14 
COMPUTATIONS OF Z. COLBURN. 
On numerical An illustration is here afforded of the combinations by which 
computations, p roc j UC f ar j ses from two quantities multiplied into each 
particmavlv 1 n . 
those perform- other ; and a familiar acquaintance with the preceding rules 
ed by. Z. Col- w jjj renc j er the extraction of the square root a process of easy 
attainment. 
To find the square root of 58'58*37‘l6 
Deduct the square of the first root 7=49 
95 
Deduct twice 7x6= 84 j and 
6 a =3,6— say 87 
8 
Deduct twice 76 x 5, say 1 5 x 5 ... . / 5 
5 
Again, the same divisor, 15x 4 6,0 
As the last root is known to be a 4 or a 6, it must obviously 
be 4, which gives the nearest product to the quantity wanted. 
If the second 8 had been operated upon, the 4 would have 
been produced correctly : but, to shorten the calculations, this 
nicety may be omitted. 
The more full explanation is the following : 
1. Deduct the square of the first root, 7* 
2. Multiply twice the first root by the number, which will 
bring the product nearest to the term brought down ; square 
also the assumed number ; subtract the two sums from the 
dividend, The new quotient, 6, will be the second root. 
3. Then twice the first and second roots (placed in iheir 
local order thus 76) must be multiplied by the number that 
will raise them nearest to the remainder of the last dividend, 
considered in its local value. This new factor, 5, will be the 
third root. 
4. Again, twice the first and second roots, 76, multiplied by 
the number, w hich will produce a quantity nearest to the remain- 
ing dividend, will shew the fourth root, 4. 
It is only in the operation, where the second root is found, 
that any nicety is required. The multiplier (of twice the first 
root) may occasionally be assumed an unit too much $ but the 
error. 
