(1

Phyllotaxy – Jun 9th 63

Begin with few things so remarkable as angles converging series 1/2, 1/3 by adding numerator & denominator could tell - other singular properties of these numbers, which as not being a mathematician I need not allude - Dr. Falconer's remark in Elephant paper led me to attend to subject.

(Variation in petals of flowers &c &c perhaps at end)

On [illeg] these angles they will be seen to be few [sketch]; when drawn as spire on flat surface ready to be rolled up - the first point which strikes one is extreme symmetry of position [sketch] so as to draw nutriment equally from all surrounding parts. No doubt this is of importance & in some cases partly to be alluded apparently of high importance,

 

(2

But any angle, & fraction on an angle will go in 360 or any [illeg] multiple of 360 will give perfect symmetry of position & this is clearly not the governing cause. Look at [sketch] & observe what singular angles some time stand out & it may well be asked what conceivable reason can be assigned, that we do not have such angles , as 2/7 1/7 &c &c or [illeg] 1/4 1/6 or 3/5 & so on almost ad infinitum. The causes can by a slight assumption be assigned, The assumption is that in all cases whatever of spiral arrangement, there are two leaves, (in [3 words illeg]) & only two leaves which form a parallel spire both with diverging at equal angles. This assumption is not great plum-twig -  cases of variation of opposite with alternate which is simplest

 

[3v]

arrangement, - If we grant this instead of taking single spire of Botanists everything follows –

I do not pretend to explain spiral arrangement (though perhaps naturally follows), nor do I pretend to explain what makes each species have its own angle this may depend on size of buds with respect to axis; but what I can explain is why we have the series 1/2 1/3 2/5 3/8 &c & not others, for these are the only angles which result from 2 parallel spires round same axis.

 

[A4]

I suppose on 2 parallel spirals, leaves must occur on the "true" angles alone, because with any other angles, a generative spire would not be formed; & I further suppose this generative spire is necessary as it implies or leads to symmetrical arrangement on rhomboidal quincunx.

Ascertain what are angles of leaves on the 2 real spires in each case - perhaps [sketch] draw from some various [fully] converging series –

We ought to know relative position of spire with various angles: perhaps this gives the key of converging series.

 

[A5]

Find out whether angle, which does not divide 360° & its multiples into equal parts can produce symmetry?

What do I mean by symmetry?