PLEASURES OF SCIENCE. 11 



you cannot get near : for instance, the length and breadth of a field on 

 the opposite side of a lake or sea ; the distance of two islands, or the 

 space between the tops of two mountains. 



Again, there are curve-lined figures as well as straight, and geome- 

 try teaches the properties of these also. The best known of all the 

 curves is the circle, or a figure made by drawing a string round a fixed 

 point, and marking where its other end traces, so that every part of 

 the circle is equally distant from the fixed point or centre. From this 

 fundamental property, an infinite variety of others follow by steps of 

 reasoning more or less numerous, but all necessarily arising one out 

 of another. To give an instance ; it is proved by geometrical reason- 

 ing, that if from the two ends of any diameter of the circle you draw 

 two lines to meet in any one point of the circle whatever, those lines are 

 perpendicular to each other. Another property, and a most useful one 

 is, that the sizes, or areas, of all circles whatever, from the greatest to the 

 smallest, from the sun to a watch-dial-plate, are in exact proportion to 

 the squares of their distances from the centre ; that is, the squares of 

 the strings they are drawn with : so that if you draw a circle with a 

 string 5 feet long, and another with a string 10 feet long, the large 

 circle is four times the size of the small one, as far as the space or 

 area enclosed is concerned ; the square of 10 or 100 being four times 

 the squares of 5 or 25. But it is also true, that the lengths of the cir- 

 cumferences themselves, 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. 



But the circle is only one of an infinite variety of curves, all having 

 a regular formation and fixed properties. The oval or ellipse is, per- 

 haps, next to the circle, the most familiar to us, although we more fre- 

 quently see another curve line formed by the motions of bodies thrown 

 forward. When you drop a stone, or throw it straight up, it goes in a 

 straight line ; when you throw it forward, it goes in a curve line till it 

 reaches the ground ; as you may see by the figure in which water runs 

 when forced out of a pump, or from a fire-pipe, or from the spout of a 

 kettle or tea-pot. The line it moves in is called a parabola ; every 

 point of which bears a certain fixed relation to a certain point within 

 it, as the circle does to its centre. Geometry teaches various 

 properties of this curve ; for example, that if the direction in which 

 the stone is thrown, or the bullet fired, or the water spouted, be half 

 the perpendicular to the ground, that is, half way between being level 

 with the ground and being upright, the curve will come to the 

 ground at a greater distance than if any other direction whatever were 

 given, with the same force. So that to make the gun carry furthest, or the 

 fire-pipe play to the greatest distance, they must be pointed, not as you 

 might suppose, level or point blank, but about half way between that 

 direction and the perpendicular. If the air did not resist, and so 

 somewhat disturb the calculation, the direction to give the longest 

 range ought to be exactly half perpendicular. 



The oval, or ellipse, is drawn by taking a string of any certain length, 

 and fixing, not one end as in drawing the circle, but both ends, and 

 then carrying a point, as a pencil or chalk, round inside the string, 



