CHAP. II. LEAVES. 95 



right angles, the angular distance {Divergence) between two 

 contiguous scales, seen from the centre, is ^^^ of the circum- 

 ference. Hence the divergence of the generating or primary 

 spiral ^. The various peculiarities of the secondary spirals 

 which result from the above arrangement, may be seen by 

 inspecting fig. 51. 



A. If any figure in this circle represent the divergence of a 

 spiral, the same will also represent the number of coils which 

 that spiral must make before the twenty-second scale upon it 

 comes vertically over the first. 



B. The figures in this circle (corresponding to the several 

 divergencies in A.) show the number of similar and parallel 

 spirals which must be coiled round the cylinder, in order that 

 every scale may range upon them. 



The same figures also indicate the height of each spiral — 

 viz.: either the comparative lengths of the vertical lines be- 

 tween scales 1. and 22. or the distance between two horizontal 

 circles through scales 1. and 2. ; and, lastly. 



These figvn'es are the common differences in the different 

 arithmetic series apparent on the consecutive scales of each 

 spiral. 



C is the arrangement of the first twenty-one scales on the 

 generating spiral. 



D shows the number on the scales which begin a secoiid 

 series of each kind of spiral, i. e. the numbers on their twenty- 

 second scales. 



N. B. The number on the scale which begins a fresh series 

 oi any spiral is found by the formula (a + 21 B) where (a) = 

 the number on the scale beginning a former series of the 

 spiral, and B the common difference of the numbers on two 

 contiguous scales. 



Ex. Gr. Considering the spiral {Jig. 52.) through the scales 

 1. 9. 17. &c., 153. 161. 169. &c. A. 1st, Its divergence (from 1 

 to 9) is 100 — 20, and, 2d, It must coil once towards the left, 

 or twenty times towards the right (of a spectator at the axis) 

 before it passes through the twenty-second scale upon it (viz. 

 No. 169.), which ranges vertically over the first. B. 1st, There 

 are seven other similar spirals parallel to it. 2d, Their height 

 (as from 1 to 169) = eight times the height from 1 to 22; 

 and, 3d, The common difference of the numbers of the scales 



