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Qa

/77J

THE

Young Mathematician's Guide :

jSmg a PLAIN and EASY

INTRODUCTION

T O T H E

MATHEMATICKS.

IN FIVE PARTS.

VIZ.

I* Sin'tl^metlCfc, Vulgar and Decimal, with all the ureFul Rules ; and a General Method of ExtrafUng the Rpots of all Single Powers.

n. £IIg;eil)d9 or Arithaietick in Species ; wherein the Method of RaifingandRefolyiiig Equations is rendered Eafy; and illuftrated with Variety of Examples, and Numerical QueAions. , Alfo the whole Bufineis of Interefl and Annuities, (sfr. performed by the Pen.

HI. The (Eleinenttf of (Beometrp contrafted, and Analytically demonftrated ; with a New and^Eaiy Method of Ending the Circle's - Periphery and Area to any afli^ned £xa£lnefs, by one Equation only ; alio a New Way of making Sines and Tangents.

^- Conic &e(tfon0, wherein the chief Properties, l^c. of the EUipfis, Parabola, and Hyperbola, are dearly demonilrated.

V. The arittmctiCk of infinites explained, and rendered £afy ; with it's Application to fuperficial and folid Geometry.

With an Appendix of fB^attlCal (Saufffng:*

By JOHN WA R D. ^^

The Twelfth Edition, Carefully Correfled and Improved by Samuel Ciatik.

To which is added,

A SUPPLEMENT, concaining the Hiflory of Log aritijms, and an Index to the whole Work.

' LONDON: ^

IWntcd for J, BitcROfrr, J. Rivikctok, L. Hawes, W. Clarki I aodR. Collins, W. Johnston, T. Longman, T. Caslqn, S. CaowDia, B. Law, T. Beckkt, G. Robinson and J. RoBsars, S. Bladon, andR. Saz.owin. 1771, ^

is'i^a To the Honourable

Sir Richard Grosvenor, of Eaioff^ in the County Palatine of Chefter^ Baronet.

SIR,

WHEN requcftcd by fbmc Bookicllers in London, to Revifc and Prepare Ais Trea-^ tife for a New Imprefpon, and once re- folved to anA^er their Demands; I was not long confidering at whofe Feet to lay it.

My Memory may indeed be impaired by Age» Misfortunes^ and Accidents ; nay, I am feniible it is fo: But it muft be entirely loft, when I am forgetful of the great Obligations I lie under to Sir Richard Grojvenor.

Your Hoipicality and Generofity make you ftand unenvied in the Abundance of Fortune. Any Up- ftart may contrive to fpend a Great Eftate; but it is a Felicity almoft peculiar to Great Birth to become One.

Were I now to defcribe Liberality, without Pro- fufenefs I Steadinefs in Principles, without any pri- vate View; Candour and Affability, Good Nature joined to.ibund Judgment, and a Serenity of Tem- per, which your Enemies will always find the Com- panion of true Courage; and then pronounce that you are poflefTed of all thefe good Qttalities in as high a Degree as moft Men living ; NfeQei^leman that knows you well, would think I fiatii^ccd you.

A 2 , JSir>

The DEDICATION.

Sir, Give me Leave to iky, I honour your Cha- radter, and love your Pcrfon : My Exprcffions arc uncourtly, my Stile unpolifhed, and therefore more proper to be prefixed to a Work wherein the Matters related are indeed clad in a plain and homely Dre(s ; but they are true, and defighed to pn^agate Ma- thematical Learning among fuch as deiire to be in- troduced into that Sort of Knowledge ; and I am extreamly pleafed they are permitted to be fent into the World under your Protedion.

That you may long live, to promote the Good of your. Country, and that City in whofe Intercfl you have fo heartily engaged yourfclf ; and that you may ever fucceed in your own private Affairs^ and live to enjoy all the Blefiings that attend a quiet prudent Life, is the carneft rrayer of.

Honoured S I Rp

Tour moji Obliged^ Humble^

and Obedient Servant,

J. WAR a

To the R E A D E R.

ITbixi it medle/s (and almoft endlefs) to run over ell ths lUefubufs and Advantages of M3LthcmaLUcks in General-^ and JbaU therefore only touch upon thofe two adndrabU Sciences^ Arithmedck and Geometrv ; which art indeed the two grand PiOars (or rather the Foundations) t^on which all other Parts of Mathematical Learning depends

As to the Ufefubiefs rf Arithmetick, it is well fnown that no • Biifinefsy Commerce^ Tradoj or Employment whatfoever^ even from the Merchant to the Shop-keeper^ &c. xan be managed and carried 19, without the AJMance of Numbers.

And as to the Vufidnejs of Geometr^^*// is as certain^ that no curious Arty oj' Mechanict-Xf^oriy can either be invented^ improved^ ' or performed^ without it's affijfiing Principles ; tho' perhaps the Artijiy 9r Workman^ has but little (nay rcarcc any) Knowledge in Geometry.

Then^ as to the Advantages that arife from both thefe Noble Sciences^ when duly joined together^ to afftfi each other ^ and then applfd to PraHice^ (according aa Occafion requires) they will readily be granted by all who conjider the vafi ^Advantages that accrue to Mankind from the Bujinefs of Navigation only. As alfo from that of Surveying and Dividing of Lands betwixt Party and Party. Befides the great PUafure and life there is from Time* keepers^ as Dials f Clocks^ fFatches^ Sec All thefe^ and a great many more very ufeful Arts^ (too many to be enumerated here) wholly depend upon the aforefaid Sciences.

And therefore it is no JVonder^ That in all Agfsfo many Ingenious and Learned Perfons have employed themfehes in writing upon the SubjeSt of Mathematicks ; but then mofl of thofe Authors feem te prtjuppofe that their Readers had made feme Progrefs in that Sort of Learning before they attempted to perufe thofe Booksy which are * generally large VolumeSy written in fuch ahjhrufe Terms^ that young Learners were really afraid of looking into thofe Studies.

Thefe Confiderations firjl put me (many Years ago) upon the Thoughts of endeavouring to compofe fuch a plain and familiar In^ troduSfion to the Mathematicks, as might encourage thofe that were willing (to fpend Ibme Time that Way) to venture and proceed on with Onarfulnefs ; tho* perhaps they were wholly ignorant of //'a frfi Rudiments. Therefore t began with their firjl Elements or Printiples.

That

The Preface.

That is^ I began with an Unit in Aritbmeticiy and a Point in Ceonutryi and from theft Foundations proceeded graduaUj.mf% lead^ ing the young Learner Step by Step with all the PUunnefs icemldj &c.

And for that Reafon I puhlijhed this Treatife (Anno 1 707) by the Title of the Young Mathematician's Guid^ i which h^s annvered the Title fo fvell^ that I believe I may truly fay (widioat Vanity) this Treatife hath proved a very helpful Guide to near five thoufcmd Per feme ; and perhaps mo ft of themfuch as would never have boieel into the Macbematicks at all but for it.

And not only foy but it bah been very wett received amongfl tha Learned^ and (i have been often tdd) fo well approved on at the Vniverjitiesy fif England, Scotland, 011a Ireland, that it is ordered to he publicity read to their Pupils j &c»

The Title Page gives a jhort Account of thefeveral Parts treated ofy with the porre£lions and Additions that are made to this Twelfth Edition, which I Jhall not enlarge upon, but leave the Book tofpeakfor itfelf\ and if it be not able to give Satisfaction to thg Recsier^ I cm Jure all I can fay here in ifs Behalf will never re^ commend it : But this may be truly faidy That whoever reads it overy will find more in it than the Title doth promife^ or perhaps ie expels : it is true indeed^ the Drefs is but Plain andHomefyy it beir;g wholly intended to injlru^y and not to amufe or puzzle the young l,earner with hard JVordsy and obfcure Terms : However y in this I /halt always have the Satisfa^ion ; That I have fincerely aimed at what is ufefuly tho* in one of the meaneft IVays ; it is Honour enough for me to be accounted as one of the Under- Labourers in clearing the Ground a little y and removing fome of the Rubbijb that lay in the fFay to this Sort of Knowledge. How well I hifoe per" formid Thaty mufl be left to proper Judges,

To be brief 'y as I am not fenfible of any Fundamental Error in

this Treatifcy fo I will not pretend to fay it is without ImpcrfeHions^

(Humanum eft errare) which I hope the Reader will excufcy and

faft over with the like Candour and Good-fFill that it was compofed

forhisVfe.

T HE

THE

CO N T E K T S.

arftSmetfclu Pani.

nRac9gnitay Concerning the proper Subje^s, or Bufmefs of Ma* ^ thematicks^ &c. ^ Page f

Chap. I. Concerning the feveral Paris of Arithmeticky and of fuch CharaElen as are ufed in this Treatife. ^

Chap. n. Concerning the Principal Rules in Jrithmetick^ and

how they are performed in whole Numbers* 5

Chap. III. Concerning Addi(iony SuhtraSiiony and Redu£iion of

Numbers that are of different Denominations, 31

Chap. IV. Of Vulgar Fra£lions^ with all their various Rules. 48 Chap. V. Of Decimal FraSiiom or PartSy with all the ufeful

KuleSj and ContraSiionSy &c. 57

Chap. VI. Of continued Proportion^ both Arithmetical and Geo*

metrical', and how to vary the Order of Things, 7 a Chap. Vn. Of Disjunct Proportion^ or the Golden Rule^ both

Dire^j Reciprocal or Inverfe^ and Compound, 85 Chap.VIir. The Rules of Fellowjbipj Bartering^ and Exchanging

of Coins, 99

Chap. IX. Of Jlligation or Mixing of Things^ with all ifs

Varieties or Cafes, ^ lO

Chap. X. Concerning the Specif ci Gravities of Metals, &c. 117 Chap. XI. Evolution or ExtraSfing the Roots of all Single Powers^

how highfoever they are^ by one General Method. 123

aiffeijja. Part 11.

Chap. I.

The Method of noting down ^antities^ and tracing

of the Steps ufed in bringing them to an Equation. 14.3 Chap. II. The Six Principal Rules of /llgebraick Jrithmetick, in

whole ^amities, J4.7

Chap. III. Of jflgehraici Frontons y or Broken ^antlties. 163 Chap. IV. Of Surd$y or Irrational ^^antities. 172

Chap, V. Concerning the Nature of, EquationSy and hfzv to pre^

pare them for a Solution^ &c. 175

Chap. VL Of Proportional J^antities, both Arithmetical and GeO"

mttric^ continued J alfo of Mfi/ical Proportion, 1 84

• Ch'4p.

I'he CONTENTS.

Chap. VII. Of Proportional ^amties Disjunifj hth Simp/e^ PupUcate^ and Triplicate j and how to turn Equation^ into AnahgieSy &c* Page 190

Chap. VIIjl, Of Suhflitution j and refolving ^uadratick Equations,

'94-

Chap. IX. Of Analyji%y or the Method of Refolving Problems^

Exemplified by Forty Numerical ^e/lions* 20Z

Chap. X. The Solution of all Kinds of jfdfe^ed Equations in

Numbers, 234.

Chap. XL Of Simple Inter eji^ and Annuities, in all their various

Cafes, ^ . -.245

Chap. XII. Of Compound Inter ejl., and Annuities both for Year sand

Lives J and of Purchafing Freehold EJiates. . .25^

€!eOm£ttp^ Part III.

Chap. I. Of Geometrical Definitions and Axioms^ tct* 283

Chap. IL The Firfl Rudiments or Leading Problems in Gea^ me try. 292

Chap. HI. A Collision of the mojl ufefut Theorems in Plane Gecrnetryy Analytically demonfirated^ 300

Chap. IV, The Algebraical Solution of Twenty e(ify Problems in Plane Geometry \ which does in part Jbew the life of, the lafi Theorems. '320

Chap. V. Practical Problems and Rules, fhr fnding tie Areas of Right-lined Superficies^ demonftrated, 338

Chap. VI. A NiW and Eafy Method of finding the Circle's Pc^ rlplury^ and Area^ t$ any affigned ExaSlnefs ; by the Solution of one Equation only, Alfo a New Way a^ making Natural Sines and Tangents a priori. 347

. Conickaecttowa* Parti v.

Chap. I. Definition of a Cinej and a'l ifs SeJfions^ &c. 36 1 Chap. II. Cof:cerning the chief Properties of the Ellipfis, &c.

Chap. III. Cmcerning the chief Preperties of the Parabola. 380 Chap. IV. C'jnceming the cbitf Properties of the Hyperbola. 386

antOmettcfe of Jnfinttesf* Part v. ^

The Arithmetic^ of 'Infinites explained^ and rendered eafy ; with />'/ Af^Ucation to Geometry^ in demonjirating the Super-^ ficifd and Solid Contents of Circular and Elliptical Fi^ gurcjy See. 397

,4n Appendix af PiatttCal C^aUffinff*

fThe^etn gll the chief Rules andFroblems ufefut in Gauging^* are applied to Practice y &C» 4^3

'. ; «.'■/// /^'' fi)!.^ - ,1 t' •' // ''V^

Parti. 1

A N,

INTRODUCTION

T O T H E

PART I.

P RM COG N ir A.

THE Bufinefs of Mathcmatlcks, both in Theory and Pradice, is to fearch out and determine^ from given Yi^iXZy the true Quantity ^ either of M^iitcr^ Space, or Motion, according as Occafion requires.

By Quantity of Matter is here meant the Magnitude, or Big^ infs of Bodies (of the fame Denfity) whofe Length, Breadth, and Thicknefs, may either be meafured^ or eftimated.

By Quantity of Space is meant the Dijiance which a Body moves 9Ver in a gruen Time^ with a certain Velocity, '. And by Q^z.x\iity of Motion is meant the Swiftncfs of any moving Body drawn into its ^antity of Matter.

The Ccnfideraiion of thefe^ according as they may he propofedj an the Subjects of the Machematicks, but chicflythat ^Matter.

Now the Conjideraticn of Matter, with r^fpUl to it's Quantity, Form, and Pofition, wUch may eitlwr be Natural, Accidental, or Defigned, may poJfiUy admit of infinite Varieties : Rut all the Va- rieties that are yet known^ or indeed can be conceived^ are wko.'ly comprized under the due Conjideration cf thefe iwo^ ^Magnitude and Number, which are the proper Subje^s of Geometry, Ariih- mctick, and Algebra. JU other Parts of the Mathematicks being only the Branches of theft three Sciences, w rather their Jpplicatim to particular Cafes^ -'h

B • (Peometcg

2 PR^COGNITA, PaxtJ.

(SZOXtlttt]^ is a Scimce by which we fearch outy and come U JhioWy either the whole Magnitude, or fome Part of any propofed "Quantity ; and is obtained by tomparing it with another known ^antity of the fame Kind^ which will always be one of thefe^ viz. A %iXiZy (or Length only) A feurfeee, {that /j, . J^ijgih tfTzi Breadth) ^ ^ ^0U&, (which i^^/^ Length, Breadth, and Depth, or Thicknefs) Nature admitting of no other Dimenjions but thefe Three.

SLVXt^XXlttlt^ is a Science by which we come to know what Number of Quantities there are (either real or imaginary) of any Kindy contained in anaihet Quantity of the fame Kind: Now this Conjiderathn is very different from that of Geometry, which is only to find out true and proper 'Ahfwers to ail fuch ^eftions as demand^ how Long, haw Broad, how Big, &^. But when we confider either more Quibtities than one^ or how often one Qban- tity is contained in another y then we have recotqfe ^^Arithmetick, Vibich is to find out true and proper Anfwers to all fuch ^ueflions as demand^ haw Many, what Number, or Multitude of Quantities there are, . To be brieK the Subje^ of Gcomttrj is that of Quan- tity, i)iiith refpe£i to tfs Magnitude only ; and the Subjeii ^Arith- inetick is Quantities with refpeSi to their Number only*

2Ig;tbca is a Science by which the mofl abflrufe or difficult ProblemSy /» Arithmetick cr Geometry, are Refolved and Denjon- firated \ that is^ it equally interferes with them both ; and tbere^ fore it is promifcuoujly nanied^ being fometimes called Specious Arithmetick, as by Harriot, Vieta, and Dr Wallis, fs^f. And fometimes it is called Mo^^ern Geometry, particularly by the late ingenious and great Mathematician Dr Edmund Halley, Savilian Profeffor of Geometry in the Univerfity of Oxford, and Royal Afironomer at Greenwich; who^ in giving the following Inflame of the Excellence of our Modern Algebra, writes thus :

• 7T?e Excellence of the Modern Geometry (faith he) is in ' * nothing more evident^ than in thofe full and Adequate Solutions

* // gives to Problems ; reprefenting all the pojfible Cafes at one

* View^ and in one general Th^rem mawf Times comprehending

* whole Sciences ; ^hich deduced at length into Propofitions, and *- demonjlrated c^ter the Manner'-^ tlfe Aft\;ients»V might well he-^

* come the Subje^s of large Treatifu: Fbr whatfoever Theorem ■ ^ folves the mofl complicated Problem of the Kind^ *does with a

« due Reduilion reach all the fubordinate Cafes.' Of ptUk he gives a notable Jnfiance in the Doftri^e V* Dioptficks for finding the-^ocl of Optic Glaffcs uniycrjally.' Cf7eif Fhilofophical Tran£ fftions, Numb. iQ^].

'4i

Chap. I. Of Cftttactecis*

Tbus you have nfiiort and ggnerd Account of the proper Subje£fs of fhofi nobU itnd ufeful Sciences, Aritbmecick, Geometry, arkl Algebra. / Jhall now proceed to give a particular Account of ittcb ; and firft of Arhhmecick, which is the B^{i% or Foundation of all Arts, ioM Marhematick and Mechanick ; and therefore fi ought to he well undoKftood before the rejl are meddled witbaL

C H A P. I.

Onc^rnhig tbefeveral Parts of ^i^tt!it% with the De^ fimtiam of fucb CbaraSiers as are ufcd in this Treatife.

^g^iXli\Xtttit1Si^of the An of Numbsring^ is fitly divided into three diftinA Parts, tivo of which are properly called JV!9/«- to/, and the third Artificial.

The firft, beii^ the mofi^ plain and eafy, is commonly called Vulgar Arithmetick in whole Numbers ; becaufe every unit or hteger concerned in it, reprefents one whole ^antity of fome Species or thing propofed.

The fecond is that which fuppofes an Unit (and confequently the ^antity or thing reprefented by that Unit) to be Broken or Divided into any Number of equal Parts^ and confidersof them cither as pure Parts^ viz. each lefs than an C/mV, or elfe of Parts and Integers intermixt. And is ufually called the DoStrint of Vulgar Pr anions.

The third, or Artificial Party is called Decimal Arithmetick ; (eing an Artificial Invention of managing Fra£fions or Broken Numbersy by a much more commodious and eafy Way than that of Vulgar Fra^imrT For the feveral Operati#n8 performed in Decimals^ differ but little from thofe in JVhole Numbers : and therefore it is now become of general Ufe, efpfccially in Geome* trical Computations,

iarij:i)^ltirtftk (in all it's Parts) is performed by the various - ordering and difpofing of Ten Arahick CharaQers or Numeral figures (which by fome arc called Digits.)

. C OneyTwOyThreeyPouryPiveySixySeveny Eight yNtnOyCypher. i I 2 3 4 567 8 9 'o -

The Ufe of thofe Cbara£bers is fmd to be frrfl introduced into England tiiore than fix hundred Years agOy viz. about the Year ^130, viie Dr WallisV Algebra^ Page 12.

B 2 The

aritSmetlClt^ Parti.

' The firft of thefc CharaSfen is called Umty^ and reprefents onc^ of any Kind of Speeds or ^antity. As one f^orld^ one Siar^ one Aian^ &c.

Viz. Umty is that by which every thing that is, is called one, (Euclid. 7. Def, I.) and is the beginning of all Numbers, That is to fay. Number is a Mukitudeoi Units. Euclid. 7. Def. 2.

For, one more one, make Two ; and one, more one, more one, make Three, tfr. TVhichis the Jirji and chief VoRuX^Xic^ or rather an Axiom in Arithmetick.

Vix I '^^^^ 1 + 1=2. 1+1+1-3. 1+1+1+1=4. ' \ 1+1 + 1 + 1+1=5. And fo on to 9.

Nine of thefe Figures were thus compofed of UnitSy and dif- ferently formed to reprcfcnt fo many Units put together into one Sum^ as was intended each (hould denote : Nine being thegreateft Number of Units that was then thought convenient to be exprefled by one fingle Character ; the laft of the Ten h only a Cyphery or (as fome phrafe it) a Nothing, becaufe of itfelf it f^nifies no- thing ; for if ever fo many Cyphers be Added to, or Subflraded from, any Number^ they can neither increafe nor diminifii that Number ; but yet, as a Cypher (or Cyphers) may be placed, the other Figures^ will become of different f^alues, from what they were before, as will appear further on.

For the more convenient ordering of the aforefaid Numeral Figures according to the feveral Varieties that happen in CompU'r tations ; I do advife the young Learner to acquaint himfelf with the Signification of the following Algebraick Signs or Chjara^ers^ which he will find of excellent U(e, as being a fhorter, better, and much more fignificant Way of denoting what is to be done, (in mod Opej^oiions) than can otherwife be cxpreiTed in Words at ieugch.

SIGNIFICATIONS.

Signs Names. ^ The Sign of Addition ;' as 8+7 is 8 more 7^ and fignifies that the Numbers 8 and 7 are to I be added info one Sum. . The like is to be un- PJus tfr^derflood when feveral Numbers are cQnneAed > together with the Sign +. 1

As 34+22+9+45, ifc. denotes thefc are i all to be added into one Sum^

+n

more.

The

cbap. I. Of c^aciKteciBL 5

Hj^^ C The Sign of SubJfra£lioni as 9—6 is 9 lefi ?V S 6, and fignifies that 6 is to be taken from 9^ ^ ^^J^* L that fo their Difierence may be fouiid.

H* • r The Sign of MuliipUcation \ as 9 x 6, is 9 in-

fh^ 1 ^^ ^' ^^ ^^Snifies that 9 it to be Miiltipiifd '^^ • L into or with 6.

{The Sign of Dtvifitni as.8-s-2, is 8 by 2« and fignifies that 8 is to be Divided by 2, alfe thus 2) 8 (4 or thus *} each fignifying the lame thing, to wit, 8 Divided by 2.

= 1 1 Ev*^*

The Sign of EquaSiy or Eqt^iw^ viz. when Lthis Sign = is placed betwixt Numiers (or f^uantitus) it denotes them to be equal ; as |9=9, or9+6s=i5, or9— 6=:3»farf. That 'is, 9 is Equal to 9» or 9 more 6 is Equal to 15, ^and 9 lefi 6 is Equal to 3, is^c*

The Sign of Proportions or that commonlf .called the GolJin Rule^ or Rule of ThrUj and

Ho • ): : is always placed betwixt the Two middte "' yTtrms or Numbers in Proportion. That 2 : 8 : : 6 : 24« To be read thus $ as 2, is to 8 ^ /o is 6> to 24*

Thefe Sigui and their Significations^ being perfeAly learnti will help to (horten the Work.

CHAP- II.

Concerning the Principal Rules in 9rft|&met(C(t> and bow they are performed in Whole Numbers.

THE Rules by which Numerical Operations ^xt performed in all the Parts of Arithmetick^ are many and varioust , ^veral of them being- formed and raifed as Occafion requires, when applied to Practice j yet they are all comprehended within the due Coofidcration of thcfc Six, viz. ^Utmuatfoil (or Jl5o«

tation)

SDmniuRCR*

Parti.

tatioti) atmftion, &uIitrait(on, S^ultCpUcatfon, a)f'

j)faon, and Solution, or Extraiiim tf Rtets.

Sea. I. oy j^ttuutattott «• jpiotatfon.

j^Utneratfon, or NotatltHt teacbeth t^Read or Expreft the tns Value of any Numbir when writ down ; and confequently .Co wrke down any propofed Numbtr according to it's true Vahte .frben it is named : And this oonfifteth of Two Parts,^ * -

X. The due Order of placing 4^wn Figures^ a. The true valuing of each Figure in it's Place.

Both which are plainijexhtbited in thd fbllqnving Table*

ft*

1^ I**

^1

6 7 8,9 8 7,6 ;-4., 3 ^ ^

J^criod of iFeriod of {Period of I Penod of

•fMjJftoiu, \

}fanA.

Umn.

Sy this iflslimterfttiOn TMe it is apparent, *tbdt the Orderof Places is reckoned from the Right-hand towards tht Left ; the firft Place of any Number being always that which is the out- moft Figufi to the Right-hand :. ,and whatever Figure Jlands in that Place, doth only fignify it's own'fimjple Value, viz. ^foJ inanv Units as that Figure reprefents.

The fecond Place is that of fens^ and any Figure (landing in that Place figaifieth fo many Tens as that Figure reprefents Units.

The

Chap. 2, Of jaumgcaiion. 7

Tbe third Place b Hunirtds^ the fourth Place Thoufandsy ice. That is, each Place ^wards the Left-hand is Tfn Times the Value of that next It, towards the Right. •

For Inftancet Aippofe 759 ^ere propofed to1>e read or pro- nounced accbrding to the Value of each Figure as they now £and« The firft F^ure in this Sum is 9, becaufe it ftands in the Place of Units^ and therefore figniiks but it's own fimple Va^ lue^ to wit, 9 UmtSp or 9. The Tecond Figure 5 ftands in the Place of T^ensy and therefore fignifics Five Tens or Fifiy. The ^gwf 7 fiands in the third Place* or Place of Hundreds^ and therefore it fignifies Seven Hundred i and the whole Sum is to bet read or pronounced thuA, Seven Hundred Fi/iy Nine^

Note, Akfaough the Figure 7 flends in the third Place (accord- ing to the Order of Numbering) yet when the whole Sum comes to be read, it is firft pronounced ; the reading of Numbers being performed like that of Letters or Words, always beginning with the ontmoft Figure towards the Left-hand, and fb many Figures as are placed together without any Point, Comma, Lme, or other Note of Diftindtdn between them, are all but one Sum^ and wkA be read as fiicb.

For Example, 763596 is but one entire Sum or Number^ not-' vithAandkig it conMs of fix Places of Figures^ and is thus read ; Seven Hundred Sixty Three Tboufand^ Five Hundred Ninety Six. ■ ,

The lilce is to be obTerved in reading or expreffiog the trae Vahjc of any Sum or Rani of Numbers confitKng of Seven^ Eighty Nine^ or more Places of Figures^ each Figure ^cing to be valued accordbg to it's Diftance ftoA tbe Place of Unity ; As in the hrtmiog Table.

Ifow Aich Values may as well arife by Cyphers^ as by other Figures ; for inftance, 6 flanding by kfielf, reprefents but Six Units : But if z Cypher be annoxt to it thus, 60, then it becomes Sixty ; for the Cypher poffeffii|g the place of £/«///, hath hereby removed the 6 into the Place of Tens ; and another Cypher more would make it 66a, Six Hundred^ &c* '^^ Whekce ifk may be noted, that' although a Cpber of itfelf ^ \vdfj nothing (as hath been faidbefoio) yet beiHg^Iaced qp tfiS Ught-hand of any figure ^ it augments the Value of that Figure 7 advancing-k into r higher Place than t>therwife it would have beeflr^^ad not die Cypher beemtkere, r •;

* Taka one Exam^e more is Numiration (if you ^{eafe, that in the Table), vis* ^t^^'jfk^^yti^ which i% jf^e^ording as is there fignifidi, '* \ » • - . « . • ■ - 1

' •' = * Six

dititlfntCtfCfc- Part I.

Stx Hundred Seventy Eight Thoufand Millions^

Nine Hundred Eighty Seven Millions^

Six Hundred Fifty Four Thoufand^

Three Hundred Twenty One Units j of any propofed Specie^ or ^amities whatfoever.

And here it may be obfcrved, that every third Figure from the Place of XJnitSj bear$ the Name of Hundreds ; which fliews that if any great Sum be parted, or rather diftinguifhed into Periods^ of Three Figures in each Period (as in the foregoing Table) it will be of good Ufe to help the young Learnec in the eafier valuing and expreifing that Sum.

seft. 2. Of anwttoru

Pojlulate or Petition. Thett any given ^UXHitV may he increafed or made more^ by putting

^ another ^umbet to it.

fllltlftf on is that Rule by which feveral Numbers are colleded and put together^ that fo their Sum or Total Amount may be known.

In this Rule Two Things being carefully obfervcd, the Work will be eafily performed.

1. The hrft is the true placing of the Numbers^ fo as' that each Figure may fland diredly underneath thofe Figures of the* fame Value, vi%. place Uiuts under Units^ Tens under Tens^ and Hundred^ under Hundredsy &c.

Then underneath the loweft Rank (always) draw a Line ta feparaee the given Numbers from their Sum when it is found.

Example. If thefe Numbers 54327, and 2651, were given to be added together, they muft be placed

543^7 2651

Thus, I

2, The fccond thing to be obferved is the doe G^Ileding or Adding together each Row of Figures that ftand over one ano- ther of the iame Value : And that is thus performed : RULE.

Always begin your Addition at the Place of Units, and Add together aU the Figures that Jtand in that Place, and if their Sum he under Ten, Jit it down below the Line underneath ifs own Place ; but if th^tr Sum be more than Ten, you mu/l fit down only the o^erplus^ or odd Figure above the Ten (or Tens) cmdji many Tens as ibe Sum of thofe Units amount tOy you mujl carry I to

chtp. 2. Of anwttotu 9,

U tie place 9f Tens \ Adding them and all the Figures that ftand in the place of Tens together^ in the fame manner as thofe rf the Units were added \ then proceed in the fame order to thi pifee of Hundreds, and fo on to each place until all is done.

The Sum arifing from thofe Additions will be the Total Amount fcqaired.

£ XA M PL E I. Let it be required to find the Sum of the aforefaid Numbers^

56978 the Sum required. Beginning at the place of Untts^ I fay I and 7 is 8, which being left than io» I fet it down (according to the Rule) under- neath it's own place of Units ; and then proceed to the place of Tensy faying 5 and 2 is 7, which being lefs than 10, 1 fet it down undern^th it's own place of Tens^ and proceed to do the like at the place of Hundreds^ and then at Thoufands^ fetting each of theur Sums underneath their own refpe£live places : Laflly, be- caafe there is not any Figure in the lower Rank to be added to the Figure 5, which (lands in the place of Ten Thoufands^ in the upper Rank, I therefore bring down* the faid 5 to the reft, placing it wnderneath it's own place, and then I find that 54327+265 i=s56978, the true S^m required.

EXAMPLE 2.

Suppofe it were required to find the Sum of thefe Numbers^ 3578+496+742+184+95. Thefe being placed, as before ^ireded, will ftand as in the Margin. Then beginning (as before) at the place of Units^ fay 5 and 4 is 9, and a is 11, and 6 is 17, and 8 is 25 ; fet down t^e 5 Units underneath it's 3578 own place of Units^ and carry the 20, or two Tens^ to the 496 piflceof 7/»f (at which place they arc only 2) faying, 2 74a ^ 9 is II, and 8 is 19, and 4 is 23, and 9 is 32, and 7 184 Is 39 ; fet down the 9 underneath its own place of Tens^ 95 and cany the 30, or three Tens (which indeed is 300) — — to the place of hundreds^ at which place they are but 3, 5095 &ying, 3 I carry and i is 4, and 7 is 11, and 4 is 15, jkid 5 » 20 i here becaufe there is no Figure overplus (as before) I fet down a Cypher underneath the place of Hundred^ ««<! carry the 2 Tens (or rather the 2000) to the place of Tboufands, iaying

C (*•

IP gtfl^mitfclu Parti.

(as before) 2 I carry and 3 is 5, which beitie the laft, I fet k down underneath it's own placie, and all is finiihed. And find th<i Svmor Tif^/ amount to be 5095=^3578+496-|-74a+i84-|-9S« • If this Example be well oonfidered, it will be fufficient i^ ihew the ufual Method of JdJitim in whole Numbers ; but to make all plain and clear, I (hall (hew thd" young Learner the Reafon of carrying the Tens from one Degree or Row of Figures^ to the next Superidr Degree, which is done purely to fave Trouble, and prevent the ufing of more Figures than are really necefTary, as will appeat by the following Method of adding together the fame Numbers of th)s laft Example.

Thus, add together each fingle Row .of Figures by itfelf ; as if there were no more but that one Row, fetting down the Sum underneath its own place.

The Sum of the Row oWntts^ is I 1 1215 "

The Sum of the Row of Tens^ is I 3J7 6 t a ja

The Sum of the Row of Hund. is 1700 ' ^

The three Thoufand brought down |3|o| o|o .

The Sum or Total Amount, as before, is 5095

;}

From hence I prefunie it will be cafy to conceive the tni^ Reafon of carrying the aforefaid Tens^ ; and alfo that Cyphers do not augment or increafe the Sum in Addition. (See Page 4.) .. I might have here inferted a Lineal Demonftration of this RuU of Addition \ but I thought it would rather puzzle than improve a young Learner^ efpecially in this place ; befides tbb Reafon of it is fufficiently evident from that Natural Truth of the WhoU being Equal to all ifs Parts taken together. Euclid u Axiom 19.

That is, the Numbers which are propofed to be added toge- ther, arc by that Axiom underftood to be the feveral Parts, an<l their Sum or Total Amount found by Addition is underflood to be the Whole.

And from thence is deduced the Method of proving the ^ruth of any Operation in Addition^ viz. By /parting or fcparatinj^ the gwtty Numbers into Two Parcels (or more, according to the Largenefs of it) and then adding up each Parcel by itfelf: For if thofe particular Sums fo fpund^ be added into one Sum^^ an3 that Sum prove £quai» 4>r the iadie with the Total Sum firft

found|^

r

Chap. 2. Of Sttttrattfon^ n

Sounds then all is right; if not^ care muft be taken to difcover aad correA the Error.

E XA M P L E.

5647 7

3289 (.The Sum of thefe Parts is 12952

4016)

Add.

2900 ^

5007 vThe Sum of thefe b 95^3

16063 > !■

The7'4i^/5tfi7iof] ^ , . The fijviM of each 1 ^^ ^^

all thefe Parts 1^2405 Paiccl put togpt'her $ ^^*^5

Sea. 3. Of ^ufttratfton.

Pofiulate or Petition.

That any ^ViXtibZt may be Uminifljtd^ by taking another leji "^WXa^SZK from it.

&tt&tt9(tf0tl is that Rule by which one Number is deduced or taken out of another^ that fo the Remainder^ Differences^ or Excefs may be known.

As 6 taken out of 9, there remains 3. This 3 is alfo the Differetme betwixt 6 and 9, or it is the Excefs of 9 above 6.

Therefore the Number (or S«/;»^ out of which Subtraction is required to be made, mud be greater than (or at lead equal to) the Subtrahend or Number to be fubtraSfed.

Note, This Rule is the Converfe or DireSf contrary to Addition.

And here the fame Caution that was given in Addition^ of placing Figures diredly luider thofe of the &me Value, viz. Uniti under UnitSy Tens under Tens^ and Hundreds under Hundreds^ &c. inuft be carefully obferved ; alfo underneath the lowed Rank, there muft be drawn a Line (as before in Addition) to feparate (heeiven Numbers from their Difference when it is found.

Then having placed the lefler Number under the greater, the Qperation may be thus performed.

RULE.

Begin at the Right Hand Figure or jplace of Units {as in Addition} and take or fuhtraSi the lower Figure in thai place

C a frot^

12 StftfltnCttClU Parti.

from the Figure that Jlandt over it, Jitting down the Remainder or Difference underneath its own place. • If thi Tw^ Figtucs chance to he Equals fet down a Cypher : But if the upper Figure be lefs than the lower Figure, then you mujl add lo to the upper Fisure, or mentally call it jo more than it li, pnd from that Sum fuoTtra^ the lower Figure, fetting down the Remainder (as before direded). Now hecauje the lO thus addedy was fuppofed to hg borrowed from the next fuperior place (viz. of Tens) in the upper Figures, therefore you mufl either tall the upper Figure in that place from whence the lO was borrowed^ one Ufs than really it is^ or elje {which is all one^ and mofl ufual) fou muft call the lower Figure in that place one more thanjt really is, and then proceed to Subtradion in that placoy as in the former \ and fo gradually an from one Hew of Figures to another until all be done.

EXAMPLE 1. Let it be required to find the Difference between 6785, and 4572. That is, let 4572 hcfubtra^ed from 6785.

Thefe Numbers bein^ wrote down, as before direded, will ftand

6785

4573^.

Thus {

2213 Beginning at the place of Units, take 2 from 5 and there will remain 3, which muft be fet down underneath it's own place ; and then proceed to the place of Tens, taking 7 from 8, and there will remain i, to be fet down underneath it's own place ; again, at the place of Hundreds, take 5 from 7, and there re^ mains 2, which fet down, as before ', lafily, take 4 from 6 and there will remain 2, which being fet down underneath it's own place, the Work is finiflied, and the Difference fo found will be 2213=6785 — 4572, as was required,

EXAMPLE 2.

The Difference between 5849 and 7496 is required.

Having placed the Numbers as in the Margin, begin at the place of Units (as before) and fay 9 from 6 cannot 7496 be, but 9 from 16 and there remains 7, to be fet down 5849 tinder it's own place ; next proceed to the place ofTens, -

where you mud now pay the 10 that was borrowed to 1647 make the 6, 16, by accountingthe upper Figure q in that place one lefs than it is, faying 4 from 8 and there remains 4, or elfe (which is the niofi pra£[ifed) fay i I borrowed and 4 is 5

froai

r'

I

Chap. 2. Of dubtrattton* 13

from 9 and there remains 4> to be fet down under it's own place (as before] ; again, at the place of Hundreds, fay 8 from 4 that cannot be, but 8 from 14 there will remain 6 to be fet down; and here I have borrowed 10 (as befbre) which muft be paid in die fame manner as the other 10 was, vi%, either by calling the 7 in the upper Rank but 6, faying 5 from 6 there remains j^ or elfe by faying i borrowed and 5 is 6 from 7 and there remains U which being fet down under it's own place, all is done, and the Difference required will be 1647^7496 — 5849.

EXAMPLE 3. . From . 830476 Take 741068

Remmns 89408

By this Example you may perceive that Cyphers in the Suh' trahendj viz. in the Numbers to be fubtraSfedy do not diminifh die Number from whence Subtra^fion is made. See Page 4.

Thefe three Examples, I prefume, may be fufficient to fliew the young Learner the Method of Subtra^ing whole Numbers ; as for the Reafon thereof, it is the fame with that of Addition^ Page 10, vi%. of the Whole being Equal tn all ifs Parts taken

[ Ugetber.

I That is, in this Rule the Number from which SubtraJiion is lequired to be m^de, is underftood to be the Whole,^ and the Subtrabendy or Number to he/ubtrd^ed, is fuppofed to be a Part

I of that Whole ; confequently, if that Part be taken from the

I Whole, the Remainder will be the other Part.

From hence is deduced the common Method of proving Sub^ traffionj by adding together the Subtrahend and the Remainder. For if the Sum of thofe Two (which are here called Parts) be equal to the Number from whence Subtra^iion was made (which is here called the Whole) then the Work is right 5 if not, care muft be taken to difcover and corre<^ the Error.

E XA MP L E. From 59435 Take 47608 1

fAdd

f 118273

Proof "J f The St/m which is equal to the Number from

C 59435 C whence Subtra^fion was made*

Or

T^-ES— B!"W»aH^

.14 Wim0t^ B»»ti>

Or from the abpvef^id Kc^fojit it vfill be cafy (9 cppctiw ^Qif to prove the Tr\ith of S^iira^lg^^ by S^tfKa£ficH4

for if from i9^31 ^'WB ^^^ ^^^ wljolc,

there be tak^n 47608 ^ ^art cf that wholc^

there will remain 1 1827 the pthi?r part (as bcfpre}

And if from 59435 the wbofc, there t)c ^u^r^fU4 the

laft part, i;/z. 11827

there will remain 47608 the yeiy Numh^ which was re- quired to be firft 'SiAtra^g4*

From 75643 Frotn 70000CO

Take 9000 T^jce 986432

Remains 66643 R^»n^ins ^91356?

sed. 4. Of jjpuiitftiKcattot.

fll^trttiplfCatiOn is a\Rtt/< by which any given Numhtr may be fpeedily increafed, according to any pr6po(ed Number of Times*

That is^ Qne Number is /aid to Multiply anpthfr^ when the Number multiplied is fo often added to it/elf, as there art Units in the Number multiplying \ and another Number is produced^ (Euclid. 7. Def. 15.)

To perform Multiplication^ there arc required, tviro given Nun^ h/rs^ called FaHors.

The Firft is the Number to be fnultipUed^ vhich is ^oerallr put the greater of the Two Numbers^ and is commonly pJ^e^ the Multiplicand,

The other is that Numberlyy which the Firft is to hje ^la/r/- pfiedy and is ufually called the Multiplicator or Multiplier i and this denotes the Number oi Times that the Multiplicand is rcquirca to be added to itfelf. For (q many Units as are contained in the Multiplier^ fo many times will the Multiplicand he really added to itfelf (as per Euclid above). And from thence will arifc % Third Number^ called the ProduSl. But in Geometrical Opera- tions it is called the Redangle of the two Numbers.

For jnftance; fuppofe it were required to increafe 6 four times, that Is, to multiply 6 into or with 4. Thcfc two Numbers arc to he fct (or placed) down as in Additianot &ubtra£ti9Uj

Thus

Chiq». 2.

Of 90mt0t^(tim

M

^^-{ I HSl^f'h^^'^^'

' Pr^dUf 24 viz. 4: times 6 is 24, kf AdaAm^ viz. By fching down 6 four tifntt, aaid thetl adding them together into dne &«) Thus

Fr^ffi tenet h is ividint that Multiplication is mdf a Compendious Way of adding arj

as plainly Sppeait |6 "

W V

24

Add

fh/n Nttmber t0 itfetf, at matif Timst at mtj be frtptftd.

Before any Operation can be readily performed in MuM^^ tstitHj the feveral Prtdu£ts of two fingle Figures muft be per> kB&j learned by beart, iflfs. That a times 2 is 4, that 3 timel t is 9, and 3 times 6 b 18, f^c according as th*y are expreflfed in die following Table ; wherein I have omitted tnultiplying with I, it being lb yttj eafy that any one may do it.

MidtifRcathn TaUe.

9X9==gt

= 9|4»<4

3x3= 3>'4=> 3x5= i_ 3x6=18

3x7=

3x8=24

3x9=27

24x5:

= 16

2C 5)4x6=24 4X7=2

x8=: 4x9=36

214

7=285

jx5=25 5;i6=3C >x|=35,

22 SX9

x«=4c

6x6=r36 '6x7=±43 6x8=48 ^X9=54

7x7=49 7x8—56 7x9=63

I thtrik it wmectSbty to give any Expionaiion of this Taile ; (or if the Signs and their Significations be well undefftood^ (viia fage 5) it muft needs be eafy. Only this may be noted, that

4x3-3x4^ or 7x5=5x7* ^c.

Thax is, 3 times 4 is the fame with 4 times 3, or 5 times 7 b the fame with 7 times 5, i^c. The like muft be underilood of all the r^ff in the Tabk.

And wheh all thefe fingle BreduSh are fo per/*e£tly learned br Hearty as to be faid withottt paUfing *, you may then proceed foot not till then) to the Bufmefs of Mwltiplicduoh ; whith will Oe fotiiid v^ eafy, if the fpilowing Rsdi (and Examples) b« . euefuHy obftrVfed.

RULE. Stiurt^ iegiH with that Figure wbieh Jlands in the Units placid (f$i^ l/bdtiplkTj and nritb it multiply tie Figure whicb Jlands

J^6 acftftm^fcit^ PartL

in the Units place of the Multiplicand j if their ProduA be left than Tenj fet it down underneath if* own Place of Units, and proceed to the next Figure of the Multiplicand. But if their Produd be above Ten (or Tens) then fet down the Overplus onlj (or odd Figure, as in Addition) and bear (or carry) the fisid Ten or Tens in mind until you have multiplied., the next. Figure of the Multiplicand, with the fame Figure of the Multiplier i then to their Produft add the Ten or Tens carried in mind^ fet^ ting down the Overplus of their Sum above the Tens, as before : and fo proceed on in the very fame Manner y until all the Fi- gures of the Multiplicand are multiplied with that Figure of the Multiplier.

E XA M P L E r. Suppofe it were required to multiply 3213 into or with 3. izx^ MMpUcmul, K,^^„, ' 3 Multtplier^ $

ProduSi 9639

Beginning at the Units place, fay 3 times 3 is 9, which, be- cavfe it is lefs than 7V», fet down underneath it's own place, and proceed to the next place of Tensy faying 3 times i is 3, which fet down underneath it's own place } then to the next place, vi%. of Hundreds^ faying 3 times 2 is 6, which fet down, as before y laftly, at the place of Thoufands^ fay 3 times 3 is 9, which being fet down underneath it's own place, the Operation is finifhedj and the true Produii is 9639=3213x3, as was re* quired.

E XA MP L E ^.

Let it be required to multipfy 8569 - into 8. Set down thefe Numbers as before, >

Thus{ 8569

68552 Beginning at the Units place, fay 8 times 9 is 72, fet down the 2 underneath it's own place of Units^ and bear the 70, or 7 Tens in mind, and proceed to the next Figure of the Multi^ plicand{3Lt which place the 7 7>»j are only 7) faying 8 times 6 is 481 and the 7 carried in mind is 55 ; fet down the odd 5 underneath it's own place of Tens^ and carry the 50 (which is really 500) to the next place (viz. of Hundreds) at which place it is only 5, where fay, 8 times 5 is 40, and the 5 car«> ried in mind is 45 ; fet down the 5 underneath it^s own phce^ and carry the 40 or 4 Tens (which is really 40^) to the

nes^

Chap. 2.

Of ^mmttiotu

'7

ocxt place, vix» of Th^ufindsj faying, 8 times 8 is 64, and 4 carried iq mind b 68. (Now thU beii^ the laft Place or Fi^ur^ to he mubipUid) Set down the whole Pr$dH^ 68> and the Work iidone*

So that 8569x6 ::z 68 5 52, the Prodm£i required.

Now the Rtafbn of this and all other jthe like Operations, may be eafily conceived from this which follows. 8 s 6 IJ

The fame Factors as before.

		7
	4	8
4	0	0
4	0	0

(th<

68552

{Here 8 times .9 is 72, as before^ becaufe the 9 ftands in the Units place.

Now here it is not really 8 times 6=48, but is 8 times 60=480, becaufe the 6 (Lands in the place of Tens. ^ Arid here it is not 8 times 5=40, but it is O 3 really 8 times 500=4000, becaufe the 5 ftands ^ in the place of Hundreds. f Laftly, becaufe the 8 in the Multiplicand &2Lnit o 3 in the place of the Tbou/andsj it is therefore 8 ^ times 8ooo=x64000, and not 8 times 8=64. The Sum of the particular Prodsi£fs^ which gives the true ProduQ^ as before.

fiy what hath been already faid, with a little Confideration bad to the Examples^ I prefuroe the Learner may cafjy under- fand how to multiply whole Numbers with any fingle Figure. And when it is required to multiply with more than one, then b many Figures as there are in the Multiplier^ fo many parti- cular ProduOjs there muftbe.

That is, all the Figures of the Multiplicand^ muft be multi^ flied with every fingle Figure of the Multiplier^ as if there were but one fingle Figure : and the Sum of all thofe particular Pro- ^Sy will be the true ProduSf required. But in thofe Opera- ^ns, great Care muft be taken in letting down the particular Produffs (which arife by each multiplying Figure) in their proper places. Which will be eafily done, if the following Dircfliona .be carefully obferved.

f Always place the firft Figure (or Cypher) of every ^iz* ) particular. Produ^f^ diredly underneath the multiplying i Figure. Or thus :

Tht Firfi Figure {or Cypher) of the fecmd particular Produft ^Mfi Jiand directly under the fecond rigure (or place) of the firft Produ^i ^^ the FirJ Figure (or Cypher) ef tht Third

D particular

i8 ' atftltmettCiU PanL

particular Product, mu^ Jiani direSfiy underneath the Third Figure^ the Firji Produd : Jndfo on until all is done.

Now the Reafon of placing the Firft Figure of every particular ProduH in their Order, will be very obvious to any one that confiders the lart Example \ wherein the Cyphers are only fet down to {hew the true Diftance of the firft Figure in each particular ProduH from the Units place. And altho' it is not utual to fet down Cyphers in this Manner; yet ikey are always fuppofed to be there : That is, their Places are always left void, as in the two following Examples ; wherein I have placed Points infiead of Cyphers,

EXAMPLE 3.

Let it be required to multiply 78094, into or with 7563.

7563

}

234282 The Firft particular ProduSf with ^

468564 . The Second particular Pr^duS with 60

3QO470 . . The Third particular Pr^^/i/^ with 500

546658 . . . The Fourth particular Pr^^/«if? with 70CO

590624922 The Totaly or true ProduH required.

EXAMPLE^.

Suppofe it be required to multiply 57498 into 6oQo5.

57498 ' 60008

459984 The ProduSf with 8

344988 .... The ProduSf with . 600CO

3450339984=57498x60608, as was required.

Here you may obferve, that I pafs over the Cyphers^ and only take care of placing the firft ProduSl of the laft Figure ^ viz, of 60000, according to the foregoing Dircfiions.

When there is a Cypher or Cyphers to the Right«hand either of the Multiplicand or Multiplicator^ or to both ; in that cafe multiply the Fiures as before; neglefling the Cyphers until the particular Produdfs are added together : Then to their Sum ?nnex fo many Ciphers as arc in cither or both the FaS!or\^ As in thefe Examples :

EXAMPLE

r"

Chap. I.	Of i^ultipltcattom	'9
EXAMPLE 5. 9538 4600	EX A MPLE 6. E XAMB LE ^. 87600 785000 79 56900
57228 = 38152	7884 6132	7065 4710
43874800	69204QO	3925
44666500000

Tah a few Examples without their Wark at large.

75649x579=43800771

687000x356=244572000

530674x45007=23884044718

790 1 375 X 30000Z: 237041 ?5oooo

537084000x590700=317255518800000

102030405x504030201=51426405540261405

98765432 1 x 1 23456789= 1 2 1 93264 1 1 1 2635269

'Sote'j If it be required to multiply any Number with 10, 100, 1000, loooo, i^c. it is/only annexing the Cyphers of the MuhipRir to the Figures of the Multiplicandy and the^Work is done.

Thus 5578x10 =5780. 578x1000 =578000

* I 578x100=57800. 578x10000=5780000, &c.

Thcfe Examples (being well underftood) are fufScient to in- ftruft the Learner in all the Varieties that can happen in Multiply' ing of whole Numbers^ according to the Method generally prac- ticed : However it may not be ^mifs to (hew here bow Multipli* catien may be performed (with many Figures) by Addition only.

EXAMPLE.

Let it be required to multiply 879654 into 79863.

In order to perform this (or any other Operation of this kind) by Addition only, you muft make a Tariffa or fmall Table of the given Multiplicand^ in this Manner :

F/r/?, Make a fmall Column, and in it place gradually down- ward the Nine fmgle Figures \ viz. i, 2, 3, 4, 5, ^c.

D 2 Then

20

^[tft^ttcttcft^

Pm I.

Then againft the Figure i, fct down the Mubtpbeand (which in thi$ ExampU is 879654) and againft the Figure 2» fet down the double of the Midtiplicandy found b^ ad^ng it to itfelf; to this double add the Multiplicand^ fetting down their Sikm againft the Figure 3, And fo proceed on by a continual Addition^ until there be Ten times the Multiplicand in the Table J which, if the Work is true, will be the Multiplicand icfelf with a , Cypher to the Right-hand of it, as in the annexed Table. This being done, it will be eafy to conceive, that the Figures in the fmall Column of the Table^ do refpeflivcly rcprefent thofe of the Multiplier: And that the Numbers againft any of thofe Figures in the fmall Column, will ^ be the true Product of the Muliiplicand agree- ing to any Figure of the Multiplier ^ as plainly appears by the Work of this Example.

^79654 J The /i^^« as before.

87965+

^['759308 2638962 3518616 4398270 5277924 6157578 7037232 7916886

^79^540

Then

3, in the r^ii^ is 2638962 =879654x3

5277924 =1879654x60

703723a =879654x800

79 1 6886 = 879654x9000

6157578 =879654x70000

{3, in

6, is

8, is

9, is

7, is

The ProduSi required 70251807402 =879654x79863

Notey This Method of tabulating the Multiplicand^ is both eafy and certain ; being neither fubjed to Errors, nor burden*-' fome to the Memory, and therefore in large Calculations it may be found very ufeful. But for common PraSice the ofual Method (as in Page 18, ^c.) is beft, and to be preferred before this.

Moft Majlers that teach (and feveral Authors that write of) Arithmeticky (hew how 10 prove the Truth of Multiplication^ by cafting away all the Nines that are contained in both the Favors and their Produii ; but bccaufe that Method is very fallible, as might be eafily (hewn, I fliall therefore omit inferting it, and leave the Proof of Multiplication to the next Se^on^ where- in (I prefumc) the Reafon and Proof, both of it, and Diwftm^ will plainly appear.

s^a.

Chap. 2. Of ZHmtm* 21

Sea. 5. 0/ 2[>ibtfion.

3)(blfl[0tl is a Rule by which one Number mzy be fpeetlflif fitkrs^id ftofgi another, fo many times as it is contained therein.

That is. It fpeedily difcofers bow often one Number is con« tamed (or may be found) in another : And to perform chat Ope- ration there ale required Two Numbers to be given.

1. The one of them is that Number which ia^propofed to be £fnJtJf and is called the Dividend.

2. The other is that Number by which the laid Dividend is to be £vidid^ aad is called the Divifor.

And by comparing thefe Two, viz. the Dividend and the Divifer together, there will arife a Third NunAer^ called the ^eiient; which Ihews how often the Dhuifir is contained in the Dividend^ or into what NunAer of Equal Parts the Dividend t$ then divided. Therefore,

DiviiioB is by Euclid fitly termed the meafuring of one Number if mother^ viz. one Number is biA to meafure another by chat Number^ which when it mubipiieSy or is multiplied by it, it pro-* duceth. Euclid J. Def. 23.

Jnd if a Number meafuring another^ multiply that Number by which it meafureth^ or be multiplied by it^ it produceth tbg Itiumher which it meafureth. Euclid 7. Axiom 9.

That is to fay. It that Number which divides another (called the Divifor) be multiplied with the Number which is produced hy Divifion (called the Quotient) their Product will be the Num^ ber Divided ot Dividend. Whence it follows, that Divifun ^nd Miitiplication are the Oonverfe and Dire£l Contrary one to ano- ther (as Suhtra£lion is to Addition) and do mutually prove the Truth of each other's Operations.

I ihall therefore make choice of the foregoing Examples in Multiplicationy in order (as I prefame) to render the Buiineis of Pivifion more plain and eafy.

Firfi, let it be required to find bow often 6 is contaifled in 24; Aat is, to diviiU 24 by 6.

N, B' Always place down the given Numbers in this Order: Firft fet down the l^ivifor^ and to the Rightrhand of it draw. a crooked Line ; then fet down ihe Dividend^ and to the Right of it draw another crook^ci Line, in which mufi be placed the ^otient Figure or Figures^ as |jiey become found.

Thus

22

Ztitfmetith

Part I.

Dividend, Thus Divifor 6) 24 (4 the ^otient.

Here I confider how many times 6 there i$ in 24, qnd find it 4, wz. 4 times 6 is 24, therefore 4 is the true ^otient or Anfwcr required. .

This will appear evident by Saifrjr- iioHy as in the Margin ; where 24 the "S Dividend is fet down, and from it 6 the . Divifor ^ continually fuhiraSled fo often as ic cap be, which is juft 4 times. Therefore 4 is the true ^^- iitru or Anfwer required.

24 j6

18 6

12 6

6 6

COROLLARY.

From hence it is evident, that Divijion is but a ^^/irci^ or compendious Method of fuhtraHing one Number from another, ' fo often as it can be found therein ; for if the Divifor be con* tinualiy fuhtraSled from the Dividend^ accounting an Unit (or i) for each time it \s fubtra^ed {zs above) the Sum of thofe Units will be the ^otient.

Ail Operations in Divifion do begin contrary to thofe of Multiplication y viz. at the Firft Figure to the Left-hand, or that of the higheft Va)ue, and decreafe the Dividend by a repeated Sttbtra^ion of each ProduSf arifing from the Divifor when mul- tiplied into the patient Figure. And the only Difficulty in Di- lijion of whole Numbers (or indeed of any Numbers) lies in making choice of fuch a ^otient Figure^ as is neither too big, nor too little ; and that may be eafily obtained by obferving the following Rule^ which hath two Cafes.

RULE.

Cafe r . As often as the Firji Figure of the Divifor is taken from the Firjl Figure of the Dividend : Bo often mufl the Second Figure of the Divifor be taken from the Second Figure of the Dividend, when it is joined With what Remains of the FirJi. And as often mujl the 7 bird Figure of the Divifor be taken from the Third Figure of the Dividend, 6fc.

But if the Firft Figure of the Divifor cannot be taken from the Firft Figure of the Dividend^ then

1 Cafe

Chap. 2. OfDftJifiOm 23

Cafe 2. So often as the Firft Figure of the Divifor is takem fr9m the Two Firft Figures of the Dividend^^ often muft the Second Figure of the Divifor be taken from the Third Figure of the Dividend, when it is joined with what remained of the Second: Andfo often mu/i the Third Figure of the Diviibr be taken from the

I Fourth r igure of the Dividend', Wf .

That is, the ^otient Figure muft be fuch, as being multiplied into the Divifor^ will produce a Produ£f equal to fuch a part qf the Dividend as is then taken for that Operation : But if fuch a ProdiiSf cannot be exadly found, then the next lefs muft be uken, and ordered, as in the following Examples : of which let that in Page 16 Se the firft, wherein there was given 8569 the

I Multiplicandy and 8 the Multiplier. To find the ProduSf 68552.

I Let us here fuppofe the faid Produlf 68552, and 8 the MuU tiplier^ both given ; thence to find the Multiplicand. That is. Let it be required to divide 68552 by 8* ,

Dividend

Divifor 8) 68552 ( ^otient when found.

I ...

According to the Rukj Cafe i. I compare 8 the Divifor with

I 6 the Firft Figure of the Dividend^ and finding I cannot take it

I from that, I then confidcr (by Cafe 2.) how often 8 can be

taken from 68, the two firft Figures of the Divideridy and find

it may be taken 8 times ; for 8 times 8 is 64, being the greateft

i ProduH of 8 (into any Figure) that can be taken from 68. I

therefore place 8 in the ^otienty and with it multiply 8 the

Divifor^ fetting down iheir Product underneath the (aid Two

Firft Figures of the Dividend^ fubtra£ling it from them, and

then the Work will ftand

Thtis 8) 68552 (8 6+

In order to a JSecond Operation^ I make a Point under the next Figure of the Dividend^ viz. under the 5, and bring it down onderneath in it's own place to the Remainder 4, which will by that means become 45c Then I v.onriilcr how many times 8 can be taken from 45, and find it may be 5 times ; for 5 times 8 is 40, I therefore place 5 in the ^otient^ and with it multiply 8 the Divifor^ fttting down and fubtra^ing their Produ^^ as bcfur^. Then tht Work will Hand

Thus

84 graSmCt^* Part I.

" Tbus 8) 68552 (85 ' ^"^ '

64.

For a Third Oiieratioit, I make a Point under the nest figwt of the DivicUndy viz. under the 5, and bring it down, •s before, proceeding in allrerpe.^$, as before ; and (hen the Work MftUfiaod

Thus 8) 685P (856 64..

45

40

55

48

7 Laffly, I point and bring down the 2, viz. the laft Figtirt of the DiviJtnd to the Remainder .fj., which will then become 172, and proceeding as in the other Operatitnst I find that 8 the Divifor can be taken juft 9 times from 79^ aod the Work it fifliibed, and will ftand

Thus 8) 68552 (8569 64...

45

40

48

(o) Hie trne ^9timt is found to be 8569, being exadiy the Eighth part of 68552, or the MuhipUcand of the propofed Example o£ Multiplication. As was required.

The Reafan of thefe Operations will be very plain to any one that will a little confider of it» aa follows ;

Divi/ar

Chap. 2.

Of iomotu

25

Dhfifrr 8) 68551 (8000. The Firft ^oitent Figure.

00

This Prodii^ of the. Dhtfar into the

uotient is 64OOO, v/z. 8 times 8000 ; the

U9tient Figure being always of the fame

alui or Degree with that Figure under

which the l/mysplaceofit'sPrd^tf^ftands.'

45 43	5 2 (500. The Second Rodent Figure. I And here the ProdiOi is 4000, viz. \ 8 times 500, not 8 times 5.
552 (60. The Third patient Figure^ jj r Alf« here the Produ^ is 480, viz. 8 ^^^\ times 60, for the Reafom abovefaid.
	7 a (9. The Fourth ^oiient Figure. Now here the ProiuH is^ut 72, wz. 729 times 8, becaufe the 9 ftands in the C place of Unite i -

SiAtraa 64

Divifir 8) SiAtTda

Divifir 8} Sabtreia

Divifir 8)

Sahra^f

; Rmainr (o o) Now the Sum of all the feveral ^oiitnts^

vSz. 8oco+50o4*6o-f 9=85699 ^ before.

If the Precefi of this Example be well conildered and compa- ' red with that of Multiplication^ Page 17, it will evidently ap- . pear to be only the Converfi of that ; for the particular Pro'^ duffs are alike in both, only that which is la/l there, is firfi here; there they are aJJed^ here they wet fidttra£led. So that i whoever underftands the true Reafin of the one, muft needs nnderfiand the Reafin of the other, and then Divijion will be- come very ea^^ although the Divifir coniifts of feveral places of Figures,

EXAMPLE.

Let it be required to divide 590624922 by 756^ • Dividend.

Divifir 7563) 590624922 (

Tis plain at the firft fight, that 7563 the Divifir^ cannot be hken from 5906, the like Number of Figures in the Dividend.

Therefore, by the Second Cafi of the Rule [Page 23.) there

Uttft be allowed Five Figures of the Dividend^ viz. 59062 for

he Fir/i Operation or ^otient 5 that fo the Firjl Figure 7 of

be D'ruifir may be taken out of the two Firji Figures^ viz. 59

P ihe Dividend^ ^c.

I E Then

26 arttfimetfCft* Parti.

' • • . _ II

Then I proceed (per Cafe 2.) and confider how often 7 may bp taken from 59t and find it may be taken 8 limes, for 8 times 7 is but 56, which I mtmMy fubirad from 59, and there remains 3 5 to this 3 1 mentally adjoin the Third Figure of the Dividend^ viz. 0, which makes it 30, out of which I muft take the Second Figure of the Divifor^ ^iz. 5, fo often as I took the 7 from 59, which was & times : But that cannot be, for 8 times 5 is 40, which is more than 30, 'therefore 8 b too big a Figure to.be placed in the ^$tient\ yet, hence I conclude, that the next lefe, viz, 7 may be taken withouf any further Trial. I therefore place 7 in the ^otieniy and with it muUiplj the Divi/ir^ fctting down their rrodu£f under the Dividend^ and fubtra£f it frojn thence, as in the other Example^ arid then the tf'ori will (land

Thus 7563) 590624922 (7*^ 52941

6j2I

• In order to a Second Operatiori^ I make a Point under the next Figure of the Dividend^ viz. under the. 4, and bring it down to the Remainder 61 21, which will then become 612,14, with which I proceed m alt rcfpefts as I did before with the 59062, and find the next ^otieni Figure will be 8, with which I mul- tiply the Divifory &c. and fubtra^ their Produ£f from the faid 61214. Then the tTork will (land

Thus 7563) 590624922 (78 52941 •

61214

60504

710

To this Remainder 710, I point and bring down the next Figure o( the Dividend j viz. 9, which makes it 71 09; no.w becaufe the Divifor 7563 cannot be taken from 7109, f there- fore place a Cypher in the patient.

And this muft always be carefully obferved, viz. That for roery Figure or Cypher, which is brought down from the Divi- dend, in order to a new Operation, there mujl always be etther n Figure or Cypher, fet down in the Q«ioticnt. Then the Work wilJ llaixd '

Tbui

r

Chap. 2. Of DittOOtU ~ 27

■ -^

Thus 7563) 590624922 (780 52941 • •

61214 60504

vtz

7109 To this 7109, I bring; down another Figun of the Dividend^ wiz. 2, and chen it will become 71092; then I confider hqw often 7 can be taken from 71, tTr. (juft as at the firft Operation,) and (ind it may be taken 9 times, therefore I fet down 9 in the \ ^atient^ and with it multiply the Divifory fetting down and fubtranlng their ProduSi^ as before j then the Work will ftand Thus 7563) 59062492a (7809 52941 •••

. 61214

60504

71002 68067

3025 To this Rimesnder 3025, I point and bring doWn the lafl Figure 2 of the Dividtndy which makes it 30252 $ then pro- ceeding in all refpeds as before, I find the ^otient Figure to be 4, with it I mubiply the Divifor^ fetting down znd/uiira0i/^ dicir Predua as before, and then the Work will ftand Thus 7563) 590624922 (78094 52941 • • • •

61214 60504

7x092 68067

30252 30252

(000^0)

ttcre the Work is ended, and I find the ^otient to be 78pg4, being the true Multiplicand of the propofcd Example of Multi^ fUcatioriy Page \%\

That is, 7563 is contained in 590624922 juft 78094 times, Vc.

£ 2 If

28

dxi^^ttitiu

Part I.

If the Work of this Example be confidered and compared with the Rule (Page 22.) the whole Bufmers of Divifan will be eafy } for indeed the only Difficulty (as I faid before) lies in making choice of a true ^oiient Figure^ which cannot well be done according to the Common Method of Divijien^ withcwt Trials, yet thofe Trials need not be made with the vihoXtDivifir (as appears by this laft Example) for by the two FirS Figures of the Divtfor all the reft are generally regulated ; except the Second Figure chance to be 2, 3, or 4, at the iame time the Third Figure be 7, 8, or 9, then indeed refped muft be bad to the Third Figurey according as the Rule direAs.

However, if thofe Trials are thought too troublefome, they may be avoided, and the fame ^otient Figure may both eafily and certainly be found by help of fuch a fmall Taile made o£ the Divifiry as was of the Multiplicand in Page 20.

Let it be

the Example make a Table Thus,

r 'Divifor. .

79863) 159726

239589 413*945^

5 3993 '5

6479178

SS90+r 638904 7^8767

EXAMPLE 4.

required to divide 7025 1807402 by 79863. See •f Multiplieaiieny Page 20| and as toere dire^ed of ibe Divifor 79863.

10

798630

Dividend, 70251807402

63890A

636140 559^4^ 770997 718767 522304 479178 431260

3^945^ 3'94S2

(000000)

^otienU

The Work of this Operation I prefuoie may be eafily under- ftood. For thofe Figures in the Table ar« the Predua of the Di^ vifor into all the 9 Figures ; con- fequendv thofe 'Figures in the fmall Column do (hew what Figure is to be placed in the ^otient ; without any doubtful Trials of the Divifor^ with the Dividend^ as before.

This Me;hod of tabulating iht Divifor may be of good Ufe

to a Learner j efpecially until he is well prafiifed in Divifi'm \

' and even then if the Divifor be large, and a ^otient jpf many

Figures be required ; as in refolviiig of hieb Mquations^ and

calculating of Jfironwical Tables^ or thofe orimcrcft, fcfr.

Hitherto

Chap. 2. Of ^fyismu

29

Hitherto I have made choice of Examples wherein the Divi^ dmJh exaAly meaftired or divided off by the Divi/or, without leafing any Rmiunder. But it moft ufually happens, that the Dhifir will not exadly meafure the Dividend ; in which cafe the Rmaimkr (after Divifion is ended) mull be fct over the Divifar^ with a final! Line betwixt them adjoining to the ^otintt.

E XAMP L E 5. Suppofe It were required to divide 379 by 5. "

k\ "270 in^t^^ Remainder.

35- ^^*^"<»»

as

Remains (4)

EXAMPLE f^. Again, Let it be required to divide 43789 by 67..

67) 437S9 (65314 ^^ ^"^^ %uoti%nt required, 402 • •

358 335

239

201

Remains ^ ^^j

Such Remainders tnus pitccd over their Divijors (which are indeed Vulgar Fra^iom) may be otherwifc managed, as fhAl be (hewed farther on,

N. B. When the Divifor happens to be an Unit^ viz. i, with m Cypher ox Cyphers annexed to it, as 10, lOO, icOCf; isTf. Divijion i« truly performed by cutting off with a Point or Comma, fo many figwres of the Dividend as there are Cyphers in the Divjfir ; then ire thofe Figures fo*^cuc off to be accounted a Remainder^ and the wft of the Figures in the Dividend will be the true Quotient re- quired, becaufe an Unit or I doth neither multiply nor divide.

E XA M P L E T.

Let it be required to divide 57842 by 100. The Work may ftand thus, xoo) 578^42 the ^otient required \ or ihuf 100) 57842 (578 T^ the fame as before.

Hence it follows, that if any Dhifor have Cyphers to the Jt'l^t-band of it, jom may cut off fo many gf the laft Fgu^es

30 artfi&mettClU Part

in the Divtdendy and Svide the other Figurts of the Divldendf by thofe Figures of the Divifor that arc left when the Cypbtri are omitted. But when Divifion is ended^ thofe Cyphers fo omitted in the Divifor^ and the Figures cut off in the Divielend^ are both to be reftored to their own places.

E XA M P L E 8. Suppofe it were required to eUvide 675469 by 54OO 540c) 675469 (125

54- •

108

274 270

Remains (4) But the true RemainJer is 469*

Confequcntly the true ^otient is 12^^^^.

As to the manner of proving the Truth- of any Operation, either in Multiplication or Divifion^ I prefume it may be eafily underfiood, by what is delivered in Page 21, compared with the three firft Examples of Divijion ; .for from thence it will be eafy to conceive, that if the Divifor and ^otient be multiplied together, their ProduSf (with what Remains aiftcf Divifion being added to that ProduSf) will be equal to the Dividend, As in the Fifth Example^ where the Dividend is . 379, the Divifor is 5, the ^uotienk is 75, and the Remainder is 4*

I fay, 75x5 = 375* to which add tht Remainder 4, it will be 379.

Again, in the Sixth Example^ the Divifor is 67, the ^otieni is 653, and the Remainder is 38.

Then 653x67=43751, and 43751+38=43789 the Divi^ * dendy ice.

There are feveral ttfeful ContraSfions^ both in Divifion emi Multiplication, which I have purpofely omitted until I come to treat of Decimal Arithmetick. Alfo I have omitted the Buiinefs of Evolution or Extradiing of Roots^ until further on j and (0 (hall conclude this Chapter: ^'w'lih sl few Examples of Divijion ^nwrought at lar^e, leaving them for the Learner's Pradice.

579) 43800771 (75649. Or 756+9) 43800771 ( 579.

45007]

Chap. 3. Of i©etfft>tja(, ^eaftiresaf, &c. < 31

45007) 23884044718 (530674.

Or 530674} 23884044718 ( 45007. 356) 244572000 (687000. '59600) 57659066400 (967434. loooo) 679543820000 (67954382. 79) 282016 (3569tI-

CHAP. III.

Conierning OfltttlOll and aubttaCtiOtt ^/ Numbers sf different Denominations, and bow to reduce them from one Dcoominacion to another.

SECT. I. I. Of Englilh Coin.

THE lead Piece of Money ufedin England is a Farthings and from thence arife the reft^ as in this Table.

Fertb. f 5^' \^2i Crown.

4= '^ ^^' An/lJ *^^' isanv/pj^//.

48= 12= iiiJW^ • ^"^1 6x. %d.zNobU.

960= 240=20=1/. PtfK»/ Sterling. 1 1 3 j. 4 ^. a ilAiri.

>/i//. When /. i. ^. q. are placed over ^^r to the Right-hand %f) Numbers^ they, denote thofe Numbers to fignify Pounds^ ^ ibitiingSy Pencff and Farthings.

/. J. ^. ;.

As 35 10 6 2. Or 35/. lox. 6rrf. Either of thefe dofigntt'y 35 Poundsy 10 ShiUiftgs^ 6 Pence^ 2 Farthings,

The fame muft be underiiood of all the folio wing Character s^ Wonging to their rcfpcftive Tables^ viz. Of If^eights^ Mea^ furtSy &c.

2. 7>^;^ Weight.

TTie Original of all JVtights ufed in England^ was a Ctfr» of 9^af gathered out of the middle of- the Ear^ and being wdl dried, 32 of them were to make one Penny Weighty 20 Penny f^tights one Ounce y and 12 Ounces one Pound Troy. Vide Siatutes of 51 /f/a« III. 31 Bow, I* 12 lien. VII.

But

32 atft^etf^ Parti.

But in later limes it was thought fufHcient to divide the •forefaid Penny freight into 24 equal PartSy called Grains^ being the lead ff^eight now in common Ufe; and from thence the reft are computed as in tlis lable.

C'- ^'^^' r By Trwr fyeighi are

24= tP.W. Penny might. j^,.. J v^cighed Jetueb^ GoU^

480= 20= lO^O^^^'ce. ^^^^'Y Silver J C^rn^ Brseul^

75^60=240=12—1 lb Pound. t and all Liquors.

Befides the common Divijidns of Troy Weighty I find in Anglic Uotitioy or. The Prejent State of England^ printed in the Year 1699, that the Moneyers (^s that Author calls them) do fubdkridt the Grain. %

Thus

3. Jpotbecaries Weights. The Apothecaries divide a Pound Troy^ as in this TaUe.

Gr, Grtm.

20= I 3 Scruple

60= 3—15 Dram 480= 24= 8= I j Ounce ;76o=r2S8=^6— la::::.! ^ Troy^ the fame as before*

By thefe Weights the Apothecaries compound their Medicines : but buy and fell their Drugs by Averdupois Weight.

4. Averdupois Weight.

When Averdupois Weight became firft in Ufe^ or by what Law jr was at firft fettled, I cannot find out in the Statute Booh \ but on the contrary, I find that there (hould be but one Weight (and one Mesfure) ufcd throughout this Realm^ viz. that of Troy^ (Vide 14 kd. ill. and 1 7 Ed. III.) iio that it fcems (to me) to be firft introduced by Chance^ and fettled by Cujiomy viz. fiom giving good or large Weight to thofe Commodities ufually . weighed by i:, which arc fuch as are cither very Coarfe and DroJJy^ or

very

Chap, j. Of aggigfttg^ $|9catttt:ei8, &c. > 33

ytry 6Ajt& to wafte ; as all kinds of Grocery Wares. And Pitck^ Tory Ro/hiy Wax^ Tallow^ Flax^ Hemp^ &c. C^pper^ Ttn, Steely IrM^ Uad^ &c. Alfo Flejh^ Butter^ Cheefty Salty ice Totbefe and the like (rprefiime) it was thought convenient to aOow a greater fFeight than the Laws had provided, which iilppened to be about a Sixth part more : For I found by a very nice Experiment, that one Pound Averdupou is equal to 14 Omuiiy 1 1 Pinny Weightty and 1 5 4. Grains Troj^ And it is now computed as in the following Table.

0r4W.

r*

16=: I Qz. o^^n. 1 i/^,:zza Stone

256^ 16= ilbPcui^ And< 28=i^C.

28672-= 1792= II2±: I C HMndrtd. I s^zz-^ofC.

573440= 35840=: 2240= 20= I *lun. V. 84=3 ;» o/C.

5. Long Meafure.

As the leaft part of ff^aght came at firft from a Wheat Contf

fe (it is generally faid) the leaft Part of Long Meafun was at

^ iiift a BarUj Corny taken out of the middle of the Ear, and being

vcD dried, three of them in length were to make one Inch i and

thence the reft, as in this Table.

^^C"^- r4 JVtfiii=i of a rorit

3_ xt±:^ Axiii liTardzziEU.

36ZI 12=: \F^_rett. 1 2 Tards-=,i Fatiom.

108= 36= 3= I r.Tardi.

594= 198= i6i= 5i=: i ^' ^^>

23760= 7920= 660=: 220= 40=1 furlmnt.

190080=63360=5280= i76on32oaa8=i Aftli.

tftty That forty Poles (or Perches) in Length, and four in Breadth, do make a Statute Acre of Land.

That is, 2%oTardsy multiplied Into 22 2^ir, =4840 Square T^»isy are a Statute Acre.

Ami according to the Tranfadions of the Frenclr Academf^ Amo 1687, a Paris Foot Royal is = 12,8 Inches Englijh j Six oftfaofe Feet make a Toifei and 57060 Toifes-zz^ ^li^ Englijb futy are the Meafure of one Degree of a great Circle upon the Surface of the Earth. So that one Degree is 69 Miles and 288 Terdsy which is very near to i>ur Countryman Mr. Norwood'^ Experiment made betwixt London and Torky Anno 1635 $ who mid that 367196 Feetzz6(f Miles and 958 Tards^ do make a

F Degree.

34 atltlWtettClU Part L

Degree. And not 60 MiUs^ according to the common received Opinion and Praflice of the Navigators or Seamen.

Hence, according to the French Account, the Circiimferencc 9f the Earth (fuppofing it to be a true Spherical Figure} ts 24899 Englijh Miles.

6. Of Liquid Meafures.

All Meafures of Capacity, both Liquid and Dry, were at firft made from Troy fTeighiy Fide Statutes 9 H, III, 51 H. III. 1% H. VII. Wc. wherein it is ena£led, that eight Pound Troy Weighty oiH^eatj gathered out of the middle of the Ear, and Well dried, itould make one Gallon of Wine Meafure : And that there fhould be but one Meafure for ffTme^ Ale and Com^ throughout this Realai^ (Vid. Stat. 14 Ed. III. 15 Rich. II.) But Time and Cuftom hath altered Meafures^ as they have done Weights (and perhaps for one and the fame Reafon) for now we have three difFer^t Meafieresj viz. one for Wine^ one fgr'Ale or Beer^ and one for Corn. , I have inferted Taifles of each, as they arc now computed by Cubick Inches^ and pradifed in the Art of Gauging^ &c. ; The common IVine Gallon feakd at Guild-Hall in London ; by i^hich all lyines^ Brandies^ Spirits^ Strong- Waters^ Mead^ P^^^j Cyder ^ Vinegary Oily and Honey^ &c. are raeafured and fold ; is fuppofed to contain 231 Cuhick IncheSy and from thence the reft are computed, as in this Table.

Gall. Cuhick Inches. r i8 = i Rundlety attd

2131::= iG^GaUont. ^^^^ \ ^ji. mBkc z Wine or

0702= 42= I '^^^^' '1 Vinegar Barrel.

14553= 63=1^ !!st± Uyi^* liJ-IlL)

19404— 84=2 =:i|-i ^*'*^'^'''

29I06=:I26=:3 =2 =rl-i— I Butt or Pipe,

58212 = 252—6 =4. =r3 =2==y Tuk.

But Dr Wybard in h'ls Teftomctry, Page 289, doth fuppofc the fVine Galhn to contain but 224, or 225 Cubick Inches at «tbe moil, and purfuant to this Account an Experiment was made

S' Mr Richard Walker and Mr Philip Shales y two General fliccrs in the Excife. They caufcd. a Veflcl to be very exai^I^ made of Brafs, in Form of a ParalUUopipedony each Side of it's Bafc was 4 Inches^ and it's Depth 14 Inches 5 fo that it*s juft Cotw- tent was 224 Cubick Inches. This Veflcl was produced at Guilds Hall in London (May 2-5, 1688O before the Lord-Mayory the CommifjiQners of E^cife^ the. Reverend MxFkmJlcady Aftr. Reg.

Chap. 3. ofi®eiB|t35,^alUrej5,&c. 35

Mr HoUey^ and feveral other ingenious Gentlemen, in whofe Prefence Mr. &haUs did exadly fill the aforefaid Brafs Vefiel with dear Water, and very carefully emptied it into the old Standard Wine Gallm kept in Guild-HalU which did fo exaa^y fill it, that all then prefent were fully fatisfied the Wine Galkn doth contain but 224 Cubick Inches. (This notable Experiment I fam tried.) However, for feveral Reafons^ it was at that time thought convenient to continue the former fuppofed Content of ' , 231 Cubick Inches to be the Wine Gallon^ and that all Computa- tions in Gauging (hould be made from thence, as above.

The Beer or Ale G^/b^r- (which are both one) is much larger than the Wine Gallon i it being (as I prefume) made at firft to correfpond with Averdupois Weighty as the Wine Galkn did with 7r^ Weight: For (as I faid before. Page 33.) one Paund Averdupois is equal to 14 Ounces^ 12 Penny tVeights Troy^ very aear.

And, as one Pound Troy is in proportion to the Cubick Inches in a Wine GalLn^ ^ is one Pounds Averdupois to the Cubick Inches in sdi Ale Gallon. That is, X2 : 231 :: 14^^- : %ti^^ vei^y near the Cubick Inches contained in an Ale Gallon^ as appears from an Experiment made by one Nicolas Gunton^ General Ganger in the Exdfcj almoft 90 Years ago, who, by fuch a VcfRl • mentioned before in the lafl Page^ did find the Standard Ale ^uart (kept in the Exchequer j Fid. 12 Car. II.) to conuin jull 70^ Cubick Inches^ coniec^endy the Ale Gallon muft contain 282 Cubick Inches^ and from thence the following Tables are computed.

Ale Meafure.

282= I C^JfV' r A Firkin of Soap and of

2256= 8=1 f^"-^"' Notc,^ Herrings are the fame

451 21Z 1 6= 2r= 1 f^^^^'"* I with that pf AJe.

902431 J2=4=:2zzi ^^[f[: 113536=148=6— 3= 14::;:= I Hojrjhead.

Beer Meafure,

282= I ^^^'

202= I ^^^ 2538r= 9=1 ^"'^'f' 5076r:i8 = 2=i tCiiderkin, 10152=: 36^:4=: 2— I Barrel, 15228=^4:^6= 3=1 j^Ti

Hogfhead.

~% N. B.

L

36 atftltmetfelt. Parti.

N. B. This Diftindion or DifFeronce betwixt JU and Be^r Meafyrt^ is now only ufed in London. But in all other Places of England the following TahU of Beer or AU^ whether it be ftrong or imal), is to be obferved, according to a Statute of Eifcifi made in the Year 1689.

CtA, Jncbtu

7. Of Dry Meafure,

Dry Meafure is diflerent both from Wme and Ale Meafure^ be« ing as it were a Mean betwixt both, tho' not exadljr fo ; which upon Examination I find to be in proporcion to the aforefaid old Standard H^ne Gallon^ as Averdupois Weight is to Troy Wagbt : That is, As one Pound Troy is to one Pound Averdupois^ fo is the Cuhick Inches contained in the old fVine Gallon^ to the Cubici Inches contained in the Dry or Corn Gallon.

Viz. 12 : I4fj: : 224: 272^, which is very near to 272^ the common received Content of a Com Gallon : Altho' now it b otherwife fettled by an Adlof Parliament made in April 1697, the Words of that Pi& are thefe : ^

Every round Bufhel with a plane and even Bottom^ being made eighteen Inches and a half wide throughout ^ and eight Inches deep^ fiouU he efteemed a Legal Winchefter Bufhel, according to the Standard in his Majefly*s Exchequer.

Now a Veilel being thus made will contain 2150,42 Ctc^/ri Jnchesy confequently the Com Gallop doth contain but 268^ Cubick Inches. .

Cnh. Jnfhe$,

ESl 04 Bufiels^z Comb.

,8=5 I GiUroM. Note,S 10 ^arters^zWey^znA

,6= ^= I P^ L XI IVeys'=LZ Lajl of Com.

'268

2150,4=: 8= 4^1 Bujbi^ T 7 203,23=64— 32 ^8 = I Quarter,

I obferved amongft the Lead-Mines in Derby/hire^ (Anno 1692} that the Miners bought and fold their Lead Ore, by a Meafure which they called an Ore Difli \ whofe Dimenfions I p^efully took, and found it

Thusi

Depth 8,4) Confer

f Length 21.37

Thus) Breadth 6. Ylnches.

( Depth 8.4)

Chap, 3. Of a&eftfttg^ ^eaCntfe Sic 37

Confcqucmly it's Content is 1073,52 CuUcit Inches^ which is retj near equal to 4 Cmt GoUmj according to the abovemen- tioocd Settlement.

Nine of thoTe Diihes they call a Load of Ore, which, if it be prttty good, will produce about 3 hundred Weight of Lead^

8. Of Time.

It is not an eafy Thing to rive a true DifinikM of Ttnut foF (acconiii^ to the Pbihfopbical Poet) ^

Time tf itfilfts notUng^ but from Thought Rearues ifs Rifty by Lbouring Fancy wrought From Things coHfider'dy wbtl/iwethinkonjome As frefentj fome as poft^ or yet to conn. No Thought can think on Time, thafsjlill confeft^ Bui thinks on Things in Motion or at Reft.

And fo on. Vide Lucretius^ Book I.

That is. Time only fliews the Duration or Mutation of Things, a Year being the Standard or Integer^ by which fuch Continuance or Change is computed. And a Tear is that Space of Time in which the Sun (apparently) complcats it's Revolution from any one Poim in the EcUftick (an imaginary Circle in the Heavens) to the fame Point again, which, according to modem Obfervatiom^ b performed in 365 Days^ 5 Hours^ 48 MinuUSj 57 Seconds^ 21 Thirds^ &c. But a Second being the Icaft part of Time that ao be truly mea/ured by the Motion of any Mechamcal Engine^ as a Cbck^ &c. (a Third being lefs than the Twinkling of an Eye) I b^^ the following Table with Seconds.

Seconds:'

60= I'MUmae.

3600== • 60= i^^* 86400= I440ZZ 24=iJDgr.^ ^ ^^

3»5S6937=525949=8765=365+s+48+S7=i3^^^»«&4 " (aSolar Tear.

But the common Civil Tear, now in ufc, confifls of 365 Days and 6 Hours^ and is divided into twelve unequal Months^ called Calendar Months, whofe Names and Nusnber of Days Vt the Sviajed of evjcry Almanack.

To

38

9ct(fKQdiciu

Parti

To thefe TabUs it may not be amifs to add a brief Account ci Bkh Coinsy ffVghtij and AteafureSy as are frequently mentioned in the Scriptures : As I have deduced them from thofe iPi^icfa feem to be the moft CorreA, inferted in the Indix to the Jargc BiUi^ Printed Anno ijpij and compared with thofe ufed in Englandy by the Lord Bi(hop of Petirborough.

Th. Hebrew mgbts, compared with {o^'J^Z'S^ains.

JGerah:=,

ID Gerahs-=za Bekah=^

2 Bebabs^za Shikel-=i

100 Shekek-nei Menahzzz

o o

o

45

o

4 9

12

Netiy A Sbekel is faid to be their Original if^eight. TheirCw^l

3

12

EngUJb Coin. L s. d.

A Silver Menabiz Taknt of Silverz^ Talent oi GQld::=. ThcGoUDnmzz

J . I . Si Weight 6o 5A/i//f.

357 . II . loi We^bt is 300 5ifr/^^/x.

507 s i^ 15 • 7t Tbe fame Weight men- 1.0:4 tioned £». ii. 19*

The Ronton Jkhney mentioned in the New Tfflament,

hDeneartMs^ or Siher Penmyzn'] d. 3 Fnrtbi^gs.

Jfes of Coffer^o

Jlpmum-i^LO

^uadrantzio

AMtezzo

3 Fartbings. It Farthing.

^of a Fartbing.

y of 0 Fartbing.

Their LongMiafure^ compared with j

A Finger's BreadthzzX oT

4 Fingers'iza Hanfs Bnadthz^. o.

2 Handszztbi Uafl Spanz=. o.

3 Ihnd^i Breadtbz=ztbe langefi Spanziz o.

2 Spansssitbe longe^ Cttbit= o.

4 Cubitszza Fatbom-rz 2.

6 Cubits— EzekiiVs Reedsss 3.

400 Cubitszsa Stadiumzz 243.

10 Stadiums^=aa Mile:^ 2432.

3 Miles=za Parafangzm. 7296,

Which is 4 JE/f^//}* ^//« and] 256.

Englijb Meafure. Yar. Feet. In.Pts,

o o o o I I I o o o

0,91a

3.648

7,296 10,944

9,888

3.55^ 11,328

o

p

Their

Chap, 3. Of mtig^ ^aatcefe &c. 39

ThcirMfsfuresof Capacity, compzxtdwiih[ CaTpt^I^d^.

A Cotyh^

4 Logr=:ta Ca6::z

10 Cotyla*s=:an Onurzt

3 Cabss=a Hin=i

2 Hinszza Seahziz

3 Seabszzan Ephazz

10 Bpba'szsz a Chmmrss^

o - oi 3,037

0 . oi 9,83 0.3. 10,458 0.6. 1,5

1 .2 . * 2,5

2 .4 . 5,

7 • 4 • i5»

75 • 5 • 5»625^

Sea. 2, ;^t)MtiOn e/' Weights, &c.

np H E foregoing TabUs being fo well underftood, its that you can -^ readily tell (without pauluig) how msmyUnits of any one De^ wminatiMf do make one of the next Superior Denominatton (ejpi^ dalfy in tbofe Tables which art moft ujefiil for your Bufinefi) it will then be as eafy to aid or fubtra^ Uiem, as to add or ftdf^ tra£l wboU Numbers j due Care being taken in placing all Num-- hers that are of one Denomnation exa£Uy underneath each other.

That is to (ay, in Money ^ place Pounds under Pounds, Shillings uader ShilUngSy Pence under Pence, &c. Underftand the like in freights and Meafures, &c. according to their (evtral Deno* mnations : Then in Addition obferve this Rule. RULE.

Always begin with thofe Figures of the lowejl or leaft Denomi^ nation, and add them all together into one Sum, then tonfider how many of the next Superior Denomination are contained in that Sum, fo many Units you mufl carry to the faid next Superior De- nomination to be added together with thofe Figures that ^and there ; and if any thing remain over or above thofe Units fo ear- ned, that Overplus mti/i be fet down underneath it*s own Deno- mination: And fo proceed on from ^k/ Denomination to another milaUbefiniJbed.

Example in Coin»

Let it be required to add 35/. i4x. odd. and 27/. 02s. loJ. and 54/. 131. 04</. and io/» lys, ogd. intooncSum.

The particular Sums being placed, as before dire£led, will ftand as in the Margin following.

Then according to the Rule, I begin with the Pence (Hiding Here the loweft or leaft Denomination) and adding them all together, I find their Sum to be 29 d. that is 2 j. and 5 d. (for

40

9^:i£^(ticlu.

Parti.

/. X.	d.
35 • «4 •	06
27 . 02 .	10
54 . 13 .	04
10 . 17 .	09

24 = 2 J. and 29— *24=5) the 5 ^. I fet down underneath Ws own Denomsnation^ and cany the 2 J. to the Place of Shillings^ aMing them and all the Shillings together, I find the Sum to be 48 s. VIZ. 2 /. 8 J. 1 fet down the 8 s. under- neath it's own place of Shillings^ and carry the 2/. tothePlaccofPtf«»A,tf/itf»f them and all 128 • 08 • 05 the Paunds together, I find their Sum is 128/. confequently the Total Sum required is 128/. dis. 05^.

Now as it often happens in keeping Books of Accounts^ (and in other Bufinefs) that it is required to add up large Sums of Money, confifting of 30, 40, or more feveral particular Sunu^ nay, perhaps filling up the whole length of a Sheet of Paper, I apprehend in thofe Cafes the beft and eafieft way will be to part them into Parcels, not exceeding above 10 or 12 parti- cular Sums in each Parcel ; that done, add together all the Sums of thofe Parcels intoone Sum^ and that will be the Toud Sum required.

Alfo to avoid the making of Points^ or other Marks amon^

Jour Figuns, it will be convenient to get the following TaUes y heart.

Tbe Pence Table.

J. s.	d.	t.
12=1 36=3 J^5	72= 10?= 120=	6 I 9 10

The Shillings Table.

s. L 20=1 40=2 60=3

80:=:4 100=5

s. I. 120= 6

140= 7 160= 8 ]8o= 9 200=10

The Ufe of thefe Tablis is f o obvious^ that I prefume it is necdleis to explain them.

Examples in Addition 9/ Weights.

Troy Weight.

lb. Oz. Pw. Gr. 3 . 09 . 00 . 10 5 . 08 . 15 . 2X

10 . 10 . 12 . 22 o . II . 19 . 23

Jverdupois ffVgbt. Tun. C. J^. lb. 0%. 12 . 15 . 2 . 24 • 12 7 . 10 . 3 . 21 • 15

0 . 18 . I . 14 • II

1 . 19 . 3 . 27 . 15

Sumzt . 04 . 09 . 04 Sum 23

05 . o . 05 . 05 Exan^les

Chip. 3. dtOitcacttan of leaglittf, &c. 41

Examples in Additi$n of Lmg-Miofure.

nrJf ^s. Nails MiUs Fur. Piles tards Feet Inch.

35. a. 3 2. 6. 32. 4,2. .9

17 . 3 . I o . 7 . 27 . 3 . I . 10

129 .1.2 i.3.39'.2.2«it

182 . 3 • 2 Sum 5 • 2 • 20 . o . X • 6

I think it oeedUrs to fet down more Examples oi this kind, l6r if tbefe 5 (efpecial)y the lafl) be well underftood, they will be fufficient to (hew how any other may be performed.

Sea. 3. ^uWratrton of Weights, &c.

^VkraQiem is the Converfe of the precedent Work, and may ^ be pcrferaied by obferving this

RULE. 'BtfiJLn with the Loweji or Leaji Denomination (as before in Addition} and Take or Subtract the Figure (or Figures] in that pla£i 9f the Subtrahend, from the Figure (or Figures) that Jiani eon tbem of the fame Denomination ; fetting down the Remainder^ (« iff Page 12.) But if that cannot be done^ then you wti/l in» creafi the upper Figure (or Figures) with one of the next Superior Denomination, and from that Sum mate Subtradion; and fa proceed to the next Superior Denomination, where you nrnft pay the one borrowed^ by adding Unity to the Subtrahend in that place ^ &c» tff in whole Numbers.

Examples in Coin. L s. d. . /. X. d^

From 386 • 09 . 08 From 569 . 10 . 06

Take 173 . 04 . 06 Suhtr. 389 . 15 • 08

R/mmns 213 • 05 . oz 179 • 14 . id

The Firft of thefe Examples is felf-evident. In the Second Bxaa^U^ beginning at the place of Pence (being here the Leaft Dmominaiion) 1 am to take 8 d. from 6 d. but becaufe that cannot be done, I muft (according to the Rule) barrow one of the next Denomination^ viz. i j. and add it to the 6 d. which makes it iSd. (for is-^iiid. and I2d.+6d.:=:i9d.) then I take Sd. from that ltd. and there remains rod. to be feC down underneath the place of Pence ; that done, I proceed to the place of SbilKngs, where I muft now pay the i r. faying one borrowed and ic makes 16 fiom 10 cannot be, but

G 16

42 arttfmtgtfck^ Parti,

1 6 from 30 and there remains 14. That is, I borrow one of the" next Denomination^ vis. iL and add to it the 10 i. whidi makes it 30 s, (for i /.=szox. and 20 j.-|-io:= 30) having fet dowa the Rimaining 14X. underneath it's own place of Shillings I pro- ceed to the place of Pdunds, where paying the i /• borrowed, it will be I borrowed and 9 is 10 from 9 cannot be, but 10 from 19 and their remains 9, and fo on as in whole Numbers until all be finished; and tht Remainder will be 179/. 14J. 10^.

Thifi Example being a little confidered wilt render all others ia this Rule eafy.

Examples in Weights. Troy Weight. Averdupois Weight.

Bf. 9z. pwt, gr.. c. qrs, lb. oz.

From 9 . 10 . 16 . 18 17 . 2 • 15 . 10

Take 5 . 09 . 18 . 22 14 . 3 . 18 • 12

Re/is 4 . 00 • 17 . 20 2 . 2 . 24 • 14

Examples in Long Meafure, yds, qrs, nails. miles fur, poL yds. feet inches.

From 78 . 3 . 2 22 . 3 . 26 . 3x- o . 9

Take 29 . 3 . 3 18 . 6 . 29 • 4 • 2 . 11

Rejis 48 . 3 • 3 3 . 4 . 36 . 4 . o . 10

Examples in Time.

days b. , x" •

From 27 • 18 . 35 . 21 Subtra^ 16 . 21 . 46 . 36

Remaitrs 10 . 20 . 48 . 45

The Proof of Addition and Subtra^ion in thefe Numbers of dif- ferent Denominations^ is the very fame with that of whole Nmm^ hers in Page 13. I (hall therefore refer you to that place, and omit repeating it here. ,

Sed. 4. 0/ laeDuttion*

DY Redu^icfij Numbers of different Denominations zrehrou^t . into one Denomination,

That is, it alters or changes any Superior Denomination pro-

pofed^ into any Inferior or Lefler Denomination Required ;

4 ftill

cfaap. 3. Of Remicttom 43

fiill keeping them eqaivalenc in Value. And by that means they become fitly prepared for Multiplication and Divijion \ which ocherwife could not {o conveniently be performed. Therefore the Bufinefe of Redulfion is very ufefiri in the RtAe of Proportion^ (oommooly called the Golden RuU^ or Rule of T'/^/^^efpecially to tbofe who do not underftand cither Vulgar or Decimal FraHiom. And it is thus performed*.

RULE.

C&nfider bow many Units of the Denomination Required^ make 0Me §f that Dcnomiiiation propofed to be Reduced (which is eafily humm by it^s refpe^hue Table) and with that Number of Units, Multiply the Denomination propofed^ and their Product will be the Number Required.

Example in Coin.

Let it be Required to Reduce or Change 357 /. into Shillings y flod thofe Shillings into Pencey which (hall Ail! foe equal in value with the 357 /.

357 liubipfy with 20 the Shillings in one Pound.

I 7140= the Shillings in 357 /.

! J^ipfy with 12 the Pence in one Shillings

1428 71+

8568o=:the Pence in 357 /. as was Required. Or 357 /• may be reduced into Pence^ at one Operation i Thus,

357'- Mdtiply with 240 the Pence contained in odc Pounds

1418 7'+

85680= the Pence in 357/. as before.

But when the Numbers propofed to be Reduced are of feveral J^enminafions^ and it is required tO bring them all to the Loweft ; you muft Reduce the higheft or greatcft Denomination to the B«t Icfs, Adding iht Numbers that are of that lefs Denomination together; then Reduce their Sum to the next lower Denominaiicv, AWiag together all the Numbers that are of that Denomination^ ^ fo prcceed gradually on till all is done.

^ G 2 EXJM^LE.

44 atftlWtttfCiU Part I.

EXAMPLE. Let it be required to Reduce 375 L ijs. 10 d. 3f« into one Dmomimttion^ viz. into Fartbhfgs*

357/. ijs. lod. 3f. ao

7500z:thc Shillings in 375 /. + 17'-

75 1 7athe SbilEngs in 375 /. 1 7 1. 12

15034 7S«7

90204=:the P^Jir/ in 375/. 171. + 10^.

902i4=:tbeP/iir/in 375/. 171. 10 i. 4

36o856=thc^jft*i;f^xin375/. 17 j. 10^. + 3

36o859/jr/A.=375 /. 1 7 j. 1 0^. 3 f . as was required. The Work of this Example^ and all other Operations of this kinds maf be fomewhat uortened by obferving the following Method.

375/. jys. lod. 3q. 20 Multiply and Jdd in the 17 s,

7517

12 Multiply and Add in the ipd.

^5034 7517

90214

4 Multiply and ^i(/ in the 3 frs.

360859 the Farthings as before. ExampU in Troy fFiigbt. Suopefe it be Required to Riduce 29 A 8 jx. x8/tttf« 9U.fr. into the Leaft Denomin0ti$ny tib. into Grains.

Thus

Chap; 3. Of RaWftfetU 45

Thus 29 ZJ. 8^2- iS pwt. 21 gr.

Mwbipif with 12 the vz, in 1 A. and add in the 8 0^

■ II I HI I I

66

29

356 = the ^. in 29 ffi. 8 9Z. MMpfy wid) 20 the pwts. in i mc. and add in the iS^tetfx,

7138 = the^j in 29/^. 8me. iS^wr. itidtifly with 24 the^j. in i /«;/• and add in ihe ii jrr*

14278

171333 the^/.=s29ii. 8«. i8pwt9.2igru

Thefe two Exanfiples at large being well underAood, mayfuf* fice to fliew how all Operations of this kind are p^ormed ; ei-* ther in fFeigbts, MeafwiSy or Tinu. I (hall ooly infert a few £3U0iples of each /brt for the Learner's Pradior.

1. In 23 C Xqrs. 7.1 lb. 9^. Averdupoit Weight; How many Ounces f Anf. 42905 Ounces, ^

2. Ill 252 Eng. Mflesj How many Jirir, /Vif , and Inches f Aofw. 443520 jrA. = 1330560/ir/^ = 15966720 inches.

3. In 1692 ooamion Years^ How many A^x, Hntrs^ and iBnutes? Anfw. 618003^/, 14832072/^1^/, 889924320 mmaes,

Nrte^ a common l/ar £= 265 -Dig'x, 6 Hours^ fee P^r^/ 37,

4. In 5786 Pounds^ 17 Shillings, 9 P/Wfr, Sterling; How ma- ny SAiiZfji^/, Pence, and Farthings f Anfw. 1 1 5737 x, 1^88853 if. «Mr 555541 2/flWfAi«/r. That is, 5786/. ifs. gd.}± 1 15737/. 91^.= 1388853//. &c.

The next thing wiU be to (bew how to brtng Nimibers from a lefler to a greater Denomination, which by moft Authors is cal- led (iho' very improperly)

WmUtimafcending.

This Work is the Converfe of the laft, and is performed by Divijitn. Thus,

R U L E- Cemfidir hew numy ef the Denomination fnfefedmah $tu tfthe DenomiiiaiMMi required, tfW M0ii# /i&e# Niimber y««r Divifer, iy vAia divUe the Denomioaitoii./n^^l omMv Quotient wiM h fh$ Numb^ regmnd.

EXAMPLB.

46 gtitlimeticlu . Part i:

E XA MP L E. Let it be required to find how many Shillings and Pounds ase contained in 85680 Pefue.

ThQ Pence in is. are 12) 85680 (7i40x. = 8568oi/. y^gain, the Shillings in i/. are 20) 7 140 (357/. the Anfwer required.

Another Example in Coin. How many Pence, SbiHingSy .and Pounds,, oxt ^contained io 264859 Farthings.

12) 20)

4) 264859 (66214^. (5517^- (ays'-

2+	62 151.
08	21 117
OS 19	94 (17)^' (10) d.

Rimains (3) ^. C Note, the Remainder is always of jthc fame

I Denomination with the Dividend. The laft S^otient 27 si* together with the fevcral Rimaindtrs give the Anfwer required.

Yvz.2ysl* ^V* lod. 2q.::z2642^g Farthings.

Example in Troy Weight. Suppofe it were required to 6od how mgny Pwts. Ozs. and/i(/, gre contained in 17 1333 Grains.

20) 12)

24) 171333/M7138/W, (356 (29/*.

168. .. 113 24

33 24	138 116 108 (18) ^x (8)«.
93 72
213 192

• Remains {21) grs.

Anfw. 29 /A. 8^. 18 ^w/. 21 grs. This and the laft Ex^ ample are the Reverfe or Proof of tbofe in Pages 43, 45.

I, In 42905 Ounces, Averdupois JFeight*, How fn^nj Pounds, i». Thus

Chap, 3. Of RmitttoU 47

28) 4)

Thus 16) 42905 (2681 a. , (95 ^rj. (23 a

109 252 15

^30 161 ~)

^ HO *

(9) (21) Anfw. 23C. 3frs, iilb.^tz.

2. In 15966720 Inches I How many £«^/f> MiUsy &c. Anfw. 252 ^Zfj, &c. as occafion may require. There are many ufeful Queflions may be anfwered by the help ^Riduaion only : As the changing one fort of Coin for another ; and comparing one fort of Meafure with another, i^c.

foxlnRzncc: Suppokonehzd^j^jRixdollars, ztj^'s. ed.per i>oUar; and dcfired to know how many Pounds Sterling they

I 347

54 = the Pence in one Dollar, viz, 4 x. 6 i; = 54 d.

1388 1735 20)

15611.(78/, 161

73 (0^-

18

{6)d. Anfw. 78/. IS. 6d. Sterl. zre zz ^^y RixdoUars..

^eft. 2. In 645 Flemfi) Ells ; How many Ells Engli/h?

^"•ote, 3 garters of a Yard Englijh make one EllFlemJh^ and ' p ^ 5 Stuart ers of a Tard^ is an Englijh EH. Therefore, 645

3 = the qrs. of a Yard in i Ell Flemijh. fJ.in I Ell = 5) 1935 (387 EngUJb Ells for the Anfwer.

•^«^. 3. Suppofe a Bill of Exchange were accepted at London^ fcf the Payment of 400 /. SterL for the Value delivered at Jmfter- ^in Flemijb Money at i/. 13 J. dd, for i Pound SterL How «»nch FUndJb Money was delivered at Amjierdam ?

Ftrft^ il. 13 J. bd. = 402^. the Value of one Pound Sterh U Amfierdam.

Then, 402^. X 400 = 1 60800 ^« s= 670/. Flemijh^ and fo ttttch was delivered at Amjierdam.

2 CHAP.

fti I !■

4« arttftmettclu * ?$ni.

■ n 11 I ■ r 111 ■ I I I I ■

C H A P. IV.

Of ©ulgar f rartiows^

Scft. I. o/Botatforu

AFraMvn^ or Broken Number^ is that which reprefents a Ptfr# or /^or/x of anjr thing propoTcd, (^di Page 3.) aad is cxpref* fed by two Numbers placed one above the other with a Line drawn betWiJtt them :

The Denominator, or Number placed underneath the Line^ denotes how many equal Parts the thine is fuppofed to be divided into (being only the Divifor in Divifion). And the Numerator^ or Number placed above the Line, (hews how many of thofe Parta ai?e contained in the Frafiion (it being the Remabder after Divifion). (Sa Page 29.) And thefe admit of three Oifiindions :

f Proper or Simple 1 Viz. j Improp:r >Fra^ion^.

C Compound 3

A proper,' pure, or Simple Fra^ion^ is that which is lefs than an Unit. That is, it reprefents the immediate Part or Parts of any thing lefs than the whole, and therefore it's Numerator is al-^ ways lefs than the Denominator.

A . C i is one Fourth Part. . j 5 t ^^ ^"^ Half.

^l-^ysontThirdPen^. ^'^^ 1 ^ is two TAirir, ifc.

An Improper Fra^ion is that which is greater than an Unit. That is, it reprefents fome Number 4>f Parts grcaur than the whole thing ; and it's Numerator is always greater than the De- nominator.

As 4. or ^ or 44 &c.

A Compound FraSlion is a Part of a Part, confifting of feveral Numerators and' Denominators cooneAed together wiA the Word [of].

As 4 of ^' of 4, fie. and are thus read. The om Third of the three Fourths of the two Fifths of an Unit.

That is, when an Unit (or whole thing) is firft divided' Into aoy Number of equal Parts, and each of thofe Parts ar«

fubdivided

chap,4> > Of oulgar JTracttottief^ ' 49

fubdividcd into other Parts, and (o on : Then thofc laft Parts are called Compound Fraifions^ or FraSitons of Fra£tionu

As for inilance, fuppofe a Pound Sterling (or 20 s.) be the Unit or Whole ; then is 8 j. the | of it, and 6 $. the \ of thofe two Fifths, and 2 x. is the 4 of thofe three Fourths \ viz. 2 j. =s ^f J of ^ of one Pound Sterling. All Compound Fra6liom are reduced into fuigle ones, Thus,

R U L E. Multiply all the Numerators into one another for a Numerator^ aid all the Denominators into one another for the Denominator.

Thus the y of -J of 4. will become -^^. Or J^j.. [ For 1 X 3 X 2=6 the Numerator, and 3 x 4 x 5 = 60 theDcnd* minator, but -^^ or ^ of a /. SterL is 2x. As above, >

Sea. 2. To aitet or Cfiailge different JTraCtlOeWl into cm Denomination retaining the fame Value ^

TN •rder^to gain a clear Underftanding of this Sedion, it wiJt . * be convenient to premifc this Pro^tofition, vi%. If a Number multiplying two Numbers produce other NumSers, the Nupfibcrt produced of them fhall be in the fame Proportion that the Num- bers multiplied are, IT Euclid'].

That is to fay. If both the Numerator and Denominator of any Fra&ion be equally multiplied into any Number, their Produ^bK will retain the fame Value with that Fradion.

Asinthefe, ^~' Or — ^=— Or — 7=— > ^^•

'3x2 6^*^ 3x3 9 3x5 15

That is, 4. and |. Or 4 and |.. Or -J and \\. are of the Ame Value, in rcfpecS to the Whole or Unit.

From hence it will be cafy to conceive, how two or more Fraftions that are of different Denominations, may be altered or changed into others that (hall have one common Denominator, and rtiil retain the fame Value.

Example. Let it be required to chanpe \ and 4 into two other Fradinns that fhall have one common Denominator, and yet re* tiin the fame Value.

According to the foregoing Propofltion, if 7 be equally multipli-

«l with 7, it will become 44, v/«. i^ =ii. Again, if |be

3x7 21

«VoHy multiplied with 3, it will become A^ ^'2. ?iLl= J?-.

7x3 21 H And

so antfjmCtiCfe* Part I.

And by this mc^ns I have obtained two new Fradlioas, ^^ and -^^^ that are of one Denomination, and of the fame Value with the tviro firft propofcd, viz, ^^ = *. and ^V— T*

And tiooi hen.e doth arife the gt-ncral Rule for brin^in; all Fractions into one Denomination.

RULE. Multiply all thi Denominators into each other for a new (and common) Denon)Inator. And each Nuriicrator into all the Deno- minators hut it's own^ for new Numerators.

Example. Let the propofed Fraclions be y, y, -J, and ^. Then, by the Rule, A new Denominator And the new Numerators will

will be thus found. be thus found.

3 I . 2 . 3 • 6 5^ 5 3 3 3

4

6o ; 7^

420 • I40 . 168 . 3^5 • ^60

Hence 420 is the common Denominator ; and 140 . 168 • 315 . 360, are the new AVw^r^/tfrj, which being placed Fraftion- wife are if -J . i^l^ , ^i-J- . i^i the New Fraflions required.

420 3 420 5 4^<^ . 4 4^0 7

^hii""'*^"- •• f - ..,•11 I ■■ IP.

Sea. 3. To bring mixed B^ttttllJCrS into jTiaCtiOnSl, and the contrary.

5	6	9	18
+	4	5	S
20	»4	45	90
7	7	7	4

M

I X'D Numbers arc brought into improper FraSiiom by ih© following Rule.

RULE. Multiply the Integers, orwMe Nambexsy «//VA/^/ Denominator efthe given FraSfion^ and to thtir Product add the Numerator, tba ium will be the Numerator of she FraSlion required.

Example, 9 J by the Rule will become y . For 9 X 5 = y , And, y+T— Vthc improper Fraflion required. Again, I3,'>; will become ^*. For 13= 'tV And i^5-f-}.j-=r Vy. And To for any other as occafion requires. To find the true Value of any improper Fradion given, is only the Cooverle of ilui> Rule. For if V = 9 t) ^ befoie is evident :

Then

Chap. 4. Of ©uljjar JTracttoniB^ 51

Then It follows that if 49 be divided by 5, the Qiioticnt will *

pvc 9 ♦. And if ao6 be divided by I5, it will give I34|, f^r,

confequently it follows, that If the Numerator of any improper Fra£iion be divided by it's.

Defx>ininator, the Quotient will difcover the true Value of that

Fn£tion«

EXAMPLES. V=5. AndV=4f And'-z:6 ' pry=3^;£sfr.' When whole Numbers are to be exprcued Fxafiion-wilc, it is

but giving them an Unit for a Denominator. Tbus45is^V>

9 is -?, and 25 is V> ^^•

Scft. 4. ro abteeware ^ IBjomz -fracttonis into

their Loweji or Leafi Dcnominaiian.. . ..

'T^ H I S is done, not out of any neceflity, but for the more con- venient managing of fuch FreSfions as are either propofed in tijge Terms j or fwell into fuch, either by Addition or otherwifc : |>efide8 it is moft like an Artift to exprefs or fet down all Fr/iSions in the ioweft Terms poffibic ; and to perform that, it will be ne- ccflary to confider thefe following Propofitions.

Numbers are either P^fntC or CORipOftlL

1. A Prime Number is that which can only be meafured by •n Unit Euclid 7. Defin. 1 1 •

Th*ti$, 3» 5, 7, II, 13* I7» fafr. are faid to be Prime Numbers, becaufe it is not poffible to divide them into equal Parts by any other Number but Unity or i.

2. Numbers Prime the one to the other, are fuch as only an Unit doth meafure, being their common Meafure. Euclid 7. J^ffa. 12.

ror inllance, 7 and 13 are Prime Numbers to each other, be- caufe they cannot be divided by any other Number but an Unit. And Q and 14 are alfb Prime Numbers to each other, for altho' 3 will meafure or divide 9 without leaving a Remainder; yet 3 will not meafure 14 without leaving a Remainder: Again^ altho* 2 will meafure 14 without any Remainder, yet 2 will not meafure 9 without leaving a Remainder, £^f.

3- A compofed Number is that which fome certain Number incafureth. Euclid y. Defin. i^^.

For inftance, 15 is a compofed Number of 3 and 5, for yx 3 ^ iSi confequently 3 or 5 will juftly meafure 15. Alfo 20

Ha is

ZtitbrntttCli, Part I-

i^Gompored of 5 and 4, viz. 5 )C 4 = 20, therefore 5 and 4 will each juftl^ meafure 20. ' 4. Numbers compofed the one to the other, are they which fome Number being a Common Meafure to them both doth nieafure: Euclid 7. Defin. 14.

That is, if two or more Numbers can be divided by one and the fame Divifor j then are thofe Numbers laid to be compofed one to another.

For Inftance, 14 and 21 are Numbers compofed the one to the other, becaufe they can both be meafured or divided by 7.. For 7 X 2 =r 14, and 7x3 = 21; therefore 7 is a common Mea- fure to 14 and 21. So that if ^ were propofed to be abbreviated, it will hfrnmc f>

, Thu,i7) I4=£

(7) 21=3

And how thofe greareft common Meafures may be found, comes from Euclid 7. Prob. i, 2, 3, and is thus:

RULE.

Dividi the greater Number by the lejer^ and that Divifor by the Remainder (if there be any) andfo on continually until there be no Remainder left : Then will that lafi Divifor be the greatefi common Mtafure (^md if it happen to be i, then are thofe Numbers Prime Numbers'^ and are already in their loweji Terms ; butifothervMfe) Divide the Numbers by that laft Divifor y and their ^otienU i/^U be their Icajl Terms required.

EXAMPLE. .

Let It be required to find the greateft common Meafure of 72 aftd 108, viz. of ./y\.

72) 108 (I

_ 7^

36) 72 (2 f Here becaufe there is no Remainder; ^2 1 36 is the greateft common Meafure.

{0) Thi.ri.fore i 3^) 7^ = ^ f Hence ^Z^V «» abbreviated, Therefore, | ^____ ^ ^^ . j^e joweft Terms.

Again, to find the greateft common Meafure of 744 and 899,

Thus,

Chap. 4. Of i^tiigat JTtacttonsu 5$

Thuj, 744) 899 (I

7A4

i55) 744(+ 620

124) »55 (i

124.

(0) ■ Hoe 31 b foimd to be the greateft common Meafiire by which 744. and 899 may be abbreviated to 24 and 29 their lowcft Terms. Thus, J4) .J^ (=1^ esTf. w//. If the propored Numbers be even, they may be brought lower by a continued halving of them, (o long as the/ can be lialved, viz. divided by 2.

E XA M P L £, It is required to Reduce 4| to it's leaft Terms.

f irft, i) ii (=^. Again, ») »» (=^4^. This done, you eafily perceive that 7 will be the common Mea- fmt to 14 and ai, viz. j.) 44 (rr y, Wr.

If the Numbers propofed to be reducled have each a Cypher, or Cyphers, annexed to them, they will be abbreviated by cutting off a like Number of Cyphers from both. Thus, 44S will be f^. And ^ will be *, &c. ThatU,.^i-=4l=i. And -4=^. And J|J=||=^z: ,V

scA. 5. auwtfott of ftastimL

\X7H AT hath been done by the Rules in this Chapter, is chiefly to prepare and fit FraSfions of different Denominations for Addition or Subtra£ii^nj as Occafion require), vi%. If they are C^mfstnui Fra^ionsy they muft be reduced to Simple or Pure Froaions^ per RuU^ Seff. I.

* If they are of different Denominations, they muft be altered or changed, per Ruky SeSf. 2.

That is, all FraSfions muft be brought into one Denomination before they can either be added or fubtraded \ and that being <!ooe, Jddition is thus performed.

RULE. Jdd iogetber all the Numerators^ and their Sum will be 6 New ^unuratvTj utuUr which fubfcribe the Common Denominator.

I Examples

54 ' arftlnneticiu Pani.

£*tfwpifj III dimple JFracttoitti^.

Let it be propofed to add y, ^, and |, together. Firft, i = ^S ^ l-^J, andi=^, ^^S^iJ?. 2.

Then ;-S+Ti-|-34=Tl» *^ Sum required, which according to Si^ion 3, is i ^, v/a. |^= i *-J.

Examples in ComiHttmll iTtaCtfOttjef^ Let it be required to add ^ and 7 of ^ into one Sum. Firft \^ of ^ becomes ^^ or |« per Sed. i. And (per Sed. 2.) ^ and 4 is ^ and ^ viz. 4=tt» «»<• T=Ai b« ^+TV=ri ^l^c Sum required, viz. 4. 4. * of | = 4^.

£xif i9i!^£f J i;i mirtQ iEhunbetfii

It is required to add 5 7 to 7 ^ thefe per Sed. 3. will be y and y. But V ^^ V w*^^ become «» and 44 per Sed. 2. Then •4 + ?^= 'tV> an<J tV = i^Tr.**'* Sum required.

Or you may bring only the Fra£iions to one Denomination.

Thus, 5 T and 7 ^ will become 5 ^\ and 7 -f^.

Then S A + 7 t t = » ^ tt- That is 1 3 Vr* As before.

Sea. 6. QiMcmion of jFmtion^^

RULE.

SUBTRACT om Nunnrator from tbi other (according as the i^ffli^f* requires) and their Difference will be a new Nume- rator^ unaer which Subfcribo the Common Denominator^ as in Addition. ,

E XJ MP L E I.

Let it be required to take ^ out of ^ Firft 4 and 4- per ScSt. 2. will become i^ and 14 i then^— ^ = ^, thatis|— *=if As was required.

E XJ M P L E a. It IS required to ^ubtraA ^ of | from |^. Firft, *. of » = 4.*. per Sea. i. Again 4^ and -^ will become ffj. and 4^. per Sea. a. Then 4fi -4^ = 4^.

EXAMPLE 3. From 6 i Subtraa 3 i J. Firft, 6 i«=s¥- and ^i^- VV P«^ Rule Seft. 3. Again, V =?= VttN *"«• Vt = VrV* pcr Rule Scfl. 2. ~ Vrr — 'tit = riV* = * ttJ = * ir ^^ othcrwife thus :

Firft,

r

Chap. 4. Of jgmgat JTracttong* ss

Firft, 6 4^ = 5 ^ then bring f and ^^ into one Denooiination,

TbenSHI — 3tH = »t4t=»U- As before. EXAMPLE 4.

Let it be required ttf Subtrad -f of -f of ^ from 7. Firft, ^of^of ^=,3^1 And 7 = 6 4||, 'n«n6^4^-4^ = 6-J4i=6^ = 7— fof »of ».. A» was requird.

If thefe few Examples be well underftood, the whole BufineA of adding and fubtrading Vulgar FraSliom will be eafy ; which is really much more difficult to perform xhaait\ii:MtAiukipUcati9H or Druifion > as will appear ia the next SeGion*

Se£t. 7. ^nif^ifcotton of ftmonsL

T N order to perform either Muhiplication or Divifion^ you muft ^ prepare the Terms to be multiplied (or divided) thus: Reduce Compound Fradfiom to Simple ones, pirSe^. I. Brinff mixed Numbers into improper Fraifionsy and exprefs whole Numbers Fra&iiM-wi/ey pir Sea. 3. . Alfo it will be convenient to abbre- fiate them to their fmalleft Terms^ when it can be done. Then UuklpGctttion may be thus performed*

f Multiply the Nimurattrs 9fu int9 another ftr a new Nu* Rule, j meratar ; and tbe Denominators one into another for a new

V Denominator » Js in thefe

EXAMPLES.

1. The Produa of * into 4^ = A* That is, r7l=r3

2. And the Produd of ^-^ into *^ = ^J. Or ^.

3. Again, the ProduS of ^ into y of 4. = ^. Or tV*

Fnr * rtf 5 ' ® Xhi^n 7 v » <> JL? — *

rOryOfy -yy. I HZU ^j^ X -jy -j-yy Ty.

4. Let it be required to multiply 6 with 3 ^. Thefe prepared for the Work will ftand thus, • x "/ •

m, 6= * and 3 * = y. Then | x V = 't% or ^o^. Or, otherwife thus 6x3=18. And 4x6=:y=::2f. •Then 18 + 2 4. :z 20 ^. As before.

5. Let it be required to multiply 7 ^ with 5 ^

Firft 7 ^- V» and 5 ?=V. Then Vx V=*-H'=40i$.

The Reafon of this Rule for Multiplying of Fra^ionSf and

confequently of thefe Operations, and aul others performed by

it| will be evident from this following.

g6 arftitmgtttfc^ ' Parti.

Fix. If 4 be multiplied with 'y according to the Rule, their Produa will be y. But V = 8.

Now ^=2; and ^*= 4 per Sod. 3. B^^t 4 x 2=8. Ergf, &c.

Sea. 8. DiWfion offt&stvam^

nr H E FraSftom being firft prepared as before direacd, Divifioh ^ may be thus performed :

r Multiply thi Numerator of the Dividend^ into the Den9» p • J minator of the dividing Fra£fl$n for a new Nttmerator : and " I multiply the other Numerator and Denominator together yfcr t a new Denominator.

EXAMPLES.

1. Let /y be di? idcd by |, vix. |) ^ [^^^ ^ * the Quotient. That is„according to the Rule 6 x 7 =: 42 the new Numerator,

ini 25 X 3 = 105, the new Denominator, &c. as above.

2. Let it be required to divide \^ by t« ^'^^ tV) tt (itt

= ^f .

For 12 X ^o = 240 the new Numerator, and 27 x 5 = 135

the new Denominator*

3. Suppofc it were required to divide ^i^ by ^ of ^.* Firft, ^of| = 4f. Then-)^V(V^- = TV-

4. Let 20 \ be divided by 3 4. ; viz. . ' ° * by y : For20|=:'4Sand3|=zV. Then y) '|* (=6 theQuoticpt.

5. Lfft h Be required to divide 40 ^ by 5 ^.

Firft^ .ir> *1 — « 5 1. 6 anff c 3 — 3 8 'Th<»n 3«\ 454.6 /r7«a»

r irn, 40 p. _ „ , ana 5 ^ .- ^ . I nen ^ ; ^7 ( ^^^ . ^^^ ^V = 7 ! ^*^^ ^"^"c Quotient required.

6. Suppofe it were required to divide 13 by ^.

Firft, 13 = V- Then 4.) V {^^ = 18 |, the Quotient.

7. Again, let it be required to divide | by 6. Fiz. 4) 4- (^5^ for the Quotient required.

jV. fl. From hence you may obferve, that when any whote Number is divided by a proper Fraftion, the Quotient will be greater than the Number propofed to be divided : But if any Fraaion'be divided by a whole Number, greater than i, then the Quotient will be lefs than the Dividend : As in the two laft M^mples.

M

Chap, s Of Derimal jTratttoniS^ 57

As to the Riafon (or Proof) of this RuU for dividing Fra^ions : It is only the Cotruerfi to that of Multiplication^ and will be vtiy evident from this following.

Let ^ be divided by 4. Which according to the Ruli is *»» t) V (It = 4* The true ^otient. Now V = 8. And ^zsi^ per St^. 3. Confequently V £vided by 4 is but the fame with 8 dividid by 2. v/z. 2) 8 (4. The ^otiint as KnMne.

I could have inferted Geometrical Demonftrationsy for the Rules of Multiplicaiion and Divijion of Fra£fions ; but fuppofing the Learner as yet unacquainted with thofe kind of Demonftrations^ I thought thefe might be more intelligible to him^ efpecially in thispl^.

CHAP. V.

Of soecfmal iTtartionfif.

Jf/'HENj or hy whont, ibis excellent Invention ^Decimal Arithmetick wasfirjl introduced^ is uncertain ; but doubtlefs it's ImprofVimiftts^ and tie Perfeifion it is now in^ are owing to later Tears,

Sea. I. Of »otattom

1 N Decimal FraSfions^ the Integer or wbole Thing (whether it be Coin J JFeigbty Meafure^ or 77m/, &c} is fuppofed to be

£vided into Ten equal Parts \ and every one of thofe Ten Parts

are fuppofed to hcfubdivided into other T/n equal Parts^ ^ &c. ad

infinitum. The Integer being thus divided' (by Imagination) into 10, lOO^

'ooo, loooo, {ffr, equal Parts J hccome%thc Denominator to the

l^eaTnal FraSlions.

Thus tV- i4^* TTtW- TbV^9V TiftVd\g» ^'*

Now thefe Denominators are feldom or never fet down, but only the Numerators 5 and thofe are either diftinguifhcd, or fepa- ratcd from wbole Numbers by a Pointy or a Comma.

Thus, 5,4 is 5 T^. and 0,7 is t^^^. 35,05 is 35 ^l^^ bfc.

But before we proceed further in Notation^ it will be conveni- ent for the Learner to confxder the following Tabky (taken out of the learned Mr Oughtred*s Clavis Mathematica) which ihews the »C7 Foundation of Decimal Fra^ions.

I moh

58

adtfjmcttciu

Parti,

lyhoh Numbertf^ Decimal Parts,
5432	1 0, 1 23456
^§ 5	*'V^ vS vS vS vS v^
,?^'" ^t
•	^"S-g.ff.-"
	§-§.

By this Tahl* it is evident, that as in the whole Numbers or Tntt^ gersy every Degree' {rom the Units Place increafcs towards the left-hand by a Ten- fold Proportion : So in Decimal Parts every Degree is decreafed towards the right-hand by the fame Proper-- fioHy viz. by Tens.

Therefore thefe Decimal Parts or FraificnSy are really more Homogeneal^ or agreeing with whale Numlers^ than Vulgar Frae- tiotis ; for indeed all plain Numbers are in cffed but Decimal Parts one to another.

That is, fuppofc any Sories of equal Number^ as 444, fc*c. The firft 4 towards the Left is Ten times the Falue of the 4 in the middle, and that 4 in the middle is Ten times the Vahie of the laft 4 to the Right of it, and but the Tenth Part of that 4 pn the Left, ^u ,

Therefore all or any of them may be taken either as Integers^ or Parts of an Jnteger : If Inf^^ers^ then they muft be fet down without any Comma or feparating Point betwixt them thus, 444. But if Integerif and one Part or FraQion^ put a Comma betwixt tbcm/thus, 44.4 which fignifies 44 whole Numbers^ and 4 Tenths of an Unit: Again, if two Places of P^r/; be required, feparate them wilh & Comma thus, 4,44 viz. 4 Units and 44 hundred Parts of an Unit^ &c.

From hence (duly compared with the Table) it will be eafy to conceive that Decimal Paris take their Denomination from the Place of their laft Figure.

That

\ ,5 = ,V is,-} ,56 m ,f\%

Parts of an Unity ^c^

Cyphers

Chap. 5- Of Decimal iTwcttonis. 59

Cyphers annexed to Decimal PartSy alter not their Value. As jjo, ,500, or ,5000, {tfr. are each but 5 Tenths of an Vniu

of the laft Chapter.

But Cyphers, prefixed to Decimal Parts decreaie their Faluiyhy Rmoving them further from the Comma. r >5 = 5 ^t»*h Parts. Tk < »^S == 5 'P<''"^' of a Hundred. I *O05 = 5 Parts of a Thoufpnd. 1 ,0005 = J P^rti of. Ten Tooufani^ ,

Confequeotly the* true Value of all Decimal Parts are fapwn b7 their Diftance from the Units Place i thb beiiq; once righdy «nderflood» the reft will be eafy.

I N fetting down the propo<ed Numbers to be added, or fub«

^ traded, great care muft be taken in placing every Figure di-

Rdly underneath thofe of the fame Value, whether they be mixed

Numbers, or pure Decimal Parts, and to perform that you muft

luve a due regard to the Comma's, or feparating Points, which

si^ht always to ftand in a dtred Line one under another ; and to

the Right-hand of them carefully place the Decimal Parts, accord-

f iog to their refpeAive Values, or Diftances from Unity. Then

r . Add enrfuhtraSl them^ as if they were allivhele Numbers ;

Rule < as^from their Sum^ er Diferencey cut off Jo mar^ Decimal

i Parts as are the moft in any of the given Numbers.

EXAMPLES in aWlttfcm.

Let it be required to find the Sum of thefe following Numbers, ^^^ 34^5 + 65,3 + "8,7 +95 + 87,8 + 7>9, which being tnily placed, will ftand

Thus,

34,S

128,7

95^0

87.8 7>9

Their Sum required, 4191Z

I 2 EXAMPLE

6o attt^mettelu Parti-

EXAMPLE 2. Let it be lequired to find the Sum of 25,8^4.-('34}578-{-9>c>76. +13,907.

34.578 9,076

' 3*907

83,415 The Sum required. When the Decimal Pans propofed to be added (or fubtraAed) hitve not the (ame Number of Places, you may for convenience of Operation fupply or fill up the void Placeti by annexing Ci- phers. As in thefc ExampUi.

EXAMPLE 3. EXAMPLE 4. EXAMPLE 5.

45,0700 574.678953 0,975641

50,7580 95.796430 »745*S7

123,0057 78,054600 ,000598 .

74,7020 54,789000 ,800700

24,0000 8,900000 .640536 ^

■ ■ II ■ • p ■

3^8,3357 812,318983 3»>6a727

EXAMP LES in fettbtraition^

Let it be required Co find the Difference between 45»375 and 74,284.

EXAMPLE I. EXAMPLE 2. EXAMPLE 3. That IS, From 74,284 From 437,5 From 75,0034

Take 45,375 Take 89,657 Take 57,875

Remains 28,909 347^^43 1791284,

EXAMPLE 4. Let it be required to find the Excefs between 562 and 93,5784

EXAMPLE 4. EXAMPLE 5.

That is. From 562, From 345,7578

Take 93,5784 Take 157,

The Excefs 468,4216 188,7578

Noti^ The two laft Examples are fuppofcd to be fuppiied with Cyphers, which if a£^ually done would (land thus, 562,0000 345>7578

93,5784 157,0000

■ II I ■■ II ■ Ml I I ■ I !■■

Remains 468,4216 As before. -188,7578

EXAMPLE

Chap. 5. Of DecftnaJ JFracrtoitf^ 61'

EXAMPLE 6. EXAMPLE j. '

From 0,547893 From 1,000000

Take 0,439758 Take 0,997543

0,108135 0,002457

The Proof of Addition and Subtradion in Decimals, is the iame with that of whole Numbers, page 13, &<•

Sea. 3. jg^ultiplicatione/^SDectmalfif.

117 H £ T H £ R the Fadors or Numbers to be multiplied are ^^ pure Decimals, ormixed. Multiply them as if they were all whole Numbers, and forthe true V alue of their Produi£l obferve this

f Cui 9ff (via. feparati with a Comma) fa many Places tf Rule.^ Decimal Parts in the Produ£f^ as thin an in both the Facr

L tors accounted together. As in the fa.

EXAMPLE I. EXAMPLE 1.

2,23 24,3

90 72 9 63 6

6048 12848

6 048 642 4

^ 6,74352 780,51 6

The Reafbn why fuch a Number of Decimal Parts mud be cut off in the Produd, may be eafily deduced from tbefe Examples. Thus,

In Example i. It is evident, that 3, the whole Number in the Multiplicand, being multiplied with 2, the whole Number in the Multiplier, can produce but 6 {vi%^ 3x2 = 6). So that of ne- ccllity all the other Figures in the Produdl mud be Decimal Parts ; according as the Rule dire^«. «

Or, the Rule is evident from the Multiplication of wholeNum* bcrs only : Thus, fuppofe 3000 were to be multiplied with 200, their ProduiSl will be 600000; That is, there will be fo many Cyphers in theProdu£i, as are in both iheFaftors, {Vide page 1 8. j Now i^, inftcad of thofe Cyphers in the Fadlors, we fuppofe the like Number of Decimal Parts ; then it follows, that there ouglu to be thefame Number of Decimal Parts in the Product, as there were Cyphers in the Faftors.

Again, the Rule may be otherwife made evident from Vulgar Fradlions, thus; Let 32,12 be* multiplied with 24,3%

and

62 atftftrogtfcfc Part I.

and their Produd will be 780,516 *& in Example 2, above. Now 32,12=: 32 -^. and 1^,3 = 24 -^ Mrhich being brought into Improper FraSions (ptr Stit. 3. piq;t 50.) wiu become

32-,^= V^'and24A = *-.V. Then W.,' X W = 'iHi'-pfrSia.j. peg* 55. But '4|^« = 780 ^yi*,' «»*• 780.5 »6, M before.

. Any of thefe three Ways do, I prefume, fufficiently prove the Truth of the abovefaid Rule, f^c.

EXAMPLE 3. EXAMPLE 4.

78*546 5745

436 .0675

471276 28725

235638 40215

314184 34470

34246,056- 387.7875

N. B. // fomttinui happens in midtipfying Parts with Parts, 9iat thrrt vnll ntt btj* many Figurts in tm ProdiUff as thtri ought tt it Plaas of Decimal Parts by the Rule : fn that Cafe you mufi fuppfy their DefeSl ij prefixing Cyphers tt the PrtduB \ as in theft Examples.

6.

EXAMPLE 5.	EXAMPLE
.1365 .*435	.0347 ,0236

I I 825 2082

709s 1041

9460 694

4730

305758775

,00081892

When any propofed Number of Decimals is to be multiplied with 10 • 100 • 1000 . lOooo, tf^. it is only removing the feparating Point in the Multiplicand, To many Places towards the Right-hand, as there are Cyphers in the Multiplier.

Thus, ,578 X 10 = 5,78. And ,578 x ioo = 57,8. Again, ,578 x lOOO = 578. Of, ,578 X lOOOO 5= 5780. I Thcfc

Chap. 5, "Of jDerimal JTcatHong^ 63

TJiefe thjuigs being cooiidered, It will be cafy to multiply Pecimals, ai^d ^jstermine their uueProdu£b. As in thefe fello w* iog Examples.

57,056 mukipUed into 0,578 will produce 32,978368 7,6543 bito 5,4246 will produce 41,52151578 0,56879 X 0,05674 = 0,0322731446 0^03246 X 0,92264 = 0,0007672544 $7649 X 0,03687 = 3231,61863 94>3S786 X 6,57869 =1620,7511100034 3,141592 X 52,7438 :;= 165,6995001296

In general, it will be needlefs to exprefs all the Figures oF the Produft at large, (efpecially, when the Fadors have each of t)icm many Places of Decimal Parts, as in the two laft Exaoi- pies) only fo many of thenr as may fuflhre for the intended Deiiign ; and yet the Produd may be as true to (o many Figures as are retained, as if theFadors had been multiplied at large. And fiich compendious Contradions are not only of Curiofity, but may alfe be found of great Eafe and Ufe to the ingenious Pradittoner ; eipecially in refolving adfeded Equations, Or in calculating of Trigonometrical Problems by the Natural Sines and Tangents, bfc. AH which may be thus performed*

Viz. Set the Vmftflacetftht Multiplier direifly underneath that Figure of the Multiplicand, who/e Place yoM intend to keep in the fr^duQ ; and place all the other Figures of the Multiplier in a quite tiutrarj Order to the ufual way. Then tn multiplying always begin et that Figure of the Multiplicand which fands over the Figure ^"herewith you are then a multiplying^ fitting down thejirft Figure if each particular trodu6t £re£ily underneath me, another \ yet oerein you mufi have a due Regard to the Increafe which would arife tut Qfthe two next Figures to the Right-band of that Figure in the Multiplicand which xou then begin with.

E XA M P L E.

Let it be required to multiply 3,141592 with 52,7438, and rttiin only lour Places of Decimal Parts in the Produd.

If the propofed Numbers were to be multiplied at large, they Ottft ftand in a dtre(5t Order as ufual.

Thu5 \ 3»'4^59^ t And would produce ten Places of I 52,7438. I Parts, as in the laft Example.

But

6+

9tftl^6tfCL

Parti.

Thus 3>»4iS92

But feeing it is required to have only four Placet of thofe Parts in the Pttxiud^ fet them down as before direAed, and they will fland

The Multiplicand placed as before.

The Multiplier in a reverfe Order.

The ProduA with 5, regard had to 5 times £•

The Produd with 2, increafed with 9x2.

ProduS with 7, increafed with 5 x 7+9 x y.

Produd with 4, increafed with i X 44-5 x 4.

'Pi:odu£l with 3, increafed with 4 ;c 3.

Produd with 8y increafed with 4 x S-f-i x 8, 165,6995' The true Produd as was required.

The Reafon of this Contradion is very obvious frool the tiiiiolc Operation wrought at large*

Thus 3.HIS9?

1570796

62832

31991

1257

* 94

5217438

25 94

12566^68

ii99i

62831

1570796

165,6995001296

'3^736

24776

144

84

0

Fr^m hince it it evident^ that M tbi Fi^ guns in the Square t$ tbi Rigbt-bandy an wholly omittidin the former Contra^iion 5 and that the ^lajl fmgle Produft here^ is tbefirfi there \ confequentljf the Reafon for placing, the Multiplier in a reverfe Chrdar^ snuft ntids appear very plain.

EXAMPLE 3.

Suppoftf it were required to multiply 257,356 with 76,48 and to have only the entire Produft of integers.

?57:356 The fame at large { ^57^356

84,67

18015

1544

103

20

19682

20 102

1544 18014

58848 9424 136 92

19682,58688.

The chiefcft Care and Difficulty that attends thcfeContra^onst IS the true fetting down of the Units place in the Multiplier un- derneath the proper Figure of the Multiplicand, according to the dcfigncd Produ<2.

Viz.

Chap. 5. Of Dectoial iTractioniSt 6s

Vn. In Example i. It was required to have four Places of Decimal Parts in the Produd ; therefore the Unit's Place of the Muldplier was fet under the fourth Place of Decimals in the Multiplicand : And in Example 2, becaufe it was required to have an entire Produd of Integers only i therefore the Unit's Place of die Multiplier was iet under the Unit's Place of the Multipli- cand. This being once rightly under(lood> will render the Method eafy in Pradice.

Sea. 4. WiWmoft;Dttimm,

t\IFISION i$ accounted the mod difficult Part of Decimal ^ Arithmetick : In order therefore to make it plain and eafy, it will be convenient to refume what has been faid in page 25.

f Tbi Quotient Figure is always of the fame Value or Degrei Viz. \ witb that Figure of the Dividendy undir which the Unites t Place ef it's ProduRftands.

As for Inftance, Let 294. be divided by 4. " ,

r This is not 7 but 7o> becaufe the Unit's 4) 294 (7 / ^^^^^ of 4 X 7 ftands under the Tens Place 28 t of the Dividend.

14 (3 But this is only 3. 12 Xemaim (2) Hence 73 J is the Quotient.

Now if to the Remainder 2 there be annexed a Cypher (thus, 2,0] and then divided on, it muft needs follow that the Unit's Place of the Produdy arifing from the Divifor into the Quotient, will taod under the annexed Cypher ; confequently the Quotient Fi« pw will be of the fame Value or Degree with the Place of that Cypher: But that is the next below the Unit's Place, therefore the Quotient Figure is of the next Degree or Place bdow Unity i IW is, in die firft Place of DecimJ Parts.

Thus 4) 2,0 (,5 So that 4) 294,0 (73,5 the true Quotient required. This being well underftood ; Divifion of Decimals may (in all At various Cafes) be eafily performed. However, that it may k rendered plain and eafy even to the meancft Capacity, if pof- I'k) let Dividon be again defined, as in page 21.

K Vi«.

66 adtfimetiCfe* Parti.

Viz. If that Number which Divides another y be multiplied with the Number which is quoted j th^ir Produff will be the Number Mvidsd,

This Definition alope (if compared with the Rule ^^i^^ 6 1.) will afford a general Rule for dircovering the true Value of the Quotient Figure m Divifion of Decimals.

{The Places of Decimal Parts in the Divifor and ^otienf being dountedtogether^ mujt always be equal in Number with thofe in the Dividend. And front this general Mule arife four Cafes.

Cafe I. When the Places of Parts in the Divifor and Dividend are equal, the Quotient will be whole Numbers.

As in thefe Examples. 8>4S) 29S>7S (35 0,0078) ,4368 (56

^53 5 390

42 25 468

42 25 468

(0) 7^

Cafe 2. When the Places of Parts in the Dividend exceed thofe in the Divifor 1 cut off the Excefs for Decimal Parts in the Quo* tient. As in thefe Examples.

24>3) 780,516 {jfl^i^ 436) 34H6,os6 (78,546

729 305^

515 3726

486 3488

291 2380

243 2i8q

486 ,534) ,30438 (,57 2005

486 2670 1744

(o) 3738 2616

3738 2616

(0) (o) I

Cafe 3. When there are not fo many Places of Part9 in the Dividend, as are in the Divifor, annex Cyphers to the Dividend to make them equal. Then will the Quotient be whole Num- bers, as in Cafe i.

examples!

Chap. 5. Of Dedtnat jFtacttonfr^ ^j

EXAMPLE. '. ~

Let it be required to divide 192,1 by 7,684, and 441 by ,7875, 7,684) >9a,ioo (25 »7875) 44»»oooo (560

»53 68 393 75

38 4ao 47 250

38 420 47 250

(o) (o)

• Cafe 4. If, after Divifion is finilbed, there are not fo many Fi* gures io the Quotient, as there ought to be Places of Part$ by.the geoeraJ Rule \ fupply thejr Defed by prefixing Cyphers to it.

E XA M P L E. Let it be required to divide 7^25406 by 957, 957) 7,25406 (,00758 the true Quotient required, 6 699

5550 Again ,^75) ,0007475 (,0013

4785 575

7656 1725

7656 1725

(o) (o)

Kote^ When Decimal Numbers are to be divided by 10. 100. 1000. looco. Wf. that is, when the Divifor is an Unit with . Cyphers; Divifibn'is performed by removing or placing the fcparating Point in the Dividend, fo many Places tovfands the Left-hand, as there are Cyphers in the Divifor.

EXAMPLE.

»o) 5784 (578*4 100) 578,4 (5,784'

1000) 5784 (5,784 10000) 578,4 (,05784

Note, Thefe Operations are the direH Converfe to thofe in page 62.

' I prefume it needlefs to give more Examples at large :. only I ftall infert a few Dividends, and Divtfors, with their Quotients, • wherein are contained all the Varieties that can happen in Divifion of Decimals,

»574) 493>o66 ( 859 5,74) 49'3o66 (8,59

574) 49i>o66 (,859 5,74) 49/066,00 (85900

574) 49i3o66 (,0859 »0574) 493»o665 (8590

5»74) 4930»^6 ( 859 ^©574) ,493o66 (8,59

K 2 There

68

^tit^itXKUCk^

PartL

There is alfo a compendious Way of contradins Dividon, like ^at of Multiplication, page 64, by which much Labour may be faved ; efpecially when the Divifor hath many Places of Decimal Parts in it : And it is thus performed.

Having determined how many Places of whole Numbers there will be in the Quotient^ if any at all ; or if none, of what Value pr Place the firft Figure in the Quotient will be : Then omit, or dot off one Figijre of the Divifor at each "Operation ;* viz. for ^very Figure you place in the Quotient, dot ofF one in the Di- vifor ; having a due Regard to the Increafe which would arife fron^ the Figure fo omitted,

EXAMPLE. Let it be required to divide 70,23 by 7,9863.

The Work contraSed. 7,9863) 70,2360 (8,7938 63 8904

' 6 3396 „ 5 5904

7492 7187

305 2.?9

' 66 64 (2)

The fame at Length. - 7>9863) 70,2300 (8,7938

63 8904
6 3396 55904	0 I
749' 7187	90 67
304 2W	589
64 61	54*0 8904
c	7506

The Work contraded I prefume is fo obvious (if compared vith the fame at large) that ij is needlefi to give any farther Explanation of it.

Sea. 5. rq Reduce ©tiigar JTtactt^njS/*/^ poftnaljEU

and the contrary.

A N Y Vulgar FraSion being given, it may be reduced, or ra- •^ ther changed, into Decimal Parts equivalent to it. Thus,

ijnnex Cyphers to the Numerator ^ and then divide it by the Denominator^ the ^otient will he the Decimal Parts equivalent to the given FraSiion ; or (?/ leajifo near, it as may be thought mceffary to appxoach.

JS; XJ MP L E.

r

Chap. 5. Of Dectmal iTractfoniS. 69

EXAMPLE.

Ic is reqiured to change or reduce ^ into Decimals* 4) 3>oo (,75 The Decimal Parts required. That is, ^=z^V==»75- Again i = ,S; thus 2)1,0 (5. And ^=,25; 4) 1,00 (,25 $uppo& it were required to change ^ into Decimals.

7) 4,0000000000 G57 142857 14 faTr. r= ^. tJ^tty When the laft Figure of the Divifor, (that is, the De- nominator of the propofed Fraction) happens to be one of theie Figures; w*. i . 3 • 7 • or 9 • (as in the Example) then the Decimal Parts can never be precifelj equal to the given Fradion; yet by continuing the Divifion on, you may bring them to be, very nei^the Truth. As in this Example; Suppofe it was required to change ^^ into Decimal Parts.

13) 1,0009 (,07692307692307 tic. cd tnJbtUum.

90 That is, 0,07692307692307 = -i^fir}.

120 And from hence it may be farther

117 obferyed, that in thefc imperfea

^- — Quotients, the Figures do return agaiii

3^ and circulate in the fame Order as be-

^^ fore : as you may eafily perceiye they

^o ^^^^ ^^ ^° '" ^^^ feventh Place of

^^ both thefe laft Examples.

10

&c. As at firft.

Thefe being underftood, it will be cafy to find the Decimal Parts equivalent to any known Part or Parts of Coin, Weights, Mcafures, Times, ^c. If you firft reduce the given Parts of Coin, ^c. into a Vulgar FraQion, whofe Denominator is the Number of .thofe known Parts contained in the Integer, and the given Parts it's Numerator.

Examples in Coirty &c.

I. Let it be required to find the Decimals of 16/. id. Firft 16/. =44 of one Pound, and 6i/. = ^ of 1/. ^ Bwt 4|. + ,V = |4- Then 40) 33,000 (,825, the Decimal Parts required : That is, ,825 = 16 j. 6^.

Again, Suppofc it were required to find the Decimals equal to

Hcre^

70

aritftmeticL

Part I.

Here 3/. is 3 Integers, and 13J. = :^^ of r /. and 44/, = ^t-j^. But44^ + TTTy = -iT&' Then 240) 160,000 (0,666666 Wf. Hence ;^i. 13/. 45.= 3,666666 (ftc. As was required.

2. What are the Decimals equal to 7 J Inches, one Foot being piade the Integer?

Firft, 7 Inches are 7^ of i Foot, and | of i Inch arc ^» But ^+-^=ii' Then 48) 31,000 (,64583 ^^. = 7 J Inches. . 3. Let it be required to change 8 Oz. 19 Pwt, 8 Grains into •Decimals ; one Pound Troy being the Integer.

Thefe being reduced into the leaft Terms, and added together^ will become ff^- of i Pound.

Then 5760) 4304,000 (,74722 Wr. The Decimals required. And thus may any propofed Parts of Coin, Weights, Mea^res, t^c. l>e reduced or changed into Decimal Parts ; which perhaps may at firft feem fomewhat tedious in Pra£tice, but being a little acquainted with them it will be found very eaiy ; and the ingenious Practitioner will (with a little Confideration) foon find how to reduce them almoft mentally ; or with the help of a very few Fi- gures, without the Ufc of fuch large Tables as are ufually inferted in Books of Decimal Arithmctickj or at moft they may be contrac- ted into fuch as thefe following, which if duly applied to thofc fables in Chap. 3. will be found very ufcful^

Decin^ol Tables.

In Englifh Coin. 0,05 s= I i.

0,0046667 . =x I </.

0,00104167 = I Farthing. I /. being the Integer.

-TT-

Troy IVeight.

Q,0S = I -Pw'^;

c,0': 208333 = I Grain^ I Oz. being the Integer.

Jpothecdries f freight.

9,125 == I Dram.

6,04166667=: I Scruple. 0,00208333 = I Grain. Oz. being the Integer,

Averdupois Weight. 0,0625 . . . . = I Ounce. o»oo39o625 = I'Dram. I lb. being the Integer.

Averdupois Great fFeight. 0,25.... . . =S-JC

0,008928^7 = 1 lb. O,coo558o3 = I Ounce.

C' being the Integer.

Time. 0,04166667 = I Hour. o,o::o6943 4 :=: I Minute. 0,00001 157 -r I Srcofid. I Day^or 2A. HcurSy being madf the Intforr.

The Ul'c of chcit 'l'ab!t3 will be evident by the following

£ A A M P L k\

Chap. 5. Of Decimal iTtactimtg^ ?i

E XA M P L E. Let it be required to find the Decimal Parts equivalent to 17 J. 9^. % Farthings^

Firft 0,05 = I /. Therefore 1 7 x,05iz,85 . . . • r: 1 7 j.

And 3004166=1^. Therefore ,004i66x9=:,037494=:9 d. Alfo 2),oo4i66(=2002o83=4.^. Confequently their Sum, vh, 0,889577=: 1 71. 94^/. Now to find the Value of Decimals in known Parts of Coin or Weights, fcf r, is only the Converfe of the former Work, and b thus performed.

Multiply the given Decimals with the Denominator of the Vui- f^r Fradion required : That is, multiply the Decimals with fuch a Nuipber of Units, as are contained in the next lower Denomina- tion of that Kind or Species which your Decimal is of; and the Produd will be the Number required.

EXAM? L E. I. What is the Value of 0,825 Decimals of i Pound Sterling ; That is, how many Shillings, Pence, £ffr. ^==,825 ? Firft, the next lower Denomination is 20, becaufe 20 s. make one Pound. Therefore 0,825 20

Shillings 16,500 and Parts of Shilling. 12

6,000 Anfwcr 0,82^'=: i6j. 6d.

Again, What are the known Parts of EngUJh Coin equal to 3,666666 Decimals ? Here the 3 Integers are 3Pounds. Then ,666666

20

Shillings 13,333320 12

Anfwcr 3,666666 = 3/.' 13 f. 4^. 666640

3^333^

Pence 3. 999 8 40 = 4 near. What is the Value of 0,74722 Parts of i lb Troy ? Firft, ,74722 Then, ,96664 Again, ,33280

12 20 ^4

1 49444 ^«''^- i9>3328o 1 33'^

7 4722 o P50

Oz. 8,96664 Oz. VwU Gr. 7^987^0

Thefe collefted are 8. 19, 81 very near.

And

72 arttftmetfelU PartL

And thus any propofed Number of Decimals may be^turned or changed into the known Parts of what they rcprcfent, wz. Whc- . thcr they be Parts of Coin, Weights, Mcafures, or Time, &g.

I have omitted inferting more Examples of this kind, becaufft I take the Excellency, and indeed the chief Ufc, pf Decinial Ffac* tions, to confift more in Geometrical Computations, than in the coitemon or pra£kical Parts of Arithmetick, as will appear further #0 } although even in thofe they are very ufeful upon feveral Ac- . counts ; efpecially in the Compuutions of Incereft and Annuities^ (^c. But of that more in it's proper Place. I (hall therefore conr dude this Chapter, with a Remark or two upon the Nature and Properties of FraSions in general.

If any given Number (whether it be whole or mixed) be mul* tlplied with a proper Fraflion, either Vulgar or Decimal, the Produd will be lefs than the Multiplicand, in fuch a ProponioQ as the multiplying Fraflion is \tb than an Unit or i.

That is ; as the Denominator of the Fra^ion is to it's Numerator^ fo will the given Number be to the ProduSi.

Therefore, whenever any Number is to be multiplied with a

vFradion, whofe Numerator is an Unit : Divide that Number by

the Denominator of the Fraftion, and the Quotient will be the

Produa required. Thus 12x^=3. And 1 2 -r- 4 = 3. Again,

12 X T = 6» And 12 -H 2 = 6, ^c.

From hence it follows, that if any Number be divided by a proper FraSion, the Quotient will be greater than the Dividend, by fuch a Proportion as Unity is greater than the dividing Fra£tion.

Thus 12 -^5: = 4^, was. i : i : : 12 : 481 ^^. But the Truth of thefe will be beft underftood after the next Chapter.

CHAP. VI.

Of Continue P|O(iO|tiOn0> ^^d bow to change or vary $he Order of Things.

Scft. I. Concerning Arithmetical Progreffion, ufualfy called Arithmetical Proportion Continued.

WHEN any Rank or Series of Numbers do either increafe or decreafe by an equal Interval or common Difference, thofe Numbers are faid to be in Arithmetical Progreffion.

Chap. 6. Of ^pffOlWAU n

J, ft •a.3.4.5.6.7&rr. M Here thr Interval or I 7.6.5.4.3.2.1 11 common Differ, is r.

rxi 2 . 4 . 6 . 8 . 10 . 12 • 14 . &r. } f Here the common ti.3-5*7.9-li'*3- ^^- J .1 Difference is 2. And fo of any other Series, whofe common Difference is

3.4.5. ^^*

If any three Numbers be in Arithmetical Progreffion, the Sum of Che two Extreams (vix, the'firft and laft) will be equal to the Doable of the Mean or middle Number.

As in thcfc, 2.4.6. Or 3. 6.9. Or 3. 7. 11

f?x.2+6=4+4. Or3+9=6+6. And 3+11:^7+7. fa^r. Lanma 2.

If any four Numbers are in Arithmetical Progreffion, the Sum of the two Extl-eams will be equal to the Sum of the two Means.

Asia thefct 2.4.6.8. Or 3 . 6 • 9 • 12..

Yl%. 2-f8r::4+6. And 3+12=6+9. iic

Corollary i.

From tbifi two Lemmds it is etffy to comoiwj that if tvir fo mmf Numhrs be in Arithmetical Progreffion^ tie Sum of the two ittftam will be equal to tie Sum of any two Means^ that are equally ^fientfrom tbofe Extreams.

Asinthefe, 2 . 4 . 6 • 8 . 10 . 12 . T4 . 16.

Then 2 + 16 = 4 +14 = 6+ 12 = 8 + 10.

Or if the Number of Terms be odd, as thefe, 2 • 4 • 6 . 8 . 10 . 12 . 14 • 16 . 18. dc.

Then2+i8 = 4+l6=s6+i4=8+i2 = io+io.

Lemma 3. Each Term in every Series of Numbers in Arithmetical Pro-

pelEon is compofed of the Interval or common Difference, (b

often repeated, and added to the firft, as there are Terms in the

I^rogrcffion, after the firft. Asinthefe, 1. 3. 5. 7. 9. ix. 13. 15. 17. fie Here the Interval or common Dimrence being two, Jt will

ke 1+2=3. 3+2=5. S+2=s7. 7+2=9. 9+2=11.

n+2=i3. 13+2=15. 15+2=^17. faTr.

Corollary 2.

Hence it is evident j that the Difference betwixt the two Extreams [ya.- I and iy}is compofiJ of the common Difference y mult if lie J «to the Number of all the Terms y excepting thefirjl.

As in the aforefaid ProgrefSon, I* 3* 5* 7* 9* xi* ^3. 15^.

I* The '

74 acttftmettrtU Part I.

The Number of Terms without the firft is 8 ) • * t*-^!^ The common Difercnce is 2 1 Multiply

The Difference betwixt the two Extreams 16

Propojkson I. In any Series of Numbers in Arithmetical Progreffion, the two Extreams, and the Number of Terms being gtven^ thence to find the Sum of all the Series.

{MmltMy the Sum of the HV9 Extnams into the Num^ ber of all the Terms ; and divide the Product fy a. ne ^tient wiU be the Sum of all that Series. Per Carol. I.

E XA M P L E r. It is required to find the Number of all the Strokes a Clock firikes in one whole Revolution of the Index, vi%. twelve Hours. Here i-f-i2=i3 the Sum of the two Extreams.

12 the Number of all the Terms.

26 U

. Then 2) 156 (78, The Number of Strokes required*

EXAMPLE^.

Su^pofe one Hundred Eggs were placed in a Right Line a

Yard difiant from one another, and the firft Egg were a Yard

from a Bafket i whether or no maj a Man gather up thcfe 100

Eggs fingly one after another, ftill returning with every Egg tp

the Bafket and putting it in, before another Man can run four

Miles. That is, which will run the greater Number of Yards ?

In this Queftion 2oo4-2Zi202 Is the Sum of the two Extr.

And 100 Is theNumberofail the Terms*

■ ' f The Number of

Then 2) 20200 (10 100 j Yards he runs that

( takes up the Eggs. Now 4 Milesrr 7040 Yards i The Yards he runs that takes up But 10100—7040=1 3060 1 the Eggs more than the other.

Propojition 2. In any Series of Numbers in Arithmetical Progreffion, the twor Extreams and Number of Terms being given ; thence to find the commoa Difference of all the Terms in that Series.

{The Difference betwixt the two Extreams^ being divided by the Number of Terms lejfened by Unity ar^ I. the ^otient will be the. common Difference cf the Series. Per CoroL 2.

EXAMPLE.

Chap. 6. Of P?OpO?ttmU 75

EXAMPLE I.

One had Twelve Children that differed alike In all their Ages ; the youngeft was Nine Years old, the cldeft was Thirty-fix and a half; what was the Difference of their Ages, and the Age of euh?

Here 36,5—9=27,5 The Difference of the two Extreams. . And J 2 — 1 = 11. The Number of Terms lefs an Unit. Then 11) 27,5 (2,5 The common Difference required, Goofequently 94-2)5= ii>5 The Age of the youngeft but one. And 1 1,5+2,5= 14 The Age of the youngdl but two. And ib on for the xeft Fir Curd. 2.

E XJ MP L E OL.

A Debt is to be difcharged at eleven feveral Payments to be made in Arithmetical Progreffion, The firft Payment to be Twelve Pounds Ten Shillings, and the laft to be Sixty-three Pounds. What is the whole Debt, and what muft each Payment be?

Per Tbiorgm I. Find the whole Debt thus : 12,5+63=75,5 The Sum of the Extreams. I I The Number of Terms.

75 5 755

2) **3o>5 (4I5»25=:4I5/. 5/* The whole Debt. Then, per Theorem 2. find the coomion Difference of each Payment. Thus 63 — 12,5=150,5 The Difference of the Extreams. And 1 1 — I = 10 The Number of Terms lefs i. Then lo) 50,5 (5,05^=5 /. i s. The common Difference.

/. J. /. /. /. J. Confequently 12. 10 + 5. 1^=17. 11 The fecond Payment.

/• J. /• /• /. /• And 17 . 11+5 . 1=22 . 12 The third Payment, bfc.

EXAMPLE z.

A Man is to travel from London to a certain Place in ttn Days, and to go but two Miles the firft Day, incrcafing every Day's Journey by an equal Exccfs ; fo that the laft Day's Journey may be Twenty- nine Miles ; what will each Day's Journey be, aod how many Miles is the Place he goes to diilant from London ?

L 2 Firft

76 gtitttmettCfU Parti,

Firft 29 — 2=27 The Diffcrcqcc of the Extreams. And 10 — 1=9 The Number of Terms lefs i. Then 9) 27 (3 The common Difference. Confequcmly ^1+3=5 The fecond Day's Journey. And 5 + Z^i The third Day's Journey, fsTr. Again 29+2Z131 The Sum of the Extreams. 10 The Number of Terms. • 2) 3x0 (155 The Diftance required.

There are eighteen Theorems more relating to Queflions^ in Arithmetical Progrcffion ; but becaufe they would require a great many Words to fliew the Reafon of them, I therefore refer the Reader to the Second Part, viz. That of Jlgibroy where he majr £nd their Analytical Inveftigation.

S«a. 2, Concerning ^eomettfcal )^^aisat^ncintiuued*^ fonQctiincs called Geomeirical PrqgreJftoB.

X17 H EN a Rank or Scries of Numbers do either increafe by one

^ ^ common Mulriplicator, or decreafe by one common Divifiir^

thpfe Numbers are faid<Co be in Geometrical Proportion continued.

.(2.4.8. 16 . 32 . i^c. here 2 is the common Multiplier. C ^4 • 3a • 16 . 8 . 4 . &r. here 2 is the common Drvifor. Q f 2 . 6 . 18 . 54v 162 . &r. here 3 is the common Multiplier. ^ I 162 . 54 • 18 . 6 . 2 here 3 is the common Divifor.

M/^, The common Multiplier (or Divitbr) is called the Ratio ; and it fhev^rs the Habitude or Relation the Numbers have to one another, vi%, whether they are Double, Triple, Quadruple, i^c. which Euclid thus defines.

Ratio (or Rate) is the mutual Habitude dr Refpe£i of two Mag» nitudes (confequently two Numbers) of the fame kind each to other^ according t^ ^antity^ Euc. 5. Dcf. 3.

Proportion (rather Proportionality) is a Similitude of Ratio% Eiic. 5. Def. 4.

So that there cannot be lefs than three Terms to form a Proportionality or Similitude of Ratio's ; and if but three Terms, the fecond muft fupply the Place of two, As in thefe 2.4.8. That is, 2:4:: 4 : 8 . (of : : fee page 5.)

Here 4 the middle Term fupplies the Place of two Terms, to wit, of the fecond and third ^ 8 bearing the faxoe Reafon,

Likenefs^

Chap. 6, Of P^OpQ^tfQIU 77

Likcnels, or Proportion to 49 As 4 doth to 2. viz. As 2 : is to 4 :: So is 4 : to 8.

Lemma i.

If three Nombers are proportional, the ReSangte or Prodtid of the two Extreams, viz. of the firft and laft Terms* will be cqtial to the Square of the Mean or middle Term. (20 Eucl. j,)

As in thefe 2:41:4: 8. Here8x2=si6 the Produd of the Extreams.

And4X4 = i6theSquaKeoftheMean« ErgiSxzss^x^

Carpi I.

Hence it follows, that if the Produd of any two Numbers be equal to the Square of a third Number \ thofe three Numbers Will be in Proportion.

Lemma t.

If ibur Numbers are proportional, the Produd of the tw0. Extreams will be equal to the ProduA of the two Means (19 Eadid 7.;

As in thefe, 2 : 4 : : 8 : i6. Here 16 x 2 ^ 32.

And 8 X 4 = 32. Confequently 16 x 2 = 8 x 4.

' Corol 2.

From hence it foUows^ that if the Produ£f of an^ twe Numiert he equal to the Product of any other two Numbers, thofe four Num* hen are Proportionals.

And from thefe two LemmJt it will be eafy to conceive, that if ever fo many Numbers are in continued Proportion, Hhe Pro- doQ of the two Extreams will be equal to the ProduA of any two Means, that are equally difiant from the Extreams. As in thefe, 2 • 4 • 8 • 16 . 32 . 64 . &r.

Here 64 x 2 = 32 x 4 = 16 x 8. &f. And if the Number of Terms be odd.

As in thefe, 2 . 4 • 8 • 16 . 32 . 64 • 128. lie. Then 128 x 2 = 64x4=: 32 x 8 = i6X 16.

Note, the Chara^er made Vfe of to fi^ifj untinued Pro^ fwtion^L is -H .

I In

r

78 Ztt^mttJOu Part L

In every Series of -i^ (viz, ef c9hUnval Proportionab) that Number which is compared to another, is called the Antecedent of the Ratio ; and that Number Co which it is compared, is called it's Confequent.

As in thefe, 2 : 4 : : 4 : 8. (lere 2 is the Antecedent, and 4 is the Confequent ; and 4 the middle Term is an Antecedent to % It's Confequent : whence it follows, that in every Series of -ff- all the middle Terms between the firft and laft are both Ante* cedents and Confequents.

As in thefe, 2 . 4 . 8 . 16 . ^2 . 64, ^c. Here 4 • 8 • 16 • 32- are both Confequents and Antecedents.

For 2 : 4: : 4 : 8 : : 8 : 16 : : 16 : 32 : : '32 : 64, Vc.

So that all the Terms except the laft are Antecedents. And all the Terms except the firft are Confequents.

.Lemma 3.

In a Series of proportional Numbers, it will be : As any one of the Antecedents is to it's Confequent : So will the Sum of all the Antecedents be ^ to the Sum of all the Confequents. (1% Su^lid s)

That is, in the foregoing Series, 2 :4::2.-f 4 + 8 + 16-1-32:4 +8+ 16 + 3^ +64- For it is evident, that 4 .[- 8 + 1 6 J|-. 32 -|- ^4 ^^^ ^^^ ^^ ^^ the Confequents^ is double to2-(-44-8-4-i6+32 the Sum all the Antecedents; as 4 is to 2, according to the Ratio, and would have been Triple', or Quadruple, iJc. had the Ratio been 3 or 4, fsi'f. , .

Note, In every Series of ^ the Ratio is found by dividing any of the Confequents by it^s Antecedent. , As in thefe, 2 : 6 : : 6 : 18 : : 18 : 54 : : 54 : i62« Here 2) 6 (3 the Ratio. Or 6) 18 (3 isfc. From the fecond and third Lemma's may be raifed two general Theorems or Rules, for finding the Sum of any Series in r^ with- out a continued Addition of all the Terms.

Let the Series 2 • 4 • 8 . 16 . 32 . 64 . 128 • begiven^ to find it's Sum.

Si'ppofea;::^ the SuTi'of all the Terms. Then will z — 128 = the Sum of all the Antecedents. And z — 2 = the Sum of all the Confequents. ' Ba 2:4:12 — llS iz-^z , per Lemma ;i. £r^* 42 — 512 = ZZ-^/^. per Lemma 2.

Confequently

Chap, 6. Of ffijOpOjttOm 79

Confequently 4% — 22= 512 — 4.

Theorem; { * = ^ — — ^ '" Words at length thus,

{/V^w /A^ ProduSf of thi ftcond and laft Terms fubtraSf the Square of the firfl l^erm^ and that Rf^ mainder being divided by tlxe fecondTerm lefi thefirjl^ will give the Sum of all the Series.

Or if the firft Term, the common Ratio, ahd the laft Term be only given; Then,

{Multiply the laft Term into the Ratio, and from their Product fubtra£l the firft Term ; divide that Remainder by the Ratio lefs Umty or ly and it tmS give the Sum of all the Series.

For 4« — 2«= 512— 4. As above. ConTequently 22s — zzi256<^2- viz. the laft <livided by 2.

Then a = ^^ "^ ? Theoremz* 2 — I.

EXAMPLE. Let 2 . 6 . 18 . 54 . 162 . 486. be the given Series. Here 2 is the firft Term, 3 is the Ratio, and 486 the laft I'erm.

But 486 J^ 3 = 1458. And 1458 — 2 = 1456. Then 3 — l = 2) 1456 (7^8 the Sum required. That is, 7^8=2 + 6 + i8 + 54+'62 + 486.

* Since in cither of thefe Theorems it is required to have the laft Term known, (the virhich in a long Series of a?, will be very tedious to come at by a continued MultipUcation) it will therefore be convenient to Ihew how to obtain cither the laft Term or any other Term, whofc Place is affigncd, without producing ail the Tcnns,

In order tathat, it'will be ncceffary to premife the Coherence or Similitude that is betwixt Numbers in Arithmetical Progreflion and ihofe in Geometrical Proportion.

If to any Series of Numbers in -^ when the firft Term is not an Unit or i, there be affigncd a Scries of Numbers in Arithme- tical ProgrcflSon, beginning with an Unit or i, and whofecoin- mon Di&rciice is i. called Indices or Exponents :

^^^'^{2 .4 J . i6 3^.e4. 1^8 45^. -

Then

8o . llCttl^etfClU Parti.

Then will the Addition or Subtradion of any two of thofe Indices (or Numbers in Arithmetical Progreffion; diredly cor- refpond with the Produft» or Qiiocient of their re^)edive Terms in the Series of -fr.

'"^^"a So I X i6 = 128 the fcventh Term in ^

Airom f As 64.4 = 10.

^gam, I So 64 X 16 = 1024. *« ^cnth Term in -«•

But if the Series of -ff- begin with an Unit, the Indices muft , begin with a Cypher.

Now by the help of the Indices, and a few of the firft Terms in any Series of -r^, it is plain that any Term whofe Place or Di- fiance from the firft Term is affigned, may be fpeedily obtain^ without producing the whole Series,

EXAMPLE!.

A Man bought a Horfe, and was to give a Farthing for the firft Nail, two for the fecond, four loi the third, CsTr. in -h*, the Number of Nails was to be 7 in every Shoe, v/x. 28 Nails in alU What muft he have paid for the Horfe ?

P^aTo • I • 2 • 3 . 4 . 5 Indices

C I . 2 . 4 . 8 . 16 . 32. Farthing in -f^

Th«i \ 5 + 5=" And I 10+10 = 20

' C 32 X 32 = 1024 c 1024 X 1024= 1048576

Which is here to be accounted the 28th and laft Term. Becaufe the firft Term in the Series is r, which doth neither multiply nor divide.

Now this 134217728 being the Number of Farthings to be paid for the laft Nail, by it, the common Ratio which is 2, and the firft Term which is I, may be found die Sum of all the Series, ftr Theorem 2.

134*

Chap. 6. Of P|0pO?tfOtt, &c. 8i

^ I I ■ M^i I T — I II 1 ■ r — "■ — '

I342I7728 '^

268435456 From this Produ£l rubtra£t i. Ftz, 26843545O— I3BS268435455. Then 2 — i:=z i the Divifor. Confequently 268435455 is the Sum of ail the Series, or Price of the Horfe in Farthings, which being brought iato Pounds, (Scc^^/ 46) will be 279620/. 51. 3^. 3fri.

EXAMPLE 2.

A cunning Servant as!:reed with a Mafter (unlkiUed in >{umbers} to ferre him Eleven Years without any other Reward for his Service but the Produce of oiie Wheat Corn for the firft Year ; and that Produd to be fowed the fecond Year, and fo on from Year to Year until the End of the Time, allowing the Increafe to be but in a ten-fold Proportion.

It is required to find the Sum of the whole Produce.

m{

1.2.3. 4 • 5 • Indices or Years.

. 10 . 100 . 1000 . icboo . 100000 Wheat Corns in -1^

f As44-a=6

I So 1 0000 X 100 ni 1 000000 the 6th Year's Produce. And J 6+5=11

I loooooo X I0OQQ0 2S looocooooooo the eleventh ot laft Year's Produce.

Then (either by Theonm i. or 2) the Sum of all the Series willbeiiiiiiixiiio Corns. Now it may be computed from Pages 31 and 34, that 7680 Wheat Corns, round and dry out of the middle of the Ear, will fill a Statute Pint. If To,

Then 7680) iiiiiiiiiiio( 14467592 Pints, but 64 Pints are contained in a Buihel.

Therefore 64) 14467592 (226056 ^ Bulhels. Snppofe itfo be fold for 3 Shillings the Buihel ; <

Thenj "^^564-

Shillings 6781684=33908/. 8x. ^\d. A very good Rccompence for Eleven Years Service.

There are fevcfal pretty Queftions refolved by Numbers in Arithmetical Progreffion ; and by thofe in -ff-, which the ingenious Learner will cafily perceive hereafter ; viz. When we come to the Solution of Qiieltions relating to Incereft and Annuities, t^c,

** M There

I

8.2 . arttjjmeticl* Parti.

There is alfo a third Kind of Proportion, called Mufical, which being but of little or no common Ufe, I ihs^Il therefore give but a ftort Account of it,-

•' Mufical Proportion or Habhuda is, when of three Nambers, 'the fird hath the fame Proportion to the third, as the Diference between the fntt and fccond hath to the Difference between tfav fecond and third*

As in thefe, 6 . 8 . 12 viz. 6 : 12 : : 8— (5 : 12 — 8 If there are four Nuiaabers in Mufical Prpportion 5 The firft will have the fame Prpportion to the fourth, as the Difference between the firft and fecond hath to the Difference between the third and fourth. '

As in thefe 8 . 14 • 21 . 84. HereS : 84 : : J4 — 8 = 6 : 84 — 21=63. That is, 8 : :84 : : 6 : 63.

The Method of finding oiit Numbers in Mufical Proportion, ^s beft exprefled by Letters -, as (hall be (hewed in the Algebraick Part.

Seft-3. How to (t^WX^t or ^SXfibe Order of Things^ Sec.

np HIS being a Thing not treated of in any coinmpn Books of ^ Arithmctick, (that I have had the Opportunity of perufing} made me tliinlc it would be acoepxable to the ybmg LcKttfttj to know how oft it is poffible to tary or change the Order or Pofitien of any propofed Number of Things.

As how tnaiiy feveral Changes may be rung upon any propo/bd 'Nuinberof Ijdis; or how many. Several VariatiokYs may be Qiade of any determined Number of Letters, or aoy other Things pro* pofed tp be varied.

Thf Aiethod of finding out thf Numtir of Changes is by a continual

Multiplication of all the Terms in a Series of Arithmetical Pr^ef

/90ns J whofe firjf Term and common Difference is Unity or i.- jtnd

'j/>e iflfl Tjcr/n the Number of Things ffopofed Jo h Varied^ viz.

1x2x3x4x5x6x7, i:fc. As will appear from what follows.

1. If the Things propofcd to be varied are only two, they ad- ' init of a dou.blc Pofuion (as to Order of Place) aud no more.

ihus { !;J }=.=

! X2

And if three Things are propofcd to be varicf*, they may;

3 ^

i

Chap. 6.

Of ]paop9atlon, &c.

H

be zhangcd iix fevecal Ways (as to their Ord^r of PIslcc) and no Biore.

For, beginning with i, there will t>^ { J ' Next, beginning with 2, there will be < Again, beginnmg with 3, it will be |

2 2 3

3 2

3

I 2

I

' Which in all make 6 or 3 Tiraes 2, viz. 1x2x3 = 6, Sttppofe four Things are prdpofcd to be varied i Then they will admit of 24. feveral Changes, as to their Order ff different Places,

ri • a . 3 • 4 \i • a • 4 • 3 I'be^* •3-^*4

For beginning the Order with i, it will 1 Here arc fix different Changes.

\i . 3 . 4 . 2

yi . 4 . 2 . J

v,i . 4 • 3 . 2

And for the fame Reafon there will be 6 different Changes^ when 2 begins the Order, and as many when 3 and 4 begins <h« Order; which in all is 24 = I X 2 x 3x 4. And by this Method of proceeding, it may be made evident, that 5 Things admit of 120 feveral Variations or Changes 9 and 6 .Things of 720, &<:• As in this following Table.

Tbt Number	The Manner how	The different Changes or Vari
of Things	their feveral	afions every one of the propo-
pr9f9fidto	VariatUmenre	fed Numbers can admit of
hi varied.	produced.
I 2 .	I 1X2	= I = 2
3	ax3	r=6
4	6x4	= 24
S	24x5	= 120
6	120x6	:= 7«0
7	720x7	= 5040
8	5040 X 8	= 40320
9	40320 X 9	= 362880
10	362880x10	zz .3628800
II	3628800 X 1 1	== 3.9916800
12	39916800 X 12	= 47000x600
fcfr.	fsfr.	l5fc.

M 2

Thefe

^1

84 a|;it|)ttiettCit4 Parti.'

Thefe may be thus continued on to any affigned Number. Suppofe to 24 the Number of Letters in the Alphabet, which will admit of 620448401733239439360000 feveral Variations*

From the(e Computations may be ftarted feveral pretty, and indeed, very ftrange, Queftions.

EXAMPLES.

Six Gentlemen, that were travelling, met together by Chance at a certain Inn upon the Road, where they were fo pleafed with their Hpft, and each other's Company, that in a Frolick they made a Contrafl to ftay atthat Place, fo long as they, together with their Hoft, could fit every Day in a diiFerent Order or Pofition at Din- ner ; which by the foregoing Computations will be found near 14. Years. For they being made 7. with their Hoft, will admit of 5040 different Pofitions j but 5040 being divided by 365J (the Number of the Days in one Year) will give 13 Years and 291 Days. A very pretty Frolick indeed.

I have been told, that before the Fire of London (which hap- pened i/««<j 1 666) there were 1 2 Bells inSt^^ryi^fi^w'sChurch in Cheapjidej Londont Suppofe it were required to tell how many feveral Changes might have been rung upon thofe 12 Bells ; an.d ^t a moderate Computation how long all thofe Changes woul4 have been ringing but once over.

Firft, 1x2x3x4x5x6x7x8x9x10x11x12 ;=:47900i6oo9

the Number of Changes,

Then fuppofing there might be rung 10 Changes in one Minute : viz, I2X io=r 120 Strokes in a Minute, which is 2 Strokes in s( Second of Time : Now according to that Rate there muft be allowed 47900160 Minutes to ring them once over in all their different Changes ; viz. 10)479001600 (47900160.

In one Year there is 365 Days, 5 Hours, and 49 Minutes % vhich, being reduced into Minutes, is 525949.

Then 525949) 47900160 (91 Years and 26 Days.

So long would thofe 12 Bells have been continually ringing without any Internilflion, before all their different Changes could have been truly rung but once over. It is ftrange, and icems al- moft incredible, that a fe^y Things fliould produce fuch Varieties.

But that which fcems yet more ftrange and furprizing (yea, even impofl^JDle to thofe who are no( Verfed in the Power of Numbers)

is.

Chap. 7. Of pjopo?tion^ &c. ^

is, that if two Bells more had been added to the aforefaid 1 2 thejr. would have advanced the Number of Changes (and confequentljr the Time) beyond common Belief. For 14 Bells would require (at the (ame Rate of ringing as before) about 16575 Years to ring all their different Changes but once over.

And if it were poilible to ring 24 Bells in Changes (and at the &ine rate of 10 Changes in a Minute, which is 2 Strokes in one Second) they would require more than 1 1 7000000000000000 Tears to ring them but once ovti in all their dtflbrent Changes ; as may eaiily be computed from the precedent Table.

CHAP. VIL 0/l?|0pOltC0n D(SSjUmt; cmmoftfy called the eiSX«a

pRop9rtion DisjunSf^ ortheOEfoI&eit IBLuU, is either Direfl or ^ Reciprocal, called Inverfe. A^d thofe are both Simple and Compound,

SECT. I.

Tilreff Proportion is, when of four Numbers, the firft bears ^^ the fame Ratio or Proportion to the fecond s as the third doth to tlje fourth.

As in thefe 2 : 8 : : 6 : 24. '

Conicquently, the greater the fecond Term is, in refpc£l to the firft; the greater will the fourth Term be, in refpeifl to the third.

That is, as 8 the fecond Term is 4 Times greater than 2 the firft Term : So is 24 the fourth Term, 4 Times greater than 6 the third Term.

Whence it follows, that if four Numbers are in DireA Pro- portion, the Produd of the two Extreams will always be equal to the Produ£l of the two Means, as well in Disjunfl as in continued Proportion j according to Z^/n^n^ 2. pogoTj.

For As 2 : 2 X 4 :: 6 : 6 X 4. Or As 3 : 3 x 5 : : 6 : 6 x 5. But 2x6x4 = 2x4x6. Or 3x6x5 = 3x5x6.

That is, the Piodu£t of the Extreams Is equal to that of the Means.

Again,

86 anttimrttClU Part I.

Again, the Icfs the fecond Term is, in rcfpeS to the fir ft i the kfs will the fourth Term be in refped to the third. Asinthefe i8 : 6 r : 12 : 4. Tbatis^ 18: 18 :-f-3:: 12: 12 -r- 3. But 18 X i2-r-3=;i8 -T-3X 12. 'Viz. 18 x+r: 6 x 12.

Confequently 2 • 8 . 6 . 24. And 18 t. 6 • 12 • 4 • are true Proportionals, per CoroL 2. pi^e 77.

From thefe Confiderations, comes the Invcintion of finding at fourth Number in Proportion to any three given Numbers*

JVhence it is called the Rule of Three.

For if the fecond Number multiplied into the third, be equal to the firft multiplied into thfe.fourth, it is'eafy to conceive, that if ahe Produft of the fecond and third be divided by the firft, the Quotient muft needs be the fourth Number. For if that Number, which divides another, be multiplied into the Quotient produced by that Divifion ; their Produ£t will be equal to the Number di« vided. See page 21.

As in thefe 2 : 8 : : 6 : 24. Here 8 x 6 = 48 = 24 x 2. Butit24X2 = 48, then will 48 -t- 2= 24. Or 48-^-24 =: 2.

Notey Any four Numbers in direft Proportion may be varied feveral Ways. As in thefe*

Viz. If 2 : 8 : : 6 : 24. Then 2 : 6 : : 8 : 24. And 6 : 24 : : 2 : 8. Or 24 : 6 : : 8 : 2, fefr.

Thefe Variations being well underflood^ will be of nofmall Ufi iu fhejiating of any S^uefiion in this Rule of Three.

When three Numbers are given, and it is required to find a

fourth Proportional ; tlie greateft Difficulty (if there be any) will

be in the right dating the Queftion, or abftrading the Numbers

' out of the Words in the Queftion, and placing them down in their

proper Order.

.. Now this will be very cafy, if it be truly confidered, that al- ways two of the three given Terms, are^only fuppofed, and iiffign or limit the Ratio or Proportion. The third moves the *Queftion ; and the fourth gives the Anfwer.

As for inftance j if 3 Yards of Cloth coft 9 Shillings : What will 6 Yards coft at the fame Rate or Proportion ?

Here 3 Yards, and 9 Shillings, arc two fuppofed Numbers that imply the Rate ^ as appears by the Word [if] viz. if ^ Vards coft 9 Shillings (then comes the Queftion) What will 6 Yards coft ?

'. KB.

Chap. 7. Of ]^|Op0;ti«t, &c. 87

" ' ■ ■ ■ * - - _

y. B. The Term, which moves the Qucftion, hath generally fomc of thore W^ords before it; v/x.'33li)at tolll ? 1^0to titan? ?

^oto tone ? 1^01B Car ? or i^olb muefi ? &c.

Then (carefully obferve this ; vh^») the firft Term in the Suppofition muft always be of the £uae.Kind and Denomination with that Term which moves the Qijieftion. And the Term (ought will always be of the fame Kind and Denomination with the fecond Term in the Suppofitic

Thus. jyj>V.yAAA

^"^ { 3:9:: 6: Then

All QuefttonB in diredl Proportion may be anfwered by three {tsts^\Tbt9rems.

i Multiply their fecond end third Term together^ and Theorem i. A divide their Product by thefirjl Term; the ^a^ t tient will he the Jnfwer required.

yds. finU yds. JUL Tins 3 : o : : 6 : 18. The Anfwer.

— — ^ I bccaufe the fecond Term

3) 54 (18 Shillings, c was Shillings. .

f Divide the fecond Term By thifbrft^ Aen nmhiply the Theorem 2. ^ ^otient into the iiird Term, and their PreASt C %mll be the Anfwer required.

yds. Jbil. yds. Jhil. 3 : 9 : : 6 : i8. Thus 3) 9 (3=53. Then 3x6=18, as before*

r Divide the third Term by the firjt^ then multiply the Tbcorcm 3. •) ^otient into the ftcondTerm^ and their Produ£f I will be the Anfwer.

yds. fhil. yds. fhU. 3 : 9 : : 6 : 18. ' Thus 3} 6 (=2. And 9 x 2 = 18, as before.

Here you (ee that all the three Theorems are equally true ; but^ % firft is moft general, sind ufually praAifed. Yet the two laft ^^1 be readilf performed^ when either the fecond or third Term ^ be divided by the iirft ; and will be found of fingular Ufe in ^ Rules of Fellowjhipy &c. as will appear further on.

Ki

88 arifttnetidu . Part I.

^uij}. 2. If 8 Pounds of Tobacco coft 1 4 Shillings ; what wW half a hundred Weight (viz. 56 Pounds) coft at the fame Rate ?

Thus 8 lb, : 14J. :: 561b : 4/. 18 j. ThcAnfwcr.

224 .56

8) 784 (=98 J. =4/. 18 X. Or thus 8) $6 ( = 7 .Then 14 x 7 =: 981. as before.

^uefi, J. If li^ Shillings will buy 8 Pounds of Tobacco j how much will 4/. 18 X. buy after the fame Rate f .

Stated thus, 14 : 81b : : 4/. i8x. ^()%s. :

Then 98 x 8 = 784. And 14) 784 (56 lb. The Anfwer.

^uejt, 4. If half a hundred Weight of Tobacco be w«rth4/. i8x» How much may I buy for 14 Shillings at that Rate ?•

Stated thus, 4/. 18 j. =981. : 561b :: 141.:—— Then 56 x 14 = 784. And 98) 784 (8 lb. The Anfyirer.

^iJi-S* Suppofe4/. lis. will buy 56 Pounds of Tobacco i what will 8 Pounds of the fame Tobacco coft ?

This Qucftion is thus ftated, 56 lb : 4 /. r 8 /. =98 x. : : 8 lb :— ' w Then 98 X 8 z= 784. And 56) 784 ( =z 14J. TheAnfwcr.

Note^ The three laft Queflions are only the fecond varied, being propofed purely to give an Inftance how any Queftion in this RuU of Thru may be varied, according to page 86.

^ejl. 6. What will three quarters of a Yard of Velvet coft, when the Price of 21 Yards and a half is worth 22/. zoi. 6 d. This Queftion truly ftated will ftand

Thus, 21 \yds. : 22/. 10 x. td.w^To theAnfwer.

Which may be found thi;ee feveral Ways j vi%. by ReduSfioni by Vulgar Fra^fions ; and by Decimals.

I. By ReduSlion. Brin» the firft and third Terms into one Denomination ; viz. into Quarters, and i educe the fecond Term info it'sleaft Denomination, perScSl. 4. page 42.

' Thus 2 1 4. =: 86 Quarters. And 22/. i o x. 6 ^/. zr 5406 Pence. Then 86 : 5406 : : 3 : 15--. 8 1-|. For 5406 x 3 =r 16218.

And

Chap. ^. Of ffilJpgjtfOtt^ &c. 89

And 86) 1 62 1 8 ( = 1881^4/. Then j88||P^«c^= 15/. %i, 2 i^ Farthings 'j the Aniwcr required.

2. The fame Queftion ftated in f^ulgar Fraffiom will fiand Thus 21 ^ = V : 22 fS= \V • : T : (See S^^J?. 3. page 50.) Then VV X I = V-^'- And V} \%^ ( = ij|4 /^.-^ 55. 5^.

Thcfe I* J-* Parts of a Pound arc brought into Shiilings by multiplying the Numerator with 20, and dividing the Frodud by it's Denominator, ^c^

Thus 5406 X 20::: 10.8120. And 6880) ic8i20 (i5i. And thefe renwna 49^. Again 4920 x 12 = 59040. . Then 6880) 59040 {8</. and'4^, as before.

3. Tlie lame wrought hy Decimal Fra^Ihns wtH be thus ; 2i|. = 2i,5 • ^^^* 'o^- 6V. 1^22,525, and ^ = 0,75 Therefore 2 1>5 : ^ZyS^S • • o»75 • ^ ^^ Anfwcr. Then 22,525 x 0,75 = 16,89375 And 21,5) 16,8937s (0,7857/. = 15X. 8i. ufar. ^?^V

%/?. 7. ViC. ^qrs. Ill lb. of Sugar coft 6/. ix. 8 J. What will 1 2 C. 2 frs. coft at the iamc Rate ? Thati5,:2 C 3^, %i lb : 6/. ix. 8i. :: 12C. 2frx. To what!

4 . ^^ ♦

II yrj. 28

1400 lb«

f^'^ 3o8+ax2:3a9lb:i46oi. : : 14001b:-—

Then 1460 X 14003:2044000. And 329) 2044000 (6212^^4 ^2^1. ijs. 8 Id. the Anfwer required.

The fame Queftion flated in Dedmals will ftand

Thus 2,9375 : 6,c*33 : : 12,5 : To the Anfwer.

Then 6,0833 ^ ^2,5 = 76,04125, which being divided by •i*9375 ^^11 giv« 25,8863, &c, tfae Anfwer in Dcciroala, which te>ag|tt into Coin, will be 25 /. 17 /• 8 f W. as before.

Note, ^en the firft %erm u an Unit or i, the ^ejlion is ^fiifired by' Multiplication only.

Example. * Siippofc I give 5 ShtlKngs 4 Pence for one Ounce of Silver, What muft I pay for 32 ^ Ounces at the fame Rate ?

That is, I Ounce : ^ x. 4^. r: 32 ^ Ounces : To, ^c. *^^h i« beft ftated thns 1 ; 04^* : : 32,5 :

N Therf

po SltftlltnCttCft^ Parti*

Then 32,5x64=12080^.3=8/. 13X. ^d. the Anfwcrrtquired^ For I neither multiplies nor divides.

When the fecond or third Term is an Unit or t, then the, Queftion is anfwered by Divifion only. As in this ]£xample.

If a Silver Tankarfl weighing 21 Ounces, coft 5 /. 191. What is that an Ounce i

Thus 21 oz. : 5/. 19^ = 119X. : : i : 5/. 8^. the Anfwer.

Thati8 2i) 119 (= 5*. 4*s=5x. %d.

The Proof of all Queftions in the Ride of Thru Direif^ may be( eafily conceived from what hath been already faid ; vm. That the Frodud of the firft and fourth Terms, muft always be equal to the^ Produd of the fecond and third Terms.

Or otherwife, by varying the Quefiion, as in the fecond^ third, fourth, and fifth Quefttons.

r ihall conclude thb Sefiion with infertine a few Queftions and their Anfwers ; leaving their Work for the Learner's Pradice.

. ^efl. T. What will the Carriage of 17 C. 3 qrs» 1 1 lb. come to, at the Rate of 7 s. the Hundred i

Anfwer, 67. 4/. ii^^.

^Ji^ 2. If 6/. 41. 1 1 -^i. be paid for the Carriage of 17 C 3 fri. 1 1 lb ; What was paid for the Carriage of i lb ?

Anfwer, 3 Farthings.

^teft. 3. A Grocer bought 3 C I ^. 141b. Weight of C&v«, at the Rate of is. \d, per Pounds and fold them for 52 /. i\s^ ^Whether did he gain ot lofe by the Bargain, and how much I

Anfwer, he gained 8/. I2;.

^eji. 4. A Draper bought of a Merchant eight Packs of Cloth ; ^very Pack had four Parcels in it ; and each Parcel con- tained ten Pieces ; every Piece was twenty-fix Yards ; he gave after the Rate of four Pounds iaattn Shillings for 6 Yardt. Wiiat came the eight Packs to, and what were they worth per Yard?

Anfw. They Came to 6656 /. And were worth its. per Yard*

^Sluefl. 5. A Merchant bought 436 Yaida. of Broad Chth for 8 s. bd. per lard-, zud fold it again for lOJ. ^d. per Yard* .What did he gain by the 436 Yards f

Jnfw^ he gained 39 A 195. 4/ ^

Siueft.

Chap, ^. Of i^fflpgjtlOn, &c, 91

J^«^. 6. A Goldfmkb bought a Wedge of (7^^/, which weighe4. 1 14 lb. 3 9Z. Spw. for 514/. 41. What did he ^zyper Ounce?

Anfw. 3/. per Ounce. ^fi. 7. What will 48 9%. 17 fw. 20 Gr£i/»j £/* Silver Place cofflc to, at the Rate of 5 x. 6 d. per Ounce f

Anfw. 13/. IS, lo^d. ^ueft. 8. If in four Weeks one fpend 13>. 41/. How long; will 53/. 6 J. laft at that Rate ?

Anfw. 6 Years, 47 Days, 2 Hours, 2' 4* i^tt^. 9* What will the one eighth Part of a Ship be worch» when the half is valued at 10 15 /. 10 j.

Anfw. 253/. 17 J. 6V. ^Ji. 10. The Sun is faid to perform one entire Revolution (or 360 D^eea) in the Space of 365 Days, 5 Hours, 48 Minutes, ud 57 Seconds of Time, called a Tropical or Solar Year ; How ouch doth it move ta one Day i , r ,.,

Anfw. 59 . 8 . 19 i^c ^Jl. II. If j. of a Y;ird of Felvet coft 4 of a Pound Sterlings What will Vr of a Yard coft of the fame Felvet at that Rate i

Anfw. ^y*^ = I f. 4^. . ^ift. 12. Suppofe 2/. and 4. of 4 of z Pound Sterling will b«V 3 Yards and 4. of 4 of a Yard of Clothy How much will i of tivd coft at that Rate?

Anfw. ^444 of a Pound = 9 ;. 4 ^d.

Sea. 2. Of Rertpiacall9iopajti0ni «/iwi^ caM-

The Rule e/" Three Invcrfc.

PEctprocal Proportion is, when of four Numbers the third (viz. that which moves the Queftion) -beareth the fame Ratio' to , ^ iirft : As the fecond does to the fourth.

Therefore, the Icfs the third Term is, in refpcfi to the firft ; . ^ greater will the fourth Term be, in refpe£i to the fecond.

EXAMPLE I.

If fncteen Men can do a Piece of Work in fix Days ; How many Days will eight Men require to do the fame Work, at the fame Rate of working ?

Here it is plain that eight Men muft needs have more Tim« ^ 16 Men to do the &me Work. Confequemly the greater

N a the'

9^ gtctt>metfefc. Parti.

the third Term is, in refpcS to the firft, the lefs will the fourth Term be, in rcfpcft to the ^cond.

Example 2, If 8 Men can do a Piece of Work in 12 Days, How many Days will 16 Men require to d6' the fame Work ? Here it is plain the' fourth Term muff be lefs ihnn the lecond^ b^aufe 16 Men undoubtedly can do the fame Work in lefs Time than 8 Men can.

From thefe Conjiderations^ compared with thoft in page 85. it ft ill he eafy to perceive^ whether the Terms tfany propojed S^cjlion art in Dire/3 or Reciprocal Proportion.

For when^ according to -the true Meaning and Defign 9/ anf ^ejiion in Proportion ^ More requires More^ or Lefe requires Lefs^ d>e Terms are in DieSl Proportion ; as in this laji Sexton.

* But if More' requires Lefs^ or Lefs reqtdres More (as above) then {be Terms will be in Reciprocal Proportion.

The Manner of placing down the propofed Terms is the fame in both Rules, viz. The firft Term in the Suppofition muft be of the fame Kind and Denomination with the third Term" which moves the Queflion ; and the Term feuprht muft be of the 6ihe Kind and Denomination with the f^cond Term in the Supppfitipn. As in the two laft Examples.

Men Dafs Men Dayl'

TK»c i^iE^amplei. ^ 16 : -6 : : 8 :--^— •

Thus, m I ^^^^^^ ^, 8 : 12 : : 16 :

The Qucftion being truly ftated, obfcrvc this Theorem.

^X Multiply ' the frjl and fecorid Terms together j and Theorem, j divide the Produ£i by the third Term^ the ^otieni t will be the Anfvber required. Thus in the fecond Example 1 2 x 8 = 96* Then 1 6) 96 (=s6Day8, the Anfwer required.

That is, 16 Men may do the fame Work in 6 Days as 8 Men ckndo in 12 Days.

Now the Reafon of this Operation (and confequently of the Theorem) is grounded upon this Confidcration ; viz. If 8 Men require 1 2 Days to do the Work, it is plain that one Man would require 8 Times 12 Days 1^96 Days tp do the fame Work ; but " if one Man can do it m 96 Days, moft certainly i^ Men can do it in one 1 6th Part of that Time. Therefolre 96 divided by 1 6 will give the Anfwer required, viz. it) 96 (fear before, ^c.

■ ^e/l. 3. Suppofe 800 Soldiers were befieged in a Town, and fheir Vi£luals were computed to ferve them two Months (or 56 Days) How many of thofe Soldiers muft depart the Garrifon, that the £ane Victuals may ferve the remaining Soldiers 5 Months ?

The

Chap. 7. Of IPlOpQItifln^ ^c. 93

• * ' ^'

The Q^ieftion truly ftatcd will ftand Months. Soldiers. Months. SoMers. Thus, 5t : 800 1:5: —— "^

2

S) 1600 (^20 : So many Soldiers may ftay inihe Garrifon. ^

Confequently, 8oo-r'320rr4So .Soldiers that muft go out a£ die Garrifon, which is the AniWer required.

^eftlon 4, A borrowed of his Friend B 250 /. for fix Months,* promiiing to do him the like Kindnefs upon Demand : Some nme after B defires A to lend him 400/. the Queftion ti, hew- long B muft keep the 4C0A to be fully fatisfied. for his former Kindnefs to A.

Thus, 25t>/. :* 6 Months :: 400/.:—^ 6

, 406) 1500 (3 Months. 12

28 Days in one Month. 4) 84 (21 Days; Anfw. 3 Moikht, 2 1* Days.

^eftton 5. If a Penny White Loaf ought to weigh eight! 0<M^ Tf'oy Wnght^ when Wheat js bid for .fix ShiUims fix PicDce the Bufliel, what muft it weigh when Wheat is fowifor'

four Shillings the Bufhcl ? ^ , . . . • . .

Thus.6j. bd.zz. yiJ. : 8 oz.^ : : 4^. =: 48 i/. : to the Anrwo*.

8 . ' ? : J

48) 624 { f3 oz. the Airfwer required.

144 144

(O) ' ' • - ,

The Proof of this Inveffe Kule is eafil)^ deduced from it't Operations 5 viz. The Product of the firft and fecond Terms, muft be equal to the Produfi of the third and founh Terms.

Nftty Any <^ftion that falls under this Invcrfc Rule or Re* ciprpcal Proporcion, may be fo ftated as 10 have it's Therms io Dtrea Proportion ; by only changing the Places of the ficft ani dW Terms in the Queftion. Thus,

^ftf/KoM

54 ' gtitftm<tfclU Partly

■f ■■-ml" «• - ■ • - -I ■■■II ■ I. II. ■.: I

^eJiton6. If a Field will feed eighteen Horfes for kvcn Weeks : how long will it feed forty-two Horfes at the fame Rate of feeding ? ...

Firfl, 1 8 Horfes : 7 WceVs : : 42 Horfes : 3 Weeks.

Here the Terms arc ftatcd invericly, as before.

Otherwifc thus, 42 Horfes : 7 Wteks : : 18 Horfirs : 3 Weeks. Then 18 x 7 =: 126. And 126 -r- 42 n 3 Weeks. The Anfwer required.

Sed. 3. . Of CompOUttH P^pO|ttot ; cofmoulf calkd The Double Kuie oi Three.

xy7m^«i/ffiPy0^0r/^ii (as it.is here meant) is, when there are ^ five Numbers given to ifihd out a fixth Proportional 5 and this IS generally performed by a Double Pofition ; that is, by ftating and working the Queftion at t^^rb Operations, either ih Direft or Reciprocal Proportion, according as the Queftion requires.

And tberefon it is calUd, The Double Golden Rule^ tr DotAU Rule of Three.

The Double Rule DirecSl is, when the fixth Termor Number fought, is found by two Operations,' both of them in Dired Pkroportion. 1 ..

Exempli I. If a hundred Pounds gain fix Pounds Intereft in twelve Months ; how much will three hundred Pounds gain in > Bine Months, at the fame Rate I

Fir(i 100/. : 6/. : : 300/. : 18/. 6

100) 1800 (18/. I The Intcrcft of 300 /.

.1 for twelve Months. Months. Months.

Then, 12 : 18/. : : 9 : 13/. loj. 9 12) 162 (13/. io#. The Anfwer required.

I fuppofe the Learner will eafily conceive the Realbn of thefe two Operations* For, firft, it is plain by Dired Proportion, that if too /. gain, 6 /. in twelye Months, 30Q /• will gain 18 /• in the fame Time, and at the fame Rate.

I And

Chap. 7. Of lS>?0p02t(0n^ &c. ^s

And by the fame Rule'it Is plain, that if ir Months will pro- duce or give 18 /. Intercft for 300/. then 9 Months muft needs give 13 ^ for the ft me Sum, viz, 300 /.