15. In the latitude of London, the distance round the earth, measured on the parallel of latitude, is about 15,550 miles; now, as the earth turns round in 23 hours 56 minutes, at what rate per hour is the city of London carried from west to east? Ans. 649 miles an hour. 16. A father left his son a fortune, of which he ran through in 8 months; of the remainder lasted him 12 months longer; after which he had 8207. left. What sum did the father bequeath his son ? Ans. 1913/. 6s. 8d.

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. 18. A younger brother received 84007., which was just of his elder brother's fortune. What was the father worth at his death? Ans. 19,2007.

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 were then exactly together." What was the time? Ans. 27 min. past 5.

20. If 20 men 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. 21. A father devised of his estate to one of his sons, and 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. 94d. 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 to be 12007. more than the other. What did the testator die worth? Ans. 40007. 23. 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 meet? Ans. 69 miles from Exeter.

24. 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. 10,100 yards, or 5 miles and 1300 yards. 25. The clocks of Italy go on to 24 hours; then how many strokes do they strike in one complete revolution of the index? Ans. 300..

26. 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 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. 3 d. 32783. 27. A person increased his estate annually by 1007. more than the part of it; and at the end of 4 years found that his estate amounted to 103427. 3s. 9d. What had he at first?

Ans. 40007.

28. Paid 10127. 10s. for a principal of 7507., taken in 7 years before; at what rate per cent. per annum did I pay interest? Ans. 57. per cent.

29. Divide 1000l. among A, B, C; so as to give A 1207. more, and B 957. less than C. Ans. A 4457., B 230l., C 3251. 30. 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.

31. 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.; worth 1857.

32. What number is that, from which if there be taken of, and to the remainder be added of; the sum will be 10? Ans. 97.

33. There is a number which, if multiplied by of of 14, will produce 1. What is the square of that number? Ans. 1.

34. What length must be cut off a board, 84 inches broad, to contain a square foot, or as much as 12 inches in length, and 12 in breadth?

Ans. 16 inches. 35. What sum of money will amount to 1387. 2s. 6d. in 15 months, at 5 per cent. per annum simple interest ? Ans. 1307. 36. 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 40007.

Ans. 95007. 37. A young hare starts 40 yards before a greyhound, 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 yds. run. 38. Divide 93607. among A, B, and C, in such a manner that A's share may be to B's as 7 to 6, and B's to C's as 4 to 34.

Ans. A's share 36407.; B's 31207.; and C's 26007. 39. If of a steam-ship be purchased for 15,360l. 13s. 4d., how much will be gained per cent. by selling half the vessel for 12,9027. 19s. 2zd.?

Ans. 127. per cent.

40. Find the cube root of 068 to eight places of decimals, contracting the work for the last four figures.

Ans. 40816551.

of of a

41. Suppose 27. and of of a pound will purchase 3 yards and yard of cloth; how much may be purchased by 9 shillings and of a shilling? Ans. of a yard. and give a

42. Divide 437. 12s. 9d. among 7 men, 9 women, and 3 boys, woman of a man's share, and a boy of a woman's.

Ans. A boy's share 17. 12s. 24 d.

1762. 274 1977.

43. A workman was hired for 24 days, at 4s. 6d. per day, for every day he worked; but for every day he was absent he was to forfeit 1s. 6d. How many days did he work when the balance due to him was 37. 18s.; and also how many days was he absent, when he had to receive only one day's wages? To be done without position.

Ans. 19 days, in the former case; and 18 days, in the latter case. 44. The interest of a certain sum for 12 years and 9 months, at 47. per cent. simple interest, was found to be 1857. more than the interest of the same sum for 6 years, at 57. per cent. Find the sum without the aid of the rule of position?

Ans. 10007.

ALGEBRA.

DEFINITIONS AND NOTATION,

1. ALGEBRA is that department of Mathematics which enables us, by the aid of certain symbols, to abridge and generalize the reasoning employed in the solution of all questions relating to numbers.

These questions are of two kinds:

The Theorem, whose object is to demonstrate certain properties and relations which exist in numbers which are known and given.

The Problem, whose object is to discover certain numbers which are unknown by means of other numbers which are known, and which bear a relation to the unknown numbers, indicated by the conditions of the problem.

2. The principal symbols employed in algebra are the following:

I. The letters of the alphabet, a, b, c, &c., which are employed to denote the numbers which are the object of our reasonings.

II. The sign + which is named plus, and is employed to denote the addition of two or more numbers.

Thus 12+ 30 signifies 12 plus 30, or, 12 augmented by 30. In like manner a+b signifies a plus b, or, the number designated by a augmented by the number designated by b.

III. The sign - which is named minus, and is employed to denote the subtraction of one number from another.

Thus 54 · 23 signifies 54 minus 23, or, 54 diminished by 23. In like manner a- -b signifies a minus b, or, the number designated by a diminished by the number designated by b.

The sign is sometimes employed to denote the difference of two numbers, when it is not known which is the greater. Thus a ~b signifies the difference of a and b, when it is not known whether the number designated by a be less or greater than the number designated by b.

IV. The sign × which is named into, and is employed to denote the multiplication of two or more numbers.

Thus 72 × 26 signifies 72 into 26, or, 72 multiplied by 26. In like manner, a × b signifies a into b, or, a multiplied by b; and a × b × c signifies the continued product of the numbers designated by a, b, c; and so on for any number of factors.

The process of multiplication is also frequently indicated by placing a point between the successive factors; thus, a. b. c. d signifies the same thing as a xbx c x d.

In general, however, when numbers are represented by letters, their multiplication is indicated by writing the letters in succession, without the interposition of any sign. Thus ab signifies the same thing as a . b, or a × b; and a b c d is equivalent to a . b. c. d, or a × bx c x d.

It must be remarked, that the notation a. b or a b can be employed only when the numbers are designated by letters; if, for example, we wished to represent the product of the numbers 5 and 6 in this manner, 5.6 would be confounded with an integer followed by a decimal fraction, and 56 would signify the number fifty-six, according to the common system of notation.

For the sake of brevity, however, the multiplication of numbers is sometimes expressed by placing a point between them in cases where no ambiguity can arise from the use of this symbol. Thus, 1. 2. 3. 4, may represent the 2 7 6 continued product of the numbers, 1, 2, 3, 4; and

3 9.

may represent

V. The sign÷which is named by, and when placed between two numbers is employed to denote that the former is to be divided by the latter.

Thus 24÷6 signifies 24 by 6, or, 24 divided by 6. In like manner a÷b signifies a by b, or, a divided by b.

In general, however, the division of two numbers is indicated by writing the dividend above the divisor, and drawing a line between them. Thus 24÷6 and a÷b are usually written and

24 6

α

VI. The sign =which is named is equal to, and when placed between two numbers denotes that they are equal to each other.

Thus 56 + 6 = 62 signifies that the sum of 56 and 6 is equal to 62. In like manner, a = b signifies that a is equal to b, and a + b = c -d signifies that a plus b is equal to c minus d, or, that the sum of the numbers designated by a and b is equal to the difference of the numbers designated by c and d.

VII. The sign 4 which is named is unequal to, and when placed between two numbers denotes that one of them is greater than the other, the opening of the sign being turned towards the greater number.

Thus a 7 b signifies that a is greater than b, and a ≤ b signifies that a is less than b.

VIII. The coefficient is a sign which is employed to denote that a number designated by a letter, or some combination of letters, is added to itself a certain number of times.

Thus instead of writing a +a+a+a+a, which represents 5 a's added together, we write 5 a. In like manner 10 a b will signify the same thing as ab + ab +ab+ab+ab+ab+ ab + ab + ab + ab, or ten times the product of a and b.

The coefficient, then, is a number written to the left of another number, represented by one or more letters, and denotes the number of times hat the given letter, or combination of letters, is to be repeated.

When no coefficient is expressed, the coefficient I is always understood; thus 1 a and a signify the same thing.

IX. The exponent or index is a sign which is employed to denote that a number designated by a letter is multiplied by itself a certain number of

times.

Thus instead of writing a ×a×a×a×a, or a a a a a, which represents five a's multiplied together, we write a3, where 5 is called the exponent or index of a. Similarly b b x b x b x b x b x b x b x b × b, or b.b.b. b. b. b. b. b. b. b, or b b b b b b bb bb; or the continued product of 10 b's is written more briefly b1o, where 10 is the exponent or index of b.

The exponent or index of a number is, therefore, a number written a little above a letter to the right, and denotes the number of times which the number designated by the letter enters as a factor into a product. When no exponent is expressed, the exponent 1 is always understood; thus a1 and a signify the same thing.

The products thus formed by the successive multiplication of the same number by itself, are in general called the powers of that number. Thus a is the first power of a; a × a = a a = a2 is the second power of a, or the square of a; a a a = a3 is the third power, or cube of a; aa aa a = a3 is the fifth power of a, and aaaa..... to n factors = a", is the nth power of a, or the power of a designated by the number n.

X. The square root of any expression is that quantity which, when multiplied by itself will produce the proposed expression, and, in numbers, is generally denoted by the symbol, which is called the radical sign. Thus the square root of 9 is √9 = 3, and √a2=a, is the square root of a2; for in the former case 3 × 3 = 9, and in the latter a × a = a2.

XI. The cube root of any expression is that quantity which, when multiplied twice by itself, will produce the proposed expression. The fourth, or biquadrate root of any expression is that quantity which, when multiplied three times by itself, produces the given expression; and the nth root of any expression is that quantity which, multiplied (n-1) times by itself, produces the proposed expression. Thus the cube root of 8 is 2; for 2 × 2 × 2 = 8,

the fourth root of a1 is a; for a a. a. a = a1, and the nth root of "y" is xy; for xy x x Y x x y to n factors = x.x x x to n factors Xy.y.y.y

to n factors x" y".

The roots of expressions are frequently designated by fractional or decimal exponents, the figure in the numerator of the fractional exponent denoting the power to which the expression is to be raised or involved, and the figure in the denominator denoting the root to be extracted or evolved. Thus the symbol of operation for the square root of a is either a or a1; for the cube root it is a, or a; for the fourth root Va, or at; and /a, or a denotes the nth root of a. Also a3, or a, denotes the sixth root of the fifth power of a; and a", or "a", signifies the nth root of the mth power

of a.

XII. A rational quantity is that which has no radical sign, or fractional exponent annexed to it, as 3 m n, or 5x2 y2.

XIII. An irrational quantity is that which has no exact root, and is expressed by means of the radical sign, or a fractional exponent, as √2 Va2, or at y

2

XIV. The reciprocal of any quantity is unity divided by that quantity; thus the reciprocals of a2, a3, y3, z13, are respectively a1, x3, y3, z*; but the following notation is generally used, as being more commodious: thus the fractions a, x, y, z, are expressed by a−2, x—3, y—3,

-3

-5

2