When the multiplier is a series of nines, annex, or suppose to have annexed as many ciphers as there are nines, from When the multiplier is not more than 12 less than any number, consisting of a significant figure followed by any number of ciphers, multiply by such significant figure, and annex thereto the ciphers as before; and from the product, subtract the product of the multiplicand and the number deficient. To multiply by any number in one line, consisting of 2 figures under 20. Multiply by the right-hand or units figure of the multiplier; taking care to add in the past figure of multiplicand. When the multiplier is 112,113, 114, 115, &c. any number may be multiplied thereby in one line, but in these cases it will be necessary, not only to add in the first, but also the second past figures of the multiplicand.” If the Multiplier be any digit with 1 annexed thereto, as 21, 31, 301,3001, &c. The multiplicand will answer for the product of the unit, and the product of the digit removed a place or more (as the case may require) towards the left, added thereto, will give the required product. Črampleå. 12702 22508 131254 2256427 * This method is of use in bringing cwts: grs. and lbs. into pounds, b thus multiplying by 112, and mentally adding in the unds of the § weight, if any, as will be more particularly noticed in another part of this work, When the multiplier is the product of two or more numbers, multiply by one of the numbers, and the product by the other which will sometimes save labour. .. o - &ramples. When some of the figures or numbers in the multiplier, are multiples or products of other figures in the same ; it is generally easier and shorter to multiply by the least numbers, and then these products by such numbers as will make the greater, particular care being taken, to place the units figure of every line, exactly under the units place of that number in the mustiplicand to which it belongs. 161. 216608 × 153 162. 374879 x 217 Multiplication may also be performed without the aid of the table, by continual additions of the multiplicand, and if the multiplier is large may shorten the work. This, though more curious than useful, yet furnishes a proof that multiplication is but a compendious addition, as was said at the commencement of the rule, In making this tablet, if the last line be the same as the first with a cipher annexed, it may be presumed to be right. -- There are other contractions, which cannot be taught without the aid of Division, and are to be noticed in that rule For a demonstration of the general rule, and the several £ontra: tions here given in Multiplication, I offer the reader to Walker's Philosophy of Arithmetic, page 16 and pages 34 to 37. D IDIVISION Is a compendious subtraction, shows how often one number is . contained in another, and teaches how to separate a given number into as many equal parts as shall be assigned : it consists of three terms, which are called the dividend, divisor and quotient. The first is the number to be divided, the second that by which we are to divide, and the third is the result of the operation, which expresses the number of times that the divisor may be subtracted from the dividend, or number of equal parts required. When one number is contained in another a certain number of times exactly, it is said to measure this number, but if it does not, there will be something left, which is called the remainder. This number which is sometimes over, is usually considered the numerator of a fraction of which the divisor is: the denominator, having a straight line drawn between them. The remainder is always to be thus disposed when a whole number without any name is divided, but when the number divided has a name as pounds, tons, hundreds, &c. the remainder is considered as so many ones left of the said number or dividend, and by multiplying this remainder, by the number contained in the next inferior denomination, to that of the dividend, we may continue the division as at first, and if anything again remain, it is to be considered as so many ones of the denomination to which the former remainder was brought by the last multiplication, which may again be multiplied and divided as before, until all the inferior denominations of the first dividend are gone through, and then if any thing finally remain, it is to be annexed to the last quotient, placed over the divisor, with a line drawn between them, as a fraction or part of the last quotient. This character -- is the mark of Division, and shows, that the numbers between which it is placed are to be divided; the number which it immediately precedes being considered as the divisor, thus 84 -- 7 = 12 signifies that the number 84 is to be divided by 7 and 68 + 9 = 7#, that the number 68 is to be divided by 9; in the first example 84 is the dividend, 7 is the divisor, and 12 is the quotient arising after performing the division ; in the second example 68 is the dividend, 9 is the divisor, 7 the quotient and 5 the remainder. Division is also sometimes represented, by drawing a line, over which placing the dividend, and under it the divisor, thus ** = 12 signifies the same as 84 +7= 12. Though it is alumost impossible to give a rule for division, that a child can understand, yet as this book may fall into the |