Objects, Advantages , And Pleasures Of Science.

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 geometry teaches the

properties of these also . The best known of all the curves'isj the circle , or a figure made by dra wing a string round one end which is fixed , 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 reasoning , 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

hues 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 isthe

, 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 inclosed is concerned ; the square of

10 or 100 being four times the square of 5 or 25 . But it is also true , that the lengths of the circumferences 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 , perhaps , next to the circle , the most familiar to us , although we more frequentl y see another curve , the line

formed by the notion 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 Way see by the figure in which water runs when forced out of a pump , or from a firepipe , or from the spout of a kettle or

teapot . 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 , if the direction in which the stone is thrownor the bullet

, fired , or the water spouted , be half the perpendicular to the ground , that is , halfway 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 farthest , 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 calculationthe 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 to different points , and then

carrying a point round inside the string , always keeping it stretched as far as possible . It is plain that this figure is as regularly drawn as the circle , though it is very different from it , and you perceive that every point of its curve must be so placed , that the straight lines drawn from it to the two points where the string was

fixed , are when added together , always the same , for they make together the length of the string . Among various properties belonging to this curve , in relation to the staaight lines drawn within it , is one which gives rise

to the construction of the trammels , or elliptic compasses , used for making figures and ornaments of this form , and also to the construction of lathes for turning oval frames , and the like . If you wish at once to see these three

curves , take a pointed sugar-loaf , and cut it any where clean through in a direction parallel to its base or bottom ; the outline or edge of the loaf where it is cut will be a circle . If the cut is made so as to slant , and not be parallel to the base of the loaf , the outline is an ellipse , provided the cut goes quite through the sides of the loaf all round , or is in such a direction