THE 



PHILOSOPHICAL TRANSACTIONS 



OF THE 



ROyAL SOCIETY OF LONDON; 



ABRIDGED. 



On the Methods of approximation in the Extraction of Surd Roots. By John 

 Jfallis, S. T. D. and Saviiian Professor of Geometry at Oxford. N° 213, p. 2. 

 Fol. XIX. 



J. HE several methods of approximation, which have been mentioned of late 

 years, for extracting the roots of simple or affected equations, give occasion to 

 say somewhat on that subject. It is agreed by all, and I think demonstrated by 

 the Greeks long ago, that if a number proposed be not a true square, it is in 

 vain to hope for a just quadratic root of it, explicable by rational numbers, in- 

 tegers, or fractions. And therefore, in such cases, we must content ourselves 

 with approximations, without pretending to accuracy. And so for the cubic 

 root, of what is not a perfect cube. And the like for superior powers. 



Now the ancients had their methods of approximation in such cases ; some 

 of which have descended down to us. But since the methods of decimal frac- 

 tions have come into practice, it has been usual lo prosecute such extractions 

 in the places of decimal parts, to what accuracy we please. 



Mr. Newton's method of approximation for extracting roots, even of affected 

 equations, I have given some account of in my English Algebra ; and somewhat 

 more fully in the Latin edition ; where I gave an account also of Mr. Raphson's 

 method. Since which time, M. de Lagny has published his Method of Ap- 

 proximation, principally for single equations, or extracting the root of a single 

 power. And Mr. Halley has since improved this method, with a further ad- 

 vantage, especially as to affected equations. 



These may all, or any of them, be of use for making more speedy approaches, 

 and by greater leaps, in many cases, than Vieta's method, pursued and improved 

 by Mr. Oughtred and Mr. Harriot of our own country, and by others abroad ; 



VOL. IV. B 



