It is deserving of notice, as one of the many proofs which the records of antiquity afford of the similarity of the results of human thought in all ages, and of the truth of the Wise Man’s saying, that “there is nothing new under the sun,” that the ancient Egyptians appear to have had the same conceptions of the solid of stability that were present to the mind of the modern Engineer of the Eddystone Lighthouse. In the admirable work recently published by Sir J. Gardner Wilkinson on the Manners and Customs of the Ancient Egyptians, he gives, in the first volume of his second series, at page 253, a wood-cut, shewing the figure of the deity Pthah, under the symbol of stability, according to Egyptian conceptions. This symbol so closely and strikingly resembles the general appearance of the Eddystone, that I willingly give it a place in the text, ([No. 1]) denuded, however, of the arms and head-dress of the deity whom it shrouds.
In applying these general notions to the design of a Tower for the Skerryvore Rock, I was, of course, guided by numerous circumstances, which modified my views and produced the individual form of Tower which I have adopted. Since the days of Smeaton, when his magnificent Tower was lighted by common candles, the application of optical apparatus to Lighthouses has greatly altered the state of the case; and the improvement of the system in modern times has, in most instances, rendered a greater altitude of Tower desirable, in order to extend, as much as possible, the benefit of a system capable of illuminating the visible horizon of any Tower which human art can reasonably hope to construct. In the particular case of the Skerryvore, also, the great distance of the outlying rocks (some of which, as will be seen from the [chart], are 3 miles right seaward of the Lighthouse) concurs with the improvement of the Lights, in making it desirable that the Tower should be of considerable height, and that the light should command an extensive range. It was, therefore, from the first consideration of the subject, determined that the Light should be elevated about 150 feet above high water of spring tides, so as to illuminate a visible horizon of not less than 18 miles of radius; and, after much deliberation, and a full consideration of the infrequency of communication with the proposed Lighthouse from the great difficulty of landing on the Rock, and the consequent uncertainty of keeping up the supplies, I found that, for the convenient accommodation of the Lightkeepers and the suitable stowage of the stores, a void space of about 13,000 cubic feet would be required. These elements being fixed, the general proportions of the Tower came next to be considered.
In the Eddystone the radius of the base, at the level of high water of spring tides, is somewhat less than one-fifth of the height of the Tower above that level; while in the Bell Rock, at the same level, it is little more than one-seventh of the height. If, again, we suppose the curve of the Eddystone to be continued downwards to the level of low water, the radius (in so far as we may judge from sketching the continuation of a curve undefined by any geometrical property) would be rather more than one-fourth of the whole height above that level; while in the Bell Rock the proportion, in reference to the same level, is a little more than one-fifth. Viewing the whole height of the Skerryvore Tower above high water of spring tides as equal to 142 feet, and finding that, in the cases of the Eddystone and the Bell Rock, the radius of the horizontal section at that level is respectively one-fifth and one-seventh of the whole height; and again, viewing the extreme height of the Skerryvore Tower above low water of spring tides as equal to about 155 feet, and considering the proportionate radii of the Bell Rock and Eddystone (in so far as the latter is ascertainable) as respectively one-fifth and one-fourth of the heights of the top of the masonry above the level of low water, I finally decided upon giving the Tower at the Skerryvore such dimensions as would not be widely discordant with these general proportions. In this view, I determined that the radius of the base should not exceed 22 feet, on the level of about 4 feet above the high water mark, where I expected to obtain a solid foundation—a base which bears to the whole height of the Tower a proportion somewhat less than that of the Bell Rock, which is one-fifth. It so happens, that the diameter adopted is nearly the greatest which the Rock affords; for, although a glance at the accompanying plan of the Rock at high water ([Plate, No. III.]) would lead one to suppose that a more extended base might have been obtained, I found, after many careful examinations of the gullies and fissures which intersect it, that some of the concealed fissures run much farther into the Rock than might at first be imagined. The adoption of a much larger base, even had it been otherwise advisable, would therefore have involved some risk of the external ring of stones of the lowest course giving way by the yielding of an unsound part of the outer portion of the Rock to the pressure of the superincumbent mass, and might eventually have led to the destruction of the Tower.
No. 2.
The height of the Pillar having been finally fixed at 138·5 feet, and the radius of the base, at the level of about 4 feet above high water, at 21 feet, I next proceeded to consider the details of its proportions. Of the whole height of 138·5 feet, 18 were to be absorbed in a suitable capital for the Pillar, consisting of a parapet for the Lantern, an abacus, a cavetto, and a belt separating these from the shaft. The internal void I determined should be 12 feet in diameter, as the size most suitable for the reception of the lantern and apparatus; and this, combined with the choice of about 13,000 cubic feet of void already mentioned, fixed the height of the solid frustum at the base of the Tower at about 26 feet above the foundation. Having farther decided that the thinnest part of the walls, immediately under the belt-course which separates the capital from the shaft, should not be less than 2 feet thick, as necessary to give due solidity and strength to the walls, and prevent, by the breadth of the joints, the percolation through the walls of the water which might be furiously dashed against them in storms, I had nothing farther to do but to determine the nature of the line which should connect the extremities of the top and bottom radii of the Pillar. As I had already concluded that this line must, as in the Eddystone and Bell Rock, be a curve line, concave to the sea, I next proceeded to try the effects of various curves traced between these points, in giving a convenient and advantageous disposition of the materials, with regard to both the thickness of the walls and the mass of the solid frustum at the base of the Tower. These two points, as will be better understood by means of the accompanying diagram ([No. 2]), are separated from each other vertically 120·25 feet, and are horizontally distant from each other 13 feet, which is the excess of the bottom radius over that of the top of the shaft, or the consequent amount of what may be called the aggregate slope of the wall. The solid generated by the revolution of some curve line about the vertical axis of the building then becomes the shaft of the pillar. For this purpose I tried four different curves, the Parabola, Logarithmic, Hyperbola, and Conchoid, figures of which, upon the same scale, will be found in [Plate, No. IV.], with the position of the centre of gravity, which was carefully calculated, marked on each. The logarithmic curve I at once rejected, from its too near approach to a conic frustum, and the excessive thickness of the walls which such a figure would produce, where the hollow cylindric space for the internal accommodation commences at the level of 26 feet above the base. The parabolic form displeased my eye by the too rapid change of its slope near the base; and I had some difficulty in reconciling myself to the condition of the exterior ring of stones at the base, too much of the outer portion of each stone being left without the advantage of direct pressure from the superincumbent mass of the wall above. The two remaining pillars, derived from the hyperbolic and conchoidal[12] frusta, are nearly identical in form; and of these two curves I preferred the former, which gives the most advantageous arrangement of materials, in regard to stability, of all the four forms. This quality of advantageous proportion exists in these forms, in the ratio of the numbers in the last column of the following table:[13] which shews a slight superiority of the Hyperbolic over any of the other forms.
| Hypothetical Towers. | Height of the Tower in feet. (H.) | Diameter | Volume of solid Tower in cubic feet. M. | Distance of centre of Gravity from Base. G. | HG | Economic Advantage. G.M.G′·M′. | |
|---|---|---|---|---|---|---|---|
| at Base in feet. | at Top in feet. | ||||||
| Hyperbolic, | 120 | 42 | 16 | 62,915 | 41·227 | 2·911 | 1·00000 |
| Conchoidal, | 120 | 42 | 16 | 62,984 | 41·336 | 2·903 | 0·99627 |
| Parabolic, | 120 | 42 | 16 | 63,605 | 43·400 | 2·765 | 0·93963 |
| Logarithmic, | 120 | 42 | 16 | 74,742 | 42·460 | 2·826 | 0·81608 |
| Conical, | 120 | 42 | 16 | 84,737 | 43·280 | 2·773 | 0·70725 |
[12] The solid, in this case, would have been formed by the revolution of the interior conchoid of Nicomedes about its directrix; and its co-ordinates were kindly calculated for me by my late revered preceptor, Dr Wallace, Professor of Mathematics in the University of Edinburgh, who employed so many hours of his latter years in labours of kindness among his friends. This act of the Professor was the result of a conversation I had with him on the subject. Before I received his friendly communication, however, I had resolved to adopt the rectangular hyperbola, whose co-ordinates I had myself determined with this view some time before; and when I found that the conchoid and the hyperbola, traced between the two fixed points by means of the calculated co-ordinates, were so nearly coincident, that it was difficult to prevent their running into each other, even when drawn out on a large scale, I determined to adhere to my original purpose of adopting the latter curve as my guide.
[13] The last column of this table is derived as follows:—Assuming that the economic advantage of any proposed tower of given height and diameter at base and top, is inversely as the mass and the height of the centre of gravity above the base, and denoting these quantities by M and G respectively, the fraction 1G·M may be taken as an indication of the economic advantage of the proposed tower. Let 1G′·M′ express the economic advantage of another tower; then the advantage of the second tower, compared to that of the first, taken as unity, will be G·MG′·M′, by which expression, the last column in the table was calculated.
The shaft of the Skerryvore Pillar, accordingly, is a solid, generated by the revolution of a rectangular hyperbola about its asymptote as a vertical axis. Its exact height is 120·25 feet, and its diameter at the base 42 feet, and at the top 16 feet. The ordinates of the curve, at every foot of the height of the column, were carefully determined in feet to three places of decimals; and the Appendix contains a tabular view of the co-ordinates from which the working drawings were made at full size. The first 26 feet of height is a solid frustum, containing about 27,110 cubic feet, and weighing about 1990 tons.[14] Immediately above this level the walls are 9·58 feet thick, whence they gradually decrease throughout the whole height of the shaft, until at the belt they are reduced to 2 feet in thickness. Above the shaft rests a cylindric belt 18 inches deep; and this is surmounted by a cavetto 6 feet high, and having 3 feet of projection. The contour of this cavetto is that resulting from a quadrant of an ellipse revolving about the centre of the tower, with a radius of 8 feet on the level of its transverse axis; and the moulds for this curve were drawn at full size from co-ordinates calculated for the purpose. The cavetto supports an abacus 3 feet deep, the upper surface of which forms the balcony of the tower, and above it rest the parapet-wall and lantern.