VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 3 



the square of a or '2 : which being subtracted, leaves n — a' =5 — 4 = 1 = 

 B = '2 A E + E^. Which divided by 2 a = 4, gives 4- : but divided by 2 a + 

 1=4+1=5, gives 4- : that too great, and this too little for e. And there- 

 fore the true root (a + e = y^ n) is less than 24- = 2.25, but greater than 2; 

 = 2.2. And this was anciently thought an approach near enough. 



If this approach be not now thought near enough, the same process may be 

 again repeated ; and that as often as is thought necessary. Thus, take now for 

 a. 24-=: 2.2, whose square is 4.84 = a^ ; this, subtracted from 5.00, leaves 



.16 2 



a new remainder b = O.16: which divided by 2 a = 4.4, gives 7^ = — =: 



' '^ 4.40 55 



16 8 



03636 +, too much ; but divided by 2 a + 1 = 4.5, it gives ^-~ = — - 



= 0.03555 +, too little. The true value, between these two, being 2.236 

 proAiiTie, whose square is 4.999696. 



If this be not thought near enough, subtract this square from 5.000000; 

 the remainder b = 0.000304, divided by 2 a = 4.472, or by 2 a + 1 = 4.473, 

 gives, either way, OOOOO68 — ; which added to a = 2.236, makes 2.236o68 — , 

 somewhat too large ; but 2.236067 + would be much more too little. 



Proceed we now to the cubic root. For which, the rule is this : from the 

 non-cubic proposed, suppose n, subtract the greatest cube in integers, sup- 

 pose a^ ; the remainder, suppose b = 3 a'^ e -|- 3 a e^ -|- e^ is to be the 

 numerator of a fraction for designing the value of e, the remaining part of the 

 root sought A + e = ^ n. To this numerator, if, for the denominator or 

 divisor, we subjoin 3 a'^, the result will certainly be too great for e, because 

 the divisor is too little. If, for the divisor, we take 3 a'^ 4- 3 a -f- l, it will 

 certainly be too little, because the divi>or is too great. It must therefore, be- 

 tween these limits, be more than this latter. And therefore this latter result 

 being added to a, will give a root whose cube may be subtracted from the non- 

 cubic proposed, in order to another step. This approach I find in Wingate's 

 Arithmetic, published in the year l630, and must therefore be at least so old; 

 how much older I know not. But if for the divisor we take 3 a'- -j- 3 a, the 

 result may be too great ; or, in case b be small, it may be too little. 



Thus, for instance ; if the non cube proposed be 9 = n. The greatest in- 

 teger cube is 8 = a^, whose cubic root is a = 2 ; which cube ^ubtracted, 

 leaves 9 — 8 = 1 = b = 3 a'^ e + 3 a e'- -|- e\ This divided by 3 a' 

 = 12, gives -fV = 0.08333 -|-, too great for e. But the same divided by 

 3A--f3A + l = l2-f6+l =19, gives -^ = 0.05263 +, too little. 

 Or, if but by 3 a' + 3 a = 12 -J- 6 = 18, it gives -pL = -J>- = 0.05555 -f-, 

 yet too little. For the cube of a -|- O.06, = 2.06, is only 8.742 — , which is 

 short of 9. And so much short of it, that we may safely take 2.07 as not too 



£ 2 



