FIGURE 11. The Hindu variant of the double-difference method of surveying.This procedure gives a value of one-dimensional ð equal to 3.1416, which is quite
accurate indeed. It exceeds the true value by less than 0.01%.
Aryabhata also knows about the double-difference method of surveying that we
discussed above. Whether this knowledge is a case of transmission or independent
discovery is not clear. The rule given is slightly different from the discussion that
accompanies Fig. 6 and is illustrated by Fig. 11.
The distance between the ends of the two shadows multiplied by
the length of the shadow and divided by the difference in length of
the two shadows give the koti. The koti multiplied by the length
of the gnomon and divided by the length of the shadow gives the
length of the bhuja. [Clark, 1930, p. 32]Trigonometry. The inclusion of this variant of the double-difference method of sur-
veying in the Aryabhatiya presents us with a small puzzle. As a method of sur-
veying, it is not efficient. It would seem to make more sense to measure angles
rather than using only right angles and measuring many more lines. But angles are
really not involved here. It is possible to have a clear picture of two mutually per-
pendicular lines without thinking "right angle." The notion of angles in general as
a species of mathematical objects—the figures formed by intersecting lines, which
can be measured, added, and subtracted—appears to be a Greek innovation in the
sixth and fifth centuries BCE, and it seems to occur only in plane geometry. Its
origins may be in stonemasonry and carpentry, where regular polygons have to be
fitted together. Astronomy probably also made some contribution.
The earliest form of trigonometry that we can recognize was a table of corre-
spondences between arcs and their chords. We know exactly how such a table was
originally constructed, since an explanation can be found in Ptolemy's treatise on
astronomy, written around 150 CE.