84 



OEGANOGRAPHT AND GLOSSOLOGY. 



indicates the generating spiral. But the direction of this generating spiral from left 

 to right, or right to left, depends on the angle of divergence ; if the fraction be f, 

 01" T 3J o^ ifj and so on, the primitive or generating spiral wUl follow the most nume- 

 rous secondary spirals ; but if the fraction be -|, or /-j-, or fj-, &c., the generating 

 spiral will follow the least numerous secondary spirals. 



Take, for example, the fraction |- (fig. 459 a), and let us examine the relation be- 

 tween the genera- 

 ting and secondary 

 spirals. Whatever 

 may be the direc- 

 tion of the genera- 

 ting spiral, the 

 least numerous 

 secondary spirals 

 must follow the 

 same, and vice 

 versa. Suppose the 

 spiral to be aright- 

 to-left one, as in 

 459 a, it follows 

 that, placing No. 

 1 where the radius 

 touches the- outer 

 end of the spiral, 

 and successively 

 numbering the 

 leaves from ^ to f, 

 the nearest radius 

 to the left will be 

 occupied by a leaf 

 before the nearest 

 radius to the right. 

 The first leaf on 

 the left radius will 



evidently be No. 4 ; for it will occur after traversing three ^ (-|) ; that is, after 

 one entire revolution, plus i, and consequently on the left-hand radius nearest 

 the one from which we started. The leaf which will be found on the right radius 

 wiU evidently be No. 6, for it will occur after five times f (y), that is, after one re- 

 volution minus ^, and consequently on the nearest right-hand radius. Now we 

 know that the number of secondary spirals is equal to the difference between the 

 numbers of two consecutive leaves on one of these spirals ; therefore, if we suppose 

 the fraction to be f, the number of the secondary spirals from right to left, that is, of 

 the secondary spirals which follow the direction of the generating spiral, will be less 



459 e. Frimitive spiral from right to left, beamig five cycles, eacli of thirteen leaves, indicated by 

 the immbered points, and inserted on five turns of the spiral. The secondary spirals, formed to the 

 right by the numbers in fives, are indicated by the finely dotted lines ; the secondary spirals, 

 formed to the left by the nimibers in eights, are indicated by the lines . 



