Mr. W. G. Horner on Congeneric Surd Equations. 47 
evident by a course of amusing experiments upon a familiar 
equation, é. g. 
w+ 2a Te? —S8rt+12=—0, 
whose roots are 1, 2, —3, —2. Applying the four axioms in 
succession, we shall perceive how the incautious blending of 
two truths, by means of rules in themselves unexceptionable, 
will produce a falsehood. 
ish To #4 22°-—7e°=—8r+12=0 
Add B= «le 10 
at} 2 —722°=—|F7a4+ 11 = 0; 
a false equation, with regard to all the values of z, with the 
single exception of 1, the value already used. Similar results 
would accrue from the addition or subtraction of any other 
divisor of the equation; the result will be false in every value, 
except those which are also found in the equation added or 
subtracted. Thus, 
From 2* + 22° —72#* — 824412 @) 
Take 62° — 1827 4- 12 10) 
z+ 22 — 132° 4+ 102 = 0; 
whose only correct roots are those also of 27— 347+ 2 = 0. 
2ndly, The given equation is resolvable into the quadratics 
2—3r+2=0, anda’ +57+6=0. 
Therefore, multiply 2 — 3r=— 2 
by e+ 5br=— 6 
GE Da ae AG a= ~ 18% 
a statement altogether erroneous, not containing a single cor- 
rect value of z. 
On the other hand, divide 
a4+20% —~72? —8e = —12 
by a’? —32e= — 23 
P+ 2a —7T«4#—8 
Hes = 6, 
xr—3 
* P+I92r*—138 2 =— 10: 
incorrect, except in respect of the roots of z°— 37+2 = 0. 
You will clearly perceive, without dwelling upon the distine- 
tion of cases, the very simple nature and origin of the para- 
dox. The axioms speak of quantities which are s?multaneously 
equal; but no two roots of an equation, unless they be equal 
roots, are coexistent: if e = 1 it is not at the same time = 2. 
Consequently, as in each of the examples v in the upper of 
the two equations has some yalues, which substituted in the 
