MATHEMATICS. 21 



tained, one greater and the other less than the area 

 required. 



If you draw a circle with a string 5 feet long, and 

 another with a string 10 feet long, the large circle is 

 4 times the size of the small one, as far as the space 

 or area enclosed is concerned ; the square of 1 or 

 100, being 4 times the square of 5 or 25. On the 

 other hand the length of the circumferences, that is 

 to say, the number of feet over which the ends of the 

 strings move, are in proportion to the lengths of the 

 strings, so that the curve of the larger circle is only 

 twice the length of the curve of the lesser. 



The Cycloid. 



A cycloid is the path which any point of a circle, 

 moving along a plane, and round its centre, traces in 

 the air ; so that a nail on the felly of a cart wheel 

 moves in a cycloid, as the cart goes along, and as the 

 wheel itself both turns round its axle, and is carried 

 along the ground. 



A body moving in a cycloid by its own weight or 

 swing, together with some other force acting upon it, 

 will go through all distances of the same curve in 

 exactly the same time ; and, accordingly, pendulums 

 are contrived to swing in such a manner that they 

 shall describe cycloids, or curves very near cycloids, 

 and thus move in equal times, whether they go 

 through a long or a short part of the curve. 



If a body is to descend from any one point to any 

 other, not in the perpendicular, by means of some 

 force acting on it, together with its own weight, the 

 line in which it will go the quickest will be the 

 cycloid, not the straight line, (though the last is the 

 shortest of all lines that can be drawn between two 

 points,) nor any other shaped curve whatever, although 

 many are much flatter, and therefore shorter than the 

 cycloid ; but the cycloid, which is longer than them, 



