Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1909 Excerpt: ...to V. In this position the angle 6 is shown which it makes with H This is the angle made with H by the plane of the giver lines. The angle with V is found in a similar manner by using a front line instead of a level line and drawing the line of greatest declivity perpendicular to it. PROBLEM 24 (converse of Prob. 23). To construct a plane which shall make an angle 6 with H and an angle I with V. 99. The construction may be deduced by reversing, I the order of procedure used in finding the angles 6 and I in Art. 96. In order to have a 'V problm whose solution is pos sible, the sum of 6 and 1 must--», JC be equal to or greater than 90 4 'vf and not niore tlian 180 'y Construct two cones tangent to the same inscribed sphere, /.w'i their elements making the an F1g. 70. gles 6 and with the planes of their bases. Either one of the two planes which are tangent to both the cones will fulfill the requirements of the problem. The cones in this case have their axes perpendicular to each other, one lying in H, the other in V. Let the sphere (of any radius whatever) be taken with its center on GL. (Fig. 70). Tangent to this sphere draw the elements of the cones, making 6 and with H and V respectively. The vertices of the cones are determined by the intersection of these elements with the line through the center of the sphere perpendicular to GL. (m,") and (mh) are these vertices. The circles (cb) and (de) are the bases of these cones, with centers at (o), the former in (H) and the latter in (V). The required plane is tangent to both cones. It must therefore contain both vertices and be tangent to both bases. Since the vertex of one and the base of the other lie in the same co ordinate plane the trace of' the required plane in that co-ordinate plane will pass t...
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