136 
18 If the object of our search be only the discovery of this 
representative fraction *, or the angular divergence of the 
generating spiral, then all that is required will have been done , 
but in order to prove the truth of the rule given above we 
must proceed to affix numbers to every scale and so put it 
to a rigid test. We have, then, to show that the first cycle of 
the spiral line passes through twenty-one scales before ir "' d ‘" S 
at No. 22, which stands immediately over No. 1. Seco J, 
the cycle must coil eight times, or complete eight entire cn- 
cumferences in so doing. . , , 
19. Method of Numbering the Scales.- Assuming there to 
have been 8 parallel secondary spirals to the right, and 
the left, as in fig. 2, the process of affixing a proper numbei to 
each scale on the cone is as follows : — Commencing with No. 1, 
affix the numbers 1, 9, 17, 25, 33, 41, 89, 97 lOo, &c on the 
scales of the secondary spiral passing through it tc .the n^h , 
these numbers being in arithmetical progression the commo 
difference bein°- 8, or the number of such parallel spirals , tl 
iltheLaleson one of the secondary (as shaded) spirals will 
have numbers allotted to them. In a similar manner, affix the 
numbers 1, 14, 27, 40, 53, &c., on-the successive scales of the 
secondary spiral to the left, using the common difference 13. 
Thus we shall have two secondary spirals intersecting at No. 1, 
and again at No. 105, with every scale properly numbered. 
From these two spirals all other scales can have proper numbe.s 
affixed to them. Thus, add 8 to the number of any ^cide and 
affix the sum to the adjacent scale, on the right hand of it. 
Similarly, add 13 to the number of any scale, “ d 
to the adjacent scale, on the left hand of it ; e.g., if 8 be padded 
to 40 48 J will be the number of the scale to the right of it, so 
that 40 and 48 are consecutive scales of a secondary sp ral 
parallel to that passing through the scale 1, 9, 17, & c. , 
13 be added to 25, 38 will be the number ot tb e adjacent scale 
i e on the spiral parallel to that passing through 1, 14, 27, Sc. 
By this process, it will be easily seen that every scale on the 
Ze can have a’number assigned to it. When this > has ; been done, 
if the cone be held vertically and caused to revolve, tbe ° bser 
can note the positions of each scale m order (l.UW ’ 
he will then find that the cone will have revolved eight times 
before the eye will rest upon the 22nd scale, and which 
immediately over the first. artificial 
20. This experiment, then, proves the rule for the artitic 
methocT of discovering ^ the fraction * which represents 
angular divergence of the “ generating spiral. 
21. We may also remember that there must 
rows of leaves. These may generally be seen 
the 
be 21 vertical 
without much 
