33$ Mr. Manning’s new Method 
h. 1 . 10. Multiply into ~ and add or subtract as many 
units as the decimal point was removed to the left or right. 
Note. The multiplication of a number by — - 1 — is very ex- 
peditiously performed by means of the Table of multiples of 
i 
h. 1. io" 
The demonstration of the above rules is obvious. Setting 
the figures of a number one place to the right is dividing that 
number by io ; 2 places, by 100 ; 3 places, by 1000 ; and so 
on. And subtracting a number, so placed, from the number 
itself is subtracting a 10th, a 100th, a 1000th, &c. (in the re- 
spective cases) of the number from itself; and consequently 
the remainders are (respectively) —ths, -j^ths, 1 ^ 9 -ths, &c. 
of the numbers subtracted from. Let 6, c, d , &c. denote as 
in the rule ; then the original number = — ! x x x 
See. x last remainder. Therefore the log. of the original 
number = b x log. ^ + a log. + d x log. ^ 4- &c. + 
log. of last remainder. Now the last remainder being unity 
followed by a certain number of decimal cyphers, its correct 
hyp. log., as far as twice that number of places, is (as is well 
known) the decimal part itself of that remainder. Hence 
the rule is manifest. 
A similar method, by addition only, by means of the ready 
computed logarithms of — , i-|i, &c. might, in some 
cases, be used with advantage. Let N denote the given 
number, consisting of unity and a decimal whose h. 1. is 
sought ; and let P denote any number less than N, and whose 
h. 1 . is previously known. Set P under itself removed one 
