[GLASHAN] ALTERNATE NUMBERS AS GEOMETRICAL INDICES 175 
.’.7 sin POP, sin POR=r sin POP; sin P}OR+ 7 sin P,OP sin P2OR 
.’. sin P}PO.h=sin POP2h,+sin P10Ph 
.’. sin P,OP.(ne) =sin POP2(me) +sin PiOP(me) 
o (sin P:0P2.n)e= (sin POP: . m)e+ (sin POP . m2) € 
Hence e being a variable independent of 7, 51 and 72, we are justified 
in postulating 
sin POP, : n=sin POP: . m—+sin P,OP - M2. 
9. In the investigations and demonstrations which follow, the 
triangle of reference will be denoted by ABC or abe, &, e2, e will denote 
the indices of A, B and C and m, 72, 73 the indices of BC, CA and AB. 
The phrases “index of P”’ and “index of p”’ will be abbreviated to 
“ind P” and ‘‘ind p’’. The word line, unless limited by a specifying 
adjective, will mean straight line. 
10. I. Given the indices of the points A, B and C, to determine the 
index of any other point D in the plane of ABC. 
Complete the tetrastigm ABC; D and let 
A:miv:A+p+v::(CDB) : (ADC) : (BDA) : (ABC), 
then will 
(C’B) (CDB) À 
(BA) (BDA) » 
(A'C) (ADC) 

(B'A) (BDA) » 

(A’D) (DBC) À 
Let e’ and e be the indices of A’ and D, then will 
pe A’Ce + BA’e; _ Men + ve; 
BC u+v 
el pue A'De, + DA’ ba Nat (utr) 
A’A ; A+ uty 
_ Aa Metres 
A+yu+v 
ZE = and = À are the Cevan ratios of ABC; D. viz. 
m v 
C’A: C’B, A’B : A’'C and B’C : B’A. 
