640 PHILOSOPHICAL TKANSACTIONS. [ANNO l6g4. 



A new, exact, and easy Method of Ending the Roots of any Equations gene- 

 rally, and that without any previous Reduction, By Edm. Halley. N° 210, 

 p. 130. Translated from the Latin. 



The chief use of the analytic art, is to bring mathematical problems to equa- 

 tions, and to exhibit those equations in the most simple terms. But this art 

 would justly seem in some degree defective, and not sufficiently analytical, 

 unless there were some methods for finding the roots, whether lines or numbers, 

 of those equations, and so the solution of the problems be completed. The 

 ancients knew little of these matters beyond quadratic equations: and what they 

 wrote on the geometrical construction of solid problems, by help of the para- 

 bola, cissoid, or any other curve, were only some peculiar effections designed 

 for some particular cases. But as to numerical extraction, there is every where 

 a profound silence : so that whatever we now perform of this kmd, is wholly 

 owing to the inventions of the moderns. 



And first of all, that great discoverer and reformer of the modern algebra, 

 Francis Vieta, about lOO years since,* showed a general method for extract- 

 ing the roots of any equation, under the title of " A Numeral Resolution of 

 Powers," &c. Harriot, Oughtred, and other authors, whatever they have 

 written on this subject, it must be acknowledged as chiefly taken from Vieta.+ 

 But what the sagacity of Newton's genius has performed in this way, we may 

 rather conjecture, than be fully assured of, from that short specimen given by 

 Dr. Wallis, in the g4th chapter of his Algebra. And we must be forced to 

 expect it, till his great modesty shall yield to the intreaties of his friends, and 

 pe'rmit those curious discoveries to see the light. 



Lately, viz. 1690, Mr. Joseph Raphson, F. R. S. published his " Universal 

 Analysis of Equations, " and illustrated his method by many examples ; in which 

 he has given indications of a mathematical genius from which the greatest 

 things may be expected. By his example, M. De Lagney, an ingenious pro- 

 fessor of mathematics at Paris, was induced to prosecute the same subject : but 

 being almost wholly occupied in extracting the roots of pure powers, especially 

 the cubic, he adds but little to the extraction of the roots of affected equations ; 

 and that rather perplexed too, and not sufficiently demonstrated. Yet he gives 

 two very compendious rules for the approximation of a cubic root, the one a 



* A little before the year 1(700. 



f Improvements are usually gradual and successive: Harriot, Oughtred, &c. added to the im- 

 provements of Vieta, as Vieta did to those of Stevinus, and he to those of others that preceded 

 him, &c. , 



