SECT. I. ORGANOGRAPHY AND GLOSSOLOGY. 



133 



Also No. 35, is evidently nearer than any other bractea 

 to the vertical line through 1 and 90- To con- 

 struct the figure which represents the projection of one 

 length of the generating spiral, we may thus proceed. 

 Place No. 1 in the circumference 

 of the circle (fig. 143.), and di- 

 vide it into 89 equal parts; place 

 No. 35 on the part nearest to 35 I 

 No. 1 : and 34 is the com- 6 9\ 

 mon difference on that secondary 

 spiral, which is more nearly 

 perpendicular than any of the 

 others. The series on this 

 spiral is, therefore, 1, 35, 6'9, 

 103, &c., of which we may 

 place 69 on the next division 

 to 35 ; but as 103 belongs to a 

 second length of the generating 

 spiral, we must subtract 89 from it, and thus we shall 

 obtain No. 14, which ranges vertically below it, and 

 is, consequently, within the first coil of the generating 

 spiral itself, and therefore succeeds No. 69, on the circle. 

 From No. 1 4 then, we may begin with another secondary 

 spiral, whose common difference is the same as the last ; 

 and, consequently, we place the Nos. 48, 82, next in 

 succession to 14; but 106 rises into the second length 

 of the generating spiral, and we must subtract 89 as 

 before, which gives us No. 17, for the next number in the 

 circumference of the circle which represents only the first 

 length. And so on until we arrive at No. 2. We shall 

 thus ascertain that No. 2 is placed at 55 intervals from 

 No. 1, and, consequently, that the divergence in this 

 example is = -|^. It may readily be understood, by any 

 person accustomed to mathematical investigations, that 

 the first term common to the two arithmetical series, 1, 

 35, 69, &c., and 2, 91, 180, &c. (and which is 1871), 

 will be the number on the bractea intersected by 

 that spiral, which is represented by the first of these 

 K 3 



