82 
L O G 
L O L 
And 5.5649027 — .0000212 = 5.5648815 = 
logarithm of 367132, nearly. 
If the number consists both of integers and 
fractions, or is entirely fractional, find the de- 
cimal part of the logarithm as if all its figures 
were integral ; and this, being prefixed to the 
proper characteristic, will give the logarithm 
required. 
And if the given number is a proper fraction, 
.subtract the logarithm of the denominator from 
the logarithm of the numerator, and the re- 
mainder will he the logarithm sought 5 which, 
being that of a decimal fraction, must always 
have a negative index. 
And, if it is a mixed number, reduce it to an 
improper fraction, and find the difference of 
the logarithms of the numerator and denomina- 
tor, in the same manner as before. 
In finding the number answering to any given 
logarithm, the index, if affirmative, will always 
shew how many integral places the required 
number consists of ; and, if negative, in what 
place of decimals the first, or significant figure, 
stands.; so that, if the logarithm can be found 
in the table, the number answering to it will 
.always be had by inspection. 
But, if the logarithm cannot be exactly found 
in the table, find the next greater, and the next 
less, and then say, As the difference of these two 
logarithms “ the difference of the numbers an- 
svvering to them * * the difference of the given 
logarithm, and the nearest tabular logarithm ; a 
fourth number ; which added to, or subtracted 
from, the natural number answering to the 
nearest tabular logarithm, according as that lo- 
garithm is less or greater than the given one, 
will give the number required, nearly. 
Thus, let it be required to find the natural 
number answering to the logarithm 5.5648815. 
The next less and greater logarithms, in the 
table, are 
5.56478447 The numbers C 367100 
5.5649027 5 answering £ 367200 
Their diff. .0001183 100 
And 5 5649027 — 5.5648815 — .0000212. 
Therefore .0001183 ; 100 ” .0000212 ; 18 
nearly. 
Whence S67200 — 18 = 377182 number 
required. 
The Use and Application of Logarithms. — It is evi- 
dent, from what has been said of the construc- 
tion of logarithms, that addition of logarithms 
must be the same thing as multiplication in com- 
mon arithmetic 5 and subtraction in logarithms 
the same as division ; therefore, in multiplica- 
tion by logarithms, add the logarithms of the 
multiplicand and multiplier together, their sum 
is the logarithm of the product. 
num. logarithms. 
Example. Multiplicand 8.5 0.9294189 
, Multiplier 10 1,0000000 
Product - 85 1.9294189 
And in division, subtract the logarithm of the 
divisor from the logarithm of the dividend, the 
remainder is the logarithm of the quotient. 
num. logarithms. 
Example. Dividend 9712.8 3.9873444 
1 Divisor 456 2.6589648 
Quotient 21.3 1.3283796 
To find the Complement of a Logarithm. — Begin at 
the left hand, and write down what each figure 
wants of 9 , only what the last significant figure 
wants of 10 ; so the complement of the logarithm 
of 456, viz. 2.6589648, is 7.3410352. 
In the Rule of Three. Add the logarithms of 
the second and third terms together, and from 
the sum subtract the logarithm of the first, the 
remainder is the logarithm of the fourth. Or, 
instead of subtracting a logarithm, add its com- 
plement, and the result will be the same. 
LOG 
To raise Po’ivcrs by Logarithms . — Multiply the 
Logarithm of the number given, by the index of 
the power required, the product will be the 
logarithm of the power sought. 
Example. Let the cube of 32 be required by 
logarithms. The logarithm of 32 = 1.5051500, 
which multiplied by 3, is 4.5154500, the loga- 
rithm of 32768, the cube of 32. But in raising 
powers, viz. squaring, cubing, &c ol any deci- 
mal fraction by logarithms, it must be observed, 
that the first significant figure of the power be 
put so- many places below the place of units, as 
the index of its logarithm wants of 10, 100, &c. 
multiplied by the index of the power. 
To extract the Roots of Poivers by Logarithms . — 
Divide the logarithm of the number by the in- 
dex of the power, the quotient is the logarithm 
of the root sought. 
To find Mean Proportionals between any tnvo num- 
bers. — Subtract the logarithm of the least term 
from the logarithm of the greatest, and divide 
the remainder by a number more by one than 
the number of means desired ; then add the 
quotient to the logarithm of the least term (or 
subtract it from the logarithm of the greatest) 
continually, and it will give the logarithms of 
all the mean proportionals required. 
Example. Let three mean proportionals be 
sought, between 106 and 100. 
Logarithm of 106 = 2.0253059 
Logarithm of 100 — 2.0000000 
Divide by 4)0.0253059(0.0063264.75 
Log. of the least term 100 added 2.0000000 
Log. of the 1st mean 101.4673846 2.0063264.75 
Log. of the 2d mean 102 9563014 2.0126529.5 
Log. of the 3d mean 104.4670483 2.0189794.2 5 
Log of thegreatestterml06. 2.0253059. 
LOGIC. The professed business ot logic 
is to explain the nature of the human mind, 
and the proper manner of conducting its 
several powers, in order to the attainment of 
truth and knowledge. 
Those, therefore, who have treated ex- 
pressly of this subject, have endeavoured 
first to define and describe the several facul- 
ties and operations of the human mind, as 
perception, judgment, memory, invention, 
&c. They next proceed to iay down rules for 
correct reasoning and argument. Every act 
of the judgment they term a proposition, and 
all propositions are either affirmative or ne- 
gative. All questions or arguments they re- 
duce to syllogisms, that is, from two axioms 
or propositions (called terms, in the techni- 
cal language) laid down, they deduce athird, 
or conclusion, and the previous propositions 
they divide into major and minor. Thus, 
let the question be, IV lid her God is an 'in- 
telligent being ? Here the major or principal 
proposition proceeds from the word intelli- 
gent, and the minor respects God. They 
would then arrange the syllogism as follows: 
Mtij. To dispose things in right and per- 
fect order is the work of an intel- 
ligent Being: 
Min. Cut God has disposed creation in 
right and perfect order ; 
Conclusion. Therefore God is an intelli- 
gent Being. 
They next class or arrange the differ 
rent kinds of syllogisms according to the 
nature of them. Propositions are not only 
affirmative and negative, but they are also 
particular or universal. Hence syllogisms 
will vary not only as the major or minor pro- 
position is negative or affirmative, but as 
either is an universal or particular affirmative. 
Sec. Hence they dispose the several kinds 
of propositions into modes, and the syllo- 
gisms into figures, according as they affect 
the subject or the predicate. The modes 
are indicated by the letters a, e, i, o, as they 
are affirmative or negative, universal or par- 
ticular. There are nineteen modes and four 
figures. The first figure is when the middle 
term is (he subject of the major, and the pre- 
dicate of the minor : as, 
No work of God is bad : 
But the natur I passions and appetites of 
men are the work of God ; 
Therefore they are not bad. 
This figure includes four modes, denoted 
by tiie words, 
“ Barbara,, celarent, Darii, ferio ;” 
referring to the vowels which each syllable 
contains. 
The second figure is when the middle term 
is the predicate of both major and minor: as. 
Whatever is bad is not 1 he work o; God: 
But the natural passions, &c. are the work 
of God ; 
Therefore they are not bad. 
This figure includes four modes, denoted 
by the words, 
“ Cirsare, camestres, festino, baroco.” 
The third figure is when the middle term 
ts the subject of both major and minor, as. 
All Africans are black: 
But all Africans are men; 
Therefore some men are black. 
This figure includes six modes, denoted by 
the words, 
“ Darapti, felapton, disamis, datisi, bocardo, 
ferison.” 
In the fourth figure it is the predicate of 
the major, and the subject of the minor, as. 
The only being who ought to be worship- 
ped is the Creator of the world: 
But the Creator of the world is God; 
Therefore G od is the only being who ought 
to be worshipped. 
There are five modes of this figure denoted 
by the words, 
“ Barbari, Calentes, Dibatis, fessamo, fre- 
sisom.” 
Such is the scheme proposed by the school- 
men as the only guide to truth and wisdom ; 
but how little it has been able to effect may 
be seen from the labours of those who have 
practised it most, those very schoolmen 
themselves. The truth is, if logic is the art 
of reasoning, the best materials to form a lo- 
gician, that is a reasoner, are a sound under- 
standing, an extensive and accurate know- 
ledge of facts, and an unprejudiced disposi- 
tion ; and every attempt to reduce the ope- 
rations of the human mind to mechanical 
rules, to bring genius to a level with dulness, 
must be futile and vain. The various terms, 
and figures of logic will be found in their re- 
spective places. See Mode. 
LOLIUM, darnel-grass, a genus of the 
digynia order, in the triandria class of plants,, 
and in the natural method ranking under the 
4th order, grand na. The calyx is mono- 
phyllous, fixed, and imiflorous. There are 
five species. The most remarkable are, 
1 . The perenne, red darnel, or rye-grass. 
This is very common in roads and dry pas- 
tures. It makes excellent hay upon dry. 
