2. Draw a line 2 inches long perpendicular to a given line 3 inches long, from a point of an inch from one end, by geometrical construction. (Prob. 4, Case 2.)

3. (a) Upon a line 24 inches long, construct a rectilineal figure of seven sides. (b) Copy the foregoing on a scale of the original, see Prob. 5, Fig. 14. Make a b of A B, and proceed as explained in the figure.

4. Reduce (a) last example, to a triangle of equal area. (Prob. 28.)

5.* Construct an isosceles triangle equal to the sum of four squares of which the sides are,,, 14 inches respectively (Probs. 33, 26). Having found a square equal to the sum of the four squares by Prob. 33, the isosceles triangle will have a base twice the side of this square, while its altitude will be equal to the side of the square. (See Euc. I. 41.)

6. (a) Divide a line 4 inches long into 7 equal parts. (b) Considering the divided line as a scale of equal parts, construct a triangle of which the sides are 5, 4, and 3 parts respectively (see Prob. 34, Fig. 50). Make A B equal 3 parts, and with A, B, as centres and with radii equal to 4, 5 parts respectively, describe arcs intersecting in c. Join C A, C B, and A B C will be the triangle required.

7. Given an arc of a circle, to find the centre of the circle. (Prob. 7.)

8. (a) Describe a circle with a radius of 14 inches, and in it draw a regular heptagon. (6) Make a rectangle equal in area to the heptagon. (Probs. 7, 8, and 30.)

9. From a circle of 2 inches radius, cut off two segments containing angles of 40° and 100° respectively. (Prob. 14, Obs.)

10. (a) Construct a triangle of which two of its sides are

3 and 2-2 inches respectively, and one angle 50°. (b) Inscribe and circumscribe the triangle by a circle. (See Obs., Prob. 34, and Probs. 22 and 7.)

11. (a) Construct a regular octagon with a side of 2.2 inches. (6) Reduce the figure to a triangle. (Probs. 8, 29.) 12. Draw two circles with radii of 92 and 64 inch touching each other externally, and about them circumscribe a triangle whose angles shall be 46°, 62°, and 72°.

13. (a) Construct an isosceles triangle upon a base of 2 inches, each of the angles at the base being 70°. (6) Make a rectangle equal in area to the triangle. (c) Make a square equal to half the rectangle. (Probs. 20, 26, and 31.)

14. (a) Make an isosceles triangle having a vertical angle of 70°. (b) Make a triangle equal to the above having an angle equal to a given angle.

(a) See Prob. 21. (6) Make the angle CBA (see Prob. 27, Fig. 46) equal to the given angle, and join c A.

*

15. Describe two circles with radii of 2 inches and 1 inch respectively, tangent to one another, and inscribe a nonagon in the first. (Probs. 17 and 8).

16. Explain the use of the line of from a line 5.3 inches long cut off Explanation of Sector.)

lines on the Sector, and

of the length. (See

17.* (a) Construct a triangle of which two sides are 15 and 2 inches long, and the included angle 58°; circumscribe the triangle by a circle. (b) Join the centre of the circle and the given angle of the triangle, and upon this line as a base, construct a regular octagon. (c) Reduce the part of the octagon that lies outside the original triangle, to a triangle of equal area. (Probs. 34, 8, and 28.)

18. Construct a scale of chords and make an angle of 70°. (Prob. 1.)

19. (a) Divide a line 10 inches long in the proportion of the numbers 2, 2·5, 1·2, 2, 2·3. (b) Divide a line 4 inches long proportionally to the above. (Prob. 16.)

20. Find a fourth proportional to 2, 2.5, 3.3 inches.

21. Divide a straight line 5.3 inches long into 8 equal parts, and through the points of division draw parallel straight lines -inch apart, making them alternately dotted and continuous. (Prob. 23.)

22.* (a) Construct three squares, the areas of which are 81, 196, and 2:56 square inches. (b) Construct, geometrically, a fourth square, the area of which shall be equal to the sum of the areas of the other three.

ON SCALES.

In the study of Engineering, Architecture, etc., it is necessary that the Student should understand the construction and use of Scales. A drawing is said to be made to scale when its various parts bear a certain proportion to the parts of the object of which it is a representation. Let A B C D E (Fig. 14, Prac. Geom.) be a drawing of an object of which A B measures actually 120 yards. Now, by applying A в to a scale of equal parts, it will be found to measure 1 or 1.2 inches, i.e., a length of 120 yards is represented by a line 1.2 inches long; or, what is the same thing, a length of 100 yards is represented by a line 1 inch long; and the drawing is said to be made to a scale of 100 yards to 1 inch. To find the proportion of the drawing to the original object,

F

PROBLEM 28.

To make a triangle equal to a given rectilineal figure. Let A B C D E, the given figure, be an irregular pentagon.

B

C

Fig. 47.

Produce C D indefinitely

both ways. Join a c, and draw B F parallel to it, meeting C D produced in F. Again, join a D, and draw E G, parallel to it, meeting C D produced, in G. Join A F, A G. Then by Euc. 1., 37, the triangle C B F is equal to the

triangle A B F. Take away a B F which is common to both triangles, and the triangle A в a will be equal to a F C. Substitute the triangle a F C for A Ba; then A F D E will be equal to A B C D E. The same reasoning applies to the triangles D EG, AEG; and, therefore, the triangle A F G is equal to A B C D E.

Since the area of a triangle is equal to the product of the base multiplied by half its perpendicular height, (see Prob. 26 and Euc. 1., 41), we can thus find the area of a rectilineal figure of any number of sides.

Obs. 1. This problem has a practical application, in fortification, in reducing the profile of a parapet to a triangle, by the area of which the dimensions of the ditch are regulated.

Obs. 2. The area of a profile or any given rectilineal figure, can be found by reducing it to triangles and trapeziums. See note on the "Calculation of the equality of Deblai and Remblai," page 215 of Captain Lendy's work on Fortification.