4 / 3 x pi x a x (b squared),
in which 2a is the major axis and 2b the minor axis.
Now, the average dimensions of the cocoons of the Three-horned Osmia are as follows:
2a = 13 mm. (.507 inch.—Translator's Note.), 2b = 7 mm. (.273 inch.—Translator's Note.) in the females;
2a = 9 mm. (.351 inch.—Translator's Note.), 2b = 5 mm. (.195 inch.—Translator's Note.) in the males.
The ratio therefore between 13 x 7 x 7 = 637 and 9 x 5 x 5 = 225 will be more or less the ratio between the sizes of the two sexes. This ratio is somewhere between 2 to 1 and 3 to 1. The females therefore are two or three times larger than the males, a proportion already suggested by a comparison of the mass of provisions, estimated simply by the eye.
The Horned Osmia gives us the following average dimensions:
2a = 15 mm. (.585 inch.—Translator's Note.), 2b = 9 mm. (.351 inch.—Translator's Note.) in the females;
2a = 12 mm. (.468 inch.—Translator's Note.), 2b = 7 mm. (.273 inch.—Translator's Note.) in the males.
Once again, the ratio between 15 x 9 x 9 = 1215 and 12 x 7 x 7 = 588 lies between 2 to 1 and 3 to 1.