893 L O G A R 
hies. Thus, —j'59462Sj = T‘4.oS 37I9. The above re- 
dnotion to a negative logarithm, if 3'on want .to lubtraft 
i'uch a logarithm, is very convenient, for you have then 
only to add the- logarithm after you have reduced it. 
Thus, if you want to fubtraft "j-597 2 648, you may add 
2-4027316, and the operation is performed; for the fub- 
t raft ion of a negative quantity is the fame as the addition 
of a politive quantity of the fame magnitude. Should the 
decimal part conlift of ciphers only, the whole logarithm 
is then negative, and therefore no reduction becomes ne- 
ceffary. 
When you have to fubtraft the logarithm of a decimal, 
and you write the logarithm by adding 10 to the index, 
you may take its arithmetical complement and add it, and 
it gives the true refult. For, if you add the arithmetical 
complement, it makes the refult greater by 10 than if you 
had fubtrafted the logarithm itfelf; but, as the logarithm 
is 10 too great, by fubtrafting it the refult becomes 10 too 
little ; therefore, by adding the arithmetical complement, 
you get the true refult. And in like manner, if you write 
the logarithm by adding xoo to the index. 
A Number being given, to find its Logarithm. 
To find the logarithm of a number confifiing of one, two, three, 
nr four, figures. —Rule. The logarithms of all numbers to 
too are put down with their proper indices ; and from 100 
the logarithms are put down without their indices ; and, 
when the logarithm is taken out, it is the decimal part 
only of the logarithm, and the index is fupplied, being al¬ 
ways lefs by unity than the number of figures in the in¬ 
tegral part of the given number. From 100 to 1000, the 
decimal part of the logarithm ftands in the column Log. 
direftly againft the correfponding number in the column N. 
and, as the two firft figures are common to feveral loga¬ 
rithms, they are not repeated till they alter ; they are 
therefore to be prefixed to the five others ill the lines be¬ 
low them, till you come to the next two common figures. 
The decimal part of the logarithm of a number confifiing 
of four figures, is found in the fecond column under o 
at the head, direftly againft the number in the firfl co¬ 
lumn. 
In taking out the logarithm correfponding to any given 
number confuting of more than four figures, the three firll 
- figures of the decimal part of the logarithm are found in 
the fecond column ; and thefe are common to all the 
figures in that and in the other nine columns, till you 
come to the next three figures; except when the firft 
figure on the left of the other four figures, found in that 
or in one of the other nine columns, changes from 9 to o; 
in which cafe, the next three figures are to be prefixed to 
the four figures thus beginning with o, and to each of the 
tour following figures in the fame horizontal line.—Ex. 
The decimal part of the logarithm of 30837 is 4890721 ; 
for, though againft 3083 the three common figures next 
■above are 488, yet from the third to the fourth column, 
the firft of the next four figures changes from 9 to o, there¬ 
fore we muft take 489 for the three firft figures. 
To find the logarithm of a number having five figures. —Rule. 
Find the firft four figures of the given number in the firft 
column of the Table, and even with them in the fecond 
column take out the three firft common figures; go on in 
the fame horizontal line till you come to the column 
which has at its head the fifth figure of the given number, 
and take out the four figures there found, and annex 
them to the right of the three before found (remembering 
the obfervaticn in the preceding paragraph), and you have 
the decimal part of the logarithm; and the index is lefs 
by unity than the number of integral figures in the given 
number. 
Ex. Let the given number be 39217. Here, in the fecond 
column, againft 3921, you find 593 for the three firft 
figures; and in the fame horizontal line, under 7, you 
have 4744 ; therefore the required logarithm is 4-5934744. 
To find the logarithm of a number having fix figures. —Rule. 
Find the decimal part of the logarithm for the iirlt five; 
ITHMS, 
figures as before, and take the difference, d, between fhat 
logarithm and the logarithm next greater; then find that 
difference in the lalt column but one, marked D, of a com. 
mon fet of logarithmic tables, and under that difference 
in the laft column marked Pis (proportional parts) againft 
the figure in the fixth place, you have the part "to be 
added for that figure. 
Ex. Let the given number be 392176. The decimal pa-t 
of the logarithm for the firft five figures is 5934744, the 
difference between which and the next greater logarithm 
is no; and in the laft column under no, and againft 6, 
you find 66 ; therefore the decimal part of the logarithm 
is 5934744 + 66 =2 5934810; and, there being fix places of 
integral figures, the required logarithm is 5-5934810. 
To find the logarithm of a number having feven figures. _ 
Rule. Find the decimal part of the logarithm for the firft 
fix figures, as before; then divide the number in that part 
of the column Pts having d at its head, correfponding to 
the feventh figure, by 10, and add it to the decimal part 
of the logarithm for the firft fix figures; rememberihg to 
place the firft figure on the right of the quantity added, 
in the eighth place of the logarithm. 
Ex. Let the given number be 3921764. The decimal part 
of the logarithm for the firft fix figures is 5934810 ; and 
in the laft column, under no, 44 ftands againft 4, the 
tenth part of which is 4-4 ; therefore the decimal part of 
the logarithm is 59348144; and, there being feven places 
of whole numbers, the required logarithm is 6-59348144. 
To find the logarithm of a number having eight figures .— 
Rule. Find the decimal part of the logarithm for the firft 
feven figures, as before; then divide the number in that 
part of the column Pts having d at its head, correfponding 
to the eighth figure, by 100, and it will give you the 
number to be added to the decimal part of the logarithm 
for that figure; remembering that, if the quotient confifts 
of an integer and decimals, the integer muft Hand in the 
feventh place; and, if there be only decimals, the firft 
decimal on the left muft (land in the eighth place. 
Ex. Let the given number be 39217648. The decimal 
part of the logarithm for the firft leven figures is 59348144 j 
and in the laft column, under no, againft 8 Hands 88, 
which divided by 100, gives o-88 ; hence, the decimal part 
of the logarithm is 593481528; and, there being eight 
places of whole numbers, the required logarithm is 
7 ' 59348 i 528. 
If the given number be not all integral, the decimal 
part of the logarithm is found in the fame manner, and 
the index is lefs by unity than the number of figures in 
the integral part. 
Ex. Let the given number be 392-1764. Here, the deci¬ 
mal part of the logarithm is 5934815; hence, the loga¬ 
rithm required is 2 5934815. 
To find the logarithm of a decimal. —Rule. Omitting the 
decimal ciphers on the left, if there be any, find the loga¬ 
rithm correfponding to the other part, as for a whole 
number, and annex a negative index which expreffes how- 
far the firft fignificant figure of the decimal is from the 
place of units. Or, if you write the logarithm with a 
pofitive index, the index will be as much below 9, or 99^ 
as there are decimal ciphers on the left. 
Ex. To find ihe logarithm of 0-0004346. Look into the 
Table, and againft 4346 you will find the logarithm 
6380897; and, there being three decimal ciphers on the 
left of the fignificant figures, the required logarithm be¬ 
comes T-638o897, if expreffed by a negative index ; or 
6-6380897, if expreffed by a pofitive index, by adding 10 
to the index. 
To find the logarithm of a proper fraQion .—Rule. Add 10 
to the index of the logarithm of the numerator, and fub¬ 
traft the logarithm of the denominator from it, and you 
get the logarithm of the fraction, expreffed by a pofitive 
index, having 10 added to the index, as in decimals. Or 
fubtraft the logarithm of the denominator from that of 
the numerator, and you get the true logarithm of the frac¬ 
tion, expreffed by a negative index. 
Ex. 
