To circumscribe about a given circle:

9. An equilateral triangle. 12. A regular octagon.

10. A square.

11. A regular hexagon.

13. A regular pentagon.

14. A regular decagon.

To construct upon a given line as one side:

15. An equilateral triangle. 18. A regular octagon.

16. A square.

17. A regular hexagon.

19. A regular pentagon.

20.

A regular decagon.

21. To construct upon a given line as one side a regular dodecagon.

22. To construct upon a given line as one side a regular polygon of 15 sides.

23. To construct an angle of 36°.

24. To construct an angle of 9o.

25. To construct an angle of 12°.

26. In a given circle, to construct a mean proportional between a chord of 30° and a chord of 150°.

27. To transform a given regular octagon into a square.

28. To construct a regular hexagon, given one of the shorter diagonals.

29. To construct a regular pentagon, given one of the diagonals.

30. To inscribe in a given circle a polygon of n sides, n being any whole number.

CH
Fig. 98.

The following construction is found in most cases to be sufficiently exact for practical purposes.

Divide the diameter AB (Fig. 98) into n equal parts (in the Figure n = 7). Draw the radius CDL to AB, produce CB to E, and CD to F, making BE and DF each equal to one of the parts of the diameter; join EF, cutting the circle for the first time in G. Then the line GH joining G and the third point of division of AB, counting from B, will be very nearly equal to one side of the inscribed polygon of n sides.

=

For n 3 and n = 4, this construction is impossible; for n= 5 it is useless, on account of its inaccuracy; but for n >5 it gives a very close approximation to the exact value of the side required.

31. To construct a regular decagon equivalent to a given. regular pentagon.

32. To construct a circle in which the inscribed regular octagon shall have a perimeter equal to that of a given square.

33. To draw through a given point a line which shall divide a given circumference into two parts in the ratio of 3 to 7.

34. To construct a circumference equal to the sum of two given circumferences.

35. Three given circumferences are denoted by c, c', c'' ; to construct a circumference equal to c+c-c".

36. To construct a circle equivalent to the sum of two given circles.

37. To construct a circle equal in area to three times a given circle.

38. To construct a circle equal in area to three-fourths of a given circle.

39. To divide a given circle by a concentric circumference into two equivalent parts.

40. To divide a given circle by concentric circumferences into four equivalent parts.

41. To inscribe four equal circles in a given regular octagon so that each circle shall touch two other circles and one side of the octagon.

42. To inscribe in a given circle five equal circles so that each circle shall touch two other circles and the circumference of the given circle.

CHAPTER VI.

THE ALGEBRAIC METHOD.

§ 25. CONSTRUCTION OF ALGEBRAIC EXPRESSIONS.

The lengths of lines, expressed in terms of a common unit, are called Linear Factors. Abstract numbers and symbols representing abstract numbers are called Coefficients.

In the following exercises the first letters of the alphabet, a, b, c, etc., are employed as linear factors, and the letters m and n denote coefficients.

I. Elementary Expressions.

1. To construct x = a+b.

In other words, to construct a line equal to the sum of two given lines. The construction is obvious.

Draw OA = a, then from A take in the opposite direction AB = b; OB represents the value of x.

If b>a, the value of x is negative. In this case, the problem has no meaning if absolute magnitude alone is considered. But if position is also taken into account, the negative value of x is represented by a line OB extending from O in the opposite direction to that of a positive value of x, or that of the positive line OA.

This exercise furnishes a simple illustration of the principle of Descartes, that contrary signs in Algebra correspond to opposite directions in Geometry. Or, stated more exactly, if a positive value of a

linear expression is properly represented by a line drawn from a certain point in a certain direction, a negative value will be represented by a line drawn from the same point in the opposite direction.

3. To construct x=a+be+de+f.

4. To construct x = na.

The solution consists in dividing a in the ratio m: n.

7. To construct x = ab.

In this case, x is represented by the area of a rectangle whose sides are equal to a and b.

x is the fourth proportional to a, b, and c, and constructed by means of No. 139 or 162 or 163.

x is the third proportional to a and b, and is constructed by means of No. 157 or 158 or 164.

x is the mean proportional between a and b, and is constructed by means of No. 157 or 158 or 164.

x is equal to the hypotenuse of a right triangle whose legs are a and b.

13. To construct x =

x is equal to one leg of a

√a2 - b2.

right triangle in which the hypotenuse is a, and the other leg b. Also, since √a2 - b2 = √(a + b) (a − b), a is the mean proportional between a + b and a – b.