
BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 353 
Also CA,=(CD,)+V—1. (D,A,) 
CA, _1=(CD,)—V/—1. (D,A,) for ©,4,)=(0,4An_1) 
OA; x CA, 7 =(CD?)+ (yA, which = =(CAy 
or its equivalent in area (CA,)?. 
XI. Cotes’ Properties of the Circle. 
Let the circumference of the circle be divided into n equal parts; and to the 
extremities of these let lines be drawn from the 
centre (Fig. 8), as OP,, OP,, &c., and from any 
other point C in the diameter. Then 
CP, = OP, — OC, CP, =OP, — OC, &c. 
CP, xCP,x CP, .... OP, 
=r OA) sua (OAS ah yy. yf eiOC* 
Where 5, is the product of all the coefficients A 
of direction for OP,, OP,, &c., 3,_1, the sum of these 
coefficients taken »—1 together, and so on. But 
these coefficients (Prop. III.) are also the values of 
Fig. 8. 

i 
1”, or the roots of the equation x*—1=0. Now the 
product of the roots of this equation with their signs changed is —1, and 3, is the 
product with the signs unchanged. 
Therefore if m be even, 3,=—1, and, if m be odd, 5, = +1; and in either case, 
fn 2, we. each =0. 
Hence CP,xCP, .... xCP,==+(OA)"+(0C)"; the upper signs being used 
when 7 is even, the lower when 7 is odd. 
But CP, CP,, &c., represent the lines considered in relation both to length 
and direction ; therefore, to change the equation into one in which the length 
only of these lines shall be expressed, we must divide the first side, or muitiply 
the second by the product of all their coefficients of direction. 
If n be even, the several pairs, as CP,, CP,,_;, are evidently of the form 
3 9 
(CP) . 127 and (CP,,_3). 1. 2* ... CP, x CP,. 7=(CP,) x (CP,=1) 
and the same is true for every pair except CA=(CA). +1 and CB=(CB) . —1 
(CP,) x (CP,) .... OP, =[—OA"+00"] x —1=0A"—0C". 
If, again, 7 be odd, the several pairs remain as before, only, no P falling upon 
B,—1 is not a coefficient of direction : 
(CP, x (CP,) x &e., =OA"— OC” as before. 
VOL. XVI. PART III. , 4uU 
