OBJECTS, PLEASURES, AND ADVANTAGES OF SCIENCE. 



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 poiut of its curve must be so placec 

 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 straight lines drawn within it, is 

 one which gives rise to the construction of the trammel*, 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 sugarloaf, and cut it anywhere 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 ia an 

 ellipte, provided the cut goes quite through the sides of the loaf all round, or is in such a direction that it 

 would pats through the sides of the loaf were they extended ; but if it goes slanting and parallel to the line 

 of the loaf's 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, being nearer the perpen- 

 dicular, the outline will be another curve of which we have not yet spoken, but which is called an hyper- 

 bola. Yon 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, however 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 point in the circumference of a circle, as a nail in the felly of a wheel rolling along, takes 

 through the air, is a curve called the cycloid, which baa many remarkable 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. Another curve often seen is that in which a rope or 

 ctiain bangs when supported at both ends : it is called the Catenary, from the Latin for chain ; and in this 

 form some arches are built. The form of a sail filled with wind if the tame curve. 



II. DirnmxNci BETWEEN MATHEMATICAL A*D PHYSICAL TBUTHS. 



You perceive, if you reflect a little, that the science which we have been considering, in both its branches, 

 bat 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 it 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 anylei it makes with them, would 

 be equally true whether there were anything 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 make* the whole area 10,000 square yards, this is equally true whatever the field is, uli.-tin-r 

 corn, or grass, or rock, or water ; it is equally true if the solid part, the earth or water, be removed, for then 

 it will be a field of air bounded by four walls or hedges ; but suppose the walls or hedges were removed, 

 nd a mark only left at each corner, still it would be true that the space inclosed or bounded by the lines 

 supposed to be drawn between the four marks, was 10,000 square yards in size. But the marks need not 

 be there; you only want them while measuring one tide: if they were gone, it would be equally true 

 that the lines, supposed to be drawn from the places where the marks bad been, inclose 10,000 square 

 yard* of air. But if there were no air, and consequently a mere void, or empty space, it would be equally 

 true that this ipace is of the size you had found it to be by measuring the distance of one poiut from 

 another, of one of the space's corners or angle* from another, and then multiplying that distance by itself. 

 In the same way it would be true, that, if the space were circular, its size, compared with another circular 

 pace of half it* diameter, would be four time* larger: of one-third its diameter nine times larger, and of 

 one-fourth sixteen times, and so on always in proportion to the squares of the diameters; and that the 

 length of the circumference, the number of feet or yards in the line round the surface, would be twice 

 the length of a circle whose diameter was one-half, thrice the circumference of one whose diameter was 

 one-third, four times the circumference of one whose diameter was one-fourth, and so on, in the simple 

 proportion of the diameter*. Therefore, every property which it proved to belong to figures belongs to them 

 without the smallest relation to bodies or matter of any kind, although we are accustomed only to see figure? 

 in connection with bodies ; but all those properties would be equally true if no such thing as matter 01 

 bodies existed ; and the same may be said of the properties of number, the other great branch of the 

 mathematics. When we speak of twice two, and say it makes four, we affirm this without thinking of two 

 burse*, or two balls, or two tree* ; but we assert it concerning two of any thing and every thing equally 



