12 OBJECTS, ADVANTAGES, AND 



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 belong- 

 ing to this curve, in relation to the straight 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 sugar-loaf, and 

 cut it any where clean through in a direction parallel to its base or bot- 

 tom ; 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 ; but if it goes slanting, and parallel to the 

 line of the loafs side, the outline is a parabola ; and if you cut in any 

 direction not through the sides all round, but through the sides and 

 base, and not parallel to the line of the side, the outline will be another 

 curve of which we have not yet spoken, but which is called an hyper- 

 bola. You will see another instance of it, if you take two plates of 

 glass, and lay them on one another ; then put their edge in water, 

 holding them upright and pressing them together ; the water, which, 

 to make it more plain, you may colour with a few drops of ink or strong 

 tea, rises to a certain height, and its outline is this curve ; which, how- 

 ever much it may seem to differ in form from a circle or ellipse, is found 

 by mathematicians to resemble them very closely in many of its most 

 remarkable properties. 



These are the curve lines best known and most frequently discussed ; 

 but there are an infinite number of others all related to straight lines 

 and other curve lines by certain fixed rules ; for example, the course 

 which any part, as the nail in the felly of a wheel rolling along takes 

 through the air, is a curve called the cycloid, which has many remark- 

 able properties ; and, among others, this, that it is, of all lines possible, 

 the one in which any body not falling perpendicularly, will descend 

 from one point to another the most quickly. 



II. You perceive, if you reflect a little, that the science which we have 

 been considering in both its branches, has nothing to do with matter ; 

 that is to say, it does not at all depend upon the properties or even 

 upon the existence of any bodies or substances whatever. The distance 

 of one point or place from another is a straight line ; and whatever is 

 proved to be true respecting this line, as, for instance, its proportion to 

 other lines of the same kind, and its inclination towards them, what 

 we call the angles it makes with them, would be equally true whether 

 there were any thing in those places, at those two points, or not. So 

 if you find the number of yards in a square field, by measuring one 

 side, 100 yards, and then, multiplying that by itself, which makes the 

 whole area 10,000 square yards, this is equally true whatever the field 

 is, whether com or grass, or rock or water ; it is equally true if the 



