252 LINK'S VIEWS. [BOOK i. 



and distinguishes them as first, second, and third archi- 

 strophe, &c. If we now draw straight lines from the points 

 of origin of the coordinates to all the points of the Archi- 

 strophes, we get the secondary strophes, and the author calls 

 the strophes described through the points of the second 

 archistrophe, protostrophes, as also those through the points 

 of the third archistrophe, deuterostrophes, &c. They are also 

 designated, according to the number of the point in each 

 archistrophe, strophes of the first, second, third order, and 

 so forth. The ordinal number of each archistrophe towards 

 which any secondary strophe runs, when diminished by 1, 

 thus determines the class, and the ordinal number the point 

 in such archistrophe points out the order of the strophe. 

 The author calls this the distinguishing point of the strophe, 

 and if its coordinates x and y are represented in general 

 by a and /3, it readily follows that the pih point of the 

 (q 1) th archistrophe will be determined by the coordinates 

 a= p?=p.a, and (3=pb. These two equations form the ground- 

 work of the investigation of quincunxes of parallel series. In 

 the second division Mr. Naumann treats of the circular or 

 concentric quincunx, where, namely, the leaves or similar 

 parts stand on a spherical disc, which, however, occurs but 

 seldom in the vegetable kingdom." 



" I have," adds Link ' ( (in my Grundlehre der Krauterkunde, 

 2nd edit., part i., pp. 446, 447 et seq.), given a mathematical 

 statement of the matter, which seems to me very simple, and 

 which rests on the fact that leaves or bracts are drawn out 

 into a spiral line from their circular position. To determine 

 this change of position, I have taken the angle at which the 

 secondary series are inclined to a directly ascending primary 

 series. Thus, from the angular distances of the whorls or 

 strophes from the principal line, we may determine the 

 number of revolutions which are made by the leaves or bracts, 

 between any given leaf or bract, and that which is situated 

 in a straight line above it. This calculation has here, princi- 

 pally, only a general application to different cases ; the 

 enumeration and measurement must be made upon the plant 

 itself, since organic digressions from the fundamental form 

 do not admit of accurate measurement." (Ray Reports, 1845, 

 p. 347). 



