subtracted from the product. Thus, in the cases of 371 = (11 × 11 × 3 + 6 and 527 = · (11 x 12 x 4) — 1. Supposing that 12 is the largest number to be used as a multiplier at one operation, the method just stated will be generally sufficient. In fact, all numbers, as far as 1420 inclusive, may be replaced by a set of multipliers each not greater than 12, arranged as in the examples given, that is to say, all but one being used as continued multipliers, and the product of the last and the multiplicand being either added or subtracted at the end. And, by proper arrangement of multipliers, each not greater than 12, multiplication by any number whatever may be effected in a somewhat similar manner. The trouble and difficulty of making these arrangements of multipliers may, however, be avoided by using a method which, though generally longer, is much more simple. For example, let it be required to multiply by 1427. Multiply by 10, and 10, and 10, this will give three lines of figures, equal to 10 times, 100 times, and 1000 times the multiplicand respectively. The sum, therefore, of (1 x third line) + (4 × second) + (2 x first) + (7 × multiplicand) will be the product required. As an example, multiply 7 dwt. 2 gr. by 1427. This is merely given as an illustration of the method just explained. It would be a shorter and easier plan to reduce the 7 dwt. 2 gr. to grains, and after multiplying by 1427, reduce to dwts., ounces, and pounds. The question is here worked out in both ways, In all cases judgment must be exercised, whether to reduce to one denomination or not before multiplying. It will be seen, further on in this book, that the above question, as well as most of the questions in Compound Multiplication, may be still more shortly worked out by a process called 'Practice.' 24. 2 mi. 5 fur. 73 yd. x 713-3 mi. 1 fur. 181 yd. × 589. 25. 26. 1 yr. 210 d. 3 hr. × 629 = 1 yr. 129 d. 21 hr. × 731. X 199 yd. 2 ft. 6 in. × 689=1 fur. 2 po. 5 yd. 6 in. × 585. 27. £3 15s. 3d. × 2527 = £5 19s. 12d. × 1596. 28. 8 hr. 57 m. 51 sec. x 2419=11 hr. 51 m. 21 sec. x 1829. Compound Division, by a divisor not greater than 12, may be performed at one operation, by dividing successively each denomination, beginning with the highest, setting down the quotient, reducing the remainder to the next lower denomination, and carrying forward the number so obtained. In division of money it will often occur that there are farthings, or fractions of a penny. In the cases of other concrete quantities also, there may be a fraction of the lowest denomination in the dividend. The method of treating such fractions has already been explained. (See p. 22.) If the divisor be greater than 12, but can be separated into factors each not greater than 12, then the division can be performed by means of two or more short divisions of the kind just given. If the divisor be greater than 12, but cannot be separated into small factors, the method is the same in principle with that used in other cases, but the form of the work is similar to that of long division of abstract numbers. If the divisor be 100, 1000, 10,000, &c., the division at each stage of the process may be most quickly performed by cutting off two, three, or four figures to the right. It has now been explained how concrete numbers may be added together, or one subtracted from another, and how they may be either multiplied or divided by abstract numbers. A question now arises with respect to the meaning to be attached to multiplication and division by concrete numbers. Suppose first the case of an abstract multiplied by a concrete number, as for instance 7 multiplied by 8 feet. If our first idea of multiplication be adhered to, this would mean a number of sevens added together, such number being equal to 8 feet, and this meaning is utterly unintelligible. But it has been stated that the full meaning of the words multiplication and division could not at first be given, and we shall find that this case of 7 multiplied by 8 feet leads to a sound and advantageous extension of their signification. In abstract numbers we found it was universally true that the order of the factors was immaterial, so that 7 times 9 and 9 times 7 were each equal to the same number, 63. If this law were extended to the present case, 7 multiplied by 8 feet ought to be the same as 8 feet multiplied by 7, and we know that the latter is 56 feet. It is clear, therefore, that we cannot adhere both to our original definition of multiplication, and to the law that the order of the factors is immaterial. From the definition it follows that 7 × 8 feet has no meaning, and from the law it follows that 7 x 8 feet means 56 feet. Being thus obliged to abandon one or the other, it is better to abandon the definition, and extend our meaning of the word multiplication so as to retain the universal application of the law. Again, in abstract numbers, it was a law that multiplication and division were contrary operations; that is, that a number multiplied and the product then divided by the same number would remain unchanged. Thus 11 x 12 = 132 and 132 ÷ 12 = 11. If this law be extended, we should have 7 multiplied by 8 feet 56 feet, and therefore 56 feet, divided by 8 feet, must be 7. Hence an abstract number may be multiplied by a concrete number, giving a concrete number in the product, and a concrete number may be divided by a |