— 
in Involution and Evolution. ~ 345 
times, except the one omitted, and six times the product of every 
two, except with the one omitted. Again: six times (a+/+&c.)c 
contains six times the sum of the products of each letter, with the 
last letter which was before omitted; and three times the square 
of the last letter makes the second side equal to the first. Thus 
suppose only three letters concerned, then 3(a+)+c)*?=3a*+ 
3b7+3c?+6(ab+ ac+be)=3(a+b)?+2:3(a+b)c4+3c?= 3(a*+ 
2ah+l*) + 6ac+6le4+3c* = 8a?+312+3c*+6(ab + ac+bce) as 
before. 
Now as the first period consists of three times the square of the 
first left-hand figure of the number to be squared with a cipher 
annexed placed in the first row, the product of the left-hand 
figure and annexed cipher, into the second figure of the number 
to be squared placed in the second row, and the square of the se- 
cond figure placed in the third row: 
And in general as the mth period consists of three times the 
square of the first or left-hand m figures to be squared with a 
cipher annexed placed in the first row ; the product of the x 
figures with the cipher annexed, into the (n+1)th figure of the 
number to be squared placed in the second row, and the square 
of the said (7+ 1)th figure placed in the third row : 
Therefore in the (z+ 1)th period, instead of taking three times 
the square of the number consisting of +1 of the first figures 
of the number to be squared with a cipher annexed, for the num- 
ber to be placed in the first row; we shall find this first row of 
the (7+ 1)th period, by adding each figure in the lowest or third 
row of the th period, three times each figure in the second row 
of the mth period twice, and each figure in the first row of the 
nth period once. 
The third or last row is ouly a mental operation; the middle 
row is easily found by multiplying first by the digit 3, and the 
new figure. 
It is not meant that the method now shown for cubing a num- 
ber should supersede the common method ; but the principal use 
is to explain the reverse operation of extracting the cube root in 
numbers. ‘It may also serve as a method of proof to that com- 
_ monly used for raising powers. For it is of some advantage to 
have two different methods of performing every arithmetical ope- 
ration, that the one may furnish a cheek to the other. 
EVOLUTION. 
Problem. 
To find a number of which its cube shall be the nearest less 
number to a proposed number;—Or, as commonly expressed, to 
find the cube root of a proposed number. 
Rule. 
