894 
LOGARITH M'S. 
t> 2 p 4 
And, if p— — -4 -— 4 - 4 — &c. be put =r, we 
2 3 4 5 
fliail have 
z 5 &c. — r , 
1 +JZ-i-|' s2 'S 2 -1-i 3 2 3 1-S 4 -Z 4 -f- 
2.3 2.3.4. 
or ,r.z-}-fs 2 z 2 4-—s 3 z 3 l—L_ j 4 z 4 + 
2. 3 . 4-5 
I 
s 6 z 5 &c=r—r, i s j 
' • 
pending only on the affumed value of r. And, as the 
form of this feries is exactly the fame as that which con- 
ftitutes the numerator, and which has been fhown to he 
the hyperbolic logarithm of a, it follows that the modu¬ 
lus of any fyftem of logarithms is equal to,the hyperbolic 
logarithm of the radix of that fyftem, or of the number 
whofe proper logarithm in the fyltem to which it belongs 
2.3 2.3.4 2.3,4.5 , 
which let be put 2= q ; then, by reverting the feries, z 
q -h? 2 4 -|<? 3 — jq* 4 ~?< 7 5 Sec. 
or - will be found = 
x 
d — — I- 7 4 +i 7 5 &c. 
F~ hp 2 + iP 3 - kP* + i /> 5 
P - 4 - jp 3 — IP 4 + T /> 5 
q — + i? 3 — i? 4 + &c ' 
Tiie logarithm of a, or 1 
P — jP z + $/> 3 — iA 4 + \P 5 Sec . 
q ~\f + i ? 3 -*? 4 +i ? 5 &c’ 
q — r — 1, the logarithm of a is == 
0 2 +f(fl—'i)3— \(a— i)4+i(a ^ 
“-K»—0 2 +H»-— i) 3 —*(»•—0 4 +K r —O 5 
and confequently x 3= 
/>, is therefore = 
or, fince p = a — 1, and 
-t) 5 
The form of the feries here obtained for the hyperbolic 
logarithm of a, is the fame as that which was firft difeo- 
vered by Mercator 5 and, if the feries of Wallis be required, 
it may be investigated in a fimilar manner, as follows: 
The general logarithmic equation being r x ~ a , 
as before,' let a zz. and z = - ; then r = a* " 
i—p 
= ;==^> and - = r — (p + P ~ + + jSec.) 
-hs(/>+~+-—■ -f —Scc.Yz 2 ~ — 
2 , 3 /L 2 , 3 W a 1 1 A 
i-/* 
T , . * 2 *3 *4 
(M-—+—+—&c.)V- 
2.3.4 2 3 4 y 
3 ' 4 
p 2 p 3 
2 . 3 . 4 . 5 ( >+' a + 7 +“ 
(r—1) 
which is a general expreffion for the logarithm of any 
number, in any fyftem of logarithms, the radix r being 
taken of any value, greater or lefs than 1. 
But, as r in every fyftem is a conftant quantity, being 
always the number whofe logarithm in the fyftem to 
which it belongs is 1, the above expreffion may be Am¬ 
plified, either by affuming r =2 to fome particular number, 
and from thence finding the value of the feries confti- 
tuting the denominator; or by affuming this whole feries 
222 to fome particular number, and from thence finding 
the value which muft be given to the radix r. By the 
‘latter of thefe methods, the denominator may be made to 
•vanifh, by affuming the value of the feries of which it 
eonfifts = 1, in which cafe, the logarithm of 1 -J -p be- 
p 2 p 3 /)4 pS 
comes —p — -J- 1 - -— Sec. or the logarithm of 
a 3 4 5 
t—(a— 1)— i(<2— i) 2 +i(«—O 3 — i(a— i) 4 +K fl —0* 
&c. and r, by reverfion of feries is found 2= 2-7182818 See. 
The fyftem arifing from this mode of determining the 
falue of the radix r, is that which furnifhes what have 
fceen ufually called hyperbolic logarithms ; and appears to 
be the fimpleft form the general expreffion admits of. 
If, on the contrary, the radix r be affumed — to fome 
particular number, as for inftance 10, the value of the 
feries q~~±q- i q3 —I} 1 See. or its equal 
(r -0 —{(r-1) 2 +|(f- 1 ) 3 — |(r—1 ) 4 +{(r - I )5 &c. 
will become =2 2,• 30258509 See. and the log. of 1 -f *p — 
~ 3 o2sV 5 o 9 X U-*P*+iP*-*P*+iP ,Scc ) orthelog.of a = 
rioi 5 w 9 * («-o-^(a-o 2 +f(a- 
1) 5 See. which gives the fyftem that furnifhes 
Briggs’s or the common logarithms. 
And, in like manner, by affuming any particular 
value for r, and thence determining the value of the 
feries q— §? 2 -H? 3 —i? 4 +-|? 5 Sec. or'its equal 
(?-—1)— £(7-—-i) 2 -H(r— 0 3 —!(r—i) 4 +K r —i ) 5 Sec. 
or by affuming the fame feries of fome particular value, 
and thence determining the value of r, any fyftem of lo¬ 
garithms may be derived. 
The feries q— ± q 2 + •§ q*— -J- 9 4 -j- ^5 &c , or its equal 
( r r , , ) r 4 ( r T I) J 2 + *^- 1 ) 3 - i ( r “ , ) 4 +i( r -"i ) 8 &c - 
which forms the denominator of the above compound ex¬ 
preffion, exhibiting the logarithms of numbers according 
to any fyftem, is what was firft called, by Cotes, the mo¬ 
dulus of the fyftem, being always a conftant quantity, de- 
&c.) 5 z 5 Sec. 
• ^ ^5 
And if p ^|- 1 - — Sec. be put we 
2 3 4 5 
fliail have 1— ■sz-i-h 2 z 2 -— s 3 z 3 4- —— s 4 z 4 Sec. r= is 
2.3 1 2.3.4 r 
or sz — |i=z*+—i3. a 3- —s 4 z 4 Sec. = 1 _ I, 
a -3 2.3.4 r * 
which let be put = q ; then, by converfion of feries, 
zori will be found = ? + * ?a 
— +ig 4 4 -T? 5 . r 
~t> + kp 2 +ip 4 + ip s d confe q uenJ y * 
_ P + + 3 P 3 + l P* + \p s See 
~~ q + 2 q 2 + 3 q 3 4 -i ? 4 &c" 
The logarithm of a or is therefore 
p + lp* + %P 3 + iP*+iP 5 &c 
~ i+Tjr+U 3 +i? 4 +i ? 5 &c 5 or > fmce ? = 1 “ ~ 
• and q = 1- 
* V 
-> the logarithm of a is = 
Which is another general expreffion for the logarithm of 
any number, a, in any fyftem of logarithms, that may be 
Amplified in the fame manner as the former, the denomil 
nator being ftill equal to the hyperbolic logarithm of the 
radix r; or, which is the fame thing, to the modulus of 
the fyftem. 
For, if the feries ? + £ + § ? 3 -H ? 4 +£ f 5 Sec. or 
its equal 
be affumed = 1, the hyperbolic logarithm of-A—will be 
1 p 
~P -ViP 2 + 4 -fi 65 Sec. or the hyperbolic 
logarithm of a 
^•CrMTMtW^ 
, and 
