3i 6 Mr. Robertson's Demonstration 
nominators of the second, and in such a manner as to make 
the parts on the first side, which have the same numerators, 
unite : thus the part-^ must be reduced to the denominator ij ; 
the parts 4 - must be reduced to the same denominator 
r 1 ccs V 
a. f ; the parts ~~ -f- to the same denominators (3 e, &c. 
Now, upon examining the two lines as 
represented in the columns in the margin, 
a general rule for this reduction presents 
itself. For the denominators, exclusive of 
v in the first column, proceed in the fol- 
lowing regular manner, which is not pecu- 
liar to the perpendicular lines now under 
examination, but is the same in any two 
contiguous lines in any period of the mul- 
tiplication exhibited in the 16th article. 
The first and last denominator, in each 
column, consists of a single letter, as f in 
the first, and ^ in the second, of these we 
have selected for illustration. The second 
denominator consists of the next lower letter to the highest 
multiplied into at, as as in the first column, and a fin the second. 
The third denominator consists 'of the second lower letter to 
the highest multiplied into (3, as (3 £ in the first column, and (3 e 
in the second ; and the same gradation is observed in the other 
denominators. Now each term in the first column has two 
members in the numerator, and to make these unite with the 
terms in the second column, the first member must have the 
same denominator with that term in the second column in the 
same horizontal line ; and the second member must have the 
First line. 
Second line. 
G -f a F 
G 
C* 
>i 
a F -f- b E 
aF 
* £ V 
JE + cD 
b E 
(3 e 
cD+dC 
c D 
yyV 
y2 
d C + cB 
d C 
oy 
e B +/A 
eB 
£ a. V 
. td 
fA + g 
f A 
TT 
g_ 
