194 ARITHMETICAL COMPUTATIONS. 



Now the penultimate figure of the number is G....6— 2=4 =E. 

 And since 27 (or A)=r4, the last figure of which is 4 

 or B. The middle figure of root is 2, and root is 723. 



thTruf ?* ° f This rule ' 1 should add ' becomes ambiguous in all cases 

 where the number proposed terminates with an even digit, or 

 with a 5 j thus, in 41,421,736 A=s3 and B=2. 



Now, as either 4 x 8=32 or 9 x 8=72, it follows that, ac- 

 cording to the rule, either 4 or 9 might be the middle figure, 

 and either 346 or 396 the root j but as 3Qfj j 3 = nearly 400 | % 

 or 64 millions, it appears on inspection of the number proposed, 

 that 346 must be the true answer. No error would, therefore, 

 be produced by this ambiguity. Indeed, the only cases of 

 ambiguity which can deceive, are in numbers terminating 

 with 5. 



The rule for The rule for the square root differs only in these particulars j 



sqiare^ltS 6 t0 determine A > take the sim P le P wer of tile last figure of 



not much dif- the root, and instead of 3, multiply by 2. To determine B. 



iercnt subtract the penultimate figure of the square instead of the 



cube of the last figure of the root. In all other respects, the 



two rules exactly agree. In the case of square, there is, how- 



but the ever, an ambiguity which does not exist in the cube. It hap- 



ambiguity is pens, that the final figure of a square number gives two figures 



greater. which may terminate the root 5 as for instance, 4 3 =1 6 and 



6 9 =36. If, therefore, a square number terminate with 6, 



its root may terminate with either 4 or 6, and, therefore, more 



mistakes will occur in the application of the rule. I believe 



this coincides with the fact j since the boy makes many more 



errors in the extraction of the square, than in that of the cube 



root. 



Reference to The principles of these rules, and the rules themselves, or a 



of P R^mcr' d" • very s *'S nt modification of them, have been known so long 



Ourmes,pub- ago as the year 1768; in that year, M. Rail ier des Ourmes 



luhed 45 years published two memoirs on the subject. They are to be found in 



ago, and con- * j j 



taining the Pp. 485 and .550 of the fifth volume of " Scavans Etrangers." 



principles of They are entitled u Methode Nouvdle, &c. or a New Method of 

 these method?. . . ... . . 



dividing, when the dividend is a multipile of the divisor, and of ex- 

 tracting the roots of perfect powers. See page 550. His method 

 only takes the last figures into account. In the extraction of the 

 higher powers, this is undoubtedly the easier way. The second 

 is, " Methode facile, &c, or an easy Method of discovering all the 



(t prime 



