Gunnery. Is the art, in a restricted sense, of determining the motions or ranges of projectiles discharged from cannon, mortars, howitzers, and other kinds of artillery; and, in a more general sense, of determining not only the motions of such bodies, but also the arrangements by which they are rendered effective instruments of war.

The use of fire-arms had been long known before any theory concerning them was attempted. Nor is this remarkable, as the theory of the motion of projectiles depends on a knowledge of certain laws of nature which were not discovered till many years afterwards. It was different as regards the improvement of those defensive arrangements which had been found sufficient in the earlier times to afford protection against the more ancient engines of war, or, on the other hand, to cover those who were called upon to use them. The architects of the middle ages (see FORTIFICATION) quickly saw the necessity of modifying the forms and proportions of the ancient walls which then surrounded fortified towns, and rendering them more suitable both for the use and for the resistance of the newly-discovered artillery adapted for the use of gunpowder. The Cavaliere Saluzzo of Turin has indeed shown, from the archives of his native town, that Giorgio Martini, Architetto Senese, undertook the task of remodelling the ancient walls of castles and of towns even before the commencement of the sixteenth century, as Martini died in 1506. In his plans the profiles of the old castle are not much altered; but the trace is greatly modified, so as to produce a more perfect reciprocal or flanking defence, either by a combination approximating to something like the bastioned trace, or by capannati—so called from their resemblance to a woodman's hut—in the ditch, a work in principle the same as the caponnières subsequently invented by Dürer. Nothing, perhaps, is more calculated to exhibit the military spirit which pervaded the Italian architects, sculptors, and others, as well as those of the military profession, than the list of Italian authors on the art of war, in the corps papers of the Royal Engineers, which was drawn up by an accomplished lady, Mrs Lennox Conyngham, from an inspection of the libraries of Rome alone. In that list appears the name of Nicholas Tartaglia, who was the first author who wrote professedly on the flight of cannon-shot. 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 incurved 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 also intermeddled in the questions which hence arose; and many disputes on motion occurred, espe-

cially 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 $7\backslash frac\{1\}\{2\}^\backslash circ$ 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 $5^\backslash circ$ elevation will be $2\backslash frac\{1\}\{2\}$, at $10^\backslash circ$ it will be $3\backslash frac\{1\}\{2\}$, at $15^\backslash circ$ it will be $4\backslash frac\{1\}\{2\}$, at $20^\backslash circ$ it will be $4\backslash frac\{1\}\{2\}$, and the greatest random will be $5\backslash frac\{1\}\{2\}$. This last, he tells us, happens in a calm day, when the piece is elevated to $42^\backslash circ$; but according to the strength of the wind, and as it favours or opposes the flight of the shot, it may be from $45^\backslash circ$ to $36^\backslash circ$. 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.

Before proceeding further with the history of gunnery, or passing the epoch at which the writings of Galileo had prepared the way for a sounder knowledge of its principles

Gunnery. it is only an act of justice to Nicholas Tartaglia to record what he actually either knew or conjectured on the subject. The second of his works, published, as before stated, in 1546, was translated into English by Mr Cyprian Lucar, and published in London in 1588. It consists of three books of Colloquies concerning the Arte of Shooting; and the motive for writing it is thus stated by Tartaglia in his dedication of the book to King Henry VIII., in the language of Lucar's translation:—"It was never my profession, not at any time have I delighted to shoot in any har-chibuse, hande-gunne, or in any other small or great piece of artillerie, nor doe intende to shoote hereafter in any of them; but one only question which a skillfull gunner in 1531 did aske of me in Verona provoked mee at that tyme to thinke thereupon, and by that occasion to finde out the order and proportion of shootes or markes neare hand, and also at markes far off, according to the variable elevation of the piece which doth shoote, whereof I should never have had any care, if that gunner had not with his saide question stirred me up to deale in the same." The idea having been thus raised in his mind, Tartaglia

was stimulated by the threatened war with the Turks to publish, in 1537, a short Treatise of Shooting in Gunnes, to the ende (as he observes) that my devises in the same might bee considered of. This book, he says, did no good; and as he continued to be asked many questions by men of station and learning, as well as by gunners, on the subject, he determined to answer all such queries in his second book, which is therefore arranged in the form of dialogues between Tartaglia and the Duke of Urbino, the Prior of Barletta, the Lord of Achaia, bombardiers, gunners, and gun-founders.

1. In the third Colloquie of the 1st Book he lays down as a proposition, that "a pellet doth never range in a right line except it be shot out of a piece right up towards heaven, or right downe towards the centre of the world."

In proving this proposition, Tartaglia assumes that the effective weight of the pellet, or ball, is diminished in proportion as the velocity is increased, and vice versa; and hence that the ball is less drawn to the earth at the first part of its flight than it is at the last. The explanation of the fact is therefore founded upon erroneous principles, but the reasoning from it is good; for Tartaglia says:—

"If now it be supposed that in any portion AB of the trajectory of the ball, the ball moves in a right line, divide AB in two equal parts at E; now, as the velocity is greater in AE than it is in EB, the ball will be less urged to the ground in the first than it is in the second half, and hence that the line EB cannot be as nearly straight as AE; or, subdividing again AE into two parts at F, FE will be more removed from a straight line than AF; and so on—proving that no part of the trajectory could be absolutely straight." Considering the imperfect knowledge of the time,

this demonstration was perhaps as much as could be expected, as it distinctly recognises the principle, that weight or gravity continued to draw the ball to the earth from the first to the last moment of its motion under the impulse of the propelling force, and hence that it could not at any moment move in a straight line.

2. Point-blank.—Tartaglia, in removing the scruples of his imaginary auditors, explains in the most satisfactory manner the different acceptations of this term, as now applied in the British and French service.

In this cut it is assumed that CD is, by a proper arrangement of sights at the breech and muzzle, made parallel to the axis of the gun; and hence, that the line of aim, CDE, is parallel to FG, or to the axis of the gun produced; in which case it is manifest that the ball could not arrive at G, but would come to the ground at I, which point, pro-

vided GI be equal to the height of the axis of the gun above the ground, marks the point-blank range of the British artillerist, or the lateral space passed over by the ball in the time it takes to fall to the ground. For convenience sake, the axis is supposed to be horizontal, and the range is also taken on a horizontal plane.

If the sights are so arranged that one shall be higher from the axis of the piece than the other, the line of sight or aim will no longer be parallel to the axis, but, when prolonged, will make an angle with, or intersect it. If the muzzle sight be the higher, always estimating from the axis, the intersection will take place behind the breech, and the line of the axis will be depressed below that of sight; if, on the

contrary, the breech sight be the higher, the intersection will take place in front of the muzzle, and the line of the axis will be elevated above that of sight. The latter is the case when the line of natural aim, or sight, is used, or that passing through the highest point of the breech-ring and the low sight, or highest point, of the muzzle-ring. The case is analogous to that of fig. 3, where FG represents the

Gunnery. line of the axis produced, CDL the visual line, and HI the true line of flight or trajectory. Now, it is evident from this drawing that the shot intersects the line of vision just at K, and hence that if the mark M chanced to be nearer the gun, though on the visual line, it would not be struck,

at least in the centre, by the ball, which, in this part of its course, would be below that line. After passing K, the line of flight rises above the visual line, but in its descent, meets it again when I and L coincide, and if the mark M is placed at that point the ball will strike it.

This is better shown in fig. 4, where the visual line CDL is represented horizontal—the axis, therefore, being elevated by the angle made by the intersection of the axis prolonged and the visual line. In this case, then, the ball rises and intersects the visual line at I, and again on its descent at L, proceeding on to N, so that the mark M would be struck if placed anywhere on the trajectory from L to N; and when the line CD is tangent at once to the muzzle and breech of the gun, or passes through any fixed and invariable marks or sights placed for the purpose on the summit of the base and muzzle rings, it becomes the natural line of aim or sight; and if placed horizontal, as in the figure, determines the point-blank range, or the distance from the gun of the second intersection L of the trajectory with the natural line of sight.

In the excellent treatise on artillery by Didion, chef d'escadron of the French artillery, the meaning of the term but-en-blanc, or point-blank, and the range corresponding to it, are stated as above, and, as observed by Didion, the point K of fig. 3, or I of fig. 4, being, from the ordinary construction of guns, so near to the muzzle (in a 68-pounder about seven feet), may be considered as corresponding with the actual point of intersection of the axis of the gun with the line of sight, the point of second intersection only being therefore of practical importance as determining the range.

It is very necessary to keep in view the two different interpretations of point-blank and point-blank range which have been here explained, in comparing the published ranges of English, as well as American and foreign guns, as will be perceived from the following statement:—

Griffiths (Artillerist's Manual, 6th edition, 1854) gives—
"The point-blank range of iron 32, 24, 18, and 12-pounders, with solid shot, as varying from 380 to 260 yards;

from which to 1200 yards every $\backslash frac\{1\}\{4\}$ degree increases the range 100; and from 1200 to 1500 every $\backslash frac\{1\}\{4\}$ degree increases the range about 50 yards; and the point-blank range of brass medium 12, 9, and heavy 6-pounders, with solid shot, at 300 yards, and from which to 700 yards, every $\backslash frac\{1\}\{4\}$ degree elevation increases the range 100 yards; from 700 to 1000 every $\backslash frac\{1\}\{4\}$ degree increases it 75 yards, and from 1000 to 1500, each $\backslash frac\{1\}\{4\}$ degree increases it 50 yards.

Captain Mordecai, in the Ordinance Manual of the United States Army, 1850, gives the following ranges:—

It may be observed, that both in the American and English service the word "dispart" is used, and means the natural tangent to the angle of natural sight or aim, the length of the gun measured from the rear of the base ring to a line raised vertically at the highest point of the swell of the muzzle, or at the permanent mark or sight fixed there, being the radius; and hence the angle of dispart is synonymous with the angle of natural sight, an angle which in English iron ordnance varies in construction from $1^\{\backslash circ\}$ to $2^\{\backslash circ\}\; 30\text{'}$, and in the French service varies also, as shown in the annexed table, extracted from Piobert.

Now, Griffiths states the point-blank range of the 12-pounder iron, with 4 lbs. of powder, as 360 yards; and in the following table, it will be observed that the point-blank range of the French 12-pounder, a gun which would be equivalent

to a 14-pounder English, is given as 650 metres, equal to 710 yards. In like manner Captain Mordecai gives the point-blank range of the 12-pounder American field-gun, with 2 lbs. 5 oz., as 347 yards; whereas the French 12-pounder

Gunnery. with a similar charge (see following table) gives 540 metres, or 600 yards nearly. The French point-blank range in the first case corresponds to an elevation of 1° of English practice, and in the second to an elevation of less

than 1°, as Captain Mordecai gives the range of 1° as 662 yards, though the angle of dispar of the French siege 12-pounder is 1° 9' 4", that of the 12-pounder field-gun being 0° 59' 39".

Range Table of French Siege and Garrison Guns fired with different Charges.

This Table is for guns in perfect condition; when much used the hausse must be augmented. The Hausse-de-Mire corresponds to the tangent scale of British ordnance, the degrees being replaced by the natural tangents of the required elevations in millimetres.

Cyprian
Lucar—
"Dispart."

The term dispart is of ancient use, and Lucar (1588) lays down as one of his maxims, that "every gunner, before he shoots, must trulie disparte his peece, or give allowance for the disparte; and when he dispartes a peece, he ought to set the said dispart in the midst and uppermost part of mettall over the mouth of the peece;" a caution equally necessary at the present day, as every gunner ought to make himself acquainted with the dispart of his gun, and with the range corresponding to it, and then familiarize his eye with that distance, which would thus become a base of comparison for ranges within and without.

3. Mode of action of gunpowder. Resistance of the air.—As the real nature of the products of combustion, as well as of combustion itself, is a comparatively recent discovery, the exact theory of its action was not to be expected from Tartaglia, and yet he gives a very reasonable account of it. In the 22d Colloquie, in which a gunfounder inquires why guns generally burst at the breech, Tartaglia answers to this effect, that the great exhalation proceeding from the saltpetre acts against the ball, and as it is difficult to put it at first in motion, though easy to keep up the motion when once given, should the gun be too weak in that part it will yield to the force of the windie exhalation and burst; but if the metal be sufficiently strong, and the ball be moved, there will be no fear of bursting, unless by any accidental cause the motion of the ball be arrested, when the gun may burst, as it sometimes does, near the muzzle; "for so soone as the pellet is in moving, that exhalation will continue with ease if no other let do happen, but so soone as the pellet commeth to the mouth of the peece, it finds all the aire without the peece, and by how much the pellet, together with the said exhalation that thrusteth it to assault the aire, commeth more swiftly, by so much the more united and with a greater force, doth the aire oppose itselfe very strongly to resist that sudden moving, and thereupon, in that place, another difficultie or strife riseth betweene the exhalations

within (which thrusteth forth the pellet), and the aire without,—that is to say, the exhalation would goe out of the concavitie, and the aire without doth resist the same; but in the end, the exhalation within being of a greater force, and getting the victorie, breaketh forth and teareth in pieces his said enemie. And then the mouth of the peece being, as it were, in the midst of the strife, doth alwaies suffer very much; and this is the cause that the peece, lacking his due thicknesse on the said place, or for some other unknowne fault, doth there easily breake."

4. The length of the gun should be duly proportioned to the charge.—It had been supposed that the longer the gun the greater would be its range, but Tartaglia in the 11th, 12th, and 13th Colloquies, points out that though the long culvering of these days had a greater range than the shorter cannon, it required a correspondingly greater charge—that of the culvering being $\backslash frac\{1\}\{3\}$ths of the weight of the shot, and that of the cannon only $\backslash frac\{1\}\{3\}$ds; and further reasons, that for any given charge there is one length only which can give the maximum range, as if too short, part of the powder will be expelled before ignition, and so much power be lost; and if too long, the ball would be in the gun after the total ignition of the powder, and be checked in its progress by friction against the bore—the proper limit of length being that which will place the ball exactly at the mouth at the moment when all the powder shall be on fire, and the windie exhalation be at its maximum, "for on that instant all the expulsive vertue of the powder begins to worke on the pellet in the chiefe of his furie or force, and after that vertue expulsive hath wrought on the pellet, the said pellet, finding nothing to let or resist his range (except the aire), will flie more farther than if the concavitie of the peece had beene more longer or more shorter." Notwithstanding the partial imperfection of the reasoning, this was a curious approximation to the truth, as regards the exact proportion of the charge "for giving the maximum velocity" to the length

Gunnery. of the gun; and though Tartaglia did not treat of the more general question of the inexpediency of increasing both charge and length beyond a certain point, he gave the explanation of the fact when he stated that the air resists the more, the more violent the action of the expulsive exhalation. Had he known the law of that resistance, he would have probably perfected the explanation by showing that ultimately the resistance would become so great as to require enormous strength in the gun to resist the concussion.

Robins (1742) explains the relation of the length of the bore to the charge and velocity communicated to the ball, by construction thus:—"Let AB represent the axis of the piece; draw AC perpendicular to it, and to the asymptotes AB and AC describe any hyperbola LEF, 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 by the table of logarithms, from the height of the line AD thus determined, and the diameter of the bore, the quantity of powder contained in the charge is easily known.

If, instead of this charge, any other fitting 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 this charge AI in the subduplicate proportion of the rectangle AE to the same rectangle diminished by the trilinear space HKE." This explanation depends upon the proposition relative to the determination of the velocity of the ball with a given charge to be subsequently referred to, but Robins' reasoning is here anticipated in order to place the result in opposition to that of Tartaglia.

Hutton (1812). In his tracts published in this year, Hutton details the experiments in gunnery carried on by himself and Major Blomfield, Royal Artillery (afterwards General Lord Blomfield), and other able artillery officers, for several years in the Warren, now arsenal of Woolwich. Some of these had been previously published in 1786 in a quarto volume of tracts, and a previous set, made in 1775, in the Philosophical Transactions for 1778—Dr Hutton having been awarded the annual gold medal of the Royal Society for his paper containing the results of the experiments, and the deductions drawn from them.

Some of these experiments were directed to the determination of the relation between the charge of powder, the length of the bore, and the resulting velocity. The experiments were made with five guns of the same calibre, being intended to discharge a ball of 16 oz. weight, but of lengths varying from 30.3 inches to 82.3; the lengths of the bores varying from 28.53 to 80.80 inches, gun No. 5 being intended to be reduced in length by cutting off successive portions after a certain number of rounds of practice, so as to test the effect, on the velocity, of a variation in the length of the bore. The deductions are thus stated by Dr Hutton:—

"1st. The law determined by the previous experiments between the charge and the velocity of ball is again confirmed—namely, that the velocity is directly as the square root of the weight of powder, as far as to about the charge of 8 oz. (half the weight of the ball used); and so it would continue for all charges were the guns of an indefinite length. But as the length of the charge is increased, and bears a more considerable proportion to the length of the bore, the velocity falls the more short of that proportion.

"2d. That the velocity of the ball increases with the charge, to a certain point, which is peculiar to each gun where it is greatest; and that by further increasing the charge, the

Gunnery. velocity gradually diminishes, till the bore is quite full of powder. That this charge for the greatest velocity is greater as the gun is longer, but not greater, however, in so high a proportion as the length of the gun is; so that the part of the bore filled with powder bears a less proportion to the whole in the long guns than it does in the shorter ones; the part of the whole which is filled being, indeed, nearly in the subduplicate ratio of the length of the empty part.

"3d. It appears that the velocity continually increases as the gun is longer, though the increase in velocity is but very small in respect to the increase in length, the velocity being in a ratio somewhat less than that of the square roots of the length of the bore, but somewhat greater than that of the cube roots of the length, and is, indeed, nearly in the middle ratio between the two.

"4th. It appears from the ranges determined by these experiments that the range increases in a much less ratio than the velocity, and, indeed, is nearly as the square root of the velocity, the gun and elevation being the same. And when this is compared with the property of the velocity and length of gun in the foregoing paragraph, it appears that we gain extremely little in the range by a great increase in the gun, the charge being the same. And, indeed, the range is nearly as the 5th root of the length of the bore; which is so small an increase as to amount only to about $\backslash frac\{1\}\{4\}$th part more range for a double length of gun."

The comparison of these results of experiments made at a time of vastly advanced knowledge, with the statements of Tartaglia, must, notwithstanding some of their imperfections, justify a very high estimate of the position which he would have held amongst writers on gunnery had he lived after the discoveries of Galileo and Newton.

In 1638 Galileo printed his Dialogues on Motion. In these he pointed out the general laws observed by nature in the production, resolution, 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 in 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 that 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, before 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. 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 reason-

able 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 parabolic. But, instead of drawing the proper inference 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 all 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 move 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 Reuss, 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 several 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 Reuss were at any time controverted, or any reason offered for the failure of the theory of projectiles when applied to use. Nothing further, indeed, 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 of almost everything else relating to the flight of military projectiles, and, indeed, advanced the theory of gunnery much nearer perfection than it had ever 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, which he determined to be owing to an elastic fluid similar to our atmosphere, having its elastic force greatly increased by the heat; and further, that the elasticity or pressure of the fluid produced by the firing of gunpowder is, ceteris paribus, directly as its density.

"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: hence Mr Robins determined, 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 were 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."

By exploding powder in a receiver connected with a mercurial gauge, Robins determined that an ounce of powder produced, on explosion, nearly 575 cubic inches of gaseous fluid possessing the same elasticity as common air; and, making allowance for the increase of elasticity due to the heat of the receiver and of the red-hot iron used for igniting the powder, that the gas, when reduced to the actual temperature, would have filled 460 cubic inches. Now, to determine the ratio of the bulk of the gunpowder to the bulk of this fluid, remembering that 17 drams avoirdupois of gunpowder fill 2 inches, the proportion $16\; :\; 17\; ::\; 460\; :\; 488\backslash frac\{1\}\{2\}$ gave the number of cubic inches of an elastic fluid equal in density with the air produced from 2 cubic inches of powder; "whence the ratio of the respective bulks of the powder and of the fluid produced from it, is nearly as 1 to 244."

"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 at 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."

The augmentation of the elasticity of air by temperature to the extent of "the extremest degree of red-hot iron, Mr Robins investigated by heating to an incipient white heat a portion of a musket barrel six inches long, closed at one end and drawn out at the other conically, to an aperture of one-

Gunnery. eighth of an inch in diameter. The aperture was first closed by a wire, and the conical end of the tube after being heated was plunged into water, and the whole left to cool to the ordinary temperature of the air, when, the wire being removed, the water rushed in to fill the space now left vacant by the again contracted air. By the average of three experiments he determined the weight of the water which entered the barrel, and knowing the quantity or weight of water which would fill the whole, the difference between the two was the weight of water which would fill the portion of the barrel occupied by the cooled air. The proportion between the space occupied by the air before expanded by heat, and the same air when expanded by an incipient white heat, was determined by these experiments to be as $194\backslash frac\{1\}\{2\}$ to 796.

"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 $194\backslash frac\{1\}\{2\}$ 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 $194\backslash frac\{1\}\{2\}$ to 796, it follows, that if $24\backslash frac\{1\}\{2\}$ be augmented in this ratio, the resulting number, which is 999$\backslash frac\{1\}\{2\}$, will determine how many times the 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$\backslash frac\{1\}\{2\}$, 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."

The method adopted by Robins for determining the elastic force of the gas produced by the ignition of gunpowder, when reduced to the ordinary temperature of the air, was independent of the actual nature of the gas, and therefore unaffected by the erroneous views then entertained respecting it. In fact, the weight of the gases, instead of being only three-tenths of the weight of the powder, is about six-tenths of that weight; and by the estimate of Gay Lussac, the proportion between the space occupied by the gases and by the powder would be nearly double that adopted by Robins. Gay Lussac obtained from 100 grammes of powder 50 litres of gas, and as the 100 grammes, of density 0.9, would have occupied one-ninth of a litre, the elastic force of the gas, when compressed in that space, would be $50\; \backslash times\; 9\; =\; 450$; and Captain Boxer, reasoning upon the known composition of gunpowder and the theoretical results of its decomposition as a definite chemical compound, makes it 317$\backslash frac\{1\}\{2\}$; but as experience has shown that these results are by no means confined to the theoretical products, it is probable that the determination of Gay Lussac is very near the truth. In like manner, the estimate of the temperature produced by the ignition of gunpowder has been variously stated, as well as the resulting elastic force: thus Gay Lussac assumes the temperature at 1000° Cent., or 1832 Fahr., and the resulting elastic pressure as 2137 atmospheres; Piobert assumed a temperature more than double

that stated by Gay Lussac, and arrived at a pressure of 7500 atmosphere; but, as observed by Senderos (1852), it is impossible to determine with accuracy in this manner the impulsive force of the gases produced from the ignition of gunpowder, though, without doubt, it greatly exceeds that stated by Robins, as will be pointed out hereafter.

Having thus determined the force of the gunpowder, Mr Robins next proceeds to determine the velocity with which the ball is discharged, adopting in the solution of this problem, the two following principles, neither of which is strictly correct,—1. That the action of the powder on the bullet ceases as soon as the bullet leaves 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 is, 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. 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."

"With respect to the grains of powder which are often blown out unfired, and which are always urged as a proof of the gradual firing of the charge, there may perhaps be some few grains in the best powder of such an heteroge-

Gunnery. neous 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."

Such were the reasonings of Mr. Robins; but however rapid the ignition of gunpowder, it is still progressive; and without doubt the ball moves before the whole impulse of the powder from its complete ignition has been received, and it is equally certain that some portion, however small, of the powder is generally thrown out unburnt. Were it not indeed for the movement of the ball before the full development of the elastic force of the gases, accidents from the bursting of guns would be frequent, as may be judged from the consequence of any impediment in the way of the movement of the ball, or from accidentally leaving it at a distance from the charge. Senderos observes—"The full force of gunpowder, with the intensity it possesses, is not used in fire-arms, but only a small part of it. It is undoubted that the transmission of any force requires time. The projectile opposes a resistance proportioned to its mass or inertia, and as soon as the force has become sufficient to overcome that resistance, the projectile begins to move, and allows the gases to expand into a larger space, thus losing density and caloric before they exert their full force on the gun."

"These postulates being allowed to be just, let AB, fig. 6, 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 H 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 itself 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 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 cylinder, to be 45 inches; its diameter DC, or rather the diameter of the ball, to be $\backslash frac\{3\}\{4\}$ths of an inch; and AF, the extent of the powder, to be $\backslash frac\{2\}\{3\}$th 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 3.19 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 $\backslash frac\{3\}\{4\}$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 $34,900\; \backslash times\; 2$, or 69,800 times its weight; whence FL to FH is as 1 to 69,800; and FB to FA as $45\; -\; \backslash frac\{2\}\{3\}$, or $42\backslash frac\{1\}\{3\}$ to $\backslash frac\{3\}\{4\}$, that is, as 339 to 21; whence the rectangle FLPB is to the rectangle AFHS as 339 to $21\; \backslash times\; 69,800$, 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 .43429, &c., is to the tabular logarithm of $\backslash frac\{AB\}\{AF\}$; that is, of $\backslash frac\{45\}\{2\}$, 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 12263, 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 $42\backslash frac\{1\}\{3\}$ inches, the velocity a heavy body will acquire in falling 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

Gunnery. 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 impelled body, will be directly as the logarithm $\backslash frac\{AB\}\{AF\}$ (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 $\backslash frac\{3\}\{4\}$ths 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 200, 300, 400, 500, 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 by means of the instrument (the Ballistic Pendulum), represented in fig. 7, 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 is 62$\backslash frac\{3\}\{4\}$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$\backslash frac\{3\}\{4\}$d, and 66 to 52, take the quantity of matter of the pendulum to a fourth quantity, which will be 42 lbs. $\backslash frac\{1\}\{2\}$ 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 lbs. $\backslash frac\{1\}\{2\}$ 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 $\backslash frac\{1\}\{2\}$th of a pound, or the $\backslash frac\{1\}\{2\}$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 $\backslash frac\{1\}\{2\}$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 $17\backslash frac\{1\}\{2\}$ inches; the distance of the ribbon from the axis of suspension is $71\backslash frac\{1\}\{2\}$ inches; whence reducing $17\backslash frac\{1\}\{2\}$ in the ratio of $71\backslash frac\{1\}\{2\}$ 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 $3\backslash frac\{1\}\{2\}$ 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 $17\backslash frac\{1\}\{2\}$ inches measured on the ribbon, no more is necessary than to multiply $3\backslash frac\{1\}\{2\}$ 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 $3\backslash frac\{1\}\{2\}$ feet in 1"; and we have before shown that this is the $\backslash frac\{1\}\{2\}$th of the velocity of the bullet. If then a bullet weighing $\backslash frac\{1\}\{2\}$th of a pound strikes the pendulum in the centre of the wood GKIH, and the ribbon be drawn out $17\backslash frac\{1\}\{2\}$ 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 differ 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 $17\backslash frac\{1\}\{2\}$th, 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."

Mr Robins then gave several precautionary rules for securing precision in the experiments, and guarding against accidents, amongst which were the two following:—

"The weight of the pendulum and the thickness of the wood necessary to prevent the bullets from being shivered by striking directly on the iron, 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."

"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 $\backslash frac\{1\}\{4\}$th of an inch; of the second (B) was the same; and of D, $\backslash frac\{1\}\{8\}$3 of an inch. In the first of these experiments, another barrel (C) was used, whose length was 12.375 inches, and the diameter of its bore $\backslash frac\{1\}\{4\}$ 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 $\backslash frac\{1\}\{2\}$th of an inch over or under. This showed an error in the theory, varying, according to the different lengths of the chord, from $\backslash frac\{1\}\{2\}$th to $\backslash frac\{1\}\{4\}$th of the whole; the charges of powder were the same as in the last. In sixteen other experiments the error was $\backslash frac\{1\}\{2\}$ths of an inch, varying from $\backslash frac\{1\}\{2\}$th to $\backslash frac\{1\}\{4\}$th of the whole; the charges of powder were 6, 8, 9, or 12 pennyweights. In seven other experiments the error was $\backslash frac\{1\}\{2\}$ths of an inch, varying from $\backslash frac\{1\}\{2\}$th to $\backslash frac\{1\}\{4\}$th of the whole; the charges of powder 6 or 12 pennyweights. In eight experiments the difference was $\backslash frac\{1\}\{2\}$ths of an inch, indicating an error of from $\backslash frac\{1\}\{2\}$th to $\backslash frac\{1\}\{4\}$th of the whole; the charges being 6, 9, 12, and 24 pennyweights of powder. In three experiments the error was $\backslash frac\{1\}\{2\}$ths, varying from $\backslash frac\{1\}\{2\}$th to $\backslash frac\{1\}\{4\}$th of the whole; the charges 8 and 12 pennyweights of powder. In two experiments the error was $\backslash frac\{1\}\{2\}$ths, in one case amounting to something less than $\backslash frac\{1\}\{2\}$th, in the other to $\backslash frac\{1\}\{2\}$th of the whole; the charges 12 and 36 pennyweights of powder. By one experiment the error was $\backslash frac\{1\}\{2\}$ths, and by another $\backslash frac\{1\}\{2\}$ths; the first amounting to $\backslash frac\{1\}\{2\}$th nearly, the latter to almost $\backslash frac\{1\}\{4\}$th 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 $\backslash frac\{1\}\{4\}$th 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 $\backslash frac\{1\}\{2\}$ths 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 $\backslash frac\{1\}\{2\}$th 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 further 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."

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; and from it we are taught the necessity of leaving the same space behind the bullet, when we would, by the same quantity of powder, 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 à ricochet.

"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.

"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 $\backslash frac\{1\}\{3\}$ths of the substance of the powder, the remaining $\backslash frac\{2\}\{3\}$ths 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."

Mr Robins then investigates the cause of these irregularities in the expansive motion of the fluid by experiments; but before referring to them, it is right to observe, as has been before stated, that in British gunpowder, consisting of 75 parts of nitre, 15 of charcoal, and 10 of sulphur, the potassium of the nitre and the sulphur are the only constituents which unite to form a solid residuum, the sulphate of potassium, and that their weight being about 39 lbs. per cent., the remaining 61 lbs., or $\backslash frac\{2\}\{3\}$ths of the whole, form gaseous elastic products.

"The experiments made use of for this purpose were of two kinds. The first was made by charging a 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. Gunnery.

"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 6 inches, and the weight of the piece, and of the iron that fastened it, &c., was 12 $\backslash frac\{1\}\{2\}$ 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 10.1 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 1 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 1 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."

Mr Robins here adduces some experiments to obviate a possible objection by showing that the confinement of the powder was not necessary to ensure its total ignition and the full development of its elastic force, and "that the push of the recoil, arising from the expansion of the powder alone, is found to be no greater when it impels a leaden bullet before it than when the same quantity is forced without any wad to confine it;" and then proceeds as follows:—

"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 the ball not 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."

However ingenious the researches of Mr Robins into this important element of gunnery—the velocity with which the gases produced by the ignition of gunpowder expand—they have not been admitted by some as satisfactorily solving the question; and yet Hutton, although, by a combination of experiment and calculation, he had deduced a velocity varying from 3000 to 5000 feet, after correcting one of the quantities in his formula, at first assumed too high, arrived at a conclusion very nearly the same as the experimental one of Robins. The determination, indeed, of the velocity of the elastic gases is attended with much difficulty, as is that (as before pointed out) of the initial force with which these gases act upon the projectile they are intended to propel. Robins considered this force to be about 1000 atmospheres; but Hutton found it to vary, according to the charge and the length of gun, from 1700 to 2300, and he therefore considers it as fairly represented by about 2000 atmospheres, and his results were confirmed by those of Dr Gregory, who made it 2250. Now, Hutton's formula for determining the ultimate velocity of the ball, and conse-

Gunnery. quently of the gas then pressing against and urging it forward, is

where $a$ represents the height or length of the charge, including cartridge, or of the space behind the ball, $b$ the whole length of the gun-bore, $h$ the length of the portion of the cylinder or bore which would be filled with the powder, $d$ the diameter of the ball or of the bore, $n$ the ratio of the first force of the fired powder to the pressure of the atmosphere as 1, $w$ the weight of the ball, and $p$ a quantity having some fixed relation to the weight of the charge; and if in this formula the weight of the ball be made 0, $V$ becomes the value of the velocity of the expanding gas. But here enters a difficulty, as it is not easy to determine what proportion of the weight of the powder ought to be assumed for $p$, from the uncertainty of the actual condition of the gases and of the solid residuum, at the moment of decomposition of the powder. Supposing an equal density to exist throughout the bulk of the gases, and that the solid residuum is diffused equally through them, $p$ should be, as Hutton at first assumed it, $\backslash frac\{1\}\{2\}$ of the weight of the powder; but, as it is more probable that the rear portion of the gas is much more condensed than the front portion, and, consequently, that the centre of gravity of the whole gas has moved through a still less space, $p$ must be taken less than $\backslash frac\{1\}\{2\}$, and in this manner Hutton found that the velocity of the gases, when $p$ was taken as $\backslash frac\{1\}\{2\}$ of the weight of the powder, became between 7000 and 8000 feet per second; and when taken $\backslash frac\{1\}\{3\}$, from 3000 to 5000—results sufficient in themselves to prove how impossible almost it must be to determine theoretically the velocity of the gases. In investigating the decomposition of gunpowder, there are two points to be taken into consideration—the velocity of ignition, and the velocity of combustion; or, in other words, the time required to burn each grain of powder, on the one hand, and the time necessary for communicating ignition, as the flame is conveyed by the expanding gases with great rapidity from one grain to another. Piobert has endeavoured to estimate the velocity of combustion independently of that of ignition by forming a kind of bar with a paste of powder, 1 foot 2 inches in length, and about $\backslash frac\{1\}\{2\}$ths of an inch square, the bar being smeared with fine hog's lard, and placed vertically on a plate with water. This bar, weighing 330 grammes, was ignited at the top, and required 29.2 seconds for combustion, being at the rate of 486 of an inch per second. By other experiments of the same author, powder inclosed and slightly compressed in a tube $\backslash frac\{1\}\{2\}$ths of an inch in diameter and open at one end, burnt at the rate of .3 of an inch per second, or, when strongly compressed, at the rate of .4 to .6 of an inch per second. From these statements it is evident that, however rapid the combustion of powder, it is not instantaneous; and that the great object is to facilitate the transmission of the flame through the powder so as to render the ignition of the whole as nearly simultaneous as possible. The process of granulating powder for this purpose was early introduced, as Luis Collado, before mentioned, expressly points out the greater force of powder when grained as compared to that of its meal; and Cyprrian Lucar explains the mode of graining or "corning" the powder by passing it through sieves after having broken up the cake which had been first formed in the incorporating process. The size of the grain is an important element, and ought to be so arranged as to reduce the time of combustion to the minimum consistent with a due rapidity of ignition; and more particularly so as the denser the powder the greater quantity of gas must be produced at the same space, and the greater therefore the elastic force developed; whilst, as regards the grain itself, anything which increases its density must increase the velocity of ignition and diminish that of combustion, whilst the rapidity of combustion in-

creases as the grains are more porous and less smooth. These observations sufficiently demonstrate that the combustion of the charge cannot be effected in less time than that required for the combustion of a grain; but in this respect it must be remembered that the combustion proceeds from the circumference to the centre, and therefore requires only half the time as compared with Piobert's experiments. If, then, each grain were $\backslash frac\{1\}\{2\}$th of an inch in diameter, the complete combustion would be effected in $\backslash frac\{1\}\{2\}$th of a second; and if $\backslash frac\{1\}\{3\}$th, in $\backslash frac\{1\}\{3\}$th of a second; but long before that time the quantity of gas evolved must have been sufficient to move the ball, its ultimate velocity depending on the time it remains in the bore, or, in other words, on the more complete combustion of the powder, as well as on the continuance of the action of the gases produced.

"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 $\backslash frac\{1\}\{2\}$ inch. 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 $\backslash frac\{1\}\{2\}$ 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 16 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 2 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 ex-

Gunnery. pansion; 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 $11\backslash frac\{1\}\{2\}$ 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 ascension 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 in the case of all other fluids) be considerably diminished; and in order to avoid this irregularity, in all 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 have 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 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 $\backslash frac\{1\}\{3\}$d or $\backslash frac\{1\}\{3\}$th 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 $\backslash frac\{1\}\{4\}$th 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, 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 $4\backslash frac\{1\}\{2\}$ 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 $4\backslash frac\{1\}\{2\}$ 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 11 and 12 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 3 to 1 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 $\backslash frac\{1\}\{4\}$ths of a second, the resistance to the middle velocity will come out to be near 33$\backslash frac\{1\}\{2\}$ times the gravity of the ball, or 2 lbs. 10 oz. avoirdupois. Now, the resistance to the same velocity, according to the laws observed in slower motions, amounts to $\backslash frac\{1\}\{4\}$ths 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 proportion than that of 7 to 11; whereas we have seen in the former experiments, that to still greater degrees of velocity the augmentation approached very near the ratio of 1 to 3.

"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 $4\backslash frac\{1\}\{2\}$ seconds, the second flew 319 yards in 4 seconds, and the third 373 yards in $5\backslash frac\{1\}\{2\}$ seconds. 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 4 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.

"From the computations and experiments already mentioned, it plainly appears that a leaden ball of $\backslash frac\{3\}\{4\}$ths of an inch diameter, and weighing nearly $1\backslash frac\{1\}\{2\}$ ounce avoirdupois, 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 to that of a leaden one reciprocally in the subduplicate ratio of the specific gravities of lead 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

Gunnery. than is just sufficient for putting down the bullet easily: whereas, in real service, either through negligence or unskillfulness, 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 be compensated by the greater heat which in all probability attends the firing of these large quantities of powder.

"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), requires 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 510 pounds, which is near twenty-three times its own weight.

"Now, the prodigious resistance of the air to a bullet of twenty-four pounds weight, such as we have here established it, sufficiently shows 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 as found on the parabolic hypothesis. 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 two 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 may be 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 parabolic theory of gunnery, would, if applied without reference to the resistance of the air, lead to great errors in practice; "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 Ufano assigns to an arquebus, four feet in length, and carrying a leaden ball of 1½ 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. Mersennus also tells us, that he found the horizontal range of an arquebus 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: or in the case of an iron bullet of 24 lbs. weight which, fired from a piece of the common dimensions, with its greatest allotment of powder, has a velocity of 1650 feet in a second, as already shown, if the horizontal range of the 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."

In a similar manner, Robins pointed out that even with very moderate velocities the path of flight materially differs from a parabola, and that the highest point of flight is much nearer the place where the projectile falls to the ground than to that from whence it was at first discharged,—the descending curve being shorter and more inclined to the horizon than the ascending, as may be seen even in the figures of Tartaglia, cuts 1, 2, 3, 4; but it is not, at the present day, necessary to urge farther that the parabolic theory, though the necessary basis of theoretic gunnery, cannot of itself be applied in practice except when the initial velocities are less than 200 or 300 feet per second, when the resistance of the air becomes comparatively very small.

Mr Robins also first pointed out the fact, that from certain deviating forces which act upon the ball, it does not pursue that simple curved path which would be the result of the combined action of gravity and of the ordinary resistance of the air, both of which forces, acting in a vertical plane, have no tendency to produce lateral deflection, or to cause the ball to pursue a path of double curvature, or, as he expresses it, "to move on the surface of a cylinder the axis of which is perpendicular to the horizon." Didion thus states the causes of deviation:—"They are," he says, "of two kinds, namely, those which act on the projectile whilst still in the bore of the gun, so as to modify its direction and initial velocity, and which also produce a movement of rotation, as first observed by Robins, which becomes the cause of other deviations in the projectile; and, secondly, those which act upon the projectile during the whole time of its flight. The first deflect the projectile from the direction of the axis of the gun in proportion to the distance; the others may be considered accelerating forces, variable as each discharge, and even during the flight of a projectile, being distinct from gravity, which is constant, and from the resistance of the air, which is tangential to the trajectory of the projectile,—these two latter forces producing by their action what may be called the normal trajectory. Amongst the causes of deviation, some act in a permanent manner and produce effects which may be anticipated beforehand."

Deviation from the ball striking against the interior of the bore.—Solid balls, as well as hollow projectiles made of cast-iron, have always a diameter somewhat less than that of the bore of the gun for which they are intended. In consequence of this, the elastic gases escaping above the ball, which rests on the lower surface of the bore, press it down, producing much friction and a rotatory motion in the ball, whilst the gases pressing it behind produce a movement of translation. This pressure in bronze guns soon produces a depression or hollow, and the ball in consequence rises at an angle a little more elevated than the axis of the piece; and should the gun be short enough, this would be the final direction of the ball on leaving the gun; but, in general, such is not the case, and the ball will strike the upper part of the bore and be deflected downwards, when the result will be a depression of the true line of direction; or should it rebound again, an elevation—an effect which must increase as the gun becomes more worn. This deviation would be one merely in a vertical direction, were there not other causes tending to combine a lateral deviation with it, such, for example, as irregularities in the form and density of the

Gunnery. projectile, as well as differences in its position as respects the charge, by which it happens that the resultant of the action of the gases is not exactly in the vertical plane, and the direction of the ball on leaving the gun diverges a little from that plane. The vertical deviation, though not always, is generally in elevation. From French experiments, it appears that with guns this deviation on an average amounts to $0^\{\backslash circ\}\; 3\backslash frac\{1\}\{2\}\text{'}$, and in howitzers to $0^\{\backslash circ\}\; 10\backslash frac\{1\}\{2\}\text{'}$—one-quarter of the shots from guns having an elevation of more than $8\backslash frac\{1\}\{2\}\text{'}$, and about one-quarter a depression below the axis of $1\backslash frac\{1\}\{2\}\text{'}$; whilst in howitzers one-fourth of the shells were elevated more than $15\backslash frac\{1\}\{2\}\text{'}$, and one-fourth $5\backslash frac\{1\}\{2\}\text{'}$ above the axis—the remaining half, as also in the guns, falling between these extreme numbers. In a horizontal direction, half of the shots deviated from the vertical plane passing through the axis more than $4\backslash frac\{1\}\{2\}\text{'}$, either to the right or left, the others diverging less.

Robins and Lombard both determined experimentally the fact of deviation; the latter by placing a board screen marked by a horizontal and vertical line, at a short distance from the gun, and the intersection of the lines corresponding to the axis prolonged—the deviations of the ball when piercing the screen being thus at once exhibited; and Robins—who may be considered the author of experimental gunnery—by paper screens, as well as by board targets. The arrangement of screens which Mr Robins deemed the best he thus describes:—

"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 50 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:—

"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, we shall in this case discover divergencies 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:—

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 is a deviation from the vertical plane of above 37 inches, and this too in a transit of little more than 80 yards.

"But farther, to show that these irregularities do not depend on any accidental circumstance of the balls fitting or not fitting 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 about 15 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:

"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 23.35 short of it; so that here is a deviation of more than 23 inches.

"From all these experiments, the deflection in question seems to be incontestably 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 $1\backslash frac\{1\}\{2\}$ 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 3 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 14 inches, as I remember, on the contrary side.

By this arrangement of several parallel screens it became evident that the ball did not always pursue the simple direction of the deflection it had received on leaving the bore, but from a cause acting during the flight was again deflected, sometimes, as in the curious experiment of the bent barrel,

Gunnery. in an opposite direction to that of the original deflection. With his usual acumen Mr Robins assigned this effect to its true cause.

"The reality of this doubly curved 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 further 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, 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. 8, 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 conducted 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

Gunnery. 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 BCDE, 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 27$\backslash frac\{1\}\{2\}$, 20 in less than 55$\backslash frac\{1\}\{2\}$, and 80 in 82$\backslash frac\{1\}\{2\}$; so that the first 10 were performed in 27$\backslash frac\{1\}\{2\}$, the second in 27$\backslash frac\{1\}\{2\}$, and the third in 27$\backslash frac\{1\}\{2\}$.

"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 3$\backslash frac\{1\}\{2\}$ lbs.; and ten revolutions being suffered to elapse, the succeeding 20 were performed in 21$\backslash frac\{1\}\{2\}$. 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$\backslash frac\{1\}\{2\}$ lb. a weight of one pound would make it revolve in less time than it did before, performing now 20 revolutions after 10 were elapsed in the space of 19$\backslash frac\{1\}\{2\}$.

"Hence then it follows, that from the 3$\backslash frac\{1\}\{2\}$ lbs. 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 2$\backslash frac\{1\}\{2\}$ lbs. in the situation M; and it appearing from the former measures, that the radius of the barrel is nearly $\backslash frac\{1\}\{2\}$th 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 21$\backslash frac\{1\}\{2\}$ (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-lb. shot, it irrefragably confirms a proposition I had formerly laid down from theory, that the resistance of the air to a 12-lbs. 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 con-

Gunnery. From 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 $\backslash frac\{1\}\{4\}$ lb. the globe went 20 turns in $54\backslash frac\{1\}\{2\}$, with 2 lbs. it went 20 turns in $27\backslash frac\{1\}\{2\}$, with $4\backslash frac\{1\}\{2\}$ lbs. it went 30 turns in $27\backslash frac\{1\}\{2\}$, and with 8 lbs. it went 40 turns in $27\backslash frac\{1\}\{2\}$. 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 $4\backslash frac\{1\}\{2\}$ 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 track, and sometimes proceeded so far as to have its direction at right angles to that in which it began 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."

Mr Robins also applied the whirling machine to the experimental illustration of the great difference of resistance offered by the air to the passage of bodies of equal surfaces, and even meeting the air at the same angle of obliquity, but of different forms; the difference being so great, "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 4 inches square, and whose planes made angles of $45^\backslash circ$ with the plane of its base, and also a parallelogram 4 inches in breadth, and $5\backslash frac\{1\}\{2\}$ 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 $35\backslash frac\{1\}\{2\}$. Then the pyramid being turned so that its base, which was a plane of 4 inches square, went foremost, it now performed twenty revolutions with the same weight in $38\backslash frac\{1\}\{2\}$. 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 $45^\backslash circ$ with the horizon by a quadrant, the parallelogram, with the same weight, performed twenty revolutions in $43\backslash frac\{1\}\{2\}$."

"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 $45^\backslash circ$; and yet one of them made twenty revolutions in $35\backslash frac\{1\}\{2\}$, whilst the other took up $43\backslash frac\{1\}\{2\}$. 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 $38\backslash frac\{1\}\{2\}$, which is the medium between the other two.

Gunnery. "But to give another and still more simple proof of this principle, there was taken a parallelogram 4 inches broad and $8\backslash frac\{1\}\{2\}$ long. This being fixed at the end of the arm, with its long side perpendicular thereto, and being placed in an angle of $45^\backslash circ$ with the horizon, there was a weight hung on at M of $3\backslash frac\{1\}\{2\}$ lbs. with which the parallelogram made twenty revolutions in $40\backslash frac\{1\}\{2\}$. 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 $45^\backslash circ$ 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 $35\backslash frac\{1\}\{2\}$, so that there were $5\backslash frac\{1\}\{2\}$ 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 somewhat larger scale, as the arms of this modification of the whirling machine were about 6 feet long, are related by Lovell Edgeworth, Esq. They confirm the accuracy of Mr Robins' statements.

These are the principal experiments made by Mr Robins in confirmation of his theory, and which not only far exceed everything 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 anywhere 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. Professor Robinson, for example, the original writer of this article, proceeded at this point to investigate another difficult and obscure point connected with the resistance of the air to bodies passing with great velocity through it.

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 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

Gunnery. 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 $Cb$ 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 $Cb$ acts as a resisting power on the piston $b$, the column pressing in the direction $Aa$ 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 $AaCB$ 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 $AB$ 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 forepart 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. Gunnery.

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 air rushes into a vacuum to be 1200 feet in a second, if a body 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 $\backslash frac\{1\}\{24\}$th 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 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 $7\backslash frac\{1\}\{2\}$ on the back part. The resistance therefore not only overcomes the moving power of the air by $7\backslash frac\{1\}\{2\}$ pounds, but there is a deficiency of other $7\backslash frac\{1\}\{2\}$ 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

Gannery. 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 as new a method of determining the velocities of bullets, by measuring the force of the recoil of the piece. As in all cases action and reaction 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 discharge in the opposite direction; 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 react with equal force against the bottom of the bore. Even the fine elastic invisible fluid which is generated from the powder in its inflammation cannot put itself in motion without at the same time reacting 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 in

the Ballistic Pendulum; 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 as stated by Robins, 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 $v\; =\; \backslash frac\{(V-U)W\}\{B\}$.

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, 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 1.4.6 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 9.05 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 47.4 lbs., to which $\backslash frac\{1\}\{3\}$ths of a pound were to be added on account of the weight of the rods by which it was suspended; thus making $W\; =\; 48$ lbs., 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 $1.8522\; \backslash times\; 579.31\; =\; 1073$ feet, which differs only by 10 from 1083, the velocity 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 consequently

as $\backslash frac\{(V-U)W\}\{B\}$, which measures the velocity of the bullet,

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 velo-

Gunnery. city 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 $0.204655\; \backslash times\; 579.31\; =\; 118.55$ 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 118.55, 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, 13.15, 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, $=\; 8.8\; \backslash times\; 118.35\; =\; 1045$ feet in a second; the velocities by the pendulum coming out 1040 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.

These differences Count Rumford partly ascribed to the possibility of error in measuring the arc of ascent of the pendulum in his experiments, as the bullet was very light, consisting of a leaden casing over a plaster of Paris nucleus, and the movement of the pendulum very small in consequence; an error, therefore, of $\backslash frac\{1\}\{100\}$th of an inch, if made in one of the experiments, being sufficient to account for 120 feet in a second of the difference in the velocity. The resistance also of the air in the passage of the ball from the barrel to the pendulum, distant 12 feet, was another supposed cause of the difference, as Count Rumford assumed, that in this passage, performed in the $\backslash frac\{1\}\{100\}$th part of a second, the ball lost 335 feet of its velocity, and therefore struck the pendulum with a corresponding diminished force; but even after allowing for these causes of irregularity between the results of the two methods, he still found large differences, and was led to discover and admit that they were the consequence of the fracture of his balls by the concussion of firing. Indeed, Count Rumford afterwards observes,—

“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.”

And hence it must be admitted that these experiments did not permit of a satisfactory comparison of the two methods of determining the recoil.

As Mr Robins considered that the whole of the powder of the charge is ignited before the ball begins to move, and that the gas proceeding from it is instantaneously produced, he came to the conclusion that the velocities of balls of the same size, though of different weights, would be reciprocally as the square roots of the weights; but Count Rumford proved that the ignition, even of the powder, was not instantaneous; and though Dr Hutton's experiments found this law to hold good, he considered this agreement to be the result of compensating circumstances, and states his own opinion that the correct law is nearly the reciprocal sub-triplicate ratio. Count Rumford also pointed out that Robins' estimate of the force of gunpowder—1000 to 1, as regards the pressure of the atmosphere—was too low, his own experiments making it 1308 to 1, an estimate still further raised by Hutton, as has been before stated, to 2000 to 1 or even 2230 to 1.

At this part of the subject it is necessary to point out the extraordinary importance of the labours of Robins, who was assuredly the pioneer of modern gunnery, and with whom commenced, as Sir Howard Douglas justly observes, a new era in the theory of gunnery. This success was due to the introduction of experimental proof; and simple as the means adopted were, they cannot be too highly prized for their efficiency. Didion quotes from the Histoire de l'Académie des Sciences de Paris (1707), a passage respecting the younger Cassini, which shows that he had adopted before Robins a practical application of the principle of “measuring the velocity of a projectile by that which it impresses upon a larger mass against, or into, which it is fired”—though with the object rather of testing the influence of the wad and of the disposition of the charge, than of measuring the velocity. Cassini's machine is described as a piece of wood armed at one end by a thick plate of cast-iron which was to resist the balls fired against it from the same musket, always placed at the same distance. The piece of wood was moveable so as to yield more or less to the shock, the extent of movement being marked or measured by the machine.

This simple plan was better fitted for measuring comparative than absolute results, and can scarcely, as suggested by Didion, have influenced Robins in his invention of the ballistic pendulum, first used in 1740, and the object of which was to measure the velocity of musket balls and the resistance of the air. To enlarge the sphere of inquiry, Dr Hutton, professor of mathematics at the Royal Military Academy, was subsequently authorized by the Master-General of the Ordnance to carry on several series of experiments, from 1775 to 1791, conjointly with Major (afterwards Sir Thomas) Blomefield, with a pendulum formed of pieces of wood clamped by iron, and weighing first 657 lbs., and then 2300 lbs., being intended to receive balls of 1 lb., 3 lbs., and 6 lbs. weight; and under the same authority, other series were undertaken in 1811, and from 1815 to 1818, the latter being entrusted to Dr Gregory, who had become professor of mathematics, and was associated with General Miller and Colonel Griffith—the weight of the pendulum being 7000 lbs., and the ball fired against it 12 lbs. These experiments deserve especial notice, not merely from their great importance, as determining essential elements in gunnery, but also from the evidence they afford of the active scientific spirit of some of the artillery officers of those days, and the example they set before the officers of the present. Nor were these experiments allowed to be carried on without turning them to account as a means of instruction—Dr Hutton carrying with him to the practice-ground his students

Gunnery. of the first class of the gentlemen cadets. His words are—
"On this occasion I took out with me, and employed the first class of gentlemen cadets belonging to the Royal Military Academy, namely, Messieurs Bartlett, Rowley, De Butts, Bryce, Wm. Fenwick, Pilkington, Edridge, and Watkins, who have gone through the science of fluxions, and have applied it to several important considerations in natural philosophy. These gentlemen I have voluntarily offered and undertaken to introduce to the practice of these interesting experiments, with the application of the theory of them, which they have before studied under my care. For though it be not my academy duty, I am desirous of doing this for their benefit, and as much as possible to assist the eager and diligent studies of so learned and amiable a class of young gentlemen, who, as well as the whole body of students now in the upper academy, form the best set of young men I ever knew in my life; nay, I did not think it even possible in any state of society in this country, for such a number of gentlemen to exist together in the constant daily habits of so much regularity and good manners, their behaviour being indeed perfectly exemplary; and I have no hesitation in predicting the great honour and future services which will doubtless be rendered to the state by such eminent instances of virtue and abilities;" and he added in a note, dated 1812, "At this distance of time, and long before, the world has had the satisfaction to find, that this prediction has been most amply and accurately fulfilled in every instance,"—a truth which will be admitted by all who still remember many of these names.

These remarkable words of a man who had raised himself, not only by talents of a very high order, but by unblemished character, to a position of great respectability and of high responsibility, deserve attention at this moment, when it is proposed to remodel the institution of which Dr Hutton was the distinguished professor of mathematics. By associating with him the gentlemen he has named, he took the first step towards the formation of a Class of Application; and this is really what should be done with the academy. Years ago a practical class was formed as supplementary to what is called the theoretical class; but such distinctions are not only erroneous but injurious, as theory and practice cannot be safely separated at any time, or at any period of a professional course. The gunner may, indeed, be taught much which is required from him practically; but the officer must be taught theoretically, and with him the subsequent training ought not to be merely practice, but the application of theory to practice. It is to men instructed at the Royal Military Academy in the highest branches of dynamics that we should look for the future improvement of our means of offensive and defensive warfare; and it is to be hoped, therefore, that the absurd distinctions between "theoretical" and "practical" classes will be abandoned; that the principle of "application" will be the characteristic of the reformed academy or college; and that its professors will be again associated with the officers of artillery in carrying on further experiments for the improvement both of the science and of the art of gunnery.

It has been already stated that Benjamin Thomson, Esq. (afterwards Count Rumford), furnished a paper to the Royal Society in 1781 containing various experiments made partly for the determination of the most advantageous situation for the vent in fire-arms, and partly to measure the initial velocity of bullets when discharged from them. In these experiments he merely employed, as Robins had done, a musket barrel; but in determining the velocity he used the recoil of the barrel itself, as well as the motion of the pendulum against which the ball was impelled. The former method Mr Thomson called "a new method;" but it had been previously pointed out by Robins, who also appears to have suggested the idea of applying it in the "eprouvette," for testing the comparative force of different samples of gun-

powder, again proposed by Thompson, and subsequently so beautifully carried into effect by Hutton, who, in 1783, commenced his experiments with five brass one-pounders, cast expressly for the purpose, suspending the guns by a framework, and by additional weights, bringing them all up, including the weight of the suspending frame, to one weight—917 lbs. The weight of the ballistic pendulum was 559 lbs., and distance between the gun and ballistic pendulums was 35½ feet, the axis of the gun being point-blank, or horizontal. The first set of experiments was designed for the purpose of testing "the comparative strength of the different barrels of powder, by firing several charges of it, without balls or wads." So that in this case the gun pendulum became an eprouvette. It was found that the pendulum was considerably affected by the explosion of the powder, and in consequence a paper screen was afterwards interposed between the gun and ballistic pendulums. The recoils were found to be in a higher proportion than the charges of powder, as will be seen from the following statement of the results of four experiments:—

So that the first proportions, as ratios between the recoils, all exceed that of the charges as 1 to 2, approximating, however, to it with the increase of charge; and that the ratios between the successive ratios of recoil are nearly equal as shown in the 5th column. Dr Hutton then carried on several series of experiments for determining the velocities of the ball, both by the recoil of the gun and the vibration of the ballistic pendulum; and in his account of the experiments of 1786, he gives the following tabular view of the results, the velocities being given in feet per second:—

Comparison of the Velocities by the Gun and Pendulum.

So that it appears that here, also, as in the experiments of Count Rumford, the recoil by the gun gave frequently a result very much in deficit of that given by the pendulum. The guns were those previously mentioned, all of the same calibre but of different lengths, the balls weighing a little more than 1 lb., and being all reduced by calculation, as regards the results, to a uniform size.

Dr Hutton concluded, from these results, that the velocities, determined from the two different ways, do not agree together, and that the method of determining the velocity of the ball from the recoil of the gun is not generally true; and, consequently, that the effect of the inflamed gunpowder on the recoil of the gun is not exactly the same when it is fired without a ball as when it is fired with a ball. The difference also does not appear to be regular, neither in different guns with the same charge, nor with different charges with the same gun. That with small charges the velocity by the gun is greatest; that the velocity by the pendulum

Gunnery. continues to gain upon that by the recoil as the charges increase, and ultimately exceeds it more and more, as the charge of powder is increased.

Experiments in 1788 and 1789 were made with brass 3-pounders—one short and the other very long—so as to observe the effect both of length of gun and of weight of powder; the charges being 4 oz. and 16 oz. The weight of the pendulum was 1426 lbs. Amongst other results it was ascertained, that firing with 16 oz. of powder, or $\backslash frac\{1\}\{2\}$ the weight of the ball, the long gun gave an initial velocity to its ball of 584 feet per second, and the short gun a velocity of 1371 feet per second; so that the velocity by the long gun exceeds that by the short by between the 6th and 7th part of the latter; the lengths of the guns being 40 inches and 69$\backslash frac\{1\}\{2\}$ inches.

In 1789 experiments were also made with a long 6-pounder, and in 1791 these were resumed with a new pendulum weighing 1630 lbs.; the weight of the gun being 1370 lbs., and the weight of the gun with its framework for suspension 1618 lbs. Afterwards a medium 6-pounder and a light 6-pounder were also used with successive pendulums weighing 1861 lbs. and 2119 lbs.; so that it may be said, without hesitation, that Dr Hutton's general experiments were by far the most important which had ever been made.

The velocities of the ball as discharged by the long gun, after some slight corrections made with a view of reducing the numbers to something like a regular series, but not such as to materially affect their absolute values, are as follows:—

A table which shows that, with equal increments in the distance from the pendulum, there is a gradual diminution in the loss of velocity corresponding to the mean velocity at the middle of that increment; thus, when the mean velocity is 1780, the loss of velocity per second is 65 feet; when 1656, 59; when 1590, 56; when 1480, 52; when 1380, 48; when 1282, 47; when 1190, 43,—all agreeing very well in the corresponding decrease of the effect of the resistance of the air on the velocity of the ball, except the two means 1380 and 1282, which are too near each other in their results.

The velocities gained by the ball fired from the medium 6-pounder, at 30 feet distance, were—for 2 lbs. of powder, 1585 feet; 1$\backslash frac\{1\}\{2\}$ lb., 1460; 1 lb., 1260; $\backslash frac\{1\}\{2\}$ lb., 877. The velocities gained by the ball fired from the light 6-pounder, at 30 feet distance, were—for 3 lbs. of powder, 1624; 2 lbs., 1558; 1$\backslash frac\{1\}\{2\}$ lb., 1440; and, by calculation, Dr Hutton determined that a 6-lb. ball, moving with the velocity of 1200 feet per second, was resisted by a force equal to 115 lbs.; and when moving

with a velocity of 1600 feet per second, with a force of nearly 222 lbs.,—results of great importance.

In France, similar experiments had been made at Metz, in 1839 and 1840, with the French 24, 12, and 8-pounders, and their howitzer of 22 c., or 8$\backslash frac\{1\}\{2\}$ inches. For these experiments, however, a considerable change had been effected in the instrument, as shown in fig. 10. Instead of a mass of wood, requiring frequent renewal, as in the English pendulum, a more permanent receiver was substituted, which, being filled with some material moderately penetrable, can be used without limitation.

The ballistic pendulum, then (fig. 10), consists of a conical vase of cast-iron, A, called the ballistic receiver, and suspended by four rods, B B, B' B', to an axis, C, 16 feet above it. The two rods, B B, embrace the front, and B' B' the rear of the receiver; and further, the two rods on the same side of the receiver approximate together at the top, so as to be connected with the one end of the axis, whilst the two on the other side are connected with the other end. The rods are bound together by four cross pieces, D, D', E, E', and by three ties, F, F', C, which render the whole system rigid. The ties, which unite together the rods below, are seen in H, K, and K', the whole being secured by a screw-bolt, L, below, and another, M, above; the bolt, L, carrying a moveable weight formed of discs of lead, N, and kept in position on the bolt by the springs, O. This weight, the position and magnitude of which can be changed, serves either to depress (if necessary) the centres of gravity and oscillation of the receiver, and to render its axis horizontal. The axis, C, is of iron, and is shaped at its extremities, P, into knife-edges, slightly curved, and resting on the steel plates or bearers, Q, Q, the upper surface of which is formed into two planes, meeting in a curve of double the radius of that which forms the bearing of the knife-edges. The bearers, Q, Q, rest on cast-iron plates, R, R, which are firmly

fixed upon two piles or piers of masonry. The receiver, SSS, is shaped within as a truncated cone, the bottom of which

Gunnery. is rounded off, and it is sufficiently long to prevent the projectile from passing entirely through the sand with which it is filled; it is formed of cast-iron, and strongly bound by hoops of forged iron. The front is covered with a thin sheet of lead to prevent the sand from being shaken out, and this sheet is marked by a horizontal and by a vertical line, the intersection corresponding to its axis; and the actual position of the shot when entering the receiver can be therefore readily determined by reference to these lines.

It will be readily supposed that this ingenious contrivance would not be effectual at the long distances required for firing to determine the resistance of the air, as the ball would frequently strike the exterior rim of the receiver instead of penetrating within it. On account of this practical defect another pendulum, fig. 11, has been contrived.

This is composed of an external hollow wooden cylinder, 7½ feet long and 5 feet in external diameter, strongly bound by iron hoops, B, and connected by four wooden suspending rods with a wooden axis, D, in which are firmly fixed knife-edges, F. The receiver, AA, within is made of double plate-iron, and its interior diameter is 4 feet 4½ inches. The bottom, H, is closed solidly with wood; and in front, at K, is an advanced bottom formed of ¾-inch board, the interval between the two, a length of 4½ feet, being filled with sand, introduced by openings from above, subsequently closed by the iron doors, I, I. The whole receiver, when filled with sand, weighed 14,000 lbs., the sand itself weighing more than the one half. Sir H. Douglas mentions that the Chevalier d'Antoni carried on similar experiments at Turin in 1761, though with a different kind of machine; and that Major Mordecai, of the United States artillery, made experiments, under the authority of the government of that country, with an apparatus constructed on the French principle—the suspended gun being either a 24 or a 32-pounder, weighing, with the suspending apparatus, 10,500 lbs., and the receiver pendulum 9358 lbs.; so that it may be now said with some degree of vexation, though with truth, that the only school of artillery without the means of carrying on ballistic experiments, namely, the gun pendulum and the receiver pen-

dulum, is that of England, the country of Robins and Hutton. It is surely time that we should awaken from this sleep, and cease to be satisfied with the results obtained, or the honours acquired by these illustrious men.

In respect to velocities below 300 feet per second, it was found difficult to test them by the ordinary form of ballistic pendulum, as the balls dropped from it without penetrating. This defect might have been avoided by arrangements similar to those of the receiver form of pendulum; but in the meantime Robins' whirling machine was applied to this object. Robins' own experiments have already been alluded to. In 1763 were published experiments made in France by Borda, with a machine similar in principle to that of Robins, though differently arranged (fig. 12). In this, a horizontal axis, AB, carries a small cylinder, round which winds a

cord, at the extremity of which is a weight, E, which gives motion to the fly. A thin rod, PGK, wedge-shaped, forms the two arms of the fly—two equal surfaces, P and K, being adapted to their extremities, the centre of which is about 4 feet from the axis of rotation. On the string two marks, distinctly visible, are fixed, and by the aid of a half-seconds pendulum the interval of time between the passage of these marks was measured to a ¼th of a second; and the number of turns corresponding to this length on the cord being known, it was easy to compute the velocity of rotation, and, consequently, the absolute velocity of the centres of the resisting surfaces themselves—care having been taken that the velocity had become uniform at the fifth turn, or just at the passage of the first mark—and the moving weight, therefore, exactly counterbalancing the resistance of the air during the 22 turns corresponding to the interval between the two marks; the proportions between the radii of the cylinder and of the fly being taken into account. By employing different weights, the resistances corresponding to different velocities are obtained.

Hutton made very extensive experiments with the apparatus of Robins; and in like manner M. Thibault, of the French Marine, has made experiments (1826) with one similar to that of Borda.

II.—APPLIED GUNNERY.

A consideration of what has been said respecting the experiments of Robins, Hutton, and many others, must have been sufficient to convince every one that the application of the theory of gunnery, or of any theory of the motion of bodies in a fluid, to the practical purposes of war, so essentially depends on the knowledge of elements very difficult

Gunnery. of determination—such as the initial expansive force of the gases produced by the combustion of gunpowder, and the complicated resisting force of the air—that it has been found absolutely necessary to determine these elements, or the proximate results depending on them, such as the velocity of the projectile, by direct experiments. The following deductions have been made from experiments, and may therefore be considered as principles of the science:—

1. Two pieces of the same bore, but of different lengths, being fired with the same charge of powder, the longer will impel the bullet with a greater velocity than the shorter; but the increase in velocity is very small in comparison to the increase in length—the velocities being in a ratio somewhat less than that of the square roots of the length of the bore, but greater than that of the cube roots of the same, nearly, indeed, in the middle ratio between the two.

2. The range increases in a much lower ratio than the velocity, the gun and elevation being the same; and comparing this with the proportion of the velocity and length of gun in (1.), it is evident how little is gained by a great increase in the length of the gun with the same charge of powder. In fact, the range is nearly as the 5th root of the length of the bore, so that the increase amounts only to about a 7th part more range for a double length of gun.

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 velocity.

4. It is easy to perceive that the velocity will not be increased for any given gun, by increasing the charge beyond a certain degree; because when the barrel is almost full of powder the ball quits the piece before the charge has given it the full velocity; and on the other hand when the charge is very small, it is too weak to give the ball a sufficient impulse. Hence it follows, that in every gun there is a charge which will give the greatest velocity to the ball, and that by either increasing or diminishing it, the motion of the ball will be diminished. Dr Hutton further gives the following results of his theoretical and experimental investigations:—

Table of Charges for the greater Velocities.

Now, then, in the 32-pounder of 56 cwt., the bore of which is about 17 calibres in length, the charge would occupy somewhat less than $\backslash frac\{1\}\{4\}$th of the length of the bore, and would weigh nearly $\backslash frac\{1\}\{4\}$th of the weight of the ball, the extreme velocity being about 2080 feet per second.

5. The resistance of the air acts upon projectiles in a twofold manner; for it opposes their motion, and thus continually diminishes their velocity; and diverts them from the regular track they would otherwise follow, under certain conditions of the form and motion of the projectiles, which will be hereafter discussed.

6. That action of the air by which it retards the motion of projectiles, though formerly 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.

7. This retarding force of the air acts with different degrees of violence, according as the projectile moves with a greater or less velocity; and within moderate limits as to velocity, 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; or as the squares of the velocities.

8. The exponent of the power of the velocity expressing the resistance gradually increases as the velocity increases; and when the shot moves at the rate of 1400 or 1500 feet per second, at which rate a perfect vacuum will be found behind the ball, as air is assumed to rush into a vacuum at a velocity of only from 1200 to 1300 feet per second, that exponent attains a maximum, being then 2.125; beyond such velocity the exponent decreases.

9. 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. It will be seen presently that the deviation is explained on a different principle.

10. 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 merlons, 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.

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

12. Hence the proper charge of any piece of artillery is not always that allotment of powder which will communicate the greatest velocity to the bullet; nor is it to be determined by an invariable proportion of its weight to the weight of the ball; but, on the contrary, it is such a quantity of powder as will produce the fitting 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.

13. Robins, following out these rules, states that no field-piece ought at any time to be loaded with more than $\backslash frac\{1\}\{2\}$th, or, at the utmost, $\backslash frac\{2\}\{3\}$th of the weight of its bullet in powder, nor should the charge of any battering piece exceed $\backslash frac\{1\}\{4\}$th of the weight of its bullet; but these proportions are not adhered to strictly in modern practice, light field-pieces being, however, fired with so small a charge as $\backslash frac\{1\}\{4\}$th.

14. If balls have equal weights but different diameters, and move with equal velocities, the resistance varies nearly

gunnery. with the surfaces or with the squares of the diameters, increasing a little above that proportion when the diameters are considerable. Hence, if the velocities are also different, the resistance is proportional to the surface and to the square of the velocity; or $r$ representing the semi-diameter of the shot, and $v$ the velocity, the resistance varies with $r^2\; v^2$.

15. If balls have equal diameters and different weights or densities, the resistances vary directly as the squares of the velocities, and inversely as the weights; or, $w$ representing the weight, the resistance varies with $\backslash frac\{v^2\}\{w\}$.

Great 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 cut 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.

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 $\backslash frac\{1\}\{20\}$th 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.

16. Deviations.—It has been shown that Robins pointed out the deviations which are occasioned in the course of

projectiles by a rotation in them, produced by any accidental cause, on an axis not coincident with the line of flight; and in the last two paragraphs reference has been made to the artificial means adopted by rifling, or grooving, the bores of the barrel of muskets, to give to the projectile an initial rotatory movement round an axis coincident with the line of flight, and thereby to avoid such extraneous cause of deflection. Robins made many experiments with rifles, to show that the accuracy of fire at great distances thus obtained is their principal advantage, and that they have not an advantage over the unrifled barrel as regards either extent of range or penetrative power; the increased force supposed to have been gained by more completely shutting in, as it were, the elastic gases, and thus avoiding the loss by windage, being counterbalanced by the great friction of the ball in its passage through the bore. Robins also suggested the use of an ovoid, or egg-shaped ball; but in this respect the indications of Newton as to the form of the solid body which, in passing through a fluid, would experience less resistance than a body of equal magnitude and of any other form, have gradually led to the adoption of the modern elongated balls. Sir Howard Douglas has given a very clear view of the subject, and his remarks will be here quoted:—

"The body is a solid of revolution, and the differential equation is $y\; =\; c\; \backslash frac\{dz\}\{dy\}\; dx$ in which $c$ is a constant, $y$ is any ordinate, and $dx,\; dy,\; dz$, are elementary portions, EF, ED, DF, respectively, in the sectional figure.

"The two ends A and B of the solid are both plane surfaces, as $DF\; =\; dz$ is greater than $EF$ or $ED\; =\; dx,\; dy$, and therefore the numerator of the fraction $\backslash frac\{N\}\{H\}$ must always be greater than

the denominator, and $y$ cannot become 0, or coincide with H. That the minimum of resistance should be obtained from an elongated shot of this, or any form approximating to it, it is necessary that the axis AB should always be kept in the direction of the trajectory, an object which is accomplished by producing a rotatory motion round that axis, the ball being discharged from a rifled bore. Were it not that such a rotation were produced, the axis would perpetually deviate from the direction of the path and even turn over."

The advantages of this form of shot are, that when rotatory on their longitudinal axis, and moving with their smaller extremities in front, they experience less resistance from the air than spherical projectiles of the same diameter. To this form alone are to be referred the long range, with the great momentum and penetrating power of the projectiles for rifle muskets, which have been recently introduced into British and foreign military services. Sir Howard Douglas describes also the cylindro-conical and cylindro-conoidal balls used in the iron-rifled guns, invented in 1846 by Major Cavalli of the Sardinian artillery, and Baron Wahrendorff, a Swedish noble. The entire lengths of these projectiles were—of the cylindro-conical 16$\backslash frac\{1\}\{2\}$ inches, and of the cylindro-conoidal 14$\backslash frac\{1\}\{2\}$ inches, their greatest diameter being 6$\backslash frac\{1\}\{2\}$ inches. They were made to act upon the grooves of the bore by two projections, one on each side, making each an angle with the axis of the shot of 7° 8'. If hollow, the weight was about 62 lbs., and if solid about 101$\backslash frac\{1\}\{2\}$ lbs.—the hollow projectiles being burst on the principle of a percussion shell. These guns are loaded at the breech. The difficulty of forming a convenient and efficient rifled cannon has indeed been so great, that it may be considered to remain as yet an unsolved problem in the ordinary way; but Mr Lancaster has adopted a different principle, and patented an ingenious invention for causing a shot to rotate

Gunnery. on its axis throughout the range, by firing it from a cannon having an elliptical bore of small excentricity. The 68-pounder gun has been bored up to the elliptic section, and the shell, as shown in Plate II, fig. 18, adopted—the transverse section of the shell being elliptical. It is well known that several of Lancaster's guns were in operation during the last remarkable campaign, though no official account of the result of their practice has yet been published; but that at Shoebury-Ness is considered on the whole satisfactory, both as to extent of range and precision in firing, more especially at the longer ranges, since the 68-pounder, one of the best guns in the service, maintains an equality in short and medium ranges. The major axis of the ellipse of the bore does not continue parallel to one fixed line, but makes a revolution of about $\backslash frac\{1\}\{4\}$th of the periphery in its length, and the elliptic shot, following necessarily the course of this helix, is caused to rotate on its long axis. Much precision is required in putting the ball into the gun, but this is attained by a very simple machine lately adopted by Mr Lancaster for the purpose, and which does away with any chance of mistake.

Turning again from these modes of correcting irregular rotation, by inducing a regular one in the direction of the line of the trajectory, to the consideration of the deviations produced by any irregular rotation, it may be first premised, that not only vertical but also lateral deflections must be the result of such rotations. Irregular rotations may be produced, as suggested by Robins, by a rolling motion in the bore; or this may be the result of unequal density in the different portions of the ball, causing the centres of gravity and of figure not to coincide. Thus, for example,

if the centre of gravity, $g$, be above the centre of figure, $C$, in A of the four circular sections, the resultant of the projecting forces will cause the front of the shot to turn from below upwards; if below, as in B, from above downwards; if to the right, as in C, from the left to the right; and if to the left, as in D, from the right to the left. In all these cases the deflection will be in the direction of the rotation, namely, it will be upwards in A, producing an increase of range, downwards in B, or diminishing the range, to the right in C, and to the left in D.

These theoretical deductions have been fully confirmed by experiments, both in England and France.

At the instance of Sir Howard Douglas, experiments were made both by the navy on board the "Excellent" gunnery ship, and by the ordnance at Shoebury Ness, with a view to ascertain not only the nature of these deflections, but also the practicability of making a useful application of the principle of eccentric projectiles as a means of increasing the range:—

EXCELLENT, July 18, 1850.—With a 32-pounder gun of 56 cwt.—the quantity of metal removed from one side of the shot being 1 lb.

In these the deflections are also given in yards, being the mean of all the rounds under similar circumstances; hence, of course, where some were deflected to the right and others to the left, the mean of each set is given. It will be observed that with the lesser charge and elevation a difference of 453 yards is produced in the range by shifting the centre of gravity from above to below the centre of figure, and with the higher charge and elevation, a difference of 492 yards. The lateral deflections, however, are by no means so considerable, amounting in the one case to $6\; +\; 7\; =\; 13$ yards, and in the other to $20\; +\; 24\backslash frac\{1\}\{2\}\; =\; 44\backslash frac\{1\}\{2\}$ yards. With an 8 inch-gun, 9 feet long, and weighing 65 cwt., the quantity of metal removed from one side of the shot being 5 lbs. 5 oz., the difference between the ranges, according as the centre of gravity was placed above or below the centre of figure, was 189 yards when fired with a charge of 10 lbs., and elevation of $2^\{\backslash circ\}\; 30\text{'}$, and 378 with the same charge and an elevation of $5^\{\backslash circ\}$. With a 32-pounder, charge 8 lbs., and elevation $12^\{\backslash circ\}$, 1 lb. of metal being removed from one side, the difference 938 yards; and with another 8-inch gun, the quantity of metal removed from one side being 3 lbs., the charge 10 lbs., and elevation $10^\{\backslash circ\}$, the difference was 1132 yards. The experiments of Shoebury Ness were similar in their results,—the eccentric hollow shot in those of 1851 giving an increase of range, as compared with concentric shot, varying, according to the absolute range and elevation, from 145 yards in a range of 1700 yards, and an elevation of $4^\{\backslash circ\}$, to 559 in a range of 2465 yards, and $8^\{\backslash circ\}$ of elevation; 621 in a range of 3184 yards, and $12^\{\backslash circ\}$ of elevation; 749 in a range of 3709 yards, and $16^\{\backslash circ\}$ of elevation; 939 in 4137 yards, and $20^\{\backslash circ\}$ of elevation; 706 in 4605 yards, and $24^\{\backslash circ\}$ of elevation; 916 in 4650 yards, and $28^\{\backslash circ\}$ of elevation; and 670 in 4866 yards, and $32^\{\backslash circ\}$ of elevation—the ranges stated being those of the concentric shot, and the range of the eccentric being obtained by adding to the other the differences here stated—the highest range being that of the eccentric shot with $28^\{\backslash circ\}$ of elevation, which amounted to 5566 yards, or about $3\backslash frac\{1\}\{2\}$ miles. The lateral deflections in the long ranges were very great—the extreme to the left being 361 yards, and the extreme to the right 255 yards; but it is remarkable that some of the deflections of the concentric shot were quite equal, and some even exceeded those of the eccentric shot.

Captain Boxer, in his Treatise on Artillery, gives more copious details of the practice at Shoebury Ness, but the above extracts are sufficient for the present purpose. Similar experiments were made at Metz in 1839, with common and eccentric shells of 11 inches diameter, and the results are given by Didion in the following table:—

All these experiments lead to the same conclusions, that when applied in such cases as the above on accurate principles for the express purpose of obtaining an increase of range, the eccentric shot, and more especially shells, may be sometimes used with advantage, but that, as a general rule, the great practical difficulties of ensuring certainty in their use in the field must negative their application;

Gunnery. and that, on the contrary, the surest way of obtaining correct practice is to take care that the shot and shell shall be correctly concentric, and thus extraneous causes of rotation and of uncertain deflections be avoided as much as possible, the direction of that deflection being always the same as that of the rotation of the front of the ball.

It has been already observed that Robins was the first to attribute the usual deviation from the true path of a projectile, to the disturbing influence of the movement of rotation which in general accompanies the movement of translation. Robins and Lombard considered that this deflection was principally the result of the friction of the surface of the ball against the layer of air of unequal density adjacent to it. Poisson investigated the effect of this rotation, in combination with a movement of translation, considering every point of the surface as subjected to a resistance; one portion of which is normal, being the resistance of the fluid or air, properly so called, and the other tangential, being due to friction. When the bullet is perfectly spherical and homogeneous, and on leaving the bore rotates round one of its diameters, the rotatory motion continues during the whole flight in the same direction and round the same axis which remains constantly parallel to itself. The velocity of rotation diminishes in the inverse ratio of the product of the diameter by the density of the ball, but by an extremely small quantity. The deviation, either vertically or laterally, so far as it is due to simple friction, must necessarily be in an opposite direction to that of rotation, as regards the anterior hemisphere; but as this theoretical deduction is opposed to the results of experience, Didion states that the friction arising from rotation does not account either for the direction or amount of deflection. Didion explains the deflection in this way,—assuming the axis of rotation to be vertical, and the moving of the anterior hemisphere from right to left, he says that all the points situated on the right hemisphere move by rotation in the same direction as the centre of gravity moves by translation; whilst those on the left hemisphere move in an opposite direction to that of the centre of gravity, and hence, that the first have a greater velocity in respect to the air than the last, and as the displacement of the air is consequently less easy, the density of the fluid and the pressure resulting from it, are greater on the right than on the left hemisphere.

This explanation appears to imply that the rotation is supposed to add to the normal resistance, whereas the force exercised at any point of the surface, as a result of rotation, can only be tangential. Captain Boxer's explanation, which indeed is also that of Professor Magnus, appears, therefore, to be the correct one, in which he deduces the increased resistance of the air on the right hemisphere as a result of friction;—for example, the air meeting the front of the ball, tends to rush past it both to the right and left, but, pressing with great force against the ball, it is resisted in its passage on the right by the friction consequent on rotation, which is here opposed to the motion of the air, whilst it is assisted in its passage to the left by the friction, the motion being then in the same direction. Of course, the amount of deflection must depend both on the velocity of translation and the velocity of rotation; and hence, when the former becomes 0—the centre of gravity continuing at rest, and the pressure on the anterior and posterior hemispheres being then the same—there can be no deflecting effect produced by rotation. In all these explanations, however, it appears to have been overlooked that, supposing a ball to acquire a motion of rotation within the bore by friction, such friction is only a fraction of the pressure exerted upon the ball in its passage, and, consequently, that the ball must receive a primary impulse of deflection the moment the centre of gravity of the ball leaves the muzzle of the gun. To remedy these evils by rifling of any description, it is necessary that the projectile should be caused to rotate

either on the greater or less axis of inertia, as these are the only axes which can remain permanent in space, and that this axis should be coincident with the line of flight. It is very difficult to ensure an exact coincidence of the centre of gravity with the axis of figure corresponding apparently with one or other of these axes; and hence it is that the centre of gravity, moving spirally round the axis on its passage through the rifled bore, receives a certain deflecting influence corresponding to the direction of the spiral groove as it leaves the muzzle—such deflection, however, being necessarily constant as to direction. The experiments of Robins and Hutton had, as before explained, for their object to determine the two most important elements of gunnery, namely, the velocity of the ball on leaving the gun, and the resistance to its progress afforded by the air. Professor Robinson basing his investigation on the deductions made from these experiments, such as that the velocities communicated to balls of the same weight being nearly as the square roots of the weights of the charges; with shot of different weights, fired with the same quantity of powder, the velocities are reciprocally as the square roots of the weights, and when the weights of shot and powder are both different, the velocities are directly as the square roots of the powder, and inversely as the weights of the shot nearly,—proceeds to the important question of determining the ranges, the velocities being known, and lays down as preliminaries the following rules of Robins, the second of which, as has been shown, cannot be admitted; for although an exact duplicate ratio does not represent the ratio of increase in high velocities, there is no such sudden increase as supposed by Robins above the velocity of 1100 feet per second.

1. "Till the velocity of the projectile surpasses that of 1100 feet 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 25 or 26 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 1", 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 thereby 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 range 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.

"If the velocity with which a projectile begins to move is known, its potential range 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 range, 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 cer-

Gunnery. tain quantity, which I 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 bullet 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.

PROB. 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 5 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 954 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 solved 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, or at least much greater, resistance described above;

and consequently the real potential random will be greater than what is here determined. However, in this case, 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, 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 1.47, 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.

PROB. 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 2/3rds of the angle of elevation. Then, by means of the pre-

Gunnery. coding 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 12½ inches diameter, and of 231 lbs. 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 3525 yards; but as the shell is only ¼ths 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 ¼ths of the angle of elevation, becomes 2544. The quote of the potential range by this diminished F is 1:354; 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 5900 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 ¼th 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 ¼ths 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 ¼ths 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 9350 yards; which, divided by the reduced F of the last example, gives for a quote 29.75; 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 for so long a time was treated as too insignificant a power to be attended to in laying down the theory of projectiles.

Schol. I must here observe that as the density of the atmosphere perpetually varies, increasing and diminishing often by ¼th 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 ¼th 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 30°. Indeed, if experiments are made with extraordinary quantities of 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.

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 $q$, the logarithm of 10 being supposed unity; then the actual range sought is $3.4\; F\; \backslash cdot\; 2^\{2l\}\; -\; \backslash frac\{11\}\{10\}\; F$, where the

double sign of $2^\{2l\}$ is to be thus understood; that if $q$ be less than 25 it must be $-2^\{2l\}$; if it be greater, then it must be $+2^\{2l\}$. 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:—

Whence it appears, that, when the potential random is increased from 3 F to 2500 F, the actual range is only increased from 1½ 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 5½ F, which is not its ¼th 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 was once considered 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 may be assigned.

These remarkable tables and rules of Professor Robinson do not appear to have been noticed by foreign writers. Hutton makes use of them with some modification in the following manner. First, he determined the greatest terminal velocity which a ball would acquire in falling through the air as exhibited in this table:—

And these form a table for determining the elevation which will give the greatest range, the initial velocity being given, and the size and nature of the shot known; which table, he states, is a modification of that of Professor Robinson, founded on an approximation of Sir Isaac Newton.

To use these tables, divide the initial velocity by the terminal velocity peculiar to the ball, as given in the third column of the first table, and look for the quotient in the first column of the second table, against which, in the second column, is found the elevation which will give the greatest range; and in the third, a number which, being multiplied by $a$ of the first table, gives the range nearly as exhibited by the following example:—

Let a 24-lb. ball be discharged with a velocity of 1640 feet per second. By the first table, the terminal velocity $v$ of a 24-lb. ball is 419, and the altitude producing this terminal velocity, or $a$, 2729, hence $\backslash frac\{1640\}\{419\}\; =\; 3.92$ nearly. Now opposite 3.9865 in the second table stands $34^\{\backslash circ\}\; 15\text{'}$ as the angle which would give the greatest range, and the corresponding number in the third column, 3.0549, being multiplied by 2729 gives 8336 feet for the greatest range, being rather more than a mile and a half. As it is not customary nor ordinarily practicable to discharge guns at these elevations, these tables can seldom be applied in service; though in the recent campaign it might have been useful to consult them when guns either damaged, or otherwise useless for their ordinary practice, were discharged with great effect at high elevations by either sinking them partially in the earth or by suspending them.

Dr Hutton therefore computed the following table for shells—making allowance for the difference in the terminal velocity consequent on the difference between the specific gravity of the filled shell and the corresponding solid iron ball—to replace the first of the former tables:—

Using this table, then, instead of the former, and assuming, in the case of a 13-inch shell, that it is projected with a velocity of 2000 feet per second, which is about a maximum, we find in the seventh column, opposite the 13-inch shell, 534; hence $\backslash frac\{2000\}\{534\}\; =\; 3.746$; and opposite 3.7330 in

the other table will be found $35^\{\backslash circ\}$ as the angle which gives the greatest range, and 2.8515 in the third column which, multiplied by 4430, the altitude opposite the 13-inch shell, gives 12,632 feet, or almost $2\backslash frac\{1\}\{2\}$ miles, for the greatest range. The French, however, fired shells at the siege of Cadiz a distance of more than 3 miles, the cavity of the shell being filled up with lead, and Hutton therefore investigated what would be the range of a 13-inch shell if so filled, and found that the range would be 16,005 feet, or 3 miles and 165 feet, the corresponding angle being $37^\{\backslash circ\}\; 20\text{'}$. Such projectiles, however, were more formidable in appearance than in reality, as they seldom burst, and when they did the explosion was inconsiderable. Dr Hutton makes also another very important reference to the discoveries or reasonings of Professor Robinson, who first pointed out that "balls of equal density, discharged at the same elevation, with velocities proportional to the square roots of these diameters, will describe similar curves; because then the resistance will be in proportion to the momenta or quantities of motion,"—thus $v$ being as $\backslash sqrt\{d\}$, $v^2$ will be as $d$; consequently $d^2\; v^2$ will be as $d^4$, but the resistance $\backslash gamma$ is nearly as $d^2\; v^2$, $d$ being the diameter and $v$ the velocity, and hence $\backslash gamma$ is as $d^3$, or as the momentum which is as the magnitude of the mass, or as $d^3$. In this case, then, the horizontal velocity at the vertex, opposite the curve, will be proportional to the terminal velocity; and the ranges, heights, and all other similar lines will be proportional to $d$—a principle which may be of considerable use; for by means of a proper series of experiments on one ball, projected with different velocities and elevations, tables may be constructed by which may be ascertained the motions in all similar cases.

Dr Hutton gives the following table, deduced partly from theory and partly from experiment, which may be applied on the above principles:—

Gunnery. Now, bearing in mind Professor Robinson's proposition if it be required to find the dimensions of the path described by a 12-lb. shot, discharged with a velocity of 1600 feet per second, and at an elevation of 45°. Here, as the curves are similar, and their corresponding lines proportional to the diameters of the shot, when discharged with velocities proportional to the square roots of the diameters, the velocity of a 24-lb. ball, corresponding to the 1600 feet of a 12-lb. ball, is first found in the table. Then, as the diameters of the two balls are 4.403, 5.546, the proportion will be $\backslash sqrt\{4.403\}\; :\; \backslash sqrt\{5.546\}\; ::\; 1600\; :\; 1796$, the corresponding velocity of the 24-lb. ball, opposite to which in the table are the corresponding ranges and height, 2386 and 692; therefore, as 5.546 : 4.403 :: 2386 : 1894 yards the range, and 5.546 : 4.403 :: 692 : 549 yards the altitude. In like manner, the table may be used for determining the ranges of mortar shells; as, for example, a 13-inch shell, projected with the velocity of 2000 feet per second, at 45° elevation. First, the diameter of the shell being 12.8, the proportion is $\backslash sqrt\{12.8\}\; :\; \backslash sqrt\{5.546\}\; ::\; 2000\; :\; 1317$; but as the weight of the shell, filled in the ordinary way, is less than that of a corresponding solid shot, this velocity must be reduced in the same manner as was done in page 132, with reference to a previous table; namely, as 178 : 149.4 :: 1317 : 1105, the corresponding velocity of the 24-pounder, to which in the table answers the range 1930; and finally, as 5.546 : 12.8 :: 1930 : 4455 yards, or about 2½ miles.

The difficulty of establishing fixed rules for the ranges of projectiles, an object of great practical importance, will be

appreciated from what has been stated; a difficulty, indeed, due not only to the mathematical perplexity of the question as regards the determination of the trajectory of the projectile in a resisting medium like the air—a question which has exercised the ingenuity of the most profound mathematicians, including Euler, Borda, Poisson, Legendre, and many others—but also to the uncertainty of the law of the air's resistance in relation to the velocity of the projectile. By the experiments of Robins, Hutton, and others, with the ballistic pendulum, the velocity close to the gun's mouth, and at short distances from it up to about 250 feet, was determined, and thence the loss from the resistance of the air in passing through such small spaces deduced; but experiments have not as yet been multiplied sufficiently to express the terms of resistance in formulae which will meet the circumstances of all projectiles. The resistances, too, at low velocities have been generally determined by the whirling machine, which does not strictly represent the circumstances of rectilinear motion; in the latter the resistances vary, with equal velocities, as the surfaces; whereas, in the circular motion, the co-efficient of resistance requires to be increased as the surfaces increase. From the present accuracy of fire of rifled muskets much knowledge might be acquired of the resistances to a small ball, by firing at a properly constructed pendulum, at different distances up to 800 yards; and though it cannot be expected that the precision of fire of larger projectiles will ever allow this distant use of the ballistic pendulum, still the comparison between the loss of velocity in great and small bodies might be founded on much better data than at present.

TABLES OF ORDNANCE.

Length, Weight, &c. of Iron and Brass Mortars and of Carronades.

Several ingenious methods have indeed, from time to time, been proposed to measure the velocity by determining the time of the ball's transit through a definite space; thus, for example, the revolving machine of Grobert, which consists of two circular discs of cardboard about 6½ feet in diameter, connected together by an axis 13 feet long, to which a rotatory motion is given by means of a band passing over a pulley on the centre of the axis, and connected with a proper combination of wheels to produce the necessary velocity of rotation. The two discs are divided into 360° each by radiating lines, and the axis being placed in the prolongation of the line of flight, the ball passes through both discs whilst rotating with a considerable and uniform velocity; and the angle between the two perforations, as shown by the lines, affords the means of determining the velocity, thus:—let $\backslash phi$ be the angle, $t$ the time of a revolution, $\backslash frac\{\backslash phi\}\{360\}$ the time of passing the distance $d$ between the two discs, and the velocity $v\; =\; \backslash frac\{360\; d\}\{\backslash phi\; t\}$. Now if $t$ be $\backslash frac\{1\}\{10\}$th of a second, or the machine revolves ten times in a second, and $v$ be assumed = 1200 feet, $\backslash phi$ would be 39°. Don Tomas de Morla had previously proposed a wheel or cylinder, rotating on the top of a vertical axis, the ball being so discharged as to pass as nearly as possible through a diameter of the cylinder when the angle between the two opposite perforations afforded the means of determining the time of passage and the velocity; but neither of these methods are sufficiently exact for the purpose. Professor Wheatstone, some years ago, proposed the electro-magnetic erograph to the artillery committee, but his proposition was not then favourably entertained, though it is now again under consideration. The mode in which it has been tried on the continent is as follows:—Two screens or targets of metallic wire twisted continuously into a net-work, which the ball cannot pass through without breaking, are set up, the wires being of course coated so as to prevent metallic contact between the parts of the net: each screen, by means of a connecting wire, forms part of the conducting circuit between a voltaic battery and an electro-magnet, which suspends a marker or pencil over a cylinder made to revolve uniformly with the necessary velocity. Now, when the ball passes through the first "target net" the circuit is broken, and the pencil instantly falls upon the cylinder, and makes a recording mark upon it. When the ball passes through the second "net" the connection of the second electro-magnet is in like manner broken, and by the fall of its pencil a similar mark, though on a different point, is made upon the cylinder. The time of revolution of the cylinder being known, the angle between these two marks, as in Grobert's and Morla's machines, determines the time of the ball's passage between the two targets, and hence the velocity of the ball. Though it is said this system has not as yet been perfectly successful, it may now be reasonably expected, from the increased knowledge of electro-magnetic phenomena, and the improvement of electro-magnetic apparatus, that it will be speedily rendered so.

In reference to the preceding tables of guns, it is necessary to consider the very important questions connected with their construction. In the first place, it is evident that there should be some relation between the expansive forces of the gases developed by the ignition of the charge of gunpowder, and the tenacity of the metal cylinder or gun which has to resist them; and here again it is necessary to know whether the gases act simply by pressure or by shock. That the action of the gases at the first moment of their development is analogous to a shock, or percussive force, is the general belief at present; and indeed, when it is considered that all the particles of matter constituting the retaining tube or gun are in a state of rest, or in a passive condition, when, by the almost instantaneous development of the gases, they are suddenly exposed to the action of a repulsive force capable of impressing on the particles of the gases a velocity of 7000 feet per second, it can scarcely be doubted that they must be affected very much in the same way as they

would have been if exposed to the action of a percussive force; though, of course, when once the shock had been transmitted through the solid substance, and the whole of its resisting forces had been called into action, the gases, if brought into a state of rest, would act by simple pressure. Timmehmans charged a musket barrel with 1.2 oz. of powder, and then in successive discharges increased the resistance each time by adding 1, 2, 3, &c., balls, until the barrel burst, when it was found that whenever the resistance exceeded a certain limit all the barrels swelled out in a narrow annular space, forming a defined border somewhat in front of the end of the charge. Now, it seems certain that if the gases had acted by pressure only, the whole of the bounding cylinder of the bore behind the charge should have swollen out equally, and not have been restricted in the extension of the calibre to the space of a simple ring; from which it was inferred that the gases, acting by shock against the ball, were thrown back by its resistance against the narrow space described. In like manner, Timmehmans reasons from Swedish experiments made in

Gunnery. 1831, in which it appeared that iron guns in which the end of the bore was hemispherical, afforded much more resistance than those ending in a plane surface; that this was due to the difference in the shock consequent on the difference of form in the bodies exposed to the action of the fluids.

Admitting, however, as has been done, the analogy in the mode of action of an almost instantaneously developed repulsive force on the walls of the cylinder, which restrains the expansion of the gases which exercise that force, to that of a shock or percussion, the experiments cited do not appear sufficient to establish a perfect identity in such actions. Without doubt, in such a case, time is necessary to enable the resisting body to bring into play its tenacity, and it resists, therefore, at first by the simple cohesive force which binds its particles together, and in this consists the analogy between the action of gunpowder and a percussive force. In considering the first experiment, it must be remembered that at two positions of the bore the destructive action of the gunpowder may be considered greater than at any other—namely, at the section transverse to the axes and immediately in front of the bottom of the bore, as the gases are then acting in two directions perpendicular to each other, namely, against the bottom of the bore or in the direction of the axis, and against the walls of the bore or perpendicular to the axis, thus tending to drag apart the breech from the rest of the gun;—and again at a transverse section somewhat in front of the charge, as the friction of the ball tends in a minor degree to produce a similar effect, and still more so when several balls or other projectiles are used, so as to check the progressive motion of the mass, and to augment the dragging effect of the friction. In this latter case, as the expanding gases would be also checked in their progressive motion, an accumulation would for the instant take place immediately behind the balls, and thus produce an augmented pressure on the sectional ring corresponding to that position, the result being, as in the cited experiment, an annular or local expansion rather than a general one.

On the principle of pressure, however, it does not appear difficult to obtain a reasonable approximation to the required thickness of the containing tube, following the reasoning of Senderos. Let $R$ be the radius of the exterior of a tube or gun barrel, $r$ the radius of the interior, $\backslash pi$ the proportion between the diameter of a circle and the circumference; then the surface of pressure against the base of the tube acting to produce fracture, at the circumference $2\backslash pi\; r$, will be the area of that circle, or $\backslash pi\; r^2$. The surface of pressure, equal to the other, but acting longitudinally, will be a rectangle, one side of which is the diameter of the tube, or $2r$, and the other a portion of the length of the tube $l$, such as will make the two surfaces of pressure equal, or $\backslash pi\; r^2\; =\; 2rl$,

and hence $l\; =\; \backslash frac\{\backslash pi\; r^2\}\{2r\}$. Now, the surface of resistance in a

direction perpendicular to the axis will be represented by the area of the annular space represented by $\backslash pi(R^2\; -\; r^2)$, and the surface of resistance in a longitudinal direction, by the areas of the two surfaces, one at each end of the diameter corresponding to the two fractures along $l$, that is by $2l(R-r)$. Now, comparing these two together, the resistances being proportional to the surfaces of fracture, they will be as $\backslash pi(R^2\; -\; r^2)\; :\; 2l(R-r)$; or substituting the value of $l$, as $(R^2\; -\; r^2)\; :\; r(R-r)$, and making $R-r=d$ as $d(R+r)\; :\; dr\; =\; d(d+2r)\; :\; dr\; =\; d+2r\; :\; r$; which shows that the resistance to fracture in a transverse direction, or perpendicular to the axes, is always much greater than that in a longitudinal direction, being indeed four times as great when $d=2r$, or one calibre—not, however, making the comparison with a transverse section close to the bottom of the bore, for the reasons before stated.

The pressures and resistances may now be compared together, and for this comparison the tenacities of wrought iron and of the best bronze have been stated by Senderos as 4234 and 3872 atmospheres respectively, and the cohesive force of cast iron as 1358. According to the experiments of Navier, these numbers should be 4164, 2475, and 1307, and by others the cohesive strength of wrought and of cast iron have been stated at 4166 and 1266 atmospheres. With these data, equalizing the pressure tending to produce fracture in a longitudinal direction—that is, to force one half of the cylinder to separate from the other in two longitudinal fractures of the length $l$—to the force of cohesion resisting fracture, $P$ representing the pressure and $C$ the tenacity or cohesion—we have, as before explained, $2rlP\; =\; 2l(R-r)C$, or $rP\; =\; (R-r)C$, or $rP\; =\; dC$. But $C$ in English cast iron is 1266 and $P\; =\; 2230$; hence $r\; =\; 2230/1266\; =\; 1.76r$, or about $\backslash frac\{1\}\{3\}$ths of a calibre, which is the thickness adopted for shell guns, but $\backslash frac\{1\}\{3\}$ths less than the thickness of the heavy iron guns.

It must, however, be remembered that the cohesive forces stated ought not to be admitted in practice, as they are liable to be affected by heat and other causes; and hence, if only $\backslash frac\{1\}\{3\}d$ be deducted from the cohesive force, $d$ would become $2.6r$, or $1\backslash frac\{1\}\{3\}$ths calibre, which is greater than the maximum thickness adopted, or that at the breech. With bronze of 2475 tenacity, and reducing by one-third, the thickness would be about $\backslash frac\{1\}\{3\}$ths calibre, the adopted thickness being $\backslash frac\{1\}\{4\}$ths. With wrought iron, taking the English estimate, and deducting $\backslash frac\{1\}\{3\}d$, the thickness should be about $\backslash frac\{1\}\{6\}$ths of a calibre—the actual thickness of metal at the breech of an artillery carbine being very nearly that deduced from theory, or $\backslash frac\{1\}\{4\}$th of a calibre. It is not to be wondered that this great saving in thickness or in weight of metal has so often led to trials of wrought iron for guns, though the great difficulty of forging such large masses has hitherto checked the extension of its use; and further, it will be shown that a certain weight is absolutely necessary for a gun, and that the use of wrought iron, therefore, is only desirable when peculiar lightness or peculiar strength are indispensable.

General Construction of Iron and Brass Guns.

Form, Weight, and Length.—Having arrived at a tolerable knowledge of the thickness of metal necessary to resist the first shock of the gunpowder, it becomes compara-

Gunnery. tively easy to resolve most of the other elements of construction. By the law of Mariotte, for example, the pressure of gases varies directly with their densities, or inversely with the spaces they occupy; and hence the pressure, when the gases have expanded so as to occupy twice the space, will be one-half; or three times the space, one-third; and so on. This law appears to have been practically followed from an early period, as the external form has been made more or less conical, and generally indeed such as to constitute three truncated cones, corresponding to the first reinforce, second reinforce, and chase of our ordnance, though sometimes the two first have been united either into one cylinder or one truncated cone, which is an arrangement more consonant to theory. In considering the portion of the bore to which the greatest thickness of metal should be extended, it is necessary to know the position of the ball at the moment when the gases developed are in the highest state of condensation, and this has been assumed by many to be when the ball has moved forward a length equal to its own radius; and farther, to assume the greatest length of charge ever intended to be used with the gun, which therefore need not exceed either that of the proof charge, or of the charge which would produce the greatest initial velocity in a ball. In the 24-pounder the proof charge is $\backslash frac\{2\}\{3\}$ths of the weight of the ball, in the 32-pounder about $\backslash frac\{1\}\{2\}$ths, the 42-pounder about $\backslash frac\{2\}\{3\}$ths, the 56-pounder $\backslash frac\{1\}\{2\}$, the 68-pounder about $\backslash frac\{2\}\{3\}$ths, the 18-pounder $\backslash frac\{1\}\{2\}$ths, and the 12-pounder and smaller guns equal to the weight of the ball. In the 68-pounder the charge for producing the maximum velocity by the table of Hutton would be 43$\backslash frac\{1\}\{2\}$ lbs., whereas the proof charge may be taken at 28 lbs. On the former supposition, the length of the cylinders exposed to the maximum action of the gunpowder would be 27$\backslash frac\{1\}\{2\}$ + 4 inches = 31$\backslash frac\{1\}\{2\}$, or 2 feet 7$\backslash frac\{1\}\{2\}$ inches, which would extend about half-way from the termination of the first reinforce to the trunnion; and it may be remarked, in reference to this point, that the fractures on the bursting of a gun extend longitudinally, in somewhat irregular or curved lines, rather beyond the trunnions—the breech being partly broken off near the bottom of the bore on the one end, and the chase being separated from the other part of the gun on the other. With the length of the 68-pounder of 95 cwt., the whole volume of the bore, as compared to that of the maximum velocity charge, is not quite 4 to 1;—so that if 3 to 1 be adopted, the thickness at the muzzle ought to be $\backslash frac\{3\}\{4\}$ths of a calibre. If, then, without regard to ornament, the gun were constructed upon the simple principle of a cylinder extending from the extremity of the breech to the point on the second reinforce above intimated, and from that point to the extremity of the muzzle as a truncated cone, the 68-pounder of 15 calibres would weigh 103 cwt., being the mean between the two heavier 68-pounders, one of which is 112 cwt. and the other 95 cwt., whilst the ratio between the weight of the ball and the weight of the gun would be 1:170.

As the weight must therefore depend, first, on the thickness of metal necessary to resist the greatest expansive force of the gunpowder intended to be applied; and secondly, on the length of gun—an element in itself depending on the charge to be used, and the initial velocity to be given to the ball—it becomes necessary to determine the length to be adopted, which is generally stated in calibres as well as in feet and inches. In the early periods of artillery every founder seemed to adopt his own fancy, and hence the variety of guns was not only great as regards the calibre, but also in reference to the length of guns of the same calibre. Some of the guns were of either monstrous size or form. Luis Collado, mentioning a culverine mounted at Naples, and firing a ball of 48 lbs. weight, which was 47 calibres in length, and the double cannons founded by order of the Castellan of Milan, Don Alonso Pimentol, which were 130 calibres in length; and he mentions that in his time at Milan no less

than 200 varieties of ammunition were required for the service of the artillery of the castle alone. The following table of Venetian guns in the beginning of the sixteenth century is a further example of this apparently capricious multiplication of species:—

Table of the different Pieces of Artillery mentioned in the 11th Colloquy, 1st Book of Tartaglia.

This great variety of pieces of artillery was classed, according to Collado, by the early writers under three heads—1st, Those for long ranges; 2d, Those for battering; 3d, Those for throwing one or more stone balls. The first, of 32 calibres, comprised muskets, falconets, medium sacers and sacers, aspics, medium culverins, culverins, &c. The second, or battering cannon, are distinguished as quarter guns, medium guns, simple common guns, reinforced and bastard guns, the serpent, double guns, basilisks, &c. These varied in length from 18 to 28 calibres. The third genus included the ancient bombards, mortars, and brass guns, with detached chambers which were fixed to the guns after being charged, as well as the guns for firing stone balls. They were heavier than those of the preceding genera, but were proportionally shorter, not exceeding 8$\backslash frac\{1\}\{2\}$ calibres in length. This attempt at classification was not, however, adhered to by all the founders of guns, and hence it became absolutely necessary to take some step to lessen the confusion. At the beginning of the seventeenth century, therefore, Christobal Lechuga attempted to reduce the vast number of existing guns to six different calibres only, and shortly afterwards Don Juan Bayarte fixed the number, which were afterwards called guns of regular calibre or of ordnance. This great progressive step of the Spanish artillery was imitated by the other European Powers, by whom the species, dimensions, and calibre of the guns to be retained were established by regulations of state, and were hence called ordnance. Cannon may naturally be divided into two great classes—namely, guns for firing solid shot, and mortars for firing shells or hollow projectiles; but an intermediate genus, partaking of the characters of both, has been interposed, namely, the bowitzer, which is manipulated like a gun, but is used for the discharge of hollow projectiles. Lechuga adopted at first the calibres of 40, 24, and 12-pounders, calling the two latter medium and quarter guns, but afterwards added the 16, 8, and 2-pounders. The 40-pounder was rejected by Diego Ufano and Bayarte, and subse-

Gunnery. quently the calibres fixed by the French Ordinance of 1752 were adopted in Spain—namely, the 4, 8, 12, 16, and 24-pounders for land service, and the 6, 12, 18, 24, 32, and 36-pounders for the marine; but the 4-pounder was subsequently abandoned. Gribeauval was the great reformer in 1765 of the French artillery, when, for siege and fortress guns in brass, 24, 16, 12, 8, and 4-pounders were retained, and for field guns the 12, 8, and 4-pounders; but in the eleventh year of the Republic the list was reduced to the 24-pounder, short; 12-pounder, long and short; 6-pounder, long, short and light or field, mountain; 3-pounder; and 24-pounder howitzer; and, as is well known, the present Emperor of the French has put upon trial a still more simple arrangement, having invented a howitzer gun something like the Licorne of the Russians, and thus endeavoured to reduce calibres to a minimum. In our own service the list of guns shows that there is still a superfluity of species, or rather varieties, and that many must be considered merely experimental; as, for example, of the 68 and 42-pounder, and still more of the 32 and 24-pounder: but some of these have already been practically abandoned.

Not only does the powder, as it progressively burns, continue by successive shocks to act against the ball, but also the gases, as they expand, continue to act upon it, though with a regularly diminishing force. This variable but diminishing accelerating force is opposed to the resistance of the ball from friction in the bore, which is uniform, and by the resistance of the air, which is variable, but increasing as the velocity of the ball increases. Whenever, therefore, the sum of the retarding forces has become equal to that of the accelerating forces, it is evident that no further advantage can be obtained by the action of the gases, and hence that the limit of length has been attained. Capt. Boxer cites the experiments of Col. Armstrong (1736), who endeavoured to determine this question by the ranges obtained. He used a brass 24-pounder, 10 feet 6 inches long, which was shortened by 6 inches after each trial, and the mean ranges he obtained were—for the length of 10 feet 6 inches, 2502; for 10 feet, 2512$\backslash frac\{1\}\{2\}$; for 9 feet 6 inches, 2564$\backslash frac\{1\}\{2\}$; for 9 feet, 2617$\backslash frac\{1\}\{2\}$; for 8 feet 6 inches, 2514; and for 8 feet, 2453$\backslash frac\{1\}\{2\}$; from which it appeared that the greatest ranges were obtained from the gun of 9 feet, or 18$\backslash frac\{1\}\{2\}$ calibres. Dr Hutton's experiments, before noticed, with the ballistic pendulum, showed that, as regards the initial velocities, they continue to increase with the length of the bore, but in a much less proportion. By Piobert's first quoted series of experiments, or those made in Hanover in 1785, cited also by Capt. Boxer, it appeared that with a charge equal to half the shot's weight, no advantage in point of range was obtained by increasing the length beyond 18 to 24 calibres; and by his second series of 1801, beyond 18 or 19—though in the trials with a 6-pounder, the 15 calibre and 12 calibre guns were very little inferior to the 18 calibre. Capt. Boxer, whilst remarking on the uncertainty as to the absolutely best length of gun, as exhibited even in these extensive experiments, points to a curious law observed in examining these ranges by Mr J. F. Heather, M.A., one of the very able mathematical masters of the Royal Military Academy—namely, that the ranges obtained from guns of 12, 15, and 19 calibres are relative maxima, being greater than those of the guns of intermediate lengths; and he adds that the Hanoverian experiments point to another such maximum in guns of 23 calibres. Reflecting on the experiments of Hutton, and on the general theory of the accelerating and retarding forces already noticed, it seems impossible to connect such alternations with the action of the gases, and they can only be accounted for by some modifying cause, such as the zig-zag motion of the ball within the bore, causing it to range further, or vice versa, according as the last rebound may discharge it in a direction passing above or below the axis of the piece. Although, perhaps, not fully

conclusive, and still meriting further and very careful experiments, it may for the present be assumed that there is little reason for exceeding 19 calibres in length, or for going below 12, so far as the effective working of the gun, as manifested in its range, is concerned. The lesser number of calibres is best fitted for guns of large calibre, and the greater number for those of small calibre. Assuming then, as has been done, 15 calibres for the 68-pounder, and resolving the gun into two parts—a cylinder and a truncated cone—with a reinforce in rear of the charge, another at about half a calibre in front of it, and a third at the muzzle, a simple and apparently an effective gun would be obtained of moderate weight; and the same would be the result with other guns. In the particular case of the 68-pounder, as the proof charge is about $\backslash frac\{3\}\{4\}$ds of the charge which gives the greatest initial velocity, the length of the cylinder, or breech section, might be diminished, and the muzzle or chase, or conical section, increased, by which the weight would be diminished; but as a general rule, the other appears the most satisfactory, as the proof charge is usually $\backslash frac\{3\}\{4\}$ds of the weight of the ball.

Impossible to diminish the weight of the gun beyond a certain limit.—It would be wrong to pass from the important subject of the construction of guns without noticing the necessity of securing the carriage of the gun from too severe a shock; and this can only be done by bestowing upon the gun itself a considerable weight, or by interposing springs, which must be very difficult if not impossible in practice. This will be readily understood from the following considerations. Let $W$ be the weight of the gun, and $w$ that of the carriage, and $WV$ the momentum of the gun, on first receiving the shock, and before it has acted on the carriage; now, in hard bodies, when one in motion strikes another at rest, the whole quantity of the motion of the two bodies after the shock will equal that of the first before the contact; hence, taking $v$ as the common velocity of the gun and carriage after the shock, or velocity of recoil of the compound body, $(W+w)v\; =\; WV$. Further, the momentum of the gun on receiving the first shock of the powder cannot be altered by diminishing its weight, as the velocity would increase in the same proportion; and supposing $W\text{'}$ and $V\text{'}$ the new weight and velocity, $W\text{'}V\text{'}\; =\; WV$; and in like manner as $(W+w)$ would then be less than $(W+w)$, so $v\text{'}$, or the velocity of $(W\text{'}+w)$ would be greater than $v$, and in consequence $v\text{'}w$, or the momentum communicated to the carriage itself, greater than $wv$, or the motion communicated when the gun weighed $W$. Every diminution, therefore, in the weight of the gun increases the shock upon the carriage. And if to preserve the amount of recoil within a reasonable limit, and to increase the power of resistance of the carriage, its weight should be increased as much as that of the gun had been diminished, say by $u$, then the momentum of the gun and carriage would be the same as before; and the velocity, as in the first assumption, $v$. But this would be composed thus $(vW\; -\; uv)\; +\; (vW\; +\; uv)$; so that the carriage would have received the shock of the additional force represented by $ru$, and have suffered accordingly, as it is impossible to transfer the momentum of the gun to the gun and carriage without some portion of it being lost in destructive action upon the carriage, a portion which will necessarily increase as the amount of the force expended on the carriage is increased. The greater, therefore, the weight of the gun as compared to that of the carriage, the less will be the loss of force in the transmission of momentum, and of course the less injury done to the carriage. This being considered, it is evident that though by the use of wrought iron, the weight of the gun might be diminished by nearly $\backslash frac\{3\}\{4\}$ds, it would be impossible in ordinary guns so far to diminish the weight without the introduction of other means to protect the carriage from injury. In field guns, where the recoil is of less importance than mobility and durability, wrought iron seems the very best material which could be

Gunnery. used; and, in like manner, where the object is not a great range but a large calibre, as in flank guns, it might be advisable to replace such guns as short 24-pounders, by wrought-iron 68-pounders, or by cast-iron guns of large calibre but diminished thickness of metal, strengthened by wrought-iron hoops, as in the gun of Captain Blakeley, R.A. Hereafter, more may be expected; and a travelling 68-pounder constructed of a wrought-iron cylinder, sliding on a cast-iron bed, but checked in its motion by springs or buffers—the bed itself being supported on the carriage—the gun, or cylinder, being carried separately from the bed and carriage; guns of such large calibre might then enter into ordinary siege equipments. The monster gun of the Mersey Foundry is of wrought iron, but as its weight will be 24 tons 7 cwt., and the weight of its solid shot 300 lbs., it is rather a triumph of forging than an example of diminished weight, the proportion of the weight of gun and shot being 180 to 1. It is only further necessary briefly to state the proportions between the weights of the ball and of the gun, or between the weights of the charge of powder and of the gun, most generally adopted. In Spain the proportion between the weights of the gun and the charge in long guns is between 800 and 947 to 1; and in short, or field guns, about 480 to 1; or between the gun and ball from 234 to 313 to 1; and in short or field guns, about 142 to 1. In France the proportion in the naval 36-pounder is as 214 to 1; in the 30-pounder, 221 to 1; and in brass guns about 160 to 1. In the United States, from 299 to 1 in the iron 12-pounder, to 201 to 1 in the 42-pounder; but in their 12-feet and 10-feet columbiads, the heaviest of their ordnance, as they weigh 15,400 lbs. and 9240 lbs. respectively, the proportion is as 137 to 1. In brass guns it is as 147 to 1. In our own service, in the three forms of 68-pounder, the longest of which weighs 112 cwt., the proportions are 184, 166, and 143 to 1; in the 42-pounder, between 224 and 179 to 1; in the 32-pounder, between 223 and 140 to 1; in the 24-pounder, 233 and 154 to 1; in the 18-pounder, 261 and 124 to 1; omitting some of the very light and bored-up guns, which can only be considered exceptional cases. In the brass guns, the proportion in the light 6-pounder is 112 to 1; and in the 9-pounder and 12-pounder medium, 168 to 1. So that, taking the guns really effective for all purposes, the weights of brass and iron ordnance are nearly proportional to their respective tenacities. In the Spanish guns, however, the weights of the garrison and siege guns of bronze are as great in proportion to the ball as our iron guns—a striking illustration of the necessity, with the present system of carriage, of retaining a due weight for the gun. In addition to the bearing of the effective action of the powder, &c., on the length of the gun, as before explained, it is well to bear in mind that there is a minimum limit in respect to those guns intended to fire through embrasures, as it is necessary that the muzzle should enter at least 2 feet into the embrasure to prevent its rapid destruction by the concussion consequent upon the discharge. If, then, the trunnions be placed at $\backslash frac\{4\}\{5\}$ths of the whole length of the gun from the muzzle, and the radius of the wheel of the travelling carriage be 2 feet 6 inches, the minimum length is given by this simple equation, $\backslash frac\{4\}\{5\}$ths $x\; =\; 2$ feet 6 inches = 2; or $x\; =\; 7$ feet 10$\backslash frac\{1\}\{2\}$ inches. So that 8 feet may be taken as the minimum in this respect.

Preponderance.—If the axis of the trunnions of a gun were fixed in the line passing through its centre of gravity, great instability would be the result, and consequent uncertainty of fire, as the shock of the ball against the bore would be sufficient to disturb its equilibrium; and the ball would be liable to disturbance, even before firing, from many accidental causes, when the front of the gun could be so easily moved vertically. The weight of metal, therefore, behind the trunnions always exceeds that before, and this excess is called "preponderance." In the Spanish ord-

nance this excess in garrison and siege guns is $\backslash frac\{1\}\{2\}$th; in field guns, $\backslash frac\{1\}\{3\}$th; and $\backslash frac\{1\}\{4\}$th or $\backslash frac\{1\}\{5\}$th in howitzers, excepting the mountain howitzer, in which it is about $\backslash frac\{1\}\{6\}$th of the weight of the whole piece. Piobert gives the preponderance in the heavier guns as 8 or 9 times the weight of the projectile; in field guns as 12 to 13 times; and in howitzers as 6 to 7$\backslash frac\{1\}\{2\}$ times that weight. Timmermans deduces from general practice a preponderance varying from $\backslash frac\{1\}\{5\}$ths to $\backslash frac\{1\}\{3\}$ths of the weight of the gun. In the British artillery the preponderance is very various, being $\backslash frac\{1\}\{2\}$th, $\backslash frac\{1\}\{3\}$th, $\backslash frac\{1\}\{4\}$th, $\backslash frac\{1\}\{5\}$th, $\backslash frac\{1\}\{6\}$th, $\backslash frac\{1\}\{7\}$th, $\backslash frac\{1\}\{8\}$th, $\backslash frac\{1\}\{9\}$th, $\backslash frac\{1\}\{10\}$th, $\backslash frac\{1\}\{11\}$th, $\backslash frac\{1\}\{12\}$th—the result in some measure of the great number of pieces of the same calibre, though it is evident that it ought to be reduced in most cases, as excess of preponderance in heavy guns greatly increases the labour of adjustment. The position also of the axis of the trunnions is, as represented by the circles on the figures of Plate II., below the axis of the gun, a position which causes a rotation on the trunnion, and hence a pressure upon the elevating screw, which is useful also in securing steadiness.

MANAGEMENT OF GUNS.

Laying a gun includes two operations—pointing and elevating. By pointing is understood the placing it in such a position that the axis of the piece shall be exactly in the vertical plane passing through 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 thus cause the ball to 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 summit of 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 from one to two or more degrees, 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, the continuation of the line-of-metal will cross that of the axis of the piece, and the shot will be thrown to that side of the object on which the lowest trunnion is. As the axis of the gun would not be in the same horizontal plane as the object when the gun had been pointed by the line-of-metal, but elevated above it by the angle of dispart, a dispart sight is placed either on the muzzle, or, according to General Millar's plan—which is now universally adopted in heavy guns—on the second reinforce; so that the visual line becomes parallel to the axis of the gun, and when laid point-blank, 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; but when elevated it is subject to the error pointed out. 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 depression, or when the axis is parallel to the horizon; it is desirable also that the platform should, if possible, be laid horizontal. The elevation required to strike any object is found by ascertaining its distance. For this purpose sets of tables have been constructed from actual practice (see Tables at the end), by reference to 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, in French, hausse, 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

Gunnery. 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 $\backslash frac\{1\}\{21\}$ 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, or between the base ring and the sight, be multiplied by $\backslash frac\{1\}\{21\}$, the product will be the tangent of $1^\backslash circ$ in inches, 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, or when the dispart exceeds the tangent of $1^\backslash circ$, and is subtracted from the natural tangent of $2^\backslash circ$ on the scale, the length of the scale at $2^\backslash circ$; when, however, a middle sight is used, the elevation can of course be given by the tangent scale from $0^\backslash circ$ upwards to about $5^\backslash circ$. 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 into degrees, &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.

A scale has of late years been sometimes used for iron guns, marked with the number of yards range instead of degrees; and this has been found very useful to men who might not perhaps understand the tangent scale. It would unquestionably be very useful to mark approximately on the tangent scale opposite the degrees the number of hundred yards of range;—thus, for example, in the 9-pounder, $1^\backslash circ=4$, $2^\backslash circ=5$, &c.; and in the 24-pounder, of 50 cwt., $1^\backslash circ=7\backslash frac\{1\}\{2\}$, $2^\backslash circ=11\backslash frac\{1\}\{2\}$, $3^\backslash circ=14\backslash frac\{1\}\{2\}$, $4^\backslash circ=16\backslash frac\{1\}\{2\}$, $5^\backslash circ=18\backslash frac\{1\}\{2\}$ hundreds, &c.; and this is indeed the more necessary where there are several varieties of the same gun differing as to length from each other, as it becomes almost impossible to commit to memory the ranges corresponding to the elevation in each species; and great loss of time, and chance of confusion, would be avoided by thus avoiding the necessity of a reference to printed tables of ranges in the field. In marking the tangent scale, regard should be had to the mean rise of the ball on leaving the gun, referred to in treating of deviations, or else allowance should be made for it. Another mode occasionally used by the French is by the depression of the breech, and is thus effected: A line being drawn through the centre of the trunnion to the extremity of the button of the cascbel, the length of this line becomes a radius for determining the lengths of the natural tangents corresponding to the degrees of a new tangent scale; and these being marked on a long rule, the zero point being the point of contact of the rule and button, when the rule is resting on the platform, and the axis of the piece is horizontal, and the degrees or tangents being numbered from $0^\backslash circ$ downwards—a very convenient mode when in night-firing at a breach, for example, it is only necessary to secure the elevation. It may be here observed, that by using the middle sight in our iron guns, it becomes impossible to use the angle of dispart. This might be avoided by perforating the sight, and thus enabling the gunner to use

the angle of natural aim, which angle, with its corresponding range, ought to be marked on each gun.

Another method, when it is required to fire continually at the same object—for instance, a breach—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 modes of pointing and elevating here described are used in guns, howitzers, and carronades. In respect to the first of these it will be observed, that since Colonel Paixhans proposed his canons à bombes, the tendency has been to return back to guns of large calibre, for a long time almost abandoned, and only partially revived in carronades; and this change is unquestionably one of great importance to the defence, and will be equally so to the attack, when the difficulty of transporting such heavy weights has been in some degree overcome. In our service at present the heaviest solid shot proposed to be used is that of 68 lbs., and there are three varieties of it, as shown by the table, of which perhaps the 112 cwt. and the 95 cwt. (or Dundas gun) are the best; but for throwing hollow shot—differing from shells by being cast concentric—or shells, there is a 10-inch as well as an 8-inch gun, these guns corresponding in their object with the canon-obusier of the French. Plate II., fig. 1, represents the 68-pounder, and fig. 2 the 8-inch gun—a shell gun, as it is commonly called, the length and weight being greatly inferior to the 68-pounder, though nearly of the same calibre. The shell guns are admirably fitted for coast batteries, as their moderate weight renders them more easily manageable than the 68-pounder, whilst the magnitude of their calibre, whether hollow shot or shells be used, renders their fire very destructive to ships. It is to be observed also, that in coast batteries space alone requires consideration; whilst in ships weight is an equally essential element, as a vessel which could carry on her broadside only eleven 8-inch guns, might be armed with fourteen 32-pounders; and as Sir Howard Douglas therefore reasons, the magnitude of the fractures being taken as the squares of the diameters, or as 704 to 506, whilst the number of shots will be as 11 to 14, the actual spaces opened or fractured will be as 7744 to 7084, or considerably in favour of the 8-inch gun; but, on the other hand, the fire of the 32-pounder would be spread over a much larger space, and be more destructive as to the men; so that it is probable Sir Howard is justified in considering that the greater number of shots from the 32-pounders more than counterbalances the greater space of fracture from the fire of the 8-inch guns. Another reason which induces Sir Howard to object to the too exclusive arming of ships with 8-inch guns is, that the 32-pounders may commence double-shooting at 400 yards; whereas the 8-inch guns cannot commence effective firing with two shots at a greater distance than 200 yards; so that between 400 and 200 yards the 32-pounder armament would have the advantage. On land, however, the number of guns would be the same in either case; and the 8-inch gun, mounted on the dwarf-traversing platform (Plate III., fig. 1), is deservedly a favourite, being associated either with the 68-pounder for longer ranges, or with the 32-pounder. (Plate II., fig. 3.) It will be observed that the shell guns (fig. 2) are chambered on the Gomer principle; but this system of construction will be further noticed in treating of howitzers. For field batteries the favourite brass gun of our service is the 9-pounder (Plate II., fig. 4, and Plate III., fig. 7), but these are associated with the 24-pounder brass howitzer (Plate II., fig. 7); so that in the same battery two different calibres are in use, and consequently two different classes of ammunition. The present Emperor of the French has done away with this complex construction of batteries, and has adopted one form of gun—namely, the 12-pounder howitzer gun—fitted for discharging three kinds of projectiles—namely, solid shot, shells, and case

Gunnery. shot, and thus requiring only one form of carriage. The principles upon which this great simplification has been adopted have been described by Captain Favé of the French artillery, whose work has been well translated by Captain Hamilton Cox of the Royal Artillery. They are the principles frequently referred to in the preceding pages—namely, that though a greater initial velocity may be obtained by a greater charge, the resistance of the air so rapidly diminishes high initial velocities, that the ultimate ranges and the ultimate momentum are very little superior to those obtained from a considerably less initial velocity. This is illustrated by a reference to Piobert's tables, by which it appears that a 12-pounder, discharged with a velocity of 1610 feet per second, corresponding to a charge of $\backslash frac\{1\}\{4\}$d the weight of the ball, retained a velocity of 1516 at a distance of 164 feet, having therefore lost in that short space a velocity of 94 feet per second; whilst, by experiments made at Metz in 1836 and 1840, a 12-pounder shot, fired with a charge of $\backslash frac\{1\}\{4\}$th, had an initial velocity of 1516 feet per second; so that the ball propelled with a charge equal to $\backslash frac\{1\}\{4\}$th the weight of the ball on leaving the gun was much in the same condition as to velocity as the ball propelled with $\backslash frac\{1\}\{4\}$d at 164 feet from it—the two trajectories from these respective points being the same. It is indeed only where great momentum is required at a short distance from the gun that great charges become effective; and for the purposes of field guns there seems great strength in the reasonings which have led to the adoption of the 12-pounder howitzer gun by the French, as, in addition to what has been stated, a French battery is equally effective when called upon to fire shot, shell, or case; whereas, in the compound British battery, the howitzers are so much deducted from its strength when solid shot firing is required; and, in like manner, the guns are a loss when howitzer firing becomes the most valuable. The calibre of a 9-pounder would be too small for fulfilling all the purposes of shot and shell; and hence the French have wisely adopted the 12-pounder, and might perhaps have gone even higher in calibre, reducing the weight of their gun by a corresponding reduction in the weight of the charge.

Howitzers.—This form of ordnance is of later date than the mortar, which will be presently noticed, and was designed for the purpose of firing shells in the field, for which object it was necessary that it should be mounted upon a carriage, so that it combines in itself the functions of the gun and mortar. As the charges for the service required from howitzers are small, so are their comparative weight; for example, whilst the 24-pounder gun of 18$\backslash frac\{1\}\{2\}$ calibres weighs 48 cwt., and is charged with 8 lbs. of powder, the 24-pounder howitzer weighs only 12$\backslash frac\{1\}\{2\}$ cwt., and is charged with only 2$\backslash frac\{1\}\{2\}$ lbs. of powder; and in like manner, whilst the 8-inch shell gun of 13 calibres weighs 65 cwt., the 8-inch howitzer weighs only 20 cwt. The use of such small charges renders it necessary to adopt chambers—that is to say, spaces either entirely or in part of less diameter than the bore itself, as the small cartridge containing the charge could not be so placed in a large bore as to prevent great irregularities in the relative position of the cartridge and ball. Chambers of various forms have been at different times contrived; and, according to the principles adopted by Piobert, their relative values may be thus stated, taking into account the position of the vent, or of the point of the charge at which the inflammation is first set on foot:—1st, Spherical, conical, or pyramidal, when the vent communicates with the surface of the first and base of the two last, and the truncated conical ranks with these when fired at its greater base. 2d, Cylindrical, when fired by the lateral surface. 3d, Cylindrical, when fired from either base. 4th, Truncated conical, when fired from its lesser base. 5th, The spherical, conical, or pyramidal, when fired at the exact centre of the first or vertices of the other two: or, merely classifying the forms in general use—1st, spherical; 2d, cylindrical; 3d, truncated.

Gunnery. conical. The Gomer form in which the truncated cone is terminated by a spherical end, is that used in the howitzers (Plate II., fig. 6), and the mortar (Plate II., fig. 9), as well as in the shell-gun. In the carronade (fig. 5) the chamber is cylindrical, being terminated by a spherical end; and such is its form in the great Antwerp mortar (fig. 10). It may also be added, that by adopting a chamber of greater thickness of metal is obtained around the charge in ordnance which are comparatively thin from their lightness. Experiments were made in France at Strasburg, Douay, and Toulouse, to ascertain whether the lodgment in the bore produced by the pressure of the gases on the ball, an important consideration in brass guns, could be lessened or obviated by a peculiar position of the vent. One was placed in the prolongation of the axis, the cascbel being suppressed so as to fire the charge in the centre of its extreme end; the second in a line, making an angle of 30°, with the vertical drawn from the extremity of the axis, and the third in the usual way. The destructive effect of the charge in producing a depression in the bore was found to be nearly in the proportion of 6 to 1 as regards the two first and the last, the difference between the two first being very small. Experiments were also made in France in 1817 to determine the best position of the touch-hole, or vent, of muskets, as regards the charge, the musket being suspended as a pendulum, and the ball being discharged against a ballistic pendulum; but the results, though exhibiting a maximum effect of the ball at a position of the vent corresponding to a distance of one line in front of the bottom of the charge, were so nearly equal to it in several other positions that they would not justify the peculiar selection of any one, unless dictated by convenience in other respects. In artillery it is only necessary to arrange it in a position which will ensure the ready conveyance of fire to the charge, and it is therefore usual to form the vent at an angle of 15° with a vertical from the axis, and terminating at two or three lines in advance of the bottom of the charge—this slight inclination facilitating the breaking of the cartridge by the prick, which might otherwise slip between the cartridge and the bore.

Howitzers were, according to Senders, first made in Germany, and subsequently improved by the English and Dutch, but were not used in France till a later period, as they did not appear in the ordnance of 1732, though afterwards adopted and introduced in the celebrated system of Gribeauval. In the British service the 12-pounder howitzer is associated with the 6-pounder gun in the horse artillery, the 24-pounder with the 9-pounder gun in the field batteries, and the 32-pounder may be associated either with the 12-pounder gun in the reserve for positions, or with the 18-pounder gun, should a brass gun of that calibre be adopted in our field service, as it has long been in that of Austria. The 10-inch howitzer (Plate II., fig. 6), and the 8-inch are used in sieges; and the small howitzer (fig. 8), analogous to the Coehorn mortar, for mountain service.

Carronades.—The carronade, invented, or rather improved, by Mr Gascoigne, was, in June 1779, approved 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. (See Plate II., fig. 12.) It is correctly bored; and the shot so nearly fills the calibre that the least possible impulse of the powder is lost by the escape of gas between the cylinder and the shot, which last is also thereby more truly directed in its flight. The bottom of the cylinder is terminated by a chamber ending in 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 cascbel, through which the breechin-rope

Gunnery. is reeved, the only rope used about these guns. This gun has some advantages over 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 throw its shot to an equal distance with a longer gun, yet, from the adaptation of the shot to its cylinder, 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; but it has its disadvantages, as for example, by adopting so small a windage, or difference between the diameters of the bore and shot, a windage which would have been impracticable in long guns, it often happened that the shot, although fitted for the long guns, when rusted would not enter the carronades; and the advantage, therefore, consequent on less windage ought not to exist, as the windage of guns and carronades should be the same, being reduced as much as possible, consistent with facility of loading and the use of hot shot; the windage in iron guns need not, indeed, exceed 15 inch, and Sir Howard Douglas recommends 14 inch for heavy guns, and 11 inch or 11 inch for the 9-pounder downwards. And further, carronades are liable in some positions to fire the rigging or the hammocks either by the flash or by the vent fire, an evil which might be remedied, as Sir Howard Douglas suggests, by giving the 24 and 32-pounder carronades a somewhat longer bore, and adding "something to the flash-rim."

The serious results of the too general adoption of carronades in our ships of war during the last American War on the Lakes, where the Americans obtained such great advantage by the use of their long and heavy guns, fully justify Sir Howard Douglas in his final remarks on this description of ordnance:—"The defects of carronades, and the danger of employing this imperfect ordnance, are now generally felt and admitted; that ordnance, however, rendered important service in its time, for it taught us practically the great value of a reduced windage, the advantages of quick firing, and the powerful effects produced at close quarters by shot of considerable diameter striking a ship's side with moderate velocity;" and these remarks are in some measure applicable, though in a minor degree, to the bored-up guns, in which it has been attempted to obtain increased calibre with diminished weight by reaming or boring out guns of originally lesser calibre, as may be seen on inspection of the table of iron ordnance.

The fire of artillery may be divided into two classes—horizontal (or at angles near the horizon), and vertical. Horizontal fire may be subdivided into horizontal direct, and enfilade. 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 enfilade fire is that which is not directed against the front of a line but along its prolongation, and the most important form of it is the ricochet fire, which 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 ricochet 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. Vertical fire is that which is thrown from mortars at elevated angles. It was much used at the siege of the citadel of Antwerp in 1832; still more, perhaps,

at the late siege and defence of Sevastopol, and became Gunnery. the principal fire at the attack on Sveaborg. Coehorn, who was opposed to Vauban, the author of ricochet fire, was a great advocate of vertical fire, as was also Carnot.

Mortars.—Mortars have succeeded to the ancient bombards, and were at first intended for discharging either one very large ball of stone or a shower of smaller stones. As they are now intended for the projection of shells only, they are designated, not by the weight of the hollow projectile, which is subject to considerable variation, but by its diameter in inches, as 13-inch, 10-inch, 8-inch iron mortars, and 5½-inch and 4½-inch brass mortars, of the British Service—the 5½ being also called the "Royal," and the 4½ the Coehorn, having been invented by that celebrated engineer, and intended to be used against "sap-heads;" and, indeed, from its portability, to be carried to any point, either in the defences of the place or in the trenches, from which the nearest portions of the approaches, or of the counter-approaches of the enemy could be most efficiently molested. (See Plate II., fig. 11.) The projectile used is the shell or "bombe" of the French, which is thickened at the end next the powder, or at the point of greatest shock, and also about the hole in the shell intended to receive the fuze, or, in the earlier periods of artillery, the match for igniting the bursting powder. It is also provided with loops to facilitate the operation of lifting and placing it in the mortar, in which respect, as well as in having the "culot," or thickening at the bottom, it differs from the "obus," or howitzer shell of the French. The object is twofold; first, as a simple projectile, in which character it acts by the explosion of the bursting powder within, which shatters the shell, and causes its splinters to fly about and act as so many distinct projectiles; and, second, as a mine. In the first case, only that quantity of powder necessary for explosion need be carried, but in the second, the effects depend on the quantity, and hence it is that in a previous article the advantage of very large shells has been strongly urged. The difficulty of constructing mortars for propelling very large shells is undoubtedly very great; but should Mr Mallett succeed with his proposed 36-inch mortar, formed of flat rings bound together by longitudinal bars of wrought iron, the problem will be solved, and the effect of the explosion of 480 pounds of powder, sunk deep into the ground, or penetrating, by the weight—probably 3000 lbs.—of the charged shell through the roof of a magazine or casemate, may readily be conceived. The weight of this mortar and its bed will be 45 tons, and the weight of the heaviest piece when asunder about 15 tons. Mr Mallett is sanguine as to its success. See CANNON for examples of the former use of wrought iron in the manufacture of ordnance in France and Spain, both at remote and recent epochs; the St Etienne Company having submitted an 8-pounder to the most severe trials, and offered, in 1813, to supply the government with 24-pounders of forged iron at the rate of eight per diem.

The mortar is made much shorter than guns, in order to facilitate loading (see Pl. II., fig. 9). It is chambered conically, or on Gomer's construction, so that the shell fits into the chamber, and does away with windage, an advantage which cannot be fully secured by that form of chamber in guns or howitzers fired horizontally, or nearly so. The great Antwerp mortar (fig. 10) had a cylindrical chamber, but it was only partially successful from the defects in the casting of its shells (fig. 17). Senderos states that the art of projecting shells was a happy invention of the latter end of the fifteenth century, and that the difficulty which first attended it, as well as the amount of subsequent improvements, may be judged by the fact that the celebrated artilleryist Don Antonio Gonzales, for a long time afterwards, deemed it necessary to set fire to the charge and to the fuze of the shell separately, which was called serving it with two

Gunnery. fires, a tedious and dangerous process. Until lately it was usual in loading the mortars to lift up the heavy shells by means of handles provided with hooks adapted to seize upon the loops of the shell; and with such weighty masses as the 13-inch shell, the firing could only be slow and gradual, which, after all, must always be the most effective fire. It is now customary to use a "derrick," as in Pl. II. fig. 14, for this purpose; this simple form being that of Sergeant Forrest, Royal Artillery, represented here as in use, but drawn back when not in use. The navy use also mechanical means, and in consequence the firing against Sveaborg was so rapid and continuous that many of the mortars were destroyed by it. Fig. 13, Pl. II., represents also another modern arrangement, namely, the suspension of the large sea mortar, according to the plan of Captain Julius Roberts of the Royal Marine Artillery. The figure explains that by the rotation of the circular platform below, ready means are obtained of pointing the mortar in any direction; and as the mortar revolves round the horizontal axis on firing, it is hoped that the ill effects of recoil will be avoided. In anticipating the results of this arrangement it must be remembered that as the mortar is at an angle of 45°, one half of its momentum will be expended in the direction of recoil, and that the other half must act with a most powerful shock on the axis. Some were tried in the Baltic, but the results are not yet conclusive, though several new mortar-boats are fitting with them.

The moment of bursting is regulated by the fuze fixed into it, as seen in figs. 15, 16, and 17, Pl. II. Fuzes may be either concussive, percussive, or time fuzes. The object of the two first is to cause the shell to burst immediately on striking an object; of the latter, to cause it to burst after a certain time, as determined by the length of the burning composition in the fuze. The concussive fuze is provided with an internal mechanism, so adjusted that though it resists the shock of firing, or even that of a short graze, it shall yield to the shock of impact, the concussion shaking the burning composition into the loaded cavity of the shell, and causing it to explode. The percussive fuze or shell depends for its explosion on a chemical composition of highly explosive character, which bursts the shell at the moment of striking, without being previously ignited. Captain Moorsom's percussive fuze is well known as the British type of this class of fuze, just as Captain Boxer's now is of the time-fuze. The time-fuze is divided into metal and wooden fuzes, the former being used in the navy, and regulated to burn according to the specific use, 20", 7", and 2"; the object of percussive or concussive fuzes being to replace especially these short-timed fuzes. The wooden fuze now so admirably constructed by Captain Boxer, and which has so entirely replaced the old fuzes, which required to be cut in the field to the proper length, has been figured and explained in art. ARTILLERY. The composition-bore is now made eccentric, so as to allow more thickness for the two powder channels. Two rows of holes are made, one into each powder channel, the bottom hole in each row being continued into the composition-bore—each hole of one row, corresponding to the centre of the space between a pair of holes of the other, admitting therefore of subdivision without too much weakening the fuze by bringing the holes of one row close together. By a simple boring-bit a communication between any one hole required for the special length or range, and the composition of the bore, is readily made. The fire, therefore, is communicated from the composition of the fuze, when it has burnt down to this perforation, to the rifle powder in the powder channel: see ARTILLERY for further explanation; and it may be observed that Captain Boxer has since extended the principle, with the necessary modification, to fuzes for mortars.

As mortars are not fired through embrasures but behind epaulements, they are directed by pointing rods, placed in the proper line upon the epaulement; the mortar being

brought into line with them by means of a plumb-line held by a gunner standing behind the mortar, on which the line of metal has been marked, the line being placed in the vertical plane passing through the two pointers. Other modes of pointing both guns and mortars will doubtless be hereafter introduced, as the great improvement of artillery practice at long ranges will require the use of telescopic sights, and the introduction of the collimating principle, a fact which has not escaped the attention of artillery officers. The elevation is given by the gunners' quadrant.

Besides the ordinary projectiles, shot and shells, grape and case shot are fired from guns; the first being a number of balls tied or quilted together like a bunch of grapes, and the other a cylindrical tin case filled with balls. They are not calculated for long ranges, but are very destructive at short ranges from 200 to 400 yards either against advancing troops, or fired from the flanking defences of a fortress along the ditch. The Shrapnel shell, or spherical case, is, however, a projectile of still greater value, as it can now be used at almost any range. They can be fired from either gun, howitzer, or mortar; but the object is to fire them from the two first, as, on bursting, the balls which fill them fly forwards with the then velocity of the shell, and being spread by the resistance of the air, deal out destruction equivalent to the action of many muskets in addition to that effected by the splinters of the shell. The practice of putting balls into shells, in addition to the bursting powder, is by no means modern; for Lucar (1588), after explaining the mode of charging shells, and causing them to burst by means of a piece of gunner's match fixed in the match or fuze hole, says—

"Also you may, if you will, put into the said hollow baule or pellet certaine square or round pieces of lead, or divers short pypes of iron like unto pocked dogges full charged with gunpowder and pellets, and fill up the rest of the concavie with fine gunpowder, and having anointed it with turpentine, and roled it in fine gunpowder, shoote it out of a peece of artillery, with a trayne laid to the mouth."

This was in every respect a Shrapnel shell, the mode of firing being consequent upon the early idea that the fuze required to be separately ignited. Shrapnel, however, revived this forgotten projectile, and by giving it an effective form, became its second inventor; as Captain Boxer may now be considered its third, as by the total separation of the balls from the bursting charge, he has done away with the failures by premature bursting consequent on the ignition of the powder by friction against the balls, and rendered it possible to use it with high charges, and for all ranges—the Shrapnel still becoming the true counterpoise to the improved Minié rifle ball, and restoring to artillery its superiority over musketry. Carcasses, or shells filled with a highly inflammable composition which escapes in several directions through the holes (3 or 4) made in them for that purpose, may be fired from mortars or guns: they are also a very old invention.

Rockets.—The history, principle, and possible importance of these projectiles have been fully discussed in art. ARTILLERY, where it was shown that almost every country but that of Congreve, who had first introduced into modern Europe war rockets, had paid great attention to their improvement. In France also they have latterly been much improved; and it has been found possible to use a short stick instead of the long one seen in Pl. II., fig. 19. In our own arsenal also, the subject has been taken in hand by Captain Boxer, and mechanical means adopted for insuring precision in the bore, and in the position and form of the vent with every prospect of success. But with every respect for these laudable and skilful efforts, it would be unjust to deny to Mr Hale the merit of having first in this country invented machinery of a most beautiful description for the manufacture of rockets, and for doing away with the stick entirely. Mr Hale's

contrivance consists in causing the rocket to rotate on its axis during its flight; and, as in the case of an elongated shot, to move steadily with the point foremost. For this purpose the burning material issues from five orifices made near the neck, obliquely to the axis of the tube, the effect of which is, that the body of the rocket is made to rotate when it is also propelled. Sir Howard Douglas adds to his description of these rockets that the contrivance is very ingenious, and may be expected to produce advantageous results. He, however, adds that, at low angles, they had been found liable to failure, being subject, on a graze, to be deflected from their original direction much more than ordinary rockets, and that unless this cause of failure could be removed they would be of little use against troops in the field, that is to say, in horizontal firing or plane battlefields. He also observes, that "in other respects also the success of the Hale rocket may be doubted; the stick rocket continues its flight, directed by the stick, after the composition is burnt out; but the Hale rocket loses its directing power as soon as the composition is consumed, because the rotation then ceases, and nothing can be expected from the rocket beyond the distance it has reached when the composition ceases to burn" (which is sooner than in the common rocket, as the composition is more powerful). This objection, however, does not appear well founded, as the object of giving a sufficiently rapid rotation has been effected before the composition has been burnt out, and the rotation then continues as in any other rotating projectiles. Mr Hale also promises to realize Congreve's anticipations, by throwing up bundles of rockets hooped together and rotating in mass, to the amount of 400 lbs. It is time that these inventions should be subjected to some decisive trial, and either adopted into the service, or, if found defective, rejected on sound and scientific reasons. The rocket stands of Mr Hale are most ingenious and effective, and strongly contrast with the rude apparatus figured in Pl. III., fig. 13,—the wagon for conveying them being represented in fig. 14.

In respect to the other figures of Pl. III., their names sufficiently explain their object—the two upper figures exhibiting the mode of firing over a high genouillère, or a low barbette, by means of a dwarf traversing platform; and the other of firing over a parapet by means of a high one; and this figure is represented of iron, as generally used in our colonies, and in times of peace—cast iron carriages being inadmissible in actual warfare, as they are so easily injured by shot, and are not repairable.

Before closing this article it is necessary to say a few words on these important points in practical gunnery, as, in fact, the main object of the science is to insure that the gunner shall so throw the shot or shells that they shall either strike or explode at some defined point, and also that the shot shall have force sufficient at that point to perform the work required from it,—as, for example, if the residual velocity of a Shrapnell shell were almost 0 at the time of its explosion, the balls would fall almost harmless on the ground, just as the balls would do when fired from a mortar, as proposed by Carnot (see FORTIFICATION). The trajectory of a projectile would be in vacuo, as before stated, a parabola, and therefore easy of computation, supposing the initial velocity known. As, however, all military projectiles are projected in the air, and are resisted in their motion by that elastic fluid, the trajectory is not a parabola; and that simple theory can only in rare cases be applied to practice, or in cases where the velocity is less than 300 feet per second. It is, however, necessary for the clear understanding of the phenomena of projectiles to know at least the formulae which represent the results of the parabolic theory, and they are therefore given as follows:—

Let $V$ be the initial velocity in the direction of projection; $\backslash phi$ the angle of projection above the horizontal plane; $x$ and $y$ the horizontal and vertical co-ordinates to the curve of the trajectory at any point $P$, estimated from the commencement of the curve, or point of departure as the origin; $t$ the time of flight to that point, $v$ the horizontal velocity at it, and $v\text{'}$ the velocity in a vertical direction; $\backslash theta$ the angle of inclination of the tangent at that point with the horizon; $X$ the whole horizontal range, or horizontal co-ordinate, when $y$ becomes again 0 as it was on departure; $Y$ the greatest height of ascent, or when $y$ is a maximum; $T$ the whole time of flight; $H$ the height, falling from which a body would acquire a velocity equal to $V$.

and as $v\; =\; \backslash frac\{dx\}\{dt\}\; =\; V\; \backslash cos\; \backslash phi$, which is constant, it is evident that $v$, or the horizontal velocity at every point of the curve is constant; $v\text{'}\; =\; V\; \backslash sin\; \backslash phi\; -\; gt$, and the velocity in the direction of the curve = $\backslash sqrt\{v^2\; +\; (v\text{'})^2\}\; =\; \backslash sqrt\{V^2\; -\; 2Vgt\; \backslash sin\; \backslash phi\; +\; g^2\; t^2\}$, or = $\backslash frac\{V\; \backslash cos\; \backslash phi\}\{\backslash cos\; \backslash theta\}$.

the same for any given angle and its complement, it is evident that there are two angles of elevation, $\backslash phi$ and $90^\backslash circ\; -\; \backslash phi$, which with the same initial velocity give the same range. The elevation remaining the same, the horizontal ranges being $4H\; \backslash sin\; \backslash phi\; \backslash cos\; \backslash phi\; =\; 2H\; \backslash sin\; 2\backslash phi$, it is evident that the initial velocities, or $H$, remaining the same, those ranges vary with $\backslash sin\; 2\backslash phi$, and such range is therefore the greatest when $\backslash sin\; 2\backslash phi$ is greatest, or when $\backslash phi$ is $45^\backslash circ$.

As the influence of the resistance of the air in modifying the motions of a projectile is less in proportion as the projectiles are greater, the true range of a musket, as compared to the theoretic, being 1 to 18, with field and siege guns the range in vacuo exceeds that in the air, as 3.05, 2.80, 2.69, 2.47 to 1 respectively, the proportion diminishing as the projectiles become greater; so that it might be reasonably anticipated, that with still heavier and larger projectiles, whether shot or shells, the approximation of the real range to the theoretical one, on the parabolic theory, would be still closer; but unfortunately there is here a difficulty, as the initial velocity of shells fired from mortars at high angles has not yet been determined experimentally, and the ranges therefore cannot be satisfactorily calculated. Supposing, however, the ranges observed, the times of flight may be calculated, and compared with the same times observed, and this has been done in France with the 8½ and 11-inch mortars. In the 8½ inch, fired at $45^\backslash circ$, the ranges being 375, 688, 1254, 1960 yards respectively (the charges varying from ¼ lb. to 2½ lbs.), the differences in time were 1.4, 1.6, 0.7, 1.6. In the 11-inch fired at the same angle, the ranges being 505, 803, 1238, 1701, 1922 yards, and the charges varying from 1 lb. to 3½ lbs., the differences in time are 1.3, 1.5, 1.8, 2.2, 4.1, or between 1½th

and an $\backslash frac\{1\}\{4\}$th of the whole time, excepting in the last case, where it amounts to about $\backslash frac\{1\}\{2\}$th. When fired at an angle of $30^\backslash circ$, the difference in the times vary from $\backslash frac\{1\}\{2\}$th to $\backslash frac\{1\}\{4\}$th of the whole times, the time observed always exceeding the time calculated; the retardation being due to the resistance of the air. In the case, therefore, of shells, the movement approximates much more closely to the parabolic motions than in either cannon or musket balls. The velocity indeed of the 11-inch deduced from the range and theoretic time, would be about 435 feet per second; and Griffiths gives 500 feet as the velocity of the 10-inch fired with 3 lbs. charge.

These examples taken from Didion, prove that a limited use may be made of the parabolic theory in respect to the motion of shells when fired with such moderate velocities; but it is totally inapplicable to the case of guns; and the labours of the most able mathematicians have been applied to the solution of the difficult questions involved in the determination of the trajectory of the projectile in the air, when moving with such great velocities.

For elevations not exceeding $10^\backslash circ$, Poisson's formula is—

$x$ and $y$ being the horizontal and vertical ordinates at any point of the curve, the origin being in the axis of the bore at the muzzle of the gun, $\backslash phi$ the angular elevation of the gun, $h$ the height due to the initial velocity of the shot, $r$ its semi-diameter, $e\; =\; 2.71828$ the base of the

Napierian logarithms, and $k\; =\; \backslash frac\{\backslash rho\; \backslash delta\}\{r\; \backslash delta\}$ the co-efficient of the square of the velocity in the expression for the retardative force of the resistance of the air, $\backslash delta$ being the density of the air or medium, $\backslash delta\text{'}$ that of the shot, and $q$ a fraction which must be determined by experiment, and which, though doubtful, may be assumed as 0.225. It can be simplified for small elevations and short ranges, by developing $e^\{2kx\}$ in series, neglecting all above the fourth term, and substituting the result, when the formula becomes—

$V$ being the initial velocity.

Didion has given other formulae of considerable simplicity, but as they require for their practical application the use of the tables of values of several co-efficients, which he has calculated and published in his admirable treatise already so frequently referred to, they will be better studied in that work.

In ranges corresponding to $1^\backslash circ$ of elevation, which in the larger natures would give about 700 yards, the cosine may be taken as equal to radius, or 1; and hence, as Captain Boxer points out, the equation becomes—

The time of describing any portion of the trajectory is expressed by the equation $t\; =\; \backslash frac\{1\}\{k\; \backslash sqrt\{2gh\; \backslash cos^2\; \backslash phi\}\}\; (e^\{kx\}\; -\; 1)$; and when $X$, or the whole range, is substituted for $x$, $t\; =\; T$, the whole time of flight.

Sir Howard Douglas also gives an empirical formula from the Aide Memoire Navale for deducing the ranges of shot from the maximum range determined by experiment, the elevation being between $10^\backslash circ$ and $30^\backslash circ$.

Let $R$ represent the range at $30^\backslash circ$, considered the maximum, then $X\; =\; R\; (\backslash sin\; 3\; \backslash phi)^3$.

It will be observed that in all these formulae, and it must be so in every formula, everything depends on the accurate determination of the initial velocity, and a correct knowledge of the nature and amount of resistance opposed by the air

to the motion of the projectile. The mode of determining the former experimentally has been already pointed out; but as it may be necessary to determine by calculation the initial velocity of one ball from the experimentally determined velocity of another, Piobert has proposed the following empirical formula:—

$V$ being the initial velocity sought of the ball whose weight is $W$, and $v$ the experimentally determined velocity of the ball whose weight is $w$, the charge $p$ being the same in each case. Major Mordecai, of the United States army, has found this rule to agree with his experiments when the charges do not exceed one-third of the weight of the ball, and the gun is at least 16 calibres in length, but does not consider it sufficiently accurate for higher charges. Taking

that with a 32-pounder, the charge being one-third, and the windage 0.16 inch, $M\; =\; 5200$; with a 24-pounder, the charge being one-third, and the windage 0.14, $M\; =\; 5400$.

A British empirical formula is $V\; =\; 1600\; \backslash sqrt\{\backslash frac\{ap\}\{w\}\}$; $p$ being the charge, $w$ the weight of the ball, and $a$ a co-efficient depending on experiment, and varying, according to Hutton, with the length of the gun between 2.1 and 2.5, as also with the ratio of the weight of the charge to that of the shot. General Millar, by experiments in 1817, has also shown that it increases as the windages decrease, being 2.8 with a windage of 0.202 of an inch, and 3.55 with a windage of 0.075; whilst from experiments made at Deal in 1839, and on board the "Excellent" from 1837 to 1847, the mean values of $a$ are 3.2 for a windage of 0.233; 3.4 for 0.2; 3.6 for 0.175; 4.4 for 0.125; and 5. for 0.09; and when the windage is 0, $a$ becomes 6.66. The velocity, therefore, with a windage of only 0.125 being to that without windage as 1 to 1.23, the loss from that small windage being one-fifth nearly of the whole velocity, a convincing proof of the propriety of reducing the windage of all balls to that quantity which will admit their ready introduction, allow for the chance of rust, and for their expansion when used as hot shot, which in the largest balls is about $\backslash frac\{1\}\{25\}$th of the calibre, or in an 8-inch shot, 0.114 of an inch; so that 0.15, proposed by Sir H. Douglas, would be amply sufficient. With caronades, in which the windage varied from 0.061 to 0.078, the mean value of $a$ was 4.5.

The initial velocities and ranges being known by experiment, the co-efficient for the resistance of the air, may be obtained from the equation $k\; =\; \backslash frac\{3\}\{2\}\; \backslash cdot\; \backslash frac\{V^2\; \backslash sin\; 2\; -\; gx\}\{gx^2\}$.

The velocity at known distances from the gun may be represented by $v\; =\; \backslash frac\{V\}\{e^\{kx\}\}\; =\; Ve^\{-kx\}$; or, according to the formula of the French School at Metz, as quoted by Sir H.

Douglas: $v\; =\; \backslash frac\{V\}\{(1\; +\; aV)\; e^\{\backslash gamma\; x\}\; -\; aV\}$. $v$ the velocity at the point sought, $V$ the initial velocity, and $x$ the distance, or horizontal co-ordinate, being expressed in English feet, $a\; =\; 0.0007$; $\backslash gamma$ varying with the nature of the shot, but in a 32-pounder is equal to 0.0001034; this formula being founded on the assumption that the resistance of the air is partly proportional to the square, and partly to the cube of the velocity.

The penetration of shot, the velocity at the point of contact having been determined by one or other of the above

Gunnery. formulae, may be thus represented: $P\; =\; \backslash frac\{2v^2\; r\; \backslash delta\}\{3\; \backslash rho\; g\}$; $P$ being the depth penetrated, $v$ the velocity of the shot at the instant of striking, $r$ the semi-diameter, $\backslash delta$ the density of the shot, and $g\; =\; 32.2$ feet, the force of gravity; $\backslash rho$ a co-efficient of resistance, to be determined by experiment on various substances.

When the resisting substance is the same, $P$ varies as $r\; v^2\; \backslash delta$, or when the density of the shot is the same as $r\; v^2$; the penetration in this case then varying as the diameter of the shot and the square of the velocity. From Poncelet's hypothesis that the resistance of a material struck by a shot is proportional to the square of the shot's diameter; and from the Gavre experiments, $P$, in respect to value, may be represented by this formula (Sir H. Douglas having modified it so as to represent $P$ in English feet):—

$P\; =\; 4.612\; \backslash \; r\; d\; \backslash log\; \backslash left\backslash \{\; 1\; +\; \backslash frac\{v^2\}\{1076696\}\; \backslash right\backslash \}$; $d$ representing the specific gravity of the shot; water being 1. For firm earth this may be multiplied by 1.64; for sand and gravel, by 1.3; for loose earth, by 3.21; for sound masonry, by 0.41. In the previous formula the specific gravity may be also substituted for the density when it becomes $P\; =\; \backslash frac\{2v^2\; r\; d\}\{3\; \backslash rho\; g\}$, as $d\; =\; \backslash rho\; \backslash delta$.

Terminal Velocity.—When a body descends in air from a state of rest, its velocity increases for a time by the action of gravity upon it; but the resistance of the air increases as the velocity increases, and hence it must at length become equal to the accelerative force of gravity which is constant, after which the body will move uniformly with the velocity acquired, or with the terminal velocity, making

therefore, $kv^2\; =\; g$, $v^2\; =\; \backslash frac\{r\; \backslash delta\; g\}\{\backslash rho\; \backslash delta\}$; or $v\; =\; \backslash sqrt\{\backslash frac\{r\; \backslash delta\; g\}\{\backslash rho\; \backslash delta\}\}$; and with shells $v\; =\; \backslash sqrt\{\backslash frac\{r\; \backslash delta\; g\}\{1.42\; q\; d\}\}$; the weight of a solid shot to that of a shell being as 1.42 to 1; $q$ being = 0.225, as before stated.

As the terminal velocities vary with the square roots of the diameter of the shot, the density being the same, and the terminal velocity of a shot 2 inches in diameter, as deduced from Hutton's tables, being 248 feet per second; that of a 24-pounder is 415.53, or

$\backslash sqrt\{2\}\; :\; \backslash sqrt\{5.612\}\; ::\; 248\; :\; 415.53$;
and that of a shell of the same diameter would be
$\backslash frac\{415.53\}\{\backslash sqrt\{1.42\}\}\; =\; 415.53\; \backslash times\; 0.8392\; =\; 348.71$.

General Duchemin has given the following formulae for the loss of velocity by windage: $V$ being the initial velocity, $V\text{'}$ the velocity lost, $s$ the diameter of the vent, $C$ the calibre of the gun, $C\text{'}$ the calibre of the shot, $C\; -\; C\text{'}\; =\; C\text{'}$ the windage, $a\; =\; 3.5$, a constant, and $\backslash beta\; =\; 0.182857$, a constant also of the same species as $C$ and $C\text{'}$—

In using this formula for English feet, it is only necessary to represent $C$, $s$, and $C\text{'}$ in decimals of a foot, and to make $\backslash beta\; =\; 0.599955$; or if these terms are given in inches, to make $\backslash beta\; =\; 7.199458$.

Duchemin has also given the following empirical formula for the charge of greatest effect—

or the number of times the calibre is contained in the length of the bore, and $C$ and $\backslash delta$ as before. This formula may be used for the charge in English weights and measures by using 0.1683, or 2.0198 instead of 0.513, according as $C$ is represented by a decimal of a foot, or in inches—and replacing $34^m\; \backslash cdot\; 0.423$ by its value 75.0973 in English pounds.

Many other usual formulae might be extracted from va-

rious authors, but those given will be sufficient for the reader, and indeed for most purposes of general study. Those who desire to obtain a still more accurate knowledge of the dynamical theory of projectiles, should turn to the admirable work of Didion; and those who wish to know still more fully its bearing on the purposes of war, whether in our marine or land armaments, should read the excellent work of Sir Howard Douglas, now arrived at its third edition, which is unquestionably the best book on the subject in our language. The work of Captain Boxer (of which the first part only has as yet been published) preparing as a course book for the Royal Military Academy, is even now deserving a careful perusal, and will, when finished, render references to writers on foreign artillery unnecessary. Sir Howard Douglas was originally an officer of artillery, and has ever since retained a feeling for his first service, which has led him to apply the powers of a highly scientific mind towards its improvement. Captain Boxer is an officer of artillery, and now occupies his right position as a man of science at the head of the Royal Laboratory: let, indeed, the system now maintained by Lord Panmure of appointing to such posts only men of science be continued, and the Royal Artillery will soon supply officers fitted to fill the posts of its manufacturing establishments with honour, and thereby to advance the interests of their country.

Of other works I need only name the excellent Spanish treatise of Senderos, so often quoted in this essay.

Valuable, however, as the investigations of science always are, and essential also as they must be for perfecting the theory, and thereby also the practice of gunnery, it is manifest that the calculation of formulae could never be undertaken in the field; and hence that the results of the labours of the calculator must be placed in such a form as to be readily consulted at the moment they are required. Tables, therefore, have been formed for that purpose, such as those of the French, and those of the English service, which close this article. The latter, for heavy guns, have been carefully digested by Lieutenant-Colonel Lefroy, R.A., from the results of both land and sea-practice; the sea being that of the "Excellent," the instructional gunnery-ship at Portsmouth. The tables for field-guns are those of Col. Burn, R.A., as printed and published by him on cards for the general and convenient use of the members of his profession. They represent, therefore, our practical knowledge up to the present day; but they must not be considered as results finally and fully determined. Far from this, as every day ought to do something for their improvement, and will do so when the important establishment at Shoebury-Ness has been rendered fully efficient in all its details. It is bad economy to stint such an establishment in anything necessary for its efficiency; or by denying those instruments which have become familiar in foreign arsenals, to make it only a school of drill, instead of an important element in the great school of instruction, which, beginning at Woolwich, should end here.

It has been already remarked that the electro-magnetic chronograph for determining the time of flight of a projectile through various points of its trajectory, was submitted by Wheatstone to the select committee at Woolwich; but as yet it has not been applied at Shoebury-Ness. It is true that this mode of determining the time of flight has not always succeeded; but the causes of failure do not appear difficult of removal; and a government establishment is assuredly the right place for experiment.

This article, then, is closed with a fervent wish that the spirit of inquiry, which is beginning to spread over our artillery, will be fostered, and the thirst for professional knowledge, which now animates so many of its officers, will be encouraged and rewarded; and that ere long our artillery will be as distinguished for its science, as it has always been for its discipline and valour.

Tables of Ranges of Iron Ordnance. From the Handbook of Field Service of Lieut.-Colonel J. H. LEFROY, F.R.S., Royal Artillery.

REMARKS.—The ranges for the 10-inch gun are the mean of the two scales of Colonel Burn, one 8 feet, the other 5 feet above the plane. The same data must be nearly correct for the shell of 57.5 lbs. The times are interpolated from the "Excellent's" Tables, 1852; which, however, allow about 0.5° greater elevation for the same ranges. Height of gun above plane B 6.5 feet.—"Excellent" not stated.

The ranges for the 8-inch gun, hollow shot of 47 lbs. are interpolated by projection, from the mean of 137 rounds fired at Shoebury Ness, 1850-2; for the sea service, hollow shot of 56 lbs. and shell of 50 lbs.; the first, with each charge, is from Colonel Burn, the second from the "Excellent."

For the Shrapnel shell, from Colonel Burn.

The data above, printed in ordinary type, are from Lieutenant-Colonel Burn's Cards; those in darker type, from the Tables of the "Excellent." The differences, like those shown in the Tables of the 8-inch gun, deserve notice, but are not attributable to a difference in the height of the plane, which, when given, is from 5 feet to 8 feet. A similar discrepancy between the naval and land service tables in the French service led to an investigation of the subject, by a committee, at GAVRE, 1843, which came to the practical conclusion, that the mean deviations at sea do not differ et. par. from those on land (Douglas, 3d edit., 146). Officers will readily decide, from their own experience, which data, under their particular circumstances, are to be relied on.

(a) Mean of two n; the guns differ effectively by 4 inches of length; the rest that follow also n.

(b) The data that follow are for the 32-pounder guns in general use in the navy. Tables of "Excellent."

(c) Mean of two n, not differing more than 15 yards from what is here given, at any range.

The sea-service 13-inch mortar weighs 101 cwt.; bed, 83.5 cwt.; and, with charge of 20 lbs., ranges 4200 yards. The sea-service 10-inch mortar weighs 52 cwt.; bed, 55.5 cwt.; and, with charge of 10 lbs., ranges 4000 yards.

1.—The 68-Pounder Carronade used as a Mortar and fired at 45°. With a charge of 8 lbs. of powder threw its shell (of 8 inches diameter and 44 lbs.), Sutton Heath, 1810...3500 yards.

2.—A 24-Pounder of 6½ Feet at 45° Ranged as follows:—
With a charge of ..... 3 lbs. 3500 yards.
Ditto ditto ..... 4 ... 3700 ...
Ditto ditto ..... 6 ... 4000 ...

3.—A 24-Pounder of 9½ Feet at 45° Ranged as follows:—
With a charge of ..... 6 lbs. 4500 yards.
Ditto ditto ..... 8 ... 4300 ...

4.—An 18-Pounder of 9 Feet at 40° Ranged as follows:—
With a charge of ..... 3 lbs. 3700 yards.
Ditto ditto ..... 4½ ... 4000 ...
Ditto ditto ..... 6 ... 4200 ...

These data may sometimes be useful in annoying an enemy from some very distant spot.

5.—Range of Iron Mortars at 45°. 13-inch mortar, land service, charge..... 9 lbs. 2800 yards.
13-inch, sea service, charge..... 10 ... 2800 ...
Ditto ditto ..... 12 ... 3400 ...
Ditto ditto ..... 14 ... 3500 ...
Ditto ditto ..... 16 ... 3900 ...
Ditto ditto ..... 18 ... 4100 ...
Ditto ditto ..... 20 ... 4400 ...
Ditto ditto ..... 25 ... 4700 ...
Ditto ditto ..... 25 ... 4850 ...
Ditto ditto ..... 28 ... 4500 ...
Ditto ditto ..... 30 ... 4500 ...

6.—10-Inch Land Service at 45°. Charge ..... 6 lbs. 2400 yards.
10-inch sea service at 45° ..... 5 ... 2800 ...
Ditto ditto ..... 8 ... 3400 ...
Ditto ditto ..... 10 ... 3500 ...
Ditto ditto ..... 12 ... 3800 ...
Ditto ditto ..... 20 ... 4500 ...

G U N N E R Y.

Tables of Brass Howitzers. From the Cards of Colonel BURN, R.A.

RANGES OF 12-POUNDER HOWITZER.

RANGES OF 24-POUNDER HOWITZER.