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rise from the resistance of the air in the flight of creased elasticity of the air contained in and beshells or cannon-shot. In this persuasion they tween the grains, in consequence of the heat and supported themselves chiefly by considering the fire produced at the time of the explosion : a extreme rarity of the air, compared with those cause not adequate to the 200th part of the efdense and ponderous bodies; and at last it be- fect. On the other hand, Mr. Robins determincarje an almost generally established maxim that ed, by irrefragable experiments, that this force the flight of these bodies was nearly in the curve was owing to an elastic Auid similar to our of a parabola.

atmosphere, existing in the powder in an After the publication of Newton's Principia it extremely condensed state, which being suddenly might have been expected that the defects of the freed from the powder, by the combustion, theory would be ascribed to their true cause, the expanded with an amazing force, and violently great resistance of the air to such swift motions; impelled the bullet, or whatever might oppose its as in that work he particularly considered the expansion. subject of such motions, and related the result of The intensity of this force of exploded gunexperiments made on slow motions at least ; by powder Mr. Robins ascertained in different which it appeared that in such motions the resist- ways, after the example of Mr. Hawkesbee, reance increases as the square of the velocities, and lated in the Philosophical Transactions, No. 295, he even hints a suspicion that it will increase and in his Physico-Mechanical Experiments, p. above that law in swifter motions, as is now 81. One of these is by firing the powder in the known to be the case. So far, however, were air thus :-A small quantity of the powder is those who treated this subject scientifically from placed in the upper part of a glass tube, and the making a proper allowance for the resistance of lower part of the tube is immerged in water, the the atmosphere that they still neglected it, or water being made to rise so near the top that rather opposed it, and their theories accordingly only a small portion of air is left in that part differed most egregiously from the truth. Huy- where the powder is placed ; then in this situagens alone seems to have attended to this princi- tion, the communication between the upper part ple; for in the year 1590 he published a treatise of the tube and the external air being closed, the on gravity, in which he gave an account of some powder is fired by means of a burning glass, or experiments tending to prove that the track of otherwise; the water descends upon the exploall projectiles, moving with very swift motions, sion, and stands lower in the tube than before, was widely different from that of a parabola. by a space proportioned to the quantity of powThe rest of the learned enerally acquiesced in der fired. Another way was by firing the powder the justness and sufficiency of Galileo's doctrine. in vacuo, viz. in an exhausted receiver, by Nor was any farther notice taken of these errors dropping the grains of powder upon a hot iron till the year 1716, at which time Mr. Ressons, a included in the receiver. By this means a perFrench officer of artillery, of great merit and ex- manent elastic fluid was generated from the fired perience, gave in a memoir to the Royal Acade- powder, and the quantity of it was always in my, importing that, although it was agreed that proportion to the quantity of powder that was theory joined with practice did constitute the used, as was found by the proportional sinking perfection of every art, yet experience had taught of the mercurial gauge annexed to the air-puinp. him that theory was of very little service in the The result of these experiments was, that the use of mortars: that the works of M. Blondell weight of the elastic air thus generated was equal had justly enough described the several parabolic to three-tenths of the compound mass of the gunlines according to the different degrees of the ele- powder which yielded it, and that its bulk when vation of the piece: but that practice had con- cold, and expanded to the rarity of the common vinced him there was no theory in the effect air, was about 240 times the hulk of the powder; of gunpowder; for having endeavoured, with and consequently in the same proportion would the greatest precision, to point a mortar according such Auid at first, if it were cold, exceed the to these calculations, he had never been able to force or elasticity of the atmosphere. But as Mr. establish any solid foundation upon them.' One Robins found, by another ingenious experiment, instance only occurs, in which D. Bernouill that air heated to the extreme degree of the white applies the doctrine of Newton to the motions of heat of iron has its elasticity quadrupled, or is projectiles, in the Com. Acad. Petrop. tom. ii. four times as strong, he thence inferred that the

Besides which nothing farther was force of the elastic air generated as above, at the done in this business till the time of Robins, who moment of the explosion, is at least four times published a treatise in 1742, intitled New Prin- 240, or 960, or, in round numbers, about 1000 ciples of Gunnery, in which he treated particu- times as strong as the elasticity or pressure of the larly not only of the resistance of the atmosphere, atmosphere on the same space. but also of the force of gunpowder, the nature Having thus determined the force of the gunand effects of different guns, and almost every powder, or intensity of the agent by which the thing else relating to the flight of military pro- projectile is to be urged, Mr. Robins proceeded jectiles; and indeed he carried the theory of to determine the effects it will produce, or the gunnery nearly to its utmost perfection. velocity with which it will impel a shot of a

The first thing considered by Mr. Robins, and given weight from a piece of ordnance of given which is indeed the foundation of all other parti- dimensions; which is a problem strictly limited culars relating to gunnery, is the explosive force and perfectly soluble by mathematical rules, and of gunpowder. M. de la Hire, in his History of is in general this :-Given the first force, and the the Academy of Sciences for the year 1702, sup- law of its variation, to determine the velocity posed that this force may be owing to the in- with which it will impel a given body, in passing

p. 338, &c.




through a given space, which is the length of the he Aght of the shot through a given space, or bore of the gun. In the solution of this problem by measuring the range of the shot at a given Mr. Robins assumes these two postulates, viz. 1. elevation, and thence computing on the parabolic That the action of the powder on the bullet hypothesis what velocity would produce this range. ceases as soon as the bullet is out of the piece; The first method labors under this insurmountaod 2d. That all the powder of the charge is able difficulty, that the velocities of these bodies fired and converted into elastic fluid before the are often so swift, and consequently the time obbullet is sensibly moved from its place : assump- served is so short, that an imperceptible error in tions which for good reasons are found to be in that time may occasion an error in the velocity many cases very near the truth. It is to be noted thus found of 200, 300, 400, 500, or 600 feet in a also, that the law by which the force of the elastic second. The other method is so fallacious, by fuid varies is this, viz. that its intensity is di- reason of the resistance of the air (to which inrectly as its density, or reciprocally proportional equality the first is also liable), that the velocities to the space it occupies, being so much the thus assigned may not be perhaps the tenth part stronger as the space is less: a principle well of the actual velccities sought. known, and common to all elastic fluids. Upon “To remedy then these inconveniences, I have these principles, then, Mr. Robins resolves this invented a new method of finding the real veloproblem, by means of the thirty-ninth proposition cities of bullets of all kinds; and this to such a of Newton's Principia, in a direct way, and the degree of exactness (which may be augmented result is equivalent to this the when the too at pleasure) that in a bullet moving with the quantities are expressed by algebraic symbols ; velocity of 1700 feet in a second, the error in viz. the velocity of the ball.

the estimation of it need never amount to its 10 a b

500th part; and this without any extraordinary u=27130V x log.

nicety in the construction of the machine.' cd

Mr. Robins then gives an account of the ma223 ad


chine by which he measures the velocities of the or = 100V

x tog ; balls, which machine is simply this : viz. a penwhere v is the velocity of the ball, a the length of horizontal axis, against which block are to be

dulous block of wood suspended freely by a the charge of powder, b the whole length of the fired the balls whose velocities are to be deterbore, c the specific gravity of the ball, or weight mined. of a cubic foot of the same matter in ounces, d • This instrument thus fitted, if the weight of the diameter of the bore, w the weight of the ball the pendulum be known, and' likewise the reja ounces. For example, suppose a= 24 inches, b= 45 its centre of oscillation from its axis of suspension,

spective distances of its centre of gravity and of inches, c=11345 ozs. for a ball of lead, and it will thence be known what motion will be

7 120 communicated to this pendulum by the percussion dfinch; then v=271307 x log.


7 of a body of a kutown weight moving with a

known degree of celerity, and striking it in a = 1674 feet per second, the velocity of the ball.

given point; that is, if the pendulum be supOr, if the weight of the bullet be w=

posed at rest before the percussion it will be 1



known what vibration it ought to make in con1115 X 189 120

sequence of such a determined blow; and, on x log. =1674 feet, as be- the contrary, if the pendulum, being at rest, is 29 X 32 7

struck by a body of a known weight, and the fore.

vibration which the pendulum makes after the • Having in this proposition,' says Mr. Robins, blow is known, the velocity of the striking body (shown how the velocity which any bullet ac

may thence be determined. quires from the force of powder may be com- Hence then, if a bullet of a known weighi pated upon the principles of the theory laid down strikes the pendulum, and the vibration which in the preceding propositions, we shall next the pendulum makes in consequence of the stroke show that the actual velocities with which bul- be ascertained, the velocity with which the ball lets of different magnitudes are impelled from moved is thence to be known.' different pieces, with different quantities of Our author then explains his method of compowder, are really the same with the velocities puting velocities from experiments with this assigned by these computations, and conse- machine ; which method is rather troublesome quently that this theory of the force of powder, and perplexed, as well as the rules of Euler and here delivered, does unquestionably ascertain the Antoni, who followed him in this business; but true action and modification of this enormous

a much plainer rule is given in Hutton's Tracts, power.

vol. i. p. 119, where such experiments are ex* But, in order to compare the plained at full length, and this rule is expressed municated to bullets by the explosion with the by either of the two following formulas, velocities resulting from the theory by computation, it is necessary that the actual velocities with

= 5:6727cg x = 614:58cg x which bullets move should be capable of being

bir discovered, which yet is impossible to be done by the velocity; where v denotes the velocity of the any methods hitherto made public. The only ball when it strikes the pendulum, p the weight means hitherto practised by others for that pur- of the pendulum, b the weight of the ball, c the pose have been either by observing the time of chord of the arc described by the vibration to the Vol. XVIII.


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P+6 birn'

radius, rg the distance below the axis of motion initial velocities within any tolerable degree of to the centre of gravity, o the distance to the cen- truth. The knowledge of this velocity is of the tre of oscillation, i the distance to the point of utmost comsequence in gunpery; by means of impact, and n the number of oscillations the it, together with the law of the resistance of the pendulum will perform in one minute, when medium, every thing is determinable relative to made to oscillate in small arcs. The latter of that business; for, besides its being an excellent these two theorems is much the easiest, both be- method of trying the strength of different sorts cause it is free of radicals, and because the value of powder, it gives us the law relative to the of the radical vo, in the former, is to be first different quantities of powder, to the different computed from the number n, or number of os- weights of shot, and to the different lengths and cillations the pendulum is observed to make. sizes of guns. Besides these, there does not With such machines Mr. Robins made a great seem to be any thing wanting to answer any ennumber of experiments with musket barrels of quiry that can be made concerning the flight and different lengths, with balls of various weights, ranges of shots except the effects arising from the and with different charges or quantities of pow- resistance of the medium. In these experiments der. He has set down the results of sixty-one the weights of the pendulums employed were of these experiments, which nearly agree with from 300 to nearly 600 pounds. In that paper the corresponding velocities as computed by his is described the method of constructing the matheory of the force of powder, and which there- chines, of finding the centres of gravity and fore establish that theory on a sure foundation. oscillation of the pendulum, and of making the

From these experiments, as well as from the experiments, which are all set down in the form preceding theory, many important conclusions of a journal, with all the minute and concomiwere deduced by Mr. Robins; and indeed, by tant circumstances; also the investigation of the means of these, it is obvious that every thing new and easy rule set down just above, for may be determined relative both to the true the- computing the velocity of the ball from the exory of projectiles, and to the practice of artillery; periments. The charges of powder were varied for, by firing a piece of ordnance charged in a from two to eight ounces, and the shot from one similar manner against such a ballistic pendulum to nearly three pounds. And froin the whole from different distances, the velocity lost by were clearly deduced the inferences we have alpassing through such spaces of air will be found, ready given. and consequently the resistance of the air, the In the year 1786 was published the first vo. only circumstance that was wanting to complete lume of Dr. Hutton's Tracts, in which is detailthe theory of gunnery or military projectiles; and ed, at great length, another very extensive course of this kind Dr. Hutton made a great number of of experiments which were carried on at Woolexperiments with cannon balls, and has thereby wich in the years 1783, 1784, and 1785, by obtained the whole series of resistances to such a order of the duke of Richmond, master general ball when moving with every degree of velocity, of the ordnance. The objects of this course we from O up to 2000 feet per second of time. In the have also enumerated in the theoretic part of this structure of artillery they may likewise be of the treatise. greatest use; for hence may be determined the These objects were obtained in a very perfect best lengths of guns; the proportions of the shot and accurate manner; excepting only the article and powder to the several lengths; the thickness of ranges, which were not quite so regular and of a piece, so as it may be able to confine, with- uniform as might be wished. The balls too were out bursting, any given charge of powder; as most of them of one pound weight; but the powalso the effect of wads, chambers, placing of the der was increased from one ounce up till the bore vent, ramming the powder, &c. For the many was quite full, and the pendulum was from 600 other curious circumstances relating to this sub- to 800 lbs. weight. The conclusions from the ject, and the various other improvements in the whole were as follow :theory and practise of gunnery made by Mr. Ro- *1. That the former law, between the charge bins, consult the first volume of his Tracts, col- and velocity of ball, is again confirmed, viz. that lected and published by Dr. Wilson in the year the velocity is directly as the square root of the 1761, where ample information may be found. weight of powder, as far as to about the charge

In the year 1755, says Dr. Hutton, in con- of eight ounces : and so it would continue for all junction with several able officers of the royal charges, were the guns of an indefinite length. artillery and other ingenious gentlemen, I under- But as the length of the charge is increased, and took a course of experiments with the ballistic bears a more considerable proportion to the length pendulum, in which we ventured to extend the of the bore, the velocity falls the more short of machinery to cannon shot of one, two, and three that proportion. pounds weight. An account of these experiments • 2. That the velocity of the ball increases with was published in the Philosophical Transactions the charge to a certain point, which is peculiar for 1778; and for which the Royal Society to each gun, where it is greatest; and that, by honored me with the prize of the gold medal. farther increasing the charge, the velocity graThese were the only experiments that I know of dually diminishes till the bore is quite full of which had been made with cannon balls for this powder. That this charge for the greatest velopurpose, although the conclusions to be deduced city is greater as the gun is longer, but not greater from such are of the greatest importance to those however in so high a proportion as the length of parts of natural philosophy which are dependent the gun is; so that the part of the bore filled on the effects of fired gunpowder: nor do I know with powder bears a less proportion to the whole of any other practical method of ascertaining the in the long guns, than it does in the short ones ;

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the part of the whole which is filled being indeed elevation of the gun, the weight and dimensions nearly in the reciprocal subduplicate ratio of the of the powder and shot, with the range and time length of the empty part. And the other circum- of flight, and the first velocity of the ball. From stances are as in this table.

which it is to be hoped that the measure of the Table of Charges producing the greatest Velocity. mined, and thereby lay the foundation for a true

resistance of the air to projectiles may be deter

and practical system of gunnery, which may be Gun. Length of Length Part of the Weight of as well useful in service as in theory.' aum. the bore, filled, whole. the powder "Since the publication of those Tracts,' says

Dr. Hutton, 'we have prosecuted the experiments inches. inches.

still farther from year to year, gradually extend28.2 8.2


ing our aim to more objects, and enlarging the 38.1 9.5


guns and machinery, till we have arrived at ex57.4 10.7


periments with the six-pounder guns, and pen4 79.9 12:1


dulums of 1800 lbs. weight. One of the new

objects of enquiry was the resistance the atmo"3. It appears that the velocity continually in- sphere makes to military projectiles; to obtain creases as the gun is longer, though the increase which the guns have been placed at many difin velocity is but very small in respect of the in- ferent distances from the pendulum, against which crease in length, the velocities being in a ratio they are fired, to get the velocity lost in passing somewhat less than that of the square roots of the through those spaces of air; by which, and the length of the bore, but somewhat greater than that use of the whirling machine, described near the of the cube roots of the length, and is indeed end of the first volume of Robins's Tracts, for nearly in the middle ratio between the two. the slower motions, I have investigated the resist

64. The range increases in a much less ratio ance of the air to given balls moving with all than the velocity, and indeed is nearly as the degrees of velocity from O up to 2000 feet per square root of the velocity, the gun and elevation second; as well as the resistance for many debeing the same. And, when this is compared grees of velocity to planes and figures of other with the property of the velocity and length of shapes, and inclined to their path in all varieties gun in the foregoing paragraph, we perceive that of angles; from which I have deduced general very little is gained in the range by a great in- laws and formulas for all such motions. crease in the length of the gun, the charge being * Mr. Robins made also similar experiments on the same. And indeed the range is nearly as the the resistance of the air, but being only with fifth root of the length of the bore, which is so musket bullets, on account of their smallness and small an increase as to amount only to about of their change of figure by the explosion of the one-seventh part more range for a double length powder, I find they are very inaccurate, and con

siderably different from those above mentioned, 5. It also appears that the time of the ball's which were accurately made with pretty conflight is nearly as the range; the gun and eleva- siderable cannon balls of iron. For this reason tion being the same.

we may omit here the rules and theory deduced 66. It appears that there is no sensible differ- from them by Mr. Robins, till others more corence caused in the velocity or range, by varying rect shall have been established. All these exthe weight of the gun, nor by the use of wads, periments indeed agree in evincing the very nor by different degrees of ramming, nor by enormous resistance the air makes to the swift firing the charge of powder in different parts of it. motions of military projectiles, amounting in

"7. But a great difference in the velocity arises some cases to twenty or thirty times the weight from a small degree of windage. Indeed with the of the ball itself; on which account the common usual established windage only, namely, about rules for projectiles deduced from the parabolic one-twentieth of the calibre, no less than between theory are of little or no use in real practice; for one-third and one-fourth of the powder escapes from these experiments it is clearly proved that and is lost. And, as the balls are often smaller the track described by the flight even of the than that size, it frequently happens that half the heaviest shot is neither a parabola, nor yet appowder is lost by unnecessary windage. proaching any thing near it, except when they

8. It appears that the resisting force of woud are projected with very small velocities; insoto balls fired into it is not constant. And that much that some balls, which in the air range only the depths penetrated by different velocities or to the distance of one mile, would in vacuo, charges are nearly as the logarithms of the charges; when projected with the same velocity, range instead of being as the charges themselves, or, above ten or twenty times as far. which is the same thing, as the square of the Mr. Benjamin Thompson (the late count Rumvelocity.

ford) instituted a very considerable course of *9. These and most other experiments show experiments of the same kind as those of Mr. that balls are greatly deflected from the direction Robins, with musket barrels, which was pubthey are projected in; and that so much as 300 lished in the Philosophical Transactions, vol. 71, or 400 yards in a range of a mile, or almost one- for the year 1781. In these experiments the confourth of the range, which is nearly a deflection clusions of Mr. Robins are generally confirmed, of an angle of 15°.

and several other curious circumstances in this *10. Finally, these experiments furnish us with business are remarked by Mr. Thompson. This the following concomitant data, to a tolerable de- gentleman also pursues a hint thrown out by Mr. gree of accuracy, namely, the dimensions and Robins, relative to the determining the velocity of

of gun.

a ball from the recoil of the pendulous gun itself. reserves to himself without giving any particular Mr. Robins, in the eleventh proposition, remarks, description of it. that the effect of the exploded powder upon the The other, or second chapter of Mr. Robins's recoil of a gun is the same whether the gun is work, in eight propositions, treats of the resistcharged with or without a ball; and that the chord ance of the air, and of the track described by the or velocity of recoil with the powder alone, being flight of shot and shells.' And of these, the first substracted from that of the recoil when charged proposition describes the general principles of with both powder and ball

, leaves the velocity the resistance of fluids to solid bodies moving in which is due to the ball alone. Thence Mr. them. Here Mr. Robins discriminates between Thompson observes that the inference is obvious, continued and compressed Auids, which immeviz. that the momentum thus communicated to diately rush into the space quitted by a body in the gun by the ball alone, being equal to the mo- them, and whose parts yield to the impulse of mentum of the ball, this becomes known ; and the body without condensing and accumulating therefore, being divided by the known weight of before it; and such fluids as are imperfectly the ball, the quotient will be its velocity. Mr. compressed, rushing into a void space, with a Thompson sets a great value on this new rule, limited velocity, as in the case of our atmosphere, the velocities by means of which he found to which condenses more and more before the ball agree nearly with several of those deduced from as this moves quicker, and also presses the less the motion of the pendulum; and in the other behind it, by following it always with only a cases, in which they differed greatly from these, given velocity: hence it happens that the former he very inconsistently supposes that these latter Huid will resist moving bodies in proportion to ones are erroneous. In the experiments, how- the square of the velocity, while the latter resists ever, contained in Dr. Hutton's Tracts, a great in a higher proportion. The second proposition multitude of those cases are compared together, is to determine the resistance of the air to proand the inaccuracy of that new rule is fully proved. jectiles by experiments. One of the methods

Having in the ninth proposition compared to- for this purpose is by the ballistic pendulum, gether a number of computed and experimented placing the gun at different distances from it, by velocities of balls to verify his theory; Mr. which he finds the velocity lost in passing Robins, in the tenth proposition, assigns the through certain spaces of air, and consequently changes in the force of powder, which arise from the force of resistance to such velocities as the the different state of the atmosphere, as to heat body moves with in the several parts of its path. and moisture, both which he finds have some ef- And another way was by firing balls with a fect on it, but especially the latter. In the known given velocity, over a large piece of eleventh proposition he investigates the velocity water, in which the fall and plunge of the bali which the flame of gunpowder acquires by ex- could be seen, and consequently the space it panding itself, supposing it fired in a given piece passed over in a given time. By these means of artillery, without either a bullet or any other Mr. Robins determined the resistance of the air body before it. This velocity he finds is upwards to several different velocities, all which showed of 7000 feet per second. But the celebrated that there was a gradual increase of the resistEuler, in his commentary on this part of Mr. ance, over the law of the square of the velocity, Robins's book, thinks it may be still much as the body moved quicker. In the remaining greater, and in this proposition too it is that Mr. propositions of this chapter he proceeds a little Robins declares his opinion above alluded to, farther in this subject of the resistance of the air; viz. that the effect of the powder upon the recoil in which he lays down a rule for the proportion of the gun is the same in all cases whether fired of the resistance between two assigned velocities; with a ball or without one. In the twelfth pro- and he shows that when a twenty-four pound ball, position be ascertains the manner in which the fired with its full charge of powder, first issues Hame of powder impels a ball which is laid at a from the piece, the resistance it meets with from considerable distance from the charge; showing the air is more than twenty times its weight. He here that the sudden accumulation and density farther shows that the track described by the of the fluid against the ball is the reason that the flight of shot or shells is neither a parabola, nor barrel is so often burst in those cases. In the nearly a parabola, unless they are projected with thirteenth proposition he enumerates the various small velocities ; and that' bullets in their flight kinds of powder, and describes the properest are not only depressed beneath their original methods of examining its goodness. "He here direction, by the action of gravity, but are also shows that the best proportion of the ingredients frequently driven 10 the right or left of that diis when the saltpetre is three-fourths of the whole rection by the action of some other force :' and, compound mass of the powder, and the sulphur in the eighth or last proposition, he pretends to and charcoal the other one-fourth between them, show that the depths of penetration of balls in equal quantities. In this proposition Mr. into firm substances are as the squares of the veRobins takes occasion to remark upon the use of locities. But this is a mistake ; for neither does eprouvettes, or methods of trying powder; con- it appear that his trials were sufficiently numedemving the practice of the English in using rous or various, nor were his small leaden balls what is called the vertical eprouvette; as well as fit for this purpose ; and it has appeared, from a that of the French, in using a small mortar with number of trials with iron cannon balls, that the a very large ball, and a small charge of powder, penetrations are in a much lower proportion, and and instead of these he strongly recommends the that the resisting force of wood is not uniform. use of his ballistic pendulum for its great accu- See Dr. Hutton's Tracts. racy. But for still more despatch, he says, he In the small tracts appended to the principles, should use another method, which however he in this volume, Mr. Robins prosecutes the sub

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