Plane trigonometry, by S.L. Loney.

PRINCIPLE OF PROPORTIONAL PARTS.

453

Now in our ordinary logarithm tables n contains
5 digits, i.e. n is not less than 10000. Hence, if h be less
u h2
than unity, we have 2- less than
2 n2
1 1
( 43429448...) x 1o,
'21714724...
i.e. less than 1, i.e. < -0000000021....
Also - - is less than one-ten thousandth part of this.
3 n3
Hence in (1) the omission of all the terms on the righthand side after the first will make no difference at least as
far as the seventh place of decimals. To seven places we
therefore have
log0o (n + h) - logo n = 10
So log0i (n + 1)- log0n =
Hence, by division,
log10 (n + h)-logon _h.
log, (n + 1) - log,0n
The principle assumed is therefore always true for the
logarithms of ordinary numbers as given in our tables.
380. We may enquire what is the smallest number in the tables to
which we can safely apply the principle of proportional parts. We must
find that value of n which makes _h2 1 so that n2>h. 107. h2.
2n2 107. 2
The greatest value of h being unity, we then have
n2>. 107, i.e. >2171472 4....... n> 1473.
The number 1473 is therefore the required least number.