three. And, moreover, the angular divergence of any leaf from the next in succession will be found in a similar manner to he that fractional part of 360°. Similarly, just as all angular divergences of the leaves of the primary series lie between 120° and 180° inclusively, all those of the leaves of the secondary series lie between 90° and 120°; the limiting point being at an angular distance from the first leaf of 99° 30′ 6"+. Lastly, it must be observed that the fractions of the secondary series are the successive convergents of the continued fraction:

1

3+1
1+ 1
I+ &c.

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14. In a manner analogous to the above, we might construct a tertiary series, commencing with the fractions,, and which would then appear as follows :—1, 1, 3, 74, 75, 7, &c. Such a series, however, does not exist in nature, as far as I am aware. Having, then, before us three analogous series, it is obvious that we might construct any number of such series, and finally all would be represented by the algebraical forms, where a is any number:

These fractions being the successive convergents of the continued fraction

1

a+1

1+1

1+ &c.

15. In all the preceding investigations, I have supposed the space between any two successive leaves on the stem to have been sufficiently developed to enable me to trace an imaginary spiral line through the leaves. But it sometimes happens that such spaces, called internodes, are so short or are practically wanting, that the leaves become crowded together, so that it is quite impossible to say which is the second leaf after having fixed upon some one as the first. This is especially apparent in the case of fir-cones, where the scales may be considered as the representatives of leaves, and which, though crowded, are arranged in a strictly mathematical order.

16. If a cone of the Norway spruce fir be held vertically, the scales upon it will be observed to run in a series of parallel spirals, both to the left hand and to the right. This is a result of their being crowded together, as well as of their definite arrange

ment. It is the object of the observer to detect and represent that order by some arithmetical symbol. This may be done by attending closely to the fol

lowing directions:-Obs. 1. Fix upon any scale as No. 1, and mark the scale which lies in as nearly a vertical line over it as possible, viz., numbered at 22. Obs. 2. Note the scales which are below, nearest to, and overlap that scale (No. 22). Obs. 3. Run the eye along the two most elevated spirals, one to the right hand, the other to the left; and passing through these scales which overlap the scale numbered 22.* Obs. 4. Count the number of spirals (called secondary) which run round the cone parallel to these two spirals just observed; there will be found to be eight such parallel spirals to the right, and thirteen to the left, inclusive respectively of the two first noticed.

17. From these observations, a rule has been deduced for obtaining the fraction which represents the angular divergence of

Fig. 2.

the so-called "generating" spiral which takes in every scale on the cone, in a manner similarly to those described above. Rule: The sum of the two numbers of parallel secondary spirals, viz. 13+8, or 21, forms the denominator, and the lowest, 8, supplies the numerator; so that represents the angular divergence of the generating spiral. From this it is obvious that the scale immediately over No. 1 will be the 22nd, and this must commence a new cycle.

*These spirals are shaded in the figure so as to render them more conspicuous; viz., the spiral 1, 9, 17, 25, &c., to the right; and 1, 14, 27, 40, &c., to the left. I have said the most elevated spirals, because, had I chosen the spiral passing through the scales 1, 19, 37, 55, &c., or 1, 6, 11, 16, &c., the object of search would not have been obtained.

18. If the object of our search be only the discovery of this representative fraction, or the angular divergence of the generating spiral, then all that is required will have been done; but in order to prove the truth of the rule given above, we must proceed to affix numbers to every scale, and so put it to a rigid test. We have, then, to show that the first cycle of the spiral line passes through twenty-one scales before arriving at No. 22, which stands immediately over No. 1. Secondly, the cycle must coil eight times, or complete eight entire circumferences in so doing.

19. Method of Numbering the Scales.-Assuming there to have been 8 parallel secondary spirals to the right, and 13 to the left, as in fig. 2, the process of affixing a proper number to each scale on the cone is as follows:-Commencing with No. 1, affix the numbers 1, 9, 17, 25, 33, 41, 89, 97, 105, &c., on the scales of the secondary spiral passing through it to the right; these numbers being in arithmetical progression, the common difference being 8, or the number of such parallel spirals; thus all the scales on one of the secondary (as shaded) spirals will have numbers allotted to them. In a similar manner, affix the numbers 1, 14, 27, 40, 53, &c., on the successive scales of the secondary spiral to the left, using the common difference 13. Thus we shall have two secondary spirals intersecting at No. 1, and again at No. 105, with every scale properly numbered. From these two spirals all other scales can have proper numbers affixed to them. Thus, add 8 to the number of any scale, and affix the sum to the adjacent scale, on the right hand of it. Similarly, add 13 to the number of any scale, and affix the sum to the adjacent scale, on the left hand of it; e.g., if 8 be added to 40, 48 will be the number of the scale to the right of it, so that 40 and 48 are consecutive scales of a secondary spiral parallel to that passing through the scale 1, 9, 17, &c.; or if 13 be added to 25, 38 will be the number of the adjacent scale; i.e., on the spiral parallel to that passing through 1, 14, 27, &c. By this process, it will be easily seen that every scale on the cone can have a number assigned to it. When this has been done, if the cone be held vertically and caused to revolve, the observer can note the positions of each scale in order (1, 2, 3, 4, &c.) ; and he will then find that the cone will have revolved eight times before the eye will rest upon the 22nd scale, and which lies immediately over the first.

20. This experiment, then, proves the rule for the artificial method of discovering the fraction, which represents the angular divergence of the "generating" spiral.

21. We may also remember that there must be 21 vertical rows of leaves. These may generally be seen without much

difficulty by holding the cone horizontally, and looking parallel with its axis, when the twenty-one rows of vertical scales will be observed, somewhat in appearance like the rows of grains in a head of Indian corn.

22. I have said that the 22nd scale will be found immediately above, but not accurately in the same vertical line, with the one selected as No. 1. That it cannot be precisely so is obvious from the fact that of 360°, or 137° 31'+, is not an aliquot part of a circumference; the consequence is, that the 22nd leaf must stand a little out of the vertical line, and of course the 43rd will be double that distance, and the 64th treble the amount, and so on. Hence it results that this supposed vertical line is in reality a highly-elevated spiral line, and instead of there being 21 actually vertical rows of scales, there will be 21 very elevated spirals (see fig. 2).

23. That the rows of leaves on any stem may be strictly vertical, the arrangement must be represented by some fraction the denominator of which measures 360°, such as, 1, 1, and; whereas,, &c., represent spirals in which no two leaves are ever in the same vertical line exactly.

24. As a general rule, all leaf-arrangements on stems with well-developed internodes can be represented by some one of the fractions, }, %, and : whereas those with undeveloped internodes, as in the scales of cones, thistle-heads, &c., are represented by higher members of the series, such as 1, 1, 3, &c.

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25. I must now turn to the other condition under which leaves are arranged, namely opposite. When this is the case, each pair of leaves, as has been stated above, stands at right angles to the pairs above and below it. Some plants have, either normally or occasionally, three or more leaves on the same level. When this is the case, the leaves of each group stand over the intervals of the group below it; i.e., they alternate with the leaves of the groups both above and below it.

26. This kind of arrangement is best seen in the parts of flowers, all of which are homologous with, or partake of, the same essential nature as leaves, and which, when complete in number, are separable into four sets of organs, called the four floral whorls; viz., calyx of sepals, corolla of petals, stamens, and pistil of carpels. It appears to be an invariable law that the parts of each whorl should alternate with those of the whorls above and below them. Indeed, so impressed are botanists with the persistency of this law, that when the parts of any one of the floral whorls stand immediately in front of the parts of a preceding external whorl, they at once infer that an intermediate whorl has disappeared. This is conspicuously the case in all primroses and cowslips, and other members of

the family to which they belong; wherein it will be noticed that each stamen is affixed or adherent to the tube of the corolla, but immediately in front of a petal, and not between two petals. That this idea of the suppression of another whorl of stamens is not without foundation, it may be observed that in the flowers. of a little denizen of damp meadows, Samolus Valerandi, and akin to a primrose, has rudimentary stump-like organs which stand affixed to the corolla, and alternate with the petals; while the true stamens alternate with the former; and therefore, as in the Primrose, stand immediately in front of the petals. In the Primrose itself, no trace of any such suppressed whorl of stamens is ever apparent. In a large number of plants which are habitualiy-normally-without a corolla, the stamens, as would be expected, stand in front of, and not alternating with, the sepals.

27. Although the organs of flowers are usually grouped in distinct whorls, yet in many are they spirally arranged; and when this is the case, they can be represented by some fraction of the series given for alternate leaves.*

28. A point now to be particularly observed, is that these two arrangements, viz. the "spiral" and the "verticillate" (or "whorled," including the "opposite "), appear to be due to forces acting independently of each other; for it is rare to find whorls passing into spirals, and still rarer for spirals to pass into whorls,-if, indeed, it ever occurs.

29. The Jerusalem Artichoke, however, furnishes many illustrations of the former process, and in some instances of the latter, though no gradual transition from a spiral to 'verticillate' or opposite conditions ever occurred in the cases examined.

30. A description of a few examples will be sufficient to enable it to be understood how a passage from opposite or verticillate leaves into spiral arrangements can be effected. Ex. 1. The change from the opposite (decussate) leaves into the divergence. This occurred somewhat frequently as follows:-A pair of leaves slightly converge to one side, the angular distance between them being about 150°. The succeeding pair likewise converge, but have a somewhat less angle, one of the leaves in each case becoming slightly elevated by the development of an internode; so that the sixth leaf now appears over the first, or the lowest leaf of the first pair that converged to one side. It must be noted that the angles between the radii drawn to the position of the converging leaves do not accurately contain 144°,

A point worthy of note is, that the free portions of the corolla of a primrose overlap one another in just such a way as corresponds to the arrangement of spiral leaves; though, of course, they are actually verticillate.