895 
L O G A R I T II M S. 
J-c. to t. Now the modular ratio is that ratio of which 
the modulus is the meafurc; hence, if we make m — y, m 
will become the meafure of the above ratio, and the ratio 
becomes the modular ratio ; making therefore m —y, the 
ratio becomes i -f- i -j" h "l" '—“ & c - 1 ^ ,e modular 
2.3 
ratio, which is the fame for all fyftems, it being indepen¬ 
dent of m and y. 
* Mr, Long's Method. 
This method was publifhed in N° 339 of the Phil. 
Tranl. and is performed by means of a finall table con¬ 
taining eight clafles of logarithms, as follows. 
Log. 
Nat. Number. 
Log. 
Nat. Number. 
9 
7 943282347 
*00009 
x .000207254 
•8 
6.3°9573445 
8 
1-000184224 
•7 
4011872336 
7 
1-000161194 
•6 
3 - 98.071706 
6 
1-000138165 
’5 
3-162277660 
5 
roooi 15136 
•4 
2 ’ j 11886432 
4 
1.000092106 
•3 
1-995262315 
3 
1 000069080 
*2 
1-584893193 
2 
1-000046053 
*i 
1-258925412 
I 
1 -000023026 
•09 
1-230268771 
*000009 
1-000020724 
8 
1-20226443 3 
8 
1-000018421 
7 
1-174897555 
7 
i’oooOi 611 8 
6 
1-148153621 
6 
1-000013816 
5 
1-122018454 
5 
rooooi 1513 
4 
1.096478196 
4 
1-000009210 
3 
1-071519305 
3 
1-000006908 
a 
1-047128548 
2 
1-000004605 
1 
1*023292992 
I 
1 -000002302 
•009 
1-020939484 
*0000009 
1-000002072 
8 
roi 8 <;qi 388 
8 
1-000001842 
7 
l - 0162-48694 
7 
1-000001611 
6 
1-013911 386 
6 
1-000001381 I 
5 
I-OH579454 
5 
1*000001 I 51 | 
4 
1.009252886 
4 
1-000000921 II 
3 
1 ’Oo 6 q 3 I 66 q 
2 
1-000000690 
2 
1-004615794 
3 
1-000000460 
1.002305238 
| I 
1-000000230 
•OOO 9 
I.OO 2 O 74475 
•OOOOOOO 9 
1-000000207 
8 
1-001843766 
8 
1-000000184 1 
7 
1-00161 3 109 
„ 7 
l -000000161 1 
6 
1-001382506 
6 
1-000000138 1 
5 
i-ooi 1519 5 A 
5. 
x *000000115 
4 
I '000Q214 59 
4 
1 -000000092 
3 
IOOO 69 IOI 5 
3 
1 -000000069 
2 
I-OOO 460623 
2 
1 - 000000046 . 
I 
1-000230285 
j 1 
1*000000023 
Here, becaufe the logarithms in each clafs are the con¬ 
tinual multiples t, 2, 3, &c. of the lowed, it is evident 
that the natural numbers are fo many fcales of geometri¬ 
cal proportionals, the lowed being the common ratio, or 
the attending numbers are the 1, 2, 3, &c. powers of the 
lowed, as exprefled by the figures 1, a, 3, &c. of their 
correfponding logarithms. Alfo the lad number in the 
fil'd, fecond, third, &c. clafs, is the 10th, 100th, 1 oooth, &c. 
root of 10; and any number in any clafs is the 10th power 
of the correfponding number in the next following clafs. 
To' find the Logarithm of any Number, as fuppofe of z 000, by 
this Table .—Look in the fil'd clafs for the number next 
lefs than the firit figure z, and it is 1-9.95262315, againd 
■which is 3 for the fird figure of the logarithm fought. 
Again, dividing 2, the number propofed, by 1-995262315, 
the number found in the table, the quotient is 1-0023744.67 ; 
which being looked for in the fecond. clafs of the table, 
and finding neither its equal nor a lefs, o is therefore 
to be taken for the fecond figure of the logarithm ; and 
the fame quotient 1 '002374467 being looked for in the 
y.Qi. XXL Ho. 3 So. 
third clafs, the next lefs is there found to be 1-002305238, 
againd which is i for the third figure of the logarithm ; 
and, dividing the quotient 1-002374467 by the laid next 
lefs number 1 -002305238, the new quotient is 1-000069070; 
which being fought in the fourth clafs gives o, but fought 
in the fifth clafs gives 2, which are the fourth and fifth 
figures of the logarithm fought; again, dividing the lad 
quotient by 1-000046053, the next lefs number in the ta¬ 
ble, the quotient is 1-000023015, which gives 9 in the 6th 
ckafs for the 6th figure of the logarithm fought; and again 
dividing the lad quotient by 1-000020724, the next lefs 
number, the quotient is 1-000002291, the next lefs than 
which in the 7th clafs gives 9 for the 7th figure of the lo¬ 
garithm; and dividing the lad quotient by 3-000002072, 
the quotient is 3-000000219, which gives 9 in the 8th clafs 
for the 8th figure of the logarithm ; and again the lad quo¬ 
tient 1-000000219 being divided by 3-000000207 the next 
lefs, the quotient 1-000000012 gives 5 in the fame 8tli 
clafs, when one figure is cut cd', for the 9th figure of the 
logarithm fought. All which figures collected together 
give 3-301029995 for Briggs's logarithm of 2000, the index 
3 being liipplied ; which logarithm is true in the lad figure. 
To find the Number anfwering to any given _._ 
Logarithm, asfiuppofe to 3-3010300.—Omit- 3 I ' 995 -^- 3 I 5 
ting the charaCteridic, againd the other 00 
figures, 3, q, 1, o, 3, o, o, as in the fil’d co- 1 1-0023052381 
lumn in the margin, are the feveral num- 0 0 
bers, as in the fecond column, found from 3 fOooo69'->8o 
their refpeCtive id, 2d, 3d, Sic. eludes; 
the effective numbers of which multiplied Q 
continually together, the lad product is 2-000000019966, 
which, becaufe the charaiteridic is three, gives 
2000-000019966, or 2000 only for the required number 
anfwering to the given logarithm. 
Mr. Bonnycafile 's method is as follows : The feries now 
chiefly tiled in the computation of logarithms were origi¬ 
nally derived from the hyperbola, by means of which, and 
the logidic curve, the nature and properties of tittle num¬ 
bers are clearly and elegantly explained. The dottrine, 
however, being purely arithmetical, this mode of demon- 
drating it, by the intervention of certain curves, was con- 
fidered, by Dr. Halley, as not conformable to the nature 
of the fubjeft. He has, accordingly, invedigated the fame 
feries from the abftraft principles of numbers; but his 
method, which is a kind of difguifed fluxions, is in many 
places fo extremely abdrufe and obfeure, that few have 
been able to comprehend his reafoning. An eafy and per- 
fpicuous demonflration,of this kind, was therefore dili want¬ 
ing ; which may be obtained from the pure principles of 
algebra, independen tly of tiled off rine of curves, as follows : 
The logarithm of any number, is the index of that 
power of iome other number, which is equal to the given 
slumber. Thus, if r x -=a, the logarithm of a is x, which, 
may be either pofitive or negative,- and r any number 
whatever, according to the different fydems pf logarithms.. 
When 12=21, it is plain that x njad be =0, whatever be 
the value of?-; and confequently the logarithm of 1 is al¬ 
ways o in every fydem. If x~i, it is alfo plain that a 
mud be—r; and therefore r is always the number ia 
every fyflem, whofe logarithm in that fydem is 1. 
To And the logarithm of any number, in any fydem, it 
is only neceffary, from the equation r x —a, to find the va¬ 
lue of x in terms of rand a. This may be driftly effetled, 
by means of a new. property of the binomial theorem of 
Newton. Thegeneral logarithmic equation being r x —a, let 
3 
I ‘ 995 - 6 - 3 I 5 
0 
O 
1 
1 -0023052381 
0 
O 
3 
roooo65-->8o 
o- 
O 
0 
O 
:i -f p, and - —z; then rz=za x . — 1= 1 4- p'F. — 1 d" 
p* 
u>— —V— 
2 3 
P 2 ,P 3 
——f — ■ 
4 23 
8 cc.) 3 z‘ 2 + 
— See.) -1- —-J--— &c)* 
4 2.3.4 23 4 
I 4)~ fi* p^ 
- -— 4--— &$.)«**, &C. 
2.3.4 -s a 3 4 
3 9 & 
2.3 a 3 
z 4 + 
And 
