STRUCTURE.] NAUMANN's VIEWS. 251 



b. 26, 1842, s. 1.) The author first adverts to the regular 

 arrangement of the scars in such fossil plants, as Lepido- 

 dendron and Sigillaria; premising, however, that he has no 

 pretension to be a botanist, and is chiefly acquainted 

 with what Carl Schimper and Alex. Braun have previously 

 done in this matter. Although it is true that nature 

 does not work in organic bodies by square and compass, still 

 it is probable that her forms have always some geometrical 

 basis. 



" A quincunxial arrangement," says the author, " is 

 always found where parallel series or rows (or also series radi- 

 ating from a common point, and inclined at the same angle) 

 of equidistant points are formed in such a manner, that the 

 points of each individual series do not correspond with those 

 of the neighbouring series, but are opposed to a defined part 

 of the space intervening between two points. If we make 

 the distance between the points of each series = #, the 

 distance or interval of the individual parallel series = b, and 

 if ^ be a fraction whose numerator may at the farthest be 

 half as great as the denominator, then the quincunx will be 

 constituted from this, that all the points of the second series 

 are removed from a direct parallelism with those of the first 

 series by -^ a. 



The first point considered is the quincunx of parallel 

 series. The whole arrangement in this case will have com- 

 pleted a cycle in m series ; and hence the numerator of the 

 fraction - is to be regarded as the proper cyclical number 

 of the quincunx. To determine the oblique lines which 

 Schimper called spirals (wendel), and which our author terms 

 strophes, he draws two right-angled coordinates through 

 a figure, which represents the superficies of a cylinder, with 

 the quincunx projected on a plane. One side of the ordinates 

 may be called the positive, the other the negative. If now 

 we join any point in the line of the ordinates with the nearest 

 point of the neighbouring series, -^ a, we obtain a line, by 

 the production of which a complete series of points is con- 

 stituted, and we have also a complete system of similar series 

 parallel to each other. These series are the first and most 

 important strophes. The author calls them Archistrophes, 



