Indian Science and Technology in the Eighteenth Century: Some Contemporary European Accounts c12.9

c12.9

The fifth and sixth columns sufficiently explain the tables in pages 253 and 254 of Mons. Gentil, but there remains a part more difficult, namely, why in calculating the *Bauja,* or the doubles of the first differences of the ascensional difference ‘20/60 of the length of the shadow is taken for the first; 4/5 of the first term for the second, and 1/3 of the first term for the third.’ The primary reason of taking differences here, seems to be that the chords may be nearly equal to the arcs, and that, by adding of the differences, the arcs themselves may be found nearly; the reason will appear from the following investigation. Let N be the equatorial shadow of the Bramins in *bingles*; then 720 the length of the Gnomon, or twelve *ongles*, will be to N the shadow, as radius to the tangent of the latitude; and radius to the tangent of the latitude as the tangent of the declination to the sine of the ascensional difference; consequently 720 is to N as the tangent of declination to the sine of the ascensional difference. Now if the declinations for one, two, and three signs be substituted in the last proportion, we get the sines of the three ascensional differences in terms of N and known quantities; and, if these values be substituted in the Newtonian form for finding the arc from the sine, we get the arcs in parts of the radius; and if each of these be multiplied by 3600 and divided by 6,28318, the values come out in *puls* of a *gurry* if N be in *bingles*, but in parts of a *gurry* if N be in *ongles*; and by taking the doubles we get the values nearly as follows: