Bombelli frequently used fractions to approximate square roots. Here is an example:
Suppose we want to find the square root of 13. The nearest square is 9, which has root 3. The approximation of the square root of thirteen is 3 + an unknown.
3 + x = √13
Its square is 9 plus 6 unknowns plus 1 power. We set this equal to 13
(3 + x)2 = 9 + 6x + x2 = 13
Subtract 9 from either side of the equation, and we are left with 4 equal to 6 plus 1 power
6x + x2 = 4
If you ignore the power and set 6 unknowns equal to 4, the unknown (x) equals 2/3
6x = 4 gives x = 2/3
The approximate value of the root is 3 2/3 since it has been set equal to 3 plus 1 unknown.
√13 = 3 + x = 3 2/3
However, taking the power into account, if the unknown equals 2/3 , the power will be 2/3 of an unknown which, added to the 6 unknowns, will give us 6 and 2/3 unknowns. This equals 4.
6x + x2 = 4 implies 6x + 2/3 x = 4
The unknown will be equal to 3/5 and since the approximate is 3 plus 1 unknown, it comes to 3 3/5.
x = 4/(6 + 2/3) = implies 3 + x = 3 3/5
But if the unknown equals 3/5 the power will be 3/5 of an unknown and we obtain 6 3/5 unknowns equal to 4
6x + x2 = 4 implies 6x + 3/5 x = 4
Then the unknown comes out to be 20/33
6x + 3/5 x = 4 implies x = 4/(6 + 3/5) = 20/33
And this process can continue.
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