ARI 



ARM 



ebtained, when that sum is divided by 9. 

 For instance, if 87654 be divided by 9, 

 there is a remainder of 3 ; and if 8, 7, 6, 

 5, 4, be added together, and the sum 30 

 be divided by 9, there will be likewise a 

 remainder of 3. 



i o 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 pro- 

 ceed with the remainder : thus, in casting 1 

 out the 9's ot 56774 we say 5 and 7 are 

 12, 3 above 9 ; 3 and 7 are 10, 1 above 9, 

 1 and 7 are 8 ; 8 and 4 are 12, 3 above 9 : 

 the last remainder is to be put down, and 

 then proceed to the other lines, accord- 

 ing 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. 5943 

 26 



Example. 



845 

 346 

 784 

 Sum 1975 



To prove Division. Cast the 9's out of 

 the divisor, and also out of the quotient, 

 the remainder of the former place on the 

 side of the cross; that of the latter on the 

 other ; multiply them together, and take 

 in the remainder, 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 the remainder of the di- 

 vidend obtained from the dividend after 

 the 9's are cast out. 



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 cast- 

 ing 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 re- 

 mainder on one side of a cross, then d 

 the same with the multiplier, and put the 

 remainder on the other side of the cross; 

 multiply these remainders together, and 

 cast the 9's out of the product ; the re- 

 mainder 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, which, if the operation be correct, 

 will be the same as that at the top. 



This method of proving sums lies un- 

 4er 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 error 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 fun- 

 dame ntal rules of Arithmetic, we shall re- 

 fer our readers to the several articles in 

 alphabetical order, for rules depending 

 on the four already treated on. See AL- 

 LIGATION, ANNUITIES, EXCHANGE, INTE- 

 REST, &c. &c. 



ARITHMETICAL complement of a lo- 

 garithm, the sum or number which a lo- 

 garithm wants of 10,000,000: thus the 

 arithmetical complement of the logarithm 

 8.154032 is 1.845968. 



ARM, a part of the human body, ter- 

 minating 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 particu- 

 larly applied to the ships by which an at- 

 tempt was made, by Philip II. of Spain, 

 to invade England, in the reign of Queen 

 Elizabeth, A. D. 1588. This expedition 

 was excited a ( s well by the injuries which 



