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ON CUBIC EQUATIONS

TRACT 28 .

the equation has no imaginary roots, but at no time else;and it may justly be demanded what can be the reason of socurious an accident. But this seeming paradox will becleared up by the following consideration. It is plain, thatthis circumstance must have happened either through someimpropriety in the manner of deducing the values of z andfrom the two assumed equations x = z an ^ ~~iP> or else by some impossibility in one of these twoconditions themselves: but, on examination, the deductionsare found to be all fairly drawn, and the operations rightlyperformed. The true cause must therefore lie concealed inone of these two conditions x = s + y ar *d zy = — fjt>. Inthe first of them it cannot be, because it only supposes thata quantity x can be divided into two parts z and y, which isevidently a possible supposition : it can therefore no w'hereexist but in the latter, namely, zy — — -\p. Now this sup-position is this, that the product of the two parts z andj/,into which the constant quantity x is divided, is equal to t s pwith its sign changed. Now this may always take place•when p is positive; for then — -5 -p will be negative, and twonumbers, the one positive and the other negative, may al-ways be taken such, that their product shall be equal to anynegative number whatever, and yet their sum be equal to agiven quantity x ; and this is done by taking the positiveone as much greater than x, as the other is negative; forthus it is evident the positive and negative numbers may beincreased without end : there is no impossibility then in thesupposition when p is positive ; and therefore then the for-mula ought to exhibit only real quantities, that is, in all thecases after the 16th in the table of forms, as we have beforefound. But the same thing cannot always happen when pis negative, or — = zy is positive : for that zy may be

positive, the signs of the two factors g and 3 / must be alike,either both + or both —, that is, both + when the signof x is +, or both — when that is —: but it is well known,that the greatest product which can be made of the twoparts, into which a constant quantity x may be divided, is