292 



Mr. E. W. K. Edwards. 



[Jan. 28, 



" A Radial Area-Scale." By E. W. K. Edwards, M.A. Com- 

 municated by Professor A. G. Greenhill, F.R.S. Eeceived 

 January 28— Eead March 3, 1904. 



This is a contrivance for finding the area of a plane figure by means 

 of a transparency. The design in the transparency consists of a 

 number of radiating lines. Each of these lines is graduated. 



There are various patterns of this design, and their respective claims 

 to convenience and accuracy form a wide field for discussion. In the 

 accompanying transparency (reproduced in the figure), which is fairly 

 simple and effective, there are eleven straight lines radiating from a 

 point at equal angles. The way in which the transparency is used is 

 as follows : — 



The figure whose area is to be found is placed under the transparency, 

 in close contact with it, so that its contour lies just between the two 

 outside lines of the transparency, i.e., so that each outside line touches 

 the contour, or passes through a cusp or angular point, or contains 

 some rectilineal portion -of the contour. Each of the radiating lines 

 thus becomes a tangent or transversal, or contains a side of the figure. 

 The graduations of the right-hand points of intersection of these 

 transversals are read and added together ; then the graduations of the 

 left-hand points of intersection are read and added together. The 

 second sum is subtracted from the first ; and the result records the 

 number of square inches in the figure. 



It will be seen that if each of the outside lines touches, or passes 

 through an angular point of the figure, there will be eighteen gradua- 

 tions to be read — those on the outside lines cancelling each other. If 

 one of the outside lines contains a rectilineal portion of the contour, 

 there will be twenty graduations to read ; if both outside lines do so, 

 twenty-two graduations must be recorded. 



If a quicker use of the area-scale, with less chance of accuracy, is 

 desired, the figure to be quadrated may be placed so as to lie just 

 between the first and ninth, or the first and seventh, or the first and 

 fifth lines, in which cases fourteen, or ten, or six graduations will be 

 read respectively. 



If the figure is too large to be included between the outside lines, 

 it may be divided into two parts by a straight line drawn across it, or 

 into three parts by a pair of straight lines inclined to one another at 

 the same angle as the outside lines, which may be done by means of 

 the cardboard slip accompanying the diagram. The second of these 

 methods of dividing up the figure may be also employed when it is 

 desired to obviate the inaccuracy that may result from the two outside 

 lines otherwise being tangent to the figure. 



