Ɛpsilon

From E. A. Darwin   [before 8 June 1858]1

Dear Charles.

To make sure that I now understand look at my diagram.2 The two diagram A, F in the hexagon the section of the Dodec.

Call CB radius of sphere (r) & CF radius of ⊙ in which the hexagon is inscribed r′, then

r′ : r :: .8165 : 1
in the bees hexagon r′ = 0inch.125 or ⅛ inch
.:.   r = r′ .8165 = .125 .8165 = 0inch.153
r−r′ = 0inch.028 distance sought.
The distance from center to obtuse ∠
= r .866 = 0in.153 ×.866 = 0in.1325
distance to ob: ∠ of Dodec = 0.1325
r′ = distance to ∠ of Heg = 0.1250
Difference = 0.0075
or ¾ of 1100 inch.

Make no scruple of asking any number of explanations, as it is very little trouble to me, & saves you bother.3

Yours E D

[Enclosure]4

CB = R = 1 diagram
CC’= R 2 = 1.414
EF = 12 FG = side of hexagon
EF = EC × tang 30o = EC × 13
EC = 22
2EF = FG = 23 = .8165
diagram

CD annotations

crossed pencil; ‘Unimportant’added pencil
Bottom of first page: ‘0.028 0.056 0.25 0.306 = diameter of sphere’; ‘sphere rather more than 120 inch larger than hexagon’ pencil . Enclosure: ‘or one side of hexagon = radius’; ‘One side of Hexagon in circle equals radius of circle.’ pencil \note5
Erasmus says Dodecahedron diameter is radius 2. In Wasps nest, hexagon = radius of sphere (on hexagonal theory) & saucer up to lower part of festoon will equal half radius.— from angle to angle of hexagon of course = twice radius. [therefore sign] saucer 14 of diameter of hexagon.—
1
Dated by the relationship to the following letter.
2
See enclosure.
3
CD was eager to revise his calculations relating to the construction of bees’ cells (see letter to W. E. Darwin, [26 May 1858]). Erasmus Alvey Darwin had helped him in the past with geometrical problems (see letter from E. A. Darwin, [May–June 1858], and Correspondence vol. 3, letter from E. A. Darwin, [May 1844 – 1 October 1846]).
4
The enclosure is in DAR 48 (ser. 2): 46 in association with CD’s notes on bees’ cells.
5
The note is in DAR 162: 48/2.

Please cite as “DCP-LETT-2278,” in Ɛpsilon: The Charles Darwin Collection accessed on 5 June 2025, https://epsilon.ac.uk/view/dcp-data/letters/DCP-LETT-2278