The theory of Schimper, which should rather be called the theory of Schimper and Braun, considering the active part which Braun took from the first in framing and applying it, was capable of further development only in the mathematical and formal direction, as was shown especially in Naumann’s essay, ‘Ueber den Quincunx als Grundgesetz der Blattstellung vieler Pflanzen’ (1845). The defects above described, but not the merits of the theory were shared by the doctrine of phyllotaxis laid down about ten years later by the brothers Louis and Auguste Bravais. Their theory makes use of mathematical formulae to even a greater extent than that of Schimper without paying any attention to genetic conditions, and yet it is less consistent with itself, for it assumes two thoroughly different kinds of phyllotaxis, the positions in which are arranged in a straight and in a curved line; for the latter without any apparent reason a purely ideal original divergence is assumed which stands in irrational relation to the circumference of the stem, and from it all other divergences should be derivable; and this ultimately degenerates into mere playing with figures which in this form afford no deeper insight into the causes of the relations of position. As regards serviceableness in the methodic description of plants the theory of the brothers Bravais is much inferior to that of Schimper[49]. The genetic morphology founded about the year 1840 had to make the best terms it could with the doctrine of phyllotaxis, which was constructed on a totally different principle; the two went their way on the whole side by side without disturbance from one another till the year 1868, when Hofmeister in his general morphology attacked the principle of Schimper’s theory, and endeavoured to substitute a genetic and mechanical explanation of the relative positions for the purely formal account of them; this attempt however, which from the nature of the case has not yet led to a finished theory but nevertheless contains the germ of a further development of this important doctrine, does not come within the scope of this history.