MATHEMATICAL SCIENCESSEVENTEENTH CENT. iy 



and this relation will be true for x and y, when these letters repre- 

 sent the length of lines drawn from p', and also when they represent 

 the same thing for any point whatever in the circumference. So 

 much is plain by mere inspection of the figure as regards any point in 

 the same quadrant as P ; and that the equation just given also truly 

 expresses the relations between the two lines drawn upon o x and 

 o Y from any point in circumference will become equally obvious on 

 reflection. Hence the equation is true for each and every point in 

 the circumference of the particular circle, and it is an impossible equa- 

 tion for any point out of that circumference. Also by putting suitable 

 figures for those which express the distances of the centre from b x 

 and o Y, and for the square of the radius, we can make -our equation 

 represent any circle whatever. 



We should recommend the reader also to find practically the curve 

 from its equation. Let him draw on a piece of paper two lines at 

 right angles to each other, and from a scale of equal parts set off dis- 

 tances, and draw perpendiculars, so that he thus may obtain as many 

 points as he chooses in the curve. He will first observe that the pos- 

 sible values of x and y lie within certain limits, thus : y may equal 

 any number between 8 and 18. When some assigned value of x re- 

 places that letter in the equation, the corresponding value or values 

 of y maybe found. Thus: suppose y to be 16, the equation becomes 



and this equation will be found to be satisfied when x = 7, and also 

 when ^=15. There are therefore two points in the curve which are 

 both 1 6 inches from o x, and one of these points is 7 inches and the 

 other 15 inches from o Y. By assigning another definite value toj/, 

 the corresponding values of x may be found by solving the equation; 

 and thus any number of points in the curve can be found by setting 

 off, parallel to o x and o Y respectively, the several values of x and y. 

 Lines drawn from any point in the same manner as P F and p D in 

 Fig. 56, are called the abscissa and ordinate of that point, and the 

 equation given above expresses an invariable relation between the 

 pairs of co-ordinates of any point whatever in the particular circle 

 represented, and a relation which is true only of points in the cir- 

 cumference of that circle. It will not be difficult to see that if, in- 

 stead of writing n and 13, which are the co-ordinates of the centre, 

 and 25, which is the square of the radius of the circle represented 

 in the figure, we substitute a pair of co-ordinates corresponding to the 

 centre, and the square of the radius of any other circle, we can 

 equally represent that circle also by an equation. Therefore, putting 

 generally a and b to stand for the co-ordinates of the centre, and r 

 for the radius of any circle, we know now that every circle may be 

 represented by an equation of this form : 



