1865.] discovery of Imaginary Roots of Equations. 269 



the Philosophical Transactions, will be found a proof of Newton's Rule 

 for the discovery of imaginary roots carried as far as equations of the 5th 

 degree inclusive. The method, however, therein employed offered no 

 prospect of success as applied to equations of the higher degrees. I take 

 this opportunity, therefore, of announcing that I have recently hit upon a 

 more refined and subtle method and idea, by means of which the demon- 

 stration has been already extended to the 6th degree, and which lends itself 

 with equal readiness to equations of all degrees. Ere long I trust to be 

 able to lay before the Society a complete and universal proof of this rule 

 so long the wonder and opprobrium of algebraists. For the present I 

 content myself with stating that the new method consists essentially, first, 

 in the discerption of the question as applied to an equation of any specified 

 degree into distinct cases, corresponding to the various combinations of 

 signs that can be attached to the coefficients ; secondly, in the application 

 of the fecund principle of variation of constants, laid down in the third 

 part of my ' Trilogy,' and, in particular, of the theorem that if a rational 

 function of a variable undergoes a continuous variation flowing in one 

 direction through any prescribed channel, then at the moment when it 

 is on the point of losing real roqts, not only must it possess two equal 

 roots (a fact familiar to mathematicians as the light of day), but also its 

 second differential, and the variation, when for the variable is substituted 

 the value of such equal roots, must assume the same algebraical sign*. By 

 aid of the processes afforded by this principle, which admits of an infinite 

 variety of modes of application, according to the form imparted to the 

 channel of variation, and constitutes in effect for the examination of alge- 

 braical forms an instrument of analysis as powerful as the microscope for 

 objects of natural history, or the blowpipe for those of chemical research, 

 the problem in view is resolved with a surprising degree of simplicity ; so 

 much so that, as far as I have hitherto proceeded with the inquiry, the 

 computations, algebraical and arithmetical, which I have had occasion to 

 employ may be contained within the compass of a single line. The new 

 method, moreover, enjoys the prerogative of yielding a proof of the 

 theorem in the complete form in which it came from the hands of its 

 author (but which has been totally lost sight of by all writers, without 

 exception, who have subsequently handled the question), viz. in combina- 

 tion with, and as supplemental to, the Rule of Descartes. On my mind 

 the internal evidence is now forcible that Newton was in possession of a 

 proof of this theorem (a point which he has left in doubt and which has; 

 often been called into question), and that, by singular good fortune, whilst 

 I have been enabled to unriddle the secret which has baffled the efforts of 

 mathematicians to discover during the last two centuries, I have struck 

 into the very path which Newton himself followed to arrive at his con- 

 clusions. 



* The above is on the supposition that there is no ternary or higher group of equal 

 roots. 



x 2 ' 



