n-1}-1

If p_{n-1},p_n, and q_n are prime numbers then

a = 2^np_{n-1}p_n and b = 2^nq_n

are amicable numbers where a is abundant and b is deficient. Using this we find the amazing pair, 17296, and 18416 first announced by Fermat in the seventeenth century.

In astronomy Thabit Ibn Qurra also made major contributions. According to Copernicus he determined the length of the sideral year as,

365 days, 6 hours, 9 minutes and 12 seconds.

This is accurate to 2 seconds!

There was much else that Thabit Ibn Qurra did - all of it original - in mathematics and astronomy. His work and that of others preserved for us the Greek legacy, but at the same time they continued its development in new and profound ways.