) 
LOG 
by logarithms. Tire logarithm of 32 =: 
1.5051500, which multiplied by 3, is 
4.5154500, the logarithm of 32768, the cube 
of 32. But in raising powers, viz. squaring, 
cubing, &c. of any decimal fraction by lo- 
garithms, 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. mnltiplied by the index of the 
power. 
Logarithms, to extract the roots of 
powers by. Divide the logarithm of the 
number by the index of the power, the 
LOG 
quotient is the logarithm, of the root 
sought. 
To find mean proportionals between any 
two numbers. 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) con- 
tinually, and it will give the logaritlims of 
all the mean proportionals required. 
^Example. Let three mean proportional* 
be sought, between 106 and 100. 
Logarithm of 106 = 2.0253059 
Logarithm of 100 =r 2.0000000 
Divide by 4)0.0253059(0.0063264.75 
Lofifaritlnn of the least term 100 addded 
Logarithm of the first mean 101.4673846 
Logarithm of the second mean.... 102.9563014 
Logarithm of the third mean 104.4670483 
Logarithm of the greatest term... 106 
2.0000000 
2.0063264.75 
2.0126529.5 
2.0189794.25 
2.0253059 
\ 
LOGIC, the art of reasoning. As the 
necessities of our existence oblige us to 
think, and to arrange our thoughts in such 
a manner as may enable us to communicate 
with each other, we are. habitually impelled 
towards a conclusion that it is unnecessary 
to teach reasoning as an art. It is hardly 
needful to combat this notion by argu- 
ments which will easily occur to most men 
of reflection ; and indeed the contrary per- 
suasion was so prevalent in the middle ages, 
that men seem to have been more occupied 
with the art, than with the proper use of it. 
In order to reason welt, it is necessary 
that the nature of our perceptions and ideas, 
and the notions or conclusions we draw 
from them, should be well understood. Lo- 
gic, therefore, is a science of extensive oc- 
cupation; which has its beginning in the 
constitution of things, and the processes of 
the human intellect, and its practical ter- 
mination in the structure, use, and applica- 
tion of language. Its objects are no less 
than the universal acquisition of knowledge, 
and that mutual communication which con- 
stitutes a large part of the employment, and 
is the most distinguishing character of man. 
The impressions made by external ob- 
jects upon the senses, are called sensations 
or ideas of sensation. See Ideology. The 
recollection or remembrance of those sensa- 
tions are simply called ideas. The genera! 
notions which are produced in the mind by 
reflecting upon ideas, have been called 
i'deas of reflection ; but as they all grow out 
of the comparison of the first-mentioned 
ideas, and do universally in the last result 
imply propositions, it appears much prefer- 
able to call them notions. 
Logical writers divide ideas into simple 
and complex ; but as we have no simple 
sensations, and can therefore have no sim- 
ple ideas but by the artificial process of 
abstraction, the division seems useless. The 
word complex here signifies compounded, 
and the compounded nature of our ideas 
will practically depend, in a great measure, 
upon our choice or determination in the 
subject of our reasoning. Thus, a lemon is 
soft, fragrant, yellow, and acid. If I throw 
a lemon at another, the attention will be 
chiefly directed to the organ of touch, and 
its fragrance, its tint, and its acidity, will 
be abstracted or left out. But the per- 
fumer, the designer, and the chemist would 
separately attend to those parts of the idea 
which were suggested by the organs of 
smell, of vision, and of taste. And in this 
manner it is that we may separate the sim- 
ple ideas of yellowness, acidity, and fra- 
grance; though, in nature, their causes never 
appear insulated and apart from those of all 
the other sensations. 
Abstraction, or the leaving out parts of 
ideas or notions ; generalization, or classing 
things togetlier, as possessing the remaining 
distinctive characters ; composition, or the 
rc-as. 3 umption of some of the abstracted or 
