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ANEW 



' CoMPLEAT and Universal 

SYSTEM or BODY 



OF 



: : • . 



; Decimal Arithmetick, , 

CONTAINING, 

I. The Whole DoBrifie o£ Decimal Number fy not only the Plata 
and Terminate, buc alfo fuch as Repeat or Circulate ad Ififini* 
turn i and a Plain buc Ferfed Management of both, laid down 
and explained ii| all the Fundamental Rules of Plain Arithmi'^ 

' tick, and by Logarithms. 

II. The Application and Ufc of Decimal Arithmetick m all the 
Parts or Branches of Arithmetical Science ; viz. Vulgar Arith' 
meticky Vulgar FraBions, Duodecimaly and Sexagfjimal Arith' 
metick ; alk> in Algebra and Logarithms. In ail which its 
Excellency and abfolute Necejpty is fully evinced. 

III. Its Application and Ufe in all fuch Parts of the Mathemd" 
ticks as abfolutely require its ABfiance ; viz. Plain Trigone^ 
metryy and the Afts depending thereon ; as, Navigationy Forti' 
feationy Altimeityy ana Longimetry s Alio the Menfuration of 
all Kinds of Superficies and Solid Bodies ; and the Arts refult- 
ing therefrom ; as, Gaugingy Surveying, &c. 

IV. A New and Compleat Seft of Decimal Tables never before 

£ubliflied, fhewing by Infpe&ion the Value of all Kinds of 
lecimals f without the tedious Methods of Redudions hitherto 
ufed) to rour or fix Places of Figures ; Alfo all the Common 
Tables very much inlargedy correBedy and improved ; whercia 
all the Circulating Numbers a^e marked. With all other Ta- 
bles of Intereft, Annoities, Exchange, 8Pc, necejfary to render 
the Work compleat. 

y. An exaft and accurate Camn hf Lot^arithms for natural Num- 
bers. And thro* the Whole, leveral Things new and ufeful^ 
not here exprefs'd. 

By B B N J A M i N ]VJ A R T I N. 

L ND Ni 

Printed for 7, Npony at the White Harty iiear Mercers Chappet^ 

inCbeapJidf. M.DCCXXXV. 






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* I II I ■ ' 

»«tii . | - >i ■» . . l l lii rn fcMT iii Mm " ■■ ■■■■■* I > « » hi-i- i - fc 



The PREFACE 



1 



Prefume *tis entirely needlifs to write a Panegyric on 
the fuperlatrve Excellency of the noble Science ^/Deci- 
mal Arithmetick, Jince the World has already been 
fuffidently apprized thereof in the extream Benefit and Ser^ 
vice it has afforded the World 0/ Mathematical Literature, 
even thtf in its Embryo &ate •, much lejs doth it need En- 
^ comiums to fet forth its Nature j Worthy and valuable Pro^ 
^ perties^ which have been difcovered and illuftrated by iMf 
\ Improvements ; of which the enfuing Work is but (as it 
^ were) an Inftance. 

"^ The Book I here prejent ibe World withal is a regular 

Syflem of this valuable Art^ according to all ^ latefi Im^ 
preuements cf others^ and many (in the feverdl Parts 
thereof) of my own 5 the two greatejl of which are^ The 
Doilrine of infinite circulating Decimals by the leartkd^ 
Mr. Samuel Cunn *; the ot&er^ A New Sett of Tabled' 
Jhewing the Vcdue of any Decimal Part of any J^ege^y 
whether Money, Weight, Meaforc, Motion^ Time, fisfr. 
iy InfpeBion only^ to a fufficicnt ExraCtnefi^ without fh^ 
tedious Reduftions hitherto necejarify ufed ; tc;(?«* tofi . 
me not a little Ttme and Pains to dakidate^ but was ne^ 
cejfary to render this Art compleat. 

The Foundation on which I have built this Supers 
flru5lure is the abovementioned Gentiematfs fmall, but 
learned^ Treatije of the DoSlrine cf Decimal circudating 
l^umhers : But that great Mafter having laid the Forni' 
dation deep^ and in a great Kfeafure out of the Vu^af 
Ken ; / thought it might be of Service to young Students, 
a little to difclofe and lay it more open to their Fiew, and 
this was att I at firft intended to do ; but hmmg do'rnr 
that^ Materials came in fo fafiy that I went on and 
. ere^ed the Syftem of Decimal Sci^ce thereup&n^ as you 
here fie it •, of which take the following Account. 

A z Jn 

122144 



The PREFACE." 

In the Fundamental Rules of the Art^ viz. Addition, 
SubftraAion, Multiplication, ^z»^Divifion, I have been 
as plain as foffibly l could without Prolixity^ and Jhewn 
the compleat and perfeSl Management of both Plain and 
Circulating Decimal Numbers in each of the Rules^ info 
eafy and obvious a Method as the meaneft Capacity^ with 
proper Attention^ may comprehend \ and have taken care^ 
in its proper Place^ to give the true Reafon, or Rationale, 
of each particular and different Procefs^ efpecially ofthofe re- 
lating to Circulating or Repeating Numbers of all Kinds j 
.as I went on^ omitting nothing that I could communicate to- 
wards the perfeSling this fundamental and important Part, 

In Redudion I have perfeSled this Art beyond %vhat {t 
hath ever been x as I have not only largely exemplified all 
iloe common ana vulgar Methods of Reducing tOy and from 
Decimals^ in all kinds of Vulgar, Fraftional, Duodcci^ 
mal, and Scxagefimal Numbers, by Arithrnetical Operatic 
ens ; but have ccmpleated the Tabular Part, which has 
been hitherto very deficient ^ both in the Tables already ex- 
tant ^ and in the Want of others. The firfl ofthefe DefeSs. 
I have endeavoured to fupply by cor reeling j inlargingy 
and compleating the common Tables for reducing the vari- 
ous Dencminaticm of all Kinds of ^antities to Decimal 
Numbers •, wherein I have taken care to mark all the 
Circulating $r Repeating Numbers, Single or Compound, 
Jo far forth as they came within the Verge or Limits of 
the f aid Tables \ which hath not till now been done by any. 
The other Defel}, and that which renders this excellent 
Art mofi lame and imperfeSiy viz. The Want of proper 
Tables to exprefs again the Value of a Decimal in the 
Vulgar Denominations, or knotvn Parts of its Integer, 
without tedious and operofe Arithmetical ReduSiions^ I 
have Mlfo fupplied by the Addition of a compleat Sett of 
fuch Tables, and Jhewn their life in Examples of nil 
Varieties. Th^ Reader tnay have a farther ' Account of 
thefe Tables in the Place where they are infer ted \ of which 
I fhdll fay no trwrCy but that thefe are the fifft Tables 
of this Kind > that were ever publijhed. 

ll^at fhave hitherto faid^ relate^ purely to tbe-pj(ftuice 



;"»• 



The PREFACE- 

and Subjlance of the Art it felf ; what follows concerns Us 
Application to Arithmetical and Mathematical Sciences. 

In Vulgar Arithmetick, I hofve applied the DoSlrine of 
Decimal Numbers, both Plain and Circulating, in every 
Part 5 andjhewn its admirable Ufe^ Service^ and Expe- 
diency thro* the Whole. Particularly I have facilitated 
the Bufinels of Prafticc by a new Table and Method of 
Working by Decimals ; whereby the Difficulty and the In- 
Iricacy of this Rule by common Arithmetick is awided. 

In Incbange / have been very particular and explicit j 
having made this Affair (the Bafis of Merchandife) more 
intelligible to meaner Capacities, than I have met with it ; 
explained the Nature and Meaning of the Par and Courle 
^ Exchange, and hanje exhibited large and compleat Ta- 
bles of both •, fuch as are very rare to be found in Books of, 
this Nature^ tho^ they (ire an indifpenfable Part of De^ 
cimal Arithmetick. ^e Rules of Intereft, Simple and 
Compound, being of fo great and general Concernment 
and Importance J and yet fo little truly underfiood^ I hofve 
taken abundance of Pains and Care 4o let young Stu- 
dents y?^ the Theory, or true Reafon and Nature thereof^ 
by a perfpicuous Method of Refolving the mojl excellent 
Theorems of the late Mr. Ward, in Decimal Numbers ; 
which in this Part of Arithmetick are abfolutely neceflary. 
jfnd to facilitate and expedite Calculations of Intereft, / 
have procured a compleat Sett of Decimal Tables of In- 
tereft, and Jhewn their Conftruftion and Ufe. In the 
ether Parts of common Arithmetick, I have been large and, 
clear i and omitted Nothing that I could contribute to per* 
fe5l them. In fine^ in every Part 1 han)e Jhewn the Ne-. 
celBty and E^xpediency, and in the Whole, the Prefe- 
rence and foperior Utility of Decimal Computations. 

In Vulgar Fradlions, I have Jhewn bow dl ^uejiiont 
etre moft eafily and commodioufly refolved by Decimals ; 
cmd for that End have calculated a Table Jhewing by In- 
fpedion the Decimal equal to any Vul^r Fradlion whofe 
tknominator exceeds not Twenty, / have extended {be 
Table no farther, becaufe thofe Jm^l Fra^ions ate moji 
fre^u^nf itnd ufeful in Bu/in^. ,. 

In 



The Preface. 

In Daodectmal afid Scxagefimal Arithmeiick^ Tbave 
^fiewn the Nature and Rules of the Arts ; and^ by marry Ex- 
am^s^ how ^vtftions are m&Ji advatitagmjly wrought by 
Drcimal Numbers, efptciaUy in Dnoderimals, fo much 
ttfidin Mtnforatiofl, 1 ha^e - alfo inferted compleat Ta- 
bles jfer turning Duodecimal and Scxa^limal Numbers 
mto Dccima! ones^ and Jitch as are mt fo be found every 
where \ wherein (as in all my other TcAles) I home noted 
the Circulating Decimals, winch no one hefides bath done. 

In the excellent Art of Logarithms, / have not only ex- 
fiain^d the Principles and Rules of the Art itfelf hut large- 
ly Jhewn bow all Kinds of Decimals are managed ctnd orr 
iered thereby ; others have tanght the ff^orld the Manage* 
^w«^ <^ plain ^ terminate Decimals by Logarithms ; But 
/ftfl/<?f circulating ^ repeating Decimals of ail Kinds j hath 
fallen to my Province only^ Jince no one before has attempted 
it. I have here explained the Method offnding.the Loga- 
rithms cf any Repctcnd, whether fingle or compound, 
p»pe or mixed with abfolute Numbers^ with more Eafe and 
Certainly than can be found for any terminate Decimat 
exceeding the Numhers in the Canon. Iba^e fhewn how 
to work all Sorts a/* Decimals in all the Rules of this noble 
Art\ and Po make this Part of the Work compleat^ I have 
ieferted a C^rvcm of Logarithms for Natural Numbers ; 
where by Rules you are taught to find the Logarithms of 
any Number not exceeding i ocooooo, and the contrary. 

In Algebra, the Ufeand Neceffity of Decimals in rmfmg " 
and rtfching Equations, / haroe dcmonjirated in a Selc6t 
Number of the maji airious and uleful ^ejlions relating 
to the Theory of Arithnlerick, to the Philofophy 'of 
Motion, &?r. and Jhewn how lame andimperfetl^ even 
this mofl perfect and perfifling Art would be without the 
Subftrviencc of Decimal Numbers. 
• Bitherto of the Ufe and Application of Dechtiak in 
the fever al Parts of Arithmetical Science \ in the next 
Place take what concerns its Application to the Principal 
Parts of MathcoiatiGal Knowledge. 

1)7 Plain . Trigonometry, / have convinced the Reader 
how fibfolutely neceffary Pacimals are in order tp exprefs 

the 



The PREFACE. 

the ^antity of the Sides of all Right-lined Triangjfes, i> 
the Solution of all the Cafes of Right and Oblique-angled 
Trigonometry. And as this Art is the Foundatioa and 
Eflence offeveral otberSy as Fortification,. Navig?rtioiH 
Mcnforation of Altitudes and Diftances^ &?f. Iba'm 
Itkewife fhewn the Nature and Rudiment* <?//)&^ Arts and 
Sciences^ and the Manner of ferforming Conckjtons iy 
them in Decimal Numbers. Sa that any Terfm may hert 
hoth learn the Art of Trigonometrical Calculation, and 
its Application to the aforefcAi ArtSy after the beft Me- 
thod^ with the fame Eqfe and te as good Purpofe^ as from 
many Books wrote purposely on the Subje£l. 

In the Menfitration of Superides and SoUd$». no> one 
wHl pretend to difpute the Superlative Ufc ^/Deciaal A- 
rithmetick ; whereof the fmallTraJ^ Ihavc bereptUifh^i is 
ik Jkffident Inflance. Ihan)e not only taught bom U fmafun 
more Superficies and Bodies than any omer one Book that I 
know of hut fhew^d how this ^t is the very Bafis Md 
Subftance of Gauging, Surveying, md all other Kinds ejf 
Meaforements ufed by Artificers^ none of wbkb can be o^ 
imrfd to any good Purpfe without it ; njor any Operations 
therein, fb well performed as by Dacimal Aridunetick ; 
enid here I have provided the Gaugjer with a Table of 
Mulriplicrs or Divifors for fining the Content of any 
Srrperncics, or the Capacity of any Fejel in Wine Gat 
kms. Ale Gallons, Corn Gallons or Corn BuIheK ^^ 
iber the Dimenfions be taken, in Inches, Feet, or Yards. 

^tbm I home given a general Account ofthtSubfiance (ff 

the Book ; it would be emUefi to defkeni ia Particulars : 

Many TUnrsof Importance in various Parts of the Book 

wUt offer tiremjilves to the View of the Reader unexpeil^ 

$JSyy and (fppear in the whole-^ with the Face, of Nofvelcy, 

Ihave fpared no Pams in Confulting the beft modern Ai*- 

thors on each particular Head as I went on, andexfraSci 

from them feverally whatever I found of value and fFortb 

and would contribute toperfeS wy Bcfign \ fo that nothing 

of Cof^equeme can be found in any other Piece of Decimal 

Fra&ions (as this noble Art has been diminudvely cd- 



The PREFACE. 

led ) hut what may be here met with amidji a great Variety 
of other novels but ufeful and curious Matters. 

So that upon the H^hole I hope this Book doth truly merit 
the Title it bears j viz. A New Compleat and Univcrlal 
Syftem or Body of Decimal Arithmetick. 

If then any Perfon be deftrous of a Good and thorough 
Knowledge of Decimal Numbers of every Kind^ and of 
their Compleat and perfe£l Management by the Rules of 
Art, they may be here fatisfied. If they would learn its 
Application, or how to ufe it to the beft Adfvantage in the 
various Arts, Trades, and Bufinefs ofhife^ they will here 
meet with plentiful InJlru5lions^ and Examples in every 
Sorty adapted to particular Cafes. Would they learn the 
true Grounds^ or Rationale, of all Arithmetick whether 
Vulgar, Fraftional, Duodecimal, Sexagefimal ; and of 
^he Mathematical Arts, Menfuration, Gauging, Sur- 
veying, Navigation, Fortification, Altimetry, Longi- 
metry, Cs?f . let them pleafe to fpend a few of their fpare 
Hours here. Are they difpofed to learn the Ufe q/" Loga- 
rithms, or the Method of Trigonometrical Calculations, 
they are here with Eafe informed. In fhort^ they may 
here find in ^«^ fmall Volume, what I have been obliged 
to turn over many both fmall and great to procure ; and 
therefore if Variety, Utility, Novelty tf«i Brevity c^ff 
pleafe^ I hope the Publick will candidly accept my Labours; 
I am not apprifed of any Faults therein^ and am very fure, 
there are but few material Ones^ having taken all the 
Care and Pains I was able^ to prevent them. 

But if the well difpos*d and inquifitive Reader^ after 
having penifcd it throughly, Jhall then judge it defici- 
ent ; Ifhould be very glad if any thing better of the Kind 
Jhould offer ^ that may afford him greater Satisfaftion ; 
and till then only^ I entreat bis kind Acceptation /tfi?^ 
candid Perufal of This. 

VALE. 



THE 






J 






THE 

CONTENTS 

TH E Ifltrodu£iioH — *Page t 

C H A P I» 
Additfon of Decimals ~ ^ 

C H A P. n. 

CHAP, m* 

Miltiplicatioii — -^ v$ 

CHAP. IV. 

Divifion of Decimals t*- a 8 

Divifion contraded — — 34 

G H A P. V. 

kedu£iioh — — — 45 

j^^udion . of Vulgar FraSkions to Decimats 

47 
Re^uftioa by Logarithms — 48 

keiiiidioix of Mixed Numbers Xo Decimals 49 

Kedufkion by various Tables of Money^ 

. Weight, Meafure, Time and Motion 5 z to 54 

jpommoQ Tables of Money, iWeights^^ Meafurc^ 

^ ancfTime , . — r -^ • J5 to 58 

Deci- 



The CONTENTS. 

Decimal Tables of Money, Weight, ^c, 

P^X^ J9 to 69 
Tlie Defciipcioa and Ufd of i Sett of New 
' Tables for riiat Purpole 71 to 80 

T^e Tablei-themfelves • — • 81 tot 25 

q HAP. yi. 

The Ufc of . Decimals in all the Rides of 
Propoi-tion, viz. The Golden Rule Direft 

.1 . 125 to 127 

Ditto Inverfe — 129 

The DouWc Rule of Three — 131 

CHAP. VII. 

r 

A N^w Method of Decimal PraSice ijt 
A large General 1 able — — 15^ 
its extraordinary Ufe • -^ 13810141 

C H A P Vril. 

V ' 

• m 

%■ 

* 

The Ufe of Decfmals in Fellowlhip 143 

The Ufe of Decimals in Tare and Trett 145 
The Ufe of Decimals in Barter -r- ,145^ 

The . Ufe "of Dedfaals. in Gain and Lofs 1 5 *i 
The Ufe 6i Diecimals in Exchange - -^ 153 
Table of the Courfe and Par of Exchange 
. r \: . ry^ to 1 5^8 

Tablt of tlie Cotrrfe and Piar in Alligarion 160 
Table of the Courie aqd Par in the Rule of 
^ :FaIfePoficiQn , *^~v:; ^ ^' i66 

jTkbles of the Courfe and Parr in- tlie Rule of 
Eitratdion of Roots ^ — ^ — 16^ 

C H A P, 



I 



h iil'ii III M ii«— ■»! mi ■ r~ • — »-*~ * T..'*t .■'- m * ' M *t.-n.- ,^< «ri- ■ »i »■ .n «iim 



1 

J 



01^ CONTENTa 

G H A P. IX, 

The Ufe of Decimals jn Intereft paee 17? 
The Ufe of Decimals in Simple J 174 
The Ufe of Decimals in Compound iy6 

Table ot Time — — 201 

The Nature, Cpnftruaion, and Ufe of the 

Decimal Tables of Simple Intereft 203 

Decimal r<j;J/f J of Simple Intereft — 210 
The Nature, Conftruaion, and Ufe of the 

Decimal Tables of Compound Intereft 2 2«i 
The Decimal Tables of Compound Intereft 

235 to ijS 
CHAP. X. . 

The Ufe of Decimals in the Arithmetick of 
Vulgar Fractions — ^ .^rp 

The Ufe of Decimals in ' the Arithmetick of 
Duodecimals — 267 

The Ufe of Sexagefimals — r— 27} 

C H A P. XL 

The Ufe and Management of D.cimals <aft6r 
a new Manner) by Logarithms — 27^ 

Addition and Subftra^ion thereby 278 

To find the Arithmetical Complement 279 
To find the Logarithm of any Repeatin*' De- 
cimal ^ — — .— " 2S0 
Multiplication by Logarithms . — iBd. 

Divifion — - — jgl 

The Golden Rule — »— . jg_ 

Extraaioa of Roots — ^ 2^9 

CHAP. 



the C O N T E K t J., 

C HAk xii. 

* * • • ' '. ■*•<•* 

^Ije tJfe of Djecimals ia Alg.ebra cixemplified 
|n the Solution of thirty four Seled, Curioucr^ 
and Ufefiil Qjjjeftiohs* ,. !P^^?pQ to 31 1 

CHAP. XIII. 

The Ufe of Decimals in Pla^i Trigonometry 

312 
The Ufe of Decimals in Navigation .318 
The Ufe of Decimals in Fortification 322 

The Ufe of Decimals in Altimetry — 327 
The Ufe of Decimal? itl. Longimetry 33 1 

• : : :C'H A^P.' XIV. ' ' •" V ' '■ 

• , - • * - t • 

The great .Ufe of peeioi.aJ$; ia all Kinds- <tff 
. Merifutaripn, viz. of Superfiqips; ; -^333 

— r-— -of Sblicjs . , "^r ; . ,-j^- --- -343 

''-^^^ of Artificers Work ' — 354 

of Gauging, ^^f:> — 355 

— in Surveying ~ , — . 35.7 

A Table of the Logarithms to all^Nmnbert^ 
iiot exceeding One tXhoufand, ■ orFoiir, f 1^- 
ces,- whether they be Intire, Broken, 6f 
iMixt Numbers — 36i"to the EncJ. 



F,ii n I s. 



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^^^^m^k 



THE 



INTRODUCTION, 



CONCERNING 




The Nature, Kinds, and Notation, of 
Decimal Numbers. 



H E excellent Art of 7)ecimffl ytrjthnJctkk de- 
rives its Name from a L^rinWor.i (^ijiz Dccetn^ 
Ten) which denotes tlie Nature of its Num- 
bers, which repr^fent the Parts of any Integral 
^antity divided in a jDecitpe^ decimal, or terifold 'Prc^ 
forticn. 

1. Any Integer, or whole Quantity, being divided into 
lo, TOO, looo, lOOOO) ^c» Partu, becaufc rbofe Numbers 
are in a decimal or tenfold ProporMon 5 therefore fuch 
Numbers as rcprefent any of tho(e Parts, are calTd 2)^- 
cimal Ninnhers^ or ^Decimals. 

5. Thus, fuppofe I divide any whole Qiiantity into 10 
Parts, and take 7 of them $ thofe are can'd 7 dcclm .1 or 
tenth Parts of that Integer 5 and are thus vu'g irly wrote 
^Vj fuppofe it divided into roo, icoo, l^c. Parts 5 then 7 
of them would be exprefs'd thus --^^j -x-t-s-% ^C, and read 
as before. 

4. Al^o many Integers and Part?? of apotber, would b^ 

cxprcfs'd thu.s 87V. iPiVr, 4757V-5V. 2 \-^ Vr, iJ?^. and 
read thus 5 8, and 7 tenths j 19, and 6"] Hundreths, or 
Parts of a roo j 475, and i54Thoufandrhs, or Parts of ^ 
Thoufand 5 2, and 2945 Parts of ten Thoufand. Thui 
the fDetJominator of Decimal Parts or Numbers, is always 
an Unite with Cyphets annrxcd. 

B 5. There- 



ft ^he InfroduBiorty concerning the 

5. Therefore if the Places of Figures in the Nnnierator 
be equal in Number to the Places of Cyphers in the 2)c7i0' 
fninafOfy (or be made equal rhercto, by prefixing Cy- 
phers:) the ^emmtnator in fuch a Cafe will be known, 
tbougn it be not written ; and rberefbre in the Notttio* of 
Decimal Numbers, is always omitted ^ and the Numera- 
tor ^ordered as aforefaid) alone is join'd to the integral 
^antity^ with a Comm*^ or Poitit, to diftinguifh it there- 
from. 

if lT#»nr#» 5 4. _4 ? . '♦ f 7 . 1 7.?, . 4 . 5-»^ 

O* XlCnCC f-y-y ♦ i-i-yT 9 T"flr5"T > JTZ-i tT * T-y^T 1 C>^» 

are thus written, .34; .04155 .0027*5 .004. Alfo 

tlTr\ 58i4t; 1^9x444?; iW-^t^ are wrote 27.5 ; 
j8»o4 5 129.0132 5 1,0017. And on the contrary, by .12 5 
I.7(r 5 .oo<J f 2|003; 5 .0015 wc underftand ^LL. ; lyVs-V 



7. Cyphers prefixed to decimal Numbers, decreafe their 
Value in a de<u|^le or tenfold Pr^ypettien j Us aPfExed to 
Integers, they increafe their Value in the fame Proportion ; 
thus. 5 5 .05 J .005 5 .0005 5 l$c* are» as they proccc(|, 
each one ten times lefs then the preceding Decimal 5 as 
IS eafy to conceive. 

8. When the 2}enominator is an Aliquot Part of the 
T^nmerator increafed by Affixing Cyphers thereto, the De- 
cimal equivalent to fuch a Fraflion, will be compleat and 
terminate 5 as, ^t^. — .55 j\^ n= .^5 J 44 = -75 5 
tV=.055 i-J-T = -o*5$ i4r='^J75 5 :?-k=3=.GJ09i25 5 

9. But if the Dcnomimtor be no aliquot Part of the 
Numerator thus increafed 5 theDeciixial equivalent to fttch 
« Fraftion will be interminaie o^ endlefs • that is, it will 
fconftantly repeat one Digit fenly 5 2« -f :=^ •?33?53» 5^<?. 
ud infinitum 5 or 4. :i:r .666666, (^c. or yV ^=^ • ^'^S 5 ^ >5» 
5?^. oc^ty rn,i;8888> 5?^. or 3y^!r::.0 52i9i^<?<J<J<f, iSc. 
fine fine. 

10. Or elfe a certain Number of Figures perpetually 
Circulate, or repeat in the Quotient. Thai >t- ct: .18 18 

18 i9,i$c,adinfipift$m'y airo,-J^±3.i85 185 185 i85,5J;£?, 
And V? =^ .95*5805152980, ££?<;. And -^l^zrz.oi 36^6^6^ 
'i^c. without End. And tbofe Numbers which thus inll- 
ii'iccly circuliite or repeat, are moll fitly termed Re/>ere72/is. 
\l bofe which fircula;6 ^ Digit o^ily, ar^ called a /v^'S^^ ^^' 

^eteijd ^ 



I 



Nature^ Kwds^ and Notation of Dsfimah* j 

fetend 5 and thofe in which feveraj Figures circulate, aro 
caird a Compound Repetend^ in the fojiowing Tra£i. 

11. For the greater Elegance and Perfpicuity, in all 
the Operations of Circulating Numbers, I have dafhed 
the firft and laQ Figure of the Refetend ; thereby xnaking 
Qne Place of the Repetend fufficient. Thus the Exam- 
ples above are thus wrote or exprefled; ^9 -^^ -S^^i 
\i3Sj .032291^. And the Comfouvd Repctends thus ^ 
.T^ ^ .Xo^'^ •^^5238,0 5 and .0^^^ j herein following the In- 
genious Mr. Cunn^ the firft Improver of this Part of 25^- 
cimal ^rithmetick, 

12. jin a Compound Jiepetend^ any one of the circula- 
tif^ Figures tray be taa^ie the£rft of the Repetend^ for 
l4[iffance, in the R^petendi.b^i:^ 325 32^, ^c. it may bo 
jinade 8.634:5^ 5 pr 8..6j2^32r. And by this Means any two 
.or nnore R-epeier/ds ma^ be made to begin and end in tbo 
fan^e Place ^ and then they are faid tp bp conterminous. 

^3. Several o^her ^^hin^^s relating to the Nature and 
Prpperiies pf circulating Numbers, J have interfperfed iiv 
the foHowing Treatife in their proper Places, where they 
i;Day be undcrjOipod^ and wjiich are pot to be found in any 
pther Book of decimal ArithmeticK 

i^. Jn all ^eqim^l Nnmhcrs^ if the Poiat of Diftinflipa 
te removed one J^lace towards the Right ijand^ every 
Tigjare, >and confeauently the Whple Exprcffipn, will bo. 
Increased in a tenfold Proportion j as in thofe Decimal 
^xprftffioos 3.756, 37.56, 275,6, 3756. which are eaph. 
one 10 times sreatg: tbsin vhe preceding one. In which 
J'xpporrioo alio, 'tis manifeft, they decreafe in Value, 
by removing the Decimal Point a Place to the Lett 
Hand. 

15. Xhe Nature and Properties of ,2)ecimal Uumbers^ 
ar« the fame with thofe of Integers .or Whole Numbers^ 
ai\d the Method pf Working botn the fame (excepting 
Repetend^. Hence ariferh the Excellency and fuperior 

'Ufefulnefs of 2)ecimaJ jirubm^tii;kf abovie all other kinds 
of Computation. 

16. To make the preceding Propofition evident, fup- 

fiofe 'twere'required toexprefs theTime.finceour Saviour's 
n<;arnatian to the Year prefent, in Centuries and Decimal 
Parts of a Ceatury 5 it would be thus 17.33 j wh^re you 
obfcrve one half of the Number coniill of Jufgers^ and 
the other jbalf of fDecimals. But firppofc the Time ex- 

B % prcli'd 






4 ^ke TntroduEiion^ concerning the 

Prcf>M in Years, the Number confiils of the fameFigure.% 
^733 » ^"^ ^^ whole or integral. 

17. Hence 'tis plain the fame Number may be either 
Integral ox Decimal^ and that either in Whole or in Parr, 
according to ahat is made the Integer 5 for in the forego- 
ing Cafe, if a Myriad be the Integer, the Time will be 
expieffcd by a pure Decimal 0.1733 j ^^ * Century be the 
Integer, by a mix'd Decimal 17.33 j if a Year be thela- 
teger, by the integral Number 1733 j as before. 

18. AH the diflferenr Species, or Parrs of different Kif^Js 
and ^c72imivaticnSy of Money^ Weights^ and Meafures^ 
and all other Qjiantiries, are to be reduced to 2)ecnnal5^ 
or may be exprcflfed in 2)ccimal ^arts of their refpeftive 
Integers, by proper Tablei calculated for that Purpofe 5 
alfo any i)ecimal may very nearly by Infpeftion only 
{"without the tedious Reductions hitherto ufed) be read in 
the vulgar Parts br Denominations of its refpeftive Inte- 
ger, by a Set of new Tables, which I have compofed for 
theEafe of thofc who are converiant in this excellent 
Science. 

19. Since then it has been fhewn that Decimals are the 
fame wirh whole NiftnbcrSy as to their Nature and the 
Manner of Operation 5 and that all mix'd Numbers, or 
fuch as confiit of divers and different Denominations, are 
reducible thereto, and vice verfa j it follows that all the 
Arithmetick of mix'd and heterogeneous Numbers is to 
be perform'd by Decimals^ with the fame Safe, Expedi- 
tion,- and Pieafure as that of 'xbole Numbers. 

20. And by Confequence, That Vulgar Arithmetick^ 
Vulgar FrafliouSy Ducdecimal and Sexagefimal Arithme- 
tick^ Cthofe Parts of the Science of Computation hitherto 
deem'd fo hard and intricate, and therefore but little 
fiudied or known) are all by this noble Art of Decimal 
Arithmetick perform'd with the utmoft Bafe and Pieafure, 
that any Arithmetick is capable of, and which I have a- 
bundantly evinced in the Sequel of the enfuing Work. 

21. The Figures of ^Decimal Number are to be nume- 
rated as thofe of whole Numbers, viz. from the Right 
Hand to the Left ; but they muft be denominated of the 
Number of Parrs the Integer is divided into. The follow- 
ing Table will make the Numeration and Denomination of 
Decimals very eafy. 



Nature^ Kinds, and Notation of Decimals. 5 

A Table of the Numeration and Denomination of 

Decimals. 



s 



•2 2 



«» 7^ 






o 



8i2 o 8 



«0 <-M 

c o 



go© 

8j 



w G S 2 c S 



.2 •o'O 

3 C G 






o. 
o. 
o. 
o. 
•o. 
o. 
o. 
o. 

0, 



0000000 




o 
o 
o 
o 
o 
o 
o 
o 
I 



o 
o 
o 
o 
o 
o 
o 

2 
2 



o 
o 
o 
o 
o 
o 

3 
3 
3 



o 
o 
o 



o 

4 
4 
4 
4 



o 



o 
o 

5 
5 

5 

5 
5 



o 
o 
o 
6 
6 
6 
6 
6 
6 



o 
o 

7 
7 
7 
7 
7 
7 
7 



o 
8 
8 
8 
8 
8 
8 
8 
8 



=: Parts of the Integef. 

= Parts of Ten. . 

^^^ Parts of an Hundred. 

=^ Parts of a Tboufand, 

= Parts of Ten Thoufand, 

= Parts of an Hundred Thoulknd. 

=: Parts of a Million. 

rr: Parts of Ten Millions, 

== Parts of an Hundred Millions. 

=: Parts of a Thoufand Millions. 



9 
9 
9 
9 
9 
9 
9 
9 
9 



Thefe Numbers arc all of 
them the refpeftive Parts 
y of a Thoufand Millions, into 
which the Integer is divided 
as above. 



Aa 



6 Addiikn if J)ecimaji. 

An Explanation of the Charaftcrs and Abbreviature* 

ufed in cbe foMowkig Sook. 

It has been of late an Expedient to avoid Prolixity in 
Writing, to make ufe of fome convenient and fignificanc 
Charaders to exprefs thofe Words wbich maft often occur, 
and occafion Tedioufnefs and Taut^lqg^ in tbe Work, the 
moft irkfome Vices that can attend it 3 and accordingly I 
have here ufed thcm$ which, with sheir Cignifications, 
are thus to be underdood. 

Figure. Names. S'^wficatio^s* 

*f" Flua, or more. As tf 4* ^» ^ ^ more h ^ inAddtticf). 
-*- Minus, or leG. As ii — ^, is^^i tda^ ; b Subica^^licsi^ 

X Multiplied into. As 4 x ^, is ^ multipl. intolr^ in Mulr. 
*r- Dividbd by. k^mr^Jf^iia divideil by j( po BiviC 
ss Equal to« Aa 4( c:^, is ^ equal to b y in Eqnaj^^ 

:) Is to. SAa^ : hM ca d ^. as ^ ia tp ^, fois 

Hj So is. c -cto ^^ in £rof>ortions. 

@- Involved. As -2 Q-, is the 2d involved. 

tu# Sv«l vcd. As 1 4UI, is -tho 2d evolved^r ;cxtraj6l«ll. 

t As Z*^*, %r^gb^^ai, ^c. is the Root 
1/ Surd Hoot. I fiqnare, cub*d, biquadrate, ifc.^f 

\ ab^ Surd. • 



ir 'igi ^y^mm 



CHAP. I. 

Addition of V n ciM A h s. ■ 

ADdition of Decimal Partf admits pf various Cafc;^, 

according to their different Kinds 5 either as they } 

are terminate and compleat^ or iuterminate^ and ' 
continually repeat either o?2e or more Figures. I Ihall il- 

luftrate all the fcveral Varieties by fuitable Examples of | 

Money ^ IVelgbty Meafure^ i^c. \ 



Cafe I 



Additkn of Dfcimdis. f 

Cafe I. If your Decimah ke ternrirmc, place Units un- 
der Units, 1 ens under Tens, ^c, in whole Numbers, and 
annex the DecuSaU in order towards tbc- Right Hand $ 
then add theao, and cut off from rb« S^m to the Right 
Hand (b many t^laces foir Decinn^ls a» are equal to tho 
greateft Nun^ber of Detin>al Pia«es m^ any ot the given 
added Numbevs; 





(I. 

1. 24,025 Then 

1 20,0125 /, s. d. qr. 

r »9.4»75 i i^, 23 . — . — 1,4 
1 i«,<»S "i p,t±(>^%^zfi 


EXAMPLE 
Add together 

• 



.lUit i\ r t •-- n, I y^^ .l_t I i 



I'he Slum 1321^3125 52f 13a — 6 ^ 3 ^ o^ 



/. Troy. 
457,825 
570^065625 
Examples 11. J ^4,5375 

Add tc^etber *> 806,253125 li 0^ f^. gr. 

I o,do3^i25 f 43 m.-^ i--o,j6» 
1.695,05 I 73.— 8— 15— 4.8 



J 



4i>.«Ai«««ai^,^a^..i..Mte«-Ma)«»<Mi 



The S^m 2553,73.43-75 s= 2553 — 8—16 — 6 



Cafe 2. If you'bave a great many feveral Sams to add, 
and their Decimals run to a great Kumber of Places, it 
will not be neceffary to add them all, but only fo many 
'Places, as are fufficient to give the Vuttie in the aggregat- 
ed Sum, which will require but 4 <$r 5 Places, or 6 at 
moft • for £0 far only the largcft TabteS go. 

Obferve ro make ihat Figure (at which you break off) 
more by a Unit, if tbe next rejefked Figure be more than 
5 5 but if the next Figure be lefe than 5, rejeft the Fi- 
gures only. 

Then add the feveral Sums, and^the cef rain Places of the 
Decimal are generally fewer by one than the Decimal 
Places retained in any of the given Sums. I fhali fubjoin 
an Examph at hrgCi and the fama thus contracled. 

EXAMPLE 



S Addition of Decimals. 

EXAMPLE. 

47.982774354 f -^ -» 4798277 
«7S-6732956 « 175-6733 



*-4375*432 



e I 



2-4375* 



97.7020o6764-^ | j* 97.70201 

276.92301762 S 1 276.92302 

30.00420999 ^^ J 30.00421 

fc— ■^M I ■ I ■! 11 W «^^— ■ I ■ — 

630.722828648 630.72283 

In this Example there are 4 Places of Decimals certain, 
or the fame, in both Operations ^ and the Rule will fcarce 
ever err above an Unit in the laft Place. Now whatever 
you fappofe the Integral Quantities to be, the Tables 
will ihew you the Value of the Decimal Parts, 

Cafe 3. Suppofe the Numbers vou are to add have re- 
featitJg Decimals 5 if they are Jingle Repetends^ make 
fbem all conterminous^ that is, end together 5 and then 
add as before, only to the laft, or Right Hand Place oF 
Decimals, add as many Units as there are tTtnes in it ^ 
and that lafi Digit fliall be one of the Repetends. 

N. S. It may be oroper to give the firft and laft Places 
of all Repctends a i)ajlo with the Pen, for Diftinftion. 

EXAMPLE I. 

/. 

124.2^3 
64.5 li^ 

0.^33 
59.80O /. s. d. qr. 

~ , — . — 2,88 



3.8a'3 5 »30= , — . — 2,{ 
45.0'^ <! ,73 = 14—7—08 



Sum -=. 297.7^3 = 297 — 14— 7 — 3,68 



EXAMPLE 



Addition &f Dtcitniih. 

E X A M P L £ II. 

Feet. 

0.0208^ 
2.5625a 

6.O4ITO r,i6r2: . — .17 
2.1^6666 1 ,04^=: ....-1.92 

Suta as 2a,04I;tf6=22— . C— 2 



iiXAMPLE m* 

4,72708a 

0.00208^ 
p.oapi/5 
4^31250 

^,03S4Jj* •«• /«>/v jjjr. 
10.7583:33 f ,34 = . — 1,58 

6.910666 1 ,08 =r t — 14^ 

45.o83'333 = 45 — 1-16 



EXAMPLE IV. 

275-252777 
47 87TI 1 1 

436.027083: 

10.677777 

ioi.2$5555 D. K 

1 27.769444 r ,20= 4_ 2,8 
243-95 8 jTS 3 l ,81 = 19 ^ 26,4 

Sum ^ 1 242.81 2C8^ rr 1 242 — Ip .— . 2p,2 



C/7/J» 



i o , Addftion ef Decimals. 

Caftf 4. If your Dedmal be a compound Refeiend^ that v^ 
confifts of feveral Places of Figures vf bich continuaUj repeat 
or return ; the Sum or Aggregate of any given Number of 
fuch Decimals will alfo repeat ; and the Number of Places, 
or of the Figures, in each repetend, wiBbe equal to theleaft 
common Multiple of thofe feveral Numbers which repreient 
the Places of Figures in the Reprctends added. Hence (the* 
it be fcarce ever rteceflary to have above five or fix Places of 
Decimals, yet) if any one be minded to fee the Repetend 
compleatj he muft obferve this Rule ; 

From the Place where aH the Repetcnds hegin together^ 
continue each CVrimal ca a Number of Places equal to the 
Multiple aforefaid\ then add, and to the laft Place add as 
many Units as there are lo's in the Place where the Repc- 
tends all bejs^in together, and the Figiiires in thofe two Plaees 
are the firft and the laft of the Repetend. The Examples 
following will make all plain. 

/. 
# 13 . 046^ /. J. d. qr. 
EXAM. L ^ 2 . i^04^/,5o -—1—1,78 

Sum 21 . ^613^—21—7—6 — I 




EXAM. 



Sum = 47 . 713338'^= 47^2—23—14— 4,82 

Sods. 

r-121 . 47237 

EXAM. lU. ^ «° • J7J55 R:^Jd. F' h- 

^ 64 . 90^34? J)07— . — . — 1,73 

C 8'9 . 07444 \ ,72=3—2 10,56 
Scm as ^55 . 728-7* =;555 -4—0 — 0,3 



EXAM- 



JdJttieH of DedmaU' ii 



Ye art, 

8#.263'4 
EXAM. IV. .^ 1*^-4*26 



7^.3 27P Y. M. W. D. H. 
^05.7^105 ^ ^6= , 1— 16,5 

Sum = 5P9.^64,(^ =:5p^_7— I— 3_ 2 



Signs. 

2.52475^^ 
EXAM. y. it'Z^fJ 



5.2x521$ S/^. ^ ' '' 
7'.07O7oy 5 ,68= . —12—14 

0.7-3261^ 1,78=23 — 24 — 

m^— — — ^— i— — — ■ •• .■ ■■,■■■ 

Sum = i.7'868c7 == 1—23— 36— 14 



Feet Square. 

^'.041 04 1 041 041 

EXAM. VL^ ^.7373737?727? F?- ^?- fi.f 

4.0O582j^a65826 r ,01= — 0,23 

4.73^473147314 1.53-7^— 5»i2 
Sum =3 16,^3010943109^ = 16 -76— 5,35 



Thefe Six Examples, I imagine, are fufScient to iliuflrate 
this laft Cafe of compound ReftUnds ; but if it chances to 
happen that a comfleaX or terminate Decimal be to be added 
with them, you moA^^&x Cyphers thereto, to efteem siiid deal 
with them as a Refeiend. 



C a CHAP. 



;x i SubftraBioH tf "Decimals. 

CHAP- IF- 



SUPSTRACTION. 



Cafe I. TT F yoar Decimals be ierminAie and comfleat^ pkce 
I asdireaedinthisCafeof ^^i/f/o«,^Subftra£l 
JL as in Whole Numbers ; ims^ining all the vacant 
Places fill'd with Cyphers. , 

EXAMPLE I. 



Z. /. u d. qr. 

From 729.47?6 f ,56= . — 1—1,38 
Su^ftraa 634*927 1 ,54—10-^ 9^M 

Remains 94'S45^ — H ' '^ - ^Q - ^>78 

EXAMPLE IL 

C, C. CI* It. oz. dr. 

From 472,07 J y^6:= . — . —8—3,88 

Subftraa 392,4354 l.63=2^i4^-3-«5,36 

■■ ■ \ ^ - ^ -' ' " • 

Remains 79,6346 = '79 : 2 : 15 : o : 9,24 
EXAMPLE HL 

/*. Ih. ox. fwt. ff. 

Frotq 147, f,52=.-I— 5.95 

5ubttaa 94)7248 1,27— ?— 4 — 19,2 

Remains 52,2752 = 52— 3-- 6-^ 1,15 



C^/> 2. If your Decimals run to «z4«)f P/^rt'i of Figures, 
do as dircfled in the Cafe oi Addition ; Sindfuijiradi as i*i the 
i7/?Cf/^;'*aiid*the laft place ofr the Decimal Remainder will 
never Err more tbnn an VwU 



EXAM- 

1 -»*' 



*i 



SfthfiraBhtt of Decimals* t| 

EXAMPLE I, 

Ftpm 2,752804624 tajfe 1475937579. 
Mlesy M. F. P. r. F. In. 



Thus i ?»7$2805 ■< ,58= — X-.4— 2— t,i4 

*°. » '\i,476j;58 6,27:^2—6—2-0—7,2 

Remains 1,275867 = i— .2-*8— o— -i — 6,57 

In this l^mple the laft Figaie 7 is a Unit too mudty baC 
that b not CO be rejgafded ; tor iq this Cafe the Value of a 
Unit in i^t Place is but ^63 of an Inch. 

* • * ' 

Caje ^. If your Dedmal refetiX Single Figures ^ proceed 
(as in tlus Cafe of Addition) to place them, and fubJhaS 
as ufiial; except that when die 'Subtrahend is the gieater 
Number, you mufi increafe the upper Figure bj p only, and 
in every iucb C^e carry one to the next Place. 

EXAMPLE I. 

£• /• J« a. qr. 

From 54i7?3?3 J'jPi^-T^^o*?? 
Subftraa rj^^if T,77=i5-4— 3>^ 

•i*i^P«"^»i^""^"^^ ^^m^ma^a^^^am ^^mm^mmmm^^^ ^i^it^mm^^mm 

Remains 36,7791(^=5 36—15—6—9,93 

EXAMPLE n. 

From 57,5289 f ,56= 1—1,38 

Subftraa 49,58?? C,94=i8— 9—2,4 

■ PI . 

Rpmains 7,9^5^^ = 7— } 8—10—3,78 



EXAM- 



14 SuhfiraBicM of "Decmahl 

EXAMPLE IH. 

Hogsheads. H. G. f. 

From 1672451/^ f ,i6z=:.>^.8 
Snbftraft 879,3000 \ ,i5rirp— 3,6 



Remains ,7P3-i5i^ = 7^9— P— 4»4 

E X A MP LE IV, 

loads. L. Q. JS. G. 

From 47jr,222 C ,60=— •.—1,92 
Subfiiaa ^6yiy^ Z P4=.— 1—4,6 



Reoudns 126,04^ = 126^0 — i — 6,72 



mm 



EXAMPLE V. 

Yem. r. M. JK D. a. 

From 47,957200 5 >88= • •— • — ?--S>o8 
Subfiiaa: ,0083x1^ Lt9f=:i2 — l-^o — 2,4 

Remains 47*948883' = 47-^12—1—3—7,48 

Cafe 4* IF the Dedmals be ean^oufsJ Refetends, order 
them as aire£ted in the Cafe of Aadiiioni Uien fuhfttacl ; 
and look if you mufi i)0|row one ta the rlace where toth 
Hefeiends begin together ; if £>, you muft add one to the 
Right-Hand place of the Subtrahend ; and the Remainder 
cither Whole or in Part, vdllfrew the RefeXend, 

EXAMPLE i 

L. U s. d^ qr» 

From 47,4^7^178 f ,2i=s . — •—^2,01 
Subftraft 15,^1^56565 4,85—17—0 — o 

Remaius 31,8:^21612: •==. 31—17 — 0^—2,01 



•iWi 



EXAM- 



. SuhfiraBm. cf Detimahi ^ 

E X A M L E n. 

Ounces. 6z. fw. gr. 
From 153.^274? U^9—-^ 2,35 
gnbftraflt 142*8^5 J l,07=i— P,6 

Remains 1 1,0^49^ = 1 1— i^i 1,95 

I I" I a t 

EXAMPLE m. 

RodsZqh Rq. Yq. Fq. 

From 75-5?33 J, 35= P4 

Subftraa 42.7597 1,77=23—2,44 

Remains 32.773^ = 32^23^^39^ 

E X A M P L E IV. 

K»v/j. r. F. /a. Q. 

From 47.8^oado' f,8i=r. l,i5 

Subflraft ^0.92^91^9 1,92=2—^—0,48 

Remains 6,9Jrtiooj'=: 6—2—9—1,64 



■» ■ 



EXAMPLE V. 

Days. D. H. M. 

From 75.2758000 r,94= . — 13>53 
Subftraa 47>^5i^'3563 "1,91=21—50,4 

Remains 27,919443)^=5 27 — 22— 3,93 



■»n-M* a^MMMa*^«>Ma«aW»»i 



EXAMPLE VI. 

Degrees. D. ' '^ 

From 49,S'2«'5285 ^,4?=.— 17>^4 
Subfiraft 38,4736000 (,05=3— o 



iw»«^*i 



Remains ii,o549ar8> =: ii—:}— 17,64 



CHAP. 



' m 



ii> mitipJicdtiSn tf Tetimah. 

CHAP. in. 

MuLTIPLldAtlON. 

Cafe L Y P your Decimals be compleat and ietmiftdte^ 
I whether they be fnire or join'd with Integers, 
-^ Multiply them as if they weVe all wboJe Num^ 
lets ; and cut off (to the Righ^-Hand) h many Places for 
Decimal Parts in the Frodud as there were ii>i)oth the MuU 
ttplier and MultifUcand counted together^ B^t if it fo hap- 
pen that there are not fo many Places in the Produfl, fupply 
the Defed by prefixing Cyphers. 



EXAMPLE I. 



Multiply 32.12 7 p^^^ 

9626 Fq* Inq. Qr^ 
12848 i ,60=. — 15,8 

6424 1 >5i^73— 7>04 
Produa 780,516 = 780—74— 4^4 



EXAMPLE IL 



Multiply 42,51 



ply 42,51 V Yards. 
by ,241 i 

4251 K?. Fq. Inq. 
17004 f,9?=.-i2,84 
8502 1,24:^=2—25,04 



PrcdJd 10,24991= 10—2—35^ 



•i*MaM 



EXAM' 



JMb^Itiplication of Decimals. ^ 



EXAMPLE IlL 



Multiply . 78,546 MtUs^ 
^ 436 



471276 U. Fq. X. t. P'. IH. 

2?*)6?8 r,6ods:.-r. 1—5—0—1,8 

314184 l>05= .—16 — — 0—0 



ProduQ 34246,056 = 34246— 0—17-5-0— 1,8 



EXAMPLE IV. 



"■"■S IS'? >°^"^- 



lifcllfc 



11&25 lAq.Aq.Rq. Po^ 

7095 J, 28*=. 2,8^ 

9460 1 ,10=6—2—22^ 



Produdl 1001028775= 6—0—2—25,25 , 

Cafe 2. When it happens that the Phces of Dedntlds rati 
rar in both Fadors, and confequently would make a very.ls^gd 
Decimal in the Prodoft, you may contra£l your Work-in 
fiidi a Cale, to as few Places ot Decimals in the Produd as 
yoa pleafe, ot is fuitable to yout Defign, . by the following 
HuUry f^/jT. fee the Units Place of^theAdiv//i^//^dire£Uy un* 
der diat Figure of the Decimal Pate of the Multiplicand 
whofe Place you would preferve in the Produd. 

Then inveri^ or piace aU the other Figures of the M»/- 
ii flier in a contrary order to the common way. 

J^aftly, in Mulcifdying always begin at the Figure of the 
Multiplicand which flands over the Figure wherewith you are 
then a Multiplying, fetting down the firft Figure of each 
particular Produa direBly under one another. But withal 
take care to fee what Increafe would anfe from the Multiply* 
ing of the two next Right-Hand Figures of the Multiplicand, 
which you mufi conftanily add to Ae firft Figure in every 
tr^duS^ 

D EXAM- 



^9 MdiifUcdiioH of DecimaU. 

•EX A M PL E t 

r" • ' * 

Suppofe I would multiply $2412031 Yards.%y 47,29195 
Yards, and to have only four Bla£0s of Dedmals in rfic 
Piodufi:. 

Fbae them as heford direae4 and they wilt (land 

ThiK J 929412031 The MttkipU^and as ufudL 
*** "' t 59192,74 The Multiplier invcrfed. 

' 6468842 
184824 

81171 

924 

. .4^ . 



457<^3345i 



The ^eafpn of, and how great a farf of the Work is 
fived hy^ this ContraBioUy will appear trom the Oferatiw at 



Thu3 \ 9M1203I 



59192,74 



831 


2050155 

708279 


924 
83170 

184824 
($468842 

36964812 


1 203 1 

062 

17 

4 


4370,3451 


4P35045 



Hence it apqpear s^ that ImI/ 
the Work is uleleft, tx. sk 
thofe Figures included in the 
Square^ whofe Sum make in- 
deed 7 places of Dedmals^ 
but are of ^ vdbi^^ and tbec^ 
fore fupeifluous, 



««i 



mm 



EXAM. 



mMfJkatioH of 



f# 



ex; AMPLE n. 

Multiply i^JW» bjr 142,1 arjj and to have thereby rcferv'd 
tvao Places or Dedmals in the Prodod, place them 



Thus i\^^H 

29'iP 
148 



The common way f 14,7^4 
atlaige. 112,125 



«Pi 



4 
29 

2958 
14794 

<75»i34 



4=582 
588 

I* 



7662 



: EXAMPLE m. 

Multiply 257,3^^6 wi* 76^, and for an Intire ProJuSi 
of Integers, place fflem as 6y the iP«ife 



ii» -[U5»« - 



The fiwwf 257,556 
at fcrrgeA 7648 



18015 

103 
2a 



30 1 58848 
102 9424 

92. 



r8oi4 



,1 



ip6&2^l 58688 



From fir^y^ Examle^^ & mMiA hoivt advantagous theie 
(§fnfendtous Contratiions arc to facilitate and Jhorten the 
Work of thofe long and 6f€tou9 Calcdations and Computa- 
tions, which the experienced Pm^itioner finds occur but too 
phcn in Aritbmeticky Algehfa, and Geometry. 

Cafe 3. If the hbdtiflicand be a Repetend only^ and the 
Multiplier a fi^g^^ l^i^U Multiply as ufual ; only obierve to 
4dd in the lap place of the Proi^fl as many Units as it con. 
iains Nms^ and thai: ^3Ce is a Repetend. 

P 2 E X A M- 



\ 



to i^ffUcation ^ Decimah,. 

EXAMPLE L 

^ffu «f ihe Zoiiaelu 



t It 



Multiply 10,701^ ^183= . — 15-P 
Pfodtia ' WoSa'ss 5 — 15— is-ro 



IB 



EXAMPLE n. 



f^. /(.f. 



• M""^ '>'°J>rw..^;?^j-^ 



Produft 65,i3« =a 65— 1— 54^1 



EXAMPLE m. 



•**§ ♦'^)- '^-c^a^i? 



Pradoft 38)08j|>4 = 38— 2-h>94 



But if the hitfti flier confift <£fivmi! IXgits or Figmo^ 
then make ea^ jptiticabr Piodufib cMtfrMMMM, by ooatmne 
iiig tlw /ffg'tf Kefettnd of cadi manb die Ri^-Han^. 



EXAMPLE L 



t24S9- 
449?ft9< 



a2459£| 



fkodkft 4741^3057^ 



U 



Multifliaaion ef Deeimalsl it 

If the MuIU flier be a ^ffefend, muldplj at ufiial ; but in 



Quotient till k becomes a Sifigle 
found B^f€%end\ and this lljall be the true Hfptl$ ^ Aor 
fwer* 



>v ',1 



EXAMPLE t 

1 

i 

Muldbly 724,35 
5)289,740 . 



TrueEmiQa .32,i93r 



EXAMPLE a 



t t 



Tnx Produa = ;^84f 



T^ 



EXAMPLE IIL 

by 8,7ji? 

y)ioo>473 

11163]^ 
176001 
201 144 



Produa 2198,6041^ 



EX AMf 



jMl M^lk^f^im^ of TkdmMlu 

. Multiply 48,7S# . 



> V 



1625^48: 
P7S08888 



Fkodofi io4^pjftr 




and the MmtipJier bat a ing'^ 

tbiFirfti bat ofafenre ta adtt tor t 

^oduft fo moftf Units as there are 72r«jln the Proda£t of die 

Left-Hand Place of the l^fpefen^. , Apd the ProduSi ihall 

contain a Refetend wbbfe yUm ate /ijffA^ to thofe in the 

hbiltiflicofta. 

EAAMPLE I 

Mulddt 582^47 
If _^ ^ 

Ptodua 4658,y7« 



* • • 



E X AM f L ^ fi. 

hy ^ 

Prodoa X7f^gi3» 

EXAMPLE m. 

Mutes'^ 37#9.«3r 
by /xjj 

IV>dii£l 26,2^46)! 



II 



If ^MultifUtr cot0»oi Vixtimre tbanwu, make 
aD the fevetal Kbdofb' eo»temmous towaids tbe R^kh, 
Haadf as taug)it in tbe/«/f C2^« 

EXAMPLE L 

Multiply 73,2^ 

*y 49*7 

5128x0^ 

2i97;75#7 

ProduS 3 202^05'?^ 



^— T" 



EXAMPLE n. 
bf 437Q>2 

805#6O25r 

1 2082^9038 ipo 

1^10^20509205 

Prodoa l760QH2,pf233ar 

■ I 4^ II , ip«i 1^ ■ i l l i rf 

But if not o«/y the iUuUifiictmd^ but (be MtdiipUer alTo 
be a Compound itepetend^ Midtiply (as has been be&re 
laagbt,) each Figuie (£ the RefeUnd^. and add the feveral 
Produaa tc^ether ; Tbea add th« Refuk to it ielf in this 
Manner, fet the firfi Left-Hand Figure fo many Places fo> 
ward as exceeds the Namber of Places in the Repetend by 
one I and the Reft of the Fig^res in order after it ; and 
thus proceed till the Refult/4/f/i^^if^ be carried ^^jro«^/ the 
frft ; Lafily, add theft fevem Refiiltl together, beginning 
under the Rig^t^Hand Place of the firft, and from thence 
dajh as many Figures for 2l Repetend^ as ihe Repetend <£ 
the Multiplier does eon^fi of. 

EXAM. 



p^ JtBibiplieatum&fTyedmMb. 

1g:XAMPLE L 

bf _^ 

151006 

67002 

705Q? 

FirfiPfodoa 7871:126 

7871526 
7871326 



TroeProdofi 787>9^^ 



EXAMPLE U. 

MukirAy 432067 
by ,02:43^ 

25^402 . 
I 296201 
1728268 
884134 



FirftPfoduft 1052515212 

1052515 Qc. 
105 Qc. 

TcueProdua 10526,20^474! 



EXAMPLE m. 
Multiply 42710,36 

12813108 
17084144 
8542072 

FitftProdua 8714^1957508 

87141957 8r- 
871 &rc 

TraePiodoa 87149281^93^ 

If 



" Muhi^ication cf Bechnahl 4 J 

If the Multiplier has any fermivaft Places join'd with the 
Refetendy and if the Repetend he fmaJt and theft manj^ 
the beft way will be to im^ijdy and add the Produds of 
Che Repetend firft ; then alter multiply by die Wimmata f%- 
guresy and add their Prodofls to the Sum of the Produfi of 
the Refetend ; and to this tafi Refult add the faid Sbm. of 
the Repetend ProduSis, as iathelafi Examples. 



EXAMPLE. 



Multiply 432^1.3 
by 23,4^4, 

172972 
43243 



The Sum 6054OZ of the Prodod of the Rxp^fends. 

72972 

S6486 



SOI249i(^l ; 

605409 

60 &c* 



10124,977^7 



T 

t 

But if the terminate Figures arc fewj and the Places 
of the Repetend are many; theflioneft way will be tofuh* 
ftraSl the terminate Figures from thofe of the Refetend^ and 
multiply by the Remainder as a Refetend, 



EXAM- 



CHAP. IV. 
Division 0/ Decimals. 

DIVISION of Dedmak is peiform'd in the fame 
manner as hivipom of Iniej^ersy both in regard of 
placiifgt)at Numbers, and the Work it felC 
The <biifeft Hiffiadij^ in general, is to difcaver the fme 
V^Mt oi the QuotUnt Figures, that is, how, to tt^tzttjufth 
tbe Integers and Decimals it contains. HowWer the BuCneis 
of Vnluhg the Quotient is rendered very jdain and obvious, 
by a due Obfervation of either of the following Rules j viz. 

Rule 1. The Quatieni Figure is always of the fime Va- 
lue with that Figure of the Dividend j under which the Units 
Place of its ProduA ftands. Or thus, 

Rule II. The Decimal Parts in the Divifbr and Quotient 
muft be always equd, in Number to thofe of the Dividend. 

Some Author^ give fme of thefe Rules, and fome the other ; 
but I have fupplied you with both, that nothing may be want- 
ing to render this neeeffary and frequent Part of the Art as 
e^i and ready as pofliole. 

^From the 'Second general Rule may be deduced theie four 
f articular and very ufefail Direftions, viz. 

1 DireB. When the Decimal Places in the Divifor and 
DividcsY} areiCqual, di^ Quotient will be whole Numbers. 

2 DireB. ' When the Places of Decimals in the Dividend 
exceed tfaofe of the Diviibr, the Decimal Parts in the Quo- 
tient mbft be equal to tharExccfs. 

3 DireQ. • If the Divifor exceed the Dividend in Decimal 
Places, annex Cyphers to make them equal, ^tM will the 
Quotient b^ Integers, by DireSi i. 

14 i^reSi If after you have finiihed Divifion and find not 
(b many Figures in the Quotient as there ought to be places of 
.-Decu^al Parts by the gcneralRulc, fupply tnat Defeft by pre- 
fixing Cyphers to the Quotient. 

• The Learnpr being thus fraught with general Rules and 
^parriciflar fe^rcfiions ; oanfiet, I think, without Impeachment 
<(f his Ingenuity,'' tc^uire »)y thing farther to be Taid or done 
• • '^ - • . . to 



'C 



( 



pivtfion of DecJmdU^ %^ 

to make Divifion of Decimals evident and ea^, except the 
Operations themfelves in aH the various Calbs ; tQ whidi I 
now proceed. 

Cafe I. When your Decimals are comfhat and [oqh. fer* 
minatey place them and work as in Whole Numiets^ having^ 
a ftria regard to the Rules and Direaions before g^ven for Va- 
luing the Quotient. 

In Divifion of Decimals there may happen Nine Varieties, 
with refpeft to the Nature of die NanAers, whidi may be of 
three Sorts ; viz. 

Firft, Integers ; or Whole Numbers. 

Secondly, Mitct ; confifling of Integers and Decimals. 

Thirdly, Pitre Decimals ■; without any Whole Numbers. 
Now the Dividend being it OX of three Kinds j and capable 
of a DiviTor of three lands, there foflows of coniequence, 
tbefe nine Diver fittes, i4z4 

C Whole Number. 
Any Whole iS^tfmft^ may be'dhrided by a < Mixt NnnAer. 

L Decimal. 

^.WhoU Number. 
A Mixt Number may be divided by a ^Mixt Number. 

LDecimal. 

C Whole Number. 
A ¥ure DMcimd mj }» divided h|^'<Mixt Nusnber. 

IDecumh 

I ihall explain and exemplify this by one Example, r/z. by 
1722 divided by 12 thus, at large. 

12) 1722.0 (145.5 
12 

• 52 • • 

48 



•42 
36 



60 
60 



.... 



Here 



^o T^ivifion of DecimAls. 

Here you fee the Divifor and Dividend are botH iobole 
i^umheri ; and becaufe there was a Remainder of 6, I bor- 
row a Cypher in order to divide ic off clean, which gives 
(by hire& 2.) (Uie Place, to wir, 5 in rbe Quocient For a 
DecimaL I UiaJl fuljoin this one Elxample varied according 
Co an the Varieties aforefaid. 

Variety i— — 12)1722.0 ( 143,5 by LifiS. 2- 

2 12)172,20 ( 14,, 5 by Dvre6i. 2. 

3— — 1 2),! 7220 (,01435 by DireS. 4. 

4 I,2)i722.o ( 1435 by DiteSi. i. 

5— -—1,23172,20 ( 143,5 yii DireSl. 2. 

6 — — I,2;,i7220 ( ,1435 by DireSi. 2. 

7— — ,i2)i722.oc( 14350 by DireS. 3- 

8 ,12)17,220 ( 143,5 hi DiTiS. 2* 

9 ^,I2),I7220 ( 1,435 by LireSl. 2. 

But notwithfianding I have given a Sfedmen oEalt the Va^ 
tieiits in the laft Example, and pointed to the Di^eSIiov^ 
by which each Quocient was form a ; yet 'twill be neceflary 
to iOuArate the j^vrrj/ ^»/^ by Elxamples wrought zxlargPy 
wherein the immediate Ufe of die particular Direfiions will 
more okvioufly aipfcwc. 

Example i. Wherein the Plates o/Decimdl Parts in the 
Divifor acd Dividend are egual. 

8,45) 2P5>7$ (35 ,0074) ,4884 (^^ 

2535 444 

• 4225 444 

4225 444 



• • • • • • • 

llere becaule die DecimJs in Divifor and Dividend are 
equal in Number j therefore the Quodents in both Liftances 
aic jxbole NumherSy by DireB. !• 



■^« 



T^ivifion. of Decimalsp 3 1 

Efcample 2. "When the Decimal Parts of die Dmdend 
exeegd ibefe of the Divilbr. 



24.3) 780,516^32,12 

72p 


,0067) ,3953 (,59 

335 


^1 . 


•603 


- 29t 


• • • 


243 


/ 


486. 

486 


« 



• • • 



In this Cafe the Excefs is cut oC in both the Quotients 
for Decimal Parts ; by DireSl. 2. 

Example 2. When there are not fo many Places of Parts 
in the Dividend as in the Divilbr. 

7,684; lp2,I00 (25 ,7875)44^100000560 

•15368 59375 

38420 47250^ • { 

38420 47250 



Here Cyphers are annexed to the Dividend, to anfwcr the 
Decimal Places of the Divilbr, that the C^ote might be whole 
Numbers ; as in DireSl i. by DireEl. '3, ' 

Example 4. When, aher Divifidnis finijlieet^ there, are 
not /o many Figures in the Quotient as tjlere ihouM be Deci'- 
maf Part so^xh^ General Rule, 

P57) 7,25406 (,00758 , ,573)>ooo7475GoDi3 

6699 575 

5550 1725 

4785 1725 



• • • • 



7656 " ' 
765^ 

In 



• • • • 



" 



31 Divifion of Decimals. 

In bodi diefe Inftanccs, I9 Direaim 4, I prefix Cyphers 
to die Quodents, diac together with thofe in the EMviibrs 
tbof oMg^ be equal to die Decimal Pkces of tte 

K* anv ITboUj Aiixtj or Decimal Number is given to be 
divided by 10, 100, looo, @r. you only remove the fepara. 
ting Point tomids the Left>hand fo many Ptata as there are 
Cyphers mibe Divifar ; as on the contrary in MuhifUcation^ 
die fefmrOiug Pdnt is moved to the Rioht-hand fi> many 
Places as there are Cyi^iers in die Muki^Lec 

EXAMPLES im 

Multiflicstiom. Divifton. 

1^25 X 10=15,23 ro>i$23(r52,:j 

1,523 X ioo=i52j3 160)1523(15,2? 

1,523 X 1000=; 152J 1000)1523(1,523. 
,0072 X 10= ,072 10)72 (7,2 

,0072 X 100— ,72 100)72 (,72 

, ,0072 X 1000= 7,2 1000)72 (,072 

^72 Xioooozr 72 roooo)72 (^072 

• 

I fhall* next give a Method vdiereby you may work any Cs& 
of Divifio» fay Multiplication, and vice verfa^ any Cafe oi 
MultipUcation by Divijion ; and this, in iQany Inflances, will 
be found very excellent and ufefuL 

PROBLEM 1. 

Suppoft I have any -sNumber, 73^15^ ^o multiply I9 any 
odier Number 125; but yet have a mind to ^/Z vA/^ the faid 
Number, and to have a Quotient equal to the Pifodua of 
thofe two Numbers ; Quere the Diviibr ? 

ktde. .Divide xUnit With Cyphers annexedh^ ^t f^ven 
Multiplier y and the Quotient is the Divifar fought. 

EXAMPLE. 

Given Multiplier 125) 1.000 (pc8 = the Divifor fought 

1000 



• • • 



Then 



pivi^on of *Ikpmak* 3| 

Then^**^^y 7?i5 ,008) 73i5>000 (91437$ 



36575 


II 


14630 


8 


73 »5 


35 


9HV% 


33 


» 


30 




24 




60 




5<S 




40 




40 



t • 



Thus I hate obtadn*d a Quotient tir^ /z«f^ as the PtoduO. 

PROBLEM a 

Suppoie I have any Number 7315 to be divided by any 
odier Number /x)8 ; but would multiply the faid Number, 
and have a ProduEt equal to die Quotient of the fame Num- 
ber divided I9 ,008 ; Quere the Multiplier P 

Ride. Divide an Vnii^ with Cjp£ers annexed^ by the 
given DtVifiTy and the Quotient will be the Multiplier fought. 

,008) 1000 {125 == the Multiplier fought. 
8 

20 

l6_ 
40 



Thus yda lee this,- and the Remainder of the Work, U 
only the Reverfe of the fbrmec ; and therefore need not be 
repeated. 

F Cafe 



34 



Dlvifion of Decimals. 



Cafe 2. If your Divifor confift of. wuviy Places <£ Oeci- 
Old Puts, die Work may be very much contraBedj and 
yet a juft Qncfiemt obtained by the following Rule. Having 
deteimin d the Value of the eiuotie^it Figures^ prooeed in 
muhiflvng the Dwilor widi the firfi Quotient Figure as 
ufual ; but for every rigure after ^ in multiplying, enit oc 
frkk off one in the Divifor ; fliO having a due regard to the 
Increafe^ which would anfe firom the Figitre and Figures fo 



EXAMPLES. 



Contrasted. 

7^9863) 70,2300 (8,7938 
638904 



• • • ¥ 



5Sg04 

•7492! 
7187 

•305 

'66 
63 

■^1 



M large. 

7,9863)70,2300(8,7938 
638904 I 

63396 p 



55904 



•■ 



7491 

7^97 

304 
239 

64 
63 



90 
67 

230 
589 

6410 
8904 

1 7506 



Tho' much Labour be this way favetl, yet it is not proper 
to ufe it unlefi the Decimals in the Quocient be lure to four or 
fix Places ; lince *tis obvious, the next Place, or all the Re- 
mainder of die Quotient in the contraBed Work would be 
three times more or greater than the faoie in the Woik at 
laige. 

If the Dividend contain many f laces of Decimals^ there's 
no occafion for ufing but a very fexD of the firft^ as appears 
l^ this fecond Example^ 



'\ 



EXAM' 



DiwfioM of Deeiitt^U* 35 

EXAMPLE )t'. 



•••-••• 15707P6 

86x99 
62831 



001296 (52,7438 






29368 
21991 



• • • 

• • • 



Here yoo may obfenre, that of Thn Decimal Plaes in the 
Dividend, I have uied oidy Fmtr ; and yet have a-Quorient 
iofcfur Platesrcf Dtcimak true ^ Hence all the Figures ivhich 
would have. fill'd die ^tfed Spoi^Ki \^ i^ ^^n worVd'at 
hxfgt^ Tixt fuferfiu^s I and thw, tisev^ht, make i^^T the 

Wort J ,:: 

Caft ^. If your Dividend contain a fingle ^efefend^ znA 
your Dinibr be %fȤe terminaie Digits divide as uiual ; and 
when you take down your Hepetemf^ the Quotient will tegin 
to repeat. r 

EX A M. L ^^ EX A M. IL 

4) i95»o* (48*7* ,6) 3»7V (5 i9ar 

16 30 



35 >7 

32 12 



50 56 

28 54 



^ V^i//«)?«{ftMR. jg W</ I»finUum. 



F 2 If 



If libit Divifor k aaf I^avbct ^iermi»ate Digit ty the 
dtutintt vfSLrefegt * Jinglt Difff i ^ not always iegimf 
when the Rtfetni h oken down. 

E X A M. OL EX AM. IV. 

312 85067 

288 1IB765 

240 J. _ 34^*355 



66 • 2»67g^^,.^^-^^^ 

48 ... 19506OJ •' 

_ ^^ ' • ; ■ ■ 

186 21673 

' • ' •:j.2 . . • . . . 

' If yoaiDivrforhtoiAj iei /l;»grr RefHt^^ mi One DM 

ting off we mcnre RMit4iafid Figwe in ih^/VtMwS, v..«„^ 
is now your new Dividend; then divide as ufial, ani tlM 

EXAMPLE V. Diiide5TM*>y^- 
The nhridend =r 572^1. 
Multiply by 9 

Divifor = ,e \ 515,16 ^= the new Dividend. 

/ 48 V643,95 =:ib? troe Quotient. 

it 

76 
72 



40 

Other. 



• • 



DifM/iofi' 



h 



Odierwife thus ; place the Dividend under if [elf^ hat oue 

Place forward to ^ ^^T^V^f'A^Q^ ^^ uibfhaa, the 
Remainder wiQ be the new Dividend, tne lame as be&ie. 

' Thcbivirfcnd^ ^ii^ ^a^i^fere/' ' . ^f^,,. 
The fiune placed 5724 one Figure foxWa'rdiJ ^ 



*•*, 






n 



From hence aUb ^JSgW^J^A^/ff^f^^ ^ one 

more Figure in the new uividend '{br^ip^i/ii^^ \^tf 

Th2X either of thefe Ways will give the iwi^l^uotient you 
have feen, apd that the (^otiene thb way' produced is the 
onh true cne will appear from the Work of^ die lad Example 
at large. ^ V".* 

%y) 5724[c^300 8r. (64 Wf^. •. 



•39011^666 8r. 

3^5 5555 8^. 




^f 






A ■ ,\ » 






#444 8^* . 






" '.X 



In this Operation, tis manifif^ tfai^tig^ Ihe RefSiends in 
every particular Step would pcocceS t^'Jkfinitij yet in the 
laft Place you &eilafxc is an \nfinite PrqduB equal to an 
ififimte Remainder ; and confequeiitly the Work mufl there 
cea(e, and the Quotient neverchelefs be true. 

If the Divifor conlifts of terminate Numbers pin A. to 
the Repetend^ and the Dividend be comfhat ; proceed thus : 
Subftra3 the terminate Numbers of the Dhifar trom the Dim 
vifor it Mfy and the Remainder ftiall be a nev^ Divifor ; 
and deal with the Dividend as in the laft Example, for a 
nev) Dividend. 

EXAM- 



|8 Uiv^it tif Decimshl 

EXAMPLE VI 



! " 



BagS^ itnguiMdto divide 8569^ fcy d^ ; Woik« 

ifi0^8%69fiS fxj^jiiac QMdfBrA if nxidc'd at hige. 
48 856968 



'/ . 



3532 
3066 

;26tf8 
a628 



• . < 1 



• • » 4 < 



**4093 

3942 



50 

IP the DMfiit mi DHrideml do ^4irA contain a Repefend^ 
Older them as befixe dtieaed ,• and the fjmtient will be eu 
ther iemuMe^ refea zJtKffr Dijpt, or dfe a compound He-- 
fetend. 

EXAMPLE VTL 
Divide x^U bf ^ 

/) 120,84(20114 the trae Qooticnt. 
12 



• • 



08 

6 

24 
24 



EX A M- 



Divtfion of *Decmah* 

EXAMPLE Vm. 

45og5 

,06)405,8^^764^ 
56 



3f 




«H 



26 
24 



g Viltf Jhfiititiitt> 



EXAMPLE 


DC 


Divide 23,41^ by 7. 

Thitt J 2M^ 
c 2346 


7)91,120^(3,01714285 

2X 


••12 

4 


7 


50 1 
4? 


' 


10 




7 





28 



30 

60 
56 



40 
35 



itf hifiitttum. 



0^ 



Ci^e. 4. If ComP<nm^ tt^peiends ^rc^ found in your D/- 
^ifoTj or Dividtfnaj of hotb ; . then obferve to fee the Diviibr 
and Dividend under them/^hfs (6 many Places forwards to 
the Higbt'-band, at th^re are Fhce$ in the RefeUnd of the 
Divifor exdufively ; next, fuhftr^ them, and the Remaia- 
der$ will be relpe£Hvelj a nm Di?ifoc and Dividend. 



EXAMPLE L 

Divide 243^/^, by iixj^. 

-11/ 34? 



111,87) 245/>63 (2,17* 
. ' 2^174 

1 1 187 



•81360 

• 30510 

22374 
•8136 



Ad Infinitum. 



Hie Truth of the Work will appear as well by the rom- 
mom Rule of multiplyng the Divifot and Quotient, as by 
the Work at large. 

If there be no tervinate Part of the Diviibr, you fubflraa 
nothing from it. 



EXAM' 



Example n. 

Divide ^^f27ifii^:H ^^i? * ' 
•Then 395273' 






<i^D 394»87S34i (1245,573 
3*7 

•77» 

i268 

*I798 
1585 



n» 



• M33 



'2314 
2219 

-IB n Wi r 

EXAMPLE III 

IMvide 70065 by t^^. 
70005 

70005 

X,48)6s»934,jy5(^a^3,j75 
5S»2 

1073 4^ 

Z035 ' 444 



■fc ll 



374 -555 

296 444 



789 x^io 

740 1036 

mammtmmmm » ■• 

74» 



• • • 



If 



42 DivifioHcf Decimals. 

If there be no ReMeni in the Dsvifor^ nvhatever the 
Dividend may be^ tnere*s no SutfiraStom to be ma^ «f 
either Divifor or Dividend. 

EXAMPLE IV. 

Divide 1761,31^401', by 4^7,64. 

417,64) i76i,3^40xC,#2i^ 
167056 



-P0804 
83528 



'«M 



•72760 
417^4 

309961 
292348 



» Ad Infinimm. 



17613 

EXAMPLE V. 
Divide 3ip28,0O7jri«, by 7645 

30580 




48557 
45870 

24871 
22935 

^■Mi— 

I936I 
15290 



\ Ad Infimtum. 



40712 
38225 



2487 

In An^iT^M it may ofinf happen that the Qaodent maw not 
refeai £> foon as is defiled^ in fucfa Cafe the Value of the 
Quotient mx/ be expcdied compleatly by a Vulgar pn^iM. 



Divifion cf Decimals. 4J. 

Biit in order to undetfland this, >viil be neceflary to pre- 
Bufe 1^ feQowiiig Lemmas* 

Lemma i. 

A Zeries of Nines infinitely continued, is equal to Vnifjy 
or One, in the next. Left-hand Place; thus 0,999, 8c. is 
equal to 1 5 and ,0999 Qc. •=. ,1 ; and popp? 8r. = ,01 ; 

and 549»9^ 8^- = 55- 
Demonjiration. Tis .evident that ,9 jss 74 want?, only 

» i of Unity ; and ^99 wants , .i ; ^999 wants , o.I of Uni-* 

ty f fo that if the Series were continued to Infiniiyy the 

Difference between that Stries of Amines and an Vntt^ would 

be equal to Unity divided hy Infinity^ that is^ Nothing at all. 

tl.RD. 

Lemma 2* ^ 

Any finite Repetend multiplied by jio, and then fith- 
JiraQed itom that FroduSl ; the Remainder will be the Jan^^ 
Nuffiier eompleat or terminate^ in the next fuperior Lett* 
hand t'lace. 

Demonftration. Let the given ^epetepd be ^6666, f^c^ 
this multiplied by 10 is 6^666 @r. 

From which SubArad ^66 @r. 

There will remain 6, • • • a whole Number. 
Thus 47,77 &f* will become 450. and ,0533 8r. will be 
,3. d.RD. ' 

CaroHary i. 

Henpe it follows that if any Compound Repetend be multi* 
jdied Iqf an Unit with 6} many Cyphers annexed as are equal 
to the Places ot the Repetend, and then &bAra£led from the 
Produd, there will be left to the Left-hand the lame Numbers 
terminate and compleatj that conflituted the Repetend; thus^ 
^25 multiplied by 1000, will be ^25r,325 from which if you 
fubfhrad ,^2^ there wUl remain the terminate Number 325 ; 
Thus 12,5^4^ will be 1 273 1 J and ,00074^ will be ,743, and 
5275^ will become 5270,1. 

Corollary, 2. 

Hence aUb if any Repetend be multipUed by fo many 
Nines as it contains Places, the Refult iviUb^ the fame as be^ 

G 2 fore: 



44 7>ivifio» pf JMfimsh. 

fqc^^ that if, tifa^ Refeteud temimige uA emflNa. Vof 
any thing multiplied by Ten^ and «fr<? iMnBd, is^tbt 
lame as multipHed ij Nine ; 
Thus ; ,666 &c. X 9 =. 5>9«9 ®f- = 6. by ImiM i. 

And *27 X 999=^52^f9?9 == 527. fcr ii«?».. 

CoroJUry 3. 

Hence it m^fi foUovr ths^t, viV^ cxrf/#, aqy Nuqibar cbnH 
ded by as many Nines as it contains Fj^r^ p eqinl to &• 

6imtiimbet ferfettcj^y nmlafir^i Thps |-5»,^<5 8r. 
AndSg==';?27. And J2 ^ =;J2^4^, 

Hence, laflly, appears the Reafon of all the dificrent Me- 
thods and peculiar Proceffes uTed in the Aritbmtick of Or* 
eul/fting N'U'nihers, call'd Refetends. 

The preceedin^ Lmmds and Corollanes l^ing well ud^ 
derftood, it tvill then be very eaft to v^ue any Jdnd cC 
Z)tfrw«/j?J in the manner of Viilgar Fraaions. For ^e Quo, 
tieot in Divilion, take this Exan^e fcon Mr, O^ff. 

E X A M P L E VL 

Divide 2^347/f, by 4rpL 

4 ^^347^ ji42^ 

417,2) 2122^265^(511176 — ^^ 
2c86o 4J7j2 




•7345 
417^ 

3173* 

29204 

• 25 J46 
25032 

"3H 



The 



Divifion of Decimals. 45 

The Reaibn why the (Rodent is thtis exprefled winbeeyi. 
dent if weconfider. ^\ * 

Firft, That 314 is not the'^fffrr/Re&iainder becaofethe 
Pivtdend is a Aefeiend^ perpetuaUy fupplying a circuJathp 
^fmai$$d0r,^w\ii(dk expr^d in i(l ^(Mf Terfa^wwddtbe, 
where we leave off, wrote thus ^i^6*iw* But thb Infinite 

fiMv« of Figofcs k .m^ v^wflo^ ;bnt| ^M &:3.fcr 

. Seooodly, It being pina dut )M^^ is the Qm R4- 

inainder^ and 417,2 the Div^{br, *cis necel&fy ibey Ihodtt 
be expreffed in the C^otient as here you fte them by the 
Rules of cmmon DiuRon^ ' / 

If iofiead of>3i 4? r^ yoo wnoe ksEpiiv^tni i}i4Mf6$B0t 

and from it iubftrad the Uiumaie part 314, ther^ will remaia 
3x4262^2 a new Nmncrator. And if to 417,2 you ^zdfi/ as 
many Cypbt^n as the Repetend confifls of Puces, thus 
4I7,20CXX)0 ; and ag^ fubArafi it, as a termiaata pair, 
there wiU iemain 417x5^28 &r a nfw HenomntAor ^ and 

then this new and noie jfiwf/r i^f 4^1^ ll^ 

equal to thai in the Quotieoi. 

The Reafm of reducing the FraSiian of the Quotient in 
this Maoaer is obvious from CorpL i. of Lenu 2. 

For 3145^58^ X looooo ^ 3142-658^26586 8r, ^ 
From which fub(ha(3 it felf 3i4af658j^ &c. 



■^ 



• • • • 



And there will remainthe new Numerator 31426271 aebefixcr 

Then 417,2 X lOOOOO sr 417,200000 

From which fubfira£l it felt^ 4172 

_ — — ^— '^■— — 

There remains the Denominator 417195828 as beforq.^ 

Thus I have fupplied you wich Rules for managing the 
whole DoSinne of Circulating Numbers \ and given the 
Tbe^ axi^Reafons for the fame ; which you may fearch for 
in vain in any other Book (chac I know of) fo fully as here 
laid down. 

CHAP. 



C tf A p. V. 

Reduction </Decima>,5.' 

IT bcais iiffiacadf cmncaoed diat dl JrHhmeHed 

mem wmtfftat Arty and aie^awoded ^tmk Per-: 
«fe«Vf7and that the w^V y<rf ofDectmd Jntbmetul abnp 
L lulcWk of an the varioos kinds rf Nwiba^ 
feme time hath all itt Opeiatiooi peifonnd by the fiaieM/» 
tod tmm» if**, and in Ae wf fame ma of A/^gra/ 

£ ff ♦erfci in the SH««f^ of'NmherSy toh jufUy tendaed 
bwW :;<«ViEr«rtif * in the greateft Bfiem among thofe 
Sho ttff^f^y^^iw/ it ; and U moft *«w«i% ufei by 4em in 
almoft dl kinds of Ntmerkal Cebalatums. 
"^The Part we now treat of is ahfolutely neceffarj to the 
tnu Ihderftaitditti and Ufe of thU «ff*&»* Art ; and teadies, 

Firft '10 reduce or exprefi any Vulgar Fratltott in Dect- 
ml P^ruoi Hit Integral Zffafttitj. . 

iT c^.o«^;- To reduce fuch Numbers as confift of variom 
Parti and Denominations, ai thofe oi Money, mighty Mea. 
Cure, *c. into D*«»m/* for more eafy Oj>eration. 

Thndly^'To reduce Decimtd Parts ^tto the comma and 
iKMun Parts of Miinet, Meafure'y &c. 
"t^ 1 To reduce Vutgar Fraclions into teeimals, the 
conitoiOn *Rule is, to divide -the Nunurator by the DenmiftA- 
Ur and the «i«>f»V»» wUl be the Decimal required ; that is, 
equWenl to i^ Vulvar Fraaion giweiv 

EXAMPLE I. 

What is tlie Dedmal ewivaUnt to the Fraaion ^ ? 
4) 3»oo (>75 *^ Decimal required. 

28 , - 

♦ ao 

EX- 



%• 



Bedu^ioM of Vulgat ft anions » 'j^x 

EXAMPLE n. 

Rediice4 of a Ponnd into Dicimal Pant oi ft Found.' 
8) 3/X)0 ( ,375 = 7 : 6 die AoTwer. 

•6o. 
56 



40 
40 






E X A M P L E 

Reduce i4 of a Pound Troy into Decimal Pmsl 

n» fwf* 
16) ^/3O0o Gi87^ 3s 2 : 5 dieAnfifW* 
16 

140 
128 



X20 

ixa 

•*8o' 
80 



• • 



EXAMPLE IVo 

Reduce tt of a IM into Dednudh 

Rod. F. In. ^r. 
87) 4,000(^48 = 2 : 5 :^ X AniWei; 



X30 
Z08 

220 
216 



E X A Mh 



4t MiifNtii^9tci'Ma DnhMlliSic. 

EXAMTftE V- 

. Ga/. Pis. 
io8 



,»> 'I 



200 



Jg ^ Ad Infinifiim. 

II 

H. Gal.:fiftts. 
Hence the Anfwer is 5^07 = 5 : 25 *: ^ 

ReduElion oiVubmt ^aSti^tst info Deeimals b alfo cod: 
___3dioufly performd by Zo^^ar/ViS^w. eipccially when the 

»omixr4f or from ^ tjegtiritkm ci the Numerator^ the iP^ 



m 



»- i. 



£ X A M P L E I. By to^jjrlvivj. 
Kedttce the Praaion ^^^ into DecimffTirts. 

412? ; 

Thus, the Logarithm of 12713 — ^^ «^ 2.lo:)8o^7 

the Logarithm of 4123 fubftraA^ ^^ 3*6i^^iii 

■> » 

Remains the Logarithm or the Dedok^^O^ =r 8.4S85904 

What is filfeil»aiteafc4EdK'lDk*aFifai(» ^2^ 

l^rom the. tiyptlnt ql the Kmmtim »» |.?5p04l4 
Subftra£t the Logarithm of the Defominaini 3.5414544 

There Remains the Log. of theDec^ 9026157 ~ 84175870 

h$.iMiqfd Qfkmtitx 5;^^ a|id the Anfwer 
IS ^.026157. 
The fame thing might as well hav4 beeft ^ne by redodng 

the mi^ft Fraction into an improper #ne, im. IZlSz 

• ^ • ' 3479 _^ 

Then, 



wiUbe compleat, thus 5.026157. 

light 



Then, as in the firft Eaatnpld. 

From the Log, of the Numerator 17486 — 4.2426^04 

Subftraa the Log. of the DcnominaDor 5479 ' — 3-5414544 

Remains the Log. of the Anfwer 5,026157 — 0.7012^60 

Note J In the firft and' fecond Examples, and in all fuch 

Cafes where you fubfttad a greater Index from a lefs^ you 

borrow Tens and as many Digits as the remaining Index want; 

of Nine^ fo many Cyphers prefix to the Decimal, 

' ' ' '•. ' - 

CAfe 2. To reduce Numbers which exprefs Quantities of 

various Kinds and Denomination Sj as Money^ Meafure^ &c. 

into Decimals^ there are three Ways or Methods, which are 

as follows. 

Method I. Reduce the different Sipodt^ to one ; that is, 

to the loweft Denomination they confift of ; then reduce the 

Integer to the fame Denomination ; i\it firft will be the Nu^ 

merator^ the latter the Denominator of a Vulgar FraSiion ; 

which Fra£lion reduced to a Decimal (by Cafe i.) will be that 

required. 

E X A M P L E L 

What Decimal Part of a Pound is 5^. yd. ? ? 

s. /. 

' Multiply 5 Thea icdpcc i the I»(ej;er. 

By i2d. 12 20 

60 20 

Add the 7^. 7 12 



67 Pence. 240 

Mul. by45 4 4 

268 960 Farthings in a Pound. 

Add the 3j 5 

271- Farthings. 

ft 

Then 960) 271^000 (,28229 Qe. The Anfwer. 

H So 



So that ,28229 ^ the Berimif^ -^ « f^m^ n one 
i>encMnioadon, ^tfu\ to'^t. fd.y^^ Ihe Pait ef -aj^ooBd in 
^//;/> -Dcndmifiadoni 

Wfet Orrfoyfl/'Pkrt of -a^a^*Wi/ Wr%*r is t ^.^iW, 
Reduce 2 : 21 : 1 2 to Ounces. And i reduce to Ounces. 



iO^ 



16 



^ 



i% >if2 

78 t6 



1 244 '(9««>iPi 1 79 2 dwrr^i in C. UK 

Then -{^^}^ i« Ac Vulgar Fraaion. 

And I792)i244>0'f«,4p4i96 the bedmal Rttt of an 
i7»Arir^/r</]j[i&f(anfweriogio 2>j.: 21/^ : i2'«z« rtquired. 

EXAMPLE m. 

Y. F. /». 
4:2:8 I Pole. 

3 5>5 



14 5>5 

12 3 



176 "fcfftt-Ji 16,5 

12 



i^ %{£f « id a PtU, 

But ^ = ^'s ,«r die (i«peatk«) DetUd'af^fp^Zr, 

equal to tbe 4 Ktr/t* 3 Feet, wa^ 8 /«r£e a 
la die flme^Mamiir'pt^Ki «ridk^ ainf bdter'f^Mn^Slpmrx. 

Metbod 



OMifod 2. Wind! ^h^iDf^mlt ««X (i» k^^: ^hmik. 

prefix the given Part of tl)#. n^t fpfie/jpr. Df4SpQ|tipn^ 
clmii ff e. ^nm^V^fm4 V^Vk^lmMi^^^^^l^ Js.qF t^ ijjpc 

it ; and thus pw^o^ t^l jfi4|, «fe.n4r. l^. th\ Img-r, iii% ^ 

E»A MP LEI 

What &e€imal' Part o£ a, Round is 12 s. &di k^ 
Firft 4> 2,(i(,5 the DecImalioC one Penny fibr ^. 
Socoodiy i^6,5'(V54ii^tke.D^. BartofyaShill4for6<£ ^'; 
Vbkd^ 20) 52:^541^ (,^270% the DecinuL Itert of a 
Poua^ as. 11193 vequttcd^ fei lIj. 6^^. •• 

|:X A M P^L E IK 

'ft 

What Decimal dF a Pbuiid 7>ay i? z oz. iSftoh 20 g^. .* 
Firft 2^ 20,000 (.875 the 0ecynal for''2C^rJ 
Steon^ $p) i9>875 C?^375 the t>ecimal fer i^pwh 2Cfr. 
ThiTcBy, 12)2,9:^375 (.2453125 the becimal Part or SI 
pound Kojjfor 2oi. \9fv3t'. 2C^^ as was required. 

5:XAM?LE III. 

What Decimal Part of a Jlt^iiiS?^. jw. 5^'. 6i&. 40^ 

5^" ? 
Pirft 60) 50,00 (,8^ Decimab for 50'* of 1'. 

Secondly 60) 40,8^ C,68o^ Dec. for 40' : 50''' of ao |?9yr. 

Th^, 24i 6,680^ (^2783 !:4€ = d/i 49' : %o" of ^ /%. 

FourthTj, 7) 5,2783^4^ (^^54"4 ^f* Decimals cf a Week. 

f ifih^j^ 4) 3,75404 8r. (SP3851 DecimJs of a I\^onUh 

Sixthly, 15) 6,93851 (,5337^ pecic^alsoF ^Year. 

So that we fee the ftx different Parrs of Time above fpe* 
cified are reduced to this fmaJJ Dfcimal ,5^373 ; which cx- 
preffeth the fame Part of a Yt^r zs thpy do ; which, ty the 
way, nu^y be an Inftance of the great Simflicity^ Eaf^j and 
Excellency of this admirable Jrf, 

In fhefe three Examples I have omitted the VVork at large, 
fitting dowii only the Drvifors. DivideTidsj aud Qtfiftie&U ^s 

H 2 fuffi* 



y i ReduBion to Decimals iy Tables. 

foflScient to give the Uamer as good a Notion of the Method 
as the Operations at lei^th, wfidi be may make his Exer* 
cife atfleafuri to good advantage. 

Method 3* The third Method for finding the Decimal 
of 2XV] given Part of Quantity conli(fing of diverfe Denomi' 
not ions y is by Tables ready cadcdated for that purpofe. 

This is not only the moft eapfj hut the moft expeditious 
Manner of working the Procefles of ^/ Kindoi Computations 
in general ; and is of particukr fervice in this Cafe of fre^ 
faring Numbers for' Decimal Operations. 

For that realbn I have here inserted a 5^ of Tatles, which, 
tho' fome are common of this kind, are the moft comfleat and 
univerfal of any I have feen extant ; and in order to ren- 
der them fo, I have not only very nmcb enlarged and t^ew 
vamped the old ones, but alio added other very ufeful ones ; 
as thofe who are read in thefe Matters, will ibon perceive. 

By the following Tables, all Ae Species of Money ^ Weighty 
Meafure^ &c. confifting of what ever Denomination, and 
be the Integer what you pleafe, are immediately tum'd into 
Decimal Parts ,- and are then work'd wkh the known Faci^ 
litj and Pleasure of Whole Numbers. 

As to the manner of tffing thofe Tables, that is fo obvious 
and natural^ even by a bare InfpeSion, that I prefume tis 
needlefs to fay any thing to a Perlbn cf Genius, though a 
Learner^ about that. The Scheme of Examples following 
being fufficient to teftify the great Vfe and Excellency of fuch 
Tables, and are both Precedents and Precepts themfelves. 

EXAMPLE L 

What is the Decimal Part of a Pound for 13 j. yd. {? 

JnmU J. jou yif^^ »65 

find anfwcring to ^ ^ Farthings f P3225>i^ 

The Anfwer is — — — — ,682291^ 

EXAMPLE n. 

What Decimal Part of a Mark is lis. 2d. I ? 

< ^T» ¥ C II Shillings — — ,6< 

In Tal'e I. un- ^ , Pence — — oi'"; 

der a Mark, againft ^ ^ p^^^^^^ _ _ \oo''^2^. 

TheAnfwet — ». — ,8+0625 

EXAM- 



EX AM PL!E -nr. : 

What Jiecimal Part of a Moidore is9i.[^d.\} . 

In Table I. under Moi- 5 ^ Shillings ^ . ,3:3^3^, , 

dore you find againft ; ">|J^"Sf-' "" '^^^^^ 
•* ° Ciraraimg. — ,000771 



The Arfwet is iHascompou/id Regetend . )343363M« 

E X A M P L E IV. 

I ^ 

% 

What Decimal of a Pound I^oj anfwers to $ 02. YjfvBUf. 
22gr.f ' • ^ 

In r^Jfe II. you 5 ^ ?"'^*«r •-!:• " '^'^o^*^ 

findagainft J17 Penny Weight- ~- • ,070%? 

^ c 22 Grains ,1^ — /xajSip 

The £»« ot which is the Anfwer — >-. 491319 

EXAMPLE V. 

What Decimal Part of an Hundred Weight is 21 1*. 
14 oz. .^ 

In 72iJfe III. you f 21 Pounds \— J187S 

find againft c 14 Ounces — i-^ ,cX378ia 

The Anfwer is . r^ ,— .>J[955i2 

EX AMP LE VI. 

What Decimal Part of 2l Tun is ^gr. 6 iujh. Tgdl.? 
In r^W^V.Z)rri 3 Q"^rt«s . _ _ ,6 

The Anfwer is — — .^ ,771875 

EXAMPLE VIL 

What Decimal Part of a Hogshead of Wine, is 2 J /f»«<^. 
14 G^//. /* 

In Table VI. Zija/i Ai^4^ 12^ Rundlets 571425 
furey you find againfi J 14 Gallons ,^2222 ©r. 

The Anfwer (repeating a ftngle Figure is ,93647^ &c. 

EXAM. 



54r Btdkatamta "Basiamii fy TaUas^ 

EXTAMPLE VDt 

How is 27.Miks, 7 EurJcfigf^ 3.^fl<^.aod ^Xardi^- 
preffed in Dedtnids.^ 

In Wle Vin. Long C xFutlongt — ,85^ 
Meajure, (oat Mile thc.<^:j,5,Rad,,or Pol^ — » j^oj^TS 
Intx^gO^. ]0ai.find ag^inib C 4 Yards ~ ,002272 

Tbe 27' Mrilf J prefixed, the Anfwet will be 27^86647 

EX^A Kf ELE IX/ 

Wlmt Vfeimal Part of a tear is 7 Mmrtbsy 3 ^^ *^ v^ 
2:0*1^? 

li' jaW^f Df. you S-'^tS^J* *^ "~ '^^?i*' 

* C.2Dax8 — "~ #°W4 

The Anfwer is ^601 585 

EXAMPLE X. 

WfcatDrtwwT Paie oTa'Sijf* «f t»e Zaiiatb 25* 46* 8" ? 

la Ibile X, you S^lS^^." "^ Mmi 

fttdl^Bnft <46Mioutes - ^2*^555. 

.*^ C o occonas .. ,000073 

The Anfwec is a fingle Refetend — ^SjSpdiir 

EXAMPLE XK 
What IHcmsi Part of a i!)^^^ is 45/ 57** ? 
In th« fame Table ^^g Minute& •-, — ,8li^M6 
you fee againi! \ 57 Seconds — .^ jO»5k%3 

TheAnfwcris -- — f*V^ 

Having ^hiis fo largelv exemplified tht IZfr of the Tables^ 
the 722WCJ tbemMvei foHow ; wherein obferve, 1. I have 
rffl/iW the ^7? Figure of att/rgft Ref^e9d$^ and the iSr/J 
and /^y? of the comfound B^PaUnds that come mihiii the 
Table. 2. I have nevertkekfa emimued cacb to fn Places 
For tbiir ftkes who wQu:d be exaS^ but Avmp m^ weH bow 
to mana^ Refeiends. 

Commtn 



Common Talste ^ Money, heights, Mea- 

fures, and Time. -s ' 

}fi= f 2 = "^i Sbittjng. 

TM^m. ^Af^tb0MtiNmeight. 

Grains. 

20 = X Scruple. 

480 s= t«4«aBB r 8 := iOuncf. 
%76o^^8:=^6 =55.12 ;=; xlt. 

Grauu* 

24 = I TemyWeigbt. 
480 = 20 = 1 0»«r^. 
7560 = 240 = 12 =5,lt^/ 

a*KfV. i M»myer4 iP&%&f, . 

Blajiks^ 

480 = 20 = I Droite. 
11520 =.i)4aors53-t24 ae? ^p.i4ite. 
230400 = p6oo =s 480 =r 20 = I Gm;i?. 

3i*Zf V. Averdtif9U9y^gbL ' 
Drams* 

28a7r>=:'I7P;jz=^tI2,— -I^»5ft»*tfrf. ' 



^ Common Talks of Money^ 

. .1.- • Tdble VI. mneAieafire.^ 

Cubic In. -' " 
231 = I Gallon. 

9702 3= 42 B=: I • TJrr ^. 

I45S3 = 63 = Ik = I Hogshead. 

19404 ^=: 84 = 2 = I, = 1 Punch. 

29106 = I?0=:3: ?=2 = I J =5 I jB»l/, 

58212 =3: 25? == 6 = 4 =3 =3 2 = I 7Jr». 



i ^ 



.* 3aii?VII, AkUeafute. , 

Cubic In. 

282 r= I Gallon. 
2256 = 8 = i Firkin. ~ 
4512 = 16 t== 2 = I KUderkim 
9024 =. 32 es 4 =2 2 tii: I Barrel. 
13536 = 48 = 6 = 3 = il — I Hogshead. 

Table yiJ^. BeerMeafurr. 

Cubic In^ : ^ 

282 = lG)alhnr ^ \^ 

2583 ^ P = I f?rAi». 
5076 -rii: lo ?i=,2 ^ iKUderkim. 
10152 = 36 = 4 = 2^=51 Barrel. • 
15228 = 54 sa 6 =^3 = II z==.,i Bogsbead. 






JiWtf K. Z)r7 Meidvte. 

268.8= LI fi^fer/ V T 

537.6= 2= I Peck. 
2150.4= 8= /^iBuJhel 
8601.6= 32=i 16= 4=s i=CboOT?^. 
I72(?3.2= A|.= 32= 8= 2= I Quarter. 
688.Ut§=256=i28==32= 8=4=1 Clb/i/^r. 
86016.0=350=160=40=10= 5=si IFeyy ot Ioa/. 
.][72032.o=640=320=oo=2ossio=2=i iLi/?. 



Jible 



"^7 



Seconds, 

Sotzz 1 Minute. 

3600= forr: 1 JffiMW^. 
86400= 1440= 2A^ : I iJrtjis, 
604800= 100802= 168= 7= I Week* 
2419100= 4j032o= 672= 28= 4= I M^nth. 

31556937=525949^8765=365=52=13+1 Dayy + 
5 Hoar/, 4-48* +57" = Year. 

Tahle XL Long Measure. 

Barly Corns. 

3= I Inch. 
36= 12= 1= Foot. 
108= 36= 3=t= ilizfrf. 
5?4=: ip8= i6i= 5i=£ I Pole, 
23760= 7p20= 660= 220=: 40=1 Furlong. 
190080—63360=5280351760=^320=8=1 Aa'//^. 

7^^/^ XIL Square Measure. 

Square In, 

144= I Feeffq. 
I2p6= p= 1 Yards fy. 
3600= 25= 2, J= a Jraces fy. 
39204= 374— 304= 10, S= 1 Poles fq. 
1568160=10^0=1210=435,;= /pzizi Rood fq. 
6272640==4356o=r4840=:i742s=ioo=4=i Atresfq. 

t^le XIH. Scripture ^eafure. 

Digits. ' 

4= 1 P/i/w. 

12= 3= 1 5^^. 

24= 6= -5= 1 C«Jif. 



96= 24^= 8?= A— 1 Fathom. 
144=. 36= 12= 6=i;= I Ezekiel^ Reed. 
192= a8= 16= 8= 2= i;= 1 Atabia^i Pole. 
l920=s48om6o=:8o=20=i3J=io=i Scbtenus otmea* 
furing Une. 

I Tahle 



58 CommoM Talks efMoney^ U^» 

m 

Table IV. Baftern Meafure. 

Cuhits. 
Aoo a= I Stadium. 

5000 = 5=1 SahhatbDays Jowmej. 

4000 r= 10 = 2 s^ I Eajttrn Miles. 
12000 = ^50= 6= 3^=1 Parafang. 
96000 ii: ^40 2= 48 = 24 = 8 = I Days j6Ufnej. 

Table XV. Hehrew Meafute. 

Gachah 

20= I Cah. 
56-= i-t== lOnar. 
120= 6z=r 3i= 1 S^^&. 
360= l8r= 10=: 5= I JB/>^^. 

1800= po= 50=15= 5=1 Letheclr. 
36oo=i8o=i=ioo±r30— 10=2=1 HomeTy 01 Giiron^ 

Table XVI. Hebrew Meajure. 

Eafb. 

If— I lo^. 

5J= 4^ iCiJ- - 
i6s= 12= -3= I H/«. 

52= 24^= 6=^ 2=: I &/2&. 

96^ 72= 18=^ 6= 3= I Safb Epba. • 
^gQ_y20:;=:l8o=:6o=3o=:lO=l Cofon Cbtmet. 

Table XVII. fli^lr^J Money. 

Gerabs. 

10 = 1 Bekab* 
20 == 2 =: i Shekel. 
1200 = 120 = 60 = I Maneh. 
60000 = 6000 = 3000 ac 50 = I 7a/^«r. 



'Decimal 



S9 



Decimal Tables efMohey, Weight, i£c. 




4,2 

■.!? 

84 

Si45 
ic,5 

i« 
12,6 
.3.«5 
14.7 
1^.75 

6.8 

.8,i> 

IP.)!? 



,0005208 

,001041^ 

,002083- 

,003125 

/x}4i« 

,0052 

,00625 

,0072(1 

,008^3 

,oo??75 

^ ,01041^ 

,011458^ 



9 il.040625 . 
10 ,d,04I*66 I 
16 iyi427083 
10 l',04575 

10 i',044791* 
I140458J? 

11 aP46875 
n ij047?l' 

-;!i>48<;583 




7.0 



,5208<r3 
,541*6 
6 4,5625 
,58?3!? 
,6041^6 
,625 
, , .«458j? 
;>c ,#66666 
■ ,6875 
, ,7o8)r;3 
4 ,7291*6 
?''75 



I 

iSo DeamalTalUs if Monty, Weight, 



J^^e* Tin^] atf^ Mfi^w, 



SE7 



/»*, 



i8 ,782^^6 



. 



1 5po:56<af5 

4, 01^5^2 

6^02193:8 

,032707 
101,0562^ 

^^vf ■« • III 

,O0OipQ5 

^001811 

31,002717 



4< Jacobus 



LD, Part: 




Q 
I 

% 

% 

4 

a 

21 
132 

24 
p. 






I 

a 
4 



7 

8 
9 

IQ 
II 

qr. 



J 

'J 

% 

,76 

3 

,84 

D. Fart. 

,01 

,013^5 
fid 

>023f334 

',03?66 



P 









4 




J 



■ ■ ^ ii^-- - 

|[>ooo8^| 

2 fioiifSk 

3 ,0025 



iZX P4prf^. 



I )0^3^037 
^ 1^74074 

te 

>9 >?7®3Ta 

13 >*8r4?i 
l3 ,*!&«;« 

<H555W| 
17 /2^2a 

H?,70870| 



21 
2i 
2^ 

24 

25 
26 

J. 



6i 

»?77777 

1)8888^. 

>*^902 
O' Parti. 



t 
2 

5 

4 
5 



■lii^pi 



^ii* 



I 

9 

10 

II 

jr. 



.oi^34j 
p^8>i95 

>02^&^4 

,030864 



I 

2 



,000771 
|OOI543 
,002314 



t-r 



t "^ ■ ■ " 



p <■ 



T A P L ^ «. ; 

I 

t 

Troy Weight ; im Po«i»/ 



i-\v .■•.»»"■ 



Pi:. I D. Far. 

1 ,083:35 

2 ,106^6 

3 »25 

4 »5'33?3 

5 ,4ij«^6 



<3z. 

T" 
8 

f 9 

10 

II 



,583-33 

,75 
,91^6 



Pwt. 



6i Decimal TaBIej ef Money f Weight, 



I 

? 

4 
7 

8 
P 

lO 

II 

12 

14 
17 

On 

I 

2 

? 

4 

I 

7 
8 

9 

10 

II 
12 

13 

H 

15 

i^ 

17 
18 



•MiMi 



i004i#5 
,oo8»?9 

,011^666 
,oao83'3 
,025 

P29U6 

»°45833 

,0541^6 
^583'33 

^6666 
»07o83'5 

»075 
fifT9U6 

D.Patt. 

^30173 
,000^47 

y0006pA 

,000868 

9001042 
,001215 

,oox?9p 

,00x562 

,001756 

fiOlpl 

,002083 
,002257 
,602431' 

,002604 

,002778 
,002951 
,003125 



19 
20 
21 
d2 

2? 



D.Fart. 

,0032^ 
,003472 
,063646 
,0038 ip 
?0035>P3 



^rOonoe 
ibe Integer. 



»•«•»• 



PnvXb.Part 



I 



I 
2 

3 

4 

6 

7 
8 

9 



I 

2 

3 

4 
5 

6 

7 
8 



,05 
I>i 
lis 
.a 

»a5 

»3 

'35 

»4 

.45 

.5 

»S5 
,6 

»7 

.7$ 

,8 

.85 
,9 

.95 



Gr. Z). Pdrf. 






I 

2 

3 

4 

5 
6 

t 



,00208^ 

,0041 iS6 

,00625 

.008^3^ 

,010414 

,012s 

.0HS8^ 



fir. 
T 

10 

ti 
12 

»3 
H 
15 



D.Part. 

flii666 

,0208^ 

P229I/f 

,025 

,02708^ 

P29ii^6 



Gr. 

i? 

17 
t8 

t9 

20 

21 
22 



*03?25 I ;23 



D. P/«ff . 

.03333^ 
,03541* 

'°575^ , 
.039583' 

/34ij$66 

.04375 
^4583:3 

/5479K 



TABLE m. 

Arerdupois Weight, aif 
Huudtei IVtigbt the kh 
teger. 



Qr. J), Part. 



1 
2 

2, 

lb. 

.1 

•2 

3 

4 

5 
6 

7 
8 

9 
10 

II 
12 

n 
14 



•25 
.5. 
75 



D.Part. 



,008928 
,017857 
,026786 

035714 

,044643 

053571 
,0625 

,071428 

,080357 

,089286 

,098214 

,107143 

,116071 

.125 

1I33928 



/*. Z). P^wt. 



t6 

47 
18 

19 

20 
21 
22 
23 

24 

25 
26 

27 
Oz. 

I 
2 

3 

4 

5 

6 



T^ 



,142857 

,151785 
,160714 

,178571 

,1875 

,196428 

,205357 
,214286 

,2232x4 

^232x43 

,241071 
D. Part. 



,000558 

,00X1x6 
,0^1674 
,002232 

1 ,00279 
^003-^48 



\ „ 



fir' 



Meajure, Time, dtii Mottm, 

"'■ "-"W-l |t^. V.Pm. Ox. D.Furti s~ 



*J 




«4 



'TT'BTrT'vir 



E 


Cp^rt. 


£ 


,025 






/=? 






,075 






,t 




', 


."^ 







.1? 




7 


M 




f. 


Cfm. 




1 


,003145 




i. 


yOO«2S 




3 


fi°9i7^ 


f 


4 


/1125 




^ 


,oi5«2S 
,0187s 








7 


,021875 




One Qtuiter 




r^f /»f<^fr. 




ft 

1 


D.Pm. 


2 


."5 




2 


.25 




1 


.575 




4 


.5 




S 


,625 




6 


,7S 




7 


.8« 





G. P. Pan . 

1 ,015625 

2 ,0)t25 

; ,046Sy5 

4 /«;5 

5 ,078125 
i ,o?375 
7 ,105575 



theUte^, 
a. AP-aft, 



.125 

,25 

>575 

.5 

,625 



,015625 

,03125 

,046875 

,0625 

,078125 

>09375 

-100375 



Liquid ^cafure. fot 
Wine, &c. Ok Tun 
tit Int^tr. 



D.ear). 



D.fm. 

.035714 
P7H'28 
,107143 
,142857 
.178572 
,214285 



,003968 
,007936 
,011904 
,015871 
,01984] 
,02380ft 
,027776 
,051744 

.035714 

,03^82 

<o+365 

,047618 

,051^6 

,055555 

.055^2? 

,063491 

,06746 



0»e Hoglh; 
the Integer 



,14285 

,25|7I 

,42856 
.57142 
,7142s 

.857«3. 
D.Part 

,01587: 
,031741 
.04761; , 
P6349; 
,07936' 
,09523! 
,*iln: 
,12658* 
.14285: 
.158^3 
,17460: 
W>47( 
.20634; 
1*2222: 
,23809. 
,25306! 
,269841 



'Jti Kuiidlei 
Ibi lnti-g?T 



Meafure-^ Thhei anH Motiom tfj 



.055555 

,232222 

,36«S88 
.*44«4 
,5 
.;5555'i 

,0^1111 

JI66666 
,772222 
^7777 

MHH 



D.Fm. 



,oo6P44 
,013888 
,0208^3 

.0347s-i 
,04ltfjc 
1,048^^1 



Offf Gallon 
thilttttger. 



,125 

,25 
i375 
i5 
,625 



TABLE VII. 

0/ Ak and Beer Mea 
fore i aiu Hoglbcad 
tht late^er. 



fr. 

1 


0. Prt«. 


a. 
? 


,l<i>66 


2 


>?3333 


3 


3 
4 


,5 

^6666 


4 


5 


.8<r331 


6 


G. 


D.Ftm. 


7 



D.Pan. 

,0625 

.08*335 
,104I^L 

.■458*; 



TABLE 

LoagMeafure; 1 
the lattgir. 



."5 

.25 

.375 

i5 

.^25 

375 



O. fort, 

,003125 

,06625 

.0051375 

,0125 

.015625 

P1875 

,021875 

,025 

,028125 

P3125 

.o?«75 

.o?>5 

,040625 

.04375 

,046875 



vin. 



D.Pai 



.053'25 

P5625 

■059375 

.0625 

,065625 

«6875 

,071875 

'^J 

,078125 

,08125 

.084375 

(=875 

,090625 

«S'375 

,096875 

,1 

,103125 

,10625 

.IOP375 

,1125 

1,115625 

',"875 

1,121875 



6iS 


'Decimal Tahies of Money, Weight 




ir. D.pMrtA 


f. 


B.Pat I ' ' " ~ 


1 — 


,002384 


3" 


,7 




[x 


,003568 


29 


.725 




2 


,ooii?6 


30 


(75 




? 


,001704 


51 


■r' 




1 


,002272 
.oozSii 


?■ 


,8 

|25 






34 
35 


'Jl, 








fjte Furlong 






r 




l).P^. 


3' 


.975 


J 


,025 




,2 


,05- 


I 


po»»45 


1 


'? 


i075 




,00,»iK 


" 


it 


.1 




iOI^I(« 

,o*«l8i 


I 


!S 


,125 






6 

'7 
'8 


.175 

2 




,0!.?27 




fT 


Z>. Part. 


, 




."5 






(7/ Time ; wm Tear th 




'3 




»3 
'4 

16 


.375 
4 


0«Pole(fc 




btiier. 


jT 


iXP/iyt. 


ST 


D.furt. 


M. 


D.Fm 


17 
i6 


.425 
.« 
475 
.5 


~" 


,iei8i8 


I 


,076923, 


II 


.846153 


»9 

30 


2 
3 


.?«3^36 
.>454i4 


2 
3 


,153846 

,307692 
,384615 

(538461 
,615384 


12 


(923076 
D.Pm. 


21 


.535 


4 


i7»7272 


4 






22 
23 

21) 


.!5 
.575 
.6 

,625 

^75 


5 

F. 
I 


,909090 


1 

7 

8 


I 
2 
3 


,01923" 
,0384* 
.05^63 


D.Part, 


,offo606 


20 

•=7 


2 


yl'21212 


9 
10 


,692207 
,76953 









Mea/urr, Time, and Sbtieiu- 


«7 


E 


x?^r 


D 

2 
1 

4 
J 


D.e^m 


f ABLE X. 


1 

2 

3 
4 
I 


,002747 

,0C54!'4 

:Sfo^ 

,01373'i 


,285714 

rfi857l 
,571428 
,714285 
357H2 


0/ Motion, jl^iz"J>*> 
Zodiac lie I»<J"'- 


D« 


D.Ptrl. 


M: 


D./tef 


6 ,010402 


H 

I 

2 

3 
4 
5 


a fart. 

P05P5! 
,011904 
,017856 
,02j8c8 
,02976 


1 

2 
3 
4 

1 


,0*6066 
,1 

.i«3333 
,i*j6d6 

,2 


3 
4 
5 
6 
7 
8 


,001*6^ 
,OCi222 

,002777 
,OOJ33: 
P03«S 
,00*444 


Ow Month 
tbe Imiger. 


W^.\t>.Pm. 


■ 


6 


,035712 


7 


.2?!333 


S 


,005 


f 


I 
9 

10 


,041664 
,047616 
,05!5<i» 
,05952 
,065472 
.071424 
,077376 


8 

10 
11 


,2^6666 

,<3!33; 

,3^6666 


10 

11 
12 
13 


,o:»555 

,006^11 

,00*666 

,007^22 


8 


11 

12 
13 


12 
'3 
14 


■4 
,4S3333 

4«6666 


14 
'5 
16 


PC7777 


; 


»4 


,08jJ2» 


15 


.5 


% 


,009^44 


6 


15 


,08pl8 


16 


.5*3333 


,0J 


4 


16 
■7 
i3 


,o?52n 
,101184 

,ll9a^ 
.124W8 


17 
18 
1? 


,5<6d6o 
.*33333 


20. 

21 


,011^66 




I? 


20 


^ii666 


22 


,OI«32 


0»» W«ek 


21 


,7 


2J 


,012777 


Ibt latter. 
D. D. fart 


21 
22 
23 


22 
23 
'4 
25 
26 




24 

2I 


,01*444 
,011 

,o»*51^ 


I ,14-2857 








27 


,p 


29 


,0)6^-11 








28 


,i>?3333 


30 


,0}jS666 








19 


,<H6696 


31 


,017^32 












32 


,Oi?777 








It'- 


D. Part. 


33 
34 


;S^i 








I 


,000^55 


35 


.01M44 








,2 . 


^ootui 


36 


,01 



'<«8 ntcimtlTalUs (f Monty, fVtIght, 



Meafure^ Time\ and Motion. 



^ 



S": b.tan.\ (S 

6 ,001^66 

7 ,001944 

8 ,caz222 

10 ,002777 

1 1 ,00:50^5 

12 ,00^333 

13 ^3d,ri 

14 fiO^%^ 

15 ,00411^6 

16 ,00*444 
,17 ^474r2 

18 P05 



20 
21 
22 
23 

24 

25 
26 

h*27 
28 

2?> 
30 

3i 
1 33 



D. Part. 



,00*555 
,0058^3 

,0061*1 1 

,006388 

,006666 

,006944 

,007*22 

P075 

,00^777 

,00805-5 
.ooi85'33 
;oc86i'i 
',0oe888 
,oo9ifS6 



1 5'. 



?4 
35 

38 

39 

40 

41 
42 

43 

44 

46 

47 



D,9art. 

,069444 
,009722 

,01 

,010277 
,0105^55 
,01083-3 
,01'! 1 1 r 

,oix3€:8 
,011066 
,011944 

filX222 

,0125 

>0I2777 

,013055 



48 

49 
50 

51 
52 

53 
54 
55 

56 
57 
58 
59 



b.Part. 

»oi?333 
,0136^1 

,oi3«88 

,0141^^6 

>oi*444 
,oi47«3 

,015 

»oi52S77 
,01*55. 

,oi5r 

,0161-1 : 
,01638] 



70 



A Crtfferal tktimal Talk. 



jip ■ ■ « " i 



III ■ 



I i in 



M Nl Nl M M M Nt ^4 ^V M I 



^l<»^,°y 



! 



■ Urn 



Jij: 









•MN4 ^&*^ 






00 






00 






NO 00^ 0\ Onn^ 4^ «m o^ 10 ^ o m 
M4^ Q\>0 w%W OvOOOvw^^>4 J3 



■5" 





QOvi x| 0\'«A<gn^uj lo is> M o 






2^.?^- 



% ^ M^ On K> 

^\ b> IN> .^ ^Is^ 



• '• '• ^w I 



^f^^ ^ "Xi^^C^^r «* ^ >« 






V|>^ M V|^^ lO 



, _O4^Q0n>^»-In/|NOOO<] hi VI I 

M |s>u«»4*>Vi^ OOVO O 1-4 K>4^vi 0\^ 00 
K>^ Q\ 00 K> go VlXl \0 M to VI Q\ 00 

0^ PO V ^ Bb^p\v^^Viife^»ieoT>H« »« O 



0\ 







00 






OS 0\ Vt N^^4^\MsM 10 »o M i-i O 

Q0v>^ ^ lo^ 10 0\i-i OvMN^ 0«»^ 

M V^ Qs4^ M 00 Ovv>^ M 00 Vt vja O ^^4 V\ to 



>o 



1 



00 OO*^*^ 0\ 0\ vwi 4^ <4^ >M sM ro K# M »N 



b jro 
-A lo 



I 

lb 
9. 

is 



'Si, 
I 



71 



T!oe Exphnation and TJ[e of^ the foregoing 

general Decimal Table. 

f . The Figjsrcs at J^, which run to 20, flicw Hie Numlet 
of Paris any QuanfHj or fnffger is divided intp ; and the 
Figtires in the fide Column are chofe Pafts ehcmfidves ; The 
Figures forming tjtie. ifrunfgular &)ace, and difpoled into 
S^q^t Cekva^ji 91^ the Pecimal P^s of l^ i^ff^al 
Quantity anfweiing thereto* Kenee any Quantity divided 
iato any Number of Parts under 20, the Ji^cimgls apfwer. 
ing to each ci ifede Pftrta are feen in o»r t^f rat in l^eic 
proper Column. 

Example. Suppofe a Quantity divjid^d ixfioJSigfa Pfud 
parfSy and you would know the PetimM Part equivalent to 
each : Look at top for ^^ under which, ^re difpoied Ae Deri* 
maUj viz. ,12^ ,25 ,375 ^ fuifweiing in otdet to the Parts 
in the j|ft/£ Gs>honm, ... 

2. The Figures in the fide Columns may be taken for the Nu^ 
merat&ry and thofe at top, for the D^OtfUMtar of a Vul- 
gar Fraftion, 

Then the Decimal correfpondhg toAokJuooJi^umiersT^ 
fpeaively, is equal to the fore&id FraSipm: ^hus the Z)^« 
cimal 5714 anf^eriog^o 4>iatlir fide, ^d 7 at topj is ejual 

to the FraBiofs 4-» So — s=:,$922. -5- :^j8^. — ,*=S'4i 

7 *3 ^^ H 

^Ib any larger FraSion whofe Parts 4te an Equimultiple 
of any (rf thefe tabular Fr/ifif/ow,, QP. nifty ;b€. reduced to 
diem, are equally anfwerd in DmWi.by-.this Table ; fee 

thU fir«»^/. ^ = ^ = |i=: ^^,3* Decimals 

in the Table. 

Cafi^. T<> reduce ztffDftimfil itttp^^^ equivalent kmmfi 
Parts of Coin^ Weights^ Meofure, Muffjffff &c. obferve 
this 

Hule. Multiply the given Decimal by the Number of 
Units contain d in the next lower Denomination of that Spem 
(ies ot Quantity, which your Decimal hi^} and- thus ptoceed, 

tUl 



7> ReduBion cf Decimals^ &:c. 

dD yoa Yoivt cofwerted your Decimalij or conic to the limefi 
Put ; and the fever al froduQs will be the fever al Parts ct 
the Quantiij required. See the following Example's; 

EXAMPLE I. 

What common Parts of ^ Poynd (vi^. Sbillings,^ Pence^ 
Qc^J are dontaiii'd in 0,73825 Decimal Parcsof a Pound? 
Firii^ Multiply by 20 Shillings, the next lower Detu. 

14,76500 ShilUngs. 
Ttten Multiply by 12 the next lower De. to the lafi. 

9,i8ooo Pence; 
Laflly, Muldplf by 4 the lowcfi Denomin. of all. 

0,72000 Farthings. 
Rence the Anfwer is i^r Shillings, p Pence, and 7 Tenths, 
or 72 Hundredths of a Farthing. 

EXAMPLE IL 

Reduce 0,7208^ to the known Parts of a Crown. 



9,604!)^ 

12 

7,24*99 

4 



mm 



Aniwer 3 s. yd^ ;^« 

E X A M P L E in. 

Reduce 5^90625 into known Parts of a Mark. 

9) 2,671875 



,2968750 

2671875 
89 0625 

11,8750000 
12 

10,5000000. 
4 

2,00000000 



Thus the Anfwer is exafl with- 
out any Remainder, viz^ 

Marks J. d. 

5 : II : 10 : 






EXAM- 



ReduBion of Decimals^ &c. 73 

EXAMPLE IV. 

Reduce ,727564 into the known Parts of a Pound Tray. 

12 



8,730768 
20 

14,615360 
24 



2461440 
I 230720 

14,768640 

The Anfwet is 8 oz. i\ fenny tat. 14 J**. \- 

EXAMPLE V. 

Reduce ,49723 into the known Parts of an C. Weight. 



1^892 
^ 

791136 
IP7784 

27,68j>76 
16 

413856 
68976 

11,03616 
16 

21696 
^616 



0,57856 

The Anfwet w'lB ftand thus, iqr. rjlh. 11 o;f. o;</r. 

L EXAM- 



74 ReduBion of Decimahy 8fC 

EXAMPLE VI. 

Reduce ,57? ot a RoX into irs known Parts. 

2^928 

3 



12 

6,558^ 



The Anfwcr is 3jrf/. oF/. 6\ In. ; and this repeat- 
ing Deciaial ,0^85' over. 

Thefe fix Examples I imagine fufficienC to (hew the ctmman 
Method of reducing Decimal Parts into the common and 
kftowii Parts of aiiy Species of Quantity. 

But as thofe Operations of Reduciion are for the moft part 
very lalorious^ tedious^ and require alundance of Figures^ 
I have fometimes wondred that a Set of Tables have not been 
compofed to facilitate this Part of Decimal Aritbmeticky as 
well as for the tontrarj Operations of reducing different Spe^ 
cies into Decimals ; efpecially lince one b as neceffary as the 
ether. Tables for that purpoie have long fiiKe been contrived, 
but none for the reverfe\ to turn /^fo, out not to turn out of 
Decimals. 

Tis tfue, Ibme Decimals (as thofe of Money) have the 
firji and fometimes the fecond Figure pretty ealily valued by 
a fmall Application of thought - but even this is tor the SkiU 
ful to do, not for any that are but Tyost or rude in the Art. 
Yet how much Tables for ReduSlion of Decimals to vulgar 
Parts are wanting, may appear from the great Indufiry many 
have uled to lay down Rules for tliat purpofe. which being fo 
prolix J verbofe, ohfcttrej and confequentlv impertinent^ that 
a Peribn wou'd fooner and with more eafe and pleafure work 
out bis Anfwer by t\itordi?/ary Method than by thofe unmteU 
liglfle and infgnificant Rules ; and according to the olaSaWy 
wuu'd find the tartbefi way about, the neareft way home* 

But 



DefcYipthn and life of New Talks. 75 

But havira for Realbns already rendered dctcnnined to 
write a ccmffeat Treatife of Decimal Ariihmeiicky I thought 
ic could by no means be vxtrihy of, or anfwer that Titlt^ uii- 
kr$ *vith many other ImfrtyvemefitSj 1 could make one more 
to tender this Part of the Art moft eafy and exfeditious ; ajid 
having imploy'd my Thoughts a little on this TopiCy I foon 
perceivi^d an Expedient tu^t wou'd do the bufijacf^^ wliich 
ivas this. 

F/z. To divide the Figures of Dedmah into Pairs j from 
the left Hand to the right ; then to tabulate the Digits ot 
every Pair from Units to an Hundred^ in proper Columns ^ 
and laflJy, to (tffix ibe true Value of every Plice of Figures 
in the Column appofttely anAvering thereto* 

So that by this means the Value of any two Plaices of Fi- 
gures (lb far as they are valuable) in any Decimal^ is feen by 
hfpeRioff oqly 5 and the Value of four or fix Places ar^ 9s it 
were placed in one vie\v, and with the greateft eafe andreadi^ 
fiefs are obtained in any common Species of Coins ^ Meafuresy 
and Weight. But it being a Contrivance of my owvy I Ihall 
not on that Account -fay anything as to their Merit ; but only 
give a IhortDefcriptiot of the Tables and th^ Manner oiufing 
them by an Example of each. 

A T^eferiftion of the following new T^ecima^l 

Talks. 

For every different Kind of Decimals in common Vfe^ I 
have compos*d a proper Set of Table s^ and according as the 
Decimal is more or lef^ valuable, it is divided into two or three 
Pairs ; and to each of thefe Pairs is a TahU exhibUiug the 
tru0 value of the Figures in eaclj Pair from Unit to lOO. 
The Tatles dE every lort are feen at the top of the Page ; 
thus reprefented Table \. Table 2. Sec. j^acb Table conlifts 
of y^ti<T4/ Columns; the fir/f Column has at Top ^^» to 
^nifie the Numbers of the Decimal Pairs. . The other Co^ 
lumqs (all but the laft) reprefent the Value of the Numbers 
in the firfi Column in the various Denominations of the 
Parts the Integer are vulgarly known by. The laji Co- 
lumn contains the Decimal Parts of the laft Denomination^ 
and in every Table is marked with P.ts. 

Having eiven a brief Explanation of the Tables in gene- 
xal, I fluU now ihew their Ufe in difcoVering the Value of 

L 2 an) 



7^ Defcription and life of New Tahks, led 

* • * • > 

any Decimal eiven, which it thus. Snppbfe you would know 
die y/due dttiiic Decimal ,689457 of a Tim Averiuf^if 
Weighty proceed thus^ firft divide thofe Namfaers iii«o Pjilri 
as here, 68,^4,57 ; then cake iixtfirfi riahc Hand Pair, ^tt. 
57y and feek inTMei. Averdufois weighty and you^tilT 

find againft •{ ^Z ^ j _ 'f ^ S that is,. 2p\ ;' then for the 

fecond Pair, 94, bek in ^aUe 2* and agaiiift P4 yob fee 
21 tt. o 0Z. ^ifwts. i and for the third Pair^ 58, ^k ifl 7b*' 
He 2. and agaiiiA 68 yob obfecve i^C. i^. 11^* •3,t0x. 
So that thefe feveral Nuaiben added in k pfoper aiannec MdU 
^land thus, 

r Pifft Pilr ,57 

Againft die < Second pair ,04 

C Third Riir ,68 



C. qr. 


U. oz. 


— 00 : •' 


00 : 02,04 


— 00; : 


21 : 00,52 


— n • 2 ; 


11 : 05,2 



,%to 



HeiKe the value of y689^^y is 13 ; ^ : 04 : 05^7<( 

Now tliat the Reader may (ee at oace both the AchoHi^gp 
and ExaSnefs of the 7Ii^/^i ; I ihall fhew the Work of Glid- 
ing the value of the £ud Decimal^ at large in ^catm^^ '^y 
01 ReduSiiony which is as follows. 

,68j?457 of a Tte. 

20 



1 5,789140 :=r Hundreef- 

3,156560 = Quarter of C 

^ <28 



•\ 



1252480 
^13120 



• » •« 



%6 

23020(80 
383680 



^,138880 = (Wiwwf 



» » t 



Th» 



C» ^« tt» . oz» 
The Value thisi^ay is 13 : 3 : 04 J 06,1388 8r. 
The Value by the Table is 13 : 3 : 04 : 05jr6 



*l — ■ ■!■ 



The Difierence only 00 : o : cx> : 00,3788 

Henee it appears how ekaS^ and yet how eaiy Und ^f^e* 
ditious thefe Thhle^ are in the buiinefs of Reducing Decimals 
to their fraper Value in the <o9»mofi and v«i^^ir Parts of their 
prq^ Inieger. I Aall next give Examples of all kinds of 
Decimal Parts j in order to render the Ufe and Emoluments of 
thefe new invented TabUsy fts f}ain ana ohnous as may be ; 
tho' they are of therafelves as eafy to be underftood as any 
^riibmetUMl W:d4es,vrk2a;Sy&fcu 

EX A MP L E t 

. What is the Value of ^725 of a Pmtnd SUrlh^g .^ 

s. d. 

Anfwet 9 : 5 : 4,6 

EXAMPLES 

What is. thctValne of ,147 of a Shilling or Ftoi f 

1 Taih^T^T 5^"' 1,14 is _ 01 ; 2 ,73 

The Anfwer in rf, and /, q|c /»• and qr. is o} : 3,05$ 

JE.X AMPLE ID. 

What is &e Value of ^7347 of a twnd TVoy .* 

««. fwt. 0. 
In ^ '^*'*' ''V affainft-J'47 is " — 00 : 01 : 3,07 

Anfwet 08 ; i g : 7^7 

PXAM- 



78 The life vj the New Talks for 

EXAMPLE IV. 

VVTiat is the Value of ,91249 of a C. We\iht AveriufoU? 

Qr. Ik. oz. dr. 
r Table !• f C >90 » — o : 00 : 00 ; 02,58 

- In ^ Tatle 2. > ag»infi ^ ,24 is — o : 00 : 04 : 04^ 
Z Table 3. 3 C ,91 »» — 3 : 17 • 14 • 1 1,53 

Anfwer 3 : 18 : 03 : 02,9 



EXAMPLE V. 



What is the Value of ,7777 of a Pound Afotbeearies 
might? 

5 5 3 gr. 
H 4 Si? '• Vag?iiift^'77is ^ 00 : o : 2 : 04^ 
*" ]1 Tdble 2. f w^" \ ,77 18 _ 09 : I : 2 : 15,2 

A)ifive4r 09 : .2 ; Z : ,00,08 



■p—— V 



EXAMPLE VI. 

What is the Value of ,8754 of » K<», Wine Meafure. 

P. Hg. T. C. 

Awftver l ; x • P : ?i 6 



EXAMPLE VII. 

What is the Value of ,7509 of a Load of Corn .^ 

2 5. Q* 

J Talle I. 7 .„_•«/ ,09 is — 0:0: 0,28 



i-isi-i »«*«•{ ,^,1: z 



3 J 6 : Q 



AniWer ^ r 6 : 0,28 



EXAM. 



Valuing all kindi of Decimals* 7^ 

EXAMPLE VIII. 

What is the Value of ,8495 of a Tear .> 

M. W. D. H. 
, f nhle 1. \ ^ A f ,95 is — 00 : : 3 : 11^2 

Anfwer ii : o : 2 : 01,62 



EX.A M P L E IX. 
What is the Value of ,889 of an Hour or Degree? 



99 



^ < Tnhh I. 'J^ ,„,:„A f ,9P is — ,— 00: 32,4 

^" i r^w^ 2. r 's^'"" v,88 is - ^ 52:48/ 



\ 



Anfwer 53 : 20^ 



EXAMPLE X. 

What b the Valiie of ,0596 of a Sigi of the Zodiac fi 

o i 9^ 

— 00 : 17 : 16 
ox : 30 : 00 

Anfwer 01 : 47 : 16 



^{Tu[:}'^^i%t z 



>«M«Mi*MW 



EXAMPLE XI. 

What is the Value of ,976305 of a Mile ? 

Ft. R. Yd. Ft. In. 

C Taile i.S C ,05 is : 00 : : : 00,31 

In ^ Table 2. C againft^ ,63 is : 02 : o : o : 02,79 

C TaHe 3.^ c >97 « 7 ^ 30 : 2 : : 07,2 

■■■■■ ■ I ■ "III ■ 

Anfwer 7 : 32 : 2 : o : 10,3 



EXAM- 



to TbeUfe ef the new Tailesy &c. 

£:XAMPLE XIL 
What is die Vdne of ,278 of a Jtadf^fuofe / 



o : 10,32 
yXj is — 2 : 61,9^2 

Anfwer 2 : 72,25 



EXAMPLE XUL 



What Uthc Value of ^9V^1 of a Milefyuiore ? 

A. R. P. 
C 7Ji>/f |. 7 ^ ,07 is — 0:0: 00,71 

^ : : 92 
505 : 2 : 16 

AnTwer 508 : 9 : 08,71 



r JJi>/* I. 7 r ,07 is — . 

^ ^ 7Ji>/* 2. >s^pdnfl^ ,50 is r- 



EXAMPLE XIV. 
WiMt i» tlM Vidiie of ,p74 of an ^^#/ 

R. /». r. 

. f KJfe I. Xaoainft^ yp — o : 00 : 19,?6 
"l7SJ/r2.r*8"°"1. ,97 — 3 t 31 ' o6/?5 

Anfwet ^ : 35 : 25,41 

EXAMPLE XV. 

What is the Value of ^79 of a Y^fdfoUtf? 

F. Ik. 

•^ \ »W* 2. r. "S"*^ 1 ,62 is -. 16 : 1278,72 

Anfwet 16 : 16^,66 

TABLE 



8i 



A Set of New Decimal Tables exprejjing 
the Value of any Decimal in the Kjiown or 
Vulgar 'Denominations of the Ditepal Qjian-- 
tity^ whether Money, Weight, Time, Mo- 
tion, or Meafure of every Kjnd. 

Table I. Of Money, One Pomd the Integer. 




M 



N<». 





8i 


A Set cf New Decimal Tahlei, 


&C. 






N». d. 


/. 


I'l. 




N°. 


d. 


f. 


h. 


N8 


d. 


f. 


Pt. 




?' 


2 





■I« 




po 


2 





M 




S"> 


2 




,12 






2 





^■i 




91 


2 





■71 




96 


2 




,21 




S7 


2 





•IS 




92 


2 





.81 




97 


2 




,?I 






2 





.14 




n 


2 





,92 




9S 


2 




.4 




H, 


2 0','i4 




P4 2 


I ,02 


,99 


2 




•■i 



Table n. Of Money^ one Pouudthe Integer, 



N^.r: 


T 


r. 


K 




"N^ 


"T 


T 


r 


W 




pa: 


r 


■^ 






























I — 


2 


I 


,6 




29 


5 


9 


2 


,4 




57 




4 


lU 


4 


3 


>2 




!=> 


6 












58 




7 


i — 


7 





s 




3" 


6 


2 


I 


?" 




59 




9 


4 — 


9 


■2 


A 




32 


6 


4 


3 


,2 




6c 







5 














3! 


6 


7 





,8 




61 




2 


6 




2 


I 


5" 




34 


6 


9 


2 


A 




6! 




4 


7 




4 


3 


,2 




35 


~ 












«5 




7 


8 




7 


o 


,8 




3» 




2 


1 


7 




6i 


12 


9 


P 




9 


2 


.4 




37 




4 


3 


,2 




«5 







10 














38 




7 





,8 




66 




2 


ti 




2 


I '.6 




V 




9 


2 


.4 




67 




4 


12 




4 


3.2 




40 














68 




7 


13 




7 


o 


,8 




4' 




2 


1 


',6 




69 




P 


H 




5> 


2 


.4 




42 




4 


3.2 




70 







»5 














43 




7 





,8 




71 


14 2 


l6 




2 


I 


/ 




4* 




9 


2 


,4 




7' 


14 4 


17 




4 


3 


,2 




45 










,— 




73 


'4 7 


i8 




7 





,8 




46 


9 


2 


I 


.6 




74 


14 9 


19 




9 


2 


.4 




47 


9 


4 


3 


,2 




75 "5 


20 







C 






48 


9 


7 




,8 




7«i5i 2 


21 




2 


1 


,'6, 




4? 


9 


9 


2 


.4 




'Z '5! 4 
78;i5 7 


22 




4 


3 


i2 




53 


10 





0' 


, — 




2? 




7 





.8, 




5' 


10 


2 


1 


,6 




79,>5 9 


24 




9 


2 


>4 




52 


io 


4 


3 


,2 




80 16, 


35 










1— 




53 


10 


7 





,8 




81 


16 2 


26 




2 


1 


,« 


54 


10 


9 


2 


,4 




82 


16 4 


27 




4 


3 


.2 


55 


II 


c 


ci 






83 


16 7 


28 




7 of^ 1 1^6 1 


n 


2 


J, 


,6 


8^1161 9| 







^ 5ff of Mm Decimal Talk 


.,&c 




8} 


R5- 


T 


J. f. 


Pt. rJ'.ii: 


Z 


r 


R 




N°. C 


d. 


r 


K 


85 


17 





J 


9o;i8 








;_ 




95 


I? 








11 


2^ 


I? 


2 I 


.<s 


91 i8 


2 


I 


,« 




P6 


I? 


2 


I 


,6 


h 


■> 


4 5 


,2 


92 i8 


4 


? 


,2 




P7 


I? 


'4 


5 


,2 


fiS 


17 


7 


,8 


91 i8 


7 o 


,8 




s8 


19 


7 





,8 


B9 17 


9 2 ,4 1 P4 l8 


P 2l,4 




99 


12. 


9 


2 


il. 



Table I. Troy Weight, one Pound the Integer. 



84 


j4 Set of New 


•Decimal Tahiti 


,&C 




w:fv.rg,.ip,. 


iM» 


p'^/.^r. 


7?: 




tf^fw.grft. 


85 


2 c],96 


i;^ 


2 


3 


S 




P5 


2 


61,72 


86 


2 


> m 


ipi 


2 


4 


.4' 




P6 


2 


7 .29 


87 


2 


2 ,11 


IP.'. 


2 


4 


.PP 




P7 


2 


7. 87 


88 


2 


= ,J8 


iPS 


2 


5 


.■)<i 




P8 


2 


8.4+ 


89 2 


=,,26 


94 


2 6 


JLi 


_ 


99 


2 


9 ,02 



Table IL Troy Weight, os? Poui/Jthe Integer. 



A Set of. New Decimal Talks, &c. 8y 




IT: 
o 

9 



pt. 

,2 



14' 54 



Table I. 



^^, Pwt.gr. 






I 

2 

4 

5 

6 

7 
8 

9 
10 

20 

40 

50 
60 

70 
80 

90 



I 
I 

2 

2 

n 


3 
4 



,048 

,op6 

^144 

,IP2 

jH 
,288 

.384 
>432 

,48 

,96 

>44 

)92 

54 

,88 

,84 
o2 



90 
91 
92 

P3 

94- 



:>z 



10 
IC 

II 

II 

til 





«'• 


/»*• 


16 


c 


*— — 


18 


9 


,6 





^9'5 1 


3 4:.8 


5 I4S4 



95 
96 

91 
98 

99 



0J&. 



II 

IIIIO 
II 12 



8 



II 
II 



IS 

17 



o 

9 

^9 

4 



,6 
,2 
,8 



14!'4 



Table II. 4 J°y ^eigh^ ^w 

t O/ifwe^ /^^^ Integer. 



I 

> 

4 

5 
6 

7 
8 

9 

10 

II 

12 

16 

17 
18 

IP 

20 
21 
22 

23 

24 

25 
26 

27 
28 



Pwt:gr. 



2 
2 
2 
2 
2 

o 



3 



4 
9 

H 
19 



4 
9 

H 

19 



4 
9 

H 
19 

o 

4 
9 

H 
19 

o 

4 
9 

14 
19 

o 

4 
9 



4 
4 
4 
4 
4 
5 
5 

5i 
5 \h 



Ft 
,8 

»4 

) 

,8 
,6 

)4 
,2 

,8 
,6 

•2 

> — 

,8 

,6 

,4 
,2 

:8 

.4 
,2 

,8 
,6 

.4 



NO. 


fwt.grA Ptl 


29 


5 


19 


>2 


50 


6 







?i 


6 


4 


,8 


32 


6 


9 


,6 


33 


6 


H 


A 


34 


6 


19 


j2 


35 


7 





5*" 


36 


7 


4 ,8| 


3^ 


7 


9 


,6 


38 


7 


H 


>4 


39 


7 


19- 


>2 


40 


8 





) — 


41 


8 


4 


,8 


42 


8 


9 


,6 


43 


8 


14 


.4 


44 


8 


19 


.2 


45 


9 





5 


46 


9 


4 


,8 


+z 


9 


9 


,6 


48 


9 


H 


4 


49 


9 


19 


,2 


50 


TO 





> 


51 


IC 


4 


,8 


52 


10 


9 


.,6 


53 


10 


H 


r4 


54 


10 


19 


>2 


55 


11 





> 


S6 


II 


4 


,8 



NJ>. 



85 


AS-it 


•^^A 


''ew 


Decimal Tables, 

A 


, &C 


• 




N*. t'vJt.gr. 


Pf. 




/'/J 




N-j 


Pwi. gr.^i't.^ 


57 


II 9 


,6 




72 


\\ 


S> 


.6 




87 


17 


9 


,6 


58 


II 14 


>4 




73 


14 


H 


A 




88 


17 


H 


>4 


5? 


II Ip 


>2 




74 


14 


IP 


.2 




8p 


17 


19 


•2 


60 


12 


?•' 




75 


15 





>•• 


1 


90 


18 





>— 


61 


12 4 


,8 




76 


15 


4 


,8 1 


91 


18 


41 


,8 


62 


12 9 


,6 




77 


15 


9 


,6 




92 


18 


9 


,6 


6? 


12 14 


>4i 


78 


15 14 


,4 




93 


18 


H 


y^ 


64 


12 Ip^,2 




19 


15 »P 


,2 




94 


18 


19 >2 


65 


13 


0,- 




80 


16 





,~ 




9S 


19 


,— 


66 


*3 


4!,8 




81 


16 


4 


i,8 




96 


19 


4S8 


67 


M 


9 


,6 




82 


15 


9,6 




97 


19 


9 


,6 


68 


1? 


14 


.4 




8? 


16 


14 


,4 




98 


19 


14 


A 


69 


H 


19 


.2 




84 


16 


19 


,2 




99 


19 


19 


,2 


70 


14 


ol,~ 




85 


17 





> 










71 14 ' 


4 )8 


86 • 17 • 4 


,8 






1 ■ I 



Table I. 



_, , , ,, C Averdupois Weight, 
1 able 11. -^ ^j,^ p^,,^ j;^^ /„,g^jy. 



I 



I 
2 

? 

4 

> 

6 

7 
8 

5> 

10 

20 
50 
40 

50 
60 

70 

80 

90 



I 
I 
I 
I 

2 

2 



■■■•wp' 



,025 
,051 
,076 
,102 
,128 

»i53 
,179 
,204 

,23 
,256 

,512 

,768 

,024 

,28 

,792 
,048 

2304 



I 

2 

4 

5 

6 

7 
8 

9 

10 

II 

12 

H 

15 

16 

17 
18 



oz. 



I 



2 
2 
2 
2 
2 
2 



i/r. Pf. 




N**.oz.| 


dr. 


Pf. 


2 ,56 




19 


3 





,64 


5 


,'2 




20 


3 


•3 


,2 


7 


,68 




21 


3 


5 


,76 


10 


»24 




22 


1 
5 


8 


,32 


12 


,8 




23 


3 


10 


,b8 


15 


,36 




24 


3 


13 


,44 


1 !,92 




25 


4 





J 


4^48 




26 


4 


2 


,56 


7 ,04 




27 


4 


5 


''0 


9:56 




28 


4 


7 


,68 


12 


,i6 




29 


4 


10 


,24 


14 


,72 




30 


4 


12 


,8 


I 


,28 




31 


4 


15 


,36 


*> 



,84 




32 


5 


I 


.92 


6 ,4 




33 


5 


4 


,48 


8 1,96 




34 


5 


7 


,04 


11 ;,52 




35 


5 


9 


.6 


14 


*o8 




36 


5 


12 


,16 



N**. 



A Set of New Decimal Tahlesy 3cc. 87 



lN". 


oz. 


«//•• 


-'M 


N".l02.Ur.j et. 


"w: 


oz. 


dr. 


v±\ 


37 


"5 


14 


>72 




5T' 


9 4 


,48 


79 


12 


I0(,24| 


38 


6 


I 


,28 




59 


P 7 


:,04 


8b 


12 


12 


>8 


39 


6 


3 


,84 




60 


P P 


,6 


81 


12 


«5 


.36 


40 


d 


6 


>4 




61 


P 12 


,i6 


82 


13 


. I 


.^2 


41 


6 


8 


,P6 




62 


P 14 


.72 


|3 


13 


4 


h8 


42. 


6 


II 


,52 




63 ' 


ro 1 


,28 


f+ 


13 


7 


,04 


43 


6 


14 


,08 




64 1 


10 3 


.84 


8s 


13 


P 


,6 


44 


7 





,64 




^5 1 


[0 6 


»4 


8^ 


»3 


12 


,16 


45 


7 


3 


,2 




66 1 


:o 8 


,P6 


87 


»3 


H 


»72 


46 


7 


5 


,7<5 




67 I 


II 


>52 


88 


H 


f 


,28 


*z 


7 


8 


'P 




68 I 


14 


,08 


89 


H 


3 


,84 


48 


7 


10 


,88 




69 I 


I 


,64 


po 


M, 


5 


.4, 


49 


7 


13 


H4 




70 1 


« 3 


,2 


Pi 


'4 


8 


,p6 


50 


8 





»— 




71 • 


I 5 


,76 


P2 


H 


II 


,52 


51 


8 


2 


,56 




72 I 


1 8 


'li 


P3 


H 


H 


,08 


52 


8 


5 


,12 




73 « 


I 10 


,88 


P4 


»5 





,64 


53 


8 


7 


,68 




74 ' 


I 13 


.44 


P5 


»5 


3 


,2 


54 


8 


IP 


,24 




75 I 


2 


)^^ 


P6 


'5 


5 


,76 


55 


8 


12 


,8 




76 I 


2 2 


,56 


P7 


•5 


8 


'3^ 


5<J 


8 i5l 


,36 




77 : X 


2 5 


,12 


P8 


15 lOl 


,88 


fs7 9I il 


,92 


178 1 1 


2 7 


,68 


99 t^li^l^fl 



Table I. 



I Table IL k Averdupois Wdght. 
1 ( 0/^^ C« or 1 1 2 lb. the Int. 



'N». 


dr. 


/'f. 


hjo. 


oz. 


«/r. , Pt. 




N'^.joz. , rf)-. |, pf. j 


• 1 




,02 




I 


» — 


2 


,86 


' 


12 


2 


2 


>4 


2 


m T 


.05 




2 


• " 


5 


.73 




n 


2 


5 


)27 


3 


-- 


,c8 




3 


p» • 


8 


,6 




14 


2 


8 


,»3 


4 


-- ,11 




4 -- 


II 


.46 




15 2 


1 1 


.0 


5 


-- ,14 




5 -- 


14 


»33 




i6 2 


13 


,87 


6 




.17 




6 


I 


I 


,2 




17 


9 





'73 


7 


m m, 


,2 


7 


1 4 
I 6 


,06 18 


3 


3 


1,6 


8 


- - 


,22 


s 


,P3 IP 


*» 

3 


6 


,47 


P 




>2'> 




p 


1 


P 


,80 


20 


3 


P 


'34 


10 


.' ,28 


.0| I 1 


12 


»^7| 


21 


3 


12 


f,2 


ho 


--',57 ill ' I • 


«5 ,53' 1 


22 3 ' IS ' 


,7 



N'. 



88 J Neva Set of DecimalTailei, !ic. 

IT-k]- IT S Averdnpois Weiglit, 



m 


4r. 


ft. 




KF 


0^. 


dr. 


fl. 




N^. 


oz. 


dr. 


tt. 


io- 




fi6 




25 




I 


•2*' 




5° 


"s 


15 


.35 


40 




.14 




24 




4 


,8 




51 


S 


2 


.21 


50 




.45 




25 




7 


,67 




52 


P 


5 


,c8 


ip 




,72 




26 




10 


.14 




')3 


9 


7 


'2' 


70 




e 




27 




13 


.4 




M 


9 


10 


.81 


80 




I29 




58 







.27 




55 


P 




,68 


90 


2 


.58 




2S> 




3 


.14 




l-s 


10 





.55 




)*' 




?n 




b 


.CI 




57 


10 


^ 


.41 








31 




8 


,87 




58 


ic 


6 


.28 








32 




II 


.74 




59 


10 


9 


.■5 








33 


1 


14 


,61 




60 


10 


12 


fx 








54 


I 


.47 




61 


10 


■4 


,86 








35 


6 


4 


.34 




62 


11 


1 


.75 










36 


t 


7 


.31 




53 


" 


4 


.62 



A Set vf New Deciimti TaHei, Zic if 



Table I. 



I Table "■ 1^ ^ c. Wt. I*. J»tii«i 





























^_ 


m 


T 


e± 


T. 


K 


"1: 





I 


^:i 


;r 


-)«, 


^ 


Pf. 


76 


~ 


\'j 


9 


.89 




Si 


25 


5'r*5 


c 








^ 


77 


^ 




12 


'75 






1 


1^ 


i'-s 


I 


I 


■4 


j7; 


78 


_ 




■ 5 


i62 






2 


% 


5-^ 


2 


3 




-M 


79 


— 




2 


,49 






3 


5-8 


3 


5 


12! 


-it 


8o 


_ 




■i 


■ 36 






4 


2? 




4 


7 


10 


3S 


8i 







8 


)22 






1 


3P 


5 


9 


9 


.6 


82 


_ 




II 


^ 






6. 


3!! 


531 


■6 


11 


s 


^f 


1' 


— 




"3 




,96 
,82 






I 


3t 
3! 


5''l3 


7 
8 


13 
■ 5 


7 
5 


.04 


85 


— 




3 


,6? 






9 


3» 


1-J* 


10 


I 


4 


.48 


S6 







6 


.5« 






10 


35 


6( 1i 


ri 


3 


3 


,2 


87 







9 


»4I 






II 


3' 


6. 16 


12 


15 




,92 


88 


_ 




12 


,I8 






12 


^i 


6 !7 


"3 


7 


c 


.64 


8s. 


-^ 




15 


><5 






13 


6, 18 


H 


8 


5 


•5^ 


90 







2 


W3 






■4 


39 


6' J9 


15 


id 


14 


,08 


91 







4 


,8p 






15 


40 


6' -?0 


16 


12 


12 


>8 


S>2 







7 


.76 






16 


4> 


i''" 


I7 


■■* 


1 1 


.52 


P3 







10 


.IS? 






;i 


42 


i' J2 


'9 





'i 


,24 


04 







1? 


Mtp 






43 


& -,3 


2c 


2 


8' 


,96 











.36 






IP 


44 


K '4 


21 


1 


T 


,08 


y5 






3 


■^3 






20 


45 


7' -"5 


22 


6. 


4 


97 






6 


fi9 






2t 


46 


7 >6 


23 


8 


5 


>I2 


P8 






8 


^96 






22 


47 


7 >7 


24 


10 


3 


.84 


PJi 






II 


.83 






23 


48 


7 >8 


J5 


12 


2 


.5« 










1 




2+l4P 


7t,'25 


j6 


Lii 


-lfc2£i 



Table I. 



TdW, II 1 AverSapoi! Welghr^ 



sr 


nr 


PI. 


N"^!*. ,«. 1 ft. 1 


N'.i7*: 


-gr-K- 


1 


_- 


.01 




1: 





3 


,58 ' : 


5 


1 


!• .J 


2 


.- 


.07 




2 





7 


.16 


6 


1 


15 >48 


3 


-- 


,1 




3 





10 


,74 


7 


1 


9 .06 


i_ 


.ijLjii. 




^ 





J± 


^' ^ 




1 


12 .64 



90 


JSetarNew DecmalTa^es, 


&C 






TF 


oz. 


ff. 




T^ 


!?: 


oz. 


*»!. 




N-i. 


It. 


~oz. 


rr 


5 


.". 


J8 




9 


,2 





>22 




^ 


13 


Tt 


'^4 


-6 


^• 


.»» 




10' 


2 




,8 




49 


10 


15 


.42 


7 




,25 




11 


-2. 




.38 




P 


II 


3 




,8 


-- 


,28 




12 




io 


,96 




51 


II 


6 


^58 


9 




1^2 




■) 




14 


.54 




52 


11 


10 


,16 


10 


-• 


,35 




'4 






.12 




53 


II 


■3 


,74 


■20 


-• 


41 




'5 






.7„ 




54 


12 




.32 


■50; 


1 


'°7. 




i« 






,28 




55 


12 


1 


,9 


4bi 


1 


.43 




■7 




12 


,86 




56 


12 


.48 


1°^ 


I 


,79 




18 






.44 




^i 


12 


2 


,06 


60 


2 


M 




'9 






,Q2 




12 


5 


,64 


70 


2 


% 




20 






.6 




59 


■ 3 


J 


.22 


8o 


2 




21 




ii 


,18 




60 


13 


5 


,8 


90 


3 


>22 




22 




14 


.76 




6l 


13 


3 


& 










23 






.34 




62 


»3 


5 


,96 










,'4 






.92 




53 


'4 




.54 










21 






.5 




64 


14 


j 


.12 










26 




13 


.08 




65 


■4 


\ 


.7 










27 


*6 




,66 




66 


"4 


1 


,28 










28 






.24 




67 


14 


) 


,86 










25 






,82 




68 


■5 


! 


.44 










.30 




II 


.4 




l!9 


15 


7 


,02 










.!■ 




14 


.98 




70 


15 


3 


,6 










32 






M 




71 


15 


t 


,18 










3i 






•H 




72 


16 




,76 










34 




9 


.72 




73 


16 


! 


.34 










'3I 


8 


>3 




% 




74 

75 


16 
16 


i 


.92 
.5 










37 


8 


J 


.4* 




76 


17 





.08 










58 


8 


.04 




77 


17 


3 


,66 










39 


8 


II 


,62 




78 


17 


7 


;^5 










40 


8 


■5 


.2 




79 


17 


10 










41 


9 


2 


,78 




80 


17 


>4 


.4 










42 


9 


7 


,3« 




81 


18 


1 


■98 










43 


9 


9 


,94 




82 


18 


5 


.5« 










44 


9 


'3 


,52 




1' 


18 


9 


.>4 










45 


10 


1 


,1 




84 


18 


12 


.72 










46 


10 


4 


,68 




85 


19 





.3 










47 


10 


8 


,26 




86 


19 


3 


,88 



"5 

88 
8p 

9.1 



lb. 



j4 Set of New Decimal Tallesy &c. pi 



19 
19 
19 

20 
20 



ezr pt. 



7 
II 

2 



^6 

,04 

,62 

,2 

.78 



92 

P4 

P5 
I 96 



20 
20 
21 
21 
21 





— — — 1' 

21 1 1.1 ',26 

21 14' ,84 

22 2,42 



Table III. Averdapois Weight, me 'fun the htegtr* 



!11- 


c: 


"5r 


/*: 


oz. 


"F? 




N< 


C. 


Q^\lh. 


oz. 


/•f. 


• 


~~— 


— •" 


—'mm 


• * 


- ^ 




■■ i» 


«.. 




•m^ 


^ ■■ 


•--MS 


I 
2 








I 


22 

z6 


6 

12 






29 

30 


5 
6 


3 




5 




p 




.^ 


3 





2 


II 


3 


>2 




31 


6 





22 


6 


)8 


4 





3 


5 


9 


,6 




^2 


6 


I 


16 


12 


5 
6 


J 











J'"* 


* 


33 


6 


2 


II 


3 


>2 


I 





22 


6 


,4| 


•34 


6 


3 


5 


P 


>6 


7 
8 


I 


I 


16 


12 


,8 




35 


7J0 
71 








J... 


1 


2 


II 


3 


,2 




36. 


22 


6 


>4 


P 


I 


9 


5 


P 


,<5 




37 


7i 


I 


16 


12 


,8 


10 


2 











> 




38 


7 


2 


II 


3 


>2 


12 


2 
2 



I 


22 
16 


6 
12 


>0 




39 

40 


7 
8 


3 




5 



9 




,6 


19 


2 


2 


II 


.3 


,2 




41 


8 


.0 


22 


6 


i" 


H 


2 


5 


5 


P 


,6 




42 


8 


I 


16 


12 


^5 


? 


a 








J""~ 




43 


8 


2 


II 


3 


>2 


16 


3 





22 


6 


'i 


44 


8 


3 


•5 


9 


»6 


17 
18 


3 


I 


16 


12 


,8 


45 


9 


1 





.—^ 


3 


2 


II 


3 


'3 


46 


9 





22 


6 


.1 


ip 


3 


3 


5 


P 


,<i 


4^ 


P 


I 


16 


12 


20 


4 


0. 





. 


>^" 


48 


P 


• 2' 


II 


^ 3 


,2 


21 


4 





22 


• 6 


u 


4P 


P 


3 


5 


9 


,6 . 


22 


4 


J 


16 


12 




50 


10 











1- 


23 


4 


2 


IX 


3 


.2 




51 


10 





22 


6 


1 


24 


4 


? 


.5 


P 


,6 




52 


10 


I 


16 


12 


25 


5 





. 





j""* 




53 


10 


2 


11 


3 


>2 


26 


5 





222 


6 


'i 




54 


16 


9 


5 


9 


,6 


27 


5 


I 


16 


12 


,8 




55 


11 





•0 





• 


28 


5 2| 


II 


3 1,2 f 1 «56 1 


II 





22 


5 


iJ 



N 2 



N" 



si 


> Set (f-Nfo, Dtcimal Talks 


arc. 




51 


TT 


«-i25 


ysj 


«^ 




T*° 


C. 


?; 


IT 


». «.| 


57 


II 


T^ 


12 


i ■ 




y 


■1 


3 


i 


9 


,6 


sS 


II 


2 


■| 


3 


,2 




go 


16 





<j 





>o 


5? 


II 


; 


9 


A 




81 


16 





22 


6 


a 


ia 


12 





2I 









82 


16 


1 


16 


12 


ISI 


12 





«' 


'i 




83 


16 


2 


It 


3 


,2 


&2 


12 


1 


ad 


u- 




i* 


>« 


J 


5 


9 


.<* 


63 


12 


z 


11 


3 


.2 






17 











■D 


«4 


12 


3 


^ 


? 


^ 




86 


17 





22 


« 


t 


4? 


ij 


a 


Q 






«7 


17 


I 


16 


12 


66 


I? 





22 


6 


^ 




88 


17 


2 


II 


3 


,2 


67 


n 


I 


16 


12 


A 




%> 


17 


i 


5 


9 


,6, 


1^8 


13 


2 


If 


3 


(2 




pa 


|3 











|0 


159 


13 


J 


1 


? 


1« 




iPI 


16 





22 


6 


i 


70 


U 















P2 


18 


1 


16 


112 


71 


■4 





22 


<> 


.? 




55 


(8 


2 


11 


'3 


,3 


72 


H 


I 


16 


112 




94 


16 


3 


5 


9 


.« 


73 


H 


a 


II 


3 


.2 




91 


19 








; 


, — 


74. 


u 


3 


5 


» 


A 




\^. 


19 





,22 


6 


4 


75 


15 











i— !>7 


19 


I 


16 


12 


,8 


76 


15 





22 


6 


,4 98 


19 


3 


II 


3 


,2 


77 


15 


I 


16 


1,2 


JH » 


19 


3 


; 5 


, 9 


>6 


78. 


-il 


2 


II 


-J, 


,2 








■ 





T»ble L 



T^T. II i Apothecaries Weight, 




try*: 



'*. 



w. 

St 

82 

86 

87- 
88 

89 

90 



s: 
52 

54 
14 
55 
56 
56 



o 

5 
2 

r 
.4 

6 
3 

6 
5 



,24 
,28 

>4 

>44 

,48 

•,56 

,0 



^'^■1. I . >l 




Tabic I. 



t • * * i 

I '■ T*hle II JT ^®* M6a(qre, W 



•¥ 



t^^. 


* • 


pt. .| 


N°: AmKI 


7T 




N6. tf . |> j 


"m 


aiB>» 


-— 


-^ 




•"^- 


.— — 1 


-— 


I 


■ 1 1 ^— j 


•mm~ 


I 


— 


,04 


« 


I 





4 


,32 




23 


12 


3 


,3^ 


2 


— 


-iD8 


" ' 


"2 


T 





,64 


^w *- 


"24 


12- 


7 


'fiS' 


? 


— 


>I2 




3 


I 


4 


,96 




25 


»3 


4 


tt^ 


4 


— 


.J7 




4 


2 


I 


#: 


• 


26. 


?♦ 


•0 


i32 


5 


— 


»21 




5 


2 


5 


,6 




27 


H 


4 


^4 


6 


— . 


,J«5 


-' 


^ 


3 


X 


»pa 


• . - 1 


^ 


11 


a. 


^ 


7 


-^ 


.3 




7 


3 


& 


»24. 




i2P 


15 


5 


,28 


6 
P 


, 


',58 


» 


:8 


4 
4 


2 
6 


'88 


i 


30 
31 


.16 
16 


1 
5 


,6 

,P2. 


10 


.— 


4? 


• 

1 


:io 


5 


3 


.2. 


t 


32 


17 


2 


.24- 


20 


'— * 


35 


« 


It 


5 


7 


»52 


• 
• 


33 


17 


i5 


:?il 


30 


I 


,29 




12 


6 


3. 


,84 


t 


34* 


18 


2 


40 


I 


,72 


i 


13 


7 


0. 


,16 




31 


18 


7 


.2 


50 


2 


,16 


{ 


H 


7 


4 


,i5 




36 


iP 


3 


,12 


60 


2 


>59 


« 


15 


8 





,8 




32 


ip 


7 


34 


70 


3 


.02 




16 


8, 


5 


,12 




38 • 20 


-4 


,16 


80 


3 


il5 


» * 


17 


P 


I 


.44 




^P 


21 





.f 


90 


3 


,88 




18 


P 


5 


'7^ 




40 


21 


4 






t 


• 
• 


ip 


10 


2 


,68 




41 


22 


9 


,J2 




■ 


• 


( 


20 
21 


10 


6 
2 


.4 
,72 




42 
43 


22 
23 


:5 
I 




mmmm 


i 1 


I22 'III 7 ',041 


mmm 


44 ,23 1 6 


,08 1 



N*, 



'^l AStt,cf-Nlnt'I>l(imolTitki,tic. 



m 


-t] 


rr.- 


TF 


■5! 


1"" 


ITT 




■^ 


'Sq 


r-m 


H- 


























ii 


•'- 


<;'' 






T— ■ 


5 


13 




1' 


22 




A4 


\t 


-- 


,f 






I- 


3 




1; 


23 




.48 


» 


-• 


>I5 






'l 


7 


)I2 




38 


,23 




(52 


1 A 




.» 






2 


4 


',16 




3S 


24 






.25 






'3 


I 


!,2 




40 


25 




.6 


■ 6 


-- 


i3 






3 


« 


::li 




41 


25 




,64 


'^l 


-•■ 


.)5 






4 


3 




42 


26 




J68 


• • 


4 






5 





.32 




43 


27 




.72 


9 


.^ 


rfl 






5 


5 


,36 




44 


^ 




.76 


10 


-- 


.5 






6 


2 


>4 




45 


28 




.8 


'20 


1 








6 


7 


'4J 




46 


28 




fs^ 


}0 


I 


.51 






I 


4 


,48 




+2 


29 


. 


%■> 


2 


f=" 






8 


I 


'■'! 




48 


30 


1 


.92 


10 


i 


.52 






8 


6 


,s« 




4P 


30; 




,96 


60 


i 


^2 






9 


3 


.4 




50 


3" 






^ 


3 


,5! 






10 





:^s 




1> 


32 




^ 


♦ 


,02 






10 


5 




52 


32 




93 


4 


.5? 






11 


2 


.72 




53 


33 




il2 






11 


7 


'Z' 




54 


34- 




,ld 






12 


4 


.8 




55 


34. 




*3 






13 

■3 


I 
6 


% 




56 
57 


35 
35 




'^ 






14 


:3 


,92 




58 


3S 




.32 




24 


■5. 





,96 




59 


37 




.36 




25 


15 


6 






«o 


37 




4 




26 
27 


16 
'7 


3 




.5 




61 

62 


38 
3? 




;^ 




28 


17 


5 


>I2 




5' 


39 




.52 




2? 


18 


2 


,16 




64 


40 




.5« 




30 


18 


7 


,2 




65 


40 




,6 




31 


'9 


4 


;:^ 




66 


41 




:a 




32 


20 


I 




67 


42 






33 


20 


6 


,32 




68 


42 




.72 




34 


31 


3 


.3^ 




rip 


43 




,76 










Ji. 


22 


° 


.4 




J2. 


44 


6 


.8 



■R^ 



7» 
72 
73 
74 
75 
76 

77 



44 

45 

45 
46 

47 

78 49 
7$> 49 
8o ISO 



5 

2 

7 
4 

2 

7 

4 
I 



G. f. Pt. 

i 

,00 

,96 




,12 

,16 

,2 






■*. 



82 

85 

86 

87- 
88- 

89 
90 



sTzr 



5« 

5« 
52 
52 
53 
54 
54 
55 
56 
56 



o 

5 

2 

r 

4 
1 

6 
3 

6 
5 



> 

,24 
,28 

»3* 

»4 

>44 

,48 

,56 



W 



sn. 

92 

93 

94 

95 
96, 

97 
98 

99 



57 

58 

59 

59 
60 



2' 

4 
I 

6 
3; 



6i| o 
6l| 5 



r 



2 2 




18. 




y9t 



Table I. 



I 



T»ble II X^®* M^ifqre, m^ 



"TO 

I 
2 

3 

4 

5 
6 

7 
6 

9 

10 
20 
30 
40 

50 
60 

70 
80 
90 



■?: 



>a4 
"io8 

>»2 

.17 

.21 

*«5 
>3 



I 
I 

2 

2 

3 
3 
3 




>43 
,86 

>29 

,16 

,59 

,021 

,45 
,88 






mmmm 



Too" 


■g: 


?h 


I 





4 


'2 


T 





3 


I 


4 


4 


2 


I 


5 


2 


5 


6 


3 


X 


: 7 


3 


& 


B 


4 


2 


•■ 9 


4 


6 


.10 


5 


3 


II 


5 


7 


12 


6 


3. 


13 


7 





H 


I 


4 


15 


8 





16 


8 


5 


17 


9 


I 


18 


9 


5 


ip 


10 


2 


20 


19 


6 


21 


II 


2 


22 


II 


7 



N«. G. |/»- 




mmmmm 



mmm 



23 
«4 
25 

26! 

27 

^9 

30 

31 

32 

33 
34* 
35 
36 

38 

39 

40 
41 
42 
43 

44 

mmmmm 



2 

2 

3 
» 

4 
5 
5 

6 
6 

7 

7 
8 

6 
9 

9 

20 

21 

2J 
22 

23 
23 



3 
7 

4 
•0 

4 



a^ 



5 
I 

5 

2 
6 

2 

7 
3 

7 

-4 

Q 

4 

9 

.5 

I 

6 



,36 

>68 

3 2 
64 



,28 

,6 

,92 
,24. 

>2 

M 

f 

,12 

lit 

jo8^ 

« 





9« 


4 Set tf Nm Diehul Taikj 


«re. 






TO 


TK 


K 


Tt. 




W^ 


T». 


ft. 


N1 


S. 


TT 


TT 




« 


S 




A 




iS4 


34 


4 


? 


5' 


44 


"6 


ii 




46 


=4 




.72 




iS5 


31 





!♦ 


45 


2 




« 


2S 




^ 




£« 


35 




.12 


85 


45 


7 


)2 




^ 


25 






iS7 


3< 




m 


66 


46 


3 


il 




4» 


0« 




i68 




B» 


3« 




'1 


p 


46 


7 




50 


27 




in- 




fej. 


37 




^ 


88 


47 


4 


■16 




^i 


27 




t32 




70 


37 




rf 


8p 


48 





■48 




5J 


28 




■^ 




71 


'S 




.72 


,i» 


48 


4 


,8 




M 


28 




t9^ 




72 


3« 




w 


191 


4? 





)12 




54 


29 




,28 




75 


3» 




'»i 


>92 


49 


5 


>44 




55 


29 




fi 




>4 


39 




,68 


»3 


50 


I 


,7« 




5^ 


50 




,92 




75 


40 




, — 


?4 


50 


6 


,08 






30 


6 


2 




'7^ 


41 




)32 


ii'5 


5' 


2 


rf 




3' 
SI 


■6 




78 


411 
42 




;^ 


■96 


51 

52 


6 

3 


.72 
Pi 




60 


32 




.2 




g 


42 




I28 


p8 


52 


7 


,3« 




ii 


?2 




.52 


1 


43 




,5 


JJ 


55 


3 


>8 




(S2 


33 


3 


M 


1 


8t 


41 




rfl2 












«3_ 


iiJ 





,ii 


u 


82 


il. 


2 


j51_ 




■ 







Table I. Cota Mearare, oiif A{/Zv/ 1£< Inttier^ 



A Set of New Declmtltdtes, &ft 9i. 



N° 


5r 


ft rt 


r- 


15? 


1 


t: 


ST 


- 


^ 


46 




1 '<i4 




4, 


5 





54 




s 


+7 




6 ,08 




65 


5 


I 


)« 




8: 


48 
49 




6 ,72 

7 ,?6 




66 


5 
5 


2 
2 


:^i 




li 


50 




V— 




68 


5 


, 3 


.52 




8( 


51 




?;ft 




69 


5 


4 


.14 




8; 


52 






70 


5 


4 


.8 




8! 


53 




I ,J2 




71 


5 


5 


•^ 




8i 


54 




J 1.56 




72. 


5 


6 




PC 


55 




5!,» 




n 


iS 


:;6 


.71 




91 


5« 




' ■?♦' 




74 


:5 


.7 


.34 




9: 


57 




4 .48 




75 


16 





r-t 




9- 


58 




5 ..■2 




76 


'6 





a 




91 


59 




5 


■74 




77 


16 


. I 




P! 


60 




6 


)4 




78, 


i6 


: I 


■'; 




9( 


63 


J 


7 
7 


fei 




g 


.6 
. 6 


' 2 

3 


>5? 

.2 




P- 
9( 


«9 







.92 




81 


I 6 


3 


■*t 




PS 












"H" 


J , 


"*" 


■. ; 


r" 





Table I. 



Table II. :.{i§'^'^ 






P:\Pt. 1 Vi'iB.pY. rt. 



,68 



5 

7 
4 

i 
6 

a .8*1 



■.:i 



^ii 



oj^ 



^i A Sit of New DechnaJTahks^ &C» 



nw: 



so 
So 

76 
80 

90. 



2 

3 
3 
4 

4 



7?^ 



55 

07 

.op 






ISJl 


JB. 


27- 


2 


2a 


^ 


2? 


i 


30 


2 


31 


2 

1 


32 


2 


3$ 


2 


34 


i 


35; 


2 


3<^ 


2 


3Z 


2 


38 


3 


39 


3 


40 


3 


41 


3 


42 


3 


43i 


3 


44 


3 


♦5 


3 


46 


3 


4Z 


3 


48 


3 


# 


3 


50 


4 


51 


4 


52 


4 


53 


4 


94 


4 


^5 


4 


56, 


4 


Vi 


4 


^8 


4 


1S^ 


4 


60 


4 


61 


4 


6:i 


4 


6i 

« 
f 


5^ 



rsap. 




N*. 



A Sit of New 'Decimal TMes^ic- •*» 



Table,, j T»b,.II.{,Xt^''-' 



'*•* 



V. 


iT 


K. 


■m-g 


rSpf 


-m 


IFH! 


577 


? 


F^ 




















--1 


1 




fii'i 


I 


- !.5S 


3* 


•- 


I 


^'^^ 


2 




PI 


2 ■ - 


- 5i04 


57 




I 


V^H 


? 




.075 


3 - - 


- 7.5^ 


J8 


■- 


I 


?2 


•'1 


« 






4 


— 15.08 


!» 


■T 


I 


J5 


|i 


5 




|l26 


5 


_t2,6 


40 




1 


J7 




» 




.■51 


6 - - 


- 15 .12 


4« 




1 « 


*o 


,39 


7 




,I7« 


I -~ 


— '7 .44 


42 


-- 







^ 


8 




,201 
,22« 


8 ^ _ 
9 


_ 20,16 
_i2,« 


4! 
■44 


;: 






li 


10 




,252 


10 — — 


-25.2 


45 


■ -■ 




.93 


10 




:Si 


II 


- 37 .72 


¥> 


■ - 




10 


p 




12 — - 


-5C.24 


47 


■ - - 




I^ 


iM 


JO 




/»8. 


13 


- J! .7!> 


48 


..T 




*1 


E 


p 




,21! 


■4 -- 


_,5,2S 


4P 


.- 




tl 


So 




.51= 


•5 


-!7,8 


50 


I 


0' 







J° 




.764 


\l -- 


-to ,32 


51 


I 


C3 


' a 


5! 


'= 




,016 


17 


I 0,84 


52 


I - 





5 


04 
56 


: 


,248 


18 - - 




53 


I 





7 






19 


; if 


54 


■I 


3 


IC 


lA 




23 


55 


1 


3 


ta 


A 




21 


I I0^>?2 


5^ 


1 


D 


15 


12 




22 - - 


I 1 3.41 


57 


t 


3 D 


17 


«. 




2J _- 


1 15.!« 
I 18,4! 


58 


I 


3 


2Q 


i( 




24 


59 


1 


5 


23 


*i 




25 - J 


c d. 


60 


I 


5 


55 


3 




26 _ I 


2,5: 


61 


1 


D 


27 


7! 




27 - I 


5U4 


1S2 


I 


3 


30 


.24 




28 _ 1 


7l.5< 


^3 


J 





33 


71 




29 _ 1 


c toj,o8 
ci5[,i2 


H 


I 





?5 


2! 




JO - I 


«5 


I 





37 


£ 




?I - I 

32 - J 


66 


I 




I 


1£ 

c 


1^ 




J 




5J - ' 

34 - • 

35 - ■ 


cioLii 

22|«8| 


68 
69 

70 


I 
I 
I 


1 


1 


i. 



Oj 



J Set of Ntw Decimal Talks, &c. 



TibliL 










-I 


m 


: 






AT 






N«. 4 


. ft.. 




N«.' 


p. 


g. /T. 




N° 


/>. 


«■ 


ft. 




• 1OO4 




^ 


_ 


- 


rtS 




22 


2 


2 


,"f6 


3 . 


- lOOJl. 






— 


— 


,96 




a; 


2 




,04 


^ - 


- ,014 






— 


I 


.44 




'4 


2 




.52 


4 ' 


- ,019: 






— 


I 


.92 




'1 


3 




.*- 


1 : 


- lOM 

- />28: 




6 


- 


2 

2 


& 




2d 
27 


3 
3 




,48 


7 - 


- fl?8' 




7 


— 


3 


'J* 




28 


3 


I 


.44 


S - 




8 


— 


3 


.84 




2P 


3 




.P2 


J - 
10 > 


::?:^ 




9 
10 








'K 




30 
51 


3 
3 




:fe 


20 . 


. />!,6 




It 




I 


,28 




32 


3 




:i: 


V ■ 


- .144 




12 




I 


.76 




33 


1 




40 - 


- Mi, 




'? 




2 


.24 




34 


4 




i: 


g . 


■■%. 




■4 

15 




3 
3 


,2 




K 


4 

4 




70 


■ %f^ 




16 




? 


,68 




37 


4 




,76 


80 . 




17 







,[6 




38 


4 




.24 


90 . 


- 452 




18 







.«4 




3P 


4 




,72 






1" 




1 


.12 




40 


4 




,2 






20 




I 


,6 




41 


4 




,68 




: 1 |2. 


2 


2 


jOS. 


_ 


42 


_5. 





,16 





A 


Set of.i 


V(?te; Dedimd Tahlef^ &t 




loi 


t \In\ 


1 


■/«. 




1 


In.\ ' 




N°. 


PM\ 


Pt.\ \ 




(?. 


/•f. 1 N«>. 


PjW 


p*. 


43 


•*•"* 


■ 


^4 


62 


1 

7 


I 


,7} 




81 


9 


2 


'J88 


44 




I 


,12 


* 


63 


7 


2 


,24 


^ 82 


9 


3 


,3« 


+5 




I 


,6 




d4 


7 


2 


,y^ 


i .^5 


9 


3 


,84 


46 




2 


,08 




^5 


7 


3 


,2; 


E 


H 


10 





>32 


'^Z 




2 


.56 


■ 


66 


7 


? 


,68 


* 


85 


10 





j^ 


48 




3 


,04 




67 


8 





,16 




86 


10 


: I 


,28 


49 




3 


,52 


4 


68 


8 





,64 




87 


10 


X 


,76 


50 


6 





1 




69 


,8 


I 


,12 


■ 


88 


10 


2 


>24 


51 


6 





rjS 




70 


8 


I 


,6 


■ 


89 


10 


2 


,72 


52 


6 





,96 




71 


8 


2 


,08 




90 


10 


3 


,2 


53 


6 


I 


,44 




72 


8 


2 


,56 




91 


10 


3 


,68 


54 


6 


I 


,92 




73 


8 


-> 


,04 




92 


11 





,16 


55 


6 


3 


'L 




74 


8 


3 


,52 




93 


II 





,64 


56 


6 


2 


,88 




75 


9 





J"^ 




94 


II 


I 


,12 


57 


6 


3 


'2^ 




76 


9 





,48 




95 


II 


I 


,6 


58 


6 


3 


,84 




77 


9 





,96 




96 


II 


2 


,08 


59 


7 





,T2 




78 


9 


1 


,44 




97 


II 


2 


,56 


60 


7 





,8 




79 


9 


I 


,92 




98 


II 


3 


,04 


61 


7 * I ',28 


^8o 1 pi 


2 


,4 




99 


II 




,52 



Table I. 



Table II 4^ Liquid common Meaf. 
• C A Hogjb. of ^i Gall. Int. 



N" 


P. 


I 

2 


..^ 


3 




4 




5 


— . 


6 





7 


— 


8 





9 





10 


— 


20 


— 


30 


I 


40 


1 



Pt. 

>03 
,07 
,11 
,'5 
,19 
,22 I 

,26 

>3 

,34 
,38 

,761 

,»5 



No.G. 



I 

2 

3 

4 

5 
6 



9 
10 

II 

12 

13 



1 
I 

2 
2 

3 

3 

4 

4 

5 

5 
6 



P. 

3 
7 



7 
3 
7 
2 
6 
2 

6 
I 



,68 



31,52 
,36 
,2 

,04 



,88 

,72 

,56 

»4 

,5>2 



NO. 


G. 


P. 


Ft. 


H 


6 




,76 


15 


7 




,6 


16 


7 




,44 


17 


8 




,23 


18 


8 




,12 


19 


9 





,96 


20 


9 


4 


.8 


21 


10 


,64 


22 


10 4 


,48 


23 


II 





,32 


24 


II 


4 


,16 


25 


13 


>"■" 


26 


12 J 


,8d 



N<». 



io% jf Set cf Neuo T>ecim^ Tahks^ &c* 



"? 


t^'l 




^f'\ 


P. 


«. 




15^: 


^g7 


P. Pt. 

5 .76 


$0 


I 


,92 




27 


12 


7 


,68 




«4 


30 


60 


2 


.3 




28 


13 


3 


.52 




«5 


3; 


I 


^ 


70 


2 


,68 




29 


13 


7 


.36 




66 


3' 


5 


38 


8o 
90 


3 


,07 




?o 


«4 


3 


,2 




67 


32 


I 


3 


AS 




31 


»4 


7 


li 




68 


32 


5 


,12 










32 


15 


2 




69 


33 





,96 


m 








33 


15 


6 


t72 




70 


33 


4 


3 










34 


16 


? 


.56 




71 


34 





,64 
,48 


■ 








31 


16 


6 


.4 




72 


34 


4 










3<i 


17 


2 


>24 




73 


35 





»32 










37 


17 


6 


,08 




74 


35 


4 


,16 










38 


18 


I 


,92 




75 


36 





J'^ 










39 


18 


5 


,76 




76 


3« 


3 


M 










43 


19 


I 


,6 




77 


3<5 


7 


,68 










'41 


19 


5 


.44 




78 


37 


3 


,52 










42 


20 


1 


,« 




79 


37 


7 


,3^ 










43 


20 


5 


,12 




80 


3^ 


3 


.2 










44 


21 





,96 




81 


38 


7 


P4 










45 


21 


4 


3 




82 


39 


2 


^8 










46 


22 





.^4 




8? 


39 


6 


,72 


' 








47 


22 


4 


.48 




^ 


40 


2 


,56 


. 


• 


- 




48 


2? 





,32 




^1 


40 


6 


»4 










49 


23 


4 


,16 




86 


41 


2 


,24 


• 








50 


24 





f^* 




87 


41 


6 


,08 










51 


24 


3 


,84 




88 


42 


X 


,92 






• 




52 


24 


7 


,68 


89 


42 


5 


,76 










53 


i5 


3 


,52 


93 


43 


I 


,6 


V 








54 


25 


7 


,36 




91 


43 


5 


*^ 


\ 








55 


26 


3 


,2 




92 


44 


I 


,28 










56 


26 


7 


.^ 




93 


44 


5 


,12 










57 


27 


2|,88| 




94 


45 





,96 




• 






58 


27 


6 


,72 




95 


45 


4 


3 










59 


28 


2 


.56 




96 


46 





^J 




a 






60 


28 


6 


,4 




97 


46 


4 










61 


29 


2 


'H 




98 


47 





.32 




■ 






62 


29 


6 


,08 




99 


47 


4 


,16 


1 








63 


30 


I 


,92 


. 








i 

























N». 



A Set e^Neio Decimal Talks, 8cc. loj 



Table I. I Tab!e 11. i ^f Meafare, a Tun 
I Itr Load the Intern. . 



N«: 


G. 


Pt. 


No.ftl 


B. a. 


Pt. 


N«.| 


«?. A i G. 


ff. 


« 


-— 






■*■«■ *^"l 


M^«* M^i^ 


— •— 




^MBW 


»m-m 


. .^ 


^^m 


.^- 


I 


— 


'^ 




I 




- - 3 


>2 




35 




6 





>■"" 


2 


— 


,06 




2 


m m 


• ^ 


b 


.4 




36 




6 


3 


»2 


^ 

A 


— 


,09 




3 


• 


I 


I 


,6 




37 




6 


•* 

6 


94 


4 

5 
6 

am 


««iA 


,12 

,t6 


4 


» • 


I 


4 


,8 




38 




7 


I 


,6 


^^^^ 


1 


-• 2 





tT^ 




39 




7 


4 


,8 


' 


>«9 




6 


-- 2 


5 

3 


,2 




40 


2 








>''"^ 


7 
8 

9 

10 
20 


"** 


,22 

,28 

.52 


: 


7 
8 

9 

10 

II 

1 


tf * 


2 

3 
3 

4 
4 


6 
I 

4 



3 


A 
.2 




41 
42 
43 
44 
45 


2 
2 

« 

2 
2 
2 




1 
I 
"2, 


3 

I 

4 




,2 

i4 
,6 

,8 


50 


I ,28 


t 


12 
'3 


» 


4 
5 


6 


»4 
,6 


1 


46 
47 


2 
2 


2 

2 


1 


>2 

>4 


50 

do 


t 

1 


,93 




15 


- 


5 

6 


4 

t> 


,8 


■ 


48 
49 


2 

2 


B 

? 


I 

4 


,8 


70 
80 
90 


2 
2 
2 


.24 

,'88 


■ 


16 

•7 
18 


m m 


6 
6 

7 


3 
6 

I 


>2 

4 
,6 




1 ^ 

50 
51 
52 


, 2 
2 
2 


4 
4 
4 


1 



3 
6 


/ 

,2 
>4 








IP 


- • 


7 


4 


'8. 




53 


2 


5 


X 


,6 




(■ 




20 










>"^ 




54 


2' 


5 


4 


,8 






« 
• 




31 




- 


3 


.2 




55 


2 


6 


' P 


. ____ 








- 


22 


^ 





6 


.4 




56 


2 


i6 


2 


,2 










23 


1 


I 


1 


,*! 




57 


2 


;6 


D 


H 






« 


1 


24 




I 


4 


,8 


• 


58 


2 


•7 


2 


,6 










25 


• I 


2 





» 


r 


59 


2 


7 


4 


,8 










26 


T * 


2 


3 


,2 


( 


60 


3 


•0 





J—* 










27 


I ' 


2 


6 


j4 




61 


? 





^ 


>2 




A 


1 


28 I 

• 


3 


r 


,6 




62 


1^ 

3 


;o 


6 


>4 


\ 


; 2^ 




' ^1 


4 


,8 




63 


3. 


t 

• 


I 


,6 


t 


30 




4 





J"~ 


' 


64 


• 

3 


I 


4 


,8, 








■ ■ 


3« 




4 


3 


,2 




65 


3 


.2 





y-n 




' 




. 


32 




4 


a 


>4 




66 


3 


:2 


3 


52. 










'33 


■!< 


5 


I 


,6 




67 


5 


2 


•6 


r4: 




i 




* — 


34 


i 


5. 


JL 


.8 




68 


*> 
>• 


i 


I 


>p; 



N«. 



«o4 


^ Srt ofNtTB Decimal Tatle 


r, 


kc 






w. 


5r 


T. 


b: 


Pt 


N1 


6-: 


S7 


B7 


7T 


W 


« 


rs. 


Pf 


i9 




5 


4 


is 


87 










;^ 


Pi 


4 


4 


3 


,2 


70 




■1 


c 


, — 


81 




c 


3 


,2 


92 


4 


4 


6 


•4 


71 




< 


3 


,2 


82 







6 


•4 


P3 


4 


5 


1 


.6 


72 




1 





.4 


83 




I 


I 


.6 


!'4 


4 


5 


.4 


,8 


73 




■; 


1 


,6 


¥ 




"1 


4 


.8 


n 


4 


6 







74 




<t 


■1 


,8 


«5 




2 


C 


,— 


•)i 


4 


6 


3 


'2 


n 




t 


c 


,_ 


8£ 




» 5 


,-2 


97 


4 


4 


6 


,4 


76 




6 


5 


,2 


87 




2 


6 


.4 


98 


4 


7 


I 


,6 


77 




6 


6 


•4 


88 




3 


1 


.<s 


9P 


4 


7 


4 


,8 


78 




7 


1 


,« 


8p 




3 


4 


,8 












7P 


J 


7 


4.8 1 


90 


-t 


4 


0,-1 










Takle IT -f ^**"S Meafurc, me PoU 
i 



I or Rod the Integer. 



(W 


ft. Pti. 1 


-w-YTzer. 


/»: pt. N-: 


fj- 


WT^ 


rr 




























1 


— 


,019 


I 


-^ 


— 


1 


,98 


23 







9 


M 


2 


— 


.039 


2 . 


— 


— 




,96 


^ 







11 


.52 


3 


— 


.059 


3- 


— 


— 




,P4 


2'i 








»v 


4 


— 


.079 


4 


— 


— 




,92 


2<S 








,48 


^ 


— 


.C9S> 


*). 


— 


— 




,9 


if? 








,46 


6 


— 


,118 


6 


— 


— 


II 


,88 


28 








.44 


7 


— 


,138 


7 


— 






,86 


29 








42 


8 


— 


■158 


8 


— 






,84 


30 






II 


»4 


P 


—^ 


,178 


9 


_ 






,82 


31 








,3S 


10. 


— 


.198 


10 


— 






3 


32 




2 




.?6 


20 


— 


.39« 


11 


— 






.78 


33 




2 




.54 


30 


-*■ 


.W4 


1? 


— 




II 


,76 


34 




2 




.32 


40 


■ — 


,792 


I? 


— 






.74 


35 




2 




'3. 


50 


— 


S9 


14. 


— 






,72 


36 




2 


11 


,28 


60 


I 


,188 


"5 


— 


2 




,7 


37 


2 







,26 


70 


1 


,386 


16 


— 


. 2 




,68 


38 


2 







.24 


80 


1 


,<84 


'7 


— 


2 




,« 


3!> 


2 







,22 


SW 


I 


,782 


18 


~ 


2 




,64 


40 


2 







,2 








I? 


I 


e 




,62 


4' 


2 





9 


,18 



A Set of New Dectmal Talks, &c, loy 



Table I. I Table 11. {^f,^'^^„^ '" 



lod 




A Sit 4Nm 


Decimal Tahks^ 


&c. 




N_& 


ei. 




TI^ 


Py 


h. 


«. '''J 




TIT 


Ft 


ft 


«• M.l 
































v> 


— 


.72 




27 


_ 


9 


2 


,88 




64 


V 


11 





,16 


&> 





:^J 




28 


_ 


10 





.32 




«5 


1 


11 




.6 


2° 


_ 




29 


— 


10 


I 


,76 




66 


1 


II 




'-i 


80 


I 


."^2 




V> 


— 


10 


3 


,2 




67 


2 







►tS 


i» 


I 


.2,6 




II 


— 


II 





,«4 




68 


2 







.92 




,"8 




69 


2 


c 




.31* 




.1= 




70 


2 


I 




.8 




,j(i 




71 


2 


1 




:S 




1 




72 


2 


I 








73 


2 


2 




,12 






74 


2 


2 




,^6 




.72 




7S 


2 


3 




, — 




,16 




76 


2 


3 




.44 




,6 




77 


2 


3 




,88 


3 


48 




78 


2 


4 




.32 






79 


2 


4 




,76 




,92 




80 


2 


4 




.2 




:!' 




81 

82 


2 
2 


5 
5 




'% 




% 




8? 


2 
2 






.52 
.96 




>12 




85 


2 


6 




1^4 




.5« 




86 


2 


6 










87 


2 


7 




,28 




i 




88 
8s. 


2 

2 


7 

e 




.72 
,■6 




.52 




90 


2 


8 




.6 




.7« 




91 


2 


8 


r4J 




.2 




92 


2 


9 






a 




93 


2 


9 




.92 






94 


2 


9 




f 




.52 




91 


2 


10 






,P6 




96 


2 


10 




.24 




.4 




97 


2 


10 




,68 




,84 




98 


2 


11 




.12 




,'8 




99 


2 


11 


3 


.56 
















2 


.72 















A Set of New 1)ecimal Talks, Sfc. '107 



Table I. 



T'l-lell-fLT^.''""' 



<Mk 



sF 


i^r. 


TfT 




Wf 


^ 


n 


F 


%pt. 


-n?- 


^ 


15 


rr 


TFt 


1 




,■3") 




I 


— 


- 




"0 .3'3 


3^ 







2 


'VSA 


2 




,126 




2 






1 


c ,66 


37 




1 


C 


C »2I 


? 




,i8(. 




3 


_ 


— 




^99 


38 




1 





6,54 


4 




.'■i! 




+ 


— 


— 


2 


I '32 


3? 




1 


I 


c,87 


% 




.3' 5 




5 


— 


— ^ 


2 


7,,6^ 


40 




1 


I 


3 3 






,378 




6 


— 




c 




.98 


41 




1 


2 


^2' 


7 




.441 




7 


— 







( 


.3' 


42 




1 


2 


7,86 


8 




.W 




S 


— 




I 


2 


M 


43 









2,19 


9 




,567 




9 


— 




I 


8 


.97 


44 






C 


f .'(2 


10 




>033 




1:5 


— 




2 


3 


'3 


45 






I 


Ms 


30 




,a66 




II 


^ 




2 


9 


,63 


4^ 






I 


J .18 
= 51 


30 




3p? 




12 


— 


2 





3 


.96 


.47 






: 


40 




,532' 




»3 


— 


3 





10 


.2? 


48 






; 


s,&t 


SO 


? 


•"^' 




14 


— 


2 


1 


4 


.62 


■v> 




3 


c 


4.17 


5o 


3 


,78? 




15 


— 


2 


I 


Id 


.9' 


50 




3 


C I 


5 


f° 


4 


i43" 




i<5 


— 


2 


2 


■5 


.!! 


51 




3 


I 


4.83 


io 


5 


^4 




»7 


— 


2 


; 


II 


.61 


5! 




3 


1 1 


1,16 




18 


— 


^ 


a 


I 


.94 


53 




5 


2 , 


l;g 




19 


— 


3 


1 





|27 


54 




3 






20 


— 


3 


I 


6 


.6 


55 




4 





',15 




21 


— 


3 


2 


c 


.93 


56 




4 


I 


It 




22 


- 


? 


2 


7 


26 


57 




4 


I 




2J 


— 


4 





I 


,59 


58 




4 


: 


1,14 




24 


— 


4 


c 


7 


92 


59 




4 2 


7,47 




25 


— 


4 


I 


2 


)21 


So 




5 c 


1,8 




26 


- 


4 


1 


8 


^? 


61 




5 c 


8,13 




27 


— 


4 




2 


91 


62 




5 > 


246 




28 


— 


4 




9 


24 


«3 




"^ 


2 79 




29 


— 


*) 


c 


5 


57 


64 


2 


o;c 


S 12 




V> 


— 


5 


c 


9 


9 


65 


2 


1 ' 


3 145 




31 


c 


5 


1 


4 


25 


66 


2 


i ' 


9,78 




32 


1 





c 


4 


i^ 


'Z 


2 


a 2 


4.11 




53 


I 


c 


c 


10 


68 


2 


c : 1 


c,44 




34 


1 





1 


5 


25 


651 


2 


I c 


1-77 










il_ 


' 


.£-L 


iI,S-i| 


■70 2 


I Cl^ 


,1 



lo8 A Set of New Decimal Tables ^ 8rc. 



^ubie in. Long Meafure, one Mile the 
Integer. 



A Set of New Decimal TMei, &c. 109 



Table I. 



Table II. P^°,"' Meafare, .». 
i Mile Square thi Meger. 



Mn^ 


eti. 




ti\Alkl 


f? 


Pf. 




^1j-^?;%i^?,p'-i 


I — 


,102 




I _ _ 


10 


,24 




6 







't'.i, 


z — 


,234 




2 


— ■ — 


20 


,48 




7 


— 


I 


9i;,68 


5 — 


.^-7 




={ 


— — 


^0 


>72 




tj 


— 


7. 


1,02 


4 - 


,4"!' 




4 


— I 





,96 




9 


— 


? 


12 ,16 


t — .Sizl 


5 -1. 


II 


52 




10 


— 


2 


22 4. 



no J Set of New 'Decmai Tallesy 8rc- 



J Set of Neva Decimal Talhs, &c. 1 1 1 



TablcIII. Sq 


uare Msafare, ow MU Square the ineg. 


N«. 


^ 


ts ff 


■ 'Ht. 


5 


?S 


a 


N% 


^S- US 


fi 
























1 


6 


;i 


34 


217 


2 


16 


67 


429 




8 


2 


12 


35 


224 





— , 


68 


435 




32 


3 


^9 


52 


36 


250 


1 


1 


69 


44! 




I6 


4 


2') 


2 1« 


37 


236 


3 


70 


448 




"> 


% 


32 


- 


38 


243 





32 


7' 


454 




1 


6 


38 


r^ 


39 


249 


2 


16 


72 


460 




7 


44 


40 


256 





-., 


73 


467 




32 


8 


51 


3: 


4' 


262 


1 


1 


74 


473 




16 


9 


57 


2 1« 


42 


268 


3 


75 


480 




-, 


10 


64 


- 


43 


'I' 





32 


76 


4S6 




1 


II 


70 


I 24 


44 


281 


2 


Id 


77 


492 




12 




3 8 


45 


288 







78 


499 




32 


13 


8g 


3! 


46 


294 


t 


24 


79 


505 




16 


'4 


89 


2 i« 


47 


300 


3 


8 


80 


512 




.-, 


•5 


(.5 


- 


48 


307 





32 


81 


518 




1 


il 


102 


3I 


49 


313 


2 


16 


82 


524 




17 


108 


5» 


320 







2' 


53' 





32 


i8 


"5 


32 


51 


52« 


I 


'i 


b 


537 




16 


19 


121 


2 i« 


52 


932 


3 


85 


544 




— , 


30' 


128 


— 


53 


339 





32 


86 


555 




24 


21 


'34 


3I 


54 


34) 


2 


16 


S 


55<i 




8 


22 


140 


55 


352 





— , 


88 


■^H 




32 


23 


'47 


3 


5ii 


358 


I 


24 


8p 


569 




16 


24 


'53 


2 « 


57 


364 


3 


8 


90 


H^ 




-, 


25 


1 60 


- 


58 


37' 





32 


9' 


582 




24 


26 


166 


I 24 


59 


377 


2 


16 


92 


,88 




8 


27 


172 


3 S 


60 


384 


o-,| 


93 


595 




32 


28 


'79 


3 


61 


390 


^24l 


94 


601 




16 


29 


■85 


2 i( 


62 


396 


3 


8 


95 


60 




-., 


3° 


192 


-- 


*3 


403 





32 


96 


614 




24 


31 


298 


n 


64 


409 


2 


16 


97 


620 




8 


32 


204 


<5 


416 





— , 


98 


627 





32 


ii. 


211 




61) 


4221 l'24l 


99 


6^ 


2 


_i£ 



ii« A Set of New Decimal Talles, &c. 

Tablet. I Table II. -J ^°T ^f""' "" 
I C Rodjquare the Intege}-. 



"m. 


TT 


TT. 




TJ^ 


¥ 


w 


tt 




N^ 


5 


F, 


TT 


I 


— 


fi2 




I 


— 


2 


,72 




36 


10 


7 


,52 


2 


— 


,01 




2 


— 


■? 


,44 




37 


11 


t 


,64 


i 


— 


,08 




3 





8 


,16 




38 


II 


4 


,36 


4 


— 






4 




1 


,88 




W 


II 


7 


,08 


5 





■13 




5 




4 


,6 




40 


12 





,8 


6 


'- 


,16 




6 




7 


,3» 




41 


11 


; 


,52 


7 


— 


,i8 




7 




I 


,°4 




4» 


12 


6 


,24 


8 


— 


|2) 




8 




3 


,76 




43 


11 


8 


,96 


. 9 


— 


,24 




? 




6 


,48 




44 


13 


1 


,68 


lo 





,w 




10 




q 


,2 




45 


13 


I 


,4 


20 


— 


.« 




II 




2 


,?2 




46 


13 


,12 


3° 


— 




11 




I 


,64 




47 


14 


I 


,84 




■3 




'32 




48 


14 


4 


'>i 




'4 




1 


,08 




49 


14 


7 


,28 




'5 




4 


,8 




50 


15 


I 


,— 




l6 




7 


,52 




51 


■5 


3 


,72 




17 




1 


.24 




51 


15 


6 


,44 




18 




3 


,96 




53 


16 





,1' 




ISI 




6 


,68 




54 


16 


1 


,88 




20 







,4 




55 


16 


5 


,6 




21 




3 


,12 




51S 


16 


8 


.32 




12 




■> 


,84 




57 


17 


1 


,°4 




»3 




8 


,16 




58 


17 


4 


>''i 




>4 




1 


,18 




59 


17 


7 


48 




15 




5 






60 


18 




,2 




16 


7 


7 


,71 




61 


18 


3 


,92 




'A 


8 


I 


rf4 




62 


18 


6 


,64 




8 


4 


,16 




IS3 


19 





.3< 




29 


8 


6 


,88 




64 


19 


3 


,08 




30 


9 





,<s 




«5 


19 


5 


,8 




31 


9 


3 


>3» 




66 


1? 


.52 




3» 


9 


< 


,04 




67 


10 


2 


<H 




33 


9 


8 


■'S 




68 


IC 


4 


S6 




34 


10 


2 


,48 




6i> 


10 


7 


JX 










-ii. 


10 


Li. 


j1_ 




I2II 


21 


I 


4. 





^ .Sf-if 0/ New 


Decimal TalJes, 


&C 


• 


113 


■Rs; 


Xl 


f? 


Pf. 




N^.y^ 


F^ />/. 


Ne. 


■^ 


"^ 


TT. 


71 


21 


4 


,J2 




81 J24 


4 


>?2 


. 91 


27 


4 


.52 


72 


2< 


6 


84. 




82 24 


7 


,04 




92 


27 


7 


>24 


7? 


22 





,56- 




8? 25 





,76 




n 


28 





,96 


74 


22 


3 


.28 




84 25 


5 


^8 


• 


9A 


28 


3 


,68 


-75 


22 


6 


> " 




^5 


25 


6 


»2 




95 


28 


6 


A 


76 


22 


8 


>72 




85 


25 


8 


,92 




96 


29 





i'^2 


77 


23 


2 


>44 




87 


26 


2 


,64 




97 


29 


2 


i84 


78 


2^ 


5 


,16 




88 


26 


5 


.36 




9» 


29 




,56 


79 


21 


7 


,88 




89 


26 


8 


,08 


■ 


99 291 


8 


,28 


80 2? 1 1 1 


,6 




90 


27 


I ,8 1 


• 1 • 1 



Table I. 



I Table II. i !^"?^ Meafure, one 

f C Tarn Square the Integer. 



N". 


it 


^f. 


^' 


TJ« 


f7 ^ 


/'t 




No.fy 


^f 


ft 

• • 


I 


-- 


,12 




I 


- - 12 


,96 




23 


2 


10 


;o8 


2 


'• * 


,2S 




2 


-- 2'5 


,92 




24 


2 


23 


,04 


3 


• i* 


.38 




3 


-- 38 


,88 




25 


2 


36 


>"" 


4 




>5» 




4 




51 


,84 




26 


2 


48 


,96 


•^ 




,64 




5 


• ^ 


H 


.8 ; 




27 


2 


61 


,97 


<S 


■i • 


.77 




6 


« * 


77 


,76 




28 


2 


74 


,88 


7 




,9 




7 




90 


.72 




29 


2 


87 


,84 


8 


I 


P3 




8 




103 


,68: 




30 


2 


100 


,8 


9 


I 


,16 


\ 


1 9 


-- 


116 


,64 




.31 


2 


113 


,76 


10 


I 


,29 




10 


.-- 


129 


,6 ~ 




32 


2 


126 


,72 


20 


2 


,59 


4 


II 


r • 


142 


,56. 




33 


2 


139 


,68 


30 


3 


,88 


% 


12 




n 


,52 




34 


3 


8 


,64 


40 


5 


.17 




13 




24 


,48 




35' 


3 


21 


,6 


50 


6 


,46 




H 




37 


.44 




36, 


3 


34 


.56 


60 


7 


.75 




15 




50 


.4 




37 


3 


47 


,52 


70 


P 


,04 




16 




63 


.36 




38 


3 


60 


48 


80 


JO 


.33 




»7 




76 


'52 




39 


3 


73 


.44 


90 


II 


,62 




18 




89 


,28, 




40 


3 


U 


.4 






. 


• 


J9 




102 


,24' 




41 


3 


99 


,3<5 


« 






• 


20 




"5 


.2 




42 


3 


112 


.32 










21 




128 


,16 




43 


3 


125 


,2« 


• » »22 1 11 


J,f » 


,12 




44 hi 


138 


.24 



N». 



114 A Set of New Decimal Tables, 8rc. 

IRS" 



45 

46 

47 
48 

49 
50 
SI 

S2 

55 
54 
?5 
56 

% 

5P 

60 

dt 

62 

6? 



4 
4 
4 
4 
4 



Ah. na : 



Tfpr, 



^1 

4t 
4 
4 
4 

5i 
5 
5 

5 

51 
5 



7 

20 

46 

5P 
72 
84 

97 

1 10 

122 
136 

5 
18 

91 

44 
57 

-2i 



,2 

,16 

,12 
,08 

,04 



.96 
,92 

,88 

,84 

,8 

,76 

»72 

,68 

,64 

.6 

1.56 
,52 
>48 



64 

65 

66 

67 
68 

69 

70 
71 
7a 
73 

74 

75 

76 

77 
78 

79 
1 80 

81 

82 



7? 



Iq. 



5 

5 

5 
6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

1 

7 
7 
7 
7 



Pf, 



109 

122 
135 

4 

17 

30 . 

43 
56 

69 
8z 

95 
ic8 

120 

133 

2 

15 
28 

41 



>44 

A 

.36 

'32 
,28 

24 
.2 
,16 
,12 
/)8 

64 

i96 

.92 

,88 

,76 



84 

85 
86 

87 
88 

89 

90 



^it H' 






5lk2 



92 

93 

94 

95 
96 

97 
98 

99 



7 

7 

7 

7 

7 

7 
8 

8 



91 8 



8 
8 
8 
8 
6 
8 
8 
8 



67 
80 

93 

106 

119 
132 

I 

14 
rr 

40 

53 
66 

79 

92 

105 

118 
131 



,68 
,64 
>6 

>52 
,48 

)44 

»4 

,36 

,28 
»24 

>2 
,16 

>I2 

,08 

,04 



Table I. 



't u\ TT S ^^^^^ Me!iuir6) mk 



IK 


!q. 4.q 


*IT 




wr 


"W 


q.q 


-PT 




*Rr 


[Iq. qjj 


-^ 




--*• . 




»■*>< 






•*%^ 




« 


• -.— 


•d^BB 


— A^ 


I 


— 


*-. 


,2? 




I 

• 


I 


7 


P4 




H 


20 


.2 


',56 


2 


*— 


■ — 


»+6 




•2 


2 


14 


/^ 




15 


21 


9 


,6 


3 


-^ 


•— 


,69 




3 


4 


5 


,12 




16 


23 







4 


-^ 


— 


,92 




4 


5 


12 


.1^ 




17 


24 


7 


5 


— 


i ,iS 




5 


7- 


3 


,2: 


' 


18 


25 


H 


,72 


6 


* 


1 ,?8 




6 


8 


10 


'2* 




19 


27 


5 


,76 


7, 


. 


I ,61 




7 


10 


1 


,2B 


K" 


20 


28 


12 


,8 


8 


— 


i 


■,8. 




8 


II 


8 


•.3P 


1 


21 


30 


3 .84 


9 


-* 


2 


>G7 




P 


12 


15 


,3^ 




.22 


31 


19 .88 


10 


,_ 


..? 


>3 




io 


14 


6 


■H. 




23 33 


I 


,92 


20 





4 


,6 




|i 


15 


«3 


.44 




24 34 


8 


,96 


30 




6 


,9 




12 17 
19 J r8 


4 


,48 




;25 36 





',84 


40 1- 


9 


Ai 




II 


iSl 


' • 


26 37 


7 



N». 



A Set of New D ecimal T ahles, Srf. 

*. I . - T,.. ZV d - rV - -I bj . ■ "Lin . , .— 



Q2 



ii6 A Set of New 'Decimal Talles, 5rc. 



■N". 


7Z 


— 




W 


Tf 


/.,. 


1*1. 


1 


ft 


i.e. 


w. 


I 


4 


;66 




1 


_" 


466 


■'56 


36 


9 


■244 


',16 


2 


9 


»32 




2 




933 


tt2 


37 


9 


17^0 


,72 


? 


15 


is 




3 


_ 


1399 


,68 


38 


10 


+49 


,28 


4 


18 


.«5 




4 


I 


13S 


'2* 


39 


10 


'1' 


.84- 


5 


2i 


■3' 




5 


I 


604 


,8 


40 


10 


1382 


.4 


6 


27 


,!>8 




6 


I 


1071 


•}6 


4^ 


11 


120 


,96 


7 


32 


A 




7 


1 


l')37 


•^l 


42 


11 


587 


.52 


8 


37 


•}• 




8 


2 


276 


,48 


43 


■ I 


■054 


,oS 


9 


4' 


,97 




9 


2 


743 


,04 


44 


11 


1520 


,6^ 


ro 


46 


>6'i 




10 


2 


1209 


>6 


45 


12 


259 


)3 


20 


»3 


>3 




11 


2 


1676 


,16 


4^ 


'2 


725 


,76 


3° 


13? 


,i>8 




12 


3 


414 


,72 


47 


12 


1 192 


II 


V 


186 


,6 




■3 


3 


881 


,28 


48 


12 


16^8 


so 


233 


>25 




■4 


3 


■347 


^4 


49 


'3 


I? 


44- 


So 


27? 


,c 




'5 


4 


S6 


'♦. 


50 


'3 


864 




70 


32« 


,62 




16 


4 


552 


,96 


5^ 


■3 


■330 


Si 


So 


37? 


J 




17 


4 


1019 


,52 


52 


■4 


69 


,■2 


i8 


4 


1486 


,08 


53 


■4 


535 


,68 


'P 


5 


224 


>«4 


54 


■4 


1002 


.24 


io 


1 


6pi 


,2 


55 


■4 


1468 


,8 


!I 


5 


"57 


,76 


5^ 


■5 


207 


.3^ 


12 




1624 


■M 


57 


■5 


673 


.92 


i) 


6 


5«2 
829 


58 


■5 


1140 


.48 


-4 


6 


.44 


59 


■5 


■607 


P4 


'5 


6 


1296 




60 


16 


m 


,6 


!6 


7 


34 


l^s 


61 


16 


,16 


!7 


7 


501 


,12 


62 


16 


1278 


,72 


18 


7 


967 


,68 


'53 


■7 


■7 


,23 


i9 


7 


■434 


■I-* 


64 


■7 


483 


,84 


JO 


8 


172 


.8 


«5 


■7 


950 


4 


il 


8 


639 


,3'i 


66 


■ 7 


14.6 


,9« 


)2 


8 


1105 


,92 


67 


18 


■55 


.52 


J3 


8 


1572 


,48 


68 


18 


622 


.08 


M 


9 


3" 


,04 


69 


'§ 


■c88 


.64 










fi. 


9 


777 


A_ 


70 


ii 


■555 


)2 





A Set of New 


Decimal Talles, &c. 


17 


N' 


W 


/.f. 


".1 


N". 


fr. 


/f. 


ff. 


W.Fc 


I.e. 


« 


71 


19 


29! 


.76' 


81 


21 


ISO? 


^6 


91 24 


984 


,91 


72 


19 


760 


.12 


82 


22 


241 


.92 


92 24 


MSI 


,V 


7? 


19 


1226 


,8B 


81 


22 


708 


,48 


9? 25 


I9C 


,0! 


74 


19 


169? 


•W 


5+ 


22 


"7'i 


,04 


94 25 


6^6 


,64 


7S 


20 


«2 




81 


22 


'% 


.6 


95 25 


112: 


,2 


76 


20 


m 


.■i6 


86 


2? 


.16 


96 25 


■ ■iS( 


,71 


77 


20 


1^6'i 


1I2 


87 


2? 


846 


-72 


97 26 


,2i 


.?2 


78 


21 


101 


,68 


88 


2? 


'in 


.28 


98 26 


794 


,81 


79 


21 


S70 


,24 


89 


24 


V 


,84 


99 26 


i26i,4J 


8d 


21 Io:id 


,8 


90 


ii 


118 






1 1 



Table I. 



Table II X ^°'''*^ "^ Solid MeaAire, 
'1 one Solid Foot the'Integer. 



w^ 


77 


.% 


■RT 




H5: 


TT. 


fe 


T 1^ 


/.rl 


«7 


pt. 


1 


'Z. 


II 


,05 




I 


17 


17 


.$ 


397 


1 


M 


2 


— 


22 


,11 




. 2 


34 


35 


,s 


414 




,08 


5 


— 


33 


,'7 




3 


51 


53 


,7 


432 


1 1 


, — 


4 


— 


44 


■23 




4 


i^ 


7 


,^ 


449 




,92 


5 


1 — 


55 


,29 




5 


86 


25 


,fi 


466 




,84 


6 




2 


.35 




6 


■03 


43 


.5 


4S3 




.76 


7 




13 


41 




7 


120 


61 


."i 


501 




,68 


8 




24 


.47 




8 


■38 


■5 


.3 


518 




,6 


? 




35 


.53 




9 


■55 


33 


,s 


";35 




,52 


10 




46 


'5? 




10 


172 


5" 


.! 


552 




.44 


20 




29 


,18 




11 


190 


5 


,1 


570 




.31S 


30 


5 


11 


.77 




12 


207 


23 


,c 


587 




,28 


40 


6 


58 


.3<S 




13 


224 


■tS 


,$ 


604 




,2 


50 


8 


40 


« 




14 


241 


58 


,s 


621 




,12 


60 


10 


23 


,54 




15 


259 


12 


,8 


639 




,04 


l" 


13 


6 


.13 




16 


276 


32 


,7 


6$6 




,96 


80 


IJ 


52 


■72 




17 


29; 


48 


.« 


673 




,88 


90 


"5 


35 


.3' 




18 
19 
20 
21 


^28 

345 
562 


2 

;8 
56 




69. 
708 
725 
743 


I 


,8 
,72 
M 
.5" 


■ 






22 ':i8o'iO 


X L 


760 ' 20 


^ 



ii8 


A Set of New 


Detlinal Xahk. 


,&t. 




wr 


\hl' 


TS 


f?r 


N" 


1. 1. |<(r 


TTT 


r 


/'... 


5? 


ft. 


45 


777 


38 




«7 


110558 


,98 




■434 


■5 


',t6 


46 


7»4 


54 




^5 


112; 12 


,s 




1451 


33 


,21 


47 


612 


10 




66 


1 140130 


>72 




1468 


■i' 


»2 


48 


82P 


28 




"J 


1157 48 


,64 




I486 


5 


)I2 


4C 


s: 


44 




68 


"75 2 


.56 




■503 


23 


.04 


50 







<%> 


llpaao 


Kt8 




1520 


40 


rf>« 


51 


S81 


"7 




70 


I20?3e 


.4 




'537 


5« 


,<8 


52 


8j8 


35 




71 


1226 s6 


.32 




■555 


12 


,8 


<; 


S"5 


53 




72 


I2«lO 


.24 




1572 


30 


,72 


54 


^JJ 


7 




73 


126128 


,16 


'?2 


I5«jp 


48 


,64 


55 


S150 


25 




74 


127846 


,08 


P3 


■607 


3 


,56 


56 


967 


43 




75 


1296 


, — 


94 


1624 


20 


,48 


57 


!*t 


61 




76 


■31317 


,P2 


S5 


■ 641 


38 


•4 


58 


1002 


'5 




77 


■3!035 


,84 


p6 


i«5l 


56 


.32 


59 


(Oip 


33 




78 


■347,53 


,76 


P7 


1676 


10 


)24 


60 


1036 


51 




7(1 


■365! 7 


,68 


p8 


■693 


28 


,16 


<i 


1054 


5 




So 


■38235 


.6 


'? 


■ 710 


46 


,08 


«2 


107 1 


23 




ei 


13W43 


>5J 










Ji_ 


ic88 


40 


ii. 


8! 


141661 ^' 


! 







TiWeL I Tabic IL {]-•■"* **"f°"''"' 



t Acre the Mtegrr. 



TP" 


^ 


T7 


Pt. 




Wl 


1 


2 


3 


TT 


TT 


TV 


I 


_ 


_ 


48 




N!. 


25 


50 


75 








)0 


2 


— 


— 


#6 




I 


26 


5^ 


76 


I 


I> 


'IS 


1 


« 


1 


»t4 




2 


27 


52 


77 


3 




^ 


4 


— 


I 


,91 




3 


28 


53 


78 


4 


24 


>2 


5 


._ 




& 




4 


2!1 


54 


79 




12 


,1 


6 


_ 


2 




5 


30 


55 


80 


8 





, — 


7 


— 




:^: 




6 


?■ 


56 


81 


9 


18 


.l-) 


« 


— 


3 




7 


12 


57 


82 


■ 1 


6 


'0^ 


9 


— 


4|>32 




8 


33 


58 


5' 


12 


24 


>2 


10 


— 




9 


34 


59 


84 


■4 


12 


,1 




_ 


9,. 68 




10 


35 




85 







,^ 


30 


— 


■4 .52 




11 


36 


61 


86 


17 


18 


.'I 


IP- 


— 


>9',36 


12 


il. 


62 


871.9 


^ 


.£L 



A Set of New Dedmal T^hles^ &r (% i la 



fp: 


P9 


Tq. 


Ft 




^o/i 


I 2 .? 

38 63I88 


^i 


1^3 


ft. 


53 


-!24 


,a 




W. 13 


20 


24 


■ 


60 




29 


P4 




14 


?9 


64 '8P 


22 


12I 


>I 


70 


I 


i 


,88 




15 


40 


65 Ipo 


24 


a 


,— 


60 


I 


8 


»72 




16 41 


d^/S*! 


25 


18 


.«$ 


P3 


I 


1 3 1.56 




17 42 


67 


P2 


27 


^ 


»05 












ig 


49 


68 


93 


28 


24 


»2 












Ip 


44 


69 


P4 


30 


>ia 


>< 












20 


45 


70 


P5 


32 





»— 






* 

• 






21 


46 


71 


P6 


?3 


18 


»i5 












22 


'^Z 


72 


P7 


35 


d 


,05 












23 


48 


73 


P8 


5^ 


24 


»2 


1 1 


' 24. '491 


H PP. 


38lM 


'I 



Table I. 



Tabtoa^?>^- One Teat tbeh^ 



I 

2 

3 

4 

5 

6 

7 
8 

P 
10 

20 

33 
40 

50 
60 

73 
8o 

P3 



Z). 






I 
f 
I 
2 
2 
2 

3 



s 



I 

2 

i 

4 
5 

7 

I 

17 



H 

)B7 
«74 

♦5 

.38 

♦25 

!i 
,,76 



II {,041 

IP ',8 

4S56 
13,32 

22 ,08 

6,84! 



N^. /H/klb;)^ WjN". 



I 
2 

4 

I 

i 

10 
II 
II 

'? 

H 

15 

16 

I18 ' 



1 

J 

I 

2 

J 

It 

I 

I 

2 

2 



I 
I 

2 
2 

? 

3 

o 


I 
I 
21 



c 
c 
I 



»— * 



.3 



3 



15 



•0.7 



22 



ai4 

4 6 
Oil 

413 



,2 
8 

4 



5 
I 



I.* 

4,8 

20,4 
12'.— 



3»^ 






IP 



5*10 

2J 2 

6'i8 

2 

6 



3 



»2 
,8 

.4 



9 
i6',8 



,6 
,1 



ip 

2(5 
2t 

»5 

16 

17 
i8 

2P 

3> 
3* 

33 
34 
35 

36 



2 
2 
2 
2 
2 

3 
3 
3 
3 
3 
3 
3 
4 
4 
4 
4 
4 
4 



^!f5:^ 



I 

2 
2 

3 

3 



I 
I 

2 
2 

? 
3 

Oi 
O 
I 
I 



3 
6 

3 
6 



3 
c 

4 
c 

4 
i 



2 I 
2 S 



8i4 

1^,6 
7»2 

22,8 

'4.4 

21 

4 

20 

12 



^19 
ic 

2 

18 

9 



,6 

,8 
4 

,^ 
,2 
,8 

4 
,6 



N*». 



jtio 


A set of New *Decimai Tahles^ 


.&c: 




N". 


M. 


y. 


Z). 


H. 1 Ff. 


N^. /I:/ 


tr.p. 1 


ft 


/•/. 


■ ■■ 


--* 


--> 


•^ 


» ! ._. 




... 


^.^ 


— . 


~ ^ 


^l 




3 


2 


I 


,2 


6p 8 


3 


6 


20 


»4 


38 




3 


5 


i5 


,8 


70 


9 





3 


12 


>~" 


39 







2 


8 


» • 

.4 


71 


9 


I 





3 


,6 


40 







6 


6 


>"~" 




72 


9 


I 


3 


19 


,* 


4* 




I 


X 


15 


,6 




73 


9 


2 





10 


,8 


4» 


^ 


1 


6 


7 


.* 




74 


9 


2 


4 


2 


'4^ 


43 


* ^f 


2 


2 


22 


,8 


• 


75 


9 


3 





18 


\ 


44 




? 


6 


H 


»f 




76 


9 


3 


4 


9 


,6 


45 




3 


3 


•6 


r 




77 


10 





I 


I 


.2 


46 




3. 


6 


21 


,6 




78 


10 





4 


16 


,8 


4Z 


6 





3 


13 


,2 




79 


10 


I 


I 


8 


>4 


48 


_.^. 


I 





4 


,8 




80 


10. 


I 


5. 





>""* 


49 


6 


I 


3 


20 


>4 




81 


10 


2 


I 


15 


,6 


50 


6 


2 





12 


> 




82 


10 


2 


5 


7 


.2 


51 


6 


2 


4 


3 


,6 




25 


10: 


• 3 


2 


22 


»8 


^i 


6 


3 





19 


»2 


- 


f+ 


10 


3 


5 


H 


»4 


53 


6 


3 


4 


10 


,8 




2^ 


II 





1 


6 


>"• 


54 


7 





I 


2 


A 




86 


II 





6 


21 


,6 


55 


7 





4 


18 


>"* 




87 


11 


I 


2 


13 


.2 


56 


7 


I 


I 


9 


,6 




88 


II 


I 

1 


6 


4 


,8 


^ 


7 
7 


I 

2 


5 
I 


I 

16 


j2 

,8 




89 
90 


11 
II 


.2 


2 

6 


20 
12 


>4 

> 


59 


7 


2 


5 


8 


A 




91 


11 


3 


3 


3 


,6 


6p 


7 


3 


2 





J"^ 




92 


II 


3 


6 


19 


,2 


61 


7 


3 


5 


15 


,6 




93 


12 





3 


10 


,8 


62 


8 





2 


7 


>i 




94 


12 


I 





2 


.4 


63 


8 





^ 


21 


,8 




95 


It 


I 


3 


18 


^"^ 


64 1 81 


I 


X 


14 


>4 




96 


11 


2 





9 


,6 


65 


8 I 


6 


6 


J 




97 


12 


2 


4 


1 


j2 


66 


8 


2 


2 


2T 


,« 




98 


12 


3 





16 


,8 


67 


8 


2 


6 


»3 


,^ 




99 


12 


3 


4 8 


4 


68 


8 


3 


7 


4 >8 


• 




1 ■ 







N<*. 



^Settf New DecitiMf TaNtf, (jrc 1 1 1 



Tablet. I 



Table U. 



{Time. Om Maah tht 




Table I. I TMrfe Ih 



S Time. 



Om Daj th^ 




-» 



it J Set of New ^Decimal TaVesyZic 

N^'lH. M' Ft. 




fl 



N«». 



J Set if- New Decimal falks, 8rr. Mj 









; 








C liime 


(tr^tiion. 


■ Tabid It; 1 I: ■tAXe-.ti.i^OftMMn oi.Df 




; I gAtthe Imtg&,^ . 


N°. 


•• 


i*,: 


.: 


-w 


'.;' 


'.^ 




N^. 


LI- 


1' ■/' 


I 


15 


;«i 




>!■ 


■0 


3» 




2? 


ii5 


;i4 


2 


— 


,72 


! 


•2! 


■ I 


JB 




?o, 


|i8 


>— 


', 3 


I 


_i^.. 




.. 5' 


* I 


48 




3r 


I18 


.3« 


4 


I 


rf* 




4 


z 


M 




32 


>!P 


12 


5 


,1 


,8. 


, , 


5 


? 






35 


■P 


48 


..6 


"2 


.!« 


''■•■ 


■^ 


3. 


if 


' 


34 


T20- 


■=» 


7 


2 


rS!' 




7 


4 


12 




3$ 


■21 


> — 


8 


2 


,88 




8 


4 


48 




36 


21 


3« 


9 


? 


'.n' 




■? 


5 


'M- 


— 


"37 


-27- 


:rz- 


10 


^ 1 


•^ 




10 


.6 






38 


22 


.■4S i 


20 


■7 


,>!' 




11 


'--6' 


%^ 




V> 


2T 


i24 


3' 


10 


,8^ 


i 


It 


7 


12 




43 


24 




?» 


»4 


A; 


i 


■1 


■7 


48 




4r 


24 


1 3' 


50 


18 






H 


18 


«4 




4» 


=5 


iia 


«0 


21 


F 




■5 


'P 


f- 




43 


125 


US 


7* 


21 


l2- 




Id 


'9 


56 




44 


;2(5 


;2it 


85 


S8 


,B 




17 


10 


12 




4V' 


12^ 


1,-^ 


W 


32 


'1 




18 


10 


48 
2+ 




,4« 

■47 


,27 
:28 


1?,^ 






■'■" 




23 


i2 


,-L 




48 


;28 


148^ : 










2i 


t2 


j« 




.4' 


29 


,24 ! 








1 


52 


J3 


n 




SO 


30 




r-j 








«^ 


l! 


4! 




51 


oO 


r?i : 


/ 








»4 


♦4 


H 




5' 


'51 


in ! 










25 


•51 


(■^ 




<3 


;■■ 


us 1 


■ 1 1 








26 


»i; 


?4 




54 


;2' 


124- ! 


^- ! 


' 






17 


»6 


t« 




55 


33' 


1'- ; 




^^'p 


1 


28 


i'j 


jS 




J£_ 


J2l 


i36' 



»*4 


> 5'r/ 4^ ASftv Deamal Tahles^ kc. 


"?r 


' »™ 


f * 




^ 


1 -'-1 


CN • 






57 


34 


la 




7» 


43 


13 


57 


52 


12 


58 


34 


48 




73 


43 


48 


88 


52 


48 


59 


35 


24 




74 


44 


24 


f8p 


53 


24 


60 


3« 


^"^ 




75 


45 


>"" 




90 


54 


^ >*r 


61 


3< 


36 




76 


♦5 


3< 




91 54 


36 


6a 


37 


13 




77 


46 


13 




92 55 


12 


!^ 


37 


48 




78 


46 


48 


• 


95 


55 


48 


M 


38 


24 




l^ 


^ 


24 




94 


5* 


24 


^i 


39 


»•■ 




80 


48 


y 




95 


57 


fT" 


66 


39 


3<J 




81 


48] 36 




96 


57 


3<J 


67 


40 


12 




82 


49 


12 




97 


58 


12 


6S 


40 


48 




& 


49 


48 




98 


58 


48 


69 


4« 


H 




!♦ 


50 


24 




99 


59 


24 


70 


4» 






85 


51 


,-| 








7« 


-i.*'SL 




86 SI 36 t 


• ■ 





T.b.,1. |T.bi.a{SJt^.«-**^ 



IC: 








"w: 


?J 


^«V» 




"Sj 


€lll 


I 


— . 


10 


f 


I 





18 




«9 


5 


4* 


2 


-^ 


21 




2 





,36 




20 


6 


f 


3 


— 


32 




3 





54 




21 


6 


18 


4 


— 


43 




4 




12 




22 


6 


3« 


5 


— 


54 




1 




52 




23 


6 


54 


. 6 




4 






^ 




24 


7 


12 


7 




15 




7 




6 




25 


7 


32 


8 
9 




26 
37 




8 
9 




24 
42 




26 
27 


7 
8 


f 


10 


J ' 


48 




10 




*" 




28 


8 


24 


20 


3 


36 




II 




18 




29 


8 


42 


30 


5 


24 




12 




36, 




30 


9 


^^ 


40 


7 


12 




«3 


3 54 




31 


9 


18 


1° 


9 


«T* 




«4 


4 '2 




32 


9 


3^ 


60 


10 


48 




»5 




52 




33 


9 


54 


70 


12 


3* 




16 




f 




34 


10 


12 


;8o 


»4 


24 


• 


17 






'1 


10 • 


30 


{90, 


16 ^ 12 1 


• 18 ' 5 


H- 


« ^«t.«0 


,i^ 


10 • 48 1 



^* 



A Set of New Decimal TahJes^ 8rc 1^^ 



37 

3« 
i9 

40 

4« 

4» 
43 
44 

M3 

H 

ill 



N». 



fr*. 



N*. ^, 



t 
I 
t 

■t 

It 

3 
3 

3 
4 t 

14 
^4 
5 
■■% 
5 



;1 

6 



6 

H 
4» 

18 

54 
la 

f 

H 
4> 

s8 

3<J 
54 

13 

6 



58 

5P 
61 

? 
64 

66 

67 
68 
69 

71 

72 
73 
174 
75 
7* 
77 
78 



»7 
17 
18 
18 
18 
18 

t? 
19 
19 

20 
2a 
(SO 
St 
21 
U 
I tt 
22 
S2 

far 

2? 






24 

42 

18 

3« 
541 

12 

30 

4ff 
6 

42 

18 

3* 

54 
It t 

6 



79 
82 

86 

«7 
88 

9t 

n 
J 94 

j>5 

99 



23 

•5- 



2 
26 

17 
47 

a8 

48 

'9 

29 






4« 

ai 

3« 
54 

12 

f 

a4 
4» 

ss 

a4 

ii. 



GH AP. 






tm »> •<* « •«•« ' 



b JH Al>i Vl. 



*•» I 







i /. \ i J r> ^>^ ""'.I I ?/!. » 

^IR'ECtlPiofiriifmM '^Hn ^ fi^tir Numbers^ 

than che y>rofe^-^! the Ivin^ s ^[^j;^r [oc, lefs tton( 
ihey(i«ft& Nombcr. • Tku ai 4 is to i 2 fois 5 ^o .18 ; con-^ 
^qiiei^tly tl^e .tw mxiclit \ lumbors tniltipli^ ^gecher, laie; 
\fqtiul !td thq ProAjft if ftjicj /^v ^ xfrtrrtP oifcs ,' afe 1 2'X; 6 :={ 
k 8 X k- '^ 12. l^ende is d^dncec the g^ne\a\ $^e for wpric-^ 

I Mc^iply lhefeci>^d^Gun1lechy he third, jao^ divide by^tii^ 
firft; Btid tpe Qtptieht wil| bfsihp fouith Number foQght, or< 
JfcLnfcrcr.- ' i • . ? ' ! : - - 

Now as thcfe Rules of Proportion have fome certain Num- 
bers siven to find others in the fame Proportion, and their Sub* 
JcS being generally Trade and Kierchavdifeythofe given Num- 
bers often coniift of diverje Parts and Denominations^ which 
therefore are to be reduced to D^rf^ff^/j, in order for the Quefti- 
on to be wrought in thefimpleft Manner, and with the greateft 
eale and expedition ; which ought to be the Aim of every 
Artift. 

But as the Manner ot Reducing the Parts of Cainsj 
Meafures^ JVeighiSy 8cc. hath been fully taught already ; I 
(hall only, here exprefi the Queftion vulgarly, but ftate and 
worlff it decimally. 



V& 



^eftionij. If 7i Yards of Cloth coft 2 1. 12S. 9d. 
Kat will 140 i Yards coft at that Rate ? 

Thus 



^ The ^-Ufi t)f-'^e'€hfials fjf 

.' '■ « ■ ■ ■-••' •'■'■ ■ '"^Z?.- V. <£; Yds *■■■'' '•■.'■ ■ 
iDefunally 7,75 : 2^6575 : :. 140,5 •• .. - • ; 



A •» 



• « 



.13^8.75 / 

2 0375 L. s. a. 

■ 7i75)l7B.^fi8y5 (47^153 -47 : 16 : } t 
9100 



•<^^— ^ 



^ 'V •< ^ 



•'6P56 

■5415-., ., 

•6518 
- 6200 

•1187 

775^ 
3»75 



' 2500 
a^25 

175 



Quefthn 2. At the Rate of $ /. 6 J. 4 i/. pei C. We'tgftt^ 
What will i^C 2^. i47^.-cofl? 

C.. /. /. </ C Q. /*. 
Seated th^ >J ^'''S^^J; ^ • ^ --8 -r 4 • • 'P ; « = H 

5,41^ 

,,'p ) 117750 . . 

i - 1308^5. . 

19625 i , 

78506 
98125 

ThcAnfwcr; ic^/. 6 j^oi = ,106^0268^ ^ , 



•I .4 •> 



I *. . • 






-r. i K 



Quefthn 



qmtfihn ^ ^HPfe ^ Higjbtiidty three P^timSf and 
f«« daoitf oC S««r coA 6/. I4«. 8A Ho» miKh u dais 



s«»4*»-{SS3ifW> '^It^ 



4 — ;) — 5 ; 6 — 14—8 i: L 

M 1 



4f5«2*) ^»76f 33 ( 
45 *73 

■ 54 

4,588 ) 6,7266 ( m66i ( ,0271 
45M (108 

21386 'IfiS 
18^52 378 

i< I ■ 11 I m 

* 90^40 "Si 
a7588 54^ 

28120 27 
27528 

' • 5J>20 

4588 



GaUm. 




^ 4. The meam Mdiivn of the Ste beit^ kmnm 

/ 0'r_.,iA. rw. *»:<> rtftrntivMl m IrtwMit in nthar time H* 



|)etfoaiif one iutiie Revolndon thnu^ 
the.Ec)iptiC| 9c 360 D^iee*.^ 







I 

OPSRAttOU 



J3»j ) 324,000 (365,2762s=s365 : 6 : 37 m 

26«< 

•5790 
5322 , 





^4, 



*artWia«*i 



Tbefe are the various Cafti which may happen in the 
Bute of Three fiireSi by which tnf ^^ inw oUocve the 
Advojitage c^ Defitmlt^ .and the abfiAtteNccdnty of under- 
Sanding the Manageipent of drtulating or re f eating Decimals^ 

3f1r^ Rule q^ l%iee Imr^dL 

Inverfe Propcrtim is, when d£ four Nun^bers, the ibird 
bears the ykme i?4fio or Kx^Sofg to the jir/^, as the fecond 
does 10 the faurtlu 

Wbehde die i?»7^ is ;' to jnuUiiply the firft and fecond of 
the given Numbers, and divides that ^PrOdud by the third ; 
fbe Quotient will be the fourth Ndmber, or Anfwer. 

To know when the Ternlf of a propofed Queflion are in 
,this reciprocal or inverfe Proportion^ obfenre this Rule ; vix. 
When the third f Bigger ^^XxA^firft^c tefsy -i r^^A 
Number is c hfwr j and requires l/Wor^, T 

S TcrtM 



J 30 tn the Golden Uule direH. 

Tenns are in the inverfe RaiiiK^; and are to be wotked by 
the Rule above ; as in the following Inflaoces. 

QueftioH I. If when Wheat b fcdd for 5i. 6d. fer 
Buihel, the Penny White Loaf ou^ht tci weigh eight Ounces 
Troy \ What ' mufi it weigh when it is at 4 ;r. fer Buflid ? 

T-u u • J r Vulgarly 6 i. 6^. : 8 oz. : : 4 s. 
Thus flated ^D^cimJly 6,5 : ,^ /*• : : 4 ^• 

4) 4,3- (l,o83f = 1': I. or 13 Ow. 
4_ 

32 



'^ > ad infitiitum. 



f^ftion 2. Two Equal Far^lograms Ay B ^▼en, 
dieLengtb of ^ is 8 Feet 8 Inches, and its Breadth 4 Feet 
and 3 Incbesj the Breads of B is 2 Feet 10 Inches, Quete 
itsljcngth? 

F. In. F. In. F. In.. 
rr^ ii i. J fVolgathr 4 — 2- ! 8— 8 :: 2—10 

■ ■ ^ 8ytf 

4.25 I ^;f I . 9) 2550 1^ 1 2,8^ 

^ ' 28?3 13. 



5400 



28/ . ?68 \.i3 Inches the Anfwer. 



t " 



i: 



2.55) 3315 
255 



76$ . 
765 . 



- ^ 



Quejiion 



The Ufe ofDecimah iji 

♦ 

Queftion 3. A Piece of Land 4 Rod broad and 40 \oafl 
being a Statute- Acre ; tis required to Knotv what Lengthy witb 
10 Rod and 2 Yards Breadth, wUl make an Acre ? 

R. X. Y. 

Stated thus i X"'?^'Vi 4Mo::io-2 
*^ { Deamally 4 « 40 : : lo^i^ 



« • * 



'^ !£ l^di/. RY.P.In. 

10,26) 158,4 (i5,458=ri5 : 2 : 1 : 2 

.1026.- 

5580 

4500 ' 
■" 4104- 

Til^f Double Rule of Three ; cr Rule of Fivf Nambets. 

In this Rule of Proportion there are Fhe Numbers given 
to find a Sixih in Proportion-; which is either Dirt(jEl ox In* 
verfe^ according to the Nature of tHe Queftion. 

Queflions in this Rule arf? performed at two Operations, 
that is, by a douhle Jiating the Queflion, moi^ genenUy. 

Queftion i. What is the Intereft of 364/. $ i. for (even 
Months, three Weeks, at the Rate of 4 /. 10 4. ,fer Cant.fer 
Annum ? 

I m: I s. I. s. M.W. 
Thus r Vulgarly 100 : 12 : 4 — 10 : : 364— ^ 5 : 7 — 3 
ftated i.E>ccimally joo : 12 : 4,5 : : 364,25 : 7,75 

Firft Operation, 100 : 4,5 :: 364,25 

4.5 



Hence the Intereft of 364,25 /. ? 1821 25 

for me Yvt, is 16,39125 /.J: . I45700 



/. 



100) 1639,125(16,39125 
S 2 Thea 



I^i 7» the QtHhn Rxk ^finpff, 

.Second Opendoa nt 16,^9x25 : 7,7$ 

57,7 



>H7?9 

1 1474 

B19 



^i* 



A 



/. J. ^. 



duefttoff 2. 

Usysy at the 
(he Anfwer .^ 




12) 127,032 (10VS86— 10 : II : 8i 

The Anfwer. 

Sqppofc it were required to know what /Vf«. 

I)/. 14 X. 8«iL sa nine Months and three 

e of 4/. 10 j; per Centj fer Amum^ Quere. 



k J. A|0. I. I s. d.M^D. 
Thus f Vulgarly 4 : 10— 12— •100— 15 : 14 : 8— p : 3 
fated I Deamally 4,5 —.1 2>— >ioo— -i 5,7^ "-Pyioj 

The Firft Operation DireSi. 

•/. • . 

too 



The Principal 
fipin^wbence 

cokties in 12 
Months is 



i; ■ I ■■ ■ 

223 

i8q 



•435 

405 

283 
270 



^ ■ ^ »i mm* 



15? 



1 



The Second Operation therf fore muA p€i wiou^ r»- 

verfely. 



12 

9,107) 4i9*,| (4fe,^5^ = 4^ •• «9 • " 

«27S 
^42 



••^3355 
•8713^ 

<!■ Il l I j I 



But atqr Qoeftion in Qiis Role may be va hm iA itt 9m Ofe^ 

ratiofgy by the following Ht^e : 

Make the 4 2d ^ Five, the feme S-sihP""** P^ 
J 3^ C ^Vi<} V!^4h the 3 6th the Nnoi. foi^. 

Then, Multiply the Tbree Numbers to the Right-hand toge- 
ther, and the tvx> prfi to the LeEt-hand ; and mvtde the j&yf 
ProduSi by this /^ and the Quotiemt will be the fidi Nusk* 
ber, or Arfnomry if the Proportion be DireS* 

Qnefiieft 3. If a 1000 Men can dig a Trendi 500 Feet 
Long in 24 Hoois, what Length of fijch a Tcfoeb eait j^8o9 
Men dig in 10 Hours ? 



Tb«8 



«34 



in the GoUen Ruk ^B. 



Men Hours Feet Men Hours 
Thus flatedy looa 24 : 500 : : p8oo. 10, 

24 ^ ro ' 



24CX)0 



98000 

500 

241000)490001000(2041 y0 

48"- 
J06 

'96 



The Anfwer is 
Feet Feet In. 
204l,i^ r=: 2041 : 8 
The Length required. 



•40 

H 

^60 

16 



/« Infinitum. 



If any Part of the Queftion be in reciprocal or inverfe Pro- 
portion ; {dace the Three firft Numbers as in the laft Queftion ; 
and of the other two, phce That Ac fourth, which is of the 
feme kind as thelecood; and confequently th^ oth^r muft 
l)e the 6fth ISumben 

Queftion 4. If 1000 Men can die a Trench in 24 Hours 500 
Feet Long, How many Men will dtg 2041,1^ Feetin 10 Hours ? 



Stated thus,' 



Rule. Multiply the. ^ 
ift, 2d, and 5th Num- 1 
bers, and the 2d, and 



Men H. Feet. 

1000 : 24 : 506 : 

10 

5000 



a Feet. 
10": 2041 ,)tf 
1000 



24 



4th; then divide the }> 
firft Produ6^ by the laft, j 
theQuotient is the An- I 
fwcr ; viz. p8co Men. J 



8r#j666,6 

4c83'?333>3 ^ 
5 1 ooo)490oo|occ(9opo 

4S 

40 
40 



•00 



But for the greater rcadinefs and cafe of the irrgenious A^ 
rithmetician^ f Ihall tranfcribc that famous ^f;5f^i? Theorem 

in 



\ 



TbeUfe of Decimals^ i JS 

in Mr. Ifard's Young MatbfmaimMS Gukkf whidi Ihews 
at once how to anfwar any QueiHon of PhuNimhers at one 
Operation, widiout regard to the Proportion of tbc Terms j 
be that Direa or Iniirea as it will 

the Theorem U ^s, TgF = Gpt. In this Theorem jm 
obferve tltfcc Capital Letlers, viz. r,;/*,G, andthcftme 
three Letters in [mail Charaae«,.^,f,A The Awe Qi/^i. 
tals figniiy ^tTbreefirft condiitondTerms of the Quefhon, 



- ▼ 



r P, Is the WtncifdCixxtt oiGain^ Lofs^ASlion^kc. 
' Thus, < r, IstheTFw^, Sface^ tiftancey &c, 

tfi,IstheG4/«, Lo/i, Abiiony &a t 

Of the I&wV /««// Letter Sy (which correfpond to, and fig- 
nify the fimc with the Capitals) two always move the Qoefti. 
on, the other (hews the Anfwer ; which, ^ the Letters aio 
three, is threefold J and anfweicd by the lame Theorem diP 
pofcd in thcfc three proper Terms. 



c/>7 

yi%. If < t >be fooghty the Theorem is 



Orthus, 33rP-TG'=^ TtP-rGp = f. Gpi^TB^sg. 

If any Arithmetician fhould complain he does not undei^ 
ftand fuch Algebraic Forms and Cha^aSlersj all that I ha?e to 
anfiyer is,. 1 hat 'tis a very necejfary Part of hb Bufinefi and 
Prdfeffion^ zni highly (pncerns him to learn if. 




> ♦.* 






<•,"■, . , . • lit - . \ ~ 






C H A P. 



T i 



n* 



C H A F. Yin 

• ; : 

A New Decimal TraSiice i or a p^ort Way 
4/ cm^i^ . all . k^ of inierchmiije if- 
DSClMAtS. 

* 

THO' dieit k fiacce mj Parf of Jbriihmticl ill 
wtidh tkiimeU sijre 6l ^ateV (oc iodc^ fo gMac) 
Service, i&pr^ctijet f&i of alt oebets Kas^bfeen 
die /fis/? imPrdeihy it ; ha»9 aih Author dth be met with 
on this Head ; and dio^ who have uodertaken it, have pre^ 
ntoMlf tJs #im otit iitspgi^^W Skittl^f^ ftiKt tctc IMb B^tttel^ tm* 
Aiiftid. Il^^diM^ak^ 
llfti€fH^ tif this xoiHt.* 
Hie Vmet(faliqitoifMtiottiRM9^ are^beTe 



J. i/. Parts. 



6 

5 

4 

3 

2 

2 

• a' 



^8 — 



2 
3 



^6:-. 8 

— 0—10 



Ifte MM or aHafot Pmfot 
a Pound Sterling. Bf whidi <& 
vidifrg,-ff,vti an Anfwei in Bemidt, 




'^> 



The ^t2M or nHguof P4rU6t 
zSBillhg ittrlfkg. • By wMeh rff- 
viding^ gives aa Anfwer in Sftf/* 
Jingi. 



ixt the 7^H^ foUowing is fiur more general^ ex f editions 
fuid ufrfttli and has not yet been applied to Decimal 
PfoRice. 



'.. 4 General Table /or- Decimal Ffaftice. 



I Price. Oivifors. 
■d: 

o 

■ < 

I 

I 

I 
I 

2 
2 

a 

I) 

3 

\^ 

I 

4 
4 
4 



5 

5 

5 

5 
6 



q. . 

1 3>4»PQ , . 

2 6,8o 

^4j8p: ,r ; 

d 4,60 

2 2,80 

3 3i49.r-r8 

0540 

1 3,40,+8 

)2 3,40»"t4: 

^ 8o<— rlX' .' 

oO» •; ■ *.■_ J ir 

1 80,-f 12 
.2 80,4-6 . 

c 60. . .., 

2 60,4*8 

3 6o,+8,2 

6,8 

1 40,-- 8 

2 40,-12' ' 

3 80,X2,— 12 

40> 



;1 



) Frice.t Divifors. 



6 
6 
6 

17 



7 
7 



8 

a 

8 

,,-p. 

•p. 

fq 

i'o 

10 
II. 

II 
tl 



I 

2 

3 





I 

21 

3 



I 

2 



I 

2 

3 

o 

I 



80,X2,+I2 
40,+I2 

40j-bP 
40,4-64 

40,+* ^ 
4Qj+4fP ' ^ 

30. ; . ; 

8o,X3>-4 
/0,X2^T^n8i 

3 8q»X3r'^^ >12 
8o,X3,+i2 

30,+4,8 

40,4-2,2,6 

4-)X2y*T7i5( 



PI 

2 



2 
3 



riff 
I 



20 
2IIO 



5 
■6 

7 
8 



Divifors. 



IOj+2 

4l^o>X« 

4 

i<S^4'' 
io»X4>+2 

;• jp io,x8, . 

17: io,x8,42 



IP 



40>X'lf,-:|2 ^ V 



1,-20 



^« Explanation e?/ the ^f^ceeding Table.' 

Thefirft Column fliewsthe Price 6P Hie Commodity, either 
in Pence and J^artbingSy or in SbilfingSy for one of a fort 5 
as one Pound, Yard, Piece, Scc^, ."'.' ■ .« . ; 

Againft the Piece, you dWerwc ^ t)ie fecond Column, fevc- 
ral Numbers, of which thofe yih\chjiand firji, and have no- 
thing prefixt to them, are Dii}iforsj bjr ivhich ar^ given 
USantity ot Number of Yards, Slh,'Pounds,^Tei(j be </i- 



< ( ' . • « ;.. . 



If 



13^ -^ New Decimal TraBice. 

If any Nuniber follow thefe with any Charter frefixt to 
fhem as 2^o,X2,+8. 8o,X3r-*i2, Qc. They aie to be 
mnderftwJ^ ana ready as in the following Examples. 

34,— 8 Prom a ^th of a sd, take one 8tb of that 4th 

« ^rv x/« J-p/ To a 40th of a ad, multiplied by 2, add 
3,40^X2,-M>^ an 8th of that AOth. 

io^ I o ^ J To at 60th add an 8th of that 60th, and 
oq,+e5,2 ^ half of that 8th. 

* « *i-o tx -f To a 3d add an half of that 3d, then fub- 
?>*r2r-i«> \ ifoa a i6th of that %i. 

' ^^ C From the given Number take one 20th 

1,^20 ^ p„^ 6 

; Theft being well onderftood, 'twill not be difficult to ufe 
kbe Tsd>]e 00 m oocaiions with eafe ; efpecially after perufing 
theEmmples cofim^, whidi are chofen fer the more dif&cuic 
Iws liiereo£ 

I Note^ When die trici confiRs of SbitUn^ only, the Nam. 
jber maybe multiplied by the Decimal^ that is, bdlf the Nurn^ 
fet ofShtVingSy and the Aidwer mil be the fame. 

Ek. 1. At If fit Yard, What coft 144 Yards ? 
^ -~ One 3d == 48 

One 4tb of that =s 12 
One 80th of that ss: 0,15=^35* Anfwer, 

. £»• 2. At 3j. fit Yard, What coft 172,5 Yards ? 

One .4th = 43,125 
One 80th of that :s 0,53907=104. ^d \. 

Bk.%. hX id. iq. What coft 1792,25 

One 4th = 448,0625 
A'^di of that^s 74,67708 
An 8th of that = 9,33463=9/. 6 s. 8 J d* 

Ex. 4. At I rf. 3 }. What coft 974*^ 

One 3d =2 3247,^ 



A 4odi of that = 8i,ie8{t8 
From whi h take an 8th = 10,1436^ . • 

Remains the Anfwer = 71,0452^=71 : 00 : 10$ 



Ex. 



A New Decimal PraSice. 139 

Ex. %. hi 2d. I q. What coft 9691^,2^ 

One 3d = I232r,09 



s. d. 



A 40th of that = 30^*0^ 
To vrhich add one 8ch = 318502$ • 

The Sum the Anfwer s=: 3496513 = 34 : 13 : o£ 
Ex. 6. At 2 d. 3y. What coft 4130,2^ ? 



One 80th = 5 1, 1^27- 
From which take a 12th = ^:goz , . 

Remains the Anfwer = 47^2^ "= 47 J 616 

I 

Ex. 7. At 4 rf. I q. What coft Ip3ar,4^ ? 

One 3d:= ^^^i^ 

A 40th of that s= i6,^o^ 
Ditto =3 i6,-r04 
Of which add an 8th = a^i^ • . 

The Sum is the AnlWer =3 34»*2^ ^^ 34 * 4 * 5 

£v. 8. At4i. 3^. What coft 94.9,^ f 

One doth = I5,8'07 
One 8th of that =5 1^7^ 
One half of that = 0^7 t • 

The Sum is the Anfwer = 18^7^ = 18 : 15 : 5 

Ex. 9* At 5 rf. 3^. What coft ioir,2 ? 

One 80th = 12,1^ 

2 



One 80th, X 2 = i5,3'5 
Subfiiafl a i2thof an8oth= 1,0^ , ^ 

Jlcmaiiu the Anftrer as 24,2^ = 34 : 5 : 6j^ 



Ta Ar. 



f 4* ^ Ntw bfcimal TraBiee. 

Ex. 10. At 8</. la. What coft 2640? . 

One 80th =s 53^ 

3 



That multiplied by 3 = ^ 
Subfinua a 4th of an Both = 85^-5 • ' , 

• Remains the Anfwcr = 90,75 = 90 : 15 : o 

Bx. IX. At pi: ij7. What coft. 96^^92:? . . 

One 8ot& = 12^24 

3 

An 80th X 3 = 36,072: 

JTo wlfich add 1 2th of an 80th =: 1 ,002: , , 

- /• $• a* 

The Sum is the Aniwer = 37,074 = 37 : i : 5:$: 
Ex* 12. At 10 d. 3 J. What coft 1600 



mm 



One 40th = 40 

Add an half ss 20 - 

. Add halt that = 10 
Add a fixtii of that = 1,^ • , 

The Sum is the Anfwer = 71,/^= 71 : 13 -'4 

Ex. 13. At 11^. 3y. What coft pOj^,i« ? 

One 30th r^ iy>yZ06 
To whidi add art half ss i^jtO^: 

From the Sum := 45,^0^ • 

Subftraa a i6th = 0,^4^ • . 

Remaiins the Aofwer = .44,:5'^=44 : 7:3f 

JSx. 14: At 75. What coft 365,15 ? 
One loth «=» 36,525 ' 

3_ 

One loth K 3 = 109,575 
Xo which add half that I oth =r 18,1625 - - . 



The Sum is the Anfwei = 127,8375 =3 127 : 16 : 9^ 

.■»<•■ • • ■ » , « I 1 ^ ^^ 



-ITT, 



A Neiv Decimal VraBice. ' 141 

Ex. 15. At 1.9 i. Whaccoft^ 257^? 

Subflraa a 20th = 1 2,88a* - 1 . j 

Remains the Anfwer = 244,78^ =s 244 : 1 5 : 8 

£;tf. 16. At 13 i. 9$^. What coft 96 1 ^9^} 
96x^9z '\ 8oth = 12,024^ 
ioth= 96,;r9^| 3' p^r the 

:! pq, ,3 80th X3 = 36,072:^9 j^. 

i of lothzr 48,09^1 ^^'' 1 ^ c ; 

The Sum:i= 625,^4^ I j ^ ^25,2:4^ for 1 3 : 00 

""""^ The Anfwer ssa 662,3*2^ for 1 3 : 9^ 

In this Example (and any other) the Anfwer for the SKL 
lingfh found with leaft troable, and Figares, by Multiplying 
ihe given Number by the Decimal of the Sbillings, 

p6x,9z The given Number. 

^^ . ,65 The Decimal of a Pound for 12 s. 

Thus, ^ ■ I ' 

480^60 
577^5-^1 
625,2:4^ The ikme as before. 

If the /'rw or f^tf/«^ confift of Pounds, Shillings j Pence ^ 
&c. the moft ready and praSicd way, is to turn the 7t>bole 
into Decimals, then multiply the given Number (turned into 
Decimals alfo, if exprcfs'd in diverfe Parts ;J and the Pro-^ 
duB will be the Anfwer. 

En.ij. At 5/. 16/. Zd. What coft 270? /. s. d. 

J5i8£=s 5:16:8 

9)8io 



900 
2160 
1350 



The Anfwer = 1 575,00 



Ex. 



X4« TheUfeof Decimals in FelloTvJhif, 

A *• d. C. Q. Ik* ox. 

Be* i8. At 1 : 17 : 3 * What coft 14 : i : 14 : 10? 

then 4'»4»^8os8_ 

*""» 1 526568,1 The Multiplia Mt»«t«/. 

143805 

I I 5044 

8628 

86 

i 



The Anfwet L 26,8285 = 26 /. 16 /. 6 4 </. 

Thefe Examplea are fufficient to the ii^enious Pndical 
Student of Decimal Arit/mettck ; who with thofe Indrudi- 
cm will eafily (profrio Marte) apply this noble Ait to «J7 
Ctfet of Cvmmm Trade and Merebandife. 



CHAP. VIII. 

TheU/e of Decimals m the ^les of 
Fcllowfhip, Tare and Tretc, Barter, 
Gain and Lofs, Exchange, Alligation, 
Rule of Falfe Poficion, Extraction of 
Roots, 

Single Fellow (hip ; or That without Time. 

THE Rulet of Fellow/hip are proper to Jdertbairts 
and thole who Tirade in Company, or PartnerJbtP ; 
where they have a common joint'Stock to tiarack 
withal ; for to every one of the Company is diftributed his 
duejbare of Gain or Lofs acquired by ijradvi^, in proportion 
to his Stock laid out, by this following 



The XJfe of Decimals in TeUowJhif. 143 

Rule. 

As the total Sum of the Stocks, is to the total Gaim of 
iJofs ;fo is each MarC% f articular Stocky to his particular Gaim 
or Lofs. 

Examfle. Sappofe Four Merchants^ Aj By Cy Dy make a 
yoint Stock of 421 L 8 s. 6d. 

r* A puts in 154 : i^ : 4 =3 1^4^666 
rp.^ J B — 110 : 18 : 6 = 110,9250 
inus, < ^ _ P5 : 00 : 8 = 95,0^35 

C D «— 60 i 16 : o z=: 6ofi 

The whole Stock = 421 : 08 : 6 ss 421,425 



The Trade and Gain 88 1 17 s. 10 d. = 88,8pii^ /. Us 
lequired to find each Man s Part or Share of that Gain. 

/. /. 



/. 7. 

Then as 421425 : 88,8pii^ 



: 154^ : 32,6192 
: 110,925 : 23,394 

: 95>0-5' : 20,0425 
: 60,8 : 12,8227 

The Sum of the fcveral Shares ~ — 88,8784 

1 I III !■ i^— i— 

Which being t}ie fame with, or equal to, the total Gains 
always proves the Truth of the Work. 

But all Cafes in the Rules of Fellowfliip are fooneft and 
eaiiefi anfwer'd by finding the Proportional Part of the Gain 
or Lofs due to one Pound ; and then iy that to multiply each 
Man's particular Part of the Stock ; for the fever al ProduSis 
in fuch Cafes, are the feveral Anfwers j viz. each Man's Part 
of the Gain or Lofs. 

I. I. I. I 

Thus, As 421,425 : 88,89ii^ : : i : ,2109 the common 
Multiplier, 

/. 1. 

A% Part of Stock = 154,^ J5's Part of Stock z= 1 10,925 
Multiply by ,2109 Multiply by ,2109 

1391^ 998325 

154^66 1109250 

309^333 221850 

Jt% Part of Gain =r 31,61921 £'s P. ot Gain s 23,394, gr. 



144 ^^ ^ ^ f)e€imaU in FeJlfwJhip. 

1. I 

Oi Part of Stock =: 95^^ Z>'s Part of Srock sz 60,8 

Muitii4y by ,2X09 Mukiply by ,2139 

8552^ 5472 

9">c?35 6o8q 

1900/^666 1216 



C*s Part of Gain = 2O3P425 ? C*s Part of Gain*= 1 2,82272 
Here every Man's Share is the fame as befote. 

Double Fellowfhip, or That with Time. 

Fellcwjhip wiiff Time confidcrs the Share of the Gain or 
Lofs with regard to the Mone% and the Time it was imploy- 
ed, and frofortionaies it to iotb by the following 

Rule. 

Multiply each Mans Stock hy the Time it was employed \ 
then fay J As the ftm of tb^e f^roduBs^ is to the whole Gain 
or Lofs ; fo is every one of the PfoduSsy to its proportional 
Part of the Gain or Lofs. 

Example. ThreeJAtrchaniXs A, By andC, enter into Pirrf- 
nerjhipj thus 5 

7. 

A puts in 65,5 lor 8 Months, 2 Weeks, and 3 Days. 

B -^ 78,.^ — 12 Months, 3 Weeks, and i Day, 

C — 84 . — 6 Months, and 6 Days. 

They traffick zni gain 140,0 if/. Tls required to find 
each Man's bhare proportional to liis Stock, and Time 'twas in, 

/. Months. . . ProduSls. 
C A% Stock 5j,5 X 8,607 = 563,7585 
Firft< fi's Stock y^^fS X 12,3357 = 970,4084 
tC's Stock 84, X 6,214 = 521,976 



The Sum of the Produ3s*;= 2056,1429 



Then 



The Ufe of Decimals in'T^rt and Tvttt. 14 j 



Then, A^ 

2056,1429 : 140,01)^ 



i: 



/. 

5^3>7585 : 58,3918 s= A 
970,4084 : 66,6846 = B 
521,976 : 35^5465 = ^ 

The whole Gain very near =: 140,0229 /. 

.Queftions in this Ru]e alio are much better aniwer'd hy 
finding the Proportional Part to one Pounds for a common 
Multiplier, as bef oj:e. 

Thu?, as 2056,1429 : l40,oij^: : i : ,0681 Coram on Mul- 
tiplier. 

The Operation for A, — 565,7585. Fojr JB = ^^704084 
The Multiplier inverted 1 860,0 1 860,0 

338255 '582244 

45100 77632 

563 970 



38,3918 66,0806 



mm 



for Cs= 521,9760 

1860,9 




a" ^±6^ f Their Icveral Parts of the 
^^ "Li^ cGain, as before. 

Thus appears the excellent Ufe of Decii^ah in thie Rules of 
Fdlowlhip. 

Tare and Trett, 

Tare is the Weight of the Hogshead, Cbeft, Bagy Cask, Sec, 
which contain the Goods hought or fold. 

Trett is an Allowance of 4/*. in 100, or 104/*. for Goqds 
whprein is Lofs by refufe^ 86c. 

Cloff is an Allowance of 2 Pound upon every Draxtght 
which exceedeth 300 Grofs Wtigbt. 



Suhiili is the Weight when the Tare is deduSed^ but not 
die Treit. 

Ntat W^i09i i$ ifae Remander when Tarey Tutty and Tfej^ 
if all asc aUowed, «ce fiitev iWD^. 

For rcfolving Quefiions in this Rule there are feveral 
Methods I but thgre ky ^ecin^gl^ ^xc iauq)i At fliortefi and 
beA, and are as follow. 

QM^Jim f • What isiihe J//4f Wei^t €f 9 C 2 jff. 7 /^* 
22r/ »c 14 ;K fir Out. to be dcduaed. 

JF?r/?, This may be anfwered by the Golden Rule in lie^- 

». C. 

if I r. illow 14, Wh9t will p,5d25 aUowfor Tate? 

H 

- ^1 . III. ■ 

582500 
95625 

Total Tare Ih. t=i 133,875 = i,ip53 C. 

c. 

^ The Gro/jr Weight = 9,5625 
rpj^^ \Thc Whole Tare ws 1,1^3 to t* fchfliaaed- 

The Difference is — 8,367;^ = 8 Cl jr. 1 3 /i'. 
the Neat Weight. 

A Second Way, is to Multiply the Grofs Weight by the 
Z)^^/;^/!/ of C Weight, equal to the Tare allow'd. 

C. 



\ 



The Grofs Weight :^ 9^'ili%% 
The Decimal of i^b. = ,125 



478125 
191 250 

P5625 



The T<ire (as b^fo^e) = 1,1953125 to be fubflrafl^d.. 

A Third Way, is to multiply the Qrofs Wehbt by the 
Zieeimaf6£ the Ale^t Part of a Hundred Weight. . 



Thus, 



Toe Ufe ofOecimaU in fare d»i tretf. 1 47 



c. 

Thaii ft«i 1,003 
Subaiaa ,125 

Thei\i«ofC. ,875 


C. 

Then' 9.^6i5 

578P Multipliet inverted 

76500 
1?f 





The N»a% Weight total = 8,3671 as above. 

A Ftm^b Way, is to work by AU^t Parts as in PraSiee. 
Thus 14 fedng rtiK" 8tH I*art at i f 2 ; if you take an Stb ' 
of P.5^af» riHTwill'betlieT.i*'*' of the Whole. 

C. 

Thu* 4" ^»^^2^ *" ^'■''■/' '^I'S^^- 
' I T=i>i953 iheTiWf, asbciore. 

For the aoA fxpidkibtls fifldlHg thfr jbrf by ^iqtiot 
Parts, thaVfriflfeitedttitf K^#uig Ta^/f of T^r^ and fro- 
ftr Divifort. 



The Cortfirtaion and Ufe of this Tabic of Tare is the 
fame with tbe Table of Prices ot Values in Praflice, which 
fee there rtt^ht. 

Having (hewn how to fhtA the Tafei the neit Bufinefs is to 
find the Tittt, or thtf" A"f If'eisbt when the Trelt is de- 
dofied- itooi'tlK-SH^rt^'; 

TKus Hulriply f. jOjS+fthe Prodna ia^tbe J>f(f. 
die SuHiUVy T >96i6 the P'rodud'ts the AVof /^/gftf. 

U 2 Quefilom 



148 The Ufe cf Decimals in Tare and Trett^ 

QuefttM 2. In 72 c, 3 jr. 12 lb. Grofs, Tare at 12/K 
//r C. Trett /^Ib. fet 104. How many C. »^4f .^ 

r. 

The Grofc =: 72,8571 
Multiply by the Decimal of 1 2 lb. inverted = 1 701 ,0 

72857 
' 5100 

72 



The Tare (fubtile) = 7,8029 

The Subtile = 65,0542 
Multiplier inverted = 4850,0 

iP5i6 

5204 

260 



The Tiett to be fubtile re 2,4980 
The Neat Wt. = 62,5562 

A Shorter Way, thus ; C 

The Grofi = 72,8571 
Mult, by the Neat Dec of 1 12 lb. inverted = 8298,0 ~ 

58^856 . 
65571 

' • H57 

582 

The SabtHe* r= 65,0466 
The Mukipliec inverted =: 6169,0 

585419 
39027 
. 650 
^9P . 

The Neat Weight =-: 62,5486 

Thcfe are the beji Methods for finding ftr/ and T^etf ; an* 
that I have here given tor finding the Trett is nevj to me, noc 
having feen it in any Author 1 have met with; 

Sarter» 



The Ufe of Decimals in Barter* 149 

Barter. 

By the J^uU of Barter^ Menbants and Traders exchange 
Goods of different Values^ Kindsy and Qmantitiesy ib as to 
fuflain no Lofs or Dif advantage by fuch a Barter or Change. 

Queftion i. Two Merchant s^ A and B barter ; A would 
exchange 'yC. ^ qr. 14/^. ot Pefper^ worth 5/. 10 s. per 
C. with jB for fof fo« worth 10 d. per Ik. How much Cottom 
mufl B give A for his Pepper .«? 

Proceed thus by Decimals to ftid the Value d^ the Pepper. 

Say, As i ; 3,5 : : 5,875 

3*5 

29375 



The Value of the Pepper z= %o>5625 =» 20 : Xi : 3 

Then to find the Quantity o£ Cotton equal to the Value of 
the Pepper ; 

I. C I. 

Say, As fi^U ; ,oo8p2 : : 20,5625 

29800,0 

1645 
185 

4 



,041 ^\ ji8^4 
041/ 1834 / 

>o?75) i<5506 (4i40»<5 

__ 1500 

Thus B muft fflTe A '\ • * 

C. C. qr. Ih. ) »^06 

of Cotton for his Pep- \ , . • 500 

. 375 

2250 
2250 



per, 



■**M 



• • ■ 



Queftion 



15^ "f^ Pf^ ^f I>fthfMis in Barferr 

Quefilon 2. ^ has ^2 Dozen of HatSy worth in ready 
Money 2s. 6J. bat tortus at 2s. 9 J. per Hat. B has 
Cotfov at 10 d. Psr Pounds ready Money. Qaere ac Wi^r^/ 
llrtir^/'tesW'fi laijft iMrArlu»CiRM aod ]k>Y# lAuch he 
matt j^vefbc die flbti .^ 

X. X. us* 

rifft; fijs A» 2^ : ,25. : 5 ^ : d»^ s= Ontf Piwnj^ 
Sa diat B!s Canbn itf^ «i be adWntedr a Peiii^ a^ y^dfld 



Secondly, to find the Vala^rf ihr (634) Thfi \3xfmneft 

Say, As i : ,1375 : : 624 : 85,8 

624 



rtkM*- 



S500 
»750 
82^ 



«lh 



A X. 



8ii,gboo = 85 : 16 the Price of all the Hats. 
Thifdif m lauiw ytthtt CoitbM at nJl fev^Poondcmbc 
had &c that Money ; /• C L C. 

Shy, ti» ,04^ : ,oc{^.r: 9%jS:i&jS9S% 

85.8 

7136 

4463 
7136 



0458 765336 

P4t25} >6888o24 (16,6982 C 
4125 



27630 
Hence it apf^cm that ^ 24750 

16,6982 = 16 : 2 : 24 > ^^T>^ 

of Co/f o« at 1 1 .d. .Per />. I • 40^ 24^ 
for 52 Dozen of Hdts at / 37125 

2s. ^d^firEUt. 



33990 
33000 



9900 
8250 

1650 Thefe 



The lift fOf Ikfcimah in Gain uni Lo{& 15 1 

Thefe tinro Quefiions well underAood, are fiifficienr for all 
qther Caie9 in ibi$ Hidff* 

Gain and Lofs. 

By ibis Rule Moa ^ Tf^in and Bufi^efg know iwlocthey 
qpt by E^etailing Goods \ and in cafe of Damngej what they 
^^> j>y feilioc it «c any fireo Hace ; and <«fiechflr chey gmm 
or ^, cokncMT at ipW ifi^f^ ^ O^rr. 

Queftioff I. If I buy Cambricks at 5 j. 6<£ jpfr Yard, and 
fell them ^ 8 1. p i/. What is the Gain fer Cent / 

Sav jVulg. 5^6 : 8-p.- : 100 : i^p— i— p^ 
^ iZ)^^ ,275 : 4375 : : 100 : 159,0^ 

s» d* s* d» /. 7. s. d* 
Or thui jV"Ig-5-6:3^3::ioo:59-i-p;7TheGain 
• ' ' iPcc:. 275 : 1625:; 100 : 5P>o^ i/>^r CenU 

By the yery (aoi^ Manner of Working you 6nd the Lofs. 

QwtftioM 2. If I buy Cambrick at 5 j. 6 d. fer Yard, 
How muft I fell it to Gain 59 /. 19* pi d, fer Cent ? 
The Converge of the /^y? Method folves this QuelUon, 

7. /• s* ' d, s, d, s. d. 

Thus f Vulg, 100 : 159— I — 9% : : 5 — 6 : 8 — p ) The Price 
fay, IDcc. joo : i^pjc^^ : :|275 • )4375/'^'' ^*'''' 

QufftloH 2. If I buy a C. Weight of Ti^/Jrr for 4 /. 1 3 /• 
4^. and fell it at ii d. fer Pound i What do 1 Gnift or- 
Zo/J*, and at v^bat fer Cent ? 

Pirft find what a C Weight will amount to at 1 1 </. fer 
B^undm 

Ih. d. Ih, L $, d. 

rVulg. I : II ::ii2:^— 2— 8; 
Thus, ^ C. L ' C. lb. K Anfw. 

^Dcc. ,oo8p2 : ,04583 : : i : 5,1^ \ 

L J. 1/. /. J. ^* /• /• 
Then f Vulg. 4^x3—4 : ^-..^^S : : 100 : no? r/r. lo/. 
fay, iDcc. 4,i^ : 5,1^ : : 100 : no \fer Cent. . 

Then 



1 5 * The life of Decimals in Exchange. 

/• t. d. I. 

Then From 5»— 2 — 8 And from no 

Take 4 — 13 — 4 take 100 



^^ Remains o— « 9—4 the Galftj at the Rate of 10 /• fer C\ 

The Cowerfe of this needs no Example. 

QueBiom 4. If I buy 5 J Loads of Wheat foit 45 /. 16 /. 
8 d. For how much muft I fell it fer Quarter, to gain 6 /• 
10 $. by the Bargain ? 

/. X, d. 

F' ft J To the given Price = 45— 16-«8 

c Add the defigned Gains = 6—10—0 

The Sum is = 52— .^—8 

for whtth the (aid Wheat muA be fold« 5 \ Load = 27,5 
Quarters. 

Qr$. /. Q. /. /. /. «f. 

Therefore fay 27,5 : 52,3f :: l : i,pa^ rr x— j8— oi 
the Anfwer. 

Thefe being the frindfal Cafes of this Rule, are fufBcient 
if well underiiood ; and the Operations at large are omitted 
for the Excrcifc of the Ingenious. » ' 

Exchange. 

Both the Nnme and Bufinefs of Emcbaftge is oftahg^us to 
that cf EarUr ; only that relates to Goods and Commodities ; 
whereas this is conccrnd in Foreign CoinSy Weigbts^ and 

Meafures. 

Excbangj then confifts in finding the true Sum or Value of 
one Country Coifi^ &c. equivalent to any given Sum or Value 
of that cf any other Country. 

The Par of Exchange is the fixt and Standard Value o£ 
Forerg'z CcinSy &c. exprefs'd in Sterling Money of our own ; 
a;)d is that in the Tables* Tis fo called becaule in Excbavgf^ 
Par fro Pari^ i. e. One ecjual Value for another, is given. 

The Qoiirfe of Exchange is the current Price of E\chang?y 
always unfectJed, being fometimes alove and fbmetimes helow 
the par ; according to the various Circumflanccs and Accidents 

of Triide and Nations. 

The 



Decimal Tables of Foreign Coins. 155 

The Courfe of Exchange is publilhed in the Weekly Pa- 
pers and Pamphlets, which compar'd with the Par in thie Tar 
hies, it appeals whether it be above or below it at any Time. 



EX A 

Courje of Exchange. 



^ Amfterdam 
^- Rotterdam 



s. d. 



5 



m 






5 



?i 



Hamburgh ^4 
Anixoerf 35 

•^ ^ Mndrid 
Leghorn 
Genoa 
yenice 



4» 
48 



: I >:s* 

• 7 



M P L B 
The Par. Difference. 
S' d. s. d, 

f-33 54U— « •■ 8 
J —33 * 4 J^ — I '- 
\ -33 •• 41 1-0 = 



•33 ' 4 



7 

7 
o 



1 






J -^54 • i U -o : 3; I J 

I— 52 : i^^— o : : 



J 



It is to be obfervcd, that when the Qourfe of Exchange,^ 
above the Par^ tis a general Indication that our Trade is 
profperous, and the Nation on the Gainful Side ; as on the 
contrary, if tis below ihe^ Par, the Trade is tad, and the 
Nation /oo/>r. 

The Par of Exchange in Coins, /kfeafures. Weights, &c. 
between Us and Foreigners, are exprefs^d in the following Ta^ 
bles, and which I have reduced to Decimals for more conve- 
nient and ready refolving of Queftions in this Rule. 



Low Country Coins. 



A Stiver — — 

A FlemiOi Shilling (= 6 Stivers) 
A Gilder (n 20 Stivers) — 

A Flcm. Pound { — 33 j. 4 </. Flemilh) 
An Emblem DoUer — 

A Campen DoUer — 

A ZelarKi Doller •— — 

A Lyons Doller ^^ «— 

A Specie DoUer -^ — 

A Duccatoon -« *— 



/. 


$. 


d. 


h 


. 


r : 


il 


0,00$ 





: . 


•7^ 


0,03 





; 2 


: 


0,1 . 


I ; 


: ; 


: 


1,0 


i 


r 2 : 


5t 


0,11$ 


; 


f 7 ; 


7r 


0,13' 


: 


• ? • 





0,15; 


: 


: 4-: 


' , 


0,2 


; 


5 • 


1 


•0,25 


: 


6 : 


2 ' 


0,31$ 



Gcr- 



J J4 Decimal Tshles ^ Fwttgn Coins] 



German C^inu 



A R» DoQer 9/ fht Empire 
A Gitte fif Norembcrg 



— P 7 I lo>354 



French C^/^j . 



i< pcQier 

4 |4«IC(::=:20S9dz) 

u< CcQim (oB 3 Li|ies> 



o 
o 
o 
o 



o 
o 
I 



o 
6 
6 



I o 



Spanifb Coins. 



Malira<Ses i3.t>D^ 
^ Rial (=»72lMalva4»> 
^ Piece c/ 8 (kUb> Nkr 
^Piece«/8Mexico 
4* Piece q/" 8 Peiii 
APieceo/SSevfflie — 



c 
o 
o 
o 
o 
o 



05 





^ f 



4 
4 
4 
4 



I 



Fortugal C^*«i. 



Ree$, X2,t ff wbich^matt 
Mill Kee (= xooo Reei> 
^TeOooa — 



a Of I 

o d 8 ^ 
o r' 3 



ItalisHi 0/>j^ 



Thf Livre 4^Lcghom 
Crown ou.rn(rt at Florence 
Duoet de Banco, ^Venice 
Tb^ Coumnt t>ii a( 
l)ucat «f Niiples^ 
A St.. Mark — 

;4 Palttrpio ^ki'M -. 



o 
o 

6 
O 

o 
o 

o 



o 

s 

4 
3 
5 



9 

4 

4 
o 



2 xo 

2 6 



0,0003* 
0/50374. 

0,075 
0,325 



0,00 f 04- 

0,02*1 2^ 
0,2281 ^ 
0,225 
0,2208^ 

0,125 



'0,004X^6 

0^354^ 

0,0625- 



o,P?75^ 
0,2625 

0,21 # 

0,13^ 

0,^5 
o,X4i6 



tD^c/w 



n 



^55 



Decimal Tahies of Foreign Long Mea/iires. 



London 

Paris 

Amfterdain 

BriU 

Antwerp 

©ore 

Leyden 

Mechlin — 
Middkburgh 

^ta»bQrjj;h — . 

thttttKtk — 

Cologn »— 
Frankfbid ttd Mcin 

Spaniih — 

Toledo -^ 

Roman <- 

fioflonHi -^ 

Mahtua «-* 

Venice — 

Dantzick -b- 

C^openhagen. ^ 

Prague ^ 

Turin — 

(jretk ^ 



Foot 



1,000 

1/368 
0,942 
i,«03 

i,i«4 



Lyons 

fioik^n 

AmmidaiB 

Antwerp 

Leaden 

Frankford 



«. lyojrt I Hamburg _ 



0,9^8 

0,919 

0,991 
6,920 

o,9«4 

0^954 

0,948' 

ipoi 

0,900 

0,967 
1,^04 

1,569 
1^162 

0,94+ 
0,965 
1,616 

1,831 

1,062 

1,007 



Leipfick 
Lubeck 
NorembHrgh 
&Afia 



Boikilbdta 
Dantxidc 

Florence 

Spanilh 

Lifibon 

Gibraltar 

Toledo 

Na{)les 

Genoa 

Mibn 

Parma 

China 

Cairo 

Tufkifli 

Peilian 



London, TTie PouAd v 
Ayerdupbis y 
Paris — 

Lyons — — 

Bolo^n ^ 

Amfterdani r^ 

Antwerp — 

L^ydea fp- 

Lorain -q^ 

Mechlin — 



It. 

1,00 

0,93 
1,09 

0,89 

0,93 
0,98 

0,96 

0,98 

0,98 



I 



EB 3,976 
2,os6 

• 2,2^ 

2,27} 

— 2,260 

- 1,826 

i,9<i$ 
2,280 

- - f,9c» 

— 2,227 
* t>,954 

- i,&5* 
-^ 2,447 

^ i,9«3 

— &ace 1,91; 

— Palm 0,751 
— Vdrfc 5,001 

- 2,7§o 
- 2,6(1$ 

- Canna 6,880 
*- Paim 0,8^0 

. Calamus 6,544 

^ Cubit 1,866 

-• j,o!i6 

— — 1 ,824 
wp. Pike a,2Do 

— Aralb 3,197 



Londofi, The PoundV 



Averdupois 
Middld>urgh 
Strasjl^urgh 
Bremeil 
Cologn 
Frankford 
Hamboroug|[| 
Leipflck 
Noremburgh 
•X -2 ■ ' 



r 



1.00 



0^8 

o;94 
Gop«n 



1^6 Decimal Talks of the Courfe of Exchange. 



London, the Pound 

Averdupob 
Copenhagen — 
Vienna » 

Caflile - 

Lisbon — 

Gibraltar *« 

Toledo — 

Rome — • 

Bononia — 

Fbreoce — 

Naples ^— 



}• 



/. 

1,00 

94 
0,85 

0,99 
1,00 

1,23 

ii43 



London, the Pound f 
Averdupob \ 

Genoa — 

Mantua — . 

Milan •*- 

Parma — 

Venice — 

Dantsick — 

Prague — 

Cairo — 

Conftantinoplc r- 



/. 

1,00 

1,42 

1,40 

1,06 
1,61 
0,86 



Having pcffenred the Reader with large Tables of the Par 
of E'^cb^^g^ ; I Ihall next exhibit a Tahle of the Courfe ofEx^ 
(bang- in Fenctj and SbiBw^s and Peme^ (into which Fo^ 
9tign Coims are reduced) iti Decimal PartsoS z Pcu^d Sterlhrg. 

♦ 

pecimal Tables of the Courfe of Exchange. 







— TT" 




16 




w 


37 


"7. 

1 

,1541* 




58 


■■ /. 


t 


,001 


,0* 


,241^' 


• 


■ 

T 


,002 




17 


•,0708a 




38 


,158^ - 


— 


59 


»»458< 




1 


,0051 




18 


P75 




3P 


,1625 




60 


,25 




1 


,0016 




19 


,0791)* 




40 


,1* 




61 


,2541 # 






,00^6 




20 


,<^8.? 




4» 


,»7o8a' 




62 


,258a' 


■ 


T 50036" 




21 


,0875 




42 


,175- 




63 


,2625 




I 


,0041^ 




22 


,091*, 




4? 


,^79X0 




64 


,4* ' 




3 


,cc8^ 




23- 


,0958,- 




44 


,18^: 




65 


,27083: 




3 


.0125 




24 


•,i • 




45 


,1875 




66 


.»75 




4 


,0I(^ 




25 


,10411? 




46 


,191* 




67 


,2791* 




5 


,02C8^ 




26 


,108a:. 




47 


.»958ar 




68 


,28a' 




6. 


•,025' 




27 


,1125 




48 


,2 ' 




69 


^2875 




7 


,02pli^ 


- 


28 


,"<*, 




4? 


,2041* ■ 


70 


,291 rf 


f 


8 


>0^ 


J 


2<7 


,I2085- 




50 


,2C8^ 




71 


,2P58j 


« 


P 


>0375 




?o 


.»25 




5» 


,2125 




72 


.3 




10 


,041^ 




n 


,12911? 




52 


,21j* 




73 


,3041/ 




II 


>^45S"^ 




32 


,1? 




53 


•,22083: 




74' 


,3^8^ 




12 


.05 




^3 


.nT-; 




54 


,225 




7^ 


>3»25 


% 


^3 


,0541^ 


« 


^4i 


,i4i<? . 




55 


,22916 


76 


,31/? 




H 


,058^ 




35 


,i4«)8^: 




56 


,2^ -• 


77 


,32-8^ 


V 


ts 


,062S 




36 


,!■! J 


17 


,2?T> I7S ' 


,^2* 



* • - 



Decimal Talks of the Cmrfe ofExchanfjp. 157 



P2,38^ 

P4 .?9»* 

96'-, 4 



97 
98 

99 

fOO 




/f 



T 



.408^ 
>4I25 



1 



5. P. I i'?<^«»- P. 

520 1,6 
I 1,6011 1« 
,608^ 
,6125 
,6l« 
,6208^ 

,6^ 

,6375 
,641* 

,64s8ar 

,65 

,6541.* 
,658^ 

,6625 

,67o8<5r 
>67S 

,68^ 

,6875 
,69 1 J* 

,6958? 

.7 

,7341'^ 
,708,v. 

>7i25 

,71^ "• 
,72c8^ 

.725 
,7291^ 



•' - 



3 
4 

5 
6 

7 
8 

9 

10 

II 

I 

2 

3 

4 

5 
6 

7 
8 

9 

10 

II 

34.0 

I 
2 

4 

' 5 
6 

7 



,625 
,623576 
,621761 
,620155 
,6i855<$ 
,616966 
,61^384 
,61^810 
,612244 
,610687 
'.,609137 

,607594 
j6o6o6o 

,604534 

,603015 I 

,601503 

,6^ 
,598503 

,597314 

.)95')33 
>594359 
,592592 

,59"33 
,589680 

,588235 

,586797 

,585365 
,">83Ci+i 

,582524 

,581113 

,5797 » J 
,5783i3_ 



S.P. 
9 

-.10 

II 

35:0 
I 

2 

3 
4 
5 



t - 



1 



6 

8 

9 

' id' 

II 

360 . 
. I 

2 

3 

4 

5 

6 

7 
8 

9 
10 

II 

37.0 
I 
2 



7^/^zw. P. 

1,7? 

1,7375 

1,741^ 

1,74583^ 

1,75 

i,754»'* 

1,758.? 

1,7625 

1,7^ 

1,7708.^ 

1,775 
fj779ltf 

1,78^ 

1,7875 
1.791^ 
1,79584^ 
1,8 
1,8041 tf 

1,8083: 

1,8125 

1,81* . 

1,8208^ 

1,825 

1,8291^ 

1,8^ 

»,8575 
1,841^ 

>,8458^ 
1.85 



1,8541^ 
1,858^ 
3 i,8j25 



,576925 

,575539 
,574<62 

,572792 

,57142* 
,570071 

,568720 

,567375 
,566057 

,564705 
,563380 
i56'20±) 

,56=747 
,559443 
,558139 
,556844 

,554272 

,552995 
,551724 

,550458 
,549199 

,547945 
,546697 

,54 

,544217 
,542986 

,541760 

,S-4<» 
,559325 

,538116 

,536912 



t ^ 8 The XJfe of Decimals in Exchange, 




4 
6 



ifiS 



p. 

»5357t4 
»5?452> 
►5? 







Big. P . 

,529801 
,528634 

>5 27472 

.526315 



I 



From dicie Sett oFTables of the Par and Courfe of £r- 
HMm^^ <wUch Me flMfe oompleat cfa^ yetfiMmio 

ttre V€9moM tMlts of ArftbmetkkJ the ingeni6us Accx)mp- 
Ufic ndU reidily caR up any jBi// 0/ B^cbange \ or cpntert tht; 
Coinsj it^eij^bts and Meafwts of any otbei Country inco ^/{p^ 
fame of opr 0«w. And qr ctmpaiii^ the Cb»r/i» with the Paf^ 
may ftewhether our Mtfion Gttint or £»/ft by csading to an)r 
1P<freigm PariSy and in mAmu Plopoition. 

^efthn I* Suppofe at Mm^# I woilld eaechalige 175 1 
k2 J; 6 J* fbr thetr /)irr/tri nSf Eane9 at 4 1. ^^ f^ Fiec^^ 
(iosv many muft I have? 

/* i. '. /* ^« A /. 

Firft, 17$— •12^*6 c= 175^2$; and 4^4 ce: |2lA 



Then ,21^) i75»^2$ 
21 17$625 

,iP5r 158,0615 
1365 

X156 

5rr5 




The Anfirer. 



1812 

1755 



•575 



^uefihn 2* The Cb^ryi^ ot Exchmge zt Muirid being now 
41 ^ rf. ^^r PiVr^ of 8 Mexico J whatlNumbet of thofc Pieces 
may I have in exchange for S?^»766 /. P 

Per Table the Firft, 0/ the Cdurfe of Exrbnnge 41 J rf. 

=r ,1744^/- 

I hen ,1744^) 533j766 (=3060 Pieces, the Anfwer. 



'TheUfe cf Decimals in Eocchat^e. 1^9 

Qpefihn 3. A Bill of Excbang(f was accepted ar 14^4$^ 
for the Payment of P93>9ar/« for the laine value 4fli^^ ^ 
£,isi0M in MiJrees ; Exchangp at 5x 4 d. ftr Pua. f^w 
many of thole Milrees was paid at la^ton .^ 

Firft 64 ^/. ( = 5 s. 4 5.) = y2fSL 

Then ,2i^) 9?^)93 ( =3502,25 Miireesy the Anfwer. 




which, IS above 240 /. more ; and coofeqiieiKly there was Tq 
much /o/i* 

Quefihn 4. In 1421 /V^r^ j of 8 /Vr», How many i&^ 
//7J Pomids $terJiftSj Excb^^e at Par .^ 

Multiply -— 14^1 
By the Par = ,2208^* 

pT^ Another Way. 

■ 'ii m ^ $4 

473^^ fFor | =t 0,220*^ 

11368 N 20 = ^i^i<966 

2R420 y 400 = 88,;?3?3) 

2842 J 1000 =5 220,8;j33J 

Anfwtt r. 313,8041/^ V- 1421 rsB 3i3,8b4i# 

Queftion 5. When the Bccbaffg^ (torn Anhjoerp to Lm^ 
don is at i/. 4^, 7^. (ssj^i. yrf.) FUmiJh\ How many 
Pounds £):(§7//Zy at London will tsattance 236 /. ptemrjh ajc 
Antwerp i^ 

^Multiply the Tabular Num-> ^^^^^ ^j 
.^ :. ,. ^ berfbr Bngfifb Voands f ^^^Sji, 8c. 
Rroceeddios, ^ By the given ^fumber — 236 

vThePtod^ is the Aofwer/. 136,4819, g^ 

Ciueftton 6. What Number of Flemijb Pounds will 
it equivaknt to 400!. SturJivg^ Ekcbaug^u i/. 13 s. 6d. 
( =r 33i,, 6td.}/ 

Multiply the Pound Flemijh l>575 

By the Number of Pounds Sterling 400 

The Produft is the Anfwef = 676^000 Pounds SttrL 

tfueftibn 7* A hufck Man fclli 2550 Plemjh EHs of flirf- 
Und to an Bi^^ijh Man, a Spaniard, a Vefee4iAMy an JtaUa% 

and 



1 6o The Ufe of Decimals in Alligation 

and a Pcrittguefe ; who are to have eactf a like QudPitiiy ; 
duere how much in their tnra Country hdenfure P 

Firft-2050 Flemijh Ells arc equal to 1230 EUs Etglijlfy 
equal to 4612,5 Feet, which dinded by 5 quotes 922,5 Jreet 
each. 

Tk*., I.* As^ f ?^75S Yards for the Brifo«. ^ 

t»^^ twLr 79^fi9 Ttetfotth^ Venetian. 1 

TaWeofMeafure} ^^g^^ J>Anf- 

P22,5 Feet are e- •{ ^g^^^ ^^^^^^ ^^^ ^^ f}^^j.^ [ 
qua! to — 1335^^ f^^^i for the Poriuguefe. J 

Queftion 8. What Number of Pounds AverJufois at P7- 
r«;7a will Equiponderate 270 Pounds Averdufois Weight at 
London ? 

Divide 270 by ,8;. the Quotient 525,3 /. is the Anfwer. 

But if 'twas required to know what Number of Pounds 
Averdupois Weight at London would equal any given Number 
at any other place, then you msffi multiply by the Tabtdat 
Number. ' 

Alligation, or Rule of Compofition. 

AlUgat'on (fo called of the Latin Word AlUgo, to hind 
or tie tcg^ib^r ; becaufe the vubar Way is to tie or conneU 
irg/tber the Numbers ccncern'd in the Work,) is a i^«fe for 
icMj>oundwg or Mixing f ever al Ingredients of different Sorts 
together^ in any Manner or Proportion. And is divided into 
Alligation^ Medial and Alternate. 

Alligation Medial is that by which the Mean Rate or 
price of anv Mixture is found when the farticuldr Qua^ti' 
ties J ( and their Prices^ are given ; and it is perform'd by thic 

Rule, 
Multiply eaib Quantity hy its Price ; then fay^ As tbe 
Sum of all the Quantities^ is to the Sum of the faid Pro^ 
duSlsy fo is any Part of the Mixture y to the Mean Price of 
of that Part. 

Queftion I. A Tohacconijl would mix lO Ih. of Tohaceo at 
pd. the Pound with 60 1 h. at 14^. per Ik. with 40/*. at 
18^. per It, and with 1 2 1 /*. at 21. per lb. Quere what a 
Pound of fucb a Mixture is worth ? 

Firft 



^he life ^f Decimals inAltigation. 

Ik Hate. ProduSs* ^ 

. r'20 XP375 produceth — 0,75 
Firft ^^^ X,058^ produceth '— : 9>S 
' y40 Xj075^ produceth ' — 3,0 
C 12,75 X>1 producedi -^ 1,275 

The Sum 132,75 of the Quan. The Sam 8,5 2^ the Prod. 

7*. /. Ih. h c ' 

Then fiy, As 132,75 : 8,525 : : X : 0642 

132,7$) 8*5^50 G0642 = I j: :5.^rf. ^ lb. htSvitt. 



S?i<^ 






•.■4p<5oo '■.■ : • ■ ., 

• •■ • ■ ■ itS<<6 \'' \ • ...'■ 

.- *' Ml r • t\^ :. r 

2450' ■• :• ••. .;:; 

• . - • -»•■'■• 

. • . •> • 

QueftioH 2. h Gpld[mth hath Gold %:'3^,09s^ worth^.4A 
f^ 02. 8 ,T ^2* at 47> 5 ^*J J (H&t M .4/.^ #• S^- 1 ^pd 
9 ox. at 4 1, 1:3 ^* 4^ ouppoie tbffCe all melted down to* 
getber, dj^iv ffhat in Ounce of ^ Mixture would be 
woithi^ ^ 

^ 12 X.4i .produceth ^8 
FirlL i 83 X 4>25 produceth 55,27$. 

C g X 4><^ produceth 42 Z^- 

Tibe Som 32,3 - l%e Sum 93$,275 of the-Frb^. 

Then fiy, As 3^302. : 138,275 txxi oz. : 4,280^. * 

32,3) 1 38,275 (4,^80? = 4 '• 5> 7lt ^' . 
1292 



• • 



907 

6^6 



261$ 
2584 

! • 3I00 

290/ , 



I' 



'••-IF » ' • W 

-'■••» A y .. • ... . , *. 

• » > t - • 



MP3 ^ Anf«r«4l. 5>;7i^. 



'j6i The Ufe »/ Decimals in Alligation. 

AlligntioH AhemaSe is that by which the particular Qucpi^ 
iities ot' every Ingredient >n any Mixture ^le, found; when the 
f articular Rates of every one of the Ingredietits ; and the 
Mifan Rates are given. 

This is (as it were) the Converfe of the formet^ and admits 
of three Cafes. * .. 

\ Cafe 1. The PMicuhr Rates and ^e Mean Rate be- 
ing given, to find tnc Quantity of each Ingredient for the Mix^ 
ture propofed. 

/ Queftion i^ hVintuer would make a Mixture oC Malaga 
91 J s. 6 d. per Gallon ; with Canary 21 6 s. 9 d. per Gallon ; 
Sherry at 5 j. per Gallon, and White Wine at 45. :^d. per 
Gallon : What Quantity of each Soit nlui! he take, that the 
Whole Meafure may be lold for 5 s. id d. per Gallon ? 

Nmte^ In all Queftions of this> Natqre, where two or four 
Things are mixt together, when one haljf of the Prices are 
Greater^ and ^t other half leffer than the AdeanRate^ you 
muft let di greater and leffer Price aHve^, and the fime helom 
the mean trice ; then take the Difference between the mean 
Rate and the pdrticuhr Rates, and place them alfemaiely^ 
end they wiM be the Quantities required. ^ * 

Ratesi ' Differences'. 
/ tf>'y A^rf/zTijil i,58^Gal. of Mahgaw 

s. ^4,25 White fi,e GaL of White. 
Mean Rate = 5,8^ y 5 Sherry \o,9iiS GaJ. ot Sherry. 

' ^ ^iJ^ Canary J oji^ Gal. 'of Canary. 

The Sum oE^ t^oie.Di&ecenees is =;= 5,0 Gallons tbeivhole 
Mixture. « 

. Note^ Tht- Differences are not only the Quantities^ which 
anfWcr^^the<JucftiQn^But any other Number s, in the fame fro* 
purtio/i as they are, wiU antwer the Qucflion as well.. 

All multlpKea ft/ - ^ . — — 3 I 

Produce the Proportionals 4,75 — 5 2,75 2,5 ! 
Thefe multiplied by — — — 4 r 

Produce thefe whole Num-1 - - . j 

%crs in rhe fame i^^^fio; and> ip. arc 11. 10. I 
fo on In infinitum* j ^^ J 

In Cafe one of the Given Rate^^ (when more than two) 

^.-i£?4"h *^*^ ^ ** "** ^cSet} ^^'^ *^ """^ 
Rate* • ^ .- i . Yhea 




The Ufe of Decimals in Alligation. i5} 

Then the mean JRatej partictitar Kafes^ and Differences 
mad iland as in the following Examples. 

P. R. Drfftrences. P. R. Differ. 

ri8 4 + 2^ C14 4 7 

Ikkan Rate 2D ^ 22 2 ^Mean Rate 20< 184 r 

^24 64-2 i 

R. Difference's. R* Differences* 

C17 2 •> -'48 30+154-4;) 

The Method is the fame for any other given Rates, or 
Prices. 

Caje 2- When the Particular Urates j the Mean ^ate, and 
the Quantity of one Ingtedient is given ; co &C[^ the QMontity 
of all the reft of the Ingredients. 

This is caird AVigafion Parti alj becaufe a Part of the 
Mix'd ingredients only are known. 

In this Cafe you mufl fet down the mean Rate^ the parti'^ 
a^ar Hates, and their Differences juft as before ; then fiy, 

Rule. 

jii the Difference^oppofte to the known QuoHtiUfy it to 
ihe known given Quantity ; fo is auy other Difference , to 
ibe Quantity of its oppopte Name. 

Quefiion 2. How much MaUga at 7 s. 6d.; Sherry at 
5 s. ; tfbite Wine at 4 i. 9 d. the Gallon, isoft be mixt with 
eighteen Gallons of Canary at p x. p d. fftr GaUoi;!, chat the 
Whole may bt fold for 5 x. 10 rf. per Qallon ? 

Rates* \ 

Mean Rate 5,% ^ l^^ f^I^^fjl's, <m^^^ces. 

C 4,25 White }0y9if$ L 

r Oj8^ to the Gallons of Mahga. . 
Then, As 1,58^ : 18 : : 5 i,i^ to the QJIons oTSh^^y. 

i 0,9 ii^ to die Gallons of W. Winep 

I leav(( the Work to exerdie the Learner. 

Y 2 Cafi 



1 6^ The Ufe ef Decimals /# 4ttigatioM. 

Cafe 3. The fgfficular RaUSy tltyt mfan EaUy z^cA <he 
Sum of all the v^uantkies of the Ingredients given; h^K^ 
to find the panicular Quantities dt the Mixture. 

Thi« iscUl'd Alligation Totd y bec^uie. tbe whole Quantity 
of the Mixture is given. 

- It is thus periorin'd ; &t dowh the pieau Rqte^ the partkU'* 
hr S^tesi and find their Difffrence$ as before. Then fay, 

Rule. As the Sum of all the Differences^ is to the Sum 
of alltbeQUMtitiesi fo is each f^a^Hular Differ ejtce^ to its 
particular Quantity. 




Galons, anc! to be fold at 5 j. 10 d. fer Gallon ^ Quere t£^ 
Quantity of each Sort fer the Mixture ? 

Mean Rate 5,8? ^ l^^ ^^ [ \f^^ ^ Dific«nces. 

4,25 Whiu J 0,91^ 3 

5 s=s The Sum. 

Gone,,. 5;^'?;?*?^??r?^* 

Note^ The Work of theft, and fuch like Proportions, 
may be very louch fltoiteoed, and eagerly peiform*d by a c$i^ 
mon Multiflicator a| in FeJlovj/hif. 

Now becaufe Alligation alternate aniwers not Quetlions 
compleatly, that is, does not give all the Anfwers fuch Quefti- 
ons are capable of; and fo perhaps not always thofe which beft 
fuit the occafion ; I Ihall Ihew ffrontMr. Ward) how this Im^ 
perfedVioii of common Aritbmeiick i% fiipplied by Alg^'hray and 
all the poffikle Anlwers' Co any C^ieitioqs may be dearly and 
cafily dilcbvei'd. : . ; . .. '-' 

Quefiion 4. A Tohacconift hath three Sorts of Tobacco, 
viz, op.e of 2 J. 8 d[per Pound ; another of 20 J. fer Pound j 
a third fort of 1 6^. per Pound fXjf. thefe he would make a 
Mixture to€ontiiiiv5^Pound-that may be fold for 22 d, fer 
JPound 'j How much of each Sort may he take ? . 






The Ufe of DecimaU in AUigation. i6y 

r ii = the Quant it V of that worth is. 8 ^/. se^ 32 /. 
Let < ^ rs that of 20 d. fer Pouna. 




that of i6</. fer Pound. 

92^-|-20^+l6jf:fI23;5 

20^4-i63f =1232—324 

4^—336 — 16^ 
^= 84 — 44 



ffcncc'tis^deiit 
from the 7th Step 
that the Quantity j^ 
niSed by a mvA oe 
lets than 21) and (by 
the 8di) Step greater 
than pf . That is a 
maybe any Number 
between 21 andp-p 

If there be more than three Quantities concerned in the 
Queflion, the Work will be more ' large ; becaufe the IS* 
^its of all the Quantities above two, ^uft be found. 

Quefiion 5. Suppofe it were leqkired to mix four Sorts of 
Jflne together ; v«. one worth 71. 4 ^. fer Gallon ; a ic^ 
cond worth 4 5. 7 J. a third worth 3 s. 8 tf. and a fourth 
wonh 2 J. 9d. Per Gallon. How much of each Son muft be 
taken to make a Mixture of 63 Gallons^ to be Ibid for 5 ^. 6 ^* 
fer Gallon, without Lofe ? 



s. d. 



1= that Quantity worth 
~ that worth — 
Firft let ^ 7 =: that worth — 

zz that worth — 
the mean Rate — 




7 

4 

3 
2 

5 



4 

7 
8 

P 
6 



d. 

88 

55 

44 

31 

66 



Thcnl 
And 
1—4 
2— 8&1 

3X33 

4-5 

3X55 
8-4 

Suppd&l 

/'Vr7*l 



a Jl^e-^j'-^tt t= 63 

88/1 + 55^ 4- 44? + 33« = 4^58- 

^-+*J^-4-» = ^3 — ^ 
55^ -f 44? 3^« - 4158 — f 8tf 
33^ + 3V + 3iy = 207P - 33^ 

22«^+llJ^ = 207P— - 55/I 

2^ -i- 7 = i8p — ^a 

55^ + 55 J^ + 55« = 3465 - 55^ 
117 j- 22»=?33ii— .P3 

jr -f- 2»= 3^1— 63 



I. 

2. 

3- 
4- 
5- 

7- 
8. 

P- 

10; I 

1 1. 1 tf =» 22. Then ^a j ic, and 34^= 65 

.12— a# 



1 1, i ii S3 22. ± ncn ^n - I IC) 

i2.| 2<+jp=i85> — 54S3 7J) 



J 66 The life of Decimals in Vofttion. 

12— 2^1 i3ljr = 7^ — 2^ 

Pet ?d.|i4U+jf \^u 5*^5 —if = 41 

From the leventh and tenth Steps it appears, that the Quaiw 
tity denoted by a^ muft be lefs than 37^, and ^eat^r than 
21 Colons ; whence 16 anfwer flow from tlie Limits o£ a only. 
Tlien if 4 be put =: 22, by the thirte^thimdfixteenth Steps 
it appears ^ £= 3p. jp = i, and u s; !• And t^us proceed- 
ing with each fingle value of a^ abpve 120 Anfwers may be 
found to this (Xieftion in whole Numbers ; in Fra£tions. in* 
fitute. 

Pofition, or Rule of Falfc^. 

This Rule of Pofnton^ or rather Suffofition^ i$ fo calFd, 
becaufe we fupfop or inake a PoftUon of fome uncertain 
Numbers, in order that by ueafoning from them we may gain 
jh^true NunAet fought ; and^ becaufe thofe Foftiipns are al- 
together at random or adjoenfure^ the RuU is aOb call'd Fdje. 

The Uie of tiiis RuU^ before the common Knowledge of 
A'gshra^ was much more confiderable than fince ; becaufe that 
Arc fupplies Theorems for refolving all kind of Queflions in 
this Ruje in a better and ^ore x:urious a manner than here ; 
Yea fome of the iefi Pieces of AritbmetH have intirely difc 
carded it, and others poft-poqe it, as ohfilefe and of littlp ufe, 
fince A^gehra. 

Queflions in this are moffly perform'd by one or two Sup* 
fojitions; if by* one, the Rule is faid to be of Single Poji* 
tion ; if two Suppofitions are neceiEary, '^s called potable 
Pofiiion. 

Single Pofition, 

fiueftion i. Three Merchants A^ B, T trade in Company, 
and gs^n 100/. of which A had a certain Parr, p had twi/ce 
as muth, ai)4 C had thrice as much as £; How ipuch had 
each Man ? 

Suppofe A had 4 /. then p muft have 8 /. and C would 
hay^ 24 A which together make 36 /. bit ihou'd liavip -be/ea 
an loo/. 

' Ther^ 



7he life ofT)ecimah in Vofition. i6j 



Therefore Reafon by Proportion I. 

Thus, As ^6 : lOO : : -^ 8 : 22>2r =r B 

C 



r 4 : 11^ r= 

: : < 8 : 22,ar =: 

624 : 66^ s=s 



Their leveral Parts added make — 100 for PiooH 

Queftifnt 2, A Schoolmafter being asked how many Scbo^ 
lays he had ; an(wer*d, if I had as many, and 4 as many, and 
4 as many, I fliould have pp. How many had he ? 

Suppofe he had 40; Thcn40-|-40 + -20 + iamio^ 
but ic Ihould have been but 99. Therefore iay . c 

As no :. 40 : : pp : 36 Scholars, the Anfwer. 

Queftiom ?• Three Men A^ B, C buy aShip^for jioi. 
15 J. of which A paid aa unknown Sum ; B paid 2\ as much : 
andC3| as much : How much did eadi Man pay? 

Suppofe A paid 48 h then B paid 48 X 2,S = 120 L and 
C muft pay 48 X 393: ^^ 160U Buc 48 4* 120 -j« i6o 
= 328 inftead of 310,75/. 

Say therefore,; As 328 : 48 :: ?io,75 : 45,4756, 8r. 

Then >< paid — — 45,4756 

B paid (454756 X 2,5 — ) 1 1 3,68p 
C paid (45,4756 X 3i? = ) '5^5853. -* 

Proof is the Sum — 310,75 

Double Pofitioir. 

In the Double H^e^ two Suppofitions are ufed^ because, here 
the Numbers cannot be parte(Uo find the Anfwer by Propor- 
tion as before, 

There&re when we mad^ two S^pfpfitions^ andintfs.in 
both, obfervc the Nature of the ErrourSy whether they., be 
Qr eater or tejfer than the Number fropofed \ and according- 
ly mark them with the Signs More ot. Lefsj viz, . -(-j — ; 
and place them prccifely ag^inft their prober Sup^opthns } 
then obfcrve the gcneraf - "^ - 

Rde^ 



i69 The Ufe of ^Deeimah in Fafition. 

Rule, 

Viz. Ai the Difference of the Ehottrs if alike ^ (ot their 
tttm if unlike) is t6\ the Difference of the Suppojitions ; fo 
is etibtft of the Bfroursjto a fomth Number. 

The/tfwrf fr Numbcx addtx>j at fMbfiraSl froipi llie Sapgo* 
flian oppofice cait ; 9nd yoa have i\k Numbtt fougbr. 

Quefiion i. Admit three Merchants build a Ship which 
09fti9&>:Soands^ ^paqf« a ccftain Part anknewn ; JBpaid 
2i as much, wanting 15,5 & and C paid as much as both 4l 
and £, and 75,25 A over ; How much did each Man pay ? 

Firft, Suppde:;^pBdd aoo /. their B moft have paid 484^5 1. 
and C paid 759^75 /•. But thofe three Sums, viz. 200 4* 
i84s5 4^7^9975 » 1 444^25/. ^^^4iicli ii more than i^6aby 
84,2^/. Wheieforetheiirft Errour is 1- 84,25/; 

Secondly, Suppoft A paid 180 /. then iB^paid 434,5 /. and 
C paid 689,75/. B"^ ^8o4-434j5 + 689,75 = 1 304,25 /•. 
vrhkh' is too little by^ 95975^- therc£)re the Suppolitions and 
their Errours will fknd thus,* 

. |Thc Pirft Suppolftion 2QO5 + 84,i5.Errour. 
The Stff o«a Suppofition 180,-55,75 Errour. 

Tht Difference of Svppo&* =z ao i40=sSuoiofErroursi 
Then Jby the General Rule, fay. 

As, f 140 : 20 : : 55,75 : 7,964 8^ > 
Or, I 140 :. 29,; ; 34,^5 : 12,035 8c. J 

Then B muft have paid ~ 454)410 =s ilTs Part. 
Ante mift hav« paid? -p* 717^624 s= C'sPait. 

The Siim of which is!. . = 1560 ""^fot. Proof. 

Not^r When the JS^rof^ri ace ^^zr^ and have wdiie Signs; 
lr^(^ f ir^ Sum of the Suppofiions, is the Number ibi^ht. 

Ex- 



The life df DeeimaU in EsitraBioxu 1 69 
Extraftiori cf koots. ' 

• • • • • 

The extream Ufk ef DetimaU in all kinds of ExtraBh 
MS is fuffictendy Iniown to all verftd in Aritbmfticdl Kntno* 
ledge ; and its abfoluU Neeeffitf in fome Farts i of Arithme* 
ticky and its Excellency beyona eren Logarithms tfacmfelves 
id others, is alio as wdl known. 

I would know what is the SpMT^ i?oof of 161,29? 

Thus 1 6 i,a9 (i 2>7 = the Anfwer« 

1 




• • • • 



What is die Sguare Hoot of J477? i. • 

In liicb Cafes as this, yoa mim add txvice as many Cyphers 
to the given Numbet, as yoa deSgp to Iiave Dteimal rbteet 
in die J(pol of the above Nuinbec to three Places of De; 
einult. 



Then ^477,000000 (58^1 the Root maptit'd. 

' , 108) 5>77 

1168) 10300 
?344 



417^) "9^600 
P4144 



117761) • 145600 

117761 



2783? 



Rcouiied tbe Sm^ ^a0! oE Ai 1i» ■($ I^KCs of Dedmals. 

X .' ... 






281) ••400 



2824 ) 1190Q 



iV^ 



2828) **6q4 HaKongte^ cot jPbcesofthe 

565 6, I j^rork by- & cfrntn^ed way 

""^ cf Dhifiotfi gnd ffin the other 

Is 3 » tridr f« i jvought at /i»j^« 

8 

"fVj^ » % ^*^o«f of ^8? f 

,4489 G67 die Rjoot requitU 

88?' 



• • « 



What is the Spi^r^ iPoof of ,00576^ 



• • • 



00576 (^24 (be RoQt.fiu#L. 
004 



^044) •• 176 
176 



• • K 



WlMt 



\ 



Wm:m.^ S^ate^t^^a&io^^l' 



• • • • , ajAJ. 



V > • ^ ^ 



,00005625 <,0075 =: the flooc ibught. 

145) i^ 

7^ 



• • • 



To ExmSt the Roots of Single 

Wh9t is the Square Roet of M" er Ihdfj infinitely repeated ? 






♦ • • » 



>rii|iiii 8r. ^ 

9 (,3^335 6^. 0Ti^ Square Root. 

189 



663) • 22H 
1989 



66&J,) 2221 1 C ^(k dbWnward 

22221 1- C 4^ infinitum* 
What is the 5^^^ i?o0/ of^ ^4,4 £^. ? 



• • • 



44*4444 8f« Ci^ ^' ^ Root* 
36 



38844 C-^ Iftftifum. 

If the Root does not r^/Mf in the frjl Figfue, 'twill be 
vera uncertain when it wiU repeat. 

MllP, Pnlrthefe fro* D/gif j l and 4 (of all the Mw*-) 
when infinitely repeated ; have theit JRootSf pure pnge 
Hefetendt. 

Z 2 70 



1 7* The Ufe cf Decimals in Extra&iqns, 



\ •. , , • . f. 



7a Eztraft the Square -Root of Compound 

Repetends. 

What is tbe Square Root of I9iifi^ } 

• I • ' ■ 

Thus, J9^fiif (I4P805686)S the Root. 
1 



24) ^96 

. 96 -> % 



^ ' t . » • 



2805) --SsdS 

S4O9 ; •^ 

280605) • 1595685 

I4P3O25 

28061 I) 192660 

168366 
■ > ■ 

•24294 
22448 



/ "^ 



A ' 



t^ 



1846 ^ f "\ 

1 40 



•V ^ 



* « 



•25 

22 

In this Maimer the sltilful Arfift may procec4 and gain the 
Hoof of any Refetend to what Number of Places he pleafcth. 
I omit the ExiraBion of the Cube Root here ; becaufe I IhaQ 
have occafioA to Ihew the Method and Rationale of that, and 
of the Square Roof, both, w|ien I come to fhtvf the Ufe of 
Decimals in Algebra. 






V Vv 



« I 



m 



TV U/e 9/ ti.i C l;M il. S !»• the !BkJi4'^ 
ne/s of Intereft, both $kiglf ^ Cvmpound 3 
65f Annuities, ;PcnfiQnsi;<7t. thorns' 
Im in ^refent Worth: ^and ht Amurs 4 
Of Rebate or Difcount v Of Free-hold' 
or Real Eftate?. i ^ 



•/ 



IATriffiC E ^Th ^p^aBSuJh nf Money paid lot ciie 
Ufc of any. greater SW/f «5()K^ng to aoy iC^it^ agrefd 
oa« 99 5 £ Per too/. &c»'{bc iTcar j and icis€icbn»' 
SfmpU ot Com founds . , .- . . . r . » 

^ pimple Inf ef eft Is ."that which a^fts phlj[ from ihtPrincU, 
fal or Stoi« ^ AfWjf 'knt ; and ^f]Ei Inierefi^ and Pri/ffifai- 
are alwayiB the fiiuc as ir firft, - » * » 

Compound Iniereft is that which arileth from the t^rinci* 
pal ana its Simple Inter^ (wh^n^^^^i^-and/or^or^) reckoned 
togjethtr ^s ^ Nevi Sumy fo that both JPri^ifal and Infertft 
here arc always ir.creafing;*; /* * ' 

AmuHieSj Penfions\ Salaries^ ice are Rents, Profits^ and 
Payments made Yearly^ or Bo/^ Ktfr/jf, .&c. and they are 
(aid to \s%mAtre$tr$i, when they sffre.due and unpaid for any 
Number of Paiymjenys. ' ^ : : - 

X/kateogc Difiottij^ i& an Afateme/tf of P^t of a £umiof 
M^jtey daeYometinie hence, in O)i)i)deration of prompt or 
prefeot Payment of the Kemaioder ; and this is dontf ac any 
Race of Incered 

In exempUfying.the wonderful Ufe of Decimals in the Af" 
fair of Intereftj Ssz. I n^ only fhew the Reader the Solu- 
tiai|s of thole admirable Theorems in Numbers ac large, which 
Mr. Ward (in his Matbemaiicians Guide) has with^^^^^f In^ 
vention contriv'd fron> die following . Lataj ^ Method of 
Realbning from thenc^. 



In 



x 



if4 IbeVfe of DeeimtU in litttnfl, 

5 R=Txtaa<lLintiaittrCnl.fttjliiiam. 
&. ^ r — Thelte of the /VwtMUt hnsdl., 

HI jvtiitltnffy, oCK ait' nniflfi DiccfelE 
r -: Tiiift-omw**iii4K At tengfliipuS ■ 



fj j< B Tbe ^tfHWrt it the fjfinKifji ind its fatetefl, 
V y — i == * l^'way of SifffiiiAivL iq ^laht* 

y — ^4-«=« BySi(*^i«tf/Malfo,inpieCwodll. 
Tl%>itfielrr nirrliae^, b't 



bwut^^rojonioi 

1 £y i(' is tfi^ 



^tlie '^^ffff o^^&e Aflfe of^ /iierS' ^ffi^i 6y i 
found (foi 'tis otily the Intcwft (f it &f>y«4i4 




tf..,< J t* *• ?• 4^ ^ * HKeifll^4rt. 
*'"■ 1 A 2ir. ^. ^. j* if. AJIamJt. 
Hencs 'tis cv^flent M'Simfi ^fettJF A tf. issSrfejr' 
of PWW-iil v#/WM«frrf ^'■(g^fjjft^ittcittfln^. 

WheKiii< T = The MimiiT of aU tIK I'eiilis. 
< re = The M 3>n» of the &ri«. 

Then 



Then, As I .'TJl:^ F : WPf islhtcrtftofp. But the 
Whence tJm Genera Tbearemy 

jtmuikiesy Sinple litter e|. 

Here 17= Ae Fw/jf RehLjii ^ ^^9Mi9J^Tif«fi^ 
Then 2U = the JRenf^ and AC/ =s Ae Interefi for the lecood 
Year 5 and thus the following Progreffions for fi^l Yeara. 

. • • • ■ 

Thus < 17. 2t/. al7. 4K «7. ®r. The Rent. 



Hence 'tis pUin, t|iat*i;4-?^i;4rj}»t^4-i^-Hl'5=4<. 
J^je SWf of sW tbe JUfrti aqd f^/kTfit^j fa^ j6rlor« 
five Years. Fioin whence it tollows 

That KV^2RU-\-iRV-\-^VjsiA -TK For here r=5. 

Eiifiip <a» *» xl'. Then jfcfi «4^ i-|r+R3= i^Iii 
Then hg SuhJ^Hfitio^j m f^i'^ii'iJ^^^^^^ 
3%ni «4*+-3»f4^ ^ *Jw -Ae J8r^ and hft Tetau 

of Ae gregrisfSoitv* i4-^^^:=iT, '■ Thctci&K ?^^ x *" 
sp^iwj .of aU .tibe Xprpw. 
Now21=^r=f !Unce2I£fJC£-z. Con- 

2 A - it * 

fequently II£:zii* = ^ ^^^ The General Theorem 

for Annuities jp Jtmem$. 

Bpt b5Ci«fe i' — thp i'rW^fur/ jj^iirf*, is n^ .in the /^ 
General Tb^.orm^ That JV^l ffifKi^ P9 ^miVHii t^tu^ 
tiioreco; Whei;^fQie f ^A^ d^r^ mi^iS )^ pppcuv'd : Noftrbe* 
ckufe A denotes the fame Thing, viz, the Amount ^ in -t^p 

iift laJifieitfralTbmms ; ^d twfpfe 4W| #w (ymffties, 

e^afto one and tte fme thmg^ H^e fqi^ fo 9¥P .(^ott^er;. 
tipA F TH+ P'=z A in the firft General Theorem j And 

Ticpe- 



TbMcfo* PTR 4-Pg ?T^^ 



PmchdJingeS Anmtities. ^i 

. - • f • 
(Mttftmnd Intereft. 

The ProP&rtiom for €ii£«g K^ die AdOo <^ the J?^^ of 
Comp&Mfuf Ifitereft^ (whidi.is only the AmauntJplE i A and its 
'Intrteft %» one Ytari) is Thk 

. .. . ■ • 

As J 100 2 105 :: I : 1,05 =5 B at 5 ^^^ C^«/. 
* i ICO : 166 : : I : 1,66 = if at 6 />^ Centy &c. 

. « • • • 

Bat asMTf pMiK^ ilto thi 4moiM of o«r Pwnd^zt me 
Years Enii fo is that Atnpunty to the Amount done Pound 
nitvioYeaf^s End; ahd'fo on contiiualiy.' 
^ Thatisy 1 : il::«'.JRS;:RR:B!RiJ;:Rj :R4 -R^-: 

ii» : : &c. 

rp, f I. 2» ?• 4» 5» = Years. 

^^ l^R. RK Rt. jR*. Ri. = The ^mtoff of |/. at 
any I{ate. 

Hence 'tis evident the i^moir^ proceeds in^a Creometrieal 
Proportion^ wherein tbeTiMe (- t>, <x Numbeir oiYearsy 
is ahvays equal to, 6t the fame with, the Index of the Power 
of the 2^^. and bigbefi'Rm of die Seriesi viz. Rf, or fin 

But^ as one Pound : is to the Amount of one Pound ^r 
any given Time : : fo is any propofed principal or Sum ; to 
iu Amount to€ tilt fame if me^ . . 

That IS, As 1 : fi« : : /* : PRt. 

BixcPRtsssA The G^ofr^/ Theorenu 

Annuities. Compound Imereft. 

g. r R = 0»r P(HWPi/ and its Interefikt one Year^ as before. 
a 1 1/ = The firft Year'i Rent without Intercft. 

Then Rt/s= The Amount of AefirfiYcar*s Rent, and its 
Intereft. 

And hence is fbrin*d the following frogreffon of Amtnmts 
in continued Geometrical Proportion. 

ThiM J ^* ^* ^- 4-^ 5- ^^ Th« Years. 
^*"* \ t/+l/if+Wi»+l7ift+t/R4, &c The Amounts. 

Hence 



Thf life af Decimals -f> Interefli -^ 1 77 

Hence U -f t/i? + C/R» + i/«9 rK/5.4 i= -rf TlicUmmf 
of any Yearly Rent or Annuity forborii five Yeai^. ' ' 
Now the laft Term in the ab^ve Series is' C/R* ;;=: t/^t- 5^ 
There foje A-^VR^ - » = The Sr//» bf all the Antectfdenfs. 
And -^ — C^ssThc Sum of all the Confe^uants in the 

So that it will he, t/:l?t/:: ^~t/i?^-«- :v^-t/. 

Therefore .4t/ — UU= RUA ~ I7(7i?t, DivMe all by U. 

Then A-^U—RA^VRi. The Gf »^r^/ Thiforem. 

Foe iIms p'^pf** Wiiftir^ :w6 mvtft 'ii«)oe«fl as mSimfltf In- 
Urefi Hxi rhjs Gafe,^ tqrgaii) an Equaticn. ovgeveral Theonumi 
wherein Ihall be /^. . » 

The Theorem for Inter eft is PR^ •=. 4. 

Ax)S^\ci the^ laftTheoritB - ^ "^ 3 ^^ . - 
Confequently, f ^^ ^=5 — ^^ ' The General Theorem^ 



> ' 



Free-hold Eftates, Comjmnd -Btterefl 

Free*bold or i?^^ Pftates are fa[)p^fe'd 'tQ be purchaftj 
for ever. And the Co-hftptitinton of the Valu^ of firch EJfates 
is grounded on a S^'f/V j of Geometrical Froportionals decrea- 
fing ^ Infinitum. - 

Let /^, Ia R, denote the facpeas bftore 5 then the Series will 

^"^ T' T-' W K^' i ^'= ^^ ^^- ^* '^^,'"'^ '^^^*« ^ 

= o. Then will > — = Sktot of all the Antecedents j and 

t/ * * ' - 

/^ — ~w- — S«//w of all the Confequents. 

Therefore, as -^ :—::/':/' — r which gives /» r 
■^ f/ =T= /^. The ^:neral Theorrm. 

Theorems Rejohing all Queflion^ comerning Simjvld : 

Inter^ft, 

Given P, iR, T; To find AP 
Theorem i* TRF+ P zr A. 
Given T, R, A; To find P .^ 

^ A a Given 



17^ TUte Vfe df Decimals in httetefi, 
Giwh u/, i», f } To ftid R > 

theorem 3. "Y^YW ~^' 

* 

Given f , *, y< J To find Tf 
Theorem 4. •\-57r = -T- 

teats, oneQmrfar, twoAftvflf/^ and dj^tem D^x^ at^^ ^ 
ftr Cent, f^er Am. 

CF = 255,5 7 
Here i$ given < iP = Qp6 f Tofifldiii^rMt^ 

2r=:?,4^5P8S ' . 

Multiply — 1546598 = r ^ 

ProdoQ — o,207P589 = r^ 
Mult isr bveifioo 5)652 s: f* 

4»59>75 
I03P7P4 

■« 24775 
10397 



ThcPtodua -.,53,34142 = 7^^ 
Add — -t-256,5 SS.P 

Anfwcr — £ 309,84142 = -4 = 309/. l6f xoA 

Quejiioft 2. What Principal or c^iw 0/ AlwfJ put to 
Interclt, will raife z Stock of (cir be wHb) 405 T 6 j. lit 
five yz^/iri, and' eight Mqntbs, at the Rate of 5 /. ftr Cent, 
fer Ann. .^ 

r^= 405,3001*7 

Here is givim <R^ 0,05 >Tofin4 P>f«' TA^* 2. 

/r = 5,613698 ^ ' 



Midci* 



tedply 5,613698 = r 

^ — , o£5= ft 

i>Kxh^ oi^6349 ssTR 

Add Unity i,. 



i'fi+ 1= 1,2806849) 4D5,300i#(3i647X44.fc=*'? 

•!••: 38420547 



'»' T 



'2109469 
1280684^ 

• 82678$ 

Hence the Princif^ j6^\o 

in common Com, i« »<,« 

216/. 9*. 5«/. fot th* li!!L 

JMTwefc > 9(4$ 

$96* 



•■■^ 



•284 
128 

5i 

5 
5 



koie. By thU TKeotem 'tis too find what frefe«i Moiuj, 
ox prom ft Paknufit, wUl fitisfy » D'^ ^^ any Tune her*^ 
after, Ahatingot JUfcotintingai mi Rate fer Ce«t. 

Q»elim 2. At what Rate of Inttrefi fer Cent. wiD 
36X mott^ to 36/. iSMx/iii in fix AtortA*, three 
Weeis, and tbiee /)tfjii ,* 

iiere is given 5 T a ,^6 STo fid «, j>«r W/. 3. 



/ 



a 3 



Muld; 



1 8o • The tXfe of Dedmah in tatere/F, 

Mulriply ,5j)«rs=F Frapi 56,<?4)J8- ■=? i< 

By — 36 = F Subft. 36, —P 

1578 ,, ill-. 

Thcj?, As 1 : 0^5 : :. 100 : 5 /. Aiiftver 5 ^^r Ont* 

Quejiion 4. Intvliat Z>'/w^ will 200 /. i ^. 8^» amount to 
250/. at 4/. 1 5-. .pfrCeTTt. Inter efi ? 



Here 



is given 4^ -=:. 0,045 VTo find F, /►^r r/r.-^i. 4. 



» ' ^ 



Molt. 200,c8^ i= P From 250,000 ~ A 
By — 0,045=^ Subft. 200,08^ = /' 

iooo4i# • 9,00375) 49>9i^66(5,54399-.r 
800^333 . 4501875 

m ' <■ ■ ■ I ..I iiiM 

Prod. 9,00 375 '« Pie 489791 

45c I 87 
The Time thch* Is — TTZT" 

7 months, and 3 3592 

i)/{yfi. 2731 

8io 

81 
.81 



. U » 



I < 



• t »■* 



'» 



Theorems refohing, all Queftfons maming Annuities^ 
"^ .P^nfionf, 0^c, in Arrears, Siniple InterclW 



y " 



Given I/, Tj i^.; to find ''yi ? 

Theorem. ^.^HLLzlII/r + tVzz:A. 



Given 



1 



\ ■• 



TheUfeof Dedmah in Intenfi. \ 8 1 

■ Qsf!taA,r,R% To fir4 V? 
Theorem 2. { jTJg JrA^ ^t ~ ^' , 

' CWen A,T,U; To find R > 
„, , J2A-^2TU „ • 

GiTeo U, R,^; To find.'r/ 

to 

" Queftion i. If 2«;o7. Yearly Rent (or Annuity^ Sec) be 
forlorn or unpaid 7 R^ri ; what will it am&uat xa in that 
2?i»^, at the ^4f^ of 6 /• per Cent, per Annum ? 

ft'- 250 J 
Here IS given <T — 7 . ^To find ^, perTbeorem i. 

Multiply -^ 250 -3 1/ 
By — 7 = r , 

Produa- — 175a rxTU 
By 7^T. 

. 12250 ~m/. 

Subflraa 1750 z= TV. 

Remains 10500 ^=sTTU^ TV. 
Halve 5250 = TTV— TV ^ 2 

Multiply by o,od-= R 

Add 17S0> = TV 

Sum 2065/. =^ ThfiAnfwer. 

If the Payment of the aforefiid Anmity had bean made 
half Yearly, then wogld 17= 125 =252, and 7= 14 = 

tfkmher of Payments ; and /f = 0,03 = 2£^ , j„^ ^fj.. 
ing as/>.»r r/&.'<»r«» j /T will be found = 2091 1. 5 *. which 



w 

iSa Tbe Vfe &f Deeimah /» merefii 

b more than the Yearly Pa|mcne ^ 76h %i. Hence tfi^ 
iffineriiait Pajmentj fine mere advantagicus. 

dueftiin 2. WHtt AnnmMy or Tearlj Paymenfj being 
nmfaulS ^ lears^ wiU w& % ofori of ^72 /. I2 j. 8 ^. at 5 
fcT CcMty fit Awtmn^ 

Here is ^ven s T =5 8»s ^ To find U^ fer Ti^or. n 

C^»o,05 3 ♦ - 

Wukipljr — 8^5 ?=r . 
By - ,05i=ie 



AfpilT. • ~ 



t,'=r 



212$ 
3400 



"^i^ 



Prodoa — 3j$iz5 = ff^ 

Subftiaft — 04250 = r^ 

Remains — 3,1875 asZTi^^Tit' 

Add 8,5 X 2 s=s iy,oooo ss 2r j^ ^_yr 

rat— JiR+ir-:=20,l8y5J 1145,2^669 &c. 522>f (56J73148 

•I558PI6 

121 I 250 

• 147666 
141312 

The Annuity there* "^^54 

/. 6055 

/ore 1156,73148 = $6 i 7--S- 

to 

i<6 



•X 



8?'A 



The.U/ttf Decimahin Ini^efi. 1 1| 

Queftion 3. At what Rateof fyUreft^ ferCent. per dm. 
ivill 40 /. 13 J* 4 d^ Yiforlj Kent^ aptouaj^ c6 450/4 13 f. 4^, 
inp Years. f ' 

C^fD45o^; 

Here is givien <r = 9 >To find ^, /^ T6forcm£ 

iy =± 40^ ^ 

Multipli ^40^2=1/ 
By - /P=gy 

Produa. — 366 ~s=s ri/ 
^fultiply apin Igr $ = iT 

From that Prod. ^294 ^mTTU 
Subftraa .. --r3o6 »r£/ 

Remains ^ ^^ b^Tftr^TU ThcDivifei. 
Ihicn from 90i>? = ^^ 

Subftraa ^ ^32 z=itttU 

flcmains — t6p,j db i>l — 2W TbeBitideiia. 
\ Th6n ^a8) t6jJ^3 (0,05784 = iE 

J4640 

2049 

•244 

< ll I ■ 

• » 

•II 
fl 



« • 



I, 2« /• /. /• X. i£ 

Therefore as i : 0,0578^ : : lOO : 5,784 = 5:15:84 
th^ Rate per Qnt. repaired 

Queftum 4. In What Thhe will 2«;o /. YearU Rent^ raift 
a Stock of 2065/. allowing 6 fer Cent. &c. for the For- 
bearance of the Payments as tbey become due ? 

\V == 250 7 
Here is givch < yl «= 2065 > To find T^ Per Theftrem 4J 

Firft 



^ 



i 94 . The life of T)ecmah fu Interefi^ 
Firfl ' — 4130 ' =2^ 

Then aultiply i— 250 rr C/ 

By m^ 0fi6 =R ' \ 

Prodiiflis — 15 !=sVR 

TBcn -^ 53,^ ^^ ^^ ' ^ '■' 

And 33,j — I := 32,^ = ^ --r i =^, 

And 32^ -r 2 = i6,if = J ;v 

The Square of it is 261^361- =3 J *x := — 

To which add a75>? =U 

Thejumis ^ ,;3g^g4 = j ^-f j 

The Squaic Root ^ _ .Uji 'jcx 

of which b J" 23,1^ =Y^^^ — 

'From whidi take i64i^ == l«r 

There remains 7 = T The Time required, 

viz. 7 Years. 

The Divifions and ExtraSion^ at large I have omitted f^r 
the Learners Elxercife ; but I have reprefented all the Numbers 
in One ; which Mr. Ward'% Method could not do as being de»f 
ficient in the DoBrine of Calculatvig Number s^ as may be 
ob(erv*d in his Work of this and other Queflions of Intereft* 

N. B* In all QueAions about Yearly^ or ftated Rents and 

PaymentSy the InXereft is reckoned for every Pny- 
ment after it beoomes due^ thro' the whole Time 
. of Forhearance^ 

Theorerus refohing all Quejlidni concerning the Prcfent 
Worth of Annuities, Penfions, (Jc. at Simple |a- 
tercft. 

Given 17, R^ T; To fwid P .^ 

^r , jTTR — TR + 2Trr p 

Trj,orem i, ^ ^fj^^Z ' ^= ^ 

Given 



The Ufe of Decimals in latereft^ 185 

Given P, iP, Tj To 6nd r/.> 
Given /», 17, T j To find R / 
Given V, P, R ; To find r.> 

Que ft ion i. What is 75/. Kr^rZjf Rent^ to continue p 
Years^ worth in ready Money^ at 6 fer Cent. &c. 

cl/ = 75 7 
Here is given Xr^ 0,06 ^ To find >, ^ ^ Theorem 1. 

Multiply — o,o5aBsriJ; 
By — 9 = r 

The Produa ' ,54 — r^ 
Again by 9^=s.T 

From which 4,86 = TTR 
Take — 54 ==riZ 

Remains 4,32 = TTR — TR 

To which add 18,00 = iT 

Dividend = 22,32 = TTK — TR +27 
Divifor = 3,08 = 2Kf + 2 



B b ' Then 



tZ6 TheUfeof Decimah in hterefi. 

Then if»} 22,^2 ( 7,24675 

2156. Ti — U 

•-76O 1623375 

616 5072725 



1i»i *♦ 



144^ ^543»5^<525 =r /• 

I 2.J2 < =: ';43 /. IQ X. I i </. 

• 2c8b 
1843 



• 1640 
1540 

•IDO 



Queftion 2* What Annuity y to ennfinue 21 Ytr^zri, will 
X5;^/. ijw ^ » ^. poKJiare, at 5f^Cr«#..^ 

192,0731 1 
Here is given -^ T = 1 1 > To find 17, ^^ The. 2. 

C *= o>05 V 




Multiply — - 

By - 

That Erodua 1,05 =^ TR 

Again by — 21 =35 r 



105 =5 

2:1a 



The Produft is 22,05 ttTTR 
From which take ip5 ssTR 

There remains 21,00 =: TTR -^TR 
To which add 42,00 =: 2T 

The Divifor = 63,00 s= TTR ^TR'-\' iT 
Then to — 1^5 =: TR 

Add Unity 1,00 — i 

The Dividend = 2,05 =: T/f + 1 

Then 



Tbs Ufjt. of Tkmmds n InUvefi. tS; 

Then f>i) 205 <,a3254 

:* 160 
126 



•340 

3'5 



250 

But — 384,1462 = ^P 
Multiply by 45250,8 invert 

I I 5244 

19^20 

Produa — 12,5 =t7 
The ^^r^r»7fy then is 12,5!. = 12 /. iQ^i. the Anfwcr. 

-A^f^, This is a very frequent and w/cy^/ Queftion ; and 
ought to be \vDrk-u -with /j*'^/?/ ExaEinefs ; and 
therefore if a Perfon be not very ready at, nor right* 
ly underfiands the Manner of ContraEled Multi- 
flication and Dhifon^ 'twill be beft to work the 
common Way. Which alfo is to be obferved in all 
Quefthns of 'Moment. 

Quefihn 3. At what Hate o( Simple Interefty will 250,;^' 7. 
or 250/. 6 s. S d. purchafe an Jnntiity of ^0/. lo/. fer 
Annum, to continue lo Y^arj .^ 



Here is given ^ t/= 30,5 ^Tofind H, per Theorem 5. 



B b 2 Multiply 



i 88 TheUfeof Decimals in Interefi, 

Muldply - 30,J = IJ And 500^-2/' 

^' - ^Jl-^ Alfo 6io%=iTU 

TheProduab — io%—TU „-.._ - 

Which again mak.by_^o«r S^'^Vr^l?^^. 

Produccth — y>^o:=iTrVL dcnd. 

From which take 305 s= TV 

There remains 2745 =« Trt/ — rl7. 

Then — 5006,1^ ss aPr 

qrhe DiflFcrence "2261^ zzTTU—TV^ 2PT 

theDivifor. 

'^'^*" "226?'?09 l&eDivifionofRepetends. 

2035,5) 8840 (,04343 = H. 
8142 

•^ 
810 

"^ 
81 

•7 
6 



/. I I. I. 

Then fiy. As i : 0,04343 : : joo : 4,343 =54/. 6 s. 
?o 4 i. the I^atc fer Cent, fought, 

Queftton 4. In what Ti;5j?^ will 7/. ^^r ^;!r«i//n paj a 
Dtfhtof 120 L Bs, at 6hper Cent. Or, For how' long a 
Time may an Annuity or 7 /. ^^r Annum be parchas'd ojc 
^njoy'd for 1 20 /. 8 ^ at the aforefaid Kate ? 

Here is givenS R = 0,06 i To fin4 7*, i»«^ Theorem 4. 

I I 



'■» 



X3i 



The Ufe of Decimals in Interefi. i %^ 

Firft «. 240,8 = 2^ f 1,0^=1 X 

And ~ 2A^ -2^ I ^^^ = i^ 

From which, take J?£_?= -^ f 34^ 

. To the Remaind. lfi0 zzz-'^^^rr- 1 

R U I 1,067 =i 

Add Unity •— i = + 1 

The Sum is 2fi0 = ^ — ^ +!=;* by Subftitat, 

Then — 1,0^ = J4f 

And — 1,067 == 4** 

Again r- 0,42 = if t/ 

^ - 573,?=^ 

Th«, ^ 5744 = ^ + ^ 

Sq. Root ot that 23^^ cs V^+^ 
To which add i/7^ = >x 
The Sum is 24,^ =3 2$ = ^ the Time Ibughf. 

Having thus, in a mod perfpicuous Manner, (hew^d the 
great and invaluabk Service 0/ Decimals in working Qti^fli- 
ons of Simple Intereftj &c. I Ihall proceed to treat of the 
£ime things, in the like Manner, in Comp0und Interefi. 

Theorems refolviag all Qucftions of Compound 

Intereit. 

Qven P, if, f; To find ^.^ 
Tk^orem i. P^:=^A.^ 

Given AjSjti To find P .^ 
Theorem 2. k-jt = P. 



Xbet^em 3. \-^'=:^ R^ 



Rates* 



i^^ The Ufe of Decimals in intereft. 

Jlat&s> logarithms. Eates* Logarithms. 

1,10 = 0,041392 1,055 = 0,023252 

1,08 ='0,033423 ; 1,045 *= o,oipii6 

1,06 = X),025305 ' 1,04 = 0,017033 

1,0$ = 0,02118? J 1,05 = 0,01283:7 

Quefihn {. What will 256 /. 10 s. dnamuitXim 7 Yeia^s^ 
it 6./. fer€em. ^c Com found Inter eft ? 

^Pt= 256,5;^ 
Here is given 2 it*s= 1,06 > To find Ay fer Theorem i. 

C^ = 7 > 

Multiply the Logarithm d£ the Rate 4>o6 =: 0,025505 
By the Index of its Power (^viz. f = ) 7 

The Produa the Logarit. of R* = 1,50363 = 0,177135 
Multiply that by P — = 2^,5 = 2,40^87 

The Prodiia is the SxsipmXt A = 385,6811 = 2,586222 
That is, 335/. 13 i. 7 i</, the Aiifwcr reqaited. 

Quejiion 2. What Principal^ or 5;iw of Money ^ will raife 
a Stock of 20,6 in 5 Years, at 5 /. ^^r Cent, fer Annum 

Compound Intereft ,^ 

r A:=: 20,5 7 

Here is giveji < ' ^ 5 > To find P^fer Theorem 2: 

I /e=i,o5i 

Multiply the Logarithm of the Rate 1,05 3=0,021185? 
By the Index 01 its l^ower i :=i -— 5 

The Product is the Logar. of ^c — 1,27628 = 0,105945 
By which divide .the Amou?^ A 5= 20,6 = 1,315867 

/. 

The ^iotii'72t h the Principal P = 1^6,1407 = 1,207922 
That u ibLjs, tlie Sura required* 

Cli^eftio^ 3. Ill wlnt JJms will 37 Z. ^r5 s. amount to 
75,05/. (or 76/. I IS.) at 4/. 10 X. fsrCentP 

< A— 7^,65 1 * 

Here is given *S /* 1= 37,75 > Totfind tyfer Theorem 3* 

( R ^ 1,^5 ^ 

Divide 



The life cfDeeimcth in Interefi* 191 

Divide the Amouut — A ^ 76^5 nr i^S&^^2 

Bjj the Principal (or Sum) P z£> 57,75 = i,574g>i6 

The Quotient is — — — R^-z^z 2,05^046 = 0,307596 

Then 2,03046 divided by (£:=:.) 1,045 ; and that Quo- 
tient again by 1,04^ ; andthus^continuaHy dividing the Quo- 
tients by 1,045, 'till nothing remains, the Number of fuch 
Dhiftons will be «qual to ( t =s ) ohe Time fought. But 
tills is fooncr, and eafier done by much, by Logarithms. 

Thus, Divide the Logarithm o^ 2,05046 ^:ssiSt^) by the 
Logarithm ot i ,045 ( =: -R ) and the Quotient is the Time. 

p,cipii6)Oj^75p6 (16,091 =f the Time fought. 

19116 

I I 6456 

1 1 4696 ^Vi%. 1 6 Years, 1 Month, 

' " ' \ and 5 Days, the Anf. 

. 1749 ^ i ^ 

1720 



20 



Queftion 4. At what Rate of Cpmfound InUreft^ wiD 
)i /• 15 j» amowit co 70 /. 18 j. ia 5 Years .^ 

rF= 51,75 7 
Here id given Jji-= 70,9 J> To find I?, fer Tbe$rem 5. 

Divide the Amount — A z= 70,9-^ i.8sc64'5 

By the Principal ot Sum ^ jp = 51,75 = i,7i:;9i :> 

The Quotient will be ^t zr ^r— 1,370048 re 0.1567:56 

The Siirfolid Root of which is iZ =; 1,065 ztl 0,027347 

/. /. /• /. /. s* 

Then fay, As 1 : 1,065 :: 100 : 106,5 =; 106— i;5=:R 
the Hate per Cent, per Amium fought. 

Note ; R^ being equal to 1,570048, of Consequence B^z=: 
f. 1,570048, which may be extrafled by t,\ A^gebraick con* 
vergirg Series ; the Ktinner of doiiij ir, fee hi Chap. 11. of 
tlie Vjg of Decimals in A'gd^ra. 

Tl«> 



19 » The Ufe cflyecmah in Inter eft. 

Theorems rtfohing all Qutftimt reJating to Anna ides, 
&C' fH Arrear, talculated at Coinpound Intereft. 

Giwn TJyRtt; To find A.^ 
Theorem t. < * =■/< 



G1t6i a, Ryii To find V.* 
Theorem 2. J^L=4=zU 



Given I/, A, /?; To find t^ 
Theorem 3. {^±t^I=li^ = iPt 

Given A^ U, t ; To find R .^ 
theonm 4, -{^ ^ — /?t = ^^ 

Queftion i. If 30/. It^jr/ir J?^«t be forbom or unpaid 9 
K^/iri ; What will it amount to at the Rate d^ 6 /. /^^ Ga/. 
&c, Comfouni Intereft ? 

Here is given i^ / 5= 9 > To find A^ fer Theorem i . 

In the firft Place, let R = -^ ip6 = 0,025305 

Be involved to the 9 Power (viz. R^) — 9 

That will be — K^ — 1,689451 = 0,227745 

Multiply by —. U= 30 = 1,477121 

The FroduS is — VR^ = 50,683530 = 1,704866 

From that Subftraft £/ = 30 

The Remainder is the Divid. = 20,68353 = IJRt — 17. 

Divide therefore VRt — [/ = 20,68353 = 1,315626 
By -. R^iz=z 0,06 = 8,77815 1 

The Quotient is — -4 = /. 344,72^7 ==^ 2,537475 
That is the Amount =s 344/. 14/. 6 4^. the Anfwer. 

Queftion 



Tbe Ufe tf Decimals ift Interefi. 19% 

Que ft ion 2. What Annuiiy :j/. lOS* per Cent Com* 

^ound Intereft^ will raile a S/o^i of 3^/* <^u being focr 
ora 8 Year^ ? ' ' 

r^ = 344,2$7 
Here is given ^ /^ = 1,035 >To find 17, fit Tbeor. % 

U =8 3 

Multijij the Amount •-; A s=a 344,25 =s 295368)^4 
By the Rate — i? =s i>035 ===^ 0,014940 

From that Produft .— R/< = 356,2987$ = 2,551814 
Subftra^ the Amount .— y^ = 344,25 

^* ■ I ■ I 11 M M 

The Remainder is RA r^Azs^ufi^lij^^ the DivicL 

Then involve — Rzn 1^35 -s 0^x4946 

To the 8th Power ~ viz. Re — 8 



That Power will be k^ = 1,316803 = 0,119520 

The fame lefs Unity is R'^ — i =s 0,316803, the DivifivCi 

Therefore divide R4 r- -^ = 1 2,04875 =s 1,080908 
By — i?t «. I m 0,316803 =r 9,500785 

_ •. - 

The Quotient is — 1/= 38,0297 «b 1^80x23 

The Annuity therefore which was fought, is found to b# 
38,0297 /. = 38 L OS. J d. fer Annum, Anfwer. 

Queftion 3. In what time will 38 /. 5. yd. raife a Stock 
of 344 /. 5 J. at 3 h 10 X, fer Cent, fer Annum^ Camfcuni 
Intertft ? 

r-i; =■ 38,0297 \ ^ 

Here is given <A = 344,25 CTo find f, fer Tbfwr. 3J 



Firft 



IP4 ^^ Tjfe of Tyecimah In Inter efl. 

r&A iTfulriply Vhfe Ambuiit A ±z ^44,^5 =s= 2,556874 
tj Ae '^vtii R-atc —=. ft ^ss 1^55 = c,oi 4940 

To that Prodi a K^ m 356,29875 = 2,551814 

Add the Aiuiuity — l/=r ^,0297 

TtbA tk€ Sim i^^ + t/ = 394,^2»45 
Take the Amount ^ c=r 344,25 

dlvidfeby 3 ^= 38,0297 =1,580125 

iThc ^doticrit wiB be ^t = '1,516805 = 0,119528 

Tiit»'B^d^ l,3l6S03 continually by the Rate 1,035 "*•' 
till iloching refiiaifiB, and the Nuitiber oftbofe E>mfioiis%vill 
be 8 s=: f = Time required. 

'BbC ^ittich betfer' by Locarkhms thus ; Divide ihe ^Loga^ 
riihai of diei\)wer by the Logkrichn of the 'Rate, the Quo* 
dene Is .-r t the Time fought* 

?ni* c.M494i>o^ii952^ (8ssrf t?hc Time, pftcX^ 

o, 1 1 95 28 8 Years ; Anfwfcr. J 



• •*••«- 



. QtufiUm 4, At what Rate per Cent. Compound Interefiy 
uriU-^/. -Kfor/jf fe«#, being'jf^riorff^ or unpaid 9 Ye^^^ a* 
tnoufit to 344/. 14 J. 6 4 d. 

Here is given $ A t:- 344,7267>To'findif,/>fr Tfo. 4. 

^ t=;=9 3 

• *Firfl^(Uvi(l^tli€ Aino^it • A -r= ^44,7*167 rir 2,537^75 
By the Annuicy — Uzi: ^0 =: 1,477121 

The Quotient is — . -j^rr: 11,4509=: 1,000354 

• Ag^iif the Amounfl j rr x 

iefs the Annuity is> ^ -:^- ?H,7»67 ~ 2,497P3? 

Which divide by — L^:=: 30 =1,477121 

^ jj — Z 

The Qaotienr is _ ^ y » 10,49014 = i,0208i 2 

■'Now RtrrzR^. Therefore the Theorem affords this 
E(iuatio?jy viz. 11,4909/? — , jR^ — I0;490i4 

This 



The Ufe ^\Q^sm^Is inlntiteft. 19 f 

ThU E^ttaii$^ is,«afily rcfol«'d by a Ca»f^f/^g Series 
(which fee iri the Ufe of Dedmals ill J^g^bra.) 

N6te^ This Queftion may be very eafily ^nd expeditioufly 
aiifwet*d, b^u\t^.}^\t'oiFalfeFofitioni thus 

Malf§ fvjo Suppojith^is. of the Rate^ which may include 
betivee;;! ihem Ae ^atif you [e^k, . 

Then find what the Aimunis of the given Annmfy would be 
at the two fupfitfed Rates of In%etitfii^ fif$r Tif^mm i • 

Laftly; Opferye the Errors o^ ih'yfe Amount frona the 
Amount here given, then ty thofe Suppojitions and their £>•- 
ror/j find t}|c true R^f^, f v/z. 1,06) 2^5 is ^there taught. 

Th^iems r^fohuing aU Qu^hm ^omerning the Prefent 
Worth of Annuities, Pcnfions, or Lfcafes in Hfver- 
6qb, at Coo^imd Interefl;. 

Given I/, % i 5 To find P ? 
Tb£oram i* v ^ — p 

LH—i 
Giyen P, R, t ^ To find I// 

Givea UyR^P-, To fijid f .' 
Theorem 3. { ^ . ff_ >.^ = -«'. 

Given C/, />, < ; To fird R /* 
Theorem 4. J ^ = -^ A' -f Rt — ^t+ r. 

Quefth^ I4 What is 30 /. Yeairjy ^ent, JVatth irt r^iia^ 
Money, for ies Coeitinuattco jYenrs^ allowMig 6L fen Cent. 
Compound tntereji to the Purchafef ? 

Here is given. < i^.sr # ,06 > To.Snd P,/4T Tbwem i. 

•. ••• : . .11 f= 7 >. . 

' ' C c 2 Firft 



Jl^5 The Vfe of DH^imaU in Interefi. 

Fitft, involve ^— JR = r,o6 = 0,025505 

To the 7th Pow^, {viz. R^) ' — 7 

That will be ^ R^ =; it§006i .5=0,17713$ 

Then divide ' — t/, =; 30 5= 1,4771 21 

By Hf, there will remain -^ =r t?,9520cr 1,299986 

/ Then from the Annuity U = 50 

Subflra£J — - = 19,95? 

/v ^— — - 

Remains the Dividend l/r- _ = 10,048 3= 1,002079 
.Wbiqh divide by the R?te Ipfs ^ \^ -, ^^ _ o _o ,* 

Tlie Quotient is the prefent^ ^^ J ^, *' -. 

Worth — P — f— ^67,471^ = 2,2x3928 

The Prefent Wptth, In ready NJoney is 167 /. 9 j. 5 ^. the 
^nfvvcr. 

N, B. Suppofc this were an An'nuiij in Reverjionj or not 
to be entered on till after 7 IV^irj are' pkft, and*thcnccto con- 
tinue 7 Years ; and you would know the f recent Worth ; find 
by the fecond Theoreoi of Comfourfd Initrefiy what read^ 
'Money will amount to 167/. 95. ^^/- in j'Years^ at the 
//7W^ fll^r/^ of Intereft ; and that will be its frefent Worth ; 
and fo for any other Annuity in Heverjion. 

QueftioH 2. What Annuiifj to continue 7 K'/zrx, may 
be fur chafed for 120/. 5^. at 6 per CentSompound Inte-f 
reft.^ 

I /* = 120,257 . 

Here is giycn-< i? = 1,06 S. To find t/, />fr Tbeor. 2^ 

^' ' ' ■ ^ f =7 • S ^ ^ ..• 

Involve the Rate — R-= ifiS = 0,025305 

To the Index of its Pcwei^ (viz. := f J 1 . — ■ - . 7 

■» ' ' ' ' 

The Power of R will be /if tr 1,50361 r=: 0,1771^5 

Which mult, by tb^ preftnt Wortji *»5=:i20,2^ tt 2,c8cc8 f 

TheProduais . ~ /"iPfp i8o,l5c87 r== 2,^57^19 
^ultiply that by the Rate — ' 7f ±=' ip6 rr 0^25 305 

That Prodna is P^^ X il= ^91,65722 m 2,282524 
JFrom which fubftrid • /'i^i » 18^,8087. - • 
5rhere Remains the Dividend jb,84852 =5= P/Jj X R 



The Ufe of Decimals fn Interefl. 1^7 

Divide thwforc./'ilXH.—'*«? = 1^584852= 1,0353^ 

By the Power of H lefi i :=. i([-^ii=o,5036i = .9,702113 

.The Quotient is the. Annity u -= 21,54057 =»; 19333256 
The Annuity fought therefore is 21,54057 /• = 21/. 

' Qy^fiion 3^ For what T/iw^ wjll i6y /. p s» 5^. purchafc 
an A^Piuitjf'dJi, 30/. /^^r Arntpmy ^^ ^.ferCenX. Qmfound 
Interefi? 

rP ~ i6747l6->' 
Hcrf is giveo 3 U^ :z= 3P V To find t, perTbeor. 3^ 

To the prefent Worth P = 157,4716 
Aj^d the Annuity — 1/ = 30 



» <■» 



The Swiis — / +!/= 197.47 J^ 

Ttien muk.thc prefent Worth F = 167,4716 ==; 2^13928 

By the Rate — fi z= ip6 = 0,015305 

The Produa is — /'/I = »77>5IP9 — 2,24^233 

Which fubft. frorijLSum of./'-}^ Ja.= IP7,47X6 

The Remainder is P+!7-P/^'= 19,9^17 the Divifon 

Then Divide the Anpuity — U z= ^oz= 1,477121 
Py the Divifor f-^U-r PJi = 1 9,^5 17=1, 299986 

The Quotient is *-* 'i?t n: 1,50361 = 0,177135 

Lafily, Divide the Jjogaritlm of iff, by the Lcgaritbrn of 
the i^^r^ ; the Quotient will be the T/m^ = f fougdt. 

Tiius, , 0,025305) 0,177135 (7 = f the Time fought,? 

0,177135 »iz. 7 Years. Anfiver,j 



.HI 



dueftion ^ Suppofe I purchafe an Annuity of 21 /7 10 /. 
p ; <j^. to onriiwe 7 K»jr5, for |2a/.' 5 j. tesdj Mwey ; at 
iviut Rate far Cent. Compound InUr/efi^ w^s the Pufch^e 
niatie ? . . ' s 

' <9 -=: 120,25 "7 

fJcre is given .-^ t/ = 21,54057 > To find if, petTbt. 4. 

it z=zT J' 

rirft. 



t^S TheUfeof Decimals in Utenfl. 

Fkfl; divide the Arwaitj V =r 21,54057 =: i,7??2«>$ 
By die pcefenc Woccb .^ . P rs 120,25 ss 2,086084 

TheC^dcnck -^ -tj— 0,17915 r= 5^253172 

Then multiply it into the given Power of the Rate^ to 
which add the Pawery &c« as fer Theorem \ and you have 
Chifli Eq^kdiBttf inz. 0,17915 V + ^ ~ ^ »Qji79i5 ; 
wlMfioe a P«Brfi>n «eady tt AigeBrakk SxtKaSw^s^ may iboii 
difeover ^ =: 1,06. Then fay, as i /. : 0,06 /. ; : ioo /• ^ 6 /• 
the Race fer Cent, required* 

Neie ; Thb (and all QueRions of this Nltore) may be 
Siofwered by the l(jile ^ Pofkhn^ in the (ame manner as was 
iiirlcded in the foprth Queflion of Annuittex in Arrearf, 

Tfaeorems ufdving all Quefiiom relating to the Ptircbar 
fing of Free-hold or Rea:l Eftates, at Compound 
inttreft. 

Gven PJP; To find UP 
Thi&rm t. P /t-^ P ^U, 
Given l/i^; To find Pr" 

' ^ Theorem 2. ir- — = P« 

A — t- 

Given P^ I/5 To find 2^.^ 

« 

Cluefiion |. Suppofe tl Free-hold Eftatc of 25 /. fer An- 
tmm were to be fold ; What is the Worthy aUoDiring 5 /• 10 9* 
ferCettt. &c. Comp^uxd fytereft to the Buyer? 

Here is given 4 ^ ^ TL^e\^^ fi"«i ^jf^^ Ti.^or. 2. 

Divide the Annual Rent — t/ =r 25 = 1,397940 

. By the Rate kftUilky • iC — i = 0,055 rr 8,740562 

The Quotient is thp Worth P =;= 4^^4,5 = 2,657578 

« 

"TheFtf/iK^of that E^l^te therefore is 454/, los. ioid. 

Qnefiion 



h^''' 



TXf Vfe of Dnimah in Inte^efi: 199 

(luefiian t. Sup^ofe a Perfan vouid fay out ^01, i:;^j. 
44/. OH a Fne^Md Eftatej and fo as ^o ibe allowed 6/. fe^ 
Ct*?it. forMsMonof, Comfwnilntereft *, What tiuift<hecfar 
Amnual Rent of fuch an Elftace } 

Bfitt i% givQB { J = JJ^^^^ }.To fiiid I/, fiT necr. u 

Molciply the prcfent Worth —./'=: 41^,6^= 2^61^789 
By the Rate -^ JR = ip6 r= 0,0.25505 



^■iiVBWV^'Wl^ 



The PtoduA is — Pi^ = 44M ^= 2,649094 

From which fubflrad the Worth P ss 41^^,6 

There remains-the Annual Rent { », — ^ — T ^ 
The Anfiver }^ 1/ = 25 Z. ^ ^«m»9. 

Quefiitm J. 'Suppofe one give 4116 Z^ 13 &. 44^. for zFr^f 
bold Eftatr^ 25 A fet .Annum ; What ^ftte fer CenLC^wt- 
found Inter eft ^ has the Pur chafer for hb Money? 

Here is fiwa ^ ^/^ *!^^ ^To findi^, /^r ISr^^r. 3. 

To the prcfent Wonh — -P =: 41^56 
Add the Annual Rent — 1/ = 25,0 

Divide tfaeir Sum — P -f" ^ = 441 >^ = 2>^5op<^ 
By the prefent Worth w— p *5. 41/^^5 = 2^J978|r 

The Quotient is the Rate fought /( =2 i,c6 =* 0^2530$ 

Then ftf, As 1 /. : ,06/. : : loo I. : 6 1.. p^ Cent, the 
Anfwer. 

Rebate 0r DifcounC. 

What-tliis is f have already defind in.rtie Bmoning of this 
Chapter ; The Interefty and DifioUnty of thtjnme Parcel of 
Money, is i«ery difierent, i)ao^ vulgarly underftood (aad actoid- 
ingly is reckoned) the fame thing. 

Jn order theretorc to have a right Notion of Difcount^ and 
how it differs from Irttereft; tve muft confider, that bifereft 
is the Inert afe of any Principal^ or'Sum of Money, accord- 
ing to a^fj i{atey or PraPortwn^ ^gretf^don; "and in compu« 
jing it, we have Regard only to che tare Principal ; But 
what is properly call'd Difcounty is the Difference between a 
Sum of Mor.ey due any Time hence ^ and fuch another Sum as, 
being fut to Utereft^ wou! J, with its hcrenf:^ by Intereft^ 
become tc^iuil co che fiiJ Sum hire after dii:. 

Thus, 



7c6 The life of*DecmaU in Inttrefi. 

Thus, for Exiokplby If I hare 105 7. due to me 12 
Mmths hence ; the Difcounthx frompt Payment thereof ac 
5/. fer Cent. Smfle Intfffffi muft bes/. and: the frefeni 
money J or Worth of that 10^/. is 100/. ; Becaufe if 1 put 
lOO/. out at the aforefaid ^te^xt would in that Time be equal 
or am6^nt to 10^ /• Wherefore the Inter eft of 105/ di& 
counted (as is the common way^ I fhould receive but ^^L 
x^s. ; the Inteteft of the 105 /. being ^ s. more tlian tftc 
ixwt Difcouni ; and confequently the reckoning tUterefi for 
Difcount is very diiadvantagious to ikoSt who iiiake fucb Dif- 
tounts. 

The Proportion for Ftfi^^^ or Difcount tfcett fe, 

' As too L and the ^^f^ : is to the Rate : : fo is any other 
Suih : to its true Dfpount for the fame Tii 



The Tbeorepi. for finding at pfKC both the bifcount and 
fYefeni tf^ortB of any Sunfi of Money, dot any titnc htreafcer, 
is the fecond Theorem of Simple and Compound Inter eft y as 
I there obfcrv'd. 

By the fecond Theorem of Simple Intereft^ it was found 
that the Prfent Worth of 405 /. 6 *. o </. due 5 Years and 8 
Mor.rhs hence ac 5 per Cent» 

/• J. d. 
Muft be ^ — — ^16-— p— 5 

Which fubflraaed from tte i>>fy^ a^ , 

leaves the fr/^i? £•//?()««/ r ~ W— 16-7 

But the liter eft of that Sum ist 

for that r/W — r — 113^15—2 



Which exceeds the trite Difiount by — 24— •iS-^y 

Above a fourth Part lois to the Difcounter of Intereft fot 
fuch a Sum* 

Note. This Theorem of Mr. Ward'sy is far more eafy, 
cdncife, and elegant, than any other extant, for finding the 
frffent Worth, or Difcount for prompt Payment of any 



Debt. 



^ TA- 



at 

A "table of Days for any pven Ttme» 



Fifftj To ktKnr drf Kftmier of J)mi from ibe Seitwiint 
D4 Xhii 



ici Tf^e Ufe of the freceeding Tahle. 

This is bbtain'd by 1nfpe£tion only ; Thus from yanuafy 
the lft| to Seftemher the 7th, is 250 Days ; To Notembttr 
^ 27th art 321, Qc. 

Secondly f To know what is the Number of Days from any 
gtfcn Diy of any Monib^ to the End of the Year. 

Suppoie SePienAer the feveiith, then from — ^ 365 

SkbftraS toe Number anfwering to 5^/f . 7 — 2 56 

I'hcvt remains the Number of Days ibughc, viz. 115 Days. 

tbhrJly, To find the Number of Dmjs between the given 
tiay of any one Movthf and any given Day of any other 
Moffby in die fame Year. 

For Inflanoey To know bow many Dayf there are between 
JfrU dte lytb, and OBohet 2^. 

Thus, From tbe Number anfwerii^ to OQcber ^:^ — 296 

SuhflraO diac anfweriog to Afril 17 — 107 

Tbe Remainder b the Number of Days Ibught -— i8p 

PoMetblyj To find the Number of Daysy from any riven 
t)ay of any Month in one Year^ to any given Day or any 
Mofrtb in the «^xt Kf^. 

How many Days is it from September the 7th^ ^7^^ to 
4^1 the i9tby 1734 ? 

From tbe Daqfs of a Whole Year — 365 

SubftraS die NambettoSr;pf^i»^^7 — 250 

Remains the Number to the End of the Y^or «^ 115 
To which 4I& the Number to April ip *» lop 

Tbe Sum is the Number of D^yj required ~ 224 

And thus is the Number of Days readily found for any /;r- 
f<TV/i/ of 71f«r given, in the fame F^<«r compleatly 5 or which 
is parr of one, or part of another Year. 

How very neceffary and mfefid a Table this is in all Parts of 
Afitbmetical Science relating to Time is fuf&deiitly evident to 
the Skilfid therein ; but becaufe it is nmt Particularly fo in 
the whole Afiair of Mereft^ I have therefore prefixed it to 
tbe other Tables. 

Havim then die Nmier of Ditfs^ 'tis eafy to fihd what 
Decimal Part of tbe Tear^ they make ; and having found dnt, 

Jou have the r, f, in the i<x^pii^ Theorems reprdenting any 
WofaKr^r. An 



The Nature^ Confiru&iony and Ufe^ &c 203 

An Example in Simple and Compound Interefty will makl 
the whole Matter eafy and confpicuous. 

Example i^ What will 65 L amount to, being lent from 
Aiarcb the 7th to Novemher the 3d, at 5 h per Cent, per 
Annum Simple Int^eft .^ 

From Afarcb the 7th to November the 3d arc 241 Dafs; 
thofe make 8 Monibsj 2 Weeh^ and 3 Z)/ijw, =?= 660273 iD^ 
rm/s/i of a Year. Then by Theorem 1. 

Multiply the 77w^ — . — r^o,66:)27} 

By the Ratio of the J?/ar^ mm ^ = 0,05 

And chat ProduEl — TR ss 0^3301365 

Multiply by the Principal ^ P ss 65 

, The Produa is -^ TF P =;: 2,1458872$ 

To which add the Principal P =^ 6% 

The Sum is the Amount fought = 67,1458 to. /. 

Example 2. What is the Amount thereof at Compound 
Inter eft J the Rate and Time, being the fa$ne P 

, The Logarithm of the Kate R 5s 1,05 = 0,02n8p) 
Multiply by the fm^ ~ ^.= ,6603 

The Produft is the Logar. cAR^ •=: R^^% r= 0,01395^1 2 
To which add the Log. of the Prin. /*= 63 rr 1,812^135 

The Sum is the Log. of Amount A =67,1 281= 1,8269045 

And thus the Tkforems ferve to anfwer ^eftions^ when 
the Time is only fart of a Year, as well as when comfleat 
Years, 

Proem to the Tables of Simple Intereft, eon^ 
cerning their Nature, ConftruQ:ion, and 
Ufe- 

The great Defign of Tables of Intereft (both Simple and 
Compound) is Ea\e and Expedition in prafiical Calculations. 
For, befides that the i?«/<f^ exprefled in J^ortf^/ for juifwer* 
ing Qucftions of Intereft arc tedious and intricate, and the 
Reafon no ways to be underfiood ; the Operations rhemfelves 
are, for the moft part, very laborious ; and confequently Taktei 
which expedite and facilitate the fraSlice are inJifpenfibly 
Q^ceflary. 

D d 2 This 






f 04 The Natun^ ConfiruSihn^ and U} 

This being und«n|)bly evident, the C^ueflion cx:curs, Whe- 
ther ibefe Tahles arc to De m^de in Dw.Tial or mxed Nuas^. 
tert (i. e. <gch as cxprefs the Money in its coniiuoniI><jno- 
mination^ of Founds^ BhiVirgSy and Pence) P '1 he Anfwer 
to this can admit of no Demurr amongft thoje who underfland 
fhc Dottiine of Decimal Number^ ; rhey all know the ^v- 
felleficyjind fuperiour Uf^ cf the fij-ft Sorty viz. Decimal Td- 
lies, ^t Intereft TMe^ cxpreffed in common Uonaj are in- 
dulged CQ thofe who underftand lyolhecimals^ as Crutcbff^ to 
jfhe Lame^ and SfeSacles to the If^eak-Jj^hted, 

1 he Nvwhers in the firfi of thefe Tables of Zimfle Inie-^ 
reft for DajSy aod in the Secoud for YearSj being in ilri fifiu 
metical Proportim^ (nakes theoa capably of thai: t^effeSHon^ 
which 9P other TaHes c^i pretend to. 

Tbefe Tables are /o contrived, That the Jnterejl ct any 
Principal Sum is eafily found for a^y Number of D^^i or 
Ye^i ac any iP^f^ frcpi ome Pound to J<?»> witji the fiahes 
and Quarters* Having followed heieiu th^ R^v. Mt« C^^^t;^ 
in his Arithnetica Infinita. 

The ConftruSlipn of thefe I2ii/^j is eafy from the T^^a- 
r^wj themfelves, (and indeed the Reafon of their ConftruBi^ 
pn can be no otherways fo' eafily conceived.) Thus oy Theo^ 
rem Kht firft of Si«7^/^ Intereft^ viz. f Af -|- Z' = -^ is the 
prfi and f¥Cond TabU conftruaed. For fince the Amount lefs 
the Principal^ is equal to the Inierefi^ therefore the Theo- 
urn will be fRP :=: htcrefl. Now if p =3= i /. / xa ,002739 
iS^. (the Decimal of a I><ar fot one Daf)^ and ^ =: any j^^j. 
tio of Jnterejij fuppofe 5 per Cent. ; then the Sitftple Interefi 
of <>«^ Pound for one [)^y, at 5/?«^ Cent, is ,002759 @r. 
X y05 X 1 — OCOI3698 Qc. whicn being multiplied oy the 
nine Di^ts feverally conftitucc that part of the Table of Intc« 
ireft at 5 p^r Cent, and thus the whole firft Table is made. 
T he [econd Table for Years is only the various Ratios of In^^ 
ierefi multiplied by the faid Nine Digits ; for fince t n=,. \ 
Yf'ar, and /* ^ I /. it will be tRP z=- R the Intereft for th^ 
j^yft Year, Sec. 

The third Table Ihews the Rebaie or Difount to be mad« 
for one Pound, at the fevefal R/ites per Cent, for Days. 
i he Manner^ Truth, and Heafon of its Conflruclion is de« 

rived from Theorem 2. of Simple Intereft y viz^ - ^ ^^ ■ =z Z', 

For fine© the Frincipal or prefent Worth fuhdu£led from the 
Amount gives the Rebate or Dijcount of that Amount ; there- 
'•■■'■■ - •"■ '■ ■ forq 



of Drcimfll Talks of Smfle Uter^fii %«b^ 

^'^5 the Djpou^t of any Ameuut fcr any Vme at aay Hats 
(without Regard of thjs ptdfeui Value ox. prwcifal ^oncy^ 

may be found by this Theorem „ ^ =D =: tUfioxntf. 

Hence if wc put ^ :r i /. / ^^ ,00^7:59 &. and if ^=r any 
jRatio of Inter fffi^ fuppofc 5 jp^r CVv*. then by this laft The* 
orem we have tfie Lif count 01 one Pound for o»^ D^y at the 
^^/^ of 5 ^^r Cf«/. />dr Amium ; For ^f/? =5 1 X ,002739 
Be. X 5O5 =^ ,000^5698 Qc. And /I? + I ■= 1,0001:1^98 
i§c. then by Divijion '^ 1,0001^698 Be } ,00015698 8c. 
( ~ 500013697 Br. the Lifcount. If f a= i r<?^2r ; then thfe 
Annual Lifcount of one fowid at 5 per Cent, will be found, 
by the above Theorem^ thus; At^ ==,05 and f^ + i = ips* 
Thttcforc by Divifion, 1,05) ,05 ( rst ,04761904 0A the 
Difcount. And thus is the Difcount of any Sj/ot at any Rate 
for any Tf/w^ abomw one Year found at once by the above 
I'hcorem ; and for any Time undei a Year by the '^bU of 
Difcount for Z)/i^j, of which J have now taught the Cow- 
/trudiion in a nev) and m«r^ ra$ional fidethod tban any I have 
yet feen. 

the Ufe of Table I, and 11. 

In order to underftand how to make thofe two Tnhles uni- 
verfally pfefijll, the Reader is to obferve, thsw: if a Number 
ponlifts of only one Di^ic w^th Cyphers affixed^ a3 lO, 50, 
70:), 0000, 80OOGO, Sf.' 'tis called a pure Number ; but thofc 
Numbers which coiilili of nioip than one^ or 'wholly of Di- 
pirs. As 370, S68, 7569, Bf. may be called Mwd Num^ 
hers, iSow every mixed Niuijber may be refolved into th^fe 
fure NumhetSy of which they are compofid ; thus the mix* 
ed Number 567, raay be refolved into the Pure Numbers 
500, bz^ and 7 \ fo alfo 15890 is refolved into loooo, 5000, 
8:0, and- c)Z. , 

Now then as to the Vfe of the Tables, obferve thefe 
R ules ; 

I. If the Number of Z)^y^, Years, 8cc, propofed, faeaw/>- 
ed Number^ let it beref Ived into ^^''^ Numbers. 

II. With the fure Numbers feverally enter the Tables, and 
take thofe htcimal Numhers which ftand againft the firft Fi- 
gure of each fure Number^ in the Column marked Num^ 
hers. * 

III. Remove ;he Dcii^nal Point iq each fuch Dednhl 

Num. 



to6 The Nature^ CotiftruBion^ and life 

Number, fo many Places to tfie RighUbandj as there are Cy- 
fbers in the refpedive fure NuniDers. 

IV. LaiUy, Add together all i^ Denmal Numhers^ and 
&iid the Value thereof by the Tables for thatpurpofc. 

Thicfc things fremifea^ the life oF the Tables will be ob- 
vious from the Exam f Us of the following Problems. 

Troilem i. 

To find the /»f«^^y? of any 5i/»!r of Monpy for ^ D/»j, ora 
Ikar^ at aoy Rate fer Cent^ per Annum f 

ExampJe I. 

What is the Inteieft of 2746 h at 5 /• 15 s. per Cent, for 

a Day? 

Decimals.^ 

In Table u under y 20C50 — ,31506 

the Rate ^l. You C 700 — ,11027 

find againfi the fur^ C 40 -^ ,00630 

Numbsrs j 6 — ,00094 

The Anfwfr U ~ — ,452«)7 := 8 ^. yld. 

Example 2. 

What is the Interefi of the &me Sum^ at the fime Rate 
for a Year P 

Decimals. 

c 20c© — 150,00000 ) 

In Table 2. 3 700 — 40,25000 C Under <^\d. 
You find againft y 40 — 2,30000 C pinr Cent. 

C 6 — - 0,34500 -^ 

r 

The Anfwcr in DenmaJs I. 192,895 

Which is in Mbney z= 192 /. yj s. 10 Jrfl 

Problem 2. 

To find the Jntereji of any S«w of Money For any Num. 
ber cf D/yfj. 

Example. 

What is the Interefi of 265 /. for 149 Days, at the RaU 
of 3^. 15 J. ^<^r Grff, &c. 

MulcL- 



of DecimalTahles of Simfle InUreft. 207 

^ Multiply the Priftcipal Stim — 2^5 /• 

By the given Number of Days -^ i^p 

The Produa is the mixed Number 3P485> with which 
leiblfred, enter the Table as before : 

Decimals. 
Thu, in Tahle 1. i 3^^^ ~ 3'^820o \ 

You find againft j ^^ " ^^92^66 . y ^ , 

the /.«r. &uai. 1 4g - WO^ ^ '^SrG.}/ 
bers »o — 0,00822 ! ^ 

I 5 — O1OO051 J 

The Anfwer in Decimals — /. 4,05648 

In Money ^l i s. 1 ^d. 
The Method is the lame for anj greater Number dC Dafi. 

Problem. 5. 

To find the Inter eft of any &«i forborne any Number of 
Years at any of the given Rates per Cent. 

Extmfle. . 

What is thtlntereft of 175/. 15 j. G}rbome 15 Years at 
the iP/if^ of 6 fefCent. &c,? 

Multiply the Principal Sum •• 175?7S 

By the Number of K?jrj given 15 



The Produa is the mixed Number — 2284,75 

Which reiblved, as before, will fland thus^ 

[2000, — 120,000^ 

j 200, — 12,000 1 

In TaUe 2. 1 80, ~ 4,800 ! Under 6 per 

You find againft I 4, — 0,240? Cent 

I - >7 — 0*042 I 

I )05 — 0,003 J 



The Anfwer in Decimals /, 137,085 
The lame in Money 137 /. is. 8 rf;. 



N.B. Tht 



to8 The Naitare^ CoftfiruBioVf emi life 

N. B. The Reader, muft obfenre^ in rcfolring a ndket^ 
Number wherein ^v^- Decimals^ to rediove rhe Point Oiie 
PJacc more to theA/f than are the Number of Cyphers in 
the Decimal fure Number, asm the laft Example. . 

ih Vft 4f Table III. Of Difcount. 

|n fcekingithe Difiouftt fot any Sttm due Ut rhe "^wi ofanf 
Number of Dfyt, if the Number of Days be a mixed 6ne, 
rcfolvc ihcm into fmre Numbers ^ before taught ; and even 
with them in the Tabk. take xhe Difcoujft of i /. which add 
and multiply by the Frincifal Sum, the Produft will be the 
r(/fO»«f thereof. ' "* ... 

PtoWefn 4*' 

To find the Difcount of any Smn^ fcr any Namtcr 6fDaysi 
at any given Hate in the Table* 



Ei/^mpi^ 



e4 



What is the Bif count of 83 Tounhy 10 ShiVings^ for 
^55 T)ays at 4 f ^r 0»f. ftr;AMmn ? 

Decim^i. 
Vou find r 200 — ,0214478 7 y . 

1« the JiW. ^ .^30 - ^L^$rC4f^t>;*#, 

even with C 5 *- PO05476 ^ ^ ^ 

' The Sum is — >C2527i5 

Which multiplied by the Sum 835 ltd 

The Produft is the Anfwer 2,1 10237 8r. ri 2 — -2— »i^ 

Problem 5. 

To &iiL»thc Jyificuni of any S«w for a JV4f. 

* — 

ExOmfle^ . 

What is the V^count of ido /. for one Tedr, ^t 5 fer 
Cent f 

- "^ In 



of Decimal Talks of Simple Intereft^ 2 09 

In the Table under 5 pen 5,^ g,^ 

Cent, and agamft 365 Days is T ' ^^ ^ ^. ' 

Which raul. by the VrindfaL Sum loO 1 » A 

The Produa is the Anfwer /.4,76i5? 8^. = 4— 15— 2^ 

•Now the Intercft of lOO /* for one Year, at 5 1 ^ ^ 

per Cent, is — • — i 



The Differ, therefore oiDifcount and Interefty is o — 4— p^ 

Whence *tis evident, he who allows Interefl for Difcount 
wrongsiiimielfconliderably, which yet is very common among 
Traders ; for fo much Money ought to be paid, as, at Interefi^ 
wonW amount to the S«/w due, in the 7ime propofed. 

Example 2. 

What is the JDz/fOttaf of P342/. at 4i fer Cent, for 
nYear? 

The Difcount of i 7. for 365 Days, at "> ^,^^^^ c^ 
4 ^ /..r Cent, in the T^.*/., is'' ' _ > '^43o62, Sr. 
Which multiplied by the Principal Sum 5^342 

The Produft is the-Anftver — /. 402,:>852 iic. 

In Money 402 /. 55. 8 ^. And thus proceed for other anr 
nttdl Difcmnts. 

I muft acknowledge this Table of Difcount gives not the 
precife Tifuth, and yet differs but little from it ; being fpffi. 
ciently exaSi for any Ofc. None but a Table of the Difcount 
for every Day, can be^psrfeSi ; becaufe every Dafs Difcount 
differs , bei»g ftill lefs as the Number of Days increafe. 

This Table is ferfeSlly true for all the Days exprefs'd 
therein, and, aslfaid, may be ufed without much Errour for 
any x)dier» 

/. s, d. 

j7..^,-w^ rThetrueD//r^a»/is — 2— i— 11 

\^^JT7 <Th« Z)//ro«;^f by this Table 2-2- 2^ 

in troi?. 4. "^^j^^ j^^^ f^^ ^^ ^j^^ ^^^ j^^^ ^_^_ ^4 

E e T.iVJS.L^E.S 



2IO 



TABLES #/ Simple IntcrefL 

Table I. Toe Intere/l of one Pound for Days. 

Table II. Toe Jnterefl of one Fpundfor Years. 

*Botb at any Rate per CenC from one to ten 
Pounds with Halves and Quarters. 



Table I. Tie Imtaeft rf em 


' Potoii per Diem. 


Nun*. 


I fer Cen, 

,00002740 
,00005480 
,00008220 
JOO010959 
,0001 3698 
,00016438 
,00019178 
,00021918 
,00024657 


lif^c. 


li ferC. 


i^ferC. 

^00004794 

,00009589 

/xx)f438? 

' ,00019170 

,00023972 
,00028767 
,00032562 
,00038356 
,poo43i5i. 
,0014583:? 


I 

2 

3 

4 

5 

6 

7 
8 

9 
Month. 


,00003^25 

/x>oo685o 
,00010274 
,00013699 
,00017123 
,0002054ft 

,0002397? 
,00027398 
,00030822 
,001041^6 


'000041 10 
'00008220 

:003I2329 
900016438 

xx)02O548 
»ooo24657 

Kxx>36986 
,00125000 



Table II. 'fhe hterefi oj om Pound per Annum. 

Numb. I f^rCent, 



I 

2 

3 

4 

5 
6 

7 
8 



0,01000000 
0,02000000 
0,03000000 
0,04000000 
0,05000000 
0,06000000 
0,07000000 
0,08000000 
q»Q90ooooo 



iXftnrC. I '^pinrC\ 



«» 



X^^pifV. 



0,01250000 
0,02500000 
0,03750000 
0,05000000 
0,06250000 
0,07500000 

10,08750000 
0,10000000 

0,112^0000 



0,01500000 
0,03000000 
0,04500000 
0,06000000 
0,07500000 
0,09000000 
,10500000 
0,12000000 
0,13500000 



0,01750000 
0,05500000 
0,05250000 
o^oyoooooo 
0,08750000 
0,10500000 
O9I 2250000 
0,14000000 
0,15750000 

Tabk 



Decimal Tables of Simfle Interejl* zii 

Table I. "fhe Inttreft oj one Pound per Diem. 



Numi- 


2 per C. 


2\ ferC. 


2 \ fer C. 


2 I psr C. 


I 


,00005480 


,00006164 


,00006849 


,00007534 


2 


,00010959 


,00012329 


,00013699 


,00015068 


3 


,00016438 


,00018493 


,00020547 


,00022602 


4 


,00021918 


,00014657 


,00027397 


,ooo3oj 37 


5 


,00027397 


^00030822 


,00034146 


,00037671 


6 


,00052876 


,00036986 


,00041095 


,00045205 


7 


,00038:556 


,00043151 


,00047945 


,00052739 


8 


,00043835 


,00049315 


,00054794 


,00060274 
,00067808 


P 


,00049315 


,00055479 


,00061644 


Month 


,ooii^6666 


,ooi87SOO ,00208?^ 3 1 


,oo22>;i^6 



Table II. 7%t Jnterefi of one Pound per Annum. 



Numb. I ifer Cent. 



I 

2 

3 

4 

5 

6 

7 
8 






0,02003000 
0,04000000 
0,06000000 
0,08000000 
0,100 DOOOO 
0,12000000 

0, 1 4000000 

0,16000000 
0,18000000 



2 i fer C. 

0,02250000 
0,0450'^iooo 
0,06750000 
0,09000000 
Ojii 250000 
0,13500000 
O5I 5750000 
0,18000000 
0,20250000 



2 i pert. 2 XftfrL\ 



0,02500000 
0,05000000 
0,07500000 

0,10000000 

0,12500000 
0,15000000 
0,17500000 
0,20000000 
0,22500000 



0,02750000 
0,05500000 
0,08250000 

0,11000000 

0,13750000 
0,16500000 
0,19250000 
0,22000000 
0,24750000 



Table I. 7%e later eft of one Pound per Diem. 




:^perCe»t. 

JOOOO822O 
JOOOI6438 

»ooo24657 
>ooo32877 
>0004i096 
,00049315 
I ,00057534 

,00065753 
,00073972 

,00250000 



-iXf^^^' 3 i?^^^'. 



,60008904 
,00017805 
,00026712 
,00035616 
,00041520 
,0005^424 
,00062328 
,00071232 
,00080137 

,002708;<-3 



,00009589 
,00019178 
,00028767 ' 

,00038356 

,00047945 

,00057534 

,00067123 

,00076712 

,00086301 

,0029ij^66 



,00010274 
,00020548 
,00030822 
,00041096 
,00051363 
,00061644 
,00071917 

,00082192 
,00092465 
,00312500 



E e 2 



Table 



2 1 1 Decimal Tables of Sim'^le Interefi. 

Table 11, The Interefi of one Pound per Annum. 



UMi^, 



I 

2 
? 

4 

5 

6 

7 
8 

9 



:> p^r LiefzT. 



0,05000000 



3 i ft^r C. 3 J />^r C. 



0,0325000a 
0,06000000 0,06500000 

OjOpoooooo j 0,0^50000 



0,12000000 

o,isoooooo 
0,18000000 

0,21000000 

0,24000000 
0,27000000 



0,15000000 
01,6250000 
0,19500000 
0,22750000 
0,26000000 
0,29000000 



0,03500000 
0,07000000 
o,ro5ooooo 
0,14000000 
0,1^7500000 

0,21000000 

0,14500000 
0,28000000 
0,31500000 



3 4 per C. 



OP3750000 
0,07500000 
0,11250000 
05 1 5000000 
0,18750000 
0,22300000 
0,26250000 
0,30000000 
[0,33750000 



Table I. The Intenft of one Ponnd per Diem. 



Lays. 

I 

2 

4 

5 , 
6 

7 
8 

9 

/Ifo^fh. 



^perCtfit. 

,oooicp59 
,0002 1 pi 8 
,00032877 
,00043836 
,00054794 

,00065753 
,00076712 
,00087671 
,00098630 



■i^^fa 



,00011644 
,00023288 
,00034931 
,00046575 
,00058219 
500069863 
,00081507 
,00093151 
,00104794 
,0035411^6 



4 i fer C. 4 I per C 



■fch 



,000123^9 

,00024657 
,00036986 
,00049315 
,00061643 
,00073973 
,00086301 
,00098630 
,00110959 
,0037^00 



,00013014 
,00026027 
,00039041 
,00052055 
,00065068 
,00078082 
,60091096 
,00104109 
,00117123 

.QQ3958>r 



Table II. The Interefi of one Pound per Annum. 



Yl ars. 



I 

2 

4 

5 

6 

7 
8 

9 



^ per Cti?2t, 

CjC400.:>ooo 
o,r ^000000 
0,12000000 
0,16000000 



^iptfrC. \ 41 per C. 



0,20000000 
0,24000000 

0,1' 8000000 
0,32000000 
0,36000000 



0,04250000 j 
0,085000001; 
0,12750000 
0,17000000 



0,21250000 
0,25500000! 
0,29750000: 
0,34000000 
0,^8250000 



0,04500000 
0,09000000 
0,13500000 
0,18000000 
0,22500000 
0,27000000 
0,31500000 
0,36000000 
0,48500000 



^iperC. 



,04750000 





0,09500000 
0,14250000 
0,19000000 
OJ23750000 
0,28500000 
0,33250000 
0,58300000 
0,42750000- 



Table 



^Decimal failes of Simple Inter eft. 21 j 
Table L The Interefi of one Pound per Diem. 



Days. 



^m— 






I 

2 

3 

4 

5 
6 

7 
8 

9 
Month. 



^ftrrCenf. 

m ii« ■ ' — ■ 

:, 0001 5698 

,ooo27?p7 
,00041096 

,000^4794 

,00068493 

3O00821P2 

,00095890 

,00109589 

,00125288 

,0041^666 



,ooor4383 
,00028767 
,00043151 
,0005753^ 
,00071918 
,00086301 
,00100685 
,00115068 
,001 29452 
,00437500 



5 i fer C. 

,00015068 
,00030137 
,00045205 
,00060274 
,00075342 
,0009041 I 
,00105479 
,00120548 
,00135616 
,00458^33 



,00015753 
,60031507 
,00047260 
,00063014 
,00078767 
,00094520 

,00110274 
,00126027 
,00141781 
,004791^6 



Table II. The Interefi of one Pmnd per Annum. 



Years. 5 ferCent. 



I 

2 

3 

4 

5 
6 

7 
8 

9 



0,0 5000000 

0,10000000 

0,15000000 
0,20000000 
0,25000000 
0,30000000 
0,35000000 
0,40000000 
0,45000000 



0,05250000 
0,10500000 
0,15750000 

0,21030000 
0,26250000 
0,31500000 
0,36750000 
0,42000000 
0,47250000 



5 \ferC. j 5 i fer C. 

0,05500000' 0,05750000 
0,1 1000000 ' 0,1 1 500QOO 
0,16500000^0,17250000 

0,22000030 0,23000000 
0,27500000' 0,28750000 
0,33000000! 0,34500000 
0,38500000 1 0,40250000 
0,44000000 1 0,46000000 
0,4950 0000 10,51750000 



Table L T'he Interefi of one Pound per Diem. 



Days, 



I 
2 

4 

5 
6 

7 
8 

9 



6 per Cent. 

,00016438 
,00032876 
,00049315 
,00065753 
,00081192 

,000986 ::?o 
,00115068 
,00131507 



1,00147945 
/^Ok >- /«* ^ /^ --\/^ 



6 % pr C. 

,00017123 
,00034246 
,00051370 

,00068493 
j ,00085616 
,00102740 
,00119863 
,00136986 



,00154109 
Month* ' ,00500000 ' ,005208^9 






6 \ far C. 

00017808 
00035616 
00053424 
00071232 
0008904 1 

00106849 
00124657 
00142465 
00160274 
00541(^66' 



6 \ pr C. 

,00018493 
,00036986 
,00055479 

,00073972 
,00092466 
,00110959 
,00129452 
,00147945 
,00166438 
,00562500 



Table 



21 14 Decimal Tables of Simple Intere/l. 



Table 11. The Intereft of one Pound per Annum. 



^ear$, 6 fer Cent. 

I 0,06000000 

3 0,12000000 

3 0,18000000 

4 0,24000000 

5 0,30000000 

6 0,36000000 

7 0,42000000 

8 0,48003000 

9 0,54000000 



0,06250000 
0,12500000 
0,18750000 
0,25000000 
0,31250000 
0,37500000 

o,43750DOo 
0,50000000 
0,56250000 



6 I per C. 

mi ■ iiWiB^awa 

10,06500000 

'0,13000000 

0,19500000 

{0,26000000 

0,32500000 

0,39000000 

0,45500000 

0,52000000 

0,58503000 



6 i p^r C. 

0,06750000 
0,13500000 
0,20250000 
0,27000000 

0,40500000 
0,47250000 
0,54000000 
0,60750000 



Table L The Interefi of one Pound per Diem. 



^ 



Days. 

I 

2 

3 

4 

5 
6 

7 
8 

9 



y per tent. 

,00019178 
,00038356 
,00057534 
,00076712 
,00095890 
,00115068 
,00134246 
,00153425 
,00172603 
_^oosa<^533 



7X>rC. 

,00019863 
,00039726 
,00059589 
,00079452 
,00099315 
,00119178 
,00139041 
,001*^8904 
,00178767 
,00604 f<^6 



7 \ per C. 

,00020548 
,00041096 
,00061644 
,00082192 
,00102739 
,00123288 
,00143836 
,00164384 
,00184932 
,00625000 



7 i per C. 

— ■■^-'^^* ■ 

,00021 ?3 3 
,00042466 
,00063699 
,00084932 
,00106164 

,00127397 
,00148630 

,0016986:} 

,00191096 

,006458^3 



1 able II. The Interefi of one Pound per Annum. 



I Ye/irs. j 7 per O nt. j 7 \ per C. 7 i per L\ \ y \ per t\ 



I 

2 

O 

4 

5 
6 

7 
8 

9 



0,07000000 
r, 14000000 
0,21000000 
0,28000000 
0,35000000 
0,42000000 
3,49000000 
0,56000000 
0,63000000 



0^07250000 
0,14500000 
0,21750000 
0,29000000 
' 0,36250000 
0,43500000 
0,50750000 
0,58000000 
0,65250000 



0,07500000 
0,15000000 
0,22500000 
0,30000000 
o,37500poo 
0,45000000 
0,52500000 
0,60000000 



0,07500000 



0,07750000 
0,15500000 
0,23250000 
0,31000000 
0,38750000! 
0,46500000 
0,54250000 
0,62000000 
0,69750000 



f^ 



Tabl« 



becimal Talks of Simple Intere/i. 2 1 y 

Table I. *ibe Inter efl of one Pound per Diem. 



8 ifer C. 

,00022603 
,0004520s 
,00067808 
,000^0411 
,00113014 

,001 35616 
,00158219 
,00180822 
,00203424 
,00687500 



8 i. per C. 

,00023287 

,00046575 
,00069863 

,00093150 
,00116438 
,00139726 
^00163013 
,00186301 
,00209589 
j,oo7o8^3^ 




~8 per C. 

,00021918 

,00043835 
,00065753 

,00087671 

,00109589 

,00131507 

,00153425 
,00175342 
,00197260 
,00^66666 I 



8^ ptrC. 

.00023973 
,00047945 
,00071918 
,00095890 
,00119863 

,00143835, 

,00167808 

,00191781 

,00215753 
,007291/^6 



Table II. 7%e Iniereft oj cne Pound per Annum. 



Years. 

I ' 

2 

? 

4 

5 
6 

7 
8 



8 pt^ Cent. 

0,08000000 
0,16000000 
0,24000000 
0,52000000 
0,40000000 
0,48000000 
0,56000000 
0,64000000 
0,72000000 



8 V per C. 

0,08250000 
0,16500000 
0,24750000 
0,33000000 
0,41 250000 
0,49500000 
0,57750000 
0,66000000 
0,74250000 



8 {. per C. 



« ^ 



0,08500000 
0,17000000 
0,25500000 
0,34000000 
0,42500000 
0,5x000000 
0,59500000 
0,68000000 
0,76500000 



8 \perC. 

0,08750000 
0,17500000 
0,26250300 

0,35000000 
0,43750000 
0,52500000 
0,6x250000 
0,70000000 
0,7375oono 



Table I. 7%e Interefl of one 

9 per Cent, 



I 
2 

? 

4 

5 
6 

7 
8 

9 
Month, 



,00024657 
,000495x5 
,00075972 
,00098630 
,00123287 
,00147945 
,00x72602 
,00197260 
,0022x918 
,00750000 



9 i per C 

,00025542 
,00050084 
,00076027 
,00X01370 
,00X26712 

,dox52055 
,00177397 

,002027^9 

,00228082 
,007708^3 



Pound per Diem. 
9 V per C. 



,00026028 
,00052055 
,00078082 
,00104x09 
,00x30x37 
,00156x64 
,00182192 
,002082x9 
,00234246 
,007911^66 



9\ P^r C. 

000267x2 
00052424 
00080137 
00106849 
00x33561 
00x60274 
00186986 
002x3699 
00240410 
00812S00 



TaUc 



a I tf Decimal Tables of Simple Jnterefi, 
Table n. 'Tbt litttrtfi of we Pomtdpn Annum. 



Tears. 


9 pin" Cent. 


9-^peTC. 


9kfirC. 


9\perC. 


1 


0,09000000 


0,09150000 


0,09500000 


0,09750000 


2 


0,18000000 


0,18500000 


0,19000000 


0,19500000 


3 


0,27000000 


0,277500°° 


0,28500000 


0,29250000 


4 


0,36000000 


0,37000006 


OijSoooooo 


0,39000000 


5 


0,4.5000000 


0,46250000 


0,47500000 


0,48750000 


6 


0,54000000 


0,55500000 


0^57000000 


0,58500000 


7 


0,63009000 


0,64750000 


0,66500000 


0,68250000 


S 


0,72000000 


0,74000000 


0,76000000 


0,78000000 


. ? 


0,81000000 


0,83250000 


0,85500000 


O,877'!0000 



T A- 



?^7 



TABLE m. 
^Simple Interest. 

TAe Rebate or Difcount of one Pound for Days, 
at the Rates <?/ 2 ; 2 4. ; 3 ; 3 i ; 4 i 4 4- ; 
5 ; 5 ; per Cent, per Annum. 



I 
2 

3 
4 
5 


2 per Cent, 

,000054ft 
,0001096 
,0001644 
,0002191 
,0002735> 


liiper C. 

,6000685 
,0001 370 
,0002054 

,0002739 
,0003424 


3 per Cent. 

,0000822 
,0001644 
,0002465 

,6003287 
,0004108 


3 i per C. 

,0000959 
,0001917 
,0002876 
,0063834 
,6004792 


6 

7 
8 

P 

JO 

20 
"lo""" 

40 

60 
70 


10003287 

,0003834 
,0004382 

,0004929 
,0005477 

,0010947 


,0004108 
,0004791 
,0005477 
,0006161 
,0006845 
,0013680 


,0004929 

,000575a 

,0006571 

,0007392 : 

,0008212 

/>Ol64ii 


,0005750 
,0006708 
,0007666 
,0008623 
,0069580 
,0019141 

,0028685 
,0038210 
,0047716 
,0057205 
,0066676 


,00l6jLII 

,0021870 
,0027322 

,0031769 

,0038210 


,0020506 
,0027322 
,0034139 
,0040928 
,0047716 


,0024597 
,0032769 
,0040928 
,0049073 
,0057205 


80 

,100 

no 
120 


,0643644 
,0049073 
,0054496 
,0059913 
,0065324 

,0070729 

,0076128 
,0081522 

,0086909 


,0054496 
,0061266 
,0068027 
,0074779 
,0081522 


,0065324 

,0073429 
,008^1522 
,0089601 
,0097667 


,0076128 
,0085563 
,0094980 
,0104379 
,0113760 

,0123123 
,0132468 
,0141796 
,01 SI 106 


'I3O 
I4O 

150 

160 


,0088255 
,0094980 

Vo 1 01 695 

,0108401 


,0105720 
,0113760 
,0121786 
,0129780 



F f 



TABLE 



/ 



SxSt De^M^i Talks of Rebate or Difcount. 

T A 5 L E lU. 
The, T}iJtount^ of ope Foun4 fpK ''D^' 



n 



aji. 






*/«• 



C^«f J 4 r /^^ C. 



I 
2 

? 

4 
5 



6 

7 
8 

9 

10 



20 

30 
40 

60 
70 



8d 

90 

100 

no 

120 



130 
140 
150 
160 



,0001096 
,0002191 
,0003287 
,0004382 
,0005477 



,0006571 
,0007665 
,0008759 
,0009853 
,0010947 



,0021870 
,0032769 
,0043644 
,0054496 
,006523^ 
,0076128 



,0086909 
,0097667 
,0108401 
,0119112 
,01 29800 



,0140465 
,0151006 
,0161725 
yOI7232I 



,000x253 
,0002405 

,0003697 

,00049x9 
,0006161 

• • 

,0007392 
,6008623 
,0009853 
,0011084 
,0012314 



Wi 



,0024597 
,0036850 

,0049073 

,0061266 

,0073429 
,0085563 



,0097667 
,01097^1 
,0121706 
,0133802 

,01457^88 



,0157746 
,0169674 
,0181574 



,0001 37P 

,0002739 
,0004108 
,0005477, 

,0006845 

^0008212 
,0009580 
,ooip947 
,0012314 
,0013680 



6 per Qem. i 



,0027322 
,0040928 
,0054496 
,0068027 
,0081522 
,0094980 



^108401 
/>I2I786 
,oi3«;i35 
,o^4^448 
,0161725 



P174966 
,0188172 
,0201342 
,P2HU77 



,0001644 
,6003287 
,0004929 
,0006^.1 
,0008212 



,0009853 
,0011494 
,0013133 

/XU4773 
,00(6411 

,0031769 

/>049073 

,0065324 
,0081522 

,0097667 
,0113760 



,0129780 
,014578s 

,oi6r725 
/>i776io 

y)i934f4 

,0299228 
,02:14960 
^0240642 



T-A-ft-LE- 



Decimal Talks of Relittp 6r 0ifci>imL t tp 
T A B L E ni. 

I 

T&e Difcount df one Tound fir Bays. 



Days. 

170 
I do 

200 
210 

220 
230 

240 
2^0 
260 



<v^ 



270 
280 

500 
310 



320 
330 

340 

350 

360 



3^1 

362 

364 



2 fer Cent. 



,0092291 
,0097667 
,0102037 
,0108401 
/>ii3759 



§nm 



,0119112 

,0124459 

,0129800 

,0135135 

,0140465 




,0145788 

,0151106 
,0156418 
,0161725 

,0167026 
,017^2321 

,oi776iof 
,0182894 

,01818172 
,0193444 



P«fW 



1 1 K ' J i 

,01 5^3971 

,oiP4499 
,oi5>5025 

• ,0J5>5552 



2\ferG. 



. 



,0115098 
,0121786 
,0128465 

»o>35i35 

ft 

,0148448 
>oi45cJ9i 

,0161725 
,0168350 
,0174966 

,0181574 

^ii8i72 
,019476a 
,0201342 
,0207914 



,01214477 
,0221031 
,0(227577 
,0134114 
,01240642 

,0(241294 
,0241946 
,0^^598 

»0!2^325r 
-4Q!24i2QDl 

ZZmXSmZZL 



3 />^r Ci?«f. 



,0137801 
,0145788 
,0153763 
,0161725 
,0169674 



,oi776id 1 
,0185534 

,0193444 
,0201342 

,0209227 



^i fere. 

,0160399 

,0169674 
,ai7893Z 

,0188172 
,0197395 

,0266601 
,0215789 

,0224959 
,0234114 

,0243z5 1 




,o!tfiS^r62 

y^?SSn7 
,0289712 

,01290487 
y>a9f2d2 



,0252370 
,0261473 

,0270558 
,^279627 
,0^288679 

.9237714 
,0:306732 

f«l3l.57W 
,oi3H7i8 
.0333686 



' 



f 



o,JJ34582 
05?54i78 

0^3726? 
Or338t6jt 



Ff » 



TABLE 



'i%6 DfcimaJ Tables of Simfk Itaerefl. 



TABLE ni. 

H^e Difctmnt of one ^ound for Days. 



Days. 



170 
i8o 
ipo 
200 
210 



220 

240 
250 
260 



270 
280 
290 
300 
310 



^20 
330 
340 
350 
'360 



361 
362 
363 
3^4 



4f*r 



Lt€9tt, 



,0182894 

,0203972 
,0214477 
,0x24960 

,0235420 
,0245858 

,0256273 
,0266667 

,0277038 



,0287387 

,0297714 
,0308019 

,0318302 

,(^28564 



,0338804 
,0349022 
,0359218 
,0369393 
,0379547 



TT^hT 



,0380561 

>o38i575 
,0382588 
,038360? 



,0205286 
,0217100 
,0228885 
,0240642 
,0252370 

,0264070 

><^275743 
,0287387 

,0299003 
,0510592 



,0322153 
,0333686 
,0345192 
,035667^ 
,0368122 

JO379547 
,0390444 

,0402314 

><Hi3^57 
,0424974 



,0426104 
,0427234 
,0428364 
,042^493 
,0430622 



<J per Cent. 

,0227577 
,0240642 
,0253672 
,0266667 
,0279627 

,0292553 

,0305445 
,0318302 
,0331126 

>03439i5 



,0556671 

,0369393 
,038^082 

j03?47a7 
,0407352 



,0419948 
,0432503 
,04450^6 
,0457516 

,0469974 



,0471218 
. ,0472462 

>O473705 
,0474948 

,04761,91 



6 fer Cent, 

,0271855 

,0187387 
,0302869 
,0318302 
,0333686 

,0349022 
,0364309 
>037P547 
,035^4737 
,0409879 



,0424974 
,0440021 

,0455021 

,0469974 
,0484880 



,0499740 

,0514553 
,05293^0 

,054^41 
,0558717 

,0560182 

,0561647 
,0563111 

,0564575 
,os66o:;8 



* 4 



1. 



7bf 



su 



The Nature, Conftru^ioii, and Ufe 9fthe De- 
cimal Tables of Compound Intereftn 



What Compound Inter eft is, I have already fhewn in the 
Theoretical rart of this Dodlrine $ and &ojn th^ (aid Theory 
it alC) appears that Tables of Compound Intereft are abfolate^ 
necefl^ for thofe who underftand not Logarithms or jilgo^ 
tra ; and therefore (though I have taught the Ufe of Logth 
Tztbms after the left Manner in this Book) yet I have fup. 
plied the Reader with a Set of Six Tables for the Purpo&s 
of Compound IntereA ; I have framed them from the n^oft 
compleat and approved Calculations of Mr. John Smart ; his 
Book (which is voboUy on Tables of Intereft) having the \x& 
CbaraSer for Ejcaflneis, and the Errata% of the rrds^ no 
more than/ofirr. 

As I intend nothing ihall be wanting in any Part of this 
Syftemj to make it compleat ; lb I have contrived thefeT2S>les 
to aiifwer any Qjtffftion of Compound Intereft ^ for the Rates 
contained therein : For thoug)i th<ey are not ^o large as the 
Lnrgeft^ yet are they larger and more univerfal than any o» 
thers, in any mixed Phcet o£ Aritbntetick I have yet ieen; 
I have choi^o. all the moft ufuat and neceffary Rates of Inte* 
Xeft ; and continued each Annual Table to <o Years which is 
farther than is generally needful^ and ihaU ihew how they 
may be ufed for any indefinite Number of Years required]; 
but firft of their Conflruiiionj whidi is thus in the mdS de^ 
nwnftrative Manner aeduced fiOm the Theorems aforegoing^ 
whence not only the Mnrnter^ but die Heafon of their Co9« 
ftruSlion (a Thing very neceffary^ though I know not where 
die to be met with) will be exceeding apparent. 

The Conftruaion of the Ftrft Table which ihews the /*- 
mount of one Pouxfd (ot Daysy as aUb of the Second Table, 
which fliews the fame for Years^ is made from Theorem i. 
of Compound Intereft j which is ?R^ =s A. Now if we puC 
f -: I /. then is the Theorem reduced to S(^ s= A. Confer 
quently, if 12 ==: 1,05 /• per Cent, (or any other Ratitk) «^ 
f;=i>2y3y4, @r. Years; 

Amount 



»H The NatMre, CnfirtiffiM, and Vfe 

Then it i^ be JP n ^ = |.os die firfl Year.' 
Wkich OMlt. by « « ipS 

The Pkodua is ^f 5= ^ = 1,1025 the /if«oW Year, 
And wffin M JK = i>o5 



tmm^m^ 



The Pkodua is ^) ss>n 1,15762$ the fi^M Yeir. 
Ao4 ag^tn bj iC ssb 1^5 



■^^ 



T)ie Piodoa is /e^ = ^::$5i,2i 550625 ibtfcurtb Yesi^, 

^^)d tins for die odier Yctrs fiibleqo^t in die Tables 

Thns alfo if JT «t 1^00013368 dit lli«rr« for t Da]f « 
Mnic as before iP :rr 1,00013368 

The Prod, is i?i ==: ^ = 1,00026738 die ^ssM fbc 2!), 
And^atobyJ? =1,00013368 

ThePlK)d.isi?> -r^;:^ x/XD040|iothei<^9foir^&r3D. 
That is foimii the Amnmif for fiO die fdbft^oeat Days in 

Here it nifbe proper l» obfiarve, that die ^imM ti fy' 
Unft of any Sum, at die lane ttfte^ is more ac Cfrnf^unS 
Inioreft diaii at Zimph^ for any time hIm^ a Kr^r ; ejmal^ 
Mr Kmt; bat /^ fbr Mf tee Ufs than « Trjr. Thou^ 
this iecasa /mmr io the kft Aflertkni^ ycc the iffafoft it 
endent to ai^ im> underflands and confidfrs that ^ple In^ 
Ureft b grounded on ArHbm^i^^^y bpt Coitpw^ Af^</f 

The QMftruSiion. of 7^/^ 3. is by Theorem 2. vi?* hr- 

8B5 P the jpnyrfff fl^oftiSr or Vebie of one Pbund; whidi is 
here to be confidered^ as the .^^jfHomrf ,• therefore if ^ =r i /• 
J(==:i,05 and f z= i, 2, 3, 4, 5, and YetB^fj as before ; 'tis cvi- 
wnt thatC/vrfy, or i, being divided 1^ the Nombers in the /^« 
0Md Table (dd^^ by 8<y ml! gitc the Numbers iq dii^ 
r^iri Table, or the frefeni Vde^nxA ih for die J^tMaf 
Years^ fuid 5 fer Cent, and fi> for any oth^ Rafe of /4rf<^ 



fc.^ "^ 1,157625 J?.a^,865837S>£s2<{5 

0-3 ^1,276281 1^ 1,783326 Jg.^ 4 Yeats, gci 

TYxOotiftruahnot Table ±. is from Ti&WftB 1. o£Jmti- 

ttiet. Sec m Arrears, viz. -2^ ii ss ^. Now as it i< 

U ta I ^ aai J|SB 1,05 a»befote, dien the netrem vJihe 

brougjit to -—J — ^, the ^«ioK»f of i7. Jmtity for 

*cJ*imlier of Yean^fc%i'd bf f ; That is, from ^^^0 the 
Nombew i» die-/*o»rf.3J[*fe; fubftraft U»jtj, or r. The 
gemmfir divided bj, ,05 (pt> H — 1) giycs the N<ndMt«- 
in the /0«rt£ Table. 

Be ample at 5 f ^ a;yf. 



And thus yoa proceed for any dfi&w Rate of latereft. 

The CcttftruaJM of T^i^/^ * i, contained in TBeoremi.i 
cf the prefiftt mrtb ot Value cX. Aatmities, which fees 
Now thefeui Vsail. and putting R ::s:i/)< and t=^' 
Jpt!i'* 4» 5y 8f'. Yeai»i th^ ri5*a>wiiaaiedbterji!iec39iaBi^ 
-j5^=i»^.tbefr^J>«f»>«Hi&fe,gj,t B»tia.Q»ntoiaUt 

oa; of ; 22di«i 4. 'twas flywn diat ^t r, ^ ,<aoiadttd ihet 
SL"^f ,'•"' Ta^'K. Therefore 'tisSnifyl, if tfteNqiM 

Dhide 



124 The Nature J Conflruciion^ and, Xffe 

Dhide the Numbers of the fourth Tahle^ ij the Numbers 
^ the firft Table ^ the ftuotients make the fifth Table. 

TaUe u Table 4. Table 5. 

1,05) 1,00000 (=0,95 238^ ($c. fat the ifl Year* 
1,1025) 2,05000 (= 1,85941, (§c. for the id Year. 
1^157625) 3,1^250 (—2,^2324, 8ci for the jd Year. 
1,21556625) 4,310125 ( — 3,54595, 0r. for the 4th Year. 
1,27628156) 5,525631 (^4,31947j8^- for the 5th Year^gc. 

And thus for any other Rate of Intereft. 

The ConfiruBion of Table 6p is to be dedocM frbin Tbe-^ 
&rcm 2. of the frefent Worth of Annuities, 6^. which fee. 
Now fince in this Cafe P is =: i/. therefore that Theorem 
voSl be reduced to this form, RRt ^ R^ :s UR^ —I/; 
whence (at 5 per Cent.) 'twill be ,05 l(f =s Ujft .^ £/,. con- 

fiqpcmly ^^^inT" ^= 1/ die Annuity required; but this be- 

ing juft the Reverfe of "^J^ , which make the Numbers 

of Table 5. 'tis plain, tbefe two Theorems which ccMiflitote 
the Numbers of Table 5. and 6. multiplied together can n^ke 

.Hience then if the Numbers of Tahle 5, be made Divi- 
:lbrs,.aod Unity or i. the conflant Dividend, the Quotients 
lihall be the Numbers which conftitute the fixth Table, at 
•J fer Cent, and after the lame Manner for any other Rate 

t>f Intereft 

. • ■ 

. ' ' Examfle at 5 fer Cent, 

ll^ 1 • ??85P4i05 i^S.S .,1378049 I ^ ^. 2 h 

a«sC ^ 2.7232480 }>|'=i_g4,3637734^^-^^'? 3 f ^ 

^ e a « I 3.54')9505 i ? «.'i=i ,,2787437 a rl , 4 | 

J -.1 U3294767i«l-f-5 l,22779i6j^.|S,l5 J '"" 

In like manner, when neceflary, may othei: Tables be con* 
flruCled from the Theoroas i tiere are as many Tables as any 
Book (that I have feen) cont;ains, and more, tbui ace in mo& 
My Aim * in the ConflrufUon of thefe Tables is mere to ihew 

the 



of the Dec. TdhJes ofCbmp. Interefli 21$ 

the young Artift the Rationale or Reafon thereof, than the 
Manner h.^w only , fince the latter has been often done, the 
former not at all that I know ot ; at leaft, not in the natural 
Method by Dedu£lidn from the Theory it felf, as 1 have here 
done ir. 

Quomodo faSltim eft P Is a Queflion proper to Mecba/iicks j 
C^r it a fit faciendum P Befeems an. Artift to enquire. 

• • 

The life of the following Tables* 

The [//> of all tbefe Tables depends on this one obvious 
and eafy Gerierd Rule^ 
^ Multiply the Tabular Number, which flands againff Xh€ 
given Number of Days or Years, and under the given Rate 
Ot Intereft, by the given Principal Sum j and the Produift will 
fatisfy the Queftion, 

Example of 246 7. at 5 p^r Cent. /of* 30 Days, or Yeats. 

In Tahle I. againft ^d Days under 5 per C ftands 1,0046182 
• Which mulcipli^ by the Principal $um» — . 246 

The Produ(ft is the Amount required ; . W* /. 247,0684772 

. • « i 4 , ' 

In fnhle II. Againft 30 Years^ at 5 fer Cent, is 4^321^424 
Which multiplied by — . ♦- — 246 

111! il -^ ■! ■ ■,■ 

The Produ£l is the Amou?zt required \ viz. h 1063,1^78 ^c. 

In fdhie III. Againft the fame Time zndJiate^ ^0,2313775 
Which multiplied b]^ -*- — 246 

The Produa isthtprefe?^ WbrWtecpiud^^l 56,918? ©r- 

In Taple IV. For the given T/W and Ildte^ Is 6634388475 
Which nJulcipiied by — — . 246 

The Prod, is the A. of fuch an Annuity ; A 116343,9565*©^, 

. * » • • 

-. . ■ . I 

In Tahle V.. for the given Time znd Ratey^ is 155374^51 

Which multiplied by ■ ^'' ' ' — 246 

The Prod, is xhtfrefent W. of that Ann. /. 3781,6229 8r« 



i-iHi I 



G g In 



i^i TkhU VL For the gitrcn Ttme and Rtttt^ is ,0650514 
Which iQdldplied by — — 246 

Thf VtodaBL is die iwichiLfcd ^^mmij. i 1/16^20'^ 

Therefore bjr die Tthles we immedijgitcljr know, that 246 L 
forborne :jo Days, at 5 ^tr Cent. ^ Awmm Ctrnf^undlnr 
iereft will ammifi to 247/. ^ j. 4 i rfl 

Ihat 246/. for^me 50 Tearsj at 5 f /?r r*»/. te. will A» 
PjfGuat^ to 1 063 /. 5 X. 1 1 i i/. 

* That the frefeni Jf^rth of 246 /. due 50 Years hence, 
%t |h^ l^f^ ot 5 fer Cent. *c. is 56 /. 18 /. 3 4 rf. 

That the Afndunf of an A?i?mitj of ^4$ /. /v?r Annum^ 
^rlfopi^ or ui^d 30 Yew, at 5 //?r C?«f. &c. is 1(7245/. 

That the frefenf Worth of an Annuity of 246 /. to con- 
Aiuc 3p Jt4ri, an 5^ />f r C^^r/, />rr -r<«w«!p is 37Sx *'. 
12 i. 5i</. 

Ta^tt t)ae .AMMTiIfy ^rf|i(cb 146 i wfll purdi^de, to contMue 
5<? Years^ nckoaiiig 5 perCknt. list^r^fi^ isi6l4fer An- 

mm- 

JSi djc Amount of any Sbtw be fought, for a Number of 
Days which are not in the firft Tahhy and Years which arc 
^a^ ill xh^jacoajy obfesy^ cbi^ 

£ule ; Divide die givcft Number olDaySj or E«jr5, into 
Ijvo fiich/Npmbers 2s asc in the Table, thc)n.0Dwlriplw the A* 
mMU Refnijiirig to eadi, into each other • then tbrn the Pro- 
duA be the Amottnt for the Time required. 

Example i. 

VVlrttt^.5?ii 0Mffiirf.CQ» i#i.l^4fltojtt, at 5 per. Cent. 
fer Annum / The two Parts ot this Number in the TaBTe^ 
arc 1^0, ^qd a ; tlwrtfore 

' ^ takkt. Againft ipo Days, und^r^ <j>er (:. i$. 1^,0257228 
An4.aggii]ft 4Days, at die lame Itate, is 1,0005348 

The PKJds 15 fhe yAw. of i /. for IP4 Days, viz. 7^5262714 
Which multiply b^ the Frincipal Sum, wz. 523 

Produa is the Anfwer „ /^ «^,7S^840 

^ * ^ * ' ■ ■■ • * 

E>:empl^ 



of the Dec, tables ofContp. Arejf.' ivf 

t 

Example 2. 

r 

What is the Amomt 6f 150 71 Ih $1 Yekr'^, kt 5 J!»^r 

^ ■ 

fa ihiU It A^infli 50 K?ar;ji un*r 5 /^^-r C* is 11,4674000 
And againft 41 Teari^ ic ^ /^r C is y»:{pi p€8i 

The Prod, is the Arn. of i /. for ^t Yii^rs, M*i:. 84^§S3J 
Which iliukiply by the Frtncifal Sita^ vfe . . 15^ 

•htfe iProdua is the Aiiftver — /. 12715,^3249$ 

In /bfo^r^jf 12715/. Oj« 7 ^^. 

Whut tHIt ^23/. rn^wa to ifr 5 y>iri^ diid 194 i}/ijus V 

lu riW^ II. againft 5 Years^ at 5 /»^ G?.vi. is 1,2762816 
And thi k^. of it iA 194 D^iy^, a^ Aovd, Is t,0i6itH 



i ^ m. wi ■ -^ 



The Pr. is the Anu of i /. in 5 K'^afr/, afid 194 /). i^JopSi 1 3 
Which multiplied by the Principal Sum — 5*3 



«| n il* * 



. TIte Pr(3di}£l is the Ai>fwer^ r/ic . --. /. 685,03 1341 3 
Ift A^<wrfjf 66^/. 3 i. 7 i /*. 

* • 

ATI A The other Tables of Compound tnierejij as thqp 
cannot in this Manner b^ ej^ended, 10 chey feldocn require it, 

I ffiaJl now pi^e(cnt Ae R^ddr wfth a fetv Quiifidtts of a 
fnorjc cbmfhk t^Atute^ and which fre^qtienftfy h^ppeo, itf ()N 
der to thew the t)iore ekUyifivi f/fi of the Tables. 

Que ft ion t. 

9«{7(^ 1 htve 706 j^. to be {Aia m^ witht^7 TeiiiF»;iit fhii 
Manna: ; at th^ End of the firft Year 90 U of two Years, ioo /. 
6f fbuf Year^ ^0 k and of feven Ycar» fOO 7. Qufcre vAnk 
the prefinf Wofib of cbofe feveral Payomity ii vxria^j Mo* 
fteyf afowifli^ 4 i ^ CenUQompvkni Mere ft f 

G g 2 In 



2^9 The Nature^ CpnflruSion^ andUfe 

In Table III. the frefent Worth of 1 7. at 4 J per Cent. 

Due at the End of i Year, is — ^5,95 69378 

Which multiply by the Princifal — r po 

The Produa is the frefent Worth of 90 I. = 86,124403 

Thus the prefent Worth of loo/. due at 1 r-r^o/t 

the End of two tears, is found f "^ P^5729P 
Alfo, if 20o7. at the End of 4 T^^rj 3- 167,71226 
And of 4C0 /, at the End of 7 lV/?rf = 299,93140 

The Sum of all thefe is .^ /. 639,341051 

Which anfwcrs the Qucflicn, viz. 6^pL 6 s. 9! <<. 

Queftion 2. 

^ owes to B 4<;5 7. to be ^ii/V in i^ Years^ viz. at the End 
of every 2 Years 65 7. But he would agree to pay him in 7 
Yecrsy hj eqnal Payments each Year ; which B agrees to, and 
at the Rate of o fer Cent. Comfound Intereft. Quere what 
the Annual Payment muft be ? 

!• Find the prefent Worth (by TiW^ III.) of the 7 Pay- 
ments which v/ere at firfl to be made, as per Quefi. i. 
which you will find to be 293 L ^s. id. 

2. Then find (by Table VI.) what Annuiijj to continue 
7 Years' at the given Rate^ 293 7. 5 *. 2 rf. will pur- 
chafe ; which you will find to be 52 /• lO s. 8 d. and is 
thf Anfwer to the Quefiion. 

Quejiion 5. 

. "^ has a Term of. 7 Years in an Efiate of 35 7. f^r ^«- 
W»;w. j8 has a Term of 14 Years in the fame Eflate in B^r 
verfion after the 7 Yicars ; and C has a farther Term of 20 
Years in Rtverjion af^cer the 21 Years. Quere the prefent 
Values of the fiveral Terms, at the ^ate of 5 />^r Cent, per 
Annum P 

By 2it*fe V. the prefent Value of 3 "J 7. /vr Annum y may' be 

' l. s. d. 

found^ for 41 Years, to be .^ .—- 605 — 6 — o» 

for 21 Years, to be 1;^- . 44S — 14 — Pi 

for 7 Years, to be -r: t- 292 *— 10 — 5i 

Which 



\ ' m % 



2000 



of the Dec. Talks of Comp. Interefl. 2 19 

Which fubftra£l from each other, it will appear, 

/. J. </. 

That the preftnt Value of A'i Term is 202 — 10 — Si 

of £ s Term 246 — 4 •— 4 
cf C's Term 156 — 11 — rj 

For thefe Values anfwer the Qiieftion /. 605 — 6 -^ o> 

Quejiion 4. 

Which is tnort advantagious a Term of t«; Years in an R 
ftate of 100/. fer Annum ^ or the Reverjion of fuch an B^ 
^<2f^ for ever after the Expiration of the faid 15 Years ; com- 
puting at the Rate of 5 fer Cent, fer Annum Comfound In;-' 
tereft P 

An Eftat« of 100 7. per Annunty in Fee\ j 

Simple at 5 fer Cent, is M'ort/j — f '' ^^ 

Li T/aW*? V. thtfrefent Value of the fameL , , ^ 

£j?^r^, at the fame Rate^ for 15 Years, is J ^- 1037,965s 

The DiiFerence is /. ^62,0542 

Now this Difference being the Value of the ReverjioVj 
it appears that the firft TerAi' of 15 Years Is better th.tij the 
Reverfxon for ever afterwards by 75,9316 7. = 75 /. 18/. 
T ^d. Anfwer. 

Queftion 5/ . ^ 

A Perfon having 1 2 Years to come, in a Leafe of an E- 
ftate of 60 1. fer Annum for 40 Years, would know what 
frefent Money he rauft pay in order to renew or comfleat 
the Leafe by adding 28 Years thereto, computing ;it O fe/ 
Cent. Comfound Intereji / . . . 

By Table V. the frefent Value of lifer \ j t< o^^-'or 

Annum^ at 6 fer Cent, for 40 Years, is j ' ^' ^ -^ 

Bv the fame Table the Value of i /. fer \ j c^Q^fi 

^«. at that Kate^ for 12 Years to come, is / ^'3^3^ f+ 

The Difference is 7. 6,66245 3 
Which multiplied by 60 

The ProduS is the Anfwer, viz. — /, 399,747180 
In Monej^ ^99^ X4 x. 11 ^; 

Queftion 



4Jd the Natkte^ ConJlruSiiony andVfe 

CiueftioH 6. 

A gives 15^0 h for xn Ammitf oF 100 /. fer Anrntm fof 
56 Years. B puts 1550 1 otit at Inter eft. It is required 
to know whteh will amount to the g" eat eft Sum at the End of 
Ae 50 Years, at the ftdte rf 6 /. ^^ Ceni. Sc Compoitju/ 
TnlereftJ? 

By T/iWif IV. the Amumi of k30 7. 7 

Annuityi in jo Y^aiS at 6 f #r 0»l» ^ it 2^ Jlt5^46 
nay be round to be --f- — 3 

By 7^f /^ It. it Qiay be foufid^ that y 

the Amount of 1550 /. for thac tW tnd V /. 28551,2388) 

jRdte will Be — — 3 . 

Hence ^V Annuity is oi(»e than S a 1550 /. by h 48^2,35161 

at the JEild of 50 Yearj. The frefefit VMe of which Xy^ 
ference is found, by Tahtt Ml. to bfe i^t. ^^. 8 i rf. kild fo 
touch was ^'s Cafe bef9&¥ tbski i^s, 

auefiion 7, 

tVhat Annuity to continQe 14 Years, iskay be purcbafed 
with icoo L d«e at the end of 5 Yeass \ the Annuity to coin<^ 
ibeiice prefently, at 5 /. fer Cent .^ 

By Tahle III. the frefent Worth of O 
looc/. due 5 Years hence at 5 ;p^r C>«f. ^ = /. 785*5262 
way be found ■— — 3 

By TM4 VI. *t may b^ fdun«, th«» the 
Aftm^ty \vhtelfi^:;,526^/, #iU fiirchaffe 
fer 14 Yearj, at the Rate ot 5 ^tf>» G^»l. is 

In Money y 79 /. 3 i. 4 ^/. f ^r Amti% th« AflfvVer. 

QtefitoH 8. 

For a f,^^/'^ (5f certain Pr^jff/ for 7 Years; -/#, mstosfi^w 
Offers, eirlier to fay 150/. 2i^^Fine^ and :^oo L fen An- 
nuni^ or 1760/* r/^^* without any Rent. 5, bids 650/. 
I'lrV?^, and 20C /. pet Annum. And C, offers 200/. Ft«^, and 
465 /. r«r Annum* Qocre which fe the^//? Offer, and what 
the t'lffi-rence^ coroputii^ at 5^/. fer Cent, ftc Cttmfounaf 

Infer eft / _ 

I- By 




of the Dec. Tailes tf Comp. Jntevefi. a 3 % 

1. By Tahk fl. th« v<«W!ir^ of x%ol iai , 

7 Years, auc ^/itr 0«/. may be found to be I ^ 211,0655^ 

By Table Iv. the Ammt kA ypLfarJ 

4nnHm\x\ 7 Ycaxs ^t the yveoj^atc a»y> i 242,602s 
o^ found — «, ^ J ^ 7 J 

Therefore A% Offer, at the End of 7 Years ^ , ~ ^ 
itOMld H ~ — -^ r ■• *4«>^4 

2. By37^J/^rII. the yfoiwa^rf of 1760/. iol - — ^ 
Years (^^s fecond Offer) at the iaid iS^^^f, ^ ^. 2392,0802 
fouiidto.be — — — J 3^*»^ 



7 
is 



< J » ~( 



mount 



:?. By TiW^ H. the 4^w©a«f of 650/, in ^ 
7 Years, at the %\yzyiRatfiy «riU be found C /. 014.^18^ 

to be — .^ y 

By Table TV. the .^iiwowf of ^pp/. ^^r> 
^»««/w in 7 Years, at that fi/rf^, will, be > /. 1,6^.40 w5 
found to be — J ?t- . 

Therefore B*s Offer will, in 7 Y^ars, a->^'T '"^ 

ount to — J* ^- 2543,0205 

4. By Table 11. the Amount of aoo /. in 1 

7 Y«rs, at the given Rat^y will be found > /. 281,4212 

co De — — f 1 ^, _ \ 

By JJzftfe JVi the jtrnminf of 405 /. /^r^ 
ii^ifuflw for the gi^en Time and i?/i/^, willf /. 2207 ^122 
be found to be _ ^ 3 

So that C*8 Offer, in 7 Years, wiB amount to /. 3578^93^ 

u'^J'^"^''*^^!'^^^^^^ ^^ ^^^ f^^^ Oi%"^ at Ae Endlf 
thefaid l^rip, beiog thus kapwn ; the Prffejff Worth of the 
fejei^ 4VIPHVU ms be found by Tahl^ Ul which arc as 

The prefect ffortb o^A's firft Offer will be 1 885— 18—05 

A^ f^tAiOSst ^ i7i)o--*Qa-^oo 
^> S?' ~ 1807-^05^06 

t< s Utter — 2S43— oo-™o8 
Thwe&te die /^-//^ ^<,rf* of what C <#rs is more 

than - — ^s fifft Offer, by 657— «_< 

-rf's fecond Offer, l^ 849— 5> 8 

Which fully anfners the Quefiioa ^^ 

N.B. This 



t$t The Nature^ CoftflruHion^ and life 

N. B. This . Queflion might be mord readily anfwered by 
finding the ftefent l^art&s of the fevdral ofFer'd y4v-» 
nuities fas fsr Tdhk V.) and adding to them the 
feveral Fines ; as the Reader may try ac his Leifure. 

Quejiion p. 

What Annuity is fufficient to pay off a Deht of 50 A////V- 
9fts in 30 Years at 4 /. per Cent Compound Intereji P 
In Table IV. againil 30 Years, under 4 per C. is 0578501 
Which multiply by the Debt — 5000000^ 

The Produft is the Annuity fought y viz. /. 28^1505 
per Annum. 

So that fuppofing the National Deht to be 50 Millions^ 
and the Intcreft paid to be 2 Millions per Annum ^ or 4 /. per 
Cent, then will a Sinking Fund cf 8^1505/, per Annuniy 
clear the whole Belt in 30 Years. 

N. B, By this Ejcample appears the Neccffity of continuing 
the Tabular Numbers to fo many Places oi Decimals* 

^eftion 10. 

Suppofe one Farthing had been lent at Compound Intereji 
at ji /'^r Cent, in the ^r/? Ff/ir of the Chnftian JEra^ or 
liirth olCbrifty and fo continued to this present Year thereof 
1734; Quere the Amount thereof? 

N. jB. Though this Queflion might be anfwered by Ta^ 
hie II. as I have before Ihewn, yet I Ihal! here ufe 
Logarithmsy as mofl expeditious in this Cafe. For 
having faid enough about the Ufe of the Tables, I 
here intend only to give the Reader a hint of the 
furprifing Nature of Numbers in Geometrical Pro- 
portion. 

I 
Therefore, Tht Logarithm oi i^it Rate i,o5:=ro,C2ii8p3 
Multiplied by. the Time — — 1734 

The ProduS is — — — 36,7422462 

To which add the Logarithm of i Far^ j 
tbi^gj or the ,0010411^ Part cf a Pound, C = 7,0177288 

The Sum is the Log. of the Amount fought z= 33,75^^750 

Now 



sftb^ Decim.T^hks of Camp. Intereft. 2ii 

Now the Index of tb& Lognritbm being 93, Ihcws the 
Number ot Figures, of which the Amount of on^ Farthing 
in the given Thne doth cOnfift^ to be :}4^ of which let it bS 
(uf^ient to express the 4firft in Figures \ the Reft in Cyphers ; 
then will the fiid Amoint be 

5754OOCDOOCXX)OO00OO0OO00CXX)00OO0O0O /. 

Now the Value of a /o//</ ^oflFy, fetfeElIy SPbericai^ 
wKofe Diameter is 8006 En^lijb Mjles^ { which is lomewhat 
^'/[^^^r than the Diameter ot the G/o^^ of our Earib*) I lay 
fuch a folid Body of fine Gold would be in F^/n^ about 

23866oodcxx)o6ooboooooooocxDOo /• 

Now if firortt ^ch of thefc^^4f Numbers^ be cut off 23 
Cyphcr^j the reniaining Figures will be 5754OCX50000 in tbt 
jimtiunt ot the Fartbtng ; and 23866 in the Value ot* th6 
iJ/a*^ of GdU. But 23866) 57540000COO ( =s 240iDoo6 
nearjiy. 

Hence it appears, That one fingle Farthing put put to 
Vfurj in the Manner af mfaid would /mount to more in V^Ue 
than two Mil/ions and four hundred Thoufand Glbkes of 
finefoUd Gddy each higger than the Glole of the E^h ! 
hftrange and furprifingy buj no lels certain Truth ! t^xA 
this immenfe Amount would be greatly increafed by inlar^ing 
the Hate of Intereft. 

I ihall now conclude this Part, by prefenfing the Reader 
with a fmall Tahje concerning the frejent Worih or Value of 
teftates upon uves^ with its Ijfe j This table was at firft 
composed by the Great and Learned Dr. Halle j^ for every 
Fifth Year of Age to the 70th, as follows. 



I 


r^^'* 


■^" 


■"I^tfr* 


50 


ttiar't 


Purebap 
10,-28 


•38 


tvrcbdfe 


Ihtrch^e 


9,21 


5 


1540 


30' 


".7^ 


55 


8,51 


io 


»5}44 


35 


It«I2 


60 


7,60 


15 


13.3? 


40 


io»57 


65 


(^M 


20 


_i2id_ 


-ii. 


9,91 


-22- 


-...,?fp „, 



H h 



tht 






• » 



^. Suppole a VtxTon of 50 Yeiil of j^ oflfeti t<h ftll 1^ 
tifc in an Inflate of 46 /• /rr Annumy what is the AW»r 
thereof kk i^4^ %$&^j, 

Hifc Age <X ^6, is Ytwrs !^h*fc .-* p,2i 

II1K PfodttCt M the Anftm 441^ S3 1* ^ 4 <^. sic 4a),66 

ffit happeh Jim a iiftands iPeoeiglfiflnr For fo many Years 
after, be offered ; 'twill be neceflary to reduce the Year's Pur- 

^h iiM YfAx94mmM^: te Y«m» tft a Li^yis ^y t^^ ^/fifr' 
}tt»%^ ttei. Si4<t)ofe 1 woftiid find what NtiKber of ^^taiit 
Vtet%«(Wt%ll>oiids «6 »0)V7 Y^itts Furehift^ »tdit^ C^^. 
ticdc &A l»/> V. Odtef tte ^» iP^Hf) and I dodth^mxe 
near efi Value ciAnnutijoi i/. fier Annum to 10,57^ tob*^ 
•b,4772^97, oppaflte o^'fiilidi is rt Year*, mhidk irf^to be 
idodi totiit YsaftiiiiiRtMrjfoj^, Md timi ^di^ CaTe i$ tim^ 

Whit fo dil^ fnk9tf W&rnr^i «A iSfet^ bf ^t «?f 
nr^MMM ttelt it«^ Ar ^ Y^aift «fter die 2)fWi& df a IPe^n 
40 Years of Aff^ at 6/w C«r»f / 

Thfe Age^ 40 Yt^ 48 to,57 Yca» t>OTchaft, tririck in 
%Vle V. ^v« 17 Yta/s oeitath tb cbme at 6 Wr C^tt. 
Then 2© + «7 i± J7 Yeats. 

Tberefoie the fw:f«rf «p^ /. ^ftr 37=14,73^804^ 
prAri>ium lor Aie:givcn j^i? -i- jrtoM7j=ic,|.7t25P7 

'fhe frgpm It^ortb'dl x I. Per Amum ifor «o= 4J2«^<;207 
Whidi ttultifly by the ^«»»*y — ^ ^^ yg 

'tlir Fvod. is *♦ Atfwft ^ajt^ 4 *. n k. -ai /. 232,^4265 



TABLES 



m 

TABLES ^Compound Injthrest. 

Table I. 

Tie Ameant ^ «M Pumi fiir Qijr< i tt tit 
Rates (f t; li; i; ii; nni;i, and 
< (ler Ceot. p?r Aomjn). - - - - 



flflj. J /«• C. J i /<r C. ,5 ftr^ } {/^n- C. 




Hb 2 



»3tf Decim* Tahfes of^Iomf^ latertfor Pays, 

T A B I, E J, 
^e Amount efone PounJ, Compound Intirefi* 




I 

2 

I 

6 

7 
8 

19 

20 

49 

U 




1OOOIO74 
,0002149] 
?/)Op9224 ! 
1OOO42PP I 
JPOO5374' 



/>oo6449. 

f)d07524l 

/X)o86oo 

1O009675 

iOoioT^l 



10021513 
iob32288 
,0045074 

,0053871 
/}o6468o 



,0075501 
/>08(5533 
,0097177 
/>Tc>8o33 
,oVj8!900 



f 129779 
>o 1 40670 

,OKI578 

,oi«2487 
i,<»f734i2 



.4 i /«• ^- 

1^0001206 
1,0002412 

i/>oo36i8 
1^0004824 
I90006031 

1,0007238 
1,0008445 
1,0009651 
1,0010859 
1^0012066 

1^0024148 
1,0036243 
1,0048354 
1,0060479 
1,0072618 

1^0084773 
1^0096942 
I10109125 
^^121324 

1,0145765 
1^0158007 
1,0170265 

»i>oi82537 
«i^oi94824 



5 fer C. 

< 

1,0001336 

fiOO^Oli 

,0005348 
,0006685 



l^er c. 



o^^^m 



mm 



»|- 



,0008023 
,0009361 
,0010699 
1OOI2O37 
^3376 



PO2677O 
,0040182 
,0053611 
PO67O59 
,€086515 



,0094009 
^107511 
^121031 

^13^569 
,0148125 



ioi6i 699 

,0175291 
,0186902 

^202571 

,0216178 



1,0001596 
1,0003193 
1,0004790 
1/X106387 
IP007985 

1^)009583 
1,0011181 
1,0012779 
1,0014378 
1 ,001 5976 

1,0031979 
1,0048007 
1,0064060 
1,0080139 
1^0096244 

I ■ ^l 

1,0112375 
1,0128531 

I.OI447I? 
1^0160921 

i>oi77i55 I 

1^193415 
Ij02097ai 

'i)022&a3 
1,0242(351 

3»025a7I5 



«( 



T A E L H 



Dei^Wi Tables &f Comf,:inte.ufor Dayu %n 

X A B: L B I. 
7%? JmottMt of one ^ound. Compound Interep. 



:i 



Dttft. 

YJO. 

ifio 



■ -^■'^■•i 



. 210 

• 

240 
250 
260 

270 

280 

290 
300 
310 

320 

330 
340 
350 

360 
362 

363 
365 



2 /?<r'Ci 



«p092658 
j,oop8i55 
31,0103^1 S 
x>bippop8 
j[/)U4584 

;I^IXP073 

•1^1310.60 
^^142055^ 

i/>i4T5^3 
1,^155070 

1,0158580 

i/>i^4093 

i/>i75i27 
j^oi8o64$> 
j,oi86i74 
1,0191702 
.1,0197233 

,1,01^7786 
1,0198^40 

1,0198893 
r 1,0199446 

1,0200000 



2 Ifer^C. 



iferC. 



, » 



,oii5$70 

,0129366 
/)I3622l 

/)i4|o8i 

|0I495^5 
pji 568514 
,0163687 
,0170565 
,0177448 



,0184336 
,0191228 
,0198125 
,0205026 
,0211932' 



,0218843 
^225758 
,0232679 

,o2g90P3 
,02465^3 



1^0138623 
^,0146837 

«iPi55057 
f ,01 63 28^ 



■rs^ 



,0247226 

,024791? 
,0248613 

,0249306 

,0250000 



^^0^79759 
1,0188006 

1,0.196260 

1,0204520 

1,0212788 

ffi22f062 

1,0229342 
?, 0237630. 
1,0245924 
1,0254225 

1^0262532 
1,0270847 
IJO279I68 
1,0287^95 
1^0295830 

1,0296664 

1 ,0297497 
,i#029833i 

1^0299165 

1)0300000 




I,t>i6x5i6 

1,0171098 

1,0180689 
1,9190288 

*,PI99897 



^-^m 



1,0209515 
1,0219142 

i>*>228778 
1,0238424 
f ,0248078 

«>025774i 
«, 026741 4 
1,0277096 : 
1^0286786 
1,0296486 

1,0306195 
1^3159x4 

1,0325641 

'»<^33'5378 
1,0545123 

i>0346p98 

i>0347073 

1,0348049 

1,0349024- 

1,0350000. 



The 



Ijf Qecim, Tahhs tfCom^* Jiair.fir D^u 

TABLE T. 



iftrC. 






/>26l24) 

/>272ftT5 



WHi* 



P505445 

/>? 27614 
,0338717 



^0349832 
(03<io96o 

jOJ7«o?9 
(0383250 
,03^4413 



,0396648 
,0397765 

,0396862 
,0400cx>o 



i/>at944< 
y/>23t774 

IP«5M( 

^f>2<8858 
k/>38i24^ 
I/)193655 



f^ 



^10350963 

i/)3559io 
k ^68406 

«/>38?9«7 

»/>593444 , 

1^405985 
1,0418542 

T/>43Mi4 

1,0443700 

1,0444960 
1,0446220 
•1,0447479 
1,04^39 
'1,0450000 






/>229t43 ' Tf>^io5 
/»4W27 

,0257228 

iPaT09*9 
^284687 



^298444 

/)3^6en 



/??fiy505 

^0395259 
,04091*1 
,0423087 

<o43yo2^ 

i04S«*P0 
^64969 
^0478967 
10492984 



ii^a9i522 
Vf>307964 

»^ja44?5 
</>14P928 

fPi739^ 
}fii90^72 
1/)407I73 

«P4X38oo 

1,0440454 

f»0457i35 
1*0475842 

1^0490576 

• I •' — :^^ 

4,0524124 
€,0540938 

»f055777P 
i/>574647 

V59I542' 



♦»-* 



Wfc»* 



;o4943e7 

,0495790 
,049715^3 

,0i^^ 

,0520000 



^0593 23 5 
1,0594924 
1^0596616 
1^0598308 

1,0^00000 



TABLE 



TABLE II. 

^/COMPOUNDlNTlREST. 

7be Amount cf one Pound for Years, ttt the 

Rates «?/» ; » T ; 3 ;„3 T i 4 i 4 T i 5» and 
^ \ ^x Ceat* per Aoaum. 



-BmBm^H 



Yearu 
I 

2 

4 
6 

9 

lO 

II 

I!2 
'? 

14 

l6 

i8 
IP 

20 



21 
22 

*3 

24 

a5 



2 ferC. 



ritairihKHAi 



1,0824:^21 
1,1040608 



» i fr C. 



1,0200000^*70250000 
1^506250 
r,07tf8po6 
1,10381*28 
1,1314082 



3 



1, 1 25162 
1,148685 
1,1716593 

I,»P5P9X5 
MW9944 



*■ fc 



»>«43374? 
1,1682417 

1,4^36066 

'.««J»4787 
»,|45«683 



^■■*Bi^a«a 



i*572r«S7 

1,4002414 

1,4282462 

14^681 1 1 

^♦»5?474 



«ji 596934 
1,1886857 
ti2i8^29 
1,2480629 

1^2800845 

If? I 20866 
1,3448888 
1,3785110 

»>4' 29738 



■**iMi 



14845056 
1,5216182 

»»«9<5587 
1,5986531 

>, 63186 164 



1 iy$t^66i\ i,679'yBt3 
*yH^9796 1,7215714 
I,57«8?92 1,7646106 
1,6384374 i,B687259 
1,6406059 r,853944i 



1,0300000 

i,b6opooo 
1,0927270 
»>> 255088 
1,1592740 

1,194852? 
1,2298733 
1,2667700 

»»3«>4773i 
«»3439«63 

'»38423?8 
1,4257608 
1^1/585337 

»>5^ 25897 
«>5175*74 

1,6047064 
^,6528476 
1,7024330 
i»7535o6o 
1,8061 1 12 



■taM 



.^ - ^fc 



1,8602945 
1^9X61034 

2,0327941 
2P937779 



3 4. pefC. 

;r,035oooo 
'^0712250 

i,»087t7* 
1,1475230 
1,1876863 

1,2292553 
1,27x2792 
1,3168098 
1,3628^^3 
1,4105987 

»yf599697 
1,51 10686 

»»5^i956o 

1,6186945 

•,6753488 

1,7339860 

«.794^55 
»»8574^2 

1,9225012 
»,989:«88' 



2,05943^4 

2r»3l5ll5 

i,io6t 144 

2,2933 2% 
2,3632449 



TABLE 



440 Deem* Talks of Comp, Iiiter. for Days* 



TABLE li. 



> i 



* - * 



The Jmountofone Pouni^ Compound Intereft. 






Yeats. 

I 

2 

4 
5 



I 

9 

10 



II 

12 

£5 

»7 
18 

IP 

20 

21 

22 

23 
24 
25 



. f t 



M •»<». 



4Pff C 

I/>40pOCX> 

j,o8i($ooo 
1,12486^0 
i>i 698586 

1,2166529 



i>265?i90 
1,51 59? i8 
1,3685691 
1,4233118 
1,4802443 

1,6010322 
1,6650735 
1,7316764 
1.8009435 

1,8729812 

i,P47P005 
2,0258165 
2,1068492 
2,191x231 

2^2787681 
2,3699188 
2,4647155 
2,5633042 
2,6658363 



4 » ^C. 5 f£TC 



,0450000 
,0920250 

^1411661 

,1925186 
,2461819 



,3612601 

,36086x8 

,4221006 

,4860951 

,5529694 



,6228^30 

,6958814 
,7721961 

,8519449 
J9352824 



:^o.22370i 
2,1123768 
2,2084787 
2^3078603 
2,4117140 



2,' 



,0500000 
,1025000 
,1576250 

,2155063 
,2762816 



,3403956 
,4071064 

^4774554 
,5513282 

,6288946 



;202III 

1336520 
2,7521662 

2,8760138 
3,0054344 



.710J393 
,7958563 
,8156491 
,9799316 
2,0789282 

2,1828746 
2,2920183 
2,4066192 
2,5269502 

2,6532977 

2,7859626 
2,9252607 
3,0715238 
3,2251000 



6 fer C. 

i,p6coooo 
1,1236000 
i,i9«oi6o 
1,2624769 
1,3382256 



1,4185191 
115036303 

1,5938481 
1,6894790 

J ,7908477 

1,8982980 
2,0121965 
2,1329x83 
2,26090^9 
2,3965582 

2,5402517 
2,6927728 
2,8543392 

3>0255995 
3,?07i355 

3,3995^36 
316035374 

3i8i97497 

4,0489346 



3>3863549l4>29i8707 



TABLE 






iDrr. T4lhs 9f<:imr^,TnPeu fw %oTeitrs, ih^ 



t A B L E H. 

'^bt Amotiiit of diie Poumi^^imfoundlmtn^. 



•aa 



2«arsitfirCetit. 



225 j 

3» 



MfttfMMh 



3* 

.34 
IL 

:j6 

i7 
38 
39 
4« 



+1 

42 

43 
44 
45 



4<J 

47 

30 



i,6734.i^st 



1 j. ^r-C. k ^ Gw/. 



, i,9*ofe9i7j^,t5^59i**>44-^9$*^ 

i-,7o58<6i4) iv947Boioi(y z^lara^^^ 2»5345*y i- 
tiif^t-idft^fe' i,'^^6^50 tt,tft7$t7iS %v^2«)i^i9i 

4»574«44^ iM64075' «>3^f*l5 2»7«i8^^9l 
4.,8'it8^.r^ft<o97^^f5J6,i|.c.7i6fe4J2y8og7^|7 



^ > fo- c: 



1,^7988^ t,i5oido67[2)5()ooSe^^)$Oigo^!i4j 
7-,964S«l»$ 1,2637^^9 fe,5750li9 5^*0676^5 
4v9>z2tt;i^S Q)<z|;8S§6S B>652))$S ^,i<ii94£3 
1-^0(6^ ft>3l5jilt ft,78i9d5.V3»22£)8663 

-S^£? **3£3f^ 22I136SM ^^3335 ^&4 

A,039«875: ft,4825S5|'M0«i78'«i 3,4^2«6i 
-^2^<»68j53oi ft,4^3:34B 5* {^98:58^66 3,57ida|;4 
2,i^Mc^87^4»55^6ea4h,€>74783«a3i60oci3 1 



a, i<)f7447 



Mmb.; 



a»6t95744h>t^70iS^3»8a53'73 7 
2,685<&638h,4^20377 3,95925.97 



B»i5iaoo4; 2,7521904 
2^9794441 a»«209952 
2,343^893:2*8915500 
^*39°053' 2<963So8o 
2-437*54243.0379032 



2,4866112; 
«.5363435; 
2^5870703; 
638(81471 






H|^to<*-. 



3»ii 38508 
3,1916971 

3,2734895 

3»35-327^S 



}»3598$894i0978338 
?,46o6958j4,2432579 

.5645 «^74»3 897*^ «» 
4>54534i^ 
4»7o^3585 



5,^714522 



^,895043 7 



69i58AaJ3»437jo87 



^13225ff 



4, 8^6944 1 



hO 1 1 8950 5^6372840' 



,-,2135889 



1^2^62 194 5,3960645. 
(^3839o6o»5,5849ae8 



MM 



■B 



I i 



TABLE 



242 Dec^ TaSJes ofC(mff. Inter, far 50 tears. 

TABLE ir. 
The Amount of one Pound j Compmnd Intereft. 



]%tf»x.4 fer Cent. 
4,7724697 



i6 

a? 

28 

29 

30 



3« 
3* 
33 



2,8833685 
a,pp87o35 
J,ii865i4 

3 »*4?3P75 



3»373i334 
3,5080587 

3,6483811 

34 i3»7J>43»<J3 

35 3»94<^o889 



1; 



4,io3P3a5 
4,268o8p8 
4,4388134 
4,6163659 
4,8010206 

4,pj>3o6i4 
5,1017839 
l5,4i!>04952 
5,616515c 
5,8411756 



3,1406790 
3,2820095 
3,4296995 
3,5840^64 
3,7453181 



6,0748227 
6,3178156 
6, 705282 



3*^138574 
4.0899810 

4,2740301 

4,4663615 

4,6673478 

4>8773784 
5,0968604 

5,3262192 
5,5658990 

5,8163645 

6,0781009 
6,3516154! 
5,6374381! 
6,9361 2 29| 
7,2482484 



5 /«r C. 



3.55y<?7»7 

3.73 345<J3 
3,9201291 

4,1161356 
4,3219424 



4,5380395 
4,7649415 
5,003x885 
5,2533480 
5,5x60154 



6 fa Cent. 



4*54938*9 
4,8223459 

5,1x16866 

5,4183878 

5.743491 i 



6,0881006 
6,4533866 
tf>84o5 898 
7,2510251 
7,6866867 






5,7918161 
6,0814069 

<J.3854773 
6,70475 1 1 
7,0399887 



8,1472519 
8,6360870 

9,i54»523 

9,7035074 
10,2857x78 



7,3919881 

7,7615875 
8 f 1^96669 

8.5571503 
8,9850078 



7,5744196, 
7,9x52684' 

8,a7»4555i 



6,8333493J8,643^7»oi 
17,106683 3 (9,0 J26362'i 



9,4342582 
9,905971 1 
0,4012696 
09213331 
1,4^74000! 



io,j>o 28609 

".55703a<5 
12,2504545 

12,9854818 

13,7646107 



14,5904873 
15,4659166 

i<5»39387i<5 

'7.3775039 
18,420x541 



TABLE 



^ 



243 



TABLE IIL 
^/Compound Interest. 

7 he Prefent "Worth of one "Pound for Years, at 
the Rates of % ; 2 4. j 3 ; 3 4 ; 4 j 4 4. ; 5, 
and 6 ; per Cent, per Annum. 



Years. 

I 

2 

3 
4 

5 

6 

7 
8 

9 
10 

II 
12 

I? 
14 
15 



■n- 



16 

17 
18 

IP 

20 



21 
22 

»3 
24 
25 



2ferC. 

,P803P2I 

.^9611687 
,9423229 
,9238454 
,9057308 

« 

,8879713 
,8705601 

,8534905 

'53^7552 
,8203483 

,8042630 
,7884931 
,7730325 

,7573750 
,7430147 

,7284458 
,7141625 
,7001593 
,6864307 
,^729713 



1 ,9756097 
' ,9518144 

,9285994 
,9059506 

,8838542 

,8622968 
,8412654 
,8207465 
,8007283 
,7811984 



■r^ 



,6597758 
,6468390 

>634i55P 
,6217214 

,6095308 



"•-i* 



,7621447 

,7435558 
,7254203 
,7077272 
,6904655 

,6736249 
,6571950 
,6411659 

^^255277 
,6102709 

■ ' '. « 

,5953862 
,5808646 
,5666972 
,5528753 

,5393905 

li 2 



^ferC. 

,9708738 
,9i2')9')9 

,9151417 
,8884870 

,8626088 

,8374843 
,8130915 

,7894092 
,7664167 

,7440939 

,7224213 

,7013799 
,6809513 
,661 1 178 
,6418619 

,6231669 
,6050164 

,5873946 

,5702860 

,5536758 



,5375493 
,5218925 

,5066917 

,4919337 
,4776056 



3t/'*^^- 

,9661836 

,9335107 
,9019427 

,8714422 
,8419732 

,81^5006 
,7859910 

,75P4"^ 
,73^7710 
,7089188 

,6849457 
,6617833 
,6394041 
,6177818 
,5968906 

,5767059 
,5572038 
,5383611 

,5201557 
^5025659 

«- — .— 

,4855709 
,4691506 
,4532856 

,4379571 

,4231470 

TABLE 



%^ Dec. Talks of Comf. iMt. for^o Tearu 



TABLE III. 



T3ie *Brefe«t. Wartb t^on^ ^oun4^ C^mf* JjOarej^, 



r*- 



*^^mm 



Years. 



[ 



I 

2 

4 
5 



6 

7 
9 

9 

to 



II 

H 
»5 



IP"™'*'" 

i6 

i» 

as 



4f«r C. 

,96153.85 
,9245562 
,5»B{>9,64 
,8548041 
^219271 

,7po;i45 
,7599178 
,7306po2 
,7025867 

,^751*42 

•64^5809 

»624597> 
>6oo574i 

'57T475I 
»5552d45 



,954a?7^ 
,9157299^ 
, $702966 
, $385613. 
,^024511 

,76:18957 

' ,754*285 

,7031851 

,6729044 



: ,58966^9 
,5642716 

,5?99729- 

«,51^7204r 



'»P5*58,1Q 

,9070295 
' ,8638576- 

',^2e^>025; 

,78^6^^ 



' ,Vl&999H 
At9St9i 
• ,7920937 
► ,-[472582 



I III 



',55390^3^ ',4944.6!a3: 

',513^7?? ;, 4731^4 

,4936281 [,452^004 



21 
22 

24, 
25 



,<!|746424 
,4563870 

■ " 1 II ■ — 

f v^38«336 

,4»»95H 
,4057263 

3961215 



I* * 



■,4^r»» 
1 ,4146429 

III 

* ,3967874 
,3797009 

,3633501 

\ ,3477®35 



3751 ♦68.^,352730 



; ,7*5'2«54, 
" ,7106813 

^ ,076839^ 
,^446c8i9 

,^»39i<?3: 

' > ■ 

,5303214 
,5050679 

,4840171 

i I i " i ■ I. 

' ,43^967 
' 4155207 
«,a957w> 

e,37:688J>5: 

,3^8499^ 
,3255713 

•'#953028^ 



»f65052i_ 
,6274124 

; .591^85 
,558^48 



••i 



,46803^0 
,4425Cfio 

,4172^1 

I 

»|93<J«53 

,}7i?&4.4. 

I ,$503438 

,J305t30 

,J"«Af7 

,?94fS54 

,?S77505.» 
,2617975 

,2469766 

,2329986 



^••lii^w^ 



■<•*■ 



■MM^ 



Write 



TAB'LH 



T A B L 9 lii; 

j|%« Pr£{eat Worth (i oma P«»«u^ ^i^^hf9h^ 



y««rv^ I* 2peeC,~ 



a»' 
3P- 



if-. 
?^ 

35) 



3/^ 

4P- 



49i 



iff' 
48. 
4a 
5$> 






« ^r-^ 





»5r4Si3^ 
:..3785?584' 






2i|!lw£ 






RMMaMft 







^ Sa^"*^ 



\ 




\ !l5<529t4 



• ■ - 






33r49j* 



f 



31H129 

30t67»2'. 






HMMtHMfl^ 



l»'i 



Vm96 



a8o9^29< 
37*97a« 



3|2i3W 
3|i0494i 

5 ,al8985t7 
t , j8oo|»6 
^ ,4705919 
H ,:|6i4V25 

c ,a440)i4 

t ,22784^9 

( ,420IO»3 



2')673^ 
a49a^* 

24&9$!88- 

234?503 

226S(»7t- 



MMHMi 



,30546^9 

M98st>7 
i,tpi8&^ 
,i853k»2 
K ,J7P0«}§4 



i^wi 



■H 



TABLE 



i4tf Dec* Talks of Comp- Itft»for jo Tears* 



TABLE ra. 

the Frefeta Wovtb efone Pounds Comp* luterefl* 



Ye»$, 


4^«'C 


4 i f^^- 


•iperC. 1 


6 ffr e. 


1 

1 


26 


,5^892 


,3184025 ,2812407 1 


,2198100 




27 


,:{468l66 


,3046914 


,2678483 


,2073680 




28 


,3334775 


,2915707 


,2550936 


,1956301 




29 


,3206514 


,27;;oi5o 


,2429463 


,1845567 


. 30 


,3083187 


,2670000 


,231377*1 

,220j595 
,2098662 


,1741101 




3« 


,2964603 ,2555024 


,1642548 




3* 


,2850579 ,2444999 


ii549574 




33 


,2740942 ,23397»» ,1998762 1 


,1461862 




34 


,2635521 


,2238959 ,1903548 1 ,1379" 5 1 




35 


,2534155 
,2436687 


,2142544 
,2050282 


,1812903 ,1301052 
,1726574 ,122/408 




36 




1i 


,2342969 1 ,1961992 


,16411356 
^1566054 


,1157932 
,1091389 




,2252854 1 ,1877504 




39 


,3166206 


,1796655 
,1719287 

,1645251 


,1491479 


,1030555 




40 


,2081890 


,1420457 
,1352816 


,0972222 


' 


4« 


,2002779 


^0917x91 ' 




42 


,1925749 


,1574403 


,1288396 


,0865274 




43 


,1851682 


,1506605 


,1227044 


,0816296 




44 


,1 1 80464 


,1441728 


,1168613 


,0770091 




45 


,1711984 


,1379644 
,13202^3 


,11129^5 

,10^9967 
,1069492 


P726501 
^5685378 


. 


46 


,16^6139 
,1582826 




47 


,1263381 


,0646583 




48 


,1521948 


,1208977 


,0961421 


,0609984 




49 


,1463411 


,1156916 


,0915639 


,0575457 




50 


* ,1407126 1 ,1107097 1 ,0872037 » ,0542884 




• 


1 * . 




1 


:abi.i 


i 



•"^ 



Hy 



TABLE IV. 
0/CoMPOUND Interest. 



tie Amount of one^ Pound per Annum, or Art* 

nuity, for Years ; at the_B,ates of2\ 2 t i 3 ; 

; 3 tJ 4 ; 4 f; /5> and 6 per Cent* per AoiMim* 



I 






Years. 



I 
2 

3 
4 
5 



6 

1 
8 

9- 

10 



II 
12 

14 
15 



i5 

17 
18 

IP 

20 



21 

22 
2? 

24 
25 



% ferQent. 



1,(^000300 

2,0200000 
9,0604000 
4,12X6060 
5,2040402 

6,3081210 

7,4?438^?4 
8,5829691 

.9,754^284. 



» i/«<f-c. I 



i/joooooo 
2,0250000 
3,0756230 

4.1525156 
5,25<53285 

6,3877367 
7,547430^ 

. 8,7361 I5P 
p,P54<;i88 



.v,7^4«'204 y,y^4>ioo 
io,p4972ioJ 1 1,2035818 



^ferCent. 



, ipoooooo 
2,03POooo 
3,6909pcx) 
4,1836270 
5,5091358 



12^1687154 
13,4^20897 

14,6803?; 5 

I7,2934j6p 

18,6392853 
20,01 20719 
21,4123124 
22,8405586 
24,2973698 

2537833172 
27,^989835 
28,8449632 
30,4218625! 32,3490379 

32,0302997. 34>i577639 



64684099 

7,6624622 

8,89*3360 

10,1591061 

".4633793 



12,4834663 

13,7955530 
15JI404418 

16,5189528 

17,9319267 

19,3802248 
20,8647304 
22,3863487 
23,9460074 

25)544^576 

27,1832740 

28,8628559 
30,5844273 



123077957 
14,1920296 

X5»6i77904 
17,0863242 

18,5989139 

20,1568813 
21,7615877 

^3*4144354 
x5,i 168684 

26,8703745 



28,6764857 
30,5367803 

32,4528837 
344264701 
364592643 



3i/^C 



rite 



x>(?ooaooo 

2,c>35oooo 
3,1069250 

5,^624659 



6,$ SOI 522 

7,7794075 
9,0516866 

10,3684958 
M»73i393i 

J3>i4^P9i9 
14,6019516 

i6,ii30J?03 

17,6769864 

19,2956809 



20,9710297 
22,7050158 

24,4996913 
26,3571805 

28,2796818 

30,2694707 

32,3289022 

34^46041 37 
36,6665282 

38,9498567 



TABLE 



f 



fHi Dec* Tabk4 of Cofuf* lnt»fw jfo teats* 



TA$LE IV. 



^9 A 



Jpbt 4mo^u>fitf'il MMityf Can^iumi intev^. 



fc N # ■ > 4 



J^^ 



MMk 






* I 

41 4 

J 

4 
* 



4 



«P '( 



*5 



*i^»>>i<fc»>» - 



-2;05ococ3a 2,o dooooo 



9,ta3dooo 9,t9?os5o| ^,|$2S000( ^j^^S^iooo 
4,^^4640 9|,i76x9>t) 4,i<oxfiSO 44748016 
•S^c^^a^ ^,|707oH '5.5a565«* 5.^70930 

«*"-i«*i*Mi*«**i** >MMiirikM4hMbii*i*» ftaai^MiirfMtai^Ai^ita '< 

«>j92W^1i «,r««8pt7 ^^pty^ 6,^55)187 

WS'6*945' "8,019151^ 4|t42bo64 6,8958578 
9>M«p<»^ 9,k8oOt)^ 9i$4J»t«S9{ 9,19746811 



■^iAHlii<*A^Ml^4k*«*.] 



CM86 



»a68^ 



i5 



ao 



a? 

25 



4 1,8245; 1 1 



-»7 - tH>6975»24 24,741706$^ 

25,6454159: a&,«5So857. 
27,671 22945 ^0635655' 
25^778o786j gijjTi 42a« 



■**■«« 



i2,0o6io7«^xs,fee8leO94ltt>$77B9s^ ' " ' 



MaM^*riki 



* ^>. 



13,8411768 



1.64.051^ T^ 



17,1599*^ 






•4ta>i^i«A«Mi.Mf«*« 



T4,$D6787t1i4,»7it5455 



^ -itei^ 



l< MllfA! 



22,7X93^67' 



^7>t«<2^8ftt 



llS,|«J9942t> 
... , ^$824^85 

«,5y8>583* »3»fe75Pyo7 



*M*I * 



■ < I ll >.J— »M^ 



^1,96^2017 55,78345^ 



16,&178886 
^,0826041 
41,645^083 



?^it05377? 

41,6891965 
44,5654101 



irfftirfh 



«3»l'574^i8 S5.<»725589 

S, $403664: 2B,Mi88o6 
,l3a3847< 40,tO5««i3 
30»?'?J'ao5?' 33»y595>^25 
63y5«'5?54>; 3^,^855 J>2o 



' ■• - - 



;5,7i925i8l3[9,P927^75 
38,S052i44]4j,59229n 

4i>*30475i 
44,5oi$r989^ 



47>7'27q?^ 



461^956285 
50,815-57^2 
54,864<i 28 



MMb 



■ <*fc 



■ • I iiH 



TABL E 



bee* Talks of Comp* lHt,for yb fearsl 24^ 

TABLE IV. 
The Amount of 1 h AMttuityy Comfomd Ititerefis 



YearS' 



26 

27 
28 

29 

30 

3« 

35 



36 
38 
40 



41 

42 

43 
44 
45 



46 
47 

48 
49 

50 



2"^^ Ceift- 



42,3794408 
44,2270296 
46,1115702 
4.8,0338016 
49,9944776 



33,6709057 
35>3443238 
37,0512103 
38,7922345 
40,5680792 



lifer C. 



36,01 17080 

37,91 20007 

39,8598008 
41,8562958 

43,9027032 



. 



46,00012707 

48,1502775 
50,3540345 

52,6128653 

54,9282074 



51,9943672 

54>C342545 
56,1149396' 

58,2372584 



57,30x4126 

59,7339479 
62,2272966 

64,7829791 



60,401983167,4025535 



;l 



62,610022870,0876174 
64,6822233172,8398078 
67,159467875,6608030 
69,502651178,5523231 
71,892710381,5161512 
-. -| 

74,5305645184,5540344 
76,817175887,6678853 
79,353519390,8595824 
8i,9405697:94,» 310729 
184,57940(5 97,4843488 



3 firCettt. 



38,55 3<3422 

40,7096335 
42,9309225 
45,2188502 

47,5754157 



50,0026782 
52,5027585 

55.0778413 
57,7301765 
60,4630618 



63.2759443 
66,17422261 

69,1594493 
72,2342327 

75,40? 2597 



3 t /«• C- 



41,3131017 

43,7590602 
46,2906273 

48,9107993 
51,6226773 



54,4294719 

57.33450*5 
00,3412101 

63.453 «524[ 
66',6740i27 



t 



78,6632975 
82,0231964 
85,4848923 
89,0484191 
92,7198614 



70,0016032 

.7^,451^^93 
77,0288947 
80,7249060 » 

84,5502778 



88,5095375 
92,6073713 

96,8486293 
i!Oi,23833l3 
105,7816729 



96,501417:^ 
100,3965009 
104,4.083960 
108,5406479 
112,7968673 



110,4840315 
"5,35C9726 
120,3882566 
125,6018456 
130,9979102 



Kk 



TABLE 



% JO Dec, Tables of Comf. fnt, for Jo teafi^ 

T A B L P iV. 
the AmouMtofi /. Annuity , Compound laterefi* 



TtMTt^ ftr Cettt.^ I ftr Cettt. 



36 

27 
28 

99 

30 

31 

33 
34 
35 



I 36 

38 

39 

40 



4« 

4* 
43 
44 
45 



44,311744^ 
47/3842144 
49^675830 
52,9661863 

56/5849377 



59»3283352 
62,7014687 
66,2095274 
69,8579045 
73,6522248. 



J 7.5983 1*8 
1,7022464 

85,9703362 

90,4091497 
95,0255157 



99,826536 

104,819597 
110,0123817 

115,412816; 



47,5706446 
50,7113236 
53,9933332 

57,4230332 
61/3073698 

64,7523878 
68,6662452 
72,7562263 
77/3302565 
81,4966180 



86,1639658 
91,041 344 



5 ^/r C*af . 



6 ferCent. 



51,1134538 59,1563827 

54,6691265 63,7057657 

58,4025828 68,5 281 if6 

62,3227119 73,6397983 

66,4388475 69^581^2 



70,7607899 
75,298829^ 
80/3637708 

85,0669594 



90,3203073111,4347799 



95,8363227 1 19,1208667 
101,6281388 1 27,2681 187 



96^ 1 382048 107,7095458 1 3 5>9b42058 



101,4644249 1 14,095023* 
107,0303231120,7997742 



121,029392c 138,8499651 



46 

48 

49 
50 



1 26,8705677 



1 1 2,8466876 
118,9247885 
125,1764040 
131,9138422 



146,0982135 



»35,23>75ii 
'42.9933386 

151,1430056 



84,8016774 
90,8897780 

97,3131647 
104,1837546 



145,0584581 
154,7619656 



1 27,8397829 165,0476836 



175,9505446 
187,5075772 
199,7580319 



159,7001559212,7435138 



168,6851637 226,5081246 

178,11942181241/3986121 

[256,5645288 

272,9584006 



132,9453904 153,6726331 178, 
1 39,2682o6o'i 61,587901 6 188, 

'45»8357342;i69,8595572'l98,4266626 _^ ,_^-_. 
1 5 2,6670836 1 78,503028 2|209,347y957f290,3 3 59046 



0253929 



TABLE 



MI 



T A B t E Vi 
Of Co MPQUND Interest. 

7 he Prefent Worth of one Pound per Annum, or 
Annuity for Years, at the Kates */ 2 *, * 4- ; 3 ; 
3 T » 4 » 4 tJ 5> ^W 5; per Cent. per Annum. 



\ 



Years. |2 fat Cent^ 



I 

2 

3 

4 
5 



■V 



6 

7 
8 

10 



0,p803P22 

1,9415609 

z,88:j8833 
3,8077287 

4>7i 34595 



5,6014309 
6,4719911 
7.3254814 
8^622367 
8,9825850 



;^ 
I? 
?4 
1$ 



16 

17 
18 

19 

20 



■»*▼. 



21 
22 

24 
25 



9,7868480 

io>575Hi2 

11,34837^7 
J 2,1062487 
12,8491635 

15,5777093 
14,1918719 
14,9920313 

15,6784620 
i6,35H333 

17,0112092 
17.65804J52 
18,2922041 
18,9139256 

i?»5234565 



%\ferC. 

0,9756098 

j.^274242 

2,8560236 

3,7619742 
4,6458285 



5>5o8i254 
6>34P3Po6 

7,1701372 
7,9708655 

8,7520639 



9,5142087 
10,2577646 

10,9831839 
11,6909122 
12,3813777 

13^0550027 

13,7121977 
14,3533636 

14,9788913 

15,5891623 



3 fer Cent. 

■ .J. . ■ 
0,9708738 

2,8286114' 
3^717098^ 
4j57?7072 



5,4171914 
6,2302829 
7,0196922 

7,7861089 
9,5302028 

9,5256241 
9,9540040 

10,6349553 
11,2960731 

M,9375^35l 

12,5611020 
13,1661185 

1317535131 
i4»3»379Pi 
X4»8774748 



^*r 



rf9V 



1^,1845486 
16,7654132 
17,3321105 
17,8849858 
18,4*43764' i7,4i1»477 



t 



15*4150241 
I5.9?69i66 
16,4436084 

i6,935542« 



0,9661836 

2,8016379 
3,6730792 
4,5150524 

5,3285530 

^1145439 

6,8739555 
7,6076865 

8;3i66o5| 

9,0015510 

10,3027385 
10,9205203 
11,5174109 

1 2^94^68 
1216513206 
13,1896812 

1317098374 
14,2124033 



I4#6979742 
15,1671248 
15,620410^ 

i6,os8:j676 
16,4815146 



lU '1 ' 



% 



tt 



TiV 91, S 



i J % Dec* Talks ef Comf, Ixt* for^o Tears, 



TABLE V. 

the Prefeta Worth ofil. Annuity^ Comf, Intetefi* 



Years. 
I 

2 

3 
4 
5 



6 

7 
8 

9 

to 

XI 

12 

»5 



i6 

17 
i8 

20 



21 
22 
23 

24 
25 



4^ferCent.\ ^^pierC. I5 ^«" Cr/rf. 



o,?6i5385 
i,886op47 
2>77509io 
3,6298952 
4,4518223 



5,2421369 
0,6020547 
6,7327448 

7>4?«3M 
8,no8p55 



8^7604763 

9,38507?3 
9,9856473 

10,56^1213 

11,1183868 



11,6522949 
12,1656680 
12,6592961 

»3>i 339385 
»3?59032*)3 



14^291589 
14,4511142 
14,8568405 
15,2469619 
15,6220787 



o>9569378 
1,8726678 
2,7^89644 

3.5875257 
4,3899767 



5,1578725 

5,8927009 

6,5958861 

7,2687905 
7,9127182 



8,5289169 

9,1185838 

9,6828524 

10,2218253 

»o>7395457 



11,2340151 

11,7071914 
12,1599918 

12,5932936 

13,0079365 



i3»40+7239 
13,7844248 

14.1477749 
144954784 
14,8282089 



0,9523809 
1,8594103 
2,7232480 

3.5459505 
4,3294767 



5.0756921 
5.7863734 

6,4632128 
7.1078217 

7,7217349 



8,3064142 
8,8632516 
9.3925730 

9,8986409 
10,3796500 

10,8377695 
11,2740662 
11,6895869 
12,0853208 
12,4622103 

12,8211527 
I3,i63cx)26 

I3>4«857?P 
13,7986418 

i4»0P39445 



6 per Cent A 



0,9433962 
1,8333926 

2,67?ori9 
3,4651056 
4,2123638 



4,9I73?44 
5,5823815 

6,2097939 
6,8016923 

7,5600871 



7,8868747 
8,3838440 
8,8526831 
9,2949840 
9,7122491 



10,1058953 

P4772597 
10,8276035 

11,1581165 

11,4699213 



11,7640767 
12,0415818 
12,3053790 
? 2,5503 576 

'2»7833562 



TABLE 



PeCf Tahles qf Comf_. Jnt^for f o 7V<»jr, »f $ 

T A B I, E V, 

The Prefext Worth efil, Jif»tat/fComP- latere^. 



Yeatf. 



26 

28 
29 

30 



31 
32 

33 
34 
35 



3« 

38 

40 



4? 
42 
43 
44 
45 



45 

48 
49 
50 



2 f^r C. 



20,1210^58 
20,7068978 
21,2812724 
21,8449847 
22,3964556 



22,9377015 
23,4683348 

23^885636 

244985917 
24rf>986i93 



254888425 

25.9694534 
264406406 

26,9025888 

27,3554792 



27*7994895 
28,2347936 

28,6615623 

29.0799631 
29,490159? 



29,8923136 
30,2865820 
30,6731196 
31,0520780 
31,4236059 



2 ipf c. 



18,9506111 

194640109 
19.9648887 

204535499 
20,9302926 

21,3954074 
21,8491780 

22,2918809 

22,7237863 

23»i45i573 



23,5562511 
13.9573181 
24,3486030 
24,7303444 
25.1027751 



3 fer Cent. 



254661220 
25^8206068 
26,1664457 
26,5038495 
26,8330239 



27,1541696 
274674826 

27.7731537 
28^713695 
28,3623117 



17,8768/^20 
18,3270315 
18,76^1082 
19.1884546 
19.60Q4413 

20^^4285 

20,388765s 
20,7657918 
21,1218367 
214872200 



21,8322515 
22,1672354 
124924616 
22,8082151 
23,1147719 

23,4123999 
23,7013592 
23,9819021 

24,25jL2739 

24,5187125 



24,7754490 
25,0247078 
25,2667066 

25,5016569 
25.7297640 



3i ferC. 



16,8903523 
17.3853645 
17,6670188 
18,0357670 
18,3920454 



18,7362758 
19,0688656 

19,3902082 

19,7006842 

20j00066l2 

20,3904938 
20,5705254 
20,8410874 

21,1024999 
21,3550723 



21,5991037 
21,8348828 

22,0626887 

22,2827910 

22,4954503 



22,7009181 

22,8994378 
23,0912443 

23,2765645 
234556179 



TABLE 



» J4 ^^^* Talks ofComff Inter* for 50 Tears, 

T A B L E V. 

Ihe frefen^ Worth of i /. Annuity ^ Comf, Inter* 



Years, 



26 

27 
28 

30 



31 
^2 

33 
34 
35 

36 

38 

3S> 

40 



4» 
42 

43 
44 
45 



46 

*2 
48 

49 
50 



4 fer Cent. 



16,5295844 
16,6690616 
16,9837132 
17,2920318 

^7,S88492I 

17.8735500 
18,1476441 

^8^.^1962 
18,6646116 

18,9082803 

19,1425771 
19,367862$ 

19,5844831 
•9,7927721 



19,9930500 
20,1856250 

20,37-7931 
20,5488395 
20,7200378 

20,8846517 
21,0429342 
21,1951289 



4 t /«" C- 

15,14166115 
15,4513028 

15,74287J5 
96/3218885 

16,2888885 

16.5443909 
16,7888909 

17,0228621 

17,2467580 

17,4610124 

l7,666odo6 
17,8622398 
18,0499902 

18,2296557 
18,4015844 

i8,s66i095 
18,7235498 
18,8742103 
19,0183831 



^ fer Cent. 



1^,2883707 
19,4147088 
19,5356066 



I4»375i853 
14,6450336 

14,8981272 

15,14107351 
« 5^37245 ic 



15,5928104 
15,8026766 
16,0025491 
i6,i5>i9039 
16,3741942 

116,5468516 
16,7112872 
16,8678926 
17,0170^06 
17,1590862 



6 fer Cent. 



13,0031663 
13,2105342 
73,4061644 
i?,59072ii 
13,7648312 

i?,929086| 
14,0840435 
14,2302x97 
14,3681412 
14,4982465 



17,294^678 
17,4232074 
17,5459118 
17,6627732 

17,7740697 



21,3414700 19,6512981 
21,48218261 19,7620078 



17,8800663 
17,9610155 

18,0771576 
18,1687215 
18,2559253 



14,6209872 
H>7367304 
14,8460192 

14,9490747 
15,0462969 

15,1380160 

IS2145434 
15,3061730 

15,3831821 
154558321 



15,^243699 
15,5890282 
15,6500266 

15.707^725 
15,7618610 



TABLE 



t AB Le VL 

O/COMPOtTKD InTBREST. 

The Annuity which one Pwnd will purchafe for 
any Number of Years jr at the Rates cf i% 

2 4^; 3 ; 3 i; 4 ; 4 tJ U andSfnCent. per 
Annum. 



dent. 



^■■■■■■^■■■■■■■■^^■■■MMaMaMiiMMaiHiHfiHBaiiNBHRM^iHMlfiMMnri 

iJ^ferQ. \i fer Cent.yi 4. fer C. 



Tears 



2 per 



I 
a 

4 
5 



6 

1 
8 

9 
10 

II 
12 

«3 
14 
15 



16 

17 
18 

«9 

20 



21 

22 



1,0200000 
,5150495 

*34^7547 
,46262:3^8 

,2121584 



,1785258 
,1545120 
,1365098 
,1225154 
,1113265 



,i®2i779 

,094559^ 
,0881183 

,0826020 

. .077^255 



1,0250000 11,0300000 
,5188272 ,5226108 



,3501372 
,2658179 
,2152469 






,1815499 

,1574954 
,1394674 

,1254569 

,1142588 



,0736501 
,0699698 
J066702 I 
,0637818 
,061 1567 

,05^7847 

,0566314 

,o$4668i 

,05285111 

,0512204' 



,1051060 

,09748?! 
,09'i0483 

,0855365 

,0807665 



,?535304 
,2690271 

,2183546 



ii*»<*< 



,0765990 
,0729278 
,0696701 
,0667606 
,0641471 



,0617873 
,0596466 
,057^964 
,0559128 

,0542759 



,1845975 
,1605064 
,1424564 
,1284339 
,1 172305 

,1080775 
,1004621 
,0940295 
,0885263 
,08.37666 

,0796109 

,0759525 
,0727087 

,0698139 

,0672157 

,0648718 
,0627474 
,0608139 
,0590474 

,0574279 



1,0350000 
,5264005 

•3569342 
,2722511 

,22 14814 



,1876682 
> I 63 5445 
,1454767 
,131446c 
,1202414 



^tlM ^i*! 



,111092c 

,103484c 

,0970616 

,0915707 

,0868251 



,0826848 

,079043 1 

^0758168 

,072940.;} 

,070361c 



,0680366 

,065932 

,064^188 
,0622728 
,0606740 



TABLE 



if 6 Dec. Tables efComf. Inter. for Jo Tearsi 

t A B L E VI. 

TW Anitutty vtAicb one Pound ioitt ficrchafef 

Compomnd Interefi* 





Tiios- 


4 ^ Cmt. 


4t/«'C 


5 ^er C. 

1,0500000 


6 /« 0«. 


I 


1,0400000 


1^50000 


1,0600000 




1 


,5301961 


.533997<S 


,J378o4P 


»J4543<SP 




3 


,3<Jo3485 


.3<J37734 


,3572085 


,3741098 




A 


,2754901 


,2787437 


,2820118 


,2885915 




3 


,2246271 


,2277916 
,1938784 


,230^748 
,IP70I57 


,2373964 


» 


6 


,1907519 


,2033626 




7 


,1666095 


,1097015 


,17281^8 


,1791350 




8 


,1485279 


,1516097 


,1547218 


,1610359 




9 


,1344930 


>I37J745 


,1405901 


,1470222 




lO 

II 


,1232909 


,1263788 


,1295045 
,1203890 


^1358680 




,1141490 


,1172482 


,1267929 




12 


,1065522 


,1096662 


,1x28254 


,1192770 




'3 


,1001437 


11032754 


y 1 0545 58 


,1129601 




«4 


,0946690 


,^978203 


,1010240 


,1275849 




Itf 


,089941 1 
,0858200 


,0931138 


,0953423 
,0922599 


,1019628 




0,890154 


,09895 2 1 




17 


^821985 


0,854176 


,0885991 


,0954448 




i8 


*o789933 


0^822369 


,0855452 


,0923 565 




19 


,0761386 


0.794073 


^0827450 


,089^209 




20 
21 


i073j8i8 
,0712801 


Oj76876i 

,0745605 


,0802425 

,0779961 


,0871846 




,0850046 




ai 


,0691988 


••7*5457 


,0759705 


,0836456 


»3 


,6673091 


,Q7o58ij 


,0741368 


,0812785 


*4 


,0655868 


^068^870 


,0724709 


,0796790 




2J ■ ,0^40121 1 


,05743 fo 


.0709545 


,0782267 



TABLE 



Dec. Tahiti cf Cbmp, Int, fw 50: Tears, a 57 

T A B L E VL 

t 

The AHmity^ which one Pound will furchafe^ 

Camfound Inti^efi, 



FedrsA 2fefCent. 
26 ,0496^9! 

2f ,048^931 
28 ,0469897 

29 >0457784 

30 ,0^4464.99 



3» 
32 
33 

34 
35 

36 

37 
38 

39 

'■40 

A-l 
42 

43 

44 
45 

46 

"47 

r'4a" 

49 



0435964 
6426106 

6416865 

0408187 

0400622 



0392329 
0385068 
0378206 

037171 I 
03155558. 



0359719 

03i54:>73 
^3148899 

0343879 
03^9096 



2 i. per'C. 

,0527688 
,0513769 
,0500 85'9 
,0488913 
,0477776! 

-- - 

,0467390^ 
,0457683' 
,0448594 
,0440068 
,0432056 






** 



0334534 ' 

0830f^9 

03ft66i8? 

^322040/ 



',04245^6' 
,0417405 
,0410701 
^04043 62 
,03.9836^ 



3 ferCenti « ^^perC, 



i*« 



,0392679 
,^387-288 
,0382169: 

^0377304 
JD372675 



^05593^ 
,0545642 

>053«932 
,0521147 

^ ,0510193 

,0499989 
,0490466 
,0481561 
,047^220 
,0465393 

,045^038. 

,«)45tjj6 

,0444593 

,0438439 

,0432624 

.,0427124 

.,04219-17 

,0416961 

,0412.299 

.',0407852 



■M^ 



.j0368;268! 
.,P3 64067 r 
:,fi)3i6o(>6o- 

»035^35' 

<»P35«58j 



^0401625 
V0399605 
♦0395778 
>039i«3i 
,0288655 



,0592054 
,0578544. 
J0566027 
,0554454 

,054371 3 

>o533724 

,05"244i5 

^0^15724 

>o5!07597 
,04^99984 

,04j92<842 

;,x>4|86i33. 

»Q4|7^82i: 

»<?4!7 ^^78 ; 

,04)68(273' 

,0462982 ' 

,0457983 • 

»04^3?54 
»0448.777 

,0444534 

I 
,04405 1 I 

,0436692 . 

,CH.33o65 J 
^4fe96j.7 I 
,04^6^37 I 



I « 



LI 



TABLE 



»y8 Dec.TMesvfCof^,TM'er»f6r%<^tnfirs. 

T A B L fi VI. 

the Atmmfy lahkb^oMe -Pdund mU furchefe^ 

Comfottftd IxPere/K 



18 
if 

90 

J» 
33 
34 ■ 

ii 

37 
38 

39 
40 



4« 

4» 
43 
44 
45 



48 
"4^ 



■■■■■■■■■■■nMW ^ 

^0(525674 

>t)dooi3o 

^0578301 

i05V58s54 
ioJ5^486 

,oj4ji48 
.0535773 

,0^ 288^9 
,0$2239tf 

>o5 iotf«i8 
,0^05255 



f 

,06)^0214 

»<»<5k7i^5 
•,od!j5«o8 

^011^24146 
,o^i3£»5 

><>dlo4435 

,0 $•87445 
,oj!7p8i5> 

»o^7f7«5. 

^5l5^oj8 
,0515^840 
,0554017 
,o5'4«557 

»c^5"4343 « 



^0500174 
,o4j>54oa 

,0490899 
^0485^45 

47 j ^^75119 



■^■■a 



^- 






A^85T« 



0,538^1^ 
0,534^87 

c>5 25807 
'0^52:2010 

i0^!i5O73 
)0$ 1^886 
.*05oS87a. 



5/vrCSf/^ 

«^»^H— ■ ■ - 

,od9f(543 

,0^7* 22 J 

90650; 14 

jOi5l4i32i 
io6^32So4 

^0^10717 

,odo434S 
>05i9«3^8, 

iOS9434> 

i05»8;»78*v 

,Q5|7t8tt2.3 

>Pn|l>47 
i05!d^33 

jOi5|5tfl53: 

,-o;|62^i7 






i05!5^28t 

»^5'5tff4» 
i<fty|yji84 
>«>550397- 
^5'477tf7- 



♦<>7*P044 
,0756972' 

,0745926 

,0735796 

, 072648^ 

,0717542 
3O710023 
,0702729 
,0695^84 
,0689739 

,0683948. 

,6678574^ 
,0673581 

,o6r58938| 
,06^49^15; 

■yo.615058 

',06568:3 

y^tf5J3^3I 

^o<^oo6i 

,06^7005 



^o<^i4i 

s,o<*3»97 
4o6§[tfif3 
:,o6ftiM 



u 



« • 



I J 



CHAP. 



»f> 



> •! *t 



Q, H A P. X. 



'^he Ufa of D.lcCiUALS in Vulgar , Dw- 
decimaly. and ^iexmfi^tl, Fra^om* 

t 

VuJgair FraBions in Decimals. 

IH A V E already (hewn the Method of finding the De- 
cimal e^Y^nc eo an^ yidgar IkaStmy V> S^r^mhn ; 
What I propofe here, is to Ihew with how much g^ter 
Eaie and Fles(fiiic any of the Oestaci^ns Q^V^ffgar ]^aEiion$ 
are wroi^l^ \yjtiecimd Nmnbers. I fhaU exemplify the \liff^ 
ter in tbe cpn^m^^ R^ki ^ &>Dow§. 

Adiitjom. 

B>tmfie\, What is the Sum of -^ and ~ of a Pound? 
. f . r «.- • 7 • 

., ^Tl»6A«f»'WQf -J =;»?777 7 B« th« eeofral 
Add^ 9 ^ V DteimalTaWe, 

The Sum of both is =5 /. 1,2062 = 1/. 4/. X i^. AnC 
ExamfU «. What is tbp Value of —|- and -^ of a 5jb/^ 

Ada < r ble ' 

To the Z)^f. of "I- :*: ;8a^3^3 ^ . * 

Thar Sum \» ' — . o^i^58; a»-zoi ^. Anf. 

L 1 2 tSxmfte 



%6o The V/e ^Decimals in the 

ExmfU 3. What u ^, ^ and -| of a Yari? 

C The Decimal of ~ = >|$?8 
Add c And the Dec of 'j^ — ^367 > By the Tab. 



1 



And the Dee. of ^ =3 ,5384 



The Sum, of Courfe, is —^ = 0,9259=^2 F.^In^lau 
ExmfU 4. What is the Value of ^, -t, ^, and -j 
ef JL of a Hundred Weiebt Averdufoit ? 

i The Detlmai of — -3. = ,,^555 1 

3 

Thfc Decimal of — -i- = ,8oco «„ », 
The Decimal ot ►— — -^ =3 ^222 J 

, ThcD^r.of-^- of 4 = 4" = j^iSO I 

I ' 4 ^ « JC.Q./*.«. 

The Sum of afl is — C. 2,:}i5€'=r2 : 1 :7 : 2 

Exmf}e 5. What Number of R^irj, do 476— , 36^ , 
2.^,75^.»iJ««dl^of^Ycar, n,ake? • 

'/'••'• m pari. 

r 21J? — 2ly^^9^, 

Thofe jpwrf FraSlions] J* 

being fee 4owir in order,: 7— = 7>9285 
and the becimafs ot At\ - :J4 '• - : ^ 
FraSiioval Parts bcing^ iH = 1,8234 
f ound in the Gtf»r4f IJiw I ^^ I7 i 

and fct do^vn oppofite there- ! ^2 ::= 0,8944 

fo • add them, and thpirj , iQ ^ — • „ 

Stfi wiU be _• "^ •' — 54S,754i J^^^^- 

Ttejt is, 545 Years, 9 Months, ^ Weels, and 2 /)^ti nearly. 

* » - . Suhfitaoii^n* 



7 



^ 



Jrifymtttck ofVulgnr FraHions. 

ExdmfU U What is -—• kfi -|- of a Pouad !SterJi»g ? 

Trom — ,^ -5- =5 ,8^33 7 By the ge*^ 

VneralDed* 
Subftraa -« .^ -|- z:^ ,375 \mal Table. 

There remains — • -. 0,458^ = ps* 2^. AnC 

Example 2. What is the Value of — of -J- Icfi i of 

9 

From ——of -2-=- -21 = .^Si' ^ 

J ! " ^^S^BytheTaUe. 

Siiftraa ^ of -l^-i = ,41/^ 
4 . 9 12 '^ 

Thcrcremains — o,i^7 = oKaJR ji;». 
Example 3. What is 14-, Icfe -S. of a Po^i IJ^jy / 

From .— 142. sss 14,4:2227 

J > By the Table. 

Take — i =s 0,7142 >^ 

^^ - '^ ■ ». 02.^. 

There remains — i 13,5083 = 13 : 6^2 

4. What is I7c*^ 

verdupois ? 

C. 

* From V ■- I7cg = I70,8P47 7 B, ehe 
.; Take-*' ' .. I5P^ = t5P^7,6 ^ ^•"«- ' 

There remaini the Anfwer 1 1,421 1 C. 



Example^ What is 170— , Icfs 159-^ C. Weigfjf^ 4. 



"^ 



t^ Tbe^ Z^ ef Dechntifr tn the 

MUHfUeMiM. 



^ - -- J^ '^ 

. 9) ."428 
The Prodiia is i Fq. 20,3 /«;• 5= ji269f 

Example 2. Smfofc a Pbc< of Hnji^ jA t^fct iq 

Length, -jlof 9,Foq% Wide, an4 -| oF^.^f ^Fc«(t thick, 
What is Che Solid CoaifMt of that Piece? 
'Mcilt^Iy the Length — 14^ = X4,4?85 

By ^eWMrh *- ^ = fz 

The Produd will be found to be -^ 1049^4^ 
That naultipBcd by the Thick- 7 

nefiy which is -^of y =5^ -^^ 3 ~- ^^ 
, . -. ■« ' '"* 

TW Produd is the i3l9»/«!rf fcx^ht aar 9,ft7Bt8, 0f. 

Th»t is, ? fept, 4^9 ^*«j 4^V ^«<»<^<? ^Cpn- 
tictA the AnAver. 

ExamfU 5. There is a G>f/r» xo^ Feet in Lcr^H, 4^ 

D^, (ind 5-i Wid«} Qa«e how many Com GMons it 
win hold?' 



'Mihit^ly the Length ~ 2c4 = to.it 

By the Width m« vm 5 Jl ^^ 5,;^ 

That Prodiia will be — ^ 54,526^ 
Multiply that by the Depth >^ 4I ^^i 4,^ 

The Produa i$ the SolidContentv TTl q 

in Feet and Parts ^ )^ = 245,3976 Sf. 



Multiply that by the Gallom in 
a Solid Foot: vix. ^ 

Produa is the I^iohiber < 



= 6,40625 



Note ; When there isjivcn any Number of pure FiaOions 

'tob«4Qfild^aiifi(o(itt)e«i9Aer^ ^11109 4wlti» 

ply the -.mmeratars Md Denominators ixAo one 

another, and ibe i^rodiSs «ritt be ft EoAion, 

' whdfc'Vauc in^D«frffflw/i you ou^ Bad asbcfote 

'.But j^ = 0/30764 'Th»Anfirer 

Dtvijion. 

Example x.. DiVide -^ by -^ 

7 *7 



i .A 



I i 



i^^' 



. . . « 



p64 '^ tJfe ef Decimals in ihe 

Thm -i s= 0,0588) ,71428* = -J-' (• 2»H754 = Th^ 
'7 . ^88 (Arfirer. 



^162 

-• • 86» 
588 



a8o5 

•^5 



■i^a* 




•562 

Example 2. What is I^ dWded by ^ of 4^? 
Thus-i of- = :^^ = ,21 ?),842io5263 == j| 

21 842105263 

ilp2 ,7578947568 (5,947368 

There is fomewhat re* 576. 

inailQible and uncoiminon in " ^ " 

ch^ Work of this Queftion, '^'J 

fF/jt. Firft, that the ift, ^d, '72g 

Sth, 8r. Remainders, with ..pop 

the Numbera taken down^ 753 

conlift of 2 Pair of the fimc — — . 

Figures ; Secondly, That the ^\H 

2d, 4th, g^. Remainders, '544 

ivith the Figures taken down, . **707 

are juft lulf the others | e-^ 

ThiidIy,That where the Re- J^ ■ ■ ' 

mainders of Pairs are prime 15^5 

Numbers, the next is a Re- '^5^ 

minder is a Pair of even ' 1616 ' 

Numbers, and then goes on .^,z 

OdOCC, 



Mu ':8c8 £rr 



Arithmetic^ cf Vulgar FraEiionsl 265 

ExamfU ij. What is 2 j2 of a pound^ divided by — ? 
Thus 12=5 ^0) 2,3«^8 = 2jJ 



** * 



.9 ) 2»365 ( 2,62^ ss 2 /^2 i. 61 i 
18 (The Anfwcar. 



.56 
54 



25 
18 



6^ 



7 
tSxmfle 4. What is it /^^^ C, ^(gl^f, when 2i-| C 

cofts 67-2 Pounds ? 

10 . • ' . 

Thus 2i-| = zi^ar) ^7»4?75 = 67i[| 

« 674375 

2142)66,765125(3,11685;. 
6426 

•2503^ :.• ;. . 

2141 

That is ^116857. S3 • 3611 . 
a 7. 2i. ^d/fer CI. itt^ ^ 2i4t 
Anfwcr. 



t .• 



X4692 
1 2852 

18405 

17136 

11 ■ ■< 

12690 
10710 



'?8o 
M 91 Ejtfracttcn 



■■-> . 

266 TbeUfeof Decimals in the 

• t w • • • 

ExtraBion of Roots. 
ExmfU I. What is Ae Zguart if(M>f oF^t^? 



• • 



Trtieb^ciiAdof gs= ,(^9*(.8^5= |-= the Root 

^4 (fought. 



1^3) • 5 



544 
40? 



1^3) • 59 
■4^ 



555 
Exmpl* 3. Reqmied dw Sjtur^ if wf of tbc Sw^ 

* » • 

» * 

The Dectmal of i3 r= ,7^47058 8f . the Root of \idiicb 
extrafied will be fij^pS QcJ the AnTw^^ 

Bca$^k 5. JExtoid ttic ^jfori? ifoof of 58^ 
The Decimal of thtmbffd FraBiom 58 J = 58,^ 



m 



Then atraa th6s; ^ 58^ (iM == if h: ^t^oot 

-49 -i^fougbr. 



876 



<5«i) 19177 



Exam* 



TheUfi of DepimaJs i/fj kc. 267 

Example 4. Extraa the Cute Root oF -j| 

This is Bcft (as being va% fotmca and e^&ft) done by 

La^^itbmsy thus , 
riom the Logarithm of the Numerator 9 :s. 04771 21 2 
Sqbflr»ft the Zi)g4r.oftheD^«oiw/»ift>r 16 =: 1,^041200 

There remains the L^F^ar. of the D^r. ,1875 = 9,27300 «^ 
A^tothe^Iff^exofv^ Logarithm 20 — >29,27500i2 
OncTbirdpf which lo^wfA^ R72i?s8 « .9,7^76670 
the ^Logaritbm of the iEoof fcughtr'^^^S^o >^>7^7 / 

The Cube Root then of -I is ,572358 ^*ich was to be 
' 10 



gxample 5. What is the Cuhe Root of 5i2j|? , 

Firft, From the Zegojcithm. of the 7 ,, ,. 12012 

i\&«^4f or ofihe Fraaional Part J — *5 - *> :i5/^^ 
Subftraa the Logaritb.^i tl)c Denominator 15 s= 1,176091 

There remains the Logarithm of the Dec. fi^ =. ,9,9378 )2 
To which add the integral Part 512 

The Sum is the equivalent w;tf^^Z)^f. 512,8/^ == 2,710004 

A Third of whole Logarithm is p 
the Logarithm of the C«*^ iPoof > 8,0045 « 0,903334 
fought — — — J 

ThcC«*tf ^<x^ therefore of 512^2 is 8,0045 

\ Notjf* The Ufe of Decimals is not only very obvious jn 

all Parrs of the Doarinc of Vulgar FraSHons^ bjit 
abfolutely neceflary in ExfraSlionof Hoots ; which 
fometimes elle cannot be done, ' 

The Ufe of Decimals in Uuodecimai 

Arkhmetick. 

Duodecitnals are a Sort of PraSliofS made uft of in M^«^ 
furaiion ; Where one Foot is the Integer j The Fo«f is di- 
vided into twelve Parts or Incb^^s ; one of thefe, into twelve 
others, and fo on dividing by twelve. Whence as ten is the 
Common Denominator in Decimals^ fo twelve is the Common 
Denominator in Duodecimal Fraaions. 

M on 9 The 



•68 the Uft €f I)ecimah iu 

V 

The Notation and Rcadii^ of Duodecimals is 

Fetft, Pfiifiis, Seconds^ Tbirdsy Fourth^ &c 



Thus 



<Fert, FfiifiiS, Eecondsj \ 
C I'i '• CQ : 10 : 



5 .: cp : 10 : cd : xi te. 



Now becaufe this kind of jfritbmetic is ufeful to Eexfops 
concernd in Buildings Meajuringy &c. and the moft ufeful 
Parts, viz. Muliiflication^ Divifioft^ and Exira£H$Uj bring 
by far the mofi difficult | I diought it very proper (and hope 
it will be very acceptable) to flicw how thofe Operatiom may 
be moft eafily and fpeedily perfbrm'd by Decimal Aritbrne-' 
tick. 

To that End I have made tht following Tahh for the ready 
converging any Duodecimos into Decimah^ «nd the contrary. 
The Ufe of which, to thofe who understand any D«4i)^1V 
bles at all, is very obvious and eaiy. 

The Duodecimal Table. 



^F 



Duode- 
cimaU* 



2 

3 

4 

5 

6 



9 
II 



The Decimal Ps^s. 



Primes. ' I Seconds 



y\iS6666 
,ar3333? 

»55^333 

y6S6t66 
>8?3?33 






it 






,006944 

,oi5Jr88 
,02c 8^3 

^027777 
,0347^2 

,041.(^66 

,04861'X 

5CS5555 

,0625 

,c6p444 
,0763^^8 



Thirds. 



tit 






,000578 
,001157 

fOOi736 

,002314 

,0028^3 

,003472 

,004051 

,0046^9 

,005208 

,005787, 

,006365 



Fourths^ 



iti» 



-•? 



a** 



«P«a 



^vm 



,000048 

,opoop6 
,000144 
,000192 
,000241 
,000289 

,000337 
^00038$ 

,000435 
,000482 
,009530 



This Table, as I /aid, being fo eafy, needs no Infiruflions 
fcr its Ufe .; nor Iball 1 pretend to fay hecimals arp ot any 
Service in the RiA^ of Ad^AXirt dLi}d ^ut^ra^ionol pffOi^ 



But 



Duodecimal APtthmeticU. t6^ 

But their extream Urility in the afoie&id Opeiations of 
AluliiplicAtip^j Divifien,' and ExSraclimoi Roots, oFD»- 
mdecimahy will be unaeniably evident by the enfuing Exam^ 
jJcs. 

MuUifUciztion. 
EftmfU X* What is pF, lo' multiplied by 8 K off ? 

F. 

Mulciply the Dmmtf/ of. p : la ^=i 9^71 
: % tiiR Pecimal d 8 : oS'se 8,^ 

9 ) 5900 

^ Vactto^htberFeef . - 78)^6$ 

Apfwicr, vix: X 8S : 2 : 8 3= 85,ir22 F^ipf. 

Example 2. Whkt is the Pcodua of 40 F. op' : 10" r 
I7 11': 09"? 

F^et / ' // 
Multiply the Derinfol of 40 : op : 10 3= 40,8194 

Bf the Decfmal 6^ ~ n : op =: 9p79i)^ 

— — — — 

I « 

In fuch Cafei where the Detimals ran fiu, axA termtiate 
in Xepeteitdsy 'cis-beft to multiidy bf ibt coittra£ied, oriw- 
pertea Way, heretofote taught, thus; 



^85736 

36757 I 

468 
272 



* p^ 



J^nX\yith 19 1. 11' : 07" ; o<5'^'' j 06 = ^9^9690^ Feet. 



Example 



^ TfrelJfe^ Ikcimah in 

» 

ft 

MuId|lytlieD^fmi4/ofx7i:pao^: 00:00:= 175,027 
Bj the Decimal dt^ — . >— 08 =5 ,000386 

1400222:2 



The 
the 






reH ' 



// - ///. 



The Pirodga is the Anfwcr 106 /?. 07' ; 03'' =106,60416 

ExmfU 5. Wh« is the Spf^r^ of 12 Fi?^f, op* ; of* ? 

/ // to 

r 12 ; 09 : 07 : ;o -> = X2,80439C( 

t^ .12 : 09 • 07 • xt> t = 934*j2' 
-A InvcfCed A j; > ? ' 

2560878 

1024351 

5121 

384 
"5 



fixSntc 163 Feet J 11' 05" Oi^ 09'^":=; 163,9523? 



DivifionA 



-^^- 



Thiu^ die De€i^<x i2 r/'MOV 07'' Jr^i2^iP4 

Pitt A 

! Then 3) 12,881^ <4ra^99i« =54 : 03 : oi^ :04 

12 

i" 



28 
li 

2^ 



* ■ I ■- 



f* V 

424 



• • 



4 

? S}» i*f Infisitttm. 



»4 

'X2 



24 J 

The z)mw of ^ -^t •:2 ' S? - '*'Jt?^ 

I T* : 04 = ,527 

Then ,$27) «4,«458C 

»47D ?3>i8i2S (27,75 = 27 F<-rf, osr 

- ♦ 3681 
-3325 



*356» 
3325 




tji The Ufe ef Decimals i» 

txMtlU 3. Divide 5 Feet, by i Foof, 02* : oj- : u'* 

•mcIVfMM/of 1 Footy 02* : o?" : ii^'ss 1,195865 
Thw 1,19^^} 5,000000 (4,188079 = , 

4 775460 (4 F. 02' : 03" : oi"' 

•224540 
119386 



•■ 



105154 
9550? 

% -9645 

II 

lO 



«> -^ IfDividc 32 :,I0 : xi : 06 r= 32,9x31^4 
BMWifie 4. -^g^ g ; Q, . 10 2 XX = 8,159x42 



Tfaus 8,159142) 32,PI3I5'4 (4.033904 « 
. , 132636568 (4^:00^: 04".: 



lor 



:* 276626 
244774 


a 

• 


• 3x55* 
24477 


• 


;7375 
7343 




32 
32 

» • 




>* 


ExtraSlton 



TheUfeof Decimals^ &c. 273 

ExtraBion of Roots. 

BKompk I. What is the Side of the Square of 

^* .. // /// ////. 
165 : II : 05 : oi *: 09 ? 

. The DmWof thofe DuodechnaU is 165,95239, thtS^uare 
Soot of which extni£led either by Logarithms or in the com- 

^* i // /// 
mon way, gives the Side 12 : 09 : 07 : 10 for Anfwer. 

Example 2. What is the Side of the Cube, whofe Solid 

*^ / // /// tf// 
Content is 1 : lo : 07 : 06 : 10 ? 

By Logarithms 

^* / If in iui 
Thus, the Decimal i : ro : 07 : 06 : 10:= 1,885898 
Then the Ijogaritbm of — 1,885898=0,2755182 
One Third of that is _ 1,25549 =0,0918394 

But 1,23549 IS the Decimal of 1 : 02 : 09 : 10 : ii 
Which is the Length of the Side of the Cube propofed. 

As in Vulgar, fo in Duodecimal FraSHofis^ the ExtraSion 
of Roots, can be performed, no way lb well as. by Decimal 
Parts ; and the other Laborious Operations are hereby ren- 
dered cafy and concife. 

The Ufe of Decimals in Sexagefimal Arith- 

metick. 

Sexag*fimaU arc thofe FraSiions which have 60 far their 
common Denominator ; and are chiefly ufed in Computations 
of Motion and Time. 

HeiKe this Kind of Arifbmetick is proper to Aftronomj^ 
which, as it is a Science oi Motiony and Ttme^ makes ufe 
thereof in all its Calculations: Hereby it is the Aftronomer^oaX- 
culates the Motion^ Place , Magnitude^ t>iftance. Time^ Af" 
feSlsy and other Phenomena of the Hefivenly Bodies ; tnc 
Suvy Moon, Planets, Comets, znd Stars. 

N n The 



274 ^^ ^/^ ^f Dectmab in 

The Notation and Rcadiag of Sexafffmals is in this Man- 
ner following ; 

5S/gw, Degf^esy MinuteSy SecondSy Thirds, &c; 
6 05 : 26 : 57 : 53 : 47 &a 

And as 60 is one Degree of Motion^ fo it is one iE?<7»r 
of f r>«^ ; hence Sdx^gefmats properly fl) caBcdV beglii ohlj 
at Alinutes^ and go to* Seconds, Thirds^ 4e. fbr^itb, inf^ 
both A/ofiov, arldr/i»if; though h\ common, it comprehends 
4*j Divijion of either.. 

But as all Aftronpmicd Calculations are tnade from SexagS'- 
fifnalT^umhet^y already computed and difpbied into TaMes'of 
various Sorts ; if 1 would ftnw or demonftratt the Ufe of D^- 
cimalsy and their Prefereiic« taSrj^tfl[^j!?«24/AbiiiJtTi hi theie 
Kind of Computations ; I rnuft firft fuppofe thofe Sexagefimal 
Tables, made into Decimal oait^ ; and if fuch a thing were 
once done. I Believe 'twould be jio vtty hard Task to make 
good the Pfopofikidn affert&i 

The Bfadermttii only judge of thb by t!he following Ex- 
ample of Addition in both Species. 

Deifmallf. 

49. = 3,73211 

58 = 11,9933* 
37 = 10,(53511 

45 = $^,843 1 7 

57 = »,9999iJ 

49 = » 1)6243? 

53 sa 8,82601 





SextigefinkJl]. 


S- 


6 / 


0? .' 


1 21 : 57 : . 


11 : 


: 29 : 4t • 


io : 


: 18 : 59 : 


09 : 


: 25 : 17 : 


c8 : 


29 : 59 •• 


II : 


18 : 43 : . 


Sum c8 : 


24 : 46 : • 



— «^ <m. 



hi this Spifimn I think 'tis eafy to obferve how condfe, 
fimple, natural ar^d ea(y the Operation by DeeimaTs is if cdm* 
pared with tnc Sexage^malProcets; which therefore 1 tkink 
muft needs prove.the i^refinrence and Eicellency of thofe Ta* 
Ues in Decimalt. 

But fince none as yet have laid any thing, abovt this Affaii^ 
nor have we any Agronomical Tables in Decimals^ t 6mi 
give a Specimen ^tteoi'm the Mean Motions of all the Pla^ 
nets for one whole Year, Daj, Hour, and Minute, in the 
Table fubjoin d. 

Planets 



Sexagefimal Arithmetkh 



?75 



Planets. 


AYtrar. 


A Day. 


4» H(j»r. 


A Minute. 




S. 


S- 


S. 


S. 


^Sm 


M,9P035 • 


0,03285 


0,001367 


0,0000 >8 


Moon 

» 


4»3»27i 


0,43921 . 


0,018292 0,000294 1 


Saturu . 


QAO709 


C,Opi*M t 


0,000046 


0,000000 


Jupiter 


1,01096 


0,00276 


0,000110 


0,000000 


Mars 


6.37574 


0,01749 


0,000728 


0,000009 


Ventts' 


7A9a^' 


OP554O : 


.0^002:22 ! 


0,00003^ 


MdrW!)\ 


i,7Poa5: 


0,1363^ : 


o,op$683 . 


0,0000p2 



Such then is the Form, and fuehiwould'be the DiflTer^uae 
of Deeimal and Sexngffimal SiUes ; fhe Numbers bete jire 
bomogenecusy all of one Sort ; in- them, th^yare beterQgener 
ousy or confift of diverfe Sorts ; b^.they are U/^iformwd to 
be lyrote as i«f ^^^ Numbers, there they are ranged in a dif* 
ferent Form and in Aivctk:CIafftfSj asjdlUmiaoed'lNumbers are ; 
befides the great ^Eafe and Facility (^ Working iimw^/ i|i 
Comparifon of Sexng^fimal ^Numbers, as I >betore ob&rved. 
'Upon .all thefe Accompts, and (everal others! might mention, 
A Set of Aftronomiedl Tables in Decimal i^umher^xm^ cer- 
sainiy be much more Ufeful, a(id'avei^ way prefei^ble to the 
prefent Sexagefimal Tables. 

After haying turn'd your Sexagefimal Numbers into Deci^ 
malsj they are to be worked in the fame Manner as Duodeci^^ 
mals throueh all the Rulesy as is there taught ; and therefore 
needs not be here again repeated. Only, I would here ob- 
ferve, that the Rules of Multiplication and Divifionj wbich 
are here often nece&ry, cannot be performed without a great 
deal of Difficulty, or a long and tedious ^Frocefs, whereas by 
Decimals 'tis done with t}ie utmofl Facility and Expedition. 

To this End, I hare taken Care that the Reader.ihouid 
not want large and fufficient Tables for the expeditious turn- 
ing of his Sexagefimals into Decimals, and the contracy ; the 
like of which are not to be found elfewber^^ that I>know of. 



N n 2 



C H A P- 



tjS 



CHAP. XL 



Doe life and Management 0/ D E c I M 4 L S 
{after a new Matimr) by Logarithms. 



THERE may chance to happen to the Reader a 
double Advantage in thb Chapter ; for firft, he may 
here perceive, not only the common, but an entire 
ntfp Management of Decimals by Lcgaritbms ; Bxid feconJlyy 
he may here as well as any where learn the whole noble and 
excellent Art pf Logmtbmical Aritbmeiiekj ik he has not 
learnt it already ; for Decimals and Integers having the (ame 
elTential Propcrtia, the Logarithms of both are the fame, and 
differ only in Axix Indexes* 

Bgt that the yomg Student may the better underfiand how 
to vary and adjuft the Index ot the Logarithm^ I have in the 
fblloiving Table given dl Variety of Cafes that can happen to 
a Number, its Logarithm, and Index^ under the various Con- 
ditions, and Denominatiojis of Whale Nwnber^ Mixed Num^ 
ter. Pure Decimal^ Repeating Decimal $y Decimals with Cy* 
fbers prefixed, Qc. as follow. 



Whole Numbers 
Mixed Numbers 
A Perfiia Decimal 



- \ 



5243 
524.3 
52,43 
5,243 
,5243 

c P5243 

Decimals with Cyphers prenxed < ,005243 

4,0005243 

^3y? Of 4? 
243,5r or 243- 

5243.3- 0' 524? 
i-,243' 



A Sivglif Repetend 

Mixed Single Re- 

ftUnds — 



Compound Mepetend^ -^ 



Index Logar. 
= 3>7»957PP 

= i,7iP5799 
=3 0,719^799 

— P,7IP57J>S> 

=* 8,7ip57PP 

= 7.7IP57PP 
= 6,7195799 

r= 0,5228787 

= 1,6368221 

= 2,3862016 

=1 3,7196075 

= 0,6378333 

= 0,3860408 

=! 0,7196234 

Com- 



The Ufe and Management of Dec. liz. zjj 

Index Logar. 

r jooooaf = 5,5228787 

Compound Repetends with i ,0004^ 3= 6,65783^:5 
Cyphers — _ ^ ,002-43' = 7.5860408 

*- ,05-24^ ^ 8,7x96254 
Mix'd Compound^ Rcpctcnds •[ ^^^f — i^vjolooo 

The fame with Cyphers \ '^fV ==^ J'^^giJ 
^^ ^*^ I >oo5^4^ = 7,7ip6ocx> 

From this general Scheme ^ the following Obfervations maj 
be made relating to the Logarithm^ and determining its Iitdex 
for any Kind of Number. ^ 

OiftfTvaiion i. That the /;!?yfAf of the Lognritbm of any 
whole Number, is always one le(s than tlie Number of Places 
of Figures in the whole Number. 

Ob/ervathn 2. That the Io^m/j&;« of any Number, whe- 
ther Integral, Mixd, or wholly Decimal, is the very fame ; 
only the Index differs and muft be adjuftcd folely in regard of 
the integral Part of the Number ; as per Ohferv. i. 

Ohfervation 5. That if there be no integral Part but the 
Number is entirely Decimaly and the iirft left-hand Figure be 
one of the nifze Digit ty the Index is ( ). 

Ohfervation 4. I'hat if the Number be e:Kirely Decimal^ 
and have any Number of Cyphers prefixed, the Index (being 
in this Cafe dotted on both tides) muft be fuch a^, when futn 
flra£led from p, the Remainder may exprefs the Number of 
Cyphers prefixed. 

Ohfervation <. That any Repetend, or Set of Cirettlatiprg 
Numbers, whctner WhoUy or Decimaly obfervc all the Rules 
of terminate Numbers aforementioned, relating to the Index ; 
but the Logarithm is different. 

Ohfervation 6. That the Logarithm varies, according as 
the fame Figures arc cither terminate or repetcnds ; and 
a|B[ain as thote Repetendi make either a Party or the Whole 
Number ; or thus, the Logarithm is bigger or lefs as the firft 
Figure of the Repetend is fo. 

As to what concerns the Adding and SuhjiraSling cf ftr- 
dexesy that maybe throughly underwood by the following Ta- 
ble of all the Varieties that can happen in chat Afl^ir. 

Addu 



97^ TbeUJi ani Management tf Decimals 



Addition. 



I. To — 



- 2,$IJ2I7tf 

— i,8< 



to6i8cx> 



Sam = 49^193976 



SubftraBion. 

X. Prom ^-. 4,jiP5976 
Subftraa 1,8061800 

Rem. s=s 2,51:52176 



2. To — ,3,3916407 
Add — ,5,2041200 

Sum = ,18,5957607 



'4M«i 



3. To — 2,2671717 
Add '— ,8,1414498 

Sum = 04086215 

4, To — ,8,5132176 
Add -* 3,8061800 

Sum = 2,? 195976 



Add 



I a: 



,9,2<57I717 
5132176 
8061800 



Sum = ,3,5865693 



a. From 
Take 



,18,5957^7 
,3,3916407 



Rem. = ,5,2041200 

5. From — 0,408621s 
Take — ,8,1414498 

Sum =3 2,2671717 

4. From — 04.086215 
Take .— 2,2671717 

Sum = ,8,1414498 



5. From -^ ,7,4086215 
Take — ,9,2671717 

Sum = ,8,1414498 



To underfiand the better ivhat concerns the Ordering and 
^Adjufliflg the Indices j in the foregoing Exampks, I have fub- 
xjoin'd the following Scheme of the Number of Cyphers, and 
:tbdr correfponding //«4f/V^ j. 

AfecwJ.ofCy.o, 1,2,3,4,5,6,7,8,9,10, 11,12, 13, 14, 8r. 
Their Indie. 9, 8, 7, 6, 5, 4, 3, 2, 1,0, 19, 18, 17, 16, 15, gr. 

Hence obferve. In cdding Indices^ i. If both be jAffir- 
mativey their Sum is jiffirmaiive. a, if both be Negative j 
and the Sumht under 10, add 10 thereto ; but if above 10, 
or jufi 10, caftaway 10; the Remainder is »^^af;W. 3. If 
]Ooe Index be affirmative^ and the other negative ; the Sum if 
under to is negative : it^y? 10, or above io> caftioaway ; 
the Remainder is affirmative. 

In 



after a new Matmer ^ Logarithms'^ iy^ 

In SuiftraSling Indices^ obferve, i.U they^ are boA ^/T^ 
firmaiivey and the Bigfier be the Greater ^ the ftentauider is 
Affirmative \ if the Lower be the Greaier^ the Remainder is 
Negative, (lO being added to the higher.) 2. If one or both 
be rfegativcy and the bigj^ fmaller than the foro^r, add 10 
to it; than if the higher be of greater Vaiue^ the l^emains 
are Affirmative ; if not^ they are Negative^ 

In order to underiliand the ^r# of Log^thms, and the dex* 
terous Managpaent of Numbers (more particularly circulating 
Decimal Numbers) thereby, 'twill be abfolutdf nece^ry to 
underftand, and' that perfedily weU, the following Logarithm 
metical Problems. 

J^roMem- 1. To find the Arithmetical ComfUme?if of any gi- 
ven Logarithm. 

Utile. Begin at the Left-hand co«fubflraa (meigtattjfj'each 
Figure from p, and the lad d all from 10. 

Ejfom. What is die Arithm* Comft. of the Log. 33^49262 

An^ec (per hde) ig = 6,1350738 

Problem 2. To find the Logarithm of any termnate Number 
under looooodOft 

Xule. Tike the Logarithm out oF the Tables to thtfour firff 
Figures of any given NinAier of above four Places, 
and aUb the next greater I/^aritbm ;. then take the 
Diflfetencc of thofe tw(^ Eogarithms^ and mtdtiply k 
by the remaining Figures ot the g^vcn Number ; from 
the Aroduft cut off £> iilany Pb^ of Figures to tUc 
Right-hand, as weoe the remaining Figures above 
fiur f then add the other Part of the Pcodu£l to the 
Logarithm of the four Figures firft taken out of the 
Gmon ; that Sum is the Logarithm fought. 

Example. Required xh^iogarirhm of 101 2659 ? 



l4aMawi*a«lhi«» •«• 



Their Difiererce — — 4289 

Multiply by the itmainfegFiguMB * 659 . 

Tie Prodtfft (wkh f Phoes cut off) 2&6,45i" 
Which add ta the L09iHttmoCiOKa:;3eoo5r8o5 ' 



\ 



The Sum h the Log^ctthKH : . i^ ^ 6^0^4631 (oxight. 

Problem 



iSo JT^ U/e and Management of DecittiaJs 

PfMem ^« A Logarithm 1)eiiig giv^n, to fiiid the Number 
belonging to the fame. 

Xmte. Seek the ikxi lefs Logarithm to the given one, in the Ta-» 
bles, and its four Figures are the firft four of the Num- 
ber required Then take the Differences of the gi^eci 
Logarirhm and the next lefs^ and alfo the next greattfr 
9m next lefs ; Add to the firfi Diflhrence, & many 
Cyphers as you feek Figures more than four. Divide 
1 hat by the fecond Dinerence, and the Quotient an* 
nexed to the four Figures firft fbuiid compleats the 
Number required. 

Example. Required the Number of the given Logaritbm 

The Logarithm next kfs is — ici2 =r 3,005180$ 
The Logarithm ff^xf^r^^/^ is — 1013 =s 5>0O56op4r 
The DiiF. of the given Log. and next lefs is =r 2826 
The Dit. of the next hfsy and next greater is :s 4289 

Then lay. As 4289 : 2826 ;. : lOOO ; 6^9% which annex 
to the firft four Figures 1012, they compleat ioi2d5P tlie 
Number fought for the given Logarithm. 

» » ' • • \-» 
Froilem 4. To find the Logarithm of any tetminaie Deci'^ 

W Number, 

Ji$i/e. Seek the Lngarithm for it as though it were a iohoU 
Number^ and then adapt the Index as before taught. 

Thus the Logarithm of 1012,659 is 3,0054631, and of 
,1512659=5,9,0054631, ^r. 

Problem 5. To find the Logarithm of zfingle RefAend^ 

or cinuUting Digit. 

Rule, To the tabular Ijogar^bm of the Di^it, ad^ the yf- 
rithmetical Comflepuntd the Logarithm of p, the 
Sum is the Logarithm (ought. 

Example. Required the Logarithm tX 6} 

To the Tkbular Logarithm ok 6 ss 0,7781512 
Add the Arith* Complement (£ the Log. 9 = o;o457575 

The Sum is tbc Logfiritbm fought of i^ ss 0,8239087 

111 



after a new Mannerly Logarithms. %%i 

I(ep. Digits. Logarithms, 

In this Manner I .r = o,04«57«>75 

have calculated the ^ ^^ 0,346787$ 

Logarithms of all ar = 0,5228787 

thclii^t Digits per' 4 = 0,647817$ 

Pttt^^lh circulating^ $ ^= 0,7447^75 

and difpofed them = 0,82^3^7 

ready for Ufe in the 7- = 0,8908$$$ 

annexed Tabk. ^ = 0,9488475 

^ == 1,6000006- 

Prqllem .6. To find the Lcgarithm pf an^ pure CompounS 

Rtifsiend. 

Jtule. To the Tabular Logarithm of the Number (as tcrmi* 
n^e,) add ih^ Arithmetical Complj^mevt (^{ the Logz^ 
rithms of fo many 9% as arc Places oi \ht JiepeUnd i 
the Sum is the Logaritjiin gf tl\e given R^petend. 

Example 1. Required the Logarithm of xht Compound 

To the TaUtlar Logarithm of ~ 24 = 1,9802112 
Add the j^iihm^Jjcal Qomflfitn^t .pf, 9p ^r 9^F^^^ z 

The Sum is the Logarithm of — -ar^ = l>3845760 

Example 2. Required tht Logarithm d£ ^6jf} 

To the Kite/iar Logarithm of — 36,$ = i, $622929 
Add the Arithmetical Complement 0^999 = 0,000434$ 

The Sum is. the Lognrithm of — ^o,5r = i, $627274 

Bxaniple . 3. Required the Log^itbm . of ^4;^ ? 

To the r^J«7^r Logarithm of — 3746 = 3>57?5^78 
Add die Arithmetical Complement of ^999 — 0,000043 4 

The Sum is the Logarithm of — ^4^ = 3,57:^6112 

Example 4. Required the Logtrithm of zoo^a ? 

To the Tabular Logarithm of — 2pp,6o rs 2,3023309 
Add the /^ri//b. Cc///^. of the Log.of 99999 2= 0,0000043 

The Sum is the Log:xrithm of z:o,6c^ = 2,3623352 

O iVb^^^ 



2 82 ' The Ufe and Management of Decimals 

Note, In all the foregoing Examples C^nd in thofc which 
follow) the Indexes of the Arithmetical Complex 
menu are emitted ; and this mufl be ob/cj^yed by 
the Learner in all Operations of this kind* 

Problem 7. To find the Logarithm of any mixed Repefendy 
either Sirg^e or Compound. 

Rule, From the given mix'd B^petend^ fubftrafl its f 6rw/- 
nate Part ; Then to the Lcgariihm of the Remainder 
add the Arithmetical Ccmfltment of the Logarithm 
of fo many Nines^ as there are Figures in the Refe-- 
fendy the Sum will be the Lcgarithm fought. 

Example i. Requir'd the Logarithm o{ 2y6 f 

From the given Repetend — ^^ 
Subflraft the terminate Fart — 2 

Then to the Logarithm of — 2,4 = 0,5802112 
Add the Aritbm. Comp. of the Logar. of jj* = 0,0457575 

The Sum is the Logarithm of -^ 2,^ = 0,4259687 

Example 2. Required the Ugarithm of 57,2^ ? 

From the given Repetend *-^ 57>23: 
SubflraS the terminate Part — 572 

Then to the Z>yj5r//i&/« of — 51,51 = 1,7x18915 
Add the ^r;f;&. Cow/>/. ot the Log. of p = 0,0457575 

The Sum is the Logarithm of 57,23' = '57576490 

Example 3. Kcquivd iht Logarithm of 2y7$'2:^ 

From the given Hep et end — 2,7^? 
SubduS the terminate Part — 27 

Then to the L'igarithm of — 2,726 = 0,4955258 

Add the Aritb. CompL of the Log. of 99 = 0^043648 
The Sum is the Logarithm of .— 2,7^^ = 0,4398906 

Example 



ajieY a new Manner hy Logarithms. 283 

Example 4. Required the Logarithm of 725,^ ? 

From . — — . 7^%^ 
Subdua' —■ — 7. 

To the Logarithm of — 72459 = 2,8602781 

Add the Arit. Comfl of the Lpg. of 9^9 = 0,0004345 

The Sum is the Logarithm of — 72:5,^ = 2,8607126 

PxamfJe 5. Requir'd the Logarithm of 26^927 ? 

From — — 266*^2,7 
SuWkaa " — — 26 

To the Logarithm of ^ 26890,1 =r 4,42959424 

Add the ^rit. Co«w/>. of the Log. of 9999 = 0,0000434 

The Sum is the Logpritbm of — z6e92,7 = 4,4^96358 

In the like Manner may the Logaritj^m of any other Mix'd 
Refetend be found, fo far as the Canf^n of Logarithms (you 
ufe) will permit. 

Prohlem 8. Between fTO,o Numbers given, to find anyNum- 

ber of i«^^« Profortipnais required. 

Rule. Subftrafl the Logarithm of the Uffer Number from 
the Logarithm of* the ^r^/if^r ; flf/V/^^ the Remainder 
by a Number greater by one than the Number of 
Means fou^t ; this Quotient add to the Logarithm 
cf the /^JT^r Number ; the 5««f is the Logarithm^ of 
the ^r/? Mean ; to which the faid Quotient is to be 
added again for the Logarithm of the fecpnd Mean ; 
and thus proceed for as many Means as you pleafe. 



O a ExdmfU 



284 The Ufe and iianagmtTn of Decimals 

Example. Between 8 and 56 to find ftmr Mem Profv^ 
tional Numbers. 

The Logarithm of 56 is — 1,7481880 

The Logarithm of 8 is •— «^ o,po2opoo 

The Remainder or Difference is — 6,84')Op8o 

One fifth Part (for four Means) is -^ 0,1690196 

To which add the Logarithm of 8 — 0,9030900 

The Sum is the Log. of the firji Mean 11,8092551,0721096 
To which add again — — 0,1690196 

The Logarithm <X xKt fecond Mean ^7,42=1, 241 1^92 
Add again — —^ 0,16901^6 

The Logarithm of ibethwl Mea»is 25,71=1,4101488 
. Add again — — 0,1690196 

Logarithm of ^t fourth and laft M^4« 37^4=1,^7^1684 

This Ptbblem I have chiefly inierted for their Sftkes who 
wculd hereby learn to calculate Tahles of Cmpemnd Iniereft ; 
The Numbers in the Table of Amounts of i /. being only 
Mean Proportionals between the Logarithm of Rdtfe and die 
laft Ytix\ Amount in the Table. 

Mtltiflication cf all KJnds of Decimals ly 

Logarithms. 

c To the lu^mthm of the Mtdtiplicandy 
i(fiJe. < Add the Logarithm of the Multiplier; 

C The Sum is the Logariibm dE the Ptodu£^. 

Exatkpte i. JWultipIy — 12,4 « 1,0934217 

By — — 3^=0,55^3025 

i The Ptodua ~ 44,64 = 1,6497242 

Example 2* Multiply — 36,5 =r 1,5622929 

By — ,00019 =r ,6,2787536 

Produa 'mm ,006935 5=^7,8410465 

Examfh 



€^er a new Manner ly Logarithmsl * 2S) 



By 

Sfiamfle 4. Mulcifly 

By 



570 






~ 43W4= 2,6378298 



,00021 



j7^98477ff 
,6,3222193 



Produa — ,000002037 S3 ,4,3089910 



Bxamfle ^ Mdtiply 

By — 

Fcodua 

JS^Mfle $• Multiply 

By •*- 

Pfodua 
Sxample 7. Multiply 

By 

Produa 

ExmpU 8. Multiply 

By 



^ a6,4 =: i,4at ^039 



i,4atJe^9 
0)8239087 



— 176 =z 2^2455126 



2,73 
— 2^ 



0,4361626 
0,4259687 



*^ 7,28 = 0fi&2l\t^ 

— ,473 =,9,6748611' 
^= 1,803705 3 

- 30,t = 1,4785664 



57,24 
2,75^3^ 



1,7576996 

0,4398906 



Produa — i77,6i23fi^ = 2,1975902 



Bxamfle 9. Multiply 

By 

Produa 

Bxamfle 10. Multiply 

By 

Produa 

Example ii. Multiply 

By 

Produa 



= 0,8239087 

— ,5r = ,9,744727 5 

— ^,7or = 0,5686362 



88,5734218 &. = 1,947:5034 



1,5627274 
0,3845760 



5 1,3265407 
= 1,6234 581 

— ^00,71^ = 2,9499988 



— 2^,2 3f 

— 4«:,0 



Example 



% tS Tie Ufe and Management of DeciwdU 

Btum^ 12. Multiply — ^0042rcr =?? ,7,62:^4^1 

By — ,ooocg = ,5,94884 75 

Produft ,0000003735 QL = ,3,5723056 

THivifion cf all Kjnds of Decimals hy Logd* 

rithms. 



ROe 



From the Lcgaritbm of the Dividend, 

SubftraS the L^aritbm ot the DiviJ^r ; 

The Remainder is ^trLogaritbrnoi the Quotient. 



Examfle i. 


Divide 


— 44,64 = 1,6497242 
— . ?2,4 = i,0934;}i7 




The Quotient — 3.6 = 0,5563025 


Examfk 2. 


Divide 

By 


— 310 2,4913617 

; — , 4.275 = 0,6309561 




Quotient 


— 72,5 »457 = 1,8604256 


tbcampU :;• 


Divide ^ 
By 


— 434,34 =r 2,6378298 
— ,762 =,9,8819550 




Quotient 


— 570 = 2,7558748 


Exam fie 4. 


Divide 

By 


— ,006935 = ,7,8410465 
— 36,5 — 1.5622929 




Quotient 


— ,00019 = ,6,2787536 


Example 5. 


Divide 

By 

Quotient 


— ,000002073 =,4,3089910 

— ,000X1 =, 6,3222 J93 




— ,0097 — ,7,>'8677i7 


Example 6* 


Divide 

By. " 


— 176 c= 2,1455126 

— . 6y 0,8239087 




Quotient 


— 26,4 = 1,4216039 


Example 7. 


Divide 
By 


*• 7,28 = 0,8621314 

_ 2,0 ~ 0,4259687 



Quotient 



— 2,73 



0,436162 
Eyamfj 



after, a new Manner ly Lcgarithins. 187 

Bxample %. Divide — 30,1 •=1,47856^4 

By — ^^, = 1,8057055 

- vQH9^J'ciit '— . r ,475 =39>67486il 

Bxample 9. Divide — ijjfi\2^ =: 2yi97')902 

By ~ . 2,7ja' ~ 0,4398^ 06 

Quotient — 57,24 = ii7576596 

Example ip. Divide . — .-^jya is= cs5|68636a 

By — - > ,* =3 ^7447275 

Quotient — ^, == 0,813^7 



- s 

' ^ __ ♦ •*■• «... N ♦ I «. » t 



Example 11. Divide . — \^0O,7l8' = 2,9499988 

By — ' 4^;^ :=r 1,6254581 

Quotient — 2T,2^ = 1,3265407 

ExamfU \2, T>W\^b ,000600243^1^ rs ,^386'l5j06 
^ ' By t — * ,WoSf7^ es, 6,5570(^2 

Quotient ,0011^,9^ =,7,0491454 

The Golden Rule in "^Decimals ly Loga^ 

rithms. , . ^ 

• ' • '. 

Example i. DireS Proportion, 
If 2C ^j'r. %ilk. q£ Sugar — 2^9:^75 = 0,4^79778 

Coft 61* IS. 8^. —^^^ 6,c8^ = 0,7841516 

What cbfts 12C 2qrs. — 12,5 r=: t,0969ioo 

1,8810416 



Anfwer, 13/. 17 j. 8^^. — 25,886458^ == 1,4130638 

Example 2. Inveric Proferiion, 

If Wheat ht 6 s. 4^. ^<?r Bufloel^ — 6,^^ 0,8016525 

And the P^». wBite Loaf weigh 7:^ 6z. 7,75 = 0,88^3017 

" f — 

What muft J: weigh, when 1 1,6909342 

/^/jFiftff is 3 i. 10^. per Eujhel} T — 3,8^ = 0,5835766 
Aufwer, 12 ox. 16 pwU 2gr. = i2;,8o43 ©r. = 1,1075576 



i88 theUJe and MaiUigemi^cfBecimaJs 

Ater, EUier of tbde Quefiions mtf (and that «K>ft cx>ii- 
. «i^caAr) be wrought at once, viz. by One Adih- 
tiofi of me two Logarithms now added, and the yf- 
rifbmethal Compkment of dot Logaritbm that is 
fubihafted; for to fuhftraS a Logarithm, ox add 
its Arttbmeti^al Cdmpjemetftf produces the fame E& 
fea, or is the fime Thing. 

itxamfU 3. Direft TVof trffav tff ^»^ Operation. 

If i C of IMyrro — — j5 = 0^01029? 
Coft 4/. i^i. -S^. — 4»6? = 2'^^58235 

^/Vhatcoft 7 Pounds? — ,10625 = ,8,795880 

Anfwer, oo7. 11 i. 7 ^- — »S79i^ = ,9,7628034 

SxpnpU 4» Inverfc Profcrtion^at one Oferation. 

If -8 «p^i in JSr;**, . — 8 « 0,9030900 

Require 20 in 2>»gf6, to make an Acte \ 20 =:: 1,3010300 

What Length does 12,5 -Rods in> _ 8,9030899 

JF^Vf* re^e^ an i<w/ / *^ JL—1 

Anfwer, I2i8 Ro4s -- , 12,8 = 1,1072099 



This Method with woric- 
ing at once with Arithmetical 
Complement is to\beadvfled 
to ?he expert Geometrician 
ill huTrigOffometrical Cdlcula* 
iionSf as ojuch the beft. 




••••«•••••••* 



fM.4 



Ais.. :^J2r,jr iC 






Example 5. 

-As the Sine of the At^le ^OC 39 : 15 = 0,1987985 
Is'tothe-Side given ^ C — 12,5 = 1^969100 

^ is the Sne of the Angle BAC .50 : 45 = 9^889612 

To.the«idefoaj^iB£* — 15,3=: 1,184^697 

ExtraSiion 



after uiiiw.Mkm9^ly Ls^^ 2%t 

ExtroBi&n afRoSts in Decitmh 4y Lo^a^ 

'to exriiS the koot of any I^umber, do tbusf . 

^ Spiof^ Rooc o Divide the Lev ^ a 

u It DC tftc < ^i^«^^^^Root f f^^ Niio*€r "S 4 

Ar^ybW Root 3 % ^ C ^ 

Then tlie iecond, third, foutth^ Br. Taitof diet.qp»iduil 
dius divided, IhaQ be the Loguritfam joTilie Root £»^fat. 

Example %. What is the S^i/zir^ Root of tbe Nomber 
2830,24 ? 

The Loguithm thereof is *— 3,4518233 

Half of whidi, is the Logarithm of 53,2 ss 1,725^116 
The H&of fougVu:. 

Example 2. Required the 5^^^ Xaot of 13,2 ? 

lidg^ithai of riie fg^ven Numbfr 12,2 r= 1,1205,73$^ 
[air, b tbe Logan of the Root 3^331 Sr. a o,5iSo2869 

JSxjimfU 3« Rcquit-d the 5f»^^ J{»o/ of I4,#l 

TTie Logaritbm of — I4>^ — f^lid:)3i4t 

Half, is the Logar. of the Root 3,8297 8^. sS: 6,585*657 

Bxemtfie 4. What is the Ciiir i^^r of 1^726^ 

The £<jf .trf f Alto of — — 1,728 «** 0,i3ffj3;||^ 
One Third, is the Log. of the Ck^ Root 1,2 ^2 0,079 1 8i i 



Enrnpie 5* Wliat is di^ BtfOhikate Ropt of i7>,6if ]• 

Thd t^ogariflm of — 17*^^ « M4*11»* 

Ooe 4ch| is tlic Log. of tbe R. 316401198^, s 0^^11394 

ixakple 6. Quere the $HrfoUi Root of 31,25 ? 

The L^iibm rf .— — 31,25 — 1,494850a 
One Fitch, is theLt)g^ <)f the Root i^^o 8r^ ::3s O|29op7bo 



apo The Ufe of Decimals in Algelra. 

Note 5 When the Index is Negative^ add to it lo for the 
Square Root, 20 for the Cube ; 30 for the Biqua- 
drate ; 40 for the Surfblid Roct, @r. and then di- 
vide as before ; as in the following general Ex- 
ample. 

ExamfU 7. What are the feveral Roots of ,2758^ ? 

The Lo(rarithm of — ,27^8^ = ,9,44071 32 

An Half, is the Sifuare Root ,52523 8r. — ,^,7203566 

A Third, is theC«*r Root ,6509 gr. = ,9,8135710 

A Fouith,isthc BiquadrateKoot ,7247 Sc. = ,9,8601783 

A Fifth, is the Surfolid Root ,7729 8^. = ,9,8881426 



C H A P. XU. 
Tie Ufe of Decimals in Algebra^ 

exemplified in the (I(efolution of thirty four 
SeleBy (pleajanty and Ufeful Al^ehraick, 
Quefiions. 

NOTWITHSTANDING ^/^^^r^t has the Glo- 
ry and Reputation of being one of the Topmoft 
Branches of tlie Tree of human Arts and Sciences ; 
yet muft it be acknowledged that ( as fublime and ufeful a Sci- 
ence as it is) it would anfwer no great Purpofe of praSical 
Knowledge, were not the Art of Decimal Arithmetickj on 
every Occafion, called in to its Affiftance. Vulgar FraBions 
and Algebra together, may be view'd as the Blind leading the 
Blind; And Whole Numbers raiferably help the lame Dog 
over the Stile. 
f 'Tis D^f;in<2/x therefore ..which in all Cafes ( not fiudied 
and fiated on furpoje^ but) which contingently or occafion- 
^lly happien, can only fpeak out plainly and intelligibly the re- 

clufe Meaning of an Algeiraick Equation or Theorem. 

~" This 



/ 



I 

The Ufe jof Decimals in Algelra. 291 

This I fhall make appear by the Refolution of the moft cu- 
rious pleafant and ufeful Qucftions, which I have felefled from 
the beft Al^ehraick Authors extant ? 



fijieftion I. The Sum ( == j = 67 ) of any two Numbers, 

and their Difference (zznd zz^o ) being gi- 
ven to find thofe Numbers, 



Let 

Then 
And 

2 — 3 



I 1 4 — the Greater, and ^ = the Lefler Number. 



2 

3 

4 

5 
6 



ii + tf s= J = 67 
2« =: J + rf =s P7 

i-ii: — ^8,5 the greater Numb.] 



a r= 



2e zn s ^ d — 37 

3 ■ ■ = 18,5 the lefler Number! 



i 



fought;. 



e 



Quefthn i. The St(m ( = * = 15 ) ^nd Prodgef 
C = f := 15 ) of any two Nuipbersgiyen, to find thofe Num? 
bcrs? • 



I 
2 



Then-{ 
I e- 2 3 

■■.2X'^ 4 1 



3 — 41 5 



5 



6 
7 

S 



4-t-*=rf=l5 



tt^*- 



ZpZ}^}-Quere4,an(J*. 



4/1 + 24^ 4" '' =* " = 225 






as — 24^ - j-eessss t ~ ^jp =: 1^5 



4 — f = v' / / — 4)> rr : 12,845 Qe. 
,24 = f - H V// - 4f = Ji7>845 

= l±4:=:*^-=i3^2i 8..Great.L^j^ 

: 1,038 @f Lefler ^0"8ht. 



4 



s~V$t—0 



) 



^ . • 



Pp. a 



duefiii 



Her^ 



) 



ap> The life ^ Dee$maif Hm Jlgf^p 

Qu ftiw :j- What twa Noinbers are . choti whofe Sum is 
r=: 40 — J, and tfa^ Greater divided by phe Z^i^r ihaU 
quote 50 = 5 .^ 

5 j^ + ^ = i = 40 

39i?iS^a^-Qwt€tJ^"S^ 

df^efii^, 4) Wb^t tv(^Nimbe*» «^ thof^> ^offe ^<^^ i^ 
?;: 8 cj ^, and the Sum of thcir^/jf^i: 37|:== z/ ^ 

Here i\ I I? :+/**** '^ ^.^ * P «5N»c ^ «fid <^?' 

^ — 2| 4 2/1^ = W — » = 26,4 

\ i \a^e =Ji z.^'^if — ^j^. 

1-7" f >IM! '« I ..,1 . «. V , 



2 X < 
I*— 9 






5 "C 2 



7 4-"2l8Ug '+Vf-*^ ;;=5./>< 



Gicatet * 



2 ; .- . iNotnba 






Quefiipn %. . ^f tbe ^»j?u>F any tVO Numl^en bp, 5Q ?= *, 

iLzfcf — .< .i/i 



Pcrc^ 



3 



1 -F-S 



4-r 2 5 



24 = 






Y 



4 s= iiXf = 27,72 die Greatet } 
f «ii^ = 22,27 the Leifer j 



fiHiglht. 



Qfl^ivt 



I^Ufitei JSfedmah m J^fifhf. ^sp^ 



30 ■=. dyiWd ttvs Prodva 5i{i :=:/. ftftere ihofe Nuad>ea? 



2X4 

6+1-^2 



« — * =s4=: 






91 ldr»::tr/l 

3 

4 ] 



•N 



4^^ = 4f = 9048 

2 



7 
8 



^-.iiiii^ [fought; 



i 



Queflion 7* Suppofc thie Di0renefi of two Nuaiben be 
the lame with the Scr^fiiMrfoTtlui ^M#4r diiadtd b^te 
/'^^y viz* = 18. Ctuete tbofe (<^ia]l)en ? 



Here 



c. 



I 



2 X^ 3 

I +r 4 

?> 4[ I 
X -7 



4 — ^rri/— 18 




— *-^*i 



e =ycrio;y^ 
a:=zqe^=s, 18^ 

^=— ^ =51,058 the Leffiir^ V\ 



5—1 ' ' L Number 

8 L== Jl,==:ti*,oiJ8FtH4QK«4r 'J fcj'g'^- . 



Queflion 8. The Diffe^)ic9^xf\\ff(i NifdlAeif =5'-= k, 

and the Sa|lv-f^dMg>r %4mr.m52L^:^p:9:i hdoBgti^il ^ tftfitl 
tbofe NumberSf 



«*« f I ^'li^.ri^'y^s }^^^^ '•' 






3 \aa — 2ae -^^ ee s:: dd =s 2^ 

4 I 2^=r X — ddz=: 20 

5 \aa j^ 2ae^-\-eesss2z^Mss8'i 



5 w 3 



994 ^ ^ ^ Dectmafj i» Algebra, 



5M2 



6 

7 



6— 1->2 8 






V 
LNambct 



e=: 



i 



and cbe Difference of tbeir Squares ::= 27^ ss^, being giv^en; 
to find thofe Numbers ? 



X 

2 



Here^ 






4 2« = 



35.25 



-^ = ^ =.'V'^ ^^" 1 Nmnber 



3-5 M 



/ = ^,^=s 5,625 Leffcr j! 



loug^ 



JiUffiiott 10. The PtcJuQ of any two Numbers, being 
':s ^7:^ -r ^, and the Qasotreitf- cf the Greater divided jt^ 
Ok Leffer z=i 7^ i d^aer^, thofe Numbers? _ ' 

• _ . * _ ■ 



u >• J 



H«e M 2 — = J = 75 { Q"^' «' 'f 

1 X 2| 3 |4<i = j/> =s 42P75 • ' . 

jiw-si 4 U aa\/^ i= ^07,304 8f. Createt ) 

1 -r 2 5 I ^* = -J- = 7,64 ' JNunajet, 

5 w 2 6 # = V<^s:.2j764 8f. Lefff. l 



' •• -• 



^> V 



« ^ 



S^eftkB 



The Ufe of Decimals in Algelrdi 195 

duefiton II. The produEl of any two Numbers =100^^, 
and the Suj^ of their Squares =; Xooo = z ; ^erd thofe 
Numbers? 



Here -{ 


I 
2 


1 X 2 

2+3 
4 u« 2 

% - 3 
6 m* 2 


3 

4 

5 
6 

7 , 



T$i ='r= .0=0 } '»'""^ • •' 



aa 



5+7+2^ 

5-7-r"2| 



2^^ = 2^ = 200 

/i4 — 2tf^ + ee — - z — 2^ - 800 
<a — ^ — v' ;5 ^ a> = a 28,284 



^: 



Queftion 12. The /Vo^is^iJ? of any two Numbers =io=r/^, 
and the Difference of their Squares = 20 = at, being given; 
thence to fand thofe ISumbers ? 



1 ©• 2 

2 ©- 2 

3 X4 

4+ 5 
6 luf 2 

2 + 7 
8 4-2 



9 uii 2 



7—24-210^2 



3 



/Z4?^ -^pp = 100 

aaaa — laaee 4- ^ ^"^^ z=ixx=: 400 



.; ' 4^^i?^ = 4P/' z=: 400 

6 aaaa 4- 2aaee-j-jeee :-■ *Ar 4- 4/>;>3b8oo 

7 [/W 4-* ^^ rrr ^xx+^pp = 28,284 

8 ' ' 



10 



II 



2^^ = Af + V£ifii2 = 48,284 



,=\/ ^+^r^^^^ 3::4^l7gr.G. 



1 



/^V^^^^ -"= 2,035 g.L. 



FN. 



>. 



Queftion^ 



nd tfie Smmdt theit ^wrw = V =^ Iwqf yn* 
CnddiafeNiimben? 






> 7 * 







8@- 2 



^ 



\ 






Qurfi/om 14. Suppofe the Quattemt of two Not 
t= 20 =f, and the Difference of their Spupres = lOO 
Thencsc to find the ftunbeti. 



^'^^ ^ 

7-r?f-iri 

9 tlM 2 

8 w 2 



4 
5 

7 
8 

10 



aa~. ff**=4oo« 

ffT' ->■<'« =s « =s 100 
ft = -^^^ = ,2506 &• 

At := -«* ... BB 100,2506^ &^* 



H L = V-^ = 0,5006 8r. Ldfcc Y'"^ 

dueftion 



The life of Decimals in AJgehral i9i 

QueRlpn 15. Suppofe the Sffto of the Sfuaifes of Aiy two 
Numbers s^ 306 == s, and the Differakce of cbe SoASpiarti 



rr 250 

{ 

1 + 2 

And 



X, to find die Numbers ? 

1 l*- + «=-«=3PQVqaert4,/> 

2 l4<l — ^<»s=;v £S 250 J ^ ' 

3 1 2/w = X + jr = 556 

I Z jr X 



5 
6 



4=s 




= 275 

■ > 

*=x6,^33 8f. Gteatetl 



=V' 



@^ Leflec 



yNiunb^ 
rfouglik. 

j 

Thps any two of <hofc 5/4f Things (viz. 5i^«r, Difftfrencef, 
Pf6du6i, Quoiienty Sum of the Squareij and Difference or 
the SquarJsy of any two Numbers) being given ; 'twill be ea^^ 
to find the Nuipbers themfelves and all the other Partictdars; 

I have chofe to give the Andtyiical Procefs oi the Work dt 
each Qaeflion at lar^e, that the Young Student may lee the' 
Manner of IilveAigaai^ Theorems; add by viewing the fre- 
quent; Divifions and Extraflions, may the more dkady per-' 
ceiyer the great Ufe^ or rather^ the abfolute Necefiity of Bed* 
fnalsy in order to exprels the Equations in Numbers. 

But in thoft Queftions which ioUow, I have ohiy eMiiMted 
the Tbswem or Equation which ainfwafs thenir^ and giv^n the 
Solution of each in Decimal Numbers. 

Quefiion 16. There are two Numbers a^ e. The Sum 
of their SquArei \» aa-^^e z=z% td^ The Qfeitiff Is tdl 
the Ltfs asi2=:^^ist0 7=^ Tlut is 4 i e 11 b i 4 
Quere 4, e ? ' 

tbeirem. e = V^Z J to ie rciavci 

titO, Multiply the Sum of the S^ua^es « = ^ 
By the Square of df .-« dd ^ ^^ 

TheProduflis - .2^4/476} Difi<}en|; 

Then to vat Squa^t <X i ^ M ^ 4? 
Add the Si;»/ir^ of ( — ^^hb -zsl 1^ 

The Sbm of both is — di'\^ii ss**iM^diviibr. 



15^ fbe Ufe of^DecimaU In Algehral 

flf-whidi Divide aw, V *dd _ 



The S^are Root of which is N-^^'TTj^ —ez= 4,966 
^° " l7 : 4,966 :-:irf%'ii3r^^^^4,966 &c, 

Queftion 17. There are three Nimbers in rf/ffwi?^ 
Proportion, rir* -^ -^ a. m. e. 

'2 
The S^OT ot the Extreams — ^ + f = 57"^ = '^ 

i!ndtlj€MM« — wsxr?. Queijetf, ^.^ 

. TBe^cm. . r= f . ^.^^" ^^^ } to be folvecL 

Firft, The Sj^^r^ of the Sum is — w =s= 1 398,7^ 
' Aivi atic Square of m X4 is r— 4w/w r± : 676 

. Which iMnOtdj thcrc'r«nMn$, ss -^ 4mm == 722,76 
- The ^Square Xo&t whcncof is ^ ss-^ ^mm =• 26,885 
. To iihich add the Stow of the Bxtrsams s = :57,4 

TtM(.Si«iais -.-:• J + V « - 4"»» ~ 64,285 
Tiw half .of thut is th« Fir^ iN^«w«*«r ^= 32,14a 

Then as -f '^ .:«»::«:? . ^^^^ 



^ - ^r Swft of. which is th» Proof a-\-.e =s 37,4 



4 • • • • 



Quefihn i8. There arc three Numbers in continued 

. ; 2 I Quere 

TheirS^wis — ^ + « + ^ = * = 3^-7 r^,«Lr> 

And the^'5«a di ^hcir S^. /rf -f »5iot ^ee=.x:=rj6l I 

r " • ' \ . 

-- ., ' . . ~ / ♦ ' ^^ . - I • ' . Firft^ 



* t • * « 



»• 



The Uff (f Decimah in Algelta. 299 

Firft, from the Square of their Sum ss =r ^78,7955 &• 
Subflrafi. the Sfl»^. of their 5fff4'i^^ s r^ 763 



■^» 



The i^^i»/7/^rfipr is — w — z z=: 115,795^ 

Which divided by twice the S«/« 2J = 62,^7i/[29 
Gives the ([uotient *— * ' jw nr 3,446 8r. 

Then 51,^85714? — 3,446 — 27,839 z=za + e 
Which may be found as in the Theorem of th^ laft Queflion. 

QueftiOTt 19,^ Suppofe three Numbers <i, i, c in Mtffical 
Proportion,- viz. As a : c : : a — h :*—.<•, and any Two 
pf them being given, to find the Third. 

eh 
Theorem i. ^ = -7 — r Finds ^, if ^, r, begiveo. 

Theorem 2. * = ^ ^ Finds ^, if a^ r, ,be given. 

Theorem 3* ^ = r Finds r, if 4, J, be giveiij 

• * 

Suppofe *rsi3, andr=io To find a? 
Multipjy ^- • i z:^ 13 

By ~ ^ f = 10 

The Produft ^ ' cb -z^ 13© ' 

Divide by twice r, lefs J, ic-^l-j) i:iO=ci (i,e,57i4jf. 

The Quotient is ^ — 1^,57142: the Number fought. And 
fo for either of the other. 

Queftion 20. Snppofe /(j^y- Numbers^ a, i, c^ d, in il&* 
^<r<z/ Proportion, viz. a: dwa — hxc — d-, an<} any Three 
of thefe given, to find the fourth. ;. 

Theorem 1.4= -^™. Find? ^ if *, c^d, be given. 
ri?»?«rf « 2. * = ■ ^"'^ J" ^ -, Finds *, if 4, </, f, b« givAi, 

. T^m-ms. f=if^rL£*,Findsf,i€4,M>giv«, 
Ti&wrm i^ 4- — ^ , Finds c|, if 4, J, <• , be given. 

Qq 2 eiuefthn 



• . 



'|o^ Tffe Vfe rf Decimals i» 4^gfha, 

Qneftion 21. Soppofe it was required to divide any Num* 
fcer £20 rr $) into E^dream and Mean Proportion ; That is, 
into jtwo fiich Pans, 4, and f , that tf a == 4^ 4- <?if = se. 
IQoeK4, r/ 

Tbeofem. 4^>Jss^ ^rs^ 4 | 

^4f ^ the given Numbejr — ss = 400 

'Add thereto J of the faid S^«^^ — i ^^ = 100 

The Sm is — *J 4- j ii ^ 500 
* Ttitfyuare foot thereof is — v' w + f w rs 22,3606 %• 
f rouHittbich fiibftrad i the given Numb, is :^ 10 

JTbere Rebuins the Grf^ier Part — a— 12,3606 @r. 
WhidlfoBftra(acdrrnmtliedycn^Jumb.> _ ^.'^tr 
Sri»ereiiiaii«d^i#^ r- 7>63Pa'@^. 



>7off, Tis impoDibic to apfwcr this Queft^on in WhoU 
Vtmktnrs. ' 

Siuefthtf ?2. What is the Canon or pfeorem for ExtraSt^ 
ingthefquarf Sfiiot^ ' 

Suppofe -f- ^ rr Root ; Then th$ C4«o» is this, p/z, 
«-p24* + *«:=:S(nate: * 



• • 



Exttaa the Sq. Root of 65*5,36 ^s^aa-^- lae^ee 
From die Numb. fuWlL 400 zziaa (azz 20 

TheKRemains — 255,36 = 2ae -{■ (e I ^ Z 'T 

Divide that by 2«"40 ^255,?6C 5 - * 'i.,-i---r;i 

f hen fubftra^ - , ^^5 = 2-^ + ee \^::^a, ;^. 
Xbeie Remains — 30,36 = 2^e + ^tf anew. 



VVhichdividby.2«=50)?o,36(,6rr^ anew r 2%^a 
JThen %aiq fubfitaft . 30,36 = lae-^fe ^ j6zze 



• • • » 



gueftion 2 3* What is the Qanan or Theorem for extrading 
^e C«fc'/Wor/ . • - ^ 



is tt^ C0nof$ or Rule for i^xtrading;! 



Rcquird 



7ie Ufe of Decimals tH Jlgelral 30J 

• • f 

Requir'd die CRoftt of 1955,1 25 = «4« + 3/ia*+ ^aee-{^ee 
&ub|baa ipc Cube 1000. = aaa («=io. ifl; 

There remains — 95:^,12$ = 5«»f + ^aee + eee 
Div. by ?«^+5«=330)P53^i25(2=# 

Then — 600 a= 3<m* 

And — — 120 = a<»* 
Laflljr — 8 =z et^e 

The Sum ot all is 728 s= J3*^-^V^^ee HSbtt. 
-- . Ittom the Remainder. 

JJiere lemains — 225,12$ = ^aae ^^aeeJ^eee^ anew. 
Di». ty 3^+3* =468)225^(,5 = ^ ^ f Then k ^ 

Tl]«n — — 216,000 = 3a<tt> J ^ =* 2 

The Sum of an is 225,1 25 =r 3^^ 4. 34^^ -f eee to be 

(fubft. from the laA Rem. 

lience the Root is 10 4- 2 4- ,5 rr 12,5 

tffte. From hence app«rs the RahnaUdl^t Method of 
eKfraams ^Cuhe Root ; for the Precepts then; 

lZf^^y^°^l expreffing the order and coMi- 
nation at the Sjmolt of this Caaov. 

Qiteftion .24. What is the Manner of Bctraaiw b, Con. 
rgivg Series, or Theorems raifed thereby ? 



^fe, |?ut Mr. ITarJs I take to be the beft, which is thus. 

Ltc aaszG. Quere 4.^ 
Let 

Then 4 |--— — — - or, -, — -- - jTheoremsfot 
\v\-e » V+i^-'^'ltheSq.Roqt. 

Lee 



I 

2 

3 



»■ + '=« The Root fought. , 
rr -{- 2re .^ ee ~ aa —G. 

»•' + '" = ^ - '^ = fi The mviJeiid. 



'■\: t> 






*3o> The Ufe of Decimals in Algelra^ 

Let aaa = G. Qaere j. 

1 Ir -h ^ fl The Ro;t fought 

2 frrr \ '\r7e --^ ^r^^ -f- eee — /7^/j = ^ 

1 ^/»*» n, _ rrr 




rr4-^^+ 



t'^tf 



V 



=:D, the Dividend. 



reA-tezsih — bcins; of finall or no Value. 
— i — = e. The TBeorem for the Cuhe. 



= Z) and 



D 



=z e 



' If aana rr Q Then — — — m^ auv* — — , — - 
The Theorem for the Biqaadrare Root. 

If aaaaa — G. Then — ^^^^-— = D. And > ^ 

The Theorem for the Surfolid Roof. 

And in the fame Manner proceed foe any othpr Root. 

Note ; To work by thcfe Tbccrcms, r muft be taken left 
than the Root ; otherwife, if r be taken greater 
than the true Root, it will beir, — G inflead of 

Q — yr, and — := e and the fame in other 

' 2r — e 

Theorems for the other Roots. 
After the fame Manner you raife Theorems for all kind of 
adfeSled Equations. 

Spppofe aaa -f- 244 = 587914 Querc aP 

Put 
. I ©- 3 

I X ^ 

2 in N'jmb. 

3 in Numb. 

4 + S 
5*-^5i392o 

7 -r^ 

8 -r 



l 



I f 4-f =: a Put r tr 80 I ^ P^^^ " f ^ fj 

24^'+ 14^ :;^ 244 

512000 -j- I920oe -|- 240^^ — ^z^ 

1920 4- 24<? 7= 24iJ 

513920 + 19224^ 4- 240^^ = 587P14 

192241? 4- -4-^^ = 73^94 

Z:>,ie '\- ee ^=^ 308,31 =:= D the Dividiend 

^ — r— ! : See the Operation 





4 

5 
6 

7 
8 



bo, I -|- ^ 



8Cj,0 



The Vje ofDecimaisin JIgelrai joj 

80,1) 3o8,?t = Z) (3,7 s^ 
4-^ = 3 2493 



t Divifor 85,1) 5P,oi 
+ ^= ^ >7 5866 

1 Divifor 83,8) ••35 



r===8a 
^= 3>7 



r + ^ = 83,7=« 

Here 83,7 is a new r for a fecond Operation ; but being 
iiivoived will be found too big, or greater thati the true Root; 
Therefore it muft be made r — e zna the Root. 



Thus 

I & ? 
I X 24 

2 in Numb. 

3 in Numb. 

4 + 5 
6—587914 

'7 -r 25'iTi 

8r- 



1 [r — ^ = ^ the Root foughc 

2 rrr — 3rr^ -f* 3^^' = <«^^ ' 
^ 24r — 14^ = 24^ 

4 586376,259— 2IOI7,07iP+25I>I^^=^'W 

5 20o8y8 — 24^ =24^ 

6 588385,053 — 2 1041, 07<? 4- 251,1^^=587 

7 21041,07^ --25i,i«*^ = 471,053 (914 

8 1 83,7955^ — ^^ = 1,87595778 =:= D 



83,7955—^ 
Operation 2» 

^ 83,7955) 1. 87^?5778C ,022392736 

— ^ = 02' 1,675510 

■■ mi n i«^ I •» - ■ — iiMi ■ 

1 Divifor 83,7755) ,2001477 

.^e =3 fioz 1675470 

2 Divifor 83,7735) ,03290078 



— ^ r=± ,0003 ,02513196 



3 Divifor 83,7731 



,00776882 
753958 

22923 
i'6754 



6168 
5864 



From fa^netf 
*tis fufBcient ta 
work by Con-* 
traSed Divifion 
to find the Re- 
mainder of (he 
Rood* ' 



3.04 
251 



Now > =2- 83,7 
And# r= 0,02^92736 

Then r — e- s= 83,677607264=4 ^^ 
•The Root of 587914 rcquir'd 



49 



4 8r. 



And 



304 7^ Vf^ (f Decimals ik AVgehtd: 

And if this Root be not exafi enough, it may be made ^ 
New r, to wori as befbr^, in a third Operation, 
And thus may the Root of ixxf A^eBed Equation be 

fiueftien 25. 

( Am Body _ ^ ^ 

i Pals over a g^ven Sfact -^ r , 

In a ^ven Time — / 

And any other Body .» £ 

^.^m^Cm i P*fi o'tt a Space •*. d 

^•K^* < In the 27«^'^ - - ^ 

Alfo their Diftame of Z'&fr^ — ^ 

And the Interval of TVivi^, w 1 ^' 

I which they begin to move, be j 

Thence to determine — te Th^ 

IXIlanc6 they pafs, *ere the htndmoft Body overtakes the 

foremoft, if they both tend the fame way i or before thej 
meet, iftfadr Motiombeoppofite. 

Firft, If they both tend the iame Way, and ^begjb to move 
M, aad is nearcfi the Place chqr tend to ; 

The Theorem is * r= /j J^ u 

But it B begin to move 7 ^^ hJc^fed 
firfl^ the Theorem is J * ~ f^ ^gc 

If they both besin to move 
in the fame Moment 
Theorem will be dius 

For if h'szo^ then the Member of the Eputtiom that bab 
it, vaniOiedi. 

Secondij* If the Moveohle Bodies meety and ^y as before^ 
be the Difmnce of the farthefl Body, from the Place of Hem* 
countery or Meeting ; then e^x will be the Di(Un0e of the 
aher Body. Call the Body at tartfaeft DifiaQce ^» and the 
ccber jS» 

Tbei^ 



^ 



they both besin to move 7 f^j 

in the fame Moment 9 the c^^^ fll^oc ^' 
rawrfl» will be riws \ J"" S^ 



Tht Vfe of Decimals in Algelra^ 3^5 

Then, if A moves firft,"> _ cdh \ cgr 
xh&Thsoi^em is — f^ — yu \ g^ ^ 

But if B moves firft, > ^cdh-^Jge 
the ThJorem is — . f ^ — /^-U oi 5* 



iame time, the Theorem ^ ^ ^=^J^ 475; ^* 



If they both move at the ^ 
fame time, the Tbiorem ^ 
is - _ S 

Thefc Six Theisms anfwer mqft of the curious (and fomt 
of them very ufeful) Queftions that are ufually propofed con- 
concerning the Motion of two Bodies. 

Qtitftion 26. Suppofe the Sm (A) in the Beginning of 
VirgOy and eight Days after the Moon f £ ; is in the Begin- 
ning of Gemini ; Qucre the Place ot the next New Moon / 



// 



r = 0,03285 = 00 : 5p : c8 

o / // 

4892 r:: 13 : 10 : 35 



Here are given <^ _ * -% 

/ e T=z :Lj > Quere ;if .** 
t- i& = 8, 3 

By Theorem i , work as follows ; 

Multiply — rf = 0,4892 > 'j given Motioil. 
By — r :=r 0,03285 ©*j given Motion* 

The Produa ii dc z=i 0,01607 

Which mult, by 6 = 8 Difference of Time. 

That Prod, is ^fi^ = 0,12856 

Again multiply d zs, 0,4892 ^*s given Motion, 

By /? = 3 Diflference of Place mul- 

^, _. , , ^ , ^ ^ (tiplicd into /• 

The Prod, is fed — 1,4676 ^ * '' 

To which add dch = 0,12856 

Sum is bicJ^fed — 1,59616 The Dividend. 

Then from —fd-=z 0,4892 
Subftraa — ^r = 0,03285 

Remains fd^gc = 0,45635 The Diirifcnr. 



5orf The XJfe of Decimals in Algelra* 

By »vhich divide, T _ , _., ^'S."^ o . if 
The Quotient ^T * = 3,4976? =. 3 : 14 : 55 = c:5 

To which add tjic Moom^ prcf. FL =r 2 : 00 : oo : CO 

The S»«i is the Place of the 1 SV 

Next New Mooyifou^t fvite. ktWf * ' ^^ ' ^ 

Que/ihn 27. If a Sib/^ fi fail from the Equator peicifely 
North, at the Rate of 7^ Degrees in 9 Days, and another Ship 
Ai 6 Dajs after fet fid) 00 the fame MefkUan the fame Way, in 

Latitude :;6 : 90, and runs 8 : 4^ iQ (we Days ; 'tis recjiii- 
tedto tell in v:hat Degree of North Latitude B will overtake 
Af 

By Theorem 7* Thusj 

Mukiply the given Motion of ^ — r rr 8,75 

^ the given Motion oi B *— «/ — 7,5 

ThcProduais — — ^r == 65,62s 

Multiply that by the Interval of Tiwe ^^ b z:sl 8 

The Product b the Subftrahcnd -— h^c z= 52S:OCO 

Then tajultiply the given /fff^^rt;^? 0/ F/^f(?5 e — 56,5 
By the given Motion of B — fl^ = 75S 



The Produft is — — ed tsz 275,75 

Which multipiie J by A's given Time — f =^ 5 

Th^ Pxjodud in ^ ^ fed ~ 1368,75 

From which fubftracl the Subftrahend — hdc z=: 525,00 



*»> I 



K^nai!RS ih^ Uivide^ — fed^bdc •== 84^,75 

.. Then from 7,5X5 — /^ ~ ?7>5 

Subftraa ?,75 X 3 — — ^ = -^>-5 

There remains the Divifor -* fd^gc c= 11,25 
By which Divide,; the Quotient is =l ;vo ^ — -^cthnpa 
the Latitude fought, viz. — > * ^ ^5^^ ueg, 

Queftion 28. ^ challenges B to run a Race with him, 
provided he will give hiqi y^ Rodin a lOO ; now the Vek* 
City of B a Jtuhning to that di Ay'u^»^% to 5i , Quere 
which of the two beat ? 

By 



The Vfe of Decintak in Aigeh^i J07 

By The^em 5, work thus | 
Multiply \!at Velocity of iB . — d :sz 7,75 

By the gvjt& Interval oi Diftance^ ~ <» = 30 

The ProduS (becaufc/^ 1) is the Dividend fed « 2:52,5 

Then from -p. --^ fd :=i 7,7$ 

Subftntft — — ^r =1 5,5 

There Rctnadns the Dlvifot fd--gr rr 2,25 



Then « 2,25) 252,5 (i-^ = X :r: 103,3' Rods. 

Hence ^ b«at JB, iiiKe ab^ve th^ 100 Rods were ]^is'd ere 
^ cajue up with, or could overtake him. 

Queftion 29. Soppofe the H^ur ar4 Minute Hand of a 
tioroJogiHm^ 6r Cfetek, be now both ii\ ConJuvSiion at 12, 
$»fr^ the Hace r f their next Conjunction ? 

if yon ptoceed by the fame Thewrtm )> yoo wilt find it to 

i^* • f^* f ft' Hi iiif i^*i 

te ac t : 0^ Hows .:s=: 1 : 5 : 27 : 16 : 21 : 49, ®f. 
the laft five Places repeating ad infinitum. Hence we itv^y 
obferve, that though there really is a ctrtain Moment of Time 
id wtilcb rtie MinUte^fJs^d is preciiely in Co^jtmBiov 
with the fhuY'fi^ff^y yet 'tis . impollibte to determiner or rt- 
prelent that Momvnt of r?^^ either in v>bole fSantherSy or 
Decimal FraBions ; But by Vulgar Fractions we know ic is 
I , : Hours, that is juft one Eleventh Part of an Hour after 
^^^ A Clock* 

Quejtion y^^ FiK>m London tc> Chicbefter is 60 Miles ; 
A Foft-Boy ^yf^ fets out from London^ and goes 8^ Miles iqi 
2i Hours ; Another i^oft-Boy (h) \j Hoor after &ds out front 
ChicheRer^ and rides 9 Miles inji Hours, I d€raand,ho\y 
far A will have gotie before he meets B ? 

This Queftion is anfwered by Tk-orem 4^ thps ; 
Multi^yj/s ^/ve-w S^jf^ *- — r :^ |},75 

By the ftfe-M;^/ of the Times -* A =«' 1,5 

Then multiply that Produft * — ^ rft .tii: 1 5,1 25 

By thegm-w S^/rr^ of B * f- aT rs p 

ThePrQieais. — W f4fr :± 1x8,125 

8^ r 2 Again 



Anf. 



308 TbeUfe of Decimals in Algelra. 

Again multiplv — -•«•■= 8,7$ 

]B^ xiA given Time of 3 ^ g ^ ^>25 

Aiid that Produft — ^.? = 28,43^5 

Mult, by the Inierv. efDiftanee — < = 63 

TothatProdaa — C?' = 1706,25 

Add that above, viz. — r</A = ii8>i2S 

The Sum is the Dividend cdb \ cge zfs 18241375 

7>cn to the Produa above, — r«; = 28,4375 
Add the Prod, of (/X^=^i>5X9>) >^ = 22,5 

The Sum is the Divifor ^fd^cg^ = 50>9?7$ 

B, whidi dividing ; the Qiioti. i ; _ ^^^ g 

ent is the Dfft. of wf s Journey j ^ ' 

^hcn the Ltfiance B will have 7 

■ pafrd is — —J ^ ^^^'^ 

And thus proceed for anfweringQueftions by Theorems the 
fifth and fixth. By tljefc Theoremi feveral other pretty Pro^ 
ihms propofed, may be refolv'd by any one verfcd in thofe 
Matters. 

Queftion 21. This frejent Year of our Lord^ the CycU 
of the ^ IS 6 =s «r, and the Cycle of the Mpon { call'd the 
prime or Golden Number ) is 5 =1 ^ ; Qu^re the Y«ar of the 
Dionjftan Era or Period ? 

r e -=1 6 7 Lfct a? be the 

Here is given ^ rf =r 5 > Year of the F^- 

C'c— ^ = s:=:ij r/V fougjir. 

TJie Theorem is 5j>,i^ z + 3,4ir ^— * a^rf s^: *. 

Fi'T, multiply the Difference of the Qfcles ^ zn I 
By the Number — ^ 59, t 

Then that Produft is — " ?=s 5Pyt 

To whiclf add /i. X 3»t — ,sr 15,$^ 

From that Sum -^ — = 74,^ 

Siibftr^aVx 2,r . — ^ — c=: 1 



Xberc remains tijc IVar of jjie P«'/'eflf r? ^^ l2"ght« 

This 



The Ufe. of Decimals in Al^ehra. jop 

■ This Tbecrem I contrived my fdf ; mid infertcd it here as 
being a Jiecimal one. 

Queftlon 152. Let A^ and B, be two fphericalBodies per^ 

feSih elafticK and let (a) denote the Vilociiy of Aj and (h) 

= the Vdociij of 5 ; then the Moti&n d£ At=i aA^ and the 

Motion ot 5 =«• i' B ; laftly let x := the Increafif of Motion 

conamunicated by the IwpaSi or Stroke, to one Body ; and 

• die Decreafe or Lofs of Motion in the fercutient or jiriking 

•Body. 

Let y{ follow JS, and let it be required to determine the 
Celerity of each Body after the Stroke or Impulfe. 

If ^ and B tend K 4's Celerity kx = ^^. '!f +-i!.^ 
both the fime Way, < .^ ■+■/, . _ 

the Theorem for ^ ffs Celerityh x - ^^ "^^ !l / 

But if they meet, the TbS' 3 ^-|- B 

0r^i8« will be akeied thus, for y _ 2ttA-\'hA-—bB 

IB, * = 5^:3— 

Exdmple. Snppofe two Bodies of the fame fort, ^ of 5 ^ 
Pounds, and p Degrees of Velocity ; and 5 of 6i Pounds, and 
4 Degrees of Velocity ; tend the ume Way ; Qucrc their Cele^ 
. rities after the Impulfe ? 

Here A =: 55?- ^ = 9* iS = 6,5. * sss 4. Then, 

From the Motion of A — — 4/^ =3 48 

Subftraa the Velocity of ^ into B aB zs 58,5 

There remains negative, — . aA-^aB = — 10,5 

To which add twice the il^of;a« of B -* 2^B = ^ 53 

There Remains the Z)iwV£f«ri/ — = + 4l>5 

Then ^ +B = ii,e) + 41,5 (^^^p r= yf's Celerity. 
In like Manner may be found 8,46 = B's Celerity. 

Note ; If either Celerity come out Negative^ it fignifics the 
Motion of that Body, after tiie Impulfe, to be con- 
frary to what it was before. 

I have 



jro TTje Ufe of Decimals in AJgehd. 

I have inferted this OueAion and Theorem^ for the Sake of 
any fuch Perfons as would wifli to have always a Theorem by 
them for the ready determining the Celmiy of Bodies in A/o- 
iha by CMleuIgtion^ and the rather becaufe this DeShfj^ of 
Motion is the principal Bafis of a good Part of the mdern 
mecbanieal fbilofrpby. 

Quefiiom m* Szys A, IVe an jiere of Land to iiKlofe.^ 
lap B, I've ten Tboi^anJ fiich Acres as thofe, which lie l^i 
a Square ; but the rprm you delign muft have tht fame F^ir- 
r/>^ as goes round all mine. Quere the Length and Breadth 
of ^'s^rr^ of Land? 

C a r=z 1 = The v^<f4 of A*s Plot of Land. 
Ltt J i/ ztr 100 =5 The Sr^^ of B's fquare Plot. 

C X — The Side of ^s /*;f to be fonnd. 

The Theortfm h nr rs ^"k^V « ; • ^- V. 

From the Square of rf ^^/ =s lOOOO 

SubftraS the given Ar^a ^^ a z=. i 



* fc> . — 



Tlicre remains — <</-- tfra ^^^^j^ » 

The Sq*Root thereof W^id-^a = 99^99$ 
To which add — ^f =: lOO 

^ The Sum is one Side of the Area JK=ip57,995r: the Length, 

And the other Side itf — . — ::=! 0,0035:=: the Breadth. 

Thus iviice their Sit'm is = 4^ =: 400 the Perimeter of 
Both. 

Qnffti^ 34. Suppofe the Tov>er A 160 Feet high, and 
a^iKtber TffvSer B 124 Feet Wg\ at the Difiaftee AM -150 
Feet ; 'ris req*rircd to fee a Ladder hi fowie Pomt (e) in the 
I ine A^ of fi'ch a le^r^th^ is from thence k may reach thcTops 
of both the Tcvjers : Q^uire the Point ^,and the Icf/^th of fuch 
a Lndhr / 



Let 



Tie Ufe of Decimals in Algelra. 3 1 x 



ft 



:i6o the Tower A 
:i 24 the Vm^ef B 
1 50 the Diflance f il 

the Point's Dift. 
: the Ladder fou^C. 




The Tbeorfm for the Dift once (0 ) ■ ^^ - ■ ^ = 



2^ 



Tbtniotx!tytLingtb<£x!tit Ladder (b) \/aa^€0 = b 

Thus, to the Squarif of F» biigbt — W r= 15376 
Add the Square of the Diftance ^-^ cc zsz 22500 

Then from that 5««i — W -j- rr = 37876 

SubftraS the S^»i»-^ of v<*s bsigbi .^ aa ^=^ 25600 

There remains the Dividend ihArccmm. aa 



11276 



Which divided by 2^ = 300, the l ^ ^^ ^ 

to fient is thtDi fiance (e) — r ^ "= ^^'^^ 



Quotient 



Q^. 



Then the Length of the Ladder is ^aa^ee x i&^i65,i45> 



CHAP. 



3" 



CHAP. XIII. 

Th Ufe of Decimals in ^/iim Tri- 
gonometry, and other Mathematical Sci- 
cnces defending thereon. 

TH C excdlent Ufe and indifpenfable Neceffity ot this 
noble Art in all Trigonometrical Calculations^ is 
evident enough to thole who are verfed therein. No^ 
thing with any ExaQnefsj Eafe, or Expedition can be done 
therein without it ; and as Trigonometry is the Foundation 
( yea the very Elffence ) dE Navigatiany FortificafioHy AltimC'' 
tfjy Lonffmetr% and is of Ufe alfo in divers Cafes oi Aftro- 
nomjy Surveyings Diallings &c. *tis manifeft the Ufe and 
Knowledge or Decimal Arttbmetick is fo requifite in all thole 
Arts and Sciences^ that without its Afiiftance a Perfon can 
make but a gloomy and firuidefs Progrefi in the Study of them. 
I fhall therefore illuftrate the Ufe of Decimals in the Re. 
folution of all the Cafes oi Rightlined Trigonometry ("for that 
only is to be unckrftood in this Chapcer) both in the Dodrine 
of Rigbt'angledy and Ohlique^angled Triangles^ as follows. 



Right-angled TriangJIes. 

Cafe I. 
rThc two Angfes 
Given < BstrdCj 

CThe Bafe B Ay 
To find the Caihetus and 
Hypothcaufc, 



C 




2.^^ 



The Analogy to fiad the Catbetus. 

o / 

As the Sine of the Angle C 56 : 15 Com. Arith. c,o8oi "156 

Is to the B^/^ B -4 24,>^ — = 1,3863818 



a i 



So is the 5/>tf of the i<«^/^ 3?>45 — ^= 95744759^ 
To ^^QAtbstusox Perpendicular 16^26 = 1,2112744 



Tloe Ufe o/DecmaJsy &c: 313 

Cafe 2. Tiie A^zalogyxo find the Hypotbemfe, 

b / 

As the Sine of the Angle C56 : 15 Com. Arlib. o,o8oi«;56 
Is to the Bafe BA 24^,^ — = 1,5863818 

So is Radius 90 — . — == 10,0000000 

To the Hypotbemfi B C 29:27 

Cafe :j. 
r-The two A/^gles B 

^'^^" "^ And the Hy pot be9ufe 

L B C ; 
To find the Bafe and CaSbetus^ 

The Analogy for fiading the l/j/^, 




As Radius 90 — 

Is to the Hypoihsnufe B ^27:5^' 



— . 10,0000000 
_ = i,4?98906 



So is the 5/^^ of the Angle C 5^5 2 '15 ^ 9,919846 4. 
To the jB^/^ B ^, 22:89 — == 1,359737^^ 

Cafe 4. The Analogy to find the Caibetus* 



As the Radius 90 — 

Is to the Bypoibenufe B C 27:^? 



— 10,0000000 

= 1,4398906 



So is the Sine of the Angle B 33 : 45 = 9,74473P^ 
To the Catbstus^ ot AC 15:29 — = 1,1846296 



.i 



Cafe «;. 

r^. f The Bafe B A 

^'^^" irhc Cat hetus AC; 

To find the ^;?^/^j, and the 
fjypotbenufe. 




2 14 Tlr Ufe of Decimals 

The Analogy to find the Avgle B. 

A9t\itBafe BA 27:% — =■ MS^S^^T 

Is to the Radhtt 50 — — = io,oooooco 

Sou^Peifendicula^iySy — = 1,1420765 

To the Tivgftt of the Angle £ 26 : 46 = 9,7027438 

Then the Aftgle C is 63 : 14. 

Gift 6. The A»dltgy to find the Hjpotbemfe. 

As die TMgeat of the Ai^Ie B 16:46 C. A. o,2972«>62 
\tXa^tCatheU$AC\7,&l — = i.HZOT^t 

So is the SfCMt of the fime Angle « = 1 0,04922 2 5 

To the Hyfaf i&^»/Jr :io:8 ~ = 14885552 

N^te. The &f/i«t of any ^ngle is the Ariihmeiicd Com^ 
flement of the Co/*^ ot the ftid Angle, added to 
Radius xo,ooooooo. 

^•— « STheBiiJtl^^ 
^"^^ 1 The Hypotbenufe BC; 
To find the ^»;gfei, and the C/i- 

The Analogy for the A^gle C ^ ^ %Q^ ~^ 

As the Hypotbenufe bC 20 — — 1,4259687 

Is to Radius 90 _ — = io,oococoo 

SoisthcB^/^B^22:8 — = 1,3579348 

o I 

To the Sifie of the Angle C 58:45 == 9,?3 19^6 1 
Wherefore the Angle B 31:15. 

G2/> 8. The Analogy to find the Catbetus. 

As iP^A»x 90 — — io,ooocooo 

Is to the Hypotbenufe 20 _ = 1 ,4259687 

So is the S/«^ of the Angle S 3 1: 1 5 . = pyJHPJJ^ 




i*«« t ij 



rtMP 



TotbeG#fM»ii<C 13:85 — = 1,1409463 

. .. Cafe 



in Trigonometrical Calculations. 



Cafip. 



~ J The Hipoibenufe B C 
Given irht Catbetus A Cy 
To find th^ Angles, and the B/ife 
BA. 




The Analog to find the Angle C 
As the Catbetus A C i66:$ t- -rfr//^. CQm. 7>7788467 

Q 

Is to the Radius 90 , — — 10,0000000 

So is die Bypotbenufe B C, 316:2 = 2,5000:569 



To the Secafit of the Angle C, 58:15 



jk— 



10,2788636 



o / 



Then the other Ang^e B will be 31:4$ 



Cafe 10. The Analogy for findii^ the Safe. 



As the Radius 90 — - 

Is to the Caib'^tus 166:5 — 

So is the Tangent of the Angje C 58:15 = 
To the Bafe B A 26^:92^ — 



10,0000000 
2,211153? 

10,2084365 



= x,4295$98 



Oblique-angled TriangleSf 



Cafe I. 

fThc/f/^/^j 
I ^,B,and 

Jo find die other 
tvooSidet, ■ 




315 



The Ufe of Decimah 

The Analogy tot the Side B C. 



As the Sine of the Angle B 57:30 Ariib.Cffn. 0,2i$«)S29 
1$ to the Side y4 C 350 — = 2,5440680 

So is the Si fie of the Angle .4 62:30 — 9,9^79289 
To the 5;^/f B C 509:97 - == 2,7075498 

The Analogy tor the Side A B. 

As the Sine of the -<<«^/^ B 37:30 Aritb.Com. 0,2155529 
Is to the Side ^ C 35a — = 2,5440680 



So is the Sinf of the AngJ« C 80:00 

To the SiA^B 566:2 — 

Cafe 2. 

Two Sides 
A C mi 

Given ^2^^ ^^ 



^5440680 
= 9,99335?5 

= 2,7529724 



lAnd an An- ^ 
I j^ CppO' c/ 




To find the other ^ 
Side and Angles* 

The Analog iox^thSA^gU C. 
As the Side /4 C340 — Aritb. Comp. 7,4685211 

Is to the Sine of the An^U B 6o:oo — 9j9375306 

So is the Side B A 2X2\1 — ^ 2,3265407 

To the Sine of the An^e C ^2:42 — = ^77325924 
Then the Angle A muftbe 87:18 therefore, 

The Analogy for the Side B C. 

As the Si^f* of the Angle C 32:42 Cb«7. Aritb. 0,2674076 
Is to the Side BA 2^2:^^ — === 2,3265407 

So is the Si^ie of the Angle A 87:18? — 9>P995j7 6 
To the &W<^ B C 39:21 *-- = 2,5934^59 



r 




f in TrigdnotHetrkaJ CakuJationSi 

C*^« ^And the Angle 

\ included A^ 
To find the other Side 

and An^Us. ^ 2.40. 41 

The Analogy for the Angles* 

As the S«« of the two Sides 661:2 Cam. A. = 7,1796672 
Is to the Difference ot the Sides 180:4 = 2,2562365 
So is Tangent of i Sier/^ of the unknown 1 

An^es jB and C 76:27 — 3 

To theX&^rg^/?! of j^ then difference 48:32 = 10,0538632 

Then, To half the Sum — 76:27 or two Ang. 5, T. 
Add half the Difference 48: 3 2 

==: Greater Ang. B. 
= Leffcr Angl. C. 



^teiMi 



The Sum is .^ 1 24:59 
SubftaQdie ^ hiffer. Remains 27:55 



The Anaiogj for the Side B C. 



O X 



As the Si»^ of the Angle C 17:55 Antb Com. 0,3295808 

Is to the 5i>f^ of the Angle A 27X)5 = 9,6582842 

So is t\iQ Side 4 B 240,4 ^ = 2,3809345 

To the Side B C 23,37 — — = 2,3687995 



Cafe 4. 

f All the three 
^. i Sides, -45, 
G^^^»U c, and 

To find the Avglf^. 




^ S73. ^ 



•-. 



V 



y-£, 



The 



3i8 7f>e Ufe of Decimals 

The Analogy for the Segments Ad^ and dB. 

As the Greater Side or Bafe AB ^jy^^ Ar. Com. 7^278258 

Is to the Sum of the other tvoo Sides \ ^ .^^ ^ 

AC+ CB = 455.2 - }-- 2,6582023 

So is the Difftrence of the two Sides\ ^^«.»^,^ 

AC -ci!= 124:1 - r - 2P94l2i<5 

To the Difference of the Segments of the 1 ^ ,q^, ^ ^_ 
B4/>-^^ = 151:4 * - }^= 2,1801497 

Then farom the Grea. Side 01 Bafe = 373:4 
SubdtheDif.of tbeSfj[nr«fi^^= 151:4 

"jTheKe will remain ^rB .^ s;? ^22 

The half of which is Bd ■:=: ed :=z iii the Leflcr Scgtn. 

Alio to Ae add ^y, the Sum is Ad = 262^ the Greater S^g* 

The whole OlUque Triangle A CB being thus reCblved into 
the two Right^anffed Triangles Acdj and fi C J, die Angles 
A, By and C are found by the feventh Caie of kigbt-angled 
Triangles foregoing. 

Having thus pafi'd through all the Cafes <J[ Xigbt and Ohti^ 
qne^angled Plain Triofigles^ in each of which the ablblute Ne* 
pcflity of Decimal Numbers to exprefs the Length of the Sides 
fought, is fufliciently evident ; I fhall next (hew, in brief, the 
Application of the foregoing DcOrine of Plain Trigonometry 
to feveral Af'ts AiatbematUal ; intending tl^er^ to convince 
thofe who purpofe to learn them, of the Neceflity of their firft 
learning Decimal Ariibmetick. 

The Ufe of Decimals in Navigation exemfJi^ 
fied in all KJnds of Sailing. 

]. Plain Sailing. 

Ill Plain Sailing, or That by the Plain Chart j die Parts of 
a Trinf?gle receive new Denominations. 

Thus, 1 he Bafe is the Difference of Longitude or Defar^ 

ture ; 
The Perpendicular is the Difference d[ Latitude ; 
The fJypotbenufe is the Difiance the Ship has run; 
The Angle at Perpend, b the Courfe ot the Ship ; 
And the Angle ac Bafe the Complement o{ the Courfe. 

Admit 



in the Art of Navigation. 



$10 




Admit a 
Ship fails 
from the 
Zat. Nart/f 

e / 
51:30 on 

the Humh^ 

A C 'ypS 

Nauitc 
Mile% 
whofe 
Covrfe is 

Querc her t ^e/jartur^ 

Departure ^^ ^ 

ana Difference of Latitude .^ 

The Analogy for finding the Departure. 

o 

As Radius 90 — — . 10,0000000 

Is to the Diftance run A C 598 — = 2,776701 % 

6 r 

So is the Sine of the Courle C 56:1 5 — 9,9198464 

To the Departure Weftward \ ^ ~ ^ , 

from her former Meridian > 4^7^^ = 2,6965476 

The Analogy for the Drffdrence of Latitude. 

o 
As the Radius 90 — — 10,0000000 

Is to the Di fiance run ^C 598 — =r 2,7767312 

So is the CO'ftne of the C()«r/> 55:45 ,— 9,7447390 
To xht Difference of Latitude 332:2 = 2,5x14401 

But 332,2 Miles are equal to 5:32,2, and the Ship*s Courfe 
being Soutb'Viefierly, 

Therefore from the Latitude failed from — 51:30 A^ 
Subftraa the Diff-^rence of Latitude — 5-*32,2 

Remains the Latitude come to — =: 45:57,8 jV. 
The fame Cafe follows in 

Mercator''& 



jid The V ft of Decimats 

Mercat6r*s Sailing. 

Afercators Sailing, or That calculated by his Chart, is 
much moce coneSt and exaS than Plain Saili»g : For in thb 
Chart At Degrees ol Latitude increafe according as the De* 

gees of Longitude decrease ; and thefe Increments of the 
egrees of Latitude are called the Meridional Pnris ; of 
' ivhich a Tabk is compofed, by Means of which Mercator*s 
Chart is conftro^cd, on which ^ Ship's Diftance, Courfe, pro- 
per and increafi^d Difference of Latitude y the Departure and 
Difference of Longitude , are truly laid down or delineated, 
as in the Scheme fubjoin'd. 




The Analogy for the Difference of Longitude^ 



o 



As Radius 90 i— . _, -* 10,0006000 

Is to the Increased Differ, of Lat. 50^6 = 2,7038071 



o « 



So is the Tangent of the Courfe 56:15 — 10,1751074 
To the Difference of Longitude 756:6 = 2,8789145 

The fame Cafe and Data follow in 

Middle 



^ "* 



ih the Art of Navigations 



331 



Middle Latitude Sailing., 

^This kind of Sailing is computed from the middle parallel 
of Latitude J which . is half the Sum of the tvio Latitudes 
of the Places, fail'd from, and come to ; and depends alto- 
gether on the following ^fi'//^^/ Theorem or Analogy. 

• 

As the Co- fine of Middle Latitude 

Is to the Taffgent of the Courfe^ 

So is the Di0srence of Latitude in Miles, 0r. 

To the Difitfrence of Lovgltude in Miles, £^r. 

The Reafon of this general Analogy is evident in the fot 
lowing Scheme and is deduced there&om. 




CMsz. 

AD = 

DE 

FG 

AE 

^AD 

Be 



Explanation of the Scheme. 



/ L Is the Middle Latitude 48.44. 
CH The Ghjine of the Middle Latitude, 
Ai F The Difference of Latitude. 

The Departure Weftward. 

The Difference ot Longitude. 

The Diflance filled. 

The Rhumh or Courfc SW\yjW. 

The Tangent of the Courfe. 



Now *tis manifcft As CM : CB :: MF : Fdy iirhlcH 
is the fame as the Analogy above in Words at length. 



T i 



iTj# 



^22 Tie Ufe &f i)f€!mah in Fortification^ 

The Annhgj For tht Lifer ence tf Lofr^itude. 

At the Co^?f^ Df Middfe Lat. 48.44 ArHb. C,o,»8o742*f 
Istotht Im^etft of the Courfc 56.15 •— ' 10^175 tC74 
So b the Ufffi-'rence of Latiiade ^35.2 as 2,521 ^-ppd 

To the Difference of £fi«'^if. 753-7 = 2,87724^7 

Abfif I. The Proportions for finding the Difference df T^- 
f /f«//^ aiid Departure in }Aecator*s ind 'Middle TjtfrWde Snil^ 
ivg^ are the lame as in /'/^/^ Sailiffgy and therefore not re** 
peattu* 

2. That Mecator*s Sailing gives the correSI Difference of 
latitude and Longitude both ; AdiddJe Latitude Sailing, only 
the correB Difference of Lorgitude ; p/<ij*« Sailing gives nei- 
ther corre£Uy ; and therefore their Merits are in Proportion. 

3. That Viiddle latitude Sailing agrees with P//7;« Sailing 
in feme llefpe^s • and with Mercator's in others, very nearly ; 
and therefore is to he ufed accordingly. 

A. That from the foregoing Inflance it ts evident no exad 
Calculations in Navigation can be made without Decimal 
Numhers; and though 1 liave exprefs'd but one Place of Dect^ 
malsy jet three Places moie may be eafily found by Problem 3. 
t)f Logarithms. 

The Ufe of Decimals in fortificoHton. 

A Fort is a Piece of Ground in Form of a Polygon rrg'iJar 
or irregular^ environ*d ^vtth a Rampier, or Wally and a ijitf/f 
to impede the-Ai&uitsof an Enemy* 

A Scheme of a regular Pentagmal Forty with its E^pla- 
mtion, is here after fubjoin'd. 



TJlre 



\: 



The life of Decimals in. lortijkatMU 523 




I 



s 



! vr 






c 



\ 






a 





The Explanation, or Nances of the feveral 

^arts. 



1. The 

2. The 
5. The 

4. The 

5. The 

6. The 

7. The 

8. The 

9. The 

10. The 

11. The 
12* The 

13. The 

14. The 



Cnrtatne ^^ 

Biilwark Of Baflion 
prant of the ^u/wark 
FliVfk _ 

Gorge of a Hul^ark 
Gorge Line •« 

Hc^a/ Line 1,^ 

Shoulder ^^ 

Planked Angle 
Irrmard Flanking Angle 
Outward Pktfrking P. ngle 

£P^J 7? t''«^ ^f Defence 
Sbortefi liac of Defence 
Palf^ Braf ^ 



ON 
NFGHT 

pa 

NF 
NT 
NC 

ca 

* F 
G 

SGF 

KMG 

PG 

PP 

EC 



Tt Z 



A Table 



3 24 The Ufe of 'Oecimah in fortification, 

J Table of the Dimenfions of the Angles olf 
Jerved in Fortifying the regular Polygons 
following. 



>< 


SOQOQOOOOOOOOOn 
A G 6 y C O «^^6 ^i;,0 r" O 


^ 


SS8°888.S.8S3.8-8a°8 


> 




> 




> 


S8§8 88S.S.8°&8S~°8 


> 


S8.48 8.8,%8,8,°%8,8°8 
o!l'8 5::r§,ta&;|SiSR5-3t 


& 

2 
1 

1 


toisi-, a-Gi. t3;omt;njls 



, The Ufe of Decimals in Fortificatiojs. 325 

Tbo' 'tis no' necd&ry the Angles in Fotts fliould be pre- 
pfely fuch as are before affign'd ; yet fiippofing them to be 
fuch, I Ihall fliew how to determine the Quantity of the 
Sides an4 Lines of the Pentagonal Fort above in Decipial 
Numbers by Tr^onometrical Calculations, having the Length 
pf the Curtaine and Front of the Bulwark given. 

Admit the Curtaine be — A?" = 140 1 Yards 
. And the Front of the Bulwark FG = p? f *■"" 

Then tfic Analogy for die Sine J F^ u 

As the If^/iins po — — I0,9000poo 

Is to the Front ol the Bulwari FG r=9^ :r= 1,9700368 

So iitheSineof theAngleSFfi 19.30 — 9,52350 00 
Jo theSine — ^f = 51.15 = iA9%'i{ ^ 

Again J As Radiut 90 — — 10,0000000 

Jsto the FfOiit of the Bulwark f'Girsp:^ = 1,9700368 



'fo the Line — 5^=187.98 .== 1,9443834 

^Tien SG = 87.(j8, and S/= 7®, Therefore the whole 
Sine KG := 315.96. 

Again ; as the'Sine of t\\e Angle TAG 30.00 A C. 0,230781 % 
Is to ; KG^lG=z 157.98 — 3B 2,i986'5oo 

So is Radius 90 — «- 10,0000000 

To the Semmameter A G e=z 268^8 =: 1,4294313 



Again; as the Sine 'AG 30.00 A. C. 0,2307813 

Is to ^ the Side of /G -157-98=2,1986500 

So is the Sine of t! ■ / 54.00 — 9)9079576 

Jo the Perpendicular A I - 217.4 = 2,3373889 

■ i" ■ ' z' 

A^iin; 



^i6 V^eVfeof Decimah i» Fortification^ 

Again, as the Sine of ^ic Angl^ FCQ 86.00 A. C. o^oo 1 0'}5>2 

So is the Sine of the^ Angle FQ C 34-^0 P,753l28o 
To the line Fr= 52.9? — gg 1,724224 

Alfo, M Ae Skie of Fcfl ?6,oo ^i<Ji. C 0,0010592 
Is to tb^ Front Fa:==:9^ — ;== 1,970^36^ 

So is the Sine of the Angle G FC 59^30 9,93'53204 

— - I ■ ■■111 ^ 1 M. 

To i\ie.HeMdUne If ^ == 8o4| "^ == >>yo»4?6 4 

Then AG^G Gr= 4Q:== |88.i5t the Sc»idiamctqf of 
. the inn^r A»«^^<wr. 

Again, « Radius 90 — xc,oooooco 

Is to the Line P C^=i <^%,99 — ;== i, 724^240 

^ So i$ *e Sine oP Ae A*>gle FC AT 40*00 9,808067$ 
To the F/4>5rit FA^^ 54.06 « = ^,^22915 

Th«n FJ^4-SF=i! /Z)=: 65.21. Andy#/-./p=T 

.0 , 

Again, as Rad'm^ 90 -*- — JO,oococoo 

Is to the Line F C = 5 2r9P -^ = 1,7242240 

Sa is the Siue oCUie Angje NFO ^o,oa 9,8842540 
To the Qorge Une NC = 40,59 ==: 1,6084 780 

Ag^in, as the fine of F/* | ig%JP -rfnV^* C. 0,4765047 
is to the Flank FN = 34? . ^J' — = 1,5322915 

So is the Sine of the A^igTq' ^^Ff7 70.30 9,97434^^ 






To the Line pN=: 96.V? \ '. -r — 1,9831428 

Then N—PN- P = ^^i^&h the fecond Flank. 
And ON-^ SG=RGz:;: 327.98, I lien in the Trian- 
gle ROG A* 



.^^ 



/ •» 



Vje tffe of LkcimaU in AUimetry^ &c. § 2 7 

As the U\K RO ( = ID) = «5.2i -*. a 8,i8568';8 
Is to the Line ^G = 227.98 — = 2,3578967 

-80 is R-jdim $0 ~. .^ ropooQdoo 



To the Tangentoi the Ahgls it 6 ^4.02 10,^^5825 

Then, asthe Sineof TfOG T+'OZ Arith.C. o,oijo%^o 
fe to liie Sine it fl —227.28 >-* t= i,3^75?67 

So is Hadius 50 — — IC,0300C30 

T» the fJOT 1^ Dtfentt <?fi3S 257.15 — t, 1749817 

Thus having the A^'g^es as iji the Tahle, you are here 
taught the Manner of liudicig die S(^<?j and Sin^s of any rs'- 
^lar Fortj in any Meafure, and Bedmal Parts thereof. 

"The Uje 6J Djcimals in Meafuring Heighths Depths, 
ajn/ DiAances ; both &Qce&b\c and inacceHIblo. 

I. At'imeUy, or the Menfurarion of ^/7;fK</f j and Z)t'^/i(, 
is thus jterfornieij. 

Let A B tcprefent a Towrr whofe Hi^igbt h required. 
C,nrv,r, J'^^^ Diftn'iee ACss 26.^7 Vards. ^ , 

Buppoie -^-j-jj^ ^ijgjg ^^^ ^gjyjjj by aftarti/rd'^O 51-33. 



}j8 the Ufeof "Decimals iit 

The Analog £ot finding rtie Height A B. 

A* Radios 90 — — — 10,0000003 

Is 10 the Diflance ^ C = 16.37 — ;= 1^211101 

So is the Tangent (£_ ACB 51:30 — io,c9?5P48 
To the Heighc of the Tower A B 33.15 =n 1,5205049 

And thus the Height 01 Attitude of any other acceffhU Ob- 
je£l may be tbimd. 

Suppofe the Steefle ABht inacce_ffhU for the River li /; 
Then with a (Quadrant at C take the Angle A C B, and m^- 
fmc a Biftojue toD, where take again the Angle A D B ; 
and let thofe AngUij and the Diftaace be as in the adjoin'd 
figure. 



Then C 2 tnufl be found hj this Analegji 

As the Sine of C B D 14.30 — • C ^. 0,6014004 

Is to the Sine of B CD 32,15 — p,727227(S 

So is the DifioKCe or Side C Z) = 24.56 = 1,3<J02284 

To the K/fiM/Z>*wZ)B=: 52.34 ^ 33 1,7188564 
Having 



'-- 



Having f^iild^£, ^9tta»n%aA. 4Bt^^\ 

As Radius 90 — •- 

ts to the Line Z}£ is 52.^4 



•w> tcLoodoooo 
di- = 1^7188564 



SoiitheSineof -4/)£46.45 5=^^^8623526 

To tbe Height w^sA A B zs ^tz *» lisSiaojd 
Thu* you find the Hei^ c£ any innceffihU Oyefls. 

Jfl6feCUea«4ior« lifei^ you woidd ptteafiare ftaad^H 
•bft, OS «M «i^ i)». w tbe Ttfttt^f* ABi then take tho 

Andeii;c£9f.do, ;! C I? to.90 ; U«n feopi C mrafiite thjS 
BijflaMte CD — 2S.6 t«n/r> and at i)» take die Ao^s ^ik 

o / 



52.^) mcAADM ^2>jp* a*i'«l6W« 




Then the Andcgj for finding the tide DS,ia 

As the 5i»r of C B D if.jo — ^W/*. Con. 0,52185821 

Is to the ^KB of MCd $5.00 ^ — 9i7'i^\9ii 
So iitist Side ot Difiamnf CD =z 28.6 db 1^56^66 

To the F^iH Xiflr p 5 p 54,5s -2 =J[>7£68i5"i 

U t» Then 



f 0^7353708 



3JO . The life of Decimals in 

Then &rt M t^e SJfwtfCofritiUirtEiif'of^^Z)^ 
(teiSto) 57.3Cr-^.C. 

' """ o / 

Is to the Sine of B D V*. 20.00 ■ — 9,';;40^ir 

80 is the Side 05 = ^4.5^ - -i- ;=r 1,7368155 

TotheHf%*rofdieObjca/iBte22.i2-= 1,3448383 

DfptbSf if Perpgviiicular, are raoft conveniently meafiired 
' ieith a /.'I'wp ■nd plumet ; but if rhe Of ?f A be flaming, fuch 
• as Valli'ys, &c and the ferpendicuhr Pti^-ouiidiij be leqoi- 
red; Jo as fellows. 

Let A Be beaFdffrf^, whole ohlique Defcents 01 Sides 
-frre ,4 fi, and B C ; aod irs perpendicular Depth B D lequirdd- 
Then meafure the Side A B, cr fi C, and take with a C/ua- 
drmt the Ai'gle A BO.. or aBC; whereby the others will 
be luTOwn ; then ( fuppoling them as below ) ufe the follow 
iiig Analc^j, 



The Amkgj for finding the Defth O B. 

As Radius pj — -^. 13,0000030 

h to the ftaat'Difcent AB — 'ziS-i^- = 1,4174717 

' So is the Sineof BAb 50.50" — 9,9106660 

. To the PerfevditularVeplb DB st 21.2B = 1,3381577 

7 he fame might have been equally f^und by the ^ther 
.Bilbt-a'.'gidd TiWngle CI>B, by 6it tiOSt Maihod. - 

■ ^.; 2. Lon ^V, 



l.ongmetr)\ or meafuwig Difiances. 3J1 

~ ■ , » 

?• hoffgtmefry^ or the Meiifurationof the Diftances of 0^> 
l<Ssy either firom Us, or from oiie another^ is t)ius per- 
formed* 

Let Aj ^, be. two Trees ; and let it be required tc> find the 
Difiance of A from C or Z) ; as alfo of B from the fame twp 
Points ; and the Biftance of A from J. 
. . Having ( by a Theoiofite or Semicinh) \t Z5, Fouiid the 
Angles B Z) C^and-^^Z) B ; and at..C th« ^gk$ A CD and 
A'CTBT i*^ mcafured the J)ifta>ice of the ftuo Stations CD, 
as below : Ufe^he following Analogies* 



» • 



'0(teami^M. 



"^. > - „ . . ^. ..^.'" >«• « 



^' 



»^|jif.:-^.....,....::i ...S'^'-^i, 



, Jhe ^V<'<?^ for finding ^C. 

As the Sine of the Angle CA D X6.15 A. C. 0,55? 1072 
Is loihtDiftance ofStatjoni C D == 25.05 = i>3P88077 

^ :So is the Sine^ of the Angle A D C76.45 — 9,9882821 
To- the Z)//?^r^ -<<C = 87.39 • •— = 1,941 IP72 






The Analogy for the Lift once A D. 



O i » 



As the 5/«^ of the AngleC A D 16.15 A. C. 0,5531072 
' k'to the Difiance6t' Stations CD := 25.05 =r 1,3908377 

Of i 

5o is the Sine of ^ CD 87.00 — =5= 9,9994044 * 

To the Pin once ^ D = 89.39 -rr = i j95 1 3 ' 93 

U u a The 



•« jj tbe Ufe 6f Vecim^Ii i» 

The Xwd^fil for tbe Hifianie BC* 

As the Sine of the Angle CB D 1S.15 ^. CJ 0,504*284. 
b to die Diftmce of Statitnt C p 2'>rf)5 == i,?jp68o77 
So is thp Sine Comp. of the ^gte B^Q-^ 

(to i8a) 88.15 . -* .. , . ^ 
To tfw D»/?«»f# »C «? 75>«P5 w «*> »iao2839t^ 

* ' , . ■Ill .m -i I I 

The Andogj for the ^\flme B P. 

A« the an? of 4he A«ig)fc 6 B'l^ ^.^ ^- ^' ®'1^?*1 
b to the Pi/ffl»f/ of Statimt CD = ^5^5 zp: J,3yBQvr*. 
$0 i*theSine.of th«? Angle ^C© 70»30 . P»?y^^"*'* 

: To the Pifiaiiee B D =714 ... •*•. - - ^ ' 








tic jtiat&ii f»the Dttonce of tVi* ^emftm 5aclr othef, 
^ ■ wz.^B. Butfoftftir 



• 1 ■••.... r 



167.28 ^r/VA. Comp. - — J '';^ ^ ^ . 

b to their DifffrencRAJO -^ €^^7.7,^..^ 0,8680564 
So is the Tang' of half the Angles A^-^-tiACj 

;,-. . \9,9m^2% 

81:45 *• — — ' J^ 

To the Taiwent of half their Diflference^ 

- ■ d. ; . . i8,<}49Q4S6 
^B^(7_B.iC 2,30 --. .J 

Then the Angle CB^ 84.15 j and the Angle B^C 75.15, 
Wherefore th^ Apalogj for the Vifiatice A By is, 

■" M 'i ■ > ' • 

A» the Sine of the Angle /^|H?. 84^5 AC- Qi90^fl'r 
|s to the Di/?«i«f«r of 4 C=:;, 87.33 — = ?'94i »Sr» 

80 is the Sine of die Angle ACB 16.30 — 9,4533118 
To thp Pi/?, of dw fro? Tr^ff /^S = 24,^3 =s hW^lV^f 



The Ufe tf Dfcimali iff^pfurati^Mi &c. 355 

Thui I hsre endcavouTcd to make it appear how abfolutely 
iwcellarr ^le Ufa tad KocnvledAt-of- ihc ooble Ait (£.pecimal 
AritbmeiUk is in tJiofe Parts or MaihemathaiSciefiee, which 

common and ufeful. Acts ; I Hi^|hi: havo goiie farther, and 
fliewn its Ufe in feveral Paits of Aftrcnomj, &c. but I intend 
onh an -ioflruaive Sfeciimn of its'nccltciit Ofe'in »)4l KM 
<^ Ltwna^ a^:£KA Iptefimw dn GbSfitb^ wilt be £Mind 

tOiW 



CHAP. XIV. 
Th gfM Uft «/ De:cimai;s' m ihe 

Mdnfiiration 0/ dlt }Qnds of Superficies 



t 



N ^ 3Wf f jvherein tb< tfc e/ Decimal Aritbrnettch 
h vkA o^viout, nftxflitn, and exccUeni ) 1 hare only 
tl^ eo atkwK^s ^le ttaaO-ef, That the Numben are ab- 
{ ^a tt\j ta k e n >■ an4i»^ ^etsefiim anjt 'KmnfilMt^ it^ hthesi 
F^et, Yarii, Rods, Miles, Acres, &c. in the Artdt, and 
2w/f</ Co«ff «f of Hodfes. And that-aftet the jfC^i*, or Con^ 
tent is ibGnd, t'fhall ftleiir th£ Manner of Reducing it to any 
^fziffVkieij^tl^s.v&d.iaSmveyfSS^iht'iff^ ate. by mBaw 
cl VecimaU-y - ^ . . ^ . 

ff Proppfejon I. i© »Du/«^ * SifMU. 

Rm, lib^i^r* Kd» i|nto!Uj£ir, the Produa u the ^f^^ 

Muluply by »k1Cj — j2,* 



8«r 

12*66 



51^ ^'i'* r?»(f)rt or /^Tf^ is = rjt^ 



fropo- 




334 ^ ^ ^f 'Decimals in the 

PropoCcion 2. T<f Mtafwru a Paralellogtaro. 

guk. MuUiply the Jjetigtb by the BrwA*, the PrtxJufl is 
Ac jlrrA, or 'ConUnt. 

■ffKV^. • Molt, the Itngtb AB:^i6,^ 
.' ; ,By..the Bt^mT/^ BD^ 8,6. 

990 
1^10 

The Produa iitheCmfraf — 141^ ^^ 

PropoCtibn 3. Tfl Meafure a Rhombus. 

KuU.- Multiply one Sde into Ae pffffepJitiAw'Heigift^ 
, th? PPOduflis the^rM.ot,Ctf«(f«*rt(iuiDed. . 

£xiifff. Mult, the Side AB = 20,7$' 
By the He jg*f BD = 1 6,4 

Jhe Ptodua u (he ^r« = ^40^4 ' , . - ji 

PtbpcStion 4. To Meafure a Rhoisbbiiles. 

Hufe. MoWply<he ifv^S^thc p^peit(firular TJei^b'tf Ot 
Breadth, the ProduS is the Content." 

By BffA^f/i or He'iht^C iSiy%6 ^^^^ 
768 

The Prod, is the .^rf4=ii,7i6S 

Propofition ^7 Ti Meafure a Plain Triangle. 

Xule. Mttlttply the BaJV into half the Perpendiatlar Height; 
or th* l^rddperpendiculnr Height into half the B<i/f ; the 
Produit will give tlje ^tea. ■ ' ■'. ■' '■ ■' ' 
. ■ " ^- 



MeaptratioJt of Superficies and Solids. 335 



Exam. Muk. the Bdfe 4 B = lo,^ (* 
ByhalftheftyfAfCDsa 4,2 ' 



The Produfl is the Wr« = 45,7? 




Prdpdfition 6. To Meajure a Trapeziani. 

Jtale. Multiply the Vitipnal into die half Sum of the two 
Perfienditulat't ; or the Contrary j and the Piodud 
will be the Arva of Superficial Content. 

£>. Mul.the Viagonal AB-=io,<^ 
By ibi i Sum qX iba i_^ 
PerfiitdUulari he dt J *'' 




The Ptottis the Co«ff«l =34,65 

Propaficion 7. 2o Measure a Parallelopleuron. 

Rule. Multiply the 'Dia^iial by the hilf Sira of the two f w^ 
' fundlculari, ihe Produfl is the Area. 

Bxam. Mult, the Vjn^Mal vf 5=14^ 
By J 5«CT of 1.4 and c^ =r 6,^ 



The Produia is tjic Content ~ ?0,^ 



Pnpofitlon 8. 7i Myafure an irregular Polygon, or 
Polygram.'' 

Rule. Divide all foch muha-gnlar and jrrc^fi/^ Figure* 
into Trap.ziamt and Triarg^^t, thea me*l'iire them by 
Prop. ^ ajid 6. 

Enamglti. 



3 j« tbeVfe if DedfMh in the 

BxoKfU. Dinde 
the irr^ar Polwt 
ABCD&G into *e 

dten ion tkc Aiaw'^ 

wluch Me to In nwa- 

Sfti-r^*.^ ih. am rf ^ J4W eo«te, -iB 
pK die &f'Tfty CnlenI or ^r« of 4e givtn Tiih/pam. 

Pmxftioil !). U ^iV"" «i3r tegoUr Poligoo, «i o Peft- 

ngon, Heiigon, Hep««oo, Oaagon, Nomgon, 8^. 
nV MuMpli half the Sob of its SW« imo the J!««»» of 
^ Cir,lt rt>i»rf in the f «"•« j « W that **/"» 
:iSo dieWS the S**, the froiiS will be the ^^a 



A«t((Mn<et, nt. — J '"' 

Muiiip^ bri ■'"Nomhet of Sides — Vi 

7670 
3068^ 

The Piodua it the i Sun of the Sidea — = 58,« 
Multipljthatljihe««?i»iCO — = '°<1 

■ 9.71 
38550 



llaltaMtktK'lr" — = 4°2i«75 



xotbeSatfyilotmUrrlf 
infiTibed. 



Tr«i«, asi;o,i8867s8t«. 

<Miaj»is w ui,ff3740frScc. 
Z)#fivw* ai l:l,f 38844 &C. 
2)aiK*£a»,ai i!i,8«320lt«- 



Menflttation of Suferficies and Solids. 337 

Plopofition 10. To fiifuih il,7>Jlfiit UuhlfUtrs far-im} 
of the Remlii Polygnns, TDnl muliiplyina rtt Square of 
mj Side thnllj, lb' Produa JhtU kc Iht Area of tht 
Polygon, 

Bxauiflt. In » Fetugon. 



0i»,the wh^leCiKlc,wi:e.56oDeg. 
By the Number of Sides jWe V 
The Quotient is the ^£CD^72*' 
The i thereotU the -^AOBrs^i' 
Whofe Comp. is the ^^«C=5;t° 



Then make this Proportioa ; ' 

. ArM.C 
As the Sine of (he Angle ACB = ^6 = 0,2307813 
li to half the Side ( = 1 , always ) s ,5 = ,p,6pep70a 
So is the Sine of the Angle ABC = H = MoKMi 
To the Perpendicular, 01 7 ,' . ' * * 

J(y,>. of the hfiri- yAC— ,688lp = ,9MV^9 ' 

icd Circle — 3 

Then (hjthtMPropofi/oi) ArUh.Contf. = 0,<88rsl 
Multiplied into i Sum of the Sides ' — = 2,5 '. 

3440?1 
1?76}8 



,'!^ % 



The Piodna is the *« -^ — = 1,720475 

And thus may the Arta for anj other Po^|io^f be found 
whole Side is i. And this Aren will be ■iit totji/inl Uulti- 
flier for that kind ciPolKOfi, A T able of fuch MUltipIierty 
o« Aria I, for the feyetal KigilAr eoljimi follow. 



358 The Vft (f DecimtU in the 



S«frx. 


Nameu 


MuhipUert. 




7i-Tgn» 


"^3^5 




Tttrogon 


I,COO0O0 




Ptntngttn 


1,720475 




ffex/igoH 


2,$?8376 




Heptagat 


3,6l^95P 




OSfgon 


4,828427 




E^nf/tgat 


6,181827 




Z>^«o« 


7^4109 




Evdtcagitn 


8,51+2^0 




Do^ecfigoi 


9,3;3iJ5 



^m u thrfe are the Jfea't of each Poijgov re'peflively, 
whoA Side it I J "Dcl as the Area's of Like fig-ir.i, are as 
the Sfiart of their bomo^ogoHt, or ^ike Sides ; tlicrefore the 
SoKtrf of a SiJe of any ot chofe PeJygont mulnplied into irs 
rcTpeflive Arra iti the TaWe, will produce the true jlrra 
theffcf. 

Example. Snppofe the SWe of a HepU^9» be 10 ; the 
SfflTijj-frf" which is lOO; but lOoX^.^J??^? — ^^hi9'i9'=^ 
Area of fuch a Brftcpity Suid the like for aiiy other. 

Propofition 11. Th Measure o Circle. . 

gfHe. Mukipl; the ^ltar^ ff the Diamtter ( if that be gi^ 

»en) hy 0,78^4; the Frodiift b (ht Area. Or, ( if Ae 

erriphL-rj be given ) Multiply the Square of the Feripbery 

by o,07?^7J thePpoAiftBihe^rw, at belore. 

ameier of a Cirfi^ be f '*" - ^'^ 



the S^aflr<f <he«of ii a= 
- Which nwihiply by — 



21612 
nie>^*iof(l'eCifyfM=*2i,2372i6 



Mfnptration of Superficiet and Solids* 335* 

Put Z) = Diamtttr ; P = Perifbery ; and ^ = 4reii, cf 

?ny Circle. 



nit will be Jo,ji 
f Vi,: 



5,l4l6D = P. And 0,785400* = A. 

,S7j2 " D. And ^i 2,5664^ = P. 

Thus by thife fix thforemi miy all the Variertei rehtin^ to 
the Diamster, Peri^biirj, and Area's of CinUt be Iblvej. 

Prop^fitioti 12, 7b A/f j/ar; f) Semidrde. 

Rule. Miilriply half the Semicircular Arch, into half the O/- 
aaieter; the Produfl is the Area. 



Byam. Multiply * /f pC = 4,;?3 ^ 
Iiitoi >1£; = £C = 2,76 8 '' 
2600 




TheSf ffl/Virrslir Wm = i i ^6 



Propofition i j. R Menfvrt iht Seflor 0/ d QrcU. 

Rule. Multiply half the Arch into the Riuliutj the Produa is 
the Area of the Seltor, 

EKOi/iplf. Miikiply ; -< B — 4,^? 
Into die i?4^i«f ifc^ 11,16 

2718 
»o.554« 



Xx a propop 



34^ 7h^ ^ff of liecmaU in the 

Propofidon 14. To Measure the Segment of * Circle. 

Rule. Compleat theSffltw AC-BO, y^"' "'x, 

and raeafure it by the lift Propo/t- / '\ 

li<m ; aiid then frid the jirea of tlie / ' 

TViavgfe ABC by Propofitiov <. I C 

Olieii fubdua Che AreM of the W- \ ,^ 
avg!e from the Area of the SuSlor, \ f' . [^ -.^j 
the Remainder u the ^rra of the -"X"' '- '^^ 
Segment. ^"cte^^-^^ 



Or thus, ( I>y the Curious Theorems of Mr. PFard.) 

5 R =Thc Jiatfiu I, or Se/nidhmeter AC. 
d r= The vAff between the verf,;d Sinr aitd Radius. 
IC x= HalftheChordot B<i/Vof the Sfj, ^ZCviz. ST.) 

ot the Segmeat. 

Propofition 15. To Me/tfmv an Ellipfis. 

J^if. Multiply the Travfverfe and Conjugate Diameterj into 
each other ; then muhiply that produSi by die Number 
0,7854, the FroduB is the ^rfa rcquiied. 

' .iE'*i7jn. Molt, the Tranfverfe CD = j6 
By die Conjitgate AB^i6 

..- I16 

f The Produfl is , . -r S?^ 
. .Multipij that ty -r 0,7854 



TicAiaofibe B«/J!j =_4^2,3?o+ 



f^o. 



Menfiiratlon of Superficies and SoIiJs, . 341 
Propcrfuton 16. To Meafum (ii? Parabola. 

JtuU. Multiply the Greattfi Oriiaaie,ai Safe, into the 
perpendicular Height, and that Producl by the ungte Refe- 
tend ,6, the ProduSi is the Area. I 

Ex. Mult, the Ordinate ^^=55,7$ 

By (he Abfdffa, or liiisht, CD =43>? 

16125 

16115 

21500 

Multiply this Ptodma j= 2327,375 
Bj the Repetetd — . ,* 

9)13966250 

The Area of the fiwa*. = 1551,83s 

Note, an ealier way is to multiply the firji ProduS by ay 



Propofition 17. 7> Mfaftire the Circular Space, tidied.- m 
Lune i {"i^iff^ A*'' tbe falcated Mooii. ) 



/■«/(■. In order to find the 
Area of the Lone aEBD, 
feek firft the Area of the Semi- 
im-le AEB, by Prop. 12- 

Then find the Area of the 
Segment A DBF o^ the Cff"- 
rleADBC, by Prep. 14. 

Laftly ; Subiiraa the -frc* 
of the Segment from the Area 
of the Semifirle ; there remains 
the jrfffii ot th? iaw requi- 
red. 





J<f» The Vfe of Dfcimah in the 

Alio "dUft- fv» iMHei A, •, we 
tog«her equal to the Triat^le C, 

Notiy 'litis is calTd the Qaa- 
JrOiirr, dr Squaiiii^ the ijivet of 

pK^fition 18. To Affttfitre the Cjdoid. 

JP«&.Find A ^ B 

Ae Area <i 
the Circle C, 
defcribed on 
ihe/#*/jDE, 
and multiply 
that by 3, the 
trotlu8 is 
die Area of the CjiloiJ ( called alfo ir»c Trtxhoid) aEBD. 



Pr(^>cfirion ip. To Mtafwrt « Spbcnca] Triangle. 

H«/?. Fmn tbeSwwof rhe/if^^^ff^W, fi)bd!!fl iSo De- 
grees, niiilripiT the ^feififits of the whok S^btrt or iS/o J» hj 
tlic P-cmaiiidiiT ; thu f'toduft divide b| 71D, the ^uotievt is 
liic Content or ^r^o of the Tria'gte. 

Extm. Suppaft die Aiiglef? 

^+B + C==: 1,7.02$ 

FmrntharSumSobdufl 180 

Nft-'lr. rill* Remainder :9 3^,0^ 

By the SiM-/<«ofi . ^ ^ 

Wv^.r^ ■*'"_!!!! 

- 7^C*6()6 
Diwde by 71c) P7842,c* ( 1 55,8?! = ^« of the T^i- 
Note, 




t 

MtnfuratUn 0/ Suferficief and Solids. 343 

Kotfj This is a very uncommon j curiottSf and uffful Pro 
poficiQi). 

Menfuratioa of Solids. 

Proportion 20. Jh Meafure a Cvbc; 

jRule. Multiply the Side of the Cube into it ft!f, and tl«t 
Prodiid ag^n by the Sidtf ; this laft Pfodud will be the Solid 
Covfentf or Solidity of the Ciic^cf. 

£jr<«Kr. Mult, the Side AB =: 5,7 
Bptfclf — — s,7 ^ 



^99 
285 



-- ts 




Tills Produft — 92^4.$^ ^ 

Multiply again By — 5,7 

2274? 
The Produa is — 185,195 = SoUdiiy tf the Gr*^. 

Propo&ion 21* To Menfur0 a Parallchpipcdon. 

Ruh. Find the Area of the End or jP^/J*, and Multiply 
that by th« Length of the Piece^ the Produd is the Solid 
Content thereof. 



E<am. Mult.. C B rr 9,^ 
fiy — BD=^ 7 

The i^rM of the > , . ^ 

End - > — ^^>^ 

Molt, that by the^ 

Levgtb^ A B 



The SoUd/ty — = ^349,95 




Tas:;* 



Exam* 



344 The Vfe of Decimals iftthe 

Exam fie 2. Of a Square Prifm^ or pardllelopiptfdoH. 

Mulc.tiitoitfelF,1 a 

or Sttuere the* ar-.r ft 
SidVof thci'*^-^*^ 
Smuare End ] 

464 

290 




The Area oFthe1_^^ ^. 
Square End /— ^^^^^ 

Which mule- > 
riplied hjV^AB= 17,6 
the iengtb^ \ 

20184 
23548 

The Produa is 5^2,064 = &W OanienU 



Example 3|, Of a Triangular Prifm. 



Mult, the £4/2^ BC:=8^ 
Vy the Heigbt VE = 6 

The -^r^ii of the 1 

Which mul-l 

tiplied py^AB=22y7 
Xht hngtij 

^008 
1008 




g^te^^^^ ^^^ ^^ 



^ 



d> 



The Prod, is ^ 1144,00 == Sc^Wifj of the Prifm. 



JExM* 



Mcnftt^a'^ioif df^erfities and Solids, 34J, 

^^gle 4. Of a Cjlmder. 
foiiare tHe Diameter I =:tf,6 



The A«a oMh5l_^.- 
cimlarBaf" >—**'* 
Which mult.by the ?_L. 

ThcSjSWrVf ofl^^ . 

Propi?fition. 22. ?S Meafure ibe Comeit S^ferficie^t ^-^ 
C^inder, 

3ul^. MuItiE^ the Petipher) of the B4/> into the len^B 
le Ciliiider ; the foodua is the Catteht. 
Example. SuppofethcC/rfsm^wiW^of theflrffrV „-„ 
( in the lait Figure ) B BCD to be — "^ / "'*^*» 
Then multiply that by the teng*k aB =- 2J 

The Superfidal Content of die CjUndet i^ 481,71^ 

Propofitioo 23. 7^ Mf^/Wtf a Pyramid. 
t^ute. Mulriply the jtfea of the Safe into one Third of itt 
Akituje or £fc?iW j c]^ iftqiju^ i,s the BoUd Content. 



Mxam, Suppofi the Atea of 1 . 

' Hi^Squate Safe A BDSf *^ 

MuUiflj that into f of Fc as 4,2^ 

6483 
25pa 
' 5'g» _- 



Thc^o/frf Cfi«v<ofthe> _,,^_ 



• - ^ 

54^ Tbe Ufe of Decimals in the 

Note^ The Role is general for any kind of Pyramid whofe 
Bife it anj regular Fdjgoff. 

Prcpofidon 24. Tb Mtafurt tfCone. 

^ The Ruk is cfae very £uiie as for the Pyramid miiie laft 
FropefidcML 

Exmuple. Soppofe ^t' Circular V ^^ - 

B^e ADBE be in Area f — i9f% 
'Aai ioTth^ Height FC, he =j3$ ^^ 

IP7S ^* 
3160 
1 185 

The Solidity of the Cd»f will be = 152,075 A 




Propofidon 25. To Meafure tbe Curve Superficies of a 

Cone. 

i?«7^, MultiiJj the Perifb^ of the B4/> into the Length 
of the Side ; Halt that Produ3 b the Content or ^^4 of the 
Orrv/ Surface. 

Example. Supps^t the Pertpberj ADBE(oty ^ 
the Co«if above ) be — — — X ^3' 

That the Length of its SiVi? i< F ~ =: 18,2 



47* 
188) 



8 



236 



*' 



42p,52 



CO 



TbeCow<^jr Superficies of the C^»f wiD be found) ^, ^ ^, 
be — — _ — / 214,76 

Vropofition 26. 7b Meafare tbe Fruftum 0/ ^ Pyranud or 

Cone, mrl parallel to its Bale. 

i?ffZf • Multiply the Area ot the Greater Bafe^ by the Wr^tf 
of the E^£irf and exttaa.t|^5jtf^^ i?o^^ of tbe Produa ; 



Menfuration of Superficiei and Solids, 347 

To that Root, add the Sum of the two Area's of the End ; 
then multiply this /d/? Skw by J of the frw^wj Hf /gW, the 
Produa ii the Solid Content. 



Sxaa^e i. Of any Pyramid. 

'ea c£ the Greater Ba 
("of a Square PyramiJ ADBC • 



Suppole the Area c£ the Greater Safe 1 , 

( of a Square PyramiJ ADBC ^i * 
Andthe^M of the Lejfer FEGH= 16 

384 

64 

Tix Produd is — . — 1024 

The Square Root Amtoi li ^32 

To whidi add the Sum of the 2 7 o„A 

^ea't - =: /" ^ 

Thii Sara — — • 112 

Muhijdy bj \ of the Beigbt DE = 4^^ - 

looe 
448 

Tlit Solid Coirteti — ^= ^52,5$' of the I^at^ 

Frufitm. 

And thus the Frujhim of any other kind of Pyramid is to 
be found. 

Extmfle 2. Of a tow, 

Suppofe the Arta of the Greater SafeX , 

(ofthcC»«*adioin*d) ^Z)BC«f"'»° 

Andthe^rMofAcZf/trFflSHsa ii.a 

2452 
1226 

»226 

The Produa is — 1 373>i » 

Yy2 1%e 



The &»«r/ it«ot thereof is — = 57,055 & 

Jo wliidi add the S«»i of the TOO Atcs's = M;,8 

This /«/!&» — , — —"70^55 

jMultipfjbjiof theaiWXS -_f£L 

854275 

11P5965 

7025150 

The SWa Cosftrt — = "53.27"! 

JVopofflSn 17. 75 M«JW-> « Spheie, »ir S»<OlE«trea^ 
^. Round. 

/ItiU Multiply the Bimfter into the ciramference, the 
Ptodua is the SapirfddaittrnC, then multiplj that bj i 
of the Biamur, the PioduS will Be the ftW CMMI of tl(e 

^anfU. Of ^thp Sapfrfirial and &/iV Content. 

the fl/pfc ot Spbete> 62,852 
^CBfl = i 
^Bj. the ^*(f ofthe V -„ 
Sfi^rt ^B = £ 

T^^^*'''"^»|"^,25»,«40 ^ 
Wiiicti mull, by ^i of / ^ 

' '<f4 ■ = f. ' 

■JlSe 5o?>i' Cmtnit= 4188,8 of the 'Sf fore. 

Ptppdfition 28. liKfaftire any Fruflum or Segment cf ^ 
J ■'■; . *■ 'Sphere pr Globe. - ' . - 

/ "■ '■ ■ 

f. Let 



Menfi^i^tf Suftrfitkt /mi Stliii. 3^9 

rD=- Diameltr of the Q 

,3 Sfhrri C E 
»• ^" •5 H = »4SI of ihe Stf 
t ment CD 

Then ?''"^-?«' = m 
Of the Po/or Segment ACSD. 




r D = Dimeter ■€£, as C. 

fl r«. ^ before, 

"C H =; the Tmckntfi DF. 

■ ThCn2^^xH=:the 

AfjA//tf Segnent HBTQ, called (he ., 

^Tottf. By chefe two Tbeoremi, 

paay the SoK^ty of any ^egnent of 

-a £/0$i? 6r Sfi/tre-hc {bniid. {B 

To find the Superficial Content bf any Segme/tt, as 

J. Asthcyi*!/, Of Dii«ifT?y of tfic'S'^Ai*^, 
Thus J' '' ^^ '''^ whole Superficies of the S;>;frf« ; 
"S-So is the Height oi mj Segment. 
41 'To the Area Xii' ill Curve §iper fides. 

Propofition 49. Tofind-theConUTA o/«SpheK»ii 

''Rult. Multifdy die Smtare of thtrDiameler of the GregtHi 
pirch^-by the Le»gtb ; ttten multiply that Produfl by 0,5236 • 
fpjis laft PTodH^ will be the Solidity of the Spheroid. ' 



INfc 



350 TTieVfeof Decimals ii tli 

^tm. Sqaare the t>i0meter\ , 

^»5 



liAikiply du* Square •^ « 
Br die Lngjib AB — = 

Thi* Ptodoa ^ 



42.25 
10 



Muldi^ bf 




— 422,$ 

«- — »523< 

25350 

iatf75 
8450 

21125 
The SolidHy — = 221,221 of the ^berold. 

Kettf The two Tbtaremst ivhich find the Content of the 
Segments of a Sfberei, find thofe like SeWRMrfi of the 
Sfberoldf if ia them D be madesa C2>, in mis SfhertH. 

Fkopofidon 90. 7& Meafurg a Ptoibolic Conoid. 

i?flr2r. M ultlpfy the 5fii^r of the Diameter of the Bofe 
by the Hrigibf ; and that Produfi by 0,3927 ; this laft Pro- 
dud is the folid Content. 

Exam. Mult, into it fclf, or) 
fquare the Diameter AB^=^T 



P|2 

184 
828 



This &«f^r — 84,64 

Multiply by the Heigjbt CD ^ 10 

And the Produa 
Multiply by 




— 846,4 

5P248 

16928 
76176 

T»lolH Cwfien^ = 332,31512? of th?P^fi>o//rCd»oH 

PipopcK 




Menfitration cf Superficies aftd SoUdsl 3 J^ 

I^opc^ooa %i. 7% Meafure tbeFtiAimi o/«FanbdiQ 
Coixnd. 

Sule. Add the Spure of die Diameter of the Leffer Baff 
to the Square of the JHameter of the Greater Bafe^ Kviae 
diat Sam by 2,5464; then multiply ^ Quotient bj the 
Hfi>M ; Ac Produa is the S«^id Coittertt. 

un 

Exam. TTic ^dr/ of ^B = 84,64 f^ '.■ ' .« 
And the S^at^ of £Fr= 40,96 V^*' 1 |;-J 

Sum of die^ium .^ = 125,6 ' 



Then 2,5464) 125,600(49,3244 

1°'^^^ 5_= CD the Heisbt. 

llltil 246,622 = the Solidity of the 

82648^. 

Propofition 32. 3% Meafure a Parabolic Spindle^ or I^ra- 
midoid. 

Hafr. Multiply the Square of the Diameter of the Greatefi 
Circie, by the Z? ir^i ; and that Produft by 0,41888 ; ims 
tall Produft is the Solid Content of the Spiadle, 
Exam. I. Square the"? 

Diamtrter of the >■ 6,05 

Greatefi Circle AB J 

6,05 

Mule, this Square 96,6025 
By the 2>i!i£f ^ CD ^ 8^ 



iS-ioiaj B 

2P28200 
This Produa ' 311,12125 
Multiply by ,41888 

Solid Cont, sas i3o,?2246i?2 of the Parahlie Sfiviffe. 

Mxampte* 



3 J» • The Vfi^ Decmais in the 



Bxamfte 2. To Meafure the Middle Segmifik of the Spift^ 

dlcy EGHPm 



Let 




the lAamier of the gteaiefi Circle j§ B. 
^the Viametir of eitfaor Bafe or &ri j^F or GSi 
2BK^ssAie EMcefi oi AB above EF^okI^H^ 
the Z>v^fiEy of the Frufium FH. 



l^ 3,82 '^ S middle fruft 



the 



Propofitioo 35. li Measure any of the Five Regular (at 
* •Platonic) Bodies. 

Thofe Bodies being on^r an Aggregate of fo many /^^ 
mids as they cx)nlift of 6/V^j, each Side being the baie 
of a Pyrmnidj n>aj, with due Confideracion, be meafured by 
ProPojfition 22. However tor the more jready and expediti- 
ous Fra6!ice, IfhaHhere fubjoin aTabJe of the 5o/iA'/Jf and 
Superficies of each Body whofe Side is i. or Unity. 



f^mfmrn 




Names. 



Tt^rahtfdron 
He^fobpdron 
OEiah^dron 
Dodecahedron 
icofahedron I 



Superficies, 

1,73^51 

. 6,000000 

3,464102 

20,645729 

8,660254 



SpUdiij. 



•&^» 



0,1178511 

ipoooooo 

0,4714045 
7,663119 

2,l8l6p5 



'^. To ufe tbepceoe.e.d|i^ Tli?& for finding 
the Superficies- of any of tjaofi^fivxBodieSi 
do thus ; 

Square tlat Gipen Side of the Body, and 
by that multiply the^Tahular Sjuperficial 
Number \ theTProdu6t is the Superficies of 
the Body, which was fought; 



Hexahedron. 




V. 



Exarn^ 



Menfurathn of Superficies and Solids, 353 

Tetrahedron, 



ExtimpU. Suppofc the Side of the 
hodeeabedroit be 8^ tlie iguare of which 
u 64. 



Then.roultip!y the Tah. Numb. 20,645729 
Ey tlie Square o£ (lie ^Je 64 




I2J87+J74 



Superpiial Content 



1321,326656 



To fiuj the SoW Co/jf^wf ; Mulii- 
pljf [lie Tabular Number ot tlic Soli- 
dily, by the Cube of the Si'^tf given, 
tlie ProJufl is the Solid Content. 



Icofahedron, 



Example. Of the aforedid Dode- 
cahedron, 

Multiply the Tabular Numb. 7,65 j i ip 
By the Cuie of the SiJe ; 
(=8,)t.«. - > ^ 

is;xo238 
766^119 
g8;i5 5 ?S 
£o!id Content =a 5S>23,5i6p28 



And thus procted foe the Cufe ficies ai:d Solidity of the 
other Bodies. 

Propoficion 34. JJ Meajtirt avy foliJ or hollow Body htm 
iircgular foever. 

Itule. Take any Vrpl in Form of a Para^Ukfipedov, 

ind fiU it with Water lu a certain Hc/jAf, aud then iaunerfe 

i * ihc 



354 ^^ ^ ^f Decimals tM the 

the irregular Body therein, and obferve how much the Water 
iaratfrd by ihe Side of the f^epl ; for that Heater is equal in 
QuoHtitj, or Solid Content, to the irregular Body ; and may 
be found by Fropi>fitiot 20. 

Extmfle. Suppofc the Vef' 
/el BCl> E, in Form of a Pa* B 
raHehpipedon, whofe Length 
u tivo /''wf, eight /aciej j ^ 
a.-id Breadth one foor, ten 
Jiibei ; aiid it be fill'd mih 
WatertaFQ, twK\va Inch's 
Deep —EF. And it is re- 
quired to meafiice the Log O 
W, of a racft irregular Form. 
In order to do ihis, 1 take and 

imircrge the leg in the f^^/ & 

of ffater (as 111 the lower t'i' 
gure) and obferve the Water 
rife from FG to B I, tha 
PfiVftf of which f viz. FH, 
pr GI) I meafure, and find to be 5,^ hcbfs. 

' Then by Proportion 20, I find a Body of Water, 2 Feet^ 
P /«ci&°/ in Length ; i Foo/, ten Inches in Width, and s,5 
Inches Deep, to contain 5520 So'/if Inches, or 1.0)7 SoZ/rf 
fff/ ; which thetefocc is the 5o/»i/ Content of the Zi^ ^ / 

Propofitioq 55. To d^» f^c Dimerfions hy vbicb the feve. 
ral Aitificcrs meafure their iforf:. 

f -(As Pavements, 

KSaperfi.ialj Chimneys, Pieces, 
/I//(/<)Wi pcaVeJ fi^f^a"43 tCornilhes, £>. 

theitWotkby I J«fi)ff \ r Colomns, and o- 

/ Solid ^ther/o.'j\/Pai[s o| 
V. (f Buildings. 

r /=''">f:Arch«,Qiioins,Coniices,Fafcia*i,Sf. • 
fricklayeri tat\-^Yardi Pavemenis, Pieces, f3c. 

fute oy [b^ j Rod j All rnanner of Walls, and Chimneys. 
*■ Sfjiiare ; AH m^nnet of Tyling, and blating. 

f>r^'-f.'fjmea- J S'^^^ of J Roofing, Paititiomng, FJocfc 
iLie Dy the t loo Feet 5 ingi Be. ' 



Menfuration of Superficies and Solids. 3 y J 

yoi^eriyPainte^SjPJa7jie''i Square Yard for the mofl part j 
reri mealiire by the \ feldom, by the Foot Square* 

Glafiets meafure their f Decimal Foat Square ; very rarely hf 

Work by the (.Inches and Quarters. 

Solid Inches. 
Gangers meafure the Area's r 2^1, For Wine GallortS^ 
and Content, of Super- < 
ficies and Solids, by c 

Survejers meafure r J^o^ ox Pole ; but moflly by the Cbaid 
Land by the iof an 100 Links =: 4 Rod. 

Propofition 56* To affign Multipliers, dni Divlfors, is^heriem 
i^ the Gau^er i!»^^ rt^adilj find any Area or Content i« 
Gallons or feulhels, whether the Dimeniions he taken ifi 
IiKhes, Feet, or Yards. 

This I ftall d6 by difpofing the Numbers in thdr prop€!< 
Order in the Table fubjoin d. 



202, For Ale Gallons. 
268^* For Corn Gallons* 



Dimenjions. 



Inches. 



^••1 



Feet 



Yards 



l^ote. 



f^immm 



Multipliers. 



0.004329 
0.003546 

0.003722 

0.0004.6 <; 

7-48052 
6.12765 
6.42448 
0.80356 




ir.G. 
A. a. 
c.a. 

C. B. 
C.B. 



mmmm 



67.32468 IV. a. 

55.14885 A, (s. 

57-82032 C. G. 

7.23204 C B. 



ftands ioi 



dkUt 



y9 



T>ivifors 

231 ~k 
282 A. G. 

268^8 C. G. 

2150.4I Ci B. 

0.13368 tv.c. 

0.1635a A.G. 
0.15565 c. Gi 
1.24446 C. Si 

0.014853 ^. 
0.01816S A. 

0.01729+ C. Gi 

Oil 38273 a. Bi 

Wine Gallons. 
Ale Gallons. 
Ct^n GdVonii 
Cord Biijhelsi 



ti» 



fk 



$S6 The Ufe of T)eciffnfh in the 
The Up of the freceeding Table. 

If by the foregoir^ Propojitiovs the. jlrea or Content 
©f any Suferficies or Solid be f^und ; and you would know 
how many If^iney Ale^ or Cern GallonSy or Bujheh ir c^n- 
fains; Miik'tply, or Divide, the given Aren rn Sohdity, by 
the TaMar Number cnrreTponding to the refpeSiive Meajfurey 
and toimenfion^ the ProdiiS b the Area or Content (ought in 
Gallons^ or Bujhets. 

Example. Soppole the Parallehpipedon \n Example i. oi 
fropo/ifiom aa reprefent a Ciftetn^ and the Vimenjions there 
ijfed be peet ; then the foUd Content of the Ciftern is there 
found to be I }49,P5 Solid or tubick Feet. 

'349» 9^X748^^2 = Content in IVine Gallons. 
ijn^ J 1 349,9^X6,1 2765 r= Content in Ale Gallons. 
1349,95X6,42448 = Content in Cofn Gallons. 
1349195X0,80356 = Content in Corn hufbeU* 

Or hj Divifion. 

,13368) 1349,9$ (= Content in Wine Gallons. 
ThiB 3 >i6352) 1349,95 ( = Content in Ale Gallons. 
^ ,15565) 1349,95 (= Content in Corn Gallons. 
1,24446) 1349,95 (== Content hi Corn Bufhels. 

And were Dimenjions taken by a Decimal Yard or Foot 
(which are by far the bcft hftruments for Menfurafion ;) The 
Biifinefi of Gauging would be eaiy, and greatly expedited by 
a Tahle not before extant, that I know of. 

I 
AW 5 M you would find the Content of Circular Area's 
at one Operation ( without reducing them by the 
common Multiplier 0,785398) do thus, 

r 359,05 for Ale Gallons. 

Divide the Square of j 294,12 for Wine Gallons. 

^e Diameter hj j 342,24 for Com Gallons* 

\ 2738, for Corn Bujbels. 

But 



Men fur at ion of Spperfcies and Solids. 357 

But in this Cafe the Dtmenjions mufl be underilood of 
Inches only. 

By what I have fiid in this FropofitioVy I fuppofe the dex- 
terous young Arti/l will be eafily apprifed of the Nature, 
Manner, and Reafon of Gauffftg ; and how to apply the fore- 
going Profojitions to that purpole. 

Propofition 37. To apply the foregohg Proportions of 
Superficial Meiifuration to Surveying. 

Suppofe a Field m Form of an IrregHlar FolfTin ; as 
ABCDEFQHj below. 




The 



358 The XJfe of Decimals in the 

The Fidd being meafured, Plotted^ and the Plot rcrolv.ed 
into Ttafetia^ and Trhnglesj as per Propofttion 8. 

Proceed to find the Contents of the Triangles by Profo^-' 
thm 5. and of the Trapezia by Preoption 6. 

Thu» for the Area of the Triangle HAB^ 
Multiply half the Safe H1S=: 18,3s 
By the Perpendicular Aa := i? 



5505 

1835 



238,5$ 5s the iir^?/!. 



For the Area of the TriangJe FBC^ 
Multiply — i FC = ip,25 

By the Perpendicular B d =: 18,5 

9625 

15400 

1925 



356,125 = the Area. 



For the Area of the Trapezium BHGFB^ 
Multiply half the Difgonal GB s= 20,8 
By the Sum of the Perpend. Hb^Fcsr^2fi 

1664 
416 
832 



890,24 r= the Area* 

For the Area of the other Trapezium CI> EFC^ 
Multiply half the Diagonal C E =: 27,55 
By the Sum oi the Perpend. De+Ff.^ 35,7 

19285 

13775 
8265 



To this laft , ^ 983,535 = the Area^ 

r 890,24 

Add the other ;^r^A'i — ^ 35^)^25 

I 238,55 

The Sum of all is the Content — 2468,45 = the Field* 

Novir 



r 



Menfuration of Superficies and Solids. 3 59 

Now if the Numbers are foppofed to be Poles or Eods^ 
Divide the Whole Superficial Content or Area of the Field^ 
by 160, the QuotieiE will be the Number oi Statute Acres the 
Field contains. See the Work. 

160)246845 (15,4278 Acres; ^ ^^ ^^ 
16 Acres Hood Hods Yards 

Or, 15 : I : 28 : 13 



■ 



86 


80 


68 


64 


•44 


32 


125 


112 


• 130 
128 



• •• 2 

But if the Field vfzs meafured with a Chain of an 100 Dnks 
( rr 4 Rods.) Then, becaufe an Acre contains 10 Square 
Chain Sf and every Square Chain containing loooo Square 
links J therefore a Square Acre will contain lOOOO Square 
Unks ; and hence the Reason of flriking off 5 Figures to the 
Right-hand from a Given Jtre^ in J^inkSj and taking the 
Ji^Ji for Acres. In this Cafe^ the Area ot the Plot above 
would be but ,0246845 of an Acre ;. i^ e. about 3 Poles and 
7 Yards J Square^Afeafure. .- 

But fuppofe the Figure above reprcfent the Plot of a large 
Common wbofe Dimenfions are taken in Chains and Decimal 
Parts of z Chain: Then the ^^^4 thereof would bp 2468,45 
fij»/?r^ CA41W, -which is 246,845 ^^r^j ; Or, 246 Acres ^ 
3 Roods y 14 -Po/^'j. 

And thus may hvf Area^ ,or Superficial Content of any 
/''Wrf or /'/of of Groiin^ (found by fome one of the firft i^ 
Proportions oP this Chapter) be turned or refolved into Acres 
hy this Propofition. • / 

Notey By the nineteenth Fropofttion you may fiud, with 
^afe, the Number of Miles or Acres contained in the Whole, 
pr any Parr, of fhe Superficies of the terreftriaJ Gkhe ; and 
pi any Province. Kin^om. Empire ^ or Nation of the World; 



36o The Ufe of Decimals, &c. 

iad it otrglit thcicFire to be well undetftaod by all wh3 wooU 
ftudy Foliticai Ariibmtthk, this being the moA certain, curi- 
001, iJiJ piiiicipal Branch uf that fat ; Thu' it be not iir- 
crcr; Tnh on that SubjeS, nor iii an; Book of Mejifuia' 
tion that I know of. 

1 bin I iiave fiiiilhed a Tr^il of Planometrj, and Sttreome- 
iry, or of the Me»furaU'ot of iuperfidtt aiiJ iolids ; coit- 
taiiiing a QreaXtr Vaiiety than I know to be iii manf Br^ois, 
wrote whoUy on the Subjcfl, of two, ihtee, or I'uur ShUHr.^t 
Price ; haviiij; endeavoured here, ax in all other Parts ot thii 
ijfitmy to obliijc the Reader with all th« could be ufifkl tot 
bull to know, ui the molt plain but Comptndioat Alatisr, 
u the eal'iell Kate 

Vivf, VaU i Si quid novijli reSftus ifti.\ 
Caiii^dzis imperii ; ft noJi, his utere mecum. 



FINIS. 



TABLE 



OF THE 



LOGARITHMS 



TO ALL 



NUMBERS, 



Not exceeding loooo, or 4 Place^^' 
whether they be Intire, Broken, or Mia 

Numbers. 

Particularly ufcfiil in ExtraAing tfte 

Square Cube, ^c. Roots, and folving 
Qpeftions ia Compound Intereft, ^f. 



As» 



)6i 

Nimbcr*. 
1 

X 

4 

7 

9 

loo 

lot 

I05 
I06 

IC7 
109 

'ni 

* 113 
114 

Ii5 

117 

Ii8 

I20 

131 
122 

125 

4 84 

127 

128 

129 
1^0 

131 
1Q2 

>33 



Jrtificial Numlers : Or, 



^1) 



oocoooo 
3010300 
477121 

602060 

6989700^ 

778iHtt 

8450980 

9030900 

9Un^5 
oocoloo 

004^214 

oo84o^ 

Ol2*t7C 

oif^3f! 
o2iii?93 
025JOS9 
0293838 
0334237 
0374265 

0413927 
0451230 

0491100 
0536784 

OS 6904 8 
0606978 
0644580 
04918^1 

'07fU7o' 

0791812 

0817854} 

0863598 , 
o%9oiti 
09342171 

iPo^7<>5 

1038037 
1072 100 
(I05897 

1159433 
Ii7i7«3 
ii0575p 
1238516 



134I1271048 



0413927 
322219J 

4913617 
6it78j8 
70757*2 

785319^ 
8512583 

90848J0 

9V904t4 
0004341 

0047511 
QP9PH7 

Ol?2^t7 

o»74f07 

OZj6027 

0257154 
029789s 

0^8i57- 

OJ78247 

0417873 

045714^ 
Q«l950f6T 

O534625 
OS72856 
0610753 

064 8 322 
o68fx69 

675911I 

0795430 
0831441 

08671^7 
0^015^0 

0937718 
0^72^73 
locgtSi 

1041455 

1075491 

1109262 

1142773 
1 I 76027 

1241780 
11742^8 



^791811 I IIJ9453 



ot§i843 'Oityoo^ 



7160033^7^4^759 

79239«n- 7^3405 

8573325 

9138138 

9^i78>8 
0008677 
0051805 

0094 5 Oj^ 

C13<^794 
01786^7 

02^20157 
0x61245 

OJ01948 
03f42273 
03&2226 
0421816 

046|P4» 

049fW8 

053t464 
0576661 

0614525 
0652061 

66892716 
072617$ 
0762762, 

0799045 
C835026 
0870712 
o^o4tb7 



1461280 
38021 12 
5^14789 



3424227 3617278 
5051500 5185139 ^ 

€232493 yB334684p4I»5*7 

7323^37 
ScthiSoo 

8633229 8691317 

9i9o}8i 

9684829 

001)009 

0056094 

00^5756 

0141003 



0224284 

0265333 

0505997 

0346184 
O386202 

0425755 
04^9^2 

05*3757 
0542*5^9 

05 80462 

0618293 

0655797 
0692980 

O729847 

0S02656 

0835^608 

0874264 
090963 1 



0941216' 09447' I 

097^043 ^97*51!* 

l»iof93 toUQJB 

1044871 'lt)4%84 

107^880 Io822'S6 

111262$ II15985 

II46IIO II49444 

II79338 II82647 

1212514 1215598 

1245042 124850I 

1^77525 1^80760 



924^793 

973127^ 
0017337 

0060379 

0x02999 

0145205 



mtmim 



0^28400 
0269416 
0310043 
0350293 
0390173 
0429691 
046^852 
05*07663 
0546150 
0584260 
0622058 
0659530 
9696C8I 

07a>5l7 

0770043 
0806265 
0R42187 
0877S14 
0913151 
0941204 

09^*975 

I0r747» 

1©5I«694 

1085650 

1119343 

1152776 

1185954 
1218880 

125155& 

I28?99? 



Lo^a't-ithns \to ^^^9^^ 



Natural 





SuiVibers. 




I 




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i 


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ics; 




i09j 




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III 




It2 




115 




114 




'«5 




115 




117 




iiS 




itp 




120 



'5 



I 



i 



121 

i?6 

127 

I2S 

129 

n' 
152 



17609^1 
J979400 
54406^0 
6532115 
7403^^7 
S129IJ3 
8750613 

9204^*^^' 

0021661 
0064660 

•6r494f>3' 
.bt9n$3 

.02734$6 

o1uo^$* 

0351297 

0394141 

o4?3§n 

04727491 

05115^5 

0549954 

0588055 

062^826 

oe6]2$9 

0700379' 
07371^3 

.0773679 
0809870 

0845753 
08S1361 
0916669 
095169) 
0985437 
1020905 
105510a 
10S9031 
1122698 
1156105 
1189257 

1222159 

1254813 

1287223 



'6 



! 



I 



4641200 

4»49753 
5^630^5 

48i%8o 

195439 
8Bo8i|d 

9544984 
^822712 

00a s' 9 So 

0068937 

6111473 

0153597 
ot953«7 

0236639 
0277572 
0318123 

0^58298 

0398105 

043755^ 
047^*^42 

05153^4 
9553783 
0591846 
0619578 
0666985 

0704073 
0740847 

07773^* 
0813473 

084933^ 
6884905 

09201^5 

0955180 

0989856 

1024337 

1058506 

109241a 

1126050 

115943* 
1192559 

1225435 

12580641 

1290450 



4304489^ 

43*3637. 
5681617 

r67io9^8 

•7^874^ 
8j^074i< 

886496^ 

93P5«9^ 

9«?77I7. 

oo3oa$5 
0073269 

oi 15704 

0117787 

01994*7 

0240750 

0281644 

0322157 
0362295 

0401066 

044147^ 

04805 j 21 

05192^9 
0557^05 

0595634 

o63?334 
o67#7e8 

^7077^5 

©744507 

071094 1 

O8I7073 

0852900 

088844^ 
o|P»3696 

0958664 

C9933H 
027766 

061909 

095785 

129400 

I6275S 

195859 

228709 

26131^ 

295676 



i^527M 
44t7i.5^ 
579783*5 
6812412 

7634280 
I 63^5089 
'8920946 
9444827 
9912261 
'oo34<^05 
0077478 

0119931 

0161973 

02036 13 
0244^^7 
0285712 
0326188 
0366289 
0406023 

0445398 
0484418 

0T23091 

0561^4*3 

©599419 
tJ637o»5 
0674428 

071145 3 
0748164 

0784^68 

0820669 
0856473 

0891^84 
0927206 
0962146 
0996806 
103119^ 
lof*53c8 

1099159' 

1132746. 

ll65o77 

1199 U 
1231981 

1264561' 

1296^90 



?3*3 



2 7875 3^! 
4625980J 
5910646- 
690196 1| 
77c8«52o| 
83884911 
8976271^ 
9493^00 

995^5* 
0^38^12 
008 17 4 It 

012^1 54 

0166155' 

0207755 
0248960 

0289777 
0330214 
0370279 
04^*99774 
044*3 I ^ 
04883 Oil 

05^<'939 
056f23^ 
060^200 
064^834 
0678145 
071^^138 
075181^ 
078819% 
0824263 
0860037 

o«9<03 
093<^7i< 
096^62 

106025 
10^461 
106870*} 

ll625?5l 

116939I 
126:44 

123525 
12678c 

IJOO^IQ 



1 



A a a 3 






til 

t}6 

137 

■3S 

«J9 
140 

«4' 
>4S 
14} 

»44 

Mf 
146 

«47 
I4> 

U9 

150 

iU 

I S3 
>f4 

«S< 

ijtf 

•59 

itfi 

164 

S^ 

1^7 

*^ 

'?• 

»7» 

«73 

'T4 

«>5 
176 

13L 



p 

I 



Artifiual NmAers : 



11^7206 

1398791 

i4)of48 
1461280 

1493191 

1522883 

1583615 
1613680 
1643M8 

1673173 

1708617 
173 186 J 

1760913 

17897^ 
1818436 
1846941 

1875207 
1903317 
191 1 M6 
1958996 
1986571 

Sol 397 < 

9O41&0O 

^8a59 

io95l5o 
»i2i876 

2148438 

* '74835 
296io8l 

22i7l6l 
2253093 

2a7H86; 

23044^9 
2319961 

33(5284 
23804^1 

>4of493 

*43«38o 

S455I87 

2479735 
2504100 



M«5JI3 

«338f8i 

«375174 
1401937 
»433«7i 
146438 1 
U95«7o 

»52|94l 
1556196 

1586640 
1616674 

1646502 
1676127 

1705550 
I734I76 
1763807 
1791645 
1811292 
1849752 
1878026 
1906II8 

19340^9 
1961/6& 
1999119 

2ol670£ 

SO439I3 
S0709S5 
1097830 

ti24f4o 
2151086 
* 177471 
2203696 
1 A 19764 

8819^77 

2281436 I 

8907043 

1332500 

2557809 
2382971 
2407988 
843286! 

«4f7593 
2482186 

2506639 



1309767 

1341771 
1J73541 
1405080 

1436391 
'467480 

'498347 
1588996 

1559430 
1589653 

1619666 

I 649474 
1679078 

1708482 

1757688 

I766690 

1795518 

1814146 

1852588 

18S0844 
190S917 

1936810 

1964515 
199106$ 
2019431 
2046615 
8075650 
1100508 
2127201 
21537B2 
1180100 
2206310 

.22 42363 
2258166 

2384003 
2J09596 

2331058 

2360 J3 I 

2385479 
^4104$ I 

24)5541 
2460059 
^48463 7 



Or. 

3 

1J12978 

1544958 
1376705 
1408222 

U395ii 

« 470577 
1501422 
855^049 
156^462 
159^^63 
1622656 
1652443 

1632087 

1711411 
1740598 

1769590 
1798389 
1826999 

1855421 
1883659 
1911714 
1939590 
1967287 
1994809 
.2022 1 5H 

2049335 
1076344 

2103185 

2119862 
2156376 
2182717 

2208922 

2234959 

*»6o84i 
2286570 
2^11146 

2337574 
*3^285l 
^337986 

2412974 

2437819 
1461523 

I487085 



1516187 

2548144 
1379867 
1411361 
1442628 
1473671 

1504494 
1535100 

15^5491 

1595672 

1625644 

i«M4»> 

16»4971; 

8714139 

1743506; 

1771478 
1801259 
1819850 

I8s8i53i 

1886473 

19145 10 
1942367 

1970047 
1997551 

I024883 
2o52o44 

2079035 
2105860 

2131521 

2159018 

1185^55 
2211533 

2237554 

2163421 

2289134 

2314696 

1340108 

2565373 

2?90491 

H*5465 

2440 2 9^5 

2464986 

2489536 j 

151^948 I 



natural 
Kottibcr. • 

136 

137 

I3« 

U9 
i4o 

141 
14a 
HI 

«44 

14s 

'.«46 . 

X4S 
H9 

M54'? 
151 

118 

119 
160 

161 
162 
165 

164 
16^ 

166 
167 
168 

159 

170 

171 

172 

173. 

174 

»71 
176 

177 
178 



Logarithms 
, 6 



ito ijtg^) 



1319593 
13515*6 
1383027 

i4H49B 

144574a 
147^763 

1507364 

J538149 
1 5685 10 

1I9W7& 
1618630 
1658376 
1687920 

1717264 
1746411 

t87$36< 
1804126 
183269S 

X8610S4 
1889285 
1917304 

1945143 
1972806 
2000293 

2027607 

20$471o 

2t8i7i1 
2108534 

213H78 
2161659 

3187930 
2214142 
2240148 

2265999 
2291697 

1317244 

2342641 
1367891 

2392995 

24179U 

244277 1 

2467447 
1491984 

2516^82 



322597 
1545^7 

386184 
417632 

[448^54 

479853 

510632 

54H95 

$71144 
601683 

631614 

6613^0 
690863 

7201 88 

7493 16 

778250 
80699 2 

863912 
892^95 

920096 
947917 

971162 

2003032 

2030329 

201-7411 

P84413 
lf>l20 3 

2164298 
2190603 
221^750 
224174a 
2268^76 
^294258 
2519790. 

2 34 1173 
237040^ 

2395497 
2420442 
2445145 

2469907 

2494430 

2$i88i4 



1325798 

1357^81 
15S9339 
1420765 

141 1964 
1482941 

15 1 5698 

1544240 

i 574568 
1604685 

1634595 
1664301 

169380J 
1723110 

1752218 
1781132 

1809856 
x838;90 
1866739 
1894903 
1911886 
1950690 

1978317 
2GO5769 

2033049 
2060159 
2087100 
21^3876 

2140487 
2166936 

2193225 

2219356 
2.245 3 VI 
2271151 
2196818 

2322335 

.M47703 

.23729^3 

2397998 

2422919 

24477 1 s 
2472^65 



8 

1328998 
£360861 

1392492 

1423891 
1455072 
1486016 
I 5 16762 
1547282 

J 1775^9 
16076S6 

1637171 
16672*50 

1696744 

1726029 

1755118 

1784013 
1811718 

1841233 
1869563 
1897709 

19^5674 
1953460 
198x070 
2008505 

2035768 
2062869 
2089785 
II 16 544 

1143139 
2169572 

2195841 
2221960 

2247920 

2273724 

2599377 
2324879 

1350^32 

S37U37 
h> 40049 8 

Mi4h 

^7482 3 



.2496374 I .^4995 17 
2521245 '^52367,- 



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932194 
364034 

395643 
427022 

^4581.77 
1489110 
519814 
150322 
530608 
[610684 

[640553 
670218 

728947 
7580I6 
786891 
815578 

844075 
872386 

9o:>5i4 

928461 

916229 

[983811 

2011139 

20384»5 

1065560 

2091468 

2119211 

2145789 
2172206 

2x98464 

2224563 

^250507 

22762.96 

2301934 
232742I: 

2352759; 
23779I0 

2401996, 

24-7898'' 
245i6s8v 
2477278*' 

2501759- 
2526103 



^ 



$66 

Nanral 
Sumb<T». 

179 
180 

181 

i«? 
194 

i85 

187 
188 

189 
190 

191 

«93 

>9S 

1,6 

1^8 

2c2 

304 

205 

207 

210 

2ll 

114 

2h 
2x5 

»I7 
2xS 

219 

?20 

221 

t22 



Artificial Numbers 



2^18^30 

*552725 
2576786 
2600714 

26:4511 

2548178 

2671717 

£695119 
2718416 

274' $7^ 
i 7646 18 

278753^ 
1810354 

1^5^011 

^^5557} 
187^017 

1900346 
2922561 

2944662 

2956652 

29885]! 

3010300 
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30')3 5i4 

3074950 

3090302 

3II7S39 
3»3»6;2 

5159703 

3180653 
320U63 

)222I93 

^241825 

3^^3359 
328^756 

33041 <S 
i3U<85 

5344537 
336^597 
?3 4^65 
34-4441 
3^24117 
3443923 
34<^3!?3o 



2530956 

25SS137 
-5791H 
260JO99 

2626883 

*650538 
2674064 
•6974(54 

2710738 
2743888 
1766915 

278981T 
2811607 

28j5274 
28578*3 
* 88025 5 
2902573 
1924776 
1^46866 
M8845 

2990713 
501247I 
3034121 
3055663 

3077099 
30984^0 
1fl96s7 
314020C 
3161801 
3182721 

3203540 
3224160 

I 3M488i 
I 32iJ54o7 

! 32S5834 
3306167 
33^6404 
^346548 
3366598 

33^6557 
34^6414 

^426100 
3445887 



i 



t 



2533380 

2i57S4» 
2581582 

2605484 
26 292 5 < 

265i99<S 
2676410 

^6g979i 
27130J8 
2746^^ 

2^^69211 
279^10$ 
28 J 4879 

2837534 
28to57I 
1882491 

2904798[ 

1926990 
2949069 
2971056 
199289^ 
30146^1 
5036; 80 
3057511 
3079237 

3J00557 

1'2177.4: 
3142887 
3163897 

3184807 
3205517 
3226527 

3M6939 
3267454 

3187872 

3303 9S 

33*8413 
i348557 

3368,98 
3388547 

3408405 

34.-S173 
3447851 
3*67441 



i 

2535803 
2559957 
25^^3978 
2607867 
163162^ 

»^iJ253 
26787ft 

2702128 

2725373 

274850^ 

2771506 

27M388 

28r7t5.o 

2839793 

t962ji8 

288472*, 

29o7o2i 

2929:0^ 
i95i>7r 

2?73227 

2>9S^73 

30384^?^ 

30599i9 

Jto26g4 
JfijSS; 

3X44992 

3165993 
3186893 
320769 a 
3228393 
3248995 

3269500 

3289909 

33ioii2 

3330440 

3305S6S 

3370597 

3^'^^37 
34iOi86 

3430145 

34498 1 4 
34^395 



2538224 

256236^ 

2 5»<^375 
2610248 

263^V93 
1657^600 

2704453 
2727696 
27^0809 
1773803 

2 79^66 s^ 
1019141^ 
2842051! 
2864^65: 

2SS6^6V 

290^124^^, 
25^1415; 

2951^7* 

2^7^! 

i997»^ii 
3018:977 

30405 9 5j 

3C(^2IO' 
3085 sOi 

3I0480J 

3 r26po4^ 
3147097 
31^^087 

3188977 
3209767 

323Q457 
3251050 

3271545 

3291944 
3312249 

333^457 
335257a 

337a';95 
339^5*6 
34*2365 
3432116 
3451776 
>47»348 



.HdgarJtbms (^to 2229.) 



* Natural 
Numbers. 

179 
.iSo 

l8t 

185 
187 

\^ 
191 
192 
t9l 

194 

193 

tQi 

xo\ 

209 
2IO 

an 

^4 

di6 

»'7 
Its • 

2t9 
' tttO 

2n 







5c 

« 54064$. 
156477^ 

ttflfl6i9 

27o67Bft 
2730013 

i75flt? 

2776092 
1798950 

i^2ld8a 

« 844307. 

18668(0 

18^9196 
29 f 1468 
29Ji62d 

29S1^7l 
i977<5oy 
^999420 

30tll44 

^04^7^ 

3064^50 

3^85^44 
3106933 
3138118 
3149100 
9170181 
3191061 
3111840 
3«^>Sii 

32?3589 
3893979 

33»4«73 

333147^ 

3^^4S7i> 

3574595^ 

3J94514 

34'434j- 

34340«6 

^4^373? 
JU£0_i 



6i 

• 

'2^543663 
25^7177 

«59iiS^ 

3^*1oo8 
^1^38737 

126^2317 
^6^^780 
S709116 
5^73 1328 
I2755417 

C778J83 
2801229 

«t3955 
284^563 

2«6j)c54. 
2891428 

^913^88 

W31835 
t29S7869 

;2979792 

3001 605 

3013309 

•3044905 
30663^ 

3087778 
3 109055 
3130231 

3151303 
5»7«273 

:3199«4) 

3213913 

i2345** 

3255 '57 
3»T56>^3 
329601 2 
5516297 
'333'64HH 
;^3>^585 
337<5589 

34i^?23 

?435e55 
345^698 



a54$48l 
256is8i 

^93S49 
26fl75SS 
2541092 
2664^69 

268^119 
2711445 

2734643 

X7577i9 
2780673 

28035*7 

282622 1 

2848817 
i^87l^ 

2*93659 
2.915906 

2,9} 8044 
2960067 

29S1979 
JOO3781 

3025474 
3047059 
3068537 
3089910 
3iiii7i 
3131343 

31 5340? 

3 174565 
3199324 

3215984 
? 2 366 15 

'3257209 

3277675 

329H*45 
3^18323 
3338^01 

33585^9 
3378584 

339848^ 
34I8301 

3438023 

34576f7 
3477202 



*547897 

2571984 

IWI939 
2619762 
2643455 

26670*0 
2790457 
2713769 
27369^6 
2760020 
2781962 
280S784 

2828486 
2851070 
*<73l^8 
289J889 
2918127 
2940151 
2962263 
2984164 

JOOS955 
5027637 
9^49^1* 

J070679 

309 104 3 

3113599 

3*34454 

3*55505 

5»7645y 

3197305 

32I8JS5 

3*38706 
3259260 
3279716 
3300077 
3310343 
33405 I4 

33M93 
3380579 

5400473 
3420177 

3439991 

8419^15 



3122iii 



1^7 



2550312 

*S74386 

2598327 
2622137 

2645817 
2669369 
2692794 

2716093 
2739868 
27623*0 

^785250 
2io8o59 
28g«75o 

2853322 

2875778 
2898118 
29*0344 

2942457 
2961458 

2986348 

100H128 

►029799 

[0^1363 

1^72820 

1094! 7 2 

1154^0 

136563 

157605 

178545 

"99384 
2^0114 

240766 
1261310 

281757 
13OJI08 

3*2364 
342526 

382572 

1402458 
^42225* 

;441957 
1461573 



3tf8 



Artificial Numhers: Or^ 



f^ombers 

au 
9at 

190 
833 

a34 

»3< 

239 
34O 

«41 
944 
a4l 

247 
34B 
249 

253 

M3 

254 

256 

»59 
-260 

261 

363 

863 

264 
365 
i66 



34»3049 
3503^0 

3U10J4 
3560*59 
3579348 

5598355 

3617378 

3636130 
3654880 

1^73559 
3693159 
3710679 

3729130 

3747483 
3765769 

3783979 
38031 13 
3S30170 

3838154 

3S 56063 

3873898 
3891661 

3909351 

3936969 

39445*7 
3961993 

3979400 
399^737 
401 400) 

403 1 305 

4048337 
4065402 
4082400 

4099331 
4116197 
413^998 

4149733 

4166405 
4183013 

4199557 
4216039 

423*459 
4348816 



34*499^ 
3504419 

35*3751 
3M3006 

3563171 
3581353 
3600251 
3619166 

3638000 

36|67«i 

3<57$43J 
369401A 

371*536 

3730960 

37493 1<5 
3767594 

378579^ 
3803932 
3821973 
3839948 
3857850 
3875678 

3893433 
3911116 

3938727 
3946268 

3963734 

398in7 
39984^7 
4015728 
4033921 
4050O47 
4067105 

408409^ 
410102I 
4117880 

41 14674 

4151404 
4168069 

4184670 

4201208 

4*17684 

4*34^97 

4350449 



348694* 

3506356 

35*5«4 
35449Stf 
3564081 

3583156 
3603146 

3621O53 
36)9878 
3^58633 
3677285 
3695869 

3714373 
373*799 

375 i 147 
3769418 

378761* 

3805730 

38*3773 

3841741 

3859636 

3877457 
3895305 
3913880 

3930485 
3948018 
3965480 
3983873 
4000196 
4017451 
4034^37 

4068807 

4085791 
4103710 

4119563 

413^350 

4153073 

4169733 

41863*7 
430*859 
43193*8 

4235735 
4*5*080 



I 

8488887 

35083V3 
3537612 

3546845 

lf«5994 

3585059 
3604040 

}6**939 

8641756 

366O49* 

1<79147 
8^72j 

3716319 

3754637 
3752977 
3771240 
3789427 

3807538 

3835573 
3843534 
386I421 

3879235 
389^975 
3914644 
3932341 
39497^7 

3967233 

3984608 
40019*); 
401917* 

40363 C3 

4053404 

4070508 
4087486 

4lo4]l»8 
4121244 

4138035 

415474* 

417*394 

4187983 
4204509 

4230972 
4237372 

42n7l3 



3490852 
3510328 

3529539 

3548764 

3507905 
35H6961 

3*>5934 
36M825 

3643633 
3663361 

I68100B 

369957^ 
3718065 

3736475 
3754807 

3773Q62 

379*241 
5809345 

38*7373 
3845326 
3S63306 

38810I* 

3898746 
3916407 

3933997 
1951516 

3968964 
3986343 
4003^53 
4030893 
4038066 

4O55171 
407**09 

4089180 

4I06085 

4132925 

4139700 

4156410 

4173056 
4189638 
4*06158 
4233614 

4*39009 I 
4255U2 I 



Logarithms ^to 2^^.) 



d 

> 



Natural 
Numbers. 

224 

225 

227 
228 
229 
230 
2?l 
232 
233 

234 

238 
239 
S40 
241 
242 
243 

244 

245 

246 

247 

249 
250' 

in 

252 
254' I 

257 
25« 

2$9 
260 

*2fi3 

»<5 



II 



•^^^ 



349*775 
3511163 

353I4€5 
35506Si 

3S69813 
3588861 
3^cJ7827 
3626709 
3<54S5io 
3664130 
3682869 
370142* 
3719909 

37383*1 
3756636 

3774884 

5793051 
3811151 

3829171 

3847117 
386495P 

388^789 

3900515 
3918169^ 

3935751 
3953264 

3970705 

3988077 

4005380 

4012614 

4039780 
4056878 
4073909 
4090874 
4107772 
412460^ 

4141374 
415S077' 
-4174717 
4191193 

4207806 

4»i4*5r 
4240645 



3494718 
3514093 

3533391 
3552599 
3571723 

J59o76i 

3609719 
3628503 

3647 j86 

3666097 
3684718 

3703280 

572*^53 
5740147 
3758464 

3776704 
3794868 

3812956 

3830969 

384S908 

386677} 

5884565 

3902284 

39 1 99 3 1 

5937506" 

3955011 

397244^ 
3 9898 I.I 

4007106 
4024^33 
4041491 

405^8584 
4075608 

4092567 
4109459 

41^6^85 

4143047 
4196744 

417^77 
1 419^947 
4109454 
4&1589S 
42422B1 

4158601 



3496660 
3516031 

35353»<^ 

35545M 
3575630 

359£66i 
3611640 
3630476 
3649160 

3^79^4 
3686587 

3705131 

3723595 

3741983 
376029(2 

3778514' 
379668P 

38147^1 

3832766 

3850698 

3868555 
3886340 

390405* 

3939260 
3955758 

597418^ 

5991541 

4008831 

4026052 
4043205 
4060189 
4077307 

4094259 
4111144 

4127964 

4144719 
416 1410 

4178937 

4194601 

4111101 

4227559 

4245915 
4260130 



8 

34PWbi 
3517963 

3557139. 
355543P 
3575537 
3594550 
361350P 
3632358 

3*51154 
3459830 

368^445 
37069^1 

372543^ 
3745817 
^761118 

5780343 

3798492 
3816565 

38345^53 
3S51487 

3«70337 
3888x14 

3905819 
3923452 
3941013 

395«5P4 
3975924 
3993175 

4-^10557 

4027771 

40449*^ 

4061994 
4079005 
4095950 
41128.29 
4129643 

4146391 
4163P76 

4179695 
4195254 
4212748 
4229(80 
4245550 
^4261858 



^69 



3500541 
351^895 

3559^^2 
355«34> 
3577441 
3596458 
3515390 

36H239 
3553007 

3671695 
369030^ 

3708^30 
37^7279 
3745551 
3763944 

3781 i6j 

3800303 

S8t83di8 
38363559 

5854275 
3872118 

3889888 

5907585 
3925211 

3942765 
3960249 
3977661 

3995007 
4011181 

4029488 
.4046617 

4063698 
4080703 

4097641 

4I145M 
4151520 
4148063 
4>54j|4i 

4181155 

4197^05 
4114294 
4230I20 

4247183 
1 4263486 



Bbb 



37© 

Natoral, 

Number^ 

967 

268 

269 

270 
271 
173 

t7^ 

a7f 

»7^ 

277 

t78 

«79 
980 

x8i 

8S2 
tS} 

2S4 

386 

2B8 

»89 

290 

£9i 

292 

293 

<H 

295 
196 

297 

298 

2^9 
3C0 
301. 

305 : 
304 
30$ 
30^ 

307 

308 

309 
310 



Artificial Numbers : Or^ 

3 



4265113 
4281348 
4*97523 
4313658 

4329693 
4345689 
43^61626 

437750^ 

4393 J»7 

440909 » 

4424798 
4442448 

44S6042 
4471580 
4487063 

4$o«49i 
4517864 

4533*83 
4548449 
4563660 

4178819 

45W9^^ 

4608978 

462J9II0 
4658930 
4653818 
4668676 
4683473 
4698220 

4711917 

4727564 
4742«63 

475^712 
4771212 

4785665 
4SOOO69 
4814436 
48*8736 
4842998 

4857214 
4871384 

4885507 

4899585 
4913617 



4266739 

4282968 

4*99137 
4315246 

4331295 

4H7285 
4363217 
437909<3 
4394906 
4410664 
4426365 
4442010 
4457598 

4473131 

4488608 

450403 1 
|4W9399 

4534712 

4549972 

4565179 
4580332 

♦595433 
4610481 

4625477 

4640422 

4655316 

4670158 

4684950 

469969 L 

4714384 
4729027 

47436*0 
4758164 

4771660 

4787108 

48OE5O7 

4815859 
4830164 

48444** 
48J8633 

487*798 
4886917 

4900990 

4yi50i8 



426836$ 
4284588 
4300751 
43 1685^ 
4332897 
4348S81 

4564807 
4380674 

4396484 
4412237 
442793 2 
4443571 
4459,154 
4474681 

449015 J 
4505570 

45*093* 

453624* 

4551495 
4^66696 

4581844 
4596940 

4611983 

4626974 

46419U 

4656802 
4671640 
4686427 
4701163 

47*5850 
4730488 
4745076 
4759616 

4774107 
4788550 

480x945 
481729* 

4831592 

4845845 
486p052 
4874212 
4888316 

45^01395 
4916418 



4269990 

4286207 
4302364 
4318460 

4334498 
4350476 
4366396 

4382258 
4398062 
4413809 

44*9499 
4445^2 
4460709 
4476231 
4491697 
4507109 
4522466 

4537769 
45530IS 
4568213 

458^356 

4598446 
4613484 

4628470 
4643405 

46583S8 
46731^0 
4687903 
4702634 

4717317 
4731549 

4746533 
4761067 

4775553 

4789991 
4804381 

4818724 

4833019 

4847268 

4861470 

4875616 

4889735 
4903799 
4917818 



4271614 
42875^25 
4303976 

4320067 
4336098 
4352071 
4367985 

4J83.841 
4399639 
4415380 
4431065 
4446692 
4462264 
4477780 
4493^241 

450S647 
452J998 
4539196 

4554540 
4569731 
4584868 

4599951 
4614985 

462996$ 

4644895 

4659774 
4674601 

468937P 
4704iojf 
^718782 
4733410 
4747988 
47^62518 
4776999 
4791433 
4805818 
4820156 

4834446 
48486^0 
48628^8 
4877039 
4891144 

4905«P3 



Natural 
Numbers. 

267 

26S 

269 

270 

271 

272 

274 

• *7J 
276 

277 
278 

279 
280 
281 
282 
283 
284 
28S 

285 

287 
288 

289 
290 

^91 

392 

?93 
294 

296 

297 
298 

?99 

.30P 

5PI 
302 
303 

30^ 

306 

307 
308 

3P9 



4J7323S 
428944} 
4305588 
432^673 
43376i>8 
435 K65 
4369573 

4585423 
440iai6 

4416951 

4432630 

444.8^ 5 i» 
4463818 

4479329 
4494784 
4510184 
4525531 
4540823 
455<^o6i 
45712^6 
4586378 
4601458 
4616486 
463146J 
4646386 
4661259 

467608 J 
46908^3 

4705575 

4720247 
4734I70 

4749443 
4763968 

4778445 



Logarithms 

4174861 
4291060 

4307499 

432327' 

4339*-9« 
43^5258 

4371 i6j 

4387005 

44Q2792 
4418522 

4434 '95 

414981 » 
44<^$372 

4450877 

4496326 < 

4511721 
45 2706 i 

4542349 

4557582 

457*761 

4587S89 
460^963 

4617986 

4632956 

4647875 

4662745 
4677560 

469^3*7 
4707O44 

47?i7M 

4736329 

4750898 
4761*18 

4779890 



(^to .31QP.) 



4792873 I 4794313 



4807254 
4821587 

4*3587 3 
4850112 

4^6430S 

487845 » 
489255? 

4906007 
491061.6 



4808689 
4813018 

4837199 

4&SI533 
48.65711 

4879863 

4893959 
4908009 
4932014 



<.. 



4276484 

4292677 

4308809 
43.24883 
4340896 
455<^35i 

4372748 

4388^87 
4404368 

44*0092 
4435759 

445*370 
4466915 

44^2424 
449786? 
4513258 
4528595 
4545875 
4559102 

4574277 
4589^99 
460446$ 
4619485 
4654450 

46493<54 
4664227 

4679039 

4693801 

4708513 

4723175 
f7j7788 

4752^52 

47^867 

4781334 
4795754 

48 10 I 24 
48 24448 
4838725 

4851954 
4867Mji 

488|»7S 
4895366 

4909412 

49254n 



8 

4278106 
4294293 
4310419 
4316487 

434^494 
4558444 
43743U 

4390167 

4405943 
4421^6; 

4437322 

445^928 
4468477 

448397 » 

449^410 

45 « 4794 

4530124 

4545400 

4560522 

457579J 

4590908 

4605972 
4610984. 

4635944 
46$o853 
4665 7 11 
4680548 

4695275 
47099*2 

4724639 
4739H7 
4753806 
4768316 

4782778 

479719* 

48U559 
4825878 

4840 X5Q, 
4854375' 

4868554 
488168$ 

489W75 
49108 I4 



371^ 



4279727 
4295908 

4312019 
4328090 
4344692 

4360035 

43759i0 

439»747 
4407517 
44*3229 
4438881 

4454485 
4470019 

4485 5 n 
4500951 

4516329 

4531654 
4546924 

45^2H2 

4577505 

4592417 
4607475 
46224S2 

4637437 
4652341 

4667194 
46S 1996 

469<S748 

47»«41o 

47i6jo2 

4740705 

4755^59 
4769765 
4784222 
4798631 

48 12993 
4817307 

4841 5 74 
4855795 
4869969 

4884097 
4898179 
4911116 
4926107 I 



Bbb 31 



HaarM 
MmiBCfs* 

313 
313 
3U 

116 
317 

319 

310 

321 

322 

3«3 
JU 

326 

3«7 
328 

3S9 
336 

33 1 
33^ 
333 

334 
?31 

J38 

3<9 

340 

34 1 

343 
344 

348 
34"9 
250 
3^^ 

3^^ 

144^ 



Artificial Numbers 
i 



4927*04^ 

494 '54^ 

4955443 
4969296 

4983166 

499^871 
5010^93 

5024271 

5057907 
505 1 500 
5065050 

5078559 
509202$ 

5105450 
5118^4 

515*176 

5'45478r 

5158738 

5171959 

5185159 
5 198 280 

5211381 

5x14442 

5237'4<55 
5*5044! 

5a'6J393 
5276299 

5289167 
5301^ 

5314789 

y 5^7544 
5340^61 

535*941 

53*5584 
yi78i9i 
5J9076I 
5403195 

5415^92 
5428254 

5453«>7i 
54^5427 
5477747 



49«9ooo 

4942938 
4956851 

4970679 

4984484 
4998245 
$011962 
5025637 

9039268 
^os*857 
506640 } 
5079907 

5C93370 
$166790 
51*0170 
5I3J508 

5 1 46 805 
5f6oo6& 

5 « 73*79 
5186455 
$199592 
5*12689 

521574'^ 
1^«765 
5*5*744 
5264685 
1277588 

5 29045 i 

53<^3278 
5516066 

J3«8»17 

5341531 
J 354267 

5J66847 

5J79450 
51920^6 
5404546 
541/040 

54*949^ 

>44i9*< 
5454368 
5466660 

54*8977 

549 '259 



3936396 

49445*9 
4958*18 
4972062 
4985862 
4999619 

'501533* 
5027001 

50406*9 

5054*1? 

5067755 
5e8tf*5f 

509471? 
5168150 

5i2r^5 
5134840 

5148113 

5l5l?86 

5174598 

5187771 
520090 J 

5113996 

j**7a50 

5340064 

S253<'40 

5265977 

5^78876 
5291736 
5304558 

5317^43 
5336O90 
554*800 

5^55473 
5368109 

5580708 

5393*71 

54OJ797 
5418288 

543074* 
5443 l^t 
5455545 

54<^7894 

5480207 

54Slti48ft 



; Orr 



493«79« 

49457*0 

4959604 

497*444 
4987240 

500099* 
5014701 
50*8f66 
5041989 
505^569^ 

5069107 
50^2605 

569*>57 
5109469 
5 I 2284 I 
5'36i7i 
5 1 49460 

,516*709 
5175917 
5189086 

5*02214 
5215303 

5228)53 
5241564 

5254355 
5267x69 

5*80163 

529^020 

556f839 

5318619 

55313^3 
5344069 
535^758 

536937^ 
53819^6 

53945i^£ 
5407<>48 
J4i951< 
543 1986 
5444401 

5456781 
5469126 
5481436 



4935186 
4947 i»df 
4960990 
4974825 
4988617 
500236^ 
5016069 
5029731 

5043)49 
5O5 69X J 

5070459 

508595* 
5097400 

5I1080S 

5224175 

5137501 
5*50787 

5164031 
5177*36 
51904010 

5805515 
521661^ 

52296^6 

5242663 

5*55651 

5268560 

5*81451 
5*94303 
5307118 

5319895 
535^635 

5545338 
535»oo3 

5376651 
5583223 

5395779 
5408*98 

54*6781 

5451**9 

5445641 

5458617 

5470359 
5482665 



JMLlUXi.U0^n7 



ffatiiral 

an 

31S 
3»o 
3tft 

3aa 



$^ 6 



I 



3 JO 

J3' 

3.3a 

333 

334 

331 

33^ 

337 

33« 

339 
340 

34» 

34» 

343 
344 
341 
34^ 
147 
34« 
349 
350 
35« 
35a 
313 

5! 



4Wf9*> 

49«»1W 
49**37 f 

49*9994 

fQ^J737 
5017437 
$031094 
50447^9 

50718 to 

SOftj«97 
10^743 
1119147 

11*11 10 

init3«^ 

1M»I»| 

5i5$3VI 
1t7«5)4 

II9171I 

ftd48f1 

1«»79»^ 
1*3095* 
1a439tft 

1*549*1 

1**9^51 
5tS»738 

J«911«y 
HO«39« 
5f*tt7l 

J333907 

f34Mod 

$ 339^^7 
53?ift?i 

53»44»» 
139703* 
$4^14« 

14*9^^ 

543447* 
5446tao 

5459t$3 

5471191 

54i3«94 
<do6i6t 



49?t9759 
4949^90 

4963nr<^t 

49775^7 
4i9t570 

loi^dof 
1^3*458 
104^68 
5059615 

5Q73tto 
1086644. 
I1QO085 

5113485 

5136844 
5 140 16& 

5 « 53439 
1166676 

5 »79«7« 
5 193018 

5»tf»41 

y 119393 
1*3**60 

5»41»19 

f*1«»l9 

5*71141 
5384oa4 

1*9*849 

5309677 
53*144* 
5331«79 
5347874 
534053* 
5373MI 
53«1737 

5398*«* 
54I0798 

J4»l*74 
5433714 
5448 119 

54*04*9 

147a«*3 

548*1*3 

14973*7 



493n» 

491 »27^ 

49tfS»4^ 

497^*7 
499*7f* 
SOQ648I 
5QaQt7« 

5940^416 
1049990 

1oi74Stt 

5<»»7990 
5*01^7 

5*U»93 
5118178 

1 14149 1 

5H4764 

3«$7997 
5181189 

1»9434* 

5**7451 
5110518 

^1335** 
5t465f? 

1*19513 

1«7*43l 

5**133* 

51981 11 

531^955 
53*37*1 
1336450 
5349*41 

1361795 
53744«3 

1386994 
5399538 

141*047 
14*4H9 

54)^16 

1449318 

54*17*4 
147405$ 

54**551 
549**1* 



8 

49989*^ 

4952^667 

48{^1*d 

4980347 
4994^)1 
5Q07&5* 

5a*ijr39 
50j5lSri 

1043785 
5062344 

5Gi7$86o 

5089535 
5loi7<S8 

1 116160 

5**»5»i 
5142810 

1 lS*o89 

5 1693 iS 
5182506 

519^*51 

5108764 

52*1833 

5*34**3 
5*47*54 
5*60807 

52737^1 
5286596 

5299434 

5312234 

5324996 

13377*1 
5350408 
5363059 

5375*72 
5388250 

5400791 
1413^ 
54*17*5 
543*198 

545059* 

5462958 

5475*** 
148757* 
5499*^* 



%73^ 



4940 If 4 
49l40f6 

4967913 

498*7*7 

499S496 

1009222 

50119051 

503*545 
5050141 

50*8*97 
50771 10 
5090680 

5 104109 

5117497 

51 30*44 
1' 44149 

1« 174*4 

5170*39 
5l8^3| 

519*9*» 
5210073 

5233138 

53*6164 

5*49151 
5262100 

S 275010 

5187881 

530071* 

531351* 

5326^70 

5338991 
5311<*75 
53643*8 

537*932 
$3*910* 

5402O43 

5414144 
1427010 

1459439 

1451*34 

1464193 

147*517 
548**06 

5501060 



374 

Nacoral 

Sotnberi. 

355 

35^ 

557 

35« 

3S9 
360 

361 

563 

3^3 
364 
3^5 

368 

369 

37* 
373 
374 

376 
378 

380 
381 

381 

383 

384 

385 

386 

3«7 
|88 

389 
390 
3^1 

39« 

393 
394 

39J 

396 

397 

M 



Artificial Numbers : Or^ 






5503183 
5^14500 
5526682 
5538850 

$550944 
55^3025 

5575071 
5587086 

5599^^^ 
561 1014 

56229^9 
$634811 

5646661 

5^58478 
5670264 

5682017 

5^93739 
57«54i9 
5717088 

5718716 
5740311 
5751878 
57^^3413 

57749x7 
5786391 
5797836 
5809150 
5810634 
5831988 
5843311 

5854617 

5865873 
5877"0 

5888317 

589949^ 
5910646 

59&r768 
5932861 

59439*5 

59549^2 
5965971 
5976951 
59875>oy 
19p883r 



5503507 
55»572o 
5527898 

5540043 

555^M4 
5564131 
5576175 
5588^85 
5600262 
5612207 
5624118 

5635997 
5647844 
5659658 

5671440 
5683191 
5694910 
5706597 

5718252 

5719877 

5741471 

5753033 
5764565 

5776057 
5787538 

5798979 
58*0389 
5811770 
5833122 

5844443 
5855735 
5866998 
5878232 
5889436 
5900612 
5911759 
593287S 
5933968 
5945030 
5956064 
5967070 
5978048 

5988999 
5999911 



504730 

516959 

529114 
541156 

553362 
565437 
577477 
589484 
601458 

613399 
615308 

637183 

649027 
660838 

672617 

684564 

696080 

707764 

719416 

731038 

742628 

754188 

765717 

777»»5 

788683 

8ooiti 
811529 

822907 

834155 
845574 
856865 

868113 

879353 
890555 
901728 

91287J 
923988 
935076 

946135 
957166 

968.169 

970145 

990091 

6poiOf3 



505952 
518158 

530330 
542468 

554572 
566643 
578680 
590683 
602654 
614591 

626497 
638569 
650209 
662017 

673793 
685137 

697249 

708930 

720580 
732I98 
743786 

755342 

766868 

778363 
789828 

801263 
811668 

824043 
835388 
846704 
857990 

869147 
880475 
891674 
901844 

9*3985 
925098 
936183 

947?39 
958268 

969268 

980241 

991 1 86 

6ooibio^ 



507174 

519377 

531545 

543680 
555781 
567848 
579881 
591881 
603849 

615784 

627685 

631^55 
651392 
663196 

674969 

686710 

698419 

710097 

7*1743 
733358 

744943 
756496 

768019 

779511 
790973 

801405 
813807 

815179 
836521 

847834 

859117 
870371 
S81596 
S93792 
903959 
915098 
926908 
937290 
948344 
959369 
9703^7 
981336 

99»»79 
6oo3f>if 

■ V" V 





Logarithms {to jpSo.) 


375 


Nitanl 


f 


6 


7 


8 


9 


Numbtrs. 












35V 


1108396 


1509618 


1110839 


lluoiS> 


5fllt89 


3!« 


S^J0S9J 


552181J 


55»303i 


J5^4HB 


5i>!4i5! 


357 


JSJJ7«0 


5533975 


55^11^9 


55364O3 


5!37«I7 


318 


5in'*9» 


554<Slo3 


S547314 


5548534 


5149735 


319 


is 56989 


5558197 


51 59404 


55*o«ii 


554.8,8 


3«0 


51*9353 


5570^17 


II714S1 


5572661 


5i73!<S9 


3<I 


558108} 


S5822S4 


S58348I 


1584686 


5(»sS86 


i«. 


S 19 3080 


5594178 


559547'S 


519667, 


S197870 


3'3 


5605044 


5006239 


5607433 


I608627 


5609810 


3l!4 


S61697I 


5618167 


J61P558 


5620i48 


!«2i>39 


3«S 


S6tS87i 


5630062 


I6312JO 


563:437 


5«33«'4 


3M 


J 640740 


5641925 


5643109 


5644293 


5«454?7 


167 


«fi?'573 


J65375S 


56^4936 


5*56117 


5657298 


3«8 


1*6437 i 


^i6^W 


S66«7Ji 


1 667909 


5««90!7 


3 £9 


J«76M4 


5«77320 


5678494 


5679669 


54«o84i 


370 


StfS7R8a 


5689054 


56902J* 


5691397 


5T9'5«8 


371 


5*99588 


5700757 


570'9i« 


5703094 


S7u4Ma 


37* 


S7ii2*3 


57i34>S 


5713594 


5714759 


5715924 


373 


6721905 


57»4069 


1725*3* 


5726393 


S7'7!55 


374 


S7I45'8 


5735678 


5736837 


573799« 


57J9154 


37S 


1746099 


5747»s6 


5748412 


5749J'58 
5761 109 


5750723 


37< 


5757*50 


5758803 


57I99S<S 


5762261 


377 


-1 7159169 


S770320 


577«470 


5772620 


577)769 


378 


5780659 


5781806 


578295J 


57S4100 


S7»5!46 


i79 


579111! 


5793»6i 


5794406 


5795550 


5796693 


380 


5803547 


5804688 


58058*9 


5806969 


5903110 


381 


58 H 


l8ifio84 


5817222 


S8l8jjB 


581949; 


391 


58! 4 


5827410 


1828585 


I829719 


58)0814 


383 


58: 14 


5838786 


5839918 


1 841050 


5»4>i8i 


j«{ 


58. Ij 


5850093 


5^51111 


585*35' 


5«5)479 


38s 


58< t4 


586137,0 


5862496 


186362a 


5864748 


38« 


J871495 


1872618 


5873742 


5874865 


5875987 


387 


58811717 


J&8j8j8 


1884958 


1886078 


5887198 


38! 


5893910 


I89J028 


5896145 


5897262 


5898)79 


!«» 


^905075 


5906189 


5907304 


1908418 


5909532 


390 


I91S110 


59' 7322 


S9184J4 


5919546 


5910617 


391 


1927)1 3 


5928427 


,1939JJ6 


59J0544 


591175! 


39«' 


5938397 


S9J9503 


5940609 


59417'S 


194(820 


39) 


.1949*47 


J95055' 


5951654 


5952757 


S953860 


394 


■5960470 


5961571 


1962671 


5963771 


5964871 


1 395 


I9714.S5 


197256} 


5973*60 


iy7475 8 


5975855 


1 39< 


198143* 


I98J527 


5984833 


59817" 


1985811 


h 


5993371 


5994464 


,5995556, 


5996648 


1997739 


«o:)4l8l 


6goJ37,J 


'6oo«i6i 


60075^1 *ot'164'> ' 



37* 


Ar^cial Numbers 


; Or, 




Mitral 






«~>ten. 










4 


in 


00Q971, 


foio8i7 


Soirjoj 


6013993 


6ojioBi 


4AO 


0OIo6o« 


6011685 


6022771 


6023856 


6094941 


401 


«°!H44 


6033^7 


6033609 


60H693 


6035774 


4M 


A043iei 


«04i34i 


60444 n 


6045500 


6046J80 


4ai 


80^3050 


60S4128 


00 J J -01 


6055382 


60573 5 9i 


40* 


«M!«I4 


6064888 


6065963 


6067037 


60(S8i,i 


OS 


8074150 
60!iMo 


607J633 


6076694 


6077766 


6078837: 


4«S 


60865 JO 


6087390 
609807I 


6088468 


608953^ 


*>' 


2??^; 


6097011 


6099144 


6cOojia 

6110857! 


^0? 


6107666 


61087 JO 


6109794 


4»9 


*I17»J) 


6li«29^ 


6u93l6 


6130417 


611I478J 


4:10 


«137«39 


6i388»8 


61J9917 


«Uioi5 


6i3i073r 


4»l 


4I3B418 


6 '39*7 S 


6l40r3l 


6UI587 


6142645 


■41* 


0148972 


6ttcx»6 


6i(io8o 


6IJ1IJ3 


6153187 


4I! 


JliP^oi 


«l6o«i 


6l6i6oi 


6i626{4 


6163709 


414 


5170OO) 


6171052 


6171101 


»»73U9 


6 174 197 


■i;i 


(SiMSi 


6lBifi7 


6i83,7j 


618361, 


618456^ 


416 


6190955 


619I977 


6193021 


6194064 


6195107 


^<i 


£itoj3<5o 


6iot40t 


630 J 44 J 


6104484 


62055 M 


6x11763 


6312802 


6213840 


*SI4B79 


6215917 


415 


tfaaai^o 


6337177 


6324H3 


6235149 


6Z26184 


4M 


62J249J 


6233127 


63 J 45 60 


*SJSS94 


6136637 


-*«I 


0141S31 


62438,3 


62448 84 


6245915 


6*4694i 


472 


6223154 


62U153 


6255182 


635(5211 


6357139 


4>3 


^63404 


62644io 


6365457 


6266483 


6a67yo# 


4«4 


6273«.S9 


6374S83 


62757^7 


6276730 


6277754 


-•!1 


628J889 


61&4PIJ 


6j859?3 


6286954 


■S^oli?! 


4M : 


<S2p4096 


6291115 


S^iH 


62971 « 


"'J 


6304279 


6305196 


rjo6ji2 


6307329 


6308345 


418 


<31443S 


/ a 


6316467 


63«748«: 


6318491 


419 


63U173 


1 5 


6326597: 


■6317609 


6328620 


4)o ! 


633468^ 


' 4 


■6136704 


'*3377iJ- 


6333723 


4J> : 


634477! 


< D 


.6346788 


6347755 


6J49801 


-4?' 


fia^-a-,?, 


' 'S; 


6356848 

.<j668a4 


635785 t. 
63 '5/88 7 ' 


6ji885ir 


■4!3 ■ 


i. 9 


1 3- 


63688!^ 


-4!4 r 


6 7 


18, 


63768^8 


.6377898' 


6378898 


■4!] ' 


^ 3 


' - . Vl 


e^ssasj* 


0387837 


63888*4 


4!« 


! , ' 


639(861 


■63968J71 


6597851 


6358847 
6408788 


-4!; i 

■45* j 


6474814' 


64Of80S 


6406801: 


640779J- 


.441+741 


.6415733 


6416714! 


64177H 


6418705 


^S9 


-^424645 


-fi4l|6j4 


642661^ 


.6437611 


64»«6ci 


440 * 


=<S4)4<»T 


■6435fM. 


■iS436(Qd 


Hi74i7 


64r847t 


441 


4444386 


«44«M 


6446ts^ 


6447329 


4448313 


-•c^ 


tlliwj 




^4g 


±^S*iii 



N. rabcif. 

399 
400 

4oI 
403 
io5 
404 
405 
406 
407 
408 
409 
4IU 
411 
411 

41? 

414 

4«5 

416 

417 
418 
419 

420 

4«l 

422 

4n 

424 

42$ 
426 

427 

428 

429 
450 

431 
41 i 
4?3[ 

494 
435 
415 

41 7 
43^ 
419 
440 

441 
442 



5 



Logarithms (^to 4429.) 

8 



6oi5i6'8 
6o266i5 
6036855 
6047659 

6058415 
6069185 

6079909 

6090605 

610127.^ 

6111921 

6122539 

6133132 
6143698 

6154240 

6164755 

6175245 
6185710 
6196150 
6206565 
6216955 
6227320 

623-7663 
6147976 

6158267 

6268534 
6278777 

628^996 
6299190 

6309361 
6319508 
6329632 
6339732 
6349808 
6359861 
636989t 
6379898 
6389882 
6399842 
6409781 
6419696 

6429589 

6439459 
6449307 

<^4S9»31 



6016255 
6027109 

6C37937 
6048738 
6059511 

6070 2 59 

6080979 
6091674 

6102342 
6112948 

6123599 
6134189 

6144754 
6155292 
6165805 
6176293 
6186755 
619719J 
6207605 
6217991 
6228355 
6238693 
6249006 
6259295 
6269559 
6279800 
6290016 
6300208 

6310377 
6320522 
6330643 
6340740 
6350<?I4 
6360865 

6370895 
6380897 

6390879 

6400837 

641077? 

6430577 
6440445 

6450291 

6460114 



6017341 
6028193 
6059018 
6049816 

6060587 
6071332 
6082050 
6092742 
6103407 
6114046 
6124660 

6155247 

6145H09 

6156345 
6166855 

6177140 
6187800 
619S235 

6208645 
62 19:) 30 
6229300 

6239725 

625C036 
6260322 

6270585 
6280823 
6291036 
6301226 
6311992 
6321535 

^33'653 
^141749 
6351820 

6361869 
6371894 
6381896 
6391876 
6401832 
6411765 

6421676 
6431565 
6441430 
6451274 
6461095 



6018428 

6029277 
6040099 
6050895 
6061663 

60724O5 
6083120 
6093809 
6104472 
6115 109 
6125720 
6136304 
6146S63 

6«57397 

61679^? 
6,781^7 

6188845 
6199177 

6209684 
6220067 

6230424. 
6240757 
6151066 
626f ]^o 
6271610 

6281845 
619^057 
630224.; 
6312408 
6322548 
6332664 

^14'7S7 
6352826 

636287^ 
6372895 
6382895 
6392872 
6402826 
6412758 
(5422665 

6452552 
6442416 

645J257 
6462076 



377 

9 

6019514 
6030361 
6041180 
6051973 
6062738 

6073478 
6084190 

6094877 

6105537 

6116171 

6126779 

6137^61 

6147918 

•6158449 

6168954 
6179434 

6189889 

62C0319 

6216724 

6221104 

6231459 
6241789 
6252095 

6262377 
6272634 

6282867 

6293076 

6p3 262 
6313423 

6323560 
6^33674 

6343765 
6353852 
6J63876 
6373896 
6383894 
6393869 

6403820 

6413749 
6423656 
6433540 
6443401 
6453240 

<^453QS7 



C c c 



378 



Artifiqial Numhers : Or^ 



K^rurai 
Nuinberk 

443 
4i4 

44 S 

446 
447 

448 

449 
450 

451 

45 a 
4S3 

454 
415 

45^ 
457 
458 
459 
460 

461 
462 
463 

4^4 
465 
466 
467 
468 
469 
470 

471 
472 

473 
474 

475 
476 

477 

478 

479 
480 
481 
482 
483 
484 
485 
4S6 



64640^7 
6473830 
6483^00 
6493349 

6503075 
6512780 
6522463 

6532115 

6UX765 

655*384 
6560981 

6570559 

6580114 

65S9648 

6599162 

66o8d55 
661S117 

6627578 
66370Q9 
664^420 
665^810 
6665180 
6674530 
6683859 
6693169 
6702459 
6711728 
6720979 
6730209 
6739420 
6748611 

6757783 
0766936 

6776069 
6785184 

6794179 

6803355 
681^412 

68214(1 

6830470 
.6839471 

6848454 

685 74 "7 
686^363 



6465017 

6474H08 

6484576 

64943** 
6504047 

6513749 
6523430 
6555090 
6542728 

6S52345 
6561941 
6571515 
6581068 

6590601 
6600112 
6609603 

6619073 
6628,5 1 i 

6637951 
6647360 

6^56748 
6666116 

6675463 
6684791 

6694099 

6703386 

6712654 

6721903 

6731,131 

6740540 

6749529 

6758700 

6767850 

6776982 

6/^6094 

6795187 

6804^6^ 

6813317 

6922354 

6851371 

6840370 

6849J5I 

6858313 

6867256 



6465997 

6475785 
6485551 

6495296 
6505018 

65U719 
65*4397 

6534055 

6543691 

6555506 

65618^9 

6572471 
6582013 

6591553 
6601062 

66 105 5 1 

6610019 

6629466 

66JS893 
6648399 

6657685 
6667051 
6676397 
6685723 
6695028 

6704314 

67155^ 
6722816 

675^053 

6741 2fo 

6750447 

6759615 

6768764 
6777894 
6787004 
6796096 

6805 168 
68142&2 
6823256 
6S32272 

6841^69 

6850248 

6859208 
6868149 



6466977 
6476763 
6486527 
64^6269 
6505989 
6515687 
6525364 

6535019 

6544653 
6554266 
6563857 

6575427 
6582976 

6592505 
6602012 

6611499 
6620964 

6650410 

6639835 

66492^9 
6658623 

6667987 
6677331 
6686654 

6695958 
6705242 
6714506 

67x3750 
675^974 
6742179 
6751565 
676053I 

676967^ 
6778806 
67 879 I 4 
6797004 

6806074 
6815126 
6824159 

6833173 

6^42168 

6851145 

6860103 
6869043 



6467957 

6477740 
6487502 

649724a 
6506960 

6516656 

6516331 

6535984 

6545616 

6555226 

6564815 

6574383 
6583930 

6595456 
66P3962 

6612446 
6611910 
6631353 

6640776 
6^50178 

6659560 

6668912 

6678264 

6687585 

6696887 

67061169 

67.15431 
671^673 

6733896 
6743099 
6752283 

6761447 
6770^94 

67^797 1 9 
67H8324 

6797912 
6806980 

6$i6o30 
6825061 
6^3^073 
684206^ 
68531041 
6860998 
6869936 



Natural. 

Nombcrs 

443 
444 
445 

447 
44« 
449 
450 
45 » 
45^ 

4^4 
451 
456 

45r 
458 

459 
460 

45i 

4<^2 

463 

464 
465 
466 

4^ 

4(S8 
469 

470 

47^ 
47* 
47 J 
474 
47^ 
47« 

477 

47« 

479 
480 

481 

482 

483 
484 
485 

486 



Lcigarithms 
5 i 6 



( to 48^9. ) 



<3468936 

<^47«7i8 
^5488477 

6498215 
6507930 

65176*4 
6527297 
6536948 

6546578 
6556186 

6565773 

6575339 
65^4884 

6394408 

660391 X 

6613393 
6622S55 
6632296 
6641717 
6651117 
6660497 

6669857 

6679197 

6697816 
6707096 
67*6356 
6725^96 

6734817 
6744018 

67^3200 

676236^ 

677 505 

6780629 

6789734 

6798B19 
6B078S6 

6816994 

68*5965 

<8}4973 

68439^5 
6852938 

68618*92 
68jro8tt 



6469915 
6479695 

648945* 
6499187 
6308901 

6518593 
6528263 

6537912 

6547539 

6557145 
65667JO 

6576294 
6585837 

6^95359 
6604860 

6614340 

6613800 

6633259 

664^658 

6651056 

6661434 

667079* 

66801J0 

6689447 

6698745 
6708013 

6717281 

6726519 
6735738 

^744937 
6754117 
6763177 
6772418 
6781540 
6790643 

67997*7 
6808792 

6817838 

"6826 86 5 

^835873 
6844863 

6853834 
6861787 



6470894 

6480671 
6490426 
6500160 
65O9871 
6519561 
6529229 
653h876 
6548501 
6558105 

6577250 
6586790 
6596310 

6605809 
6615287 

6624745 
6634182 

6643559 
6652995 
66623 7 I 

66yij27 
6681062 
6690378 

6699674 
6708950 
6718206 

67274** 
6736659 

6743856 

6755034 

676419* 

677333* 
678*45* 

679«55» 
6800634 

6809697 

681874I 
6827766 

6836773 
6845761 

6854730 
^863681 
687*613 



8 

6471873 
6481648 
649 I 40 I 

6501132 
651084] 

6520528 

6^3 Ji9< 

6539839 
6549462 

6559064 

6568645 

6578205 

6587743 
6^97261 
6606758 
6616234 
662569O 
6635125 
6644539 

6653933 
6663307 
667*661 
6681995 
6691308 
^700602 
6709876 
6719130 
6728365 
6737574 
6746775 

67559^* 
6765107 

^774^4 
678336* 
679*461 
6801541 
681060* 

6819^45 
68*8668 

6837673 
6846659 

6855616 

16864575 
1 6^7^5<?6 



379 



6472851 
6482624 

<^49?375 
6502104 
6511S11 
6521496 

633 1160 
6540802 
6550423 
6560023 

6569602 

6579159 

65^8696 

6598*1* 
6607706 
6617181 

6626624 
6636097 

6645480 

665487«' 
6664844 

6673595 
6682927 

6692239 

6701530 

6710802 

6720054 

67^9*87 

6738500 

6747693 
67J6867 
676602* 

6775 «^7 
6784*73 

^793370 
6802448 
6811507 
68*0548 
68*9569 
6838571 
6847556 
68565*2 
68654691 

42^^ 



C c c ) 



3S' 



Natural 
Sumbrrs. 

4*7 
48S 
489 
490 
491 
49 a 
491 
494 
49 y 
4y6 

497 
498 

499 
500 

501 

502 
50? 

504 

5oy 

507 
508 

509 
510 

511 

51* 

514 
516 

518 

5?o 

521 
521 

52J 

524 

556 

527 

529 

^30 



Artificial Numlers / Or, 

3 



6875190 
688419^ 
68930^9 
690 I 96 I 
6910815 
6919651 
6928469 
6937269 
69460 s & 

6yu8i7 
6963s 64 
697229^ 
^98ioos 
6989700 
0998977 
007C97 
0156S0 

014305 

031914 
041 S05 

05C080 
Os86;7 
067178 
075702 
084206 
091700 
101174 
1096^1 
118072 
126497 

13490? 
143198 

i<l674 
160033 

168977 

176705 

1H5017 

195319 
201593 
209^57 

218106 
226339 

M4557 
2^59 



6876181 
O88508S 
6893977 

6902847 
691 1699 

69205^4 
6929350 
6938148 
69^6929 
695569* 
6964438 
65^73165 

6y«|876 

6990^69 

6999244 
0075)02 
016543 
025167 

033774 
041363 
050936 
059492 

06803 1 

07^555 
085059 

093548 

102020 
1 10476 
11^915 

127339 

M5745 

14413^ 
M^Sio 
160869 
169211 

*77537 
185847 

«94U2 
202420 
210683 

218930 
227162 
235578 
243578 



6877071 
6885978 

6894864 

6903733 
6912584 

6921416 

693023 1 

6939027 
6947806 

695^568 
6965311 

6974037 
6982746 
69914J7 
7000111 
7008767 
7017406 
7026^18 

7034633 
7043221 

7051792 

7060347 

7o688h4 

7077405 
7085908 

7094^96 
7102866 
7III321 

7119759 

7128180 

7136585 
7144974 
7'y?347 
7161703 

7170044 
7178369 
7186677 
7194970 
7203247 
7211508 

7219754 
7227984 
7236198 

7244397 



I 



6877^64 
6886867 

6895752 
6904616 
6913^)68 
6922298 
6931111 
6939906 
6948683 

6957443 
6966185 

6974909 
6983616 
6992305 
000977 
0096^2 
018269 
026390 
035493 

044079 
052649 

061 kO I 

069737 
078256 

0867^8 

095244 

105713 
112165 

120601 
129031 

"37425 
I45812 

154183 
162558 
170877 
I 79200 

• 87507 

'95799 
^04074 

212334 

220578 

S28806 
237019 
245^1^ 






4 

"6878855 
6887757 

6896640 

6505505 
6914352 
69*3180 

6931991 

6940785 

6949560 
6958318 
6967058 
6975780 
6984485 
6995173 
^001843 
'010496 
roi9i32 
r02775l 
ro36352 
'044937 

05 35^5 

'062055 
^070589 

079107 
^087607 
'096091 

104559 
ll^oio 

'21444 
129862 

r 58264 

14665O 

155019 

1^3373 
1717IO 

180032 

188337 
196627 
204901 

213159 
'221401 

229628 

^37839 





Logarithm^ 


t ito 5309O 


381 


f Satoral 
INumbcrf, 

4«7 


< 


6 1 T 1 


a 


5^ 


# 




w 




^ 


6179746 


6880637 


6881.528 


6882418 


6883308 


4S8 


6S88646 


68895:55 


689641 J 


6S913U 


6892200 


489 


6897517 


6898414 


6899301 


69001^8 


6901074 


490 


6906390 


6907275 


6905161 


6909046 


690993d 


491 


6915235 


69161I9 


6917002 


6917885 


6918768 


49» 


692406s 


6924944 


6925826 


6926707 


6927588 


493 


6932872 


693375* 


69346 i I 


6935511 


6936390 


494' 


6941663 


6942541 


6943419 


6944.297 


6945 174 


491 


6950417 


6951313 


6952189 


695 5od5 


6953941 


496 


6959193 


6960067 


6960942 


6961816 


6962690 


497 


696793 t 


6968804 


6969676 


6970549 


6971421 


498 


6976652 


6977523 


6978394 


6979264 


6980135 


499 


6985355 


6986224 


6987093 


6987963 


6988831 


500 


6994041 


6994908 


6995776 


6996643 


6997510 


50t 


7002709 


7003575 


7004441 


7005307 


7006172 


502 


7011361 


7012225 


7013089 


7013953 


7014816 


503 


7019995 


7020857 


7021719 


7022582 


7023444 


504 


7ot86l& 


7029472 


7070333 


7031193 


7032054 


505 


7037212 


7038071 


7038929 


7039788 


7040647 


50<J 


7045793 


7046652 


7047509 


7048366 


704922 J 


?o7 


70^4360 


7055216 


7056072 


7056927 


7057782 


508 


7062910 


7063764 


7064617 


7065471. 


7066324 


509 


7071442 


7072294 


7073146 


7073998 


7074850 


510 


7079957 


7080808 


7081659 


7082509 


7083959 


5«i 


7088456 


7089305 


7090154 


7091.003 


7o9«.85 1 


51a 


7096939 


7097786 


709863 J 


7099480 


7100327 


5»3 


7105404 


7106250 


7107096 


7107941 


7108786 


514 


7113854 


7 1 14698 


7115542 


7116385 


7117**9 


5M 


7122217 


7123129 


7123971 


7124813 


7125655 


5«« 


7130703 


7^31544 


713*385 


7133225 


7134065 


5t7 


7139104 


7139941 


7140782 


7141620 


7142459 
7150837 


518 


71474^8 


7148325 


7149162 


7150000 


5«9 


7155856 


7156691 


7t57527 


7158363 


71S9198 


520 


7164207 


7165042 


7165876 


71667 1 


7167544 


521 


7172543 


7173376 


7174208 


7175041 


7175873 


5«J 


7180863 


7181694 


7182525 


7183356 


7184186 


5»B 


71^9167 


7189996 


7190816 


7191655 


7192484 


5*4 


7197455 


7198285 


7199U1 


7199938 


7200766 


5a< 


7205727 


7206554 


7207380 


7208206 


7299032 


586 


7213984 


7214809 


7215633 


7216458 


7217:282 


527 


722222; 


7223048 


6223 87 1; 


7224694 


7**5517 


548 


7230450 


7231272 


7232093 


7232914. 


7233736 


1 5t9 


7238660 


7239480 


7240300 


7241120 


72419^9 


; ^s* 


7246854 


724767* 72484911 


7249109 '7*50127/ 



3B» 



Jfpi/itiul Numhtrs : 



Natural 

fl4 
HI 

M4 

n6 

J4t 

548 

14> 

5^«> 
55« 

51^ 
J1} 

154 
J55 
55<5 

560 
56« 

563 
564 
5^5r 

5^^ 

569 

570 
571 
572 

571 
574 




7^S*945 

73^9116 

72^7279 

7»7J4I3 

7«8j53« 
73^1648 

7*9^T45 
75078^3 

73M888 

7}«J938 

7ii9Wl 

73479*8 

7355989 
756)96j 

7371926 

73^9^73 
7387806 

7395721 

74<>j627 
7411516 

74»939i 
74S7«s ' 
74JJ098 
744*9}<J 

74^7*8 

7458552 
74665^ 

7474ns 

74818S0 
7489*53(9 

74973^ 
7505087 

75r»79f 
75*041^4 

75*8164 

753583' 

7^4^83 
7^51 i23 

7558749 
7^66361 

7573960 

7581546 

82U2. 



7*51763 

7259933 
7S68087 

7*84^49 

7492458 
7506551 
730B630 
7316693 

73«4749 

7332775 

7340794 
7348798 

735^787 
7}6476« 

73727*2 

7380667 
7388598 

7J9^5M 

7404416 

7411304 

7426177 

7418057 

7435881 

744571* 

745*5*9 

74591^32 
7467120 

7474895 
748^56 

7490403 
7498136 

7505855 
7513561 

752i»5t 

7528932 
75^6596 

754«4248 

795i^8d 

75595 fo 

7567122 

7574r«9 
7582304 

zmzL 



7252581 

7260749 
7266901 

72770|y 

7*i5l6l 

729i26« 

7301360" 
.75094^7 

73 "7199 
7325546 

7333578 
734159$ 
734^598 

75575«5 
73<55558 

73735*7 
738146! 
7389390 

7397 J05 

7405^06 

7413092 
7420964 

7428822 
7436665 

74444^5 

745'Jio 
7460111 

7467898 

7475672 
7483431 

7491177 
7498908 
7506626 

7514331 
7522022 

7529699 
7537362 
7545012 
75 "^2649 

7560279 
75^7882 

7575479 
7J83062 

759^32 



Or, 

7253 35^ 
7161565 
7269716 

727785* 
7285972 
7 29407 S 
7302168 

7310^44 
7318364 

7316350 

7334380 

754*596 

7350997 
7358383 

7366355 

73743*2 

7382254 
7)901 »2 
7398096 

7405995 
7413880 

742»7fiT 

7429607 

7437445r 
7445277 

7453091 
7460890 

74<'^676 
7476448 
7484*06 

7491950 
7499<^8i 

7507398 
7515100 

7522790 

7530466 
7438128 

7545777 
7553412 

7561034 

7568641 
7576257 

7583819 

7591188 



J 



4 

^254115 

7261360 

7^7*531 
7278664 

286784 
294888 

♦30^977 

f3i9'«)9 

♦327153 

♦355182 

t343"97 
♦551196 

t359t8l 

♦3<^15« 

>375i07 

>383Ch8 

7 590974 

>3 98*86 

^40^784 
Hf4668 

74^2^37 

^430^92 
7438232 
7446059 

7453*71 
7461670 

74<i9^54 
7477225 

7484981 

7492724 

'7500453 

7508168 

7515870 
75^t3t58 

75J1232 

7538893 

754654* 
7554178 

7561795 
7569402 
7576996 

7584577 
7552144 



Logflrithtus ^to 5749.) 



3«5 



N^taril 


5 


Nombers- 




«l 


7?$5033 


S3* 


7,63 196 


?33 


7271344 


534 


7279477 


535 


7287595 


5?<5 


7295697 


5^1 


7503785 


538 


73n?$7 


539 


73 199 14 


$40 


75*7957 


541 


7335985. 


542 


7343997 


543 


7351995 


544 . 


7359979 


54S 


736794^ 


J46 


737590a 


547 


73?384i 


54« 


7391766 


549 


7199671 


550 


74^757^ 


551 


74^*55 


55? 


74^^3323 


55J 


1 743.1 176 


554 


7439CX15 


555 


7446341 


55<5 


7454.65:1. 


557 


74^2449 


558 


7470232 


5^9 


7478001 


560 


7485''5$ 


5«t ! 


7493498 


5«?» 


7$oi2^5 


5^^? 


7508939 


54 


75«f^39 


'^5 


75,243^6. 


566 


7.531999, 


5(Jr 


75396-59 


\6% 


75^730J. 


5«9 


7514937- 


570 


75<^?S\6. 


' 571 


7570162^ 


57* 


7577.7V5. 


573 


7581354 



Xll. 



7S9^9P^^ 



72S5?y9 
7^64012 

7a7aij8 

7210190 

7^88496 

729$596 

7394593 
7312663 

7320719 

7318760 

733^787 
7344798 
7352794 
7360776 

7368744 
7376696 

7JM34 
739*558 

7400467 

740936? 

74«$^^3 
74MIP9 

743 1 96 1 

743979? 
7447^2;* 
7455432 

7463??* 
7471009 

7478777 

•7486531 

7494271 

7501997 
75P97XO 

.75:?5»94 
7.55.27^6 
7540424 
7548069 
7555700 
7563318 

: 7,570922 

7578513 
j 7586391. 

759365^. 



■•^?^ 



7^5^67 

7?648?7 

727*97^ 
728uo> 

7*^9^;^ 

7^97316 

730s4c?9 
73 13470 

7>1956a 

71375^5 
7^4559? 

n5i593 
7361574 
7?69UO. 
73774.9' 
73^54^7 
73S933S9 
7404^57 

7417039., 

r4?4895 
743*745 
7440^8^. 

74484Q4 

7456I*U. 

^4$4QO(S 

747*787 

747955) 

74^8759$ 

7491944-! 
75Q?7$» . 
75t04So. 

7?.l8Li78- 
^^3353? 

7j4n§9 
7564979 

7$7l68$ 
7579^7* 
7586848. 



I 

V 



8 

^57483 

2i<JS042 
*7j7W 

a??9l4 

^9002 7 

306^08 

3U*7^ 
^29329 

33,8j90 
?4^?98 

4>9> 

«7i 

^703331 

37S285 

3^62:^u 

402047 
409^39 

4178^.7 
42548Q. 

433?3a 

7449U7 

45$99J 

4^47^5 
7472564 

4?Q3^9 
4«8o5o 
495817 

5P3HI 
5l*:?5i 

M?947 
5Jftf?9 
534*98 
54x954 
^4959$ 
557224 

$6494P 

57U4» 

5fQ03a 

$]^76d.5 

c.95*68 



■•^p" 



7^58300 
726^447 

7274599 
728?7?6 

729^8)8 
729*9^4 

7307«V5 
73150(2 

7323133 

733U70 

7339(9^ 

7347198 

7355191 
73<^JI08 

7371131 

7379076 

7387PI3 
739,4932 
740^837 

M 107^8 

7418^^4 

^4i.<?4^^ 

14343 M 

7442447 

7449967 
7457772 

746,5.5^4 

7473 HI 
1481 105 

^4S?8i4 
749^'^ 90 
75043^2 

75UO^l 

7^1.9746 

?5 2^7397 

753(5^5 

75V7?9 

755^o,359 

7557^7 
14756.5)590 

7573201 
75%>788 
75 883<i2 



384 



Artificial Numbers : Or^ 



Natural 
Sombrrs. 

srs 

577 

57« 

579 
580 

5S1 

583 

584 

585 
586 

588 

58J 

590 

591 

59a 

59S 

596 

5^7 

598 

599 
600 

601 

6o« 

6cB 
604 
60s 
(S06 
607 
608 
609 
610 
611 
6i2 
^13 
6t4 

615 

616 

6I7 
6s8 



7J96678 
76042115 
7<5ii758 

7619278 
7626786 

76}4^^o 
7641 76 1 

7649230 



7597434 

7604979 
7612511 

7620036 

7627556 

763JC29 

7644509 

764997^ 
7657430 



76641*8 I 7664872 

7671559 I 7^743<>» 
7678976 



7686381 

7693773 

7701153 
7708^0 

7715875 
1721117 

7750547 
7737864 
7745170 
7752463 

7759743 
7767012 

7774268 



76797*7 
7687121 

7694512 

7701890 
7709-56 
7716610 

77-395' 
7731279 

77j8$9^ 

7745899 

7753 «9» 
776047 1 

77^577^8 
7774993 



7781513 778«2J<^ 



7788745 
7795965 
7803173 
7810369 

7817554 

7824726 

7831887 
7839036 

7846173 
7853298 

7860412 

7S67514 
7874605 

7881684 
7888751 
7895807 
790*851 



77894^57 
7796686 

7805893 

7811088 

781817* 

7825443 
783*60* 

7839750 

7840886 

7854010^ 
7861113 

7868224 

78753*3 
7882591 

7889457 
789651* 

17903555 



7909885 1 7910587 



7598189 
7605 73 J 

7613163 
7620781 
76*8186 

7^^35777 
7643156 

7650722 

7658175 
7665616 

7673043 
7680458 
7687860 
7695250 

7702627 
7709992 
7717344 
77H684 
7732011 

7739326 
7746629 
77539*0 
7761198 
7768464 

7775718 
7782960 
7790190 
7797408 
7804613 
7811807 
7818989 

7826159 
7833318 
7840464 

7847599 
7854722 

7861833 
786S955 
7876021 
7883098 

7890163 

7897217 

7904*59 
7911*90 



7598944 
7609486 
7614016 
76*1532 

76*9035 
7636526 

7644003 
7651468 

7658910 

7666359 

7673785 
7681199 
7688600 

7695988 

7703364 
7710728 

7718079 

7725417 

7732743 
7740057 

7747359 
7754648 
7761925 

7769190 

777 M3 

7783983 
7790912 

7798129 

7805333 
78125^6 

7819707 
7826876 

7834033 
7841178 
784831? 

7855434 

7862544 

7869643 
7876730 

7883805 

7890869 
7897922 
7904963 

791199^ 



7599699 
7607240 
7614768 
7622283 
7629785 1 

7^37174 
7644750 
7652214 
7659664 
7667102 

7674527 
7681940 

7689339 
7696727 
7704101 

77 1 146 3 

7718813 
7726150 

7733475 
774078S 
7748088 

7755376 
7761651 
7769916 

77771^^7 
7784407 

779*634 

7798850 
7806053 

7813245 
78 104 14 

7827592 

7834748 
7841892 

7849024 

7856145 

7863254 

78703 J2 

7877438 
7884512 

7891575 
7898626 

79056661 
791*69^1 



Logarithms (^to 6iSp.^ 



iH 



I Natdral 

57$ 

57^ 

577 

578 

579 
580 

581 

582 . 

583 
584 

586 

587 
S88 

18? 
590 

591 

593 

S94 

595 

596 

5^7 
598 

599 
600 

601 
602 
^03 
604 
605 
606 
607 
608 
609 
610 
6il 
6i2 

613 

61$ 

($16 
617 
618 



760045 5 

7607993 
7615520 

7613034 
7630554 

7638022 

76454^7 
7652959 

7660409 

766784s 
7675169 

7682680 
7690079 
7697465 
7704S38 
7712199 

77'9547 

7726884 

7734207 

7741519 

7748^18 

775^1^4 

77<59579 
7770642 

7777891 

7785130 
7792556 

7799^71 
7806775 

7 « 13963 

7S2[I<fI 

7828308 

78354^3 
7841606 

7849757 
7856857 
7863965 
7871061 
7878146 

78S52I9 
7^92281 

7899331 
7906370 

79^3397 



7601208 
760^746 
7616272 

7623784 
7631184 

7658770 

7646144 

7653705 

766115^ 

76685^^8 

767601 I 

7683421 

7690818 

7698203 

7705575 
77»i9H 
7720282 

7717616 

7734939 

7742249 

7749547 
7756832 

7764106 

777»3^7 

7778616 

7785853 
7795078 

7800191 

780749^ 
78i0ai 

7821859 

7829024 

7S56178 

7843319 
7850450 

7857568 

7S64675 
7871770 

7878853 
7885926 

78^2986 

790O0J5 

7907073 

7914099 



7601962 
7609500 
7617014 

7624535 
76310^3 

7^39518 

7646991 
7654450 
7"^6i897 

76693 n 
7676752 

7684161 

7691557 
769894O 

7706311 

7713670 
7721016 
77*8.349 
7735670 

7742979 
7750276 
7757560 

7764^33 
7771093 

7779340 
7786576 

7793800 
7801012 

7808212 
7815400 
7822576 
7829740 
7836892 

7844033 
7851162 

7858279 
7865385 

7872479 

7879561 
78866^2 

7893691 

7900739 

7907776 

.7914801 



8 

7601717 

7610253 

7617775 
7625285 

7632782 

7640266 

7H7737 
7655195 

7662641 

7670074 

7677494 
7684901 

7692296 
7699678 
7707048 

7714405 

7721750 

7729082 
77^6402 

7743710 
7751005 
7758288 

7765559 
7772818 

7786065 

7787299 
7794522 

7801732 
7808931 

7816118 

7823193 
78^0456 

7837607 
7844746 

7851874 
7858990 

7866095 
7875188 

7880*69 

7887339 
7894397 

7901444 

7968479 

7915503 



-. t . 



' 



7603471 
7611005 
7618517 
7626035 

7633531 
7641014 

76484P4 
765594' 
766^385 
7670816 

7678235 
7685641 

76930^5 
7700416 

7707784 
7715140 

77224^3 
7729814 

7737i33 

7744440 
7751734 
7759016 
77662S6 

7773543 
77^0789 
7788022 

7795243 

7802453 
7809650 

7816836 
7824010 
7831171 
7858321 
7845460 
7852586 
7859701 
7S66805 

7873896 

7880976 

788S045 
7895162 

79dii48 
79<*?i82 



D dd 



iZ6 

% 

Natural 

619 

630 
621 

622 
62$ 
624 

625 
626 
627 
628 
629 

631 

634 

635 

636 

638 
679 
640 
641 
642 

643 

644 
645 
646 
647 

648 

649 
650 

651 

653 
654 

656 

<^$7 - 
658 

659 

662 



Artificial Numbers : 



7916906 

7923917 
79^0916 

79379^4 
794488a 

795is4< 
79^8800 

79^55743 

797*67^ 

79795^^^ 
7986506 

7993405 
S000894 
$007171 

•014037 

8020893 

S027737 
8034571 
8041394 
8048207 
8055009 
8061800 
8068580 
8075350 
80821 10 
S088859 

8095597 
8102325 

81090^3 

8115750 

8122447 

8129134 

5l3s8io 
814^476 
S149132 

8IS5777 
8i/$24U 
8169038 

8175654 
818&259 
8188854 

8195439 

8202015 

8108580 



7917608 
7944617 

7931^*5 

7938602 

794*^57^ 
7952542 

795949? 

7966437 
79733^^8 

7980288 

7987197 
7994097 

8000982 
8007858 
8014723 
8021578 
8028421 

8035254 

8042076 

8048887 
805568S 
8062478 

8069258 

8076027 
8082785 

8089S33 

8096270 

8102997 
8 1097 14 
81 16420 
8123116 

8129802 

8156477 
814314^ 

1^9797 
8156441 
Si 63076 
8169700 
8i.763iy 
8182919 
. 6189513 
6196O97 
8^02672 
^109236 



7918309 

79<53>8 

79323 14 
7939300 
7946274 
7953*3^ 

79^0190 

79^713* 

7974060 

7980979 
7987887 
7994784 

8001670 
800H545 

lo 15409 
8022262 
8029105 

^035937 

8042758 
8049568 
8056^68 
8063157 
806993 5 
8076703 
8083460 
8090207 

8096944 
8103670 

81103^5 
8117090 
8123785 
8130470 

8 1 37 144 
8143808 
8150462 
8157105 
8163739 
I170362 
8176976 
8183579 
8^90172 
$196755 
8203328 
8109891 

>ll Iw IT I I ■ ■ 



Or, 
3 

7919011 

7926018 
7933014 

793999« 
794697* 

7953933 
7950884 
7967824 

7974753 
7981671 
7988577 

7995473 
8002358 
8009232 
8016095 
8022947 
8029789 
8036619 

8043439 

8050248 

8057047 

8063835 
8070612 

8077379 
80841^6 
809O881 

8097617 

810434^ 

81 1 1056 

8117760 
8124454 

8131138 

8137811 

8U4474 

8151127 

8157769 
8 164402 

8171024 

8177636 

6184239 

(19-831 

?i974i'3 
1203987 

82i054> 



7919712 

79i67i>i 

7933712 

7940696 

7947668 
7954629 
7s>6i578 

7968517 

7975445 
7982362 

7982267 
7996162 
8003O46 
80C99 • 9 
8016781 
8023631 

8030472 
8037302 

8044I21 
8050919 

8057726 
80645 1 3 

8071290 
8078055 

8084811 

8091555 
8098^90 

8105013 

8111727 

8118430 
8125123 
8I31805 

8138478 
S145140 

tl5i79» 

815^433 
8165664 

81716S6 
8178297 

8 1 84898 
8191489 
8198071 
6204642 

82II2J(53 





Logarithms Oo 6629.) 


38r 


1 mtjt.i 


1 


6 


) 


f [9 


H7 


79»04I3 


7921 1 14 


79»i8m 


7922516 79»iai« 


620 


791741S 


7918118 


7918817 


7929^17 7930"7 


611 


7M441I 


793 s no 


7931 809 


7936507 79i7«=* 


622 


7SHifp+ 


7942C91 


79427S9 


7943486 7944183 


«2! 


7941*365 


794tfo6i 


79497 J7 


79S0454 


7911150 


6M 


79S5S»* 


7956010 


7956715 


7957410 


795«I05 


ill 


796««73 


7967^67 


7963662 


79'S4J5« 


7965050 


79ft9iii 


7969904 


7970597 


797II90 


797»J83 


637 


iTPTfiH? 


7976839 


797T52I 


797^213 


7978905 


6i^ 


798J053 


7983744 


79S4435 


798^125 


7985816 


6asi 


798<»957 


7990^7 


7«9IJJ7 


7992027 


799*716 


(SjO 


79968^1 


7997540 


799SII38 


7998917 


799»<So5 


6ii 


8oOi7J4 


8004431 


800W09 


8005796 


8006414 


6)3 


3^10605 


SoiU9» 


»oii978 


801 366 t 


80I3J5I 


6j3 


■8oi74S6 


9Qisi,3 


8018^17 


8oi9S2a 


8020208 


C3* 


18024316 


Soj^ooi 


8o3s68s 


8026169 


^"=51 


«!5 


I80311S6 


803lBj9 


8032^33 


»oj3aoi 


8033888 


63 S 


8037084 


8053564 


^oiSJlS 


80400)1 


8040712 


6^7 


8oil483 


I046164 


8046S4S 


8047526 


•6^8 


80,51^09 


8052289 


8052969 


eo5j649 


S054J19 


639 


8058404 


SaspuSj. 


83597^1 


806044? 


8061121 


«40 


80^^191 


306^859 


80(56147 


8057 =2 5 


806730, 


641 


8071967 


8o7*S4j 


8073320 


?C73897 


8074874 


£42 


807^71' 


S079407 


SoSooSj 


8080759 


!i;^s* 


643 


8o8l*Sj 


9aS6i6Q 


fioSaSjs 


8087OO 


8088^84 


644 


8093329 


Sou 190 J 


So^3i77 


8094^50 


8094924 


64$ 


8o989!62 


8o99'S35 


;BiP33o8 


E 100980 


8101653 


64« 


8^0^685 


^1063^7 


8107029 


8107700 


8103371 


tf4J 


8U??SS 


8ij^8 


Sl'i739 

2119*39. 


8114409 

Bi3i.o8 


8115080 


<18 


SllylOO 


8ily7tf9 


8121778 


649 


,Si257M 


8»2i4<5o 


Pf?7ii9 
8iJ38o§ 


?i!7797 


8118465 


650 


81 m7; 


»i33i4l 


8.34475 


8135143, 


<5i 


8135144. 


SijgSll 


8 14047? 


8141144 


8141810 


653 


■8 1 4530s 


8 14647 1 


^147135 
8I5J76J 


8147801 


8148467 


6f3 


8.SM56 


815)1'" 


8., 4449 
I16.087 


8,55113 


«14 


8155096 


8159780 


iiao42i 
8i*7°5» 


8161750 


«^'^ 


18 1 5< 727 
"'7^347 


8i6£i39 


Ii677'4 


8168156 


616 


8173009 


S17J67P 


!l7133' 


8I74P63' 


6$7 


8,178?; 8 


8179611 


8180178 


8t8o9)9 


8181599 


6SS 


8185118 


8..a6..7 


S.8S877 


8i875?6 


818B195 


tfS9 


8193146 


3192806 


6193455 


8194113 


81947I1 


660 


8io87rt 


8 199 186 


820004J 


8200700 


8201358 


6«i «!o;29B 


81059,55 


82065.1 


8107168 


8io7s24' 


662 Si'.jSf^ 


»2i?;,u 


81U170 


8*13815 


-iSfSsJ 



Ddda 



}«< 



Artifidal Xsmhers : Or, 



t-,7 
t.i 
f'9 

I '*' 









ic!--9; 
S?8ii>o 



7JTU*« 

TSS7197 
79«^7 

4^4 2; 7 6 
Sci8;S7 
So55f83 
to6U7S 

%'7^<:ij j 
>cii7S5 



. 


] 


4 


7;i?jo9 


79>9Cli 


791971* 


79«J" 


79Z«cig 


79^^71- 


7932j!4 


79HOU 


7935712 


'5J9JOO 


743599? 


7940tfi.e 


J?*SJ'4 


791^97' 


794"iC& 


795J'3^ 


7>Jj9?3 


79^45t9 


79:0.9- 


79fea«4 


7-61,78 


79<-"5' 


796rSi4 


79-'8ii7 


7PT4D60 


7974TS3 


'97i44S 


798=979 


7v'>i6:i 


7yiiaj62 


798:S.7 


7595,77 


79S22S- 


7994-81 


'9?147i 


7996161 


S001373 


S=c2isS 


S00JD4S 


Sdos141 


'W»92J2 


Sqc 99 19 


i0M4O9 


So.'ics^ 


Soi(S78i 


8oj2«62 


<02Ipi7 


802,631 


lci;iof 


83j;789 


8030472 


*>}'9)7 


8ojf6i9 


80J7JOI 


fc427i8 


So+Mis 


3344IM 


&049S68 


80S024? 


«3^C9!9 


Soi£;t8 


80^7047 


S0S7726 


iofjiw 


<btf?Sj5 


8064,. J 


»o«s9jt 


8070; I » 


8071390 


'076701 S077379 807805, 


S-Sji-io 8o»4ij5 80S48H 


«0:>W7 8c90SKi flopn^j 



388 



Artificial Numbers • Or, 



V 



Hatvral 

666 
667 

669 
670 

671 

672 
671 
674 

^75 
676 

677 
67S 

679 
680 
68t 

683 

683 

684 
685 
686 

687 

668 
689 

6yO 

691 
692 
693 
694 

696 
697 
698 
699 

700 
701 

7oa 
7o) 
704. 
70^ 



821U« 
8221681 

8238216 

8«a474^ 
8241258 

82477^5 
8254261 

8260748 

8267225 

8*7^695 

8180151 

8286599 

829^038 

8199467 

8305887 
8312297 
8318698 
8315089 

55J«47i 
$337844 
8}44ao7 
8350561 
8356906 

8363241 
8369567 

8375884 
8382192 
8388491 

83947S0 
840I061 

840733* 

8413595 
8419848 

8416092 
84i2328 

•438554 
8444772 

8450980 

8457180 

84633 7 X 

8469553 
8475727 

8481 «9t' 
848804^ 



8215790 

8222335 
823b869 

8235394 
8241909 

8248415 
8254910 
8261396 
8157871 
8174339 
8280796 

8287243 

8293681 

8300109 

83C6528 
8W2937 
»3»9337 
8325728 

8332109 
8338480 

8344843 
835IT96 
8357^40 

8363874 

8570199 
8376516 

8581822 
8^89120 

8395409 
8401688 

8407959 
8414110 
8410473 
8426716 

843295 1 

8439176 
8445393 
8451601 
8457800 
8463996 

8470171 
8476343 

8481507 

^8^88662 



2 

S216445 

8122989 
8229522 
8236046 
8242560 
8249065 

8255S59 
8261044 

8268519 
8274985 
8181441 
82S7887 

82945*4 
83OC752 

8307169 

831J578 

8319977 

8326566 

8331746 
8^39117 

8345479 
8351831 

8358174 
8564507 

8370832 
8377147 

8383453 
8589750 
8396057 
8401316 
8408586 

8414^45 
8411098 
8417340 

8433574 
8439798 

8446014 
8452221 

8458419 

8464608 

8470789' 

8476960 

8483123 

8489*77 



8217100 

S223641 
8230175 
8236698 
8243211 

S2497«5 
8256208 

8262692 
8269166 

827563' 
8t82o86 

828853- 
S194967 
8501394 
8?o78ii 
8314118 
832C616 
8327005 

8333384 

85^9754 
8346114 
835i4<55 
8358807 
83^5140 

8371463 

837777« 
858408^ 

8390379 
8596666 

8402943 
8409212 

84T5472 

8421711 

8417964 

8434V97 
8440420 

8446635 

8452841 

8459038 

8465227 

847 1 406 

8477577 

8483739 

848989* 



8 
S 



4 I 

8217755 
S224296 

^25o8'-8^ 

8«3735o 

8243862 

8250364 
8256»57 
8263940 

8269813 

8276177 

8282731 
8189176 
S295611 
1301036 

'308452 
,3 '4858 

^J*»-S5 
8327643 

^3340? I 
8340590 

^546750 
^353100 
^359441 
8365773 
^372095 
8578409 

8384713 

839I008 
8397294 
8403571 
8409858 
8416097 

84?2347 ' 
8418588 

^434819 

8441042 

8447*56 
8453461 

8459658 

8465845 
8472034 

8478193 
8484355 



^8490^0^ 



Logarithms ( to jcSp. )^ 3 89 



t Natsral 

Numbers 

664 
66| 

667 
668 

670 
671 
672 

674 

67 S 
676 

677 

679 
680 

6«l 
*682 
683 
684 
685 
686 
687 
688 

6^9 
690 
691 

69a 
69^ 

69A. 
695 

696 
697 
698 
699 
70O 
70I 
702 

703 

704 

705 

. 706 



821S409 
8224950 
S191481 
823 bo; 2 

8-44513 
8251O14 

8:57506 
8263988 

8270460 

8276925 
82^3 J76 
^289820 
82962^4 

8i02678 
b 309093 

8315499 
83*189^ 
8328281 

8334659 
8341027 

83473S5 

8^537}5 
8360075 
83^^405 
83727^7 
8B79039 
8385H3 
839'<^37 

83V7922 

8404' 98 
8410465 

8416722 
842297 
8429211 
8435442 
S44f 664) 

84478^7 

8454081 

8460277 

84««46^ 
8472641 

84788rio 

84849^0 

8491122 



8219064 
8225603 
H232133 

825 653 
8245163 

^251664 
8458154 
8264655 
8271107 
8:77569 

82^40^1 
8290463 

8296896 
8303310 

8J09734 

83161^9 

8322534 
85289J9 

8335296 
8341665 

8348021 

8354369 
8360708 

836703.8 
83713<9 
8379670 

8385973 

839^^66 

'B398S50 

404825 
8411091 

841M48 
8423596 

8419735 
8436065 

8442286 

8148498 

8454701 
8460696 
8467081 
8473«58 
84794*6 

8485586 
849*736 



8219718 

8226257 

82U786 

8239305 
8245)Si4 

8252513 

8258803 

826)^283 

8271753 
8278214 
8284665 
8291107 

8297539 
8303962 
8310375 

8316778 
8323173 
8329558 

8335933 
83*2299 

8348656 
8355005 

8391341 
8367670 
8373990 
8380301 
8386602 
839*895 
8399178 

8405452 
8411717 

8417973 
84C4220 

84?C458 
8436687 
8442907 

8449119 

8455521 
8461515 

8467700 
8473876 

8480043 
8486201 



• 8 

8220372 
8-^26910 
8235438 
8239956 

8246464 
8252963 

8259451 

8265931 
8272400 
8278860 
8285310 
8291751 

8)98182 
8304603 

83tiOi6 
8317418 
8523812 

8330195 
8336570 

8342937 
8349291 

8355638 

8361975 

8368303 

8374622 

8380931 

8587232 

8393523 

8399806 

8406079 

8412343 
841^598 

8424844 

8431081 

8437310 

8441529 

8449739 

8455941 
8462134 

84683 1 8 

8474493 
8480559 

8486817 



8221027 
8227563 
8234090 
9240607 
8247114 
8253612 
8260100 
8266578 
8273046 
8279^05 

8285955 
8292394 

8298824 

8305245 
8311656 

831805S 
8324450 
8330853 

8337«>7 

834^571 
8349926 
8356272 
8362608 

8368935 
8575255 
8381562 
8587861 
8^941 5 i 

8400433 

8406706 

8412969 

8419223. 

8425468 

8431705 

843793* 
8444150 

8450360 

8456561 

8462752 

8468935 

8475110 

8481275' 
8487432 

_842i£8a 



}9o 

Hatgraf | 

|07 
70S 
709 
710 
7H 
71* 

711 

7I4 
7M 
716 

7"7 
718 

7»9 
7ao 

7*1 

7tl 

7*3 

7n 
725 
796 

7«7 

7*8 
729 

750 

711 
732 

7U 

715 

7}^ 

737 
7B.8 

7^9 

749 
741 

74i 
74} 

744 
745 
7i^ 
747 

748 4 
749 

7^^ 



Aftificial Numbers : Or^ 



8494194 

8500333 
85064^2 

B5H583 

851^696 

8524800 

8530895 
853^82 
8543060 

85491 lo 

85Sli9« 
8561241 
8567289 

857}3«5 1 

857935} 

858517^ 

859«3«3 
8597386 

864) $3 $0 

8609366 
8615344 

8621^14 

8627^75 
8639229 

8639174 
8641111 

8651049 
8656961 

8668778 
867467 5 
8680564 

86)6444 

%>}«? 
8698182 

8704939 

8709$$$ 

87IJ7?9 

8711561: 

872738? 
8733*06 

-8759016: 

8744B18 



8y^o6|| 871 



8494608 
8500946 
8507075 
8513195 
8519307 
85^5419 
8531504 

8$3759« 
8543668 

8549737 

8555797 
8561849 

8467893 
85739*8 

8579956 

8585975 
859<984 

8^97985 
8603979 

8609964 

861594& 
8621910 

86t787i 

8*3^823 
8639768 

8645704 
865163;^ 

86^7562 

8663464 

86759^4 

868.1x5? 

86j70?ji 

8iJ9W04 
8698763 
870^624 
87^0473 
87*6313. 

87??*4^l 

8f5378?j 
f 739597 { 

874519?^ 
875ti9^> 



84944t3 

8501559 
85076^7 
8§4 3$07 

8519917 

8526070 
8532113 
8538498 

8544275 

8550343 
8556403 

856M54 

8568497 
8574n> 
8580557 
8586575 
8^92584 
8498585 

8604578 
861056^ 

8616539 

8682507 

8628467 
8634418 
864ol$2 

8646*97 
86|2?2$ 
8648144 
86^4055 
84^995 9 
8^75*$i 
8681740 
86876^0 
B69369I 
W99354 
8705 f<59 

P7 148^7 

87?^f6?: 
873*?f9: 
P749177: 
874597? 



8496037 

8509178 
8508300 
85 14418 

8580528 
84*66-29 
85^8722 

8538806 
8544882 

85^0949 
8557008 

8563059 

8469101 

8175134 

8581159 

8587*76 
859318.5 
8599185 

8605177 
861 1 160 
86I7I36 

86;23ip^ 

862906 ;i 

8635013 

8640956 

8646S90 

8652817 

865873$ 
8664646 

8670!{4g 

8676449 
868£329 

,8688907 

<Bf94.p77 
8499949 
^7^795 
jSf^ 1 ^4/f 
87^7460 

37?I34> : 

"8734^56 

874P?57 
.874^55/ 



849<<65i 
85027S6 
851)8913 

8515030 

85*1139 

8527339 

853333' 

85394U 

85494^9 
855M56 

85153663 
85/^9704 
8575737 

8587777 

859^785 

8599734 
8605776 
8611758 

86«77}5 

86^3^99 
8629658 

8635608 

'8641550 

8647483 
8653409 
865P«^7 
86d52)6 

867/^38 

8677Pii 
8682917 

8698794 
8694464 

8700526 

87063110 

8712926 

87*^064 

87^5^94 
8799716 

873f53i 
8741338 

8747^37 



Natural 

Aumbfers* 

707 

70a 
7OP 

710 

7ii 

711 

7M 

7i6 
7»7 
718 

719 
723 

721 
722 

72j 
726 

728 
729 

7?o 
731 

733 

734 
73^ 

736 

737 
7^8 
739 
740 

741 
742 

74} 
744 
745 

'U6 

747 

749 
7$o 



^ 



Logarithms {to 75op#) 



8497264 

8503399 
8509514 

8515641 

8521749 

8517849 

8533940 
8540022 

8546:>96 
8552162 
85582F9 
8564168 
8570308 

8'5763io 
8582363 

8588379 

8594385 
8600384 

8606374 
86U356 

8618330 

86*4196 
8630253 
8636202 
8'<64ti4} 
8648O76 
8654001 
8659918 

8665827 
8671718 
8677620 
8683505 
8689381 
8695151 
87011U 

8706965 
8711810 
871^647 
8744476 
8730198 
8736112 
8741918 

87477 »^ 
87*5 507 



8497878 

8504011 
8510136 

85I6252 

8522359 
8518458 
8534548 

8540650 

854^703 
8551768 

8558814 
8564872 
8570912 

8576943 

85S1965 

8588980 

8594986 

8600983 
8606973 

8612954 

8618927 

862489a 

8630848 
8636797 

8642737 

8648669 

865459} 

8660509 
8666417 
8672317 
8678109 
8684093 
8689969 

8695837 

870I697 

8707549 
8713394 

8719*30 

8715059 
8730880 

8736693 

8741498 
8748296 
8754086 



i\\ I in 



7 : 

8498492 
8504624 
8510748 
8516863 
8522970 
8529068 

8535*57 
85412^8 

85473*0 

8553374 
8569429 

855547^ 

857I5I5 

857754s 
8583567 

8589581 

8595586 

8601583 

86O757I 

8613552 

86(95*4 

8615488 

8631443 
8637391 

8643331 
8649262 

8655185 

8661 ICO 
8667008 
8672907 
8678798 
8684681 
86^0556 
8696423 
^702283 

8708134 
8713978 
8719814 
87I5641 
873146I 

8737*74 
8743^78 
8748875 



8 

8499106 
8505137 
85 II 360 

8517474 

8523580 

8529677 
8535765 

8541845. 

8547917 
8553980 

8560035 

8566081 

8572118 

8578148 

8584169 

8590181 

85961 86 

8602181 

8608170 

8614149 

8620110 

8626084 

8632039 

8637985 

8645924 
8649855 

8655777 
U6i6gi 

866759S 
8673496 

S679387 
8685169 

869 If 43 
8697010 
8720868 
8708719 
^7^4562 
8720397 
8726224 

8732043 
8737855 
8743<^58 
8749454 
;?755.245 



• 9 

8499719 
8505850 
85 1197 2 

1518085 
8524190 

^530286 

1536574 
8542453 
8548524 
8554586 

8560640 
8566685 
8572711 

8578750 

8584770 
8590782 
8596786 
8602781 
8608788 
8614747 

8610717 

8626679 

8632634 

8658580 

8644517 

8650447 
8656369 

8662282 

8668188 

8674086 

8679975 
S685857 
8691730 
8697596 

8703454 
8709304 

8715146 
87209S0 
8716806 

875^625 

87^435 
8744138 

875003(1 

8755.821 



t 



3P» 

I Natural 
Noinberft. 

75« 
753 
7S4 
755 
75^ 
757 
758 

759 
7^ 
761 
762 

763 
7^4 
765 
766 
767 
768 

769 
77O 

771 
772 

77? 
774 

775 
776 

777 
778 
779 
780 
781 
782 

783 

784 
785 

786 

787 
788 

789 

7VO 

791 
792 

793 

221. 



ArtificiaJ Numbers 
t I 2 * 



8756399 
8762178 
8767950 

877?7t3 

8779469 

:>78)2i8 
^7909 59 
* 796692 
8802418 
8808136 
8813847 
8819550 

8825245 

8830934 
88^6614 

8842288 

8S47954 
885J612 
8859263 
8864907 
8870544 
8876173 
88S1795 
8887410 
8893017 
8898617 
8904110 

8909796 
8915^75 

8920946 
8926^10 
8932068 
8937618 
8943161 
8948697 

8954225 
8959747 
8965262 

8978770 
8976171 

S9S1765 

8987i5t 

899273* 
^98205 



8756978 

8762756 

8768526 

8774289 
8780045 

h78$79» 

I879IS32 
879726$ 

88^1990 

SS08707 

8814417 
8820120 

8825815 
8831501 
8837182 
8842855 

8848520 
8854178 

8859828 
8865471 
8871107 

8875736 

8881357 

8887971 

8893577 
8899177 

8904765 
8910354 

8915952 

89*1503 
8927066 

89J2623 
895817Z 

89437«5 
89192-50 

8954778 
8960299 
89<5!8i3 
897M20 
897681 I 

89813 «4 

8987800 

8993279 
8998752 



8757')56 

876^^3 
876910^ 

8774865 
^780620 
8786367 
8792106 

8797838 
8803562 

8809279 
88i49H8 
88206S9 

8826384 
8832070 

8S37750 

8845421 
I 8849086 
S8S4743 
8860393 
8866o3<' 
8871670 
8877298 
8882918 
888\532 
88941^.8 
88^9736 
8905318 
891091ft 
8916489 

8922059 
8937622 

8935178 

8938727 
8944268 

8949803 

8955930 
8960851 

8966364 

8971871 

8977570 

8982863 
8988^48 

8993857 
.8999299 



: Or, 

875^1^4 

^763918 

6769680 

8775441 
8781195 

b78694i 
8792680 
87984.U 
8804134 
8S09850 

8815558 
8821259 

8826953 
8832639 

8838317 
8843988 
8849652 

8855308 
8860957 

8866599 
8872233 
8877860 

S883480 
8889093 

8894698 
8900296 

8905887 
891 1470 
8917047 

8922616 
8928178 

8933733 
8939*81 
8944322 
89S0356 

8955883 

g96i403 

8966915 
8972421 
8977920 
8983412 
898S897 

8994575 
8999846 



8758712 
l>764488 
8770156 

8776017 

8781770 

8787515 

8795259 

8798983 
8804706 

88104 2.J 
8816129 
8821829 
8827^21 
8855207 

8838885 
8844555 

8850218 

8855874, 

8861522 " 

8867163 

8872796 

8878423 

8884042 

8889653 

8895258 
8900855 
8906445 
891202^ 
89I76O4 

8925173 
8928734 

8934288 

8939836 

8945376 

8950909 

8956435 

8961954 

8972971 
8978469 
898.5960 

»989445 
8994922 ; 

90^392f 



Logarithms (^to 79^9*^ 



I Number*. 

7sa 

753 
754 

7SS 

756 

7S7 
7S8 
759 
760 

761 
762 

763 
764 
765 
766 
767 

768 

769 

770 

774 
772 

773 
774 
775 
775 

777 

778 
779 
780 

781 
782 

785 
784 

785 

786 

78-7 

738 

789 

790 

791 
792 

793 

794 



5 

8759290 
875506$ 
877o8?3 

8776592 

87**345 
8788089 
8793^26 
8799556 
8835278 

88(0992 

3816699 
882239s 
8828090 

8833775 
8859452 

884.5122 

8850784 

8856439 
8866086 

8867726 

887J359 
8878985 
888460B 

8S90214 
8895818 
8901415 
8907C04 
89W.586 

8918161 
8923729 
S929290 

8934843 
8940390 

894592.9 
8951462 

8956987 
8961506 
8968017 

8973521 
8979019 
8984509 

8989^93 
8995469 



8759868 
8765642 

8771409 

8777168 

8782919 
8788663 
8794400 
8800128 
8805850 
8811563 
8817269 
8822968 
8828659 

883:^H3 
8840019 
8845688 
885x350 

8857004 
$862651 

8868290 

8873922 

8879547 
8885165 

8890775 
8896378 

8901974 
8907 56J^ 
89ni44 
8918718 
8924185 
8929S46 
8935398 

8940944 
8946483 
8^52015 

8957539 

8963057 
8968568 

8974071 
8979568 
8985058 
8990541 

8996O17 
9001486 



8760445 
8766219 
8771985 

8777743 
8785494 

8789237 

8794973 
8?co7oi 

8806421 

83I2I54 

8817840 

8825537 

8829228 
8834911 

8840586 
8846255 
8851915 
8857569 
8863215 
8868354 
8874485 
8880109 
8885726 
8891336 
8896935 

8902533 
590811; 

8913702 

8919275 

8924842- 

8930401 

8935953 
S941498 
8947037 
8952567 

8958092 
8963608. 
8969118 
8974621 
8980117 
8985606 
8991089 
8996564 

90 J203 2 



8 

8761023 

8766796 

8772561 
8778319 

8784069 

8789811 

8795546 
8801273 

8806993 

8812705 
8818410 
8824107 

8829797 
8835479 

8841154 

8846821 

8852481 
8858134 
8863779 

8869417 

8875048 
8880671 
8886287 
8891^96 
3897498 

8903092 

8908679 
8914259 

8919832 

8925398 

8950957 
8936508 

8942^5 3 

8947590 
8953120 

8958644 
I 8964160 

8969':69 

8975171 
8980667 
8986155 
8991636 
8997 III 
9002^79 



395 



8761601 

87^7575 

8773137 
8778894 

8784643 

87903S5 

8796119 
8801846 
8807564 
8^13276 
8818980 
8824676 
8830365 
8836047 
8841721 
8847387 
8853047 

8858699 
8363345 

8869980 

8875610 

8881233 

8886848 

889*457 
889>?o58 

890365 I 

8909238 

8914817 

8920389 

89259S4 
89^512 

8937065 
8942607 
8^48143 
8953673 

8959195 
8964711 
8970219 

8975721 
8981116 
89S6703 
8992184 
8597658 
9003125 



E e e 



35>4 

, Natural 
jMomhcr*. 

795 
796 

797 

79« 

799 
8co 

801 

802 

80 i 

8oi 

805 

807 
808 

809 
810 

811 

8i2 

813 

814 

816 

817 

818 

819 
Sao 
821 
822 
823 
824 

826 
827 

828 
829* 
850 

831 
832 

833 
834 

836 
837 

\ 8:?8 



ArtificialJ^nlers 



9003^71 
9009131 

9014583 

9O20?20 

9DI5468 
9050900 

9036325 
9041744 

904715$ 
90^51560 

9057960 
9066351 

9068735 
9074 n 4 

9079485 

9084850 

9090209^ 

9095560 

9 100 905 

9x06144 

9111576 

9116902 

9122220 

9127533 
9*'^2839 
9148139 

914U52 

9148718 

9153998 
9^59272 

9164539 

9169800 

9^75055 
9180303 

9185545 

9190781 

9196010 

9201133 

9206450 
9211661 

>9Xl6865 
9222063 
92272^5 

02:; 2440 



/ 
yoo4ii8 

9009676 

901511S 

9020573 

9026011 

9031443 
9036867 
9042285 

9047696 
905J101 

9058498 
9063^89 

9069173 
907465 1 
90H0022 
9085386 

9090744 
9096095 
9101440 
9106778 
9112109 

9117434 
9122752 

9128064 

9133369 

9138668 

91439^1 
9149146 

9154526 

91^9799 

9165066 

9170326 
9175580 

9180828 
9186069 
9191304 
9196533 

9201755 
9106971 

9112181 
9t»7J8y 

9222582 

9227773 
92^29^8 



9004764 

9010222 

9015673 

902III7 

9026555 
9031985 
9037409 

9042827 
9048237 

9053641 

905903* 

9064428 
9069812 
9075188 

9080559 

9085911 

9091279 

9096630 

9101974 
9107511 
9111642 

9117966 

9123234 
^128595 

9133899 
9139198 

9144489 

9149775 
9155054 

9I60326 

9165592 

9170851 
9176105 
9«8iJ52 

9186593 

9191817 

9197055 

9202277 

9207493 

9212702 

9217905 
9223102 
9228292 

923H77 



Or, 

3 

9005310 
9010767 
9016218 
9021661 
9027098 
9031528 

9037951 
9043368 
9048778 
9054181 

9059577 
9064967 

9070350 
9075726 
9011095 
908645B 
9091815 

9097 16< 
9102508 
9107844 
9113174 

91 t 8498 
9123815 

9129126 

9 '344 30 

91397*7 
9145018 
9150303 

9'i5*i8i 
9160853 
9166118 

9^71378 
9176630 

9181877 
9187117 

9^92350 
9197578 
9202799 
9208014. 

9213222 

9218425 
9223611 

9228811 

9*33995 



♦ I 

9005856 
9011313 

9016761 
9022205 

9017641 

9033071 

9038493 

9043909 

90493^8 

9054721 

90601 16 

9065505 « 

9070887 

9076163 

9081632 
9086994 

909*350 
9097699 
91O3042 
9108378 
9115707 
9119030 
9114^46 
9129656 
9134960 
9140257 

9145547 
91508JI 

9156109 

9161380 
9166645 

9171903 

9177155 

9182401 

9187640 
9191873 
9198100 
92o;32i 
9108535 

9213743 

9118945 

9224140 
92293301 

224513 



Ni.mbcrj, 
795 

191 

798 

799 
800 

801 

803 

803 

804 
805 
806 
807 

808 

809 
810 

8it 
812 

813 
814 

815 
816 

817 
818 

819 

82^ 

821 

82& 
8«3 
824 

825 
826 

827 
828 

829 

8jo 

«3I 
892 

836 



Logarithms 
5 <5 



(f^gsSp.) 



9006402 
901 1858 
9017307 

9022749 
9028185 
90^3613 
90590^5 

9044150 
9049SS9 

905 5 26 I 
9060655 
9066044 
907 U2 5 
9076800 
9081169 

9087550 
9092885 
9098*34 

910357^ 
9 1089 II 
9114240 
9I19<62 

9124078 

9130187 

9x35490 
91407S6 

9146076 

9*51359 

9156636 

9161907 

9167x71 
9172429 
9177680 
9182915 
9188164 

9i9339<5 

9198623 

9205842 
9209056 
9214263 

92194^55 
9224659 

9229848 

92350M 



9006948 

901M03 
9017851 

902^293 

90*8728 

903415^ 
9^39577 
904499* 
9050399 
9055800 
9061195 
9066582 
9071963 

9077337 
9082705 

9088066 

90934*0 

9098768 

9104109 
9109444 

911477* 

9l2o:)94 

9125409 
9130717 
913^^019 

9I4'JI5 
9146604 

9151887 
9157163 

9162433 
9167697 

9172954 
9178205 

9183449 
9188687 
9193919 
9199145 
9204364 
9*09577 

9214784 
9219984 

922-5179 
9250367 

9*J5549 



9007494 

9012984 
9018 95 

90*3837 
9^29271 
90 J 4698 
90401 19 

9045533 
9050940 

9056340 
9061734 
9067 1 2 1 
9072501 

9077874 
9083*41 

9088602 

9093955 
9099303 

9104643 
9109977 

9115305 

9120626 

9125940 
9131248 

9136549 

9141844 

91471J3 
9152415 
9157691 
916&960 
9168223 

9173479 
9178730 
9183^73 

9189*11 
9194442 

9199667 
9*04886 

9216098 
9215304 
9220504 

9225698 
9230885 
9*36066 



8 

9008039 

9013448 
9018940 
9024981 
9029814 
9935241 

90^0661 
904.607 3 

9051480 

9056880 
9062274 
90676^9 
9073038 
9078411. 
9083778 

9089137 
90944^0 

9099837 
9105177 

9^10510 

9115837 
9121157 
9 12647 1 
9131778 
9137079 

914*373 

9147661 

9152943 
9158218 

9I63487 

9168749 
9174005 

9^79*54 

9»«4497 
9I89734 
9194965 
92COI t^ 

9205407 
9210619 

9*15824 
92210*4 
9226217 
9231404 
9?36584 



395? 



9008585 

yol4038 
90IP485 

90* 49 '-4 
905035 

9035783 

9041302 

90466 1 5 
9051020 

9057419 
9062812 

9068197 
9073576 
2078948 
9084^4 
9089675 
90950*5 
9100371 

9105710 
9111043 

9116569 
9f2«689 
912700* 
915*309 
9137609 
914*905 

9148190 

9M347X 

9158745 
9164013 

9161275 

9174530 

9179779 
91850*1 

9190258 

9195488 

9200711 

9205929 

9*11140 
9216545 

9221543 
9226736 

I 9231922 

1923710* 



E e e 



39tf 

Naniral 

S40 

841 ' 

842 

843 

«44 

«41 
846 

847 

849 
850 

8ji 

852 

«54 

856 

«57 

8s8 
859 

860 

-; 2 

^64 
865 
866 

867 
868 

869 
870 

871 

872 

873 
874 

875 

876 

S77 
878 

879 
S80 

881 
882 



Artificial Numhers 



9237620 

9242793 

9247960 

9^53121 

92cS276 

926H24 
9268567 

9275704 
9278834 

928^959 

9289077 
92941^9 
9299296 

^30439^ 
9309490 

93«4)79 
9519661 

93247^8 
9^29808 

93 34«73 
9339932 

9^44984 

^350031 

9555073 
936C108 

93^vi37 
9370161 

937S'79: 
93^c)i9i 

9385197 
939^198 

9395 »9i 

94001 8 i 

940516s 

941 0142 

94'Mi»4 
9420081 

9^^9996 
9434945 

94398^^ 
944*^27 

9449759 
9454^1^ 



9n8JJ7 
924^310 

9248476 
9255637 

9258791 
926^939 
9269081 

9274217 
9279347 
9284471 
928958^ 
9294700 
9299806 
9304906 
9309999 

9315087 
9320169 

93^5245 

93^0515 

9»n79 

9340437 
9345489 

9350S36 

93^5576 
9360611 

9365640 
9370663 
937S680 
9380692 
J385697 
9190697 
9395691 
94C0680 
9405663 

9410640 

9420577 
9425537 
9430491 
9435440 

94403^3 
9445320 

9450252 



9238655 

9343827 

9248993 
9254152 

9259306 
9264453 

9269595 
9274730 
9279^59 
9284983 
9290100 

9295211 

9300316 
9305415 
9310508 

93i559<5 
9520677 

932575^ 
9330822 

9335885 

9340943 

9145994 
93 5 I 040 

9556080 

9361114 

93<56i43 
9371 1<55 

957^182 

938119} 
9586198 

939t«y7 
939^191 

9401 179 

9406161 

941 1 137 

9416108 
9411073 
94£6o32 
9450986 

94? 59 .4 

9440877 

9445814 
9450745 
945 567^ 



Or, 

3 

9239x72 

9244344 

9249509 
9254668 

9259821 
9264968 

9270109 

9275245 
9280372 

9285495 
9290611 

9295722 

9300816 

9305925 
9311017 

9316104 

9321185 

9326159 

9531328 
933639X 
9341448 
9346499 
9351544 

935^^584 
9361617 

9366645 

9371667 
9376683 

9^81695 
9386698 

9391^97 
9396690 

9401677 
9406659 

911K535 

9416605 

9421569 
9416528 

9431481 
9436429 

9441371 
9446307 

9451238 

^g4i6l63_ 



9239690 
9244860 
9250025 
9255184 
92603 5<j 
9265483 
9270622 

9275757 
9280885 

9286O07 
9291123 
9296233 

9301336 

9306454 
9311526 
9316612 
6321692 
9326767 

9331835 
9336897 

9341953 
9347004 
9352049 

9357087 
9362120 

9367148 
93721 69 

9377«84 
9382194 
9397198 
9392196 

9397189 

9402176 

9407157 
9412132 

9417101 
9422065 
9427OH 

943 1 976 
943^923 
9441865 
9446800 

945i730 
9456655 _ 



t 



Logarithms ^to S82p#) 



I Natnral 

Sombcips. 

839 
84.0 

841 
842 
84} 

844. 
845 

847 

848 
849 
850 
8'5i 
852 

8jr4 
855 

857 

8s8 

859 

860 

861 
862 

863 

864 
865 

866 
867 
868 
869 
870 
871 

87i 

873. 

874 

87J 
876 

877 
878 

879 
8,80 

881 






I 



^ 5 

9240208 

9245377 

9250541 

9255^599 
9260851 

9265995 
9271136 
9276270 

9281397 
9286518 
9291634 

9296743 
9^0x847 
9506944 

9312035 
931^121 
9322^^00 

9327^74 
9332341 
93B7405 
9342459 
9347509 
9352553 

6357591 
9362623 

9967650 
937*671 
9377686 
9382695 
9387698 
9392696 

9397688 
9402674 

94Q7654 
94126^9 

94^7598 
9422561 
9427 J 19 

94J247I 
94374^8 
9442358 

9447*94 
9452223 

9417147 



9240724 
9245894 

9*51057 
9256215 
9261366 
9266511 

9271650 
9176783 
9281909 
9287030 

9292145 
9297254 

9?o2357 
93074*53 
9312544 
9317629 

9322708 

9327781 
9332848 

9337909 
9341964 

9348013 

9353057 

9358095 
9363126 

9368152 

9573172 

9378187 

9383195 
9388198 

9393^95 
9J98187 

940317* 
9408152 

94»3I26 
9418095 
94250.58 
9428011 
9432966 
94379«2 
9442852 

9447787 
9452716 

945763Q 



9241246 

9246410 
9251573 
9256730 

9261880 
9267025 

9272163 
9277296 
9282422 

9287542 
9292656 

9297764 
9302866. 

9 307963 

9313053 
9318157 
9323215 

9328288 

9333354 
9338415 
954.5469 
9348518 

9353561 
9)58598 
9363629 

9368655 
9373674 
9378688 
9385696 
9388698 

9393695 
9398685 
9403670 
9^8650 
^4131523 

9418591' 

9423554 
9428510 

943H61 
9458406 

9445346 

9448280 

9453208 
9<i58l^i 



9241759 
9246927 
92526S9 

9257*45 
9262395 

9267539 
9272677 
9277808 

9282934 
9288054 

9293167 

9298275 

9303376 

9308472 

9313561 

9318645 
9323723 

9328795 

933^860 

9338920 

9343974 
9349022 

9354065 

9359101 

9)64132 

9369157 
9374176 

9379189 

9384196 
9389198 
9394'94 
9399184 
9404169 
9409141^ 
94i4t20 
9419088 

9424049 
9429005 

943)956 
9438900 

9443840 

9448773 
9453701 

945862) 



J97 



9242276 

9247444 
9252605 

9257761 

9262910 

926805 3 

9273190 

9278321 

9283446 

9288565 
9295678 

9298785 
9303886 

^30898! 
9314070 

9319153 
9324230 

9329301 

93343<S7 
9339426 

9344479 
9H9527 
9554569 
9359605 

936463.5 
9569659 

9374677 
9379690 

9384697 

9389698 

9 94695 

9399685 

9404667 
94c 9645 

9414617 
9419584 
94«4545 
9429501 
9434450 

9439395 
9444333 
94492^ 
I 9454193 fc 
9459U5 i 



Nafottl 

S84 

88s 
«8tf 
887 
888 
889 



Artificial Numlers • Or, 




«94 

«98 
S99 

900 

901 

902 

903 

904 
90$ 
906 

907 

908 

909 
9to 

Pii 
91* 
9t5 

914 
91S 
9itf 

918 
9«9 

921 
913 

9H 

9^i 



9459^7 
9464523 

9469433 
9474337 
9479236 
9484130 
9489018 

9493900 

949 W7 7 
9503649 
9108$t5 

95t3575 

9518230 

9523080 
9527924 
9532763 
9537597 
954^425 
9J47248 
9552065 
955'6«77 

9561684 
9566486 
9571282 
9576073 
9580858 
9583r6s9 
9590414 
9595184 

9599948 
9604708 
960946a 
96142 r I 
9618955 

96*56^3 
96£84£7 

963315s 

9617878 
9642596 
9647309 
96520^7 
9656720 

96614S7 
966 6 i M> 



9466099 

9465014 
9469923 

94748*7 
9479726 
9484619 
9489506 
9494388 
9499364 

9504135 
9509001 

9513861 

95 187 16 

952356^ 
9528409 

9533*47 
9538080 
9542908 

9U773O 
9l<2547 
9557358 
9561165 

9571761 
9576552 
9581337 
95W117 
9590891 
9595660 

9600425 
9605183 
9609937 
9614686 

9619429 
9624167 

96289CO 
963^628 

9638350 
9643068 

9647780 
^652488' 

9657190 

9661887 
19667^9 



9460591 
9465505 

9470414 
9475317 
9480215 
9485108 

9489994 
9494876 

9499872 
9504622 

9509487 
9514347 
9519201 

9524049 
9521893 
9533730 
9538563 

954339^ 
9548212 
95530^8 

9557839 
956*645 

9567445 

9572241 

9577030 
958x815 

-95S6594 
9591368 

9596137 
9600901 
9605659 
961041* 
9615160 

96I9903 
9624640 

9629^73 
9634100 

9638822 

9643539 
964825 1 

965*958 

9657660 

96^6*356 
9^67048 



9461082 
9465996 

9470905 
9475807 
9480705 

9485597 
9490483 

9495364 
9500235 
950<i09 

9<09973 
9514832 
9% (9686 

9524534 
9$^9V7 

9534*14 
9539046 
9543872 
9548694 

9553510 
9558320 

9563125 

956"925 
9572720 

9577509 
9582293 
958707a 

9591845 
9596614 

I 9601377 
9606x15 

9610887 

9615635 
9620377 

9625114 

9629846 
963;573 

9639294 
96440 1 1 
9648722 
9653428 
965^130 
i 96^1826 

'9<6?^5»y 



9461574 
9466487 

9471395 
9476297 

9481194 
9486085 

9490971 
9495852 
9500726 
9505596 
9510459 
9515318 
9520^71 
9525018 
9529861 

9534697 

95395*9 

9544355 
9549176 

9553991 
955^801 
956369:5 

9568405 

9573199 
9577988 

958*771 

9587549 
9592322 

9597090 
9601853 
9606610 
96x1362 
961^ 09 
9620^51 

962J587 
96363 19 

9635045 
9039766 

964448* 
9647193 
9653899 
9658599 
9663295 
1 9^7985 



Logarithms ( to ^169.) 



Naroral 
Nnmbcrs. 

88s 

8U 
885 

S87 
888 

889 
890 
891 
891 

893 
894 

89J 
896 
897 

S98 

899 

900 
901 
90t 
903 
904 
9o$ 
906 

907 
908 

909 
919 

9lt 
9l2 
91} 

914 

916 

9l7 
918 
919 

926 

921 

922 
92) 

924 

925 

926 



5 



■«ki 



9462066 
946697 S 
947»8S6 

9476787 
9481684 

9486574 

9491460 

949^330 
9501113 

9506082 

9510946 

9515803 
95^0656 
9525503 

9130345 
9535181 
9540012 
9544837 

9549657 

9554472 
9559i8i 
9564086 
956888< 
957J678 
9578466 

9583*49 
9588027 

9592799 
9597567 

960*3*9 
9607086 

9611837 
9616583 

9621325 
962606 1 
96)0792 
963S5»7 
9640238 

9644953 
9649664 

96543^9 
9659069 
96637^^4 



946*557 
9467469 
947*376 

9477*77 
9482173 
94(7061 

949 « 948 
9496817 
9561 701 
9506J69 

95^1432 
9516289 

9521141 

95*5987 
95308)8 
9535664 

9540494 

9H53»9 
9550139 

9^54953 
9559762 

9564566 

9569364 

9574157 

9578945 
9583727 
9588505 
9595276 
9598043 
9602805 
9607561 
9612312 
9617058 
9611799 
96265 J4 
9631*64 
9635990 
9640710 

9645425 
9650134 
9654839 

9^59539 

966A1J3 

9668^13 



9463048 
9467960 

947^866 
9477767 
9482662 

9487552 
94924 J6 

9497314 
9502188 
9507055 
9511918 

9516774 
95*1626^ 
9526472 
9531312 

9536147 

9540977 
9545802 
9550621 

9555434 
9560243 

9565046 

9569844 

9574^3^^ 

9579423 
9584205 

9588982 

9593754 
9598520 
9503280 
9608036 
9612787 
9617532 
9622272 

9627007 
9631737 

9636462 
9641181 
9645896 
9650605 

9655309 
9660009 

9664703 
966^^92 



8 

9463540 
9468451 

9473357 

9478257 

9483151 

9488040 

9492924^ 

9497802 

9502675 

9507542 
9512404 

95 '7*60 

9522111 

95*6956 

953179^ 
953^31 
9541460 
^546284 
9551102 

9555915 
95^^0723 
9565526 

9570323 

9575115 
9579902 

9584683 
9589459 

9594230 
9598996 
9603756 
960^511 
961 326 1 
9618006 
9622746 
9627481 
9632210 

9636934 
9641653 
9646367 

9651076 
9655780 
9660478 
966517* 
9669860 



399 



9464031 
9468942 

9475847 

9748747 
9481641 

9488529 
9493412 
9498290 
9503162 
9508028 
951*889 

9517745 
9522595 

95*7440 
9532280 

9537114 

9541943 
9546766 

9551584 
9556397 
9561204 

9566006 
9570803 

9575594 
9580380 
9585161 

9589937 
9594707 
9^99472 
960413* 
9608987 
961^736 
9618481 
9623220 

9627954 
9632683 

9637406 

9642 I 2f 
9646838 

9651546 

9656250 
9563948 
9665641 



400 



Nanral 





Nttmbcri, 




m 


9670797 


918 


9675480 


9'^9 


9^80157 


9lo 


96848x9 


931 


9689497 


93* 


9<594I59 


933 


9698815 


934 


9705469 


935 


97081 16 


936 


9712^758 


937 


9717396 


938 


972202S 


939 


9726656 


940 


9731278 


941 


9735896 


94« 


9740509 


943 


9745 « 17 


944 


97497^0 


945 


9754318 


9i6 


9758911 


947 


9765500 


948 


9768083 


949 


9772662 


950 


9777*36 


95« 


9781SO5 


952 


9786369 


953 


9790929 


9U 


9795484 


9'iy 


9SC0034 


9^6 


9804579 


957 


9809119 


958 


9815655 


959 


9818186 


96o 


9822712 


961 


9827234 


962 


9831751 


963 


9836263 


964 


9840770 


96^ 


9845273 


9C6 


984977* 


967 


9854265 


968 


9858754 


V9 


9863238 


970 


9867717 



Artificial Numbers t Or^ 

3 



9671266 

9675948 
968062$ 
9685296 
9^89963 
9694625 
9699282 

9703934 
970S581 
9713222 

9717859 
9722491 
9727118 

9731741 
9736358 
9740970 

9745577 
9750180 

9754778 
9759370 

9763958 
9768541 

977JliO 
977769J 

9782262 

9786826 

9791385 

97959?9 
9800488 
980503? 
9809573 

9814108 
9818639 

9823165 
98x7686 
9851102 

9836714 
9^41221 

9845723 
9850221 

9854714 
98^9201 
9863686 
9868165 



9671734 
9676416 
958(092 
9685763 
9690430 
9695091 

9699747 

9704399 
9709045 

9713686 
9718323 

9722954 
9727581 
9732202 

9736819 

974*431 
9746038 

9750640 

9755^37 
9759829 
9764417 

9768999 

9773577 
9778150 

9782718 

9787282 

9791840 

9796394 
980094) 
9805487 

9810027 

9814562 

9819092 
9823617 

982S138 
9852654 
9837165 
9841671 

9846173 
9850670 

9855163 

9859651 
9864134 

9868515 1 



9672203 
9676883 

9681559 
96862^0 

9690896 

9^^9555 7 
9700213 

9704865 

9709509 

9714150 

97 1 8 786 

9723417 
9728043 
9732664 
9737281 
9741892 
9746498 
97<»ioo 

9755695 
9760288 

9764875 
9759457 

9774035 
9778607 

9783175 
9787738 
9792296 
9796849 

9801398 

9805942 
981048 I 

9815015 
9<I9544 
9824069 
9328!;89 
9833105 
9837616 
9842122 
9846613 
9851120 
9855612 
9860099 
9864582 
9869060 



9672671 
9677351 
9682027 

9686697 
9691362 
9696023 
9700678 

9705328 
9709974 
97I4614 

97^9249 

9723880 

9728506 
9733126 

9737742 
9742353 
9746959 
9751560 
975iSiv56 
9760747 

9765334 
9769915 

9774492 

9779064 

9783651 

9788194 
9792751 

9797304 
98018^2 

9806395 

9810934 
9815468 
9819997 
9824522 

9829041 
9833556 
9S 38066 

984257^ 
9847073 
9851569 
9 8 5606 1 
9860548 
98650JO 
9869508 



Logarithms Qto 9709.) 



Natural 
Soniberf. 

9«7 
988 
92p 
930 

931 
93* 
933 
954 
93 < 
956 

917 

93« 

939 
840 

941 
942 

943 
944 
945 
946 

947 

948 

949 
950 

95t 

952 

^54 

9^5 
9S6 
957 
958 

959 
96o 

961 

962 
963 
964 
965 

9e;5 
967 
96S 

969 
970 



9^73^39; 
9677819: 
9682494 

96S7164 
^691829 
9696488 

9701143^ 

9705793 

P7J0438' 

9715078 

97i97»3 
97^434^ 
9^728968. 
9733588 
97 J8203 

974^814: 

9747419 
9752000 

9756615 
9761206 

97^579« 
9770373 

9774950 

97 795 2 « 

9784088 

9788650 
9793-207 

9797759 
9802307 

9806850 

9811388 

9815921 

9820450 

9824974 
9829493 
9834007 
9838517 
9843022 

98475*3 
9852019 

9856^10 

9860996 
9865478, 
98<S9955 



9^73607 
9678287 
9^682961 
9687630 

9^92295 
969^954 
9701608 

9706258 
9710902 

5X715542 
97201 70 

9724805 
?72943<^ 
973405^ 
97386^4 
9743274 
9747879 

5^752479 
9757075 
976.1665 

97662^1 

977083 1 

9775407 

9779978 

9784544 
97B910O 

9793662 

97982 14 
9802761 

9807304 

9811841 

9816374 
9820902 

9825426 

9839945 

9834459 
9838958 

9843473 

9847971 

9852468 

9856959 
9861445 
9865926 
9S70403 



9674076 
9678754 
9683428 

9688097 
96^2761 

9697420 

9702074 
9706722 
97 1 1 366 
9716005 

9720639 
9725268 

9729892 

97345 u 

9739126 

9743735 
9748340, 

9*752939 

9757534 
9762124 

9766709 

9771289 

9775864 

9780435 
97S500t 

9.789562 
9794,118 
9798669 
9803216 

9807758 
9812295 

9816827 

9821355 
9S25878 

9830396 
9834910 
9839419 

9843923 
9848422 

9852917 
98574^^7 

9861893 
9S66374 

9870850 



8 

9674544 
9679222 
9683895 

9688564 
9693227 
9697885 

9702539 
9707187 

97il«30 

9716469 

9721.101 

9725731 
9730354 
9734P73 

9739587 ; 

974419^ 
974.8800 

9753399 

97,5799 M 

976^5 82 , 

9767167 

9771747 

9776322 

9780892 

9785457 
9790017 

9794573 

9799124 

9803670 

9808212 

98 1 2748 

98(7280 

9811807 

98*6^30 

9830848 

9835361 
9839869 

9844373 I 
9848872 

98<??^6 

9857856 

9861341 
9866822 

9871298 



46! 



9675012J 

967969^^ 
9684362! 
968903 oj 

9693693I 
969835 ij 
9703OO4t* 
i9707652i 
97I229d 

97«<^932[ 
9721565J 
9726193J 
973081^ 

9735435: 
9740048^ 

9744656 

9749^6a 

9753858 

9758452 

9763041 
9767615 

9772204 

9776779 

9781348 

9785913 

9790473; 
979502g 

9799579 
9804125. 
9808666 
9813202 

9817733 
9822260 

9826782 

9831299 

9835812 

9840^20 

9844823 

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9853816 

9858305 

9862790 

9867270. 

6 87 1 745 



Fff 



40? 

I Natoral 
|Numbcm 

971 

972 

975 
974 

97^ 
976 

977 

97< 

979 
980 

9«i 

98} 

984 
p8$ 

9S6 
987 
^88 
989 
990 
991 

992 
993 

994 
99^ 

996 

997 

998 

>9 



Mt facial Numlfrs : Or^ 






f 873 192 

9^76665 
9881128 
9885590 

9894498 

989S946 

990 n ^9 
99^7827 
99122^1 

99ld6^:> 
9921115 

992553^ 

9929951 

9934362 

9938769 

99i;i7« 

9947569 

19^5196^ 

9956352 

9960737 
9965117 
9969492 

9973864 
9978231 

998M93 

99-86952 

9991305 



9<?7£640 
9877109 

9881575 
9886035 

989049* 

9894943 
9899^90 
990383 J 
9908270 
9912704 

99*7133 
99*«557 
9925977 
9930392 

9934805 
9939210 

994361* 
994«oo9 
9952402 

995679* 
99^1175 
9965554 
9969930 

997430 ' 
9978667 

9983019 

99^73^7 

I 9991740 
I 9996090 



9873087 
^877556 
9882021 
9H36481 
9891)937 
9895388 
9899S35 

9904277 
99087 « 4 
9913147 

99^7575 
9921999 

9926419! 

9930834 

9935244 
9939650' 

994405 1 ! 

9948448 ' 

9952841 

9757229 

9951613 

9965992 
9970367 
9974738 
99^9104 

998 34^55 
9987823 

9992176 1 



9873>34 
9878003 
9S82467 

98^69^7 
9891382 

9895833 
9900279 

9904721 

990915% 
99^590 
9918618 

9922441 

992*'^86o 



9931275 
9935685 

9940090 

9944491 

9948S88 
9953280 

9957668. 

9962051 
996^430 

9970804 

9975 « 74 
9979 UO 
99839 1 
9^88258 
999361 I 

999^959^. 



98?<98i 

9878149 
9^82913 

98S7373 
989(^28 

9896,27^ 

9900723 
9905164 

99096:) I 
991^033 
99184O1 

59^228^4 
9927302 

995171^1 
9936126I 

994053 1 

9944931 

9949517 

9951719 
9958106 

9962489J 
9966|68| 

9971242 

9975611.; 
9V79976 

I 9984337' 
998^694 
1 9993046r| 









Logarithms (^to loooo.) 40} 



r Nat Of a' 
]Nttnibcr:>. 

97 i 

972 

S73 

975 
976 

977 
P78 

979 
980 

98 c- 

982 

985 

987 
988 
989 

990 
991 
992 
P93 

994 

99^ 
9?6 

997 

998 

999 



"9^7442^ 
9»7889<5 
9S8j3^3 
9887Bi« 
9892273 
9896722 
9901168 
990 5 60S 
9910044 



957487$ 

9879i4J 
9^S}S 6 

98sv2^4 
9892718 

9897167 
9901612 

9906052 



9914475 9914919 
9918903 991934^ 



9923726 
99*77*4 

99}ai57 

9936^66 

994097 « 

9945371 

99497^7 
9^54158 

995854$ 
9962927 
99^7505 



99237^8 
992^18$ 

993*598 

9937007 
994»4if 
9?458ii 

9959206 

99545?^ 
995S9H3 

9^63365 



9875522 9875769 
9^79789 98802516 
98T4252 9884698 
9'<887JO I 98891^5 
9893165 I v*9?6ofi 



9897^12 
99^205^ 
9906496 
9^10931 

99M36i 
9919788 

v;9242io 
9928627 

9933039 
9937448 

994185* 
9946*51 
9950645 

995503^ 
9959422 
996^803 



9967743 I 9968180 
9975^S53 



9971^79 9972116 

9976048 9976485 

9980413 99S0849 

9984773 9985209 

99891*9 99S9564 ,_ 

9993481 99939*6 9994 3 SO 

9997828 9998292 t 9998697 



9976921 
9981285 

9985645 
99900OD 



9898056 
9902500 
9906940 

9911374 

9915805 

9920230 
99246U 

992906 « 
9955480 
99)7^8 
9942291 
99466,0 

995X0S5 

9955474 
Q959860 

99^4^1 
9968618 

9972990 

9977558 
9981721 
9986080 

9990435 
999478? 
i??9l2i 



9876216 
98806^2 
9^85144 

9S8y450L 
9S94050 
9898501 

990:1944 
9907383 
99II818 

9916247 
9920673 

9925093 

9929510 
9933921 

9938329 

9H275I 

9947130 
9951524 

9955913 
9960298 

9964679 

9969055 
997J427 

<r977794 
9981157 
9986516 

9990870 
99952^0 

999^566 



looDO it's Log. 93 4,0000000 



Tbe End of the Table of the Logarithms. 



Fffz 



\ 



-J^-^_ 



•. s 



/ 



/ 



/