It will be interesting to compare the difference in the trajectory of a projectile under defined conditions, worked out with the air resistance taken into account, compared with the trajectory when the air is supposed to be non-existent. We find an example of the former problem fully worked out by many elaborate mathematical formulæ 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 at 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. [Illustration: FIG. 81.—_Diagram._ ]