A block of wood, in the form of a rectangular parallelopiped, measures along its edges 181 feet, 51 feet, and 3 feet, respectively; determine its value on the supposition that a cubical block, measuring 11 inches along the edge, is worth 3s. 6d. 10. If 36 men, working 8 hours a day for 16 days, can dig a trench 72 yards long, 18 wide, and 12 deep, in how many days will 32 men, working 12 hours a day, dig a trench 64 yards long, 27 wide, and 18 deep? 11 XI. 1. Express as a decimal; and thence find its value when unity` represents £300. 2. A parish containing 2456 acres is rated on a rental of £3070; a rate of 8d. in the pound being levied, what on the average is the charge per acre? 4. 3. Find the price of 2 tons, 16 cwt., 17 lbs of sugar at 10d. for 21 lbs. If1 cwt. of an article cost £7, at what price per lb. must it be sold to gain of the outlay? 5. Find in inches and fractions of an inch the value of '00003551136 of a mile. Explain the process employed. 6. Express each silver coin now current in England by a decimal of 23d. Ifth of 23d. be the unit of money, what decimal will express a halfpenny? 7. An American dollar is 4s. 3ğd., and is 5'42 francs; find the number of francs in £1 sterling, and express both a dollar and a franc in terms of the unit of money mentioned in the last question. 8. A and B can do a piece of work in 6 days, B and C in 7 days, and A, B, and C can do it in 4 days; how long would A and C take to do it? 9. If a sheet of paper 5 feet long by 23 feet broad be cut into strips an inch broad; how many sheets would be required to form a strip that would reach round the earth, (25,000 miles)? 10. A bag contains a certain number of sovereigns, three times as many shillings, and four times as many pence, and the whole sum in the bag is £280; find how many sovereigns, shillings, and pence it contains respectively. RULE OF THREE. 142. We may compare one number with another, or ascertain the relation which one bears to the other in respect of magnitude, in two different ways; either by considering how much one is greater or less than the other; or by considering what multiple, part, or parts, one is of the other, that is, how many times or parts of a time, or both, one number is contained in the other. Thus if we compare the number 12 with the number 3, we observe, adopting the first mode of comparison, that 12 is greater than 3 by the number 9; or, adopting the second mode of comparison, that 12 contains 3 four times, and is thus 2 or four times as great as 3. Again if we compare the number 7 with the number 13, we observe, according to the first mode of comparison, that 7 is less than 13 by the number 6; and, according to the second, that as 1 is one thirteenth part of 13, so 7 is seven thirteenth parts of 13, or ths of 13. 143. The relation of one number to another in respect of magnitude is called RATIO; and when the relation is considered in the first of the above methods, that is, when it is estimated by the difference between the two numbers, it is called ARITHMETICAL RATIO; but when it is considered according to the second method, that is, when it is estimated by considering what multiple, part, or parts, one number is of the other, or, which is seen from above to be the same thing, by the fraction which the first number is of the second, it is called GEOMETRICAL RATIO. Thus for instance, the arithmetical ratio of the numbers 12 and 3 is 9; while their geometrical ratio is 12 or 4. In like manner the arithmetical ratio of 7 and 13 is 6, while their geometrical ratio is 144. It is more common, however, in comparing one number with another to estimate their relation to one another in respect of magnitude according to the second method, and to call that relation so estimated by the name of RATIO. According to this mode of treatment, which we shall adopt in what follows, "RATIO is the relation which one number has to another in respect of magnitude, the comparison being made by considering what multiple, part, or parts, the first number is of the second, or how many times or parts of a time, or both, the second is contained in the first." 145. It is plain that, for any two numbers, the fraction in which the first is numerator and the second denominator, will correctly express the multiple or part, or both, which the first number is of the second, or the number of times or parts, or both, of a time the second is contained in the first. Thus if we take the numbers 12 and 3, the fraction, which is equivalent to the whole number 4, shows the multiple which 12 is of 3, or the number of times 3 is contained in 12. And again, if we take the numbers 7 and 13, the fraction will express the part or parts which the number 7 is of 13, or will express the part or parts of a time that 13 is contained in 7: for 1 is one thirteenth part of 13, so that 7 must be seven thirteenth parts of 13, that is, ths of it; and 1 is contained 7 times in 7, so that 13 must be contained only ths of a time in 7. We conclude therefore that the ratio of one number to another may be estimated and expressed by the fraction in which the former number is the numerator and the latter the denominator. 146. The ratio of one number to another is often denoted by placing a colon between them. Thus the ratio of 7 to 13 is denoted by 7:13. As we have shown that the ratio of one number to another may be expressed by the fraction in which the former is the numerator and the latter the denominator, we see that 7: 13 is. The two numbers which form a ratio are called its terms; the first number, or the number compared, being called the first term, or THE ANTECEDENT, and the second number or that with which the former is compared, the second term, or THE CONSEQUENT, of the ratio. 147. If the two numbers to be compared together be concrete, they must be of the same kind. We cannot compare together 7 days and 13 miles in respect of magnitudé; but we can compare 7 days with 13 days; and it is clear that 7 days will have the same relation to 13 days in respect of magnitude, which the number 7 has to the number 13, so that the ratio of 7 days to 13 days will be the same as the ratio of the abstract number 7 to the abstract number 13, and may be expressed by the fraction. If how ever the concrete numbers, though of the same kind, be not in the same denomination of that kind, it will be convenient to reduce them to one and the same denomination in order to find their ratio. Thus, if one of the numbers be 7 days and the other be 13 hours, the ratio of the former to the latter will not be that of 7 to 13, but that of 7 days to 13 hours, that is, 168 hours to 13 hours, which will clearly be the same as that of the abstract number 168 to the abstract number 13, and so will be expressed not by, but by 163. We see, then, that 7 days: 13 hours is the same as 168: 13, and that each is=168. Thus it is plain that when the num bers are concrete, we may reduce them to one and the same denomination, and then, in considering their ratio, treat them as abstract numbers. 148. PROPORTION is the equality of two ratios; so that, when the ratio of one number to a second is equal to the ratio of a third number to a fourth, proportion is said to exist among the numbers, and the numbers are called PROPORTIONALS. Thus, the ratio of 8 to 9 is equal to that of 24 to 27, for the former ratio is §, and the latter ratio is 24, which is also equal to §. The ratios being equal, proportion exists among the numbers 8, 9, 24, 27; and thus those numbers are proportionals. 149. When proportion exists among four numbers, that is, when the ratio of the first to the second is equal to that of the third to the fourth, this proportion or equality is often denoted by writing down the two ratios in the manner mentioned in (Art. 146) in one line, and placing a double colon (::) between them. Thus the existence of proportion among the numbers 3, 4, 9, 12, is indicated as follows, 34:9: 12 which is commonly read thus, "three are to four as nine to twelve," or 'as three to four so nine to twelve." It will appear from what has preceded, that by the expression 3: 4 :: 9; 12, it is meant in fact that 150. In order to form a proportion four numbers are required. It may indeed happen that the second and third are the same, in which particular case it might be said that only three numbers are required; thus 96: 6:4; but even in such a case it is better to consider the second and third as distinct numbers, and to regard the proportion as consisting of four numbers, of which indeed two are equal. The four numbers required to form a proportion are called its terms. In the proportion 34:9: 12, we have 3 for the first term, 4 for the second, 9 for the third, and 12 for the fourth term, of the proportion. 151. It has been stated that proportion is the equality of two ratios, and we have explained that the two numbers constituting a ratio must either be both abstract, or (if concrete) both of the same kind. In a proportion if one of the ratios be formed by two abstract numbers, the other may arise from two concrete numbers. For it has been explained (Art. 147) that if a ratio consist of two concrete numbers, we may reduce them both to the same denomination, and then treat the resulting numbers as abstract, the ratio of those abstract numbers being the same as that of the two concrete numbers from which they have arisen. For the : same reason, one of the two ratios constituting a proportion may be formed from concrete numbers of one kind, while the other is formed from concrete numbers of a different kind; for 7 days: 13 days :: 7 miles 13 miles, each ratio being in fact that of 7 to 13. Indeed it appears by (Art. 147) that the ratio of two concrete numbers may always be expressed by a ratio of two abstract numbers. If both or either of the ratios in a proportion be formed from concrete numbers, we may thus replace each such ratio by one arising from abstract numbers, and in this way every term of the proportion will become an abstract number; so that, notwithstanding the remark in note (Art. 26), any one of the terms may then be multiplied or divided by any other. 152. It is readily seen that if proportion exist among four numbers taken in a certain order, it will exist also among the same numbers taken in the contrary order. Thus the numbers 8, 9, 24, 27, being proportionals in the order in which they stand, the numbers 27, 24, 9, 8, will also be proportionals. For, 153. If only three of the numbers in a proportion be given, we can by means of them find the fourth, and the method or Rule by which it may be found is one of great importance in Arithmetic. We have seen that proportion exists among the numbers 8, 9, 24, 27. If the first three numbers only were given, and we were required, by means of these, to find the fourth, the method or Rule to be adopted ought to determine a number to which 24 would have the same ratio, as 8 to 9; or, which is seen from the last article to be the same thing, it ought to determine a number which will have the same ratio to 24, which 9 has to 8; this number being of course 27. Almost all questions which arise in the |