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From this expression of the time we learn that however great the velocity of o and the height to which this body will rise, may be, the time of its ascent is limited. It never can exceed the time of falling from the height a in vacuo in a greater proportion than that of a quadrantal arch to the radius, nearly the proportion of eight to five. A twenty-four pound iron ball cannot continue rising above fourteen seconds, -ven if the resistance to quick motions did not increase faster than the square of the velocity. It probably will attain its greatest height in less than twelve seconds, let its velocity be ever so great. In the preceding example of the whole
- ta V ascent vio, and the time t = T x arc. tan. 7
From inspecting the diagram p. 153, it is evident that the arch AI is to the tangent A G as the sector ICA to the triangle C GA; therefore the time of attaining the greatest height in the air is to that of attaining the greatest height in vacuo (the velocities of projection being the same) as the circular sector to the corresponding triangle. If therefore a body be projected upwards with the terminal velocity, the time of its ascent will be to the time of acquiring this velocity in vacuo as the area of a circle to the area of the circumscribed square.
2. The height H to which a body will rise in a void is to the height h to which it would rise through the air when projected with the same
u”--V ; for
wo - V2 height to which it will rise in vacuo is 7. and
- - - - --- - - . . .301.03 which it will rise in the air is 43.429 of the height
- - - - --- - - 5 . to which it will rise in vacuo, or + in round
We thought it necessary to treat of the perpendicular ascents and descents of heavy bodies through the air thus particularly, that the reader may conceive distinctly the quantities which he is thus combining in his algebraic operations, and may see their connexion in nature with each other. We shall also find that, in the present state of our mathematical knowledge, this simple state of the case contains almost all that we can determine with any confidence.
Sect. VI.-OF THE OBLIQUE PROJECTION.
We would now proceed to the general problem to determine the motion of a body projected in any direction, and with any velocity. But our readers will believe that this must be a difficult subject, when they see the simplest cases of rectilineal motion abundantly abstruse: it is indeed so difficult that Sir Isaac Newton has not given a solution of it, and has thought himself well employed in making several approximations, in which the fertility of his genius appears in great lustre. In the tenth and subsequent propositions of the second book of his Principia he shows what state of density in the air will comport with the motion of a body in any curve whatever; and then, by applying this discovery to several curves which have some similarity to the
path of a projectile, he finds one which is not
very different from what we may suppose to obtain in our atmosphere. But even this approximation was involved in such intricate calculations that it seemed impossible to make any use of it. In the second edition of the Principia, published in 1713, Newton corrects some mistakes in the first, and carries his approximations much farther, but still does not attempt a direct investigation of the path which a body will describe in our atmosphere. This is somewhat surprising. In prop. 14, &c., he shows how a body, actuated by a centripetal force, in a medium of density varying according to certain law, will describe an eccentric spiral, of which he assigns the properties, and the law of description. Had he supposed the density constant, and the difference between the greatest and least distances from the centre of centripetal force exceedingly small in comparison with the distances themselves, his spiral would have coincided with the path of a projectile in the air of uniform density, and the steps of his investigation would have led him immediately to the complete solution of the problem. For this is the real state of the case. A heavy body is not acted on by equal and parallel gravity, but by a gravity inversely proportional to the square of the distance from the centre of the earth, and in lines tending to that centre nearly; and it was with the view of simplifying the investigation that mathematicians have adopted the other hypothesis. Not long after the publication of this second edition of the Principia, the dispute about the invention of the fluxionary calculus became very violent, and the great promoters of that calculus upon the continent proposed difficult problems to the mathematicians. Challenges of this kind frequently passed between the British and foreigners. Dr. Keill of Oxford had keenly espoused the claim of Sir Isaac Newton to this invention, and had engaged in a very acrimonious altercation with the celebrated John Bernouilli of Basle. Bernouilli had published in the Acta Eruditorum Lipsiae, an investigation of the law of forces, by which a body, moving in a resisting medium, might describe any proposed curve, reducing the whole to the simplest geometry. This is perhaps the most elegant specimen which he has given of his great talents. Dr. Keill proposed to him the particular problem of the trajectory and motion of a body moving through the air, as one of the most difficult. Bernouilli very soon solved the problem in a way much more general than it had been proposed, viz. without any limitation either of the law of resistance, the law of the centripetal force, or the law of density, provided only that they were regular, and capable of being expressed algebraically. Dr. Brooke Taylor, the celebrated author of the Method of Increments, solved it at the same time, in the limited form in which it was proposed. Other authors since that time have given other solutions; but they are all (as indeed they must be) the same in substance with Bernouilli's. Indeed they are all (Bernouilli's not excepted) the same with Newton's first approximations, modified by the steps introduced into the investigation of the spiral motions mentioned above; and we still think it most strange that Sir Isaac did not perceive that the variation of curvature, which he introduced in that investigation, made the whole difference between his approximations and the complete solution. All the solutions given of this problem depend upon a particular law of resistance assumed, without proving that to be the law by which a body is resisted in its motion through the air. This resistance is supposed to be in the duplicate ratio of the velocity; but even theory points out many causes of deviation from this law, such as the pressure and condensation of the air, in the case of very swift motions: and Mr. Robins's experiments are sufficient to prove that the deviations must be exceedingly great in such cases. Euler and all subsequent writers have allowed that it may be three times greater, even in cases which frequently occur; and Euler gives a rule for ascertaining with tolerable accuracy what this increase and the whole resistance may amount to. Let H be the height of a column of air whose weight is equivalent to the resistance taken in the duplicate ratio of the velocity. The whole resistance will be expressed by H +
23345 This number 28845 is the height in feet
of a column of air whose weight balances its elasticity. We shall not at present call in question his reasons for assigning this precise addition. They are rather reasons of arithmetical conveniency than of physical import. It is enough to observe that, if this measure of the resistance is introduced into the process of inves
tigation, it is totally changed: and it is not to? much to say that with this complication it requires the knowledge and address of a Euler to make even a partial and very limited approximation to a solution. Any law of the resistance, therefore, which is more complicated than what Bernouilli has assumed, namely, that of a simple power of the velocity, is abandoned by all the mathematicians, as exceeding their data: and they have attempted to avoid the error arising from the assumption of the duplicate ratio of the velocity either by supposing the resistance throughout the whole trajectory to be greater than what it is in general, or they have divided the trajectory into different portions, and assigned different resistances to each, which vary, through the whole of that portion, in the duplicate ratio of the velocities. Thus they make up a trajectory and motion which corresponds, in some tolerable degree, with what? With an accurate theory ! No ; but with a series of experiments. For, in the first place, every theoretical computation which we make proceeds on a supposed initial velocity; and this cannot be ascertained with any thing approaching to precision by any theory of the action of gunpowder that we are yet possessed of. In the next place, our theories of the resisting power of the air are entirely established on the experiments on the flight of shot and she ls, and are corrected and amended till they tally with the most approved experiments we can find. We do not learn the ranges of a gun by theory, but the theory by the range of the gun. After all, therefore, the practical artillerist must rely chiefly on the records of experiments contained in the books of practice at the academies, or those made in a more public manner. Even a perfect theory of the air's resistance can do him little service, unless the force of gunpowder were uniform. But this is far from being the Case. The experiments of Mr. Robins and Dr. Hutton show, in the most incontrovertible manner, that the resistance to a motion exceeding 1100 feet in a second is almost three times greater than in the duplicate ratio to the resistance to moderate velocities. Euler's translator, in his comparison of the author's trajectories with experiment, supposes it to be no greater. Yet the coincidence is very great. The same may be said of the Chevalier de Borda's. Nay, the same may be said of Mr. Robins's own practical rules; and yet his rules are confirmed by experience. But we must not infer, from all this, that the physical theory is of no use to the practical artillerist. It plainly shows him the impropriety of giving the projectile an enormous velocity. This velocity is of no effect after 200 or 300 yards at farthest, because it is so rapidly reduced by the prodigious resistance of the air. Mr. Robins has deduced several practical maxims of the greatest importance from what we already know of this subject, and which could hardly have been even conjectured without this knowledge. And we must still acknowledge that this branch ot physical science is highly interesting to the philosopher; nor should we despair of carrying it to greater perfection.
Certainly the most complete set of experiments made with a view of obtaining a rational theory of projectiles are those of Dr. Hutton, which were carried on at Woolwich during the years 1775, 1783, 1784, 1785, 1787, 1788, 1789, and 1791, the objects of which were very various, and some of the results highly important. The latter are thus enumerated by the author in the second volume of his Tracts :‘1. It is made evident, by these experiments, that powder fires almost instantaneously, seeing that nearly the whole of the charge fires, though the time be much diminished. “2. The velocities communicated to the shot of the same weight, with different quantities of powder, are nearly in the subduplicate ratio of those quantities; a very small variation, in defect, taking place when the quantities of powder become great. “3. And when shot of different weights are fired with the same quantities of powder the velocities communicated to them are nearly in the reciprocal subduplicate ratio of their weights. “4. So that, universally, shots which are of different weights, and impelled by the firing of different quantities of powder, acquire velocities which are directly as the square roots of the quantity of powder, and inversely as the square roots of the weight of the shot, nearly. ‘5. It would therefore be a great improvement in artillery to make use of shot of a long form, or of heavier matter; for thus the momentum of the shot, when fired with the same weight of powder, would be increased in the ratio of the square root of the weight of the shot. “6. It would also be an improvement to diminish the windage; for, by so doing, one-third or more of the quantity of powder might be saved.” (This, however, must be understood only to be true within certain limits.) ‘7. When the improvements mentioned in the two last cases are considered as both taking place it is evident that about half the quantity of powder might be saved, which is a very considerable object. But, important as this saving may be, it seems still to be exceeded by that of the guns: for thus a small gun may be made to have the effect of one of two or three times its size, in the present way, by discharging a long shot of two or three times the weight of its natural ball, or round shot: and thus a small ship might discharge shot as heavy as those of the greatest now made use of." The objects of the latter courses of experiments are thus detailed: viz. to ascertain, “1. The velocities with which balls are projected by equal charges of powder, from pieces of the same weight and calibre, but of different lengths. “2. The greatest velocities due to the different charges of powder, the weight and length of the gun being the same. “3. The greatest velocity due to the different lengths of guns; to be obtained by increasing the charge as far as the resistance of the piece is capable of sustaining. *4. The effect of varying the weight of the piece; every thing else being the same. ‘5. The penetration of balls into blocks of wood.
“6. The ranges and times of flight of balls, with the velocities, by striking the pendulum at various distances, to compare them with initial velocities, for determining the resistance of the medium.
‘7. The effects of wads, of ramming, of windage, &c.’
We shall now quote this author's expression for the resistance of the air, deduced from these experiments, and thence determine the ranges, times of flight, &c., of projectiles according to that hypothesis.
Theorem.—The resistance of the air, to a ball
rojected into it with any considerable velocity, is expressed by the formula r = (.000007565 vo — 00175 v) d”. But, for the smaller velocities, r = 0000044 d'v' will be a sufficiently near approximation, where r represents the resistance in avoirdupois pounds, d the diameter of the ball in inches, v the velocity in English feet. See Hutton's Tracts, vol. iii. p. 232.
PRob. I.--To determine the height to which a ball, projected perpendicularly upwards, will ascend, being resisted by the atmosphere.
Putting a to denote any variable and increasing height ascended by i. ball; v its variable and decreasing velocity there; d the diameter of the ball, its weight being w; m = 000007565, and n=-00175, the co-efficients of the two terms in the above theorem. Then (m v’ — n v) d” will be the resistance of the air against the ball jo. avoirdupois pounds, to which, if the weight of the ball be added, then (m v’ — n v) d” + w will be the whole resistance to the ball's motion, and
—m m) do *—
consequently (m v’ o d" + w – (m v to n v) d” + 1 = f; the retarding force. Hence the general formula v i = 2 g fi becomes — v i = 2
cause the velocity is decreasing, where g = 16*, feet, or sixteen feet, the descent of a bodv in one second by gravity.
making v negative, be
Erample 2.-Again, let the ball weigh twentyfour pounds, and, therefore, its diameter 5.6, and velocity 2000 feet per second, as before; then V* — 231.5 V + 18600 d 18600 d = 5782 feet, the height required. In the first of these examples, where the height is found to be only about half a mile, the ball would ascend to nearly twelve miles in a nonresisting medium; and hence we may see the immense effect of atmospheric resistance to the motion of projectiles. PRob. II.--To determine the time in which a ball will have acquired its greatest height, using the same formula of resistance as in the last case. Here the general value of t, determined on principles similar to those above employed, gives
or t =
p arc. tan. **) - q
X arc. tan. –p. rejecting
After the investigation of these problems, and some others of a similar nature, Dr. Hutton proceeds to the investigation of his principal one, viz. to determine the circumstances of ranges at different degrees of elevation; which we transcribe. “Rules for the general solution of this problem would be best derived from experiments; and these should be made at all elevations, and with all charges, and with various sizes of balls, observing both the ranges and times of flight in every experiment. Such experiments would give us the relations existing, in all cases, amongst these four terms, viz. the ranges, the times of flight, the velocity or charges, and the size of the balls. Numerous and various as are our experiments, as before related, and fruitful as they are in useful consequences, we have obtained but a small portion of those alluded to ; nor do I know of any proper set of such experiments any where to be found. Such must, therefore, still remain a valuable desideratum; the few that we have been able to make afford us but very few and imperfect rules, being chiefly as follows: —1. That the ranges with the one-pound balls, at an elevation of 15°, are nearly proportional to the times of flight. 2. That the ranges with the three-pound balls, at 45° of elevation, are nearly as the times of flight, and also as the projectile velocities. Besides these inferences, it does not appear that the experiments are extensive enough to afford any more useful conclusion. “By trials, however, amongst many of the numbers in art. 24, it appears that in most of them at an elevation between 45° and 30°, the time of flight is nearly equal to one-fourth of the square root of the range in feet, in which respect it nearly agrees with the similar rule for the time of flight in the parabolic theory, at the angle of 45° for the greatest range, which time, it is well known, is equal to one-fourth of the square root of the said range in feet. Whence it is probable that, with the help of a few other ranges at several elevations, some general relations might be evinced between the ranges and the times of flight, with the tangents of the elevation. “But such experiments and enquiries as these, unfortunately, it is no longer in my power either to procure, or by any means to promote; and we can, therefore, only endeavour to render, without them, what service we can to the state, and to philosophy, by such means as are in our power. “There are some few theoretical principles which it may be useful to notice here, as first mentioned by professor Robison. Thus balls of equal density, discharged at the same elevation with velocities which are proportional to the square roots of their diameters, will describe similar curves; because then the resistances will be in proportion as the momentum or quantity of motion. For the resistance r is d” v' nearly; d being the diameter, and v the velocity. But v being as V d, v' will be as do; consequently d” t” will be as-d’; that is, r is as do. But the momentum is as the magnitude or mass, which is as d” also, the cube of the diameter. Therefore the resistance is proportional to the momentum, when the velocity is as V d, or the square root of the diameter of the ball. In this case, then, the horizontal velocity at the vertex of the curve will be proportional to the terminal velocity; also the ranges, and heights, and all other similar lines in the curve, will be proportional to d, the diameter of the ball. And this principle may be of considerable use; for thus, 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.’ We shall have occasion to advert again to these valuable contributions of Dr. Hutton.
OF THE PRACTICE OF GUNNERY, OR MILITARY PROJECTILES.
Having laid before our readers the substance of the latest and most inproved theories of projectiles, we proceed to give them a brief sketch of the most improved modern practice.
Mr. Robins, in his preface to his New Principles of Gunnery, states that he had met with only four authors who had treated experimentally on this subject. The first of these is Collado in 1642, who has given the ranges of a falconet, carrying a three-pound shot, to every point of the gunner's quadrant, each point being the twelfth part, or 7° 30'. But from his numbers it is manifest that the piece was not charged with its usual allotment of powder. The result of his trials shows the ranges at the point-blanc, and the several points of elevations as below.
which greatest random, he says, in a calm day is at 42° elevation; but according to the strength of the wind, and as it favors or opposes the flight of the shot, the elevation may be from 43° to 36°. He does not say with what piece he made his trials, though from his proportion it seems to have been a small one. This however ought to have been mentioned, 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, also published treatises on this subject. The former of these was many years gunner of Dover Castle, where most of his experiments were made, the earliest of which are dated 1611, though his book was not published till 1646, and was entitled The Gunner's Glass. His principles were simple, and within certain limits very near the truth, though they were not rigorously so. He has given the actual ranges of different pieces of artillery at small elevations, all under 10°. His experiments are numerous, and appear to be made with great care and caution; and he has honestly set down some which were not reconcileable to his method : upon the whole he seems to have taken more pains, and to have had a juster knowledge of his business than is to be found in most of his practical brethren. Galileo printed his Dialogues on Motion in the year 1646. In these he pointed out the general laws observed by nature in the production and composition of motion, and was the first who described the actions and effects of gravity on falling bodies: on these principles he determined that the flight of a cannon-shot, or of any other projectile, would be in the curve of a parabola, unless so far as it should be diverted from that track by the resistance of the air. He also proposed the means of examining the inequalities which arise thence, and of discovering what sensible effects that resistance would produce in the motion of a bullet at some given distance from the piece. Notwithstanding these determinations and hints it seems, however, that those who came after Galileo never imagined that it was necessary to consider how far the operations of gunnery were affected by this resistance. Instead of this, they boldly asserted, without making the experiment, that no great variation could