S32 Mr Watts’ Remarks on Captain Kater^'s 
and by substituting these values of and in equation (S), 
it will become. 
■< (!)•-') 
we shall have therefore 
, ja + b) (.a—b) 
( 3 ). 
16 
But as, in experiments relating to the pendulum, the arcs of 
vibration a and h, dilfer very little from each other, the ratio ~ 
will also differ but little from unity ; and this circumstance en- 
ables us to extract the ^th root of the number by approxi- 
mation. To effect this, we remark, that if any number what- 
ever be represented by a, we shall have, by the nature of loga- 
rithms, a — (10)^Oo«, the logarithmic base being equal to 10; 
consequently, a ^—(10) 
have, 
- log a 
; and for the same reason we shall 
but, since the developement of is =: 1 4- A a* -| — 
4- &c. ; the modulus of the logarithmic tables being 
A = 2,302585, &c. ; so by analogy, the developement of 
(10) A log (^0 + ^log" (0— ; 
but as the number ^ differs very little from unity, we may con- 
fine ourselves to the first power of its logarithm, and in this 
case we shall have 
q 
