# From Chauncey Wright   1 August 1871

Cambridge [Mass.]

Aug 1st. 1871

Dear Mr. Darwin

A running title would I suppose be out of place in a pamphlet. Touching these additions I have concluded that the last short paragraph which I added as an after thought, ought to be omitted and I have cancelled it in the M.S. which I return.6 To make the rest clearer would, I think, require it to be expanded beyond its worth, at least as relevant to th〈e〉 matter of the article. I think if it is not too long that it might be made a footnote to the last sentence on page 96 endin〈g〉 in “our author’s theory and criticism”. But perhaps a reference had better be made here to this additional matter (if you think it worth printing) as a note in appendix—or at the end of the article; such as “See note in Appendix”, or something like it.7

I added in pencil a note in the proof I sent you p. 89 something about the necks of camels &c being like those of the giraffe. This is not printed in the Review. If you approve it, it might be inserted in the text or added as a footnote.8

It seems to me that the list of books at the head of the article ought to be omitted altogether or at least 〈to〉 be put in small type in a foot-note. In case you decide to omit it the only reference to it (in the first paragraph) should be changed so as to read instead of “the list of publications which we place at the head of this article”, “a long list of recent publications” testifies &c. The list I gave was only of recent American publications.9

I suppose that all Americans whom it will profit to read the article will see it in the Review, so that very few copies of the reprint could be usefully disposed of here. Perhaps half a dozen copies would be all that I could usefully distribute by sending them to friends who would value the little book far more than the matter of it.10

It will give me great pleasure to meet your sons when in the course of their tour in this country they honor us with a visit.11 I hope they will come before I have made up my mind to what I have for some time been planning, a trip to England at least and, if I become traveller enough, to the Continent.12

I hope soon to publish, or to ask you to do so for me, a paper on the utility of the phyllotaxis, as you suggest.13 I have already printed two papers on this subject, one in 1856 in Gould’s Astronomical Journal No 99, and the second in 1859 in the Mathematical Monthly. A copy of the last was sent you by Prof. Gray14   In my new paper I shall avoid as much as possible all abstruse mathematics which I see has so obscured my thesis, that it is only known to mathematicians. The speciality of the phyllotactic fractions is not that they represent complete systems, so that after a time some leaf will come over the first one and be connected with the same vessels in the stem. This property would belong to any exact fractional interval. An exact proper fraction, after the number of steps represented by its denominator and the number of revolutions represented by its numerator would make a complete system, or bring the next succeeding leaf over the first. The peculiarity of the phyllotactic fractions is that the distribution is most rapid and complete within each set or system; that is, it is much more perfect than for other exact fractions. The incommensurate interval of the ratio of the extreme and mean proportion gives the best distribution of all; but here the system is infinite; that is no leaf ever comes exactly over an older one.15 I have found among old papers a proof of my first article on this subject, and for the sake of the diagram of this arrangement I send enclosed a page of the article.16 The exact fractional intervals of the phyllotaxis have the distributive character of this most perfect arrangement to this extent, namely, that they determine, as no other intervals do, that every leaf shall fall in the middle third (or not beyond it) of the space between two older ones in which it falls. In all other exact intervals there is crowding. Take for instance $\frac{4}{9}$, which does not occur in nature. Why should it not? The second leaf in this system would be placed very nearly opposite the first or very near the middle of the space, and the arrangement is so far well enough; but the third f〈a〉lls at eight ninths, $\frac{8}{9}$, that is within $\frac{1}{9}$ of the first leaf, crowding up against it, yet not near enough to get any advantage from connection with the vessels or sources of supplies which the first leaf has grown from. The fraction $\frac{3}{9}$ or the one third system would be better, for though the first and second leaves divide the circumference into 1 and 2 parts (the extremest ratio in the phyllotaxis) yet the third falls exactly into the middle of the larger interval, and the fourth is directly connected with the vessels from which the first leaf has grown. Take the interval $\frac{3}{7}$ for another instance which does not occur in nature. The second leaf falls, it is true, near the middle but the third at the interval $\frac{6}{7}$ is within $\frac{1}{7}$ of the first crowding it unnecessarily and has three times as wide a space on one side as on the other. In the phyllotactic intervals the space on one side of a leaf is never more than twice as great as on the other. In all other cases greater disproportions would occur in the distributions.

But I must not anticipate the article by too minute an account of it.

Very truly yours | Chauncey Wright

## CD annotations

2.1 Touching these] after opening square bracket pencil
3.1 p. 89 … giraffe. 3.2] scored blue crayon; cross added blue crayon
5.3 Perhaps … of it. 5.4] crossed pencil
6.3 trip to England] underl blue crayon
7.1 paper on the utility] underl blue crayon
7.4 Mathematical] underl blue crayon
End of letter: 2 words illeg pencil

## Footnotes

Wright refers to the letters to him of 13 and 14 July [1871] and 17 July [1871].
CD wished to reprint Wright 1871a, in which Wright reviewed St George Jackson Mivart’s Genesis of species (Mivart 1871a). See letters to Chauncey Wright, 13 and 14 July [1871] and nn. 2 and 5, and 17 July [1871].
Wright 1871a was published in the North American Review.
See letter to Chauncey Wright, 13 and 14 July [1871] and n. 5.
The manuscript has not been found. Wright had sent CD a manuscript copy of notes omitted from his original article (letter from Chauncey Wright, 21 June 1871); CD had returned them with his letter to Wright of 13 and 14 July [1871]. They were the source of the appendix to the pamphlet reprint (Wright 1871b).
The full sentence to which Wright refers reads, ‘The bearing of our subject upon the doctrine of Final Causes in natural history has been much discussed and is of considerable importance to our author’s theory and criticism’, the author being Mivart (Wright 1871a, p. 96). It is followed in the reprinted version by a footnote referring the reader to an appendix on the meaning of the terms ‘use’, ‘contrivance’, ‘purpose’, and ‘intention’ as used by CD and interpreted by Mivart (Wright 1871b, pp. 36, 44–6). See also letter to A. R. Wallace, 9 July [1871] and n. 3, letter to Chauncey Wright, 13 and 14 July [1871] and n. 6, and letter to Chauncey Wright, 17 July [1871].
The passage in Wright 1871a, p. 89, beginning: ‘The giraffe alone is provided with a natural watch-tower, but the others are not left without defence’ is reprinted unaltered in Wright 1871b.
The list of books at the beginning of Wright 1871a was omitted in Wright 1871b and Wright’s suggested rewording adopted (Wright 1871b, p. 3). See also letter to Chauncey Wright, 13 and 14 July [1871].
See letter to Chauncey Wright, 17 July [1871].
Wright met George Howard and Francis Darwin during their visit to the United States (see letter to Chauncey Wright, 17 July [1871] and n. 5, and letter from Chauncey Wright, 11 October 1871).
Wright visited England in 1872 (see Correspondence vol. 20, letter from Chauncey Wright, 2 September 1872).
See letter to Chauncey Wright, 13 and 14 July [1871]. Wright communicated a paper titled ‘The uses and origin of the arrangements of leaves in plants’ to the American Academy of Arts and Sciences in October 1871 (Wright 1871c; CD’s lightly annotated copy is in the Darwin Pamphlet Collection–CUL).
Wright refers to Wright 1856 and Wright 1859; see also n. 16, below. The Astronomical Journal was founded by Benjamin Apthorp Gould. The copy of Wright 1859 sent to CD by Asa Gray is now in DAR 48: B48–52; Gray had sent CD other material by Wright (see Correspondence vol. 12, letter from Asa Gray, 3 October 1864).
The angles that produce the various regular arrangements of leaves around plant stems can be expressed as fractions all falling within a mathematical progression where each numerator and denominator is equal to the sum of the two preceding numerators and denominators. An example of such a progression, also known as a Fibonacci sequence, is the series $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{5}$, $\frac{3}{8}$. Wright analysed all the theoretically possible arrangements expressed by such sequences, and postulated that those not found in nature had been eliminated by adaptation through natural selection, those remaining being closest to a hypothetical arrangement that would ensure that no single leaf ever fell exactly over any other, thus maximising exposure to light (Wright 1856; Wright 1871c). He concluded that each leaf ‘is so placed over the space between the older leaves nearest in direction to it as always to fall near the middle, and never beyond the middle third of the space, or by more than one sixth of the space from the middle, until the cycle is completed, when the new leaf is placed exactly over an older one’ (Wright 1871c, p. 399), and that the less efficient arrangements are more commonly found in fossil than in living plant species (Wright 1871c, p. 406). Wilhelm Hofmeister had drawn a similar conclusion about the origin of leaf arrangements in 1868; according to Hofmeister’s rule, ‘new leaves arise … in those positions that are as far as possible from the insertion area of the nearest existing leaf primordia’ (Hofmeister 1868; Keith Roberts ed., Handbook of plant science (Chichester: John Wiley and Sons, 2007, p. 102). For the history of work on phyllotaxis and Wright’s contribution, see Wiener 1945, especially pp. 32–4, and Marzec and Kappraff 1983. See also Jean 1994.
Wright presumably enclosed a copy of Wright 1856, p. 23, which includes a series of figures showing the angles produced by the successive arrangement of leaves around a stem. The enclosure has not been found, but the figure from the published paper is reproduced on p. 515.

## Summary

Discusses revising his North American Review article [see 7829] for publication as a pamphlet in England.

Plans to publish a further article on phyllotaxy.

## Letter details

Letter no.
DCP-LETT-7890
From
Wright, Chauncey
To
Darwin, C. R.
Sent from
Cambridge Mass.
Source of text
DAR 181: 165
Physical description
6pp †

## Please cite as

Darwin Correspondence Project, “Letter no. 7890,” accessed on 9 December 2016, http://www.darwinproject.ac.uk/DCP-LETT-7890

letter