mid Infinite Series. 4 at 
is lefs than 1 . So that in general the fxrft feries gives the 
greateft of the three roots. 
84. But it is evident, that this cafe agrees with the 
10th form in the table Art. 30; in which the middle 
root r is found to be -y~ — 2 q — - -V q.b — — 2 d/~b f 
and the two other, or greateft and leaft roots, are 
- 1/ x 1 ± v/3 = A/\b x 1 =fc -1/3. 
85. Hence by a comparifon of thefe two different 
forms of the fame roots w r e find 
2 - 5 - 8 -*i. t4 » _ 
.12 . ic. 18 &c ’ A ? 
S3 + 1 _ . _ 2 - 5-3 
2 St 3 • 6 3.6. 9 . i2 3.6.9 
and 42 = I _ JLJL + - &c. = b. 
2V2 3 3.6.9 3.6.9.12.15 
86. And by adding and lubtradting thefe two, we find 
2.5 2.5.8 „ * 
+ + — - 8cc. and 
- + + 8cc. =2 c. 
C C 
i 
1 + i + . I 
Sz 3 3.6 3.6.9 3.6.9.12 
_l_ = 1 - i + + 2 -s - a ^- 8 
S 2 3 3.6 3.6.9 3.6.9.12 
87. Alfo, becaufe — x — - is 
S3 + 1 x S% - l 
2^2 2^2 
2-^4 
, which is 
; therefore the mean proportional between 
the two feries a and b, is to the feries c, as the fide of a 
fquare is to its diagonal. 
» 
88. Moreover, to and from the two feries a and B, 
adding and fubtradting the two feries in Art. 74. 
K k k 2 namely, 
