( '9^ ) 
uig giveti, to find th& Chord of another Arc, that ftiall be 
to the firft as » to i. Let y bs the Chord given^ z the Chord 
required ; now the Arc belonging to the Chord ; is, 
6dd + |S + '^^ '-'^^ belonging 
to the Chord =^ is =^ -1- ^ + ^ + 7117^ C^r. the firft 
of thefe Arcs is to the feeond as i to » ^ therefore multiply- 
ing the Extrearns and Means together, we ftiall have this 
Equation : 
+ 6dd + + ill? = ^ + ^ + ^ 
Compare thefe Two Series with the Two Series of the 
Theorem^ and you will find i, h= c= , </= o, 
~j4^ w= c^c, hence will be = »7 + 
cJ^iT, or ny 4*^ Y^x^^d ^^PP^^"§ denote the 
whole preceding Term, which will be the fame Series as 
Mr. Nevjton has firft found. 
By the fame Method^, this general Problem may be folv'd; 
she Abfcifle correfponding to a certain Area in any Curve be- 
ing giverij to find the Abfciflb, whofe correfponding Area 
ihali be to the firft in a giv^en Ratio. 
The Logarithmic Series might alfo be found without bor- 
rowing^any other Idea, than that Logarithms are the Indices 
of Powers: Let the Number, whofe Logarithm we inquire, 
be i-t^ fuppofe irs Log. to be az^ bzz'^ cz\ &c. Let 
there be another Number i+J' J thereof its Logarithm will 
be ay-^- hyy-^ cy^^ &c. Now if 1+ ^ 1+ j''^* follows, 
that az^hzz-\'cz^ &c, ay^byy-\-cf, &c i: n, i. 
that is, az -j- hzz + ^'^^ + ^^^^ ' 
Therfore 
