14. A uniform ditch is 100 yards long and 4 feet in perpendicular depth; a vertical section of the ditch is a trapezoid, whose opposite parallel sides are 3 feet and 1 foot 8 inches respectively; find the number of tons of water the ditch would contain, it being given that a cubic foot of water weighs 62.5 pounds.

MATHEMATICS.

(GEOMETRY AND TRIGONOMETRY).

Logarithm books will not be required.

1. If a straight line be divided into any two parts, the squares on the whole line, and on one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square on the other part.

2. Describe a square that shall be equal to a given rectilineal figure.

If a square and a rectangle have equal areas, prove that the perimeter of the square is less than that of the rectangle.

3. If two circles touch one another internally, the straight line which joins their centres, being produced, shall pass through the point of contact.

4. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.

AB and CD are two chords in a circle which intersect when produced in the point E without the circle, prove that the difference of the angles subtended at the centre by the arcs AC and BD is double of the angle AEC.

5. In a given circle, inscribe a triangle equiangular to a given triangle.

6. Describe a circle about a given equilateral and equiangular pentagon.

7. The sides about the equal angles of triangles which are equiangular to one another are proportionals; and those which are opposite to the equal angles are homologous sides, that is, are the antecedents or the consequents of the ratios.

The diameter AB of a circle is produced to C, and through the line CD is drawn at right angles to the line ABC; prove that if the straight line AD cuts the circle in E, the rectangle AE, AD will be equal to the rectangle AB, AC.

8. Parallelograms which are equiangular to one another have to one another the ratio which is compounded of the ratios of their sides.

9. ABCD is a rectangle, and E, F, G, and H are the feet of the perpendiculars let fall upon the diagonals from the angular points; prove that EFGH is also a rectangle.

10. Define the circular measure of an angle.

If the number of degrees in an angle be equal to the number of grades in the complement of the same angle, prove that the circular measure of the angle

11. Define the principal trigonometrical ratios, and
each of them in terms of the tangent.

express
Find sin 60° and cot 30°.

12. Find sin (A + B) in terms of the sines and cosines of A and B, stating the limitations in the values of A and B assumed in the proof. Apply to find sin 75°.

13. Find tan 2A and sin 3A.

14. Given tan

A 2

a

=

find sin A and sin 24.

15. In any triangle, of which the semi-perimeter is s,

prove the formula sin

A

=

2

S- —b) (s ·
bc

16. From B and A the acute angles of the rightangled triangle ABC the lines BD and AE are drawn to the opposite sides of the triangle such that the angles DBC and EAC are each equal to a.

Prove that DE= AB tana.

17. Given A=40°, a=140·5, b=170.6. Find B and C.

Given that

log sin 40° = 9.8080675.

log 1405=3·1476763.

log 1706=3.2319790.

log sin 51° 18′ = 9.8923342.

log sin 51° 19' 9.8924354.

=

18. Two sides of a triangle are 85 and 75 yards respectively, and the included angle is 70°; find the remaining angles.

EXAMINATION FOR ADMISSION TO THE ROYAL MILITARY ACADEMY, WOOLWICH.

January, 1874.

ARITHMETIC AND LOGARITHMS.

1. Add together 25, 1, and 12.

2. Subtract 105 from 123.

3. Multiply together 14, 121, 513, §7, and 1.

4. Divide 111 by 251.

5. Add together 51-81035, 3.78109, 0013, 5·1005763,

and 0000037.

6. Subtract 43.86165 from 554.6103.

7. Multiply 6.0035 by 016.

8. Divide 60.105311 by 0035.

9. Reduce £2. 9s. 6d. to the decimal of £4. 2s. 6d.

10. Add together

11. Subtract 199

of 54, 13 of 73, and 21§ of 15%

from 201.

106.

12. Multiply together 813 of 11, 3% of 111, and 2 of 9.

13. Divide 6 by 1.

14. Add together 15.21435 of a day and 3.6105 of an hour, and give the answer in minutes and the decimal fraction of a minute.

15. Subtract 50-14 of a pole from 16105 of a mile, and give the answer in furlongs and the decimal fraction of a furlong.

16. Multiply 69.302 by 1·0093.

17. Divide 419-013 by 6013 to three places of decimals.

18. Reduce 9.72 grains to the decimal of a pound troy. 19. Reduce 6 miles 7 furlongs 4 yards 2 feet to inches.

20. If 17 cwt. 1 qr. of sugar cost £24. 3s., what will be the price of 15 lbs.?

21. Find (by Practice) the dividend on £3,105. 10s. at 14s. 74d. in the pound.

22. Find the simple interest on £1,576. 10s. for 8 years at 61 per cent. per annum.

23. In 3724816 ounces how many tons, cwts., &c.?

24. If 36 men can mow a field of 15 acres in 1 day, how long will it take 6 men to mow one of 7 acres 5 square poles?

25. Find (by Practice) the value of 42 oz. 8 dwts. 12 grs. of silver at 5s. 10d. per oz.

26. Find the compound interest on £6,530 in 3 years at 6 per cent. (neglecting fractions of a penny).

27. Compute by means of the tables the value of

log 3·1=4913617 and log6·122774=·7869483, find the value of 31.

350

29. Extract the square root of 14,5; also the square root of 19 accurately to six places of decimals.

30. How much money must be invested in the 3 per cent. consols at 937 so as to yield an income of £576 per annum?

M