If we conceive a slowly tapering cone (fig. 1.) to have its

A

Fig. 1.

B

E

C D

summit A at the centre of agitation of a system of spherical waves, and if we take the axis of the cone for the axis of x, it is clear that the displacements E,,, of the molecules within the frustum B C D E may be regarded as functions of x and t; and may therefore be expressed by the equations (2.), nearly. It is also manifest that the same equations will express the displacements for any other frustum of the medium, by making the arbitrary quantities to vary according to the position of the frustum. Consequently, if we suppose

= a sin (nt―k x+a)

for the frustum B C D E, the same equation may be taken to express the value of for any other frustum of the same cone, by regarding a, n, k, a as functions of x.

Let g be the radius of the sphere of influence of the mole

x

cules: then, if were infinitely small, the minute portion of e a wave contained within the sphere would be a plane wave, and a, n, k, a constant. Hence we perceive that these quantities must be functions of; and consequently, that we may write

x

the only variable quantity in these series being x.

Now when x is infinite a must be zero; therefore A = 0:

is, at all sensible distances from the centre of agi

tation, an extremely small quantity, we may reject its powers above the first; therefore a = B. The quantities n, k, a

x

approach, as x increases, towards the values which they have in the case of plane waves, which values are independent of a. And since the small portion of a wave contained within the sphere of influence of any molecule cannot, at any sensible distance from the centre of agitation, differ sensibly from the same portion of a plane wave, we may regard n, k, a as constant for all parts of the cone. If then we retain a to denote

Be

Bg, the constant part of

Ꮖ

we have

and, in general, for any cone taken as we have supposed, we have, from the equations (2.),

sin (n,t−k, x+a,)+ɛ. Bi̟ sin (n,t+k, x+b,), (4.)

x

When the waves all move from the centre of agitation, the second sums in the equations (4.) will vanish: and limiting our view to a single term of one of the first sums, we have an expression for the displacement virtually the same as that which Professor Airy, in his valuable tract on the Undulatory Theory of Optics, has partly assumed and partly borrowed from the theory of sound*.

It may be observed, by the way, that the method adopted in this paper of expressing the displacement of the molecules, is analogous to that employed so successfully in physical astronomy to express the differences between the mean and true places of the planets.

2

When the molecules are so arranged that the sums s2, s s, &c. are different for different directions of the coordinates, waves going and returning to and from a centre of agitation will not be spherical. The most simple case of such waves will probably furnish a subject for another paper. I am, Gentlemen, yours, &c.

JOHN TOVEY.

Evesham, April 15, 1836. last P.S. I perceive that throughout my paper tently called the differences Ax, ▲y, ▲z variations.

* Mathemat. Tracts, p. 271.

Third Series. Vol. 8. No. 49. June 1836.

I inadver

3 F

LXXXVII. On the former Extent of the Persian Gulf, and on the Non-identity of Babylon and Babel; in Reply to Mr. Carter. By C. T. BEKE, Esq., F.S.A.

To the Editors of the Philosophical Magazine and Journal.

GENTLEMEN,

THE opinion which was, in the first instance, advanced by

This

me in the Number of your Journal for February 1834*, was to the effect that the low lands of the Euphrates and Tigris have been formed by the gradual deposits of those rivers, and that this operation has been so extensive, that, at the time of the erection of the Babel of Genesis, it must have been physically impossible for that city to be built near the spot where the Babylon of Nebuchadnezzar afterwards stood. opinion may be considered as embracing two distinct and separate propositions:-the first is that, within the period of history, an advance of the land upon the sea has taken place of sufficient importance to affect materially the geography of the localities in question; the second is that, within the same period, that advance has been so great as (independently of all other arguments,) to warrant my conclusion with respect to the non-identity of Babylon and Babel. If the former of these propositions be untrue, à fortiori must the latter be so; but, on the other hand, even if the former be established, it does not follow that the latter is likewise correct.

From Mr. Carter's former arguments I certainly was led to consider, that he not merely disputed the correctness of the first proposition to its entire extent, but that he went yet further, and contended that the changes (if any) which have taken place, are altogether insignificant. In his present remarks he says, however t, "I much object to such expressions in the reply as, 'Mr. Carter has, in fact, asserted the opinion that, since the time of Nearchus, the encroachments on the gulf must be very unimportant,' omitting the words 'to the point in question, any later encroachments,' &c., as conveying the idea of a mere assertion without proof, and a much broader one than my remarks warrant." I am most anxious that no difference should exist between us on the score of mere misconception of each other's meaning, and I therefore give at length, in the note at foot, an extract of the whole passage from which I made my citation; and I put it to the candour

* Lond. and Edinb. Phil. Mag., vol. iv. P. 108-111.

↑ Supra, vol. vii. p. 195.

"Following the course of Nearchus, as given in his own clear account of the voyage preserved by Arrian, from his arrival at the Arosis, the river

of my opponent himself, whether I was not fairly authorized in the conclusion which I came to with respect to his meaning: indeed I would ask whether, when in his present reply he says, with respect to "the navigation of Alexander and his fleet in the delta streams," that "the ancient canal, the entire circuit, all the points of the navigation then presented by the spot, are still offered for our observation," it must not be understood as his unqualified opinion that "since the time of Nearchus the encroachments on the gulf have been very unimportant." If I am so unfortunate as still to misunderstand his meaning, I beg to assure him that I do so most unintentionally.

to say,

As regards the observation that my words "convey the idea of a mere assertion without proof," Mr. Carter must allow me that a construction appears to be put upon them which ought not by any means to be adopted in a discussion like the present. Every proposition advanced, or assertion made, on either side, must be presumed to be made upon what are regarded as "proofs;" and it is simply from the considered insufficiency of those alleged proofs that the correctness of any such proposition or assertion is questioned on the other side. For my part, I feel that I might have reason to object, not merely to some expressions, but also to the tone generally in which Mr. Carter's last reply is written; but I refrain from doing so, and I sincerely trust that neither of us will have occasion again to refer to any such unpleasant topic.

In order to prevent any future misconception, it is to be understood that the first and principal point in dispute between us is, whether a change of such importance has taken place as materially to affect the geography of the localities in at the N.E. next before coming to the streams of the Delta, in his progress to Kataderbis and the island of Margastana, in his passage through the channel over the shoals to his arrival at Diridotis (by the Khore Abdallah), on the S.W. side of the Delta, and comparing it with the present state of the country, we learn with surprise the small degree of change which the general characters of the coast have undergone during the lapse of so many ages. Dr. Vincent, in his able work on the Commerce and Navigation of the Ancients in the Indian Ocean, adverting to this remarkable fact, observes, that Capt. Howe's chart explains the journal of Nearchus as perfectly as if it had been composed by a person on board of his fleet,' (vol. i. p. 423.) and (p. 466.) the pilot on board Nearchus's ship steered exactly the same course' (along the coast of the Delta) 'as MacCluer's Karack pilot 2000 years afterwards.' The junction of the river called by Arrian the Eulæus (coming from the N. or N.E.) with the Tigris by the still existing ancient Hoffar canal, across which Alexander sent a part of his fleet while he sailed down the Eulæus to the mouths of the Tigris, and so round to meet it (Arrian, Exp. Alex. vii. 7.) further shows that to the point in question any later encroachments on the gulf must be very unimportant." Lond. and Edinb. Phil. Mag., vol. v. p. 247-8. The Italics are Mr. Carter's.

question; that is to say, a change so great as to render the descriptions of ancient writers inapplicable to the actual coastline and state of the neighbourhood generally."

Seeing that my hypothesis precludes the possibility of Nearchus's voyage being made applicable to the present coast of Susiana and the countries at the head of the Persian Gulf, it is scarcely necessary for me expressly to dispute in detail the correctness of the identifications, considered to have been established by Dr. Vincent, of the river Arosis, of Kataderbis, the island of Margastana, Diridotis, &c. &c.* That the river Karoon is not the Eulæus, nor Shuster the representative of Susa, has already been asserted by many geographers of eminence, whose voices are united in favour of Shus and the river Haweeza or Kerrah. Without intending to range myself with these geographers, I believe I am correct in saying that, as between them and Dr. Vincent, the greater show of reason is

Although I am quite willing to concede that "a few miles of addition to the Delta is not the question" between us, yet, as regards the learned Dean's identifications, I must remark, that a few miles-nay, a very few miles indeed-of addition would (I much suspect,) render it impossible that "Capt. Howe's chart should explain the journal of Nearchus as perfectly as if it had been composed by a person on board of his fleet." To establish the correctness of this position, it appears to be necessary, not merely that the coast should have remained unvaried since the time of the Greek navigator, but that Capt. Howe's chart should accurately represent that coast: it ought, consequently, to correspond in all points with the trigonometrical survey recently made by Lieuts. Brucks and Haines, of the East India Company's Marine Service.

By the kindness of Capt. Horsburgh I have been furnished with copies of the Company's chart, as also of that of Lieut. MacCluer (by Dalrymple, 1786 and 1788): Capt. Howe's he was not in possession of. Owing to the longitude not being marked in MacCluer's, I am prevented from making an exact comparison of these two charts; still differences of sufficient moment are to be detected between them. For instance, the island of Karack is represented by MacCluer as being 8, and Korgo more than 4 geographical miles long, whereas they are actually just half those lengths respectively: Buna (Derabuna), by the Core Moosah, is made as much as 9 miles long, from north to south, and 3 miles broad, whilst it is only 3 miles long, and less than 1 mile broad, its length being from east to west: Derah, adjoining this last island, is made 7 miles long and 3 miles broad, but it is in fact only a mile and a half each way: the Core Abdallah, represented in the copy of 1786 as being 10 miles broad, with 8 miles of coast between it and the mouth of the Bussorah river, and in that of 1788 as only 6 miles broad, with about 10 miles of coast, is actually 12 miles broad, and the two mouths meet at a point, without any coast intervening. These variations (which are only a portion of what might be pointed out,) may be said to be but trifles with respect to "the general characters of the coast;" still they are more than sufficient to show that MacCluer's chart would have been rather a dangerous guide for Nearchus to have placed implicit confidence in. Capt. Howe's chart, which was adopted by Dr. Vincent, is (I believe) not even so correct as that of MacCluer; but I have not at present the means of referring to the Dean's work, so as to ascertain this positively.