888 
a'itd if we now make 
B == (N — i) —i (N 
we (hall have, on the fault 
B 2 
N x =r i - 4 - B z -f- - 
LOGARITHMS. 
-0 2 . + i (N. 
principles, 
B 3 
z 2 4 - 
■i) 3 
Sec. 
N* 
i 4 - A .v z 4- 
X 2 z 2 
a . 3 
+ 
2 J r &c. 
A 3 
But, 
&rc. whence, by comparing 
feries, we have 
A „ A 2 * 2 B 2 
Al-zrBj 
z 3 4- 
i . a i • s- • 3 
the co-efficients of z in both 
A 3 
B 3 
See .; 
a i. a 1.Z.3 1.2 
each of which gives the fame refult, viz. A x — B ; whence 
we obtain immediately, 
_ B_ _ (N — O —|(N — i) a 4 - £ ( N — Q 3 — See. 
~ A (>• — 1) —| (z — 1) 2 T I ( r — O 3 &c. 
which is the analytical expreffion for the logarithm of any 
number N, in funftions of itfelf, and the radix of the 
fyftem; that is, writing a inllead of N, 
„ _ ( g — 1) —1)^ 4 - ^(a — Q 3 —& c. 
^ i0 o- fl “ (r - O -4 (r- I)* + 4 (r— x) 3 -&C. 
± b — | a 2 dr F a 3 — £ d 4 ri: &c. 
0r - ,0 --' * 0 = ( r- 0 -U'-V+rt-V^ 
■ This, however, mult only be confidered as a fimple alge¬ 
braical method of expreffing a logarithm ; but it does not 
always anfwer the purpofes of calculation; for, if a be al T 
number greater than unity, it is obvious that the feries in 
the numerator will either converge very (lowly, or other- 
wife will diverge, and the fame with regard to the deno¬ 
minator, fuppoling r to be equal to 10, as it is in the com¬ 
mon fyftem; in faff, the terms of the feries are larger the 
more remote they are from the beginning; and confe- 
quently no number of them can exhibit, either exaflly or 
nearly, the true fum. Let us, therefore, inveftigate the 
method of fubmitting thefe to calculation; in order to 
.which we will repeat again our laft feries, viz. 
± a — | a 2 ± | a 3 —• l a 4 ± See. 
log. 1 ±a— c? - _ j) _ * (r — t ) 2 +£ (r — 1 ) ; 3 — &c. 
and here, (ince the denominator is always a conftant quan¬ 
tity when the radix of the fyltem is given, we may make 
M=(r—1) — | (r—i) 2 +#(r—-i) 3 —&c. _ 
which renders the above expreffion ltill more fimple, as in 
that cafe it becomes barely 
1 
log. 1-4 - a — —x 
o 1 M 
+ i« 3 
4~ Sec. 
Or, taking a negative, 
log. 1 — a — ~x £ _a_ia 2 —i« 3 _a & c . ^ 
Whence, again by fubtraftion, 
log. = + + | 
4 - 
K0W(2~- 
a 4- 1 
(2 4* I 
a - 
going expreffion 
; if, therefore, we lubftitute in the fore- 
inltead of a, it becomes log. a : 
a 4 - 1 
the denominator in our frit expreffion a known quantity, 
which we have reprefented by M. It will/ however, be 
proper, before we proceed any farther, to offer a few re¬ 
marks upon the abfolute value of this feries, according to 
any given radix. Firft then, fince 
a —|u 2 41 a3 —d a 4 + &c. 
° S ' 1 ^ a ~ (r — 1 ) — h (r— i) z 4 K r ~ O 3 — See. 
the denominator and numerator of this fraction are totally 
independent of each other, and therefore r may be aflumed at 
plealure, and the value of the whole denominator computed 
for any particular magnitude affigned to this letter: or 
otherwife, the whole denominator may be taken equal to 
any quantity, and the value of r itfelf determined bj? com¬ 
putation. The latter method, at firll light, appears the 
molt eligible; for, by affuming the whole denominatoi; equal 
to unity, it difappears entirely, and the expreffion becomes. 
which feries muff neceflarily converge, becaufe the deno¬ 
minator of each of the fraftions is greatei than its nu¬ 
merator; Kill, however, when a is a.number of any con¬ 
siderable magnitude, the decreafe in the terms will he fo 
flow as to render the formula ufelefs for the purpofes of 
calculation. 
A't prefeat we have aflumed the feries which conftitute 
log. (1 a) — a- 
4 * 3 23 — i- a 4 -f Sec. 
There are, however, inconveniences attending this fyf- 
tem, that do not appear upon a flight view of the fubjeft, 
but which are notwithftanding very evident upon a farther 
inveltigation. In the cafe in which the whole denominator 
is aflumed equal to unity, the value of r, the radix of this 
particular fyltem, is found to be 2-7182818284, Sec. and 
the fraction A becomes =2 1. Thefe conftitute what are 
M 
called hyperbolic logarithms. In the common fyftem the ra¬ 
dix r is aflumed equal to 10, the fame as the radix of our 
fcaie of notation; and hence arifes a molt important ad¬ 
vantage, which is, that the logarithm of all numbers ex- 
preffed by the fame digits, whether integers, decimals, or 
mixed of the two, have tlxe fame decimal part; the only 
^Iteration being in the index or charafteriftic of the loga¬ 
rithm. For the radix being io'o, 1, 2, 3, Sec. will be 
logarithms of 1, 10, io 2 , Sec. that is, io°zz:i, io 5 r=io, 
io 2 2 =ioo, &c. and therefore, to multiply or divide a num¬ 
ber by any power of 10, we have only to add or fubtract 
the number expreffing that power from the integral part 
of the logarithm, and the decimal part will (till remain 
the fame, by which means the tables of logarithms are 
much more contrafted than they could be with any other 
radix; for in the hyperbolic fyftem, or in any other which 
has not its radix the lame as that of the fcaie of notation, 
every particular number would require a particular loga¬ 
rithm; and this circumftance would either lwell the tables 
to an unmanageable fize; or, if they were kept within the 
prefent limits, frequent computations would become ne- 
ceffary; fo that in either way it is clear that the advan¬ 
tages of the prefent logarithms much more than counter¬ 
balance the extra trouble in computing them. This in fa ft 
only confifts in multiplying the hyperbolic logarithm by 
a conftant faftor; viz. the reciprocal of the foregoing 
conftant denominator reprefented above by-A, the value 
r 1 M 
of which, when r— 10, is --- — 2= -434.29448, 
2-30258509, &c. * 
Sec. Hence it is obvious, that different fyftems of loga¬ 
rithms are connefted together by conftant multipliers, and 
by means of which a logarithm may always be converted 
from one fcaie to another. Thus the hyperbolic logarithm 
of a number is transformed to the common logarithm by 
multiplying the former by -4342944; and the latter is con¬ 
verted into the former by multiplying it by 2-30258509. 
Having laid thus much with regard to advantages of 
different fyftems of logarithms, and the method of trans¬ 
forming them from one fcaie to another; we will now add 
one example by way of illuftration. Let it therefore be 
propofed to find the common logarithm of 3. In this 
cafe our leries, 
log. a ; 
M 
X 
becomes log. 3=2 
j d—^jd-^d — J —+ Sec. I 
H 3. a 3 5.2 s 7.a 2 i 
