254

APPLICATION OF TANGENTS.

In the construction of doors, windows, archways, eithersemicircular or gothic, the uprights AM, CN, figs. 4 and5, are vertical, and at right angles to the horizontal ra-dius AO = OC, fig. 4, and to AC, fig. 5 ; consequentlythe uprights are tangents to the arch in A and in C.

In the elliptical arch ABCD, fig. 6, formed like thehandle of a basket, there are arcs of a circle, AB, BC,CD, the centres of which, m, O, n, are thus placed:—

1st. O, m, and the point B, where the arcs AB and BC meet, areall in one right line. 2nd. O, n, and the point C, where the arcsBC and CD meet, are also all in one right line. If XBY, therefore,is at right angles to OmB, and if ZCT is at right angles to OnC,the two lines will be at the same time tangents—the former to thearcs AB and BC in B, and the latter to the arcs BC and CD in C.As the arcs of circles thus drawn have the same tangent, there is noangle, or sharp turning, or abruptness, at the point of their inter-section.

Whenever it is required to substitute for a continuedcurve the arcs of circles which are as near as possible ofthe same form, and from which no interruption to conti-nuity arises, the circles should meet, so that at the pointof meeting they have the same tangent.

Planes, tangents to surfaces .—Parallel to a given plane,let us make, in the surface AGB, fig. 7, a succession ofplane sections, AB, CD, EF; they will gradually dimi-nish as they approach the limits of the surface, and weshall at length arrive at a point G, which alone will be ona plane MN, parallel to all the sections.

Let us draw on the surface various curves, AGB, aGb,passing through the point G; through this point, lettangents to these curves be drawn. As no right line canpass between the tangents and the curves, all the tan-gents must be placed on the plane MN.

Thus every plane, forming in G a tangent to the sur-face AGB, contains all the right lines, tangents in G toall the curves drawn through this point on the same sur-face. We must except, however, singular points, suchas the summit of the cone, &c.; but these points arealways exceptions on surfaces.