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other quality, but agrees with it only to the extent of being a 
relationship of the same genus. 
§13. Partial numbers. — Let r denote any relationship, and 
N a partial number ; in counting Nr A each rA counted must 
be different from any preceding one counted. But in such an ex- 
pression as (3 + 2 )rA, where two partial numbers are connected 
by + , it is not necessary that each of the individuals counted in the 
3rA should be different from each of the individuals counted in the 
2rA. Hence (3 + 2)rA is equivalent either to 5rA, or (3 + 2d )rA, 
or (1 + 2*2 )rA where the parts connected are now exclusive of 
one another. Similarly, (3 - 2)rA is not necessarily equivalent 
to IrA ; the expression is either irreducible or reducible, and if 
reducible may be either (2 - \)rA or rA. 
A partial number placed before a sum of terms connected by 
the signs + or - is non-distributive. For example, 
%c m A = 2 (c f B + c f C) 
is best viewed as meaning that the children of the man A are 
identical with two children either of the woman B or of the woman 
C, rather than meaning that they are identical with two children of 
the woman B together with two children of the woman C. Upon 
this view the above equation means that 
%c m A = 2c f B, or = 1 c f B + 1 c f C, or = 2 c f C . 
§ 14. Complete numbers. — If, as before, r denote any relation- 
ship, then such an expression as (3 + 2 )rA must be equivalent to 
(1 + 2 ’2 )rA where the former 2 is a coefficient. This follows 
because the sum of the rA is three. Similarly, (5 - 2 )rA = IrA. 
A definite number is similar to an indefinite number in being 
non-distributive when placed before a sum of terms. Thus 
3 (cjB + c f C) means that the children of the woman B together 
with those of the woman C amount to three. 
§ 15. The symbol 5 expresses any complete number without 
denoting what the number is. For example, the equation 
5c Henry VIII. = 5c Catherine of Arragon + 5c Anne Boleyn 
+ 5c Jane Seymour, 
