6 



And that the omiffion of the fraftion — ia the third and following terms 



n 



of the feries A will not caufe any error within the prefcribed numbers 

 of places of figures, will appear by confidering the value of the quan- 

 tity that is thereby omitted in any term of the feries C : for it will 

 appear for many reafons to be always lefs than — » the greateft poffible 

 value of which is a decimal confifting of i precede'^ by twice as many 

 cyphers, except 3, as there are places in n ; and therefore the value of 

 all the omiffions in 100 terms will not amount to an unit in the lafl 

 of the prefcribed places. 



Lemma 2. 



" The values of e and n continuing as before, it is required to find 

 a like feries that fhall give the root of the refidual i—^e true to the 

 fame number of places." 



(D) I. 1 ^ -^' '—+ -e. —X.e. —X-e. — X - f . &c. 



(E) I — - e e"- f ' e* e^ , 



n 2n 3n 4« 5« &C. 



In every term after the firft of the feries B change the fign of the 

 quantity e, and there will refult the feries marked D, whofe terms, 

 multiplied as before, produce the feries required, which is marked E. 

 This feries differs from C only in this, that all the terms after the 

 firfl are negative : for in the fecond term of D, e is negative ; there- 

 fore the fecond term of E (having one negative factor) is negative ; 

 and in every fucceeding term of D, there are two negative faftors, 

 whence in every following term of E, there will be an odd number 

 of negative faftors, and therefore they will all be negative. 



Lemma 



