be shown, by the method of limits, to be true also of plane figures bounded by curves, it follows that in any two similar plane surfaces the ratio of the areas is the second power of the linear ratio.

505. Some of the following exercises are only arithmetical applications of geometrical principles.

The algebraic method may be used to great advantage in many exercises, but every principle or solution that is found in this way, should also be demonstrated by geometrical reasoning.

EXERCISES.

506.-1. What is the length of the radius when the arc of 80° is 10 feet?

2. What is the value, in degrees, of the angle at the center, whose arc has the same length as the radius?

3. What is the area of the segment, whose arc is 60°, and radius 1 foot?

4. To divide a circle into two or more equivalent parts by concentric circumferences.

5. One-tenth of a circular field, of one acre, is in a walk extending round the whole; required the width of the walk.

6. Two irregular garden-plats, of the same shape, contain, respectively, 18 and 32 square yards; required their linear ratio. 7. To describe a circle equivalent to two given circles.

50%. The following exercises may require the student to review the leading principles of Plane Geometry.

1. From two points, one on each side of a given straight line, to draw lines making an angle that is bisected by the given line.

2. If two straight lines are not parallel, the difference between the alternate angles formed by any secant, is constant.

3. To draw the minimum tangent from a given straight line to a given circumference.

4. How many circles can be made tangent to three given straight lines?

5. Of all triangles on the same base, and having the same vertical angle, the isosceles has the greatest area.

6. To describe a circumference through a given point, and touching a given line at a given point.

7. To describe a circumference through two given points, and touching a given straight line.

8. To describe a circumference through a given point, and touching two given straight lines.

9. About a given circle to describe a triangle similar to a given triangle.

10. To draw lines having the ratios √2, √3, √5, etc.

11. To construct a triangle with angles in the ratio 1, 2, 3. 12. Can two unequal triangles have a side and two angles in the one equal to a side and two angles in the other?

13. To construct a triangle when the three lines extending from the vertices to the centers of the opposite sides are given?

14. If two circles touch each other, any two straight lines extending through the point of contact will be cut proportionally by the circumferences.

15. If any point on the circumference of a circle circumscribing an equilateral triangle, be joined by straight lines to the several vertices, the middle one of these lines is equivalent to the other two.

16. Making two diagonals in any quadrilateral, the triangles formed by one have their areas in the ratio of the parts of the other.

17. To bisect any quadrilateral by a line from a given vertex. 18. In the triangle ABC, the side AB= 13, BC= 15, the altitude = 12; required the base AC.

19. The sides of a triangle have the ratio of 65, 70, and 75; its area is 21 square inches; required the length of each side. 20. To inscribe a square in a given segment of a circle.

21. If any point within a parallelogram be joined to each of the four vertices, two opposite triangles, thus formed, are together equivalent to half the parallelogram.

22. To divide a straight line into two such parts that the rectangle contained by them shall be a maximum.

23. The area of a triangle which has one angle of 30°, is one-fourth the product of the two sides containing that angle.

24. To construct a right angled triangle when the area and hypotenuse are given.

25. Draw a right angle by means of Article 413.

26. To describe four equal circles, touching each other exteriorly, and all touching a given circumference interiorly.

27. A chord is 8 inches, and the altitude of its segment 3 inches; required the area of the circle.

28. What is the area of the segment whose arc is 36°, and chord 6 inches?

29. The lines which bisect the angles formed by producing the sides of an inscribed quadrilateral, are perpendicular to each other.

30. If a circle be described about any triangle ABC, then, taking BC as a base, the side AC is to the altitude of the triangle as the diameter of the circle is to the side AB.

31. By the proportion just stated, show that the area of a triangle is measured by the product of the three sides multiplied together, divided by four times the radius of the circumscribing circle.

32. In a quadrilateral inscribed in a circle, the sum of the two rectangles contained by opposite sides, is equivalent to the rectangle contained by the diagonals. This is known as the Ptolemaic

Theorem.

33. Twice the square of the straight line which joins the vertex of a triangle to the center of the base, added to twice the square of half the base, is equivalent to the sum of the squares of the other two sides.

34. The sum of the squares of the sides of any quadrilateral is equivalent to the sum of the squares of the diagonals, increased by four times the square of the line joining the centers of the diagonals.

35. If, from any point in a circumference, perpendiculars be let fall on the sides of an inscribed triangle, the three points of intersection will be in the same straight line.

GEOMETRY OF SPACE.

CHAPTER IX.

STRAIGHT LINES AND PLANES.

508. The elementary principles of those geometrical figures which lie in one plane, furnish a basis for the investigation of the properties of those figures which do not lie altogether in one plane.

We will first examine those straight figures which do not inclose a space; after these, certain solids, or inclosed portions of space.

The student should bear in mind that when straight lines and planes are given by position merely, without mentioning their extent, it is understood that the extent is unlimited.

LINES IN SPACE.

509. Theorem.-Through a given point in space there can be only one line parallel to a given straight line.

This theorem depends upon Articles 49 and 117, and includes Article 119.

510. Theorem.-Two straight lines in space parallel to a third, are parallel to each other.

This is an immediate consequence of the definition of parallel lines, and includes Article 118.

511. Problem.-There may be in space any number of straight lines, each perpendicular to a given straight line at one point of it.

For we may suppose that while one of two perpendicular lines remains fixed as an axis, the other revolves around it, remaining all the while perpendicular (48). The second line can thus take any number of positions. This does not conflict with Article 103, for, in this case, the axis is not in the same plane with any two of the perpendiculars.

EXERCISES.

512.-1. Designate two lines which are everywhere equally distant, but which are not parallel.

2. Designate two straight lines which are not parallel, and yet can not meet.

3. Designate four points which do not lie all in one plane.

PLANE AND LINES.

513. Theorem.-The position of a plane is determined by any plane figure except a straight line.

This is a corollary of Article 60.

Hence, we say, the plane of an angle, of a circumference, etc.

514. Theorem.-A straight line and a plane can have only one common point, unless the line lies wholly in the plane. This is a corollary of Article 58.

515. When a line and a plane have only one common point, the line is said to pierce the plane, and the plane to cut the line. The common point is called the foot of the line in the plane.

When a line lies wholly in a plane, the plane is said to pass through the line.