Front Page Titles (by Subject) APPENDIX A.: SKEW ARCHES. - An Autobiography, vol. 1
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APPENDIX A.: SKEW ARCHES. - Herbert Spencer, An Autobiography, vol. 1 
An Autobiography by Herbert Spencer. Illustrated in Two Volumes. Vol. I (New York: D. Appleton and Company 1904).
Part of: An Autobiography
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[From the “Civil Engineer and Architect’s Journal” for May, 1839.]
The usual method of obtaining the spiral courses, in drawings of skew arches, is productive of much labour.
I have been led to believe that the following plan is much simpler, more expeditious, and consequently easier of comprehension; and although the same idea may possibly have occurred to others, it may not be so generally known as to be entirely unacceptable.
A spiral is defined as being a line traced upon the surface of a cylinder, by the extremity of a revolving radius, which radius has also a uniform motion along the axis.
Let AB, fig. 1, be a cylinder, and DE any line making an acute angle with the axis, it is evident that the line DE, is the locus of a point having a uniform motion, in each of the directions DB, DF, and if the line DE be wrapped round the cylinder, it will still possess the same property, only that the motion in the direction DF, will be transformed to a motion round the cylinder, and the line will thus become a spiral.
I have said this in order to show, as clearly as possible, that a straight line, when wrapped round a cylinder, produces a curve conforming to the definition of a true spiral, and will now proceed to explain the simplest method I have found of projecting this curve.
If a piece of paper, having a straight edge, represented by the line DE, be rolled round a cylinder, it will be found that all the points H, K, &c., will approach the cylinder, in vertical planes perpendicular to the axis, the edges of which planes are represented by the lines LH, MK.
Hence it will be seen, that to ascertain the position of any point H, when the line DE is wrapped round the cylinder, we have only to wrap round the line LH; this may easily be done by drawing an end view G, of the cylinder, and taking NP equal to LH, finding NH′ the length of the curve equal to NP,* and projecting the point H′ to H″, we obtain the position that H will occupy upon the cylinder. In the same manner all the points in the curve may be found.
We now come to the practical application.
Fig. 2.—Let ABCD denote the outlines of the plan of the soffit of a skew arch, and let EFG be drawn making the proper angle† with the face.
Then by the plan I have just described, the line EG may be wrapped round the cylinder, and E′ FG′ the curve generated, will represent one of the spiral courses. Now each of the courses of a skew arch would, if produced, wrap itself round the cylinder, and present a curve similar to E′ FG′, hence every one of the courses of the arch will be a portion of this curve; if, then, a mould be cut to the curve E′ FG′, it is evident that by setting on the proper distances, along the lines AG′ E′C, and applying the mould to the corresponding points, all the courses may be drawn, as shown on the figure, with little trouble.
I believe the common practice is to project each of these joints on the soffit separately; where the arch is brick, and each course shown, this is a work of much labour.
The same principle, with a little modification, is applicable to the other views of the arch, more particularly to the outline of the development of the soffit, only that in this case the operation is unrolling instead of rolling the line.
In case any may not understand the preceding explanations, I would recommend those who feel interested in the matter to try the experiments with the paper and wooden roller, and they will quickly perceive the principle.
[*]This may be done by calculation, but measurement by compasses is near enough for practical purposes.
[†]This line would be at right angles to the face, on the surface halfway between the soffit and crown. (See Mr. Fox’s pamphlet on Skew Arches.)