2 PHILOSOPHICAL TRANSACTIONS. [aNNO l6g4-5. 



especially as to simple equations, if we suppose such extractions to be pursued 

 to the full extent. 



But if we make use of Mr. Oughtred's expedient, for multiplication, divi- 

 sion, extraction of roots, and other like operations, by neglecting so much of 

 this long process, as is afterward to be cut off and thrown away as usekss 

 which I think is generally practised, the work will be much abridged, and the 

 advantage of the other methods much less considerable. 



And if we further consider, what preparative operations are to be made in 

 some of those other methods, before we come to the prescribed division for 

 giving the root desired ; the advantage will not be so great as may at first be 

 apprehended. But, without disparaging these methods^ what I here intend, is, 

 to show the true foundation of the methods used by the ancients, and the just 

 improvement of them. Which though anciently scarce applied beyond the 

 quadratic, or perhaps the cubic root, yet are equally applicable, by due adjust- 

 ments, to the superior powers also. . 



I shall begin with the square root, for which the ancient method is to this 

 purpose : from the proposed non-quadrate, suppose n, subtract the greatest 

 square in integers, suppose a^. The remainder, suppose b = 2 A E -(- e'^, is 

 to be the numerator of a fraction, for designing the near value of e, the re- 

 maining part of the root sought, viz. a -|- e = v' n, whose denominator or 

 divisor is to be 2 A, or 2 A -|- 1, the true value falling between these two ; some- 

 times the one, sometimes the other, being nearer to the true value. But the 

 latter is commonly directed. This method M. de Lagny affirms to be more 

 than 200 years old : and it is so ; for I find it in Lucas Pacciolus, otherwise 

 called Lucas de Bergo, or de Burgo Sancti Sepulchri, printed at Venice in the 

 year 1494, if not even sooner, for I find there have been several editions of it; 

 and it seems much older, for he does not deliver it as a new invention of his 

 own, but as a received practice, and derived from the Moors or Arabs, from 

 whom they had their algorism, or practice of arithmetic by the ten numeral 

 figures now in use. And it is continued down hitherto in books of practical 

 arithmetic in all languages, which teach the extraction of the square root, and 

 this method of approximation, in case of a non-quadrate. 



The true ground of the rule is this : a'^ being the greatest integer square 

 contained in n, it is evident that e must be less than 1. Now if the remainder 

 B = 2 A E -f e'- be divided by 2 a, the result will be too great for e ; but if we 

 diminish the quotient, by increasing the divisor, adding 1 to it, it then becomes 

 too little ; because the divisor is now too great. For, e being less than 1, 

 2 a -|- 1 is more than 2 A -{- k, and therefore too great. As (or instance ; if 

 the non-quadrate proposed be n = 3, the greatest integer square is a* = 4, 



