son could never become twice as old as the younger son, after the death of the father. Let us therefore modify the general problem as follows: A man dying left two sons, the elder of which was a years of age, and the younger b years of age. How many years before the death of the father was the elder son twice as old as the younger? If we let x represent the number of years, then the solution will be as follows: Now suppose, as before, that a 30 and 18. Then by the new formula, x = 36—30 = 6, a result which will satisfy the modified conditions; for, six years before the death of the father, the age of the elder was 30—6= 24, and the age of the younger was 18—6 = 12. From the foregoing discussions we draw the following inferences: 1.- When the solution of a problem by a simple equation gives a negative result, the minus sign indicates that the problem is impossible, if understood in the exact sense of the enunciation. 2.-The impossibility thus indicated consists in adding a quantity when it should be subtracted; or in treating a quantity as reckoned or applied in a certain direction, when it should be reckoned or applied in an opposite direction. 3-In all such cases, an analogous problem may be formed, involving no impossibility, by changing the terms of the absurd condi tion to their opposites; and the answer to the new question will be found by simply changing the sign of the negative result already obtained. 182. The foregoing discussions give a more extensive signification to the plus and minus signs, and lead to a more general view of positive and negative quantities, than was presented in a former section. Let us recur to the problem of the two sons. In the solution of this problem, we employ the signs,+ and -, in the statement, mere ly to indicate addition and subtraction. But in the result, these signs have a very different use; they enable us to distinguish the circumstances or conditions of the quantities which they affect. Thus, under the first hypothesis, the period of time represented by a occurred after the death of the father, and in the result is found to be affected by the plus sign; but under the second hypothesis, the period represented by x occurred before the death of the father, and in the result is found to be affected by the minus sign. Thus we perceive that plus and minus, in Algebra, are not symbols of operation merely, but also symbols of relation, serving to distinguish quantities in opposite conditions or circumstances. It should be observed, however, that this enlarged use of the plus and minus signs is not entirely conventional or arbitrary, but is necessarily involved in the more extended signification given to the terms addition and subtraction, in Algebra. Indeed we shall never meet with a negative result in the solution of problems, so long as the language conforms, in the exact arithmetical sense, to the facts of the case. EXAMPLES FOR PRACTICE. 1. What number is that whose fourth part exceeds its third part by 12? Ans. 144. The question is impossible, if understood in an arithmetical sense. Let the pupil modify the cnunciation, and solve the new problem. 2. A man when he was married was 30 years old, and his wife 15. How many years must elapse before his age will be three times the age of his wife? Ans. -7 years. That is, their ages bore the specified relation 7 years before, not after, their marriage. 3. The sum of two numbers is s, and their difference d; what are the numbers ? 8 d Ans. Greater, 2 + ; Less, S d 2 2 How shall the result be interpreted when s = 120 and d = 160? 4. Two men, A and B, commenced trade at the same time, A having 3 times as much money as B. When A had gained $400 and B $150, A had twice as much money as B; how much did each have at first? Ans. A was in debt $300, and B $100. 5. A man worked 7 days, and had his son with him 3 days, and received for wages 22 shillings, and the board of his son and himself while at work. He afterward worked 5 days, and had his son with him one day, and received 18 shillings. What were his daily wages, and what the daily wages of his son? Ans. The father received 4 shillings per day, and paid 2 shillings for his son's board. 6. A man worked for a person 10 days, having his wife with him 8 days, and his son 6 days, and he received $10.30 as compensation for all three; at another time he wrought 12 days, his wife 10 days, and sou 4 days, and he received $13.20; at another time he wrought 15 days, his wife 10 days, and his son 12 days, at the same rates as before, and he received $13.85. What were the daily wages of each? Ans. He received $.75 for himself, $.50 for his wife, and paid $.20 for his son's board. 7. A man wrought 10 days for his neighbor, his wife 4 days, and son 3 days, and received $11.50; at another time he served 9 days, his wife 8 days, and his son 6 days, at the same rates as before, and received $12.00; a third time he served 7 days, his wife 6 days, and his son 4 days, at the same rates as before, and he received $9.00. What were the daily wages of each? Ans. Husband's wages, $1.00; Wife's, 0; Son's, $.50. 8. What fraction is that which becomes when 1 is added to its numerator, and when 1 is added to its denominator ? Ans. In an arithmetical sense, there is no such fraction. The algebraic expression, 18, will give the required results. How shall the enunciation be modified, to form an analogous question involving no absurdity? 9. Four merchants, A, B, C, D, find by their balance sheets that if they unite in a firm, receiving the assets and assuming the liabilities of each, they will have a joint net capital of $5780. If A, B, and C unite on the same conditions, their joint capital will be $7950; if B, C, and D unite, their joint capital will be $2220; and 4.3560 C. 5930 if C, D, and A unite, their joint capital will be $7320. Required the net capital or the net insolvency of each. 10. Two men were traveling on the same road towards Boston, A at the rate of a miles per hour, and B at the rate of miles per hour. At 6 o'clock A was at a point m miles from Boston, and at 10 o'clock B was at a point n miles from Boston. Find the time when A passed B upon the road. 11. What time of day will be indicated by the preceding formula, if m = 36, n = 28, a = 5, and b = 3? Ans. 4 o'clock. 12. There are two numbers whose difference is a; and if 3 times the greater be added to 5 times the less, the sum will be b. What are the numbers? b+5a Ans. Greater, ; Less, 8 b-3a 8 = 48? How shall this result be interpreted if a = 24 and b NOTHING AND INFINITY. 183. The limits between which all absolute values are comprised, are nothing and infinity; and the symbols by which these limits are denoted, are 0 and co. 184. In certain algebraic investigations it is convenient to employ these symbols in connection with each other and the ordinary symbols of quantity. They may thus sustain the relations of divisor, dividend, quotient, or factor. Such relations, however, can not really exist except between symbols of quantity. Hence, in Algebra, 0 does not always signify merely absence of value; nor does o represent infinity, in the highest sense of the word. The more complete definition of these symbols may be given as follows: 185. The symbol 0, called nothing, or zero, may be used to denote the absence of value, or to represent a quantity less than any assignable value. 186. The symbol oo, called infinity, is used to represent a quantity greater than any assignable value. INTERPRETATION OF THE FORMS A A 0 AND 187. In order to understand the signification of the expressions, we may consider the symbols 0 and co as resulting from an arbitrary or varying quantity, made to diminish until it becomes indefinitely small, or to increase until it becomes indefinitely great.. 188. Let a Ն represent a fraction, a and b being arbitrary quantities. And let it be remembered that the value of a fraction depends simply upon the relative values of the numerator and denominator. 1. If the denominator b is made to diminish, becoming less and less continually, while the numerator a remains unchanged, the value of the fraction must increase, becoming greater and greater continually, (119, II); and thus when the denominator b becomes less than any assignable quantity, or 0, the value of the fraction. must become greater than any assignable quantity, or co. Hence, we conclude that A finite quantity divided by zero is an expression for infinity. 2. If the denominator b is made to increase, becoming greater and greater continually, while the numerator a remains unchanged, the value of the fraction must diminish, becoming less and less continually, (119, II); and when the denominator b becomes greater than any assignable quantity, or co, the value of the fraction must become less than any assignable quantity, or 0. Hence, A finite quantity divided by infinity is an expression for zero or nothing. 3.- If the numerator a is made to diminish, becoming less and .ess continually, while the denominator b remains unchanged, the |