Plane trigonometry, by S.L. Loney.

CHAPTER XXX.

PRINCIPLE OF PROPORTIONAL PARTS.
378. IN the present chapter we shall consider the
Principle of Proportional Parts, the truth of which we
assumed in Chapter XI.
We then assumed that if n be any number and n + 1
the next number, whose logarithms were given in our
tables, and if h be any fraction, then, to 7 places of
decimals, it is true that
log (n + h) - log _ h.
log (n + 1) - log n
The truth of this statement we shall now consider.
379. Common Logarithms. We have, by Art.
260,
n 7- h, + h h\
logo, (n + h) - log,, n = logg, +,
where / - 43429448...
Hence, by Art. 256, we have
h - h2 h3
log,0 (n + h)-logo n = --......().
n 2n2 3n (1).