between B and D(= Hm). When the arch ant of the forces represented by (V, cot. i +Hm) in order to determine the stability 5. In single arches it is necessary to of the abutments. Take moments with know the point of application of the result-reference to the axis of abscissas MY Then if yordinate of point in question, | In this we neglect the spandrel forces above and ym and y, be the ordinates of D and D"" so far as they affect the stability of the B, we have abutment. This can be done with safety. The line of pressures and depth of keystone are determined as heretofore. Example 1. Let the assumed form of the soffit be a semi-circle, and let the loading consist of the arch and backing of homogeneous masonry carried up to a horizontal "extrados" MY (Fig. 51). FIG. 52. wards, instead of a pull inwards, we flatten | D' parallel to the tangent to the arch there, the arc D'A D. A few trials will determine and from its intersection with P draw a line this flattening near enough for practice. Thus if D'AD (Fig. 54) is to be balanced under the vertical load alone, find the centre of gravity of the section D' A and its load. Draw a vertical line P through this point, then if we can draw a line from any point in the middle third of the joint parallel to the arch at A which will intersect A L within the middle third, then the extreme points of the line of pressures in the section A D' will be within the middle third, and the line of pressures will generally be altogether within it. The new radius required for the are D'AD may also be determined roughly by under vertical load alone, the horizontal putting H. war? = HR and thence get-thrust at every point of it must be the same ting since, if D'AD is to be balanced and = HR, the thrust at D1 and D. Few men of science venture to teach that force can have existence apart from matter. To Faraday, we believe, is due the first hint at a possible conception of pure force. His utterances on the subject were brief, and intended as a species of abstract theory of magnetic influence. They have scarcely modified the current of educated thought; and at this moment no adequate conception of the existence of abstract force is to be found in the circles of science. Yet it is not, we think, impossible to form an idea of the existence of pure force; indeed, it may almost be said that we have daily evidence that force does exist apart from matter, although the presence of matter is necessary to render it evident to our senses. Gravity is, in one sense, a pure force. It is an invariable condition of, and totally inseparable from, matter, in the sense that no matter exists which is not subject to the action of the force known as gravitation. But the theorem will not work both ways. We know that masses of matter attract each other; but it has yet to be proved that a mass of matter does not exert the force of attraction, although no other mass of matter is subjected to its influence. There is probably nothing more mysterious in the whole range of material phenomena than the subtle influence which sets space at defiance, and, without any tangible link, couples the universe in bonds that cannot be broken. A force which can neither be intercepted, diverted, nor modified in the smallest possible degree, by any of the gigantic agencies of nature or the trivial influences of man's art. Gravitation is absolutely independent of all other forces, and in this respect it is to some extent isolated and unique. It is, we think, scarcely possible to realize the operation of gravitation without arriving at some crude and imperfect conception of pure force apart from matter; and however imperfect that conception may be, it will be found useful in solving some of the great problems presented to us by the operation of natural laws. Let us take it for granted, however, that force does not exist without matter, and come at once to the purpose with which pound of good coal will liberate in combusthis article is written. Our object is to call tion 14,500 units of heat. This is very attention to the fact that no one has yet easily written and remembered, but we attempted to show the relation that exists dare to say that not one of our readers out between matter and force as regards relative of the hundred has ever attempted to realize quantity. It was impossible in the earlier what it means. The British unit of heat ages that any attempt should be made to has been defined as equivalent to 772 footdetermine this relationship In the younger pounds-Joule's equivalent; the power years of the world, manifestations of great stored up in a pound of good coal therefore force were always associated in men's minds represents the exertion of a force sufficient with enormous quantities of matter. But as to lift 14,500 X 772, or 11,194,000 lbs., or knowledge, and especially chemical knowl- almost five thousand tons, a foot high, or a edge, has extended, the fact begins to ob- single ton to a height of nearly a mile. Is trude itself, that the absolute quantity of it too much to say, with such a fact before matter necessary to the tangible develop- us, that the quantity of matter required to ment of gigantic exertions of power may be render force manifest is almost infinitesimal. excessively minute. Whether it is or is not A particle of coal, indeed, so small as to possible that an accurate numerical demon- elude our senses with ease, is competent to stration of the relations of force and matter exert a very tangible power. The 14,500th in the abstract will ever be arrived at, it is part of a pound of coal is less than onenot for us to say. Possibly the ultimate third of a grain, yet it suffices to store atom may yet be found, and its measure of up a force which would lift nearly 7 cwt. possible work on another atom defined; but through a height of one foot. If a pound it is certain that at present both have of coal were divided into 11,194,000 separate eluded the grasp of the highest intellects of parts, a powerful microscope would be necesthe nineteenth century. Tyndall has gone sary to detect one such part, and yet each far to determine the work done by an atom represents as much energy as would raise under certain conditions, but he has not a pound weight a foot high. We need seek attempted to define the condition of maxi- no further for evidence that the existence mum atomic work, and nothing in the of matter in quantities tangible to our whole range of our subject is certainly senses is by no means necessary to the deknown, except that under given conditions velopment of power; and we are led to ask extremely minute quantities of matter can once more whether any point can be defined exert forces which almost baffle conception. where matter ceases and force remains; or A very crude popular illustration of the as-is it certain that without matter in some sociation of great force with little matter is afforded by gunpowder. But gunpowder really furnishes but a very indifferent example of the amount of energy which may be exerted by minute quantities of matter. A tiny drop of that curious compound, chloride of nitrogen, the precise nature of which has yet to be determined, has been known when touched to explode with such violence as to split the heavy leaf of a solid oak laboratory table and leave the apartment a wreck. Gunpowder beside this fluid is an inert substance. Fulminate of silver and nitroglycerine supply other illustrations of the power which minute quantities of matter possess under suitable conditions of rendering manifest the exertion of mighty forces. It is unnecessary, however, to seek for examples among the more recondite productions of the chemist. It is not easy to find a more noteworthy illustration of this combination of much force with little matter than that afforded by coal. One shape or form force can have no existence, manifest or abstract? The relations subsisting between force and matter have been powerfully brought home to practical engineers by the Blackburn boiler explosion. It is not our intention here to say anything of the causes which may have induced the catastrophe, nor are we about to enunciate any theory of boiler explosions. It is impossible, however, to read with care the report which we published last week, without feeling that a puzzle is presented to us, which can only be explained by the fact that very small quantities of matter may, under certain circumstances, manifest an energy apparently altogether incommensurate with the mass of matter employed. We have no means of knowing with accuracy the weight of water contained in the two boilers at the time of the explosion; probably it was about 25 tons. This water was heated to about 324 deg. Fah., the temperature |