pleted only 220 yards of the wall. It is required to determine how many men must be added to the former, that the whole number of them may just finish the wall in the time proposed, at the same rate of working. Ans. 4 men to be added. QUEST. 6. Determine how far 500 millions of guineas will reach, when laid down in a straight line touching one another; supposing each guinea to be an inch in diameter, as it is very nearly. Ans. 7891 miles, 728 yds, 2 ft, 8 in. QUEST. 7. Two persons, A and B, being on opposite sides of a wood, which is 536 yards about, begin to go round it, both the same way, at the same instant of time; A goes at the rate of 11 yards per minute, and B 34 yards in 3 minutes; the question is, how many times will the wood be gone round before the quicker overtake the slower ? Ans. 17 times. QUEST. 8. A can do a piece of work alone in 12 days, and B alone in 14; in what time will they both together perform a like quantity of work? Ans. 6 days. QUEST. 9. A person who was possessed of a share of a copper mine, sold of his interest in it for 18007; what was the reputed value of the whole at the same rate? Ans. 4000l. QUEST. 10. A person after spending 207 more than of his yearly income, had then remaining 301 more than the half of it; what was his income? Ans. 2001. QUEST. 11. The hour and minute hands of a clock are exactly together at 12 o'clock; when are they next together? Ans. 1 hr or 1 hr 5 min. QUEST. 12. If a gentleman whose annual income is 1500l, spend 20 guineas a week; whether will he save or run in debt, and how much in the year? Ans. he saves 4087. QUEST. 13. A person bought 180 oranges at 2 a penny, and 180 more at 3 a penny; after which he sold them out again at 5 for 2 pence: did he gain or lose by the bargain? Ans. he lost 6 pence. QUEST. 14. If a quantity of provisions serves 1500 men 12 weeks, at the rate of 20 ounces a day for each man; how many men will the same provisions maintain for 20 weeks, at the rate of 8 ounces a day for each man? Ans. 2250 men. QUEST. 15. In the latitude of London, the distance round the earth, measured on the parallel of latitude, is about 15550 miles; now as the earth turns round in 23 hours 56 minutes, at what rate per hour is the city of London carried by this motion from west to east? QUEST. 16. A father left his son a fortune, : Ans. 649 of which he miles an hour. ran through in after which he 8 months of the remainder lasted him 12 months longer; had 8207 left: what sum did the father bequeath his son? Ans. 1913/ 6s 8d. QUEST. 17. If 1000 men, besieged in a town, with provisions for 5 weeks, allowing each man 16 ounces a day, be reinforced with 500 men more; and supposing that they cannot be relieved till the end of 8 weeks, how many ounces a day must each man have, that the provision may last that time? Ans. 63 ounces. QUEST. 18. A younger brother received 84007, which was just 3 of his elder brother's fortune: what was the father worth at his death? Ans. 192001. QUEST. 19. A person, looking on his watch, was asked what was the time of the day, who answered, "It is between 5 and 6;" but a more particular answer being required, he said that "the hour and minute hands are exactly together" what was the time? Ans. 27 min. past 5. QUEST. 20. If 20 men can perform a piece of work in 12 days, how many men will accomplish another thrice as large in one-fifth of the time? Ans. 300. QUEST. 21. A father devised of his estate to one of his sons, and 7 of the residue to another, and the surplus to his relict for life. The children's legacies were found to be 5147 6s 8d different: what money did he leave the widow the use of? Ans. 1270l 1s 9}{d. QUEST. 22. A person, making his will, gave to one child of his estate, and the rest to another. When these legacies came to be paid, the one turned out 12007 more than the other: what did the testator die worth? Ans. 4000l. QUEST. 23. Two persons, A and B, travel between London and Lincoln, distant 100 miles, A from London, and B from Lincoln at the same instant. After 7 hours they met on the road, when it appeared that A had rode 11⁄2 miles an hour more than B. At what rate per hour then did each of the travellers ride? Ans. A 72, and B 61⁄2 miles. QUEST. 24. Two persons, A and B, travel between London and Exeter. A leaves Exeter at 8 o'clock in the morning, and walks at the rate of 3 miles an hour, without intermission; and B sets out from London at 4 o'clock the same evening, and walks for Exeter at the rate of 4 miles an hour constantly. Now, supposing the distance between the two cities to be 130 miles, where will they Ans. 69 miles from Exeter. meet? QUEST. 25. One hundred eggs being placed on the ground, in a straight line, at the distance of a yard from each other: how far will a person travel who shall bring them one by one to a basket, which is placed at one yard from the first egg? Ans. 10100 yards, or 5 miles and 1300 yards. QUEST. 26. The clocks of Italy go on to 24 hours: how many strokes do they strike in one complete revolution of the index? Ans. 300. QUEST. 27. One Sessa, an Indian, having invented the game of chess, showed it to his prince, who was so delighted with it, that he promised him any reward he should ask; on which Sessa requested that he might be allowed one grain of wheat for the first square on the chess-board, 2 for the second, 4 for the third, and so on, doubling continually to 64, the whole number of squares. Now, supposing a pint to contain 7680 of these grains, and one quarter or 8 bushels to be worth 27s 6d, it is required to compute the value of all the corn. Ans. 64504682162857 17s 3d 332779. QUEST. 28. A person increased his estate annually by 1007 more than the part of its value at the beginning of that year; and at the end of 4 years found that his estate amounted to 103427 3s 9d: what had he at first? Ans. 4000l. Ans. 5 per cent. to give A 120 more, and B Ans. A 445, B 230, C 325. QUEST. 29. Paid 1012/ 10s for a principal of 7501, taken in 7 years before: at what rate per cent. per annum did I pay interest? QUEST. 30. Divide 10007 among A, B, C; so as 95 less than C. QUEST. 31. A person being asked the hour of the day, said, the time past noon is equal to ths of the time till midnight: what was the time? Ans. 20 min. past 5. QUEST. 32. Suppose that I have of a ship whose whole worth is 12007; what part of her have I left after selling of of my share, and what is it worth? Ans. 3, worth 1857. QUEST. 33. Part 1200 acres of land among A, B, C; so that B may have 100 more than A, and C 64 more than B. Ans. A 312, B 412, C 476. QUEST. 34. What number is that, from which if there be taken of , and to the remainder be added off, the sum will be 10? Ans. 9. QUEST. 35. There is a number which, if multiplied by of of 1, will produce 1: what is the square of that number? Ans. 1%. QUEST. 36. What length must be cut off a board, 8 inches broad, to contain a square foot, or as much as 12 inches in length and 12 in breadth ? Ans. 1619 inches. QUEST. 37. What sum of money will amount to 1381 28 6d, in 15 months, at 5 per cent. per annum simple interest? Ans. 1301. QUEST. 38. A father divided his fortune among his three sons, A, B, C, giving A 4 as often as B 3, and C 5 as often as B 6; what was the whole legacy, supposing A's share was 4000/? Ans. 9500l. QUEST. 39. A young hare starts 40 yards before a grey-hound, and is not perceived by him till she has been up 40 seconds; she scuds away at the rate of 10 miles an hour, and the dog, on view, makes after her at the rate of 18: how long will the course hold, and what ground will be run over, counting from the outsetting of the dog? Ans. 60 sec. and 530 yards run. QUEST. 40. Two young gentlemen, without private fortune, obtain commissions at the same time, and at the age of 18. One thoughtlessly spends 107 a year more than his pay; but, shocked at the idea of not paying his debts, gives his creditor a bond for the money, at the end of every year, and also insures his life for the amount; each bond costs him 30 shillings, besides the lawful interest of 5 per cent. and to insure his life costs him 6 per cent. The other, having a proper pride, is determined never to run in debt; and, that he may assist a friend in need, perseveres in saving 107 every year, for which he obtains an interest of 5 per cent. which interest is every year added to his savings, and laid out, so as to answer the effect of compound interest. Suppose these two officers to meet at the age of 50, when each receives from Government 400l per annum; that the one, seeing his past errors, is resolved in future to spend no more than he actually has, after paying the interest for what he owes, and the insurance on his life. The other, having now something beforehand, means in future to spend his full income, without increasing his stock. It is desirable to know how much each has to spend per annum, and what money the latter has by him to assist the distressed, or leave to those who deserve it? Ans. The reformed officer has to spend 667 19s 1d 2·6583q per annum. The prudent officer has to spend 4377 12s 11d 3·4451q per annum, and ALGEBRA. I. Introductory explanation of the character and objects of this branch of Mathematics. 1. Ir a series of given numbers be directed to be combined in any specified manner, that specification may either be expressed in words at length, or by means of the usual symbols of arithmetical operations, and such other contrivances as have been already explained in the treatise on arithmetic. When the symbolic or abbreviated mode of expression is employed, the collection of numbers and symbols constitute what is called an arithmetical expression. Thus, if from the sum of six and seven we were directed to take three, and multiply the remainder by one-half the defect of six from ten, then the arithmetical expression for this would be all the symbols and modes of writing employed in this expression having been already defined and rendered familiar (page 6). 2. If we actually perform the several operations here indicated or directed, we shall obtain what is called the value of the expression, which in the present case is 20, and in each case is dependent upon the arithmetical conditions of the given expression. When we express that 20 is the equivalent or value of such an expression, we form an arithmetical equation, viz. 3. In the solution of any arithmetical question, we are enabled. for the most part, with a little consideration, to refer to a class for solving which rules have been already invented. These rules consist in the substitution of a series of arithmetical operations of a simple kind, to be performed in a specified order upon the several numbers given in the conditions of the question. Thus, in questions which are reducible to "the Rule of Three, terms," or simply the "Rule of Three," the answer is found by arranging the given terms in a particular and specified order, and then dividing the product of the second and third by the first of the terms so arranged. Now as this rule is the same whatever the first, second, and third terms may happen to be, it could not be expressed without some symbols to stand for those terms, which, whilst expressing the fact of their being numbers so arranged, would yet not confine them to any particular values as numbers, nor to any particular class as objects. The letters of the Alphabet have been used for this purpose throughout Europe, and those regions which have received their science from Europeans, without a single exception. Sometimes they have been so chosen as to be the initial letter of the kind of quantity whose numbers they stood in place of, as t, s, v, for time, space, and velocity: but they have been generally so selected, that the earlier letters of the alphabet, a, b, c, ... should stand in the place of the numbers which in every actual question are given, or express the given conditions ; whilst the final letters, z, y, z, w, v, ... have been employed to occupy the place in the immediate expression of the question, of those numbers which till the rule has been put into execution are unknown, and which therefore it is the object of the problem to discover. Thus, in the Rule of Three, if a, b, c taken in order be selected to designate generally the first, second, and third of the given terms taken in order, and which are all given in each actual case, and if the answer, as yet unknown, be denoted by x, we shall have the condition, when simplified from all extraneous considerations, expressed thus: abcx, and the formula or rule for solution would take this form: 4. So likewise, if in the case of the numbers 10, 6, 7, 3, which occur in the arithmetical formula in the beginning of this chapter, these numbers had been the particular values of four quantities which in the solution of some specific class of questions were always given amongst its conditions, and that by some means or other, the rule for solution had been discovered to be, that the third number (taking them in the order of their occurrence in the question, or of some arrangement to which the rule always subjected them) must be taken from the sum of the second and fourth, and the remainder multiplied by half the defect of the second from the first: then, writing as the symbols of the first, second, third, and fourth terms of the equation in art 2. the letters a, b, c, d, and for the yet unknown number, that is, the answer sought, writing the symbol x, the expression of this rule in an algebraic formula will be 5. To take another example, suppose it were proposed to find that number to the square of which we add the number b, then the sum shall be equal to a times the number itself; then the formula of solution would be and it is very obvious, at first sight, that the method of solution falls under no rule that has been given in the treatise on arithmetic; and therefore such rule must be sought for by some new method of investigation, either analogous to those by which the arithmetical rules were discovered, or possibly by some process altogether different from them in principle as well as in plan. * The student may be surprised to see that two different methods of calculating the result have been given; but he will see the reason of this hereafter, and also that often three, four, or indeed any number whatever of answers to different questions are possible. To take an instance, the values 4 and 3, as those of a and b, would find x = = 2 + √1, and x=2—√T, or 3 and 1. He can verify these by actual trial. In trying other numbers he may hit upon some conditions that will not admit of any solution whatever; as, if a = 2 and b=2, the solutions would be a = 1 + √ = 1, and x = 1 − √√ − 1, results which he is not in possession of methods capable of interpreting. Such cases are said to be impossible, and their explanation will be found in a note on "Quadratic Equations" in the present volume. |