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rounded and the other hollowed, they fitted closely, making a continuity of cup and ball joints. In the following figure

b

are shown four pieces, joined by the cord a, b, c, d; the end e fits exactly in the cup c, and so of the rest. On the cup end of every fifth piece there was an iron ring d, f, with a thumbscrew f, to pinch the cord after drawing it as tight as possible.

The cloth or curtain* which was used with this machine is an excellent contrivance for keeping the soot within the fire-place. It is made by cutting a large circular hole in the centre of the cloth intended for the curtain; this is then filled up by seaming to the edge the mouth of a bag, in the bottom of which there is an aperture opening into a sleeve one or two feet long, so that it very much resembles a large funnel protruding from the cloth, when secured to the fire-place with spikes, as seen below. Lids to emp

brush is then attached and laid on the grate, and the cloth fastened to the mantle-piece. When the hand grasps the pole by taking hold of the cloth sleeve, the aperture is so perfectly closed as to prevent the ingress of soot; and when it is necessary to bring the hands and the machine in such a position as that both shall be directly over the grate, all that is necessary to be done is to push the projecting bag inwards, when all that can be wanted will be obtained without disturbing the curtain or making the least opening for the escape of sooty particles.

I am, Sir,
Yours respectfully,
Stepney, Nov. 1829.

HENRY D.

THE BRITISH ALMANAC AND COMPANION TO THE ALMANAC FOR 1830. -MR. LUBBOCK'S DEALINGS WITH THE TIDES.

(Third and concluding Notice.) "Hast any philosophy in thee, shepherd?" SHAKSPEARE.

II. Remarks on the disquisition on the tides in the "Companion to the Almanac."

Mr. Lubbock commences his disquisition with a history of what has been already done in this region of inquiry, of which we need only say, that it is quite as full and accurate as might be expected from a gentleman whose reading has left him with those notions on the subject, which we presented to the admiration of our readers in our last and preceding Numbers. He then indulges us with about five pages of mathematical verbiage-we had almost called it investigation-which we will venture to say, will make more than the mere "groundlings" stare. We can but re

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commend its perusal to those who have nothing to learn respecting the theory of the tides, or to such as desirous of pursuing the research, would like a Will o' the wisp for their guide.

We have already noticed the levity with which it pleaseth this prince of investigators to speak of the renowned Laplace. He commences his present paper with an allusion to a something in the "Mecanique Celeste," which he does not call a blunder, but which yet he assures us "is not a misprint." Laplace's "notation," he says, "is at variance with that used by himself." Be it so; perhaps it is so. But, let us ask, is it at variance with truth? Or does it vitiate his results? As we intend keeping Christmas at Gilead Hall along with Dr. Birkbeck and some other sworn enemies of quackery and humbug, and write this while on our way thither, we regret we cannot refer at present to the "Mecanique Celeste" itself; for we have no doubt that even we should find the matter of difficulty lies very near the surface. This nibbling at the most accomplished annalyst the world has yet produced-except perhaps Lagrange-is very paltry. It never could have passed the scrutiny of a Superintending Committee who did their duty, without being suppressed.

But though Mr. Lubbock does not seriously question the substantial efficiency of Laplace's manner of notation, he makes bold to depart from it entirely; yet, as if only half assured of treading on safe ground, in less than two pages he makes a second slip in his notation, and in less than half a page more a third. He compares or rather seems to compare his results with those of Laplace, Bernoulli, and Euler; yet with all this changing, and shifting, and comparing, he cannot bring any one of his modifications to accord with the results of either of those distinguished philosophers. Really there was no more reason for thus altering the notation, than there would have been for changing his wig every five minutes during the process; and if it were not that Mr. Lubbock is an F. R. S. and all that, we should suspect, as a friend suggests, that he extracted his formulæ from some old college portfolio, and was afraid of modifying the notation to agree with Laplace's, lest he should fall into some such blunder as the musical amateur did who on transposing a tune in the major mode from the key of G to that of F, put E sharp instead of B flat.

We do not profess to be very deep in

such matters, otherwise we should seriously call in question the means by which our notable investigator simplifies his results. Thus, at the bottom of p. 53, he simplifies an expression for finding the time of high-water by saying,

66

if we suppose the luminaries to move in the equator, we have," &c. Very true,

Mr. Lubbock, you may "call spirits from the vasty deep." But will they come at your bidding? Will the luminaries move in the equator, or be there simultaneously, as you suppose? Of the three orders of phenomena in the tides, occurring respectively, twice a day, twice a month, and twice a year, the last which exhibits an augmentation of tidal flow near the equinoxes has its maximum when the moon is new and in her perigee at an equinox. Mr. Lubbock must not approximate perpetually to this state of things by an hypothesis, nor must he, however much it may serve to simplify the theorem, assume it as a fact that the sun and moon are always in the equator. With regard to a mean state of things the hypothesis might, as we hinted last week, be adopted safely in latitudes at or near 45°, as was done by Laplace. But in other la titudes the height of the tide and (though in a less degree) the time of highwater varies so with the declination, that it cannot be supposed nothing without involving great inaccuracy.

It would be easy to offer other very solid objections to Mr. Lubbock's theoretic researches; but as he applies them scarcely at all to his subsequent practical inquiries, and not at all to the table at p. 5, "British Almanac," which we should à priori have thought it would have been his main object to verity (and not to nullify, as he does in fact), we proceed to examine his practical deduc

tions,

Mr. Lubbock's object has here been to form and classify the times of highwater at the London Docks when the moon was one day old, two days old, &c., and then to deduce from them the times of high-water corresponding to every hour of the moon's southing. Now this method is from the beginning to the end objectionable. Let us attend, for example, to the results which he tabulates at pp. 57, 58. He there endeavours to infer a mean from the observations of twenty-four years, wich he says (p. 58) includes a complete revolution of the moon's nodes."" To be sure it does, and a great deal more than a complete revolution. But had Mr. Lubbock computed his means by his own method for even the mere period of the

revolution of the nodes, he would have found them to differ sometimes three or four minutes, sometimes six or seven, from those he has presented. Besides, how fallacious must be a system of means deduced from simply taking the difference between the times of the moon's southing and of the succeeding high-water! The differences, for example, on the same day of the moon's age in 1802, 1804, and 1806, were 55, 45, and 35 minutes! A truly scientific observer knowing that these discrepancies are affected by the concurrence or non-concurrence of several causes, as varying lunar and solar distances,* &c., would aim at their separation by some of the admirable methods which his art supplies. The sidereal revolution of the moon's perigee, the varying inclination of the lunar orbit, the varying obliquity of the ecliptic-all of which are periodical variations, each having a dif ferent period from the revolution of the moon's nodes-all affect the differences in question: hence Mr. Lubbock's results become vitiated throughout, and no saving clause about Bernoulli's "beautiful theorem" (p. 64) will avail Mr. Lubbock or his Superintending Colleagues; for by his own showing, and indeed confession, (p. 58) he has neglected just one-half the requisite obserrations.

Whether all this neglect has arisen from the influence of that power which Hudibras characterises as reigning over certain brains "in high spring-tides at midnight," or whatever be the causehere is an acknowledged fact, which renders what was blundering before supremely ludicrous. There is not a pilot from the Downs to the Humber who does not know, that at the Nore and in the River the inferior or secondary tide is greater than the superior, and flows in a different time when the moon has north declination, and vice versa. Now can the anomalies thus occasioned, and others of which we have not room to speak, be fairly or even feasibly reduced to a mean when half the observations are neglected? Let Mr. Lubbock, who through the kindness of that truly public-spirited individual, Mr. Isaac Solly, (Chairman of the London Dock Company) has so much information at bis command, turn to the plates which illustrate the tides for one month only in Mr. Palmer's "Report on a New Entrance to the London Docks," and he will have such evidence of the dif

The sun's distance changes in the course of half a year by its thirtieth part.

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ference of two consecutive tides as should we think raise some doubts even in his mind of the correctness of his own process.

But that these things render so excessively palpable the system of blundering which pervades the whole of Mr. Lubbock's inquiry, we know not that we should have touched upon them; for the acknowledged facts to which we briefly adverted in our first notice (No. 331), fully prove that all this gentleman's learned labour is, independently of other considerations, totally useless, because founded on observations grossly inaccurate. It is admitted that the tides are not observed at the London Docks within ten minutes, and that the register may be wrong by five minutes (p. 61). "At St. Katharine's Docks the observations are made with about equal precision." And at the East India Docks the matter is no better (p. 62). Mr. Lubbock seems to place great faith in a pretty little apparatus

with a vernier," (p. 62)—but he may rest assured that even that will fail him, for a stray wave or two, three inches higher than the rest, thrown across the Thames by a gust of wind, would inevitably spoil all. We have a plan in our minds which we are persuaded would be infallible-and it might please Mr. Lubbock because it depends on a curve and a computation-but we shall take example by the calculators of tidetables, and for the present at least keep it a secret.

But we fancy we hear some dogged admirer of the "Diffusion Society" thus exclaiming, Not so fast, Mr. Editor; you think you have made sweeping work of Mr. Lubbock and his computations: but you have overlooked the sovereign test by which this investigator checks and restrains all those irregularities to which you have alluded, the clock whence the time is taken,' (p. 62) the clock of Rotherhithe Church!!

Shades of Newton, Bernoulli, Euler, and Laplace, descend and do homage to the superior sagacity of J. W. Lubbock, Esq. F. R. and L. S.! The time as shown every day for 24 years by Rotherhithe Church clock-a clock which the observer has no other control over, than what he derives from gazing at it across the Thames-a clock which may have been as often going as not going, as often going wrong, as going rightmade an element in computing the tidecolumns of "The British Almanac !!" Woe unto those luckless mortals, the calculators of high-water for " White's

Let CPO be a cycloid; ASB, aPb, any two positions of the generator; Bb the corresponding points of contact of the generator and base; and O, P, the points of description. Join OB, Pb, and produce them indefinitely towards Q, and from B parallel to Pb draw BS: join PS. Now the angles P6C, SBb, are measured by half the arcs Pb, SB: but because SB, Pb, are parallel, and PbC=▲ SBb ... the arcs Pb, SB, are equal; and since the two circles are equal, the chords are also equal.. PS Bb: because the angles BbQ, bBQ, are less than two right angles, by the SBO..Pb and OB produced will meet on the side towards Q (Euc. 12 Ax.); and let Q be their intersection. Now in the triangles BSO, BbQ, the angles at B and Q are equal; also 4 B b Q (=26BS)=4BOS (E. 3. 32.): therefore the triangles are similar, and BO :SO=bQ::Bb; but Bb (by the mode of generation) the are SO..alternately BO: bQ chord SO:: arc SO. Let PO be diminished without limit, and we have ultimately bQ=BQ, and the chord SO the arc SO... BO:BQ, in the ratio of equality, or QO=2ÓB.

=

313

Hence, as before, the radius at the cusp 0, or is infinitely small.

Although by the methods before given the same result is obtained, yet in order to satisfy, as far as I can, the inquiries of your able correspondent, Mr. J., as he had originally limited his doubts to the cycloid, I deemed it better to give a solution for that particular case, which would admit of the different numerical values of BQ for any intersections being known; and such is the above, by the help of the common tables of sines and circular arcs for BOBS (Pb) = bQ :: BQ. Hence as bQ is already known, and BO, Pb given, BQ may be determined.

From his communication, Mr. J. seems inclined to think that the radius of curvature at the cusp is infinitely great, rather than of any finite dimensions: he says, "the nearer the ellipse approaches the right line, the less the radii of curvature at the ends become, &c. ;" and from thence infers, or at least adduces it as a confirmation of his opinion, that such radii at the cusp of cycloidal figures will be infinitely great. Now, in order that this argument may have its due weight, it should, I think, have been shown that the cycloid becomes a straight line when the radii at the ends vanish, which certainly is not the

case.

If what is mentioned in the second paragraph of his letter had held good when extended to the case of nodated lines, it would have been a considerable argument in his favour; but that, I submit, is not the fact. Let the whole series of the cycloidal curves be drawn, then the nodated lines will be entirely exterior to the cuspidated, and, I should imagine, have a larger radius of curvature; but, at all events, if the radius at the cusp be infinitely great, that part of the node next to it cannot differ much from it, if we assume the two curves very near each other. A nodated line would therefore present the following variation of curvature at the extremity of the node, almost infinitely small; at that part of the node which abuts laterally on, or nearly parallel with, the cusp, almost infinitely great; and at the vertex, finite. But as there evidently is an uniform and extremely graceful