BOTANICAL GAZETTE. 



123 



TABLE I. 



velopment of the spirals in 

 these cycles, we deduce the 

 following two fundamental 

 principles : 



I. Every cycle conta in s w itli- 

 In itself ill the form of spirals 

 all the vertical ranks of the 

 preceding cycles. 



II. The conseciitive orders 

 of spirals mount one above 



anotlter alten lately to right and left till finally they reach the vertical ranks. 



14. As» consequences of these two principles we may infer as 

 follows : 



a. The system begins with 0-1. 



b. Every cycle discloses the preceding cycles of the system. 



c. The number of spirals in the successive orders is the same as 

 the denominators of the fractions which represent the system. 



d. The spirals of the 1st, 3d, 5th, etc., orders have the same oblique 

 direction to the right or to the left. The spirals of the 2d, -Ith, 6th, 

 &c., orders have the same oblique direction to the left or to the right, 

 (•ross-wise of the spirals of the odd orders. 



e. The common difference of the numbers in the series which rep- 

 resent any order of spirals is the same as the number of spirals in the 

 order. — [To be continued.] 



Heliopsis vs. Helianthus. — Probably there are few botanists, es- 

 pecially of the less experienced ones, who are not deceived, at times, 

 in regard to Heliopsis Isevis, Pers. Even after studying Heliopsis and 

 noting its peculiarities, it is still easy to mistake it. Its external ap- 

 pearance is so similar and yet its structure is so different from any 

 Helianthus that we wonder how it was possible to be so completely 

 deceived. Were this the case only with myself, it would be nothing 

 remarkable, but I have heard other and more experienced workers 

 speak of the same trouble. It certainly is very exasperating to take 

 considerable pains to obtain neat specimens, lay them carefully in 

 papers, and then find when you reach home that you have collected 

 Heliopsis instead of some new Helianthus. The poh'morphous nature 

 of this species makes it the more difficult to recognize. In the 

 typical form I have found the stem often glaucous and the leaves 

 nearly smooth. The var. scahra has the leaves and often the stem 



