LOGARITHMS. 
903 
therefore put down greater by a unit than their true va¬ 
lue. Thefe logarithms are frequently ufeful inaftronomi- 
cal computations, as it very often happens in a proportion 
that the firft or one of the mean terms is 6o'. 
Rule. If the firft term be6o', or 3600'', and the fecond 
or third terms be each iefs than Go', add the logarithms of 
the fecond and third terms together, and the fum is the 
logarithm of the fourth term. If the fecond, or the fe¬ 
cond and third terms, be greater than Go', and the fum of 
the two lalt figures in the addition, the index excepted, 
be equal to or greater than 10, you are not to carry 1 to 
the index. For, in the firft inftance, the logarithm of the 
fecond term is 1 greater than its true value; and there¬ 
fore, by not carrying 1 to the index, you get the pro¬ 
per value of the logathrim. In the fecond inftance, 
the fecond and third terms being both greater by 
a unit than their true values, the fum is 2 in the index 
greater than its true value; by neglefting therefore 
to carry 1, the logarithm is Itill 1 greater than its true 
value; but, the fourth term being here greater than 60', 
its logarithm is 1 in the index greater than its true value, 
and therefore the above addition gives the logarithm as 
found in the table. If one of the mean terms be Go', fub- 
traft the logarithm of the firft term from the logarithm of 
the other mean, adding 1 to the index of the logarithm 
of that mean if neceffary, and the remainder is the loga¬ 
rithm of the fourth term. The reafon of here adding 1 
is upon the fame principle as that of before fubtrafting 1 ; 
that is, on account of one of the logarithms being greater 
by unity than its true value. When a unit is to be added 
or fubtrafted will be manifeft in any particular inftance. 
If the logarithms in the tables had been made negative 
above 60', the rules would have been very clear and Am¬ 
ple : for in this conftru&ion of the tables, when 60' is 
the firft term, take the fum of the logarithms of the other 
two terms (regard being had to their figns), and you have 
the logarithm of the fourth term, which is lefs or greater 
than 60', according as the fum is pofitive or negative. 
When 60' is one of the mean terms, liibtraft the logarithm 
of the firft term from that of the other mean (regard be¬ 
ing had to their figns), and you have the logarithm of 
the fourth term, which is lefs or greater than Go', according 
as the difference is pofitive or negative. Here there can be 
no ambiguity, which there may be in the prefent form of 
the Tables, as the fame logarithm may anfwer to a quan¬ 
tity both above and below Go'. 
Ex. 1. What is the fourth proportional to Go', 1' 36", and 
27' 38" 
s' 36" ... log. 1’5740 
27 38 - - - log. 3367 
Anfwer, o 44 - - - log. 1-9107 
Ex. 2. What is the fourth proportional to Go', Gf zf, and 
37 ' V"f 
65' 25" - - - log. 9625 
37 41 - - - log. 2020 
Anfwer, 41 5 ... log. 1645 
Ex. 3. What is the fourth proportional to 16' 47", 60', and 
17' z'i"? 
17' 28" - - - log. 5359 
16 47 - - - - log. 5533 
Anfwer, 62 27 - - - log. 9826 
Here, it was neceffary to add 1 to the index of the loga¬ 
rithm of the third term, or, which is the fame, to add 10 
to the laft figure 5 of the upper line; becaufe the fourth 
term is greater than 60', and therefore its logarithm in 
the Table is 1 greater than its true value. 
Ex. 4. What is the fourth proportional to 27' 19", 60', and 
5 ' 9"- 9 
5' 9" - log. 1-066-5 
27 19 - - - - log. 3417 
Anfwer, 11 ig - - ? log. 7246 
If the firft term be 24 hours, and the fecond term be hour* 
and minutes, and the third term be given in time or be 
an arc, we may find a fourth proportional by conceiving 
the head of the Table to reprefent hours. 
Ex. 5. What is the fourth proportional to 24 h. 13 h. 53', and 
24 h. ... ar. co. log. 6021 
13^. 53' - - - log. 6357 
76 34^ - - log. 8941 
Anfwer, 44 17 ... log. 1319 
We here reject 2 in the index, becaufe the firft term is arc 
arithmetical complement, and the third term is 1 greater 
than its true value. 
In like manner, whatever may be the three terms, whe¬ 
ther hours and minutes, minutes and feconds, of time ; 
degrees and minutes, minutes and feconds, of an arc; or 
two of one, and one of the other; a fourth proportional 
may be found, provided the quantities fall within the li¬ 
mits of the Table. 
Proportional Logarithms. 
In the Requifite Tables, there is a Table of proportional lo¬ 
garithms, which are analogous to the logiftical logarithms,, 
the common logarithms being here fubtracled from 4-0334, 
the logarithm of 10800, the number of feconds in 180', 
which is 3 hours, or 3 0 , the logarithm of which is there¬ 
fore nothing; and the Table goes no farther. Thefe lo¬ 
garithms are peculiarly adapted to the purpofe of finding 
the apparent time at Greenwich, by comparing the true 
diftance of the moon and fun, or moon and ltar, found 
from the obferved diftance, with the true diftance, put 
down in the Nautical Almanac for every 3 hours, under 
the meridian of Greenwich, becaufe 3 hours always forms 
the third term ; and therefore we have only to fubtraft 
the logarithm of the firft term from that of the fecond. 
Ex. 1. On July 7, 1775, the true di/lance of the Sun from 
the Moon was found to be 109 0 34' 26" ; to find the time at 
Greenwich .—By the Nautical Almanac, the true diftance 
is found to be 108 0 5' 58' / at 3 o’clock, and 109 0 37' i6' J 
at 6 o’clock ; the moon has therefore receded from the 
fun i° 31' 18" in 3 hours; now the difference between 
108 0 f 58" and i 09°34 , 26 / ' is i° 28' 28"; to find therefore 
how long the moon will be in receding i° 28' 28", we have, 
(upon fuppofition that the moon recedes uniformly from 
the fun, which is very nearly true,) i° 31' 18" : i°28'28" 
:: 3 h. : 2 h. 54' 25", which operation is thus performed 
by Proportional Logarithms; and the whole work, may be 
arranged as follows : 
True dift. of ]) from © 109 0 34' 26" 
.-at 3 h. (N. Aim.) 108 5 58 
-at 6 h. -109 37 16 
Diff. bet. xft and 2d 
- . . 2d and 3d 
Time after 3 h. 
Add 
Time at Greenwich 
I 
28 28 
pr. log. 0-3085 
I 
31 18 
—■ - ■ ■ - 0-2948 
2. 
54- 25 
»-.«■=«, 0*0137 
3 
0 0 
--- 
5 
54 - 2 5 
The third term being here 3 hours, its logarithm is o; wa 
have therefore only to fubtracl the log. of the firft term 
from that of the fecond, and the difference 0-0137 is the 
log. of 2 h. 54' 25", the time in which the tnoon recedes 
x° 28' 28" from the fun ; and, as this is the fpace through 
which the moon has receded from the fun after 3 o’clock,, 
if you add 2 h. 54' 25" to you get fi. 54' 25", the time 
at which the moon was at the diftance 109° 34' 26'' from 
the fun. 
Ex. 2. What is the fourth proportional to z° 7' 48", x° 19" 
zj", and 3 0 ? 
i° 19' 27" pr. log. - " 03552. 
2 7 48 - - - o-1487- 
Fourth prop, j 5 s 53, 
0-2065 
