Contextualizing Playfair and Colebrooke 255
99 C1817: 220–1.
100 C1817: xviii.
101 A contemporary mathematical review of the solution of Pell’s equation indicates that the
‘Indian or English method of solving the Pell equation is found in Euler’s Algebra’. However,
it is subsequently clarifi ed that Euler, and his Indian or English predecessors, assumed that
the method always produced a solution, whereas the contemporary understanding is that if a
solution existed the method would fi nd one. Further, Fermat had probably proved that there
was a solution for each value of a , and the fi rst published proof was that of Lagrange (Lenstra
2002 : 182).
Bhaskara then posits the ratios:AC = BC (^) and AC = AB
BC CD AB AD
(^) χ (^) = (BC)^2 and δ (^) = (AB)^2
AC AC
Now ( χ + δ) =
(^) (BC)^2
- (^) (AB)^2
AC AC
Or (AC)^2 = (BC)^2 + (AB)^2
And thus the value of AC is computed, and from this the value of BD. 99
Th us the procedure is reasoned again for a particular case with the sides
of 15 and 20, but clearly the procedure is applicable for any set of numbers
that constitute the sides of a right-angled triangle. It needs to be pointed out
here that Colebrooke highlights the fact that Bhaskara ‘gives both modes of
proof ’ when discussing the solution of indeterminate problems involving
two unknown quantities.
Th e instances Colebrooke has selected in his dissertation are ‘conspicu-
ous’ as he says, for as pointed out earlier his method is to accentuate the
contrast to destabilize as it were the then received picture within the binary
typologies of the history of mathematics mentioned earlier. 100 But the
task is undertaken with a great deal of caution. Th e next example chosen
is that of indeterminate equations of the second degree, wherein, accord-
ing to Colebrooke, Brahmagupta provided a general method, in addition
to which he proposes rules to resolve special cases. It is well known that
Bhaskara solved the equation ax^2 + 1 = y^2 for specifi c values of the variable a.
But Colebrooke went on to suggest that Bhaskara proposed a method to
solve all indeterminate equations of the second degree that were ‘exactly
the same’ as the method developed by Brouncker. In eff ect, Colebrooke
appeared to be suggesting that Bhaskara’s method was generalizable, that he
was aware of the problem and its ‘general use’, a feature for whose discovery
modern Europe had to await the arrival of Euler on the stage of European
mathematics.^101