LOG
or the Direction of the Veffel with regard to the Points of the Compafs. In the third, the Number of Knots run off the Heel 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, £5?c.
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 l J oint in that Radius is fuppofed to move from the Arc towards the Centre, with a Retardation of Motion in aGeometrick Proportion. As fuppofe there be a Quadrant of a Circle, BCA, {Plate Analyjii, Tig.xiJ) and any equal Divifions in the Arc, as A F = Ff=ff t &c. with five correfponding Radii, as fuppofe C A, C F, Cfi &c. whofe Parts or Portions C i, C a, C «, ckc. are geometrically Pro- portional ; then if a Line, as i, a, a, b, d, C, be drawn thro* ihofe Points, it will be the Logarithmic or Loyijlic Spiral.
LOGARITHMS (from aq>©- ratio, and £e&i*& Hu- merus) are ulually defined Ntmerorum Frobortionalium eani- differentes Comites j 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 confiderM as a Quantity ftdgeneris, 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 ealily conceiv'd from what follows.
A Series of Quantities increafing or decreafing accor- ding to the fame Ratio, is call'd a Geometricarprogref- fionj e.g. i. 2, 4.8. 16.31. &c. A Series of Quantities increafing or decreafing according to the fame Difference, is called an Arithmetical Progrtfiion ; e.g. 3. 6. 9. 12. 15. 18.24. Now if underneath the Numbers proceeding in a Geometrical Ratio, be added as many of thofe pro- ceeding in the Arithmetical one 5 thefe laft are call'd the Logarithms of the firft.
Suppofe f.g. two Progreffions :
Geomet. 1. 2. 4. 8. 16. 32. 64. 12S. 256". 512 Arhhmct.o. 1. 2. 3. 4. 5. 6. 7. 8. 9 Logarithms. o will be the Logarithm of the firft Term, was. 155 of the 6th, 32 j 7 the Logarithm of the 8th, 128, ckc.
Theor. I. If the Logarithm of Unity be o, the Logarithm of the FaBum or FroduB will be equal to the Sum of the Logarithms of the FaBors.
Vem. For as Unity is to one of the Factors, fo is the other Factor to the Product. So that the Logarithm of the Product is a fourth equidifferent Term to the Loga- rithm of Unity and thofe of the Factors ; but the Loga- rithm of Unity being o, the Sum of the Logarithms of the Factors mutt be the Logarithm of the Factum or Pro- duct, q. e.d. _
Corol.x. Since the Factors of a Square are equal to each other, i.e. a Square is the Factum or Product of its Root multiplied into itfclf j the Logarithm of the Square will be double the Logarithm of the Root.
CoroUi. In the fame manner it appears that the Loga- rithm of the Cube is triple, of the Biquadrate, quadruple 5 of the fifth Power, quintuple 3 of thefixth, fextuple, i£c. of the Logarithm of the Root.
Corol. 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 j and the Logarithm of the Root is had, if the Loga- rithm of the Power be divided by its Exponent.
Schol. Hence zve derive one of the great Vfes of Loga- rithms, which is to expedite and facilitate the i'v.jinefs of "Multiplication and Extraction of Roots ; the former of which is here perform' d by mere Addition, and the latter t ly Mtdt'tpiication, Thus 3, the Sum of the Logarithms
I and 2, is the Logarithm of 8, the Product of 2 and 4. In like manner 7, the Sum of the Logarithms 2 and 5, is the Logarithm of 128, the Product of 4 and 52. Again,
( 467 ) LOG
3, the Logarithm of the Square Root S, is half the Zogm- rithm of tf, the Square Root of 04 ; and 2, the Logarithm of the Cube Root 4, is fubtriple the Logarithm 6 of the Cube o~4.
Theor. II. If the Logarithm of Unity he o, the Logarithm of the Quotient -mill be equal to the Difference of the Lo- garithm! of the Bfoifof and Dividend.
Vem. For as the Divifor is to the Dividend, fo is Unity to the Quotient ; therefore the Logarithm of the Quotient 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 Divifton, and performing it by a bare SubftraBion. E.g. 2 the Dif- ference between 7 and 5, is the Logarithm 'of the Quo- tient 4 out of 12S by 32. In like manner, 5 the Diffe- rence between 8 and 3, is the Logarithm of the Quotient 32, out of 2515 by 8.
An Example or two will render the Ufe of Logarithms in Multiplication, Divifion, &c. obvious.
Num. Log. Multiply 58 1.83250 by 12+ 1.07918
Num. Log.
Divide 8 1 6 2.yu68 9 by 12 1.07918 9
%l6 2.911S8
9 0.9 5424 9 0-95414
0.95424 0.95424 C.95424
Sq.i
08
9 9 9
Sq.%i 2)1.90848(0.95424^^. .
Cukfjtp 3>.85272(o. 9542 4 &6< R.
The Properties of the Logarithms hitherto mention'd and their various Ufes, are taken notice of by Stifeluis : but come all far Ihort of the Ufe of Logarithms in Trigono- metry, firft difcover'd by the Lord Neper.
To find the Logarithm of any Number, and to conjiruR a Canon of Logarithms for Natural Numbers.
1. Becaufe 1. 10. 100. 1000. 10000. £?c. conftitute a Geometrical Progreffion, their Logarithms may be taken at pleafure: To be able, then, to exprefs the Logarithms of the intermediate Numbers by Decimal Fractions take 0.00000000, 1. 00000000, 2.000OC000, 3.00000000, 4.00000000, l$c.
2. 'Tis manifeft that for thofe Numbers which are not contained in the Scale of Geometrical Progreffion, the juft Logarithms cannot be had : yet may they be had fo near the Truth, that as to Matters of Ufe they fhall be altogether as good as if ilrictly juft. To make this ap- pear, Suppofe the Logarithm of the Number 9 were re- quired : between i.oooooco and io.oooccoo find a Mean Proportional, and between their Logarithms 0.00000000 and i.ooooooco an Equidifferent Mean, which will be the Logarithm thereof, that is, of a Number exceeding Three by Ti lllll Z, and thetefore far remote from Nine. Between 3 and 10 therefore find another Mean Propor- tional, which may come fomewhat nearer Nine h and between 10 and this Mean, another ftill ; and fo on be- tween the Numbers next above and next underneath Nine, till at laft you arrive at 9.0CCOC000, that is 9 tH "o°?oo -, which not being one Millionth Part from Nine, 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 laft have 0.954251, which is exceedingly near the true Logarithm of Nine.
3. If in like manner you find Mean Proportionals be- tween i.occccoo and 3.1622777, and affign convenient Logarithms to each, you will at length have the Logarithm of the Number 2, and fo of the reit.
Mean
