TRANSACTIONS OP THE SECTIONS. 153 



drawn shall be perpendicular to a plane which is a tangent to the 

 curved surface of the cylinder at that point in the spiral line ; there- 

 fore the straight line thus drawn shall be perpendicular to another 

 straight line which is a tangent to the spiral line at that point. 



When the intrados is developed, the spiral lines which form the edges 

 of the courses shall be parallel, and their distances shall be equal ; and 

 the spiral lines which are the edges of the ends of the stones shall be 

 developed in straight lines perpendicular to those lines which are the 

 developments of the spirals of tiie edges of the courses. 



It is evident that each of these spiral lines will have a certain radius 

 of curvature, and that this radius of curvature, at any point of the spiral 

 line, will be equal to the radius of curvature at any other point in the 

 same spiral ; and that the radius of curvature at any two given points 

 in two spiral lines which have parallel developments, are equal to one 

 another. 



Therefore, if two points be taken in a spiral line, and if a straight 

 line be drawn from one of them parallel to the axis, and if, through the 

 other, the cylinder be cut by a plane perpendicular to the axis, and if 

 the surface of the cylinder be developed ; the development will be a 

 right angled triangle, of which the quotient arising, by dividing the 

 product of the square of the hypothenuse and the radius of the cylin- 

 der by the square of the development of the circular arc intercepted 

 between the spiral and the straight line, will be the radius of curvature 

 of that spiral. 



By these principles the geometrical construction of an oblique arch 

 may be easily made for the use of the workmen, or calculations of all the 

 parts may be expeditiously and accurately performed by the engineer ; 

 it is only necessary to have given the angle of obliquity of the acute- 

 angled pier, the width of the arch within its abutments, the height of 

 the intrados above the level of the springing, the perpendicular distance 

 between the planes of the two faces, and the number of arch stones in 

 each elevation, in order to construct the arch. 



On an Alteration in the Construction of Wollastons Goniometer, by 

 xvhich its Portability is increased. By VV. H. Miller, M.A. 

 F.R.S., Felloio and Tutor of St. Johns College, and Professor of 

 Mineralogy in the University of Cambridge. 



In this instrument, which has a circle 4*4 inches in diameter, pro- 

 vided with two verniers reading to minutes, the branch which carries 

 the crystal screws into the end of the inner axle, instead of being in one 

 piece with it, as in the usual construction, and is taken out when the 

 Goniometer is put into its case. The distances of the milled heads, by 

 which the circle and inner axle are turned, from the collar through 

 which the axle of the circle passes, are considerably reduced. The 

 foot of the instrument is a plate of brass 4'4 inches long, and 1-6 inch 

 wide, capable of being fastened, by means of two screws, to one half of 

 the case, which is provided with three adjustable foot-screws. A mirror 



