262 9. MEASUREMENT
FIGURE 14. Finding the Sun's elevation at a given hour.In this relation, ÌF is the sine of MS, that is, the sine of the observer's co-latitude,
and MO is the radius of the celestial sphere. The line Ç Ê is, in a loose sense, the
sine of the arc RH, which is proportional to the time elapsed since sunrise. It is
perpendicular to the chord RV and would be a genuine sine if RV were the diameter
of its circle. As it is, that relation holds only at the equinoxes. It is not certain
whether Aryabhata meant his formula to apply only on the equinox, or whether he
intended to use the word sine in this slightly inaccurate sense. Because the radius
of the Sun's small circle is never less than 90% of the radius of the celestial sphere,
probably no observable inaccuracy results from taking Ç Ê to be a sine. In any
case, that is the way Aryabhata phrased the matter:
The sine of the Sun at any given point from the horizon on its
day-circle multiplied by the sine of the co-latitude and divided by
the radius is the [sine of the altitude of the Sun] when any given
part of the day has elapsed or remains. [Clark, 1930, p. 72]Notice that it is necessary to divide by the radius, because for Aryabhata the
sine of an arc is a length, not a ratio.
5.2. Brahmagupta. Brahinagupta devotes five sections of Chapter 12 of the
Brahmasphutasiddhanta to geometric results (Colebrooke, 1817, pp. 295-318). Like
Aryabhata, he has a practical bent. In giving the common area formulas for trian-
gles and quadrilaterals, he first gives a way of getting a rough estimate of the area:
Take the product of the averages of the two pairs of opposite sides. For this purpose
a triangle counts as a quadrilateral having one side equal to zero. In the days when
calculation had to be done by hand, this was a quick approximation that worked
well for quadrilaterals and triangles that are nearly rectangular (that is, tall, thin
isosceles triangles). He also gave a formula that he says is exact, and this formula is
a theorem commonly known as Brahmagupta's theorem: Half the sum of the sides
set down four times and severally lessened by the sides, being multiplied together,