Plane trigonometry, by S.L. Loney.

162

TRIGONOMETRY.

152. The proof of the Principle of Proportional Parts
will not be given at this stage. It is not strictly true
without certain limitations.
The numbers to which the principle is applied must
contain not less than five significant figures, and then we
may rely on the result as correct to seven places of
decimals.
For example, we must not apply the principle to
obtain the value of log 2'5 from the values of log 2 and
log 3.
For, if we did, since these logarithms are '30103 and
~4771213, the logarithm of 2'5 would be -389075.
But from the tables the value of log 2'5 is found to be
'3979400.
Hence the result which we should obtain would be
manifestly quite incorrect.
Tables of trigonometrical ratios.
153. In Chambers' Tables will be found tables giving
the values of the trigonometrical ratios of angles between
0~ and 45~, the angles increasing by differences of 1'.
It is unnecessary to separately tabulate the ratios for
angles between 45~ and 90~, since the ratios of angles
between 45~ and 90~ can be reduced to those of angles
between 0~ and 45~. (Art. 75.)
For example,
[sin 76~ 1l' = sin (90~ - 13~ 49') = cos 13~ 49',
and is therefore known].
Such a table is called a table of natural sines, cosines,
etc. to distinguish it from the table of logarithmic sines,
cosines, etc.