264 MORPHOLOGY. 



induction, and believed that they thus discovered that, in an overwhelm- 

 ing majority of plants, spirals were the basis of the position of leaves, and 

 that the angles of divergence were rational parts of the circumference in 

 the series of fractions ^, ^, f , -f , T 5 F , ^ 8 T . . . , the law of which is at once 

 evident, since every succeeding member originates from the sum of the 

 numerator and denominator of the two preceding members. In all 

 these spirals it naturally holds, since the angle of divergence is a rational 

 fraction of the circumference, that, after a certain number of leaves, one 

 will again be exactly vertically over the first leaf. They found a 

 number of other laws for the sequence of the individual spirals of the 

 same axis, as well as on different axes of the compound plants ; at the 

 same time they observed other aberrant conditions, which were neglected, 

 partly as exceptions, partly as independent occurrences, in turn, of a 

 peculiar regularity. The brothers Bravais started from the consideration 

 of a mathematical spiral described about a cylinder, investigated the 

 laws of position of points marked upon this at equal distances, and of 

 deviations from them, when the distances of the turns of the spiral 

 decreased and increased, when the cylinder was supposed to be an acute 

 or obtuse cone, when a plane or concave surface. Then they sought to 

 apply the laws thus found to actual plants, in instituting a multitude of 

 very accurate and well-imagined measurements, defined the limits of 

 error in these measurements, and finally showed that there was nothing 

 to oppose their assumption of a single constant angle of divergence for 

 all spirals, since the deviations of Schimper and Braun's discoveries fell 

 within the limits of the possible error in the measurements. On account 

 of the irrationality of the angle of divergence to the circumference here, 

 no leaf ever stands exactly vertically over another throughout the whole 

 axis. The spiral is from its nature infinite, and only comes to a ter- 

 mination by cessation of growth of the axis. Under this law they 

 include all the cases of Schimper's series, above given, and many others 

 besides, which Schimper could only take cognizance of through the 

 assumption of a different kind of regularity. They call these leaves 

 curviserial (feuilles curviseriees). Beside these remains a series of 

 different cases, in which the leaf undoubtedly stands perpendicularly 

 over a preceding one ; these they call rectiserial (feuilles rectiseriees\ 

 of which they have not yet given their development of the laws : they 

 intimate, however, in their published views, that transitions from one 

 system to the other occur, from whence it may be concluded that perhaps 

 both may admit of deduction from one law. 



Neither of the theories as yet possesses a safe foundation, since both 

 regard only the developed plant, instead of tracing the course of develop- 

 ment. The developed plant does not present itself as a mathematical 

 body, and none of its leaves exhibit a mathematically equal divergence ; 

 we cannot come to the point here without a certain amount of setting 

 right, and the admission of a pretty wide margin for errors of observa- 

 tion. The brothers Bravais say themselves, mathematical accuracy 

 is almost superfluous in such researches, which admit of it so little ; but 

 they are certainly too good mathematicians not to admit, that mathema- 

 tical laws which are not true to the hair's breadth are good for nothing. 

 On the other hand, the history of development would of course place in 

 our hands the power to find the mathematical laws confirmed with 

 perfect exactness by experience. It only needs to observe the leaf- and 

 flower-buds of Conifer^ Syn anther ece, &c. beneath the microscope, to be 

 astonished at the elegant and exact regularity which they here so strik- 



