BOTANICAL GAZETTE. 



14s 



20 



10 



25 1 : 



20 



li' : : 



17 



14 



I I 1 

 I I I 



. I S 



I I 



11 



. () 



I I I I t I I I I 

 I I I I I I -I I I 

 I I I I I I I 1 



Fis^s. 5 and 6 



15 



finding the cycle may be divided into three parts. 



I. To mark the numbers upon the scales. If we 



trace one set of spirals plainly to be seen wind- ' ' • • ■ 93 



inc: to the right, we find they consist of three par- '• '< 0-1 '< 

 allel lines of scales extending from the base to ','.',[ 



I > I I 



the apex of the cone. This gives us the common ^^ ; ; ; 



difference 3 (14. e.). Jn like manner from a set 



of spirals winding to the left, we get the common 



difference 5. Marking the lowest scale 1, we set 



off from it in one direction the series 1, 4, 7, 10, ,,,,,,;;,,,, 10 



&c.. and in the other direction the series 1, 6, 11. 



16, &c. Then from 4. 7, 10, 13 as starting points, 



with the common difference 5, we mark parallel 



with 1, 6, 11, &c., the series of scales, 4, 9, 14, &c., 



2, 7, 12, &c., 5, 10, 15, &c., and 3, 8, 13, &c. This 



numbers all the scales of the cone. 



II. To find the denominator of the cycle. The number of the spirals 

 in the two consecutive orders is 3 and 5 ; hence the number of spirals 

 in the next higher order is 3x5=8 (14. c). This order therefore con- 

 sists of eight spirals parallel with 1, 9, 17, 25, &:c. The sum of 5 and 

 8 is 13 and an inspection shows that the series 1, 14, 27, 40, &c., be- 

 longs to a vertical rank. There are therefore 13 vertical ranks, and 

 the denominator of the cycle is 13. 



III. To find the numerator of the cycle. If we trace the primary gen- 

 erating spiral 1, 3, 4, 5, &c., from the scale marked 1, we find that it 

 makes 5 turns around the cone to reach the scale marked 14. This 

 gives the numerator 5, Therefore the cycle is 5-13. 



17. That the cycle of the cone belongs to the common system 

 (0-1, 1-2, etc.) maybe inferred from the number of spirals in the suc- 

 cessive orders being 3, 5, 8, 13, (14. b. c). The numerator 5 of the 

 cycle may also be inferred from the second order of spirals preceding 

 the vertical rank. The numerator of any cycle in the common sys- 

 tem is always the same a^ the denominator of the second preceding 

 cycle, and, consequently the same as the next higher order of secon- 

 dary spirals. 



18. The number of cycles in a Pine cone may be indicated by pre- 

 fixing a multiplier to the fraction expressing the cycle. This will 

 also show approximately the number of scales in the cone. Thus the 

 cycle and number of cycles in a cone of Hemlock Spruce (Abies Can- 

 adensis is represented by 2 (5-13). This also shows that the number 



