I do not here enter on the subject of the general expediency of the proposed restriction, but I think I may say that it has not arisen out of any expression of public opinion of its necessity; and that if the question were asked of the public at large, whether they would prefer the risk of establishing a monopoly, or of throwing a damp on the spirit of improve ment they would, by a large majority, decide on the former. The public know that no monopoly can long exist in England, bút they cannot foresee the effects of restricting the free course of enterprise, and paralyzing individual industry.

CIRCULATING DECIMALS.

Sir, It is a singular omission in most works on arithmetic, in Hutton's mathe matics, and several others, that no mention is made of continues fractions, al though their application to questions relating to decimals and their équivalent vulgar fractions is most important, a few instances of which I adduced in a communication you inserted from me in vol. xi, p: 227, which comprises some practi cal rules and examples on this interesting subject.

The second rale given, by G. C. L. (p. 68 present volume), for finding by continued fractions the equivalent to any given decimal, is, as an approximating rule, more generally useful than the one 1 gave. I have for some time used a similar method for finding two consecu tive indices in any given decimal series; as the difference between any two con secutive indices, and the complementary indices must be the same; a means is afforded for determining the denominator of the series.

It was, I think, wise on the part of "A Country Teacher" not to sigu his

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name to the communication inserted in No. 670, p. 175, as he must, on reflec tion, feel ashamed at the supercilious tone in which he has clothed his remarks. His must be an intellect of no common order, when he can take in his mind's eye what he considers neither myself nor G. C. L. can discover by calculation.

A very short time enabled me to find out two fractions, which, if not identical, very closely resemble those referred to. 8302 19503

The one is

1177

=425678119 +, and

the other, =425678100 +. The

2765

first is a pure circulate, and the other a mixed 'one. This 1 infer from some fanciful properties which I have observed, and I do not suppose "A Country Teacher" can do more, for I dare say he has never calculated the whole of either series, or would be willing to do so; nor do I think he could discover any of the latter figures of the series without first producing all that preceded them; either of which I can readily do by some of those fanciful properties he affects to despise.

His opinion respecting the neglect of your mathematical friends to notice my question is of little importance; it can not but be acknowledged, that what is published on the subject in our books of arithmetic, and even in the Encyclopedia”, is very imperfect. Most of them direct to perform the operations of addition and subtraction to make the decimals similar and conterminous. Now this is prac-. tically impossible, except in a very few cases. Some authors, seemingly aware of th's difficulty, direct that these decima's should be converted into their equi valent vulgar fractions; and in the case of a mixed circulate, tell the learner to subtract the terminate part from the last figures of the series, and to place a certain number of nines and cyphers equal to the number of figures in the series as a denominator, and then to reduce this fraction to its lowest term. The practical absurdity of this rule must be apparent to any one who reflects upon the immense number of figures in many of these series. A really practical rale I never saw published till ny communication and that of G. C. L. appeared in your pages. I must be considered to have a very

limited knowledge of the nature of frac. tions, did I not know that there is an infinite number of exactly the same value, and that it is most advantageous to exhibit them in their lowest terms. The

doctrine of continued fractions has also long taught me that there is an infinity of fractions nearly equal to any given one, and that in any ratio it was certainly an oversight of mine not to remak, that in the practical application of my rule it is requisite that a certain number of figures of the series should be given according to the magnitude of the denominator; but, in truth, I did not consider my communication to be ad. dressed to school-boys.

The most important charge against me appears to be my doubts of the existence of decimals" that go on to infinity, following no regular law." I can assure your correspondent I have no doubts on the subject; I positively deny their existence. If any such exist, they must have an infinite number for their denominator, else the variety of remainders, from which the successive terms of the series

is produced, cannot be infinite; for if the denominator be not infinite, the series must recur or else be finite.

Although neither myself nor "A Country Teacher may know the law that regulates a decimal series equivaJent to the square root of an incommensurable number, it would be an absurdity, and contrary to all our experience, to suppose that it follows no regular law.

am willing to pay due respect to the opinions of the learned, but must be allowed to hold opinions of my own. The time is passing away when the dictum of any man or set of men shall constitute a test for truth,

I shall feel obliged if "A Country Teacher," or any other of your correspondents, will explain the reason of the following:-If 7 be set down on the right hand and multiplication be made with 5, setting down the units of the products on the left of the 7 to form a multiplicand, the result will be the equi. valent of one-seventh. Setting down 3, and multiplying by 4, will, in a similar manner, form the reciprocal of onethirteenth.

Yours respectfully,
ANTHONY PEACOCK.

PRACTICAL SUGGESTIONS ON THE WORK

ING OF INCLINED RAILWAY PLANES
AND CANAL LOCKS.

Sir,-The following plans are communicated for insertion in the Mechanics'

Magazine, should you deem them deserv ing a place.

The first plan is designed to facilitate the transport of waggon-trains on railroads over ascents of considerable acclivity.

I assume the propelling or tractive power of a rotary-engine to be in the inverse proportion of the diameter of its wheels. This is obvious from the consideration that a double stroke of the piston produces one revolution of the propelling wheels, and causes the engine to ference of its wheels. For the applicamove over a space equal to the circum tion of this principle I propose to have attached to the wheels of the engine a second series of felloes and tires, say of half the diameter, and interior to same. To receive these there must be constructed a corresponding elevated rail on such sections of the road, as, from their acclivity, require a considérably increased tractive power. This, on the assumption of the above, I presume, incontrovertible principle, must double the tractive power of the engine while it moves on the interior reduced wheels.

My next project is designed to raise or lower canal-boats from one level to an other without any loss of water. The importance of obtaining this result: be sufficiently apparent to any one who will take the trouble of referring to a paper on this subject in your Magazine for April, 1833, p. 31.

I propose to have a case or chamber constructed of sheet iron, or other mate. rial impervious to air, of the same capacity and form as the lock or basin to which it applies. That it be so adjusted as to be capable of being lowered into the lock at pleasure, and to be kept in position by grooves cut in the sides of the Jock, with corresponding projections on the case or chamber, which prevent disturbance in sliding up and down. To the upper surface of this moveable chamber the apparatus of the air-pump is to be attached, on a scale proportioned to the capacity of the chamber.

To effect an ascent of the boat from the lower to the upper level. On the ad.

vance of the boat within the lock, the chamber is to be lowered and exhausted of air till the water thereunder (which is of the lower level) rises somewhat above that of the upper level. Then air must be admitted into the chamber to effect, or rather permit, its withdrawal, when the flood gates of the upper level may be opened and the boat withdrawn.

For effecting a descent from the upper to the lower level of the canal. The lock being filled with water as above, the process adopted in ordinary locks for lowering the water must be followed. I am, Sir,

Your obedient servant,

ROBERT CAREY, Rector of Donoughmore.

Clonmel, Ireland, June 20, 1836.

THE STEAM-CARRIAGES' BILL. A Bill is now in an advanced stage before the House of Commons, to which, if we were to judge of it from its title, we should feel bound to wish all possible success. It is intituled "A Bill to repeal such portions of all Acts as impose prohibitory Tolls on Steam-carriages, and to substitute other Tolls on an equitable footing with Horse-carriages." But on looking into the Bill itself, we find clauses there which have nothing whatever to do with tolls-which no one could have reasonably expected to find thereand which are of a most debateable, or rather, we should say, of a most decidedly partial and unfair description. We refer particularly to the clause "prescribing the dimensions of vessels for generating steam"!!!

If there were nothing improper in the provisions contained in these clauses, why attempt to smuggle them in this man ner through Parliament? Is smuggling usually taken as a proof of honesty of purpose? Why not have stated in the title of the Bill that it was meant to regulate the manner of constructing steam-carriages as well as the tolls they should pay? What good reason can possibly be given for so important an omission? Was it feared that if the title had been so framed, the numerous individuals who have embarked in the projection of steam-carriages for common roads might have taken alarm at the measure that was in progress, and have insisted on seeing that they were not unduly damaged by it? And if such

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fear was entertained- -as no doubt it must-was it right to deal thus in secret with the interests of others? Either just towards those who were to be individually affected by the Bill, or fair towards Par liament which was to be made the unconscious instrument of (possibly) much indi vidual injury and annoyance? And if, per adventure, the authors and promoters of this Bill should be some knot of persons concerned in forwarding one particular steam-carriage speculation to the preju dice of all others, and that a speculation which would be favoured by the restric tions stealthily introduced into the Bill, while every other would be damnified by them-can the whole affair be considered as any thing else than a rank job?

Maiters are not mended by the fact, that the preamble to the Bill is just as silent as the title, with respect to the more material (because most onerous) portion of its contents. Every preamble should give the sufficient reason for the enactments which follow; but this gives no reason whatever for the clause introduced respecting the construction of steam-carriage boilers.

Now as to the particular clause in ques tion: it is thereby enacted" that it shall not be lawful to use any vessel or ves sels for raising or generating steam, to propel any carriage along a public street or road, any part of the transverse sec. tion of which shall exceed ten inches in diameter in any direction, if circular or cylindrical; and if such vessel or vessels shall be made of any other figure than cylindrical, then no part of the trans. verse section or sides shall exceed eight inches in any direction, under a penalty for every breach of such regulation, not exceeding one hundred pounds, or less than twenty pounds."

We shall not stop to discuss the po licy of the State's interfering with the march of science by such restrictions aš these, but content ourselves for the present with asking how it has been as certained that ten inches diameter is the limit of safety in the case of circular or cylindrical vessels, and eight inches in the case of vessels of other figures ?when and where the experiments were made by which these points were sup posed to have been established? and by whom (this especially) they have been proved to the satisfaction of the Commmittee which sat upon the Bill? Some

What if it should turn out that the restrictions proposed are calculated to protect one particu'ar boiler to the exclusion and proscription of all other boilers? And what if the inventor of that particular boiler, and the friends of that inventor, should appear to be the authors and promoters of the Bill by which they are to be so exclusively and unfairly benefitted-pretending the while. to be actuated solely by a regard for the interests of the public? Suppose, for instance; the effect of these restrictions would be to drive Mr. Hancock's earriages off the road-the said Mr. Hancuck being the only person who has now. vehicles of this description running for the public convenience, and having done more than all the other steam-carriage projectors put together to bring to maturity the application of steam-power to common-road travelling and to give a monopoly of the speculation to some competing, but less successful and meritorious inventor-a Gurney, a Dance, or a Macneil-suppose such to be the case (a supposition, we suspect, not very wide of the reality)-and who (but the favoured parties themselves) would not feel anxious to see the Legislature spared the disgrace of such a monstrous act of injustice and oppression ?^

The Bill professes to apply only to steam-carriages on common roads, but the boiler-dimension-prescribing clause, as it is now worded, is equally applica ble to railway steam-carriages; though this, we presume, can hardly have been intended. The words are:- Any vessel or vessel for raising or generating steam to propel anv carriage along a public street or road." A railway is, of course, as much a public road as any other entitled to that denomination.

Another peculiarity there is in the clause not unworthy of observation. It merely prescribes the dimensions of the vessels employed for raising or generating steam, and says nothing about any vessels in which it may be received and

stored up for use. Is it imagined that danger only attaches to the former? Oc is there such a thing as a separation in the case (Mr. Gurney's, for instance,) which it is desired to exempt from the operation of the law?

The Bill is in too advanced a stage in the House of Commons (having been committed and reported upon, and standing for a third reading this evening, Friday, to allow us to hope that it can now be thrown out, or reconsidered there; but we trust that when it reaches the Upper House, means will be taken by those whom it immediately concerns to have its merits fairly canvassed. We have done our duty in drawing public attention to it, and should have done so sooner, but that owing to the deceptive. ness of its title, we had no suspicion of its mischievous tendency, and only a day or two ago stumbled by accident on a copy of the Bill.

HISTORICAL NOTE ON THE DISCOVERY OF THE NON-CONDUCTING POWER OF ICE. BY A. D. BACHE, PROF. OF NAT. PHILOS, AND CHEM., UXIV. PENN.—

(From the Franklin Journal.)

In the fourth series of his electrical researches, Mr. Faraday devotes himself to the establishment of a "new law of electric conduction." In the course of experiments for this purpose, he says that he "was suddenly stopped by finding that ice was a non-conductor of electricity, and that, as soon as a thin film of it was interposed in the circuit of a very powerful [voltaic] battery, the transmission of electricity was prevented." This observation is made to lead to a beautiful train of research on the conducting powers of various oxides, salts, chlorides, &c., capable of existing in both the solid and Iquid states. In these experiments, a galvanic battery of two troughs, containing twenty pairs of four-inch plates, was used.

Similar results were obtained with electricity from the machine. A thickness of live-sixteenths of an inch of ice scarcely allowed the electricity to pass at all, though of this high tension.

It seems, then, that Mr. Faraday thought it necessary to investigate this fact, which he had accidentally observed in relation to galvanic electricity, in its application to electricity as evolved from the machine,