37. A circle being given, to find a given number of circles whose radii shall be proportional to given lines, and the sum of whose areas shall be equal to the area of the given circle.

38. In a given equilateral triangle, inscribe three equal circles tangent to each other and to the sides of the triangle.

Determine the radius of these circles in terms of the side of the triangle.

39. In a given circle, inscribe three equal circles tangent to each other and to the given circle.

Determine the radius of these circles in terms of the radius of the given circle.

NUMERICAL EXAMPLES.

NOTE.-The following approximate values are close enough for ordinary purposes: π = = 4, √2 = }}, √3 = }}, √/5 = 1. Radius

of earth = 3960 miles.

40. The vertical angle of an isosceles triangle is 36°, and the length of the base is 2 feet; find the base angles, the length of the bisector of a base angle, and the length of a side of the given triangle. Ans. 72°, 2 feet, (1 + V5) feet. 41. One angle of a triangle is 60°, the including sides are 3 feet and 8 feet; find the area and the third side.

Ans. 61/3 square feet, 7 feet. 42. The three sides of a triangle are 9 inches, 10 inches, and 17 inches, its area is 36 square inches; find the area of the inscribed circle.

Ans. 4π.

43. The adjacent sides of a parallelogram are 12 feet and 14 feet, the area is 120 square feet; find the long diagonal.

Ans. 24 feet.

44. The area of a right triangle is 6 square feet, the length of the hypotenuse is 5 feet; find the other sides.

Ans. 3 feet, 4 feet.

45. Obtain a formula connecting the length of a chord 7, its distance from the centre d, and the radius r.

46. Obtain a formula for the length t of a common tangent to two circles, given the radii r, r', and the distance between the centres d. Ans. (r — r')2 + t2 = d2 for external tangent.

(~° +r)2 + t2 d2 for internal tangent.

47. Through what angle does the hour-hand of a clock move in 1 hour? in 1 minute? Through what angle does the minutehand move in 1 minute?

What angle do the hands of a clock make with each other at ten minutes past three? at quarter of six? Ans. 35°, 97° 30′.

48. Two secants cut each other without a circle, the intercepted arcs are 12° and 48°; what is the angle between the secants?

Two chords intersect within the circle, a pair of opposite intercepted arcs are 12° and 48°; what is the angle between the chords? 49. Two tangents make with each other an angle of 60°; required the lengths of the arcs into which their points of contact divide the circle, given radius equals 7 inches.

Ans. 14 inches, 291 inches.

50. A swimmer whose eye is at the surface of the water can just see the top of a stake a mile distant; the stake proves to be 8 inches out of water; required the radius of the earth.

Ans. 3960 miles.

51. A passenger standing on the deck of a steamer about to start observes that his eye is on a level with the top of the wharf, which he knows to be 12 feet high; when they have steamed 8 miles the wharf disappears below the horizon; required the radius of the earth. Ans. 3974 miles.

52. How many miles is the light of a light-house 150 feet high visible at sea? Ans. 15. 53. On approaching Portland from the sea, Mount Washington is first visible 12 miles from shore; Portland is 85 miles from Mount Washington; required the height of the mountain.

Ans. 6270 feet.

54. The latitude of Leipsic is 51° 21', that of Venice 45° 26', and Venice is due south of Leipsic; how many miles are they apart? Use 4000 miles as the earth's radius. Ans. 413 miles.

55. The latitude of the Peak of Teneriffe is about 30° N.; the rising sun shines on its summit on the 21st of March 9 minutes before it shines on its base; required the height of the mountain. Ans. About 12,000 feet.

56. A quarter-mile running-track 10 feet wide, with straight parallel sides and semicircular ends, is to be laid out in a rectangular field 220 feet wide. How long must the field be in order that a runner, keeping in the middle of the crack, may have onequarter of a mile to cover? how much can he gain by keeping close to the inner edge of the track? what is the area of the field? of the portion encircled by the track? of the track itself?

Ans. 550 feet; 31% feet; 121,000 square feet; 97,4284 square feet; 13,200 square feet.

57. The fly-wheel of an engine is connected by a belt with a smaller wheel driving the machinery of a mill. The radius of the

fly-wheel is 7 feet; of the small wheel, 21 inches. How many revolutions does the small wheel make to one of the fly-wheel? The distance between the centres of the two wheels is 10 feet. What is the length of the connecting band?

Ans. 51 feet 2 inches.

58. If from each vertex of a regular polygon as a centre, with a radius equal to one-half the side, an arc is described outward from side to side of the polygon, an ornamental figure much used in architecture is formed. Such a figure formed on a polygon of numerous sides is often used as a rose-window.

The figure bounded by three arcs is called a trefoil; by four arcs, a quatre-foil; by five arcs, a cinque-foil.

Find the area of a trefoil, given the distance between the centres of adjacent arcs equal to 21 inches. Ans. 7.338 square feet.

59. A rose-window c six lobes is to be Ilaced in a circular space 42 feet in diameter. How many square feet of glass will it contain?

Ans. 1123.8 square

feet.

SYLLABUS OF PLANE GEOMETRY.

POSTULATES, AXIOMS, AND THEOREMS.

BOOK I.

POSTULATE I.

THROUGH any two points one straight line, and only one, can be drawn.

POSTULATE II.

Through a given point one straight line, and only one, can be drawn having any given direction.

AXIOM I.

A straight line is the shortest line that can be drawn between two points.

AXIOM II.

Parallel lines have the same direction.

PROPOSITION I.

At a given point in a straight line one perpendicular to the line can be drawn, and but one.

Corollary. Through the vertex of any given angle one straight line can be drawn bisecting the angle, and but one.

PROPOSITION II.

All right angles are equal.

PROPOSITION III.

The two adjacent angles which one straight line makes with another are together equal to two right angles.

Corollary I. The sum of all the angles having a common vertex, and formed on one side of a straight line, is two right angles. Corollary II. The sum of all the angles that can be formed about a point in a plane is four right angles.