"13,:tbre we enter on the solution of this question, we may remark, that as the two given right lines are supposed to intersect each other, the point, A, of the coincidence of their horizontal projections, and the point, a, of the coincidence of their vertical projections, will be the projections of the point in which they cross each other, and will, consequently, be in the same right line, a G A, perpendicular to L M. Were the two points A and a not in the same perpendicular to L the given right lines would not intersect each other, and of course would not be in the same plane.
Solution. Conceive the two given right lines to be so as to meet the horizontal plane, each in a point, and then construct these two points of coincidence. To perform this, produce the lines a It and cc, till they cut LM in the points d and e. which will be the vertical projections of these two points of coincidence. From the points it and e, draw upon the horizontal plane, and perpendicularly to L two indefinite right lines, d E and e E, which, as they must pass through one of these points, will determine their positions by their intersections, D and E, with the respective horizontal projections A D and A C, produced if necessary.
•• This done, draw the right line D E. which, with the two parts of the given lines comprised between their intersecting and the points D and E, will form a triangle, of which the angle opposite to D E will be the angle required : we hay.• therefore, only to construct this triangle. To do so, having dropped from the point A, the indefinite perpendicular A F, upon D E. conceive the plane of the triangle to turn as upon a hinge on its base D E. till it lie flat on the horizontal plane ; the apex of this triangle, during its movement, will not depart from the vertical plane described by A F, and N‘ ill at length apply itself in some degree upon the prolonga tion A in a point, 11, of which it remains only to find the distance from the base D E.
NOW the horizontal projection of this distance is the right line A l', and the vertical height of one of its extremities above that of the other is equal to a c ; hence, according to the property of Figure 3, if upon L M. n F be measured from o to f, and it' the hypothenuse af; be drawn, such hypothenuse will be the distance required. Finally, if a f be carried
from F to n, and if from the point it the two lines n D and n E be drawn, the triangle will be complete, and E u E will be the angle sought flit..
Eighth Qllestion.—The projections of a right line, and the traces of a plane, being given ; to construct the angle formed by such line and 'dame.
" Solution. Suppose a line perpendicular to the given plane to be drawn from a certain point in the given right line, the angle formed by such perpendicular with the given right line, would he the complement of the required angle, the construction of which will resolve the question.
Now, if upon the two projections of the right line, two points be taken, in the same perpendicula• with the intersec tion of the two planes of 'injection ; and if lines be drawn from these two points, perpendicular to the respective traces of the given plane, they will describe the horizontal and vertical projections of the second right line. The question will therefore be reduced to the construction of the angle formed by two right lines which cut each other, and will be of the same nature with the former.
" It is usual, in projecting a chart of a country, to imagine the remarkable points to be connected by means of right lines forming triangles, which are to be transferred to the chart on a smaller scale, but placed in the same relative order as those they represent. The operations necessary to be made on the earth consist chiefly of the measurement of angles, and of these triangles; and in order to the angles being described correctly on the chart, they ought each to be in a horizontal plane, parallel to that of the chart. If the plane of the angle be oblique to the horizon, it must not he repre sented, but its horizontal projection must be taken, which may always be found, if after measuring the angle itself, those angles which its two sides form with the horizon be also measured. Hence we derive the following operation, known under the appellation of the reduction of an angle to the horizon.
" Xinth Question.—The angle formed by two right lines, and the angles formed by such lines with the horizontal planes, being given; to construct the horizontal projection of the first of these angles.