# From E. A. Darwin   [8 June 1858]1

Dear Charles

I find I cannot draw a ⊙ round your [DIAGRAM HERE] two points, as the difference is so small only about a $\frac{1}{70 or 80}$ of an inch.

I send you 2 more projections as the more points of view you see it in the better2

I also send you some rhombs cut out, with flaps to fasten over the edges, & I advise you to begin with the flap (A)3

They are not the same size as Millers,4 & I do not know if you have any reason for that particular size.

Also a pyramid, with 2 circles on the base to be raised up & the circle on the short diameter will show you exactly how much the obtuse ∠’s are within the sphere

[DIAGRAM HERE] obtuse ∠

12 of these pyramids would make the Dodec.

You will be surprized how firm the model will feel when put together, but dont be in a hurry & let one flap dry before you paste another,

When you have got the model in your hand, & hold them in the same position as my diagrames, they will all become quite clear.

ED

Over the page I will draw another little diagram to fix the positions of the acute ∠s

[DIAGRAM HERE]

P A W N S E P N E S W – equator S P N P’ – meridian W P E P’ – East & West circle 6 acute angles at the letters 8 obtuse angles (A) in the center of the quadrants

[Enclosure]5

[DIAGRAMS HERE]

No 1 A d a c f K b C D e B No 2 A a C f k b D e B AB vertical axis passing thro’ acute ∠s A & B CD horizontal axis passing thro’ centers of 2 Rhombs K axis perpendicular to CD & AB or the plane of the paper a b e f rhomb parallel to paper, natural size diameter bf = AC = side of square in ⊙ diameter ae = KA radius of ⊙ a b c d—rhomb oblique to the paper cb is parallel to ae and = $\frac{1}{2}$ ae or Ke

Kc = eb = side of rhomb The obtuse ∠ at c is in the plane of the paper [therefore sign] Kc = radius of ⊙ passing thro’ obtuse ∠s Kb = $\frac{1}{2}$ distance of centers = distance from center of sphere to rhomb [No 2] The same as No 1 but turned round the axis AB 45o. The 4 rhombs are all at ∠ 45o with paper, but fAfB are the long diameter of the vertical plane, & fC, fD long diameters in the horizontal plane

## Footnotes

Dated by the postmark on a cover associated with the letter (DAR 162: 50).
See enclosure.
The figures have not been found.
See letter from W. H. Miller, [14 May 1858] and n. 2. Associated with William Hallowes Miller’s letter are two pasteboard figures labelled ‘Rhombic face of dodecahedron’ (DAR 48 (ser. 2): 1). The angles of the rhombs are marked ‘70o 31.7’’ and ‘109o 28.3’’. See Natural selection, p. 513 and n. 1, where the angles of a rhombic face of a bee’s cell are discussed.
The diagrams are in DAR 162: 48/5 and 48/6.

## Bibliography

Natural selection: Charles Darwin’s Natural selection: being the second part of his big species book written from 1856 to 1858. Edited by R. C. Stauffer. Cambridge: Cambridge University Press. 1975.

## Summary

Encloses projections and models relating to geometry of bees’ cells.

## Letter details

Letter no.
DCP-LETT-2283
From
Erasmus Alvey Darwin
To
Charles Robert Darwin
Sent from
unstated
Source of text
DAR 162: 48a, 50
Physical description
4pp, encl 3pp