GUNNERY,

Is the art of determining the motions or ranges of projectiles discharged from cannon, mortars, howitzers, and other kinds of artillery.

I.—THEORY OF GUNNERY.

The use of fire-arms had been long known before any theory concerning them was attempted. The first author who wrote professedly on the flight of cannon-shot was Tartalea. In 1537 he published a book, at Venice, entitled Nova Scientia; and afterwards another, printed at the same place in 1546, in which he treats of these motions. His discoveries were but few, on account of the imperfect state of mechanical knowledge at that time. He determined, however, that the greatest range of cannon was with an elevation of forty-five degrees; and he likewise ascertained, contrary to the opinion of practitioners, that no part of the track described by a bullet is a right line, although the curvature is in some cases so small that it is not attended to. He compared it to the surface of the sea, which, though it appears to be a plane, is yet undoubtedly incurvated round the centre of the earth. He also assumes to himself the invention of the gunner’s quadrant, and often makes shrewd guesses as to the results of untried methods. But as he had not opportunities of observing practice, and founded his opinions solely on speculation, he was condemned by most of the succeeding writers, though often without any sufficient reason. The philosophers of those times til so intermeddled in the questions which hence arose; and many disputes on motion occurred, especially in Italy, where they continued till the time of Galileo, and probably gave rise to his celebrated Dialogues on Motion. These were published in the year 1638; but in the interval, and before Galileo’s doctrine was thoroughly established, many theories of the motion of military projectiles, and many tables of their comparative ranges at different elevations, were published; all of them egregiously fallacious, and utterly irreconcilable with the motions of these bodies. Many of the ancients indeed indulged in speculations concerning the difference between natural, violent, and mixed motions; but when they did so, scarcely two of them could agree in their theories.

It is strange, however, that, during all these contests, so few of those who were intrusted with the charge of artillery thought it worth while to bring these theories to the test of experiment. Mr Robins informs us, in the preface to his New Principles of Gunnery, that he had met with no more than four authors who had treated on this subject. The first of these is Collado, who has given the ranges of a falconet carrying a three-pound shot to each point of the gunner’s quadrant. But, from his numbers, it is manifest that the piece was not charged with its customary allotment of gunpowder. The results of his trials were, that the point-blank shot, or that in which the path of the ball did not sensibly deviate from a right line, extended 268 paces. At an elevation of one point (or 7∣ o of the gunner’s quadrant), the range was 594 paces; at an elevation of two points, 794 paces; at three points, 954 paces; at four, 1010; at five, 1040; and at six, 1053 paces. At the seventh point, the range fell between those of the third and fourth; at the eighth point it fell between the ranges of the second and third; at the ninth point, it fell between the ranges of the first and second; at the tenth point, it fell between the point-blank distance and that of the first point; and at the eleventh point, it fell very near the piece. The paces spoken of by this author are common steps.

The year after Collado’s treatise, another appeared on the same subject, by one Bourne, an Englishman. His elevations were not regulated by the points of the gunner’s quadrant, but by degrees; and he ascertained the proportions between the ranges at different elevations and the extent of point-blank shot. According to him, if the extent of the point-blank shot be represented by 1, the range at 5° elevation will be 2'^, at 10° it will be 3-∣, at 15° it will be 4⅜, at 20° it will be 4-į), and the greatest random will be 5⅜. This last, he tells us, happens in a calm day, when the piece is elevated to 42°; but according to the strength of the wind, and as it favours or opposes the flight of the shot, it may be from 45° to 36°. He has not informed us with what piece he made his trials, though from his proportions it seems to have been a small one. This however ought to have been attended to, as the relation between the extent of different ranges varies extremely according to the velocity and density of the bullet.

After him, Eldred and Anderson, both Englishmen, published treatises on this subject. The first published his treatise in 1646, and gave the actual ranges of different pieces of artillery at small elevations, all under ten degrees. His principles were not rigorously true, though not liable to very considerable errors; yet, in consequence of their deviation from the truth, he found it impossible to make some of his experiments agree with his principles.

In 1638, Galileo printed his Dialogues on Motion. In these he pointed out the general laws observed by nature in the production and composition of motion, and was the first who described the action and effects of gravity on falling bodies. On these principles he determined, that the flight of a cannon-shot, or any other projectile, would be ip the curve of a parabola, except in as far as it was diverted from that track by the resistance of the air. He has also proposed the means of examining the inequalities which thence arise, and of discovering what sensible effects that resistance would produce in the motion of a bullet at a given distance from the piece.

Though Galileo had thus shown, that, independently of the resistance of the air, all projectiles would, in their flight, describe the curve of a parabola; yet those who came after him seem never to have imagined that it was necessary to consider how far the operations of gunnery were affected by this resistance. The subsequent writers indeed boldly asserted, without making the experiment, that no considerable variation could arise 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.

In 1674, Mr Anderson above mentioned published his treatise on the nature and effects of the gun; in which he proceeds on the principles of Galileo, and strenuously asserts that the flight of bullets is in the curve of a parabola; undertaking to answer all objections which could be brought to the contrary. The same thing was also undertaken by Mr Blondel, in a treatise published at Paris in 1683, where, after long discussion, the author concludes that the variations from the resistance of the air are so slight as scarcely to merit notice. The same subject is treated of in the Philosophical Transactions (No. 216, p.
[7:11:43]

68) by Dr Halley; and he also, swayed by the great disproportion between the density of the air and that of iron or lead, thinks it reasonable to believe that the resistance of the air to large metal shot is scarcely discernible; although in small and light shot he owns that it must be accounted for.

But though this hypothesis went on smoothly in speculation, yet Anderson, who made a great number of trials, found it impossible to support it without some new modification. For though it does not appear that he ever examined the comparative ranges of either cannon or musket shot when fired with their usual velocities, yet his experiments on the ranges of shells thrown with small velocities, in comparison of those above mentioned, convinced him that their whole track was not parabolical. But instead of drawing the proper inferences from this, and concluding that the resistance of the air was of considerable efficacy, he framed a new hypothesis, which was, that the shell or bullet, at its first discharge, flew to a certain distance in a right line, from the end of which line only it began to describe a parabola. And this right line, which he calls the line of the impulse of the fire, he supposes to be the same in ail elevations. Thus, by assigning a proper length to this line of impulse, it was always in his power to reconcile any two shots made at different angles, let them differ as widely as we may please to suppose. But this he could not have done with three shots; nor indeed does he ever tell us the result of his experiments when three ranges were tried at one time.

When Sir Isaac Newton’s Principia was published, he particularly considered the resistance of the air to projectiles which moved with small velocities; but as he never had an opportunity of making experiments on those which move with such prodigious swiftness as shots and shells, he did not imagine that a difference in velocity could make such differences in the resistance as are now found to take place. Sir Isaac found, that, in small velocities, the resistance was increased in the duplicate proportion of the swiftness with which the body moved; that is, a body moving with twice the velocity of another of equal magnitude, would meet with four times as much resistance as the first; with thrice the velocity it would meet with nine times the resistance; and so on. This principle itself is now found to be defective with regard to military projectiles; though, if it had been properly attended to, the resistance of the air might have been reckoned much more considerable than was commonly imagined. So far, however, were those who treated this subject scientifically from giving a proper allowance for the resistance of the atmosphere, that their theories differed most egregiously from the truth. Huygens alone seems to have attended to this principle. In the year 1690, 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. All the rest of the learned acquiesced in the justness of Galileo’s doctrine, and erroneous calculations concerning the ranges of cannon were accordingly given. Nor was any notice taken of these errors till the year 1716. At that time Ressons, a French officer of artillery, distinguished by the number of sieges at which he had served, by his high military rank, and by his abilities in his profession, presented a memoir to the Royal Academy, importing, that, “although it was agreed that theory joined with practice did constitute the perfection of every art, yet experience had taught him, that theory was of very little service in the use of mortars; that the works of Blondel had justly enough described the seve r ral parabolic lines, according to the different degrees of the elevation of the piece; but that practice had convinced him there was no known theory for the effect of gunpowder; for having endeavoured, with the greatest precision,' to point a mortar agreeably to these calculations, he had never been able to establish any solid foundation upon them.”

From the history of the academy, it does not appear that the sentiments of Ressons were at any time controverted, or any reason offered for the failure of the theory of projectiles when applied to use. Nothing further, however, was done till the time of Benjamin Robins, who, in 1742, published a work entitled New Principles of Gunnery, in which he has treated particularly, not only of the resistance of the atmosphere, but almost every thing else relating to the flight of military projectiles, and indeed advanced the theory of gunnery much nearer perfection than ever it had before attained.

The first thing considered by Mr Robins, and which is indeed the foundation of all other particulars relative to gunnery, is the explosive force of gunpowder. This he determined to be owing to an elastic fluid similar to our atmosphere, having its elastic force greatly increased by the heat. “If a red-hot iron,” says he, “be included in a receiver, and the receiver be exhausted, and gunpowder be then let fall on the iron, the powder will take fire, and the mercurial gage will suddenly descend upon the explosion; and though it immediately ascends again, it will never rise to the height it first stood at, but will continue depressed by a space proportioned to the quantity of powder which was let fall on the iron. The same production likewise takes place when gunpowder is fired in the air: for if a small quantity of powder is placed in the upper part of a glass tube, the lower part of which is immersed in water, and the fluid be made to rise so near the top that only a small portion of air is left in that part where the gunpowder is placed; if in this situation the communication of the upper part of the tube with the external air is closed, and the gunpowder fired, which may be easily done by means of a burning-glass, the water will in this experiment descend on the explosion, as the quicksilver did in the last; and will always continue depressed below the place at which it stood before the explosion. The quantity of this depression will be greater if the quantity of powder be increased, or the diameter of the tube be diminished.

“When any considerable quantity of gunpowder is fired in an exhausted receiver, by being let fall on a red-hot iron, the mercurial gage instantly descends upon the explosion, and as suddenly ascends again. After a few vibrations, none of which except the first are of any great extent, it seemingly fixes at a point lower than where it stood before the explosion. But even when the gage has acquired this point of apparent rest, it still continues rising for a considerable time, although by such imperceptible degrees that it can only be discovered by comparing its place at distant intervals: however, it will not always continue to ascend, but will rise slower and slower, till at last it will be absolutely fixed at a point lower than where the mercury stood before the explosion. The same circumstances nearly happen when powder is fired in the upper part of an unexhausted tube, whose lower part is immersed in water.

“That the elasticity or pressure of the fluid produced by the firing of gunpowder is, cceteris paribus, directly as its density, may be proved from hence, that if in the same receiver a double quantity of powder be let fall, the mercury will subside twice as much as in the firing of a single quantity. Also the descents of the mercury, when equal quantities of powder are fired in different receivers, are reciprocally as the capacities of those receivers, and consequently as the density of produced fluid in each. But as, in the usual method of trying this experiment, the quantities of powder are so very small that it is difficult to ascertain these proportions with the requisite degree of exactness, I took a large receiver containing about 520 inches, and letting[7:11:44] fall at once on the red-hot iron one dram, or the sixteenth part of an ounce avoirdupois, of powder, the receiver being first nearly exhausted; the mercury, after the explosion, was subsided two inches exactly, and all the powder had taken fire. Then heating the iron a second time, and exhausting the receiver as before, two drams were let down at once, which sunk the mercury three inches and three quarters; and a small part of the powder had fallen beside the iron, which (the bottom of the receiver being wet) did not fire, and the quantity which thus escaped did appear to be nearly sufficient, had it fallen on the iron, to have sunk the mercury a quarter of an inch more; in which case the two descents, viz. two inches and four inches, would have been accurately in the proportion of the respective quantities of powder; from which proportion, as it was, they very little varied.

“As different kinds of gunpowder produce different quantities of this fluid, in proportion to their different degrees of goodness, before any definite determination of this kind can take place, it is necessary to ascertain the particular species of powder that is proposed to be used. (Here Mr Robins determines in all his experiments to make use of government powder, as consisting of a certain and invariable proportion of materials, and therefore preferable to such kinds as are made according to the fancy of private persons).

“This being settled, we must further premise these two principles: 1. That the elasticity of this fluid increases by heat and diminishes by cold, in the same manner as that of the air: 2. That the density of this fluid, and consequently its weight, is the same with the weight of an equal bulk of air, having the same elasticity and the same temperature. Now, from the last experiment it appears, that ∙⅛th of an ounce avoirdupois, or about 27 grains troy, of powder, sunk the gage, on its explosion, two inches; and the mercury in the barometer standing at near 30 inches, I∣ths of an ounce avoirdupois, or 410 grains troy, would have filled the receiver with a fluid whose elasticity would have been equal to the whole pressure of the atmosphere, or the same with the elasticity of the air we breathe; and the contents of the receiver being about 520 cubic inches, it follows, that γfths of an ounce of powder will produce 520 cubic inches of a fluid possessing the same degree of elasticity with the ’ common air; whence an ounce of powder will produce near 575 cubic inches of such a fluid.

“But in order to ascertain the density of this fluid, we must consider what part of its elasticity, at the time of this determination, was owing to the heat it received from the included hot iron and the warm receiver. Now the general heat of the receiver being manifestly less than that of boiling water, which is known to increase the elasticity of the air to somewhat more than ⅜th of its augmented quantity, I collect from hence and other circumstances, that the augmentation of elasticity from this cause was about ∣th of the whole; that is, if the fluid arising from the explosion had been reduced to the temperature of the external air, the descent of the mercurial gage, instead of two inches, would have been only l£th inch; whence 575, reduced in the proportion of five to four, becomes 460; and this last number represents the cubic inches of an elastic fluid equal in density and elasticity with common air, which are produced from the explosion of one ounce avoirdupois of gunpowder; the weight of which quantity of fluid, according to the usual estimation of the weight of air, is 131 grains; whence the weight of this fluid is ⅜H or 1 ⅜ths nearly of the weight of the generating powder. The ratio of the bulk of gunpowder to the bulk of this fluid may be determined from considering that 17 drams avoirdupois of powder fill two cubic inches, if the powder be well shaken together: therefore, augmenting the number last found in the proportion of 16 to 17, the resulting term, 488∣, is the number of cubic inches of an elastic fluid, equal in density with the air produced from two cubic inches of powder: whence the ratio of the respective bulk of the powder, and of the fluid produced from it, is in round numbers as 1 to 244.” This calculation was afterwards confirmed by experiments.

“ If this fluid, instead of expanding when the powder was fired, had been confined in the same space which the powder filled before the explosion, then it would have had, in that confined state, a degree of elasticity 244 times greater than that of common air; and this independent of the great augmentation which this elasticity would receive from the action of the fire in that instant.

“ Hence, then, we are certain, that any quantity of powder, fired in a confined space, which it adequately fills, exerts, at the instant of its explosion, against the sides of the vessel containing it, and the bodies it impels before it, a force at least 244 times greater than the elasticity of common air, or, which is the same thing, than the pressure of the atmosphere; and this without considering the great addition which this force will receive from the violent degree of heat with which it is affected at that time.

“To determine how far the elasticity of air is augmented when heated to the extremest degree of red-hot iron, I took a piece of a musket-barrel about six inches in length, and ordered one end to be closed up entirely; but the other end was drawn out conically, and finished in an aperture of about ģth of an inch in diameter. The tube thus fitted was heated to the extremity of a red heat in a smith’s forge, and was then immérsed with its aperture downwards in a bucket of water, and kept there till it was cool, after which it was taken out carefully, and the water which had entered it in cooling was exactly weighed. The heat given to the tube at each time was the beginning of what workmen call a white heat ; and to prevent the rushing in of the aqueous vapour at the immersion, which would otherwise drive out great part of the air, and render the experiment fallacious, I had an iron wire filed tapering so as to fit the aperture of the tube, and with this I always stopped it up before it was taken from the fire, letting the wire remain in till the whole was cool, when, removing it, the due quantity of water would enter. The weight of the water thus taken in at three different trials was 610 grains, 595 grains, and 600 grains, respectively. The content of the whole cavity of the tube was 796 grains of water, whence the spaces remaining unfilled in these three experiments were 186, 201, and 196 grains respectively. These spaces undoubtedly contained all the air which, when the tube was red hot, extended through its whole concavity; consequently the elasticity of the air, when heated to the extreme heat of red-hot iron, was to the elasticity of the same air, when reduced to the temperature of the ambient atmosphere, as the whole capacity of the tube to the respective spaces taken up by the cooled air; that is, as 796 to 186, 201, 196; or, taking the medium of these three trials, as 796 to 194∣.

“As air and this fluid appear to be equally affected by heat and cold, and consequently have their elasticities equally augmented by the addition of equal degrees of heat to each; if we suppose the heat with which the flame of fired powder is endowed to be the same with that of the extreme heat of red-hot iron, then the elasticity of the generated fluid will be greater at the time of the explosion than afterwards, when it is reduced to the temperature of the ambient air, in the ratio of 796 to 194^ nearly. It being allowed then (which surely is very reasonable) that the flame of gunpowder is not less hot than red-hot iron, and the elasticity of the air, and consequently of the fluid, generated by the explosion, being augmented in the extremity of this heat in the ratio of 194-∣ to 796, it follows, that if 244 be augmented in this ratio, the resulting number, which is 999-*·, will determine how many times the [7:11:45]elasticity of the flame of fired powder exceeds the elasticity of common air, supposing it to be confined in the same space which the powder filled before it was fired. Hence then the absolute quantity of the pressure exerted by gunpowder at the moment of its explosion may be assigned; for, since the fluid then generated has an elasticity of 999 or in round numbers 1000 times greater than that of the atmosphere, and since common air by its elasticity exerts a pressure on any given surface equal to the weight of the incumbent atmosphere with which it is in equilibrio, the pressure exerted by fired powder before it dilated itself is 1000 times greater than the pressure of the atmosphere; and consequently the quantity of this force, on a surface of an inch square, amounts to above six tons weight, which force, however, diminishes as the fluid dilates itself.

“But though we here supposed that the heat of gunpowder, when fired in any considerable quantity, is the same with iron heated to the extremity of red heat, or to the beginning of a white heat, yet it cannot be doubted but that the fire produced in the explosion is somewhat varied (like all other fires) by a greater or less quantity of fuel; and it may be presumed that, according to the quantity of powder fired together, the flame may have all the different degrees, from a languid red heat, to that sufficient for the vitrification of metals. But as the quantity of powder requisite for the production of this last-mentioned heat is certainly greater than what is ever fired together for any military purpose, we cannot be far from our scope if we suppose the heat of such quantities as are usually fired to be nearly the same with that of red-hot iron, allowing a gradual augmentation to this heat in larger quantities, and diminishing it when the quantities are very small.”

Having thus determined the force of the gunpowder, Mi-Robins next proceeds to determine the velocity with which the ball is discharged. The solution of this problem depends on the two following principles: 1. That the action of the powder on the bullet ceases as soon as the bullet is got out of the piece: 2. That all the powder of the charge is fired and converted into elastic fluid before the bullet is sensibly removed from its place.

“The first of these,” says Mr Robins, “will appear manifest when it is considered how suddenly the flame will extend itself on every side, by its own elasticity, when it is once got out of the mouth of the piece; for by this means its force will then be dissipated, and the bullet no longer sensibly affected by it.

“The second principle is indeed less obvious, being contrary to the general opinion of almost all writers on this subject. It might, however, be sufficient for the proof of this position, to observe the prodigious compression of the flame in the chamber of the piece. Those who attend to this circumstance, and to the easy passage of the flame through the intervals of the grains, may soon satisfy themselves that no one grain contained in that chamber can continue for any time uninflamed, when thus surrounded and pressed by such an active fire. However, not to rely on mere speculation in a matter of so much consequence, I considered that if part only of the powder is fired, and that successively; then, by laying a greater weight before the charge (suppose two or three bullets instead of one), a greater quantity of powder would necessarily be fired, since a heavier weight would be a longer time in passing through the barrel. Whence it should follow that two or three bullets would be impelled by a much greater force than one only. But the contrary to this appears by experiment; for, firing one, two, and three bullets laid contiguous to each other with the same charge respectively, I have found that their velocities were not much different from the reciprocal of their subduplicate quantities of matter; that iš, if a given charge would communicate to one bullet a velocity of 1700 feet in a second, the same charge would communicate to two bullets a velocity of from 1250 to 1300 feet in a second, and to three bullets a velocity of from 1050 to 1110 feet in the same time. From hence it appears, that whether a piece is loaded with a greater or less weight of bullet, the action is nearly the same; since all mathematicians know, that if bodies containing different quantities of matter are successively impelled through the same space by the same power acting with a determined force at each point of that space, then the velocities given to these different bodies will be reciprocally in the subduplicate ratio of their quantities of matter. The excess of the velocities of the two and three bullets above what they ought to have been by this rule (which are that of 1200 and 980 feet in a second) undoubtedly arises from the flame, which, escaping by the side of the first bullet, acts on the surface of the second and third.

“Now this excess has in many experiments been imperceptible, and the velocities have been reciprocally in the subduplicate ratios of the number of bullets, to sufficient exactness; and where this error has been greater, it has never arisen to an eighth part of the whole; but if the common opinion was true, that a small part only of the powder fires at first, and other parts of it successively as the bullet passes through the barrel, and that a considerable part of it is often blown out of the piece without firing at all; then the velocity which three bullets received from the explosion ought to have been much greater than we have found it to be. But the truth of the second postulate more fully appears from those experiments, by which it is shown that the velocities of bullets may be ascertained to the same exactness when they are acted on through a barrel of four inches in length only, as when they are discharged from one of four feet.

“With respect to the grains of powder which are often blown out un fired, and which are always urged as a proof of the gradual firing of the charge, I believe Diego Uffano, a person of great experience in the art of gunnery, has given the true reason for this accident; which is, that some small part of the charge is often not rammed up with the rest, but is left in the piece before the wad, and is by this means expelled by the blast of air before the fire can reach it. I must add, that in the charging of cannon and small arms, especially after the first time, this is scarcely to be avoided by any method I have yet seen practised. Perhaps, too, there may be some few grains in the best powder of such an heterogeneous composition as to be less susceptible of firing; which, I think, I have myself observed; and these, though they are surrounded by the flame, may be driven out unfired.

“These postulates being allowed to be just, let AB (Plate CCLXXIV. fig. 12) represent the axis of any piece of artillery, A the breech, and B the muzzle; DC the diameter of its bore, and DEGC a part of its cavity filled with powder. Suppose the ball that is to be impelled to lie with its hinder surface at the line GE; then the pressure exerted at the explosion on the circle of which GE is the diameter, or, which is the same thing, the pressure exerted in the direction FB on the surface of the ball, is easily known from the known dimensions of that circle. Draw any line FH perpendicular to FB, and AI parallel to FH; and through the point II, to the asymptotes IA and AB, describe the hyperbola KHNQ; then, if FH represents the force impelling the ball at the point F, the force impelling the ball at any other point, as at M, will be represented by the line MN, the ordinate to the hyperbola at that point. For when the fluid impelling the body along has dilated itseli to M, its density will be then to its original density in the space DEGC reciprocally as the spaces through which it is extended; that is, as FA to MA, or as MN to FH; but it has been shown that the impelling force or elasticity of this fluid is directly as its density, therefore, if FH [7:11:46] represents the force at the point F, MN will represent the like force at the point M.

“Since the absolute quantity of the force impelling the ball at the point F is known, and the weight of the ball is also known, the proportion between the force with which the ball is impelled and its own gravity is known. In this proportion take FH to FL, and draw LP parallel to FB; then, MN the ordinate to the hyperbola in any point will be to its part MR, cut off by the line LP, as the impelling force of the powder in that point M to the gravity of the ball; and consequently the line LP will determine a line proportional to the uniform force of gravity in every point; whilst the hyperbola HNQ determines in like manner such ordinates as are proportional to the impelling force of the powder in every point; whence, by the 39th Prop, of lib. I of Sir Isaac Newton’s Principia, the areas FLPB and FHQB are in the duplicate proportion of the velocities which the ball would acquire when acted upon by its own gravity through the space FB, and when impelled through the same space by the force of the powder. But since the ratio of AF to AB and the ratio of FH to FL are known, the ratio of the area FLPB to the area FHQB is known; and thence its subduplicate. And since the line FB is given in magnitude,-the velocity which a heavy body would acquire when impelled through this line by its own gravity is known; being no other than the velocity it would acquire by falling through a space equal to that line: find then another velocity to which this last-mentioned velocity bears the given ratio of the subduplicate of the area FLPB to the area FHQB; and this velocity thus found is the velocity the ball will acquire when impelled through the space FB by the action of the inflamed powder.

“ Now, to give an example of this: Let us suppose AB, the length of the ι cylinder, to be 45 inches; its diameter DC, or rather the diameter of the ball, to be fths of an inch; and AF, the extent of the powder, to be 2⅛th inches; to determine the velocity which will be communicated to a leaden bullet by the explosion, supposing the bullet to be laid at first with its surface contiguous to the powder.

“By the theory we have laid down, it appears, that at the first instant of the explosion the flame will exert, on the bullet lying close to it, a force 1000 times greater than the pressure of the atmosphere. The medium pressure of the atmosphere is reckoned equal to a column of water 33 feet in height; whence, lead being to water as 11,345 to 1, this pressure will be equal to that of a column of lead 34,9 inches in height. Multiplying this by 1000, therefore, a column of lead 34,900 inches (upwards of half a mile) in height, would produce a pressure on the bullet equal to what is exerted by the powder in the first instant of the explosion; and the leaden ball being ∣ths of an inch in diameter, and consequently equal to a cylinder of lead of the same base half an inch in height, the pressure at first acting on it will be equal to 34,900 × 2, or 69,800 times its weight; whence FL to FH is as 1 to 69,800; and FB to FA as 45 — 2ĝ, or 42∣∙ to 2į, that is, as 339 to 21; whence the rectangle FLPB is to the rectangle ĄFHS as 339 to 21 × 69,800, that is, as 1 to 4324. And from the known application of the logarithms to the mensuration of the hyperbolic spaces, it follows that the rectangle AFHS is to the area FHQB as 43,429, &c. is to the tabular loga-

AB

rithm of; that is, of 3 1d i , which is 1,2340579: whence the ratio of the rectangle FLPB to the hyperbolic area FHQB is compounded of the ratios of 1 to 4324— and of ,43429, &c. to 1,2340579; which together make up the ratio of 1 to 12,263, the subduplicate of which is the ratio of 1 to 110,7; and in this ratio is the velocity which the bullet would acquire by gravity in falling through a space equal to FB, to the velocity the bullet will acquire from the action of the powder impelling it through FB. But the space FB being 42g∙ inches, the velocity a heavy body will acquire in ŕàlling through such a space is known to be what would carry it nearly at the rate of 15∙07 feet in a second; whence the velocity to which this has the ratio of 1 to 110,7 is a velocity which would carry the ball at the rate of 1668 feet in one second. And this is the velocity which, according to the theory, the bullet in the present circumstances would acquire from the action of the powder during the time of its dilatation.

“Now this velocity being once computed for one case, is easily applied to any other; for if the cavity DEGC left behind the bullet be only in part filled with powder, then the line HF, and consequently the area FHQB, will be diminished in the proportion of the whole cavity to the part filled. If the diameter of the bore be varied, the lengths AB and AF remaining the same, then the quantity of powder and the surface of the bullet which it acts on will be varied in the duplicate proportion of the diameter, but the weight of the bullet will vary in the triplicate proportion of the diameter; wherefore the line FH, which is directly as the absolute impelling force of the powder, and reciprocally as the gravity of the bullet, will change in the reciprocal proportion of the diameter of the bullet. If AF, the height of the cavity left behind the bullet, be increased or diminished, the rectangle of the hyperbola, and consequently the area corresponding to ordinates in any given ratio, will be increased or diminished in the same proportion. From all which it follows, that the area FHQB, which is in the duplicate proportion of the velocity of the

AB impelled body, will be directly as the logarithm —, (where

AB represents the length of the barrel, and AF the length of the cavity left behind the bullet); also directly as the part of that cavity filled with powder, and inversely as the diameter of the bore, or rather of the bullet; likewise directly as AF, the height of the cavity left behind the bullet. Consequently the velocity being computed as above, for a bullet of a determined diameter, placed in a piece of a given length, and impelled by a given quantity of powder, occupying a given cavity behind that bullet; it follows, that by means of these ratios, the velocity of any other bullet may be thence deduced; the necessary circumstances of its position, quantity of powder, &c. being given. Where note, that in the instance of this supposition, we have supposed the diameter of the ball to be 4ths of an inch; whence the diameter of the bore will be something more, and the quantity of powder contained in the space DEGC will amount exactly to twelve pennyweights, a small wad of tow included.

“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 discovered. The only methods hitherto practised for this purpose, have been either by observing the time of the flight of a shot through a given space, or by measuring the range of a shot at a given elevation; and thence computing, on the parabolic hypothesis, what degree of velocity would produce this range. The first method labours 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 2, 3, 4, 5, or 600 feet, in a second. The other method is so fallacious, by reason of the resistance of the atmosphere (to which inequality the first is also liable), that the velocities thus assigned may not perhaps be the tenth part of the actual velocities sought.

“The simplest method of determining this velocity is [7:11:47]by means of the instrument represented fig. 13, where ABCD represents the body of the machine composed of the three poles B, C, D, spreading at bottom, and joining together at the top A; being the same with what is vulgarly used in lifting and weighing very heavy bodies, and is called by workmen the triangles. On two of these poles, towards their tops, are screwed on the sockets R S; and on these sockets the pendulum EFGHIK is hung by means of its cross-piece EF, which becomes its axis of suspension, and on which it must be made to vibrate with great freedom. The body of this pendulum is made of iron, having a broad part at bottom, and its lower part is covered with a thick piece of wood GKIH, which is fastened to the iron by screws. Something lower than the bottom of the pendulum there is a brace OP, joining the two poles from which the pendulum is suspended; and to this brace there is fastened a contrivance MNU, made with two edges of steel, bearing on each other in the line UN, something in the manner of a drawing-pen; the strength with which these edges press on each other being diminished or increased at pleasure by means of a screw Z going through the upper piece. There is fastened to the bottom of the pendulum a narrow ribbon LN, which passes between these steel edges, and which afterwards, by means of an opening cut in the lower piece of steel, hangs loosely down, as at W.

“With this apparatus, 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 known 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 a known weight, and the vibration which the pendulum makes after the blow is known, the velocity of the striking body may from 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.

“Now the extent of the vibration made by the pendulum after the blow, may be measured to great accuracy by the ribbon LN. For let the pressure of the edges UN on the ribbon be so regulated by the screw Z, that the motion of the ribbon between them may be free and easy, though with some minute resistance; then, settling the pendulum at rest, let the part LN between the pendulum and the edges be drawn strait, but not strained, and fix a pin in that part of the ribbon which is then contiguous to the edges: let now a ball impinge on the pendulum; then the pendulum swinging back will draw out the ribbon to the just extent of its vibration, which will consequently be determined by the interval on the ribbon between the edges UN and the place of the pin.

“The weight of the whole pendulum, wood and all, was 56 pounds 3 ounces; its centre of gravity was 52 inches distant from its axis of suspension, and 200 of its small swings were performed in the time of 253 seconds: whence its centre of oscillation (determined from hence) in 62^d inches distant from that axis. The centre of the piece of wood GKIH is distant from the same axis 66 inches.

“In the compound ratio of 66 to 62£d, and 66 to 52, take the quantity of matter of the pendulum to a fourth quantity, which will be 42 lb. į oz. Now geometers well know, that if the blow be struck on the centre of the piece of wood GKIH, the pendulum will resist to the stroke in the same manner as if this last quantity of matter only (42 lb. I oz.) was concentrated in that point, and the rest of the pendulum was taken away: whence, supposing the weight of the bullet impinging in that point to be the ⅛th of a pound, or the 3 ⅛ 3 ∙th of this quantity of matter nearly, the velocity of the point of oscillation after the stroke will, by the laws observed in the congress of such bodies as rebound not from each other, be the y⅛ 5 -th of the velocity the bullet moved with before the stroke; whence the velocity of this point of oscillation after the’ stroke being ascertained, that multiplied by 505 will give the velocity with which the ball impinged.

“But the velocity of the point of oscillation after the stroke is easily deduced from the chord of the arch, through which it ascends by’the blow; for it is a well-known proposition, that all pendulous bodies ascend to the same height by their vibratory motion as they would do if they were projected directly upwards from their lowest point, with the same velocity they have in that point; wherefore, if the versed sine of the ascending arch be found (which is easily determined from the chord and radius being given), this versed sine is the perpendicular height to which a body projected upwards with the velocity of the point of oscillation would arise; and consequently what that velocity is, can be easily computed by the common theory of falling bodies.

“For instance, the chord of the arch, described by the ascent of the pendulum after the stroke measured on the ribbon, has been sometimes 17^th inches; the distance of the ribbon from the axis of suspension is 71∣th inches; whence reducing 17∣th in the ratio of 71ģth to 66, the resulting number, which is nearly 16 inches, will be the chord of the arch through which the centre of the board GKIH ascended after the stroke; now the versed sine of the arch, whose chord is 16 inches, and its radius 66, is 1∙93939; and the velocity which would carry a body to this height, or, which is the same thing, the velocity which a body would acquire by descending through this space, is nearly that of 3-įth feet in 1".

“To determine then the velocity with which the bullet impinged on the centre of the wood, when the chord of the arch described by the ascent of the pendulum, in consequence of the blow, was 17¼th inches measured on the ribbon, no more is necessary than to multiply 3¼th by 505, and the resulting number, 1641, will be the feet which the bullet would describe in 1", if it moved with the velocity it had at the moment of its percussion: for the velocity of the point of the pendulum on which the bullet struck, we have just now determined to be that of 3¼th feet in 1"; and we have before shown, that this is the 1 /505 th of the velocity of the bullet. If then a bullet weighing 1 /12 th of a pound strikes the pendulum in the centre of the wood GKIH, and the ribbon be drawn out 17¼th inches by the blow, the velocity of the bullet is that of 1641 feet in 1". And since the length the ribbon is drawn is always nearly the chord of the arch described by the ascent (it being placed so as to of fcr insensibly from those chords which most frequently occur), and these chords are known to be in the proportion of the velocities of the pendulum acquired from the stroke; it follows that the proportion between the lengths of ribbon drawn out at different times will be the same with that of the velocities of the impinging bullets and consequently, by the proportion of these lengths of ribbon to 17¼th, the proportion of the velocity with which the bullets impinge, to the known velocity of 1641 feet in 1", will be determined.

“Hence then is shown in general how the velocities of bullets of all kinds may be found out by means of this instrument; but that those who may be disposed to try these experiments may not have unforeseen difficulties to struggle with, we shall here subjoin a few observations, which [7:11:48]it will be necessary for them to attend to, both to secure success to their trials and safety to their persons.

“And, first, that they may not conceive the piece of wood GKIH to be an unnecessary part of the machine, we must inform them, that if a bullet impelled by a full charge of powder should strike directly on the iron, the bullet would be beaten into shivers by the stroke, and these shivers would rebound back with such violence as to bury themselves in any wood they chanced to light on, as I have found by hazardous experience; and, besides the danger, the pendulum will not in this instance ascertain the velocity of the bullet, because the velocity with which the parts of it rebound is unknown.

“The weight of the pendulum and the thickness of the wood must be in some measure proportioned to the size of the bullets which are used. A pendulum of the weight here described will do very well for all bullets under three or four ounces, if the thickness of the board be increased to seven or eight inches for the heaviest bullets; beech is the toughest and properest wood for this purpose.

“It is hazardous standing on the side of the pendulum, unless the board be so thick that the greatest part of the bullet’s force is lost before it conies at the iron; for if it strikes the iron with violence, the shivers of lead which cannot return back through the wood, will force themselves out between the wood and iron, and will fly to a considerable distance.

“As there is no effectual way of fastening the wood to the iron but by screws, the heads of which must come through the board, the bullets will sometimes light on those screws, from whence the shivers will disperse themselves on every side.

“When in these experiments so small a quantity of powder is used, as will not give to the bullet a velocity of more than 400 or 500 feet in 1", the bullet will not stick in the wood, but will rebound from it entire, and (if the wood be of a very hard texture) with a very considerable velocity. Indeed I have never examined any of the bullets which have thus rebounded, but I have found them indented by the bodies they have struck against in their rebound.

“To avoid then these dangers, to the braving of which in philosophical researches no honour is annexed, it will be convenient to fix whatsoever barrel is used on a strong heavy carriage, and to fire it with a little slow match. Let the barrel too be very well fortified in all its length; for no barrel (I speak of musket barrels) forged with the usual dimensions will bear many of the experiments without bursting. The barrel I have most relied on, and which I procured to be made on purpose, is nearly as thick at the muzzle as at the breech; that is, it has in each place nearly the diameter of its bore in thickness of metal.

“The powder used in these experiments should be exactly weighed; and that no part of it be scattered in the barrel, the piece must be charged with a ladle, in the same manner as is practised with cannon; the wad should be of tow, of the same weight each time, and no more than is just necessary to confine the powder in its proper place; the length of the cavity left behind the ball should be determined each time with exactness; for the increasing or diminishing that space will vary the velocity of the shot, although the bullet and quantity of powder be not changed. The distance of the mouth of the piece from the pendulum ought to be such, that the impulse of the flame may not act on the pendulum; this will be prevented in a common barrel charged with half an ounce of powder, if it be at the distance of 16 or 18 feet: in larger charges the impulse is sensible farther off; I have found it to extend to above 25 feet; however, between 25 and 18 feet is the distance I have usually chosen.”

With this instrument, or others similar to it, Mr Robins made a great number of experiments on barrels of different lengths, and with different charges of powder. He has given us the results of sixty-one of these; and having compared the actual velocities with the computed ones, his theory appears to have come as near the truth as could well be expected. In seven of the experiments there was a perfect coincidence; the charges of powder being 6 to 12 pennyweights, the barrels 45, 24∙312, and 7∙06 inches in length. The diameter of the first (marked A) was į ths of an inch; of the second (B) was the same; and of D, 83 of inch. In the first of these experiments, another barrel (C) was used, whose length was 12∙375 inches, and the diameter of its bore ∣th inch. In fourteen more of the experiments, the difference between the length of the chord of the pendulum’s arch shown by the theory and the actual experiment was τ 1 θth of an inch over or under. This showed an error in the theory, varying, according to the different lengths of the chord, from τ 1 τ th to ^ 7 th of the whole; the charges of powder were the same as in the last. In sixteen other experiments the error was -⅛ths of an inch, varying from ⅛th to ⅛th of the whole; the charges of powder were 6, 8, 9, or 12 pennyweights. In seven other experiments the error was ⅛ths of an ’ inch, varying from ^ 1 2 d to 2 ⅛d of the whole; the charges of powder 6 or 12 pennyweights. In eight experiments the difference was γ ’ υ ths of an inch, indicating an error of from ⅛d to ⅛d of the whole; the charges being 6, 9, 12, and 24 pennyweights of powder. In three experiments the error was ∕ σ ths, varying from 3 ⅛th to τ 1 2 th of the whole; the charges 8 and 12 pennyweights of powder. In two experiments the error was -j⅝ths, in one case amounting to something less than ⅛d, in the other to 2 1 2 d of the whole; the charges 12 and 36 pennyweights of powder. By one experiment the error was seven, and by another eight, tenths; the first amounting to r 1 σ th nearly, the latter to almost ⅜th of the whole; the charges of powder 6 or 12 pennyweights. The last error, however, Mr Robins ascribes to the wind. The two remaining experiments varied from theory by 1∙3 inches, somewhat more than ⅜th of the whole; the charges of powder were 12 pennyweights in each; and Mr Robins ascribes the error to the dampness of the powder. In another case he ascribes an error of -⅛ths to the blast of the powder on the pendulum.

From these experiments Mr Robins deduces the following conclusions. “The variety of these experiments, and the accuracy with which they correspond to the theory, leave us no room to doubt of its certainty. This theory, as here established, supposes that, in the firing of gunpowder, about ∕ σ ths of its substance is converted by the sudden inflammation into a permanently elastic fluid, whose elasticity, in proportion to its heat and density, is the same with that of common air in the like circumstances: it farther supposes, that all the force exerted by gunpowder in its most violent operations, is no more than the action of the elasticity of the fluid thus generated; and these principles enable us to determine the velocities of bullets impelled from fire-arms of all kinds, and are fully sufficient for all purposes where the force of gunpowder is to be estimated.

“From this theory many deductions may be made of the greatest consequence to the practical part of gunnery. From hence the thickness of a piece, which will enable it to confine, without bursting, any given charge of powder, is easily determined, since the effort of the powder is known. From hence appears the inconclusiveness of what some modern authors have advanced, relating to the advantages of particular forms of chambers for mortars and cannon; for all their laboured speculations on this head are evidently founded on very erroneous opinions about the action of fired powder. From this theory too we are taught the necessity of leaving the same space behind the bullet, when we would, by the same quantity of powder, [7:11:49]communicate to it an equal degree of velocity; since, on the principles already laid down, it follows, that the same powder has a greater or less degree of elasticity, according to the different spaces it occupies. The method which I have always practised for this purpose has been by marking the rammer; and this is a maxim which ought not to be dispensed with when cannon are fired at an elevation, particularly in those called by the French batteries a ricohet.

“From the continued action of the powder, and its manner of expanding described in this theory, and the length and weight of the piece, one of the most essential circumstances in the well directing of artillery may be easily ascertained. All practitioners are agreed, that no shot can be depended on, unless the piece be placed on a solid platform; for if the platform shakes with the first impulse of the powder, it is impossible but the piece must also shake, which will alter its direction, and render the shot uncertain. To prevent this accident, the platform is usually made extremely firm to a considerable depth backwards; so that the piece is not only well supported in the beginning of its motion, but likewise through a great part of its recoil. However, it is sufficiently obvious, that when the bullet is separated from the piece, it can be no longer affected by the trembling of the piece or platform; and, by a very easy computation, it will be found that the bullet will be out of the piece before the latter hath recoiled half an inch; whence, if the platform be sufficiently solid at the beginning of the recoil, the remaining part of it may be much slighter; and hence a more compendious method of constructing platforms may be found out.

“From this theory also it appears how greatly these authors have been mistaken, who have attributed the force of gunpowder, or at least a considerable part of it, to the action of the air contained either in the powder or between the intervals of the grains; for they have supposed that air to exist in its natural elastic state, and to receive all its addition of force from the heat of the explosion. But from what hath been already delivered concerning the increase of the air’s elasticity by heat, we may conclude that the heat of the explosion cannot augment this elasticity to five times its common quantity; consequently the force arising from this cause only cannot amount to more than the 200th part of the real force exerted on the occasion.

“If the whole substance of the powder was converted into an elastic fluid at the instant of the explosion, then, from the known elasticity of this fluid assigned by our theory, and its known density, we could easily determine the velocity with which it would begin to expand, and could thence trace out its future augmentations in its progress through the barrel: but as we have shown that the elastic fluid, in which the activity of the gunpowder consists, is only ∙⅛ths of the substance of the powder, the remaining ⅛ths will, in the explosion, be mixed with the elastic part, and will by its weight retard the activity of the explosion; and yet they will not be so completely united as to move with one common motion; but the unelastic part will be less accelerated than the rest, and some will not even be carried out of the barrel, as appears by the considerable quantity of unctuous matter which adheres to the inside of all fire-arms after they have been used. These inequalities in the expansive motion of the flame oblige us to recur to experiments for its accurate determination.

“The experiments made use of for this purpose were of two kinds. The first was made by charging the barrel A with 12 pennyweights of powder, and a small wad of tow only; and then placing its mouth 19 inches from the centre of the pendulum. On firing it in this situation, the impulse of the flame made it ascend through an arch whose chord was 13 ’ 7 inches; whence, if the whole substance of the powder was supposed to strike against the pendulum, and each part to strike with the same velocity, that common velocity must have been at the rate of about 2650 feet in a second. But as some part of the velocity of the flame was lost in passing through 19 inches of air, I made the remaining experiments in a manner not liable to this inconvenience.

“I fixed the barrel A on the pendulum, so that its axis might be both horizontal and also perpendicular to the plane HK; or, which is the same thing, that it might be in the plane of the pendulum’s vibration: the height of the axis of the piece above the centre of the pendulum was six inches, and the weight of the piece, and of the iron that fastened it, &c. was 12⅜ lbs. The barrel in this situation being charged with 12 pennyweights of powder, without either ball or wad, only put together with the rammer; on the discharge the pendulum ascended through an arch whose chord was 10 inches, or, reduced to an equivalent blow in the centre of the pendulum, supposing the barrel away, it would be 14∙4 inches nearly. The same experiment being repeated, the chord of the ascending arch was 10T inches, which, reduced to the centre, is 14 ’ 6 inches.

“To determine what difference of velocity there was in the different parts of the vapour, I loaded the piece again with 12 pennyweights of powder, and rammed it down with a wad of tow weighing one pennyweight. Now, I conceived that this wad, being very light, would presently acquire that velocity with which the elastic part of the fluid would expand itself when uncompressed; and I accordingly found, that the chord of the ascending arch was by this means increased to 12 inches, or at the centre to 17∙3; whence, as the medium of the other two experiments is 14∙5, the pendulum ascended through an arch 2 , 8 inches longer, by the additional motion of one pennyweight of matter, moving with the velocity of the swiftest part of the vapour; and consequently the velocity with which this pennyweight of matter moved, was that of about 7000 feet in a second.

“It will perhaps be objected to this determination, that the augmentation of the arch through which the pendulum vibrated in this case was not all of it owing to the quantity of motion given to the wad, but part of it was produced by the confinement of the powder, and the greater quantity thereby fired. But if it were true that a part only of the powder fired when there was no wad, it would not happen that in firing different quantities of powder without a wad, the chord would increase and decrease nearly in the ratio of these quantities; which yet I have found it to do: for with nine pennyweights that chord was 7 , 3 inches, which with 12 pennyweights, we have seen, was only 10 and 10∙1 inches; and even with three pennyweights the chord was two inches; deficient from this proportion by ∙5 only, for which defect two other valid reasons are to be assigned.

“And there is still a more convincing proof that all the powder is fired, although no wad be placed before the charge, which is, that the part of the recoil arising from the expansion of powder alone is found to be no greater when it impels a leaden bullet before it, than when the same quantity is fired without any wad to confine it. We have seen that the chord of the arch through which the pendulum rose from the expansive force of the powder alone is 10, or I0∙1; and the chord of that arch, when the piece was charged in the customary manner with a bullet and wad, I found to be, the first time 22∣, and the second 22∣, or, at a medium, 22∙56. Now the impulse of the ball and wad, if they were supposed to strike the pendulum in the same place in which the barrel was suspended, with the velocity they had acquired at the mouth of the piece, would drive it through an arch whose chord would be about 12-3; as is known from the weight of the pendulum, the weight and position of the barrel, and the velocity of the bullet determined by our former experiments; whence, subtracting this number[7:11:50] 12 ’ 3 from 22∙56, the remainder, 10∙26, is nearly the chord of the arch which the pendulum would have ascended through from the expansion of the powder alone with a bullet laid before it. And this number, 10∙26, differs but little from 10∙l, which we have above found to be the chord of the ascending arch, when the same quantity of powder expanded itself freely without either bullet or wad before it.

“Again, that this velocity of 7000 feet in a second is not much beyond what the most active part of the flame acquires in expanding, is evinced from hence, that in some experiments a ball has been found to be discharged with a velocity of 2400 feet in a second; and yet it appeared not that the action of the powder was at all diminished on account of this immense celerity: consequently the degree of swiftness with which, in this instance, the powder followed the ball without losing any part of its pressure, must have been much short of what the powder alone would have expanded with had not the ball been there.

“From these determinations may be deduced the force of petards, since their action depends entirely on the impulse of the flame; and it appears that a quantity of powder properly disposed in such a machine, may produce as violent an effort as a bullet of twice its weight, moving with a velocity of 1400 or 1500 feet in a second.

“In many of the experiments already recited, the ball was not laid immediately contiguous to the powder, but at a small distance, amounting, at the utmost, only to an inch and a half. In these cases the theory agreed very well with the experiments. But if a bullet is placed at a greater distance from the powder, suppose at 12, 18, or 24 inches, we cannot then apply to this ball the same principles which may be applied to those laid in contact, or nearly so, with the powder; for when the surface of the fired powder is not confined by a heavy body, the flame dilates itself with a velocity far exceeding that which it can communicate to a bullet by its continued pressure; consequently, as, at the distance of 12, 18, or 24 inches, the powder will have acquired a considerable degree of this velocity of expansion, the first motion of the ball will not be produced by the continued pressure of the powder, but by the actual percussion of the flame; and it will therefore begin to move with a quantity of motion proportioned to the quantity of this flame, and the velocities of its respective parts.

“From hence then it follows, that the velocity of the bullet, laid at a considerable distance before the charge, ought to be greater than what would be communicated to it by the pressure of the powder acting in the manner already mentioned; and this deduction from our theory we have confirmed by manifold experience, by which we have found, that a ball laid in the barrel A, with its hinder part llļ inches from its breech, and impelled by 12 pennyweights of powder, has acquired a velocity of about 1400 feet in a second; when, if it had been acted on by the pressure of the flame only, it would not have acquired a velocity of 1200 feet in a second. The same we have found to hold true in all other greater distances (and also in lesser, though not in the same degree), and in all quantities of powder; and we have likewise found, that these effects nearly correspond with what has been already laid down about the velocity of expansion and the elastic and unelastic parts of the flame.

“From hence too arises another consideration of great consequence in the practice of gunnery; which is, that no bullet should at any time be placed at a considerable distance from the charge, unless the piece is extremely well fortified; for a moderate charge of powder, when it has expanded itself through the vacant space, and reaches the ball, will, by the velocity each part has acquired, accumulate itself behind the ball, and thereby be condensed prodigiously; whence, if the barrel be not extremely firm in that part, it must, by means of this reinforced elasticity, infallibly burst. The truth of this reasoning I have experienced in an exceeding good Tower musket, forged of very tough iron; for, charging it with 12 pennyweights of powder, and placing the ball sixteen inches from the breech, on firing it, the part of the barrel just behind the bullet was swelled out to double its diameter, like a blown bladder, and two large pieces of two inches long were burst out of it.

“Having seen that the entire motion of a bullet laid at a considerable distance from the charge is acquired by two different methods in which the powder acts on it, the first being the percussion of the parts of the flame with the velocity they had respectively acquired by expanding, the second the continued pressure of the flame through the remaining part of the barrel, I endeavoured to separate these different actions, and to retain that only which arose from the continued pressure of the flame. For this purpose I no longer placed the powder at the breech, from whence it would have full scope for its expansion; but I scattered it as uniformly as I could through the whole cavity left behind the bullet; imagining that by this means the progressive velocity of the flame in each part would be prevented by the expansion of the neighbouring parts; and I found, that the ball being laid lli inches from the breech, its velocity, instead of 1400 feet in a second, which it acquired in the last experiments, was now no more than 1100 feet in the second, which is 100 feet short of what, according to the theory, should arise from the continued pressure of the powder only.

“The reason of this deficiency was, doubtless, the intestine motion of the flame; for the accension of the powder thus distributed through so much larger a space than it could fill, must have produced many reverberations and pulsations of the flame; and from these internal agitations of the fluid, its pressure on the containing surface will (as is the case of all other fluids) be considerably diminished; and in order to avoid this irregularity, in aH other experiments I took care to have the powder closely confined in as small a space as possible, even when the bullet lay at some distance from it.

“With regard to the resistance of the air, which so remarkably affects all military projectiles, it is necessary to premise, that the greatest part of authors have established it as a certain rule, that while the same body moves in the same medium, it is always resisted in the duplicate proportion of its velocity; that is, if the resisted body move in one part of its track with three times .the velocity with which it moved in some other part, then its resistance to the greater velocity will be nine times the resistance to the lesser. If the velocity in one place be four times greater than in another, the resistance of the fluid will be sixteen times greater in the first than in the second, &c. This rule, however, though pretty near the truth when the velocities are confined within certain limits, is excessively erroneous when applied to military projectiles, where such resistances often occur as could scarcely be effected, on the commonly received principles, even by a treble augmentation of its density.

“By means of the machine already described, I nave it in my power to determine the velocity with which a ball moves in any part of its track, provided I can direct the piece in such a manner as to cause the bullet to impinge on the pendulum placed in that part; and therefore, charging a musket barrel three times successively with a leaden ball three fourths of an inch in diameter, and about half its weight of powder, and taking such precaution in weighing of the powder and placing it, that I was assured, by many previous trials, that the velocity of the ball could not differ by twenty feet in a second from its medium quantity, I fired it against the pendulum placed at 25, 75, and [7:11:51]125 feet distance from the mouth of the piece respectively; and I found that it impinged against the pendulum, in the first case, with a velocity of 1670 feet in a second; in the second case, with a velocity of 1550 feet in a second; and in the third case, with a velocity of 1425 feet in a second; so that, in passing through fifty feet of air, the bullet lost a velocity of 120 or 125 feet in a second; and the time of its passing through that space being about ⅛d or -⅛th of a second, the medium quantity of resistance must, in these instances, have been about 120 times the weight of the ball, which (as the ball was nearly∙ τ ⅛th of a pound) amounts to about 10 lbs. avoirdupois.

Now, if a computation be made according to the method laid down for compressed fluids in the 38th Proposition of Newton’s Principia i supposing the weight of water to that of air as 850 to 1, it will be found that the resistance to a globe of three fourths of an inch diameter, moving with a velocity of about 1600 feet in a second, will not, on these principles, amount to any more than 4⅜ lbs. avoirdupois; whence, as we know that the rules contained in that proposition are very accurate with regard to slow motions, we may hence conclude, that the resistance of the air in slow motions is less than that in swift motions, in the ratio of 4⅜ to 10; a proportion between that of one to two and one to three.

“Again, I charged the same piece a number of times with equal quantities of powder, and balls of the same weight, taking all possible care to give to every shot an equal velocity; and firing three times against the pendulum placed only 25 feet from the mouth of the piece, the medium of the velocities with which the ball impinged was nearly that of 1690 feet in a second: then removing the piece 175 feet from the pendulum, I found, taking the medium of five shots, that the velocity with which the ball impinged at this distance was 1300 feet in a second; whence the ball, in passing through 150 feet of air, lost a velocity of about 390 feet in a second; and the resistance computed from these numbers comes out something more than in the preceding instance, it amounting here to between eleven and twelve pounds avoirdupois; whence, according to these experiments, the resisting power of the air to swift motions is greater than to slow ones, in a ratio which approaches nearer to that of three to one than in the preceding experiments.

“Having thus examined the resistance to a velocity of 1700 feet in a second, I next examined the resistance to smaller velocities; and for this purpose I charged the same barrel with balls of the same diameter, but with less powder, and placing the pendulum at 25 feet distance from the piece, I fired against it five times with an equal charge each time; the medium velocity with which the ball impinged was that of 1180 feet in a second: then, removing the pendulum to the distance of 250 feet, the medium velocity of five shots, made at this distance, was that of 950 feet in a second: whence the ball, in passing through 225 feet of air, lost a velocity of 230 feet in a second; and as it passed through that interval in about three fourteenths of a second, the resistance to the middle velocity will come out to be near 33ļ times the gravity of the ball, or two pounds ten ounces avoirdupois. Now, the resistance to the same velocity, according to the laws observed in slower motions, amounts to seven elevenths of the same quantity; whence, in a velocity of 1065 feet in a second, the resisting power of the air is augmented in no greater a proportion than that of seven to eleven; whereas we have seen in the former experiments, that to still grcater degrees of velocity the augmentation approached very near the ratio of one to three.

“But farther, I fired three shot, of the same size and weight with those already mentioned, over a large piece of water; so that their dropping into the water being very discernible, both the distance and time of their flight might be accurately ascertained. Each shot was discharged with a velocity of 400 feet in a second; and I had satisfied myself, by many previous trials of the same charge with the pendulum, that I could rely on this velocity to ten feet in a second. The first shot flew 313 yards in four seconds and a quarter, the second flew 319 yards in four seconds, and the third 373 yards in five seconds and a half. According to the theory of resistance established for slow motions, the first shot ought to have spent no more than 3∙2 seconds in its flight, the second 3∙28, and the third four seconds; whence it is evident that every shot was retarded considerably more than it ought to have been had that theory taken place in its motion; consequently the resistance of the air is very sensibly increased, even in such a small velocity as that of 400 feet in a second.

“As no large shot arc ever projected in practice with velocities exceeding that of 1700 feet in a second, it will be sufficient for the purposes of a practical gunner to determine the resistance to all lesser velocities, which may be thus exhibited. Let AB (fig. 14) be taken to AC, in the ratio of 1700 feet in a second to the given velocity to which the resisting power of the air is required. Continue the line AB to D, so that BD may be to AD as the resisting power of the air to slow motions is to its resisting power to a velocity of 1700 feet in a second; then, shall CD be to AD as the resisting power of the air to slow motions is to its resisting power to the given velocity represented by AC.

“From the computations and experiments already mentioned, it plainly appears that a leaden ball of three fourth* of an inch diameter, and weighing nearly 1 ļ-ounce avoir dupois, if it be fired from a barrel of forty-five inches in length, with half its weight of powder, will issue from that piece with a velocity which, if it were uniformly continued, would carry it near 1700 feet in a second. If, instead of the leaden ball, an iron one, of an equal diameter, was placed in the same situation in the same piece, and was impelled by an equal quantity of powder, the velocity of such an iron bullet would be greater than that of a leaden one in the subduplicate ratio of the specific gravities of Iead and iron; and supposing that ratio to be as three to two, and computing on the principles already laid down, it will appear, that an iron bullet of 24 lbs. weight, shot from a piece of ten feet in length, with 16 lbs. of powder, will acquire from the explosion a velocity which, if uniformly continued, would carry it nearly 1650 feet in a second.

“This is the velocity which, according to our theory, a cannon ball of 24 lbs. weight is discharged with when it is impelled by a full charge of powder; but if, instead of a quantity of powder weighing two thirds of the ball, we suppose the charge to be only half the weight of it, then its velocity will on the same principles be no more than 1490 feet in a second. The same would be the velocities of every lesser bullet fired with the same proportions of powder, if the lengths of all pieces were constantly in the same ratio with the diameters of their bore; and although, according to the usual dimensions of the smaller pieces of artillery, this proportion does not always hold, yet the difference is not great enough to occasion a very great variation from the velocities here assigned, as will be obvious to any one who shall make a computation thereon. But in these determinations we suppose the windage to· be no more than is just sufficient for putting down the bullet easily; whereas, in real service, either through negligence or unskilfulness, it often happens that the diameter of the bore so much exceeds the diameter of the bullet, that great part of the inflamed fluid escapes by its side; whence the velocity of the shot in this case may be considerably less than what we have assigned. However, this perhaps may [7:11:52] be compensated by the greater heat which in all probability attends the firing of these large quantities of powder.

“From this great velocity of cannon-shot we may clear up the difficulty concerning the point-blank shot which occasioned the invention of Anderson’s strange hypothesis. Here our author was deceived by his not knowing how greatly the primitive velocity of the heaviest shot is diminished in the course of its flight by the resistance of the air. Now, as a shot of 24 lbs. fired with two thirds of its weight of powder, will, at the distance of 500 yards from the piece, be separated from the line of its original direction by an angle of little more than half a degree, those who are acquainted with the inaccurate methods often used in the directing of cannon will easily allow, that so small an aberration may not be attended to by the generality of practitioners, and the path of the shot may consequently be deemed a straight line; especially as other causes of error will often intervene much greater than what arises from the incurvation of this line by gravity.

“We have now determined the velocity of the shot, both when fired with two thirds of its weight and with half its weight of powder respectively; and on this occasion I must remark, that, on the principles of our theory, the increasing the charge of powder will increase the velocity of the shot till the powder arrives at a certain quantity; after which, if the powder be increased, the velocity of the shot will diminish. The quantity producing the greatest velocity, and the proportion between that greatest velocity and the velocity communicated by greater and lesser charges, may be thus assigned. Let AB (fig. 14) represent the axis of the piece; draw AC perpendicular to it, and to the asymptotes AC and AB draw any hyperbola LF, and draw BF parallel to AC; find out now the point D, where the rectangle ADEG is equal to the hyperbolic area DEFB; then will AD represent that height of the charge which communicates the greatest velocity to the shot; whence AD being to AB as 1 to 2∙71828, as appears from the table of logarithms, from the length of the line AD thus determined, and the diameter of the bore, the quantity of powder contained in this charge is easily known. If, instead of this charge, any other filling the cylinder to the height AI be used, draw IH parallel to AC, and through the point H to the same asymptotes AC and AB describe the hyperbola HK; then the greatest velocity will be to the velocity communicated by the charge Al, in the-subdupli-cate proportion of the rectangle ADEG to the same rectangle diminished by the trilinear space KITE.

“It has been already shown, that the resistance of the air on the surface of a bullet of three fourths of an inch diameter, moving with a velocity of 1670 feet in a second, amounted to about ten pounds. It hath also been shown, that an iron bullet weighing twenty-four pounds, if fired with sixteen pounds of powder (which is usually esteemed its proper battering charge), acquires a velocity of about 1650 feet in a second, scarcely differing from the other; whence, as the surface of this last bullet is more than fifty-four times greater than the surface of a bullet of three fourths of an inch diameter, and their velocities are nearly the same, it follows, that the resistance on the larger bullet will amount to more than 540 pounds, which is near twenty-three times its own weight.

“The two last propositions arc principally aimed against those theorists who have generally agreed in supposing the flight of shot and shells to be nearly in the curve of a parabola. The reason given by those authors for their opinion is the supposed inconsiderable resistance of the air; since, as it is agreed on all sides that the track of projectiles would be a perfect parabola if there was no resistance, it has from thence been too rashly concluded, that the interruption which the ponderous bodies of shells arid bullets would receive from such a rare medium as air would be scarcely sensible, and consequently that their parabolic flight would be hereby scarcely affected.

“Now, the prodigious resistance of the air to a bullet of twenty-four pounds weight, such as we have here established it, sufficiently confutes this reasoning; for how erroneous must that hypothesis be, which neglects as inconsiderable a force amounting to more than twenty times the weight of the moving body?” We now proceed to state the postulates which contain the principles of the modern art of gunnery. They are as follow:

“1. If the resistance of the air be so small that the motion of a projected body is in the curve of a parabola, then the axis of that parabola will be perpendicular to the horizon, and consequently the part of the curve in which the body ascends will be equal and similar to that in which it descends.

“2. If the parabola in which the body moves be terminated on a horizontal plane, then the vertex of the parabola will be equally distant from its own extremities.

“3. Also the moving body will fall on that horizontal plane in the same angle, and with the same velocity with which it was first projected.

“4. If a body be projected in different angles but with the same velocity, then its greatest horizontal range will be when it is projected in an angle of 45° with the horizon.

“5. If the velocity with which the body is projected be known, then this greatest horizontal range maybe thus found. Compute, according to the common theory of gravity, what space the projected body ought to fall through to acquire the velocity with which it is projected; then twice that space will be the greatest horizontal range, or the horizontal range when the body is projected in an angle of 45° with the horizon.

“6. The horizontal ranges of a body, when projected with the same velocity at different angles, will be between themselves as the sines of twice the angle in which the line of projection is inclined to the horizon.

“7. If a body is projected in the same angle with the horizon, but with different velocities, the, horizontal ranges will be in the duplicate proportion of those velocities.

“These postulates, which contain the principles of the modern art of gunnery, are all of them false; for it has been already shown, that a musket-ball of three fourths of an inch in diameter, fired with half its weight of powder, from a piece 45 inches long, moves with a velocity of near 1700 feet in a second. Now, if this ball flew in the curve of a parabola, its horizontal range at 45° would be found by the fifth postulate to be about seventeen miles. But all the practical writers assure us that this range is really short of half a mile. Diego Uffano assigns to an arquebuss, four feet in length, and carrying a leaden ball of 1∣ oz. weight (which is very near our dimensions), a horizontal range of 797 common paces, when it is elevated between 40 and 50 degrees, and charged with a quantity of fine powder equal in weight to the ball. Mcrsennus also tells us, that hê found the horizontal range of an arquebuss at 45° to be less than 400 fathoms, or 800 yards; whence, as either of these ranges is short of half an English mile, it follows, that a musket-shot, when fired with a reasonable charge of powder at the elevation of 45°, flies not one thirty-fourth part of the distance it ought to do if it moved in a parabola. Nor is this great contraction of the horizontal range to be wondered at, when it is considered that the resistance of this bullet when it first issues from the piece amounts to 120 times its gravity, as has been here experimentally demonstrated.

“To prevent objections, our next instance shall be in an iron bullet of 24 lbs. weight, which is the heaviest in common use for land-service. Such a bullet fired from a piece of the common dimensions, with its greatest allotment [7:11:53] of powder, has a velocity of 1650 feet in a second, as al- ' ready shown. Now, if the horizontal range of this shot at 45° be computed on the parabolic hypothesis by the fifth postulate, it will come out to be about sixteen miles, which is between five and six times its real quantity; for the practical writers all agree in making it less than three miles.

“But farther, it is not only vřhen projectiles move with these very great velocities that their flight sensibly varies from the curve of a parabola; the same aberration often takes place in such as move slow enough to have their motion traced out by the eye; for there are few projectiles that can be thus examined, which do not visibly disagree with the first, second, and third postulates; obviously descending through a curve which is shorter and less inclined to the horizon than that in which they ascended. Also the highest point of their flight, or the vertex of the curve, is much nearer the place where they fall to the ground than to that from whence they were at first discharged.

“I have found too by experience, that the fifth, sixth, and seventh postulates are excessively erroneous when applied to the motions of bullets moving with small velocities. A leaden bullet three fourths of an inch in diameter, discharged with a veldcity of about 400 feet in a second, and in an angle of 19° 5 ’ with the horizon, ranged on the horizontal plane no more than 448 yards; whereas its greatest horizontal range being found by the fifth postulate to be at least 1700 yards, the range at 19° 5' ought by the sixth postulate to have been 1050 yards; whence, in this experiment, the range was not three sevenths of what it must have been had the commonly received theory been true.”

From this and other experiments, it is clearly proved, that the track described by the flight even of the heaviest shot, is neither a parabola, nor even approaching to a parabola, except when they are projected with very small velocities. The nature of the curve really described by them will be explained under the head of Projectiles. But, as a specimen of the great complication of the subject, we shall here insert an account of a circumstance which frequently occurs in the discharge of shot.

“As gravity acts perpendicularly to the horizon, it is evident, that if no other power but gravity deflected a projected body from its course, its motion would be constantly performed in a plane perpendicular to the horizon, passing through the line of its original direction; but we have found, that the body in its motion often deviates from this plane, sometimes to the right hand and at other times to the left; and this in an incurvated line, which is convex towards that plane, so that the motion of a bullet is frequently in a line having a double curvature, it being bent towards the horizon by the force of gravity, and again bent out of its original direction to the right or left by some other force. In this case no part of the motion of the bullet is performed in the same plane, but its track will lie in the surface of a kind of cylinder, whose axis is perpendicular to the horizon.

“This proposition may be indisputably proved by the experience of every one in the least conversant with the practice of gunnery. The same piece which will carry its bullet within an inch of the intended mark at ten yards distance, cannot be relied on to ten inches in 100 yards, much less to thirty inches in 300 yards. Now this inequality can only arise from the track of the bullet being incurvated sidewise as well as downwards; for by this means the distance between that incurvated line and the line of direction will increase in a much greater ratio than that of the distance; these lines being coincident at the mouth of the piece, and afterwards separating in the manner of a curve and its tangent, if the mouth of the piece be considered as the point of contact. To put this matter out of all doubt, however, I took a barrel carrying a ball three fourths of an inch in diameter, and fixing it on a heavy carriage, I satisfied myself of the steadiness and truth of its direction, by firing at aboard 1^-th foot square, which was placed at 180 feet distance; for I found that in sixteen successive shots I missed the mark but once. Now, the same barrel being fixed on the same carriage, and fired with a smaller quantity of powder, so that the shock on the discharge would be much less, and consequently the direction less changed, I found, that at 760 yards distance the ball flew sometimes 100 yards to the right of the line it was pointed on, and sometimes as much to the left. I found, too, that its direction in the perpendicular line was not less uncertain, it falling one time above 200 yards short of what it did at another; although, by the nicest examination of the piece after the discharge, it did not appear to have started in the least from the position it was placed in.

“The reality of this doubly curvated track being thus demonstrated, it may perhaps be asked, What can be the cause of a motion so different from what has been hitherto supposed? And to this I answer, that the deflection in question must be owing to some power acting obliquely to the progressive motion of the body; which power can be no other than the resistance of the air. If it be farther asked, how the resistance of the air can ever come to be oblique to the progressive motion of the body, I farther reply, that it may sometimes arise from inequalities in the resisted surface, but that its general cause is doubtless a whirling motion acquired by the bullet about its axis; for by this motion of rotation, combined with the progressive motion, each part of the bullet’s surface will strike the air very differently from what it would do if there was no such whirl; and the obliquity of the action of the air arising from this cause will be greater, as the rotatory motion of the bullet is greater in proportion to its progressive one.

“This whirling motion undoubtedly arises from the friction of the bullet against the sides of the piece; and as the rotatory motion will in some part of its revolution conspire with the progressive one, and in another part be equally opposed to it, the resistance of the air on the fore part of the bullet will be hereby affected, and will be increased in that part where the whirling motion conspires with the progressive one, and diminished where it is opposed to it; and by this means the whole effort of the resistance, instead of being opposite to the direction of the body, will become oblique thereto, and will produce those effects already mentioned. If it was possible to predict the position of the axis round which the bullet should whirl, and if that axis was unchangeable during the whole flight of the bullet, then the aberration of the bullet by this oblique force would be in a given direction, and the incurvation produced thereby would regularly extend the same way from one end of its track to the other. For instance, if the axis of the whirl was perpendicular to the horizon, then the incurvation would be to the right or left. If that axis was horizontal, and perpendicular to the direction of the bullet, then the incurvation would be upwards or downwards. But as the first position of this axis is uncertain, and as it may perpetually shift in the course of the bullet’s flight; the deviation of the bullet is not necessarily either in one certain direction, or tending to the same side in one part of its track more than it does in another, but more usually is continually changing the tendency of its deflection, as the axis round which it whirls must frequently shift its position to the progressive motion by many inevitable accidents.

“That a bullet generally acquires such a rotatory motion as here described, is, I think, demonstrable: however, [7:11:54]to leave no room for doubt or dispute, I confirmed it, as well as some other parts of my theory, by the following experiments.

“I caused the machine to be made, represented fig. 15. BCDE is a brass barrel, moveable on its axis, and so adjusted by means of friction-wheels, not represented in the figure, as to have no friction worth attending to. The frame in which this barrel is fixed is so placed that its axis may be perpendicular to the horizon. The axis itself is continued above the upper plate of the frame, and has fastened on it a light hollow cone, AFG. From the lower part of this cone there is extended a long arm of wood, GH, which is very thin, and cut feather-edged. At its extremity there is a contrivance for fixing on the body whose resistance is to be investigated (as here the globe P); and to prevent the arm GH from swaying out of its horizontal position by the weight of the annexed body P, there is a brace, AH, of fine wire, fastened to the top of the cone which supports the end of the arm.

“Round the barrel BCDE there is wound a fine silk line, the turns of which appear in the figure; and after this line has taken a sufficient number of turns, it is eon-ducted nearly in a horizontal direction to the pulley L, over which it is passed, and then a proper weight M is hung to its extremity. If this weight be left at liberty, it is obvious that it will descend by its own gravity, and will, by its descent, turn round the barrel BCDĽ, together with the arm GH, and the body P fastened to it. And whilst the resistance on the arm GH and on the body P is less than the weight M, that weight will accelerate its motion; and thereby the motion of GH and P will increase, and consequently their resistance will increase, till at last this resistance and the weight M become nearly equal to each other. The motion with which M descends, and with which P revolves, will not then sensibly differ from an equable one. Whence it is not difficult to conceive, that, by proper observations made with this machine, the resistance of the body P may be determined. The most natural method of proceeding in this investigation is as follows: Let the machine first have acquired its equable motion, which it will usually do in about five or six turns from the beginning; and then let it be observed, by counting a number of turns, what time is taken up by one revolution of the body P: then taking off the body P and the weight M, let it be examined what smaller weight will make the arm GH revolve in the same time as when P was fixed to it: this smaller weight being taken from M, the remainder is obviously equal in effort to the resistance of the revolving body P; and this remainder being reduced in the ratio of the length of the arm to the semidiameter of the barrel, will then become equal to the absolute quantity of the resistance. And as the time of one revolution is known, and consequently the velocity of the revolving body, there is hereby discovered the absolute quantity of the resistance to the given body P moving with a given degree of celerity.

“ Here, to avoid all objections, I have generally chosen, when the body P was removed, to fix in its stead a thin piece of lead of the same weight, placed horizontally: so that the weight which was to turn round the arm GH, without the body P, did also carry round this piece of lead. But mathematicians will easily allow that there was no necessity for this precaution. The diameter of the barrel BCDE, and of the silk string wound round it, was 2∙06 inches. The length of the arm GH, measured from the axis to the surface of the globe P, was 49∙5 inches. The body P, the globe made use of, was of pasteboard; its surface very neatly coated with marbled paper. It was not much distant from the size of a 12-lb. shot, being in diameter 4∙5 inches, so that the radius of the circle described by the centre of the globe was 51∙75 inches. When this globe was fixed at the end of the arm, and a weight of half a pound was hung at the end of the string at M, it was examined how soon the motion of the descending weight M, and of the revolving body P, would become equable as to sense. With this view, three revolutions being suffered to elapse, it was found that the next 10 were performed in 27f ff , 20 in less than 55", and 80 in 82 1' ’; so that the first 10 were performed in 27⅜", the second in 271*, and the third in 27⅛".

“These experiments sufficiently evince, that even with half a pound, the smallest weight made use of, the motion of the machine was sufficiently equable after the first three revolutions.

“The globe above mentioned being now fixed at the end of the arm, there was hung on at M a weight of 3į lb.; and ten revolutions being suffered to elapse, the succeeding 20 were performed in 21∣". Then the globe being taken off', and a thin plate of lead, equal to it in weight, placed in its room; it was found, that instead of 3∣ lb. a weight of one pound would make it revolve in less time than it did before, performing now 20 revolutions after 10 were elapsedin the space of 19".

“Hence then it follows, that from the 3ļ lb. first hung on, there is less than 1 lb. to be deducted for the resistance on the arm; and consequently the resistance on the globe itself is not less than the effort of 2į lb. in the situation M: and it appearing from the former measures, that the radius of the barrel is nearly fa of the radius of the circle described by the centre of the globe, it follows, that the absolute resistance of the globe, when it revolves 20 times in 21⅜" (about 25 feet in a second), is not less than the 50th part of two pounds and a quarter, or of 36 ounces; and this being considerably more than half an ounce, and the globe nearly the size of a 12-pound shot, it irrefragably confirms a proposition I had formerly laid down from theory, that the resistance of the air to a 12-lb. iron shot, moving with a velocity of 25 feet in a second, is not less than half an ounce.

“The rest of the experiments were made in order to confirm another proposition, namely, that the resistance of the air within certain limits is nearly in the duplicate proportion of the velocity of the resisted body. To investigate this point, there were successively hung on at M, weights in the proportion of the numbers 1, 4, 9, 16; and letting 10 revolutions first elapse, the following observations were made on the rest. With ⅜ lb. the globe went 20 turns in 54⅜", with 2 lb. it went 20 turns in 27į", with 4⅜ lb. it went 30 turns in 27⅜", and with 8 lb. it went 40 turns in 27∣". Hence it appears, that to resistances proportioned to the numbers 1, 4, 9, 16, there correspond velocities of the resisted body in the proportion of the numbers 1, 2, 3, 4; which proves, with great nicety, the proposition above mentioned.

“With regard to the rotatory motion, the first experiment was to evince, that the whirling motion of a ball combining with its progressive motion would produce such an oblique resistance and deflective power as already mentioned. For this purpose a wooden ball of 4Ļ inches diameter was suspended by a double string about eight or nine feet long. Now, by turning round the ball, and twisting the double string, the ball when left to itself would have a revolving motion given it from the untwisting of the string again. And if, when the string was twisted, the ball was drawn to a considerable distance from the perpendicular, and there let go, it would at first, before it had acquired its revolving motion, vibrate steadily enough in the same vertical plane in which it first began to move; but when, by the untwisting of the string, it had acquired a sufficient degree of its whirling motion, it constantly deflected to the right or left of its first traek, and sometimes proceeded so far as to have its direction at right angles to that in which it began[7:11:55] its motion; and this deviation was not produced by the string itself, but appeared to be entirely owing to the resistance being greater on the one part of the leading surface of the globe than the other. For the deviation continued when the string was totally untwisted, and even during the time that the string, by the motion the globe had received, was twisting the contrary way. And it was always easy to predict, before the ball was let go, which way it would deflect, only by considering on which side the whirl would be combined with the progressive motion; for on that side always the deflective power acted, as the resistance was greater here than on the side where the whirl and progressive motion were opposed to one another.”

Though Mr Robins considered this experiment as an incontestible proof of the truth of his theory, he undertook to give ocular demonstration of this deflection of musket-bullets even in the short space of one hundred yards.

w As all projectiles,” says he, “in their flight are acted upon by the power of gravity, the deflection of a bullet from its primary direction supposes that deflection to be upwards or downwards in a vertical plane; because, in the vertical plane, the action of gravity is compounded and entangled with the deflective force. And for this reason my experiments have been principally directed to the examination of that deflection which carries the bullet to the right or left of that plane in which it began to move. For if it appears at any time that the bullet has shifted from that vertical plane in which the motion began, this will be an incontestible proof of what we have advanced. Now, by means of screens of exceeding thin paper, placed parallel to each other at proper distances, this deflection in question may be many ways investigated. For by firing bullets which shall traverse the screens, the flight of the bullet may be traced; and it may easily appear whether they do or do not keep invariably to one vertical plane. This examination may proceed on three different principles, which I shall here separately explain.

“For, first, an exactly vertical plane may be traced out upon all these screens, by which the deviation of any single bullet may be more readily investigated, only by measuring the horizontal distance of its trace from the vertical plane thus delineated; and by this means the absolute quantity of its aberration may be known. Or if the description of such a vertical plane should be esteemed a matter of difficulty and nicety, a second method may be followed, which is that of resting the piece in some fixed notch or socket, so that though the piece may have some little play to the right and left, yet all the lines in which the bullet can be directed shall intersect each other in the centre of that fixed socket: by this means, if two different shots are fired from the piece thus situated, the horizontal distances made by the two bullets on any two screens ought to be in the same proportion to each other as the respective distances of the screens from the socket in which the piece was laid. And if these horizontal distances differ from that proportion, then it is certain that one of the shots at least has deviated from a vertical plane, although the absolute quantity of that deviation cannot hence be assigned, because it cannot be known what part of it is to be imputed to one bullet, and what to the other.

“But if the constant and invariable position of the notch or socket in which the piece was placed be thought too hard an hypothesis in this very nice affair, the third method, and which is the simplest of all, requires no more than that two shot be fired through three screens without any regard to the position of the piece each time: for in this case, if the shots diverge from each other, and both keep to a vertical plane, then, if the horizontal distances of their traces on the first screen be taken from the like horizontal distances on the second and third, the two remainders will be in the same proportion with the distances of the second and third screen from the first. And if they are not in this proportion, then it will be certain that one of them at least has been deflected from the vertical plane; though here, as in the last case, the quantity of that deflection in each will not be known.

“All these three methods I have myself made use of at different times, and have ever found the success agreeable to my expectation. But the most eligible method seemed to be a compound of the two last. The apparatus was as follows. Two screens were set up in the larger walk in the Charter-house garden; the first of them at 250 feet distance from the wall, which was to serve for a third screen; and the second 200 feet from the same wall. At fifty feet before the first screen, or at 300 feet from the wall, there was placed a large block weighing about 200 lbs. weight, and having fixed into it an iron bar with a socket at its extremity, in which the piece was to be laid. The piece itself was of a common length, and bored for an ounce ball. It was each time loaded with a ball of 17 to the pound, so that the windage was extremely small, and with a quarter of an ounce of good powder. The screens were made of the thinnest tissue paper; and the resistance they gave to the bullet (and consequently their probability of deflecting it) was so small, that a bullet lighting one time near the extremity of one of the screens, left a fine thin fragment of it towards the edge entire, which was so very weak that it was difficult to handle it without breaking. These things thus prepared, five shots were made with the piece rested in the notch above mentioned; and the horizontal distances between the first shot, which was taken as a standard, and the four succeeding ones, both on the first and second screen, and on the wall, measured in inches, were as follows:
[table]

“Here the letters R and L denote that the shot in question went either to the right or left of the first.

“If the position of the socket in which the piece was placed be supposed fixed, then the horizontal distances measured above on the first and second screen, and on the wall, ought to be in proportion to the distances of the first screen, the second screen, and the wall, from the socket. But by only looking over these numbers, it appears that none of them are in that proportion; the horizontal distance of the first and third, for instance, on the wall being above nine inches more than it should be by this analogy.

“If, without supposing the invariable position of the socket, we examine the comparative horizontal distances according to the third method described above, v. r e shall in this case discover divarications still more extraordinary; for by the numbers set down it appears that the horizontal distances of the second and third shot on the two screens, and on the wall, are as under.
[table]

Here, if, according to the rule given above, the distance on the first screen be taken from the distances on the other two, the remainder will be 7 and 72∙2; and these numbers, if each shot kept to a vertical plane, ought to be in the proportion of 1 to 5; that being the proportion of the distances of the second screen, and of the wall, from the first: but the last number 72*2 exceeds what it ought to be by this analogy by 37∙2; so that between them there
[7:11:56]
[table]

is a deviation from the vertical plane of above thirty-seven inches, and this too in a transit of little more than eighty yards.

“But farther, to show that these irregularities do not depend on any accidental circumstance of the balls fitting or notfitting the piece, there were five shots more made with the same quantity of powder as before, but with smaller bullets, which ran much looser in the piece. And the horizontal distances being measured in inches from the trace of the first bullet to each of the succeeding ones, the numbers were as under.

Here, again, on the supposed fixed position of the piece, the horizontal distance on the wall between the first and third will be found above fifteen inches less than it should be if each kept to a vertical plane; and like irregularities, though smaller, occur in every other experiment. And if they are examined according to the third method set down above, and the horizontal distances of the third and fourth, for instance, are compared, those on the first and second screen, and on the wall, appear to be thus.
[table]

“And if the horizontal distance on the first screen be taken from the other two, the remainders will be 10∙15 and 27∙4; where the least of them, instead of being five times the first, as it ought to be, is 45∙35 short of it; so that here is a deviation of forty-five inches.

“From all these experiments, the deflection in question seems to be incontestibly evinced. But to give some farther light to this subject, I took a barrel of the same bore with that hitherto used, and bent it at about three or four inches from its muzzle to the left, the bend making an angle of three or four degrees with the axis of the piece. This piece thus bent was fired with a loose ball, and the same quantity of powder hitherto used, the screens of the last experiment being still continued. It was natural to expect, that if this piece was pointed by the general direction of its axis, the ball would be canted to the left of that direction by the bend near its mouth. But as the bullet, in passing through that bent part, would, as I conceived, be forced to roll upon the right-hand side of the barrel, and thereby its left side would turn up against the air, and would increase the resistance on that side, I predicted to the company then present, that if the axis on which the bullet whirled did not shift its position after it was separated from the piece, then, notwithstanding the bend of the piece to the left, the bullet itself might be expected to incurvate towards the right; and this, upon trial, did most remarkably happen. For one of the bullets fired from this bent piece passed through the first screen about lį inch distant from the trace of one of the shots fired from the straight piece in the last set of experiments. On the second screen, the traces of the same bullets were about three inches distant; the bullet from the crooked piece passing on both screens to the left of the other; but comparing the places of these bullets∙on the wall, it appeared that the bullet from the crooked piece, though it diverged from the track on the two screens, had now crossed that track, and was deflected considerably to the right of it; so that it was obvious, that though the bullet from the crooked piece might first be canted to the left, and had diverged from the track of the other bullet with which it was compared, yet by degrees it deviated again to the right, and a little beyond the second screen crossed that track from which it before diverged, and on the wall was deflected fourteen inches, as I remember, on the contrary side. And this experiment is not only the most convincing proof of the reality of this deflection here contended for; but is likewise the strongest confirmation that it is brought about in the very manner and by the very circumstances which we have all along described.

“I have now only to add, that as I suspected the consideration of the revolving motion of the bullet, compounded with its progressive one, might be considered as a subject of mathematical speculation, and that the reality of any deflecting force thence arising might perhaps be denied by some computists, upon the principles hitherto received of the action of fluids, I thought proper to annex a few experiments, with a view of evincing the strange deficiency of all theories of this sort hitherto established, and the unexpected and wonderful varieties which occur in these matters. The proposition which I advanced for this purpose being, that two equal surfaces meeting the air with the same degree of obliquity, may be so differently resisted, that though in one of them the resistance is less than that of a perpendicular surface meeting the same quantity of air, yet in another it shall be considerably greater.

“To make out this proposition, I made use of the machine already described; and having prepared a pasteboard pyramid, whose base was four inches square, and whose planes made angles of 45° with the plane of its base, and also a parallelogram four inches in breadth, and 5į in length, which was equal to the surface of the pyramid, the globe P was taken off from the machine, and the pyramid was first fixed on; and 2 lb. being hung at M, and the pyramid so fitted as to move with its vertex forwards, it performed twenty revolutions after the first ten were elapsed in 33". Then the pyramid being turned so that its base, which was a plane of four inches square, went foremost, it now performed twenty revolutions with the same weight in 38į". After this, taking off the pyramid, and fixing on the parallelogram with its longer side perpendicular to the arm, and placing its surface in an angle of 45° with the horizon by a quadrant, the parallelogram, with the same weight, performed twenty revolutions in 43⅛".

“Now here this parallelogram and the surface of the pyramid are equal to each other, and each of them met the air in an angle of 45°; and yet one of them made twenty revolutions in 33", whilst the other took up 43∣". And at the same time it appears that a flat surface, such as the base of a pyramid, which meets the same quantity of air perpendicularly, makes twenty revolutions in 38∣", which is the medium between the other two.

“But to give another and still more simple proof of this principle, there was taken a parallelogram four inches broad and 8įth long. This being fixed at the end of the arm, with its long side perpendicular thereto, and being placed in an angle of 45° with the horizon, there was a weight hung on at M of 3⅜ lb. with which the parallelogram made twenty revolutions in 40∣". But after this, the position of the parallelogram was shifted, and it was placed with its shorter side perpendicular to the arm, though its surface was still inclined to an angle of 45° with the horizon; and now, instead of going slower, as might have been expected from the greater extent of part of its surface from the axis of the machine, it went round much faster; for in this last situation it made twenty revolutions in 35∣", so that there were 5" difference in the time of twenty revolutions; and this from no other change of circumstance than as the larger or shorter side of the oblique plane was perpendicular to the line of its direction.”

In the seventy-third volume of the Philosophical Transactions, several experiments on this subject, but upon a larger scale, are related by Lovell Edgeworth, Esq. They [7:11:57]confirm the truth of what Mr Robins advances, but nothing is said to explain the reason of it.

These are the principal experiments made by Mr Robins in confirmation of his theory, and which not only far exceed every thing that had been previously done, but point out the only method by which the art of gunnery may be still further improved. It must be observed, however, that in this art it is impossible we should ever arrive at absolute perfection; that is, it can never be expected that a gunner, by any method of calculation whatever, can be enabled to point his guns in such a manner that the shot shall hit the mark if placed any where within its range. Aberration which can by no means be either foreseen or prevented, will take place from a great number of different causes. A variation in the density of the atmosphere, in the dampness of the powder, or in the figure of the shot, will cause variations in the range of the bullet, which cannot by any means be reduced to rules, and consequently must render the event of each shot very precarious. The resistance of the atmosphere simply considered, without any of those anomalies arising from its density at different times, is a problem which, notwithstanding the labours of Mr Robins and others, has not been completely solved; and indeed if we consider the matter in a physical light, we shall find that without some other data than those which are yet obtained, an exact solution of it is impossible.

An objection has been made to the mathematical philosophy, to which in many cases it is most certainly liable, that it considers the resistance of matter more than its capacity of giving motion to other matter. Hence, if in any case matter acts both as a resisting and a moving power, and the mathematician overlooks its effort towards motion, founding his demonstrations only upon its property of resisting, these demonstrations will certainly be false. It is to an error of this kind that we are to attribute the great differences already noticed between the calculations of Sir Isaac Newton, with regard to the resisting force of fluids, and what actually takes place upon trial. These calculations were made upon the supposition that the fluid through which a body moved could do nothing else but resist it; yet it is certain that the air (the fluid with which we have to do at present) proves a source of motion, as well as resistance, to all bodies which move in it.

To understand this matter fully, let ABC (fig. 16) represent a crooked tube made of any solid matter, and a, b, two pistons which exactly fill the cavity. If the space between these pistons be full of air, it is plain they cannot come into contact with each other, on account of the elasticity of the included air, but will remain at some certain distance, as represented in the figure. If the piston b be drawn up, the air which presses in the direction Qb acts as a resisting power, and the piston will not be drawn up with such ease as if the whole was in vacuo. But though the column of air pressing in the direction CZ> acts as a resisting power on the piston b, the column pressing in the direction Aα will act as a moving power upon the piston a. It is therefore plain, that if b be moved upwards till it comes to the place marked d, the other will descend to that marked c. Now, if we suppose the piston a to be removed, it is plain that when b is pulled upwards to d, the air descending through the leg AαCB will press on the under side of the piston b, as strongly as it would have done upon the upper side of the piston a, had it been present. Therefore, though the air passing down through the leg CB resists the motion of the piston b when drawn upwards, the air pressing down through the leg ĂB forwards it as much; and accordingly the piston b may be drawn up or pushed down at pleasure, and with very little trouble. But if the orifice at A be stopped, so that the air can only exert its resisting power on the piston b, it will require a considerable degree of strength to move the piston from b to d.

If now we suppose the tube to be entirely removed (which indeed answers no other purpose than to render the action of the air more evident), it is plain that if the piston be moved either up or down, or in any other direction we can imagine, the air will press as much upon the back part of it as it resists it on the fore part; and consequently a ball moving through the air with any degree of velocity, ought to be as much accelerated by the action of the air behind, as it is retarded by the action of that before. Here then it is natural to ask, If the air accelerates a moving body as much as it retards it, how comes it to make any resistance at all? Yet certain it is that this fluid does resist, and that very considerably. To this it may be answered, that the air is always kept in some certain state or constitution by another power which rules all its motions, and it is this power undoubtedly which gives the resistance. It is not to our purpose at present to inquire what that power is, but we see that the air is often in very different states; one day, for instance, its parts are violently agitated by a storm, and another perhaps they are comparatively at rest in a calm. In the first case, nobody hesitates to own that the storm is occasioned by some cause or other, which violently resists any other power that would prevent the agitation of the air. In a calm the case is the same; for it would require the same exertion of power to excite a tempest in a calm day as to allay a tempest in a stormy one. Now it is evident that all projectiles, by their motion, agitate the atmosphere in an unnatural manner, and consequently are resisted by that power, whatever it is, which tends to restore the equilibrium, or bring back the atmosphere to its former state.

If no other power besides that above mentioned acted upon projectiles, it is probable that all resistance to their motion would be in the duplicate proportion of their velocities; and accordingly, as long as their velocity is small, we find that generally it is so. But when the velocity comes to be exceedingly great, other sources of resistance arise. One of these is a subtraction of part of the moving power, which, though not properly a resistance, or opposing another power to it, is an equivalent thereto. This subtraction arises from the following cause: The air, as we have already observed, presses upon the hinder part of the moving body by its gravity, as much as it resists the fore part of it by the same property. Nevertheless the velocity with which the air presses upon any body by means of its gravity is limited; and it is possible that a body may change its place with so great velocity that the air has not time to rush in upon the back part of it in order to assist its progressive motion. When this happens to be the case, there is in the first place a deficiency of the moving power equivalent to fifteen pounds on every square inch of surface, at the same time that there is a positive resistance of as much more on the fore part, owing to the gravity of the atmosphere, which must be overcome before the body can move forward.

This deficiency of moving power, and increase of resistance, do not only take place when the body moves with a very great degree of velocity, but in all motions whatsoever. It is not in all cases perceptible, because the velocity with which the body moves frequently bears but a very small proportion to the velocity with which the air presses in behind it. Thus, supposing the velocity with which the aiι rushes into a vacuum to be 1200 feet in a second, if a bod∖ moves with a velocity of 40 or 50 feet in a second, the force with which the air presses on the back part is but 5 ’ yth at the utmost less than that which resists on the fore part of it, which will not be perceptible; but if, as in the case of bullets, the velocity of the projectile comes to have a considerable proportion to the velocity wherewith the air rushes in behind it, then a very perceptible and otherwise unaccountable resistance is observed, as we have seen in the [7:11:58] experiments already related by Mr Robins. Thus, if the air presses in with a velocity of 1200 feet in a second, and if the body changes its place with a velocity of 600 feet in the same time, there is a resistance of fifteen pounds on the fore part, and a pressure of only 7⅜ on the back part. The resistance therefore not only overcomes the moving power of the air by 7⅛ pounds, but there is a deficiency of other 7∣ pounds owing to the want of half the pressure of the atmosphere on the back part, and thus the whole loss of the moving power is equivalent to 15 pounds; and hence the exceeding great increase of resistance observed by Mr Robins beyond what it ought to be according to the common computations. The velocity with which the air rushes into a vacuum is therefore a desideratum in gunnery. Mr Robins supposes that it is the same with the velocity of sound; and that when a bullet moves with a velocity greater than that of 1200 feet in a second, it leaves a perfect vacuum behind it. Hence he accounts for the great increase of resistance to bullets moving with such velocities; but as he does not take notice of the loss of the air’s moving power, the anomalies of all lesser velocities are inexplicable on his principles. Nay, he even tells us that Sir Isaac Newton’s rule for computing resistances may be applied in all velocities less than 1100 or 1200 feet in a second, though this is expressly contradicted by his own experiments already mentioned.

Though for these reasons it is evident how great difficulties must occur in attempting to calculate the resistance of the air to military projectiles, we have not yet even discovered all the sources of resistance to these bodies when moving with immense velocities. Another power by which they are opposed, and which at last becomes greater than any of those hitherto mentioned, is the air’s elasticity. This, however, will not begin to show itself in the way of resistance till the velocity of the moving body becomes considerably greater than that by which the air presses into a vacuum. Having therefore first ascertained this velocity, which we shall suppose to be 1200 feet in a second, it is plain that if a body moves with a velocity of 1800 feet in a second, it must compress the air before it; because the fluid has neither time to expand itself in order to fill the vacuum left behind the moving body, nor to rush in by its gravity. This compression it will resist by its elastic power, which thus becomes a new source of resistance, increasing, without any limit, in proportion to the velocity of the moving body. If now we suppose the moving body to set out with a velocity of 2400 feet in a second, it is plain that there is not only a vacuum left behind the body, but the air before it is compressed into half its natural space. The loss of motion in the projectile therefore is now very considerable. It first loses 15 pounds on every square inch of surface on account of the deficiency of the moving power of the air behind it, then it loses 15 pounds more on account of the resistance of the air before it; again, it loses 15 pounds on account of the elasticity of the compressed air; and, lastly, it loses another 15 pounds on account of the vacuum behind, which takes off the weight of the atmosphere, that would have been equivalent to one half of the elasticity of the air before it. The whole resistance therefore upon every square inch of surface moving with this velocity is 60 pounds, besides that which arises from the power tending to preserve the general state of the atmosphere, and which increases in the duplicate proportion of the velocity, as already mentioned. If the body is supposed to move with a velocity of 4800 feet in a second, the resistance from the elasticity of the air will then be quadrupled, or amount to 60 pounds on the square inch of surface, which, added to the other causes, will produce a resistance of 105 pounds upon the square inch; and thus the resistance from the elasticity of the air would go on continually increasing, till at last the motion of the projectile would be as effectually stopped as if it were fired against a wall. This obstacle therefore we are to consider as really insuperable by any art whatsoever, and therefore it is not advisable to use larger charges of powder than what will project the shot with a velocity of 1200 feet in a second. To this velocity the elasticity of the air will not make great resistance, if indeed it makes any at all; for though Mr Robins has conjectured that air rushes into a vacuum with the velocity of sound, or between 1100 and 1200 feet in a second, yet we have no decisive proof of the truth of this supposition. At this velocity, indeed, according to Mr Robins, a very sudden increase of resistance takes place; but this is denied by Mr Glenie, in his History of Gunnery (p. 48, 50), who supposes that the resistance proceeds gradually; and indeed it seems to be pretty obvious that the resistance cannot very suddenly increase, if the velocity be only increased in a small degree. Yet it is certain that the swiftest motions with which cannon-balls can be projected are very soon reduced to the standard; for Mr Robins informs us, that “a 24-pound shot, when discharged with a velocity of 2000 feet in a second, will be reduced to that of 1200 feet in a second in a flight of little more than 500 yards.”

In the seventy-first volume of the Philosophical Transactions, Count Rumford has proposed a new method of determining the velocities of bullets, by measuring the force of the recoil of the piece. As in all cases action and re-action are supposed to be equal to one another, it appears that the momentum of a gun, or the force of its recoil backwards, must always be equivalent to the force of its charge; that is, the velocity with which the gun recoils, multiplied into its weight, is equal to the velocity of the bullet multiplied into its weight; for every particle of matter, whether solid or fluid, that issues out of the mouth of a piece, must be impelled by the action of some power, which power must re-act with equal force against the bottom of the bore. Even the fine elastic invisible fluid which is gen crated from the powder in its inflammation cannot put itself in motion without at the same time re-acting against the gun. Thus we see pieces, when they are fired with powder alone, recoil as well as when their charges are made to impel a weight of shot, though the recoil is not in the same degree in both cases. It is easy to determine the velocity of the recoil in any given case, by suspending the gun in an horizontal position by two pendulous rods, and measuring the arc of its ascent by means of a ribbon, as mentioned under the article Gunpowder ; and this will give the momentum of the gun, its weight being known, and consequently the momentum of its charge. But in order to determine the velocity of the bullet from the momentum of the recoil, it will be necessary to know how much the weight and velocity of the elastic fluid contribute towards it.

That part of the recoil which arises from the expansion of the fluid is always very nearly the same, whether the powder is fired alone, or whether the charge is made to impel one or more bullets, as has been determined by a great variety of experiments. If therefore a gun, suspended according to the method prescribed, is fired with any given charge of powder, but without any bullet or wad, and the recoil is observed, and if the same piece is afterwards fired with the same quantity of powder, and a bullet of a known weight, the excess of the velocity of the recoil in the latter case, over that in the former, will be proportional to the velocity of the bullet; for the difference of these velocities, multiplied into the weight of the gun, will be equal to the weight of the bullet multiplied into its velocity. Thus, if W is put equal to the weight of the gun, U = the velocity of the bullet when fired with a given charge of powder without any bullet; V = the velocity of the recoil when the same charge is made to impel a bullet; B = the weight of the bullet, and v ≡ its velocity; it will be v ~ ~~ J⅞ — .
[7:11:59]

To determine how far this theory agreed with practice, an experiment was made with a charge of 165 grains of powder, without any bullet, which produced a recoil of 5∙5 inches; and in another, with a bullet, the recoil was 5*6 inches, the mean of which is 5∙55 inches, answering to a velocity of 1∙1358 feet in a second. In five experiments with the same charge of powder, and a bullet weighing 580 grains, the mean was 14*6 inches; and the velocity of the recoil answering to the length just mentioned, is 2∙9880 feet in a second; consequently V — U, or 2∙9880 — 1∙1358, is equal to 1∙8522 feet in a second. But as the velocities of recoil are known to be as the chords of the arcs through which the barrel ascends, it is not necessary, in order to determine the velocity of the bullet, to compute the velocities V and U; but the quantity V — U, or the difference of the velocities of the recoil when the given charge is fired with and without a bullet, may be computed from the value of the difference of the chords by one operation. Thus the velocity answering to the chord 9Ό5 is that of 1∙8522 feet in a second, which is just equal to V — U, as was before found.

In this experiment the weight of the barrel with its carriage was just 47∣ pounds, to which f ths of a pound were to be added on account of the weight of the rods by which it was suspended; thus making W = 48 pounds, or 336,000 grains. The weight of the bullet was 580 grains; whence B is to W as 580 to 336,000, that is, as 1 to 579∙31 very nearly. The value of V — U, answering to the experiments before mentioned, was found to be 1∙8522; consequently the velocity of the bullet = v, was 1'8522 × 579∙31 = 1073 feet, which differs only by 10 from 1083, the velocities found by the pendulum.

The velocities of the bullets may be found from the recoil by a still more simple method. For the velocities of the recoil being as the chords measured upon the ribbon, if c is put equal to the chord of the recoil expressed in English inches, when the piece is fired with powder only, and C = the chord when the same piece is charged with a bullet; then C — c will be as V — U; and consequent-

(V—U) W

ly as g , which measures the velocity of the bul let, the ratio of W to B remaining the same. If therefore we suppose a case in which C — c is equal to one inch, and the velocity of the bullet is computed from that chord, the velocity in any other case, wherein C — c is greater or less than one inch, will be found by multiplying the difference of the chords C and c by the velocity answering to the difference of one inch. The length of the parallel rods by which the piece was suspended being 64 inches, the velocity of the recoil, = C — c= 1 inch measured upon the ribbon, is 0∙204655 parts of a foot in one second, which in this case is also the value of V — U; the velocity of the bullet, or v, is therefore 0∙204655 × 579∙31 = 118∙35 feet in a second. Hence the velocity of the bullet may in all cases be found by multiplying the difference of the chords C and c by 118∙35, the weight of the barrel, the length of the rods by which it is suspended, and the weight of the bullet, remaining the same; and this whatever the charge of powder made use of may be, and however it may differ in strength and goodness.

The exactness of this second method will appear from the following experiments. On firing the piece with 145 grains of powder and a bullet, the mean of three sets of experiments was 13∙25, 1345, and 13-2; and with the same charge of powder without a bullet, the recoil was 4∙5, 4∙3, or 4*4. C — c therefore was 13∙2 — 4∙4 = 8*8 inches; and the velocity of the bullets, = 8∙8 × 118∙35 = 1045 feet in a second; the velocities by the pendulum coming out 10∙40 feet in the same space of time.

In the far greater number of experiments to determine the comparative accuracy of the two methods, a surprising agreement was found between the last-mentioned one and that by the pendulum; but in some few the differences were very remarkable. Thus, in two where the recoil was 12∙92 and 13∙28, the velocity, by computation from the chords, is 1030 feet per second; but in computing by the pendulum it amounted only to 900; in these, however, some inaccuracy was suspected in the experiment with the pendulum, and the computation from the recoil was most to be depended upon. In another experiment, the velocity by the recoil exceeded that by the pendulum by no less than 346 feet; the former showing 2109, and the latter only 1763 feet in a second. In two others the pendulum was also deficient, though not in such a degree. In all these it is remarkable, that where the difference was considerable, it was still in favour of the recoil. The deficiency in these experiments appears to have been somewhat embarrassing to our author. “It cannot be supposed,” says he, “that it arose from any imperfection in Mr Robins’s method of determining the velocities of bullets; for that method is founded upon such principles as leave no room to doubt of its accuracy; and the practical errors that occur in making the experiments, and which cannot be entirely prevented, or exactly compensated, are in general so small, that the difference in the velocities cannot be attributed to them-It is true, the effect of those errors is more likely to appear in experiments made under such circumstances as the present; for the bullet being very light,^[1. They were made of lead, enclosing in a nucleus of Paris plaster. ] the arc of the ascent of the pendulum was but small; and a small mistake in measuring the chord upon the ribbon would have produced a very considerable error in computing the velocity of the bullet. Thus a difference of one tenth of an inch, more or less, upon the ribbon, in that experiment where the difference was greatest, would have made a difference in the velocity of more than 120 feet in a second. But, independent of the pains that were taken to prevent mistakes, the striking agreement of the velocities in so many other experiments affords abundant reason to conclude, that the errors arising from those causes were in no case very considerable. But if both methods of determining the velocities of bullets are to be relied on, then the difference of the velocities, as determined by them in these experiments, can only be accounted for by supposing that it arose from their having been diminished by the resistance of the air in the passage of the bullets from the mouth of the piece to the pendulum; and this suspicion will be much strengthened, when we consider how great the resistance of the air is to bodies that move very swiftly in it; and that the bullets in these experiments were not only projected with great velocities, but were also very light, and consequently more liable to be retarded by the resistance on that account.

“To put the matter beyond all doubt, let us see what the resistance was that these bullets met with, and how much their velocities were diminished by it. The weight of the bullet in the most erroneous experiment was 90 grains, its diameter 0*78 of an inch, and it was projected with a velocity of 2109 feet in a second. If now a computation be made according to the law laid down by Sir Isaac Newton for compressed fluids, it will be found, that the resistance of this bullet was not less than 8į pounds avoirdupois, which is something more than 660 times its own weight. But Mr Robins has shown by experiment, that the resistance of the air to bodies moving in it with [7:11:60]very great velocity, is near three times greater than Sir Isaac has determined it; and as the velocity with which this bullet was impelled is considerably greater than any in Mr Robins’s experiments, it is highly probable that the resistance in this instance was at least 2000 times greater than the weight of the bullet.

“The distance from the mouth of the piece to the pendulum was 12 feet; but, as there is reason to think that the blast of the powder, which always follows the bullet, continues to act upon it for some sensible space of time after it is out of the bore, and, by urging it on, counterbalances, or at least counteracts, in a great measure, the resistance of the air, we will suppose that the resistance does not begin, or rather that the motion of the bullet does not begin to be retarded, till it has got to the distance of two feet from the muzzle. The distance, therefore, between the barrel and the pendulum, instead of 12 feet, is to be esteemed at 10 feet; and as the bullet took up about part of a second in running over that space, it must in that time have lost a velocity of about 335 feet in a second, as will appear upon making the computation; and this will very exactly account for the apparent diminution of the velocity in the experiment; for the difference of the velocities, as determined by the recoil and the pendulum = 2109 — 1763 = 346 feet in a second, is extremely near 335 feet in a second, the diminution of the velocity by the resistance as here determined.

“If the diminution of the velocities of the bullets in the two subsequent experiments be computed in like manner, it will turn out in one 65, and in the other 33, feet in a second; and, making these corrections, the comparison of the two methods of ascertaining the velocities will stand thus:
[table]

“It appears, therefore, that notwithstanding these corrections, the velocities as determined by the pendulum, particularly in the last, were considerably deficient. But the manifest irregularity of the velocities in those instances, affords abundant reason to conclude, that it must have arisen from some accidental cause, and therefore that little dependence is to be put upon the result of those experiments. I cannot take upon me to determine positively what the cause was which produced this irregularity; but I strongly suspect that it arose from the breaking of the bullets in the barrel by the force of the explosion: for these bullets, as has already been mentioned, were formed of lead, enclosing lesser bullets of plaster of Paris; and I well remember to have observed at the time several small fragments of the plaster which had fallen down by the side of the pendulum. I confess I did not then pay much attention to this circumstance, as I naturally concluded that it arose from the breaking of the bullet in penetrating the target of the pendulum; and that the small pieces of plaster I saw upon the ground had fallen out of the hole by which the bullet entered. .But if the bullets were not absolutely broken in pieces in firing, yet if they were considerably bruised, and the plaster, or a part of it, were separated from the lead, such a change in the form might produce a great increase in the resistance, and even their initial velocities might be affected by it; for their form being changed from that of a globe to some other figure, they might not fit the bore; and a part of the force of the charge might be lost by the windage. That this actually happened in the experiment last mentioned seems very probable, as the velocity with which the bullet was projected, as it was determined by the recoil, was considerably less in proportion in that experiment than in many others which preceded and followed it in the same set.

“As allowance has been made for the resistance of the air in these cases, it may be expected that the same should be done in all other cases; but it will probably appear, upon inquiry, that the diminution of the velocities of the bullets on that account was so inconsiderable, that it might safely be neglected: thus, for instance, in the experiments with an ounce of powder, when the velocity of the bullet was more than 1750 feet in a second, the diminution turns out no more than 25 or 30 feet in a second, though we suppose the full resistance to have begun so near as two feet from the mouth of the piece; and in all cases where the velocity was less, the effect of the resistance was less in a much greater proportion; and even in this instance there is reason to think, that the diminution of the velocity, as we have determined it, is too great; for the flame of gunpowder expands with such amazing rapidity, that it is scarcely to be supposed but that it follows the bullet, and continues to act upon it more than two feet, or even four feet, from the gun; and when the velocity of the bullet is less, its action upon it must be sensible at a still greater distance.”

As this method of determining the velocities of bullets by the recoil of the piece did not occur to Count Rumford till after he had finished his experiments with a pendulum, and taken down his apparatus, he had it not in his power to determine the comparative strength of the recoil with and without a bullet; and consequently the velocity with which the flame issues from the mouth of a piece. He is of opinion, however, that every thing relating to these matters may be determined with greater accuracy by the new method than by any other formerly practised; and he very justly remarks, that the method of determining the velocity by the recoil, gives it originally as the bullet sets out; whilst that by the pendulum shows it only after a part has been destroyed by the resistance of the air. In the course of his remarks, he criticises a part of Mr Robins’ theory, that when bullets of the same diameter, but of different weights, are discharged from the same piece by the same quantity of powder, their velocities are in the subduplicate ratio of their weight. This theory, he observes, is manifestly defective, as being founded upon a supposition, that the action of the elastic fluid generated from the powder is always the same in any and every given part of the bore when the charge is the same, whatever may be the weight of the bullet; and as no allowance is made for the expenditure of force required to put the fluid itself in motion, nor for the loss of it by the vent. “It is true,” says he, “Dr Hutton in his experiments found this law to obtain without any great error; and possibly it may hold good with sufficient accuracy in many cases; for it sometimes happens that a number of errors or actions, whose operations have a contrary tendency, so compensate each other, that their effects when united are not sensible. But when this is the case, if any one of the causes of error is removed, those which remain will be detected. When any given charge is loaded with a heavy bullet, more of the powder is inflamed in any very short space of time than when the bullet is lighter, and the action of the powder ought upon that account to be greater; but a heavy bullet takes up longer time in passing through the bore than a light one, and consequently more of the elastic fluid generated from the powder escapes by the vent and by windage. It may happen that the augmentation of the force, on account of one of these circumstances, may be just able to counterbalance the diminution of it arising from the other; and if it should be found upon trial that this is the case in general, in pieces as they are now constructed, and with all the variety of shot that are made [7:11:61]use of in practice, it would be of great use to know the fact; but when, with Mr Robins, concluding too hastily from the result of a partial experiment, we suppose, that because the sum total of the pressure of the elastic fluid upon the bullet, during the time of its passage through the bore, happens to be the same when bullets of different weights are made use of, that therefore it is always so, our reasonings may prove very inconclusive, and lead to very dangerous errors.”

In the prosecution of his subject Count Rumford proves mathematically, as well as by actual experiment, that the theory laid down by Mr Robins in this respect is erroneous. The excess is in favour of heavy bullets, which acquire a velocity greater than they ought to do according to Mr Robins’s rule; and so considerable are the errors, that in one of Count Rumford’s experiments the difference was no less than 2042 feet in a second. When the weight of the bullet was increased four times, the action of the powder was found to be nearly doubled; for in one experiment, when four bullets were discharged at once, the collective pressure was as one; but when only a single bullet was made use of, it was no more than 0-5825; and upon the whole he concludes, that the velocity of bullets is in the reciprocal subtriplicate ratio of their weights. Our author observes also, that Mr Robins is not only mistaken in the particular just mentioned, but in his conclusions with regard to the absolute force of gunpowder compared with the pressure of the atmosphere; the latter being to the force of gunpowder as one to 1000 according to Mr Robins, but as one to 1308 according to Count Rumford.

II.—PRACTICE OF GUNNERY.

With regard to the practical part of gunnery, which ought to consist in directing the piece in such a manner as always to hit the object against which it is pointed, there can be no certain rules given. The following maxims are laid down by Mr Robins as of use in practice.

1. In any piece of artillery whatever, the greater the quantity of powder it is charged with, the greater will be the velocity of the bullet.

2. If two pieces of the same bore, but of different lengths, are fired with the same charge of powder, the longer will impel the bullet with a greater celerity than the shorter.

3. If two pieces of artillery different in weight, and formed of different metals, have yet their cylinders of equal bores and equal lengths; then with like charges of powder and like bullets they will each of them discharge their shot with nearly the same degree of celerity.

4. The ranges of pieces at a given elevation are no just measures of the velocity of the shot; for the same piece fired successively at an invariable elevation, with the powder, bullet, and every other circumstance as nearly the same as possible, will yet range to very different distances.

5. The greater part of that uncertainty in the ranges of pieces which is described in the preceding maxim, can only arise from the resistance of the air. *

6. The resistance of the air acts upon projectiles in a twofold manner; for it opposes their motion, and thus continually diminishes their celerity; and it besides diverts them from the regular track they would otherwise follow; whence arise those deviations and inflections already treated of.

7. That action of the air by which it retards the motion of projectiles, though much neglected by writers on artillery, is yet, in many instances, of an immense force; and hence the motion of these resisted bodies is totally different from what it would otherwise be.

8. This retarding force of the air acts with different degrees of violence, according as the projectile moves with a greater or less velocity; and the resistances observe this law, that to a velocity which is double another, the resistance within certain limits is fourfold; to a treble velocity, ninefold; and so on.

9. But this proportion’ between the resistances to two different velocities does not hold if one of the velocities be less than that of 1200 feet in a second, and the other greater; for in that case the resistance to the greater velocity is nearly three times as much as it would come out by a comparison with the smaller, according to the law explained in the last maxim.

10. To the extraordinary power exerted by the resistance of the air it is owing, that when two pieces of different bores are discharged at the same elevation, the piece of the largest bore usually ranges farthest, provided they are both fired with fit bullets, and the customary allotment of powder.

11. The greater part of military projectiles will at the time of their discharge acquire a whirling motion round their axis, by rubbing against the insides of their respective pieces; and this whirling motion will cause them to strike the air very differently from what they would do had they no other than a progressive motion. By this means it may happen that the resistance of the air is not always directly opposed to their flight, but frequently acts in a line oblique to their course, and thereby forces them to deviate from the regular track they would otherwise describe. And this is the true cause of the irregularities described in maxim 4.

12. From the sudden trebling the quantity of the air’s resistance, when the projectile moves swifter than at the rate of 1200 feet in a second (as has been explained in maxim 9), it follows, that whatever be the regular range of a bullet discharged with this last-mentioned velocity, that range will be but little increased, how much soever the velocity of the bullet may be still farther augmented by greater charges of powder.

13. If the same piece of cannon be successively fired at an invariable elevation, but with various charges of powder, the greatest charge being the whole weight of the bullet in powder, and the least not less than the fifth part of that weight; then if the elevation be not less than eight or ten degrees, it will be found, that some of the ranges with the least charge will exceed some of those with the greatest.

14. If two pieces of cannon of the same bore, but of different lengths, are successively fired at the same elevation with the same charge of powder; then it will frequently happen that some of the ranges with the shorter piece w ill exceed some of those with the longer.

15. In distant cannonadings the advantages arising from long pieces and large charges of powder afe but of little moment.

16. In firing against troops with grape-shot, it will be found that charges of powder much less than those generally used arc the most advantageous.

17. The principal operations in which large charges of powder appear to be more efficacious than small ones, are the ruining of parapets, the dismounting of batteries covered by stout merlins, or battering in breach; for, in all these cases, if the object be but little removed from the piece, every increase of velocity will increase the penetration of the bullet.

18. Whatever operations are to be performed by artillery, the least charges of powder with which they can be effected are always to be preferred.

19. Hence the proper charge of any piece of artillery is not that allotment of powder which will communicate the greatest velocity to the bullet, as most practitioners formerly maintained; nor is it to be determined by an invariable proportion of its weight to the weight of the ball; [7:11:62]but, on the contrary, it is such a quantity of powder as will produce the least velocity for the purpose in hand; and, instead of bearing always a fixed ratio to the weight of the ball, it must be different according to the different business which is to be performed.

20. No field-piece ought at any time to be loaded with more than ļth, or at the utmost yth, of the weight of its bullet in powder, nor should the charge of any battering piece exceed ⅛d of the weight of its bullet.

21. Although precepts very different from those we have here given have often been advanced by artillerists, and have been said to be derived from experience, yet is that pretended experience altogether fallacious; since, from oui-doctrine of resistance established above, it follows that every speculation on the subject of artillery, which is only founded on the experimental ranges of bullets discharged with considerable velocities, is liable to great uncertainty.

The greatest irregularities in the motion of bullets are, as we have seen, 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 ent with a number of spiral channels; so that it is in reality 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 different in each barrel, according to the fire of the piece and the fancy of the workman; and in like manner the depth to which they are cut is not regulated by any invariable rule.

The usual method of charging these pieces is this: When the proper quantity of powder is put down, a leaden bullet is taken, a small matter larger than the bore of the piece was before the rifles were cut; and this bullet being laid on the mouth of the piece, and consequently too large to go down of itself, it is forced by a strong rammer impelled by a mallet, and by repeated blows is driven home to the powder; and the softness of the lead giving way to the violence with which the bullet is impelled, that zone of the bullet which is contiguous to the piece varies its circular form, and takes the shape of the inside of the barrel; so that it becomes part of a male screw exactly answering to the incidents of the rifle.

In some parts of Germany and Switzerland, however, an improvement is added to this practice, especially in the larger pieces which are used for shooting at great distances. This is done by cutting a piece of very thin leather, or of thin fustian, in a circular shape, somewhat larger than the bore of the barrel. This circle being greased on one side, is laid upon the muzzle with its greasy side downwards; and the bullet being then placed upon it, is forced down the barrel with it, by which means the leather or fustian encloses the lower half of the bullet, and, by its interposition between the bullet and the rifles, prevents the lead from being cut by them. It must be remembered, however, that in the barrels where this is practised, the rifles are generally shallow, and the bullet ought not to be too large. But as both these methods of charging at the mouth take up a good deal of time, the rifled barrels which have been made in Britain are contrived to be charged at the breach, where the piece is for this purpose made larger than in any other part. The powder and bullet are put in through the side of the barrel, by an opening, which, when the piece is loaded, is then filled up with a screw. By this means, when the piece is fired, the bullet is forced through the rifles, and acquires the spiral motion already described; and perhaps something of this kind, though not in the manner now practised, would be, according to Robins, the most perfect method for the construction of diese kinds of barrels. From the whirling motion communicated by the rifles, it happens, that when the piece is fired, the 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 rifle barrel will revolve round an axis coincident with the line of its flight. By this rotation on its axis, the aberration of the bullet, which proves so prejudicial to all operations in.gunnery, is almost totally prevented. The reason of this may be easily understood from considering the slow motion of an arrow through the air. For example, if a bent arrow, with its wings not placed in some degree in a spiral position, so as to make it revolve round its axis as it flies through the air, were shot at a mark with a true direction, it would constantly deviate from it, in consequence of being pressed to one side by the convex part opposing the air obliquely. Let us now suppose this deflection in a flight of 100 yards to be equal to ten yards. Now, if the same bent arrow were made to revolve round its axis once every two yards of its flight, its greatest deviation would take place when it had proceeded only one yard, or made half a revolution; since at the end of the next half revolution it would again return to the same direction it had at first; the convex side of the arrow having been once in opposite positions. In this manner it would proceed during the whole course of its flight, constantly returning to the true path at the end of every two yards; and when it reached the mark, the greatest deflection to either side that could happen would be equal to what it makes in proceeding one yard, equal to part of the former, or 3∙6 inches, a very small deflection when compared with the former one. In the same manner, a cannon-ball which revolves not round its axis, deviates greatly from the true path, on account of the inequalities on its surface; which, although small, cause great deviations by reason of the resistance of the air, at the same time that the ball acquires a motion round its axis in some uncertain direction occasioned by the friction against its sides. But by the motion acquired from the rifles, the error is perpetually corrected in the manner just now described; and accordingly such pieces are much more to be depended on, and will do execution at a much greater distance, than the other.

The reasons commonly alleged for the superiority of rifle-barrels over common ones, are, either that the inflammation of the powder is greater, by the resistance which the bullet makes by being thus forced into the barrel, and that thereby it receives a much greater impulse; or that the bullet, by the compounding of its circular and revolving motions, as it were bores the air, and thereby flies to a much greater distance than it would otherwise have done; or that by the same boring motion it makes its way through all solid substances, and penetrates into them much deeper than when fired in the common manner. But these views Mr Robins has proved to be altogether erroneous, by a great number of experiments made with rifle-barrelled pieces. “In these experiments,” says he, “I have found that the velocity of the bullet fired from a rifled barrel was usually less than that of the bullet fired from -a common piece with the same proportion of powder. Indeed it is but reasonable to expect that this should be the case; for if the rifles are very deep, and the bullet is large, enough to fill them up, the friction bears a very considerable proportion to the effort of the powder. And that in this case the friction is of consequence enough to have its effects observed, I have discovered by the continued use of the same barrel. For the metal of the barrel being soft, and wearing away apace, its bore by half a year’s use was considerably enlarged, and consequently the depths of its [7:11:63] rifles diminished; and then I found that the same quantity of powder would give to the bullet a velocity near a tenth part greater than what it had done at first. And as the velocity of the bullet is not increased by the use of rifled barrels, so neither is the distance to which it flies, nor the depth of its penetration into solid substances. Indeed these two last suppositions seem at first sight too chimerical to deserve a formal confutation. But I cannot help observing, that those who have been habituated to the use of rifled pieces are very excusable in giving way to these prepossessions. For they constantly found, that with them they could fire at a mark with tolerable success, though it were placed at three or four times the distance to which the ordinary pieces were supposed to reach. And therefore, as they were ignorant of the true cause of this variety, and did not know that it arose only from preventing the deflection of the ball; it was not unnatural for them to imagine that the superiority of effect in the rifled piece was owing either to a more violent impulse at first, or to a more easy passage through the air.

“In order to confirm the foregoing theory of rifle-barrelled pieces, I made some experiments by which it might be seen whether one side of the ball discharged from them uniformly keeps foremost during the whole course. To examine this particular, I took a rifled barrel carrying a bullet of six to the pound; but instead of its leaden bullet, I used a wooden one of the same size, made of a soft springy wood, which bent itself easily into the rifles without breaking. And firing the piece thus loaded against a wall at such a distance as the bullet might not be shivered by the blow, I always found, that the same surface which lay foremost in the piece continued foremost, without any sensible defleetion during the time of its flight. And this was easily to be observed by examining the bullet, as both the marks of the rifles, and the part that impinged on the wall, were sufficiently apparent. Now, as these wooden bullets were but the sixteenth part of the weight of the leaden ones, I conclude, that if there had been any unequal resistance or deflective power, its effects must have been extremely sensible upon this light body, and consequently, in some of the trials I made, the surface which came foremost from the piece must have been turned round into another situation.

“But again, I took the same piece, and, loading it now with a leaden ball, I set it nearly upright, sloping it only three or four degrees from the perpendicular, in the direction of the wind; and firing it in this situation, the bullet generally continued about half a minute in the air, it rising by computation to near three quarters of a mile perpendicular height. In these trials I found that the bullet commonly came to the ground to the leeward of the piece, and at such a distance from it as nearly corresponded to the angle of its inclination, and to the effort of the wind; it usually falling not nearer to the piece than 100, nor farther from it than 150 yards. And this is a strong confirmation of the almost steady flight of this bullet for about a mile and a half: for were the same trial made with a common piece, I doubt not but the deviation would often amount to half a mile, or perhaps considerably more; though this experiment would be a very difficult one to examine, on account of the little chance there would be of discovering where the ball fell.

“ It must be observed, however, that though the bullet impelled from a rifle-barrelled piece keeps for a time to its regular track with sufficient nicety, yet if its flight be so far extended that the traek becomes considerably incurvated, it will then undergo considerable defleetions. This, according to my experiments, arises from the angle at last made by the axis on which the bullet turns, and the direction in which it flies; for that axis continuing nearly parallel to itself, it must necessarily diverge from the line of the flight of the bullet, when that line is bent from its original direction: and when it once happens that the bullet whirls on an axis which no longer coincides with the line of its flight, then the unequal resistance formerly described will take place, and the deflecting power hence· arising will perpetually increase, as the track of the bullet, by having its range extended, becomes more and more incurvated. This matter I have experienced in a small riflebarrelled piece, carrying a leaden ball of near half an ounce weight; for this piece, charged with one dram of powder, ranged about 550 yards at an angle of twelve degrees with sufficient regularity; but being afterwards elevated to an angle of twenty-four degrees, it then ranged very irregularly, generally deviating from the line of its direction to the left, and in one case not less than 100 yards. This apparently arose from the cause above mentioned, as was confirmed from the constant deviation of the bullet to the left; for, by considering how the revolving motion was continued with the progressive one, it appeared that a deviation that way was to be expected.

“The best remedy I can think of for this defect is the making use of bullets of an egg-like form instead of spherical ones. For if such a bullet hath its shorter axis made to fit the piece, and it be placed in the barrel with its smaller end downwards, then it will acquire by the rifles a rotation round its larger axis; and its centre of gravity lying nearer to its fore than its hinder part, its longer axis will be constantly forced by the resistance of the air into the line of its flight; as we see that by the same means arrows constantly lie in the line of their direction, however that line be incurvated.

“But besides this, there is another circumstance in the use of these pieces, which renders the flight of their bullets uncertain when fired at a considerable elevation. For I find by my experiments, that the velocity of a bullet fired with the same quantity of powder from a rifle barrel, varies much more from itself in different trials than when fired from a common piece. This, as I conceive, is owing to the great quantity of friction, and the impossibility of rendering it equal in each experiment. Indeed, if the rifles are not deeply cut, and if the bullet is nicely fitted to the piece, so as not to require a great force to drive it down, and if leather or fustian well greased is made use of between the bullet and barrel, perhaps, by a careful attention to all these particulars, great part of the inequality in the velocity of the bullet may be prevented, and the difficulty in question be in some measure obviated; but till this be done, it cannot be doubted that the range of the same piece, at an elevation, will vary considerably in every trial, although the charge be each time the same. And this I have myself experienced, in a number of diversified trials, with a rifle-barrelled piece loaded at the breech in the English manner. For here the rifles being indented very deep, and the bullet so large as to fill them up completely, 1 found, that though it flew with sufficient exactness to the distance of 400 or 500 yards; yet when it was raised to an angle of about twelve degrees (at which angle, being fired with one fifth of its weight in powder, its medium range is nearly 1000 yards); in this case, I say, I found that its range was variable, although the greatest care was taken to prevent any inequalities in the quantity of powder, or in the manner of charging. And as, in this case, the angle was too small for the first-mentioned irregularity to produce the observed effects, they can only be imputed to the different velocities which the bullet each time received by the unequal action of the friction.”

Thus we see that it is in a manner impossible entirely to correct the aberrations arising from the resistance ol the atmosphere; as even the rifle-barrelled pieces cannot be depended upon for more than one half of their actual [7:11:64] 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 Mr Robins. The foundation of all his calculations is the velocity with which the bullet flies off from the mouth of the piece. Mr Robins himself had not opportunities of making many experiments on the velocities of cannon balls, and the calculations from smaller ones cannot always be depended upon. In the sixty-eighth volume of the Philosophical Transactions, Dr Hutton has recited a number of experiments made on cannon carrying balls from one to three pounds weight. His machine for discovering the velocities of these balls was the same with that of Mr Robins, only of a larger size. His charges of powder were two, four, and eight ounces; and the results of fifteen experiments, which seem to have been the most accurate, are as follow:
[table]

In another course, the mean velocities, with the same charges of powder, were 613, 873, 1162. “The mean velocities of the balls in the first course of experiments (says Dr Hutton) with two, four, and eight ounces of powder, are as the numbers 1, 1∙414, and 1∙993; but the subduplicate ratio of the weights (two, four, and eight) give the numbers 1, 1∙414, and 2, to which the others are sufficiently near. It is obvious, however, that the greatest difference lies in the last number, which answers to the greatest velocity. It will still be a little more in defect if we make the allowance for the weights of the balls; for the mean weights of the balls with the two and four ounces is 18∣ ounces, but of the eight ounces it is 18f-; diminishing therefore the number 1∙993 in the reciprocal subduplicate ratio of 18f∙ to 18f, it becomes 1∙985, which falls short of the number 2 by ∙015, or the 133d part of itself. A similar defect was observed in the other course of experiments; and both are owing to three evident causes, viz. 1. The less length of cylinder through which the ball was impelled; for with the eight-ounce charge it lay three or four inches nearer to the muzzle of the piece than with the others. 2. The greater quantity of elastic fluid which escaped in this case than in the others by the windage. This happens from its moving with a greater velocity; in consequence of which, a greater quantity escapes by the vent and windage than in smaller velocities. 3. The greater quantity of powder blown out unfired in this case than in that of the lesser velocities; for the ball which was impelled with the greater velocity would be sooner out of the piece than the others, and the more so as it had a less length of the bore to move through; and if powder fire in time, which cannot be denied, though indeed that time is manifestly very short, a greater quantity of it must remain unfired when the ball with the greater velocity issues from the piece, than when that which has the less velocity goes out, and still the more so as the bulk of powder which was at first to be inflamed in the one case so much exceeded that in the others.

“Let us now compare the corresponding velocities in both cases. In the one they are 701, 993, 1397; in the other, 613, 873, 1162. Now the ratio of the first two num bers, or the velocities with two ounces of powder, is that of 1 to 1Ί436, the ratio of the next two is that of 1 to ' 1∙1375, and the ratio of the last is that of 1 to 1∙2022. But the mean weight of the shot for two and four ounces of powder was 28⅜ ounces in the first course, and 18∣ in this; and for eight ounces of powder it was 28∣ in the first, and 18f in this. Taking therefore the reciprocal subduplicate ratios of these weights of shot, we obtain the ratio of 1 to 1∙224 for that of the balls which were fired with 2 ounces and four ounces of powder, and the ratio of 1 to 1∙241 for the balls which were fired with eight ounces. But the real ratios above found are not greatly different from these; and the variation of the actual velocities from this law of the weights of shot inclines the same way in both courses of experiments.

“We may now collect into one view the principal inferences that have resulted from these experiments. 1. It is evident from them that powder fires almost instantaneously. 2. The velocities communicated to balls or shot of the same weight with different quantities of powder, are nearly in the subduplicate ratio of these quantities; a very small variation in defect taking place when the quantities of powder become great. 3. When shot of different weights are fired with the same quantity of powder, the velocities communicated to them are nearly in the reciprocal subduplicate ratio of their weights. 4. Shot which are of different weights, and impelled by different quantities of powder, acquire velocities which are directly as the square roots of the quantities of powder, and inversely as the square roots of the weights of the shot nearly.”

The velocities of the bullets being thus found as nearly as possible, the ranges may be found by the following rules laid down by Mr Robins.

1. “Till the velocity of the projectile surpasses that of 1100 in a second, the resistance may be reckoned to be in the duplicate proportion of the velocity, and its mean quantity may be reckoned about half an ounce avoirdupois on a 12-pound shot, moving with a velocity of about twenty-five or twenty-six feet in a second.

2. “If the velocity be greater than that of 1100 or 1200 feet in a second, then the absolute quantity of the resistance in these greater velocities will be near three times as great as it should be by a comparison with the smaller velocities. Hence then it appears, that if a projectile begins to move with a velocity less than that of 1100 feet in P ’, its whole motion may be supposed to be considered on the hypothesis of a resistance in the duplicate ratio of the velocity. And if it begins to move with a velocity greater than this last mentioned, yet if the first part of its motion, till its velocity be reduced to near 1100 feet in 1", be considered separately from the remaining part in which the velocity is less than 1100 feet in 1", it is evident that both parts may be truly assigned on the same hypothesis; only the absolute quantity of the resistance is three times greater in the first part than in the last. Wherefore, if the motion of a projectile on the hypothesis of a resistance in the duplicate ratio of the velocity be truly and generally assigned, the actual motions of resisted bodies may be there· by determined, notwithstanding the increased resistances in the great velocities. And, to avoid the division of the motion into two, I shall show how to compute the whole at one operation, with little more trouble than if no such increased resistance took place.

“To avoid frequent circumlocutions, the distance to which any projectile would range in a vacuum on the horizontal plane at 45° of elevation, I shall call the potential random of that projectile; the distance to which the projectile would range in vacuo on the horizontal plane at any angle different from 45°, I shall call the potential range of the projectile at that angle; and the distance to which a projectile really ranges, I shall call its actual range.
[7:11:65]

“If the velocity with which a projectile begins to move is known, its potential random and its potential range at any given angle are easily determined from the common theory of projectiles; or more generally, if either its original velocity, its potential random, or its potential range, at a given angle, are known, the other two are easily found out.

“To facilitate the computation of resisted bodies, it is necessary, in the consideration of each resisted body, to assign a certain quantity, which 1 shall denominate F, adapted to the resistance of that particular projectile. To find this quantity F to any projectile given, we may proceed thus: First find, from the principles already delivered, with what velocity the projectile must move, so that its resistance may be equal to its gravity. Then the height from whence a body must descend in a vacuum to acquire this velocity is the magnitude of F sought. But the most concise way of finding this quantity F to any shell or bullet is this. If it be of solid iron, multiply its diameter measured in inches by 300, the product will be the magnitude of F expressed in yards. If, instead of a solid iron bullet, it is a shell or a bullet of some other substance; then, as the specific gravity of iron is to the specific gravity of the shell or bullet given, so is the F corresponding to an iron bvulet of the same diameter to the proper F for the shell or bullet given. The quantity F being thus assigned, the necessary computation of these resisted motions may be dispatched by the three following propositions, always remembering that these propositions proceed on the hypothesis of the resistance being in the duplicate proportion of the velocity of the resisted body. How to apply this principle, when the velocity is so great as to have its resistance augmented beyond this rate, shall be shown in a corollary to be annexed to the first proposition.
[table]

“ Prop. I. Given the actual range of a given shell or bullet at any small angle not exceeding 8° or 10°; to determine its potential range, and consequently its potential random and original velocity.

“ Solution. Let the actual range given be divided by the F corresponding to the given projectile, and find the quote in the first column of the preceding table: then the corresponding number in the second column multiplied into F will be the potential range sought: and thence, by the methods already explained, the potential random and the original velocity of the projectile is given.

“ Exam. An 18-pounder, the diameter of whose shot is about five inches, when loaded with two pounds of powder, ranged at an elevation of 3° 30 ’ to the distance of 975 yards.

“The F corresponding to this bullet is 1500 yards, and the quote of the actual range by this number is 65; corresponding to which, in the second column, is ∙817; whence, 817 F, or 1225 yards, is the potential range sought; and this, augmented in the ratio of the sine of twice the angle of elevation to the radius, gives 10,050 yards for the potential random: whence it will be found ’ that the velocity of this projectile was that of 984 feet in a second.

“ Cor. 1. If the converse of this proposition be desired; that is, if the potential range in a small angle be given, and thence the actual range be sought; this may be solv ed with the same facility by the same table: for if the given potential range be divided by its correspondent F, then opposite to the quote sought in the second column there will be found in the first column a number which, multiplied into F, will give the actual range required. And from hence it follows, that if the actual range be given at one angle, it may be found at every other angle not exceeding 8° or 10°.

“ Cor. 2. If the actual range at a given small angle be given, and another actual range be given, to which the angle is sought; this will be determined by finding the potential ranges corresponding to the two given actual ranges; then the angle corresponding to one of those potential ranges being known, the angle corresponding to the other will be found by the common theory of projectiles.

“ Cor. 3. If the potential random deduced from the actual range by this proposition exceeds 13,000 yards, then the original velocity of the projectile was so great as to be affected by the treble resistance described above; and consequently the real potential random will be greater than what is here determined. However, in this ease, the true potential random may be thus nearly assigned. Take a fourth continued proportional to 13,000 yards, and the potential random found by this proposition, and the fourth proportional thus found may be assumed for the true potential random sought. In like manner, when the true potential random is given greater than 13,000 yards, [7:11:66]we must take two mean proportionals between 13,000 and this random; and the first of these mean proportionals must be assumed instead of the random given, in every operation described in these propositions and their corollaries. And this method will nearly allow for the increased resistance in large velocities, the difference only amounting to a few minutes in the angle of direction of the projected body, which, provided that angle exceeds two or three degrees, is usually scarce worth attending to.

“Of this process take the following example :—A 24- pounder fired with 12 pounds of powder, when elevated at 7° 15', ranged about 2500 yards. Here the F being near 1700 yards, the quote to be sought in the first column is 147, to which the number corresponding in the second column is 2∙556; whence the potential range is near 4350 yards, and the potential random thence resulting 17,400. But this being more than 13,000, we must, to get the true potential random, take a fourth continued proportional to 13,000 and 17,400; and this fourth proportional, which is about 31,000 yards, is to be esteemed the true potential random sought; whence the velocity is nearly that of 1730 feet in a second.

“ Scholium. This proposition is confined to small angles, not exceeding 8° or 10°. In all possible cases of practice, this approximation, thus limited, will not differ from the most rigorous solution by so much as what will often intervene from the variation of the density of the atmosphere in a few hours’ time; so that the errors of the approximation are much short of other inevitable errors, which arise from the nature of this subject.

“ Prop. II. Given the actual range of a given shell or bullet at any angle not exceeding 45°; to determine its potential range at the same angle, and thence its potential random and original velocity.

“ Solution. Diminish the F corresponding to the shell or bullet given in the proportion of the radius to the cosine of į ths of the angle of elevation. Then, by means of the preceding table, operate with this reduced F in the same manner as is prescribed in the solution of the last proposition, and the result will be the potential range sought; whence the potential random, and the original velocity, are easily determined.

“ Exam. A mortar for sea-service, charged with thirty pounds of powder, has sometimes thrown its shell, of 12∣th inches diameter, and of 231 lb. weight, to the distance of two miles, or 5450 yards. This at an elevation of 45°.

“The F to this shell, if it were solid, is 3825 yards; but as the shell is only ∣ths of a solid globe, the true F is no more than 3060 yards. This, diminished in the ratio of the radius to the cosine of fths of the angle of elevation, becomes 2544. The quote of the potential range by this diminished F is 1∙384; which, sought in the first column of the preceding table, gives 2∙280 for the corresponding number in the second column; and this multiplied into the reduced F, produces 5800 yards for the potential range sought, which, as the angle of elevation was 45°, is also the potential random; and hence the original velocity of this shell appears to be that of about 748 feet in a second.

“ Cor. The converse of this proposition, that is, the determination of the actual range from the potential range given, is easily deduced from hence by means of the quote of the potential range divided by the reduced F; for this quote, searched out in the second column, will give a corresponding number in the first column, which, multiplied into the reduced, F, will be the actual range sought.

“Also, if the potential random of a projectile be given, or its actual range at a given angle of elevation; its actual range at any other angle of elevation, not greater than 45°, may hence be known. For the potential random will assign the potential range at any given angle; and thence, by the method of this corollary, the actual range may be found.

“ Exam. A fit musket-bullet fired from a piece of the standard dimensions, with ∣th of its weight in good powder, acquires a velocity of near 900 feet in a second; that is, it has a potential random of near 8400 yards. If now the actual range of this bullet at 15° was sought, we must proceed thus:

“From the given potential random it follows, that the potential range at 15° is 4200 yards; the diameter of the bullet is f ths of an inch; and thence, as it is of lead, its proper F is 337∙5 yards, which, reduced in the ratio of the radius to the cosine of ∣ths of 15°, becomes 331 yards. The quote of 4200 by this number is 12∙7 nearly; which being sought in the second column, gives 3∙2 nearly for the corresponding number in the first column; and this multiplied into 331 yards (the reduced F) makes 1059 yards for the actual range sought.

“ Exam. 2. The same bullet, fired with its whole weight in powder, acquires a velocity of about 2100 feet in a second, to which there corresponds a potential random of about 45,700 yards. But this number greatly exceeding 13,000 yards, it must be reduced by the method described in the third corollary of the first proposition, when it becomes 19,700 yards. If now the actual range of this bullet at 15° be required, we shall from hence find, that the potential range at 15° is 9850 yards; which, divided by the reduced F of the last example, gives for a quote 2975; and thence following the steps prescribed above, the actual range of this bullet comes out 1396 yards, exceeding the former range by no more than 337 yards; whereas the difference between the two potential ranges is above ten miles. Of such prodigious efficacy is the resistance of the air, which hath been hitherto treated as too insignificant a power to be attended to in laying down the theory of projectiles.

“ Sehol. I must here observe, that as the density of the atmosphere perpetually varies, increasing and diminishing often by ⅛th part, and sometimes more, in a few hours; for that reason, I have not been over rigorous in forming these rules, but have considered them as sufficiently exact when the errors of the approximation do not exceed the inequalities which would take place by a change of 3 ⅛th part in the density of the atmosphere. With this restriction, the rules of this proposition may be safely applied in all possible cases of practice. That is to say, they will exhibit the true motions ’ of all kinds of shells and cannon-shot, as far as 45° of elevation, and of all musket bullets fired with their largest customary charges, if not elevated more than 50°. Indeed, if experiments are made with extraordinary quar. ìes '^ i ' powder, producing potential randoms greatly surpassing the usual rate, then in large angles some farther modifications may be necessary. And though, as these cases are beyond the limits of all practice, it may be thought unnecessary to consider them; yet, to enable those who are so disposed to examine these uncommon cases, I shall here insert a proposition which will determine the actual motion of a projectile at 45°, how enormous soever its original velocity may be. But as this proposition will rather relate to speculative than practical cases, instead of supposing the actual range known, thence to assign the potential random, I shall now suppose the potential random given, and the actual range to be thence investigated.

ii Prop. III. Given the potential random of a given shell or bullet; to determine its actual range at 45°.

Solution. Divide the given potential random by the F corresponding to the shell or bullet given, and call the quotient q, and let l be the difference between the tabular logarithms of 25 and of q i the logarithm of 10 being supposed unity; then the actual range sought is 3∙4 F=p2ZF —
[7:11:67]

F, where the double sine of 2ZF is to be thus understood; that if q be less than 25, it must be — 2ZF; if it be greater, then it must be + 2∕F. In this solution q may be any number not less than 3, nor more than 2500.

“ Cor. Computing in the manner here laid down, we shall find the relation between the potential randoms, and the actual range at 45°, within the limits of this proposition, to be as expressed in the following table.
[table]

Whence it appears, that, when the potential random is increased from 3 F to 2500 F, the actual range is only increased from 1⅜ F to 7 F; so that an increase of 2497 F in the potential random produces no greater an increase in the actual range than 5∣ F, which is not its τ ⅜ 5 th part; and this will again be greatly diminished on account of the increased resistance which takes place in great velocities. So extraordinary are the effects of this resistance, which we have been hitherto taught to regard as inconsiderable.

“That the justness of the approximation laid down in the second and third propositions may be easier examined, I shall conclude these computations by inserting a table of the actual ranges, at 45°, of a projectile which is resisted in the duplicate proportion of its velocity. This table is computed by methods different from those hitherto described, and is sufficiently exact to serve as a standard with which the result of our other rules may be compared. And since whatever errors occur in the application of the preceding propositions, they will be most sensible at 45° of elevation, it follows, that hereby the utmost limits of those errors maybe assigned.
[table]
[table]

It now only remains to consider very shortly that part of practical gunnery which relates to the construction of guns, the proportional length, weight, calibre, and charge of the different kinds of ordnance, the methods of pointing and elevating them, and the purposes to which they are applied.

Formerly guns were made of great length, and upon that account were found extremely troublesome and unmanageable. The error in this respect was first discovered by accident; for some cannon which had been cast by mistake two feet and a half shorter than the common standard, were found to be equally efficacious in service with the common ones, and much more manageable. This soon produced very considerable alterations in the form of artillery throughout Europe; but in no country have greater improvements in this respect been made than in our own^[2. In the article on Cannon-Making, we have explained the process of casting guns, whether of iron or other metal, and the composition of the metal commonly called brass. In the same article there are tables exhibiting the dimensions of, and various particulars relating to, the guns employed by Britain and by France. ]

Guns are made either of iron or of brass. Those formed of iron are better adapted for continued and heavy firing than those made of brass; they also last longer, but this depends chiefly upon the purity of the metal. Brass guns, when frequently fired, droop at the muzzle, and become quite useless; but being lighter than iro∏∙guns, they are better adapted for service.

The guns mounted on the works of San Sebastian were of brass, and towards the end of the siege they became quite useless; whilst the " guns of the besiegers, though much oftener fired, and considerably enlarged at the vents or touch-holes, still continued serviceable. An iron gun, if overcharged or over' ...tsd, bursts and flies in pieces, which a brass one does not,

The best length for field-guns has been found to be about seventeen calibres, that is, seventeen times the diameter of the bore. Long guns are more easily pointed than short ones, and will to a certain extent throw shot farther with equal charges.^[3. It is a maxim of Mr Robins, that neither the distance to which a bullet flies, nor its force at the end of its flight, are much increased by very great augmentations of the velocity with which it is impelled; and, therefore, that in distant cannonade, the advantages arising from long guns and great charges are but of little moment. Sir Howard Douglas, however, contends that this maxim relates only to random ranges, and overlooks the advantages arising from the superior accuracy of long guns at moderate distances; that comparing the powers of the long 24-pounder, length 9⅜ feet, with those of the short 24, length 6 feet 6 inches, though the extreme ranges be nearly the same, yet for practice at 300 yards, the long gun might be laid point-blank, whilst the shorter one would require nearly half a degree of elevation; and that, at 1000 yards the former would only require an elevation.of two degrees, which, with the short gun, would only give a range of about 800 yards. ] There are also a great many guns of the same weight, but of different lengths. This is occasioned by the several classes of vessels, &c. for which they are intended requiring an equal weight of shot, but different lengths of guns.

Very long guns are found to be disadvantageous, as it [7:11:68]is useless to project shot at a greater velocity than fourteen, or at the most sixteen hundred feet per second, since air cannot rush in with greater rapidity than at the rate of 1340 feet per second to fill up the vacuum caused by the ball. In the case of a higher velocity, the air in front would be considerably compressed, and the increased resistance thus occasioned would soon reduce that velocity. It is therefore useless to augment the charge beyond a certain limit.^[4. The increased resistance of the air, occasioned by an increase of velocity in the projectile, has been fully treated of in the preceding part of this article, under the Theory of Gunnery ]

The charge of powder commonly used in practice is, for heavy guns ∣d, for light guns ∣th, and for carronades τ 1 2 th the weight of the shot. The thickness of metal for a 42- pounder is 1 calibre at the breach, for a 32-pounder 1⅛th calibre, for a 24-pounder 1-, ⅜ths calibre; at the muzzle the thickness of metal is one half that of the base ring.

The proportion of the weight of the metal in a gun is to that of the shot, in heavy guns 2 cwt. to 1 lb., in medium guns 1⅛ cwt. to 1 lb., and in light guns 1 cwt. to 1 lb.

I.— The Length, Weight, Calibre, and Charge of the various pieces of Ordnance, and the Purposes to which they are applied.
[table]
[7:11:69]

L The Length, Weight, Calibre, and Charge of th e various pieces of Ordnance, &c. — continued.
[table]
[7:11:70]

General Construction of Eon Guns. ^[5. This is descriptive of the 24 and , 9-pounders, Plate CCLXX1V. figs. 1, 2. ] Fig. 1.

AB, length of the gun, 24-pounder.

AC, first reinforce, tths of AB; 9 feet; weight 47 cwt.

Ĳ, distance from hinder part of the base ring to centre of trunnions D ŷths of AB, AL ⅜d of AC, rings and astragal included.

CE, second reinforce, ⅛th of AB + 1 calibre, fillets ⅜th AC. Lr, chace, f ths EB; fillets ith of their rings, ogee at B ⅛th of its ring.

FB, muzzle, ⅜th EB; NB = ⅜th of FB; muzzle mould·, mgs = ⅜d NB. Trunnions s lv = 1 calibre, (diameter = tv.

Cascabel,

Fillet a 1∣ calibre, (d = 6 ) GH=⅜ calibre.

Neck b 1 calibre, le = 3 < sixteenths. IK = ⅜ GH.

Diameter c, fillet α, (∕= 12 ) lm — GH.

GH the thickness of metal.
[table]

Pointing and Elevating a Gun.

By pointing a gun is understood the placing it in such a position that the axis of the piece shall be exactly in a line with the object aimed at; and by elevating a gun is understood the placing it at such an angle above the horizontal line as will counteract the force of gravity, and may strike the object aimed at. When a gun is both pointed and elevated, it is said to be laid. The line-of-metal is a visual line extending from the base-ring to the swell of the muzzle: its position is ascertained by placing the trunnions perfectly horizontal, and then finding the highest point both on the base-ring and the swell of the muzzle, when the line joining those two points will be the line-of-metal∙ But in consequence of the conical shape of guns, this line has an inclination to the axis of about one degree, which is called the dispart. In pointing a gun, the line-of-metal is first laid in a line with the object; then, if the trunnions are horizontal, the axis of the piece and object will be in the same vertical plane; but if the trunnions are not perfectly so, on account of the dispart, the continuation of the line-of-metal will cross that of the axis of the piece, and the shot will be thrown considerably to that side of the object on which the lowest trunnion is.

In order to counteract this, a dispart sight is placed on the muzzle, which makes it of the same diameter as the breech; and then, however much one trunnion may be lower than the other, the shot cannot be thrown more than the thickness of metal to the right or left. A gun is said to be point blank when the axis of the piece is in a line with the object fired at, without having any elevation, or when the axis is parallel to the horizon. The elevation required to strike any object is found by ascertaining its distance. Sets of tables have been constructed from actual practice (see Tables II. III. IV. and V.), by which the different sorts of shot and shells may be projected with the greatest accuracy.

A scale made of brass, and called a tangent scale, being marked with the different lengths of the tangents for the several degrees, slides up and down in the breech. By means of this the elevation may be given without any reference to the difference between the level of the gun and the object fired at, and it may be elevated and pointed at the same time. In guns which have disparts, the tangent scale only comes into use at a greater angle than that of the dispart of the gun. Degrees are therefore marked upon the base-ring, beginning at the quarter sight, by means of which the gun may be elevated at any less angle than that of the dispart.

The ’ 21 of an inch is the tangent of one degree to every foot of the gun’s length, from the base-ring to the swell of the muzzle; and therefore, if the distance in feet between those two points be multiplied by ∙2l, the product will be the tangent of 1°, which, when the dispart is subtracted from it, will give the length of the tangent scale above the base-ring at one degree of elevation for that particular gun. If the scale be applied to the quarter sight of the gun, of course the dispart need not be subtracted.

Elevating guns at sea has always been attended with difficulty and uncertainty. To effect this, the following method has been proposed: Let the trunnion of a gun be divided by lines passing through its centre, parallel and perpendicular to the axis of the piece, and the lower limb be divided intodegrees, &c.; a plumb suspended from the centre of the trunnion will cut the degree of elevation or depression the gun is pointed at, which of course is always varying, from the motion of the ship. If the axis of the piece therefore be parallel with the deck, the degree of the inclination of the deck and gun will at the same time be ascertained, and the gun will be fired at the moment when the plumb-line cuts the proper degree marked upon the lower ring of the trunnion. Great accuracy may thus be attained at sea.^[6. “It is not usual (at sea) to discharge cannon-shot at great elevations, on account of the uncertainty of such practice; nor indeed is it practicable to do so. This is only practised with mortars.” (See Major-General Sir Howard Douglas’s Treatise on Naval Gunnery, art. 115.) ]

A scale has of late years been used for iron guns, marked with the number of yards instead of degrees; and this has [7:11:71]been found very useful to men who might not perhaps understand the tangent scale.

When it is required to fire continually at the same object, for instance a breach, the best way is, after discharging a few rounds, to observe some object which the gun points to when at the proper elevation, and always point at that object. This is called pointing at a false object.

The fire of artillery may be divided into three classes; the direct, the ricohet, and the vertical. The direct fire is that used in the field or at sieges, where the gun is discharged directly at the object with a full charge. The ricohet fire is not confined to any particular charge or elevation; each must vary according to the distance and level of the object to be fired at, and particularly the spot on which it is intended it shall make the first bound. Firing en ricohet was first invented by Marshal Vauban, at the siege of Ath; and it is principally used in sieges for enfilading the face of a work, by sweeping or bounding along it. A work is said to be enfiladed when the enemy place a gun on the prolongation of a line of works, so as to fire along it. Vertical fire is that which is thrown from mortars, howitzers, or guns, at elevated angles. This was much used at the siege of the Citadel of Antwerp in 1832.

By the assistance of good tables of practice, and of amplitudes, sines, tangents, and secants, all cases in gunnery in a non-resisting medium are easily solved; and perhaps the solution may be sufficiently correct for practice, if the initial velocity of the projectile be not so great as to make the resistance of the air considerable. (Vide Tables II. III. IV. and V.)

The following rules maybe observed: 1. The greatest range is at 45 degrees nearly; 2. The ranges with different elevations at the same charge, are as the double sines of elevation; 3. Any angle and its complement give the same range nearly; 4. The times of flight are as the sines of the angles of elevation; 5. The altitude of the curve, at any elevation, is found by this proportion :—as radius: tangent of angle of elevation:: altitude: range. 6. The time of flight at 45° is equal to the square root of th e range in feet, divided by 4; or, more nearly, = √ quotient of the range in feet, divided by 1∙61, or the space that would be passed through in the first second by the force of gravity alone.

Having the first graze of shot, with a given charge and elevation, to determine the charge for any other first graze and elevation. Multiply the known charge and elevation into the proposed first graze; also the proposed elevation into the known first graze; and divide the first product by the last, for the charge required.

Mortars are generally placed in fortified places, at some distance from the parapet; and in the attack in batteries with high parapets, so that from the mortar itself, the object or place to be fired at can seldom be seen, two ramrods are stuck in the superior slope of the battery, in line with the object; the centre of metal is then taken on the mortar, and chalked; the officer next lays the mortar by placing the centre of metal thus found, in the direction of the ramrods, by means of a plumb-line, and he ascertains that the elevation is correct by the gunner’s quadrant. The distance being ascertained, the following rules are to be observed.

General Rules for Mortar Practice.

In a 13-inch mortar, 3 lbs. of powder give a range of 1100 yards; whilst every - 1 - lb. of increase adds 100 yards; every į lb. of decrease diminishes 150 yards. In a I0-inch, half the quantity calculated for a 13-inch is used; in an’8- inch, one-third the quantity for the 10-inch; and in a 5⅜- inch, a charge of 1 oz. 8 dr. gives 150 yards. The following table shows the weights of the different kinds of mortars, empty and filled :—
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A shell is a hollow globe of iron filled with powder, which will burst at any intended distance by means of a fuze which has been cut to a certain length, and communicates to the powder within the shell. The fuze is a wooden tube made to fit the hole in the shell, and filled with a composition of sulphur, saltpetre, and mealed powder; it burns about ∙2 of an inch in a second. The time of flight of a shell being known, the length of fuze, and consequently the bursting of the shell, are easily ascertained. Shells are generally made to burst just as they would strike the ground; sometimes the fuzes are cut rather long, so that a shell may fall into a house, or a breach, and, by its exploding under ground, act as a small mine.

Shrapnell shells, or, as they were called by the inventor, spherical case shot (a name which they still retain), are filled with a quantity of musket-balls, which, when the shell explodes, are projected about 150 yards further. These shells are fired from guns, mortars, and howitzers, and have been found most efficient, especially against skirmishers or working parties. A six-pounder spherical case contains twenty-seven musket-balls, which, when the shell explodes, do as much injury as the same number of muskets (besides the splinters of the exploded shelD, at a distance far beyond the range of musketry.^[7. This invention, which has hitherto been imperfectly comprehended by some, even in our own service, had absolutely baffled the inquiries of the French; but it is understood that Captain Glünder, of the Hanoverian service, has recently published a minute account of it in a Continental military journal. In future wars, therefore, it is probable that Shrapnell shells will be more prized by us than they have hitherto been, because they will be adopted by our enemies, and we shall then practically feel their effect. (See United Service Journal for 1834, part ii. p. 398.) ]

Case and grape-shot are also fired from guns. They take their names, the one from having a number of small balls confined in a tin case, and the other from having the balls tied or quilted together in a manner resembling a bunch of grapes. They are not calculated for long ranges, but are very efficacious, at three or four hundred yards, against advancing troops, or in the flanks of a fortification for firing along the ditch.

The common shot is a solid sphere of iron, designated according to its weight, and is something less than the diameter of the bore, to allow for windage, which in English guns is ∕θth of the <. ũ , so that, in this way, between ⅜d and ith of the torvi, is lost.

Great changes have recen√.y bee. made in howitzers by General Miller of the royal artillery. He has superseded the useless light 51-inch howitzer by a piece of ordnance which, from its weight of metal and length of bore, authorizes such charges as, with the diminished windage, will produce due velocity and effect. In Plate CCLXXIV. figs. 3 and 4 represent a 12-pounder brass howitzer, and a 10-inch iron howitzer.

During the siege of Cadiz the French fired shells into the town from two large mortars at a distance of4000 yards. Their calibre was much the same as that of our 13-inch mortars, but in their internal construction they differed materially; the largest took forty pounds of powder for its charge, and was nearly eight feet in length. A great deal of lead was run into the shells; they seldom burst, and [7:11:72]when they did so their explosion was inconsiderable. One of these mortars is to be seen on the parade of the Horse Guards, St James’s Park.

The carronade, invented, or rather improved, by Mr Gascoigne, was in June 1779 instituted as a standard navy-gun, and ten of them were appointed to be added to every ship of war. The carronade is made so short that it is worked with its carriage in the ship’s port. It is correctly bored; and the shot being perfectly round, fills the calibre with such exactness, that the least possible impulse of the powder escapes upon explosion, between the cylinder and the shot, which last is also thereby more truly directed in its flight. The bottom of the cylinder is a hemisphere, to which the end of the cartridge is not liable to stick, and in which the smallest charge of powder envelopes the shot, exhausting upon it nearly the whole of its impelling force. There are sights cast upon the vent and muzzle, to point the gun quickly to an object at 250 and 500 yards distance; and there is a ring cast upon the cascabel, through which the breechin-rope is reeved, the only rope used about these guns.

This gun has many advantages over the others of light construction. It is so extremely light, that the smallest ships can carry almost any weight of shot (the 12-pounder weighing under five hundredweight, and the other calibres in proportion), and that without being attended with the inconveniences imputed generally to light guns, since it cannot injure its carriage, or jump out of its station in the port upon recoil; and it never becomes heated.

Though the carronade cannot, strictly speaking, throw its shot to an equal distance with a longer gun, yet, from the adaptation of the shot to its cylinder, the powers of this gun will greatly surpass the expectations of such as are not intimately acquainted with the effects of the elastic force of fired powder; since, with a charge one twelfth part of the weight of its ball, at very small elevations, it will project its shot to triple the distance at which ships usually engage, with sufficient velocity for the greatest execution, and with all the accuracy in its direction that can be attained with guns of greater lengths.^[8. At the same time, viewing the matter purely as an artillery question, there is little doubt that, in the armament of some ships a preference should be given to long guns. Sir Howard Douglas recommends {Nαvαl Gunnery, art. 89) that a frigate which cannot carry eight-feet 24-pounders, should be fitted with long 18-pounders rather than with six or six and a half feet 24-pounders, or indeed with any kind of carronade exclusively. “The chief objection that maybe made to this,” says he, “is, that short 24-pounders are more easily worked. This I admit; but surely, with a good strong crew, the difference would be very inconsiderable; and, at all events, not worth the sacrifice of vast superiority of power and practice at long ranges. Perhaps, also,” he adds, “expedients may be devised to lessen this, the only disadvantage that can be objected to long guns, for vessels that can receive them.” Sir Howard admits that at close quarters carronades are very formidable; but he contends that, at long ranges, they are no match for long guns, and that any vessel fitted exclusively with this description of ordnance, might be destroyed or captured by a vessel of very inferior rate, mounting long guns, if her commander knew how to avail himself of the great superiority of his weapons. “The very mortifying situation,” says he, “in which the gallant Sir James Yeo found himself, in September 1813, on Lake Ontario, shows the danger of the carronade system of armament. Sir James states, in his letter of the 12th September, ς the enemy’s fleet of eleven sail, having a partial wind, succeeded in getting within range of their long 24 and 32-pounders; and having obtained the wind of us, I found it impossible to bring them to close action. We remained in this mortifying situation five hours, having only six guns in the fleet that would reach the enemy. Not a carronade was fired. At sunset a breeze sprung up from the westward, when I manoeuvred to oblige the enemy to meet us on equal terras. This, however, he carefully avoided.’” Captain Barclay, in a letter also dated the 12th September 1813, makes a similar statement. “The other brig of the enemy,” says he, “apparently destined to engage the Queen Charlotte, supported in like manner by two schooners, kept so far to windward as to render the Queen Charlotte’s 24-pounder carronades useless, whilst she and the Lady Prevost were exposed to a heavy and destructive fire from the Caledonian, and four other schooners, armed with long and heavy guns.” ]

There have been two seeming disadvantages imputed to this gun, which it does not merit, viz. the nicety of fitting the shot to the bore of the gun, and its incapacity to admit more than two shot at one charge. But as seamen have few opportunities of confirming themselves in just opinions by experiments made on shore, and cannot, in that case, be fully conversant with the subject, the following hints may be useful towards removing these objections.

It is an axiom in projectiles, that a shot cannot be impelled from a gun to any distance in a direction truly parallel to the axis of the cylinder of the piece, or what is commonly called point blank, arising from several well-known causes. For, however just may be the cylinder, and however perfect and smooth may be the sphere of its corresponding shot, and admitting that the impulse of the powder acts through the centre of gravity of the shot, and that the shot consequently leaves the piece in a direction parallel to the axis of its cylinder; yet the shot is no sooner discharged, than it becomes more or less inflected by its gravity, and deflected, according to its velocity, by the resistance of the air and wind. But these irregularities are of little importance in close sea-fights; and being the effect of physical causes, they are common to all. The deviation of a shot from its true direction, however, is further augmented by the windage between the cylinder and the ball; but the greatest uncertainty in the flight of a shot, making allowance for the action of its gravity, and the resistance of the air, is occasioned by the defects of the shot itself.

If the direction of the flight of a shot to its object, then, is affected by so many seeming trivial causes, the result must be much more uncertain when two or more shots are discharged together from one gun; for the shot next the powder being impelled with more celerity than that immediately before it, strikes against it after the discharge, sometimes shivers itself to pieces, and never fails to change obliquely the direction of both; and this happens with round and double-headed shot, and with all double charges, which, from their various figures, cannot reach an object at the same elevations with the round shot; especially when these other shots are of greater weight than the round, which is often the case. However frightful a broadside with double charges may appear at sea, more confusion is created by them, and more time lost, within board, by the strain and excessive recoil, than real damage done without board by the additional charge; for upon a trial on shore, where the effect can be traced, it will be found, that at one hundred yards distance, more shot will take place within a small compass by single than by double charges; and the charges will be oftener repeated in a given time, without heating the gun. .

The other pieces of artillery commonly made use of are mortars, royals, and howitzers. The mortars are a kind of short cannon of a large bore, with chambers for the powder, and they are made either of brass or of iron. Their use is to throw hollow shells filled with powder, which falling on any building, or. into the works of a fortification, burst, and with their fragments destroy every thing near them. Carcases, which are a sort of shells with five holes, filled with pitch and other materials, in order to set buildings on fire, are also discharged from mortars.

Mortars are chiefly distinguished by the dimensions of their bore; for example, a 13-inch mortar is one the diameter[7:11:73] of whose bore is 13 inches, and so of the rest. The land mortars are those used in sieges and in battles. They are mounted on beds, and both mortar and bed are transported on block carriages. English mortars are generally fixed to an angle of 45 degrees. This custom, however, does not appear to have any foundation in reason. On the contrary, when shells are thrown upon magazines, or any other buildings, the mortars should be elevated as high as possible, that the shells may acquire a greater force in their fall, and consequently do more execution. The chamber in mortars is the place where the powder is lodged. They are of different forms, and made variously by different nations; but the cylindrical seems to be preferable to any other form.

Royals are a kind of small mortars, which throw a shell whose diameter is 5∙5 inches. They are mounted on beds in the same way as other mortars.

The howitzer is a kind of mortar mounted on a fieldcarriage like a gun; and it differs from the common mortar in having the trunnions in the middle, whereas those of the mortar are at the end. The construction of howitzers is as various and uncertain as that of mortars, excepting that the chambers are all cylindrical. They are distinguished by the diameter of their bore; for instance, a 10-inch howitzer is that which has a bore of ten inches diameter, and so of others. They were much more lately invented than mortars, and indeed are plainly derived from them.

The canons à bombes, or a la Paixhans, are howitzers of greater length of bore than ordinary, having the shells appropriated to them cast concentric, instead of a eulot, or reinforcement of metal opposite the fuse-hole. The bomb-cannon are, in fact, only an application of a principle incontestibly proved by Mr Robins and Dr Hutton, and well exemplified in the case of carronades; namely, that the velocities of shot are increased by diminishing the windage. Colonel Paixhans has acknowledged (Force et Faiblesse de la France, pp. 396, 415) that the most effective proof of the value of his improvement against fortifications was the practice carried on by Sir Alexander Dickson in 1824, with carronades and howitzers, at a range of four and five hundred yards, against Carnot’s wall, covered by his counterguard, and strengthened for about twenty feet with an additional buttress at each end of four feet square. But this practice has nothing to boast of, or to record, in a treatise like the present. “Certainly it would never have been spoken of as extraordinary in the British service, that the charge and elevation should be so apportioned to ordnance as to pitch their projectiles over a counterguard, securing their effect against, and, as it proved, in three hours and a half firing, causing a practicable breach, with eight 68-pounder carronades, and six 10-inch howitzers, in a wall of the same height with the counterguard, and ninety feet from it; but it was asserted at the time, by foreign officers of distinction, that the wall could not be breached, and that it had been attempted in vain by the artillery of the principal powers of Europe.”^[9. United Service Journal for 1834, part ii. p. 398. ]

“The ∙ propositions of Colonel Paixhans,” says another writer in the same journal, “have been some years before the public; and his ideas as to the construction of guns by which to project shells horizontally have been noticed and acted upon by most if not all the maritime powers of Europe. In England we have guns constructed on his plan by General Miller, but after all it appears to be little more than applying to ordnance with trunnions the well-known principle applied to carronades, that of diminishing the windage to increase the initial velocity, thereby affording an op-portunity of reducing the charge. Μ. Paixhans’ new guns, as well as our own, may be best described as iron howitzers, the bore being longer than usual, and the windage decreased; the construction of the gun is, however, of little consequence compared to its application. Now the practice made by the French marine at Brest against a ship of the line at different periods, proves that an eightypounder, the hollow shot of which is 8 , 4 in diameter, may, by a single discharge, cause the destruction of a ship of the first class, and that such result may be produced, either from the bursting of the shell, and the instantaneous injury thereby caused, or from fire; and it further shows, that in every case in which a shell struck the vessel, the injury sustained was very important. The committee of naval and artillery officers, in the report of Μ. Paixhans’ eighty-pounder, observes, ‘ II est evident que ľéffet produit a été terrible, et tel qu’on pense qu’une ou deux bombes de cette espèce, éclatant dans un batterie, y causeraient un désordre capable de faire abandonner, du moins de comprometre, la défense du batiment atteint.’

“In our own service, a twelve-inch gun of ten diameters, windage ∙15 inches, charge eleven pounds, elevation 1°, range 400 yards, was fired six times against the section of a ship’s side, the fac-similé of a ship of war. Four shells struck the section, and produced an effect such as to convince every body present that no vessel could have floated with such injuries; the whole of the interior was covered with splinters, one weighing fifty pounds was picked up at a considerable distance; the knees of the vessel were broken to pieces, one shot alone broke several; the knees protruding beyond the ship’s sides, and leaving a tremendous opening. As to the range of these projectiles, it appears, that notwithstanding the reduced charges, it exceeds that ordinarily obtained with the common guns of the largest calibre, and with a charge equal to one-third the weight of the shot.”

The largest mortar ever employed against the defences of a place was the iron 24-inch mortar used at the siege of the Citadel of Antwerp in 1832. The dimensions of this “monster,” stated in French weights and measures, are as follow:
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“The monster mortar hitherto appears remarkable only for the extremely bad shells which were cast for it, and the time taken in loading it. Our thirteen-inch shells stand the explosion of twenty or thirty pounds of powder, whilst the monster was served with only twelve and a half. It is remarkable that Colonel Paixhans’ guns were not used at Antwerp, particularly as the attack appears to have been carried on for display, and to have been spun out, particularly in the attack of the Lunette, as a siège du polygone"

As illustrative of the practice at the siege of the Citadel of Antwerp, and in the Peninsular war, and also as serving to show generally the effects of artillery, we subjoin some remarks extracted from the journal above referred to.

“The lieutenant-general (Neigre)commanding this arm, pertinaciously adhered to the principle that it is at pointblank that a cannon fire is most efficacious, and that nothing [7:11:74]is gained by that of mortars at a close range; that is, that the impetus of the former does not increase by reducing the distance between the point-blank and a given object; and were it not for glacis and intervening objects, revetements could as easily be breached at full point-blank as at fifty yards, whilst, in regard to the latter, their execution is rendered less certain by the necessity of diminishing the charge and fuse.

“By full point-blank appears to be intended that which the French term le but en blanc, distinguished from the but en blanc primitif ou naturel, which the English term the line of metal elevation, namely, the elevation of the axis of the bore resulting from the dispart of the gun; a plane passing over and touching the superior part of the breech and muzzle being coincident with the object at which the gun is laid. If the definition now to be assumed be admitted to convey the phrase point-blank, as applied to the opinions of General Neigre, the distance in question will be, when referring to a 24-pounder English, about 700 yards, such being its line of metal range.

“Now, at this distance, it is very certain that a breach may be effected against an ordinary revetement, if not covered by any intervening object, without any difficulty. Sir John May, of the royal artillery, in his memoir of the sieges in the Peninsula, remarks, that at Ciudad Rodrigo, breaches of 130 feet in extent were rendered practicable at ranges of from 500 to 700 yards, in thirty-two and a half hours, the greatest number of guns employed being thirty 24-pounders and two 18-pounders. Sir John Jones, royal engineer, says, ‘ at Badajoz the extent of the front of the three breaches open was about 500 feet, the greater part of which was as good as can be formed.’ This was effected at distances of from 600 to 700 yards with fourteen 18- pounders and twelve 24-pounders, in 104 hours; firing, on the average, nine rounds in the hour. At St Sebastian, twenty 24-pounders caused a practical breach 100 feet in extent in thirty-three hours; and these guns rendered a smaller breach at the same siege practicable in fifteen hours and a half; on an average, twenty rounds a gun were fired in an hour.

“Thus, at these three sieges, assuming the data they afford, and supposing fifty guns employed, a breach of 100 feet might have been effected at a range of from 600 to 700 yards; at Ciudad Rodrigo in fourteen hours twelve minutes, at Badajoz in ten hours, at St Sebastian in twelve hours fifty-six minutes; and the time might have been reduced to ten hours, as it has been proved that English battering guns will admit, without injury, twenty-four rounds in an hour.”

With reference to the opinion that “nothing is gained by the fire of mortars at a close range,” and that the execution is rendered less certain by the necessity of reducing the charge and fuse, we are of opinion that, if the primary object of mortar batteries be to force through bomb proofs, and to spring the arches of magazines, the distant mortar may be more efficient than one more near. The greatest penetrating effect of a vertical projectile must obviously be obtained by projecting it to such a height as would be requisite, in its descent, by gravity, to attain its terminal, that is, its greatest velocity; but as with the heaviest projectiles the altitude necessary to generate the terminal velocity is such as can only be produced at protracted ranges, or at elevations above 45°, the principle must be modified, so that in endeavouring to procure the greatest momentum for the shell, too much may not be surrendered in point of accuracy of fire. The altitude to produce this terminal velocity with an eight-inch shell, is 2678 feet, with a ten-inch, 3335, with a thirteen-inch, 4340, and with the Antwerp “monster,” 6030 feet, supposing its diameter to be twenty-four inches, and the corresponding solid to weigh 2447 lbs. English; but these altitudes require, at 45° elevation, raħges to which shells cannot be thrown with great accuracy. Very satisfactory practice may, however, be made with thirteen and ten-inch mortars at 800 and 1000 yards, the ranges of the principal mortar batteries at Antwerp. Therefore we may say, that decreasing these distances would lessen the efficiency against permanent works, in a degree not compensated by the superior accuracy of the fire.

If the intention of the mortar battery be to annoy troops or working parties, much less velocity is required; and as the accuracy of the practice would increase by lessening the range, it would be desirable to do so.

It is highly probable that the great effect of the vertical fire in destroying the shelter of the garrison in the bombardment at Antwerp, is attributable to the distance of the mortar batteries.

The French use brass guns for their battering trains, perhaps “because they have no iron founderies to compare to that of Carron, or indeed to many English establishments.” Still it is remarkable that, with the experience they must have had of the comparative inefficiency of guns made of this metal, the French should still adhere to them. If the rate of firing at Antwerp in 1832 be compared with that at any of the Peninsular sieges, as stated by Sir John Jones, the great superiority of iron guns will be at once perceived. The French iron is no doubt very inferior to ours; but, in a matter of so much importance, one would imagine that the interests of a monopoly would not outweigh those of the service.

In the British service, the practice with the Paixhans guns has not been carried to an equal extent as in the French; but with the twelve-inch gun of 90 cwt. 3 qrs. 4 lbs., weight of hollow shot about 125 lbs., windage ∙15 inches, charge 12 lbs. only, a range of 1300 yards has been the result. The number of men employed with the guns of Colonel Paixhans was fifteen, being that required ior a thirty-six pounder. The English gun mounted on board ship was worked by six men with ease; and it recoiled from four and a half to five feet.”

II.— Charges, Range, &c. with the under-mentioned Iron Ordnance.
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HI.— Calculation io fire Spherical Case Shot with Service Charges.
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IV.— Ranges with the under-mentioned Brass Ordnance.
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V.— Elevation and Range with 68-Pounder, or Eight-Inch Gun.
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The above tables are constructed from actual practice, with good powder, at Woolwich. According to the present system, much more care is taken to preserve gunpowder from deterioration than was formerly employed.

The description of figs. 10 and 11, Plate CCLXXIV., which, from want of room, could not be engraved on the plate itself, is here subjoined: Fig. 10, AB bracket, BC trail, o trail-plate eye, E eye and bolt keys, D and c trun-ion-plates, T cap square, H lock-plate, L traversing stay, M trail bolts, P traversing loop, Q elevating screw, b nare or stock, a nare hoop, e garnish nail, f fillies, d spokes, g nuts, i match or shot box, x whefel chain, w traversing handspike. Fig. 11, A ammunition box, B guard iron, C axle-tree stay, D lip 0 f the bed, H splinter socket, k shaft, L limber hook.
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