For ascertaining the exact length of each of the six beams, which formed by their union a pyramid of about 21° 30′ of inclination, and, at the same time, for determining its exact place, in reference to the centre of the hexagon, both of which elements necessarily varied with the level of the irregular surface of the Rock, I used the following simple arrangement:--Each beam being of the greatest length that could be required, the level and distance from the centre were ascertained for the longest beam, which, of course, had the lowest seat or rest, by means of a wooden frame, shewn in the diagram (No. 4), in which _a a_ is a vertical rod of iron firmly batted into the rock, so as to coincide with the centre of the pyramid to be formed by the main beams, and of sufficient length to exceed the greatest variation of level between the different points where the beams are likely to stand; _c c_ is a horizontal board which can be freely turned about _a a_ horizontally, and resting upon a small shoulder _d_, and which is equal in length to the radius of the hexagon, on the horizontal plane at the level of the lowest beam. On this board is a spirit-level _s_, which regulates its horizontality; _e e_ is the approximate position of the lower end of the beam; _f f_ is a _pitched board_, representing the section of the permanent beam, on a vertical plane passing through the axis of the pyramid, and also shewing its inclination towards the centre of the pyramid. As this _pitched board_ is capable of being moved up or down by sliding through a groove at _g_, it may be successively applied to the rough surface at _e e_ in the course of cutting it down, and thus be made truly to represent the position of the beam, and, at the same time, give the inclination of the surface _e e_, which must be at right angles to the axis of the beam _f f_. In this way, by repeated trials, the surface was truly dressed to its proper inclination, and the length ascertained which required to be cut from the beam, so as to make it rest on that surface when in its true position. Hence, also, in the case of all the other beams, the length which the pitch-board _f f_ was moved upwards through the groove _g_, beyond the level _c c_, indicated the quantity to be cut from the end of any given beam.[17] The surface of the Rock, dressed for the seat of the beam, being thus brought to its proper inclination, the sliding-board correctly set and the centre line _a a_ of the beam carefully marked on the Rock, a square board (see fig. No. 5) representing the cross section of one of the beams, was then put down at the proper distance, so as to cover the space indicated by the _pitch-board_ as the site of the beam, and with its centre coinciding with the radius already traced on the dressed seat or bed. When so placed, the small round knobs, or ears, _d_ _d_ (No. 5) on this board, shewed the position of the holes to be bored for the bats or side