dicular to it, the angle of these two lines is equal to the difference of the two acute angles of the triangle.

9. In the base of a triangle, find the point from which lines extending to the sides, and parallel to them, will be equal.

10. To construct a square, having a given diagonal.

11. Two triangles having an angle in the one equal to an angle in the other, have their areas in the ratio of the products of the sides including the equal angles.

12. If, of the four triangles into which the diagonals divide a quadrilateral, two opposite ones are equivalent, the quadrilateral has two opposite sides parallel.

13. Two quadrilaterals are equivalent when their diagonals are respectively equal, and form equal angles.

14. Lines joining the middle points of the opposite sides of any quadrilateral, bisect each other.

15. Is there a point in every triangle, such that any straight line through it divides the triangle into equivalent parts?

16. To construct a parallelogram having the diagonals and one side given.

17. The diagonal and side of a square have no common measure, nor common multiple. Demonstrate this, without using the algebraic theory of radical numbers.

18. To construct a triangle when the three altitudes are given.

19. To construct a triangle, when the altitude, the line bisecting the vertical angle, and the line from the vertex to the middle of the base, are given.

20. If from the three vertices of any triangle, straight lines be extended to the points where the inscribed circle touches the sides, these lines cut each other in one point.

21. What is the area of the sector whose arc is 50°, and whose radius is 10 inches?

22. To construct a square equivalent to the sum, or to the dif ference of two given squares.

23. To divide a given straight line in the ratio of the areas of two given squares.

24. If all the sides of a polygon except one be given, its area will be greatest when the excepted side is made the diameter of a circle which circumscribes the polygon.

25. Find the locus of those points in a plane, such that the sum of the squares of the distances of each from two given points, shall be equivalent to the square of a given line.

26. Find the locus of those points in a plane, such that the difference of the squares of the distances of each from two given points, shall be equivalent to the square of a given line.

27. If the triangle DEF be inscribed in the triangle ABC, the circumferences of the circles circumscribed about the three triangles AEF, BFD, CDE, will pass through the same point.

28. The three points of meeting mentioned in Exercises 28, 29, and 30, Article 337, are in the same straight line.

29. If, on the sides of a given plane triangle, equilateral triangles be constructed, the triangle formed by joining the centers of these three triangles is also equilateral; and the lines joining their vertices to the opposite vertices of the given triangle are equal, and intersect in one point.

30. The feet of the three altitudes of a triangle and the centers of the three sides, all lie in one circumference. thus described is known as "The Six Points Circle."

The circle

31. Four circles being described, each of which shall touch the three sides of a triangle, or those sides produced; if six lines be made, joining the centers of those circles, two and two, then the middle points of these six lines are in the circumference of the circle circumscribing the given triangle.

32. If two lines, one being in each of two intersecting planes, are parallel to each other, then both are parallel to the intersection of the planes.

33. If a line is perpendicular to one of two perpendicular plancs, it is parallel to the other; and, conversely, if a line is parallel to one and perpendicular to another of two planes, then the planes are perpendicular to each other.

34. How may a pyramid be cut by a plane parallel to the base, o as to make the area or the volume of the part cut off have a given ratio to the area or the volume of the whole pyramid?

35. Any regular polyedron may have a sphere inscribed in it; also, one circumscribed about it.

36. In any polyedron, the sum of the number of vertices and the number of faces exceeds by two the number of edges.

37. How many spheres can be made tangent to three given planes?

38. Apply to spheres the principle of Article 331; also, of Article 191, substituting circles for chords.

39. Discuss the possible relative positions of two spheres.

40. What is the locus of those points in space, such that the sum of the squares of the distances of each from two given points, is equivalent to a given square?

41. What is the locus of those points in space, such that the difference of the squares of the distances of each from two given points, is equivalent to a given square?

42. A frustum of a pyramid is equivalent to the sum of three byramids all having the same altitude as the frustum, and having for their bases the lower base of the frustum, the upper base, and a mean proportional between them.

43. The surface of a sphere can be completely covered with the surfaces either of 4, or of 8, or of 20 equilateral spherical triangles.

44. The volume of a cone is equal to the product of its whole surface by one-third the radius of the inscribed sphere.

45. If, about a sphere, a cylinder be circumscribed, also a cone whose slant hight is equal to the diameter of its base, then the area and volume of the sphere are two-thirds of the area and volume of the cylinder; and the area and volume of the cylinder are two-thirds of the area and volume of the cone.

TRIGONOMETRY.

CHAPTER XII.

PLANE TRIGONOMETRY.

811. TRIGONOMETRY is the science in which the relations subsisting between the angles, sides, and area of any triangle are investigated. The science was originally called Plane Trigonometry or Spherical Trigonometry, according as the triangle was plane or spherical.

PLANE TRIGONOMETRY has now a wider meaning, comprising algebraic investigations concerning angles and their functions, and the methods of calculating these functions.

MEASURE OF ANGLES.

812. In Elementary Geometry, the unit for the measure of angles is usually the right angle. The frequent fractions which the use of this unit gives rise to, render it inconvenient for calculation. It has been divided into degrees, minutes, and seconds (208).

This sexagesimal division of angles has been in use since the second century. Efforts have been made to substitute for it the centesimal division, making the right angle contain one hundred grades, each grade one hundred minutes, and so on; but this plan has never been generally in use.

B

813. There is another unit which has been called the circular measure of an angle. It is used in trigonometrical investigation, and is also called the analytical unit. It is that angle at the center of a circle whose intercepted arc has the same linear extent as the radius. Thus, if the arc AB has the same linear extent as the radius AC, then the angle C is the unit of circular measure. Hence, this с unit of measure is equal to

A

814. Various instruments are used for the measure of angles. A protractor is used to measure the angle of two lines in a drawing. It is usually shaped like a semicircumference with its diameter, the arc being marked with the degrees from 0 to 180.

Let it be required to measure the angle ABC. Place the center of the straight

edge, which is marked by a

notch on the instrument, at

the vertex B; let the edge lie along one side of the

A

B

angle, as BC; then read the degree marked where the other side BA passes the arc of the instrument. This gives the size of the angle.

The same instrument is used for drawing angles of a known size. One side of the angle being drawn, place the center of the protractor at the point which is to be the vertex; then the required number of degrees, on the