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Darwin Correspondence Project

To W. H. Miller1   [15 April 1858]2

Bees can make apparently true cylinders & spheres. (2) They never begin one cell at time always several (3) they can judge distance to certain extent, & (4) those that make their spheres or cylinders so that if completed, would intersect make an intermediate flat wall. Then assume perfect judge of distance, I thought that all angles might follow, for I cd see they would in hexagonal prism.— My notion modification of Waterhouses. Ld. Brougham sneers at it.3

(1) Question ‘planes of intersection’ ‘all the points of intersection united into an intermedial plane.’4

(2) Distance in mere circle or section of cylinder = 1 side of equi-lat △ in the circle—each circle or mathematical cylinder being after the first two drawn at that distance (called τ) from 2 others.— Can this not be applied to mathematical spheres, saying from 3 others, after three have been described.5

(3) Must I say ‘rhombic dodecahedron’ of crystallography;6 must I say math-ematical or ideal spheres & cylinders.7

(4) About the angle of 120o. 8 Are the rhombs equilateral.9

4 (bis) May I quote you as authority about the rhombs &c, produced by intersection of the spheres?10

(5) Show my statement of spheres in two planes.—

(6) About the rhombic bases holding most. Minimum of Wax.11

(7) About Hexagons being reduced in size & their first commencement against a plane surface.—12


The recipient of the questions listed in this memorandum is identified by the reference in Origin to Miller as the authority for statements about the geometry of bees’ cells (see also n. 10, below).
The date is inferred from CD’s statement in his letter to J. D. Hooker, 10 April [1858], that he was going to London on 15 April ‘to meet Miller to get wisdom on the geometry of Bees cells’. See also the letter to H. N. Shaw, 16 April [1858], in which CD stated that he ‘was in London yesterday’.
‘My notion … it.’ appears to have been added at a later date, presumably after CD’s meeting with Miller on 15 April. CD refers to the theory of the formation of bees’ cells put forward by George Robert Waterhouse in [Waterhouse] 1835, which was criticised by Henry Peter Brougham in Brougham 1839. Brougham had pointed out flaws in Waterhouse’s geometry (Brougham 1839,1: 270, 279). These pages of Brougham 1839 are among those marked in CD’s copy (Darwin Library–CUL), which he recorded having read in 1840 (Correspondence vol. 4, Appendix IV,119: 7a). In his notes on Brougham’s account of bees’ cells, CD wrote: ‘It is curious as Ld B. shows that W. makes the [’he‘ del] bottoms of cylinders intersect, whereas the cylinders do not intersect.’ (DAR 48 (ser. 2): 13). The theory that CD later expounded was a modified version of Waterhouse’s theory, but developed from a more accurate geometrical model (Origin, pp. 225–7). See letters from W. H. Miller, [14 May 1858], and from E. A. Darwin, [May–June 1858], [before 8 June 1858],[8 June 1858], and [after 8 June 1858].
The phrase ‘planes of intersection’ has been double underlined in pencil then ink; ‘all … plane.’ has been deleted in ink.
Before this paragraph, CD wrote ‘letter (2)’ in pencil. Presumably this refers to a letter Miller wrote to CD, which is now lost.
The passage ‘of crystallography’ was deleted in pencil, then in ink; above it CD wrote in pencil ‘Afterw dodecahedron’, and over this he wrote, in ink, ‘& afterwards only dodecahedron’.
CD deleted ‘mathematical’ in ink and double underlined ‘ideal’.
‘Yes’ has been added by CD.
‘Yes’ has been added by CD.
‘Send M.S. to him’ was added. For the passage to which CD refers, see Origin, pp. 226–7.
‘Yes.’ has been added.
CD wrote: ‘—Must be unequal’.


Brougham, Henry Peter. 1839. Dissertations on subjects of science connected with natural theology: being the concluding volumes of the new edition of Paley’s work. 2 vols. London: C. Knight.

Correspondence: The correspondence of Charles Darwin. Edited by Frederick Burkhardt et al. 29 vols to date. Cambridge: Cambridge University Press. 1985–.

Origin: On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. By Charles Darwin. London: John Murray. 1859.

[Waterhouse, George Robert.] 1835. Bee. In The penny cyclopædia of the Society for the Diffusion of Useful Knowledge, edited by Charles Knight, vol. 4, pp. 149–56. London: Charles Knight.


A set of questions CD prepared for his meeting with WHM to discuss the geometry of bees’ cells.

Letter details

Letter no.
Charles Robert Darwin
William Hallowes Miller
Source of text
DAR 181: 24a
Physical description
Amem 1p

Please cite as

Darwin Correspondence Project, “Letter no. 2255A,” accessed on 1 April 2023,

Also published in The Correspondence of Charles Darwin, vol. 7