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Britain. We have no wish to depreciate the labors of Bernouilli, Euler, and others, of whom we have already spoken in terms of commendation; and upon whose genius and attainments we often reflect with pleasure. Yet had it not been for the practical turn given to the investigation by Robins, and so incessantly kept in mind, and so skilfully and elaborately carried out to its professional applications by Dr. Hutton, the principles of gunnery would at this moment have been little better than a collection of barren speculative rules, calculated to mislead, rather than direct, the intelligent engineer.

In the notice we have taken of Robins's experiments, we do not perceive that we have described his celebrated ballistic pendulum. It consists of a large block of wood, annexed to the end of an iron stem, strongly framed, and capable of oscillating freely upon a horizontal axis. This machine being at rest, a piece of ordnance is pointed directly towards the face of the block, at any assigned distance, as twenty, thirty, forty, sixty, &c., feet, and then fired; the ball discharged from the gun strikes and enters the block, communicating to it a velocity, which is to the velocity with which the ball was moving at the moment of impact as the weight of the ball to the sum of the weights of ball and pendulum. Referring this velocity to the centre of oscillation of the pendulum, it will rise through an appreciable arc of vibration till such velocity is extinguished. The measure of that arc will lead to the determination of the velocity, because it is evidently equal to the velocity which a body would acquire by falling freely through the versed sine of the arc shown by the experiInent. Robins's largest ballistic pendulum weighed only ninety-seven pounds; i. employed to ascertain the velocities of balls weighing about an ounce each. The smallest pendulum constructed by Dr. Hutton weighed 600 pounds; and, as he pursued his experiments, the new endulums were made successively larger and arger, till at last they reached the weight of about 2600 pounds. He also made several improvements in their construction, especially in their manner of suspension, and in that of measuring the semi-arc of vibration; employing this curious apparatus in ascertaining the velocities of balls varying in weight from one pound to six, and propelled with nearly all possible modifications of charge. It appears farther, from Annales de Chimie et de Physique, tome 5, that in recent experiments at Woolwich, conducted by Dr. Gregory and the select committee of artillery officers, a ballistic pendulum, weighing 7400 pounds, was employed in determining the velocities of six, twelve, eighteen, and twentyfour pounders. Of Rifled-barrelled Guns.—The greatest irregularities in the motion of bullets are owing to the whirling motion on their axis, acquired by the friction against the sides of the piece. The best method hitherto known of preventing these is by the use of pieces with rifled barrels. These pieces have the insides of their cylinders cut with spiral channels, as a female screw, varying

from the common screws only in this, that its threads or rifles are less deflected, and approach more to a right line; it being usual for the threads with which the rifled barrel is indented to take little more than one turn in its whole length. The numbers of these threads are not determinate. The usual method of charging these pieces is this:—The proper quantity of powder being put down, a leaden bullet rather larger than the bore of the piece is forcibly driven home to the powder; and in its passage acquires the shape of the inside of the barrel, so that it becomes part of a male screw, exactly answering to the indents of the rifle. The rifled barrels made in Britain are often contrived to admit the charge and shot at the breech; and the ball acquires the same shape in its expulsion that is given to it by the more laborious operation of driving it in at the muzzle. From the whirling motion communicated by the rifles, it happens that, when the piece is fired, that indented zone of the bullet follows the sweep of the rifles, and thereby, besides its progressive motion, acquires a circular motion round the axis of the piece; which circular motion will be continued to the bullet after its separation from the piece; and thus a bullet discharged from a rifled barrel is constantly made to whirl round an axis which is coincident with the line of its flight. By this whirling on its axis, the aberration of the bullet, which proves so prejudicial to all *: in gunnery, is almost totally ". : and accordingly such pieces are much more to be depended on, and will do execution at a much greater distance, than the other. But as it is in a manner impossible entirely to correct the aberrations arising from the resistance of the atmosphere, even the rifled barrelled pieces cannot be depended upon for more than one-half of their actual range at any considerable elevation. It becomes therefore a problem very difficult of solution, to know, even within a very considerable distance, how far a piece will carry its ball with any probability of hitting its mark, or doing any execution. The best rules hitherto laid down on this subject are those of Robins. Of Carronades.—Mr. Gascoigne's improved gun, called a carronade, was, in June 1779, by the king in council instituted a standard navygun, and ten of them appointed to be added to each ship of war, from a first-rate to a sloop. The carronade is mounted upon a carriage with a perfectly smooth bottom of strong plank, without trucks; instead of which there is fixed on the bottom of the carriage, perpendicular from the trunnions, a gudgeon of proper strength, with an iron washer and pin at the lower end. This gudgeon is let into a corresponding groove cut in a second carriage, called a slide-carriage; the washer supported by the pin overreaching the under edges of the groove. This slide carriage is made with a smooth upper surface, upon which the gun-carriage is moved, and by the gudgeon always kept in its right station to the port; the groove in the slide-carriage being of a sufficient length to allow the gun to recoil and be loaded within board. The slide-carriage, the groove included, is equally broad with the forepart of the gun-carriage, and about four times the length; the fore part of the slide-carriage is fixed by hinge-bolts to the quick-work of the ship below the port, the end lying over the fill, close to the outside plank, and the groove reaching to the fore end; the gudgeon of the guncarriage, and consequently the trunnions of the gun, are over the fill of the port when the gun is run out; and the port is made of such breadth, with its sides bevelled off within board, that the gun and carriage may range from bow to quarter. The slide-carriage is supported from the deck at the hinder end, by a w; or step-stool; which being altered at pleasure, and the fore end turning upon the hinge-bolts, the carriage can be constantly kept upon an horizontal plane, for the more easy and quick working of i. gun when the ship lies along. But see Sir Howard Douglas's remarks on this piece, already given. Of Rifled Ordnance.—In 1774 Dr. Lind, and captain Alexender Blair of the sixty-ninth regiment of foot, invented a species of rifled field

p. They are made of cast-iron; and are not ored like the common pieces, but have the rifles moulded on the core, after which they are cleaned out and finished with proper instruments. Guns of this construction, which are intended for the field, ought not to be made to carry a ball of above one or two pounds weight at most; a leaden bullet of that weight being sufficient to destroy either man or horse. A pound-gun, of this construction, of good metal, need not weigh above 100 lbs., nor its carriage above 100 lbs. more. It can therefore be easily transported from place to place, by a few men; and a couple of good horses may transport six of these guns and their carriages, if put into a cart. But this kind of ordnance has never been extensively used, we believe, in the British service. See our article ARTILLERY, for the latest official regulations for the proportion and disposition of the ammunition attached to the field pieces of our army: as also for the guns attached to the brigades of artillery. See also CANNoN.

The Projection of THE SPHERE is a perspective representation of the circles on the surface of the globe; and is variously denominated, according to the different positions of the eye and plane of projection. There are three principal kinds of projection; the stereographic, the orthographic, and gnomonic. In the stereographic projection, the eye is supposed to be placed on the surface of the sphere; in the orthographic it is supposed to be at an infinite distance; and in the gnomonic projection the eye is placed at the centre of the sphere. Other kinds of projection are, the globular, Mercator's developement, &c. The chief application of the doctrine of these projections is to the constructing of maps and dials. In our article MAPs we have, therefore, entered at length into the principal projections; i.e. 1. By development; 2. The orthographic; 3. The stereographic; 4. The globular; and 5. Mercator's. In that of DIALLING the gnomonic is involved. See that article. It may, however, be thus exhibited more formally. The eye, in this projection, is in the centre of the sphere, and the plane of projection touches the sphere in a given point parallel to a given circle: the plane of projection will represent the plane of a dial, whose centre being the projected pole, thesemi-axis of the sphere will be the stile or gnomon of the dial. Prop. I. Theory I.-Every great circle is o: into a straight line perpendicular to the ine of measures; and whose distance from the centre is equal to the cotangent of its inclination, or to the tangent of its nearest distance from the pole of the projection. Let BAD, fig. 1, be the given circle, and let the circle C B E D be perpendicular to BAD, and to the plane of projection: whose intersection CF with this last plane will be the line of measures. Now, since the circle C B E D is perpendicular both to the given circle BAD and to the plane of projection, the common section of the two last planes produced will therefore be perpendicular to the plane of the circle CBED

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produced, and consequently to the line of measures: hence the given circle will be projected into that section; that is, into a straight line passing through d, perpendicular to C d. Now Cod is the cotangent of the angle C dA, the inclination of the given circle, or the tangent of the arch CD to the radius AC. CoRol. 1. A great circle perpendicular to the plane of projection is projected into a straight line passing through the centre of projection; and any arch is projected into its correspondent tangent. 2. Any point, as D, or the pole of any circle, is projected into a point d, whose distance from the pole of projection is equal to the tangent of that distance. 3. If two great circles be perpendicular to each other, and one of them passes through the pole of projection, they will be projected into two straight lines perpendicular to each other. 4. Hence if a great circle be perpendicular to several other great circles, and its representation pass through the centre of projection; then all these circles will be represented by lines parallel to one another and perpendicular to the line of measures, for representation of that first circle. Prop. II. Theor. II. If two great circles intersect in the pole of projection, their representations will make an angle at the centre of the plane of projection, equal to the angle made by these circles on the sphere. For, since both these circles are perpendicular to the plane of projection, the angle made by their intersections with this plane is the same as the angle made by these circles. PRop. III. Theoil. III. Any less circle llel to the plane of projection is projected into a circle whose centre is the pole o projection, and its radius is equal to the tangent of the distance of the circle from the pole of projection. Let the circle PI (sig. 1) be parallel to the plane GF, then the equal arches PC, CI, are projected into the equal tangents GC, CH; and therefore C, the point of contact and pole of the circle PI and of the projection, is the centre of the representation G, É Cohol. If a circle be parallel to the plane of rojection, and 45° from the pole, it is projected into a circle equal to a great circle of the sphere; and therefore may be considered as the primitive circle, and its radius the radius of projection. Prop. IV. Theor. IV. A less circle not parallel to the plane of projection is projected into a conic section, whose transverse axis is in the line of measures; and the distance of its nearest vertex from the centre of the plane of projection is equal to the tangent of its nearest distance from the pole of projection; and the distance of the other vertex is equal to the tangent of the great distance. Any less circle is the base of a cone whose vertex is at A, fig. 2; and this cone being pro

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duced, its intersection with the plane of projection will be a conic section. Thus the cone DAF, having the circle DF for its base, being produced, will be cut by the plane of projection in an ellipse whose transverse diameter is df; and C d is the tangent of the angle CAD, and Cf the tangent of CAF. In like manner, the cone AFE, having the side A E parallel to the line of measures dif, being cut by the plane of projection, the section will be a parabola, of which f is the nearest vertex, and the point into which E is projected is at an infinite distance. Also the cone AFG, whose base is the circle FG, being cut by the plane of projection, the section will be a hyperbola; of which f is the nearest vertex; and GA being produced gives d the other vertex.

CoRoi... 1. A less circle will be projected into an ellipse, a parabola, or hyperbola, accord

ing as the distance of its most remote point is

less, equal to, or greater than, 90°. 2. If H be the centre, and K, k, l, the focus of

the ellipse, hyperbola, or parabola; then H K =

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2 Prop. V. Theor. V. Let the plane TW, fig. 1, Plate PRojection of The SPHERE, be erpendicular to the p.ane of projection TV, and oD a great circle of the #5 in the plane TW. Let the great circle B E D be projected into the straight line be k. C Q S perpendicular to bk, and § m parallel to it and equal to CA, and make Q S equal to Q m; then any angle Q St is the measure of the arch Qt of the projected circle. Join A Q: then, because C m is equal to CA, the angle QC m equal to QCA, each being a right angle, and the side QC common to both triangles; therefore Q m, or its equal Q S, is equal Q A. Again, since the plane AC Q is perpendicular to the plane TV, and b Q to the intersection C Q ; therefore b Q is perpendicular both to AQ and Q S : hence, since A Q and QS are equal, all the angles at S cut the line b Q in the same points as the equal angles at A. But by the angles at A the circle B E D is projected into the line b Q. Therefore the angles at S are the measures of the parts of the projected circle b Q; and S is the dividing centre thereof. CoRol. 1. Any great circle b Q tis projected into a line of tangents to the radius SQ. 2. If the circle b C pass through the centre of projection, then the po point A is the dividing centre thereof, and Cb is the tangent of its correspondent arch CB to CA, the radius of projection. PRop. VI. Theor. VI. Let the parallel circle G LH, fig. 1, be as far from the pole of projection C as the circle FN I is from its pole; and let the distance of the poles C P be bisected by the radius AO; and draw ba D perpendicular to AO; then any straight line b Q t drawn through b will cut off the arches h, l, F, n, equal to each other in the representations of these equal circles in the plane of projection. Let the projections of the less circles be described. Then, because B D is perpendicular to A O, the arches BO, DO, are equal; but, since the less circles are equally distant each from its respective pole, therefore the arches FO, O H, are equal; and hence the arch B F is equal to the arch DH. For the same reason the arches BN, DL, are equal; and the angle FBN is equal to the angle LD H ; therefore, on the sphere, the arches FN, HL, are equal. , And since the great circle BNLD is projected into the straight line b Qn l, &c., therefore n is the projection of N, and l that of L: hence fn, h l, the projections of FN, HL, respectively, are ual. *... VII. Theor. VII. If F n k, h l8, fig. 2, be the projections of two equal circles, whereof one is as far from its pole P as the other from its pole C, which is the centre of projection; and if the distance of the projected poles C, p, be di.

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