Plane trigonometry, by S.L. Loney.

454 TRIGONOMETRY.
381. Natural Sines. Suppose we have a table
calculated for successive differences of angles, such that
the number of radians in these successive differences is h.
[In the case of our ordinary tables h=number of
radians in 1'
60x180= '000290888..., i.e. h< 0003.]
60 x 180
Also let k be less than h. Then our principle was that
sin ( +k)-sin c k
sin (8 + h) - sin 0 h '
We shall examine this assumption.
We have
sin (0 + k) - sin 0 = sin 0 cos k + cos 0 sin k - sin 0
=sin0 1- + _-.. +cos 0 — +... -sin0
(Arts. 279 and 280)
k,2 k3
= k cos 0 - sin 0 - cos 0...
The ratio of the third term to the first = k2 and this
is always less than 1 (0003)2, i.e. always less than '00000002.
The third and higher terms may therefore be safely neglected, and we have
k2
sin (D + k)- sin 0 =kcos 0- - sin 0...... ).
The numerical ratio of the second term to the first
term
= ktan 0.................... (2).
2 (2).