Mr. Seers's Method of solving adjected Quadratic Equations. 125 



which however must be either modern or accidental, as they 

 could hardly have withstood the waves for so many centuries. 



I have nothing to add to the received opinion, that it must 

 have been built before Roman arts and civilization (and in 

 particular the use of mortar) travelled so far north. 



Dr. Macculloch, after describing in the Geological Trans- 

 actions, vol. ii. the Fort of DunMacSniochan near Oban (which 

 I had intended to visit, but being hindered, visited this in Bute 

 instead), combats at length and successfully the opinion, that 

 the vitrifaction was the effect of natural causes ; but I think 

 the opinion could never have been held by one who had seen 

 this fort in Bute, where the traces of art are so evident and 

 so undeniable. 



The wall must have been first built, and then made compact 

 and solid by vitrifaction, which must have required a conside- 

 rable fire to be moved from place to place, as the work pro- 

 ceeded. Samuel Sharpe. 



XXII. Method of solving adfected Quadratic Equations. By 

 Mr. Joseph Seers ; in a Letter to Mr. Peter Nicholson*. 



Dear Sir, 

 f" BEG leave herein to submit to your inspection, &c. the 

 A method I discovered and mentioned to you, about two 

 months ago, of solving adfected quadratic equations. I flatter 

 myself it is quite new ; and I think it inferior to none in pre- 

 sent use. It is as follows : 



Whatever be the original form of a quadratic equation, it 

 must always be reduced to this formula of three terms; viz. 

 x 2 ± p x ± q — 0. 



In this formula, it is to be observed that p is the sum of 

 the root, and that q is their product. And having their 

 sum, and substituting (d) for their difference, we have, by a 



well-known theorem, the two roots in this expression + p — . 



in which expression the sign of p is always contrary to what 

 it is in the above formula. Moreover, we have, as before ob- 

 served, £i- x £j^- = ± q. In which equation d = + 



V p~ + 4 q : here the sign of q is contrary to what it is in the 

 formula. Hence, l£±± = ±JL±.^UJ^ : an expression 



containing the two roots of the given equation in terms of 

 known quantities. 



* Communicated by Mr. P. Nicholson. 



Ex- 



