152 



HISTORY OF SCIENCE. 



The reader will still more fully appreciate the connection between 

 an equation and a curve if he will practically construct some other 

 curves for himself. Any equation whatever expressing a relation 

 between two variable quantities which admit of an indeterminate 

 number of solutions, corresponds to some curve which may actually 

 be traced on paper. Let us suppose that the two lines o x and o Y 



FIG. 57. 



have been drawn as in Fig. 57, and that the reader as provided with a 

 scale of equal parts, and that he desires to construct the curve cor- 

 responding to the equation 



x. 



Assigning to x the successive values o, i, 4, 9, 1 6, 36, etc., the correspond- 

 ing numerical values of y are plainly o, i, 2, 3, 4, 6, etc. These co-ordi- 

 nates being set off by means of the scale of equal parts, the points o, 

 A, B, c, D will be obtained, and any required number of co-ordinates may 

 be obtained by solving the equation for other values of x; thus, when 

 x = 2, we have y = \/2, etc. The curve thus obtained is one branch 

 of a parabola, and the other branch, which would be symmetrically 

 situated below o Y, is also given by the equation, since y"- has always 

 two roots numerically equal but of opposite sign. 



The ellipse and the hyperbola are represented by equations which, like 

 those already given for the circle and the parabola, include x 2 andjy 2 . 

 And, conversely, an equation of the second degree can represent no other 

 curve than one of these four. Equations of the first degree represent 

 straight lines. By equations of the third degree nearly one hundred 

 different varieties of curves may be represented ; while those of the 

 fourth decree are capable of representing some thousands of curves of 

 different kinds. The number of possible curves is, in fact, infinite ; 

 for no equation between two quantities of the kind called indetermi- 

 nate, can be proposed, but a corresponding curve can be traced, and 

 its properties deduced from the equation. 



The principles of co-ordinate geometry are by no means confined to 

 lines in one plane. Equations can be used to represent curves which 

 do not lie in a plane as, for instance, a spiral line, like a corkscrew. 



