nature, let the foregoing figure represent a transverse vertical section of a magazine arch balanced in all its parts, in which the span or width AM is 20 feet, the pitch or height na is 10 feet, thickness at the crown DK = 7 feet, and the angle of the ridge LKN 112° 37, or the half of it LKD = 56° 18′4, the complement of which, or the elevation KIR, is 33°41' the tangent of which is = 4, which will therefore be the value of t in the foregoing investigation. The values of the other letters will be as follows, viz, DK7; AQ=b=10; DQ=h=10; AL=C=10} = 3; A = log. of 7='8450980; x log. of £+√√(22) = log. of (c2—a3) 31+/520

€ =

1

ს

a

21

log. of 2.562070408591; cy + A = 0408591y + 8450980 log. of n. From the general equation then, viz,

=

or by squaring, &c, a2 m2-2amw + w2 = w2 ~ a2, and hence

M =

m2 41

2m

x; to which the numbers being applied, the

very same conclusions result as in the foregoing calculation and table.

PROBLEM VIII.

To construct Powder Magazines with a Parabolical Arch.

It has been shown, in my tract on the Principles of Arches of Bridges, that a parabolic arch is an arch of equilibration, when its extrados, or form of its exterior covering, is the very same parabola as the lower or inside curve. Hence then a parabolic arch, both for the inside and outer form, will be

.

very

very proper for the structure of a powder magazine. For, the inside parabolic shape will be very convenient as to room for stowages 2dly, the exterior parabola, everywhere parallel to the inner one, will be proper enough to carry off the rain water: 3dly, the structure will be in perfect equilibrium: and 4thly the parabolic curve is easily constructed, and the structure erected. to rett 4T

Put, as before, a = KD, h = Day tu boga b = AQ, x = DP, and y = PC or RI. her do Then, by the nature of the parabolars

rebwod

ed

hy?

hence

tity a, what it is at the vertex; that is, ci is everywhere equal to KD. b pilicet de

=

Consequently KR is DP; and since RI is PC, it is evident that KI is the same parabolic curve with DC, and may be placed any height above it, always producing an arch of equilibration, and very commodious for powder magazines.

19

IN the 2d vol. of this course have been given several particulars relating to this subject. Thus, in props. 19, 20, 21, 22, p. 451 &c, is given all that relates to the parabolic theory of projectiles, that is, the mathematical principles which would take place and regulate such projects, if they were not impeded and disturbed in their motions by the air in which they move. But, from the enormous resistance of that medium, it happens, that many military projectiles, especially the smaller balls discharged with the higher velocities, de not range so far as a 20th part of what they would naturally do in empty space! That theory therefore can only be use ful in some few cases, such as in the slower kind of motions, not above the velocities of 2, 3, or 400 feet per second, when the path of the projectile differs but little perhaps from the curve of a parabola.10 9273eb od or 1 to 19ng some ent Again, at pa. 160 &c, are given several other practical rules and calculations, depending partly on the foregoing parabolic

theory,

theory, and partly on the results of certain experiments performed with cannon balls. ad litw sqdde oflödetsq ebian: eng

Again, in prop. 58, pa. 219, are delivered the theory and calculations of a beautiful military experiment, invented by Mr. Robins, for determining the true degree of velocity with which balls are projected from guns, with any charges of powder. The idea of this experiment, is simply, that the ball is discharged into a very large but moveable block of

Whose small velocity, in consequence of that blow, can be easily observed and accurately measured. Then, from this small velocity, thus obtained, the great one of the ball is immediately derived by this simple proportion, viz, as the weight of the ball, is to the sum of the weights of the ball and the block, so is the observed velocity of the last, to a 4th proportional, which is the velocity of the ball sought. It is evident that this simple mode of experiment will be the source of numerous useful principles, as results derived from the experiments thus made, with all lengths and sizes of guris, with all kinds and sizes of balls and other shot, and with all the various sorts and quantities of gunpowder; in short, the experiment will supply answers to all enquiries in projectiles, excepting the extent of their ranges; for it will even determine the resistance of the air, by causing the ball to strike the block of wood at different distances from the gun, thus showing the velocity lost by passing through those different spaces of air; all which circumstances are partly shown in my 4to vol. of Tracts published in 1786, and which will be completed in my new volumes of miscellaneous tracts now printing.

Lastly, in prob. 17 on Forces, near the end of volume 2, some results of the same kind of experiment are successfully applied to determine the curious circumstances of the first force or elasticity of the air resulting from fired gunpow der, and the velocity with which it expands itself, These are circumstances which have never before been determined with any precision. Mr. Robins, and other authors, it may. be said, have only guessed at, rather than determined them. That ingenious philosopher, by a simple experiment truly showed that by the firing of a parcel of gunpowder, a quan tity of elastic air was disengaged, which, when confined in the space only occupied by the powder before, it was fired, was found to be near 250 times stronger than the weight or elasticity of the common atmospheric air. He then heated the same parcel of air to the degree of red hot iron, sands found it in that temperature to be about 4 times as strong as before whence he inferred, that the first strength of the m *.Vroeds

Blamed

Hamed fluid, must be nearly 1000 times the pressure of the atmosphere. But this was merely guessing at the degree of heat in the inflamed fluid, and consequently of its first strength, both which in fact are found to be much greater. It is true that this assumed degree of strength accorded pretty well with that author's experiments; but this seeming agreement, It may easily be shown, could only be owing to the inaccuracy of his own further experiments; and, in fact, with far better opportunities than fell to the lot of Mr. Robins, we have shown that inflamed gunpowder is about double the strength that he has assigned to it, and that it expands itself with the velocity of about 5000 feet per second.

Fully sensible of the importance of experiments of this kind, first practised by Mr. Robins with musket balls only, my endeavours for many years were directed to the prosecution of the same, on a larger scale, with cannon balls; and I having had the honour to be called on to give my assistance at several courses of such experiments, carried on at Woolwich by the ingenious officers of the Royal Artillery there, under the auspices of the Masters General of the Ordnance, I have assiduously attended them for many years. The first of these courses was performed in the year 1775, being 2 years after my establishment in the Royal Academy at that, place and in the Philos. Trans. for the year 1778 I gave an account of these experiments, with deductions, in a memoir, which was honoured with the Royal Society's gold medal of that year. In conclusion, from the whole, the following important deductions were fairly drawn and stated, viz.

1st, It is made evident by these experiments, that gun powder fires almost instantaneously. 2dly, The velocities communicated to shot of the same weight, with different charges of powder, are nearly as the square roots of those charges. Sdly, And when shot of different weights are fired with the same charge of powder, the velocities communicated to them, are nearly in the inverse ratio of the square roots of their weights. 4thly, So that, in general, shot which are of different weights, and impelled by the firing of different charges of powder, acquire velocities which are directly as the square roots of the charges of powder, and inversely as the square roots of the weights of the shot. 5thly, It would therefore be a great improvement in artillery, occasionally to make use of shot of a long shape, or of heavier matter, as lead; for thus the momentum of a shot, when discharged with the same charge of powder, would be increased in the ratio of the square root of the weight of the shot; which would both augment proportionally the force of the blow with

which it would strike, and the extent of the range to which it would go. 6thly, It would also be an improvement, to diminish the windage; since by this means, one third or more of the quantity of powder might be saved. 7thly, When the improvements mentioned in the last two articles are con sidered as both taking place, it appears that about half the quantity of powder might be saved. But, important as this saving may be, it appears to be still exceeded by that of the guns: for thus a small gun may be made to have the effect and execution of another of two or three times its size in the present way, by discharging a long shot of 2 or 3 times the weight of its usual ball, or round shot; and thus a small ship might employ shot as heavy as those of the largest now used. Finally, as these experiments prove the regulations with respect to the weight of powder and shot, when discharged from the same piece of ordnance; so, by making similar experiments with a gun varied in its length, by cutting off from it a certain part, before each set of trials, the effects and ge neral rules for the different lengths of guns, may be with certainty determined by them. In short, the principles on which these experiments were made, are so fruitful in consequences, that, in conjunction with the effects of the resistance of the medium, they appear to be sufficient for answering all the inquiries of the speculative philosopher, as well as those of the practical artillerist.

Such then was the summary conclusion from the first set of experiments with cannon balls, in the year 1775, and such were the probable advantages to be derived from them. I am not ware however that any alterations were adopted from them by authority in the public service: unless we are to except the instance of carronades, a species of ordnance that was afterwards invented, and in some degree adopted in the public service; for, in this instance, the proprietors of those pieces, by availing themselves of the circumstances of large balls, and very small windage, have, with small charges of powder, and at little expense, been enabled to produce very considerable and useful effects with those light pieces.

The 2d set of these experiments extended through most part of the summer seasons of the years 1783, 1784, 1785, and some in 1786. The objects of this course were numerous and various but the principal articles as follow: 1. The velocities with which balls are projected by equal charges of powder, from pieces of equal weight and calibre, but of different lengths. 2. The velocities with different charges of powder, the weight and length of the guns being equal. 3. The greatest velocities due to the different lengths of guns,

to