250 NAUMANN'S VIEWS. [HOOK i. 



Ex. GR. Considering the spiral (fig. 55.) through the scales 

 i. 9. 17. &c., 153. 161. 169. &c. A. 1st, Its divergence 

 (from 1 to 9) is ] 00 20, and, 2nd, 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. 

 2nd, Their height (as from 1 to 169) = eight times the height 

 from 1 to 22 ; and, 3rd, The common difference of the num- 

 bers of the scales is also eight. The position of the several 

 beginnings of the 8 spirals (viz. on Nos. 1. to 8.) is shown in 

 C; and in D we have the numbers (169. 106. 43. &c.) which 

 respectively begin the second series of each spiral. 



To discover the primary spiral, we may fix on any scale 

 as a point of departure (No. 1 .), and then, by numbering the 

 scales on two of the secondary spirals (as 1. 9. 17. &c., and 

 1.6. 11. &c.) which proceed in opposite directions, we may 

 afterwards very readily place the numbers on all the scales. 

 The easiest method of obtaining the common differences (viz. 

 8 and 5), for the purpose of numbering the scales in the two 

 cases selected, is to draw a circle round the cone, and count 

 the number of each of the two kinds of spirals intersecting it 

 (which will be 8 of the first and 5 of the second). When a 

 secondary spiral perfects a complete coil (as 1. 9. &c. 161. 

 169.), the number of the spirals of the same kind is readily 

 seen ; but the former mode for obtaining this number will 

 apply equally well to cases where the cone is too short for the 

 coils to be completed." 



Professor Link has the following additional observations 

 on this subject : " Since Schimper much has been written 

 upon the position of the leaves and bracts on the stem, 

 although few mathematical researches have appeared upon 

 the subject, which is, nevertheless, well adapted to such 

 inquiries. A natural philosopher of considerable merit in 

 the department of mathematical crystallography, Herr Nau- 

 mann, has published an essay ' On the Quincunx, as the fun- 

 damental basis of the arrangement of Leaves/ (Ueber den 

 Quincunx als Grundgesetz der Blattstellung im Pflanzen- 

 reiche ; in Poggendorf 'a Annal, d. Physik. u. Chemie, 2 Reihe, 



