884 
LOG 
any miftake being made in obferving the diftance run by 
the other circles. The reckoning by thefe circles, without 
fear of miftake, may therefore be continued to nearly 12,000 
miles. A communication from this machine may eafily 
be made to the captain’s bedfide, where, by touching a 
fpring only, a bell in the head A B C D will found as many 
times in a half-minute as the (hip fails miles in an hour. 
Log, a Hebrew meafure, which held a quarter of a 
cab, and confequently five-fixlhs of a pint. According 
to Dr. Arbuthnot, it was a liquid meafure, the feventy- 
fecond part of the bath or ephah, and twelfth part of the 
bin. Calmet .—A meat offering, mingled with oil, and one 
log of oil. Lev. 
LOG'-BOARD,yi A fort of table, divided into feveral 
columns, containing the hours of the day and night, the 
direction of the winds, the courfe of the (hip, and all the 
material occurrences that happen during the 24. hours, or 
from noon to noon ; together with the latitude by obfer- 
vation. From this table the officers of the (hip are fur- 
niffied with materials to compile their journals. 
LOG'-BOOK,y. A book into which the contents of 
the log-board is daily copied at noon, together with every 
circumftance deferving notice that may happen to the 
tlrip, either at fea or in a harbour. See the article Navi¬ 
gation. 
LOG'-HOUSES, f. A name given to certain houfes in 
America, which are generally the firft that are erefted on 
any new fettlement, and. which are cheaper than any 
others in a country where wood abounds. The fides con- 
(ift of trees juft fquared, and placed horizontally one 
upon the other ; the ends of the logs of one fide refting 
alternately on the ends of t’nofe of the adjoining (ides, in 
notches; the interftices between the logs are (topped with 
clay; and the roof is covered with boards, or with fhin- 
gles, which are fmall pieces of wood in the (liape of dates 
or tiles, See. ufed for that purpofe, with few exceptions, 
throughout America. Thefe habitations are not very 
fightly; but when well built they are warm and comfort¬ 
able, and laft for a long time. 
LOG-I'SLAND, a fmall idand in the Chefapeak Bay. 
Lat. 37. 14.. N. Ion. 76. 23. W. 
LOG'-LINE. See Log, p. 880. 
LOG’S TOW'N, a town of Pennfylvania, on the Alle¬ 
gany : eighteen miles north-weft of Pittfturg. 
LOGARITHMS, f. [from the Gr. Aoyot, ratio, and 
number; q. d. ratio of numbers.] The indices of 
the ratios of numbers one to another; or a leries of arti¬ 
ficial numbers proceeding in arithmetical progreffion, cor- 
refponding to as many others proceeding in geometrical 
progreffion; contrived for eafing and expediting calcu¬ 
lation.—The addition and fubtraClion of logarithms an- 
fwers to the multiplication and divifion of the numbers 
they correfpond with; and this faves an infinite deal of 
trouble. In like manner will the extraction of roots be 
performed, by differing the logarithms of any numbers for 
the fquare root, and trifecling them for the cube, and fo 
on. Harris. 
There are various ways of expreffing the relation be¬ 
tween a number and its logarithm. We (hall fir ft ftate 
that which fubfifts between the correfponding terms of 
two feries, the number being exprefied by a feries in geo¬ 
metrical progreffion, and the logarithm by a feries in 
arithmetical. Thus: 
Logarithms, 01234.567 8 
Nat. Numb. 1 2 4. 8 16 32 64. 128 256 
Or, 
Logarithms, 01234 5 6 
Nat. Numb. 1 10 100 1000 10,000 100,000 1,000,000 
In either of thefe feries it is evident, that, by adding any 
two terms of the upper line together, a number will be 
had which indicates that produced by multiplying the 
correfponding terms of the lower line. Thus, in the firft 
two feries, fuppofe we with to know the produCl of 4X 32- 
In the upper line we find 2 (landing over the number 4, 
and 5 over 32 5 adding therefore 5 to 2 we find 7, the fum 
LOG 
of this addition, (landing over 128, the produCl of thetw® 
numbers. In like manner, if we wi(h to divide 256 by 8, 
from the number which ftands over 256, viz. 8, lubtraCl 
that which ftands over 8, viz. 3; the remainder 5, which 
ftands over 32, (hows that the latter is the quotient of 
256 divided by 8. Let it be required to involve 4 as high 
as the biquadrate or 4th power: Multiply 2, the number 
which ftands over 4, by the index of the power to which 
the number is to be involved; which index is 4: the pro¬ 
duct 8, (landing over 256, (hows th^t this laft number is 
the biquadrate of 4 required. Laftly, let it be required 
to extraft the cube root of 64; divide the dumber 6, 
which (lands over 64, by 3, the index of the root you wi(h 
to extracl; the quotient 2, (landing over 4, (hows that 4 is 
the root fought. 
Thefe examples are fufficient to (how the great utility 
of logarithms in the mod tedious and difficult parts of 
arithmetic. But, though it is thus eafy to frame a table 
of logarithms for any feries of numbers going on in geo¬ 
metrical progreffion, yet it mu ft be far more difficult to 
frame a general table in which the logarithms of every 
poffible feries of geometricals (hall correfpond with each 
other. Thus, though in the above feries we can eafily 
find the logarithm of 4, 8, See. we cannot find that of 3, 
6, 9, See. and, if we affumeany random numbers for them, 
they will not correfpond with thofe which have already 
been affumed for 4, 8, 16, Sec. In the conftruClion of 
every table, however, it was evident, that the arithmetical 
or logarithmic feries ought to begin with o; for, if it be¬ 
gan with unity, then the fum of the logarithms of any two 
numbers muft be diminiffied by unity before we could find 
the logarithm of the produCl. Thus: 
Logarithms, 1234567 8 9 
Nat. Numb. 1 2 4 8 16 32 64 128 256 
Here let it be required to multiply 4 by 16 5 the number 
3 Handing over 4, added to 5 which (lands over 16, gives 
8 which ftands over 1.28 ; but this is not juft ; fo that we 
muft diminiffi the logarithm by r, and then the number 7 
(landing over 64 (hows the true produCl. In like manner 
it appears, that, as we delcend below unity in a logarith¬ 
mic table, the logarithms themfelves muft begin in a ne¬ 
gative feries with refpeCl to the former; and thus the lo¬ 
garithm of o will always be infinite; negative, if the lo¬ 
garithms increafs with the natural numbers; but pofitive, 
if they decreafe. For, as the geometrical feries muft di- 
miniffi by infinite divifiens by the common ratio, the 
arithmetical one muft decreafe by infinite fubtraClions, or 
increafe by infinite additions of the common difference. 
The invention of logarithms, it is well known, forms 
a very diftinguifhing sera in the hiftory of mathema¬ 
tical fcience. How much they expedite every kind of 
calculation; to what an altoniffiing degree they facilitate 
the explication of trigonometry to every fcience that is 
connected with it; and to what an extent they have con¬ 
tributed to enlarge the fphere of our refearches and know¬ 
ledge in this way ; it is altogether needlefs to mention. 
Thofe who have any acquaintance with this branch of 
mathematics will not helitate to allow its peculiar utility 
and importance; and they will trace with pleafure the va» 
rious Heps that have been purfued by mathematicians, of 
all countries, in the cultivation and improvement of it. 
This property of numbers was not unknown to the an¬ 
cient mathematicians. It is mentioned in the works of 
Euclid; and Archimedes made great ufe of it in his Are- 
narius, or treatife on the number of the lands; and it is 
probable that logarithms would have been much (boner 
invented, had the real neceffity for them been fooner felt; 
but this did not take place till the end of the fixteenth 
century, when the conftruflion of trigonometrical tables, 
and folution of perplexed aftronomical problems, rendered 
them abfolutely indifpenfable. 
According to Kepler, one Jude Byrge, affiftant aftro- 
nomer to the landgrave of Heffe, either invented or pro¬ 
jected logarithms long before baron Napier, and compofed 
a table of fines for every two feconds of the quadrant; 
though. 
