Google
This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project
to make the world's books discoverable online.
It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject
to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books
are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover.
Marks, notations and other maiginalia present in the original volume will appear in this file - a reminder of this book's long journey from the
publisher to a library and finally to you.
Usage guidelines
Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the
public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing tliis resource, we liave taken steps to
prevent abuse by commercial parties, including placing technical restrictions on automated querying.
We also ask that you:
+ Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for
personal, non-commercial purposes.
+ Refrain fivm automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the
use of public domain materials for these purposes and may be able to help.
+ Maintain attributionTht GoogXt "watermark" you see on each file is essential for in forming people about this project and helping them find
additional materials through Google Book Search. Please do not remove it.
+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other
countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner
anywhere in the world. Copyright infringement liabili^ can be quite severe.
About Google Book Search
Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers
discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web
at|http: //books .google .com/I
^^o^..^. J I
■^f"
x/
^\
7r'^'
as-
■}
-J
v\
L%
V
\ \
11
i
ARITHMETICAL
Collections
AMD
IMPROVEXfENTS.
B B I N O
ACOMPLETE SYSTEM
o r
PRACTICAL AHITHMETIC
BY** :•:
ANTHpM% and JOHN BIRKS,
Late Maften of a Boarding-School at GadtrUWt and now
of Ae Pice Writiiig-Scbool at Damnprn, LmteUfiirt,
LONDON:
Printed for, and Told by the Authors ; alJbby MeC Ha WES,
CoLLiKS and Clarkb, at the Red Listi, Pater-mjitr-
Jtto i and Meir Hollihowortm, Bot^ellen, at Lyt'
' MDCCLXVI.
Ifc
> r
y %■ '- ^r
*■ • *
\ »
• • • -
* , • • •
I-.
liJ^uu^y^ T O TH E
UIGHT HONOURABLE
TX^ Lord Brownlow Bertie,
One of the Reprcfentatives in Parliament
I for the County of Lincoln, 6cc. &c.
' 4
My LORD,
TH E utility of the fcience treated
upon in the following iheets,
muft be well known to your Lordfliip,
who has {o often been a^itnefs of the
advantage and uncommonVight, argu-
^ ments drawn from arithmetical compu-
• tations carry with them in that houfe,
5 whereof you are io worthy a member ;
, lib that if this performance be found
"^ equal to the ufefulnefs of the fubjcA,
it may juftly be entided to your Lord-
fhip's patronage.
Your adiduity in parliament for pro-
moting the drainage of, and making
roads through a lately inundated, though
rich country, is and will be of fiich great
benefit, that fucceeding ages, as well as
the prefent, muft reap the advantage of
A 2 thefe
iv P E P I C A T i-O N. : :
thefe iklutary works, and pofterity ble($
the time when a Bertie graced the;
Briti(h fenate.
That' your J^xdOdp mgy ftil) fucr
ceed in promoting the good or your
native country, aiid long live the great
ornament thereof, is the hearty wi(h
of,
t
4
My
•
•
LORD,
•
«
•
Tour Lordjhif^
*
1
•
*
ptoft ohedier^t
ap4
• •
•\
t
tfiofi humble Servant ^
Th^ Authors-,
/•
-« «
PR E F A C E.
TH E tx)ok here prcfcntcd to the world is
a regular fyftcm of common arithmetic,
adapted to the ufe of the gentleman and the
fcholar, to well as the man of bufinefsi
BOOK I.
Numeration, and the next four general
rules, are enriched with many compendious me-
thods and examples -, and the rules of Pradtice
very copious and exfenfive.
Th6 dodrine of Vulgar Fraftions is here ren-
dered more eaiy, concife, and u&ful, by the means
of an eaiy axiom ', Decimals are purfued through
all the late improvements, in tne management
both of plain and circulating numbers ; and the
Extradlion of the Roots, particularly the Cube, is
done in a more eafy manner than in any book of
arithmetic, which has / ever yet come under our
in^)e£kiOn.
BOOK II.
In which Proportion is treated on in a fcientific
manner, and adapted as well to the ufe of the
young mathematician, as together with the rules
of f ni&ioe vappfied to all branches of bufincfs;
A 3 and
vi PREFACE.
and the mercantile rules are exemplified and dir
yerfified with great variety of curious examples.
In Exchange are exhibited Sir Ifaac Newton"^
tables of the aflay and real and (landard-weight,
arfd value, of moft of the gold %nd filvef coins in
Europe ; together with thofe of the ebnformity
of weights and meafures, by the Sieur John La*
rue, merchant, at Lyons ; alfo the method of
fblving queftions in the artntration of exchange
hy a numerical equation. '
BOOK III.
Contains the lefs uieful, though moft pka^ht-
and delightful parts of arithmetic ; viz. Alligation,
medial, partial, and total ; the Specific Gravity of
Metalsi the Rule of Falfe, or Pofitioui Pro-
greflion both arithmetical and geometrical ; 'alfo
Variations, Combinations, and the method of
fiinng Magic Squares : thefe, though they are
done and accounted for better by Algebra^ &e.
yet may ferve to open the mind, and excite the
curiofity of youth to proceed to the moH fublinie
and abflrufe fciences.
^ ' ft ^ _ ^
To theie are added Compound Intereft, with
the method of calculating tne value of freehold
edates at any rate of intereft; alfo annuities in re-
veriion^ according to that late ingenious mathe-
matician Mr. Thomas Sympfon, F. R. S. from a
fet of tables calculated by him for that purpofe.
'Alfo a colledlion of quedions in Menfuration,
with fuch diredions as may enable any peribn to
perform the meafurement of mod forts of fuper-^
ficies and folids ; and alio fotne curious mifcella-
neous queftions.
This
p. R B V' Ace, vU
1^ patik concludes . with an Appendix, by
fAt: 'j:fMXi^JMltn, . teacher of the. mathematics
^ Sp^din^: containing rules ah(i exai^oipies for
^dingtbe &uxi of any given number of terms in
pertain prcgrefiions. A coUeAion of problems
toncerriiftg the maxima and mn^ma of quantities,
yfith ^he meotemsr annexed. And the inveftigs^
tK)n of the fums of certain in^nite feries.
.. This jvprk contains feyeral hundred queflions
rationally fblved ; among which are all tbofe in
Clare's IntroduAion to Tracle, &c. feveral ^rom
the Palladiums, Ladies Diaries^ a|id other perbdi-
jpal book^/ as well as the n^Ofl valuable and enter-
taining that c6i;ld be found in other authors.
/ N.B. We were favoured with the Cribbagc
Qucftiori l>y Major Watfon.
The atgehraic demohflration of the rules iiere
laid down are omitted for theie three reafons ;
firft, as aritnmetical computations often carry their
rationale along with them, the offering to prove a
fclf evident . truth renders it ftiore obfcure. Se-
condly, the mathematictans are already acquainted
V^ith tffefn. And thirdly, the young ft^dent is
as little benefited by them as a pure £ngli(h. fcho->
lar would be by an expofitioh of the Bible in
Greek.
What overfights may have efcaped the authors,
cither in the prefs (from whence their refidence
is more than loo miles) or other wife, hope their
worthy fubfcribcrs will generoufly excule.; they
haviriglriihie whole endeavoured to remove the
difficulties, ; and render the paffage eafy and plea*
fant through this ufeful and delightful fcience,
..;^ - A 4 "The
neceflary to be underftood, as being a muck
fliorter^ better, and more fignificanc way of ex-
pfefl^Mi, than by W)rds at kogth. '
}The fign of addition \ ^^ 9 -k- f^
is 9 more 5, and fignifies chat
the numbers 9 and 5 are to be
added together.
r J The fiffn of SubftraAion ; 8 — g,
— - \ Minus or > is 8 lels three, and fignifies that
I 3 3 is to bis taken from 8.
5 Multiplied 7*"* ^°.^ MukipUation; as
^ \ ii,to<£by } l^^; » 7 """^"P^"* "^*°» °^
The ugn of Divifion ; 8 -r 2^ is
8 divided by 2 : alfo thus ^ or
2) 8 (4, which fignifies the fame
thing.
The fign of EquaKty ; 9 = 9» or
p + 6 = 15, or 9 — 6 rr 3, that
IS 9 is equal to 9, or 9 more 69 is
equal to 151 and'9.1eis 6, is.
equal to 3.
The Signs of Proportion, or Rule
of 'Dtfee; thus, 2 : 8 : : 6 : 24
areto be read, as 2 is to 8, fo is
6 to 24.
I Continued Proportionals in Geo-
> metric Progrelfion.
r r Thus 270* 3 or 27', fignifies that
^ \ Involution. \ 27 is to be involved to. the third
' ' powen
r"] Thus ^4 rr 2, fignifies that the
, Extraaion i ?«f^^ root of 4 X"^' ^,?J
1 of the roots ^ V243 = 3i viz. tfc* furfoW
I or root» or the 5th power of
$U B-
V
r
S'U B'S eft J BE R S . NAMES.
Mil
A ■ •
K . Thomas Allen* teacher of die oiatbciaauicl at
Iding, 12 copies ./ -t-
Arnold, merchant, in ditto
Mr. William Adkin, of Leak, chief conftaUe^ lincolflfliirt
Mr. Satnuq^ Aifttup, of Whapioad, dkto.
Mr. Robert Allea^ of Gofterton, ditto '
Mr. Samuel Ay re, of Frampton, ditto
Mr. Richard Alko^^f Ketton in Rudandflure, quarcr^iaaii.
The right-honourable the Lord Brownlowe Biertid, member
of parliaoient for Lincobiihffe
The Rev. Mr.Everard Bockworth, rcaor of WflflilncbitMisrh
T%e Rev. Mr Bryant, of Norwich * * .
Mr; Stephem Biee, ftewsrd and land-furreyor, of ATwaifcj
Theophilus Buckworth, of Spalding, gent.
Mr. John Berridge, of London> watchmaker
Mr. William Benton, of Thprney, Cambridgihire
Mr. Thomas. Brodcrick, watchmaker, of Kirton^ Lmcoln-
fhire
Mr. AmoTBulO', tof Donnington, Lincolnfhirv
Mr. Charles Boyles, fon of colledor Boyles, of Wells,
Norfolk >
Thomas B»ley^ of DantGit, Norfolk, eiq.
Mr. Jpfcph.fiUckitb, of Fjn^pton
Mr. Jofeph Barker, of Lincoln, writing'-mafter
Mr. Bramptpn, attorney at law, at Oundle, Northampton-
fhire '^
Mr. Black
Mr. Henry Bruce, of Holbeach,' Lincolnihire
Ml^. Joha Burton, of Partney, ditto
Mr. George Bland, fchool maAer, of Bafton
Mr. Heoi)i Bcnnet, of Spalding.
Mr. John ff^ter, of Wigtoft
Mr. Abralom ]^tes, of Donningtoa
Mr. Thomas Mick, of Melton-Mowbray, Leiccfterihire
Mr. Sutton Banks, of Sleaford, Lincolqfliire
"^ Mr,
SUBSCRIBERS NAMES. -
Mcr John BdU. of Gofbertxm) felatonger ' • i
Mr. Matthew Bacon, of Norwich, grocer
Mr. Chriftopher Berry, ftacioner^ of Norwicha 64:opiet.
Mr. George Bell, merchaat at Leed«, Voridhtse
Mr. WUliam Butterworth, of Ecclesfield, ditto.
Mr. Daniel Bullivant, of Sproxton, Leicefteiibire^ z copies
c
The Rev. Mr. Jobn Calthrop, vicar of Bofton'
Mr. Thomas Cole, of Donington
Mr. William Cole, of Donington, mercer
Mr. Robert Cole, of Stamford, canrer
Mr., Richard Clay, of Swinefhead, Lincolnihire
Mr. Robert Clifton, of Donington, ditto
Mafter Richard Calthrop, pupil
Mr. George Cha|Mnan, of Surfieet, Lincolnfhirc 1
Mr. Maxamillion Clifton, of Donington
Mr. Thomas Cheibire, of Spalding, furveyor of the xoa4^'
Mr. Edward Crane, of Quadring, fchool-ni«fter
Mr. Charlton^ of North Chariton
Mr. Clifton, of Swarby, malfter
Mr. Timothy Corney, fchoolmafter, of Sarfleet
Mr. John Chapnun, of Bofton, chief conftable
Mr. Thomas Creafy, of Hecktngton '
Mr. William Crofts, of Nottingham
Mr. Crawford, of Crowland
William Crowe, of Norwich,, efq.
Mn Jofhua Chown, .riding officer in the cufloms
Mr. James Crowe, merchant, at Norwich
Mr. Thomas Coleman, ironmonger, ditto
Mr. Samuel Louis Chriftin, of Bern, /n Avj^
Mr. Thomas Morley Clyatt, of Norwich, 2 copies
Mr. Benjamin Cole, haberdafher, of ditto
Mr. John Carman, comptroller at Yarmouth
Meflrs. Chambers and company, merchants in NewcaiUey
6 copies
Mr. Thomas Cole, of Norwich
Mr. Hugh Clark, of Spalding, maltfter
*D
Mr. John Daverfon, cfa; of Yarmouth, Norfolk
Mr. John Dubois, of Wilmineton, South Carolina
Mr. Valentine Dennis, of Wells, Norfolk
Mr. Thortas Deverill, of Nottingham
Mr. Dodds, fchoolmafter, of Elton
Mr. William Dcnqis^ fchooknaftrr, of Bitchfield
Mr.
Mr. George Dale^ «t'tAie Red Lioh*»inn, Alderrgate^-fitee^
London '
Mr: Ci^}c% DaMb,' of Donington
Mr, Robert Doubleday, of Wlgtoft
Mr. Ofl>ert I>entoni mercRant, of Weetinj, Noifolfc •
Mf. ThofikM tkiveyj Ichbolihafter, of Norwich
Mr.
Mr.
arrat Daibwood, of ditto
Mr. John Dawfon, merchant, at Becdea
ohn Dale^ merchant m London
B .
Mn Langley Edwalds) land-fitrreyor-and engxneeri Boftott
2 copies
Mr. John Everard, ichoolmifter, of Spalding
Mr. Thomas EQis, of S winefliead, merter
Mr. Fnuicis EafUand, of Goiberton
Mr. Edward Edwards, merchant in London
Mr. WilUayn Elftobb, lMi(Ufurvejon
F
The honourable the ladjr Fraifer» of Crefly-hal!, Lincdlo*
fliire
Mr. WBliMft Franklin, efq; of Yarmotith
WiUiam Foftec, efq; merchant in Newcaftle
TTie Rev. Mr. Fell, of Beetles
The Rev. Mr. Wyat Francis, reftor of Burton^ nearLm^
coin
Mr. Jephtha Fofter, df Lincoln, prodor
Mr« John Flindeiv, of Donington, apothecary
Mr. John Flinders, of Spalding, furgeon
Mr. Robert ForCe#, -dancing-mafter
Mr. Nathan Forfter^ of Nottingham
Mr. Jofeph Featherftone, of SpdMing;
Mr. Steplien Fox, fchool-mafter, of lEdenham
Mr. John FrankKn, of Norwich, merchant
Mr. WilKam Fenton, merchant in London
Mr. Zachariab Fincham, of Difs.
G
John- Grundy, efq;-of Spalding
Charles Galtmeau, of London, efq; 6 cdpies
The Rclr. Mr. Gibfon, of Holbeach
I^nuM^is Green, gent. Southampton-buildings, London
Mr. Gari!t, of Bofton, merchant
Mr. Thomas Gough, attorney at law, at Gedney
Mj:., William Golutng, of Donington^ baker
^4 Mr.
M *
■k •
Milfer- Jdhft <JUAknm, ^f WhtfdcMkl, LlncolbQ^ p
Mr. Joftphr Gchnilk) of Gofberton
Mr, J onadiAA Gleed^ atM-nev M iMr, Donington
Mr* William Gee, of SwinelJieft^ *
Mr. Bracebridge Greefii df Donington
Mr. kichard Gill, of Scrediiigton, X/incoinfliire
Mr. James Gibbons, of SurAeet
Mr. John Green, of Swinefhead i
Mr. Robert Goodall, of Nottingham
Mr. Edward Gibfon, of Donington ^
Mr. Nicholas Gilbert, of Norwich
Mr. William Gymer^ merchant at ditto
Mr. John GrlAths -
H
The Rev. Mr. Hardy« fchoolmaficr of NfelCon-^Mowbray I
Mr. Thomas HoUingworth and fon» bookfellers at Lynn^
6 copies
Mr. Edward dare, of .Walton^ aear PcterbMMghi iandi
furveyor
Mr. Jonn Hickfon, of Bofton, writing-mafter
Mr. John Hanley, of Swineftead
Mr. Samuel Harvey, of Long Sutton, Lincolnfliixe
Mr. Thomas Hibbms, of Stamford, mafon
Mr. John Hinch, of Donington
Mr. Richard Hall, of London
Mr. John Hilton, of Nottingham
Mr. William Holmes, of DoaingtXHi
Mr. Thomas Holdernefs, coachmaker, in the Old Bailef ^
London
Matter William Hanley, of Swinfliead, pupil
Mr. Matthew Halls, officer in the excife. Pinchbeck
Mr. Stephen Hall, ditto, in Spalding
Mr. Thomas Ho|gard, of Deeping, Lincolnihire
Mr. Robert Holhnfhead, of MeJton-Mpwbray
Mr. William Hill, merchant at Wells
Mr. William Hawks, ironmonger at NewcJdHe
Mr. Handley, fchoolmafter, Ofbornby
Mr. John Button, jun. of Northfomercotes.
Mr. Henry Hind, jun, of the Slate-quarries, Switblaftd
J, I
Matter Fairfax Johnfon, fon to counfellor Johnfon* of
Spalding
Mr. James Jones, furgeon, of Fakenham, Norfolk
Mr. James Jones, jun. of ditto* •'
Mr.
Hr* to^n Jackfon» of Skafosd, Attorney 9X law, a
Mr. Johifj Jesury, Ismd-wnitrr a tb« port of Wfi(»
Mr. John Jackfon, of Bicker
Mr. Richaird Jackfoii« of Ponington
Mr. John Uckiony of Meltoi^'^Mewbrey
Mr. Job|i Johnfon, of Nottii^bsQii
}ohn JzrviSj of Bicker, geot»
Mr. Jepfon, of Lincolnt pro£lor, 6 copies
Captain William Ivory, of Norwich
Matter Thomas James^ pf ^oft^n, pupil,
K
Mr. Richard Knowles, Smithfield-Bars^ ]l^ndpi|
Mr. Thomas Knowles, of Spalding
Mn Samoel Kent^ of ppningtoi>
L
Cartarct Leadiesy efq; of St. pdmond's-Bury, Suffolk
Mr. John Landen^ matfaematicj^ni ileward, and Iand<^
fufvcypr
Mr. Thomas Ladd, of Go(berton, Lincolnfhire
Mr. Samuel Lane, clerk to the col)e£h>r at the port of WeQ^
Mr. George Law; of Sl^Mnferd
Mr. William Lawfon, of Wigtof^
Mr. James Limbird, of Bicker
Mr. George Langley, fchoolmafter , of Leak
Mr. ICiac Lafigton, braiier, at Spalding * .
Mr. Johii^Xee; of Nottingham
Mr. Samuel Linton, of Brome-Hall, 6 copies
Mr. William Leodh, of LMdon, 6 copies. . .
. . ■ M. ■
Mafter Mumy,- fion to Dr. Murray, of Wells
Mr. Robert Maeiarlan, fif the academy^, Tower-ftrtets
London . ^
Mr. Stephen Marfton, fchootmafter, of Bofton
^ Mr. JoTcfh Mafon, of Swinelhead, Lincolnfhirc
i
I
i
Mr. ifaac Mugeleftone, of Nottingham
Mr. John AffoUon, of Hc^beach
[r. William Millington, of Donington
Ir. Anthony .Manby
J jjr. Jobii'Morfe, merchant, of Norwic|t ' *
r \V/ ^ Mr,
Mi^ Tbopa^ MttleK imcrchant; ^kto V > '. : '/"
Mr* John Marks, uphojdfteror, ditto
Mr. Johq MbiM» glsizia^ ditto '. ' *"
Mr. George Man, merchant, ditto, 6 copies ■ ^
Mn Thomas Mode, dr^ier> air YanooitUi ^
~ >hn MafoAv'Of Bimxiog^am, 3 copic»
>feph Moore, attorAey^ London
[abbot, furveyor of the turnpikc^voods, Boiton
N ' •
Noah Neale, of Stamfesd, efquire
Edward Northon, of Holbeach^ efquire
Mr. Jfrfm Clfve Ncale, of thf city of Lincoltt, Knenwiraper
Mr. Henry Neale jof London* ditto /
Mr. Jofeph Nixon, of LincolA^
Mr. John Nclfy, of Surfleet . \
Mr. Nott, of Stamford, bookfeller, 6 copies
O
Mr. Jplui Overtoil, of Qtiadring, Lincolnfhire
*^r. John Olding, of South Charlton, fchoolmafler '
Mr. John Orrne, fchoolmafter, of Mdbourn, Derbyibire,
6 copies . . J
P
Mr. George Prieftly, of Spalding, grazier
J^n Pattifon, £fq; of Norwich
The Rev. Mr. Powell, of Walfingham
The Rev. Mr. Lewis Powell, of Donineton
Mr. William Price, of Briftol
Mr. William Putteril, of Lbcoln, writing-mafier
Mr. John Pattifon, of Pointon, Lincolnihire
Mr. John Pakey, of Donington, mercer
M'. Samuel- Penibn, of Algaiiiirk, Ch. conftable
"Mr. James Brecknock Palmer, of Holbeach
Mr. John PsBv, of fioum, Liocoliiihirf, 3 copies
Mr. William Pedder, of Ripptngale, JTchoolmafter
Mr. John Lilly Parker, of Wolverhampton, 6 copies
Mr. William Price, merchant, m the High-Areet, BriftcJ,
50 copies
Mr* William Prioe, linen^rap^r, Stapdorcr Walls
Mr. Palmer, of Stilton, apothecary
Mr. Parkins, of Bpfton
Mafter James Parker, of WiOfingham
Mr. Richard Pick, ^of Gofberton
Mr, Thomas Qiiwborough^ idioolmafler, at Grantham,
Mr« John Qui^cey* of Poniag^a% gromr - : ^ .^-
R
Mr. John Richarik, junior, of Spalding, gMk * '
Mr. Charles RmUifli, of Nottingham
Mr. Riehard RuflH» cf Loi^-Sutton
Mr. Tboiliaa KnJkp m^rcbsinfi^ u Worthaai
Mr. Robert Raft, merchant* at DifSf NorMk
Mr. Williaiii Robinfon* of Donington
Mr. Philip Riche«, merchant
Mr. Peter Mich«el JUhath, of Norwich
S
SpcfauHr Swaiif , €&}; of Liy^ngtos, Cambridgefhire
Mr. John Spur, of Bicker, gtnt.
Mr. Henry Smithy attorney at laVy Stamford
Mr. WiUiam Shilton, ditto
Mr. John Sprijigihq»9 of Hetpringbam
^ Mr. Oliver Shepherd, of Doninffton, merchant
^ Mr. John Seivcll, watchmaker, Ludgate-hili, Lohdon
Mr. Robot Seiprii, of Norwich, manttfaAurer '
Mr. John Sands, of Spalding, Ibymaker
Mr. Jkhn StaYcly, of Melton^Mowbray, ftone-cutter
Mr. Thoma3 Sands, of Holbeach, mercer
Mr. Benjamin Smith, of Walcott, attorney at hw
Mr. Edward Stattard, of GoSbtrton
Mr. John Smith, ditto
Mr. Thomas Shield, of Ri{^in«Ie, Liooolialhtrr
Mr. Samuel Stone, draper, m Norwich
Mr. William Snear, ditto
Mr. Robert Stafford, Mafoo, ditto-
Mr. John Smith, of Manchefter
Mr. Charles Simpfon, of Derebant
Mr- William S'penfe, iuQ. merchant, at Leeda
Captain John Springold, of Wells, Norfolk
Mjr. Tboipas Slator, of Doniiigton
T
Aathofiy T^lor, of Picckiogtoii, efquire
The Rev. Mr, Townfiiend, of Pinchbeck
Mr. Jokn.Tiltfnari, of'tfae 'city of Norwich, litmmonger
12 copies
Mr. William Thompfon, of I/ondon, merchapt
Mr. JoTeph Teefd^le, of Sattefton, Lincolnfhire
Mr. John Torner, of Whapioad, fchoolmafter
Mr. Thomas Tinkler, of dolflerworth, Lincplnfhire
Mr« John Turner, of London ,:\
Hfi Jolw Tcn«ant, pf Bicljpeir Mft
SUBSCRIBERS NAMES.
Mr. WMam Ti|tor^ mmkmt^ hi Norwich
Mr* Edward 't^hornhillf merchant, in I«ondon
Mr. lohn Baptifta Tunna, hiftory painter
Mr# Xood, <k Ladd-^lane ' , * ^ ,
Mr. Taykr, of GoibertoA •
M^Api* David Trimnd^ fon to the Rev. Mr. Trimnel, of
Bicker, pupil
Mr- William WeBtj^ of Goiberton, fteward
Mr. John Vanniel, of Lfncoln, dancing-0|after
Mr. Emisf ry Viall, of Waliingham
Henry Watfon, of H^lhoMcYif clqiftrc
Edward Wileman, of L^g-Sutton, efquirc
William Watfon, of Kirtoq, ^fquire
Mr. John David WagenkncAv <h Frankfort, Germany
Mr. Henry Woojley, oiafterof the academy, Northampton,
' 50 copies
' Mr. William Williamfon, of Lynn, writing-mafter
Mr. Robert Whitting, of Wells, Norfolk
Mr. Jeremiah Whitehead, of Waliingham, fchool-mafter,
12 copies
Mr. John Ward, of Donington
Mr. Nathan Wilfon, o£ Nottingham
Mr. Thomas Walker, of Threekingham, Ltncolnfliire
Mr. Jofeplr Wright, fcfaoolmafier to the qoaker^^workhoufe,
London
Mr. Samuel Wells, of Bofton, bricklayer
Mr. Graham Wilkinfon, of Spalding, merchant
. Mr. Bartholomew Wells, of Bofton
Mr. James Whitehead, of Swaton
The Rev. Mr* Wright, redor of Ktrby, Lincoloihirt
Mr. White, park-keeper and clockmaker, Afwerby
Mr. Wythers, of Stilton, fchoolmafler
Mr. Wefton, of Yaxley, wciting-mafter
Mr. Thomas Weeks, of Donington, grazier
Mr. Samuel Wife, of Nottingham '^
Mr* J^mes Wiltiet, attorDCjr at IfW, ^ofrtficfl
Mr. Watts, of Norwich
Mr. John Wells, merchant, ditto
Mr. Robert Watline, merchant, of Sotterly
Mr* John Wpod, fdiooifnaft^r, of Wfilnole.
Mr. Richard Worley, bopkfeller, at Boftop, 6 copies
Yexlnirgh, of Frampcon, gent*
•f.
i:
book! 1
Tkfl deoicataiy part of Arithmetic,
' . CHAP.- Iv
The gmenl Rtneri^;
Page
<i t Xr T. I. Hme^attik - - - * -^ •*• - - - I
II. Ad£tin rf Inttgirt .', ----- e
nii ^i-:— E^Ji'C^n - - - - 8
ir. MiibipScattm -------34
: . -VL m^ r ^ - - * ."' r\- -4»
C » A P.- B»
TatW «f WjueWf s; M^AStrtifiSb ahd- Tliks, with the
J^enoouhadon-to^aifotbe?,
in. Sum-ailun tffSK' - - - - - 63
. iv; itiOiaUn - - - - - , - - 65
c' ^' A' p: m.
RVSB*. of P>ACTI€«.
SECT 1/ PraffialyMuhMiatUH . - . - 75
l£ H . ■ Diyifion - - ^ - - 90
III. AUfti(a Parts - - - - io6
r - C it A f. IVi
VuLGAH Fractions.
SECT. I. Naatim -.-•-•- - - - '- 14$
11. BUitlHtK*'' - -• - - - -;,r '47.
» Ul. JtU
\
I
C O ^ T E N T 8;
Page
-> -* in. Addition - - * - • * • f. irg
- Ci W. SubtyaOion ^ • - - - . '. - ica
'\Yi MukipGcaiiofi - , . - . - - x6o
* • VI. Dmfim ---•--•. xtt
CHAP. V.
Decimal Fractions.
■ • •
Sect I. Notation ---.•... |gg
IL ReduiHon ----••*« jjl
^r in. Jdfition -----^-'. 182
: t IV. StibtraSfion -i - - - ^ . . « jg^
V. Multiplication - - - - ^ . - ig5
t VL Divijwn --.---,.« jnj
C H AP. VL
Evolution or cxtraaing the Rdor of all fingle PowB&s.
8iE G T; J. To ixtraH the Square Root - - - 207
11. SomeVfesofthoSjuariJtoot - - 211
III. T'ff Ay^^^ £&# C«^^ ttaot - - - - 221
IV. Some VJis of the Ctdn Ro9t - • - 228
V. To extras the Biquadrate Root - -^ yjr
VI. To extraa the Surfolid RoH ... «]
VII. Ttf ^4f/n7^ /A^ ^(7/ of the Square Cuhed^
^, or fifth Power - - - ^ - . 235
^* VIII. To extraif the Root of the fecond Sur-
folidy or feventh Power - - . 236
IX. To extract the Root of Biquadrate fquaredy
^" or eighth Power ----.- 237
X. To extract the Root of the Cube cubed, or
ninth Power ---««« 239
XI. To multiply feveral Figures by feveraly
and the produa to be produced in one
fmgle Urn - 24a'
B O O K II.
ContainLng Proportion, with its Ufe j alfo the Ufe of
Rules of Practice in various Branches of Merchan-
dize and Trade.
CH A.P, 1. ,
Proportion Disjunct, ortheGotOEi^RuLE.
SECT.
t "^ 'In T E N t ^^
*«'E C T. I. DirtaPrtpmmiytriiMUttf'nirteDlreBTlx
(\ ■ n. Rtdprtad Prifirtiai^ or RmU t/Tbru
^ hverfit ----,-> a^T
''■ - m. SaUt/ftve a-i
IV. Ruk rf nrtt Rtpeattd - - - - vjZ
- -C HA P. H.
t
FA"
" :• ■:•
iSlMPLK InTEKKST. ' ' * " "
SECT. I. Simple Intireft^\.\^. - - - 291
''^ " n. Infuranci^ --.--*.., • jj?
in. Broieragi «....... ^l5
• IV. Pwrcbafing St$cks ----.-. 318
V. ^Rihate 9r Difcount - - - . . ^21
VL Efuaiim of Pofmnus • • . - - 324;
CHAP. m.
Tan and Tret ------- <%2g
C HAP. IV.
"!- . FSLLOWSHIP.
SfECT. L Single FelUwJbip ^..•. .* . : 34,
..II. ,.DauhIe FeUowftfip - .* • .. ^ • 360
III. FaHorage .- - - ^ - • - . 36^
• - - IV* Lofi and Cain - - - - - - 37 j
C H^A P. v..
Barter ^--•..... 380
C ft A P. VI,
Exchange.
^^ • >
SECT. I. Exchanging Coins -> • • '• • .. 380
11. ^//»^/(f Arbitration of Exchange - - 435
III, Compound Arbltratton of Exatange - 441.
IV. Comparlfon of ffVgbts and Meafwres 445
BOOK III.
CoBtsiiniiig the more abftmfe and curious part of Arith-
• 1 M£TrCK.
CHAP.
c 0 a If M it 'p §.
CHi^P. I.
JkK»ta>tiriami,
.X
- Page
i S C T. I. Jm^iAutMtiia. \ A . » A .- - 455
II. Aliigatim AhtnuU ^ . - • - - 456
.. ~ ' - , 46*
IV. AlUgatitn Total ..«,..-> 466
III. AlUgalAnFMbU
• *.
c H A P. a .
.♦ • -7
^ ^ ^ " p H A, Fr fit..
PosiTioK. or The ftm^n oitALtB.
11. Double Pofttion 48^
. ^ C H A p. mi
'^ " ^ ~ Concerning Divtfon - - - • - - 499
C H A P. \r.'
Progressions, Variations, Combinations, &c.
S Z G T. I. Tfrhbmrtical Rhsgrel^ - ^ ^ .^ 494$
- IK Ge9mitricld Bf^effkn' ^ • .^ . . ^05
IH. -Variathns' ---*.-—-- ^xi
i ■ IV. -ComiintttionM' - — --* - .-.- - ett
• Magu Sjttara -- --.--- j2i
C'H A P. VI.
$tCT. I. Vompumd fnmeji --.--. 517.
V. Pureh^g Freehold Efiates . - • - 538
III. Purcbe^g FritehoU BJikes in Rmrfan 539
IV. Purcbajmg Akrmties ----- ^41
V. 7^/ yaluatUn of Jnntdtiet upon Lives . 55^
CH A P. vn. .
A* GoLLeCTtON of Qj7BST10,NS;
;j E C T; r. Superfidat Meafure -' ^ ^ • - - 57b
II. Meajuremeut of Solids ----- 572
in. Mifielianeous ^uefliotts - • - . - 572
App^dix^^ • 5gf^
a Arlthx»,ctifal
Arithmetical CoUedlions
AND
IMPROVEMENTS.
B O O K I.
*
CHAP. I. SECT. I.
t
NUMERATION, or Notation, tcachcth to
read or exprefs any number when wrote down ;
and confequently to write down any propofeA
number, according to its true value.
TABLE.
•
<^
n3
1
•
•
• >
•
•
B
9>
§
•
1
1
•
<ta
S
Vm
5
o
B
0
O
••s
•
«
*§
o
CO
•s
J
•s
•
•
•o
(O
TJ
•)
2
•o
§
£
c
C
ZHS
c
C
O
c
:i
o
• •^
3
-*>
JC\
3
y
a
a:
H
S
s
h
H
a
h
D
3
6
7
4
2
8
7
2
5
t
3
I
O
6
8
5
3
-
-
0
4
6
I
3
2
7
-
-
3
9
2
8
7
5
-
-
-
« tfV
6
2
4
3
8
-
-
-
*
- ■
4
I
3
7
•
r
-
«
•
-
7
!
S
4
7
9
8
7
6
5
4
3
2
I
B AU
a NUMEftAiPION. Bookl.
All figuj%s in the iirft row towards tbe ri^bt-hand)' aic
units I thofe ifi the fecond row tens $ thofe in the third hun-
dreds ) and thofe iiKthe fourth tboufands, &c.
A cypher, although by itfelf, it iGgnifies nothing ; yet being
placed on the right-hand of any figure, augments the value
of that figure ten timesi, by advancing it into a h^her
place than if the cypher had not been there. As 6 fix, 6q
fixty, 600 fix hundred, &c.
To every three figures are orderly repeated the deAomi-
nations of^ units, tens, hundreds j fo that he that can read
three figures, may, with a little mpre it)ijbru£^ion, be quickly
able to read any number how large foever* And to every
three figures, the names units, and thoufands, are alter-
nately applied.
Lixewife to every fix figures from the right-hand a new
general name is given. As to the iirft fix fu^urest t|^e-gei^
ral name of units are given ; to the fecond ux, the general
name of millions ; to the third fix billions ; to the fourth
trillions; to the fifth quadrillions, &c.
The whole art of figural notation is comprehended in the
following table.
Quadrillions. Trillions. Billioll9< MilliotiS. Units.
Tbouf. Units. Thouf. Units. ThouC Unit*, Thoof. Units. ThooT.Unitt,
jl2 348 634 23s 314 527 625 284 123 714
n t u httt h t a htu htu htia fatu iitu htu ht«
Read thus. Seven hundred twelve thoufand, three hun-
dred and forty-eight quadrillions.
Six hundred thirty-four thoufand, two hundred thirty-
five trillions.
Three hundred fourteen thoufand, five hundred twenty-
fevcn billions.
Six hundred twenty-five thoufand, two hundred eighty-
four millions.
One hundred twenty-three thoufand, feven hundred
and fourteen (units.)
The following numbers are alfo exprefTed in words at
length, 3700841 three hundred feventy thoufand, and
eighty-four.
4184279CO, four hundred eighteen million, four hun«
dred twenty-fcven thoufand, nine hundred.
6210003745, fix thoufand two hundred tea million,
three thouiaad ieven hundred forty-£ve«
3 4*^27308751,
ChapTL NUMERATION. 3
4x027308751, foitv«one thoufand twonty-feven millioiii
three hundied eight thoufand, feven hundred fifty-one.
2934.176047 12, two hundred ninetv-^three thoufand four
hundred ferenteen million, fix hundrea four thoufand, feven
hundred and twelve.
62800203069471^, fix hundred eighteen billions, two
thoufand thirty millions, fix hundred ninety*four thoufand,
feven hundred and thirteen.
47038066250433251889411, forty-feven thoufand thirty-
eight trillions, uxty-fix thoufand two hundred fifty billions,
four hundred thirty-three thoufand two hundred fifty-one
millions, eight hundred cighty^nine thoufand, four hundred
and eleven.
A TABLE of Numerical Charact£M ufid
by the Romans*.
I One.
V Five.
X Ten.
L Fifty.
C An nundred*
D or ID Five hundred*
M or CI3 A thoufand.
IDD Five thoufand.
CCIDO Ten thoufand.
IDOO Fifty thoufand.
CCCJpOD A hundred thoufand.
13003 Five hundred thoufand.
CCCGIODDD A million.
A line drawn over any number lefs than a thoufand, in-
timates fo many thoufands 1 as LXX is 70,000 i C is
too,ooo ; and M a million.
I and X are fometimes placed before charaSers of greater
value, namely, I before V or X, and X before L or C,
in which cafe the value of I and X is to bejubtrafted
finom the value gf the following charaftcr, as IV four,
IX nine,* XL forty, XC ninety.
V and L are never repeated, and none of the other
chara£lers above four times. Thus, IIII or IV ; but V five,
XXX thirty j but XL forty, LXXX eighty j but XC ninety,
CCCC four hundred j but D five hundred.
. B 2 la
4 NUMERATION. Book L
In figures exprcTs : a million and a half South>lea bonds.
Ninefcore, and fourteen thoufand, eight hundred {beep.
Threefcore and twelve thoufand, thirteen hundred pounds
of lead, fifteen thoufand, ^d fonrfcore million of Itivcrs.
One hundred and twenty thoufand, two hundred and fix
millions, fevent/ thoufatid^ feven hundred, and feven rials
of plate. Three millions, and thirty-three thoufand, and
thirty pieces of eight. Four thoufand, and forty hundred
pounds, thirty-four (hillings, and fourteen pence five far-
things.
South-fea bonds .------ ijocxioo
Sheep .--__--.- 194800
Lead ----- _-_-. 73300 lb..
Stivers ------- 15080000000
Rials of plate ----- 120206070707
Pieces of eight ------- 30^3030
]. s. d.
404001 15 3;-
Chap/I. ADDITION. §
SECT. II.
A D D ,1 T I O N.
jADDITION is a rule whereby fcvertl number^
Lc\ are fo conneAed and piit togethe|-, that their aggre-t
gate Aim, or tot^l amount, m^y be known.
Obferve to place your numbers fo, ths|t each figure may
ftand dire£bly underneath thofe of the fame value ; viz.
units under units, tens under tens, and hundreds under
hundreds, &c. Then^
RULE,
Always begin your additioi) at the place of units, and
gdd together all the figures th^t ftand in that place ; and if
their fum be under ten, iet it down be)ow the line under-
neath its own place ; but if their fum be more than ten, then
you muft fet down only the overplus, ot odd figure, above
the ten (pr tens) and fo- marty tens as the fum of thofe
units amount to, you muft catry to the place of tens; add-
ing them and all the iigu|^es that ftand in the place of tens
together, ^ in the fame m^per as tfiofe of the units were
added ; then proceed in tl^e iaqie order to the place of hun-
dreds, and fo from place to pla^:e till all is finiflied.
• .|. 'Ifi-ihi6 biflcxtile, of leap-year, how njany days and
I hours ?
Hour9»
January 31 , == ^744
February. 29 -------•=: 696
March 31 1- -.----1-= 744
April '30 ---.---=s 720
May '31 1- =744
June 30 ------*-.=: 720
^j 31 ==744
Auguft 31 =744
September 30------- =r 720
October 31 ---.-..-- =744
November 30 ------- =s 720
December 31 --------z^s 744
Anfwcr 366 days. ----- = 8784
B 3 2. Find
6 ADDITION. BookL
4. Find the number of chapters in the fjvc books of
Mpfes ; after that the number of verfes, and give their
joint fum.
Ch.
Geo.
Vcr.
Ex.
Ver.
tef.
Vcr. I
Kdm.
Ver.
Dea.
vi.
Ch.
]
3«
22
«7 1
54
46
I
*
»S
»5
16
■
34
37
A
3
H
»2
»7
3»
^
3
4
ft6
3»
35
49
49
4
5
3»
»3
»9
V
33
1
6
aa
30
30
»7
*j
7
8
ti
»5
3»
$
HI
26
20
7
8
9
*9
35
«4
*i
29
9
lO
3*
29
20
s«
22
lO
II
3*
10
47
!l
3»
II
t%
10
5»
1
8
3»
12
»3
s8
22
5*
•
33
18
'3
H
»4
»
3»
57
45
•
»9
M
;i
.
11
»7
33
4f
*3
II
16
36
34
50
22
17
*7
16
16
»3
•
20
'Z
i8
11
*7
30
3*
22
18
M
»5
37
22
21
»9
ao
18
26
«7
»9
20
so
ti
34
36
»4
35
a3
2i
12
*4
3«
1 J3
4«
30
*»
»3
*o '
33
44
30
»5
»3
a4
^7
'
18
»3
\i
22
H'
'*7
34
4^
5|
*>
:i
*7
,
il
1
37
•21
46
34
65
*3
11
i8
, U
;i
{"s^s^
3«
<8
28
*9
35
*?
a9
49
30
43
16
2«
30
31
$5
18
■
54
30
3t
3*
3»
35
4»
5*
3*
33 .
20
»3
5«
«9
33
34
V
35
»9
12
34
35
^
18
34
— — *
•—
3«
^h
-
»3
959 1
37
36
*f
,
— —
. 1
3«
30
»3
31 ; »ii88»
39
4} ■ ■"■-
40
«1
3»
41
57
4»
3« 1
121 1 Chapter!.
1 Verfet.
43
J4 |— — — * Cenefif - - • - 50
»533
44
34
£xo^t .. « • ^ 40
1213
:i
08
X«vitJcoa .«.•.• ;i7
859-
34
Numbcrt • • • 36
1288
47
3«
Pttterooomy • - 34
95^
48
n
■ " ■■
* ■ ■■
49
5«
•^
t»7
'l^y
'533
Joint Amd • 6039 1
•
■
•
3-
Dec
ipher
Chap.l, ADDITION. 7
3. Decipber the follovnng numerical letters, and find
their fum ; viz.
JV - - - - ♦
VI -- 6
IX - - Q
XIX Hfieoaimon -------8
xm 13
XLV 45
LXXXI - ^ - r 81
XCVI -- »--96
CXC _-_----•- 190
CD ttncoimneh - - - - - - 400
bCC 700
. MCL 1150
MDCXLVIII - - - - - - - 1648
TSCCM uncommoa - -. • - ' 1000800
lODIDCCCLVIJ . * - - - 5857
TlCCLXXXX - -•- - -: - 6290
I ■ I II I ■ m
Anfwcr, 1017297
' t would advife die young ^ccomptant, in long operations
in addition, to point at every 60, carrying on the overplus;
and when he hadi caft up the whole liiie, carry 6 to the
next place for every point.
And to prove idle work, begin at the top and caft it down^
wttrds, in the fame manner as was done upwards^ pointing
pn the other ^de the iigt|res : and if the amount be the
taant both w«y$» it may be prefumed that the ^ork is right,
Or^ when you kave -caft up the whote, divide it into twa
^r^more parts, which caft up feparately ; then ^dd the fum^
#f the faod parts together, -which, if like the firft fum, thef
work may be adjudged to be I'ight.
4* How much is A (bom iixteen years ago) elder ihza
. By who will come into the world fourteen years hence I
|6 -f- 14 ~ 30 years, the anfwer.
5-
A peribn was 17 years of age 29 years fince, and he
wiQ be drowned 23 years hence ; pray in what year of his
jfge will this happen ?
17 '^ 29 4^ 23 c= 69 years, the anfwer.
B 4 SECT.
*8
ADDITION.
Book I.
SECT. III.
w» •
Addition of English Coins.
The leaft piece of money ufed in England is a farthing.
And
Farth.
4=1 penny. . -
48 = 12;= J {hilling.
960 = 240 =: 20 = I pound fterling.
s.
And 5
6
10
13
N. B. 1.
d.
- is a crown*
8 is a noble.
^ is an angel.
4 is a mark.
-* Pounds.
s.fftand (Shillings:
^or I Pence.
J Farthii
Farthings.
d. But moft commonly.
d. s.
Pence Tables to be got by heart.
d» ' s« d« d. $• d«
12 = I
2J. = 2
•30 - 3
60 =: 5
d. s.
72 = 6
84= 7
96 = 8
108 = 9
120 = 10
20 = I S J 70 = 5 IQ
80 = 6 8
30 == 2 6
40 =r 3 41
50 = 4 2
60 = 5, -
90 = 7 6
ipo =84
no = 9 2
Having placed the numbers to be added in this order,
viz. pounds under poufids, {hillings under {hillings, ^nd
pence under pence, &c.
^ . \*
R U L ^E,
, Begin with the farthings, and for every four carry one
penny, fetting down the overplus under the farthings ; then
proceed to the pence, calling up to 60, where make a
prick, and fo proceed to tl)c top, fetting down the odd
^^' ^ pence
%»
Chitp.l. AI>DIT10N. §
pence in their prpper place ; and carrying one for every
12 pence, and live for every pricic (asoo pence make five
{hillings 0 fo proceed to the units place of the fhillings,
fetting down the overplus above lo, for each of which
carry one, and fix for every pricic to the angels ; and as two
angels make one pound, cai'ry half the number of angels to
the pounds, fetting down the^odd one,, if ^ it fo happen;
then caft up the pounds,, as before. dije<5i;ed,^ in addition of
whole numbers. - . .
I. A nobleman going out of towp, [s informed by his
fteward that his x^ornch^ndJer's bill ?omes to 123 1. 19 s.
His brewer's 41I.10S. His J)utch€r's 212J. 6d. To his
lordfhip's baker i§ owing 24J. . To his .tallowchandler
131.8 s. To, his taylor .137 1. .9 s. 9 d. .. To his draper
74I. 13s. 6d. Hiscoacbmaker'sdema|id^asjZi4l. i6s.6d.
His winemexchant's 68 L 12s. . His cpnf^Aioner's 16 1. 2s.
His rent 86.1. 2s. And his ftrvants..w^es.for half a year
came to 46 1. 5 s. What nooney muft he fend to his
banker for, in cafe he would carry with him 50 1. to defray
his expences on the rdad.
1. s. d,
Cornchandler ------ 123 19-
Brewer -------- 41 10 - ^
Butcher -------- 2iz - 6
Baker -*.-----•• a4«-«*
Tallowchandler - i^' - '-r*-r - 13 8 -
Taylor ^ .----. • 137 9 9
Draper ---»----.- 74 13 6
Coachmaker ------- 214 16 6
Wincmcrchant -- - - -- 68 12 -
Confedioner ------»- 162 —
•Rent --------- 86 2-
Servants wages *---.-- -46 5-^
-For expences ------ 50 ~^
-Anfwer, ^ iioi 18 3
2. A colleftor of cafh.hath been out with bills, and
ives an account that A p^^id him 13.I. and half a crown ;
I 2L 13s. C 14 s. .and a groat; D il. 9s. Si-d. ;
£ II 1« 64<I«; F- 17 s. and a tefter; G 12 s. 2d. ; Ha
Sound and half a guinea ; I a moidore, and 13 s. ; K two
road- pieces ^23 flxillings each, a Jacobus of 25 s. and a
i&illing j^
f
•».
•
s
ADDITION «ookl.
AilHng; L nine pounds and a naik; M iil. 12 s.}
K a Bank note <)f 15 1. ; and O three crow?i-pteces and
in angel. What ca(h ha^ he in ch^^ ?
1. s, d.
A T * -'--■'- 13 » ^
B-- 2 13-
C-----T- -14 4
D --.-.- - I 9 8i
E -r IX -6i
F ------- -17 6
Q. -.-.-- -12 as
H ----- . - I to 6
I -...--- 2 - -
K - - •■ r r - - 3 ?2 -
L ------- 9*3 4
M-------I2I2-
N ,5 - -
JC76 a 6|
> * *
^. A cornfadoF buys feventy quarters pf . oats fof
46 1* 7 s. 6d> thirty-«igbt quarters of beaps for lool.)
twelv^ quarters of peafe^ which coft i61. i6s. eightyreight
quarters of barley, for 73 1, id, fixteen ditto of wheat for
56 L 9S» lOd.) and fix quarters of rye for 4U is. 6d.
the water carriage of all comes to 13 1. 2 s. 7 d* his riding
charges to i L 13 s*- and if he clears eighteen"^ guifieas by
the Mrgain, what do his bill of parcels amount to i
Beans • . •
-Pcafe - - .
Barley » • •
Wheat . .
Rye . . -
water carriage
Riding charges
CommiffioQ «
I. s.
4.
46 7
6
100 -
. f
16 16
-
73 -
8
56 9
10
4 »
6
13 2
7
I 13
—
18 18
-
£ 330 9
X
4* A
Chap. I. ADpItrOK, U
■
4« A of Amfterdam is debtor to B of Briilol, for
xnei^ceiy wares as per fnflory, 418 1. 2s. 6d. ;. for forty
C weight of Cheihire cheefe 52 1. ]8s.; forEnglifh broad-
cloth nfteen pieces, 317!. 12 s. lod. ; for 19 fodder of
lead 320 L ; for 12 tons of bar-iron 1731. 3d. ; for eight
tons of copper iiiol. 10 s. id.; for his acceptance of i
bill drawn 881. 14 s. ; for another paid for honour 50 K ;
10 4<^ca Morocco (kins 28 1* 15 s. 4d. ; paid convoys, in-
furances, and port charges 43 1. ; warehpufe room, poft*
age, fledage, boatage, and incidental charges 5 1. 5 s.; the
factorage of all came to 112I. 6 s, For what fum muft B
draw to clear the account I
•»
, 1. s* d*
Mercery wares * -•-*•-. 418 z 6
Cheihire cheefes --.--• 52i8«*<
Broad-cloth ------- 317 12 xo
t^ad --------- 310 *- -
Bar-iron -------- 173-3,
Copper - - -, - - - - - mo ID I
Accepted bin .'. 88147.
Ditto on honour .---- ^o--
M orocco fkiiis ------ 28 15 4
Convoys, infurances, &c, - - - 43 •-^ —
WarehOiile room, &c. -.-.-..- 5 5 •"
Fadorage - - - .' - - * m 6 -
i^— **—«—— —1^
^2720 4
(
5* A rate or afleffinent, for and towards the relief 'of At
po6r of the pariih of Goiberton, &c.
h $. d*
Francis Fane, Efqi - •-*..- ir 3 7.
John Robinfon, Gtent. ----- 3175-^
Kichard Calthorp, Gent. ----- 413 6.
Thomas Baley, Qent* -.--• 57 4^.
Mr. John Torry - * 3 x8 6 '
Air. John Turver ------- I 17 44
Henry Worley --.-.-•- -32
Jonathan Cheavm ------- -ir loi
' William Trickett -••-... - 5 9
Amnoiiy Birks ••-•---- -6 10}
■•
Carried orer 32 9 5^
*< A D P I T I O N. Boqk I.
. '1. 6. d.f.
Brought over .32 9 54- -
Mr, Robert Cole .'-----. ^ i 2 7 •
John Weight • ^ ..--•- . - 10 3^
. Mr* John Shaw -.--.-- — J9 11^ >
Thomas Ladd -------^ •"5 9
'. JohnLambfon -------- ^' ^ (^
Thomas Hopper ------,- *- 2 9^ . '
Francis Maftin - - - - - • - -. . *. jq 3'
. William Crawforth ----.- - 15 loi
Thomas Oldgate ------^ -lysi'
William Wifeman ------.- -311
Mrs. Margaret Parkinfon ----- -117^
Samuel Lane -------- -244
Mh. y^lice Sharp* -•-'---•-- -'23
William Curtis ----'--- - 9 71-
John tinfey -*---_*--« -_i|
Mrs. Silv. Flear ------- - |8 9
John Pattrflbn - -"^ ^.' -*--*- - - 2 4^
John Genpils -" ^■* -' -" _' - - ,- ^ jq 3
Mr; John Pacy "-"-*---.- ^ ig ii|.
William Bilton ■*-:--*.--- 14 14^
MA Robert Allen -'-'----- 2 i:f 3
Mrs. Eliz* Wilcox * - " - - - .- - 2 o 6^
iohn Mafon ---"-'..... . .: i jg y^
If. J6hn Thimbteby -" - -'-*-".. 57 9'
JohnSifilth " 3186
Henry ^Ward -------. 5^7 3-
Alexander Co*31ing ------ 1132
William Lambfon ------- 154
John Gibbons ---. --^-, "•??: l^
jC69 12 7, V-
7
6. The Right Wohourable Ihe Lord Bolfover Dejbfor
To Paul Purfeproud, upholder. '
;i758- - 1 .... * 1. s, d. '
April 19. A rich crimfon damajk bed,, laged, )
complete ------. -J 75 "" ""
May • 5. A fet of window curtains and. va- ? , . ^
lences, ditto I I - - - -J ^^ H t"»
7. A fine carpet, counterpain, and an ?
otter-down quilt * - - .- -1 '?'?."
' T" Carried over 104 i ^
Chap. I. ADDITION. 15
I. s. d.
Br9ughtover 104 i 8
June 6. A crimfon velvet cafy-chair, and I
two ftools, ditto f 13 70
13. A wrought dimitty bed and fiirni- 7 « «
turc, complete -----j ioi<>4
Ang. 20* A down bed, bolfter, pillows, and >
quilt -.--.-. .f 'S"^
Chairs 10, with two armed ditto, > >
walnut-tree framed - • - -j 34 ^^^
Nov. 20. A iire-fcreen, bed, table, and dref-
rich frame -------j 21-
finjt-riafs -.-...•} 8 14 .6
The lady Wanton's pidure, in a
^225 14 6
7. A peribn faid he had 20 children, and that !t happened
there was a year and half between each of their ages ; his
eldeft was born when he was 24 years old, and the age of
his youngeft is now one-and-twenty : What was the fa-
ther's age ?
When the eldeft was born * 24
Then 10 + 9^ - - - = aSi
Youngeft -'- --- -.21
Father's age - - • - - 73t
8. A flieepfold was robbed three nights fucceffively ; the
iirft night half the fheep were ftolen, and half a fheep more ;
the fecond night half of the remainder were loft, and half a
iheep more ; the laft night they took half what were left,
and half a iheep more, by which time they were reduced to
20 : . How many were there at firft I
20 left
21 > f3d
42 S taken the'
m "f'-
_
167 (beep in all.
SECT.
.T
f4. SUBTRACTIOff. fiqokl.
SEC T. IV.
SUBTRACTION;
«
SUBTRACTION takes a lefler number from a
grditer, by which the excefs^ diflFerence, or remainder^
msur be known*
In fetting down numbers for Work» always place the
greateft number or fum uppermoft, in fuch order* that unks
may ftand under units, tens under tens, &c. alfo pounds
unaer pounds^ fhillings under fliillings, pc&ceuhder pence^
RULE.
Begin with the loweft or leaft denomiiiation^ (as in addi-
tion) and take or fubtradt the figure, or figures, in that
place of the fubtrahend, from the figure, or figures, that
nands over them of the fame denominationf fetting down
the remainder. But if that cannot be done, increafe the
lipper figure, or figures, with one of the next fuperior ^e-^
nomination ; and from that fum make fubtra£lion ; and fo
proceed to the next fuperior denomination, where you mufl
pay the one borrowed, adding unity to the fubtrahend in
that place*
I. If a perfon.hath 105 miles to travel, and hath gone
99, how many mil^s hath he yet to go i
Miles.
99
Anfwcr 6
2. If a perfon be 49 years of age this prefimt year X765»
what year was he born In ?
1765
Anfwcr 17 16
3. In
CIiiip.l. SUBTRACTION. 15
3^ In fiAecn hundred nine-two there died a noble prince 1
how maojr years is that ago i
176s
1592
Anfwer 173
4. A cplle(dorof excife has received 2479 1. 129. 6^.9 and
Eaid into the office, by fcveral remittances, 1977 1. 17 s. j^i, |
01^ much remains in his hands ?
1. s. d.
Received 2479 12 6 J
Remitted 1977 17 74-
Ifi hand £ 501 14 io|:
•^■v"
j(. Having a piece of ground 127 feet in front, let oiF
to A 57 feet, to build on at one end ; and to B at firft,
274^ feet; which be afterwards, byconfent» extended to 4a
feet i what ground was left me in the center ?
Feet-
127
57+42 = 99
Anfwer 28
6. Your grandfather, if living, is 119 years of age;
your father a<3ually 63 ; you are not fo old as your grand-
nre by 83 years ; What is the difference In years between
your father and you i
, Years. Years.
Grandfather's 119 Father's. 63
' -83 -36
Your age 36 Anfwer 27
7. In the city jb£ J^ekin in China, is a bell weighing,
it IS faid, 1 2oooo> pounds; at Nankin, in the fame coun-
try, is another ivfii^uog 50000 pounds. The firft exceeds
the
t6 SUBTRACTION. Book L
tht great bell at Erfurd, in Upper Saxony, by 94600 pounds ;
how much then is the German bell inferior in weight to
the fecond ? .
Pekinbell 120000 Nankin bell 50000
— 94606 — 25400
£rfurd bell 25400 Anfwer 24600
' 8. Mifs Kitty told her fitter Charlotte, whofc father had
before left them twelve thoufand twelve hundred pounds
a piece, that their grandmother by will had raifed her for-
tune to fifteen thoufand pounds, and had made her own
twenty thoufand ; pray what did the old lady leave between
themr
Mifs Kitty / 20000 Mifs Charlotte jT 15000
By father £ 13200* £ 13200
^Grandmother^ 6800 -}- '£, 1800
'' ■ ^i ■ ■ s=86oo Anf.
9. What is the difference between the ages of A, born
in the year 1693, andB that will be born 13 years hence;
the queilion being put in the year ,1758?
Anno 1758
— 1693
«
65 + 13 =: 78 the Anfwer.
To prove fubtra£tion, add the remainder to the lefler
number, which ought to make up the greater, if the work
be right.
10. A horfe in his furniture Is worth 35 K 10 s. out of
it 12 1. 12 s. how much does the price of the f^imiture
exceed that of the horfe ?
1. 8.
Horfe and furniture 35 10
Horfe - - - - 12 12
MWMl
Furniture - - 22 18
12 12
Anfwer, ^10 6
XI. A
Chap. L SUBTRACTION. 17-
II. A merchant at his outfetting in trade owed 280 !• ;
he bad in caih, commodities, the ftocks, and good debts,
XX 505 1. IDS. he cleared the firft year by commerce.
393 1* 13s. td. What at the years end was. his net
balance ?
L 8. d.
Tocafli, &c. • - - 11505 10 -
Commerce- - - - 393 '3. '
Debts
11899 3 X
. 280 - -
Anfwer ^11619 3 i
12. A trader failings w^s Indebted to A 71 1. 12 s. 6d. ;
to B 34 1. 9s. 9.d. ; to C 16 1. 8s. 8 d. ; to D 44I. ; to
£ 191. 19s.; tb F ill. 2s. 3d.; to G661. 17s. 6d. ; toH
a fine of thirty marks. At the time of this aifafter he had
by him in cam 3 1. 13 s. 6d.; in commodities 23 1. los. ;
m houihold furniture 13 1. 88. 6 d. ; in plate 7I . 18 s. 5 d. )
iti a tenement 56 1. i < s. $ in recoverable book debts
87 1. 13s. lod. Suppofing thefe things. faithfully furren-
dered to his cteditors^ what wiU they then lofe by him I
Creditor*
1. s. d.
ByCafh ... - 3 13 6
Commodities - 23 10 -
Houihold furniture 13 8 6
Plate '. - - 7 18 5
Tenement - - ^6 15 -
Book debts ^ -- §7 13 10
Debtor.
To A
fi
C
D
£
F
G
H
I. s* d*
71 12 6
li 1 1
19 19 -
II a 3
66 17 6
JtO - -
Debtor
Creditor
284 9 8
19a 19 3
Anfwer jf 91 lo 5
£ 19a 19 3
13. You were born 34 years after me ; ht)W old fliall I be
when you are 17 ? and how old will you be, when I am 70
years of age?
70 — 34 = 36 You. - - 34+17 = 511.
C 14. A
)8 SUBTRACTION, Book I.
14. A made a bond for 1141. 10 s. the intereft came
to 19 1. He then paid off forty guineas, and gave a
frefli bond for what was behind. 6y that time there -was
13 1. 4 s. 8d. due of die fecond for intereft. He paid
on 37 I. 14s. 2d. more, took up the old bond, and figned
a new one ftill for the refidue. The principal again ran oa
till there was 9 L 1 1 s. 3 d. more due, and then he determined
to take it up : Pray what money had his creditor to re-
ceive ?
1. s. d.
Firft bond -s--- 11410-
Intereft ------4-19--.
raid -------- ^2 _-
Second bond ----- 91 lo -9
Intereft ------^1348
Paid 37 14 a
Third bond ---._- 67-6
Intereft ---.. — 1-9113
Anfwer - - - . --jC?^" 9
t
N
15. Received from my facSlor at Alicant, on account of
fales of tin, to the value of 197 1. 12s. ftcrling; of bees-
wax 71 1. 7 s.. 6d, ; of ftockings 47 1. 35. 6d.j of tobacco,
the net proceeds whereof were 94^ 1. 15 s. 10 d. ; of cottoa
123I. 3s. 7d.; and of wheatto theamountof 116I. qs.6d*
He at the lame time advifes, that he has per oider mipped
for my account, and rifk, Alicant wines, to the value of
226 1. 16 s. 6d. J figs 157 1. IIS. 3d.; fruit 90 chefts,
cofti04l. 6s.; olives 136 1. los. j oil 193!. 17 s#; rafin*
143 1. 4d, ; and Spanifli wool to the value of 731, 13s. 8d.
The commiffion of the whole confignment came to
71 1. 18 s. I id. which of us is to draw for the difference^
and how much ?
Debtor.
Cli4p. I-
Debtor* .
! ....
^ To Tin • -
f BecG-wax
Stockings
nTobacco
Cotton -
Wheat 1
SUBTRi
F^i^Qr.
]. s. d. <
- - 197 12 -
- 71 7 6
- 47 3 6
- - 943 IS 10
- - 123 3 7
- - 116 5 6
V C T 1 0 K.
By Wines - -
Figs - - -
Fruit
Olives - -
Oil
Raifins - -
Wool - - -
Commiffion
19
Creditor*
1. s. d.
226 16 6
^57 ii 3
104 6 -i
136 10 -
193 17 ^
143 - 4
73 ^3 8
71 18 II
Debtor
Creditor
j£ »499 7 "
- 1107 13 8
f
1107 13 8
( Balance
IZ^^ »4 3
1
•
i6* A, B and C open an accowt frith a banker, Jan.
2J, 17399 sind put into his hands; viz. A ¥7!. n^.
B 34 1. IIS. 6d. C 28 1. 18 s. iod< On the 21ft A
withdrew 9 1. 10 s. and C ^vanced 12I. ^d a cro^vn*
The 24Jtfa B called for 6 1. 10 s. The jotb C wapted
19!. 8s* 4d. On the 12th of February B depofited with
him eleven broad pieces, and th^'ee moidores. On the 19th
A fent fot 5L and a noble {nor^; but on the 24th re-
turned him. 42 1. On the 24 of March C paid in twenty
guineas, and B drew' for (ix. T^p 14th B fent in
17 1. 8 s. 6d. ; and ]|tie 171)1 A had cafh 12 L 2s. 6d.
On the 19th they fent for five guineas a man ; and oh the
8^th they returned that fum, and ten marks a piece more,
ow much did their faid banker owe xhfin> jointly fnd
feparately, at Lady-day \
Debtor. Banker's account current.
Creditor*
1. s. d.
nth Jan. - - - 17 17 -
24th Feb. - - 42 - -
j4^hMar. - - 11 18 4
Debtor . 7, 15 4
Creditor - 32 4 *
I. s.
21ft Jan. - r 9 lo
igth Feb. - - 56
17th Mar. - - 12 2'
a.
8
6
19 - - - - 5 5
••
/32 4
%
C 2
Debtor.
46
Debtor.
xith Jan.
1 2th Feb.
14th Mar.
a4th - -
SUBTRACTION.
Banker's account current.
B
L s. d.
6
Debtor -
Creditor -
s.
34 "
16 14
17 8
II 18
8
4
80 12 6
18 I -
ToB - ^62 II 6
24th Jan.
2d Mar. *
19th - -
Book I.
Creditor.
I. s. d.
— 6 ID -
-66-
-5 5--
^18
X -
nth Jan. - -
2ift - - - -
2d Mar. - - •
24th - - - -
Debtor -
Creditor -
ToC - ,
].
8.
d.
28
18
10
12
5
-
21
—
—
II
18
4'
74
2
2
24
»3
4
49
8
10
30th Jan.
19th Mar. -
1. 3. 4.
- 19 8 4
-55-
ToA -
B -
C -
1 ^^'^
1. 8. d.
39 II 2
62 II 6
49 8 ID
^151 II 6
17. B born 161 years ago, died when C was 47 years of
age ; who it feems came into the world 180 years lince, and
outlived B 43 years. The fum of the ages of thefc two
perfons is required I
180 — 47
161 — 133
47+ 43
28 + 90
133 Y^^^^ l^nce B died.
25 B's age. .
90 C's age.
1X8, theanfwer»
* 18. Sam. was born 28 years before Toby, '^rho died at 12,
and lived 19 years after him. Rachael came to light when
Sam. was 16, and died 1 1 years before him. Jofhua (when
Rachael was fcvcn years old, being himfelf then 14) went
abroad, where he continued nine year^; and returning,
furviv«l Rachael four 3rears. How old was each of thete,
and what is the fum of their slges i
28
Gfajip- 1. SUBTRACTION*
28
Toby - 12 years old.
19
Zt
, Then 59 4. 12 -I- 32+ 43 =£ 146.
Sam. - 59 years old.
— 27 = 16 -f II
Rachael 32 years old.
— u = 7 + 4
Jolhua 43 years old.
19. A chaife, horfe, and harnefs^ were together valued
at 50 1. The horfe in harnefs was worth 381. 16 s. 6 d. ;
the chaife and harnefs were eftimated at 131. 13 s. Their
ieveral valuations are required i
1. s. d. L •• d«
Together ... 50 Chaife and hamefs- 13 13 -.
Hone and harnefs —* 38 16 6 11 3 6
Chaife - - - /ii 3 6 Harnefs alone - £ 296
■ ■ ■ ■ ' I
L s. d«
50
Horfe alone - ;C 3^ 7 "•
20. From the creation to the flood was 1656 years;
thence to the building of Solomon's temple 1336 years $
thence to Mahomet, who lived 622 years after dhiifty 1630
years. In what year of the world was Chrift then bom {
Prom the creation to the flood .' - - 1656
To the building of Solomon's temple ^ 1336
To Mahomet -------- 1630
4622
Mahomet after Chrift — 622
Anfwer A. M. 4000 '
C'3 21. A
22
SUfeTRACtldN. ia6k t
21. A is 13 years younger thah B, and 17 vears older
than C, who in the year 171 j Was known to ofe '24 years
of age. How old was each of thefe perfons in 1733 i
1733— 1711 = 22 + 24 5= f6C'sT
i7 4.i4 = 63A'4 Age.
i3 + 63==76B's)
22. W, X, Y, Z fend in their mohey to fhe Baillr, and
draw upon it in the following manner ; viz. Jone 4th, I758»
Z fends in 70I. 8 s. Y had 116 1. 14s. 10 d. remaining
on balance ; and the 14th fent in 120 1. more. W paid in
47 L 18 s. 2d. in ca(h, and delivered in a Bank note for
20ol. X paid in a bill of exchange on a. sood man^^ for
33 1. 14 s. od. and in caCl made it up 100 1. Y dn the
16th drewK)r 43 1. 12 s. 6d. ahd on the 20th Z for eleven
guineas. W on the 24th added 14 1. 12 s. 10 d. and X
withdrew 47 1. 10 s. 8d. Y on the 28th paid in 18 1. 5 s.
atfid tw6 days after drew for 881. 139. 4 d. W fent for 6j
guineas on the 30th) and in five days after for 15 1. 10 s. gdi
ihore. Z on the 7th of Jaly demanded 12 K 8 s. 3 d.
and X 7 1. 3-8. id. Z on the 15th remitted them
?|il. 12 s. 4U. and per affieniheftt they received for him,
he fame time, double that lum. Y received on the 12th'
81 1. 19 s. 8d. arfd W 10 1. 10 s. Y three, days after
fent in 42 1. and W " C2 1. On the 19th X fent for
31 1. 18 s. 10 d. arfd the 24th paid in 10 1. 19 s. The
queftion is, how ftood thefe gentlemen's cam feverally, and
what money can they jointly raife ?
Debtor.
• To cafli received.
4th J une - -
24th' .- - .
15th Jaly' - -
Debtor -
Creditor -
The Bank.
W
I.
s.
d.
47
200
18
2
H
12
10
52
— _
—
314
92
II
3
1
9
'222
7
3
Creditor.
By cafli paid.
1. s. d.
30th June - - 66 3 -
5th July - - - 15 10 9
I2th - . - - 10 10 -
£9^ 3 9
Debtor.
23
Creditor.
Cteip, I. SUBTRACTI ON.
Debtor* The Bank.
To cafli received. X By cafli paid.
1^58. I. 8. d. I 1. s. d.
4th June - -• 100 ^ ^
a4thjuly - - 19 19 -
Debtor - 119 19 -
Creditor - 86 12 7
ToX . £i3 6 5,
24th June - - 47 10 8
7th July - - - 731
19th - - - - 31 18 10
£»(> 12 7
4th June - -
14th - - -
28th - - -
J5thJuly - -
Debtor -
Creditor -
ToY - -
I.
s.
d.
116
14
10
120
—
-
18
5
-
4*
*■
—
296
19
1^0
ai4
5
6
£i^
14
4
1 6th June
30th - -
1 2th July
Z
4th June
. J5th July -
Debtor
Creditor
ToZ -
1. s.
- 70 8
- 94 »7
i.
165 s
23 19
3
jC»4I 5-
9
20th June
7th July -
1. s. d.
- 43 12 6
- 88 13 4
- 81 19 8
jC"4 S 6
To W
X
. Y
Z
222 7
33 6
82 14
141 5
3
S
4
9
1. 8. «l.
ii II -
12 8 3
jC*3 19 3
In all - ^479 13 9
C 4
23. Mofo
24 SUBTRACTION.. Book L
23. Mofes was born anno mundi 2433. Homer 832
years after him. Julius Caefar lived 40 years before our
Saviour j and Alexander 312 years before Cxfar. Now as
Chrift was incarnate 4000 years after the creation, the fum
of the intervals between Homer and the direc great per^p
fonages laft mentioned is required ?
Mofes born A. M. - - - 2433
Homer born A. M. - - - 3265
Chrift born A. M. - - - 4000
— 40 .
Cxfar born A. M. .... 3960
^312
».
Alexander born A. M« <- w . 3648
•av
3960 — r 3265 = 695 from Homer to Csfar.
4000 — 3265 = 735 from Homer to Chrift.
3648 — 3265 = 383 from Homer to Alexander.
Anfwer 1813
24. A merchant, taking an inventory of his capital,
finds in his vault 28 pieces of brandy, which coft him
874 L IDS. 6d. Bourdeaux claret, 40 tons, which ftood
htm in 7541* 4 s. ; 22 lafts, four bulhels of corn in his
franary, worth 675 K 17s. 3d. j with two lafts of
Canary feed, worth ii^l. In his warehoufe were locafles
of indigo, worth 6321. 12 s. a parcel of fafFron, worth
253 L 5 s. W. P. of Stafford owed him 384 1. 10 s. In
the hands of F. G. of Lynn, he had wines to the amount
of loii 1. 10 s. Pepper m the keeping of S. Q* of the
Cuftom-houfe, value 1552 1. j6$. 8d. l^fides whidi R. O.
owes him on bond 300 1. ; and T. M. on note 260 1. 14 s.
He has in India bonds to the value of 459 1. and the intereft
of thofe fecurities made 25 K 14s. 6d. He had Bank-ftock
to the value of 2134 1. 4s. 6 d. There lay in his banker's
hands 1892 1. 17 s. 6 d. He was at this time indebted to
D. E. 7131, 13s. To M. F. 352 1. IDS. 8d. To L. P.
the foot of his account 172 guineas^ To J. B. on balance
57 1. 12 s. 10 d. To an infurance 190 K The prefent ftate
of this perfon'is fortune is required i
Stock
Chap. L subtraction: 25
Contra creditbr.
Stock debtor.
I. s« d..
M. F. - 352 la 8
L. P. - 180 12 *-
J.B. - 57 12 10
Infunuices - 190 - -
^1494 8 6
mm
By Brandy -
Claret .
Corn - -
Canary feed
Indigo
Saffron -
W. P. -
Wines per F.
Pepper per S.
A bond on R.
A note on T,
India bonds
Intereft -
Bank-ftock
Banker -
G.
O.
M.
1. s. d,
874: 10 6
754 4 -
67s 17 3
113 - -
632 12 -
253 5 -
384 10 -
loii 10 -
1552 16 8
300 - -
260 14 -
459
25 14
ai34 4
1892 17
t
/
)C"324 I:
— 1494 1
II
6
The merchant*s prefent worth, £ 9830 7 5
^5- Seth was bom when Adam was 130 V^^^ of age, and
800 years before our.faid grandfire's death. Seth at the age of
105 years had Enos. He at 90 was father to Canaan, who at
?ro had Mahalaleel. This man at 65 got Jared i who having
ivcd 162 years, waB father to Enoch. This patriarch at 65
years of age had Methufelah ; and by the time he was 187
years of age, his fon Lamech came into the world ; who at
182 years old was father to Noah ; and when Noah was 600
years old^ the flood fwept away the bulk of mankind* In
what year of the world did this happen, and how long after
the death of Adam f
Adam at -
Seth at
£nos at
Canaan at
Mahalaleel at
Tared at
Enoch at -
Methufelah
Lamech at
Noah at -^
Year of the flood
Adam 130 + 800 =: 930
After his death - 726
130 years had Seth«
105 £nos.
90 Canaan*
yd Mahalaleel.
65 Tared.
162 Enoch*
65 Methufelah.
187 Lamech.
182 Noah.
600 entered the ark.
1656
^ 6
-A^ ^
26. In
tS SUBTRAGTIONl Book L
26* In a company S had 3 1. 17 s. 2d. more than T, who
< had fix guineas lefs than R, who had within 16 s. 8 d. *
as much as W, who was known to have 100 guineas^
wanting 10 marks of 13 s. 4d. each. Pray what money
had they among them I
]. s. d« !• s* d* I. 8* d*
IOC 6 13 4= 98 6 8 W had.
98 6 8 — - 16 8 = 97 10 *- R had.
97 10 - — 6 6 -^91 4 -T had.
91 4-+317 2= 95 I 2S had.
£ 382 I 10 the anfwer.
2^. If the mean diftances between the earth and fun be
81 millions of miles, and between the earth and moon 240
thoiifand ; how far are thefe two luminaries afunder in an
eclipfe of the fun, when the moon is lineally between the
earth and fun ? And in another of the moon, when the
earth is in a line between her and him ?
Diftancc of the fun from the earth - - 81000000
------ of the moon - - .j- 240000
From each other in an eclipfe of the moon 81240000
81000000
— 240000
^ •«.»-- in an eclipfe of the fun - 80760000
28. Hipparchus and Archimedes, of Syracufe, about 200
years before Chrift ; Poffidonius 50 years before the faid
grand period 5 and Ptolemy 140 years after it ; all advanced
the fcience of aftronomy. How long did each of thefe per-
^ V - ifons Poutifh before the year of Chrift 1758 ?
5^ ' • \ r * 200 50
' : Voi \ + '758 +1758
\ . •• * ^ ^ ■ *-^ — —
Hipparchusand Archimedes 1958 Poffidonius 1808
1758
— 140
Ptolemy - 16 18
29. A
€hap.l SUBTH ACT16K. 2^
29. A grant was made by tire crown, .anno 1239, wKch
was forfeited I37 years before the Revoluiion in 1688 ; hoW
long did the fame fubfift ?
Revolution A. D. 1688 1551
— 137 Granted, A. D, 1239
Forfeited A. D. 1551 Anfwer 312
30. The building of Solomon's temple was in the year
of the world 3000.- Troy was^ b^ computation, buik 443
years before the temple, and 260 y^ars before London. Now
Carthage was built 113 years before Rome) fouhded 744
years before Chrift, born anno muhdi 4000. Is Lohdon or
Carthage the ancicntcft city, and how much ?
Solomon's temple built A. D; - - 3000
Troy before ------- — 443
A. M. 2557
+ 260
London built ----- A. M. 2817
Chrift born ----- A. M. 4000
Rome built before ----- -" 744
A. M. — 3256
Carthage before 113
A.M. 3143
2817
London built before Carthage - - - 326 years.
31. A public edifice was finifhed towards the clofe of the
loch of king John, who began his reign 154 years after
the Conqueft in 1066 5 and it flood till withm 70 years of
the peace of Utrecht, in 17 13. Of what duration waa it?
Conqueft - - A. D. 1066
134-1- 10 = +244
Edifice finifhed - A. D, 1210
Peace of Utrecht A. D. 1713
Demolition - - A. D. 1643
^ — 1210
Duration - - • . - 433 years, the anfwen
32. A,
«g SUBTRACTION; Bookie
32» A, born anno 1438, died at 48 years of a^. B died
anno 15029 aged threescore and feventeen. C, m the year
1577» ^^ 22 years of age« and furvived that time 54 years*
D, anno 16 lo^ had livea juft half his time, aoa died in
1648. £ was 13 years old at the death of D, and four-
teen years after that wa8 father to F, who was 31 when his
fon G was born ; who, at his grandfire's death, was feven
years of age. The years of Chrift, wherein thofe men were
Dorn*, and the year wherein the firft live of them died^ are
feverally required ?
A born 1438 -f* 48 = i486 died.
B died 1502 — . 77 = 1425 was born.
C in - 1577 "^ 22 = 1555 was born.*
And - 1577 -|- 54 = X631 died.
D died 1048 — 1616 = 32 half his age.
And -^ 1648 -^ 64 =: 1584 died.
E in - 1645 — 13 = 1635 was born.
And 1635 + 13 + ^^4 + 3* + 7 = ^700 ^^^*
F in * 1635 4* 27 = 1662 was born.
G in - 1662 -)- 31 = 1693 was born.
33. The powder-plot was difcovered 88 years after the
Rrformation in 15 17. The murder of Jcing Cnarles the Firft
was committed 43 years after that difcovery. The acceifion
of the Brunfwick ramily to the crown was in I7I4» juft 54
? rears after the return of king Charles the Second, who had
ived in exile ever fince the death of his father Charles the
Firft. How long was that ?
Reformation ----- A. D. 1517
Powder-plot difcovered - - A. D. -|r 88
1605
+ 43
King Charles murdered - ^ A.*D. 1648
1714 — 54= 1660 — 1648 =: 12 years, theanfwcr.
34* Arphaxad was born to Shem two years after the
Deluge, and 500 years before his Cither's death ; but at 35
years of age he had Seth, who at 30 Was father to Eber ;
who at 34 had Peleg, and he lived 430 years after that.
The queftion is, whether Shem «r Eber died the firft ; and
at
Chap.l. SUBTRACTION. ty
at ninefcore and fourteen years after the death of the longer
liver, what interval might be wanting to complete the term
of looo years after the r lood i
Selah 35 + Eber 30-------^ 65
£ber had Peleg at 34, lived after 430 - - = 464
529
Shem died after the birth of Arphaxad -> - 500
Eber was furvjvor - > - - - * - - 29
1000 *— 502 -|- 29 4* '94 = 275, interval required.
35. B was born 14 years after C, who came into the
world 19 years before A, who was 23 years of age eight
years ago. What then is the age of D, who is within 22
years ^ being as old as thofe three together t
23+ 8 = 3iAi
i9+3i = 5oC^ 117
50 — 14 = 36 B J — 22
- — — — •
117 95 the anfwer.
36. Of the noble fiunily of Cornaro^ the grandfire's age.
was 134 jrears ; and he was 93 years older than the fopy at
the time when the fon and father's age together made 1 12
years. Diftinguiih their ages ? . .
Grandfire's age - • =134
~ 93
Son's ----- = 41
112—41 =3 71, father's age.
37* K ^is 19 years older than L, who was 27 years of
age in th^ South-fea year 1720. How old is M, in 1740^
who in the year 1738 was within 24 years of being as old
as bodi of them together i ^
19 -|- 27 = 46 K's age in 1720.
1738 — 1720 z;: 18.
46 + 18 XI 64 K's I ,^^. ^^^jj
27 +18 = 45 L's {*?"*" ^738-
64 -|- 45 = 109 + 2 = iir.
1 1 1 — 24 =s 87* the anfwer.
38. If
50 SUBTRACTION. 9ook L
38. If Sampfon wa3 born 17 years after Timothy, and
Timothy 26 years before Jacob, who 28 years hence yyill be
jiift 50. In what year of Chrift were they feverally born^
the ^({fieftion being propofed anno 1758 i
JfiS -7- 50 = 1708.
1708 + i8 = 1736 Jacob.
1736 — 26 = 1710 Timothy.
jjiD -f. 17 = 1727 Sampfon.
39. A, born anno Chrifti 318, lived 207 years before B, who
lived 104 years after C, who was fucceflor to D 84 yodr^. £
was alfo uz years after D, but predeceflbr to F, by 47 ye^s.
In what year of Chrifl ^id each of thofe gentlemen flouri(h i
318 + 207 = 525 B
421 — 84 = 337 D y flourifhed,
337
449
40. A was born whenB was 18 years of a&;e. How old
Ihall A be, when B is 41 ; and what will be the age of B,
when A is 72 ?
41 — 18 = 23 A.- - - 72 + 18 = 90 B.
41. B, born anno 1108, lived 48 years before C, who
was 113 years fenior to D ; and X was 114 years before Y,
who was 74 years after Z, born anno 1527. In what yean
of Chrift were thefe men feverally born?
Bborn -------A. D. 1108
+ 48
Cborn -------A. D. 1156
+ "3
Dborn - *--•-- A. D. 1269
Z born -------A. D. 1527
Yborn ------- A. D. i6oi
— 114
V^a
Xborn ----- w - A. D. 1487
42. A, born' 445 years before the year 1733, died anno
1362. B born 37 years ago, will die 18 years hence. C,
born 256 years ago, died 197 years fince, D, born anno
1578^
T
C3up. L SUBTRACTION. jy
1578, lived till wttUn 75 years of die laid 17^3. The length
of thofe people's lives is ieverally required x
'733 "■ 445 = '288 A born.
1362 — 1288 = 74 his age.
37+ 18= 55B'a
256— 197= SQCsf ^**
^733 — 75 = '658 D born.
1658 — 1578 = 80 his age.
43. If I am 42 years older than you now, what will
be Ac difference of our ages 14 years after my deceafe,
in cafe you fliall dien furvive i
42 — 14 = 28, the fMafwer.
44. A, born anno J44T, lived till B was feven years
of age ; which was 23 years before the Reformation, in
1517. B furvived this remarkable aera juft 49 years. C,
born nine years after the death of A, lived but till B was
36 years of age. The fum of the ages of thefe three per-
fons is required i
Reformation ----- A. D.
1517
^3
A died -------A, D. 1494
born - - - ' - - ^ A. D. 1441
A'sage ------*-.
1517 4. 49, B died - - - A. D.
1494 — 7, B born - - - A. D.
»
B's age - - . - - . * \. - . 79
36 -r- 7 + 9 = C's age r y - - =20
53 + 79 + ^^ = 152, the anfwer.
45. A fnail in getting up a Mjiy-pole, only 20 feet high,
was obferved tp climb eight feet jevery day ; but every night
it came down ^^n four &et. In wlvittime by this method
did he reach the top of the pole ?
20 — 8 + 4 si: 16 tjo go the 2d. morning.
16 — 8 + 4 = 12 to go the 3d.
12 — 8 + 4 = 2 to go t\^p 4th, a^d at^ight got to the top*
46. The femidiameter of the earth's orbit, or annual
path round the fun in the renter of the fyftera, is about
SiQOOpoo miles J that of Venus 59900000. When they
* 2 arc
jt SUBTRACTION. Book L
lure both o^ the fame fide the fun, they are in periga^o ;
when on different fides, in apogaeo. What is the difference
of their diftance in both thefe circumftances i
81000000 — 59000000 = 22000000 miles in perigaeo.
81000000 4- 59000000 = 140000000 in apogseo.
Then 14000006 *-^ 22000000 =: 1 18000000, the anfwer.
47. B was 14 years old, when C was 25. How old
Ihall C be, when B comes to be 25 ?
25 — 14=11
25 -j" 1 1 = 31^9 ^^ anfwer.
48. A, born 17 years after C, and 13 before B, died 4a
years before the late king's inau^ration in 1727, aged
47 years. C died anno 1712, and B exadly eight years be-
fore him. D, born 23 years before C, died at 64. £,
born II years after B's death, will die 12 years after th6
year 1733- And F, born juft in the midway of the Inter-
val between the births of A and D, is not to reach the time
of E's death by 14 years. What is the fum of all their
ages, and which of them lived longeft i
1727
— 42
A. died A. D. 1685
Aged— 47 . 1704
Born 1638 -— 13 s= 1651 B born,
^^'7 Aged 53
Cborn - - 1621 "
Pied - - 1712 — 1621 = 91 Cs age.
1621 — 23 = 1598 D born.
Bdied - • 1704 4-64 age.
"^11 1662 died.
^7'S
1733 + ^^ = ^745 E <!««<'•
174s — '7^5 = 32 E'» ^g«-
1638 = 1598 = 40 it's half = 20*
1745 — 14 = 1731 F died.
16^8 — 20 ^= 1618 born.
Aged 113
A B C D E F
47 + 53 + 91 + 64 + 30 + 113 = 398 fum.
And 113 — 91 = 22 F, oldeft.
49. Three*
Chip. I. SUBTRACTION. 33
49* Three and thirty years before the Reftoration in 1660,
the crown granted demefnes, to certain ufesy for' 2 10 years
then to come. The proprietor, in 17 159 procured a rever*
fionary grant of 99 years, to commence after the expiration
of the firft. In what year of Chrift will the fecond term
end?
Reftoration - - • -^ - - A. D. 1660
Grant before ,- --.,•--- — jj
Firft grant made - - • - A. D. 1627
-Duration ------- — f- 210
End of the firft grant - - A. D. 1837
Reverfionary grant's contiinuaAce - -f. 9^
Its expiration ----- - A. D, 1936
, 50. A young fellow owed his guardian 74 1. 18 s. 2d.
'on balance. He paid oiF 41 1. 14s. 8d. and then declared
his fifter owed the gentleman half as much again as him-
felf: on hearing this, (he paid off in a pet 13I. 12 s. 10 d.
and gives but that her uncle William was not then lefs
in arrear than her brother and ftie together. " The uncle
hereupon pays 24 1. 7s. 3d. And then the uncle's bro-
.ther, who, by the bye, was not the uncle of thofe children,
for 1501. undertakes to fet them all clear, and has
35 1* '5 ^* 5 ^* ^^ ^^7^' ^^ fpare. Can that be true ?
L s. d»
Brother debtor to guardian at the firft - - 74 1 8 a
Paid -- —41 H 8
Remains debtor -.-..--.--- 33 36
.{- 16 II 9
Sifter debtor at firft ----^---49 153
Paid -..-.-----— 13 12 10
Sifter remains debtor »----*•- 36 25
+ 33 3 6
Uncle William debtor at firft - - - 69 5 11
Paid —-24 7 3
Remains debtor -------- 44 188
Then 3}I.3s. 6d. +36I 2 s. sd. 4-44I. i8s. 8d. == 114I. 4s. 7d.
... 1501, — 1141.4s. 7d. = 35l. iss.sdjaswaspropoifed.
D 5l..Fiv«
34 .SUBTRACTION. BookL
51. Five notable difcoveri^s were made ifi 21$ Tws
time ; viz. ift* The invention of the c^odipafa. 2d. Gun-^
powder. 3d, Printing, 4th. The diCwXrery ol Amefica^.
5th. Truth in the Reformation, The laft wat brought
about anno 1517 • the jd 77 year* before : the 2d 42.
years after the ift: and the 4th 148 vears after the. ad..
The queftion is^ in what year of Chrift did each of thefc
happen to be found i
The Reformation - - - - - A. D. i J17
Invention df the compafs - - - A. D. 1302
+ 42
GumK>wder -------A-D. 1344
+ 148
M**
America difcovered ----- A. D. 1492
— TJ
Printing invented ----- A, D. 1440
«■■— M*«
)aO9C)9C)9C)9C3e()eC)0O9QeOsO0(:)eOQOeOeOe()eO^^
S E C T. V.
MULTIPLICATION.
M
ULTIPLICATION iff a rule, by which the
_ greater of twa numbers may be fpeedily incrcafed as
often as there are units in the lefler, and in a concife man-
ner performs tfre o'ffice of addition.
In every operation in multiplication^ are two given num-
bers, called faftors ; viz, *
Firft, The multiplTcand, or number to be multiplied,*
which is generally the greater of tlft two.
Secondly, Thp multiplier or multiplicator, or number
by which we multiply, whicR denotes the number of
times the multiplicand is increafed by, or added toitfelf ;
and from thence^ will arife a third mimber) called the pro-
duei.
This
Chap.t MULTIPLICATION. 35
This in geometrical operations is called the Rectangle,
or Plain.
By addition
5
liiis^ \ f'^-
J5 produ^
2
Z
%
%
2*
2
2
2
add
35 futt*
MutTlMiICAtlOW TABLt.
X
X
X
X
X
X
X
X
X
X
^ = 4
3^:s= 6
4= 8
5 =3 10
12
^ s u-
4
4
4
4
4
4
X
X
X
X
X
X
\z
9
10
ir
28
36
40
44"
48
L
7
7
7
7
X 9
X 10
X II
X 12
63
70
V
84
8 X ft
8x9
8 X 10
8 X II
8 X la
64
2o^
88
*M^
9 X 9 ~ 81
9 X 10 = 90
9 X II = 99
9 X 12 = 108
10
X
10
-5.
100
10
X
II
=5
no
10
X
12
=
120
II
X
II
•— -
121
II
X
12
=
132
12 X 12
= 144
A*i
N. B. TTiis fable is to be pcufeAly leaned by heart, fo a^
t» be readily remembered vridiout paufing.
Tbeh i^ukii^ication tMf\t eafily performed, obfervinf
the following
D a^ R U L JE*'
» •
36 MULTIPLICATION. Book L
RULE.
Always bjegin with that figure which ftands in the units
place of the multiplier, and with it- multiply the figure
tv^hich ftands in the units place of the multiplicand ; if
their produft be lefs than ten, fet it down underneath its
own place of units, and proceed to the next figure of the
multiplicand. But if their produdt be above ten (or tens)
then fet down the overplus only (or odd figures, as in ad-
dition) .and bear (or carry) the faid ten (or tens) in mind,
until you have multiplied the next figure of the multipli-
cand with the fame figure of the multiplier ; then to their
produft add the ten or tens beared in mind, fctting down the
overplus of their fum above the tens, as before ; and fo
proceed in the very fame manner, until all the figures of the
multiplicand are multiplied with that figure of the multiplier.
394786 8643597 796534^89
7 9 "
2763502 7^793^373 8761877179
2. When the multiplier is any number between 12 and
20 ; multiply by the figure in the units place ; and as you
multiplv, aad to the produdi of each fingle figure, that of
the multiplicand, which ftands next on the right-hand.
4721217 4713^76 94713761
15 16 18
7081825s 75410816 1704847698
»mtm*,mmmmim>mmm mmmm^^^i^^m Mill ■■■ ...■iMi^H^MM
12345, 72453 6729004
13 17 19
160485 I23I7OI 127851076
3. But when the multiplier confifts of feveral figures,
the multiplii^and muft be multiplied with every fingle figure
of the multiplier ; always placing the. firft figure, or cypher,
of every particular produS, dire<My uridernelath that figure
«if the multiplier you then multiply with,
4739284
€h»p,l. MULTIPLICATION.
4739*84
94785
23696420
37914272
33»74988
.»8957>36
4*653556
44^213033940
6247386495
27356
37484318970
71236932475
187421594&5
43731795465
12494772990
^ ' ■ ' ■ I ■■
170903504957220
37
+. If there be a cypher, or cyphers, intermixed with the
figures, move for every figure, or cypher, ^onc place to-
ward the left-hand, and take care that every firft figure of
the ftveral produ£ls ftand direftly under its refpedlive mul-
tiplier.
630700325 50710984
* 6072008 4P50607
504560260Q
126140^650
44H902275
3784201950
38296I74I9002600
354976888
304265904
2^5355+920
202843936
205410266767288
5. Cyphers placed at the end of either or both faflors, are
to be omitted till the laft, produft, and then the number of
pyphers as are at the ci^|l of both muft be apnexed to it.
4260Q
2200
852
85?
93720000
429OOQ
5600
2574
2145
24O24OOPOO
6- Any number given, being multiplied by 1, undergoes
no alteration ; but if by 10, a cypher is to be annexed ; if
by. 100, ai]uiex two cyphers; by 1000, annex three^ &c*
I?3
7157
3* MULTIPLICATION. Book!
7157 x^
- -I :p -- 7'571
- ip == -1- ^ " 7*570 I And thusfor
- ICO
1000
- 715700 1 asmji
y 157000 *' phers
asmim7cy«-
asyou
10000 = 71570000
1 00000 = - - - ^ - 715700000.
pleafe.
7. In geometrical progreflions, converging ferieS| &c,
when multiplications have been very operofe, I haye fre^-
quently ^ded, AibtnuSted, or divided i or multiplied a pro-
du6t by a fmaller, when the former happens to.be a multiple
of the latter ; as I fl^all endeavour to explain in the example
following.
84964^7
874359 rBy fubtra&ing the right-hand figure from t
•——*—% cypher,, and each preceding figure froa
76467843 t that following.
42482 1 35 - By dividing the multiplicand by 2.
2548928 f ^ By dividing the product of 9 by 3.
33985708 - - By add. the laft prod, to the multiplicand,
59474989 - - By adding the two laft produfis together.
67971416 By multiplying the produfi of 4 by ^.
7428927415293
But before the learner attempts to perform operations by
this method) he ought to be acquainted with divifion.
8. If the multiplier be any number near 100, 1000, 10000,
&c. increafe the multiplicand by as many cyphers as there
are figures in the multiplier ; and fubtradl the multipli-
cand from itfelf thus increafed, as often as the multiplier
wants units of that by which the multiplicand was in-
creafed.
Let 7943628 X 999 And 437^845 X 9997
7943628000 43728450000 *
7943628 1 3 1 1 8535 = multiplic. X 3-
; 7935684372 437 '533^65
9. If the multiplier be a repetend of the fame figure,
multiply by one of die repeating figures ; and the figures of
that
Cluf>.L MULTIPLICATION. 39
that ipfodofi added, as if they had been wrote down In as
iiuui|r produAs «3 the mukiplier repeated the fiune figure,
give the produ£l required.
547856789
22222
54018
3333
»0957»3578
12174473565158
162054
180041994
10. When the repeating figure is a high digit, colled the
produd of as many ones as there are digits in the mutti-
plier, from the multiplicanc^ according to the rule in the
faft cx>ntradion ; which produd being multiplied into the
cr^eod, will give the true produd* '
m
784325634 into 7777777.
$7x47283^519374 Produ^ colle^ed for iiiiijl.
7
6100309876635618 Prodwft of 7777777.
1 1. Find the (Nrodud of the given multiplicand by the
like ni mber of nines» and divide that product by 9 ; the
quotient multiplied by the di»t which repeats in tne glvea
mifltipUery will be the produll required.
Ex. Let 4538769 be multiplied by 7777777.
45^7690000000
4538769
9 J45387685461231 N. B. Diviiion muft be learned
^ > ■ ^efoie examples of this kind
5043076162359 be attempted.
X 7
3530I533X36S«3
D 4 I2« When
40 MULTIPLICATION. Bookl;
12* When the multiplier can be parted into period$f
which are. multiples of one another, the operation may be
contracted in the following manner*
8649347864
132576961^
103792174368 produftof 12.
■ 830337394944 - foregoing produ£l: X 8.
4982024369664 - - laft produS X 6.
II41713918048 firftproduft'x 11.
1146704256(708308768
13. To mi^ltiply by a fafior, confiding of as many
cyphers between two aigits as there arc places in the mul-
tiplicand, multiply by a fingle digit ; and the produd ty
the fecond figure will fall direftly to the left-hand of the
produft by the firft figure ; but if the produdl of the firft
figure be Icfs than lo, then a cypher muft be put down
between the two produfts.
84629
7060003
592403^3887
14. The proof of multiplication, is by making the mul-
tiplicand to be the* multiplier j then if the product comes
out the fame as before, your work is right.
15. Or by cafting away the nines, which, thoi^gh not
infallible, fervcs to confirm the other. Thus, in the laft
example, make a crofs, and add all the figures, or digits, of the
multiplicand together,' as units, thus, 8 -f 4 -|» 64* ^ +9=== ^9*
caft away the nines, apci fet the remainder two on one
fide the crofs. Do the farpe with the multiplier 74-3 = 10;
fct the remainder i oh the other fide the crofs. Do the
like by the produ<H:, and fet the remainder at top. Laftly,
multiply the figures on the fides, and fct the remainder
at the bottom, after the nines (if any) are caft away ;
which muft be the fame with the top, if the work is
right.
Questions
Chap. I. MULTIPLICATION. %t
Questions to exercife Multiplication,
1. A is 17, B 7; what will their aged feverally be,
when the elder is double the age of the younger ?
A ,7
B— 7
10 ^A's age when B was bom.
X 2
20 A*s age, the anfwef*
2. Trajan's bridge over the Danube is faid to have had 20
piers to fupport the arches, every pier being 60 feet thick,
and feme of them were 150 feet above the bed of the river;
they were ajfo 170 fijet afunder : pcay what was the width
of the river in that place, and how much did it exceed the
length of Weftminfter-bridge, which is about 1200 feet
from fhore to ihore^ and is fupported by 11 piers, making
the number of arches Z2 ?
»
* Arches - 21 X 170 = 3570
Piers - - 20 X 6p = 1200
Width of the Danube 4770
of tjie Thames - 1200
Difference - 3570 the aniwer.
3. By God's bleffings upon a merchant's induftry^ in ten
years time he found himfelf pofleiTed of 13000I. It appeared
from his books, that the laft three years he had dear^ ^731*
a year ; the three preceding, but 586 1. a year $ and before
that, \kit 364 1. a year. ^ The queftion is, what was the
ftate of his fortune at every year's end that he continued in
prstdcy and confequehUy what had he to begin with ?
Merchant's whole ftock r ' £ 13000
Gain per annum 36^. x 4 = - 1456
Ditto - - - 580 X 3 = - 1758
Ditto - - - 873 X 3 = - 2619
Whole gain - - -I - = £ 5833
Original ftock - - - =: )£ 7167
r
4*Whft
r
Jpi MULTIPLICATION. Bookl
4. Wh^t diiGsreaQe is there between twice eiglit*and-
twenty, and twice twenty-eight -, as alfo between twice five*
ttnd-ii%, and twice fifty-five ?
28 X a =5*
a X 8 + 20 = 36
Anfwer 20 difFerenct
Alfo 55 X 2 = 110
ax 5 + 50= 60
Anfwer 50 difference.
5. What number taken from the fquvt pf 54^ will learo
§9 times 46 ?
54 46
X 54 X 19
216 414
276 46
2916 874 = 90429 the aofwer*
6. The remainder of a'divifion fum is 423 ; the quotient
423 ; the divifor is the fum of both, and 19 more. What
then was the number to be divided i
423
4^3
19
865 divifor
X423
2595
3460
36589s
+ 423
366318 the anfwer.
7, There
Chap.l. MULTIPLICATION, ajj
y. There arc two numbers; tbc biggeft of them is 7?
times 109 ; and their difference 17 times 28 j I demand
^eir ium and produd ?
109 X 73 =? 7957 the greater number;
a8X 17= 476
7481 leflTer number,
79S7 + 7481 = 15438 their fum.
7957 X 7481 =: 59526317 their produd:
8. There are two numbers, the Icfs is 1879 the difference
34 ; etve the fquare of their produd, ditto of their fum
and difference, and the fum of thofe fquares.
187 + 34= 221 greateft ; then 221 x 187 s= 41327 product.
41327 X 41327 c= 1707920929 fquare of their prod^A,
221 4- 1 D7=;4o8;and 408 X 408 ^ i66464fquare of their fum*
221 *• 107 ss 34; and 34 X 34 = 1 1 56 fquare of their difiei". -
LaiUx9 17079209^ + 1 66464-^ 1156 = 1708088549 anfw.
*
9. A perfon dying left his widow the ufe of 5000 1. To
a charity he bequeathed 846 i. 10 $« To each of his three
nephews 1230 1. To each of his four nieces 1050 1. To
twenty p#or houfekeepers five guineas each ; and 200 guineas.
fo bis executors. What mail be have died poffeffed off i
I. s. d.
. To bis widow * - - 5000 ^ -
To a charity - ^ - - 846 10 —
To nephews 1230 X 3 - 3690 - -
To nieces 1050X4- 4200 — -
To 20 poor houfelce^rs 105 - —
To executors ---- 210--
I4O5I 10 -
lO- In the partition of lands in an American fettlement, A
had 757 acres allotted to him, Bhad 2104 acres, C X641O9
D 12881, £1x008, F98i^, H 13800, and 1 8818 acres $
now how many acre^ did me fettlement contain, fmce the
allotments made above want 416 ac/es of j-tb of the whole?
Firft 757 + 2104 4" 16410 + 12881 + 11008 +9813 +
13800 4- 8818 + 416 5= 76007.
Then 76007 X 5 = 380035, the aafwer*
SECT.
4f
P I V I S I p N.
BooI^I.
SECT. VI.
DIVISION.
DIVISION is a rule by which we fpccdily difcQvcr
how often one number. is contained, or may be found
in another ; or by which any number may be decreafed ; or
divided into as many parts as there are upits in the number
you divide by.
To perform diyifion, two numbers are always given.
J, The dividend, or number to be divided.
11. The divifor, or number by which the faid dividend is
to be divided.
Ai)d frQm thence will arife a third, called the quotient,
which ibew§ how often the divifor is contained in the divi-
dend.
Laftly, If the divifor doth not exadtly xpeafure the divi-
dend, a fourth number occurs, called the remainder ; which
is always lefs than the divifor, and confequently a fraflional
part of th^ quotient.
Diviiion by a fingle figure, or not exceeding 12 in the
^iyifor, is performed by the following
Jl U L E.
Firft, obfervehow often the divifor is contained in the firft
figure of the dividend (or in cafe the^ firft figure of the
dividend be lefs than the divifor, in the two -firft figures)
and fet the quotient figure under it j and if any thing re-
mains, carry it to the next figure in tl)^<iividend, where it
muft be reckoned as fo many tens ; and fo on, bearing the
Remainder of each figure to the next in your mind, until
you have finifhed your operation.
Divifor 2
Quotient
Dividend.
S738473
I rem.
11 8579475321079
779952301916 - 3
12 2i5796305t7.3i
■ r n I
_ ^7983^54894 - 3
2. JBut when the divifor confifts of many plai
guifh by a point fo nfiny of the foremoft places of the
r : I ffi dividend
2869236
18647279 r/ /
372945s ft 4
\
/
CMp. 1. DIVISION. 45
dividend towards the left-hand, as are cither equal ta the
divifor, or clfe being greater, it comes neareft to it ; then
confider how often the divifor is contained in this firft period
of the dividend, and afTume that number for a quotient,
which multiply into the divifor; and whenever it proves
{(ceater than the dividend, ftrike that figure out, and put a
els in the quotient : thfen fubfcribe the produd of the,
quotient figure into the divifor, under the dividend, and
draw a line under it ; fubtrad it therefrom, fubfcribing the
remainder under the line: then prick and brine down
another figiu-e, proceeding, as before, till your diviiion be
iiniihed ; always obferving, that for every figure Or cypher
you bring down, you put a figure or cypher in the quotient*
Example.
59157 ) 252070573915549 ( 4261043898
236628
• «544as
"8314
.361x17
354942
*ii m
(
••61753
59»57
. 259691
236628
• 230635
/
f
^
531645
473*56
583894
53»4»3
. 5i48i(
/ ■ 473«5<
I
41563
3. Many
f
4£ D t V I S I O }9. Book t
3. Many figures may be fayed, if yon work by the flioit
Italian meUiod i that is> omit felting down your multipli-
cations, and multiply and fubtrad togeVheri always re-
membering to carry to the next figure as many as you
borrowed. ,
873469)43*756284563574(495445498
8330868
4756474
3891295
3974196
4803203 ' .
' 4358585
8647097
7858764
87IOI2
4* When the divifor conilfts of feveral cyphers after a
figure, or figures, cut them all off by a daih of your pen
underneath them $ and alfo cut off at many cyphers, or
figures, in the dividend ; but when divifion is finiflied,
bring down the cyphers, or figures, cut off from the divi*
dend, to the remainder.
35000 ) 2962875496? ( 846535
162
228
187
204
29965
5. As unity, or i, neither multiplies or divides, any
number may be fpeedily divided by 10, 100, 1000, by only
cutting off by a comma fo many figures to the right-hand
of the dividend, as there are cyphers in the divifor ; thofe
to the left-hand bring the quotient, and thofe to the right
the remainder.
Dividend.
Quotient.
Rem.
10
905672417^
9056724X
7
100
90567241 t'
9056724
. 17
xooo
905672417
905672
- 417
1 0000
905672417
90567
- 2417
6. To
"1
6. Todifridc b^ aiij nuoiber confifttng of ninek ; ri^ 9^
99> 999> &c« This nuy be pcrforoKd by addiciotr, as tba
multiplying by tho'fe numbers was by fubtra£lion.
RULE.
Divide tbc given dividend into periods of as many places
of figures as there are rimes in the divifor, beginning from
the left-hand ; and anAex as many cyphers to the right-
band of the number, as may he wanted to complete a period.
Then write the figures of tnc left hand period under thofe of
the fecond period, which is next thereto, towards the right-
hand ; add thefe two together, and place their fum under
the third period ; obfervmg if the fum of the two figures in
the higllM ptdce «9tf€Md mAdy to place the figure that would
(in common addition) be carried under the loweft place of
ilie fecofi^ peitiod. Add the third period to tboife /(gures
^i4iidi ftand oiKfor it^ coitcluding the carried figure; and
place thetfi mKter the foartb period ; and fo proceed tiU you
knre pfeced figures under the r^ght-haiMl period ; and under
dle«i plactf fuch ^ figure as Wotdd hare been there pldcedf
kail the work proceeded ar period further. Then add the
whole together ; aAd beginning at the right-h^hid, cancel
as many figures as there were cyphers annexed to the divi**
dend ; and from the figures that remain, cut off from the
righc-band a# rikUiy figures as the diviibr contained nines ;
fo (hall the figuws^ to the left be the quotient^ and thofe cut
off the remainder; only if the remainder be all niiles, add
one to the quotient.
Let 8765806137663 be divided by 9999.
The N** with three cjrphers Innexcfd 8765,8061,3766,3000
8765,6826,0593
I. ■ 2.
By addition arifeth this N* - - 8766682805943,592
From which the three laft' figures 7
being left out for the fbree cy- ? 876668280,5943
pbers annexed to the dividiend ^
Thefe properly feparate4 - ' - 876668280 the quotient.
And - - 5943 remainder.
7. But if your diviior be il,ixi,iizr, &c. 22,222,2222,
&c. or 33»333>3333, &c. divide the given dividertd by
the ^igitf which repoau- ia the divifor, and multiply the
V. J quotient
' ^
■ •
r
48
DIVISIDNi
Book I.
quotient hj 9 i then divide the produd by 99,999,9999,
and the xefult will be the quotient required.
Let 222671883 be divided by 777.
7)222671883
■ ^
318IO269
X9
286,292421
286,578
i
286579,000
Therefore 286579 is the quotient required.
8. If the divifor be large, and ii quotient of manv figures
be required, as in refolving of high equations, ana <^cu«
lating aftronomical tables, or thofe of intereft, under the
divifor fet down its double ; to this double add the divifor^
fetting down their fum againft the figure 3 ; and proceed on
by a continual addition, until there be ten times the divifor
in the table ; whkb, if true, will be the divifor itfelf, with
J^' cipher to the right-hand of it.
Let it be required to divide 70251807402 by 79863*
79863 ) 7025i8q7402( 879654
159726 638904
I
2
3
+
5
6
7
8
239589
3^945*
399315
479178
559041
638904
718767
798630
•636140
559041
• 770997
718767
. 522304
479'78
• .431260
39931S
31945a
31945*
9. Dlvifioiy
CIiiip.T, D r V I S 1 o w.
^ DMfiof and tt^tijpllkadoiiioteichanaeablf prore^c^
other I for in divifion^ if vou multiply the 4ivifor by che
quotient, and to the ,fxpiu£t add the rcmaincTcr (if any)
their fum will be the djyaertd. So to xvovc nifltipHcation*
if the produa be divided by the multiplier, the quotient
«riU t)e tht aoikifttcaiid } or if the produft be divided^ by
the multiplicand, the ^ttOtkiit wlU be tfui wiMplier. • ^
10. Or caft away the nines in riic Jivifor, "and quotient,
ftnd (et the remainders on the JUes of a crofs. Do the
fame with your diTi4md» iOld M the remainder at top.
Multiply the figures on the fides, ^raft away the nines, and
iet the re«uu;ideriU.tbe bottom, which muft he equd to the
top. Note, If there be a remainder, it muft be added t^
the produa, on the fides of jhc crofs, 2^id the nines throwA
out as before.
QozsTiOHf ferfofwed ly DMGloti in cdftjunOien vfi/b.
the refi of the foregt^ general rules.
t. What is the difference, and what the fum of fix
dozen dozen, and half a dozen dozen ?
12 X 12 X 6 = 864 s 6 dozen dozen.
12 X 12 = i^ -i- 2 s 7a s= T dozen dozen.
936 litm.
792 diiE^rence.
■ * ■ •
2. Siibtrad 30079 out of fottrfcore and thirteen millions
as often as it can be found, and fay what the laft remainder
exceeds, or fallsftorttrf'2tfSo?
' 3QO79Ji9j0oooo©(3O9X
276300
55^90
i*.MMiMa
Rem. 25S11
— 2IJ[8o
4631, the zvif9ftx»
c, Wha:
50 DIVISrOH- BooklL
3. What hmnber added to the forty-third part of. 44^9^
wiil make the fUm 240 ? -
_ 43)44*9(103 '^ r ■ ' "■
Then 24© — 103 1= 137, oie anfwer.
a - *
^ 4« What number dedu&ed from the 26th part of £262^
will kave the 87th part of . the^ iame i
26 ) 2262 ( 87
— 26
61, the dtifwer*
5. What number fhultiplled by . 720849 will produce
5190048 cxa£Hy?
72084 ) 5190048 ( 72, "the anfwer.
6. What number divided by 410844) will quote 9494»
^^nd leave juft a third part of the divifor remaining i
3 ) 419844 .
139948
419844
9494
1679376
3778596
1679376 ....
3778596
3985998936
'39948
3986138884, the anfwer.
7. The fum ofrtwp number^ is 360; the Icfs is 114 s
what is their difference, produA, and larger quote ?
360
— "4
246
246 — 114 = 132 difference.
246 X 114 = 28044 produa.
1 14) 246 {= 2tV quotient j vix. 6 ) tVt ( A*
8. I
Chap. I. DIVISION. 51
8. I would plant 2072 elms in 14 rows, 25 feet afundcr •
how long will this grove be ? *
dift.
14 ) 2072 (148 in each row. 148 — i = 147.
147 X 25 = 367s feet = 1225 yards.
9.. A brigade of horfe, confiding of 384 men, is to be
formed into a fquare body, having 32 men in front j how
many ranks will there be ?
32 ) 384 ( I2> the anfwer.
10. What number is that, from which if you dedufl the
25th part of 22525, and to the remainder add the 16th part
of 9696, the fum will be 1440 ?
25 ) 22525 ( 901
16) 9696(606
1440 + 901 — 606 =s 1735, the anfwer.
11. There are two numbers, whofe produft is 16 10 ; the
greater is given 46: what is their fum, difference, and
quotes ; what the fum of their fquares, and what the cube
df their difference ?
46) i6io( 35 lefler.
46 4- 35 == 81 their fum.
46 — ^5 = 11 difference.
35 ) 46 ( = 144. quotient.
46x^46 r=:2ii6
35 X 35 = 1225
3341, fum of their fquares.
XiXiixii=: I33X> cube of their difference,
la* What number multiplied by 57^ will produce juft
what 134 multiplied by 71 will do r
134 X 71 ==-9514
57 ) 9514 ( ^^Hh ^^ anfwer,
381
394
S^
E 2 13. Thctf
SZ P I V I S I O K Bodi I.
y 13. There arc other two flumbers, die greater 7050,
"^ which divided by the Icfs, quotes 94; what is the difFcreiice
of their fquarcs, and what the fquare of the produ£l of their
fum and difference i
94)705o(75leffer.
7050 X 7050 = 49702500 fquare of the irreiitar.
75 X 75 = 5625 iquare of the kfler.
49696875 diffl of their fquares.
7050 + 75 = 7 J^5 f^"" • • : 7^5® — 75 = ^975 ^*^*
'7125 X 6975 = 49696875 prod, of their fum and diflF.
49696875 X 4969^875 = a46977938476s625» a«fwer»
H* Six of the female cricketers, t^at play^^ latdy ia
the Artillery ground, fetched in company ftrokes as fol-
low; Viz. ABCDE 207, ACDEF 213, ADEFB 189,
.AEBCF 234, ABCDF 222, BFDCE 250 : How many
did they fetch on the othpr fide, itnce thefe fix perfons
wanted but fourfcore and thirteen notclies to decide the
game?
207 + 213 -f 189 + 234 + 222 + ^50 s= «3I5*
They being each mentioned five times - 5)i3l5( 263.
Then 263 + 93 = 356 ;
And 356 — I == 355^ the anfwer.
15. In order to raife a joint ftock of looool. L, M
/ and N together fubfcribe 85001* and O the reft. Now
M and N are known together to have ifet their hands
to 6050 1. i and N has been heard to fay, that he had
undertaken for 420 1. more than M* What did each pro-
prietor advance i
Firft 6050 — 420 = 5630
8500 — 6050 = 2450 L's
2 ) 5630 =z
2815 +420
icoco — 8500 sr 1500
j£ roooo
fubicriptiop.
j6. One of the fmarts in the accomptant's ofijce making
his addrcflcs in an old lady's family, who had five fine
daughters ;
Chap. I. DIVISION. 53
daughters ; (he toU him their father had made a whimfical
vvill, which might not foon be fettled in Chancery, and till
then he muft refrain his vifit. The young gentleman un-
dertook to unravel the will, which imported,. That the firft
four of the girls fortunes were together to make 25000 1. ;
the four laft 33000 1. ; the three laft, with the firft, 30000I. ;
the three firft, with~dse laft, were to nuke 2B0000I.; and
the two laft, and two firft, 32000 1.' Now, fir, if you
can make Vppcar what eaph is to have, and ^ you like,
feemingly, my third daughter, Charlotte, who I am fare will
make you a good wife, ^and you aH-^welcome } what was
Mifs Charlotte's fortune ? ' '
25000 + 33000 + 30000 4* 28000 + 32000 = 148000 i
each be mentioned times 4 ) 148000 ( 37000.
Then 37000 — 25000 = 12000 youngeft,
37000 — 33000 = 4000 eldeft.
37000 — 300CO = 7000 fecond.
37000 -i- 28000 =: 9000 fourth.
37000 — 32000 := 5000 Mifs Charlotte,
17. A father dying worth 5460 1, left his wife with child,
to whom he bequeathed, if ihe had a fon, -Jd of his eftate,
and ^ds to the fon ; but if flic had a daughter, -]-d to her,
and 4ds to her mother. It happened that flie had both a
fon and a daughter ; how {hall the eftate be divided, to
anfwer the father's Intention ?
It is plain that the father defigncd the fon's fortune to
be double the mother's, and that the mother fliould have
double the daughter's fortune.
For every pound the daughter had, the mother muft have
two, and the fon four.
Then i -f- 2t. + 4 = 7 divifor for the daughter's portion,
7 ) 5460 ( 780 1. daughter's -j
Alio 780 X 2 =1 1 560 mother's - l part.
And 1 560 X 2 = 3 1 20 fon's J
18. Fair ladies of you I muft yet enquire,
How the poll ftood for the knights of our fliire ;
The number of votes, as I have fcen,
Were five thoufand, two hundred, and nineteen ^
Which among four was juft fo divided.
As one the fecond, and the third exceeded,
B? twenty- two, and fourfcore baring fevcn ;
TIk fourth by no more then fixfcore and ten :
5 3 Then
^ D I V I S« O N. • Book L
Then how many votes ha|^ach candidate ?
You need not in finding wch trouble your pate.
5219 * f ' L.'Duaj.
22
73
130
I
4)5444( - -W i36ifirft
1 36 1 — 22=1 3^9 fecond ^ candidate*
1361 *— 73 =: 1288 third
1361 — 130 = 1231 fourth
5219 proof.
19. A general difpodng his army into a fquare battle,
/finds he has 284 foldiers over and above ; but increafmg
each fide with one foldier, he wants 25 foldiers to fill up
the fquare : How many foldiers had he ?
Since the< number of foldiers exceeds tralefler fquare by
284, and wants 25 to fill up the greater^ ^m.
284 + 25 = 309, and 309+ I ^ 310.
2 ) 310 ( 155 fide of the greater fquare*
1 55 X 155 = 214025.
Anfwer 24025 — 25 = 24000 the number of foldiers required.
20. What number is that, which multiplied by 20, and
' that product divided by 6, gives 140 in the quotient ?
Firft 140 X 6 =: 840. And 840 -f- 20 = 42, the anfwer.
21. A man being 100 years of age upon his birth-day,
had his three fons with him at dinner, viz. William, James,
and Thomas ; the Father faying to them. Well, fons, I am
this day juft loo years old ; the youngeft, William, faid.
Father, my brother Thomas is four times as old as I am,
and my brother James is three times as old as I am, and all
our ages together are juft 100 years : How old was each of
the three fons ?
4 Thomas.
3 James.
I William.
8) ioo( 121- William's y
Alfo 12^ X 3 = 37I James's ( age, the anfwer.
And 12^^ X 4 = 50 Thomas's )
22. A
Oiap-i: D I V IS I O M. ' 55
aa. A man dies and leaves a legacy of 060 1. to be dif-
pofed of among four of his relations, viz. A, B, C, D,
in thu manner; Bis to have twice as much as Ai
C twice as mueh as A and B } and D to have as much
and half as much as C What muft each perfon have ?
Ai
C6
18 )9op( 5o = A's:|
50x2 =ioo = B's I ^ . ;.
100 X 3 = 300 = C's f P*"» *^ anfwer.
300 X It = 450 = D's J
900
_ . »
/ 23. A labourer, after 40 weeks working, lays up 28
/ crowns — three weeks wages, and finds that he has expended
36 crowns + ii weeks wages. What was his weekly pay ?
Firft 11+ 3 = 14 weeks wages wanting.
Alfo 40 — 14 z= 32 weeks. And 28 + 36 = 64 crowns.
• . • 32 ) 64 ( 2 crowns, his weekly pay. Q^ E. F.
C H A P T E R 11.
Cwtaimng Tables of weights^ meafures^ and time ; with
Addition, Subtradlion, and Rtdu&ion thereof fpcm
one denomination to another.
w
5 E C T. I. TABLES.
TROY WEIGHT.
BY this weight arc weighed jewels, gold, filver, and all
liquoR. • • N
Grains.
24 = I pennyweight.
480 = 20 = I ounce.
5760 :;= 240 s= 12 = I pound.
E 4 The
5& Ta^mt 9f WEtoifTS, fS€. Book H
. The inon«yer« alfo sit tte MiiU fixb4ivide a gtatn..
!%^ Uai^s =s I peaofe.
Ibd pedoita sas t droite^
24 droit4& = imtte.
20 mites = I gnuii.
The carat is a weight which goldfmiths axu} jewellers
life to weigh precious ftones ana pearls \ it weighs four
grains, each of which is fubdivided into \^ ^ \^ -^^
&c.
Carat, or carad, is alA> the name which reprefents what
degree of finenefs gold is of ^ as fine gold,' in its purity oi"
perfe^ion, is 24 carats \ and ftandard gold, of which our
coin is made, is 22 carats of fine gold, and two of alloy,
(or a bafer metal, as copper or filver.)
Whence we may obferve, that this carat is -^^h part of
any quaiitity or weight.
APOTHECARIES WEIGHT.
Apothecaries compound their medicines according to the
following divifioa of an ounbe Troy 3 but buy and fell
their drugs by Avjcrdupcdfe weight.
Grains.
20 as. I ferujple.
60 = 3=1 dram,
43o = 24 = 8 z= I ounce.
AVERDUPOia? WEIGHT.
By- Averdupoffe weight are weighed ftich cortim^dities as
are cither very coarfe and droiiy, orfubjeft to wafte j as "all
kind of grocery wares ^ and pitchy tar, rofin, wax, tallow,
foap, flax, hemp, &cv ceppet, tin, floel, iron, lead, &c. ;
alfo flefh, butter, chccfe, fait, and moft other common
neceflaries of life. * .
N. B. 68rj grains of barley hath been found to. weigh
cxa(9Iy dnc ounce Averdupoife weight ; therefore a. pomii
containeth 10896 grains.
The pound Averdupoife is greater, but the c^nce lefs,
than thofe of Troy weight ; mf> pouild Avenhipojfe bein^
equal 10 14 oz. ix pwt. icvi grains 4 and one oui^ce equal
to iSpwt. 5f grair.sTroyi
Drams.
pop. n. T4mS9 ^ W^|CBT$» iSff. fff
l6 = I ^^«ilO0.
256 := 16 = I pound.
7168 = 448 s s8 z= s quarter.
28672 ^ 1792 = 112= 4=s I hundred.
57144^ = 35^40 =: 2240 = 80 = 2Q =s ( tun.
tl ■ — — — — .— ^— «— — ■— I I T —■■>■■
N. B. A ftone of flefli iiieac in London 19 8 lb. Aver*
dupoife, but in moil other places 141b.
Alfo 28 lb, of wool madces a tod in NorfeHCy and fat the
fouthem counties 1 but 30 Ib^ in Yorkftinr, and ochet
northern ones.
A ftone, horfrman^a weieht, i» 14 lb.
Afotfaerj or fodder, of lead, iQ^-cwt.
L I a U ID MEASURE.
As the original of Troy weight was a com of wheat
taken out of the middle of the ear, and being well dried, 32
were to make a pennyweight ; fo eight pound Troy weight
of wheat (or 6x440 grains) were enaSed by iererai ftanites
to oiake one gaRon wine meafure. This gallon, by which
^1 wines, brandies, fpirits, ftcong- waters, mead, perry,
cider, vinegar, oi!, &c. are meafured and fold» containctlj
2^ cubic inches. /
WINE MEASURE.
Cub. In. r Note, 31 J gailone is ^
281 i=: I pint, j wine or vinegar barrel.
231 = 8 =£ I gallon. ] and 236 gallons a ton
9702 = 336 = 44 = I tierce, ^of fwect oil.
14553 = 5^4 = 63 =: 14 = I hogflicad.
^9404 =r 672 = 84 = 2 =1 14 = I puncheon.
29106 = 1008 = 126 £= 3 =: 2 2:is.=arbuttorptpe.
58212 = 2016 = 252 = 6 =4 =3 =:2=j ton.
> mi »■ >■
■««^ta^<tei
A ton of 252 gailDns, atyl^lb. to the* gallon, wei^
|89o!b,=:i6cwt. 3qrs/ i4lb.
' The beer or ale gallon (which arc both one) is much
larger than the wine gallon ; it' being probably made at firft
to correfpomi with Avcpdupotie weighit, as the wine. galkn
did with: Troy weights For one pound Averdupoife bein^
Deafly equal to 14 tru lapwts. Troy; and as one pound.
Tioy is in prof)Ortio» to the 'cubic inches in a wine ^lotn,
• . • 1 . •••• fo
^ Tables tf/ Measures. BoorkL
^ is one pound Averdupoife to the cubic inches in an ale
gallon, viz. 12 : I4il- : : 231 : 282 nearly.
J.
Cub. In.
. 2256 s=
4512 =
9024 =
^3536 =
ALE MEASURE,
I pint.
is =: . I gallon*
64 = 8 = 1 firkin.
I28=i6:=2=i kilderkin.
256 =32r=4:^2=±i barrel.
304 ^48 = 6 = 3=: 14. = 1 bogOiead-
BEER
MEASURE.
Cub. Ir
1. '
-
35t
= I pint.
282
= 8 =
I
gallon.
2538
= 72 =
9
:= I firkin.
5076
= 144 =
18
= 2 =: I kilderkin.
10152
= 288 =
36
= 4 = 2 =: I barrel.
t
X5228
= 432 =
54
= 6 = 3 = 14. = I
hogfl^ead*
30456
= 864 =
108
= 12 = 6 = 3 =2
= 1 butt.
N. B. This diftin£tion, or diflEerence, between ale and
beer meafure, is only ufed in London ; but in all other
places of England the following table of beer or ale, whe-
ther it be (Irong or fmall, is to be obferved according to
a flatute of excife made in the year 1689.
BEER and A L E in the country. \
Cub. In.
35^ = I pint.
=: 8=1 gallon.
= 68 = 84: = I firkin.
^ 4794 = 136 = 17 =2=1 kilderkin. '
' 9588 = 272 = 34 =4=:2 = i barrel.
1^82 = 408 =51 =6 = 3= 14- =1 hogflicad.
DRY MEASURE.
By |n aA of parliament, made in 1697, it was decreed,
That every round bufhel with a plain and even bottom,
being made eighteen inches and a half wide throughout,
and eight incheg'deep^ ibould be efteemed a legal Winchefter
3 L bufhel.
202
2397
!
i
Chap. II. Tables of Measures. 59
buihel, according to the ftandard in his majefty's Exche-
quer. Now a veflel thus made, will contain 2150,42 ci^
inches; confequentl/ the com gallon doth conuin 268^
cubic inches.
Cub. In.
33.6 = I pint.
268.8 = 8=1 gallon.
537.6 = 16 = 2 = I peck.
2150.4= 64= 8= 4= I buihel.
8601.6= 256=: 32=: 16=: 4= J coom.
17203.2= 512= 64=: 32= 8= 22= X quarter.
172032 = 5120= 640 = 320 =8q = 20 = 10=1 laft.
But the fanner generally delivers to the merchant jo^
quarters of oats, colefeed, and fome other grainy for a
laft, in confideration of wafte, &c. by exportation.
The miners in Derbyfhire have a veflel called an ore«
difli, by which they buy and fell their lead-ore.
Its length 21.3
breadth 6 \ inches;
depth 8.
•n i,
i.43
confequently its contents 1073.52 cubic inches, very nearly
equal to two pecks, or four corn gallons.
Nine of thofe diihes they call a load of ore ; which, if
pretty good, will produce about three hundred weight of
lead.
LONQ MEASURE.
Inches.
12 =
I foot.
«
36 =
3 =
I yard.
1
72 =
6 =:
2 =
I fathom.
198 =
i6i =
54 =
2^ = I pole.
•
7920 =
660 =
220 =
no = 40 =
I ftrlong.
633602=
5280 =
J 760 =
880 =320 =
8 = 1 mile.
LONG MEASURE.
The navigators, or feamen, reckon 60 Englifli miles 'to \
degree ; fo that the circumference of the earth, according
t% tbeoi, is 360 dcgfccs X 60 ;=: 21600 mMes.
But
0c Tjibles ^M£A»Ai^*'&fc BdBkf.
> But Mr. VicrwoiKl, ^ sft esf periment iilaile beMeen
LondttA ud York, in the year 1635, foimd, that 367x96
fBetnr69-mileiMd-958 ysmb make gt degive; ^ccefding
t;o whom the circumference of a great circle = 25035 mile?.
And accordingto the Tranfadtons of the Royal Academy
of Sciences at raris, anno 1687, 57060 toifes =z: 365184
Engliih feet =: 69 miles 288 yards . make a degree on thii
ierniqueous globe.
Sqr. Inchei.
144 s=
39S04S
1 568 1 60
S QLU ARE MEASURE.
1 f^» foot*
9 rtt I fiif. ywi;
272l=s 3ois - I perch.
Z0890 s= ^ 1210 s= 40= I rood.
\ ^27»t64»BB 4|9^o ea' 4^ 3s t6c£=- 4S1 f acre.
5=17^6400 =s
40144^9600 ss 17^78400 = 30976CO ^=1094:^0 = 2560 ss 640=1 fi). mil^
a !■
■ illll
. This t|dile will be ufeful in menfur^don of Atperficits.
N. B. The leaft p^t of long meafure wi$ at iirft a barley*
corn taken out of the middle of the ear ; and being well
idried, three of them ii) length were to make one inch.
CLOTH MEASURE.
Inchps.
24 = ? naj],
o = 4 =: I quarter.
3S = 16 = 4 == I yard.
■45 • = 20 =; 5 = X ell EngTifii.
27 = 12 = 3 = I ell Flemifli.
54 ;:s 24 =x 6 =c I ell Frenclu
■
Note, All Scotch and Irifh lineps are bought and fcid b}|
the yard; but all Dutch linens axe bouglK'byxhe ell Flemifhi
and fold by the cll Engliii. . '1
■ •■ • TIME.'
Timr onir fliews the duration; or mutiCtTra of 'things, s^
year being tne ftandard, or integer, by which fuch con-
tinuation or change is opmputed. And a year is that
ipace of tit|ie in which the fun (apparently) completes; its
revolution from any one point in the ecliptic (an ioiagi*)
nary circle* in tjjc heayensj to tbc^afnf .point again* ,-
.. -^ . 5econ49-
Seconds.
60 := i ijnioute* «
3600=: • .6ojc= . f hour«
86400=: M40£= 24=} « day.
31556937 = 53t5949P= 8766= 365-s* 48' 5/' = 1 year.
SZC T. JI.
« *
i
A U L E.
ALWAYS 3>e^n wfth thofe figures of the loweft orjctft^
denominad^nj. ang add them altogether into one iassx ;.
tfaen^onfider how m^O/ of ihr aext fuperior denoininatk|^
are contained in tliat fum, fo many unils you muft cgsry to Wi
fiud next fuperior denomination, to be added to|;ether with
thofe figures that 9and there 1 and if any thing remajft
over, that overplus; muft 1>e fet down underneath it#
own denomination i^ but if jou cannot otherways difcovec
how many of th^ ntx% fuperior denomination are con**
tained in thgt fi^ia^, ^divide it by the. p1iml>ar of unit^
contained in one ^f .the next denominatipp fuperior thereto^
and fet down ^the^r^xuainder, and carry the quotienf • And
fo proceed o;^ fro^ pne denominatio^i ro another^ until all
be finilhed* ' , / .
«
f u A 4iiercbant Isi^s |ip fix b^ of Canterbury hops,
I N^ I. of whifilL weighed 2 cuit/ 2 qrs. 10 lb. ; N® 2.
2 cwt. I qr. 16 lb. ; N* 3. j2 <A¥t. 0 qrs. 24 lb. j N« 4»
2>wt. 3qrt. orf¥; i^*5. 2C^i t qr. I2lb.$ N«6. 2CWt
t qr. i%lb. ; befioes a couple «j^£ket«, ditto, that wei^ie4
|. 5Bf Ib» ead( ^^w many hra^eds weight has he to pay
i carriage for, on bringing them 'fo town I
f-
Bags,
^ Addition rf Wbiohts, Gfr. Book L*
C, qrs, lb.
Bags, N» I ----- 2 ft 10
4 23 -
5 21 12
6-----21 16
Pockets -i-^--.--2' 2i
2------aai
Cwt 15 2 27, the anfwer.
■ ■ p"> — — i^
2. In a gentleman's fervice of plate, there are fourteen
difiies, weighing 19302. X3dwt. } plates thirty-fix» weir-
ing 421 oz. iidwt. ; four doa^n of fpoons, weighing
104 oz. 6 dwt. ; fix falts, chafed, weighing 32 oz. $ knives
and forks, weighine 83 oz. 9 dwt. ; fourteen prefenters,
weigh]ngii3oz. 4dwt. ; in mugs, tumblers, beakers, and
other odd pieces, weighing 264 oz. t8 dwt. ; filver tea-kettle
and lamp, weighing 126 oz. 9 dwt. ^ and the reft of that
4|bipage 93 oz. 2 dwt. What quantity of plate had the
butler under his care ?
oz. dwtl
Fourteen diihes - - - - wt. 1*93 13
Thirty-fix plates - ^ - - wt. 421 ii ^
Four dozen of fpoons - - wt. 104 6
Six falts ------- wt. 22 -
Knives and forks - - - • wt.^ 83 9
Four prefenters ----- wt. 113 4
Mugs, tumblers, beakers, &c. wt. 26a xo
A filver tea-kettle, &c. - - wt. 1 20 9
And the reft of that equipage - wt. 93 2
oz. 1432 t2, the anfwer.
3. The diftance betwixt two places is fuch, that if thiee
miles and five furlongs is taken from itf what remains »
equal to eight miles, K>ur furlongs, and lOO jards i what
is the diftanc^ of thc^e two places i
M. F. Yds.
5 4 100
12 I 100, thcanfwer«
4. la
Ciu^i II. Subtraction of Wbiohts^ 6fr. ^
. 4. In my furvty of the rivex Glen, from Bafton-
iicdges to the Oatml, I found the feveral diftances as fot^
low:
M. F. P.
To Clarke's houfe -----«. 16 4
To the toll-bridge ------- ^ (y ^
To the falling in of Bourn-Eau - - - -. i ^2
To Gutram-cot -------- 2 5 28
To Pinchbeck-bar ---.-«« 2212-
To Money-bridge -----'.. 1- 4.
To Herring-bridge - - - -•.- - 12 \
. To.New-bridgc -.--.«.. — 5- 12 '
To Bondman-bridge - - - - - - . ^ j^.f
To Stone-goat ------... '*4 -
To Surfleet-bridge --^-6« -7 2
To the Halfpenny-toll ••- - *.- i j 28
To the Outfall -----*. .^ 11 32
From Bafton-hedges to the Outfall* - - 14 5 1%
i«*i
5. A father was 18 years four months old (reckoning 13
months to one year, and 28 days to one month) when his
eldeft child was born* Betwixt the eldeft and fecqnd, were
II months, 10 days. Betwixt the fecond and third, were
three years, eight months. When the third is 12 years,
ix months, 20 days, how old is the father ?
Y. M. D.
18 4 •*
- II 10
3 8 -
.12 6 20
Years 35 4 2, the anfwcr.
00000<>000<><>0000<>00<
SECT. Ill,
Subtraction of Weickts, Measures, &?r.
I refer the young fludent to the general rule for fub-
traaion. Chap. I. Seft. III.
1. Having
l4 iStfBTiiAmoN 9fiFf%muTZt&c. BookH
f. Having bought tw« hundred vmgiktj and Cbeeequaucters
of Atga^t Afid Ibid thereof one hundrbdt two quartcis^
14 pounds i what is yet unfold i
cwt. gn lb;
23-
•dm
Cwt. X "- 149 the susfwer;
r
a. A filler was 24 v«ats, nine nmitbS) to ^i^ old^
when Ms eldeft fon was oorn \ and js now 56 years, riiree
moothst and 22 days. How old is fht fioa i
Y. M. D.
56 3 211
24 9 iO
31 7 t2, the ahfwer.
mm
3. %e-eived in lieu of two TOld repeaters, fent to Jaihaici
in 1756* the five chefts of indigo foHowuig ; and on a ijke
adventure in -I758» the fubfeqiicnt £ve <iieflB. The^uef*
tion is, how mnch indigo i had lefs the fecbni time dwui
Cbefirft?
Anno 1756. ,Ahno 1758.
cwt. qr. lb. qr. lb. cwt. qr. lb. qr. lb.
N*i 2 I 16 Tare i 15 N*i 1 3 7 Tare 1 4
222 II I 19 21317 14
32- 12 I 13 312 10 12
4 2 - 19 I 14 4 1 - 13 - 27
5 2 3 17 I 21 , 5 2 - n cwt. I 6
Grofsi2 - 19 X 3 26 GrofsB 22 i i 15
— I 3 26 Tare - i i 15 Tare
Net r° ~ " ^f A lvent«r.. 7 " ^5
(7-15 fecond J ' I ■? ■ . ■ ■» *
3-6 diiFccence,
4. Jacob, by contra^^, was to ferre Lad>an for Kis two
daughters 14 years ; and when he had accomplifhed 11 years,
II n^onths^ 11 weeks, ix days? ^ houn, and 11 minutes;
pray how long had he yet to feiye i
II Y,
Ch«p. II. REDUCTION; €5
Y. M. W. D. H. M.
Y. M. W. D. H. M. 14 . - -
XI II II II II II =: 12 2 - 4. II It
«i
Anfwcr i 9 3 2" 12 49
5. When the air prefleth with its full weight in very fair
weather, it may be demonftrated, that there prefs upon a
human body about 307 cwt. 2 qrs. I5 lb. of that fluid mat-
ter ; and in very foul weather, when the air is moil light,
273 cwt. I qr. 20 lb. What difference of weight lies on
fuch a body, in the two greateft alterations of the wea-
ther ?
cwt«qrs. lb*
302 2 25
273 I 20
Cwt. 29 I $i the anfwer.
S E C T. IV.
REDUCTION.
REDUCTION alters or changes any fuperior de-
, nomination propofed, into any inferior or leifer deno-
mination required ; ftill keeping them equivalent in value.'
And the contrary.
RULE.
Confider how many units of the denomination required,
make one of that denomination propofed to be reduced
(which is eafiW. known by, its refpe£kivc table) and with
that number or units multiply or divide the denomination
propofed, and their produ£l, or quotient, wQl be the number
required.
r
f J. Re-
€6 REbUCTIOK. feoakf.
I, Reduce 753 1. into pence.
16
15060 fhilling9w
180720 pence.
r
Of) 753 1, may be reduced into pence at one .operation*
thus 240
3012
J 506
180720 =r pence In 753 L as before*
But when the numbers propofed to be reduced are of
feveral denommationsy and it is required to bring them
^l to the loweft) vou muft reduce the higheft or greateft
denomination to tne next lefs, adding the numbers that are
of that lefs denomination thereto ; ,then reduce (heir fum
to the next lower denomination^ adding thereto all the
numbers that are of that denomination, and fo proceed
gradually on until all be liniflied.
2. Reduce 3751. tj 6. io|d. into farthings^
L 8. d.
375 17 19J * /
20
^517 = the fluUings in 375 1. tj $•
12
90214 ss the pence i^ 375 1. 17 $« xo d.
4
360859 s farthings = 375 1. 17 $• io| d<
3. In 384627 farthings, how many pounds fterluig i
4
12
20
3846317
96 1 56 j: pence*
8013 Ihillings oj d.
T^oo 13 old.
Note,
Chap. II. reduction; «^
Not9, The rcmaiiidwr i« always of the fame denomination
-with the dividend. '
4. In 648385 grains, how many pennyweights, ounces,
and pounds ?
10)
24 ) 648385 ) 27016 pennyweights.
168 ^ . ■ .*
38 12) 1350 oz. i6pvm. 1 grain.
'45
112 lb. 60Z. i6pwts. I grain*
igr.
5. Iniycwt. 3qrs. 141b., how many ounces ?
4 cwt. qr.lh. ,
— • 17 3 14
71 quarters. 1 12 84 + 14 — 98.
28 I
— - 2002 lb»
568 16
142
2002 pounds.
16
320329 anfWer, as before^
32032 ounces*
i^^fc-**
6. Reduce 93 tuns, 15 cwt. 2 qfs. I2 lb. 13 oz. 14 dr^
into drams.
tuns cwt. qrs. lb. oz. dr.
93 15 2 12 13 14
20
1875 hundreds.
22568 1875 X too rs 187500
210068 lb.
16
1875 X too rs 187500 )
1875 X 12= 22500 >fum4<po6Jf
56+12= 68)
336jior ounces.
16
53777630 drams.
F 2 7- ^^
6t REDUCTION. Bookl.
7. In 17 tanksirdsy each weighing 27 oz. 14 dwt. 16 gr.
how many grains i
oz. dwt. gr.
27 14 16
20
554 pcwiy-wcights.
2231
1 108
13312 grains in one taxikard.
17
226304 grains, the anfwer.
8. In 53777630 dram's, how many tuns ?
drams. i6)ozs» 28 ) lb. 4)
•6 ) 5S777630 ( 3361 loi ) 210068
57* 16 140
97 no 68 20
17 141 -
16 12
30 13
— ^ Tuns cwt. qrs. lb. oz, dr.
14 Anfwer 93 15 2 12 13 14.
750^
1875 cwt. 2 qrs»
93 tun*. IS cwt.
9* The filk mill at Derby contains 26586 wheels, and
97746 movements, which wind oflF, or throw 73726 yards
of wk every time the great water-wheel, which giv^s
motion to all the reft, goes about, which is three times in a
minute. The queftion b, how many yards of filk msty be
thrown by this machine in a day, reckoning ten hours a
day's wore ; and how many in the compafs of a year, de-
dttcHng for Sundays and great holidays, 63 days, provided
Qo part of it ftand ftill ?
737^6
Ciiap.II. R E D U C t I ON. %
73726 yards at i circumvolution of the wheel.
221 178 yards in a minute.
X6o
13270680 yards in an hour.
Xio
132706800 yards in a day.
302=365—63
2654136
40077453600 yards, dieanfwer*
10. In 4712 nails of Holland, how many yards, ella
Engliih, and eUs Flemilb ?
4
4
47"
I J 78 quartersi
294^ yards.
1 178
235 ells Englifb ^ yaids«
1178
392 ells Flemiih i yard«
Sometimes multiplication and divifion are both required
to anfwer the queftion 5 as in the following.
II. In 491 barrels of beer (London meafure) how many
hogibeads, gallons, and pints ?
491 barrels.
3 ) 982 ( 327 hogiheads, and i barrel ss 36 gallons.
54
17694 gallons.
1308 X 8
1635
17694 gallons.
141552 pmts.
F3
iirl
7* REDUCTION. Aook V
12* I defire to Juiaw how many days, hours, minutes,
and feconds, fince our Saviour's nativity, it beiivg accounted
J 758 years ?
31556937 feconds in a year.
1758 years .
252455496
157784685
220898559
31556937
601 55477095246 feconds'm X7518 years.
6o| 924618254 minutes 6".
1 5410304 hours 14.' b".
\Q\ 1642096 days 14' fi" = If 58 jFMn. •
50
230
'44
o '" ■ .
1 3. In how long time would a millton bf millTon of
money be in counting, fuppofing 100 1. to be counted every
minute, without intermiffiom and the year to confift'of
365 days« five hours^ 45 minutes ?
100 ) 1000000006000 s= loooooooooo minutes.
Years days h. m*
in I year minutes 525945 ) xoooooooooo ( 1901 7 144 ^ 55
4740550 Uie anlwer required.
704500
1785550
1440 ) 207715 Rem. 100 pounds.
Sii
60 ) 355
55
A GiQgrApbical Qubstiok.
14. There Is a city in a certain ifland 708 miles more
diftant from the tropic of Cancer, than another under the
fame meridian is from the artic polar circle. What cities
are thofe, what are the diflances 4>{ thofe cities ftom the
equator, and what from each other ; remembering die polar
circle n about 23I degrees from ^e pole, as is the tropic
T 2 from
.Chap. 11. REDUCTION. 71
from the equator ; and in this pleafe to coniider 60 geogra-
phical miles as a degree ?
Tropic of Cancer 23° .30' from the equator.
21 30 half the temperate ^Kone.
60 } 708 (11 48t 2 = 5 54 h^f 708 miles.
50 54 latitude of the firft city,
23* 30' polar circle from the pole.
21 30 half the temperate zone as* before.
45 —
— 5 54 the half of 708 miles.
90—39 6 =: 50 54 latitude of the fecond city.
• Confequently both ftand on the fame fpot> and anfwer to
Chichefter in ouflex, Tongeren in Germany, and Upres in
Flanders.
15. I would put 60 hogflieads of London beer into 30
wine-pipes, and would know what the calk muft hold that
receives the difference ; 231 folid inches being the gallon of
wine, and 282 that of beer,
60 X 54 =: 3240 gal. of beer. 30 X 1 26 = 3780 gal. wine meaif.
2S2 231
■ V ■ I ■
648 378
259a "34
648 756
913680 inches of beer. 873180 f
913680 — 873180 %s 40500 inches diiJNerencc*
282 ) 40500 (143 gallons, two quarts, and almoft a pintir
Remains 33 inches, 357th being a pint.
Chronological Qv e s t i o n s.
16. England was conquered by William I. Od. 4, 1066;
his fon William II. came to the crown, Sept. ^, 1087 ;
and left it Aufi;. 2, iioo. William III. received it Feb. 3,
1689; and died March 6, 170 1. How manv days did
each of thefe princes govern, refpedi being had to the
intercalary days (added to February ever leap-year) as they
rofe in the courfe of time ?
Note, JBvery fourth year is leap-year, or biflextile ; to
find which are fuch, divide the year of our Lord by 4, and
when nothing remains, thofe are the l^p-years, and to
fuch you add one day more than 365.
F 4 1066
72 REDUCTION. Book L
—— re^nalns 2) fo that 1068 was leap-year.
And in the reign of William I. were five intercalary days.
Between September 9, and Odober 4, are 23 days.
1087 — 1066 =1 21 years ail but 25 days, William I* reigned.
365 X 21 = 7665 days.
Therefore 7665 -f- 5 — 25 = 1645 days, William I. reigned.
-^ remains 3, therefore 1088 Was leap-year.
And in the reign of William Rufus 4 intercalary days.
Betwixt Aug. 2, and Sept. 9, are 38 days.
HOC-— 1087=13 years all but 38 days, William II. reigned
365x^13 =4745 days.
Therefore 4745 -f- 4 *— 38 = 47 1 1 days, William II. reigned.
remains i, fo that 1692 was leap-year.
' And in the reign of William III. 3 intercalary days.
From Feb. 3, to March 8, are 32 days.
J731 — 1689= 12 years and 33 days, William III. reigned.
365 X 12 = 4380 days.
Therefore 4380+ 33 + 3 = 4416 days, William III. reigned.
17. Richard I. fucceeded his father Henry II. July 7,
1189; John, his brother, fucceeded him April 6, 1199;
Richard II. fucceeded Edward III. on the 21ft of June, 13779
and was depofed.by Henry IV. on the 20th of Sept. 1399.
The third Richard caufed his nephew, Edward V. and his
brother, to be murdered on the i8th of June, 1483 $ and
was flain himfelf on the 22d of Aug. 1585 ; how many days
was the nealm governed by the three Richards, refped be-
ing ftill had to the intercalary day; as they haj^encd f
— - remains i, fo tha,t 1192 w^s leap-year.
4
And in the reign of Richard L' were 2 intercalary days.
Betwixt April 6, and July 7, are 92 days.
1199-^11 89 =10 years all but 92, day«, Richard I. reigned*
365 X 10 = 3650 days.
And 3650 4* 2 — 92 = 3560 days, Richard I reigned.
-^ remains r, fo that* 1380 was leap-year.
And in Richard II^s reign were 5 intercalary days.
From June 21, till September 30, are 101 days.
1399 «-« 1377 ;;= 22 years, loi days, Richard II. reigned.
365 X ?2 s:: 8030 days.
^ ^ ^ And
thap. II. REDUCTION. 73
And 8030 4-ioi-f-5 = 8i3& ^^7^9 Richard II. reigned.
^ISl. remains 3, fo that 1484 was leap-yean
And in Richard IIPs reign was i intercalary day.
From June 18, till Auguft22, are 65 days.
1485 — 1483 = 2 years, 65 days, Richard ^III. reigned
36s X 2 = 730 days.
And 730 4- 65 + I =^ 796 days, Richard HI. reigned.
Laftly , 3560 + 8136+796=11 2492 days, the ajii wer.
18. The Bitk queen Mary came to the crown, Julir8>
1553 » ^^ reigned five years, four months, and nine days,
ller fifter Elizabeth fucceeded ; and James I. came to her
throne the 14th of March, 1602 ; he left it to his fon
Charles L on the 27th of March, 1625, who was forced
from it Jan. 3O9 1648. The queftion is, how many days
did thofe princes reign ; and at the death of Charles 1. how
long had England been under an uninterrupted fucceffion ^
Protdiant princes (Mary I. being the laft profefled Papift
that enjoyed the crownl not negle£ting the intercalary days
in February, as before r
•ill remains i^ fo that 1556 was leap-year.
4 * *
Therefore from T553, till 1 602, were 12 intercalary days^
and 01^ one leap-year in queen Mary's reign ; fo that ia
queen EJizabeth's reign were 11 intercalary days.
Froqi July 89 to Nov. 17, being four months and nine
days, are 132 days. Then 365 x 5 = 1825 days.
And 1825 + 13^ + ^ = 1958 days, Mary reigned.
1602 — 1553 = ^9 years.
Betwixt March r4, and July 8, are 115 days.
Theii 365 — 115 = 250.* Alfo 365 x 49 = 17885 d^ys.
And 17885 4- 2^0 -J- 12 =: 18147 days to the oegirining of
James s reign.
Then 18147 — '958 = 16189 days, Elizabeth reigned.
— remains 2, fo that 1604 was leap-year.
Confequently from 1602 till 1625, were fix intercalarv days.
J625 — 1602 = 23 '9 but as the date altered at our Laay-day,
the interval was no more than 22 years, and 13 days.
Alfo 365 X 22 =? 803Q days.
Therefore 8030 -f- 134-6^= 8049 days, James I. reigned.
•-ircmj^ins i, fo that 1628 was leap-year,
* . There-
^4 RE jyVC T 1 1) ^. Book L
Therefore from 1625, till 16481 were A;c inteccalary day««.
1748 — 1625 = 2? whole years, ChaNes reigned.
From March 279 ttil Jan.. 30, are 309 days.
Alfo 365 X 23 = 8395 days.
Therefore 8395 + 309 + 6 =: 8710 days, Charles I. reigned^
Then 16189 + 8049 4. 8710= 32948 days, the anfwer.
19. A grant was made Dec. 14, in the loth of Henry h
who tegan his reign Aug. 2, 1 100 ; it was refumed Nov, 19^
in the 4th of Henry IIL viho came to the crown 06t» 19,
I2i6 ; it WAS revived the 16th day of Juiy^ in the 13th of
Henry VII. who afcended the- throne Aug. a2> 1485 : but
It was a fecond time revoked^ and finally fiipprefled in the
i6th pf his fucceiTojr Henry VIII. on the loiii of May.
tiow as this man's fadier died July 21, 15099 the queftioA
isy ihow (many days was this gran^ in force,, and how manj
4id it lie dormant t
Hdnry I. began his roign Aug. 2, izoo.
Thea iioo + 9 = ^109, when the grant began, Dec. 14.
Henry. III. began OSt. 19, 1216 + 3 ^^ ^219, No¥. 19.
The firft continuance of this grant no years :waatijiig
26 days ; and in that period are 27 intercalary days,
• . • 305 X 1 10 =: 40150, and 40150 -f- 27 -^ a6 = 40151
days, firft continuance.
Henry VII. began his reign Aug. 2i2, 1485-
1485 4- 1 3 = 1498, July i6, ijrant rcaffumed.
Henry Vll. died, and Henry Vlu. fuccc^d, July 21, 1509.
1509 4" 15 = 1524, May 20, ^rant ended'.
1524 — 1497 = 27 years, wanting 57 days. ';
And in thole 27 years, are 7 intercalary days.
365 X 27 = 9855 +7 — 57 = 9805 days, lall in force.
Then 40151 + 9805 = 49956 days, the grant was in force.
Q.E. F.
Again, Henry III. began his reign, OA. 19, I2i6.
1216 -j- 3 = 1219, Nov. 19, grant reaflfumed.
Henry VH. began his reign, Aug. 22, 1485.
1485 + 13 = J498» JwJy 16.
1498 ■«— I2ig =: 279 years, bating 126 days.
In which period are 69 intiercalary days.
••• 279 X 365 = 10183^, and 101835 -|- 69 r— 126
• .=!= 101778 days, iuperfeded. Q^ E. F.
N. B. This queftion was taken from Clare's Introduftion
to Trade, &c. who makes the time of the continuance of
the grant nine days lefs than found by the folution above ;
fo would advife the }*oung accomptant to try which is in
tlie right, C H A ^P.
Cha,f. in. PRACTICE.
^5
f.
CHAPTER III.
RULES e/" PRACTICE.
TH £ rules of Pra^Eice* from .their great and frequent
uk, derive, thfir o^me, and ore contrived ifwedily
and compendiouQy* to caft i^p any fort of goods or mecf
chandize.
SECT, J,,^
PRACTJCE i, UULTlFhlCATlQH.
C A S £ I.
To multiply hf a lolxed number; that is, a fradioa
joiodi wiA ft fvtele twwber.:
RULE.
Whco you have muUipliod by the whole number^ take
h h h h h ^^ Whatever ipart it may be of the multi-
jdioaoii, whicbf added tp the produd, will give the true
vifwer.
I. In 57 fodder of lead,
each i9J-cwt.howmanycwt.?
1083 i
I
ii^i^tmi^. in zXiw
t^m
3. In 27 hogfheads of fugar,
each containing 7^.cwt. bow
many hundred weight I
Ji
189
J3|=f
ao9i
i
2. In 359 pieces of Nor-
wich ftutf^i eacii ^^i yards,
how many yards I
359
1077
7r8
8346^ yards.
4. In 354 pieces of Kerfey,
each 27|. yards, how many
yards ?
. 3S4
2478
. T"* , ....
44i: = l
9867^
CASE
7«
PRACTICE.
Bookli
C A S E IL .
To caft up any number of things, not exceeding 12, zi
any given price*
RULE.
Multiply the price by the quantity or number of things^
always obierving to carry from one denomination to another,
as in addition of money*
r. Whatcoftfevenfton^of, 2. What coft 9 cwt. of
.beef, at 2 8. 7 d. per ftene ?
s. d»
27
7
18 I, the anfwer.'
3. What coft £ve fiieep, at
1 1. 17s. 6d. each?
1. s. d.
I 17 6
S
£976
5. What coft II geefe, at
IS. 7^d. each?
s. d.
II
18 li
treacle, at 1 1. 17 s. 4^d* per
cwt?^
*
L s. ;d«
I 17 44percwt«{
16 ifr ij,
the anfwer.
4. What coft 10 yards of
broad-clothj at 17 s. loidii
«
s« d.
17 loi
10
^818 9
6. What coft 12 cwt of
fugar,at3l. 17s. 7i<L?
1. 8. d#
3 17 74?
12
X f6 II 9
Note, If the given quantity is 13, ntultiply the price by
J 2 ; and as you multiply, add to it the price or one, and the
refult will be the anfwer.
1. s. d.
What coft 13 cwt* at 4 13 7^?
13
£ 60 17 4|.
Thua
PRACTICE.
7?
Chap. HI.
Thus performed, 3 X.12 = 36 -|- 3 = 39 farthings,
3 and carry 9*
Then 7X12 = 84 + 9 4- 7 = rood. - 4 and carry S*
Alfo 3Xra=36 4-^4-3 = 47«* - 7 and carry 4.
And I X 12 = 12 + 4 + I = 17 angels, i and carry 8.
LaftIy4X J2=:48 4-8 4-4= ^-
CASE iii;
When the quantity exceeds twelve.
RULE.
Find two numl>ers in the multiplication table, which
being multiplied together, will make the quantity $ then
multiply the price by o/ie of thofe numbers (it matters not
which you multiply firft by) and that produd by the other
number, and the laft produ£l will be the anfwer.
2f What coft 27 ounces of
filver^ at 58. g^ i. per oz« I
.3
I. What coft 15 cwt. of
Ueade, at il. 78. gd. per cwt. ?
3
4 3 3
5
£ao 16 3
3- What coft 56 lb. of Hy.
ba tea^at 15s. 9^d. per lb. i
8
664
_7
£44 4 4
5. What coft 108 lb. of
autmegs, at 12 s. 3^d. per lb.?
12
7 7 9
9
iC66 9 9
«7 ti
9
jC7»6iii
4. What coft 77 cwt. of
madder, at 3I. 15s. 6d« p. cwt.?
7
26 8 6
II
^29013 6
.6. What coft 132 gallons of
wine, at 58. 4d. per gallon i
12
3 4-
II
lis 4 -
m^m
To
j8 PRACTICE. Book L
To find the price of a grofs ; firft find the price of %
dozen, by multiplying by la; which produd multiplied
idfo by iZfi gives the price of a grofs*
• What coft fevtn gf oft of buckles^ at I s. 1 1{: d. per pair ?
»
8. d«
I
i»
«
' * 3
3 price of
12
I dozen pair*
'3 19
97 13
^ price of a grofs.
^f theanfwen
■
And to find the price of i cwt. at fo much per pounds
multiply by four ; which produd muitipled by feven, gives
the price of a quarters theit four times the laft prodttft
will be the answer.
8. d.
What coft jcwt. of tea, at 4 5{: per lb. ?
4
m
17 9
7
643 prioe of a qvutex^
4
24 17 > pries of a art.
3
j^74 II -, theanfwen
CASE IV.
When the quantity is a prime number^ viz. fuch an one
as no two numbers in the multiplication table can be found
to anfwer it,
RULE,
Multiply by fuch numbers as come neareft the quantity ;
and for what is wanting, multiply the price by that number^
and
^hap. III. P R A C T I C E. 79
and add to it tht other produft, and the total will be the
anfwer* '
. i» What coft 29 lb. of Bo-
hea tea^ at &8. gd. per lb, i
s. d.
6 9
4
170
7
9 9-
6 9
iC 9 »5 9
3. What coft 68 yards of
Holland, at 5 s. 4-^ d. per yard ?
6 •
1. What coft 38 lb. of loh
ther, at ii^d. perpoand f ,
d.
[I-
4
"i
3 "
9
1 IS 3
I 1 if = price X z.
;C I 17 2
^^mm^^a
X 12 2
II
17 14 9
* 10 • 9 ^ price X ^.
iCi8 5 6
5. What coft 1 17 fheep, at
iL 7$. 6d. f
J. s. d.
176
to
MtaH««itf
13 IS -
II
151 s -
9 12 6s=pricex7-
/ 160 17 6
4* What coft 76 quarter of
wheat, at il. 17s. qd. perqr. i
1. s. d.
« 17 9
12
,a.3 -
'3S '» - . .
7 II - = pncex4.
jC »43 9 - . .
III! aw^
6« What coft 135 yardaof
broad-doth, at 17 0. y^i*!
•• d.
17 71
ta
10 XX 9
II
116 9 3
2 12 11^
119 a 2^
= pricf X 3.
To
8o PRACTICE. Book I.
To find the amount per annum of officers, ialaries at fo
much per diem, multiply the falary or wages per day by io»
and that produ£t by 9: this laft produ6l multiplied br
4 gives the falary for 360 dap ; to this add oner day^
falaiy multiplied by 5> which give the anfwer.
If an oflqer's falary be 17 s. io| d« a day,, what is diat a
year? *
s. d.
1740J '
10
8 18 lit
9
80 10 ft
4
322 2 6
4 9 5^
^
j£ 326 II ii|, d)eanfw«r<
C A SE V.
When the quantity h r^ Xj 3> 49 5f 6^ 7» or more
hundreds. *" • i •
RULE.
Multiply the price by 10, and that produA by 10 alfo,
which gives the value of one hundred ; then multiply the
produft by the number of hundreds ; then multiply that
produd, which gave the price of 10, bv 2^ 3, 4, or 5, as
the tens happen^ which place under the laft prddud without
drawing a line ; and for the units always multiply the price
by them, and fet that down under the former produds ; fo
you will have throe lines, the fum of which will be the
anfwer.
J. What
Chap. Ill, P R A C T I C E;
I. What coft 795 yards of
brocade, at il. 7 s. lojd. per
yard?
1. 8. A. .
i ^ to%
10
"6^
13 18
10
'39 5
S
7
974 17 "
125 6 loj^ .
_^ 19 3^
£"oy
4 H
2. What coft 168 lamb-
ho^, at 17a. qd, each?
s. d.
17 9
10
8»7
6
10
88
15 -
3
266
53
7
^m
£2^(>
5 -
5 -
2 -
12 -
CASE VI.
To multiply weights and meafures.
RULE.
Place the multiplier under the loweft denomination of the
multiplicand •, then multiply the loweft denomination by the
multiplier, and divide the produa by as many of the loweft
denomination as make one of the next fuperiorj fetting
down underneath ^he remainder, if any, and carry thequotient
to the produa of the multiplier and the next fuperior deno-
mination, and fo proceed till all is finifhed.
I. In n pieces of Iter fey,
each 17 yards, three quarers,
three nails, how many yards
in all ?
Y. qrs. n.
»7 3 3
II
i97_i_£> anfwer.
3. In 38 pieces of tapeftry, .
each 37 ells Flemifh, 2 qrs. 3
nails, how many ells Flemi/h T
E. F. qrs. n.
37 2 3
6
2. In 4a pieces of Holland^
each 27 ells Englifh, two
quarters, three nails, how
many ells in all ?
£1. £• qrs. ji.
27 a 3
6
165 I
2
7
227
2
6
1365 - -
75 2 2 = 1 piec« X 2.
1*40 2 Zy anfwer, .
1157 - a
As the next fupericr deno*
mination contains only 3, 4,^
or c of the inferior one?,
questions of this kind may be
perfornied without divifion*
G 4. Whac
82
PRACTICE.
Book L
4. What is the weight of
feven tankafds, eich weighiag
iioz. i6dwt. 2fgr$*f
oz. dks. gr,
IX 16 2t
82 18 %
5. What IS Ac net weight
of 39 hogiheads of fugar, each
weighing 7 cwt. 3qrs. 17 IbJ
Cwt. qn lb.
As 20 pennyweights make
one ounce, we carrj^ as in
multiplication of fhiUin^s ;
and if pounds troy be uied^
we carry the fame as in pence.
7 3 ^7
6
47 I 18
6
284 X 24
23 2 23
2S)X02(3
iS)ioS(3
«S)5i(i
»3
308 - 19, anfwer.
6- What is the weight 6f
29 parcels of tea, each 25 lb,
70Z. 13 drams I
lb. oz. dr» ^.
25 7 '3 **^!1^'
Cwt. ■ ' ■ ■■ ^,x ,
I 2x0 6x1. '•^fff^
4 «
jiS}i7S(6
10
MM
7, What is the weight of
37 fmall parcels of tea, each
weighing x 3 ounces x 2 drams ?
oz* dn
13 12
X2
10 5 -
I a 15 -
I 3 12 12
16)144(9
6 X 13 10 X2
25 7 13
6 2 If 2 9
IS*
All the needful divifions are
here put down,
S.WhatistheweightofiOj
ingots of filver^ each wei^h-'
ing 21 oz. i7dwt. i9gi'. r
oz. dwt. gr.
21 17 19
xo
218
>7
22
10
2188
»9
4
109
8
^3
Ozs. 2298 8 3
H) 190(7 s4)**o(^
" »4)95(f
as
QtfCs-
^,
fchip* HI. ^ R A C T I C R 83
QjEiTio^s to exercife the foregding RuL£$r
I. A pcrfon dyin| left hi« widow 1780I. and 1250 1. to
each of his fotir children ; 36 guineas apiecb to 15 of his
poor relations^ and 150 !• to charities: he 'had been 25^
vears in trade, and at an avera|;e had cleared 126 1* a year i
What had he to begih with \
Tb wtdow -- • -^ 1780 -
/ 1256 X 4 children - - - 5000 -
^31 10 X 15 poor relatiop* - 472 10
^ To charity - - 150 -
. Wohh - * -jf7402 10
£ 126 cleared yearly x ajli - ^213 -
j[^ 4189 10, anfwer.
2. Suf^pofiilg that foi^ a quarter's rent I paid in moitejr
^I. ^s. 6d. and was allowed for a fmall repair 18 s. 9d«
and for the king's tax 8 s. 9 d. What did my tenement go
at a year ?
jf 7 7 6 quarterns rent.
ii 9 repairs.
89 tax.
fclB I ■■! ■ ■ mm .
«i5 -
4
III Ml *
§5 - s the anfwer.
3. At Leicefter, and other places, they weigh their coals
i)y a tnachine, in the nature of a fteelyard, waggon and
all: three of thefe draughts together amount to 117 cwt.
2qrs. 10 lb. J and the tare of the waggon was I3-Jcwt4
How many coals had the cufioiiier to pay for ?
• Cwt. qr, lb.
^ Cwt«lqr. 117 2 io
•^3 I X i = 39 3 -
*)—*■
77 3 10, the ahfwcn
4* /bigentleman hath 536 1. per annum, and his expendci
are, oA day with another, z8s. io|^d. 1j deflre to kooW
how much h^ layeth up at the year's end )
G 7^ x8$.
^•-
U PRACTICE. Book I,
s. i.
i8 loi
4
3 is
7
»3
49 a
7
7
343 i8
i8
I
lOf
3H «6
"l»
4x13x7 = 364- ,
Ycar*« rent - S36 - -
Expended - - 344 16 ix^
Laid up - - <- 191 3 -^9 the anfwer.
5. A ^ntleman expendeth daily iL 7 s. lOfd. and at
the vear^ end layeth up 340 1. I demand how much is his
yearly income ?
1. 8. d*
I 7 lot expences per day,
J. s. d.
Year's expences 508 14 4t
To lay up • - 340 - -
Anfwer X 848 14 4i
5
IJ 6
X13
72
9 6
X7
507
I
6 6
7 >oi
508
X4 4i.
^
Her
I *
Chap. III. P R A C T I C E?
-3 1 I I I I O •4> I I so
t
• oe I I o 1 <^o» coo >o
N
OO
8*
86 PRACTICE.
•J I vO 00 V> ^00 ^ I **^ I
^ I
5 " ^ I
Book I
«>
I
£
ft
G
a
o
b
I
I
I
I
•3
JO
o
a«
I t •
I
t
I
I
I
I
I
I
I
*
I
I
X
09 H
9Q«
go
M
S2
o
w
•
13
d •*
00
r**
oo
•
nX
X
00
*
fOO O M M t| -^
•^ N^ M M - •
e* ^ M M *^ CO w
4J 4^ 4J 4J «j 4^ -M
ft CS Ct (« C< C« Of
1 » 1 I I I I
S2
»
w
1 1 1 1 1 1 1
X
u
1 1 1 1 1 t 1
<
1
p
quarters
bufhcls
quarters
bufliels
bufliels
bufliek
pounds
V> O^ t^ C>^ 00 t>»
9m 9m t\ *m
•2
■fi
1 1 1 1 t 1 1
V
St
>
1 1' f T 1 1 1 '
«>
(4
t 1 1 1 t 1 1
The
■ 4a ■
1 1 I 1 1 '^ 1
«1 M .. B
Oats
Bean
Bran
Tare
Peafe
Pale
Hops
-rt •^ ^
CO
O
« — , ■■
H • M '(^
< -O HI
CO
CO
M
o
«x
00 '^
00 0^
*o
o *o
M M CO
X ^
^
t»g«^l
00 t^
00
/
^x
c<
s?
X
10
» '^
CO
1^4 2.-
00
X
I
o
SI
Chap. III.
PRACTICE.
o
s
o
U
s
M
<
p
§
hi
2i.
o
H
a
o
t>
o
M
O
%
9
P4
o
I I I 1 I
•"• -^ « C< M IH
c«
p-l
e
H ex.
4
I
f
I
l-t
t
CO
'*^ M M M f^
tT W 4^ <W 4J 4^
«} C« C« C« 01 <V
O I I I I I
M ^
■ I I
^ 8 ' • • '
«.S2 I • I I
to I
«2 < »
-SIS ' e •
OS fA I V
bD V ^ to.
*• r H u ^
N
^ »
1
1
M
1
SI
•d >
1
1
x^
SI
mm
1
1
>0 tv
fC
p^
M
4 «*lX
^
SI
^ H
o
04
I
s?
1
*
i
I
p4 t^r*
X
9
«7
u
u
3
Q
PRACTICE,
•^ c«>ooo I 1 t
Bookt.
" Ooo Ooo "^r*
m
c
&
E
o
U
•o
c
c«
CO
o
a
<
<
•J
O
h
u
«>
Q
O
04
O nOOO I ) 00
(A
CO «^ ^00 00 \0
• HI M M
"m I C<>< HI I
*J *i *i « 4J 4rf
c< c« c« t< C« fl
I I • I I I
I I I t I I
II III
8
•b 1 8 I i I
tu '&
I
I
I
t
§2 s
S e C
€ E o
""^K
c/a
o
3
•- c 3
•73 .-;. Q<
S c 5
Jus-
r^ o^ vo 00 o^
m O
o
o
o
6
;! ^
S2
Vi
TO I i
CO
00 cn
I
r
O VI
o
•n
Nl
xsX
M
M
s?
•d 1
I
• «o
I
-4 ''X
2"
s?
TJ 1
I
I '
^ IN X
I «
1
s?
^00 «o
^^->o
OOO
00
.v,X
00 X
o y>
^ ^
M
00 d
so
o
s?
^sOJO
VO
«:?x
v>
00
s?
CO- *^^
00 00
,^x
'^hO
M
H fO
d
^ •-• *
d
o»w
O
d
Chj^p,
«^
in. FRAP
• o^cnioci o o in
m M M « »4
r^ r» « n p^ M
TICE,
<
s
o
•4
%
o
U
o
<
a
e
o
c
o
«C3
I
H
o.
H
o
9
o
IS
6
s
I
t
I
1
c
vS2
o
•s
l-l
o
«
^ I
I I
I I
I I
I I
u ^ u
1
•a
•* Kl **
tS «2 ti w ^ <k* ^^
C« ft Cf c« (tt c« (<
' » I I I
» I I I ,
* I I I I
=* ^ c«
d M CI IN o^^^
I I I I I i4 I
lilies
5 J
' • • • sS-
00 O."^ C<00
IP
- 1 X
00
00
^sO ON
i
«X
»o
•
N IN
M
M
Ss^
3 00 c^
1 i
«o o
I
o
S2
r^ I
I
O OS
^ O M I 00 ON
^
00 r^
00
X
coX
»o
-
; fo
•
r^
•
«0
S'x
CO
M
o'
s?
S2
-6 ^^
i>*
« ^^
o^'
; ^
ON
8^
SECT.*
90
? R AX TIC E,
BookL
S E C T. n.
PRACTICE ij DIVISION.
C A S E L
HA V I N G the price of any number pf things iK>t
exce^in^ 12, to find the price of one.
RULE.
Pbce the divifor under the hi^heft denomination (9^ to
the left-hand) of the dividend. Then divide the higjbeft
denomination of the divide^id by the diyifoi-) and bring the
remainder^ if any, in your mind only, into the next in*
feriof denomination, Ridding thereto what is in the dividend |
divide this mimber s^ above, aiid Co proceed till. the whole
is finilhed.
I. If 3 yards of velvet coft
5I. 3S* ii^d. what poft 1
yard ?
!. sv d.
5 3 "J
-5- 3
£ I 1+ 7h anfwer.
3. If 7 cwt. of le^ coft
61. 5 !• 1O7 d. what coft
I cwt. ?
1. s. d.
6 5 10^
-^^
£-17 il|r, anfwer.
5. 1 1 eeefe coft 18 s. i-J-d.
what coft one i
s. d.
18 ij^
r 7|, anfwer.
2. If 4 yards of filk eoi^
3K lis. what coft lyard?
I. s. d.
^-17 9, anfwer.
4. If Q elk of linen coft
1 1. 8 s. 8^d. what coft X ell. I
1. s. d.
t 8
wp*«
^ - 3 at, anfwer.
^•"•"♦i
6. If I2cwt. of loaf-fug^r
coft 46 1. 12 s* 9d. what 90ft
one ?
L 8. • d.
46 12 9
-7- 12
£ 3 ^7 8 J, anfwer.
CASE
Chap. in. PRACTICE-
^f
CASE IL
IVhen the divifor exceeds 12.
RULE.
Find two or more numbers whofe produfl is equal to the
given quantity ; then divide the price by one of them, and
that quotient by the other ; the laft quotient will be the
aciwer.
I. If 21 fat l)pifers coft'
264.1. IIS. 3|d« yirbat coft
one?
1. s. d.
-^ 7
23 10 at
r^ 3
^ 7 16 8f, mifwtt.
3* ^^ 35 dollars are wordi
j\. 77s. 6d. what is the
worth of one i
L s* d.
7 17 6
-5-5
I II 6
-^7
£- A 6,tmfveK
5. If 72 flieep coft 82I. ios«
yrhstt coft one^ f
1. s.
82 IP
2. If 33 lb. of butter coft
15 s. o^ d« what coft one
pound r
s, d.
10 6 3
-r 9
t 5i
-^3
- 5^, anfwer.
J^ If <4CWt. of tPSMClb
coft 101 1. 14 $• i^ac per
cwt. ?
L s.
loi 14
-^ 6
16 19
•^ 9
j^ I 17 8, anfwqf.
6. If X 32 lb. of tobacco coft
5 1. 18 s. 3d. what per lb. i
1. s« d.
5 18 3
-^ 12
^•mm^
- 9 lol
IZ
j^ I a II, anfwer.
£" " io|, anfwer*
I ■ I
LIf
s
$% PRACTICE. BookL
J. If there bg ^ remain4er in jjie firft operation, and
none in the laft, fjace It over tbT^ whole divifor for a frac-
tional part ; but if there t^e two remainders, multiply the
laft remainder by the firft divifor, and to the produd add
the firft remainder, which will be the true remainder or
numerator of the fradtion, under which place the whole
divifor fof a denominator.
7. If 28 tod of wool coft
25 1. what coft one tod ?
1. s. d.
25 - w
•^ 7
3 "
4
5t
8. If 77 lb. of tea coft 26 1.
what per pound (
1. 8. d.
26 ^ -
-H 7
«M^
3 14 2^, I farth,
' u ■■
^-17 lOt+^-orffkrth. - 6 8|;44farth.
17X10 + 3 = 73
jAnfwer 6 s. 8^d. + 5|farth.
II. If you have the price of any number of grofs, to find
the price of one, divide the whole price by the number of
grofs, which gives the price of one grofs ; this quotient di-
vided b^ 12 gives the value of x dozen; which laft quo-
tient* divided by 12 alfo, gives the value of one.
o. If 7 grofs pair of buckles
cod <9 1. 17 8. what coft one
L s. 4«
59 17 -
pair
-5-7
8 II - pergrqfs,
*M2
*~ 14 3 P^f dozen.
-r- 12
- I 2^ per pair.
10. If9 grofs pair of gloves
coft ox 1. x6s. what coft one
pair r
1. s. dt .
91 x6 -
•^9
10 4 - per grofs.
-^ 12
- 17 - per dozen.
-^ 12
- I 5 per pair.
III. Alfo having the value of a laft of oats, cole-feed, or
other grain, cuftomary meafure, to find the value of one
buihel ; divide by 3, and that quotient by 7, which gives
the value of a coom, this laft quotient divioed by 4, gives
the value of a bufhel.
II. If
Chap. 111. P R A C
II. If a hft- of cole-feed
coft i6 1. what will one buihel
coft at that rate ?
1. Sm d,
i6
-^ 3
mmt
5 6 8
7
— 15 ^ity percoom.
-5- 4
- 3 9i-»>-
T I C E. 99
12. If a laft of oati coft 5
guineas, what coft one buftel
at that rate ?
1. s. d. .
5 5-
I IJ -
7
- 5 - per coom.
-^ 4
* Wl
j^- I 3, theanfwer*
«M»M«aa^v«i
7X 3 + 3 ==3 H remaindef .
Anf. 3 s. 9t d. + tJ > Of T f*^'
or 3 9^, very near.
IV. Having the price of an hundred Weight, to find the
price of one pound, divide the price of the hundred by 4^
which gives the value of -^ of a cwt. which divide again by 4,
gives the value of yl. ^ this laft quotient divided by 7 gives
the anfwer.
13. If I cwt. of hops coft
5 1. 9s. 8d. what coft lib.
at thit rate ?
1. 8. d.
.598
-r 4
* 7 5
-i- 4
14. Sugar at 3 1. l>s. UK
per cwt. what per Ibf?
1. s. d.
5 TT —
- 6 lo^
•i- 7
- - I J |» the anfwer.
3 17 -
■i- 4
- »9 3.
4
- 4 9i
7
- - Sf.'theanfwer,
V. Having the price of i c. great weight, vix. laolb.
per cwt. to find the price of i ftone, or of i pound, divide
by 8, which gives the price per ftone ; which quotient di-.
Tided by 3, and that quotient again by 5^ will give the
price of i pound.
15. Cheefe
PRACTICE;
9+ .
15; CheeCe at 1 1. 15 s. per
cwt. what per flone, attd per
pound?
1. s. d.
I IS -
8
3
4 4f per fteiie.
I Si
3^ per poun<l.
BcokL
t6. Ditto itil. i;s. per
cwt. what per ftone^ and per
pound i
1. s. a.
15-
8
3
S
3 jf per ^one;
I ^i; per pound.
2| per pound.
VI.. Or by confidering, that as i20 pence make jo s: every
IDS. per cwt. fives i penriyi and every 2 6; 6d. gives i
fartbiilg a poun
So that 1 1. 10 »« gives 3 d;
and 5 s. gives i d. viz. by in^
fpediotl, 3td. per pound.
17. Ditto at i I. ys. 6d.
per cwt. what per pound i
Here 1 1. »ves 2 d. and 7 s.
6d. gives ^d.
Or z|: d. = per pound.
In the t{ucftion abov«^ i I;
gives 2 d. and 5 s. gives -J;d« ss
zj-d, per pound.
18. Ditto at li. lis. 6<J.
per cwt. what pei* pound i
Here iK 10s. gives 3d;
and 2s. 6d. zi-^d.
Or 3^d. per pound*
^.^jLIL Having the price of a ton, to find the pf ice of a hun-
dmf, a>c|uarter» or a pound* divide bv 5 and by 4 ; Which
laft quotient is the price of an hunored 3 then proceed as
per remark 4th.
19. Carriage at 7 L per ton, 20. Carriage at 2 1. 6 s. 8 d.
What per pound t per ton, what per pound ?
I. s; d. I. s. d.
4268
5
4
7 -
■■
^
I is
0
4
' 7
-per cwt
7
- I
9 per qr.
4
- -
:?
^ d. pet lb.
•—
II
8
-
2
4 per cWt
-
*f per qr.
B^B
-
I
>
^ di per lb.
VUL As 252 gallons of wide, ica make a tun, to find thcf
Erice of a bogfliead or gallon, divide the price of the tun
y 4, which gives the price of an hogfhead i then divide
the
diap. 111. P"R a C t tC fi7 95
the price of a h6gQicad by 7, tfid that qtxbtietit diVJd^d'by
9, giTcs the price of a gallon f - .
9, giTes tne pnce or a ga
2 J. Port wi&e at 60 If. F^er
ton, what per gallon f
L s. d.
4|6o -• -per ton. .
23. Madeira wine at 05 U
per ton, what per gallon r
1. t. d.
9
7
15 - - perhogflicad.
I 13 4
4
9
7
- 4 gfpergallon. I
95 -r* - per ton.
1 I hill
^3 ^S *" P««' hog(hcad#
2 12
9f
- 7 6i neariy p. gal.
IX. Having the value of a wcy*(viz. 2561b.) ofchecfc,
&c. to find the value of I lb. or of an hundred weight ; di-
vide the value of a wey by 8, and that quotient by 4, which
gives the value of 8 lb. ; which divided by 89 gives the value
of I lb. ; or the vahie of 8 lb., multiplied by 7, gives the
value of i cwt. ; which multiplied by 2, gives the value of
an hundree weight.
23. Cheefeat2l. 18 s. 8 d.
j^r wey, what per lb. and per
cwt. :
I. $4 d.
8
4
8
2 18 8 per wey.
-74
^ 1 10 price of 8 lb.
- - - 2-I- per lb.
8. d.
Or I 10 price of 8 lb.
X 7
12 10 per t cwt.
X 2
1*^
£ I 58 per cwt.
24. Ditto at 2 1. 8 s. per
wey,. what per lb. and per
cwt. ? M
L' s. d.
81
8
2 8 - per wcy»
■^i>«.«M«i^
- 6 -
- I 6 price of 8 llr.
A
- - 2-J: per lb. .
I. d.
Or I 6 |>rice of 8 Ik.
X 7 ^
«*w^Mi*aft*
xo 6 per ^ cwt*
X 2
i*a«a
^1 I -
X. A ftone of wool or locks, in feveral manufafturing cotf f»-
tfes, is 15 lb. axki x6fu€h ftones^ 8 tods, or 2401b. make one
3 pack ;
9^ i^ R A fc T I C £; iJook h
rack ; thoxfore having the price of a pack of wool, &c. to
mid the price of a tod, ftone, or pounds divide the price of
a pack by 8, which gives the price of a tod, which divided
by 2, giv^ the price per ftone ; vriiieh divided by 3, ahd
that quotient again by 5, gives the price 6f one pound.
25. tVooi at 7L 5 s. fer
pack, what per lb. i
L s. d.
7 5-
- i8 li per to(l4
- 9 -4 per ftone.
- 3
- - 7i P^^ ^^'
Or as 240 1. = 20 s., by in^
fpc&ion, 7 1. 5 s. per pack,
gives 7^ d. per ibi
ft6. Locks at 4 1. 15 s. ^er
pack, what per lb.
1. 8. d.
8U15 -
- II lot P^r ^od.
- 5 1 1^ ?^^ ftonci
- t III
- - 4| per lb.
Or by infpeAion, 4L 15$.
per pack, gives 4|d. per lb.
CASE III.
When the divifor is a prime number^ or hot compoftd of
two or more numbers, -
RULE,
Take the whole divifor, and divide as in divifion of in--
tegers ; multiplying the remainder by that number of the
next inferior denomination which makes one of its fuperior,
adding to the produA what there is in the dividend of the
fame denomination you are then reducing the remainder
to ; divide this fum as above, and fo proceed in this manner
till all be finiOied.
X. If
QapJEO. FRA.CTMCE:
97:
average
I. If 53 fat LincolnfhireJ 2. If 6j^ I. I2 s. 6 d. be
Iheep be, (old for oo 1. i6 1« 4d. equally divided ampngft Z38
what waa each ioid for on an | men, what is ' each mzn*s
(hare ?
1. 8. d, 1. s, d.
»38 ) 67s "^(+ 17 ^'x
123
.. . •• J . ao i
1.
8.
d. !• s. d.
53)99 16 4(1 17 8
46
20
936
406
35
12
4*4
3. At 315 1. 3 s. to^d.per
jtatf , Hrhet per day i
0 •
!• s. d« 1. s. d«
365)315 3 iot{- 17 3t
20
6103
11
98
12
1186
4
36$
t
» _■
n» »
247a
JOOA
226
25X8
»38
4. A priicof9475l. i6s. 8d*.
beind'divided equally amongft
1747 lailort^ what is eaph man*s:
ihape, tftat deducing | for
the captain ?
L s. d.
5)9475 t6 8
1S95 3 4 captain.
1. 9. d. f.
747)7580 13 4(J0 2 112
1 64
TVT
an.
. iio
20
a2i3
719
12
8634
415
4
1660 remainder i66«
H CASS
^ P'R'AGTtCE.. !BoakL
CA'SE IV.
• » ■ * *
To divide weighta ani meafiiMB*
. R u ;. E.
iVirt^btd $ni ^tmaSutet jsti dlrided exa&ly at the (kaie
manner as moneys due regard being only had to the num*
ber of thofe of ax^ iaferior denomination contained in a
fuperior one of thoL&me fpecits.
2« If 20 pieces of cloth
contain 438 yards, 3 <{aartcrs»
What does i piece contain i
Yds, qr,
-438 3
• 4
109 2 3
m ■ 0 » 0
[Yards 21 . 3 3, anAirer, :
. 4. If 322 cwt.. 2 qfs« s IW
is die Tiireight of 2 j[ hc^;(haidsy
igTfaat is the weight of i hog-
jChead ?
; Cwt. qr. lb.
I 322 2 5
-?- 5
I. If 6 pieces of lapeftry
contain 227 ells Flcmiib, i
quarter, 2 nails, wliat Is the
length of one piece? ^
ELF. qr. m
227 I -*
Ells Flc. 37 23, anfwer
- 3. Whajt is the kiiglhy f.
piece of^Iinen». if .2;t pipew;
are 7 jf4 elk Engtifti, 4 <iy^*
ters, 3 iiails?
£1. E. qi^ n.^
754 4."" 3fV. .-.
-^ 7 . ..
107 41-
• • •
■ 35 4 3» a«lwer.
5. If 28 parcels of tea
weigh 6 cwt. iqr. ij-lb. loox.
12 dr. what the weight of i
parcel ?
Cwt. q. lb. oz. dr. ^
6 I 13 10 12 -
-5-4
I 2 10 6 II *
•i-7
I
64
-^ 5
•<■
- - 25 713, .anfwer.
Cwt. 12 3 17, afrfwer.
6. If 10 filver punch-bowls
||of 4Ln equal weighty Weigh
1478 oz. I9dwt. 14 gn what
[is the weight of one ?
oz. di
478 19
-r- 10
Oz. 47 17 23,l^^v
~
Iwer.
«
7. If
ICW4?.Tiii P R AC
7< If Sttj ingMt df fflrer,
ta25<a. i2awt. >3 gr. wMt
is th* inig^ of CM Ingot ?
ofe. «m. gir. oa. dk. it.
t03)Maj la J3(ii 17 iy^
»9S
20
822
toi
X 24
24*4
364
55
t I C E. '^
S. If 19 fHOreth^cf iet^ of
an 6qwtl weigbt^ Veig^ 2 qr*
i Ib^ ^OB. i44r. wbai is iht
weight of ofie |a>ctl^
Qr. lb. oz. dr. oz. dr.
59 )2 I 9 14 ( 15 10
57
921
36
X 16
59d
9. If |3<c#fc fqr. lite,
b t&e weignt of ^hogflieads
of fugar, what is the weight
t^ One bogfhead ?
Cwt. qr. lb. cwt. qr. lb>
'^f3ii I 22(12 3 17
if
*3
X^
93
. '^
k28
442
182
ro. A common paitur^cbH*
tains 53 acres, i rood, 27
ipercHds ; another 65 acres, 2
roods, 19 perches; a tbird
47 acres, 3 roods j thftfe be-
ing jncTofed, are to te (livid^
•6d amongft 59 pariftioner^ |
haw much is each maii'a
ihare?
A. R. P.
ij I 27
65 2 19
. A. R. P.
59)166 3 6<2 3 12|J.
I
48
4.
195
18
X 40
726
18 H 2
XI
.1^
t<)0
P R AC tlCE.
Book).
II. IF 117 pieces of Bol-I 1%. The SpeiEbtor's club 6f
l^d cfMitaia 4440 ellsEng-
liihy and 3 nails^ what doth
each piece coiitam ?
. £1. £. qr. n. el.Etqr, n.'
117)4440 - 3 (.37 -4- 3
930
III
X 5
S5S
87
X 4
fat people, though it confided
but of 15 perfons, is faid
(N* ^ ) to weigh no lefs than
three tons ; hpw much at an
equality was that per man?-
Tons,
15 ) 3 «(4cwt.anf.
X 20
60
CASE V.
To reduce great hundreds (and quarters) 120 lb. to the
hundred weight; to hundreds, quarters, and pounds,'
112 lb. to the hundred weight. | ,
^' RULE.
Confider thci^ as fmall hundreds, and quarters, dividing
hj 2, and that*quottent by 7 i which laft quotient added to
the great weight^ gives the fmall weight.
In 5 cwt. 2qr, 2 lb. great In 17 cwt. i qr. 21 lb.
vcigbt> how ' much fmall great weight, how muchihiall
weiirhf ? ureiffht ?
weight i
Cwt. qr. lb.
5
2 2
23-
Add - I 16
Cwt.5 3 18, fmall weight.
'^■■S*
weight
Cwt. qr. lb.
«7
X 21
8 2 14
r - 26 add.
18 2 19 fmall weight.
C A S E VI.
To reduce fmall weight to great weight.
RULE^
jChkp.Tn. P R A C f 1 C E.
Tm
RULE. '
DiVide the liuiKlreds and quartfvsy cpnfidered as great
weighty by 3,^aad that quotient bv ^ > which lai( quotient
fubtra^ed from the fmaU weight, leaves the great weight.
Cwt. qr. lb. • ^ i
5 3 i8 fin. weight.
I 3 za
Cwt. qr. lb.
j i8 '2 ID fin, weight.
Subtrad I - 28 . ^.
17 I 12 gr. Wcijght,
3
5
Subt. - I 16
V «•
i«.
"S. ^ 2 gn weight.
N^ B. The pounds Tn the firft divifion^ are taken no
notice of} as beii^ the fame both in gre^t andfmall wfight.
' A wRcy being 2c6Ib. of cbeefe, woo|» &c. in fonie
counties of England, is compofed of 8 X 8 X 4 = 2 j6rlb^
= 2 cwt. iqr. iflb. ...
• ■ > *
Cheefe at 2| d. per lb. what per w^ ?
L s. d.
8
I 10
8
14 8
4
AtifWeft^ t8 8 fcrvKf,
• Oras2<6farthings'rs64d.
= 5^. 4d. =: ^ per Ibu
s. d.
5 4
4
120
lb.
256 -
1
1
228
10 8
S 4
£2 18 8 ai before.
•0
Cheefe at 2L 8s. perwe]r»
what per lb. i
I I 4= id. per lb.
2
2 a 8
16 -=:s$.4d. X3'
2 18 8=:2|d. perlb
. H
8
8
4
I. s. d.
28-
- 6 -
- - 9
- - »i per lb.
H3
la
JOB PRACTICE.
In 5i-}.we7sof cheefC) how
many hundreds?
C. qr- lb.
2 14=1 vref.
18
? 4
6
f09 2 24
i) 6 3 ;2
I 2 24
/ Book !•
pttihi
1
110
I256
^28
2 B •ipcrwpjr,
19 4
6
IIS 4
i) 7 4
I 16
118 I 4=:|i|wcys.
* jjT 124 4, the anfwtr.
Qt;E$TiONs to Mcrcije Practice ly Divis^ok,
|.' A draper bought 420 Tards ^f kroa^-^lod)
^^ iH^* P^^ ellEnglifh, {low much did he pay for thewhole}
, a. d.
5I 14 IQ|: pq: ^ £li{^tflv
II II per
7
■ ■ ly
■w^
• ' £ *S® S *•» ^*^* anfivcr required^
2- A draper bought of a mefcbant 8 pac):s of bro^d^clotfa ;
^^ry pack had 4 parcels ia it, and each parcel contained 10
pieces ; every piece beine 26 yards : he gave after the rate of
four pounds, fevemeen Jnillings^ as<iiixpef)cet for 6 yards ^
what came the whole tOy and what did it coft per yarc^?
6)4l« 17s. 64* ( 16s. 3d. per yard.
' ' ■ X 2
1
12 6
, '3
21
2 6 fef P^oe.
Xio
211
5" - per parcel.
X4
Hs
- ' - per pack.
X8
£ 6760 - -, the anfwer. 3. An
3* An oamati bouglit jtuiksof ^il, iirtikh ccrflkiai 151 1.
^4^ whkii luippentd ^t»^leA -^dnit ^5 gdkns y tUs he k
williM to frfl aeabi) fo^ a«>le %e Ao lofer : I demio^d flow be
ft ftil it a gidUon r
252 Jpllons in a ^Un. ^ '
3 tun. . . !
^^■^
756 gdlbm. . ' ' .
8< leaked.
^ 1 . . . '
671 ) 151 14 ( 4S* 6^d..^i^ the znfwtr.
X 20 - • -
3034
350
X 12
4200
>74
X4
I
696
^
I
*5 - • ■' .-.
4. A draper bought 242 yard/of brdad-cloth, which coft
him in all 256 L 10 s. ; for 86 yards^.^hich he gave 1 1. is,
per yard » what did he give jLyard for the remainder ?
1. 8, d. 1, . s. d, -v
114 256 xp - ^ 242
X II 91 14 $ . —^86.
12 ]6 - 164 15 ' 4 priceof *. 256 yards.
X 7 • 1 ^ ^
_«-». I. 8. a.
89 12 - 156} 164 15 4 ( 4I I s. i^d. -J^anC
228 8
— ^ X ^o
^91 14 8 priceof 86yds. ,7^ ^^ ^
— T^ 19
X la
232
X 4
321
148 H 4 Jh A
r
;ia+ FRACTICB. .: BookL
5. A tcntlemaiif at bi^ cl«atl|, left his eldtft fen once
^nd a bw what he aUotted his daughter ; and to the young
Jaidy 13831* lefs than her mother, to whom ht bequeathed
four times what he^eft towards the endowment, of Hertford
college, Oxon, viz. 1640 guineas : I require what he in-
tended for his yoimgeft fon, who claimed, under the will,
half as much as his mother and lifter ; how milch left than
30000 1. did the teftator die worth, his debts and jfuoeral ex-
pences being 988 1. los. ?
20 ) 1640 guineas. ^ . .
+ •82 . ,
• • . •
£ 1722 Hertford-college. ..'..
X 4
6888 wife.
- 1383
a ) 5505 daughter.
+ 2752 10 8.
^8257 10 s. eldeft fon..
.6888
+ SSOS •
• • •
2)12393
■> ■ ■ h
£ 6196 10 s. youngeft fon.
L s;
Then mother - 6888 -r
• Eldeft fon 8257 id * *
Yoirngeft 6196 la - '
Daughter 5505 -
College- 1722 ^
Funeral « 988 10
30000— 29557 io*= 442 1. 10 s., the anfwer^
6. My purfe and money, quoth Dick, are worth 12 s. 8 d.
s but the money is worth fevcn of the purfe j pray what was
there in it ?
s. d.
8 } 12 8 purfe and money.
1 7 purft.
/ • Jt I, money, the anfwer. y. A
jCi»p. in- P R A C T I C K 105
. 7. A dealer bought two lots tif Cittiff, ^t to^dier
weighed 9 cwt. 3 qrs. i61b.» for 97 1. 179. 6JL Their
difference in point of weight was i cwt. 2 <jrB, 16 ]b. ; and
nf price 81. 13 s. 3d. Their refpedive weights and valuea
arc required?
Cwt. qr. lb. .1. 9. i.
Wdgbt - * 9 3 16 . Coft - 97 17 6
' . o 1$ 3
Difference - ^ j 2 16
a)8 I - 2)89 4 3
Leflerlot - - 4 - 14 44 X2 li
Greater - - 5 3^ 53 5 +r
. . ^. A tradefinan increafed his eftate annuallv a third pait^
abating icx>l., which he ufually fpent in his ifaoiily ; and at
the end of j^ years^ found that his net eftate amounted to
3179 1. IIS. 8 d. . Pray what had he at out-fetting?
1. s. d.
Worth at the end of 3J years - 3179 11 .8.
4)100= 25 - -.
••^
3^ years = 13 quarters «• • J3 ) 3204 11 8 •
246. io li
Worth at the end of 3 years - - 2958 i 6^ -
- » ...
4)3058- I 6^ -
V 764 10 4i
Worth at the end of 2 years - - 2293 11 2 •
+ J^oo - - .
4)2393 " ^-
598 7 9i
Worth at i year's end - - ,• 1795 3 4J
4. 100 - -•
189s 3 44
— 473 15 w.
Anfwcr, £ 1421 7 6J.
9« A
jc6
FE ACTICR
BooklL
- 9. A certain ferfinllK>i»iiittwo1ioifestwithAeti^piM
which coft iooi.;«i4iidH trappings, if lakl 00 che £ft
horfe A» lioth will W of eqval value ; but if the £ad tiap^
fwgs be iaid en the 4M)Kr korfe, he wil) be 4ottb!le the value
of oie firft i faow much did the faid hoifes coft i > ^
1. 1. ». d.
a ) too ( 50 — ^ — pTic6 of the bcft horfe. ,
3*) 50 ( 16 13 4 prio^ of the trappings.
Difierefiee^ 33 ^ S P^^^ of the .otfier horfe.
^ S E C T. III.
PRACTICE ly Aliquot Parts.
THIS rule is only a coytitt'adion of the Golden niYe i
for ^en the value orprice of one yard, ell, bttndred>
fcc. is given, and the price or value of any other quantity
of yards^ ells, hundreas, &c. required ; the firft number
or term being unity, the queftion may be performed by
aliquot, or even; parts of numbers. Aa aliquot part of
any number is fucfa, that if the faid part be taken certaio
times, ^ ft aJljnft make the number whereof it is a part.
TABLES^ AiicipoT Pahts.
7
i
I
"8
10- =
68 =
4--
3 4 =
a6^
18 =
Tenths of ft!
- = ■/,
Of a pound,
s. i.
« 3
— 10
- 8
ssiof
= |of
I
7l = iof
6 =iof
7
I
I
TV
— To
TV
2
i8-=ty
5 -»
4 =i , .
I — TTT of Iff
4=1 of V, of A
_» — > of ■ Ckf «
— -r — . -y or TT?*T7y
S. d.
- 9= V«
^ 3= Vi
« 4 =
I 6 =
9 =
Of a ihilling.
7 6
8 -
8 4
u S
12 -
T2 6
= 4
— T
— T$
= -k
— 3
— y
»34 =
168 =
176=:
184=:
1
i
T1
I
1
2
2
2
2
3
3
3
4
4
S
5
7
7j
t
10 = ,v.
3 =
4 =
8 =
9 =
6 =
8 =
6 =
8 =
4 =
6 =
^^
Z I
ft
7«
7
t 1
6 = f
4 = 1
3 = 1
» = i
«f=*
I =/»
4=fo^i^
-i= I of •_
• And ^^
io|=i
10 = .f
9=-i
8 = f
7l= I
4f = I ^ .
5J= I of i
«i= f of I
C A$ £
aap.lll
• • •
C A B £ I. .
•107
7
When the pijoe af the integer is an aliquot part of a
pound]) or -of ^ fliiiling.
RULE.
«
For the aliquot part 4>f a pound, dhride the ghrcn quan-
tity bv that pait ; the, quotient will be pounds, and the re^
malfi^ler fo manv times that part* But for the sdiquot pcbts
#f a ^illin^, diyide as before, and thctquotient by ao ; the
faft quotient vriii he peunds, and the rvnaainder Mlings.
Bttt if it be a compound aSiquot part, take the firn efi-
llttCCpart, and then dA»e:aliqiio( part of diatpar^ vhichlaft
quotient will anfwex the quellion.
1. What coft 737 yards of
Holland, at 10 a. per yard ?
•— — !• d.
2. 873 yards ditto, at
^s. 8d.?
^.. ^
£ 291 - -> anfwer.
3* 3711 yards ditto, at 5 s. ?
' s. d.
£ 9^7 JS -> anfwer.
4. 1 171 yards ditto, at 48« ?
;£ ^3+ 4 -ftnfwcr.
5. 743 yards of Irifli doth,
at 3 s. 4 d, ?
' S* Qa
j^ 123 16 8j anfwer*
6. .275 yards ditto^ at 2 s.
6d« ^
*■ ■■■ s, a«
X 34 7 6, anfwer.
7* 7^ yards of ditto, at
as,?
"in.
jT 76 10, aiiA^r.
S. 1761 yards «titto» at««.
8d.?
t^^mm^t S*
j^ 146 15, anfwer.
9* What coft 757 yards of
dowlas at I s. ?
Tiy|757
£ 37 i7» anfwer. •
^ '
JO. What
xoS
la What coft 957 janb of
dowhS) at IK 3d.r .
4
X
4
1957
•— «. a.
239 » S
II. 1713 yards of ribboiw
*7»J
X
4
►
t. , d.
I428 -. 7f
. _£ 59 16 3^ aufwcr.
J 5 s. -{- 1 s. 3 d« = 1 6 K 3 d*
Hefe> and in fiEveral queftioiu below, the remainder k
Imked upon to be of tbe iaiae denomination with the divi-
dend ; and the fecond remainder is always added to the firft.
£ S3 " 7i>««fwer*
10 s. +7id. = ios.7td,
I au 737 ]rards.ditto>at, lod.?
737
X
4
8. d.
184 - 10
£ 30 14 2, ahfwcr.
13- 757 yards ditto>at 7^ J
t
t
T
7S7
— s. d.
94 J fr;
II ■«
j^ II 16 6-lf anfwer.
13s. 4d. + lod: = 14$. 2d. 1 15SV 4. IS, 6|d. s= 16s.. 6|-d<
14. 1511 ydurd&ditto, at 5d. ?
1511
s. d.
if .231 2 1
/; 3r 9 7, anfwcr.
Queftions in pniSice admit of various ways of workings
equally fhort, which, ferve as a proof to each other; and
indeed pvaSice is heft proved by pradice^ though it may be
pioved by feveral other methods^
45« What coft 151 1 yards of ribbon, at 6 d. per yard ?
T
1511
s.
i.
■
•
377
I
6
£ 37 15 6,wifwer.
Oi
I
1
X5II
8. .d*
7SS 6 -
■i*w
£ 37 » 5 ^»*nfwcr.
s. d.
£ 37 IS ^
^■^^••^^
Remains* 31 fixpeoces s 15 s. 6 (i<
i6. What
CIisp.tII. P R A C T r C E: to^
xt. What coft u<)9 yuds of Cstaii ribbon> M^d.per
iip9
— f • d.
2ai I 4
1
£ i8 9 8, anfwcr.
1
T
1109 ,
r i.
I
369 8
/; 18 9 8, aiifircr.
,Vlxi09
* I ■ 8* d. *
£ Mi g Si anfwer.
^-««
_ . ft
Remains ^9 groats^ or 4 times as. 5 d.
17. What coft 75.1 jrards ditto, at 3d. per jrard i
j^ 9 9, anfwer.
t f
187 9
£ 9 7 9, aafwer.
Remaiiis 31 threepences ss ^s, ^H. \*»
i8. What coft Xj;ii jards of ditto* at ad« per yard j
• »
251 10
iC W II 10.
xiw|J5" •
*. a.
12 II 10
^ m^
Renudns 31 twopences =5 iis. ibSj».. \'^
19. 1 I73yards of fectethqg, I 21 : >0^ . ymt^ of flUetifigy
at lid,? (at :Jf,?
i|"73
d.
X
4
7 6 H
X
4
107 1
267 2^
^MMMi
66 ut
^ 3 6 1 1^» anivrer^
•«*
20. 713
tto
Pft A C t ICE.
iJookl;
2o» ir^yuiM^SuMtti i ^^
— - ». A
59 5 «
1 19 5, anfwen
WMkMMM-BB^-dh
■as. 1975 ]F««d« (Utto, at 4 f
•^ 107s
TT
537*
«. tf.
44 ^ 9i
Irritate
iC a 4 9i
43. jijyaxist «id. ?
M)
4>7r7>'79T
14 s. ii^d«| anrw«r.
dHWM-kaM
C A S E IL
When the numeratot of the frafUon is moi^ than unif/i
ftnd the denominator xo,
R tJ L E^
Multiply the giirai: ^fotmiff by itaS' HHmerator, and
double the figure in the units place of the produft fot^
Ihillings^ the %ttres to Ae left-hand tipifl ftepounds.
I. What coft 7^ dfe of
fineHolIandy at it's, per ell i
1757
s.
£ 6Si 6» aniwer.
a. 6f7clla.ditt»,alri6a^i
I617
s.
j£ 493 "> anfwcn
•• « «•
S* 577 ells. cIi(l•»^l4s«^
|577
I 403 18
717 ells of loiig-4gnifii^ at
8c. pet' ell ? !
7*7
4
— ^ *•
j^ 286 i6> aniwef.
AIM
99;dla 4ittO) «t6*<^
933
3
» — »• •
jC 279 18, anfwff.
714 ells dkto, at 4s.?
2
— • 8*
£ 142 t6^ ahfwer.
4. 118:?
r
OAp-in. JR AC TIC 5.
4. ti87tU»dittOr« 12 ■> ?
att
1187
6
s.-
^ 71X 4, anfwer.
^BttB,:
£ 93 6 s., anTw*.
CAS Era.
< .
When the rramerattor Is any number under 12^ and fhe
inator 12, or odder, with or without a cj^rfrer ;Ui-
R tJ L E, i
Multiply the given quantity by the numeratio:^ ai4 di^
vide the produA by the denominator ; the quotient Will be
{KMinds or fhillings^ according to the satuie of riie qaeftiion^
and the reminder £> anny times the aliquot part ezpreiled
hy the dcaoniiiator.
*
3. 931 yards dittD, ati6s«
*
5
I. What GOft 737 yards of
hnMid-cioath, at 18 s* ^.cL ?
737
II
12 8107
— 8. d.
£ 675 XI 89 aiii!wer*
■ ■ ■»■
Romitimesia* ftd.= lis.
8d.
2. 371 yards ditto, at 17$.
*d, ?
8
37*
7
^597
— s. d»
£ 324.12 6^ anfwer.
6|+65S
£ 775 16 8, «fffWcr,
S* Q«4
III Hi I *i^
Retn. 5 timer 3^* ^^* ^
4« 573 yards ditto, at i6s« t
573
4
2292
8.
£ 458 8, anftrcr.
^wMAiaB
Um. 5 tim$s as. 6d. ^ r a «. j Rem, a times -4 y. = 8 s.
5. 746
Itt
PRACTICE. Book D
5- 7463rarcl$ditto,atis».?j 9. What cdft 137 dU.of
|Holland» at lis. 8a. f
746
_3
2238
-— s. d.
£ 559 >o -* anfwcr.
6. 865 yards ditto, at 13 s.
4d.f
865
2
— •" t. d.
£ 576 13 ■ 4, aafwer.
Rem» ' 2 times 6 s. 8'd. s:
13 s. 4d.
.* r
7* 713 yards ditto, at 12 s«
»37
7
" 959
— — s. d.
£ 79 18 4* anfwer.
Rem. II times is.8d. ^^
18 8. 4d.
10. 537 ells ditto, «t 8t.-
4d.?
537
5
la
^685
«.
8
7»3
S
3565-
s. d;
£ 445 I a 6, anfweh
« •••
£ 223 15, anfwer.
Reifi. 9 times is. 8d. .=
155.
II.. 537 ells dittO) at' 8 s. ?
Rem. 5 times -2 8. ^d. =
las. 6d.
8. 783yardsditt6)at I2S. ?
f - •
783
3
1349
8.
£ 469 16, anfwer.
Rem. 4 times 4 s. ss i6 s.
537
2
1074
— s. d.
£ 214 16 -, anfwer.
Rem. 4 times 4s =: i6 s.
12. 719 ells ditto, at ys'i
6d. f
719
3
8i»57
' s. d.
£ 269 12 6,. anfwer. ^
Rem. 5 times 2s. 6d. :=:
I2S. 6d.
13- 157
Chap. HI. P R A C T 1 C £i
13. What cqft 157 ells of
Holland, ac 7 s. 4 d^ ?
30
'57
II
1727
— — s. d.
£ 57 II 4, anfWcr.
Rem. 17 times 8 d, =: i| s*
4d«
14. 737 lb. of bofacA tea,
at 5 s. 6d.?
"I
17. 713 yard? ditto, at 4 s,
6d. ?
40
6417
s. d.
40
737
II
-
8107
s. d.
1^ 160 8 6, aniWer*
Rem. 17 times 6 d^ =: 8 s. 6d.
18. 379 jrards ditto, at 3 s*
8 d. ?
60
379
II
4169
d.
3^ 202 13 6, tnfwer. •
Rem* 27 times 6jL =;: 13s. 6d.
15. Wtot coft 871 ounces
of plate, at 5 s. 4 cf. ?
30
871
8
6968
jC 69 9 8, anfwer*
Rem, i9 times 4d* iz 9 s*
8d.
19. 7 19 yards ditto, at 3 s»f
20
s. d«
j£ 232 5 4, anfwer.
Rem. 8 times S d. = 5s.4d.
16. 837 yards of'kerfey,
at 4 s. 8 a. ?
719
3
2157
d.
3?(s85?
)C ^95 6, anfwer.
s.
;^ 107 17 -, aiifwen
20, 173 yards ditto, at 2 «#
9 d. ?
80
173
II
1903
s. d.
»■! >■
^ ^3 '5 9» anfiVen
" '
Rem. 63 times 3d. c=:i5Sf
9d.
I 21. What
114
PRACTICE.
Book I.
21. What coft IJ2S yvds
of keriey^ at 3*. 6d. r
40
1735
7
1214s
■^ 8* d.
J^ 303 12 6, anfwrer.
Rei9« 25 times 6 d. = 12s. 6d«
22. 031 yards ditto, at 2s.
8d.?
25. 795 yards ditto, at i s.
10 d.?
795
II
I2Q
8745
30
4
' s. d.
£ 72 17 6, anfwer.
Rem. 105 times 2d. =: 178.
6d.
26. What coft 173 yards of
dowlas, at I s. gd. ?
4d3
37H
■ ■ ■ «. d.
j^ 124 2 8, anfwer.
23. 107 yards ditto, at 2 s.
80
'73
7
60
107
7
_,7+9
s. d.
£ 12 9 81 anfwer.
MM
I2II
—— ^ 8. d.
£ 15 2 9, anfwer.
Rem. IX times 34* ■= 2s. gd.
27. 7 1 3 yards ditto, at is.
6d. ?
713
3
40
2139
8. d.
Rem.29times4d. z=:g$. 8 d.
24. 713 yards ditto, at 2s.
3d»?
713
9
80^417
' s. d.
^ 80 4 3, anfwer*
Rem. 17 time3 3d. =543. 3d.
£ 53 9 6, anfwer.
28. 913 yards ditto, at I s.
4d. ?
60
913
4
3652
8. d.
I
£ 60 17 4, anfwer.
Rem. 52 times 4d. = 1 7s. 4d.
29. What
Chap. III. P. R A C
29. What coft 783 yards of
dowlas, at 1 s. 3 d. ?
783
5
80 3915
» a. d.'
£ 48 18 9, anfw^r.
Rem. 75 times 3d. = i8s. 9d.
30. 713 yards ditto, at i s.
2d. ?
7
120 4991
— s. d%
£ 41 II 10, anfwer.
Rem. 7 1 times 2d. = t is. lod.
31. What coft 7371b. of
tobacco, at 10^ d. i
737
7
T I C E. „5
33. 971 lb. ditto, at9d. f
971
3
80
*9'3
d.
8
20
5»59
— ^— d*
64+ I of
s. d.
j^ 32 4 lot, anfwer.
32. 673 lb. ditto, at 10 d. ?
673
5
jC 36 8 3. anfwer.
Or 971
3
4
20
2913
■ — d.
728 3
36 8 3
34. 1713 yards of ribbon,
at ^i d. ?
8
20
1713
5
120
3365
8. 4-
28 .- 10, anfwer-
Or 673
5
6
ao
3365
s. d.
560 - to
:£ a8 - 1O9 axifwcr.
8565
■ d.
X070 ^i
— — $. d.
£ Si ^^ 7i:> anfwer.
35. What coft 1735 yards
of ribbon, at 8 d. ?
'735
33470 .
— «— d.
201156 8
— — s. d.
£ 57 16 8, anfwer.
Or 30.1735
|— s* d.
jC 57 * 6 8, anfwer*
Rem. 25 tii^ei 8d. =s l6s. 8d.
I 2 ^. What
irS
PRACTICE.
Book I.
36. What coft 753 yards of
ribbon, zt^id.i
753
3
8 2259
d. • .
20 282 4i
— ^— a. d* < '
£ 14 2 44« anfwer.
37. 783 yards ditto, at si^i
783
1
8
20
548X
68s_
342
£ 17
d.
H
s. d.
2 6|, anfwer.
38. 575 yards ditto, at 3| d, ?
575.
5
8
20
287;;
— a,
359 4i:
179 8t
— 8. d,
8 Y9 8^, anfwer.
ttmmimmm
CASE IV,
When the price is IcTs than a poundy or a ihilling, by a
fingle aliquot part of either,
RULE,
Take that aliquot part of the quantity, which fultraA
from the quantity ; the remainder will be the price of the
whole in pounds or {hillings. |,
i^ What coft 787 prds of | 2. What coft 1135 yard*
velvet, at 19s. per yard? ditto, at i8s. 4d. ?
787
t
39 7
£ 747 13, anfwer.
■MM*
•rr
"35
94 II 8
£ 1040 8 4, anfwer.
3. \yjlat
Chip. m. F R A C T 1 C E;
3. What coft tSs yawlg of
velvet, at 17 s. 6d. ?
a.
8
785
98 2 4
8
jf 686 17 6, aafwer.
4- 937 y^'' ^^tto, at i8 i.?
^f937
93 14
JC843 6, anfwer*
c. 731 yards ditto, at 16 s.
tiy
9. iSiiprds<ittto>att9s.
TOd.?
II 10
i|73i
121 16 8
jC 609 3 4, anfwcr.
6. 573 yds. ditto, at 16s.?
573
114 12
£45^ 8, anfwer.
7- iS37yds.ditto,at 15 s.?
i 1 1537
I 384 5
£iiS^ i5» anfwer. .
8. 536 yards ditto^ at 13 s.
4d. ?
536
178 13 4
£ 1488 8 2, anfwer.
10. II 09 yards ditto, at
igs. 9d. ?
I 1
1109
'3 17 3
£ 1095 2 • 9» anfwer.
II. 1 109 yards ditto, at
19 s. 8d. ?
^|M09
18 .9 8
jC357 6 8, anfwer.
£ 1090 10 4, anfwer.
I
12. 15TI yards ditto, at
19 s. 6d. ?
I 37 ^5 6
j£H73 4 6, anfwer.
13. 737 yards ditto, at
19$. 40. t
TC
737
24 II 4
j£^7i2 8 8, anfwer.
I 3 14. Whsit
ii8
14. What coft 1736 yards
of ribbon, at 8 d. ?
578 8
PR A C T I C E. Book I.
16. 13711b. ditto, at lo^il.?
1
"57 4
S
T
Tff
£ 57 17 4^ »«^^^^-
15. 737 lb. of tobacco, at
1371
171 4t
"99 7t
xi^cf.'?
TT
f
737
61 s
jC 59 19 7t» anfwcr.
17. 783 lb. ditto, at 10 d. ?
1 1
7
675 7
.*M«bi
£ 33 15 7» «f^«r.
783
130 6
652 6
^ 32 12 6, anfwer.
18. 11731b. ditto, at 9 d.?
"73
^93 3
Tff
879 9
L 4-3 19 9» anfwcr.
CASE V.
When tbe pnce is not an aliquot part of a pound, or of
a (hilling, but may be divided into fuch^
RULE,
Find two or more numbers, which are aliquot parts^
'whofe fum makes the given price ; proceed with them as
before dire&ed ; then add the quotients together, which
fum will be the anfwer.
I. What
Chap. m.
I. What coft 731 yards of
broad-cloth, at 18 s. 4d»?
PR A C T IC £• 119
4" 377 7*"^^^ ditto, at 15 «•
4d.f
i + t + i
731
365 10
182 IS
121 16 8
• +4
T^
£ 670 I 8, anf.
rff T
Or,
tsTtc
I73I
657 18
12 3 8
£ 670 I 8,anfwer.
2* 957 yards ditto, at. 17 s.
Sd. ?
i- + 4 + ^^*957
478 10
{ 47 J7
jC ^45 7> ^'^f'^^^^'
377
226 4
62 16 8
£ 289 - 8, anfwcr,
5. 3x7 yards ditto, at 14^.
3d.?
7 J. «
TB"
317
221 18
3 19 3
£ 225 17 3,anfwcr.
6. 1 01 yards ditto, at Z2 $•
2d.?
6 l_ _t_
xffT" t>CS
?c +
Or,
TT
[957
765 12
79 15
£ 845 7» anfwcr.
3. 107 yards ditto, at 16 s.
4d. ?
lor
60 12
— 16 10
^ 61 8 10, anfwr.
7. What coft 713 gallons
of rum, at I IS. 4d. per gallon?
285 4 ^
118 16 8
i^ + T^
TS
107
85 12
I IS 8
£ 8y J 8,anfwcr.
£ 404 — 8, anf^en
8. 571 gallons ditto, at los.
8d. ?
i +
t
571
285 10
19—8
£ 304 10 8,anfwcn
u
9. What
I20
• , 9- What ctA 109 gallons of
rum, at 9 s. 8 d. per gallon ?
7V + TT «09
P R.A CT I C E.
Book
43 ^* ,
9 I 8
iC 5^ '3 8,anfwcn
* 10. 137 gallons . ditto, at
9 5. 2 d. ?
t+ii37
45 ^3 4
17 2 6
^ 62 15 10, anfwcr.
11. 7x9 gallons ditto, at
8 s. 3d. ?
iV + 'T
TTT
719
287 12 .
8 19 9
14. 703 ells ditto, at6.s,
II d. per ell ?
4+
1 1
T^
703
234 6 8
g 15 9
jC 3'43 » 5,anfwer.
15. 959 ells ditto, at 5 s,
lod. ?
k + i 959
159 16 8
119 17 6
£ 279 14 2, anfwer.
x6. 371 ells ditto, at4S«
2d. ?
S + TTS- 37'
74 4
3 I 10
jf 296 II 9, anfwer.
12. 473 yards of Holland,]
>t S.s. 4d. f
-*- 4- -'
473
118
78 li
8
£ 77 S 10, anfwer.
17. 873 yards of kcrfey^
at 3^. 10 d. per yard ?
i + TV'873
145 10
21 16 6
^197 I 8, anfwer.
13. 157 yards ditto, at
7 s. 4 d. r
157
5*
S
6 8
4 8
£ 57 II 4, anfwer.
Tr
£ j6j 6 6, anfwer,
18,. 379 yards ditto, at 3$^
8 d. ?
I- + tV 379
63 3 4
664
^ 69 9 8, anfwer.
19. What
Chap. III.
S. What coft 891 yards of
7> 2it 35- 74* per yard ?
P R A C T t C fc
+t'i
Tff
891
t2t
24. 173 yards ditto* at 2 s«
9d.?
148 10
II 2 9
. £ 159 12 9,anfwpr.
• *73S y^s ditto, at
n a. r
20
3$. 6d.'f
* + w|i735
|»73
21 12
2 3
6
3
jC 23 15 9, anfwcr.
^5* 931 yards ditto, at2S<
8 d. ?
8 T Trs"
216 17 6
86 IS -
jC 303 12 6,anfwer.
21. 907 yards ditto, at 3 s.
2d. ?
93'
116 7 6
7 15 2
Tff
[907
113 7 6
30 4 8
£ 143 12 2, anfwcr.
22. 719 yards dittO) at
3 s. ?
71 18
35 19
£ 107 17, anfwer.
23. 873 yards ditto, at 2 s.
^124 2 8, anfwer.
16. 107 yards ditto, at 2 s.
4d. ?
TC-
VC
107
lO 14
I 15 8
£ 12 9 8,iuifwer.
27. ^3C yards of Irifli doth^
at 2 s. 3 d. ?
735
73 '^
9 3 ♦
13.
10 d. ?
* +
3VJ873
109 2 6
14 II -
jT 123 13 6, anfwer.
**«
j^ 82 13 9,tefwer.
28. 317 yards ditto, at 28.
2d. ?
2 12 10
j^ 34 6 10, anfw.
29, What
no
• . 9. What coft 109 gallons
rum, at 9 s. 8 d. per gallon ?
^V + TT 109
PR.A CT I C K HookL
14. 703 ells ditto, at6.s,
1 1 d.. pi5r dl ?
1 43 " „
9 I i
£ 5^ ^3 8,anfwer,
■ 10. (37 gallons, ditto, at
9 s. 2 d. ?
t+i|i37
45 »3 4
17 a 6
234 6 8
« 15 9
£ 5143 2 5> anfwer.
15. 959 elU ditto, at 5 s.
lod. ?
X i' x\
£ 6z 15 io, anfwcr.
— — ■»■ ■ ■
ti. 719 gallons ^itto, at
8?. 3d.?
tV + T
TV
719
28
M
2 .
9 9
^ 296 It g, anfwer.
12. 473 yards of Holland,
>t^.s. 4d. ?
•^ 4-. I
959
159 16 8
119 17 6
£ 279 14 2, anfwer.
16. 371 ells ditto, at 4$,
2d. I
^4. ■
rs"
37'
74 4
3 « 10
473
118
78 II
8
iC ^97 ' 8> anfwer.
*3* ^57 y^'^ds ditto, at
7 s. 4 d. r
T T Ti
^57
5*
5
6
4
8
8
£ 57 n 4, anfwer.
£ 77 S 10, anfwer.
17. 873 yards of kcrfey,
at 3^. 10 d. per yard i
6 T* 4 <
873
145 10
21 16 6
£ j6j 6 6, anfwer,
18,. 379 yards ditto, at 3$^
8 d. ?
6 T" -z'
379
63 3 4
664
^ 69 9 8, anfwer.
19. What
Qiap. ni.
lo. Whatcoft 891 yards of
kcTMy, at 3s. yd. per yard i
PRACTICE
+ r^
T5-
891
148 10
ll 2 9
. £ 159 11 9, anfwpr.
20
3 s. 6d'.l
t2f
24. 173 yards ditto> at 2 s«
9d.?
X^ I
f»73
21 12 6
^33
£ 23 15 9, anfwcr*
. 1735 yards ditto, at 25. 931 yards ditto, at as*
ftA.r I n a. ?
Txi
"KT
1735
8d.?
_ I
T'iO
216 17 6
86 15 -
£ 303 12 6,anfwen
21. 907 yards ditto, at 3$.
2d.?
-'--J- ■
[907
113 7 6
30 4 8
jC 143 '^ 2, anfwcr.
22. 719 yards dittO) at
3 s. ?
1^ + ^1719
71 18
35 19
j^ 107 17, anfwer.
23. 873 yards ditto, at 28.
xod. ?
i + 1^1873
931
116 7 6
7 15 2
j^ 124 2 8, anfwer.
Ii6. 107 yards dkto, at 2 s.
4d,?
tVt T5^
107
10 14
I IS 8
jf 12 9 8, anfwer.
27. ^35 yards of trifli dotl^
at 2 s. 3 d. f
tV + tV
735
73 '<>
9 3 ♦
i^^mm
tm^tf
109 2 6
14 II -
^ 123 13 6, anfwer.
£ 82 13 9,tefwer.
28. 317 yards ditto, at 2 ».
2d.?
tV 4" xrc 3^7
3^ 14
2 12 10
£ 34 6 lo, anfw.
29, What
ISC
PRACTICE.
Book I.
29. What coft 137 yards
of IriJh cloth, at I s. i j d. ?
TTTlTr
'37
.11 8 4
I 14 3
713 yards ditto, at i s.
713
■L »3- * 7» *n^^'
30. 795 y^^^ ditto, at X Si
TT + i4ir795 .
66 5
6 12 6
{^ 72 17 6, anfw.
3 J, 713 yards ditto, at i s.
6d.?
713
35 13
17 16 6
^ S3 9 6, anfw.
32* 913 yards ditto^ ajt 1 s.
I I >
913
45 »3
»5 4 4
35 »3
; 18 I
j^ 41 II 10, anfw.
35. 757 yards ditto, at
Hid. ?
» "rT"r T
T5
757
378
252
94
6
4
Ik
7*5
Si
jC 36 - St^anf-
I III
36. 371 yards ditto, at
9td.?
t + i+i37'
C *
18s 6
61 JO
46 4i
X
Tff
293 8t
l^ 60 17 4, anfw.
33. 873 yards of dowlas,
at I s. 3 d. ?
^ + t'o 873
43 »
:i
10 i» 3
jC 54 " 3, anfwcT,
jC '4 13 8t, an.
37. 1713 yards of ribbon,
at8f d. per yard?
856 6
214 t\
I >4* 9
■•»
Tif 1213 4f
jC 60 13 4i
38. What
Chap. III.
38. What coft 587 yards of
ribbon, at 7 d. p«r yard ?
PRACTICE. iaj
41. 373 yard* ditto, at id.?
i + i
587
19s 8
146 9
«
342 5
£ 17 2 s^ anfwcr.
39. 713 yards ditto, at
6Jd.?
TT
178 3
118 10
89 li
' -1- ■
I
373
93
62
3
2
tSS
S
£ 7 15 5, anfwcr.
42. 715 yards ditto, at
3td-?
4 + i|7i5
386 2t
^ 19 6 2i,anf.
40. 731 yards ditto, at
5id.?
T + T73I
243 8
9^ 4t
T'ff 335 "*»
•s:"?r
119 2
89 4t
208 6t
jf 10 8 6^9 anfwer.
43* 757 J^^^ ditto, at
2a d. i
94 7i
63 1
157 St
j^ 7 17 8t, anfwer.
jf 16 15 —J, anfwer.
7-
C A S E VI.
*
i
When thc^4)ricc of the integer is a farthing, or farthings
joined with pence, or with (hillings and pence,
RULE,
Work for the fijillings and pence, as before direded ;
then obferve what part of ^S^of the foregoing lines the
farthing or farthings are, wbick take, and then add all to-
gether. , \
. X I. What
124 , ^ R'A C T I C E.
J. What coft 715 yards of
tape, at i^d. per yard ?
TT
X
715
59 7
14 lOj
74 5i
£ 3 14 si, anfwcr.
2. 495 yteds ditto, at z^d. ?
X
TV
495
61 lOj:
10 3l
^, 7i
ai
jC 3 12 2^, anfwcr.
3* 351 yards of finall rib-
bon, at 2^d. j
I
T
351
58
7
6
31
6S
95
jC 3 5 9T.anfwer.
4. 741 yards ditto, at 2^d.?
741
i
Satt.
185 3
J5 St
»Vi69 9t
;£ 8 9 9^,anrwen
BookL
5. 143 7^^ ditto, at 3^ ^
243
J-
4
xr
fi5
60
65 9t
£ 3 5 9l» anfwen
6» 747 yards dittOi at 3|4«?
4
747
4
186 ,9
46 8t
t
■5'd
^33 5t
;£ " 13 5ti anfwcr.
7. 714 yards ditto, at 4^.?
7H
4
T
X
4
178 6
59 6
14 lOi
VtsUs^ 10 r
. j^ 12 12 lolt anfwer.
■<»M«a
8. 291 yards ditto, at4jd.?
X
4
291
I
X
1
~6
72
36
6
_3
4
1
"5
2t
iC 5 15 ^4' anfwcr.
9. What
CEap. 111. PRACTICE.
t). What coft ;>47 yards of
«bbTO, at jtd. pef yard?
I
T
747
4
249
62
15
h'.
1
326
9i
19$
13. fc^t yards dHto, at
7td.?
^16 16 9|, anfwer.
10. 2ioycls.<litto,at5Jd. ?
i + i2IO
4
I
"5^
52 6
35 -
13 't
100 74:
I
jC 5 - 7h anfw.
II. 737 yds, ditto, at 6^,?
737
1
T
TT
245 8
122 ID
15 4?
383 JO*
jC 19 3 i^^abfwer.
12. 1 1 73 yards ditto, at
"73
X
6
X
4
TO
1 131
23 6|.
^83 31
)£ 34 3 3l> ^fw.
14. 7ify4b*.4ittQ>at7|<].?
+ t
X
4
7"
237
177 9
44 Si
459 a:
j£ 22 19 2;,
anfw.
15. 495 )r<k.clkto>9t8j^.?
I
495
I
T
I
T
82
10
6
6
3i
1
5^
340
31
I IT - 3h «nfw.
•^^i«i*-«H
t|d.?
X
s
■s-ff
5^6 6
73 31
659 9l
,j£ 32 19 9t» anfwer.
16. 1 157 yards ditto, at
8id.f
T
TS
"57
578
192
72
6
10
31
84J
7^
/; 42 3 74, anftr.
II. What
17. What coft 527 yards of
ribbon» at 9^ d, ?
15^7
I
I
T
t
•1
TV
PRACTICE- Book L
2t. 41 5 yds. ditto, at 11^.?
263 6
K) Hi
406 2i
j^ 20 6 2^, anfwer.
18* 7i57ds. ditto, at 9|d.?
71S
I
1
JL
-♦J
I
TV
357 6
178 9
4+ 8i
580 IIj:
^ 29 - iiJ, anfwer,
19. 785yd8.ditto,atxo^.?
'785
X
X
I
T
X
4-
392 6
196 3
65 5
16 4i
670 6J
jC 33 10 6^9 anfwer.
20. 91 13 yards ditto, at
J.d. ?
4
X
4
415
207 6
138 4
34 7
.8 7i
389 H
19 9 -|, anfvir.
22. 797 yds. ditto, at i i|d.?
X
t
797
398
265
6
8
16
7i
7i
780
41
£ 39 - 4i» anf-
23. 371 lb. of tobacco, at
IS. -Jd.?
371
I
T
I
TT
956
4 12 9
4 12 9
7
H
lo^d. J
X I- I
* XT
f
T
TC
9"3
4556 6
3037 8
569 ei
8163 8}
^ £ 408 3 8|, anfw.
£ 18 18 8|, anfwer.
24. 171 lb. ditto, at IS.
TV
171
10 fi^h
jf 9 I 8 J^, anfwer.
25. What
Chap. III. P R A C
25. What coft 9071b. of
tobacco, at I s. i^d/i
1*5
I
X
4>
[907
45
7
3 «S
7
18
lOj
/ 50 I 5^,aiifwer.
^6. 175 lb. ditto, at IS.
lid.?
TV
I
JL
'75
I I KJ lU 127
29. 9081 lb. dlttOa at I s.
3t<i.?
xffr908i
1.
4
1
454 I
113 10 3
9 9 2i
;C
577 - 5i»an'w.
ft
30* 173 lb. ditto, at I s.
8 15
1 I 10^
3 7:^
TT
4
X
4
^■MM*M
J^ 10 - 6^,anfwer.
17. 137 lb. ditto, at IS.
a^d. ?
^|i37
173
8«3
a 3
10
3
9i
TT^
X
6
I
T
6 17
I 2 10
2 10
^ 11 7 -l^anfwcr*
31. 957 lb. dittO) at I (.
i%l957
■*«
X 8 2 Si^anfwcr.
■Wi^
28. 713 lb. ditto, at X s,
2|d. i
lit
ry + T2
7»3
23
15
4
17
16
6
2
4
6*
i£ 43 »6 4i»anfw.
X
4
I
T
X
4
47 »7
II 19 3
3 19 9
19 lit
£ 64 15 ii^,anfwer.
32. 875 dls of Irifh doth,
at I s. 4^d. per ell ?
. ^875
X
4
I
T
i
"/J
43 15
10 18
9
5 9
18
4i
2i
jf 61 I 4^,anfwcr.
33. What
^2?
J 3. What coft 879 ells of
1 cloth, at I s. 5^ d. per
dl?
^879
PRACTICE. Book 1.
37. 875 ell« 4itto, at I V
7td.? • '
J.
3
X
•4
JL
4
43 ^9
14 13
3 13 3
^11 ■■
jg 63 3 64,anfwcr.
34* 871 ells ditto, at i 3.
S|d.J
871
t
•5T
6
r+it43 "
Z4 10 4
5 8 xoi
x8
f 3
I
jf 64 8 4j, anfw.
35. 171 ells ditto, at IS.
6id.?
1
171
1-
T
8 II
J
VI
T
6
3
6|
jC *3 - -4* *nf«^'
36. 137 eUs ditto, at I s.
^d.f
f
I
I
T
4
875
43 »5
21 17 6
3 12 II
18 2|
iC 70 3 7l»an'^cr*
38. 173 eUs ditto, at 19.
•5^
I
»
X
4
I
»73
t
8
4
'i
6
1
I
7i
3
7t
;f 14 4 8i,.anftr.
39. 375 ells ditto, at i tf;
Sjd.?
I
T5
375
I
T
18 ij
4
9 7
6 .
1
T
3 a
6
7
91.
t
137
I
T
t
617
386
8 6^
£ 10 »4 --I* ^^WT'
m*i
^■W"
40. 7M ells ditto, at i s.
8id.?
721
f
■T"rT
■ 36 I
18-6
6-2
25-1
jC 62 6 8|,anAver,
41. What
«•
Chap. III.
PRACTICE.
4i. What coft 307 ells of
Irifli cloth) at I s. gid. per
eU?
307
129
1
T
I
T
X
4
7 »3 6
3 16 9
19 at
^ 27 16 5J:,anfwer.
42. 317 db ditto, at IS.
9td.
X
V
X
317
■
15 17
7 18
6
3 »9
3
6
7i
^ 28 I 4^,anfwer.
43. 107 ells ditto, ^t I s«
lajd. ?
107
44. 199 ells 4itto> at i %.
-rVi99
T
1
16 II 8
' 5
4 li
2 1 5t
3
iC ^8 ^7 3j:,anfwcr.
45. 147 ells ditto, at i s.
Hid.?
TT
1 H
*47
la 5
I 16 9
, 3 -4
;C 14 4 9J, anfwr.
■^■^-•i
46. 175 ells ditto, at i s.
ilid.i
X
8 18 4
17 10
2 21
«* «
;^ 9 18 4I, anfwcr.
t+tV
TT
J.
4
17s
14 II 8
a 3 9
10 11^
j^ 17 6 4j-» anfiir*
c A s E vn.
When the integer ik pounds, {hillings, pence, and far-
ditngs,
R U L E^
Multiply the given quantity by the pounds ; and proceed
with the (hillings, pence, and farthings, as in the tore*
going cafes.
K I. What
PRACTICE.
1.3.0
I. What c(A 137 yards of
"brocade, at iL 17 a. 6Jd.
per yard ?
14s. -d. == 95 18
34 =; 22 16 8
2(i) == "^I 2 10
2 loj:
Book).
Anfwer ^257. -* 44
2. 2710 cWt. of fugar, at
3tl. 3s. 7id. ?
i + Ts I271
4* 947 cwt. of bops, at
4L «s. iqi^dr.? I.
A + XT
1 4s. -d. ^
I 8(-,V)=
947
X-4
3s.4d.=
3(i) =
66» 18 -
78 18 4
J-J V 7 17 »o
19 H
Anfwer^ 4538 13 loj
» I ' , U V *
5420
4S1 13 4
33 17 6
5 12 II
Anfwer y(; 59 1 1 3 9
3. 741 rwt. ditto, at 2I.
13 s. 7|-d. f
TC
13s. 4d. =
^3(A) =
741
X 2
1482
494
9 5 3
15 5t
5. 457 laft of colefeed, «t
14I. 17 s. 9Td. perUft?
T5 TT
457
6398
i^s.-d. == 319 18
3 4tt=. 7^ S 4
•iV 9 10 S
19 -i
Anfwer ^ 6804 10 9|.
6. 375 cwt. hops, at 3 I.
7 8. 11^4. f
Anfwer £ 1986 - 8^
7 s. 6d. =
4
I
TO
375
X3
t
T
H25
:=
140 12
6
X
4
- 6 5
-
4
I II
3
7
9i
Anfwer j^. 1273 16 6|-
CASE VIII.
When the given quantities are of feveral denominations*
RULE,
Chs^ m* PRACTICE ^31
RULE,
Find the value of the integers, as in the foregoing cafes ;
and for the leier denomination in the given quantity, if
they are the aliquot part of an integer, divide the glveii
price €bstthj ^ but if thev are not aliquot parts, divide
Acm incofuch, or of*eacn other, as you can mdft con-^
Teaiiendy^ theA add all together, their fum will be tho
AiiiOjtTOT Paats iM CtoTit MbAsuHK.
One yard the mtei-gen
Qrs. n.
2-3!
— Ti
tyot quarter the integer.
Kai}.
2 * =
Oao ell Engliih iritegen
Qrs* n»
2
I
2= i
1= i
One French eH integer*
Qrs. n.
3
- ?= i
2
I
2 ==
J.
4
X
8
3 =
2 =T'y
One Flemifh ell integei*!
Q«. n.
12 =
- 3 =
- 2 = i
- 1 =1^
X
X
3
X
4
■fci«M
1. 713 yds. 3qrs. 2n* of
kerfey, at 7 8« g-^d. per yard ?
6 8 =
Qj> 2 s:
Nails 3 =
^Anfw«r £ 278 2 9
713
3
3
?37
13
4
35
13
—
4
9
li
3
J04
I
"i
I
5^
2. 1 7 ells E* - qrs. 2n. 0/
gold brocade, at 3I. 10 s. 9 d« f
17-2
3 . .
3. d*
jl -. - s, d.
8 10 - =r 10 -*
8 6 = - 6
4 3 = - 3
7 -4= T-5 price*
TS
2 nails
^60 9 9 J, anfwer.
K 2 3. What
^ <
V
IS*
3. What coft 19 French
ells, --qrs. 311. of Brufiels
lace, at 3!. 19 s. 11 d.?
3 J9 "
3
PRACTICE!. Book L
6. What coft 719 ells FIc-
mifhi % qrs. 3 n. of fine HoU
land, 9t il. 10 s. <i\AA
3n-=i
ti 19
I
71 18 6
3 »9 "
9 "i
jf 76 8 4|, anfwer.
4. What coft 71 French
ells I qn ditto, at il. 17 s.
8td. ?
F. £• qr.
£• Flemifli.
I
T
I
qr.n.
I 2 ==
719
359 ID - ssios.
23 19 4 =: - 8d.
2 19 II =b - I
- 14 iii=: --
- IS 44=4
- 10 3 =s:^
- 2 6| =
i}
^+7
•IT
iqr.
71
2
s. d.
- =: 10 -
4=68
- 2 ni= - -1
- 9 7i=:i price.
142
35 10
23 13
^ II
^ 205 6 10^.
5. What coft 709 French
ells, 5 qrs. 311. of ditto, at
14 s. 7id. ?
F. £• qr. n.
•iV+TV709 5 3
jf 1 107 12 5, anfwer.
7. What coft 4 pieces of
riboon, each 17 yards, iqr.
3 nails, at i s. i^ d. per yaid ?
Y. qr. n.
17 I 3
4
•5-cr
qrs. J
69 3 -jatijjd.
3 9-
- 8 7i
X
6
X
4
^ 3 18 5I, anfwer.
8. What coft 1 3 ells, 2 qrs«
2 n. of Holland, at 3 s. 7^ d.
per ell Englifli ?
3- =
-3=-i
496 6
- =14-
X
9
17 14
6 = -6
2 19
I = - I
- 14
9;-= - -t
- 7
- 4
3--=' I p
j(
- I
9i
jCsiS 8 3i
3
7\
qr.
= 2
anfu
a
7
I
It
91'
n.
2
2
8
"i.
rer.
AtiquoT
\
\
Chap. m. PRACTICE. 133
AtiQpoT Parts i« Troy Weight.
One ounce the integer. One pennyweight integer.
awt. gr.
J
'
gr.
10 - =
»2 = i
6 16 =
1
T
1
8= 4
5 — =s
1
6= t
4 • =
I
T
«
4= i
3 8 =
I
T
3= t
2 12 =
t
1
2 — tV
2 — SS
N. B. 4I. per
I 16 =
I
01. it ad.
I - =
I
ITS
«
pet g'un.
9. A filver gilt punch bowl, weight 49 oz. 2dwt. la gr.
what comes it to at 8 s. 1 1| d. per ounce ? ^
oz. dwt. gr.
1%+tV +
1
J.
4
49
2
12
dwt gr.
2 12
'9
I
12
12
12
3
I
- at8s.-d.
8 at- 8
3 at- 3
i \ •
i^^-J: price
iC
22
I
X, anfwer.
10. A pair of chafed filver
ialts, weight 7 oz. 5 dwt. at
8 s. 9^ d. per ounce ?
X
4>
8
9i
7
I
2
8t
iC 3 3 f^y anfwer.
II. I demand the value of
a fervice of plate, weight 971
oz. 15 dwt. 16 gr. at 3!. 19s.
ii^d. per ounce?
oz. dwt. gr.
971 15 16
3
X
dwt.*
10 -
4 -
I 16
2913
12
3
I
18
18
2
J5
6
4
9
'4
nl = i
oz.
7i
;C3386
K3
4[,anfweit
Aliquot
3. What coft 19 French
ells, -^qrs. 3 n. of Brufiels
lace, at 3I. 19s. iid. ?
3 »9 "
3
PRACTICE. Book I»
6. What coft 719 elhFle-
mifh; a qrs. 3 n. of fine Hol-
land, at 1 1. 10 s. 9j^d. i
3n-=i
II 19 9
6
71 18 6
3 ^9 "
9 "i
£j6 8 4 J^, anfwcr.
4/ What coft 7 1 French
ells I qr. ditto> at 2h 17 s.
8td. ?
F. E. qr.
TV
I
T
qr. n.
I 2 ==
£• Flemifli.
719
359 10 - =rio«.
23 19 4 = - 8d.
2 19 II zh— I
- 14 11^= - -
• 10 3 =^
- 2 6| =
1}
TT
iqr.
3. d.
- =:io -
4=68
- 2 Ili= - -1
- 9 7i=i price.
142
35 10
23 13
3 «'
^ 205 6 10 J.
5. What coft 709 French
ells, 5 qrs. 3n. of ditto, at
148. 7-Jd. ?
F. E. qr. n.
•7i>+t\j709 5 3
jf 1 107 12 5, anfwer.
7. What coft 4 pieces of
riboon, each 17 yards, i qr.
3 nails, at I s. i| d. per yaid ?
Y. qr. n.
'7 I 3
4
qrs. ^
69 3 -,ati3id.
s. d.
496 6 - =14 -
i 17 14 6*= - 6
t 2 19 I = - I
- 14 9;= - -t
3- =
-3=;
£ 3 iS si, anfwer.
8. What coft 1 3 ells, 2 qrs*
2 n. of Holland, at 3 s. 7-Jd.
per ell Englilh ?
- 4 io.:=t \ P-
91
- 3 74
»3
7 it q**' n.
I 9J = 2 2
- I
i£5i8 8 3J
^2 8 I li, anfwer.
Aliquot
^
Chap. m. I* R A C T I C E.
'33
Alkicot Parts in Troy Wbicht,
One ounce the integer. One pennyweight integer.
dwt.gr.
lO - =
6 i6 =
I
T
1
T
4 - =
3 8 =
212 =
1
T
I
T
1
T
1
I 16 =
1
TV
1
TT
I
To-
gr-
12
:r:
1
"4
8
6
z
1
T
X
4
4
r=
i
3
=
J.
8
2
—
1
TT
N.B
- Al
. p«r
Ol
. M
2d.
pergr
am.
9. A filver gilt punch bowl, weight 49 oz. 2 dwt. la gr.
what comes it to at 8 s, 1 1| d. per ounce ? v
02. dwt. gr.
Tff "7"
Tff
49 2 12
«
»9
12
- at 8s,--d.
1
12
8 at- 8
dwt. gr.
J.
4
-
12
3 at- 3
2 12
-
3
H
—
I
ii = i price
I
22
I
I, anfwer.
lo. A pair of chafed iUver
lalts, weight 7 oz. 5 dwt. at
8 s. 9^ d. per ounce ?
8
9k
7
3
I
8t
2
2t
jC 3 3 ;^> anfwer.
II. I demand the value of
a fervice of plate, weight 971
oz. 15 dwt. 16 gr. at 3I. 19s.
ii^d. per ounce?
oz. dwt. gr.
971 15 16
3
X
dwt.*
10 -
4 -
I 16
I73 »8
80 18
12 2
3 -
I 19
6
4
9
8i
1 14 = f oz.
Hi
7i
^3886 4',anfwert
K3
Aliquot
IS4>
PRACTICE,
9ook I,
Aliquot Parts i» Averdupoise Weioht.
<
One tun the integer*
Cwt. qr.4b.
10 T- - 5?
5 ^ - =
4
2
2
2
3 n
2 -
X
.. X
— 4
... X
— s
*-^ 7
-^- J-
-C 8
-_ 1
— TXT
^_ «
One hundred integer*
Qr- Ib^
2 =Z 56 =5
; = 28 ;7
1 I •.
X
s
X
16 = I
14== t
J Cfrf. = 56 lb. iiiteger.
Qr. Ik
1=28 ;p
8 x=
7 s= .
^Cwt* or 28 Ibw integer,
7 =
4 =
2 8 = e^
One pound theanteger.
X
ft
X
t
X
7
X
8
X
%
4-
X
7
X
oz.
8 =
4 =
2 =
X
4
Op^ ounce the integer.
dr.
» = t
4 — t
2 =:
:jl
n» Whatcoft jjcwt. i qr-
of lugar, at 3 !• 153, yd.
oer cwt. ?
per cwt. i
Cwt.
3
I qr. =
13. WhaA cofl 731 cwt.
3qrs. of hops, at 3I. 18 s.
yi d. per cwt. ?
i^ + V
f'c
>i9
43 16
12 3 4
■- 1.8 3
- 1.8 io| = tp.
^276 16 sh *"'"•
TV**^"*^**
1
Cwt, qri
731 3
3
^193 „
657 1.8
18 5 6
4 U 44
' '^ ^^=||P'
- 19
j^zSjd 13 JO, anfw.
Ctnap. III. P k A C
I4k What coft Jr cwt. 1 qrs.
iblb. of treacle, at il. 17 s.
S d« per cwt. ?
]. $• d.
ft
X
7
JL
4
I
>7
8
7
13
3
8
—
18
10
-
2
5"
—
-
8
jf 14 5 loj, anfwer.
15, lycwt. iqr. 12 lb. at
il. 19s. 8d. percwt*?'
ITT
Tt^
TT
qr. lb;
X -
7
4
I
184
- 9 II =?i:Pricc.
- 2 5-J=:Joflaft.
-J I 5 =|of dit.
- - 4:{piJof laft.
1 1 c e; 135
1 7, What coft 1 7 hogiheads
of treacle, at 1 1. 12$. 7d.
per cwt. each hogfhead weigh*
ing5CWC. 2 qrs. 8 lb. i
Cwt. qr. lb.
£ 34 8 &9 anfwen
I
qr. ]b.
2 -
16
8
5
2 8
3
16
a 24
6
100
I 4'
S
2 8
94-
2 24
56
8 -
2
7 -
—
7 10
-
16 31
—
4 7J
. —
2 3l
=1?
;{^i54 6 I, anfwer.
16. What coft die freight of 7 ton, 13 cwt. 3 qrs. iglb.,
alt 14 1. 17 5k 9 d. per ton i
•y + T
I
\ 14 17 9
7
104 4 3
7 8 roi
2 2 6J
1 1 =: {• of ^ cwt.
£ I f 4 ri 5tj anfwer.
T
Jl.
s
14 voi
r si
- IS d.
18. Whrt
15+
PRACTICE,
Book U
Aliquot Parts h Averdupoise Wiiqht.
One tun the iptcgjcr
Cwt qr.4b,
5 ^ - =
4
2
2
2
3 n
2 -
•»- X
_ A.
— 4
_ X
— $
-— X
*-- 7
•^ X
___ t
— T-B-
__ I
One hundred integert
Qr. lb,
2
HH
S6
=s
X-
%
1
zaz
28
;;:
X
4
i^
=
X
T
J _
^^
14
==
X
t Cift. = 56 H». integer.
Qr. Hk
J = a8 ;s
H *=
8 s=
^Cwt. Of a8 lb. Inte^.
7 =
. J= I:
One pound tlie4ntegeF.
oz.
X
%
X
•
X
7
X
•
X
X
4
8 =
4 =
a =
X
%
X
4
On^ ounce the integer.
dr.
» = f
4 = i
2 =
UL
rz. Whatcoft73cwt. iqr.
of lugar, at 3I. ?5S^ yd*
per cwt. ?
Cwt.
+ tV 3
I qr- =
13. Whajt cofl 7JI cwt.
3qr8. of hops, at 3!. 18 *•
74. d. per cwt. I
- Cwt. qr^ .
A+T^ 731 3
3
219
43 16
12 ^
- I.
4
3
> io|=tp.
^276 16 sh ^nf-
1
I2I93
6S7 18
18 5 6
4 U 4t
I 19 34=i?D
^19 7i=ir
^2876 13 10, anfw.
' Ti. What
Chiep. III. P k A C
14. What call ^ cwt. 1 qrs.
id lb. of treacle, at i L 17 s.
8 d. per cwt. i
]. s. d,
I 17 8
7
ft
X
7
13 3 8
- 18 10
- 2 8t
- - 8
^ 14 5 loj, anfwcn
15, 17CWL iqr. 12 lb. at
1 1. 19 s. 8d. per cwt.?*
9 _L ■
qr* lb.
I -
7
4
I
I 8 4
- 9 II =i: price.
- 2 5^^^oflaft.
X f 5 rz-^of dit.
il h III
£ 34 S ^9 anfwen
t I C E; 135
1 7. What coft 1 7 hogflieads
of treacle, at 1 1. 12 s. 7d.
per cwt. each hogfliead weigh-
ing 5 cwt 2 qrs. 8 lb. i
Cwt. qr. lb.
5
2 8
3
16
2 24
6
•
lOD
I 4'
S
2 8
« r 1
tsTtw
9+
2 24
56
8 -
1
2
7 -
qr. 10.
-
7 10
2 -
—
16 3l
16
-
4 7J
8
! •-
2 3t
sf}'-
j£ 154 6 I, anfwer.
16. What coft the freight of 7 ton, 13 cwt. 3 qrs. iglb.,
at 14 1. 17 Si 9 d. per ton ?
4^ + T
f H 17 9
7
X
TV
■■
I
104 4 3
7, fr roi
2 2 6i
r4 loi
iissyorl-cwt.
jC I r4 ri 5t, anfwer.
I
X
f
14 lOj.
7 5i
- lid.
.- i^4
18. Wh»t
156 PRACTICE. . Book I.
i8. What coft the freight of 37 tons, 19 cwt. 3 qn. at
19 1. 19s. id, per ton?
T. cwt. qrs.
37 19 3
■ 1. s.
d.
- per ton.
price.
£ 758 3 4> anfwcr.
Alkuuot Parts irt Land Measure.
One rood the iiftegen
Poles.
' One acre the intjrger.
R. P.
2
-
Z3
s
X
-■
=:
—
32
=
1
T
—
20
zzz
i
—
16
=
1
TV
20 =
1
T
10 =
t
8 =
i
5 =
t
4 =
B
TIF
2 =
•t
TV
19. What is the rent of 7 13 acres, 3roods> 39 perches
of flax-land, at 3 L 17 s. 6 d. per acre i
At Xvk Jl •
X 4. i
+
8
7'3 3 39
X3
R. P,
2139
356 lO -
89 2 6
2 -
I 18 9
J -
20
10
5
4
X
%
1
T
- 19 4i
- 9 8i
- 4 ">
- 2 5
- t lit
= t? Pnce.
^2766 14 69 anfwer.
20. What
Chap. III. PRACTICE.
m
io. What is the rent of
17 acres, 3 roods, and 35
perches of flax-land, at 4I.
per acre ?
17 3 35
4
R. P.
2 -
I -
- 20
- JO
-^ S
^8
2 - -»
I - -
- 10 -
- 5 -
- 2 6
jC 71 17 6, .aniwpr.
21. 17 acres, -roods, 10
perches, at 2 1. 138. 6d. per
acre?
A. R. P.
"A + 1:1
•JTT
ft
Per. 10
17 - 10
2
34
10 4
- J 6
- 3 4 = iofi
jC45 12 io,aiifwer.
N. B, This belongs to Example 2n
• «
X
4
2 13
6
1
»3
4i
3
4
Capttia
13*
P R A C T I G ft.
Bai»kl.
. W O 000 09 00 00
O , • I I I
w
M
a^ W M M ^ I ^
r* t; ti 15 "W t; w
*" « rt « rt cs ^
O
X
o
C4
CO
§
I
I
I Pi
» o^ q M
S « *-•
•g «i
c »-^
U9' V U ^ ^ ^ <^
09
to
c<
•
i5i
9
"*
M
1)
<L1
*
b
U v;s
*a
^
4>
«i
4>
4-*
s
r^
its
^
^•O
is
U
o
E
8
c
4.J
2
o
^
00
00
o
S?
£^_: « «o IN o o o «
JJ^ •>« 1^ »i« M
2» ^ 1%. t^ r^ <*voo' xj- ^
.g CO M m M M
ON
s
*r Si
II I I I » I I ^*
f; s I I • • I f
O
o ..
^ £ <| Ml I I fOM H
2 V t^« '♦•^ « « «<
o
<
X
<
s
<
X
o
«o
<^ » O
^•*5 iirf t'* *. .•
CO (^ h P< a. pq pq
Cwt
^^
cup. IIL F R A
Cwt. qr. IV. 1. 8. 4t.
S ? 04 at I 17 4
S
C T I C E.
Cwt* <|r. lb,
12 I 19 at
«99
- 18
*- 5
- a
8
8
4
8
;C 10 J3
4
% IQatz 19
6
4
6 10 •••
• - 16
2 3i
T- ^ 3
- - 6i
iC 7 9 »i
t
3^2}atqt x6 lo
2
f"i
8
J - 14 2i
- 3 6i
4» ^"t
»7
2
17 at T 13
10
4
-
' w
6 IS
4
4
27 » ♦
I 1} 10
y - 16 II
a 5
- 2
f-T
4.1.
X
14 S
12
so
T
fco 13 -
- 8 7t
4 "
l*i 7 S
4 - l» at 4 19 4
- 4
'9 17 4
t) 2 9 » i - 7 I
3 6f
jg 20 7 itj
I - 15 at 2 16 4
1, t) ll. 8s. 2d. 4 •
- 3 _
43_l£2s
I 3 iQatTj 12 4
6t
I 16
■f - 10
2
I
4
3i
li
£ 6 18 I0{:
I
2 I xgat^i 19
I
2
3 I? 2
i,|)i9$.6id. - 9 9i
- 1
X
%
ii
iC 4 '4 H
2 2 laatTi 18
T
4
_ 2
3 16^ 8
19 2
- 2
- f
^41911
The
I
13^
PR A^ TIG ft.
Beibkl.
jj foo^ rod t*i'*
V o opooooooo
t
O
O^
• •II
ffi^ IN M M ,^ J QQ
P< <»-• *i -W 4j ♦. ^ '
^ c« a c« ci 6< «i
u
O
X
o
N
in
CO
03
0.3
■ ,^*9
M 1^
J3 JO
uu;
I
I
I 0)
^C M
aj •• r*
u •"
»-^ •— Ti
00
00
o
Srt
►>-4 Ct «0 m o G iT m
,e CO M m M M
p^ c4 M d ^ ^ ^
I I I » I I
I I • I I
H
o
H ^ ti ^ ti ct tt H
^^ M M >^ •« INI M 14
o •
g ^ t^« "♦« M c« e<
S
<
X
<
s
<
X
o
€4 C«
*» ^ CJ
O ^ M fcja
r>. ^ « y ►^
bO
.^ Q
CO
■a
c/3 0< h P^ 0^ PQ cq
Cwt
CbafL IIL P R A
Cwt. qr. \h. 1. (. <!.
5 2 24 at I 17 4
5
- 18
- 2
8
8
4
8
iC 10 J3
4
;i IQ a( I 12
6
4
6 10 *
3
4- 2
iC7 9 U
3,2jaia x6 10
2
s
I
'1
8
5
X
-
14
2i
1
T
—
8
H
•^
3
I
6i
»7
£
a
8-
"t
2
17 at
y
13
10
4
■ 1
^
6
IS
4
«
,
4
a? > 4
I IS 10
j, -f - 16 u
t - 2 5
- - 2 ij
- - 7t
jC 29. X7 2i
c
TICE.
Cwt- qr. lb*
12 I 19
4.1. s. (L
at I 14 5
12
•
«o 13 -
- 8 7^
T- 4 "
- l»at 4 19 4
4
t)2 9 »
t
T
19 17
- 7
2
4
I
£ ao 7 »'T
I - 15 at 2 16 4
i, t) il. 8s. 2d. 4 -J
I 3 i9atT3 12 4
I 16
f - 10
i- I
2
I
4
3i
2 I I9at:»i 19
£ 6 18 ic{
I
2
1 —
i, I) 19s* 6id.
X
ft
2 2
3 18
2
- 9
9i
- a
9t
- 2
5t
- I
4f
,,4 14
6i
Tl 18
4
2
"3I6-
8
•f-»9
2
- 2
H
- r
Ji
jC4ii"
The
i4» PRACTICE.
Book I.
a d d I-* -^
S Si I > * •
o g
C9 o I ■ I I
<
o
s
. JL' • ' '
n ts«>^ ^ ^ ^
O 4J ^ 4-1 M «4
^ f« e« i« cl ci^
o ^ ►^ HI « »H a
1^ M M »4 Nl a'
o
M
•4
5
S
s
CO
o
a
o
S
i
1
.M
•a <H
W ^O ♦J •*f > OS
s "S t s?/ § €
M
oc dwtB, gr* •• d«
S5 lo lo tt 7 9 (eronace*
I tS 9
9 13 9 OS.
i»H7 9 1 "I $
« 6\ 4
• 4} » r.
iC 9 >7
t5 14 15 at 6'
1%
3
I
1
6
3,
It
dwC. gr.
10 -
4 -
- t%
t
- 3
jC *7 17
•9 i€ 15 at 6
4 parouw.
X9
J
i 11"
t !
t
ft
7
9
iwt. gr*
10 •
5 -
I -
- 1%
£ 9 < 'ot
^7 • t6 at 6 10 pcrottoef.
6
ft I -
It
•s
II
m
4*»t. f
6
10
QWta
r-
I Si
J
•
ift
4
,C^5"
iS
3J ««
10 at
ft
"TT
"5
•
II
awt.
.
10
3
T
r*
-
-«^f
3
s
10
-
1
«^
3i
1
#
I
m
8
£'° *
i
The
Chap.
.m
&
■
4^
jO
u
0
Q
r
CO
J
a
P4
>«
«»
•a
«»
«
s
.4^
«•
'1
•
o
X
■^
.9
•>
3
:i5
»4
ft
<
S
n
•2
i4
5
at
^
<
^
O
o
g
"S
«S
Ui
A
«<
96
<
X
D
m
•
w
1 .
pa
9
S
^
o
s
^
60
(1<
• I t
. • • •
I I
• X*
s «
P R A C T I C R
« m *
^MM mm m m m m
I I f I 1
.|. . .
;uii
Hut
^e«Nif>oiM| ft iiN.1
t , I I* • I I I I I I I •"
I I I * > I I I x^ I .1 •
III .l||g'1|.f
«< ci
2L&fi«^
to
^ CI en ii^
d
o
'3
t
09
lA
141
I. What
f4* PRACTICE. fiookt
I. What will the carriage of 17 cwt. 3 qrs. ti lb. coitie.
(09 at the rate of 7 s. the hundred i
■
78, percwt*
iC5 ^9 ""•••• CarriAg* of 17 *• ' -*
- 3 6 -k --*-.-.-, - - 1 *
- - St.- - - - ^- 7
iJAi
jC6 4 11^, the anfw#n Cwt 17 311
56 pieces
3)
5s. 4d. per ell Fltmifli*
34
'
I gi
aa4
I
7 It per yai:d.
•-
z68
1
• •
)i904 cUsEBg
'
476
4 »
2380 yards in
lall.
. 1 '
r
Pence.
s^ d« 1. s.
d.
*)238o =
198 , 4, = 9 18
4'
595
. ' ' '
238
»
0 IB
4
V
3 6
li
S46 4
5}, the anftjirer.
3* If one ounce of filv^ (plafey bullion coft 55. 4i(L
what will be the value of 14 ingot5^ each weighing 28 os«
15 pwts. la gr^ I
Ok.
Oz. dwt. gr.
28 15 J2
2
37 II •-
PR AC TtCE;
Hi
• .
KJ
i
5
X
I
T
I
I
T
Wi^»
402 17 - at 5s. 4i.d. per o7..f 4J5 44
80 8
20 2
6 14
16
4
If
'"*
■t-
I 4
^— VWM^Vrt
4 6|-»p«qf i^dwt*
f
.1.
108 5 3i, thp wfwr rqciw^. . . A
CASE IX.
DUODECIMALS.
t . . .' ., It
Duodecimals arc fo called, becaufe tkey dccrcafe by
twelves from the place of feet, tow«irds the rigbt-ljahd \
the inches I call praies, the next fecon^s, thirds, &c« ac*
cording to their diftance from ^t. ■ . : ,
This rule is fom^times called croTs^muMpIication.
Rules for multiplying Doodecxm als.
Having under the multiplicand written the corrc^oniifig
denomination of the multiplier ; multiply each term in
the multiplicand, b^ginnins at the Jov^, by the feat iathe
multiplier; write each refult under its refpectivc teim^ aad
carry an unit fpr ever 12, from each lower denomination to
its next fuperior.
2. In the fame manner multiply each term m the multi-
plicand by the prime (or inches) in the multiplier, and
write down the refult of each term, one place remo\:ed to
the right-hand of thofe in the mukiplicand.
3. In the like manner multiply with the feconds for parts
of an inch) fetting down the refult, one place ftjlt further
to the right'hand ; and the fum of all thefe give the pro-
dud required.
Let it be required to multiply 9 feet 7I inches by 7
feet 104 inches.
Thus
144 PRACTICE. Book I.
F. inch. f. / //
Thu$ 9 7is= 9 7 9
And 7 10^ =s 7 10 3
67 dj--! ry feet.
It - 5 6 - > multiplic. x < lo primes.
2 4 II 3 3 L 3fccond^
75 9 Jf 5 3» Acprodua.
It will often happen, that the feet in the riven mukipli*
cand are fo many, that to multiply them by toe lefler deno-
nunations, and take i^ of the produd as before direded,
will require fome work to be done on fpaK paper, which
may be avoided by obferving the following
RULE.
Multiply the feet firft ; then inftead of multiplying by
Ae primes or inches, take an aliquot part of the multipli-
cand, according to their correfpondin^ inches ; thus, for x
prime or inch, take ^^ of the multiplicand, for 2 inches^
take 4, forjtake^, for4takc|, for 5 takej + iort +
TT, for 6 take f , for 7 take | + i» or J + ,!„ for 8 take
4+tvori + T» for 9 ^J^c i + J, for 10 take i + t,
and for 1 1 inches jtake 4- + 4 + i > ^nd in like manner for
ieconds or parts, only obferving ^at the laft quotes are only
^ part of the foregoing, and muft accordingly be put one
place further toward the right-hand;
•
Let it be required to multiply 368 feet 7I inches, by 9
feet 4| inches i
Feet. / //
368 7 6
9 4 9
3317 7 6
122 10 6 - - = 4^ or 4 primes.
*S 4 3 9 " = T ®^ TT» ^r 6 feconds.
7 8 I IQ 6 =t^^^helaft, or3feconds.
• I
3463 6576
279 Feet.
I "i
Chap. m. PRACTICE. t45
Feet. / //
279 5 3
796
1956 - 9 - -
139 8 76 - zz: i or 6 primes,
69 10 3 9 - = J- of the laft, or 3 primes.
II 7 8 7 6 = i of the laft, or 6 feconds,
2177 3 4 10 6
• _
But if the multiplier alfo be a large number,, multiply the
feet into each other ; then for the primes and*feconds in the
multiplier, proceed as in the laft examples ; and for the
primes and feconds in the multiplicand, take aliquot parts
of the feet in the multiplier ; the fum of all will be the
anfwer required.
Feet. / //
187 10 3
73 7 9
561
1309
9311 I 6 - = t for 6 primes.
15 7 10 3 - == 6 of the laft for i prime.
7 911 I brr-J-ofthe laft for 6 feconds.
3 10 1 1 6 f)i=: -^ofthe laft for 3 feconds.
36 6 - - - = t<>f 7 3 feet for 6 primes J in the
24 4 - — — :=:tof73 feet for 4 primes > multi-
I 6 3 - — = -Jof73 primes for j'" J plicand»
13834 8153
CHAPTER IV.
VULGAR FRACTIQNS.
SECT. I.
NOTATION.
A FRACTION, or broken number, is that which
_£\ reprefents a part of any thing propofed, and is ex-
prefled by two numbers, placed one above the other, with
a line drawn betwixt them.
L Thu«
14^
VULGAR
Thus J 3^ Numerator.
C 4 Denominator.
BookL
The denominator, or number placed imdemeath die line,
denotes how many equal parts the integer or whole thing is
fuppofed to be divided into, being onlv the divifor in divi-
fion ; and the numerator or number placed above the line,
fhews how many of thefe parts. are contained in the frac-
tion«
A vulgar . fraction is either proper, improper, fimple, or
compound.
A proper frafiion is fuch, whofe numerator is lefs than its
denominator, as 4> 79 79 or
13 19
&C.
An improper fradlion is fuch, whofe numerator is equal
to, or ereater than its denominator, as s *• ^9 ~9 &c.
Here note, that if the numerator and denominator are
equal, the fradion is equal to an integer.
A fimple fra£tion hath only one numerator and denomi-
nator, whether it be proper or improper, a» 79 79 79 V»
&c.
A compound fra<^ion, or fradion of a fraAion, hath feve-
ral numerators and denominators conneded together by the
particle of, as -f of •{- of ^9 by which is meant firft, that
the integer or whole thing is divided into five equal parts,
three of which parts make 4^ which fra&ion is divided into
eight equal parts, and feven of thofe parts taken, viz. ^ of
\ ; then this fra^ion is divided into (even equal parts, and
tvvo of thofe parts taken, viz. -^ of •{. of |^
Suppofe, for inftance, a pound fterltng to be fo divided,
20 s.
4s. X 3= 128. =5:4^,
8
I2S.
IS. 6d. X 7= lOi. 6d. = f of ^of £ u
10 s. 6d.
IS. 6d. X 2=:3S.= ^ofiof -f ofapoundfterling.
A mixed number is a whole number with a fradion an«
ndced, as 5^, which is read five and thrce4evenths s 2I7 is
twentv-one and one half, &c.
^ S E C T,
v^
'^
Chap. IV. :Ptt ACTIONS. 147
I
S E C T. IL
R£DitCTiON if Vulgar FractIok^.
N order to facilitate the dodrine of vulgar frafllons^
I fhall premife the following
AXIOM*
If both the numerator and deiiomlnatof of a fhidion ht
multiplied or divided by one and the fame number, the
iraflion will retain the lame value.
Viz. f X 4 = fh *»^ tJ ^ T = ii ^^^^ is, if the nu*
tnetator 7 and the denominator 9 be each muhiplied by
the fame number, vi^. by 3, the produced fraction, viz*
ijj ^^ ^ prqpofed one ^ are equal, as the numerator
and denominator of the fifft are in the fame proportion ai
the numerator and denominator of the fecond.
Alio if lihe numerator 12, and the denominator 16, be
each divided by the fame number, vis^. by 4, the fra^ons
i and 14 fof ^^ f^^°^^ reafen are equal*
CASE I*
To reduce z compound fraftion into a fingle one*
RULE.
Multiply all the numerators into oneanother for a t\W*
merator, and all the denominators Into oneanother for the
denominator.
1. Reduce 4 of 4 of ^ of V^- into a iingle fra<^ion.
V^ L^4 ^ i- =fi ^, the fingle fraftion required*
- If a numerator of one term in a compound fradion he
equal to a denominator in another term, cancel or reje£l
both, and dfvide thofe numerators and denominators whiclt
are divifible by each other, or by the fame number; which
quotients multiplied into the remaining numerators and de-^
Mminators, reduce the compound fra^on to a fingle ona
in its loweft tenni.
t ft let
148 VULGAR Book I.
Let the laft example, viz. 7 of 4 of 4 ef ^ be reduced
into ji fingle £ra£tion, and its loweft terms.
?6f /of .5of l = i. = i^
y 3 f Ji 33 792*
3
^ 2. Let f of |. of 41 of 4 be reduced into a fingle fradion
in its lowed terms.
;f 4 g I
io( ^ of ^ of £^— , as was required.
^ 0 If 12
CASE IL
To reduce mixed numbers and integers into improper
fra&ions.
I fliall divide this cafe into three parts.
L If the integer have no affigned denominator.
RULE.
An unit fubfcribed muft be the denominator.
Thus 7 = ?, 12 = i?. 56 = ^, 248 = ^, &c.
IL If the integer have an affigned denominator.
RULE.
Multiply the integer by the affigned denominator, the
produ£l is the numerator to the affigned denominator^
Reduce 17 into a fraction whofe denominator fliall be I2.
Thus 17 X '3t =s 2C^. numerator, ••*- — =: 17.
III. If the integer have a fra£Hon annexed.
RULE.
Multiply the integer by the denominator, a<id to the pro-»
du£t add tne numerator ; the fum is the numerator to the de-
nominctor of the annexed fra^on.
Chap. IV. F R A C T I O t^ S. 149
Let 7|, 214^ and 11914 ^^ reduced into improper
fra£lions.
Firft, 7 X 8 + 7 = 63, 21 X^7 + 19 = 586, and 119 x
38+35 = 4557-
Therefore ji = ^, '2144 = ^, and iigfj = li|Z.
CASE III.
To reduce an improper fraction into its equivalent, whole,
or mixed number.
RULE.
^ Divide the numerator by the denominator, the quotient
- gives the integer, and under the remainder (if any) fubfcribe
the denominator,
»
Reduce-^, i- , i-iZ, into their equal, whole, or mixed
o 27 3* • •
numbers.
8 ) 63 ( 7i = i^ 27)586( 2iii = ^, and
7. • ^9
38)4£i7(ii9|i = lgl.
2. Let — — , i-i, and — , be reduced into their equiva-
12/ 7.3
lent, whole, or mixed numbers.
12)204(17 = ^, 7)364(52=4^, and3)io8(36
^ C A S E IV. '
To abbreviate or reduce fraAions into their loweft or leaft
denomination. '*' «
If the numerator and denominator are even numbers,
take half the one, and half the other, as often as may be ;
and when either of them fall out to be an odd number,
then divide them by any number that you.can difcover will
divide both numerator and denominator without any re-
mainder.
Or, by iinding the greateft common meafure by the follow-
ing •
L 3 R U L g.
ISO VULGAR Book I.
RULE.
pivide the greater number by the lefler, ^nd that divifof
by the remainder (if there be any) and (o on continually
until there be no remainder left. Then will the laft divifor
be the greateft common ineafiirt) which if it happen to be
J, then are they prime numbers, wd are already in their
loweft terms ; butif otherwife, divide the numbers by the laft
divifor, and their quotients will be their leaft terms x^^
quired.
102
I, ^t -2y be reduceil ipto its loweft tcnu.
330
3)
By finding the cenunon aeafure.
I
%. What js — |-. in its loweft tcqna ?
1 1 04^
2) 37)
*/ 1184V 59»y 296U ~ 1184
By finding the common meafurc*
CASE V.
To alter or change different fraAions into one denomiaa*
tion, xMiining the fame value.
R U L E-
Multiply all the denominators into each oth^ for a new
^d common denominator, and each numerator into all the
denominators but its own for ^ new numerator.
I. Reduce |, f, and ^, into fradions, having one com<^
inop denominator*
Firft,
Chap. IV. FRACTIONS. 15*
Firft, 4x9X5=: zSo, common denominator.
-Alfo 3 X 9 X 5 = 135 )
7X4X5= 140 ? numerators.
2x4X9= 7^^
Therefore ^ = ip, ^ ;=: ^, and |.= ^.
♦ 180' * 180 ^ 180
2. Reduce 4, t of | of |^, and 3I, into fradionsy hav-
ing one common denominator.
Firft I of ^ of 1= tV, and 34 = ^
Z 4 S 7
ft
The firaAions reduced to fingle ones will be 41, ^ and
Firfty 3 X <o X 7 =s 2'0» conunon denominator.
Alfo 2 X 10 X 7 = 140 )
iX 7X 3= 2i> numerators.
a6 X 3 X 10 =s 780 J
•. -4 = 1:^2, 3| = ^^=:Z!2. And'of ^ofisi.
^ aio' ^^ 7 210 T ^ T lo
XI
VN» ' ' '■•
210
If there be two denominators already alike, yeu need
multiply but by one of them, as in the following example,
3. Reduce ^ \. of {■ of -f , 5, and 254 into fradions,
having one common denominator.
Firft f of iofi = 1, 5=4, and 254= ^5.
I«t
In fingle ones, ^, f , 4, ii
8 X 1X5= 40, common denominator.
7X I X5= 35 J
SX8x|= 200 f "^^^^^'^-
128 X o = 1024 J
Viz.i = ^ iofiof|= • — J, 5 = 1 = —.
■ 40 ^ "^ ^ ^ 40 "^ I 40
. J _ 128 1024
And 25I = -» =9: -—^^
^ li 4 . [jt. When
t52 VULGAR . Book L
2. When there arc only two fradions to be reduced, if
one of the denominators is a multiple of the other, divide;
and by the quote multiply the numerator and denominator
of that fradion which hath the leaft denominator, and the
fraction thus found will be equivalent to the given ones.
Reduce -^ and 4J- ^^ ^ common denominator.
• • • 45^ and -J^J- are the fra£lions required.
3. Or if both of the denominators have a common mul-
tiple, divide each of the denominators thereby, and mul-
tiply the contrary numerators and denominators by each
contrary quotient.
.Let 4 ^nd -J* be fra&ions propofed to be reduced.
As 2 will meafure 6 and 8, their refpedlive quotes be-
ing 3 and 4.
Then 3X 8 =4x6r=: 24, the common denominator.
Alfo 5 X'4 = 20, and 3x3 = 9, the numerators.
• . • -J == >|, and ^ =z ^\j the fradion required.
Reduce -^ and ^-t ^^ ^ common denominator.
Divide by 5. . . 4 and 3 are the quotes.
Then 20 X 3f or 15 X 4 = 60, the common denomi-
nator.
Alfo 7x3 = 21, and II X4 = 44> numerators.
Therefore ^j,^ = *^, and ^l = |>, are the fradions re-
quired.
CASE Vf.
To reduce a fra£lion to an eauivalent one of any other
affigned denominator ; viz. to find a numerator, which,
with the affigned denominator, will make a fraftion equi-
valent to the propofed one, when poiBble.
RULE.
Multiply the affigned denominator by the numerator of
the propofed fraftion, and divide the produft by the deno-
minator 5 the quote (if there be no remainder) is the nu-
merator fought.
Reduce -J to an equivalent fra<aion> having for its deno-
minator 18.
Thus
Chap. iV. F R AC T I O N S- 151
Thus 28 X 3 = 84 > then 84 -f- 4 = 21, the numerator ;
that is, . -Jt =5 ▼•
Whenever the denominator affigned is divifible (without
a remainder) by the denominator of the given fra^on, the
thing is poffible^ otherwife not.
CASE VIL
To find whether one fradion be greater or lelTer in vsdue
than another.
RULE.
r
Multiply the numerators into each other's denominator,
and if the produds are equal, the fractions are fo } other-
wife the numerator of the greateft fraction multiplied by the
denominator of the other, will be the greateft produiSl.
Which is the fradion of the greateft value, viz. |^, or 4 ?
Thus 7 X 6 =r 42 ; but 5 X 9 rr 45, confequently 4 is die
fra^on of the greater value.
Let -I and ^ be fra£lions propofed.
Then 3X28 = 84; and 4 X 21 = 84. Here the pro-
ducts, and alfo the ^yalue of the fractions, are equal.
CASE VUL
To reduce coins, weights, meaftires, &c. into frac-
tions.
RULE.
Reduce the coin, weight, &c. into the loweft name
mentioned for a numerator ; and put the number of thofe
parts contained in an unit of the integer, to which the pro-
pofed fradiion is to be reduced for the denominator ; then
reduce the fradtion into its loweft terms.
Reduce 7 s. 3d. into a fradion, a pound being the integer.
12
87 pence, the fraSion will be ,!^ k
3)a^fe( = T§1* in its loweft tftrais = 7 s. 3 d:
Reduce 48. 7^d. into a fra£Uon, a pound being the integer.
12
^1 H fli^'* = 48- 7^'y *8 was required.
253 farthings.
Reduce
^\
154 VUtGAR Book I.
ILeduce 44< d. into the fradion of a flulling.
4
1 8 farthings.
i8 ^
6 ) -(=•} = 44<]. as was required.
Reduce 3cwt. 2 qrs. 21 IK into a fn^on, jr cwt being
the integer.
3cwt. 2 qrs. a lib,
4
2" ^^^ ^ '^ass:cwt2qn2ilb.Mwa«req.
413
Reduce 27 os« 17 pwt. 18 gr. into a fradion) one ounce
troy being the integer.
OS. pwt, gr^
•7 17 18
20
557
24
2228
1114
4)
Y3386 ,,3, „g
6 i— — - = -5^ == — = 270Z* 17 pwt. i8gr.
^ 400
CASE IX.
To reduce a fraAion of an unit of a higher denomination
to an equivalent fradtion of an unit of a lower fpecies of
the fame kind with the higher.
RULE.
Multiply the numerator, of the given fta£Uon, by the
number of units in ^he next inferior fpecies that make aii
unit of the denomination of your fradion^ and that produ£i;
multiply
r
t
- Chap. IV. PVLACTI ON S. 153
multiply by the number of units in the next Inferior deno-
mination tpat make an unit of the laft denomination, and
thus proceed till you come to the loweft you dei^ ; then
make the laft produd a numerator to the denominator of the
fraftion given.
t. Reduce -^ 1. to ao equivalent bsiSdon ia the <leno)iu-
nation of i d.
Ftt&, 3X2o=i6o* and6oxi2 = 720, numerator.
D. D. £.
Z^ == 3-^ »1. as was required.
I 2. Reduce ^ of a (hilling to the fradion of a farthing*
Ftf^ 4 X xa ps 48t and 4B X 4= <92» numerator,
qr.
IQ2
• . • Ju :^ ^ of a fhilling) as was required*
S %
m
3. Reduce ^ cwt. to the fra^on of j lb«
Thus 4 X 2 s 89 and 8 X 28 = 224, numerator,
lb,
% * — ^ K ^ cwt. as was requif»d«
CASE X.
To reduce a fradion of an unit of a lower denomination
to an equivalent fra^on in the denomination of an higher.
RULE.
Multiply the denominator by the number of units in the
given frisson that is equal to an unit of the next fuperior
denomination) and the produd by fuch a number of units
of its denomination, as is equal to an unit of the next above
it ;. and thus go on till you come to the higheft fpecies re*
quired, and the laft produd is a denominator to the nume-
lator of die fra^on given.
I I. Reduce j^ of a farthing into the fradion of i L
8 X 4 X 12 X 20 = 7680, denominator.
SoAatiof a£wlwg^s»r^=; ;;^ I
0i
«56
VULGAR
Book!.
Or by compound fraSions, | of a farthing ^ I
J. of i =
8
of f
of
12
4
1536
2. Reduce 4 oz« into the fra^on of i cwt.
7X 16x28x4= 12544, denominator,
cwt. cwt. oz. '
... 4 _ » _ 4
•»S44 3*36 7*
cwt. oz.
Thatis,^ of i-of 'of I = -L, = 1, as before.
7 16 28 y 3136 7
CASE XL
To find the value of a fradion in coin, weight, meafure,
time, &c.
RULE.
t
Multiply the numerator of the given fraSion by the
number of units of the next inferior fpecies that nukes one
of the denomination of your fradion, and divide the pro-
duct by the denominator ; the quotient is fo many integers
of that lower fpecies ; and if there is a remainder, proceed
as before, ftill reducing and dividing, till you come to the
loweft fpecies; and the feveral quotients, with the xiemainder
(if any, which is always the numerator of a fra^on of the
loweft fpecies) a)'e the anfwer.
I • What is the value of -f
of a pound ?
5
X20
— s. d. qr.
7 ) 100 ( 14 3 i|> anfwen
X12
24
3
X 4
T2
(5)
2. What is the value of
•yxc cwt. r
17
4
68*
X28
544
136
lb. oz. dr.
178)1904(10 II 2J4«aAf.
124
X '6
1984
204
26
X16
416
(60)
\6o /50
V i7«U9
What
Chap. fV. FRACTION S.
3. What is the value of ^
of a (hilling i
4
X12
d. f.
5)48(9 2^ aufwer.
3
X4
12
«57
4. What is the value of \
of a degree? .
3
X6o
8)i8o(a2' 30", anfwer.
4
60
240 '
5. What is die value of 4 hundred weight i
6
4 .
■ qr. lb.
7) 24 ( 3 12, anfwer.
3
X28
17
6. What is the value of 4 of -f of a year ? •
Seconds in a year = S^SS^gj? X 10 = 315569370.
60)
-21 ) 315569370 ( 15027 1 12 TT == T Seconds,
105
250451 minutes 52''4'
4174 hours ii'52''4*
173 days 22 h. 11' 52'"!, anfwer.
56 60
149
23 24
27
60
IF
If the fra£lion to be valued be an improper one, divide
the numerator hy the denominator, and the quotient is an
integer of the (ame fpecies with lie fra^on j then reduce
the remainder as before.
7, What is the value of ~ of an ounce troy ?
?)
jt;i
^
VULGAR
27\ oz.dwt. gr.
8 / 77 ( 9 12 12, the anfwer.
5
X20
100
4
"96
fiookf.
SECT. Ulf
Addition cf Fractiohs.
iN order to prepare fnEdona for addition or fubtnidioii^
all compound fradions muft be reduced to fii^^ ones ;
d if diey are of diifSerent denominations, th^ muft be
brought into the fame denomination, and reduceq, fo as all
^ fndions (hall have one common denoniinator.
RULE.
Add together all the numerators, for a new numerator*
under which fublcribe the common denominator.
I. Add ^ ii and 4 together.
Firft, | = 4^fs=44and| = 44tper reduaion.
Then 10 -{- 15 + 12 =: 37, the new numerator.
* . • -f Hh T + f » fi == *Vo tl>« f">n i«4vii«<i*
2. Add 3f 4- {- + 4-^ -} + 7 ^*o oi^* ^um.
Fiiii,
5 — i5'
8
7
8
40
=: 35 I per xedudion.
40
^f7=28
5 ^ 40J Then 25 + 35 4- 28 = 8S, and fj = 2f
*•' 3 + 7 + >T = ^^1 ^ 'vn required.
2 3, Add
Cliap- IV. FRACTIONS. jgg
3. Add ^ of IS 1. + 3|1- + 4 of 4 of i «f * pound
«4- 1. of I of a ihilling into one fum.
Firft, I of J5I. a ^ s^4f^
3t I Reduced into pounds and
J of /of I s: ' f fradions of a pound
3 7 i ~ ^ ftcrling.
^of ^s — ^s — '
And T = TV9 -^ ~ ivf T ^ 7^' *nd T^ with one com^
mon denominator.
Then 20 -f- 30 + 10 + < = 61, numerator.
•••4 + 3+ tJ=7tJ=71*I7»-5t^*> thcanfwef.
SECT. IV.
Subtraction of Fractions. ^-
THE fraftiona being prepared, as before direAed in
addition, then,
RULE,
Subtrad one numerator from the other, and their differ-
ence will be a new numerator, under which fubfcribe the
common denominator,
1. Subtraa | of 4, from f .
Firft?of ? = 1^ and ? r= !f .
3
••• ^ — — = ii = i, the anfwer required.
21 21 21 7
2. What number is that, from which if you dedud the
_ of I, and to the remain^ add ^ of 4-^ the fum will
be 3' :,,
Firft,-Iofi2-5iI. Then3 = 2ii;and2:Z-iZ.
16 19. 304 ? r?4 304 304
^ 865 _ 2162;
*** 304 7600 Alfo
I
7T
t6o VULGAR Book I.
AIfolof? = ^ .. .1^5+^ = 2411^ theanf.
as 8 7<30o 7600 7600 7ooo»
3* What number is that, to which if you add iV of 1 2
4- ^ of 27, and from the total fubtrad y of 7^ — - 1^ of
1 1, the remainder {hall be 8 ?
, Firft.iof7f = ^,and||-ofii=I? ... 62_f2-
- Then 8 + 14444= 9mi- Alfo ^V of y = |^ ; and
V,of2Z = 22Z.
'^ I ao9
. . *** 1 ^ Hi tlAAi Lafllv o'*«» —
• J55 + ;S5 - iS5 - *T^* *-a»"y» 9T<nT -
»|» 14 = 6f ifj, the number fought.
«
SECT. V.
Multiplication ef Fractions.
TO prepare frafUons for either multiplication or divi-
fion, reduce compound fractions to ungle ones ^ bring
mixed numbers into improper fradions, and exprefs whole
numbers fraSion-wife i alfo reduce fradions into their
loweft terms. Then,
RULE,
Multiply the numerators into one another for a new
numerator, and the denominators one into another for a
new denominator.
I. .Multiply f ijBtD 4, Firft, 3 X 5*= '5' and 7X 6
= 42.
Anfwer, }Xi = il = iV
a. Multiply tV into 4 of 4..
■
Firft, * of ^ = i. Then 7 x a = 14, and n X 7= 77.
ill
Anfwer, i^X|of | = fj = -^
3. Mul-
H • 4i^U|W««<
Chap. rV. FRACTIONS.
3. Multiply 7f into 5|. Firft 7 J = ^, afi^ 5I s ^,
9 ♦
Then 67 X 23 = 154.1, and 9 X 4 =5 36.
Anfwer 7^X 5| = '-4^ = 4^x1'
4. Multiply 2i by |, and this produd by 2^ and this
again by 7 of -^^
Firft 2i = 4, a = ?, and lof ^ = ^
144
Then ix-rX-rX-i^;!: — ^t the anCwcr.
Hence it may be obrerved, that if the multiplier be n
proper fra£tion, the produ(Sl will always be lefs than th9
multiplicands
«
SECT. VI.
Division 0/ Fractxoks.
TH £ fractions being prepared as dire£ted for multiplU
cation, diviiion may be thus performed.
«
RULE.
m
Multiply the numerator of the dividend into the denomi-
aator of the dividing fraftion for a numerator, and the
other numerator and denominator together for a new de-
nominator.
1. Divide -i hy — . . . . — )— f -^ rs i-J> anfwer.
4 ^ 5 5 A V 8 ^
2. Divide 1 J. by - of a fhilling. Firft - fcil, ^ 1.
Thcn^ = JL)LfIS2=2i|K =2x1. 8 s. e^d. +
60 30/7 \ 7 ^ ^ -r
fferth.
3. Divide i- by 7, Thus ^j— (~> the anfwer required.
4. Divide 4I by 5^. Fiift 44 = ~, and 5f =: 1^.
M Thea
1& VULGAR Book I.
Tien ^]i^(2l — 42, the anfwer required.
7 / 3 V>«4 57
<, Divide ». of a. by 1 of 3.. Firft 1 of 1 = ^, and
' 37 '6 4 3 7 1
F 4 8
Then ^ )~-( — < the quotient fought*
If the divifor and dividend have both the fame denomi-
nator, the quotient may be found, by dividing one numera-
tor by anouier.
6. Divide 3^ by ^. . . -)^(5. the anfwer.
7» Divide i- by ^. . . -)-(-, the anfwer.
% If the divifor and dividend have each the fame nume-
rator ; divide one of the denoininators by the other, which
will give the quotent required*
8. Divide 1 by ^. . . .^"i-i f -^ , the anfwer.
Q. Divide Z. by -Z.. • . . Z. )I.f a^, the anfvrer.
^ 9 ' 25 25/9 V
3. If the numerator and denominator of the dividend
can be divided without a remainder, by the numerator and
denominator of the divifor, their quotients will anfwer the
queftion.
10. Divide -I by 1- -)^(K the anfwer.
28 ' 7 7 /a8\4
4. If a number can be found, that will divide both the
numerators, or both the denominators (viz. thofe of the
divifor and dividend) without a remainder; ufe thofe quo-
tients inftead of the given numerators and denominators,
which will give the refult in its loweft terms.
35 H 7
3 4
1 1. Divide 1^ by 1. ... 3 1 Wl, the anfwer.
QUES-
thap. IV. FRACTIONS. 163
QyssTioKs to exercife Vulgar Fractions.
i. A lad hwng got 4000 nuts, in his return was met by
Hiad Tom, who took from him ^ of * of his whole ftock.
Raving Ned lights on him afterwards, and forces 4 of | of
the remainder from him. Unlucky pofitive Jack foun4
him, and required ^^ of 44 of what he had left. Smiling
Dolly w?s by proBiife to have ^ of a quarter of what nuts
he brought home. How many then had the boy left ?
4- of — of 4000 = 16665. Mad Tom took*
- of i. of ^ = 5834. Raving Ned took*
f 17 '75(3 left.
io **^ M ® '750 =: 1041^ pofitive Jack took.
3 „P > f ^'^y 70841 'eft;
- of - of -Jii re ,3247 fmiiingDoUy had.
57SH* the anfwrer.
4. There is a number, which if divided by I? of i.
5, 3. 16
Will quote ~ ; pray what is the fquarc of that number I
16 3 I .
3 16 ^^ 7"> wl^ich neither multiplies or divides.
••• F'^f = ^^95^AV»tbcanfwer.
3. There is a number^ which if multiplied by ^ of Lof
IJ, will produce no more than i ; what is the cube of
I that number ?
' loflofIi = ZZ)l^48,
Thenl?xl?xl^ = il^, theanfwer.
77 n 77 4><>5»3
1. Four figures of 9 may be fo placed and difpofed of, as
enote and read for 100, neither more nor lefs 5 pray how
is that to be done ?
Anfwer 99^ ssr loo.
Ma 5. Kitty
\
l64 VULGAR . Book I.
5. Kitty told her brother George, that thou^ her for-
tune on her marriage took 19312 1. out of the family, it
was but \ of two year's^ rent, heaven be praifed, of his
yearly income ; pray what was that ?
i.j22S_/'2-i—— 16093 1. 6 s. 8 d. per annum.
6. A merry young feDow in a fiasdl time got ^e better
of 4* of his fortune ; by advice t>f his friends he then gave
2200 1. for an exempts place in the guards j his profufion
continued till he had no more than S80 guineas left, which
he found by computation was juft -J^ part of the money
after the commiffion was bought \ pray what was hit for-
tune atfirft?
810 guineas =? 924 1.
Thenl)K.+(iil22 = 6i6oI.
20/ I V 3
2200 + 61 60 = 8360 = ^ of his whole fortune.
5 '
~ j~— [^^-^ = 10450I. the anfwer.
y. A certain captain fends out 4. of his fddiers + 10,
and there remained 4. -4- 15 ^ how many fotdiers had he ?
r- or 7- 4* ^^ — what he fent out.
And — or i- 4- 15 si what remained.
Their fum|-^ 25 = number of foldiers.
Hence 25 n — <>f ^hc foldiers.
• - • 25 X 6 = 150, the anfwer required.
8. A certain gentleman hu-cs a fcrvant, and promifes
him 24 pounds yearly wages, together with a clcak. At
eight months end the fervant obtains leave to go away,
and inftead of his wages receives a cloak -j- 13 pounds ; how
much did the cloak coft ? AJhbfs Analjfi*
Am
Chap. IV. F R A C T IONS. 165
As 8 months = ^ year ; therefore, at 8 months end,
his due is ^ of 24 K (=: t6 1.) «4- f <>f the cloak. ,
Then 10 1. — 13 1. = 3 1, s value of •} of the cloak*
* • * 3 !• X 3 = 9 1* the anfwer required,
9. If a man gain 33 crowns a week, how much muft he
fpend a week to have joo crowns, together with the ex-
pence of four weeks, remaining at the year's end ?
4/bifs Jnalxfl.
Firft 30 X 52 = 1560 crowns gained in a year.
Alfo 1560 — 500=: 1060, the dividend.
And 52+4= 56, the divifor.
• . • 56)io6o( 184-J: crowns fpcnt = 4I. 14s. yjd. per week,
^d 30 — 18 li == I i-r'^ crowns =; aU 15s. 4^4. laved,
10. A country fpark addreft a charming (he.
In whom all lovely features did agree !
But he not (kill'd i'th' art (you may prefage,}
Was too follicitous to know her age.
The lady fmiVd at his prepoft'rou9 rul^
Of courtfhip ; but to fatisfy the fool.
Made him this anfwer with a genVous air
(A lofty charm peculiar to the fair. )
My age is that, if multiply^ by three.
And two-fevcnths of that produS tripled be.
The fquare root of two-ninths of that is four ;
And now farewel, Pll never fee you more.
Your fond impertinence has caus d this rage ;
'Tis clowniih fure to a(k a woman's age.
So you're dcfir'd to aflift him, or perchance.
The fpark muft ftill remain in ignorance. LaJiis Diarj^
Firft 4 X 4 = 16. Then l) j(^^ = 1%.
- Alfo 3 ) 72 ( 24, and jY^i}^ = 84.
3 ) 84( 28, the anfwer required.
• . •
ir. A perfon having iabout him a certain number of
crowns, faid. If 4 + 4 + |, of what he had, were added
together, they, would make jvift 45 i *^ow many crowns had
he about him i
L~l L-i andi. = i.
4 "^ 12*3 "" 12' 6 la*
M 3 , Then
VULGAR *P* U
Thcni-+ ^ + - = f =f = 45-
T ' 1« ^- 12 '- 12 12-4 • ^-*'
... i-)li[iL2 ss 6o, the anfwer.
4^ » V 3
12. A fchoolmafter being aiked how many fcholars he
ha(l> anfwered : If I had as many, and \ as many, and ^ aa
many9 I Should have 99 ; how many had he i
rirfti=l„ l=p?-.
• 4 » 4 •
Then — + — -I f- — =s— =:Qa, per que((.
^ ^4 T^^ ^4 4 ^7* r- 1 »
. . . li)^(2^ = 36 fcholars. Qj E. F.
•
13. When I wrote this, if to my age you add^
T> 4» T (thereof) With j-more.
The num^r 25 will then oe had ;
Ingenioiii Tyro's, pray my age explore.
Firft I = — — — -2 — — ^ and — = —•
30' 2 30' 3 30 5 3d
Alfo22+li + i3+l = ^,
30 '30 30 ' 30 30
And 25 — r - ^ 244 = — = ^^.
. S 5 30
•.• — J— |i2yevs, the anfwer required.
14. What number is that, which added to its ^ -|- its 4
4- 3, maizes 108 ? ^
Firft — = i-, i- = —. Alfo 108 — 3 = 105.
14*24 ^ f
Then Will 1 + 1 + L = L = 105,
4 4 4 4 ^
... Ljiii/'li^ — 60, the anfwer.
4>' » V 7
15. Admit there is 212 1. 14 s, 7 d. to be divided amongft
a captain, four men, and a boy ; the captain to have a fhare
and half; the men each a fhare, and the boy-j of a ihar^^
what ought each perfon to have I
Chap. IV. FRACTIONS. i6y
It ^ 4- = |- capt. 4. = ^ men, and ^ bojr, ,
Then 2. + ^ + i.=;M.=: 2,2!. j^,. yd. rssiOJSd.
1. s. d.
And 2)36 9 4t 1 — cj. , . ^ cantain 1
36 9 4tX4= 145 17 5i men. ( ^ *" *•
3)36 9 4t( = 12 3 I J boy. -I,
£212 14 7
16. There is a ciftern with three unequal cocks, contain^
ing 60 gallons of water ; and if the greateft cock be opened^
it will be empty in one hour ; and if die fecond cock be
opened, it will be empty in two hours i if the third be
opened, it will be empty in dirpe hours : now I demand la
what time it will be empty, if ^U run together ?
And the third - |i '^^""^ = l|j "^'^g-
•. • ~J— /^ — = 3Zt^ minutes, the anfwer,
, 17. A gentleman has an orchard of fruit trees, one-half
of the trees bearing apples,, one-fourth pears, one-lixth
plums, and one-fifty of them bearing cherries ; how many
fruit trees in all grow in the faid orchard ?
Ffrft i = —apples, — = —pears, — s= — plums*
2 12 "^"^ 4 12*^ 6 12 '^
Then i + l + 2 = il. Alfol'-ii=i. chcrrie|
12 ' 12 * 12 12 12 12 12 « •
= 50.
• . • 50 X 6 = 300 apples. I
Alfo 50 X 3 = 150 pears.
Again 50 x 2 =: lOO plums.
And - - 50 cherries.
In all - 600. Qi E. F.
NT ^ tS* Fiv^
r
leBt y U L GAR Book I.
1 8. Five perfons difcourfing about thetr ages, faid the
fecond to the £rft^ my age is the double <if yoiir's ; and
faid the third (o the iim, my age is as much, and ^ at
much as your's '^ then faid the fourth to the fecond and
third, my age i» as much as both yours added together |
but faid the fifths oiy age is three times as much as the aee
of the ^rft, and the mm of all our ages niake juft t6^
years ^ what wits the age of each i
X
a
3i
3
lOt Si — diviAMT far the firft peifon'a age«
... -J)i^^M!=: ,6firft
AlTo/- I6x" = 32fecondJ j^.
Again - i6 X i^ = 20 third « p«»^« • •e^«
Likewife ^2 + id = 52 fourth
And - * x6 X 3 = 48 fifth
^ Sum 168
CHAPTER V.
DECIMAL FRACTIONS.
SECT. I.
. NOTATION.
THE word decimal Is derived from decern (ten) and
denotes the nature of its numbers; becaufe the inte-
ger, or whole thing, whether it be coin, weight, meafure,
time, &c. is fuppofed to be divided into ten equal parts^
and every one of thofe parts into ten other equal parts, &c.
ad infinitum,
3 The
Ch^V. FRACTIONS. i6^
The integer being thus divided by imagination into lo,
10O9 ioo<l> 1 0000, ice. ia the denominator to the decimal
Thus -yV, -reTyj TccreJ i<)</S6y ^^«
Thefe denominators arc feldom or never fct dof^n, but
pnly the numerators; and when the numerators do not
coniift of lb many places as the denominator hath cy-
phers, the faid places in the numerator muft be fupplied
by cyphers prefixed on the left-hand. So-^ is wrote .3, ^4^
is .05, ^Ht^ is .017, and t-s'sW is -0051, &c.
Alfo mixed numbers are expreifed thus, viz. 8.7 is 8 and
7 tenths, 59.017 is 59 and 17 thgufandths, or parts of a
thoufand, &c.
Cyphers at the end, namely at the right-hand of a deci-
mal, do neither augment or diminijQi its value ; for 5, .50^
•500, .5000, and .50000, are decimals having the fame
value, being each equal to f , as may be found by abbrevia-
tion of vulgar fra£^ions.
C3rphers prefixed to decimals, ddcreafe their value in a
tenfold proportion, by removing them farther from the in«
tcgcr.
^ '5 :^ 5 tenth parts.
I .05 := 5 parts of an hundred.
Thus< .005 := 5 parts of a thoufand.
I .0005 = 5 parts of ten thoufand.
1.00005 =r 5 parts of an hundred thoufand, &c«
In whoI6 numbers, the firft place above (that is, on the
left-hand of) the place of units, fignifies tens of units ; but in
fradions, the firft place beneath (that is, on the right-hand
of) the pl^e of units, denotes tenth-parts of I, or unity,
and is called the iirft place of decimal parts, or place of
primes ; likewife the fecond place above the place of units,
figniiies hundreds of units ; but the fecond place beneath
the place of units, exprefles hundredth parts of unity, and
is called the fecond place of decimals, or place of feconds ;
fo that as the value of the places in integers afcend in t ten«
fold proportion from the place of units towards the left-
hand, fo the value of the places of decimals defcend in a
tenfold proportion beneath the place of units towards the
nght«-hand.
A TA.
i^a
DECIMAL,
BoqIcI.
A TA8I/E for Notation of Integers and DECiMAL^t
872365 ^82353785
c rt tr c n 3
D » o 5 a t;-
c e-
o p* 3 no
o §•« •
"n O
O 9
s
CA
^
A
S 3 o ^7
S* ^ ST
W
^ 13 *^ ^
p p p g
CO ^ CA W
0000
*^ ^'^ **> ^^
p p r-»> »a'
3gSa
• S
S
CA
1
It may be obferved by the foregoing table, that the places of
integers, or whole numbers, are feparated from the decimal
parts by a point, that the numbers on the left-hand of the point
- expredes 872365 integers, or units ; and that the number oii
the right-hand of the point fhews -82353785 parts of i (or an
integer) fuppofed to be divided into 100000000 equal parts.
Hence, if the feparating point, in any mixed or fradion^
number, be moved one place towards the left-hand, then
every figure, and confequently the whole expreflion, is but
a tenth part of what it was before ; that is, sit is divided by
10 s if it be moved two places, it is divided by 100*, if
three places, by 1000, &c. But if the feparating point' be
moved towards the right-^and, then the whole expreffion i&
multiplied by 10, 100, 1000, &c. according as it isinoved
one, two, or three places.
There arc feveral ways of reading or exprei&ng a dec^»
mal, as fuppofing the decimal parts in the table were to be
read in words, viz. -82353785.
Firft, They may be reduced tp, and expref$ ^s vulgar
fra£lions, viz. ^'^^'^'^ ^^
1 oooocooo
^ Secondly, By calling them primes, feconds, &c. accord*.
ins to their diftance from the feparating point, viz. 8 primes^
2 feconds, 3 thirds, 5 fourths, 3 fifths, 7 fixths, 8 fevenths,
and 5 eighths.
Thirdly, Thus 82 million?, 353 thoufand, 785 eights.
Fourthly, Or thus, 8, 2, 3, 5, 3, 7, 8, 5 of a decimal.
SEC T,
^h^p, V. F 5. A C T I O If .^^ I5^,
S E C T. 11.
REDUCTION of DECIMALS,
C A S E I.
To reduce a vulgar fradion into a decimal.
RULE.
Annex cyphers to the numerator, till it j>e equal to^ of
greater than the denopiin^tor ; then (Hvide'by the denomi*
nator, and the quotient will be the decimal (ought.
If, after yoi^ have made ufe of all the cyphers annexed to
the numerator, there- be a remainder, annex cyphers thereto^
and continue your divifion, till it divide off or arrive to what
degree of exadnefs you think proper.
Always obferve to fet a point betwixt the numerator and
the ciphers annexed thereto^ and that the quotient have as
many places as you annex cj^phers to the numerator and re-
mainders ; and if it be deficient, let the want be fupplied by
prefixing as many cyphers to the quotient as it falls (hort«
EXAMPLE.
Reduce f , ^, \j f, and i^ into decimals. Thi^s,
2) i.o).5. •.4( i.oo(.25.. '4) 3-oo(«7S-
Alfo 8 } i.ooo ( .125, and 16 ] i.oooo(.o625.
»
^ Reduce f , ^ and ^\^ into decimals.
Jf)l.C(.2 = f . • 25)2,00(.08 = iV* • I2S)3.000(-024 = -x^^
Thofe decimals that are reduced from fuch a vulgar frac«
tion, whofe numerator with cyphers annexed is an aliquot
part of, or can be meafured by its denominator, are finite
or terminate decimals $ as the decimals refulting from the
foregoing examples.
No fr^ion Will produce a finite decimal, but fuch whole
denominator is 2 or 5, and their multiples*
But fuch as are produced from a vulgar fraction, whofe
pumerator with cyphers annexed is no aliquot part of, or
pinnot be meafured by its denominator, will be indetermi*
pate, or endlefs.
In circulating decimals, if one figure only repeats, it is
failed a fingle repetend; as ioif example^
3. Sup^
»7< DECIMAL' Book I.
3. Snppofe the decimal of 79 fy f, I") |, |> ^ &c. was
•required.
9)1.0000 9)2-0000 9)3.0000
.nil, &c. =J. .2222, &c. = |. .3333 = | = f.
To avoid the trouble of writing down unncceflary figures,
a fingle repetend is denoted by the repeating digit daflied ;
that IS, the decimal .iiiii, &c. = ./ = ^ .22222, &c. =
4 = ^. Alfo .33333, &c. =r .^ = ^ = 4, .y =: I, .i
ss I, .^ = « = ^ .y = ^ .f 1 1, and .^ = 1 = I.
4. Suppofe it was required to reduce ^^^ ^ and ^^'^,
into decimals.
12)1.000 36)5.000 960)31.0000000
.o8j = tV .13^ = -5% .032291^= ^Vtp
The decimals refulting from thefe laft examples are called
mixed fingle repetends,
5. Let ^j ^ and .^J^ be reduced into decimals.
11)2.000000 7)3.00000 286)l7.O0OO00Q
.181818, &c. = ./f •p'^Slf *^i^w>i*
Thofe decimals in which two or more figures circulate,
are called compound repetends ; and the manner of diltin-
guifliin^ them, is by daihing the firft and laft figure of the
repetend, by which means we make one place of the repe-
tend fufficient, as in the laft example.
In a compound repetend, any one of the circulatihjg fit
gures may be made the firft ot the repetend ; for inftancc^
in the repetend 8.6^2^325325, &c. it may be made 8.63/5^1
or &.632jf3]i!. And by this means any two or more repetends
may be made to begm and end in tne fame place \ and theli
they are faid to be conterm]nou3.
I ^
r. Let — ^ be reduced to a decimal.
^ 373*
373O
Chap. V. F H A' C T I O N S*
27 ix ) 1 3.000 ( .0034843*04, &c.
18070
31460
161 ao
1 1960
7670
20800
17J
1145
The decimal rerulting^from the laff example, is called aft
approximate decimal, having fome places true, and the reft
uncertain ; thefe ^proximating decimals are fometimes
wrote with the fipia ^ or — , to denote whether the laft
figure is greater ^or )efs than juft : thiis, .0034843205 4-, or
.0034843206 "*- s tike firft fignifies that the decimal is greater
than .0034843205, hy fome vncertain figures ; and the fe-
cond, vis. •O034£432o6 — , denotes that the true decimal
exceeds .0034843205, and is lefs than .0034843206.
CASED.
To reduce coins, weights, meafures, &c« into decimals.
RULE I.
Reduce the different fpecies into one, viz. the loweft de-
nomination they confift of, for a dividend ; then reduce the
integer into the fame denomination for a divlfor; the refult
will be the decimal required.
R U L E IL
Write the given denominations or parts orderly under
each other, the inferior or leaft parts being uppermoft ; let
thefe be the dividends.
Againft each part on the left-hand, write the number
thereof contained in one of its fuperior ; let thefe be divi«
fors.
Then beginning with the upper one, write the •'quotient
of each divifion, as decimal parts on the right-hand of the
dividend next below it; and let this mixed number be divid-
ed by its divifor. Set. till all be finilhed, and the laft quo-
tient will be the decimal fought.
.RULE
y
vj4
EC I M A L
RULE III.
lioQlt
The decimal may be readily found by the rule of pradice^
iiamely, by confidering the next inferior denomination as
aliquot parts of the integer ; and thofe ftill lower as aliquot
parts of the fuperior ones, or of each other ; the fum of all
thofe aliquot parts will be the decimal required.
Ex. I. Let 3^d* be reduced to a decimal, a pound vfter*
ling being the integer.
By Rule I.
4
960 )is.o( .015625!. = 3^d.
600
240
4«o,
By Rule IL
.1
12
20
3-00
3-75
0.3125
0.15625
The decimal as bef6re
By RuLB III.
3d. =: -^ of 1 1. = .0125
^d. ss ^ of 3d. =: .003125
Sum jf .015625
The decinud fought.
^*««M
2. What decimal of a pound is 5 s. i\ d.
By Ru LS II.
5 s. 7|d.
12
67
4
96.0) 27. 1 (.28229, &c.
790
220
280
88q
~6)
4
12
20
3.00
5.6458^
0.282291^ = 51.. 7|d,
H
GHap.V. FRACTIONS.
^7*
By RULB IIL
5 8. = ^i of X 1. = .25
6 d. = ^ of < s. ^ .025
14. = i of od. = .00625
7 = i of ifd. = .ooi04i|{
Viz. 5 s. 7jd. ssjf .282291$^.
3. What decimal of a pound is equal to 19 s* 11 d. ?
10 s. II d.
By RvLfi II.
12 11*000
20 19.91^
0.9958^ = 198. lid.
12
24.o)23.9(.9958^ = 19s. i id.
230
140
200
80
By Rule IIL
198. -d. =: .95
- 6 =: t of I s. =s .025
- 3 = i of 6 d. s .0125
- 2 = 4. of 6d. = .0083^
198. II d. =^.9958^-
4. What decimal part of an hundred weight is 2 qrs.
12 lb. 12 oz.?
qr. lb. oz.
a 12 12
28
68"
16
cwt.
1792) 1100.0 (.6 1 38393
2480
6880
15040
7040
16640
512
By RuLB IL
16 viz* 4
28-4
4
12
• .
4)'
3.0
12.75.. 3.187s
2-455357' +
0.61383 —
:'t
B/
?7«
1> E C IM AL
fiookt
By RoiB IIL
qn ibk 4sB«
T=S ^ JT -
f of 2 qr. = .0714^86 s: - 8 -
i of 8 lb. = .0357143 =K - 4 -^
f of 4 lb. = .00446428 t= - - 8
i of 8 oz. := ^0223214 =s - - 4
.6138393 = 2 12 i2> sis befiore.
5. What docimal f^rt of a pound troy are looz, x8 dwt*
l6grs.r
oz. dwt. «r.
10 18 ID
218
24
872
436
— 02.dWt. JT*
576.o)524.8(.9/aio 18 16
640
By Rvi.« II.
^l 61 4.0
2o|i8.^
X2|j0.9^
* 0.9/ lb. troy«
6. What decimal part of a degree of a drde are 48* 37^
54"' i
48' 37^54'^
60
2917
60
■■■I II T '■
at6.ooo ) 175.074 ( .9t<iS2^
1140
600
1680
60
60
60
By Rots It
54-0
48.631^
o.8io5ay=48'3/'s4'".
> Wlut
Chap. V. F R A C T I O 1^ S. iyy
7. What decimal part of a foot s=: 1O7 inches ?
joiinches. By Rule IL
1. 00
10.25
48) 41.0 (. 8541^ 12
260 ] 0.8541^ =? I o^ inches.
200
80
8. What decimal part of a gallon of ale = 133 cubic
inches-?
i82 ) 133.0 ( .47x6312—
2020
460
1780
5880
340 ^
580, «c.
9. What decimal part of a year =: 217 days, 7 hours^
18 minutes?
d* h. m*
217 7 18
24
1b68
434
5215
60
.. 312918
60
3^^556937 ) 18775080.0 ( .594958883367
299661 150
150487170
302504226
185817870
&c.
C A S E III.
»
To reduce any decimal into the equivalent known parts
of coin^ weight) meafure.
N RULE.
178 U E dl M A L Book L
RULE.
Multiply the given number by the number of units contain-
ed in the next inferior denomination, cutting off as many fi-
gures from the produd as the given decimal confifts of; then
multiply the remaining puts (if any) by the next lower
denomination, cutting off as before i and thus proceed till
you have converted your decimals, or come to the loweft
part I and the feveral figures to the left-hand of the fepa-
rating points, will be the feveral parts of the quantity re-
quired.
0
What known parts of coin are equal to .015625 1. ?
.015625
20
312500
12
3750000 Anfwcr, 3|d.
+
3^000000
What known parts of coin ait equal .282291^1. ?
•282291^
2p
s> ■ »
12
d
/ 7.7500000
4 Anfureis 5 s* 7^d.
3.0000000 ^
What known coin equals .9958^1. 1
•99S8j
20
8.
19.9166)^
12 Anfwer^ 198. 11 d.
dt "■ -
11.00000
What
Chaip, V, F R A C 1^1 O W S. r;^
What knoirn ureiglit is .6x38393 of a cwt. ?
•6138393
4
qr. > ■
2.455357^
28
36428576
9 1 07 144 Anfwer, 2 qr. 12 lb. 12 oz«
12.7499916
16
1 1 .9998656 x: 12 oz. very near.
CASE IV.
To reduce a dfecimal into Its leaft eqotvalent vulgar
ift. If the decimal' be fihite,
RULE,
Under the given decimal write an unit, with as many cy«
phers as the decimal confifts of places ; then divide both
the ntimerator and denominator by the greateft common
meafure, which gives the leaft equivalent vulgar fradion re-
quired.
1. Required the leaft vulgar fractions equivalent to
•5, .259- .75 .1259 and,o625?
Anfwer .5 = — =4, .25 = — = i, .75 = — =
-* lo ^ -^ 100 * '-^ 100
X^ .125 = — ^ =3 4, and .0625 = = A«
^^ ^ 1000 * "^ lOOOO '**
2. What is the leaft vulgar fra£lion equal to .625 and
.5625 ?
Ahfwcr, .625 = -^ =: I, and .5625 =: -^ — ^ = t\.
How to find the greateft common meafure is taught be-
fore in vulgar fra^ions, fo ihail give only one example to
refreih' the learner's memory,
. N 2 Let
i8o DECIMAL Book I.
Let ^' ■ be reduced to its \<yfrfA or leaft equivalent
10900 ^
fradtion.
5625 ) 1 0000 / I
4375 V 5625 / I
i2soU37SA3
625 \ 1250 / 2
5^5)^ (a, as before.
ad. If the given decimal be a repetend»
RULE,
The decimal is the numerator of a vulgar fradion,
whofe denominator confifts of as many nines as there are
recurring places an the given decimal ; both which divide
by their greateft common meafure (as before) and their
quotient will be th^ leaft equivalent vulgar fraflion.
I* Required the leaft vulgar fra£Hon equivalent to ^ ?
Anfwer, o.jl = |^ = -}.
2. What is the leaft vulgar fraction equal to •^6923^2^ ?
The greateft common meafure to '' is found to
. ^ 9999*^9
be. 76923.
Therefore, 7602? ) — 2iL / — the anfwef required.
' "^z 999999 V 13 ^
3. "Vyhat is the leaft vulgar fra<ftion equal to /6/ ?
The greateft common meafure to -^ is 27.
• 999
Therefore, 27 ) — [ — , the anfwer required.
W 999 V 37 ^
3. When the given decimal is part final, and part a cir-
culate,
RULE,
To as many nines as there are figures in the repetend,
annex as many cyphers as there are finite places for a deno-
minator \
Chap. V. FRACTIONS. iSi
minator ; then multiply the nines in the faid denominator
by the finite part, and to the produd add the repeating de-
cimal for a numerator ; thefe divided by their greateft com-
mon meafure, will give the leaft equivalent fradion.
What is the leaft equal vulgar fraction to .53^ -^9x5
4- 3 == 48 numerator ; .90 being the denominator i
aS 8
• . • — =r — , the leaft vulgar fradion required.
90 15 . ' ^ ^
What is the leaft vulgar fradlion equal to .5^2/ ?
Firft, 9990 = denominator, and 999 X 5 -jr 923 "s 59:^9
numerator.
^ ^ ^Q20 16 . ,
• . • 5^2if =r =2 — , as was required*
^ ^ 999? «7 ^
What is the leaft vulgar fradion equal to .008^9713^ ?
Firft 999999000 is the denominator.
Likewife 8 X 999999 + 497^33= ^49Zi^5> numcifcor.
Ai\il 10237s the greateft common meanly.
• . • 102375 J — ^^^'^^ I — ^, the.anfwer.
/ 999999000 V 97^3
jf general rule for reducing decimals inU vulgar fraSfims.
Under the given decimal fet an unit, with as many cy-
phers as there are places in the given decimal ; then fet the
finite decimal as a numerator even under the loweft figures
of the firft numerator, with its proper denominator i laftly,
fubtrad the under numerator from the upper one, and the
under denominator from the upper one, the remainder will
be a vulgar fra£Hoh equivalent to the given decimal, which
reduce to its loweft terms.
\ I. What is the vulgar frafbion equivalent to .13^ ?
>38
1000
13
100
' 125
900
=
-n^
—
5
N 3 q.. Rei»
i$2 DECIMAL Badnh
. 2> Required the vulgar fhi&ion equal to .008^9713^?
From :S^U3L •
I 000000000
fake ... ^
1000
Leaves = .ooSigyi-j* = — rs» as was rc-
9999000 ^^^ ^ 9768' ^^^j^
SECT. in.
Addition of Decimals.
WHEN decimal fra£lions are to be added together^
obferve that the commas, or Separating points in each
expreffion, b^ placed diredly underneath each other ; for then
primes, feconds, thirds, &c. will fall under thofe of the
fame name; apd in mixed numbers^ units will fall under
units, tens under tens, &c.
CASE i;
To add finite decimalf.
|l U L E.
Add as in whole numbers, and from the fum or difference,
cut off fo many places for decimals, as are equal to the
greatdl number of deciipaj places in 9fiy of tbf given x\uin-
bcrs.
Let .3746 M- 137.5 + 1.347 + 375 + 1.85+ .0736285
+ 87396.4 + 8.7386429 + 127 + 5.375, be added toge-
ther.
3746
• Clte^. V. FRACTIONS. jgj
•3746
137-5 • • .
• • 1-347 •
375.. . . . .
« • I«o^ • •
. . . ..0736285
87396.4
• •..8.73864:29
.127
88053.5588714
CASE IL
. To add decimals wherein are iingle repetends.
RULE.
Make .every line end at the. fame place, filling up the va-
cancies by the repeating digits, and annexing a cypher dt
cyphers to the finite terms ^ then add as before, only in-
creafe the fum of the right-hand row with as many units
as it contains nines j and the figure in the fum, under
that place, will, be a reipetend.
Let 3.^ -1- 78.347^ + 735.^ + .275 -I- .2^ + 187.^, be
added together.
3.^66jf
78-347^
735-^333
3750
• • • •2777
i87'^44#
1005.444!
c A s E m.
Tq add decimals, having compound repitends.
k
R U LE.
t
Make the I'epetends fimilar and conterminous ; then add
as in Cafe L only increafe the ri^ht-hahd figure by as many^
N 4 units
n
184 DECIMAL Book L »
units as arc carried from the column of figures, iv herein all
the rcpctends begin together ; laftly, dafli ofF for a rcpetend
as many places as were fo in the numbers added together.
Let + X62 + 1 74.^(5; + i^sfg + 97-2^ + 3/^6923/2^ '+
99.08 J + I 5 + *f J^> be added together*
Made fimilar and
conterminous,
162.16/1621^ '
1 34-09^^9090^
^•9SS'3939J .
97.26j?6666jJ
3-76^2307? .
99-08^3333^
i.wpoooo^
.8i^8i48X
501.62^5107^.
<>0<>00000>>00<>00<>000<><>^
SECT. IV.
Subtraction of Decimals.
C A S E I.
To fubtra£t finite decimals.
RULE.
HAVING firft fet down the. greater of the two num-
bers given (whether it be a whole number, mixed
number, or decimal) fet down the lefler under it, according
to the diredlions given in addition ; then JTubtrad as in whole
numbers, imagining all the vacant places filled with cyphers.
From 375.5 take 8647284. Alfo from 87.569245
take 29.87.
Minuend 375-5 87.569245
Subtrahend 06.47284 19*^7
Remainder 289*02716 67.699245
From
Chap. V. FRACTION S. t»5
r
From I take .732594. And from 684 take 9.3275.
From i.o • • . • 684 -
Subtn .732594 9-3^75
Rem. .267406 674.6725
Let 375.5 l^e diihiniihed or niade lefs by •976373ft7V vA
fhew their difference.
Minuend 375*5 .••••.•
Subtrahend •976373^7 '
Difference 374.523626139 or remainder.
1 •
C A S E U.
To fubtrad decimals that have repetends.
RULE.
Make the repetends fimilar and conterminous, and fub*
tra£t as in the laft cafe \ /obferving only, if the repetend of
the number to be fubtraded, be greater than the repe*
tend of the number it is to be taken from, then the right-,
hand figure of the remainder muft be lefs by unity, than it
would be, if the expreffions were finite ; and the repetend in
the remainder will confifl of as many places as there are
in the other two numbers. . .
Let 57.7J be Ifeffened by 18.9541J?, and 51.52^ by,.J>.
From 57733J» S^^S^^
Take 1 8.9541 j? .66^
• Rem. 38.7791}^ 50.86/.
htt 47.4^7817^ be made lefs by iS-jf^, and 49.^2^ by
38.4736-
From 47.4/7817^ 495^85/85'
Take is^s^s^S^i Z^^AU^^S^Ofi
Rem. 31.8^2161^ ii.0549iif85
From
/
m d:ecimal bookt.
From 43.8^4026;^ take 20*^2^% and from 49.5^ take
42.;!%^«
From 43-8/4026jzf 49-S-J3J
Subt. 20.9^925^ 4^*7^9^
Rem, 22.9/8100;^ ^•yyj/*
t^tmm^m
SECT. V.
Multiplication ijf DtciMAts.
CASE I.
WHEN both hBtors are finite decimals, whether they
be pure, or joined with integers, '^
RULE,
Muldply them as if they were all whole numbers, and
from the produd (towards the right-hand) cut ofF fo many
places for decimal parts in the pr^ud, as there were in botn
the multiplier atftd multiplicand counted together. But if it
fo happcfn that th6re are not fo many places iii the produ&»
fuppfy the dd^ by pfefixing cyphers.
«-7537 864 27.576
437685 6912 82728
6x2759 6048 55152
175074 : — 165456
262611 673.92
2.86683675
171.79848
•57386 .27345
•8237 -273
401702 82035
r72r5fe 191415
114772 54690
459C>88
.472688482
.07465185
^fm
CASE
Chap. v..
P* AC T ION S.
- C A S E JI,
xHj
Tw9 decimal fraStions being given, fo /eftiwc in their
prqdud any number of places,
RULE,
S^ the MJik*s place of the multiplier, dtfeAty under that
figure af tht 4«cimal part of the multipllwwd, whofe place
you wMld refenre in the produd, and inv^jt the order of
, all its otiier plaX96 £ that is, write thcdecimals on the left-
hand, and the JMHgers, if any, on the right.
Then in m^iltiptying, always begin at that figure of the
multrpHcan^ whicti ftands over the figure Wherewith you
are then multiplying, fctting down the firft figure of each
particular produft diredly underneath one another, due re-
gard being had to t\\% ilicr^ifc whidu would arife out of the
two next figures, to the rigjit-hand pf that figure in the
• multiplicand, which you then begin with ; carrying one
/rom 5 to 15; two frop) 15 %q ^^ ; three from 25 to 35,
&c, and the fum of tbcfe lines will give the produdt.
Let 73.8429753 be muUiplM mi«o ^(>%%7iA^ if&rvlDg
only five places pf decJRial p^r^^ ij| c^ piodud.
73.8429753, the multiplicand as qfual.
457826.4, the multiplier inverted, with the unit's place
29537190
4430579
1476W
■ 59074
5169
369
30
341.80097
fct under the, cth place in <tecimals, denoting
that there will be five places of parts in the
produ£l«
The work at large,
73-8429753
4.628754
•mmmmmm ■ ■
29(5371901^
369 2148765
5169 008271
59074 38024
1476859506
4430578518
29537*9012
> w t\
341.80096172917762
Let
%n DECIMAL Book I.
Let 843.7527 be multiplied into 8634.875, referving only
tile integers in the produA.
843-75^7 5*3-7p7
578.4368 8634,875
6750022 4
506251 59
^5313 675
2187635
962689
00216
3375 3375 0108
675 25312 581
59 506251 62
4 67500216.
7285699 72856699
0954125
CASE III.
If the right-hand figure of the multiplicand be a circulate,
RULE,
In multiplying, increafe the right-hand figure of each re-
fulting line by as many units as there are nines in the pro-
dud of the nrft figure in that line, and the ' right-hand
figure of each line will be a circulate j and before you add
them.together, make them all end at the fame place.
.I72<
6
835-2W
..7484
1.036/rf
m
6^18^6
334i09i?33
58469i«33
625.118562^
CASE IV.
If the right-hand figure of the multiplier be a circulate,
RULE,
Multiply by it as by a finite digit, fetting the produd
•ne place extraordinary towards the left-hand 5 then divide
2 that
Chap. V. FRACTIONS. *«9
that produd by 9, continoing the quotient (if needful} till
it arrives at a circulate ; then beginning at the place under
the right-hand figure of the multiplicand, cut off for deci-
mal parts,
63-274 -47375
.113^ .874
9)379644 9)284250
■ ' <
421822^ 315833;
189822 331625
63174 379000
63274
7.1921442^
4153^8*
CASE V.
When the multiplicand and multiplier are each a fingle
circulate,
RULE,
The firft line (or that produced by multiplying by the
circulate in the multiplier) muft be managed as in Cafe III.
only the right-hand figure muft be increafed by ais many
units as there are nines in the produd of the firft figure of
that line, the products of the reft muft be managed as di-
reded in Cafe IL
372^ 8574.^
.2} Bj.i
9) "34^ 9) 42871^
24822 47635/^^
74466 6OO2OJ333
68594^66
.992^8
750730.5/81
G A S E VI.
If the multiplier be a compound repetend, and the mul-» -
tiplier a finite number,
RULE,
\^ D £ C I M A I4 Book I,
ft UL j;
In multiplying, otiferveto add to the right-hand place of
the produd, fo many units as there are tens in the prod u A
of the left^hiaind place of the repetend ; and tHe produ6^
ihall contain a repetend, whofe places are equal to thofe in
the multiplicand ; and if there are more places of figures in
the multiplier than one, make all the ftv«r&il produds
contermimms towards the right-hand, as in Cafe II. and
IV.
8 .005-
■ * » pm
a*'***^ S'QSjf'fS**
Mn^
43-Z 37-135
,^^__^g^i
5128/0^ «i^036|r
2i97^5«^7 258^621^9
a93»-?4^34 86^207^32
r 604/451^245.
32024.0^3 J 25^21^9621
CASE VII.
If the multiplier be a compound repetend,
RULE,
Multiply each -figure of the repetend, and add the fevcral
produds togedber ; then add the refult in this manner ; fet '
the left-hand fignre fo many places forwards as exceeds
the number of placed in the repetend by one, and the reft
of the figures in ofdfir after it ; and thus proceed, till the
refult laft 'added be carried beyond the firft; laftly, add the
feveral refults together, beginning under the right-hand
place of the firft ; and firom thence dafli as many figuces
for a repetei\d, as the ' repetend of the multiplier confifts
ot .
83+-
Chap. tr. F R A C T t a W S. t|^
83475 49640-54
g.T^ .^050^
5843M. »4M«6*
166950 24820270
250425 3i«48378
*A« .-^
27296325 34©9?o699i62
272963 3^*998069
272 349
a732fc-3^4^ iW9^4i|f900^
If the multiplier hath iuiy terminate places joined with
the repetend, and if the repetend be finall, and thefe mznjy
multiply and add the produds of.the.,rQp^fnd firft; then
multiply by the terminate figures, and add.their produAs to
the fum of the produ^s of tl^e regetqid ; and tp this laft
refult, add the faid fum of the repetend prpd\i£b.
8-74-37
J49748'
61205^
mmm^mm^
Sum 6470338iOf die produdi of dit-repetenil« -
174874 .
87437
1 1 13.94738
647033
6470
64
u 14.600^^
But if the terminate figures are few, and the places
of the repetend many, iubtra^ the terminate figures from
tfaofe of the. repetend, an4 multiply by the remainder as
a rqietcnd.
I73S'
l^i DECIMAL Book 1.
J 735.8072
Remains 324704, the new miHtipHcr.
69432288
~ 121506504
69432288
34716114
52074216
5636235410688
5636235410
• 563623
56367.990259^1^
CASE vm.
If both fadors have compound repetends^
RULE,
Proceed as in the two laft cafes ; for as the places of the
repetend in the product will be uncertain as to their num*
ber, they can only be determined (in any manner fit for
pra&ice) by continuing and repeating the firft produd,
which ^iU contain a certain repetend equal in places to that
of the multiplicand.
Multiply 67.^2^
into 5.2^jf
S3
•5.223
203474
135^4^6
135^4^^64
339X2jfi24 .
354-3^49^6/2^
35424906
354^49
3542
35
357.827^33, Sec. Here the fourth
place of parts comes out a fingle repetend, viz* g*
Again,
ChaprAT. FRACTIONS.
»93
Again. Multiply 3.1^^
into 4./93^
4293
125^/818
^"*i
« 35034^1^36363
I35034393636
«'35034363
135034
»3S
»3SX69S33i69533
Examples of this kind,
though very accurate^
yet are more curious
than ufeful ; as they
may be eafier done
exad enough for bu-
finefs, by the con-
traced way of multi-
plication taught Cafe
*
S E C T. VI.
»
Division of Decimals.
IN any x>f the following cafes in divifion, if the dividend
be greater than the di^ifor^ the qu<kient will be either a
whole or a mixt nuoaber^ but when the dividend is lefs than
the divi&Mr, the quotient muft necefTarily be a fraflion ; for
a leiler mmiiber is contained in a greater once at the leaft,
fettt thc^-gi^ater is not contained once in the lefier.
c A s E r.
When the divifof and dividend are both finite decimals^
RULE,
Divide as in whole numbers, and from the right-hand
of the quotient point off for decimals fo many places is the
decimal places in the dividend exceed thofe in the divifor ;
and thofe to the left, if any, are integers j but if the places
of the quotient are not fo many as this rule requires, fup-
O • ply
194 DECIMAL Boot t
ply the dtftSt by prefixing cyphers to the quotient $ but if
the decimal places in the divifor be more than thofe in the
dividend, annex cyphers to the dividend to make them equals
and the quotient will be integers until all thofe cyphers are
ufed.
' 87-364) 7U-02597a( 8- r73
15.1139
637757 -7875 ) 441 -0000 ( s6#
262092 47250
••••••
• • 1
i • •
179 ) .48624097 ( .00*71643
1282 • • • •
294
.2628 ) 27
.0000 ( 100.55865
1 150
1500
769
15750
537
23250
• *
17700
• • •
15900
24750
5«S
CASS U.
To ODntra£l the work of divifion* when the divifor con-
fifts of many decimal places.
RULE.
Having determined the value of iitit quotient figuies, let
each remainder be a new dividend ; and fbf every iheh di*
vidend, point oiF one figure from the right-hand of the df-^
svifor ; obferving at each mukiplicatioa to have reglrd' to thet
increafe of the figuMs fo cut off* as in ooAtniAed snuMpli-
cation.
.67268479 ) 56.00000000 ( 83.2485
218521680
I 671 6243
3262U7
571808
33661
• • • ay
384.672158 )
4
384.674158 ) 14169.206603851 ( |t8i^$
t629Q4i863
3210089158
1327 I I 894
i/3ioa47
1923361
»w
• * •
• • • • 4 •
^*3*S407 ) 87.076316 { 9^2976554
2787663
914582
71696
6138
4
If ahv whole, faiixed, or decinut intfnber ts given to be
divided uy 109 loo» 100O9 &c. only remove the feparating
)M>int towards the left-hand (o maAjr places as there are cy--
phers in the dtvifor j alfo in multiplication the feparating
point is moved to the right-hand fo many {daoes as there
are cyphers in the mmtiplien
£zAHinKs In
Mdt^icacioo. Divifion.
.7865 X 10—7.865 10)7865(786.^
•7865 X 100 =: 78.65 100 ) 7865 ( 78.65
•7865 X lodo =2 786.5 1600 ) 7865 ( 7.865
•7865 X 10000 =x 7865 »oooo ) 7865 ( .7865
•7865 X looooo rs 786^0 100000 ) 7865 ( .07865
C A S £ m.
If th^ dividend be a repetend.
R UL R
If it be a fingje repefend, bring down the cirotdaling ti¥
ipsre mitil tli» quotienc eid^ repeats, or is as exaSt as re-
4pitre(i ; but if the repitend in the dividend be a compound
ono^ then htinz down the circulating figures in the faihe
<Mrder diey ftaod in ; and when vou have got through them
nUf bring down the firft figure m the repetcnd over again f
O 2 aixd
ja$ DECIMAL' Book t
and fo liroceed until your qkotient either repeats, or be
Amv^Xl «ic (C! n^r^flarv.
'(«
, ^ . , is
cxaft as is ncccflary.
.7484 ) 6251 18.5621? ( 83527^.3
26398
39465 / »37 ) -5^^ ( •0^(4136253
JL04C6 186
54882 . . 496
24946 856
346^
2494 726
416
41.764 ) I76x.3/?4t)/ ( ^2.1^, &c. ad infinitum.
90804
72760
30996^
176*3
764-5 ) 3» 9- 28007/1 ;f ( .4176^/
13480 .
58350
48357
24871 • • • 1
J 936 1 .L aiiHfimtwn.
2487
C A S E ly.
If the divifor be a Tingle repetend.
RULE.
If the divifor be only a fmgle repetcrid, place the divi-
dend under itfclf, but one place forward towards the right-
hand, which fubtracft from the dividend; the remainder
wrili be a new dividend, which divide by the divifor, in the
fame manner as if it was a terminate number. But if the
divifor confifts of terminate numbers, joined to the repetend,
fabtra£b thofe terminate numbers from the divifor, and fub-
trad the dividend as before direded, and the remainder wiU
be a new divifor and dividend.
Divide
Chip. V. FRACTIONS. 197
r
Pivide 134.2^ by .^
, -. . ly^ .
m
.6 ) 120.84 ( 20i.4» the true quotient,
pivide 234.^ by .if.
. .23+
.7 ) 21/.2 ( 301.^1428/, Sccdulinfimtmn.
t, '
Divide 6.25118562/9 by 875.27^. ,
875.27^)6.25118562^
875^7 625118562
787.746) 5.626067064 (.00714198
1118450
3307046
.1560624
7728780
' 6390660
■ ■ ■■ ' ■»
88692
\
Divide 856.988 by 4.8(;.
4.86 ) 856.988
' 48 856988
4.38 ) 77»-^*9a( 176-0934247
2668
4092
1500
i86q
1080
2040
3080
14
C A S E V.
If a compound lepetend is fomd in your diviTor only, or
In both your 4ivifor and cKytfend,
O 3 R U T, E,
Set the divifor and dividend mider t|iemfelves, eadi (b
many places towards the fig)it-hand^ as there are places in
the iepi:tend of the divifor i which fu^trad as in the laft
cafe, and the remainder will be a new divifcNr and djvidend.
But if die divifor is a compound hspetend without any ter-*
minate figures, divide by it as a tennin^te number j firft
fubtni^^lig the dividend fipfn itfi^j^ fts boffin direfMr
Divide 13.5X69538 by 4^/9/.
4 I3S^%33
4.293 ) '3S«343636|&< 3;»^^
$244 . ■
234.16 > &e. adb^mtimu
195"
Piyide sa644S745;f93*
44-^43^ ) 5»64.4S745/93^
4a 52044574s
42'3394)5263.93i097i94( t24.3«70i, tru« Quotient.
1029991
1832030
1384540
J'4|S87 . >
290799*
423394
Divide 395.273^1^ by .fi;f.
395273
■17 ) 394.878341 ( ?24S-673
778* • • •• •
14*7^
>798
ai33 •
«JI4
951
• • «
The
r^
Qh^>. V- FRACTIONS. 199
The following lemmas and corollaries may be of ufe in
Oluflradilg die different liiethods and peculiar procefles ufed
in tl^e arithmetic of ci/culating pumbers.
t E M M A L
A feries of nines inifuiiteiy continued, is equal to unit}/^ or
one, in the next left-band pUce. Thus, 0.9^99, &c. = i,
and •099Q» &c. =s fi| al{d^ •00999, ^^* = *<^^ * ^"^
73-^99» «c- = 74.
Dbmonstkation; It is curident that .9 = ^ wants
^7 -n ^ vntitjTy •99 w»its only tv7» 2A<1 -999 wants only
•ttW S fo that if the feries were continued to infinity, tho
dii&nnce between that feries of nines and an unit, would
be equal to unity divided by infinity ; that is, noU^ing at
LEMMA IL
Any fingle repftend divided by 10, and the quotient f)4b^
traded from the faid repetend, the remaiiider will be the '
iame number complete or terminate.
Demonstration, Let the given repetend be 3*3339 &c.
3,335 -f- JO = .333, and 3.333 — .333 = 3. Alfo,
^4445 &c. -i» 10 = J.444» &c, 5M3d 54.444 — 5,444 = 49^
COROLLARY L
Hence it follows, tKaf if a compound repetend be di-
vided byaniinit, with fq many cyphers annexed as are equal
CO the places of the repetend, and the quotient fubtraded
from the faid- repetend, the remainder will be the fame
number complete or terminate, that conftituted the repe-
tend. Thiw, ^2^.325 -s- 1000 = yii 5 and g2izis —
ij2/r$ 325 ; and 42.^43/8 -f- |oooo, will be .0042^431^4
apd 42.^43^ -r- -0042^43^, will be 42.3394-
C OK O L L A R Y II-
Hence alfo, if any repetend be divided by an unit with as
many cyphers as it contains places^ and the quotient multi-
pliea by as many nines as the repetend contains places, the
refult will be the fame as before ; jthat is, the fame number
terminate or complete ; for any number divided by lo, and
tlie quotient fubtradled, the remainder is the (ame as the
quotient multiplied by nine.
I. Thus,
200 DECIMAL, Boofe I.
' r. Thus, 6.666-7- lo = .6666; and .6666, &c. x 9
= 5*999« tec. = 6,
2. Again, 2l2g X 999 = 324-999f *^c. = 325 = 32^-3^5
3. And jz/.jas -f- 1000 = •^i/.
COROLLARY IIL
It is evident from the laft corollary, that a fingle repetend
is to the fame number termjnate or complete, as 10 is to 9 ;
a compound repetend of two places, as 100 to 99 ; and a
compound repetend of three placet is to the faaae number
terminate or complete, as 1000 to 999, &c. And by the
converfe of the faid corollary it muft follow, that any num-
ber multifrfied by i, with as n^any cyphers ag k contains
figures, and the produd divided by as many nines, will
give the fame number perpetually circulating.
Thus, 6 X 10 = 60, and 60 -f- g == 6.66^, &c.
And 325 X 1000 = 325C00, and 325000 ~ 999 = 32/.
COROLLA R Y IV.
Hence alfo, if any number be divided by as manr nine^
as it contains figures, and the quotient added to tne faid
number, the refult will be the fame as before ; for any
number multiplied by 10, and theprodud divided by 9, the
quotient muft be equal to ^ of the fame number aulded to
itfclf.
Thus, the quotient of 6 -7- 9 ^. 6 = 6.666, &c.
,And the quotient of 325 -^ 999 added to 325 = ^2/,
LEMMA III.
Any number divided by 9, 99, 999, &c. will be equal to
the fum of the quotients of the fame number contmually
divided by lo, ibo, icoo, &c.
Thus, 717 divided by 10,100, &c.
7*^7 ' 9)7»7(79^
7»7
7?7
• 717
717
79 666, &c.
■ Alfo,
v"^
Chip. V. FRACTIONS. idi
Alfo, -
336847 -r 1000, &C. 999 ) 236847 { i37.;«^
■ 3704. .
236.«47 7077
.236847 8400
236847 4080
436 -.
237.^4^084
84
From thefe lemmas and corollaries appears the reafon of
mtidtiplication and divifion of fingle as well as compovmd
r^p^tends.
I fliall here add the following ufeful propofition, viz.
To perform the work of multipltcatioA by divifiony or
of divifion hy multiplication.
RULE.
Divide an unit, with cyphers annexed, by the given
moltiplier or divifor, the quotient will be the divifor or
multiplicator fought.
Let 27576 X 625. I would alfo have a divifer which
will give a quotient equal to the product of fhofe num*
bers.
27576 625 ) 1.000 ( .0016 ) 27576tOOOO ( 17235000
625 3750 IIS
37
138880 •••• 56
55^5* 80
165456
• •
17235000
Let 67392 fte divided by 78, and find a multiplier,
which being multiplied into the fame niunber, ihall pro-
duce a number equal to the quotient.
78)
xo*
nRClMAU
tioAh
78 ) 67392 ( 8|4
499
• • «
7^) 1. 00 (.0/2820/
640 '
160
400
10
67 39*
X .o;'282o/
*i mm
336960
» 34784
S39»36
134784
67392
■■- r
963.999136b
8639991260
^'^m
mmm»»«0atm'm0i
863'9999999999999 = W4,
l^?-*
tf^mmam^^mtfm
QiiiaTi^Mt tp mtrxift Decimals,
I. A grocer bought two chefts of fugar, the one weigh-
ing net 18 cwt. 2qr8. .i^lb« at2l. 9 s. 8d. per cwt. ^
the other weighed net i^ cwt. i qr. 21 IbV at 4^: d. per
pound, ^ich be mingied togcdieri now I defire to know
how much a hundred weight of tb|9 mixture is worth ?
K s. d*
2 o 8 per cwt,
6
4id-
X4
14 18 -
i
44 14 -
X
T
I 4 10
1
- 12 s
- 6* 2-L
12
10
20 It. 8^
MiiMI^
46 17 St
38 14 4t
jC 8s-S9»?
I qr. =: .25 .
7 lb. = .0625
I 6
xo 6
X4
2 - a cwt.
6
jC 85 II 10 whok coft<
37.3125 12 12 -
— 3
37 |6 -T
10 6
2- 7^
3814 4i-
37 cwt,
CliaikV. FRACTIONS.
»<*f
18 8 14
18 I 21
37 « 7 !
[09TOo6^{ ao
3504166 j^^iaso
34JO2
522 '^•SI'J^o
149 *
2.157440
Anfwer» 2). 58. io|d.
%. Whm V^a06»f ct w^mt will jOH add to a pipe of
mountaiQ iruic» talur ^I. toralu€6tbcfirftcQftto4$/6d.
the gaDon ? * .
12
20
6
45
0225)33.000/146.^
1950 . V196 ^OQS m a pipe
J500 —
(150} 20.^ gal. theanfwer.
3. If a cubic inch of oil olive be •52835 decimal part»
of an ounce averdupoife* what quantity of oil, weidbins
7-{- poundi per gf]iffti9 wBl be contained m a o^, albwea
to nor ' ' ' ^ ^
iold 13 {• gallons of water, each 282 iblid isdies?
38
66
1.
40
$0
7.5 ) 124.16225 ( 16.555 galloni,
491 . itut anfwei;
416
412
372
4. A perfon was poflefied of ^K t ft^^ ^^ ^ copper-mine,
and fold ^ of his intereft therein for 17101. $ what was the
reputed Talue of the whole property at 4ie fiune rate ? *
•45) 1710.00 (38001. the anfim required*
360 5. If
*4 DECIMAL FRACTIONS. Book I.
5. If I buy 14 yards of cloth for 10 guineas, how many
ells FlemUh can I buy foc/ 283 1. 17 s* 64. at the fame
rate?
1. s.
10 10 . • . 1
I 10
15" *.d. per yard.
3 9
II 3. = .^625 per ell Flemifli,
Byinfpefiion^ 283 1. 178. 6 d.s 283^75!.
• .5625 ) 183.^75 ( 504.? = 504 ells5 2 qrs'. the anfwer.
262^0
37S©o
37500
• • • • 'J
6. GoEa^h of Gath k fald to ^ave been fix cubits and a
half, or a fpan, high,; tills anfyrers to' 10 feet, 4.592 inches ;
pray what iv^s the ferigth at the cubit in Britiib mes^^re i
12) 4.592 (.382^* V * '' ^ . *. / . .
io.382(»feet, theheigl^t ofGplial^ * , . .
6. j ) 10.382^ ( 1.5973241 s:^ I ^t,^7;i68 i^iches, thqanfvip.^
y. A faftor bought 84 pieces of ftulfy' which coft 537 L
12 s. it 5 s. 4 d. per yard^n I dehiani! the number of yafds
in all^ and how manj^ yards/iti -each piece? ^ v - ^
. • I"
r
4 w
• . I I — r-r-
jC -2^ = S ?• 4 «!• 537-6 '=s 537 1. w «,
-r
•2^ )537-6( 2Ql6.^<1s in ally and 7 X is s 84.
* 53-76 . - , -
•34 ) 48384
J44
• • •
i 4
24 yards in eft^ piece.
J
^ /
C HAP.
CHAPTER VI.
EVOLUTION.
PR,
Extraaing the KooT % cut of tf// Single Powers^
EVOLUTI.ON is.thc uhravcUing or unfolding iny
propofed number into the parts of which it Was made
up of or compofed.
If any aun^bttr is multiplied into itfelf, that product is
called a fauare niiml>er. .
Thus tfie fquare numbers 4, 9^ 25^ 36, &c. are each of
them compofed of two equal numbers, viz. 2x2 = 4,
3 X 3 = 9' 4 X 4 =3 i6-
If any number be multiplied into itfelf, and that produdl
be multiplied into the fame number, the fecond produ(5l is
called a cube number,
Thus the cube numbers 8, 27, 64, &c. are each com^
pofed of 2 X 2 X 2 := 8, 3 X 3 X 3 = 27, 4 X4 X4 =>
64, &lc. - ^
' Thefe powers exift in nature, viz. a root is repr^fcnted by
a line or fide, Jiaving but one -dimenfipn, vie. only kngth )
the .fquare is .a pl3n figure of tWo dimenfiqn^, viz^
length and breadth ; and the cube of three, viz. lengthy
breadth, and thtcknefs*
All the fuperior powers ha^^ no exiftcnce in natuhe, but
are compofed of a multiplication of any number four or
more times intoitfelf.
Thus, 2X2X2X2c=i6, the biquadrate, whofe root
is 2. ■ » '
Or 3 X 3 X 5 X 3 X 3 = 2^43, the furfolid, whofe root
*is 3, have no exiftcnce in nature, but may be underftood as a
feries of numbers in geometrical prqgremon*
When any number is propofed to have the root extracted,
Che firft work is to prepare it by points kt over (or under)
their proper figures, according as the given power^ whofe
root is fought, doth^require ; tvhich for the fquare is 2, f^r
the cube is 3, for the biquadrate 4, ' &c. always beginning
thofe points over the place of unity towards the left-hand,-
if the given numb^s be integers, and defcend towards the
ri^t-Kand in decimal parts*
^v 3 ^ Thu»
I
/
so6
EV6L0TIOR Bookr.
' Thua r*r the fifuaic not 5827414643847
Citw 58374^43147
Biquadratt 5837429643847
SnOm 5837*»9*4384r
Or in decEmals*
Thus Pot the (<fUirt root 0.33^794384728
Cube fe.S3*794j847««
Biquadnte 0.532794384728
Surfolid 0.53x794384718
J Tails rf Powbks.
tnd
St
,6
Index. Index. Index.
hs-s
8-
. S "s
llill
■6 8
■Si,
.1^
n
rsa;:
1024! 4096! 16384
Tndwi
65536! 2621^
625
3'^S 156^5 78*2^ 3906^5 _ 1953J
,6^^i6
296
777b 466J6 279936
10077696
ibSoy
'7P49
"^3543
5764^01
40353607
*^ ii3 i2?^ 3^768 z6ar44 2097152 167772 16 13421772^
981 729I6561 590491531441 478^969 43046721 387420489
^ 8 E CTi
\
S E C T. I.
• 4
fa Extrait $be. Sq^VAUS RooTi
• R, U L E.
«
HAVING pointed the eiven lefolvend as before di*
reded, find the greateft Tquare that i^ eontivned in the
firft period towards the left-hand, fetting down the foot as
a quotient; and fubtraft diat fquare out of the iirft period.
2. To the remainder bring down' die two ^fttirts under
the next point for a dividend. N. BJ TUs is always to be
repeated.
3* Double the quotient for a di vilbr, eoquiring how often
it may be had in that dividend ^nxcepting the lauft %;urcs}
and fet down the quotient figure, which ^nnex to the din-
for. This muft alfo be rep^lt^j 9s zvKm diviior muft be
found to every figure.
4. Then multiply the ixAoIe ineMuied divifor, and fub-
traft the produA tiom the diviileAd. Proceed thus till all
the periods are brought down*
5* Inftead of doubling the quotient every time for a divi-
for, always add the laft quotient figure to the divifor for a
jKW divifor.
6. If there be a remainder after you have finifhed your
piBti#4i» biMg fifirm §. m ^ ^pheis for declntialil 3 pro-
ceeding as before dircded,. utt tjbf ^qoI ts ae cttcaft as k ce«
quired.
N. B< Ym W>A ^4ira|rs» in: wimtd mmhett, cut 4^ as
manv whole numbers in the root, as there are periods of
whole'humbers, ^nd as many decimsds as th6te are periods
of decimals.
I. Extrafl the fquare root of 393129.
. •
393129 ( 6»7
36
122 ) 331
244
1247) 8729
8729
• • • •
2. £xtraa
lo8.
EVOLUTION.
Book I;
a. Extrad the fquare rootof 758734J9337865039195105089.
• •
7S87341933786503919SWS089 ( ^71053496^783
167). 1187
1169
i74ifl5;
J21£
. 174295 ) 931933
I 742 I 03) 6090878
17421064 ) 86456965 '
65684256 -
»742io689) 1677^7090;^
ISP789620.'
174210698 ) 109374702
4848283
1364070
14489s
5227
t
The five.laft figures in the example above^ art foitM by
the contraAed method «f divifion*
3. What is (he iquare root of 1850701^764025 i
1850701.764025 (.1360.405, the root required. ,
23)85 •
266 ) 1607
1596
J72O4) IIOI76
^ IO8816
2720805) 13604025
13604025
4. Extraft
Chtp.Vl. 8<^DARE ROOT.
4. ExtraA tht^uBte'roet of .001434.
• ' • • •
0.00123+ ( .0351483362, tie root v«iy
325
701) 900
jot
» I •
^ * p *
n
70248 ) 585600
561984
23616
as4a
435
14
5% Extraft the fquare root af t.
2 ( 1414213^27, roQt betclf,
24)100
atSi ) 400
281
2824 j 11900
1 1296
f »
» •■ •*
< «
4*282 ) 60400
56564
482841 ) 38360*
282841
** •
io^
N >
\
7022)19900 ... 'J
14044 \ \ '
1 • .•>
282842 ) 100759
)0D
*59<
764
198-
aie * V O L U.T I ON. , Boek.t
♦ - . V
Firft deinSt the grcateft fquarei ^teing the root in the
quotient as before.
2. Divide the whole remainder by at and point it a-new :
this may be called a new dividend.'
3. Mal^e the root of the firft fquare a divHbr, enquiring
how often it may be found in a new. dividend, to the next
figure, reserving the figure under the next pointy for half
the fquare of the quotient figure. ^ ^ ^
4. Multiply the divifor into it, adding to that produft
the tens of the half fquare; If any.
5* Annex the. quotient figure to the laft di^or for a new
4ivifor, with which proceed as yriih the laft until all be fi-
iiiihed. • ^
> . ' '^ »
6. ExttzEt the fquare root of 3Z7286968l«
3272869681 (5« firft fingle root,
as
2) 772869681
5 ) 386434840.S ( 57*09
+ 7 374'J = S >« 7 + T fiiw« of 7» vU.^ssa4-5^
57 * 193 _— .
+ 2)1142 = 57x3 + 4 «!»»«« ©fa
572 5148405 ^ . ^ ^
+ 09 5148405 = 572X9 + i fquare 9 sp4as»
7. What is the fquare root of J = .|^ ?
• • • .
o-77777^(-8
64
a ) »3777/ -
.068888, &c.
; . . ^ '..
.8)
CI»p.Vl. SQ^UAUfi ROOT. ku
.8} .o688i4< .^1917401688207
' +8 672
«. • I
.88 ) 16888
+.X 8805
MpAia*
.881 ) 808388
+ 9 793305
IV
.8819 ) i£o8388
4- t 881905
.88191 ) 62648388
+ 7 61733945
.881917) 91444388
4- I 88191705
.8819171 39152683888
4-0$ 264575x345
,881917103 ) 606932543
77792281
7238913
182576
6193
»;•;*■*•♦■# ■*:->-.^-*z*-t'e-*-*z*zr-.*
ft ■ * ■s .^
► s »
« . I
SECT. IL
Som U s £ s of tie S q^u a r z Hoot.
C. A S £ L
TO ^nd a mean proportional between anjr two given
numbers.
RULE.
Extrad the fquare root of the produd of the two num-
bers, which root will be the meah proportional fought.
Required 2 mean proportional between 16 and 256*
Firft t6 X 256 = 4096 ', sdfo, ^ 4096 35 64
36
• . •
16 : 64 : : 64 ; 256. 124) 496
496
? % CASE
/
M2^ EVOLUTIONt BoakL.
C A &fe It.
To find the fide of a fquarc equal in ^tta to any given fu-
perficies. • v, .
^ RULE...
Find the fquare root of the given fupcrficies, which i»
the fide of the fquare fought. • j
Suppofe I have a circular eliptticd polygdftaJ^ or irregu-
lar fUhpond, conteining in furfaC^e ft acreS, 2 roods>l5 perches^
and would have a fquare one of the fame content j defire
You*d tell how many yards eadi fide Inuft Be ?
A. R. P.
9 2 IS
±
38
40
^•5 X 5.5 = 30*^5 fipttff* yiirds iH t flcrch.
767$ . ( .:.
3070^ *
!■ ' ( yds. ft. inch.
41)64 . .
41 ^
425 ) 2333
2125
/ »
i*^^0m
4304 ) 20875
I72I6
43088 ) 365900
3447<5+
430964)2119600
. 4 .* * ! ^
■ .395744
i »
k A
7876 \ . • ^
^^6 CASE
Xluip..VI. SQUARE ROOT. aij
CASE III. .
Having the area of a circle, to find Its diameter.
1^ U L E.
Multiply the iquare root of die area by 1.128379 the
produd will be the diuneter.
In the inidft of a meadow, well ftored with grafs,
I took j lift an acre to tether my horfe :
How long muft the cord be, that feeding all rounds 1
He may'nt gra?^ lefs or ix^^xo than this acre of ground.
4840 iS|uare yards in an acre.
V'484p =?: 69.57 ••• i,ia$3j
2f> X 75*96
n\m V
129 ) 1240' 6jJ022
1161 '«>i5S3
5642*
1385)7900 -790
6925 m —
a ) 78.5007
■^r*-
13907)97500
97349 39-^ yards, length of the te-
ther and horfe.
■ * j ■ '
The periphery of any circular figure may be found by
multiplying thfi fquare root of the area by 3* 5449*
Thus 69*57 X 3.5449 33 246 yards, i foot, loj: ihchcs,
the perimeter of the before^-mentioned acre of land.
C A S E IV.
Any two fides i>f a right-angled triangle being given, to
find the lemaining fide.
RULE.
As the Oju^re of the hypothcnufc, or longeft fide of a
fight-angled trianglt, is equal to the fum of the fquares of
the other two fides; confequently the difference of the
fquares of th.e Jiypothenufe, and of either of the other
^ieSy b the fqu^.e of the remaining fide.
|. As I was walking out one day.
Which happened on the firft of Mav ;
P3 ■ A:*
4l4 EVOLUTION. Book t
As luck would have it, I did 'fpy
A May-pole raifed up on high ;
The which at firft me mudh forpri^'d^
, Not being beibre*hand advertized
Of fuck a ftrange ifticoihmon fight ;
I faid I would not ftirdiat night,
^or reft content, tuntil I'd found
Its heifht exad from oiF the eround :
But when theff wonls I juft ha4 fpoke,
A blaft of wind the May-pole broke i
Whofff*broken piece I found to be,
TxzSt in length yards fixtyrthree ;
Which by its fau broke up a hole.
Twice fifteen yards from off thfi pole |
But this being all that I can do.
The May-pok npw being bsoke in two
Unequal parts, to aid a mend.
Ye ladies pray an apfwer fend*. Laiiis Diarjm
Firft 63 X 63 = 3969
Alfo 39 X 39 = 900
tjt 3069 =: 55*3985 pieces {landing.
••• 63+S5-3985 = M8.39«5y*^4« = il87anl«>?f<»t»
a^ inches, height of the pole.
2t A caftle wall there was, whofe height was found
Tq be an hundred feet from th' top to th' ground :
Aeain|l the wall a ladder ftood upright,
. Qf the fame length the caftle was in height*
A w^ggiib fellow did the ladder Aide,
(The bottom of it) ten feet from the fide*
Now I would know how far the top did fidl.
By pulling out the ladder from the wall. ,
••*.■•■
100 X xoo = xoooo
xo X 10 = xoo
v^ 9900 = 99.49874
* • ' %oo — ? 99*49874 =? •$0125 =: a very little snore thaai
6 inches.
3, I want the length of a ihoar, that being, to ftj^ %\
feet froqi the upright of a buildings will fiipppf t a jamb
?3 i^^h V> iAcheS) from the ground.
Chap/Vl. SQ^UARE ROOt^ nfs
If X II = I2ff
■ "^ ■ ■ ■ ftst* ioehet.
^ 689.02^ s 16 , 3U99iS. Q. E.. F.
^ 4* The height of an elin» growine in the middle of a
circular ifland to feet in 4iauiicter9 paunbs C3 feet ; and
a line ftretched from the top of the tree,- ftraight to the
hither edge' of the water, iia feet: what then is the
breadth of the moat, fuppofing the- land on either fide of
the water to be level ? .
112 X 112 = 12544
53 XS3 = 2809
sf 9735 = 98.666-
* • * 30 «f- 2 :s 15, and 98! — 15 = 83I, breadth of the
moat rec^uired*
j; Two {hip8 let fail from the fame port, one of them
goes due eaft co leagues, the otheyr due north 84 } how far
are they afundk^r ?
50 X 50 = 2500
84 X ^4 == 7056
• 9S5^ = 47-7S» ^ 97I leagues.
6. A liM 17 ^ds long will exa£Uy reach from the top
of a fort, on the oppofite bank of a river, known tp be 2]
yards broad: the height of the wall is required*
27 X 3:;^ 8< feet, 23 X 3 = ^ ^^^^*
81 X 81 =: 6561
69x69 = 4761
*/ j8oo = 42.426 = 42 feet, 5 inches. ^
^ v. ^fiiipp^ a Itght^houfe buflt on the top of a rock ; the
diitance between the place of obfervation, and that part of
the rock level with the eye, and diredhr under the building,
b 'gi^n*'3io ftthoins; the diihknce from the top of the
rdek to tMptaceoP obfervation, is 423 fathoms, and from
the top of the building 425 : the height of the edifice Is
Inquired.
^4 4^S
^
ai6 fiVOXUXIOK. f^ooiiiU
425 X 425 = 180625
310x310=: 96100
^94525 = 290^73x87) Jight-houfe and rock«
Alfo 423 X 423 = 178929
— 96100
4/82829 = 287.80^27, rock.
•.* 290*73187 -r- 287,80027 = 2.9316 =17.59 feet. Q»
^. F. the height of the light-hpufe.
8. A ladder 40 feet long may be fo planted^ that it fhatl
reach a window 33 feet from tne etousid^ on one fide the
ftreet j and without moving it at me foot, will do the fame
by a window 21 feet high, on the other Qde : the brcadtl\
of the ftreet is required.
40 X 40 = 1600
33x33=1089
^511 = 22.tii
21 :^ 21 = 411
* 1
♦^1189 = 34.48
•B^
Anfwcr, 57*08 feet, Q. £• F.
9. An ancient bath was found, of a triangular^ form,
fhe fum of whofe three equal fides was 125 feet: thearej^
pf the bottom is required.
3 ) 125 ( 41.^, each fide.
2 ) 41.^ ( 20.8^^4 half the fidf •
Then 41. J? x 41.6 = 1736./11
Alfo •20.83^x20.83= 434.027
^1302.0845:36.0849 perpend.
Then 41.^ x 36.084 = 1503.5.
2 ) 1503.5 ( = 751.75 feet, the area required*
to. The paving a triangular court, at i8d. per foot,
eame to joo 1. the longeft of the three fides was 88 feet :
what then was the fum of the other two equal fides i
C:b^.VI, SQUARE ROOT, fti^
x8d. = .oysK). jDO.ooo ( 1333.^ feet, the $r€z^
V = 44) «333-J ( 3f^'3j perpen4. *
44 X 44 = ^936 ^
v' 2854,//=: 53.42s, one of did two
♦ • • 53-425 X 2 =;: 106.85 feet, the anfwcn lefler fidts,
jl» I imuld plant 10 acres of hop-ground, which muft
be done, either in the fquare ordei*, or as the number 4
^Rnds on jtiie dice, or in the quincunx order, as the num-
ber 5 ; the three neareft binds, in both cafes, mnft be kt H-
peall^ juft 6 feet afunder : how many plants more will be
Inquired fcfr the laft order than for the firft, admitting the
form of the plat to lav the moft aahranugeous for the plan-
tation in ckher cafe r
6 X 6 := 36 fquare feet each plant in the fquare order.
3X3= 9
^ 27 = 5.19615 perpend.
Alfo 6 X 5*19615 = 31-1769 feet, each in the quincunx
in one acre are 43560 fquare feet« order.
JO X 43560 == 435600.
31 . 1 769 ) 435600 ( 1 3972 plapts in the quincunx 7 ,
36)435600 ( i2|oo plants in the fquare. J ^^^^^*
Difference = 1872. Q. E. F.
\ m
12. The quarry of glafs is 2i inches on every fide, and
^B much crofs the middle, coft id.; the fquare is 5^ inches
by 3t> and coft i^^d. j what will be faved, glazing 1000 feet,
the cheapeft of the two ways, fuppofe the leading of the
lights be nearly equal in either kind of work ?
3-75 ^3-75 =14.0625
^.875x^.875= 35^5625
i/ 10.546875 = 3*2476 perpend.
Then5 25 X 3.5 = 18.375 i j fqpare
Alfo 3-^76x3.75 = 12.1785 r"^''^*i quarry.
1000 X 144 == 144000 fquare inches.
^8-375 ) 144000.000 ( 7836.7 fquares,
J a. 1785 ) 144000,0000 ( 1 1824 quarries,.
' 7836.7
tit E y O L U T I O 9T. B6dt
iz8z4
f I
T
7836-7 _.
979 7 -TF
98s 4
' J^, j^ 1$ y cpft in fquares. 49 5 4 coft ifl qunriw.
*-*49l* 5'* 4d. — 48I. igs. 7d, 2:5S*9<1. advan-
tage in fquares.
13. A fiunmer^houfe is a cube of lo feet in the dear,
the cornice of which pcojeds juft 15 inches on a fide, and
being of .timber and ftucco, the fides aie 6 inchas Hlidc,. fo
that the' whole front of the roof, from out to out, is ij^
feet. This is hipped from each of the comers to the c^n-
ter^ and beii^ troly pedonent pitch, it raifes |. bf the
front, or three feet ; I would, hj the help of thefe dimen-
fions, meafure the flating, without venturing to clhnh for
more, and compute the coft M 34-d. per fquare foot.
Fufft 10 + 24^ + I = 13J =: *- =; whole breadth.
Z
, Then — x -y = 3 feet, the rffe of the roof. - . -
> ) J3'S { 6.7s =s half the breadth.
Alfo 6.75 X 6.75 =2 45*5625 ^
3x3 =9
t
• 54.5625 = 7-38664.
Then 13.5 x 2 x 7.38664 = 199*43939 area of the roof.
Tff
J994393
2.49274
•4154s
£ 2.90819 ^ aU iffs. 2d« the anfwer.
♦ »
14. There are two columns in the ruins of Perfepolis
left ftanding i!rpright ; one is 64 feet above the plane, the other
50 : between thefe, in a right line, ftands an ancient ftatue,
the head' whereof is 97 feet from the fummit of the higher,
and 86 feet from the top pf the lower coluitin ; ^the bafe
whereof mcafures juft 76 feet to the center of Ac figure'a
bafe : by thefe notices tne djfiance of the top of die columns
may be, by numbers, e^^ily fouad%
^ ' . f irft
Ctaifain, SQJJARE ROOT.
Firft 86 X 86 ss 7396
Alfi> jfB X 76 =: 5776
i/|6ao
Then 50
•-i-40« 34922
*«f
^
■^:: — :.
84-
l«aMtlMMi«(»i
^■■■ii*
9.75078,' height of the ftatue.
Alfe 64 —» 9.75078 =5424922.
Affiih 97 X 97 . , • 9 9409
54.24922 X 54-2492* = 294a-97787
r^- — ^^^^^^gm
^6466.02217 =5 80.4121.
TC.4121 -h 70 =a I50.4121, dUbnce of the column*.
64 — 50 =2 14) difference of their heights.
80.4121 -f\ 76 = Ii6*412!
156*4121 X I56»4I2E s 24464.745
14x14 .. ss 196
4/ m66p-74S == 157; Qz. E- F.
But if the ftatue he m enormous CoIofTuf, higher than
jthe towers ;
By working-as before, the ftatue will be found 40. 29888
• feet higher than the lower coliinm* -
Aifo 50 + 40.29888 = ^^,?.z r ••••!?<r
90.20888, the height of ^p— t ■••>... ^
tne ftatue* ^Ai.....t»w».«... ...» ,.., •...rr^itzjiai
90.29888*^ 64 SB 26.
29888, hieher than the
higheft coaunn.
7^
97 X 97 =: 9409
26.29888X26.29888= 694.63 n
§
v^87i7,3689 = 93.1668
V 07 1 7? y>o9 = 93- jow
76 + 93-3668 = 160.3688, diftance of the columns.
^69.3668 X 169.3668 = 28685. U29
14 X 14
= 196
i/ 2888.i.ii29=i69.94.Qi£,f «
15. The femidiameter of the earth being 3984«c8. miles,
and the perpendicular height of a mountain three mile? ; how
hr will it be fcen at fea, or on plain ground, fuppofing the
rye of the fpc£fator to b^ qi\ th« furfocc of the ground or
F4tP^ ? 39?4 58
lio E V ^ L U 1 1 O N. Bwdtf.
: 3984.58 femidrametcr. .. * *
3 . height. • '
.' ■ ' . . . •/ .*
^ .3987-^8 X 39?7S8 = 15900794.2564
3984. iS^x 3984-58 = 15876877.7764
v^ 23916.4809^ I54-64 miles.
• a E. F.
C A S E V.
Given three (ides of a triangle, to find the area.
RULE.
From half the fum of the thriee. fujes, fiifctia^l e^^ch'fide
feverally; let the half fuoi, and the thr^e differeitces, be
multiplied continually ^ the fquare root of the product will
be the area required.
• • •
I. Having a fifli-pond of a triangular form, whofc. three
fide$ moafure 400 yirds, 348, ;and 312 j virhat quantity
y of ground docs it cpver ?
* . Firft 4C0 + 348 + 312 = .1060.
Alfo i^— = 530 yards, half the fum of the three ildei.
And 530 — 400 =5^0")
530 — 348 = 152 J* diffeeences.
530 — 312*= 118 3
. Alfo 530 X 130 X 182 X 218 = 2733676400.
^/ 2^733676400 = 52284.57 1 3 JB -fiiuare rarfs.
4840) 52284.571338 ( 10.8026 =: loa. 3r. 8p. <^ E. F.
-2. A field of a triangular fprm, whofc fides are 380,
429, and 765 yards, lets for 55 s. per acre ^ bow oMich
does the whole bring in per aonum \
Firft 380 + 420 + 765 2= 1565.
Alfo -" z^ 782.5 yards, half die fum of the tbreiB fides.
And 782.5 — 380 es 402.5^1
782.5 — 420 == 3625 \ differences,
782.5 — 76522= 67.5 J
Alfo 782.5 X 40Z.5 X 362.5.x 17.5 = »998oq37*o.9375,
V^ 19980037x0.9375 =: 44699^034 fquare yards,
4840)44699.034(9.2374 = 9 acres, 38 perches.
' Anfwer, 25 1. 8s^ -44* per annum.
. - ^ ^ ^ SECT.
Cli^K^. C:U B i: iR,0 1> ,T; 2tt
^ r •
H
SECT. Ill
. ■
To Extras the Cube kooT.
R U L £ I.
A V I N G pointed the givtn feFolrfthd int6 -periods
of three figures, as before direifted,' '
Seek the greateft cube in the left-h^nd period \ Write the
root in the quotient, and the cube under the period i which
fubtrad, and to the remainder bring down the nejct {Seriod;
call this a new rtfolvend, under which draw a lihe. '
t
2. Under this refolvend write the triple fquare of die
root, fo that the units in the latter ftand under the ji^ace of'
hundreds in the former ; and umder, t)ie faid triple iquare
write the triple root, removed one place to thcrijjxt^lthc
fum of thefe is the di^ibr^ uade^.wfakl^ draw a iiae.
3. Seek how oft this divifor' may be had in the.. new
refolvend (its right-hand place e^f4|ft^) and write the
iefult in the quotient*
. / ' ; • ' .••• ••' " ) < .
4. Under tha divifiv <^K ^^ fNi^dnft of ihe triple
iquare of the root by the laft qu6tjlDit^^pire, fttttngdown
the units place of this line under that of 4wi» -in-the
divifor; under this line write '- the |MAi£1 of 4ikt triple
root by the fquare of the laft quotient figure'; let this line
be remoued one place bevond the-riglit iit dtetelkSir!^ and
under this line, removea one place forward ta the Tight,
write down the cube h( the ■klt^u'dtieiil^gqYe, the
fum of thefe three lines call th^ fubtrthiniel, «inder which
draw a line,
5. SubtnuEl the Jubtrahend from the new' refdlvehd ; to
this remainder brinj; dowh' the h^t period ibr another
refolvend s the dhrHot ^ttioft^ t!He tsfJ^le Y^ttait of the
quotient, added to the triple thcf eo?, '&c. ' as before dr-
re^ed. -. . , .
It
Extraft
tit
EVOLUTIOl^. Boolcl.
Extnd the cube root of i2alSi$jlji^
• • •
122615327232 ( 4968
64
{8615 new refblvend.
48
49*
972
53649
4966327
72OJ
147
72177
43218
5292
tripIefi]uareof 4*
triple of 4.
divifor.
fubtrahend.
refdvend.
triple fquare of 49.
triple of 49^
divifon
triple fquare of 49 X 6*
triple ot 49 X fquare of 6<
cube of 6.
4374936 fubtrahend.
591391232 refolvend.
738048 triple fquaie of .496,
1488 triple of 496.
7381968 divifor.
5904384 triple fquare of 496 X 8.
95232 triple of 496 X fquare of 8.
512 cube of 8.
591391232 fubtrahend = laft refolvend ; fo that
— ' 4968i8thetruecube root of 122615327232.
RULE
Chftibvyi. . C U JET E R O O T. ti^
!• The refolvend being pointed into proper period^, find
the neaieft lefs root of the figures of the firft punftition on
the left-hand; fubtraA its cube from the number given ; t»
the remainder annex the next figure for a new refglvend.
2. Take 4 of the refolvend for a dividend.
^« And for a divifor t^ke the fquare of die root added
to half the root (or rather added to the produd of the root,
and the next quotient figure, leaving out the laft figure of
the produ£t.)
4* Divide the faid dividend by that divilbr, the quotient
is the fecond figure of the root.
5* Begin the operation a-new ; v\z, cube the two figures
of the root, and fubtrad the cube from the given number^
suinexijig another figture for the refolvend.
6. Take the third part of the reMvend for a dividend,
and the fquare of the root added to half the root (or rather
added to the produ£l of the root and the next quotient fi-
gure, ftriking ofF the laft figure of die produ&) for a di-
vifor*
7. Tlie divifion gives anodier figure of the root ; but
the divifion is to be continued on to two figures, by the
contra£tion in divifion of decimals, or otherwiie.
8. Repeating , the operation with four figures in the
root, you will get four more by a new divifion, which
gives eight figures in the root; and from 8 to 16,^ &c. al-
ways double.
9* Note, when the cube exceeds the number given, a
lels figure muft be writ in the quotient ; and obferve every
divifion gives one figure, and the reft are found by con-
tinuing the divifion, aftd dropping a figure of the divifor
every time.
ID. If after all the periods, both in whole numbers and
diecimals, are brought down, the.extradUon may be continued
as far as you pleafe, hy ftill adding ternaries of cyphers*
At laft cut off as many places of whole numbers as there
are points in whole numbers, and the like for decimals.
1 1. If you jdcfire the laft quotient to go true to more
places of figures, add half the laft quotient to the laft
root, and iquare the fum for a divifor, and divide over
again. .
Extract
324 £ y O I» UuT. I-O N, . B«9k C
t r -
- * • •> ,
E^tr«ft4M cube sMt Qiitdf ^1^4756621 7*
I'
M..,^ 16 iqiuLce*
3 ) 283 64 cube. '
4*2)90 ,
, .1 • • • ■ «
18) 4 ^
92398
91125
3 ) 12736 fquMt tf 45,« . Ma5*
3025) 4245(208 cub^ 4591125.
9 4068 45 X 2 s 99.
2034) X77 root 45208 in wbol^,niiipfaei% ««
163 its fquarc 2043763264^
. <;ube 92394449638912.
14
92398647506217 ,. , 452PS ^
923444496389^^2 6
3)4x97867305-0. . 27U48
« •
2043763264) 1399289101.6(684^539^
-f- 27124 12262742328
' ■, root 452Q8.68465.
2049760388) 1730148688
1635032310
95^16378
I • ■ '
R u L E m. ' .
'»
•< •<
1.. After. the {iv«n reTolvQiiA is tnily poihtod, ftek the
greiiteft cube ip the left-^iand period ; wijte che root in the
Juotient^ fubtra£t the cube ftom.liie period, b^ direAed in
le other rule ; and tQ the remainder bring down all d)#
remaining periods in the given number, for a new re-
folvend.
^ I - a. Ta
Chftp.ti. eCUBE root: Izi
2. To the root {or^ quotient} annex as many eyphers as
there are remaining periods^ multiply . this by 3; by this
produfl divide the refolvend, and point the Quotient into
penoda of two places (beginning at units) ooferving that
there be ^o more points than there were periods brought
down to the refolvend.
3. Make the root, ^found in the firft . period of the given
numbers) adiviforj let how often it may be had in the
left-hand period of the quotient (excepting the place under
the point )^ and the figure refulting write in the quotient (to
the right-haiid of the root firft found) and on the right of
the divifor ; multiply this increafed divifor by the laft quo*
dent figure; to the remainder bring down the next period |
divide thisi>y the'Iaft divifor.
Extraft the cube loot of 8302348oocpOQ.
8302348000000 .
8
6»0Ma> 3ot348l>o»OQQQf .
2oa)$039i5j( 024.
■- ' 404 2000
aoo^) 9991 2024
The cube of . 2024 3s 1^14^9824 . •
8302348000000 ) 20240 X 3 = 60720^ dIvifor«
8291469824
6072,0 ) io878i76o<>,'o ( 179153,42556
20248 ) i79'SS'4^55^ ( 8.8479
101984
20248.8) 17 16942
16 19904 20240
20248.84) 970385f
+ 88479
8099536 20248.8479, root, Q^ £. F.
^10248.847 ) 160431960
141 741 929
18690031
Q. R U tE
%i6 E T O L U T I p V; Boql^I^
R ULE IV.
Divide the gmn refelTen4* by thtto times the (lippofed
toot, and from die quotient fubtraA one-twelfth of the
fquare of the fappofed root ; the fqutre root of the re*
mainder^ added to half die fuppofed root, wiH give the true
root required.
^ . W^t Is the cube xoq% of i467o8«483 i
Suppofe the root co*
Then 50 X 3 s 150 ) 146708483 ( 978.8565
50 X 50 s 2500 i alfo 12 ) 2500 ( 208.3333
■ ' " ■
770.5232
a)5o(ss25) bait tm fiipfouM noot. .
52.7, the nMt.
But for greater exadnefs I proceed to another operation.
Thus, 52.7 X 3 = 158.1 ) 146708.483 f 927^7»
52.7 X 52.7 = 2777.29 ... 12 ) 2777.29 ( 231.44083
^ 696.50656
^ 696.50656 =S 26.39141 ■!■
2 ) 52.7 ( 26.35
• - .52.741419 the root more exaA*
Exttad the cube root of 2.
1X3= 3)2.o(.66^
I X I andi2)i.o{.o9j
s^ -5833 ( -7
2)ao(.5
1.2, root.
By a fecomLoperation 1.2 X*j = 36 ) l.00{ .^55
1.2x1.2=1.44 I2)l.44(.l2
\/ 4355 (•659 —
2 ) 1.200 ( .6.. .43/
1.259, root J
Ch*. YI. ^ ^ B E R O-O T. %2jt.
. • *
By a cJurdopenrtipo v^sgx 3«» 3.777 ) a.QQa( ;y295ib78'66.
I.259X 1.259= 1.585081 ... 12 ) 1.585081 ( .1310900833
• '3974307033 = -630+i»» 3 -'
2) 1.259 ( .6295 , -39743070^3
Wltat is the ci^c root of .0001357 ?
Suppofed root .05*
•01 X 3= -'5) -oooissj (.00094^6
•05 X 05 r= 0025, and 12) 0025 ( .000208 j
V^ .ooo738;af ( .027 '00073^
,952 root, which by involution I find too
* ..* I take *05i Jror the fuppofed root. oiuch»
•051 X 3 353 .153) .0001357 ( .00088751 ,
, .051 %ost S3 j9oa6oi> ana^ia ) *om6o 1(0002x758
» * • •
< .00066993
^ 00066993 fls «02588
2) .051 ( .0255
Pmfmmm^mtT
*05i38 root.
Then .05138 X 3 = •15414 ) .000 1 35700 ( .0008803685
•Qjfi Jl >C »^i:^ SB .0026399044
12 ) .0026399044 (.0002199920
,» til I»>»^^^PW
.0006603765
I if
V' .0006603765 = .02569779
2^2? as .02569
.05138779, the root. Q. E. F,
mm^m
What is the cube root of 13I?
In decimals 134 =^ '3'^
.Suppofed root 2x3=;: 6). 13.6 ( 2.2^
2 X 2 ;= 4 > alfo J2) 4.0 ( 0 J3
Q.2 i/i94
n
»t E y O tr U T t O H; to6kU
* = I.OO
,1
a.39, root
».39 X 3 = 717 ) >3*^ ( 1.90609019
^1.4300818601.1958
t»sst.i95
•
i.3908,
toot.
a.3908 X 3 = 7->7H ) 13-666^ ( <-90544S4«as;
2.3908 X 2.39<'8 = 5-7 »59H64
12 ) 5.71592464 ( 0.4763170533
1.4291183593
^ 1.4*91183592 =r 1.I954S7397
2)2.3908= 1.X954
2.390857397, the root. Q.E. F.
The fecond method of extrafUng; the cube root is diat
ufed by that great matbematrcian Mr. Emerfon, in bis
treatife of arithmetic, and doubles the figures in the root at
each cperation.
The third is the method Mr. J. Rob^itfon, F^. S. ufea
In his menfujation, by which each operation triples the
figures in the root.
. But the fourth and laft I take to be the eafieft, 9S th^
operations are performed by e^fy divifions, and an extraAion
of the fquare root.
N. B. This method only double* the 6gures in the root
at each operation.
T
S E G T. IV. ,
Some Us£S of ibe Cv».z Root.
H £ cube root is of very great ufe iii mathematics, but
I ihall only exhibit a few ci(i|#»
^ CASE
Cfai^vi. cirB£ ROOT* e^
G A s E r. • ^ ' - -
To find (he fide of a cube th^t ihall be eaual in foliditjr
to any given fblid, as a globe, cylinder, priim^ cane» &c.
K U; L E.
'EittX2& the cube root of the folid coiitenf of the ^iven
body, wbicU will be the fida of a cube of an equal folidity,
♦ '• ■ ^ > . .
. S(ippqje a dieft, whofe length is 4 feet 7 Inches^ breadth
a fee^f incites, and depth i foot 9 inches j required the fide
oF' a c«H>e of equal folidity i
. F. I.
IBjrSthJ 3 = I7 } ^'^^'^««-
38s
110
Depth I 9 =: 21
1485
2970
* 1
3n85 ibiid inches.
3«>>< 3= 9^ ) 3"85 ( 34^*5 271.5(16.4
joX ^ = 9<^«")90o(;7s I
271.5 26 ) 171
156
^^
16.4 324) 1550
, 31.4, root.
t
For a fecbnd operation, 31.4X 3=94.2)31 i85(3;ji.oac6
31.4X314 = 985.96 . .. . 12) 955.96 ( 82.16333
^ 248.88773 ( 15.7762 24^88773
a)3'''4('S-7
31.^47629 fide of the cube required.
"" Q.3 C A S R
«•**
ij^ £ V O L U t I b N. ^otfti.
C A S £ II.
Having the dimenfions of any folid body, to find'tkofe
cf a fimilar folid, any number ot times, greater or kft than
the folid given, ^ ^ ^ E.
i
Mtihiply the cube dP each of the given dimenfioiis by the
difference between the folid giren, and that required, if
greater ^or diride by the diHerence, if lefer) than the fo-
fid given ; then cxtraa the cube root of each produft or
quotient, which will be the dimenfions of the folid required.
Suppofe the length of a fliip's keel be iz« feet, the
breadth of the midfiip beam 25 feet, and the depth of the
hold 15 feet; I demand the dimcnf|Oms of anothei^(hip, of
the fame form, that fhall carry three times the burthen ^
125 X 1 25 X 125 X 3 = 585937s
25 X 25 X 25 X 3 — 4687s
15X 15X "5X^3.= , lOi^S
••' ^s/ 5859375 = 180.28, keel. ^
Alfo 'v^ 46875 =: 36.65, midfliip beam. >Q«E.F.
And *^ 101^5 = 2i.6,depthinthehoId. J
Or fuppofe the (hip was to be but of half the burthen of
that whofe dimenfions are given as above,
m ■ I ■ ^ ^ . ^^^^*
^ ss 976562.5 ^
"'^ ''^ '^ = ,687.5
• . • ' v' 97^562.5 = 99.202
Alfo ^^ 7512.5 = 19.84 -
And '^ 1687.5 = ri.906
CASE -HI.
Having the dimenfions an^ capacity of a. foUdy t^ find the
dimenfions of a fimilar folid of a different capacity,
RULE,.
uivil* the cube of the -dimenllons given, multiplied into
the capacity of the veflel or body reguired 5 the cube root
-of the quotient will be the refult.'
• I If
Cliip.VI. BIQUADRATE ROOT* %^,
If a fliip of 100 tuns be 44 feet long at the keel, of what
Icngftb ibill die keel of that ihip be^ whofe burdien is aao
tuiw? ^
Fixft 44 X 44 X 44 X 220 = 18740480
100 ) 18740480 ( 187404.8
'^ 187404.8 =: 57.22592, the anfwer required.
C A S E IV.
Between two given numbers, to find two mean propor<^
tionals, ,
ROLE,
^ Multiply the lefs extreme by the cube root of the quo^
tient of the ereater extreme, divided by the lefs ; the pro^
6u& is the lefler of the two mean proportionals, which
multiplied by the faid cube root gives the greater mean
fought.
xind two mean proportionals between 7 and 15379*
7)15379(^197
»» X3=?33)ai97( 66.575
12} 121 ( 10.083
.
^ 56.492 s= 7.5
5*5
13. cube root.
and 91 X 13 == 1183, fccond J '"^^'^ P""^^' ^^^ ^-^
Fora&7 : 91 :: 1183 : 15379.
SECT. y.
To BxtraB the Br<ivADii'ATE Root.
«
RULE.
EXTRACT the fquare root of the given loTolvend,
and the fquare root of that firft jroot will be the bi«
quadrate root ccquired,
Q^ 4 Extraa
9 mm, f^ «*»4\«
Extraa the biquadratc root of 3343^59122491 3ff<..V
• •••••••■**•••
334815812249x3441 ( 182979729 ( 13527, biquadrate
I ^ I . , foot rcqiiirei^.
• ^
48)234 23) 8*
224 69
362)1081 265)1397 •
724 »3*5
3649 ) 35758 4702 ) 7297
34841 5404
36587 ) 291712 27047 ) 189329
256109 189329
^m
3^949 ) 35603*4
3*93541
3659587 ) 46678391
2;6i7io9
SECT. VL
To ExtraSt the Sursolid Root. *
RULE.
HAVING pointeil the fftven refolvcnd into periods
of five figures, feek fuch a furfolid number in the
table of powers (or otherwife) as comes neareft to the firft
period of the refolvend, whether greater or lefs; and call
the rcfpefUve roqt, either more than juft, or lefs than juft>
as it falls out \ annexing fo many cyphers to it as there
are remaining periods of whole numbers in the refolvend.
2. Find the difference between the refolvend abd tike fur-
folid number, fo taken, by fiibtra&ing the lefler from the
greater.
3. Find the cube of the forefaid furfolid root, with its
annexed cyphers, which alfo may be done by the table of
powers, and multiply that cube into five, the index of the
furfolid, and divide the difference between the refolvend and
the
ChafnoVL SI^RiSOJLIiJD ROOT* ajr
the furfolid number by that produd ; bv. which it will ^e
dcpreflSif io a iquare, and whin pbintea into period^ of' iWo
figures eachy call it the new r^olvend«
4. M^e the firft root without cyphers a divffor, en-
quiring how often it mav be found in the firft period of the
new refidvend ; with this, cohfideration, if the root, now a
divifor^ be left than juft, annex twice the quotient figure
to it ; but if more than juft, fubtrad twice the auotient
^ure irom a cypher, either aiuiexed, or fuppofed to be
annexed, to that divifor or root, multiplying it, fo increafed
or diminiflied, with Ae faid quotient figure ^ fcttihg down
the units place of the produ^ under the pointed figure of
that period, fubtra£Hng it as in divifion.
£xtra£tthe fiufolid root of 3076828^1 io67i5625«
• • • •
307682821 106715625 (3 - ■ •
64682821 IO67I5625
3000 eubad = 27000000000
17000000000 X 5 = 1 35000000000, divifor.
135000000000 ) 64682821 1067 1 5625 ( 479132
/ 3) 479*3^(14
-f 1X2= 2 32
32) 1591 3000
4-4X2=: 8 1312 140
3140
By a fecond operation* .
307682821 I 067 I 5625 [ .
3140 ^ 5 = 305244776182400000
2438044924315625
314 •• 3 = 30959144000
5
»*m
" ^54795720000, divifor.
*S479S7aoooo ) 2438044924315625 ( 15750
314 ) 15750 ( 5 Firfl: root 3140
5X2= JO 15750 + 5
tm
315a 0 True root 3145 .Extraft
1^4 EVOLUTION* Boakl.
ioa4
976379602989073960279630298S
4762039 701 092603972036970 1 2
400000 f^* 3 :s 64000000000000000
5
320000000000000000
32 ) 47620397010 ( 1488137406
400 ) 1488137406 ( 037
^^--06 1 182 400000
394) 29613
— 14 27482 3963<^> root. ,
\
m»
. 39^6
But I only take 306 for a iecond operation, Which I find
by involution to be lefs than juft*
V* 9763796029890739602796302988
396 ©^ 5 =x 97381381 i0976oo<
236579 1 88 147 396o27963(>2988
369000 ^ 3 = 6200Q1 36000000000
62099136) 2365791881479602,7963 ( 38970160.5871
396000 ) 38970i6o'.587i ( 098.3889
18 356562
t
39612 3322060
16 3169088
396136 15297258
6 I 1884098
3961366 34x3^^6071
16 316909408 .
39613676 34406663
31690941
3715722
396000
098.3889
396098.3889, the root fought.
■ ■ i ■ SEC it
Clwp^FI. EVOLUTION, 435
SECT. VII.
70 Extras tkeRoor of the Sq^uare Cubed; or^
Sixth Power.
RULE.
EXTRACT the fifuare root of the given refolvend;
then extrad the cube root of that fquare root, which
will be the root of the fixth power required.
Or you may firft extrad the cube root of the refolvend,
aftd then the fquare root of that cube root, and that will be
the root required.
Pxtrad the fixth power of 43572838x009267809889764416.
435728381009267809880764416 ( 20874107909304.
4 1 find by pointing the fquare root
— into periods of three figures each,
403) 3S72 that there will be five figures in
3264 whole numben in the loot of the
— — fixth power.
4167) 30883
^ 29169
41744) 171481
166976
« _
41748X) 450500
417481
41748207) 33P199267
2922374+9
&c.
• • • • •
aooo X 3 = 60000 ) 20874J 07909304 ( 347901798
12)400000000 ( 33333333
314568465
314568465 ( 17600
10000
27600, which by involution I find too much,
therefore take 275. 27500
%3^ £ V O L U T I O K ^osk L
■ 47500x3 = 8*500) 20874107909304; (25301948$
V5 X 275 = 75625 . . 12 ) 756250000 ( 63020833
189998656
• • • •
189998656 ( 13784
1 2)27S(»37S
■
• ■
23 ) 89 275349 true root of the fixth power*
69 ■ *
267 ) 2099
1869
2748 ) 23086
21984
27564)110256
I 10256
SECT. VIII.
^0 ExiraS the Root of the Second S tr r sO l i tr ; or^
Seventh Power.
RULE.
HAVING pointed the refolvend into periods of fereii
figures, feek out fuch a number of the feventb power,
by the table> as comes neareft to the firft period of the re-
folvend, whether greater or lefler, calling its root more
than juft, or lefs than juft, annexing a proper number of
cyphers. ...
' 2. Kind tlie diiFerence between the refolrend, aiidthac
number of the feventh power, by fubtrading the lefler
from the sreateri
3* Find the furfolid, or fifth power of that root, whhits
annexed cyphers, by the table of powers ; and multiply
that furfolid number into fcven, the index of the refolvencC
4* Make that produd a divifor, by which the forefaid difie-
tence muft be divided ; ib that it may be deprefled to a
fqoare, and pointed as fuch. *
5* Make the firft root, without cyphers^ a divilbr, work»
ing with it and the new reffdvead, as in the furfolid i only
here
Cha^tl. EVOLUTION. 43^
Jiere ^ anift' ificf«are or diminiih the divifor with thrice
the quotient figure*
£xtnfi the 7tk power of 344877i74673075i3i82492t53794673«
34487717467307513182492153794673
3 #. 7 ss 2187
1261771746730751318249215379467^
Firft root 30000 o* 5 = 24300000000000000000000 .
243, &c. X 7 r^ 1 70100000000000000000000
Contraded 1701 ) 126177174673(74178233
3) 74178233(^0
-)-aX3=:6 72 3000a
20
360) 21782 •■
32000
Second operation.
3^
344877i7467307Si3i8*492'S3794673
7=34359738368
1X7979099^7513182492153794673
32 0. 5 = 33554432* which X 7 = 234881024
23488*024 ) 127979099207513 ( 544868
320)544868(017
+ 1x3= 3 3203
3203 ) 224568
+ 7X3= 21 224357 31000
32051 211
+0*7
320179 true foot.
SECT. IX.
T^oExtraStbe Root of the BiqyAPRATs Sq^arsdi or^
Eighth Pow£Jl.
RULE.
EX T R A C T the fquare root of the giren reToIvendf
which will reduce it to a biquadrate number, which
call a new refolv^nd i the., iquare root of which will be a
. • y fquare
438 EVOLUTIOK Book f>
iquaie number; of which extnA the kpmt imc, wfaidi
root will be the refult required.
Let uaioi628i32Q47623624649794a46o48i be the mm
rcfolvcnd, whereof the root of the eighth power is to be ex-"
traded.
112101628132047623624*457942460481
-2. ( 334^J58iaa49i344i»Mquad, refol.
63)221 Then ^ 33481581224913441 s; 182979729
JL2- ••• ^ 182979729 = r3527, root
664)3201 of the eighth {KHecr. Q. E- F.
2656
6688)54562
53504
66961)105881
66961 , ^
669625)3892032
3348125
6696308)54390704
■5?57Q4^4,
66963161) 82024076
66963J61
669631622)1506091523 .
1339263244
6696316242)16682827962
13392632484
66963i62444)3a90i:9547846
267852649776 .
6696316x4489)6x166891307049
6026684620401
6696316244981} 0000518664879
0696316244981
66963162449823)230420241989842
2008894873494^9
66915316244982^4)2953075464037346
2678526497993056
6696316244982684)27454896604429004
26785264979930736
66963i6a4498s688K}66963i624498628r
6696316244986281
0 .
SECT.
^faKwTL B y O L U T I O N; jij^
SECT. X.
Tt Extras tie Root ^ the Ccbb Cubsd> ar
Ninth Powek,
R U L E.
EXTRACTS cub* root of the gkren refolvend, and
the wfult will be a cubic nfaivend ; of which ex-
trad Ac cube root alfo, which will be the loot of the ninth
power re(}uired.
L« 976379§oa08oO7'396oa7963o49889o be the refdvend
given } out of which the root of the ninth power is to be
extraded.
99 X 3 = «97 ) 9763796029890739602^
99 3»7473+i07
11)9805(8170833333
891 ■
«9« , 1^847039007740 = 4970000000
— ^ =495
^5 —
Root 9920000000
99M y 3 =: 19760 ) 976379604989073960179630
9920 •• 2 = 9840640 ( 3280845440151 ,
^^^:^= 8a0053333333
^ 24607921068180 = 4960637000
^• = 496
And, Root 991^637000
9910637 X 9910637 = 98419038485769000000
XJ
«976i9u ) 97637960298907306027963028890
24604761244077639421.99
• ^604761244077639421.99 = 49603^8663,5616
'^ "^ * •9910637163.5616,
Then
>4^ C V O L V t t on; Boqkll
Then,
2100 X 3 = 6300 ) 9920637 163.5616 ( i574543.<
Stioo X 2100 = 44I0000. 12 ) 4410000 ( 367666*1
ij 1206877 = 1098
*i^ = 1050
Root 2x48 in whole ntanbers* *
Again,
1148 X 2148 = 4613904
2148 X 3 = 6444) 99^0637163-5616 ( i539S»S-3%
12 ) 4613904 ( 38449^
1155023.3885
^ 11550^3-3885 = I074,720i
-H^=i074. :
The root of the 9th power 2148.7201. Q. £. F.
■ ■ ■ •
Thus have I endeavoured to make plain the metlml of
extrading the root of all the powers, Whofe index is- any
number not exceeding^ the nine dieits, which by a littl^
confideration may be extended to mil higher powers. £
now conclude this chapter with the following method pro-
pofed by Mr. Halliday and others, viz.
*
*
SECT. XI.
TO multiply feveral figures by feveral, s^dd the produft
to be producel in one line only.
RULE.
Multiply the unib of Ae multiplicand by the urtits of
the' muhtplier, fetting down the units of the produ&, and
carry the tens ; next multiply the tens in the multiplicand by
the units of the multiptier, to which add the produd of
the units of the multiplicand, multiplied bv the tens io the
multiplier, and the tens carried ; then multiply the hundreds
in the multiplicand by die units of the muItiplioTf adding
the produd of the tens in the multiplicand multiplied by
the-tcAs in the multiplier, and the units of the multiplicand ^
ky the hundreds io th« oHil(iplier % and ib proceed till you
have
•
CkMft VL E V O L U T I O N. 241
have multiplied the multiplicand all through, by every figure
in tbe multiplier. ^ ^
321434
2132x3
68533481016
ExPtANATIOI^.
Firfr, 3x2=6; fecondly, 3 X 3+ I X 2 = n, /.,. j
said carrv i. ^
pirdly, 3X4 + Ho + 0<3 + i = 20, that is, b
and go 2. . -> . ' ,
f o-rtUy, 3,,X « + 3 X 2 + I X 4 +ixl + 2 = 2i»
». *. I and go 2. " ' *
FiftWy,J3<T+ 1 X 2 + HTi + JSTi + r^^ +
2 1= 28, t, e. 8 and go a; j 1 -->. -r t-
_i«*'y» 3x1 + «"x2 + UTi + r^+73^ 4.
3 ;^ 4 + 2 = 34^ t.t. 4 and go 3.
Seventhly, 1X^ + 2X3 + 2X2 + 1^+3x1 +
3 = 23, /. e. 3 and go 2. "* ^
.EighlJiW, 2X 3 + 2 X 4 + 1 X I + iin + 2 = 23»
I. /• 3 and go 2. "*
Ninthly, 3X3+ 2x1 + 7x1 = 15, ,'. t. 5 and go i.
V ^<»»ly, 1 X 3 + 2 X 2 + I s= 8, to fct down.
Laftly, 2x3 = 6, which finiflies the work.
35234
52424
187107216
Firft, 4x4 = 16, that is, 6 and go i.
3x4 + 4x2 + r == 21, that is, i and go 2.
2x4 + 3x2 + 4 X 4 + 2= 32,/. #.2 and go ;^.
5X4+2x2 + 3X4 + 4X2+3=2 47, ».<..
7 and go 4.
3x4 + 5x2 + 2x4 + 3X2 + 4X~S + 4 =
■ 60.
3x2 + 5X4 + 2x2 + 3X5 + 6 = 5
3X4 + 5><2+2X5+s = 37,
I.
1^ 3X»
,4a EVOLUTION. Book I.
7x2 + 5X5 + 3 = 34- \
r^r,3X5 + 3=»8-
Mr. Hani<lay faye, Aat Ais it not only performed very ex-
peditioufly in (mall figures, but alfo in great figuret may be
done readily enough by any pcrfon who can add one num-
ber to another, not exceeding «i ; but I for my part think
it a hazardous puzzling operation, and only nt for the
praflice of another Jedidiah Buxton.
Tbe Enb of the FmsT Book.
Arithmetical Colledions
AND
IMPROVEMENTS.
B O O K IL
^ontuning PROPGRtioN, widi its Use ; alio the
Use of the Hules of Practicc, in varioas
Srancfacs of Merchandize and Trade.
CHAPTER I.
PROPORTION DISJUNCT,
eALLED THE
GOLDEN RULE} «r, RULE ^ THREE.
P
ROPORTION Disjuna, or the Golden Rule,
are either dire£t Or reciprocal, called Inverfc, and thofe
%rt both iingle and compound.
S K C T. I.
DIRECT PROPORTION.
DIRECT proportion is when of four numbers the firft
beareth the fame ratio, or proportion to the fecond, as
the thtfd doth to the fourth ; as in thefe :
5 = 35 = 2 17 : 119; or, 65 : 13 : : 20 : 4.
Bjr ratio is here meant the common multiplier or divifor ;
and it fliews the habitude or relation one number hath to
another, vis. whether it be double, triple, quadruple, &c.
ib that proportionality is a fimilitude of ratio s.
R 2 Tliat
244 GOLDEN RULE; or, BOok 11.
That is, the greater or lefs the fecond term is ih refped to
the firft, the greater or lefs will the fourth be in refped to
the third.
Thus the ratio or common multiplier is 7 in the iirft four
proportional numbers, viz. 35 ; the fecond term in the pro-
portion is 7 times greater than 5, the iirft term; fo is 119,
the fourth term, 7 times greater than 17,' the third tenn.
Alfo 5 is the ratio or common divifor in the fecond four
proportional numbers; for 13, the fecond term in the pro-
portion, is 5 times lefs than 6^, the firft term ; fo is 4, the
fourth term, 5 times lefs than 20, the third term.
If four numbers are in direct proportion, the produfi of
the two extremes will always be equal to the produ& of the
two means, viz. 5X119 = 35 Xi 7, each being equal to
59c, and 65 X 4 = 13 X 120 = 260.
if four numbers are proportional, they will alfo be fo in
alternation, inverfion, compqfition, fubtrad^ion, converfion,
apd mixtly. Euclid 5. Def* 12, 13, 149 159 i6.
That is, if 65 : 13 : : 20 : 4 be in direA proportion. \
Then 65 : 20 : : 13 : 4 alternate.
And 13 : 65 : : 4 : 20 inverted.
Alfo 65 -f- 13 : 13 : : 20 + 4 • 4 compounded.
Or 65 4- 20 : 20 : : 13 -}- 4 : 4 alternatelv compo^nd•
Again, 65 — 13 : 13 : : 20 — 4 : 4 fubtraded.
Or 65 — 20 : 20 : : 1 3 — 4 : 4 alternately fubtraAed,
And 65 : 13+ 65 :: 20 : 4+ 20 converted.
Laftly, 65 + ^3 : 65 — 13 : : 20 -|- 4 : 20 — 4 mixtly.
When three numbers are given, and a fourth proportional
is required, in order to ftate the queftion right, obferve the
following directions; viz.
Firft, That always two of the three given terms are only
fuppofed, and affign or limit the ratio or proportion ; the
third moves the queftion, and the fourth gives the anfwer.
Secondly, The term which moves the queftion, hath
generally fome of thefe words before it, viz. W^at will ?
How many ? How lone ? How far ? Or how much ? &c.
Thirdly, That the firft term in the fuppoiition be of the
fame kind and denomination with that term which move«
the queftion, and the term fought will be of the fame kind
and denomination with the fecond term in the fuppoiition.
. AU queftions thus prepared may be anfwered by three
feveral rules, but the firft is moft commonly ufed.
% RULE
Chap, L RULE OF THREE.
245
I
R U L E L
Multiply the fecond and third terms together, and divide
their produd by the iirft term i the quotient will be the an-
fwer jreqwred.
R U L E 11.
Divide the fecond term by the firft, then multiply the
quotient into the third term, and their piK)dud will be the
anfwer required,
RULE IIL
Divide the third term by the firft, then multiply that
quotient into the fecond term, and their produd will be
the anfwer. . * #
I. If 3|. yards of kerfey coft 8 s. gd. what will 257^
yards coft at that rate ?
By Reduction.
yds. s. d.
As 34 : 89:: 2574:
4 12 V
IS 105
1030
—
103
12)
15)108150(7210 pence.
31 20} 600 10 d.
30 - lod.
f4|3
3-75
By Decimals.
3^^ yards.
12
20
9
8.75
£ 4375
257.5 yards.
R3
As
SM^ GOLDEN EULEibR, Book
As 3.75 : .+375 : : 257-S
21875
30625
21875
8750
3.75) 112.65625 (30.0411^ =5 30l. -s. 10 d. as before.
• • 1562
• ^'625
ajoo
250, &c.
2. If i| ounce of firer |U^ coft 10 s. 11^ (U what will
I fervice, weighing 327 oz. 12 pwt. 9 gr. coft at that rate I
ez. pwt. 8. 4. oz, ]fw%. gr.
As I 15 : 10 11^ : : 327 12 9
20 12 20
35
24
J40
70
840 gr.
4
6552
24
525 farth. 26208
13104
FT-*
157257
5»5
r.^
•^r*
786285
3»45»+
786085
4)
840) 82559925 (98285 faraiinga.
695
339
719
12
20
245714^ pence.
ao47 7t
^y2 )£ 102 7 7i, anfwer,
5^5
By Decii^als.
20
oz.
15
'•75
I
ir.25
10-9375
0.546375
I if
4x6=
44
20
OZ.
6)
9> 2.25
ia.375
327.61875
Ckaf.L BULE Of THREE.
*47
A# J.75 : .54687; : : 317.61875
♦S46873
«6380937i
399333125
262095000
19657 1250
131047500
"63809375
«'75) 179.1665)0390625 (104.380859 =s
416 102 L 7S.-7^<1. asMfoit.
666
I5aj
1039
1640
3. If 2cwt. 3qrs. 21 lb. of liigar coft 61. is> 8d,i
what will 12 cwt. 2 qrs. coft at that rate i
By DECiMAts.
7J«
Cwt.
J-7S
2.9375
la
20
8
6.e»z
2
"5
Cwt.
2-9375
: Tw
Cwt.
12.5
08a
ii.5
3041^
73000
2.9375) 76.041^(25.8865 =: ^ !• 17s. 8|d. the anfwer.
172914
26041
2591
»9^
»5
R4
By
848 QOLDEN RULE} Ml, Bool^ II.
t
By Reduction.
cwt. qrs. lb. h s. d. cwt, qrs*
As 2 3 ^i : 6 I 8 : : 12 2
4 ap 4
II
28
329 lb.
121
12
1460
1400
5^
28
1400
»)
329) 2044000 ( 62 1 2|-
700 20)517 - 8|:
420 >
910^25 17 Sj^sasbeforCf
4
X008
By 'Vulgar Fractions.
pwt. qr. lb. cwt. 1. s. d. 1. cwt. qr- cwt,
2 3 21 = ig, 6 J 8 =: ^ ; and 12 2 = ^.
12
16 1« 2 • ' 12 2 24
Then i^) ^ fi^=:25l. 17s. 84d- anfwcr as before.
When any one term in the proportion is an unit, the
anfwer will lomettmes be moft readily obtained by prafiice^
as in the two following examples.
4. If I give 5 s. 4d. for one ounpe of filver9 what
|nuft I pay for 32} ounces at that rate ?
SS'
Chap. I. RULE or THREE.
949
s. d. s. oz. ,
5 4 = 5.;; and 32^ ozt = 32.5 By Practic«,
oz* s. oz. 8* d,
I : S'3 ' ' 3^5 t) 5 4
Si 4
9)975
1625
I73-3-?
= PL 13s. 4c|,
r I
t
8 10 8
2 8
Anfwer, jf 8 13 4
5. If a filver tankard, weighing 21 punces, coft 51, iqs*
firhat is that an ounce.? >
oz. L s« oz.
21 : 5 19 : : I
20
21)119(53. 8d. tbeanfnrer*
14
3
7
I. 9.
5 19
M»
I 19 8
168
As before, j£ - 5 8
6. If a piece of cloth coft lol. 165. 8 d. I demand how
many yards it contains, the ell Engliih being worth 88. 4d.f
8
208.3
— 8. d. yd. yd.
£ .41^ =84 li = I.2S-
12
20
16.^
I. 8. d*
io.8j = 10 16 8
As .410 : 1.25 : : io.8j
1.25
5416
21666
108333 7^'
•41^) 13 541^ (32i'5 = 3^7^' ^ anfwcn
_£i i3>54i
.375) 12.187s
937
li7S
It
• « « t
250 GOLDEN RUL£i m, Bookll.
It is to be obferved, that there mav be fuperfluous tenns
in a qiKftio% which pnuft be omittea i as the 1 2 months in
the niext queftion.
7.. If 100 L in 12 months gain 4 1. fo«. what will 74].
10 s. gain in the fame time, at the famr rate of interdl i
L h s. 1. 5.
noo : 4 10 : : 74 10
20 20 20
2C0C 90 1490
90
«ooo) 134100 (67.0I = 3 !• 78. -4 <!• f> «^wcr.
100
li
1200
.4
4800
800
Sometimes this analc^ pr proportion will not bear, until
fome operation in addition, fubtradtion, multiplication, or
diviiion, are performed ; or, perhaps, an operation in one
Off more of thoib xules may be requiisdaft^ the proportion,
IDl itrdrc tp fiod^out tbti numbei faughlb a3 in fosM of the
following examiples.
8. If 19 3raids of yard- wide ftufFexa^y line 14 yards oTfilk
of another breadth ; how many yards of the latter will line
i94 picf es of the former, each piece holding 284 yaurds ?
184*
147a
368
_9?
19 : 5244 : : 14
l±
19) 73416 (3864 yards, thc^ anfwer.
164
121
o ' • 9* **
Chap. I. RULE of THREE.
«5'
2. If 244- U>* of raifins coft gs/ 2^d. «4iatwill i8 frails
, each Mreighing 3qr$. 19^ lb. ?
1 4*
14x7
4
Cwt
24.5 = 6. 125
•87s
.21875=: 244 lb.
4
12
20
I
2.25
9.187s
£
•4S937S
I 4
28
4
C.
19.25
• 3-6875
.921875
1.
.ai875 : -459375
.9*1875
18
737500a
921875
»6.5937*
S739S4.
6637soa
8296S&
'49344.
497»
1161
83
.21875) 7.6*2754 (38.846875
joooa5 I 20
185254
10254 »6.9375oo
1504 f "
T6
■•*
11.2^50000
4
1. 000000
Anfwer, 34L 168. ii^d.
10* The globe of the earth, under the litie^ is 3^0 degrees
in circumference; each degree 6gi miles ; and this hody being
turned on its own axis, in th« fydefeal day, or 13 hours,
<:6 minutes ; at what rate an hour are dbe ii^iabitants of
Bencoolen, fituate in the iiiidft of the burning zone, car-
ried from weft to eaft, by this rotation ?
360
25* GOLDEN RULE; or. Book 11
360
h. / 3240
23 56 216
60 180
X436 : 25020 : : 60
60
Miles. F. P.
1436) 1501200 ( 104s 3 9 fff, the anfwcr,
6520
7760
580
X8
*
464a
•33*
X46 ,
13280
354
1 1 . A fiaftor bought 72 pieces of Holland, which coft
537 1. 12 s. at5s» 4d. per ell Flemifh ; I demand how
many yards there were in all, and how many ells Engliih
in each piece ?
537 1. i2 8. = 537.61,..5s,4d. =.2^1. and|yd. =: .ysjd.
1. yd. 1.
•2^ • -75 •• 537-6
75
26880
3763^
.2^) 403.200
2 40320
70
15 1 2 yards in all*
21 yards in a piece.
.24} 362.88
122 1.25) 21.00(16 ellsEngliib, and
28 850 I yd. in a piece.
48
12. A
ChajJ*!. RULE ot tHRfeE* isi
12. A hStoT bought a certain quantity pf tabby and
brocade, which togemer coft him 126 1. 14 s. 10 d.: the
quantity of tabby he bought was 48 yards, at 4 s. 4d. per
yard, and for every two yards of tabby he had fivQ of
brocade; how many yards of brocade had he, and what
did it coft him a yai^ i
^148
I
9 12
- 16
1. s. d.
126 14 10
10 8 -
JO 8 price of the tabby, 116 16 10, price of brocade.
Then as 2 : 5 : : 48
S
2) 240
120 yards of brocade*
I20)ii61. 6s. iod.(i9s. 4T»vfV- theanfwcn
X 20
2326
46
X 12
82
5«4
328
88
13. If I fell 24 yards of Holland for lol. ross; how
many ells Flemifc ihall I fell for 283 1. 17 s. 6d. at that
rate ?
3)24 .
8
32 ells Flemiflx = 24 yard*.
As
ft54 GOLDEN RULEj OR, Book
L s. ell F. 1. s. d«
As 10 10 : 32 : : 283 17 6
20 20
' _ _ ^^^^^^^^ *
210 5677
12 12
2520 68130
13626
20439
2520) 2180160(8654 ells Flemifli, the znfwcr.
1641
1296
t» ■
360
14. There are two numbers, 75 is the lefs, to which the
greater is in proportion as 8 to 5 ; what is their fum, and
Sie produd of their fum and difference, the dtiferenc^ and
pl-odu^t of their fquares, and the fum of the fquares of their
two quotes, the greater divided by the lefs, and again the
lefs by the greater ?
As 5 : 8 : : 75 : 120, the greater number.
75 + 120 = 195, their fum.
1 20 — 75 = 45> difference.
195 X 45 ^= 8755, produ£l of their fum and differ.
X4400— 5625 = 8775, diff. of the fq. of their fUm and dif.
X4400 + 5625 = 200259 fum of thofe fquares.'
75} 120 (].6, quote of the greater divided by the lefs.
1.6 X 1*6 =: 2.56, its fquare.
120} 75.00 (•625, quote of the lefs divided by the greater.
.625 X •625 zr •390625, its fquare.
2.56 X -390625 = 2.950625. Q. E. F.
15. There are two numbers more, the greater is 224,
tearing proportion to the other, as 8 to 7 ; what is the
fquare of their fum, difference, and either quote ; what
is the refult of the fquare of the fum of their difference,
added to the produft of their fum and difference i
8:2
^
C3uip.i. tLULE or THR£& i^^
< • 7 = • ^^ • ^96» ^cffcr number.
aa4 H" J96 s 420 X 420 =: 1764069 f<)U2Fe ^ their fute,-
004 -^ 196 s a8 X 28 s 7849 £|uaret>f their difiereace^
,*^ X tIt ^^ T7» ^4* of the quoteof the left dind^bythe gr«
^|j^ X 28+ 28 = 1 17B8, prodttft of their fmn, and diSe^
rente added to tiheir di€erenee.
Laftly, 11788 X 1 1788 = 238956944. Ct E. F*
i6. In a feries of proportional ntmben, the Srft is 5^ th«
third 8, and the prodttd of the fecond and third 78.4;
vrhat is the dilference of the fecond and fourth f
8)78.4(9.8, fecond.
Then 5 : 9.8 : : 8 : 15.68, fourth.
•,• 15.68 — 9.S = 5*88, the anfwcr.
17, A May-pole 30 feet 11 inches lon^, at a certain
time of day, i^ll caft a ihadow 98 feet 6 inches long ; I
would hereby find the breadth of a river, ^ that running
within 20|^ nset of the foot of a fleeple 3^ feet 8 inches
high, will at the fame time throw the extremity of its
ihadow 30 feet 9 inches beyond the flream ?
F« In. Feet. Shad. Shadow.
50 II = 50.91^ : 9*5 • ' 3P4 * 581.6515.
20.5 + 30.75 = 51.25.
Therefore 58 1. 65 1 5 — 51.25 = 530.4015 = 530 f. 4.818 in. .^r^^.
the anlwer required.
iS. Sttppofe the fea ailowatice for the commoa mtt to
be 5 pounds of Jbeef, and 3 pounds of bifcuit a day, for a
mefs of four peojde ; and that the price of the firii barrelled
be to die king 2>^d. per pound, and of the fecond i^A ;.
fuch was the fliip's company, that their jRefh coft the go-
vernment 12 1. 126. per day; pray what did they pay for
their bread a week' ?
lb. d.
5 beef value 11^: = .046875 1 perday, C • 3^8 "S l^cr week
3 bifcuit 4j = .01875 J or 1 .13125 , J P^^ ^^^^•
12 1. 12 s. =r 12.61. per day, or 88.2 1. per week.
lb. worth lb. worth
beef, bifcuit. beef, bifcuit. «
.328125 : .13125 : : 88.2 ; 35.28 = 35 1. 5 «. 7i d* anfwcr.
19. In
iS6 GOLDEN RULEi OR, Bboklf.
19. In the year 1581, pope Grcrgory reformed the Julian,
kalendar ; ordaining, that as the year is found to conhft only
of 365 days, 5 hours, and about 49 minutes^ in order to
prevent the inconveniencies of carrying the account of time
too forward, by taking die folar year at 365 .days and 6
hours full, which in a feries ^ f years muft bring Lady-day
to Michaelmas, that the chriftian ftates for the future mould
drop 3 days in account every 400 years j that is to fay, for
each of tne iSrft three centuries in that fpace of time, the
intercalary day in February fhould be omitted ; but retained
as formerly in the laft or fourth century, beginning with
the year loop, when 10 whole days were funk at once: by
which artifice the variation of time will not, at leaft fo^ a
long fpace, be ' very confiderable. According to this f^u-
lation, it is reauired to know in what year of Chrift the
• new ftile, as it is called, will be 20 days, as now it is only
II, before the old ftile, infhich makes no fuch allowance i
20 — 11 = 9 days to be funk.
D. Y. D.
As 3 : 400 : : 9 days : 1200 years to com^.
•• • 120a -f- 1700 = 2900, the year required.
20. If the fcavengcr's rate, at i^d. in the pound (^omes
to 6 s, ji d. where they ordinarily aflefs ^ of the rent ;
what will the king's tax ^r that houfe be, at 4 s. the pouxld,
rated at the full rent i
6s. 7Yd, = t2X1. .. i^d. = TTjl- • •• 48. = t'-
^ Then ♦^ 53 / 53
As -r^ : -tW • : t - V =^ '3^- 5^* theanfwer required.
21. Agreed for the carriage of 2^ tons of goods j three
fiiiles wanting tV> for | of -f of a guinea ; what is that per
'hundred for a mile ?
i of I of 1^ = ^Viy I and 1^% miles st *|.
Then as J tons : -^^ : : ^ : -^4^ I.
And ^^) ,~,V^r (tt4^ 1. = f|4 of a farthing, or
little more than i a farthing, the anfwer required^
22. A father dying left his fon a fortune, -^ of which
he ran through in Ax months ; y of the remainder held
him a twelvemonth longer, at which time he had bare
348 1.. left; pray what did l|is fiither bequeath him?
I f
Ch^fcL RULE OF THREJE; ^sf
-i- — 4^= -I remaining at the end of fixmontiha.
16 16 16 *
] i6 24 ' .
Pot 4 of jl =3 '-| = 348 - - by the qu«ftion.
•J : X::.^: 140 18 5^
1284 18 5^:, the anfwcr reqaircd.
Ma
23. A perfon dying, left his wife with chM, and making
bis will, ordered, that if (he vent with a foo, y of the eftatc
ibould belong to him, and* the remainder to his mother;
and if (he went with a daughter, he apf^inted the mother
7, and the girl 4- } but it happened that ]||ie was detivert^
both of a ion and daughter, by which ihe loSt in equity
MOol. more than if it bad been only ^ ffid; li^t would
her dowry have been, had fhe only had a Ion ? ^
As the fon was to have twice as much 'ii the'motheri
imd the mother twice as much as the daughter, let the
eftate be divided as follows, viz. 4 + 2 -f* ^ ^ 7> . ^^
whole eftate, fo that as (he had both a fon and a <uiugh«*
ter, the mother mufi have but \ of the whole eftj^ %
whereas, had it htpn only a daughter, i&e would have
had ^ . .
— = — , and -a:—. ••• -2 — — ;-5— — 2000 1.
7 ai 3 21 ai 21 .21
". 8 aooD I 14000 ^ , . - *
— : :: - : *-^^s— ^s J750'» the anfwer.
21 t 3 8 ^^
24. A younger brother received 2200 1. which was jutt
j\~ of his elder brother's fortune ; and 3^ times the elder'^
money was half as much again as the father was worth ^
what was chat i
^ j 2^ /26400 _ ^^^ ^ _ ^jj^^ brother's forturie,
5280 X 3y = 16500.
li- =: I : i-S£2 : : ^ : 11222 — iioool. father's fortune.
I 5
. 25. A perfon making his will, gave to one child -^ of
his eftate, to another 44 S 2nd when thele legacies came t^
^ S ^ be
be |Miid# me .tinned out 540K' TDt;>ino0e3faafv tib^^oAtt^
I. ... # . - » . » , • • ^
li = ili. andi?=2+i. Then 2il - i21 „ itil
39 1170 30 1170 .. .M7Q r*'!"^^. * **7^
ss 1538 1. 12 s. xi4d. + -2-q, tlie suiiwer. " . - ^
26. If I of I of 1^ of a (hfp bt vrotfh i'br ^ o^ ff ^f
'At cargo, valued at 12060!. i yrhal did both fliip i^cifr^',
ftand the owner la r " .^ ,. .......
r Pf f.of ^ as iT-t wid - of - q£ r: ~ Hl^ - -^ -
7 5 8 10 9 7*3 */j • "
♦-,, I ti 12COO 880000 , J. ^ , 45
Then ^ : -^ : : »■ ■ ■ : .i i ar 3223I. 8s. loii 22,
10 273 I 273 ^ '^ ^91
« • * 3223 U &s. fo^d* ^ + 12000 L =s 15223!. 8't, ipj^d*
27> In (bme partfhes m the coantrjr t{^ t^e^ 3L 1*
year in 17 Frovi the 'rents, in aflbffing Che fiumerft} ^ae
iMldle'landlbM receive net out of a farm^f T40L a^year
in thofe placesj^ when the king*9 tax is as now 4«i in 4lli
pMJidf? * . ' •
45* = •2L As 17 : 3 :: 140 : 24.706 L, abatement.
Then 140I. — . 24.706 !•,= iii^2M.U . ^ .
Alfe 115*2941. X .2 ss 23.05881^. tax.
••• * 140 1. — 23.0588 1. =3 116.9412 K =: il61. 18$. iqi^umf^
^8. If I leave Exeter at ten o'clock on Tuefday morning
ftUFXQhdoOy aiid ride at the rate of two miles an hoar withf-
ckit intermiffion; you fet off froin London forlExeter at fix
the lame evening^ and ride three miles an hoi^r conftantly ;
the queftibn is, whereabout on the road ypuandl ihall
meet, if the diftance of the tw^ cities be i3om^es?
S X 2 =2 16 miles, I had travelled before you (et out«
rjo — 16 si 114 i and 2 4- 3 s^^milesi bothgofli t hftar.
Then 5 : i :: 114 : 22-J^ hours, they wifl ipeet.^ ' *'*
224 X 3 == 68j ^ 1 t diltant from Lbmton,
28f X 2 + i6 = 6i| J °"^^^ I diftant fitrra Exeter.
And 28f X 2
•» '
1^. A fets out from London to Lincoln, at the very fam«
time tfiat B fets forward for London from Lincoln, diftant
•* .• 1 '- " . • 100
)0O
IPO flvlest at dsbt hours cndthnr meet ontfaeioait, 'tf|id!
it ttaii aff^aMd nt A'hid fti « Jr;kite78«riiMrM<tte ctaov
Ml Itt tt^EociaCg an iMT dU tacb bf tfaoh travd ?
•r .. ' • .., "^' ... '
' ' ' . hours, miles, hour, milfes*
_ . ^"J. S •: iocr ':; t : in^y both travelled,
Tlicn 124. — 2^ = 10* and 2) 10 (5 rfiile^, B toit.
And $. 4« Ir ^ ?T f^^^» A rOde ^n bour«
30. A T«favoir for water h^s two cocks to fap^jly at>
fey-ijfep J5c^ H, »Y \t.^ filled alone in 44. mioutes^ by the,
ftdlmlirjnriSi llouri dhd It hat& a dffchafgihg cock»
by ^hich it lAaV) when full, be emitted in half an hour:'
Aow fu^fe itoeft thf£6 edctts, bv sittlAettty fhouLd all of
them he left open, and the water mould chance to cbtne 1a ;
^«i4^;ti|fte^^fi|ppifiiig th^ lAfltfx aiid iAtnc of the water ta
be always alike, would this dftern be ih filing I
In one minute is filled hy ^^'^^^fX^f tla^ cifteni.
Alio 4h i" w =0 y\\ of tte dffertt ffllef lA tf minutcj
both rbttitn^.
; la one muittte 'run^ out ^ ;:= -^ of the whole ciftern.
AM vSr8»1^»^ aitirik B T8T ^^ ^ ^^^^ ininttte,, alt
* • * rit M • ^ ^ • 165 minuses :=: 2^ hours, the aafuFff
required*
JX* ,A can io a prece of work in lo days^ B alone in i^ ;
fill UteMi bVAabbul ii cbiilllldr, 'ki what time will it be h-
nahed ? -
• • « • »
^^tit^ V^\ i^ {f^ of *e ivorik iA t diy. ' •
T™*T3V • « «y '• r Wdtk': »^* ii Si-J dajfs, th<J^rt|(er.
^a. B and C together ran build a boplf in rS dajft ; with
tksaift^Aceof A, they can do it in 11 da/si in what
time wiU A do it by hnnfelf ?
. MX t can pecform ^ =5; ^ ^ of % woi^K. i'ti 6tit
A + B f C.canpelfetm ^ tt^VV V ^ i«y^ , ^ ^ ,
. in^A rA* '^ -tV# ?= rfri Acan perform pf the Whole
worii All ^ive day , .
• . • -rl^ : a iay : : I work : * V su aSf <»yi» lh» anftrcr.
«. f"»~ ••,.'•♦
6 « J3- I?
. 1
26o golden: RUL£ro]|, B«o^<afj»
i.^1. If A aloi^ can do a, piece of.vocic.io i^daysy^^A.
and 3 together Jn feven doys-f in^wimt tuxiuRjpu;.B 4i>:iti
ajone? . ,
-A + B «an do^=?f§; A ildne Vir =^'of tRir
work in one day. ■ . "^ ' \ ' * • ^
Theh 41 — 7^ ?= J^, 3'5 day*3 work. . - /
• . • ^ : I day ; : X work : y == 2i3t ^*y4 tktznfyfct
required, . . , ,
34, X, Y and Z can, working "togethcTA^.^poqipIeat a
ftaircafe in I2 days ; Z is man enou^ to do it alone in %^
days, and X in 34 j in what time could Y get it done t)y
,« ^- y
* X Vt = Tro^» Z ^ =^, and X ^-^Y+^ZiJo-
^•. t=: -^{^g-, all working one day.
Then ^oV + tVt = r^-^y ^^'^^ '^^ <>"« ^^7 by X attd Z
working together.
A«d :^ — ^ ss ^-o done in one day bv Y aIoi;ie»
* '•• tJt • I ^*y» •• X work, : *!* a «| days, the
anfwer. '
35. Three workmen can do a piece of Work in certaia
times, viz. A can do it in three weeks, B c«| do thrice
the work in eight weeks, and C five times in' 12 weeks ;' in
what time can they finifh it jointly ?
Ncufton^l Umverfal Arithmetic.
* A c^ndo T = fJl their fum'^»=f work, allvork.
p. * " ? H U I tag together one week.
I week ss 6 working days, and x day =12 working hour^.
• . • \ work : 6 days : : i work : ^* days, tr 5 diys
44iours, the anfwer.
36. If a cardinal can pray a foul out: of p<urgatory, by
'himfelf, in an hour, a bifhop in three, a pri^ in five, and
a frier in feven ; in what time can they pray out three fouls, *
all praying to{^tber ? . . fidkuthm^
Whilfc thfe cardinal prays I =s 4^^
■■ Thebifliop- - - -•^=svSl*e5rfam4:i|,lflaij:
The pricft - - - - 4 =s ^Vt I •><»"' together.
.. And the frier, - -:- I'tsVoV"' • , *
fouh.hour. fouls, hours, hour. / // ' '
''' \'ly • I •••■ Z- 4fl' =36 » 47 23tV» ti»e anfwer.
37« I
^ ■
Edimiit^H^ diftaht by'tbmputation fif 350 iritles, aiid my;
rout is fettled at 22 miles a day ; you four days after are
lent after me with freib orders^ and are to travel 32 miles a
dsy ; whereabout: on the road ihail 1 be overtaken by y^u i
22 X 4 =2 ^8 miles you have travelled before! fet out. '
' 32 *» 2i =s 10 mHes f gain each day of you.
10 : I : : 88 : 8.8 days.
Then 8.8 X 32 = 281.6.
. " •^.* ijfo — 281.6 = 68.4 = 68 miles, 3 furlongs^ ^8.
poles. 00' this fide Edinburgh.
•
38. If the fun moves every day one degree, an4 the
n^cion: ttttrteen i and at a certam time the fun be at the be-
ginning of Cancer, and in three days after the moon in the
bf^nmn^ of Apes ; the place of their next followixi^ con-
jnndibn ss required i Newtoffs Univerfal jfriwmitu,
26r X 3 =i 90 degrees, from the firft of Aries to Cancer,
" ^-4* 31S 93 degrees, the fun before the moon.
13 — 1 =5 12 degrees, the moon gains in one day.
12 : X : : 03 : 74: days, in which time the fun will be
uvciiattn.
" ^ • *' 71 + 3 — ^o| degrees of Cancer, the anfwer,
39. If the half of fifteen be feyeh,
Wliat is- the fourth of eleven ?
As -i :. 7 J,: T. : T.s^ ^ih ^^^ anfwcr required.
2 ' 4 5*^' ■
I m "
- ' ■ ' . . J. ; : . »*
. In, niechanjc»9 a lever .of Jthe fecond order is, :$yhen the
power z&s at one end, the prop fixed dire^y at .^C.qtb^^ .
and the weight fomewhere between them.
• In fhi^' oYder'br Icvehrj thfelr force are in a cohtj'a-pro-
pdniotn to theii"leAgths; - » .
- ^40i^4fi lever be |oo inches long, what wstgb^t Jynig
7y inches from the end, refting on a pavement,, may be
m^<^ with jth^ force qf 168 lb, lifting at the p^hejr.c^d 44
thiereyer? ;. . - '
"100 — y.f :p 92,5,. longed end. . , , .
inches, lb. inche;!i. lb.
S- 3 ■ • Ih
• *'-
2^2 GOLUBN ROLEjiiR, Ikx^H
» ." •-. - ■ ■
In a lewf of the thW order, tke prop » planted at «ne
end of the T»ar, the weight at th« oth« eUd, and tM fMring
.|t)rcf foni««*ere beweep.
«
il» A w*«-whe«l tuFM a crank, working lik«« |>\OT|>-
fods, fijied tuft fuc feet from th.«? joint of pin i. by wh>di
their fevcrat levers, each nin? fec^ in length, a»MfteMd,
for the fake of the intended nwtion, at one ei^ i the ftxaera
, of the pumps being worked by the other, ^Pifs, them to be
kveca of the thud Qider : now I would know Wjiat the
leneth of the ftroke in each of the barrel* wllV* »f »«
crank be made to play juft nine mches round its cent« f
Q w - _ j'8 inches, the diameter of dw ctaak.
feet. mch. feet. inch. . ^ ^. - .
6 : i8 :: 9 t 27, the length 0f th« ftroke,
41. With what fopce ought that watcr-t^riio^ ijo W Mirn^
which, circumftanced as in the laft qucftion, raif^thrce
cubic feet pf water at every revolution of the wbecU eath
* experimentally wei^in^ 6^^ lb. avcrdupoife ^ the fti^oa
of the machine rejc^ed f . , ,
62{- lb. X 3 = iSyilb. = weight of 3 folid feet o£ vatei^
. jcQing the friaion, Q^ £• F.
The magnitude of fphcres arc dircAly ia {rcfWtJon to
the c^bes of their diameters.
43. If the diameter of thef earth fe 7975maea, ofAe
moon 2x70 tniles, fuppofing them to be cxaa fphares, as
l^ojr aM natt what comparilbii is t^€r« between them ^
point of magnitude ?
Cube of the earth V diameter 55 50616 fS73pW»
'Cube of that of the moon = 102^8313000-^ .^^^
. • . • 10218313 : 506261573 : : \ : 49S445* Qi *•' *^*
The lefs |)brous a body is, the greater ;s its ienlity. '
44-^ The compaftaefs or denfity of the mfOeo it tp tbet^
the earth, as 132! is to ?oo ; what proportion theft is ihofe
between the quanti^ 0/ matter hi th0. earth/ aj|d that ui.tho
mpow ^
The earth in the foregoiog quemon is feWf* -^o ^
4Q'^44S times bigger than the moon^
. %•. 123,5 • ^^ • -495445 : 40.M7. Q^ E..F.
'i /ihs^ hi ftit eartfi contains 40.x 17 times moft^Bh^ttef.
The velocity of found is found by experiment to %e
ppiform; viz* about 1150 feet in one .feoond of tinie» if
A,JPS^Xi with nothing to retard or obftrud its motion.
m
45* If I fee the t^fli of a ptece of ordnance fired by a
.vefl3 in diftrofs at fea, which happens, we will fuppofe^
,Uf2x\y at die inllant of its going off, a;id hear the report a
. miiivte juid three feconds afterwards ; how far Is ihe ofT,
reckphiiilg for the pafiageof found as before?
I minute 3/econds.= 63 feconds.
As 1 fecond : 1150 feet : : 63 feconds ': 72450 feet =213
miles, 5 Aff^n^S) 30 poles, 5 yards, the anfwcr required.
■^ ^
46* How long after firing the warning-gui) in Hyde-Park,
^«My the fame 1^ heard at Highgate, taking the diftahce at
.~5j»milc^f ' . .
.- r - jj miles =r 29920 feet-
TTien 11 50 feet : i fecond : ; 29920 fect ; 26 fecpnjcU^ v^^
thirdj^ the anfwer required.
i ...
' 47. Stippoft' a thaid carrying apples to market ^w^js met
by three boys, and that the firft took half thattbebady but
returned 10 ; tha£ the fecond took one-third that fbe then
had, but returned two; laftlv,. the third topk fovay.half
that ihc had left, but returned her one ; and when fhe had
* got clear,, fta l^ad 12 apples left 5 what number . of .apples
faadfiM atffirft^f . Einerfon*i AntBmtic^
Flrft 12 — I is TX ; and it X 2 = 2*j before Ihfc'Mct
thelaftboy.
Alfo wticr^ ^tx^ >- and 4- ^ f'^ - v t * 3^) ^^^ ntsmber
flie had before A^ naet with me fecond boy 1 and before the
fteft. bo]r:-ret)i|iied her xo, ihe had but 20> equal to what
the bo)- tpok«
• . • 20*X ^ =? 40 apples, it the firft. Q^ E. F,
Froof AO -?• 2 = 20 ; alfo 20 + JO = 30, when (he met
^;,tl«fecdn^
' .* 'Likewifc 10 -?^ ■? i;?: id j and 30 + 2 -* ro = 22, whpn
^'*ixii^bvthcla& ^
, I^wly,^ 22 -J- ^t^: it; and rr + i :;= 12 left, -per
'*'-qi«fitin.
4^4 (S^caiOEW BUL£j oh, Hooklft
aS. a tradcfipan begins the world with loool. and finds
that he jcan gai$ lOOoI. in s years by land tradealonc} and
that he can gai* looo 1. in 8 years by fea trade alone j and
likcwifc that hc{ fpehds lOOol. in 2 J^ years by gatoing j how
Jong will his dlatc lafl^ if he foUows all three ? ^
Efrierfoift JrUbmeiicm
io©o
= 2()o 1. gain by land trade in one year.
lOCO
— 125I. gain by fea trade in one year*
315 1. his whole gain.
i2!?£ = 400 1. loft by gaming in one year.
DiiSerence 75 1. lofs at the year*s end.
••• 75I. : I year :: 1000 1. : 134- years, the anfwer.
«
49. A clock hath two hands or pointers ; the firft^ A^
goes round once in 12 hours ; the fecond, B, onceinatt
hour ; now, if they both fet forward together, in what
time will they meet again ? Emerfen's Arithmetic^
As A goes only Vr ^f ^be circumference in an hour^
And B goes the whole, or \\ ;
Then W — yV =^ tI> B gains in an hour.
C h. C h. h. / //
The velocity acquired by heavy bodies falh'ng near the
furface of the earth, is 16^ feet in the firft fecond i, and as
l6j^ feet are to the fquare of one fecond, or 1 ; fo is the
given diftance, to the fquare of the feconds reqiured.
Or by multiplying 164-, the defcent of a heavy body in
one fecond of time, by as many of the odd numbers, be<>
ginning from unity, as there are feconds in the ^iven time ^
viz, by I for the nrft, 3 for the fecond, 5 for tne third, 7
for the fourth, &c. the fum total will give the fpace it ham
paflTcd.
50. Suppofe a ftonc let go into an abyfs, Should be ftbpi-
ped at the end of the eleventh fecond after its delivery,
what fpace would it have gone through ? •
J* : j6.o8j :: ix xii =;;; i2j ; 1946.08^. Q;E.F.
Or,
^fi^M .i^t^tE 0f fTHRiBtE.^
%'S§
rbni .'.It -r-
5fir ,-. oil
■«■ »I , r. " ■
3 =
5 =
7 =
9 =
«1 S3
15 =
17 =
19 =
.48*250
80.41 19
112.58^
144.750
176.91^ >'inthe-<
219.06J
241.250
273'4i^
305-58^
337-75 J
r ift **
2d
4th
5th
6th
7th
8th
9th
loth
^iith
A ,-1^
' .'* :
fecondy'cftitne.
1946.083:, as before,
•M
51. If a ftonc be I9{. fecondsin defccnding from the top
c^ a precipice to the bottom^ what is the height of the
funer
I* : 16.083 :: 19.5 X 19-5 = 380.25 : 611 5.6875.
• . • 6) 61 15.6875 (= 1019 fathom, i foot, 8^ inches. Q.E.F.'
52. Tf a hole could be bored through the center of the
earth, in what time, after the delivery of a heavy bqdy on
its furface, would it arrive at its center ?
The femidiameter of the earth 3980 miles = 21014400 feet*
]6.o8j : 1* : : 21014400 : 1306594.82.
feconds* mill. // ///
: %/ 1306594-82= 1135-554= 18 55 33. Q. E. F.
■ " x. If the quantities of matter in any two or more bodies^
put in motion, be equal, the forces wherewith they are
moved will be. in proportion to their velocities.
' 2« If the velocities of thefc bodies be equal, their forces
."wffi be dtredlly as the quantities of matter contained in.
them. . • *
' ' jf. If bt)th the quantities of matter and the velocities ly
unequal, the forces wfth which bodies are moved, will be in
a proportion compounded of the quantities of matter they
tontiUh, and of the velocities wherewith they move*
53. There are two bodies, the one contains 25 times the
matter of the other (or is 25 times heavier) ^>^t ,the Icffer
movesVith rooo tfmes the fwiftnefs of the greater ; in wha^^
proportion ^c the forces by which they are moved ?
As
* As i'if ! idoo : : c : 40, the Idfs is moir^ witji a fbti^
fo much greater that AcDrficr.' - }--.;. . :^il)
54. There are t^o bodies, one of which weighs 100 Ih.
the other 60; but the kfler bo^y is impelled bjr*a''fdi*ce 8
times greater than the bthef ; f ne proportiisii' o^ the veIo«*
<»tww. livfiftrewith thcfe bodies mpve«r u nequir^ i ^
AsfSa: roo : i t : ifssi*': ••
So that the velocity of the lefs to the greater :mU im
•3- X T — T — *3i ' ^' t I • • *
So the velocity of die lefs to the greater will be^ as 13^
to I, or as 40 t^' 3.
55. There are two bodies, the greatei* contains fftiitves
the quantity of the matter in the lefs, and is moved with a
foir^e 48 times greater ; the ratjio of the velocit]^ of thefe
tW6 wdtes is required ? '' - * ^ y^
If the forces were equal, the velocity of the lefler would
be 8 times that of the greater.
But a» the force the greater is moved with is 48 thnea
that ,which moves the lets,
jAs 8 : 48 : : I : 6 ; fo the velocity of the lefs to' that
of the greater is as i to 6.
. I. In comparing the motions of bodies, if their velocities
beequai, the fpaces defcribed by them fliali be in the dlreft
proportion of the times in which they are defcribed.
, 2. If the times be equal, then the fpaces defcribed 1»|II
• be as their velocities. * >
3, If the times and the velocities be unequial^ the :.fpaces
„ . be in a proportion compounded of the times and vdo«
cities^,
56. Theiie are two bodies, one of which moyea 40 cinea
fwifter than the other 4 but the fwifter body- has moved but
one 'mintite, whereas the odier faaa been, in motion two
hours ; the ratio of the fpacea defcrihed by thefe two bodiea
is required ?
In two hours are 120 muiuta* -
As 40 t 120 : : I : 3, fo is the fpace tbfi fmfterhath
moved to that of the flower. . >
57. Suppofe one body to move 30 tiaies fwifter thfm an-.
others as alfo the fwifter to move (2 mi|xutes> the other
^ - only
Ck^ t It U t Q' <ft T H IfiK'E. ftS^
<l^ I S. wbat dilEmnce wiU there be between ^e fpocet by
thm inforibedy {uppofing thtf i&ft JU3 moved 66 inches i
. ^ iiM^M 0 5 feet, mcnred hf the fecmd*
And I : « :: 3pxi2 = 360 : ifoo, feythci^fl^'
•. ' 1000 — 5 =: 1795 feet, the ahfmr. . '
t& Theft ate two bodies, one iyfanpf haa defaibcdjo
wUis^ thft other mIv ^5 ^ but the Sift hath oMred Mlh five
Ifapet^tha vetpcii^ oi the ftc^nd ; what is die ntio thai of
the times they have been defcribing thofe fpacea I
— ^ . - : : I ; 21 fo that the firft bpdj hath b^en in
jDotioo double the time of the feeciad*
^.¥^:¥^¥::¥i:¥-.¥I^i»<¥^i*^-:¥^r^.
, S E C T. n.
RECIPROCAL PROPORTION,
CALLED, THE
R£ C I P R O C A L prop^nioii is, whan of fear numbdv
, the third (viz. that which moves the queftion) beareth
the &me ratio or proportion to th^ firft,^ as (be fecop4 does
lOthefcurth.
Therefore the lefs the third tenn.is, in refped to the firft,
the greater will the fourth be in re(peft to the fecond.
And the greater the third term is, in rtCfeSt to the firft,
' the left will the fourth term be in refpe& to the fiecond.
' Therefore, obierve that in anjz: queftion in proportion,
when MoRB requires Moaa, n L£8s requires less, the
terms are in direct proportion^
Bt« if jtftw rnqimea &€ss^ oc i,ES% re^ww 4«oile,
tbtn ther terms will be in reciprocal proportiof^ . '} :
Tlie fame direfiions for ftating the queftion are to be
9hkmi»i k^T^ as in tiseft ptnpartion.
*" Thie iq[nU9ift)a tnog tvuly ftatedi» oMbcve (jbis gi^nend
f0^. ^ . . '.
- . RUI-E.
■
W RULE OT TH^EE tiltokQy'
RULE. ' /
Muldply the firft tnd'fecoiid terms tdgethttc^ miLfUride
their produ6l bv the third term, the quotient will bft the
anfwer requirea.
I. If a penny white loaf ought to weigh* 6 ounces and
12 dnuns tfreidifpoife, when wheat k fold at 6s. 6d. per
iMdhel^ what muft it wei^i when. wfattR is fold at 4s..
the bttflieH
s« d. L oz« dr* • s* I«
6 6 = .325 - - 6 12 = 6.75, and 4 =: .2. '
Then as .32^ • 6*75 '- •^ i
•3^5
3375
. I3S0
2025
' ■ oz. oz. or.
.2) 2.19375 (10.9687^ ss 10 15I, the anfw.*
- -
Here it is plain, that the lefs the price of wheats '^the
bigger Uie loaf ought to be.
a. A, general is belieged in a town, in which are 1569
foldiefs, with provifion of viduals for three months 5 how
many muft depart the garifon, that the fame viftuals
may laft the remaining foldiers j^ mouths i
mon« fold. mon.
Reciprocally 3 : 1569 :: 7.5 •
7.5) 4707 (627, or 628, may ftay»
207
570
■4S-.
''CMftqumitly f 569 -^ 627 :£ 942, or at the leaft 941
nttift.depaft.
3. How many yards of dl-wide flannel is fuflicient t6
line a cloak, conuining i8|- yards of camblet, ^ yard
wid€ f I
Qm^ 3.ir*;Y*il-
«»
yd. jrdf. » , yd.
I = .75 i8f = 18.875, ^^ li » I,a5^
&icd(tfD«atfy» .75 : 18.875":: 1.25
- • .75
94375
132125
fis. yds. qr. tt«
X.25} 1415625 (11.325 SIX X 14, th»
165 anfwtrr
406
312
'••••* 625 *'"•'* / .
• •
4.; How many yards of mattii^, that is zi feet wide, will
cover a floor that is 17 feet long, and 15 feet 3 inches
.^rqad i
£ett. in. yds. feet. ydt. f^. yd.
'15 3 =s 5.08^ - - 17 = 5.^ - - 2^ =1 ,8^
yds.
Reciprocally, 5.0^5 : 5.^ : : .8^ .
'9)jt0500 ^ , .
3488*
2541^6 . ^ . . .
■ yds. yds. ft. in.
.8j) 28.90/5 (34.56 == 34 I 84, the anfwer.
8 28905
.75) 26.0150
5. A borrowed of his friend B 250 1. for 7 months, pro«
ii^$ng tOr do him the like favour ; foine time aliH^r B halh
an bccaiion for 300 1. ; how long may he keep it t^ be mnii
full amends for the favour i
-. <-•
Reciprocally^
A9^> RULfi o» tHltE£ BtolirdKr
■ It moil* Ik
" '_7 - :: ■ .
300] 1750 (5 months nA 15 days, tbetnf* required.
250
X30, dm in-antoiidi.
■■1MB
75 . . •
6* A regiment of foldiers; confifting of 976 men, are to
be new clothed, each coatta rontain 2*- yards of clotli,
that is. If yards wide, and lined with fh^oon | yard wide ^
how many yards of Ihallooft^wH] line them ?
. ' — ^76 * ' ti is 1*625, «rt* i » *t75 *
*7r •
■ >
48S
I i
«*•
'KeciprocsJly, 2440 : 1.625 • • '^75 '
'•625 . _
1220
1464
■ yw,. qr. nls. .
•875) 3965-000 (45J^-42S5^ = 453» » . if. **
465^ anfwer..
2750
1250
3750
2500
7500
<ooo
12J
7« ]tf a tailor can make a coat and waiftcoat with thfe^:
Vards and three quarters of btoad-doth, of oa« yard ;uul a.
. halfs
I < • •
lulf 3^ breadtb i bovir jn^py yard^ of ftuflF^ of ^ yard's breadth,
win he requb-e fo'fit thf i^me perfon I ;
There being three orders of levers, or three varieties,
whereia die ipeMds,. Ip^ps*^ oc namHg poms. Hay be
difibicttdy aiipliea tv the . veAb, or inflexible bar, im ofdtr
to tftA myhanigat ^petsAon ki a cootquouc nuufi^er^ *
A lever of the firft order hath the power placed at one
of ,ica Aidls, and Ae wiij^ts to be rafei k put at the tcher,
mt '^ fdkvuOk br .p«op ibmewbere JbttWim then. -
lA-tbis order, tbe power ^t^^ ^ oaeiaid wiUbe ffe^i-
procaUy'pi>Dportioiiat to tbe-Aftancea of tboie endufrom tke
Culerem, or poiiit fupported; er in the fteetyards, a& ibe
diftance of the Wiei^ irem the p4iiit of fu^prttfioiu
t« What weight ^iK a taUm ^ aUe to raift, w&d
prcfles with the force of a hundred and « half en the end
of an equipoifed hand^ike ibo inches long, which i& to
jneet with a conTeniem prop exaftly 74 indies above the
oclies end of die machine r '
M<>^-^ 7.5 s£ 9a.^5, tke kngeft end ^ tike levier 'fiom
the Mereai.
inck Ui« inch. lb» . ewt.
]^e<ifimad]y, 92, j : 168 : : 7.5 : 7iOj2 s l^9 thfian^mr*
9. What weight, hung at 70 inches diftance from the fut«
<n^ of a fleelverd^ will equipoife a hogfhead of tohoirco
9^ cwt. freely fuipead^d at two inches diihmce on the can*
9^ cwt» » 1064 lb*
ii). lb. in. lb.
ReciprocaUy, 9 : 1064 !: 70 : 3O7) tbeaafWer.
The eiFeds or degrees of light, heat, and attca£tioa, are
reciprocally proportional to the (qiiares of their diftahces
from the center whence they are propagated.
10. Suppofe that in a room where twa men, A and B are
fitting there is a fire, from which A is three feet, and B fix
Gfet -diilaht; it Is required to find how' much hotter it is at
A's'&at, than at B's ?
Reciprocally^
-> ,/ i.
tfji RULE 6f T»REE. 'BUkiL
• Reciprocally, 6x6 = 36:i::jX3 — 0:4; (odm
A'« place is tour times as hdl U BX - ?.*
1 1. Suppoiing die earth to lie Siopooop mtlcf^^^iAMt
from the funj^ 1 would kno^ at w)iat cUBange irpiii Jaim
another body muft be placed, fo as to xece^rc vligh^ ^ ^d
Beat double to that of the earth ?
1
Sxoooooo X 81000000 =5 656x000000000000.
>BffoipL 1 1 65610000000000PO • • %i 31805000000000004
* • • ^ 3280500000000000 5&-5ya7)6494iMBdai A^anlWl
12. Mercury, the neareft of the planets t» tfao looocc
of heat, light and life^ in our fyftro)«-tbs liia it dbout
.^.millioti of oiilct ficooi him; and Sfttuai» thefvmoteft
0f ^.planets, is ufuaUy diflaitt alMmtT^^aiillioiis of
swlei.r tvhat oomparifon or pitnportson :ta nqre ^ ^
the folar influences on thefe .two bodies i
r i ' ./
■
32 X, 32 5= 1024, and 777 X 777 s= fiojyagr C|¥ares of
fuftaoc^, ^phers omitted. ^ .-; -
.. Wn.. Mercury, ^ i ., ii.'^,i:;>'
Recip. .603729 : 1 :: X024 : 589^^. , ••*' Thejio^
influence on Mercury to that bf Saturn, is as 589-1^,*^ to u
i 13. The diftanoe between the t$ith anil fim i&4iocMmtcd
Itioooooo of miles ; the diftance between Jupicer an^ tht
fun 424000000 of miles ; the degrees of 4itht and heat re-
tmtihf Jupiter, compared with d»t of m earthy, ii^ re-
quired I
»
' .8x X 81 s 6561, and 4^ x 424s ^7977^^ fiiuaitesof
tbdf diftancea, the cyphers being omtttedi* *^
. Recip. 179776 : I :: 6561. : 27.4; fothat.tfa« fan's in-*
fluence on the earth to that on the planet JiqMter* is 27.4 to i.
.•%40 A OirtatB body on the fiirfipe of.tbc.earA wdghs
i 12 lb. ; the qveftion is, wthitbci'tfais body muft be carried,
that it may weigh but 10 lb. I
lb. fij. r. lb.
Rectp. tt2 : 1 : : 10 : ix.2, fquare'femidiim^ter. '
' Theff ^ ti.iL = 3.34664, femidiairtetfcrof the earth from
its center 1 or 9351I miles from its furface. ^ -
* I5. A Gf oQtAjPHiCAL Paradox. /
^ There' b a vaft country in Ethiopia fuperior^ tOMi^hofe
^^i^hiUBififs cbembon doth always appear* t6 be mpft
«iil%)it«sneidi when ttit Is leaft crfiglitcncci ; ahd to bo
• lealK '^bei» fiioft( admitting the mean dtftance of the
earth and moon's^ centers 240000 miles, in what proportioa
is this illumination i
; . Smi j^m tb« earth Siqoqoqo ^ a400dp ;=::.8j::i4000<4
fm bom a full moon.
81000000 -— 240000 =: 80760000 miles, the fun froQi a
junv moon.
•i • Kia4 9^^124..:=: 65999376 rfquares of d^r diffet^nt M^
9iBffir^ 8076 rs 65221 776 ) tancesy the C)^ers omitted*
. Jteupt 652^x776 : X : : 659993^76 : .9882^ (o tt»t thp
fBvpbrtioa of hght and heat a new moon hach to that of a
full one is.
As 1 to .98829 or as 45832910452929, in whole numbers*
1 6. If a body weighs 16 ounees upon the furfitce of the
earth, what wiil its w^gfac be co miles above it, taking the
' earth's diameter at 7970 Engliu miles i
' 7970 •4*> 2 =: 3985 miles, the earth's femidiameter.
^ 39S5 '^31^ ^ i^iSoaas^ 10 fiiuare } at(b 39S5 4- 50 =^
4035-
And 4035 X 4^35 = 16281225.
Recip; 15880225 : t6 :: rdiSr 225 : 15 oti '9J]!^|4^.
It hath been found by experiment, that a pendulum ^3
hichetieng, in our latittide, vibrates bo times m' cite mi^
nute ; and that the length oip pendulums are.to one another,
as the fqsare: of the namber of their ribratiotis madt in
the £une ijpaoe of time*
171 What is the length of that pendulum which fittings
half' feconds, or vibrates 120 tiaaes in a mimitef
- *
Recip. 3600 : S9«2 : ; 14400 : 9^ inches. (^' £. F»
1 9* What difference will there be in the number <^ vi-
J^taiions made \>y a. pendulum of 6 inclies lon^ ^n4 another^
bi 12 inches lonff, in an hour's timef
f I
R^ecjprocalljr,
R«dpwcally, J9.* : I6oo.: i { 'J : ;*^^ : . ,
V 11760 = 10^.444 Yi3S20=;s3^.36z.
'Then 453.362 X 60 =5 0201:71 , . '
■-■ And 108.444x60=6506.64 *. • • '"^ ;
■ »
■ . ■ ■ 2695.08 Q. E. F.
In comparing^ the motions of bodies, the ratio or pifopor-
tion between their velocities will be compounded of th^
dire£l ratio of the forces wherewith they are moved, aad
the reciprocal of the quantities of matter they contain.
, Xg. The battering ram of Vefpafian Weighed, fuppofe
1 00000 pounds, and was moved, let us admit, wHh fuch a
velocity, by ftren^ of hands, as to pafs through 20 feet
in one fecond of time, and this was found fufficient to de-
moliih the walls of Jerufalems with what velocity muft a.
bullet that weighs but 30 lb. be moved, in order to do the
fame execution ?
' Recip. 1 00000 : 20 : : 30 : 66666^ feet, in one fecond.
20. A- body weighing 20 lb. is impelled by fuch a force,
as to fend it 100 net in a fecond ; with what velocity would
* body of 8 lb. weight move, if it wc^ injpelled by )tbe
fame force t
Reciprocally, 20 : 100 : :'8 : 250 feet. Q; E. I?.
S E C T. III.
COMPOUND PROPORTIONi
OA, TH£ . * ■
RULE pr five:
THE rule of five is fo called, from having five numbers
given to find a fixth ; three of which five given num-
bers, are only conditional, or fuppofed : and thcL other two
move the queftion.
.1 AU
All queftions in this rule include two in the rule ^f diree,
either both djred^or on^ ind^^e^ tntkdie olla^rmtecipro-
cal {M-oportionj /vrtiich fo depend upon each other> that the
anfwer of the firff heuig made the middle term of the fooond,
the fourth temi ef the fecond will be th& final anfwer of the
qiicftkm.
Yet here obfenre,- chat many queftions, though they majr
be wrought bv two (pr iQore) operations in th^ rule of
ibtvcj cannot be anfwered by the rule of five.
In ofdet'toMre my qtie^on in the rule of 6vty obferve
the following di|^^on0«
Always place thc^ thre^condftinnal t^rms ii this order, let
that nttftiber whitti is the principal c^ufe of gain, lofs, or
aAioD, &c. be put in the firft place 9 that nuiiber which
denotes ^e fpace of tjme, or dUlaoce of place, &c. be put
tn the (econd place ; and that number which is the gain>
Icfy^ or adibn, $cc. be pii^ ki the third place : that ttone^
place tlfc other twcr termsi which move the queftion, imdeif
tfao(e of-the fame name.
Then if the blank or tenh fought fall under the third
place,
RULE,
Multiplf the three laft terms together for a dividing, and
the two firft together for a divlfor ; the quotent arifmg from
them will be the fixth term.
But if the hlzstk or term fought ££U under the firft or fe«
cond place, '
R U L E IL
lildftffif't]leftrlf,'fecohd^ and laft terms tejgethcr foi? a •
dividend, and the other two together for a diviior ^ the
quotient arifing from them will be the fixth term.
I. If the cattiage of S cwt. 3 qrs. weight, 150 mHes, coft
3L 73. 4d. ; what mim be paid for the carriage of 7 cwt«
2 qrs. 25 lb. weight, 64. miles, at the fame sate i
cwt.qrs. lb. L s, d. 4. cwt. qn« lb« lb.
331= 644 ' 374 = 808, and 7 2 25 = 865.
lb« miles. d.
- '-^ ••- -64^1.' . 250 . «o8-
8% . 64 . **"
'•' toS X 865 X 64 rs 44730880, dividend.
T 2 - 644
644 X 150. = 96600) 4^880 (463.
I948 ao) 38 7
5080 j^ I 18 y^Abeann
' 1; If 2 men can do 12^ rodls of ditching in 6i d9f$i
how. many rods may be done by 18 men in 14. days f ^
men. days. rods.
s . 6.5 . 12.75
18 % 14. .
12-7S X x8 X 14 = yS^ dividend.
2 X 6.5 n 13) 3150 (242^ rods, the an^wec^
•
. 3. If a regiment of foIdier9> confifting of 939^ can eai(
up 351 quarters of wheat in 7 months ; now many ibidieni
willeat up 1464 quarters in«5 months, at that rate ?
foldiers. months. qrs« wheats
939 • 7 • 3S;
5 • 1464
939 X 7 X 1464 = 962287a» dividend. ,..
•3?! X 5 = '755) 96M872 (S483iVr foldiers, the anfw?^. .
8478
14587
547a
*\
9; TTTT \-r5T*
4. If 30 men can perform a piece of work in iiubysi
how many will accomplifh another^ four times as lHg> io.
^ne-fifth of the time ?
men. dayt, wdfk^
30 • 11 . 1
• '" • ' 4 "' ' ■
30 X II X 4 s= 1320, dividend.
ii\H20/66po , . r :
-r j— -( -— - = 000 men, the aniwer.
5. If 9 men in 21 days mow to8 acres of. ground ; in
kow many days will 5 men mow 72 acres, at jhe fame rate
of working?
9 men.
€hsp:t Titz RULE OF FiVe. 2ff
9 • 21 • to8
5 ' • • • 72 . . ^
9 X 21 X 7i = 13608, dividend.
108 X 5 — 540) 13608 (25! days, the anfwer.
When the terms in proportion are more tl^n. <» as ^aj
foiiietiines happen, the following rule of Mr. Emerfon^
may be ulefiil.
RULE.
1. Here, as in the fingle rule of three, pirt that term in-*
to the fecond place', which is of the fame denomination
with that fought n and the terms of fuppofition one above
another in the firil place ; alfo the terms of demand in tho
lame order, one above another, in the third place; then the
ftft and third of evcfry r6w will be of one name, and mtift
be redttced to t^e fame denomination, yiz. the lowe^ft coli*
cerned.
2. Then proceed with each row, as with (b many fepa-
rate queftions in the fingle rule of three, in order to find
out the ieveral divifors, ufing the fecond term in common
for each of them ; that is, in any row, fay. If the firft
tenti gives the fecond, does the third require more or le& i
if more, mark the leiTer extreme i if lefs, the greater for a
divifor.
3* Multiply all thefe divifors together for a divifor, and
all the reft of the nimibers togetner for a dividend ; tho
Quotient is the anfwer, and of the fame name with the
Mcond .term.
4. To c6ntra£l the work, when the fame number^ aret
eoncerned in both divifor and dividend, throw them out'^of
both; or divide any numbers by their greaieft common di-
vifor, and take the quotients -inftead of them.
v6. If the carriage of 150 feet of wood, that weighs j ftone
a foot, comes to 3 L for 40 miles ; how much wiU tho
carriage of 54 feet of freeftone, .that weighs 8 fione a foot^«
coft for 25 mITcs ? '^ N '
41 150 feet. 3L • .54 feet.
■^ ii 3 *one.\ 8 ftone.
* ' # 40 ttwles/ 25 miles; .
»*
T 3 54 X
%jA CoMPpvicp PROPOHTfOMt. «^,^ Bbqkrrll,
54x8x25x3 54X I X>5.X I g<.a<5'^Sf _9
150x3x40, 150x1x5 ""lio "*3o''s
5)9(11. i6s. theaiuwer.
4
x»o
8q
Or by an arithmetical equation further infifted upon and
eatplained in ejtcbaiige.
# g ftone, ^ ftone*
# ^f^ miks. /l miles.
Divkk bodi the divirors and dtridends by Aeif" gjettfft
common meafure, cancelling as you ha^ done with them,
and rettine down the quotients^ till you have brat^t the
dlvifor ana dividend to their loweft terms.
■ 5 ■
.".♦ 5)9(1!. 16s. theajifwet.
4
' X 26
80
7. If 248 men in 5^ days, of 1 1 hours each, dig a trench
of 7 degrees of hardnefs, and %%2i yards long, 3|.wide, and
2-J deep ; in how manv days of 9 hours will ^,m^n ^ig a
trencKof 4 degrees of hardnefs, and 337^ yanls long, j|>
wide, and 3Tdeq^?
24^ men. 5^ days* 24 # men.
IX hours. 9 # hou^.
7 degrees. ^4 # degrees.
« . 2;^ deep* . 3I. doep«
• • V »
Mow
1
Qhip^l. Th^RULE of FIVfi. ' ^79
Ntf# I tfaflfTpoTe alf ^e divifors to the Idft^ aftd aSl Ae
dividends to the right-hand, freeing all the termv from
fira^BooSy by multiplying each fide by the^ denominators.
2 II
Then dividing each fide by all their common meafures^
and cancelling the numbers done withal,
z »
2 7
— II
Then 1X2=4, divifor ; and 11 x 7 X21 = 1617, dividend.
*• * 4) 1617 (404^ days, the anfwer.
Thofe who want a further ocplanation of this method,
may find it more fully treated upon in arbitration of ex-
change, which I had written fome time before I had the
peruial of Mr* Emerfon^s book.
S EC T. IV.
COMPbUNDPROPORTIONv
OR, THE
R U L .E o F THREE rkpiated.
AL L queftions in the foregoing rule of five (as hath
been before obferved) may be refolved by two4>r more
operations iiiilth^ rule of three repeated ; a few examples
whereof we ftall give : alfo feveral qucftions that cannot be
V T 4 folved
■ • • * •
folvpd by the rule of five, may be anfwered by two o^fliore
riS^tions of'^he r^k of three ';, variety whercpf fbUoy^,^
" i: If 1 men can do 124 rods of ditching in 6i days j
hoti^ many rods may be done by 18 men in 14 days'?
meiit rods. men.
As 2 ; X2i :: 18
"^
2) 225 (fi2i rods.
AMb 6; days =;-^ : 1124 rods = i^ : : ^ days,
; f. If a regiment of foldiers, confifting of 939^ am ett
ut> 351 qUAiters 6f wheat in 7 months; how many foldkf*
will etit up 1464 fiMUters in 5 naontfat, at that rate i
qrs. wh. fold. qrs. wh. 1-,-^^
Dire£Uy> 351 : 939 : : 1464 : *> \
Reciprocally, J : i^^ : ; I : 5483^^ foW>^«t ^
anfwer as before in the rule of five.
3. If 9 men in 21 izy$ mow 108 acree of ground ; in
how many dsgr^.will 5 men mow jz 4cre5» at the fame vat«
fii working i^
acres, days, acres, days.
Dircftly, 108 : 21 ; : 72 : 14
m^fi. days, men*
2Uctproc^ly» 9 : 14 : : 5 : 25! days, the anfwer.
N, B. Tic firft queftion is what is gencrsdly called by
autbors the double rule of three dire£i , and the fecond and
toircj^ the dpuble of t^ree ipverfe.
4. By felling 240 oranges at five for 2d. half of which
coft two a penny, and the other half three a penny, I enf-
dently loft a groat ; pray how comes that about i
ora. d. era. d. s. ora. d. ora. d. ' s. d»
As'2 : I : ^ ^20 : 60:=: 5. Ag^in, 3:1:: I20 : 40^3 4.
Them 58. 4- 3'' 4d. =r 8s. 4d. the coft.
ora. d. ora. d. s.
And 5 r 2 : : 240 : 96 s= 8, confequently loft 4d.
' ' 5. tf
Qkipkl^H T^v^A^^ o^rTanEB EiVATftft: ttg
-'^E'^i^^^ ^1^}^^ be worth 21 pears, and S.pca^cotti
•hS^cfcny; ^^iit wUl be the price of fourfcorc aiid four
apples?. ,
3 : .5 : : 21 : 3.5, price of 12 apples. '
apples, d. apples, d. s. d. '
12 : 3*5 : : 84 : 24.5 = 2 -4, the anfwer.
6. A gay voung fellow had 18200 1. left him by aif old
uncle, to wnofe memory he expended 3 per cent, of his
whole fortune in a fiimptuous funeral and monument ; 9 per
cent, ot the remainder he made a prefent of to his coufins^
forgottep for his fkke by the old man ; with ^ of what w^
left he bought a fine feat ; with j- of the refidue a ftud of
horfes ; he fquandered away 550 1. upon one miftrefs ; and
after he bad lived after the rate of 2000 1. a year for 19
nu)nth» tcf ether, he had both ruined his health, and
impaired .his fortune ; pray at his death what was there left
for his fifter, who was his heir at law i
. 100 : 3 :: 18200 : 546, funeral and monument.
18200 — 546 = 17654.
100 : 9 : : 17654 : 1588.86, coufins.^
17654 -— 1588.86 =. 16065.14.
16065.14 X T =: 4590*04, fe»t.
16065.14 -^ 4590*04 = II475.I-
8) 11475.1 (1434.3875, hories.-
1 1475. 1 — 1434.3875 = 10040.7x25,:
12 : 2000 : : 19 : 3166.)}, riotous living.
+ 55OJ miftrefs,
10040.7125 — 3716.6 S2 6324.0418^'.
• 6324.0458^1.. X5 6324I. -s. lid, the anfwer.
7. If a fack of coals be die allowance of 7 poor people
fy¥4L week ; how many poor belonged to that parifh, which,
when coals were 1 1. 16 s. per chaldron, had 41 1, to pay
in 6 weeks on that account?
*. 8. 1 chal. I. chal.
I 16 =?:-:: 1- : — = 22^ chaldron*
5119 .^
Here i z^^cks of 3 bu(hel each are accountod 1 clnldron*
Then Vr X t- =% i chaldron burnt by 7 in 6 weeks.
As - : -^ : : ^ ; ^-^ 5= 3l84 poor, the anfwer. .
J. M 9 9 ^ » F
. Mt
0it ^OMPO0H9 Pnepi^KTitfN^ ^, ^k]|«
i. It is a rule in fome pariihes to iBk& the inhabitants in
f>rop6rtion to ^V ^f ^^9^^ ^^^^ » ^'^^^ ** thcVetrly rent pi;sgr
of that houfe, which pa^^ 81. lo »• to the ^ing under this
limitation, at 5 s. in th^ ^und ^ . .
.tax. rent. tax. iti^
.25 : I ;: 8.5 : 34-
.8 : I ;: 34 : 42*5 :;m 42I. los., the^nfwes.
9. 'A and B on oppofite fides of a w,6od IJ4 toifes abouj;»
they hegin to go round it both the fame ^y at the fame
lAftant of time } A goes 11 toifes in 2 minutes, andBjj
ih 3 : the queftion is, how many times will they furround
the wood before the nimbler overtakes the flower I
min« toifes. m*
2:111:3: i6t toifes.
Then 17 -r- i6J^ = t toife B gains of A in going 17.
toife, toifes. roufld»
••• i ; 17 : : t : 17 rounds gone by Aj and i6iB.
10. A ciftern holds 103 gallons, and being brimful, has
2 cocks to run off the water ; by the lirft of which a pail
of 3 gallons. wiU be filled in i minute, by the other in i
minute and 15 liDconds ; in what time will this ciftern be
emptied through both thefe aperture? together, fuppofing
the efflux of water all abng the fame i
^ " 1 c
Firft cock runs off z gallons a *^ = --^ of the ciftern
^^ 103 515
m I minute.
As r min. ic fee. =: - : -^ : : i : — , run off by the
_ . , . 4 'oi 515 J
fecond cock in i minute.
And -^ + — 3= -i-» run off by both in i minute.
5«S 5'5 S'5 '
27 C I C
• .' --^ : 1 : : I : ^-^ =^ 19 min, 44 feconds, the anfwer.
515 27 ^ "^ * ^
XI. If, when Port wine is 17 guineas the hog(bead, a
company of 45 people will fpend 30 1. therein, in a cer-
tain time ; what is wine a pipe when 13 perfons more
will fpend 63 1. in twice the time, drinking with equal
moderation i
45 men
' / 45 men : 20 1. : : «j8 men : l^.y.
And 25.^ X 2 Si 5t./rs. worth, at i7jB;uineas per hogiheaul.
5I./ 1 tj^i^s I : ^3 : 21.8x25 r per behead.
*•* 2i.8i25LX2=43.625l.=r43}. I2s*6£theanrwer«
12. In diftrefs at fea they threw o«t ijho^htitiii 5f fu-
gar, worth 34 1; per hogfliead, the worth of which came
up to bttt ^ of the indico Ihey csft orerboird ; bdides which
they threw out 13 iron guns, worth 18 1, ids. a piece i
the yalue of all thefe amounted to ^ of -^ of th^ mip and'
UiXatg S pray what of this value came into port f
1. •• d.
17 bogfiieads of fugar, at 34 1, per hogOiead, 578 -^ ^
As ^ : S2L : : 2 : 22fli indico, value * - loil lo *
7172
13 irMguas^at i81. los. each, is « • - ^140 xo -•
Value of the whole caft overboard - - - 1830' - -
Then 1830 !• = 4 of ^ of the fbip and lading, or f f -{•
••• fi'-^-ft : 22212, ,rmed.tpon4337l-iS«.6|d'
13. A, B aad C will trench a field in la dajrs ; fi, C
sad D in 14 ; C, D and A will do it in 15 ; and D, A.
and B in 18 ; in what time will it be done by all of them ;
and each of them fingly i
f ^ rr .0833333-1
c-an do) -n: = .0714*85 (part of tb«
can ao< ^ ^ .0666666 (whole work.
I^tV s= .0555555 J
All working three days will do .276084 part of the work.
Then, .27^84 : 3 days ; : x work ; 10.8309505 days^
all working*
B, C, D
4 C, D H '
1*^
3.1690S : 10.83095 : ; 14 i 47.848 by A"|
A,C,D 15
' ',*-tjoJ3D^S
4^1690$ : 10^3095 •• : 15 : 38.969 *5r B
— 10.83095
.7.16905 : 10.83095 :: 18 : 27.194. bfC
A,B,C 12
„ — ia83095
u
'
nmmm
1.1690J : 10.83095 : : la : 111.176 by D
^ ■ J *
14. If 4wii1g the' tide of ebb a wherry fets out f roin
jLpindon weftward^ and at the fame inftanc anotber ihould
put ^ at Cbertfey for London, taking, the difluice bfc
watef 34jaiUe6j the ftream forwards this, and retards the
Qt)Kr» a^ fnilea in jm hour ^ the boats are equally laden^ the
rowers equally good) and the ordinary way of working ia
fiill water, would proceed at. the rate of five miles an hour:-,
the queftion is, where in the river the two boats would
«cer?
^ Xt IS {llain from the queftion, that he that rows
froctt "^ ^ \ London goes | ^| j miles in an hour.
"" ■■ h* m. h'.
Sum 10 : I : : 34 ; 3.4
h. milt*. * ■«—
••• I : 2.5 :: ^4 : Bj- ? ., ^^ 't^'Londdn.
And! : 7.5 :: 3.4 : 25I h''" ^'''^{chertfejr. .. .-
15. A young hare ftarts 5 rods before a greyhound, and
ii ifot perceived by him till flie has been up 34 feconds ; ihe
feuds away at the rate of 12 miles an hour, and the dog- lit
view makes after her at the rate of 20 5 how long will the
eourfe hold, and what ground will be run, begrnfting with
the out4et«ing of the dog ?
5 rods or poles — 82.5 feet - - i hour =s 3600 faconds. ^
l^ miles =1 63360 feet 5 and 20 miles =s ^056^6 feet.
Then
Then 3600'' : 63360 feet : 34'' : J98.4 feefc "^ M
82.5 4" 59^*4 ^= 680.9 feet, the hare had ttut.
8 : 20 : : 680.9 ' '702:^9 run by the greyhound.
105600 ««- 63360 s: 42i40.
42240 : 3600 : : 680.9 : 58^^% run by the greyhounds
16. A lent his friend B fourfcore and eleven guineas
firom the nth of December to the loth of-MavHfollowing ;
^9 on aiMber iiccitftont Jet A have a 100. marcs from Sep-
tember the 3d to Chriftmas following ; quere, < how long
ought the penon ohUged to let his friend U&-40I. fellyt to
^etahatt the favour?
Flrft» 91 guideas ss 95 L 11 $• for 150 days.
And 1 00 marks =661. 13 s. 4d. 113 days.
Rec^rocally, 95.55 : 150 days :: 66.jS^ : 215 days.
215 1. — 113 = 102*
. Recip. 66.^1. : 102 : : 40 : 170 days nearly. Q. £. F.
17. There are two pieces of clock-work, moving with a
tbfi tiviU each of tiben lower a weight uolfonnly to the
depth of 35 feet ; the firft weight, or A, defcends 44 >nc!ktt$
in an hours and when it is lee down 12 foit, the fecond^
or B^ ia |>ut off; and the train of wheels belonging to thii
9Wichiii^ ifi ib. ordered, that the weights will be> ihrthe AttUf
levels 100 inches before the^ come to the bottDiil|.'the ire«
Jocity of B's defisent is required i
As -^inches : i hour : : 12 feet = 144 inches : T^.^^
10 ^^ . . 13
35 feet as 420 inches -^ 100 inches ::s 320 incbei, wKtre
the weights will be level.
As ^ inches : i hour : : 320 inches : ^ — , timeAde*
10 ^ 14
ibends 320 iaches.
Then ;i22 .*. iilS - I!^ hours, time B defccids to a
. , »3 i3 »3
level wHh A.
/ ill . iZ— -hours : 320 inches ; : . i hour : 4-t =s 2^ in-
chea, theanfwer.
Ij8L Mv water-tub hoMs 147 gallons, the pipe ufually brvigs*
ill 14 gallons in 9 minutes, the tap difcharges, at a'mediuhi,
40 jgallons in 31 minutps; fuppcmng thefe both carelefsjv
t^ be left open, ;^nd the water to be turned on at two o'clock
' * , ' in
in the morniiig ; the ftnraiits at fiv% iiadiogdi^waitoriiiii-
ning, IhuU the tap, and jis Jblicitous in what time thf tab
IviU b^fillod lifter thk ^pod^n^ in-c^'^^vatm* ^ntiguea
flowing from the main I
» •
Firft, 9 minutes : 14 gallons : : 31 minutes : 48I gaUoos,
ffl& in 31 mimitae. . ^ ^
Then 48J.*-** 40;s: ^ gal. in the tubat tlK end 0^-31 mill.
AUb 31 mia. : 8| gal. ; : 3 x 60 s i8e : 4714 gal. ta
hours.
. Farther, 14^^^^}^ s 99rr E^* the tub wants of bctngiutt.
And 14 eal. : 9 min. : ; 99^ : 63 min. 4844 ieGoad6»
the tub will be full.
• . ' The tub will be full at 3 minutes 4)!^ feconjs ^er 6»
19. One being afked what hour of the day it was, an-
fwered» tbp day at this time is 16 hours fongj if now ^ of
the hobfv paft be ^dded to |. of the ranai)<ider, you wiM
haye the hour de|ired, reckonmg from fun-riiing.
FMy.^^ai the bwirs paft-f-4 of theft te cone s hours
— — — »B
* » * f hoursr paft ziz^ct thefo fS^ eoeie^ ' -*^
J And 4 + a s 4 of the hours to come ca hours pall, <»
time of the day ; confiiqiientiy, the ratio of Ac houm paft
eie to- th()fe to come, :
As ♦ to 4, or a$ 14^ to i.
\ Then 4^ + 4 = .}, the fum of thofe ratios.
^•* -f • T • • '6 : 9f, hours from fun-riiing. ^^i v t^
-M* i 4 =: |6 : 6^ hows le^fun-ictttng. J^^-^v
I
20. A triangular bath 6 feet deep, is exaflly inclofed bf
3 fquare paviUons, aujj re^l^gular ; the fum of whofe plans
togetfacF make juft 50 poles j the area of A» the lefs, is to
that of B, the middle one, as 4i to 8 ; and the fum qf iihe
■Teas of A and C, the biegeft, is to tljat oT Bi as 8i to 4;
how many wine hogfheads of water will this hath jecdtrei
As 8B : 4tA : : 4B :, 2|A.
8i — 2^ = 6^- =t C's proportional part.
Alfo A 2t + B 4 + C 6t s= lat.
As 12.4 : so :: | 4^^ i itl^^\^^^ ^^ t^ef^;
• ' And
t
^ Afut'by^Hi^icAtoit the fidci of db9 triangle A ss 3» B.3: 4,
.And 04 i6|iApet^wric»«a^ pole^ j6i$ K 4 as 94.fftl» thr
perpendicular. ...
Alio 16.5 X 43= 66^fcet V = 33, half duibafe^
Ani
[7^ ss 16936128
^. • ^f)i^ij8i;GS»6i -^. 63 =. 1163 hogPuwb,
'4 21. A certain oian hiret a labourer on thia condition,
that for every day he worked he fhould receive 12 di. kul:
for evc^ day he w%^ idle he (hould be mul&od 8 d/ when
390 days were paft,- neither df-them were fndebtcd to one
another ; how many d^ys did be WQ^ky ^ how many days
was he idle ? * ' ' ^
As for every;(^y he }fo|iE:^ h« received . - '. lad.'
DO for every day he played he paid. « ^ - ' 8
. their fum ao d. \
. And as his idle days canie to the fame money as thofe he
w^ed» tlmefgj:e tto proportioD.wU) be.co|itra.| ^
^^a. 16: 8 :: 39^ : '56{^s,,4 worked. jQ^g;^;
and 20 : la : : 390 : 234 1^^ *^C played, S ^ *** **
22# A niao hired a labourer for 4a d^ys^ 00 conditipf^ that
he fltf>uld have 20 d. ibr everyday be wrought, and fqrfisi^
10 dl for eve^y day beJdled, at b^ he received 41$^ 8,4^
for \iA^ labour; bow many day$ did be workf and how;
ma^ w^ be i^e ?
41 s. 8 d. t=: 500 pence.
20) 500 (25 days wages.'
Then 40 — • 25 r= 15 days ihore.
for every of vrhich days he worked he had 20 d. *
And for every day be played be paid - - lo
* ' , 3<>
By coytra proportion, -v^— -
As 30 ;. 10 : : 15 : ^
And 30 : ao :: 15 : 10
* . * 10 =s days be w«s idle.
And 25 •{- 5 s= 30 days he worked*
' •* 22. There
♦ ' «
23. There 18 an ifland 73 nulea nmnd, and dlree ficMmiefi
«D fttft tMedier» to travel the fine wa^ about it ; A travels
5 miles a day» B 89 and C xo s when will they all oomp
eogether again f Mmerfim's Jritbmeiic.
Alfo io-s = smile*Cj8^*^^^**^*'*y-
m. d. m*
' Then 3 : X : : 73 : 24^* days, when A and B meet.
And 5 : I : : 73 : 14I- days, when A and C meet.
And B nor C can never meet with Af but at the ead* of
4iKfe periods.
Thent4t = 21, andx44 = ^.
21:21 i: 219 : 365.
The following machine being accounted a lever of the
fecond order, whofe force is dire£Uy, and its prefliire in a
contni'^proportion to the length.
24. In giving dire^ons for making an Italian chair^
the ihafits whereof were fettled at 11 feet between the
axletree, whereon the principal bearing is, and the back*
band, by means of which the weight is partly thrown
upon the horfe. A difpute arofe whereabout on the ihafts
At center of the body of this machine ihould be fbced ;
die ooachmaker advifed this to be done at 30 inches from
the asdetree ; others were of opinion, that at 24 it would
be a fufficient incumbrance to the horfe. Now^ admitting
the two pafiengers, with their baggage, ordinarily u\ weigh
2 cwt. a-piece, and the body of the vehicle to be about
^olb. more, piay what will the beaft in bQt^ thefe cafes
e mjde to bear more diaa his harnefs ?
Firft, II — 2.5 = 8.5; alfo IX — 2 = 9t and4cwt«
70 lb. = 518 lb.
Diredly, xi : 518 :: 8.5 : 400^x9 fo'^-
Contra, 8*5 : 400^^ : : 2.5 : i I7tV9 prefliire in the former^
Alfo IX : 518 :: 9 : 423^-. force Ji„ the2dcafe.
• . • 9 : 423 : : 2 : 94A, preffure I* "**'*«*'**'=•
Anfwer, the beaft beaks iX7Alb, in the fonaer, and 94^b.
in the fiscond cafe.
as- If
Chip. 1« Tab Ruti op Three llE^rATED• 289
a$* IF * levtr 40 efii^ve inches long will, by a certain
power thrown/ fucctffivdy thereon, in i^ hoars ntife a
weight 104s feet ; in what time will two ether levers^ each
IS.effe&ive inches long, raife an equal weight 73 feet i .
As 40 inches. : 104 feet : : 18x2: 9^.6.
Then ^3.6 feet ; 13 hours : : 73 feet : 10 hours 8^ minutes^
tiie anfwef.
26. A weight of i-^ lb. laid oh the Ihoul^ers of a man,
j^<no greater a burden to him than its abfolute weight, or
' 24 ounces ; what difference will he feel between the faid
weight applied near his elbow, at J2 inches from the fhoulder,
and in the paTm of his haiid^ 28 inches therefrom ; and
how much more muft his mufcles then draw, to fupport it
at right-angles, that is, having his arm extended right out i
As I : 1.51b. : : 12 in. : x8 lb* weight 12 in. ) from the
And I : i.<; lb. '. : 28 in. : 42 lb. weight 28 in. ) (houlder*
• • * 42 ^^^ 18 = 24 lb. the tolwer required.
27« A ball t^eighing fdUr pounds Upon the furface 'of the
tertn, to what height in the air muft it be carried to weigh
but iHttc pomsds, and how long would it be falling (o the
ground ? L^ies Diary.
Taking the earth^s iemidiam^er at 4cod mlleSi
Then 4000 X 4000 = 16000000.
. As the weights of bodies decreafe as the fquare Of their
lUftance from the earth's center, we have,
« < «
Recip. 4 : 16000000 1:3: 2 1 333333.^*
Then V'21333333.^ = 4618.8021.
Then 4618.8021 — 4000 = 618.832! miles = 3267275 feet#
%r 618 miles, 6 feet, 16 poles, '3 yards, i foot, the height of
the ball tauft be carried.
Again, i6.o8j :.i fquare feeohd : r 3267275 : 203837.48*
/203837.4J8=B4si.48'''«s=7' 31'^ 29"^^ the time of
falling.
But V that great diftance from the earthy when the ball
wUl hav6 lofl ^ of its weight, its velocity will alfo be dimi'^
niflied.^
Viz. 16.08^ X »75 = 12.0625.
Then as 12.0625 • ' = • 3267275 : ^70862.
^ 270862 = 520 = 8' 40'% the time of falHng.
U 28. A
290 Compound Proportion i *;s •60^'fi.
. 28* A ball defcetidii^ by the fbrce of-gntviiy from^the
top of a tower, was obrerved to foil balf the-w^yin the laft
fecoiid of time *, required the tower's height, ana the whole
time of defcent ? JuaJiis Dktfy^ ^76S*
The fquare roots of the diftances being as the times,, viz.
as the v^ I : ^2 : : is the time of falling thfou^h the lirfl:
half, to the time 0/ 'falling thrbugh^the whole required
height*
• . • As ^ 2 — r = .4142 : ^ 2 = 1-4142 'y^ - 3*4H
feconds, the time of defeeht.
And I* : 16.08^ : ; D 34142 *= ii«6574 : 1^7.48 fefet,
the tower^s height.
29. Sappofe that in etery'fingle revolution of the u^pcr
Hone of a water-mill, it evacuates or grinds one^^eighch o0a
pint of meal; and ^ppoiing there be eight ftaaiards^or
pinions in the rounds that turn it once, and that thefe
rounds arc driven by a wheel of 45 teeth 5 -alfo* the'ttean
* circumfetehce of the water-wheel on the (aihe ^xis'^be*40
feet, Which requires one hundred and a half to move it,
or put it into motion : now, if a -floodgate, whofe breath
is a foot, and height half a foot, and the 'hei|Kt *6f the
water be three feet above the furface of the hole, be let run
dire£Hy againft the upper furface of the wheel, it is Tc-
quired to hnd the quantity of meal ground in an hour ♦y
the faid niill ? Genthtmif rDiitiy^ ^7S^*
Firft, I X .5 X 3 :;: r.5; alfo 62;5 lb. Vdght of a cubic
foot of water. Then 62*5 x i»S ^=s 93*75 ^^^ <hf iiiftan-
taneous preffure of the water. J
Alfo, i6tV s= — :• 1* : : 3 feet : ~ = fimare of the
12 ^ 193
time.
And /— 's=: the time. •»
6 '^^
••• ^-^ : 6 :: I : ^193^ 13:89244 feet, velbcity
per fccond.
Alfo licwt. = 168 lb.
93.75 X 13.8924 -f- 168 = 7.75248 feet/ any ?-pbint of
the wheel moves in one fccond.
7.75248 : I : : 40 : 5.1595 feconds, moving rdund.
5-'59S : I :: 3600 : 697*742 rounds, the wheel moves
m an hour.
Ou^.'I. Th« Rul£ or Thrjea >R£PsiT£0. 291
J =,Si^«- As'^ • i-62S : : .69(7.742 : 3924.799 ^'O^^^i
file jftone moyes in an4ioiir.
* . • -8) 3914.799 r490.6 -pints «c 7 biHhoIs, 2 pecks, io|^
pints, the miU grinds in an hour. (^ £• F.
30. Obfeived, Aat >wh&e a Aone .was defoending to
meafure -the depth 0raweA,'4i.fbihg4UKl pilummet (that from
the point of fiifoeftfion, or the place where it t^s held, to
the center of -oibtUation) or 'that past of the ibob, which
being divided by ^a tcWouk^ 'line, Ariiok ifiro^i the center
abovefaid, wo«}d -divide it anto ^twopansof .ei]ual weight
metfiMacf •jtA'it'Jfi^bes, had 4nade ^ei^t ivibration^ ; pray
what was the depth, allowing 1 150 feet per ileoond, for the
return of ibimil to the. ear ?
39^ : 3600 :: 18 : 784x5.
1/7840 = 88154378 -vibrations iji,pne4ninutc.
60) 88.;5^i8 (,J^^T&7'^ ^vihr^tlan^ in .a jfecpnd.
i*47572ft>o.pQOPpo,(i5yf(u fccQAds .in !?igh.t yibcations.
1* : itf.08/ : : g :^,^i pn.zg.iSj : f^72^c?40^lj? feet.
As 1 150 feet: .1 : : ;472^|6(4.P9?^ • vH 099% time the
found .was j:etv];iii{^.
5.421 -r^ ^P99' i^ 5»on, true time of the bod/s de-
fcont.
• • • 5QI rX iS^fti f;? «^iOQi X ,i,6.oi83.= 403-6932 feet,
* ^y. Afpire my-^nius ! help my rhimin? mufe.
In themes I in my , native country chufe :
WWfift others plow .the .waves, and .tread Acftrands
-Of 'dJftant oceans, and of -foreign lands ;
To^fijl the mouth .of fame with fomething new
(No tBsvttcr 'tis how. much of it-i^^twie)
Ti-om i\lps.or.mountains, ftories-ftrange they bring,
^©f dtfcrtSy eaves, orhorrid monfters fing.
Tejl how Vefuyius* fulph'rous darts do fly.
Or JEtnaVfiuoak obfcures- the -azure Iky j
Or magnify the hazards they have run,
^ylla^s axid £3iarybdis' pointed rocks to Qiun.
Such tales we take ^n tiruft, from thofe who rove,
Tho' none give rule^*, by wTiich the truth to prove.
But this by numbers may explained be.
By thofe who never did the cavern fee :
In Derbyfliire, a wonder of the Peak,
Is £ldon-hole, as poets often fpeak ;
U 2 Whofe .
•
1
292 Compound Proportion, (^c. BodcH.
Whofe depth exadly none could e'er defcry,
Tho' atheift Hobbes his utmoft (kill did try.
And wrote De ABraHUus Pec0.
And burlefque Cotton, does ftrange tales rehearfe.
In ruftic words, and Hudibraftic verfe.
How he this mighty orifice did plumb.
But could not at the bottom of it come.
With iixteen hundred yards of rope let loofe ;
And tells a ftory of a womaa's eoofe :
Fab'lous the one, fo muft the ouier be
Erroneous too, without philofophy ;
Extenfion of die rope might him ifeteive.
Or fmall proportion which the plumb would have
To fuch a length ; and part in water drowned.
When in this vaft abyfs, within the ground.
But I the depth have found exadly true.
By gravity, a method fomething new.
As heavy bodies do accelerate.
In fpaces known firft to our Newton great.
Four ponderous ftones into the well let fall.
In meafur'd time, agreed in numbers all.
A pendulum, fixty-one inches long.
By which the time I meafured (was not wrongj
Vibrated freely (whilft that each ftone fell)
Eight times ; by which the depth Pd have you tell^
Allowing rightly for the approach of found.
That your own works may not diemfelves confound*
LadiisDiary^ 1722*
Firft, 39,2 : 360Q : : 61 : ^313.4426.
4/ 2313.4426 = 48.09826 vibrations in one minute.
60) 48.09826 (-8016377 vibrations in one fecond.
.8016377 ) 8.0000000 (9.097957, feconds at eight vibrations.
Alfo i» : i6.o83f • • □ 9-97957 = 99»59i82 : 1601.7584.
1 150 feet : i'' :: 1601.7584 feet : 1.39283, time of the
founas afcent.
Then 9.97957 — 1^39283 r= 8.58674, true time of the
ftone*s defcent.
• . • I* : i6.o8j : : D 8.58674''= 73.7321 : 1185.88 feet,
the depth of Eldcn-holc. ^ E. F.
CHAP.
i
C 293 3
CHAPTER II.
S E C T. I.
SIMPLE INTEREST,
INTEREST is a fmall fum of money paid for the
ufe of a larger fum, at any rate agreed upon ; which
according to law muft not exceed 5 L for the intereft of
100 1. principal, for one year.
CASE I.
The principal, rate of intereft, and time, being given ;
to find the intereft.
Firft, When the yearly intereft of any fum is required,
RULE,
Multiply the principal by the rate of intereft per cent.
ner annum, dividing the produ£b by 100 ; which is done
Dy cutting ofF the two right-hand figures, thofe to the
left being the intereft in pounds. Then multiply the re-
mainder (if any) by 20, cutting oiF as before for ihil-
lings; and that remainder by 12, cutting off as before
dire&ed for pence } and find the farthings (if any) after
the fiune manner; the figures to the left of thofe cut off
being the intereft.
I. What is the intereft of 8731. 16 s. 8d. for a year,
at 5 per cent. ? *
1.' s.
873 16
jC 43-69 3
20
1
8
5
4
f
•
By Practice.
1. s. d.
5 per cent = -jV) 873 16 8
£ 43 13 10
13.83 s.
12
0
^ 10.00 d..
Ai^lrer, 43 1. 13 s. 10 d.
•
. ■ U 3 2. What
v.
294 SIMPLE INTEREST. Bookll.
2. What is the intcrcft of 1437 1. 17 s. for a year, at
44 per cent ?
I. s. d.
1437 17 -
4i
57SI 8 -
718 t8 6
I I I .1^ ■■ iiiii *
;£ 64.70- 6 6^
20
By Practice*
1. 8. d*
20 per centc=l4 b43^ ^7 ^-
4.»-
2»7 rr 4k
fa < ■ ■■ -■ >
S7 w 3i
7 3. 9f
14.06 s.
12
i£64 14
78
4
3.12 qrs.
Anfwer, 64L 148. ^A,
3* What is the intercft df '178I. r6s, for si year,r at 3^
per cegc. i
1. 8.
17ft 16
^
538 18
133 14 6 d.
1 6.68 ii 6
20
. j9jr Paj^CTicE,
13-72 s.
12
8.70 d*
4
2.80 qrs,
2d pef ctnit xs •{*
178 6 -
iV
35 13 4*
1
T
1
T
3 " 34
I 15 7*
- 17 9i
- 8 loj
iC 613 8
Secondly, When the intereft of aey fiim is requucd for
feveral years,
RULE,
r
CteJ*. SIMPtf: INrTEiLEST. 2^5
RULE,
^ftcr the. yearly intereft is found, multiply that by the
nuqjiber of 'years, the produfl will be the.anfwer.
A. What i$ the intereft of 74 1. 15 s. for five years, at
4| 1. per cent* per annum I
By the Rule of Five,
100 . I . 4.375
74-75 • 5 •
4.375 X 74-75 X 5 = 1635.15625 dividend, an4 100 X I
== loodivifon
••• 100)1635.15625(16.3515625= 16I. 7s. -^d. the
anfver. ' '
By the Rule.
Sjf Practice.
L s*
1. s. d.
T 74 »S
20 per cents 1-174 15 -
4f
1 ■ ■
A^/% ^ ^
tV|h 19 -
290 - -
18 13 9
•
I 9 lOj
1
T
T
I 9 <io|
3 8|
l3'V - 4i
I 104
X20
Intere^£:)riye;ir3 5 4I
5.40 s.
5
X 12
Anfvirer,/ 16 6 ii|
4.84 d.
4
The fmall defe£l In the pradical methods, is owipg to
th(& parts p{ ^ farthing oi^it^.
5. What is 'the intereft of 963 K 7 s. 6 d. at 3I per cent,
for 13 years ?
U4 t963^
296 SIMPLE INTEREST* Booklt
, \* '• 1* By Practici,
i 963 7 6 \
3I u 8« d.
^im
2890 2 6
481 13 9
120 8 5 J
20 per cents f
963 7
X
£34-92 4 8J i
20
192 13 6
4
/; 18.44
19 s 4
9 12 8
4 16 4
I 4 I
» •-
12 Year's intcreft £ 34 18 $
5.36 s. Year's int by the ift method £ 34 ^8 si
4 XI3
1.44 <*• Anfwcr, £ 453 19 %i
JSjf FivB Numbers.
100 . I . 3.62^^
, 963-375 • 13 •
. 963375 X 3,625 X 13 r= 45399046875
100) 4^399-046875 (453.99046875 = 4531* '9»- 9i^
■ anfwer.
Thirdiv, When the intereft of any fwi is required for
years and months,
RULE,
Firft fiqd the yearly intereft, which multiply h]r the num«<
^er of years, as berore; then for the months divide the
yearly intereft by the part or parts the s'iven mionths are of
a year, which add together with the reft, and their fum will
be the anfwer.
6. What is the intereft of 56 1. 10 s. for 7 mondis, at4}
per cent, p^r annun) i
• •
..i J $61,
Ch»p.n. SIMPLE INTEREST. 297
1.
$•
56
10
4i
«
226
••
M
U
2
6
9.40
2
6
20
L
8.02*
12
4
1* s* d.
^]2 8 -4 per suinuni.
t
i« «
I 4 -
h 4 -
jC I 8 -^^ die anfwer.
1.20 qis.
By Fivx NvMBUs.
P. T. a
100 . 12 . 4,25
56.5
50-5 • 7 .
56.5 X 4-25 X 7 = 1680.875, dividend.
loox 12 = 1200) 1680.875 (1.400729 = il. 8s. the
anfwer,
7. What is the intereft of 759 1. 16 s. 7 d. for 12 yearly
4 months, at 5-J.I. per cent per annum ?
L s. d.
Year's intereft 38 18 9I
'2i
I. 8.
T 759 16
d.
7
5i
3799 2
94 19
II
iC 3^-94 2
20
54
18.82 s.
12
9.89 d.
• 4
467 5 9
12 19 7i
Anfwer, £480 5 4"
3-59 qrs*
Bf FlVB NUMBK&S*
P. T. G.
100 • I « 5.125
' 759.8291/8 . 12.^ .
S"5 X 7S9'829i^ X 12.^ = 48027.534232/
100)48027.534(480.27534=480^ 5s. 5|.d. theanfwer.
3 %. Lint,
2^ 9i M[PUE rSlTM ESGT., Bbok,i&
8. Lent» at Chriftmas 176O9 the fum of 5<iooIl at 44
per cent. ; after which time I lent feveral fumsat ti|e fame
rate, and drew updn the Sorrower, as bufti|ars required ;
viz. ou.l<a4)[*dfiy; ^ly. I ^V^ for 185 guineas ; on Mid-
fummer-day 1761^ iUA^506»moidore3, anddrew for-Tool. ;
on Michaelmas-day* 1761, I lent 569 1. 17 s. : H demand
what caih the borrower owed me at that time ^
i !• « T^JS<^o !• principal, at44percen^.
250
— 25
295, intereft for x year at 44 por cent
4} M5
8.
S6 5> int/ereft due at Lady-day lji(U
5000 -f principal.
I W, 'I
1.
5 =
5P56 5;, ampi;r\t at Lady-c^y«
194 5 drawn: =r 185 guineas.
I
t
^2 -, new principal.,
243 2
—34 6 2^
m 'w
4) 2f6 15 9|, intereft for i year at 44
1.
5 =
54" 13 11^, intereft due at &$dfumn^«
4862' - -, laft principal.
4V^ ^ Hi
— 2^ •• ' - = 700 !• — 500 moldores, pr 675 U
VB-
t
I
4891 13 ii-^-, new principal at4| pcy ^^t^
244 II 8J:
4) ^^0 2 6^, intereft for i year.
'55 1 r 7ii intereft due at Michaelqu^A
. 56% *7. -r I
^:^ £ S5'6 II 6|, the anfwcr.
Fourthly,
CJUixBL SriMPLE INTEREST
Fontthijr Whed die i
days.
of any fam is xtq}ur€d for
ll U L E,
Firft find tte intereft^for a year, then b^ tbc rule of
three dix9St^ viz. 3^ : one i«ar's intereft : : t&e day« the
money is at IhtereS : intereit required.
9. What is the inteieft of 375 1. 15 s. for imp d4j»» at
4| per cent, per annuarl
I. s.
i 375 IS
4i
days. I. s. d. days**
As 305 : 17 16 10^ : : 127
12
^503 -
18717
93 "
6
9
I 17-84 16
20
3
16.86 8,
12
10.35 d.
4
i«4o qrs*
214 2 3
SO
2141 2 d
124 17 Ilf
365)2266 •*• 5{(61-4s»i|d« aaf.
76
X20
1520
60
XX2
7^
360
X4
»443
348
10. What is the mtereft of 284 L xos. for Quro years-,
four months> and 25 days, at 3^ per cent per annum i
iZ^l
Joo SIMPLE INTEREST. Bookll.
]. s. davs. 1. »4 d. il^ys.
a8i lo 305 : 9 ?9 H --as
3i L
853 10 49 IS 8i
142 5 5
£9.95 15 36s) 248 18 7t ( »3 »• 7i <».
20 20
I9.IS8.
12
1.80 d.
/
497«
1328
233
X12
4
3.20 qrs.
-
2796
221
X4
1. s. d.
3
year's
i intereft.
884
19 18 2h ^^^"^ years.
3 6 4I9 four months.
- 13 7t3 twenty-five days.
£ 23 18 34) the anfwer required.
Or as five per cent, is ftatute jntereft, multiply the given
fum by the number o{ days, and divide the produfl by 7300
jvias. ' ^ M ; the quotient will give the intereft at five
per cent, but if the intereft at a higher or lower rate is re-
quired, take aliquot parts of the quotient for a diiference,
which add or fubtrad accordingly*
II. What is the intereft of 5471.. 15 s. at five per cent,
for 320 days ?
547-75
32a
109550
164325
7300) 1752.8000 (.24*0109 rr 24I. -s. 2-J-d.
' '■ 12. What
Chap.n. SIMPLE INTEREST 301
12. What is the inteieft of 24$!. 19$. for 171 days, at
^ per cent, i
248.9s
17s
12447s
174265
— ^ A
73) 435*^25 (5*9679o> lit five per cent*
7ofr 596798
496 — — —
582 5.371182 = 5I. 76k 5d. at 4i per cettt«
7,5 , —
580
13. what is dteintereft of 713I. 17s. 6d. for i93dayftt
•t 3^ per cent, i
7?3-875
193
2I41625
642487s
713875
4)
73) 1377-77875 C 18.87568, at 5 per cent
647 4.71842, at it per cent.
637
537 14. 15526 s 14 1. 3 s. i^d, the anfi
268
497
59S
II
^QE STMPLB TNTCXEST. Socik^O.
It .bctag Joined t^iT n iinccfMUe m fttie aa)a))atio« of
intcreft ro take ^iquot parts for months, becaufe ,tbe yojU
is divided into months confifting of an unequal number of
days ; I have therefore, to find the number erf days from
one time to another, inferted the foUowing^IVBLS.
_
"n
S >
s
._
^
J.
g"'
Q
2
Q
1
}
f
Hi
1^
i
•1*
<|
■f
§•
g
I
3*
c
*.
1
J
I-
|-
-^
I
32
60
91
121
152
a»J
^44
SH 30s
33^
3t
«3
)5l
»»
UW
'S3
^%t
2«
*45
WJ i 306
3
34
62
93
■ 23
■54
IM-
*4»
»76,307
Ig
**5
4
3?
63
94
124
'55
216
-^
»77'3o8
i.78 309
I
•I
34
^
125
156
186
2;^
339
g
'5
126
;i5
M
249
Sao
ffi'i;?
340
34'
39
ii
98
'59
189
220
2»
>ei
a«2
g4a
9
4°
68
99
129
160
190
221
la
282
3"3
343
o
41
69
100
130
161
191
222
.83
3'4
344
I
42
70
101
■31
162
192
223
254
284
3'5
345
2
43
71
102
132
163
■93
224
%
%
316
346
3
44
72
103
133
164
'94
225
^:2
^
4
4S
73
104
'34
.6j
'95
226
1
f&
i
46
74
S,
'35
166
196
%
3'9
349
*l
,5^
nK
i«i
fi
259
2%i
320
350
7
48
\%
%
HI
x6o
28o;
321
35'
8
49
J?
165
'.9.9
2J0
261
w
322
352
i!
JO
10,9
'39
'"^
S
231
26^
JS2
323
353
0
5"
v
IK3
140
171
232
263
^3
324
354
I
52
80
111
141
■72
202
233
294
|2^
355
2
S3
81
tI2
142
'73
203
234
2?
2^
29s
356
3
S4
82
113
'43
■74
204
235
296
3'2
357 .
358
+
5S
o^
"4
'44
'75
205
236
5^1
2l^
328
S
S6
;♦
"5
;:2
176
206
2II
329
359
6
57
»s
116
■77
207
269
299
330
360
I
ss
81.
"7
■47
178
208
239
270
300
33'
361
59
ll
118
148
'Z'
209
240
271
301
332 362
9
88
719
149
l8o
210
241
272
302
333 3'3
0
89
120
150
iSl
211
242
273 303
334 304
I
90
'5'
212
243
304
365
(
ChajL/fl. S TM P L E TN T:ER £S T« «|iq
iT'Hr UsB $f •At fbngoitig Twbvb.
'firft> l^o find the number.of <la^s. from the «iid«ofllip
.year to' any ^iven -dgy dn^any month 4A. the y.ear follo«rii|g.
OppoCte tfae^ven clay in the inargin'Imk under ^hq^voo
month, which will 'fltfew the nuftiber of days ^equired^
Thus, from December 31, till Auguft 18 following, are
And till ufldber 36, 'a»,303 d*y$,. » &c.
Secondly, To find the -days -from a^y eiven day of any
month, to the end of the year. - *- - Suppofe Jdly 27.
From 365 days in a year.
Take the number anfwering July 27, vh&. 208
Refh. 1 5 7,days required.
^' . * ""^
Thirdly, To find the number -of days between a given
' day in any^given inoi)tl>, «Bd any. given day of any other
month in the fame year.
Si»pp«re the days between- April ^5, «ik1 .November 28,
.be required.
The number anfwering November 26 - ,33^
Subtraift that anfwering April .^ - - 95
\
Rem. 237, days fought.
Fourthly, To find the number of days -fiiom any given
day of any month in one year, to any given day of aily
month in the next year.
Suppofe the riumbcr of days from the -aift of Auguft
17589 till the 27th.of May XJS9y ^^^^ rOquirdd.
From -•'---— - '- - - 36*5, days in a year.
Take the N® anfwering Aug. 21 - 233
. Rem* 132^ to the #nd of the
'Add- the !!• anfwering M ay 27 - 1 47 ; year.
279, days required.
But
:^
304 SIMPLE INTEREST. Book IL
/
But in the biflextile, or leap-yealr^ if one of the given
days be before the 29th of February, and the other after,
bne dwf muft be added on that actbunt. Thus-, if the
ntiinber of days in the laft example had been from the 21ft
of Auguft 17599 till the 27th of May 1760, it would have
been 280 days.
13. What is the intereft of 1501. from the i8th day 6f
January, td the tith df November, at 4^ per cent, f
314 November tt.
18 January^
297
IS*
*o)'
73)445-50(6.102^
5-7976 =i: Sl- IJ(«- 4j<i-
14. What is the intereft of 384]. 16 s. from the ^t6 6^
May, to the nth of December, at 5I per cent. ?
345 December ii,
127 May y.
2lg
3848
30784
3848
7696
73) 838.864 (1 14^128, It 5 pit fceiil.
I.i49i2t at ^ per cent.
12.6404 s= 1^2 1. 12 s. 9{.d.
15. What is the intereft of 537 1. 15 s. from Noveni^
bcr the nth, 1764, till June the 5th, 1765, at 3^ per
cent. I
537-75
Chap. IL SIMPLE INTEIteST. 30^
537-75 36s
106 31S
322650 To the year's end 50
ia7550 . 156, to June t}ie 5th,
L 5) _^
73)1107.765 (I5«i7486»at5p. cent. ao6, days in all.
T
3.03497 :=: I per cent.
4.17318 = il per cent.
11*00x68 = <i I. -^s* ^d. theanfwer.
Intereft i$ to be calculated in the fame manner on caih
accounts, accounts current, ice. where partial payments
are made, and partial debts contraded.
• " ^ ' . - , : ' i . ','
z6. On the ift of May I lent Ralph Newlands, per bill,
at ohq day's date, 500K, which I received back in the foK
lowing partial payments ; viz. on the 13th of May 50 1.;
on the ^th of June 561. ; on the 14th of July 44 1. 1 on
the 23d ditto 50 1. i on the i8th of Auguft 87 1. ; on the
30th ditto I13L; on the 21ft of September 30 1. ; on
the 1 8th of Q&ober 30 1. ; on the 29th ditto 40 1. ; on the
nth of November 50 L ; on the 28th of December^joL :
what intereft is. due, at five per cent. ?
I .
I
I
Mr.v
, 1
j66 :SlMPLE.ilLNT£ItE6T. BatkU.
lUlpb NewUndt. • - - Oab«or ,
May I. Lent pef Bill at one day's date - "'506
13. ReceiviA in JJirt '■ ^'' - ': *• - ' Sd
■
« •
July 14. Rcceivtd in fart
* • ■
Bal* 456
' —
lUl. 394
- 44
Bal. 350
23* Received in part '- - i ^ ^ 50
- - .1 ftd. 300
Aug* I S. Received in part «.«>'-•• 9y
Bal. 21
to. ^liccived in part . - - - - i
Septa 21. Received in part ..-•-> 30
t •
. Bd. 176
O^ i<. Heofltcd ih p4tt vi i t . ^ 3D
Bit t40
19. Received in part ^ & «. • ^ .1. 40
Bill. k6a
Ni^.li.ltttceiVfcdlnplIt ^ *'^ * - $^
Bal. 50
Dec. 284 Received in full of principal - 50
22
<
40
^
12
22
27
ki
i^
47
9900
15760
3150
7800
2556
1)66
2350
59846
73)598.46(8!. 3 s. 11^ d. intereft on this account.
17. Lent John Jamefon, per bill, dated i8th of January*
payable one dav after date, 878 1. 19 s. 10 d., which I received
back in the following partial payments ; viz. on the 27th of
February 57 1. 15 s. 7 d. ; on the i8th of March ^7 1. 14 s. ;
on the 29th of April 34 1. x i s. ( on the 12th of May 136 1. ;
15 s. 7d.; on the 19th of June 67). 13 s. 4d. | on the 15th
•f
J07
Ckt^M, 9IMf[LB IKTBRE3T.
•f J«Jy iSl; iX** 6 d. ; *n the 45* clkto if 1 1, ti (. 11 d. t
on ihc^iauEMkfjftl p. 4«l. J db the 19^ of No-
vonber too l.j on the ajd ditt* J«ol. » and on die 30th
of J>cc«aber rwnved the b^anoe of the pfindpal: how
flittdk infei^ okl^t I M dainl, at five per ceat. f
Cab ttiUaaiont wit]t Jbmr jxnsolr.
. Sxtended. ^ ProduA*.
L s, d.
Jar. IT/ClAtperbillatoaels^o , ^
d^date - - - - r?* '9 »o 40
feb." 27.- Red^lred in part " - ; if 15 7
B«l« 8jr f iTjb
Mar. itt^ ReMfdd MpaSi -i |7 14
I
^ -• - BalJ 7! J. so jJ4»
iffril ;29. JLsdeived in part -{ |4 xt
w_ ^ ». ^' 748 »9 3f«3
M^iiff KMeived in part -' i j6 ij 7 .
i«*[
. 6u J f iS
sal. 612 t V
JmW 4i^ BMBivtd iA feM -' ^ 13 4
■ * _
Qal. 54]^ xa f
July 15. Received Im part « 15 ^5 6
DittS'2]^. Received iti part - tii 11 is
>* I a fc
Bal/ 417 2
Oa, 3. Receive in part - 78 7
2.ii
4
wmi*i
10
70
Bal. 33B iij 7147
Nov. 19. Received ih pSLVt - ido
Bal. 23S i; 71 4
Ditto 23. Receive in part - 100
wmma
Bal. 138 15^ 7 ^
Dec 30. Receiveditn full of I .^- ., ^
principal - . - - . J'^ *^ ^
15603 .9
3^907 !• 6
•73* »o 3
23364 «| 4'
"psr 8 8
|a8y 84
29200 4 2
15922 ts 5
955 »4
i»3+ 16 7
I
---X 187327 *8
*a
CoMr
3oa
SIMPLE INTEREST. BodttH:
Computation tf Example 17.
.!
By the Table.
25 Feb. 58
igjan. 18
18 March 77
a> Feb. 58
«9
29 April 11^
18 March ^7
4*
taMay 13s
29 April 119
i3
19 June 170
12 May 132
38
19 June 170
20
25 July 10
3 Oa. 276
25 July ao6
70
J. s. d.
878 19 10
— . £
4394 "9 «
^_^ 8
3';«59 IJ 4
821
4 3
6
49^7 5 *
14.781 r6 6
821 4. ft
1 5603 - 9
7«3 10
i
*7o> «
• *
6
JL
3390.7 10
6
- 748 19
3
»3
9736 10
3
6*2 3
ft
8
12
7346 4
0
22038 12.
1224 7
0
'23262 19
4
544 10
4
»3
7078 14 4
2-
14157 8 8
J28
H
10
10
5*87
8
^
4*7
2
It
7
2920
5
10
2Q2pO
4
»
19 Nor. 325
^OBu 276
23 Nov.\ 4
30 Dec. 364
J27
37
4. s. d.
3*8 »5 7
«o^ 13
1
7
14228 14
'693 >7
6
It
1^9x2 IS
_i
238 15
.7
4'
955 *
Ji
.138 IS
1
832 13
"8
6
■
4996 I
'38 15
7
$^134110
7
The fum of all thefe mul-
tiplied into diei£ refpeAive
times are 187327 L 6 s* 8d«
Then
u S* d. I. • S. d.
73)1873.27 6 8(25 13 2{.
413
intereft*
48.27
20
:.•>
96546
23J
•
« • *
16.46
11
•
1
*
197.60
51.60
4
••
206.40
CASE
\
CIu^.n. SIMPLE INTEREST; 309
CASE 11.
»
The junounCt rate per cent, and time given ; to find the
nrlncipal.
RULE.
As die amount of loo 1; at the rate* and time given : is
to lOO K : : fo is die amount given : to the principal re*
quired*
Ti What prmcipal fum, being put out to intereft at diree
percent, per annum, will amount to 39981. irs. 10^ d.
in 3^ years, and 54 days t
years, days.
Time 3.397945^ F 3t^ 54 , ^
3 mtereit of 100 1. for a year.
10.1938356 =: intereft of 100 1. for 3^ years, 54 days.
+ 100
iioa938356, amount of 100 L for the faid time.
As 110.1938356 : 100 : : 3998*644791^ • 36^8.737275.
•.• 36^8.737275 ss 36281* 14s. od. theanfwer.
And 3998I. I2S. lold. -^ 3628L 14s. 9 d. ==: 369I.
18 s. 1-1 d. difcoun't.
a. .What is 309!. 16 s. 10 d. due three yeirs, one quar-
ter, two months, 18 dap hence, worth in ready money,
abating or difcbunting 44 per cent, per annum ?
3.465982, time.
573.4, rate reverfed.
1386393
X03979
24262-
'733
15.16367 -j- 100 = Ii5«i6367.
115.16367 : 100 :: 309.841)} : 269.044627.
269.044627 1. ^r= 269.1. -s. io|d. the anfwer. '
And .300 1. 16 s. io d; '^ — 269l#-s. lofd. r: 40 1.
158. ii^d. difcount.
CASE III.'
The account, principal, and time given \ to find the rate
of intereft,
X 3 RULE.
019 SIWI^LB flJT?M5Tr fMnlh
I
/Utke^^p^oHillipttid bMo thm tec : k to t^ wMe
intcreft : : fi> is iooI» : to t|ie rate per cent, per amium?
J. M whMt me «f ifl^ficft, per emu gfr 4i)uiim»> «rill
y^^^t ^n^ 54 ^7' ^
36a3*ri7*7i» principal. 3?9|'64479«9, nnnwit.
4MP*
10886211825 3^90751^ intercft.
1088621Z83
«WV4MPi^MnMk^MMw»
316586355
25401x01
3265863
18144
12330.249661 : 369.9075 r<7 :: 100 i 3{ier€eiit»
369'9P75t<^ K i<» » 3699a.75i/».
12330.2497) 36990.7511^ < 3 per cent, per annun, i)ie an-
fwer*
m
%. At wbgt mte of intereift» per i^ent; per mni49» will
269 1. -> s* lol^d. amount to 309!. 163. io4. ig 3^ J^^n^
two monthff and 18 days. <
260.0^46279 principal. 309*8416661 amount.
2895643 ^ 269 044^^7
8^»339 40.797039t i»»Vtft.
1076178 '
161427
'345*
242t
ai5
5
^325037 ' 40-797039 i©lP«* - J^^ - 4'375 « -H
per cent, the anfwer.
C A S E IV.
The principal, amountt a^nd rate of intertft ^ing jjiftnt
to find the time.
RULE.
t
As the intereft of tht iriiole princi{}al for one year, at
ttie given rate : is to one year : rib is the vHtoti : to the
tune required. ' . " '
I. In what time will 3628 L i4ar 9 d. • uamMi to 3998 1.
I2S. io|d. at tbrfc {iTf cent Bn *Aiiun ^
3^^*7375 '. 3998.63483 ai||ount.
X 3 — w8.7J7i |irincipal.
io8.862i25» year's intereft. 36^9075, Jptereft.
^mm
108.862-: I ;; 39^.9q73
wSMt) 396 9^73 (3-3979^ == 3i Y^^h S* <**y«*
326580 — 25
433^ '3
326586
147938
365
106627 739690
'i
97970 887628
-— — 443814
7620 5jr9e7?7 ?= |4«i*y»'
^02]
.979
42
33
2. In what time will 269 1. - s. io| d. amount to 309L
1^ t» lOii. ;^ 44 per cent, per annum ?
269.044627, prittctpal. 309.84i66(?, amo^n^•
^73.4, rate reverfed. 269\o44627, principal*
'*^»^
J0761785
807134
J88330
•H
11.779701 ycwr'a intcas^.
>
40.797039, intereft.
X4
As
jia SIMPLE INTEREST. Books.
As 11.7707 : I year :: 40.797039 : 346598276313.
I K7707 ) 40.797039 ( 3.46598a =31 years, 2 monriis, 1 8 dar».
253121 —.25 s^year.
548493 21598a
470828^= 166666 s: 2 mondu. ,
776659 .049316 B 18 iMft,
70624a ,365
70417 146580
58854 295896
— T '47948
I 1563 .
10593 i8<o0034O
970
942
28
23
Questions iV» tbe three lafi Cases refihed if tie
Rule of Five.
1. What money, at 34. per cent., wiU clear 38 1. los. in
a jrear and quarter's time f
P- T. G.
100 .1 • 3-5
f.25 . 38.5
'°° K '.,^ 38-5 = 3850* dividend i and 3.5 x 1.25 =
4.375, divifor. "* "" ^
Then 4.375)3850(8801. the anfwer.
2. Put out 384 1, to intereft, and in 8^ yean there we»
54*1. 8 8. found to be due; what rate of intereft could
then be implied ?
542 1. 8 s 384 1. 5= 158 1. 8 s. intereft.
P. . T. G.
384 . 8.25 . 15.4
100 . I .
3i6?'diJifo?° ^ ' ~ '^^'*°* **"***"<* '' «"<• 3*4 X 8.25 =s
3168) 15840 (5 per cent, per annum, the anfwer.
3. Lent
.Ciaf.lL SIMPLE mTERBST. 313
3. Lent 109 guineas. at 4 per cen^ which, by clie igth of
Auguft 1760, was raifed by the iutereft to to many moidores,
bUing a 8. 6 d. -, pray on what da)r4id the bond bear date ?
109 guineas =s 114L 9 s. = 114.459 principal.
I09inoidores =: 147 U 39* which — as. 6d. =: 147 L
-'B* 6d. amount.
Then 14^!. -s. 6d. ~ 114I. 9s. = jal. us. 6d. =
3a.S75. intereft.^ ^ ^
100 .1.4
114.4c' . . . 3a.57s
100 X I X 33t.S7S == 3*57-5» Avidendj and 1 14.45 X 4 =
'457.8.
457-8) 3^57-5 (7- "555 = 7 y«^» ^^ 4^ ^^ays.
July hath 31 days«
TriUAugufti8
Sum 49 **> 4a> gives July the 7th, the anfwcr.
4. If 100 1. in la years be allowed to gain 39 1. 195. 8d.
in what time will any other fum of money double itfelf by
the iame rate of intereft ?
P. T. G.
100 . la • 39*98j
I • -I
39.98j\ laoo /
3998/ 120
35*985} 1080.000 (30.0ia5 = 30 years and 44. days, anfw.
5* In what time will the intereft 0/ 49 1. 3 s« equal the
proceed of 19 1. 6 s. at ufe 47 days, at any rate of intereft f
Reciprocally, 19*3 !• : 47 days : : 49.15 1. : 18.45 days,
die anfwer.'
5. A bond^was made on the 7th of Auguft 1713, at 6 per
cent* per annum, for 1114!. 10 s* ; on the nth of May,
17189 140 1. was paid off, and a frefh bond entered into for .
die remainder,, at 5^ per. cent, per annum ; it the time
die intereft for this laft was 21 1. 16 s. 8 d. there was paid
oflF 87 1. 1 1 s* 9 d. The 6ld bond being then taken up, a
new one was giiren for the refidue, which being paid ofF
September 11, 17249 the bond-owner took no more thaa
314 SlMffLE f KTBItilST. «Q|kll.
14P9L 108. tit in fuU payment. At what nte tt^ did
ke t^}ft ihteieft par cwt p^r anauqit upoa itie^ltft fpitar^
of tjbelKml'
To t|ie' iith of May, 1718, ^ ^ years 277 days =s
4-758904 i*i4»S X ^ BM87
78.66, rear's kitereft xeiwi.
a8c53424
381712
3^3"
318.2279, iAtoi«ft|br4ye«yt a77ilaya»
140 • « • • paid off*
^mrsap^mm^t
1 78.2279, furplus.
■ ■ ' •
1 1 14.5 K 4- 178.2279 sr 1292.7279 1. Mvr prindpd.
P. T. O.
IQO • I • 5.25
U9» • • • ^'-^^
6787.8215) 2183.33354 (.321 7 = 117 days, which an*
fwers to the 5th~of September 1718.
Then 1292.72791. + 21.8^1. = Z3145612I.
— 87.5075 paid off.
- £w6.gfyj9 newprincip.
From September 5, 1718, 'till September 11, 17^4, are
fix years, ux days, z: 6.01644.
And 1409.8^— 1226.9737 1? = 182.8596 If inter^
P. T. G.
1226.9737 • 6.01644 » I89V-850
100 • I •
« • * 7382.0(96) 18285.96 (2.477096 = 21.9$. 6^d. the
anfwer requirocL
S E C T. II.
INSURANCE.
INSURANCE is iiccunty givpn im coii^^er^ti^ fff
fo much per cent, paid in Mand t^ make good ihi9$»
iMrcbattdiaea, houfes* l(c« t» the valu^ ^f t)v>( ^ wM$^
1 the
die pregjjum is r?feiv_e(l, ip c^p pf }^^ by %b^ fljIMf*,
fire,'ortiie.lik$.
This, a$ well as broK«ra^ WjJ cwpiTW^o** I? «M»-
puted in the (ame manner as fimple intereft for a ymrt
■ * ■
Wha^ is thf mfurance gf 737 1, ig s, at j| p«r «;pn; f ,
J. ». Or,
1FJ7 »« lArrST «« -
3f
4wwn*»w*i
as.s ^ 73 15 4i
368 I* 11. a: 4 1.4 IJ «♦
92 49 i = i 7 7 H
£16.74 17 9 .r if 5'
26
14-97 »•
■••■•i
2.92 qrs. Anfwer, 261. 14 s. ii|4.
What \$ the infurance of 874!. 13s. 64l» at 13I percent. ?
1. s. d. Qr,
874 ^3 ^ !• 8. d.
I3i 10 per cent = tV 874 13 6
li^
11370 IS 6 2 = i 87 9 4
437 o 9 I = J 17. 9 loj
I := ^ 8 14 II
/
118.08 23 4 7 St
— ^ n8 t 6|
1.62
12
7«4r
4
|.89 AltfPfr, liSl. »f. 7^d»
Primage
Si6 INSURANCE. Book IL
» .
Primage is an alloHranoe paid to maiinen at their
firft (ailing out of port for their loading the (hip*
Stowage is the money paid for flowing the goods in
a Tefiel.
Avera^ is the quota or prc^rtiont ' which each pro-
prietor ot a (hip, or the goods therein, is adjudmi, on a
reafonable eltimation, to contribute toward the loles which
are fu(buned by hmp of the goods being-caft overboard for
the prefcrvation of the reft, and of the (hip.
What is the infurance of an Eaft-India (hip and cargo,
valued at 35727 K 17 s. 6 d. at 17^^ percent. I
1. s* d* !• 8» d*
35727 17 6 tV 357^7 17 6
i7f
— — 10 per cent. = 3572 15 9
10 times 357278 15 - 5 per cent. = 1786 7 loj
7 times 250095 26 2i per cent. = 893 3 1 17
4. - - 17863 18 9 t per cent. SB 89 6 4t
i - - 8931 29 4i i percent.= 44. 13 2}:
I 6386.35 15 3l
20
7.14 s.
12
1 1| per cent. ^ 6386 15 3!;
1.83 d. Anfwer, 6386 1. 7 s. i^d.
4
3.35 qrs.
yoo^imy^^
SECT. III.
B R O K A G E.
BROKERAGE, or Brokage, Js the fee or reward paid
unto a perfon called a broker, for affifting a mer-
chant or faflor in buying or felling goods, &c. This bu-
fincfs
Chap. IL B E O K A G E. 317
finefs was formerly carried on by broken merchants, or
traders, from whence their name derived; and In Lon-
don they are not to aft without licefice from the lord
mayor*
What is the brokage of (561. &s. 8d. at 6s. per cent«{
L S« . d* j I r'
.^ 8 II 3 > i . L
•f ' 1
• . ^.a a 9k '
1. ■
S*
d.
8.56
6
8
. ao
i.
*
11.26 1
•
12
3.20 qrs.
Anfwer, £^ 11 4f
u^
What b the brokage of 737 1. 138. at 4 s, 9d. pef cent. ?
L s« r 1. s. d*
7.37 13 T 7: 7 H
i 1 9 6
753 ». ^ • i 3 '8i
la I lor .
t «
6.36 d. Atifwer> £i iS ~i
4
MMMMh
1.44 qrs.
What is the brokage of 2572 1. 15 s. at |- per dv^tt I
fc ' I 1
1. 8. - 1...1L- d.
45.72 IS i *S »4 6i
20 I III
i 6 8 7i
i+'SS «• -3 4 3i .
12 • .■
Anfwer, ^9 12 ii^'
64od.
4
2.40 qfS.
( 4
SECT.
• f *
Q »t )
< • <
.« ' •
• . *<
PURCHASING S T O C K &
• • -' H © I^ Irf .
MULTIfLY the fum to be pdrchafed bjrtbeei:.
€|6 iboA 'ioo» and then prpceel as-beterd direfied
ki comwiting tmereft» the produft of which adlfed to the
given fltckf^ ^vAs tfie purchafe; or yon majfind it bjr
praAiid^* if inoit convenient.
What, it x^e imrcbaft ^ 987 K 15 8. Soutk-fiea ftock^
at 113^ peclcent.r
I. 8.
ill I I
0840 M -
Xfp H 9
:i2| 9 4t
■ II ^ '
t
X
«
X
4
X
«'
1. 8« d.
a 9
I 4
,^1124 15 ii|, aafwer.
r •
1.00 8.
iC987 IS
+ 137 . '
jl^ tiai. 16, UteanTw^
a* 769 L India fibck, at 117I per ceiil« ?
St 8. d.
. 7«9 1.
' tS "^ «• » - - - 5
T »I 7 7J: - " - - 2 *. *.
I iS St . - - - - 5 r
- 19 2i - - ,- - - a 6
^ 902 12 3» die anfwcr.
3. What
^ I. s. •
•fc • • ' ^ * ^
III.
» f • • I •
^ - •^ 93f : r,. 9 : : J^ i0]^
V ^ -I iV aa 13 d.
jS 1696. 9 i4t.th»iil^«pr^«(^red.
5/Wniif j^ the piirdiafe of 17271. Bank flo^ at 1^19
ftretnuf
*• \ By PBAcnci,
■iV, 17*7 -, -
M «6 7 -
34 «o 9i
2»«ot. >( to 9«
»
4L s. d.
If t7 - r»
S«« a 7
jC aofs a 7, tht aDTwar.
£ 2055 2 7t as before.
i .
6* Juw
/
^zo FURC HAVING; STOCKS. Bo^D.
6. June the ajd^ 1745, bought 900 1. of new South-Seji
anouitiest at 11 if percent tix. the day before thcJdq^big
the books, the brokerage vriiereof is alway 2 s. 6 d. peroont.
on the capital, whether you buy or fell. The Midfiimmer
dividend, 2 per cent, became due and payable on the loth
of Auguft following ; by which time the re^Uiofi srowing
confiderable in iht north, ,the faid ^nuiti^^ were down at
92-^ per cent. In the general alarm, fold 400 1. capital at
t: at pricey but continual the remainder MI) i fecond, thiid,
f u*th, and fifth diyidend,. as before, caune due: and on
G|,cning the books oh the ioth of *Au£uft IJ47, ibid out at
I02|- per cent. Now reckoning I might Save made five per
cent of my money,, had I kept it out of thp ftocks, how
ftood this article in point of profit and lofs ?
"' • J* a* d«
100 : mf : : 900 : looa 7 6
Brokerage of goal, at 28. 6d« per cent. • 126
•^^ '.J - - • £ looj 10 -
1745,! Mi^Tunimer dividend, at 2 per c^t.* ig .. «
Interr-of 1003 h for 45 days, at 5 per cent, peraim. 6 14 8
Brokerage of 400 Lkt 2 8.6 d. per centt • -^ «^' ib -
Sold 400 1. at 92t per cent, ----- 370 -^ -.
1 / 622 14 8
Intereft for f a year^ due the loth of Feb* 1740 1511 4^
Dividend recdved at that time .•>.io~-
6a8 6 -^
Intereft- due the loth of Auguft, 1746 - - 15 >4 li
644 - 2
Dividend feceived at that time • - - *' lo — -*
634-2
Intereft due the loth of February 1747 ^ 15^17 **
649 17 2
.dividend received then n -.- --- 10 - -
Carried over 639 17 2b
t.1iap.IL REBATE, on PISCO UN T. gaf
Brought over - 639 17 s
Imenft the roth of Auguft, 1747 - - - 15 19 li
■ ■ > -IMI 1
65s 17 r
Midfummer- dividend received Aug. 10, 1747 10 - -•
645 17 1
Sold off 500L at idf per cent. * ^ -^ - , 512 2 6
'^rokerag^ - --•-.-*-. --12 6
To my damage in the whole - * * * ^ 132 2 i
SECT. V.
REBATE, OR DISCOUNT.
REBATE, or Discount, is an abatement of a fum
of money due Ipme time hence, in confideration of the
prompt or prefent payment of the remainder.
The ready money that will £itisfy the debt, is called the
prefent worth ; becaufe if it was put out to intereft at the
given rate per cent, per annum for the time the difcount is
computed, it would ambunt to the given debt.
The true method of finding the difcount of any fum, is
by Cafe II. in fimpie intereft > or,
R U L £,
Firft find the intereft of 100 1. for the time mentioned 1
then, as 100 1. with the intereft is : to the intereft : : fo is
the debt or fum propofed to be difcounted : to the difcount
required ; which, fubtraded from the debt, leaves the pre.*
feat worth, or money to be paid down.
I. What is the difcount of 57 1. 18 s. due 12 months
hence, at five per cent, per annum ?
w. Y 105 ;
522 REBATE, OR DISCOUNT. BookH.
105 : 5 :: 57.9 I. s. d.
5 57 18 -^ debt.
105) 289.5 (2.75238 = 2 15 ^ difcount.
£ SS ^ i^i> prcfent worth.
2. What is the difcount of 573 !• 15 s, due three years
hence, at 4*- per cent, per annum ?
4.5 X 3 = 13-5 = intcreft of 100 1. for three years.
M3»5 • 13s • -373-75
286875
172125
5737S
573 15 -
I < 3*5) 7745*6^5 (^S*H339 = ^8 4 io|, difcount.
505 10 i|, Q.E. F*
-
3. What is the difcount of 725 1. 16 s. fix five months,
at 3{- per cent, per annum ? ,
M. G. M. G.
As 12 : 3*875 : : 5 : 1.61458J
101.614583: : 1.61458^ : : 725.8
725r8
1291^666
80729 1 JJ6
32291^666
1130208^333
ioi.6i458j\ 1171.86458J/
10x61458/ xi;i86458V
91.453185) 1054.678125 (11.53^35 =^ i^ '• i^ s. jl d.
140146275
48693090
2966497
222901
39995
3259
510
Chap. IL REBATE, qr DISCOUNT, 323
Or to find the prcfent money, obferve the following'
RULE.
Firft find the intereft of 100 1. for the time mentioned
as before ; then fay. As 100 1, with the intereft added : is to
lool. : : fo is the debt or fum propofed to be difcounted :
to the prefent money.
4. What ready money will difcharge a debt of 543 1. 7 s*
doe four months and 18 days hence, at.4-|. per cent, per
annum?
4 months =y .^33333
18 days - = .049315
Time t=z .382648 X 4*62$ =: 1.769748, int. of lOol.
As 101.769748: 100:: 543.35:533.9012 = 5331. 18s. -id.
1. s. d.
Debt 543 7 •
Prefent worth 533 18 -^, the anfwer.
Dtfcount ' £ 9 ^ ^ti
5. What ready money will difcharge a debt of 1377 1.
13 s. 4d. due two years, three quarters, 25 days hence, dif-
xount 4j- per cent, per annum ?
2j years = 2.7c
25 days = .068493
2.818493x4.375 =£12.330907 = int. of lOOl.
112.330907 : 100 : : 1377./^ : 1226.4359858.
Anfwer, 1226.4359858 c= 1226 1. 8 s. 8^^. prefent money,
6. What difference is there between the intereft of 500 1.
at five per cent, per annum, for 12 years, and thedifcoimt
of the fame fum, at the fame rate, for the fame time ?
P. T. G.
100 . I .' 5
500 . 12
5 X 500 X i^ = 30000, dividend.
iCo X I = 100) 30000(^00, the intereft.
Then 12 X 5 = 60, the intereft of 100 1. for 12 years.
And 100 -f- 60 = 160, its amount.
As 160 : 60 :: 500 : 187 1. 10 s. the difcount.
*•* 300 1. «^ 1871. 10 s. ss 112I. los. advantage to the
jjitcreft.
Y 2 SECT.
[ 324 ]
S E C T. VI.
EQUATION OF PAYMENTS. •
WHEN finreral debts are payable at different times,
and it is mutually agreed between debtor and credi-
tor^ that all thofe fereral turns be paid at fuch a time, that
neither debtor nor creditor may be wronged- thereby, this
is called the equated time of payment* The rule given by
Mr. Cocker and others for finding this equated time is,
RULE,
Multiply each payment by its time, and divide the Turn
of all theie produos by the whole debt ; the quotient was
by him accounted the equated time.
A perfon dying, bequeaths to a youneer fon loool. to be
paid as follows ; viz. 300 1. at one years end $ 300 L more
at a year and a half; and the remainder at the end of two
years and a half. Now the executor agrees with the legatee,
to pay the whole at one payment ; how long from the death
of the father muft this payment be, fo that neither party
be wronged, or fuffer lofs ?
By the foregoing rule 300 X i = 300
300 X li = 450
400 X 2^^ = 1000 -
1000) 1750 (.75, or i| vear,
the aniwer.
Mr/Kerfej finds fiiult with the fi>regoing method, a9
no intereft is thereby implied, and thinks that a diicoune
(ftatute intereft) fliould be allowed for each pavment ; «nd
to find the. equated time, gives a rule to the following pur-
port.
RULE.
Find the prefent worth of each pavment, according to
its refpe^live time and rate ; then add all the prefent worths
together, and call their fum the principal j laftty, having the
principal, amount, and rate of intereft, find die time by
Cafe IV. of fimple intereft.
Now allowing a difcount of five per cent, the fiidution of
\ the foregoing queftion will be as fpllows ;
I vi2.
Oiajp. n. EQUATION of PAYMENTS. 325
viz. 105 : 100 :: 300 : 285.7142
107.5 2 iQO : : 300 : 279.0697
1 12.5 : 100 :: 300 : 355S5SS
920.3394
5 per cent.
. j^ 46.0 1697, inc. for a year.
1000) amount.
9^o-3394> principal.
46.01697 : 1 :: 79.6606 : 1*7311 = i year, -8 months^
23 days, the equatea time.
Sut the learned Mr. Alexander Malcolm juftly obferves,
that though the debtor gains the intereft of what he keeps
after it was due, that he lofes onlv the difcbvint of what he
pays before it was due, which is lefs than the intereft ; and
that theKfore the creditor n>ay juftly except againft Mr.
Cocker's method ; and I apprehend, that for the fame
reafon the debtor may have as juft an exception againft
Mr. Kerfe/s.
The beforeHnentioned Mr. Msdcohn, frooi an algebraic
way of reafoning founded on the principles of Ample in*
terdSt, njfes ana demonftmtes a theorem, from whence it
deduced the following
RULE.
Find onejrear's intereft of the debt that is (irft payable,
by which divide the fum of the debts (of the firft and fe-
cond payments) and to the quotient add the fum^of thei
times ; call this the firft number found.
Then multiply each debt by its time, and divide the fum
of the produ£ts by one year s intereft of the firft payable
debt i which quotient added to the produ<Sl of the two
times, call the ftcond number found.
Subtrad the fecond number from ^ of the fquare of thd
firft number^ and out of the difference extra^ the fquare
root ; which root being added to, or fubtrafted from half
the firf^ number found, the fum or difference will be the
time fought.
N. B. As diis rule is ambiguous, if you take the fum, if
that happens to be greater than the time to the term of the
laft payable debt> the difference will be the time fought. Or
Y 3 if
326 EQUATION of PAYMENTS. BooklJ.
if you take the diiference, and that be lefs than the time
to the term of the firft payable debt, tKe fum will be the
time fought. But if both the fum and difference happen
between the two given terms, it muft be examined which of
them -will make ah equality of intereft and difcount.
I fhall here reaiTume the foregoing queftion, allowing the
intereft and dividend at five per cent, per annum i
D bt ^ 300, Its intereft for a year 15 1. 300 X r =: 30Q
C 300 300 X 1.5 = 450
15)600 15)750
+ 2.5, fum of the times. + 1.5
42.5, firft N* found. Second N» 51.5
Then 42.5 X 42-5 = 1806.25, which -=- 4 = 451.5625.
Alfo 451.5625 —^ $''5 = 400.0625.
^ 400.0625 = 20.0015624 ; and 2) 42.5 (21.25.
• . ' 21.25 *— 20.0015624 = 1.248437 years, the equated
time for the two firft payments.
Then to find the true equated tinie when the whole 1000 L
muft be paid together.
Put 660I. for the firft payment, intereft for a ye^ir 30 1.
400 1. fecond payment.
3^)«ooo »'H8437ltimes.
— 2.5 $
33.333333 ■
3.748437 3-748437» ^^'^ ^^^
» - • ■ .
?) 37*08x77, firft number found.
18.54088, its half.
Alfo 600 1. X 1.2484376 = 749.06256
400 X 2.5 = 1000
30) 1749.06256
58.302085
1.2484376 X 2.5 = 3.121094
Second number found =: 61.423179
And
Chap- 11. EQUATION of PAYMENTS. 327
And 37.08177 X 37-08177 = 1375.057666
4) « 375-057666 (343-764416
343-7644'6 — 61.423179 z= 282-341237
-/282.341237 == 16.80254
• . • 18.54088 — 16.80284 = 1.73804 = I year, ^ mon.
26 days, the true equated time required.
2. At Michaelmas, feventeen hundred nineteen.
My writings will (hew (which are yet to be feen)
That to me were three hundred and twenty pounds due.
And half of that fum, beiides forty-two, viz. 7,0% L
Juft five years after, I then might demand.
But would fain have the whole fomewhat fooner (in
I agree to rebate for the latter fum too, hand.)
The fame rate (fimple) intereft our ftatutes allow.
But then I exped fome ufe will accrue
From my fixteen-fcore pounds, that laft year were due.
Now to know on what day, I fliould be very fond.
To receive my five hundred and twenty*two pound.
Ladies DioTf^ 1720.
Y^, t 320 1. its Intereft for one year is 16 1.
\ 202
16) 522 (32.625 + 5 tlm^ = 37.625, the ift number.
The 320 being accordingto thefecurities to be paid down*
So 202 X 5 = loio
And 16) loio (63.125, the fecond number.
Then 37.625 X 37.625 =s 1415.640625
Alfo 4) 14J 5.040625
353.91015625
-— 63.125 18.8125, half the firft number.
V 295-78515625 = i7'Q5H ,
1.7601 = I year, 277 days,
which anfwera to July 4, 1721. Qi.E. F.
As. there are only a few days difference between this and
the other method, and that this method will be operofc, par-
ticularly when the payments are to be made at many diffe-
rent times, either of the former methods may do with-
out any confiderable wrong to either party ; yet, in my
opimon> truth is worth enquiring afte^.
* - . Y 4 CHAP.
C 3*8 ]
CHAPTER III.
r
, TARE AND TRET.
•
TARE is an allowance in merchandize made by the
king to the importer, or to the buVer by the mer-
chant, for the. weight of the bag> calk, cheft* wrappers, &c.
in which any goods a^ put j feveral forts of goods hate their
tares afcertained in a table annexed to the book Of rates.
Grofs weight is the whole Weight of goods, With ths
cheft, caft, bag, &c. that contains them.
Tret . is an allowance, in wti^able goods, of 4 lb. in
104 lb. made by merchants in Lbndon to their trad^finen
and retailers for break, waftfe or duft, yet himfelf U onlv
allowed tare in paying cUftom; fo that ke payetfa as wefl
for the bad as the bcft commodity.
ClofF, doughy or draught, is a fmall allowance m^de by
the king to the importer, or by the feller to the buyer,
to caufe the weight to hold out when goods are weighed
again. The king allows ilb. draught for goods under
I cwt. ; 2 lb. from * to 2 cwt. j 3 lb. from 2 to 3 cwt.
4lb. from 3 to 10 cwt. j 7 lb. from 10 to 18 cwt. 5 and
9 lb. from 1 1 to 30 cwt. and upwards.
Subtile, or futtlc, is the weight of the goods when the
tare is deduded, but not the tri&t.
Net weight IS the remainder, wheh both, if both be
allowed, are taken away.
Aliquot Parts.
lb. lb.
8==ttoffcwt. 3i = iJ
7 = il
1.
CASE I.
When the given tare is the aliquot part of an hundred,
as 14 or 16 lb.
R U L E,
Divide the given weight by the denominator of the frac-
tion reprefcnting the part, the quotient will be the tare.
I. What
attp.Zil. TARE. AKD TRET. jat^
I. Wlut is the net weight of four barrels of figs t
Cwt« <)rs. lb.
vi*. N* I - - - 3 « ^« ^
tare 141b. percwt^
I - - - 3 « i»-|
2 4 324 I
3 * - - 6 *- 20 1
4---S a 26 J
8)20 2 4» grofs.
2 2 4f tare.
Cwt. 17 3 -^ net.
2. What Is the net weight of five bags of cinnamon t
Cwt. qrs. lb.
vii. N* X * - - I 3 24 'J
2---3 I 181
3 2 2 i6^tare 16 lb. per cwt*
4-^-22 2^ I
5 . - • I 3 22J
7) 12 3 22, grofs.
I 3 ii» tare.
Cwt. II - II, net*.
CASE 11.
When the tare is not an ali<{uot pan of an hundred, but
the aliquot part of t or ^ of a cwt
RULE,
Take firft the aliquot part of aft hnndfed, and then part
of that part, a^eeaUe to the nature of the queftion, until
jou have found the true tare.
Another way of finding the tart wlraa' it is not an
aliquot part of 112 lb.
R IfL E,
. Multidy the hundreds by the tare, to be.allowed for x cwt*
and for Ae quarters and pounds, in the grbfs weight, take a
propor-
S30. TARE A^D TRET. Book IF;
proportional part of the faid allowance, and the fum is the
tare in pounds ; which is either to be reduced into hundreds,
and deduded from the gfOfs weigbt^^orthegrofs weight into
pounds, and dien deduct the tare. . ; .
What is the net weight of igcwt. aqrs. 12 lb. grofs,
tare 7 lb. per cwt. ? . .
Cwt. qrs. lb. . Otherways.
4
19 2 12; grofs. 19 X 7 =^ 1331b.
4 3 ^7 •
i - 25^, tare.
I
4 —
Cwt. 18 I I4|, net.
lb. 137^ = I cwt. 25ilb.
tare, asoytheothermethod.
r
2. What is the net weight of ,15 qwt. 2 qrs. 21 lb;
tare 8 lb. per cwt. ? , .
Cwt. qrs. lb.
15 2 21 Or, 15 X 8 = 120
4:X8= 4
lb. 14 = I
7= -4
7
2
2-27
I — I3tj tare.
14 2 74, net.
Tare 125^= icwt. ijJJb.
as before.
3. What is the net weight of four bags of hops, tare
4 lb. per cwt. ?
Cwt. qrs. lb. Otherways thus,
viz. No I - - 4 I 18 17 X 4 = 68
2--33 24 4:X4=2
3-. 4 2 16 iX4=i
4 4 3 4
4
7
17 3 6, grofs.
4 I ^i
2 15, tare.
17 - 19, net.
lb. 7 1 cr 2 qrs. 15 lb;
tare, as before
CASE
Chajp^III. TARE Avtk TRET,
33'
C A S E m.
When the tare is no aliquot part of an hundred weigbf,
quarter, &c«
RULE,
Divide the given tare into aliquot parts of an hundred,
quarter, &c. the fum of which will be the anfwer.
I. What is the net weight of I9cwt. 3qrs. 141b, of
antimony, tare 6 lb. per cwt. ?
Cwt. qrs. lb. Or thus*
7 19 3 H
2
2
I
3 10
I 19
4
I I
I
- 7t» t^^
18
3 Hy n«t'
19 X
ix
lb.
6 =
6s=
6c
14 =
"4
S
«
7ilb.
"9i
tare,
= I cwt*
as before.
2. What is die net weight of 71 cwt. 3 qrs. 21 lb* of
pot-afhes, tare ip lb. per cwt. i
Or thus.
71 X zo c= 710
^Xio= 5
i X 10 z= 2i
Cwt. qrs. lb.
71 3 ^J
7
4
10 I 3
2 2 7^
12 3 I0|
6 I 1979 tare.
Cwt. 65 2 j|, net.
14 =
T
Tare 719^ = 6cv^
I qr. 19^ lb. as before.
^.What
83*
TARE. AOTD TRET. Book n.
3. What IS the net weight of £ve cafks of alum, tare
12 lb. per cwt. i
Cwt. qr. lb.
riz, N* I - - 2 X 27 ^
2 - - I 3 25
3-- 3 - la
4 - - 2 2 21
5 - - 3 I IS
13 2 22
6 3 ir
- 3 »St
- J Jt6|
Tare i i 24)^
Net I ft - tsi
Otherwayi.
, I3 X 12 == ij6
I X 12 = 6
lb.. 14 = 14^
lb. 8= i
Tare 1647, as before s
xcwt. iqr, 2441b.
4. What is the net weight of five caiks of oil, weighing
9» foQowSy tare 18 lb. per cwt. ?
OtherWajrt.
21 X 18 se 378
i of 18 = 9
|ofi8= 44
* ^ofi8= 2t
Tare 394=r3cwt. aqrs.
2 lb. as before»
Cwt.
i{r».Hi.
N» I - - 3
3 »9
2 - - 4
I 25-
3- -3
it 2i
4- r S
2 18
5 -
^ 4
7
21
3 16
8
3
-Ht
~
I »sl
Tar
Ket
63
2 a
18
I »4
5, What
i
Chap. III. TARE and TRBTk 33I
5. What is the net weight of 27 cwt. i qr. 21 lb. of
prunes in caiks, tare 20 lb. per cwt. i
Cwt. qrs. lb.
7
4
27
I 21
3
3 «9
3*54
4 3 I
tare.
Cwt. 22 2 4*^9 net.
Otherwajrs*
27 X 20 = 540
iof20 3i 5
iof2o=c 2t
lE70f20s' If
i6|1d. at before*
6. What is the net weisht cf feven fats of hogs hriftles^
each containing 3 qrs. 19 lb. tare 17 lb. per cwt. ?
Otherways.
6 X 17 =S 102
i X 17 s 4f
X 17= 2
Cwt. qrs. lb.
- 3 »9
7
2
6 I 21
h 3 H
- -I2J
- - H
lb. 7= z
——' qrs. lb.
Tare 109J = 3 25*
E
3 25t, tare.
Cwt. s I 23I
In many commodities the allowance for tare is not reck-
oned by die hundred weight, but fo much of the grofs;
this is called invoice tare.
CASE IV.
When the tare of raw filk from Smyrna or Cyprus is to
be deduced,
RULE,
For 3 cwt. and upwards allow i61b. tare; from 3 cwt.
down to 2 cwt* 14 lb. tare; and from a.cwt. dowawasds,
I a lb. tare.
Likewife in Virginia tobacco :
Tor all hogfiieads under 3 cwt. allow 70 lb. tare ; from
3 to 4 cwt. 8olb, ; from 4 to 5 cwt. 9olb. ; and, from
5 cwt. upward, 100 lb. tare.
I. What
334
TARE AKD TRET. Bookll.
I. What is the tare of eight hogflieads of tobacco )
Cwt« qrs. lb. , qrs. lb.
viz. N* 1 - - 2 z i8-^ f- 2 14
a - - 3 I 21
3--4 a 8
4 - - 3 3 "
6 - - 4 ^19
tare
I _
- - 5 2 27
v.:
5 3
IJ
- 2 24
- 3 6
.- 2 24
1- 316
1-36
I- 3 16
U. 3 16
Grofs 36 2 "5 Tare 6 i 10
tmm
I
Net cwt. 30 - 23
X. What is the tare of fix bales of raw filk I
lb. •
viz. N** I -» - 325^ ri6
2 - - 185 J 12
3- - ^74 V tare J ^+
4--377[''' ]i6
S - - 129J I i^
6 - - 215J
14
Grofs 1505
Tare 84
Net 142 1
C A S E V.
When allowance is required for tare and tret,
#
R U L E,
Find what is to be allowed for tare, according to the
foregoing rules -, which having dedufted, the remainder is
futtle, which divide by -*-^ =: 26, and the quotient is what
4
is to be allowed for tret, which dedu6l from the futtle, and
that remainder is net.
X. Wha«
Chap. 01. TARE and TRET.
'33$
X. What IS ihi net weight of a 4punch^on of prunes,
grofs " 13 cwt. I qr. 21 Ibu tare 14 lb. , per cept. tret 4 lb.
in 104 ?
• Cwt. qrs.lb; . ^
8fi3 I 2t
104 -^ 4
26
I 2 209 tare.
XI 3 I, futtless 1317 lb.
I 22, tret =50 lb.
II I 7, net = 1267 lb*
2. A merchant buys fix hogiheads of tobacco, each con*
taining 9 cwt. i qr. 141b. grofs;. tare i cwt. - qr. 18 lb.
per bo^ead; tret 4 lb. in 104; and clofF 31b. in every
336 grofs ; what wilf the net weight come to, at 6^ d. per
pound?
Cwt. qr. lb.
9 > 14 X 6
I - 18 X 6
Cwt. qrs.lb. ]b.
= 56 I -, groli s 6300
=s 6 3 24, tare 5s 780
3 '
f
49 I 4t f«ttlc
= SS^o
.
I 3 16, tret
= 212
47 I 16
= 5308
112)6300(3
■
=56H>
.- 2 -, doff
= S6
•
46 3 16, net
= 5*5*
40
5252
)
12
131 6 -
10 z8 10
»
■
•
142 4 10, chcanfwer.
Mr.
336 TARE ahD TRET. Book It
Londoiiy March io> 1758.
Mr* Jambs Dektoh,
Bought of John S ands, fix calks of Barbadocs fugar.
Cwt. qrs. lb. qrs. lb.
N* I - weight 8-16 Tare 3 7 each.
2 - - - 7 3 20 x6
3 - - - 8 I 16 —
^ . . - 8-12 Cwt. 4 3 «4
?- - - 8 2 21 - ' ■■
- - - 8 323
Grofs 50-24
Tare 4 3 14
Net 45 t 109 at2l. 7s. 6d. pertwt. - 107L
— -^ — - — 13s. 7d.
Computation*
Cwt. qr.lb. 1. s. d.
4S I 10, 4t ^ 7 ^
1. s. d.
% 7 6
5
II 17 6
9
106 17 6 1 f 49 cwt.
4. II lOif 1 4 cwt.
I 8*>price6» 41b.
i I $i\ 1 4lh.
*. 10 J L 7, lb.
iC^Q7'3 7
N. B. Below is the computation of the bill of parcels on
the next page.
lb.
8. d.
20
6
25199 at I A 2
lb. d.
3 1621, at 4
125 19
20 xp 10
20
£ 146 18 10
540 4
27-4
i
Sir.
Chap. III. TARE and tRET. $3;
Mr. P.TEK Mason. '''"'°'" ""P"' 3> 1758.
Bough^ of Henry Eustace Johnson, Elqi for rea^y
money, cotton 13 bags.
Cm.qrs.lb.- Cwt.qrs.Ib.
viz. N» I . 3 I 7 N* e - 2 3 16
Z-23-. .9-3-- 27
3 - ^ 3 S i^ - ^ 3 4
4 ^ 3 -• ' J II - 3 I lo
ti 3 ^1 Cwt. 12 I I
12 I f '
24 I <-> tetfti firofs.
-•— 3 13,- tare aUowefl.
^3 I i5» futtlc =s 2619 lb..
— — . tret - 100
net - 2519 lb. 1. su ' d.
at IS, ad. per lb. - - • - 1^6 18 lo
More, viz. Cwt. qrs. lb.
N* 17 - - 2 2 7
18 - - 3 2 8
ig - - 3 i 26 ^damaged.
20 - - 3 - 10
21 - - 2 3 12
Grofs 15 2 7
Tare — ±2
15-5 fifttle a= 1685
'* ti-et = 64
net != 1621 lb.
jat 4 a.* {)cf lb. --■•--27-4
£ 173 19' a
• 1
1 *
Sir
1
338 TARE AND TRET. Booklfi
March 24, 1758*
Sir Andrew Buckwoeth, and Company,
Bdught of the United Eaft-India company, at 4 months.
Pepper, 2 lots, viz.
Cwt^^qrs.Ib. lb..
N* IS - 10 bags - wt. 27 3 18 Tare 150
19 • 10 ditto- V - 24 I H - - 138
—— — -— cwt.qrs.lb.
Grofs 52 t 14 Tare 288 = 2 2 8
Tare 228 — ^
Net 49 3 6=SS78">* _ ,
-■ ■ I L s« d»
atiof d. per lb. • • . - 241 2 7I
Redwood, 2 lots, viz.
Ton.cwt.
N* 42 - 120 fticks - wt. 10 13J.
43 - 100 ditto II J2
220 fticks - wt. tA St
at 3I. 75. per ton. - - - - 74 13 3
Wonnfeed, 3 bales, viz.
Cwt. qr. lb.
K* 16 - - - - wt. 3 I 19
19 42-
27----- 23 10
Grofs 10 3 I
Tare 1-15
Net 9 2 14 = 1078 lb.
at IS. i|d« perlb. - - - - 60 la 9
iC376 8 7i
5578 lb.
C;ha{>.m. TARE AKD. TRET;
H>. d, cwt. qrs. lb.
339
1
I
T
1
I
T
5578, at lof . Or, 49 3 6, at loj d. per lb,
^7*9 4
1394 6
464 10
116 U
58 li
3 5i
7
^■i""«a«i
4822 7i
t 4 2|^ per qu
4
4 16 10 percwt«
— * ■ ■• 33 17 10
21
237 4 10
3 " 7i
. 5 2f
As Wore, £ 241' 2 7^
32tun5|jCwt at 3 7
. 7
23 9 -
3
70 7 -
tun. cwt»
3 7-/ \ ' -
- w 9>priceof'{ - 5
- I SI ) - j
jC 74 13 3 2a 5i
Cwt* ({ts. lb* s* d«
^ 2 14, at I if per pound.
7
7 lojr, price of 7 lb.
8
3 3-
'2
6.6 -, per cwt.
9
56 14. -
3 3-
'5 9
j^6o 12 9
Z 2
The
340 TARE AND TRET. looklL
The net proceeds of a hogfliecu} of Baibadoefe fugur were
4!. 14 s. 6(k.; the cuflom and fees 2I. 8 s. 6d. ; freight
22S. 8d.; fa^orage 4s. 6d. ; the grofs weight wa$ gcwt.
94 lb. ; tare ^ : pray bow was the fugar fatei in Ae bill
of parcels ?
1, s. • d.
Net proceeds 4 14 6 ;
Cuftom, &c. 286
Freight - - I 2 8
Faaora|e - - 4 6
8 10 2 = 8.5083-1.
Cwt. qrs. lb. cwt
9 3 10 d: 9-839a8;S77
Tare .98395»«S7
Net 8.8553572 •
88553572) 8.50833333 (.960812 =: 19s. 2^d« the anfwen
53851 185 .
719042
10613 • . -
- '75».
I have imported 80 jars of L«Kea x)il, each containing
1 180 folid inches ; what came the- freight to, at 4 s. 6al
per cwt. tare i in 10, counting J^lb^ of oil to the wine
' gallon of 231 cubic inches?
1 180 X 80 5^ 94400 inches.
231) 94400 (408.658 gallons.
408.658 X 7-5 ^ i064«93S pounds.
10—1 = 9) 30^4.935 grofs.
340-548
1 12 J 3405.483 (30.406 cwt*
4 s. 6d« =3: .2^ 1.
30.406 X .225 = 6.84135 ;?. 6t, 18 s. 9^d. the anfwer.
G H A P,
[ 341 3
«
CHAPTER IV.
' a
F E L L O W SHIP.
TH £ rule of fdlowlhip is that by which the accompts
of feveral partners trading in company are adjufted,
made up, or divided ; fq that every partner may have his
luft part of the gain (or lofs) In proportion to the money he
liatb in the 'joint ft9ck, and (o the time of its continuance
thereia*
S E C T. I. '
SINGLE FELOWSHIP.
BY iingle fellowihip is adjvtfted the accompts of fuch
partners that put al} their feveral, and, perhaps, diife-
jrexit £uins of money into one oommon ftock at the fame
time ; ^d therefore jLt is ufually called the rule of fellowihip
mthout time^
RULE.
As the whole ftock : is to the whole gain or lofs : : fo is
^ery man's particular part of that ftock : to his particular
^are of the gain Qf lofs.
I. Three inerchants, A, B, C, enter upon a joint adven-
ture; A puts into the common ftock 175 1* 13 s. 4d. ; B
Z17I. 16 s. 8d. ; and C 981. 17 s. 7d. : with this ftock
they trade and gain 2641. i I demand each merchant's fli^re
cf the gain i
A's ftock 175 1. 138. 4d. = 175.6666^
B,^8 - - 117 1. j68. 8d. = ii7.833j3f
^^s - - 98 1. 178. 7d, = 98.8791/?
392.3791^
392«S79i6I.: l64t ::
f 175.6666^ : 118.191802= ii8 3 10 =A'sl
i "7-8333^ •• 79-^80459= 79 5 7t = B;«fg^a-
4 98.8791^ : 66.527743 ?= 66 10 6|: = C s i
£ 264. - - -^ = whole gain.
Z 3 But
34» SINGLE FELLOWSHIP. Bpokll.
But aueftiofii of the (gme kjiid firith the; foregoing, and
thofe relating to bankruptcies, the readieft way of folutioii
will be, by dividing the whol^ gain by the whole ^ock, or
the bankrupt's whole eftate by the fum of his debts ; the
quotient will be a conunon multiplier, or fo v^^ci^ ^ pouqd
as die bankrupt's eftate will pay*
^. A merchant |}reaking ow^ hif creditoj;; ^ (oUf^ws :
I* S« a* !•
riz. To Mr. Truft - 372c 17 3 = 3725.862^
Mr. Credit 1 796) 1+ « 5= 79677375
Mr. Gnpe - 5674 12 6 = 5674.625 *
Mr. Covet - 967 10 4* ^ 967.51875
Mr. Squ^ze 734 6 2^ = 734- 3^9375
Mr. Hard - 873 |8 6 = 873.925
Mr. Near - 382 14 3 = 382.7125
Mr. Dunn - 125 416 7! = 125^83125
Mr. DiiEdent 637 18 6|. = 637.928125
* « • • • ^ «
In all, j^ 21090 9 - =21090.45
His whole eftate is i750ol« what is each creditor's part
of th&t in proportion to his debt i .
whole debt, whole eftate. 9« d*
21090.45 : 17500 : : I : •829759441 =s i6 7^ per lb.
1. 8. d.
3725.8625
7967-7375 ■
5674.625
^67.51875
734-309375
873.925 •
382.7125
125.83125
637.928125^
•1091.569586=309? M ♦IT.
0611.305415=6611 6 liC.
4708.573669=4708 II 5jQ-
802.807817= 802 10 i|C.
k82975944I-( 609.300137= 609 6 - Sq.
= ' 745.1)1.7520= 725 2 IlfH.
317-559351== 3^7 V ajN.
104.409669= 104 5 2^1),
I 529-3^6885= 529 6 6tD:
1 7500.001802 P= 1750? - -
In cafes of bankryptcy, when there are many creditors,
firft find what the bankrupt'^ eftate will pay in the pound ;
and then each particular' part may be found by the rule 6f
praaice, very near the truth : and here note, that the fmall
Redundancy in th* larger fums, in this queftion, is owing to
x6s. 7|d. being taken a fmall tnatter more than the bank-*
Chap. IV. SINGLE FELLOWSHIP. 343
rupees eftate would allow 1 and the deficiency in the finallef
ones to the fra&ion of a farthing omitted.
Truft.
1. s.
d.
f
T
37*5 17
3
1
1862 18
74
1
T
931 9
3i
1
180 . 5
lOt
1
93 2
II
1
T
15 10
51
2 4
4t
I
3091 II
64
Covet.
1. a.
d.
1
967 10
4i
t
483 »5
2t
I
T
241 ^7
7
t
T
48 7
6
I
24 3
9
fl
T
4 -
Z»
- 11
6
I
802 16
»?
Near.
1. 8«
d.
a
382 14
3
191 7
li
J.
5
95 13
^^
X
19 2
8t
9 "
4t
X
7
I II
lot
- 4
6i
iC
317 11
2
Credit.
X
ft
I. s.
7967 14
d.
9
X
ft
X
5
X
ft
i
X
7
3983 »7
1991 18
398 7
'99 3
33 3
4 H
4i
8i
8.
lot
"i
10
;C
661 1 6
5i
Squeeze.
Xi
2
1. t.
734 6
d.
2i
X
ft
X
i
X
ft
X
6
X
7
367 3
183 II
18 ^
3 «
- 8
I
6i
3i
8|
;£
609 5
"1
Dunn.
X
ft
1. s.
125 16
d.
74
X
ft
X
s
X
ft
X
6
X
7
62 18
31 9
6 5
3 2
- 10
I
34
94
10^
54
54
/:
104 8
14
Gripe.
1. S.
d.
X
ft
5674 12
6
X
ft
2837 6
1418 13
3
X
s
li
X
ft
283 14
74
X
6
141 17
34
X
7
23 12
laj
3 7
64
;C
4708 If
84
Hard.
1. 8.
d.
X
ft
87318
6
X
ft
436 19
3
X
s
218 9
7i
X
ft
43 13
II
X
21 16
I It
X
7
3 *2
94
;C
- 10
44
72s 2
"t
Diffident.
1. s.
d.
X
ft
637 18
6J
X
a
318 19
3t
X
5
»59 9
7i
X
ft
y '7
II
X
6
15 18
"i
X
7
2 13
*4
- 7
7
,c
529 6
,6
As there is no general rule for folving all queftion$ that
may oc^ur or be propofed in partnerfhip, or other bi-snches
Z 4 of
344 SINGLE FELLOWSHIP. B<x*II.
of trade^ the aofwer muft be found by the ingenuity of the
arithmetician ; whfls by this time, may he iuppoied to be
pretty well grounded in figures.
I A hath in ftock 35 1. B 20 1.; they trade and gain
and agree that it (hall be divided fo, that A is to have
10 per cent, and B only 8 y what muil each Ivtve of the gain ?
• 35 1 ^t 10 per cent. \s 3. c U ^ j ^ ,
20 1, at 8 per cent, is 1.6 1. $ '««**:>•*
]. s. d.
•..5.1:40 :: 3.^:27.45098 = 27 9 Ht=A;s?« '
5.1 140 :: 1.6 : 12.54902 = 12 10 |i^ = Fs J ^^
4. A, B and C put in money together ; A puts in 20 1. $
3 and C piit in. 85 1. ; they gained 63 1. of which B to<^
up 21 1. ; what did A and C gaiii) and B sind C put in.^
?os : 63
'' """"'Ar' .. ^ ^MgainO Which
= 33,and63— 33=3^, CsJ^ * t^ere ^
And63 : 105 : : 21 : ^^ B^s J^ ^ J^ ^ J
63 : 105 : : 30 : 50, Cs J ?
5. Some others advance in trade as follow ; vix. W, X
and Y raifed 350 1. 10 s. -, W, X sind Z 344. 1* 10 s ; X,
y and Z macte up together 400 1.; ancM^^ Y and Z
(contributed 378 1. 4 s. in the conclufion they parted with
their joint property for 450 guineas ; what did ^^y gain #f
lofe by their adventure i
W, X, Y 350 10 -,
W, X, Z 344 10 f each partner being mentioned three
X, Y, Z 400 - \ timeji.
W, Y, Z378 4J
3)1473 4 ^
d.
49 1 I 4, the wbglp ftocif .
20) 450 guineas. .
22 10
472 10 m^de pf their joint property.
Then 491 1, IS. 4 d. -rf472l. fos, s f 8 1. II s. 4d. lofe^
the anfwer, 6. A,
Ckip.IV. SIUOLE FELLOWSHIP. 345
I
6i ▲, B and Cptt in tmrie 360I. aii4 gained 'a70rl. ;
ft wkick as oftea is, A took up 3 1. 9 to6k up 5 1. 1 and as
pflcn as B tool^ up 5 K C took up 7 1. ; what did each gain
gnd put in ?
3
+ 5
+ 7
25) 360 (249 the G^wnon multiplier for the %ck« ' .
3 X 24 = 72 = A's ^
5 X 24 ss lao s: B's > ftock.
7 X 24 s= 168 = O^sJ
l|) 270 ( 189 the common mukiplier for the gain,
and 3 X 18 =* 54 = A'si
5 X 18 = 90 = B*s 5 gain,
7 X 18 = 1265= Cs)
7« A, B-and C put in money together | A puts in 20 1. ;
B 30I. ; C a fum unknown : tnej gained 36 1, whereof C
took |6 1. ; What did A and B gain, and C put in i
20 36
+ 30—16
50 : 20 : : 10 : 8, A's gain.
20 «~ 8 = 12, B*s gain*
. 8 : 20 : : 16 : 40, C's ftpck. •
8. A, B, C and D put in money together, and gained
fi fum of money, of which A, B and C took 60 1. ; B, C
and D took 9ol.s A, C and D took Sol.; .and A, B
fuid D took up 70 1. i wh^t diftin^ gain did each take up i
I. '
A, B, C 60
5,C,D 90
A, C, D 80
A, B, D 7d
3)300
the anfwer.
54^ SINGLE FELLOWSHIP. BookIL
g^ A and B clear by aii adventiire at fea 50 guineas, with
which they agreed to buy a horfe and chaifn wherectf they
l¥ere to have the ufe, in proportion to the fiuns adventured,
which was found to be, A 10 : toB 7 ; they cleared 45 L
per cent, what money did they each fend abroad ?
50 guineas = 52 1. 10 s. ; and 10 + 7 = ^7*
45 : 100 : : 52.5 : 116.^ =: ^116 1. 13 s. 4 d.whole fimu
17: 10 : : 116.^ : 68.6274 = 68 1. 12 s. 64d.A's ?n^
17 ; J III ii>4 I 48.0392 = 4« 1. - s. 94: d. B's J ^^^^•
10. A father divided his fortune amoneft his fons, giving
A 7, as often as B 4; to C he gave as often 2, as to B 5 ;
and yet the dividend of C came to 2166}^ 1. : what was the
value of the whole legacy I
C. ]. s. d. B« I. s.
2 : 2166 7 6 :: 5 : 5415 18
B. L s. d. A.
4 ; 5415 18 9 ,• 7 • 9477 '7
. and, as above, 2166 7
17060 4 -J-, the anfwer.
11. Part 1500 acres of land-, give B 72 more than A,
and C 1 12 more than B.
fum 256
Then 1500 — 256 = 1244
3) 1244(4144 = A> 7
Alfo 4»4t + 72 = 486^ = B's > (hare.
And 486* 4- i|2 = 5981 = Cs3
12. Divide 1000 frowns j give A 129 more than B, and
B 178 'fewer than C. *
looa
129 + 178 = 307
3)693( 23ir=B's.
231 4- 12Q = 360 = A*S.
?3' + 178 = 409 = C's.
^3. Part
Chap. IV. SINGLE FELLOWSHIP. 347
13. Part 250 1. give A 37 more than B, and fct 0 have
a8 fewer than B.
Firft, 37 — is = 9 - • - Alfo 350 -^ 9 *s 24^.
. 3)241(801 ^VbI . ' —
8o|+ 37 = "71 = A> Spam
805. — 28 =s 521 = Cs 3
14. In an article of trade, A gaina 14 s. 6 d. and his ad-
venture was 35 s. more than B's, whofe fha're of profit is!
but 8 8. 6 d. ; what are the particulars of their &»(ii ?
Firft, 14 s. 6 d* — 8 s. 6 d. ss 6 s» difference of their gaiiu
S. 8, S« S» * . 1. I« d«
Thenas 6 : 35 : : 14.5 : 84,58af = 44 7=A's IgQ^j^^
6 : 35 : : 8-5 : 49-583 = ^t 9 7 =B s 1
15. Three perfons entered joint trade, to which A con-
trived 210 1. B 3121. V they' clear 140). whereof 37 1.
IDS, belpn^ of right to C ; that pcrfon's ftock, and the
feverat gains'of the other two are required?
210 1. -f- 312 = 522 1. = A's ftock + B'^s.
140 K — 37.5 = 102.5 ^* = A'^ g^^*^ + ^*^*
As 102.5 • 5" •• 37*5 : 190.^75^/2^1. = 190 1. I9s.:^j
C's fhare.
210I. 4- 312 +I90.^56|z<=7i2.^756^ whole ftock.
712.^756^ :« 210 : 41.2357 =141 1. 4s. 8^d. +A's }^^^
140 : : 1 312 : ^i.2043 s 61 K 5 s. 3id. — B's J ^ •
^6. A apd B venturing equal fums of money, clear by
ioint trade 154 1. ; by agreement A was to have 8 per cent.
TCcauf"^ he fpent time in execution of the proje£^ ) and B
was'bnly^to nave 5: thequeftion is, what was allotted A for
}iis trouble ? -
. !. s. d.
A»8 + 5= 13 : 8 :: 154 : 94 I's . 4^, AjsJ .
13 5 5 - 154 : 59 4 71^^ Bsje
Anfwer, £Z5 10 g^
17. ^y B and C play a concert at' hazard, and making
up accomptS) it appears that A and B togetUer brought off
' "I ' •• - 13I.
34« SINGLE FELLOWSHIP. BookU.
13 1. 10 s. } B and C together 12 1. 12 s. ; and A and
C togetber won { i !• 16 s. 6 d. j what -did thej* feverally
2) ^' 18 6
l," •• d. 1. «. d.
1819 3— 12 12 -X56 7 2,»A*s>
18 19 3 — II 16 6 = 7 29, B'slfharc.
J? 19 3-* 13 JO -= 5 9 3>C'sJ|
x8. A, B and C «ie tbrte hodb bekm^ing ta diSertut
men, and are employed as a team to draw a Toad of wheat
•from Hertford for 30 s. ; A and B ard deemed to dp f of
the work; A and C | ; and B and C 1%. of it : they are to
be paid proportionably, and you know hqm to div^ it as.
it mould be.
.2 l^,^ J^ d^ —
7 "^ 56 "" 560* 8 56*
211 — ^4 = -^ = C^s fhare moflp than B'$p
56 S^ 5^
2=;:22,andX = 14
8 80 10 80
560 560 5 5b' 1/ 560 V 56
560 560 560 1/560x56
560 ' 560 560 560 ^ 500 560
. Then rejeding the common denominatori^
roi + 59 + 109 = 269, fum of the numerators.
s. s* d.
269 : 30 : : 101 : II 3A5V = A^si « ^^^ - .
59: 6 gil|=.B'sP^^/*^°^<>^
109 : 12 1^14 = C's) ^^y*
269 : 30 :
269 : 30 ;
j^ I 10 -, proof.
19. Three perfons purchafe together a Weft-India iloop^
towards the payment whereof A advaiiccd |, B |, and C
3 140 l.i
Chap.IV. SINGLE FELLOWSHIP. 34^
140 1. } how mudi piaA A and 'Bj and iHiat part of die
veflel had C ?
B's part.
i* ». li — II = Cs part of the veflel.
56 56 56
it 140 1 ab40 [' *' "*! - ,■«
and 140 - -, C J
Whole coft, £ 712 14 6t T^
ao. A, B and C have lool. to be divided amongfr
them in fuch manner, that two timbs A's fliare be equal
tb thre^ times B's Ihare ; and four times B's fliare equal
to five times C's*
Here it is plain, that A gets 3I. to B's 2I.
_ And that B gets 5 1. to C*s 4 1. *
As 5 : 4 : : 2 : 14= i-6> to B's 2 1.
iTherefere tbeit fliares ^1^ A 3 1, to B's 2 K and C's 1.6
And 31* -f" ^ 4" 1*6 = 6-6, Aim of thofe parts.
1« s. d.
As 6,6 : 100 :: 3 : 4W5 = 4S 9 ' 7 ^i f = A's^ ^
Alfo 6.6 : 100 : : 2 : 30.3;z^=s 30 6 -^ ( fe ) =: B's > I
And 6.6 : 100 : : 1.6 ; 24.24 = 24 4 loj 3 **^ I— C*s j ^
21. A and B join their ftock, and veft them in brandies ;
A's ftock was 19 k 19 s. 8 d. more thto that of B ; now
by felling out their commodity at 55 s. per anker, A deared
74!. IIS. and B juft 5O guineas: the quantity of brandy
d^alt for is required, and the gain upon the anker i
1. s.
5a 10
^^'■'\^{^a^
-(
^22 . r, difference of their (nmu
\ 22 K
r
8
iso SINGLE FELLOWSHIP. Book IL
221.18.: 19L 19s. 8d.:;i27Kis.fum: iisl.is^iidicoft.
115 I II = 2301 II d.
55) 4^42* (88 ankers, ind 2 s. xt d. enter.
88) 127 I (i L 8 8. 71 d. gain per anker.
-
22. In raifing a joint ftock of 400 1. A advances -^ ; B
14 of -}- ; C 7 more $ the difference between A's adventim
snd JB's, and D the reft of the money 1 what did every one
fubfcribe ?
± = || = i23l. i8.6d.i«=A';
^ofl=-2- = m=si63l. 128. 8d.4i4= B'
II f aa 858 ^ *"
sii«_2^— iz. l?
» ^ '43 I JI— j3?^
6 858 "^ 858 "" 858 ""
858 ~ 858 "*" 858 ~ 858*
Sp'^Sjs-p- ^^- Is.2d.:f44=:DsJ
23. A father devifed fj: of his eftate to one of his fons»
and 1^ of the refidue to another, and the furplus to his re-
lid for life ; the children's legacies were found to be 257 K
3 s. 4d. different =1 -^ $ pray what money did he leave
the widow the ufe of J .
.-49_ji666
83 ~ 6889* •
-or:r=^ + ^=5g = i07l. 4s.6d.|*«=Cs
And
2822
s 1666 * 1156
257 1. 38. 44. Alfo
2821
6889
6889 6889 6889
i666_4488
"^89— 6889*
^'"^ ^ - 5555 ^ S^ :^ Widow's part of the eftate.
As
Chap. IV. SINGLE FELLOWSHIP. ^5^
ddeft Ton.
. 1156 , IJ43 , , 1666 . ia853»9-^--,. ,,, ^ . ,.^
yoiingeft.
1156 . 1543 _ 2401 . 3704743_ .., , I ^, oj .„„,
^ 6855 ■ ^^ •• 6885 • ^9i6"-534l- « s. 8d. ^near-
ly) widow.
^ 24. A father, ignorant in numbers^ ordered 500 1. to be
divided among his five fons, thus : Give A, fays ho^ 4, B ^9
C f , D f , and E f ; part this equitably among them ac-
cording to the fiither's intention.
FiiftL — lis x-.i2S x-ii .-70 ,^ X —
420*
Then lis
420
+ ^ + J2. 4. 21 + ^ - li2,thcirfu«l.
]21
420
A, 459. 500.. >40
420 I 420
±i?:i22:: i££
420 * 1 * ' 420
1S2: S22:: il
420 ' I " 420
459. 500_ 70^
420 ' I ' * 420
^9 . 50Q . . ^Q
420 ' I * ' 420
84 , ro . 60
420 • 420 ■ 420 420
L s. d. farth.
152 10 * *:^
114 7
I 1444 = A'$.
6 3^ = B'*^
91 " - 3^ is C's»
76 5 •- ai4 J = D's.
6s 721^ = F«.
^500 - -
^ 25. A in a fcufle feized on * of a parcel of fugar-
plums ; B catched 4. of them in his hands ; and C laid
hold on T?5 more ; D ran off with all A had left, except
f, which E afterwards fecured flily for himfelf ; then A
and C jointly fct upon B, who in the conflift (bed half
be had, which were equally picked up by D and E,
who lay perdue. B then kicked down C's hat, and to
work they all went a-ncw for what it cpntaincd ; of
which
i$2 SINGLE FETLnOiWSHlP. Baokll*
which A got :J:> B f , D ^ and C tai E ^equal feaves of
vOat was left of that ftoclt. D then Atack ^ of w^it A
and B laft acquired out of their hands ; they widi difii-i
cuftjr recovered J- of it in etftial fliai^ again, -but the other
three caiTied otf I a*piece of the fame; Upon this (hey
called a truce, and agree that the f of the whole left by A,
at firft Ihould be equally divided amongft them ; how tfiuch
of the prite, after this diftribtition, remained with each of
the competitors i
Though A at the firft feized j., he loft all again this heat.
lof^-ri-B'sl.n
lofi=i = Cs\ ^
10 3 s •*
3 20 60 420
7 X 60 ^ 420 I
420 420 420 ^
iTitaen. Their fum tst
firft acquifi^ion*
420 420
Thus endfed the firft heat.
Again, - of -=;-=: B's
a 4 S .
Retained . - - - = C's
ll + ±.= 122 = D's
70 * 16 560
H + JL.-ilL-Fs
420 ~ 16 1680
Proceeding, ^ of - «: —
° 4 5 20
part at iHrc end of the fecond
5^ fcuffle.
;A'i
120
part after the third
fmufs*
7 5 ^ 560 560 J
Theni- + i+l.= 21-.
20 ' 15 * 35 420
II
ri
II
i_21 = Ji.,andior-li::=~ = Cs1^^_.,^
5 420 420' ' 420 840 f psirt M the
157 , II _ i7<) __ r^> T third 1'mufo.
4.
i^6Bo ' 840
1680 ^
Further,
CmKp*
FitrtbATj
B.
6a
fc^^'^y
A and
so
JLofi+iof-iofiir ^^Ws
16 80 ^ 4 15 8 3840
8 80 ^ 840 13440
• 80 ~ «40 +480 -* ** •
f 80 ^^ ibio 13440
put after the
laft fcufl[e«
Tliea
384P ' 15 a688o
1. ■ —143?. r». I**'^ carried
+ n — TTSsi 5s ^ « yoflF at cbel#.
«3440 ' «S
s688o
t02^
a688o
D's
4480 - IS
JLS2SL + ^ __
13440 ■ 15 at88o "J
So that if the number of fitgar-plunibs were a6l8o.
Ag«t - - 2863
B . - - 6335
C - - - a4?8 S* film 26880.
I
D' - * 10^1
E - . . 4950J
■6. B* At havi^ | of 4 af the half of a tradii^ Hoop and
cargo worth 161 31 L 148. fells his brother B 4 of 4 of his
intereft therein at prime cof^ ; what did it gok the brother,
nd what did hircoiifin P pajr at the (ame tiine for -^ of the
remainder?
. I of 4of ^of 16131.7 =4234.57125!. = 42341 II s/^d.
n at uiftm
J of ^ of 4234.57125 = 2032.5942 = 2032 1. 1 1 s. to^d»
B.
16131.7 -• 4234,571252=2 2 1897.X2875, remainder.
, Aof ii897,i2875 =9734-01443 = 97341. -8. Sid.
eeufinP.
A a 27. Two
y
154 «IN6^'B REitiLiOWfiJ&tlJPv l^Mb
, ,2^ T^04p|rchan^t^ cqmpanj, A put in ao 1, and B^put
m 135 ducats*}" they gain 67 1. 1018.' of whid A tgok 30 K
what is the value of a ducat ? - -^
...
6/1. ios."— 30I. = 77I. ios..±: B^s galpj
30 : 20 :i: 37,5 : 25!* iTs ftock sx ijf dwtatau
I2ki\ 25 '• = 590 5* (3 s« 87 <!• y^.u^ Pf"^ ducaty the anfwer.
'f.8. Three merchants, A, B and Cy* freight ihips to
Lifl^QO with fugar, to the value of I577B1. 2 s. 6d. fler-
linjg. A, b<)ught 2<o cwt. x qr. %%\h. |it 2K s6s. percwU
B paid 2 1. 6 s. 8 d. per cwt. for his ; but meeting with a
ftorm at Tea, the mariners were conflrained) for 'the fafety
of th«ir lives,; to caft out pare of the4ihip's lading. A"^
proportion ejeAed part was j^th part of the fhip's-^lading,
^nd 34 time$ the quantitv^ caft o\tt board, nm jjrttmes die
whoie freight erf* A and B. When they cam6 to land, A
fold his remaining .part for 4 1. 4 s. per cwt. and found him-
felf d lofer ib per cent, beffdes cbargesT B acfvanced the
remaining part of his commodity 20 p^r cent, and C gained
^sJ' 84«. percwt. by thequttitity l>e feved«^ -Queie, what
did each merchant lofe by this voyage, the charge of . the
ftftne amouBting to 560 guineastf «^ - -^J ^* FUtmr.
* * ' ' J ^ " ' * I ^02 (
f cwt. : 2 1. 16 It. = ^ : : 25(ycwt. t qr. a^lb. ss -^-g-
:70iL 5«. A's.coft beCdegdiarges. '^
19Q 10 ^ ^ 2805 c6i , , J A,
-i- : Y •- 7011. 5s. c=— — 2 ti-*..ss7ol. 2C. 6d.A'i
lofs beCdes chaiges. ;
70x1.5 s* — 70 U 2s. 6d. SS631L 2s. 6d, =s ^^t
value of A's remaining part. ' * "
^ 4I. 48. = " : « : : 5^ : 2:*5^ si ijocwt. I qr. lib.
A's remaining part.
250 q^t. I qr. 22 lb.. •*• i jfo cwt. i qr. 2 Ib. =^ lOO c^
2olb.= -— i, A*s ejeded part. - --^
V -iS^f ^^^ 70125 ' , . .
— ^X 100 = i = iooi7cwt. 3qrs. I2lb. whole
cargo.
$Sr^ "3" •• ^^ • 4007 €Wt. 16 lb. whol^ cjeaed
part — -^^^ »^
~ 7 * 3
Clla^.nr..SlNOLB FELLOWSHIP. 355
!?: £S :^^l-2S2: -1-iiii ss 4508 cwt. 41b.' A's cargo
+ K* . ^
4508 cwt. 41b. — 250 cwt. iqr. 22 lb* ss: 4257 cwt.
2iq«. r6 IbJ^ te *45±!^j B's <argQ.
10017 cwt. 3V*- '^'^- — 45o8cwt. 4lb, = 5509cwt.
3 qrs. "8 lb. as 'iiiZ£, C's cargb.
7012? z$9«;o 238415 ^ 4708c ., -.,
--— : — r*" •' • ' 1"^ • *^ =» 1703 cwt. 4 lb. B't
qe£led (MNrt.
100 cwtt . 20 Ibt + tI 703 cwt* 4 lb. ss 1803 cwt. 241b.
400jr Gwc i61b. — 1803 <^^* 3^4 Jb. =5 2203 cwt 20 lb,
v's ejected part.
» • ^ • = ^tP- ^^a9934l. 7». 6d.B's coft,be.
fidci charges*
701 1. 5»w + 9^1. 73. 6i s 1063s 1. Mts* 6d. A*t
+ B'«coft.
15778I. 2$. 6tl. -*- 166351. 12B. 6d« s 5142 1. los^
Q*^ toft, beiides charges.
4257 cwt. 2 qjfs. 10 lb. — 1703 cwt% 4 Ik s jt554 cwt.
Bqrs. 61b. = !i^^ B's remainder.
. 5509 cwf. 3qfst 8 lb. — a203 cwt» 2 lb« s 3305 cwt»
3 qrs. 16 lb, ss 22^\ C^s remainder.
' t:.l :: IM^ : 5960I. 12s. 6d. value of B's remain-
tug part at prime coft.
value of B*s. ,
, §9341- 7s-'6d. — 71521. 15$. =27811. I2S. 6d. B's
lolsy charge excepted.
^'; —--T^^ -^ = 77''- 7«- 6d. C gained by,
whathcfavcd.
>54^7«r , 1028? 0256? ^ .
value of Cs remainaer.
w
4
Aa 2 3085 1.
355 SINGLE #ELl6vrSHIP. Ibofcfli
, 3o8ql- 10S.+571I. 7s. 6d. = 38s61. 17s. 6d. ad-
vanced, value of C*8,
5142I. 10 8, — 3856 1. 17 s. 6d. = 1285 1. 12 »• Od*
13I. as. 6d. +2231.28. 6d.ssa36l.5t.-a.. I ' Si .
525 1. — 236T, 58. -d. =s 2881. 158. - d. C'sJ «
29. The* were at a (a& 4o wkh sBd jot^omenj and 15
fervants, who fpetit a+1. and fat every 10 s. thafc a tote
paid, a woniati ^ 6s. and a fetvant %i.^ vhat did eacb
peribn pajr?
r 4o y 10 = abo, 30 X^ :fci8o, a*Us X a?* 30» *^«r
fum 410.
1, J. d.
410 : 24; : 200 : 11.^073^ = II 14 .»i» ">^ ? •©
. 410 : 24 :: i4b : ra^pSf =» ^^ 10 M, women V^
420 : 24 : : 30 : 1.^560^ = i 15 li, Icnrants J
20) III. 14s, i^a. (lis. Si*, licarty cichman.
aoyioU loi. 8^4. ( 7a. ^d* 4» each woman.
J5) iL 15s. lid. ( 2s. 4d. ^ eachftrvant.
30. It is propoTcd to divide 300I. amongft three perfena»
fe that A gets 6 L morethan^B* iftt mote than 4, and
C 81. ie& than 4; what is the fliafe of each?
AoDotding to the moft obvious meanitog of this qtieftion,
die (blution u^ follows:
lofjooLsisol. f of 300I. = iool. and^of 300L
sr2ooC ,
Alfo 150!. + 6 = 156; 100 + 12 = 112; and 200—
8 = 192.
And 156 + 112 4- 192 = 460, their fum*
•..• 4te : 300:: 156: lOiU i4«* 9** A ~^* 7 ^
Alfe 460 1300:: 112:731. -». iod,4^ = F« >|
And 460: 300:: 192: 1251.4a. 4d.^ = C8 J •
Others
9
iiator
Others taking the ^ueftioa in a different fenfe, folve it
thus ; • . : , -^ „ • ' • '
6 + 12 -^ 8 == 10 ; ^and 300 — ^ io x: 29a
Then4.=5i, x^J' and * = $ ; their ftnix J.
» : *|^ : : I : 96J 4- 6 =f 102!. 13s. 4 J, ftr A.
: »f« : : ^ : 64^ 4- ^^ = 76I. 8 s. lod. 4 - B..
: 'I'* • -ll • I3ic>| — 8 = i»I, 17s. ^d. ^.- C.
31. jt being amed.t^at the French Kingy Pope, and
Pretender, aip to mare lo^ooo acres in the ^iteriul regions,
in the proportion of -J-, -j^^ and -^ refpeftively ; t^t the Pre-
tender relinquifhing his right, how is the territory to be di-
yidfid bct^i^ tk^ p^9f tw^^ withQHt j^e Mppr ^ lawyer i
^ ^ and -^'reduced, fo as to have one cDmcQoa denonir-
tor, will be ^ ^, and ^ ^ receding, t^e denominator
20, iCi 1^ l#. *
Then fto 'f 15 + 12 SB 47.
r 20 : 4^55 j/f» French King.
As4f:ioo4b0::^«5 (319141^, A)f)e.: : * ^•
(.12 : 25531^^, Pretender.
But ^SJSf 4f ^P*^ t^i^K relinquifhed by the Pretender,
jnuft be Avided between uie French King and tfie Pope,
viz.;4 + 3 = 7: 2553141 : : 4 : 1458^-
^;- • 4*553^ + 14589x4* = SV^^h -^^^ *• French
And 3191441 + to^^^^■9 = 42t8s7TV^» ^^^ «*^? ?^^-
Q. £• F. ..
32. B^ueht 100 quarters of malt,, jneal 4|id oatmeal,
together ' far 14a I. lor ^erj five bu&eb of malt, I had
three of meal ; and for every ^igbt of mefti,' I had -feVen of
oatmeal : pray what i\i tjbu^ cqft me (everally a hulhel, the
|nalt biding hfXf ^s dpar again as the ^eal, and the meal
double the price of the os^meal }
3 meal : 5 malt :: 8 meal : 137 = ^, malt.
Then 8 + I3t + 7 = 284. s: -7, their fum.
8s 40 'oo 800 V*-. " ^f{- , 64ro
3 * 3 ••T' 77* 47 -IT =376.7^1=—.
85
Aa 3 3.
ik)6k
bdh.
gc 8 loo Ato Y*: ■™f*. ' • 384a
3 1 I 17 '^ . *7
31 i *7 '7
. . 800 i izool
And — . X - = I
17 3 *7 I
480
~ >for the price of the*
4«o ^ 1 _^ i*rt
malt,
meat.
■
oatmeal.
The denominaton may be omitted, and enh numera-
tor divided by 30^ the quotients will ftill rec^in the fame,
proportion ;
viz. 40 + i^ -f 7 = 63.
As 63 : 142 ; : 40 : ~-^ = 90 3 ^tt
63 t 142 : : 16 ;
. »^7«
^3
'malt.
63:142:: 7-^
80'
36 » SA^Z-S^nw^J-
a
IS »5 6}4
K
oatmoaL
>\ 7X/^ .
17/ 63X5040
17/ 63 V 7560
malt
3s. »»»d. meal
per
S. bufhel.
(i257-=si«. 7^4. o»ti»ed I
33. Three men. A, B, C, tuy a tip for 310 1. 15 ?• o^
whibh A paid an unknown 'fum> B paid af as much i
M C 3t as much : how much did each man pay f
A' — §• * »~* 6* "*^ 3 * .
6 ■ 6 ^ 5 6
And 310 1. ifs. = 310^ ss — ^
41
J
Then
And
$ % >part.
•dta
jC 310 IS -
^. Ther4 were 25 cbblers, lo tdf/lotsj 1 8* weavers, and
12 combers, ;fpent 133 (hillings '«t a ineoting; IQ which
i(eckoning. fiv^ oobleis .paid as much as focir (ajrlors^ 12
taylors as much as nine weavers^ and fix weavers as much
as eight combera; .how much did eaeh company pay, and
^vhat each, man?
Per queftion^ 5 cobler^ -> ▼
4 combers -J
Then i cobler f
t coitaber ^
« :}
for the
1 • •
coalers.
taylors.
weavers*
combers.
^rl * > cobler.
' 9 (for each r»y'«'"-
2 ' 4 t ; .1 weaver.
I Q J I comber.
n
35, Once as 1 walked upon the banks of Rve, *
To fee the purling ftrfams glide gentlv by.
And hear the pretty birds to chirp and fing.
Making the ffmves with melody td ring ;
I in the meads three beauteous nymphs did 'fpy.
That for their pl«ifure came as well as I j
And unto me their fleps they did direi^,
Saluting me with moft benign refpei^.;
A a 4 Saying,
96^ .!SmehE FELLOWSHIP t<MkOi
'Savins^ Well met, we've bufinefs to impart,
Whico we cannot decide widmut ypar asc i
Our grannum's dead, and left .a Ug9fY>
i^hich is to be divided amongft us diree :
Ip pounds it is two hundred twenty-niac ;
AUo a good daark^ being fterli^gxoin.
Then (pak« the eMeft of the lovely three,
III tell you how it muift divided be s
Likewiie our name? 1 unto you will tell,
, Mine is Moil, die other Anne and Nell:
As oft as I live and five-ninths do take,
Anne takes four and three-fevenths ber fiartto.m«ke|
As oft as Anne four, and one-nmth does tell.
Three and ttMHtbirds nuil be took up by NelL
L. D. 1717.
Firft 229I. 13s. 4d. =s %29.f1 51 xz $./ 1 4f =4.^a857/.
And 4j. :i2 4./ J and 3I = 3.^.
As 4./ : 3,^ :.: 4.^2857/ : 3.949809
Hence as often as Moll ukes*5.5^c555
Anne ulf s 4.'j^8c7/
And.NcB i«949o09
^0*-
I3«933»3S
. . . n.gttQ?« : f S-555SJS 5 9»-S69 «= 9' " 44* MoU,
MQ-J: : i +'^*857/ : 71.995 = 7* ^9 'Oh Anne,
/ 229.66^ ,
SEC^. II.
DOUBLE FELLOWSHIP!
OR.
FELLOWSHIP WITH TIME.
DOUBLE FELLOWSHIP it a rule whereby
we compute the gain or lofs of fevenU merchants who
-employ difFerent Aims of m<Mtey4iffi9reot tiaies In partnerihip.
RULE,
RULE.
As ^e fum of tlie produ^s of eadi 'manS -ftodk' and
time : is to the whole gain or lofs : : fo is the partitidsdr
produd of each man's, ftock and. tlfi^e : to ea^ man's
particular gain or Iciis.
V Throe ptfrfonib Ay B and C, enter into paatheifliip Aus :
A puts in 6< U for e^ht months ; B 'puts In 'jiS 1. for
12 months; anil C puts in 84 1; for iix months. With
this they traffic^ and gain 11% k^ ra^ 1 demand each
Moi's (hare of the ^n in proportiM to his iook W
time of employing it ? 1
5:50*
i960
r52o:44L 4s. -d. ss A's|q^
i960 : i66::<936 : 79L H9^ aid. k B's I g.
(504 : 4»L 16s. 9id» «: C'sJ ?
Or by finding tCfMttttOtt multiplier ; viz. i96o)i66.6(.o8j«
Then.520 X •oSs = 44.2t for A^
Alfo 936 X •085 s;: 79*{6, for B > as before.
And 504 X •oSs r= 4x^049 for C 3
2. Three perfoos. A, By C9 hired a certain pafture for
S4 L in which A keeps 40 cows for four months ^ B
Joeeps 30 cows for two months; and C l^eeps 36 cows
for five months 1 hpw much pf the jet^ oudbt each of
them to payr ^
A 40 X 4 — l6o
B 30 X 2s= 60
C j6 X J = 180
400 : 24 f:: 160 : 9 ^2=: A's^pait of
\:: 60 : 3 12 = B*s}' tl>e
1:^80: xo t6 ssC'sJ rent.
3« Six
iSdi DOW'BLE
t 'in'moi^b^
. 3» Six nRicliajits* wB. ■ A^ B-,' C, -D^ Eadl F> ^iter
.into. pUtMciihtp, «nd -compoftt & joint'^ii^.in dus itam-
•Kit ^ . •' ■* ' *H»*»
I. ». . .
rA puuin 64 10 = 64.5 •> ^, r 4J>
B... 7815= 78-75 6 I
D- . - 80 xo = 8Q.S p|ia77"'^*'-
E- . . 74 1^=5 74.6 , I 9i|.;
'viz.'i
J 1;
<•
K
^ 7
4hty traCic and gain 258!^ 18 6« 4^d._ It is required
to find every man's fhare of the gain^ accofdih^ t^'bia
ftock« and the time it was employed ? '- '
U months*
A's ftock 64.5 X 4*5 ~ 3590.25. ^ ^
Fs - - 78.75 X 6. :;= A72.5
Cs - - 100. . X 8.25 == 825.
D*s - -' -80.5 ^ X 12. == q66.
E^s - - 74-6 X
F'a - - MS.75 X
\
966.
9.5 = 72»-7^
7* £7 OIIO«25
'i'-
Svia St. 4t44*>
♦ !•»
I ^
.» 4
■WW
The whole £ain is 258 L 18 s. 4|d» t=: 258'9i875.
Then 4142*7] 258-91875 (.0625, common muitipueiv
I. s. d* . .
966. X •0625 s=
708.7 X .0625
S80.25 X '0625
Their whcde gain, j^ 258 18 4J:
»•
4. A and B in partnerfliip equally divide the gainj A's
monev, which was 84 1. 12 s, 6 d. lay for 19 months ^
and B's for no more than ieven : the adventure, pf the lat^
ter is fought.
Reciprocally, i9mon» : 84,625 : : 7 mon. : 229.696 1. =s
JS129U 135. II d. anfwer.
' • • V
5- A,
€SUlfbW. FflXOWSHiP WITH TIKC& ^g
5* Ay By C hd^ a common Hock of loool. A pins
tool, in nine nuMnhs ; B80I. ki 11 montht} MdiC izol;
in eight months : what was each of their particular ftadcai
la) 8o{ 6.0 .
8o)x2o(x5.
I. s. d. .
3a.^: 1000 ::iu;t: 32^9^3P5 = 338 19 8 A's 7 «»
3a.y:i^o:: M:- 203.^8^3= 203 7 9l3'«^S.
31.^: 1000 : : 15. : 457.^2712 =457 la oiC's J r
,i^,» A hat)i 200 1. more ilock than B | but A continued
Us only five mondis, and B nine, and drerw equ^l gains:
what arc their ftocks ?
m* L m* 1«
9 «— 5 = 4 : 200 : : 5 : 250 rs A's ftoct.
^ 4 : 200 1:9: 450 St: fi's flock.
700 ss whole ftock.
7. A and B paid equally for a horfe, February 7, ^756 j
A on the lOdi took him a journey into the weft, and re*
turned on the lodv of June fbllowi(ig ^ B on the 2d of Au-
^ft took him into Scotland, anid ftayod till November 13,
aqd then concluded his fervice this year. From January 17
following, A ufed him 10 days > and in' fix weeks after his
return, employed him till April 30 ; B then rode him frmm
May-^y -to Mi^Kumilier ; A hftd him from Jviiy^i^ till
X4 days after St. Jameses tide; B, on September 30^ took
him intb-Norfblk, and .oune bafrk. 0£bob6r 19. Ho- then
wa3 fold foryl. ids. and they would have the. money
equitablf-farttd bet#sefl then») vic&l in prdportlon to fhe
ufe each made pf their fteed.
From Feb. 10 till June 10, are ^^^ ? A in all*
Between Jin. ry -and April 30 - r - 61 >^^o^^C.
• From July 14 till 14 after i5c. James^ 24 J ^^^^^^^^
FromAug.2 till Nov. 13 ...--. 104^3 in all
May I till June 24 - - ^ - - 55 \ ^„^j^„^
St^. 30 tiirOaober 19 ^ . . 2o3 ^79days.
Then 208 +'X7'9 =i 387 days, the horfe was in Ufe.
1. s. d.
As 387 : 7.5 :: I79> B'stimc : 3 9 4J4, A's i fliareof the
317 : 7.5 a : 108, A*^ time : 4 - 7^ B's J money.
• 8. A
f^ A for a m«e montbn a^vMtux^ f^^tm^ 90 h B for ^ne
or feireji months rec^ivod 9,5 guiotas j 41^ Q ,£n: lying -^v^
•f \ua cMtribudoo fiye mpaths b^ f titlifi.t^ gsi T. Tm
total of their adventures multiplied into the^- ^eipe^ife
times was 640 1. what then was the particulars?
25 guineas = 26.25 1. ' ■
Tl^ Jio 4* 26.2 j «{-. 3^ = 7^'^i? ^^^ whole gain.
' "^ rio : 163.57837
; 7?*^S • 640 :;j 26-25 : 214.69649''
• '"^ ' 1.31. : 261.72524
9 163.c7837l18.175 =18 3 6 ^a;sj
7 2 14.69648 130.670^ = 30 13 5 tr B*8\^4ventate^
5 261.72524 1 52.345 =53t 6io|s=C>J
9. Ten pounds a quarter is allowed the five aucjitors of a
lire-office ; they attend about feven time# m a. quaver, and
the abfentecfs money is always divided -equally among fuch
as do attend. A and B on * thefe occafions never mift ; C
and D are generally twice in a quarter abfent, and £ only
once; at die payment what ha4 e^ch map to receive/
5) ao (iL /each a^m's «i|iuri Aain^
7) 1 (5.8« ib^d* each m^ f^w Uf 493r*> ; > -
5s. B^d. K2= IIS. 54.4. Caodl^ m.^i^ €9^ for
Ipro daj» abfeoccL .
atftendmoe.
1 1.^ 58. 8^d. a I K 14s. 3f d« «o V'S^i^^mw^m^.
«l s. 5f d. X 2 =s 1 1. 2 s. 10^ d. C's moity D'« 4e^lts»
j)il. 28. iofd.(78. 74d. A»Baiidi;'sai«^firf*t^9
denultB*
4)58. ^A.^ IS. 5fi. A, B, CmAm Aw of F§
default. ,
Then 2I. *f- 7 s« 77d. -|* i *• 5t^* =s 2 1. 9 s. -^d. A*s
and B's each.
1 1.% s. 64d. 4- < ** 5t = < 1- 'OS. ^ poll P each)
1 1. 14a. ^d. -f* 7s. 74. d. s 2I. I 8. ip^d* £*sih^«>
10. A. B and C enter into partnqrihip } Aputs in on tbo
tft of March 60 1. B puts in^e i^ of :May 960 yarda of
broad-doth ; and C pi^ in on the ift of Jupe 4054)99^^
On the ift of January fcdlowing they accoontecl their
Cain» of which A and 6 took up 456 1. B and Q took
up 431 J. and C ^d A4oo]^ up 375 L I deoon^ ^bat
I * * wao
>v
Ok^ IT. FiTLUOWSIflP wiTx Tills; ^
was gaAn«d #8 ^ell ih the utiekas a part; what B va*
luei a yard of bia cloth at, and' what was C's ducats
L - *
A*s gain + B's — 456
jl's * - + Cs ±1 431 ^ . . .
A%^ - + Crt :fc375
2) laiSi fitnif each bdng aaoned twice*
631, whofegalim
^cn (531 — 431 = 200L = A's'^
Mfo 631 ~ 37 J = 236 1- = B's I gain*
And 631 — 456 =s 1751- = C's 3
'60 • 10 . loo 60 X Id X 256 = 153600
• • 5 • 256 8 X 206 = 1600
1606) I C3B00 (96 1, value of B*s cloth. .
160) 96 L = 1920 s. (i28« B*8 doth pef yard.
P* T. G.
te ^ iK> • aoo 60 X x6 X <75 ^= 105000
7 . 175 . aoo X 7 cs 1400
9400) 105003 (75 1. yalue of C's ducats*
(i|QS)75L 2s 1500 s, (3s. 8|>d. value of one ducat*
285
^ * -la •
II. Adeantgl. in fix months; B 18 1. in five months ;
and CaaL iadsiae months, with a^ftockof jzl. 10 s. what
then die ihc general ftock amount to ?
p.
T.
G.
7*S
■ :•
' 7*.5 X 9 X
ft
• 9
. 6
• «3
• «3
•
tlividend.
23 X
^«( DOUBLE FEULOVfSHXPi 4r, BbelrH^
3$ X tf 7= t^) S481.5 (61.46739 s 61 9 4 te' A'l
P. T. G.
72-5 K 9 X 18 = 11745* dividend. . » '^
23x5= 115)11745(102.130435=: loa a T^sBsB't
7a 10 - =Cj
Anfwef, the whole fiock ^ 236 i XI7 !
12. A, B and C enter partnerihip ; A puts in the ift of
January lool. and the ift of May puts in 156 1. more ; and
^n the ift of September. takes out 30 L the lemainder flays
in till the year's end.
B puts in the ift <^ January 250 1. and on the ift of June
60 L morei and on tj^e ift of November 100 1, more ; which
continues till the yekr's end. ,
C puts in the ift o^ January 300 1. and tKe ift of April
takes out 200 1, and on the ift of Augiift tiketf out 50 1.
more ; the remainder ftays in till the year's end i Whiit muft
each have, of the gain, which was 1331* ?
1. 1.
A from 1 January 100 X t2 a: noo
I May 150 X 4 = 60O
I September - - • 120 X 4 = 480
2280
B from I January 250 x 12 =: 3000
I June --*-. 60 X 7= 420
I November - - - 100 X 2 = 200
from I January 300 x 3 =i 9<^
I April • . • • - 100 X 4- s^ 400
I Auguft 50 X 5S3 450
1550
v»
Then 2280 + 3^20 + 1550 rs 7450.* • f
1. s. d. qts.
f 2280 : 40 14 -4 . V^ A*rl
74SO • 133 •• ^3620 : 64 12 6 /^V» B's IgaiA.
(1550 = 27 13 5 . -^^ C's)
13- A,
A'sjqdoney was in. three months, fi?s .moi|ey was. 'In ^ye
W3i|ths, and C's money was in feven months ; they ^ned .
^j(4;*l« which was fo divided, as^ of A's ^in was equal to
4 of B's gain ; and 4 yf B*s gain ,was equal to ^ ^ C'a '
Mia ^ Itrhtt jdid each^ merchant gaiA jmd put in ^
^Supjisfe AVgain to be 2")
Then will B s be - - 3 > by the qucftiw.
Anise's - 43
. ,, Their fum ssg ".-
1. f 2 : 52 = A's J
Alfo 9 ; 234 : :< 3 : 78 = B's J gain,
14 : 104 sr C'sj
And 52 X 3 B '56
78 X s = 390
104 X 7 = 728
1274, their fum.
1274) 3822 (3f common multiplier.
1^ X 3 = 468 =s A's
••• 1^ X 3 = 408 =5 A'S)
390-X 3 a 1170 =: B'sVftock.
yaS X 3 = ^184 :s C'»)
3
-' S^iZi' whole fiock.
: . : . SEC T. IIL
FACTOR A GE.>
WHEN a peifon does not tranfafl bufinefs himfelf,
but comimffions another to ad for him, tbtf peribn
lb comniiifi9ned is caHed a fador, and the hufinefs he
tranfa^ is caij^d fadtorihip or fa£lorage»
A , 2. What
3W
!»ACTORAOE.
Booill.
t. Whftt is est
pfer£eat. f
i 793 >7 6
1587 I? *
18 Q
99 4 «^
i396
j£ aa83 18 5^
20
16.78 •.
12
<f 7$ji' i78...6a.r«t«f
Of) 1. <. d.
loptfcut or^ 793 17 6
T 79 7 9
- 19 10
9,41 d. anf. 20 1. 16 8. 9^d«
4 '
1.64 qr«
« -m
%. What it the commiffion of 967 U i|t. 4d* at 3f
per cent J
1. i. d. ^»
T 967 13 4 L 8. d. '
3# -iV 9^ «3 4
2903 - -
4 483 16 8
^ 241 18 4
fSO 19 2
jC 37-49 14 «
20
9-94 »•
12
4
1.20 qr.
T 96 »S 4
iPi«i«
i 34 3.>«
IS I H
I 4 ai^^
^ 37 9 nit anfwer.
J. A
Ck9p^W. F A C T 0 R A G Er 3^9
3. A nriduiit's lead ftock Mng xooL uH the ftftor's
30 1« wko icccwcdi -^ of the gain } what was his fervicca
valued at? :
f : 100 : : t : 50; therdSaic j[0 — - 30 = 26, the anfwer. '
Othcrways.
100 + ,30 = xjol. whole flock: 3) 130 ^43 1. 6 s. 8 d.
*•* 4ti« 6f. 8<l. — 30K = 13L 6a. 8a. value of the
4. A owdMMt ddireta tt> kt$ fiidor zooL allowing him
to join f» it JO L and ndues his fervtce at 40 1. what ihare
id the gain cinght the fiiAor to have, the whole gain being
There are two wajra of iblving qneftions of this kind;
but if the oMfchant and fiiAor pievioully agree (as to pre-
vent difputei diejr always fliould) the method is deter<t
mined.
The moft common method :
30 -|- 40 =5 7Q> fador's ftock.
xoo, merdiant's fiock«
% •
4
1. s. d.
170 : 75 : : 70*: 30 17 7i> faftor's I (hare of
170: 75 : : 100: 44 a 4}f merch. 5 thegain
75 —
But if the ^in be made upon the real ftock 730!. and
not upon the imaginary one 170I. the fa£ter ou^ht fo be
gratified for hta fervioe» by being allowed the proik of 40 U
of the real ftdick ftiore than-what he adimlly put in.
In confiderttion whereof the above queftion muft be folved
as follows : '
100 — 40 := 609 merchant's ftock.
30 4- 40 = 70, fador's ftock.
1. 8. d«
130 : 7^:: 60 : 34 " §1' ?»"«^^ {ihare.
150 : 75 : : 70 : 40 7 8J, faaor's J
£ 75 --
'^ b 5- A
his perfeti it 200 L-mkeii thqr'iafadh'T^ mdt aecdait|^lie|r
find they have gained ao per cent. i^Wrir'tftt'i'iiillilt^
l^flie.Aft;*^B»6^
Firft, 560 1. '+ 26&tkT[bo\-tM ioo r*io i ;;joo : 14^
V 700.: 140 ;v900 : /fiihStot^t^xtt; *
By the other tiietho<i9 t^ wlwlc A^ l^ilg^
As 100 :. ao : : joo : i<bl. fbe wbok gain.
Alfa too I ^atD : : 360 : 4ot< for "die mefclNuit. •
*«* too : ao :{-tcO': 14b K -W fbe AiAor.
6. A ttettahttt's ucai ftick feifee kx>L and the ftaor't
ittl flock?
Firft iopi, -- 20 := 80, wliif^ ibfi h^.m^pat hu
But by the other methods iboL ^^20 :»8o ss^in tbe
real ilock. -.it, .11 ->' ';
*• * 460 -«- 100 = 6b» vbiriLtiw liApr iqthii.csUb.imts
in* ^ ^
7. A merchant'^ refl flock looU asid the jfadqr beioj^
aDowed I ot the gain fer bia fcnrice; what' teal flock iniitt
be join to have |.or die-gftfai?
««
•By ^tt firft method.
4) joo (25; and ioo 4- 25 = t^sl* imaginary flock.
3) J?5 (♦' I- J3 »' .4<»t tJwj f^^ bciM to haw f.
41 K 139. 4 d. •« 25 X£^i6 h 13s. 40. thefaaor muft
put in. *
By the other netbod.
3} '00 ( ^31. 6 s. fti. s ^' miagaBai%'flock«
Alfo lool. 4- 331* 6s. So. s 1331. 68. 8d. whole
imagiiuiiy flock. ^ . . (J
3) 1331* 6s. 8d.(44rl;^8s; lo^^d. t^Boi^f.
*•• 44K 88« io}d. i--^lr^s;r8^;i=ti:ii.'^Si( ^f-d.
theanfuper. - •'*-
8. A merchant's rea! ftock being 120 1. and the fadorV
6p ler they agreed, that S|C the year's end the Mtw flioul^
have
m. W ^ ^ *?*: fnd . oia i but. tbw broke up at
|rBt. months codi iut)jag ^uoed 1501. bow tnuch oug|ic
the £idQf to iuve /
Firft lao + 60 =: rto» iffliole flock ; and 2) 180 (90 1.
the (hare of ea^ dr t0^ t«<^ tlAt ydkN end : To that the
lEiiAor was to havie^oL of the mmhant's Ooocky had it
continued in trade For |a m^othi. ^
But la : 30 1 1 8 t-Ml. the feAor^t due of the merchant*
Alfo taoi. — 20 s 100 L merdiant's ) ftock at eight
Allil6«'+Mt:-5 8bI; AlfeciPt - - '^ | moitcfaf.
L t. d.
L f. d. L •• d. ^
And ^t +66 ij 489146 13 4,fiiaor'i -5*^'
9. It it propoflsd bjr ,an elderly ^Hbn in trade, defirous
9f a little P6(rm$ to'Hiimt 9 wtftt ixA induftrious youn^
idlaw to k wM jirthe bufinefs, and to encourage him
oilers, diat if htf Cfltumftances will allow him td advance
looh hh pay^IkMdbd 4^!. • year ; if he fliall be able to
put 20cL Into die flodc, he isJXi have 55L a ytit\ and M
300 1. he Ihall receive 70I. annually : in this propofid what
wsts ^owM fbr hi^ «ttendaftoe fimpTy ?
•
Firft 70 L *^ 551* ss 151* )hence it is plain he propofed
And 55 1. ~> 40 1. 9^ I JL J . to allow him 15 per cent.
for his money.
• . • 40L ^ 15 ss 2^1. the anfWer.
SECT.. IV.
Las S AHB GAIN.
BV this rule We difcover what is got or loft by any fUfcA
' of goods, Of'hotiF much per cent, is got or loft aeoord*
ang to the price h«ugh^ and fold at 1 by which we jire in*
ftrufied to ' raife or fall the price of commoditieii in fuch
proportion, that neither our gain may be lb exorbitant as to
injure ouc cuftomcrs^ nor our l^fs fo great as to impoveriih
oiirfelves*
B b a n At
.
m
L.OSS AND GAIN. ' B9Qk
f
1. At what price muft I ttjXxQWtf bf,riigu» which coft
2 1. 6 s. 8d. tp gain lo per cent.?
10 per cent, is i^} a 6 8 . . .. ,
Anfwer, 2 ii 4
2. A Mancheifter man t^iy^th yarn for 6 s. fbr a bundle^
which not proving fo gopd as was expe^^ed, would put it
off again, .fo as but to lofe 6 per oent« by it $ what is the
felling price ?
. s. d.
10 per cent,, ia tV (i - / ;
I
5 per cent, is 4 - 7'^
X per cent, is |. - 3.6
^immmt
Lofs i- 4.^2
4.32 SB 5 s. 7*68 d. the felling price.
3. If a tun of wine coft 45 1* 19 s. 10 d. how muft I
fell it a tun to gain 267 per cent, i
— a
!• 9. d.
20 per cent, is j.
5 per cent, is ^
X per cent, is 7
4 per cent, is ^
jC 58 3 6i, the anfwer.
45 19
10
9
3
li(
a
5
"i
— ,
9
H
•^
4-
7-
4. If I buy broad--cloth for 11 s. 6d. a yard i how muft
I fell it f o gain 20 per cent.^
s. d.
20 per cent. i$ 7) i i 6 «
' ^ «
w^
Anfwer, 13 9f
m
5. If
Cho^^. VaSS Auu GAIVJ 373.
iimift it be foMjita pomi
10 per cent," a ^
<'per ctnt. k i
t per cent, n -^'-^
t ' 6
- 13
- I 3i
iC »S - 3t> *8 ^o^l' =*** !• fterling,
equal to the pound's weight in a pack, it will Ec ^i s, 3-d.
z pound, and ^ d. over m the whdleV' ' * '"' ' ;; •
6. If tpin2 i* in iht fk^in^f what is my gam per
cent/ "^ . ^
Id. iri ioo~ " • " - ^^ '
AnTwet,"" / 16 It" 4"pef cent' "* ' ^
*7* fiv^a qtialftity of damaged lump-fugir loft 5{d. in
the ihilling, what did I lofe per cent.t ^
.1 m» • • 1 ■« ■
•I "J'i
»6 13 4
,^6 s -
■• » '■ *
TiVnfwer, jf 47 18 4 per cent,
. " .'< ' >* y •-•'*'.,.• ,
'8. If by tlie fale of a cheft of lemons x gained 4s. in'
ihe pound, what is my gain per cent. ? -^ ,
48. uf 100 ; J : . :
t> Anfivcr, ^ 20 perceat- —
fo «*
9. A grocer bougnt 3 cwt. i qr« 14 lb. weight of
clevefy/^ttherate of :2^« .44 per pound, and fofd them
for 52 I. 14 s. whether did he gain or lofe by the bar-
gain, and iWiOttc^J \ .*- 'A
; : ' * Bb 3 2s.
m
LOSS A«tt GAIN. JMbU
7
»* ^
<» .- ^
l6 4
3 5 4
'4
' 1% .3. A» *
itidtot gt 14 «•
Coft-«44 2 -
Ga^ /J Si 12 '-, tb^ 9i4wcr«
* ^
t«J t 4* |>rice of I €Wt<
3
1
X
' • - - -, -
3^ '4 - -
tfiX, an. IK
- 3 - -
1 . »■ "
U
3 5
I IS
4 « ^ -
1 Ji«.
.J £
£44 a -
• - * I
. 3, < t4
10. A merchant boughf 436 yards of IWMl-sdodi fer
9. d. >
• < /J
i<x 4» ItAA at a 3rani.
8
t
eeft.
•*•. X
J*- »
X 105 giiiMd per yard.
43^
i4$ 4
ft
TV
799 4
^— *
39 X9 . 4» tfaeaafw.
n • Sold goods tctr CO 1; il s.'ITd. and gained 3*^ d« in tho
iUjngfWhat didrl gala ft^ cept and^xrM^ ni^MO^I
"^ 1. I. 6. d» .
to per^ccnt. J 50 12 H - • " '
3d. i${: too
id. i» i 17
•■*■
434
Per cent if 29 3 4
• r
4
ice. 2
fr^
2 -
6
tr
- 10
1
6
It
» f
ibbtdiAi
»» * -
Prive coil / 35 ly ^f^l '
IWM*
X^Jlf
CIniklT. LOSS AKd GAfK 375
12. If I buy I cwt. of tobacco for4l. 13 s. 4d. and fell
ikBpiti^ fi^r w. a-fOttUd ^ ifrim A^l g^t^^ Mt^ and wkat
irx 7'gg^ »5ypift»rf7lb.
M ^V
4 S »
•4
« #
£5 ^ 8 — 4I. 138. 4d. =^ai-4d% _
As<4^|f : {^2 ; : ipQ : mo r.'^ hia^mms iofter coif.
13. AManchcfter man bi^'s ao ton of chcefif, with which
,he wtet iatolrfl^^i >^<^oft hiov 400 1. the Trelght'dnd
cuf^n came to 50 !• bis own expoiees and ch^gef Were
rfit 13 s. 4d; how ititiff &e feB it st 'pound to gaift idL,
per cent. ?
- - - - 4QO
J^IMl-ClrflDbt
•-**■
^ofof^u.^^iW'i i
•i-4'- J it4>.- "■■■
93
price.
112
1.
001569.
4f - I*.
«
* * Anfwer, - - 3d.p.lb*
Ii.# A ftafionerfdd qwBs atit s. « thoniand, bv ix^ich
he'tiiMrrid4 xtAtcmiouaf^ bK growing fcim^ mUed «btin
{3j«.6ii. a.tfmifittid }. what m^t he claar ptr tent, b/
the latter price ? ^~ x.<.
f-X^ sa^s4i«^tfd> gained per liibufand by the
133. 6 d/«rr- 6 8. 104^. =s 6$t 7j[i. 2^-^» g^^ P^ hx>o
bp Aa fecon4 lale.
37^ LOSS AVD GAIN. .B{k4;1I.
i5» .B9iight M^t in XiOiMlofi ^i ^u^^^fkiti^ fui^otnd
fold them afterwards in Dublin at 6ft* thrp^r ^^Mwitttf ttf
the chargeB'4it an average to be ad. the pair.^ Mi €Ofl£der«
ing I muft lo{b xa per ceot. bj remitting m^r money home
a^ain } what muft I gain per cent, by the article of trade i
68. = il.
As *ii : 1 : : 122 : ?J22. Then iooU.^i =tS8.^'^
240 lo I 53 . ,
^•..Vijok 10s. xxy^-d.---* 100 1. £=: TO K lot. 1144^4.
theanfwcr, ' . ^ o - «
« w • 1
16. If. my fai^or at Leghorn returns me 800 barrels of.
anchovies, each weigjiing 14 1. net, worth lajjd. per pounds
, in lieu of 7490 poun^ of Virginia tobacco ; ana if 1 . find
that I'ha/e.gaioed Sifter the rate of" ,17'per cent« by thif faidf
confignmeiit, ^pray hew was mv faid tobacco invoiced per
pound to the fador, tnat is^ whtt was' the prime e6ft7^ ^ •
Barrels 8od-x~ t^^^- = iiaoo> it raf.d. = -^«
Sid.
, 17, If by felling bops at 3L xoiicpercwt. the pbotcr
^If^r^ 30. p^r cent, wlut was his gain per ceht. wmn tbo
Tame goods fold at 4 U and a crown i
■ • . i
L ?o^ + 30 == »3Q • lOP - 3 SJ- « ^^^3 =^coft per^^
4.25 — 2.6923 z= 1-5577 1- gain pc* qwt.-* , .
2.6923 : 1.5577 :: xoo : 52.58414 =^5^*';*^f»J
8.29 d. the anfwcr. . . : ...l r -.*.«. r
- ■ • . * t. •
10. Sold a repeating watch for 50 gumctSy'Wd Wib-
doing loft 17 per cent, whereas I ought in^ddaling to foLWft
cleaf^.^O'pa- cent., then how inudi ws^ it :^(»14 on^rdke
juft* valuer? .... • »♦- . * . 4i ' ' i . i 'i* ^ -'^ t-»^
J.- r-j.v^ too
I
Ouprnr. LOSS AWDGArK. ^y
4 - .
, ^T^n^too :> laoii 65.'2530iii i ^5.9636141.
*ii^fif5.^j6si4'-i^5a.5=ai3'.4036i4. = 23^h 8s. -d. jf.
«»
19. If.by leiliittingfoHoIIiand, at3t'8. gd. Flemifllper
pound flerliog* 'fiv& per cea^ is gamed \ '\Mm goes the ex-
change^ when by remlCtaHCd I clear 10 per cent. ?
lojlr : 31.75 :: rioL : y^.i6^^. =s rl/ ijs.
3.143 d. the anfwer. " '-
. • - '••.■• \
10. If by fending pewter for Tuvtyy and paMing ;.:«nM[
it at i^^. ytt pound, the merchant clears cent, per cent.
^irfeanr'dMa he ctear iniIoliaiid»::vd»rc he difpbfesiof the^
cwt. for 81*? . . ^' r r
3a.o
M
W
^4
*:»38
And as his gain was "cent, per cent, it coft iiiai 5.j9f L
per ciiiftc . . -
Therefore 8 1. -^ 5,94 =.a,p/ = ah -^s^.^J^ tiiA ia*
iWcr. \ .
2i« Bought comfits ^.to the value , of ^41 1. jsw.4d.. £9r
3s« I d. per pimnd ; it happened tliat fo many of them were
damaged ja'carriagesr that jly felling what remained'good at
4S. 6d. t^e^und, my returns were no more t^aj):34>^
2s. 6d. pi^y how much of thefe goods were fpoUedij and
wlUtt^ this part ftand me in ? «
34K 28, 6d. =: 34.125 - - 3s. id. = .1541^- .
4 St 6d. spt .22j[« - . • ' ' i t ^
As .22<j;L : lib. ;: 34*125 ; I5i*^> remaini9d.good»
i.-^:.i54rji ::'j5i.i6.';''23. 38^941. S3 23l.'7f.;7f d.
the gbodt coft. -1.' ^. ,, ..- ^
•Then 41 J. /as, 4d,-^23|. 7s, 7|d. =£181.58. -^J-d.
thd diin^ c^tt.
. ttv ▲ thai* 45 pifM lOf: Malaga win^, ^wfaich he-partM
vkhrio fi at 4^ pet cent* profit, wbo iWd them cc» Cfbr
i^v 38I*
jr^t LOSS A90,&AIN^ Bte^IL
38 K lis* 6 d. advantage ; C made them orer^ ta D for
50a t. trs.6d. wiid dw»ixim^y'6^pti^
the wine coft A a ^dlozv ?
the iii:iAe coft A a gfldlt
500 16 8 = 500^8^ 1 '
^ ^..+ 3J u6g 38,575 ?=;&vi^!?rj .
560.8^ — 71.1291/ s 429.7041]^, Jjrrcoft.
429-76416 — 18.62051^1 = 4ii*9S3&2^ Afs i^dft
i26 X 15 = 1800) 41 1.08365^ (.2175045 = 4s. 4^d;
them again immediately for 188]. 10 s. with fiMrmftllte
lit; what 18 gained per cent, per anntmiY ' - v*
1*
• mon« @^*
180 .4 . ^.5 • f*r / ':
' • 100 . 12 ■ . - ' - *' ■-* . ^^
f|i» K 4 <3i 720) fo2oo(f4L 9s^ 4<« the anllMn
'-^04
24. Having bought 160 gallons of French bcattdr^ at
6^. 6 d. a g^lorf, diere chanced to Icelirout it ^€lta\ at
what race pw gallon may I fell the remainder, with c%^
months credit, fo as to gain upon the whole prime coft^ at
dw rate of 12 per cent, per annum? ^
,^» •• n t • I ,
>»1 •#
I, mosk faiiu
15^ • O t ' *- ' ■
#^
iC 5^> prime coft.
. foo y i^'s tioo) 4991 (4.16.!. ^in W the wkolcf.
Alio 521. -|- 4.16 = 56.161. and 160— li ^3; 14a gajjpi
%$. Havini paid 14 s. a yard for loo yards: of cloih, I
^VDpofe'to cjin^s per<cettC.' rradf cnoDey ; and if 1 idi it
lipoid time, tB. have snortpver 10 per ctat. perwoMt.ftv
• ; At
L, 70f prime coft.
Chap^nr. LOSl iMtf GAIN. 379
the fbfMiiriid^, /iji^ai n^ilbfae Aejoriarof phe yai^ wit^
firittoiUbs ereaity -raauKe 56di tlSfe ^n$ f
» . p L' ■ • -^
.25 per cent. i&^ 75^ ^
•■'.'• "'* '' .' •' " i.M' • •'
f ♦ I . • < •
; too) |^J!4^^(^i|fe$i =p. ?»$• 4H the iMpfjiireK^ , ;
a6* I/aid out Ia a lot of midUn 480 L 12 $• upon exami-
nadoii of which two parts in feven proved daoAged ; fo
that I coidd make but 58. 6d« a wd-of the fame; and
bv €0 doings find I loft 48 K 18 s. mit ; at what rate per
ell am I to part with the undamaged muflins to make up
myfaidlofi? 1 ^
-, , aiiot 2 4806 d6i2 '• *; ^:
48oLi2s.«^X- = — :?R^=ri37 6 3|,
a»ft.i^ jlb^ iama^ goods.
0 M ^ ^ * ' '
'■■•■■ • • " * ^^* 4818 -»
loft by the daihaged goods*
• i^^ 88 8 3|,
made of thfftunaged^oofti.,
waged.
2 24*56 8664& ' , . . „ ^
1-: -?r^: I i:">**^^: iigjft vMlsmall.
.7 77 77 ^^ -^
•^ ^ f^.a;,..^;^ 803^^ yards undam^gedr
^^^^^^ 2152= 59iU js. 8f«.inade
**80Ol ^ »*^4^J 5 100661 - rer^ *
t8« "'Qi IE* F, • ■ ^ — ~— . .
CHAP.
• . ^ ^ '.^ ?;-.T ? r*\j: rY?:' .- •* ..-. .r . .
■ p - > .' A ' K • • Xr ' xS ' • ~ Jew - ^
W(|£N^inerchant$ or"*tra^^eh (bcdianeje 'o0£ c6in-
mcMlity for anodier, it ils caDtfd bftftefing-' stud by
the rul^fof ^rbpordoo, the price and quantity of thcrgdods fo
exchang«i are determined, fo that neidber f arty flMdi fufbdn
a lofs by fuch traffic. ^:
In folving all aueftions in truck,^ the intrinfic value of
the thinjr received ought to .tally wit(i a like value of die
thing deliyered, where theyvdeal upon a par $ for if there
be any difference, fome one of the paidfes faa^^thefuhraatage
of the other by the vakie of tixuwSwmo^ ^> <
I. How inany pounds of fuear^ ^t 4j-d. per pound muft
hegivth iji'bstri{^for6bgrofsofinkle, ^ihT8'£ per^ofe.?
' 8 «•, 84* , 1 ' ' r '
i iV.
I t
104. .
ao8
6o — '
^ . tf *• »
4f s 9) 12480 halfpence.
r.i '
1 3867 pounds, the anfwer^ , . . . ;
r* f ;^ Or, 1 J86 lb. and ^d. in money.
2. Two merchants, A and B,, b^?r j,, A.i^H|f .fXr^
change scwt. 3qrs. I4^1b* of pepper, at 3!; ids.* per
Ofii. )mSk B'fer cotton,'^ worth 10 dj per'pann9; l^w
much cotion mufl B give A for his pepper 1^'^
1. 8. 1. «• 4» ^ .r. . > ;
4. 3 10 / ^ : !ao IX .^ :value of A'spcpper.
5 20
17 10 411
l^ 6 — — - 112X cwtqr^lb.
8 9 io}4935{493isc:4 i i7cpttoii9the.Mfwdh;
3- A
»
Chip. V. W M' 11 T E K : '38?
3. A and B Varttf ; A gives 120 yards of kerfey, j|
yards whereof coft tss. gd. for ftdCMngs, ^iit 7 s. a pair,
and b«Q.at 6j$f fi (L <icfa». .^ foual Ainmbec pf ^s, and a
pair Qt lloc)dq^j; A9W maay of each muft ^ give A for
hjskerfi^i. S.'.^S'
i* y3*. t. d. I. • yds.
3.5 : 15 9 = .7875 :: lao
. 120
04^5000 (27 !• value of die keriey.
. Thcn^a.'.^ 6«« 6d. sii r3s* td. =: .675) ay«ooo {40
pair of flockinga and hafs^ the anfwer. >
J^i^ Xwp ipcathant^, A and B^ barter i A would exchange
lOcWt. of cheefe, at i 1. i s. 6 d. with B for ei^tit pieces
of Iriih dothy at ^ L 14 s. apiece ; 1 demand which muft
ieceiv« money, and how much ?
B's 8 pieces of cloth, at 3L 148. per piece, come to 29 12
A's 20 cwt. of cheefe, at 1 1. i s. 6d. per cwt* - - 21 10
So that A is debtor to B .•...^82
5. Two merchants, A and B, barter -, A hath 86 yards
of broad-cloth, worth 9 $• 2 d. per yard, readv money ; but
in barter he will have 1 1 s. per yard ; B hath inaloon, worth
28, I d« per yan]« ready money j it is required to find how
many yards of ihaloon B muft give A for his doth, making
his gam in barter equal to that of A ?
. ,.,The moA. common method;, in authqra of folving thia
^queflioA is as follows.:.
AS9S. 2d. = nod, : IIS. ±1 132 d.:: 2s. i d. = 25d.
: 30 d. =2'a^.6d. the advanced ► price of a yard of B*s
ihaloon. '
Alfo 86 X II = 946s. =471. 6s. advanced value of
the cloth.
2.5) 946.0 (378I- yards of ihaloon, the anfwer. *
Wh«il the price of each quantity are raifed proptr^*
tionally, the quantity fought may be found by the Teady
mondy values, without having any regard to tot advaiTtsed
prices.- *
So
^ B" Ai ir 1! m K Bofirll.
Sa d)at.<he foc^ing (lueftkmjBBV ht^W!Mi«jW*«w '
> C 4; 9.4. ■• ' .:-.-. r. J"- ih.'i.?
5 ro - ^ , .09375) 35,475po (178.4 ya«I«,
• 7 tbf Biuwer-aft before.
39 8 4 = 39-41/9 value of the dotlu
J >l » ^ L
f* A hiii ^MMiits ifTdrib 4d, a p^waii hat till bnkik
cbaiges 6 d, and alio r^uires f dP tteit In rea4^ atebejr j ' B
lU9 oandlo^at 6 Sw 8d* thedMcti, ^ttd litt4i}4>lirtiiv MftuBft
0ian, ckarges but 7 9. ihovM tbefe perfons deal togiikcttot
the vilue of Tfi !• bow Qiucb will A hay^ goi: of S f
6d. : 4 d. : : 20K : i3l* 60. td. worth of A^8 (ilitiliitK
I}I. 68, 8d. -^lol. = 31* 6s. 8d.
^ ^ 7fc ae.S5» 111*61* 6db » .^jl.
L ik db f/ '
•B^-jaS-- 10: 9.^8571^ =9 s 8 af
368
A's advantage s= ^ 5 19 -« 2^f anfwen
I <ii
f^ 1 my fflei'ehaim hare Tarious kinds of |joods to battdr^
A ftalh 73< jards of Indian filk, at 8 s. 6 d. per ^aitl readj
tbimtfiMoi m barter 10 a. Mo 53a eanet, at 3 s. s^ieee ream
moncjy and in barter 39 • 4 d. and 16 pieces of maiilin, at 4L
^pitcfr rrady itoney, in boater 4L r^a. B baili fauiec clotb/
at 1 1. per yard ready money *, glab nuHinfiiAQfe^ at:i a. 84^
per pouna ready money, and a finer kind at as* 4d. per
fotind ; how many yards of cloth, and pounds of each
ind of riafs, of aul a like number, muft^B give A, ad«
vancing his goods proportionally alfo in barter f
L a. d*
A's Indian &k 735 yarda^ at fts. £ d. is 312 7. 6
Canes 532, at 3 s. - - • 79 16 «
¥ IMliii i6pl€«iS| sr;4L - 64 ^ -
L I' /'..
£ 456 3 6«4s6-i7S
OofOfr, w M K n % m .^
Gidb auadEiAwe fcr Jb« •• • •it ft 8
DiMt ^pr^ftrt -----«•»♦
^
•.* x.2^,4£^i75{38p^ of each ibrt of Fs diiof^ tke
4nfwer#
8. A wd fi iNtfter; A katfa xoo pur^t ol IwMcUcWth^
worth 12 s. a yard ready niooeyi but in barter be will
iMive i3a^44^Mil will aUbbavejr«f the bttsw vjdue in
»»4y«Nif|r » R iMttfc f«^.iit8d« apoiuidi ham mich
^^A!0irpili^B.|»4elivar» aadlMmiiiitobesaiiedtM^pial
•thakMtcff?
f 00 fanb 9r do^, at 13 s. 64. per ]«rd» it 6j I. 5s.
4>67 ^
^■i^
16 t6 3n^«soDe]r, s= i6t8isU[I«(U)d-ns.9:.6L
.6) 16^125 [2$^ yatds. Alio 8 d. s .Oj
Thenioo— »^s:7i^yd«. v^ikhatias. is 43L 38.
.03l)4a-««75 9<*-=!43-W7J
93 43 187s
.03) j)8^687f (i2&2f fere, pounds of fugar.
Airoi2s. : 8o. ;: 13*5 <• : 9d. advanced jpirice of die
i^gar. . ' *' -
r» 1 .^
9. Abas ker(e]rs at4L 51. a^cce leadjr iMQty pin
liansr tbey are duugod hvbui at jL 6u aadhf. and half
of that bequi^ed <iowa ; B has flax at 3d* a jpoiiiidi how
oi^tfae t0 nUBitia tsudk^ 9^t to be hurt if die otor*
tion.of A? -
FiH(» cli 4s* r^4l- 5s,;c:liU ts.^B«2)5U6au(2l. I3».
paid.
Then 4L 5 s: •— 2 1. 13 8. = i U 12 s. =s 384 pence,
value of the half remaining.
And il. I2S. 4- il. I s* s ah 13s. si: 636 pence,
siade of the half remaining.
••• 384 : 63d : : 3 : 4J4, die anfwcr required.
16. A hath 40 pair ^f ftockings, at ^s» a pair ready
lixmey, or 3 s. 8 d. in barter 1 but he is willing to di^unt
I three
^ K A It T m m BbflkE.^
thnt j^r oiiit«'<of his batter price, to hfitc :^ gfif paid in
ready money ; how many yards mnft he detir^ wiHEtbe
idoney'«A itqitifefy and wbtrii Ae.r«l»\4(liii doth to
equal Ac Iteiarf'- - - e - : i.: ; ;:t ;: '..
Firft, 100 — 3 = 97' ^^ ^' ^^^ ^ •tSfV sdlb lO 9v=s J
.iodT97 :: -183 : .1778^ = barter price ^erthe^
dMcount, . - .»
And 40 X •1778^ = 7«M3'I. value of the ftoclBftgs"**^
the fai^ pri9C. , ' , . ,
* 4) 7.1ij(i;77«^= il; 15 «. ill d/ ready nio^-f' *^
.15) 1.778^ (1*1.8/ pair in value. ' ' ^ *
40 — 11.8/ = 28. Impair, at 3s. is 4.211^.
" .^) 4,5tiif(o.44J =s 8 yards^ i quarter, i iMuly, "ne*!^.
^15 : .178^ :: .c : .59^ r=r 11 s. lOb^d. flieadvaiicdl
price of the broad^cloth.
II. 'A let B have a hogihead of Aig^* dfjS hundred
weight) worth 31 s. for 42 s. the hogihead, ^ of which he
is . to pay in cafb ; B hath paper wotfii 14 s. the ream,
which it is agreed &all bear no more than 15 s. 6 d« ^auid at
that rate truck for the reft ; how flood the account ?
1. s. I. s. . *
•Q ^^^ r..»«r •♦S ^ ^ I :• 5 37 '^t advanced value.
x8cwt. fugar, at| , „ J*a | ^ jg^ real value.
A jf 9 18 advanced his fugar.
' 3) 37 ifiCi^l* i4s. rcady^ money, and 25!. 4s in pap^r.
Then 15 s. 6d.=:^ 14s. = -*,aad2<1.4s. = — f.
... , 40 ^ 49 ^ ^ 5
And^^^sri--ts.6d.
^ 4X? 40
Asl': -3. ::iZ^: M-al. 8s. 9Xd. B adifanced
his paper.
*. • ^1. i8s. — 2I. 8s. 9^d. = 7I. 9S..ay|d. the
anfwer in A's favour.
12. A barters with B 40 lb. of cloves, at 6 s. a pound
to equal the barter, and how much was delivered i
Firft, 100 — zo =r 90, and 7S* 6dt s •375*
As iQO : 90 : : .375 : .3375 = 6$ ^d. ss A's barter
pncpy when 10 per cent, is deducted. 40
Chtf.t, i A tt "r E tL |«5
4f^lt 'fJTt iss J»5 sx 13 1. ib k ihdae of tU cldTct at
6»75 : 6 t: Bi : i9.f s tts« 8cL tii^ le^f *ttaMfi
prte cf tli0 vdrci* (^ £. F.
I ten bOoMett <b ^e fiigacibtla Aldlttlder Malcolm^
tcadier of tfie mathematics at Aberdeen, for die foregohig
as well aatte 71b, 8rh» and loth queftiont, and fone others
in this work^ liriio, after exf^dditig tMH liUfbkei of their
firft ftopoStn^ points out, and fully demonftntes the above
nettod tb he true.
I2> A his 50 broad-cloths^ at xjL tds* iptec^} butia
diange reqUiBS i^^ ddLing wool at 2 il. iS i. pet done of
B in return, that was rewy WortK Vmi 4s. ad. a tod;
die queftion is, hoir fnnjr fades of wool will pay for the
cloth, and wfakk df tSe icAih has tKe be^b Ih die bar^
gain?
L s. 1.
13 -'X5Os£6«0}jiJvu)ce(i valoeof thAhroad-dotlu
U 10 X 50 =i iJi, tid V^u&
' t ' ^ fiV ^ ^ tiw bria^-cfodl,
l" ti^ tod ^fn, Mr Jd; per ft6n^.
is. frd. £:: .iii% : 1 : : A^o : <id6 lloAe.
tf Ml = a67^ioo (aucrfi^lcs cif woo) for ffve pieces of
^ [i$2bo ftoni^, it 5(f. pit i6M profit
s. d.
4lJ I 8
-•MM-^irtiik4
^ ,jo9 6. ^ gaibi by die wool
Therefore loSk Ss^ .8d. — 75!. ^sii^ *»• td. B*s
elear gain by dOfd!^.
' ff; A &a^ r66 IWUff 6f pi^, rf ffs. ready moilcy, which
in barter he fets down if id's.- B, fennble of this, has
paa^ets at 6d% nfi^^i k^iy moWly, which he ade-
C c quatelf
|f^ B : A R T ; E>.r;* -Jhqiffm
quately charges, and infifts to hare over an4»b^ye Jfof the
price of thofc he parts with In fpccie* \^hal numbCFof
books is he to deliver in lieu of A's papeiv whut„ cafti will
make good the differenceyiaad howtOMich ds H (thl^: gainer
Kw t-Kic affair? •, -.:. -i •
« ^ ^W w > <» •
by this affair I
■
As 8 s. rios. ;: 6d. : 7id. advanced* jrice^^f B*«
pamphlets. _ •! -'^ .v •
T> T
100 reams ojf paper.
+ 20
f 5°; rdvan;ed H« «f ^'P»P«^
/ 12 los. B to haVe in caih*
40, value alfo of B's pamphlets.
X 40, fixpences in i h r
160O9 pamphlets to be delivered.
• .• ^.ol. i — 12L 108. =: 271. 108. what they then ftood
bim ins fo that the advantage to B was 12 1. 10 s.
' * 45* A and B barter ; A h^th 140 lb. 1 1 oz. of plate, a€
6 s. 4 d. per ounce, which in truck* he rates at f s. 2 d* an
ounce, and allows a difcduixt oh his part to have -f of that
in ready money; B has tea worth 9s. 6d. the pound,
which he rates at lis. xd» When they con^e to ftrike the
balance, A risceived but 7 cwt. 2 qrs. ,i81b« pf tea : pray
what difcount did A allow B, which of them had the ad*
vantage, and how much, in an artlde of trade thus circum-
ftanced?
f I
1691 at 6 4
oz*.
8... d*
^, fibqi^tj z
I
281 x6 8
169 2 -
84 II -
563 13 ,
.4^ 5
i
£ 605 18 10^ its adyan-
£ 535 9 8, real value » ' . ^ ced value*
— *——— ^ of A's plate,
6os^ l8s» lod. — 53^1. 9s, 8d, pz 70!. 9s. ;^d.
i
519 7 6? 72 i8^ ftt9 6 per lb.
— ' 8
cwt. qr. lb. 8. d,
7 2 r8» at II 2
8
4 9 4
17-
■ "I
3
.16
^^^^^^^^^ •
•
-
.7
a6
12
ft
■
# »-
_*
"53
:-4
• •
,.
-7
i 62 10, 8 . ^ 3^V 8-
7 . «6 ?2 -
' ■ 7 .12 --.
437 H 8 - 19 -"*
8 18 8"" *' -- - 407 II' -, fcatvaluc •£ BS
• ' • -^Xi a-4-' -• - — ^~~^ '^^^•.
» I
-479. I --9-advabce4 valuet>f B'sfta^
-— *— — ^^ -^
^519 7 64 -• 479 1 1 s. = 40L 69. 6^: iitednrti
' :'— 407rt ;^ir " - V . < : ■ aUo#c* b^iV.
'1' . * * "^ .'• ' . !- . . .:::.•-•: , • • • ' -
III 16 '6§-' ' ^. ,*. ' • • • f 1- •;; '^
*r7<^ 9 2| A,'sadvwtag^b]C^tlfd'of hi»pl«l««»
£ 41 7 44,< B't urbdJ^ adnatagt, Q, E. f .
' »Mj •
• . 1 • •
t6«^-A, widi ijitetition to clear ^ guineas 6h a b^o^iii
ivtth B, rate^ boos at 16 d. oer lb% that ftood him in 4b d.
B^ apprifed bf that, Kt9 Mim mal^ which coft 20 s. a
quarter at an adequate price ; how much malt did they con«
trad for f
• • I
5
As icd« : 16 J. : t Id s. : 22s. a«iyanced value of the m^t«
32 *«» 20 :;= 12, B^s gaiii per quarter*
Guineas 30 ^ 21 fa;: 630 ihtUings*
12} 630 (52f quanefs = 4^0 bufliels, the ai^i^r.
Cc 2 17. A
0^ t.KjK T\ .Ef If 9o^m
ip^^A faiA%.ttwky A. has Hcwt. SiVa^ oC^Farnhaiii
hops, at 2 1. 19 s. percwt. but in barter infifla ^H^ffit
miAcas > B ha$ wme worth 6 **P^ SpUpi^ whith^ht-riircs
in proportk>R to A's dtttnA OndiiWlinca A«|«c^4l
but a hogflieaul and a half of^ wiat^ praf whai b»^ he m
ready money K
* •«. r-
<mti lb« ^ h $«
14 8i at 3 3
"7
aa I ^
a-
n CWt. ip9, Ik
4+ a - 14 - -
I II -6 - a -
- - ^ — t
1* *-25: ^VA ' '. -3, : •aaoSW* «^n^ tAm of .4M,
uton 01 wine«
li bog(bea4 of wine is: 94^5 gal. x •p^^ » SP^a^a* .,
• 46U 75. (S|d. -^3oir $1, 5^4 s i6i«- aa<.H^
/f ^i Ajim erder us put off 7^0 dti of dteMoi HeBandt^
wordics. an ell, at 68. 8d. propofey^ kr eafe he has half
the vabe m money, to give B tbereoflVdifbduni of to per
of t^e ^f^^^ management^ turn im jdlke at 30 s, the
pomd-i' pAy wtei %te> k<»iNny4of«^ hi fdUlv/ money;
and what quandty of fafiron was he todeHnroHtlfe change 1
. -5? tJZ^o dlsv
^ {440' aavattrt*V*"*^ ^ SfllMW' I
24, dt(cottnt« -'^
216,^ remains. ' » ''
108^ paid io ready motien
*.>
''-• ■ #■ •
itfe
OMftofl. lit <2 R ^ R '« ft, ^H
^'AM'i|ji4li'« f ft; ;: i$8 t 72ft. ihe'Duantlty deilveredi
o; ^;. ^ .-. ; ■'- ; . «.•.•..•.": ! .;. • . r-f' •>
A
C H A P T B-A VI.
SEC T, I.
E X e H A KG E.
*
EXCHAN04lfG the coins rf m« country iMo
thrfe of aitodieit }s Itk^ Ac kufiMla 6f bartering com-
modities (that is)i it confifts Of finding what fixtn of OM
country eotn wBi ke ^nal iti vsfluc to any propof^ fnm of
another counts coift ; and In order to- pdfotm that, it
wai be nteeeflSuy to have a true account at all tim*s of the
J*»^tra!b^ of thew fofetgn coins which are tol* exchanged^
as they arc compared in value of our Engliih coitt ; for thtf
pwrrf4ftxaange ^as the merchants call it) differs almoft
di*tf 4ty*oiif londmi tb other countries ; Aat irf, k rifes
aM.ftI}9,^coMiAg » money is p^nty or fcarce, or^cord^
ing to the time allowed for payment of money in exchange,
^jK^^f *f«b^<^^ftd payments in foreign countries tta^y
IWiUiqe ^eir |)tif chafes and payments in ours, ttyro witt bo .
j«^ ci^ougK or bills on the one tp clear account* wicb the
^IjfCi. ft» ^at in this.c^ the exchange, oo bolfc lidcs will
*^^P^» ^atis> pne who fiive^ i^pney io& one jcountry^
wiK rcc^i ;>^ much m the oth^r in-wt^ghtand ftandard.
V a naoon, Supplies ,us wif]^ inarc tbaa it takes from us,
itf IT we pay that nation more money than it pays to us,
Iftcre will be a balance againft us, which we muft ncccflarily
prfjr ; in order to which, the demand for the money of that
nation, or it^b^ of oxckaiUM, becomes greater ampng us
than the quarttity' fo fupply that demand, which raifcs the
value of their money or biHs, and lowers om'e ;, or, la other
words, pi|t9 the price of their money above par, and ours
below it, which conftitates what we caU the courfe of ex*
change. From hence we may naturally infer,
I. That the courfe of exchange betwixt two nations is a
bc»ld, wl^ich prodaim J pwblickly the ftate of commerce and
C 9 3 mopcy- .
»90
BXCH AN0B.
lMAvy-«ii^ociat»iiS'brtiriait diemi and
is indebted t» the other..
of the two
II. That thr. Nation which is lAdebfeed hath ^ difad^
"vantage in commerce and inoney*traiiia£kiojis ; and thai the
one which hath the balance in its favour^ hath in evoy
lefpe^ thcr iidyMiitagp*
IIL That ^ balance of trade natundly imports fpecie,
and renders, mphey at home more y^uablc ab(o$ul j; whereas,
on the other hand, when the balance is aninft a natiottj
their fpecie is exported, and becoq;ies thereby Icfs valued.
The £ng}j(h fl^ndacd for gold coin is ^z c^ats of fine
gold, and two carats or ^^ of alloy.
In the royal mint a pound of Ibndard gold is divided into
*44i P^u^ '^^ A guinea, at which ra^^ a guineea.will weigh
i pemiy weights, 0-4382 grains.
The Eiiglifli ftandard for (ilver is xiozs. .2pwts* of
4me fUver, and iSpwts. of alloy, in the pound =: •^-.
The pound weight ftandard .filver is divided into 62
|>arts, each a ibillingj fothat a fhUIing will weigh 3 pwts*
'ao«9 grains*
• ■ *
^^Taqle cf the proportUu rf ih wilue in Jiveral nations rf
the world between gold and fdver^ taken from PofiUtbwajt s
Vmverfal Dieiianwj of Trade and Commerce.
In Japan one ounce of gold is -
China -------
Mogul empire - - - -
France - - ^ - - •
Spain and Portugal ------
But as they re(|uired a premium of fix per J
cent, on payments in fuvcr, it reduces it to j
England ^-----.--
^jfplamation and ufe of th^ follonving
One pound troy
one ounce
one pennywei
qnegr^
gold.
ITS
I
I
I
I
>.s<
filver,
- 8
10
IS
16
.U
i5i
&f coins (viz. J
12 ounces.
yweights.
troy -| g f 12 ounces.
- - /•« Vac pennywei
eight I g 1 24 grains.
- - J w ( 20 mitcj.
The
Ctu^;>Vl. B X e H K N 6 & .391
The ffvft colttfnn cxpfdfo the finencb af'the ^fiayeri
piece ; the B. fisnifying better, w4 the W« mffe-tUti the
EngHlh ftandard.
The^^dtid colmitn' is the abfohite tedgbt'bP Ae pieae*
, 7^e third column its ftaaddid w^ghf, or ltd fnamity of
iUndiffd metal-.
' The fourth column its value in Engltfii momjn
Ex. gr, in the fecond article of iilver coin, fhe ^lew
Sfcvfllc piece of eight is i-J pwt. In the pound worfe than
Englifh ftandard weights 1 3 penny weights^ 2( grains, 15
mites of'ftcrlihg filver, is in value 43 pence, anq 11 deci-
mal parts of a penny.
An^ in the firft article of gpld ^oin, th^ old Lous d'ors
is half a grain worle thaq Englilh ftandard ; its weight is
fpur pennyweights^ 7 grains, 8 mites of English ftandard
gold, and its value 16 (hillings, OtIj. pence.'
' The par of exchange between fengiifli and^Dutch money
]& eafily found, thus : as by Sir lfaa<rs table, the dttcafbcrn.
of Holland is worth intrinfically 65.59 d. Englifh » which
is received at the Bank at 60 ftivefs, or three guilders, and
confequently is equal to 10 (hillings Flemifli ; therefore by
the rule of three, as 65^9 d. Engliih is to ^os, Flemrfhi
fo is. 240 d. in a pound £ng]i(h, to a fourth number^ which
will be found to be 36.59 s. Flemi(h ; and fo mugh Bank
money at Amfterdam fhould be received for o)ie pounj^ of
^o p^n9p ftcrling.
u ♦ ■ . ■ -
I
•%
« «
« *
^ ."»
■ >
9
Cc4 Sir
■. w
V
if
l-s
O O
It
^ J5
« o
pr:t^:Hi4^i*or'm
.^>K -^ 11^ ni» v^ w^ I m^ I c^
1
a.
U» 1-4
►5.5
< ^
■I
I 00 H TOO <^^*<« «1
^ir.iin ■, tf J .1*1, > 1 • ».l '^^ * - • T* Tir^—
<
4J M|rt . ••((« Mb*
09
O
>.
•J
0
(X)
O
I'
■c
at
OS
•a
4>
3^
I* f
I I
t I
I I
I I
^
2 "5 « a i:
•» •» 5» pis O '-
I I
I 1
CO v^ o ^<« o «
«5
t, - o
8i« VIS'S o o^
•» «r
«* « g fcT » '*■ -
- w o^ a ^^«^i 'tx^u^K-? ^.s-Sr:
£.1
9e ^
iS 3 fl
w'5 "•
s.^^ 1-5 a-s «^£^
j3 js js -c -c jc -« ^*J3 js ^ jc xr -3 jb ,4- -fi'^ ^
«*'«.
HXC^HANaa 3M
1 Si's"?*?? a &£-tq-e fs s..
•8395
f.a
,8^2
•3
a,
■'■=■=■=■=■=!!
t-PRFl-P
-►•■«■
•isSs
= •!'
If] H*
. U O o bOJS 00?
394-
EXCHANGE.
B6okID
• • • • •
ON
I
f^ w^ -^ t^ fN^oo ^ v> o ^n
<^ 60 O c4 cs r^.p4 >D to N
lit
r^ I ^ %o w^ v^ to >/> lo tn ^ H 1 I I
r< o «a I -r^* I '^'^n"^ ^ ^ roo «^ r* oo
I
M M M I
ero**^ I ro | ^eo «^c««oo toco hi « |
r<.0 ^ C 0^ I
M M »i4 Cl I
rsOOHD C^ ^0
111
I I f
v^
I
00 "(l-wi
O VO^ 00 00
00 OO 00 00 00 oo
00 Vi M'QOO I O N
%0 t4 C tA I - ^
3
CO
MM MC«MlHM|«fMl«4 mU
O o O O t^ro <*>
M M *4 »^ M C^
r^ 00 . Cv Q o *A
I
I I I
o
^^
o o
ctf (^ q> C C
c c 5t V 5
• t3
I rs.
o
I
I
M
u
N
c
Q
ca
c
o o
i*4 «J 4)
o o >^
CO
*^ G* S ©
HN Ih ^ O
c s ^^
3 W W G
^ *i w c
;s « « C <3
O «^ t^ ^ v»«
«, o o o o
o
•- ^
«8 «.
— c *'>
I
bO
c
c:
o
Ok
!•■■
3 •
5-.
i '
o
'c
1)
>
o
M
♦J
o2
^ ** o
.ss Z
*- o 8
o g c V
^O -13 .> "O
•3 S s^ rt =S
8
o
CO
^ "S -»
2 g S *^2 §22 X « X X ;•
V3 O X
V V %•
4> V
^ 5
^ J5 Qt
CI#o«l. EXCHANGE.
3fr
: ?: I MM
•S.I 1 1 ( 1
• • • •
M n M M
<^^N.l-• e« '^hK f O O I- O ^0 Ov f I I I f «^ •^ *^0*
f«i>COC^VO NMfioOlJ^ ^C M>l I 1 I I tNOOOO O
-
» |4> »|30 M K -Jh -4 rt *•[ H
t I
r^ r^ ■* t>- N •* o *A o t« t^ ^ ■* I tNfc p eo •« tsk d. >♦ ei
^ sO *^ <^ ^^' ^
i MOO I M M
^«»»' ^'w ^ ^ «
I I I I I ^ IC^^
8
c ^ ' I 8
to JS
(J
o
>^
JD
O
O
CO
c
0
o
IN
o
§
Urn
o
u
u
o
O «) c4
'c -3 c .
o 5 §
w V ♦J
•5 '^•^-'^
8 0 4^ O
CO
O
CA
O
2
o
U
3
O ^
•^ «» ^
4^ (3 u, g #v
« a o "S G
^ S c te ««
u § !i S
w o o o
•S^ 3-S y^
C M
•"* 3
^ a
U4 P
M M o '*:s o S
^ o
— X o « X
3.J> ^^8^
3 O
O
c
g ^^00 o 5 ^ S ©^
V ui S o C O
5 o "• .§ ,*i{ *i
& C « <5
o.s
oWo£Pj3**0
g
CO
Cm
o
c
'o
u
u
4.J
3
Q
Wi
O
V 3
•^ o
«>
«>
V 4) 4)
o 4>
^iS
4J
4J
^Ht-r-HHhtr^Hhri^
a> o 4> t> O ^o ^
GOLD
1 1 r
t « w ♦«.,<» «♦
% -S'+ot v^M t^ ^u * I •O'O <*)M m'm fi^ «.*tt„ai„*»
I 'I I t I I I I I
.jrrrr«
1 '. '. '. '.
ni
ncJcQAmoimaq
I, »i
'•!
Jill 111 !5i|!=vS"f.
^m mlnmmm
,0qiM$' B X c M A 17 e e:
90
« « "i"t?"fj;r ] I . I j » I "S" T- - « « t ,
"-■—■■;-■ --.---"■**'T*. -*■■■-.■ . '. ' ' i-'.^..-
-r—T.-.-
a S ■ « h J^-S
^^^,11^^553
S4
IJ
I"
EltClAHB,
I
3^ K X C H A*^ N G Ei Bdoklt?
Encsand,' with Holland, FtAvpitM apd QBRMAkY.
The bank of Amfterdam il t&eltti6ft'cOnitdMi6le in'Etiw
rope'; and as bufinefs therein is negotiated by transfers,
millions may be paid in i day, with(^uc the interventionr<of
any ca(h ; whkhisijof the ^i^ateft c^nfequence ijpia^oaWe
in expediting trade ; and is produfli^e of fo great t^mityp
that batik payments is reckoned from 3 to 6 per cent. Ketter .
than pavmonts in cafh, although a premium is alfo allowed
Ae baAK for every depofite.
Th^otlanders keep( their accounts in fiorici^ or guildeiv
ftiveK and pennings;* or in pounds, fliiQings> am pence
FlemiA, dhrided as'the pound tDtrtiti^, '. . ' '
8 peMngs :^
% grots
6 ftiVers
2^ flo](in^
6 florins
5 guilders .
ftiver*
(hiiting.
20 ftiiwert * Vinake one<( * florin or guilder*
rix-dollar.
pound Flenufh*
dilcat. *
Exdome is 'made witK Lood6n from 30 to jS Aitlu^
Flemjflv mr one pound fleriing.
' - ! ; ' 'CA S E I.
Git^ea the* fum due in ond country coin, and diat payable
in an6t]^er country (Join, ' tp find" the, iate^f cxchtoge. *
- RULE-
*-,' ^ ^ ■ ' « ^ , . » J - I ,
As the fufla due : is to that j>aid of pay:^ble :; fi> is an,
unit oof the firft : ta the tnalue of an unit of a ftcond.
•■ .
t. A merchant at Aniftierdsim paid 150 guilders tor i^U
15s. received fcy his t^refpondent nt^Londpn'^. whtt is the
value of a guilder ?
Asi50guil. : 13*751. :: ignih : is, idd. t]ieanfwer« '
2. If I receive in London 678 L 15 s. g|d. for 1173U
14s. 10 d. Flemifh, due at£.6tterdam ; What is the fate of*
exchan|;e ?
As 678.790625 l.fteif. : 1 173.741^ UFlejn. :: i : il. 14$.
7 d, Flemifli, for 1 1. ilberling,
N. B,
..N. J3. T^^t the coMAtiy fin wboft monev the CQurfe cf
exchange is reckoned, has always the greatefi advantage the
I '» «
C A S E 11.
To reduce Flemifli pounds, fhillings and pence mto guiU
iteirs atid ifiVe'rs, ' "
- •' ■ ' ' • H tJ L E,* ■ - :ul
Bring them into pence Flemifh,. then divide hv 4<1 (tedufit
4Dp^cQn|i^oneguilder)aodthe quotient wift be guUders^
and tf any thing remain, divide it by 2 (becaufe tHM .peace
make one ftiver J and the quotient will be ftivert «
3. In 1 1731* 14 s. 10 d. Flemiiby how many guilders ?x
1173L 148* lod. = 281698 penee.
40) 28169^ (7042 guilders, 9 men^
By Practice'.
ffiv. '
ti73'X ^ 3} 7038 •* 6 guilders being X LtSlemiAu
IDS. s 3 - 4of 6 guikleu»oriL
4$. = 14 4of il« * i
-.lod.= - 5 4.0/ lod.
Guilders 7044 ' 9 ftivcrj.
4. In 7042 guilders and 9 ftlvers, how many FlemUh
pounds?
guil.' fliv.
6)7042 9
", . ;
1173 •*, reniains 4 guilders, 9ftiver9«.
3 guilders =:; lo -
I guilder =34
9ffivers z:: 1 6
^£ 1 173 14 10, theanfwer.
i»i»»<
-•■..h.
CASE
igm ftXCMAKGE. Bo«(kIt
C A S E m^
. • • • .
1*0 reduce fterUog iai* Ftadifli
As <iM ^lllid ftetUllg t is to the given race ^Oidnage.: ;
lb it Che givcii fterling : %m the Flottiik foui^c
9. if K Mr ift Lott^ 67tL t^t. ^<^ iUtt ayr^ I
MP ii^ i jR (Mr «tt Amfkfdfltt^ eMhaHf^ IL t48. 74!^
Ai |L : i.;^i9i6K Flem* : : 678.790695 : Wfyj^f'sk
tt7jk 1^4^ 16<t4 Fleririfti
1678 '$ 9f
339 7 lol
4.i
<
T
2 16 61
ti73 14 9t Fletnifli vaXKBBf^ ite aiifiwai*
. C A S E IV»
T9 reduce Flemifli OMrnej ittto iertitig«^
k tJ L E.
As the given rate of exchange : is to ftm pritind icrliog : :.
fo is the given Flemiih : to die flirling lequtiod*
6« Chang<^
HklB^.yi- EJC CHANGE.
«f<#
6. Change 1173I. 14.3. 10 d. FlemMh into fterline ex-
change, at 34$. jd. per pound fterling.
34 7 = Ji •• XI73 14 w
X it2 20
415 *
43474
■ «
— — r !• «• d.
415)281698(678 IS of
3648
20
6560
2410
335
12
>••» «•'•
4020
285
>
1 140
Di
pjsk
7. In 1036 1. Flemilbi exchange at 345. 4td. how much
flerlingr
8* d. i*
34 44: I :: ioj6 \
It P
412 6216 gil4fint
2 20
825 12430 ftJverst
4
■ 1. t. d.
825)497280(602 ij 3i ftcrling.
2280
630 ,
20
12600
4350
225
12
2700
225
4
900
8. tn 5875 florins 17 ftivers banco, how many pounds
ftcrling, exchange at 32 s. iod« ?
8. d. 1. flor- ft*
32 10 : 1 :•. 587s 17
12 20
394
1 1 75 1 7 ftivers.
2
10 s.
•
394)235034(596!.
3803
^ 2574
8d.
210
—2?
4200
"260
12
3*20
»
Agio
>•
Clsp.Vl fiXCHANGlL 403
Agio fignifies the diflFeneocc of Ae i^^lfip <^ ^lUMot
money and bank notes in Hollaod, Venice, &c. which in
Holland is from 3 to 6 per cent, ia Ikvour of the notes ; alfo
the reward given for the changing one coin, or fjpecies of
money, for another*
C A S 5 V.
To turn current money into banco.
RULE,
As 100 with the agio added to it : is to icx) : : fo is any
given fam current : to its value In banco.
9* In 3758 florins, 15 ftivers current, agio 5I per
cent, bow many pounds fterliny, exchange at 35 s. 1 1 d. ?
As 20 ftivers make i florin *« * | florin = laj- ftivers.
flor. ftiv. flofw 4or. ftiv.
Then as 104 124 : mo :; 3785 15
20 20
>09ft 75^75
2 -z
44«5 imtite iS^SS^
€or. 4»r.
■^■F
4185)15035000(1592 12 banco.
245QP
38750
10850
49600
77S9
04 9. Alf#»
.
404 EXCHANGE.
s. d. 1. flor. ftiv.
AI(b,as 35 ii : i ii 359* **
Sookll.
12
43*
20
71852
2
1.
430143704(333
14+0
1474
181
20
'3620
172
12
2064
340
4
1360
67
s. d.
8 4^ anfwer.
C A S E VI
To turn banco into current money.
As 100 : is to 100 with the agio added :: fo is any
given banco : to its value current.
10. In 456 1. 8 s. fterling, how many rix-dollar$ ctirrent»
agio 4|, exchange 366. i^d. i
1.
s.
a
a
J.
5
JL
t
•
456 8
228 4
114 2
22 16 4|
2 17 -4;
824 7 5i
X 6
i
i
Rix-doUars i
W46 4 7^
2
9892
C978 24 ftiven banco.
As
Chap. VI. EXCHANGE.
405
As 100
50
5000
R. D. ft.
I04i :: 1978 24
8 50
837 98024
837
• *
692468
296772
791392
\
5.000)82799.388
8) 16559.8776
2069.9847 =
R. D. ft.
2069 49 current.
II. If by remitting to Holland, at 31 s. 9d« Flemiih per
pound fterling, 5 per cent, is gained ; how goes the ex*
change, when by remittance I clear 10 per cent. ?
31 s. 9d. = 1-58751.
As 105 ; 1-5875 :: no
no
105) 174.6250(1.663095 = 11. Flem. 13s. 2r grots,
anfwer.
V/hat is faid above may b? fufficient for reducing the
coins uf any country into fterling, and to render the fol-
lowing examples, and buiinefs of exchange in general, obyi-
ous to every common capacity*
HAMBURGH.
2 deniers gros
12 deniers gros or 1
6 fols tubs 3
16 fols labs
2 Marks M*. '
3 Marks
7i M*. = 21 R. D, J
make
one
fol lubs,
fol gro5.
- Mark M^. ^
dollar.'
rix-dollar =: 48 fols lubs.
^livre gros or pound Flemiih,
or 120 fols lubs =: 20 fols gros.
' Dd 3
12. Re-
r \
406 EXCHA^&E. BDofcH^
12. Reduce 1541 M*^. 144- fok lfll)$, bank money of
Hamburgh, into fterling money of England^, exohaog^.s^t
32y fols gros per pound fterling.
1.
32^ ; I : : 1541 I4t
6 i6
194 fols lubs, 9246
% 1541
388 deniers. 2467O7 fols luU^
3 '2 .
1 164 thirds of i^en. 49S40|' deni«n.
3
^ ^ I. ». d.
1164)148022(127 3 4, thejnfwer.
8342
194
3880
4656
'3- ^^*^7l« 3«« 4d- fetlli*^, fe6v many Hatoburgfr
I-
32t
254
381
42 4, being. 4. of 127,
5 S» being ^ of 324 nearly.
ao)4 r f • I 91, deniers gros.
»!.■■■ III!
Liyre9
thap, Vi: E X C rt A N Ci 6, ^f
Lines 205 tt 9
ji m^ks in one iivrc.
MfiL
1435
102 8, for |. iivrds.
3 12, for i 6f 2t mark^.
- 6, for ,1^ or 60 fols.
«- 39 for ^ oF 6 dkaiers.
- It» for 4 of 3 deniers.
M^ 1541 14I:, fols lub», the znfwiri ihd proof of the
■ foregoing example.
14. In 750 1. 14 s; 7d. ftefling, exchange at 32 s. 8d.
how many rix-dollars bari6o-of Hamburgh i
1. s. d.
750 14 7
375 7 3t,
75 I 5t
^5 - 51
1226 3 9^Flemi{U.
2i And 3 fols gros = 18 7^ .
9| deniers = jj*^^-
2452 —
hi 23
R. D. 3065 23 fols lubs^ thtf ^fn^en
15. In 3065 rix-doUars, 23 fdls tubs, how many ^unds
ilerling, exchange at 32 s. 8d. f
8.> d. 1. ft» R* Dt S.L.
As 32 8 : I : : 3065 24
392 deniers. 9195
16
147 '43 fols lubs.
392) 294186 (750 L 14 s. 7d, ajifwer.
^ 1988
~256
20
5720
1800
232
12
2784
D d 4 1^*1^
4o8 EXCHANGE. Book II.
l6. In 584 rix-doUars, 9 fols grofs flight money, agio 4^'^
percent, exchange 35 s. 8^d. how many pounds fterling ?
I rix-doUar befng 48 fols lubs, • . • ^ — 27 fols lubs.
R. D. S. G.
J04 27 : 100 : : 584 : 9
48 8
832 4681 fols gro6.
416 6
5019 fols lubs. 28086
100
5019)2808600 (559 R. D. 28 S. L. banco.
29910
48150
2979
48
23832
IT91D
J42992
42612
2460
8. d. 1. E^D". S. L.
35 8i : I :; 550 28
12 48
428 4472
2 2236 .
§57 26860 fols lubs.
4
857) 107440 (125I. 7 8. 4dr ftcn anfw,
2174
4600
X 20
6300
301
X 12
3612
184
J 7. In J075 M*^. 14 fols lubs current, agio 8^ percent,
and 384 dollars, 2 fols gros flight, agio 4J. per cent, ex-
change 35 s. 7 d, how many rix-doUars tapco, and pounds
Chap. VL EXCHANGE- 409
M^ S. L. UK io84 = 108 ft. 6 fob lubt.
As 108 6 : 100 : : Z075 14 foh lubs.
16 16
173+ '7^14
X 100
Mk. S.L.D.
1734) 1721400(992 II i banco.
. 16080
4740
Dol. Dol. S.L. —
lo^l == 104 28 ' 1271}
2iols gro5# s= 12 i6
30352
3012
1278
Dol. S.L. Dol. S.L.
104 28 : 100 :: 384 la
768
1x52
12300
100
»T>oT «^«T 3356) 1230000 (366 16 banco.
R.D. S.L.K.D.S.L. 22320 2
As 74 : I :: 575 ii^J 21840
16 48 ■ 733 M'^. ban.
— - — ■ 1740 992 iii
120 4600 32 -— «
2 2300 3)1725 IIt
-— - 3408
240 2761 1 5112 575 iij^banco.
2 -
■ 1. F. 8. d. 54528
24*0) 5522.3 (230 I 6 ban. 20968
s. d. L l.F. 8. d. 832
35 7 • J •= 230 I 6
12 20
427 4601
12
427)55218 (129 1. 6 s. -^^d. fterling, anfwtr.
Amfterdam,
< V>nO
I
1 I
I I
^ p^
i
Vi
a ^
^ M
%P M
CO
n
00
to
&
U
•
60
0
CO
09
-o
p<
J
^■4
•<
•
:g
«x
S
OS
4-f
v*«
U
0
f S 8 is-
SO
poc^roi
o o
CO CO
o« a.
btO
•a
I
o
c
^ i d ^ o
o o o O o
O
I •
a
1?
c
o
o
u
6
o
h
•
H
Z
M
O
Q
<
o
s
o
I
&
11
U
i? pS
4>
>
t2
18. What
Chip^Vl BXCHANO& 411
x8. What fterling doth the i Avowee oa the other fide
amount to; viz. ^23 gilders^ 6peAiiiii§i» A^ifi, 6 4 Fie-
miih for 1 1. fterling ?
8. d. 1. fleoau
34 6 : X i: Sii
12 20
4x4 1.6460
414)32921(791. 10 s« 4id« die an-
394»^ fwer,
ao
4306
t6o
1920
264
FRANCE^
England exchanges with France on the drdwft of^ diree
livres Tournois, or 60 fols Fi^cb^ and give» (WMt, fter-
ling', more or lefs, for this exchange crcfwM^
2o fols i make one < livf e f^ Frarite*
3 livres 3 ( ecd Or crown.
The exchange between France' and other countries are
more variable than any, owing to the frequent alteration of
their coin ; which is fo great, that Mr. Poftlethwayt affirms
he has known, in the fpace of a few years,* the crown or
ecu of three livres from 5d. to near 60 d. Englifli ; but
that the iirft indeed was payable in their kank-notes then
(viz. anno 1720) rn great difcredit : fo that there can be
no other way of afcercaining th^ par of exchange with that
kingdom, but hy all wB^d ^Jf add weighing their fpecies
at the times.
9 19. In
4X» E X C HA N G E. Book U.
• •
19. In27l. 16 s, 8d. fterling, exchange at 314^ d. per
ttu 5 how many livrcs Tournois ?
d. liv. 1. s^ d.
As 3ii : 3 : : 3^7 16 8
20
556
12
6680
a
13360
3
63) 40080 (636 liv. 3 fol. ^ den. anfwcr.
228
390
12
20
240
5^
12
612
45
20. In 5731 crQiirns;.45.1bIs, how much ftcxline, ex-
change at 3 i-J^ per crown ? ®
crown. crowns, fols.
» • SIt' • : 5731 45
Z 2? ^ *^''^WJ^ = 60 dcniers,
M9 343905
H9
309SHS
1375620
687810
60
8
12
85632345
1427205
1784004, pence fterling.
20 j 14866 81.
£ 743 6 Jij fterling> anfwert
2It Sup*
Chap. VI. EXCHANGE. 413
21. Suppofe Paris owes London 4186 livres^ 7 fols^ 5 dt*
niers, and remits the fame fum to London at 3 1| per Crown.
liv. fols. den.
3 : 3i|. :: 4186 7 5
X26
8
253 837^7
X 12
1004729
3014187
502*3645
2009458
80
8
12
20
254196437
3177455
353050
44131^, pence fterling.
3677 7t
£ 183 17 7t fterling, anfwer.
22. What comes 175.96 quintals to, at 2 liv. 17 fol. 7 den.
per quintal, of 1 00 lb. per invoice on the other fide r
quint.
175.96
2
X
1. 1
87.98
I
43-99^
JL
4>
17.598
X
6
4-3995
•733^5
lIv. fols. den.
506.62075 = 506 12 5
•— •
Bourddaux,
s
•I
I
I
BXOHANflfE. aeokIL
9
O
60
d
I
o.s
s s
CO
•-a
cS o
i
4'
9
I
o
I
cr
&
"3
«2 M
O^ C^ 0^ O^ P^ jO^ O^ 9^ 1^ p I
w H C^^iAvO r^OO
O O ChONO^O
M M M *i«
•CI C3
I
O O
>
r
Chap. VI. JB^CHANg^^fe
xn
o
s
m
^ m
M
"*
NO t^
?
♦^ 1
CO
6
I vo 1 t I <^o
c
2
I.
S
>
o
u
C
9
I
3
CO
S >>o § o 8 I
i|l>|ia|'
O O O O O O O
>
S
•J?
§
o
a
t2
4f€
-ac
M
<
CO
no
8-
i
23. Three
4i5 EXCHANGE. Book H.
23* Three hbgiheads of Graves claret, at 50 crowns per
tan^ periAvoiGe,
crowns.
X 3 livres in one crown.
2)150
20 punchions of prunes .506 12 5
Charges ----- 133 4 i
What comes livres 714 i6 6 to, at 2t per cent.
at
1429 13 -
357 8 3
■ I r
17.87 I 3
20
17.51 .
12
■ liv. ibis, deiu
*.I5 Anfwer, 17 17 6
24* What ought the 175.96 quintals of prunes to weigh
in London, one quintal at Bourdeaux being iiolb.^
quint.
175-96
no
cwt* qr. lb.
112) 19355.6 (172 3 7, M^ight in London.
815
—
28)91
25, What
ehap-Vi. EXCHANGE.
417
, 25. W^at comes 732 livres, i3fols, 11 deniers to ia
London^ at 57^ d. per crown, at Bouideaux ?
llv. cr. deil,
60 : S7t :: 732 13 ik
12 20
720
H653
12
175847
S7J
1230929
879435
87923.5
12)
720) loi 1 120.25 (14043^
291
.»
312 ip) 1 170 3j
240 " . — "
24 iC 58 »o 3t
4
96
24
SPAIN.
4 Maravedis yellbily or 7 •<)
oj- Maravedis plate - }
8t Qpartas, or - - - 1
34 Maravedis vellon -^ J
16 Qjiairtas, or 1
34 Malravedis plate - - )
8 Rials df plate -• - ^
one
I
Quartas*
Rial vellon*
Rial of plate*
.Pffo, piafter^ pitoe of
s
' or dollar.
N. B. A rial vellon is j^^ of a riat of plate, and ^ of
piafter. ^
£e
26.. Re-
I
I
4i9 £X€»tAN€£. BookB.
26* ReduQC 1387 pistftersy 3 n^s, 3 maiavedU of Spain,
into pdundSf kc, VLttYfXify ^f Eo^laray c;tdiaiige at 41^
fterling piafter )^
piaft. ri« man
1387 3 3
45*
693s
5548
t93| = 1^,387 pidfen.
f = 3 maiavedisf
12
20
642994
5374 "t
■w^*"
£ 263 14 Xl\x ^k% MfwV*
27. In 572I. 18 Sn 9 d* l|ow ftdHf pieces of j^ exchaage
at 42f per pialltr ?
4H • « *• 57* »8 9
8 20
341 "458
12
13750$
8
piaft. ri, mar.
341 ) 1 1 00040 ( 322 J 7 13, tjie anrwer«
884
133
534
399
4522
11(2
89 28.1a
r
Chip. VI4 fe X t H A N G E; 419
aS. In 27A ddlan, a mis, jr quartat^ how msUuriieiilMls
leriiiif » cjccluuftge St 4S d. p« dollu f
dot. ti. qr;
9192
1096
24 =s 4 rud« or t eoedUAg^
?i = 7 qutrtM.
t2
i3»78f
lojl aj
iC 54 i« 2i
riMM*
2^i I«* 58794 quartM, how fiyuiV Mtinds fttrlinft
dttngeat40fd. perpiaftdr?
I ps.-} d. quartat;
8 : 4of : : 58794
16 8 3il
*a8 321 58794
117588
176382
147444
i843o{.
118) i«7«874 (147444
6ti7
952 . .
56^
567
554
42 Arifwer, £ 76 is »0J
12
2)3
*535 iof
30. What IS the brokerage of 15066 Hals of plate in the
invoice foI16vrihg> at ^ per cent. ?
rials.
4) 150:66
37 J, brokerage.
Iti a
%u What
420 E X C H AN G E. . BookH.
31. What is thie commiffion of 151475 ri^s of plate^ at
2i per cent, i
rials.
, I5M7S
2.5
757375
302950
378.6875» or 3784- rials commiffipn.
32. What fteriing money does the whole 15526 rials of
plate in the invoice following amount to, exchange at 52<i«
flerling per piece of ^i
15526 rials of plate*
i*a
31052
77630
8
12
20
807352
100919
■ d*
8409 II
s. d*
420 9 II fterlingy the anfwer.
I4US
r
Chap.' VI. E X C ». A K: G E.
4*t
n
>
o
.s
o
Q
en
h4
«3 ^
•-4
it
t-l tg
3b
du'V
u ^
Urn ^
II
o
c
«
c
9
•
o
o
c
argi
o
^
S
i4
1
V
M
Ik
cu
T3
U
#^%
fc
S
M
§
^
E
i^
o
o
43
««-«
•%
♦4
o
&
•^
t>
4-*
5
o
ca
o
1
1
1-^
o
c
^^T^
**oo
NO
00 r^
M
CO
«o
•
oft
«o
P4
I
SB
o
M
as
o
MM
CO
o
go g
a
M|f4
s
a
s
o
u
o
h
8
§
5
a
u
3*8-5 .
e u 2 -<3 H
e o o o d
Ohhhh
Ee 3
POR-
4t* fiXeTHANGfi, BpcAn,
? O % T U & A L.
Portiftal exth^ges with London on the milrea, ai^4>
liondon ^ives from ^ to 66 pence fterliog for the feme.
400IWS r - - - "Imakconci^"*^^^'
looaieasy or zi crufiidoes 3 c miliea.
13. In 2729 crifftdoesi^ 372 reas, l^ow much ftnii^t n<^
^tfy exchange 62 d« per milrea I
cruf. reas.
2729 372
400
1091.972
58. 2d.
5455
|8i 10 d. beuK j of 1091 tHilreas*
2 7 = s*!. being i«^gc for soft I
2 «-4 = 4 or douUe* exchange for 400 i - ^
Xr = TV of 40Q =^ 4^)
it = * of 40 = 2airea8.
i= ipf 20s: la)
■**■
10)5641 IC
j^ 2^1 I lod. fhrling, the «if«er.
34. In 754I. 18 s. 6i. fterling, ho«r many cciifiidi
exchange 64t <!• per viilrea ?
d. mil. b •• d.
64f : I : : 7^ i« 6
2 «0
»5*
J
?29) 362364 (tSflsi iQilin».
2809 1043
ii . 1164
5618
i404i-
fPijLi crufadoes, the aatpff*
WkH
amp. VI. EXCHANGE. 423
35» W&at flerCiK Aioney doei the Uivoice followingv
iz. 187 nulreasj 616 reas, amount to, ak 40 reas for sdrr
viz. 187
mil. tea.
3187.686 Reiq« ^ :p rv of 3 d. or A of H.
*f
w *
52 13 <^ ftcrling) the aitfwer.
Aj^ 2) i75-iS^ Ati pc^ wax.
IS .875, brokerage*
And 176.025, at 3 per cent,
is 5.280* comtnifloR*
£e 4 Oporto,
4*4 ^
»4
8
o
o
EXCHANGE.
Book II,
oo o o
00 Ct CO
Ou
a
o
§
c
<
60
ft. "2 "* • -=
.S X ^ ^ c ft' -ts
00
Pi
o
U
' to
M
'<
a
^ J8 ^^ .^.^
8 ^^'^ "^ ^"^
'^ i 6 g s i "
O 9 *r? M *^ O O
O U £3 O,^ U D<
O O O O O O O
hhHhhhh
•8
M O
555
t> £ N O A-
0af. VI. E X Cf H A N G E. 42^
E N O A.
In St. George's bank at G^noa, accounts are kept iii
piafters, or pezzoes, which are divided into fblidi apd de->
nari^ as the pound ^erling.
'^ But foai^ merchants keep their afQoiints in lires, or lira«
folidi, and denari, divided as before, which money is only
j- in value of the bank money.
The exchange vufts from 45 to 54 d. per piafter.
36 In 784 pez. 19 8* 6d. lire'inoney,*^ how much mo-
pey of exchange ?^ •
. pez. '6. ' d« .
5) 784 19 6, lire money.
156 19 10^, exchange money, an(Wer.
37. Reduce 156 pez. 19 s. 10^, exchange monfy,>^<o
ljvre§.' • .
• pez. s« ^»
156 19 10$
764 19 6, life money, anfwer.
• *
38. London is indebted- to Geno|i in 17x01. 16 s. 4d«;
for how many pezzoes may Genoa value pn London, the
exchange at 47^ d.i.
d. -J- P. pez. 1. s. d. : ■};-?.
47t = 95 : I •• 1710 16 4 =^ 821192
■ ■■ s. d.
pezzoes 8644 2 6,anfwer.
39. Genoa is indebted to London in 8644 pez. 2 8. 6d.
for how miich ' flerling may London y^lue on Genoa, the
^change at 47^ per pezzoe i
I. s. d. »
6} 8644 2 6
8) 1440 13 9, f0r4pd.
2) 180 I 8|, for 5.
90 - lof, for 2 J.
1710 16 4, anfwer.
40f Lon«
4a6 ISXCHAffQB; JMnU.
40. London dnws on Genoa for 1710K 16 s. 4d. fter-
liiig ; how much lire InoMy vHi pty (be drajigbti exchange
at. 48 d. . per piaftcr i
1. !• d.
1710 16 4
5 timc$ 4 ^ in a pound ^Ibg.
8554 1 8 of elodMige,
S liww in a pdsMBoe,
4a79a ^ 4 Uie mm^y^ aafwer.
LEGHORN.
N. B. At Leghorn a dollar is valued a| 61i¥vet» at Genoa
but five*
*X2 denarii -| f- foldi.
2t4 greA ^ tducait.
In L^hom accounts are kept in piailers, foldi, and denarii
divided as at Genoa. Some likewife keep their accounts in
liras, or lircs, divided as the piaftcr^ Irut this money is
^ only i of the money of exchange.
41. In 278 1. 17 s. 9d. teling^ bow many peuoes of
Leghorn, exchange at 47|d. per peseoe^?
i. pez. 1. s. d,
47f : I r: 278 17 9
379
379) 5354^4 (1412 P^^- <& f^ 9 ^l^* anfver.
15*4
486
1074
316
xao
6320
2qQO
3H
257 4^ Lja
Clm^
EX change;
4*7
42 London is indd»Ud to Leghorn in 74C61MI. 9 s. (d»
lilt monejr ; what fte4ing ftands as an ea ui valent in the Lqo«
fiM merchant's bool^^ tk^ exchange Deing ait 49{-d« pet
piatt* 8« o*
6)7456 '9 9
cxchsuuei
I
2
2
181 10 91, at 404
36 6 H
4 M 95:
2 4 4t
I 2 Sf
- II 4
ifti
j£ 226 7 ft» tnfwer.
I
I
faftoiy
4^1
E X C H A N G E; ' Book IT.
• %
I
f
NO
dvo
^4
o
Q
o
• s
I vD NO
I f f I I
I I r. «<s r
tor*
On m
• I
I
1
M
i « «
' Id
I'.
CO
o o
6(0 :)
cv o
Ohm 2
•ft "-a
o 55
It rt C
« CO
9 «» I*
8
® c.ti
•s-S
o s
o
o
8
se is
2
* c
bOCi
«^ 1-
o c^
•^ -frt
^ fc b
a
%
s.
o
• •4
Of
>^
S
o
<
•4
<£
>o
M
O O o o o o o
HI
43- At
Chap. VI. EXCHANGE. ^9
43. Atiij fols per -piece of 4» viz. the foregoing inw
' voice, . hoi¥ • much .fterling may Mr. Hourd credit hi^
faftor^ exchange at 4 s. 6d. fterling per piece of 4?
f
m
liv. fols. den.
1781 26
20
115) 35622.5 (309.76087 pieces of |.
1 122
875
700
1000
800
309.76087
61.95217
7.74402
69.6962 =:69l. 13 s. II d. fterJing, the anfwer.
. (
I
Provifion of 1729 liv. 5 CcABf at 3 per cent.
Hv.
17.2925
3
51.8775 ^ 51 liv. 17 fols. 6 den. provifion*
VENICE.
Money of exchange is always underftood to be that of
6ucats in bank, whidi is imaginary, loo whereof make 120
ducats current money ; fo that the difference betwixt bank
and current money is an agio of 20 per cent. Though the
brokers have invented another agio to- be. added, which it
more or lefs, according to bargain.
The courfe of exchange of a ducat of the bank of Ve-
nice is from 45 to 50 d. fterfing^
44. Ve-
j^ SXCHANGE. IkMkll.
44^ Venke draws oh London' for aSso ducats i^lblsi
io|4 dcnicrs biijco, exchange at 45|4. f^ 4tt(at | ]i9»
much fterling will pay the dran^t i
due. fol. den.
6)2850 10 lOf^
8) 475 I 9j, at4od.
8) 59 7 9, at 5d.
78 Sit at J.
£ S4' »8 -"» at 45|, aiCwer.
45. Reduce 1459 ducats^ iSfolsy i denier d'or bank mo*
ney of Venice, into fterling amucf^ exchange at 47^ d.
fterling per ducat f
due. fohden.
1459 18 X
izl
I02I3
5836
23f) for 10 folidi.
a-J, for I ^ ' 20
4, for I denier.
697 1 of pence fterling. £'^90 9 24, anfwer.
46. Venice is indebted to London in 4789 ducab, 19 s.
2 d. current money ; how much fterling may London draw
lor, agio at 20 per cent wheli the exchange is at 4 s. i d.
per ducat banco ?
duc« 8* d.
6:51: 4789 19 3, tuRutit mency*
SSoj at
") 399L^3
*) 399 3 3i* ** *
16 fa 74- at - I
/ 814 '9 a4, at 4 i
WbeiJ
CJiap. VL EXCHANGE. 43<
When Lpodpn exchanges on the piece of foreign monev^
as tbcFronch cro^m, Venetian ducat, &c, London ougnt
to* remit' wbeA &C exchange is ^o'Vfj, and draw when it is
high, to negotiate with advantage. The reafon will be
obvious, for 100 !• will go farther in purcbaiing ducats,
crowns, milreas) &c. when the courfe of exchange is at
40 d. than when it is at 50 d. and 100 crowns will go far-
ther in paying xMt due by France to Londoqt wheh the
exchange is at 32 d. than when it is only at par.
P O t A K D AND PRUSSIA.
make
"gfi»fcb.
diikin«
fixer.
tymph.
^Qoe 1 ach dt hatbeiii*
ftiviii or gilder,
""*"*} dollar.
ipecie
3 fhiUingtt or 18 pbfciiai^genl
3 MoTck - - -
2 citkins - - - i
3 fixers - - -
^^^iJiJ. ^
4 adi de kuhcn -
3 fiodns or gildora
4 gilders - - -^
Dantsuck and Konin|;fl)erg exchange with London by
way of Amfterdam and mmburgh ; 070 Poli£h erofch being
s 1 1. erob banco in HollMd, «I0 roliih grolch being =
I rix-ddhr banco of HanbMilgh.
47. Let 5850 florins be changed into fterling money, 27O
grolchi Poli per pound Flemifl), and- 33s. 4d. Flemifh per
pound fterling.
G. p. flor.
30
30
175500
S«50
j^ 650 Flemifh.
33s. 4d. : I :: ^50
1* sto
400
13000
400) i566fOO
■t^a
I 390
•MiMi^
, the anfwer.
R U S-
4$s EXCHANGE. Book it
RUSSIA.
3 copccs - -"J , raltine.
10 copccs -' - j I grievcner.
ascopecs - - l^makc oncJ P^JPP"^"-
2 poltins - - I I rubble.
2 rubbles - -J Lducat.
•
The Ruffian rubbles are converted into florins current
money of Amfterdam, and the current into bank moneys
according to the agio of three or five per cent, and bank
money into fterling, according to the courfe of exchange
between England and Amflerdam.
•
48. In 4675 rubbles j 46 copecs, exchange 122 copecf
Kr rix-dollar current, agio three per cent, and 34s. yd.
emilh per pound fterling, how much fteriing money I
rub. cop'.
4675 ' 40
106
■ ■ ■■ — rix-dollart;
122)467546(3832.34426 -
X2.5
■.«■
1916172130
766468852
9580.86065 florins current.
103 : 100 :: 9580.86065
I03) 958086.065 (9301.80645 florins baxKcd^
,86 ^
830 372072.258
66s
470
58(5
34'S. 7 J.s 415) 372072.258(8961. ii8.2id.thc
4007
2722
232.2
20
cyp.TL EXCHANGE. 4^
IRELAND.
In Ireland accounts are tept hi pounds, Khillings^ and
pence Iri(h, divided as in Etighhd ; ' but having no coins of
their own, (hev &re fupplied by the different countries, with
which they tiraffic/
The par of exchange beti^reen E6gtin(d and Ireland is
lOoL fterllng for 108 1. 6s. 8d. IrHh, or t s, Englifli =
13 d. Irilh.
The courie of eScK&'nge is from five to 12 per cent* acr
' cording to the balance of trade«
49. London remits tb Ireland yij 1. ^5 s. fterling ; how
much Iriih muft London be credited, exchange at ix-f per
ctrii. ?
1. s.
AI787 ?i
78 15
I
I
III ollH ijp^-^"^-
£ ^79 ^ ^y ^^ anfwer*
50- Dflblin dra^s updh Loridon for "879 1. 6 s. 6Jd,
Irilhr cxchanei at ii|p^rceiit. ho^ Aiuch ftetfing miift
London ^y Uublin, to dfftharge this bill ?
lii*62jf : 100 ::. 879.326041^1*
111.625)87932.6041^(787.75 s= 787!. t^s.
979510
865 104
837191
559166
A M E R I C A aAd THE W E S T-I N D I E S.
In exchange witA our coldfties in America and the Weft-
Indies, accounts ard kept In pounds, {hilling;s, and pence,
divided as in England, and their rnoney is caned currency.
The fcarcity olF tsdOi obliges them to fubftitute a paper
currency for carrying on their trade ; which being fubjcA
to cafifalttes, fuffers a very g/eat difcoimt for flerling in the
purdiafe of bills of exc&Jk^. . .
Ff SI- PW-
434 EXCHANGE. Book II.
51. Philadelphia is indebted to London 1575 1* 14 s. 9d.
currency ; what flcrling may London Reckon to be remitted,
,when the exchange is 75 per cent, i
As 175 : 100 :: 1575!. 14s. gd.
By dividing the two firft terms by 25.
cur. ft. I. s. d.
As 7:4='- 157s H 9
4
7)6302 19 -
Anfwer, 900 8 5^, ftcrling.
'52. London receives a bill of exchange from Philadelphia
for 900 1. 8 s. 5fd. fterling; for how much currency was
London indebted, exchange being at 75 per cent. }
1. 8. dk
' 900 8 5t
450 4 2U at 50 per cent.
M5 2 I7, at 25 per cent.
2
X
a
Anfwer, £ 1575 14 9, currency.
• 53. London -coniigns to Virginia goods, per invoice,
amounting to 578 1/ 19 s. 6d. which are fold for 847 1. 15s.
6 d. currency ; what fterling ought the factor to remit, de-
ducing five percent, for commiflion and charges; and
what does London gain per cent, upon the adventure, fup-
pofingthe exchange at 30 per cent, i
130 + S = 135 = 5 X 9 X 3
'35 : 100 : : 847 15 6
10
8477 IS -
10
5
3
84777 10
16955 10
5651 16 8
£ 627 19 7t, to be remitted.
— 578 19 6, corfigned.
£ 4g - I7 fterling, gained.
578-975)4900.625 (8.46403 s 81. 9S. 3id. per cent.
54. Vir-
Chap. VL Simple AtiBiTRATiON ^/Eicchaicge. 435
54* Virginia is indebted to London 5751* 19 s. 6d4 fter-
ling ; with how much currency will London be credittd at
Virginia, when the exchange is 331- per cent. ?
r
T
L s. d.
575 «9 6
191 19 10
Anfwer, £ 767 19 4, currency.
.^X %K. SS ^2 ^X 8S ^X ^3 ^2 ^9 'Or ^f ^2 ^S 9S ^S »» ^3 gp ffS »» ^X ^^ ^B
SECT. IL
Simple Arbitration c/ Exchange.
WHEN a faSor has orders from his employers toxemit
a certain fum of money to any place, ana then draws
upon the laft place to fome other ; as the par of exchange
is continually fluSuati^g, there may happen to be a lofs in
the executing one part of the commiilion, and a gain in the
other part thereof; which the ikilful fa£tor (hould endeavour
(if poffible) fo to improve to. the benefit of his employer,
to make the gain fuperlor to the lofs ; or in cafe the nego-
tiation would be to his conftituant's lofs, he may write to
him for new orders, or wait till the courfe of exchange be
more in his favour.
Arbitration of exchange may be performed by one or
more operations in the rule of three*
I. V, of A9ifterdam» draws upon X, of Hamburgh, at
67 d. Flemiih per dollar of 32 fols Lubeck; and on Y,
of Nuremberg, at 70 d. Flemiih per florin of 65 crutzers
current. If V has orders to draw on X, in order to remit to
Y at the faid prices, how would run the exchange between
Hamburgh and Nuremberg i
67 d, : 32fter. ::, 7c3d. : 30^^ fols Lubeck per florin.
2« M, of Amfterdam, orders N, of London, to remit to
O, of Paris, at 54 d. fterling, and to draw on P, of Ant-
werp, for the valuta, at 33 j (hillings Flemiih per pound fler-
ling ; but as foon as N received the commiffion, the exchange
was po Parts at 54^ d. per crown : pray at what rate of
Ff2 ex*
43^ SiMFLB AnBiTftATfci^ ifExciiAnot. Bodi If.
Exchange oueht
and be no loieir ?
exchange oueht N to draw on P, to execute his orders»
lofa
Reciprocally, 54 : 33.5 : : 54.5
1340
167s
54.5) 1809*0 (.33s. i^^d. Fkm. dieanfwer.
1740
105
12
1260
170
3* London changes with Amfterdam on par at 33} s.
Flemifli, for one pound fterling; Amfterdam chanecs on
Middleburghy at 2 per cent, advance : how ftands die ex-
change between London and Middleburgh ?
Flemw K
As 100 : 102 i I 33t = i»f
it
9)612
6^
102
100) 170.0(1.7 l.=s I k I4S«F1. perlb.fier.
4« Amfterdam changes on London at 34 s. 4d. per pound
fterling) and on Lifbon at 5a d. Flemiik for 400 reas } how
then oug^t the exchange to go between London and LKboo ?
8. d. d» ' d»
As 34 4 = 412 : 240 : : 52
-J?
480
1200
412} 12480 (sOt^tt P^ce for 400 ieas«
120
•'* SoAn- X- *» ^ 75t%t^* ftcrling for looo reas.
5. Q^
Chap.yi. Simple Abbitaatiov^Exchancb. 437
5. Q2 of Amfterdam, remits to R, of PariS) sooocrownss
91 pence Flemiih per crown, at double ufance, or two
months, and pays -^^ per cent, brokerage ; with orders to
remit him again the value at 93 d. per crown, allowing at
the fame time -^ per cent, for provifion : what is gainca per
cent* per annum, by a remittance thus managed i
100^ : 100 :: 91 : 90^544 = go^i^^U:^
loof : 100 : : 93 : 927^! == 9mitlh
Thus 9zji;4jj — 90144^5 = i^u-H*
••• a : itUUi :: 12 : lofil^^f, theanfwtn
6* A9 of Paris, draws on B, of London, izoo crowtas,
at 55 d, fte^Iing per crown ^ for the value whereof B dniws
ag»n on A, at 56 d. fterling, befides reckoning half per
cent, did A get or lofe by this tranfadHoo, and what t
cr. s. d. 1. 1. 1.
f 1200 at4 7 As 100 : 5 : : 7.ys
X
t
X
6
5
240
30 xoo) 137.5 = il. 7s. 6d. com.
5
£ 275 - - d. cr. d.
176 commiffion. As 56 : % : : 66330
56) 1^6330 (ii84H- .
£ 276 7 6 = 66330 pence. )
» • * I aoo — 1 1 84ff := 1 5I4, A's gain by this tranfaAion.
7. A, of Amfterdam, owes B, of Paris, 2000 florins of
current fpecie, which he is to remit him, by order, the ex-
change 9oJ- d. per crown of 60 fols Turnois, the agio of
the bank being four per cent, better than fpecie ; but when
this was to be negotiated the exchange was down at Sof d.
per crown, and the ano raiied to five per cent, what aid B
get by this turn of aitairs ?
Florins 2000 X 40 =r 80000 Flemiih pence.
crowDS.
As los : 100 :: 803.8558 : IS^'^lllU^ozccountcd.
104 : 100 : : 883.9787 : 849.9787 J ^ .
!■■ cr. foL dea.
PliFerence 1.3125 = i 18 9 infii-
vour of B. QiE.F.
Ff 3 8. But
438 Simple Arbitration <>/ Exchange. BookIL
8. But arbttration of exchange may commonly be more
readily performed by a numerical equation; viz. Let us
fuppofe that the exchange between London and Amilerdam
is at 34 s. 6d. for 1 1. iterling ; and between London and
France 3iid. fterling, for 1 ecu or crown.
To find the proportional arbitrated price between Amfter-
dam and Paris,
Majce the following numerical equation ;
viz. I crown Paris = 3i|d. fterling.
And 240 fterling =z 34s. 6d. = 414 d. Flemifli.
The right-hand numbers conftitute (by being iDultiplied
continually into one another) a general dividend ; the left-
hand a general divifor, the quotient of which will give a
true folution to the queftion.
. But thefe may be feduced in lower terps, or lefs propor-
tional numbers, by obferying the axiom in redu^ion of
vulgar fractions,
240 d. = 34 X 6d. = 414
I X4
2-7-6
3
4
1^
4=^27
4 X 40 160 J'ri6oy
This operation is thus perfcxmed :
31^X4= iZJf which place under the line on the
faipe iide) and plaoe 4 on the other fide to balance it.
Divide 240 and 4x4 each by fix, and the quotes will be
40 and 69 ; which place op the fame fide with their divi«
dends, cancelling all numbers as they are done with.
The reft are fo plain and eafy, it needs no explanation.
9. Again, fuppofe the exchange between Paris and Amt
fterdam is at 54444* and on London 31^5 the proportion^
arbitrated pri^e between London and Amfterdam is required?
l.fter. d. •
I = ^0
^ = ^rli
/^^ 876;$
127 4
4 6
• f~ ^^7" ^ ^^^ ^^^^' P^"^^ = 34 s. 6 d,
Chap. VL Simple Arbitration of ExcHAUon. 439
The foregoing operation is performed thus :
54n-74 X 160 = 8763 placed underneath, and 160 fct
on the other fide to balance.
Then 31^ x 4 = 1279 to balance which place 4 on thf;
other fide.
Then I perceive, that 160 and 240 are each divifible by
4O9 the quotes whereof are 4 and 6. >
Lafily, finding 4 on each fide, thev cancel each other.
You are deiired, as before diredeu, to cancel ' every fi«
gure as it is done with.
10. Laflly, exchange Amflerdam on Paris at S^Hi^ ^"^
Amilerdam on London at 34 s. 6d. what is the arbitrated
price between London and Paris i
I crown Paris =: //444
h% 2921
4.0 I
The common meafure of
the fradion being 23.
II. London exchanges on Amflerdam at 34s. gd. for
1 1. fterling, and on Lifbon at 5 s. 54 d. per milrea ; .what
is the arbitrated price between Aipfterdam and Lifbon ? .
J crufado Lifbon = 4^ ''^^^*
ff^^^iezA = ^/id. flerling.
J4^d. fterling :=: z4 i'=^ 417 Flemiib pence.
f Hi
if
4 if
16 I
7
• . • ^'J ^ 5s -2-2 s= 451 9 d. Flemifb, for ow crufado
Lifbon. ^
Ff4 12. Am-
I
12. Amft^rd^m exchanges on Ljfbon at A.$\i Flemifh I
pence, and' 6n London at ^4 s/ 9 d. what is the arbitrated \
gripe pf exchajige between LondoA ^d l/i(hpji ?
*
I milrea =: ^fifi^ reas.
X c<u&do = ^jftfjEi reas = ^j^ d, Amfterdgor. 1
^/;^d.Fl= /^^d, fterling.
w 2919
139 ♦»<
... 22!2iLiJL5 - ^ =, 65id. ft^li^gger Mr*,-
N. B* The common meafure of the fradion being 139.
13. Liibon excliang(;s on Amft^dam at 4|-|| bct cryfad^
on London at c s. 54 d. p^r milrea > wliat la me arbitrated
gfic? b^^i^ top4oo W* Aniftcrc^ffi ?
il. fterling = /^/^d. fterling.
^^j-^^'^'^g = /^^i^ reas PortugaK
^^pf reas ;= 45^! Flemiih pehc^«
r
Here the anfwer co^es out exa&ly 417 Flemi^ pence,
or 34 s. 9d. of Am^erdam; and this will frequently hap-
pen, and the operatioA performed on the thymb-nsqi. oy the
expert accomptant.
Thefe examples prgvc the truth of this method in regard
to each other.
i4.^Amfterdam hath orders to remit a c<utsM9 (um to
Csutbft'; 4t> thA U|D$ <& this ocd^r Amilerdam can r^mit to
Cadiz at 94|d« per ducat of 375 maravedis, and Londori ta
Cadiz at 38 d. per piaft^ of 272 maravedis. Quere, which
will be moft advan^eous to Amfterdam, to remit direfily
a * ' ' -. . ; to
*%
Ch«jp.VL CoMrpviTP A^^lTt AT. ^ExeaAKoB. 441
tp C^dl^, or b^ ^ndon, the exchange bf twfen Amflf idam
9b4 L«n4pn bieiag 35 A< 10 guU. ner pomw} fterling ? ^
1^^/ fnaravedis =: 38 d. fierling.
^ 4. ^iliftg :s 1/ s. iQ gnM. ;?= ^^ d. Amtt%sinn.
13.^
? 43
IOX4.tXl2C I0212C
fgr 375 ijwr^vedis ; whick is i8«. ^Jd, % ^voty 100 1.
^flifig ja ftfiom of Am^dJ^i^ } vi^
As 94.75 : 9386489 : : 100 : 99.06584 = 99!. is. 3|d.
Then 100 1. — 991* is. ^4. = i^s. ^d. as above.
S E C T. IIL
'Compound Arbitration (/£«ch4Ngi.
WHEN the price of exchange is given betwixt one
coiyitryand another, betwixt that Tecond and a third,
and betwixt that third and a fourth, &c. to find the arbi-
trated price between the firft and the laft, obferve the
following
RULE.
» . •■
Place the antecedents in one column, and the confe*
quents in another, to the right of the antecedents, lb as to
^m a nuqfierical actuation in the algfebraic way of analyfis,
in wY^fk the i^ antecedent and the laft confequent, to
wluck ai| antecedent is r^uised, muft always be of the
fiwe denonunatioo or fpecies ; the firft consequent muft be
•f die fiuae dcnemiaat^n with the lecond antecedent ; the
fecond
442 Compound Arbitr AT, it/* Exchange. BookIL
fccond confequent with the third antecedent, &c. through*
out. If a fra£iion is annexed to any of the numbers, both
the antecedent and confequent muii be multiplied into the
denominator of that fra6Hon, and the proportion will ftill
be the fame. The terms. being thus difpofed, cancel the
quantities that are die fame on both fides of the equation,
and. abridge fuch quantities as are commenfurable ; then
multiply all the antecedents into one another for a general
divifor, and all the confequents for a general dividend,
and the quotient will be the anfwer, or value of the ante-
cedent required.
I. Suppofe London to remit 500 1. to Spain, by the way
of Holland, at 35 s. per pound ; thence, by the way of
Frarice, at 58 grotes per crown ; thence to Venice, at 100
crowns per 60 ducats banco ; and from Venice to Spain, at
360 maravedis per ducat banco ; how many piafters of 272
maravedis will it amount to in Spain, exclufive of charges I
anteced. confeq.
I pound r= ^/pd. Flemifb.
^ grotes = I crown.
^fifi crowns =; ^fi ducats.
I duc^ =r 2J^f^ maravedis.
j^^jf maraved.=: i piafter.
How many piafters for 500 1. ?
Thefe reduced, will be.
I
- -
21
29
- -
I
I
- -
3
I
- -
45
17
■ ■
I
500
21
2L?JL
4*; ^
1^00
287 c
= ^87544?- =: 2875!:, nearly, the anfwcr.
29 X 17 /-r+y J /-*♦ /
2. A banker in Paris remits to his fador in Amflierdam
455 crowns Tournois ; firft to London, at 30 d. per crown;
from London to Rome, at 65 d. per ftampt crown ; from
Rome to Venice, at .100 ftampt crowns for 140 ducats
banco ; fromVentce to Leghorn, at 100 ducats banco for lOO
piaders of Leghorn -, and from Leghorn to Amfterdam, at
86 Flc-
Chap. VL Compound Arbitrat. ^Exchange. 443
86 Flemiih pence per piafter, how many guilders banco
will be received at Amfterdam f
^uiteced. confeq,
I crown Paris = gpi d. fterling.
fg d. fterling =2 i crown Rome.
^1^^ crowns Rome = /^^ ducats Venice,
ipo ducats Venice = //ef^ piafters Leghorn,
1 piafterLeghorn = 86 pence Flemiih.
4ii crowns Tournois.
13 6
i . 7
9i
^— — = ■ == 25284 Flemiih pence»
86 X
3.
And 40) 25284
Anfwer, 632 guilders, 2 ftivers.
3. A merchant of London hath credit for 1360 piafters
ofXeghorn, from which there is advice that a remittance
can be made at 50 d. per piafter. The London merchant,
Ending he could mate no more hy drawing for them,
orders them to be remitted in the rollowing manner; viz.
firft to Venice, at 94 piafters for 100 di^cats banco ; thence
to Cadiz, at 320 maravedis per ducat ; thence to Lifbon, at
630 reas per piafter of 272 maravedis; thence to Amfter-
dam, at 50 grotes per crufado of 400 reas ; from thence to
Paris, at 56 grotes per crown 5 and laftiy, he brings them
home at 3i|d. per crown: what will be the arbitrated
price per piafter between London and J^ehorn, and how
much will be received at London, without recKoning charges ?
anteced. confeq.
g^ piafters =. /^fi ducats banco.
I ducat =r ^ffi maravedis.
j^y^ maravedis = ^;gfi reas.
jjfj^f^ reas =: jffi grotes.
g^ grotes = I crown.
g crowns = 0^ pence fterling ss 3i|^ X 3.
What = I piafter ?
I •
^4 C«|fPOUVP ARflTftAT.^fEx^iiAllO^BOQklL
I 1- - - - I
I - -.- - 5
34 .... 15
I ..... 25
I . - - - I
1 - ■*• - -r I
. . . SJLLLiil - l!ZS ^ 5^^^ d. p,r piafter.
1. s. d.
1360 piafters, at SS-^d. per piafter =312 10 .^
Ditto at 50 d. per piafter = 283 6 8
Gained by the negotiation jC ^9 3 4
4. Amfterdam being to remit to London 750 !• Flemifliy
he firft fends it to France, at 54. d. per qrown; from thence
to Venice, at 100 crowns for 56 ducats banco ; from thence
to Hamburgh, at too grotes per ducat ; from thence to
Portugal, at 45 grotes per cnifado of 400 reas ; and from
Portugal to London, at 5s. 3d. for 1000 reas; and fuppofe
the commifEon, &c. at each place be half per cent, quere,
how much fterling money muft be received in London ; and ,
whether .more or lefs, than if it was remitted dire^y from
Amfterdam to London, at 35 s, 6<1. Flemifh per pound
fterling?
^ pence qp 1 crowo*
f0(i crowns =; jfj^ diiCHts b^co of Venice.
I duc9t V. ass /^ grotes Hamburgh.
4^ gr. Ham. 5? 4f^jt rcas Portugal.
/rtsWreas = j^j
£ yj^o Flemiib) 9t 35 J^ s. per pound fterl.
/a
9 i»
10
••• 2X
••• C.-^ 1= ^3435.51. tterling. ,
X '005 = i pcf cent at Port.
2Si^* = 1.775 FlcmHh ~ 2.iy
433. 3^
1.77s) 750.000 (421.535^^ T X -065 s± f per rtnt. Hamb.
had it been remitfciJ di- > -^—
redly to London - - 5 2*1^^ f
43i.2Jof
X .005 =s t per cent. Venice.
2.156
429.0548^
X -005 ai i per cent, Paris,
^•1453 -
426.9095 I'eceived in Irondon.'
4*2^.5352 diredb I'emittante.
■^
Anfwer, 4.3743^^^4:1. 7 s. 5|i. Ld*.
don gains by the remittance above.
SECT. TV.
Conipdrifai^ of yf BiaHTS and Mbasurb^.
IT is a very neceflTary ^ay (of great imporfdnce to the
merchant) to be acquainted with th^ Weights andmekfures
of the different countries ^hert he deals ; td* facilitate which
knowledge,! have in the* fflloi^n'g; pages exhibited* ^then-
tip tables of the conformity which weights: and meafttfes in
the moft noted trading places in f^urope have with one
another.
I. Suppofe 100 lb, of Amfterdam be equal to 100 lb. of
Paris; and 100 lb. of Paris to be 1501b. in Genoa ; and
100 lb, of Genoa to be 70 lb. in Leipfick ; and 100 lb. of
Leipfick
44^ Weights and Measures. Book IL
Leipfick to be |6olb. in Milan ; how many Milan pounds
will equiponderate 54$ lb. of Amfterdam i
lb. lb.
/fifi Amfterdam =1 //^^ Paris.
;(fffi Paris =i /^^ Genoa;
/^^ Genoa - - = jffi Leipfick.
/^^ Leipfick - = /^fl Milan.
Quere, Milan - = 548 Amfterdam ?
/ 3
^S 4
7
2
548 X 3 X 7 X 2 =: 23016
25) 23016 (92o||- Milan, the anfwer.
2* If 7 aunes of Paris make 9 yards of London, 36 yards'
of London 49 aunes of Holland, 7 aunes of Holland 9 braces
of Milan, 3 braces of Milan 2 vares of Aragon, 5 Vares
of Aragon 2 canes of Montpelier, 9 canes of Montpelier
10 canes of Thouloufe, and 4 canes of Thouloufe ocanes
of Troyes, in Champaigne ; how many aunes of Troyes
jHrill meafure 100 aunes of Paris I
^ aunes of Paris - r= jg^ yards of London,
j^ yards of London = jfg aunes of Holland.
. if aunes of Holland =: ^ braces of Milan*
2 braces of Milan = f vares of Aragon.
^ vares of Aragon = f canes of Montpelier.
^ canes of Montpelier ^ /p canes of Tbouloule.
4 canes of Thouloufe = ^ aunes of Troyes.
How many aunes of Troyes =: loe aunes of Paris ?
10 X 3 == 300 dividend. 2} 300(150 aunes of Troyes.
if Table
1^
i I
1-
VI. Weiohts and MfAJOils. 4+7
'It
ss's.tessjs ;8 JS.S s.ii'Si-ff-
"Jr-
si-at^sKtsi-WMKSWss
'ih.ii
:s:';s?2l = : = :--"*'-:a:
=liilii
J?EsJ?iSs = ^^^^=?=
-d
s,as!-8J.2:s5Sssis5,:ss
-lit
°l|i|
s'S.s.ss susis's 3;"S,5.JS KiiajJ J
"ifii
%y8ys,?^'-':rfeRS.s-iSJ8
i;s:|^s*-s?.8 5S^iiasjrs
'tirhi
8 Js ; 3;? S.S is: SKSi'i'Jrsj
.-si I
Ml
_lls U«uQMmOX-li-'2206-o'o!'"HO
44«
VflttfRTi akd MiAsvt^i.
s3^
««^
M MM
M^M.MMMM M
0«
» J t I ' ■
M M M M M
0M0*^O^M0«O«MMMdt^ to^ «o >o o o
M
O
M
1(3 Ji
,*«n eo w d tntt •<
H Ok
ro.^6 00 o M »<^ a\ «/^ Ok t^oo MOo^t^«^MO«iO
Ok^OMo^oooooMooors. rvNO vo so o on O
MMMM M*4MM MM
i
O M<«>OM<iti^«^ONO M O\00 kO «l O
M MMMMM W' MM
<*• OS O «o v> o •'WO OkV%«|M4*sOO»*9 OsOQ O
u^so ko t>. ^ !«>. ^ tooe \o r«\o m m o O o «a ^^
=^1
>
O ooeeMvtHMQOM ♦♦•*
M M MM MM
f* ^ t^ f« •! ^ Q ^so «i o\oo o <^oe Q 00 ve v% t^
>«#i«o «n r« ^>o ^ f«>oo >o>o «'%m m o^O o^iA^tt
i
M v»M00MSOMfn0ko «ntifl« 0^
M MM MM MM
«^QP Ok ^ ^ 0« II ^ oo^"*Oll<OQ9 O^oo ^ ^
O i.9
00 MH
-J.flL MMMMMMMMMMMM,*" MmM
a
tr M «0c«ooo\ moo^oo«« ^oo
M MM
r S:; Skips' r^ S^SS $^^d ?<l'3kS ;s^?> 9.
M 00 ^ «n 00 00
•OM
t*« rt
okoovd osoooooMao^t^ t^^iS ^^o S% ok o^
iss
M
Okoe f«
oo a% Ok
fl M «0 W^ Ok «/^ t*»'tn
o «
Okoe fl «4^dk«ndkO OkO anir KOO eO tN ti lO cm
Oo8 okoo r-^Moo 5 o\kO ko ^ in «a 9mo 9?
4-
00 «« ^ m 0\kO M lO M to
•^ « M M
►« o r*^
J« 2 J^!^*^io •too tntsM ^fnu-»vk«n^ fi^m*
S 2 2 2 2 •" S ^^ «• o M 5 IX t>*« kO ^ o ov d
^V^MM M MMMM M M
g 8-8 3i 8 8 8 8 8 8 8 8 8 8^ 8 8 8 8
mmmmmmmmmmmmmMmm^Mmm
^f'-'"
r 449 3
Gg
^Ta-
450
Weights mi Measures. Book IL
J Ta
with
B L E reprefenting the conformity which the long
each other, taken from Po/ilethwayt^s Commercial
^ o
The ^lls of Amftcrdam, Haer-| A
lem, Leydcn, thfe Hajgue, Rotter-
dam, and other cities of Holland,
as well as the ell of Nuremberg, are
equal among themfelves. They are
alfo comprehended under the ell of
Amfterdam, as that of Ofnaburgh
is under that of France and Eng-
land } and the ell of Bern and Bafil
under that of Hamburgh, Frank-
fort, and Leipfick.
A loo yard! of Eogl. Scotl. and Ireland
B loo ells of France and England - -
C loo ells of Holland or Amftcrdam -
D loo ells of Antwerp and BraiTcls - -
£ I oo ells of Hamburgh, Frankfort^ &c.
F zoo ells of BreHau, in Silefia - - •
G IOC ells of Dantaick - - - - -
H loo ells of Bergue and Drontheim •
I 100 ells of Sweden or Stockholm
K i&oeUf of St. Gall, for linen - -
L 100 ells of St. Gall, for cloth - -
M 1 00 ells of Geneva -----* i ^
N ICO canes of Marfcilles and Montpelier f g
O 100 canes of Thoul. and Upper Lang-
P 100 canes of Genoa, of 9 palmos .
Q^ioo canes of Rome - - - - .
R xoo Tares of Caftille and Bifcay ,- -
S 100 Tares of Cadia and Andalufia
T xoo vares of Portugal or Li (boo - .
V loocovedos of Portugal or Lifbon .
W xoo brafTes of Venice . - - - -
X 100 braiTes of Bergamo, ftc. ...
Y 100 brafles of Florence, Leghorn, &c.
Z 100 brafTes of Milan «... -J
• o
o
#-►
I
B
100
76
62i
60
66|
67i
(>5i
87
67
I24«
2I4t
i99i
245*
27.7 1
93-J
9H
123
74
73i
72t
6si
58il
w
1^ CO
a. **i
n
W
I
78
100
57t
60
481
464
52
S2t
5't
67T
S2i
97t
167^
156
177I
735
7i|
96
58i
57;-
554
50
45t
s
? o
>
3
I
i33i
i73t
100
roi-J
83t
80
89
90
"/a
116
895^
166 j^
286
2661
327
303
125
I22f
164
100
98
95
85t
78^
D
S
M
>
s
•o
s
I3IT
i66|
98T
100
82t
79
87i
89
86i^
"4t
88^
i64i
282f
26jt
323
299t
I23t
119
162
98^
96I
93f
84f
77
Chap. VI. Weights and Measures.
45«
nuafures of the prinafal trading dtiis $/ Europi baoi
HiAianarj.
£
F
G H
I
K
L
M
3*
w
2
w
w
W
M
H
^ ST
•5' a:
fT ST
ST
0
If
P CO
IsofS
holm.
Is of
linen.
Is of
Is of
cloth.
'
ambure
fick and
0^
0
o
1
a.
0
CO
•
CO
•
0
I
■if-
S'
CO
•
1
CO
I
5*
V
r 1
1
1
1
., .
A
i6o
i66f
150
't^
»54
"44
1494
80
B
205t
2»3i
192^
188
195I
'^J
1914
1024
C
110
1264
1I2|
110
"4i
86
112
60
604
P
I"i
"4
"14
116
87
"34
E
too
10+t
924
§'*
95t,
1^\
014
50
F
96
100
894
88
9ii
684
894
48
G
9H
lilt
100
98
102
764
994
534
H
108
ii2i
lOlt
100
103
774
100^
54
I
los
?09t
974
96i
100
75t
98
52i
K
i39t
»45
»30t
1274
98i
'33
100
130
694
L
1074
M I-J lOOj-
102t
764
100
534
M
200
208t
i87i
i83t
191
H34
130T
100
N
3+3 s^
357i
32 li
3144
327*
246
320t
1714
O
3*0
333f
300
i93t
304
229 f
2984
160
P
3921^
4of4
3674
3594
374i
2814
366t
iQ6i:
Q,
363f
378i
3404
3337
347^
a6o4
3394
18 1 4
R
ISO
i56t
1404
»37i-
»434
1074
140
75
S
1464
1S2|
138
»34i
'394
105
'37
734
T
196^
205
i84i
i8ot
1874
141
1834
944
V
120
125
II2i
110
ii4i
86
112
60
W
"7i
I22f
i04t
1074
II2t
!*♦
1094
584
X
"4
ii8|
1064
i04i
io8|
8x4
1064
57
Y
1024
io6f
la
94
98
73t
954
Si\
Z
934
97il
854
89t
67
874 4641
l> »■ I ■■
-r
Gg %
452
Weights and Measures. Book IF*
A Table nprifenting the confirmty which the long
with Mcb otbery taken from PoJlLsthwayfs Cmmeraal
Continued.
N
O
9
O CA
So
A lOo yards of England, Scotland, &c.
B iGoclls of France and England • -
C loo clli of Holland and Amfterdam -
T> loo elli of Antwerp and BruflTels - -
E I60 cUa of Hamburgh, Frankfort, &c.
F lOO ellt of Brcflau, in Silefia - - -
G loo cUt of Daptiick
H I oo ells of Bcrgue and Diontheim
I 100 ells of Sweden or Stockholm - -
K lOo ella of St. Gall, for linen - - -
L lOO ells of St. Gall, for cloth - - -
M lOO ella of Geneva - - - - t -
N loo canes of Marfeilles and Monipelier
O ICO canes of ThouJoufe, &c. - - -
P ICO canes of Genoa, of 9 palmos - -
O 100 canes of Rome ------
R 100 vares of Caftille and Bifcay - -
S 100 varea of Cadix, &t. - - - -
T 100 vares of Portugal or LUbon - -
V I JO covcdos of Portugal or Liibon
W loo brafles of Venice - - - - -
X too braffcs of Bergamo, &c, - - -
Y 100 braffcs of Florence, &c. - - -
Z 1.0 brafles of Milan . - - - -
o
n
CA
Q- ^
^2
I
Is
3 "
Q o
CO
•-I «-*
n o
?» C
>
464
35
35t
28
3't
3ii
30i
V>r
3it
584
100
93r
Ii4i
116
43i
57f
35
344
33t
30
27I
5°
64t
r:
3ii
30
33t
33J
3H
43i
33t
62i
I07t
100
1224
"3t
46^
45
6ii
37i
36i
354
3*
29 1
Q.I
CO
C
O
o
404
5>t
30t
304
254
24t
27t
274
26^
354
27t
504
87t
814
100
924
38i
37t
50.
29I
29
26i-
234
O
»>
s
n
CD
50
o
3
44
564
33
334
27-
26^
294
291
284
3&4
29i
55
944
88
io8
ICO
4it
404
54i
33
324
3H
28f
»5t
chap. VI. Weights and Measures.
453
meafures of thi principal trading cities of Europe have
Dictionary.
R
S
T
V
w
X
Y
Z
<
<!
<
0
ee
►=."
u
Cd
ares<
cay.
r§
§,
r
t3 CO
u
' n
1
§2,
s
B
TO
§s
• 0
►tJ
0
"-1
0
0
•
\9
S3 0
It) •»>
- w
g.tp
peg
*• 0
2 S
• r
1
s:
i"
•
107
i09t
8i|
1334
136
1044
1544
1714
i36t
140
io4f
171
1744
•79
»99t
2194
•80
1'^-
61
r
100
102
1054
1 164
I28t
81
84 '
614
101^
1034
1064
118
130
654
684.
504
!34
^5
884
97
107
64
f>sf
484
80
8,4
844
934
1024
7it
^^\
54i
89
904
93t
1034
ii4f
72
741
55
90
914
944
105
"54
70
7 It
534
87-»-
o9t
92
102
ii2i
924
95t
704
116
n8|
122
»354
149
7it
73
54i
!?^
914
94
104
"44
i^>
»3H
1014
1664
170
1724
>934
2I4t
228^
234
1744
286
2914
301
3334
3674
2131
218
1624
2664
272
2804
3094
3424
2614
268f
mi 327 1
3334
3444
381
420J-
2424
2454
1844
303
309
3>9
353
3894
100
102-f
76i
125
1274
i3>i
H54
1594
97^
100
744
122^
"54
J 79
142
»57
»3It
»34
ICO
164
i67t
1724 :
191 2104 1
80.
814
61
100
102
1054
1164
I28t
784
8ot
594
98
100
1034
1144 ' "6
76
^^
58
95
97
100
1004 122
684
704
524
854 874 1
95
100 1094
624
634
47i 78 .1 794'
824'
91 100
Gg3
3. Sup-
454 Weights and MsASukss. Book IL
3. Suppofe you owe looanees of wheat at Lyons, and
would know what quantity you would purchafe at Macon
to replace them, and have no other means of knowledge
but the following ; viz.
g ances of Lyons — =s ^ fetiers of Paris.
I feticp of Paris - - = 2 bulhels of Bourdcaux.
2f bufliels of Bourdeaux = ^^ muds of Amfterdam.
4(f muds of Amfterdam = ^^ fanegas of Cadiz,
, /j^ fanegas of Cadiz - = 0 anees of Macon.
How many of Macon - = ft^^ of Lyons ?
9 - I
I 19
jr • ^
r ------- 20
2 X 19 X 20 = 760
9) 760 (841^ anees of Macon, the anfwcr.
4. Suppofe a 'merchant of Hamburgh, not knowing; die
Sroportion between'the ell of that plac« and yard of Lon-
on, and having orders to procure 8i yards of cloth, of
which 7 elts of Hamburgh muft be had for 3 1. fterliQg ;
how ihall he difcover how many pounds fterling the 81
yards will amount to, only by knowing that 7 ells of
France ma|^e 9 yards of London) and 7 ells of Holland
make 4 elk of Prance, and that i ell of Holland make
I f of Hamburgh ?
Note, fince z is 14 ; confequently^ 5 =: 6, which dif-
patches the fradlion*
g yards of London = ^ ells of France.
^ ells of France - rs 7 ells of Holland.
5 ditto of Holland =: f ditto of Hamburgh.
^ ditto of Hamburgh == g pound fterling.
How much fterling for 81 yards?
2 Z
81x7 = 567, divifor.
2x5= 10, dividend.
*•* 56.7 s= 56I. 14 s. the anfwer required.
The End of the Second Book,
Arithmetical O)lledions
AND
IMPROVEMENTS.
4|ogQ3(%>o{3(4(dgj{oo$ooQooj{oo^^
BOOK III;
Containing the more dbftrufe and curious part of
Arithmetic K*
CHAPTER I.
ALLIGATION.
WH£Ncorn> wine, fpices, metal, &c. are required
to be mixed together, the method of proportioning
fildi mixhires is called the rule of alligation.
SECT. I.
ALLIGATION MEDIAL.
BY alligation medial the mean rate or price of any mix-
ture is found, when the particular quantities and their
prices are given.
RULE.
Firft find the fum of all the quantities propofed to be
mixed, and alfo the fum of their particular rates ; then
as the fum of all the quantities : is to the fum of all the
rates ; : fo is any part of the mixture : to the mean rate or
price of that part.
I. A vintner mixeth 314 gallons of Malaga fack, worth
7 s. 6d. a gallon, with 18 gallons of Canary, at 6 s. gd.
a gallon ; 13L gallons of cherry, at 5 s. a gallon ; and 27
gallons of white wine, at 4 s. 3d. a gallon ; what is a gal-
lon of this mixture worth ?
gal. s., d. 1. s. d.
3i|. - - fack - - at 7 6 - - = 11 16 3
18 - - Canary * at 6 9 - - =s 616
I3t - - cherry- .ats---=:376
27 - - whitewine- at43-*=s5i4 9
Gg4 271.
45^ Alligation Medial: BookllL
1. 10 X 9 = 90*
) 27
■—8. d* .
2 14 -
10
9
- 6 - per gallon, the anfwtr.
2. With i^ gallons of Canarjr, at 6 s. 8 d. a gallon, I
inixed 20 gallons of white wine, at 5 s. a gallon, and to
thefe added 10 gallons of cyder, at 3 s. a gsulon ; at what
rate muft I fell a quart of this mixture, fo as to dear 10 per
cent. I
gal« s« d* L 8. d.
13 - - Canary, -at68--468
20 - - white wine at 5 - - - 5 - -
10 - - cyder --at3---iio-
43 = 172 quarts . - - * • ^ 10 i6 8=io«8j
xo) 10.8^
172) 11.91^ (.06928== IS. i)..627d. tbeanfwer.
SECT. IL
ALLIGATION ALTERNATE,
IS when the particular rate of evenr ingredient, and the
mean rate, are given, to difcover the particular quantity
of each ingredient concerned in a mixture.
RULE.
Place the mean rate fo, that it may be eafily compared
with the particular rates ; fetting down the differences be-
tween the mean rate anc^ the particular rates, alternately,
and they will be the quantities required.
I. A
Chapel* Alligation Altbrnati. 4.57
I. A grocer would mix a quantity of fugar, at lod. per
pound, with other fugars of y^d, 5d. and 44-d. per pound,
intending to make up a commodity worth 6d. per pound i
in what proportions is he to take of thofe fugars ?
4
I
4
When bne branch is h'nked to two or more other branches,
the differences ought to be as often tranfcribed as it is fo di-
verfly Unked.
2.. A proveditor for the army intending to mix wheat at
4 s. abuihel, with rye at 3 s. abufhel, with barlevat2s. a
bufhel, with peafe at i s. 4d. a bufliel, and witn oats at
1 2d. a bufhel, is defirous to know in what proportion to
mix them, fo that the mafs may be worth is. 8d, per
bufliel?
There are divers ways of alligating or linking thefe num-
bers together ; viz.
or,
2
8
4
4
16 + 4
08
8
4
4
20
28
4
8
4
28 + 4
t6
4
8
4
|i6
I
2
I
8
4
W
R
B
P
O
or,
Alligation Alternate. Book ITT.
tli.//-ja8 + 4l3al8| . L>»_^U8 i8
4 48|i2l (.iiif^ l28+.6+4|48
|4 + 8 liij 3| r4S-X I '
,8 8 , \!6-%3\ 1
'28 |'8 yl / i6jy)J\i[
U8+16+4I48I1JI l_\zjy \z'.
r48>^ 4+8 |IJ| )| r48-^| 8
V56^^J^ 4+8 ..(3 IjeO, 8 + 4
{•eJ/U ,i + .6 4J,, /,5j)(l),6 + 4
Ci2>^ U8+16+4I48I12I Ll2>^'|s8+i6+4
8 z
'^ 3
32 8
48,.
8 3
12 3
'2 3
20 S
48I12
Chap. L AiLioATJON Altk&natb.
459
SO
4+8
12
3
4 .
4
1
4+8
12
3
ii+i6+4
48
12
28 + 4
32
8
4 I 4M
8+4 UA 3
8^4 lizl 3I10
28+16+448 12
16 + 4 I20I 51
4+8
4+8
4 + 8
28+16+448
I28+16+4I48
12
12
12
W
R
B
P
O
W
R
B
P
O
Here you have 24 different anfwers by the various wsys of
alligadng or linking the prices together, which may be in-
creafed infinitely by doubling, tripling, &c. the quantities;
or they may be leli^ned by making the pecks, pints^ or any
leiler quantity.
The reafon of thefe combinations, and the alternate
placing oT their differences, will appear from this plain con-
flderation, viz. that i^hatfoever is loft by felling any quan-
tity whofe price exceeds the mean, is gained again on the
quantity alligated thereto, whofe given price is lefs than the
iDeai|«
When two kinds of things only are given to be mixed^
the rule of ^ligation will give but one aniwer*
3. Suppofe it is required to m;x brandy^ at 8 $• per gallon^
with cyder, at i s. per gallon ?
5 8\ I 4 gallons of ^brandy.
^(1/13 gallons of cyder.
If three kinds of things are given to be mixed, the rule
qf alligation will give but one anfwer ; but then (as might
have been obferved in mixture of two things] all numbers
that are in the fame proportion between themlelves, and thei
number which compofes that anfwer, will alfo fatisfy the
queflion.
But by an artifice explained by the ingenious Mr. Jamea
Dodfon, in the 1 8th edition of Wingate's Arithmetic, innu-
merable other anfwers may be obtained, compofed of num-
bers in a different proportion.
4« Let it be req^ired to mix brandy, at 8 s. per gal-
lon, with wine at 7 s. per gallon, ana cyder at i s. per
3. gallon J
460 Alligation Alternate. Book III.
gallon ; fo that the mixture may be worth 5 s. per gal-
lon ? \
8-N
4
4
3+2t
4 brandy,
4 wine.
5 cyder.
Now fuppofe, that if it be determined to ufe five gallons
of cyder in the mixture ; but to ufe any quantity of brandy
and wine that will anfwer the queftion.
Then may the quantity of brandy be increafed or dimi-
. niihed by 2 ; the difference between the prices of the wine
and mixture, if at the fame time the quantity of wine be
diminiihed or increafed by 3, the difference of the prices of
the brandy and mixture.
Thus, 44-2=6 brandy, and 4 — 3 = i wine ; fo
that fix gallons of brandy, one gallon of. wiiie, and five
gallons of cyder, will alfo anfwer the queflion, as may be
eafily provea by alligation medial.
Again, 4 — 2 = 2 brandy, and 44-3 = 7 wine.
So tihat two gallons of brandy, feven ^lons of wine, and
five gallons of cyder, will alfo anfwer the queftion, as may
be proved.
But inftead of the numbers of the firft ianfwer, 4, 4 and
5, larger numbers in the fame proportion, viz. 12, 12
and 15 were taken, the follbwing eight anfwers would be
found by increafing and diminifhing the quantities of
brandy and wine, as above dire£led, the quantity of cyder
remaining conflantly 15. *
Brandy 18 . 16 . 14 • 12 . iq
Wine 3 . 6 . 9 • 12 . 15
Cyder 15 • 15 • 15 • 15 • 15
And if inflead of thcfc ftiU larger numbers in that propor-
tion, or in proportion to any of the laft found anfwers, be
afTumed, a greater number of other anfwers may be found.
But if inftead of fuppofing the quantity of cyder invari-
able, the quantity of brindy be taken for fuch 5 then an
infinite number of anfwers maybe found, by continually in-
creafing the quantity of wine oy 4, the difference between
the prices of the cyder and mixture; and the quantity of
cyder by 2, the difference between the prices of the wine
and mixture.
Thus, afTuming the fecond anfwer, 6, i and 5, and
making the fix gallons of brandy invariable :
Brandy
8 .
6 .
4 •
a
18 .
21 .
•
24 .
27
15 •
15 •
»5 •
»S
Chap. I. Alligation Paktial. 461
Brandy 6 .6.6. 6 . 6 • 6 . 6 . 6 • 6, &c.
Wine I - S . 9 . 13 . 17 . 21 . 25 . 29 . 33, &c.
Cyder $ . 7 . 9 . 11 . 13 . 15 . 17 . 19 . 21, &c.
Or by taking the third anfwer, 2, 7 and 5, as the bafis,
and making the feven gallons of wine invariable ; increafing
the quantity of brandy by 4, the difference between the
price of the cyder and mixture j and the quantity of cyder
by 3, the difference of the prices of the brandy and mix-
ture:
Brandy 2 . 6 • 10 . 14 • 14 • 22 • 26 . 30, &c.
Wine 7*7« 7- 7- 7- 7- 7- 7>&c*
Cyder 5 . 8 . ii . 14 . 17 . 20 • 23 • 26, &c.
When there are four kinds of things to be mixed, and
two of them of greater value, and the other two of leffer
value than the mixture, the rule of alligation will give
feven anfwers, as may be obferved by queflion i, in this
rule ; with any of which, or with any numbers in the fame
proportion, innumerable other anfwers may be found, con«
fitting of numbers in different proportion among them-
felves, by making any two invariable, and changing the
reft in tne manner as above, obferving alfo the following
RULE.
The numbers by which the quantity of any fimple is to
be varied, is always the difference between the price of the
mixture and the price of the other fimple, which in any
operation is confidered as variable.
Secondly, That if the fimples, which in any operation
are confidered as variable, be both of greater, or both of
lefs value than the mixture, then, while the one is ip-
creafed, the other muft be diminifhed ; but if one be of
greater value than the mixture, and the other of lefs,
then they muft both be increafed, or both diminifhed.
5. Let it be required to mix brandy, at 8 s. wine, at
7 s. cyder, at i s . and water, at nothing per gallon, toger
ther; fo that the mixture may be worth 5 s. per gallon ?
I fhall only alligate the feveral values of the fimples
together by tne following method:
brandy,
wine,
cyder,
water*.
Now
5 + 49
5 + 49
3+^ 5
3 + ^5
v^««
AiLiOATtoK Alternate, fiook I
Now making the wiAC and cycler invariable :
Brandy 9 • 14 • 19 • 24 • 29 • 34 • 39, &c.
Wine 9« 9. 9. 9* 9* 9« g* &c«
Cyder 5 . 5 • 5 • 5 • S • S • 5> *^c.
Water 5 • B . 11 . 14 • 17 • 20 • 23, &c.
Making the brandy and Cyder invariable :
Brandy 9. 9. 9. 9. 9. 9. 9, &c.
Wine 9 • 14 . 19 • 24 • 29 . 34 t 39i &c.
Cyder 5 . 5 • 5 • 5 . 5 . 5 . 5. &c.
I Water 5 . 7 . 9 • II . 13 • 15 • 17) &c.
Making the wine and water invariaU^ :
Brandy 9 . 13 • 17 • 21 • 25 • 29 • 33, &c.
Wine 9* o. 9. 9. 9. 9. 9, &c.
Cyder 5 • o • 11 . 14 • 17 • 20 • 23, &c.
Water 5 . 5 . 5 . 5 • 5 . 5 . 5, &c.
Making the brandy and wine invariable :
Brandy 9*9*9
Wine 9.9.9
Cyder 10 • 5 • -^
Water 1,5.9
Or taking four other numbers in the fame proportion :
As 9 . 9 • 5 and 5, viz* 36 • 36 • 20 and 20.
Brandy 36 • 36 • 36 • 36 • 36 . 36 • 36 • 36
Wine 36 . 36 . 36 . 36 • 36 . 36 , 36 . 36
Cyder ' 40 . 3^ • 30 • 25 • 20 • 15 • 10 • 5
Water 4 • 8 • 12 • 16 • 20 . 24 . 28 * 32
Laftly, making the cyder and water invariable :
Brandy, &c. 44 • 42 . 40 • 38 . 36 • 34 . 32 . 30 . 28, &c.
Wine, &C. 24 • 27 • 30 • 33 • 36 . 39 . 42 • 45 . 48, &c.
Cyder, &c. 20 • 20 • 20 • 20 • 20 • 20 . 20 • 20 . 20, &c.
Water, &c. 20 • 20 . 20 • 20 . 20 • 20 • 20 . 20 • 20, &c«
Not only the fets of numbers thus found, but their fums
and diiFerences^ will alfo be anfwers.
. 'Thus
Chip*!. AtLIGAtlON ALTEtlKATt* 46$
bran. wine. cyd. wat.
Thus from or 1042 • 27 • zo . 20
Take or add - 9 • 9.10. i
The remainder . 33. 18 > to. ig?^;;, ^^ ^^^^^^ ^^
And film - . SI . 36 . 30 . 2i3 *M"eftion.
Thefe anfwers may all be proved by alligation medial ; I
(hall only prove the laft, viz. the difi^rence, and leave the
rtft to exercife the young arithmetician )
gaU s. K s. d.
viz. 33 brandy, at 8--^- 134-
18 wine, at 7---- 66-
10 cyder, ati---- -10-
19 water, at----- -.--
80 £ ^o ' -
t*»m»
80 gal. : 20 1. : : I gal. : 5 s.
SECT. III.
ALLIGATION PARTIAL.
»
A Litigation Partial is when, having the feveral
rates of divers ingredients and the quantity of one of
them given, we difcover the feveral quantities of the reft in
fiich fort, that the quantities fo found, being mixed with
the quantity given, that mixture may bear a certain rate
propoftd.
Having fet down the mean rate, the particular rates and
their differences, as before, fay,
RULE,
As the difference oppofite to the known quantity is to :
the known quantity, fo is : : any other difference : to the
quantity of its oppoflte name.
I. Let it be required to mix brandy, at 8s. per gallon,
and wine, at 7 s. per gallon, with 10, gallons of cyder, at
IS*
464 Alligation ALTEkNATS. Book III.
I s. per gallon^ fo that the mixture may be worth 5 s. per
gallon.
©
4
4
3 + a_
13
4
4
S
5 :
S •
10
10
4 : 8 gallons each of brandy and wine.
13 : 26 gallons, the whole mixture.
Now, having found one anfwer by the above proportion^
others may be found by the method before delivered.
Brandy 12 . 10 • 8 . 6 . 4 . 2
Wine 2 . 5 • 8 • II • 14 • 17
Cyder 10 . 10 • 10 • 10 • 10 • 10
By which means five other anfwefs are obtained.
2. A tobacconift has by him 120 lb. of fine Oroonoko
tobacco, worth 2 s. 6 d. a pound ; to this he would put as
much York- river ditto, at 2od. with other inferior tobaccoa
at i8d. and 15 d. a pound, as will make up a mixture an-
fwerable to 2s. a pound; what will this parcel weig^ i
6
6
6 + 9
■I
6
6
• •
Then 19 + 6 + 6 + 6 = 37.
19 : 37 : : 120 : 233449 the aniwoT required.
But as fomq anfwers in whole numbers may alfo be ob-
tained by the JForegoing method, putting 38, 12, I2andia
inftead of thofe found by alligation, the two laft being in-
variable.
38 . 42 . 46 . 50 . 54 . 58 . 62
12 • 18 . 24 . 30 . 36 . 42 . 46
12 • 12 . 12 • 12 . 12 • 12 • 12
12 • 12 . 12 . 12 • 12 • 12 • 12
Now, taking the fum of the two laft fets of numbers*
at 2s, 6d. at I s.8d. at is. 6d. at is. 3d.
viz. 58 • 42 • 12 • 12
62 • 48 • 12 • 12
120 4- 90 + 24 + 24=^258^
being a fecond anfwer.
By
Chap* I. Alllioation Partiai. 4(55
By making the fecond and laft invariable :
3^ •
38
44
•
50 •
12
12
12
«
12
6
12
18
*
24
i%
12
12
•
12
at as. 6d. at
I s. 8 d*
at 18. 6d.
at
ISr 3d.
3^
12
^ 6
•
12
38 *
12
.12
•
12
50
12
• 24
•
12
126 4.
36
+ 4»
+
36 = 234>
the third anfwer,
Laftly, making the fecond and third invariable :
11 • 20 • 29 . 38 • 47 • 56 . 65 • 74 . 83 . 92. lOI
12 . 12 • 12 . 12 • 12 . 12 . 12 •' 12 . 12 . 12 • 12
12 • 12 . 12 . 12 • 12 . 12 . 12 • 12 . 12 • 12. 12
•<-6 • o . 6 • 12 • 18 . 24 • 30 • 36 • 42 . 48. 54
2s. 6d. • IS. 8d. • i8« 6d. • IS. 3d.
loi . 12 .12 • 54
loi • 12 .12 • 54
47 • . 12 • 12 . 18'
II . 12 . 12 . — 6
260 • 48 • 48 .120
fubtr. 20 • 12 • 12 • *
2)240 . 36 . 36 . 120
120 4. 18 + 18 + 60 S= 2l6,
the fourth anfwer.
Thefc laft three anfwcrs mav each be made the bafis of
divers others in different proportion, by making the firft term
with any one of the others invariable ; and the other two va-
riable to their utmoft limits, which I fliall leave for the prac-
tice of young ftudents in arithmetic i having (I think) been
copious enough upon this fubje£t.
H h SECT.
•••
C 466 1
SEC T. IV.
ALLIGATION TOTAL.
•
ALLiaATioN Total is* fo calkd, when die particular
rates, the. mean rate, and the whole quantity of the
ingredients to be mixed, aregiven, and the particular quan-
tity of each ingredient is required. To find which, ob-
ferve the following
R U L K
Having found the feveral* difFe|«nces as'before dlreded^
fay, as .the fuai of all. the difierencea : ifr to the whole
quantity of the mixture : : fo is each particular difference : to
its particular quantity.
I* Let it be required to mix brandy, at S^s. ^wine, at 73.
and cyder, at i s. per gallon together. ; fo that the mixture
may cqntain 26 gallons, and be worth 5 s. per gallon.
li
8
7
4
4
3 + 2
'3
26 ::,|4
41
4
5
13
8 brandy.
t wine.
ID cyder.
26
One anfwer being thus obtained, the reft may be foun/d
by the following
RULE.
I. Let the quantity of that ingredient, whofe value alone
is greater or lefs than the value of dne mixture, be in*
creafed or diminiihed J[>y the difference or differences b^
tween the prices of the other two ingredients, and the
price of the itiixture.
II. Of the remaining two ingredients, let the quanti^
of that ingredient, whofe value is fartheft firom the value of
the mixture, be increafed or decreafed (according as the
former is) by the fum of the differences between the prices
of the other two ingredients, and that of the mixture.
I lU. Let
Chap. I. ALLJd^ATION ToTAt.' 467
III. Let the quanti^ of the remarning ingredient be de-
creafed orincreafed, alfo, by thefum of the differences be-,
tween the prices of the other two ingredients, and that of
the mixture } but obferve^ that the quantity of this ingre-
dient is to be decreafed,- when thofe of the two former
are increafed ; and the coilttary.
1. The value of the cyder alone is lefs than the value of
the mixture.
Alfo, 8 — 5 = 3^ 7 — 5=3 2, and 3 — 2 = i, the
difference of thofc di^orences.
• . • 10 + 1 = XI9 and 10 — I i= 9,* are the quantities
of cyder.
II. Of the other two the value of the brandy is furtheft
from that of the mixture.
Alfo, 7 — 5=3 2, 5— i=s4, and 2 + 4 = 6, fum
of their aifFerences.
••• 8 + 6 = 14, and 8 — • 6 = 2, ard the quantities of
brandy.
Lafthr, 8 ~ 5 ss > 5— i = 4, and 4 4-3 = 7.
• . • 0 — 7 = 1, and 8 -f- 7 =1 15, are the quantities of
the wine.
Thus We have obtained two arifwers more, which make
in all three different ahrwers to this queftion.
Cyder ii . w . 9 J gat»°n ™«1"1-
But if there be four or more ingredients out of which
the mixture is to be compounded, thCn <3ne or more of
^em muft be confidered as invatriable; fo that there may
be only three variable, and thofe fo, that one of thenl will
be of a contrary value, with refpe£l to the price of the
mixture, from the ether two.
2. It is required to mix fuch a quantity of brandy, at
8 s. wine, at 7 s. cyder, at.i s. and water at nothing per
gallon, as will make a hogfhead, or 63 gallons of the mix-
ture, worth 5 s. per gallon.
Then by the proceft in alligation alternate, queftion 5th,
the two following proportions may be found -, viz.
bran. wine. cyd. wat.
Among the firft found anfwers 9 . 9 . 5 . 5
Among the third -r - - - 9 . 14 . 5 . 7
18 -|- 23 -I- lO-f- 12= 63.
Hh 2 Then
r •'
II .
10 .
• 4
39
. 10
II .
> 10
468 Alligation Total. Book IIL
Then making the water invariable, wc have, by the fore-
going rule,
Brandy 36 . 30 . 24 • 18 • 12 . 6
Wine - 2 . 9 • 16 . 23 . 30 . 37
Cyder 13 • 12 • il • 10 • 9 • 8
Water 12 • 12 • 12 • 12 • 12 • 12
II. Making the cyder invariable, produceth
Brandy 32 • 25 . 18
Wine 7 • 15 • ^3
Cyder 10 • 19 • 10
Water 14 . 13 . 12
IIL Making the wine invariable, gives
Brandy 19 . 18 • 17
Wine 23 . 23 . 23
Cyder 2 «^ 10 • 15
Water 19 . 12 . 15
Laftly, making the brandy invariable, we have
Brandy 18 . 18 • 18
Wine 24 • 23 • 22
Cyder 3 • 10 • 17
Water 18 . 12 . 6
If you are defirous to find more anfwers, you may, for
W^ter makes any number invariable from 5«to 19
Cyder from ---^-----2 to 15
Wine from --------- 2 to. 39
Brandy from ---------4 to 36
But if inftead of gallons you mix by pints ; viz.
inftead of 24 . 16 , 11 • 12 gallons,
you take 192 . 128 . 88 . 96 pints for the bails
of the operation, a flill greater number of anfwers may be
produced; viz.
B. 192. 191. 190. 1 89. 188. 187. 1 86. 185.184. 183.1 82. 181. 180. 1 79
W. 1 28.1 28.128.128. 1 28. 128^28. 1 28. 1 28.128. 1 28. 128. 128.138
C. 88. 96.104.112.120.128.136.144.152.160.168.176.184.192
W. 96. 89. 82. 75.. 68. 61. 54. 47. 40. 33. 26. 19. 12. 5
6. 192 . 193 • 194 . I95 . 196 . 197 . 198 . 199 • 20O • 201 • 202
W. 128 . 128 .128 . 128 . 128 . 128 • 128 . 128 . 128 . 128. 128
C. 88. 8q . 72. 64. 56. 48. 40. 32. 24 . 16 . 8
W. 96. 103 . no. 117 ♦ 124. 131 . 138 . 145 . 152-. IC9. 166
C H A A •
[ 469 ]
CHAPTER IL
Spxcific Gravity ^f Metals^ G?r.
THE fpecific gravity of a body^ is the relation thb^
weight .of a body of one kind hath to the weight
of an equal magnitude of a body of another kind.
Gold is the heavieft of all knoWn bodies, the moft male-
able and dudile of all metals ; is incapable of ruft, and
not fonorous when ftruck upon ; requires a ftrong fire to
melt it, is the moft diviftble of all bodies ; and its dudlility
is fuch, that wire-dfawers can extend a leaf of gold to the
xaooooooth part of an inch in thinnefs, over a flatted iilver
wire, which will be perfectly covered, though viewed with a
microfcope; by which means an ounce of gold may be
made to reach more than 1554^ miles.
Silver is the fineft, purefl, moft dudile, and moft preci-
ous metal, according to its natural properties, except gold.
Lead is the heavieft of all metals next to gold ; it is the
fofteft of all, is leaft fonorous, except gold, very dudtile,
and the moft ready fulible of all, except tin.
Tin is a white fhining metal, of fo pliable a nature, that
it may be bent into any form \ its hardnefs is between (liver
and lead, and' is the lighteft of all metals.
Copper is an hard fonorous metal, difficult in fufion, and
is mixed widi gold and filver, in order to harden them,
and render them moire ufeful either in coin or uteniUs,
which would otherwife be too foft and flexible.
Iron is the leaft heavy of all metals, except tin, but
confiderably the hardeft of. them all ; Are renders it more
dudile, being moft of all maleable when hotteft; when
wrought into fteel, is lefs maleable ; it is more capable
of ruft than any other metal ; it is very fonorous, and
requires the ftrongeft fire of all the metals to melt It.
in the compariion of the weights of bodies, i^ will
Ve the moft convenient to confider one body the ftandard
or unit to which others are to be compared.
Rain-water is nearly alike in all places, a cubic foot
of which hath, by repeated experiments, been found to
w^igh 621 pounds averdupoife.
Hh 3 A Ta-
470 Specific Ghayity cf Metals. Book III-
A Table fl)twing the fpfcific g^qvify t$ rain-^watir ,of
inch of each In parts of a pound averdupoifey taken from
of an ounce from l^ard ; fhf deficiencies m both authors
Bodies.
Fine gold - - -
Standard gold . -
Coaftgold ^ -
Quiclclilvcr - -
Lead - - - -
Fine fJver - -
Standard filver -
Caft filver - -
Copper - - -
Plate brafs - -
Caft brafs - -
Steel - - - .
Bar iron - - -
Block tin - - -
Caft iron - - -
Loadftone - -
Blue flate - - -
Veined marble -
Common glafs
Flint ftone - -
Portland ftonc
Freeftone - - •
Brick - - . .
Alabafter - - -
Ivory \
Horn J
fp- gra.
19.640
19.520
18.888
"•313
11.091
10.629
10.528
8.769
8.350
8.104
7.850
7.764
7.238
7-*35
5.106
3.500
2.702
2.600
2.582
2.570
2.352
2.000
1.888
1.832
wt. lb. av. wt. oz. tr.
0.7103587
0.7060185
0.6828703
0.4976574
0.4091696
0.401 1501 I
0.3844400
0.3807870
0.3171658
0.2942C93
0.2929532
0.2839265
0.2808159
0.2617901
0.2580647
0.1846788
0.1 2649 14
0.0977286
0.0940393
0.0933883
o.09295«
0.0915788
0.0723379
0.0683061
o. 0662606
10359273
9.962625
9.911707
7.38441 1
5.984010
5.850025
5.556769
5-503967
4.747121
4.404273
4.272409
4.142127
4.021 36 1
3.861510
3.806568
2.724083
1.867272
1.429A11
1. 360841
i-3Si4«9
1-345139
1.231038
i»04Moi
0.988456
0.958489
Chap. U. Specific Gkavity of Metals. 471
metals and other bodies ; and the weight of a cubic
RohinfoiCs Menfurationy and of ounces iroyy and parts
fuppUed.
Bodies,
Brimftone - -
Clay - . . .
Lignum vitse - -
Coal - - - -
Pitch - . - -
Mahogony wood -
Dry box wood -
Milk I .
Sea water 3 " '
Rain v^ater - -
Red wine - - -
Bees wax - - -
Linfeed oil - -
Proof fpirits 1
or brandy 3
Drjr oak - - -
Olive oil - - -
Beech - - - -
Dry elm \
Dry afli 5 " "*
Drywainfcot - -
Dry yellow fir -
Cedar - - - -
Dry wliite deal -
Cork . - - -
Air . . - .
fp. gra.
1.800
1. 712
1327
1255
1. 1 CO
1.063
1.030
1-033
1 .000
0.993
0.995
0.932
0.927 .
0.915
0.913
0.854
0.800
0.747
0.657
0.613
0.569
0.240
0.0012
wt. lb. av.
0.0651042
0.0619213
0.0479862
0.0453921
0.0415943
0.0384475
0.0372^530
00372530
0.0361690
0.03591C8
•0-0359881
0-0337095
0-0335503
0.0330946
0.0330222
0.0308883
0.028935a
0.0270182
0.0237630*
0.0221715
0.0205801
0.0186805
0.0000434
wt.02.tr.
0.949424
0.902498
0.699936
0.661959
0.606576
0.560691
0.543282
0.542742
0.527458
0.523766
0.524820
0491591
0.489268
0489008
0.481569
0.450449
0.421966
0.394011
0.346539
0-323332
0.300123
0.126590
0.000633
When a heavy body is weighed in any fluid, it lofes
therein fo much of its weight, as an equzJ bulk of that
fluid is found to^ weigh : as for inftance,
A cubic inch of lead = 5.284010 ? ^^^^^^ ^^^
A cubic mch of water = 0.542742 > -^ '
Their difference is = 5.441268, the weight of a cu-
bic inch of lead in the water, &c.
H h 4 I. An
47* Specific Gravity of MgXAU. Book IIL
1. An irregular piece of lead ore, taken from the
Yorkfhire pit, weighs in the fcale juft 12 ounces | but
weighed in water lofes 5 ounces of that weight ; fo that a
quantity of water of the bignefs of the ore weighs juft
5 ounces : from the Derby (aire pit a rough fragment of
ore weighs, out of water, 14^ ounces; and in water 9
ounces : the comparative or the fpecific weight of thefe
two ores |s required ?
144- '-*- 9 = 5t lb. weight of water of an equal bulk.
Then 14 J x 5 = 724 DerbyOiirc 1 ore's gravity.
And 12 X St = 06 to Yorklhire J E. F.
2. An irregular fragment of glafs in the fcale weighs
171 grains ; another of magnet 102 grains : in water tho
firft fetches up no more than 120 grains, and the other 79 :
then 51 and 23 ^e the feveral weights of their compara--
tive bulks of water : what then will their fpecific gravities
turn out to be \
IV X 23 = 3933 glafs to J g
102 X 51 = 5202 magnet J ^37 ^^ i/«*
The folidity of any body, multiplied by the tabular weight
correfponding, will giv^ the weight in pounds averdupoife,
or ounces troy. t
3. What is the weight of a piece of oak, of a redangu*
lar form I whofe length is 56 inches, breadth 18, and depth
1 2 inches?
Firft, 56 X 18 v' 12 = 12096 cubic inches.
Then 12096 X .0330946= 400.3122816 lb. Q. E. F,
4. What ia the diameter of jm iron (hot, weighing 42
poinds averdupoife ?
Firft, .2580647) 42.0000000 (i62.7499«
Then .5236) 162.7499(310.84778457.
^0/ 310.84778457 = 6.7743, the diameter required,
5. What is the weight of an iron bombflicll of three
incites thick, the greateft diameter being 16 inches ?
•
Firft, 16 «f?-p 6 35: 10, the diameter of the concavity.
Alfo 16 X 16 X 16 =4096.
Apd 4096 X .5236 = 2144.6656,
• Ag^in? 10 X 10 X 10 == 1000,
Alf^
Chap. II. Specific Gravity of Metals; 473
Alfo 1000 X .5236 = 523*6.
Then 2144.6656 — 523.6 = 1621.0656, the folidity of
the flielL
••\ 162 1. 0656 X .2580647 s: 418.33981b. the weight
required.
6. In the walls of Balbeck, in Turky, there are three
ftones laid end to end, now in fight, that meafure in length
61 yards ; one of which in particular is 63 feet long,
12 feet thick, and four yards over : nqw, if this block was
marble, what power would balance it, fo as to prepare
it for moving f
Firftly, 63 X 12 X 12 = 9072 folid feet.
Alfo 9072 X 1728 =: 15676416 cubic inches.
Then 15676416 X .0977286 =s 1532034.18871b.
• . • 2240) 1532034 (683 tun, i8cwt. 981b. Q. E, F. .
7. Required the weight of one of the Portland key-ftoneg
to the middle arch of Weftminfter-bridge ; the diameter of
the arch being 76 feet \ the height of the key-ftone five
feet ; the chord of its greateft breadth to the front of the
arch three feet four indbes ^ and its depth in the arch four
feet ?
Firfl, 76 + 5=1 81 ; alfo 3 f. 4 in. = 3-i» greater breadth.
As 81 : 3.^ :: 76 : 3.127572, its leaft breadth.
Here the chords and their arches being nearly equal, viz.
fo fmall a part of fo large a circle differs very little from
a right line, the figure of the key-ftone may be reckoned
a prifmbid, ^nd meafured accordingly ;
viz. 3.^ X 4 = ^Z^Z 5 *J^o 3-'a757* X 4 = 1^.5x0288.
rr»L I3J+ 12.510288 ^ o
Then ^^ ^ ^ = 12.9218105.
Alfo 12.92181 + 12-510288 + 13.^ = 38.76543.
And 38.76543 X f = 64,60905 folid feet.
Thcii 64.60905 X 1728 =: 1 1 1644.4384 i:ubic inches.
••• 1 1 1644.4384 X .0929543 = 10377.831b.
Anfwer, 10377.831b. =: 4 ton, 12 cwt. 2 qrs. 17.831b,
The weight of any body in pounds averdupoife, or
ounces troy, being divided by the tabular weight corre-
fponding, the quotient will be the folidity in cubic
inches,
8. Wh4t
474 Spscim Gt^AtfTV 4f Metals. Book IIZ.
8. What will a block of marble, weighing 8 tQiis»
14 cwt. come to^ at 6 s. a foot folid ?
% tont 14 cwt. nr 194.88 !b.
•0977286) 19488.0000000(199409.4 inches*
1728) 199409.4 (115.4 cubic feet.
1^5-4 X .3 = 34^62l. = 34I. 121. 4|a.
The abfolute weight of a body floating in a fluid, is
prccifely equal to the weight of fuch part of the fluid as
ihall be thruft away thereby, and difplaced ; or, in other
words, to the immerfed part of the body.
9. Suppofe that a man of war, with all its ordnance,
rigging, and appointments, draws fo much water as to
difplace 1300 tons of fea- water, London beer meafure ;
the weight of this Ye&l is required?
Firft, 1300 X 4 = $200 hogfheads.
Alfo 5200 X 15228 = 79185600 cubic inches.
And 70 185600 X -037253 u= 294990^ lb. averdupoife.
Annver, 2949901ID. = 26338 cwt. iqr. 171b.
10. How many inches will a cubic foot of dry elm fink
in common water?
1728 X .0289352 = 50.0000256 lb. is the weight of a
foot of elm, or of the water difplaced.
O.J6169) 50.0000256 (1382.4 cubic inches immerfed.
* • * 144) 1382.4 (9 6 inches, the anfwer.
«
iz. Suppofe a feaman hath a gallon of brandy in a glafs
bottle, that weighs 3^ lb. troy on board ; and to conceal it
from the king's ofllcers, throws it into the fea \ if it will
finky how much force will juft buey it up ?
Firfl:, 3J. lb. troy = 42 ounces.
Alfo 1.360841) 42.000000 (30.864 cubic inches.
Then 231 X .489268 s= 113.020908 ounces brandy.
And 42 4" 113.020908 = 155.020908 ounces in ^11.
Again, 231 + 30.864 = 261.864 inches of water.
Alfo 261.864 X •542742 = 142.12462 ounces of water.
*• ' 155.020908 -— 142.124621= 12.896288 ounces heavier
than the fame bulk of fait water.
12. Another of the mariners has half an anchor of
brandy ; the calk fuppofe meafures -J- of a cubic foot ;
what
Chap. II. Sfioific Gravity 9f MtTAts. 475
what quantity of lead is joft rcquiiite to keep the cade and
liquor under water \
Firft, 8) 1728 (216 cubic inches, the caflc*
Alfo 231 X 5 = 1155 cubic inches of brandy.
Then 216 X ^489008 = 105.625728 oz« wt. of the calk.
•Alfo 1 155 X .489268 =: 565.104540, weight of the brandy.
Again, 2x6 -f- i'55 = 1371 <^ubic inches.
Then 1271 X .543^74^ = 744-099282, weight of w^tcr
of an eoual bulk.
Alfo 105.625728 + 565.10454 = 670.730268.
And 744.099282 — 670.730268 = 73.268914.
Alfo one inch of lead 5.98401 — .542742 =: 5.44x2689
weight of one inch of leaid in water.
•. • Recip. 5.9840X : 73.268914 : : 5.44x268 : 80^ ounces
troy of lead to keep the cafk, with its contents, juft under
water. Q. £. F.
13. How thick muft be the metal of a concave copper
ball, fix inches in its outfide diameter, fo as to fink to its
center in common water ?
Firft, 6x6x6 = 21 6, cube of the diameter.
Alfo 216 X •5236 == X 13.0976 cubic inches, the folidity
of the fphere.
2} XI 3.0976 (56.5488 cubic inches to be immerfed, or
of water to be removed.
*•* ^6*5488 X -036169 = 2.04531b. weight of the cop*
per ball.
And .3171658) 2.0453000 (6448678 cubic inches x>f
copper in the ball.
Again, 6 X 6 X 3«i4i6 =: 113.0976 fquare inches, fu-
perficies of the ball. N. B. The folidity and fuperficies of
this ball are equal.
••• 113.0976) 6.44870 (.057, or about i^^ of an inch
in thicknefs. Q. £. F.
14. What will a chain of ftandard gold weigh in water,
that raifes a fluid an inch in a vefiel three inches fquare, when
put into it ? And fuppofing the workman had adulterated
the faid chain with 14^ ounces of filver ; how much higher
would the water, upon its immerfion being raifed in the
veffel ?
Firft, 3 X 3 ::= 9 ^olid inches in the gold chain.
Then
'47^ Smcific Gravity «/ Metals. Book IIL
Then 9 X 9*962625 =: 89.663625, its weight in air.
And 9X0.527458=1 4.747 1 22, wt. of its bulk of water.
Weight of the gold 84.916503 in water. Q. £. FJ
. A folid inch of filver is 5*556769 ounces trov.
As 5-5<6769 : i : : 14.5 : 2.6094 inches of ulver.
Then 89»663625 •— 14.5 = 75.163625 ounces of gold.
9.962625) 75.163625 (7»|546, fpace taken up by the gold.
2*0094, by the filver, as above.
Sum 10.164, by both.
Then iai64 — 9 = 1.164.
•.• 9) I.i64(.i29^. Q: £• F.
15. Hiero, king of Sicily, ordered his jeweller to make
him a crown, containing 03 ounces of gold ; the workman
thought fubftituting pait filver therein a proper perquifite ;
which taking air, Archimedes was appointed to examine it,
who, on putting it into a veflel of water, found it raifed the
fluid, or that itielf contained 8*2245 cubic inches of metal ;
and having difcovered that the cubic inch of gold- more
critically weighed 10.36 ounces, and that of filver but
5.85 ounces } ne, by calculation, found what part of his
majefty's gold had been changed ', and you are deiired to re*
peat the procefs i
10.36)63.00 (6.08108 inches in folidity, had it been gold.
5.85} 63.00 (10.76923 folid inches, if all filver.
Ti»„br.ii.g«i».,8.«^s|4:^fj U:f^
4.68815
4.68815) 2.54473 (.5428, part|:old.
4.68815) 2.14342(4572,, part hlvcr.
••• .5428 X 63 = 34.1884 = 34 oz. 3dwt. 22i gr. of gold.
And .4572 X 63= 28.8036 =: 28 16 i^ of fiiYcr.
(i E. F.
Since gold and filver are always weighed, bought and fold,
by troy weight, which weights are feldom in the pofieffion
of gentlemen in the country, graziers, or farmers ; there-
fore to prevent their being impofed on by Jews, and other
itinerant traders, I fhall give an example concerning ^be
rcdui^ou of troy into averdupoife weight.
In
Chap. HI. SINGLE POSITION* 477
In the year 1696 an experiment was made by authority
(by SI balance which would turn with fix grains put into
either fcale) when it was found, that 15 pounds averdupoife
weight, were equal to 18 lb. 2 ozs. I5dwts. troy =s
105000 grains.
* . * 15) 105000 (7000 grains troy = 16 oz. averdupoife.
Alfo 16) 7000 (437i grains troy = j oz. averdupoife.
And 48o)437.5(.9ii458^ oz. troy = i oz. averdupoife.
Firft, fuppofe a filver tankard weighs 2 lb. i x oz. 8 dr.
averdupoife, its .weight in ounces troy is required?
Firft, alb. ti oz. 8 dr. = 43,5 ounces.
And 43.5 X '9^HS^$ — 39-6484375 = 39 <»• lapwt.
23J-gr. by the experiment above.
CHAPTER III. .
POSITION; OR, THE RULE OP FALSE.
THE rule of poiitidn, or fuppoiition, is fo called,
becaufewe fuppofe fome uncertain number, in order,
that by reafoning from them we may gaia the true num-
ber; and becaufe thofe fuppofitions are taken at adven-
ture, it is alfo called the ride of falfe.
B
SECT. I.
SINGLE POSITION.
Y Cngle pofition are folved fuch queftions as require
only one fuppofition to difcover the true refult.
RULE.
. When you have made choice of your pofition, work
it according to the nature of the queftion as if it were
the true number ; and if by the. ordering your pofition
you find the refult either too much or too little, you may
then find out the number fought by this proportion;
I viz.
1
St7» SINGLE POSITION. B6ofc III
yjz. as tbe reddt of your pofitbn h Co : iIk pofition,
ib is : : the given number tio : tbe number fought.
I. Three perfons, viz. A, B, C, thus difcourfe togc-
tlier concerning their age; fays B to A^ I am as old- and
half as olcf again as you ; then fays C to B, but I am
twice as old as you ; A replied, I am Aire the fum of all
our ages is 165 : now I demand each man's age ?
• Suppofc 2i. =s A.
Then will 24. + 12= 36 = B.
Aitd 30 X 2 = 72 = C.
••• 132 : 24 : : 165 : 30, A*s
And 2, C's i ^- *^-
And 90, C
2. Three perions^ Andrew, Benjamin and Charles, ate
to go a journey of 235 miles 5 of this journey Andrew is
to go a certain number of miles uaknown^; Benjamin is to
go four times as many miles as Antfrew, and three miles
more ; and Charles is to go twice as many miles as B^iiUi-
min, arid five miles more : How Qiiany miles muft each of
thefe perfons travel feverally ?
miles*
Suppofe - - - - JO, Andrew.
Then lo X 4 + 3 = 43» Benjamin,
Again, 43 x 2 + 5 = 91, Charles.
144
Alfo 3 + 3 X 2 + 5 = '4
130 and 235 — 14=221.
••• 130 : ro : : 2ii : 17 -j r Andrew.
Alfo 17 X 4 — 3 = 71 [miles) Benjamin.
And 71 X 2 + 5 = H7 J (.Charles.
23s
3. There #ere in company together four perfons, Adam,
Edward, Charles and William ; Adam told Charles that he
was older than him by two years ; Edward told them, thai
he was as old as both of tbem together, and four jreaTs older ;
William, hearing them| iatd, I am juft 96 yes^r^ old, and
that
Chap.m. SINGLE POSITIONS 47^
that is equsd t» att your ages : how oU -ras each of them
feveraliy i
Suppofe Charles aa
Then Adanr be 22
And Edward - 46
Then 24-2 + 4 = 8.
W— 8 = 8o-
And96 — 8 =: 88
••• 8t) : 20 :: 88 : 22 = Charles's
Al(b 22 4- 2 =^ ^4 = Adatn^s
And 22 4- 24 + 4 =jo = Edwafd^s f^^'
Their fum == 96 =: William's
4. The captain, lieutenant and comet of a troop have
t^d^en among them from fome enemy 478 crowns, which
they agree to iharr in this manner ; the captain k to have
24 tim^s as much as the cornet, wanting only feven crowns ;
and the lieutenant h to have five times as much as the cor-
net, wanting^ three crowns 5 what is each officer's ihare ?
^in> 7 + 3 = 10
Alfo 478 4. 10 = 488
Suppofe the cornet to have 8
Then 8X5
And 8 X :^4
230
Then 8 X 5 — 3 = 37
— 7 = i8
And 2304^ 10 ::= 240
••• 240 : 8':: 488 : 16-5^ = cornet's -7-, ^
Alfo i^ X 5 — 3 = 78tV = lieotenant's i ^%^' J^
Laftly, i6vV x 24— 7 = 383^ = captain's - J ^" ' '
5. Let 273 1. be divided amongft four perfons, iviz. An-
drew, Bennet, Chriftopher and iJaniel ; Andrew is to have
a fhare unknown; Bennet is to have twice as much as An-
drew, and 30 1. more; Chriftopher is to have three times as
much as Andrew, wanting 52 1, and Daniel is to have five
times as much as Andrew, and 20I. more; how muft this
273 1, be divided amongft them, fo that every one may
have his true Ihare i
I
Suppofe - - - 20 A
Then muft 20 X 2 4" 3^ = 70 B
Likewife 20 X 3 — 5^ = 8 C
And 20X 5 4" ^0 = '^0 ^1
Again, 304-aa — 52=2
Alfo 2734- 2^= ^75
Lilc:ew.2i8 4- ^= 220
218
220
;^o SINGLE POSITION. BookllL
asio : 20 :: 275 : 25, Andrew's -%
Then will 25 X 2 + 30 = 80, Bennet's 1 1,
Likcwifc 25 X 3 — 52 =s 23, Chriftophcr's v'^'
And 25 X 5 + 20 = X45» Daniert J
£^73
6. Admit three merchants build a fhip) which coft 1360L
A pays a certain part unknown ; B paid %i as much, want-
ing 15.5 !• and C paid as much as both A and B, and 75. 251*
over^ now much did each man pay?
1.
Suppofe A paid 100 t
B 100 X 2.5 — 15.5 = 234.5 75.25 — 31 = 44-25
C 334-5 + 75*^5 — 409-75 I 1360— 44-25 = i3iS-7S
744.25 — 44,25 = 700
]. s. JL
••• 700: 100:: 1315.75: 187.96428^=187 19 34, A
Then
187.96428/JX2.5— 15.5=454.41071^=454 8 24,
And 642.375 + 75.25 =717.625 =717 12 6, C
1360.
7. As I walk'd forth to take the air.
The heavens and nature fmiling were }
A grave old (hepherd there I '(py'd^
Clofe by a chryftal fountain's fide.
Unto this (hepherd I did fay,
How many ibeep have you I pray i
But he reply'd, add to one half of thefe.
One fourth, ^, 4> 2nd, if you pleafe.
One tenth, ^, and -^ too ;
Thefe being made one fum by you,
Exa&ly to my age will be.
In this proportion, as 15 to 3.
What is your age, good Sir ? faid L
To whicn the &epherd made reply ;
One-half, one-fourth, one-fifth do take.
One-tenth, one-twentieth, they will m^e ;
If added, five fcore and ten more.
And now my age. Sir, pray explore i
And
Chap.m. DOUBLE POSITION. .481
And now methinks his age I'd know^
Which I muft beg of you to Chow ;
Likewife the number of the fheep.
Which this crabb'd fhepherecl there did keep ?
«
, Suppofe 20 = fhepherd's age.
Then i = 10, i = 5> 7 = 4> ^= 2». and ^^ =' x.
Alfo 10 -{- 5 -j- 4 -}- 2 -f- I = 22. •
As 22 : 20 : : no : 100 = fhepherd's age.
Again, fuppofe 40 = number of iheep.
Then will 4. = 20, ^ = 10, 4 = 8, ^ = 5, V^ = 4,
^= 2, and^= I.
Alio 2o + io+8+5 + 4H-^ + x=z5o.
And 3 : AiS • • lOO • S^o* pcr queftion.
• • • 50 : 40 : : 500 : 400, the number of flieep. (j^ E, F.
00<M>000<>0000<>00<K>00<>0
SECT. II.
DOUBLE POSITION.
IN double pofition two fuppofitions are ufed ; and if we
mifs in both, obferve the nature of the errors whether
they be greater or lefs, and accordingly mark them with the
figns -4- or — i then,
RULE,
As the difference of the errors, if alike, or. their fum, if
unlike : is to the difference of the fuppofitions : : fo is
either of the errors : to a fourth number ; which added
tOy. or itibtraded from, its proper fuppoiition, gives the
number fought.
I. A young gentleman walking in a garden, and meet-
ing with a bevy of young ladies, began thus to addrefs
them : Blefs you all 10 fair ladies ! Sir, replies one, you
are miflaken, we are not to j but if we were twice as many
more as we are, we (hould be as many above xo, as we
are now below : what was their number ?
Suppofe 4, then 4 x ^ + ^ =^ '^*
Now as 4 is 6 lefs than 10, and 12 but 2 above xo j
••• 2 «— 6 = — * 4, the firft error,
I i Again,
* *
482 DOUBLE POSITION. JJookllL
Again, fuppofc 7, then 7 x 2 + 7 sc 21.
As 7 is 3 lefs than 10, and zi greater by ii;
• . ' 1 1 -^ 3 =: + 5, the fecond error.
Then 4 -I- -8 sr 12, fum of the errors.
And 7 — - 4 =? 3> difference of the fuppoittions.
Alfo 12 : 3 : : 4 : I. '.•4+1=5. Q. E. 'F.
Or 12 : 3 : : 8 : 2; and 7 1— 2 3= 5, the anfwer, as
above.
2. A gentleman hath two horfes of good value, and
a faddle worth 50 L which fet on the back .of the iirft
horfe made his value double that of theiecond; but if fet
on the back of the fecond horfe, liiake's iiis worth triple
that of the firft horfe : I demand-the value of each horfe i
Suppofe the firft horfe to be worth 24 L
T4ien 24 -f* 50 =: 74; alfo ^=:37L value of the fecond.
And 37 + 50 =: 87; but 24X 3 = 72, lefs than 87 by 15.
So that — 15 = firft error.
Again, fuppofe the firft horfe to be worth 341.
Then 34 + 50 = 84 j alfo -i= 42, value of the fecond.
And 42 -|- 50 = 92 ; alfo 34 x 3 = 102, more than 92
by 10.
Hence -f- 10 rr fecond error.
Then 10+ 15 = 25 = fum of the errors.
And 34 — 24 = 10 == ditFerence of the fuppofitions.
25 : 10 :: IS : 6. ••• 24 + 6 = 30. Q^E. F.
Or 25 : 10 :: 10 : 4. •.•34 — 4 =: 30, a$ above.
3. A lady bought tabbv'at four (hillings a yard, and Perfian
at two (hillings J the wnole number of yards ihe bought
were eight, and the, whole price 20 fillings 5 how ipan/
yards had (he of each fort ?
s.
Suppofe four yards of tabby at 16
Then muft (he have, four of Perfian at 8
Sum of ihefr values 24
«
So that the firft ^rror is -f- 4*. $•
Again, fuppofe (lie h^d three yard^ of tabby, value 12
Then muft (he have five 6f Pcifian, value - - 10
The fum of their values =: 22 s.
So
Chap. III. DOUBLE l»OSI'riON. 4S3
So that the fccbnd error is ^ 2.
Then 4 — 3 is i, difference of the fuppoiitions.
Alfo 4 — 2 zsL 2, difference of the errors.
As 2 : I : : 4 : 2. *. • 4 -^2 ^ 2 yards of tabby*
And 8 — 2 = 6 yards of Perfian*
8. s.
For two yards of tabby, at 4 = 8
And fix yards of Perfian, at 2 =: t2
Sum 20, as was required*
4. A and B having a certain number of crowns, fays B
to A, give me one of your croyrns, and I (hall have as
many as you 3 but fays A to B, give me one of your crowns,
and I (hall have twice as many as you ; how Cnany had
each ?
Suppofe A to have 5^
And B 3.
Then 3 — i = 2 ; and 5 + i == 6#
The firft error being -*- 2.
Again^ fuppofe A to have 9,
And B 7.
Then 7 — 1 = 6; and 94-1 = l^-
The laft error being -j- 2.
Then 2 + 2 = 4, fum of the errors.
And 0 — 5 = 4, difference of the fuppofitions.
As 4 : 4 : : 5 : 5. • . • 5 + 2 == 7.
Alfo 4 : 4 : : 9 . : 9. •.•9 — 2 = 7.
' z= 4 i and 4 + I = S> B's crowns.
z
For 5 -f I = 7 •— I ; and 7+1=5 — 1X2, as
was required.
5. There is a certain fifli whofe head is nine inches long,
the tail is as long as the head, and half the body, and the
body is as long as both the head and tail ; I demand the
whole length of the faid fifh i
Suppofe the body be 20 inches*
io
Then^ — + 9 =: 19, tail.
Alfo 19 4- 9 = 28 — 20 = 8.
So that the firft error is — 8.
li 2 Again,
484 eIOUBLE POSITION, Book III.
Again, fuppofe the body 24 inches.
Then — + 9 =: 21, tail.
Alfo ai + 9 = 30 ; and 30 — 24 =: 6.
So that the iecond error is — 6.
Then 8 — 6 =: 2, difiFerencc of the errors.
And 24 — 20 = 4, difference of the fuppofitions.
Alfo2:4:: 8 : 16. * • ' 16 + 20 = 36 ? body.
Or 2 : 4 : : 5 : 12. • . ' 12 + 24 =r 36 V ^
Likewife ^ + 9 = 27, tail.
And 36 + 2^7 + 9 = 72 inches. Q. E. F.
6. When firft the marriage knot was t/d.
Betwixt my wife and me.
My age did her's as far exceed,
As three times three doth three :
But after ten and half ten years
We man and wife had been,
Her age came up as near to mine.
As eight is to fixteen :
Now tell me (you who can) I pray.
What were our ages on the wedding-day f
Suppofe the wife's age 12 years.
Then muft the hufband's be 36.
And 15 years after \ T^^Z j ^^'
-' ^ i nulpand 51.
Twice her age greater than his by 3.
• . • The firft error is -|- 3»
Again, fuppofe the wife's age 18 years.
Then muft the huft)and's be 54.
Alfo .5 years after \ hufband's 69.'
And the fecond error — 3.
Proceeding, 3 +3 = 6, fum of the errors.
And 18 — 12 = 6, difference of the fuppofitions.
* . • 6:6:: 12 : 12 ; alfo 12 + ? = ic 7 ., ^ -c > ^ ^
Of 6 : 6 :: 18 : 18 ; and 18 - 3 = 15 Jthewifesage.
7. A man that was idle, and minded to fpend
Both money and time, went to drink with his friend;
He faid to his hoft, if you'll now to me lend
As much coin as I have^ then my fixpence I'll fpend.
His
\
Chap.IIL DOUBLE POSITION. 485
His hoft lent the. money, his fixpence he fpent, "j
And, having fo done, to another houfe went, i
Where the ^me he requefted, and the fame fum he fpent. 3
He went to a third houfe, where, Landlord, cries he.
Lend me as much money as I've left here you fee j
Which having receiv'd, his fixpence he fpent.
So,, all being gone, home the fuddle-cap went.
To caft up his reckonings'; but his head aching fore, "i
He begs you to do't, and he'll do fo no more ; >
Whi^t had he at firfl, and how much on fcore ? J
Suppofe he had 8d.
Then 8 + 8 = 16
Alfo 16 — 6 = 10
10 + 10 = 20
20 -^ 6 zi: 14
14 + IJ. =: 28
And 28 — D = 22, which fliould be no-
• . • The firft error is 22. thing,.
Again, fuppofe he had jd.
Then 7 + 7 = 14
14 — 6 = »
8 + 8 = 16
16 -— 6 = 10
10 + 10 = 20
And 20 — 6 = 14
• • • The fecond error is I4»
Then 22 •— 14 = 8, difference of the errors.
And 8 — 7 = 1, diiFerence of the fuppofitions.
As 8 : I : : 22 : 24. • . • 8 — H = Si- ^J- I n ir F
Or8: I :: 14: i|. ••• 7 — ii = si<J-J^
RULE II-
Proceed as direfted in the firft rule, till you have found
the errors and their figns ; then
Multiply alternately the firft fuppofition by the fecond
error, aijd the fecond fuppofition by the firft error ; and
divide the fum of the produds by the fum of the errors,
when the 'errors are of different kinds ; or the difference
of the produds by the difference of the errors, when the
errors are of the fame kind, and the quotient is ,the num-
ber fought.'
8. A pcrfon finding feveral beggars at his door, gave ejich
of them three pence a-pioce, and had five pence remainbg :
li 3 he
\
48^ DOUBLE POSITION. Booklff.
be would have given them four pence a-piece, but he wanted
{even pence to dq it ; how many beggars were there I
Suppofe 15 b^gars. 15
J 1
t 4S 60
+ 5 —7
50 53
Then 53 "— 50 =: + 3, the firft error.
Again> fuppofe 13 beggars. 13
39 5»
+ 5—7
4V4M Vi^BiM
44 45
Alfo 45 — 44 = 4* i» Ae fecond error.
Then 13 X 3 = 39
Alfo 15 X I = 15
And 3 -» I = 2) 24 (12 beggars. Q. E. F.
9. A labourer agreed to thrafh 6o bufhels of corn» part
of it wheat, and part oats, at the rate of 2 d. per bufliel
for the wheat, and i^d. for the oats; at laft he received
'8 8. for his labour i how much of each did he thralh i
d.
Firft, I fuppofe 30 bufhels of wheat, price • • 60
^ Then muft there alfo be 30 bufhels of oats - - 45
105
Which fhould be but - - 96
Therefore the firft error is^*-«---^9
d.
Again, I fuppofe 18 bufhels of wheat, price * - 36
Then alfo muft there be 42 bufhels of oats * - 63
99
-^ 96
Thf n will the fecond error bcT----4-3,
• Alio
Chap. ISj double position. '487
Alfo 18 X 9 = 162
And 30 X 3 = 90
• .• 9 _ 3 zr 6)72(12 bulhels "of wheat, at2<L 2s.
Thea will their be 48 hufliels of oats, at 14. d. is 6 s.
Sum 6obuflieIs .--...- 8$.
10. Two merchants, A and B, lay out an equal fum.
of money in trade; A gains 126 1. and B lofes 87 1.
and A's money is now doublfe to B's -, what did each lay
out?
Suppofe each lays out 220 .... 220
126 87
346 133
266 X 2
Firft error -|- 80 266
Again, fuppofe 350 ----- 350
126 87
476 263
263 X 2 = 526 — 476 = — 50, the fecond error,
350x80 = 28000}
220 X 50 = 1 1000 1 '>^^*~^
••• 80 + 50 ;= 130)39000(3001. Qi E. F.
The following rules and examples I had from the in^
genious Mr. Emerfon's Arithmetic, pdge 146, &c.
RULE III.
** Proceed as direfted in the firft rule, till you have
found the errors and their figns ; then,
I- Multfply the difference of the fuppofed numbers by •
the leaft error, and divide the produft by the difference
of the errors, if they are alike ; or by the fum, if un-
like : the quotient is the correSion of the number be-
longing to the leaft error.
11. Obferve whether this be the leffer or greater num-
ber, as alfo whether the errors have like or unlike figns.
III. If it is the lefs numberj and like figns, fubtradt
the correction ; if unlike fiens, add it.
• Ii4 • IV. Ii
488 DOUBLE POSITION. Book IIL
IV. If the greater number, and like figns, add the cor-
reftion; if unlike figns, fubtrad it: fo you'll have the
true number required.
II. A certain man being afked what was the age of his
four fons, anfwered, that his eldeft was four years older
than the fecoqd, and the fecond five years older than the
third) and the third fix years older than the fourth, which
was half the age of the eldeft j how old was each ?
1 6 eldeft. ^24 eldeft.
fecond. * _• i 20 fecond.
S-PP°fM 7 Thir' Again, - —
^ I youngeft. L gyoungeft*
i eldeft =: 8 — I = — 7, firft error.
i eldeft = 12^ — 9 = — 3, fecond error.
Then 24 — 16 = 8, difference of fuppofitions.
Alfo 8x3 = H-
And 7 •— 3 =: 4, difference of the errors.
4) 24(6, the corredion.
* • * 24 -f- 6 ;;=: 30> the age of the eldeft.
Alfo 26, fecond.
21, third.
And 15, youngeft. Q^ E. F.
12. There is a crown weighing 60 lb. which is made of
gold, brafs, tin, and iron ; the weight of the gold and
brafs together is 401b. of the gold and tin 45 lb. of the
gold and iron 361b. Quere, how much gold was in it?
Suppofe 32 lb. of gold ------ 28
8 brafs -----»•- 12
i3*'« '^ " " 'Z
4 iron -^--f.--- 5
57 ^
Then 60 . — 57 = — 3, the firft error:
And 65 — 60 =2 4" 5» ^^^ fecond error.
Alfo 54-3 = 8, fum of the errors.
Likewife 32 — 28 = 4, difiercnce of the fuppofitions.
Again, 4 X 3 == 125 and -^ = It» the corredion.
•.• 32 — 1} n 30;-, the quantity of gold. Q^ E. F.
13. Three
Chap. III. DOUBLE POSITION. 489
I J. Three companies of foldiers paffing by a {hepherd,
the nrft takes half his flocif, and half a (heep; the fecond
takes half the remainder, and half a fheep i the third takes
half the remainder, and half a fheep ; sifter which the
fhepherd had 20 ibeep remaining ; how many had he at
firft?
Suppofe 60 ------- 80 I
Firft took 30.5 ------ 40.5
, . 295 39-5
The fecond took 15.25 ------ 20.25
•
14.25 ------ 19-25
The third took 7.625 ------ 10.125
6.625 ------ 9-125
Then 20 — 6.625 = '3-37S» ^^ I error
Alfo 20 — 9.125 = 10.875 = fecond 3 *
Ag^iH) 80 <*- 60 = 20, difference of the fuppoiitions.
Alfo 13.375 — 10.875 = 2.5, difference of the errors.
20 X 10.875 = 217.5.
2.5) 217.5(87, correction.
•. • 80 -I- 87 = 167 (heeep. Q^ E. F.
Scholium.
Bv fuppofin^ one of the numbers o^ and the othd: x, the
wore is (ometimes ihortened.
14. A fador delivers fix French crowns, and four dollars^
lor 2I. 13 s. 6d. and at another time four French crowns,
and fix dollars, for 2 1. 9 s. 10 d. what was the value of each I
Suppofe o = value of a French crown.
Then will 4 dollars = 53.5 s.
Alfo4)53i(i3f
And 6 crowns -}- 6 dollars ^ 8oi«
Then 80^ — 49^^ = -f- 30x'|^, firft error.
Again, fuppofe i s. =. value of a French crown.
Then 6 crowns and 4 dollars = 53^.
Alfo 53J — 6 = 47 J. = 4 dollars.
4) 474-(ii-i>9 value of a dollar.
And 4 crowns -)- 6 dollars := 75^.
Then 75 J — 49fi = + 25tt> fecond error.
Alfo 30tV — ' 25 -it- = S> difference of the errors^
•••5)30,^(6x^ = 68. I d. = value of a crown. Q^E. F.
And
'49C^ CONCERNING DIVISORS. Boqfertk
And 6 crowns = 36 s, 6 d. '
53s. 6d. — 363^ 6d. z^ 1-7 s. valueof 4 dollars^
• , • 4) 17 (4 s. 3 d. as value (if a dollar, Q^ E. F.
I» this rule it is genemlly prefumed, that the fiift error
is to the fecond, as the difference between the true and fiiflf
fuppofed number is to the difference between the true and
fecond fuppofed number. When this does not happen, the
rule of fiilfc does not give the exad anfwer, except the two
fuppofed numbers be taken very near the true one.
The errors are the difference between Ae true refult, and
each of thefalfe refults; fo that if the errors are unlike,
' the true number lies between the fuppofed numbers ; bat if
alike, the true number lies without both of the fuppofed
ones.
A great many queftions may be refolved by this rule,
which cannot be refolved by any other rule in arithmetic j
but there are many queftions where it cannot be certainly
known, whether they can be refolved by it or not, tiu
they be tried."
CHAPTER IV.
CONCERNING DIVISORS.
IT being often neceflary, in arithmetical calculations, to .
find fuch multiplieris, or numbers, which may |>e divided
by any number of given divifors without any remainder, or
to leave any affigned remainder, or remainaers ; by which
means many pleafant queftions, not reducible to any of the
foregoing rules, may be folved.
Firft find the leaft number that can be divided by any
number of given divifors without a remainder.
RULE.
Multiply all the prime numbers, and the roots of fuch as
are fquare or cube numbers, continually ; the produd will
be the leaft number required.
I. Shew me how to find what*s the leaft number
That you can divide without a remainder.
By
Chap. IV. CONCERNING DIVISORS. 491
By giving divifors, as the digits nine.
For a true canoA I'd give a pint o' wine ?
Ladies Diary ^ ^7*9«
Divifors I. 2. 3. 4.5. 6.7. 8.^.
But as ^ 4 = a, that 6 may be cancelled, being com-
pofcd of 2 X 3 ; ^i/ 8 = 2 ; and ^ 9 = 3.
..- 1x2x3X2X5x7X2x7x2x3 = 2520. Q.
E. F.
«
2. What particular Icaft whole number is that^ which be-
ing divided by 2, 3, 4, 5, 6, 7, 8, 9, ihall leave a re-
mainder of I » 2, 3^ 49 5, 6, 7, 8, refpeftively?
It is plain by the queftion above, that 2520 is the
leaft number that can be divided by nine digits, without
a remainder. *•* 2520 -« i = 25199 the number re-
quired.
3. A country girl to town did go.
Some walnuts for to fell,
A gentleman fhe chanc'd to meet.
And thus it her Befel :
My pretty maid, fays he to her.
What number have vou here ?
I can't tell. Sir, fays me to hira.
But this ril make appear ;
I told them o'er ere I came out.
By fix's, five's, four's, three's, two's.
And, ev'ry time I number'd them.
One remain'd overplus;
I told them o'er by feven's at laft.
And there were no remains ;
If you can find the number out.
Pray take it for your pains.
Firft, the leaft number that can be divided by i, 2, 3,
4, 5, 6, without a remainder ; viz. i X2X3X2X5 = 6o,
Then 60 + i = 61, will leave i, when divided by each
number; but 7)61 (8, and 5 remains.
Alfo 60 X 2 -{. I = 121 ") r u- U ^ A'- rXA V.
. \ Q I none of which aredivifibleby
^QX 3 + I = 181 ? 7 without a remainder.
60 X 4 + I = 241 J
But 60 X 5 + I == 30I9 is the leaft number which ad-
mits of the conditions of the queftion.
Then
492 CONCERNING DIVISORS. Book III.
Then to find the next Icaft number which admits of the
fame conditions j viz, 60 X 6+1 = 361*1
60 X 7 + I ^ 4ZI none of which
60 X 8 + I = 481 I are divifible
60 X 9 + I = 541 j by7,without
6oxio+i = 6oil a remainder.
60X II + X = 661J
But 60 X 12 + 1 = 721, is the next number admitting
the conditions aforefaid.
Alfo 721 — 301 = 420, the common difference of all
numbers anfwering the fame conditions.
'•• 30i> 721, 1141* 1561, 1981, 2401, 2821, &c. ad
infinitum^ will anfwer the conditions of this qucftion.
4. To find the leaft number of guineas, which being di-
vided by 6, 5> 4j 3 and 2 refpeftively, fhall leave 5, 4, 3^
sand I, refpeftively remaining ? L. Diary ^ I748«
As by the foregoing queftion, 1X2x3x2x5= 60,
the leaft number, which divided by ij 2, 3, 4, 5 and 6>
leaves no remainder.
•. • 60 — I = 59. Qi E. F. as may be eafily proved.
. 5. Required the leaft number, that being divided by 9,
Ihall leave for a remainder 6 ; if divided by 8, the remainder
will be 5 J if divided by 7, the remainder will be 4 5 and (o
on, each time leaving for a remainder three lefs than the
divifor, till, divided by 3, the remainder will be nothing?
Af 2520 is the leaft nun:iber which can be divided by the
nine digits, or by the feven higheft of them, without a re-
mainder,
•.' 2520 — 3 5= 2517. Q. E. F, as may be eafily
proved.
6. Required the three leaft numbers, which divjded by 20
fliall leave 19 for a remainder ; but, if divided by 19, mall
leave 18, if divided by 18, ftiall leave 17; and fo on (al-
ways leaving one lefs than the divifor) to unity ?
GentlemerCi Diary j 1747.
Firft, I, 2, 3, 5, 7, II, 13, 17, and 19, are prime
numbers,
Alfo v^ 4 =: 2, V 8 = 2, v^ 9 = 3, and ♦y/ 16 = 2 5
And all the reft are compofite numbers.
••• IX2X3X2X5X7X2X3XIIX13X2X17X
19 = 232792560, the leaft number that can be divided by
the
Chap. IV. CONCERNING DIVISORS. 493
the given divifors without a remainder ; alfo 232792560 X
2 = 465585120; and 232792560 X 3 = 698377680, being
divided by the given divifors, will leave no remainder.
••• 232792560 — I = ^32792559 7 the three leaftnum
465585120 - I = 465585159 f birs O E F
and 698377680 — I = 698377679 3 Vii A. !• .
Agreeing with the algebraic procefs, by Mr. Robinfon,
in the Gentlmen*s Diary, 1748.
7* A jolly fine girl did ride on the way,
With' plums in a bafket, it being market-day;
She rode on but foftly, the weather being hot.
So I aik'd her what number of plums fhe had got :
She faid, the juft number fhe did not well know.
But I'll tell you which way will the true number (how.
If you count them by two's, there will then remain one;
If you count them by three's, there refts two when
you've done ;
If you count them by four's, the remainder is three ;
If by five's, then juft four the remainder will be ;
If by fix at a time you do count them again.
You'll findjwhen you've done, that juft five will remain j
But if feven at a time, you do count them o'er all.
The remainder will be then juft nothing at all :
Now what is the number, and .to what do they come.
At fourteen a penny, I'd fain know the fum ?
By the third queftion it appears, 60 is the'leaft number
that can be divicied by the firft fix digits, without a remain^
dcr, 60 — I = 59, the leaft number that can be divided
by the faid fix digits, leaving each divifion one'lefs than the
divifor ; but 59, divided by jy leaves a remainder.
Then 60 x 2 — i = 1 19, the leaft number that anfwers
the conditions of this queftion.
Alfo 420 is found, by Queftion 3, to be the common dif-
ference of numbers, anfwering the fame conditions. '.'119,
539> 9S9> '379> ^^* ^*'^ admit of the fame conditions.
n '^V'^^^Ht o. o^A J their value.
8. Once old mother Gripe td a market went.
Some butter to fell it was her intent ;
At a certain rata^per pound ftie it fold.
What fhe got for it all, as I have been told,
Were
t
'4j94 AniTHMfiTiciAi Pitod&tssiON. Book llL
Were two (hilUagt and two pence farthing juft^
Now how much butter bad the old toaft.
And how {he might fell ber butter per pound>
Is what is required to l>e found ?
Of various anfwers this queftion will admit.
Find them all out, and they will wbet thy wit.
Firft, 2s. a-^d. ss 105 farthings, wbich is compofed of
thefe odd numbers j viz. 1x3X5x7x2 105*
d.
3
5
7
35
Sq that 1051b. at
105 35
105 21
105
105
105
105
4
3
T
- - - I
15
7
5
3
per pound, alf'anfwer thecondi*
« >- - li [ tions of this queftion. Q^
3t
- - - 54 I
- - - 8|
Or lib. at 2s. 2^
This queftion, and the foregoing, was taken fromTapper's
Delight for the Ingenious, for July, Attguft and September,
171 1 ; the folutions my own.
CHAPTER V.
Progression^ Variation, Combinatpon, &c.
S E C T. I.
ARITHMETICAL PROGRESSION.
ARITHMETICAL PROGRESSION is a rank or feriesof num«
bers increafing or decreafing by a common difference,
or bv a continual addition or fubtra£bton of fome equal
numoers.
As { ![ • o' 3 • ♦ • 5 • 6 • 7 • 8 • 9 {common difference i.
Or 1.3. 5.7.9. II. 13, coiQmon difference 2.
Alfo 42 . 35 . 28 . 21 . 14 . 7^ common difference 7.
In an arithmetical progreffion are five things ; any three
of which being given, the other two may be found, which
admit of twenty different propofitions*
3 J. The
i
X. The firft term, commonly the leaft \
2. The laft term, commonly the grcateft J ^^^^^^»
2. The number of .terms.
4, The common cxcefs, or difFerence.
5. The aggregate, or fum of all the terms.
We (hall only concern ourfelves with fome few of them^
but let us pretnife, that,
1. If any three numbers are in arithmetical progreffion, the
fum of th« two extremes, viz. the Jfirrt and laft, will be
equal to the double of the mean or middle number.
As in thefe, 3 . 8 . 13 j viz. 3 + 13 = 8+8
Or I . 7 . 13 - - - I + 13 z=: 7 + 7
And 7 , 14. . 21 - - - 7 + 21 ^= 14 4- 14
2. If four numbers are in arithmetical prc^reffiqn^ th^
fum of the two extremes will be equal to the fum of the
two means.
As I • 3 . 5 . 7 f viz. I -f- 7 s= 3 4- 5
And 5 . 8 - 11 . 14 ' - - S 4- J^4 = 8 4- i'
3. Alfo if many numbers be in arithmetic progreffion, the
fum of the two extremes will be equal to the fum of any
two means that are equally diftant from the extremes.
7 . 9 . n . 13 . 15 . 17;
viz. 7+17= 9+^5 = " + i3-
Or if the numbers be odd :
i .3.5.7 . 9 . II . 13;
m. l + i3=:3+"=S + 9 = 7 + 7-
4. Every feries of numbers in arithmetic progreffion is
compofed of the cxcefs or common difFerence, fo often re-
peated as there are terms in the progreffion, except the firftk
As in thefe, 2 . 5 . 8 • 11 • xi4 • 17, &c.
Here the common difFerence being 3.
Then will 2 + 3= 5.5 + 3 = 8.8 + 3 = 11.
II + 3 = 14 . 14 + 3 = 17, &C.
Hence may be obferved, that the difference between the
two extremes (2 and 17) is compofed of the common dif-
ference, multiplied into the number of all the terms, except
the firfl. ^
In
49^ . Arithmetical Fhogr^ssion. Book III.
In theaforefaid progrefSon, 2 . 5 . 8 ii . 14 • 17.
The number of terms without the firft is 5 ) " i.. 1
The common difference - 3 J'»^«P**
The difference of the two extremes - 15
PROPOSITION L >
The two extremes, and the number of terms, being giiren,
to find the fum of all the feries,
R U L E,
Multiply the fum of the two extremes iftto the number
of terms, and divide the produ£l by 2, the quotient will
be the fum of all the feries.
I. How many ftrokts do the clocks of Venice (which go
on to 24 o'clock) ftrike in the compafs of a natural day i
I 4- 24 = 25, fum of the extremes.
24, number of* terms*
100
50
2) 600 (300 ftrokes. Q^ E. F.
2. The length of my garden is 94 feet ; now if eggs be
laid along the pavement a. foot afunder, to be fetched up
fmgly to a bafket, removed one foot from, the laft, how
much ground muft he travel that does it ?
2 + 188 = 190, fum of the extremes,
94,, number of terms.
Feet in a mile 5280
660
. 16.5
2) 17860 (8930 feet.
8930 ( I mile, 5 furl. 21 poles, 3^ feet
Qt E. F.
3650
350
PRO-
Cilflp. V. AlLITHMSTICAL PROOHESStOII. 497
PROPOSITION 11.
The fixft terin9 the common excefs, and the number o£
tenm being given, to find the fum of all the feries.
RULE.
From the produ£l of the number of terms in the com-
mon excefs, fubtrad 'the common excefs, and to the re-
mainder add the double of the firft term ; half the prod|i6l
of that fum multiplied by the number of terms, gives the
fum of all the feries.
3. A gentleman bargains with a bricklayer to fink him a
well twenty fathoms deep, upon thefe terms ; viz. to pay him
three ihillings for the firft fathom, five for the fecond, feven
for the third, &c. raifing two ihillings every fathom; what
will be due to the bricklayer for compleating the fame }
Firft, 20 X 2 = 40 ; alfo 40 — 2 = 38.
Again, 38 + 6 == 44 ; and 44 x 20 = 880.
•• • 8to -f- 2 = 44ofliillings = 22 1. Q^ E. F.
PROPOSITION III.
The firft term, number of terms, and fum of all the
feries given, to find the common excefs.
RULE.
Divide the double fum of all the feries by the number of
terms, and from the quotient fubtra£t double the firft term;
divide the repiainder by the number of terms leflened by
unity, the quotient will be the common excefs.
4* A gentleman travelled 100 leagues in eight days, and
every day travelled eaually farther than the preceding day ;
it is known that the firft day he travelled two leagues, how
many leagues did he travel each of the other days i
aoo -f* 8 = 25 J alfo 25 — 4 = 21 ; and 8 — 1 = 7.
7) 21 (3, the common difference fought.
Then 3 added to 2, and every other term refpe<Stively,
gives 5 for the fecond
« third
II - - - fourth
14 - - - fifth J-day's journey. Q. E. F*
17 - - - fixth
. ao fevenf h
23 — - eighth .
100 leagues*
Kk ? R O-
49«
Arithmetical Progrhssiom. Bookltt.
PROPOSITION IV.
The two extremes; and number of terms being given, to
find the common difference.
RULE.
The difference of the two extremes, divided by the num-
ber of terms-lefs unity, the quotient will be the common
excefs.
5. One had 12 children that differed alike in their ages,
the youngeft was nine years old, the elder 367 ; what was
the difference of their ages, and the age of eadi i
»
Here 36.5 — 9 =1 27.5, difference of the extremes*
Alfo 12 — I z= 11) 27.5 (2.5, common excefs.
Which add to the age of the youngeft, and fo on con-
tinually to the reft.
VIZ.
youngeft
nth -
loth -
9th ..
8th -
7th -
6th .
5th .
4th -
U :
eldeft -
11^
9
J
Z
2li
26i
29
3'i
34
36i:J
►years old.
6. A debt is to be difcharged at i x feveral payments in
arithmetic progreffion ; the firft payment to be 12 1, ics.
and the laft 63 1. what is the whole debt, and what muft
each payment be ?
Firft, 12.5 + 63 =: 75.5, fum of the extremes.
II, number of terms.
2) 830.5 (415.25 :=4i5l. 5s. whole debt
Then 63 -— 1 2.5 = 50.5, difference of the extremes.
II *^ X s= 10) 50.5 (5*05 =: 5 1. 1 s. common difference.
Therefore
Chap. V. Arithmetical Proqrbssion. 499
1.
8.
Therefore 12
10
»7
II
22
12
27
»3
32
14
37
15
4a
16
47
17
52
10
$7
»9
63
415
5
ment.
firft
fecond
third
fourth
fifth
fixth
fevcjnth
eighth
ninth
tenth
eleventh
415 5, whole debt, as before.
^m
'7. A man is to travel from London to a certain place in
ten days, and to go but two miles the fir£k day, increafing
every day's journev by an equal excefs, fo that the laft
day's journey may be 29 miles ; what will each day^s jour-
ney be, and how many miles is the place he goes to diftant
from London ?
Firft, 29 — 2 := 27, difference of the extremes.
10 — I = 9) 27 (3, the common difference.
Which added to each day's journey,
firft.
gives 2
IX
14
20
26
29.
►miles for the-<
fecond.
third,
fourth*
fifth,
fixth.
feventh.
eighth,
ninth,
^tenth.
155 miles from London.
PROPOSITION V.
The two extremes, and the common excels given> to
find the number of terms.
RULE.
Divide the difference of the two extremes by the common
txcefsj the quotient plus unity is the number pf terms.
Kk 2 8* A
50O Arithmbtical PaooiLSSsfOK. BookllL
miles.,
every day
Fi^9 35 "^ 5 :=^ 3^> diflFerence of the extremes.
Then 3) 30(105 and 10 + 1 = 11 days journey* Q. £. F.
PROPOSITION VL
The common excefs^ number of terms, and fum of all
the fenes given, to find the firft term.
RULE.'
Divide the fum of all the feries by the number of terms,
and from the quotient fubtrad hsdf the produft of the
common excefs into the number of terms le(s unity, the
remainder will be the firft term.
9. A man is to receive 300 1. at 12 feveral payments,
each payment to exceed the former by four pouiub | he is
willing to beftow the firft payment on any one that can tell
him what it is i what muft the arithmetician have for his
pains i
Firft, 12) 300 (25 J alfo 12 — I = It.
Then 11 x 4 = 44; Md 2) 44 (22.
••• 25 — 22 = 3, the artift's reward. Q. E. F.
10. Suppofe it 100 leagues between London andCarlifle,
two couriers fet out from each place on the fame road 1 that
from London towards Carlifle travelling every day two
leagues more than the day before ; die other from Carlifle
to fet off one day after, travelling every day three leagues more
than the preQcding one ; and that they meet exadly half
wav, the firfl at the end of five days, and the other at the
ena of four $ how many leagues did each travel each day?
Firft, 5) 50 (10 ; alfo 5 — i = 4«
Then 4x2 = 8; and 2) 8 (4.
• . • 10 — 4 == 6, his firft "J
6 + 2 = 8-- fecond /
10 - - third ^day's journey*
12 - - fourth I
14 - - fifth J
Sum 50
Agato»
ChsLp.y. Arithmetical Progression. 501
Asain, 4) 50 ( 12.5 5 alfo 4—1 = 3.
- Tl
• _ •
hen 3X3 = 95 «»d *) 9 (4-S«
12.5 — 4.5 = 8, his firft 1
8 + 3 ss II - - fecond I . . .
^^ 14. -third J<Jay » journejr.
17 - • fourth J
50 leaguei.
PROPOSITION vn.
The laft term, number of terms^ and common excefs
gircn^ to find the firft term.
RULE.
Multiply the common excefs into the number of terms
left unity, the produd fubtraded from the laft term leaves
Che firft.
II. A man in fix days went from London to Man*
chefter, every day's journey was greater than the pre-
ceding one by four miles, his laft day's journey was '40
miles, what was the firft?
Number of terms 6 •-• i rr 5
Common excefs 4
ao
Then 40 — 20 = 20, the firft day's journey. Q^ E. F.
I fliall now add one propofition more, exclufive of the
10 above-mentioned.
PROPOSITION vm.
When one perfon or diing moves with an equal, and
another the fame way by a progreffive motion, to find in
vhat time the firft will be overtaken.
RULE.
Add the common excefs of the purfuer's* day's journey
to double the fpace gone each day by the purfued ; from
tl^t fum fubtra£l double the fpace that the purfuer travel*
led the firft day, and divide the remainder by the com-
Kk 3 mon
502 Arithmetical Progression. Book IIL
mon excefs, the quotient will give the number of days in
which the purfued will be overtaken by the purfuer.
12. A noted highwayman having committed a robbery,
not fufpeSing a purfuit, fled northward at the rate of eight
leagues a day; Jonathan Wyld, upon the fcent, follows
him, in a progreffive motion, only three leagues the firft day,
five the next, feven the third, and fo on, increafing every day
two leagues ; in how many days will the highwayman be
overtaken? ,
Firft, 2 + i6 = 18 ; alfo 18 — 6 = 12.
••• 2) 12(6 days. Q. E. F.
For 6 X 8 =: 48 leagues, the fpace travelled by the robber.
Then, by Prop. fl. 6x2=12; alfo 12 — 2 = 10;
and 10 -j- o = 16.
Alfo 16 X 6 =s 96. -•* 2) 96 (48 leagues, wheb' the
thieftaker comes up with the highwayman.
13. Y. Z. made the following bet for 1000 guineas, to
be decided the Monday, Tuefday, and Wednesday, in
■ Whitfun-week, on Barnham Downs, between the hours of
eight in the morning, and eight at night. The propofer has
ten choioe cricketers in full exercife, who on this occafion
are to be diftinguifhed by the firft 10 letters of the alphabet*
Thefe are to ri|n an4 gather up, and carry fmgly, 1000 eggs,
laid in a right line, juft twp yards afunder, putting them-
l^ntly into a bafket placed juft a fathom behind the firft.
They are to work one at a time, in the following order :
A is to fetch -up the firft 10 eggs, B the fecond, C the
.third, and fo forward to K, whofe turn it will be to fetch
up the looth egg. After which,' A fets out again for the
next 10, B takes the next, and fo forward alternately, till
K (hall have carried up the iQOOth egg, at 100 eggs per
man. The fellows are to have 300 K for their three days
work) if they do it, and it is to be diftribiited in propor-
tion to the ground each man fhall in his courfe have gone
over. It is required, fiFft, How many miles each perfon
will have run i Secondly, What part of the 300 1. will
come to his (bare? Tnirdly, Whether if the men had
been pofted at [)roper places, they had not better have run
from Lrondon to loj-k twice, and back in the time, taking
the meafure 180 miles?
F|rft» forA's race, 4, firft term, 40, laft term» their fum 44.
N°.of terms 10 x 44 = 440.
Chap, V, AniTHMETiCAL Progression. 50|
• . • 2) 449 (220, A's firft race.
Then 901 X 4 = 3604^ firft term of the Uft r^ce.
Alfo 910x4 = 3640, laft term.
3604 + 3640 = 7244> their fum,
• • • ^^^ ^ =: 36220, A's laft race.
Then to find his whole part in this expedition^ put
220> firft term* 362220 laft terira, fum 36440.
... 3 44Q X lo _ 182200, fum of A's races.
2 .
For B's part in the expedition :
Firft, II X 4 = 44 > 1^ ^^i^ 20 X 4 = 80 ;
Alfo 44 -|- 80 = 124; which X 10 = 1240*
• •* 1) 1240(620 yards, B's leaft race.
Laft race 911 X 4 = 3^44 Lkp:. r^^ -^, .
. Alfo 920 X 4 = 3680 V^^"^ ^""^ 73H-
• . • ^^^^ ^ s: 36620 yards, B*s greateft race.
Then 620 4* 36620 = 37240, fum of the extremes.
. ^ . ^7 40 X _. 1 85200, fum of B's races.
z
Again, 186200 -« 182200 = 4000 yards, common dif-
ference ; which added continually to each of their 0iares»
Ihew^ that
yards.
A in all ran 182200 ;=
B - - 186200 =
C - - 190200 =1
D - f r942oo =
E • - 198200 =
F - - 202200 =
G -. - 206200 ;=z
H - - 210200 ^=
I - - 214200 =
K - - 218200 =
miles, furl
. poles.
103
4
6
40
80
120
no
2
160
112
4
200
114
117
7
I
20
60
119
3
100
121
123
5
7
140
180
2002000 == 1 137 4 — Q. E. F.
As looio : 911 : : 300 : 27 6 -AVt» A's )
Alfo looio : 931 :: 300 : 27 18 -i^^j^y, B*s J P*"^"
Then 27 1. 18 s. -TTjirr <!• — 5^7 !• 6s. -TVirrd. = lis.
iItWt^^* common di^erence; which added to each man's
ibare,
Kk 4 gives
504 Arithmstical PROG&Mtiow. Book in.
gives
A
B
C
D
£
F
G
H
I
K
I.
30
30
31
3*
3^
8.
6
18
10
2
14
5
17
9
I
13
""r5%T
— lO'OT
_-. X I 6
II 9»^
1 06 1
**Tr5T
ft •♦'
► Q. E. F.
From Lonidon to York^ fuppofe 180 miles*
Miles in the whole expedition 11377
And 180 X 4 :s 720
Short of the prefent undertaking 41 7^ miles.
The following queftion I was favoured with by mj efteem*
ed friend, Major Watfoni chief engineer ifi Lord Clive's ex*
pedition to the Indies.
14. Suppofe a man to have a calf, which at the end of
three years begins to breed (and afterwards) a female calf
every year; and that each calf begins to breed in like man-
ner at the end of three years, bringing forth a cow calf
every year, and that the^ laft breed in the fame mannei^
&c. &c. to determine the owners whole ftock at the end
of 20 years ?
By nature of this queftion, the number of cows that
calved at the end of thefe years will be as follows i
is? 4» 5* 6. 7. 8.
I. I. I. 2. 3. 4*
15 .16 .17 .18 .i§ .20 y
60.88.129.189.277, and.
9
6
10 . It . 12 • 13 • 14
9 • 13 . 19 • 28 • 41
years.
406 refpeArvely, which are
found by adding the laft to the laft but two.
Then of the whole feries t-f.i-|.i^24-3 + 4 + 6
+ 9 + 13 + ^9 .' ••• +D4-E + F-f*Gl>c repreicntcd
by S, when D, £, F and G denote the four laft terms, we
(hall then have 14.1^1^2 + 3+4+6 + 9 + 13
+ i9 + 28.,. + D = S — E— F — G, which being
jtaken from the above, we have i+i + i + ^ + ^ + S
+ 4 + 6 + 9 + I3--*.. + F = E + F + G5andby
3 ^ding
Chap^V. Geometrical PnooRBSSioif. gog
adding 6 to both fides of tbe equation, we then get I + i
+ 1 + 1+2 + 3 + 4 + 6+0 + 13. •. + F+CJ =
£ + F + aG ', which confequendy will be the man's ftock
of cows and calves at the end of any number of jEears,
which, in diis cafe, £ = 180 + F = 277 + aG =s Sia*
wiU be 1278. Q. £. F.
From the above folution it appears, that the whole ftock
of cows and calves, at the end of any number of years,
will be equal to the number of cows that would calve at. the
end of three years after the given time.
S E C T, IL
GEOMETRICAL PROGRESSION*
GEOMETRICAL Progrbbsion 18 when any rank or feriea
of numbers increafe by one common multiplier, or de-
crtafc by one common divifor ;
At 2 • 4 • 8 • i6 • 3a • 64$ here i is die common nnil<*
tiplier.
And 1215 * 4^5 • 135 * 45 * 15 * 5' ^^*^ 3 ^ thecemmoa
divifor.
Note, The common multiplier, or divifor, is called the
ratio.
Here note, that if three numbers are in geometrical pro*
greffion, the product of the two extremes will be equal
to the fauare of the mean or middle term, as in thefe,
2 • 4 • 5.
Here a X 8 = 4 X 4* each being = i6.
Alfo if four ntmibers are in geometrical progrefSon, the
produd of the two means will be equal to the produdl of
the two extremes, as in thefe, 135 • 45 • 15 • 5*
Here 135 X 5 = 45 X I5» each being 675.
Hence, if ever fo many numbers are in geometrical
progreiHon, the product or the two extremes are equal to
the produd of any two means that are equally diftant
from the extremes.
As in thefe, 3 • 9 • 27 • 8i. 243 . 729.
, Sere 3 X 729 = 9 X 243 = 27 X 81 = 2187.
Oi
So6 GsoMETRicAi, Froorbs3ion. Book HL
Or if the number of terms be odd, as in tbefe :
3 . 9 • 27 . 8i • 243, &c.
3X243=9X81 = 27X27 = 729.
In any geometrical progrei&on, the fame things are to be
taken notice of, as in arithmetical progceffion -,
viz. Firft, The firft term, commonly the leaft.
Secondly, The laft term, commonly the greateft.
Thirdly, The number of terms.
Fourthly, The ratio, or common multiplier, or dirifor.
Fifthly, The fum of all the feries.
Any three of thefe being known, the reft may be found.
If to any feries of numbers in geometrical progreffion not
proceeding from unity, there be afligned a feries of num-
bers in arithmetical progrefion, beginning with an unit or
I, whofe common difference is i, called indices, or ex*
ponentss
thus 5x-*'3*4-5«6. 7, indices.
1 2 • 4 . 8 • 16 • 32 • 64 . 128. &c.
then will the addition or fubtra£lion of thofe indices
(or numbers in arithmetic progrei&on) direi^y correfpond
with the produ£l or quotient of their refpedive terms or
feries in geometric progreffion ;
Aat is ^ !1^ 3 + ♦ = 7i
"^^ "» I fo 8 X 16 = 128, thefeventh term in tt.
Again, |^^ ^^J ^ ^ J — i6384,the 14th term in -ff.
^^' I fo 128 ^ 8 = 16, "
Q ( as 6 — I = 5,
^^ C fo 64 -f- 2 = 32, &c.
But if the feries begin with unity, the indices muft
begin with a cypher.
Thus 50.1.2.3. 4 . s . 6 . 7, &c.
' I I • 2 . 4 . 8 . 16 . 32 • 64 • 128.
Now by thefe indices, and a few leading terms, the laft
term, or any diftant one, one may be fpcedily found.
PROPOSITION I.
The firft term being unity, the ratio and number of tems
being known, to find the laft or any remote term.
RULE.
Chap. V. Geometrical Proorbsstoh. 507
RULE.
Find a few of the leading tefms, over which place their in-
dices, as before directed ; then multiply the laft found term by
icfelfswhich will produce a term double thereto ; and fo proceedf,
till you Either arrive at the term fought, or bn« that falls
m little fliort of it i if fo, multiply the term laft foimd by
that term, anfwering the difference of theindice of the laft
found term, and that; fought $ which laft produd will be the
term required.
1 . A country gendeman going to a fair, meets with a
crafty youth who had a drove of 25 very good oxen ; upon
^ifldng their price, was anfwered, he Ibould h^ve them- for
16 pounds each, pne with another*^ the gentleman offers
him 15 pounds each) and take all: the youi^g (bark tells
him it would not be taken, but if he would eive him what
the 20th ox would come to by beginning at the fird with a
(ingle farthing, and doubling only to the 20th, he (hould
have them all ; what did they come to a head ?
0 • I • 2 • ^ • 4 • 5) indices.
1 . 2 . 4 • 8 . 16 • 32, terms.
Then, i 5 + 5 = io» alfo, I 1 + 5 = 9»
^^^ I 32.x ^^ =? 1024; ^^^' 1 16 X 3^ = 512-
10 -f- 9 = '9-
1024 X 512 = 524288, which is the 20th term^
as the indices are lefs than the term by one.
And 524288 farth. = 546 1. 2 s. 8 d. price of the whole.
••• 25)5461. 2S. 8d. (2il. 16s. lojd. Qi E. F.
But if the firft term of any feries be greater than unity,
that and the ratio being known, to find any remote term
without producing the reft,
RULE,
Find a few of the leading terms, as before directed ; then
multiply the laft term fo found by itfelf, and divide the pro-
duct by the firft term, and this again multiplied by the term
as is wanting, and divided by the firft, gives the term re-
quired.
•
2. A nobleman dying left ten fons, to whom and to his
executor he; bequeathed his eftate in the manner following ^
viz. to his executox for feeing his will performed T024
pounds;
Firft, {
5o8 Geometrical PftOORtssioN. Book III,
pounds; the youngeft fon to have as ipuch, and half aft
much, and every ton to exceed the next younfi;er in the
fiune ratio of i-^ : what is the (hare of the eldeftf
o • I • 2 • 3 • 4 ? 5f indices.
1024 . 1536 . 2304 • 3456 . 5184 . 7776, terms.
7776 X 7776 _ j^^i^ ddeftfon's fortune. Q^ £. F.
PROPOSITION n.
The firft tenn» ratio, and laft term given, to find the
fmn of all the feries.
RULE.
Multiply the laft term into the ratio, and from the pro-
dud fubtra6l the firft termj divide the remainder by the
ratio lefs unity, the quotient will be the fum of aU the
feries.
3. On New-year*8 day a gentleman married, and received
of his father-in-law a guinea,* on condition that he was to
have a prefent on the nrlt day of everv month for the Bi&
Tear, which (hould be double ftill to wnat he had the month
before s what was the lady's portion ?
Firft io.i.2.3. 4* S, indices.
' C I • 2 . 4 . 8 . 16 . 32, terms.
term.
Agam, 2048 X 2 =s 4096s alfo 4066 — i = 4095.
20) 409s
ao4 15
^^4299 15 s. the lady's fortune. Q* £• F.
4« One at a country fair had a mind to a ftring of 20
fine horfes ; hut not caring to take them at 20 guineas per
head, the jockey confente^ that he , (hould, if he thought
good, pay but a fingle farthing for the firft, doubling it
only to tne 19th, and he would give the 20th into the bar-
gain : this being prefently accepted, how were they fold ?
Firft, { °
1.2.^. 4 • 5
2 . 4 • 8 . 19 • 32, &C.
Thea
Chap. V. GsoMiT&iCAt P&OG&issioir. 509
Then \ 5+ S = Wt gifo{ 5 + 3 = 8
*"^" C 3* X 3^ = 1024; ***" 1 32 X 8 = 252*
C 10 + « =18
^"S*«*> 1 1024 X 252 = 262144, the loih term.
Then 262144 X 2 = 524285;
alfo, 524288 — I = 524287 faith. = 546 L 2 s. 7' i.
••• 20)5461. 2s. 7|d.(27l. 6a. i^d. each. Q.E.F.
5. A cunning fervant agreed with a mafter (unfkilled in
numbers) to ferve him eleven yeafs without any reward for
his fervice but the produce of a wheat-corn for > the firft
year i and that produd' to be fowed the fecond, and fo on
from year to year, until the end of the time, allowing the
increafe to be in a tenfold proportion : it is required to 6nd
the fum of the whole produce f
Firft, r • ^ • 3 • * • 5 years.
* CIO. 100 . 1000 . 1000 . looooo corns of wheat.
Then i 4 + 2 = 6
C loooo X 100= loooooo, the 6th year's produce.
And ^ ^ + 5 = "
I loooooo X lOoooo = loocpooooooo, the nth
or laft year's produce.
Then ratio 10 x 1 00000000000 = 1000000000000 ;
Alfo loooobooooooo -— 10 = 999999999990.
Ratio 10— I = 9) 999999999990 (iiziiiiiiiio corns
in all.
As hath been before obferved, 7680 wheat-corns will fill a
ftatuteagi^
'111611 7680) iiiiiiiiiiio (i4467<9t pints. *
In a bufliel 64) 14467591 (226056^- bulhels, which fup«
pofe at 3 s. 4 d. per buiheh
■J) 2260561-
■ s. d.
Anfwcr £ 37676 - 4i, a very ample reward. Q^ E. F*
6. It is reported that one Sella, in India^ having; firft in-
vented the game at chefs, ihewed it to his prince ohekram ;
the king, who being highly pleafed with it, bid him aft:
what he would for the reward of his invention ; whereupon
he aiked, that for the firft little fquare of the chefs-board
he might have one grain of wheat given him ; for the fe«
cond two, and fo on, iloubling continually according to the
number of fquares on the chefs-board, which were 64:
the king, who intending him a noble reward, was dif*
pleaied that he had aik^ fo trifling a one i but SefTa de*
daring
5IO GsoMBTRicAt Prog&bssiok. Book III.
daring be would be contented with this, it was ordered
to be given him -, but the kins was aftoniflied when he found
that mis would raife fo yaft a a ^uantjty, that even the
whole earth could not produce it ! fo you are defired to
repeat the operation.
Firft 5o. 1.2.3. +• ^' in<Jiccs.
riro, 1 1 . 2 . 4 . 8 . 16 . 32, terms, &c.
Then i 5+ 5 = io, jf 5 10+10= 20
I 32 X 32 = 1024 ; C 1024x1204=1048576
AMin i ^o + 20 z=i 40
'^K"*^* C 1048576 X ^48576 = 109951 1627776.
Alio
40 4* 20 = ' 60
109951 162776 X X048576 = 115250695006846976
. Laftly,
60 ' + -2 =;: 63
115250693006846976 X o = 922005560054775808
Now, 922005560054775808 X 2 = 1844(5 1 1 120109551616.
*.- 1844011120109551616 — 1= 1844011x20109551615
wheat-corns.
7680) 184401 1 i20i0955i6i5(240i056i4597C97 pints.
64) 240105614597597? 375 1650228087 buihels.
8) 3751650228087 (46895627851.875 quarters, which, at
lU 7 s. 6d. per quarter, amounts to 644814882961. which
is more than would pay one year's rent of all the dry land
on the face of the eartn, at 1 1. 10 §. per acre, which may
thus be proved :
The circumference of the earth 360 degrees (6^^EngItlh
miles to a degree) = 25020 Englifh miles, circumference of
the earth.
Alfo, 25020 X 25020 X'.3i832 = 199268447.328, area in
fquare miles of a perfed globe.
Alfo, 199268447.328 X 640 = 127531806289.92 acres of
land and water, 4 of which is fuppofed to be water.
•.' 3)127531806289.92(42510602096.64 acres of dry
land, which, at lU 10 s. per acre, h 63765903144.96
pounds a year I which, compared with the valuation of the
wheat, as above, will be found 7 15585 151 1. lefs.
PROPOSITION III.
i Of any decreafing feries in rf, wfaofe laft term is a
I cypher, to find the fum of thofe feries,
RULE.
Chap. V. VARIATIONS. 511
RULE.
Divide the fquare of the firft terih by the diiFerence be-
tween the faid firft term ahd the fecond term in the feries^
the quotient will be the fum of all the feries.
7. To find the fum of i + ^ + ^ -f Vti ^* ^ infinitum.
Thus, iXi=ij alfo, i — 4 =i. »
• « • ^) .^ (4 :^ I, the fum of the feries required.
8. To find the fum of 4. + 4^ + ^V + to &c. adinfinitum.
Firft, ^x4 = ^; alfo, 4=4.; andj— .J=».
9. Suppoie a ball to be put in motion by a force which •
drives it 12 miles the firft hour, 10 miles the fecond, and
fo on continuallv, decreafing in proportion of 12 to xo, to
infinity ; what (pace would it move through ?
Firft, 12 X 12 = 144 ; alfo 12 — 10 = 2.
• . • 2) 144 (72 miles. Q. E. F.
it may appear ftrange to fome, that it fliould be
poffible to give the fum of an infinite progreffion in num-
bers ; whereas, if the terms were continued, it would, after
a thoufand years labour, sind after producing thoufands of
millions of terms, be never the nearer finiming.
SECT. III.
VARIATIONS.
BY variation is meant the different ways any number of
things may be altered or changed, in refped to their
places.
To find the number of different changes that may be
rung on any propofed number of bells, ~
RULE,
Multiply all the terms in a feries of arithmetical progref-
fionals continually^ whofe firft term and common difference
is
512
VARIATIONS. Book IIL
is unity or i, and the laft term the number of things pro-
pofed to be varied; the laft produd will be the number of
variations required*
The changes on any nundier of bells not exceeding 12,
are exhibitedin the following Tabli.
The.mio»ber of
The manner how
The different changes
' things propofed
their feveral va-
or variations every
to be varied, «
riations are pro-
one of the propofed
duced.
numbs, cah admit of.
X
1 X I
a
IX 2
= 2
3
ax 3
= 6
4
6x 4
= a4
S
«4X 5
= 120
6
120 X 6
= 720
I
720 X 7
5040X 8
= 5040
= 40320
9
40320 X 9
=: 362880
10
^62880 X 10
= 3628800
XI
3028800 XII
= 39916800
12
39916800 X 12 1 s: 479001600 1
Let it be propofed to find the number of changes that
may be rung on 12 bells, and to compute how fong ail
thefe changes would tdte ringing once over.
Firftt 1x2x3x4x5X6x7x8x9x10x11X12
s 47900 1 6oo» the numl>er of changes.
And fuppofe 10 changes to be rung in a minute ; viz 12 X
10 =: 120 ftrokes in one minute, or two ftrokes in a fe*
cond :
Then 10} 470001600 (47900160 minutes.
Alfo I year =: 365 days, 5 hours, 49' =: 525949 minutes.
525949)47900160(91 years, 26 days, 22 hours,
564750 41 minutes ; fo long would
In I day ait 1440} 38801
looo""
60) 1 36 1
12 bells be ringing, without
any intermiffion, to ring their
different changes bitt once
over.
1. A
Chap.V. combinations; 513
2-. A young ichohu-, but an arithmetician, coming into a
town for the convenience of a good library, demands of a
rendeman, with whom he lodged, what his diet would coft
for a yeitf ? The gentleman aflcs him 10 1. the fcholar an^
fwcredy he was not certain what time he might ftay, and
would know what he muft give him for his diet fo long as
he could place his family (coniifting of fix perfons Beiides
himfelf) every day at dinner in a contrary pofition ? The
1 gentleman confldering of itf and thinking it could not be
ong, tells him, he would allow him his diet fo long for five
pounds ; to which the fcholar aiTents i vAiaX did he give him
for his table per annum ? - ^
_ *
Firft^ 1X2x3x4x5x6x7= 5040 variations, or
Then 36<.26) 5040,00 (13 years, 291 days, theanfwcr.
13.70725; 5*000000 (.36239 =: 7s. 3d. per annum nearly,
theaniwer.
>zo.
S E C T. IV.
COMBINATION S.
COMBINATIONS are the various conjunftions
which feveral things may receive without any regard
to order, being (aken 2 and 2, 3 and 3, &c.
To find the different combinations in any number, or
quantities* ^
RULE.
Having placed the given quantity by xtfelf, decreafe it
gradually by an unit, fo often as there are quantities in the
combination ; placing them one above another, with a fign
of multiplication between them, which number muft be
multiplied into one another for a dividend : .then placing an
unit with the like number of places, increafing by unity
till you arrive at the number to be combined ; which mul-
tiply continually for a divifor, and the quotient will be the
number oF combinations fought*
!• A famous general having ferv'd his king
Long time in wars, and had victorious been -,
^ L I for
514 COMBINATIONS- BookllL
For which his fervice f with a pleafant fmile)
Aflc'dof his king one nuthing for each file
Of ten men in a file, which be could then
Make with 9 bodv of one hundred men.
. The king, confiaering his brave aftions paft.
And feeming modefty of his requeft.
Gave his confent^-^-To what will it amount
In fterling money i take your pen and count.
I 2 3456789 10
^'"'''aS^r""^ = 17310309456440 farthing,.
*.* 17310309456440 farth. = 18031572350L 9$. 2d.
Q. E. F.
' The. number of combinations^ of 2 in any number of
things, are 2 raifed to the. power of the number of things
to be combined ; for inftance, if it was required to find
the different ways 11 halfpence, hufled in a hat, may
turn up i as a halfpenny hath two faces, *.' 2X2X2X
2x2x2X2x2x2X2x2, or 2*' = 2048, the dUferenc
combinatipns required.
Now to find the diiGBrent chances for any number of heads
or tails, put a for the heads, and b for the tails ; then,
. Here it is to be obferved, that tf", or all heads, hath
but one way of turning up i the fame may be oUerved
for II tails. ' '
But 10 heads and one tail, and the contrary, may
come up II different ways each.
Alfo nine heads and two tails, or the contrary, may
each come — = 55 different ways.
Again, eight heads and thrbe tails, or the contrary, may
each come -^ — ^rs 165 difierent ways ; which numbers (b
found are called unciaes, or coefficients.
From hence may be deduced this general rule.
RULE.
If the index of the firft letter of any term be multiplied
into its own uncix, and that product be divided by the
number
Cfap. y. COIVFBINATIONS. 5,5
number of terms to that place, the quotient wlU be the
uncise of the next fucceeding xerxn forward.
Let us proceed to find the reft of the coefficients, or
chances, by the rule a^bove^
viz. fcven heads and four tails, or the contrary, may
each come ■ ss 33P dwierent ways.
Again, for fix heads and-five tailis, or the contrary, each
may coine ' •' =;:46z.'
It may be obferved, by. proceeding as above, 'that the.
unciaes, or coeiEcients, do only increafe until the indices? of
the two letters becoaiie equal, or change/ places, and then
the reft will decreafe in/ the fame order.
Thus, tf«»4. If 12*^*+ 55tf9i* ^ i65^«*» + 33ca7b* +
liizA?* -f. iu, are all the different combinations, or way«,
II halfpence can turn up;
viz. I + II + 55 + 165 + 330+462 + 462 + 33<>
+ '65 + 55 + " + I = 2" = 2048, as before.
Alfo a-dic, having fix faces or fides, the number of conv
binationa or different ways 11 dice may turn up, are 6'^
= 362797056. ^
From confiderations like thefe I compofed the follow-
ing queftion, which was publi(hed in the Palladium 1753,
and the next year was anfwered by three ingenious gentle-
men ; but they not confidering, that in the fame web of a
goofe*s foot, the punch mark and flit mark cannot exift
together, the laft mark naturally deftroying the firft, which
caufed i miftake in their calculations ; for both may com-
bine together as well as with all the reft, and bring out the
fame number of combinations as eight halfpence, and
four triangular dice, with three faces each, fhaken toge-
ther, could produce.
2. In Lincolnshire, here bounteous nature yields
Fat flieep and oxen, and luxuriant fields ;
Our generous clime, replete with rofy health.
Choice friends afford, bright, fair, and plenteous wealth.
Some fenny ground have we, with fl6cks of geefe.
Yielding five times a year their feather'd fleece ; •
On which, devoid of care, fwains fleeping lie.
After repaft of favoury giblet-pye.
L 1 2 One
5i6 COMBINATIOJ^S. BookHL
One day, at Bofton, o'er a jug of ale^
A golTard offered all his flock to (ale,
At fifteen pence a-piece; but I proposed
A different price, with which he quickly closed.
(The geefe are mark'd, b^ cutting toe or heeU
The webs are pierc'd or flit with marpen'd fl:eel)
An hundred pounds for juft as many geefe,
As may be different mark'd : what's that a-piece ?
Each goofe having three toes, two webs, and one heel
on each foot, in all 12 different things to be marked.
But as the four webs may be either flit or pinched, but
not both together, ••• 2* x 3* ^ 20736, the number of
combinations.
Alfo, 20736 — I = 20735 = number of marks.
And 100 1. =:: 24000 pence.
••• 20735) 24000 ( i-^Vy^ d. the price of a goofe. Q.E. F.
3. A perfon, P, bets, fix pounds with another perfon, Q^
that in throwing up three halfpence^ they (hall all come up
the fame way, viz. all heads, or all tails, once at leaft in
three trials: at the fame time Q^ bets 10 guineas with R,
that in throwing up four halfpence, they (hall not all come
up the fame (i. e, all heads, or all tails) once in four trials :
required each perfon's advantage or difadvantage, with the
odds in each cafe, by an arithmetical computation only ?
Since there are but two chances for three halfpence com-
ing all one way at a fingle throw, and fix for the contrary,
it is evident the probability of miffing all the three throws is
iXiX i = fi» and that of the contrary i — -J^ = |^.
'•* The odds in favour of P, are as 37 to 27.
In the fame manner |-Xj^Xfx{-=: 1^77= probability
of miffing all the four throws with four halfpence, and that
of the contrary H^. The odds in favour of Q^ are
as 240 X to 1695 ; hence P*s advantage in the firft cafe is
i8s. 9d. and Q^ in the fecond cafe is equal 2I. 6 s. 2^d»
4* .Two defperate gameflers, A and B, agreed to throw
each of them two guineas at a particular point or mark ;
and then to tofs up the four guineas fairly, and each perfon
to take up the heads in tty: following manner as they arife;
that is, A*s two guineas happened to light the firfl and fe-
cond, or the two neareft to the faid point ; fo that if either
one or two heads arife, they muft fall to A's lot ; but B's
two guineas being the third or fourth, or the two farthefl
from the point, fo that if either three or four heads arife,
they are to be B's property : moreover, if two heads hap*
pen
Chap. V. COMBINATIONS. 517
pen to arife in the iirft tofs, then A gets all the four guineas ;
and if three headt come up firft to S, then will the other
guinea be A*s of courfe. Quere, what is each perfon's
probability of winning, and how much is the value of A's
chance before they begin to tofs ?
Martins Maga^m.
Firft, let a reprefent heads, stnd b tails ; then in the bi-
nomical a '{- b raifed to the 4th power, the powers of b be-
ing rejie^ed (as no winning chances on either fide) will re-
main a^ -|- 4^1 -f- 6aa -f* 4^ » ^^^ indices reprefe'nting the
chances, and coefficients the number of different wavs
thofe chances may happen ; the two laft terms being As,
and the two firft B's; in all, 15.
TT + TT ^f T = iV> ^^"^ ^^ A's 7 firft winning tofs;
« 4- TT o*" T = TT = value of B's \ fumlr*
Then 44 — tt ==^ fT> whole value of the fecond tofs.
Now B's chance of getting four heads muft ceafe, fo there
can only remain feven winning chances, viz. fl^+ 3^* + 3^*
whereof
Alfo,
And
Alfo, ^,V + ^ = tV7 i and ^V + 1^7 = t'^V
Ccnfequently, A*s chance to that of B's, is as 73 to 32 ;
viz. more than 9 to 4.
••• tV + TOT = tVt = ^'' 8®' 44^* == value of A's 7 ,
and tV + t-St = tVt = i'- S^- 74^- = value of B's J ^^^^^^^
5, Two gamefters met the other day.
The one call'd B, the other A 5
But having neither cards nor dice,
" They get to hotch-cap in a trice
Witn 16 halfpence fair and flat.
All which they hufled in a hat.
Says A to B, all thefe are mine.
And I will lay a pint of wine,
That in two trials there will be
Nine heads or tails, as here you fee.
No matter which, but on they play'd.
Till filver, brafs, and gold were laid i
But as to B, his chance was bad.
For he got broke of all he had.
What were the odds, I pray declare.
Groom-porters fly ; ladies fair ?
By the late Mr. Jofeph Smithy of. Fleet.
LI ^ 2«<=^
ily remam feven winnine; chances, viz. a' 4- yr + 3^;
jf only one, viz. the firft, belongs to B.
5i8 COMBINATION?. BooklH.
2'^ ::?65536, number of different diuices' on 16 hatfpenee*
Let a reprefent the heads, and b the tails.
Then a'^ + i6a««A + i20tf'*A* + 56otf» V + i82tf"**«
4- 4368^"*^ + 8oo8fl*^^ + ii440tfW7,
Then 11440 x 2 = 22880 chanees for nine heads or .
ta'ls to come.
%• 65536 -^ 22880 = 42656, not to come thefirft time ;
vizr. 22880 to 42656, that they come nine heads or uils
the firft tofs.
As 65536 : 22880 : : 42656 : 14892^
Then 22880 + 14892/y = 37772^, for > nine heads
Alfo 65536 — 37772^V = shhlh againft' J or 9 tails
tiH'ning up once in two trials.
From hence A's chance to that of B,
Is fomething nK>re than four to three.
6. Two gamefters one day at dice they would play.
And being foil merry in wine,
Says B unto A, what odds will you Iay»
I Vaft not all the fix faces this time?
Say A then to B, ten to one I'll lay thee.
With fix dite the fix faces you caft not.
Pf ^9 gentlemen j {hew, and next year let them know,
^r or the odds on the caft, firs, they do not.
6^ = 46656 different combinations.
And 1x2x3x4x5x6= 720 variations.
Then 46656 -— 720 = 45936 chances againft A.
But as A laid 10 to x *.* 7200 chances for B.
••* A's chance to that of B, is as 4^1936 to 7200, or as
6.38 to I.
A Table Jhewing the probability pf winning w hjing mq
nnmho' rf games Ugnhfr^ whin the gameften are equal.
Powers.
Odds.
2 - -
2->
-2-^1:= l-»
2* =
2» =
2* =
2* =
- 2« =
2' =
2» 3=
4
8
16 I .
3*
.64
128
256 J
4— I = 3
8^,= 7
16 — 1 = 15
1 32 — I = 31
64 — 1 = 63
128 «- I = 127
L256 — I = 2SS-'
7. To find how many holes a pcrfon can make at crib*
bage, that has the whole pack in his )und,
Fifft, for the fequcnces :
As
CHuikV. COMBINATIONS; 519
As 4 is the number of ways an ace can be fhewn, jc
foHows that 4^ will be the number of diiFerent ways that
one ace and one duce can be (hewn ; and 4' the number
of fequences with an ace, duce and tray, &c. &c. .
Whence it appears, that 4** will be the number of fe-
quences of 13 in each; which multiplied by 13,
viz. 4«' X 13 = 872415232, the number of holes to
vrlrich all die fecjuences amount.
Secondly, for the number of fifteens.
The determination of thefe depends upon the following
cafes, according to the feveral way^ by which 15 can be
tBado by a, 3, 4^ 5, &c. cards, the number correfponding to
which cafes are found from the following
Theorem.
Let a c= number of cards of one fort, £ of a fecond,
and c of a l^ird (all the cards together making 15) then
will 4: x4 r^J X 4 X 4 (b) X 4: X 4 (c) &c be the
number of i^'s correfponding.
For example. Let there be twa 5*3, one (tray) or 3,
and one (duce) or 2, then will the number of 15*8 cor-
refponding be :
♦ Xf X4X4-=96-
rs
^t^m^m*-
IO+-<
4+1 - - -
3 + ^
3
2
+ I + I-
+ 2 + 1-
12 + I + I + I
N« of
ways.
64
.AS6
256
384
384
256
Sum 1600
re
5+ I
4 + 2 - - -
3 + 3
9-PS 3 + 2^., _
2 + ft + 2 -
16
64
64
24
3+ i+i+»
2 + 2+ t 4-1
U+i+i+i+i
96
256
16
64
144
16
Sum 760
7+
5-
4H
- I - - "
- 2 _ - .
-3
1-4
k .
N«of
ways.
24
64
64
24
256
256
96
5-
4 -
4-
.3-
l-i + i- -
. 2 + 1 - -
h3 + i- -
|- a 4> 2 - -
1-3 + *- -
15H
4H
3H
3-
2-
l-i+i+i
-2 + 1 + 1
-3 + 1 + 1
L4 + 2+ I
-2 + 2+ *
384
144
384
4
16
256
96
44
4+1+1+1-
3+2+1+1-
2+2-J-2+1-
-I
-I
-I
1 »+» + 1+1+1+11
Sum 2348
LI 4
N»
5a»
COMBINATIONS. Book III
r?
N»of
ways,
i6
6 + 1 '. ' - -
54-2 - . - -
4+3
8 +
S+I + I - -
4 + 1 + I - -1
< 3 + 3 + » - -
3+ 2 + a - -
4+I+I +1
3+ a + x + i
2+2+2+1
3+« + ' + » + *
L2+2+I + I + I
64
64
64
96
256
96
96
64
384
64
16
96
■**r»
Sum 1376
^ I » *
6 + 3
6 + 2+1 - -
5 + 3+ » - -
5 + a + 2 - t
4 + 4+ I - -
4 + 3 + a * -
3+ 3 + 3 - '
6 + I + I + 1
6+J5 + a+i + <
^l4+3+i + »
4+ 2 + 2 + I
3 + 3 + * + J
3+2+2+2
S+i+i+i+i
4+2+1+ 1 +1
3+3+1+ «+i
3+2+2+1+1
2+2+2 I 2+J
24
«4
96
256
96
96
256
16
24
384
.384
384
384
64
3-J.t+i + i + i-l-i.
16
256
576
_i6
64
rs + 5 - - - -
5 + 4+ »
5+3+a - ^
4 + 4 + 2 - -
4 + 3 + 3 - -
S+3+I+*
5 + 2 +2+ I
4+4+I+'
4 + 3+ »+»
4 + a + 2 + a|
3+3+3+
+^3+_J±l+f
5+2+1+1+1
4+3+i+'+»
4+2+2+1+1
3+3+a+»+*
3+2+2+2+1
IP' —
I
4+»+l+l+«+«
i+»+» + i+i+«+«
N«of
ways.
4
96
96
96
144
1441
»44
1024
64
64
144
96
256
576
576
256
64
*4
3S4
H
t6
r4 + 4 + 3 - -
4+4+2+1
4+3+3+»
4+ 3+* + 2
3+3+3+2
4+1
Sum 3616
zzc
44.44-I + 1+I
4+3+2+r+i
4+2+2+2+1
3+3+3+I+I
3+3+2+a+i
3+2+2+2+2
3 +2+a-fMli 4-»
64
»44
64
16
576
96
576
«4
144
J«4
ti
■
Sum 2856
" 3+
Ch«p. V. MAGIC SQUARES.
''3 + 3 + 3 + a + i
3 -i- 3 + * + *+ ^
13 + 3+3+ I + « + ! - -
3+i3 + 3 + » + * + >+« .--
3 + 3 + 4+i + > + » + >
L3+» + 2 + a+i + i + i
ways.
l6
i6
4
»44
96
Sum 320
5«t
Then 1600 4. 760 + 1376 + 2348 + 36164. 4388 -f
2856 -I" 3^0 = 17264^ different ways to count 15 ; and
17264 X 2 = 345289 the number of holes.
LafUy, The number of prials will be 13; and 13 X X2
^ is6y number of holes.
'.* 8724152J2 -f 34528 + 156 = 872449916 holes the
pack Will make. Q. £• F.
S E C T. V.
MAGIC S Q^U ARES.
A Magic S<^are is a fquare figure compofed of ^ feries
of numbers in arithmetical proportion, fo difpofed in
parallel and equal ranks, as that the fum of each row taken
either perpendicularly, horizontally, or diagonally, are equal.
In ignorant ages, when mathematics pafied for magic,
thefe fquares were made ufe of by conjurers, for the con-
ftrudtion of tailifmans.
However, they have fince become the ferious refearch
among mathematicians ; not that they are of any real or
folid ufe, or advantage, but only as a kind of play, where
the difEculty makes the merit, as it may chance to produce
fpmc new views of numbers, which mathematicians will
not lofe the o^cfr^fion of.
7. The numbers r, 2, 3, 4, 5, 6, 7> 8 and 9, beijig
given to form them in a magic fquare, viz. counting ,each
rank perpendicularly, hprtzontally, or diagonally, that thofe
ranks may be equal to ^ach otl^er.
Suppofe
§n^ MAGIC SaUARES. Book IB.
Sufpdfe it done, and reprefented in its proper toxmy by
the {(fOtming fymbols thus placed, viz.
a h £
^'* f.
-^ ' g h i
Fiift, the funi #f the profreffianal aumbers are 4^.
Then 3 = nuoiberjoF rows.
ASb ^ db ic £= fuflti of each fideor rank.
3
And -^ = < = r, the middle number.
3 ^
Agahi, to find the corner figures, and firft to find tbc
figurd reprefented by a.
Beginning widi i, rfinrd the comer letter a^ or any other
corner letter, cannot be i ; for if a was t= i, then i miift
Ik 9 ; and ^4»£s:i5— -i=ci4; as aHo ^ 4* ^ =^ '5 "^
I ==; 14. But there remains no two numbers after 5, i,
and 9, whofe fum is 14, but 6 and 8 : *.* if any of thofe
^ures were i, the other would b^ c \ and then no figun»
would remain for the value of either dox g\ wherefore a is
not equal i, nor any corner letter oqual i or 9.
J cannot be =r a^ for if It were, then / (hould be =z 7 ; ,
5 + f3r 15*— 3=s \X\ as alio d \ g zsz 12 : but
there remains no two numbers after 5, 3, and 7, whofe
film 16 la, but 8 4" 49 which cannot anfwer to b and c^ ajsd
d and g \ w)|erefore a, or any other coroAr letter, is not =2
3 1 nc icber is 1, nor any other corner ktter, ss 7.
From what bath been faid, it is plain, that (if the quef-
tioB propofed is capable of being folvad) the corner letters
are all even numbers ; wherefore, \i a :zz %y i will be 3s
8, and i muft be either 4, or 6. Let ^ 3=; 4 j then / r= 6,
A s:; 9, d zsz jf / s= 3* ^uid h.zsn % and fo the fquare is
csQOipleted as required.
294
7 5 3
618
But if € were equal 6 (a being := 2.) ; then ; = 4> h z=z
i, d zz ^y f zz i^ and /^ =1 3, and then the fquares will
and thus :
Chap.V. MAGIC SQUARES. s^3
9 ^ i
438
Or Aty may b^ found medntii^atl/ thus : fet them aU
down progreffively, about which draw a fquare cornerways.
Then fet the four ang;ular figures at the corners^ and put
the outennoft alternately.
2.6 276
• 5 • 9 S I
4.8 438
1. Let it be required to form At i)umbers i, 2^ 3, 4, 5»
69 7) 8, 9, 10, II, 12, 13, 14, 15 and J 6, into a magic
fquare, viz. fo that counting each rank from one hand to
the other, as alfo up, down, and diagonal-wife, thofe rimka
tsfay be equal to each o^r.
* Suppofe it done, and reprefented in its proper fEHrm by
the following fymbols, vis.
a
k
c
d
4
f
i
h
•
t
k
I
m
n
0
P
9
Thefumof the faid progreffional figures are 136.
' Alfo 4 =s number of rows.
Then -^ =: '^4 =; fum of each fide, or rank.
4
Now beginning with the leaft of thofe numbers, put azsx*
Then the other corner letter n cannot be =1 2 ; for
if it was, b '\' m az a '■\- n'si i-f-^ would be 3 $ but
there are no remaining two numbers of the given ones,
whofe fum is 3, therefore n cannot be = 2, /i Seine; =s i.
Neither can n =: 3 ; for fuppofii^ n := 3, then b -^^ m
zs,a ^ n z^i «|-3=:4; but there ve no remaining two
numbers, whofe fum is 4.
Now putting » = 4 J th^n A4"'" = ^ + ^=' + 4
s= 5 s that is, i&, m =: 2, 3^ which are the only two num-
bers
n
524 MAGIC SQUARES. Book IIL
bers remaining, whofe Aim is 5 : *•' i -f" ? ^ 34 *"* 5 =
29; that is, dj q are 13, 16, or 14, 15, for no other
couple amounts to 29. ^
Suppofe again f ss 13; then mxifti= x6} when we
defign the fquare in part^ viz.
I b € iS
f i
4 ^ ^ 13
As the four corner figures are fixed, and e^ i s: x^ 15 f
alfo £, m 2, 3 ; it is plain f cannot be 5, 6, or 7, for if it
was, / would be i c, 14 or 13, which numbers are already
difpofed of; therefore, fuppoflng it 8, and then /=; i2«
Again, ^ -f'i 3= 14, and as there remain no two numbers
whole fum is 14, but only 5 + 9 ; but i + ' ^ ^ ^ + '4
= # + *. 17 = 2 + IS, or 3 + 14, or i8 = 3 + 15 5
confequeiitly i =: 4, 5 or 6 (not equal to 9} *.* ^ s= 5*
And ^ + / = 5 + X2 ::p 17, muft likewifc be = e + * i
which may be effected two diilerent ways, either by putting
/=s 15 or 14, and then i^;^ 2 or3, bychufingthe former;
i = I4» ^nd m ss 3, and then the fquare will be farther
deiigaable, viz.
1 h i lb
15 8 9 2
14 5 i^ 3
A ^ P n
It remains to difpofe of four numbers, 6, 7, 10 and 11,
and inftead of ^, r, 0 and ^, fo ^ A -f* ^ '"^y ^^ = 1 7 ; as alfo
• + \^ s= 17; which may be done by coupling 6, lis as
alfo 7, 10 : but c ^ p mufl be =s i 4" /^= ^3) which will
be eiieded by 6 -f" 7 > ^^^^ whence p Seing s= 6, ^ will
be ;:;: 7 ; ana then 9 =5 11, and confequently ^ = 10 ; and
then the fquare will be fully completed, thus :
I
10
7
10
»5
8
9
2
H
5
12
3
4
11
6
»3
Or putting^ 5^7 J then r :=: 6, 9 = 10, and ^ =: 11 ;
and then the fcjuuie will ftand thus :
Chap. V. MAGIC SQJJARES. s'^S
I II 6 i6
15 8 9 2
14 5 121 3
4 10 7 13
Or by fctting down the numbers progreiEohally', rcfcrving
the diagonal numbers, the fquaie may be filled up by aa
cafy tranrpofition of the reft, as follows :
• ■
I . . 4 I 15 14 4
.67. 12 6 7 9
• 10 II • 8 10 II 5
13 • • 16 13 3 2 16
3. S^ppofe a fquare form .of fet numbers there be.
In their natural order (as i, 2, and 3)
Amount to the fum, when they're added together.
Of 62 juft, in rank and file either :
If alfo from 'corner to corner you count.
Yet ftill 62 (hall be their amount f
What numbers are they, and how muft they be put.
When fixteen there be that completely will do't ?
The lum of one row 62 X 4 (number of rows) =: 248^5
fum of all thofe progreifional numbers.
And tlieir common difference is =; r.
16) 248 (15.5. Then 15.5 — 7:5 = 8, the firft term.
And 8' + 15 = 23 = lallterm.
Then obfcrving the dire£Uons given in the foregoing
queftion, this magic fquare may be filled as follows j
8 . • XI 8 22 21 II
, 13 14 . 19 13 14 16
. 17 18 . 15 17 18 12 .
20 . . 23 20 10 9 23
4* 'Tis to you, lovely ladies, I fue and fubmit,
iWho outvie Sidrophel in magic and wit)
>*or folution of this knotty problem proposed.
By which undertaking my fenfes arc doz'd ;
To find by what method thofe fquares you may fill,
Which are magical call'd, and by that try your ficill ;
8, 9, 10, II, 14, 15, 16, 17, 20, 21,22, 23,26,
27* 28, 29.
^%6 MAGIC SQUARES. BookllL
To place all thefe numbers, fo that the apiiounty
Tuft half a fcore ways, feventy^four you may counC
If youll anfwer but this, now yourfelf do atture,
I will meddle witfi what they caul magic no more.
« . . II ' 8 a8 17 II
» 15 i6 . ' 23 'S i6 20 !
• 21 22 . 17 21 22 14
26 • • 29 26 10 9 29 I
5. T« form a magic iquare of the numbers i> 2, 4, 5,
tic to 25, indufive;
Firfty 5 are the number of rows;
alTo 3259 the fum of thofe progreiEonal numbers ;
and 5) 325 (65 ^ fum of each fide or rank.
<
j(6 14 S 2 25
3 22 20 II 9
15 6 4 23 17
24 18 12 10 I
7 5 21 19 13
6. To form the progreffional numbers from i to 49, both
inclufive (their common difference being i) into a magic
fquare.
Firft, I + 49 =: 50 X 49 = HSO*
Alfo, 2) 2450 ( 1225 ^ fum of thofe progreffional numben.
*•* 7} 1225(175 xs fum of each row.
I 9 17 25 33 41 49
24 32 40 48 7 8 16
47 6 14 15 23 31 39
21 22 30 38 46 5 13
37 45 4 12 20 28 29
" >9 ^7 35 3^ 44 3
34 42 43 2 10 18 26
7. You that delight in figures, try your ikill, ,
A magic fquare with numbers for to fill i
One to a hundred numbers juft muft be, i
Which to the numbers of the fquares agree :
But farther, you muft them fo iuft contrive.
Twenty-two ways, make five hundred and five. 1
2 No
Chap* VI. Ceupomn Ivterut* §zj
No two fquares alike in numbers muft be,
But ten in fareaildi^ ani ten in length, lec's^ fee,
XI 92 12 88 14 15 16 84 85^ 90
100 82 26 27 67 35 59 58 50 ^ I
99 «9 7.5 74 33 W 4* 43 J' 1
2 20 76 73 34 36 60 67 49 98
4 81 25 28 68 65 41 44 52 97
94 2« 77 72 2% 17 6x 56 4^ 7
5 fe> «4 29^69 64 4^ 45 53 96
6 79 ^3 30 70 38 62 55 47 95
93 22 2« 7* 31 63 39 46 J4 «
91 9 .S9 '3 87 86 S5 17 a8 10
Aoy one- of tbe fonegoiiig {quares oiay ))e dUpofiM many
other difFerent ways, as may he tried by tiioft who have
time and inclination for fuch operations.
CHAPTER VI.
SECT. L
COMPOtJND INTEREST.
C)m POUND Interest is that which arifeth n^t only
from the ufe of the principal, but alfo from the uie of
the intereft a6 it becomes due ; the inlereft being added to
the principal at the end of every year, making a new princi-
pal for' the fucceeding vear; fo that thepriacipal and intereft
are continually increanng.
C A S E I.
The principal, rate, and time given, to find the intereft.
R U L E I.
To the principal add the intereft for the firft year, which
will be a new principal ; to which add the intereft for an-
other year, for a fre(h principal for the fecond year ; and fo
proceed for any number of years.
R U L E II.
Multiply the principal by the amount of one pound for
one year continually for all the propofed years ; the laft pro-
duct will be the amount as before.
I. What
\
5x8 CoMFQ«if» ln'KftEtr* Book, lit
I. What is the coo^xmnd mtcreft of 500 1, tor four
9 at five per cent, i
jreus
I. I.
5 = 3^) S«>o» princiMl - - - - - - 1^^
25t intereft for the nrft jrear )
20) 525f amount for the tfty or principal for ad year*
26 5» lAtereft for the fecond year.
20) 551 59 amount for ad. prind|)«l for the jd.
27 1 1 3> intereft for the 3d year. '
20) 578 16 3, amount for 3d. principal for the 4di«
26 18 9|», intereft for the 4th year.
607 15 *4» 3<noimt for the 4di year.
500 - ^ principal.
£ 107 IS -I* intereft.
By RuL5 n.
The amount of il. for oneyear, is 1.054
Then 500
1-05
5259 amount for the firft year.
i>05
2625
551,25, amount for the fecond year*
1,05
275625
5S"5
578.8ia5, amount for the third year.
i>05
28940625
5788125
607.753125, amount for the fourth year.
500. principal.
107.753125, intereft,
2. What
idirapu Vf. GQMffVKD rlff7$«««'&
<S^
2. What IS the Alnips)miii .Intertft of 760 1 10 s. for
four years, at four per cent, per an^ium t
1. 1. s.*
20
I
T
760 xo, principals
k . i A
152 2
d.
30 8 4|, intcrcft th^firftycaN
TJ790 18, 4|,«nrottnt5 principal for the 2d year.
T
158 3 8
31 12 8|, intcrcft the^yeir.
t
T
t
T
822 II . 1 1,, amount i princlpahfor the 3d year.
164 10 2j-
32 18 -I-, intcreft for 3d year.
f 1^55 9 2> amount j principal -for the 4 th year,
t
r
171 I 10
34 4 4t> intereft for the 4th ^ear.
889 13 6^, amount for -the -4th year*
760 Id -, pfirlcipaL . . _ . i
•
• ^ ■«__
^129 3 6 J, intereft, - —
Mm
By
^30 CoMVOVMD XHTtMyp. Boq1| HL
By Rott n.
The amount of 1 1, for * jtu, at four per cent, is i^i^
Then 760.5 .
1.04
304M
760$
79o.92» amount for .the £rft year*
1.04
3I63IS&
79092
82a«5568» amount for the 2d jcear.
1.04
32902272
8225568
855.459072
1.04
3421836
855459
■ I J ..'
889.67736, amount for the 4th ycaf«
760*5 principal.
129.17736, jntereffcs X29I. 3s. 6fd.
CASE II.
The amount, rate per cent, and time given, to find the
principal or prefent worth.
RULE.
At the amount of 1 1. compound intereft, at the rate
and for the time given : is to 1 1. : : fo is the amount
given : to the prefent worth required*
3. What
• 3; Wii'it h thd 6kfenr«r6rtliof 8I59I. ^3S. S^i. 'due
four yeinrs h^ilbe/aifcfuf p.6r (^m. pe( Annum, tOApeund
SnteTCnr
. - • • • - ^ ' . ' i .»
Firft, X.04 X 104 X x«04 X 104 =r f.16985856 :
Therefi^t i.i6985856 i t :t 869.677^6 : fi^.'p \
AitMcfp^jboh f6s« prefeat wojthi
* • ' ; • ■ . ' .
C A S £~ |IL
Tfte {iriDcifMl, titci and kneunt gtVcii) to find tke tifi^.
RU LE.
t>ivide the amount by the principal, and that Quotient by
the amount of i )• for a year, and the next (Quotient by thd
inhe ; and to on continuallyy tiU the laft quotient be unity |
the nvmbcr of whtth-divifiont wilLbaihe time required*
»# • - • •
4* In what time will 7601* los. amount to 889I. 12s.' 6id.
compound intereft, being allowed at four per cent, i
760.5) 889.67708^ (1. 16
1.04
1.16 - - - - lib
X.C4
1.12 - - - - 2d
1.04
1.08 - - - - 3d
1.04
1.04 - - - -4thJ
^divifion.
Hence the term is four years.
The 4th Cafe is to find the rate per cent, the principal^
amount, and time given $ but this requires the extradion
of the roots of very high powers, or the ufe of logarithms ;
which (as my booic is f welled to a greater bignefs than at
firft intended) I am obliged to omit.
And for the folving queftions in compound intereft with
more facility, have inferted the following tables.
The conftnidtion of the firft table following, (hewing the
amount of 1 1. for years, is only by the involution of the
amooBt of 1 1. for years, to the power of the number of
years.
' Mm a Thus^
f
\^'
531 Compound iNTB&Ysx^ Book II£
Tbu8,>the,;vnount of x K for two ^cars, at five per c^L
compound intereft, will l>e 1.05 x 1-05 = i«I025« «
* Alfo, 1.05 X I 'OS X 1*05 = 1.157625 '=: the amouni
of 1 1* for three years, at five per cent.
And ^^ conftrudion of the fecond table is by die .con-
tinual multipliactiQn of the. amount of iL for a day ; the
amount of i K for a day being the root of its amount for
a year, extraAed to the 365th power.
.T%e amount of iL ibr a day, at five per. cent."* biduig
1,0001336, its amount for two days will be 1.0001336 X
1.0001336 ss 1.0002672 &c« and 1.0001336X 1*0001336
X Ir000i336 ;= 1.0004011, the amount of x 1. at com-
pound intereft, for three days, at five per cent.
^ And thus by continually multiplying bv the amount for
a day* at each rate per cent* the iepond. table is conftruded^
and the 364th produd will be the amount.
DlCIMAL
E §n }
DxciMAL /TAELES of Couwovv'AmtKtnr
At the rates of 3, 3}^ 4* 4i> >n<l 5 P^ cent, per annum.
The
■
aaieunt of one pound for years.
■Ui
■*i lii^
I
3
4<
5
3
9
)o
'^•T
|.
^perctati
Wi*.
x
9000
I.09d7ft7a
x.ias^oSt
J.159X74P
x.i940$ftf.
Ma93738
X.t^7700
1.3439163
i'.3M338
»*^8jij7
M"5»97
X. 604^64
i.65aa476
l^7<»433o
.1.7535060
X.S06XX12
i.S6oft945
i,9i6>034
1.9735*65
••93»7.94i
^ »;OW7779
»fi J a- 1565911
a^aifiaS^o
^.Ji ,».a%rgi276
».4ae7a|dt4
m. 5090803..
a. 57508*7
ft.654335a
18
XSi
so .
*3
»4
34
1^
37
38
39
3* P.'€Vt.
i*i *i
I i& .A.8j^36x4
ae8ft8x78j
ftr9 ^5x266
^.9Ml8}4
3.1^4269
• 3'*^<^77
ix.oQjogoo
^ x,.07 iat5o
.x;xD^i7S
: J.i4r:5:i1o
.J:)it7l»8«t3
, iutt9t553
. 1^72x79%
1.316B090
. 1:36x8973
1.410591^7
"•4599^
' ik5Uf68o
.1.5639560
X. 6 1 86945
1.675I488
1.7339860
« -794^755
.1.857489*
i:^X2$oi3
X. 9897888'
»o5943»4
1.13151XS
ii.206)i44
i.x83)284
'X.363*449
*-44S9585
»'53»S^'
2.6ac{7i9'
2 7"t779
.2.8067937
^a.^050314
3.^.067075
3.111x4*3
3.2208603
'3 3335954
.3^50X66^1
3.5714254
*a/696«ii3
3-8x517*7
Y 3-959*597
4per ccot.
I 0400000
X.O816OOO
X. 1348.640
»x.z698586<
x«ft]f65X9
x.2<53i9o
X.P59318
X. 3685 69 X
>*4X 33118
X.4802443
»-5394S4«
1.60x0 jis
1.6650735
1.7316764
1.8009435
1.8729812
I 9479005
'X.0X58X65
X.I06849X
2:1911231
2.2787681
2.3699188
»'4*47»55
2.5633042
X.6658363
XJ7724697
2.8833685
2.9987033
3^186514
3.*433975
3-3731334
3.C080587
3.648381 I
3.7943163
3.9460889
4.1039325
4.2680898
4.4388^4
4.6163659
4.8010206
4|:p. cent.
1. 0450000
1.0930x50
I.i4ii66r
X. 29x5 186
X.X461819
1.3022601
1.3608618
1.4221006
1.486095 1
1.(529694
X. 0228 J 30-
i*6958»X4,
l'772l96i
1.85x9449
1.935x824
2.oa237ox
X. 1 13376*
X.41084787
X. 3078615
2.4117140
2.520x411
2.63365x0
2.7521663.
2.8760138
3.005x344
3.1406790
3.282C095
3.4296999
3.5840364
3.7451181
3.9138574
4.0899810
4.X740J01
4.4663615
4.6673478
4.8771784
5.C968604
5,5658990
5.816364s
i^i^
5 ptt CCBt.
r
i.o50«.oo
x«iM|oeo
i.i^7ix^
x.xt55o6j
1.276x816
'•3409956
t.4b7|co4
i'477f5J4
i.55T]28x
1.6x88946
't.7«03393
1.7958563
1.885^9) '
1.9799316'
X.078928X
x.i82|74:6 '
2.2920183 ■
X'4o66i92 ^
2.1X69502
2.653297^
2.7859626
*9*S*^07
307155138
3.2x51006
3-3«63549
3.5.556727
37334563
3.91(:I29I
4.II6I356
4.3219424
4-5380395
4.7649415
5.0031885
5x533480
5. 5 f box 54
.7918x61
0814069
6.38«;4|'7S
6.7047^11
7.0399887
i
Mm
DSCIMAI*
t '514 1
P^eiMAi. TABLES afCoufDa^D Iktbbxsv.
T A B L E 11.
The amount of one pound for days.
I
4
IIP
•i<iP
•190
240
•250
]290
300
3 IP
■330
1350.
360
itex*
1.0000809
1.000x619
I.OOCRZ40
fjQOcjio^O.
^jOoc(|^86o
1.0000391
lfPpI09p9
•110^24334
«t«>3>445
*iOC)4§7o8
1^0056849
l<o«6|996
f«o«8o479
I^O|o|834
I.Qliloax
-1AI3J415
1.0138623
1.0141837
t-ot5|ow
f.9l6|a84
1.PI7I518
I-OI7J759
i.o]8|oo6
1.0194260
1.0204520
-1.0212788
1.0221062
1.0229342
f»«23t630
l.02459%4
1.C254225
1.0262532
l-o27!}847
1.0279168
1028^4^5.
1.0295830
m^mmt^
1.0000942
1.0001085
t.Q0^66oo
t.ao07542
J.oo484ft6
s.Qo<io429
> x«QDi8|67
9.0048315
1.Q047236
^.QD|6ftO
1.0066193
x.ooa5tw
1.0094696
i.oiqif^24
2:01x3742
X«QS 23(279
1.0X321825
1.0140379
x.oflU943
.2^016115x6
X.017XCA8
1. 0x8068^
X. 01 9028s
1.0199897
1. 02 09 51 5
1*0219142
X.Offzi778
1*0236424
1. 02460178
'•o*3r74i
1.0267414
X.01177096
140286716
1.029^486
i. 0306 195
1.0315914
1.0325641
»-3345«3
■P1M#
.0001074
.0002149
£003324.
tfmm
0004199
«i 9005374
PPq644^
0007024
QOtiSSoO
0009^75
CO 1 0751
002x513
003298$
««ci74
fooij^
«7S50«
.0080335
.0x08033
.01x8900
^129779
.0140670
.0151579
•oxOMtf
«ol7|4lfc
.0184350
^195299
•0106261
.0217239
.022|sxl
.0239215
.XMI(02*3
•026X243
.027^2^5
.o$8|3X9
.0294375
•030J443
.0316522
.032^614
.0338717
.0349832
.036^960
.0372099
409832^0.
039441-5
"vrr-
.0001206
000241^
AQO9&U.
3>'
.0006445
.000965^
»oex^oi6
^0024148*
-^>3$4
.0060470
.O0726oi
.008A773
•0096942
x> 109 125^
•0121324
•0S33537
•OH|7^5
*Qi58o^
.0176265
;oi82597
•0194824'
^0207x26
.02x94^2
•«*3«7n
^024^x20
•0256481
.026I858
.0281249 1
03it|i»
.0130963
OH84«9
vOj5e9io
.0368406
03809x7
0393444
HPC985
0418542
1.0443700
.000X336
•0002673
I
OSpSotl
IX
■001337*
.0026770
.004oiSo
.oo536ts
•0067059
.0080525
0*075 fx
:vOi2io3X
.0«4Si>5
.0x0x099
.0x7099
.diYi9da'
.Oio^5§y
.o^o6t7^
02^
.03260XJ
•03S9HJ
•03^75^
.031 x^
039511
O409x<
.04^3087
.0437029
.0464969
.047S967
.04999^
The
j
The ufe qf lite fraegeulg ttbksi
CASE i.
Pftsicipily ntt> aiid4uiirgivc|ii»^tefiiiit]ie aoioiuit
: & U t £.
. * 1
p r* •* • '
Multiply the ambiintof il. found in die firft table, at
thp^. rale and (qr t;he timfi giyen> by ihcpropofpi princ^,
and the pr<>du£E gives the anfwer. .-
• ^* What will niKamount to in ax yeart, at Aur per
conit, pet annum 7 . .'
The tobularjnumbfqr 9gainft,ax veaii at feurp^ccnt^
2.2787681, ^ Then 7Jii x^i-i787W a 1641.99x7 s
164a !• 19.3*^ 10 d. the ampunt'required. .
6. What win 358 1* amount to in 4Q4?ys, at five per cent,
per "anntim, compound intereft ?
In the fecond table, againft 40 d^, at five per cent, is
1.00^3611.
'•' 36^ X 1^^611 t&359^9X9a73|ftss359l i8l. 44dw
the anfwer* .
If the amount be required for any number of years ex*
cieieHing thtfe in (tetd^ie, divide the given number of years
into two m aHmW' fusil aufubeie a^ are in tfao sabM^
and multiply the amounts anfwering thereto into one ano-
tbdr 6(»ftinialIv».'aQd the fatfr pt^oft by thtr prfncipial,
wJacb.wiU beiAeamouixcrequired.^
7« What is the amount of Sal. .10 s. for 75 years,, at five
per cent, per aniium^ compound intereft i
Firft, 40 4- 35 = 7S«
TUeimouat of ilk ht^Of^taSf a|:5 percem^ is 7«b399t87.
^. Dittofor 35 years j - - - -'5.5160154.
Then 7-6399§87 X 5.5160154 ==; 38.8326861.
Alfo 82.5 X 38.8326861 = 3203.0966.
Anfwer, 3203 1. 13 s. n d.
If the amount be rtquired for any number of days which
are not in the ubles^ proceed as with the years in the laft
escaimplei
M m 4 8. What
8. What is the amoun&oif«5ft2'^^^75 ^ays^^'ae^iisitr^yer
cent, compound intereft i '
I d t- '* O
The amount of 1 1. fofifod2Ljs\\tjLpcrctnuiB^*^y^2*
Then 1.0339963 X i.ooo6o?i = 1.0337x94.
And 1.0337194 X.#2#=s 51(^.6 «2463i-- r -
*' Td-flhi the- a&5^t<fef reaw iAtf daysy'obfcjr»'ttd Al-
lowing example. •'*■ ' •*''• ''"' '"'^ - -:-:-■ -
» • • • • - «
•* 9.'^Wfiat wlff 35y^. "t^. iimoaitt -to % ; four t^Kirs "iml
274 days, at 3^^ per cent/ per aimum, coid|M)tttiJuiterdlf'^
-fcooOSTtro:
-And ^7.7 j X 1.1775431 = 421-2^6044=^ ^ll. is. 3|d,
<be anfwcr. ' * ' ..'•:.« v-:- *■ , • r ,
• . ' • . *
.' jAjaountt ratc^jai^ time giv^n^a 3^. thc-princtpatt -
R U .L £.
i iDmde the amonot giveaby ith^ aotount of i L found in
^bt^&itmb}Q» and thfi^u6tki¥t wiUibeith^ anfwcr.' ^t
( i(^« What is the pr^ent wonh orx64a.h/i9tR. lo^d. due
21 years hence, at 4 percent* per^anoumy 4:oaipoiKnd.4B«v
tereft ?
The amt. of 1 1, ix\ ii years, at 4 per cent, is 7.2^87681^
Then 2.2787681) 1642.9918 (72i>' the an^wer.
t
0
Iti. What is the prefent worth of 3203k 13$.. iid. due
^5 ye^^ hence, at 5 per'cej;it. compound intereft^
The amt. 6f 1 1. for 49 years, at 5 p6^r cent.is 7»03O9887^
Ditto for J5 years, - - - - S-Si6oi54.
Then 7.0399887 X 5.5160154 =1 38.8326864.
V 3&8326864) 3203.6966 (82*5 =8^1. lO^ the anArer.
12. What is the prefent worth of 421 L 59. ^Jl dtte^
fpur years and 274 days hence, at 3^ per cent, per annum?
\
By Tabu firfl artd-feodnd; '^
The amount of x 1. fop-four fedsvj is i. • w ^ ^•'4'TS.*2r
Ditto for four days, - ^ * 1.000377, .t
Then 1. 147523 X i.oa|7jLiJj<i;ooa377JE: 1.1775431.
.--r •^.-' .-• - <? .^-CA S^/E' ill.- ' - ' '-'' " *^ - - -^
c -Any f nndpal, rao^'ailit moMt 1>6iifg^^tr^ j.' to-iind tho
X'* >- •• ■» "Rl 11'^ Tj* E ' " ' ' •••
»•>>!, «•
Divide the amount by the principal, and die' quotient'
will >b^i ^.'•inouAt of -i L at ,the -given* j^te,-^ which wUL^
Be fouhd^in the firft ialxle uiider that fate, even Avith the*
time required.
• iJ3^)Iawbft%^^«c'wHl7?|j. ai»oi»nt tOTi6t?l.;;«9s.:iod;
at 4 per cent, per anmim^- compound intereft \ *'
^ 71;-) x%2-99|8 (2.^78765, the ai^ount of il for th^
t^me^ op|K#5 to which^ jiftder 4.per c^in,.iii .the (iMjorid"
t^We,: 4?. :%^y.qaf?,, the anfwer- required. ....
Bqt }f the quotient oannpt-be truly found in tlie table,
take out the next number,, suid make it: a diyifory by which
divide the fifft quotient, ' and * fcelc the fecbrid quotient ifi
table the fecond; but if ^t oanyot-be truly found in that
^ table, .ta|Le out ^he nextteafi number there, ^x\d divide the
* l?cond quorieht by it, and then feek ^ain' for the third;
quotient, and the number thus found in the table is the '
dumber of days. . . ' •
a
i ^4. In what time will 3^7!. 15 s. amount to 421 1, cs.
4d. at 2i per cent, compound intereft?
357.75) 421.2^ (i. 177543, the number next to which,
ynder 2i per cent, ftands againft four years, and is 1. 147523.
Then 1,147523) 1-177543 (i. 0261608, the neitt lefs
fiumber to which, under 3^ per cent, ftands againft 270
^ys, and is as folV>W8| *- ~
V . »
VIZ.
7
viz. 1.025741 } 1. 0261608 (1*00377 ftands agtinft four days.
Anfwtfy feur y<W^ mi. 274. ^y^*
.. . C A»«'W» • >
. Principle time, and tSMint ginm tD find the rate of
iBtcrcft*
.E U If.l^ /;. .
Divide tM am^tuit by theprirfdi^^ 4ti4 Ae 4(iKytkiif
wiU be the amount of 1 K which being found in the firft
taUe^ even with the giteli tifte^ iH.UOder the rate required.
. 15. At wha* tale jiet<:eM« fir 4MMP will TijiL beeolde
1642I. 19s. lodl in 21 yearsf
721) 1642.991S (2.278768) the amount of il. for 21
years> which wUl be found uidU 44per cent, the anfwer to
the qucftion^
SECT. .IL
— t»OUND Iir*BR*ST*. ' ^
FRteliol(^ of ltd eftaies; 4re fiieh as are p^fthaibl td
contimicf for ever ; qtieftiohs lelating to viiikk (ejiorat'
in rcverfion) are (blved in tht tnoft eaiy^ manner only fy
the rule pf d)ree.
CASE r.
When the yeartf income \& required.'
As tool. ! is to die propofed rate per cent. ; : fb is Ae
fmn to be laid out ; to the yearly income.
' I, A perfon dcfirous to lay out 17601* in the pi^icfaafe
of a freehold eftate, fo as to get 4^ per cent, for the money»
compound intfjccft \ what muft b< the aiUiual income of
fttch an eftate?
• 100 : 4.5 : : 1760
4-S
880
704 .
^i
zoo) 7020.0 (74.2 s 79l> 4S. die anfwer.
CASE
GDUlpr¥K CaMPMMf hntmnn $^
C A a fi IL
II the ^ue pf At tftMh If fiquitnlc
. • , '. • ■ ■ . I ■ - ' * r * - r
* .' K: V t E»
As Ae rate per cent, r is to lool. ::to Is Ae yeerfy
ipiit > to Yiie vilMe VHMi^*
a« Af^elbte brinf) in jrnrtr 79L 4$)i i^tt wofild it ftll
for^ allowiog the purchafer 4^ per.cMit.. Ar hb inooey i ^
4«5 : SCO : : 791^ : Vfiph the aaTwer.
C A 8 E Iff.
I. ^ *-f
* V
To find th(»nite per eent. on money laid out on die por-
^IjAfe of .fti^oid eftsMM
» • -*.• *•<■•
iUthc n^ejr K»4 out ^itthi: purch^ zM tQ the yeftly
jcm, : r fo Is lool. : to the .xate per cent,
J. Suppbfe X76oI. be paid for a freehold, efts^e^ idiicji.
(b yearly 79 1. 4 s. what rate of intereft hath the pur-
tSxsAr for his money ?
1760 ; 79.2 :: lOO : 4.5 = 4i: per (;ent» aiifwer*
4. Suppoie an eftate of 79 1. 4 s. per aniium be fold at
2X} yean purchafe; how- much per cent, hath the pur-
chafer for hb money?
^ 24.y X 79-36 :=: 1760 : 79.2 : i roc : 4.5 per cent.
SECT. III.
Twcln^g Fribholjx Estates m Retersion^
C A S E I.
THE yearly rent of a freehold eftate being known,
to find the prefent worth of the reverfion of the faid
cftstes after the expiration of a certain number of years.
2 RULE.
1
Find the^tatttAie' dfuAc C(^aPb)^'4l^afifcdiSd cafe of
^ laft fe^oii. Tbeny br cafe .the fecend of compound iit^
terefty find wh^ principS 4r %^ will amount to the full
Suppofe the reverfion of a*^ffMl0M #ftkc 79^.^4. $. M
annum, to commence feven years hence, is to be fold ;
4Mati ^* :wm^M&if\f^ |>ea^ ifSwt^x -^tflMtlt^^thS'iparchtfer
4J j^tKm.'lik^tis-'Mttty/I^ Initio-- q V , ^^\'i.z\ . ,* !
*• •
X'The fwiPJ^n l^or' the iltVf^oii ^of a ftreHoia'ejfcite,^^
to commence aftbf'i certain Aiithb^W years^ S^ihg khoPwii/
to find the yearly income,. a^Iq^ying ;he, purchaser fo.xnuch
rfet4^«it. for^Hismott^jf. • ' -'^ -i /'^ • " '"'
•»»■•• f' * ■ 't J 11.7 ' ' f
RULE, * ' '
Find • th'd kfhduhf o^ the ^^SiiHe mofiey^ to"^ the tiiic
wb«n Oie rcyerfio(i b to cpmiMjcic^ hy the firft -cafe of com-
pound lotQref^, then fi^d'yi^ ve^ly .lACoinQ Wbict^i that
artiowt will purchafe. * , , . .. ^.^.
Ai« A*'/ » * •
Suppofe x!c^ j-eyerfion of a^reehold eftate, to commence
feven years heflce, is /old for 1293?. 5 s. ii^^. allowing
tt^e purchaf^r 4^ {7Sr,cer.t. compound ^intereft, for his money ^«
what ought thie* yearly rent to ' be ? •' • '
• . "• %
The amount of 1 1, for / fcmi ; yea^s, at 4 J- per cent, is
J.3608618. . •' * ^ , ^
TbejA 1293^2981147 >^ t.^ddSfoft s 1760U aknouMr-
And 100 : 4.5 : : 1760 : 79.2. .
Anfwer, 79I, 4 s. pferUmnum.
• k^.
.A S B C T.
M
G&apr 3iS. CoMPOUnd bntKnO f4|
. . S E C T. IV.
P IT R C H A' S^^ »^G -A N !? tT'l'f J E 4.^ ~
1 ' . • f ' ' '.
ANiiuitEcs^; penittms) i£Uneiesi t&cJ a^ ,t¥4tt^> prDfit$».
.and paymeatSs/ina^e' yeafljr or :halfi7^iiy;.^Q;(4oA;
they are JgiAlto: be JA.«rrfeaM:wllta thay«aMj.4ae: ga4.l|iH
^hereof follow. .,.,.. ,
: C^i0ru£fton }f tie jEr/? Table «/• ASNtiitiES; ' '^
• ThU'taitf" flttwfi" tSfe priilfciit Wth er'taliie (pf- lU'
paysbhf'at'any period, from one to forty years inclufive,
and is conftruded, b.y.divi5ling .f 1- l>y i^s amount found
in the fecond' table ofconipbimdintereltioi^ the time and
<atB affigliaJM ' / .1 1 • • * • . ''^ ^.''^j ,::♦
- Agabift the firft taWo of tompouhd intcreft, theprefciif
worth of 1 1! for three years, ' at 3 per cent. 1^ ^Ipgttjij*'
1.09272^5 T.0000600 (.9151417, for theprefcht worth of
rl. three years hence, compound intereft, ai 3 per .cent* •
.. Conflru£fion tf the ficmd Tabu,
This table Ibews the amount .of i !• per annum, . mk
is conft^-udled from, the firft. table of compound intereft,
thus : To 1 1. the firft year of this table, add the "firft
year of the. table for years in compound intereft, and th6
amount will be the fecond year Jn this ^ table *$ to which
add the fecond year in the table of compound intereft^
and the amount of it will be the third year in this tabic,
&c.
Thus 1. 000000
add 1.030000 the amt. of 1 1. for i year, at 3 per cent*
2.030000 the amount of 1 1, for 2 years.
1.060930 amount of the fecond year.
3.090900 third year of the fecond table.
Cm/iru£iiQn
f •
p
14^ QwranM iRttAUT. BookaL
CittftmSim ^ftbi Mrd Table.
The third table fliews the praent value of 1 1. per anniun,
and Is conflruAM v follows^ ynu Ae prefent Take •£ tbe
firfi: year in die &ft table» is the (anie as die firtt year in the
third table ; the firft and fecond years in the firft table^
added tocvitert make the Yeodnd year in the third tabte^
attid the mird year in the firft table, added to die fetond yeas
ill die diird table, nuke the tUrd year in tbethird.
Thttt^ ift yeari tables ift and jd, at 3 per cent b 19768738
-J-hc ad year, itt ublc ift, ■94?595^
Their rum, ad year ia the diird tables is - - 1.911469^
Third year^ in die firft table, ^iSH^I
Tbiid year^ in die du|d taUe» .^ . • ^ 9-82^114}
OMftnafUncfAe/nirthT ABIE. J
This table Ihews what annuity 1 1. will piirchafei Im^
wd is conftru&ed, by finding the prafimt worth of x L per
annum ia the third table at the affigned rate and time»
^d dividing imity therebv, and die quotient will be the
annuity that iL will purcnafe at the ume rate for the iaac|
dme..
Example. What annuity will iL purchafe, to c6tt-^
dAue time years, at 3 per cent. ?
In die diird taUe, under 3 per cent oppofite to dire^
years^ is 2«8286ii4
2.8286114} I.00MOOOO C«353S2<^4» ^^ annuity for thre^
years^
DsCIMAf.
I 54J 1
DsCIMAl TABLEil y COMTOOHV iKTt&ESY.
T A BL£ I.
Ttie piCfott wocth of one pouad for yua.
1
t
^
1
S
3
4
i
J
5
II
»4
ft
«^
so
ai
a»
*S
»4
*5
a6
•7
s8
»^
30
3»
3«
33
34
II
P
39
40
•f7d»73*
•94*6959
•;37»4«43
.Silo»i5
•7«94<»9*
.9644ii7
.744093,9
.7o^s799
.eSQ95t}
;d4|8<t9.
.€231^9
.60 jo 164
,587394^
.5702860
•553*75«
•5315493
.5118925
.50^69x7
.49^9337
^71005^
.4636047
.450x891
.43707^*
.41413464
w|,x i<986t
.3929>7<
.3883370
.3770263
.3660449
•35$3«34
•34503*4
.3349829
•3252262
.3t575j6
.3065568
^33S»«7
^»94»7
.97x4422
•f4»97i;
•|t35Po6
.7859910
.759^1116
•7|373«o
.708^x88
•6$4^57
.6617833
-<39404«
.6177818
.5968906
•5767059
•557*038
•5383611
.5101557
.50*5659
.5855709
.469x506
4532856
•437957*
.4211470
UI088378
.3950x23
.3816543
.3687482
.3562784
.3442304
.3325897
.32x3427
.3x04761
.J099769
»2898327
.28003x6
127056x9
.2614x25
.2525725
4per«eBt. 1 4|pcr<ct.
.9^15385
-5*4556*
.1189964
.854So4t
•82x9271
^7903145
•759?i7«
.7306902
•7«85*67
.6755642
.6495809
.624597 X
.6005741
•S77475*
•555»^5
•5339082
•5*33723
.4936*91
•47464*4
.4563870
.4388336
•4*«9554
.4057263
.390x2x5
.375"68
.360689a
.3468x66
•3334775
.32065x4
.3083x87
.2964603
.2850579
•2740942
.263 5 52 X
.2436687
.2342069
.2ac2854
.2x66206
.2082890
•9^37t
«l7S$96i
.83856x3
.1l02A5X«
•7«7*957
.7349*w;
.7031852
j67»9044
•^439*77
.€16x988
.5896639
•^6427x6
"•53997*9
•5167104
1 •^944693
I •473*764
•4528004
•433$o«*
.41464^9
.3967874
.37970C9
•3633501
•347703s
•33»73<^6
. 3**40*5
.3046914
,29x5707
.2790 t50
•267^000
•*555024
•^444999
.23307x2
.2238959
.2141544
#1050281
.106x992
.X877504
•*79665^
.X7X9i«7
■
.7835262
.746*854
-^67683*4 .
j6446oS9«
.613^33 ;
-54S793:
•5568374 !
.5303214
•505«*?9
.48x0071
•45*^^5
•436*967 .
•4555207
•395734«
.3768895 .
•35394*41
.34x84^
i3*5S7*3
^32oa679 ■
•^953058 *
.28x2407 .
.4678483 '
•*5S0936
.2429461 :
.2313775
.2103595
,2098662 ;
.199^726
.1903548
.x8x»903 .
.172Q534
.I5W054 •
•149*479 I
.'4*<y57 I
Decimal
1^1
D&ciiMAt TA9L£$ flf Compound' iNTBRss-xt
TABLE 11.
The amount of one .pound per annum, 6r annttity for
years.
\-
ite«M
ft
3
4
5
6
I
9
10
II
12
»3
14
»5
t6
»7
i8
«9
•o
SI
22
»3
27
28
«9
30
J»
3*
33
34
3
3
i
I per eeiit.
I
I
4PI
1.0000060
2.0300060
3.0909000
4.1836270
5.3001358
6.4684099
7.6624622
8.8923360
10.1591061
n.4638793
12.8077957
14.1920296
15.^177904
17.0863242
18.5989139
20.1568813
21.7615877
•3-4«44354
25 X 168684
26.870374s
28.6764857
30.5367803
}2 45*^837
3H»H?o2
3;4Sf^3
J«-55104**
40-?o»^335
4».9|OQ22|
45.2t88502
47'S7S4tJ7
|o.ooa6782
5»|o»75«5
S4-^7»4i3
•|*ri9443
fi. 1742226
%•» 594493
722342 3«7
75'4«"i97
m»
3I per cent.
1
I.OOOOOOO
2.0350000
3.1062250
4.2149429
J. 3624659
6.550I522
7.779407s
9,0516860
10.3684958
I1.731393X
13.1419919
14.6019616
16.1130303
17.6769864
19.2956809
20.9710297
22.7050158
24.4996913
26.357I8o^;
28.2796818
30.2694707
32.3289022
34.4604137
36.666<282
38.9498567
41.3131017
43.7590602
40.I906273
48 9107993
51.6226773
54.4294710
57-3345025
60.3412101
^3-453"5*4
66. 6740127
76.0076032
73 4578693
77 0188947
80.7249060
84 SS0277*
4 per cent.
I.OOOOOOO
2.0400000
3.I2I6000
4.2464640
5,4163226
6.6329755
7.8982945
9.2142263
10.5827953
12.0061071
»3-48635i4
»S'02«|o55
16.6268377
18.2919112
20.0235876
21.8245311
23.6975124
25.6454129
27.6712294
29.7780786
31)9692017
34.2479698
36.6178886
39.0826041
41.6459083
44«3"7446
47.0842144
49.9675830
52.9662863
56.0849377
59 3183351
62.7014687
66.2095274
69.8579085
73.6522248
77.5983138
81 7022464
85 970336*
90.4091497
9502551574
4{ per cent.
I.OOOOOOO
1.0450000
$.1370250
4.278 19 XX
5.4707097
0.7168917
8.019I5I8
9.1800136
10.8021142
X2.2882004
13.8411788
'5-46403x8
17.1599133
18.9321094
20.7840543
22.7193367
24.7417069
26.8550837
29.063562 c
31.37I4A28
33-783x368
36.3033779
38.9170299
41.6891963
44.5652101
47-5706446
50.7x13236
53-993333*
57-4»3033»
61.0070097
64.7523878
68.6662452
72.7562263
77.0302565
81.4966180
86.1^9658
91.0413441
96.1382048
101.4644040
107^0^113^31
4
5 pec cent.
I.O0OQO09
2.0(0000d
3. 15*5000
4.3101250
5- 5*563 «;
6.8019x28
3. 1420084
9.54910SI
^1.026(643
I2.577i|2|
14.2067871
15.917126$
I7.7I29828
19.5986320
21.5^85636
»365749x8
2 c. 8403664
28.1323847
30.5390039
33.0659541
35 7«9»5'8
39.5052144
4x.43<H75«
44-5019989
47-7»70988
5' "34$l8
54.6691265
58.4^5828
62.3227110
66.438847*
707607899
75-»988294
8o.o6377c^
85.0669594
90 1203073
95.8363227
101.62813)88
X07 7095458
1x40950231
Decimal
t 545 ]
Decimal TABLES ^/ Compoxjnd iNTERiist*
TABLE III.
The prcfcnt wotth of one pound per annum, or annuity
for years*
H5
A
Sr
I
%
3
4
5
0
7
%
9
ig
II
12
'4
I?
i8
»9
ao
21
as
24
as
26
»7
28
30
3»
3*
33
34
3
3
38
39
40
3 ptr cent<
0.9708738
I. ©134697
2» 82861 14
3.7170984
4.5797672
5 417 10 14
6.2302529
7.0196922
7.7861089
8. 5302C28
9.2526241
9.9540040
0-6349553
1.2960731
»-937935»
2. 56 II 020
j<i66ii85
3'753S«3i
4.1237991
4.8774748
5.4150041
5 9369166
6.4436084
6.9355411
74131477
7.8768424
8.3270315
t«764ioS2
9.2884546
9.6004413
20.000^285
20.3887655
20.7657918
21.1318367
21.4872200
21.8321525
22.1072354
22.4924616
22.8082151
23.1147719
3 J ftx cent.
0.9661836
1.8996943
2 8016370
3.6730794
4 5»50SM
5.328553J
6,1145439
6.8739555
7.6076865
8,3166053
9.00155^0
9-6633343
0.3C27385
0.92c 5203
i.5i74i«9
1.0941168
a. 6 5 13206
3.1896817
3.7098374
4.2124033
4.6979742
5.1671148
5.6204105
6.0583676
6.4815 «46
6.8903523
7.2853645
7,6670128
8.0357670
8.3920454
8.736*758
9.0698656
9.3902082
9.7006842
20.0006612
10.2904938
20.5705»54
ao.84i«>874
11.1024999
2i.355C>7*3 '
4 per cent.
0-96153^5
1*^860947
a.7750910
3.6298952
4*45*8223
5.2421369
6 0020547
6.732744*
7-43533'4
8.1108955
8 7604763
9-3850733
9.9856473
0.5631223
1.1183S68
1.6522949
2.16566S0
2 6592961
3*^339385
3,5903253
4.0291589
4,4511142
4.8568405
52469619
5.62207S7
5.9S27678
6*3295844
6 0630618
6.9837132
7.2920318
7. €884921
7 87355*^0
8.1476441
.8.4111962
8.6646 1 16
8.9082803
9.1425771
9 3^:^625
9 5844831
9.7527721
4} percent.
0.9569378
1.8726678
2.7489644
3-5875^57
4.3899767
5.1578725
5.89270Q9
6.595S861
7.2687905
7.9x27182
8 5289169
9.1185808
9.6828524
0.2228253
07395457
1.234015]
1.7071914
2.1599918
2.5932916
3 0079365
3.4047239
3.7844248
4»477749
4.4954784
4.8282089
5.1466115;
5451J038
5-74287?5
6.0218885
6.2S88SS5
6.5443909
6.7888909
7 0228631
7.2467580
7 4610124
7.6660406
7.8622398
8 C490902
82296557
8.ii-:uS44
■*«■
5 per f <rnc.
oj 9 523809
1.&594IC4
2.7232480
3- 5459 505
4.3294767
5.C756921
5 7863734
6 4632128
7. 1078217
7.7217349
8. 3064 142
8.8632516
9-3935730
9.8986409
10.37965^0
10.^377695
11.27406^2
1 1 6895869
12.0853208
12.4622103
12 821L527
T3.1630026
» 3 4^*5739
•134^6418
I4.cf3«)445
14.57 5*S 5 3
146430336
14 89^72
15 141. 735
•i5-37245»o
i5.592Si04:
15.8026766
16 C925491
16.1929039
i6i374i942
X 6; 546^1^ to
i6.7ixii72
I6.867J426
I7.oi7«4o6'
17. lSQ08%s
*-^— ^^ W#
Nn
Decim/x
[ 546 ]
Decimal TABLES (?/ Compound Interest,
TABLE IV.
The annuity which one pound will purchafe for any number
of years.
<
n •
I
2
3
4
5
6
7
8
9
lo
II
12
»3
>4
Ic
ID
>7
i8
^9
20
21
22
»3
24
^5
26
27
2S
29
30
3»
3*
33
34
35
36
37
38
39
40
3 per cent.
1 .030C000
•5226108
•3535304
.2690271
.2183546
.'^45975
.1605064
.1424564
.1284339
.1172305
.1080775
.1004621
.0940295
.0885263
.0837666
.07961c 9
.0759525
.0727087
.0698139
.0672157
.0648718
.0627*74
0608 1 39
.0590474
.0574279
.0559383
.0545642
.0532932
.0521147
.0510193
.04999S9
.0490466
.04S1561
.0473220
.0465193
.0458038
.0451116
.0444593
.0438439
.0432624.
1 .
3 1 per ct. 4 per cent. I 4^ per ct
1.W350C00
.5264005
.3569342
.27S2JII
.2214014
.1876682
.1635445
.1454767
.1314460
.1202414
.1110920
. 1034840
.C970616
.0915707
.0868251
.0826848
.0790431
.0758168
.0729403
.0703611
.06S0366
.0659321
.0640188
.0622728
.0606740
.0592 54
.0578524
.0566027
•05^4454
.0543713
.0533724
.0524415
.0515724
•0507597
.0499984
.0492842
.0486133
.0479821
.0473878
.C468273 I
1.0400000
.5301961
.3603485
.2754901
.2246271
.1907619
.1666096
.1485279
.1344930
.1232909
. I 141490
.1065522
.1001437
.0946690
.0899411
.0858200
.0821985
.0789933
.0761386
.0735818
.0712801
•0691988
.0673091
.0655868
.0640120
.0625674
.0612385
.0600130
.0588799
.0578301
.0568554
.0559486
.0551036
.0543148
.0535773
.0528869
.0522396
.0516319
.0510608
.0505235
1.0450000
•5339976
•3637734
.2787437
.2277916
.1938784
.1697015
.1516097
.'375745
•1263788
.1172482
. 1096662
.1032754
.0978203
.0931138
.08901 54
.0854176
.0822369
.0794073
.0768761
.0746006
.0725457
.0706825
.0689870
.0674390
.0660214
.0647195
.0635208
.0624146
.C613915
.0604435
.0595632
.0587445
.0579819
.05-72705
.0566058
.0559*40
.0554017
.0548557
.0543431
5 per cent.
1.0500000
.5378049
. 3672086
.2820118
.2309748
.1970175
.1728198
.1547218
.1406901
.1295046
.1203889
. I 128254
.1064558
.1010240
.C963423
.0922699
.0886991
.0855462
.0827450
.0802426
.0779961
•0759705
.0741368
.07247C9
.0709525
.0695643
.0682919
.0671225
.0660455
.0650514
.0641321
.0632804
.0624900
.0617554
.0610717
.0604345
.0598398
.059284:
.0587646
I .0581782
CASE
Chap. VI. ' Compound Intehest^ j|4^
CASE I.
Principal, rate, and time being given, to find the annuity^
RULE.
Multiply the annuity which iL will purchafc, flt 4h^
Irate and for the time given (found in the fourth table) and
the quotient will be the anfwer.
1. A gentleman hath 1760 1. which he would fell for an
annuity, to continue 21 years, at 5 per cent, compound in-
tereft ; I demand what will be his income per annum ?
1760 1. X .0779961 = 137.273136;
Anfwer, 137I. 5; s. 5^d.
2. A fine for the leafe of a tenement is fettled at 153!.
tinder a referved rent of 16 1. a year : now the tenant cannot
conveniently pay more than 50 1. but for twefve years to
come of the term is willing rather to pay an adequate rent<
computing 5 per cent, compound intercft ; what ought that
rent to be i
Firft, 157 — ^0 = 103.
Then, by the fourth table, 1 1. will purchafe for 12 years,
at 5 per cent, an annuity of .1128254 per annum.
Then 103 X .1128254'= 11.6210162 = III. 12 s. 5d.
advance rent.
• . • 16I. + II 1. i2s. 5d. =1 27 1. i2s. 5d. Q, E. Fi
3. A fon, previous to his marriage, is minded to have
50 1. a year, freehold eftate, fettled on his fi^miiy ; and to
have immediate pofleifion of it, offers his father in lieu, atl
annuity for his life, valued at twelve years purchafe, dif-
counting 4 per cent, thereon ; whereas he is content th^
cftate (hould be valued at a difcount of 3 per cent, which
is 33-j. years purchafe j pray what had the father for his life ?
Firft, J3,j X SO = 1666.^ = i6661. 13 s. 4d. value of
the annuity.
Then 1 1. by the fourth table, for 12 years, at 4 per cent. ^
will purchafe .1065522 per annum.
•*' 1666.J? X. 1065522 =: 177.58699= 177 1. lis. B^d.
N ft 2 C A S fi
5.4S CoM^ouKD Interest. Book IIL
C A S E IL
Principal, annuity, and rate given, to find the time*
RULE.
Divide the annufty by the principal, and the quotient
will be the annuity which 1 1. will purchafe at the given
rate, which will be found in the fifth table under that rate,
and even with the tim^ required.
4. If an annuity of 137 1. 5 s. 5id. is purchafed for 1760!.
at 5 per cent, compound interefl, what time ought it to
continue?
1760) 137-^73^36 (.0779961;
which under 5 per cent, in the 4tn table> i$ oppofite to
21 years.
CASE III.
Principal, annuity, and time given, to find the rate.
RULE.
Divide the annuity by the principal, and the quotient
will be the annuity which 1 1. will purchafe for the given
time, which will ftand even with the time, and under the
rate required.
5. If an annuity of 8.0 1. 4 s. 10^ d. to continue 20 years,
be purchafed for loooh what rate of tntereft hath the pur-
chafer for his money ?
1000) 80.2425 (.0802425, under 5 per cent, which is the
anfwer.
CASE IV.
Annuity, rate^ and time given, to find the amount.
RULE.
Find the amount of 1 1. per annum, at the rate for the
time given, by the firft table ; by which multiply the an-
nuity, and the produ(5l will be the amount required.
6. A minor of 14 had an annuity left him of 70I. a year,
the proceed of which, by will, was to be put out, both prin-
cipal apd intercft, yearly, as it fell cjue, at 5 percent, till
he
.Cftap. in. Compound Interest. 549
he fhould attain to 21 ^ears of age ^ the utmoft improve-
ment being made of this part of his fortune, what had he
then to receive i
The amount of 1 1. annuity, at 5 per cent, forborn feven
years, by the fecond table, is 8.1420084. '^
Then 8.1420084 x 70 = 569.940588.
Anfwrer, 569 1. 18 s. 9^d.
CASE V.
*
Annuity, rate,- and amount, being given, to find the
time.
RULE.
Divide the amount by the annuity, and the quotient will
be the amount of 1 1. at the given rate, which will be found
in the fecond table, under that rate, even with the time
required.
#
7. In what time will an annuity of 70 1. amount to
569 1. 1 8 s. 9|d. compound intereft, at 5 per cent. ?
70) 560.940588 (8.1420084, even with feven years in the
(econd taole, under 5 per cent,
CASE VI.
.Annuity^ time, and amount given, to find the rate*
RULE.
Divide the amount by the annuity, the quotient will be
the amount of 1 1. per annum, for the given rate ; which
will be found in the fecond table, below the required rate.
S. At what rat6 per cent, per annum will an annuity of
137 1. 5 s. 54- d. amount to 1760 1. in 21 years.
^37'^73^36) 1760.000000(12.8211523, in the fecond
table, below 5 per cent.
CASE vn.
Amount, rate, and time being given, to find the annuity.
RULE.
Divide the amount given, by the amount of 1 1. found in
the fecond table^ at the rate and time given, the quotient
will be the annuity required*
N n 3 9. What
^50 Compound Intirest. "Book III.
9. What annuity will amount; to 569 1. 18 s. gjd. in
Jcvcn years, at 5 per cent. ?
8.1420084) 569.940588 (70 1, the annuity required.
CASE VIIL
Annuity, time in reyerfion, and rate bring given, to
|if'>d the prefcnt worth.
RULE.
In the third table find the prefent value of 1 1. per annum,
^t the given rate, both for the time being, and alfo for that
^nd the time in reverfion added together, then fubtrad the
time in being from the other, and multiply the remainder
by the annuity, the product will anfwer the queflion.
10. What ought a man to give down in ready money,
for the reverfion of looo 1. a year, to continue 20 years, on a
icafe which cannot commence till five years are at an end,
allowing the purchafcr compound intereft at 5 per cent ?
The prefent value of il. per annum, by table 3d, for 25
years, I4-093944S
fpr fivp ye^rs, 4.3294767
9.7644678 X 1000 = 9764.4678.
Anfwer, 9764 1. 9 s. 4^d.
' II. Suppofe I would add five years to a running Icafe of
15 years to come, the improved rent being 186I. 7s. 6d.
per annum ; what ought I to pay down for this favour, dif-
counting 4 per cent, compound intereft ?
Fifftj ?5 + 5 == ^^ years, 1 1. is worth - 13-5903253
Alfo 15 years is worth .--•-• 11.1183868
^.47 19385
Then 186.375 X 2.4719385 = 460.70753.
•.•460 1. 14s. i^d. the fine required.
12. Held of a college 486 1. los. a year, on a refervcd
rent of Q4I. money being at 5 per cent, intereft.; what fine
ought feverally to be paid on a 7, ar 14, and a 21 year§
icafe ?
486 1. los, -^ 94 = 392I. IDS. annuity,
^'he prefent worth of 1 1, for the time an4 rate is 5.7S63734.
2 Ihen
Chap. VI. Compound Interest. 551
Then 392.5 X 5-7863734 = 2271. 15057.
%• 2271 1. 3 s, its worth for fcven years.
A!fo.the prcfcnt worth of 1 1. for 14 years, at 5 per cent.
is 9.8986409.
Again, 392.5 X 9.8986409 = 3885.21655,
• . • 3885 I. 4 s. 4d, its worth for 14 years.
The prefent worth of iL for 21 ys. at5 pcrct. is 12.8211527.
Alfo 392.5. X 12.821 1527 = 5032.30243.
Anfwer, 5032 1. 6 s. for 21 years.
C A S E IX.
An annuity, feveral times in reverflon, and rate given,
to find the prefent value.
RULE-
In the third table find the prefent value of 1 1. per an-
num, at the given rate, for the feveral given times, which
being feverally multiplied by the annuity, the produfts will
be the feveral prefent values of that annuity for the feveral
times given: then fubtrad the feveral prefent values one
from another, and the feveral remainders anfwer the 'quef-
tion.
1 3. A has a term of feven years in an eftate of 50 1. per
annum ; B hath a term of 14 years in the fame eftate ;
and C hath a further term of 10 years afterB in the fame
eftate; what is th6 prefent value of their feveral interefts
in the faid eftate ?
Firft, 7 4. 14 + 10 = 31.
The prefent worth of il, at 5 per ct. for 31 years, is 15.5928104
For 21 years --.-«---- 12.8211527
And 7 years --r 5-786373+
1. s. d.
Then 50 X 15.5928104 =: 779.64052 = 779 12 94.
Alfo 50 X 12.8211527 = 641.057635 =• 641 I i|
And 50 X 5-7863734 = 289.31867 3=289 6 4^
1. s. d. ■)
• . • 289 6 4i I "S
1. s. d. 1. s. d. ' >-3 -< y
Alfo 641 I i| — 289 6 4i = 3Si H 9l ^ I ^'^ '
And 779 12 9'- — 641 I 11=138 II 7iJ Lt^'^ J
14. Which is moft advantageous, a term of 19 years of
an eftate of ico per annum, or the revcrfion of fuch an
N n 4. cfiatc
1
A's j
g
552 Compound Interest. Book III»
^ftate-fer ever^ at the expiration of the faid 19 years,
compating at the rate of 4 per cent, compound intereft ?
Fir{^, 4 : 100 ; ; 100 : 2500 1. value of the eftate for
ever.
And by the third table, the pre^ -j 1. s. d-
fcntworthof lool. annuity > 1313.3938=1 1313 7 lo^
for 1 9 years, at 4 per gent, is 3 »
Value of the reverfion is - - 1186.60621^1186 12 ij-
The firft 1 9 years better than the reverfion by ^^ 126 1 5 9
1 5. For a leafe of certain profits for feven years> A offerf
to pay 150 1 . gratuity, and 300 1. per annum ; B offers 400L
gratuity, and 250 1, per annum; C bids 650 1. gratuity,
and ^06 1. pipr annum } and D offers 1800I. for the whole
purchalb, without any yearly rent : query, which is the
beft offer, and what the difference, computing at 4 per cent, f
By the third table, the prefent worth of 300 1. 1
per annum, for feven years, at 4 per cent. > 1800,61641.
viz. 6.002547 X 300 is - ,-...,..,. J
I. s. d.
Then 1800.61641 + 150 =51950 12 4, val. of A'soffer.
Alio 6.0020547x250 4- 400 =1900 10 3^, val. of B's offer.
Again, 6.0020547X200+650= 1 850 8 2t, val . of C's offer.
1800 value of D's offer.
Hence it appears, that A's offer U better f ^^ 7 ^i,_ \ r''**
by above il^oj Id^sI
SECT, V.
The Vaj-Cation 0/ Annuities u^on Lives.
THE value of an annuity for life, depends not only
on the intereft that mooey bears, but alfo on the pro*
bability of the continuance of life, as it is evident that there
n-ult be a groat difference in the value of an annuity for
t he life of a man of 20, and a like annuity for the life of
4 mm of 6p»
Chap. VI. Compound IntSrest. 'S5i
The late Mr* Demoivre and Mr. Simpfon have both
bandied this fubjed in a very (kilfuf manner ; from the latter^
of which I have extracted the following tables and problems,
whereby an annuity on any life or lives may be valued ac-
cording to the probability of the continuation thereof.
The u/e of thi Table of Lives*
If it was required the probability that a perfon of 36
lives 30 years longer :
Look in the tabic againft 36 years, and oppofite theretQ
is the number 331.
Alfo againft 66 is the number 93, which (hews, that out
of 331 perfons living of 36, only 93 arrived at 66.
• . • -^ is the meafure of the probability required.
Let it l^ required to find the value of an annuity of 100 !•
for a life of 20, intereft at 4 per cent.
By the fecond table in the foregoing fefiion, the prefent
worth of 100 1. difcount 4 per cent, due at the expiration
of one year, was it fure to be paid, is 96.15365.
But the probability of the continuance of the faid life
one year, by this table, appears to be only |4t-
• ' ' 96*15385 X Tffi- = 94-697> ^c vJue of the firft
year's rent.
In like manner the value of the fecond year's rent may
be calculated s the probability of his living two years 19
ti. = ^; and 92.45562^ the prefent worth of 100 1, at the
4^2 33
end of two years.
••• 92.45562 X — = 89.65393, value of the fecond year's
rent.
And bv a like way of proceeding, the values of the third,
fourth, nfthf &c. years rents, to the utmoft extent of life,
may be determined $ and the fum of all thefe will be the
required value of the annuity 5 which will be found to
i:Qme out 1480 1, very near.
jl Table
/
[ 554 ]
'^ Tab LB fiewing tie Probabilities of Lit n, 6fr.
Mum. of
Ago
Num* of
AgM
Num. of
Ages
Nam. of
Ages
' pCTrofit*
eurr.
born
peribni
curr.
20
^erfons.
curr.
40
perfons.
curr.
60
1280
462
294
130
—410
"~ 7
— 10
— 7
870
I
455
ai
284
4»
123
61
— 170
— 7
t
— 10
— 6
700
a
448
22
274
42
117
62
-6s
— 7
— 10
— 6
. 635
3
441
23
264
43
III
63
— 35
-~ 7
— 9
— 6
600
4
434
24
255
44
105
64
— 20
— 8
— 9
— 6
5»o
5
426
25
246
45
99
65
— 16
— 8
— 9
— 6
564
6
418
26
237
46
93
66
— 13
— 8
— 9
— 6
55 »
7
410
27
228
47
H
67
— JO
— 8
— 8
— 6 1
54»
8
402
28
220
48
81 68 1
— 9
— 8
— 8
— 6
532
9
394
29
212
49
75
69
.-r- .8
■- 9
— 8
— ■ 6
524
10
385
30
204
50
69
70
— 7
— 9
— 8
— 5
5^7
II
376
3'
196
51
64
7>
~ 7
^ 9
— 8
— 5
. 5»o
12
367
32
188
52
59
72
— 6
— Q
— 8
— 5
504
13
358
— 9
349
33
180
8
53
54
— 5
49
73
498
14
34
- V
172
•54
74
— 6
1 '
— 9
— J
•— 4
492
»5
340
35
165
55
45
75
— "6
— 9
— 7
'
— 4
486
16
33»
36
»58
56
41
76
— 6
~ 9
— 7
— 3
480
17
322
37
15^
57
38
77
-^ 6
— 9
— 7
— J
474
18
313
38
144
58
35
78
— 6
— 9
— 7
— 3
*
468
19
304
39
137
59
32
79
~ 6
— 10
— 7
— 3
462
20 1 294
4^
130 1 60
29 80 1
N. B. Thofe marked with the lign — are fuppofed 10 die
off yearly. Problem
Clu^«VI. Compound Interest. g^g
Problem !•
To find the value of an annuity for an affigned life.
R t L E.
Look for the given ?ige in Table I. and againft it, under
the affigned rate of intereft, 'will ftand the number qf yean
*purchafe.
1. Suppofe one of i8 years of age would fell an annuity
of lool. during his life, what ready money would the
annuity be worth, allowing a difcount of 4 per cent, com-
pound intereft ?
Firft, oppofite to 18 years. Table I. under 4 per cent, is
15.2 years purchafc. ••• 100 X iS-2 = 1520!. the prefent
worth,
2. A widow lady with 200 1. a year jointure, aged ^o
years, marries a young merchant, who, to enlarge his ca-
.pital, propofes to fell the jointure ; what ready money
ihould he receive, difcounting intereft at 5 per cent. ?
Oppofite to 30, under 5 per cent. Table I. is 11.6.
••• 200 X 1 1*6 = 23201. the anfwer required.
Problem II.
To find the value of an annuity upon two affigned joim
lives.
C A S E I.
If the lives are equal.
RULE.
Againft the given age. Table II, under the given rate
per cent, will ftand the number of years purchafe.
3* Let the two given ages be each 18, and the intereft
5 1. per cent, and annuity 50 1.
Table II. againft 18, under 5 per cent, is 10.5.
• . • 50 X 10.5 = 525 1. the anfv/er recjuired,
C A S E II.
If the given ages be unequal, but neither of them Icfs
than 25^ or greater ;han 50. RULE.
55^ CoMPOUKD Interbst. . BocdcIIL
RULE,
Take half the fum of the two for a mean age, and
luroceed as iii Cafe I.
4. There are two joint lives upon an annuity of 250 1,
one of 34, the other of 48 ; whac is the prefent worth of
that annuity, compound intereft, at 3 per cent. ! .
Firft, 5 i- 1= 41, half fum of the ages.
2
Table 11. againft 41, under 3 per cent, is 8.9 jrears
pnrcfaafe,
* • • 250 1. X 8.9 == 2225 1. the anfwen
CASE III.
If one or both ages be within the limits, but fo that the
difference of the values correfponding to thofe ages be not
more than j- of the lefier.
RULE.
Add ^ of that difference to the faid lefler value, and the
ium will be the value fought*
5. Let one age be 15, and the other 29, annuity 150I.
intereft 3 per cent, the prefent value is required.
Againft 15, under 3 percent, per Table IL is 13.9
And againft 29 .------.-- n.o
. Difference 2.9
Alfo 2.9 X ^s: I*l6 and zi -f* ^*'^ = I2*i6, the years
purchafe«
••• 12.16 X 150 = 1824.
jf general Rule, be the differemt of the values what they will.
Multiply the difference of the values byhalf of theleiferof
the two values, and divide the produA by the greater ; then
to the lefier add the quotient, which will give the true
anfwer very near.
6. Let one age be 1 1 years, and the other 68, annuity
l6ol. and intereft at 4 per cent, the prefent value is required ?
Againft ii years, under 4 per cent, is 12.9
Alfo againft 68 ------ 4.6-7-2 = 2.3
Difference 8.3
Thw 8.3 X tf3 = 1909 ; alfo 19.09 -f- 12.9 == 1.48.
And 4.6 -{- 1.48 = 6.08 years purchaie.
«•• 160 X 6t08 = 9721. j6$« the anfwer required.
.4 Probleiia
Chap^VI. Compound iMTERfisf. 557
Problem IIL
To find the value of an annuity upon two lives that is to
continue as long as either of them is in being*
C A S E I. .
If the lives be equal*
RULE.
Find, the given age ip Table III. and againft it, under
the- propofed rate of intereft^ is the nomber of y^'ars
purchafe.
7. Let the given ages be each' 50 years, and the rate of
intereft 4 per cent, required the value of an annuity of 30 1.?
In Table III. againft 5c, under 4 per cent, is 13.3
years purchafe.
* • * 30 X 13.3 =: 399) the value required.
C A S E IL
If both ages be between 25 and 50.
RULE.
Take half their fum for a mean age, which proceed
with as in the laft cafe.
8. Suppofe one age to be 30 years, and the other 46,
rate 3 per cent, and annuity 70 1. required the prefe;it value i
76
Then 46 -f* 3^ = 1^^ ^"^ = 38> half their fum.
Anfwering to which, .under 3 per cent, ftands 17.7 years
purchafe.
••• £^o '\' 17.7 = 1239, the anfwer.
CASE III.
If one or both ages be without the limits mentioned in
the laft cafe, but the difference of the values correfpond-
ing to thofe ages, as found in Table III. be no more
than 7 of the lefTer,
RULE,
§$$ CoiiPOUND Ikter£St« jBook Ilt^
RULE.
Take half the fum of thofe values for the value required.
o. If the two propofed ages be 6 and 21 years, the an-
nuity 25 1. and interetl 4 per cent* its prefent value is '
jrequired ?
Againft 6 years is - 19.7
And againft 21 years is 18.2
2) 37.9 (18.9^ years purchafe.
••• 25 X 18.95 = 473-75 = 473^- ^S®? *« anfwer.
C A S E IV.
Let the given ages be what they will.
RULE.
Find the value of the two joint lives, by Cafe IV.
Prob. II. which fubtrad from the fum of the values of
the two (ingle lives, and the remainder will be the re-
quired value upon the longeft life.
10. Let the propofed ages be 10 and 66, the rate of in-
tereft 4 per cent, and the annuity 70 1. required its prefent
worth ?
Table II. againft 10 years, ^^^^^7*^^
4 per cent, is - - - - - - j^3*"
Alfo againft 66 ------ 4.9-^^2=2.45
8.1 difference.
Then 8.1 X 2.45 = 19.845; which -s- 13 = 1.5; alfo
i-5 + 4-9 = 6.4.
Againft the two fingle lives, per Table I. vis. 16.4 •{-
7-3 = ^3*7-
Laftly, 2^.7 *— 6.4 = 17.3 years purchafe.
- . • 701. X I7'3 = I2tii 1. the anfwer required.
Problem IV.
To find the value of an annuity upon three joint lives.
CASE I.
If al] the lives be equal.
R U !.£<
Chap. VI. CoMPouKD Interest. £$^
RULE.
«
Find out the given age in Table IV. and againft it, under
the propofed rate of intereft, will be the number of fcaas
purchafe.
II. Let each age be 27, the mte of intereft 3 per cent,
and the annuity 65 1. its value is required ?
Table IV. againft 27^ under 3 per cent is 8.8 years
purcbafe.
* • * 65 X 8.8 r= 572 1. the anfwer required*
C A S E II.
If all the three ages be between 15 and 55 years, and the
diiFerence between the greateft and ieaft not more thaa
I c years.
RULE.
Take 4. of their fuin for the mean age, and proceed as
in Cafe I.
' 12. Let the propofed -ages be 21, 27, and 33, intcreft
5 per cent, and annuity 50 1. its value is required i
r irit, ■ zz: 27, mean age.
Alfo iTable Iv. againft 27 years, under 5 percent, is 7.3
years purchafe.
' • ' 73 X $0 = 365 1. the anfwer required.
C A ^ E III.
If one or more of the propofed ages be without the
limits mentioned in Cafe II. but the diiFerence of the va-
lues, anfwering to the greateft and Ieaft of them, be not
greater than half the Ieaft.
RULE.
To the fum of the two greateft values add twice the
Ieaft, and take ^ of the fum for the mean value required,
r
13. Let the three ages be 7, 15, and 33, the annuity 50I.
and intereft 3 per cent, the prefent value is required ?
I«
's6o CdMPOuKb luttnnr. Book III.
In Table IV. agaiflft { 15 f ftands J 11.2.
133^ t 7-9-
Alfo 11.9 + 11-2 = 23* J and 79 X a = 15. 8/
Then 23.1 + 15.8 = 38-9 J and 1^2=9.725 years ral.
•. • 50 X 9-725 = 486.25, the value required,
CASE IV.
Let the ages be what they *ill.
RULE.
Multiply the fum of the three corrcfponding values by
the fquare of the leaft of them, leferving the produft j mul-
tiol? the two greater values into each other, and to th*
double of the produa add the fquare of the leffer v^uc;
divide the referved produd by this fum, and fubtraS the
fluotient from twice the leffer value, the difference will b«
tine value fought.
14. Let three ages be 13, 31^, and 53 years, annuity
60 1. and intereft 4 pet cent.
Againft thefc in Table IV. j 31^ J ftand | 7.3.
Then ics + 7.3 + 5° = ".8 ; alfo c X 5 = »5'
Which 22.8 X 25 = 570» ^° be referved.
Again, 10.5 X 7-3 = 76-65 5 which x 2 = 'S3-3-
Alfo 153-3 + 25 = '78-3) 576 (3-2. nearly.
Then 5 X 2 c= 10, double the leaft value.
Laftly, 10 — 32 = 6.8 years purchafe.
• . • 60 X 6.8 = 408 1. the value required.
Problem. V.
To find the value of an annuity upon the longeft of thre«
lives. '
CASE L/
If the lives be all equal.
RULE.
' Seek the common age in Table V. and againft if, <irtd«
the coinmoir rate of intereft, will be the number of years
purehafe required. j^^
Chip- VL CoMpdu^TD ItttiiLitsv: s6t
15. Let the three ages be each 45 years, the annuity
275 1. and the intereft 4 per cent, required its value ?
In Table V. againft 45, Under 4 per cent, ftahds i5«^
years purchafe.
275 1. X 15.9 =: 4372!. los. the value required*
• . •
CASE II.
If none of the ages be lefs than io> nor greater than 60
years, and the difference of the greateft and kaft of them
Hot more than 15 years,
R U L £,
To twice the fum of the two leaft add the greateft, and
take y of the fum for a mean age.
16. Let the propofed ages be 16, 24, and 30 years, tho.
annuity 1701. and intereft 4 per cent, the vilue is re-*
quired ?
Firft, j6 + 24 X 2 =: 80 } alfo 80 + ^<> = "^*
Then — =:.22, mean age, againft which, Table Y. .is
19.4 years purchafe.
••• 170 X 19.4 = 32981. the value fought. '
Q h S E IIL
If the difference of the greateft ^nd leafl values found
againft the propofed ages, in Table V. be no more than \
of the leaft,
RULE,
To twice the Aim of the two greateft values add the leaft,
taking ^ of the fum for a mean value.
17. Suppofe the three ages be 28, 35, and 44, the rate
4 per cent, and the annuity 60 1.
By Table V. the value of the three ^^^^ f '^'3'
»«««» ^'=^; t44
Then 18.3 -|- 17.3X2=71.2; alfo 71.2+ 16= 87.2.
And 5) 87.2 (17.44 years purchafe.
* .' 17.44 X 60 =1046.4 = 1046 1, 8 $. the anfwer.
C287 n8-3»
Oo c A s e
V
I
CASE IV.
Let the given ages be what thej wiU.
RULE.
Find the value anCwering to the greateft of the riven
ages in Table lU. Ind.the values correipondtng to all the
three fevepal ages in Table V. and let this dif&rence of the
values, anfweruig to the greateft age, be taken and refekved^
}et the fquafe of the greater of thefe two be divided by the
f rodud of the other two lemaining values, and mtrltiply the'
figuare of the quotient by the leierved difference; then this
laft produd added to tne vafne of the annuity for the two
youngeft lives, will be the value required^
18. Suppofe the given a^es 20, 369 and 60, the mtttrtt
4 per cent, and annuity 751. the prefeiK vakie k r^^ired?
By Table IIL the value found againft 60 year» is i r.2«
By Table V. thofe agathft<- 36 > aite -j r7.2>
160 J C12.7.
Then ia.7 — 11.2 = r.5, the referved diiFerence.
Again, 12.7 X 22.7 •=, 161.29$ aUb i9«7Xf7>xs&
338.84.
Then 338.84) 161.290 (.5 nearly; and .5 X -S = JtS*
And the difference referved 1.5 x -'^'S ^ '375» nearly .4.
20 *4*^ 46
Alfo ■— • ■? = 2t, mean age,, by Cafe 4. Probleni JSL
z
die value of which, by Table III. is 16.9) or nearly ij
years.
And 17 -{- .4 s 17.4 years purchafe.
• »» I7*4X 75913051. vdue of die anltticy MfedtaL
Tabu
t S6S ]
Ta4Ls t' For tht valtiBtitn oftnuuiiies upoHone Kfi.
Is
u
11
f i
it.
ti
k
14. 1
sh
1^
11.4
ii
%
^
£a
T'
IO.Z
'30
I
14.1
18J
'iS-i
42
10. 1
li.2
12.8
.4.3
1«.4
19.0
43
10.0
11. 1
12.6
i
143
19.0
44
1?
tl.o
12.5
to
ill.
12£
i|
10.8
ii3
II
'♦•J
10.0
i§,9
46
9.7
10.7
12.1
1ft
14.4
tl
9.5
10.5
11.9
■J
14- 1
■ 8,7
9'4
10.4
ii4i
M
14.0
1S.5
49
9-3
10.2
11.6
11
IIJ
■ 4^
5°
9.2
10. r
■ '4
>6
'3-7
1 18.1
3'
8.1
9.9
H.2
\l
■3-5
17.9
52
9.8
1 1.0
■3-4
17.0
53
96
10,7
'9
■3.1
17.4
54
8.6
94
JO.5
ao
lis
, ,j^
55
±i
-W
.■°.3
If
ll»
1 17.0
56
8.4
ti
lO.I
la
12.7
16.8
57
8.2
9.9
»}
1X6
.6.5
58
8.1
l-I
9.6
44
• 2-4
16.3
e
80
8.6
9.4
iJ
■iJ
16. 1
79
84
_H
lt>
ii-i
■5?
15.6
61
7-7
8.2
89
'f
iz.a
(2
7.6
8.1
8.7
a
II.S
15.4
J3
7-4
7.9
1-5
»9
1 1.7
IS.2
!+
73
7.7
8.3
3?
11.6
ili
Is
7.1
7.5
8.0
3'
11.4
148
66
6.9
7.3
7.8
31
11.3
14-6
ii
6.7
1'
7.6
33
li.S
144
68
6.6
6.7
7.4
34.
if.o
142
69
6.4
6.7
7.»
T0.9
14.1
70
6.2
±i
•6.9
36
10.8
■3.9
71
6.0
^3
6.7
P
10.6
'3.7
72
5.8
6.1
6:5
10.5
13.5
73
5-6
5.9
6.2
39
10.4
■3.3 ■
74
54
5.6
59
40
.0.3
,. 13.2
75,
5.»
5.4
5.6
[ 5^4 ]
Tablb n. For the vabunicn 9/ annukrei upon two joint I!ae£^
m
'
wi *i c: • .
v^ ^
-
•>a .
V-" .
>-• -""I
(« ^ M trf 1
rt t:
•
M s
rt *f -
'« *: 1
•
to
ca
c4
9 i«
3 u
D »«
*
a k
3 i?
S «.
B
V) flu
^s.
c
£-8.
^£"JL
S-8.
CO
is ^
i! •«
;[3 -*-
:i m
s
c ^
JU ^
U «rf
W •'
0 «<
.«/ «<
>• ce
t5- ««
>^ «
>■ ««
>. *
>> ••
76
11.3 1^-7
1 1.5 12.9
■ 14«4
41
?•*
S.Q
y.9
7
14.6
42
7-1
7.8
8.7
^
11.6
13.0
14.7
43.
7.0
7-7
8.6
•
9
II.-6
13.0
14.7
44
6.9
7.6
8.5
8.3
82
1
10
.11
1 1.6
1 1.5
13*0
12.9
14.7
14.6
45
46
6.7,
6.6
7-4
7-J
\
12
11.4
12^
14-5
^2
6.5
7.2
8.1
•
13
n.a
I2.Z
14-3
48
64
7.1
7?
14
II. 2
12.5
14.1
49
6.J
7.0
7.8
15
If.O
12.3
139
59
.6»a
6.8
7.6
1
16
xo.U
12,1
^37
5*
6.1
^l
7-4
\i
10.7
II.9
*a*5
52
6.a
6.6
7-3
10.5
11.7
13.2
53
59
^*
7-a
*
'9
10.3
11.5
130
54
5-8
6,3
7.0
23
10.1
i'-3
12.8
55
5^7
6w2
6.9
2f
io*.o
J1.2
126
56
5.6
6.x
6.7 ;
22
9-8.
II.jO
12.4
57
5-5
6.0
6.6
*3
9-7
10.8
S2.2
58
5-4
5-8
6.4 '
24
9-S
10.6
12 0
5*
51
S-7
6.3
as
>94
10.5
I 1.8
60
5,2
S.6
6.1
26
9.2
X0.3
1 1.6
6i
5^1
55
6.0
27
9.1
lO.I
11.+
62
5.0
5-4
Sr9
28
89
9-9
11.2
63
4.a
4.8
5-3
57
29
8-8
9-8
II. Q
64
5«
5-5
♦ ^
30
8.6
9>6
10.»
65
4.7
5.a
54
31
8.5
9-4
ia6
66
4.6
4-9
5-3
32
^•3
9.2
10.4
67
4 5
4.8
5«
33
8.2
g.r
10.2
68
44
4.6
4,8
1
34
8.1
8.9
1,0. a
69
43
4-S
3S
8.0
8.8
9.9
70
_+.?
4 4
4.6
3<i
7.8
■ 86
9'7
71
"4.1
4-3
4-5
37
7^
8.4
9^5
72
39
4-»
4-3
38
7-5
8.3.
9^
73
38
4.0
4.2
39 .
7-4
8.2
9.2
■ 74
37
3-8
*o
4a. 7-1 I ».i
9 '
75 ' 3-6
J-7
381
TAauR
Tablk III. Fcr the valuation of annuitus upon the bngefi (f
two lives.
<^ «: t '■i >i 1
>- «j
•
^ a
S a
IS a
9
3 e
C< C
&
•§8
■§8
•?8
-ss
•§8
w
a h
S »«
s h
Ctf
9 ^
3 t:
3 h
9
S-Ji,
S-Si.
^S.
■ 2"^
2
sr
S*"
> ;$
>• ti
;Ss
> tJ
>• t«
"6
16.9
19.7
19.8
23-3
41
13.2
14.9
17.0
5
17.0
23-4
42
»3»
147
16.8
17.1
199
23-5
43
13.0
14.5
16.5
9
17.1
19.9
*3-5
44^
;i?
14-3
16.3
10
17.1
I9-?
235
45 !
14.2
16.1
"
17.1
19.9
23-5
46^
12.6
14.0
15.8
12
17.0
19.8
23-4
%
12.5
13-8
15.6
»3
19.7
23-3
12.4
13.6
»5-3
«4
'H
19.5
23.1
49
12.2
134
151
15
16.6
J 9- 3
22.9
50
12.1
^Z'l
• H-9
i6
1^.4
19.1
-22.6
5«
II.O
131
14.6
\l
16.2
18.9
22.4
52
II. 8
12.9
14.4
i(j.i
18.7
22.1
53
11.6
12.7
14.1
»9
15.9
18.5
-21.9
54
11.5
12.5
139
20
15-7
;8.3
21.6
55
"•3
12.3
13.6
21
IS-6
18.2
21.3
56
11,2
12.1
13-4
22
15.4
18.0
2I.I
57
II.O
1 1.9
'^i
23
153
17.8
2a8
58
10.9
11.7
12.8
24
15.1
17.6
2d.6
59
10.7
11.5
12.5
as
15.0
»7-4
20.3
60
10.5
1 1.2
12.2
26
14.9
»7-3
20.1
61
10.3
II.O
12.0
;i
14.7
17.1
19.9
62
10. 1
108
1 1.7
14.6
16.9
19.7
63
9.9
lo-s
11:4
29
14.5
16.8
19.5
64
9-7
10.3-
II. I
30
14.4
166
19-3
65
-9.4
lO.O
■ 10.8
3«
14.2
16.4
19.1
66
9.2
9-7
10.5
3*
14.1
16.2
18.9
\l
8.9
9-4
I0.2
33
14.0
16.1
18.7
68
i7
92
9 9
34
139
15.0
.15-8
18.5
69
8.5
8.9
95
35
138
18.3
70
8.2
8.6
92
36
»37
15.6
j8.i
71
8.0
8.4
8.9
37
13.6
>S-5
17.9
72
7-7
8.1
8.6
38
.'3-5
»5-3
'7-7
73
7-5
7.8
8.2
1 39
»3-4
15.2
175
•74
7.2
7-5
7-9
1 40
»33
15.0 17.3 1
75
6.9
7.2
7.5,
Oo
Table
[ 5«« 1
t tV. farihevahtatim efannuiiiti uffH ihret jntO Hoer.
1
6
ii
9-7
I
JO-O
Ii
J»-7
1
41
5-5
1
it
7
9-9
10.8
11.9
4^
S-4
6.0
5'
8
lO.O
10.9
1Z.0
43
54
i?
t-J
9
10.P
10.9
11.0
44
5-3
6.4
lo
11
10.0
9-9
HI
JI.O
±5
46
5"
ii
^.
»2
9f
10.7
%
i'>
35
».«
13
9.6
lO.s
11.6
5.0
5-4
ii
H
95
10.4
11.4
49
4-7
4.7
5-3
9.2
?'2
10.2
10.0
9.8
11.2.
n.o
lo.S
J2
J'
1'
1
8.8
9.6
10.6
{3
4.6
J"
it
19
8.6
94
10.4
»4
4.5
Jl
S3
10
8.4
-2i
10.2
JS
4-4
-ti
21
8.2
|:°
100
fi6
4.4
*l
J»
?2
8.1
9.8
57
4-3
4.6
!■<>
23
7-9
S-7
5.6
S8
4-2
45
tl
2+
'•I
5-s
9*
1?
4.1
4-4
25
7-6
_y
9.2
4.0
.iJ
-t!
26
7-t
8.J
u
61
3-9
4-»
♦-S
%
7-3
60
62
3-8
4-1
4-4
7.I
7.8
8.6
'3
3-7
4.0
4-3
29
'•S
7-7
8.5
64
3-7
S-7
4-a
3?
31
6.8
6.7
7*
41
is.
66
3-5
4.1
IS
32
6.5
7'2
8.0
67
34
S-6
33
6.4
?■'
7-9
68
33
3-S
'!
3+
6.2
t.§
'•I
69
32
3*
3.6
2^
6. J
_76
70
JJ
-H
-3-4
36
6.0
^'
7'4
7'
3.0
3'
8-3
H
^1
65
7-^
7»
il
3-0
S-J
38
5-8
6.4
!■'
73
2.9
S-o
39
'•Z
6.3
7.0
74
2.5
=•7
2.8
40
5.6
6.2
6.9
75
»s
2.6
2-7
Table V.
1567]
For the valuatim of annuitus upon the hmgtfi tf
three iivee.
too
S
■
-S 8
I
u%
6-
18.0
7"
16 <
8
18.2
91
1S.2
10
<8.2
II
18.2
12 '
18. 1
»3
18.0
14
17.9
«5
178
16
17.6
17
'7-5
18
if.j
»9
17.2
20
17.0
21
16^
22
16.8
23
16.6
*4
16.5
a5
16.4
26
16.3
^
16. 1
16.0
29
15.9
30
15.8
3»
15.0
3a
15-5
33
15.4
34
»5-3
25
15.2
36
15.1
3Z
15.0
38
14.9
39
14.8
140
H-7
568 CoMPOimp Interest. Book IIL
Problem VI.
To find the value of the rcverfion of one life after another,
RULE.
From the value of the life in expe£iation take the value
of the two joint lives, or from the value of the {ongeft of
two'^livcs take the value of the life in pofleijion; there*
ffiainder in either cafe will be the value of the reverfion.
19. Suppofe the life in pofle0ion be 68 years, the life iq
expedation n years, and intereft four per cent, and an-*
nuity 50 1. the value of the reverfion is required ?
Againft 1 1 years, under 4 per cent, iz.o. Table II.
Alfo againft 68 years ^ ^ ,. ^ - 4.6 -?■ 25=2.3, ,
Difference 8-3.
Thcn8.3X 2.3= 19.95 alfo ^^ =; 1.48.
And 4.6 -I- 1.48 = 6.08, value of the two joint lives,
Alfo by Table I. againft xi year?, is 16.4.
And 16.4 <— 6.08 = 10.3 years purchafc.
'•• 50 X 1Q.3 = 515 •' value of the reverfion.
But if the younecft life be in poffeiEpn,
By Table I. agajnft 68 years, at 4 per cent, is 6.9.
And 6.9 t— 60.8 =;; .8 years purchafe.
••• 50 X •8 = 4Pl* valMe, if the youngeft life be ia
pofleflion.
pRopi-EM VII.
Tp find phe valMC of the reverfion of two lives after ope»
RULE.
From the vglue of the three lives fubtrad the value of
^he Ijfe in pofflbflion, ^e remainder will be the value of the
two Jives in reverfion.
20. Let the age of the life in poffeffion be 50 years, and
thofe pf the two lives in rcverfion 45 and 56 years, the an-
nuity 75 1. and intcrpft at 4 per cent, the prefent value is
required ?
Firft, 56 + 45 X 2 :?: 190; ^'fo. 190 + 56 3= 246.
Then 5) 246 (49, mean age, againft which, Ta^)leV^ isi5.i.
Alii?,
Chap. VL Compound Interest. 569
Alfo, by Table I. the value of the life inpofleffionis zo.i.
Alfo i5.i-^io«i=:5 years purchafe*
75 X 5 = 375 !• ^^^^ required.
•_ •
Problem VIII.
To fin4 the value of a reverfion of one life after two.
RULE.
From the value of the three lives take the value of the
two lives in pofleffion, the .remainder will be the value of
the life in reveriion.
ar. Suppofe 18 and 26 be the ages of the two lives in-
pofleffion; and 32 that of the life in expe<5lation ; the an-'
nuity 120 1. and intereft 4 per cent. ?
Firft, i$ + 26 X 2 = 88 ; alfo 88 + 32 = 120.
Then 5} 120 (24, againft which, under 4 per cent»
Table V. is 19 years.
Againft { jj { TablelU. under 4 per ct. { \^'J ' ' .
2)36(18 years. .
And 19 — 18 =; I yearns purchafe, or i2ol. theanfwcr.
What is above obferved, hath regard to fuch annuities as
^re paid yearlv ; but if the payments are made h^f yearly^
which is moir commonly the cafe, the above-mentioned
Mr. Simpfon judicioufly obferves, that the value at which.
Ae annuity is eftimatea ought to be increafed ^ of a year's
{lurchafe; and if quarterly, •{• of a year's purchafe; as the
ife, upon whofe failing the annuity ceafes, has nearly the
fame chance to drop, in the fecond, third, or fourth quarte/,
as in that foregoing i in which cafe the purchafer hath a
chance to receive ■^, 4, or ^ of a year's rent more than the
annuitv, when the annuity is paid yearly ; and intirely lofes
the laft payments, if the aeath happens but one day before
the annuity becomes due.
As my Book is fweiled far beyond the limits at firft in«'
tended, 1 hope my worthy fubfcribers will excufe my proceed-^
ing in Geometry and Menfuration, as propofedin (omeof my
Advertifements ; fo fliall add a coUe^ion of (jueftions, witn
m Appendix, a^d conclwdp,
CHAP.
C 570 3. '
CHAP T E R VH.
A COLLECTION 4 JQ^UESTIONS.
SECT. I.
SUPERFICIAL MEASURE.
SyS-FAC£jS« /iK^ «s Ifind, flooiSng, pabid^^, tjrJuigf
P9^jng» pUi^rifg, j(c. if it be A joijr«4ded agiuc«
whofe oppofite fides are c^n^ b]r mvlcipIyJctt; the Jemgdi
into the perpendicular height, |ives the fupecficial. content,
and eithqr of th^ dynenfioos be»og gii^ea} the other oiflj be
Cpnod bjr <liirifioo.
I. The higgfeft pf tKe Egyptian p^faioide, w^ Grand
Cairo, being Square, and meafuring, accotding to Mr.
Greaves's -acG^int, 693 feet Englifli on a fide; how many
auMa Jthcn pf ground doth it ftand on f
FirR, 693 X 693 = 480249 fquare feet.
Ap acre 3= P feet 4356^)43024^ (11 ^Km.
A p^ch ;;;: 4;] iect 272.;is) 1089(4 perches.
;t. W^at ^iSStxtni;x, J3 .these ,heiweQn ^ Joor ;3J ieet Jo<^
bjr ;tp birof^y ^Q^ tvo others ,th^t f^ie^uce 44 feet ?^eoc
by JO; ^fA wbitt dpaU theic come to at 45^. j)er fqiiarej
ltt;i. 40 fpet J)j 10 i
Ki#fl> «8 X ao a: 56P ; alfo 14 X 10 X 2 =:: I80.
Tfcen 560 ^— 2S0 ;;= 280, difference.
tAto 560 -f- a8o r= 840 I and 45 1. := 2*251.
•10 X 10 = 100) 840 (8.4 fquares.
•8*4 X st.^5 = ^^-9 == 18I. 18 s. amount.
j.»
3. A redangular fpur-fided rooip meafurcs J29 feet 6
ifiehts ab<HJt, mkI is to be wainfcoted, at 3 s. 6d. per yard
fquare : nfter the due allowance for girt of cornice and
meRvbeft, -k is 16 feet 3 inches high j the door is 7 feet by
3 'feet 9, the window-Siutters, two pair, are 7 feet 3 by
4 feet 6i the cheek- boards round them come 15 inches be-
4 low
low the fliutters, and are 14 inches in breadth ; the. Uning*
boonoU fovnd the door^wajr 'ave jf> indifls t>rcMi^ ; the door
And vviiubw-AtitlerB, hemg wsoug^t on jboth fides, ^
Mcjuawi as aroik aad half, aitd paid for aooordihgly ; the
chimney 3 feet Q hj 3 feet, not being indpfed, ^ to be
deduced from tne lup^rficial cpntent of |b^ rpom ; ^d the
cftimate of die charge is required ?
F. I. F. L F. J. F,
Firft, 129 6^X 16 3 - = 2104 4 6, roonu
2
73x4 6 - = 32, 7 fib ilmtters.
8 6 + 46X2=^26 X I 2 X 2 = 60 8 -^ chttluboards,
14 + 39=^7 9XM =313 8 -. 4wr-Ji|>injs.
2234 5 6
3 9 X 3> to bededuacd =: i| 3 -, chi^nfy.
^•i
Square ftec 2223 2 6
■M
^) 2203(047 iquamjNuds.
JO 17 6
12 7 -
jC 43 4 6» coft*
II I I 'p* 'PI
%•
4, WJien ii roof is of a trjgwe pjtid\, tfe? fiAers «« |-sf
^e breadth of die t>uildiMj3[ ; opv fwpofiAg the eavs-boftisds
to projea ^o JAcJw on n fid^> iv^ wUl dir jbmbw ripping itti
out-houfe coft, that meafures 32 feet 9 inches lohg, by 22
fyet ^ iDcbciS broad uppA the fiH^^ at 15 s* p«r i^vsure f
F, I. F. L J?. -
Breadth za 0 « ^ ^ of which js 17 -«- g
F. LP. L F. I. P. F.X?
Alfo J7 - ^ + 10 «=2 17 lo 9, which X 2 :;;: 3^ 9 6
35 9 6 X 32 9 t= ii7;ifeet 2in. | 6
100) 1172(11.72 fquarcs, and 155. =.75!.
IJ.72 X .75 ;=;: g.79 =s 81* 15s. 9^ d. the ai^fwen
5. If
I i
\
ff% SvpsRFiciAL Measure;. .fcbklH.
5« If my cpurt-»yard be 47 feet 7 inches iquare, «id I have
laid a foot* way of Purbrac-flone, 4, feet wide, along one
£de of it; what will paving the reft with flints come tOy at
6i* per yanl fquare i
Firft, 47 f. 7 in. — 4 f. = 43 f. 7 in. breadth.
Then 47 7 X 43 7 *= 2073 ^o >-
-And 9} 2073 (230 yards 3 f. 10 inches.
^1230 ^ 2^4. vadne of the 3 feet 10 inches.
'— 8. d.
£ 5 152a>the anfwer required.
6. A fquare cieling contains 114 yards 6 feet of plaiftcr-
ing, and the room 28 feet broad i what was the length of
it?
Firft, Ii4yards 6 feet = 1032 fquare feet.
Then 28} 1032 (367 feet, the anfwer.
7. An elm plank is 14 feet 3 inches long, and I would
fcave juft a yard iquare flit off; at what diflance from the
<^ge muft the line be ftruck ?
Firft, 14 feet 3 inches = 171 inches ; alfo 36 X 36 =
1296 inches in a fquare yard. *.* 171) 1296 (7-14- uiches,
the aafwer. (99)
8. Having a refbtngular marble flab, 5& inches by 27, I
would have a foot fquare cut off, parallel to the fhorter
edge ; I would then have the like quantity divided from the
remainder, parallel to the longer fide ; and this alternately
repeated, till there fliould not be the quantity of a foot
left : what will the dimenfions of the remnant be ?
Firft, 12 X 12 = 144; alfo -^ =r 5.^, breadth of the
firft cut.
Then 58 — S^f = 52.^, the remaining length.
Alfo iM = 2*7349 breadth of the fecond cut
^ 52.JI ^^
Then 27 — 2.734 = 24.266, the remaining breadth*
«44 _
.2i
= 5-934> breadth of the third cut.
S^'f
52-^ -- 5.934 = 46 7321*
•^T == 3.Q814, breadth of* tke fourth cut*
^ 24.166 — 3.0614 = 21.1846;
— ^-^ = 6.7074, breadth of the fifth fe^oo.
21.1846 l^f-r:f , ,
46.732 — 6.7974 = 39.9346.
= 3. 6059, breadth of the fixth fc^doa,
39.9341 J 3^> ^
21.1846 — 3.6059 = 17.5787-
— ^^ = 8.1917, breadth of the feventh Ye^on*
. 399346 — 8.^917 = 3«-74a9^
^—11^ — 4'5364, breadth of the eighth fefiion.
. 31.7429 ^^^^ ^
. ^7 5787 — 4-5364 = i3-04a3«
^ =s 1 1. 04x1 9 breadth of the ninth fedlon.
12.0423
Then 31.7429 *— 11.04x1 =: 20»70i8> remaining length
atlaft
•' Alfo ' '^ ^ = 6.g<6, breadth of the tenth fedion.
20.7018 ^
'•* ]3,»C423 — 6.956 = 6.0863, breadth remaining at
the laft. Q. E. F.
9* Being about to plant 10584 tre«s equaHjr diftant^
the length of the 2i:ov& muft be fix times the breadth i
how many of the morter rows will there be f
6) 10584 ( 1764 ; then v^i764 =3 42 long rows.
*•* 42 X 6 =^.342 ihort rows» 42 in a row.
10. A common joift is 7 inches deep, and 24 thick; but
I want a fcantling juft as bis again, that fhall be 3 inchei
thick : what will thq othes dimenfion be ?
Firftj^ 7 X 2.5 ^ 17.5 inches^ area of an end.
Alfo 17.5 X 2 = 35> double area.
'•' 3)35(1 It inches. Q^ E* F.
XI. I have a fquare girder, 19 inches by 11, but one of a
quarter of the timber in it, provided it be 9 inches det[t^
will fervc ; how broad will it be .^ ^ .
Firft, 19 X II = 209, area of an end.
. Then 4) 209 (52J:, arqa of an end of the piece wanted.
V 9} 52.25 (5.80^. Q,E. F. '
11. l^
la. I have a wooden trough, iS/mt^ at 6 d. per ]ar4» eoft
me 38* ad. painting within ; the length o( ft is loaincliesy
the depth at inches ^ wi^ fe rt^ GMuMb i^
Firft) 36 X 36 = li^ (i|«are indket }» » yaid.^
Alfo 38. ad, = 38d»
d; in. d*.
As 6 : ia96 : : 38 ; SaioS^ ai<e»ef diewli^le trough
Then xoax ao X a = 4a84^ area of the two fides.
39349 area of the bottom and ends.
Then ipa + 4a s 1 44} ^24 (a;^ inches. Q. £^ F.
13. My plumber has put a8 ib. per foot f^ttasb intor
a ciftern, 74 inches andf ts^Scid tbr nMlMtfs or tbr I^mI
]ong, 26 inches broady and 40 deep; he has put three
flays wichin acrofr it, 16 inches deep, of die fame ifaviigtb,
and reckons aas. per cwc £0r Workanimfliterialtf : Xbekig
a mafony have payed him a wor]&-fhop, aa feet lO inches
broaJ, with Purbeck-flfone, at 7 i. per foof, and opon the
tasdaaek I find there is jsv 6xL <hir to him 9 whus wae die
length of his work-ihop f
Fii^y a6 + 4or -4* 40* ss 106, breadth df the bottom
and fides.
Alfo 106 >e 74 "S^ 794fy irei^oT <h# battbi» a«l fides.
Then 40 x a6 X 2 2= 2080, area of botH eMda,.
And a6 x 16 x 1= 1248, area of the ftays.
tii 72, ^ whofe aMi in inc&cs.
144) nr7^(77^3S^fi|iiavef«ei.
19.39583
£ 2i*3354i^=itl. 6rf« tj^^d. vschieof tfredffcm.
•029i^\ aia6dai^ (725.5 f<|uare Mcaa faU ill#pi
291/ 2.1 1 6041
.©«6^5) 19.044.37j 21 ft. ro in. fs aa.f^
i2.8j\ 725.5
2.28y 72.5
I » mi iJi
20.55) 653.00
22.8^ \ 725.5 (3i'776s 31 feer 9^ faicbe^, the afiftfer.
74. The
14* The area of a refiangular powdering-^trough of a man
of war tn€a(ure» 7ff i^uAffr feet^; iki NMaea^ thir depAi is
ao fAch«9, the bieadtb i6 ; th0^1efligilkis fov^tf
Phrft, 27 feet I'fa itfdfesr ±z 4000* fijlrafe^ irtAa.
Theh 20X 16 X i ' — 64c, iirao/ b6tfx dncis.
3360, bottom andhotb fid^s*
• . • 20 + 20 + 16 :^ 56) 3360 ^60 iiuchce^ theaoAiirer.
15. In 110 acres of ftatutc-mcafurc, in which the pJafe
is 16^ feet lon^, how manv Che(hire acres, where the
€\3t9co^^^ fole is 6 vai<ds M% i an<f hfow many YWkfl&e^
iUfkffie the pole* ii^ iric is 7 yard* in- IdngAr ?
f4'.yds.
PJrit, 5.5 X 5,5 rr 30.^^5 J f'Sfafufe ^ |
Mfo 6 X 6 r=i 3(6 J == 1 perch |CReniife $ £
Arid 7 X 7 =t 49 > C Torkffiire J f^
'.•^ Kec?procaifIy, i. r. p.
As ^o 2c • 110 ••( 36 : 9^ 435X== 92 i 28,.Chelhire 1 9
As 30.2s . 110.-.^ ^^ . g^^^^ _ l^ 2 ^^^ y^^^ J P
r. t wottfd At 3584 plants in st>w^y e^cB 4 f^
afunder^ and the |dantft 7 &et apart', ui a reSaogular plot
of ground ; what luti mYt tfai» take up }
Firft, 7X 4^ =i^ 28;-fqaarefcct (aVea) between the plants*
Then 35S4 X 28 = 100352 fquare feet.
In aii acie are 43560 (quare feet.
%* 43560) i0035a-(2^u:res, 1 rod, 8^ perches, the anfwtn
i<rf9er> fsajz:
ia72»a5)I 2342; . •
A triangle, or ifaree-'dded figure^ (bemg die ha^T of Sk.
faur-fided one of the fame height and' kngth) if vou^ mui*'
tiply the bafe, or longeft fide, ..fay the fhoctecAc-beigfat,. you^
have double the opnt^nti
17. A triangular field, 73S links ion^ and 5S3 in- the
perpendicular, brings in 1 2 1. a year ; what is it fet at an acre i
Firft, ^^ X 583= 215127 = 2 acres, 24 perches.
Alfo 2.15127 : 12 :: t.ooooo : 5-578x ss jl* xzs^ 6|d.
tlKanfwer, x8. A
576 SvpttCTiciAL MEAStjRS* Book ItK*^
. i8. A piece of gardeil«»box lies in form of a regular pen-
tagon, or figure of five equal fides, each 48 feet } and from
the center of the figure to the midle of one of thefe, it
meafures 41*57 feet nearly : the area of the figure will be
the content 0/ thefe five triangles ; pray what is that ?
Firft, 2-. X 41.57 =^997-68, area of one of the triangles.
Alfo 997.68 X 5 = 4988-4 fqtiare feet, area of the pen-
tagon.
19. The end-wall of an lH>ufe is 24 feet 6 inches in
breadth, and 40 feet to the roof; 4- ^^ which is two bricks
thick, 4. more 1 4 brick thick, and the reft one brick thick :
now Jthe gable rifes 38 courfe of bricks (four of which
ufually make a foot in depth) and this is but 4 inches, or
half a brick thick ; what will this piece of work come to,
at5l. los. per ftatute rod, the dimenfions of which are
given ?
4) 38 (9-S> height of the gable. :
Alfo 3) 40 ( 1 3./, height of each floor.
Again, 24.5 X 13 3 =.326.^ = 43S-^> ground-floor^
Alfo 326.^, firft ftoiy.
Then 326.^ ==: 217.7, g^^ret.
And ^ X 9-5 = "6.375 = 38.79, gable.
1
' xoi 7 feet, ftatute meaf.
272.25) 1017.00 (3.7355 rods, ftatute meafure.
••• 37355 X 5-5;= 20.54525 = 20 1. los. lofd. the anfwcr.
20. A four-fided figure, whofe fides are equal, is called a
trapeze : I have an orchard of that form, containing 3^ acres,
which being divided by a diagonal, or a line from corner to
corner, the perpendicular of one of the triangles is 430
links, and die other 360 : the length of the faid dfa<^onaI,
or common bafe of thofe triangles, is required i ^
* Firft, 430 + 360 = 790 5 alfo 22? -: ^95*
And 34 acres = 37^000 links.
••' 395)375000(949^1 links, theaafwer.
The
ClU^. VII. Sup£itrioiA& MiASt^ns^ j7^
The areas of circles are found, either by multip]yin| half*
Che circumference by half the diameter, or b^ multiplying
t^e fquafe of the diameter by •7854, that being the area of
the circle whofe diameter js i. T
And if the diameter be i, thecirciimferen^t will be 3^1416
nearly.
21. Give the al'ea b^ a circular bowling- glreen, tli^t Is
t6 poles a-crdfs the middle, the circUmference being 3*1416
times the diameter of a cirde i
i6 X 3.1416 == 56.26^ poles cifcumferehc<i.
16 . q'?.26c6 ^ ^ ,
— X ' r-^-' = 20 1 .062 j fquare poles.
tn ap acre are 160) aoi (i acre« 4.1 poles, the anfwer.
Or x6 X li X -7854 as 101.0624^ as before.
22* The furveying Whed is fo cdntrived^ as tb tfirn juft
^ice in the length of a polei or 164. feet } what then is its
diameter?
One round, pet qudlioti, is 8^^ feet.
3.1416} 8 2500 (2.626 feet = 2 ft. j[ in. the atlfWer*
23. I would turf a found plat, ttteanirlng i ^6 feet aboUfi
and would know the charge at 4d. per yard fquare i
m
3.14.16) 1300000(41.38, diameter*
65 X 2(5. 69 = 1344-85 fquare fteu
9) '344-^5 (149-428 fquare yards,
-rs) 149.428(2.490^ = 21. 9 s. 9|d. theahfvsref*
In an i acre are 2420 fquare yards;
14. I want the length of a line, by which iliy gardehfif
tnay ftrike a round ok-ange^yi that ihall conuin juft half
an aCrt of land ?
Firft, .7854)1446.0060 (308 1. 5 ji
And ^^3081.23 (5S'54 diameter.
*•' 5t) 53.5 (27.75 = 27 J yardsj the anfwtf*
25. Agreed for an dakeri kerb to a found well ^ ^t^i*
per fobt fquare 5 it is exaftly 42 inches in diameter, with«»
in the brick-work, and the breadth of the kerb is 16 t>tf
144: inches j what will it conl& to ?
J^ifft, 14,5 + 41 4- U-5 — ^rj ghsater Jlaihettf. ,
P p Tma
Then 71 X 71 =3 5<^+i 5 alfo 504' ^c .7854 = 3959-20I4
Then 41X42 = 1764; alfo 1 764 x. 7854= 1385-4456
DifFcrencc of the areas arc •.---- ^573"75S*
Then 144) ^573-7558 (17.8733, area of the kerb.
Alio 8 d. == .tf fliiUing,
17.8733 X •<*= i*s. 1 1 a. nearly, tbeanrwer.
26. It is obferved, that the extreme end of the minute:-
hand of a public dial moves juft 5 inches in the fpace of
2i minutes -, the queftion is, what is the length of that
index?
As 3.25 : 5 : : 60 : 92.307, circumference.
Alfo 3.1416 : 1 :; 92.307 t 29.38, diameter.
%• 2) 29.38(14.69 inches, the anfwer.
27. A, By C join for a griitd-ftone 26 inches orert
value 20 6. towards which A paid 7 s. 58 s. and C5S.
the wafte-hde, through which the fpindle pafled, was
5 inches fquare ; to what diameter ought the ftone -to be
worn, when B and C begin feverally to work with it i
Begin your calculations from the center.
Firft, 36 X 36 = 1296, which X .785+ = 1017.8784,
the area of the whole ftone.
5 X 5 = 25 + 25 = 50 ; alfo ^5^ = 7071^8, the
diameter of the circle circumfcribing the fpindle-hole.
Then 50 X -7854 = 39-a7> area of the circle circuo^
(bribing the fpindle-holc.
Alfo 1017.8784 — 39.27 = 978.6084, area to be di-
tided.
s. Cjs. : 324 5129 = A'si
As 20 : 978^6o84 : : 4 8 : 391.4433 = ^'s V
(5 : 244.6521 =C's J
Then 244.6521 + 39 ^7 = 283.922 J.
.7854) 283.9221 (361.5.
Alfo ^361*5 = I9«03, diameter where C begins to grind.
And 391.4433 + 283.9221 = 675.3654.
-7854) 675.3654 (859-9-
**' ^859*9 =^ 29.324 inches diameter, Where B begins to
^rind. Q^ E. F.
4
28.1
Ghap* VII. SupiREiciAL Measuhe: 579
. 28. I demand what difference there is in the area of the
ie<aiQn of a round tree, 20 inches over, and Us inlcribed
and circuqircribed fquare3^^
Firft, 10 X 10 == 100; alfo 100 -j- 100 = 200.
Then y^200 = 14. 142 135, fide of the infcribed fquare^
Alfo 14.142135 X 14.142135 = 200, its area.
Again, 20 X io = 4C0, area of the circumfcribed fquare»
Laftly, 400 X -7854 = 314.16, area of the circular
fedion. •
Hence the infcribed fquarc is 1 14.16 ? • u 5 *^^ little.
And the circumfcribed - - 85.84!* "c too much,
29. Having paved a femicircular alcove with black and
white marble, at 2 s. 4 d. per foot, the mafon's bill was
jufl 10 1. what then was this arch in front, confidering that
as .7854, the area of the circle, the fquare of whofe diame-
ter is r, fo is the area of any other circle to the fquare of
its diameter ?
Firft, 23. 4d. == .ti^l. : I : : 10 1. : 85.7143 feet area.
Then 85.7143 X 2 = 171.4286.
Alfo .7854} 171.4286(218.269.
*.* 4/218.269 =: 14.7739 = 14ft. 9i in. the anfwer.
^O. What proportion is there between the arpent of
France, which contains 100 fquare poles, of 18 feet each, and
the Englilh acre, containing 160 fquare poles, of i6-^feet
each ; confidering that the length of the French foot is to
that of the EngUfh, as 16 to 15 ?
Firft, 18X18X100= 32400 French feet, the arpent.
Then i6j x i6i X 160=143560 Englifhfeet in an acre.
Alfo 16 X 16 = 256; and 15 x 15 == 225.
Recip. 256 : 32400 : : 225 : 36864 Englifh feet, an arpent.
So that the Englifh acre is to the arpent of France, as
605 to 512, or nearly as 13 to 113 or as i to .84628,
the anfwer.
31. In turning a one-horfe chair within a ring of a certain
diameter, it was obferved, that the outer wheel made two -
turns,while the inner made but bne ; the wheels were equally
high, and fuppofing them fixed at the ftatutable diftance, or
P p 2 5 feet
580 Superficial Measure. Book IIL
5 feet afunder on the autletree ; pray what was the circum*
ference of the track defcribed by the outer wheel I
3.1416 X 4 =: 12.5664, the circumference of the wheeL
%• 12.5664 X 5 = 62.832 bv.the greitef.
And 31.416 by the lefTer.
Multiply half the arch by half the diameters ; alfo find
the area of a fedor ; that is, any part of a circle cut through,
from the center to the circumference.
32. The afea of a fe^or (fuppofe one of the divilions of
A wildernefs) which being ftruck from a center with a line
3b yards long, makes the fweep, or circular part, 63 feet,
is require<^ ?
63 feet = 21 yards is half, being 10.5 yards*
Then 10.5 x 30 =: 315 yards, the anfwer.
33. If the choM or line drawn through the two ends of
the curve be 15 inches ihorter than the arch line, I demand
tlie fegment ?
Firft, 15 inches = .41* yards.
Then 21 — .41^ = 20.583, whicn -7- 2 = io.29i|(^
30 X 30 = 90O'
The D 20.291^ = t05.9i8!4
■ I iT
^^794.0816 ±z 28.1^9 per pendicular*
Then 10.291^ X 28. i8=:290.oi9i/$, area of the triangle.
315 — 1290.0191JJ =: 24.98, the anfwer.
« a
An ellipfe, or oval, is meafured, by multiplying the pro-
dad of the long and (hort diameters by .7854, as in the
circle, and this will give the fuperficial content.
34. The ellipfe in Grofvenor-fquare meafures 640 links
the longeft wav^ and 61 2 acrofs, within the rails ; the walls
arc 14 inches tnick ; what ground do they ftand upon ?
. Firft, 8.40 X 66 is: 554.4 1
Alfo 6. 1 2 X 66 = 403 92 > fcet4
And 12)28.0 =: 221 J
Then 556.7^ x 406.25^ x .7854 == 177637 66
. And 554.4 X 403 92 X .7854 = 175877.17
Area covered by the wall - - ^ 1760.49 fquare ft<
••• 4840
Chap. VII. Superficial Measure. 581
••• 4840 >< 9 = 43560) I75877»i7 (4 acres, 6 perches^
Us area. Q, £. F.
The dimenfions of all fimilai^ figures are In proportion to
their areas, as the fquares of their refpe£tive fides^; it contra.
35. If a round pillar, 7 inches over, has 4 feet of ftone
in it ; of what diameter is the column, of equal length,
that mtafi^e^ ten times as much i
4 X 10 ;=: 40 feet.
4 feet : 49 : : 40 feet ; 4^0.
^490 == 22. 136 inches. Q^ E. F.
36. A pipe of fix inches bore will be 3 hours in running
pflT a certain quantity of water ; in what time will 4 pipea,
^ach 3 inches bore, oe in difcbarging double the quantity t
6 X 6 = 36 ; alfo 3x3x4x2= 72.
••• 36 : 3 :: 72 : 6. Q^ E. F.
37. A yard of rope 9 inches round weighs, fuppofe 22 lb*
what will a fathom of t)iat weigh, whic)i nieafures a foot
round ?
9X9 = 81; alfo 12X12 x 2 3= 288.
81 : 22 : : 288 : ^^. Q^ E. F.
38. If 20 feet of iron*railine (hall weigh half a ton,
when the bars are an inch and quarter fquare ; what will
So feet of ditto come to, at 34-d. per pound, the bars being
lit f of an inch fquare i
1.25 X 1.25 X 20 = 31.25.
i = .87s X .875 X 50 = 38.28125.
As 31.25 : 1120 :: 38.28125 : 1372.
'37*
T
To
I
■8
'7 3
2 17
/ 20 - 2, the aiifwer.
39. A looking-glafs is 16 inches by 9, and cpn^ains
a foot of glafs > what will the content of the plate be,
that (las twice the length, and three t^mes the breadth?
2 X 16 s= 32; and 3 X 9 = ^^7.
Then 32 X 27. = 86 fquare inches.
••' 144) 864(6 fquare feet, the anfwer.
P p 3 40. A
^iz SupERFiciA]^ M^AsyasV 3dQk III,
' 40. A fack that holds three buihels of corn is ^%\ inches
broad, when empty ; what would the fack contain, that,
being of the fame length, had twice its circumference, of
fwice its breadth ?
22.5 X 22.5 = 506.25 J alfo 45 X 45 =s: 2025.
506.25 : 3 bum. : ; 2025 : 12 bufh. Q* £• F.
41. My plumber hks fet me up a ciftern, ^nd his fliopr
book being burnt, he has no means of bringing in th9
(charge, and I do not chufe to take it down to have i(
iveighed ; but by meafure he finds it contains 64 fquare
feet -f^, and that itwas ^ of an inch precifely ih thicknefs.
Lead was then wrought at 21 1. ]!>er fodder. Let the ac-
comptant, from thefe items, make out the poor man's bill;
confidering farther, that 4 bz. y^. is the weight of a cubic
ipch of lead.
. Firft, 64.3 X 144 = 92592 fquare inches.
Alfo 9259.2 X .375 = 3472.2 iolid inches.
And 3472 2 X 4.3^ =: 15151.4/^ ounces.
Likewife 1515I. 4/^ oz. =; 8.455 ^^^*
•.' J9.5 cwt. : 2il. :: 8.455 : 9.10535.
Anfw^r, 9I. 2s. x^d.
SECT. II,
MEASUREMENT */ SOLIDS,
MULTIPLY the area by the depth, to find the
folidity of uniform bodies, pr fuch lis are equal from
top to bottom,
I What is the difference of a foUd half foot,. a|id half
^ foot folid i
^irft, 6x6x6=r2i6, folid inches in | foot folid.
And 2} 1728 (864, folid inches in 4 ^ folid foot.
r,* ^f ^) 864 (4 times 9s miich as the firft.
2f Whjit
Chap.Vn. ^EASURBMBHT ^ Solids. 583
a. What is the proportion^ in point of fp^ce, between a
tbata 25^ feet long, 20 feet 12 inches broad, 14 feet high,
and tvro others of juft half the dimenfions f
F- I. F. I. F. I.
Firft, 25 6 X 20 2 X 14 = 7199 6«
Alfo 12 9 X 10 I X 7 X 2 = 2799 10 ^ which
is evidently juft ^ of the firft*
3. Another room is 17 feet 7 inches long within, 13 feet
10 inches broad, and 9 feet 6 inches high } it has a chimney
carried up ftraight in the ande, the plan whereof is juft
half of 5 feet 6 inches, by 4 feet 2 : the queftion is, bow
many cubic feet of air the fame will contain, allowing the
content of the fire-filace and windows at four folid yards i
F. I. F. 1. F. L F. I.
FirA, 17 7 X 13 10 K 9 6 = 2310 811
Then 5 6 x 2 i X 9 6 = 108 10 3
Rem. 220T 10 8
And 4X27 ---- = 108--
Anfwer, feet 2309 lOy inches.
4. A (hip's hold is 1 12 feet 6 inches long, 32 broad, and
5 feet 6 inches deep ; how many bales of goods, 3 feet
4 inches long, 2 feet 4 inches broad, arid 3 feet deep, may
be ftowed therein, leaving a gang*way the whole length of
4 feet and 4 broad f
Firft, 1 12.5 X 351 V 5.5 = 19800
Qiti^-yrtf 112.5 X 4.5 X 5-5 =: 2784-375
Remaining capacity 17015.625
«— M*^
Alfo Z'9 X 2.3: X 3 = 23;. = ^.
» J /: < M6125
And 17015.625 = 17015I = -j--^'
*** 3) ■»■ ("1^ ~ 729rVj» Ac anfwer.
PP4 5- I
i
384 MlASVUBMBNT $f S0LID8. Book HI,
5. I want a redangular ciftern, that* at i61b« to the foot
fquar^, ihall weigh juft a fodder of lead > it muft be 8 f<pet
Jong, and 4|> over \ hpw many bogfheads, wine meafure^
wiU this contain, taking it at ^ of an inch from the top \
A fodder of lead weighs I9^cwt. = ^ 1841b.
16) 2 1 84 ( 1 36.5 fqiiare f^et.
Then 8 X 4.25 = 34, arej^ of the bottom,
Alfo 136*5 — * 34 = 102.5, fides and ends.
P + 8 + 4-^5 = 4-^Si> round.
114,5) I0i«5 (4-183673 feet ;;= 50.204 inches deep,
8 feet == 96 inches, and 4<^feet ^51 inches,
Alfo 50.204 — .75 ?= 49-454-
Then 96 X 51 X 49*45^ = 242126.78^1 cybic inches,
£$2) 242126 (8^8 gal. = f6 )ids. 42 gal. tne apfwer.
6« A log of timber is iSfe^t 6 inches long> 28 inches
broad, and 14 thick, die fquare all through ; now, if 2 fo-
}id feet and ^ be fawe4 off the end, how long will the piece
then be ?
Firft, :| 8 inches ^ \.^\ alfo 14 inches = i.i^.
i!5Xi.i^= i.75}2.^( 1.42857^ length of the piece cq^offii
Then 18.5 •»— 1.42857 = 17*07143 feet, theacnfwen
7. The fpli4 content of a ftiuare ftone is found tp be
1264 k^ty its length is 8 feet 6 inches; what is the area of
pne end, and what the depth, if t^e breadth affigned be
j8i inches ?
Length 8.5) I26,as (i4«8S3 ^% =5 ai38»8234 inches,
^t% of a[n C(id,
^8,^^ 2138.8234 (55.55 inches deep, the anfwer.
' 8. The dioieniions of the circul^^ Winchefter biifhel are
]8| inches over, and 8 inches deep ; how m.^7 quar^rs of
grain then will the fquare bin hold, that meafures 7 feet
|0 long, 3 feet 10 broad* and 4 feet 2 deep within I
Pirft, 18.5 X 18.5 X .7854 = 268.80315.
Then 268.80315 X 8 = 21^0,4, cubic inches in a bufliel.
Alfo 7ft. loin. = 94 in., 3ft. 10 in. =: 46 in., and 4 ft,
^ in. zs 50 Inches.
Then 94 y 46 X 50 = 2162DO cubic inches, contend
of xhc bin.
2150.4) 216200.0 (x 00 bufh. 2 pecks ac I2<lcs. 4bu(h«
} pecks, C^ E, F,
9. Taking
Chap/Vn. Mbasurzmbnt i^ Solids;
S«5
9. Taking the dimenfibns of the buflid, as above^ wba^
piuft the diameter of tht circular meafure be, which «(
l^ inches deep will hold 9 bufhels of fea-coal ftruckl
firft, 2150.4 X 9 =: 19353.6 inches, the content*
Then 12) 19353.6 (1612,8, area of the circle,
Alfo .7854) 1612.8000(2053,47-
••• i/2653.47 = 45.3 inches. Q. £'• F,
' 10. A prifm of two equal bafes, and fix equal fides, that
ineafures 28 inches acrois the center, from corner to corner j*
the fuperficial and the folid content is required, taking
the length at 134 inches f
Radius 14 X 1+ ;=; 196 ; alfo 7 x 7 ?= 49.
Then 196 -^ 49 1= 147.
Alfo v^i47 =5 I2.I243S57> perpendicular.
Then 12.I243557 X 7 a* 34.8705, area of one triangle,
Alfo 84.8705 X 6 = 509.223, arq^ of the bafe.
And 14 X 6 X 134 = 11256.
Other bafe 7=^ 509.223
fiie^ of the prifm 12274,446 inches.
•;• 1296) 12274 (9 yards, 4 feet, 34 inches, its area. Q.
j;. F.
Again, ^09.223 X 134 = 68235.88 foltd inches.
••• 1728) 6823s (39 folid feet, 843 cubic inches. Q^ E. P.-
II. I have a rolling- ftone, 44 inches in circumference^,
and am to cut off thre^ cubic feet from one end ^ wherer
abouts muft the it&Xon be made ?
Firft, 3 cubic feet = 5184 cubic inches. .
If the circumference be 44, the diameter is 14^
Then 22 X 7 = 154, area of an end.
••" 1 54) 5 ^ 84 ( 33.66 inches, the anfwer,
T2. I would have a fyringe, an inch and ^ in the bore^ to
hold a pint, wine meafure, of any fluid ; what muft the
length of the pifton fufficient to make an injedion with it,
be r
Firfl, 1.25 X 1.25 X .7854 = 1. 2271875, area of the
fircle.
In a pint are 28.875 cubic inches.
t,' 1,2271875)28.8750000(23.5294 inches, the anfwer.
58^ MsASV'i.SMENT ^ SOUDS. BMk III«
13. I would ha^ a cubic bin toAde capable of receiving
joft 13^ quarters of wheat, WincbcAer txnnfure ; what will
be the length bf one of its fides ?
In a bu^I are 2150.4 cttbk inches.
Then 2150.4 X 8 X 13.5 =r 232243.2 cwbicin. I3J. qrs.
••' '\/232243.2 = 61.4678 inches/ Q. E. F.
14. A bath-ftone, 20 inches long, t5 over, and 8 deep»
weighs 220 H>. how many cubic feet thereof will freight a^
i&ip of 290 tons ?
Firft, 20 )? 1$ X 8 = 2400 bilbic inches.
Alfo 2400 inches : 220 lb. :: 1728 : 158.4.
Alfo 200 tons = 649600 pounds.
*.- 158.4) 049600.0 (4101 feety the anfwer.
. t5« The common way of meafuring timber being to
girt a round ftraight tree in the middle, and to take ^ of the
girt for the fide of a fquare, equal to the area of the
fedion there ; if this be not cenfidered in the price ap-
poinOsd, pray on which 6Ae lies the advantage?
A. piece of timber a foot long, and 4 feet round, is a foot
fuftomary meafure.
Alio if a circle be 4 feet round, 3.1416} 4 (1*2732 dia-
meter.
And a circular piece of timber i foot in length will con-
tsin 1.2732 feet;
V* 1.273a X 50^ the feet in a load, is 63.66.
So that, in a load of timber, there is gained by the buyer
nearly 13^ feet.
The. circumfcribing cylinder is in proportion to its
greateft infcribed globe, and cone of the fame bafe, and
perpendicular altitude, as 3, 2, and i.
Therefore the cube of the diameter of any cylinder, of
tquai height and breadth, multiplied by .7854, the area of
H cirde, wbofe diameter is i, will be the folidity.
The cube of the diami^ter of a globe, multiplied by
5. of .7854, vi». .5236, gives its folid content.
And the faid cube, multiplied by 4. of -5236, ^or .2618,
fives the folidity of any cone, whofe breaddi and height
gre equal,
Alf#
1
Chap. Vn» MsAsulteuiNr vf SoliiA. 5B7
Air6 th^r fuperficiai content may be fotnri, by confidering
fhe cylinder as a fquare furface, multiplyino; the neight by the
circumference, and adding a double area for the two bafes ;
the globe, as a rectangle of the diameter and circumference 1
and the cone as a triangle, whofe bafe is the circuit, and
perpendicular the flope height, adding the area of the bafe,
16. The fi>lid content of a globe 20 inches in diameter ;
|i cylinder of the fame diameter, 20 inches long ; and a cone
%o inches diameter at the bafe, and 20 inches liigh, are fe^
.yerally required \ and alfo what they will coft painting, at
9 d. a yard ?
cub. inches.
r.7854 =: 6283.2, cylinder's^
io X 20 X 20 X < .5236 =: 4188.8, globe's \ folidity.
(•2618 z; 2094.4, cone's J
Alfo 20 X 3*1416 = 62.832, circumference,
. Then 62.832 x 10 = 628.32, area of the two bales.
And 62.832 X 20 =^ 1256.64
Cylinders 1884.96, fuperficiai content,
Again^ 62.832 X :$0 =7 1256.64, ditto of th6 globe.
1015.24, ditto of the cone,
4156.84, fttm of their areas.
Alfo ao X 20 ps 400
And 10 X 10 = 100
^500 =5 22.31614, flope height of the
31.416 X 2^2.31614 = 7oi,o8385 cone*
Area of the bafe y/. ^ = 314* 16
Area of the cone, as above 1015. 24385
As 1296 : 8d. :: 4156.84 : 25.6=1 2s. i|d. theatlfw,
17. Our fatellite, the moon, is a globe in diameter 2170
miles ; I require how many quarters of wheat fhe would
contain, if hollow, 21501% ^^^id inches being the biifhel;
and how much yard-wide ftufF would make her a waiftcoat,
was fhe to be clothed \
Firft, 2170 X 2170 X 2170 X .5236 = 5350308686.8,
1id miles in the mooi?.
Then
^ '
\
588 Measuremsnt of Solids. Book ITL
Then 1760 X 1760 X 1760 = 545177600, folid yards
in a mile.
Alfo 5350308686.8 X 5451776000 s 291^868449128-
#^756800, folid yards in the moon.
In a /olid yard are 46656 cubic inches.
29x68684401287756800 X 46656 s 1 36089414362552 1-
»>58 1 260800 K)lid inches.
17203.2) 1360894143625521581260800.0 (7910703494-
1-8470144000 quarters of wheat the moon would hold» if
hollow, Q: E. F.
Again, 2170 X 3.1416 7 68x7.272, circumference of
the moon.
Alfo 68x7.272 X 2170 ^ 14793480.25 fquare miles.
1760 X 1760 ;;;: 3097600 fquare yards in a fquare mile.
Then 14793480,24 X 3097600 ^ 4582428439x424
fquare yards, Q^ {). Ff
18. Suppofing the atmofphere, or body of the air and
vapours, furrounds the globe of the earth and fea to 6q
iniles above the furface, and the earth is 7970 miles in dia^
meter s how many cubic ^ards of air then hang about
and revolve sdopg with this planet?
Firft, 7970 4* 190 =: 8090, diameter of the earth and
atmofphere.
Then 7970 X 7970 X 7970 X .5236=; 265078559622.8,
folid mile$ in the globe of the earth.
Alfo 8090 X 8090 X 8090 X .5236 = 377233x77544.4,
miles folid in the earth and atmofphere.
> And 277233x77544.4 — 265078559622,8 = 121546x7-
»92X.6, folid miles in the atmofphere.
Alfo X2154617921.6 X 545x77600 = 6626425427414-
ir376i6oo, folid y^ds in the atmofphere. Q. £. F.
19. A gentleman bargaineth with a mafon for a piece
pf marble in th$ form of a tetraedron, op whjch he in-
tends to have four fun-dials ; the fide of each triangle is
a4 feet, or 30 inches; I demand its value, at 2d. a folid
inch, and what it will coft polifhing, at is. 3d. per foot
fuperficial I
Firft, 30 X 30 = 900; alfo 15 X 15 = 225.
Then goo — ^25 == 675 j alfo ^^675 = 25,98, perpen-
dicular of each triangle.
^ Ag.in^
— '-f-
Chap. Vlt. Measuriment ^/8oLiDS« si§
Aeain, 25.98 X 25.98 = 674.96044 alfo ijt.99 x 12.99
za 108.7401. <
Then 674.9604— 1687461 = 506.2203.
^506^2203 z=, 22.4993> perpendicular of the tetraedrom
Then 25*98 x ~ = 389.7, area of a triangle.
And 389.7 X 4 s: 1558.8 inches =: 10^825 fquarefeeti
22.4003 ^ t
Then 389.7 X — ^^^^ = 2922.65910^ inches*
' L 9. d^
Anfwer, the marble comes to 24 7 i-^
And poliflung to - - - — 13 6^
To find the fblidity of a pyramid, or cone t multiply die
ftr«a of the bafe by 4 of its perpendicular altitude.
20. A fquare pyramid, whofe fides at the bafe meafure
30 inches a-piece, and is 21 feet high by the flope in the
middle of each fide of the bafe, is to be fold at 7 s« per
folid foot ; and if the poliihing the furface of the fides will
be 8 d. per foot more, I would know the coft of this ftone
when finiihed i
f irft, 21 X 21 =: 44t ; alfo i.ijf X Mj; r= 1.5625.
Then 441 -^ 1.56^5 = 439-4375-
^439.4375 =: 20.9627, perpendicular height.
Alio 2.5 X 2.5 = 6.25, area of the bafe»
Then 3) 20.96276 (6.9676, nearly.
And 6.25 X 6.9876 = 43.6725 folid feet.
Then 21 x 1.25 := 26.25, area of one triangulaf fide.
Alfo 26.25 x 4 = 105, area of the fides.
1. s. d.
Anfwer, at 7 s. per folid foot, 15 5 8|.
And poliihing, at 8 d. pei* foot, 310 -^
£ »8 IS 8i
<mm*
When figured nlii uniformly taper, but not to a point,
(hey are to be confidered as fruftums, or portions of
the cone or pyramid ; by fuppofing, therefore, whdt is
Wanting to make the figure entire, and then deducting the
part cut oflF, we find the folidity of the pan propofed.
In
In order to compl^ti^ the cone, ufe thi9 analogy; as balf
the difFercnce of the top and bottom are to the depth, (o '»
half the greater dUmeter to the altitude of the whole cone.
Or elfe, to the ar^a3 of the top and bottom add the fqi^re
roots of the produfts of thofe areas, and this multiply by
4. of the height of the fruftum for the folidity.
2T. A round mafli-vat meafurcs at the top jt inches ovcr^
within, at the bottom 54, the perpendicular depth being
42 inches, thp content in ale-gallons is required i
As 9 : 42 : : 36 : i68.
72 X 7^ X -7854 = 4071.5136, area of the top of the tun*
168
— = c6 :s T9 altitude of the cone.
3
Alfo 54 X 54 X .7854 = 2290.2264, area of its bottooi.
1 68 — 42 = 126, which — =; 42, altitude of the ptecv
3
wanting.
' Then 4071.5136X56 = 2288004.7616, the wholepyramid.
Alfo 2290.2264X42= 96189.5088, piece wanting.
13x815.2528 cubic inches.
••• 282) 131815. (467 gallons 3t^V P"»ts, the anfwer.
Or,
4071.5136 + 2290.2264 = 6361.74, fum of tl^e areas.
Alfo 407 1.5 1 36 X 2^90.2264 = 9324687.9346.
And -•9334687.9346 = 3P53-63S2-
Then 6361.74 + 3053-6352^ = 94«5-375a-
9415.3752 X — = 131815.2528, cub. in. as befoie«
• .
22. The (haft of a round pillar, 16 inches in diameter at
the top, is about eight of the bottom diameters in height,
4. whereof is truly cylindrical, and the other 4 fwelling 5
but we will fuppofe it tapers ftraight ; and that it is ^ lefs at
top than ^ bottom -, the price of the ftone and workmanihip
is fought, at 3s. 6d. per cubic foot) and farther, th^
fviperficial content, including both ends I
5)
^h^p* V{I. Mea^vremevt tf Soi.ip9. vf^M
5) i6, top diameter.
+ 3a
19.2, bottom diameter*
X8
3) lS3^f>y height.
51.2, cylindrical*
102.4, ^ conical fruftum.
Firft, 19 .2 X 19.2 X .7854= 289.525085, area of thcgrcajcr,
• Alfo 16 X 16 X .7854 = 20 1 -0264, area of the lelfcr bafe.
Sum of the areas 490.588385
Then 289-526 x 20i.o;i64=; 5^212-7924.
v^ 58212.7924 =: 241.273.
Then 49Q.5B8 -J- 241.273 == 731.861.
••' 731-861 X — ^ = 24980.84546, conical fru^-) ^
And 201 .0624 X 5 1 .2 =:: 10294-39488, cylinder i ^
Solid content of pillar = 35275*25034 inches.
Then 1728) 35275.25 (20.414 folid feet.
Anfwer, 3I. 11 a. 5^0. coft, at 3s. 6 d. per foot.
16 X 3.1416 = 502656, circum. of the cylinder*
X9.2 X 3.1416= 60.3187, circum. of the bafe.
2) 110.5843(55.29215.
Then -2^ — = i.6i alfo 1.6x1.6 = 2.56
22 ''
Alfo i02«4 X IC2.4 3= 10485.76
10488.32
i/10488.32 = 102.412 4125, flope height.
Then 102.4125X55.29262=5662.6534, conical. fup^j[f»
Alfo 51.2 X 50.2656 =2573.5987, cylinder.
289.526, bottom area.
201.0624, top area^
8726.8405
Superficial content £726.94 inches 3 ^o.^ fetf, Q. E. F.
23. A
^0i Measurbi^bnt bf SoLibft. Bo(^ Itt.
> •
23. A ftick of fquare timber tapers ftraight \ the fidl^
lit the greater end is 194^ inches, at ^e lefs 13^ inches \ thd
length 16 feet 6 inches \ the value^ at 2 s. 6 d. per foot fo^
lid) is demanded 7
Firft, 19J- -^ 13I *3fe 6» difference of the fides.
3) 36 ( 1 2, the third part of the fquare df that difference*
AXfo i6 feet 6 inches s£ 198 inches, the length.
Then 19.5 x 13.5 + ^^ X 198 =54499*5 cubic inches 73
31.539 feet.
••' 31539 ^ I'S^S = 2-942375 = 3!- 18 s. lod. the
anfwcr.
To meafute a common calk : find the afeas at bead and^
bung ; add \ of the lefs, and \ of the gi'eatef*, for a mean,
area \ this multiplied by the length of the calk is its folidity
in inche^^ which reduce. Or, to double the fqulre of the
bung diameter, add that of the head ; then multiply by th^
length of the caflc^ and divide by 1077.24 for boer, or by
682.42 for wine gallons*
24. What quantity of brandjr will thd diftiller$ tun con-
tain, that meafures xo inches within at the head^ 52 at
bung, and \% 100 inches long ; ani hOw many barrds of'
London ale would fiH it .^
Firft< 40 X 40 X .7854 = 1256.64.
Alfo 52 X 52 X .7854 ss 2123^7^x6.
Then i^i-lii— 418.88; alfo 2123.7216 X^:==: 1415*8144*
Then 14T5.8144 4-418.88 r= 1834.6944.
".' 1834.6944x100= 1 83469.44 cubic inches^ the content.
Alfo 231) 183469.44 (z: 794 gallons of brancly.
And 282) 183469.44 (r= 650} ^Uons =: 20 barrels^
io'~ gallons of London ale. Q^ £. F,
25. The famous tun of Heidclturjgh, that being heieto*
fbfe annually replenifhed with Rhenim, had in it fome wine
that was many ages old, beford the French demoliflied it in
th^ late war : it was 31 feet in length, and 21 feet in diame- .
tcr, jjfid pretty nearly cylindrical j pray how many tuns of
wine would the fame contain t
Firft, 21 X 21 X ^7854 = 346.3614, a^a of one end.
Then 346.3614 x 31 = io737.2(>34?folid feet.
Alfo
chap. Vn. MlSCSLLAN^OUS QufiStlONS.' 593
Alfo 10737.2034 X 1728= 18553887.4752 cubic inches-
231) 18553887 = 80319.8 gallons = 318 tuns, 183^8
gallons, the anfwer.
SECT. III.
MISCELLANEOUS QJUESTIONS.
I* A Detachment of four regiments confifted of 4600
X\ men; Col. A's regiment exceeded Col. B*s by 33,
Col. C*8 by 95 men, and Col. D's by 200 men, how
many men were in each regiment i
4600
33
95
200
4)4928(1131^
113^- 33 = 1199 1 Colonel
1232 — 95= 1 137 I
3232 — 200 =s 1 032 -*
»
2. There are 8000 men in garrifon beiieged, whofe daily
allowance is 24 ounces of bread for 7 weeks ; but the go-
vernor finding the fiege is likelv to continue a longer time,
who can hold out 14 weeks at leaft, though he h^ by this
time loft 1500 of his men, whereby he finds himfelf obliged
to fhorten that allowance of provifions -, how much bread
muft each man's daily allowance be reduced to ^ ^
Recip. 7weeks : 24 ounces :: 14 weeks : 12 ounces.
Then 8000 — 1500 = 6500 men left.
Recip. 8000 men : 12 oz. :: 6500 : i44y. Q; E, F.
3. Required to find the leaft three whole numbers, fo
that i of one, -ry of another, and ,^ of a third, Ihall be
equal ?
Firft, taking | and ,*,.
Then 3 x 14 = 41 ; alfo 5 X 8 = 40.
And I : 7»y : . 42 : 40.
Ctq Then
59*4 MiscBiLANBOvs QuBSTioKS. BooL III.
Then taking ^'^ and -f^v then 5 X 10 = IQO \ and
7x14 = 98.
Alfo 7V : ^ :^. too : 98.
And 98 : 42 :: 100 : 42j.
*•* 40.42 and 42y are numbers in the iame ratio, which
X 7 gives 2809 2949 and 300, whole numbers; thefe'ntti»-
bers -f* 2, gives 1409 147> and 150^ the leaft whole numbers.
Q. E. F.
For 280 X ^^ Alfo i^o x |i
294 X ^ V = 105. 147 X A f = S^s-
300x^^3 ijo X A>
4. An uitfrrer dying, had left the whole fum of hi^ fortune
to be difpofed of in the following manner : To A |> to B .^
to C 4, to D ^j to £ ^9, and to F ^ ; which fums bcuQ{
all paid, the remainder he ordered to be paid to C^ which
was 800 1. Quere the ufurer's whole fum, and what each
had to their (hare? ' *
2 80 ^ 60 I 2C I 10 1
5 "^ 200 10 ** 200* 5 SCO* 20 fOQ* 4Q. ^
— , and — =s -i- their fum beins — 5S — •
aoo' 50 zoo ® aoo. t^ . . tv
..•ii-^=i.=:8ooK
*S *5 ^S -
2 : 800 : : 25 : loooal. whole eftMu, >
: 4000^ A's. , . ^ . ^s
: 3000, B's, L ^
200 : loooo :: ^^5 • W50, Cs + Sop ^OQSft. ic
: 250, Fs^
: 200, F's,
5; A worthlefs mifer, as I'm told,
Had hoarded up vaft ftore of goM;' ""
Laree fums put out to iifury,.
'Tin aged fburfcore years and threcj
, When Death deprived him of his pclf^
And took him from his (econd felf :
Of wives it happen'd he had three> •
I'hree fons, and daughters two had heft
His thiixi wife did furvive him ftiU»
But marie the tenor t»f his will :
or rufty gold)«ten thoufaad pound.
>Vas in this mifer's cofiers fwnd i
Each
f!ach foil muft be paid down in ftore^
Each daughter's fortune three times o'er |
£ach daughter's, as the will was made^
Muft twice the widow's part he paid :
Now the old mifer's in his grave,
Tell oie the fortune each muft have^
I widow I ihare, z 4^ughter$ 4 ihares, and 3 foni
18 fliares j x -f- 4 + ^^ 39 23, diviibr fpr ttie widow's pari*
L 9m d. ^rs.
aj) '^0000(434 J5 7 3^ widpw's pact*
669 It 3 2l^i each daughtex^s part«
;* . 46o8 13 10 3i4, each Ton's part.
6. A ffotie, weighing 40 pounds, is by accident hro]dd ^
JQto/aur pieces^ by which may be weighed anv quantity or
nnmber of pounds, from | to 40 • Qjjiere, tne weight of
taf:h piece ?
jf gitund lkvi.% fir tbe /olution ^Qubstions cf ihit
nature.
To double the firft or leaft ¥rdght, which always con-*
taking die pound; add 1, and it gives the fecond weight :
again, to double the fum of thefe^ two weights, add i, it
produces .the third weight ; and aeain, to double the fum
of thefe three weights, add i , aha we ihall have the fourth
Weight.
Thus I lb. zr firft of leaft weight*
Then will 2 4^ i = 3 s* next leaft weight.
Alfo 3 + 1X2-4* 1=9 = third.
And 9 -I- 3 + t X 2 + I =8 27 = the fourtb4
The fum of which, viz. i + 3 + 9 + 27 = 40-
7. A lovely pair, delight of human race.
Collateral thus their forightly lineage trace ;
A thoufand years are iince their ancient ftem.
Which branching forth, fupply'd the branch to them %
J£ach .male and female, as by what appears,
Liv'd to the age of threefcore and ten ^ears ;
And each iair female brought forth children feven.
In feven iUcoefive years ; the gifts of hcav'n y
(^q 2 From
, » .*
SgS Miscellaneous QyesTiONS. Book III.
From twenty-one to twenty-fcven of age,
A boy, andv girl each year, by turns engage ;
I The teeming mother views them with a fmilcy
Their pleafing innocence her cares beguile :
No jealoufies the parents joys moleft.
But each fond couple is with virtue bleft.
Happy for thofe, who, to no vices blind.
Can virtue choofe, and fuch relations find.
1^0 what amount did alt this kindred thrive^
How many dead, of each fex what alive ?
And of the living, let it next be told.
How many virgins but juft twenty old ?
Firft, 70) 1000 (14 generations.
Then if in '70 years 1 woman be increafed to 3, in 70
more (viz. 140) z will be increafed to 9 ; alfo, in 70 moie
(viz. 2io) 9 will become 27.
••• The number of women after 1000 years will he 3** =s
4782969.
And the number of perfons, men and women, then Kvixig,
will be -J. X 3**, or 7 X 3'' = 11 160261.
Laftly, fuppofing an equal number of all ages to be
living at that time, then ^| x 3'^ = 1366562 women
living under 20.
And yi X 3'* = 1434890, women living under ai.
, or y^ X 3** = 68328 women liviag bt-
• •
7° A
tween 20 and 21.
8. A petticoat of filk, 3 yards, 2 feet, and i inchlong, and
half a yard and 10 inches wide, is fent me to be quilted in equal
fquares of four-tenths of an inch to each fquare fide, and 58
ilitches to be taken in o inches length ; it is requiretd to find
the exa6i number of nitches the petticoat will, take, and
what the work will come to, at 5 s. per thoufand ftitches }
Firft, 3 yards, 2 feet, 1 inch = I33l;nrhcc
Alfo 4 yard, and 10 inches = 28 r"^*^**
Then 133 X 28 = 3724 fquare inches.
And .4 X '4 =-i6, area of one quilted fquare.
.16) 3724.CO (2^275 quilted fquares in all.
Now .4 --f" -4 = -Sj inches of work in evenr fquare.
Alfo 23275 X '8 = 18620, inches of worlc in all the
fquares, bcfides 133 4- 28 = 161 inches for the half border;
viz. 18620 -f- 161 = 18781 wrought inches.
As 9 inches : 58 Ititches :: 18781 : 1 2 1033-5- ftitches.
1000 ftitch. : .25!. :: 121023./ : 30I. 5s. 2d. nearly.
Q.E. F. ^^ o ^ APPEN.
I 597 ]
APPENDIX;
Omtatning the Method of finding the Sums of certain Pro-'
grejjions^ fome Problems in Maxima -and Minimal and
the Inveftigation of the Sums of cert (un Infinite Saries.
SECTION I.
Of finding the fans of any number cf terms in certain pro*
grijfiotu.
PROPOSITION I.
'^O find the Aim of any given number of terms of the
f* feries
1,1,1,1- 'I'l^i'*
: 1 1 1 &c. or h T H 1 &c.
.1.3 a-S • 34 4»5 2 , * 6 ' la " 20
Divide the given number of terms by the fame more i ,
the quotient will be the fum required.
The fum of three terms will therefore be =: -^, that of
4
five = — * and that of ten := — , &c. &c.
0 11
PROPOSITION II.
To find the fum of any given number of terms of the
feries
1.2.3 '. 2.3.^ 3.4.C 4.5.6
Multiply the numSer of terms more i by the fame more
2 \ divide unity, or i, by twice that produdl, and fubtraft
the quotient from ^, the remainder will be the fum re-
quired.
EXAMPLE I.
Let the fum of five terms, viz. 1 -4 1-
1.2.3 ^ a.3.4 ^ 3.4.5
I I
-f* — T— be fought.
4.5.6 ' S'^-7
Then 6 X 7 = 42, and — — ^ = 7^ = 7;, the fum
M T ^ 4 84 3^6 21
required.
• Q^q 3 E X A M-
^98 APPENDIX.
EXAMPLE n.
het the Aim of eight terpos of the above feries be required,
Hcrt 9 X 10 t: 90, and i.^^ * ^ * ij' ^*^
feqtiired.
P R O P O 8 I T 1 O N in.
To find the fuii> of any given number of terqia of th9
1.2.3.4^1.3.4.5 ^3.a*5.6 ■^4.5.67'
Let the number of terms, added to i, 9 and j, te^
fpeSively, be continually multiplied together ; divide unitjr
by 3 times that produ6t, ai^d (bbtrad the quotient from i^y^
iht remainder will be the fum of the terms reqiuitd.
N,
EXAMPLE.
I^t the fum of 20 terms of the above £;rie^ be ^oiig^tr
Then 21 :m: 22 X 23 =^ id(ii6, and JL jl -^^ ■''1.=^ s
18 " 3 X IC626
i^—T — =5 — 2i-, Mual the tuxfi required.
PROPOSITION IV,
To find the fum of any given nuoiber of terms of (he ferie«
> I ' I ■ J * •
1.2.3.4.; ^ 2.3.4.5.6 •^ 3-4m-o.7 ■ 4.5.6.7.8'
Let the number of terms be tncre^fed by i, 2, 3 and 4^
irefpedively, multiply thofe fums contifiually together, di?]de
pnity by four times their produd, and fubtrad the quotient
from -^ the {remainder will be the fqm required.
EXAMPLE.
Let the value of 96 terms of this feries he fought.
Then 97 x 98 X 99 X 100 X 4 » 376437^> and 4 •^
96
I 1764280 ir684C • , ^
*-k ■; = -T — z Z =? y ^^ *€ "»n required.
^764376 361380096 15057504 ^ i—i^.
; It may not be iippropcr to obfcrvc, that the fum of the
t
APPENDIX.
-L 4. -L
1,2 ^ 2.3
+ — + -
3-4 +-5
-L-: 4. J-. 4. -L. + '
i.a.3 * ^ a.3.4 ^ 3.4.5 ^
4 5-^
•^ o
1.2.^4 2.3.4.J 3-4-5«6 4-5-^-7 J
I I ' ; » I « ft • s"
4» >.H r- + — r «c. I C
1.2.3.4.5 • 2.3.4.5.6 * 1.4.5.6.7 ' 4.J.6.7.8 J §
599
t
4-
T
t
T5
PROPOSITION V.
To liiiil the fum of any number of term^ of the feries
j*4-2* + 3*4.4» + 5* &c. or, 1 + 4 + 9+16+25
Let 4, half the number of terms, and 4. of the fquare of
the faid number, be coUe^d into one fum, multiply that fum
hy the number of terms, and the produd will oe the ag-
gregate of the terms required.
EXAMPLE L
Let the fum of fix terms of the abovefaid feries be re-
quired.
^^"m
Then ^ + 3 + ^ X 6 = 91, the fum required.
EXAMPLE XL
Conceive a pyramid to be conftituted of geometrical
iquare flabs, each a foot thick, and fuppofe the bafe, or
greateft flab, to be 20 feet f<juare, the next 19, the next
j8, the next 17, and fo on, it is required to find the folid
content of fuch a pyramid ?
Here - + 10 +
6 • -3
the folidity required.
400 861
X 20 = -^ X 20
2870 ktt^
PROPOSITION VL
To find the fum of any number of terms of the feries
i» + a' + 3' + 4' + 5' &c. or, i + 8 + 27 + 64 +
125 ice.
Let the number of terms more i be fquared, and mul-
tiplied by the fquare of the nurobej: 6f terms ; ^ of this
product will be the fum required.
(Iq 4 EXAM-
6po A P P E N D I X:
EXAMPLE.
Let the value of 8 terms of this feries be required.
Then 9x9x6^ = 5184, and ii-i = 1296, the va-
lue fought.
PROPOSITION VIL
To find the fum of any number of temis pf the fcriej
r+.'^* + 3* + 4* + 5**^c. or, 1 + 32 + 81+256 +
625 &c.
Let ^ of the biquadrate, 4 of A^ cube^ and ^ of the
fquare of the number of terms, be coUedled into one fum,
from which fubtrad X j multiply the remainder by the
number of t^rms, ^nd the prodyiStwill be the fum required.
EXAMPLE.
Let the value of eight terms be required : then +
= ^ — '— ; and i — ^ -^ — X 8 23- . xs
.2*3 30 ' 30 30 "^ "^^ 30 •
8772, the fum required.
PROPOSITION vni.
To find the fum of any number of terms of the ferie«
1' + ^' + 3* + 4* + S* &c. or, I + 64 + 162 +
512 + 1250 &c.
Let 4 of the biquadrate, i the cube, and ^ of the fquare
^f the number of terms, be colled^d into one fum, from
which fubtraft ^ ; multiply the remainder by the fquare of
the number of terms, and the produft "will be the fum of
the terms required.
EXAMPLE.
Let the fum of 10 terms be required : then — + —M
_ 6 * 2 ^^
J X 10* i;qoco J i;;oooo 6 ic8oq4.oo
r-— — = — -- — ; and X 100= 2z2_ -—
12 2 72 72 72 ^^
l?.2o825> the value required.
SECTION
APPENDIX^
6ox
S E C T. IL
A Collection of Problems concerning the Maxima and
Minima of Siuantities.
PROBLEM I.
GIVEN the pofition of the points D and C, in refpefito
the j|iven right line AB, to find the point P, fo that
DP + PC (hall he a minimum.
BC X AB «„ D
Theorem.
= BP,
AD + BC
when DP + PC is leaft poffible.
N.
V
v
c
A
V
i^
EXAMPLE.
Let AB = so, AD = 40, and A! V B
BC = 30, required P B in the above circumftances ?
Then 2£JLi2 — l^z^ %i.i^2%si = PB* and therefore
40 X 30 70 '^'
A? = 28.57143/
PROBLEM n.
Through the given Point P, placed within the right
angle CAB, to draw the ihorteft line (CB) poffible*
Theorem. AD^x PD*I^ = cL
PB, where DP is perpendicular to
AB. \p
EXAMPLE. K
J> B
Suppofe AD = 10, DP = 8, A
required the pofition of the line
CB when a minimum ?
Then 10 x 64 =; 640, whpfe cube root is 8,618 nearly =3
PB, from which the pofition of the line is determined.
PROBLEM III.
Two right lines AC and AB, making the given angle A,
it is required to cut oflF a given area ACB with the fhorteft
J(ne (CB) poffiWe i ^
Theorem.
&>»
APPENDI3C.
Theorxii. x.414^ X v/^ -^ —
AB s: AC9 where a =: the eiven area^
and 1 =: the natural fine of the given ^
angle A.
EXAMPLE.
' Let th< ttiffle A be s 54* aoT, ta
ifM the lengtn of the ihorceft fence
BC, fo as to inclofe juft 50 acres.
Then will s =: .812423 to radius I, and tf =£ 500 fqwe
chains. Therefore \/ g^?^ X f .414a =« 35-07»l6 ^
AB is AC ; wherefore, by Trigonometry,
As fine ^B - - 62* 50' Co-ar. 0.050765
To log. of AC = BC 35 07 »-54493*
So if une Z.A -^ 54 20 9.909782
To log. of BC 31 02 chains i .505481 required,
P R O B L E M IV.
Of all the pyramids AFBDE of a given folidity, to find
that of the leaft fuperficies, excluding &e fquare ba(e ABD£.
' TMtoxtm, Height CP
- 3^^\\ And AB
s= BD = DE ss £A =s
m
EXAMPLE.
To find the dimenfions of a A
fquare pyramid, made with the leaft furface, to contain juft
one malt buftiel, or 2150.4 cubic inches.
Then ^ ^ ^]^^'i ^ 3225.6, whofe cube root is 14-775
2 «
inches, the depth CF ; and therefore ■- y'"' ■ = 436.629,
1 4*77 S
whofe fquare root^is ss 20.89 ~ ^^t ^^ ^^^^ ^^ ^^
fquare bafe. '
PROBLEM
APPENDIX.
fi0$
PROBLEM V.
Td dettaniii6 the tfimenfidns 6f A eylkkbr ABCD, opon
«t the top, (o ta to contain any duantitv of liquors irairt,
&c. and to htv6 th6 leaft internal fupernciesy or, which is
the Tame things W be made of th^ leaft metal of a giTeo
Ihidufiefi.
Theorem. ABs=2X
ioHdi
fy
3.1416
MdAD(==BC) = 4AB:
EXAMPLE.
To find the dimenfions of a cyi-
lindric btifhel, made of the leaft quantity of metal» of %
(iven tbidcnds.
Here r-j^ » 684.492, and 6^4.492!^ X a = 17.626
jncbes = AB the diameter; -.- the depth DA zz 8.813 incbei*
PROBLEM VL
To find that fruflum of a eofie, of a girto bafe and
latitude, which moving In diredioA of its axis, with ita
lefler end againft the parts of an homogeneous fluids (hall
fuScT the leaft refiftance poffible from it.
'TMaonart. EA**g o
EO + v/lSVCB^
^, by fiipilar triangles,
AE : AD : : BC : FG.
EXAMPLE.
'•••r..
•••..^
.^
Let the bafe CB s 6,
imd altitude ED ss gjB"^
then, pcrTbe^nm^ y/ ^ '^ s= 9 = EA; V 9
(ssEA) --» 8 (s ED) SB I ts DA. Therefore) a : x :: 6 :
7 S(» T (? GI^' Whcacf tJi« fruftMm CGFB it deMrmined.
PRO.
6o4
APPENDIX.
PROBLEM VII.
Let P be a pulley hanging freely at the end of a cord AP,
fattened at A j and let W be a weight connected to the cord
CPW, put over the pulley P, which cord is fattened at C,
fo that .the points A and C lie in the fame horizontal line
AC. Now, if the pulley and cords be fuppofed to have no
weight, it is required to find in what place the pulley will
fettle or come to reft i
TheOhem.
4AC —
AB ; from which point B, if a
perpendicular be let fall, will be
a tangent to the pulley P, and pafs
thirough the center of gravity of
the weight W> when that and
the pulley come to reft.
w.
EXAMPLE.
• ' >
Let AC = ID, AP s= 8, required ABf
required.
PROBLEM VIIL
To find a point P, in a right line conneding the centers
a and b of two fpherical bodies T and L, of given diameters
Attd denfities ; at w^'ch, if a third be placed, U ihall be the
leaft fubje£l to their joint attradion*
P
^HEoftEM. Let the quantity of mattter in the body T,
be to that hi the body- L, (which will always be found by
their givn diametera and denfities) ^s M to i> when
A P P E N D I X. 605
I
Will ^*^^y = aP, the diftance of the point P from the
I + M^
center a of the greater body T.
EXAMPLE.
Suppofe the mean diftance of the Moon and Earth to be
equal to 240000 miles, and the quantity of matter in the
Earth to that in the Moon, as 40 to i ; required to find
- where a body muft be placed in a right line conneding their
centers, fo ai to be the leaft attra^ed by thefe two planets?
,. Firft, 40^;= 3.42 very near; and by the Theorem,
' , 4 Q Q .^ 3-4 ^^ 185701.^ miles, the diftance of the re-
^ quired point from the Earth's center ; and, confequently,
54298.7 miles, equal its diftance from the Moon's.
PROBLEM IX.
The latitude of Sthe place and fun's declination being
given, to find what time of the day the fhadows of perpen-
. dicular objef^s move the floweft.
Theorem.
Ffom the natural fine of the siven latitude fubtrad the
. Xquare root of the difference of the fquares of the natural
fines of the latitude and declination ; divide that remainder
, by the natural fine of the declination, and the quotient
will be the natural, fine oT the fun's altitude at the time re-»
quired ; from whence the time itfelf will be readily found.
EXAMPLE.
At what time, on June loth, 1765, will the (hadowof a
perpendicular obje£l move the iloweft at Spalding * fn Lin*
colnlhire ? * Latitude 52* 46 N. i.
The fun's declination, on Jiuie 10, is 23* 5' N. its nat.
* fine = .39207, to rad. i. The nat. fine of 52* 46% the
gjyenlat. is = . 796178. Then by the Theorem 796178* "^
.39207*=: .480180522784, whofe fquare root is = .69295
nearly, and^^ — * 9 95 ^ 2632807 =: the natural fine
' .^9207 •* ^'
of 15* 16/ nearly, the Sun's altitude at the required
time, whence the time itfelf is found to be 40' paft five in the
morning, or 20' paft fix in the evening, very near.
PRO-
€o6
A J* ? JE N 0 I X;
\.
P R O B I. E M X-
Bii^g tbi PftizB-Qu&STiON in the MatrematicaI. tAjk^
GAZiNBy Numb. IV«
Let AC and CB be
two eiven incIinM planes^
mnd let the given weight
iv be fiippofed to deicend
dong CB, whilft %f (be-
ing conneded br a firing
iDovts^ ' paratla to the
planes over the pulley P)
aicends along AC $ it is itquired to determine die weigfa€
of 9» io that its momentum^ i^ tbefe prcyioftaactii miff
be iht gri^aj^ poffible ? ^ -'
PtttCBs=:«»CA = ^, andCDs/j tbtft^ |«t 1(Icc&|k
«tef,* ftf '• c : : w :) — vvffl dxpfcfe the fcjcc ^lofth ni^
the weight tc; tendd to defceacf along i:be|»)i|»^ ^ $. jpq4 f A^^
c : : «r :) -^^ thfat of v along the plane CA i therefore — — •
•-•will be as the efficacious forp6 nrflerewithAe wqditt
ire accelerated: this divided by w 4. «, thux quMsity pf
«natter, and there arifes _■' ',.-36=5 «>' «icif ' GomoHp
itf X w+*^
wfaichy
imw
••■•-■•i
sc
ihaximum.
In Fli*aions it will be wV/ + ^awvy-^hl^vzii 0^ wJU^
equation fohrcd, gives v =5 ^ _!_ — i X <^;<
COROLLARY. _
If the inclinations of the places be equal| then ^I v
V^2««- 1 X W 3 '4142 W.
• \
SCHO-'
ScHOtlUM,
^ If wrr the weight of the amofphere, preffing upon the
pifton in the cylinder of an engine for rkifuig water hf
fire, and tr = the quanthy of water raifed at one ftroke of
the great beam ; then will v = VT — i X tc;, likewife ;
or w : tf : : I : .4142, when die engine prodaces tb«
greateft tSk& in a given time.
^H.,,,. . • , S E C T. III. •
Of tie Inv^^atim ef the Sum rf eertMii.ia6»i»
Scries,
♦ y f
£'..'", i Vj . -T. -, !
PrUT;^.= 'J-r^^J :53 * + ^ + jta;r + ^iUfC. WhW^
t^itiM^^^^ + ^ + tjt.^ &e. terftiply
each Sde of the laft eqiutmi by /*, and there arifai.
K.
^-•■+i.\ i-^"*"** / + ^i •+n
i . • - « 34
faints ate 21^1. -»the fluent of t Z, or^ ; ^
jr + 5 '^'' ,; + * »^+3 S^-^-f;'^
&c. Butf ., *
4'« + 5 . < -H ;^ ;y -^ 1 — T^Ti X * i' -h
r ^
^ 608 A P P E N D i X^
•+* «+i
to Xj is = the fum of the infinite feries ^ U ^ , 4.
«+4 «+5
X X
— -| — ' &c. Suppofe^ noWf ;r = i» and the
3.» + 4 4» + 5
above feries's will become — L— v ■' ■ 4-i.+ — ? —
* « + l n + i^ m^ n—t
+ '■ . . to I, equal to — -— +
n
I
» + * a-w+j 3*^+4
&c. ad infinitum*
4-« + S
Therefore, taking ir rs o, or any pofitive integer, we
can obtain the fums of as many infinite feries's of this kind
as we pleafe. Thus,
Suppofe ff =: o, then the fum of the infinite feries
— -I \r 1--^+ "" &c. will be equal i, as ob-
ferved in Chap L'
. Taking n r= i, then the fum of the infinite feries
•^ + -^+^ + -^ + — i^ will be eqiial i
12. ^2.4 ^ 3.5 46^ 5.7 ^ 4
Taking n. = 2> then the fum of the infinite feries
J.4 : 2.5 " 3.6 4.7 ' 5. (J ih
Again, taking » s= 3, then the fum of the infinite feries
.— + -^+-^ + r5 + — ^' wUl be = ^.
1,5 ' 2.b » 3.7 4,8 * 5,9 4g
Affume V = — ^ = ^ + Jf**^ + ^ X fcc. then v^zx
-I h — + — &c. Multiplying each fide of the laft
equation by ;ir'Ar, we fhall have vx^x z=zx*'*'^x + — +
&c. whofe fluents ate ««• the flu«
^■^
7 ^ «•+•!
ejit
1
^9
or — — — 4- the fluent 4>f —-— v
*""'5x tox---v. Therefore the fluent o*^
— : — X >^ "~
• + 1 1
• . .'to JT — Vi coafequendy, -^ 4. -.^p. X i~ 4-
*•-• . «—♦
« — i*^«^ir4 * • * • ****" = *•»« ^'"" of the infinite
I.+.2 3^+4 5.jrfS 7^+r"**
Tttiag t-s. 1} then wHI ' \ + - + -i— +
• to I cs the fum of the infinite fcrics .— 1 -
+ ■ + ■■—■„>» &c. vdiere « may be any number
• >« + 4 5^+8
ui this progreffion, I, 3> 5> 7» 9> &c.
Suppofe IT s i^ then the fum of the infinite feries
Taking n s 3, then the fum of the ihfinite feries
— +~ + ^-+~+~» &c. wiUbesx'
1.5; 37 S-9^ 7.1 1 T^g-^S ^
Takmg n s 5» then the fum of thf infinite feries
Again taking « ss 7^ then the fan of the infinite feries
^+3":^+5n3+rr5+^**'*=-*'"'«»=^^v
And proceeding thus, by taking » = 9, ii, 13, &c.
agreeable to what is above fpccified, the reader may fum
as many feries's of this kind as he pleafes. Moreover, if
we afiume other values of 'v, we can, with equal facility
tend this met hod much further.
R r .And
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11