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•In the 97th Scdioa of his Analyfis, Doftor Hales flievvs the 

 method of taking quadratic furds out of an equation, provided the 

 number of terms be not greater than four, but if there be a fifth 

 term, vfhether rational or furd, he is of opinion that the equation 

 cannot, by that method, be cleared from furds. Were this the cafe, 

 Ave would have no other alternative, than to recur to the method of 

 Monfieur Fermat, by feigning the furds equal to an affumcd letter, 

 and thence by means of as many fimple equations, as there are 

 furds, to take thefe letters out of the equation ; but this is a v\'Ork 

 of fo much labour, that it is fufEcient to deter a perfon flightly ac- 

 quainted with algebra. However, by confidering the nature of the 

 furds that arife after involution, it will readily appear that there is 

 no fuch limit, nor any need of recurring to an operation fo labo- 

 rious. 



Let the equation propofed confift of five quadratic furds, if 

 thefe can be rendered rational, four quadratics and a rational may 

 be reduced at leafl with the fame eafe. 



It is plain, that any equation, confifling of five furds, may be 

 reduced to this form, thence after involution will arife the feweft 

 number ai furd redangles (Dodor Saunderfon fhews in his algebra 

 that if two quantities be irrational, their produd will be fo too) for 

 if the equation was divided into parts of one and four furds, the 

 refult would have fix furd redangles, and from the former po- 

 fition there will arife but four. 



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