Fig. 135. - To Erect a Perpendicular at or near the End of a Given Straight Line. Second Method.
Fig. 136. - To Draw a Line Perpendicular to Another by the Use of Triangles.
7. To Divide a Given Straight Line into Two Equal Parts, with the Compasses, by Means of Arcs. - In Fig. 137, let it be required to divide the straight line A B into two equal parts. From the extremes A and B as centers, and with any radius greater than one-half of A B, describe the arcs (d f and a e, intersecting each other on opposite sides of the given line A B. A line drawn through these points of intersection, as shown by G H, will bisect the line A B. or, in other words, divide it into two equal parts.
8. To Divide a Straight Line into Two Equal Parts by the Use of a Pair of Dividers. - In Fig. 138, it is required to divide the line A B into two equal parts, or to find its middle point. Open the dividers to as near half of the given line as possible by the eye. Place one point of the dividers on one end of the line, as at A. Bring the other point of the dividers to the line, as at C, and turn on this point, carrying the first around to D. Should the point D coincide with the other end of the line, the division will be correct. But should the point D fall within (or without) the end of the line, divide this deficit (or excess) into two equal parts, as nearly as is possible by the eye, and extend (or contract) the opening of the dividers to this point and apply them again as at first. Thus, finding that the point D still falls within the end of the line, the first division is evidently too short. Therefore, divide the deficit D B by the eye, as shown by F, and increase the space of the dividers to the amount of one of D E. Then, commencing again at A, step off as before, and finding that upon turning the dividers upon the point F the other point coincides with the end of the line B, F is found to be the middle point in the line. In some cases it may be necessary to repeat this operation several times before the exact center is obtained.
9. To Divide a Straight Line into Two Equal Parts by the Use of a Triangle or Set Square. - In Fig. 139, let A B be a given straight line. Place a T-square or some straight edge parallel to A B. Then bring one of the right-angled sides of a set square against it, and slide it along until its long side, or hypothenuse, meets one end of the line, as A. Draw a line along the long side of the triangle indefinitely. Reverse the position of the set square, as shown by the dotted lines, bringing its long side against the end, B, of the given straight line, and in like manner draw a line along its long side cutting the first line. Next slide the set square along until its vertical side meets the intersection of the two lines, as shown at C, from which point drop a perpendicular to the line A B, cutting it at D. Then D will be equidistant from the two extremities A and B.
Fig. l87. - To Divide a Straight Line into Two Equal Parts by Means of Arcs.
Fig. 138. - To Bisect a Straight Line by the Use of the Dividers.
10. To Divide a Given Straight Line into Any Number of Equal Parts. - In Fig. 140, let A B be a given straight line to be divided into equal parts, in this case eight. From one extremity in this line, as at A, draw a line, as either A C or A D, oblique to A B. Set the dividers to any convenient space, and step off the oblique line, as A C, eight divisions, as shown by a b c d, etc. From the last of the points, h, thus obtained, draw a line to the end of the given line, as shown by h h2. Parallel to this line draw other lines, from each of the other points to the given line. The divisions thus obtained, indicated in the engraving by a2 b2 c2, etc., will be the desired spaces in the given line. It is evident by this rule that it is immaterial, except as a matter of convenience, to what space the dividers are set. The object of the second oblique line in the engraving is to illustrate this. Upon A C the dividers were set so as to produce spaces shorter than those required in the given line A B, while in A D the spaces were made longer than those required in the given line. By connecting the last point of either line with the point B, as shown by the lines h h2 and h1 h2, and diaw-ing lines from the points in each line parallel to these lines respectively, it will be seen that the same divisions are obtained from either oblique line.
Fig. 139. - To Bisect a Straight Line by the Use of a Triangle.