65° 14′ 30′′ and 47° 32′ 10′′ respectively, and the difference of level of C and A is known to be 500 feet; find the distance AB, and the difference of level of A and B.

824. Two sides AC, BC, of an equilateral triangle, subtend angles of 30° and 45° at a point D; find the distances DA, DB, DC, supposing DC to fall between A and B, D and C to be on opposite sides of AB, and the side of the triangle to be 10.

825. The angle subtended by a diagonal of a square redoubt from a point in the prolongation of the other diagonal was found to be 21° 32′ 10′′, and at a point on the same line, 200 yards nearer to the redoubt, the corresponding angle was 28° 17′ 15′′; find the side of the redoubt.

826. A circular reservoir subtends at a certain point an angle of 34° 22′ 18′′, and advancing 150 yards directly towards its centre, I find it subtends an angle of 72° 18′ 30′′; find its dia

meter.

827. From the top of a tower 113.786 feet high the angles of depression of the top and bottom of a column standing on the same horizontal plane as the tower were found to be, 32° 15′ 20′′ and 68° 54′ 33′′ respectively; find the height and distance of the column.

828. The distance between two horizontal parallel telegraph wires running N.W. and S.E., and vertically above one another, is 4 feet, and at noon the perpendicular distance between their shadows on a horizontal plane is 6 feet; find the altitude (i. e. the angle of elevation) of the sun.

MENSURATION.

829. The lengths of the perpendiculars let fall from points in an irregularly curved line of fence upon a straight line of 5.86 chains, at equal distances from each other, are found to be 93, 84,

72, 68, 43, 54, 37, 29, and 23 links; find the area included between the extreme perpendiculars which fall upon the ends of the straight line.

830. The sides of a quadrilateral, taken consecutively, are 2416, 1712, 1948 and 2848; the angle between the first two is 30°, and that between the last two 150°; find the area of the figure.

831. Find an expression for the area of a triangle whose sides are a, b and c.

832. Find the area of a triangular field whose sides are 7.32 chains, 4.57 chains, and 5.48 chains.

833. Find the number of acres in a triangular field whose sides are 10.42 chains, 8.74 chains, and 12.63 chains.

834. Three sides of a triangle are 6, 6+ √2, and 6-√2; find its area.

835. The area of a right-angled triangle is 84.5 square feet, and one of its sides is 39 inches; find its hypothenuse.

836. The area of a triangular field is 14 acres; find its sides, which are known to be in the ratio of 3, 5, and 7.

837. A quadrilateral field ABCD has its sides AB = 6 chains, BC= 8 ch., CD = 8 ch., AD=9 ch., and its diagonal, BD = 12 ch.; find its area in acres.

838. A mahogany plank 24 feet in length, is 18 inches wide at one end and 15 at the other; the plank is cut across at a distance of 3 feet 6 inches from the broader end: how many square feet are cut off the plank, and how many are in the whole plank?

839. The floor of a room is a regular octagon, the distance between any parallel sides being 20 feet; find the area in square yards.

100

No3

840. The base of an isosceles triangle is 20, and its area is ; find its angles.

841. The parallel sides of a trapezoidal field of 25 acres are 10 chains and 15 chains respectively: find the perpendicular distance across the field.

842. A trapezoidal field of which the parallel sides are 579 links and 854 links, and the perpendicular distance between them 723 links, is let at £4. 10s. per acre; what income does it produce?

843. The sides of a triangle are 2 + √2, 2−√2, and 3; find

its area.

844. Find the area of a regular polygon in terms of its side s. 845. The sides of a right-angled triangle are 3, 4, and 5; find the area of the space contained by the segments of the sides 3 and 4, and the arc of the inscribed circle included between these segments.

846. Express the area of a regular plane figure in terms of a the length of the side, and n the number of sides: and apply it to the determination of the area of the equilateral triangle, the square, and the regular pentagon, whose sides are each 10 units in length.

847. Deduce the expressions for the areas of the regular polygons of n sides circumscribing and inscribed within the circle whose radius is r; and thence show that they have to one another 180°

the ratio of 1 to cos2

848. Show that the area of a regular polygon of n sides is equal to

na

4

180
n

cot ; where a is the length of its side.

849. Find the area of an isosceles triangle, each of whose equal sides is 50, and each of whose equal angles is 75°.

850. What must be the diameter of a carriage-wheel in order that it may make 500 turns in a mile?

851. A regiment, advancing in open column of companies, is wheeled by successive companies to the left; show that the distance, in paces, lost by each company during the wheel, is twofifths of the number of files in the company, supposing the ratio of files to paces to be 10 to 7, and that the right file of each company takes exactly the full pace.

852. The driving-wheel of a locomotive engine being 6 feet in diameter, determine the number of strokes made per minute by each piston, when the train is running at the rate of 30 miles an hour; two strokes of each piston causing one revolution of the driving-wheel.

853. Prove that the area of the largest circle which can be cut from a regular hexagon is three times the area of the circle described on one of the sides as a diameter.

854.

What is the area of the sector of a circle whose arc of 24° measures 10 feet?

855. Ten persons dine at a circular dining-table; what is the area of the table-cloth, supposing each person to occupy 2 feet of the circumference of the table, and that the cloth overlaps 15 inches on each side? Find also the area unoccupied when a dinner plate of 10 inches diameter is placed before each person.

856. Find the area of the remaining portion of a circle whose radius is 20 inches, when a segment having an arc of 25°, has been cut off from it.

857. Find the area of the segment of a circle whose height is one-half of the radius, when the radius of the circle is 1 foot.

858. Find the area of a segment of a circle whose base is 54 and height 10.

859. A regular hexagon is inscribed in a circle whose radius is 10 inches, and another is circumscribed about it; find the area of the latter, and show that the area between the boundaries of

the hexagons is equal to one-third of the area of the inscribed figure.

860. If a regular hexagon, a square, and an equilateral triangle be inscribed in a circle, the square described upon the side of the triangle is equal to the sum of the squares described upon one side of each of the other two figures.

861. Show that the area of a regular polygon inscribed in a circle is a mean proportional between the areas of two polygons of half the number of sides inscribed within and circumscribed about the same circle.

862. The exterior diameter of the outer ring of a circular target is 5 feet, and it is divided into 6 concentric rings of equal breadth, alternately white and black, the outer ring being white; find the number of square feet of white paint in the target.

863.

Find the area of the sector of a circle whose arc of 20° is 18 inches long.

864. Find the area of a circular segment whose height is 7 inches and its base 3 feet.

865. The angular points of an irregular pentagon are each at a distance of 100 yards from a certain point within it; and at this point the sides taken in order subtend angles of 45°, 60°, 90°, 45o, and 120°; find the area of the pentagon and the length of each of its sides, without using tables.

866. If a regular hexagon be placed within an equilateral triangle, so that three of its sides are upon the sides of the triangle, show, analytically, that the areas of the figures will be as their perimeters, and that the areas of their circumscribing circles will be as 1 : 3.

867. A and B are two points in the circumference of a circular pond of water, and a dyke from A to B is to be made so as to cut off the smaller part of the pond. If the circumference of