1818.] A new Quadratic Theorem*, 31 



the waters depends upon the oxide of iron which they contain ; 

 and there can be no doubt that its .joower would be extremely 

 lessened, if not totally destroyed, by boiling the saline and 

 chalybeate contents of the water to dryness ; for it is well known 

 that the protoxide of iron which exists in the waters, would be 

 converted into peroxide by the operation ; and whenever per- 

 oxide of iron is exhibited in a solid form, several times as much 

 of it are required to produce an equal effect as a given quantity 

 of the protoxide of iron in solution : it is, therefore, evidently 

 improper to convert any part of the small quantity of protoxide 

 of iron which these waters contain into peroxide, because the 

 diminished power of the oxide of iron cannot be compensated 

 without greatly increasing the purgative effect ; and by this 

 also the resemblance between the salts and the waters from 

 which they are produced is destroyed. 



I have already stated Mr. Thompson's opinion of what he 

 terms the artificial and spurious Cheltenham salts : I have never 

 examined any of these preparations ; but I have no hesitation in 

 asserting that, by proper management, such a combination of 

 purgative and chalybeate salts may be effected as would be much 

 more efficacious, and more nearly resemble the Cheltenham 

 waters, than any compound that can be obtained by evaporating 

 these waters to dryness. 



Article VI. 



A new Quadratic Theorem : an improved Method of extracting 

 the Square Root from Trinomials, Quadrinomials, Pentana- 

 mials, and Hexanomials. By Joseph Reade, M.D. 



SIR, 

 Should the following method of extracting the square root 

 from compound quantities meet your approval, you will please to 

 insert it in your Journal. That the mathematical reader may the 

 better be enabled to estimate the advantages of this theorem, I 

 shall work an example, first according to the method at present 

 in use, and, secondly, I shall work the same sum according to 

 my own method, that a comparison may be made. In this paper 

 I shall confine myself to the extraction of the square root, as 

 much more necessaiy and useful than the cube, biquadratic, or 

 higher powers. 



Rule at present in Use. 



1. Range the quantity according to the dimensions of some 

 letter, and set the root of the first term in the quotient. 



2. Subtract the square of the root then found from the first 



