2 ANALYTIC GEOMETRY [Cn. I. 



if it involves in any way wliatever an indicated even root "I 

 a negative number; otherwise it real. 



.cry imaginary number may be reduced to the form 

 a + b V^l, where a and b are real, and b * 0. 



Constants and variables. If AB and AC are two given 

 straight lines making an angle a at 

 point A, and if any two points 



A' ami )", MM these lines, respectively, 

 x are joined by a straight line, then 



Area of triangle AXY = \-AX- AY-sin a, 

 f.*., A = J -X'i/ -sin a, 



w here T is the length of AX, y is the length of A F, and A is 

 irea of the triangle. 



now the points X and F are moved along the lines AB 

 and AC in any way whatever, then A, x, and // will each pass 

 through a series of different values, they are variable num- 

 bers or variables ; while and sin a will remain unchanged,-- 

 they are constant numbers or constants. 



It is to be remarked that \ has the same value wherever it 

 occurs, it is an absolute constant; while , though constant 

 fMr this series of triangles, may have a different constant 

 value for another series of triangles, it is an arbitrary 



Because x and y may separately take any values what- 

 ever they are independent variables; while A. whose value 

 depends upon the values of x and y, is a dependent variable. 



The illustrations just given may serve to give a clearer 

 conception of the following more formal definitions. 



An absolute constant is a number which has the same value 

 wherever it occurs; such are the numbers 2, 7, f, 6*, TT, e 



