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only imposing on the parties giving the orders; for a great many seasons must pass, and an infinite number of observations must be made, and registers must be kept, both of those observations and of the temperature of the atmosphere, daily, at various times of the day, before a compensation pendulum could be adjusted; and without adjustment, I do not hesitate to say that it would be worse than a common one. It may be here necessary to inform many of your readers, that all persons who make up chronometers have the conveniences of hot and cold closets to deposit their work in, for experiment, during the time they adjust the balances for change of temperature; and by these means, and keeping a register of the temperatures to which the machines are submitted, they can more accurately regulate the compensating powers in a few weeks, than could be done in many years by the regular annual and diurnal changes which take place. It has lately been much the practice, in the metropolis, for tradesmen whose only object is gain, to apply compensation balances, that is, balances which are made like compensation balances,-to lever and duplex watches, without ever submitting the watch to the least test of change of temperature. Now, there is no person of science but would say, that a simple balance would be infinitely superior to a compound one, applied under such circumstances, which is only used with a view of getting a few pounds more out of the pocket of the customer.

The application of cycloidal checks to the pendulum would require as much attention to adjust as the former principle; and I am only quoting a generally received opinion, when I say that it never has been found practicable to make a pendulum move in such a perfect cycloid as to produce isochronism. I consider it, therefore, as not dealing fairly with customers, to persuade them that such perfection is attained, when it is well known that it is so impracticable, that no honourable and scientific tradesmen are in the

habit of making up work on such principles, even in small time-keepers where they have every means and convenience of attending, in their own shops, to the necessary adjustments and corrections. But to talk of applying such principles to a church clock, which is to be fixed even a very few miles from its maker, so as to prevent his observations daily, and many times in a day, is altogether an imposition on the persons who contract for the work. I am, Sir, Yours, &c. W. WYNN.

ON THE PRODUCTION OF MOTION IN BODIES, WITHOUT THE AID OF ANY EXTERNAL FULCRUM; AND THE PRACTICABILITY OF DISPENSING WITH PADDLEWHEELS IN STEAM-VESSELS.

Sir,-Your correspondent "J. O." (page 32), requests me to inform him whether, if a person who stands in one scale, with his weight counterbalanced by weights in the other, by drawing up a piston from a cylinder attached to the scale in which he stands, will cause the scale, on his side to descend: at least, I suppose that to be his meaning; but his expression is, "Will the weight of the atmosphere acting upon the piston cause the scale to descend ?" In answer, I should say, that the scale will descend, not from the pressure of the atmosphere upon the piston, but from a cause which will be presently explained.

According to the law, that action and reaction are qual and contrary, if a person raise a weight from the ground, his exertion tends, in an equal degree, to depress his body; and consequently tends, in an equal degree, to depress whatever be stands upon. It is also a law, that

a weight cannot be raised, unless by a greater force upwards" than the weight possesses downwards; for if one force overcomes another, that which overcomes the other must be the greater. Therefore, if a person desires to raise from the ground a weight of fourteen pounds, he cannot do it while he exerts only a foree of fourteen pounds; but as soon as he employs a greater force, the weight will rise.

Now, suppose a person standing in a scale in which a fourteen pound weight is also placed, and weights put into the other scale, until the man and the weight are exactly balanced. Let him now raise up the weight which we supposed placed in the scale with him, and his scale will instantly descend, because, as was before observed, while he exerted only a force of fourteen pounds, the weight would not rise; he was, therefore, obliged to employ a greater force, say a force of fifteen pounds, -to raise the weight employed; also a force of fifteen pounds, to depress the scale; and thus, though seeming to take a weight from the scale, he, in effect, adds to it, and the scale descends just as if an additional weight had been placed on his side. What applies to the weight applies equally to the piston. Suppose the pressure of the air upon the piston to be fourteen pounds. In order to raise it, be must employ a greater force, say of fifteen pounds, as before. Here, also, the reaction upon the scale is equal; and though be seems to take off pressure from the scale, he adds to it, and the scale will descend, as before, something being allowed for friction. The reason, then, of the descent of the scale is not the pressure of the air upon the piston, but the reaction of the greater force employed to raise it up.

It was noticed by one of your correspondents, a few weeks back, that if a person be sitting in a chair in a scale, with his weight balanced, he, upon rising, causes the scale to descend; a fact which may be explained in the same way. I have tried the experiment with a weight,

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but not with the piston; but I do not know what should make any difference between them, except the friction, for which something must be allowed.

Thus, it appears plain that mo. tion may be generated in a body, without the assistance of any external 'fulcrum; and if it can be generated in a sufficient quantity, in a convenient manner, there will be no occasion for the employment of paddles of any kind in the propulsion of steam-boats.

Since my communication upon this subject was inserted, an extract from an American Magazine has appeared in your Journal, by which it seems an American had invented and patented the same thing; and still more recently, a correspondent has said that he had put it in prac tice a year and a half before, though with what success was not mentioned.

Now I am upon the subject of propelling, I wish to add an observation upon an erroneous remark of "Chelmeriensis," in his communication respecting a paddle-wheel invented by an acquaintance of his; and which I only notice, because, as it appears to be the best plan which has yet been published, it will, most likely, be brought to the test of experiment, and it is very desirable to have correct notions of it before experiments are made. "Chelme riensis" remarks, that his paddlewheel will produce the greatest effect when immersed up to the axis. But it is obvious that if any paddles enter the water before they have acquired a motion backwards equal to the velocity with which the boat is moving forwards, they will drive the water before them. This would, I think, generally be the case with the paddles, from that part of the circumference on a level with the axis, to about 45 degrees downwards. This applies equally to those paddles on the other side, which are receding from the water; so that, unless the wheels were revolving with great velocity, no more than about one-fourth of the paddles could be used at once. This applies

the tube are placed between the bars of a fire-grate, the internal air will very soon be rarified, escaping at d, and its place supplied by cold air at the lower end of the tube c. In this way, without trouble, or making alterations about the fireplace, the temperature of rooms may be increased at pleasure. In using an apparatus of this kind, it should be so placed between the bars of the grate as not to acquire a red heat, and decompose the air in its passage through the tube. I am, Sir, Yours, &c.

HENRY D

Finsbury, March, 1829.

ANSWERS TO THE GEOMETRICAL

QUESTION (P. 159, VOL. X).

Question. It is required to divide, geometrically, a circle of a given area into two equal parts, by a straight line, without reference to its centre, or to a perpendicular from any given tangent, or in any way bisecting the circumference of the circle.

J. U.*

"G. S." having remarked (vol. x. p. 389), that the 31st Prop. of the 3d Book of Euclid would show how the above problem "is to be constructed, without infringing on any of the forbidden conditions," Mr. Utting (the proposer) has since written to us, to the following effect:- When I proposed this question, the 31st Prop. of the 3d Book of Euclid did not occur to me at the time. I now beg to say, that the solu

Ans. 2.-Let K be the circle to be divided. From any point within the circle, as F, draw a line to any part of the periphery, as F G and draw F H of the same length as F G, to where it meets the circumference. Then, by bisecting this angle, and continuing the line through the circle, it will be divided in two equal parts, as before.

Both these methods are, in reality, the same; only the one is outside the circumference, and the other inside. These are the methods I always use to find the centre of a circle, as there is less trouble in them than in the problems usually given for finding the centre.

[Answers have been also received from J. O. B.-Y. de L.-Mr. P. Devey-M. H.-and R. A.; but none of them so satisfactory as the above.-EDIT.]

By the Proposer, Mr. Utting. If a circle of 1-4th of the area of the given circle be made to rotate on the interior circumference of the

given circle, any point in the periphery of the smaller circle coinciding with any other point in the periphery of the given circle, will de scribe a straight line passing through the centre of the given circle, to the opposite extremity of the diameter. And if the smaller circle is made to rotate on the interior side of the opposite semicircle, the same line will be retraced back again alternately, ad infinitum.

PUZZLE RINGS.

Sir, I have lately seen, in a recent publication, a description of the puzzle rings,-a toy with which most of us were acquainted when young. It consists of a number of rings connected by wires; and an oblong piece of wire is made to pass through the whole of them. The puzzle lies in disengaging the rings from the wire; and every additional ring increases the difficulty. puzzle is of great antiquity; and is said to have been treated of by Cardan, the mathematician, in the beginning of the sixteenth century.

This

As there have been various opinions concerning the principle upon which it depends, I have taken the trouble to investigate it; and as the result may not be unacceptable to

may have occasionally amused them

An expert hand will take off nine rings in six minutes; which is made the basis of the above calculations. I have in my possession one with thirty-six rings, which was made for a gentleman who vainly hoped he could disengage them in a comparatively short time. He did, I believe, persevere till he had taken off half the number (for it remains with me in that state), after working at it, at intervals, for about six or eight weeks, when he was obliged to give it up. Any one unacquainted with the power of numbers could scarcely be made to believe, that if nine rings can be taken off in six minutes, four times that number would require nearly 3058 years and a half, working twelve hours in the day, without intermission, or that the average time required for taking off each ring would be about eighty-five years!

DEMONSTRATIONS OF THE PRO

POSITION OF "O. C. F." (VOL.
X. P. 447.)

Sir,-In No. 287, p. 447, it is required to demonstrate, that the difference of the squares of the tangent and sine of any arc is equal to the

product of the squares of the said tangent and sine. Your correspondent "O.C. F." must be aware that this cannot be the case, except the radius be equal to unity.

Put t tang., s sine, r-radius of the circle. Then by the 47th Prop. B. 1, of the Elements, 12 + p2

square of the secant of the given arc. Again, by similar triangles, (t2 + r2) : t2 :: 72: s2; by multiplication, our equation becomes 12 s2+ 2 st 2... by division and trans12 $2 position, t

22, because

12 82 ; but

1., I am, Sir,

Yours, &c.
JOSEPH HALL.

Brazen-nose-street Academy,
Manchester, March 1.

Another Demonstration of the same Proposition.

Sir,-Your correspondent " O. C. F." obviously mean his proposition to apply to cases where the radius is unity; that is, the radius of the tables of natural sines, tangents, secants, &c,

In that case, then, taking the obvious abbreviations of the words: If tan.-sin.2 tan. sin.", then (since