of the twins dying before the period at which the eldest would be of age, his share is divided among the rest in such a manner that their properties may be still equal, when they come to age; and, in consequence, each is found at that period to have £1000 more than he would otherwise have had. What was the value of the entire property bequeathed? Answ. £7367 2 114.

48. A man owes a debt, to be paid in four equal instalments at the end of 4, 9, 12, and 20 months respectively; and he finds that, discount being allowed, according to the true method, at 5 cent. annum, £750 paid at present will discharge the whole lebt. How much did he owe? Answ. £784,9781845.

49. If a merchant commence trade with a capital of £5000, and gain so much, that, after paying all expenses, his capital, each year, is increased by a tenth part of itself wanting £100, how much will he be worth at the end of 20 years? Answ. £27910.

with

50. A man borrows £500, and agrees to pay simple interest at 5 cent. annum. At the end of 11 months he pays one part of the principal with its interest; 8 months after he pays another part its interest from the time it was borrowed; and 11 months after that he pays the remainder of the principal with its interest in like manner from the time it was borrowed. What was the amount of each payment, each of the two last being double of the first? Answ. The first payment £10888317, and each of the others £217331178.

51. Mercury revolves round the sun in 87 days, 23 hours, 15 minutes, 44 seconds, and the earth in 365 days, 6 hours, 9 minutes, 12 seconds. Required the seven first approximate ratios of these periods. Answ. 1, 15, 20, t4, 137, 151, and 17.

52. Venus revolves round the sun in 224 days, 16 hours, 49 minutes, 11 seconds, and the earth in the time stated in the preceding question. Required the first eight fractions approaching the ratios of these periods. Answ. †, 1, †, †, rb, 311, 474, and 11:35.

53. 35 lbs. of tea being mixed with 20 lbs. of a better quality, the mixture is found to be worth 7/4 per lb. Required the value of each kind, the difference of their values being 1/10 lb. Answ: 8/6 and 6/8 lb.

54. Express three hundred and fifty seven in the scale of nota tion whose radix is 33. Answ. 21101.

55. If a person lend £7000 at 6 cent. annum, compound interest, and allow the interest to accumulate in the hands of the creditor, except £240 annum, which he lifts for family expen

diture; how much will the creditor owe him at the end of 16 years? Answ. £11621 1 14.

56. If a boy read each day 2 lines more of Virgil than he did the day before, and find, that having read a certain quantity in 18 days, he will read, at this rate of increase, the same quantity in the next 14 days; how much will he read in the whole time? Answ. 4032 lines.

57. Two men, A and B, are on a straight road, on the opposite sides of a gate, and distant from it 308 yards and 277 yards respectively, and travel each towards the original station of the other. How long must they walk till their distances from the gate will be equal, B travelling 2 yards, and A 23 yards, per second? Answ. I minute, 33 seconds, or 2 minutes, 15 seconds.

58. Every thing being supposed to be as in the preceding question, at what time will each be at the same distance from the original station of the other, as the other is from his? Answ. In 4 minutes after starting.

59. Suppose a person to mix 11 lbs. of tea with 5 lbs. of an inferior quality, and to gain 16 per cent. by selling the mixture at 7/3 per lb.: it is required to determine the prime costs, when a pound of the one cost a shilling more than a pound of the other. Answ. 6/63 and 5/63 per lb.

60. If a person borrow £1100 at 6 per cent. per annum, com pound interest, and agree to pay both principal and interest in eleven equal annual payments, how much must each payment be, the first being made at the end of the first year? Answ. £139 9 54.

64. If a farm of 84 acres be held at £1 7 6 per acre, on a lease of which 48 years are unexpired, what fine must be paid at present to reduce the rent to 10/ per acre during the last 30 years of the lease, compound interest being allowed at 6 per cent. per annum? Answ. £354 8 113.

62. A man aged 45 has a pension of £300 a year during his own life; but he wishes to exchange it for another to continue not only during his own life, but also during that of his wife, aged 40. What will this pension be, money being supposed to be improvable at 6 per cent. per annum, compound. interest? Answ. £236 14 73.

7

63. Required the converging fractions approaching to the ratio of 5 hours, 48 minutes, 48 seconds, and 24 hours. Answ. 1, 2, 3, 1, and 17.

64. If the acting partner in a mercantile concern contribute £1000 to the original joint stock of the company, and annually increase this sum by £150 saved from his salary; to how much will his share of the joint stock amount, at the end of 11 years, on the supposition, that, after all expenses are paid, there is a clear gain of 10 per cent. per annum on the entire capital? Answ. £5632 15 10.

65. Suppose 17 gallons of spirits, at 10/6 per gallon, to be mixed with 7 gallons at a different price. What was the price of the latter per gallon, when 20 per cent. is gained by selling the mixture at 13/ per gallon? Answ. 11/74.

66. If a grocer mix together 13 parts of better and 7 parts of worse sugar, the price of the latter per cwt. being only of that of the former; what were their prime costs, if by selling 17cwt. 3qrs. 20lbs. at £4 6 6 per cwt., he gain £12 11? Answ. The better £3 16 11, and the worse £3 4 19 per cwt. respectively.

67. A merchant bequeaths £1000 among six clerks in proportion to their salaries, and the periods they have held their situations. Now, one of them has held his situation five years, and his salary is £120; two of them four years, with salaries of £75 each; and the rest two years, with salaries of £60 each. Required their several shares. Answ. The share of the first £384 12 3; of the two next £192 6 1†} each; and of the rest £76 18 5 each.

68. If a shopkeeper each year double his capital, except ant expenditure of £240 per annum, and, at the end of 4 years, be worth only three-fourths of his original capital; find without Position, what he had at commencing trade.

Answ. £236 1 34.

69. What sum will amount to £l more at simple than at compound interest, in four months, at 5 per cent. per annum ? Answ. £3699 9 2.

70. Required the sum of the infinite series, —+&-+ -&c. Answ..

71. Given the extremes of three numbers in harmonical propor tion 10 and 18: required the mean.

Answ. 12.

72. Given the first and second of three numbers in harmonical proportion=70 and 126 respectively: required the third term. Answ. 630.

73. Given the first, second, and third of four numbers in harmonical proportion=6, 7, and 8 respectively: required the fourth. Answ. 9.

74. Given the first, third, and fourth of four numbers in harmonical proportion=5, 9, and 10 respectively: required the second term Answ. 5.

75. If a bank borrow £10,000 at 4, and employ it in discounting bills at 6 cent. annum, and afterwards borrow money a 2, and discount bills with it at 4 cent. annum; how much must be borrowed, and so employed, in the latter case, that the gain may be the same as in the former, the bills being discounted in both cases six months before they would be due? Answ, £10,4991117.

CONTAINING

AN INTRODUCTION TO MENSURATION.

DEFINITIONS.

I. AN ANGLE is the inclination of two straight lines which meet one another in a point, which is called the VERTEX of the angle: or it is the degree of their opening or divergence.

II. When one straight line standing on another, makes with it two angles which are equal, each of these angles is called a RIGHT ANGLE; and the straight line which stands on the other is said to be PER

Right Angle

Right Angle

PENDICULAR, or AT RIGHT ANGLES, to it, or to be a PERPENDICULAR to it.

III. AN OBTUSE ANGLE is greater than a right angle: an ACUTE ANGLE is less than a right angle.

An angle is usually named by three letters, the middle one being placed at the vertex, and the other two somewhere on the lines which contain the

B

Obtuse Angle

Acute Angle

angle. Thus, the obtuse angle in the last diagram, may be called the angle ACB, or BCA, and the acute one the angle ACD, or DCA... When there is only one angle at the same point, it may be named by a single letter placed at that point.

IV. PARALLEL STRAIGHT LINES are those which have every where equal perpendicular distances from each other.

V. A SURFACE, or SUPERFICIES, has length and breadth, without thickness.

VI. A BODY, or as it is often called, a SOLID, has length, breadth, and thickness, or depth.

VII. A FIGURE is a portion of space inclosed by one or more boundaries.

• The small introductory tract on Mensuration which is here given, is intended for the use of such pupils as may not have time or opportunity for studying a more extended course. For this reason, the easiest and most useful parts are selected, and the def nitions and illustrations are delivered in plain and familiar terms, rather than with a view to Mathematical precision. The rules are also given without demonstrations, as the pupils for whom this abstract is intended, are not supposed to have read a preparatory course of Mathematics. Those who may wish to prosecute the subject more extensively may have recourse to the treatises of Bonnycastle, Hutton, and others on the subject, Hutton's larger treatise, in particular, is a very extensive and valuable work.

VIII. A FIGURE in said to be EQUILATERAL, if it have equal sides; and EQUIANGULAR, if it have equal angles.

IX. If a figure be contained by three lines, it is called a TRIANGLE; if by four, a QUADRILATERAL; if by more than four, a POLYGON.

X. An equilateral and equiangular polygon is often called

a REGULAR POLYGON.

XI. Polygons of five, six, seven, eight, nine, ten, eleven, and twelve sides, are often called respectively, PENTAGONS,

HEXAGONS, HEPTAGONS, OCTAGONS, NONAGONS, DECAGONS, HENDECAGONS, and DODECAGONS.

XII. A quadrilateral which has its opposite sides parallel, is called a PARALLELOGRAM: and a parallelogram which has its angles right angles, is called a RECTANGLE.

XIII. A quadrilateral which has its sides equal, and its angles right angles, is termed a sQUARE; and a quadrilateral which has its sides equal, but its angles not right angles, is called a RHOMBUS.

XIV. A quadrilateral which has two sides parallel, and the other two not, is called a TRAPEZOID.

XV. Any quadrilateral, except a parallelogram or trapezoid, is called a TRAPEZIUM.

XVI. A DIAGONAL of a figure is a line passing through two of its remote angles.

XVII. A CIRCLE is a figure contained on a flat surface, by one line which is called the CIRCUMFERENCE; and is such that all straight lines drawn to the circumference from a certain point within the figure, called the CENTRE, are equal to each other. Any of those equal lines is called a RADIUS: and a line drawn through the centre, and terminated both ways by the circumference, is called a DIAMETER. Hence, a diameter is evidently double of a radius.

XVIII. If the ends of a thread, FPf, be fastened to two pins, F,f, fixed at a less distance asunder than the length of the thread, and if the point of a pen A or pencil, P, be carried round in such a manner as to keep the thread constantly stretched, the figure inclosed by the curve line thus described, is called an EL

D

P

B

F

E

LIPSE; the points F and f, wher, the pins are fixed, are