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JOURNAL OF THE ROYAL HORTICULTURAL SOCIETY. 



spirals just observed ; there will be found to be eight such parallel 

 spirals to the right, and thirteen to the left, inclusive respectively of the 

 two first noticed. 



From these observations a rule has been deduced for obtaining the 

 fraction which represents the angular divergence of the so-called 

 " generating" spiral which takes in every scale on the cone, in a manner 

 similarly to those described above. Rule : The sum of the two numbers 

 of parallel secondary spirals, viz. 13 + 8, or 21, forms the denominator, 

 and the lowest, 8, supplies the numerator ; so that t? t represents the 

 angular divergence of the generating spiral. From this it is obvious that 

 the scale immediately over No. 1 will be the 22nd, and this must com- 

 mence a new cycle. 



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 arriving at No. 22, 

 which stands almost immediately over No. 1. Secondly, the cycle must 

 coil eight times, or complete eight entire circumferences in so doing. 



Method of Numbering the Scales. — Assuming there have been 

 8 parallel secondary spirals to the right, and 13 to the left, as in fig. 2, 

 the process of affixing a proper number to each scale on the cone is as 

 follows : Commencing with No. 1, affix the numbers 1, 9, 17, 25, 33, 41, 

 89, 97, 105, &c. on the scales of the secondary spiral passing through it to 

 the right ; these numbers being in arithmetical progression, the common 

 difference being 8, or the number of such parallel spirals ; thus all the 

 scales on one of the secondary (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 

 numbers affixed to them. Thus, add 8 to the number of any scale, and 

 affix the sum to the adjacent scale, on the right hand of it. Similarly, 

 add 13 to the number of any scale, and affix the sum to the adjacent scale, 

 on the left hand of it : e.g. if 8 be added to 40, 48 will be the number 

 of the scale to the right of it, so that 40 and 48 are consecutive scales 

 of a secondary spiral parallel to that passing through the scale 1, 9, 

 17, Sec. ; or if 13 be added to 25, 38 will be the number of the adjacent 

 scale ; i.e. on the spiral parallel to that passing through 1, 14, 27, &c. 

 By this process it will bo easily seen that every scale on the cone can 

 have a number assigned to it. When this has been done, if the cone be 

 held vertically and caused to revolve, the observer can note the positions 

 of each scale in order (1, 2, 3, 4, &c.) ; and he will then find that the cone 

 will have revolved eight times before the eye will rest upon the 22nd scale, 

 which lies immediately over the first. 



This experiment, then, proves the rule for the artificial method of dis- 

 covering the fraction , which represents the angular divergence of the 

 " generating " spiral. 



