5. Supposing the perpendiculars in the above example to fall at the distances 300 and 450 respectively, from one extremity of the base, construct the figure, and measure by the scale its internal angles.

6. Name the standards of measure in the different descriptions of artificers' work.

7. What is the cost of building a wall measuring 29 feet 6 inches by 8 feet 4 inches, and 2 bricks thick, at £2 10s. per rod of standard thickness?

8. A rectangular floor measures 27 ft. 6 in. by 15 ft. 10 in.; what is its area, and what is the cost of paving it, at 3s. 4d. per square yard?

9. The position of an unknown point on a plane may be determined, either by two straight lines measured from known points, or by a straight line and an angle, or by two angles. Illustrate these methods, and explain their utility in the measuring of large estates.

10. The length of a rectangular cistern is 7 feet 6 inches, its breadth 6 feet 4 inches, and its depth 6 feet; what weight of water will it hold, and what is its content in gallons?

11. Find the weight of a cone of granite, of which the slant height measures 15 feet, and the diameter of the base 9 feet, the specific gravity being 2500.

12. Explain what is meant by specific gravity, and show how its value is determined in a solid, and how in a fluid body.

13. What are the dimensions of a hollow globe of copper, the weight of which is one ton, the diameters being as 2 to 3, and the specific gravity of the metal 8788 ?

No. 4. Given at Haileybury College.

ARITHMETIC.

1. Find the value of 39 cwt. O qrs. 10 lbs. at £1 17s. 10d. per cwt. 2. At the rate of 11s. 7hd. in the pound, what is the sum paid by a bankrupt for a debt of £2735 10s.?

3. Exhibit 2015 and 37454 in fractional forms.

4. What is the sum of 143 and 3 of 1⁄2 of 8?

5. The aggregate of and of a certain sum of money is £133. What is that sum?

6. Required the values of of a pound sterling, of a shilling, and of a guinea.

7. Represent the vulgar fractions,, and decimally.

8. Reduce 12s. 6d. to the decimal of a pound, and give the value of 416836 of a guinea.

9. Give the simple interest of £237 10s. Od. for 2 years, 8 months, and 29 days, at 5 per cent. per annum.

10. Find the present worth of £725 17s. 6d. for 8 months, at 5 per cent. discount.

11. A grocer gave £50 for 16 cwt. 2 qrs. 18 lbs. of sugar, and he lost £8 by retailing it; at what rate did he sell it per pound?

12. Find the insurance of a ship and cargo valued at £35,727 17s. 6d. at 177 per cent.

No. 5. Given at Addiscombe.

ARITHMETIC.

1. Find yard + foot + inch as the decimal of a yard.

2. Divide 0186534 by 00086534.

3. If the salary of

persons for 21 weeks is £120; what will be the

salary of 14 persons for a year, at the same rate?

4. Find the cost of 1875 lbs. at 3s. 7d. a pound; by Practice.

5. At 5 per cent. per annum, compound interest, what will £1000 amount to in 4 years?

6. What is the discount to be taken off a bill of £500, which has 3 months to run, at 3 per cent. per annum?

7. When the 3 per cent. stocks are at 72, what income shall I derive from the purchase of £5000 stock?

8. Extract the square root of 0176534, and the cube root of ⚫0006859.

9. Reduce to its simplest form

*005

4921

of
i's of 114 of 2.25 ÷ (1⁄2 + 1⁄2).

10. of a calf is worth of a sheep, and 7 sheep are worth an ox, and 3 oxen are worth 2 horses, and 1 horse is worth £30; what are the prices of a calf, a sheep, and an ox?

No. 6. MENSURATION.

1. How many square yards are there in a parallelogram whose two adjacent sides are 33 feet 6 inches, and 24 feet 3 inches, and the included angle 45° ?

2. Supposing that the ratio of the circumference to the diameter of a circle is as 3.1416, or π to 1; it is required to show how the following multipliers are deduced: 3183 to find the diameter from the circumference; 7854 to find the area when the diameter is given; and '07958

3. Find the area of the segment of a circle whose chord is 32, and diameter of the circle 40.

4. Find the area of an irregular figure when the breadths at six equidistant places are 20·4, 22·6, 23·4, 24·8, 26, 27-2, and the length 40.

5. The length of a sack is 4 feet 6 inches, what must be its breadth to contain 2 bushels of flour?

6. Suppose the ball on the top of St. Paul's is 6 feet in diameter, what would the gilding of it cost at 3d. per square inch?

7. Each of the sides or edges of a triangular pyramid is 20 inches; required the whole superficies, and solid content.

8. The circumference of a cask at the bung is 7 feet, and the circumference at the end is 5 feet; how many gallons will it contain, supposing that its figure is two equal frustums of a cone, and the length of the cask is 40 inches?

9. What will be the expense of painting the new cylindrical pontoon at 6d. a square yard, the length of the cylindrical part being 19 feet 4 inches, and the diameter of this cylinder and also of the two hemispheres at the ends being 2 feet 8 inches?

10. What will be the weight of a quantity of water equal in bulk to this pontoon, supposing that a cubic foot of water weighs 1000 oz. or 62 lbs. ?

11. When the roof of a building is of a true pitch, or forming a right angle at the top, it is usual to add half the breadth to the breadth itself for the girt over both sides. Supposing that I paid my bricklayer £20 for tiling, how much has he received more than he ought to do, according to the true measurement of the roof?

12. If 30 cubic inches of gunpowder weigh 1 lb., prove that the cube of the internal diameter of a shell in inches must be divided by 57.3, to find the number of lbs. which will fill the shell; and also that the square of the diameter of a cylinder multiplied by its length must be divided by 38.2, to find the number of pounds that it will hold.

13. Required the number of shot in an incomplete square pile, the number in a side of the base being 40, and in a side at the top being 20.

14. The length of a wall is 50 feet 8 inches, and its height 7 feet 4 inches; required the expense of building this wall at £13 the rod of standard brickwork, the wall being 1 brick thick. Do this by duodecimals; and explain the meaning of all the different terms of the product. N. B. A rod of standard brickwork is a square, each side of which is 5 yards or 16 feet, and 14 bricks in thickness.

ADDENDA.

Page 37. Least Common Multiples.

ANOTHER method of finding the least common multiple of two or more quantities is performed by dividing successively each of the quantities by any common divisor, and then taking the product of the divisor and indivisible quantities as the least common multiple; thus

The following is a neat method of performing the operation of finding the greatest common measure of two quantities.

Required the greatest c. m. of 645029 and 527751.

117278 527751 645029 1

In this example, if ≈ be the minutes that the hour hand has passed beyond 12 o'clock, and the minute hand has made a revolution and

K

overtaken it, the minute hand moving twelve times as rapidly as the hour hand, a minutes will be the minutes the hour hand has passed over, and 60+ will equal the minutes the minute hand has passed over. .. 12x = 60 + x, and x = ff= 5 minutes past 12 by the hour hand;

the

30027".

.. the time is 5′ 27′′ after one o'clock.

And if it be required to find

when they are at right angles after 5 past one o'clock, then if r

=

minutes passed over by hour hand,

then 12x = 15+x, and x =

=

1. the motion of the hour hand;

.. 5+ 1 = 6o, and 1.5′ added to this gives 21 the same way they will be in a straight line, at 36

past one o'clock. In past one o'clock.

but although like quantities are properly compared with each other in a proportion, yet it is the numerical values only which admit of legitimate comparison; and the denominations cwt. and pipes in reality imply the numerical values, or the x and the 3 ought to be considered as merely multipliers of the number of pounds.

There is a fallacy in equating the equation of payments when a part of the payment is to be cash down, because present payment involves no risk and admits of no casualty. For practical purposes, however, £100 cash and £100 at two months may be considered as equivalent to £200 at one month, by considering the discount on the one equivalent to the interest on the other; or we may reckon the accumulation of interest. We may therefore say £200 at two months, £200 at four months, and £100 at six months;

or 100.0 + 300. 4 + 100. 6 = x. 500,

Examples in Practice might have been introduced, but then they ought to extend to every species of Mental Arithmetic, which is not the object of the present work.