Discoveries and Inventions of the Nineteenth Century

in Messrs. Lloyd and Hadcock’s treatise on Artillery. The problem is thus stated:—“An 11–in. breech-loading howitzer” (a howitzer is a piece of ordnance used for firing 176at high angles) “fires a 600–lb. projectile with an initial velocity of 1,120 foot-sec. at an elevation of 20°. Find the range, time of flight, and angle of descent.” We shall calculate these points on the suppositions adopted with regard to Fig. 80, and with no higher mathematics than common multiplication and division. It will have been observed that we supposed two motions that really take place simultaneously to take place successively and independently: one in the direction of the line of fire, due to the initial velocity; the other vertically downwards, due to the action of gravity, the final result being the same. This affords an excellent illustration of another of Newton’s laws of motion, and should be considered by the reader in this connection. The law itself admits of being stated in various ways, as thus:—“Whenever a force acts on a body, it produces upon it exactly the same change of motion in its own direction, whether the body be originally at rest or in motion in any direction with any velocity whatever—whether it be at the same time acted on by other forces or not.” Or again: “When two forces act in any direction whatever on a body free to move, they impress upon it a motion which is the superposition (or compounding) of those that it would receive if each force acted separately.” The law is given also in the following form (Thomson and Tait):—“When any forces act on a body, then, whether the body be originally at rest or moving with any velocity and in any direction, each force produces in the body the exact change of motion which it would have had had it acted singly on the body originally at rest.” In all of these expressions the word “forces” is used, and a very convenient word it is, but it may be noted in passing, nothing but a word; for it stands for no real self-existing things, since, apart from observed changes of motion in bodies, forces for us have no existence. Nevertheless, it is useful for the sake of abbreviating statements about changes of motion, to regard these actions as produced by imaginary agents—imagined for the time and for this purpose, and therefore vainly to be sought for in the realm of reality. Fig. 81.—Diagram. In dealing with the trajectory of the howitzer’s projectile through airless space we have no concern with its diameter nor with its weight. We use the little diagram, Fig. 81, to represent the motions,—c being a horizontal line, a, a vertical one, the angle at B is therefore a right angle, and we assume that at A to be 20°. Now, the most elementary geometry teaches us that every triangle having these angles will have the lengths of its sides in the same invariable proportions one to another whatever may be the size of the triangle itself, and it has been found convenient to calculate these proportions once for all, not merely for angle 20°, but for every angle up to 90°. Besides this, distinct names have been given to the proportions of every side of the triangle to each of the other two sides. Thus in the triangle before us, if we take a, b, and c to represent the numbers expressing the lengths of the sides against which they are placed, a divided by b, that is a ÷ b, or a/b, is called the sine of angle 20°, while c/b is named the cosine of that angle, etc. These therefore are numbers which are given in mathematical tables, and we find by these that sine 20° = 0·3420201, and cosine 20° = 0·9396926, and these with the 177initial velocity give us all the data we require. We may first find the time the projectile would take to reach the ground level, or strictly that of the muzzle of the gun at B. Taking t to stand for this time, we know that AC = 1,120 × t, but CB will be the distance that a body would fall from rest at C by the influence of gravity in that same time, t, and it is known by experiment that this distance is 16·1 feet multiplied by the square of the time from rest in seconds. We have now therefore the length of the line CB, and put ab = CBAC = 16·1 × t21,120 × t = sine 20° = ·3420201, and dividing numerator and denominator by t and multiplying the above 3rd and 5th expressions by 1,120, we have 16·1 × t = 1,120 × ·3420201 1,120 × ·3420201 and therefore t = = 23·7927 secs. 16·1 Having obtained the time, it will be easy to work out the lengths b and a as 26,648 ft. and 9114·1 ft. respectively; and as c/b = cosine 20°, we have c = 26,648 × ·9396926 = 25040·8 ft., which is the range. The trajectory will be a curve (parabola) symmetrical on each side of a vertical line half-way between A and B, and the length of this line within the triangle will be equal to half of a, and in half of 23·7927 seconds the projectile, supposed to move only along the line AC, would reach the point where this vertical axis intersects AC. If during this half-time it had been falling from rest at the same intersection, it would have reached a point below by a space just one quarter of CB (the spaces fallen through being as the squares of the times), and therefore at this its highest point its distance above AB would also be one quarter the length of a = 2278·525 ft., which distance is called the height of the trajectory; and the descending curve being in every respect symmetrical to the ascending branch, the angle at which this would be inclined to AB would be 20°, but in the opposite direction to BAC, while the velocity would be the same as at A. We may now compare these results with those calculated when the air resistance is taken into account:— Without air With air resistance. resistance. Difference. Time of descent 23·7927 secs. 22·61 sec. –1·18 sec. Angle of descent 20° 23° 49´ +3° 49´ Velocity of descent 1120 foot-secs. 868·8 foot-secs. –251·2 f.-s. Range 25040·8 ft. 20,622 ft. –4418·8 ft. Height of trajectory 2278·5 ft. 1989 ft. –288·5 ft. With the air resistance the trajectory will no longer be a symmetrical curve: its highest point, instead of being on the vertical line midway between A and B, will be on one 1,050 ft. nearer to B than to A, and the descending branch will be steeper than the ascending. The total time, it will be observed, is less, although the final, and therefore the mean, velocity, is also less; but this shortening of the time is due to the trajectory itself being much less in length. The range of the projectile is decreased 178by 4,418 ft., or 1,473 yards, or more than four-fifths of a mile. The loss of velocity at the descent is very notable, and the reader will find it interesting to calculate the corresponding loss of energy by the formula already given. The reader should now easily understand that the projectile from a rifle or gun discharged horizontally through airless space at the height of 16·1 ft. above a level plain would strike the ground in one second at a range or distance from the gun exactly equal to the initial velocity, or if the gun were on a tower and its axis 64·4 ft. above the plain, the range would then be 2V. It will be seen therefore that, corresponding to the range intended, there must be in general a certain inclination given to the axis of the piece in aiming, and this is done by means of the sights, one of which near the muzzle is usually fixed, while that next the breech is adjustable by sliding along an upright bar, which is graduated so that the proper elevation may be given for any required range. These graduations are made from experiments, and of course have reference only to some standard quantity and quality of ammunition and a standard of weight, shape, and material in the projectile. Sometimes large pieces of ordnance are laid by elevation in degrees, etc., marked on their mounting, the angles being taken from a table prepared for that particular gun and ammunition, from experiments at different ranges. After these generalities about fire-arms we may enter upon certain particulars about the construction of some varieties, beginning with