more than half of that required for an uniform stable dome of the same size. In round numbers, a stone dome 100 feet wide and I foot thick would weigh something like 1000 tons; from which any others may easily be calculated, remembering that the weight increases as the square of the diameter, and directly as the thickness-assuming that to be small compared with the diameter, or else these rules do not hold good. Therefore we might build a dome 200 feet wide, covering twice the area of the largest masonry domes in the world which would be stable without any ties if only 3 ft. 3 in. thick at the bottom and anything less than I ft. 7 in. at the top, standing on a drum of the same thickness and of any height, provided it contains a slope of 12°; for the tapering makes very little difference as to that. The drum need not display the slope, which would be ugly beyond a small amount, but it could be concealed by arcades outside towards the top and inside at the bottom, which would look very well in themselves besides. Moreover the slope of the drum may diminish downwards, so much that if it is as high as the dome the slope need only be half as much at the bottom as at the top.

In practice we should certainly build iron ties in the lower quarter of the dome, and thereby the thickness might be much more diminished. The calculations all assume the absence both of iron ties and of the advantages of bonding and cement; but on the other hand they assume the stones to be or act as throughs and to be strong enough not to crush at the edges.

Some of the architects who were present at the lecture suggested that it would be better to make the beds of some of the lower courses horizontal, as is some times done in arches and generally in spires. The

may be unobjectionable, though I doubt if it is expedient, as it makes acute angles in the stone just at the most likely place to chip. But domes differ essentially from arches in walls, in having nothing on or against their haunches; and the fact is that the lower courses ought not to be more horizontal than the radial direction to the centre of the base, but less; as is clear at once from this: with the radial inclination the few lowest courses would not differ sensibly from horizontal, and consequently there would be absolutely nothing to resist the bursting pressure except 'friction and sticktion,' which would be far too little to be safe. The thrust outwards of a dome is enormously more than of a spire, and even they are safer if the beds are square to the face (p. 245).

The nearest approximation to a hemispherical dome which will stand without sensible thickness anywhere is a hemisphere with its shoulders at about 20° from the top pushed outwards by about a 50th of the diameter, and the haunches about 20° from the bottom pulled in as much; so that it does not begin to rise quite vertically like a hemisphere, but a little leaning inwards, and is rather flatter at the top. And as things which are mechanically right generally look so, I think it probable that such a dome would look better than a perfect hemisphere, though the difference is hardly visible in a small drawing. A paraboloid would also be practically stable with a very little thickness, and a dome of which the section is a catenary would be quite stable, but they are both ugly shapes for solid looking bodies.

Domes with eyes.-Domes, like that of the Pantheon at Rome, with a large hole or 'eye' cut out of the top, are obviously more stable than others, and therefore

Domes with Eyes and Ribs.

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require less thickness, because it is the weight of the upper and flattest part which is far the most oppressive or bursting, and that is lost by the 'eye.' I found that an eye with a radius of 20°, which covers only a 25th of the floor, takes off 06 or a 17th of the weight of an uniform dome, but would enable it to do with a fifth less thickness. Semicircles cut out of the bottom on the contrary diminish the stability, partly by the loss of their weight leaning inwards, but much more, by the contraction of the base at those places, or the loss of so much support as would be cut off by vertical planes just covering each opening like a shutter.

Ribbed domes.-It was taken for granted in the former discussions at the R.I.B.A., as requiring no proof, that domes must be stronger for being made with ribs. But exactly the contrary is the fact, unless the thickness is greatly increased, as may be easily proved thus: Take a pure dome of the proper thickness for stability and turn it into one of ribs and panels of the same total weight, and see what would happen. Unless the ribs or arches are three times as thick as the dome they will not stand themselves, as we saw at p. 301, much less bear the intervening panels too; and the panels will themselves be quite unstable by losing so much of their thickness as is thrown into the ribs. Or again, begin with a dome as thick as the ribs are intended to be: then if you thin the intervening spaces down to panels you take away far more weight near the bottom where the weight tends to stability, than near the top where it tends to instability. In the Pantheon the ribs are not uniform arches, but are themselves of the shape of lunes, or the slice between two meridians of the dome, and therefore are as stable as the dome itself; and the whole is a vast deal thicker than is

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requisite for bare stability, the panels being filled up so as to show very little dome outside. Another objection to a ribbed and panelled dome of small thickness is that you lose the benefit of horizontal ties or bonding; in short this, which was assumed to be the best, is in every way the worst mode of dome-building; though of course adding ribs enough to be stable to a dome already stable increases the stability, especially if they are either thinned or narrowed upwards.

Pointed domes. It is evident that pointed domes are more stable than round ones, and I calculated in the paper referred to in what degree they are so, and found that a dome of equilateral section, or one containing an arc of 60°, only needs a thickness of ‘0137 or a 73rd of its width; which is also in that case its radius of curvature; or such a dome 100 feet wide need not be quite 17 inches thick. And one of 70°, of which the radius of curvature is very nearly of the width, requires a thickness of about a 60th of its width, or a dome of 100 feet requires about 20 inches. Tapering does not make much difference in the stability of pointed domes, because they have already lost the top of the hemisphere, which is the most oppressive part. These domes too require no tie at the bottom if they stand upon a drum with a slope of only 9°, or about 1 to 7. But pointed domes are never built, except where there is a lantern to carry, which is the real difficulty of dome building on a large scale, and those which carry lanterns are generally of rather pointed section, though of course no point is seen.

Lanterned domes. Just as cutting out an 'eye' increases the stability, so adding a lantern decreases it in a much higher ratio than the weight cut out or added. I calculated these results, first for a hemispherical

Lanterned Domes, hemispherical.

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uniform dome. Let M be the weight of the hemisphere, or say 2500 tons for a dome 100 ft. wide and 24 feet thick, the least thickness that will carry any considerable lantern without help from ties: L the weight of the lantern as a fraction of M, t the thickness in feet. The tons in the third column are the weight of lantern which can be carried by a dome 100 ft. wide and of the thickness t. The table then is this:

The (2t)3 in the last column is only multiplied by 88 to show at once that the weight of the lantern increases very nearly as the cube of the thickness of the dome; and this proportion is so near that it would doubtless have been exact if all the calculations were not necessarily approximate and tentative, the equations being such as cannot be solved directly. The result is also probable à priori, because each lune must increase in strength with the square of its thickness as a curved beam, and it increases in stability besides by its lower part being wider than the upper, and therefore gaining more by the increase of thickness.

The lantern also materially increases the thrust or bursting pressure R at the bottom of the hemisphere by exactly the weight L; so that if L = 1 M,