Contextualizing Playfair and Colebrooke 249
off ered both algebraic and geometrical proofs. In a contemporary idiom
these rules are stated as:
2 ab + (a − b)^2 = a^2 + b^2 i
(a + b) (a − b) = a^2 − b^2 ii
§134 of the Lilavati is translated from Sanskrit as:
Th e square root of the sum of the squares of those legs is the diagonal. Th e square
root, extracted from the diff erence of the squares of the diagonal and side is the
upright; and that extracted from the diff erence of the squares of the diagonal and
upright, is the side. 88
§135 that follows is translated as:
Twice the product of two quantities, added to the square of their diff erence, will
be the sum of their squares. Th e product of their sum and diff erence will be the
diff erence of their squares: as must be everywhere understood by the intelligent
calculator.^89
And this theorem came in for much discussion from the 1790s when
Playfair fi rst wrote about it in his discussion of Davis’ translation of the
Surya-Siddhanta.
Now §135 is marked with two footnotes: the one indicates that §135 is a
stanza of six verses in the anustubh metre and the next importantly indi-
cates that Ganesa the commentator on Bhaskara’s Lilavati provides both
an ‘algebraic and geometrical proof ’ of the latter rule, the one marked as ii
above (my labelling), and an algebraic demonstration of the fi rst marked as
i above (my labelling). Colebrooke is not just translating from Bhaskara II ’s
Lilavati : in the footnotes he intercalates a translation of Ganesa’s commen-
tary. Th e latter demonstration is taken from the Bija-Ganita §148; and it is
in §147 that the fi rst of the rules is given and demonstrated.^90 C o l e b r o o k e
renders the term Cshetragatopapatti as geometrical demonstration and
Upapatti avyucta-criyaya as proof by algebra. 91 We come to one of the geo-
metrical demonstrations of rule labelled ii as given in the Bija-Ganita §148
and §149 of Bhaskara to which Colebrooke refers as such.
§148: Example: Tell me friend, the side, upright and hypotenuse in a [triangular]
plane fi gure, in which the square-root of three less than the side, being lessened by
one, is the diff erence between the upright and the hypotenuse. 92
88 Ibid.
89 Ibid.
90 C1817: 222–3.
91 C1817: 59.
92 C1817: 223.