272 PHILOSOPHICAL TKANSACTIONS. [anNO J 738, 



his method of extracting any root of the binomial a -\- \/ — b, this induced 

 Wm. Jones, Esq. F. R. S.* to desire him to do the same by the possible bino- 

 mial a -\-\/+ b; a request which Mr. D. here complies with, though he is 

 sensible that this has been done already, by Sir I. Newton and others. 



Pbob. I. — To reduce the Binomial V'a + ^b to Simpler Terms. Suppose that 

 this binomial, including its general radicality, can be reduced to the other 

 binomial x + ^y, freed from that radicality. Now to find such quantities x 

 and 3/, try whether the sum of the binomials V'a +^^ + v^a — Vb makes 

 nearly an integer number, which may be readily done by a table of logarithms ; 

 if it do, then put 2x =: to this whole number. Next try whether l/aa — bhe 

 an integer ; if it be, put jn = this new integer ; then will y ■=. xx — m\ and 

 therefore the given binomial will be reduced to the given form. But be- 

 fore proceeding to the demonstration, it may be illustrated by two or three 

 examples. 



Example 1 . — Let the binomial v' 54 + v/QSO be proposed. 



Put a = 54, b =■ 980; then will ^/ b = ^gSO = 31.3049 nearly ; which 

 gives a + Vb = 85.3049, and a — Vb =. 22.695 1. Now the root of the 

 first number is 9.236 nearly ; and the root of the latter is 4.763 ; the sum of 

 which roots is 1 3.999, which is very near the whole number 14. Therefore 

 putting Ix = 14, or ar = 7 ; then since y :=. xx — m, and m ='1/ aa — b =: 

 >/ 1 936 = 44 ; therefore is y = 49 — 44 = 5 ; so that the binomial reduced 

 will be 7 + -v/S. 



Example 2. — Let \/ A5 -{■ V l682 be reduced simpler. 



Put a = 45, b = l682; then is i/ b = 41.01219 nearly; hence a + ^b 

 = 86.01219, and c — ■//>= 3.89781. 



Now the cube root of the former number is 4.4142, and the cube root of the 

 latter number is 1.5857; the sum of which roots being 5.9999, which is nearly 

 6 ; therefore put 2x = 6, or ar = 3 ; but it being y = xx — m, and m = 

 1/ aa — b =i 1/343 = 7 ; therefore 3/ = 9—7 = 2; and hence the bino- 

 mial reduced is 3 -|- \/'l. 



Example 3. — Let \/ J70 -|- V 18252 be reduced simpler. 



Put a = 170, b = 18252, then will \^ b = 135.1 nearly ; which gives a -\- 

 i/b z= 305.1, and a — \/ b =■ 34.Q. 



Now the cube root of the former number is 6.73 nearly, and the cube root 

 of the latter is 3.26, the sum of which roots is 9.99, nearly equal to the whole 

 number 10. Put therefore 2x = 10, or a; = 5 ; then since y = xx — m, and 



* See an account of Mr. Jones, who was the father of the late Sir William Jones, in the intro- 

 duction to Dr. Hutton's Mathematical Tables, p. I19. 



