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of it. If the former, as at C, in the line AB, and if AB be near the edge of the material, proceed as follows : Set off from C, on either side of it, equal distances, as CD, CE, and from D and E as centres, with a radius greater than half the distance between D and E, draw arcs cutting each other in F, then a line drawn through F and C will be perpendicular to AB. If the given point is near the end of the line and the edge of the material, as A in BD (Fig. 44), then from any point a, above BD, and with a radius equal to a A, describe an arc CAT, passing through A, and cutting BD in T. Draw a line from T through a, and produce it till it cuts the arc in C. A line from C through A will be perpendicular to BD at A. Problem 3 (Fig. 45). From a given point A, above a straight line BC, to let fall a perpendicular to that line. Here the point may be nearly over the middle, or over the end of the given line. If in the first position, with any radius greater than the distance from the point A to the line BC, describe an arc cutting BC in D and E, and from points D and E as centres, with a radius greater than half the distance between D and E, draw arcs cutting each other in a and b ; then a line drawn through the given point A and the intersections of the arcs in a and b will be the required perpen- dicular. If the point is nearly over the end of the given line, as b in Fig. 46 is over AB, from b, draw a line intersecting AB in C, and bisect it in S ; with SC as radius and S as centre, describe an arc cutting AB in D, join b and D, and the line will be perpendicular to AB. The student will notice that the construction in the second cases of Problems 2 and 3 is similar. This arises from the fact that the line drawn to the given point has in each case to be at right angles to the