824 
MATHEMATICS AND NATURAL PHILOSOPHY. 
‘Art. III. Tracts on the Resolution of Cubic and Bigquadratic Equations. 
By Francis 
Maseres, £sg. F. R. S. Cursitor Baron of the Exchequer. 
THIS volume consists of six tracts, 
besides a preface of sixty pages explana- 
tory of their contents. The first and 
fourth tracts are supplements to the ap- 
pendix to Mr. Frend’s “Principles of 
“Algebra,” containing farther remarks on 
cubic equations and Cardan’s rule. The 
second is a very valuable comparison 
‘made between the methods of Ferrari 
‘and Descartes for resolving certain bi- 
quadratic equations, and as a preference 
is with reason given to the former of these 
methods, the tract is not unaptly stiled 
“ Ferrarius redivivus.” ‘The third tract 
obviates some difficulties, and the appli- 
‘cation of Ferrari’s method to the resolu- 
tion of four forms of biquadratic equa- 
tions. The fifth contains remarks on the 
doctrine of the generation of algebraic 
equations ; and the last, a comparison 
between the resolution of the biquadratic 
rx —qx* + px —xt+= 5, by the me- 
thod of Dr. Waring and that of Ferrari. 
+ All these tracts are written with the 
accuracy, diligence, and skill for which 
the author has long been distinguished. 
He maintains his opinion on the injury 
done to science by the introduction of 
negative quantities with firmness and 
dignity ; and if any thing could produce 
a restoration of the ancient doctrine, it 
-would be, one would think, the example 
of perhaps the oldest writer on algebra 
in Europe, who has, without the least 
necessity of applying to quantities Jess 
than nothing, investigated the most dif- 
ficult problems in analytics: 
** Tt is,” he says, ‘* awing to the doctrine 
of the generation of equations one from an- 
other by multiplication, and to that of nega- 
tive quantities, or quantities less than no- 
thing, that algebra has sunk from the dignity 
ofa science or object of the understanding 
and reasoning faculty, to the condition of an 
art or knack of managing quantities by the 
eye and the hand, pli little or .no interfer- 
ence of the understanding.”* 
This technical process is supposed to 
be the excellence of the modern art; for, 
according to Montucla, they relieve us 
from the trouble of thinking. ° 
We cannot enter into the investigation 
of each several tract, though all may be 
studied with advantage by the algebraist. 
But the remarks on the doctrine of the 
generation of equations “are too import- 
ant to be cursorily passed by. It is well 
known that Harriot was the first inven- 
tor of this system, and that it was adopt- 
ed by Des Cartes; but it is not generally 
known, that without the supposition of 
negative quantities, Vieta had discovered ° 
and demonstrated the most important 
properties of equations, which are sup- 
posed to have been first pointed out by 
the new method. Inthe new method x 
is supposed to be equal successively to 
the roots a, b, c, d, &c. of an equation ; 
and upon this supposition the following 
equations are formed, namely, x«—a=9, 
x—b=0, x—c=o0, &c. and these equa- 
tions, multiplied together, produce, by 
certain changes of signs, any equation 
that can be proposed. But if x is equal 
toa, it cannot be equal to 4, ¢, d, conse- 
quently the whole system falls to the 
ground. This is as evident as any pro- 
position in Euclid. A mathematician 
must not take a second step till the first 
has been fairly established. But the 
conclusion drawn is true, and this the 
Baron proves, for without making »—a 
=o, x—b=0, and multiplying two no- 
things together, which is impossible, and 
an insuperable objection to the sys 
tem, he shews how the equation may be 
produced in a simple easy manner, upon 
true principles. _ We will shew it in the 
case of a quadratic equation. 
Let x be equal to a, and. less than 5. 
Therefore two equations may be formed 
and multiplied together, namely, _ 
rey 
b —_—* = b —xX 
- bx—x=ab—ax 
ao bx+ax—x = ab, 
orr xb6+a—x22=ab, 
Now let « be made equal to 4, and 
consequently be greater than a. There- 
fore two equations may again be formed 
in the same manner, namely, 
= 
eB, Ch 
w—am=x—a, 
ald eon neg HRP is 
Set ax Sb x — ab, 
a BF -- ar—a’= ab, 
orz xXb+4a—2 5,46, 
If for 6 + awe substitute p, and for 25 
the term q, then pr —.a* = ¢; a general 
form for equations of the second order, 
in which p must represent the sum of the 
‘roots, and g their products. In the same 
manner, if it were necessary, other equas 
tions might be produced by multiplica- 
tion. But the plan*does not seem to be 
of any use, ds the properties of equations 
* as - > 
: ~ 
