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That this method may always be ufed when there is occa- 

 Con, it is neceffary to folve the following 



PROBLEM. 



A NUMBER, not a cube number, being given, it is required 

 to find a cube number, which multiplied into the given num- 

 ber, fhall give a produd nearly equal to fome other cube 

 number. 



Suppose n a given number, not a cube, it is required to find 

 two other numbers, a the greater, and b the lefs, fo as that a^ fliall 



be nearly equal to n h, or that the fradion ^ ~" , or j — 



fhall be very finall. 



When the given number is fmall, the numbers a and 5 

 may be found by a few and eafy trials ; but if it be 

 great, the folution by trials is very difiicult. But a general 

 and dired folution of it was fuggefted by the folution of a 

 problem, nearly of the fame nature, by Dodor Wallis, and pub- 

 lilhed by him in the Commercium Epijiolicum, at the end of the 

 fecond volume of his works. An example will fufticiently fhew 

 the method ufed in the folution. 



Let the given number be 13 ; and a^ will be nearly 

 equal to 13^^ Call the difference j', and we fhall have this 

 equation, ^'=i3^^+j'. We are next to find the limits of a 

 with refpcd to h ; that is, to find the multiples of b which are 

 next lefs and next greater than a. And fince the cubic root of 

 I 2 13 



