VOL. XVIII.] PHILOSOPHICAL TRANSACTIONS. 641 



rational, the other an irrational formula, viz. that the side of the cube a' + b, 

 is between a + -r-^-r-, and ' a + ^ 4. o'^ + -— . And the root of the 5th 



3 a' + ' ' * '3a 



y y — =:= 



power c' + ^ he thus expresses, -J- a + • v x '^^ + 7 t «'• These rules 



were communicated to me by a friend, as I have not seen the author's book ; 

 but having by trial proved their goodness, and admiring the compendium, I 

 wished to discover the demonstration. This being accomplished, I soon per- 

 ceived that the same method might be accommodated to the solution of all 

 sorts of equations whatever. And I was the rather inclined to improve these 

 rules, as I saw that the whole business might be explained in a synopsis ; and 

 that in this way, at every repetition of the calculus, the figures already found 

 in the root would be at least tripled, which all the former methods only 

 doubled. 



Now the fore-mentioned rules are easily demonstrated from the genesis of the 

 3d and 5th powers. For, supposing the side of any cube to be a + e, the cube 

 of this is a a a-\-3 aae-{-3aee-\-eee. Consequently, if we suppose aa a the 

 next less cube, to any given non-cubic number, then e ee will be less than 

 unity, and the remainder b will be equal to the other members of the cube, 

 3aae-\-3aee + eee: hence, rejecting eee on account of its smallness, it is 

 b = 3aae-\-3aee. And since a a e is much greater than aee, therefore - — 



will not much exceed e ; so that, putting e = - — , then the quantity -— — , 



to which e is nearly equal, will be found = — -,, or -,, that 



3 a a -< 3aa + — 



3 a a a 



■=■ e; and therefore the side of the cube aaa-\-b will be a-j- 



is. 



3aaa + b ' 3aaa + 6' 



which is M. de Lagney's rational formula. But i( a a a were the next greater 

 cube than the given number, the side of the cube aaa — b, after the same 



manner, will be found a r. And this easy and expeditious ap- 



3 aa a — b •' ^ '^ 



proximation to the cubic root, is only a very small matter erroneous [n defect, 

 giving the quantity e rather less than the truth. 



The irrational theorem is also derived from the same principle, viz. b = 3aae 



-{■3aee, or — :=ae-\-ee; so that \/ ^ a a + —- = ^ a -\- e, and 

 \ a a ■\- - — \- \ a =■ a -{■ e, the root sought. And the side of the cube 

 a a a — b, in the same manner, is found to be -l a -j- \/ \ a a — — . And 



x/ 



VOL. III. 4 N 



