682 



SCIENCE 



[N. S. Vol. XXXIX. No. 1007 



forming any diameter one cell in thickness 

 would be as a:(a + b)/2:b; the ratio be- 

 tween the cells making up the surface (epi- 

 dermis) of the respective fruits would be as 

 a^: l(a-\-l)/2y:h^; while the ratio between 

 the number of cells constituting tlie entire 

 Tolume would be as a=: [(a + b)/2]= : b^ all 

 of which ratios are different. It is evident, 

 however, that any cell might be cut by a diam- 

 eter and that the surface cells are part of the 

 volume. The size inheritance of an epidermal 

 cell might fall under either of these three 

 ratios, depending on which aliquot part of the 

 fruit was under study. If there is inheritance 

 of the nature x = (a -\- h) /2, it is not clear 

 why the diameters and not the surfaces or 

 volumes should fall under that principle. If 

 one does, the other two can not. 



If it be assumed, under like conditions, that 

 X = \/ab, the ratio between parents and Fj 

 of the number of cells forming a diameter one 

 cell in thickness becomes a : ^/ ab : h ; that be- 

 tween the surface cells a'^ : db : h- ; and that 

 between the total cells in the volume of the 

 sphere a^ : abyjab : b^. These three ratios are 

 all equal to a : \/ab : &, if a and h mean the 

 respective parents, so that no matter what 

 aliquot part of the fruit is under study, the 

 ratio of size inheritance of lines, surfaces, and 

 volumes would be the same. Surely this ap- 

 pears the more probable. 



But how can a plant extract the square root 

 of a product? For the sake of convenience in 

 demonstration let us assume that the fruits 

 are cubes instead of spheres, that all cells are 

 equal and of size 1, and that parental char- 

 acters of size are 4 and 9. Then the length 

 factor of the F^ would be made up of two 

 forces, one tending to build strings of four 

 cells in the direction of the longitudinal axis, 

 the other tending to build strings of nine cells. 

 Similarly, the breadth factor would be made 

 up of two such forces building at right angles, 

 and in the same way the factor for thickness at 

 right angles to both. Constituent 4 of the 

 length factor, coming from one parent and 

 constituent 9 of the breadth factor coming 

 from the other parent may then be imagined 

 as building at right angles to each other, in 

 strings of nine in one direction, in strings of 



four in the other. The respective partners of 

 the two factors would be similarly engaged, 

 and likewise the two length factors with those 

 for thickness and the two breadth factors with 

 those for thickness. Each set of two would 

 tend to build rectangles of the dimensions 

 9 X 4 = 36. If now there were some force 

 stronger than the tendency of the size factors, 

 which would prevent the formation of rec- 

 tangles and permit only squares, while having 

 no influence on absolute size, area, or volume, 

 the sets of size factors would be forced to 

 modify the shape of their structures, making 

 squares instead of rectangles. Since the modi- 

 fying force did not influence area, the result- 

 ing squares would also be of the area 36, their 

 sides 6, and the cubical fruits 6X6X6 = 216. 

 This modifier of size is the factor for shape. 

 When both parents carry only the factor for 

 cubical shape, the F^ fruits are cubical, no 

 matter what the tendencies of the size factors. 



In spheres the diameters are directly pro- 

 portional to the sides of equal cubes; so that 

 what applies to cubes in this respect, applies 

 to spheres as well. The factor for spherical 

 shape is the modifier of the interaction of the 

 factors for absolute size. 



At first sight it may seem as if the fact 

 that the size of the F, is the golden mean be- 

 tween the parental sizes can be of little value 

 beyond furnishing an explanation of partial 

 dominance in the F^. However, the recurrence 

 of the action of the modifier (shape) upon the 

 various size combinations in the F. interferes 

 greatly with the chances for the appearance of 

 certain visible size characters. 



We know that size characters do segregate 

 in the F„, but we admit that with them the 

 simple Mendelian ratio of 1:2:1 is never 

 realized, though in large populations the par- 

 ental sizes may reappear. Mendelians com- 

 monly try to account for the complicated 

 ratios by assuming the presence of multiple 

 factors; non-Mendelians point to the same 

 ratios as quasi-evidence against Mendelian 

 inheritance. I here offer a different explana- 

 tion. 



In the Fj fruit 6X6X6 the size character 

 6 is not the result of a factor for size 6, but of 

 the three forces exerted by two size factors and 



