160. What is the length and breadth of a rectangular field whose perimeter is 10 rods greater than a square field whose side is 50 rods, while its area is 250 square rods less than the area of the square field? Ans. Length, 75 rods; breadth, 30. 161. A rectangular piece of land was sold for $5 for every rod in its perimeter. If the same area had been in the form of a square, and sold in the same way, it would have brought $90 less; and a square field of the same perimeter would have contained 272 square rods more. What were the length and breadth of the field? Ans. Length, 49; breadth, 16 rods. 162. A starts from Springfield to Boston at the same time that B starts from Boston to Springfield. When they met, A had travelled 30 miles more than B, having gone as far in 12 days as B had during the whole time; and at the same rate as before B would reach Springfield in 5 days. How far from Boston did they meet? Ans. 42 miles. 163. The product of two numbers is 90; and the dif ference of their cubes is to the cube of their difference as 13: 3. What are the numbers? 164. A and B start together from the same place and travel in the same direction. A travels the first day 25 kilometers, the second 22, and so on, travelling each day 3 kilometers less than on the preceding day, while B travels 14 kilometers each day. In what time will the two be together again? Ans. 8 days. 165. A starts from a certain point and travels 5 miles the first day, 7 the second, and so on, travelling each day 2 miles more than on the preceding day. B starts from the same point 3 days later and follows A at the rate of 20 miles a day. If they keep on in the same line, when will they be together? Ans. 3 or 7 days after B starts. 166. A gentleman offered his daughter on the day of her marriage $1000; or $1 on that day, $2 on the next, $3 on the next, and so on, for 60 days. The lady chose the first offer. How much did she gain, or lose, by her choice? 167. The arithmetical mean of two numbers exceeds the geometrical mean by 2; and their product divided by their sum is 3. What are the numbers? 168. A father divided $130 among his four children in arithmetical progression. If he had given the eldest $25 more and the youngest but one $5 less, their shares would have been in geometrical progression. What was the share of each? 169. The sum of the squares plus the product of two numbers is 133; and twice the arithmetical mean plus the geometrical mean is 19. What are the numbers? 170. The sum of three numbers in geometrical progression is 117; and the difference of the second and third minus the difference of the first and second is 36. What are the numbers? 171. There are four numbers in geometrical progression, and the sum of the second and fourth is 60; and the sum of the extremes is to the sum of the means as 7:3. What are the numbers? SECTION XXV. LOGARITHMS. 241. LOGARITHMS are exponents of the powers of some number which is taken as a base. In the tables of logarithms in common use the number 10 is taken as the base, and all numbers are considered as powers of 10. By Arts. 119, 120, 10° 1, that is, the logarithm of 1 is 0 Therefore, the logarithm of any number between 1 and 10 is between 0 and 1, that is, is a fraction; the logarithm of any number between 10 and 100 is between 1 and 2, that is, is 1 plus a fraction; and the logarithm of any number between 100 and 1000 is 2 plus a fraction; and so on. Therefore, the logarithm of any number between 1 and 0.1 is between 0 and -1, that is, is 1 plus a fraction; the logarithm of any number between 0.1 and 0.01 is between —1 and 2, that is, is -2 plus a fraction; and so on. The logarithm of a number, therefore, is either an integer (which may be 0) positive or negative, or an integer positive or negative and a fraction, which is always positive. The representation of the logarithms of all numbers less than a unit by a negative integer and a positive fraction is merely a matter of convenience. The integral part of a logarithm is called the characteristic, and the decimal part the mantissa. Thus, the characteristic of the logarithm 3.1784 is 3, and the mantissa .1784. 242. The characteristic of the logarithm of a number is not given in the tables, but can be supplied by the following RULE. The characteristic of the logarithm of any number is equal to the number of places by which its first significant figure on the left is removed from units' place, the characteristic being positive when this figure is to the left and negative when it is to the right of units' place. Thus, the logarithm of 59 is 1 plus a fraction; that is, the characteristic of the logarithm of 59 is 1. The logarithm of 5417.7 is 3 plus a fraction; that is, the characteristic of the logarithm of 5417.7 is 3. The logarithm of 0.3 is -1 plus a fraction; that is, the characteristic of the logarithm of 0.3 is -1. The logarithm of 0.00017 is -4 plus a fraction; that is, the characteristic of the logarithm of 0.00017 is -4. 243. Since the base of this system of logarithms is 10, if any number is multiplied by 10, its logarithm will be increased by a unit (Art. 50); if divided by 10, diminished by a unit (Art. 54). Hence, the mantissa of the logarithm of any set of figures is the same, wherever the decimal point may be. As only the characteristic is negative, the minus sign is written over the characteristic. TABLE OF LOGARITHMS. 244. To find the logarithm of a number of two figures. Disregarding the decimal point, find the given number in the column N (pp. 268, 269), and directly opposite, in the column O, is the mantissa of the logarithm, to which must be prefixed the characteristic, according to the Rule in Art. 242. Thus, the log of 85 is 1.9294 The first figure of the mantissa, remaining the same for several successive numbers, is not repeated, but left to be supplied. Thus, the log of 83 is 1.9191 As, according to Art. 243, multiplying a number by 10 increases its logarithm by a unit, therefore, to find the logarithm of any number containing only two significant figures with one or more ciphers annexed, we use the same rule as above. Thus, the log of 850 is 2.9294 66 "750000 " 5.8751 The principle just stated is applicable also in the cases that follow. 245. To find the logarithm of a number of three figures. Disregarding the decimal point, find the first two figures in the column N, and the third figure at the top of one of the columns. Opposite the first two figures, and in the column under the third figure, will be the last three figures of the decimal part of the logarithm, to which the first figure in the |