VOL. XXVII.] THILOSOPHICAL TRANSACTIONS. 6ll 



of the proposed number a -\- l, will be obtained by lemma 1. For, by the 

 hypothesis, b' is produced by the multiplication of numbers, the greatest of 

 which is less than a + 1 ; and by hypothesis there are given the logarithms of 

 all numbers less than the proposed number a -{- I ; therefore also there is given 

 the log. of b% or the number produced from them all, and thence, by lemma 2, 

 the log. of b also is given. 



Exam. 1. — Assume at pleasure b = a; hence z = /. . Then, by art. 2, 



^ a + 1 f J J 



take 1/ = 2a -\- Ij by which exterminate a, it will be z = 



/. ^ ~ , , the fluxion of which is z = — — ; the fluent gives z = 



— 2 ( — l~3~3"i"7^~l"7^ ^^•) expressed in a series; hence, by lem. l,cc=il.b 



+ 2 ( - + -i-, + -^ + -^ &c. ) 



Exam. 2. — ^Take b •=. ^ aa ■\- 1a\ hence 

 2 = /. ^°" + ^ ; assume also y = 2a + 2, then z = /. ^^^ ~— ; its fluxion 



a + 1 "^ ' y 



is ir = 43/ (/ - 4y)-', the fluent of which isz = -2(^^, + |. + |, + ^&c.); 

 hence ^ = /.Z; + 2(1 + |1 + ^^ + ^ &c.) by the 1st lemma. 



Exam. 3. — Again take b = l/aa -f- 2a; but now assume y^ = laa -f- 4a -f- 1 » 

 then, by these two equations exterminating b and a from the general canon, it 



will be z = /. — ^H-L ^ the fluxion of which is i = '^yy^'lt "~ 0"'j ^"cl its 



^yy + 1 



fluent in a series z = r — — --rr &c; hence 



^ = /.Zj -}- - -f — g -{- ~— -I- ^— - &c. by the 1st lemma. 



Hence it appears that the logarithmetechny now explained, is very easy and 

 genuine, and so general, that by these two methods innumerable series may be 

 found exhibiting the log. of any proposed number. For we may assume in- 

 numerable equations at pleasure, expressing the relation between y and a, every 

 one of which will give a new logarithmic series. Yet care should be taken that 

 the equation be assumed so, as to cause the terms of the series to converge as 

 fast as may be, that the log. may be found with the least labour of calculation. 

 To perform which, the series exhibited in the last example will be very proper, 

 and which is the same as that first given by Dr. Halley, in his elegant method 

 of constructing logarithms. 



Here, by the bye, I would desire it may be observed, that the curve, which 



4 i2 



