SECT. I. ORGANOGRAPHY AND GLOSSOLOGY. 129 



may perhaps be rendered more evident by an inspection 



of the annexed figure ]<5S 



(138.), which shows the 



relative position of the 



scales on one length of 



the spiral, seen in the 



direction of the axis. 



(125.) Number ofse. 

 condary Spirals. Al- 

 though the number of 

 secondary spirals which 

 are readily distinguish- 

 able, is limited, yet it is 

 evident that we may really 



establish the existence of any number, however great, 

 by merely passing a line successively, from No. 1 to 

 any other scale, and so on to that scale next beyond 

 it, which has the same relative position towards it, as 

 it has to No. 1. In other words, we may have arith- 

 metical progressions with all possible common differ- 

 ences, which shall represent different secondary spirals ; 

 and these spirals may be coiled, some to the right, and 

 others to the left. We proposed to show (what we 

 took for granted in the last article) that the number 

 of parallel spirals of the same class, was always equal 

 to the common differences, of the progressions on these 

 spirals. It is clear that the generating spiral, passing 

 successively through 1,2, 3, &c., must be unique : 

 but the secondary spiral, which passes through the 

 odd numbers, 1, 3, 5, &c., leaves the even numbers, 2, 

 4, 6', &c., which form a second spiral, of the same 

 class ; that is to say, there are two secondary spirals, 

 where the common difference is 2. There are three 

 spirals, in the same manner, which pass through 1, 4, 

 7, &c., 2, 5, 8, &c., 3, 6, 9> & c -> where the common 

 difference is 3 ; and so on of all the rest. Several 

 other properties, of a mathematical /nature might be 

 mentioned; but sufficient has been said, to show the 

 simplicity of the investigations necessary for obtaining 



