y'1, &c.; for y', y'', &c. may approach, by continued bisections, nearer the point G than by any assigned difference, how small is the geometrical limit

soever, but can never pass that point.

i

GF
EF

of 4, 4, &c. which are successive values of, while i is diminished by continual bisections; as n x-is the algebraical limit.

FG

EF

other, may be proved thus:-If they be not equal, let be the FG

greater, and let D be the difference between EF

-}

and n x11;

, or its equal, &c. is always greater than

1

cannot approach nearer to n x-1 than by more than

approaches nearer to na1 than by any assigned
1
x

D, but

2

difference; therefore

FG
EF

-1

is not greater than n x- ; and in the

same manner it may be proved not to be less; wherefore equal to n x". Q. E. D.

If & be put for E F, and ý for F G, then

4. Observations.-This reasoning seems to me to remove Lagrange's objection lately mentioned, and to do so by employing an incremental fraction, &c. equal to an approximating fraction

y'

i

x

, &c. of which approximating fraction the denominator is always i = = E F, the numerator continually approaches to FG, and the vanishing quantity is the difference between the numerator y', &c. and F G = j.

This approximating fraction, &c. is always greater than when the curve, as in fig. 3, is convex to the axis; if a curve concave to the axis be drawn through the point E in the figure, another approximating fraction less than will approach from below

the tangent, and is, in the strictest sense of the word, the limit

which the fractions approximating from above and from below the tangent can never pass.

Thus another objection of Lagrange is removed. This objection is, that what in the fluxional or differential calculus are called limits, are not so, in the strict ancient meaning of the word limit. Whatever be the magnitude of each of the quantities E F and F G = ✯ and j, the fraction is of the same value at the point E of the curve; the fluxions or differentials, therefore, are not zeroes, though Euler and others have endeavoured to show that they are zeroes only; but they are indeterminate quantities with a determinate ratio, They are, therefore, common algebraic quantities; and may be sometimes zeroes, sometimes infinite.

x

I have taken, as a function of x, y=x". But I might have taken another function of x. Perhaps it would have been better, because more general, if I had taken y = ƒ x; y' = f (x + i) = ƒx + pi + q i2 + 7 is + &c., i being an indeterminate quantity; the reasoning, however, would, with certain conditions, have been the same, but probably less intelligible to a learner. The conditions are, that from x to x + i the curve turns its concavity to the same side, and that the ordinates continue to increase.

From what has been said, it is easy to teach the learner the formulas for tangents, quadratures, cubatures, complanations, and rectifications of lines, whether of single or of double curvature.

It is likewise so easy to teach him to calculate a few logarithms, by the series derived from the asymptotic spaces of the hyperbola, that I think the calculation and demonstration should always precede trigonometry. All this he may do without hearing the hyperbola even named.

From the 47th proposition of the first book, or from the 14th proposition of the second book, of Euclid, he may derive an algebraic equation to the circle; and from that equation he may, without a previous knowledge of the conic sections, obtain equations to the ellipse, hyperbola, and parabola; he may, indeed, obtain an equation to the parabola without any knowledge of an equation to the circle. Thus he would have various subjects to which he might apply the fluxional calculus.

Might not those authors who in algebra treat of variable quantities, subjoin the rate of variation, and thus make fluxions a part of common algebra ?

ARTICLE III.

On the Methods of detecting Carbureted Hydrogen Gas in CoalMines without setting it on Fire. By John B. Longmire.

It is not generally known that miners are in possession of means which, if properly used, will detect the presence of carbureted

hydrogen, or inflammable gas, in coal-mines, before it can do them any injury. For this purpose they make use of the smell and the sight simply, and the changes induced on the top of a burning

candle.

A tabular body of mist-like gaseous matter is sometimes observed under the roof of the mine, and every motion communicated to the air gives it a corresponding one. Miners have reasons, deduced from dear-bought experience, to suspect the inflammability of a body presenting this appearance. Such a body, however, does not always possess this property: for it may consist of a mixture of carbureted hydrogen gas and the vapour of water, which on entering the mine ascends to the roof, and remains there till it is displaced by a current of air, or accidentally set on fire by the miners; or it may be the tobacco-smoke of the hewers and the vapour of water which proceeds from the breath and bodies of the hewers, and the persons employed to remove the coals; and which, along with the tobacco-smoke, is dispersed through the air by the people at work, but which during their absence ascends to the roof, and collects into a stratum of vapour, so similar in appearance to the mixture of gas and vapour, as not to be distinguished from it but by the most experienced eye. In this view of the subject, this method is fallible; and as the carbureted hydrogen gas is very often pure, or unmixed with any other gaseous matter, its body is not visible. Hence in either case a miner who trusts to his sight only will be very often deceived.

Å miner who uses the smell leaves his candle in a working at a sufficient distance from its forehead, and advances slowly towards it; he then traverses the working in many directions; and if he finds no smell indicative of the presence of the inflammable gas, he brings his candle to the forehead. But pure carbureted hydrogen gas is odourless, and of course when its presence can be known by the smell, it must be mixed with other kinds of gaseous matter. It is said sometimes to have a sulphureous odour. I have not, however, been able to detect it by this; but I have often done so by an empyreumatic odour, which probably proceeds from a subtile oily substance that sometimes leaves the coal along with the gas. As carbureted hydrogen gas in coal-mines is often pure, its existence in them cannot always be known by the smell. Neither the sight nor the smell, therefore, are to be implicitly relied on; but they are often very useful; for a man may be travelling through a mine hastily, or inattentive to the state of the circulating air, with his head in a mixture of inflammable and common air, and his hands and candle in the common air only, and be advised by the sight or smell of the presence of the inflammable air, before he accidentally elevates the candle and sets it on fire.

But the miner can place the greatest reliance on certain changes induced by the inflammable air on the top of a burning candle. The miner's term of the candle-top will not be understood without an explanation. It is that faintly luminous yellowish-brown body

which closely surrounds the flame, and which is largest above it, and at last passes into the smoke. This body is rather obscurely seen round the flame of a candle in the night, in a room not much illuminated by any other body; but it may be distinctly seen by using the following means:-Hold the candle in one hand at about a foot or 18 inches from the eye, interpose the other hand between the flame and the eye at about one inch from the candle, and let only a very small portion of the upper part of the flame be seen. By this arrangement the vivid light emanating from the flame will be intercepted, and the body in question observed for one-eighth to a quarter of an inch, and sometimes higher, above the flame. By changing the position of the hand, so that a small portion of the flame round either side may be seen, this body will have, from the top to the middle of the flame, a yellowish-brown colour, about one-sixteenth of an inch broad; below the middle, the same colour inclining a little to a purple; and downwards it gradually passes into an ultramarine blue, which is the colour of the flame at its bottom. At that place too, this body joins the flame. Such is that body which the miner calls the candle-top, as it appears when a candle is burning in pure atmospheric air; and as he uses only that part of it which is above the flame, his term of top is very appli

cable.

Now there are certain differences in the top of a candle between its burning in a mixture of atmospheric air and carbureted hydrogen gas, and in the air only. These differences are proportionate to the quantity of gas in the mixture, and point out to the miner when the quantity is such that a very little more would cause the instantaneous combustion of the whole body of gas. The alteration in the top consists of a change in colour, and of an increase in its dimensions. These changes become greater as the quantity of gas increases. The change of colour which is universally considered as a sure indication of the existence of carbureted hydrogen gas, and of course the most dangerous change, is from yellowish-brown to greenish-blue; especially when it is accompanied with a rapid assension of very small luminous bodies, generally called points, which appear to be propelled upwards through the flame and top with a very faint crackling noise. The precise dimensions of a dangerous top is not the same in all coal-pits, nor in every situation in the same pit, nor in every candle in the same situation: so that the top which might be considered very dangerous, in point of size, in one situation, is not so much so in another. Hence a great latitude is left for the miner's judgment and prudence; and if he be experienced in his candle symptoms, and well acquainted with local differences, he may, if necessary, venture much further in a mixture of airs, than a stranger to that particular situation with the same degree of judgment. I have seen the greenish-blue top one inch and a quarter long above the flame of the candle. Other miners say they have seen it longer; but others say the inflammable air has fired when it was shorter. With respect to this variation in

the size, one general rule may be adopted, which is, not knowingly to get past the limits of certain experience, unless the object to be obtained is much greater than the risk run. The size of the top appears to me to depend in a great degree on the heat of the mine. I have examined the top in parts of a mine where the air, was stagnant but pure, and I always found it greatest in the hottest part. In making these experiments I have often seen it three, sometimes four, and once five, inches above the flame, without any difference in the colour except in intensity, which increased with the size of the top. I once advanced towards the forehead of a working in which the air was still, with a candle having a deep yellowishbrown top, four inches long, till I observed longitudinal streaks of bluish-green were mixed with the yellowish-brown. I then began to retreat, being convinced that my situation was dangerous. I have often thought the moisture in the air of the mine affected the top; but I never met with proper situations to make so many experiments as would enable me to come to very accurate conclusions.

When a miner is about to use his candle he snuffs it, and then waits till the top of the wick is inflamed. As on such an occasion as this he generally carries his candle as near the ground as his arms will permit, he never raises his candle, but depresses his body till his head is nearly on a level with the candle, and probably he may kneel on one knee; he then, holding the candle in the manner before described, and having his eye steadily fixed on the top, rises slowly, till the candle is nearly as high as his face when he is standing upright. If he finds the top has a favourable appearance, he gently sinks his candle again, and advances a few yards; but if the top indicates the presence of the inflammable gas, he experiments more cautiously, and advances a less distance between every trial; if the quantity of gas appear to be increasing, he advances very slowly, with his eye constantly fixed on the top, and his body rather bent forward; and if the quantity be still increasing, and the appearance of the top rather alarming, he advances still more slowly, and pauses at every yard in length, to be more exact in his examination. And if the quantity of gas reach its maximum before he considers the top to be dangerous, he advances slowly to the end of his journey; but if it continue increasing, he will be forced to return.

Much steadiness is required in these trials; for sometimes the top suddenly assumes so alarming an appearance, that a person may be intimidated so much as to make a precipitate retreat; and in doing so sets the gas on fire, by bringing a quantity of it, or a sop, as the miner calls it, upon his candle, by his rapid movement. Before he tries, he should first consider what he ought to do in every probable exigency; so when the top suddenly appears dangerous, he will sink the candle slowly to the ground, and bend his body as much as he can at the same time, then turn about and retreat as slowly; avoiding, if possible, the direction that he advanced in during the dangerous part.