“8 minutes and 3 seconds.â€? Techniques of this kind were known in ancient cultures such as the Babylonian, but the point is that, at least as manifest in this treatise, we can see that “Hindu calculationâ€? was not limited, in al-KhwÂŻarizm¯Ĺ’s views, to the decimal positional system. Within the sexagesimal contexts, the arithmetical procedures for dealing with fractions appear as well seated and clearly understood. Al-KhwÂŻarizmÂŻÄą relied on these ideas in order to go on and handle fractional numbers in the decimal context as well. Curiously, he did so in a somewhat roundabout way by ďŹ rst translating the decimal into sexagesimal terms, as in the following example: Multiplying one and a half by itself is equivalent to multiplying 1 degree and 30 minutes by itself, and, since one degree is 60 minutes, this is equivalent to multiplying 90 minutes by itself. This is 8100 seconds, which is the equivalent of 135 minutes, or of two degrees and 15 minutes. This, in turn, is the equivalent of 2 and one quarter.
If calculated directly in decimal terms, we would obtain the same result, of course, namely, 1.5Ă—1.5 = 2.25. So, the reason for using this roundabout approach is not that it yields more accurate results. Similarly roundabout is al-KhwÂŻarizm¯Ĺ’s method of dividing decimal fractions. His explanation of addition and subtraction, on the other hand, is quite sketchy and at times unclear. Al-KhwÂŻarizmÂŻÄą also explained a procedure for extracting roots. His method applies equally to integers, to sexagesimal fractions, and to reciprocals and common fractions. He also introduced some ingenious techniques for making all calculations more accurate, and here again the sexagesimal representation plays an important role. A simple example is the calculation of the square root √ √ of 2. Al-KhwÂŻarizmÂŻÄą calculated it as the square root of 7200 seconds (i.e., 2 = 2Ă—60Ă—60), which yields 84 minutes plus a remainder (because 84Ă—84 = 7056). This, in turn, is equivalent to 1 integer and 24 minutes plus a remainder. Yet more eďŹƒcient methods where the decimal and the sexagesimal are mixed are based on adding zeroes and working out the results with the help of various types of sexagesimal fractions. To see how this approach works, consider the example of the square root of 2. Al-KhwÂŻarizmÂŻÄą did it as follows (I write the procedure here in modern symbols in order to make it easier for the reader to follow the steps):
√
√ 2.000.000 1 1 = Ă— 2.000.000 = Ă— 1414 + remainder 106 1000 1000
1 414 414 Ă— 60 24840 ≈ Ă— 1414 = 1 (unit) + =1+ =1+ 1000 1000 1000 1000
400 Ă— 60 50400 = 1 + 24 + = 1 + 24 + 1000 1000
400 Ă— 60 = 1 + 24 + 50 + = 1 + 24 + 50 + 24 . 1000 2=
The text continually establishes analogies between reciprocals, common fractions and sexagesimal fractions, and suggests that all of these may be treated similarly. The analogies are stated sometimes implicitly and sometimes explicitly. Sometimes they help clarify the underlying concepts, and sometimes they obscure them. The text is uneven
102 | NUMBERS IN MEDIEVAL ISLAM