Page:Cyclopaedia, Chambers - Volume 2.djvu/113

 LOG

(4*8)

LOG

A C B

B D C

~B

E D B

F

B

G

F

G H

_F

G I

H

I K H

K L H

T,

M H

X

N M N O

M

O P

M

Mean Pro- portional Numbers.

I.COOOOCO

3.1622777 10.0000000

io.coccooo 5.6234132 3.10-^^777

io.oooocoo

7.4985411 5.5234151

io.oocoooo

8.559*432 7.4989421

io.oooocoo

9.3057204

8.5595432

9.3057204

8.9758713

8.5595432

9.3057204

9.1398170 8.976S713

Logarithms.

o.oooocoo 0.5000000

I.OOOCOOO

1.0000000

0.7500000

0.50000000

1.0000000

0.87500000

0.75000000

I.OCOOOOO

0.93750COO 0.8750000

I .OOOOOOOO C.96875000 0.93750000

0.95875000

0.95312500

0.93750000

C.96575000 0.95093750 0.95312500

9.139S170

9-0579777 8.9768713

9.0579777 9.0173333

8.9758713

9.0173333 8.9970796 8.9758713

9.0173333

9.0072008

8.9970795

9.0072C08 9.C02I388

8.997079*

9.0021350 8.9996088 8.9970796

0.96093750

0.95703125

0.95312500

0.95703125 0.95507812

0.95312500

0.95507812 0.95410156 0.95312500

0.95507812 0.95458914 0.95410156

0.95453984

0-9543457° 0.95410156

0-95434-570 0.95422363 0.954101 56

Mean Pro- portional Numbers.

9.0021388 9.0008737 8.99960 88 9.0008737 9.0002412 8.999608S

9.0002412 8.9999250

9.0002412 9.0000831 8.9999250

9.00002412

9.0000831

8.9999^50

9.0000041 8.9999550 8.9999250

9.0000041

8.999845

8.9999650

9.0000041 8.9999943 8.9999845

9.0000041

8.9999992 8.9999943

9.0000041 9.0000016 9.999999^

9.0000015 9.0000004 ".9999992

9.0000004 S-999999^ 8.9^99992

Logarithms.

0-95434570 0.9542S467 0.9542236 3

0-95434570 0.95428467 0.95421363

0.95428457

0.95425415 0.9542236;

0.95425415 0.95421889 0.95423889

0.95424552 0.95424271 0.95423889

0.95424271 0.95424080 09 542 3889

0.95424271 0.95424217 0.95424080

0.95424271 0.95424223 0.95424217

0.954271

0.95424247

0.95424223

0.95424271 0.95424259 0.95424247

0.95424259 0.95424253 0.95424247

0.95424295 0.95424250 0.95424247

0.95424253 0.95424251 0.95424250

4. There needs not, however, be fo much Pains taken in inveftigating the Logarithms of all Numbers 5 fince thofe that confiil of aliquot Parts being divided, and o- thers mutually multiplying each other, their Logarithms are eafily found. Thus if the Logarithm of the Number 9 be bifiecled, we ftiall have the Logarithm 0.47712125 of the Number 3.

Schol. The Indices or Characreriftics of Logarithms cor- refpond to the denominative Part of the natural Num- bers, as the other Member of the Logarithm does to the denominative Part of the Number : i. e. the Index iliews the Denomination or Place of the laft (or left Hand) Fi- gure of the Number, and confequently of all the reft. Thus o, affixed to a Logarithm, denotes the laft Figure of the Number to which the Logarithm anfwers to be nothing diftant (/. e. is in) the Place of Units. The Index 1 JTiews the laft Figure of its Number to be diftant 1 Place from the Place of Units, i. e. to be in the Place of Tens, and confequently the Number itfelf to be either 10, or fome Number between that and 100, and fo of the other Indices. Hence all Numbers, which have the fame deno- minative, but not the fame numerativc Parts, as all Num- bers from 1 to ic, frcm 10 to ico, £S?c. will have Loga- rithms whofe Indices are the fame, but the other Members diffctent. Again, all Numberswhich have the fame nu- merativc, but not denominative Parr, will have different Indices j but the reft of the Logarithms the fame. If a Number be purely decimal, to its Logarithm is affixed a negative Index, fhewing the Diftance of its firft fignifica- tive Figure from the Place of Units. Thus the Logarithm of the Decimal ,2561s 1.40824, of the Decimal ,0256 is 2.43824, &c.

Schol. The firft Canon of Logarithms for natural Num- bers, from 1 to 20000, and from 90000 to ioocco, was conflructed by Hen. Briggs, with the Approbation of the Inventor the Lord Neper, and the Manner of conilructing them fhewn. The Chafm between 20500 and pooco was filled up by Adrian Vlach. In the common Tables we have only a Canon from 1 to icoco. There are various

other Methods of conilructing Logarithms hv n- w„;;«. M, Cotes, Dr. W, Taylor, L ihichle Ladef wS

rr/zf y that way - wiU find in *• *fe*3

To Jindthe Logarithm/or a Numler greater that, anyinthe Common Canon hut lefs than : 1 0000000. Cut off four Figures

uTthe e Tahl fd g,Ve " NU r t, ber > "* feek th « 2*KE m the Table ; add as many Units to the Index as there are Figure, remaining on the right; fubftraft &JS3C found from that next following i, in the Table = then a s the Difference of Numbers in the Canon is to the Tabu- larDiftanceoftheLopmtfeBianfwering to them fo are the remaining Figures of the given Number to "he Loga- rithmic Difference ; which if it be added to the zZ- rsthm before found the Sum will be the Logarithm re- quired ; v.g. the Logarithm of the Number 8 « 7 j£ re- quired Cut off the four Figures 9237, and toVhe Cha- raaer.ft.c of the Loganthn correfponding ,0 them add an Unit ; then s ™^i Mla

From the Logarith. of the Numb. 9238=' o6< ,,<?,-> Subi.raa.Z^.Numb WM-mShS

Remains Tabular Difference

10— 471— 5

4/1

-4.9655309 -235

Now to the Logarithm

Add the Difference found

The Sum is the Logarithm required— .5.965 5

To find tie Logarithm of a FraSKon: Subftract the Lo-

anT mthe fo ^T™ f° m, *" ° f ^e Denoi^nafor,

Thu fuooof mder ?'f* V he, Si 8 n of Subftraflion-

Fr»,I" P . P ° ren,Sre 1 u,rcd tofind thc logarithm of the

Frail

Logarithm of 7=0.8450980 Logarithm of 3=04771215

Logarithm of 4=0. 5<5"79 75^7 TheReafonoftheRuleis, That a 'Fraction being the Quot.ent of the Denominator, divided by the Numera