My dear Waterhouse

I have looked through my collection of combs & have not a piece with drone cells! I had plenty, & one choice piece, but some months ago I threw away a lot, & I suppose I did not notice that this was a selected piece. Nothing easier, of course, than to replace the piece. I am very sorry that I cannot answer your query. From memory I am convinced that the shorter diameter of oval Queen cell is larger than Drone cell.—

I cannot see how the relation which you wish to establish will help you, as you have to account for so fine a gradation in the two sizes.—   I did indeed most carefully think over subject & came most deliberately to conclusion that distance at which each Bee stands from the others must be the governing element.² How they instinctively judge this I cannot conjecture. I got comb & Bees from W. Indies solely for this object.— ³ I think you overrate the difficulty of this judgment of distance.⁴ The jumping spiders can judge distance accurately at which they spring on prey.— One has only to perfect this instinct. I am convinced you are not right in speaking of cell-construction as due to excavation alone; the Bees can certainly build rough wall in proper position for any particular side of cell. They do not require to make whole cell or sphere, but begin to work at one face of the pyramidal basis alone. All that is required is for each Bee to stand in proper relative position & distance with respect to the other Bees.—

I venture to caution you (not from my own knowledge) about the mathematical part of the question; for on the theory of spheres (& spheres must be used to make the pyramidal basis) not one angle or side of the hexagon touches the sides of spheres; the points of intersection lie quite within—   I have had models made.—⁵ The intersection of cylinders to make simple hexagon does not suffice as guide with respect to the intersection of spheres.—

The tenacity with which your Osmia stuck to same position⁶ does not apply to Hive Bee for a score of different individuals work one after another at the very commencement of any one cell.—

I much wish I could have answered your query.—

My dear Waterhouse | Yours very sincerely | C. Darwin