xviii INTRODUCTION TO MECHANICS. 



ever, sufficiently plain that this resistance increases with the velocity of 

 the ball, for the particles of air re-act on the ball in proportion to the 

 stroke they receive from it ; so that if the force of projection be doubled, 

 the resistance of the air is doubled also : nor is this all, for in doubling 

 the velocity of the ball, it passes through twice the quantity of air in the 

 same time, arid receives twice the resistance from each particle ; the whole 

 of the resistance must, therefore, be four times as great as in the first 

 instance. And if the velocity of the ball be tripled, it will pass through 

 three times the quantity of air ; will strike each particle with three times 

 the force, and receive three times the re- action ; which summed up will 

 make nine times the resistance. 



The shortest mode of calculating the resistance is to multiply the 

 velocity by itself; thus, if the velocity be three, multiply it by three, and 

 the product will be nine. The product of a number multiplied by itself 

 is called its square. 



The curve-line which a ball describes, if the resistance of the air be not 

 taken into consideration, is called in geometry a parabola. But when the 

 ball is thrown perpendicularly upwards, it will descend perpendicularly ; 

 because the force of projection, and that of gravity, are in the same line 

 of direction. 



We have noticed the centres of magnitude and of motion, but we have 

 not yet explained what is meant by the centre of gravity. It is that point 

 about which all the parts of a body exactly balance each other, in every 

 position of the body ; if, therefore, that point is supported, the body will 

 not fall. Were any other point of the body alone supported, the surround- 

 ing parts no longer balancing each other, the body would fall on the side 

 at which the parts are heaviest; therefore, whenever the centre of gravity 

 is unsupported, the body must fall. This sometimes happens with an over- 

 loaded waggon winding up a steep hill, one side of the road being more 

 elevated than the other: let us suppose it to slope as described in jig. 14. 

 We will suppose that the centre of gravity of this loaded waggon is at the 

 point A. Now the eye will tell you, that a 

 waggon thus situated will overset ; and the reason Fig. 14. 



is, that the centre of gravity, A, is not supported ; 

 for if a perpendicular line be drawn from it to the 

 ground at C, it does not fall under the waggon 

 within the wheels, and is, therefore, not supported 

 by them. A perpendicular line thus drawn from 

 the centre of gravity to the earth, is called the line 

 of direction. Let us in imagination take off the 

 upper part of the load ; the centre of gravity will 

 then change its situation, and descend to B, as 

 that will now be the point about which the parts of the less heavily laden 

 waggon will balance each other ; and the waggon will no longer upset, 

 for a perpendicular line from that point will fall within the wheels at D, 

 and be supported by them. You have heard that it is dangerous, when a 

 boat is in any risk of being upset, for the passengers to rise suddenly ; 

 this is owing to their raising the centre of gravity, and thus increasing the 

 chance of throwing it out of the line of direction. When a man stands 

 upright, the centre of gravity of his body is supported by the feet. If he 

 lean on one side, he will no longer stand firm. A rope-dancer performs 

 all his feats of agility, by dexterously supporting his centre of gravity; 

 whenever he finds himself in danger of losing his balance, he shifts the 

 heavy pole, which he holds in his hands, in order to throw the weight 



