300 PHILOSOPHICAL TRANSACTIONS. [ANNOI7I7. 



desired nearly true to one or two places in decimals (which is done by a 

 geometrical construction, or by some other convenient way) and correcting 

 the assumption by comparing the difference between the true root and the 

 assumed, by means of a new equation whose root is that difference, and 

 which he shows how to form from the equation proposed, by the substitu- 

 tion of the value of the root sought, partly in known and partly in unknown 

 terms. 



In doing this, he makes use of a table of products (which he calls speculum 

 analyticum,) by which he computes the co-efBcients in the new equation for 

 finding the difference mentioned. This table, I observed, was formed in the 

 same manner from the equation proposed, as the fluxions are, taking the root 

 sought for the only flowing quantity, its fluxion for unity, and after every 

 operation dividing the product successively by the numbers I, 2, 3, 4, &c. 

 Hence I soon found that this method might easily and naturally be drawn 

 from Cor. 2, Prop. 7, of my Methodus Incrementorum, and that it was capable 

 of a further degree of generality; it being applicable, not only to equations 

 of the common form, viz. such as consist of terms in which the powers of the 

 root sought are positive and integral, without any radical sign, but also to all 

 expressions in general, where any thing is proposed as given, whicli by any 

 known method might be computed ; if vice vers^, the root were considered as 

 given : such as are all radical expressions of binomials, trinomials, or of any 

 other nomial, which may be computed by the root given, at least by logarithms, 

 whatever be the index of the power of that nomial; as also expressions of lo- 

 garithms, of arches by the sines or tangents, of areas of curves by the abscissas, 

 or any other fluents, or roots of fluxional equations, &c. 



For the sake of this great generality, it may not be improper to show how 

 this method is derived from the foresaid corollary. Therefore z and x being 

 two flowing quantities, of which the relation to each other may be expressed by 

 any equation whatever; by this corollary, while z by flowing uniformly be- 

 comes z -j- V, -r will become -3? + ^ »^ + i;^*^"* + f^p ^^ + &c. 



^^ ^ + T + ^ + iS^ + ^''- P""^"^ ^ ^°' ^* 



Hence if y be the root of any expression formed of y and known quantities, 

 and supposed equal to nothing, and z be a part of y, and x be formed of z 

 and the known quantities, in the same manner as the expression made equal 

 to nothing is formed of y ; and let y be equal to z -f f; then the difference 

 V will be found by extracting the root of this expression 

 ;r + if 4- ~ 4- -^ -f &c. = 0, For in this case z being become z 4- 



~ I ~ 1.2 ^ 1.2.3 ' ' 6 1 



