1444

THEORY AND PRACTICE OF ENGINEERING.

Book II.

indicate the proper curve of the groin over the diagonal line, at the points of intersectionof the two perpendiculars already mentioned, draw others at right angles to the diagonal,on which if the heights are set, a line may be drawn through them, which will expressits curvature. This latter method may serve to set out the stones of the greater arch, andless stone is used or wasted than by the first.

Groined Vault, on an irregular Plan .—The primitive form is a semicircle , drawn on theshortest side; on this is marked all the voussoirs, after which the others are set out,having an elliptical form. The diagonal line on the plan shows the position of the groins ;each side is then divided into two equal parts, and lines drawn from them through the centre :from the primitive semicircle perpendicular lines are dropped from each joint, until theyintersect the diagonals ; at these points of intersection, lines parallel to the centre line onthe plan are drawn, until they intersect its several sides; perpendiculars are then raised

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2543.

rig. 2544. GROUND ARCHES ON AN IRREGULAR PLAN.

from these points, on which is set out the various heights ; a line traced through themindicates the form of the respective vaults. The several voussoirs on the diagonal C I areshown to the left, and those on the diagonal DI to the right of the plan; the figurein isometrical perspective shows the stone K L.

In vaults of this description there are no groins or diagonal ribs, as was the practice ofthe Romans: in the concrete vaults which remain of their construction, as the Pantheon,Minerva Mcdica, and the Coliseum, a course of tiles or stone is generally introducedat the groin, which performs the office of ribs, and it was probably discovered at a veryearly period that there was a necessity for strengthening this part, and providing moresecurity than could be attained by the arris joint.

Before we can construct vaults composed of stones, where each is to be cut into aparticular form, we must be thoroughly acquainted with the rules by which figures maybe projected, or with the principles termed by the French engineers descriptive geometry,by which every point in space is represented, and perpendiculars drawn from them toeach of the planes of projection, the point where any perpendicular falls being the pro-