[cLAsHAN] ALTERNATE NUMBERS AS GEOMETRICAL INDICES 173 
larly for the other trigonometrical functions... In the case of the triangle, 
A will be used for (4 BAC), B for (4 CBA), C for (ZACB), a for 
BC, b for CA, c for AB and A for ABC. 
8. Geometrical indices are entities subject to the laws of com- 
mutation and association for addition, to the law of the distribution 
of a multiplier of a sum over the addends of that sum, and to the law 
of the association of the factors of a product all of whose factors 
are absolutely independent of each other; but they are not in general 
subject to the law of commutation for multiplication. Being hetero- 
geneous with arithmetical numbers, they obey all the ordinary laws 
of addition and multiplication in their combinations with these. 
The indices of points and lines in any given plane are further assumed 
to be subject to the following postulates of combination and inter- 
relation and their derived corollaries. 
(i) Let à, & be the indices of 
any two points P;, Pe, and 7 WE oe Ge À 
the index of the straight line 
P,P;, we shall postulate 
e1¢ = P, Pon 
Cor. 1. Hence e¢;= P2Pin = —P,Pon 
EE — = €1€9. 
Cor. 2 Also € —P:P17=0 
*, for every point =0. 
(ii) Let m, nz be the indices of any two 
intersecting straight lines PP,, PP, € the 
index of P, their point of intersection, and 
6 the numeric of ( ZP,PP:), we shall postu- 
late 
nin2=% sin 0.e 
Cor. 1. Hence m=} sin(2r—@) . e= 
—+sin0.e 

3 € 721 = — 1192 
Cor. 2. Also m=3 sin 0. e=0 
: m° =0 
.. for every line n?=0. 
(iii) Let e be the index of any point 
P, 7 the index of any straight line 1, and 
h the numeric of the perpendicular dis- 
tance of P from 1, we shall postulate 
en=ih 
2. 
Cor. Let « and « be the indices of 

