पाटीगणित (श्रीधराचार्य - प्राचीन संस्कृत टीका एवं प्रो. कृपाशंकर शुक्ल सविस्तार व्याख्या)

90 DETERMINATIONS
(iii) AREA OF THE QUADRILATERAL WITH UNEQUAL
ALTITUDES
Rule for finding the area of a quadrilateral with unequal
altitudes, when its sides are given :
117. Set down half the sum of the (four) sides (of the
quadrilateral) in four places, (then) diminish them (respec-
tively) by the (four) sides (of the quadrilateral), (then) multi-
ply (the resulting numbers) and take the square root (of the
product) : this gives the area of quadrilaterals having (two or
more) equal sides but unequal altitudes and also of quadri-
laterals having unequal sides and unequal altitudes.¹
That is to say, if a, b, c, d be the sides of a quadrilateral with un-
equal altitudes, and s half the sum of those sides, then the area of the
quadrilateral
= √((s - a) (s - b) (s - c) (s - d)) .
Our author, like Brahmagupta and other early Hindu mathema-
ticians, has committed here an error in declaring the above formula as
applicable to all quadrilaterals (with unequal altitudes), when in fact it is
applicable to cyclic quadrilaterals only. The formula which is applicable
to all quadrilaterals, in general, of which it is only a particular case, is
as follows :
area of a quadrilateral = √{(s - a) (s - b) (s - c) (s - d) - abcd cos²α}
where a, b, c, d are the sides of the quadrilateral, s is half the sum of those
sides, and α is half the sum of a pair of opposite angles of the quadri-
lateral.
The earliest Hindu mathematician who pointed out the inapplicabi-
lity of the above formula to a quadrilateral having only its sides given was
Âryabhaṭa II (c. 950 A.D.). He writes :

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¹ According to Brahmagupta and Mahâvîra, this rule gives the
accurate area for a quadrilateral in general. See BrSpSi, xii. 21(ii) ; and
GSS, vii. 50 (ii). The same rule occurs also in SiŚe, xiii. 28. Âryabhaṭa II
has pointed out that this rule gives the accurate area in the case of a
triangle only and not in the case of a quadrilateral. See MSi, xv. 69. So
also write Bhâskara II and Nârâyaṇa. See L (ASS), p. 156, vs. 169 ;
and GK, II, p. 39, lines 13-14 and p. 40, lines 1-2.