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Letter 2283

Darwin, E. A. to Darwin, C. R.

[8 June 1858]

    Summary Add

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

Transcription

Dear Charles

I find I cannot draw a ⊙ round your [DIAGRAM HERE] two points, as the difference is so small only about a 170 or 80 of an inch.

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

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

They are not the same size as Millers, & 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.

I think I have answered all your questions.

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]

[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 = 12 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 = 12 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 Add

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    f1 2283.f1
    Dated by the postmark on a cover associated with the letter (DAR 162: 50).
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    f2 2283.f2
    See enclosure.
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    f3 2283.f3
    The figures have not been found.
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    f4 2283.f4
    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.
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    f5 2283.f5
    The diagrams are in DAR 162: 48/5 and 48/6.
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