CHAP. II. 
LEAVES. 
95 
right angles, the angular distance [Divergence) between two 
contiguous scales, seen from the centre, is 2T 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 figures 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 second 
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 
of 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) i= eight times the height from 1 to 22; 
and, 3d, The common difference of the numbers of the scales 
