or × 360°. But as the spiral arrangement is continued up the stem and into the terminal bud, the leaves seem to "right themselves, as it were; so that the appearance of the spiral in the neighbourhood of the summit is more accurate than at the point of departure from the highest pair of opposite leaves. Ex. 2. Change from the tricussate arrangement into the divergence of the secondary series. A change from verticils of threes into the was frequent. It takes place in the following manner :-The first step is to cause the three leaves of the different whorls to separate slightly by a development of their internodes. Then, if any two consecutive whorls be examined, the order of succession of the six leaves (No. 1 being the lowest) is thus:

In which it will be noticed that the fourth leaf, instead of being over the interval between the first and second, is over that between the third and first, so that the angle between the first and second leaf, or between the second and third, is double that between the third and fourth. These latter, it will be remembered, are separated by a long internode. The same order obtains with the succeeding whorls; the nodes, however, are now much more widely separated, while a true spiral arrangement, with the same angular distance between all its leaves, is ultimately secured, and is henceforth continued uninterruptedly into the terminal bud, and represented by the fraction .

or

31. From very many observations on stems of the Jerusalem Artichoke, it appears that to resolve opposite and decussate leaves into spirals of the primary series and tricussate verticils into those of the secondary series is more easily accomplished than any other kinds of transition. To reverse the process, to bring back spirals into verticils, seems quite contrary to all nature's tendencies to change. Stems of the Jerusalem Arti. choke occasionally had their leaves arranged spirally below, and verticillate above; but then the change was abrupt. The spiral suddenly terminated, and the last leaf was succeeded by three in a whorl.

Conclusion. I have now endeavoured to give a brief and as clear account as I can of the main facts and principles of Phyllotaxis. But, if we venture to search for a cause of such definite and exact arrangements of leaves, it will probably be fruitless, for as yet no satisfactory explanation has ever been given. It is not enough to say that it is a wise arrangement that leaves should not all be over one another, so as to exclude

the light and air, and impede each other's functions; or that the alternate arrangement is an obviously wise method of securing a larger development of "blade" and conditions equally favourable to all. Nevertheless, it is fact that when leaves are crowded, or verticillate, they very often appear less capable of sustaining much development of surface.* But this might presumably have been attained without the strictly mathematical positions which alternate leaves have assigned to them.

We may, then, ask two questions, both of which are at present unanswerable. First, why does a leaf of any spiral amongst ordinary plants stand at an angular distance varying from 120° to 180° from the next to it on the stem? Secondly, why does it take up an accurate or definite position on the arc between those limits, and is not to be found anywhere along that arc?

All that can be said is, that such is the case in nature, and that when the angles between any two successive leaves of all the different generating spirals are measured, and represented as fractional parts of the circumference, they are found to bear such relations to one another when written down in succession, as obtain between the successive convergents of a continued fraction of the general form:-1

a + 1

1+ 1

1 + &c.

Whatever our speculations, as to the cause of Phyllotaxis, may be, the fact nevertheless remains, and, like the beautiful structure of the bee-cell, testifies to the truth that "God's ways are past finding out," though bearing witness the while by its general invariability to the prevalence of law, and by its exactness and functional value to the power and wisdom of the Law-Giver.

A discussion ensued, in which Mr. J. Reddie, Mr. A. V. Newton, the Rev. C. A. Row, Dr. J. A. Fraser, Mr. Hubert Airy, and the Chairman took part: the Rev. G. Henslow replied.

The Meeting was then adjourned.

*

e.g. The orders" Conifera, Galiacea, and in the genera Hippuris, Myriophyllum, and Callitriche.

ORDINARY MEETING, MARCH 4, 1872.

CHARLES BROOKE, Esq., F.R.S., VICE-PRESIDENT, in the Chair.

The Minutes of the last Meeting were read and confirmed; and the following elections were announced :

:

MEMBERS :-JOHN ELIOT HOWARD, Esq., F.L.S., Lordship Lane, Tottenham; Rev. G. S. Rowe, 20, George Road, Edinburgh.

ASSOCIATES:-Rev. W. BRODIE, M.A. (Trin. Coll. Camb.), the Vicarage, East Meon, Petersfield; Rev. C. A. BURY, B.A., Sandown, Isle of Wight.

Also the presentation to the Library of the following books :

Proceedings of the Royal Society. Part 131.
Proceedings of the Society of Biblical Archæology. Part 1.

From the Society.
Ditto.

The CHAIRMAN.-As the paper which is now about to be read specially refers to the published opinions of Sir John Lubbock, the Council invited him to attend this meeting; and I have received a note in which he thanks us for our invitation, but regrets that a prior engagement renders it impossible for him to be present. Therefore we shall not have the pleasure of hearing what he has to say in defence of his own views.

The following paper was then read by the Author :

PREHISTORIC MONOTHEISM, considered in relation to Man as an Aboriginal Savage.* By the REV. J. H. TITCOMB,

M.A.

NIR JOHN LUBBOCK, in his Origin of Civilization, lays down certain assertions respecting the religious characteristics of the races of man which are so clearly contrary to the experience and testimony of many trustworthy witnesses, that I shall devote this paper to a refutation of them.

The

* The proceedings of this Meeting are inserted here, as the paper read thereat takes up some points in Sir John Lubbock's theory which were not dealt with in a paper "On Civilization, Moral and Material." See

page 1.-ED.

object of that eminent writer was, as we all know, to establish the aboriginal degradation of the human family; to prove that its first stock was not only savage in manners, but without one spark of real religious knowledge, such knowledge being gradually acquired only after ages of successive development. In order to test this, he very properly makes the whole question turn upon the nature of a belief in Deity. Thus, on page 136, he says:

Hitherto it has been usual to classify religions according to the nature of the object worshipped: Fetichism, for instance, being the worship of inanimate objects; Sabæism that of the heavenly bodies. The true test, however, seems to me to be the estimate in which the Deity is held.

2. Let us adopt this test. Nothing can be fairer, as a matter either of philosophical or ethnological inquiry; nor can anything, in my judgment, be more thoroughly decisive of the controversy. According to the theory of Sir John Lubbock, the first stage in the religious thought of man is one of total ignorance, and unconsciousness concerning God. This he calls Atheism, p. 136. The next stage is that in which "man supposes he can force the Deity to comply with his desires.' This he calls Fetichism. The third stage is that in which "natural objects, trees, lakes, stones, animals, &c., are worshipped." This he calls Totemism. The fourth stage is that in which "the superior deities are far more powerful than man, and of a different nature, and whose places of abode are far off." This he calls Shamanism. The fifth stage is that in which the gods take the nature of man, but are still far more powerful, being regarded, however, as amenable to persuasion. These gods are a part of nature, but not creators, and are represented by idols. This he calls Anthropomorphism or Idolatry. The sixth stage is that in which the Deity is considered, "not merely a part of nature, but the author of nature; and, for the first time, becomes a really supernatural being." The last stage is that in which morality is associated with religion" (p. 137).

3. Now, according to this sort of graduated religiousness, the measure of which follows an ascending, not descending scale of growth, we have a right to demand some visible exposition of such line of growth wherever we prosecute historical or ethnological research. For example, we have a right to expect that no morality will be found associated with religious belief in any nation before the development of Totemism and Anthropomorphism, or Idolatry. Again, that no idea of one Supreme and Supernatural Deity will appear until after all the previous lower stages have been first traversed. In view, therefore, of

the fact that some of these higher and lower stages of religious belief often actually co-exist in the same age, and among the same people, we have to trace, as far as possible, which of them has had historical priority. Sir John Lubbock-assuming what he desires to prove, rather than proving it-slips into the following easy observation :—

Where man, either by natural progress or by the influence of a more advanced race, rises to the conception of a higher religion, he still retains his old beliefs, which linger on side by side with, and yet in utter opposition to, the higher creed. The new and more powerful spirit is an addition to the old Pantheon, and diminishes the importance of the older deities; gradually the worship of the latter sinks in the social scale, and becomes confined to the ignorant and the young.

4. Remarks like these glide easily from the pen of a ready writer; but you will observe that they consist entirely of assertion. The fact of a contemporaneous mingling together of higher and lower beliefs in certain countries justifies us in making no à priori conclusion as to which came first in point of order. According to the dogmatic statement of Sir John Lubbock, a transfer from inferior to superior faith has been universal. But is this opinion historical? Do the evidences furnished by ethnological research confirm this view? Taking the higher or lower estimates of belief in a Deity as the crucial test of this great question, what do facts proclaim concerning it?

5. Such are the inquiries which I propose to prosecute in the present paper.

6. One instance of a contemporary co-existence of higher and lower religious belief is to be seen in Madagascar, where the natives, though they were found in the 17th century worshipping their departed ancestors, and reverencing charms and idols, yet possessed the knowledge of a Supreme and Supernatural Deity, whose attributes directly connected religion with morality. Robert Drury, who was shipwrecked upon Madagascar in 1702, and remained there as a slave till 1717, and whose narrative is universally received as trustworthy, tells us that the name by which this Supreme Being is known signifies "the Lord above," between whom and mankind there are four mediators. Now this, according to Sir John Lubbock's theory, marks a high and later development of religious belief, which could only have been reached after a passage through the lower stages of savagery. We have a right, therefore, to expect some historical proof of this order of sequence; or evidence, at least, of some sort, beyond the bare assertion of such a statement. So far from this,