206 



In substituting the terms of the natural series, one of the 

 terms being cypher, the process of involution is easier, not 

 only because of the smulhiess of the numbers, but also the 

 same process of involution will serve for the substitution of 

 affirmatives and negatives. For, in general, in an arithmetical 

 series, when the next terms to cypher, or the tv,-o lowest terms, 

 are the same, but with contrary signs, all the terms below 

 cypher are the same with those above it, excepting the differ- 

 ence of sign. If an affirmative quantity be substituted for x, 

 and if we make one sum of the terms in which the indices 

 are even, and another sum of the terms in which the indices 

 are odd, both sums, added together, will he the result of the 

 substitution of the affirmative, and the sum of the terms con- 

 taining even powers, less by the sum of the terms containing 

 the odd powers, will be the result of the substitution of tlie 

 same quantity, with a negative sign. 



Yet we would not always substitute the same number of 

 affirmative and negative terms. For, if ail the terms of the 

 given equation are affirmative, we shall have lesser results by 

 substituting negatives for x : for the roots being all negative, 

 if we diminish them by negative quantities, we shall bring 

 them nearer to cypher. And if the terms are alternately af- 

 firmative and negative, the roots being all affirmative, we 

 should diminish them b}' affirmative quantities. If the roots 

 of the equation are great, and the absolute term thereof con- 

 siderable, by considerably diminishing those roots, that is, 



by 



I 



