46 MORPHOLOGY, OR COMPARATIVE ANATOMY. 



reference to the adjoining diagram (fig. 51). Assume any scale as No. 1. 

 Select the scale over it, in as nearly a vertical line as possible, such as 

 that numbered 14 in fig. 51. Secondary spirals parallel to each other 

 will be seen running to the right and to the left hand. Such are indi- 

 cated by the numbers 1, 6, 11, 16, &c. to the left, and by 1, 9, 17, c. to 

 the right of the reader. Or, again, very depressed spirals are formed by 

 the scales marked 9, 11, 13 (not seen) 'to the left, and by 6, 9, ]2 (not 

 seen) to the right. Of all such spirals, select the two most elevated, which 

 pass by and overlap the scale immediately over that chosen as No. 1. 

 These will be the spirals indicated by the numbers 1, 6, 11, 16, &c. to 

 the left of No. 14, and by 1, 9, 17, 25, &c. passing to the right hand of 

 that scale. Count the number of secondary spirals parallel to these two 

 respectively. There will be found to be eight such parallel spirals in all 

 sweeping round to the right; andy?^, such as 1, 6, 11, 16, c., to the 

 left. Take the lowest of these two numbers, or 5, as the numerator, 

 their sum, 5+8, or 13, as the denominator, and ^ will be the fraction 

 required. 



To prove this, numbers must be assigned to every scale of at least the 

 first cycle, *. e. those included between No. 1 and' that numbered 14 in 

 the figure. 



Starting with the scale assumed as No. 1, add 8 (that is, the number of 

 parallel spirals to the right) to 1, and write 9 on the next scale, as in 

 fig. 51. Add 8 again, and write 17 on the next, and so on. 



Again, add 5 to 1, and write 6 on the adjacent scale on the left-hand 

 spiral ; add 5 to 6, and write 11 on the next, and so on. 



Two entire secondary spirals intersecting at No. 1 will thus be num- 

 bered. 



To number any other scales, we may start from either of these spirals, 

 always adding 5 to the number of any scale on going from right to left, 

 and 8 on going from left to right : thus, 



6+8 = 14. 14+8 = 22, 

 or, 9+5 = 14. 17+5 = 22. 



So that we can assign by either method the numbers 14 nnd 22 to the 



S'oper scales. Similarly all the scales of the cone can be numbered, 

 nly those of a lower number than 6 and 9 are obtained by subtraction of 

 8 and 5. Now, it will be seen that 14 will be the number of the scale 

 directly over No. 1. This proves that the denominator is correct, for 

 there will be 13 scales in the cycle. 



Secondly, having, we will assume, numbered all the scales of the cycle 

 between Nos. 1 and 14, if the cone be held erect, and is made to revolve 

 while the eye passes from No. 1 to No. 2, then on to No. 3, &c., up to 

 No. 14, the observer will find that he revolves the cone exactly five times. 

 In other words, a spiral line passing through the scales 1, 2, 3, 4 . . . up 

 to 14, which constitutes one cycle, will coil five times round the axis. 



The perpendicular distance between the points of origin of 

 successive leaves is dependent simply on the degree of development 

 of the internocles of the stem. These may be so short that, as in 

 the common Stone- crop (Sedum acre), Araucaria imbricata, &c., 



