150 



HISTORY OF SCIENCE. 



fectly independent of the other curves. Hence the invention of a new 

 curve and the investigation of its properties required a very large 

 amount of research and intellectual effort. But when Descartes 

 brought geometry under the dominion of algebra, the former science 

 at once overleapt the narrow bounds which had circumscribed it for 

 ages, and acquired a range of unlimited extent. Instead of the few 

 special curves which had hitherto been the subjects of study, the 

 geometer could now investigate at one and the same time the common 

 properties of large classes of curves, and the number of various curves 

 that might be studied became infinite. The equation to a curve was 

 found to be a compendious formula embodying all the properties of 

 the curve, and by general rules, which were the same for all curves 

 whatever, these properties could be deduced from the equation. 

 In illustration of the manner in which, on the principle of Cartesian 



geometry, an algebraical equation 

 may represent a curve, and vice 

 versa, we offer these following con- 

 siderations for the benefit of our 

 non-mathematical readers. Let 

 o Y, o x be two straight lines at 

 right angles to each other, and in- 

 definitely prolonged towards Y and 

 x. We say (purposely limiting the 

 proposition) that any circle in the 

 same plane as o Y, o x can be re- 

 presented by an algebraical equa- 

 tion. Suppose a circle placed as 

 in the diagram Fig. 56, its radius 

 FIG. 56. being 5 inches, the distance of c, 



its centre, from the line o Y being 



say ii inches, and from o x, 13 inches. Through c draw line E c per- 

 pendicular to o Y, and line c B perpendicular to o x. From P, any point 

 in the circumference (which for simplicity in the first instance we may 

 suppose to be situated in the quadrant of the circle opposite to o), draw 

 1 ines P F and P D perpendicular to o Y and o x, the last cutting E c at A. 

 Join P c. Then P A c being a right angle, it will be clear that, granting 

 the truth of proposition mentioned on page 13, CA 2 +PA 2 =cF, which is 

 evidently the same as saying that (PF Ec) 2 +(pD Bc) 2 =CP 2 . If in- 

 stead of P we take some other point in the circumference, as for ex- 

 ample P', and draw lines from it in the same way as for P, the same 

 relations will hold, the only difference being that P' F' will be of a dif- 

 ferent length from P F, and P' D' different from P D. If we express these 

 distances, which vary as we pass from one point in the circumference 

 to another, by the letters x and y, and express E c, B c, and CP 2 by their 

 fixed and known values, the equation just given will be written thus : 



