66 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



V. Ideal Angles. 



The "Ideal Angle." 



As previously indicated, the mechanical problem set the plant 

 in building up a system of lateral members is primarily dependent 

 on the fact that the phylogenetic tendencies limit the apex to the 

 construction of one member at a time ; but, with this restriction, 

 radial symmetry is required in the structure as it progresses. The 

 corresponding metaphor would thus be the one of bidlding a 

 cylindrical chimney, placing one brick at a time, and yet keeping the 

 top always level. To meet such a difficulty, it is clear that growth 

 must oscillate from side to side, and that Hofmeister's law is a very 

 good expression of the phenomena observed. 



From his fractional series of divergences, J, J, etc., Schimper de- 

 duced the " ideal angle," and the brothers Bravais suggested that 

 this angle, 137° 30' 27"'936, an angle irrational to the circum- 

 ference, might be regarded as the sole angle of normal phyllotaxis, 

 and the same line of argument has been followed up by C. de 

 CandoUe. With the formation of other fractional series, other 

 "ideal angles" were added, and the importance of the first one 

 proposed became much impaired, while the possibility of there 

 being several " ideal angles " appeared very like a contradiction in 

 terms. All these angles followed from summation series express- 

 ing values of continued fractions of the type 1 



x+l 

 1-1-1 



1+1, etc., 

 where x might be any whole or fractional number. 



It has been noticed that a remarkable interpolation of the theory 



