My dear M^r. Darwin

I ought to have written to ⁠⟨⁠you⁠⟩⁠ long before this, (and have wished to do s⁠⟨⁠o⁠⟩⁠ but have been prevented by business,) ⁠⟨⁠    ⁠⟩⁠ tell you the fate of my paper on Leaf Arrangement which you were kind enough to communicate to the Royal Society in the early part of this year.²

The paper was read, but did not meet with the approval of the Referees, and will not appear in the Transactions.

I cannot say that I am sorry for th⁠⟨⁠i⁠⟩⁠s, for I have since met with a number of very striking facts which have shown ⁠⟨⁠me⁠⟩⁠ the incompleteness and limited applicability of my theory, and have given me I believe more truth and wider ground to go upon.³

The closeness of th⁠⟨⁠e⁠⟩⁠ le⁠⟨⁠a⁠⟩⁠ves ⁠⟨⁠    ⁠⟩⁠ about the perfect ⁠⟨⁠regu⁠⟩⁠larity ⁠⟨⁠    ⁠⟩⁠ ⁠⟨⁠  ⁠⟩⁠ment— The angle of divergence ⁠⟨⁠    ⁠⟩⁠ primary spiral (old style) was $\frac{2}{7}$ ⁠⟨⁠    ⁠⟩⁠ the seven ranks as vertical). ⁠⟨⁠  ⁠⟩⁠ow ⁠⟨⁠    ⁠⟩⁠ arrangement could not by any mechan⁠⟨⁠ical⁠⟩⁠ process of condensation have been derived from a two-ranked order. It might, ⁠⟨⁠    ⁠⟩⁠ remote possibility, have come from an alternate 4-ranked order ($\frac{1}{4}$), which coul⁠⟨⁠d⁠⟩⁠ not in reason have been derived from a 2-ranked original; but a 4-ranked order would almost certainly have assumed the crucial arrangement.— Anyhow, the universality of my theory of an original two-ranked order is knocked on the head by the apparition of that 7-ranked Sedum (which I find is abundant on rock work in the garden of a new house here which we have lately moved into).⁴

I cannot avoid the conclusion that the seven ranks of that Sedum have an original value of their own, and have not been derived from any simpler or⁠⟨⁠d⁠⟩⁠er, e⁠⟨⁠    ⁠⟩⁠

Up to the 5^th. ⁠⟨⁠of Se⁠⟩⁠pte⁠⟨⁠mber⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠ all the facts that I had obs⁠⟨⁠erved⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠ favour of my old theory,—th⁠⟨⁠  ⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠ leaf-orders were derived, by ⁠⟨⁠    ⁠⟩⁠ condensation with twist, from an origi⁠⟨⁠nal⁠⟩⁠ two-ranked order; (—the refractory mors⁠⟨⁠  ⁠⟩⁠ were the presence of 3 ranks of rootlets in Polygonaceæ, 4 in carrot, &c.; the exact or nearly exact ver⁠⟨⁠t⁠⟩⁠icality of the 3 ranks in sedges &c. the great variety of vertical ranks in the cactus tribe; the origin of whorls of 3, 4, 5 &c; and ot⁠⟨⁠he⁠⟩⁠rs of less consequence;)—and I had established to my own satisfaction the existence of a separate series of leaf-orders (of which I have nowhere seen mention before), most conspicuous in the heads of the Dipsaca⁠⟨⁠ce⁠⟩⁠æ manifestly derived by condensation from the crucial order, and therefore giving stro⁠⟨⁠n⁠⟩⁠g confirmation of the principle of condensation as having played an important part in the history of leaf-order.⁵

—But on the 5^th. of September I was scrambling about the old walls of Kidwelly Castle,⁶ and plucked a Sedum of whic⁠⟨⁠h⁠⟩⁠ every shoot had seven steep spiral r⁠⟨⁠    ⁠⟩⁠ by a leap of primar⁠⟨⁠y⁠⟩⁠ var⁠⟨⁠  ⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠ number of leaf-⁠⟨⁠r⁠⟩⁠anks.

This view is confirmed by an e⁠⟨⁠  ⁠⟩⁠ of specimens of Sedum sexangula⁠⟨⁠re⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠ Cambridge Botanical Gardens, ⁠⟨⁠w⁠⟩⁠here ⁠⟨⁠I found⁠⟩⁠ the same plant bearing some shoots with six vertical ranks (in whorls of 3) and others with seven oblique ranks (in alternate order), (the obliquity of the latt⁠⟨⁠e⁠⟩⁠r, as well as the verticality of the former, being due, as I shall show reason for believi⁠⟨⁠n⁠⟩⁠g, to the conditions of mutual accommodation among the close-packed leaves.)

Being thus compelled to entertain the notion of a variability in the original number of leaf-ranks, I find that it explains all the difficulties which resisted former attacks. It explains the astonishing variety of vertical ranks in the cactus tribe, where a new rank may often be seen a⁠⟨⁠p⁠⟩⁠pearing between two old ones: it explains why Haworthia Viscosa may have three exactly vertical ranks, while Haworthia Pentagona has five:⁷ it explain⁠⟨⁠s⁠⟩⁠ the exact or almost exact verticality ⁠⟨⁠o⁠⟩⁠f the three ranks in Sedges and ⁠⟨⁠    ⁠⟩⁠ of the five ⁠⟨⁠r⁠⟩⁠anks in ⁠⟨⁠    ⁠⟩⁠ and some others (though these latter are ⁠⟨⁠    ⁠⟩⁠ well explained by the former theory) ⁠⟨⁠    ⁠⟩⁠ explains why there may be 3 ranks of ⁠⟨⁠    ⁠⟩⁠ on the roots of Polygonaceæ, 4 on those of carrot, and other variations from the preval⁠⟨⁠e⁠⟩⁠nt 2-ranked form.— Most of these cases certainly demanded some better explanation than the former theory could give.

This supposed variability in the number of vertical ranks of leaves would be only of the nature of a multiplication of similar parts, and this (is it not?) is a well-known form of primary variation.

A consideration of the probable behaviour of different numbers of vertical ranks under the need of mutual acc⁠⟨⁠o⁠⟩⁠mmodation and economy of space, will show that we have here a probable explanation of the origin of the various whorled orders (which I once thought might be derived one from another by some effort of condensation, but I have since recognized the difficulty of ⁠⟨⁠    ⁠⟩⁠

I will not consider the ⁠⟨⁠    ⁠⟩⁠ case of a plant with only one ⁠⟨⁠    ⁠⟩⁠ for I do not know that there is any ⁠⟨⁠    ⁠⟩⁠ suggesting that such a form ever exi⁠⟨⁠sted⁠⟩⁠

Two vertical ranks would first find mutual accommodation in assuming the alternate order $\frac{1}{2}$ (Fig. 1.):

and this under further condensation with twist (—proneness to twist is well seen in the two ranks of Gasteria spiralis &c.) would give approximations to the more complex orders of the phyllotactic series $\frac{1}{3}$, $\frac{2}{5}$, $\frac{3}{8}$, $\frac{5}{13}$ &c., as shown in my former paper:⁸ (only a distant approximation to $\frac{1}{3}$, but a close approximation to $\frac{2}{5}$, and the subsequent terms.)

In my former paper I supposed that two original ranks might produce two divergent types,—alternate (Fig 1.) and opposite (Fig. 2), and that the oppo⁠⟨⁠s⁠⟩⁠ite form would pass into the cr⁠⟨⁠  ⁠⟩⁠ for sake of economy, and, ⁠⟨⁠    ⁠⟩⁠ likely to survive in the ar⁠⟨⁠    ⁠⟩⁠ shown in Fig 2. But ⁠⟨⁠    ⁠⟩⁠ this form (Fig. 2.) does exist ⁠⟨⁠    ⁠⟩⁠ ever⁠⟨⁠y⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠ of Mesembrianthemum;⁹ and I feel th⁠⟨⁠at⁠⟩⁠ it is not sufficiently explained by what I wrote before. It is the only form which fails to exemplify the principle of economy of space, which I have reason to regard as having played the chief part in the course of natural selection among primary variations of leaf order; and therefore it is natural to regard this simple two-ranked opposite order as the nearest to the original.

It is plain that this form would admit of condensation in two ways, either by lateral economy (producing Fig. 1.) or by longitudinal economy (producing the crucial order). But I shall presently show that the crucial order may, with less dislocation, have been derived from a 4-ranked form.

Three vertical ranks.— Suppose a 3-ranked form to have appeared as a primary variation from th⁠⟨⁠e⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠ the three ranks would g⁠⟨⁠  ⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠ by falling into the altern⁠⟨⁠ate⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠ and if it abandoned the se⁠⟨⁠    ⁠⟩⁠ vertical, it would gain still more by taking the oblique-ranked form (Fig. 4).

(These and the following figures are intended to represent the described arrangement as it would appear if unwrapped from the axis and laid flat.) Examples are frequent of 3 ranks with various degrees of obliquity.

The oblique 3-ranked form (Fig. 4.) under further condensation would give successively the orders $\frac{2}{5}$, $\frac{3}{8}$, $\frac{5}{13}$, $\frac{8}{21}$ &c., just as well as an original order $\frac{1}{2}$ would have done. Thus there arises ambiguity concerning the origin of the more complex orders, unless other considerations come in to decide the point.

Four vertical ranks (supposing them to appear by primary variation from the 2- or 3-ranked form) would naturall⁠⟨⁠y⁠⟩⁠ fall into the crucial order (Fig. ⁠⟨⁠    ⁠⟩⁠ without needing to abando⁠⟨⁠n⁠⟩⁠ their verticality or make any sacrifice, and would produce a form of exceptional stability (though it is one which has become wonderfully condens⁠⟨⁠ed⁠⟩⁠ in the heads of the Dipsacaceæ.)

⁠⟨⁠Fi⁠⟩⁠g. 5.

In what cases the crucial order may hav⁠⟨⁠e⁠⟩⁠ been derived from 4 ranks, and in what from 2, would need great search to say; but if we found one species with the crucial order and an allied species with 5 ranks, we should be inclined to think that the crucial had come from a 4-ranked form, as nearest to the 5-ranked and requiring least violence of primary variation.

Five vertical ranks (supposed as a primary variation from 3 or 4) would find lateral economy in the order $\frac{2}{5}$ (Fig 6.) and still more if they took a slight obliquity, as in Fig. 7.

Under further condensation we should get $\frac{3}{8}$, ^{⁠⟨⁠  ⁠⟩⁠/13}, $\frac{8}{21}$ &c., just ⁠⟨⁠    ⁠⟩⁠ had begun with a 2-ranked or ⁠⟨⁠    ⁠⟩⁠ form. Thus there is a th⁠⟨⁠    ⁠⟩⁠ attaching to the origin of the ⁠⟨⁠    ⁠⟩⁠

Six vertical ranks would accommoda⁠⟨⁠te⁠⟩⁠ one another by forming whorls of 3. (Fig ⁠⟨⁠8.⁠⟩⁠

(In any of these arrangements, the neighbouring ranks would get mutual accommodation by fitting alternately in one another’s intervals; and if the whole number of ranks is even, this alternation allows the ranks to retain their verticality, which is one of the striking features in whorled orders, but if the number of ranks is odd, the alternation fails, and it becomes necessary either to sacrifice a little of the economical fit between neighbouring ranks or to abandon their severe verticality. The latter seems the favourite alternative, and thus the odd numbers of ranks take alternat⁠⟨⁠e⁠⟩⁠ order with obliquity of ranks, and ⁠⟨⁠the⁠⟩⁠ even numbers take ⁠⟨⁠    ⁠⟩⁠ with verticality of rank⁠⟨⁠s)⁠⟩⁠

Seven vertical ranks ⁠⟨⁠    ⁠⟩⁠ economical obliquity, in Sedu⁠⟨⁠m se⁠⟩⁠xang⁠⟨⁠ulare⁠⟩⁠ with angular divergence $\frac{2}{7}$. Unde⁠⟨⁠r⁠⟩⁠ further condensation this order would give successively $\frac{3}{11}$, $\frac{5}{18}$, $\frac{8}{29}$, $\frac{13}{47}$ &c.

Eight vertical ranks would give whorls of 4.

Nine would probably give $\frac{2}{9}$ (? non-extant).

Ten would give whorls of 5, as ⁠⟨⁠s⁠⟩⁠een in a species of Linaria.

Eleven would probably give $\frac{2}{11}$ (? non extant)

Twelve would give whorls of 6.

—I have gone far enough to show my notion of the origin of the whorled orders, viz. that they have been formed directly from 2 or 4 or 6 or 8 or 10 or 12 &c. original vertical ranks,—in short, from the even numbers of vertical ranks, while the alternate orders have been formed from the odd numbers and from the alter⁠⟨⁠nate⁠⟩⁠ form of the 2-ranked ⁠⟨⁠    ⁠⟩⁠ (The whorled forms ap⁠⟨⁠pear⁠⟩⁠ ⁠⟨⁠to have⁠⟩⁠ a habit of primary variati⁠⟨⁠  ⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠ the whorled type: thus: maple ⁠⟨⁠    ⁠⟩⁠ horse chestnut, and many other⁠⟨⁠s⁠⟩⁠, in w⁠⟨⁠hich⁠⟩⁠ the normal order is crucial, are fo⁠⟨⁠  ⁠⟩⁠ of varying—not to a 3-ranked or 5-ra⁠⟨⁠nked⁠⟩⁠ alternate order, but to a 6-ranked whorl⁠⟨⁠ed⁠⟩⁠ order with whorls of 3. So a sedum in whorls of 3 often varies to whorls of 4.)

This theory which I now put forward is consistent with all that I know of leaf-orders, but I must confess it abounds in ambiguities, and after my Kidwelly surprise I shall hardly be surprised if I find still more anomalous forms.— My surprise is, that I had nowhere seen mention of this 7-ranked sedum before.

(If this theory falls to the ground, I shall be driven to believe that a bud-axis can and will develope a lateral shoot and leaf at any point where there is room for it: but that would be an extravagance akin to despair.)

You will perceive ⁠⟨⁠    ⁠⟩⁠ views are, in the main, ⁠⟨⁠    ⁠⟩⁠ my former theory, embra⁠⟨⁠  ⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠ which the former theory failed ⁠⟨⁠    ⁠⟩⁠ They also involve a limitation of the ra⁠⟨⁠nge⁠⟩⁠ of application of the former theory, rescuin⁠⟨⁠g⁠⟩⁠ from a forced allegiance a number of ci⁠⟨⁠  ⁠⟩⁠ which were only uncomfortable there. But the former theory is not hereby superseded the evidence in favour of it still stands as valuable as ever: only its domain is rightly restricted by the establishment of a new province on its borders. The principles brought out in the former theory, namely that the advantage of leaf-order belongs to the bud (and similar close-packed forms) and not to the adult leaf,—that the essence of this advantage is ecomony of space,—that economy of space in complex orders has probably been gained by condensation from simpler orders,—these stand firmer than ever. The only new principle required in the new theory is the possibility of different numbers of vertical leaf-ranks originating by a leap of prim⁠⟨⁠ary⁠⟩⁠ variation,—the possibility of nature acting per⁠⟨⁠  ⁠⟩⁠

If you can spare a few ⁠⟨⁠    ⁠⟩⁠ very much encouraged by ⁠⟨⁠    ⁠⟩⁠ to tell me what you thin⁠⟨⁠k⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠ of front; but do not ⁠⟨⁠    ⁠⟩⁠ annoyingly on your time an⁠⟨⁠d⁠⟩⁠ ⁠⟨⁠    ⁠⟩⁠

I enclose a copy of the extracts which Stokes sent me from the critici⁠⟨⁠sms⁠⟩⁠ of the Referees who condemned my pap⁠⟨⁠er⁠⟩⁠ in case you should care to read them.¹⁰ I have presumed to add a word or two of comment in the margin.

I am doubtful how to bring out my new notions, for I hesitate to offer them to the Royal Society to undergo a four months’ incubation resulting in such uninstructive criticism as this.¹¹

I was glad to hear a pretty good account of your health from George Darwin whom I chanced to meet at Cambridge the other day and from Leonard Darwin whom I found practising observations of the transit of Venus at the Observatory.¹²

Believe me, my dear Sir, | yours very sincerely Hubert Airy