35 



To express the relation between the vernier and circle divisions, let d- 

 df one division of the circle; d / =the value of one division of the vernier; d-d'= 

 the least count of the vernier, or, in other words, the smallest reading of the circle. 



n= the number of spaces of the vernier which correspond to (n 1) spaces of 

 the circle. 



We then have the three formulas ; 

 (1.) 

 (2.) 



(3.) 



Thus, for example, suppose the circle was divided to 15', and it was desired to 

 read to 20". Here, d = 1 5' 



d <Z', or, the least count =20" 



Then, by formula (1) 



15 X 60" 



=45 



Therefore, 45 spaces of the vernier must correspond to 44 or(n-l) spaces of th<* 

 circle. 



Suppose again the arc to be divided to 20% and to be read to 30". In this 

 we have 



20 X 60 .. 

 =40 



Therefore, 40 spaces of the vernier must correspond to 39, or (n 1) spaces o f 

 the circle. These are the graduations which Messrs. C. L. Berger & Sons usually 

 adopt for engineers' transits. 



The cut shows a portion of the circle and vernier, to illustrate the method of 

 reading to thirty seconds. 



The lines marked 130, 140, and 150 denote 10 each. The shorter lines half way 

 between them denote 135 and 145. The next shorter lines denote whole degrees, 

 while the shortest lines are one-third of a degree, or 20' apart. 



The vernier comprises the upper series of lines. Of this series only that half 

 lying to the right of the vertical arrow, or zero, and having the figures 10 and 20 

 inclined in the same direction as the 130, 140, and 150 of the arc, is to be used in 

 connection with these figures. The vernier is double, one half to be. used with 

 one set of graduations of the arc, the other half to be used when angles are laid off 

 in the opposite direction, and then the lower set of figured, 210, 220, and 230 are used. 



It is to be especially remembered that the figures on the vernier are inclined in 

 the same direction as the figures on the arc to which they belong. 



To read the vernier, first note the whole degrees, and 20" spaces lying between 

 sue last 10 degree division and the zero division of the vernier. 



Thus in the cut, using the upper line of figures, the zero of the vernier has passed 





