Publisher's Synopsis
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1821 edition. Excerpt: ...B and C will be the points where the normals to the two surfaces, at their common point A, pierce the horizontal plane; and, consequently, BC, the trace of the normal plane which passes through them. The perpendicular AD to this trace BC is then the tangent required. It may appear, at first, difficult to conceive the horizontal trace of a plane, which does not, in fact, differ from the plane of projections; but the doubt will be easily dispelled, by imagining the operation to be executed upon a point at an evanescent distance from the real point A; a supposition which will convince the mind, without altering the result. This case occurs frequently in practice. We will here conclude the considerations on the intersections of surfaces, which belong to the elementary part of this treatise; and resume the determination of tangent planes, "This horizontal projection is a parabola. CHAPTER VII. OF PLANES DRAWN, THROUGH A GIVEN LINE, TANGENT TO A SURFACE. 189. The line through which the plane must pass being given, there remains only to determine the point of contact. This may sometimes be done by a direct method; but, more generally, it is necessary to use an auxiliary tangent surface, known to pass through the point of contact. The nature of the auxiliary surface or surfaces used depends on the data of the operation; in general, however, cylinders and cones are employed, on account of the facility they present in constructions. 190. Let us conceive a cylinder circumscribed to the given surface; it will touch it in a certain curve; and a plane tangent to this cylinder must also be tangent to the surface, since it coincides with the cylinder along the element of contact. But, this element having in general one single point in common with...
