53

required, whose ordinate can never exceed AB or thelength of an arc of 90 degrees of its osculatory circleat the vertex. From the point H draw HN paralleland equal to RC, HN is the radius of curvature tothe point H, which is always equal to c Sec <p puttingc for the radius of curvature CA at the vertex, and<p for the angle AHI, the curve makes with its ordi-nate, and in like manner find another radius of cur-vature ZG, through CGN draw the involute of thecurve AH.

At any point H draw HP parallel and equal toAV, the height of the key of the arch at the vertex,and draw the horizontal line PF, cutting the radius ofcurvature HN in F. In like manner find any otherpoint O. Through YFO draw the intrados of the archrequired, which is a curve having a maximum ordi-nate, which takes place when its absciss = cm Log

when n represents the thickness VA at the ver-tex, c the radius of curvature C A , and m — 2.302585.*

* This catenary AHU may also be supposed the intrados of anarch, in which case the extrados becomes a curve of contraryflexure, according to the dotted line, which takes place when its

absciss = cm Log. ”, or it may be supposed any line be-

tween the extrados and intrados, in which case the extrados willbe a curve of contrary flexure, and the intrados a curve having amaximum ordinate. The key of an arch may be supposed to bethe aggregate of the keys of numerous arches riding one uponanother, each increasing in thickness from its vertex as the secantof the angle, its extrados, intrados, or some intermediate linemakes with its ordinate, in which case the joints of each subjectedarch may approximate very nearly to the mathematical deside-ratum, “ that a due curve should pass through the centre ofgravity of, and at right angles to, each lamina considered as avoussoirbut as this vision can never be realised, it is the archi-tect’s duty to assume the thickness to be at right angles to the