SCOT. 2.

OF THE ARCHES.

35

passing through the middle of every second wedge, the firstoab passing through the middle of the key-piece. Then,on these radii produced, set off, from the arc of the semi-circle, ab, gh, &c, every second number in the last columnof the table, when multiplied by 6, the assumed length otab ; then, drawing with the hand a curved line through theextremities of all the exterior lines, it will be the extradosrequired, exhibiting theform and limit of the wallbuilt of uniform materials,above the circular soffit, soas to constitute an arch ofequilibration nearly as inthe annexed fig.

Where it is seen that the extrados follows nearly a courseparallel to the intrados for about 30 degrees on each side ofthe vertex; after which, it begins to bend the contrary way,having there a contrary flexure during the rest of its course,going off to an infinite distance on each side parallel to thebase, making the voussoirs at last of an infinite length, andcomposing all together a form of arch very unfit for adop-tion in practice.

We shall now show, in the next proposition, that, byanother very strict and genuine construction, an exteriorcurve is derived exactly similar to the curve here obtained:in the determination of which, some part of the mode ofreasoning in the demonstration of the last prop, is here againnecessarily repeated.

PROP. VII.

If acegi fife, be an arch , supporting a wall abki, formedof the voussoirs or arch stones ad, cf, fife, lying aslope, onsmooth surfaces , and having the joints ab, cd, fife, everywhere perpendicular to the curve of the arch ace fife. It isrequired to find the lengths of these arch stones, so that thewhole fabric may be balanced, or kept in equilibria.

d 2