2nd. With any other radius, as ED, draw another arc DKC, cutting the first arc in D and C.

3rd. From D draw DC, and DG is the perpendicular required.

NOTE.-The points E and F, from which the arcs are drawn, should be as far apart as the line A B will admit of, as the exact point of intersection can be more easily found; for it is evident that the nearer two lines cross each other at a right angle the finer will be the point of contact.

PROBLEM 8, Figure 8.

To Erect a Perpendicular Line at D, the End of the Line CD, with a Scale of Equal Parts.

1st. From any scale of equal parts take three in your dividers, and, with one foot in D, cut the line CD in B.

2nd. From the same scale take four parts in your dividers, and, with one foot in D, draw an indefinite arc toward E.

3rd. With a radius equal to five of the same parts, and one foot of the dividers in B, cut the other arc in E.

4th. From E draw E D, which is the perpendicular required.

NOTE.-1st. If four parts were first taken in the dividers and laid off on the line CD, then three parts should be used for striking the indefinite arc at A, and the five parts struck from the point C, which would give the intersection A and arrive at the same result.

2nd. On referring to the definitions of angles, it will be found that that side of a right-angled triangle opposite the right angle is called the hypothenuse; thus the line EB is the hypothenuse of the triangle EDB.

3rd. The square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of both the other sides.

4th. The square of a number is the product of that number multiplied by itself. Example: The length of the side DE is 4, which, multiplied by 4, will give 16. The length of DB is 3, which, multiplied by 3, gives for the square 9. The products of the two sides added together give 25. The length of the hypothenuse is 5, which, multiplied by 5, gives 25.

PLATE 11

CONSTRUCTION AND DIVISION OF ANGLES.

PROBLEM 9, FIGURE 1.

The Length of the Sides of a Triangle AB, CD, and EF being given to Construct the Triangle, the Two Longest Sides to be Joined Together at A.

1st. With the length of the line CD for a radius, and one foot in A, draw an arc at G.

2nd. With the length of the line EF for a radius, and one foot in B, draw an arc cutting the other arc at G.

3rd. From the point of intersection draw GA and GB, which complete the figure.

PROBLEM 10, FIGURE 2.

To Construct an Angle at K Equal to an Angle at H.

1st. From H, with any radius, draw an arc cutting the sides of the angle as at MN.

2nd. From K, with the same radius, describe an indefinite arc at 0. 3rd. Draw KO parallel to HM.

4th. Take the distance from M to N and apply it from OP. 5th. Through P draw KP, which completes the figure.

PROBLEM 11, FIGURE 3.

To Bisect the Given Angle Q by Right Line.

1st. With any radius, and one foot of the dividers in Q, draw an arc cutting the sides of an angle as in R and S.

2nd. With the same or any other radius greater than one-half RS, from the points S and R, describe two arcs cutting each other at T. 3rd. Draw TQ, which divides the angle equally.

NOTE.-This problem may be very usefully applied by workmen on many occasions. Suppose, for example, the corner Q be the corner of a room, and a washboard or cornice has to be fitted around it. First apply the level to the angle and lay it down on a piece of board; bisect the angle as above, then set the level to the center line, and you have the exact angle for cutting the miter. This rule will apply equally to the internal or external angle. Most good, practical workmen have several means for getting the cut of the miter, and to them this demonstration will appear unnecessary; but I have seen many men make sad blunders for want of knowing this simple rule.

PROBLEM 12, FIGURE 4.

To Trisect an Angle.

1st. From the angular point V, with any radius, describe an arc cutting the sides of the angle, as in X and W.

2nd. With the same radius, from the points X and W, cut the arc in Y and Z.

3rd. Draw YV and ZV, which will divide the angle as required.

PROBLEM 13, FIGURE 5.

In the Triangle ABC, to Describe a Circle Touching all its Sides. 1st. Bisect two of the angles by Problem 11, as A and B, the di