INVOLUTION, 14^ 



Note i. The number, which exceeds the multipHca- 

 dons by i, is called the index, or exponenty of the power \ so 

 the index of the first power is i, that of the second power 

 is 2, and that of the third is 3, &c. 



Note 2. Powers are commonly denoted by writing 

 their indices above the first power : so the second power 

 of 3 may be denoted thus 3*, the third power thus 3^, the 

 fourth power thus 3"*, &c. and the sixth power of 503 

 thus 503*^. 



Livolutioji is the finding of powers j to do which 'we 

 have evidently the following 



RULE. 



Multiply the g>ven number, or first power, continually 

 by itself, till the number of multiplications be i less than 

 the index of the power to be found, and the last product 

 will be the power required.* 



Note. "Whence, because fractions are multiplied by 

 taking the products of their numerators and of their de- 

 nominators, they will be involved by raising each of their 

 terms to the power required. And if a mixed number be 



proposed, 



* Note. The raising of powers will be sometimes shortened 

 by working according to this observation, viz. whatever two or 

 more powers are multiplied together, their product is the power, 

 whose index is the sum of the indices of the factors ; or if a 

 power be multiplied by itself, the product, will be the power, whose 

 index is double of that, which is multiplied : so if I would find 

 the sixth power, I might multiply the given number twice by it- 

 self for the third power, then the third power into itself v/ould 

 give the sixth power ; or if I would find tlie seventh ])cv. :r, I 

 might first find the third and fourth, and their praduci: v/onii be 

 the seventh ; pr lastly, if I would find the eighth povvcr, I might 

 first find the second, then the second into itsr^f -would be tlse 

 fourth, and this into itself would be the eighth. 



