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the ufe of infinite feries, before the great inventer of 
fluxions was known in the world : but then, the bu- 
fmefs of logarithms being purely arithmetical, the 
Hyperbola was foreign to the fubjeCt ; and the nature 
of infinite feries, tho’ well adapted to the purpofe, 
was at that time but little underflood. 
I am fometimes induced to believe, that, if the lat- 
ter (/. e. the nature and ufe of infinite feries) had 
arrived at any degree of perfection, before the in- 
vention of fluxions, molt of the feries, which are 
given for the above-named kind of calculations, and 
have been deduced from fluxional procefles, would 
have been difcovered without the afiiflance of them : 
and I am of this opinion, becaufe I am certain, that 
many of them might have fo been : to inflance in 
both the cafes above quoted j viz. the feries, for 
making logarithms, and for rectifying the circle. 
And fir ft, the terms of one of the moft Ample 
feries, for expreffing the logarithm of a given num- 
ber, is compofed of the powers of the excefs of that 
number, above unity, divided by their refpeCtive indi- 
ces ; of which the firft, third, fifth, &c. terms are af- 
firmative, and the fecond, fourth, fixth, &c. terms 
are negative j and the difference between the fums of 
the affirmative and the negative terms, is the Neperian, 
hyperbolic, or (as fome call it) the natural logarithm 
of the given number. 
Now a mathematician, who underftands the nature 
and management of feries (altho’ wholly ignorant of 
fluxions, or what the juftly celebrated Dr. Halley, in 
his admired inveftigation of this very feries, publifhed 
in N° 216 of the Philofophical Tranfaffions, calls 
ratiuncula , &c.) might arrive at the fame conclufion, 
in the following manner : 
Since 
