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THE

,

P R E F A G E

WEREthe ConflrtSimand Oftof the fiUowing

Tables kôtPttg e^/ery body theyfhouldcome

ffirth into the World withontany IntrodnBoryD^k

toHrfebnt

as

the

Cafe ftands,and that

Knowledgeis the /barebut cffem^ it, may

be proper to addfonte^

thingHpon both th^fe'HeadsAnd iphai is here prefeas-ed

is Gathered frontthe mofi(êkkraud Abhors that

haê ImprovedthofeSubjectantongflur [fives viz.

Mr. Briggs,Dr. Wallis and Mr, Hall^y, the Three

Frofefforsf Geom^y in the Savilian Chair at Ox-*

fordr

To lead the Reader on from the begpining^

give him the ii^th QhapterofDr* WaUis'i Algebra,

v^hich treats of the Originalof Logaricbois,nd giver

a fullHtfiory.ofth^ fro^tfs.To this is fnbjoynd

Mr. HalleyV CompendiousMethod of makjngLogti^

tithms^'Ofbifbrocfedsabjâ^^fyfi^m the natnre of

Numbers tPitboutanyregardo the Hyperbola j frimt

iphich is damped,for ^raîce the making of th^

matural,

and Jir. BriggsV :ffîth' the fnrihen'

ffofeciftioiff the fanfe SnbjeSijgeneronjlyom*tnunieat^ by the ingeniousandtmmearied Mr. Abr

iharpyvith his Table- of LogapicHiqsto above ffi^

Bgnrej



The P R E F A C E.,

-- i

figures as alfoheHyferboUckogarithmcf to t

ôândits Reciprocalyo 6^.Then followsythefame

Hand^theConfimBionfthe Sines^

Tangentsnd Se

cants ; with the whole Frocefsf theQnadratwreofthe Circle to yi Figures Which Quadratureas in^

vented nd here denionjlratedyythe above-mentioned

Mr, Halley.To come to theprincipalart ofthe Performance

as to the Tablesê may venture to jay ( fpithout

Partialitythat we offerere a more compleatet vfthem than can befoundin any

other Boo\now Extanty

and donbt not^ but upon carefulerufal,heywill be

founds Zffefuind CorreS. The Method we obferved

in PrintingheLogarithmss accordingo that Excel*lentAbbreviation Z r,JohnNewton i his Trigo-'nometria BrittaAnica ^ to which is added heDiffer'cnce between each Logarithm,nd the proportionaParts

yin the lafkolumn ofthe famePage, bywhich

the Logarithmof anyHumber ( contra)under

10,000,000 may bereadiljoundwithouturningo

any otherPage,

\ The manner ofplacingheTables ofnaturalSines,

Tangents /Secants,i ^/tibe/rLogarithmsis ab^

folutelyew, and very Advantagiousfor to each

Logarithmn thofeables are placedthe differencwhich are common to the Column of Logarithmsn

bothfidesBy which the Seconds may be eafilyoundmAnd forthat end,between them and the TableofLo-

garitnms,/ placeda fmallTable to convert Sexa *

gefimalsntoDecimalsnd contrarily.

The



The PRE FAG R'

The next and chiefthingconfiderahlen tbt Tahfes,

ti theirCwreUnefs and here we will^we a particu .

JarAccountof

theMeafures

e

took o Mai$them

f(f^'4isortheTablecfLogarithms,t was examined fromi

ii to 300c6 and from poooo to loiooo by Mr,

Briggs*/rith. LxyûFruaed London i62^yand

from I to lOOyboo byAdriajiVliccj^sTableyPrmt^

Condae itflS.And to fhew our Care hereinâs,ell as.

forFublickjiervice^weereplace TableoftheErrors^

ipe fonnd(whenorreîngur own) in MrSnggssand

Vucq*sabove-mentionedaffns i.andbecaHfelacqV

9Wit Errata TableisfmndinfewfhisBookfQnfomucbthai Dr, Newcoh^j above mentionedCanon iiPrinted,

*

Wftb all bis Errors ) thereforee thoughtt neceffa'9) to give it herewith our Additions.

.

Where Note^that fuchof his Erroks as we how foundare marked

Vfith(2i)^ndjinglene in Mr,Bip^'gssith (b); antf

tbofewhich are Common to Mr, Brjijggsnd Vlac |

aire markedwith (B.)

A 3-97^8902870

B 21^98B 18190

6 11490

93WB 97775

96402

98331J 1 130

5^f8B ^0145

4.05952

407^*7

95805

25257

16320

181338QI36

^93^5'45198B 07019

m

80

169

X*3|

9 B .99 7C

I3

13^4

13S9

J62S

%x67

MM

V34

f4^

4T99

o

^7046

59571

V7634*

.77W5

2.98587

5 X30^4B 4^9*5

9545e;2817^

8^43

45^

^

39403

B 05413

m 89113

.

05739

B 55x05

71070

3795*

.394*5

7KA79

[mm, Lo ar. \2i m

850181 9482

19453 I 970^

68675 I 997*

145^1 1 9973

17454 hoo58

5.796351IC061

95904 110095

20444 10292

B3.8424210847

389070 10859389376 11003

B 00829 11332

3.93227 1 1440

42620 11469

97176 11920

90604 11955

3. /525512328

34049 1*398

3 95293 13274

2.95917 14Q

73656145*7

3.9744^ H763r9768o|l4786

ttm, I J-ogar. 17înt .

15305

1584316461

17509 IB

17773

17780

1900^

19107

19113

19195

2083*

^4862

28413

33800

35560

38780

39844

39^45

405984ioi9

4149041505

44656

17090

74222

51149

134*7

4-*497^

17^66

91707

25036

88598

811654 3i873

60621

4.45^66

4.52S9151I71

78047

.a 4.6CQ35a 4-^0037

4*6085044807

343 |802379

78lPI

0135315090

483751a 4-9955 Jl49502 1a 27458

a

a

a

i

a

a

497171a

49880

5^359 a

5775^

60400

^1999

5209065160

5^7 J9

67050

73^55

74832787CO80112

95^66

97828

9909099910

Ioroo8

49148

54448

32785a 4-7^159

69385

a 4-79*38

a 165981I075^r

a 4.82450J

4.815a 4.8671a 7352

473*41

.

93451

54.98075

a 159251a 4.9904

B

9818S959a 47422

Tie



ttet^REFACE;

the Tables of H^wdi Sines,Tangentsdhi'^t-

tants srere examined iy theftdf Van Shboten, Frin.

ted at Amftendam t6ij Qghkh are faid to he with*-

wit one Fauk) and Sir JonasMoOr*s new Syfiem--,the

Tables ofLogarithtnic\SmeSyangents and Secants

vpere examined by a Table f the faidVkcqs, irt

largeO avOy FrinVed at Gouds 16^6^ is ilfo

the faid Syfiem,And in all theftExamination^there w e never fefsthan Two to hark^n whilftne

read over the Frinted Sheet to be Correêd* The

Table of VerfedSines was Frinted fromândexamitÂ

by the Syfiemabove-mentioned^theybeingto be found

no where elfehat I fyow if.

The Tr aversT^k is netfi Calculated h -4 target*

Radius thanriny Bxtint and was examined with tbt

ffrâteflare,

After the Tables foUow the various 't/fes

Logarithms made plainto the meaneU Capacity\ Ttp

which is addedy tbt Sobttion (f Flain and ^herieatTrigonometry ty Logarithms,from Mr 6r1

Englifh Edition efbisLogarithmicalAririituetU.^

and the 'Vfe ftbeVerfedinesfrom Sir Jona$MoOi*âbovcmentioned Syfiem,

The DemonftrationfCompound Interefl,fthf^m^

Frrpojitionsf Navigation,ere both of them btjifm?^

hy Mr. Hallcy,and revifedby him5 as were ntofi

%be Sheets of the whole Difcourfe wherein be

pltafed to make many advantdoious Alteratitrnt



O-: r '^f



^

tkRORS in the Difcourfes before the TABLES.

PAGE 24.line ii and

ri,for 485^939 read 485^9^5. v* 26. line

i2|

after Natural Log. infert [of loj p.28 in Loa. of

14latfer end, for

14(^^24 r. 140(^24, p. ^6' 1.40. at the latter end of tne Log. of ^ r. ^295 1

.

p. 411. lo.

makethe 29th Figure[o] inftead of 6^ vk. for 923^3 r, 92303.

{ .43. 2,for ii99*^3392 r. 1199.^3302. . 23. r. 0000580528751^. p. 47.

.

19.of the 2d. Table for %o62i. r. 20922*

in the Title of the 4th Tabic

r. [FraSions.] p. 47. at the beginning of the laft Jiine infert [you have'1

|^i. (cj ***** * L*-iwg jinxiiiiaj. ^ ^ * 1. o. lut \^L\j -*v^'j L*-^ i*i***j. ^ ^.K

1.2.

for [and] r. [ c.]p. 39.1,

9.for. 33^391. r. 3) ^33i. and.l.

17,for

After the Tables, Page 5.Line itf. for

5.7342957R d 5.734^997*



T-

Kn

^Pi'^ - ? ^îW'^ ' I I ' ' I 'f

O F

T HM

. . ,

T H E I JRL

Invention and Ufe.*

.

,

.

.. .

.

tit XliidChapigr-/ tbat Excetlfntreatifeof AlgebraWritten h the lateReverend and Leaned Dr.Jdiiiallis,SavilianAiof^ bfGeometn in theUfdverjki-Qx^d|.^ a Jdernhffthe Jbj/afSKiet/irLondobf

Logarithmsaifirftof allInrentedîthootany Example

of any beforehlflithat Iknow of)by JohnNeper^Baron

: of Mirdfifiohn Seotlmd \ and by him firft blifliedat

EditAurghyn the Ye^r 1^14 : And Toon afterby himfi^

(withthe Afflftanceof Htmj Bri^s^rofdObrofGeometry firft9t Lî^m in Grifhamîedtt^noaftcrwirdst Oxford)reduce^to a tietterForm^ add perledî

The Iflifeationatf greedilymbraced (iindd^fcnredly)fliaro^d fifen.Mr* Briggsypon the firftPobllcationof itiwas lbjplealedith

itythat he preientiyepair^ into Seotloffdo coniult the Au^

tiiofydviie with hinf,tuid be afiKteato turn,in the perfeSin^'Cffii^nd in Qilculatiiigables for it \ whiqh vras a*Wo^k cif

great LahooTt M well as ibbclleInvention^

And it was mibr^ce4and promoted abroad by BiefMn^n X/rt

JfifimsiohnKefltr Mm f^tacq etrms CragernSflyloih^

AflA ^t hqrne bv HenryGeMrJtndy who perfeftedhe Trigono*

'mrUôfâv((nkhMr. BrigpbÔ) tfnt fedTjs^fofewtt

that,in a Ihortttme, itbsecamegenerallynown, and gree ^

dilyimbraced in allFarts as cifi^tnfp^kabledvantage)eipeî^

fillyor^t and fbi;peditfonn Ttigon/mctrknlalcalacionsu

I



9 Tie i7th Cb^tr ifiDr*^2\Mt Algdfra:

The FoaQdation of it is this :

If to a Raoicof Goatioaal Pcoportioiia.l$r 9 Gepmetrical Pro^.

greflionrpm i : Supfiofc^^

.

I. 2. 4* 8 16. 11. 6^ O^f*

We accommodate a RaoJc of Exjpoaentsia an AritboRtical

-

*

^

* '' / ^ )

.

^

o. I. 2. 3. 4*' 5* tf cl

It is manifeft,that for eêry Mnltiplicatbar DiviCon oftbofe Termsône bjraiôther,here is^ anlwerabl;Addition

or Sobdudioi)of the Expai|qQCs. ;'

; i .

For as (lA theêrm^) VMoltipliedby 9 makes ji^ lb (iathe Exponents)f to 2 we add 3^ it maizes 5 ) and as 32 divided

by 8, jgives : So iffrom 5 weSabdad 3^ there remains aî AaA

ib every wh^re.

'*7Vn9 r..

I. 2. 4* 16. ix* 64^

fygfnumi. c I. 2t 4* 5. i^

2+ 3^S5 $. S^^sel*.

;

Aad dieiaiiie holds, ifbotw^ra au^ two ci thofe Terms,,bi*

terpoieone or more Means Proportionid and betwoea their

Exponents,as many Arithmetical Meam._

I As if between 4 and 8 (or between 2 aqd 16) we interpoie

a Mean PrqpdrtionalV'3^9 that is 4 V % i^ between 2 imd ^

( r1 and 4) an Arithmetical Mean, 2^} then as 4 V ^ l yomioes 32 V ^ (^ Mean Proportionalbetween 32 and 64 :) So

adding thoir Exponents 2i and 3, makes 5 , an Ariibm^ticu

Mnn between 5 and 6 : And fo everywhere.

^ And oniverfiJIywhatsver be the Values of r. 0 f^PPP^

. /

Tl ^Tirtf$$ l r. n% m* r\ rK r*. c^

MMmimsj ff. 2#. %i. 4^ S^ ^^ ^^*

Tmit, 41 rrKrâsrS and rr^rHrrr sr' ^rs

2#+3r=5#iand M* + ^^ SK

And fo every where*

And cooAqacatly whatever Term w interpofebetweeal^

^ thofe Cominnal Proportionalsif we aliointerpofeetiwoeatheir Exponents, a likeArithmetical Meant, as that is a Propoor'*tional Mwu (as if that be the Firft or^Second of two Means



Of Ug^tnOm^. tteir'Imtemm MndMfi.^ 3

fkfi^HWtSoMl,hisaccordiqdytiieFlrjEbr Second of twt Means

jbitboiedcal if that tHe Second of Five Means Prpportionaly

thisthetSecond of as oiany Arithmetical Means, c^c.) Then to

fveryr Additk n or. SiiMoaiba of thefe one with another, will

uiWer adike Multiptkatioar Diyiiionf tôfe.And' if for o^ #. 2#, sr, c. (takinge=51 ) we piit,', 1, 2, 3:

c;. then. doth thisExponentalwaysgive us tne Number of

RatioQs.gr Dimeniions in the Term to wihicht belongs

is,how many Rationsor Proportionsf r to I9

are comppuodedin r^ to .1, to wittf..

To which the Name L^^

^latkhmm.t^ 9:^He %shat is, s6^dfAfmy the Number of

Now tUs Fouidatioik /beinglaid,thetr Defign ih t6e Loga^

rithrasjs this : Having .ieleded(asdiioft

oonvenient)a Rankof Continnal PropoMiDaals,n a DecupleProgreflionto wit, .

t lb. roo . Tooo. IOOO0U tooooo. loooooo. c^r.

TJbeyfitherMma bMt EzpooeBt*)n AriHrn^kÛ PregrtjIIwi

*

(And cooieanently,he Logarithmof any Fraftions leis than {fcito be a.lêgEitiveumber.) Acid the% for each,of the Num

bets interoofra between i and lo, between 10 jtnd ioo| and fo

of the reft% (as2, 3, 4, Ck 1 1^ 12, 13,.^'^'.)heyfcek out (be- *

tweenio mid i^:etween i and 2, c.) :ad Exponent (to be ex

prefledin Decimal Parts)which isfnch a Mean Arithmetical,as

the other isa Mean Proportional.

And th^ Enonents tbOTjEsIlagdrhbms hich are Artificial

Nnmbers, fo anlweringo the Natural Nnmoers, as tbatthe Ad-ition

and SubdudJon of thefe,anfwers to the MnlUpllcationnd ,Divifion of the Natural Number5u

a

By this means, (the Tables being once nnide) the Work of

Moltiplicationod Divifion i^ performedby Addition add Sub-*

doflkioh( and conCbûentlyhat of Stfiiaringnd Cubing,by Du *

idatioaand Triplationand tliat ot Extradiogthe Square and

Cubick Root, by Biiedion and Tiifection j and the likein higherPowers*

i i or



Of thefeLdgatithnise have Printed TaUes, IbrdI

Mwftl

be found in thole Printed Tables) b agacoordingljddM at

fiobdnfted*in ve the Lo^thm of that NatOfai mmbef (tobe found by thofe 'fablM)which is the PrtfdoQ or Qjiodentel

Inch Mnltiplicadonr Diîfion. And the doable or Treble of

f itsS^^darer Cabe. Ami

Logarithmof itsQpadraÔr Gnbick Root*, and the Ifte of Hiriier Powtrs which In largfNomberst is nuttter ^ greatferbediticHi.

And (tecanfeattiam End ifthbOe^ tras to ftcilitiite

'Aftronomicalnd other TrigonometricalalculatiOBs) beinlti

thofe Logarithms for Nombers in thebr lâtoralrder^ we hare

Alio Tabfes of Artificialor Logaritbnicalinet TMngentt,ndStcaifs;the AddidcMi and Sabdodion dF which,oniwers t6 13x6

JMttlpIicitionnd D^vilkn of the Nmmd Situs,Ttammt, and

S C4aiti: Which isa veiy GompendioasAdvaafkgp lirpeditingfilchCalculations

) and is mtt left accofate than the Opefstioii

by Tables tfNttiard Simtf TmâHh And Sntmtt*

Thus it atrtaittriaa^ ; fyp^tiiB%the Angles^ven,A ^ Dtegtces,ifo

Degrees,and coalfaqotttfy,70 De*

arees) and die Sde AB 31321 facta

Bm widiitgtheses AC, or AB wfr

^

have thb Proportioa:

3n99

I

As the Sine lfC, 'ôDegrees, ^3ptf92tf

To the Sine of B, .50 D^reesi *t66Qj\^

So is the Side A B, stji} fmai -

tb tbe Skte AC^ 3-Si)S^fû

|r Mrfindiiffihich,we are tx^ Moldply 16604^ by 3f3I),tnd then Divide by 9i96siS i which gitesfor the Side AC (ak

^ft) ^5535 f^^^'

Andt As the Sine of Q 70 Degrees 9i969 t

To tht Sine of A^ tfq Degrees, 8tf k ; 54

So in the Side AB, lîl? f*''**

To the Side BC 2 8tf7{*^

For fifidlAghich,we are to Moktply 8^^0294 by 313234

and



iHd Oiwideli| 39^92^ wkkh givesot the Side BQ 38857*'

Now 4ti prevent theft tedideftMalt^rfkatbnsnd DWifiofls1^LQOttkka% we pfbccedthas V

^ '.

i ''

.

log. StfieQ la Degrees'^

9^91^9^$'ir; Co. 6.0270141

Log ;SiBe;B^5d egrees 4-9*8842546 9.8842540

Log. AB Miiitt.3132^ : + 4*49581533 4.4958633

tog. A Cf Mam. 15 53 5^ + 4^07 *3 15 '4 407 3 J

Where Sabdoftiogthe ^Firfl;Logarithm fi'om tBe Som ef the

Second and Thirdsgivesthe Fourth } which (the Table tellsas)aafirers to the Namher

2^5535^ feri. So manyPaces therefore

i$the Side A C Again^

Log, S|n6*G,oDegreelf '9W9729858Ar.Co. 0.0270142

Log* Sloe At 60 Degrees^ .

+ 9.9375306 9*9i7i3otf

Lo AB Nam* 30^3 t 4^4958633 4 495863^

Log. BG tlom. 28 tf7* 4 448^081 144604081

Where Sohdoftinghe firftLogarithm,from the Sum of thdSecond and Third, givesthe I?ourth \ which (theTable tellsas)titfWersto the Number iJ iSîr^ximi: So maay Paces there-

fere is the Side B ; which Operationsre mucji more xpeditioosythan Multiplyingnd Dividinguch largeKumbers.

Aad in like maimer^ in Spihericalriangles,ave th.ic there All

tJieLogarithmfare to be taken out of the Tables of înes Tasf^

jtentsând Secmns ; whlch^ in this Example^ are cakeu partlynrom thence^mrtty from the Table of Numbers 9 but tha Ex *

pedttbn is ahke in both.

- This was firftPubltfhedbythe Lord Ntfer(thefirftInventor of

It)in the Year i6t4t under the Title of Afirificnsoârirhm9ruMCsman^ with itsDelcriptioniid life ^ but reiervingthe Manner

of Conftro tion,aadtsDemon(h^tion,tobe afterPublifhcd : This

beingbut an ^f^h^^tforth^to fee the Judgment of Learned Men

coacerningthisDeiign,and how it wair like to be received.

In tlus (ve have a Canon or Tsbte of Katurai and Logarith*

nical Smes^ for each Degree and Minute of the Qjiadrsnc.

And whereas it was at his Ghotcc to giveto wiiat Number he

pleafedthe Logarithm o, and whether to proceedby way of In^

creafe or Decreale, he chofe to make o the L^âfithmof the

whofe Sine tooooooo, that fo the Multiplicationr Divifion bythe whole Sine (frequentn TrigonometricalCalculatiott)mi^htbedilbatched without trouble,rcquiiiaghere but the Addition,

or Sttbduction of o

And



Andbeomre the lift of LeflferSines andfiombcrSi lefiitluut

the Radias or whole Sine,,

were likelyto be of more fttfpeat

U j than of TangentSt Scicants and other Numbers greater

tnan the Radias, hechofe to giveto titoii leflerIômbers*Affirm

inative Logaridims (increafinghe Logarithms from o, as the

SigQSdecreafe)hich he calls^^miMrî : And' cosftqucnflyNegativeL ^rithms (which he callsBtfiOiva) toigreaterNiiiki-

bcrS. Deligningthole by +, thclc by-r-Ând by tliismeans, he aireds how this T2k^l^M .Sims Cwith

the t ifferenceshere inserted)ay fenre alfi for a Table of

Tkngtmsand (^ SecMtfi So that tfaisCanon^s CofUpAeataaoft

of Naturdl SUftSyand of LegarkbmiCMlSitus Tm^9nt$ aiid S a9m4

He fhews aliohow thisTable may be apjêdo the Logartthomof Abfolate Numbers *,but b^:aafe with foni troable,be refenres-

the fulleraccount hereof to a irther Treatift.

In the Year 1519, the Lord iSTrpfreingthen dead^ thefidie

was againPublUhed by his Son Rahrt Neftr\ with fornixPbft*

humous Treatifes of his Father,'concerninghe Coidhlidioa of

this Lqârithmicalanon, and concerninghis Deiiga (afterGom-i

mnnication had with Mr. Briggs)of changinghe Form of Lo*

garithms,making o to be the Logarithm of ^ (of which he

I^ before given notice in the.Premce to his lUkdal0giayub-

lilM in the Yearidi? 0

and

concerningfome

thingsperuimugto Trigonometry with fome Lucubrations of Mr* Briggisk the

feme Subjeft.But the Lord JNtperbeingdeadt the whole Work was devolved

bn.Mr. Briggs who (accordingo then:'jointAdvice) makiagthe Logarithm of i to be o, and of 10, 100, 1000, c. to be

f.^ h ^^ which he calls Indkesjor Charji erifiichjnd wbicli'

we may repute as /;yre^frumbers, with Fourteen CiphersiHflexed,which we may repute as (b many placesof Decimal Fia-

oioAS, below the placeof Unitsôr of the Charaderiftick: Aad^K^tween thefe he fits.the Intermediate Logarithmsfor.the lo-

tfrmedjate Numbers*

And confequently,he Logarithmof 1 being o, the Logarithmof Fra ions le(sthan i, or of Numbers intermediate* becween t

and o, muft be NegativeNumbers, or Numbers lefs than o,

(which he calls Defedive Logarithms,denoted by-- (the Note

of Negacion)prefixed^ôw thefe Derive Logarithmsmay be two ways expreflcd

eithero as that the Note of Negation Iball afiea the whole

l4 ganthm, or fo as to aSed only the Chara eriftick,(leavingthe;rtft of the Logarithm to be underltood as Affirmative.)

As for Example,.The Fraftion I, or ( which is equivalent)

0.375. Tî^ VnlBdonfoppofetihhe Numerator ^ so be Divided

by



ntbms is to be performedby Sabtradina Log- 3* o*477X2i3

the Logsrkhm of 8, from that of 3^ ana Lor* 8 0.9030900

the Reraainder wjll be the Logarithm of Log. 1 --o.A2xotf84

| which win them be the NegrtiveNam-^ V ^^^^^

ber,-0^5 9(^87*

Or thBsVf Mr asuch as the Logarkhm of 375, (fiippoiinik t be aalnteg^r âmbo*) is 2.57403i3 And the deprefllxiLthb to the Fkftf Second^ or Third, or further place of Decii

nul Eraôo, doth (without akerin the Figures)ivide tbo

Value h^ 10^ ioo 1000^ x which m Logarithmsis done b^

Sabtrading i, ^ 3, e. from the Cham rifticktr placeof

Integets, I9 2^ 3, ^.în that place,

being the Logarithms of lo, loOf Log. 3750 3.57403111000^ ^r ) Such Alteration,of the Log^

'

375 2.574(^3tf

Value (the Figures remaining) is Log* 37;$ i.S7403ij

done by altei:iijighe.

Cbarafterimck Log^ 3I75 0^574^313

ftf.the Logarithm, wklioat varying Lpg, 0I375 ^^$740^1^

the other Figures, in this man- Log. 0)0375 ^ Sl-^^^Sner.

Whijch'

two Forms, tha' they feem diSerent,and (bmemay

rather choofe the one, fbme the other ^ or in fbme Cafes tht

one, aAd,

in .fi meCa s the other *,yet they are in Subftaocv

or Value the iame* For

+ 0.57403^3

= 0.4259^

^,

...

Aad every one is left to Jiis liberty,hether of the two ways(br what other eanivaleot thereunto) he (hallpleaieto ufe.

bi this MethoQ Mx.Bri^s hath calculated a Table of Log8#

rithms, (Piiblilhedin the year 1624) for 20 Chiliads of AbftM

lute Numbers, (from i to 20^000 h) and againfar 10 more (frooi

9oyx o, to 100,000, ) and x m Chiliad Supernumerary ( to wi|^the Hundred and Firil:Chiliad) that is 31 Chiliadsiu alL

Before which is prefixed, largeAccount of the Nature an |

Con/hru^pnof this LogarithmicalCanon, and the Ufes thereof|aAd.Dlredion how to Supply the intermediate Chiliads,wMcJf

ire here wanting* The whole Intituled,rithmitica LogaritbmicsmThe iame was again Publiflied iu 1^28, by MriM fîdc^

(or tUci^) with a Supplemeut (as Mr. Briggsîreded) of th^

Chiliads befwe omittecf that is,h^ all, x ico Chiliads, witli

one Sapemumeraty* But in iborter Number, extended bot tq

10 placesbe)9W tbsitof the.Integers,r the Chara teriltk;k*And

liefubjpinsifi jLX gîtbmicalannon of S'm$s^ Td^t^tn^ aod

Snouts^



^ The i^Chtfiircf th.VhSaSii^j^^

SicMts^ ( for D^rees aid Miaotet tfth QuadnaC) 6f at larf

ay places. ..- ^

.

Mr. Briw proceededto Calculate a Trigonometrfcalmoii

LogarithmicaUTttitedto that foi Abfolute Numfabrs to tte

Logarithmsxtending(asinthat other)to 14 places,elide th|Charaderiftick. And havingbefore Csdculated a Table of Mn

tioralSinhj Tanffents nd Ssvat^s,(forDegrees and Qeotefmes of

Degrees) in ^fumlb rsatendiagto t f places he fitted there^

unto a Canon of UgarithmUM Sims and Tkitfr/, (becaofeth(A

of SecMts might be iparod})nd a Treadft prefixedconcerningthe CoaftniAion thereof,ith other thingspertinenthereunto |intendinga ^rdier Treatile concerningtne life of it.

But dying before this laftwas finifliMt*r the ireftpubliOiediMr. /fi9tryiettibrandSuppliedhb latter,bd PublUhed thtfwholeswidi the Title of Trit^nsmetrU Briisnicd n the Year 163,3.

To which isliib|oinednother Canon Cf LogarkhfkifMlmes and

TmgMtt^ hyMriM f^jfUflyor Degrees,Minutesând Te^th Se

conds, extending(as his former did to. to places, Hde the

Chara eriftick',and Mr* J9ri^/s0 Chiliads tor LogarithludF

Abfolute Numbers.So that the whole Dodrine of.tiogarit as hj tlifeime

fufficientlyerfefted,ith convenient Oll^s or Tabl^ fitted

thereunto,m larêNumbers: Of Which iVh Pettiu Cri^$ugivesan Account in the Pre ce to his TrignwmihidL^MtitmtkMiPrinted in the Year i6^^\ w|th his Lo^rithmicalablestbatIn (horter Numbers.

And the Tables of Logarithmsabove mentioned,({for00 Chi*

liadsof Abfolute Numbers, and for Sines and Tangents to Pe^

grees and Centefmes were the ^me Year 1633^ contraAed ioxt^

a Lefler Foimi and more Manageable (but in fliorterNumbersgthe former not extendingo above 7 places,befide the Ghara*

^riftick,hut tl^latter to 10) by NathanMRaei with DiredfioiU|

|br the life of them (inTrigonometry,eometry, AftroQofiiyGeography, and Navigation)by Edmund WMga$0.

In the mean time^ SenjaminVrfims did silfopuhUih Tables oflogarithms,n the Year 1^18} and againin toe Year 1625, ia

|iisrig9nometri4and J^lu^mesKfpierHilfo in the .Year iâ4

In his Chiiias Ijdgarkhmorim(which hp 9ppUes alio tO his JEmM*.

fhiniTabkijpublUhedin 1627-,)and QaMiui BafctuHsabout the

fimc time, or foon aftert And dfiîms Lni^vkui Ir ki^my ia

the Year 1^34, C^nd perhapsfome others^)iut all or moft of

fhem, in fhortNumbers 1 and ^nformable to the Lord Nefn^s

firftDeCgn \ uot to that Form which, uj|)onecond ThouKhts^he and Mr.Bri^s aêedup taas moft Eiigtbleia4 V^hichhath

^flcebeen received m coini (traî ^



0/ tSgsritktis^heirinoefAim and Vje..

\ Âce which timfe much hath not been added to the Dofhrine

0fLogarithms; nor was itnecellary,hat Work haYmg olttainedr

fidfidettterii^îAa, But inê Lo^kbois,n any emergent occaiion,be deCralê

with greaterfoaonels,and inlargerNombers than thofe Printed

Taês do al^d t Mr. NicÛs Merc^tor in a finallTreitjfecalled

Z^m-ithmaticbmdyrinted in the Ytar i66%jih,ews(withgreatJEbUUtyhow itmay be effefted n Noitobtfrsf whatever lengthddxrabitpith mtich rnore ea(e than heretofore,

,

Thoiie that would iiêeore of the Gonftrndion and life of

Logîdun^ ay confult the forci^mentioned AuthorsêlpeetallfrigesArithmetka L^gdrithmicavd the Tri 9nometriaBritamtici

of Briggsand iSettibrandj as alib,what Ailrim FUcq and Peter

OrugiTHsanre.writnpon this SubjeS.

But one thiftKet feems to be wantingto th more cdnipteaManagement ofthe LogmithmicdCanon : For

tho* ttere be aCapondf Logarithmsor 2V * Note,ThdtwhUB

tmrdNimAersy beginningrom i to 100,000, foms. f^ hHng i^

fo that the logarithm may be had by the gl:r^{*' ^*?'bare Infoeaion of the Canon , yet it is not f* r^ ^âlike eafieto find the Number agreeingto a ^ rt;l^% J

n both fides being fo^nd,jtbeyare to be ^^ iy^.Ditttw.Correaed by the PaartsPi^pmanMy that 6

there may be found ibmt Intermediate Nambcr that mny dgreeto the Logarithmg^ven: To ptcvent whiqh Inconvenience^

there feems to be ftec^ flaryn Anti^LogarithmkkCmm ; in

Which, the Logarithmsbeiagplaceddown in order,from i tcf

too,ooo,tht Mttnral Nnmiers^ anfêrlflfgo them, Ihould b*fpla*ced by theoii So that by this Ganon we inightfind the NnnAer

forany Legdrithm ith the iriieEafe that we findthe Logarithm

for any Namber by the Onon that we hare.

And indeed, fuch a Canojn hath been Jonffn ^dfor mixifyears^but neirer

yet

made Public):. 1 don't know whether Mri

Thomas Harriot begad that Canon, or no ; but Jilr.Walter War-^

r had his Papers,and fro them put forth his AlgebraA. 2 .

itfSf,nd gave hopes of Pubtilhingany thingsmdre. And.

the iîrieamer did, not longafter,finih the faid Canon (if

At leaft he did not firftbeginit) and made it rc (dyfor the

frrefe;nd all this,I bplieveâbout FiftyYears ago,,if not more.

And this t was told Idtelyby Dr. John Pelfiwho was inti-

hâtelyacquaintedwith Nb. Warner^ and had^flcfbed him in thrCalcuimion, I rMiember alfo,that I iaw that Work ( and did

but fce it)aoidng other Papers of Mr- tUnlgt or li\irner and

c tbit



'5875/95250

lis

I

8z75Q

0096%

flfiitow almoft thirtyYears ago ':'^Vhattocatiie of them afe^;

f knew not, tillI heard latelyfrom Dr. P^*, that they were' in

the Hands of the Celebrated Dr. Ktchard Bushtyîller Af Wif^mihfierSchool for ihariyYears, and now wry Old-; who alfo

gave nie hopes of its coming forth in a littlewhile^ by ttcr

Gare of Pr. Ptllîf at leaft {to which I yieldedwithout nWc^

difficulty)ould fuc^ced Dr. P^ff in that Carrey if he fliould

happen to die before th^ Work was finiflftd:'BGt Dr. Ai? is

dead, and that very Oldi he dying abdut the Iftfir1(585,th^

feditionof that Work beingnot fp'much as begun. And I fear,

left {Ht.BUsbtydying alfo) this thing be iuitcoftjcfpeciaMjfinrt fherdfJs none that will bfefillingto be at the Expence of

the Edition. , .

As to the life of Logarithms,altho^ they were invented cWefly

to facilitate Trigonometricalalculations \ yet they are of life,

where-ever there isany occafion for Multiplicationr Div^lfion.

Thus in t6e Biifineftf Jnatocifm^r Owi*

fonnd Inter efi^x. Gr. At the Rate of 6 in

loo, for on* Year. For then it will be, as

loo to o5, or as i to i.c6, thit is to 1,4^:

So is the'

Principal o the Principal n-reased

by the Intereft for one Year, And

confequentlythe Principal is to be Mlilti-:

pliedby 1.06^ for the itrftear 5 and tke

Pr6dua of this muft be Multipliedby 1.06^

for the fecond Year ; and fo oxiy according

to the Number of Years, thattthe Sum may

1120250934^'be found that rifes at the end of ib manjr~

Years. Inftead of which continued Multi-'

plications,he Matter * proc ds thus by

Logarithms j To the Log. of the Priveifolj

add fo many times the Log. of i.o5, as is

the Number off Years i*this givesthe Log.-

of the Sum, arifingfterthat Term of Yeats ;

and the Abfolute Number anfweringto this

Log. is the Sum defired.ExGr. Let the Principal:e 15/-.17 s/ 6 Js

EnglijhCk)in,'hat is (in Decimal Parts)

15.8^5/. and let the Intereft be at the Rate

or 6 in the 100, for one Year, and to be con-inued

in that fame Proportionfor 1 2 Yearsi

Therefore the Number 1 5.875, s to be con-inually

Multipliedby 1.06 twelve times ?

from whence arifes î. 9^362 1* that is 31 L

1 8 s, toi d, nearly*,

.

^

Now

2 17I83715

3 181907379- 1113444^7 +

4 2o|o4i82i74-

'5. 2112443310 +

il2745$9

.6 22

t

.51 89909 +

3511395

7 23

I

870 J 304*-

4322078 -]-*

9 25130^33821I5181403

5 2^182,04785

; 1 5092287 -}-

fO 28[4297072il7057824-h

11 30)135489^1 1808 1 294-^

U~3^943î90~



http://www.forgottenbooks.org/in.php?btn=4&pibn=1000024611&from=pdf






13 Tbe t2th CbafttrofDr.VfdUys Algebras

Reward of his Invention ; whereopon he askM, That fef tho

rft little Square of the Che02^-board, he might have one Grain

of Wheatgiven

him ;for the iecond, two v and fo on doubling

continoaDy, according to the Number of Squares In the Qie^Board, which was 64^ And when the King, who intended to

]l;ive very Noble Rewaid, was much difpleasd^ that he had a^d

trifling one j Sefa decWd, That he would be contnted

with this finall one. So the Reward he had fiic'dupon, was

)rder'd to be given him 1 But tbe King was quicMy AftonifliM.

when he found that this would rift to fb vaft a QtiaQiity, tha^the whole Earth it felf could not furnifli out lb much Wheat

Bot how great the Number of theft Grains is, may be found

by doqbling one continually 6$ times, fo that we may get the

ûmber that comes in the hft place ;and then one time m u:e

êt,o have the Sum of aH. For the double of the laft Term

(1^ by one) is the Sum of aH. Now this will be more Expedl*

tioollyd me Djr Logarithms, and Accurately enough too fonr this

purpofe. For if to the Log. of i, which is o, we add the Log.

of 2 (which is aôioaoo) Multiplied by ^4; that is 1910559203;the Abfolute Nuniber agreeing to this, will be greater than

t%4tf ooooo.ooooox)oooo,and Im than i mL***i,*A** .. _*- . . M# # 9 a

mm WHM ^mmmmmmfm^mtmi^mmmmmmmam

wtf



(lO

ifc, I. ^b^w^'^ 111 lyiiii i ^^B fii I ii I I

^^^^ ^ -^^^^^...-

^ ^ .

P' ' ^ M

Pbilofapbkdranfa m,Number 2 id.;

A Moft CompeiMlkmsnd FactleMETHOD

Conftruaingthe Logarithms,

Exemplifiednd Demonftrated

From the NatureoF Numbers.

Without any Regard to the Hypfrbota

With a SpeedyMETHOD for Finding theNUMBER

fiom the LOGARITHM given.

fy Mr. EdftSiHalley,?r^^ Savilian Profe/farfGeemetrjt

in the tlmverfityf Oxfordj

Md Feflovof the Rnyal

Societyn London.^t^Jm^mmam^mmmmmmm^îa^^mimâiîî^mm^lmi

TH Ifivention of the Logarithmsis juftlyfteemed ono

df the moll: Ufefbl Dilcoveries in the Art of Numbers^

and acc H*diaglyas had an Uaiverfal Receptioa andAppiauie: And the great Geometricians of this Age have not

Imfl waotuiH to Galtivate this ûbjedwith allthe Accuracyan4

Subtiltyt IVbttcc of that Coaiequencedoth requireand they

have demonftratod ieveral very Admirable Propertiesf the .

jirtifieidikmb s^ which have render'dtheir Conltra ioa much

more FacUc, than by tbo operofeMethods, at firftufed by

their truly Noble Inventer, the Lord Nefairj and oar worthy

Coofltry-MaiisAx.Briggs. / t

But sotwithftandingll their Eacfeavours,fiad very few of

thofe who Blake coiiftaat Ufe of Logaritimsyo have attained^

an Adequate Motion of them ) to know how to Make or Exa*^.

mine them, or to underftaud the Extent of thQllfe of them r

Gontentidghemfelvea with the Tables oi them, as they fiadii

them, without daringto Queftioathem or caringo know l^wf.

to

Redifie th0m,ihoald

theybe found amifs ^ bemg, I (iippofe^

under the Apprehenfioaf fome great Difiicultyherein, ôr*

ê fake,of wch^ the foBoivingr ict is principallyntended i' '

'

but'



i4 Mr. Halley Cmpen^t and facileMethod

but not vyithouthop^s,however* to produceipmjpthlnghat may

pe acceptable,o the molt knowinglixihefe mattep; .

/.

But fii'ft,t may be reqaifiteo premifea Definition of Lpgd--

rithms n prder fo rejiderthe cnfoin ifcdorfe..jnore ciWTi

the rather, becanfe the old one, Nmnerorum frhfrniondiifiitqUMferenttsomites^feetnstog fcantyto define them fully*They

majt much more properlybe iaid to ^\a Nibntri'Rsiionvm jB^fo^nentes : Wherein we confider Rstio as a QiiantitasiUgeneris e-

ginningfroiQthe Ratio ofEtfuMtfyr i tq iô^ beipgAffirtna-

thre,:when the Rdth is increafisg,,s c^.Upttytoa^grâtaNumber i but Negative,hen decreafing Arid theft R^lmifs

we fttppofeo b^ meafured by the Number of Rdtitmcnld con-ained

m each. Now thefe KatlimcHla are fo to b^ under(tood9

a^ in a cQntinuQiSqile of Proportions,qfimteia dumber be-ween

the two Terms of iht Riuic j which infiniteNumber of

Mean (Proportionalss to diat infiniteNumber of thfe like and

qualR^tinncuU between any otê^rwo Terms, as the LoârithiO'of the OMRsuio isto the Logarithmof the other. Thus if there

be fuppoledbetween i and lo an infiniteScale of Mean Propor-ionals,

whofe Numbq: is loooood'c. is

iiîritt2t.v.betVE fiaand^

khercihall be 3010a c. of fuch Proportionals,od bi^tw6en i aod.

^ .thtreWillbe 477 1 2 c^r. fthem, which Numbers therefprerethe Logarithmsof the Rationes of i to 10, i to 2, and. i to 3 ^ and

not Q^ properlyto be called the Ij gaithmsf 10, Zand 3*.

Bup if infteadof fuppofinghe Logarithmscompofed of a Num-

Bcr of equalRatitmcuU proportionalo each Rdtid v,we flialltakat

the J?j^r;Vf Unity to any Number, to confilt alivayiôfhe fiiric

infiniteNumber of RmnneuU^ tlieiragnitude in ttiisfare,,47iil^feas their Nmnber in the iformer.Wherfbrt if between Mtatfând any Number proposed,here be takfen any Infinityf Mean

Ifrroportibnals,he infinitelyittleAugment or Ddcêmebt of the

fitft'ofhofe Means from Unity,will be a X^rfiM^^î,that is.the

Mîmum or Bnxion of the Ratio of Unityto the faidNumber^

And'fteinghat in theft contintial Ptopordonals11the Ratiim^

ckU are eqnal,their Sum, or the whoIe^K^l , will be ^s the faid

JlfomctttHm is îrc^y i tliatis,the Logarithm6f eaCh Rîd wiU

fce as the l^laxioffthereof Whferfore,if the Root of any In-

writ fbwer bt extrifted out bf any Number, the- DifnemtioU f the Taid Root from Unity^(Hallbe as the Logarithm of that

^timber. So that Logarithms,thus produced sLj be of as

jânyorms as yon pleafe,o aflame infiniteInditesot the Power

uHtoieRoQt^ you feek : As if the Index be fiippofedooooa ^â

iiifii|lteiy,he

Roots flialle

the Logarithmsinvented by the Lord*jVy)jrrbat if the faid Index were 2301585 c. Mn tî gs^so-

Ha^rrthnis'WonM.mmediatelye produced* And if you-pleaic;to



toJsiroJit any ISamber of Figures,and not ta continue them biu

itwill fufficco aflame zn Index oi a Figure or two more than

i'b*'int?iidedLogarithmii to hare ; as Mr. Bri^sdid, who, -to

gM hisLogarithms true tp 14 places,by continual ExtraîpA

of the ûare Root, at laft came to haye the .Root of thq

146^374863553 28fi ?Power; but how operofethat ExtraftibAwaj, will be calilyjudgedby i*hofo ftiallndertake to Examine

his CalcidHs*)

Now, tbo' the Notion of an Înfiniteowef maf feed very

ftrange,and ( to thofe that know the Difficultyf tht Ex-

tradion of the Roots of Hich Powers) perhaps impraflicable

yet by the help of that Admirable Invention of Ur.NiwtM^

whereby he deterpunes the Vncia^ or Numbers prefixedo the

Members compofingPowers (on which chieflydepends the

Doftrine of Series ) the Infinityof the Index contributes to

render the expreflionuch more eafie : For if the Infinite

Power to be Refblved be put (afterMr. Newten^s Method J

I.

I J

t +? fi fTFii w 1 4-4^1inftead df i + 9 +^ awW^^*

ri^ : 9'+- 24 m^ i^ ' (which IS the Root

when m isfinite)ecomes x-j^ ^ ,^4- ,?4,-^4 j^^s c. mm beingmîVr infinite,nd coniequentlyhatever is di

vided therebyyanilhing.Hence it follows,that - Multipli^^

into f-^ f f + f f 9 f ~* 4f*+ f?/-^c. isthe Augment of the,firftof our Mean

Proportionalsbetween

Unityand

.

i

-^^,and is

therefore the Logarithipof the Rdth of i to 1 *^- ; and where--

as the InfiniteIndex m may be taken at plealure,he feveiaf

Scides of Logarisfamsto fuch Indices will be as or Recipro^

callyas the Indices. And if the Index be taken looeo c. ai. inf

the cafe of Neur^s Logarithms,they wiU be fimplyf^ f q -|-

Again, if the Logarithm of a decreafingUtio be fought,theI

II I

Infinite Root of i~tf, or i ^| is i ^ ;; 4' - 5^--

-7*- q^ :r q^ c* whence the Decrement of the firlt

4 If 5 ^ o m*J

of our Infinite Number of Proportionalswilt be intoq \ \.

ff + f ^'+47*+ f f

^ + 1 ?* *^^- which therefore will be as tbo^

Logarithm of the Ratio of Unity to i ^, But if m be put icoooi

t^c. then the faid Logarithmwill be j-f1 99 +.t ^M^ 4 ?*-^W-

+ i 9* c. Hence



Mr, Hallfy*mj^mmSIwW Faa^ Methft

--^ f

.

tience the Terms ci aiqr Rmi^hcmg m and ti 9

or the Difference divided,jr.he IcflcrTierfflhw tft

ta iacreafiog?4fi ; or j^When 'ds decreafing,t uiîo^

Wfaexice the Logarithmof the iaoieSati^ may be dbublgrx^rdtffor pattiag- fortibeiêace of t}ieTerms ^s a/idf it Fillbf

either

1 XI

X^ X^ ' x^ x^ x^

I X X*.

x^ x^.

x^ x^

But if the i^ju^ of 4 to j be fiippbfcdivided iato two parts#t/iz..into ê Xatio of 4 to the Arithmetical Mean hetw.eeo tbc

Terms^ and the Rstio of the 'faidArithmetical Mean to tbe

other T^m b^ then will tht Sum of the Logarithipsf thofo

ô Rations be the Logarithmf the Rmo ot d to hi and fub-^

ftituting ., inftead off ^^Ht the laid Arithmetical Meant

the logarithnisf thofe Rdtiones wiQ be, by the foregoingole^

f X xx x^ X^ X^ X^'

1.

jr XX,

x^ x^ * x^ x^

:rm^ .+3X^~4^* ^fee Sum i. IX zx^ ix^ ^^^

^ . i.a

-whereof ^ ii * + 5 -^ -J--^:j;-îcc.willbe th^

Logarithm of the Ratio of a to t, whofe Difference is at, andSnm t.. And this 5mV/ convcfrgcswice as fwift as the fbrmer^

and therefore is morejproper for the Praîceof making of Lo-arithms

: Which it performsith that Expedition,hat where

X tbe Difference is but the Hundroth Part of the Sum, the firlS

ftep fofficeso Seven l^lacesf the Lc^rithm, and tliejecond

ften to Twdvt. But if ^rij^/sflrftTwenty Chiliadsof Loga*'ritnms be fappofedmade, as he has very carefullycomputedthem^ tQ Fourteen Places,the firftftepalone iscapableto givethe Logatithmf any intermediate Flumber true to all the pjaptôf thole Tables.

After the fame manner may tticDifferencc of the laid twd

Logarithmse very fitlyppliedo find the Logarithmsf Prime

^timbers,havingthe Logarithmsof the two next Numbers above

and below them : For the Difference of the Ratio of 4 to

f 1:ând of ^ z. to ^. is the Ratio of ab to i zx^ and the half of thae

Ratio is that of y 4 1^to i z^ or of the Geometrical Mean to thef

Arithmetical.



f(fron/ht0ii^bieLogarithm, contraJ.

tj

;fc^ ifeL ^^ confequentlyhe Logarithmhereofwillethe Half Difierencc of the LogarWims orthofe RMthttest .

*,

** ** ** X* *-

Wh 5htsa 7lrj - fgood difpatcho findthe LowritHmof**;2

I^^f^ * *r^ rauch more advantageouflyerfbrra'dhi a

L? *.+?*+J^*^ ' ' J**^Difference of Its Terms

tti^î* A 5-^^*'^'^ f ^4-^ *=$*;,, Which in the

K?.ciw ^ ^*H tbî-ogwithm.f Prime Numbers, isalways

Unity;and

caffinghe Sunôf the Terms i i ;t+**= y uthe Logarithmof tte A.r/, of v-* to i^b orAViUbe found

ffie^rS IV^^*^^ ^-

-y t^.-. hitherto

Here note, tbats aflalonga^died to ada^tthefe itolei

M^Â ^11f Logarithtas.f m be loooo c; it may be neg^

^^lî 7 V''' ?*''*^'Logarithms,. wL'hintlk

Jj%e but if yoa defireJfrÂ.ogarithms;hich are nb ^generallyeceived, you muft Divide your Series,by

'

Or Multiplyt by the Reciprocalhereof -vit,

^

all^thV'ifrnf5 îgft*?.MoWplicatiottwhich is more thatl

SlcatorT^l^'l?'= ^), expedientto Diride this Multi-

SISSf %2*|P^^ of X. or V continnaUyaccotdingto the

^^r.**^.***'^'*^'^'I^Ûy where *iVfmaHandWer,

barJî.*^ ,***^ ^^ *** '' a** d together,,rhes yoi

ofwhSJlSltK^f /ogarithmo as many i?igareis you defirf

,

7J'' fnethod I wiU girea Specimen. ,

*

nroL?. ^'yf * y Gentleman,that hat kifcr*,wotild 3 .' n^Jw'take to do the Logarithmsof all PrimeNumbers under laxoo to 25 or 30 Figures,I dare aflTurehini

2n ii.? '5^ ^** ^ '' * f H i ^te l n thereto 5.-norS /Sf fK^,' * '^.*^ê dcflred.

.

And to encourage ;him^ I

t^cB ^}^ ogarithmsof the firftPrime Number^ under 20^

Û^f^rJ^^^K^^^'^^^f^^ ê accurate Pen pf Mr. Abrâm

tiÎ^^

T ^^^^

Induftrynd Capacitythe World may inurae expert great PerformancesVastheywere communicated tome by oar eommon Friend,UxhuclidSpeidaS.

d Kunt.



J 8 JIfnHalleyxo?npen wsnd Facik Method

Hum. Logarithm.

0^771 î 1(47 1 9^6iwi37t9.50 79-on(f1 1^|a92O0X.ft88tf4-tO^ft^-^^tô^4To9-8043ai4i5^.82O7x.uxî.f859^^^î9*3483T-7M9^-3M9i*54^'^^3f

9

7*

o4{4fo9.8043ai4i5^.8jO7X.uxî.f859^^3^l9*3483T-7M9^-3 39^*54^'^3^3fT X 1.041 3f.1^8 5 1

. 58115.0407^.ox 999-71 ^3*oi4 4-X70^-oiX90b4M4îff .f^ 39

13 x.xx394.t35 3.o^}^.7^9L(x6fon*5794^3^43-o 97- ^x^ 3870^-83b7X

X |844*9 i378 7|^9i854.ox^98.943* -3370S 07 *^37 4 f04^397M7fen-)M9-92SaS.9^XT3.tf3)M.7575^-9293l*79(tM9337 m4^7f9H^

The next Prime Namber is 13, which I will*take ôran Ek

ample of the êgoing Doftrine ; and by the firftRoles, the

log^ritnmf the lUtio of 22 to 23 will be found to be either

22 9 8+3ip44 9j7024T 257 8itfo**

23+10^8+3(550i+ixi93tf4'32i i7i5As likewife that of the Rstio of 23 to 241 by a like Procefs*

A ^

J^

_

'

t

^ftr

23 1058''5501 1119354*18171$ ^

24 ' 152' 41472' 1 327104**'9813^^0*^

And, this is the Refolt of the Doftrine of Mncatw^ as im ^

proved by the Learned Dn JKîKA But by the fecond Tbcanm^

mix^ ^+1X5+7^ ^- The lame Logarithmsare obtained bf

fewer ftcps: To wit,212 2

45T273375TP22640525T615585171875*^*

22 2 2

47+3114^9+14*725035''54*3^1843241^*

Which was invented and demonflxated in theHyperbolickSpaces

Anatogous to the Logarithms,by the Excellent HLv^JamesGnianîn his Exirci$atim$s Geometric^ând fince further profecutedSythe afordaid Mr. SfiiJUttn a late Treatife in Englifity him

puUiihedon this Subjeft.But the Demonftration,as I conceive,

tirds never tillnow perfefted,ithout the Confideration of the

IfyferbcU hich^ in a matter purelyAridimetical,s this is,an*

Jiot b properlye applied. But what follows,I think, I may

more juftlyaim as my own, vU. That the Logarithm of the

Jisiia of the Geometrical Mean to the Arithmeticai, between 22

and 24, or of ^ 528 to 23, will be found to be either

iop+ 1 1 19354+8821 5334+525487882248^* *^

i0S7+3S4i7j JS7P+tf55KJ7 J58485^85



30 Mr. HaUey' Qmfen^mi 4nd Facik Mftboi

Froin the Logarithmg;ivexi,o find what JUr^ itejcpreSt, h

a Problem that has not be^n to much (onfideredas the former*

but which is (blved with die like eafe,and demonftrated \yj:like Procefs,from the lame generalTheonm of Mr. JtTtwt^nt

For as the Logarithmof the.ti^th of i to r-^ was proredto be

i4-9h-î,andthatoftheie rfoofto i^^f^tobei i-^'frSothe Logarithm, which we wiU from hraoeferth call L, bein^

giv^n,i+L will be equaltoi-f^l*n the one cafe; and i I,

will be equal to i f| *in the other : Confequently+L| will

be equal to i-f^,and i LI to i f i that is, accordii^go

Mr. Newtonh faid Rule, i+wL+t * LM't w^ L*-^^ * *L* -îtiiw^ L^ c. will be

= i\qând i w L+i w* L' i iw L'-^4* *L^ Yfeifi^ c. will be equalto i gf, w beingan^ infiniteIn-ex

whatlbever,which isa fulland generalProportionfrom the

Logarithm givento find the Number, I

rithm what itwill. But liNevMrh Logaritiplicationy m is fav'd (whlcn Multiplici

_

than the reducingthe other

5ffew to his)and the Series will bo

more ample, viz^ i+ LrH L L+i L*+ L*+Ti3 L^ c. or i^L

+i LL iL' +,^4L^ ri,L' c. This 5m /, efpeciallvn grâkNumbers, converges lb flowly that it were to be wiflicait could

l?eContrafted.

If one Term of the J?4f/^,hereof L isthe togarithm,e given,the other Term will be had eaClyby the feme Rule : For if L

vfcreNepalr\ogarithmof the Jf^n^ of ^ the lefler,to the greater

term, b would be the ProduS of a into i -fL^K LL^-i LLL i Ci^==^4 +t ^ LL+ i 4 L* c. But if b were given,a wouldLbe =b b L-Yi y LL~i b V c. Whence, by the helpof thft

OiilUdsjthe Number appertainingo any Logarithmwi^ be e*-

aSly had to* the utmoft extent of the Tables. If you feck the

neareft next Logarithm,whether greater or letter,nd call its

Number a if lefler,r b if greater than the givenL, and the

Difference thereof from the iaid nearelt Logarithm you call / ^

it vfrillfollow,that the Number anfweiingto the Logarithm Lwill be either 4 into i + '+ i'^ + t'^' +w/Hto'^ c. or

dfeînto I Z+i// 1/// + ^;^/* r^cij/'c. wherein as /

is lefs,the5m will converge the fwifter. And if thefirft

aoooo Logarithmice given to fourteen places,thefe is rarelyoccaiion for the three nrit Itepsof this Series to find the Number

to as many places.But as tor Hdcq's great C^non of loooop

Logarithms^hich is made but to ten places,there is fcarce ever

need for more than the firfbftep -j-^,or a-^vna I in one cafe,

or elfe b /,or b m bl ia the other, to have thfe Number true

to as many Figuresas thofe Logarithmsonfiftof. If



p^ findinghe ^Mber frm the L g^hm lipin;, 31

lifbtare Induftryfliallver produce LogarithmickTables to

imanymore ); kcev^ô# 1^ have tftei%tiieaforefaidTbetn-ems

will be of more Ufc to deduct the correfpondentNatural Num-

l)erso alltlieplace;thereof.

la order to make the firil;hittedferve al}Ufe^^ I wa? defirous to cpatrad this Series wherein all

the Powers of I are prefeai:,oto one^. whereia each alternate

Fower jnightbe.wantingbut found itneither ib Simpleor Uni-orm

as the other. Yet the ix Itepthereof .is Icoaceiv;e|oft

Commodious for Pradiê, and withal exadt enough for Numt^rs

not exceedingourteen Places fuch as are Mr. Briggs^targeTa^

ble of logarithms,and ttiereforerecommend It to conimon l^fe-

It isthus:,4^7;^ ^^'^Tllil/ ^ V ^^^ Kumber anfwering

to the JLcâfthm given^difieringkom the truth bat by one half

of the th rd $tep of die forinerSnio*, But that w hich rondos it

yet moi îgibte is thatith equal ciiicytfervesiocJir^'SôF

any other fort of Logarithms^With tHe only variation of ^vtitlt^'

înfteadof i, that is,^-^-j^jjand^-

. . . . .

m x*

4^-*/*and ^\'jr*j,T%hich are eafilyrefolvcd into Analogies;ife.

.

As. 43429 8rc- Jf ro 43429 -^î/::So is ^ ^ to the l^Jiim-;.

Or,As 43429 c.r^-t/ to 43^29 t''' So is b 5 ber fought.\

If more ftepsf this Series be defired^twill bt found as foUowsj^

44-\l^xi \ZiTi,^rl^ ^* ^^y ^^ jv^ demonHratcd,by

working oi^tthe Divifions iaeach ftep,nd coUedingthe Qôtesy

yi^hofeum wiU bf fiound to agr^ with oor. former Series..

-

Thixsy I ho]^)1 have deaf cd upthe DoArine of Log^triehms^

and fliewn their G nftruAioii and life independentfiromthe Hjf--

ferboUywboie Afiedions ba^vehitherto been,made ufe of for this

purpofe,tho' this be a matter purelyArithmetical nor properly'

demoaftrable from the

Principlesx

Geotnttrj.Nor have I bcea

obligedto have recoarfe -to the Method aflndivifibles,r th^

Arithmetick of Infinitess the whole beingno oiher than an eafie

QaroUryto Mr; Newton^^ General Thegrem for formingRoots and

Powers.

A.



13 JnEaSeandQmpenSottMuhdifmakit^Lojgaritb,

*

Of making Natural Logarithms.

Tothe making pi L g4ritbmsy

he Firft

thiqgrcquifite.is to find the Natnral Logarithmsof Two or Three of

the leaft and firftPrime Numbers^ viz^ pf 2^,3 and 5^

or rather 10, by the Reciprocal{ y(fh\ch'rigg/s(thatare th^moft ufefiilogarithms)are compos'd.

The Logarithmof i, isalwayso. That of 2, the next Prime,then is firftreqair'd bat to attempt to raUe that direAlyand

immediately,wonld be (b rery laboriousand tedious a Taslc

(mach more the greater Primes)that 'tis mcAre.expedienttq

nfe fuch Fradionate Numbers as liebetween i and 2, by the

Multiplicationhereof 2, s and 5 may be produc'd of which,in the defign'dethbd, thofe are molt convenient,whole Na-^

merators exceed the Denominators onlyby an Unite, fincehere*

by Multiplications whollyavoided.The Rule for makingthe Natural Logarithmsf luch impro-

^r Fradions,'ay be this:

t i.e.Thi To the Double of the Denominator^ add an Unitef; this (hall

2^5^be the Devilbr : The Excels of the Numerator above the'Deno^'

mm minator, in this Cafe always 1,000 c^^. is the Dividend : The

j^ntmt^^^^ ^f ^î* fimpleFraoion, composedof this Divilbr and

MttfT ividend, muft be raisM by a continual Diviiôn,illthe Series.

run out at fuch a Number of Figuress aye required but ( be-

caufe none but the Odd Powers are of life)alter the firft H-v

vifion lef that Qbotient,and allthe reft fiicceflivelyie divided

by the Squareof the firftivilbr. The Powers beingthus rais'd,

divide each rcfpedivelyy its proper Index, i. e. the i^*by i,'the z^ by 3, the 3* by 5, c. The Sum of all thefe Qpotientswill be the Natural Logarithmof the Fradion proposM.

Becatife the Logarithmsof three Prime NumMrs, 2, 3 and $,

.

are fought,hich are mutuallyfubfervient to the compoflngeach'

other, no fewer than Three Series can fuffice: Thercfdre Three

improperFraftions muft be cholen,in each of which two at lealk

of thefe Prime Numbers are ingredientof which, foch as come^

neareft 2, as ij^9i^ requirethe greateftabour in*raifingheirleveral Series ^ but from tbofe,once compleated,the Logarithmsof the deiired Primes are moft eafilyrauced : For thofe Fra^

dions,that approach nearer iôooc^c. (i.e.whofe Denomioa*

tors are greater)the Series are rais'd with lefs labour,tho' the

deducingthe Logarithmsof the Primes therefrom be a little

more intricate,nd infer many more Additions of Logarithms^but that beinga Trouble Icarce at allconfiderable,omparative-^

lywith that of makingthe Series hefe muft be fappos'dore

eligible.

I. The



HaU^' 23

e Seriesfer makingthe LogarithmDeviibr* the Saiure of =2 the

,5) i.ooocx .ooooo.ooooao

( 30CX30.0dOOOb C.

${}toScc(8oo oooo.

S5)Sooax(3a ooooo.

25) t38o8ic(xlt6.obooo.

25) 5 no ta 104.80CXX).

9$) 20480 8cc( 9.191O0.

II5;819I0 8ic( .3276800000.

i;}^i768ta.( ijiayiooo.3$) 131073 dec( 52.4288aoooooo

2f) 524288 fcfrC2.0917 .2d0000

25) 20917^2 to 8388.608000

25) 8388608 te 335*544320

25} 335544320 ( 134*21773

25) 13421773 (

25) 536871c

25) 21475 (

I

536871

21475

859

2^

TheOiiPtmcrs divided by x,3i5,fte

30ooo*ooQoaooooo.ooooo.qooooo

3) 266.66666^666*66666.666661

5) 6.4000a;0C

7) ,i828 .7428. 7i4?.857f41 9) 5688.88888.88888.t88889

iO i8.6i8i8.i8i8i.8i8i8a

13) .63015.38461.538461

15) 2i84.53333'333333

17) 77.I0117.64705919) 2.75941.052632

2U 9986*438005

23) 364.722087

25) 3-42X773

27)^ 497 o?

29} ft85i3

3O 693

33) 26

35).

%

4^ MMi

SttinIt tligNatfl Lofr or4s:a20273.25 540.54082.19098.900657

IL The Series for making the Logarithm of ^ ^ twice 3-fi =7 the

firftDiyifor^the Squareof 7=49 the Diviibr for the reft. N

7) ljOO00O.OO00O.O000OiO0000 000000-

(.14285.71428.57142185711^285714

49)H* (29 -545 8.95043.73 77 842566

49)29i5flu(5.9499a 18266.I986a772297

49) 5949 c C 21421^5678.90201.24025

49) 12142 ftc( 247.80932.22249.004903

49)24780980^ 5 05733*3 o66 306223

49) 5057^3 te-C .10321.08797.271556

49) 103210 c( 216.63444.842277

49) 210624 8k 4.29866.221271

49) 429866 c( 8772.780026

49) 877278oo26( 179.036327

49) 179036327 C

49) 3653803(49) 74567 (

3.65380J

7456?1522

The Odd Pcwers^ivUed by i,3,5 8cr

.14285.71428.57142.85714.285714

3) 97.i8i72.98347.9io59.28o855

5) 1.X8998.0365.23972. 154459

7) I734^525 55743*034322

9) 27.53436.9i36i*x 545

lO 45975-7555 W82384

13) 793 92984 405504

The Sam h Che katuntLogl0/?=5^

17)

X9)

- 21}

25)^7)

29)

14.04229.656152

.25286.248310

461.725264

8-525539

.158861

t

~

IftSeries t= |Ud Serie s f

4 r= ill -^ 2d :si 2

5 = ift 4- 4tb s= 3

6 =:: 2d -i-5th.or=4-^4=:^

iflttpaS-ntd Series = i6 4-3=5

5C 4 -h 7 =i

hO.J4384.i0362 25890.46371

a90279 2554a54oS2 19098.900657

0.14384.1036x25 89a4637 1.960949

o*34 ^57-85902.79972.65470.86i0.54930.61443.34054.84569.76226}

o693i4.7i8ov5^945-?094t.7232i2

0.11157.17756.571047788.31474o.8o47u89562.i705o.i8730.037967

1.15X29.25464.970224200.899571



^4 ^^ ^fi andCmênAoiiMitl^bdtfakingiogdnibms,

trriThe Series for the Logarithmofi of V ; twicp4+1 ^=9 the

firftDivifor,the Squareof 9=8 1 the Divifor for allthe reli.

p)*'

f.OOCO0.O0OOO.OO0O0.0OOO0,OC}OC5OO

^ C.titiMuii.ttiii.tijiiaiiiii

Ir)ii*c. i37.i742i.^482.853^2.3^939680i37i c.(i.69350.878o8*43o28.6Jiio48f)16935 c( ^090.75 158112876.8^7174*fii)20907 C.C5.81174,79171.349718* 80 2581 17 c( 3lS66ij'J54') 3l493- 1) 5186635 c ' 3?H 79'5li9i3'*

8p3934 7 c.f 4-85^93*9l}49^2

81) 4856939 c. 5996.216975'81) 599^5216975C 74.oa737i

81)74027371 C .9t39i8

809 39t8C 1128380 11283C 130

^tVddfomcrs divided by i.3|,5;te..iiiii.iiiii.iim/iiju.iuin

3) 45-72473-7o827.6i774.i9799

5) .33870.i7S 5i.686o5.7342C7 2 8.67879.^.j26li28i689) 2-86797.iy9o7.9244ilO ?896.94ii4b.484o85i%y 36.2tf:^44-58245*V 32379.57i 5 $4

.17). 352.718646

-

1-9). J.896177

21J43520

*3), 490

^

thr Sum 1. the Natural Log, oi i=--^~.iU i7,nTi6.'i7io^pii.,,'

^

.

~~~

; ,Thofc Thfee improperFraatons,whofe Denominators, pre-

fijme,re the greateU that Can be found* whichare. canable ofeffeaingthis are ^S ^, j^j which ihallbe pitchedupolas

ano-herInftance.

.t

1. The Series for i,V,dr |^.Twice 15-30, 30+1=51 the firftDivifor,the ikiuareof 3 1 3:9 Ji is Pivilbr for the reft.

. 31) 1*0000 00000.00000.00006.000000 ften Powers divided br i.a Ae:

)60o3a c( 3 3567r4474o.8i75i.^3^tf999tft)335 c( 349* y432*59M7.733i289 0 34909 *cX 36346.9^19487756961) 363469 c( 37.82202.309 6i961) 378220 C.C 3935.694391

Sf60 393569 c.( 4. Jo 4i6960409S4l6f '

A362

032l5.8otf45.t6i29 j3225.3) i 89a 6i573'40583.84 2,5), 69.85886.51825.54662

7) 5192^2345.541108

9}

4 20244.7ojo6a ^

- 357*790399

3J

31503*M 28A

W4 ^54I0C-

A262 ,4- '-gT

The S m i theHatu ltx.iôf4^'^^,j-i6

IJ.The Seriesfor i^r,or H. Twice 14^8, 48+1=49 the firftDivifor,and.the Squareof 4^ra:24oi is Divifoff^he reft.

4P) i'00ooo.ooooo.o o ,poo90rf oxx 4heÂWoiwr* aividedb? 1*9 *e. (. '2040.8i632u553o6;i2 44.8979590id4O,Si4â^ r^,iJl:SJ:

T^ ft ?i^,l''^* '*7^^J*3^5^ 7'08626^3;al82.

a40 J,3540i c( t474.44 3 895936 .7) aio fiilliRôIU8401) 14744 ikftC X44460182 ..%), Is^,^

..

2401)61409 e( 85 ;766i8 ^ i

IIHÎJ,., 2401 a55766.8f -

.

^\o6 2 f, '''325147

,^Î oe um ., m Mawrt Loit. o^ If = O20î.oo ,f^0, 27- S6477.7288. ',



m. TKeSerie fbr if^dtji. TwifieScvor i^+irsiî isthe

firftDivilbr,the Squareof i6i =25921 is Diviibr tor the reft.

161) t.ooooo.o6o6o c (f

.*( ôo6aMtSox.l4i25.66245.* J72b$

i592i)oo6 c(396.19618.25323^53X4.5 25921) 2396i]8cc( 9144422739*220060

: 959^2.1)9244?Sec .$5663.081641

2592O 35^563 c.,

. 1.375837

The OU ftmert divided by 1,3 ^fia^/x 62i.i i8ei.^42l3 6o248*4472o{

3 ) 798.7320-5.07774.59461%5) i848.84547*8440i2

1)^

5094.725948

9 .152871

2^y20i37^S37.

'53 ii).

-^

liie Soin. Is the NJitgratLog, of rf ^00621 12599.99278.57665.564656

I rf: tirii ^ -:.*0322(f.^0 5.68785.58583.5461 95'

2\ :*:i| =^ sfSc= .O204i.09972-tfot i7-5 ^477-7i8853

a =ji = flj= .00511.12599.99278.57555.55455

lHi^+3^=4 = J = r?8 = .05889.15178.28191.72725.93970

i-l-2^+4' '=5 ^=i=i|S=.tii57.i775^-57i04.87788-3i4754

1+5'*' =5=:^=:|^= .14384.10352.25890.45371.950954 -^ 5^*

r= 7 nt i = i4 = .20273.25540.54082.19098.90055$

5'*.47'^ = 8.= ife =t; 2 = .34^57-3S902.7997:^- ^S470.85i5o

3 in 8'^= 9 ;=:

8 1.03972.07708.39917.95412.584817

jikJ. ^th^ ,^_. fj Log.to 1.151294514^49702^.84200.899571

If greater Exadaers be defir'd,art Seriet maft be adtoitted;

No fewer than Four -wiU be fuificieatf the leaft Nambers be

(i'')S,hen the (2*)pay be f^,he (3O ^, (4 ) I i then the

going Operation.

Odier Four f, may be. (i)Jf,2)|4,3)^, (4)fg.

* Five Srrjf/ will be requirM, if tht?Icaft Numhcrs be (i^ )

If.'^)|;.3')^. )JS.d );::thea (6. )|i^- | +

c.

Six Stries will be neceflary,when the leaft Numbers are :

(,) 2?,(a) l i,(3)ii ,4) ?51,(5) HI, (5) ?| ,7)il + il99 2o' ^

^S^ 224' ^^^ 124* ^ '^ 9^0' '^ 125^224

-:^8o* ^^ 80 ^99

99* 120-^24* ^^^

1^4 Sii* 960 -^307*'^ -^

24

4.a5_32ilî6c. j^, before.'24 3072 15'

.....



j An fniiuleatfaiitotiiltthoitfiitMigtj

M the tiWiibn IH M^iMaMd, 1 lllM#lfe* lb tbe MHuiM

6f Siriet.

.

i^m, TharaUthe ngaiiffiOperatkms, repreibitedl rFnSioBs, are Apim'd to oe peitomrd br Liwiiitlims.

But this work hUTiiigeen ilrtKlj

aecomplillKd by the Indaftryof Mr.

jihr, Siitrf,need not be further iA-

fiftedon; he has commnnicated th4

HyperbolkfcLoprithm Of lo, wbkh

is the double of this,bf Um compi*

ted and eoaSrmed 1 a tnpple mof

to82

Figures;and its

Reoipracalfbeing

Tlf Reciprocal*/ic A' r4 ^. vi^

o.8is858.8/S38.0CjO3.C5)30.2l578

Miltillfijtit Ktttrtt Ltg. /i,Tiz.

.

Ill .i7'?5 i?i04.87788.3l4'y;5

/ 8 Sj88si 3.8o so.3 553.0gf8.s88)l .380 5.O3 sSJ 2?;8

68 .8{88$i 38as,sa35;.j3ot43.4JS44.8iil03.2Jl82.7(Ssij

tf,o8ol2.274tftf4525.5871Iff

. 8 85.88si 3.8o 5o.3 5 30

ffo8o.l2274.fftf455.25587l

o8.oi227.4 45.5255S4

4}.42?44'8i;o3-25 28

9.21153.37828.390219

j|.342j.44 l(i.0325ltfoSa 1 2274.5^45 J2

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dculateobyhatIngenioos SfrMMn,andIiKlefattgabIefathematUMm,

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and for aH^P R 1 ME SUM B E R^ from loo to 2oo.

Mn jtifr^jSlMpt littU*H$rtmirear Bradfordin Tori/hire.

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|0- \hEalkaudOaiêndimuMftk icfmal^

Ths.Mst Work is taflieir,howMr.Srigg/tLogarithmsiboftt

$jc re immediatelycomposMf for vfhich fe?erai Rnle^ pixf be

laid dovra : That which is lAx^ mqers^ aad eafie,and comeft

lieareftth former^ for th^ Nat0ri(rLogarithffl89s this :

I. Let the Numbert wholb Lo^rithmis fimghttbioAeiêr of

t^ other two Mambers jêxt it,jgêater or le(s by an Unite

( the Logarithms of one pr both ox which are given) he madt

an improperFradion ; to the Efeaomimitor doubiled,add i^thdt (hall be the firftDiviibr, and the Square of diat mnft di^

vide the firftQiiotient,and all the reft : The Qividend mult

always be th^ Reciprocal of the Mataral Lcpeir^t^mdl

ioss8 858.89^38.o$^ n.tf553o,22 78.37t23-c. AU lê re^

nwft be as isdireded tot mông Natural Logarithms i imly th^Sum of the Series,or Logarithmof the Fraftion,when the pwAlilnmberisleftthan that ibaght,moft be 4dded ^ itsLogarithm(when greater, iUbtra^ from ic* x. Or.

,The Soriesto maktBrin/s Loôf251,. take 250, die tiettlefi a

inake the Fradion |^ whofe Logaritlmiisfirftibught9 x 250^

I te5ot b the firftPivUbr, the Squarethereof S25 loor divides

an oie reft,as in the feflowingperation.

The odd Ptmrs dtvtded bf 1 13 1Ac^01) 86858.89638.o 503{)3o.22{784( 00173.37105.06599.80769.521409

^51001) 733 8a (^9^7185.65503.646478

ts loox) 6907185 c:C 27.51855.831266

251001) 275i85583i266( 10.963525251001) i09635a^( u

.O0i73,37io5 p6599-8o7^.5 4o4

3) 23/ ij95.2i834 548m

7) M ^S^9)

.

Tff;;*= 00173 97^28 O9O00.52976 bo2fii

950 SI t.39794j00o86 72O37 6o957 252A2l

251 = 2.399^y^17g 4*8ioi8*i19î OJi9n^^

11. The fecond method may be this : If two Nombevs, nezi;^

that whoie L0g. is ftmght feitfaerne greater, and the other left;

or both greater, or both lefs;have known iJgarithmi,he Squarqof the middle Number Ihall be the Mamerator of the improper

l^radioQi the Produd of the othw two, on each fide,die Oeno^,

mnator.

For ^4i f/f.Take 2399 its Square^7121, and the Prodoft of

258 and 240=57120^ whence theFra^n J^îHI,nd their Sum

114241, the firftDiviibr,and itsSquare 1 3^05ioo((o8ipitUoi:%an the reft,asin the foUowingSeries.

.1

114241)86858.89638 o6503.6553a225784 1 Tke Odd Peven divided bf 1. 1^



teiuedfm Mr. fialBefsectSi Difiotnfi,tittte ^. of the middte Nondter be ibdgiit,he Z r. of tii6

Foîoii,addfed to theJLi^.f thecreateft and'leaft,filbe the

fig.f the Squaref the middle Namber ; diirhalf of which ii

ml^gMrUmif as in die feUowingoric i

^40 k a8 = jjrito (3)= S7 = 4 75 78 8t9 7'68n7'97740.23(4)= 4^H* =aooooo 76o3i.a8t 7.j936a

|9X S39 =: 57t3t (5)s= 5^181= 4.75^9 5 oiMi7^370oo.3ft8aii Ac Log. of $7t2x =s Log.of 2^9 = a.37839 7900948i37'^85oawi65ttf

Ifthe I^. of the greateftr leaftNumber be ibaght,fafatraf^tiieLtg.ofthe FraSao^ from the

Z^.of the Sdaare of the mid-le

Mnmber,the Remainder mall be uie Lig.of theProdti^f the

tdiHt two ; firom which iiibtraddie L g.of the known Nomberdie Remainder will be die ^.of the other.

Soppofehe L^g. dF 259) 240 given,and 238 iboght,then

(5)-\4)=(3 -(0î i if the I^x.of 139, 138 given,and 240

Iboght,hen r5 - '4;=('3 H'aXi;-

ni. The Third Method may be : I^d fnch a Prodad of th^Momber^(wkofeLogarithm isioDght)hoTe Fadors have knowa

Logftritbois,hich mafl be greater oi^ lefiby an Unite iJian aao '

ther Namber compofed df inch as haT known Logarithms thefe

two fliallmake the Fradion whofe Logarithmss to be direftlj^lomgbttacGordiio the prelcriptioasn the former,and tw

L( arithmdefir'd deduc'd thence, as in the ficlbmethod. For

Infcinceâke 2271 which drawn into 27 and 3 if produces

8M99/the Fraftion is USSiSj the firftDirHor 2^1^9999^'tab37999i^ the Squareof that 14439924XDO01 divides alltbe t^ '

if9999}96i$2.i96^to6^m.6^^ô.2i^^^Hie Odd PMm dividedbt if ih

( ooooa2a857.6tf44624460499970 I ooooa92857 6tf44 944^0499I44 9i4 x t)a g^7 cfi^8a.949077' 1) i;.^:gtt

fiHH ^ .axxxikS2S57. d44 S49 8-49^^

19000Q 3: 5.2^ 75.36oc9. 282 96t^3.ît89999 5= S.a7 7f i3 S^^382'7i^054

2^ H 31 = 837 = WS272.54579-93a59*99i55- 787 t

327 :: 2.35692.585713ii2.7Wcoi3Q489V

Aaodidr InftanceibaO be to IUi the UigariUwiof au 3

an 2iiJixiix2ii =i^82n5 i41TTheam 39tf42388 K whtck

28x55Xii3Xi97=:si98aii9440j ^ '^ * : y iHiwai, nuiw

litthe mft Divifion Qpotes the L#(.of the Fradion M 15;places.

39 4t}888t)868588963806503'e (OiiOoeooôoooft.i9io6 i087.180802571

6 rt a 5ttii3Xi97==gt98aH9 40g=9 297i2 98209 7i6644944*84^44^8396

TH4A Psnr of 2n=:i982i 1944%= 9*897 ^98 i190770.66031.62^25.1969

f of vUEh Is tbe L of sii = 2.32428.24552.97692.665



^2 An EafieidCmpendiousttBbd dfmdi^ ÎjogarkSms^

' This laftMethod may ordiaarilybe rfender'd as .Udiwi'ial as

tfaefirftvnd moft xa ^ and EXipedittotishaa thb. condi aD

the difficultyeingin findingat.proper Nambers..

-Thfemetbod

I commonly us'd,which rarelyhih, is.here fubjqjaULietter.

tjaderftoody the Performance,than exprefs'd*

^

Fof 223 X 7 = M^^ Again, 3 ^ 2^3*

= ^^9

223 X 87 = 19401 223 in 13 .=. 3^92'

=.

446 7.c?

iSgi

223 X (7x41=^)287:=:4001 131 ^3 71.3 = i5 JV9

For 229 (t = 229 in 9 = 2otfi */

3 68t -; 6 r= 1374..

31 c=T099 ^?== ^S8o.

I = 229' 8 = 1832

^31=29999 8 J9=s: 199001

.0 *2061

^^

.7ixi39==98^9 2260001

H?re two convenient Fradions are difcovercd,or makingthe

u

)

^ i^40oo i58W9=3iX23Xasf3

Logarithmsorgopoo . 2260001=71x139x229

(^^^ 9999^131x429 226oooo=2Xii3k oooa

jtntthtr Expedienterfindingumbers^ acceimm^ddte to the Third

RjUc or Method efmakingBrlggs'/ogdrithrtts^ . *

Find fttchaProdnd of theNoinbet, whpfe

Logarithmis ughe(thegiâtir the better)which hath two Humbert neareft it,oa

both or either fide,compos'dof fuch whofe Logarithms arc

known i Square the Middle, that flialle the Numerator} the

Produft of the two on either fide,is the Denominator of the

Fraftion , the Logarithmwhereof is to be made accordingo

the Rules there laid down, onlyobfervingin which the fought

Number is ingredientif in the Numerator, the Logarithm of

the Frsuôn mult be added to the Logarithmof the Dedomioa*

tor -,butifin the Denominator, it mull be fubtraftcd from the

Logarithmof the Namerator, c.

Convenient FraiHons foundforRdifinghe Logarithmsf iu^ 223,

227^ 229^,235, 239 241, asr, and 257..

Of 211 211 XI 1=2321 75387041 121X11 1X211

2321t=:S4X43,80x29=23^0 5387040=^0x54x29X43

Of 223 100X29=2900 84042OT itf9H223X223

2898rri4XjX23i223HI 3=^899 5 8404200=2900^125^23.Of



CfmMaig hogarittmlyMukipKcatim. ;)

Of UT 20x193=38^0 y i48996oos=4JQo 93X 1 93

3851=27x153, 17x227=3859 J i4899599=Z7^J43*ti7X227

Of 229 229X3 1 =7099 ^ 5 W80 #3X3 1 X229X229

7100=71x1009 14x39x13=7098 S 50395800*100x71x42x1 3x1 J

Of 133 20x19x29=11020 %tai44Q4QQ 4O0Xl9XI9X29X29

11021=103x107x233x43=11019 5 i2i4403P9 43XiO3XiO7X233

Of 239 700x14=9800J9604000O =c49X49X40OOQ

S^80i=l2ix8l9 239x41=97995 96039999=81x121x41x239

Of 241 241 X41 =9881 ^97^^416141 X41 X241 X241

9882=9x18x519520x19=9880 397634i6o=i).oxi3Xi9X9x;8x5Of 251 24oxi37 3288o 0 108 1 C944Qca40X240X 1 37X 137

3288i'2Si)CT3i nX49X5w32879Siogio94399 ix49x51x1 31x25?

Of 257 257x47*^2079 J 1^002241 47x47x257x257

12080^80x1519 18x1 ix5i 12078 i 145902240=80x18x11x51x151

And for a Proof of the Ijogarithmsf 251 and 257,

43X151=10793 ^116488849::=;43X43X2^ 1X2^1

10794*257x42, 8x19x7^=^^0792 5 116488848=8x19x71x42x257

TiKe ] fiEaqpedieSiCs,prcfiimesina;e found for any other Prime i

^ ^ IIIII I I 1 1

^Sffatherfftrtntethod nf ntiikiniLogarithms) communicdted byMr. Abr. Sharp, derivedfrom Dr. WaHis'j IHuftrationof Mer-

catorsQsiadrature of the

Hyperbola,in Philof.

Tranladions*N*' 38. whirein thi greatffiart qf the Warkj (viz. R^fi^X att

t^ht tmerf) is ftrfrmfd JyMuIlfipUcaUQa,keiu eafierd jf

picker JUffotehhm Divifion.^

I'ET any three Numbers in Arithmetical Progreffione pro^.

. pos'd^the kaft =A, the middle -cB the grcateft=E-

if the Logarithmof any one of thefe be given,he Lo^arithma

f# th other twxii ms^ be thus ohtatn'd,by an infinite Series i

I. Let th firftTerm of the Series be C=5i:^=^=^=^

H. TheSenes=-rTT ~ l-'~~*T+'~--r:r4'--- +

X 2*3 4*5 6* 7^^g 10 *

m. The Sum of aft the Odd Powers ( each b?bg (Jivfeiedy

i.ts.properndex) v/*..y+ j+ f+T* ^* *=^ ^^ *^ Hjper-



34 0/ ^^^^g LogarithmhyMultipUcatien.

IV. The Sam of the Even Powers, (eachdivided bf itsUdex)

viz. 7+^Vf+|-+^p'+c. = X is the HypcrbolickLoga*

rithm of^ gi

V. The Sum of all the Powers, or 2+^ ^^ ^^ Hyperbolicfc

.Logarithm of

-jr

VI. The Difference of the Odd and Even Powers, or 2 X is

Ethe Hyperbolicktogarithmf g^.

If B be= i^ TO, 100, looo, ioooo,c^r.all the Powers will be

rais'dby the Multiplicationf C continuallyor the HyperbolickLogarithms; or of C into the Number 0.43429448 c. and into

the feverai Produfts for Bri^s^sogarithmsj all which Powers

muft be divided by their refpcaiveIndices.

This Method hath this,eculiarAdvantageabove others.That

a Series one? rais'dfor the loweft Numbers in that Progrellion,willgenerallyerve for findinghe Logarithms of Eight or more

Prime Numbers, without any more labour than Addition or Sub-tradion^therefore isundoubtedlythe moft Expeditiousfor Com-

pofinga Table, efpeciallyor making the Logarithmsof the firlfc

Primes, tho' poffiblyot for railing fingleLogarithm.The Logarithmsof the firftPrimes, viz.. 2, 3, 5, c. muft un-voidably

be the Hyperbolickr Natural^ fince in allMethods of

railingogarithms,thefe ofier themfelves firft*,and from hence

inuft be deduc'd the Number .i^34Z9448c. which reduces them

to Brifg/s Amqrigft^varietyf Expedientsfor cffe ingthis,here ûpon one that is eafie,nd capal)lef a competentExad-

nefs,by Three Series,fince fewer will not performit to a toler-ble

Accuracy,wiôut greattediouihefs an4 difiiculty*

The Rrfi^hree Numbers are 96== A, ioo=B, i04=Ei thca

B A 100 ptf^^ 4 E B 100-7x04 4 E A

T loo^^^'T '^ 100 ^^^^^Ê+A^'I2 l2f=A=JL-C: So that ,04 =C is the Firft Term,

200 200 100 ^

wbofe Powers,e^c. nvtketh; ipirfteriqs.

The Secortd are 92=A, ioo=B, io8=E, and-~-

= =

' B '

1 00

B - A2{The T%ird are 975=A, TOoo;;pB,o25=E, and

g-= 2-=

^Qjj^cC,b PirftT^ of the Third îes.. ^ j^^^



s OfmJ i^Lagatkhm tyMuhtfticaiibit,the III* Series.

h^m^

C* = 625C^-zz 1562*;

O =: 97 55*i5 = 24414062s

C^ == 610351^^2'}C = 152587*90625

C'rr: 38146972656 - =:: 953 5743l6C 12=23841858Cm 596046C'*== 14901

atmx^

iC= 025

|C* = 0052083333333333333333

fC' =: 1953125

4c == 8719308035714

iC'=: 4238552517

By thdc Three Scries^tic

Loearithflisof allthe Pr^

under 29, eaccept11, may be

madi t and feveral abore :

For the Logarithm of 7 is

got from (IP^ looS =: tX)C

1 2x7 ; of 1 3 fit m(F*)1 04=:

8k 13, or .fttim(III*)75=275x13 1

of 17 frdm u'*)999tf=5x 7 X14H 17} of 19

from (III*)975=3 X 7 X25)c

19J

of25

from (II* 92=

4x23 i of 31 from (II*)92=32x31 ;

of 41 from (lU*)

1025=125x41, c. The Log.of 1 1 is had from a Series of

,01 9for 99=9x11, or from

TrC*'= 2167442 '--.'

^C* = 1I47 ^^ ^r 1001=27X13X11 }

the Log. of 29. is got from a^ 444=0250052102873306882090153

^|C*i= 0003125

^c^ = 9765625

rC^ = 406901041666667

JC = 1907348632^1

r5-C = 95367432

iVC = 49670-AC'^^^ 27

3C^A/ *Mfr ?ooo3i259769695#i87i947077 ing and provittgthe LOM-

?1i*Z?^79*'^^*9875403723oitnmi of all Primes under

1 1 00, and many above.

fpX 'i^o253

Z-Xr{444- 0246926125903715010143076

Series of 9O005 for iooo =i

3x5x23x29 i and 37 is had

from ,00 1 for 999=27 X 37,

c.

Expedientsof this kind

I have usM, both for find^

1

3

4

a

b

c

d

Jto4TT7

m .=

=^ 04o82a9945.SK:)a5 .i295 4577= 07696.i04ii.36i28.32498^2i70

= 03922,0713 1 53281.29626.92009

.0253

2,07131. Ji.78o79,84289.87540i27230

$704.t

\^\^ tl=|*S=- ^

8232.i5567.93954 6262i,i7iSo.

9 = 7^t-^ = 3ai-2b;-h20t^:.- ^-.^^., r.3 .yf :i-

10=9- 5 = 2a-^b+- ac^--2d=H=t== .28768.20724.5178a9t743.929aii=9rio=5a+3^r4c-l-4d=f=Log.2= .69314,71805.59945.30941.72321*2= 9 -r = Log.3 = 1.09861.22886.68109.69139.5245213 I ^ Log. 8 -zz. 2.07944.15416.79835.92825.16964

**^^b *lr*^ - ^^'* ^^ 2.30258.50929.94045.68401.79914. 54684-36420.76ou.oi488.62877.2976o,33327.9cx 96.7572lUe Reciprocalwtercof It .43429.44819^3251.82705.11289-*I89i6i6o5o8 22943.97oo5*80366 6566i,i4453.783ij6.i36

Shafl



âlt here ofier One of the Three Eit^HMlitilt^,hereby theft

iirftrimes, together With many others, were cottiputedto the

Exaftneis of 82 t hces which was bv Six Seriesjwhereof jOdi

wks the lowed:) whence the'Loprithniitf theieTea FraâjBS

were made%the

followingOjâtlons beinglht posMto be

per *fbrm'd by Lx^rithms.

^ ^100a

V*y99x2 )iR3t

v5^soooDOo

W 99^9^8=62X127X127v5)

999,5:2:14X42X17W.iooooo

/; 9975t=rt5X3SXi9 W 99875=125X17X47^^

9997 5=^7? X3 X43

/^1000008=72X17X19x43

.

iy S.

lOoS tQOQ^39 9= ^^ '^^

^^^îproooo ^^^)iooo iboo* 992 3*75^125X31

V ^^ l0ooooo1000000

999998 3961:15128X3*^^3*75'~*39 * ^

128-

^

1008,1008

,10000 43ac=j6Xa7

r V

locoitf,

100000Uz i \

OQof ô^ \

10000^432=101127 0lOTOio

Ilouow .

425 ^' ^ 1000 1 1000 I 9996 425=25x17 ^^J^joooooT 99875 ^-

icooD_256=i6Xi6.

^4? ^^_8i '. 100000 .\io^^

9975 255=15x17 ^-'4^5 255 ^'^ 999yt~ loooooo

3875 12X17x19'

J.looo 3876^^^jj^562p::i25Xi2$^

. ISîl^

5875=^25x31^*^ 995'^3875^ 550*=H8 yx*9^^ 5504

256_^2il21=^25X625 - ^78125 10000^65525=15x7x625^

^55 '^77824=64x64x19^^^^

77824 9975 65536=256x256^**/

128 .128 ^656^5 _2i,

21 iOo8_^25f^^^

8I25^^16ii?544i ?l=^ /j3,xE _ 22 =2i(^3î2;x2i= {j^

125t

1251 65536 20 v ^20 1000 24 ^' ^''r25^24 15

v*^

IS+if^ll^^c. as ia the former Operations.

It renttins to Exemplify b the making rWs Logarithni*

immediatelyy this method i Take the Three Nombers in the

II* Series,w;t.92=A, ioo= B^ io8=E,and--^=^=,o8=Ci

by which multiplyingthe Reciprocalof the HypetbolickLoga^

rithm of 10 t^, 43429448AC'=N continaally,nd dividing the

tefpeftiTeiPowers by their proper Indices, the followingSeriet

arc made#

OlNs3^H29f ^^



38 Of mdlaiôgatitbrn

CiH743 c*C* tli^27l9A iA6Uîoiii697

C X27794fcc*C Nsaaaj S 7747344^493^

Cxa2235Msc-C^xNn778 70 9757?7i95C xx77897*CiC*N=t4*3096i 58^0057$

Cx

Cx

C xV829( e==*'*xN= ^

Cx466j20i492=C**xN= 373056119

C X 3730561i9=C x N= 29844489

Cx 09844489= C ^ N:s= 2387559

C X 2387559 = C X N = 191005

C X 191005 =C'*xN = i$ 8oCx 15180 ssC'^xN =:= 1222

Cxi222gC'^xN = 98

,C xN = O347435S S5 6oi40 i2

*C X N = 74 959 578i54978

fC*x N = 2846192316601x5

4C^xNr=: t30l 6487589

iC* X N = 6476668738

,VC*-vN= 33954193

TVC' xNr= 183658.,4C*'xN=: ioi8

tVC'^xN^ 6

fc=V^'||= 034817964070697216507

4C xN*43429*c.* X389742342090405848

4C-*x N = 4447*75494689299

iC* X N = 18974615444008

jC^x N = 91078x54131

jVC xN = 466320x49

xN= 2487040 , *xN = 13643

tVC'*xN=: 76

y-4-8t = 0013942085837475x4194

^g===i'5#=.0334237554869497023i3Z+X'A^^^ .O362X2I7265444473070X

,C=:.oo xJ*=oo3474355855226oxA62i

JC xN=: 74XX959X578155

jC^ xNr= 2846x923174C'xN s X30112

*CixN=_6

1

^Z=:y'm = 00347442997^g39iS2xx

^fP% N = 0000x3897423420904058 4C^xN = 444717549469

iC*xN = X89746X5

jtC xN.= 911

Z X= t8* ^^ 0034605 3 2 X09500486X 5 8

Z+-X H^= 00^488327845*2x344264

tflt'^-99 = -9Wii^72i54i78655736

Lof. 92 =s i*9 37878273455S5^9299

f MtlttfScatteiC

Hence 'tts evidenti that

afterthe firftSeriesfor 1 08

and 91 is Compos'd, with

how much eale all the o-

thers are thence derivM*even with littlemore labour

than tranfcribingHave in*

ftanc'd in the next higher

i. e. making C:::.oo8)hence

the Log *of ioo8=:i44 7*

and of 992=32x31, are ob*

tain'd. Putting 0= .0008

the Logarithmsof 10008 s

72x139, and of 9992= 8 ^

1249, are got ; if C be put

f 00008, the Logarithmsof xoooo8= 2i6 X 463^ and

of 99992=8 X 29 X 431, are

made ^ and ifC be .000008,

the Logarithmsof iooco68t=72 X 17 X 19 X 43, and of

999992 = 8 X49 X 2551, are

found : So that the Loga^

rithms of Ten Primes are

obtamed from thb One Se *

ries,viz.. 3, 23, % 3^1 ^39^

1249, 4*3 43^ 43fând2551. Many other Senes

are as proUfickas this %

none I have met with,

whence fewer than Six, or

Four at leaft,ufeful Lo-arithms

may not be de-uced*

So that, though

the labour in

railingthe

firft Series may be Coa-

fiderable,et the advan-age

of gaining fo many

Logarithmsthence^with fo

great eafe,makes abundant

Compenfation.

Shan



0/ mfifahoganthmJyMvItifUcation*

*fi

FB(

P .

ill

CO

39

1900) N 4342 ftcrr 00922857604310697464^132171900) 22S576 Cr= i2 ^03i8o5826x82348c

1900J120303 ace = ^33174^34^4535911900)633174 accrr 3332498077081

Ji9a))335249 * -= 1753^*3$^

1900) 1 75394 c=: 923 120

1900)923130lfc = 486

I 00022857604310697464613217

3} 6391 8cc= 21105821154845305; 17539 cp 35078927*

7)486= 6j W MN Mpl

1 = Zr= ^f}^ n ,00022^^57606421279930887087

2)iao3 c.-ooooooo6bi15902913091 1746

4)33324fcc,= 833x345192^70

2= X = v^|4i*aff'-ooooooo6oi5i598622 55848653= Z X X= fiH^=.O0O2286362i58ii42i8647i952

4^= 1900=3 .27 7536oo95282896i536i33475=4-3= 1899=3 .278524964737017539^71^1^=

^

9= .95424250943932487459005J817=15-6^=211 2 .324282455297692665081558148=:i-2*Z-X^4|$4^=.000228515912614176753022229=4-f-8= 1901=3 .278982116865443 i3828935 69

19000) 4342 c.:^ 0000228576043106974646132^211058211548A

J50f1 Z=i/TrSvig=^ 000022857^0433180^28580314

60151590201309117

2= X=: 000000000601515903746215693 i-V^Zi-X*Wi^ 00002285820584770703201883

4= 19000= 4-27875360095282896153633347

5*4-3=18999= 4 .278730742746981254504314646= 9= 954242509439324874590055817rr5-6=2iii= 3 324488233307656379914258838=i-2 Z-X:= 000022857002815899539587459=4''t-8:ri9ooi:4 #27877645795564486107592092

190000)4342^3= 00000228576043106974646132

3)= 21 105821ij

1 Z=v^4|J5J4=^ooooo228576o43io9o8522) QOOO0000000601515902913091

4) 8331

2 x= 0000000000C601 5 15902921422

3=

z l-XffJ||5|oQ000228576644624988i496694= 190000= 5 .27875360095282896153623347

5=4-3 i89999= 5 .2787513x5x8638271165483678

6=27x31.^37= 2 92272545799325999 55X7 78i

7r=:5.6=r227= 2 q56o25857t93X22720iQ3Q489:[Z-X= ;fIl4-i=ooooo22857544i593 182306825

190000= 5 .27875360C95282896153633347190001 5 .27875588670724489335940172

i 7rr=^ $45O98o400i425683g7 22i6it6

i 27143= 4 .43365784669298806264718546Tlic



40 Of mMng LogMrttimsyMukipHemm.

The way to find convenient Nambers by whkh Logarithms

may be made withOat any other Divifion tlianhf the Indices of

the Powers. If the Predion be intended to beginwith an Unit

and CyphersfoUowingtake fuch a Prodad: of the Number pro-

pofedas begins,ith an Unit ( and Cyphers ifit may be ) iixb^

trad foch Prodnds from that fuccefllvelys beginwith the fiimc

lîgure;r the next le($th^n thjitwhich immediatelyows the

Unit or Cyphets iibtradinghjisFador alwaysfrom the former9

Bmltiply*dy lo, lOo, c. till a convenient Number befoood.

Ifit be delTgn^do cooSSi as mncfa as majbe of nines chode fiich

a Fa as ofthe propofedomber as begms withôr isdie neareft

to nine, but lefs,dd fuch Produds to that fucceflively,s willmake the foUowingFiguress near as poflibleinest c. Ex. gr*

Eaak i

forftîla4noo4.

Tubt. X 2fT

aixi94fl'iooi49

lUnk 3

251x3*7^3

9a22 9

39 97 9

8 2oo8

lUak 3

te3{7ia4 xo^8

39 ioc 23 o8 20^6

aak4

V1T\

8 ao5tf

38-97^

3*9P'99975

5= X2$$

5X797P 3985 iooo2]{,

398*99898 28zt39 389a000244

Ofcs 1506 .0 r= 19|6 X 23 z 2887 398406 999999Q^ 5 5PXI3 i9pc?3B9ici=99999S'

I

Rtoki*

^1004=4x251J 1000

/ôoU9=aiXi9H25l

V.100000

tank2^ 99898=2x199x251

v 999984=48x83x25110023 = 39x257

( lOOOO

Jj1000244=28x1 39X257

I*** J\ toooooo

, I y 10000127=1*7x233x257

' 00*

0QI4t

TheFDwen of

\jSxSktie RaiifedI

(iplicadoiif

looooooo

.00102

000016

oba3

^00244

ooooi^$7

^5

,-5

Tho' this Method feem to be more confined,ot adniittiagf

f^ great varietyas the other, yet at the beginningof th^ Table

where the other is deficient,tis moft Q mmodious aqd Expedi-

tious,and performs well to 1000 or upwards, an4 affi rdsex-

cellQptExpedientsfor many great Primes \ but where itfaiU,tbere

the other becomes more convenient, and performswith greater

lê.Expeditionand Exaftnefs than before : So that both togetherrender thisArt, Fix.. Loiaritbmcte^hni^,olj;compktt*



The Vfi ofthe Tahk ofLogarithmsn pag. 28. 29.

O^

H

5JSZ^

0 S- 2 Sum

Tit



'42 neVfe ofthe TMe of Logmtbas in pag.ii.

29.

I I III mil isIP iii III

^ a^ I fid's

s-ISSISa

t^ t-O

S, 0.0, 1=1? If

1?'

3 ' V

S ji 3 I

2s l5 4 ? t5 il = iB = s^u



4 0/Confiru^hiieTUgtrntMl

Definitions.

A Chord ar Subtenfe is a Right-LineoaJ^ ne iag the Extremities of an Arc, aa

Iv^arf. F 0 is the Chord of the Arcs F 0 and

F D O. A Sine ( ss s ) is half the Chord,or a Right-Linedrawn from one end of an

Arc fallingerpendicularlyn the Diameter that terminates in

the other end, F R is the Sine of the Arcs F E^ F D. The

Radius ( ri r ) is the Semidiameter,or Sine of 90* i.ooo. c.

thfegreatcftof allSines. The Co-fine ( = c s) is that part of

the Radius which is interceptedbetwixt the Center and the

Sine, or the Sine of the Complement, / r. the differenceof the

Arc from a Quadrant or 90^, as C R=sF W isthe Go-fine of the

Arc FE, or the Sine of its Complement FB, and alfo of its

Supplement or Difference from a Seimicircle A ^ E ^ is the

double of the Co-fine of the Axi D /8 or the Chord t fits Sup-lement

of iSE,fo Eo', ES ,E^, Ew, are the double Co-finesof the Arcs D7, D^, Dft,Dh, The Vcrfcd Sine ( = ) of

any Arc le^ than 90 *,is the Excefs of the Radius above the

Co-fine of the Arc, as E R E C C R is the Verfed Sine of the

Arc F B A Veried Sine of an Arc greater then 90* is the

Sum of the Radius and the Co-fine, as D R=D Gf C R is the

Veried Sine of F D The Secant of an Arc is a Right-Line

drawn from the Center through one end of the Arc tillit meet

with the Tdngentjwhich is a Right-Line touching the Circle

at the Extremity of that Diameter which cuts the other end of

the Arc, lb C H is the Sceofxtjand E H the Tsn^entof the Arc

F E. The CQ-ftcam or dt^tMigwt of an Arc are the Secant or

Tangent of the Complement of that Arc to 90, lb C I is the

Co-fttmt and B I the CQ-^t^gtntf the Arc F .

A Mitbod #/comfittinghi NiUand Sin*^Tangentor Secant efanyjtrck immediatelyprem the lengthf the Arch beinggiven.

THE lengthof any Arch is readilyobtainM from the Pro*

portion of the Diameter of the Circle to its Circum**

ftrence exhibited by Fen Cenlen fipce prolonged and con*

firm'd to 74 placesby Abraham Sharf which is as 1,00, c. to

i.i4i5*atf535 9793i3Mitf43383a79502884l97itfp$9937S^^5^

^^974944592307816405 z |-.This Number the Radius beingi.oooo^ diCf is the juitlengthof the SemichvH or Arch m

i8o^



Tdi^tnisidSec4iitStyMr. Sharp. 45

tU^ ocs whence any leftArch iseafilyot by DiTifion,jig

part hereof ebo29o8882o8tftff72T961 539484^14 ^ is the-

feDgthlf the Arch of i* Minote, which beingmoitiply'4y the

Kumber of Minates contaiaM iaany

other Arch (erves

readilyto give Its length,hence by Mr. Nmt t^% Series publifiiedyMr. HiUUy in fhi. TrsaifN* 2 1 9. The Sine,Co-fine,Tangent,C^. of any Arch are iud. If the lengthof the Arch be putsA,then is the Na^ral

Sine_

A^ I A. 1 A'

-^A'

.A?

_

An.

'^6

120 5040' 352880 399HJ800

A J__

A y _i A 7___

6227020800 1307^74368000 '355587428096000 X2164510-

A 9.

0408832000

r^fin.. I

A,

I A* A*. A A.,

U)-unc_iJ -t-^^

720^*40320

3628800 T^7po.A'*

_

A',

A'*__

A

01600 87I7829120O T 20922789888000 6402373705728000

^ 2432902008 I 76640000.

Tan nt- A4- A

,2 A' .17 AV62 AS 1382 A''

.

Tangent == A i-J -f^^ +315-^2835+15592$ '+

21844 A' . 9295^9 AI 6404582. A

. -L,

6081075 ^^^38512875 ^10854718875 T c^ ^'

r'A-i.- r- *

i. a A' 2 A' 1 A' a A

Co.tangent=j~-A-^ ~945 47*5 ~W5SS

_

1382 A}*_

4 A _ 3 S'7 A''_ rf.-

638512875 18243^25 162820783125 '

Se i - ^ 1 4- I.A*4, sAVtfi A*. 277 AV 50511 A*'rsecant as x^~ i-^ -^^^^ -1-8064+3 S288oo +

540553 A*I 199350981 A'* 4,3878302429A** 12404-

95800320 * 8717829120 '4184557977600 ^^6402-

879661671 AJ. ^.

173705728000^ ^ '^*

r^frran^ -J.*Al 7 AV 3 A*. 127 A', 73

Co-ftcantx:^+ +350 +15120 +6048;o +3411440A'l 1414477 A'*

.8191 A'V 118518239

653837184000 '37362124800 '5335311421440000

Let the Sine and Co-fine of o* 05* be foagbt,the lengthof the

Arch of 0 05' is =5:ODi45444i0433286o798^77=A.9^



4^ OfConftruHinghe Natural Sines^

Ifthe Sine and Co^fine.oix95'be required,he nttm.cfMinutcontained therein is 1795'which drawn into ,60190888, tfr.

make the lengthof the Arch = A =,522i443J45'549702(J509j

A*=

A

A*

Q01454441043 32860798077

aix$39874.8$88097130767227628517$

The n wen of the Arth of

29' 55'

A'nT . K^ 2726.3470.6107.8527.152946622

aS .00 14544410433 2860798077

x20)\ 5423716?

.00145444104332915035539

51278712714196j--6)A^

The Sine of?odeg.05'^-00145444053054202321343

*h^ f'24^ .00000000000018645466105

2)A*= .00000105769937425940486720)A*

1^25^

1O0000105 769937425941801Co-finc o dcg. 999998942300821 9 524304

05'or Sine ot 89 deg. 55'^

A*r .i423-54^^-7197-2746.43276A^* .0731.00206837,695871A^x.0388.10822854.4101.535A** .0202.6485.1272.8468.215A][*0105.8117.7282.7x60.4

A *.oo5i.2490.i77i.Oyi8.A'r .0028.8479.6158.7483.7A' ^.ooi5.o627.997o.6364A *.ooo78649.5552.92159A'*t=: 4M 066.4io7 .A''= 2.1442598404A ^=

X'1196.1312.69

A;y= $845-995

K?7 3052454A ^=:

1593.82

The Seriesfor the Sine. The Seriesforthe Co-fiac;the Pwifs dfdvm im the Co^egicimsrftteSerbs.

A= 522x44334554970265095323423523786751280

36288o)A'

7949724864276622762o8oo)A** 34434763

355 ^87428o9 c)A 3 45 V .5224^77^60285x63x5459

6)A = 023725777866212440546504o)A^= 2099439937046834399x68oo)A' = 19703371838X307674368oo)A' = 44705

'^

023727877325852903923

^^6^1?-4987398887026634XX536

i-^AS 1.003097070123938x3427624

40320)A 1 37026333608431

47900x6oo)A 85723366

2092278988800^'* 1459

j- 1.00309720715x129077831

,

^^'^*=f,;'363X7353053926' 72o)A* 28145626767842807

36288oo)A'4150903799x5

87i7829i200)A*'*x284^8o

SubCrafttbeSloeof

There remains the Sine of

- 36345499q957S58646i4

The Co-fineof 29* y' or the Sine of6bdeg.5'

.86675x7080553422x^217Out of which fubtra^he Sine of o de^05i00x4544405305S?2xI

One of the Sine of 89deg.55'= 9999989423008x2x^29^* 55'= 49873988 7026634ii53630 deg.05'= 501259053598148783707

Since thefe Series Ck)nvergethe fwifteft near the beginningtnd end of the Quadrant, for raifingTable, no more than the

firftand lafl:thirtyDegrees need be calculated,he intermedi-teare obtained from them by Subtraftiononlya^ above.

'

Tab. i



\

w -

^ .

Tangentsnd Secants yMr. Sharp. 47

Tab- 1. 7te Qjj^tkfiesftbefeowers divided

by the Co-^kients/tbe two Seriesfarthe

Sine Md Co-fine.

Jib.t. The Towers ofthe Arthofodeg.oi'==^ .ooo29bS88c =5 a.

*f 3437-74^77 ^7S493925a6ic4

t a, .0002908882086657215965395% a, (7)8461594994075^3^^7074293.a (10)246137821028138337774644 a* (14)7159^5^98437591417400705 a* fi7;2o82785542642824237546^ a* (11)605838269404798592936037 a^ (^4)176231208928302709019428 a*

( g)5i26358o676i48497209095^ ** (nJH9ii97 52 ^75537447U ':,..,

^^^' (3043377i65762767O273O969^ 8-\-)a *(40ii9535840395 3224oi6i

.00029088820866572x59615395

(7 ) 423079749703761943571 5

(11) 410229701713563896291(15)2983274576823297572500

(19) 173559S795220235353125

72o)Vf24) 841442040839998045744

5040)a^ (28) 349665097079965692499

40320)4*32) 1271418171531460744273

362880)4^(37)41093394931314862895

a*

ja^

a

iia (383126179060457273697238i2a '(41)367040008675401318839*3*''(45)106767610^32238403745614a '\49)30574^900033 10804620

15*' ^(53)903424279655122726604

^'^*' t5 ?J26279547037399856i429X7a'^(6o)76444io,622558i5i3279

*^ * *(^3}2223668836582273969S7ipa

* '(67)64683904459206586781

2oa***(70)i88i5785096io56798222

399i6-l-)a*(46)31610514985488239849

47yo|-)a' (5i)6626050659413521550

6ii7J-)a'55) 171458573949581802801

87i7-r)a'^C6o;5625198168980720354

i307^-|-)i'Hj65)69oS63338582409io6ii-t-)a*(69)125602499370660678801.

355681-)**(74)21491932968x581817614

,64oi3f '*f79)3473i943788i7388o4oiii^4-j-)a'\84)31742784843x80879995

a43i54-)a (89)77338853076977

.Tab.3. The Logmthmsofiheje

Qu^tiefiP.

X

T 3.536273S827928158479613

a 96.4637261172071841520387

^1*92.6264222387503871088637Ja 88.61 30^710 12379088236073a^j 84.4746932271171305852186

a* s 80.239449339988295932470

7^.915024206818364520009

71-543652294^47637733274

67.1042884 1 42200043 397249

6261377202198786^617173558.0774981391950477692023

53.49983x571244006880500848.884376442403566204817044.2341592073039135876492*6,,^, pgpO

39 ' 1i7572888328597i3762Jj,,,7 ; .ioo34-839392i469843626237i94',j.7674i i o3ox)9899828i53562i994903 io#il7i9^c.

Tab. 4. The Logarithmsftbe Coefficientar*

Bions of the two Senes oftbe Sine and Co*

fine^iz. oftbe Heeiprocalsf theDiviforSf

which are to be Med to the Logarithmftbe

Powers*

r

14

I

7101

a

yo4o

I

4o9 20

t

261 XSeO

I

} 99 I 6 So o

479ooi 0O'

99.6989700043360188047863

99.221878749616356367491298.6197S87582883939770638

97.9208187599523751722775

97.1426675035^8735397687

96.2975694635544747090565

95-394479476562531123415594440236967X232062488252

93.4402569671232062488252

92.398844281964981208075091.3196630359173563803525

90205719683610519611146089-0595916479322815852201

.87 8855003888766003431388

86#67038o4c622c6755622838

i5,33227' 4773 545322i840i6,, l,4 c.54489314848424016337436

2o.540729o894684i030o6365' 4..}7,7 c.^i936589797j9o955 J9398IV72 70I60572276549 3 9....4,... - 82,9i490'i5787862 6 pa4O35

lo.8883977272659684479 539a4jIi' .i 8i fii3875383ia2285407 8?

A littleto facilitatehe Operationn the precedingTable,

riie Powers of the Arch of one Minatc, and their Quotients,

being



41 Of CmftruSiy the Natural Sines^

beiagdivided b^ the re^MctiFe -effitieatsof each Member of

both the Series for the Sine and Go-fine,and the LoîthiM of

thofe Quotients and Co-efficients,o that the Powers of the

Nnmber of Minutes contained in zny Arch beingmnitiîedre*

Ibedivelyby thole Qpotiehts, produce the fereral Members of

tne Series,whereby the Sine and Go-fine are Gompofed.Thefe Tables need no Explicationtach Table and Number

as fiiras needful having a proper Title perfixM,nly the fmall

Figuresin thele two Tables, enclofed in a Parenthefis denote

the Number of Cyphers that mnft precede the Srft Figure of

the followingumber.

The lift of the Firft Table isprincipallyo compoft the Se^cond, though therebyin the Tangent, Go-tangent,Secant,and

Co-ftcant of i' may be ealilymade from their proper Series but

the Sine and Go-fine moft readilyrom the 2d to 23 Figures: Bat

the chiefDefi^n and Ufe of the Second Table is exprefi'dbove :

Shan exemplifyn makingthe Sine and Go-line of^^ 37 which

beingfo very near 4^ o muft neceflarilye as Troublefom and

Laborious as any that need be pro];)os'd.

The Number of Minutes contained in 44* 37' are 2^77, caH

this Number4, the Powers hereof muft be raifed,which (fince

i(sconfiftsonly of Four Figures)isperformedith much more

Eale and Expeditionhan the Powers of the Arch can be, which

mift confift of ib many Places as

are intended in the Sine, a due

Account muft be kept of the

Number of Figuresevery Power

extends to, tho' no more need

be exprefledn any, than are re-

Ottired in the Sine,fewer will

luiEce in moft j the reafbn is,

that after Multiplicationwith

the reipedlveNumbers, viz.

tîofewhich anfwer the (ame

ôwer in the SecondTable,

the

Number of Cyphers precedingthe firftFigureof eacn Produa

may be rightlyetermined i in

the adjoyningfmallTable of the

powers of 2tf77 the Number of

Figuresin each isexprefs'dyiball Figuresbefore it,enclosed

in a Parenthefis.

The Powers of S677 1= a bdag the

Namber of Miauut coocaia'ais

44 i^* 37 w'a.

(4) 2^77= ( 7 ) 7166^29

=: fii) 191842^2733= (14) 51356271336241

(i ) 137480718367117157(21) 3 ^9035936^8772689389(24J9S523220230168432 607(28; 26374666o5f6i6o89476a

(3O 706049810308842715294

I (35)^89009534219677

194S84; (38J 50597852310^75850705

,

(42) 135450450635496505234

C4j)36260085635x22413^

C45) 97068249245222700^ C52) 25985170922946117

(55) 695623009545267557 (59) 186218279655268

(62) 498506334637152

{66) 133450145782366

C69) 357246040259393

The



Tangentsnd Secants y Mr. iSharp; 49

Tfe P mm of 26jj dram into the re/j^Sivenmbers in Table iK

The Seriesfor the Sine. The Series for tlieCo-fine

1877 ^*A*,7787o;734598i367i29042i2a)A^ 2^%6ii403^7);956y2439

3^2880)/^' m 29015983696201226227020800)^ 1^:62171025743

3$5^^7.42So96o90)/V'^::^4002i9

j- 78i0941^^78295038675 59

6)A* .07869954378553230096215040U' 344501313064126849

399 68oo)A' ' 15994241686980

1307674368000JA'^179522015

1216451 C.A ^ 709

. 0787339955163408346174Taenneof 7023601432666095521385

44^4.37

i-\*^)A'**.oi532098586378468i623240320)A'r=: 33533220691 184292

4790oi6oo)A' * 103790330922}20922789888ooo)A 16

737199

24329 AcOA*** 28

+ I 01 '?3^^^9290545004^74

2)A* =:: .303 19286796148 io625Qtf720)A* 309880909602545790636288oo)A'* 22593413515736187i7829i200)A'^

,

34580756226402^7^7 /SccQA'^ 17:^1 j

,

,3035^25714648429333269C^/i^f44i.37711821767825702071370*

or tikifwtf o/'^5dcg.23'

A confidcrable part of the Labour both in raifinghe Powers and mul-iply

iiiglicm by the Numbers in ibc Second Tabic in tliisMethod, or byIbc Co-efiittentsf ihc TiVo Scriesin the former,may be faved,in worlc^

ingby the Logarithms,cr which the Third and Fourth Tablesmay be

VeryIcrviccable,lpeciallyhen the Powers afccnd high,as in the pre-eding

Example. The Charadlerifticksf the Logaritlimh the Third and

Fourth Tables confiftof Two Figures,

Jtc S'eHesf9rtheTgngenu^

The Serle$fortbe Oi iA^tni,

.

tVA 5I7477 38 rA=i74048in5.8|457i047 ^

6308902ITT

irJirfA'* 897090A'* 77025

A'^ 85ir

,

A*' 940

A 4-A^' ir6

Jfc r4 tt f^9 te 5. '575464016

+[A' 3i 534370488j

yfr^^,

8213930763

T7nA^ ^2239^026TTTlV^ 6167059

rr^-J+rfrrA.* i76229

470a

A'^-l-1^

rriTiYirA* '

Ateho* theSeriesforthe Tan- \semStiecam comrcrgc io flow-

Ty,hat except near the begin-ingof the Quadfant C where

theyare ofexcellentVfe)'twere

betterto make the Sin: and Co-

fineirfl,nd fix)m thence de-uce

them by thcfe known

Proportions,s the ^:^^'

lladius to

to fb is the

17729599069861m Ce^xmg.Df29drg. 55 737S8326879i6i

#r xhtTjvg.of60 deg.05*

lie Seriesfor the Cojccam of29 dcgi55'

i79S^C**''-^03437 7 c.:'in15 1791:5^4902*

'\-4A:=:ô8702405575932lJ,-A := 276800741772TTfr?A^ 7957245427

;rTVrIv,y22190727-jTrrwA'. 6155014

^mrH^^JÂ''.. 170146

iTrrrrrfr^Â** 4700

TfoCO'fecMftf29deg, 5 5'-g2.oo5051 803277';

or Secint if6(}d g'5'~

h fill



5 0/ ConfirvBijighe Natural Sines^

fo the Radius : to the{^J^'J.Yet the Two Scriesorthe O^tdnênt

and the Cs-fecdntre of much quickerifpatcb^s in the inftanceof (he

TMZ^nt and Qhungentand Ghftcum of 2^ ifjf.5'. The Powers of the

Arch of29 dig.55*

ot

1795'=

*S2214433455497ftc

sA

maybe feen

bcfote,nd need not be tepeatcd.o obtain A multiplythe Number of

Minutes, viz..1795 by ^'^ reciprocalf ,6002908 c. =3437,74677, to:-

TJtfVitrfedine of4 jr ^rîo under 90 dcg./ Âf differencefthe Radius and

theQ- finefthefM Arch;theSum oftheRadius and the Sine ofany Arch is

theVerfedine efan Arch fomuch exceeding0 dec. fothat theSeriesfortheSine and Ghfineay be ea/Sypplyto thefindinghePêrjedine ofar^ Arch

immediatelyhich isJoplainnd obviotuas^ needs no Illujfratioh.

I. The Sine of an Arc fR being given to find its

Figure Co- fine tR CFq^FRq=CRqj therefore ^ CFq^FRq :

= CR i. e. ^rr'^^ss: in ex.

2. The Sine of an Arc FR being given,to find Ef^ the Sine

of half the Arc. CR is found by i^ and confeqiientlyR ; then

^FRqÊRq : = F , but i f = EF^ therefore i j^SS-j-rr ;

1= S i Arc.-

3- To find the Sines of the double,triple,

uadruple,quin-uple, c. of

any Arc whofe Sine isgivenfiicceffively.et the

Chords DiS,^y^ y^^S^ i be allequal,draw the Chords Dy^B^, Da, Dw, and E$, , 3^,B, Bi extended, draw the

Jladius OS, and irake ^^ = D , 5^9= D^^ )t ^ Ih and 0\

t= /S,y/ut. ^, S^TT 3^,e^ == Eg, w(T t=i Efiythen are the

Triangles c^, D/S7,D C, D^^, S^ jEqâ^,SS^Tr,6^,

iSiij,ll Ifofceles and

-/Equiangular,a the

Triangles0\ Eyju^^nrhich being latrgeft,ll their parts are nioltdiftinguifhabie,he

the Angles EX^S,E/Jt^yre equal,the Angles D/S, Ey be*

ingfubtended by the iame diagonal )S do both togethermake

two rightAngles,fo alfodo the AnglesED ^ $D\ therefore the

AngresEy0, DX are cfqual,ut /Sa ^ Ey. and D = y by

Conftruftiori,herefore the TrianglesEy0^MD are equal,coci-

fequentlyhtn Eyi in the fame manner may be prov'dB/Jtî=

EK y-rr= a, ^^ = En likewife S^^ D/S, eO ri D , w = DS^.

Therefore QS : E8 : : D0 : D i-c-Radius : to double the Co-

fine of an Arc : : fo is the Chord of the Arc : to the Choid of

double the Arc, and halving the two laft Terms, : : \ D :

^Dy Co is the Sine of the Arc : to the Sine of twice the Arc,

againi) C(2: E^ : : Dy:Dî: iD :fDCi-e. r: ir^-rc::

s litre: s arc \s i arc^ (3)ck : E0: :\ D^ : i D0 i.e. r : 2 ex

arc ; : / 3 ^-^^' T^(irc ^ s kârcy 4) (/S : f /8 : : i Da : i Dk^

u e -



52 OfConftmBinghe Natural Sines^

^MQ^'i but becaufe GK =r ^, iGKq = A/j^, therefore CITh

^3 = ^4 and r (i.e.?^) -YGKxî -MPy that is,if.

to the Sine of an Arc left then 30^ the Sine of itsaefefl:aulti-

pliedby V3 ^^ added, the Sum wiH be the Sine of an Arc as.

much exceeding30^.The Sineof 1 1^ b min s= i 90808995375544multipliedy V3~h73 2.0508075688773roduces33049087453-

3349, which added to the Sineof ^9**o min.=,325568154457 155

makes ,656059028990504the Sine of 41*'o min. In the Trian-le

ha^ (^ aq^ zr:)/êg^r^ ahq^zie ^, but becaufe eg zâh^

3 e^q-rzehnd egx îz=:e(ji.t,ek^kh)g''g ^l^ el r=zkh

â zzifn^hat is,ifthe Sine of au Arc greater than 60 be mul-iplied

by V3? 2iJ^dut of the Produft the Sine of an Arc wantingfo much of 90 *be fubtrafted,he remainder isthe Sine of an Arc

fomuch exceeding30* Ex.Gr. ifthe Sine of 83'',9925461 5164-

1321 bedrawrfinto-y/3=i,732 c. and from the Produft fiTi9i-

4636349973 T,the Sine of 67'^=,92050485452439 be taken,there

remains ,798635510047292the Sine of 53 * min.

5. HavingaU the Sines under 60^ to find all the reft by AcWi-i

tion onily,oravingallabove 30**o find the Sines of the firft30*,

or havingthe Sinesof the firftand laft30**o findallthe interme-iate

by Subtraftion only. In the TriangleiWQfi,MK^QG (by

4) thereforeZMA['MK:=zSG^ that is,ifto the Sine of aay Arc2Mit{s then 60'*the Sine of the defeftKM be added, the Sum is

the Sine of an Arc fo much exceeding60**,x. gr. Sine 41**

=6560-

590289905C6-I-ine 19 * min. = 325568154457 1 55, makesSiae

79**omin.crSjSi627i8344766i.n the Triangleffe4,=r^ A (by

4) thereforeeg e i (i.e. A / )=j5^ the Sine of the Arc B ;?,that

is,from the Siiêf an Arc exceeding0^ fubtraft the Sine of the

Excefs,andthere ^yillremain the Sine of an Arc wantingfo muchof 60^ ejt;^r.Sine67^ ,920504853452439---Sine7'*=,693434.

05i47t=:S 53*^ 798635510047292. By a continued Bifedion

( byt|;ieeciDind the Sineof a^n Arc a littlelefsthen o^ i min. may

he found,and from that by Propprtionthe Sine of o* i mia But

the Sine of o^ i min. may be obtained from the lengthof itsAr? by

th^ Seriesin the other Method with incomparablyeftlabour and

greater accuracy, frona which (by the third)the Sines aikl Q -

finesofallArcs under 30^beiiigomputed,the reft^re had ( byihe 5) bySubtradionjor havingthe firftso^.made(bythe tkird)allto 60^

may be got (by4*^)and all the reftby Addkioa ( by

5th)-ôr thelaft30**beingobtained (bv third tc reftabove 3c?

are made (byfourth)and the firft30^by Su otrafticHiby fijfth.)The Sinesbcmg made the Taments Secants airehus obtained. The 7ri-

emglsFJtyVFy CEH, CBlm ifcquiangular,hen CR : RF:: CE : EHj i.a

Gh/he : Sine : : R^liw : Tangent,nd (f , i.c.)CtV : (^C,i.e.)WFxCBi Bl^

i.t.Sint:Cihfineixad: : QhtMngîCBCF: : CEiCH^ i. e. C^fineiRdd::

iW:5if4w,andCtV:Qr'*CB: 3i. Sine: Rod ; : Rnd : Qhfecant.A. M

-;

f



(53)

' I m

ill

AN EASY

Quadratureof the CIRCLE,

from 7 V 3 ^ V ^^ Cmmnunicated by Mr. HALLEXt

ProfeJJdrfGeometryn the UmverfitjfOxford.

MAoyare been tbe Attempts

in allAges(o

es^hibit

IqiiareEqual to the Ana of a Qrckt or which is all one, to find

th of the Diameter to (be Circtimfereace: This Arcbimen

dcs near looo Years fince Ihewed to be nearlya 7 to 22 ; and

confeqcientlyhe Area to the ctrcnmfcrib^dSquareas 11 to 14,

contentingimfelf in fmall and integerumbers : thoughbis Me-hod

were capable of extreme Exadnefik as has been once made

appear by the moft elaborate Calculus of Ludolffâneulen. This

Gentleman iyftbe continual

Bifedion of an Arch performedbfExtradion or the Square Root ( Analogous to Mr. Briggs^se*

tbod for making the firftLogarithm) carried his Work fo far

as to allureus, that the Diameter being i,

the Circumference was

3.i4i5.9265.3$89.7932.384^.2tf43.3832.7p50.288 fhe laft Fin

gure beingnot an Unite le than the Truth. And this was lookM

upon as fo valuablea Performance,hat it (lands Engrarenon hit

Tomb-Scone to perpetuatethe Memory thereof. However \t

liii|hte queftioned,hether it were reallyfo,unlefi by htm thaCr

had takcQ the painsto examine it throughout. And moft lovem

of thefe Matters have chofen rather to take it upon Credit,tha

givethem(elves that TrouUe.

Now fince his time,as there have been many abortive Eflayt

towards a perfedQjiadrature,y thofe that knew not enough ta

fee tbe impofllbiUtyhereof: So very much has been done towards

facilitatinghe Caknlus

byMethods far difieringrom that of Ar^

ibim^. And particularlyhe Dodrine of Fluxions^nd of infinite

Series,fajchmay not improperlye calledthe Geometry of Curve*

Vnes c bc^h invented by the moft Uluftrious Sir Ifaaceman

doth a R rdos many Solutions of this Problem. Amongft them ic

may not be improper to producethat,which of all performsic

with the leaft Work, derived from the Tangent of 30 Degrees

TzsV. ^^^ Fluxion of the Tangent of an Arch, beingto the

Fluxion of tbe Arch felf,as the Squareof the Secant to the

SqoareofttêKâdiuss isthus demonftrated.

Let



54 ^^ ^4y Qj^^^^^^F^f'^^ Circle rom^12.

Let C be the Center of a Circle,R the Riditi$=r,f R any Arch,

RT its Tangent == I, and CAT

its Secant. Draw the Line Csf

infinitelyear to C ^ T, and the

Line TPwill be the Fluxion of

the

Tangent = itand A a the p rrc

fpondentFluxioA of the Arch x:*ir.With the Center C, and Radios

C P draw the infinitelyittleArch

^P.Now ob fynitiarian ulaPiQ^Pii CT\ C jf,and again,JP :

jiai: dTx Ck ^CA. Wherefore tx 4i\m TPi As ::C T^z

Ci;^:cbatis. rr^-tt,istor r.asr to^. If

thereforer r

Vbedi-

vided by r r + ^ ^ the QiioticntiB be t --r^f ^ + lL

^* ^

+ Jll, c. == a the fluxion of the Arclv. \^ integralr flow-J

iDg Qiiantityill be the Arch it fclf,rfc.t

il-+21 _LL

jrr ^ri 7r*

+

^ , c. Now the Radios being nity,and theTangentof 30^

Degrees ^ ^, 'tisevident that ^yf

^ is thf Cnbe thereof,and

1 V-^^^J^f^^Po^cr- W^ '^^ 7^*^ Ppwc?, ift.in in^mn.

Whence 'cisobvions,that the Arch cf 30 gr. is = V -^-^ ~V -

'

v** 3 93

'

4$^

3 1S9^

3 ^3 9^

45 ^ 189 i 7^9

^c* Six times this Arch is the Semi-circomfererioeof theCircle,

whofe Radius is Unity, or the whole Circumference,hen cbq

Diameter is Unity. Therefore ^/mot^ jî î'^ L-J-J-

~

4 -7-1

*'^- 1*

^^1to the faid

CircoiQference.enêh?Rule

Divide the fquareRoot of 1 2 continuallyy 3, and the feveral

Quotientsagainby allthe Odd Numbers fucceffivelf,fis. the firft

quote by 3, the fecdod by 5, the third by 7, 6c. Then |^ the ^ 12

add the - of the fecond Qjiote,^ of the fourth, -i-of thp fixthJ

c^'c.in infimtum nd from the Sum fabftraA -ôf the firftQpoce^3 -^-^

1 of: the third, of the fifth. rr.in hfinitHnt^znAhe Remain-

dcr ihallbe the Circnmferencefought, Aa



( 55 )

An Exampleof tbis Procefs take as follows,he (qnareRoot of

12 being3 4 54i,oi5i,5i38

This Work beingto be perÂi*dn littlemore tbanh^lf an hours time

js more than fuffiaentto ocbibit the Circumference of the Globe of the

Earth to truly,as not to err the lêadtb of a Grain of Sand in the wholes

and the Compendium of this Method has temptedthe readyPen of the

8DftncomparablelAx.SHARPto continue it to double the famous

umber of Fan Cndtn. Which is a deêeof Exadhiefifar futpalTmill

belief;foritismore than (iifliciento givethe Number of Grains of Sand

that maybe comprehendedwithin the Spheref the fis'dStars,itbefne

greaterthanthe Cube of 1 2000 x 5280k 8000 x 1 00000 x ioooqo whicli

confiflsbut of 65 Places,takingall the Dimenfions with the moii So

that here y u have the DimenfioGrcnlijand the Artnarius of Archimtie,

both in one. Hence itapnears ,

that Fan CeuUns Number is true. And all

future

SjMorersf the Grch

may pleafeo fquareheir Work by tbisRule,

aixl not opoie Uiemfelves by obtrudingtheir falfercafoningn the

Worli*^

The



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Let the Tingent of 12 { Degrees = l /2*- i1)e^'i,41411^

35^23^7309596488o,t688'7* use Squarehereof ^4 is = i za^

the Cube a^^a-^zaa^ i.e. -= 5J z, the ^di Power a^r= 5a*

2 tf 4^ i.e. ::=: 5 a^ -^4 if 2^ t e. = 6 4* 4^therefore the yth

Power a^ rziSa^-^d^^ and the pcb Power 0? = tftf^ 4% ^ fo

that an theOdd

Powers maybe

raifed by multiplyioghe aext

lefiby 6^ and fiibtraftioghe preceding,this the folIowiaK were

made, then diridc each Power by its proper Index,

and lobcraft

every fecond,Vtt^ the 3dy7th,nth, 15th,19th, 6^. the remain-*

def is the lengthof the Ardi of 22 i Degrees,ri4. \ part of the

*Iht Odd Fawers of ibe Tmgtnt of

ii\ Degrees.

s:jp=: y-x=.4i 41 3^62573095048801a* =^ 121933088197564152490

a' 35 89374986230696634

a * 1056613342795 33064

a'^ 3110379273427542

a** 91561017003375

a*' 2695304687217

s' ' 79342362008all 2335621075

a ^ 68754265^41 2023937

^' J9579a^' 1754

The Affirmativeowers divided bjf

i) =.4142.1356.2373^50.48 80*17

5) = 24.3806.1763.9512*8304^98

9} 3988.t944.2914.5218.48

13) 81.27794944-5794.66

17) 1.8296.3486.672^.08

21) 436.0048.4287.a2

25) 10.7812.1874.89

29) 2735.9435' ^33) 70.7763-9^

37) 1.8582.23

41) 493.64

45) J 3*24

-

49) 3^

'Y 4166.9293.7604.2478.3371.19 239.93S5-5905-5236-7890 41

The Arch of 22| Degrees 3926.99o8.i698.7^i^548a78

Which drawB ioto8 itthe ihai^ri;pferlr=3.i4i5.9265,3589.7932.384

^ Odd Towers of the Tangentof-

22 1 Degrees.r

a = 710678118654752440084

a' 20920410530632474854

a''^ 615839386751704950

a' 18128618925493434

a' ' 533656715071820a' 15709386948432

a*^ 462444174871

a'* 13612997179

a' ^400729223

a** 11796367

a^^ 10222

a^ '

301

tboDgh theft Powers are not fo ca61y

aboYo biifiheNambcr m yeqaifrf.

The NegativeTowers divided if

3f 7, ,iS i5 s23 c

3)s: o236.8927.o62i.825a8i33t

7) 2.9886.3007.5804.6392.65

II) 559*%9*8795 6o95.15) 12.0857.4595.0328.96

19) 2808.7195.5300.96

23) 68.3016 8238.45

27J 1.71274509.21

31) 439 i289.4i

35) 11.4494.06

39) 3024 7r

43) 80.75

47J ^J^7

^___^

SI) 6

.- .0239.9385.5905.5236.789041

riiiMbU thofe V 1 2 fee not maoy



(^3)Let the Tangent of i^ deg.be 4-r 2 ] j ^0,167949x91^^

311227054725s + its Square aa is5=4 4~ i, the Cobc aaa^

âa-'-a^ It. =154 4; The 5th Power 4^= 15 4^

444^ c. = 154^ 5 4 -|-4,i.e. = iâ^ ^ : The 7th Power a^

144^

4%the

pthPower a^

= 144^ --4% c. fo that none bnithe Odd Powers need be made, any of which are raisM by multiplyûg the nextle by 14, and dedoaingthepreceding,re.Ht/Hpra.

The Affirmativeoimrs divided

^^5i9,i3,i7i2i,25,c.

2679.4919.243.i^27,o647 23*^$S

5)2.7624.3620.9291.3055.2742.925

9) 79. 1096.681 5.4774*3 1 20.40513) 2823.1797^.667,1116.338

17) 11.1286,5734 5 3^.46j

21) 464.3893.0263.968

25) 2,0108^x12^452

29) 89.3558.774

33) 4047 78r

18.610

7be Odd Powers.

a^2V3'*267949i9243i122706472553658a^ 1381218104645652763714625

a^ 7119870133929689083644a 36701336706724512389

a '. 189187174867319875a*' 97521753^543331

a** 5027028086313a 259131204444

a** 133576768a' 7 688558

a^' 3549

^ 2682.2622.9983.0S84.2969.368260

^ M. U t H^

64.2684.2183.9389.9315.5128.8

UfhM - . . t.*

* A'L*f \'it ?* 2^^7.9938-779M494-3$53-8553-JVUch drawn into 12 htticSemfer^benc:3-1415.9265.3589.79323846.2643.

37)41) 86

IP

The Negative Powers divided by

3) 0064.t262.8822.2801.9902.9437} 1416.6714.0283.6308.6242

3911) 4-^47 2433 i582.o86i.oi315) 175 669I.6l34.09l3w^87

19) 7148*96212040.83923) 30 4423-7775.247

27) 336.75^9-550

m 3O 6.0015.62535) 274.011

39) J.268'

43) 6

.0064.2684.2183.9389.935.5128.844

The railinghcfe Powers is ftillfomewhat more tronblcforo,etnot above one third part are reqnirM.

Tho' no other Method of obtaininghe Qpadratnreof the Circlecan be expcftcdcqoallinghat of Mr. HaBey^sby î 1 in FacilityndExadnefs,yet the Three

preceedingeduced from the ratnc

Prin*ciplcsmay perhapsexceed any other yet difcover'd,nd fcrve foraConfirmations far theyextend.

The

Tie Odd Powers.

%} 19237886466840597088304877a

'99 166998I98541603699876

a' 511183676474029471145a'* 2635037420113702305

13583028028775943a*' 70017468830689

a*7 360924377845a* '

1860484372

a'* 9590380a ' 49436

a'** 255



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ERRATA f^rOi ficmtEJuhiKfShsrviu'sMaihtnulbiMl'fshbt.

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(he dcg.Ih( miD.

X have

J Mall

thcDiffi

have carefullyxamined theLogarithmsnfwerin?

o allthe common Numbers from I to ioiooo,witE

. Differences,nd the Proportionalarts;and alio

the whole L(^;3Hthmickanon of J'jm/,Tdf^mtiand

StcAnti,iiththeirDiferences ; and have colledcd this

ErrattL,herein I believe I have not omitted the Cnr-

iC^on ofany Figureerringbove Unityn the lowcfi

Place;1 have alfocitamined the whole Canon of Natu-al

Sititt,an^ms and Secamt,and do believe the fame,

when thus corrcaed,willave fewer erroneous Figuresthan even rau Schoeten's,rimed at JmfierJamiz'i;altho'laidtobewithoutoneFault.

IVilliamGardmrr,Land-Surveyol

Note, Itmay henbfeTvtdlhatreat Care hat hern taken

in cerrtfUnghtPreflforhil Editim,bythefmallSur.

her cftiKVErrors;forin the Logarilbmsifthe commi

Numhert,thofecnlyre new, that have an jfltrifmefothem,tberefire Erreriin thefrfiEdition;andtberta

feveralether Emm in that Edition,ejideshefeetimed in it'sErrata,that were earreHed in the printirigf

.

fethat thereare cmflderablyevjerrrors pinteAin

thisthan in theformtTdition

NOBLEMENSand GENTLEMENS

ESTATESSurveyednd Plotted

AFTERaNEWMETHOD of his own Inven-ion,

which farexcelsin Eraftnefs allother hi-herto

ufed,and Books of Particularsdrawn therefrom.

By iVILL I ^M GARDINER, Land-SurvcyorWho may be heard of at Richard'sCogep-hnfeith-

'fen^le-bar,t Mr John Gardintrf. a Peruke-maker,

juflwithout 'fem}le-iarat Mr- Upton'sn iherrard-

fireei,ear Golden-fifuare,r at Mr Fotvke'san Engh

maker, in King-fireetfrfimin^trat allwhich Places

may be I'ecn Specimenof his fiiirMAPS.

N B. He takes Levch and caknUtes Rcveriions,=( .

The Publifhcrswill givea Book in Sheets to any Peribn that fhallfind

Two mcterial Faults in the (aid Tables.



Sum 1

05J002.^ 5i 2095 252$ 2957 338906 5980 6411 68437275 7706

07003.0295 0726 1 157 1588 2019

08 4605 5036 5467 5898 6328

09J 891219342772 0203 0633

1010 004, 3214 3644 4074 4504 4933

II 75 12 7941 8371 8800 9229

12D05. 1805 223426633092352113 6094 6523 6952 7380 7809

141006.0380I0808236 1664 2092

X5 4660 50885516 5944J637J16 8937 9365 9792 0219 06471 7 007. 321Q 3637 4064 4490 ^^yj18 7478 79048331$7^79184

19JJ08.742 2 168 2594 3020 3446

102J 6002 64276853 72J97JO^

21 009*0257068311081533 1959

22 4509493453595784^208

23 8756 9181 9605 0030 0454

24J0I0.OO0I3424848 4272 4696

25

27

28

29

7239

26011.1474

XO^O 83728794921596370059

3 1 013. 2587 3008 3429 385042732 6797 7218 7639 80598480

33 014. 1003 1424 18442264 2685

341_

5205 5625 6045 6465 6885

7662 8086

18972320

8510

2743

57046127655069737396

993103540776 1198 162

01 2. 41 541457^998420 5842

35] 9403 98230243 o66i 1082

36pi5.598 4017 4436 48555274

37 7788 8206 86:^5044 9462

38 oi6 1974 2392 2810 3229 3647

39 615565736991 74097827

1040 017. 0333 0751 J 168 15862003

41 450749245342 5759 ^i7^

42 8677 909495 1 1 9927 0344

43 018. 28433259 3676 4092 4508

441 7005I74218378253 8669

45 019* 1 163 15781994 2410 282546 5317 57326147^55626977

47' 946798820296 0711 1 1 26

48 020. 3613 40274442 4856 5270

49j 7755 8i69|S583997 9411

1000 000.0000 0434 0869 1303 X737

01 4341 4775 52085642 6076

02 86779111954499770411

03 001. 3009 3442 38754308 47404 73377770820286359067

N. icxxxD. L.OCX).

3l4

8933

3166

21712605

6510694308441277

5174I5607

9499993203640796

42534685821

8138I8569I900

245

6759|71063

12882

5363^

96590088

3950823S

2521

^7997227

10741501

5344|577i|6i98624 7051

9610003704630889x31638724298I47245150I5576

8x30

23846633

0878

5x20

0480

4692I5890

3105

7305

9881

4065

8245

242 X

6593

076X*

4925

3240

7392

198^

5

X90

U93

6223665270820517094713761429808

5237 56668666)90941952395 X

5793

0088

4379K8666

2949

7655

X928

8556

28097058

X303

090 X

XX3

21

3525

7725

193

X501 X920

5693 6XX2

^300|07

4483

8663

771X

5341

9108495009916

36564

7807

15401955

56846099(65

f238

3039

7377

1710

3473

78108244

21432576

6039I6472J69051228

1

33X3

762080515481

1924

3377

8981

323436597483

1727

55445967

9357 97800204I0627

359040134436

781882418664

2043 2%/i^

62646685I7107

X323

9742

3945

8144

x8

490.1

9080

28383256

70x0 7427

594

5757

071

8222

2369

13

LogaritEini8

3907

51x6

9432 986343x6-259

37441x74

235427844309-389

380542334284-171

48 9[52828664J90865092887133103732

75297951

5534 5955

4321

1 43

2 86

3-130

I4324--173

5548

8082I85XO

23552782I427

94079832

4084

79078332

2x512575

63916815

10501^23

X744 2x6$42i

0x620583

436585648984

2343 2759

653169507369

1x37

5319

9498

3673

7844

20x0

6173

4486

6927

06521x066

8

6376

4785420

3x78

D

344

433

7-302

8-346

426

425

42*

Pts.

5-2x6

428

1-43

2 86

3-128

5-2X4

6-257

7-300

8-342

9-385

424

X 422-8?

3-127

I4-170

422

42c

1 42

2-84

3-X26

4-168

4x9

57374x8

99x6

4090417

8260

24276589

03320747

4902

86379052

27843198

7341

1479

416

415

414

D

5-212

6-254

7-297

8-3III

9-3H2 P

5-2x0

6-252

7-294

8-336

9-3 7H

4x6

X 42

2 83

3-125

4-166

5-208

6-250

7-291

8-333

9-374

Pro.

MMWI



toioiooo.

1050021.189323072720.31343547

51 6027 6440 6854'726768052 022.01 57 0570 0983*1396808

53 4284469 55io9[552i933

54 8406 88 18 9230 9642 0054

55 023* 2525 293 53348 3759 417

56 6639 7050 7462178738284

57024.0750 1 161 1572.19822393

58 4857 5267 5678*608849859

1060025. 3059 3468 3878I428861

^i^

^

68

69I

896093709780*0190

_ .

^---46977154I7563972 S382 8791

62b26.i24516542063124722881

^3 533357416150^65586967641 9416 9824 0233b64i1049

65|o27496 3904 43 1 2

66 757279798387

67028.1644120512458

5713

9777

3838I4244I4649055 5461

91119516

1070029

71 7895

8300J870672030.194823532758

73 $99764026807

74 031.0043 0447 085 1

40854489^812385268930

77I032.21^72560 2963

76

L78 618865906993

79 033.02140617 1019

108081

82034.227383

84035.0293

1090

91

9S

96

97

989P

6285

8^1 42974698509886 '82988698 9098S7 036.2295 2695 309488 628966887087

89 037.0279 0678 1076

92 038.2226 2624 3022

93 620265996996

94 039.0173 057c 3967

J^7/7/i\ O

6119

0183

65266932

0590

4238 4640I542825786599060

2674

6686

0693

426546635062

8248 8646 9044

414145384934

810685028^^98

040*2066 2462 285813254

(602364196814977I0372I0767

2 I 3

W, 10500. L 021

47195127

87949201

2865 3272

0996

52965700

9333 9737

33 ^7377o

73967799

1422 1824

3075

7087

1094

0600

7339

1402

31633568

7211 7616

1256 1660

544458469462 9864

34773878

7487 7S88

1495 1895

54985898

949S9898

34943893

74^6 7885

1475I1874

54^05858

94429839

U193S17

73937791

1364 1761

5331

9294

721c

1162

5727

9690

3650

7605

1557

J;

553559426350

960900160423

367940864492

774581528558

180822142620

396143744787

8093850689192221 26343046

6345675871707582

0466087812891701

458249941540558176228

869591069517.99280339

28043214362540364446

69097319772981398549

10101419 182922392649

51075516592663356744

920c 9609 00180427 0836 409

3289

7375

1457

3698

7783

1865

581J7J6272J6678

9922032707323973 437S 4783802G 8425

2064 2468

6248665070520265 0667 1068

427946805081

828986909091

229626963096

6298 6698 7098

02970697 1097

429346925091

828486839082

625766557053

02370635 1033

4214 46125009

818^85858982

215S 25542951

61246520

8

5201 561441

93329745

34593S71

4107

81928600

2273

2872

61046508691273157719014005440947

----'^^-

417345764979,.8201 86049OO7J94092226 26.293031

2680

2113

D Pts.

2-82

79944123-124

411

410

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6757716-.

08301237

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30263432

7Q847489

11381543.

.51885592

88309234963^;

32773681

13501754

53825785

9812

3835433

22722671 307034.6S

6917

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404544414857

8001 8396'879i

9522347)2742

7453

1470

5482

9491

3497

78551871

5884

9892

7498

1496

5491

9481

7451

1431

5407

937y

334^

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1274

52-2

3^ .7

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789S

1896

988c

867

7849

1829

5804

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745

5-206

6-247.

7-288

8-330

405

404

40?

38974004-160

589c 3997-281

B98

397

412

4-165

9-367J

3-120

5-200

6-241

8-321

9-361

7709

1670

5628

91879582

;532

909im ?^

397

1 4Q

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396

395

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6-238J7-^78

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.



Nttm\

041. 5i)27 4322

7875

02 342-1816

9^1 x 84

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3786

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3239 3653

S';io8904

24442837

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343. 3623 4016

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1398

9311

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1084

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2835

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7-375

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9-314

34^3230

7141

12 346 IO4S6903 7292

S7SC

98763267 3657-

-

4,(5i84^2

378 . .

7682 8072

333

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r.2749 3i3S

7420 7809

1698

S^83

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S-234

7-27 J

S-3.2

g-3M

j88

i 120 049;ii8a2^6S 29^6

9929

23 D^c.379h

7663

6^ ,6S444 iSj

80498436

5384 577c 51^5

J624 DOIC

J476 ,861

2683

654'

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4.246463

8478

4894 5a8

9154

5504 6890

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387

38a

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42274612

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1936

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1321

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2706

63400

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355.3783

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56. 1423

5237

7987 8369 875c 9132

1421862567

.9S000638

9896 3278 0659

34013

7^23

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41 357(2856

4a

43 358.

i)8icoi9i 3572

J6-S33

8-309

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3-115

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7-268

8-3otf

?-34S

426c 464c

80,5

46 359.1846

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5634 5o:

2604

3639 67707148

3554 D932

^H

9 +495'

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2066

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3-1.4

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5^227

7-26^Si.

503

9-j4



Hiis Table of Proportionalarts is a Supplementto the

laft Coiumns (imitledPt5.Pro.)n the 5 precedingPages.PST^

109 to8

217217

ti? ta 83

3^6

.07106

143 143

178

Loraiithnu ofiiuf .ti]459andi }4f s. WeOkeitii

tavamage of thefeNtmben,

The Nuraber iimj bF^-^H-U^fbandifint,-t4k.iulnrhe *Hfi^ ^^

head--,_^._

Log. . 0914911, it thtM

Fjguiti)In

35

2^9 its

Ji8ji7

70

1Q5

14c t4C

'75 74

lie 09

144

180379

'HEbidexInthbTi-

UeorLosaiithiBtbo-4.34(3

li]||e ouc,itiii ]'''~tatfiidrupplied,n'a. Tlie lo-t

the Lotlbuiln^

lacainthe w[43|^

n r me man givei tbe Ol-

Sance of the hi^wft Place

in the Afafiilute Number

tmaUahy.

Hi*fn)tere a littlexni,we'll direft how n find the

169 II

4.33 13 8jia7 169 11

4^41 S3

IfdieLo^ofiiufftlw

ielaa,iiid far the

7 e (wUch in lUiE liiS) Aid it fai

Col. of PelPid*

ItainftwUdi Icnmftbadlviiled

By lOtinil

ne If. of

j?^w:3Si,S;

.

I2 166 IJ7 248 29c 33.,.,

^1 93 t24 i6f ao6 24 )289 330 )72

te403 *o Bi

JW7368-

367

j 5

34121

84:

Sa 123

ill

?3cTf, and^aioft 1134^

i*Hf. udoctt

ihenffl DthBCo-

fii wWdib

Ki,wfc ntheCb-Itutan tA n. luder

whidi J iJiij.'5y39|79|iL%i |w|H | Lfe

39 10 79

380 jfi76 1

d8c [30

.

8c

78

r

7Si

S9 72

8 77

77i'

.

7 1

7 .

7JII571215

7C in 2(5 197 ;4C s82

i 8 109 25

167 109 2;

,9i(;

l i|i^

S3

146287328)60- 28^327)68

244 384 325

iS i^t 197 2J7

57 i9 S335

.561

[48184184

36c 3^4

260303

2;6 299

256 198

290 33 m

5i

33^377

J7fi

;60 199139 79 3I9J5?

; ) 199 239 379 3183^839379318

23* 77

37a 3 10 U9

[369308 J46

199 230 169 307 34^

91 139 167 jaS 344

9C 139 267 J05 343

120338 166 J04 243

132765 JOS 34Q

,

a 6 264 J03 339

|87 32^3tia]oo]}7

,21150187334263399337

1^298335

Sc 397 134

358 395 339358 394 J31

194 13=

3 4J53

J53

303



N. 14000. L. 146.

i\ttm\

743c 7790 Bioo 1409 8719

r.0577388tf

1196 150^iSi^

3671 398a i29J \^99 t9oS

9p3j8?6+8b9^83267

^836*14.^ 54.^4

8308 8617 892 S?23'; 544

8 f 9 1 Pra.

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309 i n

6763

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07 148.394

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325c 3^58)867

769c 7999

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564359^2 726c: 7^697^77

09 9tic ?4i8 9726 303s J343

4484 *793

: }2324 263; 5-165

7-216

8-247

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83471^0.1433

5886 *193

f039 *347 t*?5 li

71167424779261

92 3499 : 8o72*5

,

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87

?2S7

7564 7871 8178 B485 ,9099 ?4od97t2 '9 532*

tf7 S2

9834

7069 ?37S

879:.

.,r. ..

1859 |2i fi47a 1779 Jo*5

9a4

79871048

)i89 349138

5246 6 ;2

9301 9^7

153. 2049 2354 26'i9

51005405

537 5843 5i'

i6oO 8906 321

i|U*66o t966 it\

Ui2 47t8 o24

7469 7774 ^

0523 8828

662 ,ip297234 me 7844

7-81 5

8-246

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154.1195

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15001804 I109 1413

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196729977 0381 586 69i

M74 5978

327

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29 155 0323 0626 ?93q 1234

l8o3 io6 9410 ^714 3018

53S 1843 2145I2449753 J0S7

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336c 3664 3968 1271 t57i'

1373077610

7 5340 D643

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5794 5097 i40C S703

18^4862178520

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J2804583 *886

7006 730876

8519 882a 9134 ?437 97?9 0032 3334;o*37^939

33157.1544 18472149*45

4568 487

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5474$77(S 6079*381615835985

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ibme Knowledge of Decimal Fradions ^ which, of all

Pradions, are mofl:Natqralt whole Numbers beingnothing elfe

in effed : So th^ Aritbmaick in whole Fiombers being uA46r-

ftood9 the Ufe of Decimal Fraftions is very eafilylearnt.

I. That the Nature of a Decimal Fraôn maybe conceived,

imaginea Foot Rule (or any other Meafiir?)to be divided intolo equalpart^êach divifion wiU be ^\ then imaRiAe every of

thofe Tenths to be divided into \o ciquaLarts^ then the Foot

(or- other meafore) will he divided V^ i ^^ equal parts^^

every fii DivifioA will be-^,or fSsi and every fecond pivifion^

ift refp^ to the whole, wiH beti$; fo that if 5 tenths and a

lialf were to be ^xpreft,it'sy^. By this means, an Hou^, i

Fathom, a Founds a Shilling,c. may be divided into 10, loo^

1 000, 10000, c. equalparts, at pleafore.IL A Decimal Fra ion hath alwarys.or itsD i\ominator aa

Um^ with Cyphers,visL. 10,. 100, 1000, ioooo, :^r.r\dfeeingthe tffe of a Denominator tn a Friîon is to^ (hiew into hoir

inanyparts an Umte is divided, it may be quite ojiiittcdnd

Jetnown by this Bûle, vItl. The Penomipato^ of a Dcchnai

^r^ ionisan Unite, îthi many Cyphers as tl^cireje places19

th^ Numerator^ and is jôwn from whole Numbers by a point

Se qM,hus; .4isT%,.34isTiJ S^7isTi55 -oo^Pisj^^, e^t*

ferve the fame of mixt Numbers^ for $78^ is^7^^, ^7*89.



d Of Deciml FraSums.

III. Cy]^er$tt the righthand c^ a Decimal FhidioA att r

not the Value ^for ;$ is^ .50 Ut^S^ ^Soooo ist S8l8 ^^ ca^

of them is one bal

IV. Therefore Decimal Fraftions are ealilyigeducedto a

|ommoA Deoomiiiator,by making all their Homerators to coo-

liftof the fame number of pbcesi ib .3 .45 .o6^ .0089, may be

writ thus .3000 .4500 .0^70 ^989 ; allwhich confiftingf fisur

places,heir common Denominator is an Unite,and four Cyphers,

vit. loooo

V. Addition and Snbtradion are the (ame as in whole Nato

bers, the placesof the fame Denomination being(tt one under

another, it will be a good Guide to placePoi^t under Point :

See three Examples ofeach*

r Addition. T Subtra^on

3

.06789 .9375

I2I.$

45.5605

75*9375

9.75

8.5

J.5

3.75

89

73 497

i .81789I 30000 \ 242.9980 I 1.25 I a.75 \ 15.503 I

1

VI. In Multiplication,ork as in Whole Numbers, and from

the Produft Separatewith a pointib many placesto the right

hand as there are decimal placesboth in the Multiplicandami

Multiplier,hen allthe placesui)on the lefthand of the point

are whole Numbers, and on the righta decimal Fraction.

VII. If there be not fo many placesin the Itoduft, as ought

to be ftparatedy the precedingRule, then placeCyphersat

the left to compleatthe Number, as may be leen in the Sixth

and Seventh Examples.

VIH. In Divilion,work as in whole Numbers, and from the

Qjiotient(eparateith a pointib many placesto the righthand

(fora decimal Fradion) as there are decimal placesin the Di-idend

morethan in the Diviibr

)

for there muft be ia many

decimal placesin the Divifor and Quotient9as are in the Di*

IX. Another



Of Dedmal Brdfun^l $

IX. AMtlier êthodof findinghe Valae of the Quotientbefbre

the DivilioQbegin,iz.. Set theDiviibr under the Dividend,then

will the Unit's placein the Divifor ftand nnder fach a placein

the DiridttMl,s isat the fiime diitance from Unity with the firft

ligpificantimres of the Qpotlexit,s in the \^ô-^ ^/ ^ .- ^

femples firftand lailrÂH^S. the^H 9o.oo (.347.98

Unites placein the Diviibr,ândsunder 8,'^

the third Figureabove Unityin the Dividend;denoting,that ^l

the firftFigure of the Qpotient, is of die fame Value, and that

here are three Integersand two Decimals in the Qjiote i. In

refts under a in the Dividend \ (hewing,* '^

that 3 in the Quotient fliadbeof the fame value,vItl.two peacesbelow Unity.

X. Therefor^, ifthe Divifor be a whole Numbpr, theQtio-

tient will have die fime number of decimal placesas the Divi-

dend^ as Ex. i

If the Dividend confiftsof fixdecimal places,nd

the Divifor but of two, there will be .fourdecimal placesin the

Qpotient, as. v 3*XL If there be more decimal placesin the Dividend, than

are in the Divifinrand Quotient, paace Cyphers at the lefthand

of the Qpodent, to coinpleatjêumber. ;See die 5thExample,

where one Cypher is pi^'d.XIL Annex what number of Cyphersyctiplealeo the right

hand of tte Dividend, putung a

point vrhere the Fraflion bcgms, l itf#yjr.ividefU. Qmtient:

f fee Rule 3-)But if the Dividend S75; ti^^.oo ( 347 9*

*e made to have three decimal 6'^.%) 1348.9000^ J4-79*

placesore than in the Divifor, 6.75; 23.489600( 3'4798

there wiH be three decimal places .675). ii^%90cio.3479*In the Quotienti which, in moft .6n{).023489000(' 034798

cafes is fnlScient,xcept it is to

be multiply'dfterwards.

XIIL

Vulgarractions

arereduced to Dedimkls of the fime

value,by dividingie Numerator by the DenoininaWF.

ExdmfU^ What is the Decimal of 9 /^ (^^ 4 ^* ' '***^

fee the Work 240) 9.0000 f.0375..

'^

DecifnMs being well undcfftoodiwill fltakethe fbttowiagTes

of Logarithmsvery eafie

Qo 2 CHAP;



4 A Number heiitggiven^o fnd us LegdrithitL

C H A P. 11.

Offin^tfii^^ Logarithmto any Kum^r, cf the Nptoberdafijrikgarkhm ekpLondin the {cUawtiraft^kas^

t. T^fifulhi L^gmrkhmfa whoU ifimAtr nder lO#o

INhe four,Wt Pages of tbc Table, art placedall ablblnte

Numbers from t to ppp,* in their natural Order, and

ainft tvcry Number its*Logarithm*, ix^ t}ieLog. of 43 is

i^33458s,Note^ The Index, or CJharafteriftickf the togarithmtfcvcfy

Camber, imm 100 to 999 being2, it isplacedover each Cohunn

of

Lo^rithms,nd are to be prefixedo every Logarithm in the

fame Column^ fo the Log.of 430 is 2.5334585,nd the Lo^6f 999 is 2.99955S? fo of the rej^.

IL T0 fatd$kr UgarithmrfjtfyNumierhsff^nfAsoffimrfluftsl

Find die Number propounded in the fir^Gofaima(of theT^**

j^jy Me) iotitled Nmu agaunftwhich in tlie

\ a\. i^w^ ftcoftd Gotama (fignSt the Uerid o) it

iSiJiS??^ fomidtheLdg.fougK,wheV^ t f ^J;? Z In tec of allSlbiuteumbers cbnfiffingf

??JJI f'KoolJJfcttfptocesXiaprefix'diasiiaybcifewii

IIL Tfffiithe LegdrUhmefenj/UmAer hdexen0sofficefUeefi

Find the four iirlFigures0^ the Number elven by the Uft

firop#and i^sUnftit in the next Column are vxt three $rftFi^

ginres.of

the4 ^-ibu^t,which noto in

your Paptf\then ibek

th^ l j|Figureof thelflua^berive amongH:he Figures at the

liead^ftl^ able, and vol the cofmnonq^eetingf theft two

Sinestate the four laftFiguresof the Lcffi.fought,which mnft

ho asmtxed to the three befturefound,(oefortfluch a properIndex beingprefix'd)sthe Log*fought. J^Mjm/r,5423^ bei^ggiveiii find 5423 (the four firftFiguresxktht firu dolnmo,

jrgahifthkh in the :n t Coiiimn toe tnree common Figures

are '1^34and the laftFigureof the Niimbpr givenf uiL% 1 find

9t the head ot the Table, under whichjna againft54239 aire

theft four Figures,iz.. 2957, ^^^^^ bemg )oy^dto the three

figuresbefore.found, 'x.734^ it will ftand thus 7342957 ; before

which the proper Index beingplaced,he Log.of 54297 is found

to be 4.7342957,

Nete^ That when the four laftFiguresof the Logarithmbe^trith a

Cypher,

hen prefixo them the three common Figures

(in the iecond Column) that follow in the next line below,

.thw 7 Fjcsrthe Logarithmof 54453, infteadof the three common

-^

.

Figwes



A^kmber bdn iven d fniitsLogarithn. J

^goresboire,iJL. î].ake the three commdn Figuresin the

line below,wc 7355 fo t ie,Loe.f 54453Vit n6t4.735ozT8ibut 4.735oiii . The fimc is to oe oblerrcd iiiall that follow,in the fame line.

IV To findhe Logarithmf d NknAericoH0$H offix Uctsl.

Find the Log. tfthe liyefiriiFigês by the laft Prop.amC

note the common Difference in the laft Coktmn bat one^ then

look that Difierena In the laft Golamn, iign'dt the^head

tts. and at bottom Vr: (whicb ftands for t^ns Froforticn^}

Apînfthe ixth Figureof the Number ^tenîs a Number^wrnch being dded to the Log. ^ the fivepla(;esbefordouadf

is the Log*fought,when the proper Index, vix^ 5, is prefix'd.Example^ 54^375 beinggiven the Log. of 54231 i found ( bythe laftProp.) to be 7^42957, and the common Difference.s

So9 and in the laft Column againft, the laftFigure,I find 40^

Which beingadded to 73439579 makes 5.73429979 which is the

Log.of 5423759 which was iou^t '

*

Noto^ Tlmt the Proportionalart may be found (withost the

Table of P^rts

Proportionaln the laft

Odnmn)thus :

Multiply9ot the common Difference,by the fixth pl ce,(Which in uii^

Examples 5) then divide It by 10, wiQ eive 40, as before. Bythis Proportionis made the TablesintitiedPts. Pro. in the laS *

Column.r

.

,'

.

*

.

* *

.

y. T^pni the logarithmfANmAtr coofjiftingffevtmflofoi.

Find tht ^ fiveplacesy Prop.III.and of the fixth pladf

liythe kft } and for the ieventh place, divide the Part Pro^

portionaly to, (that is,fet it one placefiirtherto the right

nand, than die laft Figureof the Logarithflieaches)

tiien add it to the Logaridimof the fix placesbefore

found, their Sum is the Logarithmfoughtfo the Lo^

grithmf 542375^^'^^^^ to

tf.734}oo3: oee the Mar*^

54*57

Logarithm.i|.7542957Diff:

40 80 M

Man 5.734Z997

4-7.4^957

-

..

*

.

542375a 6.73+3OC3

tin, where is rcpr^fentedkt

Sum of the Illd, IVth, andthis Vth Prop*.

Noto^ That the Part Prot*r-'

tional may be found ( witnotit

the Table in the laftColumn )

for the two laft:places,y mul-iplying

them ' by the common

Dmerenceândthen dividinghe

Produflt by ibo) fo in the laftExample^ 80 mukijply'd^y'tHtwo laftKigtttes,iJL. 5S,produces/ftf4o|-theiidivided bv 100

52ês 4^ .4/as be re/ VkA

i



t A VnnAer beii iven 9 finditsLtgimthu

Vt A Fradion beinggivcn to find its Logarithm^*iiil^

traa the Logarithm of tlieenominator from the Log-of the

Numerator,he Remainder isthe Log.fought)nd isalwaysheLogarithmof a Decimal Fra bioa.

^ Note^ That the eaCeft and moft ufefiilway to find the U^^rithm of a Fradion, isthis ; fuppofehe Index of the Log-oTyiNnmbers from i to lo to be i6 or lOo froni lo to too to be ii

or ibi from too to toco to bo 12, or 102^ from 1000 to 10006

to be t B on 03 and fo upwards. This beingallowed the Index

of the Log. of a Numben bat p\aceelow Unity, muft be 9 of

99^ iftwo placesbelovÛnity,it malt be 8 or 98, ifthree pkces,then the Indeac muft be 7 or 97, iffour placesthen 6 or^, the

latter of theie ways is often conveiuent to diftingnilhhe Indet

of a whole Number, from that of a decimal Fradion, and of*

ten necel rywhen the Power or Root of a decimal Fradion is

required,s in the next Chapter.

ExsmfAtj The Log. of i is found thust 3 Ug. 0.47712 1^

From wnich fubtrad the Denominator, vix^ 4 i^y ojSSioSoo

The Remainder is the Log. of. .75Xi5f.-p.875otf3

Noti^ thatthe

Denominator of a properFradidn is

always

Eeater than itsNumerator

;fo the fuppofinghe Index of the

g of 3 to be 10 or loo, the Index of^theRetisainder will be

$ or 99 (that is one placebelow Unity) and the reft of the

Log.except the Index, is found in the Table of Logjirithmsoanfiver to 75, 750, 7500, .00755 .075^ .75, or ai^ other Nnmbfetfwhofe twa fignificantiguresare 75, and thole which fellow or

precedeallCyphers.It was the former of thefe ways, Mr. Briggt

and Mr. Gmntr made the Charaderifticks of theirTables ofLegM-^rUhmetUl Sinu zryitntigtnts:,here it may be noted,when tbe

KdU$rdSmt or Taaginiis a decimal Fradion only,tjieIndex is

tinder 10 } but where itis a mixt Number, there the Index is 10

or more. For Example The Natural Tangentf 5 Dêes Is

10874887,the Artificialis 8.9419518 and the Natural TaoEent

of 85Degrees is 11 430052, the Artificialiît. 0580482.It is

needlefs to ufe thefe new Indices,xcept Ibme Term given or

foughtbe lefs than an Unite.

VIL Tc findthe LogMrkkmof d mhet Number.

Reduce the Number givenjnto an improperFradion, then*

fttbtrad the logarithmof the Denominator from the Log.tifthe Numerator,the Remainder isthe Logarithmfoueht.

ExmfUy Let 4fr be the mixt Number given\ this,reduced

to an improperFraftion,s 4/.The Logarithm of the Numerator, vit.

579

is i*75$8748

1^ Logarithm of the Denominator, viz^ it^ is 1*0791812

Th$ Logamhm of 4^ eotiaito 4^^^ whoTe Log.is ^^61669^6

If



I (^LogMfithdednthmtlMk

,Vt* If the Logarithmgireiie aoC found estdljul ^

Ttlile,ake the ncareft that is lefs,and fahtraftit nom the

Ipg.ibugHt,hat reiaains look fcr in the Parts Proportional(of its common DifRreiicejor the nearefl;Namber lefi than

the Remainder,agsînftwUchis a fizthFigureto be placeda(the rkht hand of the fivefiguresefore foand \ and ia ca the

Part Proportionale not found exa Iy fiibtradit from the firft

Remainder, then place a Cypher at the righthand ci the

h Remainder (in the manner of 4 decimalFraôn;^^ laftof

atL,aînfthe neareft Part ProportionaleiAer lyggeror leis;

is a ieventh Figure to he placedat the righthsindof the fir

Figuresbefore found*

Ex4mpUy Let the Logîthm îreneThe' neareft* lels is the Logarithm of 54x37

The Remainder faqd comiaon DiE is80^The neareft lefs,in the Patrs Pro. give^ y

The. fecond Remainder is

.The neareft' in the lame Parts Pro.; gives8

Ani^ ^-7343003 is the Lo^garithmof S4i37$8

Niote^ That without the 't^hlcof Parts Pfoportioîaliho

Number anlWeriqgto any Logaridmi,ot exceeding 95(9999^

may be thus found,viz^ Find Ckjthe foreîng^redioivs)he

neareft Log. that islefstand fUbtrad it from the Log. uvtuithen annex two Cyphers to the righthand of the Remamder,and divide it by the common piffer^qce,he (^tiex^tivestwoFiguresto be placedon the righth^ndof the NuQ^her anfiiferiqgto the firftfound Logarithm.

.

^

^ 7345005,

734*9$7

I

4^

40

q6.o

tf.4

rrrr

CH A P. IIL Of LcgsritHnmcalntJwticJu

l.T \^ MdiMienHon^ add the

X Lo^.of the Multiplicandand Multipliertogether,heir

Sum is the Log. of the Produd.

Ex.'hAtA. 8.5 Log- 0.9294189

by 10 Log. 1 0000000

Produft 85 Log. 1.9294189

IL In Z)jv^ ,fabtraftheLog.of the Divifor foom the Log or

the Dividend,the Remainder ia

the Log. of die QjMtieiit.

Exsim. 97 1 i^ l4 g.'-9873444Divid. by 45f Log. 2^^589648

Quotient 21.3 Log. t.%l%i^9 ^

ni. In the Rnte of Threejadd the Log. of the ficoad and third

Term together,nd frtfm their Sum fubtra^the Log.of the firft.the Remainder is the Log.of the fourth. ExMrnfUÎf Four CDs

coft 9I. what will Twelve ps coft? jiiff.jt



Of Lfigarul/l/ucaltUbnetwlLf

TheXi9tIof4 o.6Qiafoo Mff^ Thatifthc Arith. Goin of

Tliei^.^9 p.j))4H2$ the Log.ofcbefirlt Term (which

The lAjg.f 12 1.07918U illthis Example is p.3979400 ) be

TheLog.

of 1 08 2.0334iS7

^^^4. ^fe.^^r^^^^.f^^?*The t4* of 27 143 1 3tf37and third. The Refult will be the

*lame.

IV. To find the Complement Arithmetical of a Logarithm.

B^glh at the left hand^ liid take ihit Complement of each Fi-

^re to 9, only under the UA take itsComplement to to, which

IS aU one with iobtradiiigiiefame Ltgmthm from lo.ooooooo.

If there be two or mort Logarithms

to be ihbtra^tedâketheir Arith. 8

ocoooooCo.

Ar. 100

ComplemeBts. Thus the Double Rale 7437707 1 Go,Ar. 365

ofThreeînLorsrkhms^m^LyhcViTOVL^t.7781512 Leg. of tf

bjone Atklition. amm. Ifthe Intereft 3.7 1 37425 Log.of 5 1 73

Of loc/.for 3^5 Days, is */. what isthe 2506^0^0 Log. of 321

Itttcrcftof5i73/.fiw:32i,A/ 272.9tf47?2.435io58= 272. 96^.

See the Wox1(.

Of Hdijmgof famers hyLogarithms.V. Multijjlyhe Log. of the Number givenby the Index of the

Power rcquixid,he Produd will be the Log, of the Power

ibaght)lb the Log. of 32:r:t.505i5oox3t==4 51 54500 the Log%of 32768 which is the cnbe of 32.

VI. In the M ldplication,orailinghe Powers, via. Squaringor Cubing) c. ot any decimal FraAion by Logarithms^he h^

dox of the Logarithmof the Produft or Power moft confift of fo

manyUnitsâs the nmnber of

Cyphersinterceptedetween the

place of Unity^ and the firll:igmficantigure in the Natural

Number, wants of 9, 99, 99^ c. only to the Index of the Lo^

farithmf the power, (i.t. the Squareor Cube, c^c.)there will

e Inch a FigureprefaM, as wants an linitof the Index of that

Power or Number by which the Logarithmwas multipJvM ,for

Example, let the Cube of .009 be required,he Log of .009 is

7.9542425 H3rr23.8527275=r:.oooooD729he Cube ofôopandhe

Index of the Logarithmof the Power or Produft is 3, thercfortSix Cyphers muft precede the firftfignilicantigureof the f4a-

taral Number i and 2 is prefixM,ince the Index or Number,

moltiplyiagas 3. But when the Number of Cyphers,preced-ngthe fignificantiguresof the Power or Produft,exceeds 10,

'tisneceflaryo admit another Figure into the Index of the Ljo-

garithm,and make it the Compleinentto a Hundred : As fup-pofe the 6th Power, or the Cnw *Cube of the Sine of o*^ 1' be

required,t% Logarithm in the Tables is J.4t5372tfi but in this

Cafe muft be 96.4637261, which multiDly'dby 6, the Index of

R r

* ^

the



to Of L^drkbmcalJrithmetich

the Poif^cr propasM^ecomes $^S.^92i$66^v9ha ndex being7iubtraded trom.99, leaves 21 for the Number of Cyphers that

mufl: precedethe firfl:Figureof the Natural Number or Pbwerf

ivhich 15.0000000000000000000006058383.ere the Figureprc-cediagthe ladex,as the Refult of the Multipliqition^s j^lelsby

an Unit than the Number multiplying,eing6^ the Index of the

Power.

VII. This fuggeftscertain Rule for Extradingthe Roots oi

Fradions by the Logarithms Viz^ Prefix a Fi^reto the Index of

the Logdritbmof the Number, whofe Root is to be eztradedf

lefs by an Unit than the Index proper to the Root reqairM,which is to be the Divifbr ^ then divide the whole L^garitlnm^

togetherwith iti Index and Number prefix'd,y that Indext

the Quotient is the Logarithmof the Root deiired.

^x. Cr. If the Cubo-cube*Root.or Root of the Sixth Power of

*ooooooooooooooooooooctf 058383, whole Lo^.is 78.7823s^tft be

demanded, prefixf i i e. 5 to itsIndex, it is then 578.7823566^which being divided by 6, the Index

proper to the Root

fought,the Quotient is 964637261, whole Natural Number is

*oqo29o8882, three Cyphersprecedingthe firftFigure,becanle

the Index 96 wants 10 much of 99. But when the Root of an

AblbluteNumber isrecjuir'd,here need no Figure be prefixedo

ti;iendex of its Logarithm j fince it isalways lupposM,that the

Index of the Power (which mufl: be the DivUbr) precedesit. Ejt*

Cr. Ifthe Cube-Root of 6751269, whofe Logarithm is6.82993854,be required,tisan indiiferent thing,whether 3, the Index of the

Root to be extrafted,be prefix'dr not, fince that alters no-hing:

For 3j 36.829938J4 Qpotes 12.2764618,the Logarithmof 1

89,the Cube-Root

fought.VIIL Another Method to Raife dhy Tower of 4 Decimal Fraction.

Multiplythe Arith. Coifip.of the Log. of theFraftion given bythe Index of the Power required,the

Arith. Com. of the Produd isthe Log..^^j^ i f. 7,5051500

of the Power fought; Fpr Inftance,the jirith.Com. 2.4948500.625 power of .0032 is found in the Mar-

mHttiply^dy j6i%

gin to be .0275879. V 2^-7x2^00

NoteJThat fo many Cyphers mufl: ISotooo

precedethe Fraftion,as the Index ofiao6qiooo

Its Logarithm wants Units of 9 or 99* ,

^^\

i as in pag- 6 and 7 ) which in this-P^*^*^ 1.5592812500

JExarapleis one, and in the next 15. itsjir.Co.8.4407187500

being always the feme Number with the Log^of.o^^s^^9

\ the Index of the ProdiK^.

...

*

. ,

Again,



Of Lcgaritkmcirjritbmetick. 1 1

Agaiiifet the 6.1% Power of .0032 be fought, the Log. of

0032 (asbefore)is 7 5051^00, and its Arith. Comp. 2.4948500

^*tf.25=i5.5928i2'5, 1tsrrith.'Comp. is 84.407 1^5, which aa- .

Tw^rs to oQ0OQ,Q00OQ|0O0Q0 a44ag .wjikh is the 6.25 power ^

of .0032.

IX. To Extraft any Root of a Decimal Fraftion,divide the

Arith. Complement of the Log. Qf the Eraftion given by the

Index of the Root reqbired,the Arith. Comp. of the Quotient

is the Log. of the Root fought'.For Inltancc, let the .525 Root

of .0275879 be required,its Log. is 8.4407188, aqd itsAvithp

C5omp.=i.55928i2,ivided by .525 the Quotient is 2.4948500,*

and Its Arith. Comp. is 715051500, the Log.,of .oa3 2, which is

the Root required.

Again, Let the tf.25 Root of ,00000,00000,00000,25538.

be requiredits Logarithmis 844071875, and its Aritlj..

Compkî|.$928i25, divided by tf. 25, the Qiiotient is 2.4948500, aad

its Arith. pomp. 7.5051500 the Log. of .0032,. the Root re-ired.

X. To find as many Mean Proportionalss are def^r^dbetween ^

any two Numbers given,fubtraft the Log. of th^ i?aftermfrom the Leg.of the greateft,nd divide the Remainder by a :

{dumber more by one then the nuinber of Means defircd^ then

attd the Quotient to the Log. of the leaft Term (or fiibtraftit

from the Leg. of the greateft)continually,nd it wiH give the

Logarithmsof all the Mean Proportionals,equired: Example^

Let Three Mean Proportionalse fought,bctWeen lo^ ^nd

ioa

TheZflf.ofiotf 1.025305^.

The Leg.of 100 2.0039090

Divided by 4 0.oa5}O59(O'0ctf154.75 the Log.QviotYcnt^r^},

ThtLog.ofthêaft term. 1 00 1.0060000 (2)

The

Thetbird^Xhe greateft.erm, 105.2.025305^ ( i)=5i^-l

If of II Mpaa Proportionals,etween i.od and* too,, the 9tfw.

Mean was required,divide the Remainder by 12, and multipl|i^^'

tlêQpotient by 99 âd add it,to the- lealt term \ or multiplyic^^by 3,and Iiibtra t from the greatelb,t will eive the Log. of thc.'^

9th Meim Proportionalequired,nd is the umQ with the IjhiiÂ^

iBithe foreg6wgEjcaftij^le^. .

*



CiO

Wiii^MMl* mmm

mmmmm

m$^

immm^m^

CHAP- IV.

The Refolutionf the Cafesof KighÛnc

Triangle^ by Logarithms,

THUSht wc have (hewed the

Ufc ot theLogarithnwf the

Chiliads : Now wc wifl

difw tbe Ufe of the iame, twether

with tbeLogaritbfm

of tbe Guieo

of Triangles,and that in the Refi -

lution of Rigbt^KncTrianglei.

.WiereiBfthis is gtmridlyo hi

fhfirtfedh^ VfSn we jitfbe

Sine,Tangent| c*i meanThc

logarithmsoifthe ftimSine.

Tangent, c, m the iAnnfmCinon,

Fi^re H.

Prop.I. Hdving tbe threeAnglesytti

0tfe fUifH pmt either efwe her

fdiS.

Add the Logarithmof the givenfide to the Sine of the

Angle, op*poledto the (ide required,nd fiom

the Sum fubtrad the Sine of the An-le

oppofcdto the givenfide,tbe

remainder will be the logarithmf

tbe fsderequired.For Bcimplc : In the Triai^le

BCE, having the Angle CEB J ograd.C E 51 grad.56min. BC

3S grad.4 min. and the fide B E

197.3 i ^^ would know the fideC

t*s9fV ^71 ^ log.of l^.jff*9 ^l^^ tiltfiW of f I gM. %49S

.ii.i9xs^4p Che (Un.

^78998 8o chelineoi )tg(td.4mia.

aôK^rô die hg.rf ayi.ya7gfcrCB

Or you may add tbe Aritfattttieal^

Complementf tbeSne oftbc Atôppmed to tbe givenfide,o the two

other Logarithms,nd tbe Stun (hall

be tbe Logarithmf the fiderequired.as we mive fliewed in the HU

Chapter)FrppotipnV. And ito

to be noted^that tbe Arithmetical

Complementsof the Sines in die Ca-on

are to be found in the Columns

of the Secants : For (negleArngthefirftnit) tbe Secantsof the Com-lements

of tbe iame Arcbs,whereof

the Sines are cxprefledn tbe CanoiH

are tbe Arithmetical Conrotements

of the fame Sines. I^ Example:The Sine of

^8grad.4mia baiK

9.789^^80,be Secant of 61 fft56 mm. the Complement thereof is

laaiooiao, which (nKle6Hiagthethe /irftUnit) u the AritbiMtieal

Complement of tbe faid Sine.

asiooT.so tf1eAr.C0.0fthelhejegr.4fa.a9ri .7f ^te. of 197*1

^

g9 î^. 9he ibs of sipMiL %$tx f^

iHotx7 ^ thelog.of sf1.9178,asbotire'

But



angle, and tK Angles h laioiro,

and youwould have thf other fide,

as in the fermer Example,the Q|^

ffationwiUtefatrifr,tiii]|i:

Add the Tangent of the Anrie

oppofiteo the flicrqqyircd,o tnc

Loc;aritbiiif the giw^Wji w

(tom the Sum fubtrad the Radius,

the Remainder flÛ be the loga-ithm^

the fiderequired-

xo.to^t4*M tfirTanaaatofignitf

Sft40X37.59 the logâfa5 J 9t * ^Wî

Prop.IL ArMJfsnw /Uri,^Mirf/'J

lie Ofibrrva Ji^lts stultbttoird

fiU.*

Add t^cSne ^f the Angle gim,

to the Logarithmof ^hc fideoppofed

to an Angle required,nd ftom the

Sum fubMA the Logarithmof the

fideoppofedo the Angle giiren.he

remainder (hall be the Sine of the

Aiê oppositeo the other fide

given..for Esampla i

In the TriMsle

ABC, the fideAC being800, BC

?20, and the Angle A BC 1 ?f grad.

4 min. wc would know the Angles

B,AC, AC B, and the fide A B.

9.89^x3.^9chefineof iiSgrad.4xidpi .

%,^^\y fhe log.of 310.

i ^ti8.^ the fum.

a.yoiaf.00the Jog,of 8q .

9^9819.^9 thefineiSgraiip.forBAC.

Having BAC and ABC, the

Angle ACB is tfieirComplement

180 grad. vix^ 33 grad.35

min.

and the fide A B you may hnd by

tt^ lirftPropofition.

^. wwLogarithm

Ude, the Remainder (hall be the

Tangent of the Angle npppfedn

jhclS fide,, , -^

. ,

: For Example : In the Triangle

BCB. theUdeBE

being ip7-5and CB 2Si-p9 we would know the

LloiBCfi; QBE, and the Btfc

. |t.S9nMi di^Rai-adiiodfologpfxnM

i ,JDU 78tllC of Mt*9

; 9J9I89J9beTai^jS pr.4 otferQCS^

' But if the Angle yKkided be 6b*

lique,dd tbeLcârsthmf the Dil^

forcnccof the givenfides to the

Taiêntof halfthe Snm of the An*

;Ugnknown,and from the Sum

ubtraft the Logarithm of the Sum

pf Ae fpven fides,the Remainder

jballbe the Tangentof the halfof

their Difference.

For Example : In the Triangle

ABC, the fide A B being 623C 330, and the Angle ABC

128 grad.4mia we would know

the AnglesBAC, ACB and the

fide AG^

The Sum of the gjvenfidesis88 ,

and the Difference242, the half Sum

d the Anglesunknown is 25 g^.

58 9iin. (iidakkeot

x^fit.u Ae log.of 141 theDtfT. of the

fiHH^^ tbe can. of 15 gr. 58 ID. the half

fum of Che Anjêunknoivab

4 the iiun*

tbelog.or8aft,hefomofiha^

fideiglfea.t^5

9.IM8S-70the tangentofr |r^. 37 spia.

Thefe,7 grad,37 min. beinc a4^

dcd to 25 giad.58 min. the halfS0n)

pf the Angles unknown, the Sum i$

32 grad.25 min. for the greaterAfi

^leCB; and thelame, 7 S^^wL

^rop.HI* Hîvb9g$ /Ues^ nd the Imin. beingfubtra^ from 23 âd.

jlfiÛemeentfoem;tofmdtheotherj58 ipin.tjipnw**; ^^ ^'*tJ?^tt oAngUsyni the thirdfide.

'

* '^^' ' '^ * '- *

If the Angle included be right,

add the Radius to the togatitbmof I

21 min. for the lafiAngle C A

laftlVjnowing three AmIcs, and

two iides,the third m.?y be fQUA4

by di fitltiVy /iriM C9 pi



14 the Cafesf^H findTrundle*olv'dyLogsnthna,

I

Pmp.W.W4vlng the three jUes;to

fold0ff Mgle.

Add the three fidestogether,nd

take half the Sum thereof,uid the

Differences betwixt the fame half

Sum and each fide. This done,add

the Logarithmsf the half Spoi,aiyiof the Difference betwixt the fame

half Sum and the Baici togetinar.Add aliothe Logarithmsf the Dif-erences

of the other two fides,nd

the the doubled Radius ogetherThen out of thisSum iubtnA the

firftSum, and half the Remainder

will be the Tangentof halfthe An-^

gle rqqiied.

'For Example: In the TriangleABC, the fide AB being 562,AC 800, and BC320, we wchiU

know the Angle ABC.

A C 800 die baft.

A B %6% the fide.

' BC |xo ifaefidt.

film r^ft

halffum 84Z kC' i^H79^.

difôTAC 4; log, i.^Ti7S*39Thcfim^ 4 f1717-99

TbediCofAB 179. log.x,aa%6o^

'Doublod Radiuf toaoyxxioo

The Suqi ...* ^ 2$ i6a44.i9i

The firftSum fuboafied 4*5SiZdThe Rttnainder

. . . 10^148^.10Ha^ciie Ronaioder i .ni43.io Tul

4qria..fiMfBC

^ .

; Or add the ArithmeticalComple*mcnts of the Logarithmf half die

Sum, and of the Difierence betwixtthe iame half Sum and tbciBê, tB

the Logarithmsf the two other Dif-erences,

and half the Sum Ihallbethe

Tangentof half theAnglerequired.

7-07^1040the Ar. 6,

of log.8418.^87^1.^1cbeAr.Ok fIaôr4i1.440041 the log^ 171^

^71^3-77 the kg. of fu.

TbeSom iatfi48^.xoiêSam laji 243.1^ is the Ttngentsof ^

mr^^mtmmtttmmiWMMMvvi

C H A p. V.

Th( Refolutionf the Cafesof SphericalrianglesbyLogarithms.

THE Reibhition of the Sphe^

ricalTrianglessto be done

by the Omon f Triahgles,which we (hall{hew by 2S Propo-ficionsfollowing,hereof 16 are of

liedangle,nd 12 of Oblique Tri*

anês ;and firftof the lêaangle.

Figurem.

Prop.L Hjtvingthe tm fidestofindtie'Bdje:

Add theCo-Sineof one fide,o

theCo^Sineof 0c i^cr.f^^ci^

(irom the Sum fobtraA the Radius^the Remainder is the G -Siiieof the

Baft required.

For Example : In the Re6bngleA C B, havingAC, 27 gr. 54 mm,

aqd BC 1 1 grad.20 min. we would-

know the de A B.

^^1x9.27 the Go^ of iz gr. |o m. t

.42^3121. Go-Shie of 17 gr. ^4 m.

19*9375^*9fth Co-Sue of ^gr. omia.

.forAB requifoL

^^



S the Cxfisf bmdjtlIHat^ti

ACB hivinz the (kinAC tymA.

54liifo.zM Che AiigieBAC ^

gad.o miiu we woiUd know the

IcBC

fiPfot8.07he gneoF ifiefidei^ gr.U ^*.

Itop.VIIL Hipk^me pfikefiiit,

tofmitheuhar (AUquejhî

Add ttieCoSittftbe fidegiven15 the Sine of the Angle given,a dfidiftthe sum fitbtnTatheRfldiiu^theRemaindetisthe Co-Sine of the

Araletipquired.ForEnntdee in the Rc ngk

ACB^ hwriM tfae deBC II load.

|o muL and the

AngleABC

69 mL ^s mill. t would know die

An^BAC

r99iiM7 Ae Co-fineof Che fideII gr.Min

..^Ztacg he Taa. ofthe ai^. i$9|r.i m

i^pâiiau dieCo- ieof u giad.)omiB.for BAC Quired.

ftop,IX. tUvmg 0fte 0f the fidis^

^jh Angleofpojiim it^ td

findtheBafi.

^

Add the Radius to Ae Sineof the

giircnfide,nd from theSiitofob-ma the Sine of the Anjglegiven,m Renuinder it the Sme of theBale r^qiined.

ForCxaai|dct Ihe RedangleACB, havmg the fideBC 11 grad.Somiti. Md ehe Angle BAC 23Rtid* 40 laHt wt irould krow the

t^M^ t) tf 1ladhisdded to tfie9iie of

^^ ^

dbefideugnuL jomift.9^09^9^ dieSiae of die Aiftlex^ âd.

% 0irh1^ ttrlteilrfjOStid.crA B fo-

Rop.X.HmdHimtftk/Uex, 4aJthe Mt tMjH Mm Ui t9fmi

m ptberfiie.

Add the Taggtntf Ac |^'fide to the Co-'ungentf theiiYctt

Angle,and from the Sam fobcraft

the Radius,the Remainder k the

Sineof the fiderequired.For Example:In the Reflangle

ACB, havii the fideBC 11 ukL

30 nun. and the Angle BACî

grad.30 min. we wouU know thefideAC

^3oM*a^ ^ Tan. of the fide1 1 if. aoQL

ia|6i#âi cheQh'niLof iteMiJaup. Italia.'tf4yoi6^ tbeîieofx7gnid.4Wku fa

A C lequudL

Piron.XI. Hfvbi^.^ ^ ^

,

ttcAMgU9p6$fiimt it; t$ fidthi mbcr Obliquengle.

Add the Radius to the Co-Sne of

the Angle given,and from the Sum

fubtrad the Cb-Sioe of the fide

given,the Remainder is the Sine of

vat Angle required.For Example In the Roftai^te

ACB, havingthe fideBC 11 grad.

30 min. and the AngleB AC 23 gr.

30miin.we would Know the AisjUeABC

^

i9 ^l^7V Ae Kidias added to dieCo-fiae

oftheAnpIe ijend. |om.

9^9m9-^7 die Co-fineof die fiSte1 grad. JO min*

^97tio.(o dieSioeofîitd.nlib fiirA B C

reqoired.

Prop.Xn. HMing m ^ lir fdts^4mi the B4feitofM the Ohliam

Ai^ts4d/4cemm9ibefiatttiiu.

Add the Tangentf the fide^vcato the Co-Taugentof the Baie,anafrom the Sum fubtraA i;heiGidius,

the Remainder isthe Co-Siaeof the

Ai^fe rec|ukcd.

Toe



ifhencMlTtiai^k

IwrEfiunple: In the Reaangfe

ACB Iatmgtlie{ideAC27gpid.

9481111. and the Bale ABjOgîre would know tljeAngleUAC

^LTi^Sk.^ thbT ^*ofthe fideftrcr.4n^^j|| oîh Co-Tai^.f tbe Aag.josr.

I^9ds4a4 'AcCo-S ne of 13 yad. loxnin.jbr BAC legaii^.

Md the Bafii m ftd tk At^

^ff^fd 0 tinfdmejUa

Add the kadiftvb the Sineof

the iide given,nd tsom the Sum

fubtia^^ the Sine of the Bafe, the

Remainder QaW be the Sine of the

Anele^'equired.For Example : In the Rcftangic

ACB^ hairiqghe fide BC 1 1 gr^-

35 min. and the Bafe A B :;ograd.

we would know the Angle BAC

19.199^5.^1the Ha m a4ded totfaeSineof

fide II ^d. JO min.

p'i^97 theSine of che 0aKe.

'V.^00^.53heSfneori^giad;)dmhL for

BAC reqalfed*

Ttop.XlV. nnAHg 9tk îk^ fides,M the B4fi; t$fmdthembirfidx.

Add the R9diii$to the Co-Sine of

the Bafe and firoalthe Sum futn

tiaft the 0)-Sine of the fide given,the Kematnder isthe k)^Sineof the

fiderequired.For example : In the Redbngle

AGB^ hsTingthe fideBC u giad

36 min; and the Baie A B :{ograd.

we would know the fide A C

'M^3 o^ cheltadiusaddedco(fieC6-Sine

r the Bafe jofsrad. 9*99xi9-'a^he Co-Stoeof the fide 11 gnd.

' ' JO min.

9.94^33.79dicCo^fneef i5t^cli:54miii.

lor AC required.

Prop.XV. Having ths two Ohli^

AnglesI t9 findthe Bdfe,

Add the Co-Tangentof one An-ic

givento tbt 0 -Tangenrf ihe |

other Aiê ^ten atid trom the

Sum fubctad the Radius; the Re-ainder

isthe Co-Sine of the Baie

rcmiired.

tot Bcamplee In the Reaangte

ACB, havingthe Angle BAC at

grad.50 min, and the Angle ABC

6q gtad.22 min. we would know

tbeâfeAB.

imHS^M the Co-tangentf 1*3f.30 miiW

9.g58K94 the Co^tuBgentf ^ gr. %i mim

19^^^5085 dieCo-fibeof 361^ forAft

required*

Prop.XVr.Havbigthe two CAflufte

Angles to findeitherfthefides.

Add the Radius to the Co-Sinc

of citherAngle, and from the Sum

fubtraa the Sine of the other Ancle,

the Remainder fliallbe the Co-Sne

of the fideOppofiteo the Angle,

whofe Co-Sine was taken.

For Example: In thfRe6bngle

AGB, havingth*AngleBAC 22

grad.0 hiin.and the Angle ABC

(5pgrad. ai min. we would know

the fideBC

IMSU9T7 theRadhe added to the Co-finely y 3y 7/

^^ 3 ^^ 13 grad.30 nain.

9.971x0.84dKSfcie ABC69grad. iiqiin.

9.99118.^1e Co-fine of 11 gnjd.0 aaa

ftr BC reqttired.

FigureIV.

Prop.XVII. Havingthe threefides

to findanj oftheAngles.

Add the three fides,nd takehalf

the Sum, and th? differencebetwixt

the feme half Sum and the Bafc.

This done, add the Sines of two

fidestrgeiherAdd alio the Sine of

half the Sum of the three fiJea,he

Sine ot the laid Difference,nd the

doubled Radius, together;then out

of thi?Sum lubtrad the firftum,

and lialfthe Remainder flhalle the

Co-Sine ofhalfthe Angle required.

St Ac



8 The CafesfSphericalAdnglisMJitytogarithms^

For Example : In the TriangteSZP^ havingthe fideZS 40 grad.P S 70 grad.and P Z 3Sgr.30 inin.

vve would khow x\k Angle 2 P S.

TheBafe 2S40. 6

The fide P S ya o the Sine 9-9^^9S*5SThe fide PZ jiS.ia the Sine 9 794i4 91

The Sum i48T3o^theSmpi^t^?The half Sum 74*1 f the Sine 9.983^^^tlie Difler. )4.i the Sine 9 7503^79The dogbled Radtns

....tObOooeoyQ^

The Sum... 39-7?37M4

The Renudnder i9'9^^^|iHalf the Remainder 9'9^î% '

whidi is theCoSihe of is grad 47min.ThedouUe whereof is for the Angle ZPS

îgnuL ^min.

Or infteadof the Sines,hich arc

to be fubtvadted,ate theirArithme-

tkal Complement^ and add them

to the Sines of half the Som and of

the fatdDifferenceithen halfthe Sum

Aiallbe the Cb*Sine of half the An-le

required.

oa70x.4i the Ar.Co. of 7ogrtd die fide.

o.2058T-oftheAr.Co.of jSstad.the Hie.

9.98338.0heSineof74gr.if m^lialf fom

9 7 oy.79 the 5ine of Hg i y m. the diff

19.96660,^1the Sum.^

.

^ 98330^x5tbehalf Sumis theCcKSte of

tivr.^ffoin.gndtbQdoafale

a luad. 34 min. die AngleZr S

requaied.

Prop.Vm. Hdpbig tirehee An-

g^ts; 19 find4fiy rfthefides.

If for the greaterAngle we take

his Complement to iSo grad.the

Angles(hall be turned into fides,and the fidesintoAngles,and the

Operation(hallbe the fiune,s inthe former Propoficion.

PropbXK. Having me Angles id

A fide(ffojedo one QJtberfiI to find

thefideoppofcdo theotherAngle.

Add the Sine of the fide giren

to the Sine of the Angle oppofed 0 the fide required,nd from the

Sum fubtraftthe Sine of the Angle

cppofedto the fide giveuithe Re-

mainde

hallb6 the Sine of the Ode

remnred.

For Btample ; In die Triii^SZP, haringthe Angle SZP 130

grad.îniti.2 fee SPZ

xtg/ajL34 min. 26 fee and the ude S

40 grad.we would know the fids

PS.

9.8080^75 he Sine of die fide 4ognd.

^ 88391^5he Sine of the Ang. xjbgr.3.ia

19^^9x98.10the Sum.

9*71899.76theSinceftheAnlejiyu^t^

9.9729841 die RemaindeTjwlddi fs the

StaeofTOgr.wrPSreqairei.

Prop.XX.kvingtPto Angles,nidfideoppofedo one ofthem ; findthefideeween theAn^s ghfcn.

Let a Perpendicularl\ from tbe

Angleunknown^ upon his qppofitefide: Then

Add the Co-Sine of the ^vcn Anr

gleadjacentmto the j^venfide,othe Tangentof the given fide and

ftom the Sum lubtiaA the Radias,

the Remainder fhMl be the Tangentof the firftrdi*

This Arch'flballbe comprehendedbetween the giirenAngle adjaccoc

unto the

givenfide,dxni the

Seg-entof the fidfcwhere the Perpen-icularfills.Now the feoond Aieb

comprehended between the fine

Segment and the other Aiigle,s to

be found thus:

Add the Sine of the Aicfalbiinl,

to the Tangentof the ^vtn Aii^

adjacentnto the givenfide amd

firomthe Sum fabtiad the Tanoenc

of the other givenAnê, Ae w-

mainder fkall be the îne of the

iecond Arch.

The firftand feouad Arch beingadded together,r elfe fuboaAed,

you fhallhave the fide required.For Example: In the Triangle

SZP, havmg the Angle ZPS 31grad.94,26,ZSP 90 grad.28, 12,and the fide PZ jS^td.)omiikwe would know the me SP.



The CafesfSphericalria^lesoWd byLogarithms. 1 9

For Example; In the TriangleSZP, havingtheWinj^lcZPS ;

grad.34, 26^ZSP 30 gnd. 28,12ând the fide P Z 38 grad. tô niitu

we would know the Angle S Z P.

9*993^4^4the Co-fine of jS gra^.30 m n'

the givenfide.

94^%7'S6 the tancsnt of 31 gnid*34, u$^the actiioencngle.

'

jî04i.tx the G -Shie of 31 grad.iX^'uStthe adlacentAngle*

9DQtef 1 the tiageiiK of 38^cL 30 nUa.

.

- the fide oven..

t9.S3fox.74the canBeqi cs 14jrad. and a

MWMta

W ^.3o tte Sine of |iini4.7aalaalfmin. the Arch tcRmd.

$Jt9Z^7-^4the tangent of 31 giad.341^Che adiaoncAdbIa.

i9*^nM cbe Sua.

^/jSgiSaJiiht taqgeac of jogiad.i8,xa, the odier givenAngle.

9.7^91.0? theSnc of 35 gr. % nUa. and

Now in tbb Example,adding

PR 34 gtad.71^ min. (the nm

Arch) to SR Vi^?A. 52 V min.

(the Iccond Arch ) the Sum is 70

grad.forS P required.n likemaa-

QCr you may findthe fiderequired,when the Perpendicularellsut of

the Triangleropounded.

Prop.XXI. lUvU^ tff9 ^l^hani

s fide fed / i f fftbem^f$ fmdthe tbird^ngle.

Let a Perpendicularallfrom the

Angle unknown, upon biaoppofitefide: Then

Add the Coê of the giFtn fide

to the Tangentof the adjacentn-

rie,and from the Sum fulHrad the

naîtls,he Remîndf r ilullbe the

Co-Tatig^tf tl)efirlt^lc ^ ^foimd.

Tbia Angle fouqd {hM, be com-rehended

by the given fide and

Ibe Perpendicular.oiy the iecond

jAngl^,omprehendedby the Por^

peodicularand the fij^Hnknown,IS to be found thus :

.Add die fine of theAngle found,

to the Go-Sine of the givenAngleoppoSbd to tfa giyenfide,and from

the Sum fubtra the Cd-Sine of the

Qtbqr Angle g^veo,the Remairuier

i}uU be the Sine of the fecond

Ai)g{cThe firftand lecond Angle bein

X9.Û2.QO theCo-caneencof^4fr.8t50. ferP Z R, the firftAngle.

9 9$4fiv%6the Sine of 4\2,u i8 fo the

Angle toundl -

f 93l4$*^ the Co-Sine of 30 grad. , 12^die o[iôIediê,

1^890x^.93Che Sum.

9*91Q4Û the Co-Mne of 31 grad.34.1^,f the adhceac Angle.

9*9^984*7rthe Sine01 ^f erad. 44, 23, fer

S 2 R Che fecondAngNow in this Example,adding

PER 64 grad. 18,50, ( the firik

Angle) to SZR 65 grad. 44, 2^

(the fecond Angle) the Sum is 1 30

grad,9,1

J, for the Angle SZP re-uired.In like manner we may

nnd the thirdAngle,whcp the Per-endicula

lla out of the Trianglopropounded*

Prop.XII. Having tm fidfSyndiie^ê between them

\ to padeitherej the $tberAngteu

Let a nerpendkularallfrom the

^nfleunknown which you require

not, upon bisoppofiteide,thenAdd the G - fine of the givenn -

gle, to the Tangent of that given(ide which is oppofixlo the angle

required,nd from tlie fum fub-

tradthe

Radius,tiieremainder fhallbe theTangentof the firltrch.

This arch found fhaU - be compre-ended

between the îvenangleândthe fcgmcntof the pvcn fidewhero

the perpendicularalls.Now the iecond ardb is compre-*

hended between the iameiegmentand the anglerequired.Then

Add the Sine of the firftarch,tone nrit ana lecona Angle oemg Aaa tne Mne ot the tint arcn, to

added t(êther,r elfe (ubtra ted,the Tangentof the givenangle,and

yoi) il:ialire theAng^crequired.from the film fubfraflhe Sine o(



30 The CafesfSpiral Triai^ksi h*dyLogarithms.

tbc ftoond archyafaeemainder fliall

be the Tangentof the Anglerequi-ed.

Fmt Example,In the TriangleS Z P, baringthe fideP Z

3S gr.$0 min. P S 70 grad.and the An-le

Z P S ^t gr. :;4,26, we would

know the Angle P S Z.

^9je4ft,i^ the Cb-fineof ji gr. ^4 ^

the^MtlegivoL

9y9Q0^f the Tsm,oifift,yi. the fide

cne^nj;lrreq.DpQIcd to

iM) o^74 tf'.eTW.ofj*r.randlahalf

mitt, die nrftarch feond.

9,7489^10

liioSineof uGr.vaodahalf

mm. die am arch fimnl

9 7Wf7.y^ the '^^ of jr Gr- 34 ^% **^

tf^WSlJ^ ^ Sum

I .faalfm.m id.aicb found.

firîtn ther^^.of |oGr.a8.aia.lbr

XSPreguked.

7 /bifo/ibheJingUsniJamwn.

Addtb^ Singôf halfthe differ-nce

of the givenfides,o the Co* |

tangent ofhalftheAngle ên, and

fipmthe Sum fubtiad the Sgiiofbatftbe Sum of the givenfidca^heremainder (hall be the Tangentofhalf the differenceof

tbeÂsqgkse

quired.Addalfo the Co-fine of halftlie

differenceof themven fides,o the

Co-tangentf halt the Ancle gtveo,

and from the Sum fubtrafttheSine

of halfthe Sum of die fides^ven,theremaunder (hall be the Tangentof

half the Sum of the Angles requi*

red.Then add the balf Diftrenoe of

the AMles found,to the half Sunt

of the lame, aod jpou fiuU have the

greaterAngle ; KxA the une half

difierenoebeingfubtiaded fiom the

halfSum, ;rou (hall have the kfi

Angle requiredA| in ifaelinKr

Ezsmiple.

TbcfidePZ

ThefidePS

Thefom

The halffmn

ThedifiloftheSUei

ThehalfdiC

ThettaleZPS...

Thehalf^^

Thefisn . . . .^

}8Gr. JO, o.

70 Gr. o o

108 Gr. 30, o.

14 Gr. lU a

jiGr.|0| 0.

x5Gr.4f. o.

]i Gr. yîs.

15 Gr. 47,15. Co-lin|.i'î^}Ji%The Gku. xûI^mu

V - ^98 iaf8 -Thc/ W3 iiTr

TheSiae M093M1 TheCo^faio Mitftff9j|)b

TheSine 9All^M TheCojbie 9îfi/ f.

3719^7ân. t^t5,410,7^1^

p^^^11

Andt betwetf^Jhem; tofindtbc

thirdMf^

Let a perpendicularalfftom eft

tha of the Anglesunknown, upon

hisoroofitefidc h^

Add the Co-(jneoft^ riven an-le,

to the Tangentof theTi4^ firomwhde end the perpendiculars let

^1,and firomtoe

lumfiibtrafthe

?:adiu8,he remainder (hallbe the

^gent oftl)cirftrch.

^lllftheiiimlt.

; SoGr. is^.

Thefom ijoCjt.2,1a. ftrSZP.

Thediff. |oGr. ,ia. fv2SP.

This arch (hallbe comprehendedbetween the ' angle given,and the

fegmentof the fide wheretfaeper:

ptmicularfalls.Now the feoond ardi (hall be

comprehendedbetween the fione

%(nen^ and the end ofthe(iderer'quired.Then

A^ theCo-finecfthe fecoodarch

found,o the Co-fineof thefide (omwbofe end the pmodicular Ueth, and from thefum fubda^ dK Co-

fine oftbefirftardifeutidydie le*

mainder (hallbe theCo^ne cf the

)fideequired. For



'

CrfeifSphericalrioj^lForEyanmle,n theTmnflttZP,

bayinghefideP Z 38gr.30 intn.

P S

70gnd.and the

angleZPS

91 gr,)4 269e would know the

iideZS.

Add the C^m ofthe Angle9-vcn, to the Tangentf ^hcgivenJutehich isadjacentnto the ^me

Angle,and from the fum fiibtraft

the Kadiua,he remainder(hallbothe Taneent ofthe RtR arch.

fhefi PZ.

i9J^îo%^4ie TWvMt ofu Gnul. 7 and' ' abtlfm.lbrFRcbeiurftardi.

mla. the fim ardi

9i9fM4M*e

O/ii*rf Gr. f* and a

halt in. forKS the ad.arrh.

9^1^3^4,44heO/jWf j8Gr.oin, the

chei^gtraL IThisfirftarch fliallbe compre*

9#30tfDbfatlieTiî |fTjiGnd.|oiii.|hendedetween the givenAnglesand the fcgmentf the fiUwhertthe perpendicularfalls.Now theiecood arch bet^veenthe fameiej^^ment and the end ofthe fideeqm*.red,(hallbe found thus.Add the O-finef tlieftrftrcb

found,0 the Qzfinef thatgiven(Idewhichisoppofedo 'theAngle

giveind firomthe Sum fiibtro^

the0 -fioe of the otherGde given,the remainder flialle theO^finethe iecondArch,thefirHand lecondArcb beingdded together,r elie

lubtradted,ou fhillliavethe fidereouired.

Por Example.In the Triang^S Z P^havingie fideP Z 38grad.30 mm. SZ 40 grad.nd the angleS P Z it grad.4, 26, we would

know thefideP S.

s^JoaiS^ the fum.

%wif,SiSi CheC^fiît jf/oGr.aZS the

fideitqoiied.

Vtop.yXVf , Hmfwg tm pi i 4uul

9f e jingle9pp0fedo cm rftbem

tpjimlhejinffeppofed theether

Add theSineof theAnglegiven,to the Sine of the fideoppofedotheAngle required,nd from the

fiunfubura theSineofthefideop-ofed

to the Angle given,he re-ainder

(hailbetheSue of thean-le

required. W J04* n the d jRi f 31 Gr . 3;,t^gr

FortKunple.n theTriangleZP,f..^^-^

ri^^JL/rfaar ^nOn.

havingtheYidePS 70 gTad.S ^ *'t 2S2L^ lL^'' '^

40 grad.and the Angle SZP 130 79.83101.74|ie^. of 24 Or. 7 and half^ ^

- - mhi, forP R rhefirftaich.

'^ **

haJfm.fcrPRihefirfttn*.

9.9842Tô the Offintf40 Or. the.JU0 tmpcHisdo Che jtm^glvea.19.80218.7^he Sum.

9-S931^4 44bcC^fifif jftGr. 30 mitt.

the other)?^gi^

gra^:,1 2, we would know thean-hiesPZ

938391,41he Sittef x}o Or. 3,11,the

9;8o8ai6;7the Sim of 4:Gr. thcfiUcp-pofedo the UMgierequired.

IM9X^t'2othe Turn.

9,97298,58he Sim of 70 Gr. theJUtop-I I '

pofedto

the ^ngU yvcn.9i7l899i the Sime of n Gr. 34 min. ot

S P Z re^nifed*

Prop.XV. j4^h$ rv0 fidesjttdone

Jinglefpofedoatie ofthetttI tofindthethirdfidi.

Leta perpendiculari]lfi'omtlie

Anglebetween the fidesgiven,p*an biioppofiteide,hen

9.908^.3 i the Co-fintC3tGr.fa and a ^ ^ ^

halfmlo. forS R the fecood

arch.

Now in thisSample,P R 34 gr.

7 and a halfmin. (theHrftarch )

beingdded to SR 35 gr.5aa d a

half min. (the feconl ardi)theSum is 70 girad.orP S thefideicr

quired.

t^_.-.T .



93 The Cafes/SphmcalTriaiêifilv^dyLfiprntbrn.

Vkm in this Eiainplc ASmt

P Z R 54 md. iS,50, r the firft

Aqglt) to SZR 6 graoL44923^([tbeieamdAngle)foutyL tfacSuoi

is 130 giad.3,13. ferS Z r tbe Axk*

fit p.XVI. H^Kvlngmfihf^md$m[Angle emfed to om i4$hcm; t

'.findbejingleetweenthem.

let t Perpendicularallfiromthe 'glereqiurcd.

Sde which isadjacentnto the gi-en

Angle, to the Tangent of toe

lame Angle, and from the Sum fub-

trad the jRadius,he reminder fhall

be the G -tan nt of the firftn-le

to be found.

This firftApgle (bund /hallbe

comprehendedby that given fide

whigh is adiacent unto the Angle

SVen,nd by the Perpendicular.bw the fccondArch comprehen-ed

by the Perpendicularnd the

otherjuvenHde,is to be found thus,

Add the Co-fine of the firftn-le

finjod,o the Tangent of tbe

^vcn fidewhich is adjacentnto

theAngle given nd from the Sum

ftbtrad the Tangent^

of the other

givenfide,the Remainder fiialle

the Co-fine of the fecond Angleto

be fbufid.

The Sum, or theDifferenceof thefirftand feoond Angle, (hall be the

Jknji;leequired.

.

For Example. In the Triangle

SZP} havingthe fide PZgSgr.

%o min. S Z 40 giad.and the An-le

SPZ31 gnia 34,2^,we would

know the Angle S Z P.

^ 9354^ ^^^J2

1Gr.

^m.the fide diaceBC.

4^788f7-f^.tf y-**-rfS P Z 31 gr. 34,1^,Che jingUgiven.

Angleetween the fides given,up- . Mm Isto U noted,Jfj^îmUM

' Add the Co-fine of that givengent to tbeothersitêni SiAtrJS^

on^ thatthe}ore exfireffedn tbe Cf-

nm: Br CmgleSingwfirfiAut i

the leftUtnd) tboCihunigemseftArchs 'orAt^sMs then49 grduLarg

tbeAridmettcd dmfUmemstberetfbut the C/hUngemsf tbe Ar^bsor

An^grester tbenAKgrdd. io

tkkenforrhbmencd umplemems,if

from the Sum be (Atrdled 2 om tbe

lefisni infieldf41$^

-r- ^ P 2 R the firftjtmgh/ glefound,to the Tangentof the fide

9j6z69i^96 dieO-Awof^Gr. i8.fo. die civen. and firom the Sum fiibtc^

jifijrUand.

^9QOtfOb5athe Tsngeuoof |8Or. 30 oUii.

the fideadiacenc

19*^37^^48 die Sum,

9^1381^35te Tarn, of 4oGtad. tiieoOer

I giveafide.

S ;^ the feooodyî g/Sr.

PcOlKXXVIL Otmig two AngUs^4mtbe fidebetween them

; tofaieitherof tbeotherfides

let a Perpendicularillfrom that

Anglegivenwhich isadjacentuntothe fiderequired,pon his oppofitefide. Then

Add the Co-fine di tbe f^ymfideto the Tangentof that nvtn An-le

whichIS oppofedto nie fidere-uired,

and firom the Sum fiibtn (the Radius

;the reinainder (hallbe

the Co-tangentf the firftAngle E0

be found.

ThisAqglefound fliallecome

prehended he givenfide mi,

the Perpendicular.fôwthe feoond

Angle IS comprehendedby the Poh

oendicular and the fide Tequiicd.Then

Add the Co-fine of the firftn-

given,and fiom the Sumfiibtiaft

the Co-fine of the icoond Angle

'fi)und,he remainder (hall be the

Tangent of the fide required.

For Eiample. In tbe Triai^ÎeSZP, haviiftbeAngle SPZ 31 9.

l^^2 ZP 130 gr 3^ i^ and tte

f4



d4 The UfecftbeT

9 7WyMg Ae T4 . of ji Gr. 34* ^-

fer P A die firttAngk.

9i9y4tx.a die Sin o(6a Gnd. xliô.the

liritAngle ImiimL

Chap. Vl. the Vfe rfthe TahU offiêrfedines.

THe Uibof the TabkofXHrr/iriiliirfIt i32;3f5,tdh^ imMe b: S4U7io

are too ooaeront to be bereatl Tie Trmr'ai Verîacr ^^j*Qo'.20t^6

hieofVerfiiketl_

9nsm^^ ctiei e;of^5.Qr;42.itofcoMiil iUKLefooiiiL

y 9yHi at. die g-jBitl 31 Gtmî^

I9 a9aa5.7tfheSom.^ ^ ,

9.û4f.fo d Ci-Mfof |oOnd.if ani;

* 'i IbrPSZieqtdnL

.rtM*^

tmiedof: Iflialliiowpalfbewliowby

Oon OMê eafilyo (biverome of the

oft ofenil Cales of SpberudtrimUi,

vkkh akoe b caoogh to merit oeir

7iiU(rtffif t has beeo a loog time the

Vocaaad Defireiof naoy able Me in

the JfjilaMfrftrihit IMi a f^tlraOghtbe coUetel aad paUUh'd,bat cfpedally f that la^nriootaod aadeat Student

lto.9iloMBtef,ho hat esfCcCEcdis do-

firathereof more than OBce in Us elabo-

ratepiereiandfrom hom I had the mi

offome VacdgoTMca* vhidi did aflift

omch conardi die oompofiagf thefe.

ffef.. Too fidesof an OiOfaeJjpfe-fifislfjisglg^kh the Angk comofe*

headed, bowg^Ten^tofiodhe 3d.nde^

Jt deCwte ef $k ^fdki i Je te Oe

MfBetfkefdkskeseftkeemfnkBdediFAf,: : Ais tbeSfure efmSheefkafjk temdmi JtsgU,\ T$taft e

tmereme eftheVested Sines eftte^L

JUelAni efthe Afk ef Viffefêee*

msiMiilt tm kukM^fHtuWhfch it dttf^dooUe the Leg.Sine

of halfthe AnglegiYcn,and thereto add

he Log* Sines ofthe oonoinfaigides,

nnd from the leftHand ofthe Som^th

ont 3 for die Cnie of the J^îMipforeih

the Log. of half die differenceof diofe

two Verfeiines.

Whidi halfdHfereocedonUeMod hd-

ded to the Verfedcar of the difiifnae

ofthe Uff or contahringfidcsifcsthe

VerfedSinef the fidefonght.Sxmu u Inthe Triangle

I^gure5 B ^^ kt there be given

the fide BF 77*oo',the fide

^ 1 40* 00% and the conoined Angle

JPL52^ 30; CO (od the fideS .

Tie l^. Sine 4h* 00'.

9 8oto675

theisf.ef^Ptsîdesu7kVeriedSineigf57:53'44^51

^fidsfsnghuIf jod make the tUfd 1Mb the

Sjnaraof the Sine of halfthe Comple-ent

of the cottOhmd Angk to iSodOi

sl rdW

Ikejjog.ine ef77* 00' 9*9^^^Z9if06'

dsMed.

thel^. Sine if^^ H' S9bft9 4

aih tfisiâsiffe 4it/l t 39 o8S2030

grees, foa willfind the

ofthe Veffedines of the dihdfidetail

ofthe SomofthetwoindadhKlidesaa

be dodUed. nnd rnbtraAed inm dm

Verfedlae of the (kid Sam.

Bvt if bifteadofdieleoQndTermbe

taken into the Propordon*thedodhle of

theReSangle of the fines of tM^con-

tainhgfidesthatb, if theLo^ ofdto

UxmSf 2beaddfid to the Lofi of the

other middle TeroMt m wiB bnae the

Log.ofthe whole diffmioe In the laft

ptaoe; havii fimnd it,take tfabNa

Coral Sine diat Sands agatnllt.midadditen the Atenril VerfedSiaeof die dif-erence

of theLeg^ and the Som Isthe

Nmira Verfedie 9f Oit 6dc fimght-Bxm.^. LetdietwoeODtMningfidm

be 38* 30ând 66* 30; nal thecan*

cdned Anglebe 70* co\

Th'H'S^^ 3f 10* . . . . ^7Wi4ff7* Af.Sm 9f66* 30' 9^1397^

Thiior.ifamtlmnher. ..0.3010100

rte /if.; mVt5* 00* imUtd is iragf99 4

Tbt vforeft( r SiM ^nimft3SJ8)5^9t

to' li 77009'

s8 to Whkh fJmnfrwm^tleih UTOfHiU- FerfdSfnt^^^ oo*

18 00 TfanPftnuums .., ... 3004}J

dieNaaur. Verfei Shie of 53* 10

ThlMTfeP. UoffieatUfecoGdcnfafecleDifhunes of Places on Eardi. acoordfavs

d) ewirffyf a great Cutle, hydietrlM*

andl4tit. ghfcni the Dlflancei of Son, by

baviw didr Dediaadons and RiihtafosA*

ons, or Ungiaideiand Udtodesgivoft,I7

means whoeof die Mdiudes of two Snes,

or of the Sun wiriidie IXfieiencBof dme ^

^mimnsh bdne oUenred ata y cUne off

die AMdidn^^c Ltdcnde mj oc foand.



^' *

.- * V- MW ' J 'J

',*'A .

'

,

G H A P. Vlt.

..me^nî in,th Vntveriiiy/ Ox fcf,.aria

*' V -.'

.A* ' *^ ' . .

'*

* ^^-- -. -

,

-^-

'-

w*

a^ V

of Compound Intereft,hich are not withour great

jnetick. ut.before we prpceed to the prafticalart}

oto6tHf}6j;that is -a*'*^- V*f5f n^ iftftAlL* q^ fiCQijnt fter

the Sirm j^^fdrbdrAVtiinfesr.-ipfe\*fib di(lre4 oneear or Time

unity becomes r, by the fame realbn r will in another time be-

i^'^^fir raffed to* stkq hme^ty i/vibisriMdnXjit'theWuinbec igtf

Timt^^ÛMthe: amorait ^:6ric: tbBadciorb ^ ndl

tMr4Kii;trtfM;i.î]l:^te:jm^ amdi e.i .W^tfrfor:^

multiply*he Logarithm of .r h;.x.flQdr.tak.ail(Ltî^lôf J

,the Sum Ihall be the Logarithm of t;

,which is the $olu^

tionofltl fi.t fcfrnlilllmi.lV'l^ 7-; :;Ij:;T ..:../d h

II. r' is equalto -:;^refoi fSSJiiPthe-ogarithmof thg

r]the Qapt^îsJ e,Wgarlth|nôf^j^::..;i,,i: ,,;^;:^::,.,

t|ieaîthmj .'tiaJiG^Jhiii2g|5Eitt3I iSe2ll i* '.bt -fintgwheii eift1tc'Sum / wîtameait to Âtpag-fjt :âf.

t^qgffecfermkii'petn. Log. ar y ffle^FHui'itiaiaui- '



i6 Of CompoundInterej(and Ammitia,

Agato, all QnefHons.coaceraing the Rebate of Money are

jlblvMwith the iame fafe, and after the nw maimer : Fortf

in aay time, r becomes i,ia the fame time i becomes

-?and

in the fi^nd time

becomes ând ia the third t, ip r rr rrty

that the Valae or Frefcnt worth v,of any Sam /, afterany tlilb

multiplythe Log- of r by r* and fubtraft the Prodnd fctm. the

Log. oft, the rsmaiiider will be the Log. of v : which finds the

Value ofany Sum of Money payable after any time alCgncd.

]1.1 ss r*. Tberelbre from th^g* of iiibltrafttê Lc^ of

V, and divide the remainder by t : the (^te will be the Legof ^

III. Divide the afbrelaid difference of the to^ of ^ and

by the Log. of r,the Quote fliallbe the Nombcr of Years

_

IV. To find what Sam,payableafter ( dmes,

Motr.Tl^b^M aaybepurchafedforvatthe.rateoflntcreft

7 'li^'i4S.Mthe-rteoremtaMsthas: multiplythe Ug.

jMi te Pfr tt;of r by t 9iid to the ProduA add the Log. of

V : the Sara OwU be the Log- of t fought

lîNott^ becanfe the Money is to be valued in'Founds and

ertsof I /. and the time in Years

and paruof

a Year,it wiS

moft Commodious to reduce thofe Parts into Decimals (then

the Work is the fame as in ivhole Numbers ) for which porpofe

;AeTwo decimal Tat Ics are annexed.

A decimal Table for everjrFarthingin ji.and

every 3 J. m i /.



'

OJ CoK^mitiIiOtrtHnd AtamAes, 37

.

The dtcimal parts of /

nuT bejralaedy the precedta^j -

i/r,r at fightthHSj vie. the nrft Figure doablecfis Shill gSy

the Tecoiidand ttiitdjo^n'dzreF^rchia^s,batingone jforevery

a5forvo25 is 6 d. .050 \t. asd o'75s iZi*

A kecimalabic cfDaysand Months in a teir.

tt ne. Thit every Orfsma beû with the Dednal of an tvtt 10 Diyr, u^

iacrurrt downwirds, To tbe Dcdnil of so Days ii 05479ôf 31 .057s 34.

The fbHoiiVingTrffr/rliews the exaft number of Days from any-

Day pn^fed in any Month, to the fam6 day of any other Mcmh

throughoQtthe Tear :ôr Inltante,from the i, 10th ur 2ottt

ofjme to the 1, loth oriothofiW-trcA is 273 Days. I find7 v

at the Head and look down that Column, and over agiinrt4rch

in aright Une is 273, fbif it was from the 15 c^Jivieto the 1^

of March, t eonfidcr,that

the.Number of Days is one more

then 273, vit 274 days. The fame way is found any number

of Days in any time under e Yeai' by Infpeftion.

Tt Tin



29 ofCompomdlntetdJf''k^^^ Tfie Lbgarltfim'src

alfojifv|^^%3folve-ali iie KM$

concerningthe Amount brPrefeht worth' of jAnnoitiesot paidas due,'or piiichafedo T)epaidfor thnc ior-come. Let tlfoe-*

fore a be any. Annuity or yearl7Pefafibri,hofo faceefliv^ a*

mounts for times p^ftare 4 rS and ^^lofe.prefeiitalues are

-ij.fucceffiv^ly,y what goes^befpi'sAnd th^ ^ries, c^ 4r%

r

ar\ ar^^ ar\ r, 4, -r ^ w \_. _;, ijîr.ill ^ be a ranko^

mean ProBortioials c6ntulii idnfinlteLyn the uitip of r to i xi

now the ^mn of all ^jtie?tonftquenti,r of the whole infinites

Series,ill be to the^faidSum cncrâfed, ty the next ereater

Term ( or the Sum of all the Aatcctdents ) as i to. y (by 12.5

J?/ ? ?, /#r/;)Wherefore puttingjr forthfefaid Sum of the Con-

feqjuentsy wiU be equalto. ^ -f ^^\ the Sum of the Antecei**

Mr'

4entsi-andry y =ar^ : and therefore^

~ wilHic equal to*

y, the Sum of allour mean Pfbporfioaaisyhereof ^r.^' is

the greatcft.And by tHe lame Role ^-îH telhe Sum of

aH the Terms whereof 1 isthe feriateft,b that'if we fabftraft

*^ from

the difference will,e the Sum 6f allthe terms

whereof ^r*-i is the greateftnd a the lealt^theirumb, beingr ;

which Sum we will call z.. a therefore-is thus to be expreScdfJLLl^

r= to the amouni: f the Annuity a fbcbora t times

at the rate r. Wherefore from the Logarithmof 4^'fiibitrad

the Log. of r* 1 and to the fenqinde da the Log.of r*. From

the Number anfwering,

tXK this ^ hft Sum^ ifublbi^ the Number

aofweringto tlêi enlai^der^tljie )i0erehceihalie thcamoanit

fought.^.

. -

, r

Examfle Wbat will an ^nuity .of34. 4/.forbora 12 f Ye4rt

amount to -at;S fn C^î /jf jimnin.

I I.

;'

5 \Remainêrl.^%h4CC]AJNuinb. 573-333 C^^.

3 074? 308 -2S5ifjygilr187.7^^1* *' ^^' ' ^'^f4.4328 ts: ^

it** ^ The



} '

By ihe foregoing le

,= x thetcforc,;c r-^=

^_

r-^;-c tQthc Hatoral Nprnber.fow4 J^ythe^ Remainilepadd ^he^mouat prppofed,and iram.the jUog^ofie Siipfiii^flradber

afore-fbund Remainder, which ffiallbe the tog.of r*. Thi$i

divided;by the Log. of* r' ihallQpbteihttin^^f^ (f.-

.

xaml^^ In .what tiipejill,anf Adouity of- aiwi .tftWlint tfr

^^ 43*8 at the r ite pf 6.fn C^r.'

,-. . ;-.

;i.: r ;

.

\ - -aI

';i 5f*iar=:34.4\rt:a;53d5584 ; : - T^b

: JLy. r-* ir=o.odgsfl,7^gi yi .'

.

'; '\'V-

-

*

Remainder 2.7584Q7%''^\N?iwkfc*7j33je?-(^T. 4 *

^l'^ li. 6i^4,.J^.S^ c:

r-*

r

rz. z.

,9253059)0.3 1#3237 (12* Years == T.

411 Tfie iaiiieEquation oiWerto find tvTi'atAntiuttycl5g^

= 4, whence the Rule. To the Log:etthe'amount' *

*'

*

v

X add the Lo of r i^ and from the Sum lubtrf^^theog.

o ^^- ijthe Number

'

aniWcx;lngtojtheReiftainaefis-tfteLo-arithmof jr.

*

Example.An Annuityforborn 1 2 f Years amounts ar.5 pr ^Sw*^^to the

Sum of 614 /. 4328. how much is that Anauityii *

Log. X ==2.7884744.O9O253059 :- .: / Z

^/ r- 1 =8.778 IS 13 I2-J

;i.5(JC)(^2i7^125530.,': ( --.

'0.0300^710,303^708

f Log.a 1.53(^5 tf-0,31^3^38^ i^ ?-r'-rr.2,071^$

/.

4 = 34^4.

''

Xi f.Q.^ciSJ\=1,07 1585 = ^' ' I

VL In orde):to findr, the (kme Equationis reduced to t

= ~ r rV or in our pxefcntCafe 16.^614 = i7,?tfi4r^rK

Which is fo affeded as not readilyto be refolved by the Gene-*

' tal



30 6fCompoundhUnfi dni Amkitiet.

ral method for Relblation of Eqaations,nn|efs wc on finft^api

proachit by fome other means. For wmch porpofeake the

rottowingule (which wiQ fafficewhere extream Exa neis i$

gi-_ijt

not demaiided. ) Let ~r .Z. s 1 4* y.^d let -r- ^h, I lav.

that^bh'\'iky'-^hiiexceedingear the locraifeof the Rate

orr 1.

WherefoW from Log.of the Amount, fubfhaftthe Sum of the

Log**of the Time tod Annuity,and the Remainder divide by

~-^ The QiiotefliaHbe the Log. of i -{-p From the Logi

of tf,fabftraa the Log. of ^4* i, and to ^ the Number an-

iWerins to the Remainder,add twice y. To the iaid Remain-er

add the toe. of * -|-y^ and half the Sam ihallbe Log. of

^kh'{'Zhy, from which SquareRoot fubdad h. the reOdoe

willbe very n r the Incr le,or r i i add addingi, the rate

r is feond. If extream Ezadnds be defired let r thus fotmd

be $SSfm^dfand r--- r* comparedwith -. i ; willalwaysbe

greaterthan it:and dividinghe Excefs bjr r'-x ^ ^ the Qpote

added to r fliattverifys manjr more Figuresin the rate ii

trere true in the aflumed r.

Example.

An Annuity6f 34. 4/.forWn 12 { Years amounts to 6i^t^

'432$It b

requiredwhat rate of Interelt isallowed..

2.7884744L g.X.

1.0969100 L g.t.

1.5365586L0 . 4.

2.6334^86

nl\ 0-1550058(,026957$= ^/- J +V ^ 1,0*404

z y .^ = ,12808

. -2= -^ sr * = 0,44444f ^c.Lor. 9^47^17%

iii r+i zy = 0,12808

*+ 2jf =;f 0,57252 Lijf, 9*7577936

hM*\'ity Log. 1940561 II

o S 4435 = V- 9-7028055mm

,059991 = r I wherefore

1.059991 =:r.

I. After



^

ft 4 A A i

1^ OfCiiînmdlTitm^MnijKiadHtf.

19437877. f. 9*97487^

.: ''- :- C v/fr.i.faogpo

.

r * i -

iîX 311210622 sift IJMjJOlt

..

:iL. If the Time.

be ftrnght,arangth Aamufift.^ts aiw^nk

t^driteof intereftgiven,r'wJU beeqiisdo ^ orthefice^

aiîdfcdy'thtalue of the Ilwerlion,te is'by';:^*--.

^^h^ncctfieRiilc.From L05.f the Armuity,abirafffceLAgJffthe Ji^jrcft9r rw i^^with what rceigiins.rreckf^lfetjinilNumber, which will be the Value of the Fee j from it fubftraS

the prefwt worthy the refiduc is the Wlue of the- Rjey^rfiW:Take thelLog.hereof from tne Log.of the Fee,'andthe'relidad

Villbe Lo^.of r^ Wherefore divide that refidueby Log.ofx^theQpfitieutitt be t the ntihilierf Years Ibfught.

...... ^

.

Example.Iti-what ^me will an Annuityof Totferjtmumyaj^tlfPsL

Debt of 1321 /. 3028. allowinghe Creditor 5 frr C^.perJnn^

1.8450980 Log.jo^zd* * ' ^ - '

Loi rev. 1.8959593 ::'

i32 ,^dt8-tag;fe- -. -

0,0211893)1.2501587(59. ; ^ J97i = HeverfisH. ? F

ingto the Suta^fttbtnafrrtmtheÂtn ity-i^'^'llfe'LagJ'rtRemainder taken trom Logarithm tf^deavetî^^ê^ before.

-r

.

- -

rr- -v .

^ ^

i jr- . 3 1210643 - _.c: .: Log. fp ?n ; r.?4fe9?5'

a, yr l .598970O70 i^/'3 ^4^. .594 yi

3 934^.

r

pro^

wiHTbemsri i^ ^ to .x,. 6 rvi ta -i the AaAiity foaghc.

\ ^*cre Ve'rom the Siiitif the Logariftoisf^x/afld4l(a^traathc Log.of i-'^,he Remaindtr.ftime tog.of.*. *[



0/Cmfomd Intereftnd AmuitiJesl 31

ExMHfU,

What Annuityto cdntinae 59 Years can bejporcluifedpr1321.3028. at the rate of five/ Cent,fer jimtHm,

Log. r' = 1.2501687

O, jir. Log. -jr=8.7458313 = ^056213

1 - ~r =,9437877 l g'''974874a

Log.Z = 3.1 21 0012

3Log 1 = 8.6989'yQo

1.8199^22

9,974874^

Log,a 1.8450980=70

IV. The Annuity,itsprcfentworth,and time of G ntiau

knee beingpropofed.It isrequiredo findthe Rate of Inte-

reft,that is Mj z. and t beinggivehto feek r.

This Prt blcm beingof more Difficultyhatt appears at firft

Sight,nd requiringhe fblutionof thisEquation= i r*

*^ ^^, to which it itreduced ; there Ihuft be appliedomc

Method of approachinghe Root r, which is by no means evi

dtnt : And that Approximations the Number of Years and

Rate are greateror left,annot properlye obtainedby one ge-eralRale ; but ratherbytwo, accordings the Value ofthe Re -

verfionis greater or leflTer.

Ifthe number of Years be great ( as fuppofe0 or upwards)felpeciallyfthe R^te of tnterefte high, -|- willbe nearly

the Rate, or more accuratelydlf ^ x i. Call it r :

and tt: w*'* be exceedingear the Value tifthe RererCoar

r*xr-r

Let itbe at, then i -|- i(hallapproachthe true Rate fuf-

ficiently.ut ifgreateraccuracy be defitM,by repeatinghisI rck:efsou cannot failof your Defire. Hence this Rule : Frooi

Log.of 4, as aliofrom Log.of x. 4*4, take Log.of i^ this lat *

,

ter Rdmainder Ihallbe nearlyhe Log.of the Rate Multiplythat Log. by r, and to the Arithmetical Complimentof cbe

Produd, add the firltRemainder* The Decimal anfweringothe Sum taken from the former Rate ihallgivea more corred

V V Rate.

]

I. ,.....' -?A



94 CfCdmptndIittmfMi AmAia^

Kate. With that Rate^ fedc the Rarerfioa after the Time ^

ven =? x^ which add to z. From lx e.of 4 take Log. of x,4-jr:

iThe Remainder fliallbe the tog*oftne Inteitftorr^ i faSr

cientlyear.

ExMtfte.

till. 3028/. ispaidfor an Aniniltyf 70/. f$r Amtm he 59

iTears to come, i demand the fUce of Intertft aUowed the

Torchai^.

4 Log. 1.84^0980 X, 4-Î^^S*3- 4}4î7K L0g 3 I2I0022 X L9 . 3.1210022

8,7240958 09012419S i. : $2978

_^

59 002^20

i.050458ae:Log. 0.021 3787 1 .322750. 1.050458ri t

%9 8.^77250 Co.-i4r.^

lk)g.r i.2 5i3433 ^'7H09tf V

yO)0458srîs 8.7029300 740134^.1^.001520.9.9 J4i733

Co. Ar. 0.03571^7

^ ^^ ^'^4^0980 1311.303 sx. ^tf ^ ^-v^S^vSd

1.^08247 7^.002 as Revii^ = ir

1 397.30c a-Frr t+jf2^. 3-if452912

Intereftfoog ,050096Log.8. 998otf8

if the munher of Yedrs be finallfhe aforefiud Ralelrfll a

Vail little. In this Ode it w$ be r q[aii2teo approach the

Hate thus. Let llLl be the Indei of a Root of Hifirooia t

Which Root take Unity, and the remafaider calljr,and let

~Y^^^^ * * fiy,that 1 H-bî/h k^zky h ioffici^

tatlyequalto r the Rate fca^tjand will be ItiQ nearer the

Truth, as the Number of Years is unaller } and the Error tiiatb

ivillbe always in Excels. Hence the Rule : Diride the Logaritfan

6f t LL by L:JL ud from the Number anTweriiigo the Quote

take Unity : Double the Remainder, and fubtraftitirafflk ; that

is from the Quote of fix divided by r*^ 1 : To the Loearitfcin

of what remains, add the Log. ot h. Then the Nnmoer an-

fwering to haU'the Sum of thoft Logarithm (akenfrom i ^b'Willleave the Rate ibught*



Of Chmpand htenftand AmttnMs^ 35

An Annnlty of 20 /. f^ Amnm^ to continue 2^1 Years is iol |

fi r 220/. i demand the rate of Intereftallowed the Purchafer.

=: 20 Log. 1.9010300 tf

^t^

ot

*

4X r XgZ32493''

X 55220 l^g 2*3424227 IOg

Vt.\ H #a8o8:^ 5tf(o.o2$52p7

2ys:o*t2io8

_

g? Log.9477i2i3

* -2 Jf 0.17892 Log,9;25Ztf5g9^ 02$itf8 l8772978o:

s+ ^;;^s:;r i.o S8a2 V P^?tf48j *

7lr# Rnî fought.

The Rate r thns found is alwa|s fiune ihuillmatter too big^the true Rate being i otf8i4;bat iS the Nnmber of Years arc^

fewer, the Error becomes infenfible. If greater Ezadnefs be

required,'twin be eafyby the generalMethod for the Refok^

f ion of Equation^ having to near an Approximation to proIe-

cnte this enqniry as far as yoQ pkafe. Bat this ieems aban^

dantly fafficiMt6r Ufe^ which ia our priiîpdlDellignin thii

Place.

tafityyBy way of Corollaryto th^fmaer^ Let itbe reqoiredo

fiod the kate of Intereft allowed the Porchafer when he pays a Som

r;tK^faxmkwniBf 9 0t wbcveiotK Ims almaily Tens ai; ^ to

hare itproloAg'dfor oartaia Tim ap A for ExaoipK I baT

im AoiuutF of 2o f. ^ Jmmn bm tha Term of ^% Yffara,and for

402. patd4owa, |(M ) T y Tempr(4oDgedfor is Years morca

( r to 31 Y ci. ideomd, plwt fUte f lotereft n allowed v .

Utile. Call 2 1 ^ 4- 1 by the Nameof T, and I Tlkall be th fU

dical Sign of a Root of ^. Let * 5 ^ ^ C9 *1^ ^ +/ * i

liJILlsft. I (ay the Rate fooght is very near to t-V^*- '

VTi~-TF7. As in the aforôiogExample



^6 OfCompoundInterejlnd Aitmties^

Log. nts 1.3010300 2 + x-i ^ =7*= 53

Xjfg. = i ooooooo

2.3010300 J[jii .~ai4, t*^

I .(Joiodoojc

~ ^

JT ss 2tf{)0.698^700(0fil6ij6i. Log. 1 ,0626 1 tf

.125232 = /

Log. b. 0.5105450. 3 24 b^

Log. 04934^57-3,1 i4768=b.2

1.PP39707

-v' 0-5019855 3-*7^768 V** zbf

The Rate = r = i .0^3232

or tf/. 61. 5^^ percent

Af witlbe readilyproved by feektoghe Valae of the Reverfioa

of an Annoky of 20/. ftr Jmum for ten Years i^Kr 21 ^ at the

rate of 1,063232 fcr Cent. The Work ftaodsthis.

f = 1.053232 Log. 0.0166280. Log.r

Zo^.7^0,5591880 Log.f^

8.80093^9 i^fr I

9*3601249 Sum

0.6398751 Comf. ArUb^

1.3010300 Leg.a

Roverlm. 8712781 1*940905 1. Log. Rewfjm ifi f1

0*2662800. io^.r*

Rtwrftott.7.2743 1.6746251 Log.Reverjm[iB^tr1

Fahic 40.0038 fo9^bt.

Thus it appears that 40 / and about one Penny; is the true Va

lae of the Difference of the Reverfions : by which the Reader may

judgebow near an Approximationthe foregoingole afords,to-ards

findingthe Rate of Interelt,hen the Vdneof an Annoicyfor a Term of Years to comnsence after a certainTime ispro^

pofed.

Tbt



(37)

I I II J..

t .

The Propoitons of Navigatiothat occurr in the Pr0ice of Sailingy

Mercator.

M thii CoDeftion of Tables^we (honld by no ineaBS have onrit*

^

ted that moft neceflaryne of the Meridiooal Pares,defigoedfor

Service of Naviptors, ifits UCbs were not foliyfupply'dby the

Table of LogartchmickTai^ents As is demonftrated in N 219 of

the Philofophicalranfii ons. It isthere proved, t^ That the

Meridional Line, or Scale oS Mireator's Cbart^is a Scale of the Lo-

garitbo-tTlngentsf the half Compiimentsof the Latitudes, zdly^

That fach Logarithai'-Tangeatsf Mr. Brig s*sorm, are a Scale

of the Diflfereacesof L^n^tnde,pon the Rhumb which makes an

Angle of 5 1 ^ 38'.9 ' with the Meridiaa And idfy That the Dtf

ferences of Longitnde,on differinghnmbs, are to one another as

the Tangents of the Angles of thole Rhumbs with the Meridian.

Hence it follows,that the Differenceof the

Logarithm-Tangentsof the half Compliments of the Latitudes,s to the [Mfierenee of

Longitude a Ship makes in Sailingon any Rhumb, from the one

Latitude to the other,as Tangent of $I^38^ 9 .( whofe Logar*

mathematical tables contrived after the most comprehensive method 1000024611

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