CH. V] MECHANICAL RECREATIONS 91 



be upset until their owner orders them to lie down. Such a 

 bottle is made of thin glass or varnished paper fixed to the 

 plane surface of a solid hemisphere or smaller segment of 

 a sphere. Now the distance of the centre of gravity of a 

 homogeneous hemisphere from the centre of the sphere is 

 three-eighths of the radius, and the mass of the glass or 

 varnished paper is so small compared with the mass of the 

 lead base that the centre of gravity of the whole bottle is 

 still within the hemisphere. Let us denote the centre of the 

 hemisphere by 0, and the centre of gravity of the bottle by G. 



If such a bottle is placed with the hemisphere resting on a 

 horizontal plane and GG vertical, any small displacement on the 

 plane will tend to raise G, and thus the equilibrium is stable. 

 This may be seen also from the fact that when slightly dis- 

 placed there is brought into play a couple, of which one force 

 is the reaction of the table passing through G and acting 

 vertically upward, and the other the weight of the bottle 

 acting vertically downward at G. If G is below G, this couple 

 tends to restore the bottle to its original position. 



If there is dropped into the bottle a shot or nail so heavy 

 as to raise the centre of gravity of the whole above C, then 

 the equilibrium is unstable, and, if any small displacement is 

 given, the bottle falls over on to its side. 



Montucla says that in his time it was not uncommon to 

 see boxes of tin soldiers mounted on lead hemispheres, and 

 when the lid of the box was taken off the whole regiment 

 sprang to attention. 



In a similar way we may explain how to balance a pencil 

 in a vertical position, with its point resting on the top of one's 

 finger, an experiment which is described in nearly every book 

 of puzzles*. This is effected by taking a penknife, of which 

 one blade is opened through an angle of (say) 120°, and sticking 

 the blade in the pencil so that the handle of the penknife is 

 below the finger. The centre of gravity is thus brought below 

 the point of support, and a small displacement given to the 



* Ex. gr. Oughtred, Uathematicall Recreations, p. 24 ; Ozauam, 1803 edition, 

 vol. h, p. 14; 1840 edition, p. 200. 



