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or the Direaion of the Veffel with regard to the Points
of the Compafs. In the third, the Number of Knots
run off the Reel each time of heaving the Log. In the
fourth, the Wind that blows: and in the fifth, Obferva-
tions made of the Variation of the Compafs, &tc.
LOGARITHMIC, or LOGISTIC        CURVE, a Curve
generated by the equable Motion of the Radius of a Circle,
thro' equal Arcs of the Circumference ; while at the fame
time a Point in that Radius is fuppofed to move from the
Arc towardG the Centre, with a Retardation of Motion in
a Geometrick Proportion. As fuppofe there be a Quadrant
of a Circle, B C A, (Plate Analjis, Fig.t r.) and any equal
Divifions in the Arc, as A F- Ff-ff; U!c- with five
corresponding Radii, as fuppofe C A, C F, Cf, &c. whofe
Parts or Portions C I, C a, C a, &c. are geometrically Pro-
portional; then if a Line, as I, a, a, 6, d, C, be drawn thro'
thofe Points, it will be the Logarithmic or Logi~fic Spiral.
LOGARITHMS (from Aot(£ ratio, and det9u- nu-
merus) are ufually defined Numerorum Propertionalium equi-
dtfferentes Comites; but this Definition Dr. Halley and Stife-
lius think deficient, and more accurately define them,
The Indices or Exponents of the Ratio's of Numbers ; Ratio
being confider'd as a Quantity fui generis, beginning from
the Ratio of Equality, or I to i - o; and being affirma-
tive when the Ratio is increafing, and negative when it
is decreafing. The Nature and Genius of Logarithms will
be eafily conceiv'd from what follows.
A Series of Quantities increafing or decreafing accor-
ding to the fame Ratio, is call'd a Geometrical Progref-
lion ; e.g. I. 2. 4. 8. 16. 32- &c. A Series of Quantities
increafing or decreafing according to the fame Difference,
is called an Arithmetical Progreftion ; e. g. 3. 6. 9. I 2. I 5.
IS. 24. Now if underneath the Numbers proceeding in
a Geometrical Ratio, be added as many of thofe pro-
ceeding in the Arithmetical one ; thefe laft are call'd
the Logarithms of the firfi.
Suppofe v. g. two Progrefflons:
GeomeI. 2. 4. 4. 8. i6. 32. 64. Iz8. 256. 512
Arithmet.o. I. 2. 3. 4. 5. 6. 7.  8.  9
Logarithms.
o will be the Logarithm of the firfl Term, viz. I i 5 of the
6th, 32t 7 the Logarithm of the 8th, i28, &c.
Theor. 1. if the Logarithm of Unity be o, the Logarithm
of the FaFum or ProduHl will be equal to the Sum of the
Logarithms of the Fa.ors.
pem. For as Unity is to one of the Faaors, fo is the
other Fadtor to the Produc&. So that the Logarithm of
the Product is a fourth equidifferent Term to the Loga-
ritbm of Unity and thofe of the Fadors: but the Loga-
rithm of Unity being o, the Sum of the Logarithms of the
Faacors muft be the Logarithm of the Faaum or Pro-
du&. q. e. d.
Corol. i. Since the Faaors of a Square are equal to
each other, i. e. a Square is the Fadum or Product of its
Root multiplied into itfelf; the Logarithm of the Square
will be double the Logarithm of the Root.
Corol, - In the fame manner it appears that the Loga-
rithm of the Cube is triple, of the Biquadrate, quadruple;
of the fifth Power, quintuple; of the fixth, fextuple, tc.
of the Logarithm of the Root.
Carol. 3. Unity, therefore, is to the Exponent of the
Power, as the Logarithm of the Root to the Logarithm of
the Power.
Corol. 4. So that the Logarithm of the Power is had, if
the Logarithm of the Root be multiplied by its Expo-
nent; and the Logaritbm of the Root is had, if the Loga-
ritbm of the Power be divided by its Exponent.
Schol. Hence we derive one of the great Lfes of Loga-
rithms, which is to expedite and facilitate the Bufinefs of
Multiplication and ExtradJion of Roots ; the former of
wbicb is here perform'd by mere Addition, and the latter
by Multiplication.  Thus 3, the Sum of the Lomarithms
i and 2, is the Logarithm of 8, the Produ& of 2 and 4.
In like manner 7, the Sum of the Logarithms 2 and 5, is
the Logarithm of i28, the Produdt of 4 and 32. Again,
L OG
3, the Logarithm of the Square Root 8, is half the Lng*-
rithm of 6, the Square Root of 64 i and z, the Logarithm
of the Cube Root 4, is fubtriple the Logarithm 6 of the
Cube 64.
Theor. II. If the Logarithm of Unity be a, the Lozarithm
of the Quotient will be equal to the Dijference of the Lo-
garithms of the Divifor and Dividend.
Dem. For as the Divifor is to the Dividend, fo is Unity
to the Quotient; therefore the Logarithm of the Q(uotient
is a fourth equidifferent Number to the Logarithms of the
Divifor, the Dividend, and the Logarithm of Unity. The
Logarithm of Unity therefore being o, the DifFerence of
the Logarithm of the Divifor and that of the Dividend,
is the Logarithm of the Quotient. q. e. d.
Schol. Hence appears another great Advantage of Lo.
garithms, viz. their expediting the bufinefs of Div zon,
and performing it by a bare SubftraFlion. E.g. 2 the Dif-
ference between 7 and 5, is the Logarithm of the Quo-
tient 4 out of 128 by 32. In like manner, 5 the Diffe-
rence between 8 and 3, is the Logarithm of the Quotient
32, out of 256 by 8.
An Example or two will render the Ufe of Logarithms
in Multiplication, Divifion, Zfc. obvious.
Num.    Log.
Multiply 68  1.83250
by I2+ 1.07918
8X6   2.91i68
9   0.95424
9
0.9 5424
Num.    Log.
Divide 816  2.91168   9
by I2   1.07918    9
68   1.83250 Sq-.8
9    0.954'4
9    0.95424
-99 0595424
Sq.8 1 2)1 .9g848(0-95424 Oq.R.
Cube 729  3)2.8627z(o0934Z4
Cube R.
The Properties of the Logarithms hitherto mention'd, and
their various Ufes, are taken notice of by Stbfebus: but
come all far Ihort of the Ufe of Logarithms in Trigono-
metry, firfl difcover'd by the Lord Neper.
To find the Logarithm of any Number, and to conjlruFe a
Canon of Logaritbms for Natural Numbers.
i. Becaufe I. ro. zoo. ioco. 10000. sc. conflitute a
Geometrical Progrefflon, their Logarithms may be taken
at pleafure: To be able, then, to exprefs the Logarithms
of the intermediate Numbers by Decimal Fraclions,
take o.oooosooo, i.00000000, 2.oooooooo0 , 3.C000000,
4.00000000, TOc.
2. 'Tis manifefI that for thofe Numbers which are not
contained in the Scale of Geometrical Progreffion, the
jufl Logarithms cannot be had: yet may they be had fo
near the Truth, that as to Matters of Ufe they Ihall be
altogether as good as if flridly juft. To make this ap-
pear, Suppofe the Logarithm of the Number 9 were re-
quired : between i.oooaooo and i0.0000000 find a Mean
Proportional, and between their Logarithms o.oocooooo
and i.ruijocco an Equidifferent Mean, which will be
the Logarithm thereof, that is, of a Number exceeding
Threeby I 00007-7a7, and therefore far remote from Nine.
Between 3 and lo therefore find another Mean Propor-
tional, which may come fomewhat nearer Nine; and
between io and this Mean, another fill; and fo on be-
tween the Numbers next above and next underneath
Nine, till at lafi you arrive at 9-00000000, that is,
9 f00fOOoo which not being one Millionth Part from
N ine, its Logarithm may, without any fenfible Error, be
taken for that of Nine itfelf. Seeking then in each Cafe
for the Logarithms of the Mean Proportionals, and you
will at lafi have 0.9542 51, which is exceedingly near the
true Logarithm of Nine.
3. If in like manner you find Mean Proportionals be-
tween I.ococo0o and 3.1622777, and afiign convenient
Logarithms to each, you will at length have the Logarithm
of the Number 2, and fo of the rea.
Mean
L O G