Mr. C. Fox on the Construction of Skew Arches. 301 
the section would appear as in fig. 4; if cut at right angles to 
the courses, or as nearly so as the case will admit of, as they 
are really cut to form the face of the bridge, the section would 
appear as in fig. 5. 
In order that these principles may be understood, it is ne- 
cessary to have aclear idea of the nature of a spiral plane; and 
perhaps, the best definition of it is, to consider it as being 
produced by the twofold motion of the radius of a cylinder, 
2. é. let a radius revolve upon its axis at an uniform velocity, 
and at the same time impart to it a progressive motion along 
the axis itself, and then by apportioning these two motions to 
the particular case you will obtain any spiral you may desire; 
hence it is apparent that the outer edge of a spiral plane is 
produced bya straight line wound round a cylinder every- 
where forming the same angle with the axis, while the inner 
edge actually merges into the axis itself, which of course is a 
straight line. The question which now naturally suggests it- 
self is how to decide at what angle to place these spiral stones 
with respect to the axis of the bridge, or in mechanical lan- 
guage, what traverse must we give the screw ? 
In entering upon the investigation of this subject, my first 
idea was to develop upon a plane surface all the superficies 
connected with a skew arch. 
If a semi-cylinder be cut obliquely, the section is a semi- 
ellipsis, and if the semi-cylinder be then unfolded, the edge 
of the developed ellipsis will not be a straight line but a spiral 
one; and some builders not being aware of this fact, have 
squared a course from the face of the centring, and having 
drawn in the remaining courses parallel with this, have taken 
it for granted that all the courses would be square with the 
face, which it will be seen is impossible by referring to the de- 
velopment of the intrados, or under surface of the arch, which 
is the development of the centring itself: they have hereby 
been led into very serious and perplexing difficulties. 
Having shown the impossibility of making all the stones 
square to the face, I will now give the mode of deciding in 
what direction they should be placed. When the soffit is 
developed, the edge which formed the face of the arch gives 
a true spiral line: my first plan was to lay the courses of 
stone at right angles to a line extending between the two ex- 
treme points of the spiral line of the developed soffit (see 
fig. 6); this line I shall afterwards speak of as the approxi- 
mate line, as it is the nearest approximation to the line of the 
face that can be obtained by a straight line. 
On further consideration I discovered a far more eligible 
mode of laying out the lines, 
