ARI 
ARM 
To cast out the 9’s of any number, add the 
figures, and when the sum is equal to or 
more than 9, pass by the 9, and proceed with 
the remainder : thus in casting out the 9’s 
of 56774 we say 5 and 7 are 12, 3 aboye 9 ; 
3 and 7 are 10, 1 above 9, 1 and 7 are 8 ; 
8 and 4 are 1 2, 3 above 9 ; the last remainder 
is to be put down, and then proceed to the 
other lines according to the following rules. 
To prove Addition. Cast out the 9’s of 
the several articles, carrying the results to 
the following articles, and cast them out of 
the sum total ; if the operations be correct, 
the two remainders, if any, will agree. 
Example. 845 
346 
'784 
Sum 1975 
Here, in casting out the 9’s of the three 
lines to be added, I find a remainder of 4 ; 
there is also a remainder of 4 upon casting 
out the 9’s of the sum. 
To prove Subtraction. Cast the 9’s out of 
the minuend ; then cast them out of the 
subtrahend and remainder together, and if 
the same result is obtained in both cases, 
the operation may be regarded as accurate. 
Example. 59876 
34959 
24917 
In casting out the 9’s of the upper row, I 
find the remainder 8, the same is found in 
casting out the 9’s of the two lower lines. 
To prove Multiplication. Cast the 9’s out 
of the multiplicand, and put the remainder 
on one side of a cross, then do the same with 
the multiplier, and put the remainder on the 
other side of the cross ; multiply these re- 
mainders together, and cast the 9’s out of 
the product, the remainder place at the top 
of the cross ; cast the 9’s out of the product, 
and the remainder place at the bottom of 
the cross, wiiich, if the operation be cor- 
rect, will be the same as that at the top. 
Example. 5943 
26 
25658 
11886 
15451 8 _ 
To prove Division. Cast the 9’s out of the 
divisor, and also out of the quotient, the re- 
mainder of the former place on the side of 
the cross ; that of the latter on the other : 
multiply them together, and take in the re- 
mainder, if any, cast out the 9’s, and the 
remainder put at the top of the cross ; this, 
if the operation be correct, will agree with 
remainder of the dividend obtained from 
the dividend after the 9’s are cast out. 
Ex. 264) 87655 (332 
792 
. 845 
792 
This method of proving sums lies under 
disadvantages. 1 . If an error of 9, or any 
of its multiples be committed, the results 
will nevertheless agree, and so the error will 
remain undiscovered. This will be the case, 
when a figure is placed or reckoned in a 
wrong column, which is a frequent cause of 
mistake. 2. When it is known that an er- 
ror has been committed, it is not pointed 
out where the error lies, and of course not 
easily corrected. 
Having given a full account of the funda- 
mental rules of Arithmetic, we shall refer 
our readers to the several articles in alpha- 
betical order, for rules depending on the 
four already treated on. See Alligation, 
Annuities, Exchange, Interest, &c. 
&c. 
ARITHMETICAL complement of a lo- 
garithm,, the sum or number which a loga- 
rithm wants of 10,000000 : thus the arith- 
metical complement of the logarithm 
8.154032 is 1.845968. 
ARM, a part of the human body, termi- 
nating at one end in the shoulder, and at 
the other in the hand. See Anatomy. 
ARMADA, a Spanish term, signifying a 
fleet of men of war ; it is more particularly 
applied to the ships by which an attempt 
was made by Philip II. of Spain to invade 
England, in the reign of Queen Elizabeth, 
A. D. 1588. This expedition was excited 
as well by the injuries which the king had 
sustained from the English arms, as with a 
view of transmitting his name to posterity, as 
the defender of the true faith. In the preceding 
year, a whole fleet of transports was destroy- 
ed at Cadiz by Drake, who ravaged theSpanish 
coast. Cavendish, another sea commander, 
committed about the same time, great de- 
predations on the Spaniards in the South- 
Sea, taking 19 vessels richly' laden, with 
which he entered, in triumph, the rivei 
