IRREDUCIBLE case, in algebra, is used for that case of cubic equations, where the root, according to Cordon's rule, appears under an impossible or imaginary form, and yet is real. Thus, in the equation, x3 — 90 100 = 0, the root, according to Cordon's rule, will be x = 50 + %, — :l4aUU V 50 — — 24500, which is an impos sible expression, and yet one root is equal to 10 ; and the other two roots of the equation are also real. Algebraists, for two centuries, have in vain endeavoured to resolve this case, and bring it wider a real form ; and the question is not less fa mous among them than the squaring of i the circle is among geometers. See lloraTioN.
It is to be observed, that as, in some other cases of cubic equations, the value of the root, though rational, is found under an irrational or surd form ; because the root in this case is compounded of two equal surds with contrary signs, which destroy each other ; as if ,r = 5 + 5 + 5 — V 5; then x = 10 ; in like man lier, in the irreducible case, when the root is rational, there are two equal imaginary quantities, with contrary signs, joined to real quantities ; so that the imaginary quantities destroy each other. Thus the expression : 50 + ? —24.50u= 5 ; and .1/ 50 —?— 24500 = 5 5. But 5 V— 5+ 5 — — 5 10 x, the root of the proposed equation.
Dr. Wallis seems to have intended to show, that there is no case of cubic equa tions irreducible, or impracticable, as he calls it, notwithstanding the common opi nion to the contrary.
Thus in the equation ri — 63 r = 162, where the value of the root, according to Cordon's rule, is, r 81+ V— 2700 1/81 — —2700, the doctor says, that the cubic root of 81 — 2700, may be extracted by another impossible binomial, viz. by ; V — ; and in the same manner, that the cubic root of 81 — V— 2700 may be extracted, and is equal to 4-iv-s; from whence he infers, that 1+ is V— 3 + ; V— 3 = 9, is one of the roots of the equation pro posed. And this is true : but those who will consult his algebra, p. 190, 191, will find that the rule he gives is nothing but a trial, both in determining that part of the root which is without a radical sign, and that part which is within : and if the original equation had been such as to have its roots irrational, his trial would never have succeeded. Besides, it is certain, that the extracting the cube root of 81 + — 2700 is of the same degree of as the extracting the root of the original equation r3 — 63 r = 162 ; and that both require the trisection of an an gle for a perfect solution.