EVOLUTION. 151 



from the top of the sign of the root, or radical sign, over 

 all the parts of it : so the third root of 47 — 15 is 



3 



>/47 — ^5* ^^^ sometimes roots are designed like 

 powers, with the reciprocal of the index of the root above 



the given number. So the 2d root of 3 is 3"^; the 2d root of 50 



is 50"=^ ; and the third root of it is 5o"3' j also the third root of 



47 — 15 is 47 — 15I . And this method of notation has 

 justly prevailed in the modern algebra ; because such roots, 

 being considered as fractional powers, need no other di- 

 rections for any operations to be made with them, than 

 those for integral powers. 



Note 3. A number is called a complete power of any 

 kind, when its root of the same kind can be accurately ex- 

 tracted ; but if not, the number is called an Imperfect pow- 

 er, and its root a surd or irrational number : so 4 is a com- 

 plete power of the second kind, its root being 2 ; but an 

 imperfect power of the third kind, its root being a surd 

 number, 



Evolution is the finding of the roots of numbers either 

 accurately, or in decimals, to any proposed extent. 



The power is first to be prepared for extraction, or evo- 

 lution, by dividing it from the place of units, to the left 

 in integers, and to the right in decimal fractions, in- 

 to periods containing each as many places of figures, as are 

 denominated by the index of the root, if the power con- 

 tain a complete number of such periods : if it do not, t|[ie 

 defect will be either on the' right, or left, or both ; 

 if the defect be on the right, it may be {supplied by 

 annexing cyphers, and after this, whole periods of cyphers 

 may be annexed to continue the extraction, if necessa- 

 ry *, but if there be a defect on the left, such defective peripd 

 must remain unaltered, and is accounted the first period of 

 the given number, just the same as if it were complete. , 



Now 



