Reflecting a few days ago upon the manner of ascertaining the initial velocities of Military projectiles, by means of the ballistic pendulum, it struck my mind that this method1 is not altogether accurate. I take the liberty therefore to state to you what appears objectionable in it, requesting if I am in an error you will have the goodness to rectify it.

The method above alluded to seems to be founded upon two mechanical principles; one is, that when one nonelastic body in motion2 strikes upon another at rest, the quantity of motion after the stroke, is the same as that before it; the other is, that if two bodies of unequal quantities of matter have equal momenta, the velocity of the larger body, must be as much less than that of the smaller, as its quantity of matter is greater—

Now to apply these principles; a ball in motion strikes the ballistic pendulum (the weight of which, as well as the centres of gravity, of oscillation, and of gyration are supposed to be known) at rest, and causes it to vibrate; now if we can determine the velocity with which the pendulum vibrates it is evident from the second of the above mentioned principles that we can determine the velocity of the ball—Now when the pendulum (after the first vibration) descends down again to its vertical position, it will have acquired the same velocity with which it began to ascend, and from the laws of falling bodies, the velocity of the centre of oscillation is such as a heavy body would acquire by falling freely through the versed-sine of the arc described by the same centre. But by the nature of the circle, the versed-sine is a third proportional to the diameter and chord of its arc; knowing therefore the diameter of the circle which is twice the distance of the centre of oscillation from the point of suspension; and also the chord of the arc through which this centre vibrates we can ascertain by a single proportion the versed-sine of the arc; and from the laws of falling bodies the velocity acquired in descending through it, that is the velocity of the centre of oscillation—

Having determined the velocity of the centre of oscillation we can from that ascertain the velocity of the point of impact (the point where the ball strikes the pendulum) for the velocities of those points will be to each other directly, as their distances from the point of suspension—Now since the ball in motion strikes upon the pendulum at rest, and thereby puts it in motion, by the first of the principles above stated; the momentum of the pendulum ought to be equal to that of the ball; and since we know the weight of the pendulum and of the ball, and have also determined the velocity with which the point of impact moves, we can (from the second principle laid down) by a single proportion ascertain the velocity of the ball—

But the query in my mind is, whether the ball striking the pendulum and penetrating it, communicates to it as great a quantity of motion as it would if it struck it with the same velocity without penetrating it, supposing them both perfectly non-elastic—

It appears to me that it would not, but that by the yeilding of the fibres of the wood before it, its force could be (as it were) deadened in such a manner, as to partake more of the nature of pressure than of percussion—There is also another consideration which appears to me to affect the accuracy of this method which is, that the motion of the ball, and of the point of impact are never in the same direction—for suppose the ball to strike the pendulum in a horizontal direction then by the vibration of the pendulum, the point of impact must describe the arc of a circle, and therefore must move between a horizontal and a perpendicular direction … A part of the force of the ball must therefore be spent upon the axis of motion of the Pendulum—From these considerations it appears to me that this method gives the velocity of Military projectiles less than the true velocity—

I will now Sir, take the liberty of proposing for your consideration a method by which I am induced to believe that the initial velocities of cannon balls of any size may be accurately determined. It is found by experiment, (and the same conclusion is also deducible from theory) that bodies descending freely by the force of gravity fall through 161⁄123 feet during the first second of time. we also know that if a body continues to descend for any number of seconds by the force of gravity, that the spaces through which it would fall during each of those seconds would be as the odd numbers 1, 3, 5, 7, and also that the whole spaces through which it would have fallen at the end of each of those seconds would be as the squares of the times,4 that is as the squares of the numbers 1, 2, 3, 4, 8 &c. Now from what has been said, it is evident that if a second be divided into four equal parts, that a body falling freely by the force of gravity, would fall through spaces during each of those parts, that would be to each other as the odd numbers 1, 3, 5, 7, and also that the spaces fallen through at the end of each of those parts would be as the squares of the numbers 1, 2, 3, 4, that is as the numbers 1, 4, 9, 16. consequently at the end of the first part it would have fallen 11⁄192 feet; at the end of the second part 41⁄48 feet, at the end of the third 93⁄64 feet, and at the end 4th 161⁄12 feet—Now to apply what has been said to determining the initial velocity of cannon balls; suppose a cannon was placed and accurately levelled on a plain sand beach at the different altitudes,5 11⁄192, 41⁄48, 93⁄64 & 161⁄126 feet above the surface of the ground; then if it were fired at those heights7 it is evident that the ball ought to come to the ground in the same time it would if it were to drop from the muzzle of the gun; for the force of the powder acting in a horizontal direction does not counteract the force of gravity upon the ball, which therefore by the composition and resolution of forces must come to the ground in the same time that it would if it were not acted upon by the powder at all; the ball therefore when fired from the first elevation would fly before striking the ground ¼th of a second; when fired from the second, ½ a second—when fired from the third ¾ of a second, and when fired from the fourth 1 second. and if the distance over which it passed during each of those times were accurately measured we should obtain the velocities required—

By this method I am inclined to think may be determined with a considerable degree of accuracy the ratio of the resistance of the air upon cannon balls of the same diameter moving with different degrees of velocity; for suppose we have ascertained from experiment the charges of Powder that will give initial velocities that are as the numbers 1. 2, 3. “& so on; and suppose the charge of No 1, to give a velocity of 600”8 feet per second then it is evident Nos. 2 & 3 would give velocities of 1200 & 1800 feet per second respectively supposing they met with no more resistance from the air than the first one—as much therefore as they fell short of those velocities, so much more must they be resisted by the air than the first one—

I will thank you Sir to examine what I have written, and if I am in an error I hope you will have the goodness to rectify it and give me notice—