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rise from the resistance of the air in the flight of shells or cannon-shot. In this persuasion they supported themselves chiefly by considering the extreme rarity of the air, compared with those dense and ponderous bodies; and at last it became an almost generally established maxim that the flight of these bodies was nearly in the curve of a parabola.
A. the publication of Newton's Principia it might have been expected that the defects of the theory would be ascribed to their true cause, the great resistance of the air to such swift motions; as in that work he particularly considered the subject of such motions, and related the result of experiments made on slow motions at least; by which it appeared that in such motions the resistance increases as the square of the velocities, and he even hints a suspicion that it will increase above that law in swifter motions, as is now known to be the case. So far, however, were those who treated this subject scientifically from making a proper allowance for the resistance of the atmosphere that they still neglected it, or rather opposed it, and their theories accordingly differed most egregiously from the truth. Huygens alone seems to have attended to this principle; for in the year 1590 he published a treatise on gravity, in which he gave an account of some experiments tending to prove that the track of all projectiles, moving with very swift motions, was widely different from that of a parabola. The rest of the learned generally acquiesced in the justness and sufficiency of Galileo's doctrine. Nor was any farther notice taken of these errors till the year 1716, at which time Mr. Ressons, a French officer of artillery, of great merit and experience, gave in a memoir to the Royal Academy, importing that, “although it was agreed that theory joined with practice did constitute the F." of every art, yet experience had taught
im that theory was of very little service in the use of mortars: that the works of M. Blondell had justly enough described the several parabolic lines according to the different degrees of the elevation of the piece: but that practice had convinced him there was no theory in the effect of gunpowder; for having endeavoured, with the greatest precision, to point a mortar according to these calculations, he had never been able to establish any solid foundation upon them.” One instance only occurs, in which D. Bernouill applies the doctrine of Newton to the motions of projectiles, in the Com. Acad. Petrop. tom. ii. p. 338, &c. Besides which nothing so was done in this business till the time of Robins, who published a treatise in 1742, intitled New Principles of Gunnery, in which he treated particularly not only of the resistance of the atmosphere, but also of the force of gunpowder, the nature and effects of different guns, and almost every thing else relating to the flight of military projectiles; and indeed he carried the theory of gunnery nearly to its utmost perfection.
The first thing considered by Mr. Robins, and which is indeed the foundation of all other particulars relating to gunnery, is the explosive force of gunpowder. M. de la Hire, in his History of the Academy of Sciences for the year 1702, supposed that this force may be owing to the in
creased elasticity of the air contained in and between the grains, in consequence of the heat and fire produced at the time of the explosion: a cause not adequate to the 200th part of the effect. On the other hand, Mr. Robins determined, by irrefragable experiments, that this force was owing to an elastic fluid similar to our atmosphere, existing in the powder in an extremely condensed state, which being suddenly freed from the powder, by the combustion, expanded with an amazing force, and violently impelled the bullet, or whatever might oppose its expansion. he intensity of this force of exploded gunpowder Mr. Robins ascertained in different ways, after the example of Mr. Hawkesbee, related in the Philosophical Transactions, No. 295, and in his Physico-Mechanical Experiments, 81. One of these is by firing the powder in the air thus:–A small quantity of the powder is placed in the upper part of a glass tube, and the lower part of the tube is immerged in water, the water being made to rise so near the top that only a small portion of air is left in that part where the powder is placed; then in this situation, the communication between the upper part of the tube and the external air being closed, the powder is fired by means of a burning glass, or otherwise; the water descends upon the . sion, and stands lower in the tube than before, by a space proportioned to the quantity of powder fired. Another way was by firing the powder in vacuo, viz. in an exhausted receiver, by dropping the grains of powder upon a hot iron included in the receiver. By this means a permanent elastic fluid was generated from the fired powder, and the quantity of it was always in proportion to the quantity of powder that was used, as was found by the proportional sinking of the mercurial gauge annexed to the air-puinp. The result of these experiments was, that the weight of the elastic air thus generated was equal. to three-tenths of the compound mass of the gunpowder which yielded it, and that its bulk when cold, and expanded to the rarity of the common air, was about 240 times the bulk of the powder; and consequently in the same proportion would such fluid at first, if it were cold, exceed the force or elasticity of the atmosphere. But as Mr. Robins found, by another ingenious experiment, that air heated to the extreme degree of the white heat of iron has its elasticity quadrupled, or is four times as strong, he thence inferred that the force of the elastic air generated as above, at the moment of the explosion, is at least four times 240, or 960, or, in round numbers, about 1000 times as strong as the elasticity or pressure of the atmosphere on the same space. Having thus determined the force of the gunpowder, or intensity of the agent by which the projectile is to be urged, Mr. Robins proceeded to determine the effects it will produce, or the velocity with which it will impel a shot of a given weight from a piece of ordnance of given dimensions; which is a problem strictly limited and perfectly soluble by mathematical rules, and is in general this:—Given the first force, and the law of its variation, to determine the velocity with which it will impel a given body, in passing through a given space, which is the length of the bore of the gun. In the solution of this problem Mr. Robins assumes these two postulates, viz. 1. That the action of the powder on the bullet ceases as soon as the bullet is out of the piece; and 2d. That all the powder of the charge is fired and converted into elastic fluid before the bullet is sensibly moved from its place: assumptions which for good reasons are found to be in many cases very near the truth. It is to be noted also, that the law by which the force of the elastic fluid varies is this, viz. that its intensity is directly as its density, or reciprocally proportional to the space it occupies, being so much the stronger as the space is less: a principle well known, and common to all elastic fluids. Upon these principles, then, Mr. Robins resolves this problem, by means of the thirty-ninth proposition of Newton's Principia, in a direct way, and the result is equivalent to this theorem, when the quantities are expressed by algebraic symbols; viz. the velocity of the ball.
10 a l b
X log. 7.
where v is the velocity of the ball, a the length of the charge of powder, b the whole length of the bore, c the specific gravity of the ball, or weight of a cubic foot of the same matter in ounces, d the diameter of the bore, w the weight of the ball in Outices. For example, suppose a-2; inches, b= 45 inches, c = 11345 ozs. for a ball of lead, and - 7 120 d=# inch; then v = arisov/ 226 x log. 7 = 1674 feet per second, the velocity of the ball. Or, if the weight of the bullet be w =
“Having in this proposition,’ says Mr. Robins, “shown how the velocity which any bullet acquires from the force of powder may be computed upon the principles of the theory laid down in the preceding propositions, we shall next show that the actual velocities with which bullets of different magnitudes are impelled from different pieces, with different quantities of powder, are really the same with the velocities assigned by these computations, and consequently that this theory of the force of powder, here delivered, does unquestionably ascertain the true action and modification of this enormous
“But, in order to compare the velocities communicated to bullets by the explosion with the velocities resulting from the theory by computation, it is necessary that the actual velocities with which bullets move should be capable of being discovered, which yet is impossible to be done by any methods hitherto made public. The only means hitherto practised by others for that purpose have been either by observing the time of
the fight of the shot through a given space, or by measuring the range of the shot at a given elevation, and thence computing on the parabolic hypothesis what velocity would produce this range. The first method labors under this insurmountable difficulty, that the velocities of these bodies are often so swift, and consequently the time observed is so short, that an imperceptible error in that time may occasion an error in the velocity thus found of 200, 300,400, 500, or 600 feet in a second. . The other method is so fallacious, by reason of the resistance of the air (to which inequality the first is also liable), that the velocities thus assigned may not be perhaps the tenth part of the actual velccities sought. “To remedy then these inconveniences, I have invented a new method of finding the real velocities of bullets of all kinds; and this to such a degree of exactness (which may be augmented too at pleasure) that in a bullet moving with the velocity of 1700 feet in a second, the error in the estimation of it need never amount to its 500th part; and this without any extraordinary nicety in the construction of the machine.' Mr. Robins then gives an account of the machine by which he measures the velocities of the balls, which machine is simply this: viz. a pendulous block of wood suspended freely by a horizontal axis, against which block are to be fired the balls whose velocities are to be determined. ‘This instrument thus fitted, if the weight of the pendulum be known, and likewise the respective distances of its centre of gravity and of its centre of oscillation from its axis of suspension, it will thence be known what motion will be communicated to this pendulum by the percussion of a body of a kirown weight moving with a known degree of celerity, and striking it in a given point; that is, if the pendulum be supposed at rest before the percussion it will be known what vibration it ought to make in consequence of such a determined blow; and, on the contrary, if the pendulum, being at rest, is struck by a body of". known weight, and the vibration which the pendulum makes after the blow is known, the velocity of the striking body may thence be determined. * Hence then, if a bullet of a known weight strikes the pendulum, and the vibration which the pendulum makes in consequence of the stroke be ascertained, the velocity with which the ball moved is thence to be known.' Our author then explains his method of computing velocities from experiments with this machine; which method is rather troublesome and perplexed, as well as the rules of Euler and Antoni, who followed him in this business; but a much plainer rule is given in Hutton's Tracts, vol. i. p. 119, where such experiments are exits. full length, and this rule is expressed by either of the two following formulas,
radius, rg the distance below the axis of motion to the centre of gravity, o the distance to the centre of oscillation, i the distance to the point of impact, and n the number of oscillations the pendulum will perform in one minute, when made to oscillate in small arcs. The latter of these two theorems is much the easiest, both because it is free of radicals, and because the value of the radical Vo, in the former, is to be first computed from the number n, or number of oscillations the pendulum is observed to make. With such machines Mr. Robins made a great number of experiments with musket barrels of different lengths, with balls of various weights, and with different charges or quantities of powder. He has set down the results of sixty-one of these experiments, which nearly agree with the corresponding velocities as computed by his theory of the force of powder, and which therefore establish that theory on a sure foundation. From these experiments, as well as from the preceding theory, many important conclusions were deduced by Mr. Robins; and indeed, by means of these, it is obvious that every thing may be determined relative both to the true theory of projectiles, and to the practice of artillery; for, by firing a piece of ordnance charged in a similar manner against such a ballistic pendulum from different distances, the velocity lost by passing through such spaces of air will be found, and consequently the resistance of the air, the only circumstance that was wanting to complete the theory of gunnery or military projectiles; and of this kind Dr. Hutton made a great number of experiments with cannon balls, and has thereby obtained the whole series of resistances to such a ball when moving with every degree of velocity, from 0 up to 2000 feet per second of time. In the structure of artillery they may likewise be of the greatest use; for hence may be determined the best lengths of guns; the proportions of the shot and powder to the several lengths; the thickness of a piece, so as it may be able to confine, without bursting, any given charge of powder; as also the effect of wads, chambers, placing of the vent, ramming the powder, &c. For the many other curious circumstances relating to this subject, and the various other improvements in the theory and practise of gunnery made by Mr. Robins, consult the first volume of his Tracts, collected and published by Dr. Wilson in the year 1761, where ample information may be found. In the year 1755, says Dr. Hutton, in conjunction with several able officers of the royal artillery and other ingenious gentlemen, I undertook a course of experiments with the ballistic pendulum, in which we ventured to extend the machinery to cannon shot of one, two, and three pounds weight. An account of these experiments was published in the Philosophical Transactions for 1778; and for which the Royal Society honored me with the prize of the gold medal. These were the only experiments that I know of which had been made with cannon balls for this urpose, although the conclusions to be deduced rom such are of the greatest importance to those parts of natural philosophy which are dependent on the effects of fired gunpowder: nor do I know of any other practical method of ascertaining the
initial velocities within any tolerable degree of truth. The knowledge of this velocity is of the utmost comsequence in gunnery; by means of it, together with the law of the resistance of the medium, every thing is determinable relative to that business; for, besides its being an excellent method of trying the strength of different sorts of powder, it gives us the law relative to the different quantities of powder, to the different weights of shot, and to the different lengths and sizes of guns. Besides these, there does not seem to be any thing wanting to answer any enquiry that can be made concerning the flight and ranges of shots except the effects arising from the resistance of the medium. In these experiments the weights of the pendulums employed were from 300 to nearly 600 pounds. In that paper is described the method of constructing the machines, of finding the centres of gravity and oscillation of the pendulum, and of making the experiments, which are all set down in the form of a journal, with all the minute and concomitant circumstances; also the investigation of the new and easy rule set down just above, for computing the velocity of the ball from the experiments. The charges of powder were varied from two to eight ounces, and the shot from one to nearly three pounds. And from the whole were clearly deduced the inferences we have already given. In the year 1786 was published the first volume of Dr. Hutton's Tracts, in which is detailed, at great length, another very extensive course of experiments which were carried on at Woolwich in the years 1783, 1784, and 1785, by order of the duke of Richmond, master general of the ordnance. The objects of this course we have also enumerated in the theoretic part of this treatise. These objects were obtained in a very perfect and accurate manner; excepting only the article of ranges, which were not quite so regular and uniform as might be wished. The balls too were most of them of one pound weight; but the powder was increased from one ounce up till the bore was quite full, and the pendulum was from 600 to 800 lbs. weight. The conclusions from the whole were as follow:— ‘1. That the former law, between the charge and velocity of ball, is again confirmed, viz. that the velocity is directly as the square root of the weight of powder, as far as to about the charge of eight ounces: and so it would continue for all charges, were the guns of an indefinite length. But as the length of the charge is increased, and bears a more considerable Fo'." to the length of the bore, the velocity falls the more short of that proportion. “2. That the velocity of the ball increases with the charge to a certain point, which is peculiar to each gun, where it is greatest; and that, by farther increasing the charge, the velocity gradually diminishes till the bore is quite full of powder. That this charge for the greatest velocity is greater as the gun is longer, but not greater however in so high a proportion as the length of the gun is; so that the part of the bore filled with powder bears a less proportion to the whole in the long guns, than it does in the short ones;
“3. It appears that the velocity continually increases as the gun is longer, though the increase in velocity is but very small in respect of the increase in length, the velocities being in a ratio somewhat less than that of the square roots of the length of the bore, but somewhat greater than that of the cube roots of the length, and is indeed nearly in the middle ratio between the two. “4. The range increases in a much less ratio than the velocity, and indeed is nearly as the square root of the velocity, the gun and elevation being the same. And, when this is compared with the property of the velocity and length of gun in the foregoing paragraph, we perceive that very little is gained in the range by a great increase in the length of the gun, the charge being the same. And indeed the range is nearly as the fifth root of the length of the bore, which is so small an increase as to amount only to about one-seventh part more range for a double length of gun. ‘5. It also appears that the time of the ball's flight is nearly as the range; the gun and elevation being the same. “6. It appears that there is no sensible difference caused in the velocity or range, by varying the weight of the gun, nor by the use of wads, nor by different degrees of ramming. nor by firing the charge of powder in different parts of it. ‘7. But a great difference in the velocity arises from a small degree of windage. Indeed with the usual established windage only, namely, about one-twentieth of the calibre, no less than between one-third and one-fourth of the powder escapes and is lost. And, as the balls are often smaller than that size, it frequently happens that half the powder is lost by unnecessary windage. ‘8. It appears that the resisting force of wood to balls fired into it is not constant. And that the depths penetrated by different velocities or charges are nearly as the logarithms of the charges; instead of being as the charges themselves, or, which is the same thing, as the square of the velocity. ‘9. These and most other experiments show that balls are greatly deflected from the direction they are projected in; and that so much as 300 or 400 yards in a range of a mile, or almost onefourth of the range, which is nearly a deflection of an angle of 15°. ‘10. Finally, these experiments furnish us with the following concomitant data, to a tolerable degree of accuracy, namely, the dimensions and
elevation of the gun, the weight and dimensions of the powder and shot, with the range and time of flight, and the first velocity of the ball. From which it is to be hoped that the measure of the resistance of the air to projectiles may be determined, and thereby lay the foundation for a true and practical system of gunnery, which may be as well useful in service as in theory.' ‘Since the publication of those Tracts,’ says Dr. Hutton, “we have prosecuted the experiments still farther from year to year, gradually extending our aim to more objects, and enlarging the guns and machinery, till we have arrived at experiments with the six-pounder guns, and pendulums of 1800 lbs. weight. One of the new objects of enquiry was the resistance the atmosphere makes to military projectiles; to obtain which the guns have been placed at many different distances from the pendulum, against which they are fired, to get the velocity lost in passing through those spaces of air; by which, and the use of the whirling machine, described near the end of the first volume of Robins's Tracts, for the slower motions, I have investigated the resistance of the air to given balls moving with all degrees of velocity from 0 up to 2000 feet per second; as well as the resistance for many degrees of velocity to planes and figures of other shapes, and inclined to their path in all varieties of angles; from which I have deduced general laws and formulas for all such motions. “Mr. Robins made also similar experiments on the resistance of the air, but being only with musket bullets, on account of their smallness and of their change of figure by the explosion of the powder, I find they are very inaccurate, and considerably different from those above mentioned, which were accurately made with pretty considerable cannon balls of iron. For this reason we may omit here the rules and theory deduced from them by Mr. Robins, till others more correct shall have been established. All these experiments indeed agree in evincing the very enormous resistance the air makes to the swift motions of military projectiles, amounting in some cases to twenty or thirty times the weight of the ball itself; on which account the common rules for projectiles deduced from the parabolic theory are of little or no use in real practice; for from these experiments it is clearly proved that the track described by the flight even of the heaviest shot is neither a parabola, nor yet approaching any thing near it, except when they are projected with very small velocities; insomuch that some balls, which in the air range only to the distance of one mile, would in vacuo, when projected with the same velocity, range above ten or twenty times as far.” Mr. Benjamin Thompson (the late count Rumford) instituted a very considerable course of experiments of the same kind as those of Mr. Robins, with musket barrels, which was published in the Philosophical Transactions, vol. 71, for the year 1781. In these experiments the conclusions of Mr. Robins are generally confirmed, and several other curious circumstances in this business are remarked by Mr. Thompson. This gentleman also pursues a hint thrown out by Mr. Robins, relative to the determining o velocity of 2
a ball from the recoil of the pendulous gun itself. Mr. Robins, in the eleventh proposition, remarks, that the effect of the exploded powder upon the recoil of a gun is the same whether the gun is charged with or without a ball; and that the chord or velocity of recoil with the powder alone, being substracted from that of the recoil when charged with both powder and ball, leaves the velocity which is due to the ball alone. Thence Mr. Thompson observes that the inference is obvious, viz. that the momentum thus communicated to the gun by the ball alone, being equal to the momentum of the ball, this becomes known ; and therefore, being divided by the known weight of the bali, the quotient will be its velocity. Mr. Thompson sets a great value on this new rule, the velocities by means of which he found to agree nearly with several of those deduced from the motion of the pendulum; and in the other cases, in which they differed greatly from these, he very inconsistently supposes that these latter ones are erroneous. In the experiments, however, contained in Dr. Hutton's Tracts, a great multitude of those cases are compared together, and the inaccuracy of that new rule is fully proved. Having in the ninth proposition compared together a number of computed and experimented velocities of balls to verify his theory; Mr. Robins, in the tenth proposition, assigns the changes in the force of powder, which arise from the different state of the atmosphere, as to heat and moisture, both which he finds have some effect on it, but especially the latter. In the eleventh proposition he investigates the velocity which the flame of gunpowder acquires by expanding itself, supposing it fired in a given piece of artillery, without either a bullet or any other body before it. This velocity he finds is upwards of 7000 feet per second. But the celebrated Euler, in his commentary on this part of Mr. Robins's book, thinks it may be still much greater, and in this proposition too it is that Mr. Robins declares his opinion above alluded to, viz. that the effect of the powder upon the recoil of the gun is the same in all cases whether fired with a ball or without one. In the twelfth proposition he ascertains the manner in which the flame of powder impels a ball which is laid at a considerable distance from the charge; showing here that the sudden accumulation and density of the fluid against the ball is the reason that the barrel is so often burst in those cases. In the thirteenth proposition he enumerates the various kinds of powder, and describes the properest methods of examining its goodness. He here shows that the best proportion of the ingredients is when the saltpetre is three-fourths of the whole compound mass of the powder, and the sulphur and charcoal the other one-fourth between them, in equal quantities. In this proposition Mr. Robins takes occasion to remark upon the use of eprouvettes, or methods of trying powder; conemning the practice of the English in using what is called the vertical eprouvette; as well as that of the French, in using a small mortar with a very large ball, and a small charge of powder, and instead of these he strongly recommends the use of his ballistic pendulum for its great accuracy. But for still more despatch, he says, he should use another method, which however he
reserves to himself without giving any particular description of it. The other, or second chapter of Mr. Robins's work, in eight propositions, treats ‘of the resistance of the air, and of the track described by the flight of shot and shells.’ And of these, the first proposition describes the general principles of the resistance of fluids to solid bodies moving in them. Iłere Mr. Robins discriminates between continued and compressed fluids, which immediately rush into the space quitted by a body in them, and whose parts yield to the impulse of the body without condensing and accumulating before it; and such fluids as are imperfectly compressed, rushing into a void space, with a limited velocity, as in the case of our atmosphere, which condenses more and more before the ball as this moves quicker, and also presses the less behind it, by following it always with only a given velocity: hence it happens that the former fluid will resist moving bodies in proportion to the square of the velocity, while the latter resists in a higher proportion. The second proposition is “to determine the resistance of the air to projectiles by experiments.” One of the methods for this purpose is by the ballistic pendulum, placing the gun at different distances from it, by which he finds the velocity lost in passing through certain spaces of air, and consequently the force of resistance to such velocities as the body moves with in the several parts of its path. And another way was by firing balls with a known given velocity, over a large piece of water, in which the fall and plunge of the ball could be seen, and consequently the space it assed over in a given time. By these means Mr. Robins determined the resistance of the air to several different velocities, all which showed that there was a gradual increase of the resistance, over the law of the square of the velocity, as the body moved quicker. In the remaining propositions of this chapter he proceeds a little farther in this subject of the resistance of the air; in which he lays down a rule for the proportion of the resistance between two assigned velocities; and he shows that when a twenty-four pound ball, fired with its full charge of powder, first issues from the piece, the resistance it meets with from the air is more than twenty times its weight. He farther shows that the track described by the flight of shot or shells is neither a parabola, nor nearly a parabola, unless they are projected with small velocities; and that “bullets in their flight are not only depressed beneath their original direction, by the action of gravity, but are also frequently ão. to the right or left of that direction by the action of some other force:' and, in the eighth or last proposition, he pretends to show that the depths of penetration of balls into firm substances are as the squares of the velocities. But this is a mistake; for neither does it appear that his trials were sufficiently numerous or various, nor were his small leaden balls fit for this purpose; and it has appeared, from a number of trials with iron cannon balls, that the penetrations are in a much lower proportion, and that the resisting force of wood is not uniform. See Dr. Hutton's Tracts. In the small tracts appended to the principles, in this volume, Mr. Robins prosecutes the sub