Projectiles

are such bodies as, being put in motion by any great force, are then cast off, or let go from the place where they received their quantity of motion; as a shell or shot from a piece of artillery, a stone thrown from a sling, or an arrow from a bow, . This line is commonly taken for a parabola, and the ranges are computed from the properties of the curves. The a umption would be just, in case the ball, in its motion, met with no resistance: but, the resistance of the air to swift motions being very great, the curve described by the shot is neither a parabola, nor near it: and by reason of the resistance, the angle which gives the greatest amplitude is not 45 degrees, as commonly supposed, but something le , probably 43¹⁄₂. Hence the sublime mathematics are absolutely nece ary in the investigation of the track of a shell or shot in the air, known by the name of military projectiles. Gallileo having discovered that bodies projected in vacuo, and in an oblique direction to the horizon, do always describe a parabola, he concluded that this doctrine was not sufficient to determine the real motion of a military projectile: for, since shells and shot move with a great velocity, the resistance of the air becomes so great with respect to the weight of the projectile, that its effect turns the body very considerably from the parabolic tract; so that all calculations, grounded on the nature of this curve, are of little use on these occasions. This is not to be wondered at, since Gallileo, in his enquiry, paid no regard to any other force acting on bodies, than the force of gravity only, without considering the resistance of the air. Every body, moving in a fluid, suffers the action of two forces: the one is the force of gravity, or the weight of the body; and it is to be observed, that this weight is le than the natural weight of the body, that being diminished by an equal bulk of the fluid in which the body moves. The other force is that of the resistance, which is known to be proportional to the squares of the velocity of the body; and when the body is a globe, as is commonly supposed, the direction of this force is diametrically opposite to that of the motion of the body. This force changes continually, both in quantity and direction; but the first force remains constantly the same. Hence, the point in question is, to determine the curve which a body projected obliquely, must describe when acted upon by the two forces just now mentioned. Although this question is easily reduced to a problem purely analytical, the great Newton, notwithstanding his ingenious endeavors, did not arrive at a complete solution of it. He was the first who attempted it, and having succeeded so well in the supposition, that the resistance is proportional to the velocity, it is almost inconceivable that he did not succeed, when the resistance is supposed proportional to the squares of the velocity, after solving a number of questions incomparably more difficult. The late Mr. John Bernoulli gave the first solution of this problem, from which he drew a construction of the curve, by means of the quadratures of some transcendent curves, whose description is not very difficult. This great problem was, therefore, very well solved long ago; yet the solution, however good in theory, is such as has hitherto been of no use in practice, nor in correcting the false theory grounded on the parabola, to which the artillerist is still obliged to adhere, notwithstanding he knows it to be insufficient. It is certain, that that solution has been of no real advantage towards improving the art of gunnery: it has only served to convince the student in that art, of the error of his principles, drawn from the nature of the parabola, although he is still to abide by them. It is indeed something to know, that the common rules are erroneous; but unle we know how much they err in any case, the advantage is very little. One may think it a work of infinite labor to establish rules for the flight of cannon shot, agreeable to the real curve which a body describes in the air: for although, according to the hypothesis of Gallileo, we want only the elevation of the piece, and the initial velocity, and it is therefore not difficult to calculate tables to show the greatest height of the projectile, and the point where it must fall in any proposed case; yet in order to calculate similar tables according to the true hypothesis, care must be taken, besides the two particulars already mentioned, to have respect as well to the diameter of the projectile as to its weight: therefore the practitioner will be reduced to the nece ity of calculating tables, as well for the diameter of each projectile, as for its weight; and the execution of such a work would be almost impracticable. We therefore refer the curious to Mr. Euler’s True Principles of Gunnery , translated, with many nece ary explanations and remarks, by the very learned and ingenious Hugh Brown.