[deville] 



ABACUS OF THE POLE STAR 



tions to the figure. The number of the township corresponding to the 

 latitude is marked on the divisions of the graduation. 



The cartesian co-ordinates of the points defined by (12) are given by 

 (8) and (9) : 



z I, - k 



(13) 



X =^ Ô 



y =0 



z I, + /, 



y being equal to zero, the line AB is the bearer of the z scale. 

 The values oîx might be calculated from (13) and laid out from the centre 

 of J. B, but the graduation can be constructed in a more simple manner. 

 In the fii-st place, we observe that for z = o, x = — c^; so the zero 

 of the graduation is at A. For z = oo, a; = (J ; so the figurative point 

 is at B. Now the scale defined by (13; is a linear scale ; therefore it is 

 the image of a regular scale and as its figurative point is at B on the line 

 Bv, it is obtained by laying out a regular scale on a parallel to Bv and 

 projecting it on AB from a projection apex on Bv. This is done as 

 follows : 



Fig. 6. 



Join township 84, (FiiT. 6) on the v scale, to 7^26'" (t = G'') 



of the u sctile. The intersection C with AB is the end of the use- 



p 



ful pai't of the z scale. The value of » in this case is let us 



^ cos 56°20" 



say 129'. 5. With a suitable scale, measure from A on Au a length AD 

 of 129.5. Select on 5 1; a proper projection apex 6^ ; join AG and GO. 

 Through D draw a parallel D M to AG and through the point 31 where 

 it intersecis <x(7 produced, draw iliiV parallel to J.y. The scale used for 

 measuring AD if laid on MN with its zero at iV. has its point 129'. 5 at 

 M , therefore its projection from G on AC gives the required :: scale. 



