Islam,Neglect andDiscovery 115
6. Algebra—the next steps
We have heard the great eastern mathematicians have extended the algebraic operations beyond the six types and
brought them up to more than twenty. For all of them they discovered solutions based on solid geometrical proofs. God
‘gives in addition to the creatures whatever He wishes to give to them’. (Ibn Khald ̄un 1958, III, p. 126)
Not much later than Th ̄abit’s text, the Egyptian ab ̄uK ̄amil wrote what is commonly considered
the ‘second-generation’ algebra after that of al-Khw ̄arizm ̄i.^9 The work of al-Khw ̄arizm ̄i is explicitly
referred to, and many of the examples are the same; but much else has changed. The simple
geometrical diagram has been replaced by a reference to Euclid’s book II (as it was in Th ̄abit’s text),
but with numbers included. For the first time, as far as we know (and our knowledge is as usual
limited), numbers have been introduced into Euclidean propositions as a matter of routine, and
proposition II.6 is being interpreted more in the ‘algebraic’ sense referred to above. If this was done
by the ancient Greeks, or by any of their successors, they were much more discreet about it than
abu K ̄amil.
However, what abu K ̄amil did next was even bolder, as an innovation. Again, it may arise from the
study of Euclid, in this case of his book X; but this is not made clear, and the language is completely
different. He develops a set of rules—not complete, but useful—for calculating with roots, and uses
them freely in many of his examples as if they were numbers. The result is an enormous expansion
of the collection of equations you can solve, and of numbers you can name. Oddly, this appears
not so much in dealing with whole number examples leading to square-root solutions (such as the
simple one given above), as with examples where roots are part of the data of the problem. Here is
the brief, but quite ‘hard’ problem 39:
If one says that ten is added to an amount, and the amount is multiplied by the root of five, then one gets the product
of the amount by itself. For the solution, make the amount a thing and add ten to it to give ten plus a thing. Multiply by
the root of five to give the root of five hundred plus the root of five squares equal to one square. Halve the root of five
squares to give the root of one and a quarter. The root of the sum of the root of five hundred plus one and a quarter,
plus the root of one and a quarter, equals the amount. (Ab ̄uK ̄amil 1966, p. 148)
Notice that though the problem deals with numbers like√
5, they are still expressed in words;
there is no notation for them, and there will not be one for a long time (symbols for roots began
to be used in the sixteenth century). For us, abu K ̄amil’s problem needs a considerable amount of
‘unpacking’. In modern terms, settingxfor the amount, it is:
( 10 +x)√
5 =x^2This abu K ̄amil solves (roughly) by our usual prescriptions for quadratic equations again. In a
slightly roundabout way he changes the left-hand side into
√
500 +
√
5 x^2. He halves√
√ 5togive
114 , and arrives at the (correct) answer
x=√
√
500 + 1
1
4
+
√
1
1
4
All these numbers are still expressed in words, as is done in the above quote. The ‘formula’, if you
like, is exactly the same as had been used by al-Khw ̄arizm ̄i; but the way in which it is applied has
- This can be found in Levey’s translation of Mordecai Finzi’s medieval Hebrew version (Abu K ̄amil 1966), again not easily. There
is an extract in Fauvel and Gray 6.B.2., and a very interesting and complicated equation is discussed in Berggren, p. 110–111.