|
ARTES
1837
SCIENTIA
LIBRARY
VERITAS
OF THE
UNIVERSITY OF MICHIGAN
TCEBOR
SI QUERIS PENINSULAM AMINAM
CIRCUMSPICE
RECEIVED IN EXCHANGE
FROM
Prof. L. C. Karpinski
في
Richard Came His Brock
March the 18 day One Moufad
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A
NEW and COMPENDIOUS
SYSTEM
O F
Practical Arithmetick.
Elizabeth Came Her
Book Apn
25-1789
Ann Came Her Book.
Sariguest 6.. 170 9 123456
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Ann Frrowell
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1738
4
A
NEW and COMPENDIOUS
SYSTEM
O F
Practical Arithmetick.
Wherein the DOCTRINE of
Whole NUMBERS and FRACTIONS,
BOTH
VULGAR and DECIMAL,
I S
Fully Explained, and Applied to the feveral
RULES or METHODS of Calculation ufed
in TRADE and BUSINESS:
And by fhewing and comparing
The natural Dependance upon, and Agreement of,
one RULE with another, the Whole is render'd
more eaſy than heretofore, and the LEARNER is
inſtructed in the Vulgar and Decimal Operations
together, which at the fame Time demonftrates the
REASON as well as the PRACTICE of both.
By WILLIAM PARDON, Gent.
LONDON:
Printed for RICHARD WARE, at the Bible and Sun in Amen-
Corner, near Pater-Nofter-Raw, M.DCC.XXXVIII.
[Price Four Shillings and Six-Pence.]
3
1
Prog. L. R.
C. Karpinske
4-30-1924
THE
PREFACE.
HE great Number of Books, that are
already extant upon this Science, might
be a reaſonable Objection against this,
if the Author could not affure the young
Student, that the Method this is done
in, renders it much easier to be under-
food, and confequently fooner to be acquir'd, than by
any other whatever. The Doctrines of Vulgar and
Decimal Arithmetick are here fo adapted and com-
pared together, as to shew the Excellency and Uſefulneſs
of both; and to make every Branch the clearer, the
fame Question is often wrought various Ways; by which
Means the Learner is let into the Reafon of the fe-
veral Contractions made Ufe of, and fees the Bafis up-
on which all fuch like Operations depend; and by be-
ing effectually inftructed in the Doctrine of Vulgar Frac·
tions, and the Ufes they are or may be apply'd to, is
taught from the fame Principles to make as many more
himfelf, as his Occafions may at any Time require. But
above all I would feriously recommend the Study and
Practice of Decimals, whofe fuperlative Excellence may be
feen by the feveral Applications exhibited in this Book,
and particularly the Contract Method of Multiplication,
I
zubick
iv
The PREFACE.
which for its Eafe and Expedition deferves to be uni-
verfally practis'd by all Perfons, where it can be ap-
ply'd. Another Reason for the Ufe of Decimals, is,
that in all fix'd Cafes, you may turn your Divifor into
a Multiplier, which will generally expedite the Work
fo much, and render it fo eafy, that every one that
tries it will foon find its extraordinary Uſefulneſs, eſpe-
cially now the Methods of managing, repeating and
circulating Decimals are fully known.
It must be acknowledged indeed, there are many
curious Treatifes upon particular Branches of this Art,
for which the feveral Authors deferve both the Thanks
and Encouragement of the Publick, but I think I may,
without the leaft Diminution of another's Glory, ſay,
none has handled this whole Subject in fo natural, full
and plain a Manner, fuited to the Capacities of the
Ignorant, as this now prefented to you; from whence
I prefume I may promife the Learner, that if he will
but carefully study the Rules, and mind the various
Methods of doing the fame Thing here exhibited, be
muft, in a little Time, be a much greater Proficient
in this Art, than he can be by using any other Book
whatever.
THE
THE
CONTENTS.
HAP. I. The Definition of Arithmetick, and
from Page 1 to 7
C Notation of Numbers,
CHAP. II. The Addition of Whole or Abstract Num-
bers, and alſo ſeveral Methods of finding the Total
Amount of Sums of diverfe Denominations, and the
Proof of thofe Operations; under which Head Mul-
tiplication is alſo taken in, from Page 7
66
CHAP. III. Subftraction of Whole Numbers, and alſo
of diverfe Denominations, under which is con-
tain'd Divifion, and the Proofs of the feveral
Operations,
from Page 67
105
CHAP. IV. Reduction, or feveral Methods of
changing of one Kind or Species of Money, Weight,
Meaſure, &c. into another, ſtill retaining the
given Value,
from Page 105 - 120
CHAP. V. The Rule of Proportion, commonly called
the Rule of Three,
from Page 120 130
CHAP. VI. The Definition of the various Sorts of
Vulgar Fractions, and the ſeveral Ways of uſing
them,
from Page 131 165
CHAP.
¡
1
The CONTENT S.
CHAP. VII. The Definition of Decimals, and the
various Manners of Operation by them,
CHAP. VIII. The Rule
verfe Ways, both
from Page 165
211
of Practice wrought di-
Vulgarly and Decimally,
from Page 211-
- 252
CHAP. IX. The Inverfe Rule of Three, and the
Application of the Direct Rule of Three, in what
is called Fellowship, Barter, Exchange, &c.
Progreffion,
from Page 252-324
from Page 325
334
Combination,
from Page 334
341
CHAP. X. Of raifing Powers and extra&ting Roots,
Simple Intereft,
Compound-Interest,
from Page 341
359
from Page 359
390
from Page 390 - 397
OF
O F
ARITHMETICK.
CHAP. I.
Containing the Definition of ARITHMETICK, and
the NOTATION or Manner of expreffing Numbers,
both in Words and Figures.
RITHMETICK is the Science, Art or
Knowledge of Numbers, with their various
Applications in Numbering.
ly be decided, viz.
Number is that by which we find out,
know, or exprefs how many equal Parts
any whole Thing or Quantity is made up
of: And here the great Queſtion fo long
agitated among the Learned, will proper-
Whether Unity or One be a Number?
Euclid is frequently brought in as a Party favouring both
Sides; but that in the Nature of the Thing is impoffible:
For nothing is clearer than the Affirmative, or that Unity is
abfolutely a Number, and not, as fome imagine, the Begin-
ing of Number. For,
First, Whenever we exprefs the Quantity or Aggregate of
any whole Thing, we call it Unity or One, let it confift of e-
ver fo many Parts; as one Shilling, tho' it contains Twelve-
B
pence;
2
Definition of ARITHMETICK.
pence; or one Penny, tho' it is made up of four Farthings;
or one Pound Sterling, or Engliſh Money, which is made up of
twenty Shillings, or two hundred and forty Pence, or nine hun-
dred and fixty Farthings, &c. Under thefe Circumſtances
no Body will fay that Unity or One is not a Number; for fure-
ly, if the Parts 12, 4, 20, 240 or 960 are Numbers, the
Total compounded or made up of thofe Parts, muſt be a
Number, as well as the Parts compounding: For, according
to the fyllogiftick Method of Reasoning:
The Whole is of the fame Nature, Matter, Quality or
Subſtance with the Parts 'tis compounded of.
But Unity may be confidered both as a Part of a Multitude
of Unities, and alfo as an Aggregate or Total.
Therefore Unity is of the fame Nature or Quality with a
Multitude, Total or Aggregate of Unities.
But the Matter or Subſtance of a Multitude of Unities, is
Number.
Therefore Unity is Number.
Now if any one denies this, let any Number be given, as
fuppofe 3. From this let Unity or I be fuftracted or taken
away, three diftinct Times. And if Unity be nothing, or
no Number, then 3, or the given Number, will ſtill remain;
but, this is ſo evidently falſe and abfurd, that 'twould be idle
to ſay any more about it. Therefore we conclude that Uni-
ty or 1 is a true, real and abfolute Number.
And indeed, let any Number given be ever fo great, it, pro-
perly ſpeaking, is but Unity or one Total, Aggregate, Sum, or
whole Thing, made up of a certain Quantity of leffer Uni-
ties or Parts; as a Thoufand, a Million, &c. And for the
eafy Expreffion of the Quantity of Parts, any Whole is made
up of, both in Words and Symbols, many different Methods
have been invented; fome by Letters of the Alphabet, and
others by Characters appropriated to Numbers only; which
latter, as they are both more commodious, and alfo univerfal-
ly received, fhall be here explained, viz. 1, 2, 3, 4, 5, 6,
7, 8, 9, 0. which virtually contain all that is capable of be-
ing performed in or by Numbers, according to the common,
and indeed beft and moſt univerſal Scale now in practice:
For as all Operations in Numbers are only the adding
or fubftracting one or more of theſe Quantities or Numbers,
to or from one another, fo you are to take Notice that the
faid nine Figures exprefs fo many different whole Things,
'confifting of fo many Parts as are reprefented by the Charac-
ter made ufe of, viz. I fignifies fome one whole Thing or
Part,
Of NOTATION.
3
Part, abfolutely, without any Subdivifion or Parts of which
it is made up; as one Shilling, one Pound, one Yard, &c.
2 fignifies ſome Thing or Number made up of two equal
Parts, as two Sixpences make one Shilling, &c. 3 reprefents
a whole Thing of three equal Parts, 4 fomething of four e-
qual Parts, 5 of five equal Parts, 6 of fix equal Parts, 7 of
feven equal Parts, 8 a Thing or Number made up of eight
equal Parts, and 9 fomething of nine equal Parts. The other
Mark or Character, o, fimply of itſelf fignifies nothing, but
in plain or pofitive Arithmetick is uſed to increaſe or repeat
the Value of the fignificant Figure, Ten, a Hundred, a
Thoufand, &c. Times, according to the refpective Place or
Places it occupies; and in artificial or negative Arithmetick,
commonly called Decimals, it decreaſes the Value of the fig-
nificant Figures in a decuple or tenfold Proportion.
In the Practice of Arithmetick there are two principal Parts,
the firſt called Notation, the ſecond Numeration. Now the
common Method in Schools is (tho' falfly) to call Notation
by the Name of Numeration, which you will preſently be
taught, is quite another Thing.
NOTATION is that Part of Arithmetick that teaches
you to note or write down in Characters, or expreſs in Words,
how many equal Parts, or Unities, are contained in any whole
Number or Thing whatever.
You are to obferve, that the commonly receiv'd Method of
operating Numbers, is by the decuple Scale, or that wherein
the fame Figure is increaſed ten Times, by being placed or
fet one Stage or Place more backward, toward the Left-hand,
than it or another of the fame Value ftood in before; but for the
eafy Expreffion of Numbers in their feveral Places or Stages of
Value, there are appropriated particular Words or Names,
viz. Units, Tens, Hundreds; that is, all the Figures that ftand
fingly or alone, are called Units; when uſed, applied, or con-
fidered either abftractedly, as Numbers only, or apply'd to Bu-
finefs, when ſome Adjunct is joined to the Number to fpeci-
fy the particular Species or Goods intended: Thus, 1, 2, 3,
4, 5, 6, 7, 8, 9, are fimply Numbers, or they may be fo
many Crowns, Kingdoms, Men, Sheep, Plumbs, Apples,
&c. Some call theſe fingle fignificant Figures Digits; but
when they are compounded, either by repeating the fame Fi-
gure, prefixing one or more Noughts or Cyphers before them,
or mixing them indifferently together; the firſt Place to-
wards the Right-hand, is called the Units Place, or the Figure
B 2.
fimply s
4
Of NOTATION.
fimply; the ſecond towards the Left-hand, is fo many Tens
as the Figure expreffes fimply Units; and the third Place to
wards the Left-hand, is fo many Hundreds. The fourth,
fifth, and fixth Places, are a Repetition of Units, Tens, Hun-
dreds again; only there is added the Word or Term Thou-
fands. So the feventh, eighth and ninth Places, are again a
Repetition of Units, Tens and Hundreds, with the Word or
Term Millions added to them. Now this Variety of Terms
or Names, is a modern Invention, and is not abfolutely ne-
ceffary, but is now generally us'd, becauſe in Practice 'tis
found more commodious than the Repetition of the former
Terms or Expreffions many Times over, as was the Cuſtom
of the Antients in large Numbers; who us'd to fay a Thouſand
Thouſand for a Million, or ten Thousand times ten Thousand
for a hundred Millions: For 'tis the fame Thing if you fay Tens
of Tens, inftead of Hundreds, and Tens of Tens of Tens inftead of
Thouſands. Or you may fay Tens of Hundreds inſtead of
Thouſands, and Hundreds of Hundreds inftead of Tens of
Thouſands; but the commonly receiv'd Method is both
fhorter and clearer, and therefore you ſhall fee it fully ex-
plain'd by what follows.
But, first, I muſt inform you, that all Numbers are made
by a continual Addition of Unity to itſelf; and the Numbers
fo arifing are called as follows.
I one, 2 two, 3 three, 4 four, 5 five, 6 fix, 7 feven,
8 eight, 9 nine, 10 ten, II eleven, 12 twelve, 13 thirteen,
14 fourteen, 15 fifteen, 16 fixteen, 17 feventeen, 18 eighteen,
19 nineteen, 20 twenty, 21 twenty one, 22 twenty two,
&c. till you come to 29 twenty nine; then you ſay 30 thir-
ty, 31 thirty one, &c. till you have gone thro' all the figni-
ficant Figures; and then you ſay 40 forty, 41 forty one, &c.
and then 50 fifty, 51 fifty one, &c. 60 fixty, 61 fixty one,
&c. 70 feventy, 71 feventy one, &c. 80 eighty, 81 eighty
one, &c. 90 ninety, 91 ninety one, &c. then 100 one hun-
dred, 200 two hundreds, &c.
la la la la
I ---- 2 2 2 ---- 3
5.0·7
ة 8 9
5---7 0 0
Here the firft Ternary or Article confifting of three Figures
are all Ones or Units, and as they ftand compounded, and
each
Of NOTATION.
each of them reprefents one Sum, they are to be read thus
beginning at the laft or third Figure towards the Left-
hand, I find it to be an Unit or One, and over it is wrote
the Term, Name, or Word Hundreds; therefore I call it one
Hundred. The fecond Figure is alfo an Unit, and the Name
Ten being over it, I call it one Ten; which compounded with,
or added to the Hundred makes one Hundred and one Ten, or
an Hundred and Ten. The next Figure towards the Right-
hand, ftanding in the Units Place, and being alfo One is fo
called; and being compounded with or added to the other
two towards the Left-hand, and fo confidered as one whole
Number or Sum, it makes one Hundred, one Ten and One;
which in common Language would be called one Hundred
and Eleven. And this Number expreffes any whole Thing
or Quantity divided into fo many equal Parts: As, fuppofe
a Purfe or Sum of Money of an Hundred and Eleven Guineas;
a Ship whofe Complement of Men was an Hundred and E-
leven, or the great Guns in a firſt Rate Man of War may be
one Hundred and Eleven: And fo of any thing elſe.
The fecond Ternary or Article of three Figures are all Two's,
and in like manner are to be read, two Hundreds, two Tens,
or twenty, and two; which is read thus, two Hundred twenty
two; and this repreſents any whole Thing, compounded or
made up of fo many equal Parts.
The third Ternary is made up of all different Figures,
which are to be read or valued thus; the firft Figure to-
wards the Right-hand ſtands for its own fimple Value, viz. five,
becauſe it ſtands in the Units place; the ſecond towards the
Left-hand is four Tens, or Forty, becauſe it ſtands in the Place
of Tens; and the third is three Hundreds, becauſe it ſtands in
the Place of Hundreds; and the whole taken together, as
one Number or Sum, muft be called three Hundred forty five.
The fourth and laft Ternary towards the Right-hand has
but one fignificant Figure, which is that which ftands in
Place of Hundreds; and therefore that Number or Sum muſt
be called feven Hundreds only.
There are two other Ternaries put underneath the fecond
and third Ternaries already explained, v.z. 507 and 980;
the firſt of which muſt be read five Hundred and feven, be-
cauſe o fupplies the Place of Tens in that Ternary; and the
other muſt be read nine Hundred and Eighty, becauſe o fup-
plies the Place of Unity, and therefore there are only Hun-
dreds and Tens, and no Units. If you would number more
Figures than three, do as follows; firft, part them into
Ternaries,
6
Of NOTATION.
Ternaries, and obferve what additional Name or Term is
put to it, befides the Units, Tens and Hundreds ; as here:
Of Thouſands of
Millions.
Of
Millions.
Of Thou-
fands.
Hundreds
Tens
Units
7 9 5
86
324
The fourth, fifth and fixth Places, which is the fecond
Ternary towards the Left-hand, are fo many Units, Tens and
Hundreds of Thouſands. The feventh, eighth and ninth Places,
which is the third Ternary towards the Left-hand, are fo
many Units, Tens and Hundreds of Millions. The fourth
Ternary is fo many Units, Tens and Hundreds of Thouſands
of Millions: And fo you may go on to as many Figures
as you pleaſe; but theſe twelve Places or Figures are more
than fufficient for any Thing that can poffibly happen in
Trade or Bufinefs; and therefore I fhall forbear going fur-
ther, and only fet down in Words the twelve Figures above,
viz. Seven Hundred and ninety five Thoufand eight Hundred
and fixty fix Millions, three Hundred and twenty four Thoufand
fix Hundred and ninety eight. --- So 60,000,000,700 ftands for
fixty thousand Millions and feven Hundred; and 9,210,000,000
ftands for nine Thouſand two Hundred and ten Millions
815,900,035 ftands for cight Hundred and fifteen Millions,
nine hundred Thousand and thirty five. Thus by obferving
the Place of any Figure you will preſently know its Name,
and Vice Verfa, by the Name its Place: As fuppofe 'twas
demanded what Figures would exprefs the Number forty
Millions forty Thoufand four Hundred and forty; by comparing
the Names and Places in the above 12 Figures, I find that
40,040, 440 will do what was required.
So if fifty
feven thousand Millions were wanted, I muft write down
57,000,000,000. Thus may you with Eafe and Certainty
exprefs both in Words and Figures any Number you can
poffibly want.
Now follows Numeration, which comprehends all the
Practice of Arithmetick, befides that of Notation; but more
particularly what is called the four principal Rules, viz. Ad-
dition, Subftraction, Multiplication, and Divifion; all others
;
being
Of ADDITION, &c.
7
being but a particular Application or Compofition of theſe.
And as Frade or mercantile Buſineſs has a more immediate
Demand for the Affiftance of this Art, therefore I fhall par-
ticularly have regard fo to accommodate every Branch, that
they, whofe former Education has not enabled them, or whoſe
prefent Leiſure will not permit them to make deep Searches
into the hidden Myfteries of this excellent Science, may be
profitably improv'd in the eafier and more common Parts
thereof.
CHA P. II.
Of the Addition of whole or abſtract Numbers; also the
Method of finding the Total or Amount of feveral
Sums or Numbers of diverfe Denominations.
DDITION is that Rule or Method of handling Num-
bers, by which the Amount, Total or Aggregate of any two
or more Sums or Numbers, either of the fame or of a different Value,
may be found: And this may be confidered under two Names,
viz. Addition and Multiplication, the latter being only a par-
ticular Mode or Manner of performing the former, as fhall
be ſhewn by and by, and therefore ought to ſtand in the ſe-
cond Place, or next to Addition, having no Dependence up-
on that commonly taught in the fecond Place, viz. Subftrac-
tion. And as my Buſineſs ſhall be to fhew you how you may
work, prove, and know the Truth of every Rule indepen-
dently, fo I fhall not fcruple to go out of the common Road
both of Schools and Authors, when 'tis for the Advantage of
communicating Knowledge to the Learner more eafily and
convincingly. This Rule depends upon, and is an Illuftrati-
on of the following Axiom, viz.
The Whole, Total, Sum or Aggregate, of any Number or Thing,
is equal to all the Parts whereof 'tis made up, or compounded.
But every Number confifts either of one Unit or a Multi-
tude of Units; therefore the Adding or Collecting of Units
into one Total or Sum, muft neceflarily produce all other
Numbers above Unity; as I and I make 2; 1, I and I, or
2 and 1, make 3; fo 1, 1, 1 and 1, or 3 and 1,
and 1, make 4;
and I, I, I, I and I, or 4 and 1, make 5; ſo I, I, I, I, I,´
and 1, or 5 and 1, make 6; and I, I, I, I, I, I and I, or
6 and 1, make 7; and I, I, I, I, I, I, I and I, or 7 and I,
·
make
8
of ADDITION, &c.
make 8; fo I, I, I, I, I, I, I, I and 1, or 8 and 1, make
9, &c. For let the Number be ever fo large, 'tis only the
Sum, Total, or Amount of fo many Units added together; as
75 is the Sum of feventy five Units, 659 is the Sum of fo
many Units. The like 'tis of all other whole Numbers what-
ever; but as 'twould be too tedious to work large Sums in
this Manner, tho' 'tis ftrictly true, I will fhew you another
Method which is more expeditious. And as the Knowledge
and ready Uſe of Numbers, like all other Arts, comes by re-
peated Acts; ſo the firſt Thing to be learnt will be to know
the Amount of two fingle Numbers; as of and
as of 7 and 5, 6 and
9, &c. And here the eaſieſt Method will be to write down
progreffively all the arifing Numbers from Unity to Fifty,
thus: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, &c. for you may go on to 100, or more, if
you pleaſe. Begin at 7, and count five Places or Num-
bers from feven towards the Right-hand, and you will come.
to 12; and that is the Sum or Total of 7 and 5. Again,
What's the Sum of 6 and 9? Here I begin and count nine
Places forward from 6, and then I come to 15, which is the
Anſwer, or Sum of 6 and 9. In the fame Manner you may
find the Sum of 3 and 8 to be 11; for counting 8 Places from
3, you will come to II. And here you may obſerve, that
you may begin from either of the two Figures; as in the laft
Example: Suppofe I count three from the eighth Place, I fhall
find 8 and 3 to be 11, as well as 3 and 8. So in 6 and 9,
if I count fix Places from Nine, I fhall come to 15, the fame
as when I counted nine Places from Six. Again, What is
the Amount of 5, 6, 8, 9.6? Here I begin to count fix Places
from five, and find the Amount to be II; then I count 8 Places
from 11, and find that Sum to be 19; then I count 9 Places
ftill forwards towards the Right-hand, from 19, and find
that Amount to be 28; from that Place I count fix Places more
towards the Right-hand, and find that Amount to be 34;
and that is the Sum, Total, or Amount of all the five Figures,
when collected into one Aggregate: For the Proof whereof you
may begin and go backwards with the Figures, thus: Begin
at the Right-hand fix, and find what 6 and 9 is, and the
Table tells you 'tis 15; then counting 8 Places forwards, to-
wards the Right-hand, from thence you'll come to 23, which
fhews that 15 and 8 are 23; from thence count 6 Places more
for the next Figure, and you'll find it come to 29; from
thence count 5 Places more forwards, towards the Right-
II
I
hand
Of ADDITION, &c.
9
hand, and you'll find it come to 34; which is the Sum of
all the five Figures or Parts taken together, as was found be-
fore, when you begun at the Left-hand and went forwards
to the Right-hand: Or you may change or vary the Order of
the Figures in any other manner, thus 8, 9, 6, 5, 6: Or
thus, 6, 6, 9, 8, 5, and they will always make the fame Sum
or Total, as you may eafily try. You ſhould practiſe ſome-
thing of this Nature till you can readily tell the Amount of
any two Figures or ſmall Sums whatever, as foon as you ſee
them or hear them propos'd, which you will quickly be able
to do, it being very eafy: However, if you are doubtful at
any Time, you may try by the Table whether you are right
or no; and the Table you may enlarge as far as you pleaſe.
When there is no Name affix'd to the Numbers given to
be added together, they are called abſtract Numbers; and when
they have a Name appropriated to them, they are called con-
crete Numbers; but the Manner of working in both Cafes is
the fame, as you will fee by the Examples following.
Example
Example 2 Example 3 Example 4 Example 5
Houſes.
Perfons.
Sheep.
Pins.
7
36
234
6287
92547
9
29
567
2587
--176
82
890
1234
9
2
65
156
5678
51476
8
79
728
9012
72300
I
68
987
3456
-5216
5
95
437
7892
--418
38
454
3999
36146
222142
I
The first Example, or that ftanding fartheft towards the
Left-hand, is a Row of fingle Figures, which I add together
after this Manner, beginning at the Bottom and going up-
wards: I fay 5 and 1 is 6, and 8 is 14, and 2 is 16, and 6 is
22, and 9 is 31, and 7 is 38. Now as there is but one Row
of Figures, I fet down 38 for the Total or Amount of the
Whole; and to know whether I am right, I begin at the
Top and defcend gradually, Figure by Figure, faying thus,
7 and 9 is 16, and 6 is 22, and 2 is 24, and 8 is 32, and 1 is
33, and 5 is 38, which fhews the Total to be the fame as be-
fore; from whence I conclude I have truly added the Parts
together, and that 38 is the Sum or Total of the particular
Lines or Figures given; or you may fet the fame Figures in
C
any
ΤΟ
Of ADDITION, &c.
any other Order, and they will always amount to the fame
Total, as was noted before: Now thefe are called abſtract
Numbers, not having any particular Appellation, but you may
call them Yards, Miles, Feet, Tons, Pounds, Gallons, or what-
ever you pleaſe, and the Work will be always the fame; as
appears by the 2d, 3d,4th, and 5th Examples following. Where
in the fecond Example 'tis fuppofed there are feven Streets in
a Pariſh, and that the first had 36 Houſes, the fecond 29, the
third 82, the fourth 65, the fifth 79, the fixth 68, and the
Seventh 95, How many Houfes are in the whole Parish? Firſt
fet down the Numbers one under another, either in the Order
as you fee them, in the Example before you, or in any other
Order, provided you fet down the fame Numbers, and mind
always to keep the Units under the Units, and the Tens under
the Tens, &c. This done, draw a Line under the Figures
fo fet down, and begin at the Bottom, and go upwards, fay-
ing, 5 and 8 is 13, and 9 is 22, and 5 is 27, and 2 is 29,
and 9 is 38, and 6 is 44: Here I have 4 Units, which I fet
under the Firſt or Units Place, and add or carry the 4 Tens
to 9, the firſt Figure in the fecond Row, or Row of Tens,
faying, 4 and 9 is 13, and 6 is 19, and 7 is 26, and 6 is 32,
and 8 is 40, and 2 is 42, and 3 is 45, which I fet down, the
under the ſecond Row, or Figure of 9, and the 4 one Figure
backwarder or more towards the Left-hand, becauſe the laſt
Total 45 is 45 Tens, which is 450; but as there is already a
Figure in the Units Place, viz. 4, I only fet down the 45,
and find the whole Amount or Total to be 454, which is the
Number of Houſes in the Parish: As you may prove by be-
ginning at the Top and going downwards, thus, 6 and 9 is
15, and 2 is and
17, 5 is 22, and 9 is 31, and 8 is 39, and
5 is 44; fet down the 4 Units under the Place of Units, and
add or carry up the 4 Tens, faying, 4 and 3 is 7, and 2 is
9,
and 8 is 17, and 6 is 23, and 7 is 30, and 6 is 36, and 9 is
45, as before; fo that the Total is 454 both ways, which
hews the Work to be right.. Here 'twill be proper to
take Notice, that in the Addition of all Sums that are of one
Denomination, if they confift of ever fo many Lines, they
are always added, and the Totals diſpoſed of in the fame
Manner as the Firft or Units Line, where the Totals or A-
mounts are all fet down, or only the Units fet down and the
Tens carried to the next Line towards the Left-hand, which
Method muſt be continued till all the Lines, be they ever
fo many, are gone through; as you ſee done in the third,
fourth and fifth Examples following. The Reaſon is, thạt
5
every
Of ADDITION, &c.
I I
every Place or Figure towards the Left-hand, is ten Times the
Value of the fame Figure in the Place next to it, on the
Right-hand; for which Reaſon you need not regard whether
they are Tens, Hundreds, Thouſands, Millions, &c. as you will
fee by what follows. Now fuppofe again, That in all the
Houfes of the firft Street, in the Example foregoing, there lived
234 Perfons; that is, Maſters, Miſtreſſes, Children and Ser-
vants, &c. taken together; in the fecond Street 567; in the third
Street 890; in the fourth Street 156; in the fifth Street 728; in
the fixth Street 987, and in the feventh Street 437; Query, How
many Perfons are in the faid Parish? Here, after you have ſet
down the Numbers, as in the Example, you may begin at
the Bottom, faying, 7 and 7 is 14, and 8 is 22, and 6 is 28;
and as the next Figure is o, and therefore does neither in-
creaſe or diminiſh, I paſs it over, and go to the next Figure,
which is 7, and fay, 28 and 7 is 35, and 4 is 39: Here I fet
down the 9 and carry the 3, which is 3 Tens, to the next
Row, which is the Row of Tens, faying, 3 that I carry, and
3 is 6, and 8 is 14, and 2 is 16, and 5 is 21, and 9 is 30,
and 6 is 36, and 3 is 39: Here I fet down 9 under the 3,
becauſe they are 9 Tens, and carry the 3 forward to the next
Row, becauſe they are Hundreds, and fay, 3 that I carry
and 4 is 7, and 9 is 16, and
is 16, and 7 is 23, and 1 is 24, and 8 is
32, and 5 is 37, and 2 is 39: Now theſe being 39 Hun-
dreds, I fet the 9 under the Hundreds Place, and the 3, one
Figure more to the Left-hand, and then the Total will con-
fift of 4 Places, which in all amount to 3999, which is the
Number of Perfons in the Pariſh; as you may eafily prove
by beginning at the Top and adding downwards, as you
have been already fhewn. ---- The fourth Example may be
the Number of Sheep, that are in Poffeffion of feven Grafiers.
Query, How many are there in all? By operating in the
fame Manner, as you was directed in the laft Example, till
you have gone through all the Lines, beginning either at the
Top or the Bottom, you will find the Total come to 36146.
But that you may be perfectly inftructed in the Reafon
of fetting down the firft Figure of the Amount of each ſingle
Line, and carrying the Second forward to the next Row, to-
wards the Left-hand, you are to obferve, 'tis only to fave.
Trouble, and not abfolutely neceffary; for in this fourth
Example, you may find the Total thus: Firft, fee how many
Places or Rows are in the Example, which here is 4; there-
fore I make fo many Dots, thus, . or fet down fo
many o's, to reprefent the Places; under which I draw a
C 2
Line,
12
Of ADDITION, &c.
36
- 410
2700
33000
011 46
35
•
36146
3 un-
Line, as in the Example here annexed; then I caft up the firſt
Line, and find the Amount to be 36, which I fet
down, the 6 under the Units Place, and the
der the Tens, becauſe the Number 36 confifts of
3 Tens and 6 Units: Then I caft up the ſecond
Line as if it ftood alone; and I find the Total of
that Line to be 41. Now becauſe theſe are all
Tens, I make another Line, and fet the I under
the Place of Tens, and the 4 which is 40
Tens, under the Place of Hundreds, becauſe 10
Tens make one Hundred; and confequently 40
Tens are four Hundreds. Then I caft up the
3d Line, and find that Total to be 27. Now theſe
being 27 Hundreds, I fet down in another Line,
the 7 under the third or Hundreds Place, and the 2, which is 20
Hundreds, under the 4th or Thouſands Place, becauſe 10 Hundreds
make one Thouſand, and confequently 20 Hundreds make two
Thoufand; then I go on to the fourth Line, and in the fame
Manner find the Total of that to be 33; and becauſe theſe are
Thouſands, I fet one 3 under the Thoufands Place, and the
other one place more backward towards the Left-hand,
becauſe it reprefents Tens of Thoufands; and filling up the
void Spaces with Dots or o's, I add theſe four Lines toge-
ther, and find the Amount to be 36146, as before. And thus
you may find the Total of any Number of Lines or Figures
whatever; only you must be very careful to put the Units
under the Units, and the Tens under the Tens, the Hundreds
under the Hundreds, and the Thouſands under the Thouſands,
&c. as you fee in the fifth Example, where, to prevent Con-
fufion, you may fill up the void Spaces with Dots or o's,
which neither increaſe or diminiſh the Value of the other
Figures, but ferve only to make up the Number of Places
any Figure is fuppos'd or defign'd to ſtand in; as in the 5th
Example, the two oo's in the 3d Line, have no other Uſe
than to fill up the Tens and Units Places; and ſo of any o-
ther, only you are to Note, that in whole Numbers they are
always put on the Right-hand, to fill up the Number of
Places, and to determine the Value of the fignificant Figures
on the Left-hand: Now this fifth Example, you may ſuppoſe
to be ſeven Parcels of Pins, and by proceeding in the fame
Manner as you did in the fourth Example, you will find the
Total to be 222142, which is two Hundred twenty two Thou-
fand, one Hundred and forty two. Thus you may go on with
any Number of Sums, be they ever fo large; as fuppofe I had
fifteen
Of ADDITION, &c.
13
fifteen Sacks of Malt, that had the following Numbers of Grains,
in each, How many Grains are there in all?
In the firſt Sack
fecond
third
fourth
fifth
fixth
-
feventh-
<
Grains
295068
- 316845
-
1827961374576
278868
300999
341628
288977
eighth
- 2912341501714
ninth
279876
tenth
299999
eleventh
311122
twelfth
300096
thirteenth
3149441506189
fourteenth - 296682
fifteenth
- -
-
283345
The Total is
4382479--4382479
Here ſetting down all the Sacks, and the Number of
Grains in each, you may begin at the Bottom, and ſo go on
from Figure to Figure, till you come to the Top, ſetting
down the Units of each Total under the Line laft added, and
carrying or adding the Tens to the Line next towards the
Left-hand, till you have gone thro' all the Lines: Or you
may begin at the Top and defcend gradually to the Bottom,
in the fame Manner. Or (which perhaps may be eaſier) you
may break ſuch large Sums into two, three, or more Parts at
your Pleaſure, and add each Part by it felf, and then add
thoſe particular Totals together, and their Total will be the
Total of the Whole, according to the foregoing Axiom, as
appears by the above Example, where you have the Work
done in one Sum, and by three particular Sums, the Total
of whofe three Totals being found, make both the fame, viz.
4,382,479. Or you may add continually only two Lines to-
gether, till you have taken in all the Lines given, and
'twill produce the fame with any other of the Methods
here propofed, and muft indeed be very eaſy to effect; but
will require more Time to perform, upon account of
the Multitude of Figures neceffarily increafed by the great
Number of particular Totals that will arife, eſpecially where
2
the
·14
Of ADDITION, &c.
鲁
​the particular Line or Sums are many: However you ſhall
fee one Example wrought in this Manner, viz. the fourth
aforegoing, where the
Firſt Line is 6287 Sheep, beginning at the Top.
Second
Third
Fourth
Fifth
Sixth -
Seventh
3587
8874 Total of the firſt and ſecond Lines.
1234
10108 Total of the 1ft 2d and 3d Lines.
5678
15786 Total of 1ft, 2d, 3d and 4th Lines.
9012
24798 Total of 1, 2, 3, 4 and 5 Lines.
3456
28254 Total of 1, 2, 3, 4, 5 and 6 Lines.
7892
36146 Total of all the 7 Lines as before.
Thus much may fuffice to fhew you how to add any
Number of particular Sums or Numbers together, that either
have, or have not any particular Name or Denomination
affign'd them.
There are other Methods us'd for the Proof of Ad-
dition, but the common one at Schools is as follows: Suppoſe
5876.
2793
152
80
9
7946
8238
5279
30373 firſt Total
24497 Second Total
30373 third Total
theſe Numbers in the Margin, were given
to be added, viz. Firſt add up all the
eight Lines, as has been already taught,
which comes to 30373; then ſtrike or
part off the upper Line, and in the fame
Manner find the Total of the remaining
Seven Lines, which is 24497; this latter
Total place exactly under the former
Total, and draw a Line under it, then add
the upper Line that was cut off to this lat-
ter Total, and it produces 3037 3, the fame
with the firft Total; from whence 'tis con-
cluded the Work is right. I will now fhew
you another Method, which fome Per-
fons greatly eſteem, tho' I think it ought
not univerfally to be made ufe of; but
for
Of ADDITION, &c.
15
II;
for Variety's fake, 'tis as follows: Suppoſe the firſt Example,
as it confifts of but one Line, I fay 5 and 1 is 6, and 8 is 14;
I confider how much that is above 9, and find it to be 5; re-
jecting the 9, I go forward with the 5, and fay, 5 and 2 is
7, and 6 is 13, which is 4 above 9; fo rejecting the 9, I go
on with the 4 to the next Figure, which being 9, I fkip it,
and go with the 4 to the 7, which makes 11; here 9 being
rejected, I fet down the remaining 2 by it felf, and then I go to
the Total of the faid Figures, which was before found, and add
the Figures of which it confifts together, which are 3 and 8,
whoſe Total is 11: Now if you reject 9, the Remainder
will be 2, the fame with the Figure you fet by itſelf; but this
Method is not abfolutely certain, for if by miftake I had made
the Total to be 83 inftead of 38; this Method would not dif-
cover the Error, but it would be eaſily found out by the other
Methods of Proof before taught: For which Reaſon, I do
not recommend the Ufe, tho' I fhew you the Manner of
proving your Work by this Method.
My next Work ſhall be to fhew you the other Part of Ad-
dition of whole Numbers, called Multiplication, which is on-
ly a Repetition or continual Addition of the fame Sum or
Number, a certain required Number of Times: For the eafy
and ready Performance whereof, the following Table is ne-
ceffary to be got perfectly by Heart, to fave the Trouble
of looking upon it ſo often as the Figure changes, whofe
Sum or Total you are defirous of.
Multipli-
16
Of MULTIPLICATION.
Multiplication TABLE.
A
213 45
3t
6
に
​∞
B
81-9
2 4 6 8 10 12 14 16 18
3 6 91 1215 18 21 2427
4 812162024 283236
121
5 10 15 20 25 30 35 4045
4
1
61218243036424854
7142128
35 42 49 5663
81624 32 40 48 566472
9|18|27|36|45|54|63|72|81
The Uſe of this Table is to
find the Product, Total, or A-
mount of any fingle Figure, re-
peated any Number of Times,
not exceeding Nine: As if I
wanted to know how much 4
times 7 amounted to, firft I
look for 4 in the first Column on
the Left-hand, A, C, and then
I look for 7 in the top Line A
B, and carrying my Finger
from 4 in the firſt Column, till
it meet the ſeventh Column,
and there I find the Number
28 wrote. in that Space or
Square, which is the Product,
Total or Amount of 4 Times
7;
or you may look for 7 in the
firft Column A C, and 4 in
the top Line A B, and car-
ry your Finger from 7 in the
firſt Column, till it come to
the fourth Column, and there you alſo find 28; for 'tis the
fame Thing to ſay 4 Times 7, as to fay 7 Times 4; as ap-
pears by the following Examples of Addition.
7
4
7
4.
4 Times 7
7.
4
7 Times 4
7
4
4
}
4
- 4
Total 28
28 as in the Table.
In the fame Manner you will find 9 Times 8, or 8 Times 9,
to be 72; and 3 Times 6, or 6 Times 3, to be 18, &c. Thus
you muſt go backwards and forwards till you are perfectly
Maſter of the Product or Total of any two fingle Figures,
which in an Hour or two's Time you may eaſily be; and there-
fore ſuppoſing you fo, I fhall fhew you fome Examples where-
in the Excellency of performing Addition by Multiplication
confifts: As, fuppoſe I wanted to know how many 4 Times
960 amounted to. In this, and all other Cafes, you are to
obferve
of MULTIPLICATION.
17
ebferve, that there are always two Numbers given to find a
Third, which Third, when the Operation is perform'd by
Multiplication, is called the Product; but when 'tis done by
Addition, is called the Total, Sum, Amount, or Aggregate; but
whether the Work is done by Addition or Multiplication, the
Total or Product is always the fame, and is only a Repetition
of one of the given Numbers, fo often as there are Units in
the other: And for Diftinction fake, that Number or Sum
that ſtands ftill, and has the Work performed upon it, is
called the Multiplicand; and the other which operates, or
does the Work, is called the Mutiplier. Now either of the
given Numbers may be the Multiplicand, or the Mutiplier,
for the Product will always be the fame, as you fhall ſee here-
after; but for Conveniency or Expedition fake, 'tis com-
mon to make the largeſt Number, when there is any Diffe-
rence, the Multiplicand, and the ſmaller Number, the Mul-
tiplier. Theſe Matters premiſed, now to the Queſtion :
Firſt, by Multiplication.
Here I make 960 the Mutiplicand
Secondly, by Addition.
960-
and 4 the Multiplier 960 Being fet down
960
Then 3840 will be the Product 960
which is the fame with the other.
four Times.
3840 will be the Total,
Explanation 960 being the largeſt Number, I make that
the ftanding Number or Multiplicand, and 4, the working
Number or Multiplier, faying, 4 Times o is o, therefore
I ſet down o under the Units Place, and go on to the
next Figure, ſaying, 4 Times 6 is 24: Here, as before
in Addition, I fet down 4 under the fecond Figure or Place of
Tens, and carry the 2, faying, 4 Times 9 is 36, and 2 is 38;
and there being now no more Figures to be multiply'd, I fet
down the 8 under the 9, or Place of Hundreds, and the 3 one
place more towards the Left-hand; and ſo I have 3840 for the
Product, which appears to be the true Amount, by perform-
ing the fame Work by Addition; for having fet down the
960 four Times, and added the feveral Lines together, the
Total is 3840, the fame with the Product. Now theſe Num-
bers or Sums given and produced, not having any particular
Name apply'd to them, may repreſent any Number of Things
they are applicable to: As for Inftance, there are 960 Far-
things in one Pound Sterling, or Engliſh Money; confequently
D
3840
18
Of MULTIPLICATION.
3840, are the Farthings in 4 Pounds; and fo of any other
Species or Thing.
How much is 8 Times 63?
Here I make 63 the Multiplicand
and 8 the Multiplier
fo 504 will be the Product.
Having fet the given
Numbers down, one
under the other, I
fay 8 Times 3 is 24;
here I fet down 4
my Mind, and goon
under the Units Place, and carry the 2 in
and fay, 8 Times 6 is 48, and 2 I carry'd is 50, which be-
ing fet down to the Figure 4, already fet down, makes the
Whole 504; the Truth of which may be try'd, by ſetting
down the 63 eight Times, and adding up the two Lines, as
before taught, and you will find the Total to be 504 alſo.
Now this Example may be apply'd to any Buſineſs or Matter,
circumſtanced with the like Numbers: As, fuppofe I wanted
to know how many Pints of Wine were in a Hogfhead; by
examining into the Tables of Weights and Meafures, I find
that 63 Gallons is one Hogfhead of Wine, and that 8 Pints.
are one Gallon ; ſo that there are 504 Pints in one Hogfhead,
or there are 504 Gallons in 8 Hogfheads; and fo of any
Thing elfe. Again, fuppofe I wanted to know how much
7 Times 8766 amounted to:
Here I make 8766 the Multiplicand For having put
7 the Multiplier
and
and find 61362 is the Product.
·
down the Num-
bers, as above, I
begin and ſay, 7
Times 6 is 42:
then I go to the
Here I fet down 2, and carry 4 in Mind;
next Figure on the Left-hand, and fay again, 7 Times 6 is
42, and 4 I carried is 46 Here I fet down 6 in the ſecond
Place, and carry 4 in Mind, to be added to the Product of
the third Place; there I fay 7 Times 7 is 49, and 4 is 53:
Here I fet down 3 and carry 5; then I fay, 7 Times 8 is
56, and 5 is 61: No more Figures being to be multiply'd,
I fet down 61 to the other 3 Figures that are already found,
and they make 61362 all together for the Product; as you
may find by fetting down 8766 feven Times, and adding them
together, you will find the Total to be the fame, viz. 61362,
which will repreſent any Number of Things, agreeable to
the Conditions of the Multiplicand and Multiplier: As for
Inftance, there are 8766 Minutes in a common Year, and
confequently 61362 Minutes in 7 Years. Again, if I aſk
how much come 9 Times 63360 to?
Here
Of MULTIPLICATION.
19
Here I make 63360 the Multiplicand the two given
and 9 the Multiplier Numbers.
Here I find 570240 will be the Product; for having fet
down the Numbers as you fee, I begin, faying, 9 Times o
is 0, then 9 Times 6 is 54; here I fet down 4 under the 6,
or ſecond Place, and carry 5; then I fay, 9 Times 3 is 27,
and 5 is 32; here I fet down 2 under the firſt 3, or 3d
Place and carry 3; then I ſay 9 Times 3 is 27, and 3 is 30;
here I fet down o under the fecond 3, or 4th Place, and
carry 3; then I fay 9 Times 6 is 54, and 3 is 57; and as
there are no more Figures to be multiply'd, I fet down 57 to
the four Figures already down, and the Product is 570, 240,
as you may try by fetting down 63360 nine Times, and ad-
ding them together, and you will find the Total to be the
fame; which may in like Manner be applied to any particu-
lar Name the given Numbers will agree with; as in this
Example, 63360 are the Inches in a Statute or meaſured Mile
of England; and confequently there are 570240 Inches in 9
fuch Miles. If I multiply 567891234 by 2, 3, 4, 5, 6, 7,
8, 9: What will be the feveral Products? Here I fet down
the greateſt Number, or
Multiplicand 567891234 In like Manner 567891234
under it the firſt Mutiplier 2 being multiply'd by
6
whofe Product is 1135782468 the Product is 3407347 404
when the Multiplier is
7
then the fame multiplied by 3
the Product is 1703673702 the Prod. will be 39752386 38
when you multip. the Multip. by 4 the Multiplier being
8
the Product is 2271564936 the Product is 4543129872
and when the Multiplier is 5 fo when the Multiplier is g
9
the Product is 2839456170 the Product is 5111021106
Now I prefume you have a fufficient Number of Examples
of one Figure, from whence you may eaſily perceive the Advan-
tage of working by Multiplication before that of Addition, in
reſpect to Conciſeneſs, and Expedition. Now you fhall fee how
much more the Advantage increafes, as the Multiplicand is to
be repeated a greater Number of Tinies: As, fuppofe, I want-
ed to know how many Inches were in five hundred Miles ; here
D 2
I
20
Of MULTIPLICATION.
I have nothing more to do but to fet down the Inches in one Mile
for the Multiplicand, which is 63360, as in the former Ex-
ample; and for the Multiplier
500
the Product will be 31,680,000 Inches in 500 Miles.
For as o neither increaſes nor diminiſhes, but only fupplies
or fills up Places, to make the fignificant Figures have their
due Value; in this Example I put down oo to ſupply the
Places of Units and Tens, and then begin with the 5, and
fay, 5 Times o in the Multiplicand, is o, which muſt be put
in the third Place; then 5 Times 6 is 30; here being only
an o and 3 Tens, I put down the o again, having now 4 of
them, and carry the 3, faying, 5 Times 3 is 15, and 3 is
18, I fet down 8 and carry 1; and then fay again, 5 Times
3 is 15, and 1 is 16, I fet down 6 and carry 1; then I
fay, 5 Times 6 is 30, and I is 31, which I fet down, no
more Figures being to be multiply'd, and the whole Produce
is 31,680,000 Inches in 500 Miles. Now by common Addition
the Inches in one Mile, which are 63360, muſt be fet down
500 Times, and then added together, which if truly done,
the Total would be 31,680,000; but how tedious, perplexing
and ſubject to Error this Method would be, any one may
eafily judge, and proportionably, much more fo, if the Mul-
tiplier was 50,0cc, becauſe under all fuch Circumſtances
the Operation would be done in one Line by Multiplication,
and confequently in one or two Minutes Time, which might
be fo circumſtanced, that a Man's Life cannot be long enough
to perform it by common Addition; though I will fhew you fome
Methods of fhortning the Labour of it, by and by. And now,
I think, I have been fufficiently large upon any poffible Cafe
with one Figure; and therefore will proceed to give you fome
Examples of more Figures than one. As what comes
586 to, when multiply'd by 23? Here having fet down
the biggeſtNum-
ber uppermoft,
Firft fet down
586 for the Multiplicand
23 for the Multiplier
and then
1758
11720
and 13478 will be the Product.
and the leffer
under it, I mul-
tiply with the 3,
as though it was
a fingle Figure,
and find the Pro-
duct to be 1758. Now the 2 being 2 Tens or Twenty, and
virtually has a Cypher prefixed before it, though as 'tis com-
pounded
Of MULTIPLICATION.
21
pounded with 3, to make up the Number 23, 'tis not
exprefs'd, I put an o or Dot
or Dot. to ſupply the Place of
Units, and then go on with the 2, as though it were a
fingle Figure, and find that Product to be 11720: Theſe
two Numbers or fingle Products being added together, pro-
duce 13478, for the whole Product or Answer to the Questi-
on; which let be ſuppos'd to repreſent a Ship's-Crew, confiſt-
ing of 586 Men, who having taken a Prize at Sea, the
Owners offer 23 Dollars per Man to have it again, without
being plunder'd. Query, What was the whole Amount, or
how much did they pay for the Redemption of the faid Ship?
Anfw. 13478 Dollars. Or you may ſuppoſe there was 23
Purſes with 586 Guineas in each; and the Whole will be
13,478 Guineas, &c. Or you may fuppofe there was 23
Casks or Hogfheads, that held each of them 586 Pounds of
Sugar, Tobacco, Rice, &c. then the Whole will be 13,478
Pounds of Sugar, Tobacco, Rice, &c. Again, fuppofe I afk'd
what 69 Times 8766 comes to? Here I multiply the
biggeſt Number by
the leaft,
69
78894
52596.
and find 604854 to be the Product, which may be confi-
dered as the Anfwer to the following Question, viz. If a Man be
69 Years old, How many Minutes hath he lived? At a former
Queftion, 'twas allow'd that 8766 Minutes were in one Year,
and confequently 604,854 will be in 69 Years; for the Product
of every Multiplication, great or ſmall, is a Repetition of the
Multiplicand, ſo often as there are Units in the Multiplier ; and
the Name affix'd being wholly arbitrary, it may be as well
Time, as Meafure, Money, as Corn, &c. I will now give you
fome Examples of 3, 4, 5, 6, and more Figures, viz.
Example 4.
Example 5.
Multiply 905068
Multiply
57984632
by
4567
by
829
6335476
521861688
5430408.
115969264.
463877056..
4525340..
3620272...
48069259928
4133445556
Example
22
Of MULTIPLICATION.
Example 6.
Example 7.
Multiply
981726354
Multiply
1234567
by
478596
by
54321
1234567
2469134.
5890358124
8835537186
4908631770.
•
3703701..
4938268..
6172835.
7853810832.
6872084478...
3926905416..
67062914007
469850306118984
Example 8.
4000092386000075
5642800009999847
28000646702000525
16000369544000300.
32000739088000600..
36000831474000675.
36000831474000675 ·
36000831474000675
36000831474000675
320007390880006000000..
8000184772000150.
16000369544000300.
4
24000554316000450
•
20000461930000375..
22571721355721535055865691988525
Example 9.
5642800009999847
4000092386000075
28214000049999235
39499600069998929.
338568000599990820000..
45142400079998776....
16928400029999541
11285600019999694
50785200089998623
225712000399993880000.
•
I
8 X 8
1
22571721355721535055865691988525
Of
Of MULTIPLICATION.
23
Of the Proof of MULTIPLICATION.
In all Arithmetical Operations or Calculations, the Truth
or Certainty of the Work, is as neceffary to be known as
the Manner of doing the Work itſelf; for which purpoſe
diverſe Methods have been devifed, recommended, and us'd;
fome for their Eafe and Expedition, and others for their ab-
folute Certainty: Some of thefe Methods are as follows. Sup-
poſe I wanted to know whether 31,680,000 be the true
Product of 63,360, multiply'd by 500, as 'tis faid to be in
Page 20. The common Method us'd in Schools is thus:
Add the Figures of the Multiplicand 63360 together, and they
make 18, add these two Figures 1 and 8, which compofe the
Total 18 together, and they make 9; then having first made a
+ or a large X, Ifet down an o or Cypher, on the Left-hand
fide of the X, because all the 9's in the last Total are to be re-
jected, and only the Figure, under or over 9, are to be fet down
thus, oX; then I go to the Multiplier, and there being
but one fignificant Figure, viz. 5, the Total will be 5, which being
lefs than 9, I fet it down on the Right-hand of the X, then
they will ſtand thus, X5; theſe two being multiply'd together,
O
O
the Product is 0, which I fet at the Top thus, X5; then I
add the Figures that compofe the Product together, viz.
31,680,000, and they make 18, which being also added
ther, as above, make 9; for which I fet down o
toge-
at the
Bottom of the X, and then the Whole will ſtand thus, o
О
О
which fhews the Top and Bottom Figure to be alike, from whence
'tis concluded the Work is right.
In like Manner the Product 13,478 of 586, multiply'd by 23,
will be prov❜d to be true; for the Total of the Figures 586, of
the Multiplicand, is 19, which two, viz. 1 and 9, being added
together, the Sum is 10, which is I above, or more than 9,
which I ſet down on the Left-hand of the X thus, IX; the
Sum of the Figures 23, the Multiplier is 5, which being less than
9, Ifet down on the Right-hand, overagainſt the I thus, 1X5;
then I multiply thefe two together, and the Product is 5,
which
24
Of MULTIPLICATION.
1
}
5
which being less than 9, I fet at Top thus, 1X5; then I add
the Figures of the Product together, viz. 13,478, and they
make 23, which being likewife added together make 5, which
5
I put at the Bottom thus ; 1X5; then looking on it, I find
5
the Top and Bottom Figures to be both alike, viz. both
5's; from whence I conclude the Work to be truly per-
form'd, and the Product to be right. So if I wanted to know
whether 604854 was the true Product of 8766 multiply'd
by 69, as mention'd, Page 21. After having done, as be-
fore directed, I find the Figures in the Crofs, or X, will
О
ftand thus, o X6. So in the next Example where 48069259928
О
is faid to be the Product of 57984632, multiply'd by 829;
8
the Proof by this Method will be thus 8X1: In the next
8
Example, where 4567 is the Multiplier, the Proof will be
I
4
1X4, and in the Example, where 54321 is the Multiplier, the
4
6
О
Proof will be 1 X6, and in the next oX3, and in the large one,
I
О
8X8; from whence 'tis concluded that the ſeveral Products
I
are right; but by this Method you may be greatly deceived, for
innumerable other Numbers will anſwer to this Method of
Proof, though the Work is entirely falfe: As for Example, in
the firſt Example here produced, where 31,680,000 is faid to be
the true Product of 63,360, multiply'd by 500, becauſe the top
and bottom Figures came to o; whereas if the fame Figures
had ſtood in any other Order, as 86,130,000, or 16,380,000,
or 63,180,000, &c. the Anfwer would be the fame. So
likewiſe innumerable other Numbers greater and leffer than
the true one, may be found out to give the fame Anſwer,
2
as
Of MULTIPLICATION.
25
as 54, 63, 72, 18, 81, &c. leffer than the true Number or
Product: And 671,400,000, &c. greater than the true Num-
ber; and fo of all the other Examples, or of any others
that may be propofed: There is another Method of proving
Multiplication fimilar to this, viz. by dividing the Multipli-
cand and Multiplier, each of them by 7, and fetting down
the Remainders, if any, the one on the Right-hand, and the
other on the Left-hand of the Crofs; then multiply thoſe
Figures together, and if the Product is less than 7, fet it
down at the Top; but if the Product of theſe two Figures
is more than 7, divide it by 7, or take out all the 7's and
fet down only the Remainder; then divide the Product by 7,
and fet the Remainder, if any, at the Bottom of the Cross,
and if the top and bottom Figures are alike, the Work is
ſaid to be right, otherwiſe not; but this Method is liable to
the fame Objections with the proving the Work by caſting
out the Nines as above, and is alſo more troubleſome to per-
form, therefore I fhall fay no more of it; nor would I re-
commend either of the Ways for the Reafons already men-
tion'd: Indeed this latter one by 7, cannot properly come
in here, becauſe it takes for granted, that the Perſon who
ufes it, has fome Skill in Divifion, which I do not ſuppoſe;
and though almoſt all Authors are of Opinion that Divifion
is the beſt Proof of Multiplication, yet I am very ſure they
are much miſtaken, as I could eafily demonftrate; for
Diviſion is much more tedious and operofe, and confequently
more liable to Errors and Miftakes from its own Nature,
which is much more compounded than Multiplication; there-
fore I will fhew you how to prove this Rule infallibly by it-
felf. Let us take the Example above, viz. 63,360 multi-
ply'd by 500; here you may fet down the Multiplicand, and
under it the Multiplier, and work as before :
63360
Or thus,
500
31130000
0055....
15.
31680000
30.
63360
500
30000
15....
..
•
31680000
Here I multiply each fingle Figure of the Multiplicand by
the fignificant Figure of the Multiplier, and fet down its
whole Product in a Line by itſelf, regarding only the due
E
placing
26
Of MULTIPLICATION.
AJ
-
placing of the Figures, in refpect to their Order, viz. whe-
ther they be Thoufands, Tens of Thoufands, &c. and this
very naturally preſents itſelf; for as the Figures go gradually
towards the Left-hand one Place, and ſo increaſe their Value
by a tenfold Proportion, the feveral Products ariſing by their
Multiplication, muſt do the fame, as you fee by the two Me-
thods in which this Example is expreffed, and the Reſult
or Product is the fame with that in Page 20; from whence
you may conclude the Work is right: Or you may ſet down
the Multiplicand five Times, and add them together, and
the Total will come to the fame, if you add two o's or
Cyphers to it, for the two o's or Cyphers that ſtood before
the Multiplier, to make it five Hundred inſtead of 5, as ap-
pears by the Example in the Margin, where
63,360 five Times 63,360 comes to 316,800, which
having two o's added to it, makes the fignifi-
cant Figures be increaſed in Value an hundred
Times; fo that inftead of 316,800, it is
31,680,000, as before; from whence it may
be concluded the Product was right. In this,
and all other Cafes, if the Multiplier is the
Add two oo's Product of any two or more Numbers, little or
big, if you multiply the Multiplicand by thofe
31,680,000 Numbers continually, the Product will be the
63,360
63,360
63,360
63,360
316,800
fame as when you multiply by the common
Method, which may be another Method of Proof, whoſe cer-
&c.
tainty may be depended on : Or if you want 1, 2, 3, you
may add the Multiplicand fo many Times to the laft Product
of the compofing Numbers, and that Total will come to the
fame with the common Product: As in this Cafe before us,
500 being the Multiplier, I need only regard the 5, and add
two Cyphersor o's to the Total or Product, and the Work will
be the fame: As for Inftance, 2 Times 2 is 4; here I fet down
the Multiplicand
63360 and multiply it
by
The Product
2.
126720 is twice 63,360 the Multiplicand;
then multiply again by
and the Product
to which add
and the Total
2
253440 is twice twice or 4 Times 63,360;
63360 or the Multiplicand once,
316800 is 5 Times 63,360; to which add
two o's and it will be 316,8000,000, the former Product.
I
Or
Of MULTIPLICATION.
27
Or you may make
and
500 the Multiplicand
63,360 the Multiplier,
30000
1500..
1500...
3000 ..
and the Product will be 31680000, as before:
From all which different Methods you will always find the
fame Refult, Total or Product, if your Work be right; and if
at any Time there is any Difference, you may be fure there
is an Error fomewhere, which you muſt find out before you
leave it. I have been very full upon this Example, on pur-
pofe that you may the more eafily underſtand what follows.
The fecond Example fhall be
586 multiply'd by 23, which I'll do
thus: Set down 23 under the faid Multiplicand 586:
Here you fee I mul- . 1218 Or make 23 the Multiplicand,
tiply each Figure in.. 54.
the Multiplicand, by 1112.
each Figure in the . 06..
Multiplier, and fet
down the whole 13478
Product of each,
without carrying any thing,
regarding only the Order of
Places; which being done,
and the ſeveral Products
collected into one Total, I
find it to be 13,478, as be-
fore in Page 20.
Multiply
by
and the Product is
then this by
586
7
4102
3
and the Product is 12306
to this add
586
twice
586
and the Total is 13478
and 586 the Multiplier
138
184.
115..
and 13478 will be the Pro-
duct, which is
the fame with ei-
ther of the for-
mer Methods; or
you may do thus:
Here the Multiplicand 586 is
multiply'd by 7, and confequent-
ly the Product 4102, is feven
Times 586; then this multipli-
ed by 3, makes 12,306, which
is 3 Times 7, or 21 Times 586;
but I want 23 Times 586, there-
fore I add twice 586 to 12306,
and it makes 13,478; or 23
Times 586, as before.
E 2
In
28
Of MULTIPLICATION.
In like manner if you multiply
Or
by
69
8766
414
414.
8766 This being done
by 69 in the fame
Manner as you
.76554 did the laſt Ex-
..234.
ample, Page 27,
.4336. you will find the
483..
4826..
Product come
552..
to the fame, as
Here the
604854
in Page 21, by
Product is
604854 as before in Page 21. the other Me-
thod, as appears by the Example here annexed.
The fourth Example 829
Fifth Example 4567
by 57984632
The Reverſe of
1658
this Example is
2487.
in Page 21,
by
905068
36536
4974..
3316..
6632...
7461.
5803.
4145..
•
48069259928
27402.
27835.
41103.
•
4 13 8435556
See Page 21.
Thus you may prove any Sum in Multiplication whatever,
by changing the Multiplicand into the Multiplier; and if
the Product comes out the fame with the firft, you may be
fure the Work is right in both, otherwiſe not; for in this
Method no poffible Trick or Artifice can be us'd to make it
otherwife, without knowing you have committed a Fault, at
leaſt, in one of the Operations; the 8th and 9th Examples in
the large Numbers are a further Proof of the Certainty of this
Method. I fhall only add fundry Examples more, with their
Products, for your Practice, by operating of which, you will
find your felf render'd perfect in this Rule, and ready to exe-
cute any Thing that can be propofed in plain or whole Numbers.
1. What is the Product of 8650652672, multiply'd by
46875? Anf. 405,499,344,000,000.
2. What is the Product of 79278509, multiply'd by 362179?
Anf. 28,713,011,III,III.
3. What
Addition of MONEY.
29
3. What is the Product of 634070839, multiply'd by
735200839? Anf. 46,616,950,222,222,222.
4. What is the Product of 981237645, multiply'd by
546732189? Anf. 536,474,205,580,054,905.
5. What is the Product of 546732189, multiply'd by
981237645? Anf. 536,474,205,580,054,905.
I fhall now go on to fhew you the Addition and Multiplica-
tion of Sums or Numbers of diverſe Denominations; and firft
of all, as is ufual, the Addition of Money, or the common Coin
current among us, where you are firſt to take Notice that,
4 Farthings make, or are contained in one Penny; 12 Pence,
make, or are contained in 1 Shilling; and 20 Shillings make,
or are contained in 1 Pound. Tho' we have no Piece of Coin or
Money of twenty Shillings, or one Pound Value; yet all thoſe
we have, whether Gold, Silver or Copper, are rated or valu-
ed by the Pound, or rather by the Shilling, which is the
twentieth Part of a Pound; for in Gold, we have now in com-
mon ufe only the Guinea, valued at 21 Shillings or one Pound
one Shilling; and the Half-Guinea, valued at ten Shillings and
Six-pence; there are indeed double Guineas, valued at two
Pounds and two Shillings; and five Guinea-pieces, valued at
five Pounds and five Shillings; but theſe are very ſcarce, and
may rather be call'd Medals than common Coin. The Silver
is coined into Crowns or five Shilling-pieces, half Crowns, or
two Shillings and Six-penny-pieces, Shillings, Six-pences, Groats,
Three-pences, Two-pences, and Pennys; but the leffer Pieces,
or thoſe under Six-pence, are now but feldom current.
Since the coining Copper-half-pence and Farthings, which
of late Years have been coined very plentifully for their great
Uſefulneſs, in the more eafy and ready changing the Silver-
Coins, where odd Pence are concerned in the Price or Value
of any Commodity, the getting by Heart the following Table,
which is commonly called the Pence-Table, will be abfolutely
neceffary, for all fuch as defire to be ready in cafting up any
Sums of Money, either great or fmall; and therefore I
earneſtly recommend that Labour to you.
The
30
Addition of MONEY.
The PENCE-TABLE.
20 Pence is 1 Shilling and 8 Pence Some rather chufe this
30 Pence 2 Shillingsand 6 Pence
40
50
60
70
80
90
100
1
3
4
5
5
6
70
8
1
}
1
1
1
}
1
4 &c.
2
!
ΙΟ
8
42
ΙΙΟ
9 -
120
IO
130
ΙΟ
P
ΙΟ
140
150
-
}
I I
12
13
14
8
Form.
1 Shilling is 12 Pence
2 Shillings is 24 Pence
3 &c.
56
68 9
10
}
1
1
1
-
36* &t
48
60
72
78
84
96
108
120
132
I I
-
6
12
144
4
13
- 156
2
14
15
· 168
- 180
IO
16
J
1
8
17
192
204
6
18
1
4
19
2
20
216
228
240
160
170
180
- 15
190
-
- 15
200
- 16
210
220
17
18
230
240
- 19
-
20
The Uſe of this Table, is to know how many Shillings,
&c. are contained in any Number of Pence, from 20 to 240,
or how many Pence are in any Number of Shillings, from
I to 20, and the Contrary.
Now we will fuppofe the following Particulars to be what
a Servant-Maid may have laid out for her Mafter or Mistress,
viz.
26 Sept. 1737. For Small-Coal -
27
28
-
29
30
1
For Sand, a Broom, and a 2
Pound of Butter
Pence Farthings.
2
2
nda?
2}
IO
I
2
- For two Pounds of Stakes
For three Pounds and a
half of Mutton -
nd a}
8
IO
- For Thread, Tape and Pins II
For a Letter, and Hoop-39
1
Shills. 4 and 5 Pence.
2
3
2
Having
Addition of MONEY.
3 I
-
Having made fome Mark of Diſtinction over the ſeveral
Columns, as here I have writ Pence and Farthings, you
muſt always be very careful to keep every Name in its own
Column or Rank, and then add up the Column of the leaſt
Value firſt; as here I add up the Farthings, and they make
12. Now as 4 Farthings are one Penny, by the former
Table, called Multiplication-Table, which I fuppofe you to
have by Heart, you know that 3 Times 4 is 12: So that
12 Farthings make 3 Pence exactly; for which Reaſon I only
make a Dafh thus under the Column of Farthings, and car-
ry the 3 to the Column of Pence: And here your beſt Way will
be to add up only thofe Figures that ſtand in the Units-Place
firſt, as you did in the fecond Example of plain Addition, in
Page 9, and then the fecond Place or Tens, and fet down
how much they come to, upon a Bit of Paper by itſelf, or
elfe bear it in Mind, as in this Example; I fay 3 that
I carry from the Farthings and 9 is 12, and I is 13, and 8
is 20, and 2 is 22; fet down 2 and carry 2, which being
added to the 3 Tens, or Is, that ftand in the Tens Place,
makes 5, which being fet down to, or before the 3 on the
Left-hand, or in the fecond Place, the whole is 53 Pence.
Now the Table tells me that 50 Pence is 4 Shillings and 2
Pence, confequently 53 Pence is 4 Shillings and 5 Pence. So
the Bill was a little larger: As ſuppoſe this that follows.
if
Expences this Week, to 22 Octob. 1737. Shill. Pence Far.
Paid the Wafher-Woman
Paid the Milk-Woman
the Baker
the Butcher
the Tallow-Chandler
Butter, Eggs and Flower
Gave fundry poor People
1
J
5: 10: 2
13: 10: 3
14:
19
II :
8:
8: 8:
2: II: 2
I: 9:3
L. 3:07 09: 2
Here the Farthings being caft up, amount to 10, which
is Two-pence and Two-farthings over: I fet down the Over-
plus 2 under the Farthings, and carry the two whole Pence to
the Pence, and find that Column comes to 69; by the Table I
find that 60 Pence makes 5 Shillings, confequently 69 Pence
muſt be 5 Shillings and 9 Pence; therefore I fet down 9 under
the Pence, and carry the 5 Shillings to the Shillings; and here
I add as before directed for the Pence, viz. I fay 5 that I
carry and 1 is 6, and 2 is 8, and 8 is 16, and 9 is 25, and
I
4 is
32
Addition of MONEY.
4 is 29, and 3 is 32, and 5 is 37: I fet down 7 upon a Bit
of Wafte-Paper, and carry the 3, and fay 3 and 1 is 4, and 1
is 5, and 1 is 6, which being 6 Tens, I halve, by faying the
Half of 6 is 3, which is 3 Pounds, becauſe 2 Ten Shillings
make 1 Pound, confequently 6 Ten Shillings are 3 Pounds; fo
that I fet down the 7 under the Shillings, and carry the 3 Pounds
a Column further toward the Left-hand, as you fee in the
Example, where the whole Sum, Amount or Total of the
Bill is three Pounds feven Shillings and Nine-pence Half-
penny.
Again, Expences paid Sundry this Week to 22 Octob. 1737.
Mr. A. B. the Carpenter
C. D. the Linen-Draper
E. F. the Milliner
G. H. the Taylor
7. K. the Hatter
L. M. the Brewer
N. O. the Hofier
J
1.
S. d.
5: 16: 6
8 13:4
7: 19:
2178
5: II: 6
9: 15: 9
4 12:
L. 45: 59
Here the Pence amount to 33. I find by the Table
that thirty Pence is two Shillings and Sixpence; fo that thirty
three Pence, being Three-pence more, must be two Shillings
and Nine-pence; I fet down the Nine-pence under the Pence,
and carry the two Shillings to the Shillings, and find the firſt
Line of the Shillings comes to 35; I fet down 5 and carry
the 3 to the Tens, and find they make 10, which are 10
Times 10 Shillings, the Half of which being 5, are fo
many Pounds, which I carry to the Row of Pounds, and
adding it up, I find it amounts to 45; and there being
but one Row, I fet down 45, and find the whole Amount
or Total is Forty five Pounds, five Shillings and Nine-pence.
Again,
Addition of MONEY.
33
Again; Suppofe a Merchant or Banker fends his Man to
collect the following Bills, viz.
1.
ift for
7°
24
3d
4&c.
5
9
56 78
*
I
9
ΙΟ
1
2 6
: IO'
d.
S.
9.
3 * 6.
*
:
*
15
9'
8. *
*
*
in in
å min ämin
2
9
*
*
*
8
1 - 3•*
18. * II
19' :06
*
17. : 09*
16• *
14'
12
13'
:
об
: II'
*
*
18. *
:08
: 04°
: II'
: IO
: 09*
-
*
*
*
*
"4
24109 11
222: 6:63
1
3
2
: 15
: 08 -
I 8 9
5: 19
: 09 -
2 6 3
2
: 15
: 08 -
2632:15:8
Here you will obſerve, that the Farthings are put down
in a different Manner to what they were in the former
Examples, viz. is one Farthing; is two Farthings or an
44
Half-penny; is three Farthings, which are to be added to-
gether in the fame Manner as they were before. As in
this Example. I begin at the Bottom, and fay 1 and 2
is
3, and i is 4, and 2 is 6, and 3 is 9, and 1 is 10, and
2 is 12 Farthings; which being 3 Pence, I make a Stroke
thus -- under the Column of Farthings, and carry the 3 to
the Column of Pence, and add as before directed, and find
the Pence come to 92, which is 7 Shillings and 8 Pence;
wherefore I fet down the 8 odd Pence, under the Column of
Pence, and carry the 7 Shillings to the Shillings, and find the
firft Row comes to 65; for which I fet down 5, and carry
fix to the next Row, and find that comes to 15. Now
this being an odd Number, I fet down I to the 5 before fet
down, and it makes 15, which I being abated out of
the 15, the whole Amount of the Row, the Reſidue, which
is 14, I halve, and carry its Half, which is 7, to the firſt
Row of the Pounds, which amounting to 72, I fet down 2
under the Units Place, and carry the 7 to the next Line,
which amounts to 43; for which I fet down 3 under the
Tens Place, and carry the 4 to the third Row, and find that
F
amounts
34
Addition of MONEY.
!
amounts to 26; and as there are no more Rows of Figures
to be added, I fet down 26 to the 23 already fet down, and
the Whole makes 2632 Pounds 15 Shillings and 8 Pence;
for the Proof of which you may begin at the Top, and add
downwards, and you will find the Amount to be the fame :
Or you may part it into two Parts, and find the Amount of
each, and then add thoſe two Totals together, and you'll find
them come to the fame; as you may fee done by the Example
above, where you fee it parted into two five Lines; the
Amount of the uppermoft 5 Lines is L. 2410: 9: 14
and the Amount of the undermoft
5 Lines is
The Total whereof is
}L.
222:
6:63
L. 2632: 15: 8
as before, when the whole ten Lines were added at once. The
common Method in Schools is to add up
the Whole, and fet down that Total L.2632: 15:08
then to draw a Line under the uppermoft Line, and add all
under that Line together, which in this
Example amounts to
L. 1895: 19:091
Then this laft Total being added to the L.2632: 15:08
uppermoît Line makes as before
I
*
By the foregoing Method any Number of particular Sums
of Money may be truly and expeditiously added together in-
to one Total. Yet fome think the following Method eaſier
and propereft for thole who are not frequently practifing
fuch like Operations, viz. to begin and add the loweſt Species
together, and to make a Point thus. or thus
every Time
the Sum of the leffer Species is equal to, or greater than one
of the next fuperior Species; as in the laft Example, begin at
the Bottom, and fay 1 and 2 is 3, and 1 is 4, which being
4 Farthings, I make a Prick or Dot, as you fee in the Ex-
ample. Then, as there is no Overplus to be carried forward
to the next Figure, I go on and fay, 2 and 3 is 5; which
being 5 Farthings, is 1 Penny, and 1 Farthing over; I make
a Dot at the 3 Farthings, as you fee in the Example, and
carry the Farthing, which is over, to the next Figure; and
fay 1 that I carry, and I is 2, and 2 is 4, which being al-
fo 4 Farthings, which are equal to one Penny, I make ano-
ther Dot at the uppermost 2. And now all being added
together, and no Överplus remaining, I make a Daſh thus
under the Column of Farthings; then I count how many
Dots there are in all, and I find 3, which reprefents 3
Pence
Addition of MONEY.
35
F
Pence, which 3 Pence I carry to the Column of Pence, and
fay, 3 I carry and 9 is 12. Now 12 Pence being exactly
a Shilling, I make a Dot at the 9, and carry nothing for-
wards becauſe there was no Overplus Pence; but go on a-
freſh, faying, 10 and 11 are 21. Now by the foregoing
Table, I know that 20 Pence, are one Shilling and 8 Pence,
confequently 21 Pence is one Penny more, viz. 1 Shilling
and Nine-pence; for which Reaſon I make a Dot at the
Eleven-pence, and carry the 9 Overplus Pence to the next
Number of Pence, viz. 4, and fay 9 and 4 is 13; and by
the aforefaid Table I find that 12 Pence is 1 Shilling; to
that 13 Pence is 1 Shilling and 1 Penny over: I make ano-
ther Dot at the 4, and carry the Overplus I to the next
Sum of Pence, and fay 1 and 8 is 9, which being less than a
Shilling, I go forward and add 9 to the next Sum, viz. 11,
and it makes 20 Pence; which being 1 Shilling and 8 Pence,
I make another Dot at the 11, and carry the 8 Overplus
Pence to the 9, which makes 17; and as 17 Pence is 1 Shil-
ling and 5 Pence, I make another Dot at the 9, and car-
ry the 5 to the 6; and that making 11 Pence, which being
lefs than a Shilling, I go forward and add that 11 to the
next Sum, which being alfo 11, I fay 11 and 11 is 22 Pence,
which is 1 Shilling and 10 Pence; wherefore I make another
Dot at that II, and carry the Overplus 10 to the uppermoft
Sum of Pence, and it makes 20; which being 1 Shilling and
8 Pence, I make another Dot at the uppermoft 10; and becaufe
there are no more Sums of Pence to add the lait Overplus
8 to, I fet down the 8 under the Column of Pence, then I add
or count up the Dots in the Column of Pence, and find the Sum
71 which ftands for, or reprefents Shillings, which I carry
to the next Column or the Raw of Shillings, faying 7 that I
carry and 18 is 25. Now 20 Shillings being i Found here
is 5 Overplus Shillings to be carried to the next Number
of Shillings; therefore I make a Dot for the Pound and
go forward, faying, 5 and 6 is 11; and this being under a
Pound, I go on and add that to the 13 above, and that
makes 24, which is 1 Pand 4 Shillings; therefore I make
another Dot at the 13, and carry the Overplus 4 forward,
faying, 4 and 12 is 16, which being less than 20, I go for-
ward to the 14, and it makes 30, which is 10 more than 20;
fo I make another Dot at the 14, and carry forward the
Overplus 10 to the 1, and that makes 26; here I make
another Dot for the Pound or 20 Shillings, and carry the
Overplus 6 to the 17, which makes 23; I make ano-
is
F 2
7
ther
36
Addition of MONEY.
ther Dot at the 17, and carry the Overplus 3 to the
next Parcel of Shillings 19, and it makes 22; I make
another Dot at the 19, and carry the Overplus 2 to the
next Sum, 18, and it makes juſt 20; therefore I make ano-
ther Dot at the 18, and carry nothing; and becauſe the
next Sum of Shillings, 15, is the laſt, and leſs than 20, I ſet
it down as the Overplus, under the Column of Shillings, as
you fee in the Example; then I count the Dots, and find
them to be 7, which ſtand for 7 Pounds; theſe I carry to
the Units-Row of the Pounds, and add as you have been al-
ready taught in the Addition of whole Numbers; or make
a Dot at every 10, and carry the Overplus forward: Thus
fay, 7 Pounds that I carry from the Shillings, and 3 is 10;
here I make a Dot and carry nothing forward; then I go
on, and ſay, 8 and 9 is 17, which being 1 Ten and 7 over,
I make a Dot at the 9 and carry the 7 to the 2, and it
makes 9, which being leſs than io, I go on to the 7, and
it makes 16; here I make another Dot, and carry 6 to the
5, and it makes 11; I make another Dot, and carry 1 to
the 8, which being but 9, I go on and add it to the 9,
and the Sum is 18; here I make another Dot, and carry
the 8 to the 8, and the Sum is 16; I dot again and car-
ry the 6 to the next Figure, which being the laſt, makes
12; I dot again for 10, and the Overplus 2 I fet down
under the Units Place, and count the Dots, which amounting
I carry it to the Row of Tens, faying, 7 and 1 is 8,
and 5 is 13; here I dot and carry 3, faying, 3 and 3 is 6,
and 9 is 15; dot again and carry 5, which being added to
the next Figure, which is 5, they make 10; here I dot
and carry nothing; then go on, faying, I and 9 is 10; dot
again and carry nothing; now there being but I Figure
more, which is 3, I fet it down as the Overplus, as you
fee in the Example; then I count the Dots, and find their
Sum is 4, which I carry to the next Row, or Line of Hun-
dreds, faying, 4 and 1 is 5, and 9 is 14, dot and carry 4
to the 5, which is 9, and 7 is 16; dot again and fet down
the Overplus 6, becauſe there are no more Figures to be
added; then count the Dots, which are 2, and there be-
ing no more Lines to be added, fet down 2, for the 2 Dots,
one Place more towards the Left-hand, and your Work is
done. So will you find the hole, Total or Amount to be
2632 Pounds 15 Shillings and 8 Pence, as before; and here,
as before, you may alfo prove your Work, by beginning at
the Top and defcending gradually to the Bottom, only 'twill
be proper to make fome other Mark inftead of a Dot, as
to 7,
2
* left
Addition of MONEY.
37
Ι
*
* left you ſhould miftake the Dots in Deſcent, for thoſe
made in aſcending; for you will frequently find them not to
happen in the fame Place, or at the fame Figure you had
dotted before; as you fee by the Example, where beginning
at the Top, I fay, 2 Farthings and I is 3, and 3 is 6 Far-
things, which is 1 Penny and 2 Farthings: Here I make a
* and carry the 2 to the next 2, which being 4 Farthings
or 1 Penny, I make another *; and having nothing to carry,
I go on, faying, I and 2 is 3, and I is 4 Farthings, or I
Penny; here I make another *, and all the Farthings being
added, and no Overplus remaining, I make a Dafh at Bot-
tom under the Farthings, as before; then counting the *'s,
I find them 3, which I carry to the Pence, as I did before
the Dots, and going to the uppermoft Figures in the Column
of Pence, I add them to the 10, which makes 13 Pence
or 1 Shilling and I Penny. I make a *, and carry the
Overplus I to the 11, and it makes 12 Pence; which being
juſt 1 Shilling, I make another * and carry nothing; then
coming downwards, I fay 6 and 9 is 15 Pence, which being
1 Shilling and 3 Pence, I make another at the 9, and
deſcending add the Overplus 3 to the II, and it making 14
Pence, I make another, and defcend with the Overplus 2
to the 8, which making but 10 Pence, and that being leſs
than a Shilling, I add it to the 4; that making 14 Pence or
1 Shilling and 2 Pence, I make another *, and defcend with
the Overplus 2 to the II, and their Sum is 13; here I make
a* again, and add the Overplus I to the 10 below, whofe
Sum 11, being leſs than a Shilling, I add it to the laſt Fi-
gure 9, and the Sum is 20, which being 1 Shilling and 8
Pence, I make another * at the 9, and fet down the Over-
plus 8, as before; then counting the *'s, I find them 7,
with which I go to the uppermoft Parcel of Shillings, fay-
ing, 7 and 15 is 22 Shillings, which being 1 Pound 2 Shil-
lings, I make a for the I Pound, and defcend with the
2 Overplus Shillings, and add them to the 18, which mak-
ing juft 20 Shillings, or a Pound, I make a * and carry no-
thing; then deſcending, I fay, 19 and 17 are 36 Shillings,
which is 1 Pound and 16 Shillings, I make another *
at 17, and add the 16Overplus Shillings to the 16, and they
make 32 Shillings; which being 1 Pound and 12 Shillings, I
make another * at the 16, and carry the Overplus 12 to the
14; their Sum being 26 Shillings, or i Pound and 6 Shillings, I
make another*, and add the Overplus 6 to the 12, whofe Sum
18 being less than a Pound, I add it to the 13, and it makes 31
Shillings, or 1 Pound 11 Shillings; here I make another *, and
*
add
38
Addition of MONEY.
and
*
9
>
add the Overplus II to the 6, and go on with their Sum 17 to
the 18, and that makes 35 Shillings, which is 1 Pound 15 Shil-
lings; and now all the Figures being taken in, I make another
*at the 18, and fet down the Overplus 15, under the Column of
Shillings; then I count the 's, and find their Number is 7,
which I carry to the Units-Row of the Pounds, and defcend
from the Top, faying, 7 and 6 is 13. Now, becaufe in the
Pounds, and all other whole Numbers, every Row is counted
by Tens, I make a and carry 3 to 8, whofe Sum being
II, I make another * and carry 1 to 9, whoſe Sum be-
ing 10, I again make a *, and carry nothing, then defcend-
ing, I add 8 to 5, which making 13, I make another *
carry 3 to the next 7 below, whofe Sum being 10, I
make a*, and carry nothing, but defcending add 2 and
9, whofe Sum being 11, I make a *, and carry 1 to the 8,
whofe Sum 9 being less than 10, I defcend and add it to the
laft Figure 3, whofe Sum being 12, I make another *, and
fet down the Overplus 2, as you fee in the Example; then
counting the *'s and finding their Sum to be 7, with it I go
to the uppermost Figure in the ſecond or Tens Row, ſaying,
7 and 3 are 10; here I make a *, and having no Overplus
to carry, I defcend, and fay, 9 and 1 is 10; I make ano-
ther, and having nothing to carry, ftill defcending, I add
5 to 9, whofe Sum being 14, I make a at the 9, and
add the Overplus 4 to the next Figure 3, whofe Sum
7, being leſs than 10, I defcend and add it to the 5, and
it makes 12; here I make another *, and add the Over-
plus 2, to the laft Figure 1, and the Sum being but 3,
which
is lefs than 10, I fet down under the ſecond Row or Place
of Tens, then counting the *'s find them to be 4, with
which I go to the uppermoft Figure in the third Row or
Place of Hundreds, and fay 4 and 7 make 11; here I make
a *, and carry 1 to 5, whofe Sum 6 being less than 10, I
add it to the next Figure 9 below it, and their Sum being
15, I make another *, and carry the Overplus 5 to the laſt
Figure 1, and the Sum 6 being lefs than 10, I fet it down,
then counting the *s, I find their Sum to be 2, and as there
are no more Rows of Figures to be added, I fet down 2 one
Place more towards the Left-hand, and the whole Sum is
2632 Pounds 15 Shillings and 8 Pence, as before; from whence
I conclude that the Work is right.
*
II
Having now explained the various Ways by which any
Sum in Addition, properly fo call'd, may be done; there re-
mains little more to be faid upon that Head, but only to
practife
Addition of MONEY.
39
practife what has been taught, and by various Examples to
make your felf perfect; for which purpoſe I will infert more
Examples wrought by all the foregoing Methods, which I would
adviſe the Learner to go over, and alfo to fet the fame down
in various other Methods or Orders; and if the Work is
truly perform'd, the Total or Amount will always be the fame,
as has been before noted, and as you may obſerve by
the following Example, which I have varied three different
Ways, to make the Matter quite plain to you; and I
would adviſe the fame Practice to be done by other Examples,
which in a ſmall Time will render you capable of doing any
thing of this Kind with Eafe, Readineſs, and Certainty:
And without fome fuch like Application, Expedition and
Certainty are not to be acquired. One fuch Example as is
here given you, and varied according to different Modes,
will be better for your Practice than ſo many different ones,
becauſe you may always be fure of the Truth of your
Work, by its agreeing in the whole Total with what went
before; whereas if you operate Sums whofe Amounts are ab-
folutely different from one another, and if you have not
a Mafter by you, to infpect your Work, you may not per-
haps be fo well fatisfied of the Truth of your own Perform-
ance, and therefore will not be ſo proper for your Improve-
ment, as the Method here laid down, which will anfwer all
the Ends of duly informing a Learner, till he is fufficiently
fkill'd, to depend upon his own Judgment.
17
8.
I
1.
*
*
6. *
**
No in HO domain in
5°
2
9°
2
6 * 9
3 2
9°
グ
​2
9°
2
6
4.
*
*
**
*
*
9°
*
*
in mão tão a
8.
8.
953
**
**
Example 1.
3
oow
8.
5*
*
***
*
5.
: 18. *
: 12*
*
d.
: 10°
: II
*
* *
* * *
: 16. *
: 09*
: 08.
: 06 *
: I I*
: II'
16. *
:
I I'
: IO'
****
8.
7
2°
: 15
: 19*
: 18. *
: 17°
:
: 13°
: II
: 19'
IO
4'
9°
**
*
: 09*
* * *
q.
* *
**
18.
*
: 08 *
: 07°
*1
7
ли
3
9
6
2
: OI
: 09
6
7
9
3
6
5
5
: 02
: 10
3
4
N
༢
5
2
: OI
: 09
Here
6 8
60
о
6 8
2
40
Addition of MONEY.
Here you may obferve there are fmall Figures put over
the Figures of the Total, the Defign of which is to help
the Memory in the Carriage; as for Inftance, when I work
one of theſe Sums by the Method of adding up the whole
Line, either from the Bottom to the Top, or from the Top
to the Bottom, after I have found the Amount of the whole
Line, I fet down the odd Money, and over it, how many
of the next Line or Denomination, are contained in this I am
now about, as in the foregoing Example. Having caft up
the Farthings, I find it comes to 25, which is Sixpence and
one Farthing; therefore I fet down under the Line of Far-
things, and 6 over it, as a Memorandum what Pence I
carry to the Line of Pence, that I may fee whether I a-
gree, or am the fame when I caft that Line up the fe-
cond Time, by way of Proof; for tho' you begin at the
top Figure, and fo come gradually down to the Bottom one,
'tis call'd as much cafting up, or finding the Amount or
Total of the Line, as if you began at the Bottom, and fo
went gradually up to the Top: The fame is to be obſerv'd
in the Pence, Shillings and Pounds; for coming with the 6
from the Farthings, I find the Amount or Total of the Pence
is 117; by the Table I find 110 Pence is 9 Shillings and
2 Pence, confequently 117 Pence is 9 Shillings and 9 Pence;
for which, fetting down the 9 odd Pence, I carry the 9
Shillings to the Shillings; and to prevent forgetting my
Carriage, I write a fmall over the 9 fet down; then go-
ing to the Shillings, I find them, with the 9 I carried, come
to I fingle Shilling; and 20 Times 10 or 200, which are
10 Pounds, which I likewife fet down, and carrying this 10
to the Units Place of the Pounds, find it come to 82; there-
fore I fet down the 2 and carry the 8 to the next Row,
which comes to 75; here I fet down the 5 and carry the
7 to the third Row, which comes to 72; here I fet down
the 2, and carry the 7 to the fourth Row, and that amounts
to 68; and becauſe there are no more Rows to add, I fet
down 68, and the whole Sum is L. 68252: 01:09;
which you may prove by beginning at the Top, and fo
come down Figure by Figure to the Bottom, and you'll
find the Total to be the fame: Or you may part off the
top Line, and add the other 11 Lines together, whofe Sum
or Total is L.60655: 02: 101, which added to the top
Line that was cut or parted off, makes L.68252:01:09
the fame with the whole Amount before found; but if you
ufe Dots or *s, you need not make the 6 over the Far-
things,
9
Addition of MONEY.
41
things, the 9 over the Pence, the 10 over the Shillings, &c.
because the Dots being counted come to the fame Numbers.
Example 2, which is only the First tranſpoſed.
1.
S. d. q.
1234 : 15 c8
6789: 19:
2847 18 II
:
ΙΙ
6985 : 17: II
3214: 16: II
9989: 13: IO
7298: II:
2657: 18: 08
5432 19: 07
7596: 18 : 10
8273 12: II
5928: 16: 09
68252 : 01
OI : 09
: 09
4
17858: 12 : 2
23161 : 01 :
27232 08:24
68252: 01: 94
9:
}
Note, Where the Mark. is us'd, you are to underſtand
the Work begun at the Bottom and went upwards; and
where the Mark is us'd, the Work is begun at the Top
and comes down; though you may ufe either Mark for either
Method, as beft pleafes you; but the Method ufed by the best
Artifts, is to add up the whole Line, if it be not too long,
and if it be long, to divide it into two, three or more
Parts, as you fee done above.
Example 3, which is a fecond Variation of the firft Example.
9.
2
5.
5.
I
6.
2
3.
68
5 8
1.
* 8. *
8.
*
J.
d.
*
13.
: 10.
*
II.
*
** **
**
*
No cócó thaisin
*
*
dinmain choo tod
淞
​6.
*
**
****
aad ting og mod on
*
*K
****
*
*
6.
樂
​***
*
7.
ذا
{
7
5
A
6
2 5
*
4.
8
2
2
2
典
​: 18. *
: 1Q
: IS. *
: 12.
: 16.
: 15.
19
: 18.
I
: 17.
*
派
​IĆ. *
: ΟΙ
: 09
CS.
: IC
II.
:
CO.
: 08.
**
: 06 *
: 11.
*
II.
*
*
II.
0
: 07
: 10
: 09
cm 114
3
: OI
G
*
This
42
Addition of MONEY.
This Example is the fame Figures with the firft, but only
the Lines do not ftand in the fame Order, and is wrought
in all the different Ways like the firft; by which you fee
the Total or whole Amount is exactly the fame, purſue
what Method you pleaſe.
Here follows another Example, prov'd another Way.
1.
S. d. q.
3792: 12:09 ž
6285 13: II
ΙΙ
1233: 14: 10
4564 15 07
465
7 {
6 3
I W
16:08 - 40366: 01: 7
7895: 16:08
126: 17 : 06
37: 18:05
काय बोलाव ।
6548 19 09
9879: 11: 11
350: 15: 10
929 : 10: II
848 : 16 : 08
617:09:07 1
86: 18 : 06
5 : 08: 04
6: 17:07
8719 07:09
:
9928: 14: II
9 8 12 13 II
J
I 2
61860 : OI : 10
426 66
21494:
wa
13 1/1/10
-~
61860 : 01: 10 1
53047: 9
5 /
8702 12:
12: 5
.II. :
61860 : 01: 10 Total.
Here you fee this Example, confifting of eighteen Lines,
divided into two Parts, and each Part added as if it were
a feparate Sum; then thoſe two Tetals are added together,
and their Total comes to the fame with the Total of the whole
eighteen Lines, caft up all in one Line; from whence may
be concluded that 61860 Pounds, 1 Shilling, 10 Pence and
1 Half-penny, or 2 Farthings, is the true Total of the whole
18 Lines. You will obſerve that I have put over the parti-
cular Totals, the Carriage-Figures, both in the whole Sum,
and in the 2 Parts, for the Eafe of the Memory. Under-
neath
3
Addition of MONEY.
43
neath I have begun at the laft Row, or Row of Thouſands, and
having added it up, I find it come to 53; the 3 I fet under
the Thouſands, and the 5 a Degree backwarder to the Left-
band; then I add up the next, or third Line, and find that
comes to 80; I fet the o under the Hundreds, and the 8
under the Thouſands; then I add up the next Row, and find
that comes to 74; the 4 I fet under the Tens, and the 7
under the Hundreds; then I add up the Units, and find that
Row comes to 107; I fet down the 7 under the Units Place,
the o under the Tens Place, and the I under the Hundreds
Place; then I add up the Shillings, and find they come to
12 Pounds, 9 Shillings; the 9 I fet down under the Shillings,
the 2 Pounds I fet under the Units Place of the Pounds, and
the I under the Tens Place of the Pounds; then I add up the
Pence, and find them come to 12 Shillings and 5 Pence; I
fet the 5 under the Pence, and the 12 under the Shillings.
and then add up the Farthings, which come to 5 Pence Half-
penny; I fet down the 2 Farthings or Half-penny, under the
Farthings, and the 5 Pence under the Pence. Now having
gone through the feveral Lines of the Sum, I add the Par-
ticulars together, and find the Total to be the fame as be-
fore, according to the Axiom, viz. That the Whole is equal to all
its Parts taken together, let the Mode or Difpofition of them
be how or what you will. I think I have now fufficiently
explain'd this Part; I fhall only add, that if you change the
Order of the Lines of this laſt Example, and work it over
all the feveral Ways you have been before taught, 'twill
make you fo expert, that you will need no other Inftruction
for the Performance of any Thing of this Sort of Work;
only I would adviſe you always to break any Sum that
confifts of many Lines, into 2, 3, 4, or more Parts, as well
as caft them up in the Whole, which will make you much
lefs liable to Miſtakes.
I will now fhew you how you may ufe what you have
learned in Bufinefs: As, Suppose I have bought twelve Yards
of Cambrick, at 6s. 8 d. a Yard, What is the Amount,
Value, or Price of the
hole?
ཇ
G 2
Here
44
Addition of
of MONEY.
Here I may fet down the Price of one Yard three Times,
thus:
S. d.
6:8
6:8
6:8
3 Yards is L. I :
3 more is L. 1 :
3 more is L. I :
3 more is L. I :
12 Yards is L. 4
4:
Add theſe together, as be-
fore taught, and they will
come to twenty Shillings, or
one Pound, for the Price,
Coft, or Value of 3 Yards;
which being fet down four
Times and added together
will come to 4 Pounds, which
is the Price of four Times 3,
or 12 Yards.
Or you may fet down the Price of 1 Yard 4 Times:
S. d.
Thus, the Price of
Yard is
6:8
I more is
6:8
I more is
6:8
I more is
6:8
Total Coft of
4 Yards is
L. 1 :
6 : 8
4 more is
L. 1 : 6 : 8
4 more is
L. I : 6 : 8
So that the total } 1
} 12 Yards is L.4:-:-
Coft of
-
Or you may fet down 1 Yard 6 Times;
S. d.
Thus, I Yard coſt 6: 8
T
I ditto
6:8
I ditto
- 6:8
I ditto
-6: 8
Į ditto
6 • 8
I ditto
- 6 : 8
The Coft of 6 Yards is L. 20:0
6 more is L. 2 : 0 : 0 more
12 Yards is L. 4:0:0
But
1
Addition of MONEY.
45
But the beſt and eaſieſt Method is as follows, viz. To
fet down the Price of one Yard, and multiply it by 12 at
ence, or by two or more Numbers, that, multiply'd con-
tinually together, will produce 12, the whole Number of
Yards bought: As 6 Times 2, or 2 Times 6 is 12; or you
may fay 2 Times 2 is 4
Times 4
3
is 12, &c. Or
3 Times 4, or 4 Times 3, is 12; as you may fee by the fol-
lowing Examples.
S. d.
>
and
First fet down 6: 8 the Price of one Yard
and multiply by
0
6
and L. 2: o: o will be the Price of 6 Yards.
Then multiply by 2
and L. 40: o will be the Coft or Price of twice 6
or 12 Yards.
S.
6
Multiply by -
d.
: 8 the Price of one Yard
2
13: 4 is the Coft of two Yards
Multiply by 6
L. 4 oo is the Value of twelve Yards.
Thus you may find the Amount, Coft, or Value of any
Number of Yards, Pounds, Pieces, Hogfheads, &c. the Charge,
Coft or Price of one being known. But to make the foregoing
Example quite plain and eafy to be underſtood, obferve the
following Explanation; firft, the Price of one Yard being fet
down, which in this Example is 6 s. 8d. I confider what two
Numbers multiply'd together, will produce 12, the whole
Number of Yards bought, and I find 6 and 2, or 2 and 6,
3 and 4, or 4 and 3, will do it; then fetting down 6 un-
der the Price of one Yard, I begin, and fay,
First Method. 6 Times 8 is 48, which being 48 Pence, by
d. the Pence-Table, I find 48 Pence is juſt 4
Thus 68 Shillings, therefore fet down o under the
Pence, or make a which you pleaſe, and
carry the 4 Shillings to the next Row or Column
of Shillings, faying, 6 Times 6 is 36, and 4
I carry is 40, which being 40 Shillings is juft
2 Pounds; therefore fet down o under
the Shillings, or make a as before in the
2:
A:
5.
6
2
Pence,
46
Addition of MONEY.
Second Method.
6:8
2
13:4
6
4 00 Co
Third Method.
6: 8
3
I : 00:00
4:00:00
Fourth Method.
6:8
4
I
Pence, and then one Column further I fet
down the 2 Pounds; and this is the Coft,
Price or Amount of 6 Yards; then I fet down
2, and multiply thereby, and the Product is
4 Pounds; for as there are no odd Pence,
nor Shillings, the multiplying o by 2, or
any other Number, will always produce o;
and when I come to the Column of Pounds,
I fay 2 Times 2 is 4, which are 4 Pounds,
for the Value or Coft of the 12 Yards. Or
you may multiply by 2, and that Amount by
6, as you fee in the fecond Method, where
I begin and fay, 2 Times 8 is 16, which
being Pence, is I Shilling and 4 Pence, I
fet down 4 Pence under the Column of
Pence, and carry the 1 Shilling, faying, 2
Times 6 is 12, and I is 13; which being
Shillings, and leſs than a Pound, I fet them
down under the Column of Shillings, and the
Product or Coft of 2 Yards is 13 Shillings
and 4 Pence. Now 6 Times 2 being 12,
I multiply this Product 13 Shillings and 4
Pence by 6, faying, 6 Times 4 is 24, which
being Fence, is juft 2 Shillings; therefore I
fet down oo's under the Pence, and carry
the 2 Shillings forward, faying, 6 Times 3
is 18 and 2 carry'd is 20, for which I fet
down o, and carry the 2, faying, 6 Times
I is 6, and 2 is 8, which being 8 Times 10 Shillings, I
halve it, and find the Half is 4; therefore I fet down ano-
ther o, and the 4 being 4 Pounds, I fet it down one Column
further, towards the Left-hand, becauſe there are no more
Figures to multiply; and fo I find the Product, Amount, or
whole Charge of the 12 Yards is 4 Pounds, as before. In the
third Method I begin with 3, and fay 3 Times 8 Pence is
24 Pence, which being juft 2 Shillings, I fet down oo, and
carry the 2 Shillings forward, faying, 3 Times 6 Shillings is
18 Shillings, and 2 carry'd is 20 Shillings or 1 Pound; and
there being no more Figures to multiply, I fet down oo un-
der the Shillings, and I for the Pound in a Column by itſelf,
a little further to the Left-hand; then confidering that 4 Times
3 is 12, I multiply this Line or I Pound, which is the Product
of 1 Yard multiply'd by 3, or the Coft of 3 Yards, faying,
4 Times co Pence is co, and 4 Times oo Shillings is 00, and
4 Times
I : 6 : 8
4:
3
Addition of MONEY.
47
4 Times I Pound is 4; ſo I find the Cost of 12 Yards is 4
Pounds, as before. In the fourth Method I multiply the Price
of 1 Yard by 4, and that Product is the Value of 4 Yards,
faying, 4 Times 8 is 32 Pence, which is 2. Shillings and 8
Pence; I fet down 8 Pence under the Pence, and carry the
2 Shillings forward, faying, 4 Times 6 is 24, and 2 carry'd
is 26 Shillings; which being i Pound, 6 Shillings, I fet down
6 under the Shillings, and the I Pound a little further to
the Left-hand; fo the Coft or Amount of 4 Yards will be
found to be 1 Pound, 6 Shillings and 8 Pence; and 3 Times
4 being 12, I multiply this Line L. 1:6:8, by 3, faying,
3
Times 8 Pence is 24 Pence, which being juft 2 Shillings,
I make a Daſh - under the Pence, and carry the 2 Shillings
forward, faying, 3 Times 6 Shillings is 18, and 2 carried
is 20 Shillings, which being juft 1 Pound make a Daſh - un-
der the Shillings, and carry the I Pound forward; then fav,
3 Times I Pound is 3 Pounds, and 1 Pound carry'd is 4
Pounds, which is the Amount of the 12 Yards, as before. Or
you might have multiply'd by 12 at once, faying, 12 Times
8 Pence is 96 Pence of 8 Shillings exactly; fet down co, and
carry the 8 Shillings, then going on and faying 12 Times 6
is 72, and 8 carry'd is 80, fet down o, and halve the 8,
which is 8 Times 10 Shillings, and it makes juft 4 Pounds,
as before.
2
If the Price or Quantity was any other Value or Number,
you may perform it after the fame Manner; as, If I buy 42
POUNDS of TOBACCO at 13 d. į per Pound, Ihat is the
Value of the hole? Here I confider that 6 Times 7, or 7
Times 6 make 42; fo that I may multiply the given Price
by either of thefe Numbers first, and then that Product by
the other, and the laft Product will be the Amount of the
whole Quantity bought or fold; and here, and always, when
the Number of Pence exceeds a Shilling, fet them down in
Shillings and Pence, as follows, viz. 13d. is one Shilling, one
Penny, and one Half-penny, or 2 Farthings, which I fet down
thus:
S. d.
I: I :
9.
2 and firſt I multiply by 6
6
and 6:
9:
7
Then I multiply by
and the Product L.2: 7:3
is the Value or Coft of 6
Pounds.
is the Price of 6 Times
7, OF 42 Pounds.
The
48
Addition of MONEY.
The Second Method.
S. d. 9.
which I multiply by
-
and
then I multiply by
II the Price of 1 Pound,
-
-7
7 10 is the Price of 7 Pounds;
6
and the Product L. 2: 7: 3 is the Price of 6 Times
or 42 Pounds.
7
By both thefe Methods or Ways of doing the Question, the
Total Amount, is 2 Pounds, 7 Shillings and 3 Pence; for hav-
ing fet down the Price of 1 Pound, viz. I s. Id. 2q. or I
Shilling I Penny and 2 Farthings, I fay 6 Times 2 Farthings
is 12 Farthings, which is 3 Pence, therefore I make a Daſh
- under the Farthings and carry the 3 Pence forward, fay-
ing, 6 Times I Penny is 6 Pence, and 3 carry'd is 9 Pence,
which being lefs than a Shilling, I fet it down under the
Pence, and carry nothing; then I fay, 6 Times 1 Shilling is
6 Shillings, which being less than a Pound, I fet it down,
and that Product is 6 Shillings and 9 Pence, which is the
Coft of 6 Pounds of Tobacco at this Price per Pound: Then
as 7 Times 6 is 42, I multiply this Line by 7, faying, 7 Times
9 Pence is 63 Pence, which being 5 Shillings and 3 Pence,
I fet down the 3 Pence under the Pence, and carry forward
the 5 Shillings; then I fay, 7 Times 6 Shillings is 42 Shillings,
and 5 carried, is 47 Shillings. Now there being no more
Figures to be multiply'd, I fet down the odd 7, and halve
the 4, the Half of which being 2, is 2 Pounds; fo that the
Coft of 42 Pounds of Tobacco, at this Price, is 2 Pounds, 7
Shillings and 3 Pence. And for a Proof of the Truth
of the Work, you ought always to do it at leaft two diffe-
rent Ways: Therefore you fee it is fet down thus, I s. Id.
in the Second Method, and multiply'd by 7, by faying,
7 Half-pence is 3 Pence Half-penny; I fet down and
carry 3 to the Pence; then fay, 7 Times I Penny is 7 Pence,
and 3 carried is 10 Pence, which being lefs than a Shilling,
is fet down, and nothing carried; then 7 Times 1 Shilling is
7 Shillings, which is alfo fet down, and the Amount of 7
Pounds of this Tobacco is now found to be 7 Shillings and 10
Pence Half-penny; and as 6 Times 7 being 42, I multiply this
Line by 6, faying, 6 Half-pence is just 3 Pence; therefore
I make a Dafh - under the Farthings, and carry the 3 Pence
forward,
2
Addition and Multiplication of MONEY. 49
forward, faying, 6 Times 10 Pence is 60 Pence, and 3 carried
is 63 Pence, which being 5 Shillings and 3 Pence, I fet down
3 Pence and carry the 5 Shillings; then I fay, 6 Times 7
Shillings is 42, and 5 carried is 47 Shillings: Here ſet down
the 7 under the Shillings, and halve the 4, which are 4
Times 10 Shillings, and the Half being 2, I ſet it down a
little further, and the whole Amount is 2 Pounds, 7 Shillings.
and 3 Pence, as before; from whence you may conclude
both the Operations are right. After the fame Manner you
may find the Value of any other Quantities, at any other
• Rates: As; Suppofe a Butcher buys 24 Calves, at 3 Pounds,
5 Shillings and 6 Pence a Piece, What do they all come to?
Here firft, I fet down the Price of one, and confider that
4 Times 6, or 6 Times 4 is 24; or 3 Times 8, or 8 Times
3 is 24; or that 2 Times 12, or 12 Times 2 is 24. Now I
may take any of thefe 2 Numbers; and if the Work is
rightly performed, the laft Product or Total Charge will al-
ways be the fame; as you may fee by the feveral Operations
following.
1. S. d. l. S. d. l.
3: 5:6
S. d. 1. 5. d.
3: 5:6
3
56
4
6
3
3: 5:6
8
13: 2:
19: 13
9: 16: 6726: 4:
6
4
8
3
78: 12:
78: 12:
-
1. S. d. 7. 5.
d.
3: 5: 6
I2
3: 5: 6
2
6 : II :
39: 6:
12
2
78: 12:
78: 12:
78: 12: 78: 12
Here you fee this Question
folv'd fix different Ways,
and tho' the firſt Product in
each varies greatly, yet the
laft Product is the fame in
all of them, by which you
may be affured they are all
right. I have ſo fully ex-
plain'd the two foregoing Queſtions, that this cannot want
any I fhall only take Notice, that the first nine Rows of
Figures, in the fecond Part of the Pence-Table, Page 30,
will teach you to multiply by 12, at once, all manner of
Sums, that can require fuch an Operation; for 'tis the fame
Thing to fay, 12 Times I is 12, as to fay, 12 Pence is 1
Shilling; and 12 Times 2, is 24, as to fay, 2 Shillings are
24 Pence, &c. Now this Question may be done by Addi-
tion,
H
50 Addition and Multiplication of MONEY.
tion, if you fet down the Price of 1, two, 3, 4, &c. Times,
and double that Product, &c. till you make up the Number
24, as you may obferve by what is here under done.
1.
S. d.
3:
3
5: 6 the Coft of 1 Calf.
5:6-
I more.
The Total
6
6:
II:
II is the Coft of 2 Calves.
of 2 more.
The Total 13:
2: - is the Coft of 4
Calves.
13:
2:
of
4 more.
The Total 26: 4 is the Coft of 8 Calves.
26: 4:
26: 4:
8 more.
8 more.
The Total 78: 12: - is the Coft of 24 Calves.
56 the Price of 1 Calf.
I more.
I more.
1.
s. d.
3
3:
5:6
3:
5: 6
The Total 9:
9:
16: 6 is the Price of
16: 6
3
Calves:
-
3 more.
The Total 19 13 is the Price of 6 Calves.
19: 13:
-
6 more.
06 is the Price of 12 Calves.
The Total 39: 06:
39 : 06:
The Total 78: 12:
12 more.
is the Price of 24 Calves.
Thus you fee, Addition and Multiplication always produce
the fame Total Amounts, let the Particulars be what they will.
I ſhall now go on to fhew you how to act in fuch Cafes,
where the given Number, or whole Quantity bought or fold,
cannot be exactly found in, or by the Table, viz. fuch as
theſe that follow. What comes 23 Ells of Dowlafs to, at
16 d. per Ell? Here I confider, and find the Table has
no Numbers, but what are either too much or too little; and
becauſe
2
Addition and Multiplication of MONEY. 51
becauſe I have not yet taught you Subftraction, you muſt not
take any Numbers too large; and therefore you muſt take
two or more Numbers that are too little; fuch as 3 Times
7, or 7 Times 3, or 4 Times 5, or 5 Times 4, or 3 Times
6, or 6 Times 3, &c. and under this Circumſtance 'tis al-
ways beſt to take 2 Numbers, whofe Product comes the
neareſt to the given Number; as in this Example, 3 Times 7,
or 7 Times 3, which is 21, is the neareſt leaft Number, or
Product in the Table, to 23, the given Number or Quantity,
whofe Value or Amount is required; and therefore I work with
the 3 and 7, or 7 and 3, as before directed, thus:
S.
d. q.
I : 04the Price or Coft of 1 Ell.
Multiply'd by
3
Gives 4 4
Then multiply by
01 the Price or Coft of
ΟΙ
7
3 Ells.
7 Times
21 Ells.
I Ell more.
I Ell more.
L. I : 08 : 10 is the Price or Coft of
3, or
the Price of
:
To which add or 04
and OI: 04
2
2
The Total L. 1:11: 07 is the Price of 23 Ells.
S.
d.
4.
the Price of 1 Ell.
Or thus 1 : 04 12/2
Multiply'd by
I
7
Gives 09:07 the Price of 7 Ells.
This multiply'd by
3
Gives L. I : 08 : 10 2/2/2
Multiply 1 Ellby 2
and it comes to
2}02:09
21 Ells gives L. I: II: 07 for
before.
the Price of 21 Ells.
the Price of 2 Ells,
which being added to
the Price of 23 Ells, as
Here you are to obferve, that as Times 7 makes but
21, there was 2 wanting to make it 23; therefore I ſet
down the Price of one Ell twice, and then adding the
Whole together, find it comes to I Pound, 11 Shillings
and 7 Pence Half-penny. Now to be fatisfied that the Work
is right, I begin with the 7, and then multiply that Pro-
duct by 3; and as 2 are ftill wanting, I multiply the Price of
H 2
I by
52 Addition and Multiplication of MONEY.
I by 2, and add that Product to the laft Product, and the Total
of thoſe two Products gives the Amount of the 23 Ells fought
after, and which is the fame with what was found before.
Now if you take any other two Numbers, and add to the
laft Product the Price of fo many fingle Ells. as are want-
ing, to make up the Number required, you will find the
Total always to come out the fame, if your Work is right;
as for Inſtance,
S. d. q.
I
Multiply'd by
4 the Price of 1 Ell.
Gives 5: 6
This multiply'd by
Gives I : 7: 6
1 Ell multiply'd by 3 gives 4:
Total L. 1.: 11
Or multiply
by
and
This multiply'd by
1
4
for the Price of 4 Ells.
5
- for the Price of 20 Ells.
the Price of
3 Ells:
is the Price of 23 Ells.
I
7
S.
d.
q:
I
: 4 ½
the Price of 1 Ell.
2
2:
9
9 -
4: 9
for the Price of 18 Ells,
for the Price of 5 Ells.
gives L. 1: 4:
and 1 Ell by 5 gives 6 10
The Total is L. 1 : 11:
:
7
will be the Price of 2 Ells.
for the Price of 23 Ells.
And fo of any other two Numbers, whofe Product is
lefs than the given Number: As, fuppofe 4 Times 4,
which is 16; then I muft multiply the Price of 1 by 7,
to add to 16, to make up the 23 required. Again, What
comes 39 Sheep to, at 18 s. and 9 d. a Piece? Here 6
Times 6, or 4 Times 9 is 36; and this is the neareſt
Product of any 2 Numbers in the Table, fo that having
us'd either of thefe, then you muſt add the Value or Price
of 3 more to the laſt Product, and you have the true
Anfwer, if your Work is rightly performed; as you fee
done in the following Page.
18: 09
Addition and Multiplication of MONEY. 53
1.
5. d.
the Price of 1 Sheep.
I
: 18 : 09
Multiply'd by 6
Gives 5: 12:06 for the Price of 6 Sheep.
This multiply'd by
Gives L.33
1 by 3 gives 2:
15:
16
Total L.36 11
1. S.
6
- for the Price of 36 Sheep.
03 for the Price of 3 Sheep.
:
03
is the Price of 39 Sheep.
d.
- 18 09 the Price of 1 Sheep.
4
3 15: the Price of 4 Sheep.
L. 33 15:
-
9
- the Price of 36 Sheep.
2: 16:03 the Price of
3 Sheep.
L. 36 11 03 the Price of 39 Sheep.
If you make Choice of 4 Times 8, or 8 Times 4, whoſe
Product is 32, then you muſt multiply the Price of 1 by 7,
to make up the 39, &c.
Again, What is the Value of 58 Pieces of fine Holland, if
1 Coſt L. 7: 13: 8? Here
138? Here you may take 7 Times 8, or
8 Times 7. which is 56, and to the laft Product, add the
Product of 1, multiply'd by 2, and the whole Amount will
be L. 445:12:08; or you may take 6 Times 9, or 9 Times
6, whofe Product is 54, and to the laft Product, add the
Product of the firft Line, or Price of 1 Piece, multiply'd by
4, and you will find the Total of both Ways come alike,
if your Work is right; as you may ſee done here under.
5. d.
1.
7 13 08 the Value of 1 Piece.
7
53 15 08 the Value of 7 Pieces.
8
430 05 04 the Value of 56 Pieces.
07:04 the Value of 2 Pieces.
15
L. 445: 12:08 the Value of 58 Pieces
7.
パ
​54 Addition and Multiplication of MONEY.
1.
5.
d.
7 13 08 the Value of 1 Piece.
6
the Value of 6 Pieces.
46: 02:
9
30 14
08 the Value of 4 Pieces.
414: 18:
the Value of 54 Pieces.
L. 445: 12: 08 the Value of 58 Pieces.
So if the Value, Coft, or Price of 67 Any-things were re-
quired when the Value, Coft or Price of 1 was, L.58: 17 : 104.
Here you might take 8 Times 8, and add the Value or A-
mount of 3 fingle ones to the laft Product; you will find the
Total come to L. 3945: 19 : - ; or if you take 7 Times 9,
or 9 Times 7, and add the Price of 1 multiply'd by 4, to the
laft Product, you will find the Total to be the fame. Now
I prefume you have been fufficiently inftructed in whatever
is neceffary, for all Numbers, under 100: I will only add
one or two Examples more, and then fhall go on to fhew
you how to find the Value or Number fought, when it 'ex-
ceeds 100, and then fhall go on to the Addition of Weights
and Meaſures. As, Suppofe I buy 8 Yards of Muslin at
3 s. 7d. per Yard, 9 Ells of Holland, at 5 s. 9d. per Ell,
18 Yards of Silk, at 11 s. 6d. per Yard, 24 Yards of Lace, at
17 s. 8d. per Yard, 10 Yards of Cambrick, at 9 s. 9d. per
Yard; and 75 Ells of Sheeting, at 2 s. 4 per Ell, What do
they all come to? Here I fet down the Particulars, and
find their Values as has been already taught; then I fet down
the feveral Amounts or Values under one another, and the
Total is the Anfwer to the Question, as follows,
S. d.
1.
S. d.
3:7 per Yard comes to 1:08:08
5:9 per Ell, comes to 2:11:09
11: 6 per Yard, comes to 10: 07:
9
8 Yards of Muflin at
Ells of Holland at
18 Yards of Silk at
24 Ditto of Lace at
10 Ditto of Cambrick at
75 Ells of Sheeting at
17: 8
Ditto
9: 9
Ditto
2: 4
per Ell
21:04:
4:17:06
8:18:01/
Total 49:07:1
Now
3
Addition and Multiplication of MONEY. 55
1
Now for Numbers that are larger than 100: As, What is the
Value of 320 Bushels of Corn, at 4 s. 9d. per Bufhel? When-
ever the given Number confifts of 3 Figures, as in the Ex-
ample, fee if the 2 Figures towards the Left-hand are the
Product of any two Figures in the Table; and if they are,
multiply the Price of 1, by thofe two Figures, as you have
been already taught, and the laft Product multiply by 10,
and you have all but what ftands in the Units Place, by
which you muſt multiply the Price of 1, and add that Pro-
duct to the Product of 10, and your Work is done; the
Total being the Amount of the Quantity fought after, if
your Work is rightly performed; in the Example given,
the two laft Figures are 32, which is the Product of 4 and
8, or 8 and 4; therefore I fet down the Price of 1 or 4:9;
and work, as here under.
5.
d. q.
49 the Price of 1 Bushel.
19:
4
2 - the Price of 4 Bufhels.
8
7: 13 4 - the Price of 8 Times 4, or 32
IO Bufhels.
L. 76: 13: 4 - the Price of 10 Times 32, or
320 Bufbels.
S. d.
Or thus,
4: 9
-1200
the Value of Bushel.
I: 18: 4 - the Value of 8 Bufhels.
4
7 13 4 - the Value of 4 Times 8, or
ΙΟ
32 Bufbels.
L. 76: 13: 4 - the Value of 10 Times 32, or
320 Bushels.
So
56 Addition and Multiplication of MONEY.
So if the Price of an Ox was L. 8: 13: 6, What would
189 Oxen come to? Here I take 3 Times 6, or 6 Times 3,
or 2 Times 9, or 9 Times 2, for the 18; then multiply
that Product by 10, and the laſt Product will be 180; then
I multiply the Price of 1 by the g Units, and add that Pro-
duct to the Product of 180, and the Total Amount is
L. 1639: 11:6 as you may fee by the Example.
1.
S.
I Ox Coft
9
8 13:06
d.
9
9 Oxen Coft
78 01 06
2
18 ditto Coft
156: 03:
10
: :
180 Oxen Coft L. 1561: 10:
Oxen Coſt - 78: 01:06
9
189 Oxen Coft L. 1639: 11: 06
4
3
+
II 4
9, or
Again, What comes
635 Pounds of Tobacco
to at 11 d. per Pound?
Here the 2 laft Figures
are 63, which is the
Product of and
7
9 and 7;
7 ; when
you have
found the Amount of 63,
multiply the laft Product
of theſe 2 Figures by 10,
and this third Product,
gives 630; and the firſt
Line, or II d.
the
Price of 1 Pound, muſt be multiply'd by 5, and that Product
added to the laft Product or Amount of 630, yields
L. 29 15 3 for the whole 635 Pounds; and fo on for any
other Number whatever, whofe 2 laft Figures are the Pro-
duct of fome two other fingle Figures; as, 728 Dollars, at
4s. 7d. per Dollar, comes to L. 167: 11: 10; and 817
· French Crowns, at 5 s. 2 d. per Crown, comes to
L.213: 12:23. When the 2 laft Figures are not the exact
Product of fome two other fingle Figures, take the Product, that
is the next leaſt, and work as above; only work with the
10 firft, and you will have the Total, all but fo many Tens
as the Product of the 2 Figures you chofe, was fhort of
the two laft Figures of the given Number to the Left-hand;
for which multiply the firft Product, or Reſult of the firft
Line, multiply'd by Ten, by fo many Units as there are Tens
wanting; and add this laft Product to the laft, or third
Product found before, and their Total will be the Amount of
the whole Sum required, if your Work has been rightly
performed. See an Example; Ihat comes 516 Portugal-
Pieces of Gold to, each 36 Shillings Sterling? Anfwer,
L. 928: 16. Here I confider that 5 Times 10 is 50;
therefore
Addition and Multiplication of MONEY. 57
therefore I multiply L.
duct by 5, and that
Product by 10, and
I have got the
Value of 500; then
as 16 are ſtill
wanting, I fet
down the firſt Pro-
duct L. 18 for 10
Pieces, and multi-
pły L.1: 16 by 6,
and add theſe
three Lines toge-
ther, and find the
Total is L. 928: 16.
I: 16 by 10, and then that Pro-
ΙΟ
18: 00 the Value of 10 Pieces.
5
90: 00 the Value of 50 Pieces.
ΙΟ
900: 00 the Value of 500 Pieces.
18: 00 the Value of 10 Pieces.
10: 16 the Value of 6 Pieces.
L. 928: 16 the Value of 516 Pieces.
Now for the Proof of this Operation, firft multiply by 10,
and then take 6 Times 8, which will be 48; fo that 3 are
wanting to make up 51, for which multiply the firſt Product by
3, and ſet that Product down under the third or laſt Product
and then multiply the Value of 1 by the Units, and add theſe 3
Sums together, and their Total will be the fame with that a-
bove, if your Work is rightly performed; as you will fee by
what follows.
L.
Multiply by
1: 16 the Value of
ΙΟ
I Piece.
ΙΟ
the Product 18: oo is the Value of 10 Pieces.
That multiply'd by
the Product 108:
This multiply'd by
6
6
00 is the Value of 60 Pieces.
S
8
30 Pieces.
the Product 864 00 is the Value of 480 Pieces.
The firft Product by 3 is 54: 00 the Value of
The Value of I by 6 is 10: 16 the Value of
as before.
Total L. 928
6 Pieces.
16 is the Value of 516 Pieces,
So if you fuppofe that a Seaman's Wages comes to
L. 14: 19: 6 per Annum, What will pay a Man of War
of 684 fuch Men? Here 8 Times 8, which is 64, is the
neareft Tabular-Product, that is lefs than the Sum required;
wherefore I multiply by 10: 8, and 8; and the firft Product
by 4, for the 4 Tens wanting to make up the 68; and then
I
multiply
58 Addition and Multiplication of MONEY.
multiply the Charge of 1 Man by 4, for the 4 Units; add the
feveral Products together, and the Total is the Anſwer re-
quired: Or you may take 7 and 9 for 63; and then the firſt
Product muſt be multiply'd by 5, to make up the 63, 68, as
you ſee done in the following Examples.
Multiply'd by
1.
S.
d.
14: 19: 6 the Pay of
The Product 149: 15:
This multiply'd by
The Product 1198 : 00:
This multiply'd by
The Product 9584:
00:
Firſt Pro. mult. by 4 is 599: 00:
I Man by 4 is 59: 18:
The Total 10242: 18:
S.
ΙΟ
I Man.
IO
- is the Pay of 10 Men.
8
8
8
is the Pay of 80 Men.
8
- is the Pay of 640 Men.
the Pay of 40 Men.
the Pay of 4
-
-
Men.
- is the Pay of 684 Men.
d.
14: 19: 6 for
1
1.
Or thus,
1 Man.
IO
IO
149 15- for
10 Men.
7
7
1048 : 05:
for 70 Men.
9
9
9434 05:
-
for 630 Men.
748
15:
for 50 Men.
59: 18:
for 4 Men.
10242: 18: for 684 Men.
-
And thus you may find the Value of any 3 Figures what-
eyer, which is fufficient for all common Bufineffes; and there-
fore I fhall go on to the Weights and Meaſures. And firſt in
CLOTH MEASURE.
You are to obferve that 4 Nails make 1 Quarter of a
Yard; 4 Quarters, 1 Yard; 5 Quarters of a Yard is an Eng-
ご
​lifh
Addition of diverfe Denominations.
59
lifh Ell; 3 Quarters of a Yard, a Flemish Ell; and 6
Quarters of a Yard a French Ell, or Aulm.
CLOTH-MEASURE.
Yards Qrs. Nails
(5) (4)
Ells Qrs. Nails
367
582 :
842 :
976 :
789 :
555 :
666
1232 HON OH SI23
: 3
: I
:
796 3 3
:
371: I :
3
67
32
81
96 :
74
83
4 : I
2 :
132 H ♡ ♡ ♡~
I
2
62
3
2
71
58
I
:
I ♡ - 2431
216 +MN HOO
:
I
:
Ι : 2
3
2. N
2
213 I
: 3
II
2
:
I2
: I
:
3 2
13 : 3 I
675: 4
I.
(4)
: I
0 :
848: 3 :
921 :
316
827
7 7
8862
:
1
69
63
(3)
Ell Flem. Qrs. Nails
5827 :
165
I
I :
ΙΟ :
1
227
Aulm Qrs. Nails
58
36:
75
5.
2
I
2.
*
382 :
:
I
71
I
44
3.
*
58 : I :
6:2
3
56 : 2
3
6
:
I
**
..
**
19
:
89
112 2 H2H 2 H
18: I:
752: -:
643
9836
333- MM2 2 MIT ME
3
67
78
I
3.
5. 3.
5. I
: 4.
91 : 3
ΙΟ
86
: 2
*
2
3.
Ι
3.
3.
1
Ι
2
1
3
77
:
5.
3.
8271: 2: 2
94
:
4. : I
**
*
2606I : I : 2
869 : 4:3
In theſe Examples, you may begin at the Bottom and go
upwards, or at the Top and come downwards, as beſt pleaſes
you; and when you have found what the Sum of the Nails
is, then confider how many Quarters are contained in them,
that is, how many Times 4 is in that Number, the fame
I 2
as
60
Addition of diverſe Denominations.
as you did in the Farthings: As here, in the first Example to
the Left-hand, the Nails come to 25, which is 6 Times 4,
and I over; therefore I fet down the Overplus I, and car-
ry the 6 Quarters to the Quarters, and having added them
up, find them come to 27, with the 6 carried. Now 27
Quarters is 6 Yards, and 3 Quarters, becauſe 6 Times 4, is
24, and 3 is 27; therefore I fet down the 3 odd Quar-
ters, and carry the 6 Yards to the Yards, and add them, the
fame as if they were Pounds of Money, Sheep, Houfes, &c.
as in the Examples of whole Numbers and Money; and the
whole Sum comes to 8862 Yards, 3 Quarters and 1 Nail,
which you may prove by breaking it into two or more Parts,
and finding the Total of each Part, and then adding thoſe
Totals together, and you will find their Total come to the
fame as above: Or you may make a Det.or * at every
4 Nails in the Nails, as you was fhewn in the Farthings,
and as you fee done in the fourth Example, where you'll
obſerve that the Dots or. 's go upwards, and the *'s come
downwards. In the faid fourth Example the Dots are made at
6, in the Quarters, becauſe 6 Quarters is an Aulm, or French
Ell; the like must be obferv'd in the other Examples, viz.
to dot at 4 in the Quarters, when the whole Numbers are
Yards, as in the firft Example; by 5, when they are Englif
Ells, as in the fecond Example; and by 3, when they are
Flemish Ells, as in the third Example; and if you think
'twill be any Advantage to your Memory, you may ſet
down what you stop or count by, over the Head of the Line
or Column of the inferior Denominations, as in the ſecond
and third Examples above, where over the Nails is ſet 4,
becauſe all the Divifions above, are proportion'd to the Eng-
lifh Yard, and 4 Nails make a Quarter of a Yard; and in
the fecond Example, over the Quarters, ſtands 5, becauſe
5 Quarters of a Yard is an English Ell; and in the third
Example, over the Quarters, ftands 3, becauſe 3 Quarters
of a Yard is a Flemish or Dutch Ell. I have ſet nothing o-
ver the English or Flemish Ells, becauſe all whole Numbers
whatever, whether Monies, Weights, Meafures, Time, &c.
are always counted by Tens; as in the Addition of whole
Numbers, has already been fully explain'd.
LONG MEASURE.
This is only us'd in, or apply'd to know the Diſtance of
one Place from another, and to fettle the Rates of Poft-
Horfes,
Addition of diverſe Denominations.
61
Horfes, Coaches, Letters, &c. wherein a Mile, being the
Integer or whole Thing, the ufual Subdivifions are as follow;
3 Barley Corns make 1 Inch, 12 Inches 1 Foot, 3 Feet 1 Yard,
5 Yards, or 11 half Yards 1 Pole or Perch, 40 Poles or
Perches 1 Furlong, and 8 Furlongs I Mile.
I
LONG-MEASURE.
This is uſed to meaſure Roads, or the Diſtance of one
Place, Town, or City from another.
Miles Furlongs Poles
(8)
(40)
Yards Inches
(11)
(18)
Here you will
38 :
5.
17.
*
29
4.
*
: IO. 13
*
IO. 17.
obferve, that as
*
*
*
07
14.
*
*
12.
63:
82
:
98
76
:
:
58
36
:
27
:
65
579 :
6.
intron + MNO in
37.
*
2 H
*
16
25.
18
ம்00
28.
* *
IO.
09.
08.
**
13.
*
14
38.
*
07.
15.
29.
об
:
16.
*
*
39
IO
17.
5.
32.
09.
* 08
7.
6.
*
A
: 07:05: 13
3 Feet, or 36
Inches make I
Yard; fo 18 In-
ches, or I Foot
and make
Yard; fo I put
over the Column
of Inches 18, and
I make Dots or
*
s at every 18;
and in the Column
of Tards, I do
the fame at every 11; and in the Column of Poles, I do the
fame at every 40; and in the Furlongs, at every 8; and in
the Miles go by 10, as in Addition of IVhole Numbers
; for
the laft Column, let it be called what it will, is always
eſteemed whole Numbers; and after the Whole is gone
through, it amounts to 579 Miles, 7 Poles, 5 half Yards, and
13 Inches: You may vary the Order of the Figures of this
Example, and then 'twill be another Example, of which you will
find the Total always come to the fame, if the Work be
rightly performed.
LAND or SQUARE MEASURE.
This is peculiarly adapted to Surveying, it being the Con-
tents of the fuperficial Meafures us'd for that Purpofe; and
though 'tis here carried to a Mile, yet 'tis feldom in Practice,
carried farther than an Acre.
144
62
Addition of diverfe Denominations.
144 Square Inches is 1 Square Foot.
9 Square Feet is 1 Square Yard.
30 Square Yards, and is 1 Square Pole.
40 Square Poles is 1 Square Rood.
4 Square Roods is 1 Square Acre.
10 Square Acres is 1 Square Furlong.
64 Square Furlongs is 1 Square Mile.
DRY MEASURE.
This is, or ought to be us'd in Grain, Salt, Coals,
Sand, &c. and ought all to be ftruck, as unground Corn is;
but Cuſtom prevails both for the Mode of ufing and Bignefs
of the fame Meaſure, in different Places, however in Lon-
don they are moſt regarded.
I
2 Pints is 1 Quart; and 4 Quarts, or 8 Pints, is 1 Gallon ;
2 Gallons is I Peck; 4 Pecks is 1 Bufhel, Sand-Meafure; and
5 Pecks 1 Bushel, Water-Meafure; 8 Bushels is 1 Quarter;
4 Quarters, 1 Chaldron; 5 Quarters, I Wey; and 2 Weys,
i Laft.
N. B. In London, 36 Bufhels of Sea-Coal is a Chaldron.
WINE MEASURE.
By this Meaſure all Wines, Brandies, Rum, Arrack, and
all other Strong-Waters or Spirits, are meafured; as alfo Mead,
Perry, Cyder, Vinegar, Oil, Honey: And in retailing them,
all very strong or Fine-Ales, fuch as Burton, Nottingham,
Yorkſhire, Welſh, &c. Ales. Now the Subdiviſions are as
follows, viz.
2
2 Pints is 1 Quart, 2 Quarts r Pottle, 2 Pottles 1 Gallon,
18 Gallons 1 Rundlet, 31 Gallons Barrel or Hogfhead,
I
2
42 Gallons 1 Tierce, or half Punchion, 63 Gallons 1 Hogfhead,
84 Gallons 1 Puncheon, 2 Hogfheads 1 Pipe or Butt, and 2
Pipes 1 Ton.
Here you are to Note, That this Gallon ought to contain
231 Solid Inches in Meaſure, and which, of pure Rain-
Water, fhould weigh 8 Pounds, 1 Ounce, II Drams, A-
voirdupoife-Weight, or 9 Pounds 10 Ounces I Penny Weight,
and Troy-Weight.
BEER
Addition of divers Denominations. 63
BEER or ALE MEASURE.
Here 2 Pints make I Quart, 4 Quarts 1 Gallon, 8 Gal-
lons I Firkin of Ale, 9 Gallons i Firkin of Beer, 2 Firkins
of Beer a Kilderkin, 2 Kilderkins a Barrel, and 1 Barrel and
a Hogfhead, and 2 or 3 Barrels Hogfheads a Butt.
I
In this Meaſure 282 Solid Inches are a Gallon, and of pure
Rain-Water weighs 10 Pounds, 3 Ounces and 7 Drams,
Avoirdupoiſe Weight. Note, "Tis only in London, and with-
in the Bills of Mortality, that the above Diviſions are ob-
ferv'd; in all other Parts of the Kingdom the Beer and Ale
are gauged by one Meaſure, viz. 34 Gallons to a Barrel,
17 to a Kilderkin, and 8 to a Firkin.
TROY WEIGH T.
This is the oldeſt, and the original fettled regular Weight in
England, and taken from a Grain of Wheat, full ripe,
and well dried; for which Reaſon the fmalleft Subdivifion
was, and ftill is call'd a Grain; 32 of theſe Grains at firſt,
made another Weight, call'd a Penny Weight, or weigh'd
ſo much as the Silver, that at that Time went for a Pe
enny.
See the Statute of 51 Hen. 3. But 32 being not fo fit a Num-
ber, to be exactly divided into a large Number of Aliquot
Parts, as,,,, &c. the 32 was chang'd into 24, as it
now ſtands; that is, the preſent Grain, is I and a third Part
of the original Grain, or three of thefe prefent Grains, is
equal to 4 of the original Ones. Now by this Weight all fine
Goods, fuch as Gold, Silver, Pearls, phyfical Potions and
Bread were weigh'd; but by a late Statute, Bread is weigh'd
by the Common or Avoirdupoife Weight.
In Troy Weight 32 natural or 24 artificial Grains make ī
Penny Weight, 20 Penny Weights make 1 Ounce, and 12
Ounces make x Pound. Now although Medicines are weigh'd
by this Weight, yet the Apothecaries divide it as follows,
viz. 20 Grains make 1 Scruple, mark'd thus ; 3 Scruples make
I Dram, mark'd thus 3; and 8 Drams make 1 Ounce, mark'd
thus 3 and 12 Ounces make 1 Pound, mark'd thus ; and
this is particularly call'd APOTHECARIES WEIGHT.
AVOIRDUPOISE WEIGHT.
This is faid to come from the French Words, AVOIR
DU POISE, fignifying, to be heavier or more than Weight;
and
64
Addition of divers Denominations.
and this is us'd in all fuch Goods, where there is any Wafte
or Lofs by Drofs, Duft, Skins, &c. whofe Subdivifions are
as follows, viz.
16 Drams make 1 Ounce, 16 Ounces make I Pound, 28
Pounds make 1 Quarter of a Hundred, 4 Quarters 1 Hun-
dred Weight, and 20 Hundreds are 1 Ton; the Marks us'd
in this Weight are Tons -€; 2; tb, Oz. Dra.
ΤΙΜΕ.
By this we meaſure the Length of our Lives, the Periodick
Revolutions of the Planets, and the Seafons of the Year, and
compute the moſt remarkable Accidents of ancient as well as
modern Hiftories: To perform which, we fuppofe a Circle,
and in the Center of that Circle, an Index fo fix'd, that it
moves orderly, regularly and fucceffively, equal Spaces con-
tinually; and for vulgar Ufe, we go no lower than Seconds,
though you may conceive in your Mind, Thirds, Fourths,
&c. But 'tis common to fay or reckon 60 Seconds for a
Minute, 60 Minutes for an Hour, 24 Hours for a Day, and
365 Days for a common Year, and once in 4 Years 366 Days;
though this is about 10 Minutes, and 50 Seconds too much in
each Year, yet as our Superiors have not thought fit to recti-
fy it, we ſhall not contradict the commonly receiv'd Me-
thod, or vulgar Computation as now us'd, in which are the
following Subdiviſions, viz.
60 Seconds is a Minute, 60 Minutes is an Hour, 24
Hours is Day, 7 Days is a Week, 4 Weeks is a Month,
and 13 Months is a Year.
But this wants a Day and 6 Hours of the common Year,
which, by our Almanacks, is thus divided.
60 Seconds is a Minute, 60 Minutes is an Hour, 24 Hours
is a Day; and fometimes 28, 29, 30 and 31 Days is a Month;
and 12 of thefe unequal Months is a Year, that is to fay,
January, March, May, July, Auguft, October and December,
have always 31 Days; April, June, September and Novem
ber, have always 30 Days; and February for 3 Years fuc-
ceffively hath 28 Days, and the fourth Year 29 Days.
An Example or two will make all plain.
LAND
I
Addition of divers Denominations.
65
LAND MEASURE.
Acres Roods Poles
27
46
:
74 :
13
22
∞ H
87
16
94
:
(4) (40)
I
1231 2
I
#
:
25
17
35
16
28
: 19
24
32.
Here you may obſerve, that this
Example may reprefent 12 Fields,
or Parcels of Land, of which an
Eftate, or Part of one, may be ima-
gined to confift; fo that upon
Surveying and Cafting them up, they
amount to 631 Acres and 12 Poles
of Land, which may be worth
much or little, as they fhall be fi-
I
tuated, or naturally fruitful.
have fet 40 over the Poles, which
in cafting up muft not be reckon'd,
that ſtanding only to remind you
how many Poles make a Rood; and
fo the 4 over the Roods is only to
keep in your Mind, that 4 Roods
make an Acre. The Acres being
here the laſt Column, or whole Numbers, muft be reckon'd by
10, as muſt always the laft Column, let the Name be what
it will; as was obferv'd before.
76
58
27 :
85
:
631 :
32
1
I
18
17
29*
3: 32
: 12
DRY MEASURE.
WINE MEASURE.
Quarters Bush. Pecks Hogfbeads Gallons Pints
29
92
76
:
:
(8)
4
84
58 6
79 56
48 745
67
34
27 :
18
7
23+5O NDHON SM
7 :
:
I :
6
•
(4)
: I
1 32 1 2
: I
H 2
64 :
ཆབ
122
*
7
4
(63) (8)
28 : 18
4
95
32
2
46
I
38
:
19
5
7
:
35
3
9
: 16
I
I
6
28
*
3
18
: 27
:
42
32
•
I
85
36
I
67:
48
3: 3
34
49
91 : 52
657
: 3
591 : 04
2
K
99
15
51:
2
5
3
**
56 776
* * *
BEER
66
Addition of divers Denominations.
# Oz. Druts. Gr.
(12) (20) (24)
92: 11: 16: 21.
84: 10: 17: 16
09: 13: 18.
07 18: 19.
08: 19: 17.
77
BEER and ALE MEASURE.
TROY WEIGHT.
Barrels
Gall. Quarts
(36)
(4)
34
:
27. : I.
16
:
18 : 2
18
:
29. :
3
98 7
37
:
32. :
I.
245
a∞ 36
92
:
35. : 2
38
:
84
35
62 :
18.
:
3.
17
:
2
10 : 2.
71
34 :
09. :
3.
I I
:
I
16 : 16
:
2
7
505 :
I I
*
2
AVOIRDUPOISE WEIGHT,
9: :
16. I 19.
Tons C. Qr.
2r. # Oz. Dra.
(20) (4) (28) (16) (16)
: 19. 10. : 09.
:* 2 2 : 18 : II : 10
: 3. : 16. : 12. : 15.
: I : 15.: 13.: II.
: 2.: 19: 14. 12.
: II.: 15. : 14.
: 13: 14.: 13
: 27.: 08 : 09.
: 3.: 21.: 07.: 08.
8 :
7: 10
6:
5:
4: 12. :
3: 15.:
2: 19:
I :
9
8
2: 16.:
•
7: :
ク
​6: 18. :
I. :
3.
: 19. 09: 10
15: 14. II.
: 24.: 13.: 10
2.: 16 : 07:07.
: 18. : II. : 09
82: 12 : 3:09:07: 04
66
42: II : 14
II: 14: 16
98: 10: 12 : 18.
89: 07: 13: 12.
76: 08: 16: 13
67: 09: 17: 15.
54: II: 18 19
9
9
792 : 02:00: 16
TIME.
Days Hours Min. Sec.
(24) (60) (60)
16: 16.: 36: 47.
31: 13: 29.: 32.
42: 12. : 18 : 18
71: 18. 19: 29
34: 19.: 48.: 39.
18: 21 : 29. : 49.
16: 14.: 31: 59.
:
15: 22. 39. : 21
19: 18.: 42. : 31.
87:23. 54: 41
24: 21 : 16.: 51.
36: 18.: 39: 18
418 05: 47: 15
You may divide thefe Examples into two or more Parts,
and add the Parts feparately, and then the Sum of thoſe To-
tals will be the true Total of the Whole; as was fhewn
in Money: Enough has been faid of Addition. Now I fhall
go on to Subſtraction.
you
CHAP.
Of SUBSTRACTION. 67
CHA P. III.
Of the Subtraction of whole Numbers, and alſo
thofe of various Denominations.
S
an or
UBSTRACTION is that Rule, by which the Dif
ference between, or Excess of, any two Numbers or Quan-
tities, are found out, or difcovered.
And here, as in Addition, you are to regard the Order of
the Places, or Degrees of Value, of which the Sums or Numbers
given confift, and mind to put exactly under, and take the
Units from the Units, the Tens from the Tens, the Hundreds
from the Hundreds, &c. as you fee done in the following
Examples.
His prefent Majesty King GEORGE II. was born the 30th
of October, in the Year 1683. How old is he, on his Birth-day
this Year, 1737 ?
;
Here, as in all other Cafes is common, I fet down the
largeſt Number uppermoft, and exactly under it the ſmalleſt
as you fee in the Mar-
gin: The Numbers bc-
1737 the prefent Year.
1683 the Year the King was born in. ing thus fet down,
•
54 the King's Age.
1683 Proof by Subftraction.
1737 Proof by Addition.
take or ſubſtract the
Units Place of the low-
eft Number out of the
Units Place of the
highest Number, faying,
3 out of 7, remains 4;
which accordingly
fet down under the Units Place, and go on to the Tens Place,
and fay, 8 out of 3 I cannot, therefore I take 1 from the
next Figure on the Left-hand, or Hundreds Place, which
being 10 Tens, makes the 3 that ftands in the Tens Place
13; then, I fay, 8 out of 13, and the Remainder is 5, which
I fet down under the Place of Tens; then I go on, and ſay,
6 out of 6 (becauſe I had before taken 1 from the 7 to make
the 3, 13) and there remains o; then I go on, and fay, I
out of 1, and there remains o: So that the whole Diffe-
rence is 54, which was the King's Age, the 30th of October,
1737. Again,
K 2
Suppoft
68
Of SUBSTRACTION.
Suppofe a King borrows of his Subjects 632568 Dollars,
and pays them again
587694
How much is still owing? Anfwer, 44874 ftill owing,
Proof by Subfiraction 587694
Proof by Addition
632568
Here the Numbers being firft regularly fet down under one
another, as above directed, and as you fee in the Example,
I begin at the Units Place, and fay, 4 from 8, remains 4;
then, 9 from 6 I cannot, therefore I borrow I from the
5, or next Figure on the Left-hand, which makes the 6, 16;
then I fay, 9 from 16, remains 7; which I fet down under
the 9, or Place of Tens, and go on, faying, 6 from 4 I
cannot; (Note, the Figure itſelf is 5, but I took away, or
borrow'd I from it, to make the 6, 16; ſo that the 5 is now
depreffed, or become equal to 4) and therefore I take or borrow
I from the next Figure towards the Left-hand, which here
is 2, and the 4 becomes 14, becauſe every Unit or I on the
Left-hand makes, or is equal to 10 Units or Ones, in the next
Place to it on the Right-hand. Now 6 taken from, or out
of 14, the Remainder is 8, which is here fet down under
the 6; then I go on and fay, 7 from 1 (becauſe I was be-
fore borrow'd of, or taken from the 2) I cannot, and there-
fore I borrow I of, or take it from the 3, and the 1 be-
comes II, and the 3 becomes 2, for the Reaſon above.
Now 7 taken out of II leaves 4 for the Remainder, which
I fet down in the fourth Place, under the 7, and go on,
faying, 8 out of 2 (becauſe I borrow'd or took 1 out of
the 3, laft Time) I cannot, therefore I borrow, or take 1
from the next Figure 6, to make this 2, 12; then I fay,
8 out of 12, remains 4; which I fet down in the 5's Place
under 8, and go on and fay, 5 out of 5 (becauſe I being
taken from the 6, makes the Remainder 5) and there re-
mains o; fo that I find the Sum ftill owing is, 44,874 Dol-
lars, the Truth of which may be prov'd by fubftracting
this Remainder out of the whole Debt, and the Remainder
will be the Sum paid; or you may add the Sum paid, and
the Sum remaining due, and their Total will be the whole
or original Debt; as may be feen by the two Examples a-
bove.
Again,
Of SUBSTRACTION.
69
Again, Suppoſe I wanted to know the Dif-987654321
ference between
and
123456789
By working as before, the Anfwer will be 864197532
found to be
Again, Suppoſe an Army to confift of 140000 Men, and that
by Death, Sickneſs, and putting into ?
How
Garifon
82345 were taken
°} 82
many ſtill remain in the Field? } 57655
Anſwer,
from it:
Here I begin, and fay, 5 out of o I cannot, therefore
1 is borrow'd of, or taken from the next Figure or Place of
Tens; and then I fay, 5 out of 10, and there remain 5;
then I go on to the next Figure, which is made 1 lefs, by
borrowing from it, to increafe that on the Right-hand 10;
but the Figure itfelf being a o, I go on to the 3d Figure,
which is alſo a o; fo I go on to the 4th Figure, which is
alfo ao; therefore I go on to the 5th Figure, towards the
Left-hand, which is the firft fignificant Figure, which here.
is the Figure 4; from which taking 1, it will be depreſs'd
to 3, and the 2d, 3d, and 4th Figures will become 9's; for
when I borrowed Ten to add to the first o, I could not take
it out of o, the fecond Figure, therefore I was forc'd to
go on to the next Figure to borrow 10 from the Hundreds
Place; but that being likewiſe a o, I went to the next
Figure or Thouſands Place, to take the 10 from thence; but
that being alſo a o, I was ftill forc'd to go on one Figure
farther, which was the 5th Place, or Tens of Thouſands;
and there taking 1, or 10 Thoufand out of the 4, or 40
Thouſand, the Remainder of that 1, or 10,000, was confe-
quently 9990; from which, taking the Figures underneath
the 2d, 3d, and 4th Places, and fetting down the Remain-
der, I come to the 5th Place; and though the Figure that
ftands in that Place in the uppermoft Line, is a 4, or
40,000, yet I must call it but 3, becauſe it has lent one of
its Parts or ten Thoufands to the foregoing Figures; therefore
I fay, 8, which is the 5th Figure towards the Left-hand,
in the under Row, out of 3, (viz. the Remainder after I is
taken out of the 4) I cannot; therefore I borrow I
which is 10, from the 6th Place, and it makes 13; then
I fay,
70
Of SUBSTRACTION.
I fay, 8 from 13, and the Remainder is 5; then I go on
to the 6th Place, but find no fignificant Figures left, be-
caufe the I that ftands in the 6th or laft Place, was taken
away or borrowed to be added to the 3, that remained af-
ter I was taken from the 4 in the 5th Place.
!
Again, Suppofe from 785000659000123 What Re-
I take or ſubſtract
697123000456789
Anfwer,
087877658543334
Proof by Addition
785000659000123
mains?
Proof by Subfraction 697123000456789
Here I begin, and fay, 9 from 3 I cannot, but 9 from 13,
remains 4; and as I borrowed or took away I from the
2, which is the next Figure towards the Left-hand, that
becomes 1; then I fay, 8 from 1 I cannot, therefore I bor-
Tow I from the next Figure, towards the Left-hand (which
is always 10 in Value, of thoſe next to it on the Right-
hand, let it be the 2d, 3d, 4th, 5th, or any other Place or
Figure, as has often already been noted) which being put,
or fuppofed to be put on the Left-hand, makes the 2 that
muft now be called 1, become II, and taking the 8 out
of it, there remains 3; then I go on and fay, 7 from o
(becauſe the next Figure which is 1, had I taken from it,
to add to the former Figure to make it 11) I cannot ; fo I bor-
row 1, which is ro, as above, and fay 7 from 10, there
remains 3; then I go on and fay, 6 from 9, and there re-
mains 3; then 5 from 9, and there remains 4; then 4 from
9, and there remains 3, becauſe the three o's that ſtand in
the 4th, 5th, and 6th Places cannot lend any thing, therefore
Iwas oblig'd to go to the firft fignificant Figure, and which
here is, to lend 1, or 10, to the 1, that was depreffed to a
o, and thereby it is deprefled to 0, (and the 3 Cyphers be-
come all g's, for the fame Reafon, as above in the laft
Example) fo that I go on, and fay, o from 8, and there re-
mains 8; and o from 5, and there remains 5; and o
from 6, and there remains 6; then from o I cannot,
therefore I borrow I from the next Figure, and fay, 3
from 10, and there remains 7; but as the two next Figures
are alſo o's, I go on as before, to the first next fignificant
Figure on the Left-hand, which here is 5, and borrow of, or
3
take
*
Of SUBSTRACTION.
71
A
take 1 from it, fo that you muſt call it but 4, when you
come to it; and the two Cyphers that ſtand next it, on the
Right-hand, you muft call 9's, as before; fo that I go on,
faying, 2 from 9, remains 7; and I from 9, remains 8; and
then inftead of faying 7 from 5, I fay, 7 from 4 (for the
Reaſon above) I cannot, but borrowing I from the 8, I
fay, 7 from 14, and there remains 7; then 9 from 7 (be-
cauſe the 8 has I taken from it) I cannot, but borrowing
I from the 7, 9 from 17, and there remains 8; and now
I am come to the laft Figure, I fay, 6 from 6, (becauſe I
was taken from the 7 before) and there remains o; fo that
I have now got 87877658543334 for the Remainder or Dif-
ference between the two given Numbers. Now to know
whether my Work is right, I fubftract this Remainder from
the biggeſt given Number, and find the Remainder to be
697123000456789, the fame with the leaft given Number
or you may add the Remainder, firft found, to the leaſt
given Number, and the Total will be the fame with the
biggeſt given Number; as appears by the Example above:
However, to take away all Doubt, let the biggeſt given
Number, 785000659000123, be fet down again, and
under it the
;
Remainder 87877658543334 Substract the leffer from
firft found
the greater, and the Rem.
will be
697123000456789
I
For beginning at the firft Figure, or Units Place, I fay, 4
from 3 I cannot, but 4 from 13, and there remains 9; then I
go on and ſay, 3 from 1, (becauſe I took 1 from the 2, to make
the 3, 13) I cannot, but 3 from II (borrowing from the
I on the Left-hand) and there remains 8; which being fet
down, I go on and fay, 3 from o (becauſe 1 taken
from the I makes it o) I cannot, therefore I borrow I of
the next Figure on the Left-hand, (which here, being o's
for 3 Places together, I must go on to the firft fignificant
Figure, and take I from thence, which here is 9; fo that
the 9 becomes 8, and all the o's q's, as is faid above) and
it makes the 1, that is now become o, 10; then I fay,
3 from 10, and there remains 7; and 3 from 9, and there
remains 6; and 4 from 9, and there remains 5 ; and 5
from 9, and there remains 4; and 8 from 8 (becauſe the
9 has had I taken out of it) and there remains o
Ι
; and 5
from 5, and there remains o; and 6 from 6, and there re-
mains o; then 7 from o I cannot, but borrow as before;
then
4
72
Of SUBSTRACTION.
then 77 from 10, and there remains 3; and 7 from 9, and
there remains 2; and 8 from 9, and there remains ; and
7 from 4 (becauſe I was taken from the 5, to make good
the Deficiency of the o's) I cannot, but 7 from 14, and
there remains 7; then 8 from 7 (becauſe I was taken from
the 8 to make the laft Figure 4, 14) I cannot, but borrow-
ing I from the next Figure, which is 7, the 8 that is now
depreffed to 7, becomes 17; ſo that the Remainder will be
9; and as there is now no more Figures to ſubſtract, I
fay, 0 out of 6, (becauſe the laſt Figure 7, in the upper Line,
had I taken from it,) and there remains 6; fo that the Re-
mainder is now 697123000456789, the fame with the leffer
given Number; which fhews that the Work is rightly per-
formed. I have now fully explained the Method of fub-
ftracting one whole Number from another; I fhall only ob-
ferve to you, that there is no abfolute Neceffity for putting
the biggeſt Number uppermoft; but if you fhould have any
fuch Example given you, where the leaft Number ſhould be
uppermoft, you muſt be fure always to take the leffer Num-
ber from the greater, let them ftand how they will; for
'tis impoffible to take a bigger Number from a leffer; as
you may readily fee by the Examples following.
23
87
Or 72
98
Or 732
918
Or 1234, &c.
5678
26
186
4444
64
Here 'tis very plain, that neither of the under Lines can
be taken out of the upper Lines; and therefore if any thing
is to be done, you muſt take the upper Lines out of the
under Ones; the feveral Remainders or Differences will be
as you fee in the ſeveral Examples.
Suppofe from London to Berwick is 265 Miles by Com-
putation, and I travel as follows, The firft Day to Ware,
which is 20 Miles; the fecond Day to Huntingdon, which
is 28 Miles more; the third Day to Grantham, which is
35 Miles more; the fourth Day to Doncafter, which is 38
Miles more; the fifth Day from thence to York, which is
computed 29 Miles more: How far am I from my Four-
ney's End? The Queſtion ſuppoſes the feveral Places men-
tioned to be in the direct Road to Berwick, and confequent-
ly every Day's Journey diminiſhes the Diſtance to be travel-
led; and therefore to anſwer this, or any fuch like Queſtion,
I
you
Of SUBSTRACTION.
73
you may fet down the feveral Diſtances travelled, and add
them all together, and ſubſtract their Total from the whole
Diſtance; and what remains is the Anfwer fought, as
follows,
Miles.
First Day's Journey from London to Ware 20
Second Ditto from Ware to Huntingdon 28
Third Dit. from Huntingdon to Grantham 35
Fourth Dit. from Grantham to Doncaster 38
Fifth Ditto from Doncafter to York
29
150 Travelled
265 the whole Jour.
115 ftill to go.
Or, you may ſet down the whole Diſtance
265, and fubftract every Day's Journey
thus; 20 the firft Day's Journey.
245 remains.
28 the fecond Day's Journey.
217 remains.
35 the third Day's Journey.
182 remains.
38 the fourth Day's Journey.
144 remains.
29 the fifth Day's Journey.
115 remains ſtill to go.
By the working this Example both Ways, you ſee the
Refult is the fame, and in all fimilar or like Cafes, muſt
neceffarily be fo; for 'tis the fame thing to continually ſub-
ftract the Parts or Numbers from the Whole, as to add
them up, and ſubſtract their Total from the Whole; as ap-
pears by the Work before you. Now I will go on to the
Subſtraction of Money, Weights, Meafures, &c. in which
you will note, that the fame Numbers you ftopt or reckon'd
by, in Addition, as Occafion requires, you muft borrow in Sub-
ſtraction.
L
Example
*
74
Subſtraction of MONEY.
d. q.
:
Example 1. MONEY.
1.
316
S.
13: 10
159: 18 : 08
From
Take
Remains
156: 15: 01 3
Proof by Subſtraction
159: 18 : 08 3
Proof by Addition 316: 13: 101/2
From
Take
Example 2.
1.
S. d. q.
·2008: 17 : 04 J
1234 05 06
Remains
774 II : 10
Proof by Subſtraction 1234 : 05:06 4
Proof by Addition
2008: 17:04
Here I begin with the firft Example, and fay, 3 Farthings
out of 2 Farthings I cannot, therefore I take a Penny
from the Pence, and fay, 3 Farthings out of a Penny leaves
1 Farthing; which with the 2 Farthings that are already
given, makes 3 Farthings, to be fet down under the Farthings,
as you fee done in the Example; then I go on to the Pence,
and fay, 8 out of 9 (becauſe I took I from the 10 Pence,
to add to the Farthings) and there remains 1 Penny to be
fet down; then I go on to the Shillings, and fay, 18 from
13 I cannot, and therefore I borrow 1 Pound or 20 Shillings
from the Figure that ftands in the Units Place of the Pounds,
viz. the 6, and fay, 18 out of 20, and there remains 2;
which with the 13 Shillings given, makes 15, to be fet down
under the Shillings; then I go on to the Pounds, and fay,
9 from 5 (becauſe I was taken from the 6 to add to the
Shillings,) I cannot, but borrowing I from the ſecond Fi-
gure, or I, the 5 becomes 15; fo that the Remainder is 6;
which being fet down, I fay, 5 from o I cannot, (becaufe
1 being taken from the 1, leaves o) but borrowing 1, 'tis
then 5 from 10, and the Remainder is 5; then 1 from 2
(becaufe
Subſtraction of MONEY.
75
(becauſe I was borrowed or taken from the 3) and there
remains I; fo that the whole Remainder is L.156: 15:01 3.
Note, That as in Addition, the Figures of the laft Column
or whole Numbers, were always added, by confidering them
as if they were fimply Numbers, without any Name; fo
in Subftraction you muſt always do the like, as you fee by
the latt Example.
For the Proof of your Work, you may do as was before
directed in whole Numbers, viz. fubftract the Remain-
der found, out of the greater Number given, and the fecond
Remainder will be the fame with the leffer Number given,
as is done in this firft Example; for I fay, 3 Farthings out
of 2 Farthings, I cannot, but borrowing i Penny or 4 Far-
things from the 10 Pence, it makes the 2 Farthings 6; from
which taking the 3, the Remainder is alfo 3; which being
fet down, I go on and fay, 1 Penny from 9 Pence (becaufe
I was taken from the 10 to add to the Farthings; fo that
there is but 9 left) and the Remainder is 8; which I fet
down, and go to the Shillings, faying, 15 from 13, (be-
cauſe I did not borrow a Shilling to add to the Pence; ſo that the
13 Shillings were not diminiſhed) I cannot, therefore I borrow
1 Pound, or 20 Shillings, from the Figure that ftands in the
Units Place of the Pounds, which being added to the 13,
makes 33; then 15 out of 33, leaves 18 Shillings to be fet
down under the Shillings; then I go on, and fay, 6 from 5
I cannot, but 6 from 15, remains 9; which being fet down,
I fay, 5 from o I cannot, but 5 from 10, remains 5 5; and
I from 2, remains 1; fo that the whole Remainder now is,
L. 159: 18:08, the fame with the leaft given Number
from whence 'tis evident the Work is right, for the greateft
given Number may be confidered as fome whole Sum or
Number; and the leffer given Number, and the Remainder, os
two Parts of the faid biggeft given Number; which two
Parts, when added together, will make, or be equal to the
faid biggeſt given Number, as you fee done in the fecond
Proof of this Example, where the two Remainders being ad-
ded together, the Sum is L. 316 13: 10, the fame with
the biggeſt given Number. The fecond Example is done in
the fame Manner; for beginning at the Farthings, I fay, I
Farthing from 3 Farthings, and there remains 2 Farthings;
which being fet down under the Farthings, I go on and fay,
6 Pence from 4 Pence I cannot, therefore I borrow 1 Shil-
ling, or 12 Pence, from the 17 Shillings, and fay, 6 from
12 and there remains 6, which added to the 4, makes 10;
:
L 2.
I
Ι
or
76
Subſtraction of MONEY.
or you may add the 12 Pence that are in the 1 Shilling
you borrow'd, to the 4, and they make 16; then take 6 out
of the 16, and the Remainder will be 10, as before; which
being fet down, I go to the Shillings, and fay, 5 from 16,
(becauſe I was taken from 17 to add to the Pence) and there
remains II; now as I borrow'd nothing, I go on to the
Pounds, and fay, 4 from 8, and there remains 4; then 3
from o I cannot, but 3 from 10, and there remains 7;
then 2 from 9, and there remains 7, and I from 1 (be-
cauſe I was taken from the I to help the o, as has been
taught you before) and there remains o.
Now as a o on
the Left-hand of whole Numbers, neither increaſes nor di-
miniſhes the Numbers, therefore 'tis needleſs to ſet a o down
in that Place; fo that I find the whole Remainder to be
L. 774: II : 10; for the Proof whereof you may ſub-
ftract this Remainder from the biggeſt Number given, ſay-
ing thus; 2 Farthings from 3 Farthings, and there remains
1 Farthing; then 10 Pence from 4 Pence I cannot, but bor-
rowing I Shilling or 12 Pence from the 17 Shillings, I fay,
10 from 12, and there remains 2, which added to the 4,
makes 6 to be fet down under the Pence; then I go on
and fay, 11 Shillings from 16 Shillings (becauſe I is taken
from 17) and there remains 5 Shillings; here borrowing
nothing, I go on and fay, 4 from 8 and there remains 4;
from o I cannot, but 7 from 10, and there remains
3; then 7 from 9, and there remains 2, and o from I
(becauſe I has already been taken from the 2 to help the
o's) and there remains I; fo that the whole Remainder
is now L. 1234 : 05:06 4 which is the fame with
the leffer given Number; from whence I conclude the
Work to be right: Or, if you add the 2 Remainders
together, their Sum or Total will be L. 2008: 17:04 1,
the biggeſt Number' given, which is another Proof of the
Truth of the Work; after the fame Manner may the Dif-
ference between any bigger and leſſer Number be found. A-
gain,
then 7
49
Suppofe
Subftraction of MONEY.
77
Suppoſe a Merchant has bought or contracted for Goods to
the Amount of
And at feveral Times has made theſe
Payments following
How much is still unpaid?
Here you may
you may add the feve-
ral Sums paid together and
fubftract their Total from the
whole Amount, and the Re-
mainder will be the Sum ftill
L. 5816: 13: 10 1/
1239: 18:
571: 16: 10
48: II: II
65: 17: 10
2796: 10:
18: II
419 :
Total paid 5142 : 13: 7/20
Remains 674: : 3 unp.
due; as you fee done in this Proof by Ad. 5816 : 13: 10
Example, where the Total of
the Sums paid is L.5142: 13:7; which being fubftra&ted
from the whole Sum due, the Remainder is L. 674
Or, you may fet
down the whole
Sum
and fubftract each
particular Sum paid,
L.5816 13: 10
· 3.
1239 : 18:
the ift Sum paid.
thus:
I would adviſe you to 4576: 15: 10
fubftract the particular
2
remains due.
571: 16: 10
Sums paid, as you fee
done in this Example, 4004: 19:
which will render you 48 II II
quite perfect in any
7:
thing that you can 3956: 7
poffibly want of this 65 17 10
Kind, efpecially if you
tranfpofe your Numbers 3890: 09:
and go backwards, fub- 2796: 10
ſtracting your laſt Pay-
ment first, and your
firſt Payment laft, &c.
by which means you
will have as much Va-
riety, by this one Ex-
19:
2
<<
--
2
1093: 19:
419: 18 : 11
674:
www
the 2d Payment.
remains due.
the 3d Payment.
remains due.
the 4th Payment.
remains due.
the 5th Payment.
remains due.
the 6th Payment.
: 3 rem, due as before.
ample, as if you had ever fo many; and the laft Remainder
always coming to L. 674 3,
Truth of the whole Work.
:
:
In the Subſtraction of Weights,
will be a Proof of the
Measures, Time, &c.
the fame Method is to be obferv'd, regarding only how
many
78 Subtraction of diverfe Denominations.
}
many of the leffer Names on the Right-hand make 1 of
the bigger Names on the Left-hand; as follows,
CLOTH-MEASURE.
(4) (4)
(5) (4)
Yards 2. N.
E.E. Q, N.
From 25866 2: I
Take 1789232
279: 1:3
186: 4:
92: 2:3
Remains 7973 2:3
2 : I
Proof by Addition 25866: 2:
Proof by Subſtract. 17892 : 3: 2
E.F.
(3) (4)
2. N.
279 13
186: 4:
(6) (4)
A.F. 2. N
(3)
From 678 I: I
927: I: 2
Take 592: 2:3
672: : I
Remains 85: 1:2
255: I: I
!
Proof by Addition 678: I: I
927 I: 2
Proof by Subſtract. 592 : 2 : 3
672:- I
Here I have fet 4 over the Nails, viz. fo many as make
a Quarter, and over the Quarters 4 alfo, becauſe ſo many
make a Yard; over the English Ell 5; over the Flemish-Ell
3; and over the French-Ell 6; and then fuppofe I begin
with the firſt Example on the Left-hand, which is Yards,
2. N. there I begin at the Nails, and fay, 2 out of 1 I
cannot, but borrowing I Quarter, which is 4 Nails, I fay,
2 out of 5 (which is the Sum of the 4 Nails I borrowed,
added to the I that is given) and there remains 3; then I
go on and fay, 3 Quarters out of 1 Quarter (becauſe I
Quarter was taken from, or borrowed of the 2 Quarters
that are given, which reduces the 2 given Quarters to 1)
I cannot, fo I borrow I Yard from the Units Place of the
Yards, which is 4 Quarters, and the I that remains out of
the 2 Quarters that are given, make 5 Quarters; then 3 out
of 5, and there remains 2, which being fet down under the
Quarters, I go on and fay, 2 Yards out of 5 Yards (becauſe
I Yard was taken from the 6 Yards that ftands in the
Units
Subſtraction of diverſe Denominations. 79.
*
Units Place of Yards, to add to the I Quarter that was left
out of the 2 given Quarters) and there remains 3; then 1
go on, as in whole Numbers has already been taught: The
Proof may be, by adding the Remainder to the leffer given
Number, and the Sum or Total muſt be equal to, or the
fame with, the biggeſt given Number; as you fee by the
Examples above: Or you may fubftract the Remainder firſt
found, out of the biggest given Number; and the new Diffe-
rence or Remainder, thus found, will be equal to, or the
fame with, the leaft given Number; as alfo appears by the
fame Examples.
TROY-WEIGH T.
(12) (20) (24)
Њ Oz. Dwts. Gr.
From 98762: 10: 19: 17
Take 47158: II: 16: 22
Remains 51603: 11:02: 19
Proof by Subſtraction 47158: 11: 16: 22
Proof by Addition
98762: 10: 19: 17
(12) (20) (24)
#b Oz. Dwts. Gr.
From 50000 : 07 09: 15
Take 31791: 09: 13: 17
Remains
Proof by Subftraction
Proof by Addition
18208: 09: 15: 22
31791 09: 13: 17
:
50000 : 07 : 09: 15
AVOIRDUPOISE-WEIGHT.
From
Take
(20) (4) (28) (16) (16)
Tons C. 2: t Oz. Dra.
17001 : 13 : 1 : 16 : 05 : 09
9123 14: I : 27 04:07
17001
Remains
7877: 18: 3: 17: 01 : 02
17001 : 13: 1 : 16 : 05 : 09
:
Proof by Addition
Proof by Subftraction 9123 14: I: 27: 04 07
2
(4)
80 Subſtraction of diverſe Denominations.
(4) (28) (16) (16)
C. 2; t
From 271: I : 13 :
Take 159 :3: 17:
Oz. Dr.
12 : 05
15 : 09
III: I: 23: 12: 12
Proof by Addition 271: 1:13: 12: 05
Proof by Subſtraction 159: 3: 17: 15: 09
APOTHECARIES
WEIGHT.
(12) (8) (3) (20)
Њ 33 ЭGr.
From 715 05: 3 1: 13
Take 615
05: 4:2: 17
Remains .99
11 : 6 : 1 : 16
Proof by Subſtraction 615
05: 4:2: 17
Proof by Addition 715
05
3: 1:13
(12) (8) (3) (20)
#
UN
3
Э Gr.
From
200:
: I : 12
Take 19810 : 2 : 1 : 16
•
Remains
I : 01 :
5
:
2 : 16
Proof by Subſtraction 198: 10 : 2 : 1 : 16
Proof by Addition 200 :
-
: I: 12
LONG MEASURE.
(8) (40) (11) (18) (3)
Miles Fur. Poles Yds Inch. Bar.C.
From 6273 07: 16:05: 13 : I
Take 6263 07: 16:05: 13 : 2
Remains
Proof by Addition
9:07: 39: 10: 17: 2
6273 07 16 05
: :
Proof by Subſtraction 6263
07 16:05
13 I
13:2
LAND
Subſtraction of diverſe Denominations. 1
LAND or SQUARE MEASURE.
(4) (40)
Acres Rood Perches
From 7152: 1: 15
Take 6252 : 1 : 27
:
Acres Road Perch.
915: 3:27
827: 1: 15
Remains. 899 3 28
:
.88 2 12
Proof by Subſtract. 6252 : 27
:
827: I: 15
Proof by Addition
7152: I: 15
915: 3:27
Chal.
Bought 5927 :
Sold 4629
DRY MEASURE.
(12) (3) (4)
Sacks Bush. Pecks
08 : I
2
II
:
2
:
3
Remains unfold
1297 : 08
: I : 3
Proof by Addition 5927
5927 : 08
I :
2
Proof by Subſtraction 4629 :
II
:
2
..
3
(36)
Chal.
(4) (16)
Bufh. Pecks Pints
Bought 574 :
17
I : 13
Sold 489:
Remains unfold .84
84 21 2 : 15
31:
2:
14
Proof by Addition 574
: 17
17
:
: I : 13
Proof by Subſtraction 489 : 31 : 2 : 14
BEER and ALE-MEASURE.
(3) (36) (8)
Buts Beer Bar. Gall. Pints
Bought 58: I : 17: 4
|
(32) (4)
Bar. Ale Gal. Quarts.
671: 18 : 1
Sold 58: I : 13: 7
527: 19: 3
Remains oo: 0:03:5
143: 30: 2
Proof by Subſtract. 58
i
13: 7
527: 19: 3
Proof by Addition 58
1
17 4
671: 18 : 1
M
WINE-
82 Subſtraction of diverfe Denominations.
WIN-E-MEASURE.
(4) (63) (4)
(2)
Tuns Hogfb. Gall. Qts. Pints.
17 : I : I
42 : 3 : 49 : · 3: I
Bought 69 : 2
Sold
Remains unfold 26
: 2 : 30
30
:
: 2:0
17: I: I
Proof by Addition 69: : 2:
Proof by Subſtract. 42.
: 3: 49: 3: I
TIME.
(24) (60)
Days Hours Minutes.
From 719 : 13 : 52
Take 493: 17: 27
Remains 225 : 20 : 25
Proof by Subſtraction 493
Proof by Addition 719
17: 27
: 13:52
(365) (24) (60) (60)
Com. Years, Days, Hours, Min. Seconds.
From 712 107: 17: 13: 16
Take 691: 218: 19: 58 : 42
Remains .20 : 253 : 21 : 14 :*34
Proof by Subſtraction 691: 218: 19: 58: 42
Proof by Addition 712 : 107: 17: 13: 16
Having now gone through the feveral Weights and Mea-
fures, &c. the fame General Rule muft be obferved in all
of them, in taking or fubftracting the leffer given Line or
Number out of the greater; and always when the Subftracti-
on cannot be made out of the given Figures, as they ſtand,
you muſt borrow I of the next fuperior Denomination, and
adding that to the given Figures, in the lower Name, takę
or ſubſtract the given Figures in the leffer Line from that
I
Total,
Subftraction of diverſe Denominations. 83
Total, and fet down the Remainder, as you fee done in all the
Examples above; and when the Number of Parts run high,
'twill be best to add and ſubſtract upon a Bit of Wafte-
Paper, to prevent Blots and Miftakes; as in the laft Exam-
ple of Time, I begin and fay, 42 Seconds out of 16 I can-
not; therefore I borrow I from the Minutes, which being
60 Seconds, I fet it down, and under it the 16, that is al
16 ready there; and add them together, and their Sum or
Total is 76, from which
76 Total
42 being ſubſtracted, leaves
34 for the Remainder.
Or, you may ſet down the 60 which is borrow'd, and
ſubſtract 42 the Number given,
and to
18 the Remainder,
add
16 the Number given,
der fought, as before.
and 34 the Total, is the Remain-
Then I go on and fay, 58 Minutes
can-
from 12 (becauſe I was taken before from the 13) I
not, therefore I borrow 1 Hour, which is 60 Minutes; and
do as before directed:
60 borrowed;
12 the Rem. of the given 13,
72 Total
Or, 60 Minutes borrow'd
58 Min. given to be ſubſtract.
58 given to be Subftracted.
2 Remains.
12 the Rem. of the 13 giv.
14 Remainder fought.
14 Total, is the Remain-
der fought. Then I go on and fay, 19 Hours from 16,
(1 being taken from the 17 before) I cannot, fo I borrow
i Day, and do as before, and the Remainder is 21; then
I go on, and fay, 218 Days, from 106, (1 being taken or
borrow'd from the 107) I cannot, fo I borrow I Year,
365 Days.
which is
To this add
and from the Total
Subſtract
and the Remainder is
106 the Remainder of the 107,
471
218
253
M 2
Or,
84
Of DIVISION.
Or, fet down 365 the Days borrowed,
and ſubſtract 218 the Days given,
and to the Remain. 147
add 106 the Remainder of 107,
and the Total 253 is the Remainder fought. Then I go
on and fay, 1 Year out of I (1 being taken or borrowed
from the 2 before) and there remains o; then 9 from I I
cannot, but borrowing I from the 7, makes the given 1,
11, and there remains 2; then 6 from 6, (1 being taken
from the 7 before) and there remains o: So that the whole
Remainder is 20 Years, 253 Days, 21 Hours, 14 Minutes,
and 34 Seconds; as you may prove either by Addition or Sub-
ftraction, or both, as you fee done in all the Examples above.
After the fame Manner may any Weights, Meaſures, &c. be
done, regarding only what it is you borrow. And now hav-
ing fully explained all the poffible Difficulties of plain or
common Subſtraction, I now go on to the other Part, called
Divifion, or compound Subtraction: In which you are to
Note, that as plain Subſtraction undoes what plain or com-
mon Addition does, fo compound Subftraction or Divifion
undoes what Multiplication or compound Addition has done,
or made up.
DIVISION
Is that Rule or Part of Arithmetick, whereby you are
Taught to find how many Times one Number is contain-
ed in, or may be fubftracted from, or out of another.
་
As in Addition, you might make a new Total, with every
particular Sum, before you made or found the general Total;
fo here you may repeat the fubftracting the leffer given Num-
ber, or Numbers, out of the greater given Number, till no-
thing Remains, as you will fee done in the following Examples.
How many Times is 6 contained in 24?
24 Here you may fet down the 24, which is the grea-
6 ter given Number, and is called the Dividend, or Num-
ber to be divided, and ſubſtract from it the leffer giv-
18 en Number, called the Divifor; and again, fubftracting
6 the Divifor from the Remainder, and fo continue
fubftracting the leffer given Number or Divifor from
12 the feveral Remainders, till there either remains o, or
6 fome Number or Figure that is lefs than the Divifor;
as may be feen by the two Examples in the Margin,
lool
How
0
Of DIVISION.
85
48
How many Times is 9 contained in 48 ?
In the firft Example, 24 being fet down, and 6 fub-
9 ftracted from it, the Remainder is 18; from that
take 6, and 12 remains; from which take 6, and
39 6 remains; from which take 6, and o remains fo
9 that it appears there are 4 Times 6 contained in
24; becauſe there muſt be made 4 Subſtractions of
30 6, before the Remainder is 0. In the fecond Exam-
9 ple, 48 being fet down, and 5 Subftractions of 9
being made, the Remainder is 3, which fhews that
9 is contained 5 Times in 48, and 3 over; and
9 fo you may do with any other Numbers, bigger or
leffer, both in Whole-Numbers, and alfo in Diverfe-
12 Denominations. As,
21
9
3 remains.
5728 2148
716
716
5012 1432
716 716
4396 716
716 716
2964
716
2248
How many Times is 716 contained in 5728?
Here, after having Substracted 716 continual-
ly from 5728, and its Remainders, I find o
Remains; from whence I conclude, that 716
is contained juſt 8 Times in 5728.
So if it were required to know how many
3680 000re-Times L. 679: 17: 6 was contained in
716 mains. L. 2419: 18: 11. By Working after the
fame Manner, the Anſwer would be found to
be 3 Times, and L. 380: 5:5 ½ Over, or
Remaining; as you may fee by the Example
following. In the fame Manner any other
Denominations may be performed, re-
garding only the feveral Subdivifions
of the Sum given; where you are
always to obferve, that you can ne-
ver Subftract, Add, Multiply or Di-
vide different Names or Species
from one another, but only thoſe of
the fame Name.
d.
L.
2419: 18: II
679: 17: 6
1740:01: 5
679: 17: 6
-
1060 : 03: II
17 6 1
679
380
5
5:
5 remains,
But
86
Of DIVISION.
But what is properly called Divifion, proceeding by different
Steps, that render the Work more expeditious; for as Mul-
tiplication performs Addition more concifely, though the A-
mount or Refult is the fame; fo in like manner Divifion,
generally ſpeaking, produces the Anfwer to the Question,
much fooner and eafier than Subftraction: In this Rule or
Part of Subftraction, there are, and have been many diffe-
rent Methods of Operation made ufe of; but the moft fure,
eafy and plain one is this following: Firft fet down the big-
geft or containing Number, then on the Left-hand there-
of, fet down the leffer given Number, with this Partition,
or Mark) between them, and on the Right-hand thereof,
this Mark, or Partition (as you fee in the Example following.
How many Times 5 are contained in 35?
Divifor Dividend Quot. Here I confider what Number
5) 35 (7 multiply'd by 5, will make or come
35
to 35, or to the next lefs neareſt to
35, which by the Multiplication-Table,
oo remains fet down, Page 16, is found to be
35
40
4 remains.
7;
for 5 Times 7, or 7 Times 5, is 35;
therefore I fet the 7, fo found, with-
in the Partition on the Right-hand,
and Multiply the 5 by it, and fet down the Product 35, un-
der the given Number 35, and Subtract the one from the o-
ther, and the Remainder is oo. So if it was afk'd, How
many Times 5 are contained in 44? The Answer will be 8
Times, and 4 over; for having fet down the Numbers, as
before directed, thus, 5) 44 (8 you will find 8 Times 5
makes 40, which being
fubftracted from 44, the
Remainder is 4. After
the fame Manner any
other Number may be Divided; and from this fmall Spe-
cimen, you may perceive how much Labour is fav'd by doing
it this Way, and by Common-Subftraction, though the Anfwer
will always be the fame, let the Operation be performed
either Way; but if your Number to be Divided, be very
large, in refpect of the Number that Divides, then the La-
bour would be almoft infinite the one Way, over what
it would be the cther; as, Suppofe 'twas required to divide
581764, by 8: Here you muſt make 72720 Subftractions of
8, before you could give the Anſwer; which how much
Time and Paper 'twould take up, I leave you to judge; and
fhall go on to fhew you, how you may come at the
Artf-
Of DIVISION.
87
Divifor Dividend Quot.
8) 581764 (72720
56....
21
16
57
56
16
16
04
O
Anſwer more eaſily, and full as furely. First fet down the
given Numbers, as in the Mar-
gin; which being done, I afk
how often 8 is contained in 5,
the firft Figure on the Left-hand,
which being o Times, I do not
fet down the o, but I go on and
take two Figures, viz. 58, and
the neareſt Number of Times
that is lefs, is 7; with this 7 I
Multiply the Divifor, and fet down
the Product 56, and Subftract it
from the 58, and fet down the
Remainder 2, to which I bring
down the next Figure 1, and it
makes 21; then I afk the Question,
How often is 8 contain'd in 21?
and the Anſwer is twice, or 2
4 remains Times, which I fet on the
Right-hand of the 7, and Multi-
ply the 8 by the 2, and fet down the Product 16, under
the 21, and Subftract the one from the other, and the Re-
mainder is 5, to which I bring down the next Figure 7,
and it makes 57; then I enquire how often 8 is contained
in 57, and the Anſwer is, 7 Times, which I ſet in the
Quotient on the Right-hand of the former Figures 72, be-
fore found; then I Multiply the 8, by this laft 7, and fet
down the Product 56, under the 57, and Subftract, and the
Remainder is I, to which I bring down the next Figure 6,
and it makes 16; then aſking how often 8 is contained in
16, the Answer is 2 Times, which is fet down on the
Right-hand of 727 before found, and Multiply the 8 by this
2, and fet down the Product 16, under the 16 and Sub-
stract, and the Remainder is 0; to which I bring down
the laft Figure, and it ftill makes but 4, or 04; and
afking how often 8 is contained in 4, the Anſwer is o
Times, which o I fet in the Quotient on the Right-
hand, and Multiply the 8 by 0, and the Product is o,
which being Subftracted from the 4, or 04, the Remain-
der is 4, becauſe o on the Left-hand of a Common-
Number, when there are no more Figures towards the Left-
hand, fignifies nothing; for which reafon the o was not
fet down in the Quotient the firft Time, when 'twas afk'd,
how often 8 was contain'd in the firſt Figure 5; but the
o muſt
88
Of DIVISION.
o muſt be put in the Quotient, at dividing the laft Figure 4,
becauſe the ſeveral Figures that were in the Quotient before,
ftood on the Left-hand of the faid o, and were thereby in-
creas'd Ten-Times in Value; fo the true Answer to the
Question is, that 8 is contained 72720 Times in 581764,
and 4 over; as may readily be proved, by Multiplying the
faid 72720, by 8, and adding to that Product the Remain-
der 4, and the whole Dividend 581764, will be again pro-
duced; or you may Subftract the faid Quotient 8 Times
from the Dividend; and the fame Remainder 4 will be
produced, as you fee by the following Example.
581764
72720
After the fame Manner may any other
Number be performed, as, How many Times
is 4 contained in 71562834?
Divif. Dividend Quotient.
4) 71562834 (17890708, Anſwer.
509044
72720
436324
4
4
72720
31
363604
71562832
28
2
72720
35
290884
71562834 Proof.
32
72720
36
218164
36
72720
028
145444
28
72720
034
72724
32
72720
4
2
How
Of DIVISION.
89
How many Times is 5 contained in 17156425 ?
5) 17156425 (3431285
5
15
21
17156425 Proof,
20
15
15
Inal
14
IO
42
40
25
25
о
How many Times is 6 contained in 267198467?
6) 267198467 (44533077
24
27
24
31
30
19
18
18
18
046
42
6
267198462
5 remains.
267198467 Proof.
47
42
Remainder.
5
N
Divif.
?
Of DIVISION.
Divif. Divid. Quot.
9) 5872006 (652445
54.....
47
45
22
18
9
5872005
I Remainder.
5872006 Proof.
40
36
40
36
46
45
1 Remains.
Divide 8263475 by 23
23) 8263475 (359281
69..... 23
136
115
213
207
46
In this, and all other Cafes,
where the Divifor confifts of ſe-
veral Figures, you need not ſay
how often is 23 contained in 82,
1077843 but how often is 2 contained
618562. in 8, which fimply is 4 Times;
but as you are to carry the A-
7263463 mount of the Tens, arifing by the
12 Multiplication of the 3, to the
Product of the 2, 'twill come to
64 7263475 too much, or more than that
Part of the Dividend, with which
you are then concerned; for 4
Times 23 amounts to 92, which
is more than 82, therefore it
muft go but 3 Times, which is
69; and after the Subtraction is
made, the Remainder will be 13,
to which taking down the 6, 'tis
then 136. Now as here are 3
Figures, and but 2 in the Divifor,
you muſt fay, how many Times 2 in 13? and allowing for
the Addition of the 3, 'twill go but 5 Times, and the Product is
187
184
35
23
12 Rem.
1155
Of DIVISION.
91
115, and after Subftraction made, the Remainder is 21; and
the next Figure being taken down, this new Dividend is
213; and as there is one Figure more here, than in the Di-
vifor, I fay, how many Times 2 in 21? which by reafon of
the 3 being with it, will go but 9 Times, and the Product
is 207, which being fubftracted from the 213, leaves 6, to
which bringing down the 4, it becomes 64; and as here
are no more Figures in this new Dividend, than there are
in the Divifor, I take but one Figure, and fay, how oft 2
in 6 and the Anfwer will be but 2, becauſe of the 3 being
added to it, and the Product 46 being fubftracted the Remain-
der is 18, to which bringing down the next Figure 7, the
Divifor will go 8 Times, and the Product 184 being fub-
ftracted, and to the Remainder 3 bringing down the laſt
Figure 5, the Quotient will be 1; by which multiplying the
Divifor, the Product is 23, and this being fubftracted from
the 35, the Remainder is 12; and now the Work is finiſh-
ed: But to know whether it be truly performed, multiply
the whole Quotient, 359281, by 23, and to the Product
7263463 add the Remainder 12, and the Total 7263475 is
the fame with the Dividend given; or you may take all the
particular Products, and the Remainder, and add them toge-
ther; and if the Work be true, their Total will be the
Dividend, thus:
Or you may divide the given Dividend by
the Quotient, and the new Quotient will be
the fame with the former Divifor; and the
Remainder, if any, will be the fame as be-
fore, as you will fee by what follows,
69..
115
. 207
...46..
184.
.23
359281) 8263475 (23
718562
1077855
1077643
12
12
8263475 Proof.
Thus you may ſee what a natural
Harmony there is between all the
Parts of Arithmetick; how they de-
pend on, and prove the Truth of
one another; for as formerly you.
were taught to prove Multipli-
cation by Multiplication, fo here
you fee Divifion proved by Divifion; for 'tis undeniably cer-
tain, that if 23 is contained 359281 Times, and 12 over, in
8263475, that 359281 muſt be contained 23 Times, and
12 over in the fame Number.
N 2
89
92
Of DIVISION.
*
87.
348
.87...
Divide 123456789 by 87
87) 123456789 (1419043
87.
87
.783..
1
348.
364
993330I
261
348
11352344.
...48 Remaind.
48 Rem.
165
123456789 Proof.
87
123456789 Proof.
1419043) 123456789(87
786
11352344.
783
9933349 Proof.
0378
9933301
0348
48 Remain.
0309
261
048 Remainder.
Divide 928765483 by 918
918) 928765483 (1011727
918....
918....
1076
..918.
918
918...
.6426..
1585
1836.
918
6426
...097 Remains.
6674
6426
928765483 Proof,
2488
1836
6523
6426
097 Remainder.
1011727
Of DIVISION,
93
1011727) 928765483 (918
9105543..
1821118
1011727
8093913
8093816
97 Remainder.
9105543..
• 1011727.
8093816
97 Remainder.
928765483 Proof.
4072495-3215862471628 | 789654
36511597
28507465
32579960
32579960
36652455
3931637 1
24434970
36652455
20362475
16289980
26639166
504898 Rem.
24434970
3215862471628 Proof.
22041962
20362475
16794878
16289980
504898 Remains.
789654
94
Of DIVISION.
789654- | 3215862471628 | 4072495
3158616
| 4072495
3158616
5724647
1
5527578
5527578
1579308
3158616
1970691
7106886
1579308
3948270
504898 Rem,
3913836
3158616
3215862471628 Proof.
7552202
7106886
4453168
3948270
504898 Remains.
By the feveral Examples, in Pages 88,90,91, 92, 93 and 94,
you fee the Truth of the Work proved fundry Ways, eſpecially
by making the Quotient a Divifor to the fame Dividend, and
you find that which was before the Divifor, then becomes the
Quotient, and the Remainder (where there is one) is always the
fame with the firft Work: You may likewife fee, that adding
the Remainder, and the feveral Products of the Divifor, multiply'd
by the particular Figures of the Quotient, always produces the
Dividend, as well as multiplying the whole Quotient by the whole
Divifor, and adding the Remainder to that Total. And now
you have had Examples of common Diviſion fufficient to
answer any Question that can poffibly be put in whole Num-
bers: I fhall only add one Obfervation, which you muſt care-
fully obferve, let your Work be apply'd to what it will,
viz. That the Divifor, whether it be a ſmall Number or a
large one, muft be eſteemed fome whole Thing broke or divided
into fo many Parts, as that Number contains Units; and if
any thing remains, that Remainder is fo many of thoſe Parts,
or fo many Parts of that whole Thing or Unit, the Divifor
reprefents: As for Example, in Page 86, 44 is divided by 5,
and 4 remains, which 4 is or four fifth Parts of a Thing;
that is, the Anſwer to that Queſtion, is 8 whole Times or Things,
and # or four fifth Parts of a Thing: So in the next Question,
Page 87, where 581764 is divided by 8, the Anfwer is 72720,
4
3
and
of DIVISION.
95
and or four eighth Parts of a Time or Thing; fo again, in
Pages 88, 89, and 90, where the Divifors are 4, 5, 6, and 9.
In the Example, where the Divifor is 4, the Quotient or
Anſwer is 17890708, and or two fourth Parts; at 5 the
Remainder is o, fo that the Answer is abfolutely 3431285
Times or Things only; at 6 the Remainder is 5, fo that the
Quotient or Anfier is 44533077 and or five fixth Parts;
at 9 the Remainder is but I, fo that the Answer or Quo-
tient is 652445 and or one ninth Part; fo where the Divifor
is 23, the Remainder is 12, which is twelve twenty three Parts;
but where the fame Dividend is divided by the whole Part of
the Quotient, and the Remainder is the fame as before, that
is 12; but that 12 is not near fo great in Value; for 'tis
but 12 Parts of three Hundred fifty Thouſand, two Hundred
eighty one Parts of fome whole Thing or Unity; and ſo in all
Cafes whatever, let the Divifor for the Time being, be what
it will, it always reprefents Unity, divided into fo many Parts,
and is the Ordinal-Number, and the Remainder is the Car-
dinal-Number, as you may plainly fee by what is faid above.
So in the next Example, Page 92, where the Divifor is 87,
and the Remainder 48, they are called 48 Eighty-Sevenths of
fome whole Thing or Unity; and when you invert that Quef-
tion, and make the Quotient 1419043 the Divifor, the Re-
mainder is the fame in Figures, viz. 48, but not the fame
in Value; for in this latter Example, 'tis but 48 Parts of
one Million four hundred nineteen thousand and 43 Parts of
Unity, which is but about the one three hundred and thirty
thousandth Part of Unity; whereas the fame 48, when 87 was
the Divifor, was more than the one half Part of Unity. In
the next Example, the Remainder is 97, which is 97-918
Parts, or 971, 011, 727 Parts, according to what the Di-
vifor is. By what has been already taught all Questions in
whole Numbers may be truly performed; but as there are
fome Methods by which the Work may be fhortned, or at
leaft eas'd, and which in Buſineſs may be very proper to be
known, my next Work fhall be to fhew you how this
may be done. And first, in all Cafes where there are one
or more Cyphers in the Divifor at the End of, or towards the
Right-hand of fignificant Figures, it or they may be parted
or cut off from the fignificant Figures by a, or I and then
cut or part off fo many Figures in the Dividend, beginning
at the Right-hand, or Unity, as you fee done in the follow-
ing Examples, working with the remaining Figures, as if the
faid Cyphers did not belong to the Divifar, and as if the 3
Figures
96
Of DIVISION.
Figures cut off from the Dividend had no relation to it. As,
if 5764825 were to be divided by 7000; here I cut off the
3 Cyphers from the Divifor, and three Figures from the Di-
vidend; and then it becomes 5764, divided by 7, and the
Quotient is 822, and the Remainder is 825 feven thouſand
Parts of Unity. So if 159876007200 was to be divided by
890000; here I firft cut off 4 Cyphers from the Divifor, and
four Figures or Cyphers, or both mix'd, as they may hap-
pen to be, and then it becomes 15987600 divided by 89,
tho' the Figures will ftand thus, 8910000)1598760017200,
and the Quotient will be 179635, and the Remainder 857200
Parts of 890000; for you are always to fet down the
whole given Divifor, and if any thing remains in dividing
by the fignificant Figures, you are to bring down what you cut
off from the Dividend, and place it on the Right-hand of
the faid Remainder, as in the two Examples above; in the
firit of which, there remained o, in dividing by the 7, there-
fore the Remainder was only the 3 Figures that were cut off
from the Dividend, viz. 825; but in the fecond Example
the Remainder was 85, and the 4 Figures cut off, were 7200,
fo that the whole Remainder was 857200; the like is to be
obferved in all other Cafes of this Kind. But, fecondly, if it
ſhall happen that the Divifor fhould be 1, with one or more
Cyphers, then the whole Work is done, by only cutting off
as many Figures or Cyphers from the Dividend, as there are
Cyphers in the Divifor, and the remaining Figures of the
Dividend, towards the Left-hand, will be the Quotient, and
the Figures or Cyphers, fo cut off from the Dividend, will be
the Remainder: As, fuppofe 527600 was to be divided by
1000, in this Cafe 527 will be the Quotient, and 600 the
Remainder; but fuppofe the fame Dividend was to be divid-
ed by 10 or 100, in both theſe Cafes the Remainder
would be nothing, that is o or oo, and the Quotient would
be 52760, or 5276; and this, I think, is fufficient for Di-
visors that have Cyphers on the Right-hand of the fignificant.
Figures. My next Work fhall be to make the Operations
both of fimple and compound Numbers, fhorter and eaſier than
by the univerfal Method above taught; only you are to ob-
ferve that the following Methods are not applicable to all
Numbers; and firſt, when your Divifor is 2, or indeed any
even Number whatever, you may halve your Dividend and
Divifor; and if by fo doing, your given Divifor becomes
Unity, your Work is done, and the faid Half of the Divi-
dend is the Quotient fought: As, if I afk how often 2 is con-
tained
Of DIVISION97
A
.
tained in 578, the Anfwer may be found thus, 21578; fay,
the Half of 2, the Divifor, is I; and the Half of 5, the firft
Figure of the Dividend, is 2, and I is left, to which bring
down in your Mind the 7, and it becomes 17; then fay,
the Half of 17 is 8, and I is left, to which bring down in
your Mind the 8, and it makes 18, the Half of which is 9;
ſo that the whole Work will ſtand thus,
I
1578 and the
1/289
Quotient is 289; fo if I was to divide 96584 by 2, the Quo-
tient would be 48292. If 618 was to be divided by 3, the
Quotient would be 206; for I fay, how often 3 in 6? and
the Anſwer is 2, which being fet down, I fay, twice 3 is 6,
which is equal to, or the fame with the firft Figure of the
Dividend, fo that the Remainder is o; therefore I go on and
ſay, how often 3 in 1 and the Anfwer is o; which being
fet down on the Right-hand of the 2, I go on and fay, how
often 3 in 18 and the Anfwer is 6, which being ſet down,
and the 3 multiply'd thereby, the Product is 18; fo that the
Remainder is 0, and the Quotiont is 206. So if you divide
59278 by 3, the Quotient will be 19759, and the Remain-
der will be 1. If 62870 be divided by 4, you may fay the
4's in 6 is 1, and 2 over; the 4's in 22, is 5, and 2 over;
and the 4's in 28, is 7 Times, and o over; and the 4's in
is I, and 3 over; and the 4's in 30 is 7 Times, and 2 over;
fo that the Quotient is 15717, and the Remainder is 2, or 2
Fourths: Or you might fet down your Work
as in the Margin, where the Divifor and
Dividend being fet down, and firſt halv'd,
then the Divifor becomes 2, and the Divi-
dend 31435; theſe being again halv'd, the
Divifor becomes I and the Dividend 15717,
and the Remainder in this laft Work is 1,
which is equal to the foregoing Divifor, which was 2,
this I is therefore to be eſteemed 2 Units in Value; for every
Quotient is a Repetition of the Divifor, fo many Times as
there are Units contained in it: Now 31435 is the Quoti-
ent of 62870, divided by 2, that is, as the original Dividend
confifted of 62870 Units, and this latter one confifts but of
half fo many 2's, which being divided again by 2 leaves one 2.
7
462870
2131435
1 | 15717-2
Note, That whenever you break a Divifor into two Parts,
for the more eafy dividing any given Dividend; if there re-
mains any thing in performing the Work by the first Figure,
they are so many Units; and if any thing remains in dividing
by the Second Figure, they are fo many 2's, 3's, 4's, &c. ac-
cording to what your laft Divifor was.
O
This
98
Of DIVISI O N.
This being premiſed, I now go on to the other fingle
Figures, and require the feveral Quotients of 7156485, di-
vided by 3, 4, 5, 6, 7, 8, 9: Here I will fhew you how
to do them by each fingle Figure, and then how to break
fuch Figures as are compounded by the Multiplication of two
Figures together into two Quotients.
Divifors
3
7156485 Dividend.
2385495 Quotient, and o Remains.
4
7156485 Dividend.
1789121 Quotient, and 1 Remains.
5
7156485 Dividend.
1431297 Quotient, and o Remains.
6 7156485 Dividend.
1192747 Quotient, and 3 Remains.
H
7
7156485 Dividend.
1022355 Quotient, and o Remains.
8 7156485 Dividend.
894560 Quotient, and 5 Remains.
9 7156485 Dividend.
795165 Quotient, and o Remains.
Here I have chofe to divide the fame Number or Dividend,
by various Divifors, on purpoſe to fhew you the Difference
between a Smaller and a Greater, and between where there
are Remainders, and where there are none. As for In-
ftance, when I divide by 3, the Quotient is 2385495, and
no Remainder; but when I divide by 6 the Quotient is
1192747, and the Remainder is 3; as may be eafily feen by
the following Work. Firſt, I ſay, how often 3 in 7? Twice,
and
Of DIVISION.
99
and 1 over, fet down 2, and bring down in your Mind the
next Figure 1, to the Remainder 1, and it makes II; then
how often 3 in II? Thrice; fet down 3, and the Remainder
´´is 2; becauſe 3
Times 3 is 9, which taken out of 11, the
Remainder must be 2; to this 2 bring down in your Mind the
next Figure 5, and afk, How often 3 in 25? and the Anfwer
is 8; now 8 Times 3 is 24, which taken out of 25 leaves
1, which with the next Figure 6, makes 16; then the 3's
in 16 go 5 Times; now 5 Times 3 is 15, which being fub-
ftracted from 16 leaves 1; fo that the next Number to be
divided, will be 14, in which the 3 is contained
is contained 4 Times,
and the Remainder is 2, to which bring down the 8, and
it becomes 28, in which 3 is contained 9 Times, and I is
left; then the 5 brought down to that, makes 15, in which
3 is contained juft 5 Times, and nothing remains. So that
the whole Quotient, when all the Figures are gone through,
is 2385495; but the fame Dividend, when divided by 6,
has 1192747 for the Quotient, and 3 for a Remainder, as
appears by the following Work: First, I fay, How often
is 6 contained in 7? Once, and I over, to which bring
down in your Mind the next Figure, which is 1, and it
makes 11; then how often 6 in II? Once, and 5 over,
then the next Figure 5, being brought down to this Re-
mainder, makes 55, which contains 6 nine Times, and
I over; then the 6's in 16 is twice, and 4 over; in 44,
ſeven Times, and 2'over; in 28, four Times, and 4 over; in
45 feven Times, and 3 over. So that the whole Quotient is
1192747, and the Remainder is 3. The Truth of this Work,
other of the like Sort whatever, may be eafily prov'd
by multiplying the Quotient by the Divifor; and to that Pro-
duct add the Remainder, and that Total muft always pro-
duce the fame Figures with the Dividend.
or any
Or, you may break the Divifor into two ſuch Parts, that
when multiply'd together, the Product may be equal to the
Divifor; and divide firft the given Dividend, by one of thofe
Parts, and that Quotient by the other Part; as in the laft
Example 6, is compos'd of 2, multiply'd by 3, or 3 by 2; fo
that according to the firſt Rule, 6 being an even Number,
you may halve it, or divide it by 2; and that Quotient be-
ing divided by 3, will give the Anfwer, as above; or you
may divide firft by 3, and then by 2, which you pleafe.
02
Here
}
}
.}
*
100
Of DIVISION.
|
Here, after ſetting down the Divifor,
217156485 and dividing it by 2, the Quotient is
3578242, and the Remainder I, which,
33578242-1 as was obferv'd before is 1 Unit, or fim-
ply I; then dividing the Quotient 3578242
1192747-2 by 3, that Quotient is 1192747, and I
Remains: Now this I is to be eſteemed
as 2 Units, becauſe the former Divifor was
2, which with the I that remain'd before, makes 3; fo that
this laft Quotient and Remainder, is the fame with the Quo-
tient and Remainder, when you divided by 6 at once. So if
9 was the Divifor, I fay, How often 9 in 71? Becauſe I can-
not have 9 in 7, and the Answer is
and the Anſwer is 7 Times, and 8 over;
then 9 in 85 is 9 Times, and 4 over; in 46 is 5 Times, and
I over; in 14 once, and 5 over; in 58 fix Times, and 4
over; in 45, is 5 Times and o remains: Or you might have
divided by 3, and 3 and the Quotient and Remainder will be
the fame, as you may fee in the Margin,
37156485 where having divided by 3, the Quotient
is 2385495, and o remains; which being
32385495 again divided by 3, the Quotient is 795165,
and o remains, the fame as when the Di-
| 795165 viſion was made by 9, and this will always
happen, if your Work is performed right:
So that you ſee theſe different Methods may
ferve as Proofs to one another. I fhall only recommend
the Practice of this Way in fundry Examples, on purpoſe to
make you ready and perfect in the Difpatch of Bufinefs, I
would have you alſo practiſe dividing by 12 at once, in the
fame Manner as you fee done in the Example following, viz.
In 54768904 Pence, how many Shillings? Here I be-
gin, and fay, the 12's in 54 is 4, and 6 over; in 67 is 5,
and 7 over; in 76 is 6, and 4 over; in 48 is 4, and o over;
in 9 is 0; in 90 is 7, and 6 over; in 64 is 5, and 4 over:
So that the Answer is 4564075 Shillings and 4 Pence. For
the Proof of this you may divide the given Number by 4,
by 4 and 3
41 54768904
3113692226
Proof 4564075-4
О
and that Quotient by 3, or firft by 3
and then by 4, as you fee done in the
Margin, where firft dividing by 4,
the Quotient is 13692226, and O remains,
which in refpect to the Numbers now
concern'd, are ſo many 4's; then divid-
ing this Quotient by 3, the laft Quotient
is 4564075, the fame with what was
produc'd
Of DIVISION.
IOI
by 3 and 4
31 54768904
4 18256301. I
| 4564075.4
produc'd when the Divifion was made by 12
at once, and the Remainder is 1, which is 4,
becauſe the Divifim before was made by
4. In the fecond Method the Divifion
is firft made by 3, and the Quotient is
18256301, and I Remains; this Quo-
tient being divided by 4, gives 4564075
for the new Quotient, the fame with the
former, and the Remainder is 1, which is once 3; and the
Unit that remain'd when the Divifion was made by 3,
being added to it, makes the whole Remainder 4, as when
the Divifors were 4 and 3. Thus you may perform any o-
ther Numbers whatever, by breaking them into fuch Parts,
as when multiply'd together, will produce the whole Divi-
for exactly, but not otherwife. As, ſuppoſe you was to di-
vide by 15, 16, 18, 21, 24, 35, 42, 54, 63, 72, 81,
84, &c. for 15, you might take 3 and 5, or 5 and 3; for
16 you may take 4 and 4, or 8 and 2, or 2 and 8, &c.
as you have been already taught in Multiplication; but you
are to obferve, that if you can't find 2, 3, 4, &c. Num-
bers whofe Product, when multiply'd together, will exact-
ly produce the whole Divifor, then you muſt uſe the gene-
ral Method before taught. As, fuppofe you was to divide
by 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 51, 53,
57, 59, &c. no two or more Numbers multiply'd together,
will exactly bring out thefe Numbers, but the Products will be
either more or less, and therefore you muſt divide by the
general Method first taught.
Divide 792864 by 27.
I
Here I confider, and find that 3 Times 9, or 9 Times
3, make 27; fo that I work both Ways, and find the Qus-
tient to be 2936533, and the Remainder 21.
3179286412
9 | 26428804
| 2936533-21
Or,
9|79286412
3.8809601.3
1.2936533.21
In the firſt Method, having divided by 3, the Remainder is
o; then dividing that Quetient by 9, the laft Quotient is
2936533, and 7 Remains; which being 7 Times 3, the
whole Remainder is 21, as you fee fet down in the Example.
In the fecond Method, after dividing by 9, the Remainder is
3,
102
Of DIVISION.
3, and the Quotient 8809601, are fo many 9's, which being
divided by 3, the new Quotient is 2936533, and the Remain-
der is 2, which being 2 Nines, is 18, and 3 that remain'd
before, make 21 for the whole Remainder, as in the firft
Method; and thefe two different Ways, or taking the Parts,
are Proofs to one another, without any further Procefs;
but you may divide by 27 at once, according to the general
Method first taught, for your further Satisfaction and Im-
provement, as is here done.
Divifor. Dividend. Quotient.
27) 79286412 (2936533
54
252
243
098
081
176.
162
144
135
91
8x
102
81
21 Remainder.
Thus you may go on to any other Number, that you
can find the exact Parts of, let it be as big as it will; as
fuppofe 'twas afk'd,
How often 729 was contain'd in 10105109?
Here 9 multiply'd by 9, makes 81, and this by 9 again,
makes 729 the exact Divifor; there if you divide by 3
Times 9, the laft Quotient will be 13861, and the whole Re-
mainder
Of DIVISION.
103
9 10105109
9.1122789-8
91..124754-35
mainder 440; for having divided by
once nine, the Quotient is 1122789
Nines and 8 Units; then dividing theſe
9's by 9, the Quotient is 124754
eighty Ones, and 3 remains, which be-
ing 3 Nines or 27, this added to the
firſt 8, makes 35; dividing again by
9, the Quotient is 13861 Times 729,
and 5 remains, which is 5 eighty Ones,
or 405, which added to the 35 makes 440 for the whole
Remainder; fo that the Answer to the Question is 13861
Times, and 440 Parts of 729 over. Again,
| ... 13861-440
How many Times is 1728, contained in 9158426?
Here I confider, and find that 12 multiply'd by 12, makes
144; and this again by 12, makes 1728; fo that if I divide
by 12 three Times, the last Quotient or Answer will be 5300
Times, and 26 the Remainder. Here,
after the first Divifion 2 remains, which
is 2 Units; after the fecond Divifion 2
remains, which are 2 Twelves or 24; and
this added to the first 2, makes 26;
then dividing a third Time by 12, the
Remainder is o; fo that the Anfwer is
5300-26 5300 Times, and 26 Parts over.
12 | 9158426
12 | 763202-2
12 63600-26
A RULE to find the Diviſors.
If the given Divifor be an even Number, you may halve it,
or divide it by 2; and if the Quotient that arifes be alſo an
even Number, continue halving; but when the Quotient that
fo arifes, is an odd Number, then try to divide it by fome
fmall odd Number (for it is impoffible to divide an odd
Number, by an even Number exactly, without leaving a Re-
mainder) as, 3, 5, 7, 9, &c. and fo go on till your Divifor
is reduced to Unity, without leaving any Remainder in any
of the fubordinate Divifers; as,
What are the Divifors to 336?
Here I halve 336, and the Quotient is 168; and the Half
of 168 is 84, the Half whereof is 42, and the Half of 42
is 21; now this being an odd Number, I divide it by 3, and
the Quotient is 7; fo that the Divifors may be 2, 2, 2, 2,
3 and 7; but as you may think fo many Operations tedious,
you may divide your given Quotient by 4, 6, 8 or 12; as here,
in 336 divided by 8, the Quotient is 42, this by 6, gives 7;
fo that the Divijors may be 8, 6 and 7. Again,
What
104
Of DIVISION.
What are the Divifors to 567?
This being an odd Number, I divide it by 3, and the
Quotient is 189; this by 3, gives 63; this by 3, gives 21;
this by 3, gives 7; fo that the Divifors to this Number
567, may be 3, 3, 3, 3 and 7: Or you may try any o-
ther odd Number at firft, as 5, 7, 9; and if any of them
fucceeds and leaves no Remainder, that Number may be us'd;
only obferve, that 5 can exactly divide no Number whatever,
but what ends either with 5 or o, for which Reaſon I omit
trying this Number with 5, and try it with 7, and the Quo-
tient is 81; and by the common Multiplication-Table, I
know that 9 Times 9 is 81; fo that 7, 9, 9 may be the Di-
viſors, inſtead of this Number 567: After the fame Manner
you may try any other Numbers. Now follows the Divi-
fion of mix'd Numbers; as,
If L.216
16 8 be divided among 8 Men, How much
is that a Piece?
First fet down the given Numbers,
8216 16 8 both Divifor and Dividend, as you
ſee in the Margin; and then fay, the
| 27: 2 : 1 8's in 21 twice, and 5 over; in 56,
feven Times, and o remains; then go on
to the Shillings, and ſay, the 8's in 16 twice, and o remains;
then go to the Pence, and fay, the 8's in 8 once, and o re-
mains; fo that the Answer is L. 27: 2 1 for each Man.
Again, Divide L. 65: 17: 6 among 5
Men. Here I fay, the 5's in 6 once,
in 15 three Times, and all the Pounds
or whole Numbers are done; then I go
on and fay, the 5's in 17 three Times,
and 2 over; which 2 being 2 Shillings, or 24 Pence, I add
them to the 6, that ftands in the Place of Pence, and the
Total is 30 Pence; then the 5's in 3 is juft 6, and o remains;
fo that the Anfwer is L. 13 3 6 per Man. Again,
36
If 7 Pieces of Broad-Cloth coft L. 197 : 19 : 5, What's
5165 176
113: 3:6
I
that a Piece?
7 | 197: 19:5
| 28: 57
57
2
The Numbers being fet down, I ſay,
the 7's in 19 twice; in 57 eight Times,
and I Pound over, which is 20 Shil-
lings; which added to the 19 Shillings
given, makes 39; then I fay, the 7
in 39 is five Times, and 4 Shillings
remain which are 48 Pence; theſe added to the 5 Pence,
in the given Dividend make 53; then the 7's in 53 is feven
Times,
-I
Of DIVISION.
105
7
Times, and 4 Pence over, which is 16 Farthings; the 7's in
16 is twice, and two Farthings are left; fo that the Anſwer
is L. 28 5 7 and Parts of a Farthing; for the
Divifor, let it be little or big, fhews into how many Parts
the Species of Coin, Weight, Meaſure, &c. is divided that
was laſt us'd: As in this Queſtion we went to Farthings, ſo
that the Remainder is 2 feventh Parts of a Farthing, for each
Piece, or 2 whole Farthings upon the whole Quantity; for
if you was to aſk, What will 7 Pieces come to at L.28: 5:7½
a Piece? the Anſwer would be L. 197: 19: 4; fo that
from hence 'tis plain, that at the given Price in our Queſtion,
a Piece must be fomewhat more to make out the whole
Money. The fame is to be obſerved in Weights, Meaſures, &c.
As, in 2564 Bushels of Coals, How many Chaldrons?
Here I confider that 3 Bushels make
32564 a Sack, and 12 Sacks make a Chaldron ;
therfore I divide by 3, and the Quotient
is 853 Sacks, and 2 Bushels over; then I
divide the Sacks by 12, and the Quotient is
71 Chaldrons, and 1 Sack over; fo that
the Anſwer is 71 Chaldrons, 1 Sack and 2
12 | 853-2
I 71-1
Bubels.
In 564230 Nails, How many Yards?
4564230
4 | 141057:2
Here I confider, that 4 Nails make
I Quarter; therefore I divide by 4, and
the Quotient is 141057 Quarters, and
2 Nails over; then 4 Quarters make
a Yard; therefore I divide the Quarters
by 4, and the Quotient is 35264 Yards,
and I Quarter over; fo that the Anſwer
Yards 2: N.
to the Question is 35264: I: 2
| 35264: I by 4, and the Quotient
In 5726 Penny-Weights of Silver, How many Pounds?
Twenty Penny-Weights being an Ounce,
I divide thereby, and the Quotient is 286
Ounces, and 6 Penny-Weights over; then
I divide by 12, and the Quotient is 23
Pounds, and 10 Ounces over; fo that the
3 Drots.
10 6. And ſo of any
20 1572-6
12 | 286:6
| 23:10
other Name whatever.
Њ
Anſwer is 23
P
CHA P.
106
Of REDUCTION.
CHA P. IV.
Of the reducing or changing one Name or Species
of Money, Weight, Meaſure, &c. into another,
Still retaining the fame Value.
N Schools 'tis ufual to teach a RULE, called
IN
REDUCTION.
Reduction is that Rule by which the Number of Parts, of
one Name or Denomination, are turn'd or chang'd into thoſe
of another, ftill retaining the fame Value; and it is com-
monly divided into two Parts, the one called Aſcending and
the other Defcending. Afcending is when you go upwards, or
change a leffer Name into a greater. This is done by di-
viding the given Number, by fo many of the leffer, as makes
one of the greater: As, if you would know how many Pounds
were contained in 516 Shillings; here the Anfwer is pro-
duced, by dividing 516, the Number of Shillings given, by 20,
the Number of Shillings in a Pound Sterling: Or, if they were
516 Ounces Troy-Weight, to know how many Pounds. Di-
vide by 12, becaufe 12 Ounces make a Pound; but if it had
been Avoir-du-poife-Weight, you muſt have divided by 16;
and fo in all other Cafes whatever, regarding carefully the
Number of Parts of the leffer, that make up one of the
greater. Defcending is when you want to know how many
of the leffer Name or Names, are contained in 1 or more
of the greater; in this Cafe, you muſt multiply the given
Number of the greater Name, by fo many of the leffer, as
make one of the greater: As, if you would know how many
Quarters of a Yard were contained in 516 Ells Engliſh:
Here you muſt multiply by 5, and the Product 2580, is the
Anfwer. So if you would know how many Pecks were con-
tained in 812 Bufhels, you muft multiply by 4, and the Pro-
duct 3248 is the Anfwer. But if it happens that there
fhould be two or three intermediate Subdivifions, between
the given Name and that required, you may multiply thoſe
intermediate Denominations together, and make their Pro-
duct a Multiplier or Divifor to the given Number, as the
Occafion requires: Or, you may multiply or divide continual
ly by the refpective Number of Parts of each Subdivifion.
Thus:
Of REDUCTION.
107
Thus: How many Farthings are there in 54 Pounds?
Here are 3 Subdiviſions, viz. firſt 20 Shil-
lings make I Pound, and therefore I mul-
20
tiply the given 54 by 20, and the Pro- 1080 Shillings.
duct 1080 are the Shillings, contained in
54 Pounds; then 12 Pence making 1 Shil-
12
ling, I multiply the 1080 Shillings by 12, 12960 Pence.
and the Product 12960, are the Pence in
54 Pounds; and as 4 Farthings make 1
4.
Penny, I multiply the 12960 Pence by 4, 51840 Farthings.
and the Product 51840 is the Farthings in
54 Pounds: Or, you might have multiply'd 20, 12 and 4 to-
gether continually, and the laft Product 960, would be the
Farthings in I Pound; by which multiplying the 54, the
Product would be 51840, the fame as before; as appears
by the Work before you.
54 Pounds given.
960 the Farthings in 1 Pound.
3240
486
51840 the Farthings in 54 Pounds.
In 549 Portugal-pieces of Gold, each L. 3: 12 Sterl.
72 Shill. in 1 Piece.
1098
3843.
How many Farthings?
39528 Shillings in all.
12 Pence in a Shilling.
474336 Pence in all.
4 Farthings in a Penny.
1897344 Farthings in all.
P 2
In
108
Of REDUCTION.
In 158 Tons Avoirdupoife Weight, How many Drams?
20 Hundreds in a Ton.
Or,
20
4
3160 Hundreds in 158 Tons.
4 Quarters in a Hundred.
80
28
12640 Quarters of a Hund. in all.
28 Pounds in a Quarter.
2240
16
101120
25280
13440
2240
1
353920 Pounds in all.
16 Ounces in a Pound.
35840
16
2123520
353920
215040
35840
5662720 Ounces in all.
16 Drams in an Ounce.
573440 Dr.
33976320
158 in a
Ton.
5662720
4587520
2867200
90603520 Drams in all.
573440
90603520 Dr.in
all.
By the Examples above, you fee the Anfwer is always the
fame, whether you multiply the whole given Number by the
firft Subdivifion, and that Product by the next, and ſo on
till
you come to the Name fought after; or whether you
multiply thofe Subdivifions continually together, and that laft
Product by the given Number of the greater Name: But
fometimes it may be done more eaſily and expeditiouſly
one Way, than the other, which by Practice you will rea-
dily find out; and your own Choice or Judgment will
dictate to you which Method to ufe, as the Occafion may
require. Ágain,
In
Of REDUCTION.
109
In 259 Miles, How many Inches?
a
8 Furlongs in a Mile.
2072 Furlongs in all.
40 Poles in a Furlong.
82880 Poles in all.
II half Yards in a Pole.
82880
82880
911680 half Yards in all.
18 Inches in a half Yard.
7293440
911680
16410240 Inches in 259 Miles.
In 89 Tons of Wine, How many Pints?
2 Pipes in a Ton.
178 Pipes in all.
2 Hogfheads in a Pipe.
356 Hogfheads in all.
63 Gallons in a Hogſhead.
1068
2136
23428 Gallons in all.
4 Quarts in a Gallon.
93712 Quarts in all.
2 Pints in a Quart.
187424 Pints in 89 Tons.
In
110
Of REDUCTION.
In 567 Chaldrons, how many Bufhels?
36 Bufhels in a Chaldron.
3402
1701
20412 Bufhels in all.
In 516 common Years, How many Minutes?
365 Days in a Year.
2580
3096
1548..
188340 Days in all.
24 Hours in a Day.
753360
376680.
4520160 Hours in all the Days.
3096 Hours over, viz. 6 in each Year.
4523256 Hours in all the Years.
60 Or, 365 Days in a Year.
24 Hours in a Day.
271395360 Min.
1460
730.
8760 Hours in 365 Days.
6 odd Hours over in a Com.
Year.
8766 Hours in a Year.
60 Minutes in an Hour.
525960 Minutes in a Year.
516 Years given.
3155760
525960
2629800
271395360 Minutes in 516 Years.
Or,
Of REDUCTION
III
Or, 516 Years.
365 Days in a Year.
2580
3096.
1548..
129 for the of a Day in each Year.
44
188469 Days in 516 Years.
24 Hours in a Day.
753876
376938
4523256 Hours in all, as before.
60 Minutes in an Hour.
271395360 Minutes in all.
REDUCTION ASCENDING.
In 4142564820 Farthings, How many Pounds?
12 | 10641205 Pence.
| 21018867617 Shillings and 1 Penny.
Anſwer
44338 Pounds, 7 Shillings and 1 Penny.
Here the given Number is divided by 4, the Farthings in a
Penny, and the Quotient that arifes, viz. 10641205 is Pence;
then theſe Pence are divided by 12, and the Quotient 886767
is Shillings, and I being left or remaining, is 1 Penny; then
thefe Shillings being divided by 20, the Quotient 44338 that
arifes is Pounds, and the Remainder is 7 Shillings: So
the Anſwer is Ĺ. 44338 7: I. Now if we multiply 4,
12, and 20 together continually, the laft Product will be
960, by which you may divide the given Number, as
you have been before taught; and as you fee here done.
12 Pence in a Shilling.
L.
4 Farthings in a Penny.
48 Farthings in a Shilling.
20 Shillings in a Pound.
960 Farthings in a Pound.
9610
I 12
Of REDUCTION.
961014256482101444338
384
416
384
324
288
368
288
802
768
Here, after the Divifion is per-
form'd, the Quotient is L. 44338
as before, and the Remainder is
34, to which bringing down the
o that was cut off, it makes 340,
which are Farthings; and there-
fore I divide by 4, and the Quo-
tient is 85 Pence, which being a-
gain divided by 12, the Quotient
is 7 Shillings, and 1 Penny over :
Or, you might have divided by
10, 8 and 12, and the Anſwer
will be the fame; as appears by
the following Work.
340 Farthings remains, or 7s. Id.
10 | 425648210
{
84256482 Two-pence half Penny's.
12 | 532060 Twenty-pences, and 2 Two-pence Half-penny's,
or 5 Pence remains.
44338 Pounds and 4 Twenty-pences, or 6 Shillings
and eight Pence remains, which with the five Pence that re-
main'd before, makes 7 Shillings and 1 Penny, as before.
Explanation of the laft Method.
The whole Divifor 960, is first divided by 10, by cut-
ing off the firft Figure in the Units Place, viz. the o, and
then 96 is left; the Dividend is alfo divided by 10, by
cutting off the Figure in the Units Place, which in this
Example is o, though it might have been any other Figure
whatever. Now this 10 is 10 Farthings, or Two-pence
Half-penny, confequently the Quotient 4256482, are Two-
pence Half-penny's; then I confider what two Numbers
multiply'd together, will produce 96, and find that 8 and 12
will do it, becaufe 8 Times 12 is 96; then I divide by 8,
and the Quotient is 532060 Twenty-pences, becauſe 8 Times
2dis 20 Pence, and the Remainder is 2 Two-pence Half-
penny's, or 5 Pence; then I divide thefe 20 Pences by 12,
and the Quetient is 44338 Pounds, becauſe 12 Twenty-pences
is a Pound, and the Remainder is 4, which is 4 Twenty-
pences, or 6 Shillings and 8 Pence, which added to the 5
3
Pence
Of REDUCTION.
113
Pence that remain'd before, makes 7 Shillings and 1 Pen-
ny.
I
If you had divided by 10, 12 and 8, the Work would
have ſtood thus:
110 | 425648210
12 | 4256482. Two-pence Half-penny's.
| 81354706 half Crowns and 10: 2, rem, or 2 s. 1 d.
44338 Pounds and 2 half Crowns, or 5s. remains,
which added to the 2 s. 1d. makes 7 s. 1 d. as before.
8 | 42564820
Or, you may do thus :
12 5320602 Two-pences, and 4 Farthings or id, remains.
| 11014433813 Two Shillings and 6 Two-pences or I s. rem.
44338 Pounds and 3 two Shillings or 6 s. remains,
which added to the two other Remainders, viz. I s. and I d.
makes 7 s. and id. as before. So you ſee, that uſe what Me-
thod you pleafe, if your Work is rightly performed, 'twill
always produce the fame Anfwer, proper Care being taken
of duly valuing the Remainders. Again,
In 5864 Nails, How many Ells Flemiſh?
415864 Nails.
3 | 1466 Quarters of a Yard.
I 488 Ells Flemish, and two Quarters over or remaining.
Or,
12 | 5864 Nails.
| 488 Ells Flemiſh, and eight Nails, or two Quarters
remains.
Here you may first divide by 4, the Nails in a Quarter,
and that Quotient by 3, the Quarters in an Ell Flemiſh :
Or, you may divide by 12 at once, which is the Number
of Nails in an Ell Flemish; and the Anſwer both Ways, is
488, E. F. and 2 Quarters over.
In
114
Of REDUCTION.
i
In 56425 Penny-Weights of Silver, How many Pounds?
210 | 564215
12 | 2821 Ounces and 5 Penny-Weights.
235 Pounds, 1 Ounce, and 5 Penny-Weights.
In 62754 Grains of Gold, How many Ounces?
3162754
Here 24 Grains making 1 Penny-Weight,
I break it into 8 and 3.
8 20918 three Grains.
2|0|261|4 Penny-Weights, and 18 Grains.
| 130 Ounces 14 Penny-Weights, and 18 Grains.
In 6258436 Pints of Wine, How many Tons?
86258436 Pints.
7 | 782304 Gallons and 4 Pints, or 2 Quarts remains.
9 | 111757 feven Gallons, and five Gallons remains.
| 412417 Hogfheads, and 33 Gallons, viz. 4 Sevens now,
which is 28; and 5 before, which is 33.
1 3104 Tons and 1 Hogfhead, 35 Gallons and 2 Quarts.
Page 62, you are told that 2 Pints is 1 Quart, and 4
Quarts 1 Gallon, confequently 8 Pints is a Gallon; there-
fore I divide firft by 8, and the Quotient 782304 is Gal-
lons, and 4 Pints or 2 Quarts remains; then 63 Gallons being
a Hogshead, I divide by 7 and 9, becauſe when multiply'd to-
gether they produce 63: When I divide by 7, the Quotient is
111757 Times 7 Gallons, and 5 Gallons remains; then I divide
this by 9, and the Quotient is 12417 Hogfheads, and 4 remains;
which is 4 Times 7 Gallons or 28 Gallons, and 5 that re-
main'd before, is 33 Gallons. Now 4 Hogfbeads being I Ton,
I divide by 4, and the Quotient is 3104 Tons and 1 Hog-
fhead, which with the former Remainders makes the whole
Tons Hogfh. Gall. Quarts.
Anſwer to be
4104 : : 33: 2
In
Of REDUCTION.
115
In 56402650 Minutes, How many Days?
6 | 0156402510
31 94004 Hours and 10 Minutes.
| 8/31334-2
A. 3916 Days 20 Hours and 10 Minutes.
Here 60 Minutes being an Hour, I divide thereby, and the
the Quotient 940004 is Hours, and 10 Minutes remains, 24
Hours being a Day, I divide by 3 and 8; when I divide by
3, the Remainder is 2 Hours; and when I divide by, 8 the
Quotient is 3916 Days, and the Remainder is 6, which is 6
Times 3, or 18 Hours, which added to the other 2, makes
20 Hours; fo that the Anſwer is 3916 Days, 20 Hours and
10 Minutes.
In 62584362 Drams, How many Tons AvoirdupoifeWeight?
4 | 62584362
4| 15646090-2
| 4|3911522 Ounces and 10 Drams.
141 977880-2
| 41244470 Pounds and two Ounces.
761117-2
14.8731 Quarters of a Hundred and 2 Pounds.
20/2182 Hundreds and 3 Quarters.
Answer 109 Tons, 2 C. 3 Qr. 2 tb 23 10 3.
Here 16 Drams making an Ounce, I divide by 4 and 4,
inftead of 16; and 16 Ounces being 1 Pound, I do the fame
again; and 28 Pounds being a Quarter of the long Hundred,
I divide by 4 and 7; then 4 Quarters being 1 C. I divide
the Quarters by 4; and 20 C. being a Ton, I divide the
Hundreds by 20, and the Anfwer to the Queftion is as
above.
Q 2
In
116
Of REDUCTION.
In 64287564 Barley-Corns, How many Miles?
61 64287564
91 10714594 two Inches.
| 111190510 half Yards, and 8 Inches over.
| 4lol 1082218 Poles, and 2 half Yards over.
182705 Furlongs, and 28 Poles over.
338 Miles and 1 Furlong over.
Miles F. P.
Yds. 2 Inch.
So that the whole Anſwer is 338: I: 28: 2 : 8
I
Here I confider'd that as 3 Barley-Corns is 1 Inch, fo 6 is
2 Inches; therefore I divided by 6, to bring them into 2
Inches, and theſe by 9, to bring them into half Yards, becauſe
18 Inches is half a Yard; and confequently 9 Times 2 Inches
is half a Yard; then by 11, becauſe 11 half Yards make a
Pole, and then by 40, becauſe 40 Poles is a Furlong, and then
by 8, becauſe 8 Furlongs is a Mile, as may be ſeen in the
Table, Page 61. We now fhall go on to mix'd Reduction,
viz. where both Multiplication and Divifion are us'd in the
fame Queſtion. And here 'tis proper to confult how the leffer
Names may be brought into the greater Ones, or the greater
Names into the leffer Ones, with the moſt Eaſe and Expedi-
tion; for frequently the fame End may be attain'd by diffe-
rent Methods, of which fome may be much preferable to
others.
:
In L. 564 15: 6, How many Moidores?
20
3| 11295 Shillings.
9 3765 three Shillings.
418 Moidores, and 3 three Shillings, or 9 Shillings
and 6 Pence over or remaining.
Here I first multiply the Pounds by 20, and take in the odd 15
Shillings, and the Product 11295 is Shillings, which I divide.
firt by 3, and that Quotient 3765 by 9, and the Quotient 418
is
Of REDUCTION.
117
is Moidores, and the three that remains, is 3 Times 3 Shillings,
or 9 Shillings; and as there was 6 odd Pence in the given Num-
ber that will neceffarily remain, there is a Method of divid
ing mix'd Species by mix'd Species; but as that is fome-
what difficult, and, as I fuppofe my Learner to know no
more than what he has been taught in this Book, I will
forbear faying any Thing upon that Subject at preſent, and
content my ſelf with going on in the common Path a while
longer.
In 1218 Moidores, How many Guineas?
3
3654 nine Shillings.
9
31 32886 Shillings in all the Moidores.
7 | 10962 three Shillings.
1566 Guineas.
Here you are to obferve, that I multiply first by 3,
and then that Product by 9; becauſe as 27 Shillings make
a Moidore, fo 3 Times 9 is 27. Now, after multiplying
by 3, that Product is fo many 9 Shillings, becaufe 3 Times
9 Shillings is a Moidore, or 27 Shillings; then I multiply
3654 by 9, to bring them into fingle Shillings, which
Product 32886, I divide firſt by 3; and then that Quotient
10962, by 7, becaufe 7 Times 3, or 3 Times 7 makes
21, the Shillings in a Guinea; and the Anfwer is, 1566
Guineas, and o remains. Now, whenever it fo happens,
that you have the fame Figure for a Multiplier and a Di-
vifor, you may omit both the Multiplication and the Divi-
fim, and the Work will be the fame, by only operating
with the other Figures; as here, in this Example, 3 is both
a Multiplier and a Divifor; therefore I only multiply by
9, and divide by 7, and the Anſwer is exactly the fame as
1218 Moidores.
9
7 | 10962 three Shillings.
| 1566 Guineas.
before, as you fee in the Margin:
For when you have multiply'd the
given Number of Moidores by 9,
that Product is 10962 three Shil-
lings. Now 7 Times three Shillings
make a Guinea, or 21 Shillings.
And
118
Of REDUCTION.
And fo you may work in all other Questions whatever,
let the Name or Denomination be what it will.
In 2159 Pieces of 36 Shillings each, How many Guineas?
12
725908
370r Guineas, and 3 Shillings remains.
Here I confider that 3 Times 12 is 36, and 3 Times 7 is
21; therefore I multiply by 12, and divide that Product by
7, ommiting the 3 both in multiplying and dividing, for
the Reaſons above.
In 1566 Guineas, How many Moidores?
7
9110962 three Shillings.
1218 Moidores.
In 156 Packs of Cloth, each 47 Pieces, each Piece 27 E.
47 Pieces in a Pack.
1092
Eng. How many Yards?
624.
7332 Pieces in all.
27 Ells English, in a Piece.
51324
14664.
197964 Ells Eng. in all.
5 Quarters in an Ell English.
41 989820 Quarters in all.
| 247455 Yards in all.
Tr
Of REDUCTION.
119
7
In 5628 Hogfheads, How many Puncheons of 84 Gallons
9
12 | 50652 feven Gallons.
4221 Puncheons.
each?
Here I confider, that a Hogfhead is 63 Gallons; and that
Times 9, or 9 Times 7, is 63; and that a Puncheon is 84
Gallons, or 7 Times 12, or 12 Times 7; fo that I omit the
7, and multiply by 9, and divide by 12, and the Quotient
4221, is the Anfwer fought, as may be feen by the Work
at large.
5628
63
16884
33768
84)354564(4221
336
185
168
176
168
84
84
oo remains.
You muſt always obferve, that your Dividend and Di-
viſor is, or muſt be reduced to the fame Name, otherwife
you can't perform the Work: So in the laſt Example, when
you multiply'd by 9, the Product was 50652 feven Gallons;
becauſe the given Number being Hogfheads, each Hogshead
contains 63 Gallons, or 9 Times 7 Gallons, and the Divifor
12, is a feventh Part of a Puncheon, or 84 Gallons. You
may alſo obſerve, that when the Multiplier and Divifor can
both be divided by fome common Number, without leav-
ing any Remainder, you may multiply and divide by thofe
Quotients, and the Anfwer will be the fame as before: As
in the laſt Example, the Multiplier is 9, and the Divifor
2
12,
120 The RULE of THREE.
12, both which may be divided by 3, and the Quotients will
be 3 and 4; with which, working as before, the Anſwer
will come out the fame, as follows.
In 5628 Hogheads, How many Puncheons?
3
4 16884
4221 Puncheons.
The like is to be underſtood in all other Cafes, where
Contractions may be made; otherwiſe you muſt work with
full Numbers, as you fee this laft Example done; and if any
Thing remains, it is fo many of the leffer Parts, as makes
one of the greater. Enough has been faid upon this Head,
fo that I will now go on to the Doctrine of Proportion, com-
monly called the Rule of Three.
CHAP. V.
Of the Rule of Proportion, commonly called the Rule
of Three, and by way of Eminence the Golden-
Rule.
A
LL the compound Parts of Arithmetick, are in reality but
this Rule, under different Names; for at leaſt three
Numbers are always given to find a Fourth, even in plain
Multiplication and Diviſion, as will appear by confidering what
follows; for there Unity is always fuppos'd, tho' not exprefs'd,
and the Product or Dividend contains the Multiplier or Di-
vifor, as often as there are Units in the Multiplicand or Quo-
tient; as if I fay, multiply 57 by 98: Here I virtually tho'
not actually fay, as often as 1, or Unity is contained in 98,
the Multiplier, fo often 57 the Multiplicand is contained in
5586, the Product; or more concifely, as I is to 98; fo 57
to 5586. Or if I ſay, divide 5586 by 57, 'tis the fame
Thing, as faying, as 57 is to I; fo is 5586 to 98. Now
to effect this, or to find out the fourth proportional Number
or
The RULE of THREE.
121
or Quotients fought, you are to obferve, always to put the
Number or Quantity that afks the Question, What? How
many? &c. in the third Place, or that next towards the
Right-hand, and that Number or Quantity that is of the fame
Nature, Name or Kind with the demanding Number, firſt to-
wards the Left-hand, and the other Number or Quantity,'
concerned in the Question, in the Middle: Having thus dif-
pos'd of the Numbers, multiply the two that ſtood toward the
Right-hand together, and divide their Product by the firft
Number: As you fee done in the following Examples.
If 1 Yard of Cloth coft 6 Pence, What coft 8 Yards ?
Here, the Numbers being difpofed, as was juft now di-
rected, they will ſtand thus:
1--6--8 then multiplying the
6
fecond and third
together, the Pro-
48
duct is 48; but the
firft Number being
Unity, or I, neither multiplies nor divides; therefore the
Product 48, is alfo the Quotient. Now you are to obſerve,
that when the given Numbers are difpos'd according to the
foregoing Rule, whatever Name the fecond Number bears at
the Time of multiplying, the Quotient alſo bears, after the
Divifion is perform'd. So here if you divide the Product 48,
by 1, the Quotient will be the fame, viz. 48; fo that it is
48 Pence, becauſe the 6 or middle Number was Pence, which
may be brought into Shillings by dividing by 12. Again,
If a Sack of Malt, Quantity 4 Bufhels, caft 12 s. 6d.
What comes 85 Bufhels to, at the fame Price?
Here the Enquiry is, to know how much 85 Bushels coft;
and therefore that Number is to be put in the third or laft
Place, and the Number that is of the fame Name, Kind, or
Quality with it, muſt be the firſt Number; and the other
Number, or Price of the first Quantity, will be the ſecond
or middle Number.
R
Thus
122
The RULE of THREE.
7
Thus,
4
1
Bufb. s. d. Bufh.
12 685
12
150
85
750
1200
4 12750
1213187-2
20/2615-7
L. 135-7
Now forafmuch as the
2d Number is a mix'd
Number of Shillings and
Pence, 'twill be proper
to bring it all into
Pence; which being
done, is 150; that
multiply'd by 58, makes
12750, which divided
by 4, the firft Num-
ber, the Quotient 3187
is Pence; becauſe the
fecond Number was
brought or reduced in-
to Pence, and 2 re-
mains.
In Page 94, you were told that the Divifor, let it be
fall or great, always reprefents fome whole Thing or Unity,
fuppofed to be broke or divided into fo many equal Parts,
as the Divifor contains Units: Now in all Cafes where
Names are put to the working Numbers, this Divifor always
reprefents one whole Thing, or Unity of the fame Name,
fo divided as the firft Quotient denominates. Now here, in
this Example, this Quotient is Pence, and therefore a Penny
is here divided into 4 Parts, becauſe the Divifor happens to
be 4; and confequently the Remainder 2, is 2 Farthings,
becauſe a Farthing is the fourth Part of a Penny. But as
this Cafe may be frequently varied, and both the Name
of the Unit, and the Parts, may be alfo infinitely differenced,
therefore obferve this general Rule following, for finding the
Value of fuch Remainders.
Set down the Number remaining, after the first Divifion,
which on this Example is 2; then confider what it is 2 Parts
of; and in this Example 'tis 2 Parts of a Penny; then con-
fider what is the next inferior Denomination, which in this
Example is Farthings; multiply the faid Remainder by the
Number of Parts in the next inferior Name, which here is 4,
becauſe 4 Farthings make a Penny; and divide that Product
by the Divifor, which in this Example is 4, and the Quotient
is the Value of the faid Remainder, in that next inferior
Denomination, which here is 2, and remains: But if there
fhould
The RuLE of THREE.
123
fhould be a fecond Remainder, you may contiuue the fame Procefs,
till you come to the lowest known Subdivifion, in that Part of
Money, Weight, Meaſure, &c. you are then concerned about;
and after you have gone to the lowest known Subdivifion, com-
moly us'd; and after dividing ftill by the fame Divifor, there
Should be yet a Remainder, that laſt Remainder will be fo
many Parts of the laft Subdivifion you multiply'd by. The
foregoing Queſtion may be wrought as follows, viz.
If 1 Sack or 4 Bushels coſt 12s. 6d. What's theValue of 85 Buſh.
2 Sixpences in 1 Shilling.
I
6
4/6/1
4
2
4
418
2
1
25 Sixpences in 12 : 6.
85 the third Number.
125
200.
4 | 2125 (1 remains, which is of 6 Pen,
4 | 0531 Sixpences.
13-5-6
I in the Remainder.
L. 13-5-7 Anfwer, as before.
Here I only bring the middle Number into Sixpences, by
multiplying the Shillings by 2, becauſe 2 Sixpences are I
Shilling, and I take in the odd Sixpence; fo that there are 25
Sixpences in 12 s. and 6d. then I multiply by 85, and divide
by 4, and the Quotient is 531 Sixpences, and I remains, which
1 I multiply by 6, and divide that Product by 4, the for-
mer Divifor, and the Quotient is I Penny, and 2 remains,
which 2 I multiply by 4 the Farthings in a Penny, and di-
vide the Product 8 by the former Divifor, and the Quotient
is 2 Farthings and o remains; fo that the Value of this
that remain❜d after the first Divifion is 1 Penny and 2
Farthings, as will evidently appear by what was faid above,
and at Page 94; for that I was the one fourth Part of
Sixpence, which is three Half-pence: The 531 Sixpences I
divide by 40, becaufe 40 Sixpences are 1 Pound, and the Quo
tient is 13 Pounds, and 11 remains, which are 11 Sixpences,
or 5 s. and 6 d. to which add rd. the Value of the Re-
mainder, and the Anſwer is L. 13:57, as before.
R 2
<
ཙ, '
Again,
¥24
The RULE of THREE.
Again, Suppoſe I fhould fay, If L. 13: 5:7 buy 85
Bushels of Malt, What must I pay for 1 Sack of 4 Bufhels?
Here, by obferving the Queftion, as 'tis now put, I find
that the Sack, or 4 Bushels is the afking Number; and
therefore must be put in the third Place; and then, accord-
ing to the former Directions, the whole will ftand thus:
Buſh.
85-
L.
5.
d.
13:57 1/
Bush.
-4
20
265
85
12
3187
4
12750 Farthings.
4 third Number,
51000 41 600
510
оо
12 | 150
J
12:6
Here the middle Number being a mix'd Number, confift-
ing of Pounds, Shillings, Pence, and Farthings, 'tis brought
into the loweſt Name, viz. Farthings; and then thoſe Far-
things are multiply'd by the third Number, and that Product
divided by the firſt Number, and the Quotient 600 is Far-
things; becauſe the middle Number was reduced into Far-
things, which being divided by 4 and 12, produces 12 s. and
6d. as before. So that by varying the Queftion in this Man-
the different Works will be Proofs to one another, if
the Work be performed truly, and according to the fore-
going Directions: I have been very full in explaining e-
very Puntillio neceffary to be obferv'd; and therefore hope
the Reader, by carefully obferving what has been ſaid, will
be able to anſwer any other Queftions in this Rule, without
further Help, though, as Occafion offers, 1 fhall explain
what may occur hereafter, that carries any Face of Diffi-
culty at the Time of its fo happening,
ner,
:
Suppoſe
The RULE of THREE.
125
N
1
Suppoſe I buy a Bale of Linen Cloth, Quantity 58 Pieces,
each Piece 29 Ells Engliſh, at 15 d. per Yard, What is
the whole Worth, and how much do I gain per Cent. if I fell
it at 21d. per Ell Engliſh?
Here are feveral Questions or Demands in this one, and
therefore I confider firft, what is given, and then what is
fought, viz. the Price of a Yard is given, and the Price of a
Bale is fought; therefore I fay,
If I Yard coft 15 d. 1, What coft i Bale or 58 Pieces each.
1
4
4
4 Quarters 62 Farthings
Piece 29 E. Eng.
522
116
โ
1682 E.Eng.in
5
all.
8410 Quarters.
62
16820
50460.
d.d.
4 | 521420Pro.of 2,3
4 | 130355 Farthings
1.
12 | 32588 d.&3ov.
2|0|27115 Shil. 8d.
135 L. 15s.81.
EXPLANATION.
After ſtating or fetting down the Numbers, as above, I
multiply the Pieces by 29, the Ells English in a Piece, and the
Product 1682 are the Ells English in all; but as the firſt
Number is a Yard, I muft reduce both the Yard and the Bale
into Quarters, to make them both of one Name, viz. Quar-
ters; then, becauſe the middle or fecond Number is Pence
and Farthings, I reduce it into Farthings, and then multi-
ply the 3d Number or Bale, reduced into Quarters by it, viz.
62.9
4
The RULE of THREE.
126
7
62; and that Product 521420, I divide by the first Number
or Yard, reduced into Quarters, viz. 4, and the Quotient
130355 is the Anfwer, in the Name the fecond Number was
reduced into, viz. Farthings, which are reduced first into
Pence, then into Shillings, and then into Pounds; and the
compleat Anſwer to the firft Query, viz. the whole Coft of
the Bale is L. 135 : 15 : 83.
The next Thing to be enquired after, is the whole Mo-
ney it was fold for; to find out which, the Price of 1 Ell is
given to find the Value of the Bale; therefore, I ſay,
If 1 Ell is fold for 21 Pence, What is the Value of
1682 Ells?
21
1682
3364.
12|35322
2 | 0129413-6
Anfiver L. 147: 3:6
By this Proceſs, I find that the Bale, when fold at 21
Pence per Ell, will amount to L. 143: 3: 6. Now the
third Query is, What is my Gain per Cent. or for
laying out one hundred Pounds? To find which, I fay, If
the Money the Bale coft, yield or give the Money it is fold
for, What will L. 100 yield or give? Thus,
If L.135:15:83 give L.143: 3: 6, what will L. 100 give.
12
40
5727 Sixpences,
96000
20
20
2715 Shill.
2000
12
32588 Pence.
34362000
4
51543
24000
4
130355 Farthings. 549792000
96000
1
130355
The RULE of THREE.
127:
130355) 549792000 (4|0 (421 |7 Sixpences.
521420
105:8:6
283720
260710
3 & Quot.
arifing
L. 105: 8:9
from
230100
130355
the Re-
mainder..
997450
912485
84965 Remainder which are Parts
of a Sixpence.
6
509790
391065
118725
4
474890
N.B. The Quotient 3d.
is fet above to be ad-
ded to the L. 105:8:6.
391065
83825
Here the firft Number being compounded of L. s. d.
and Farthings, 'tis brought into Farthings, which occafions
the third Number, viz. the L.100 to be brought into Far-
things alfo, to be of the fame Name with it, although that
be not compounded; the fecond Number is only brought
or reduced into Sixpences, by multiplying the Pounds by 40,
the Sixpences in a Pound, and taking in 7, the Sixpences
in 3s. and 6 d. fo that after multiplying the fecond and
third Numbers together fo reduced, and dividing by the firſt,
the Quotient 4217, which is the Answer to the Question,
is Sixpences, which are brought into Pounds, by dividing by
40, and the Remainder 84965, is ſo many Parts of 130355
of Sixpence, for which Reafon the Remainder is multiply'd
by 6, and that Product 509790 is divided by the former Di-
vifor, and the Quotient 3, is Pence, and the Remainder is
Parts of a Penny, which is multiply'd by 4, and that Product
divided again by the former Divifor, the Quotient 3 is Far-
things,
I
128
The Ru L´E of THREE.
things, and the Remainder is Párts of a Farthing; ſo that the
Anſwer is L. 5: 89
83825
130355
and
Parts of a Farthing gain'd per Cent.
For the Stating is, If the whole Sum laid out, produce the
whole Sum fold for; which, as it comes to more, fhews the
whole Gains to be fo much as is the Difference (which in
this Example is L.7:7:9, and 46530
130355
Parts of a Farthing.】
So is the third Number, or L. 100 to 105: 8: 9 4
and
83825
Parts of a Farthing:
130355
The Difference between which, being L. 5 : 8: 9 4, &c:
is the Gain per Cent.
If I lay out L. 560 in Wheat, Malt, or other Corn, when
'tis fold for 4 s. 3 d. per Bufhel; and fell the fame again for
38 Shillings per Quarter, What do I gain per Cent?
4:3
Here the whole Quantity Bought, at 4s. 3d. per Bufbel,
is Sold at L. 1: 18 per Quarter; therefore the firſt
Query will be, to find the Coft of a Quarter at 4s. 3d.
per Bufhel. Now as 8 Bushels make a Quarter, I
multiply the Price of a Bushel by 8, and the Product
34 Shillings is the Coft of a Quarter at that Rate; 34
by which it appears that 4 Shillings is gain'd by
every Quarter Bought; therefore I fay, If 34
Shillings gain 4 Shillings, What will L. 100 gain?
After bringing the third Number L. 100
into Shillings, which makes it of the fame Name
with the first Number, viz. Shillings, I multi-
ply the fecond and third Numbers together,
and divide their Product 8000 by the first
Number; and the Quotient, which is the
Anfwer to the Queftion, is 235 Shillings ;. be-
20
2000
4
8000
caufe
The RULE of THREE.
129
caufe the fecond or middle Num-
ber was Shillings, and the Re-
mainder 10, is Ten 34 Parts
of a Shilling; for which Reaſon
I multiply the 10 by 12, the
Pence in a Shilling, and divide
the Product 120 by 34, and the
Quotient 3 is Pente, and the Re-
mainder 18, is 18 thirty four
Parts of a Penny, which I mul-
tiply by 4, the Farthings in a
Penny, and divide the Producť
72, by 34, and the Quotient 2,
is Farthings, and the Remainder
4, is four 34 Parts of a Far-
thing; fo that the true Anfwer
to the Question, that wants to
know the Gain per Cent. is
L. 11: 15:3: 2, and Parts
of a Farthing.
P
3+
Now to know what is gain'd
by laying out the whole Sum
L. 560, I ſay,
341 8000 | 2012315
68
120
102
180
170
L. II: 15
10 Parts of a Shil
I2
34) 120 (3 Pence.
102
18 Parts of a Pen.
4
72 (2 Farthings.
68
If 34 s. gain 4s. What will L. 560 gain?
4 Parts of a Far.
20
II200
4
44800
S
Or
130
The RULE of THREE.
Or thus,
Here, after dividing by
the first Number 34, the
Remainder is 22 Parts of
a Shilling, which being
wrought as before, comes
to 7 Pence 3 Farthings,
and 2 Parts of a Farthing
remains; ſo that the true
Anſwer to this Question
is L. 65: 17:7:3 &
Parts of a Farthing gain'd
by laying out L. 560 in
this Manner.
34
34) 44800 (210 | 131|7
34
108
L.65 1773 & 324
Parts of a Far.
102
60
34
260
238
22 Parts of a Shilling.
12
264 (7 Pence.
238
26 Parts of a Penny.
4
104
102
(:
3 Farthings.
2 Parts of a Farthing,
I might now go on to more curious and difficult Questions,
and alfo to the Method of Practice, or working by Con-
tractions, and breaking the given Price into Parts, &c.
but as all theſe Matters entirely depend upon underſtanding
the Doctrine of Fractions, my next Buſineſs fhall be to
teach you what a Fraction is; their diverfe Sorts, and the
beſt and ſhorteſt Methods of uſing them; by applying them
to a great Variety of Cafes, in all the Parts of Mercantile
Affairs in particular; the like of which has not (as I know
of been fo fully and plainly done by any Body elfe.
1
CHAP.
131
CHAP. VI.
Of Fractions, their Definition, various Sorts, and
Manner of using them.
Fraction is a Part of Unity or fome whole Thing,
A whether it be Money, Weight, Meaſure, Time, &c.
There are two Sorts of Fractions, the one called Vulgar (or
Natural) Fractions, the other Decimal or Artificial-Fractions;
and as the Vulgar or Natural-Fractions, are the firſt in order,
I fhall firft handle them, and then fhew the Agreement be-
tween the two Sorts; and in the Application, endeavour
to give you Hints to affift you, when to uſe the one Sort,
and when the other; becauſe each have their peculiar Ex-
cellence; and in particular Cafes, the one may be much
eaſier and more expeditious than the other.
Of Vulgar or Natural-Fractions.
A Vulgar or Natural-Fraction, is that which exactly ex-
preffes the Quantity of the Part or Parts of any whole
Thing, whether it be actually or imaginarily fubdivided
into any Number of Parts whatever. Every Vulgar-Fracti-
on is expreffed by two Numbers, fet over one another, with
a Line drawn between them, thus, or, or, the
lowermoſt whereof expreſſes the Number of Parts, the whole
Thing is really or imaginarily divided into, and is therefore
called the Denominator; and this is the fame with the Di-
vifor in Divifion, let it be a fmall Number or a large one.
The uppermoft Number is called the Numerator, from its
numbering or fhewing how many Parts of the whole Thing
is expreffed by the Fraction then in ufe; as 1 Numerator,
9 Denominator,
and is the fame with the Remainder in Divifion, where
there happens to be one, as was hinted in Page 94; for
here the whole Thing is fuppos'd to be divided into 9 Parts,
and the Number expreffes 5 of thofe 9 Parts.
Of Vulgar Fractions there are three Sorts, viz.
A proper, fimple, or pure Fraction.
An improper Fraction,
And a compound Fraction.
S 2
A
132
Reduction of FRACTIONS.
7194
65
A proper, fimple, or pure Fraction is expreffed by one
Numerator and one Denominator, and is always lefs than
Unity, becauſe the Numerator is always lefs than the Deno-.
minator, as, or, or, or 13, or 1, &c.
&c. An im-
proper Fraction is likewife expreffed by one Numerator and
one Denominator, and is greater, or at leaft equal to Unity,
having the Numerator, always either greater or equal to the
Denominator; and is the fame with the Dividend in Divifion,
as 12, or 12, or 1$, &c.
A compound Fraction is always expreffed by, at leaſt, two
Numerators and two Denominators; and fometimes by 3,
4, 5, or more, with the Word OF between each Numerator
and Denominator, which may confift of the fame Figures, or
different ones, as,
of 4, OR of ½ of §, OR 2 of 1 of 3, &c.
2
Having now defin'd the ſeveral Sorts of Vulgar Fractions,
before you can add or fubftract them, you muſt learn the
Reduction of Fractions, which teaches you to bring compound
Fractions into fingle ones; Fractions expreffed by large
Numbers, into thofe expreffed by ſmaller ones, and Frac-
tions of various Denominators into thofe expreffed by one
common Denominator: Of which in their Order
; and
firſt,
To reduce a compound Fraction to a ſingle one.
Becauſe in all Cafes whatever, before any Work can be
regularly done in Fractions, if a compound Fraction is con-
cerned, it muſt be reduced to a fingle Fraction, by bring-
ing it into a proper, or an improper one; which is done
thus:
8
40
Σ
Multiply all the Numerators together, for a new Numerator,
and all the Denominators together, for a new Denominator; and
this new Fraction thus produced, fhall be equal to the compound
Fraction given, and will fometimes turn out a fimple Fraction,
and ſometimes an improper Fraction; as in the Examples
above, where theofproduces, and the of ½ of &
gives; and the of of gives t; for in the firft
½ }
Example, multiplying 1 by 1, the Product is 1 for the new
Numerator; and 4 by 4, the Product is 16 for the new Deno-
minator; ſo that of is. In the fecond Example, multiply
the Numerators 3, 5 and 6 together continually, and the Pro-
duct is 90, for the new Numerator; then multiplying . 5, 7
and 9 together continually, and the Product is 315 for the
new
Reduction of FRACTIONS.
133
42
I
new Denominator; fo that of of become. And in
the third Example, 8, 1 and 5, multiply'd continually toge-
ther, gives 40 for a new Numerator, and 2, 2, and 3, gives
12 for a new Denominator; fo that the new Fraction is in-
ſtead of of of Tho' the Operations in the Fractions
given will always give the true Anfwers to the Questions, a-
bout which they are concern'd; yet very often the Work
may be fhortned, by reducing the given Fractions into others
of the fame Value, tho' exprefs'd by ſmaller Numbers, or lower
Terms; and therefore the next Rule fhall be,
2
To find the greatest Common-Meaſure to any two given
Numbers whatever.
Divide the greater Number by the leffer, and if any Thing
remains, make that Remainder a Divifor, and the laft Divi-
for a Dividend; and fo continue dividing the laft Divifor by
the Remainder, till o remains; and that Number that divides
or is the Divifor, when o remains, is the greateſt Number
that will divide the two given Numbers, without leaving a
Remainder. As for Inftance, Suppoſe I aſk, What Number
will bring to its lowest Terms? Or, What is the greater
Number that will divide both 340 and 380, and leave no Re-
mainder in either?
380
To effect this, I do as order'd above, thus:
340 | 380 | I
340
40 | 340 | 8
320
20 | 40 | 2
40
00
And fo find that 20 is the biggeſt Number, or greateſt
Common Meaſure to the Fraction given, or the two Numbers
that compoſe the Numerator and Denominator.
N. B. If Unity Should happen to be the laft Divifor, then
are the given Numbers already in their loweſt Terms, and
are called, or ſaid to be Prime to one another. As, What
is the greatest common Meaſure to
17 $
316
Į
175
134
Reduction of FRACTIONS.
175 | 316 (1
175
141 | 175 | I
141
34 | 141 | 4
136
53416
30
4 15 (I
4
144
4
绑
​N.B. The Quotients, be they ever fo
many in theſe Operations, are not regard-
ed.
O
Here, after five Divifions, the Remainder is 1, which
becomes the next Divifor, and confequently, muſt have
no Remainder; becauſe all Numbers whatever contain fo
many Units or I's as express themſelves; therefore are
&
3 I
Numbers prime to one another, or fuch as cannot be brought
into lower Terms.
What is the greateſt common Divifor to ?
1
104
504
188
504
588 | I
504
845046
504
00
Here it appears that 84 is the greateft common Divifor to
What
Reduction of FRACTIONS.
135
218
54
What is the greateſt common Meaſure to 2? Anf. 5¢
1998
918 | 1998 | 2
1836
162 | 918 | 5
810
108 | 162 | 1
108
54 108 | 2
108
00
What is the greateſt common Meaſure to}?!? Anf. 1751
1751 | 3502 | 2
3502
310
What is the greateſt common Meaſure to 11? Anf. 2
316 | 426 | I
316
110 | 316 | 2
220
96 | 110 | 1
96
14|9616
84
12 | 14 | 1
12
2126
12
lol
Ta
136
Reduction of FRACTIONS.
To reduce or bring Fractions to their leaft or loweſt Terms,
Still retaining the fame Value theỷ had in the given or larger
Numbers.
+
RULE.
You may find the greatest common Meaſure, as was taught
in the foregoing Rule, and divide the given Numerator and
Denominator thereby, and the Quatients fo arifing will be a
new Fraction, equal to the given one, and in its leaft e-
quivalent Terms.
4
As, What's the lowest equivalent Terms ef? Anf. 1.
Here, working as was directed in the laft Rule, I find
117 the Numerator, is exactly contained 4 Times in 468 the
Denominator, fo that o remains; by which I find 117 is the
greateſt common Meaſure to the given Fraction; and that die
viding the Numerator by 117, the Quotient is I; and alfo di-
viding the Denominator by the faid 117, the Quotient is 4,
and o remains in both Divifors; which Quotients being put
Fraction-wife, will ftand thus, ; fo that working any
Question with will produce the fame Answer, when ap-
ply'd to Coin, Measure, Weight, &c, as working with
117
462
What's the loweft equivalent Terms of 182 ?
234
12
204
Here I find the greateft common Meaſure to be 26, by
which, dividing the given Numerator and Denominator, the
Quotients arifing are 3, and o remains; fo that is reduced
to, which is the lowest equivalent Terms this given Frac-
tion can be reduc'd into. After the fame Manner you may
proceed with any other Fraction, or a Number of Fractions;
but as this is oftentimes very tedious and troubleſome, though
abfolutely true; therefore you may (if you like it better)
ufe the following
CONTRACTIONS.
1ft. If the Numerator and Denominator of the given
Fraction end with a Cypher, or Cyphers, cut off the faid
Cypher or Cyphers, if the Number is equal, from both the
Numerator and Denominator, and work with the remaining
Figures: Thus is equal to, and 2 is equal to 2, and
1000 to 2, &c.
魚
​70
980
2d. If the Numerator and Denominator given, end both
with 5, or one with 5, and the other with 0, you may di-
vide by 5. Thus is, or; fo is, and 23 is
74
300
22
40
جوع ولی
3dly. If
Reduction of FRACTIONS.
137
3
3dly, If the given Fraction ends with even Numbers, you
may halve or divide the Numerator and Denominator by 2,
and theſe Quotients again by 2, if they come out even Ñum,
bers; and fo on till one or both of the Quotients becomes an
odd Number: Thus is is is ; fo is 1 is
&
is is, &c.
#i
32
4
16
45
&
1 2
4thly, If the given Numerator and Denominator be both
odd Numbers, or the one even and the other odd, you may
try 3, 5, 7, 9, &c. always taking odd Numbers, becauſe an
even Number will not divide an odd one without leaving a
Remainder; and if any of them fucceed, you may try the
new Fraction ſo arifing, as has already been directed in the
four Articles here taught you, till you can't find an odd Num-
ber to anſwer, and your Work is done. As is by di-
viding by 3, or 4 is . And Note, When you bring either
4.
the given Numerator or Denominator into Unity, you can go
no further.
Suppofe
72
9
2+
the Fraction above given.
134,
2
1 2
91
3609
Ty
Here dividing by 3, the new Fraction is 22; this again
by 3, and it produces; and this by 13, is, as before.
So the fecond Example 4, being firft halved, 2 arifes;
this divided by 13, is, as before; and fo in all other Cafes
whatſoever, you may divide by any Numbers, even or odd,
big or little, that will exactly divide both the Numerator
and Denominator, without leaving a Remainder. As 22, is
and by dividing firft by 5, according to Note 2d ; and
thofe Quotients by 3, according to Note the 4th.
$7
So 148
being given, I divide by 8, and find it fucceed, the Quotients
being; this I try with 3, and find come out. Thus
you may do with any other Numbers or Fractions whatever.
The reducing Fractions to their loweſt Terms will be found
of great Service hereafter, in greatly fhortning the Labour
which is otherwife required, efpecially in Addition and Sub-
ſtraction, where the Fractions muft always have a common
Denominator before you can uſe them; though in Multiplica-
tion and Divifion you may work both when the given Frac-
tions have, and have not a common Denominator.
To reduce, or bring Fractions that have different Denomina-
tors, to others, having a common Denominator, ftill retaining
their original Value.
RULE.
Multiply all the given Denominators together, for a new
and common Denominator, and each Numerator into all
T
the
138
Reduction of FRACTIONS.
the Denominators, excepting its own, for fo many new Nu
merators, as bring and into equivalent Fractions, having
a common Denominator.
3
10
+2
6
will
Thus and being given, 12 and, will arife, which
are equal to the given Fractions, as may be eafily try'd; for,
'tis plain, is equal to ; for abbreviating, you
find it come to, the original Fraction; and by abbreviating
, you will have come out. So if and were given,
you would have, and come out; for by multiplying
7 and 9, the given Denominators together, 63 comes out,
for the common Denominator; and by multiplying 2, the Nu-
merator of by 9 the Denominator of, the other given
Fraction, the Product 18 is a new Numerator to the com-
mon Denominator 63, inſtead of 2 the Numerator to , and
inftead of 6 the Numerator, to the given Denominator 9, you
have 42 arifing by multiplying 6, the given Numerator, into
the contrary given Denominator.
2
Bring,,, and 1 into Fractions, having a com-
mon Denominator.
500
7
6
86
9
13
12
6
8
40
42
72
156
12
12
6
8
48
12
480
504
432
1248
15
15
15
6
576
15
2400
2520
2160
7488
480
504
432.
2880
576
7200
7560
6480
Y
8640 common Denominator.
2,
Here becomes 38, and 3 is 328, and is $48, and
11
소속 ​is 금속​무용​.
2488
86401
86
86407
N. B. When any of the given Fractions can be abbre-
1 2
viated, or expreſſed in leffer Terms than thoſe given, 'tis the
best way always fo to do. As above, might have been re-
duced into ; and then the Expreffions would have been
180 for ; 128, for; and 250 for reduced to 4, and
1426 for 13.
중국
​2880
2880
%/
12
If
Reduction of FRACTIONS.
139
If any of the other Fractions could alſo have been reduced
lower, the common Denominator would have been expreffed
in ftill leffer Terms, as in the
Bring, and, and
Example following.
22
6
to a common Denominator.
Firſt working according to the general Directions, the Mat-
ter will ſtand thus:
26168
8007408
306
171
$57008
8007408
785040
56051856
8007408
8007408
513
1369266768
24022224
8007408.
40037040..
4107800304 the common Denominator.
513
26168
306
513
3078
78504
15390
26168.
130840..
156978
6542.
13424184
153
313956
627912.
40272552
784890.
67120920.
941868...
13424184..
1026950076
2053900152
1 3 6 9 2 6 6 7 6 8
1369266168
4107300304, and 28163 is 122210074, and
Here, is 41078003049
12 is 2012
2053900152
4107800304•
But by reducing theſe given Fractions to their loweſt Terms,
they become, and, which is 2, 24 and 12. Do the
fame upon all Occafions, whenever the given Fractions will
admit it.
T 2
T
140
Reduction of FRACTIONS.
To reduce an improper Fraction to a whole or mix'd Number.
Divide the Numerator by the Denominator, and if o re-
mains, the Quotient is the whole Number, equal to the given
Fraction; but if any thing remains, the Quotient is the whole
Part, and the Remainder is the Numerator of the Frac-
tional Part, and the given Denominator is the Denominator
to this new Fraction.
What is the whole or mix'd Number equal to
3 whole Numbers.
249
What is the whole or mix'd Number to
+ 1
$13? Anf.
17 1
? Anf. 27
14, or 12, that is 27 whole Numbers, and 12 Parts of a
whole Number, which being reduc'd is 2.
Note, Upon all Occafions, where there's a fingle Fraction,
'tis best to exprefs it in its lowest Terms.
I.
To reduce a whole Number into an improper Fraction.
This Rule confifts of 2 Parts: Firſt, when there is no De-
nominator affign'd: Secondly, when there is one affign'd.
In the firft Cafe, draw a Line under the given Number,
and make it the Numerator, and put I or Unity under the
Line for a Denominator. As,
Bring 28 into an improper Fraction.
Here doing as above directed, the Work will ftand
thus, 2$.
In the fecond Cafe multiply the whole Number given,
by the Denominator given, and make that Preduct the Nu-
merator, and under it write the given Denominator; and this
Fraction will be the Anfwer required.
As,
What is the improper Fraction to 48, when 9 is the De-
nominator ?
Here multiplying 48, the whole Number given, by 9, the
Product 432 is the Numerator, and the Fraction fought, is
alz;
41; and fo proceed with all whole Numbers, whether
great or fmall.
To reduce a mix'd Number to an improper Fraction.
Multiply the integral or whole Part of the given mix'd
Number, by the Denominator of the Fractional Part, and
to that Product add the Numerator of the Fractional Part,
and this Sum will be a new Numerator to the Denominator
of the Fractional Part given; and this new Fraction will be
the improper Fraction fought, as in the following Examples.
What's
Reduction of FRACTIONS.
141
What's the improper Fraction to 26?
Here multiplying 26 by 8, the Product is 208, to which
add 5, the Sum is 213; to which put 8, the given De-
nominator, and the new Fraction is 213, the Fraction re-
quired.
To express the Proportion that any one, two, three or more
Fractions have one to another, in whole Numbers.
First, If it be required to know what Proportion the Nu-
merator bears to the Denominator of any Fraction; if it be a
Compound one, you muſt reduce it to a fingle one, and then
reduce it to its loweft Terms, and the Numbers expreffing
the Fraction fo reduced, will be the Numbers fought. But if
the Fraction given be a fingle one, whether proper or impro-
per, you have nothing to do but reduce it to its loweſt
Terms, and the new Fraction is the Anfwer fought. So if the
Question was made by a mix'd Number, firft reduce it to an
improper Fraction, in its loweſt Terms, and the Numerator
and Denominator fo reduced is the Anfwer.
Secondly, If the given Fractions have a common Deno-
minator, then the Numerators are the whole Numbers that ex-
prefs the Proportion; and if they can be reduc'd lower, you
may do it, as in the following Example.
What are the whole Numbers that express the Proportion
of 2 to 5/ ?
3
6
7
Here, rejecting the common Denominator, the Anfwer
will be 3 and 6, that is, the two given Fractions will have
the fame Proportion to one another, as the whole Numbers
3 and 6 have to one another. And as 3 and 6 may both be
divided by 3, the Quotients I and, will fhew the Propor-
tion to be as I is to 2, or what is called Sub-duple.
Thirdly, If the given Fractions have not a common De-
nominator, then reduce them to Fractions that have a com-
mon Denominator, and work as above. As,
What Proportion has to?
Here I find the two given Fractions, when reduced to a
common Denominator, will be 1, and; fo that their
Proportion is as 10 is to 3, which are Numbers prime to one
another; becauſe they cannot be reduced any lower, or have
no other common Divifor but Unity.
9
What Proportion is between of 3 of 2/2 ?
Anfwer. The Numerator is to the Denominator, as 5 to
13; for having redue'd the compound Fraction to a fimple
one, it comes to 234; which being reduced to its loweſt
Terms
20
I
142 Reduction of FRACTIONS:
Terms is; ſo that the whole Numbers 5 and 13,
are the whole Numbers that exprefs their true Proportion in
the lowest Terms.
Express the Proportion of
Numbers.
Here abbreviating
tion is as 4 to 9.
18
18
in its lowest Terms, in whole
it comes to ; fo that this Frac-
What's the Proportion of in whole Numbers?
Z 9
This being a Fraction, whofe Numerator is prime to its
Denominator, the Anſwer in whole Numbers is as 7 to 9.
Express in whole Numbers the Proportion of 22 to 32.
3 2
16
Firft reducing each Fraction to its loweſt Terms, I find
comes to, and ½ to 12; and theſe new Fractons being
죽을
​reduced to a common Denominator, they are 48, and 13.
Here rejecting, or throwing away the common Denomi-
nator, the Numerators not admitting of a common Diviſor,
or of being reduc'd to lower Terms, the faid Numerators
are the Numbers fought, viz. the given Fractions are in Pro-
portion to one another, as 16 to 39. After the fame Man-
ner you may do with 3, 4, 5, &c. given Fractions.
To find the Value of any given Fractions in the known Parts
of Coin, Weight, Meaſure, &c.
Multiply the Numerator by the Number of Parts in the
next inferior Denomination, and divide that Product by the De-
nominator, and the Quotient is the Anfwer; and if any thing
remains, multiply that Remainder by the Number of known
Parts in the next inferior Denomination to that laft found
and fo continue multiplying and dividing, till either nothing
remains, or you have gone through all the Subdivifions of the
Coin, Weight, Meaſure, &c. mentioned in your Question: As
in the following Examples will fully appear.
What's the Value of 3 of a Guinea ?
3
21 Shillings in a Guinea,
4163 Product.
15 Shillings and 3 remains.
12 Pence is a Shilling.
;
S.
d.
Anf. 15 9
4136
9 Pence and o remains.
What's
Reduction of FRACTIONS.
143
What's the Value of 3 of a L. Sterling?
7
20 Shill, is a L.Ster. 12
4 remains.
8140
8148
17 Shillings and 6 Pence.
What's the Value of of a L. Sterling?
5
20
91100
II Shillings and
I remains.
12
9|12
s. d.
Anf. 17 6
I Penny and 3 remains.
4
s. d. f.
912
Anſ. 11:9: 1 and of a Far.
1 Far, and or
What is the Value of 13 of a Day?
13
24 Hours in a Day.
52
26
17 | 312 | 18 Hours.
17
142
136
6 Parts of an Hour.
6 Parts
144
Reduction of FRACTIONS.
>
6 Parts of an Hour.
60 Minutes in an Hour.
17 | 360
(21
21 Minutes.
34
20
17
3 (Parts of a Minute.
60
180 (10 Seconds.
17
10 Parts of a Second.
Anfwer, 18 Hours, 21 Minutes, 10 Seconds, and 1 of a
Second.
What's the Value of 12 of a Troy?
131
12 Ounces in a Troy.
216 | 1562 | 7 Ounces.
1512
50 Parts of an Ounce.
20 Dwts in an Ounce.
1000 | 4 Penny Weights.
864
136 Parts of a Penny Weight.
24 Grains in a Penny Weight.
544
372
216 | 4264 | 19
216
2104
10
17
1944
160 Parts of a Grain, or 22 of a Grain.
216
So
Addition of FRACTIONS.
145
So let the Coin, Weight, or Meaſure, be what it will,
the Law of Procedure is always the fame.
To reduce or turn Fractions or whole Numbers of one De-
nomination into another of a greater or leffer Value, ftill re-
taining their true Value.
If the given Fractions or whole Numbers, are to be chang'd
into Parts of a higher Species, or Name, you muſt put thoſe
given firft, and the feveral afcending Parts of the Species re-
quired, following, in the Form of a compound Fraction, mak-
ing Unity the Numerator to every one of the Subdivifions,
and the Number of the Subdivifion the Denominator. As,
What Part of a Pound Sterling is 3 Farthings? Here the
given Number, of itfelf, is a whole Number, viz. 3 Far-
things; the next Subdivifion upwards is a Penny. Now 4
Farthings making a Penny, confequently the given whole
Number, is of a Penny, and a Penny is the of a Shil-
ling, and a Shilling is the 。 of a Pound Sterling: So that it
muſt be expreſs'd thus: of 12 of, which is equal to
90 or fo that 3 Farthings is the of a Pound Ster-
ling. Obferve the fame Method, let the Names be what they
will.
3
32
4
2
20
320
I 2
Secondly, If the given Fractions or whole Numbers, are to
be chang'd into the Species of the leffer Name, you muſt
make the Number of the defcending Species, the Numera-
tor, and Unity the Denominator. As,
What Part of a
Farthing is of a Pound Sterling? of
1920
22 of 12 of 4. Aní. 122 or 640 Farthings.
What Part of a
32
Pound Sterling is
2 1
147
3 of is 1.
20 160
What Part of a Guinea is
140
of a Guinea?
of a Pound Sterling?
ofis 12 or .
165
Addition of FRACTIONS.
If the given Fractions have not a common Denominator,
they muſt be reduced to Fractions that have a common De-
nominator; or if they relate to any particular Species, of
Money, Weight, Meaſure, &c. they muſt be reduced to all
of one Name; and then to Fractions of a common Denomi-
nator; then add the Numerators together, and to their Sum
fubfcribe the common Denominator, and that Fraction is the
Anfwer, which if an improper one, you muſt reduce it to a
whole or mix'd Number, as formerly directed.
U
EX-
146
Addition of FRACTIONS.
EXAMPLES.
What is the Sum of and? Anſwer.
7
What is the Sum of and? Anfwer or I and,
or 1.
What is the Sum of and? Anſwer, or 1 ¿.
What is the Sum of 3 and 3, and { ?
2
Here may be reduced to, which added to, makes
or I; fo that the Anſwer is I and §.
What is the Sum of,, and {?
똑
​Note, Always reduce any or all of the given Fractions to
their loweſt Terms, before you go about to reduce them to a com-
mon Denominator.
As in this Example: I reduce to 2, and then I find the
common Denominator to be 1080, and the ſeveral Numerators,
600, 810, 864 and 900, whofe Sum is 13, which being
reduced, the Anfwer is 21, or 2 182.
169
What is the Sum of of of and ½ of ½ of } ?
6
16
7
3
4
Here the compound Fractions being reduc'd to fimple
ones, they become, and 2, or and ; which reduced
to a common Denominator, is 180 and 126, whofe Sum is 205
or 2.
་་
63
567
1679
Sometimes the Labour of reducing may be fhortned by
multiplying one of the given Fractions, by a Number that ſhall
make its Denominator equal to the other, as above, be-
ing multiply'd by 7, becomes, which added to the other
3 comes to 3, the fame with the common Method, but
much fhorter. You may obſerve the like in all Caſes that
will admit of it.
34
639
What is the Sum of of a Guinea, and of a L. Ster-
ling?
Here you may firft make the 2 Fractions both Parts of
a Guinea, or of a Pound, as you beft like: See it done
both Ways: of a Guinea is of 24 of a Pound, which is
192 or 2 of a L. which added to is 12, or L. 1 12
20
3
I
or
J
equal
Subftraction of FRACTIONS.
147
4
23
좋은
​equal to L. 15: 0. The other Way of a L. is of 21
or of a Guinea, which added to of a Guinea, makes a
Guinea, or 12, or 4. Now a Guinea is L. 1: I
I
63
and is
14
21
as before.
639
4
L. 1:5:0 Total
What is the Sum of 3 of an Ell Engliſh, and 3 of an
Ell Flemiſh?
Here
which is
36
of an Ell Engliſh is of of an Ell Flemish,
Ell Flemish; this added to
or I
9
}
60
20
6
is, or
13 Ell Flemiſh; but if you bring the of an Ell Flemish
into Parts of an Ell Engliſh, twill be of which is 2; this
14
added to is, and 2, or 3, which is 1 and Ell Eng-
lifh. Now I Ell Flemish, is 5 Quarters of a Yard; and
and I of an Ell Flemish is the fame, which fhews both
Anſwers to be the fame. You muſt go on after the ſame
Manner, let the Name or Denomination be what it will.
1,
I 2
60
3
6
Subftraction of FRACTIONS.
Here the fame preparative Work muſt be had, as in the
laſt Rule, viz. if the given Fractions have not a common
Denominator, they must be reduced to fuch as have, whether
they are fimple, compound or improper Fractions; then taking
or fubftracting the Numerator out of the Denominator, the
Remainder is the Anſwer fought.
As, What is the Difference between 3 and? Answer, or 4.
What is the Difference between and? Anfwer 2.
When reduced they are , and 3, whofe Difference is
or 14.
½
12
36
Subſtract of from, and tell the Difference.
Thefe reduced, are and, whofe Difference is 7.
From & take 3. Reduced they are 3, and 4, Difference is 1
From 5 take.
8
+3
Here being but one Fraction, fubtract the Numerator out
of the Denominator, and abate one or Unity from the whole
Number, and the Remainder is the Anfwer. As here 4 is
the Remainder or Difference.
Subſtract 6 out of 8.
2
Here the Fractions, being of a common Denominator, Ifay
2 out of 1 I cannot, but I borrow Unity, which in this In-
ftance is 5, becauſe the common Denominator is 5 ; and fay,
2 out of 6, the Sum of Unity, and the Numerator of the
Number given, from which the Subftraction is to be made,
U 2
and
148
Multiplication of FRACTIONS.
and the Remainder is 4; then I fay, 6 out of 7, becauſe I
was borrowed from the 8 to fupply the Fraction that was
deficient; fo that the Anfwer is I and .
# 、
4
Or you might have reduced both the given Fractions to
improper Fractions, and have fubftracted the leffer from the
greater, and the Remainder would have been or 1 ; for
the Fractions fo reduced will be 4, and 2, whofe Diffe-
rence is or I, as above. Do the fame in all like Cafes.
2
Take 165 out of 318 .
into a
7
Here I reduce the Fractional Parts of the given Number
common Denominator, and then they will ftand
318 1/1/20
165 1/1/00
thus:
Whole Difference is
Subſtract of a Moidore out of
Piece worth 42 Shillings, and tell
of a Moidore is of 2 of a
7
9
24, or ; this out of
29
49
I
14
2 I
1532
of a double Guinea, or
the Difference.
double Guinea, which is
leaves of a double Guinea for
11
2 7
42
7
the Difference, which is equal to 10 s. 1d. 2 f. 9.
What's the Difference between
Day?
24
ř
2
3
of an Hour, and of a
of a Day is of 2 of an Hour, equal to 8 Hours;
ſo that the Answer is 7 Hours, and of an Hour.
Take L. 2: 17: 6 out of 5 Guineas, and.
Here I find the Value of the 5 Guineas and to be
L. 59:2 21, from which
taking 27: 6
the Difference or Anſwer
is 2: I: 8: 3
Multiplication of FRACTIONS.
Here, if any or all of the given Fractions are compound-
ed, they muſt be brought into fimple Fractions; or if they
are whole or mix'd Numbers, they must be made improper
Fractions; and then multiply the Numerators by the Numera-
tors, for a new Numerator, and the Denominators together
for a new Denominator; and the new Fraction fo produced
is the Anſwer fought.
What's the Product of multiply'd by ? Anſwer
What's the Product of of of
6
3 12/
17
1 1
28.
multiply'd by 8?
Here
}
Multiplication of FRACTIONS.
149
228
Here the compound Fraction reduced, is 3, and the
mix'd Number is 2; multiply'd together, they make 122 4
or 2,3,.
7
65249
Here may very properly come in the famous Queſtions of
multiplying Money, Weight, Meafure, &c. by Money, Weight,
Meafure, &c. For though, properly ſpeaking, no ſuch thing
is poffible, in the Nature of Numbers, all Multiplication being
only a Repetition of one Number or Thing, a certain Num
ber of Times; yet as many Smatterers in this Art, imagine
they are capable of doing ftrange Things, when they are
only able to do fuch-like Queftions mechanically, without
knowing the Reasons or Demonftrations of the Process, I
will here fhew you how you may do any thing of that
Nature, and upon what Foundations fuch Operations are built.
Firſt, You are to confider what Species of Money, Weight,
Meafure, &c. you will make the Integer. As, fuppofe I fay,
What's the Product of 3 Quarters by 3 Quarters in Cloth
Meafure? or 5 Yards by 5 Yards in Long Meafure? or 2
Pounds by 5 Pounds? or 3s. 4 d. by 6 s. 8 d. in English or
Sterling Money, &c. Here, where there is only one Name
mention'd, you need only confider them as plain or whole
Numbers: As in the firſt Inſtance of 3 Quarters by 3 Quar-
ters in Cloth-Meaſure, the English Yard being the common
Standard, I fay the Product is 9 Quarters: That is, Suppofe
a Piece of Cloth 3 Quarters wide, and 3 Quarters long, you
might cut 9 Pieces out of it, each one Quarter broad, and
one Quarter long; but though the fame Method is us'd in
Money, yet the Anfwer cannot really be the fame; for if I
was to fay, multiply 3 Shillings by 3 Shillings, the Anfwer
would be called 9 Shillings; the Anfwer is fallacious, becaufe
the Queſtion is fo; for 'tis really the Number 3, applied to
that Species of Money called Shillings, repeated 3 Times, and
to the Product the common Denomination is applied, becauſe
as there are 3 Units in the Multiplicand, ſo there are 3
Units in the Multiplier, which being collected by Addition or
Multiplication into one Sum, produces 9 Units, to which you
apply fome common Name of Money, Weight, Measure, &c.
But if you confider the Quarters as Parts of a Yard, or the
Shillings as Parts of a Pound, they will ftand thus, by
equal to in the first Inftance; or 20 by 2 equal to 40, in
the fecond Inftance; that is, fuch a Piece of Cloth would
be of another Piece of Cloth, whofe Length and Breadth
was each one Yard, or ſuch a Sum of Money would be 430
of twenty Shillings, repeated 20 Times. The fame is to be ob-
9
9
2
3
3.
+
9
4009
3
+
2
ferv'd
150
Multiplication of FRACTION s.
ferv'd of all the Species of applicate Numbers whatever,
whether Time, Weight, &c. As ſuppoſe I aſk, What is the
Product of 3 s. 4d. multiply'd by 6s. 8d? Here you muſt
confider whether you will make a Shilling or a Pound the
Integer. To make the Matter plain, I will do it both Ways;
and firft, by confidering a Shilling, as the Integer, 3 s. 4 d.
will be 3s. or 4 s. and 6 s. 8 d. will be 6s. 3, or 40s. whoſe
Product is 20° s. or 22 s., which may be called 22 s.
and; but if you make a Pound the Integer, 3s. 4d, will be
and 6 s. 8d. will be of a Pound, whofe Product is of
a Pound, or 20 Shillings repeated 20 Times, or of 400,
as will plainly appear by making it a compound Fraction:
Thus of toe is 402 equal to 223, as before.
So if you
was afk'd, What was the Product of L. 3, 4, 9 multiply'd by
L. 7, 8, 6? Here I firft turn L. 3 49 into a mix'd
Number, and it is 3; then L.7: 8: 6 is 74; thefe
7:8:6
reduc'd to improper Fractions are 222, and 227
S
18
40.
297
80
259
40
2673
3200
1485.
594..
76923 new Numerator.
3200 new Denominator.
Here the Anfwer is L.24 312, or, L. 24:--: 093220, or
2 of a Penny.
40
You have been formerly taught to multiply Money,
Weights, Meaſures, &c. by whole Numbers.
Now you
fhall fee the Reafon and Demonftration thereof: In the
Example above, where 'tis plain that the Name or Species of
Coin, called a Pound or twenty Shillings, is the Integer, and
the 4s. 9 d. is, are Parts of that Integer, as appears above by
making it a mix'd Number, confifting of 3 whole Numbers
or Integers, and Parts of an Integer; and the L.7:8:6
alfo, are 7 Integers, and 13 Parts of an Integer. Confequent-
ly, either of thefe multiply'd by each other, muft neceffarily
12
४०
40
produce
Multiplication of FRACTION s.
151
produce the fame Amount: As here,
firſt by
742 multiply'd
40
3
The Product is
22 11
19
17
40
Then 4 s. is of a Pound; and as multiplying any Number,
whether whole or mix'd, by a Fraction, whofe Numerator is
Unity, is the fame Thing as dividing by the Denominator,
confequently 7 43 multiply'd by, the Product muſt be
I 18, and of; then 6 d. is L. and the Product ari-
fing from multiplying 7 by is 4 and 13 of 1。, or
103; then 3 d. is the of a Pound, or the of 6 d. So
that multiplying 7 3 by, is the fame Thing as taking
theof, and 3, both coming to and 322. Now
collecting theſe ſeveral Products together, they will ftand
thus :
16
22 40
19
40
40
40
3
I more 23。 equal to
40
1
40
3
40
200
17
more 163 equal to
ΣΤ
more 32 equal to
and
32
3200
34
3200
17
3200
0123
3200
$7
3200
L. 24
40
7 1995
More neatly thus: 7
2245
I 18 more
7
40
more more
3
42 more more more
Σ
24 more more more Total.
Now the 24 are fo many whole Numbers or Pounds, and
the feveral Fractions are to be thus read;
4 of a Pound, equal to
£2
3207
3.3
more
of。 of a Pound, or
200
equal to 3200
more of 2
&
more
of a Pound, or 160% equal to
3208
12
of is
of a Pound, or
3200
50
Total 12}
3200
which
152 Multiplication of FRACTIONS.
3200
which 3222 brought into the inferior Denominations of a
Pound, comes out 9 Pence, and 3230 or
which is of a Farthing.
10
of a Penny,
I will do the fame Example again, and change the Mul-
tiplier, thus:
L. 3: 4 : 9, or
by 7: 8 8 : 6 is
Note the Mark+ftands for the
Word more, and
4
for the Words
4
equal to ; fo 21, is more ;
but here this is not of Unity,
but of, being the Remainder
after the Divifion by 5.
±
3 multiply'd.
22
скраброста
3/0 +10 +1D) o
80
243/。
+++
Hrtir alr
+0+ s
5/5/3/3/2
Note, In this Example you fee L. 3 4 9 made 3
: 49
becauſe the 3 Pounds are confidered as fo many Units; and
there are 19 Three-pences in 4s. 9d. and 80 Three-pences in
a Pound; and in the former Example L. 78: 6 was
turn'd into 7 43; becauſe the 7 Pounds are confider'd as
7 Units, and the 8 s. 6d. are 17 Six-pences, of which 40
are a Pound. In multiplying this laft, you fay thus: 7 Times
19 is 133, which are ſo many 80ths of a Pound, which re-
duced is 1 Pound £2 then 7 Times 3 is 21, and I is 22
Integers or whole Numbers; fo that the first Product is 22 22
as above; then 8 Shillings being no exact aliquot Part of a
Pound, I take 4s. which is, and dividing by 5, I have
+of。, as above; then I fet the fame down again for the
other 4 s. then I fay, 6 d. is the Part of 4 s. and divide
that Product by 8, and there comes out more of
more, of of of a Pound.
3
So
•
6
3
I
Then adding the feveral Fractions together, as they ſtand
one under another, and the Total is 24 ++, that
is, 24 Integers or Pounds, of a Found,
ول
And ½ of ¦ of §. of a Pound, that is, 320%
383
Now of a Pound is equal to
as before.
&
So that the Total is 24 Pounds and
I
3200
123
320›
Note, All the Fractions beyond the firft, or that next the
whole Numbers, are compounded, and the best Way to read
them is backwards, thus: Begin at the Fraction next the
Right-hand, and read towards the Left-hand; fo in the firſt
Part
Multiplication of FRACTIONS.
153
Part of this Example, you may read the Total thus, of
of of
equal to
then of of, equal to too, or
then of equal to 20, equal to
40
2
2009
3200
322 0205
322 323 323235
then is equal to
12
32
and the Total is
123
3200
And the fecond Part of this Example is read in the
fame Manner, as you fee above.
Or you may do thus:
L. 786 Multiplicand.
L. 3: 49 Multiplier.
L. 22: 5: 6 the Prod. of L.7: 8: 6 multip. by
f.
L.
L.
L. 3
1:9: 8 : 1+3
ditto by 4 s.
0:38:2 + 3/
0: 1:10: 1+0+
ditto by 6 d.
ditto by 3 d.
L. 24:
90++1 or 2 of a Farthing
Or thus:
3: 4
4 9 Multiplicand.
86 Multiplier.
7
22
24:
13 3 the Prod. of L. 3: 49 mult. by
f.
12: II: I dit.
12: 11: 1+ 3 dit.
L. 1
by 4 s.
by 4
I 1: 7:1++ or dit. by 6 d.
90+ ÷ + as before.
X
OL
154
Multiplication of FRACTION S.
Or you may do thus:
L. 7+ + 1/2
++,
9
By 3+ + ½
10
1 2
22 + 10 + 1/2
이
​9
20
8
1 + ½ + ½ + 2 or 33
12
25
20 +
6
1/2 + 1600
66
So
24 + 0 + 12 + 38.
Which are thus explained in the firft Example of the laft
three Methods. I multiply the 6 Pence by 3, faying, 3
Times 6 Pence is 18 Pence, which is 1 Shilling and 6 Pence;
fet down 6, and carry 1; then 3 Times 8 is 24 and 1 is
25. Now 25 Shillings is 1 Pound 5 Shillings; fet down 5
and carry I; then 3 Times 7 is 21, and I carried is 22
Pounds; fo that the whole Product is 22 Pounds, 5 Shillings,
and 6 Pence; but now I am to multiply by 4 Shillings, which
being but the Part of a Pound, I divide the 7 Pounds by 5,
and there remains 2, which is 2 Pounds, or 40 Shillings, and 8
in the Multiplicand make 48; this divided by 5, gives 9 and
3 remains, which are 3 Shillings or 36 Pence; and the 6 Pence
in the Multiplicand being added, make 42; this divided by
5 gives 8, and 2 Pence remains, which are 8 Farthings; this
divided by 5 gives 1, and of a Farthing remains; 9 Pence
being no exact aliquot Part of a Pound, or of the 4 Shillings,
I break it into 2 Parts, viz. 6 Pence and 3 Pence. Now
6 Pence is the 。 of a Pound, or the of 4 Shillings; and
the being eafieft to work with, I divide the Line
L. I : 9 : 8 1, which belongs to the 4 Shillings, by 8;
thus the 8's in 1 Pound is o; then I Pound being 20 Shil-
lings, this added to the 9 makes 29 Shillings; and the 8's in
29 are 3 Times, and 5 over, which are 5 Shillings or 60
Pence, and the 8 Pence being added makes 68; this di-
vided by 8 gives 8 Pence, and 4 Pence remains; which be-
ing 16 Farthings, and the I added makes 17, the 8's in
which is 2 and 1 Farthing left. Now the next Subdivifion
in that Line is; fo that the Farthing is 5 Parts, and which
added to the 3, makes of a Farthing; this divided by 8
gives exactly of a Farthing: The whole Product or Quo-
tient
Multiplication of FRACTIONS.
155
ठ
IO
tient is 3 s. 8 d. 2 f. for the 6 Pence. Now this multiply'd
by 3d. which is the of 6 Pence, produces Is. 10 d. 1 f.
and Part of a Farthing; for I fay, the Half of 3 Shillings
is 1 Shilling, and I Shilling over, which is 12 Pence, and 8 is
20 Pence, whofe Half is 10 Pence; then the Half of 2 Far-
things is 1 Farthing, and the ofis ; fo that the feve-
ral Products being collected into one Sum make
: 9 : 0 + + +, that is 24 Pounds and q Pence
more of a Farthing, and of a 5, which made into one
Sum is of a Farthing. The like you may obferve of the
two under Methods of working the fame Question.
L. 24:
9
이
​2
IG
9
Thefe Sort of Operations are us'd in nothing fo much
as in what is commonly called Cross Multiplication, or that
by which Painters, Plaifterers, Joiners, &c. caft up their
Work; and tho' many can perform the mechanick Part of
fuch Operations, yet very few know the true Reaſon of fuch
their Work; therefore you fhall have it fully explained
here: And firſt the common or vulgar Method appears by
what follows.
Feet Inches.
What is the Product of 9: 6 multiply'd
by
Here you first multiply
the Feet by the Feet, that is,
the 9 by the 6, and the
Product is 54 Feet; then
you multiply the Inches by
the Feet, that is, the 6 by
the 6, and the Product is
6: 5? Anſ. 60 Feet 11 Inc. A
54
3:
39
: 2
60: 11 /
36 Inches, which divided by 12, gives 3 Feet; then you
multiply the 9 Feet by the 5 Inches, and the Product 45 di-
vide by 12, and the Quotient is 3 Feet 9 Inches; then you
multiply the 6 Inches by the 5 Inches, and the Product 30
divide by 12, the Quotient is 2 Inches and
or; theſe
added together make 60 Feet 11 Inches ;
•
12)
X 2
Or
156 Multiplication of FRACTIONS.
Or thus,
Feet Inches.
Feet
9: 6 is the fame with 9
9 5 is the fame with 6
which is 12
1 2
22
which is
2
11
19
12
77
IZ
2
I
24 | 1463 | 60 22
144
23
23
693
24
|
Here ſee the Product is 60 Feet 2
you
and above 'tis 60 Feet 11
Inches,
2+
24
77
1463 Numerator.
24 Denominator.
Parts of a Foot;
which is
Inch, which is of 2, or of a Foot:
is equal to 2 of a Foot, which added to
22
Now
2
and half an
of a Foot
12
4 makes 4.
24
Note, In Meaſuring, as well as Coins, &c. you cannot
properly multiply Feet and Inches together; but you are to
confider the above Question in this Manner, viz. either as
a Piece of Wainscoting, Glazing, Pavement, &c. 9 Feet and
6 Inches, or half a Foot, long, and one Foot broad; and this
Breadth is repeated 5 Times, and, of a Time: That is,
fuppoſe it Glazing Work, which you may imagine to be
any Number of Panes to make up the whole Height of 9
Feet and 6 Inches, or a Foot, and every Pane one Foot
broad; then there would be 6 fuch Panes, and a narrow
one of 5 Inches broad to make up the whole Breadth; fo
that there would be in the whole Window as much Glafs as
comes to 60 Square Feet, and 2 of a Foot, to be paid for;
though the Number of Panes be more or fewer, according to
their particular Heights and Breadths. The fame may be faid
of a Piece of Pavement laid with Marble-fquares, or of Wainf-
coting, &c.
2
The last Example may be done thus:
6 Z
9/2/20
57:
3 1/2 + 1/
3: 十三
​60+
2
=
60
Here
Multiplication of FRACTION s. 157
1 2
Here I do as before in the Money Queftion, and firft
multiply 5, the Numerator, by 9, and divide by 12, and t
Quotient is 3 and; then 9 Times 6 is 54, and 3 is 573
fo that the whole Product is 57 + 2; then I divide 6 by
1/2
Σ
2 the Denominator of, and 32+ comes out for the
Quotient of 62, divided by 2, is the fame with the Pro-
duct of 6 multiply'd by, the Product of every Multiplica-
tion being the Multiplicand repeated, fo many Times as there
are Units in the Multiplier; but every proper or fimple Fraction
being less than Unity, confequently the Product arifing from
fuch Multiplication, muft bear the fame Proportion to the
Multiplicand, as the Numerator of fuch Multiplier bears to
the Denominator, which in this Inftance happens to be, as I to
2, that is, the Multiplicand is fuppofed to be multiply'd by 1,
which never makes any Alteration (every Number, whether
whole or mix'd, containing itſelf once) and divided by 2; for
every whole Number is fuppofed to have Unity for its Deno-
minator; and then by the general Rule the Numerators
are to be multiply'd together for a new Numerator, and the
Denominators for a new Denominator; and if an improper
Fraction arifes, you have already been taught to divide
the Numerator by the Denominator. As, fuppofe I fay, Mul-
tiply 55 by; here 'twill be 2 and 11; fo that
in all Cafes where Unity is the Numerator, you may divide
the given whole, mix'd, or Fractional Number by the Deno-
minator, and your Work is done. From what has been faid,
plainly appears the Reaſon of dividing by 12, 20, or any o-
ther Number, your Subdivifions of Coin, Weight, Meaſure,
&c. confift of: As, Suppoſe I ſay, What is the Product of
8 Feet II Inches multiply'd by 11 Feet? Anfwer 98 Feet
12; for the Question being put down according to Art,
'twill ſtand thus:
1
Feet.
I 2
11
2
+ Then multiplying 11, the Nu-
merator of the proper Fraction
, by 11, the Numerator of the
2+2 improper Fraction, the Pro-
12
duct is 121, or 10; then f
by is or 88; to which add or carry 10, and the
whole Product is 982; to which add the common Ap-
pellation or Name, and they are 98 Feet and of a Foot;
which you are to note, are fuch Feet and Parts, as the Mul-
tiplicand ftand for or reprefent; which in the Cafe of
meaſuring all flat Works, fuch as Painting, Paving, Wainf
coting, Plaistering, Carpets, Hangings for Rooms, Tapestry,
&c. is fuppofed to be Yards, Ells, or Feet Square, that is,
a
158
Divifion of FRACTIONS.
4
a Yard, Ell or Foot long and broad. I fhall now conclude
this Head, fuppofing myſelf to have faid enough to fatisfy
the most diffident Perfon.
Divifion of FRACTIONS.
Here, as in Multiplication, you muſt prepare your given
Fractions, by reducing whole or mix'd Numbers into im-
proper Fractions, and compound Fractions into fimple ones,
&c.
Then multiply the Numerator of the Divifor into the De-
nominator of the Dividend, for a new Denominator; and the
Denominator of the Divifor into the Numerator of the Divi-
dend, for a new Numerator; which new Fraction is the An-
fwer fought.
Divide by, and the Quotient is 2, or I and. De-
monftration. To divide any Number or Quantity by another is
the fame Thing, as to afk how many Times that Number
or Quantity is contain'd in the other, or what is the Propor-
tion between the two: As here, is contained 1 in ; or
is to, as 8 is to 9.
3
Note, In the Quotient or new Fraction arifing by this Work,
the Denominator reprefents the Divifor, and the Numerator the
Dividend of the given Fraction.
Divide of of
Quotient is $16, or 3.
vidend as 77 is to 90.
7209
by 2 of 11 of 1, or by, the
of, 16
That is, the Divifor is to the Di-
Z
3
א
2
J.
é
Note, You may abbreviate a compound Fraction thus: If
any Number whatever will divide the Numerater of one of
the Parts, and the Denominator of its correfponding Parts, or
any other Denominator; you may do it, and work with the
remaining Figures in the fame Manner, as if no fuch Re-
duction was or could be done. As above, of of reduced
to a ſingle Fraction, according to the General Rule is 12;
this reduced to its loweft Terms is; but if you reduce to
1, which is in the fame Proportion, 'twill be of of,
which is; or you may divide the Denominator of and the
Numerator of by 2, and they will be and; then the Sum
&
will ftand thus, of of, which comes to %, as before,
for the Dividend; and of of by the General Rule, will
be 22, or 1, when reduced to its loweſt Terms, as is done a-
bove. Now of of may be reduced too of 1 of
by dividing the Numerator 9, and the Denominator 12 by 3;
and this again will become of of, by dividing the Nu-
merator of 2, and the Denominator of by 3; for their
the given Fractions will become of of, which re-
duced
9
ΙΟ
I 2
୨
ΤΟ
I 2
I
+
1
9
4
(1
Divifion of FRACTIONS.
159
In
I
10
I 2
10
I 2
&
1
duced to a fingle Fraction, is, as above: Or you may re-
duce the given Fraction of of by dividing the Nume-
rator of, and the Denominator of by 3, and then they
will be of of. Now the Numerator of may a-
gain be divided by 3, and it will become, and the
Denominator of will become ; fo that the Fraction
is now of of, as before; which reduced is for
the Divifor; and thus you may do upon all Occafions
where compound Fractions are concerned; for 'tis much
better to have the Fractions always in their loweft Terms,
tho' the Anſwer will be the fame whether they are or no;
but the Proportion between the Numerator and Denominator
of the Anfwer, cannot be fo well, nor fo eafily determined,
when they are in high Numbers.
6
Divide by 11, and the Quotient is & or 13.
That is,
7
is
of 1, which is equal to 1, which re-
duced to its loweft Terms, is, the Dividend given.
Σ
Divide 8 by 7, and the Quotient is 12:
For 8 reduced is
ftand thus, ) (
7
2, and 7 is; thefe difpofed rightly
equal to 1 24.
26
26
Or you may do thus :
Divide 8 by 7, by faying, the 7's in 8 is 1, and I re-
mains, which is, and thefe added to is; the 7's in
is o Times, and of remains equal to 1, as before.
Divide 17 of
2
}
by 216 of 12
216 1/2/2
16
17
15
93
1303
216
17.
Quotient
263
3463 | 263
(
2608
15
3463
15
16 | 15 51945
163
16
17315
3463
978
163
51945
2608
Note,
160 The Rule of Three in FRACTIONS.
Note, When you have prepared your Fractions, and fet
them in order for Divifion, if either, or both the Divi-
dend and Divifor can be reduced lower, 'twill be proper to
do it, that the Quotient may be in its loweft Terms; but if
that can't be done, then you may try if the two Numera-
tors, or two Denominators of the Divifor and Dividend can
be divided by any common Meaſure; and if they can do it,
as in the Example above, where is divided by, there
the Fractions that compofe the Dividend and Divifor are
already in their loweſt Terms, and therefore nothing can be
done with them, confider'd fingly; but when they are placed
in due Order for Divifion, they will ftand thus, 14 | (. Here
I look, and find that nothing can be done with the Nume-
rators, no common Divifor but Unity being to be found be-
tween 13 and 6; then I try the Denominators, and find 7
is a common Divifor to 7 and 14; therefore I divide them
by 7, and then they will ftand thus, (Where working,
according to the General Rule, or I comes out. The
fame may be done when there are any other Figures or Num-
bers concern'd, whether they be pure Fractions or whole
or mix'd Numbers. As,
Divide 18 by 9.
* S
I may be
I
I
13
2
I }
21
J
3
Here may be made; and then
; and this gives 22 or 27 for the
Quotient, the fame as if you had work'd with the given
Numbers unreduced, and then have reduced the Quotient; as,
348 (262 or 27; and fo in any other fimilar
Cafe.
18
The RULE of THREE in FRACTIONS.
After the Question is ftated as formerly directed, Page
121, you muſt prepare your feveral Numbers, as you have
been taught in Multiplication and Diviſion of Fractions, viz.
If any of the given Numbers are whole or mix'd Numbers,
they must be reduc'd to improper Fractions; or if they are com-
pound Fractions, they must be brought to fingle ones.
Then multiply the Numerator of the firft Number con-
tinually into the Denominators of the 2d and 3d Numbers, and
the laft Product must be the Denominator to the fourth Number
fought, which is the Anfwer to the Question, in the fame
Species the fecond Number was propos'd in, or reduc'd to;
this done multiply the Denominator of the firft Number con-
tinually into the Numerators of the 2d and 3d Num-
bers,
The Rule of Three in FRACTIONS.
161
bers, and the laſt Product muſt be the Numerator to the fourth
Number, which new Fraction is the Anf. to the Question fought.
EXAMPLE.
If Yard of Cambrick coft 5s. What will a Piece comé
to containing 11 Yards?
Here, obferving the Directions above, the Question when
ftated, will ftand thus:
As Yd. is to L. fo is 22 Yd. to L. 3, or L. 3: 16: 8.
Note, If you can divide the Numerator or Denominator of the
firſt Number, and the Numerators or Denominators of either the
fecond or third Numbers, by any common Diviſor, 'tis beſt to
dofo; or you made divide the Numerator of the ſecond or third
Numbers, and the Denominator of the other, and then the
Anſwer will be in lower Terms, than if you do not ſo ab-
breviate them, as appears by this Example, whofe Expla-
nation follows. Firft, according to the Rule, Page 121, I
find that 11 Yards muſt be the third Number, becaufe 'tis
that whick afks the Question; then Yard muſt be the firſt
Number, becauſe 'tis of the fame general Species with that
Number that aſks the Question, viz. Measure; then of courſe
the other Number, which is the Price or Value of the firſt
Number, muft ftand in the fecond or middle Place, thus:
3
If Yard coft 5s. What's the Value of 11 Yards and ½?
4
The first Number being already a fimple Fraction, needs
nothing to be done to it, and eſpecially as 'tis already in its
loweſt Terms, nothing can yet be done with it, in its fingle
Capacity; then I confider that 5 Shillings is the of 20
Shillings, or the of a Pound; and therefore I fet down
inftead thereof; then the third Number being a mix'd Num-
ber, viz. the whole Number 11, and the Fraction, I bring
it into an improper Fraction, and 'tis then 22; ſo that the Stat-
ing of the whole Work, when fo prepar'd, will ſtand thus:
Now this wrought ac-
cording to the general
Rule, without further
Reduction, will be as you
fee in the Margin.
13214
+
+
23
3
92 Numerat. 12
2
22
Now 2 of a Pound is
equal to L. 32, which
24 Denomin. 24 Denomin. is equal to L. 3: 16: 8.
But forafmuch as the
Denomin
162 The Rule of Three in FRACTIONS.
3
A
Denominator 4 of the firft Number, and the Denominator 4
of the fecond Number, may both be divided by 4; that done,
the Fractions will then ftand thus: --22; and then
working by the general Rule above, the Denominator of the 4th
Number will be 6, and the Numerator 23, which are 22 of a
Pound, which is equal to L. 3 & or L. 3: 16: 8, the fame
with the other, only in lower Terms.
You may proceed in the fame Manner, upon all other
Occafions, and work even all the common Queſtions that
are wrought by the former Methods, contain'd from Page
120 to Page 130; for in reality, all that is done there is
virtually, tho' not actually doing what is here taught; as
fhall appear plainly, by working fome of thoſe Queſtions
over again, here, by the Method now taught.
Page 121, the Qucftion there is, If one Yard of Cloth coft
6 Pence, What coft 8 Yards?
This prepared according to the Directions given in this
Rule, will ftand thus:
8
44
J
Or reduc'd
11
1
which comes out
4
Here I fubftitute Unity or 1, for
the Denominators to the whole Num-
bers, 1 Yard and 8 Yards; then I
confider that 6 Pence is the of
20 Shillings or a Pound Sterling ;
all which being prepared, will ftand as in the firft Line a-
bove; then in order to abbreviate the given Fractions, fee
that the Denominator of the middle Number 40, and the
Numerator of the third Number, may both be divided
by 8, which I do, and when fo reduced they will ſtand as
in the fecond Line above. Now here no poffible Reduction
can be further made, and therefore I multiply the Nume-
rator of the firſt Number or Fraction into the Denomin-
ators of the ſecond, and third, and the Product is 5, for
a Denominator to the fourth Number fought; then I
multiply the Denominator of the first Fraction into the
Numerators of the fecond and third Fractions, and that
Product is only Unity, or 1, for the Numerator of the fourth
Number fought, which fo compleated. is; and to know
the Name of the Species to be apply'd to this, I look and
fee what the fecond or middle Number 4 was call'd, and
find twas called of a Pound Sterling; therefore I call the
Number or Fraction now found out the of a Pound Sterl-
ing, which by the Rule, Page 142, comes to 4 Shillings;
the fame as you fee in Page 121,
40
1
Again,
The Rule of Three in FRACTIONS. 163
Again, the fecond Queftion, Page 121, is, If a Sack of
Malt, Quantity 4 Bushels, coft 12s. 6d. What comes 85
Bushels to? This reduced to Fractions will ftand thus
&
ΩΣ
4 - 2 - 12.
I
Now theſe cannot be reduced any lower, becauſe no Num-
ber but Unity can divide the Numerator of, and the Nu-
merators of either or 8; and fo 'tis likewife impoffible to
abbreviate the Denominators, as 'tis alfo to reduce the Nu-
merator of the fecond, and the Denominator of the third,
or the Numerator of the third, and the Denominator of the
ſecond; and therefore the Work muſt be performed by the
general Rule, and the Amount will be of a Pound
Sterling, or L. 13, equal to L. 13: 57.
Again, the 3d Question, Page 125, is, Suppofe I buy a
Bale of Linen Cloth, Quantity 58 Picces, cach Piece 29 Ells
Engliſh,.at 15 d. per Yard, What is the Whole worth, and
how much do I gain per Cent. if I fell it at 21 d. per Ell
Engliſh?
As there are various Methods of doing fuch like Questions,
we will here purſue one different from that in Page 126,
on purpofe to fhew the Harmony and Agreement of all the
different Proceffes in this Art.
That Part of the Question that afks, what is the Whole
worth, may be applied to the buying or felling Price; but
'tis moft natural to mean the latter, for the Question of
Profit per Cent. may be anfwer'd without finding the whole
Worth or Coft, at the buying Price, as appears by the follow-
ing Work.
+
If Ell Eng. coft 15 d. 1, What coſt
Ell Engliſh?
21 Sold for
Bought for
31 Anf.
1d.
d. or 19
8
per Ell
I
gain'd.
If 19 d. 3 gain 1 d. §, What 100 L. or
24000
120 gain?
11 2 - 12
13
240.00
reduced, twill be
115
111111
13
24000
Theſe reduced again, will be 212-400
1
Theſe
multiply'd by the Rule produces 62420 or 2012 Pence 3
which is L.8:7:8:3 and 2 of a Farthing.
3
Y 2
So
164 The Rule of Three in FRACTIONS.
So that the Gain per Cent. or by laying out L. 100 will
be L. 878 3 and Parts of a Farthing.
:
31
And here I am to take Notice of a Miſtake in Page
126, where L. 143: 3: 6 is fet down and work'd with,
inſtead of L. 147: 36; and therefore the Anſwer there is
falfe, although the Work is right, owing to the faid Mif-
take; and therefore 'tis repeated here again, as it ought to
be, viz.
3
If L.135:15:8 the Money the whole coft give L. 147: 3:6,
the Money the whole was fold for, What will L. 100 give?
Theſe Num. reduced, will be 40
130355 Farthings
5887 Sixpences 96000 Far.
96000 Farthings in L. 100.
35322000
52983
(4,0)
130355) 565152000 (433,5
521420
108 : 7 : 6
437320
: 2:3
391065
108: 78: 3
462550
391065
714850
651775
63075 Parts of Sixpence.
378450
260710
117740 Parts of a Penny.
4
470960
391065
Remains 79895
Parts of a Farthing.
So
130355
DECIMAL S.
165
130315
: 8 3
+
So that it appears, the Amount of L. 100 will be
L. 108 7: 8: 3, that is, the Gain is L.
83, 8: 7
and Parts of a Farthing, which reduced, will come
to Parts, as above. By which it appears, that uſe what
Method you pleafe, if the Work be right, the Reſults will
be always the fame. I prefume enough has been ſaid to
make the whole Doctrine of vulgar Fractions fufficiently
plain, and therefore I will now proceed to explain the Na-
ture, Ufe, and Advantage of Decimal Fractions, commonly
call'd fimply Decimals, becauſe they are wrought as whole
Numbers without regarding their Denominators.
CHAP. VII.
Contains the General Definition of DECIMALS, and
explains the Methods of using them, in the feve-
ral Rules or Operations of Addition, Subftraction,
Multiplication and Divifion.
DEFINITION.
Omputation by Decimals is called Artificial-Arithmetick,
becauſe it fubftitutes an artificial fractional Number
in the room or ſtead of a true and natural one, to repreſent
the Part or Parts of any whole Thing; and theſe are ſome-
times perfectly exact, and at other times only nearly true. The
Name Decimals is given to it, becauſe all the Parts of any
whole Number whatever is exprefs'd or fet down in Tenths,
Hundreds, Thoufands, &c. always going on by a Tenfold
or decuple Proportion; and all theſe Fractions have for their
Denominator an Unit, with ſo many Cyphers annexed to them,
as there are Figures in the given Decimal, which is the Nu-
merator to the ſaid Denominator; and becauſe the Denomina-
tor is always known and fix'd, therefore 'tis feldom or never
fet down; but the whole Proceſs is in every refpect wrought as
whole Numbers; only to diftinguifh the Parts, from the whole
Numbers, a Comma, or Mark of Diftinction is made be-
tween that Part of any given Number that repreſents the whole
Number, and that Part which reprefents the Fractional Parts,
where you are always to obferve, that whole Numbers, when
mix'd with or join'd to Decimal Parts, always ftand on the
I
Left-
166
Notation, &c. of DECIMOLS.
1000
ΙΟ
2
100
4
Left-hand of the Comma or Mark of Distinction, and the
Decimals on the Right-hand as follows; where thefe Figures
675 may reprefent fo many whole Numbers, or 67 whole
Numbers and 5 Parts, which is of a whole Num-
ber; fo that together it is 67, or it may ftand for
6 whole Numbers and 75 Parts, or 12 of a whole
Number, which is 6; or it may ftand for 675 Parts on-
ly, that is, of a whole Number, which is 22. In all theſe
Cafes they are diſtinguiſhed thus: 675, without any Mark,
ftands for whole Numbers; but 67,5, in the fecond Cafe a-
bove, repreſents 67 whole Numbers, and ; and 6,75, in the
third Caſe above, repreſents only 6 whole Numbers, and 2;
and ,675, only reprefents which is of a whole Num-
ber: And if you put one or more Cyphers on the Left-hand
of the fignificant Figures, it increafes the Denominator Ten,
a Hundred, a Thoufand, &c. Times, and confequently de-
creaſes the given Decimal to a Tenth, an Hundredth, a
Thousandth, &c. Part of what it ftood for without fuch ad-
ded Cypher or Cyphers, as the ,675 or 22 above, of any whole
Thing, becomes, by making it ,0675, and 430, by
making it ,00675 and by making it ,000675; and
the like is to be obferved of any other Decimal Number, as,
,5 is 1,05 is 2,005 is 25., fo ,0005 is loo, &c.
The Agreement between whole Numbers and Decimals,
are fully expreſſed by the following Table.
675
1000
27
40.000
200
27
40
I
20099
21
40009
The Notation and
Numeration of
Decimals.
Whole Numbers.
C Millions
X Millions
XQ
C Thouſands
X Thouſands
Thouſands
Millions
Hundreds
Tens
Units
Tenths of Unity, or Primes
Hundredths or Seconds
Thouſandths or Thirds
98 7 6 5 4 3 2 I 1 2 3 4 5 6 7 8 9
Here you are to obferve that as in whole Numbers, the
Figures of every compound Number increaſe their Value
ten Times, every Place or Remove they are diftant from
X
Thoufondths or Fourths
C
Thouſandths or Fifths
Millionths or Sixths
X
Millionths or Sevenths
∞C Millionths or Eighths
Thouſa
fandths Millionths or
Ninths, &c.
Unity
Notation, &c. of DECIMALS.
167
2
9
100)
Unity, towards the Left-hand; but in Decimals they decreaſe
their Value ten Times, by the Number of Places they are
diftant from Unity, which here is juſt the Reverſe to whole
Numbers, being computed towards the Right-hand. Now
the Figures, 9, 8, 7, 6, 5, &c. above, may both ftand for
abfolute Numbers or Parts of Numbers; and alfo, as Indexes
to fhew or exprefs the Place or Value of any Figure ftand-
ing there: As for Inftance, in whole Numbers the Figure 3,
may fignify the 3d Order or Degree, which is Hundreds ; and
it may alfo exprefs fo many Hundreds; but in the contrary
Scale the 3 reprefents Thirds, or three Thouſand Parts of
Unity, as was above taught; becauſe the I or Place of Primes
expreffes fo many tenth Parts of Unity, as the fingle Figure
ftands for; as, if 'tis I, then 'tis,I or; if,2 'tis 2 if
,3, 'tis, &c. So in Second's Place,or is
1 02 is
100,03, is 180 907, is 130, &c. So in the Third's Place,001
is 1000,002 is 1000,009 is 1000, &c. only you are to ob-
ferve, that if you would exprefs only the Figure or Parts re-
prefented by the 2, 3, 4, 5, &c. Place, viz. 1, 2, 3, 4,
&c. Hundredths, Thousandths, ten Thousandths; you muſt ſup-
ply all the vacant Places towards the Left-hand, &c. with
Cyphers or o's. As, fuppofe I was to fay, fet down 8 ten
Thoufandths of Unity; here I look in the Table, and find the
fourth Place is the ten Thousandths of Unity, and therefore I
firft write down the required Number 8, and then towards.
the Left-hand, write 3 Cyphers or o's thus, 0008, before
which, towards the Left-hand, next to the firſt o, I make a
Comma, and then they will ftand thus, ,0008; the Ufe of
the Comma is, to fhew the Number fo expreffed is a Frac-
tional Part of Unity; but without that, or fome other di-
ftinguiſhing Character, they would ftand abfolutely for 8
Units, becaufe Cyphers on the Left-hand of whole Numbers,
neither add to, nor diminiſh from their true Value.
If feveral Figures are fet down for a Decimal, they repre-
fent the Aggregate or Total of fo many diftinct Decimals as
there are Places in the given Figures, as, ,123 ftand for
one Hundred and twenty three thouſandth Parts of Unity, that
is, the 1 is one Prime, or to Part of Unity, the 2, is two
Seconds, or Parts of Unity, and the 3, is three Thirds
or, Parts of Unity.
3
ΙΟ
100
10
2
100
20
Now is equal to, and is equal to 1; and
theſe added to 10%
3 will make 123 ; but, as has been al-
ready obſerv❜d, the Denominator of every decimalFraction, being
Unity, with fo many Cyphers annexed, as there are Figures in
the
168
Reduction of DECIMALS.
the given Decimal, the Denominators are feldom or never
fet down, unleſs it be purely for the fake of demonftrating
the Truth of any Propofitions, or explaining the Nature of
Decimals, &c. as above. From what has been faid, twill be
natural to conclude, you now know how to exprefs the Value of
any given Decimal, whether great or fmall; as of ,2 of ,02,
of,202, of ,0056, of ,470096, &c. the fame Law or Rule
being always to be obſerved in every Decimal, whether con-
fifting of many or few Figures, viz. that you number the
given Decimals as if they were whole Numbers, and then
imagine a Denominator to be fubfcribed to the given Figures,
which are always ſuppoſed to be the Numerator to fuch De-
nominator, that confifts of an Unit, with fo many Cyphers
annexed, as there are Figures in the given Decimal; as in the
Examples above.
1009
,2 is or two Tenths,02 is, 2 or two Hundredths,
,202 is 222 or two Hundred and two Thousandths; ,0056,
is, or fifty fix ten Thousandths; 70096
1000
100000
is 096 is feventy Thousand and 96 one Hundred
Thoufandths, &c.
The next Bufinefs will be to fhew you how Decimals
are generated, for the true and eafy uunderſtanding whereof
you are to confider that every Number is immediately cal-
led or underſtood to be an abfolute Number, repreſenting
fome one whole Thing, or an Aggregate of feveral or many
whole Things, or elſe is a Part or Parts of fome other Num-
ber that repreſents fome whole Thing; and theſe are as infi-
nitely various, as the Figures they are, or may be reprefent-
ed by; and when expreffed in its full Perfection, always
confifts of two Parts; the one called a Numerator, and the
other the Denominator, as was taught you very fully in
the laſt Chapter, with the feveral Ways of managing fuch
Parts or Fractions; but here, only one Part, or the Nu-
merator, being only us'd, we fhall now teach you how
To change, reduce, or convert Vulgar Fractions into Decimals.
First, If the given Fraction be a fimple one, add any
Number of Cyphers to the Numerator, and then divide the
Numerator, with the Cyphers fo added by the Denomina-
tor, till o remains; but if after dividing 4, or 5 Times,
fomething ſtill remains, you may ceafe, and reckon the
Quotient of fuch Divifion for the Decimal, rejecting the
Remain-
Reduction of DECIMALS:
169
Remainder, as being of too fmall a Value to be of any
Significancy: As for Inſtance,
What is the Decimal to 4? Anſwer 75.
Here I fet down the 3, and annex 2 Cyphers to it, and
then divide by 4 thus :
4 300
>75
and after two Divifions, I have
275 in the Quotient, and o remains; therefore, I fay, 75
is the Decimal to of Unity, which, if you want it for
any particular Ufe, you may call of a Ton, of a Pound,
of a Yard, of an Hour, &c. So if I afk, What is the Deci-
mal to? Here I do the fame, and find the Anſwer to be
,66666, &c.
Again, If I ask what is the Decimal to ? I find
the Anſwer to be ,25, as you may fee by the following
Work.
6811700 (,25
136
340
340
00
1
2756
842
and the Decimal to 11 is
3842) 275600000 (,71733, &c.
26894
6660
3842
28180
26894
12860
11526
13340
11526
1814
Note, So many Cyphers as you add to the Numerator, fo many
Places or Figures muſt be in your Decimal, as you ſee in the
four Examples above, where, in the first Example two Cyphers
only are added to the Figure 3; and the Decimal confifts accord-
ingly of 2 Figures, viz.,75; but five Cyphers being added to
the fecond Example, and the Divifion being made, there comes
out,,66666, and the Remainder is 2, which is rejected upon
Ꮓ
account
.3
170
Reduction of DECIMALS.
account of the Smallneſs of its Value, as fhall be fhewn
by and by: In the 3d Example only 2 o's are added, and
the Quotient is ,25, and o remains; in the 4th Example 5
Cyphers are added, and after the Divifion is made, 1814 re-
mains, and the Quotient is ,71733, for the Decimal.
But it may fo happen, and that very often too, that the
Quotient may not have ſo many Figures as you was obliged
to add Cyphers to the Numerator; in that Caſe you muſt
add as many Cyphers on the Left-hand of the Quotient, as
is the Difference between the Number of Figures in the
Quotient, and the Number of Cyphers added to the Numerator;
as, What is the Decimal to 6181?
6181) 300000000 (,00048519
24724
52760
49448
32120
30905
12150
6181
59690
55629
4061
Here, before I can have a fignificant Figure in the Quo-
tient, I must add 4 Cyphers to the Numerator; and therefore
I muſt add 3 Cyphers towards the Left-hand, to make up
that I Figure 4 Places; and then go on as before.
What is the Decimal to? Anſwer ,04.
25 | 100 (,04
100
Oo
And fo in all other Cafes, whether
your given Number or Decimal found
confift of few or many Figures.
Note, After you have carried your De-
cimal to 4, 5, or 6 Places, and the Re-
maninder is of any confiderable Value,
dividing by adding 1 or more Cyphers
to the Numerator, and the Quotient will be increaſed by 1
you may continue
I
ΟΙ
Reduction of DECIMALS.
171
1
or more Figures; and tho' the Remainder fhould continue
the fame in Figures, yet the Value of it will be but
10
I
I
{
I
1
1
Jo9 100)
, &c. Part of what it was before, according as you
increaſe the Number of Places in the Decimal; as you may
fee by finding the Decimal to 2 which is found to be
,083333, &c. For if the Decimal had been carried on
to but three Places, viz. ,083, the Remainder would then
be of 1000, or 3000 of Unity; but making it ,0833, 'tis
then but of 100, or goooo; fo when 'tis ,08333, then
100009 30000;
the Remainder is but; and when 'tis ,083333, 'tis
then but 300000 and ſo on; from whence 'tis cafy to ob-
ferve, that according to the Ufe the Decimal is to be
apply'd to, it may be proper to make it confift of more or
fewer Figures, when they are interminate, that is, either have
the fame fingle Figure conftantly remaining, or a certain pe-
riodical Number of Figures remaining; for 'tis the Property of
fome Decimals to terminate, and have no Remainders, as is the
Cafe of all aliquot Parts of 10, 100, 1000, 10000, 100000,
c. there are called perfect Decimals; and 'tis the Property of
others to have one conftant Figure remaining continually; as,
when you require the Decimal of Here the Quotient or
Decimal will be always 3's, and the Remainders I's, and if
you wanted the Decimal of, the Quotient will be always
6's and the Remainders 2's. So again, if you afk'd for the
Decimal to, the Anfwer will be as many,I's as you
pleaſe, and the Remainders will be alfo 1's, and the De-
cimal will be always 2's, and the Remainders 2. So will
have always 4's both for the Decimal and Remainders, & will
be always 5's, &c. Other Decimals there are that go on 1,
2, 3, 4, 5, 6, &c. Figures, before they fall into this fingle
continual Repetition. As, begins at the 2d Figure, be-
ing,0666, &c. fo is,1333, &c., is,2666, &c. and
begins at the 3d Figure, the two first being ,08 and all
the reft are 3's, and is,416666, &c. beginning likewife.
at the 3d Figure with 6's, &c. Others there are that con-
tinually repeat 2 Figures, even from the Beginning, as
is ,090909, &c. fo is ,181818, &c. and, is,272727,
&c. is ,363636, &c. Others begin at the 2d Figure,
and repeat 2 Figures; as is ,0454545, &c. likewife
22 is,31818, &c. Others go to 3, 4, 5, 6, &c. Places;
and then repeat the whole again, as is ,142857,14,
&c. is ,285714,285, &c. is ,076923076923, &c.
So is ,465138465138, 46513, &c. to Infinity.
thers there are that only approximate, and never cir-
2
1 3
±
2
I
I 2
1 J
1
I i
22
1 3
0-
Z 2
culate,
172
Reduction of DECIMALS.
S7
So 44 is
211
318
culate, as is 724137931034482, &c.
63
,6761006289, &c. Now thofe that circulate in any of the
Manners above, may be commodioufly wrought with in the
various Operations of Addition, Subftraction, Multiplication,
and Divifion, as correctly as if they were abfolutely perfect
and terminate; and the feveral Totals, Differences, Products,
and Quotients, will be alfo circulating Decimals; but thofe
that neither terminate nor circulate, viz. thofe called Ap-
proximates, muſt be proportion'd, as has already been taught,
Page 169, and fhall hereafter be further explained in order
to leffen the Error that will neceffarily arife by working
with them.
Secondly, If the given Fraction is a compound one, reduce
it to a fimple one; and then work as above.
EXAM P L E.
6
S
What is the Decimal to of of? This reduced to
a fimple Fraction, is 22, the Decimal whereof is ,40178.
Again,
7
1%
+
19
What is the Decimal toof of of 234? Answer
,002834. Here the compound Fraction, when reduced to a
fingle one is, 320, which requires 3 Cyphers to be added
to its Numerator, before there will be one fignificant Figure
in the Quotient; fo that there must be 2 Cyphers added on the
Left-hand, as you fee done in the Example above.
Likewife when a compound Fraction is given, inftead of re-
ducing it to a fimple Fraction, firft you may find the Decimal
to any one of the Parts, and then take the other Parts of
that Decimal, and 'twill come to the fame Thing. As,
3
What is the Decimal to ½ of of } ?
Here I find the Decimal to 3 to be
I find the
I find the
of this to be
of this to be
-
,875
,65625
then
and
,328125
,5
which laft is the true Decimal fought. Again,
I find the Decimal of to be
theof which is
the of which is
the fame as before.
Or the Decimal of 3 is
the of which is
theof which is
,375
,328125
>75
,65625
,328125.
Now
Reduction of DECIMALS.
173
Now this given compound Fraction reduced to a fingle one,
is equal to 4, which reduc'd to a Decimal, gives 328125,
as before.
So that work which Way you pleaſe, the Refult or Anſwer
is always the fame, if the Work be truly perform'd.
Thirdly, If the given Fraction be an improper one, reduce
it to a whole or mix'd Number, and if it fhall fo happen
that there comes out a whole Number, by the Reduction,
there can be no Decimal, becauſe a Decimal always reprefents
a Part or Parts of fome whole Thing, and can never re-
prefent the whole Thing it felf; but if after the Reduction is
made, the given improper Fraction becomes a mix'd Num-
ber, then you may let alone the whole Part, and work with
the Fractional Part, as has already been taught you.
EXAMPLE.
What is the Decimal to 27 ?
Here upon reducing, I find 9 whole Numbers; confequent-
ly there can be no Decimal, but only 9 whole Somethings,
whether it be Coin, Weight, Meafure, &c. Again,
IVhat is the Decimal to? Anſw. 5,4.
Here upon reducing, I find 5 and, I work with the,
and find 4 to be the Decimal, to which I add the 5 whole
Numbers, and they ftand thus 5,4, that is, 5 whole Things,
and,4 Tenths of another whole Thing: and thus you muſt
proceed continually; for all the various Cafes that can hap-
pen, are included in one of the three above mentioned.
Note, 'Twill be the fame Thing whether you reduce an
improper Fraction to its whole or mix'd Number, before you
go about to find the whole Part and the Decimal Part, or di-
vide the Numerator of the improper Fraction by the Denomina-
tor, adding fo many Cyphers to the Numerator, as occafion may
require; for in all fuch Cafes there muſt be cut or parted off
from the Quotient towards the Right-hand, fo many Figures
for the Decimal Part, as you added Cyphers to the Numerator;
and the remaining Figures on the Left-hand, are whole Num-
bers. As in the Examples following.
IVhat whole Number and Decimal Parts are equal to $?
24
24) 578,000000 (24,083333, &i.
the Remainder being in this Example always 8, if you con-
tinue the Work ever fo long. So that 24 is the whole
Number, and ,083333 the Decimal Part, which is the fame
as if you had reduced it firft; for 578, divided by 24, gives
242, or 24 12. Now the Decimal to is ,083333 and
1 2
4
Ι Σ
174
Reduction of DECIMALS,
4
12
or
remains, which is the fame with the other; for
is equal to or . The like may be obſerv'd in all o-
ther Caſes.
24
4
12
To reduce any Part or Parts of Money, Weight, Mea-
fure, &c. to a Decimal.
Here you muſt confider what is, or may be your Integer,
and order your Subdivifions accordingly, in the Form of a
compound Fraction; and then work as before directed.
EXAMPLE.
What is the Decimal to 13 Shillings, a Pound being the In-
teger ?
Here 13 Shillings is the 13 of a Pound, the Decimal to which
is,65; for adding 2 Cyphers to 13, the Numerator, and di-
viding by 20, the Denominator, the Quotient is 65, and o
remains.
What is the Decimal to 9 Pence, a Pound being the Integer?
Here 9 Pence is the 2 of the 2, which is the 20 of a
Pound, the Decimal whereof is ,0375.
What is the Decimal to
Integer ?
Here 3 Farthings is the
or of a Pound, whofe
320
3 Farthings, a Pound being the
I
20
of 2 of 40, which is the
Decimal is ,003125.
3
o
What is the Decimal to 15 Pounds, an hundred Weight,
or C. being the Integer?
I 12
Here 15 Pounds Weight is the of C. whofe Decimal
is,1339, &c.
of C.
What is the Decimal to 5 Ounces, C. being the Integer?
Here 5 Ounces is the % of ri, which is the
whofe Decimal is ,00279, &c.
16
What is the Decimal to 13 s. 5 d., a Pound being the
Integer ?
Here, and in all fuch like Cafes 'twill be beſt to reduce
the given Sum or Number into the loweft Name; as here,
into Farthings or Half-pence, thus:
C
13:5
2
24 Halfpence in a Shilling.
52
26.
11 Half-pence in 5 d.
• 2
I
the Half-pence in 13:5
is 323 for the Numerator.
the Half-pence in a L. Sterl. is 480 for the Denominator.
480)
Reduction of DECIMALS.
175
480) 3230000 (,6729 the Decimal fought. The Re-
2880
3500
3360
1400
960
4400
mainder 180, being leſs than
half the Divifor, viz., you
may reject it upon account
of the Smallneſs of its Value,
being but of a L. Ster-
ling, or the 20 of a Farth-
22
ing.
3
4320
180
Days Hours Min.
What is the Decimal of 17: 13: 25, a Year of
365 Days, being the Integer?
24
1460
730
8760
17: 13: 25
24 Hours in a Day.
71
35
60
Min. 525600
Days Hours.
421 Hours in 17 and 13
60 Minutes in an Hour.
in a Year of 365 Days
25285 Minutes in all.
5256/00125285000oloo(,048 106 the Dec.requir'd
21024...
42610
42048
5620
5256
36400
31536
4864 remains.
Nate,
176
Reduction of DECIMALS.
I
2
Note, If the Remainder is equal to or more of the
Divifor, you may make the laft Figure of the Quotient ▾
more than it naturally comes to: As here the Remainder
is. more than the Divifor, fo that you may make
the 6 a 7, and the Decimal will then be ,048107, which
is nearer the Truth than the other, being but
of a Year too much; whereas ,048106 is 668 of a
Year too little.
2
49
6570000 0
657000000
To find the Value of a Decimal in the known Parts of Coin,
Weight, Meaſure, &c.
RULE.
Multiply the given Decimal by the Number of Parts of
the firft Subdivifion of the given Integer; and if the Product
comes to more Figures than is the Number of thoſe given,
cut off the Surplus Figures towards the Left-hand, and they
will be the Answer in that Part of Coin, Weight, &c. as
anfwers to that Subdivifion; and if there are any Figures
left, you may do the like by them; and fo continue multi-
plying the remaining Figures by the Number of Parts in the
next inferior Denomination, either till the Product of the
Decimal comes to all o's, or that you have gone through
all the Subdivifions commonly us'd or known, always cut-
ting off the Surplus Figures, as you fee done in the following
EXAMPLES.
What is the Value of 75 of a Pound Sterling?
20
15,00 Anf. 15 Shillings.
Here, after multiplying by the firft common Subdiviſion of
a Pound, which is, into 20 Shillings, the Product comes out
1500, which being 4 Figures, and the given Decimal con-
fifting but of 2 Figures, I cut off the 2 Surplus Figures to-
wards the Left-hand, which are 15 Shillings; and the other
2 Figures towards the Right-hand being Cyphers, I conclude
that 15 Shillings is the full Value of that Decimal.
What
Reduction of DECIMALS
177
What is the Value of ,378125 of a Pound Sterling ?
20
Shillings 7,562500
12
Pence 6,750000
4
Farthings 3,000000
Here I multiply firft by 20, and from the Product, which
confifts of 7 Figures, I cut off 1 Figure towards the Left-
hand, which in this Example is 7 Shillings, and leave 6 Fi-
gures, the Number of Figures in the given Decimal, for the
next Operation. Now as 12 Pence make a Shilling, I multi-
ply 562500 by 12, and from this Product, which alfo con-
fifts of 7 Figures, I cut off one, viz. that next the Left-
hand, which is 6 Pence; then I multiply the remaining 6
Figures by 4, the Farthings in a Penny, and the Product
comes to 6 Cyphers and a 3, which being cut off, as before,
is 3 Farthings and o remains; fo that the true Value of
this Decimal is 7 s. 6 d. 3.
Note, There are abundance of Cafes more, where there
will be Remainders, after you have gone through the feveral
Subdiviſions, than there is where there are none, as follows:
What is the Value of,
51296 of a Pound Sterling?
20
Shillings 10 25920
12
Pence 3 11040
4
44160
Here I find 10 Shillings and 3 Pence, and the Remainder
is 11040, which being multiply'd by 4, gives but ,44040,
which is but 442 Parts of a Farthing, which is not quite.
half a Farthing; and therefore in common we fhould call
100000
A a
the
178
Addition of DECIMALS.
the Value of ,51296 of a Pound Sterling but 10 Shillings and
3 Pence.
What is the Value of, 174 of a Troy Weight?
12
Ounces
888
20
Penny Weights 17 60
24
ཀླ།
12
40
Grains
14 40
2
Oz. Dwts. Gra.
Anf. 8 17 14
and ,40 remains, which
is equal to of a
Grain. As you may
eafily prove, by con-
fidering ,40 as the
Numerator, and 100
as the Denominator ;
and then 'twill be
to of a Grain, which
reduced comes to o 2.
You muſt proceed in
100
the fame Manner in all other Weights, Meafures, &c.
Addition of Decimals.
Here you have nothing more to do than to put the
Primes, Seconds, Thirds, &c. exactly one under another,
as you do the Units, Tens, Hundreds, &c. in whole Numbers;
and then add up each Row, as if they were whole Numbers,
and the Total is the Anſwer fought. From (which you muft
cut off as many Figures towards the Right-hand, as are in the
biggeſt of the given Lines to be added, and if there are any
left towards the Left-hand, they are whole Numbers. As,
,562 Here adding up the feveral Figures given,
,345 they come to 4189; but as the given Deci-
,786 mals confift but of 3 Places, the fourth is fe-
,928 parated by a Comma; and fhews that the
94I
Total of thofe feveral Parts is 4 Units or whole
,627 Things, and 189 thoufand Parts. The like is
to be obſerv'd in all other Caſes.
4,189
Suppofe, the Sums given were
theſe in the Margin. Here be-
ing whole Numbers, and Parts mix'd together,
I add them as if they were all whole Numbers;
only obferve to keep each refpective Figure
in its due Place.
Note, You are to obferve, that here, and in
all other Cafes, the Decimals are fuppofed to
be Parts of the fame Integer, viz. all Parts of
a Pound in Money, or C's in Weight, a Yard or
Mile in Meaſure, &c.
As,
56,27845
9,152..
719,3....
42,00009
5,712..
6,8271.
5,00287
844,27251
The
Addition of DECIMALS.
179
The Decimals in the Examples above, are a Mixture of
pure or compleat Decimals, Approximates, Circulates, &c.
and therefore the feveral Totals can be without Diftinction
really, but an approximate Sum, tho' they are confider'd here
as compleat or terminate Decimals only, and confequently
the Totals the fame.
But when you are to add thoſe that circulate, or repeat a
fingle Figure, you need only continue them fo far that they
all end together; and then to the Sum or Total of the first
Line towards the Right-hand, add as many Units as there are
9's in that Line, and the Figure fo produced fhall be a Re-
petend for the Total; and that the Repetends or circulating
Figures may the more eafily be known 'twill be proper to
mark them thus: A, 4, 3, &c. as in the following Exam-
ples.
(3)
(1)
(2)
5 77
27,066
134,252777
6,83
13,416
618,958333
8,76
17,333
720,888888
5,24
62,185
517, 777777
6,99
31,866
316,027083
8,80
25,888
123,666666
41,97
177,527
2431,571520
In the first Example, the Decimals are continued only tỏ
two Places of Figures, becauſe the Repetends do not run any
farther before they begin; as is the Caſe, at the 3, 4 and 6
Lines, where the Repetends are 6, 4 and ø; the firſt, ſecond,
and fifth Lines beginning with the firft Figure, and are there-
fore mark'd upon the firſt #, 3, and g. The ſecond Exam-
ple is continued to three Places, becauſe the Decimal of the
fecond Line, which reprefents, does not begin to repeat
till it comes to the third Figure, which is 6. The third Ex-
ample is continued to fix Places, becauſe the Decimal of the
fifth Line does not begin to have the repeating Figures, till
you come to the fixth Place. You are to obferve the fame
upon all other Occafions.
To add Decimals, that have compound Repetends.
Here you muſt make the Repetends all begin and end to-
gether; and that you may be accomodated the more eaſily,
you may imagine your Repetend to begin at what Figure you
pleaſe in the Repetend. As for Inftance, Suppofe the Decr-
A a 2
mat
180
Addition of DE CIMALS.
22
mal of was given, which naturally is ,0454,0454, &c.
but for Convenience you may make it 04,8404,5404,5404,
&c. or 045,4048,4045, &c. or 0,4540,4540, &c. as may
beſt fuit your other Numbers; and when you have ordered
your Numbers, add as before; only you must add to the Total
of the firft Line on the Right-hand, as many Units as are in
the Line, where all the Repetends begin together.
076923 Here you will obferve, that the Total of
461538 the firſt Line is 34, to which I add 2, the
$923078 Tens in the laft Line, where all the Circula-
gogog tions begin, and then 'tis 36; wherefore I
X4285 fet down 6, and mark it, as you ſee in the
428571 Example; then the whole Total will be two
whole Numbers, and a circulating Decimal
2,123878 that confifts of 123876,123876, &c. the De-
monftration of which follows,
076923 is the Decimal of
,461538
of £
i+
6
1 3
923078
of
øgogog
of
I I
1
I I
34285
of
942857x
of
1
2,123876 is the Decimal to
-
2
4
7
124 the Total.
Iool
The Reaſon why in fingle Repetends, you add as many
Units as there are 9's in the Line, where all the Repetends be-
gin together, is, that the Total may be alſo a Repetend; the
Truth of which appears plain from what follows,
, is the Decimal to
4
- to
-
- to
- to
- to
to
40
Ha Ha Mia Ho wa ela na
3, the Total.
to
3 the Total.
The Sum of the Decimals is but 2,8, which without the 3,
for the 3 Nines, would be too little, but with it is right;
as plainly appears by the Example wrought by vulgar Fractions.
IO
Sub-
Subftraction of DECIMALS.
181
Subftraction of DECIMALS.
You have nothing more to do, but to put the whole Num-
bers (if there be any) under one another, as in whole Num-
bers; and the Primes, Seconds, Thirds, &c. of the Decimals
under one another, and then fubftract, as in whole Numbers.
(1)
EXAMPLES.
(2)
(3)
From ,62758 From 627,912 From 16758,--
,49128 Take 159,939 Take
Take
Remains,13630
Rem. 467,973
7598,1234
Rem. 8159,8766
In the firſt and ſecond Examples, you ſubſtract all the fame
as if the given Figures were whole Numbers; only in the firſt
Example you make the Comma, to fhew they are all De-
cimals. In the fecond Example you likewiſe ſubſtract, as if all
the Figures were whole Numbers; only in the Remainder
you
make the Comma between the 3 and 4 Figures, becauſe there is
the Commencement both of the whole Numbers and the De-
cimals. In the third Example, the upper Number is only a
whole Number, but the under one that is to be ſubſtracted
from it, is a mix'd Number, viz. 7598, are whole Numbers
and
,1234 are decimal Parts; fo that I borrow one Integer,
which is 10000 Parts, and then ſubſtract the 1234 from it,
as if it was a vulgar Fraction, and the Remainder is 8159
whole Numbers, and 8766 Parts.
More EXAMPLE
From 965,219876
From 29,17...
Take 16,19875
Take 482,36...
Rem. 12,97125
Rem. 482,859876
S.
From 673,75892
Take 218.
Rem. 455,75892
Note, If the Number from which Subftraction is to be
made, have not fo many Figures in the decimal Part, as the
other, fuppofe Cyphers to be added, and then go on as be-
fore. As in the firſt Example of the laſt three
To fubftract Interminates that have fingle Repetends.
You must make them end together, as in Addition; and
then ſubſtract, as in whole Numbers: But if the Number that
is
182
Multiplication of DECIMALS.
is to be ſubſtracted, have the biggeſt circulating Figure; then
inſtead of ten, you only add or borrow 9, and then go on as
before.
(1)
(2)
From 18,5166
Take 12,9240
From 715,113 3 3
Take 127,6536
Rem. 5,5928
Rem. 587,45963
To fubftract compound Repetends.
Prepare them as in Addition, and fubftract as in whole
Numbers; only if you borrow one at the Place where both
the Repetends begin, add one to the Right-hand of the fub-
ftracting Number, and then the Remainder is the Anfwer,
whofe Repetend muſt be mark`d as in Addition,
(1)
From ,46153$
Take ,076923
Remains ,384615
(2)
From 58,09c909
Take 27,14285#
Rem. 30,948051
Here the first Example is the Decimal of, from which -
the Decimal ofis fubftracted, and the Remainder is the
Decimal of, as you may very eafily try: So the fecond
Example is the circulating Decimal of, from which is taken
the circulating Decimal of ; and the Remainder, after an
Integer is borrowed and added, is , whofe Decimal is the
fame with the Remainder, to whofe firſt Figure towards the
Right-hand 7, Unity is added, becauſe the beginning Figure of
the Circulation in the fubftracting Line is bigger than the
other.
Multiplication of Decimals.
In the working this Rule, there is no manner of Diffe-
rence between it and whole Numbers, all the Art lying in
the valuing the Product; for which purpoſe obſerve this
General Rule.
Count how many Decimals are in both the Multiplicand
and the Multiplier, and part off is many from the Product
but if the Product has not ſo many Figures in it, make them
up by adding fo many Cyphers as are wanting towards the
Left-hand.
3
(1)
Multiplication of DECIMALS,
183
(I)
(2)
52,0003
by
..,COIZ
Multiply ,57 Multiply
by ,84
228
,06240036 Product̃.
456
Product ,4788
Here you fee the first Example confifts of Decimals only,
and the Product is 4 Figures, which being the Number con-
tained in both the Multiplier and Multiplicand, they are fo
many Decimals; but in the fecond Example the Multipli-
cand is a whole Number and a Decimal, and the Multiplier
a. Decimal only; and after the Work is done there comes
out but feven Figures, whereas the Number of Places con-
tained in the Multiplicand and Multiplier are 8; wherefore I
am oblig'd to put a o on the Left-hand, to give the Product
its true Value, as you fee above. The like is to be obſerv'd
in all other Cafes.
But as it will frequently happen, that in multiplying by
Order, to have the Product fufficiently near the Truth, viz.
to 4, 5, 6, &c. Places at leaft, your given Mulitplicand and
Multiplier will confift of many Figures, and confequently
the Operation will be tedious and troubleſome; therefore, to
prevent this Inconvenience, you may make ufe of the fol-
lowing Contraction.
Confider with yourſelf, whether 3, 4, 5, &c. Places are
neceſſary in the Product, to give a near Value to the produced
Number, whether all Decimals, or mix'd with whole Numbers;
then put the Unit's Place of the Multiplier, under that Place
of the Multiplicand, you would have the Product be continued
to, and invert the Order of all the Figures in the Multiplier,
and begin every Line with the Figure that fands over the multi-
plying Figure, and add to that Product fo many Units, as are
Tens in the Product of the foregoing Figure; and obferve, that if
the faid Product amounts to 15, 25, 35, or upwards, that
you carry 2, 3, 4, &c. for them, as if they were 20, 20, 40,
&c. As you fee done in the Examples following.
i
Suppoſe I wanted to multiply 276,38256 by 7,985,
and would only preferve three Places of Decimals! I fet
down the Multiplicand as 'tis given, and then invert the
Multiplier, as you fee in the annexed Example, which
is wrought both at large, and by the antracted Method, on
pur-
184
Multiplication of DECIMALS.
purpoſe to fhew you the Agreement between them, fo far
as they go; and alfo to fhew you how many Figures, and
how much Labour is faved, by working by the contracted
Method.
Work at large.
276,38256
Contracted.
276,38256
7,985
589,7
1381 | 91280
1934678
22110 6048
248744
248744 304
2211I
:
1934677 92
1382
2206,914 74160 1
廴
​2206,915
Here you fee that in the Contraction, all the Work at
large, beyond the Line towards the Right-hand, is fuperflu-
ous, and the Product is nearly of the fame Value, there
being only in the third Place of Decimals, a 4, in the com-
mon Way, which is 5 in the contracted Way; which 5 is
much nearer the Truth than the 4, becauſe 74, &c. follows,
which is almoſt of another; fo that the Error is but a-
boutof ooo, or 4000 Part of Unity, which, according to
what your Unit is, will be of greater or leffer Value. But
if it had been required to have the Decimal continued to
4 Places, 'twould then have been exactly the fame, for
thoſe 4 Places with the common Way, and confequently a-
bout Part of Unity too little, as appears by the Work
following.
I
26000
276,38256
5897
19346779
2487443
221106
In working thefe Contractions, I begin
multiplying with the 7, the Units Place
of the firft Multiplier, that being the
Figure that ſtands under the 2, in the firſt
Example above, which is in the 3d Place
of Decimals, and fay, 7 Times 2 is 14;
and becauſe the Amount of the foregoing
Figure amounted to upwards of 35,
carried 4, as if it had amounted to 40,
and then it makes 18, for which I fet
down 8 and carry 1, &c. as you ſee
in the Example. That Line being done,
I go to the next Figure, which is 9,
that ftands over it, by this 9, to whofe Product 72, I add
13819
2206,9147
and multiply that 8
2,
Multiplication of DECIMALS.
185
2, for the 2 Tens contained in the Product of the next
Figure 2, towards the Right-hand, which would have amount-
ed to 23, had the other Figures alſo been multiply'd by the
faid 9, as appears by that Line in the Work at large; but
as it does not amount to 25, I carry but 2, and fet down
the 4 under the 8, and go on till the Line is done, which
falls fhort of the Line above it by a Figure, as you fee in
the Work at large, where 'tis the 2d Line from the Bottom,
as 'tis here the 2d Line from the Top. The Reaſon of ſet-
ting all the firſt Figures under one another, is to have the
Total or Amount of that Line, where, in the Work at large,
the 3d Place of Decimals meets in all the Figures, as appears
by comparing the two Methods together. Then I go on to
the 8, and multiply the 3 that ftands over it, and to the
Product add 7, for the Reaſons before mentioned, and fo go
on till all is done; as you fee by the Work in both the Ex-
amples, where the Decimal is continued to both three and
four Places. You are now to Note, That if no whole
Numbers were mix'd with the Multiplier, that you muſt put a
Cypher in the Place where the Units of the whole Number
would have ſtood, fuppofing any to be there; as in this Ex-
ample, multiply'd by ,985 Decimal Parts only, and requir'd
to have left three Places of Decimals in the Product thus :
276,38256
589,0
248744
221 II
Or thus for 4 Places. 276,38256
589,0
2487443
221106
1382
272,237
13819
272,2368
But if you have more than one Place of whole Numbers
in your Multiplier, you muſt fet the Units Place of the whole
Numbess and the Decimals, in the Order you have already
been ſhewn in the foregoing Examples; and the other Figure
or Figures, towards the Right-hand, in an inverted Order;
and then multiply as before, as you fee done in the follow-
ing Examples, where they are done for 3, 4, 5 and 6 Places
of Decimals in the Product.
B b
Multiply
186
Multiplication of DECIMALS.
Multiply 9563,287468
I by
897,028794
2 by
89,7028794
3 by
9,87028794
4 by
987028794
And let the firft Product have three Places of Decimals
only; the 2d four Places; the 3d five; and the 4th fix:
Places.
(1)
9563,287468
497820,798
(2)
9563,287468
4978207,98
7650629974
860695872
66943012
7650629974
860695872
669430 12
191266
76506
6694
861°
38
85785445223
0191166
76506
6694
861
38
857854,4223
(3)
(4)
9563,287468
9563,287468
49782079,8
497820798,0
7650629974
7650629974
860695872
860695872
66943012
66943012
191266
76506
669 4
861
191266
76506
6694
861
38
38
85785,44223
8578,544223
If both the Multiplicand and the Multiplier arc Decimals,
having no whole Numbers in either, you muſt put a o under
that Figure of the Multiplicand, that you would retain Places
in the Product, for the Units Place of your Multiplier, as
you fee done in the Examples, Page 185, and at the 4th
Example above, and work as before.
The
Multiplication of DECIMALS.
187
The next Thing to be taken Notice of is, the multiplying
fuch Decimals as have fingle or compound Repetends, in
which, if you would come to Exactness, you muſt obſerve
the following Notes.
1. If the Multiplicand be a fingle repeating Interminate,
and the Multiplier a ſingle Digit, either having or not hav-
ing one or more o's before it, towards the Left-hand, you
muſt multiply as in common Numbers; only to the laſt Place
of the Product, towards the Right-hand, add as many Units
as it contains g's, and that ſhall be the true Product, re-
peating that laſt Figure fo produced, as in the following
Examples.
76,20g 25,878
457,260 |
27865
,004
20,6986,00111462
2. If the Multiplier be a Repetend to the Product, add a
o to the Right-hand, which is the fame as multiplying by 10,
and divide that Product fo increafed by 9, continuing fuch
Divifion till you come to a fingle or compound Repetend
in the Quotient, which Quotient fo found, fhall be the true
Product of fuch fingle multiplying Repetend; as follows.
48,734
3
9)146,2020
16,2446, &c.
72,435
,006
9),4346100
,482900, &c.
3. If the Multiplier confifts of ſeveral Figures, whether
mix'd with whole Numbers, or Decimals only, among which
one is a Repetend, find the true Product of the Repetend
Figure firft, and then go on with the other Figures, as in
whole Numbers, and as in the Examples following.
487,65
5,06
67,89263
,896
9 | 2925900
9140735580
32518 the Prod. of 6
4526178
2438250 the Prod. of 5,0
611033677
5431410444
2441,5010 whole Prod.
60,46970797
Bb 2
4. C
188
Multiplication of DECIMALS.
4. If the Multiplicand and the Multiplier have both fingle
Repetends, you multiply, as has been before fhewn ; and the
Product of the particular Lines muſt be continued till they
arrive at the Figure, where the Repetends all terminate toge-
ther; as you fee done in the 2d Example above.
5. If your Multiplier be a 3, that is, a Repetend, you may
divide the Multiplicand by 3; and the Quotient will be your
Product, as appears by the 4th Example, Page 187.
6. If the Multiplicand have a compound Repetend, and
the Multiplier be a fingle Figure, multiply as in common
Numbers, and add to the Product of the Right-hand Figure
of the Repetend, fo many Units as there are Tens in the Pro-
duct of the Left-hand Figure of the ſaid Repetend.
Thus,
627,8645
8
5022,9165
7. If the Multiplier has likewife many Figures, multiply
by each, as before taught; and continue the Repetends till
they all end at one Figure; and then add as before taught in
Addition, Page 180.
Thus,
417,6323
76,45
20881626
167053013
2505795 95
29234277277
31928,007117
8. If the Multiplier confifts only of a compound Repetend,
without any terminate Figures, multiply by each Figure of the
Repetend, as in whole Numbers, and add the feveral particu-
lar Products together, and to that Total add itfelf in the fol-
lowing Manner. Firſt, fet the first Figure towards the Left-
hand, fo many Places forward, towards the Right-hand be-
yond itſelf, as the Repetend confifts of Figures, and the reſt
in order following; and fo go on till the laft Total reaches
beyond the firſt Figure of the Product on the Right-hand;
then add theſe ſeveral Lines together, beginning under the
Right-
Multiplication of DECIMAL S. 189
Right-hand Place of the first Product; and if the next Figures
beyound come to Ten, carry I, and daſh off from thence fo
many Figures for a Repetend, as the Repetend of the Multiplier
confifts of.
Multiply
by
4271,36
7438
2562816
1281408
1708544
854272
1040,503296
1040500,&c.
104, &c.
1040,607356
gain, as you fee in the Margin.
Here the Multiplier, con-
fifting only of a Repetend,
after the Multiplication is
finish'd, I repeat ſo many
Figures of the Product as
will reach one Figure be-
yond the Product, fetting
down the firſt Figure under
the Fifth, becauſe the Re-
petend in the Multiplier
confifted of four Places;
and then I count the fame
Number a 2d Time from
that, and do the fame a-
9. If the Repetend of the Multiplier, confifting of many
Figures, have one or two terminate Figures added to, or join'd
with it, fet thofe terminate Figures under the Figures of
the Repetend, from the Right-hand to the Left, and ſub-
ftract, and efteem the Remainder as a Repetend, with which
work as above.
Multiplicand,
Multiplier.
Multiply
by
Subftract
789,4378
6,3436
6 the terminate Figure,
and
6,3438 is the new Multiplier.
236831340
23683134
31577512
47366268
5007,40396540
500740396, &i.
50074, &c.
5007,9047858# the true Product.
Here
190 Multiplication of DECIMALS.
Here you fee that there is only one terminate Figure in
the Multiplier, viz. the 6, which being fet under the 6,
towards the Right-hand, and fubftracted, the Remainder is
6,3430 for a new Multiplier; and then the Work is per-
form'd, as you fee in the laſt Example.
10. If the Multiplier has feveral terminate Figures join'd
with the Repetend, and the Repetend confifts of but a few
Places, work the repetend Figures firft, as before in Numb.
9; and then to this Product, add the Product of the other
Figures, and the whole Product will be the true Product
fought.
Multiply
432,43
by
23,4x4
172972
43243
60,5402
60540, &c.
605, &c.
6, &c.
61,1517 Prod. of the Repetend 14
1729720
129729
86486
10124,977, whole Product.
II. If both the Multiplicand and the Multiplier are in-
terminate, or have compound Repetends, the Number of
Places of the Repetend in the Product, will be various and
uncertain, which must be determin'd by continuing and rc-
peating the first Product, which ftill contains a Repetend e-
qual to the Number of Places in the Multiplicand.
The Managing of thoſe Multiplicands and Multipliers, that
have large Repetends, confifting of many Figures, requires a
great deal of Pains and Trouble, and is more curious than
ufeful, and therefore I fhall not profecute them any further;
for 'tis very rare that any Product can require above fix Places
of Decimals, and commonly 3 or 4 are fufficiently exact;
therefore what has already been taught, will fully anſwer
all neceffary Occafions, for which reafon I will now go
on to
Divifion
t
Divifion of DECIMALS.
191
Divifion of DECIMALS.
Which is the fame with Divifion of whole Numbers, re-
fpect being had to diſtinguiſh the true Value of the Quo-
tient; which may be done by this General Rule, viz.
If the Divifor be large, add to the Dividend as many Cy-
phers as you think convenient, in order to make the Quotient
have a fufficient Number of Decimal Places, when it ſo hap-
pens that there will be a continual Divifor, to render the Answer
tolerably exact; and in thofe Cafes, where you cannot render it
compleat, you must continue it to 2, 3, 4, 5, 6, &c. Places,
more or less, according to the Value of the Integer; and to
know when you have fo done, you must count the Number of De-
cimal Places in the Dividend, reckoning the Cyphers that were
added, if any; and then count the Number of Decimal Places
you have in the Divifor, the Number of which fubftract out of
the Number of thofe of the Dividend, and the Difference is the
Number of Decimal Places that must be in the Quotient; and
if there are not fo many Figures in the Quotient, as the faid
Difference fo found amounts to (as in fome Cafes there will not)
then add fo many Cyphers on the Left-hand of the ſaid Quo-
tient as will make them up, and then that Decimal will be.
the Answer fought. Likwife if the Dividend have not ſo many
Decimal Places as the Divifor, you must add ſo many Cyphers
at least, as to make them up; and then the Quotient will be
all whole Numbers; therefore if the Remainder be any thing
confiderable for Value, continue adding ftill more Cyphers, and
divide as before to have 1, 2, 3, 4, &c. Places of Decimals.
The following Example may ferve as an Illuftration of the
whole Work.
Suppoſe 'twas requir'd to divide 3215862756384 by
789654.
The Quotient would be 4072495, and the Remainder will
be all o's.
Now, according to the Number of Decimals and whole
Numbers, you will fuppofe in either the Divifor or Dividend,
the Quotient will be of a greater or leffer Value. As, fup-
poſe the Diviſor all whole Numbers, and the Dividend con-
fifted of fix Places of whole Numbers, and feven Places of
Decimals; then the Quotient, which confifts of feven Figures,
will be all Decimals: But if the Divifer was all Decimals,
and the Dividend all whole Numbers, you muſt imagine fix
Cyphers to be added to the Dividend, and then the Quo-
I
tient
192 Divifion of DECIMALS.
tient will be exprefs'd in its true Value, having thirteent
Figures, which are all whole Numbers, viz.
4072495000000.
Again, Suppofe the Divifor confifted of three whole Num-
bers, and three Decimal Places, viz. 789,654, and the
Dividend was as before 321586,2756384, the Quotient would
be 407,2495; and fo on in all the Cafes and Varieties that
may, or can happen, with many or few Places of whole
Numbers or Decimals.
The following Notes may be of great Ufe, in many
particular Cafes, viz.
1. If you are to divide any Number by 10, 100, 1000,
&c. you have nothing more to do but to make the ſeparating
Point 1, 2, 3, &c. Places more, towards the Left-hand,
than it ſtood before, and you have the Quotient or true An-
fwer fought. And this in many Cafes of Trade and Bufinefs
may be very uſeful, as fhall be hereafter fhewn in the Rule
of Practice, &c.
2. If you would have a fix'd Number, and fo multiply
inſtead of dividing, add Cyphers to Unity, and divide by the
given Divifor till o remains, and the Quotient fhall be the
Multiplier fought.
As, Suppoſe I would multiply the Number 345 by an equi-
valent Factor, to make the Product equal in Value to the
fame Number divided by 5. In this Cafe my Multiplier
muft be,2; as appears by the following Work.
5 | 345 1
69 Quotient.
345
,2
69,0 Product.
Again, Suppoſe I was to divide by 125; inſtead thereof,
I multiply by ,008: Thus,
125 15000 | 120
125
250
250
15000
,008
120,000
Again,
Divifion of DECIMALS.
193
Again, Suppoſe I was to divide the Number before
mentioned by 8, my Multiplier would be 125; as appears
by the following Work.
8 | 15000
1875 Quotient.
15000
,125
75000
180000
1875,000 Product.
After the fame Manner you may go on with any other
Figures, regarding only the Value of the Divifor or Multi-
plier, which will determine the Value of the Quotient or
Product, by the Rules heretofore taught you.
And contrarily you may convert a Multiplier into a Di-
viſor, by dividing Unity with a proper Number of Cyphers,
by the given Multiplier, and the Quotient ſhall be the Di-
vifor fought. As, ſuppoſe, inſtead of multiplying any Num-
ber by ,03125, I would divide the faid Number, ſo that
the Quotient fhould be of the fame Value with the Product;
What must be the Divifor?
Anſwer, 32. For by dividing Unity by ,03125, there
comes out 32 whole Numbers for the Divifor. Now mul-
tiplying any Number by a Fraction gives the Product lefs
than the Dividend, in the fame Proportion that the multi-
plying Fraction bears to Unity; as was obferv'd to you in
the Multiplication of Vulgar Fractions, Page 157, and will
evidently appear by the following Example.
Multiply 3456 by ,03125, and the Product will be 108.
Likewife if you divide 3456 by 32, the Quotient will be
108; as appears plainly by the Work at large here-under.
3456
,03125
32 | 3456 (108
32
17280
6912
3456 ·
10368
256
256
108,00000
Cc
But
194
Divifion of DECIMALS.
But if 'twas to be in Questions, where many Statings in
the Rule of Three was concern'd, as in Fellowship, &c. then
you muſt divide the fecond Number by the firft, and that
Quotient will be the common Multiplier; as fhall be ſhewn
hereafter.
3. As in Multiplication, where the Factors were large,
fo in Divifion, where the Divifor and Dividend confift of
many Figures, the Work may be very much ſhortned,
and the Quotient ftill retain the true Value to 3, 4, 5, 6,
&c. Places. Firft determine how many Figures you will have
in the Quotient, and then multiply the Divifor with the firſt
2, &c. Quotient Figures, as in common; but for every Fi-
gure after, fkip one, adding to the next Figure what would
have been carried, if it were not fo cut off; as in the fol
lowing Example, where I am defirous of having three Places
of Decimals in the Quotient.
Be the Divifor. The Dividend. The Quotient.
Let 21.234 (1236.97286790 (58,254
1061.70
:
275.27
16987
540
425
115
106
09
8
I
Here I only multiply the firft Line whole, and drop the
4; the next Line the 3, the next the 2, the next 1, &c. but
carry 1, 2, 3, &c. Tens, if the Product of the foregoing
Figure comes from 5 to 10, from 15 to 20, from 25 to
30, &c. The whole Work at large will ftand thus:
Whole
Divifion of DECIMALS.
195
บ
Whole Work at large.
21,234 | 1236,9728: 6790 (58,25435
106170
175 272 :
169872 :
540 08 :
424 68:
115 40 : 6
106 17:0
923: 67
849: 36
74: 319
63 702
10 : 6170
10: 6170
Remains.
Whole Work contracted.
21,234 | 1236,9728.6790 (58,25435 ·
(
106170
175272
169872
54008
42468
11540
10617
.923
849
15 | 20
Remains.
C c 2
As
196
Divifion of DECIMALS.
As you have been taught to manage circulating Numbers,
in the three foregoing Rules; fo you fhall be here alfo
taught what may be generally neceffary in Divifion, and
therefore I will proceed gradually, and fhew you,
1. To divide any given Number that has a fingle Repetend,
by one or more terminate Figures.
Here you muſt divide as in whole Numbers, and continue
your Divifion till you have a fingle or compound Repetend in
your Quotient, one of which will generally come out pretty
foon; and tho' fome Divifors will produce a Repetend much
fooner than others, and fome will never come to a Repetend,
as appears by the following Dividend, having the terminate
Digits, 2, 3, 4, 5, 6, 7, 8, 9, for Divifors.
2 | 3279.643 Dividend.
| 1639.8216, &c. Quot.
3 3279.64% Dividend.
| 1093.2144, &c. Quot.
43279.643 Dividend.
In theſe two Examples the ift
repeats from the taking down.
of the 2d Figure of the Repe-
tend in the Dividend, and the
2d from taking down the ift
Figure of the Repetend in the
Dividend.
In this Example, the Quot.
begins to repeat at taking
819.910833, &c. Quot. down the 3d Figure of the
53279.643 Dividend.
| 655.92886, &c. Quot.
613279.643 Dividend.
1 546.60772, &c. Quot.
7 | 3279.643 Dividend.
Repetend in the Dividend.
In theſe two Examples the
Quotient begins to repeat at
the taking down of the 20
Figure of the Repetend in
the Dividend; theſe 5 Ex-
amples all produce ſingle Re-
petends in the Quotient.
| 468.52047619047, &c. Quotient.
913279.643 Dividend.
| 364.4048x, &c. Quotient.
In theſe two Examples, where the Divifors are 7 and 9,
the Quotients contain the compound Repetends, 4761g, and
.481, both beginning at the taking down of the firft Figure
of the Repetend in the Dividend.
I
8
Divifion of DECIMALS.
197
813279.643 Dividend.
【 409.9554166, &c. Quot.
In this Example the
Quotient repeats a ſingle
Figure, beginning at
the 4th Figure of the
Repetend in the Dividend.
Again, Suppofe 79.26 divided by 4,8 and 23,4
4,879,26 | is 16,531388, &c.
But if the fame Dividend was divided by 34,5, the Quotient
would be
2,297584541062801932367149758, &c.
which has a Repetend of 22 Figures.
And 195,04 by ,48 has 408,29, &c. for the Quotient.
'Tis needleſs to expreſs theſe Divifions at large, the Work
being the fame with common Numbers; only you muſt
mind to take down the Repetend Figures of the Dividend, e-
very time you make a Figure in the Quotient, as in the
three laft Examples; in the firſt and laſt of which 48 is the
Divifor; and in the firft, you continually take down 6; and
in the laſt you continually take down 2, &c.
2. If your given Divifor be a fingle Repetend, and your
Dividend a terminate Number, multiply the Dividend by ,9,
and divide that Product by the given Divifor. As,
Suppoſe,
789,1234 was to be divided by 6, f, &, &i.
9
,710,21106 the new Dividend.
1183,6851 Quotient by 6.
| 1014,5858 Ditto by f.
$| 887,763825 Ditto by 8.
ΤΟ
Or you may fubftract of the given Dividend from it
felf, and the Remainder divide by the given Repetend, thus:
789,1234.
78,91234
710,21106 the new Dividend, as before.
The
198
Divifion of DECIMALS.
The Reaſon of this Proceſs is, becauſe all fingle Figures
that are Repetends, are but fo many Ninths of Unity; tho'
when they ſtand without that Mark, they are ſo many
Tenths; therefore when you multiply the given Dividend by
99, 'tis the fame thing as fubftracting of itſelf, from it
felf; as appears by the foregoing Work: Or you might have
continued the Work at large, having regard to the multiply-
ing and ſubſtracting repetend Decimals, as has before been
taught you.
ว
Thus ) 789,12348000/1183,6851
666,66666666
12245673333
6666666666
5579006666
5333333333
245673333
200000000
45673333
40000000
5673333
5333333
340000
333333
6666, &c.
6666, &c.
0000
If the Divifor has terminate Figures join'd with the Re-
petend, fubftract thoſe terminate Figures from the whole
Divifor, and call the Remainder a new Divifor; and pre-
pare the Dividend as above taught, and work with the new
Divifor and Dividend, as in whole Numbers.
67,48
Divifion of DECIMALS.
199
67,48 | 3469,2875.
674
34692875
60,74 | 3122,35875 (51,405)
3037 O
85 35
6074
24618
24296
32275
30370
1905, &c.
If the given Divifor has a compound Repetend, join'd to
one or more terminate Figures, fet it under it ſelf ſo many
Places towards the Right-hand, as the Repetend confifts
of Figures, and fubftract as before, and the Remainder fhall
be a new Divifor; likewife do the fame by the given Divi-
dend for a new Dividend, with which new Divifor and Di-
vidend work as in whole Numbers: But if the given Divifor
has a Repetend, and no terminate Figures, you must not fub-
ftract any thing from it; but the Subftraction muſt be made
from the Dividend only, even tho' it fhould confiſt of all
whole Numbers, obferving always to put the Dividend fo
many Places forwards as the Repetend of the Divifor confifts
of Figures; and if the Divifor has no Repetend in it, then
you must not fubftract from either the Divifor or Dividend,
although the given Dividend fhould have a Repetend.
Examples of each Sort follow. And firft the Divifor joined
with Terminates.
Divide
200 Divifion of DECIMALS.
Divide 10124,97717 by 23,414
23,414) 10124,977x7 (
234 10124977
23,180) 1002372740 (432,43
9272
7517
6954
5632
4636
1
9967
9272
6954
6954
Here in this Question, the Anfwer is compleat, leaving
no Remainder, and the Quotient is terminate; but fuppofing
the fame Numbers given, and only let the Divifor be all of
it imagined a Repetend; then there will be a continual Re-
mainder, and the Quotient of courſe can only be an Approxi-
mate.
#3.41410124,977**
10124
) 10124,87593 (432,42
9365 6
759 27
702 42
56855
46 828
10 0279
9 3656
66233
19405 Remains.
To
46828
Divifion of DECIMALS.
201
to add as many Cyphers as you pleaſe, fuppofe 10, and
continue dividing, the Quotient will be increaſed to
432,428287776543, and the Remainder will be 11254;
and fo you may go on as long as you will.
But if the fame Divifor had been either a compleat De-
cimal or an Approximate, and fo had no Repetend, then the
Quotient would have been 432,43, &c. and the Remainder
6115, &c.
Before we leave this Head, it may be proper to take
Notice of the Reafon of the laft Rule, Page 199, and of the
laft Example in Page 200, wrought in confequence of that
Rule. The Example is 10124,977x, divided by 3,414:
Here Subftraction is made from the Dividend only, for if
the Divifor was imagined to be repeated thus, 3414,
#3414, &c. then it would ſtand thus :
#3414,73414, &c.
#3414, &c.
#3414,00000 Remainder.
For removing the Repetend of the Divifor, under its pro-
per Place, it would ftand under the 2d, 3d, &c. Repeti-
tion, and the Remainder will be 23414, only; for Cyphers on
the Right-hand of a Decimal neither increaſe nor diminiſh its
Value, as has been already obſerv'd. You may obferve
likewiſe, that the Quotient of this laft Example is a fmall
Matter lefs than the Quotient of the Example above it, by
fuppofing the whole Divifor in the laſt to be a Repetend, and
only a Part of it to be fo in the other; and this will al-
ways be more or more lefs, in Proportion to the Largeneſs or
Smallneſs of the Repetend given; for if you continue the
Work of the laft Example, Page 200, the Quotient would
be increaſed to 432,4282877765, &c.
The Deſign of the feveral Operations, in the various Sorts
of Repetends, is to render the Anfwer by Decimals, as per-
fect as by Vulgar Fractions, which in many Cafes, with-
out fuch Allowances, could not be, as appears from the
feveral Examples themfelves, when compar'd together; for
without the foregoing Subftractions, or confidering the Di-
viſors as Repetends, they would then be the fame as if 'twas
10124123 divided by 231, which according to the Rules
10124160000
D d
for
202
Queſtions in DECIMALS.
for dividing Vulgar Fractions, would make the Quotient to be
101249 7 7 17
2341400000
Enough has been faid of the preparatory Rules of Deci-
mals. We ſhall now proceed to fhew the Ufe of them in
working Questions.
Decimals are exceedingly uſeful in all fix'd Cafes, viz.
where particular Quantities or Prices are conftantly us'd;
for by making a fmall Table, you may with very little trouble,
tell the Amount of any Parcel of Goods whatever, and in
many thouſands of Inftances by Infpection, as fhall be pre-
fently fhewn; but the firft neceffary Thing will be, to teach
you the Method of making proper Tables for all manner of
Purpoſes, for theſe fort of Tables are generally defign'd for
particular Ufes; but the Method of making them is uni-
verfal. As, fuppofe I wanted to have conftantly by me
the Decimals to the feveral Parts of Coins, Weights, Mea-
fures, Time. &c. I fet down the Parts as a Vulgar Fraction,
and find the Decimal thereof, as has been taught you, Page
168. Thus :
What is the Decimal to a Farthing
480
960
may
be
A Farthing being the 9. Part of a Pound, the Decimal
to this Vulgar Fraction is, 0010416, which being multiply'd
by 2, as taught Page 187, the Product will be ,002083 for
the Decimal of two Farthings: Or you may confider it as
the of a Pound, and divide as above for the Farthing,
obferving, that whenever you come to a fingle Repetend, you
need go no further than the firit Figure. The fame
done for 3 Farthings, and the Decimal will be ,003125.
In like manner the Decimal of a Penny will be ,00416, of
2 Pence ,0083, of 3 Pence ,0125, of 4 Pence ,016, of a
Shilling ,05, of 2 Shillings,1, of 3 Shillings,15, &c.
Thefe being regularly fet down in a Table, as hereafter
ſhall be done, you need only look for what you want, and
if you can't have it at once, take it at twice, regarding the
Value of the reſpective Places, and add them together;
the Sum or Total will be the Decimal required. As, Sup-
poſe I wanted the Decimal of 3s. 3 d. I look into the
Table and find the Decimal of 3s. to be .15, of 3 Pence 3
Farthings to be,015625; thefe added together make
,165625, being both terminate Numbers, or perfect De-
cimals, as you may try by bringing the whole into 3
Farthings's, and they will make 53 for the Numerator or
Dividend;
2
1
Queſtions in DECIMALS.
203
"
Dividend; and then 320, the 3 Farthings in a Pound, will:
be the Denominator or Divifor; the Quotient arifing from the
Work will be,165625; as appears from the Example fol-
lowing.
3210153
32
210
192
180
160
200
192
80
64
160
160
(,165625
The like may be done in
all other Cafes, whether
they be Coins, Weights,
Meafures, Time, &c.
regarding the Propor-
tion that the lower
Species bears to the
upper Species, or that
which is the Integer;
for here, if a Shilling
or a Crown, &c. were
ſuppoſed to be the In-
teger, then the Decimal
of a Farthing, a Penny,
&c. would be different
in Figures, tho' not in
its real Value, from what
'tis here made to be:
For Inftance, Suppoſe
a Shilling was to be
efteemed the Integer, then the Decimal of a Farthing would
be,02833, viz. 20 Times bigger, or nearer to an Integer,
than'tis now a Pound is the Integer; becauſe the Price or Value
of the Species laſt mentioned, is but Part of the former. The
like will always happen in all Cafes whatever, in Proportion
to the Integers being confider'd of a larger or fmaller Value;
but whenever Money is concern'd, I would always recom-
mend a Pound Sterling to be the Integer, and then you may
value the Amount of your Work, either by the Directions
given you, Pages 176 and 177, or by the following Rule,
which with a little Practice will be much readier, ſhorter and
cafier, and at the fame time fufficiently exact for all com-
mon Buſineſs.
The RUL E.
If the Primes Place, or the firft Figure on the Left-hand
of your given Decimal, be any fignificant Figure, double it,
and reckon the Amount of that Double fo many Shillings; then
Dd 2
go
204
Queſtions in DECIMALS.
go to the two next Figures, and for every 25 reckon Six-pence,
which add to the former Sum; and if any thing still remains,
and is under 15, reckon them for fo many Farthings, and add
them to the Sum before found; but if above 15, abate 1, and
reckon the Remainder as fo many Farthings, which add as be-
fore, and the Work will be fufficiently exact for all common
Cafes. As, fuppofe I take the fame Examples, as are in
Pages 176 and 177, viz.
What is the Value of 75 Decimal Parts of a Pound Ster-
ling? Anfwer, 15 Shillings.
5
First, I double the 7, and it comes to 14; then to the
I add o to make it two Figures, and it is 50, which con-
tains 25 twice; which by the above Rule is two Six-pences, or
one Shilling, which added to 14 Shillings make 15 Shillings,
as in Page 176. Again,
What is the Value of ,378125 of a Pound Sterling? Anf.
7 s. 6d. 4. For doubling the 3 it makes 6, then the two
next Figures being 78, is 3 Times 25 and 3 over; which is
3 Six-pences and 3 Farthings; fo that the whole is 7 s. 6 d.
1, the fame with Page 177. Again,
3
What is the Value of ,51296 of a Pound Sterling? Anf
10 s. 3d. For 5 doubled is 10 Shillings; then the two next
Figures are 12, which are 12 Farthings or 3 Pence. So if
I enquire the Value of ,06932, &c. of a Pound Sterling,
the Anſwer by the above Rule will be I s. 4 d. ; which be-
ing wrought by the Directions given in Page 176, comes to
the fame; as appears by the Work at large here-under.
I
,06932
20
38640
12
41 63680
4
2 | 54720
The Parts which remain at laft, being
lefs than a Farthing, are always rejected,
becauſe in Buying and Selling no Body
can pay less than a Farthing; but in
Computations where the Price of one
Thing may be given at the Parts of a
Farthing, and the Price of many is re-
quired, you muſt then work with the
whole Number and only reject the
Parts in the Total Amount. Now that
no Cafe may happen wherein you may
be at a Lofs, I will give two or three
Examples more of this Kind.
What is the Value of ,00976, &c. of a Pound Sterling?
Anſwer, 2 d. 4. Here the firſt Figure is a o, confequent-
the Amount, when doubled, is ftill o, and the two next
1.
Figures
Queſtions in DECIMALS.
205
9
9
Figures are 09, which ſtanding but for 9 in Numeration,
muſt be deemed 9 Farthings, or 2 d. 4. So the Value of
,8023, &c. is 16s. 00 d. 1, of ,01741, &c. is 4 d. Note,
If it exceeds 15, the Value of the given Decimal will be a
Farthing too little by this Rule, when the fourth Figure is an
8 or a 9; and alfo a Farthing too much, if the given Deci-
mal is under 15, when the fourth Figure is 0, 1, 2, 3 or 4;
as will more fully appear by the following Table of Decimals,
for Shillings, Pence and Farthings, a Pound Sterling being
the Integer.
A Pound Sterling the Integer.
Farthings Decimals.
| www
Pence
I
,0010418
,002083
,003125
Decimals.
,00416
I 14,0052083
Pence.
7.2
-ja me
~+~/c mi
770∞ ∞ ∞
8
8/1
81
Decimals. Shillings. Decimals.
,0385418 19 95
,03125..
,0322916
,033.
12
6
13
,65
14 7
,034375.
15
75
,035416
16
,8
,0364583
17 ,85
,0375... 18
›9
1 1/1/10
,00625.
9/2
,039583.
I
,0072918
94
,040625.
2
,0083
IO ,0416
EXPLANATION.
2
,009375·
10,0427083
2
,010416
ΙΟ
,04375..
By this Table you
2
3
,0114583
,0125
II
,04588..
30135418
II
+
,046875.
3,014583
I I
,047916.
3,015625
II
4
,01666..
4
,0177083 Shillings.
4
,o1875.
4,0197916
5
02083
,021875
112
5
6
6
,022916
,0239583
,025....
,0260416
,027083.
736
6
ང་ ང་
7
1,028125.
,02918.
7 ,0302083 II
1234 NO NO σ
,05
, I
,15
,2
6
,3
,35
,4
9
,45
IO
,5
10,0447916 cimal to any Part
ling. Thus, fuppofe
I wanted 9d., I
,0489583 look under
Decimals.
,25
Title of Pence, and
at 9 ftands
039583 for its De-
cimal. Again, If
I want the Decimal
to 13s. 10 d. 1
firft look for 13
Shillings, which is
,65 then 10
is,0427083
Tot. is,6927083
may
find the De-
of a Pound Ster-
the
,55
for
· 206
Queſtions in DECIMALS.
for the Decimal requir'd. But ordinarily,6927 would ſerve,
the other three Figures being of but ſmall Value; only in
the Tables I have put down all the Figures, till they ei-
ther terminate or repeat; but in all common Cafes, 3 or 4
Figures are fufficient, becauſe the Defect, if any, will be too
infignificant to be taken Notice of.
TROY WEIGHT. A Pound the Integer.
Ounces.
Decimals.
Dwts.
Decimals.
1234no n∞
I
,0883
17
90708%
,1866
18
,075..
,25..
19
,07916
>8333
4186
Grains.
Decimals.
6
,5 ...
7
,5833
8
,8666
9
IO
II
75..
8333
,9186
Dwts.
Decimals.
123456 7∞
I
,0001736x
,0003477
,00052083
,000694
,00086805
,00104166
7-0
,0012152
8
123
4
56 78
,00416
,00883
,0125.
,01666
,02083 13
9
,00138
,0015625
ΙΟ
,00173611
I I
,00190972
12
,002083
,00225694
,025..
14
,0024308
,02916
15
,00260416
,03333
16
,00277
9
,0375.
17
,00295138
ΙΟ
,04166
18
,003125
II
04583
19
,00329861
12
,05...
20
,003472
13
,05416
21
,00364583
14
,058%3
22
,0038194
15
,0625.
23
,00399308
16
,08666
EXP LA.
Queſtions in DECIMALS.
207
EXPLANATION.
Here, as in the former Table of Money, each Decimal
is carried on till it either terminates or repeats, which
you may perceive by the mark'd Figures, 3, 6, &c. for which
Reaſon they are frequently continued to 7 or 8 Places, tho'
ordinarily 3 or 4 would do; but as it fo happens, that in
the Grains eſpecially, they don't circulate till the 8th Place,
they are for that Reaſon continued fo far: In fome Places
they are continued but to the 5th Place, the Reaſon whereof
is, that there they either terminate or begin to repeat, and
are fo marked. The Use of this Table is the fame in Troy-Weight
as the former was in Money, as appears by the following
Example, viz.
Oz. Dwts Gra.
What is the Decimal to 9: 13: 17 Troy?
Anf. >75-- for 9 Ounces.
,05416 for 13 Penny-Weights,
and ,00295138 for 17 Grains.
The Total 8071180g is the Decimal to the Whole, as
you may try by turning the
Oz. Dwts. Gra.
given Quantity 9 13 17 into a Vulgar Fraction; and
then 'twill be 2, whofe Decimal is ,80711805, as above.
The fame is to be obferv'd of all others.
276
Avoirdupoife Weight. One Hundred the Integer.
7 ,00024414
8
,00027902
9
,00031390
Drams.
Decimals.
Ounces.
Decimals.
I
12345O D
,00003488
I
,00055803
,00006975 2
,00010463
3
,00111607
,00167411
I
,00013950 4 ,00223214
,00017438 5 ,00279017
,00020926 6 00334821
9 ,00502232
,00390625
,00446428
IO
,00034878
ΙΟ ,00558035
II
,00038365
I I
12
,00041853
12
,00613839
-,00669642
13
,00045341
13
,00725445
14
,00048828
15 ,00052315 15
14 ,00781249
,00837053
Avoir-
208 Queſtions in DECIMALS.
Avoirdupoife Weight. One
Hundred the Integer.
Pounds.
I
2
34no 7∞
Decimals.
,00892857
,01785715
,02678572
,03571430
,04464287
505357145
,0625...
One Pound the Integer.
Drams.
Decimals,
I ,00390625 -
,0078125.
,01171875
,015625..
,01953125
,0234375.
,02734375
1234 no 7∞
,08035715 9 ,03515625
07142857
8
,03125...
9
ΙΟ
,08928572 ΙΟ
,0390625.
II
,09821430 II ,04296875
12
,10714287 12
,046875..
13
,11607145 13
,05078125
14
,125.
14
,0546875.
15
13392857 15 ,05859375
16
,14285715
Ounces.
Decimals.
17
15178572
18
,16071430
19
,16964287
2
20
,17857134
21
1875.....
3
22
,19642857
23 ,20535715
24
,21428572
25
,22321430
*~~+ 56 .780
I ,0625
,125.
,1875
4
,25..
,3125
,375.
,4375
26
,5...
,23214287
9
5625
27
,24107145
ΙΟ
,625.
Quarters.
Decimals.
II
,6875
12
>75..
1 2 3
I
,25
13 ,8125
,5
14
,8750
75
15
9375
EXPLANATION.
Ι
Here in thefe Tables, firft one Hundred Weight, or 112lb.
is made the Integer, and the Decimal is carried to eight
Places for the Sake of Exactnefs; but I Dram coming out,
only an Approximate is put down the neareſt to the Truth,
fometimes a ſmall Matter too much, and fometimes too
little;
Queſtions in DECIMALS.
209
little, and now and then corrected to make it ſtill more´ ex-
act, becauſe the Decimal for 1 Dram is taken a little too big.
In the fecond Table, a Pound Avoirdupoife is made the Integer,
of which a Dram being an even Aliquot Part, comes out
exactly to ,00390625, which is a perfect Decimal: So that
the Drams and Ounces are all of them perfect or terminate
Decimals. In moft Bufineffes a C. is the higheſt Name made
ufe of, and therefore 'tis here made the higheft Integer; but
if you would convert any of thefe Decimals into Parts of a
Ton, 'tis but dividing the Decimal here fet down, in the firſt
Table, by 20, and your Work is done.
In the fame Manner you may calculate Tables of other
Weights, and of any Sorts of Meaſures, &c. I fhall now
proceed to fhew you how to make or calculate ſmall Tables,
by which you may compute the Value of any Quantity of
Goods whatever, let the Name or Species of them be what
they will.
First turn the Price or Value of one Thing, let it be Meaſure,
Weight, &c. into a Decimal by the Table of Coin before ſet
down, Page 205, and then multiply that Decimal according to
the Rules given in Multiplication of Decimals, by the feveral
Digits, viz. 1, 2, 3, 4, 5, 6, 7, 8, and 9; and thoſe ſeveral
Products fhall be the Amount of fo many Yards, Pounds,
Hundreds, Pieces, &c. at that Rate: And if you want the
Value of 10, 20, 30, 40, &c. or 100, 200, 300, 400, &c.
or 1000, 2000, 3000, 4000, &c. you have nothing more to
do but to fet the feparating Point, 1, 2, 3, 4, &c. Places more
towards the Right-hand of your Tabular Product, and all
the Places on the Left-hand of the feparating Point will be
Pounds Sterling, and the remaining Parts towards the Right-
hand will be the Decimal of a Pound Sterling.
EXAM P LE.
Suppoſe I would know the Price of any Number of Things at 4s.
gd. each. By the Table, Page 205, I find
the Decimal of this Sum to be ,2375, which
being fet down and multiply'd by the ſeveral
Digits, the Products will be as you fee in
the Margin; to uſe which, fuppofe the
following Question was aſk’d,
123
,2375
475.
57125
4 95..
5 1,1875
6 1,425.
7
1,6625
81,9...
912,1375
What is the Amount of 670 Yards,
Gallons, Bufhels, Pounds, &c. of any
Thing, at 4s. 9d. per Yard, Gallon,
Bushel, Pound, &c?
Ee
First
210
Queſtions in DECIMALS.
Firſt look for 6, and fet down the Figures 1425 belong-
ing thereto; and becauſe the Question is Hundreds, I make
the Point two Figures more towards the Right-
,
hand, than it ftands in the Table; thus - - 142,5
then I look for the 7, and becauſe the Number
V
is 70, I put the Point 1 Figure forward; thus - 16,625
add theſe together, and they make 159,225,
or 159 Pounds 4 Shillings and 6 Pence. In like L. 159,225
Manner may any other Number be done. As,
EXAMPLE 2.
What comes 1358 Bushels of Wheat to, at 4 s. 9 d. per
Bushel ?
Value
237,5
71,25
11,875
1,9
322,525
II II II II
Quantity
: :
1000 Anf. L. 322 10 6, as appears
300 by the Work in the Margin. And
50 fo in like Manner for any other
8 Quantity, great or fmall, at this
Price.
1358
The like may done for any other Rate or Price, great
or ſmall, by only calculating fuch a ſmall Table at the fix'd
Price. As, fuppofe 'twas L. 12 : 12 : 6. A Table made
for this Price, will be as here under.
EXAMPLE.
If I buy 113 Hogfheads of Wine, at L. 12: 12: 6 per
Hogfhead, What is the Amount? Anf. L. 1426: 12: 6, as
per Work hereunto annexed plainly appears.
1 2
I
3
12.625
25.25.
37.875
Value
Quantity
4
50.5.
•
1262,5
100
5 63.125
*75.75.
7 88.375
8 IOI....
9 113.625
ΙΟΙ.
Total 1426,625 = 113
126,25
37,875 =
ΙΟ
3
Again,
Of
211
PRACTICE.
Again, Suppoſe the Price 16 d. 1, the
Table will be
EXAMPLE.
I |,0677083
2,1354166
3,203125.
What is the Value of 13618 Yards of Iriſh- 4,2708333
per Yard? Anfwer, 53385416
Linen, at 16 d.
L. 922 I: 00, as appears by the 6,40625..
following Work.
Value
677,0833
203,125.
40,625.
,677.
5418
II II II II II
7473958%
81,5416666
Quantity 9,609375.
10,000
3,000
,600
IO
8
922,0520 = 13,618
I prefume enough has been faid upon this Subject to
make it entirely plain, from whence any Body may calcu-
late fuch Tables as the feveral Commodities they deal in may
require, with very little Trouble and great Certainty, and fo
have them always ready for Ufe. We fhall now go on to
fhew the Uſe of both Vulgar and Decimal Fractions, in the
feveral Rules ufed in Trade and Bufinefs, together with the
Method of computing Tables of Intereft, and Rebate, both
Simple and Compound.
CHAP. VIII.
Containing the Rule of PRACTICE, wrought after
feveral different Manners.
P
RACTICE is that Rule, by which Merchants com-
monly caft up the Value of their feveral Commodities,
and is in Effect only a compendious or fhort Way of per-
forming the Rule of Three.
In this Rule, the Price or Value of one Thing is given, to
find the Price or Value of many.
Ee 2
In
212
Of PRACTICE.
In order to perform which, diverſe Methods have been in-
vented; and indeed the Number of Cafes are fo many and
fo variable, that though one Univerfal Method may be laid
down, that will Anfwer all Cafes, yet that Method will not
always be the ſhorteſt, nor indeed, in many Cafes, the best.
Therefore I will fhew you two General Methods, by which
all Questions may be wrought; the one by Decimals, the other
by common Numbers; and alfo the beft particular Methods
for working any Propofition both Ways; by which
you will
gain not only a Readiness in the Execution of any Thing
that is to be done by this Rule, but alſo be affured of the
Truth and Certainty of what you do.
Note, Whatever your Species of Goods are, whether Horſes,
Sheep, Cloth, Grain, or any other Commodity, you muſt
imagine them to be chang'd or converted into ſo many Pieces or
Sums of Money, as is the Price or Rate of one given Thing,
whether it be Pound, Yard, Bufhel, Carcafe, &c. As for
Inſtance, Suppoſe Iafk, What is the Value of 265 Ells of Hol-
land at 5s. per Ell? This is all one, as if I aſk'd, What comes
265 Crowns to? So in like Manner, if I afk, What comes
95 Sheep to, at 27 Shillings per Sheep? 'tis the fame Thing
as if I had afk'd, How many Pounds Sterling are there in
95 Moidores? In like Manner, if I aſk, What is the Amount
of 195 Truffes of Straw, at 9 Pence per Trufs? 'tis the fame
Thing as afking, What comes 195 Nine-pences to? And fo
in all other Cafes, whether the Price be above or under a
Pound Sterling.
The first General Rule for all Cafes and all Prices.
I. If the given Price be under a Pound Sterling; find
the Decimal for that Sum, and multiply it by the Quantity, or
the Quantity by it, and the Product is the Anfwer fought.
2. If the Price given be whole Pounds only; multiply
the Quantity by thofe Pounds, or thofe Pounds by the Quan-
tity, as you fhall think beft, and the Product will be the
Antwer fought.
3. If the Price given be one or more Pounds, and fome
Shillings, Pence, or Farthings; find the Decimal to the odd
Money, and add the Pound, or Pounds Sterling given, to it,
and then either multiply the Quantity by the faid mix'd Num-
ber that repreſents the Price, or the faid Price by the Quan-
tity, as is moft convenient, and the Product will be the
Anſwer fought; as appears by the following
I
EX
Of PRACTICE.
213
EXAMPLES.
At 4 s. a Pair, What comes
Pair of Stockings to?
The Decimal to 4 s. is,2, by which I mul-
tiply the whole Quantity given, and the
Product is 63,2, whoſe Value is 63 Pounds
and 4 Shillings, according to the Rule
316
,2
63,2
given you, Page 203, to find the Value of a Decimal in the
known Parts of a Pound Sterling, by Inſpection.
What comes 718 Pair of Shoes to,
at 5 s. 6 d. a Pair?
718 Shoes.
,275 Decimal to 5 s. 6d.
3590
5026.
1436..
The Decimal to 5 s. 6 d.
is 275, and the Product is,
197,450
Or, L. 1979 :
What comes 3156 Yards of Iriſh-Linen to, at 18 d. ½ per
Yard?
The Decimal to 18 ½ is
The Quantity is
,677083
3156
462499
3854186
7708333
231249999
243,275000
The Product is
Or, L. 243 05: 6
But the laſt Example, and all others under the like Cir-
cumſtances, may be ſhortned by the Rule of contra&ing Mul-
tiplication, given you, Page 183.
Note, In all fuch Cafes as thefe, three Places of Decimals
are fufficient to keep in the Product; and therefore you may
fet down the last Work as follows,
077083
214
of PRACTICE.
077083
6513
Here the Product is 275
231250
in the Decimal, which by
7708
the Rule, Page 203 comes
to 5 s. 6 d.
3854
463
243,275
More EXAMPLE S.
At 1 d. per Yard, What comes 38 Pieces of Ferret to,
each 36 Yards?
Here I multiply by the
uppermoft Figures, re-
membring the firft Fi-
gure of the Multiplicand
is a Repetend.
36
38
288
108
1368 Yards in all.
,0052083 Decimal to 1 d. 4.
.416666
3124999
15624999
52083333
7,125…….. Or L. 7 : 2 : 6
Or thus contracted.
,0052083
8631
•
..5208
. 1563
312
42
7,125 as above.
In this, and fuch like Contractions, as there is much La-
bour fav'd, fo there is fome Judgment required; for in the
fecond
Of PRACTICE.
215
fecond Line, in multiplying by 3, I take 3 for the foregoing
Carriage, tho' the Product immediately before it comes to
but 24, becauſe the Figure before it is a Repetend, which if
multiply'd, would have made it 25; but I carry o the
next Time when I multiply by 6, though the o would have
5 for its Product, if the whole Line was multiply'd; ſo that
this fets it even by taking an Unit too much one Time,
and too little the other Time.
At 3 Farthings a Piece, What comes 4617 Oranges to ?
4617
Anf. L. 14:8: 64.
,003125
You may contract this,
but the Figures fav'd,
23085
being but few, I work
9234.
it at large.
4617..
13851...
14,428125
At 11 d. per Pint, What comes 27 Pipes of Wine to?
1008 Pints in a Pipe.
216
27..
27216 Pints in all.
Decimal.
,046875 to 11 d.
61272
937500
328125
4 44
This Work is contracted, and
the Anſwer is, L. 1275 : 15.
9375
469
281
1275,750
Here are only 3 Decimal Places referv'd in the Product,
and therefore the Units Place of the Multiplier is put under
the 6, which is the third Place of the Multiplicand; and be-
cauſe the Mutiplier confifts altogether of whole Numbers,
they are fet even beyond the Multiplicand; and in fuch Cafe
thofe Places are always fuppofed to be fupply'd by Cyphers
or o's.
At
216 Of PRACTICE.
At 13 s. 3d. i per Piece, What comes 968 Pieces to? Anf
L. 64364.
13 s. 3d. is ,664583 in Decimals.
869 Pieces inverted:
598125
39875
5317
Product is 643,317 Or L.643: 6:4
Three Places of Decimals being fufficient to keep in the
Product, I put the Units Place of the Multiplier under the 4,
and work as before; and parting off 3 Figures, towards
the Right-hand of the Product, all the other Figures to-
wards the Left-hand are whole Numbers.
In 716 Moidores, each 27 Shillings, How many Pounds
Sterling?
716 Moidores.
1,35 the Decimal to 27 Shillings, L. 966, 60;
or,
Anf. L. 966: 12:
3580
2148.
716..
966,60
In 849 Portugal Pieces of Gold, Value 36 Shillings each,
How many Pounds Sterling?
849 Pieces.
1,8 the Decimal to 36 Shillings.
6792
849.
1528.2 Anf. L. 1528:4:-
In
Of PRACTICE.
217
:
In 629 Portugal Pieces of Gold, Value L. 3: 12 s. each,
How many Pounds Sterling?
629 Pieces of L. 3 : 12 each.
3,6 the Decimal to L. 3: 12.
3774
1887
Product 2264,4 Or L. 2264 : 8.
In 1629 Pieces of Portugal Gold, Value 4 s. 6d. each,
How many Pounds?
1629 Pieces.
,225 Decimal to 4 s. 6 d.
8145
3258.
3258..
366,525 Anſ. L. 366:10:6.
In 519 Pieces of Portugal Gold, Value 9 Shillings each,
How many Pounds Sterling?
519 Pieces.
45 Decimal to 9 Shillings.
2595
2076.
233,55 Anfwer, L. 233: 11:-
Bought 365 Quarters of Malt at L.1: 3:6 per Quarter,
What come they all to?
365 Quarters.
1,175 the Decimal to L. 1: 3: 6.
5875
7050.
3525..
Here I multiply by the upper
Number.
428,875 Anſwer, L. 428: 17: 6.
Ff
·
As
218
Of PRACTICE.
At L. 8:7: 11 a Piece, What come 95 Oxen to? An£
L. 797: 12: 1.
1. s. d.
II
8,39583 Decimal to 8:7: 11
59 Oxen.
755625
41979
the 95 is inverted and the
Work is contracted.
Product 797,604 equal to L. 797: 12: 1.
If I have 5 Wedges of Gold, that
11 Dwt. 17 Gr. worth L. 3: 19: 10
the Whole worth? Anfwer, L. 1332:
Oz. Decimals.
333,585416
573 993
1000756
300 227
30023
I OOF
234
weigh 27 lb. 9 Oz.
per Ox. What is
5 1 1.
3
:
You will obferve, that in mul-
tiplying the 3d Line, or 2d 9, I car-
ry 8, becauſe if the proper Allow-
ance was added to the foregoing 8,
'twould come to 78; and in mul-
tiplying by the 7, if the due Allow-
ance was made to the foregoing 3,
'twould come to 25. You will alfo
obferve, that I bring the Pounds
Troy-Veight, into Ounces, and make
an Ounce the Integer, and fo take
the Decimal Parts of 11 Dwts. 17
Gra. only; but you may multiply
the given Price of the Ounce by 12, and fo reckon a Pound
Troy-Weight for the Integer, and take the Decimal for 9
Ounces 11 Dwts. 17 Grains, as is done below.
16
1332,257
E. 3
: 19: 10 the Value of one Ounce.
12
47: 18: 6 ditts of twelve Oz, or 1 lb.
*
ib's.
Of PRACTICE.
219
lbs. and Decimal
27,798784
529,74
11119 51
1945 92
250 19
556
I 39
1332,257
Oz.Dwts.Gra.
[Integer,
of 9:11: 17 a lb. Troy being the
is the inverted Value of 1 lb. viz.
of L. 47 and 925 the Decim. of
18s. 6 d. whofe Product comes to
the ſame with the former, as
and
appears by the Work;
which is an infallible Proof that
both Operations are truly
wrought.
Here you may plainly ſee the
Uſefulneſs of thefe Contractims,
for if the two laſt Examples had
been wrought at large, they would have ftood thus:
The firft Operation of the laſt Question wrought at large.
Oz. DecimalParts. Dwts. Gra.
333,585416 of II: 17
3,99375 = L. 3: 19: 10 1.
16167 927083
23350 979166
1000 75 624399
30022 68 749999
30022687 499999
1000756/24 999999
1332,2567 5781250
Here you fee all the Figures on the Right-hand of the
Line are left out in the Contraction, and the Product of the
Contraction is only of the 3d Figure bigger than the Work
at large, or about the Part of a Farthing in Value.
The fecond Operation of the fame Example at large.
lb. Decimal Parts Oz. Dwts, Gra.
27,798784 of 9: 11:17
47,925 of L. 47: 186
138919 3920
5559 7568.
250189056.
194591488..
111195136..
1332,256723200
In
Ff2
220
Of PRACTICE.
In the ſecond Operation at large, you perceive that the
Product is the fame with the firft, to the fifth Place of
Decimals, where it differs; which Difference arifes from
the Parts of the Ounce being a Repetend, and work'd fo;
whereas the other is only an Approximate, which occafions
the Anſwer to be about the Part of a Farthing lefs in
Value than the other.
20
All other Examples whatever, anfwerable by this Rule,
are wrought after fome of the foregoing Methods; fo that
I will not repeat any more to be wrought by Decimals,
but propofe the fame Queftions again to be wrought by
The fecond General Rule or Method for working Practice in
common Numbers.
Here 'twill be proper to bring the given Price into the
loweſt Name mentioned, and multiply the Price (fo reduced)
by the given Quantity, and the Product will be the Anfwer
to the Queſtion in the Name you reduced your Price to, which
muſt be brought into Pounds, Shillings, Pence, &c. as Occa-
fion may require.
EXAMPLES.
Queft. 1. At 45. a Pair, What comes 316 Pair of Stockings to?
4 Shill. Price of 1 Pair.
I
2 | 0 | 126 | 4 whole Value in Shill.
L. 634 Anf. as in Page 213.
Q.2. What comes 718 Pair of Shoes to, at 5 s. 6d. a Pair?
I I
718
718
2
II Sixpences.
40178918 the Anſwer in Sixpences.
L. 1979, as before in Page 213.
Note, When the given Price is 6d. 4d. 3d, &c. as 5s. 6d.
75. 4d. or 8s. 3d, &c. you need not bring the Money into
Pence, but Six-pences, Groats, Three-pences, &c. as you will
find done in fome of the following Questions.
Q+3+
Of PRACTICE.
221
}
Q. 3. What comes 3156 Yards of Irish Linen to, at 18 d. per
37 Price of 1 Yd. in Halfpence 2
Yard ?
22092
9468.
37 Pence
2 | 116772 Price of the Whole in Halfpence.
12/58386 Price in Pence.
2|0|48615:6 Price in Shillings.
L. 243:5:6 Anſwer, as in Page 213.
Q.4. At 1 d. per Yd, What comes 38 Pieces of Ferret to, each
4
5 Farthings.
36
228
114
36 Yards?
1368 Yards in all the Pieces.
5 Farthings that a d. coft.
4 6840 Value of whole in Far.
12 | 1710 ditto in Pence.
20 142: 6 ditto in Shill. Pence.
Anſwer,
L.7:2:6, as in Page 214.
Q.5. Atd. a Piece, What comes 4617 Oranges to?
3 Farth. Price of 1 Orange.
4 | 13851 Farthings all coft.
123462: 3 Pence and Farthings.
21012818:6 Shillings and Pence.
Anſwer, 14:8:64, as in Page 215.
1
Q. 6.
222
Of PRACTICE.
Q.6. At 11 d. a Pint, What comes 27 Pipes of Wine to?
4
1068 Pints in a Pipe.
45
216
4 1224720
2700
12 | 306180
27216 Pints in 27 Pipes.
45 Coft of a Pint in Far.
2 | 01255715
136080
Anſwer
L. 1275:15 108864
as before in Page
215.
1224720 Value of the whole
in Farthings.
s: d.
Q. 7. At 13: 3 per Piece, What comes 968 Pieces to?
12
1592 Pence.
968
319 Value of 1 in ÷ d
319 Half-pence.
8712
968.
2904..
2308792 Value of all in d.
12 | 154396 Pence.
2|0|12866: 4 Shillings and Pence.
Anfwer, L. 643:06: 4, as in Page 216.
Q.8. What comes 716 Moidores to, at 27 Shillings per Moidore?
27
5012
1432
2|0| 1933 | 2 Shillings.
L. 966: 12, as before in Page 216.
A
+
Q.9.
Of PRACTICE.
223
Q.9. What comes 849 Portugal Pieces of Gold to at 36s. each?
36 Shillings in one Piece.
5094
2547.
2 | 01305614 Shillings in all.
Anfwer, L.1528:4, as before in Page 216.
Q. 10. What comes 629 Portugal Pieces of Gold to at L.3: 12?
Shillings in 1 Piece 72
1258
4403
2 | 0 | 452818 Shillings in all.
Anſwer, L. 2264: 8, as before in Page 217.
Q. 11. 1629 Pieces at 4 s. 6d. each?
9
410 | 1466.1
2
9 Six-pences.
L. 366:10:6, as before in Page 217.
Q. 12. 519 Pieces, at 9 Shillings each?
9
2 | 0467/1
L.233:11, as before in Page 217.
20
72
Q. 13.
224
Of PRACTICE.
Q. 13. 365 Quarters of Malt at L. 1: 3: 6 per Quarter?
Six-pences I Qr. coſt. 40
47
2555
1460
47 Six-pences.
410 [171515 Six-pences the Whole coft.
L. 428:17:6, as before, Page 217.
Q. 14.-95 Oxen at L. 8:7: 11 each?
20
167
12
2015 Pence one Ox coft.
95 the Number of Oxen.
10075.
18135
12 | 191425 Pence all coſt.
20 | 15952: 1
| L.797: 12: 1, as before, Page 218.
I have gone thus far to fhew the Agreement of theſe two
General Methods, from whence you may make fome fort of
a Conclufion, which to prefer; but as this Rule is com-
monly wrought, by fuppofing the Integer, and the Price
given managed fo as to make it one, two or more Parts
of the faid Pound, 'twill be proper to fet down the follow-
ing Table, which contains the feveral Aliquot Parts both of
a Pound and a Shilling.
IO S.
Of PRACTICE.
225
S.
10 :
d.
6: 8
5 -
4
1
3:4
2:
2 :
I: 8
I : 4
6
is of a Pound Sterling.
1
d. Far.
1 1 1
-le almalt uf'~ -1 1/2"
t
1
I
2
་
3
T
I : 3
16
4
1
6
I is
2 is
of a Penny.
3 is % of a Shilling.
1
16
I 2
I :
As Avoirdupoife Weight is that which is us'd for Sugar,
Tobacco, ail Sorts of Drugs, &c. and the C. Weight is the
general Integer, I will fet down the Aliquot Parts of a C.
and a Quarter.
Quar.
lb.
I is of a Quarter of C.
2 is of C.
I
2
14
lb.
4
16
of C.
7
14
14
8
7- 1/6
By the two foregoing General Methods any Question
whatever may be wrought, belonging to this Rule; but as
Merchants and other great Traders generally work their
Queſtions, or caft up the Value of their Commodities, by fup-
pofing the Quantities of their Goods to be fo many Pounds
Sterling, when the given Price is above a Shilling, and confe-
quently the Amount or Value thereof more or leſs, as the Price
of I is nearer to, or farther from a Pound: And accordingly
they take one or more Parts of a Pound, as the Price may hap
pen to be: But if the given Price is under a Shilling; then a
Shilling or Six-pence is generally made the Integer, and the
Parts taken in Proportion thereto, as you will fee in ſeveral
of the following Examples. And indeed the Variety of Ways
the Prices may be taken, or the Number of Times it may
be varied are fo many, that it is impoffible to lay down any
pofitive determinate Rule for the breaking the given Price
into Parts; but every one muſt be left to his own Choice
G g
and
226
Of PRACTICE.
and Judgment: Only I will give feveral Examples done in
different Methods, that the Learner may fee what is or may
be done; and 'till Experience ripens his Judgment, ſo as to
vary or alter, as the Cafe may require, how he may do
in any fuch like Cafes. And firſt, I will repeat the fore-
going Questions again, that fo by comparing ſo many diffe-
rent Methods together, he may form a Judgment, which is
beſt, and when one Way, and when the other may be moſt
expeditious; for in Bufinefs, where People have a great deal
of Work to do in a little Time, the fhorteft, plainest and
eaſieſt Way muſt certainly be the beſt.
Bought
EXAMPLE 1.
4s. is L. 316 Pair of Stockings at 4s. per Pair ?
Anf. L. 63: 4. as in Pages 213 and 220.
Here you may obſerve that the whole Quantity bought,
viz. the 316 Pair, are ſuppos'd to be 316 Pounds; but as
1 Pair coft but 4 s. and 4 s. is but the of a Pound, con-
fequently the Value of the Whole can be but of 316 Pounds;
therefore dividing by 5, the Quotient or Anſwer is 63 and ;
but being 4 s. the compleat Anfwer is L. 63: 4, as before,
Sold.
EXAMPLE 2.
718 Pair of Shoes at 5 s. 6d. per Pair?
5 s. is
6 d. is to 5s. | 17 19 Ditto
L. | 179: 10, the Amount at 5 s. per Pair.
-
at 6d. Ditto.
Total L.197: 9 as before, Pages 213 and 220.
Here (as before) the Whole is ſuppoſed to be Pounds Ster-
fing, but as 5 s. 6d. is far fhort of a Pound, ſo the Value or
Amount of the Whole muſt alſo come proportionally ſhort
of L. 718 Pounds; but 5 s. 6 d. in one Sum is no Aliquot Part
of a Pound, therefore I muſt take it at twice or thrice, &c. as is
moft commodious: And therefore I take it at twice, viz. firft
I fay, 5 s. is the 4th of a Pound, and divide the 718 by 4 ac-
cordingly, and find the Quotient or Anſwer come to Ĺ.
179,
and 2 remains, which are or of a Pound Sterling,
4 2
which is 10 Shillings, as you fee fet down in the Work of
the laft Example: Then fay, 6 Pence is 1 of 5 Shillings,
and therefore I divide the L.179: 10 by 10, and the Anſwer
1š
or
Of PRACTICE.
227
or Quotient is L.17: 19; which two Lines being added to-
gether make L. 1979, as before, by the two General
Methods, Pages 213 and 220.
Bought
EXAMPLE 3.
3156 Yards of Iriſh Linen at 18 d. į per Yard.
Is. is 2% L. 157 : 16
6d. is Shil. 78 18
is
1:61/1
of 6d. 6: 11:6
L. 243: 56, as before, Pages 213 and 221.
Here I am forced to take the Parts at three Times, viz.
firft 1 Shilling, for which I take 2 Part of the whole Num-
ber of Yards, they being imagined to be turned into Pounds;
then 6 Pence being the Half of a Shilling, I divide L.157: 16
by 2, and fet L. 78: 18, the Quotient found, under it; and
then d. being 2 of 6 Pence, I divide L.78 18 by 12,
and the Quotient comes out L. 6: 11: 6, which being put
under the two foregoing Sums, the whole Amount is
L. 243:5:6.
I
:
-
Note, For the greater Certainty of knowing the Truth of
your Work, 'twill be proper always to do your Questions twice
over by different Methods; and if both the Totals agree, you
may depend upon their being right. As for Inftance,
Set down the above Queſtion again, thus:
3156 Yards at 18 d. per Yard.
1578
ź
į
Here I fuppofe the firft Line to be Shil-
131:6 lings, then 6 Pence being of a Shilling, I.
divide by 2, and fet the Quotient under
12|0|4865:6 the firft Line; then d. being of 6d.
1 2
I divide the ſecond Line by 12, and fet
L.243:5:6 down the Quotient fo found under the for-
mer; then I add the three Lines toge-
ther, and they make 4865 Shillings and Six-
pence; I divide the Shillings by 20, and then they come to
L. 243: 5: 6, as before.
Gg 2
BX-
228
Of PRACTICE.
Id. is
1 Far.
1
I 2
A
d.
EXAMPLE 4.
Pieces Ferret Yards d.
38 each 36 at 1 per Yard.
36
228
114.
1368 Yards at I per Yard,
Shill. 114
.28:6
20142:6
L. 7:2:6, as before, Pages 214 and 221,
1 2
Here the 1368 Yards are fuppofed to be converted into Shil-
lings, then I Penny being the of a Shilling, the whole is
divided by 12, and the Quotient is Shillings; the 1 Farthing
being of a Penny, the Line that was found to be the Value
of the Penny is divided by 4, and the Quotient 28 s. 6 d. be-
ing added to it, makes 142 s. 6d. and the Shillings divided
by 20 gives L. 7: 2:6 for the Quotient or Anſwer.
8 | 1368 five Far.
3 | 171 Ten Pences.
8 | 57 half Crowns.
L.7:2:6
1
Or you may fet down the Yards and
divide them by 8, and that will bring
them into Tenpences; becauſe 8 Times
Id. is 10 Pence; then 3 Tenpences.
is 2 s. 6 d. and 8 Times 2 s. 64. is one
Pound; which being done, as in the
Margin, the Answer is the fame as
before.
Note, Tho' in Page 224, you were told, that commonly
the given Quantities of Geeds were chang'd, or fuppos'd to
be converted into Pounds Sterling, yet when the Price is
fmall, as in the Example above, 'tis frequent to fuppofe the
given Quantities to be Pence, Shillings, Six-pences, Ten-pences,
Half-Crowns, or any other Sub-divifion of a Pound Sterling,
as you fhall think moft commodious for your prefent Purpoſe.
EX-
Of PRACTICE.
229
EXAMPLE 5.
4617 Oranges at 4 d. per Orange?
3 Far. of 6d. 5717: ·
1
6d.
of L.
L.14:8:64
Or thus 4617 Pence.
2
2 Far. d. 2308 2
1 ditto 12 F. 1154: 1
12 | 3462: 3
2'02818:6
L. 14:8:63
Here first, I fuppofe the Oranges fo many Six-pences, and then
I fay, 3 Farthings, the real Price of 1 Orange is Part of Six-
pence, and divide by 8, to find the Value of the Oranges fa
changed into Six-pences; then I divide the Six-pences thus found
by 40, becauſe ſo many Six-pences make a Pound Sterling;
and the Anſwer comes out L. 14:8:63. In the ſecond
Method I fuppofe the Oranges to be Pence, and fay, 2
Farthings is of a Penny, and divide by 2; the 1 Farthing
is the of 2, and divide that Quotient by 2; then fetting
the two Quotients under one another, and adding them toge-
ther, their Sum is 3462 Pence 3 Farthings; the Pence di-
vided by 12, brings them into Shillings, and the Shillings
divided by 20 brings them into Pounds; fo that the Whole
comes to L.14: 8: 6 both Ways, as in Pages 215 and 221.
Or you may do it thus:
4 | 4617
1154: I
31034612 3
8 | 115:1:0
Here I confider the given Quanti-
ty as Pence; but the Price being but
3 Farthings I divide by 4, which
gives 1154 Pence 1 Farthing. This I
fubftract from the faid 4617 Pence,
and the Remainder is 3462 Pence
and 3 Farthings; the Pence I divide
by 30, and the Quotient is 115 half
Crowns, and 12 Pence, or I Shil-
ling over: The half Crowns I divide
by 8, and the Quotient is 14 Pounds and 3 over, which is 3
half Crowns, or 7 Shillings and 6 Pence, and the Shilling over
before, makes 8 Shillings and 6 Pence; fo that the Whole is
L. 14:8:6
I
L. 14
230
Of PRACTICE.
2
L. 14: 8 : 61, as before. And thus you may vary your
Proceſs both to pleaſe and inform your felf; for if your
Work be done right, the Anfwer will always come out
the fame, let the Method be what it will.
EXAMPLE 6.
27 Pipes of Wine at 11 d. per Pint.
126 Gallons in a Pipe.
Or thus:
882
252
8 |
27216 Pints at 11 d. 4.
3402 Quot. by 8. Sub,
23814 Rem. for 5d. 4.
3402 Gallons.
8 Pints in a Gal.
401510310 Total for 114,
Six-pen. 27216 Pints in all.
L. 1275: 15
3d. 6d. 13608
2d. 6d. 9072
1 F.2d. 1134
4101510310 Total in Six-pences.
L. 1275: 15 Anf. as in Pages 215 and 222,
13
In working both theſe Methods, I fuppofe the 27216
Pints to be Six-pences, and in the firft I divide them firft
by 2, becauſe 3 Pence is Six-pence; then I divide them by 3,
becauſe 2 Pence is of Six-pence; then I divide the laft Quo
tient by 8, becaufe 1 Farthing is the of 2 Pence; then I
add all together, and it makes 51030 Six-pences, which I di-
vide by 40, to bring them into Pounds, and the Anſwer is
L. 1275 15. In the fecond Methad I divide by 8, becauſe 3
Farthings is the of 6 Pence, and fubftract that Quotient
from the whole Dividend, and the Remainder is the Amount
at 5 per Pint; this Remainder added to the Dividend,
which is the Amount of the Whole in Sixpences, at 6d.
per Pint, the Sum is 51030 Sixpences, which divided by
40 comes out as before.
4
*
EX
Of PRACTICE.
231
EXAMPLE 7.
S. d.
10Shill.968 Pieces at
24 10s.193: 6s. d.
1 ½ 25. 96: 8: —
44
13: 3 per Piece.
L. 968 Pieces at 13:31.
0: 3is 24: 2: -
4
L. 198: 12
: is 4: 4
4
ditto 193: 12
I
4
ditto 193: 12
I of 5s. 48: 08
13:31|1286: 6:4
L. 643: 6:4
:
0:34 Is. 12:02
0:03d.2:00:4
13:3 L.643: 6:4
In the firſt Method the 968 Pieces are fuppos'd ſo many
10 Shillings; the Parts are taken for 2's. I s. 3 d. and
d. which being all added together, make 1286 ten Shil-
lings, and 6 s. 4 d. the 1286 halved or divided by 2, gives
L. 643 6: 4. In the ſecond Method the 968 Pieces are ima-
gined to be Pounds; and 4s. being the of a Pound, the Whole
is divided by 5, and the Quotient L. 193: 12, fet down three
Times to make 12 Shillings; then one of thofe Lines are di-
vided by 4, becauſe I Shilling is of 4 Shillings, and that laſt
Quotient is alfo divided by 4, becaufe 3 d. is of a Shilling,
and that laſt Quotient is divided by 6 becauſe d. is % of 3d.
and the Sum of all theſe Quotients, in both Methods, amount
to L.643: 6:4, as before, Pages 216 and 222.
EXAMPLE 8.
716 Moidores, 27s. each.
IL.
5 s. is 4 L.
2 is
179
L.
71: 12
L.1:7 L.966: 12
Here you fee as the
Price is above L. fo
the Amount comes to
fo much more than
the L's. which is fup-
pos'd to be the Inte
ger, viz. L. 966: 12; as before, Pages 216 and 222.
-
•
Or,
232
Of PRACTICE.
716
Or, 105.
IO
716
5
2
358
Here I ſuppoſe the given Quanti-
ty to be 10 Shillings, and take the
Parts, as above, and the Total is
143: 2 1933 10 Shillings,and 2s.over, which
divided by 2, makes the Whole
27 | 2 | 1933: 2 come to L. 966: 12, as before,
L. 966: 12
EXAMPLE 9.
I L.
849
Pieces at 36s. each.
45. is
169: 16
Or, 849
4
169: 16
4
169:16
4 s. is
L. 169: 16
4
169:16
9 Times 4 s.
9
L.1:16L.1528: 4 is 36 Shillings or L. 1528: 4
Anſwer, L. 1528: 4, as in Pages 216 and 223.
By the first Method, I fuppofe the 849 to be Pounds &
and then take for 4 s. and fet down the Quotient 4
Times, and add all together, whofe Total is L. 1528: 4 at
L. 1: 16 or 36 Shillings. In the fecond Method, the 849 are
alfo fuppofed to be Pounds Sterling, and after having taken
the for 4s. I multiply that Quotient by 9, as taught Pages
45, 46, &c. and the Product comes to the fame as before.
EXAMPLE 10.
649 Pieces at L. 3: 12 each.
:
10s. is L.
25. is of 105.
649
649
324: 10
64:18:0
L. 2336:8:- Anf.
as in Pages 217 and 223.
Or, 649
3
1947
45. L. 129: 16
&
8s. L.
&
259: 12
L. 2336: 8
Here in the firft Working, I fuppofe the 649 Piece to
be Pounds, and fet them down 3 Times for the 3 Pounds
then I take the Parts for 10 s. and 2 s. then add all toge-
ther, and the Total is the Anſwer. In the ſecond Working
I mul-
Of PRACTICE.
233
I multiply by 3, for the 3 Pounds; then I take the Part
for 4s. and multiply that by 2 for 8 s. then add all toge-
ther, and the Total is the Anfwer, as before.
i
EXAMPLE II.
1629 Pieces, at 4 s. 6 d. each.
25. is 10
2: 6 is L.
+ L.
162: 18-.
Or,
1629
203:12:6
45. is
L.
325:16:-
325: 16:-
4:6
L.
366: 10:6
0:6. is
of 45.
40:14:6
as in Pages 217 and 223.
4:6
L. 366: 10:6
In the first Method of this Example, both the Parts are
taken immediately from the Pound Sterling; and confe-
quently the firſt Line is the Dividend to both the Parts, and
is fuppofed to be fo many Pounds, which must be divided
first by 10, for the Amount of 1629 Pieces, at 2 s. per
Piece, and then by 8, for the Amount of the fame at 2s.
6d. each; then thoſe 2 Lines or Amounts added together, is
the Value of the Whole at 4 s. 6d. à Piece. In the ſecond
Method 4s. is taken as a Part of a Pound, but the 6d. is
taken as a Part of 4s. and therefore the Quotient of the firſt
Work becomes the Dividend to the ſecond Work.
*
EXAMPLE
519 Pieces at 9s. each.
45. L. 103: 16
5
L. 129: 15-
9 s. is L. 233: 11 Anf.
as in Page 217.
12.
Or, 519 Ten-Shillings.
io is 51: 9 to be fub. for I s
467 1 Rem. for 9 s.
:
L. 233: 11 Anſwer.
Here, in the firft Method, I fuppofe the Quantity given
to be Pounds, and then I break the given Price 9s. in-
to 4s. and 5s. and find the Amount of each; then I
add, thofe 2 Lines together, and the Total is the Amount
of the Whole. In the fecond Method, I fuppofe the whole
Number of Pieces to be fo many 10 Shillings, from which
I ſubſtract of it felf, and there remains 467 Ten-Shil-
Hh
lings,
go
234
Of PRACTICE.
lings, and 1 Shilling: I divide 467 by 2, becauſe 2 Ten
Pound; and the Anſwer is the fame as before.
Shillings is
EXAMPLE 13.
Quarters.
365 at L.1: 3:6 Quarters.
45: 12: 6
18: 5:
At 1 L. 365 is L. 365 :
at 2s. L. 036:10
L.
2:6 is L.
I 2 L.
at I s. or ½
of 2 s. 18: 5
L. 428 17:6
at 6 d. or
of Is. 09:02: 6
10
Anſwer, 365 Quarters at L.1: 3: 6 is L. 428: 17: 6
as before, Pages 217 and 223.
In this Example, as the Price is 1 Pound and odd Money, I
fuppofe the Quarters to be changed into Pounds, for the
one Pound; and then take the Parts for the odd Money, and
add what they amount to, to the Pounds, and the Total is
the Anſwer, as appears by the different Ways by which
they are done, here, and Pages 217 and 224.
EXAMPLE 14.
95 Oxen, at L. 8 7 11 each.
8
Value of 8 Oxen,
67 3: 4
I2
806 :
12
Value of 96 Oxen,
I
ſubſtracted 8: 7
7.: II
95 remains, L. 797 : 12 : I
Here I first multiply the given Price by 8, and that
Product by 12; and the laft Product is the Amount of
96; from which ſubſtract 1, and the Remainder is
L. 797 : 12 : 1 for 95, the fame as in Pages 218 and 224.
Or,
of PRACTICE.
235
Or, 95 at L. 8: 7: 11
4 s. is
4s. is
of a L.
of a L.
95
760 Value at L. 8 each.
19 Value at 4 s. each.
19 Ditto.
L.798 Value at L. 1: 8.
95 at 1 d. each is 0:07: II to be fubftracted.
95 at L.8:7:11 is 797: 12: I the true Value.
Here I multiply by 8, and take twice 4s. which is one
Penny two much; therefore I fubftract 95 Pence, or 7 s.
II d. from the Whole, and then the Anſwer is the fame
as before.
By comparing thefe 14 Examples, wrought by the two
general Methods, and by theſe particular Methods, you may
perceive that fometimes the Decimal Way, and fometimes the
others are the ſhorteſt: Only you muſt obſerve that in the
Decimal Way, you uſe only Multiplication, and have the
Anfwer always in Pounds and Parts, whofe Value is imme-
diately known; and if you uſe the Contraction, 'twill ge-
nerally give the Anſwer in the feweft Figures.
As Iron, Copper, Tobacco, Sugar, Lead, &c. are the com-
mon Commodities weighed by Avoirdupoife Weight; I will now
fhew you a particular Method of working Questions of that
Kind.
RULE.
In all Goods fold by the Ton Weight (let the real Price
be what it will) fuppofe them at one Pound per Ton,
and then the given Quantity of Tons will be fo many Pounds;
and if there be any odd Hundreds, they will be fo many
Shillings; the Quarters will be fo many Three-pences; and for
the Pounds Weight, you may reckon either fo many of a
Penny, or fo many of a Farthing, as you please (both being of
the fame Value;) this done, multiply the Whole by the Pounds in
Money one Ton cost, and take the Parts for the odd Money;
add thefe Quotients to the given Product, and the Total will
be the Value or Amount of the Whole,
7
Hh 2
23
EX-
236
Of PRACTICE.
EXAMPLES.
T. C.
C. Q; lb.
What comes 56: 17: 3: 15 of Iron to, at L. 15. 15 5.
per Ton?
105. is L.
4
5 s. is ½ IOS.
Anf. L. 896: Ĭ: 8 and 4 of a Farthing.
L. S. d.
56: 17 : 10/1/2
5
284: 9: 5 - 1 Value at L. 5 per Ton.
3
853: 8: 3 3 Value at L.15 per Ton.
28: 8: 111+
14: 4: 5½ 4 +0 +
4:54+0+ 1.
I
896: 1 8 1+1+1=✯ or of a
3
3
Ι
Far:
Here I fuppofe the Price to be one Pound per Ton, and
fo I fet down 56 Pounds for 56 Tons, 17 Shillings for 17
Cs, 9 Pence for 3 Quarters, and I d., and of a Far-
thing for the 15 Pounds; 15 Times of a Farthing coming
to fix Farthings; and over, which added to the 9 Pence,
make 10 Pence Half-penny, and of a Farthing: Then as
the given Price is 15 whole Pounds, I multiply firft by 3, and
then by 5, and the laft Product is the whole Amount at L. 15
per Ton: Then I take the of the firſt Line, which is the
Value of 1 Pound per Ton, for Ten Shillings, and that is the
Amount at 10s. per Ton, and then the of that is the
Value at 15 s. per Ton; thefe laft 3 Lines added together
is the Amount at L. 15: 15 per Ton: And that I may
be certain of the Truth of the Work, I fet down the
Amount at 1 L. again; only now I reckon the odd Pounds
at of a Penny per Pound: So that the whole Amount at
1 L. is now L. 56 : 17 : 10 ††4.
28
2
L. 56
Of PRACTICE.
237
L. 56: 17 10 22*
4
and 227 11 6 12 is the Value at 4L.p.Ton
:
4
Deduct
and 910:
14:
6: 1
4: 5 -
1
20 ditto at 16 L. per Ton.
18+ Value at 5s. p. Ton.
The Remainder 896;
1: 8 -
1 +Value atL. 15:15.
per Ton; which is exactly the fame with the former, for
when the remaining Fraction is valued 'twill come to 2 of
a Penny, or of a Farthing, according to the Directions
given, Page 154.
{
TI
You fee the above Quantity in this laft Method is wrought
at L. 16 per Ton; and therefore the Amount at 5 s. per Ton
is fubftracted to reduce it to L. 15: 15 per Ton.
* N. B. The 28, or Denominator of the Fraction is here
fet at Top to fave the repeating it fo often as the Work
would otherwiſe require; and fo only the Numerators are ſet
down in the Work. Now you fhall fee the fame Example
wrought by Decimals.
T. C. Q; lb.
At L. 15: 15 per Ton, What comes 56: 17: 3: 15 to?
Tons. Decimal Parts.
56,8941914
5751 Price of 1 Ton inverted,
568942
28 447
+
39826
2845
L.896,084
The Decimal to 17 C. 3 r. 15 lb. when a Ton is the
Integer, comes to ,8941914, and the Price is L. 15,75 ;
but as 3 Decimal Places are fufficient to be in the Product
for the Answer, I do the Work contractedly, and it comes
to
238
Of PRACTICE.
to 896 whole Numbers, or Pounds Sterling, and ,084 Parts;
which by the Rule, Page 203, comes to I s. 8 d. Farthing;
but by the Rule, Page 170, 'twould be but Is. 8d. and fome
Parts of a Farthing, the fame with the firſt two Methods.
T. C. Q. lb.
What comes 61: 13: 1: 7 of Copper to at L. 87: 13:
per Ton? Anf. L. 5406: 10: 8.
Whole 2. at 1 L. per Ton is L. 61: 13: 3
6
At L. 3 per Ton is L. 493: 6: 6
II
At 88 L. per Ton is L. 5426:
II :
6
Subftract L. I per Ton and
87 L. per Ton is
L. 5364
18:
2 - I
10 s. per Ton is
30: 16:
2: 6 d. per Ton is
7: 14:
Is. per Ton is
3: 1:
7-3+
1-3+1+1
7-3+1+3+3
4
AtL.87:13:6p.T. Tot.is L.5406 : 10:
8 and 13 of a Far,
Decimally the Work will ftand thus:
T. Dec. Parts. T. C.
2. lb.
2:
Quantity 61,665625 for 61: 13: 1:7
for L. 87 13: 6
Pr. p.T. invert. 5 7678
4933250
431659
36999
4317
308
5406,533, or L. 5406: 10: 8.
Wherever the Quantity is whole Numbers, you may break
the faid Quantity into any of the component Parts, and
work as formerly taught you, Pages 51, 52, &c. or you
may work by any of the latter Methods, as beft pleafes you;
I
but
Į
Of PRACTICE.
239
but in general the Decimal Way will be both the plaineſt
and eaſieſt.
Bought 816 Pieces of Dowlafs at L. 1:6:6 per Piece,
What come they to?
This Example is
wrought by the
Rules in Pages
51, 52, &c.
L. 1: 6: 6 the Value of 1 Piece.
9
I
11: 18: 6 the Value of 9 Pieces.
9
107: 6: 6 the Value of 81 Pieces.
IO
1073: 5:
7: 19:
Value of 810 Pieces.
the Val, of 6 Pieces.
the Value of 816 Pieces.
L. 1081 : 4:
The fame Decimally L. 1,325 is the Value of 1 Piece.
816 Number of Pieces.
7950
1325
10600
L. 1081,200 the Value of 816 Pieces,
which is L. 1081 : 4
Or thus:
At 1 L. 816 at L. 1: 6:6
I
5s. is L. 204-
4
Is. of 5s. 40 - 16
6d. of 1 s. 20
1 I
Total L. 1081
8
W
4
Or
you may do thus :
816 Pieces.
53 Sixpences I coft.
2448
4080
410 | 43248 Sixpences all coft.
L. 1081 - 4
If you was to compute Ingots of Standard Silver, you
might do it thus:
What
240
Of PRACTICE.
What comes 8 Ingots of Standard Silver to, Quantity
lb. Oz. Dwts. s. d.
315 7
10 at 52 per Ounce ?
lb.
L. 315 11 8 for 315
10 for
Oz.
7 at 1_L per lb.
10 Dwts. at ditto.
L. 315 : 12 : 6 for 315 7: 10 at 1L. per lb. Weight.
L. 946: 17:
31: II:
3
6 for ditto at 3 L. per lb. Weight.
3 for ditto at 2 s. per lb. Weight.
L. 978: 8: 9 for ditto at L.
9 for ditto at L. 3: 2 per lb. Weight.'
EXPLANATION.
་་
At 1 L. per lb. Weight, 'tis plain the Money and the
lb's Weight will be the fame; and for every Ounce you
muft reckon I s. 8d. or 20 Pence; which for 7 Ounces,
comes to IIS. 8d. and for every Penny Weight you muſt
reckon one Penny in Money; becauſe as one Ounce comes
to 20 Pence, a Penny Weight muft come to one Penny; there-
fore for fo many Penny Weights, as you have, you must fet
down ſo many Pence, as in this Example, 10 Pence; ſo that
the whole Quantity at 1 Pound or twenty Shillings per
Pound Weight, comes to L. 315: 12: 6; which multi-
ply'd by 3 for the 3 Pounds, gives L. 946: 17:6; and 2 s.
being of a Pound Sterling of L. 315: 12: 6, which
is 31: 113, being added to the L. 946: 17: 6 makes
L. 97889 for the Total Amount, as may be try'd bŷ
working the fame Queftion Decimally thus:
10
15833 Decimal to 7 Oz.
,0416 ditto
to 10 Dwts.
!
lb. Parts.
315, 625
3,1 Price.
315, 625
9468 75
L. 978, 4375, or L.978: 8: 9, as before.
Here
Of PRACTICE.
241
Here taking the Decimals for the Oz. and Dwts. out of the
Table, Page 206, which being added according to the Rule
of fingle Repetends, produces a perfect Decimal, viz. 625,
which added to the whole Number or Pounds, makes 315,625;
this multiply'd by 3. 1, viz. L.3 for the Pounds, and 1 for
2 Shillings, produces as above L.978 Pounds, and 4375 Parts;
which by the Rule, Page 203, comes to 8 s. and 9 d. as by
the foregoing Method.
After the fame Manner you may go on in any other
Weight and Meaſure, regarding only the feveral Subdiviſions:
As for Example,
C. Q. lb.
What comes 5 Hogsheads of Sugar to, Quantity 47 : 1 : 12
at L. 2: 179 per C?
L. 47:
5:
C. Q. lb.
for 47: 1:-
2: I for
12
Whole at 1L. C. is 47: 7: 1:5 for 47: 1 : 12 at I L.
Dit. at 3 L.per C. 142: I :
Dit. at 2 s. per
C.
4 14:
14
Dit. at 3 d. per
C.
:3
5:1 at L. 3 per C.
8:4
: 11: 10: + 1
Tot. to be ſubſtract. 5: 6: 6:4+1
Rem. is Whole3136: 14: 10:3+1
at L.2:17:9p.C.3
Or thus:
The Whole at 1 L. is L.
47 : 7: I
I : 1/
Ditto I
47: 7:
I : S
Ditto o 10 S.
23:13:
6:6
Ditto o: 5 s.
Ditto o :
2 6d.
II : 16:
9:3
5 18
4 5
Ditto :- 3
: II: IO:
At L.2:17:9 Whole is L.136 14 103+
:
1444
I i
ar
242
Of PRACTICE.
,25 is the Dec. to
C.
,10714 and dit. to 12 lb.
,35714 dit. to 1 Q. 12 lb.
to which add the 47 C. or
whole Numbers, and it is
47,35714, &c.
7
2
Or thus:
C. Parts.
47,35714, &c.
57 882 Price inverted.
94 714
37 886
3789
331
24
Total Amount is L. 136,744 or L. 136: 14: 101.
OBSERVATION S:
4
L. 47,
Above, in the first Method, 47 C. at 1 L. per C. is L.
and confequently 1 Quarter must be 5 Shillings; then 12 lb.
being of a Quarter of a hundred Weight, of 5 s. is 2 s.
Id. and; but for convenience, I generally put the Denomi-
nator of the Fraction uppermoft; and then the Whole at IL.
per C. ftands thus: L. 4771, and the given Price be-
ing near 3 L. per C. I multiply by 3, and have L.142: 1:57,
from which I fubftract the Amount of the whole Quantity
at 2s. 3 d. per C. which is the Difference between L. 3 and
L.2: 179, the given Price, which being L.5:6:61+212,
the Remainder is L.136: 14: 10+, or L.136:14:10;
for of a Penny is juſt a Halfpenny; and that I may
be fatisfied of the Truth of this Work, I fet down firft the
Amount at 1 L.per C. twice; then the Amount at 10 Shillings;
at 5 Shillings; at 2s. 6d. and at 3d. and adding theſe all to-
gether, I have the Amount at L. 2: 179, which comes to
the fame as before: And for a further Trial, I do it Deci-
mally; and by the Table, Page 208, find the whole Weight
comes to 47,35714, &c. as above; for as three Decimal
Places are fufficient for the Anfwer, I don't take down all
the Figures in the Table; then the Price will be found per
the Table, Page 205, to be L.2,8875, which is inverted,
and wrought as formerly directed, in Multiplication of De-
cimals, and the Product comes out L. 136.744, or
L.136 14 10, the fame with the two foregoing Me-
thods, by which it plainly appears all is done right: So that
you may uſe which Method you like beft; but the Decimal
t
葛
​2
Way
Of PRACTICE.
243
Way is the eafieft. The common, and indeed general Me-
thod of doing fuch like Queſtions, is as follows.
I L.
· 1 L.
1
C.
2: lb.
47: 1: 12 at L.2: 17 9 per C.
47
10 s. ½ L. 23: 10:
5 IOS. 11: 15:
55. 5 : 17 : -6
2:6
3:26
11: 9
: II
145 is IQ.
2:
4 : 1 //
L.I: 00 : 7 ž
C.
2.
-
2 L. 17 s 9d. · L.135 : 14 :
14: 3
for 47:
I :
: .7
7
for
:
lb.
I : 12
L.136 : 14 : 10½ for 47: I: 12
.2.2.9
is 4 lb.
is 8 lb,
In this laft Method the 47 C. is wrought alone, and the
Parts are alſo done by themſelves and added to the Amount
of the 47 C.
I have now fo fully explain'd this Rule, that I think
there can remain no Difficulty, but what fome or other of
the foregoing Methods will eafily maſter; and indeed I would
recommend the Decimal Way to be uſed univerfally, both for
its Eaſe and Certainty, not being incumber'd with variety
of Denominations; for after the Parts of any Quantity
whatever are found, either by the Table, or otherwife, then
the Proceſs is the fame with plain fimple Multiplication, tho'
the Product is applicable to any Species of Coin, Weight,
Meaſure, &c. Before I quit this Subject, I will apply it
to what is commonly made a diftinct Rule, call'd,
TARE and TRET, &c.
To know the nearest and beft Methods of working all
Questions of this Kind, is abfolutely neceflary for all Mer-
chants and great Traders; and in my Opinion, 'tis beſt done
by the Rule of Practice, in which I will fhew you feveral
Ways of performing it, that you may chufe for yourſelf
that which you think beft, moft cafy, and expeditious: But
before we treat of the Methods of Working, 'twill be pro-
per to explain the Thing; to exprefs which, there are fève-
I i 2
rai
244
Of PRACTICE.
ral Technical Terms or Words of Art us'd; fuch as Grofs,
Tare, Tret, Clough or Cloff, &c. the Meaning whereof is,
that all Goods that are liable to an eafy Separation, and
confequently a Lofs or Detriment, and are fold by Weight;
fuch as Tobacco, Sugar, Rice, Tea, &c. are put into Casks,
Bags, &c. but as 'twould be inconvenient to ftrip them of
theſe Coverings, the faid Casks, Bags, &c. are weigh'd along
or together with the Goods; and this Weight of both Cask,
Bag, &c. and Goods, is called Grofs-Weight; but as the faid
Wrappers, Casks or Bags, are efteemed of little or no Value,
therefore more or leſs per C. Weight, or per Cask, Bag, &c.
is allowed to be deducted out of the faid Groſs-Weight, for
the Weight of the faid Cask, Bag, &c. and this Allowance
is called TARE; and in fome Goods there is no further
Allowance made, and in others there are, which the par-
ticular Cuſtoms of particular Dealers and Commodities muft
determine, both what Quantity, and upon what Sorts fuch
further Allowance is made, which is called TRET, and
is ufually 4 lb. in 104 lb. or 2 Part of the Nett Commodi-
ty, after the Allowance for the Cask, Bag, &c. has been
deducted, which is fuppofed to be fpoil'd, or at leaſt much
injur'd by the rolling of the Casks, Chefts, Bags, &c. at
Sea, &c. as in Tea, where by fuch Motion, fo much, or
more is ſuppoſed to be broken to Duft or Powder, &c.
26
CLOFF is yet a further Allowance after the other two
are deducted, made in fome Commodities, by the Great or
Wholeſale Dealers to the Retailers, to enable them to make
many Draughts or fmall Parcels out of one large one;
and this is 2 lb. upon every 3 C. Weight; when all, or fo
many of thefe Allowances, as are cuftomary, are made, the
Remainder is called Nett Weight, and is what the Buyer muft
pay for at the Price agreed upon.
EXAMPLES.
What is the Nett Weight of 8 Casks of Indigo, Quantity
Gross 37 C. 32; 15 lb. if 30 lb. be allowed for Tare to each
Caſk?
Q2.
30 lb. is 1
2:
I
lb.
2 Tare of one Caſk.
8 Number of Caſks.
: 16 the whole Quant. of Tare to be deduct.
37
Of PRACTICE.
245
C. 2; lb.
37 3 15 Grofs Weight.
: 16 Tare.
2:
35 2 27 Nett Weight.
What is the Nett Weight of 16 Casks of Argol, Quantity,
Grofs, 123 C. 2 2. 17 lb. Tare at 14 lb. per C. Weight?
14 lb. is
C.
1232: 17
Grofs.
15 : 1 : 23 ¦ Tare.
108 : : 22 Nett.
Here as 14 lb. is an exact Aliquot Part of a C. viz.
the, I divide by 8, faying, the 8's in 12 is 1, and 4 over;
in 43 is 5, and 3 over, which being 3 C. is 12 Quarters;
which with the 2 Quarters in the given Quantity is 14; the
8s. in 14 is 1, and 6 over, which is 168 Pounds, to which
the 17 being added, makes 185, the 8's in which are 23
and i over; but in all theſe Grofs Goods, fo fmall a Matter
as of a Pound or 2 Ounces is rejected; but if it had come
to 'twould have been eſteemed as a Pound, no Parts leſs
than a Pound in this Work being taken Notice of; tho'
for Exactneſs and Curiofity fake, you may do as you pleaſe
in Calculation, tho' not in Buſineſs; for there the cuſtomary
Methods must be followed.
Suppoſe I have the following Invoyce or Bill of Parcels of
5 Hogfheads of Tobacco, upon which I am to be allowed Tret
after the Tare is deducted, How much is the Nett Weight?
Anf.
C. Q. lb.
lb.
N° 1 Quantity Grofs 6:1:17 Tare 99
7
9
6
15
5:3:18
82
5:1: 9
76
8: : 7
104
7:2:14
98
5 Casks Grofs 33: 1: 9 Tare 459
I
5 Casks
246
Of PRACTICE.
5 Casks, Grofs 33: 1: 9 Tare 459.
33
33.
33..
..37 the lb's in 1 Q. and 9 lb.
3733 Grofs Pounds.
459 Tare.
26) 3274 Suttle Pounds.
126 Tret.
C.
Pounds 3148 Nett, or 28 :
2. lb.
: 12
Here, after adding the Quantities together, the Grofs
Weight is 33 C. 1 Q9 lb. which is brought into Pounds,
by fetting down the 33 firft under itſelf, then I Place
more to the Left-hand, and then 2 Places; and for the
1 Quarter and 9 Pounds, which is 37 Pounds, 37 is fet down
under the Units and Tens Places; and the Whole added toge-
ther is 3733 Pounds Grofs.
The Reaſon of this Proceſs is, that 112 Pounds being
C. Weight, the 33 twice fet down under one another, is in-
ftead of multiplying by 2; and fetting the fame Figures one
Place towards the Left-hand, is multiplying by 10; and fet-
ting the fame Figures two Places towards the Left-hand, is
multiplying by 100, which collected, is 112; to which add
the Pounds of the Part, and you have the Pounds in the whole
Quantity. Note, This Method is always uſed at the Water-
fide, upon Account of its Eafinefs, nothing but Addition being
required in it. From which the Tare 459 being deducted,
when there is a further Allowance of Tret, the Remainder is
called Suttle-Pounds or Weight, which is divided by 26, be-
caufe 4 Pounds is 2% of 104, and the Quotient is here made
126, tho' the true Quotient is but 125, and 3; which Fraction
being near an Integer, 'tis common to make it a whole Pound,
the Seller giving the Buyer this fmall Advantage, that when
the Calculation comes to a Pound or upwards, to make it a
whole Pound; which 126 being fubftracted from the former
Remainder or Suttle-Pounds, leaves 3148 Pounds, or 28 C.
12 lb. Nett-Weight to be paid for.
13
Sup-
Of PRACTICE.
247
{
Suppofe I am allowed only lb. Tare for every feere Pounds
C. Q. lb.
Groſs, What is the Nett-Weight of 42 : 3: 12 Groſs ?
C. Q Q.
lb.
ང་
42 3: 12 Groſs.
42
42.
42..
96
40 | 480,0 Grofs.
0120 Tare.
4680 Nett lb's or 41 C. 3 Q. 4 lb. Nett.
37 Frails of Raifins, Quantity each Grofs 2 C. 3 Q. 12 lb.
Tare 20 lb. per Frail, What's the Nett-Weight of the Whole?
Anf. 99 C.
B
12 lb.
1 Frail 2 3: 12
6 Ditto 17:
Q. lb.
1 : 9
6
4
: 16
I : I :
8
6
i
5
36 Dit. 102: 3 : 12 ·
I Ditto 2:3: 12
•
37 Dit. 105: 2: 24 Grofs.
6 2 12 Tare.
99 :
: 12 Nett.
6:2 12 Tare.
Or thus:
23
12 Grofs.
20 Tare.
22: 20 Nett, p. Frail.
6
16: - : 8 Six Frails.
6
96: 1: 2036 Ditto.
2: 2:20 I Ditto.
99 ፥ : 1237 dit.N.
Here
248
Of PRACTICE.
Here in the first Method the Frails being 37, I fet down
I Q. 9 lb. which is the fame with 37 Pounds, and multiply
by 20, viz. 4 and 5, and the Product is the Tare in C. Q
Ib's; which fubftracted from the whole Grofs Weight, leaves
the Nett Weight. Or you may bring both Parts into Pounds,
thus:
1
2:3:12
2
2.
2..
96
37
20
740
320 Pounds Grofs in a Frail.
37 Frails.
2240
960
11840 Grofs Pounds in all.
740 Tare in all.
C. 2
16.
: 12.
11100 Nett Pounds in all, or 99:
Or Decimally thus:
2:75 for 2 C. 2.
,10714287 for 12 lb.
2,85714287
,17857134 deduct for 20 lb. Tare.
2,67857153 Nett Weight of 1 Frail.
37
18,75000071
80,347 1459
99,10714661 Nett Weight of the Whole.
C. Q. lb.
Or, 99:
: 12, as before.
Note,
Of PRACTICE.
249
Note, Tho' I have taken down the whole Number of Figures
that are in the Table, in the Example above, yet you need not
be fo very curious, for in fuch Sort of Examples as thefe, five
Figures would be fufficient, as you fee by what follows.
2,75 for 2 C. 2.
,10714 for 12 lb.
2,85714 for 2C. 3 and 12 lb.
,17857 deduct for 20 lb. Tare.
2,67857 Nett Weight of 1 Frail,
I
37 Frails.
18 74999
80 357 I
99,10709 Nett Wt. of the Whole.
Here the Anſwer is a Trifle leſs than the Tabular Number for
12 Pounds, but much larger than II Pounds: So that in ſuch
rough Commodities as thefe, the Difference is infignificant.
At 8 lb. per C. What Nett in
8 is of C.
4
C.
516
36
:
3
Q. lb.
2: 16 Groſs?
17 Tare.
Or thus:
lb's. 112 Groſs in C.
479
2 27 Nett.
3:
8 Tare in ditto.
Nett
per C.
lb's 104
516 C's in all.
624
104.
Here in dividing by 14 there
remains lb. which being
lefs than Pound is reject-
ed.
520..
52 for 2 Quarters.
15 for 16 Pounds.
C. 2 lb.
53731 or 479 2: 27 Nett.
Here taking of 104 for 16 lb. it comes to but 14,
which being more than Pound, is made one Pound, viz.
15.
h
K k
What
250
Of PRACTICE.
What is the Nett Weight of 49 Hogsheads of Tobacco, Quan-
tity Grofs 197 C. 3 Q. 16 lb. Tare 80 lb. per Hogfhead, Tret
4 lb. per 104, and Cloff 2 lb. per 3 C
Note, 'Tis not ufual to allow Tret and Cloff upon Tobacco,
but a pretty large Tare only; but I have here put them all,
to fhew the Manner of Calculation, which is the fame, let
the Species of the Goods be what they will.
80 lb. is of C. Weight; therefore the Number of Hog-
fheads, multiply'd by 5, and that Product divided by 7, gives
the C's, &c. that all the Tare comes to.
Or thus:
Hogfh.
lb. Q. lb.
Thus 49
80 is 2 : 24
5
7
C. Q. 16.
1973: 16 Grofs.
7 | 245 | 5: ~
35
Tare.
7
162: 3: 16 Suttle.
6: 1 2 Tret.
156 2 14 Suttle.
3: 20 Cloff.
C. 35 Ta.
C. 2. lb. C. Q. lb.
26)1623 16 (5: 1: I
156
35:0:0
155: 2: 22 Nett,
64
4 3 | 156
27
52
26
52
I
104 Cl.
28
44
26
18
For finding the Tare of the Whole, you may work by
either of the two Ways above, and you fee the Reſult is the
fame, viz. 35 C. which being fubftracted from the Grofs, I
divide the Remainder, 162 C. 3 Q. 16 lb. by 26, as you fee
above at large; where, after the 162 C. has 6 Times 26,
which
Of PRACTICE.
251
I
26
1
which is 156, fubftracted from it, the Remainder is 6, or
Parts of C. This 6 multiply'd by 4, the Quarters in a C. gives
24, to which adding 3, the Quarters in the Sum, the Total
is 27; this divided by 26, gives I Quarter, and t remains,
which is Part of a Quarter; this brought into Pounds,
and the 16 Pounds in the Sum added to it, makes 44, which
divided by 26, gives 1 Pound and Parts of a Pound;
which being more than a Pound, I fet down 2 Pounds in-
ftead of 1 Pound and 8 Parts, as will always be allowed to be
done in actual Bufinefs (tho' not in ftrict Arithmetical Calcu-
lation) then fubftracting the 6: 1: 2 for the Tret, the Re-
mainder is 156 C. 2 Q. 14 lb. And as the Cloff is to be 2
Pounds upon every 3 C. Weight, I divide the C's by 3 (re-
jecting the odd Parts) and the Quotient is 52, which ſetting
down twice, the Sum is 104 Pounds, or 3 Quarters and 20
Pounds for the Cloff; which being alfo deducted, the Nett
Remainder or Goods to be paid for is 155 C. 2 Q. 22 lb.
26
There are befides thefe cafually other Allowances, fuch
as Breakage, Damage, &c. of which take an Example.
If 12 Casks weigh Grofs 56 C. 1 Q. 17 lb. and the Tare
is 16 lb. per C. Breakage 7 lb. per Cask, and over and above
all 1C. i Q. 19 lb. is allowed for Damage, What Nett Weight
muſt be paid for?
C. Q. lb.
lb.
756: 1: 17 Grofs 16 is
3
3
lb.
of 112, or C.
rejected.
18:06 + which being leſs than lb. is
I: II Suttle.
48 :
3:
47: 2
i
Breakage at 7 lb. per Cask is C.
: II Suttle.
1: 1: 19 Damage.
46:
-
: 20 Nett.
في
Kk 2
The
252
The Inverſe RULE of THREE.
C. Parts.
The fame Decimally.
7 156,40179 Grofs.
8,05739 Tare.
48,34440
—,75 — Breakage.
47,59440
1,41964 Damage.
46,17476 Nett, or 46:-: 20.
Note, The laft Decimal does not, by the Table, come to
20 Pounds; but as 'tis nearer 20 than 19, I put down the
bigger Number, no less than a Pound being taken notice
of in any Grofs Goods, where fuch large Quantities are fold.
I now fuppofe this Matter of Tare and Tret, &c. and
the whole Rule of Practice fully and plainly handled, fo
that any thing may be wrought thereby, properly belonging
thereto.
CHA P. IX.
In which is contained the Inverfe RULE of THREE,
and the feveral other Applications of the Direct Rule
of Three, commonly called by the Names of Fel-
lowſhip, Barter, Exchange, Lofs and Gain, Alle-
gation, Intereft Simple and Compound, &c.
TH
The Inverſe RULE of THREE.
HIS is called Inverſe, or Inverted, becauſe the Rule
for working, given you, Page 121, is inverted or
turned (as it were) backwards; for in all Queſtions that
come under this Head, after your Queftion is ftated, as in
common, you multiply the firſt and fecond Numbers toge-
ther,
The Inverse RULE of THREE.
253
ther, and divide their Product by the third Number, and
the Quotient is the fourth Number fought.
To know when a Queſtion is in the Rule of Three Direct, or
Inverſe.
You must fee whether the third Number in the Stating of
the Question before you, is bigger or leffer than the first
Number.
If it is bigger, and requires a leffer Number to anſwer
the Conditions of the Question, than the ſecond given Num-
ber now is :
Or if it be leſs, and requires a bigger Number than the
Second given Number now is, then it is to be wrought by this
Inverje Method; but if the third Number is bigger than the
first, and requires a bigger Number than the fecond, or is lefs
than the firft, and requires a leffer Number than the fecond
now given; then the Queſtion is Direct, and muſt be wrought
as before taught you, in Page 121.
EXAMPLES.
Suppoſe 6 Men could build a Wall in 20 Days, In what
Time would 12 Men do the fame, at the fame Rate of Work-
ing?
Here I firft ftate the Queftion, according to the Directi-
ons in Page 121, thus:
If 6 Men require 20 Days, What Time will 12 Men require?
This done, I fee the third Number, which is 12, is bigger
than the first Number 6; and by confidering the Question,
I perceive plainly that the Time or Days, which is the fourth
Number fought, muſt be leſs than the ſecond Number given ;
and therefore I conclude this Question is to be wrought by
the Inverfe Rule, thus:
6
20
6
12
12 | 120
Anfw.
10 Days.
Again, Suppose I could go a certain Journey in 18 Days,
by travelling 16 Hours every Day, How many Days fhould I
2
want
7
254
The Inverſe RULE of THREE.
want to do the like in,
Day at the fame Rate?
If 16 Hours require 18
if I travelled but 12 Hours every
Anf. 24 Days.
Days, What will 12 Hours require?
Here I fee that the third Number 12, is lefs than the firft
Number 16; but 'tis plain 12 Hours travelling daily, muſt
require more Days to perform the fame Journey in, than
16 Hours daily travelling at the fame Rate; and therefore
this Queftion alfo falls under this backward or inverted Me-
thod of working the Rule of Three, thus:
16
18
12
16
108
18
12 | 288
24
But if the Question run thus: If 8 Yards of Cloth coſt 32
Shillings, What will 50 Yards amount to?
Here the Operation must be Direct, and the fecond and
third Numbers must be multiply'd together, and their Pro-
duct divided by the first, as directed Page 121; for when
ftated, the Queftion will ftand thus:
8
32
50
Here the third Number is bigger than the first, and alfo
requires a bigger for the fourth than the fecond here given;
for 'tis plain 50 Yards of the fame Cloth muſt be worth or
coft more than 8 Yards.-And fo if the Queſtion was thus :
If 50 Yards of Cloth coft L. 10, What will 8 Yards of
the fame come to?
This Question ftated, will ſtand thus:
50
L.
IO
8
Here the third Number is lefs than the firft, and the fourth
Number muſt be leſs than the ſecond; becauſe 8 Yards of
the fame Commodity cannot coft, or be worth as much as
50, therefore this Queflion must be wrought by the Direct
Method.
The Inverfe RULE of THREE.
255
!
Method. And now I prefume I have made this Part of the
Matter quite plain. My next Buſineſs fhall be to teach
you how to abbreviate or fhorten the Work of fuch Questi-
ons, as are to be wrought by either of theſe Rules.
We will take the last Question for an Example.
If 50 Yards coft L. 10. What will 8 Yards coft?
Firſt fee whether the Question be Direct or Inverfe; if
Direct, after your Question is ftated, and before you have
done any Work, fee whether there is any common Number
that will divide the first and fecond Numbers, or the firſt
and third Numbers, without leaving any Remainder; if there
is, divide thereby, and then work with the Quotients, as you
would have done before they were fo abbreviated: As above,
the firſt and ſecond Numbers may be divided by 10, and then
they will be 5-1-8; for 5 and I bear the fame Proportion
to one another as 50 and 10 do; then multiply the ſecond and
third together, and divide by the first, and the Quotient will
be the Anſwer fought.
As here, -- 8 Times I is 8, and this is to be deem'd Pounds;
divide this 8 by 5, and the Quotient is 1 Pound, and or
12 Shillings.
The fame may be done in the Inverfe Rule, only you muſt
mind, that whatever you divide the firſt or fecond Number
by, you muſt alfo divide the third by the fame. By thus
abbreviating your given Numbers, you will frequently reduce
your Work into a fmall Compafs, which, without it, would
many Times be very operofe.
EXAMPLE.
If 6 Men, in 20 Days, can build a Wall, In what Time
will 12 Mẹn do the fame?
Men. Days. Men.
If 6 20
12
I
I
2/20
2
10
Here the first and third Numbers may be divided by
6, and they they become 1 and 2, with which, working as
before, the Anfwer is alfo the fame.
Ог
256 Of FELLOWSHIP.
Or you may divide the fecond and third by 4, and
then the given Numbers will ftand thus:
6
5
3
6
3130
IO
Or, after you have divided the fecond and third, as be-
fore, you may divide the firſt and third by 3, and then
they will be
2 — 5 — 1
2
10 the Anſwer as before.
COMPANY or FELLOWSHIP.
The common Schools, and moft Books wrote upon this Sub-
ject, make this a diſtinct Rule; and therefore thoſe Lads that
learn at School, and other young Perfons that ſtudy this Sci-
ence by Book, think themſelves not fufficiently ſkill'd to an-
fwer fuch Questions as are reduc'd to this Head, till they
have formally gone thro' a Set of fuch Questions, and learnt
the particular Methods of working the fame, as are ufually
given, altho' they can readily work common Queſtions in
the Rule of Three, and Practice: Therefore to conform to
Cuftom, I will here exhibit fuch Questions as I judge will be
proper for the Exerciſe of this Part of the Rule of Three.
But first I must take Notice that 'tis ufually divided into
two Parts or Rules, viz.
FELLOWSHIP
or COMPANY without Time,
And, FELLOWSHIP or COMPANY with Time.
Tho' 'tis to be obſerv'd that in actual Trade and Buſineſs,
the latter is never us'd; but as poffible Cafes may happen,
and it can be no harm to have an Example or two of the
laft Sort, I will fet down fome of both Sorts.
The
Of FELLOWSHIP.
257
The Name or Title of theſe Sort of Questions naturally in-
dicate, that more than one Perfon is fuppos'd to be con-
cern'd; therefore I will only hint here, that 'tis by this
Method the Shares in Ships, Proportions of Bankruptcies, Di-
vifions of Sums left by Will, and actual Copartnerſhips are
or may be adjuſted.
Examples of each follow.
1. A and B enter into Partnership, and make a Common-
Stock of L. 1000, of which A has L. 400, and B L. 600;
at the Year's End they find their Nett Gain in Trade is
L. 217 16 6, What is each Man's Part thereof?
:
:
Note, Whenever Time is mentioned, whether it be Days,
Weeks, Months, or Years, and all the Partners Stocks are fup-
pos'd to be all the Time concern'd, then no Notice is taken of
the Time; and all fuch Questions are underſtood to be in that
Part of this Rule, called Fellowship without Time: As is the
Example now before us, in which, and in all fuch like, you
muft fay, As the Sum or Total of all the Partners Stocks is to
the whole Gain or Lofs; fo is each particular Man's Stock to
his Share thereof. As here,
L.
If 10 | 00
5
L.
S. d.
217: 16 : 6
| 435: 13:
2
L.
4100
2
L. 87: 2: 7 for A's Part.
L.
If 10 | 00
P
L.
217 : 16:
S.
d.
16: 6
3
5
1
L.
6100
3
|653: 9: 6
L. 130 13 10 for B's Part.
Here the firft and third Numbers are firſt divided by 100,
which is done by only cutting off 2 Cyphers; then the Remain-
ders, 10, 4 and 6 are divided by 2, which reduces the 1000
to 5, the 400 to 2, and the 600 to 3; by which 2 and 3, mul-
tiplying the middle Number, or whole Gain, and dividing the
feveral Products by 5, the Quotients are the refpective Anfwers
LI
for
258
Of FELLOWSHIP.
for each Partner's Share of the faid Gain: The Truth of
which may be known by adding the Parts thus found to-
gether, and if their Sum make up the whole Gain or Lofs,
you may conclude the particular Shares are right: Thus,
L. 87: 2: 7 A's Part.
L. 130
13: 10
4
B's Part.
Total is L. 217: 16:
6 which is the whole Gain.
!
2. Five Perfons freight a Ship, with five different Sorts of
Goods, and agree to divide the Gain or Lofs of the respective
Goods, in Proportion to their feveral Stocks, viz.
7
which gains L. 89: 13:
which lofes L. 42: 19: 4
which lofes L. 198: 17:
which gains L. 27: 12:
which gains L. 216 :
A L. 179: 13: 6
B L. 281
C L. 312
D L. 419
19: 4
14 8
E L. 494
II :
7:
L. 1688: 5: 6
Tot. Gain L. 333:
5: II
Tot. Lofs L.241: 16: 10
6
4
Balance Gain'd 91 09: I
What is each Man's Share of the Lofs or Gain, according
to this Agreement?
Here the Value of the feveral particular Goods amounts to
L. 1688: 5: 6, and the Balance between the Lofs and Gain is
L. 91:9: 1 Gain; then fay, As the whole Stock is to the
whole Gain, fo is each particular Stock to its Share of the
Gain; and as the two first Numbers are always the fame,
turn the odd Money of both into Decimals, and divide the
Second by the first, and the Quotient refulting will be a fix'd
Multiplier, by which every particular Share or Stock being
multiply'd, the feveral Products will be the refpective Shares
of the Gain.
The fix'd Multiplier thus found, will be ,05417, by
which each Sum that compoſes the whole Stock, being mul-
tiply'd, the contracted Way, the feveral Products will be as
follows.
L. 179,675
Of FELLOWSHIP.
259
L. 179,675
714,5..
L. 281,986
L. 312,733
7145..
898 4
719
I 8
I 2
7145..
14098
15637
1128
28
20
1251
31
22
L. 9,733 A's Part. L. 15,274 B's Part. L. 16,941 C's Part.
L. 419,550
7145
20978
1678
L. 494,350
7145
24718
197 7
42
29
L. 22,727 D's Part.
A
Proof.
49
35
L. 26,779 E's Part.
8
5
9
44
m/+m/+
=]c
∞ na N
9,733, or L. 9: 14:
B 15,274, or L. 15: 5
C 16,941, or L. 16: 18:
D 22,727, or L. 22: 14:
E 26,779, or L. 26: 15: 7
L. 91,454, or L. 91: 9: 1
Decimals are in nothing more ufeful than in fuch like
Questions as thefe; for if this Question was wrought by the
common Method, the finding of one Man's Share would be
more Work than by this Method: The finding all the Shares
is, as appears by one Share here following, found by the
common Method. Therefore I would always adviſe you,
when there are feveral Parts or Shares to be found, firſt to
find a common or fix'd Multiplier; and do with it as you
fee done in this laft Example.
L12
If
260
Of FELLOWSHIP.
If L. 1688 5:6-L. 91:9: IL. 281: 19: 4
20
:
20
33765
I2
405186
20
1829
5639
12
12
21949
67672
21949
609048
270688
609048
67672
135344
(12)
405186) 1485332728 (3665 Pence.
1215558...
cimals. The like may be
ſeen by trying any of the o-
ther Parts or Shares.
Here after all the Work is
done, the Anf. is L. 15: 5:54,
for B's Share of the Gain, in
Proportion to his Stock of
L. 281: 19: 4 which is ex-
actly the fame with the ſe-
cond Operation in the De-
-21013015-5d.2
2697747
2431116
15 - 5 5.
2666312
2431116
2351968
2025930
326038 Prts of Pen.
4
3. Suppoſe a Bankrupt's Effects,
1307752
1215558
92194 Prts of a Far.
after all Charges are
deducted, fhould amount to L. 1739: 13: 8 ½, and his ſe-
veral Creditors were as follows, viz.
F L. 313
Of FELLOWSHIP.
261
F L. 313: 7:3
G
H
J
290 : 4:
700 :
6
486: 13: 8
K
600 :
I
L
500 :
M
381: 10:
N
418:
Total Debt L. 3689 15 5
What must each Man have; and how much did he pay in
the Pound? Anfwer,
F
muſt have L. 147,747 or L. 147: 14: 114
G
H
7
K
L
M
N
1
រ
1
136,838
1
136: 16:
9
330,042
330 :
: IO
229,466
229:
9: 3
282,893
235,744
179,873
197,083
282: 17: 10
235: 14: 10
179: 17:
197 : 01:
500
8
ml+=+¬le -let
Total L. 1739,686 or L. 1739: 13: 81/1
Note, That as the Decimal Places are carried but to three
Figures, they will not always be quite perfect, but may be a
Part of a Farthing too much or too little; but you fee that
in the whole Total or Sum of all the eight Parts, there's but
one Farthing too much by the Decimal Part, and o in the o-
ther Part, which is exact enough for all Buſineſs.
2
Here, according to the Directions, Page 192, I find a com-
mon Multiplier, which is uſed inſtead of a common Diviſor ;
which Multiplier alfo anſwers that Part of the Question, How
much per Pound does the Bankrupt pay? viz. 9 s. 5 d. and
almoſt of a Farthing: For ftating the Question, I fay, As the
whole Debt L. 3689: 15:5, or L. 3689,77083, is to the whole
Credit or Value of the Bankrupt's Estate, L.1739: 13: 81,
or L. 1739,685416; fo is 1 Pound to its Part or Share, or
what he pays in the Pound, viz. 4714, &c. or 9 s. 5d. &c.
I
Viz.
262
Of FELLOWSHIP.
L. 3689,77083
368,97708
Viz.
·L. 1739,685416 - L. I
173968541
Anfw.
New Div. 13320,79375 |N.Div. | 1565,716875 (,47148874
1328 317500
237399375
Or,
232455563 L.0:9:5
and ,6554 of
4943812 a Farthing,
3320794 which is al-
moſt 3.
1623018
1328317
294701
265663
29038
26566
2462
2325
137
133
4
Here, as the Decimal of the Divifor has a fingle Repetend,
I work according to the Directions, in Pages 194 and 199,
and the Quotient ,47148874 is both the common Multiplier,
and the Anfwer to that Part of the Question, How much does
he pay in the Pound? Viz. L. o: 9:5 and about of a
Farthing.
2
And to fhew the Agreement of all the feveral Methods
heretofore taught you, I will alſo find the common Multi-
plier without fubftracting the terminate Figures, and only
contract the Work as taught, Page 194, as you fee here fol-
lowing, and the Quotient is the fame as above.
3689,77083
}
Of FELLOWSHIP.
263
3689,77083 | 1739,685416 (47148874
1475908333
20
263777083
9142977480
258283958
12
5493125
5/13729760
3689771
4
1803354
53919040
1475918
Or 9 s. 5d. and near
327436
of a Farthing; as
295182
will appear by work-
ing this Stating by
32254
the commonMethod,
29518
as is done below.
2736
2583
153
148
5
L. S. d.
L.
S. d.
If 3689: 15:5
20
1739: 13:8 ±
L. I
20
20
73795
34793
20
12
12
12
885545
417524
240
4
1670098
240
66803920
3340196..
885545) 400823520 (452 Farthings.
885545
264
Of FELLOWSHIP.
1
885545) 400823520 (452 Farthings.
3542180..
113 Pence.
4660552
4427725
9:5
2328270
1771090
557180
885545 Remains, which is almoſt
3
of a Farthing.
Here the fame Anſwer is alſo found by the common or
vulgar Method; and now, as in the former Queſtion, I
multiply the common Multiplier by each Man's particular
Stock, and you'll have the feveral Anfwers as below.
,47148874
5263,313
141447
4715
1415
141
28
I
F's Part L. 147,747, Or L. 147 : 14 : 114.
,47148874
522,092
94298
42434
94
9
2
G's Part L. 136,838, Or L. 136 : 16:9 4.
47148874
Of FELLOWSHIP. 265
1
,47148874
007
H's Part L. 330,042, or L. 330
: 10
47148874
386,684
188596
37719
2829
283
38
150 H
I
7's Part L. 229,466, or L. 229: 9:34.
47148874
006
K's Part L. 282,893, or L. 282: 17: 104
47148874
005
-L's Part L. 235,744, or L. 235: 14: 101.
,47 148874
5,183
141447
37719
471
236
M's Part L. 179,873, or L. 179: 17:5
Mm
,47 148874
A
:
266
Of FELLOWSHIP.
,47148874
814
188596
4715
3772
N's Part L. 197,083, or L. 197 : I: 8.
'Tis needleſs to obferve what abundance of Labour is faved
by this Method of working, efpecially where the Sums are
5, 6 or 700 Pounds only; to demonftrate which, one of
theſe Sums are here wrought by the common Method, viz.
H's Part, which is done above Decimally with leſs than 20
Figures in all, whereas the other takes up more than 200:
In the one, there is only a ſmall Multiplication of 5 Figures
by 1 Figure; in the other there is 8 much larger Multiplica-
tions, and I very large Divifion, and three finall ones; by
reafon whereof 'tis more than fifty to one that there is an
Error committed in the Vulgar Way, before the Decimal
Way.
If L.3689: 15:5 give L.1739: 13:8 what will L.700 give.
20
20
73795
12
885545
20
34793
14000
12
12
417524
4
168000
1670098
168000
13360784000
10020588
1670098.
885545) 280576464000 (4 | 316840
885545
Of FELLOWSHIP.
267
885545) 280576464000 ( 4 | 316840
2656635.....
1
12 | 79210
1491296
885545
210166010: 10
6057514
330:00: 10
5313270
7442440
7084360
3580800
3542180
386200
Remainder is
Parts of a Farthing.
885545
By this Example you plainly fee the Excellence of De-
cimals in all fuch like Cafes, and therefore I would recom-
mend the Study thereof to your ſerious Confideration; and
if it fhall fo happen, that you shall have occafion for large
Calculations often, you will find your felf continually eas'd
of a vaſt Load of Trouble thereby, which will fufficiently
recompence the Labour you'll be at to gain the Maſtery of
this moſt exquifite Branch of Arithmetick. By the like Me-
thod Ships Parts, &c. may be valued; as for Example, Sup-
pofe a Ship divided into 32 Parts, and the Total Value was
L. 1856: 16, which were purchafed by the following Perfons,
viz.
32
A 6
B
What is the Value or Coft
of each Man's Part?
IO
C
JAH
4
D
3
E
9
Total 32
As
M m 2
268
Of FELLOWSHIP.
I
As 32 Parts is to L. 1856,8, fo is 1 Part to L. 58,025
which multiply'd by the Number of Parts
or Shares each Perfon has, gives the Value
or Amount thereof; as you fee done in the
Margin.
3
D L. 174,075
A L. 348,150
E L. 522,225
C L. 232,100.
B L. 580,250
Total L. 1856,800
In the cafting up Divifions of Bankruptcies, 'tis com-
mon to caft up the Amount per Pound, and to pay fome-
what less than the whole, to defray ſmall contingent Ex-
pences, fuch as Meetings at the Tavern about the Divifions,
fmall Preſents to the Attornies or Clerks for adjuſting the
Claims, &c. and therefore commonly pay even Shillings,
Sixpences or Threepences. As, fuppofe in the former Que-
ſtion, as the Sum to be divided, would not come to 6d. they
paid 9 s. and 3d. per Pound: Some would chooſe to caſt up
the feveral Claims by Practice, as follows,
5 s. & L.
45.
3 d. 16.4s.
L. 313
78
7 3 at 9 s. 3 d. per Pound Sterl.
7:
6:9
62: 13
13:5 54
3 18:4
I
16
144 187-4-12, or L. 144:18:712
4s. L.
1/4/
༨
ditto
Or,
L. 313
7: 3
!
62: 13:52
I s.
of 4 s.
3d.
3 d. 4 of 1 s.
15
3:18
62 13 5.2
मालाल
13:4 4 I
18: 4-I
(of a Far.
Anf.
144 18 71 or L.144:18:7 or 12
19
४
Which
Of FELLOWSHIP.
269
Which by the Decimal Way would ftand thus:
L. 313,3625
52 64
1253 45
188 02
627
I 57
144,931, or L.144:18:7
which is about a Farthing too much; but as both the Deci-
mals are perfect, if you multiply both the Factors together at
full length, without contracting, the Product will come out
exactly the fame as above, thus,
313,3625
I have found the Value
4625 of this Decimal in the Pro-
duct, by the Rule, Page
15668125 176, for the fake of Exact-
6267250
18801750
nefs, and not by the Rule,
Page 203, which is not
12534500 quite fo perfect.
144,93015625
20
1860312500
12
7123750000
4
and
195000000
100
Here you ſee that it comes to L. 144: 18: 7 and
Parts of a Farthing, which is equal to 12, the fame as above.
Thus you ſee the Objection made by fome, that Decimals
will not anſwer fo correctly as the common Method, is only
for want of due Skill to know how to manage them: In-
deed ſometimes the Contractions will be a Trifle defective or
abundant, if you retain only three Figures in the Decimal
2
Part
270
Of FELLOWSHIP.
Part of the Product; but if you retain 4 or 5 Places 'twill
feldom differ half a Farthing from the Truth; a Matter too
fmall in Buſineſs to take Notice of, as appears by the laſt
Example, keeping four Figures in the Product.
313,3625
5 2640
1253 450
188 017
6267
1 567
L. 144,9301, or L. 144: 18 7 4
which is but the 20 Part of a Farthing too much by the
Rule of Inspection, Page 203, and ſcarce enough by the exact
Rule of Multiplication, Page 176.
I have already obferv'd, that in practical Bufinefs, Fel-
lowſhip with Time is feldom or never ufed; yet to obviate all
Objections to this Work, I will fhew you how to perform
fuch like Queſtions, if they fhould be infifted on, viz.
In
FELLOWSHIP or COMPANY with Time.
You must multiply each Man's Stock by his Time, and add
thofe Products together, and fay, As the Total Amount of the
Products is to the whole Gain or Lofs, fo is each particular Pro-
duct to the Gain or Lofs of each Partner, for the Time his Stock
has been employ'd.
EXAMPLE.
Suppoſe A puts in L. 100 for 6 Months, and at the End of
that Time makes it up L. 169, for the whole Voyage; B puts
in L. 320 for 8 Months, and at the End of that Time is per-
mitted to take out L. 120 by Draft upon their Factor, and
leaves the Remainder in for the whole Voyage; C puts in L.95,
and at three Months End he puts in L. 75 more; and 9 Months
after that, puts in L.127 more, they all agree to divide the
Lofs or Gain of the whole Voyage in Proportion to their re-
Spective Stocks and Time. Query, What must each have, fup-
pofing that at the End of 21 Months their Factor makes them
a Remittance of L. 1000 for the Balance of their then Account
Currant ?
I
Of FELLOWSHIP.
271
I have made this Question as complicated as moft are, on
purpoſe to ſhew you as much Variety in one, as are uſual-
ly found in many; and I will explain every Part neceffary
very fully, and then every Purpoſe, I prefume, will be
anfwer'd by this one Example, as well as by feveral.
A L. 100 for 6 Months.
6
600
169 for 15 Months.
15
845
169
2535
3135 A's Total for 21 Months.
B L. 320 for 8 Months.
8
2560
200 for 13 Months.
13
2600
5160 B's Total for 21 Months.
C L. 95 for 3 Months.
3
285
175 for 9 Months.
9
1575
302
272
Of FELLOWSHIP.
302 for 9 Months.
9
2718
4578 C's Total for 21 Months.
5160 B's Total.
3135 A's Total.
Their whole Total is 12873 for the whole Time.
By the Queſtion, you fee A has L. 100 for 6 Months,
which multiply'd by that Time, and the Product, is 600;
then he put in L. 69 more, which remains to the End of
the Time, viz. 15 Months longer; the Total of the two
Sums put in, is L.169, which continued in Stock 15 Months;
and this Product added to the former, makes 3135 for A's
whole Stock and Time. B puts in L. 320, which multiplied
by 8, the Months it continued in Stock, gives 2560 for the
firſt Part of his Stock and Time. Now it may be fuppos'd
that the Ship or Cargo was in Whole or in Part difpos'd of,
and that B might want L. 120 to be employ'd by their com-
mon Factor, upon his own private Account and Riſque,
which he has the Confent of the reft of the Partnerſhip he
ſhall withdraw; and therefore his remaining L. 200 is mul-
tiply'd by 13 Months, which is the Refidue of the Time
before the Account is made up; and this Product added to
the other, makes 5160 for his whole Stock and Time. C's
firft Part, or L. 95 must be multiply'd by 3, the Months
'twas employ'd alone; then L. 170, the Sum of 95 and
75 muſt be multiply'd by 9 Months, the Time the firſt in-
creaſed Stock continued; and then L. 297 the whole Stock
muſt again be multiply'd by 9, the remaining Part of the
21 Months; and thefe 3 Products make 4578, which three
Totals make in all 12873. Now the Sum receiv'd is L.1120,
viz. L. 120 by B at 8 Months End, and L. 1000 the Ba-
lance of the whole Account; then, I fay,
As 12873 is to L. 1120, fo is 3135 to A's Part.
5160 to B's Part.
4578 to C's Part.
Here
Of FELLOWSHIP.
273
Here you may find the common Multiplier, which will
be,0870038, which multiply by the refpective Parts of
A, B and C, and you will have the following Anſwers.
0870038
3135
4350190
2610114
870038
2610114
272,7569130 A's Part.
,0870038
5160
52202280
870038
4350190
448,9396080 B's Part.
?
0870038
4578
6960304
6090266
4350190
3480152
398,3033964 C or L.398: 6:03
448,9396080 B L. 448: 18:9
272,7569130 A L. 272:15:14
Total 111919999174 or L. 1120
This Total wants only about Part of a Farthing in the
Decimal.
N n
12873
274
Of FELLOWSHIP.
12873 | 1120, &c. o (,0870038 common Multi-
102984
90160
90111
49000
38619
103810
plier.
102984
826 Remains.
3135
,087
272,745 A
21945
448,920 B
25080
398,286 C
272,745
1119,951 Total.
5160
,087
Here wants almoſt
a Shilling; fo that
its plain the Multiplier
was not large enough.
41280
36120
448,920
4578
87
32046
36624
398,286
272,756
Of FELLOWSHIP.
275
272,765 A
448,939 B
398,303 C
1119,998 Total.
Or you may do it thus, by the contracted Method.
,0870038
5313
261011
8700
2610
435
272,756 A's Part.
,0870038
0615
435019
8700
5220
448,939 B's Part.
,0870038
8754
348015
43502
6090
696
398,303 C's Part.
Here, by this contracted Method, the Total wants al-
moſt two Farthings; but each particular Man's Share wants
leſs than a Farthing: 'Tis frequent alfo to work with a
Part of the common Multiplier, eſpecially when 'tis either
very large, or has Cyphers in the Middle of it, as before, in
Page 274.
Nn 2
OF,
*
}
Of FELLOWSHIP.
276
Or, you may do it by the Vulgar Method thus :
Stock and Time. Prin. and Gain. Stock and Time.
As, 12873 is to L. 1120 fo is A's 3135 to
3135
62700
3135
3135
L. s. d.
12873) 3511200 (272 : 15 :
1 ½.
25746
93660
90111
35490
25746
9744 Parts of a L.
20
194880
12873
66150
64365
1785 Parts of a Shilling.
12
21420
12873
8547 Parts of a Penny.
4
34188
25746
8442
Remains
Parts of a Farthing.
12873
*
If
Of FELLOWSHIP.
1
277
If 12873 - L. 1120
5160 B's Part.
5160
67200
1120
5600
L.
S. d.
12873 5779200 (448 : 18 : 91
51492
63000
51492
115080
102984
12096 Parts of a L.
20
241920
12873
113190
102984
10206 Parts of a Shilling.
12
122472
115857
6615 Parts of a Penny.
4
26460
12873
614
Remains
Parts of a Farthing.
12873
If
278
Of FELLOWSHI P.
If 12873 L. 11204578 C's Part.
-
1120
91560
4578
4578
L. 3.
d.
12873) 5127360 (398: 6:02
38619
126546
115857
106890
102984
3906 Parts of a L.
20
78120
77238
882 Parts of a Shillings.
12
10584 Parts of a Penny.
4
42336
38619
3717
Parts of a Farthing.
12873
L. 272: 15: 1½ 8442 A's Part.
448 18 9
0614 B's Part,
398: 6 : 03717 C's Part.
L. 1120: -
Total.
Here the Whole is done by the common Method, which
exactly agrees with both the contracted and full Decimals, fav-
ing that A's Part in the Decimals, by the Rule of Inspection, is
made 3 Farthings, which by the common Way is but 2 Far-
things, and about of a Farthing; fo that in all Cafes where
many Figures are to be us'd Decimals are by much the beft,
BAR-
Of BARTE R.
279
•
BARTER
Is another Branch or Part of the Rule of Three, by
which the Merchant proportions the Quantity or Quantities
of one Sort of Goods, that are, or ought to be given in Ex-
change, for the Quantity or Quantities of another Sort of
Goods.
And here the Method of Computation is by confidering the
feveral Sorts of Commodities, to be of fome certain Value, in
ſome known Species of Coin, us'd by the Exchangers; and
theſe Queſtions may be propos'd both eafy and fimple, and
alfo very much compounded, as follows, viz.
Q. I. A has 50 Pieces of Cloth, each 30 Ells, allowed to
be worth 3s. 4 d. per Ell; B has Silk worth 16s. 6 d. per lb.
How much Silk must A receive for his Cloth?
Anfwer, 303 lb. 33.
50 Pieces.
30 Ells in a Piece.
3: 4 is L. 1500 Ells in all.
L. 250 the Value of the Cloth.
If 16 s. 6d. buy 1lb. of Silk, What will L. 250 buy?
2
40 lb.
33 Sixpences.
33) 10000 (303
90
lb.
Anf. 3033
100
Silk muſt be given
for all the Cloth.
99
I
323
Here you firſt find the Value of all the Cloth by Practice,
which amounting to juſt L. 250, by the Rule of Three, you
fay, If 16 s. 6d. buy 1 lb. Silk, What will L. 250 buy, which
produces 3033 ? So that a Bale of Silk that weighs 303 lb.
, and a Pack of Cloth containing 50 Pieces, condition'd
as in the Queſtion, are exactly equal in Value to one another
the like is to be obſerv'd in all other Cafes, regard being
had to the particular Circumſtances attending them. As
ſuppoſe,
;
Q. 2. C has a Parcel of Shalloons, worth in ready Money,
L. 2: 6 6 per Piece, for which D offers a Parcel of Stock-
ings at L. 3:7:6 per Dozen; but fays C, as I can't rea-
:
2
dily
280
Of BARTE R.
dily difpofe of the Stockings, they being too dear, I will not
comply, unless I may be allowed to rate my Shalloon at L.4:04
per Piece, which the other agreeing to, Query, What was the
Stockings rated at in ready Money? Anf. L. 1: 17: 44 per
Dozen.
As L. 4,2 is to L. 2,325, fo is L. 3: 375
3:375
1,4
775
Here I fay, As the Barter
- Price of the Shalloon is to
the ready Money Price, ſo is
the Barter Price of the
Stockings to the ready Money
Price: In which all the odd
Monies being made, or
turn'd into Decimals, I ab-
breviate the firft and fecond
Numbers, by dividing by 3,
and then working with the
remaining Numbers, the
Anf. comes out L.1:17:44
by the common Computa-
tion of Decimals by Inſpec-
tion, which is above half a
Farthing too little; the true
and exact Anſwer being
L. 1: 17:4: 1 Far. and
# of a Farthing, which is
fomewhat more than half a
Farthing more than the a-
bove Answer; but, as alrea-
775
16875
23625
23625
1,4) 2,615625 | 1,86830
1 4
I 21
I 12
+
95
84
116
I12
42
42
5 Remains.
14
dy has been faid, no Buſineſs can be tranfacted nearer than a
Farthing in actual Payments; therefore fuch firſt Anſwer is
ſufficient for any real Bufinefs.
There are a great many Ways of varying theſe Sort of
Questions: As, fuppofe this laft, where you may imagine
Dgave the Price mention'd, viz. L. 37: 6 per Dozen
for the Stockings, and that C could eaſily fell them for ſo
much, and that the Shalloon was really worth, or could be
fold only for L. 2:6:6 per Piece; then you may en-
quire the whole Gain or Lofs of each Perfon. After you
have afcertain'd the Quantity of each Sort of Goods, or you
may
Of BARTER.
281
may enquire the Gain or Lofs per Cent. or per Dozen, &c.
As under theſe Circumftances D will lofe L. 1: 10:1
per Dozen by his Stockings, and C muft gain L. 1: 17: 6
per Piece by his Shalloons, and by every Hundred Pounds
worth of Stockings, that D fells or barters at this Price or
Rate, he muft lofe L. 80: 12: II, that is, the Goods that
he receives by this Barter, and fells for L. 100, coft him
really L. 180: 12: 11, as may be tried by the two diffe-
rent Ways following, viz.
If L. 1,8683 lofes L. 1,5067, What lofes L. 100?
Anf. L. 80,645, &c. or L. 80: 12: II.
Or, If L. 2,325 gain L. 1,875 L. 100?
Anf. L. 80: 12: 11.
by
By the foregoing Work it appears that D lofes L.1,5067
per Dozen by the Stockings, and confequently that for
L. 180: 12: 11, he receives but L. 100; fo that he lofes.
L. 80: 12: 11 by every L. 100 he fo receives; and on
the other hand, Ċ gains fo much by every L. 100 of his
Goods he delivers, more than the prime Coft, becauſe for
what is really worth but L. 100, he receives the Value of
L. 180: 12: II. Now this laft Part of the Work is what
is properly call'd Lofs and Gain ; and is by fome made a diſtinct
Rule.
Q. 3. Suppofe A has 50 Dozen of Hats, worth 7 s. 6d.
per Hat, for which B offers him Handkerchiefs, worth 22 s.
a Dozen; but A will not agree, unless he may rate his Hats
at 8 s. a Piece. Query, How must B rate his Handker-
chiefs per Dozen, to advance the Price or Rate equally?
Anſwer, at L. 1: 3:5½ and the Part of a Far-
thing per Dozen: For,
If 7 s. 6d. or,375 be 8 s. or ,4, What will L.1,2 or L.1, I
,375) 4,4 (1,1783, &c.
be ?
575
650
375
2750
2625
1250
1125
Or,
0°
125
282
Of BARTER.
Or, 15
8
44
8
S. d.
Far.
15352 23 5
2 and
30
52.
45
7
I2
84
75
9
4
36
30
6
15
or &
If the Value of the Decimal,1733, &c. be exactly found
by the Rule, Page 176, and proper Allowances made for its
being a Repetend, both Ways will perfectly agree; as ap-
pears from what follows, viz.
,1733
20
3 | 4866
12
5. | 6000
4
2 | 4000
Here the first Product is 3,4666, &c. for as the Repetend
3 would go on infinitely, the Cypher in the Multiplier, in a
finite Number of Figures, will not take Place; but the Pro-
duct
Of BARTER.
283
+
the
10
2
duct muſt be all 6's, from the Beginning of the Repetend to
the End of the Number of Figures the Decimal is to have
Places; then by the Rule, Page 187, the Product of the
12 will be 5,6000, a Finite Number, and that multiply'd
by 4, gives 2,4, a Finite Number, for the Product, the o's
fignifying nothing, the 2 being cut off for 2 Farthings;
+
is
4 or of a Farthing, as per the Vulgar Method.
Q: 4. Suppofe A has a Parcel of Wool, worth L. 5 per
C. ready Money, and B has Broad Cloth, worth 13 s. 4d.
per Yard, which he offers A for his Wool; to which A re-
plies, I will not confent, unless you allow me L. 6 per C. for
my Wool, and pay me Part ready Money; if you comply
with this, I will take your Cloth at a Price proportionable.
Query, What must B rate his Cloth at per Yard, and what
Quantity of Money and Cloth must В give A for a Ton of
Wool, upon theſe Conditions?
B
Here I firft find the Value of a Ton of Wool
at L. 6 per C. to be
from which fubft. for ready Money, which is
and the Remain. is the Amount of the Cloth, viz.
L, 120
L. 40
80
But before I can know the Quantity of the Cloth, I muft
know the Rate it is to be valued at. Now it appears by
what is above, that A gains L. 20 by his Ton of Wool, and B
muſt gain L. 20 by felling L. 80 Worth of Cloth, becauſe he
pays L. 40 ready Money. Therefore deducting L. 20 from
L. 80, the Remainder is L. 60: Then I fay, As L. 60, the
real Value of the Cloth, is to L. 80, the advanc'd Value; fo
is 13 s. and 4d. the ready Money Price of a Yard, to 17 s. 9d.§
the advanced Price, which being wrought by the con-
tracted Method will ftand thus:
:
610
31
8 | 0
"- "
13,4
4,5 //
3
17,9
I
Or you may fubftract the Money paid A both from the
true and advanced Price of his Wool; and fay, As the leffer
Remainder 3, iš to the greater Remainder 4;. fo is the ready
Money
Oo 2
ม
284
Of BARTER.
3
Money Price of a Yard, 13 s. 4 d. to the advanced Price,
17: 9, as before.
Viz.
L. 6 the advanced Price of C. of Wool.
2 the ready Money paid.
4 the advanced Price for the Remainder.
L. 5 the true or ready Money Price of C. of Wed.
2 the ready Money paid.
3 the ready Money Price exchanged.
As L. 3 ready Money is to L.4 exchange, fo is
dy Money Price of a Yd. to 17 s. 9d., the ad-
vened Price.
s. d.
13:4 the rea-
4:53
179
In this, and all fuch like Cafes, you may reduce the firſt
and fecond, or firft and third Terms, as low as poffible; as
you fee done by the 60 and 80, in the firft Method, which
are reduc'd to 3 and 4; and in the fecond Method, they
are in the lowest Terms, viz. 3 and 4; then divide the fecond
Term by the first, and the Quotient is the Multiplier of the
third Term, whofe Product is the Answer to the Question:
As in both the Methods above, the Quotient is I, by which
multiplying the third Number, 13 s. 4 d. and the Product is
17 s. 9d. .
Now then to find the Quantity of Cloth to be delivered,
I fay,
If 13 s. 4d. or L. buy 1 Yard, What will L. 60?
3
2 180
Anf. 90 Yards.
So that B muſt give A L. 40 in Money, and 90 Yards of
Cloth for a Ton of Wool. Now to know the Truth of this
Anfwer, you are to obferve, that the Quantity of Cloth and
Money muſt be the fame, whether you confider the Wool
advanc'd in Price or no; becaufe 'tis agreed, that in Con-
fideration of paying fo much ready Money, the Cloth was to
be
Of EXCHANGE.
285
be advanc'd proportionably, fo as to make neither Party
Gainer or Lofer; fo that if you fay,
If 17 s. 9d. buy 1 Yard, What will L. 80 buy? You
will have go Yards for the Anfwer, as you may eaſily try;
for the more ready Money is paid, the more muft the Price of
the Cloth be advanced; becauſe by Money, nothing is to be
fuppos'd gain'd or loft.
Q. 5. A has Tin worth 50s. per C. ready Money, but
in Barter he will have L. 3: 6: 8, and moreover infifts up-
en gaining L. 10 per Cent. over and above the Advance, and
that he will have ready Money; B has Lead worth 14 s per
C. ready Money: To how much must he advance his Lead, to
comply and barter upon equal Terms with A?
2
Anſwer, B muſt rate his Lead at L. 1: 18: 6 per C.
Another Part of the Rule of Three, is that Part of Barter
commonly called
EXCHANGE.
;
And this more immediately relates to negotiating the Mo-
nies of one Kingdom or Country, for, or with another, either
according to the real Values that they bear to one another
or, according to the Advantage or Difadvantage that one
Country is in, with relation to its Trade, War, &c. to ano-
ther; for though all Countries, that keep or make their Coin
or Money of one certain Fineneſs of Gold or Silver, bear an
exact Proportion to one another, and this Proportion is cal-
led the Par; yet the Exchange to or from one Country and
another, very feldom is at that Rate; and 'tis obfervable
that the moft Trading Countries have commonly the Advan-
tage of the Exchange in their Favour.
Note, 'Tis a fure Sign of a profperous Trade, when the
Courſe of Exchange is above the Par, and the contrary
when 'tis below it; and as the current Coins of almoſt all
Kingdoms differ, both in their Names and Species, fo do alfo
their Weights and Meaſures; tho' in the true Nature of the
Thing, they muſt all neceffarily be reduced to fome one com-
mon Meaſure, Standard or Par, before the real intrinfick
Value can be afcertained. As for Inftance, a Dutch or Flemish
Ell is found, by comparing them together, to be of our
Yard, or of our Ell; and a French Aulne or Ell is
I of our Fard, and or I of our Ell; which Proporti-
ons must be known before any one can pretend to trade
with the People of thofe Countries, for the Commodities mea-
fured by thoſe Meaſures, with any Degree of Regularity or
Certainty.
I
or
286
Of EXCHANGE.
Certainty. The like is to be obſerv'd of the Weights and Coins.
Having premis'd thus much, next ought to follow fome
Tables, that exprefs the Par or Proportion between the Weights,
Meaſures and Coins of the moft noted publick trading Cities;
for which purpoſe I have extracted a few from the beſt
extant, but find fo many Differences and Defects among
them, that 'tis with much Caution any of them can be de-
pended on. However, the moft correct, I think, are thefe
that follow.
▾ Pennick is
Dutch Money.
English Money. Dec.
L. S. d. f.
0:0:04² 1,0003125
:
8 Pennicks is 1 Groot, which is L.0 0:03
2 Groots is a Stiver, or
6 Stivers is 1 Shilling, or
20 Stivers is one Guilder, or
6 Guilders is 1 Flemish Pound, or
10 Guilders, or 33s. 4d. Flem. is
A common Dollar is
A Specie Dollar is
Ι
تمامی مدام داری
0:0:
о I //
0:0:7
0: 2:
O: 12:
I :
:
0:3:
0:5:
[ ]
111
,0025
,005
,030
,
I
.6
༨.
,15
,25
This Table exhibits the intrinfick Value of the Coins of
the Places where they are uſed, expreffed in British or Ster-
ling Money at the exact Par; and upon account of the
Fractional-Parts the Dutch feldom or never ufe Decimal
Arithmetick, tho' the Calculations will never vary from the
Truth, if rightly and judiciouſly perform'd, as you fhall
hereafter fee: But tho' the above be the true Proportions of
the faid Coins to one another, yet the Exchange is ordinarily
made higher upon account of having the Money paid in
Bank, and not current Money, the Trouble of keeping Ac-
counts, the Safety of Conveyance by the Poft in Bills of
Exchange, and faving Charges of Carriage from one Place
to another, &c. But if you are minded to convert any
Species of English into Dutch Money, or contrarily, you may
work after fome of the following Manners. As,
Suppoſe I would know how many Pounds Flemish were in
100 Sterling?
L. Stér.
Flem.
L. Ster. Flem.
Say, as 3 is to 5, fo is 100 to 166: 13:4
5
500
L. 166 13:4
By
Of EXCHANGE.
287
3
3
2
>
By the above Table you fee, that 10 Guilders, or 33 s. 4 d.
Flemish is one Pound Sterling, that is, L. 13 Flem. is equal
to L. or L. I Sterling; wherefore reducing the mix'd
Number 1 to a compound Fraction, 'twill be ; then mak-
ing L. 1 Sterling a Fraction, that has the fame Denominator,
viz., and rejecting the common Denominator, the Numera-
tors are the whole Numbers that exactly exprefs the Propor-
tion between a Pound Flemish and a Pound Sterling, as above,
viz. 5 and 3, or 3 and 5.
Or you may say,
As L.1. Sterling is to 10 Guilders, fo is L. 100. St. to 1000Guil
And as 6 Guil. make a L. Flem. 100 divid. by 6 gives 166 L. Fl.
You are likewife to obferve, that the Dutch do alfo compute
their Money like the English, by 20 Schillings to the Pound,
and 12 Pence to the Schilling; fo that in reality their Schil-
ling is equal to of our Shilling, or 8 of our Pence, &c.
when computed at Par.
2. In L. 365 13 s. 6 d. Sterling, How many Guilders Dutch
Money? alfo, How many L. s. d. Dutch or Flemiſh?
L. 365,675 Sterling.
10 Guilders in a Pound.
Anſwer 3656,750 Guilders, or 3656 Guilders and 3.
Or 3656 Guilders, 2 Schillings and 3 Stivers.
L. 365
10
Or you may do it thus:
3650 Guilders.
I s. 6d. rem, or 18 Pence.
6 ditt. in 12 Shillings.
Guilders 3656 : 2 : 3 add the 2 Schil-
lings and 3 Stivers, found to be 18 Sterl.
and the Whole will be as above,
5
36 | 90 | 2Schillings.
72
6 | 18
3 Stivers.
18
Or, by the foregoing Table, 'tis plain, that 5 Stivers is
equal to one of our Six-pences. Now as the Remainder a-
bove is I s. 6d. or 3 Six-pences, which is 15 Stivers, and that
is 2 Schillings and 3 Stivers, the fame as above.
But
288
Of EXCHANGE.
But tho' at any Time the Exchange fhould be either a-
bove or below Par, the Subdivifions of the National Monies
remain the fame as when the Exchange is at Par; only there
is a greater or leffer Number of Pounds Sterling or Flemish
given, for the Sum exchanged, according to the Rate the
Courſe runs at. As, fuppofe,
3. I want to remit L. 500 from London to Amfterdam, and
the Courſe of Exchange was at 34 s. 6d. Flemish for a L.
Sterling, How much must I receive in Amfterdam?
This Queſtion may be done either by the common Rule
of Three, Practice or Decimals as follows.
If L. 1. Sterl. is L. 1: 14 6 Flemish L. 500 Sterl
40 Six-pences in a L,
69
500
4101345010 Flemish Six-pences.
L. 862: 10 Anfwer.
Or Decimally, thus:
L. 1,725 equal to L. 1: 14: 6 Flemish.
500 Sterling.
{
L. 862,500, or L. 862: 10 Flemish.
By Practice, thus:
is L. 500 Sterl. at L. 1: 14: 6 Flem. per
L. 1:
!
10 s. is L. and is 250
2
4 s. is L. and is roo
6d. is ĝ of 4s. and is 12 : 10
L. Sterl
L. 1:14:6 Total is L. 862: 10 Flemish.
!
Any of thefe Ways will perfectly answer the Question;
but you may perceive the Decimal Way is both the ſhorteſt
and eaſieſt; for after a little Practice, you will readily be able
to turn any odd Money, or s. and d. whether Dutch or Eng-
Liſh, into Decimals, without uſing any Table; for as the Sub-
divifions are the fame, the Decimal Figures will be the fame,
tho'
of EXCHAN G É.
289
tho' not of the fame intrinfick Value, yet of the fame relative
Value; for 'tis the Property of Decimal Fractions, as well as
Vulgar Fractions, to exprefs the fame Part of any Unit by the
fame Numerical Figures, tho' the Application may be of di-
verfe Species and Values: As for Inftance, may ſtand for 2
Farthings, 4 Furlongs, or 2 Quarters of a Yard, &c. accord-
ing as a Penny, a Mile, or a Yard may be the Integer; and fo
Decimally, the fame Part, viz. one Half, let the Integer bear
what Name it will, is always expreffed by ,5; and tho' I
have formerly intimated this before, yet I thought it conve-
nient to repeat it here again, left the Learner fhould have for-
got it, or perhaps, might not be fo well qualified then to un-
derſtand the feveral Obfervations before made, as by this Time
his further Practice, and greater Skill in this Science, may
have render'd him able to take in ſuch Obfervations: And
another Reaſon is, that ſome Perfons may read only fuch Parts
of this Book as they may immediately want, without deliberate-
ly going thro' every Branch of it: And as Decimal Arithme-
tick has not been fo much ſtudied, and univerfally practifed,
as it deferves, fome of theſe Hints may be uſeful to thoſe
who are but meanly ſkill'd therein, tho' otherwiſe good
Artifts.
But as the Denominations of Guilders, Stivers, &c. are
more commonly us'd than L. s. d. to anſwer the laſt Quef-
tion in thofe Species, you may work thus. After you have
done as before, you may fet down the Anfwer, viz.
L. 862: 10 Flemish.
6 Guilders is a L. Flemifb.
5175 Anſwer in Guilders.
Or you may do thus for the Whole.
Ster.
S.
:
d.
L. 500 at 34 6 Flem. per L. Sterling.
6 Stivers is 1 Schilling.
207
Stivers in 34: 6 Flemish.
500
210 | 103500 Stiv. in all.
Anf
5175 Guilders, as before.
Pp
290
Of EXCHANGE.
Or, you may multiply the given Number of Pounds Ster-
ling by 10, the Guilders in a L. Sterling at Par; and then
$. d.
d.
Guilders.
S. d.
fay, As 33: 4, or, 4l00 is to 50l0o fo is the Courfe 34: 6
12
414
As
d.
Or, Decimally thus:
50
4120700
To 5175 Guild.
Guild.
50 is the
-2dN°
33,4 or 4100 is to 34,5 fo is L. 500 to 5175 1725,0 abbre-
3
IO
3 viated.
I
103,5
50100
5175 Guild
50
5175,0
The Reaſon of the above Procefs is this, viz. 33 s. 4d. or
400 d. Flemish is a L. Sterling at Par, and 5000 are the
Guilders in L. 500 Sterling at Par; then faying, As 400 d.
Par is to 34 s. 5 d. the Courfe, fo is 5000 Guilders Par, to 5175
Guilders per the Courfe; and cutting off ,00 from the firſt and
third Numbers, the Remainders are 4 and 50: And in as
much as the ſecond Number muſt be multiply'd by 12, to
bring it into Flemish-Pence, to make it of the fame Name
with 400 d. the firft Number, inftead thereof I multiply by
3; becauſe the firſt Number 4, and this 12, being divided
by 4, the Quotients will be 1 and 3; and by this Means I
have nothing more to do with the firft Number. The like
may always be done in all other Cafes, and generally the
Work will be very eafily performed by Decimals, let the
Courſe be what it will; as fhall be made more fully to ap-
pear by what follows.
Suppose
Of EXCHANGE.
291
Suppoſe I have a Bill of Exchange from Rotterdam for
4575 Guilders 19 Stivers, which is to pay L. 429; 13: 4
Sterling, What is the Exchange reckon'å at?
Anfwer, 35 s. 6d. Flemish for a L. Sterling.
To do which, I ſay,
S. d.
Guild. Stiv.
As L. 429, or 13: 4 is to 4575: 19, fo is L. 1St. to 35 s.
20
6d
3
1289
As L. 429,8
42,9
3
91519
3
3 (6)
1289) 274557 (213 Stivers.
2578
Sch.35 : 3 St. or 6d.
1675
1289
3867
3867
ooo Remains.
Decimally thus:
Guilders.
4575,95
I
457595 (3)
386,7 ) 4118,355 (10,6,5
3867
35,5
25135
23202
Or 35 s. 6d.
19335
19335
Remains.
Explanation of the above.
In the firft Method, as 13 s. 4 d. is
of a L. I only
brought
multiply or bring the L's into Thirds, and the Guilders are
Pp 2
292
Of EXCHANGE.
brought into Stivers; then multiplying and dividing, as in
common, the Anſwer is 213 Stivers for a L. Sterling, which
divided by 6 gives 35s. 6d. Flemish.
19
20
In the ſecond or Decimal Method, I turn the odd Parts or
Money into Decimals, viz. 429 L's and 6, for the Repetend
belonging to the 13s. 4d. and 4575 Guilders, and 95 Parts
for the 19 Stivers, which being 12 of a Guilder is the fame
as if they were Shillings, and confider'd as Parts of a Pound
Sterling; and ordering the Decimals, as directed Page 198,
I multiply and divide as in common, and the Anſwer is 10,65
Guilders per L. Sterling; and as 20 Stivers make a Guilder,
and 6 Stivers make a Schilling, theſe two Numbers fet Frac-
tional-wife, thus, and abbreviated, produce 2; fo that
removing the feparating Point to 106,5, and dividing by 3,
the Quotient or Anfwer is 35,5 or 35 Schillings and Six-
Pence Flemish per L. Sterling, as before: And thus you fee the
Decimal and common Way perfectly agree, and the Work one
Way, is much the fame in Quantity as the other Way; but
if the Sum had been extended to more minute Parts of the
Engliſh Coin, or were thoſe which were not Aliquot-Parts
of the L. Sterling, fo as the Abbreviation could not be made;
then the common Way would be much more troubleſome
and tedious than the Decimal Way.
Suppoſe I am to remit_by Bill L. 785,15 Sterling from
London to Amfterdam, Exchange at 34 s. 7 d. Flemish for
a L. Sterling, How many Guilders, Stivers, &c. must be
paid in Amfterdam for the faid Bill?
Anſwer, Gl. 8152: 3 Stivers and 2 Pennings; which may
be wrought thus, Decimally.
L. 785,7,5
19 6,4375
98,21875
34,7 Exchange.
33,4 Par.
1,3 Difference.
Anfwer, 815 2,15625
Or,
Guild. Sti. Pennings.
8152:32
Here, after the Sterling-Money is exprefs'd in whole Num-
bers and Decimal-Parts, you only remove the Point one
Place more towards the Right-hand, and then they are Guil-
ders, and Decimal-Parts of a Guilder, becauſe 10 Guilders
I
is
Of EXCHANGE.
293
is 33 s. 4d. Flem. or. 1 L. Sterl. at Par; then fubftracting
the Par, 33 s. 4d. from the Courfe of Exchange 34,7, and 1,3
or 15 d. is the Increafe or Difference. Now as 400 Flem-
ifb-Pence is a true Flemish-Pound, or 33s. 4d. Ten-Pence
will be the Part thereof; therefore I divide the Guilders,
&c. by 40, and the Quotient is 196,4375; and 5 being the
of 10, I divide the laft Quotient by 2, and this Quotient is
98,21875; which two Quotients being added to 7857,5, the
Guilders and Parts in the Sterling-Money at Par, makes
8152 Guilders, 15625 Parts, or 8152 Guilders, 3 Stivers, and
2 Pennings; and if you don't know how to Value the,15625
by Inſpection, you may multiply them thus:
,15625 Parts of a Guilder.
20 Stivers in a Guilder.
Stivers 3,12500 and Parts of a Stiver,
16 Pennings in a Stiver.
Pennings 2,00000
Or you may do thus: for 7857,5 Guilders.
Guild. Stiv. Pennings.
33
fet down
7857: 10:
at 33,4
10d. is
40
L.
196:
8: 12
5 d. is
of 10d.
98: 4: 6
Total 34: 7
8152: 3: 2
That is, inftead of putting down the ,5 Parts of the
Guilder Decimally, put it down in Stivers, &c. and work as
in common Practice, having regard only to the Subdivifins
of 20 Stivers making a Guilder, and 16 Pennings making a
Stiver; and this muſt be your Care, let the Subdivifions be
what they will, and apply'd in any Manner, either to Coins,
Weights or Meaſures.
Now as the Low-Countries, and England, have much Ne-
gotiation by Bills of Exchange, I will exhibit a Table by
which you may convert English into Dutch Money, or
Dutch Money into English Money, by one fingle Multiplica-
tion; as you may obferve by what follows.
s. d.
294
Of EXCHANGE.
5.
d., Flem. L's Guild.
English L's.
Guilders.
32
1,6
9,6
,625
,10416, &c.
01,60208%
9,6125
,6241873
,1040312
I
1,60416
9,625
6233766
,1038961
I
1½ 1,60625
9,6375
,6225681
,1037613
2 1,608%
9,65
6217617
,1036169
21,610416 9,6625
,6209573
,1034929
3 1,6125
9,675
,6201551
,1033258
31,614583 9,6875
,6193549
,1032258
4 1,616
9,7
,6185567
,1030928
421,61875
9,7125
,6177606
,1029601
5 1,62083
9,725
,6169666
,1028277
1
5
1,622916
9,7375
,6161746
,1026957
6 1,625 9,75
$153846
1025641
61,62708% 9,7625
,6145967
,1024328
7 1,62916 9,775
,6138109
,1023018
7 21,63125
9,7875
,6130268
,1021711
8 1,63
9,8
,6122449
,1020408
81,6354169,8125
,6114650°
,1019108
9 1,6375
9,825
,6106870
,1017812
91,63958% 9,8375
,6099110
,1016518
10 1,6416 9,85
,6091371
,1015228
101,64375 9,8625
,6083650
,1013942
II
11
1,64583 9,875
6075950
,1012658
II
1,647916 9,8875
,6068268
1,65
OHHN 2 mm
33
01,65208%
1,65416
I
I 2
1 ½ 1,65625
1,6583
21,660416
2
9,9
3 1,66458% 9,9875
4 1,6
10,-
41,66875 10,0125
5 1,6708% 10,025
,, &c.
9,9125
6052963
,1011378
, 10, &c.
,1008827
9,925
,6045339
1007556
9,9375
,6037735
,1006289
9,95
,6030150
,1005025
9,9625
,6022584
,1003764
3. 1,6625 9,975
,6015037
,1002506
,6007509
,1001251
ან
,I
>5992509
,0998751
5985037
,0997506
51,672916 10,0375
,5977584
,0996264
6 1,675 10,05
5970150
,0995025
61 1,677083 10,0625
,5962733
,0993789
7 1,67916 10,075
,5955335
,0992556
71,68125 10,0875
>5947956
,0991326
8
1,68%
10, I
$940,&c.
,099.0,&c.
811,685416|10,1125
›5933254
0988876
s. a
Of EXCHANGE.
295
·s. d. Flem. L's. Guild.
ΙΟ
English L's.
9 1,6875 10,125 5925926
91,68958% 10,1375,5918619
1,6918 10,15 ,5911330
1,6937510,1625,5904, &c.
1,69583 10,175 ,8896805
1,697916 10,1875,5889571
10,2 ,5882353
10
II
II
34
1,7
00
1,702083 10,2125,5875153
I
1,70418 10,225 ,5867971
Guilders.
,0987654
,0986436
,0985222
,0984, &c.
,0982$, &c.
,0981595
0980392
,0979192
0977995
2.
11,70625 10,2375,5860805,&c., 0976809, &c.
1,708% 10,25 ,8853658,&c.,0975609, &c.
21,710416 10,2625,5846529
0974755
3 1,7125 10,275 ,8839416
0973236
311,71458310,2875,5832321
0972054
4 1,716 10,3 ,5825243
0970874
+556
77∞ ∞
4 1,71875 10,3125,5818182
1,72083 10,325 ,5811138
51,722916 10,3375,5804111
0969697
,0968523
,0967352
1,725 10,35
5797102
,0966184
61,727083 10,3625,5790110
,0965018
1,72916 10,375 5783134
,0963856
1,7312510,3875,5776174 ,0962696
8
1,73
10,4 5769230
,0961538
81,735416 10,4125,5762305
,0960384
2
35
9
9
ΙΟ 1,7416
10 / 1,7475
II
II
1,7375 10,425 5755396
,0959233
1,739583 10,4375,5748503
,0958084
10,45 5741627
,0956938
10,4625,5734767
,0955794
1,74583
10,475 ,5727924
0954654
1,747918 10,4875,5721096
,0953516
1,75
10,5 5714285,&c.,0952381
00
1,75208% 10,5125,5707491
0951248
I
1,75416 10,525 ,5700713
,0950119
11,75625 10,5375,5693950
,0948992
2
1,7583 10,55 ,5687204
0947867
21,760416 10,5625,5680473
0946745
3
1,7625 10,575 5673759
C945626
31,76458% 10,5875,5667060
0944510
4 1,76 10,6 ,5660377
41,7687510,6125||,5653710
1,77083 10,625
5
54 1772918 10,6375,5640423
,0943396
0942285
5647058
,0941176
,0940070
s. d.
296
Of EXCHANGE.
s. d. Flem. L's., Guild.
English L's.
Guilders.
6 1,775 10,65 5633802
641,777083 10,66255627198
7 1,77918 10,675 5620608
0936967
,0937866
0936768
8
7 1,78125 10,68755614035
1,783 10,7 ,5607477
811,785416 10,7 125,5600933
0935673
,0934579
0933489
9 1,7875 10,725 8594405,&c.,0932401
91,78958% 10,73755587893
,0931315
ΙΟ 1,7916 10,75 5581395
,0930233
102 1,79375
10,7625 5574913
10,7625,5574913
,0929152
I I 1,7958% 10,775 5568445
,0928074
II
1,79796
10,7875 5561993
,0926999
36.
1,8
10,8
55, &c.
,09259
001,802083 10,8125,5549133
,0924856
I
J+
~ 22 mm+tnno O 7 N∞ ∞
1,825 10,95
II 2
1,85
00
37
1,80416 10,825 ,5542725
11,80625 10,8375,5536332
1,8083 10,85 ,5529954
21,810416 10,8625,5523590
3 1,8125 10,875 ,5517241
1,814583 10,8875,5510907
3
4 1,816 10,9
41,81875 10,9125,5498281
5
6
61,82708% 10,9625,5473204
7
1,82916
71,8312510,9875,5460751
8 1,83
81,835416 11,0125,5448354
9 1,8375 I1,025 ,5442177
91,839583 11,0375,5436014
1,84.18 11,05 ,5429864
10 1,8437511,0625,5423729
ΙΟ
I I
11
1,8458311,075
1,847916 11,0875,54115
1,85208% 11,1125,5549133
I 1,85416 11,125 ,5542725
11,85625 11,1375,5536332
2 1,858% 11,15 ,5529954
21,860418 11,1625,5523590
,0923788.
,0922722
,0921659
0920598
,0919540
0918484
5504587
50917431
,0916547
1,8208% 10,925 ,5491991
0915665
51,822916 10,9375,5485714
,0914286
5479452
0913242
,0912201
10,975 5466970
,0911162
,0910125
II,
84
,0909, &c.
,0908059
,0907029
,0906002
,0904977
0903955
,5417607
,0902934
0901916
II. I
>
,8405, &c.
,ogoog, &c.
,0924855
0922121
,0921722
,0921659
,0920598
Of EXCHANGE.
297
s. d. Flem. L's. Guild.
English L's.
Guilders.
3344nnoo 7 7∞ ∞
1,8625 11,175
31,86458% 11,1875
1,86
11,2
,5517241
50919540
,5510907
,0918484
5504587
,0917431
1,8687511,2125
41,86875 11,2125,5498282
,0916380
5 1,8708311,225 5491991
,0915332
51,872916 11,2375
5485704
0914284
6 1,875 11,25
,5479452
,0913242
61,877088 11,2625
5473204
,0912201
1,87916 11,275 5466946
1,88125 11,2875 5460751
11,28755460751
,0911158
,0910125
8
1,88% 11,3
,$4, &c.
8
1,885416 11,3125
,5448354
9
1,8875 11,325
5442177
91,88958% 11,3375
,5436014
ΙΟ
1,8916 I1,35
5429864
,0904977
101,89375 11,3625
,5423740
,0903957
I I 1,8958311,375
5417607
,0902934
,0901916
38 1,9
II,4
,840, &c.
1,0googoog
Øg, &c.
,0908059
,0907029
,0906002
111,897916 11,38755411499
The Ufe of the foregoing Table.
This Table extends from 32 Schillings to 38 Schillings
Flemish, for a L. Sterling, the extreme Limits whereof very
rarely happen; but the common Courfe of Exchange be-
tween England and Holland, is from 34 Schillings to 36s. 6d.
per L. Sterling, tho' there may be Occafion upon fome extra-
ordinary Emergencies to ufe fome of the other Rates: You
are likewife to obferve, that or 4 of a Penny, are feldom
or never us❜d, and generally even Pence, tho' I have put
throughout the Whole.
EXAMPLE.
1
If I was to remit L. 675: 12: 6 Sterling from London
to Rotterdam, Exchange at 35 s. 9d. Flemish,per L. Sterling,
How many Guilders, &c. must I receive there?
2
Note, Tho' the Computation or Courfe is expreffed in Schil-
lings and Pence for a L. Sterling, the Bills are always ex-
preffed in Guilders and Stivers, &c. Therefore I look for
the Courſe in the firft Column, and againft 35s. 9 d. I
find in the third Column, under the Word Guilders, the
fix'd Number 10,7375, by which multiplying the given Sum
Qq
(having
298
Of EXCHANGE.
(having firſt turn'd the odd Money into Decimals) by the con-
tracted Method, and it will ſtand thus:
675,625
5737,01
67562 50
4729 38
202 69
47 29
338
7254,524
Or 7254 Guilders 10 Stivers and §.
As all Monies and Bills in the Low-Countries are negoti
ated in Bank-Money, that amount to more than 300 Flo-
rins or Guilders, and not in common or current Monies, fuch
as Schillings, &c. and as the Bank-Money is finer than the
current Money, and as there muſt be Books of Accompt kept
with the ſeveral Parties, fo an Agio or Allowance for fuch
Trouble and Intereft, is made from 3 to 6 per Cent. therefore
it may not be amifs to have an Example or two of reducing
Common to Bank-Money, at 2 or 3 different Agio's or Allow-
ances: For fuppofe I fhould fend a Parcel of Goods to Am-
fterdam or Rotterdam, &c. and fhould chooſe to have a Re-
mittance for the fame in Money, and not in Goods; then
my Correſpondent will charge me, over and above all
Charges of Weighing, Meafuring, Warehouse-Room, Com-
miffion, &c. the Agio at 3, 4, 5, &c. per Cent. As for
Inftance,
Suppoſe I have Goods that amount to 3865 Guilders, or
Florins, current Money, How much Bank-Money will that
come to Agio, at 5 per Cent?
Say, As¹2 current is to 42% Bank; fo is 3865 current to
3680 19 21 Bank.
3865
Of EXCHANGE.
299
5 | 105: 100
3865
21
20
20
21) 77300 (3860: 19,1
63
143
126
170
168
20 Remains.
20 Stivers.
400
21
190
189
I
21 Remains.
Or you may do it thus:
(5)
If 101 100) 3865 Guilders.
S
773
100
Bank
21) 77300 (3680 or 19 Stivers
Or thus:
If 12:00 21) 3865 Guilders
18421
3680 or 19 Stiv. 2 Bank.
Q92
Again,
300
Of EXCHANGE.
Again,
Current
Flor.
In 2459
Stiv. Pen.
17 8 Agio at 4% How much Bank?
Flor. or Guil, Stiv. Pen.
If 104 12% -100- 2459: 17: 8 to 2352; 10: 13fer.
16
633
16
104
14758
2460
1673
)3935800 (2352
904
St. Pen.
Remains, which is 10:13 fere
1673
Or thus:
Guil. Dec.
As 104,5625 is to 100, fo is 2459,875 to 2352,540
But if you want to know only how much the Agio comes
to, do thus:
As 100 is to 41% or 4,5625, fo is 2459,875 to
Or Decimally thus, by Practice.
at 1p. Cent. 24,59875
at 4 per Cent. 98,39500
at
8
16
at
52654
9839500
1229938
147593
12,299375
1,537422
112,231797
4920
1230
112,23181
Or,
112: 4: 11, fere. &c.
Or
"
Of EXCHANGE.
301
Or thus in common Numbers.
Flor. Stiv. Pen.
2459: 17: 8
4
I per Cent.
4 per Cent.
16
3 is of 4
1229: 18
1/6 is 3 of 18%
16
153 14:13 11
9839: 10:
18: 12
11223: 3: 91
20
4/6383
16
1027
And fo for any other Rate whatever, by fome of thefe
Methods, you will have the Anfwer requir'd.
We now paſs on to give fome Examples of other Coins.
and Countries, and firſt for France, that being the next
nearest to us.
FRANCE.
Here they keep their Accounts in Livres, Sols and Deniers,
but exchange by the Crown, which at Par is worth 4 s. 6 d.
Sterling; fo that 3 Livers 60 Sals, or French-Pence, is equal to
4s. 6d. or 54 Pence Sterling; I Liver or 20 Sols, French, is
equal to 1 s. 6d. or 18 Pence, Sterling; 1 Sol, or 12 Deniers,
French, is equal to 3 F. or of a Penny Sterling; 1 De-
nier French, is equal to of a Farthing, Sterling.
EXAMPLE 1.
1. Suppose I take up 500 Crowns, at Lyons in France,
and draw a Bill upon my Correfpondent in London at 52d.
per Crown, How much Sterling Money must be paid for the
Jame? Anf. L. 109: 7:6.
You may work this, and fuch like Queftions, by any of
the following Methods.
Crowns
302
Of EXCHANGE.
!
*
Crowns
500 at 52 d. ½ per Crown.
500
26000
250
45.
L.
4 d. 12 45.
12 26250
20 | 2187:6
L. Sterl. 109:7:6 Anſwer.
Crowns.
Or thus:
500 at 52 d. or 4 s. 4d. per Crown.
100
8:6: 8
jd. & 4d.
I
: 10
L. 1097 6 Anfwer, as before.
Or Decimally thus:
52 d. or 4s. 4 d. is,21875 per the Table, Page 205.
½
2
500 Crowns.
L. 109,37500
Or, L. 109,7:6 as above.
EXAMPLE 2.
Suppoſe Iremit L. 613: 8: 9 Sterling to Paris, Exchange
at 56 d. 4 Sterling for a French Crown, How many Crowns
must be receiv'd at Paris? Anf. 2594 Crowns 1 Liver 7 Sols
4 Deniers.
Here working by the Rule of Three, I fay, Cr. Sol. Den.
As 56d. is to 1 Crown, fo is L. 613: 8: 9 to 2594: 16:4129
Theſe Numbers reduc'd and work'd by any of the former
Methods will come to 2594 Crowns 16 Sols 4 Deniers,
and
Of EXCHANGE.
303
and 148 Parts of a Denier; but in Bufinefs they never
take Notice of Parts of a Denier.
227
89
In Italy they keep their Accounts in the fame Species with
France, but the Exchange is by the Dollar or Piece of
which is worth 4 s. 6d. Sterling; fo that 'tis in all Refpects
the fame as France, only the Crown is called a Dollar or
Piece of
SPAIN.
Here they keep their Accounts in Rials and Mervadies,
but exchange with the Piece of Eight, whofe Par is 4: 6
Sterling; and 1 Mervadie is - 4 of a Farthing Sterling;
372 Mervadies is 1 Rial, which is 6d. Sterling; 8 Rials is
1 Piece of, or 4s. 6 d. Sterling.
-
EXAMPLES.
1. How many Pieces of must I receive for L. 300 Ster-
ling, Exchange at 53d. per Pieces of § ?
Ri. Mer.
Anfwer, 1358 3: 343 3.
Here fay, by the Rule of Three, As 53d. is to fo is L.300.
Then work as in common, and you'll find the Anſwer
come out as above.
2. Deliver'd at Madrid 2000 Pieces of, Exchange at
56 d., What must be receiv'd in London?
Anfwer, L. 470 16 8.
:
This Queſtion is done like the firſt, in French Money.
PORTUGAL.
Here they keep their Accounts in the nominal Species
called Rees, which are the Subdivifion, and a Thousand of
thefe is the Integer; one thoufand Rees are about 6 s. 9d.
Sterling at Par, and is called a Mill-Ree. Upon the Portu-
gal Coins called Moidores, double Moidores half Moidores,
&c. is ftamped 4000, 8000, 2000, &c. or 4 Mill-Rees, 8
Mill-Rees, &c. But on the 9's, 18's, 36's and L. 3:12
Pieces the Rees are not ſtamped.
EXAMPLE.
How many Mills and Rees must I have for L, 250 Ster-
ling, Exchange at 6:5 per Mill-Ree?
As
I
304
Of EXCHANGE.
As 6,5
12
| |
77
Mills Rees.
1,000- L. 250 Anf. 779 - 220 $9
59
240
Mills, Rees.
77) 60000,000 (779, 220 $9
If I have a Bill for 1758 Mills 340 Rees, Exchange at
5 s. 10d. Sterling, per Mill-Ree, How much Sterling-Mo-
ney must be receiv'd for the fame ?
This Example may be wrought after any of the follow-
ing Methods.
Mills Rees. S.
s. d.
1758,340 at 5: 10½ per M. R.
5's. L.
2/
439: 10
10 d. / 5 s.
½ 20 10 d.
2
6
for 320 Rees
73: 5
3:13:3
: 2:
I: 20 for 200
7:0:1
2
100
1 : 1:3 + 20
I: 1:3
+ 20
L. 516: 10: 3 Tot. 1: 11:3:4 + for 340
Here, after finding the Value of 1758 Mills, then I take
the Parts of 1000 for the 340 Rees, which comes to
1 s. 11d. 3f. 23, which wanting but 2; of a Farthing of 2s.
I fet down 25. for it, and then the whole comes to
L.516:10: 3. Or you may find the Value of 340 Rees by
faying, As 1000 is to 5: 10, fo is 340 to 2 s. almoft: Or
you may do the Whole Decimally thus:
1758,340
573 920
351668
158 251
5275
1231
-88
516,513
Or
L. 516: 10:3
Or thus:
1758,340
Pence St. in a Mill 701
123083800
879 170
10
12 | 123962 | 12 Ps of a Pen.
201103310-2
L. 516-10-3
Note, The firft Line arifes by multiplying by 70, and the
fecond Line by dividing by 2, or taking the of the given
Line
Of EXCHANGE.
305
Line, and three Figures parted off upon account of 1000
being the Denominator of the Mill Ree; then the Product
123962 is whole Pence, and the 970 are ſo many 1000 Parts
of a Penny, which wanting but of the whole, I fet down
a Penny for it, which is added to L. 516: 10:2, the A-
mount of 123962 Pence; and the Whole is L.516: 10: 3,
as before.
ΙΟΟ
As Portugal and England have now, and in all probability,
will long have a confiderable Trade together, and in Con-
fequence thereof, much Negotiation of Bills between them,
I will give another Example, wrought by all the foregoing
Methods, to make them more evident.
Mills Rees,
What Sum Sterling must be receiv'd in London for 2756,675
Exchange at 6 s. 2d. Sterling per Mill Ree?
Anfwer, L. 858
3
19
92.
( 1 )
2756 Mills and 675 Rees at
6s. 2 d. per M.R.
4s. & L.
551: 4
25. 4s.
275: 12
125 is
250
2d. I
1725.
22:19: 4
200 is
1000
2d.
5: 14: 10
100 is
- 200
250 is of 1000 1:6:2:
I: 2:3:3
924
7:1:31: I
9:1: 12/2
I
Σ
id.
½ d.
2:17:
5
675 Rees is
4:2:1:3 3-3
4 2 675 is
L. 858 11 92
:
(2)
Decimally thus:
,3114583
576,6572
622917
218021
15572
1868
187
22
2
858,589
Or, L. 858: 11 : 91/1
RI
(3)
306
Of EXCHANGE.
(3)
Or thus:
2756,675
Pence in a Mill-Ree 74
11026700
19296725:
1378337/
689168
12 | 206061,456
2/0/17171,9
L. 858:11:9½
To find the Value of the Fractions in the firft Method,
obferve the Directions given in Page 154; and in the third
Method, you are to obferve, that the 456 Parts of a Pen-
ny are found by multiplying them by 4, and cutting off 3
Figures from the Product, and then 'twill be 1,825 which
is I Farthing, and 825 thouſand Parts of a Farthing, which
reduc'd will come to of a Farthing, the fame with the
firft Method; but as it is near, we put down a Half-penny.
for it, which exactly agrees with the Decimal Method,
computed by the common Rule; from whence may be ob-
ferv'd, how eafy the Decimal or fecond Way is, above either
the 1 or 3, which requires fo many different Operations.
33
40
Now we will only touch upon the Exchange between
London and Dublin, &c. where, tho' the fame Species of
Money is current, and the Methods of keeping Accounts
are the fame, yet the Value of Money, in the two Places, is
very different at different Times; therefore you ſhall have
an Example or two.
IRELAND.
If I remit L. 250 Sterling, from London to Dublin, Ex-
change at 8 per Cent. more to be receiv'd in Dublin,
What is the Sum to be paid at Dublin?
Anf. L. 270 12 6 Irish.
+
L. 250
Of EXCHANGE.
307
4
2 of 100
42 ditto
1 of 4
L. 250 at 8 per Cent.
ΙΟ
10 -
=
0:12:6
270: 12:6
Or thus:
L. 250 at 84
5
of 100
12,5
2 1/2 1/15
6.25
2/2/20
1.25
-625
L. 270.625
St.
Or thus Decimally.
Irish
As L. 100 is to 108.25-250 Sterl. to L. 270: 12 : 6
250
541250
21650
270,6250
Or
L. 270: 12: 6
This latter Method may be full as well or better than the
upper one, when the Sums have odd Money, upon account
of the Eafinefs of the Divifors; as,
Suppoſe L. 618: 13: 8 to be remitted to Dublin from Lon-
don, at 8 per Cent.
4 ditto
You may do thus:
13 8 at 8
II
24: 14: 11 22
24 14 II 9
100 is
L. 618
4 2 of 100
1
4
16 25
108 is
I: 10: 11 5+ ÷
22
L.669: 14: 5 100.
Rr 2
Or
308
Of EXCHANGE.
100 is
5 is 2!
Or thus:
L. 618 13:
8
20
30: 18:
8 100
20
2 - 11/0
1/1/0
1=2=+
15: 9:
4 10
3:
I
I : 10 42
I : IO : II 21
1084 is
L. 669: 14: 5 93
Which in Buſineſs may be called 6d. for it is 3 Farthings
and 22 Parts of a Farthing.
1280
Decimally thus:
108,25
618,683
Or thus:
618,68%
108,25
9 | 324750
3093416
12373666
36083 for the z
494946666
86600
6186833333
64950
86600
669,7247083
10825
64950
669,7247083
Or, L. 669: 14: 6 almoft.
Theſe two Products are exactly the fame; and if the
Value of the Decimal be found, according to the Rule, Page
176, 'twill come to 14 5 9, and 22, as above.
The like is to be obferv'd in all other Rates whatever,
only that the Parts of fome Rates, are more troubleſome
than the Parts of other Rates, to work by Practice; and
therefore, how fond foever fome People may be of that
Rule, 'tis frequently in a Manner impracticable, as in this
Example following.
If
Of EXCHANGE.
309
If I remit L. 795 Iriſh, from Dublin, to receive the Value
in London, Exchange at 7 per Cent. How much Sterling
Money must I receive?
Anſwer, L. 739: 10: 84,
Sterl.
If 107,5 give 100 795. (739,5348
7525
Or
4250
3225
L. 739; 10: 81
10250
9675
5750
5375
3750
3225
5250
4300
9500
8600
L. 107 ½
Or thus:
900, &c.
Mac IOO
795
2
215
2
215)159000(739:10: 8 444
1505
850
645
2050
1935
115 Parts of a L.
ས
115 Parts of L.
310
Of EXCHANGE.
L. 107
is
A
2 is of
43
115 Parts of a L.
20
215) 2300 (10 Shillings.
215
150 Parts of a Shilling.
12
215) 1800 ( 8 Pence.
1720
80 Parts of a Penny.
4
215) 320 ( 1 Farthing.
215
105
or
215 of a Farthing,
By Practice thus :
I
L. 795
107 12/2
5 is 2 times 2, which is
43
18: 09: 09:0: 36
36: 19: 06: 1: 29
71, this Total fub. L. 55
09 : 03: 2 : 22
100 is the Remainder L. 739: 10: 08: I: 21
for the Sterling Money, as above.
But there are fome Rates from Dublin to London, that
the Aliquot Parts cannot be taken at all; and in fuch Cafes
Practice cannot be made ufe of. As, Suppofe at 74, 84,
9, &c. but let the Rate be what it will from London to
Dublin, you may always do it by Practice, as appears by the
firft Example of this Exchange.
I fhall not continue any longer upon the Subject of fingle
Exchanges, but only take notice that the Weights and Mea-
Jures of different Countries, vary as much as their Coins, and
that thoſe who are expert in working the foregoing Examples,
will eafily apply the former Methods thereto. The next
Thing
Of EXCHANGE.
311
Thing we ſhall touch on, is what is commonly called the
Arbitration of Exchanges, that is, adjusting feveral Rates or
Courfes of Exchanges and Coins of different Countries, and
reducing them fo, as if the Exchange was made directly to,
or from fome one Place to another; by which the Profit
or Lofs of negotiating Bills of Exchange will appear. As
for Example,
Suppoſe I remit from London L. 300 Sterling to my Cor
refpondent at Amfterdam, at the Rate of 18 d. Sterling for a
Guilder Flemiſh, with Orders for him to remit the Nett Pro-
ceed to Bourdeaux in France, after deducting his Commiſſion of
per Cent. for his Trouble, &c. at the Rate of 3 Guilders
per French Crown, How many Crowns must be receiv'd at
Bourdeaux? Anſwer, 1326 Crowns and 40 Sols.
L. 300 Sterling.
40
6 Pences in a Guild. 3| 12000 Sixpences.
If 100 -
Guild. in a Crown
Or, Suppoſe the
4000 Guilders.
20 ditto for Commiffion.
3 | 3980 dit. to be remit. to Bourdeaux
1326: 40 to be paid in Bourdeaux.
Crowns and Sols.
Sum in the foregoing
L. 376, and the Commiſſion but
work as follows.
Question was,
per Cent. then you may
Guil.
100
-
Stiv. Pen.
or 6: 10/2/
}
If 100
L.376
40
3115040
Stiv. Pens.
50,136
10
1 of 100
2 -
I
: 10:2
ΤΟ : 2:
-
2
2 : 1: I
10:16:2 Pen. is:
16: 13: 5:1
: 0:12
14:
14:0
16: 14:
3:1 Commiffion.
3 Guild. is a Cr. 34996: 12: 7:0 Guild. &c. to be exchanged.
11665
: 32 ½ Anſ, Crowns to be receiv'd at Bour.
Here
312
Of EXCHANGE.
Here the Commiffion for the odd Parts, viz. 13 Guilders
6 Stivers, 10 Pennings is fomewhat more than 14 Pennings;
but in afmuch as the Amount is infignificant above 14 Pen-
nings, omit it, and adding the Amount of the Parts to the
Amount of the 50 Hundreds, the Total is 16 Guilders, 14
Stivers, 3 Pennings, &c. which fubftracting from the
Whole, leaves 4996: 12: 7 to be exchang'd, which pro
duces 1665 Crowns, 32 Sols, for which I put down, it
wanting fo fmall a Matter.
J
Now if to the above Queftion, was added the following
Claufes. How much is gain'd or loft in all, and per Cent.
by the above Circulation? And what is the direct Exchange
between London and Bourdeaux ?
17
In the first Question the Lofs in all is L. 1: 10 which is
per Cent. and the direct Exchange from London to Bourdeaux
is at 54 d. 4 Sterling for a French-Crown, which is
I Farthing above Par, or Lofs by every Crown, as may
be eaſily tried by ſaying: As,
199
199
'
199
3
I Crown is to 1 Far. ; fo is 1326 to L. I : 10.
In the fecond Example the Lofs in all is L. 1: 54,
which is L. per Cent. and the direct Exchange is 54973
Sterling per French Crown. The like Methods are to be ob-
ferved, let the Places or Rates be where, or what they will.
7218
Suppoſe a Bill for L. 500 Sterling, negotiated as follows,
viz. first from London to Amfterdam at 38 s. Flemish for a
L. Sterling, then from Amfterdam to Frankford at 6 s. Flem.
for 66 Cruitzers; and from Frankford to Paris at 54 Cruitzers
for a French Crown, How many French Crowns must be
paid for the laft Negotiation, and what is the direct Exchange
between Paris and London ?
A
Anfwer, 3870 French Crowns 22 Sols for the Whole, at
the Rate of 31 d. 29 Sterling for a French Crown, directly
between London and Paris. Found thus by the Rule of Three,
As L. 1 Sterling is to 38 s. Flemish, fo is L. 500 Sterling
to L. 950 Flemifh.
Then,
As 6s. Flemish is to 66 Cruitzers, fo is L. 950 Flemish,
to 209000 Cruitzers.
And,
As 54 Cruitzers is to 1 French Crown, fo is 209000 Cruit.
to 3879 Crowns, or 3870 Crowns 22 Sols
Then,
4.P
As 3870 Cr. 19 is to L. 500 St. fo is 23 to 31 d. 239 Ster.
27
I
OF
Of EXCHANGE.
313
Or you may do this, and all fuch like Queſtions, more
eafily thus:
Set down the feveral Proportions given, and multiply thoſe
that stand one over another continually together, making the
Product on the Left-hand the Divifor, and the Product on the
Right-hand the Dividend, and the Quotient will be your
Anſwer.
L. I Sterling is equal to 38 s. Flemish.
6s. Flemish is equal to 66 Cruitzers.
54 Cruitzes is equal to 1 French Crown.
2508 Divid. (722 Quotient,
324 Divifor -
•
which is 7 French Crowns and for a Pound Sterling, which
multiply'd by L. 500 gives 38702, and by ſaying,
If 7 Crowns 2 be 1 L. Sterling, What is 1 Crown?
I
The Anſwer will be 31 d. oo, as above.
By the fame Method may Weights and Meaſures be ad-
juſted. As,
Suppofe 35 Ells of Vienna make 24 Aulns of Lyons, and 3
Aulns of Lyons 5 Ells of Antwerp; and 4 Ells of Antwerp
5 Ells of Frankford: How many Ells of Vienna are equal to
700 Ells at Frankford ?
Ell
Anſwer, 490 Ells at Vienna.
Here, following the Directions laft given, I fay,
35 at Vienna is equal to 24 at Lyons, and
3 at Lyons is equal to 5 at Antwerp, and
4 at Antwerp is equal to 5 at Frankford.
420 Diviſor
600 Dividend.
1
Quotient, that is
1 ½ at Frankford, is equal to 1 Ell at Vienna.
Then,
As 10 is to 7, fo is 700 to 490.
Suppoſe A of Amſterdam ſhould order B, his Factor or Cor-
refpondent at Paris, to remit 4000 Crowns to London, Ex-
change at 49 d. Sterling for a Crown; and for the Amount
thereof to draw upon himself at Amfterdam at 99 Deniers
per Crown; but it fo happens that B remits to London at
50 d. Sterling per Crown: How must he draw upon Amſter-
dam to comply with his Orders, fo that his Principal may
neither gain nor lofe by this Variation in the London Course?
Anfwer, At 101 Deniers per Crown, for if
S f
you fay,
As
314
Of ALLIGATION.
As 49 d. & St. is to 50 d. St. fo is 99 Den. French to
IOI Deniers French.
For as he had limited the Courfe of the Exchange upon
London, at 49 d. 4, and receiv'd 50 d. therefore he muft
pay proportionally more at Paris for the Crown.
2
More Queftions may be given, but the fame Laws of
Proportion are to be obferv'd, and fome of the foregoing
Methods will folve all the Occurrences of Buſineſs that can
happen; ſo that we'll defift faying any more upon this Sub-
ject, and go on to what is called Alligation; for tho' there
is indeed a ftrong Objection against the common numeral
Way of performing the Calculations, upon Account of the
fmall Number of Anſwers that are ordinarily produced in
whole Numbers, whereas by an Algebraick Procefs, the fame
Queſtions might be fo managed, as frequently to give a Va-
riety of Anſwers in whole Numbers, limited to the particu-
lar Quantities in the Queftion, and by properly increafing or
decreaſing the given Quantities, an innumerable Number of
Anſwers in whole Numbers may be given; but as this Rule
is applicable to many particular Bufineffes, where Mathema-
tical Exactneſs is not required, I will give fome Examples
of it in the common Way, and make proper Obfervations
thereon; ſo that any indifferent Perfon may ſee the Uſeful-
nefs of that Part of common Arithmetick, call'd
ALLIGATION,
From the binding or mixing ſeveral Commodities of dif-
ferent Fineness, Rates, Values or Prices together in one
Mafs, Quantity or Heap, and to difcover the particular
Quantities or Values, &c. of each, that is neceffary or required
to make up the Whole of any particular Fineness, Rate or
Value: And this is ordinarily divided into two Species, viz.
first,
Alligation MEDIAL,
Which is when the Rates and Quantities of feveral Sorts
of the fame Species of Goods are given, to know what the
Mean or Common Price, Value, or Fineness, &c. is, of the
whole Mafs, or Quantity, taken together; as in the follow-
ing
EXAMPLE.
Suppofe 25 Bushels of Wheat at 4 s. 8 d. per Bushel were
mix'd with 47 Bufhels, worth 3 s. 9 d. per Bushel, How much
is a Bushel of this Mixture worth?
2
Anf
Of
315
ALLIGATION.
Anſwer, 4s. od. 3 Far, 1.
"Tis very plain, that the feparate Values of the two
Sorts of Grain here mentioned, muſt be firſt found by Prac-
tice, or otherwiſe.
As thus:
s. d.
4
At 4: 8 per Buſhel
At 3:9 per Bufhel.
5
Value of 4 Bufhels.
I: 3: 4 Value of 5 Buſh.
5
L. 5: 16 8 Val. of 25 dit.L.9:
L. 8:16 3 V. of 47 dit.
15:
12
o: - the Value of 48 dit.
3: 9 Subftract. for 1 dit.
L.8
:
16 3 remains for 47
L.14: 12 11 V. of 72 mix'd
together.
:
And then ſay,
If 72 Bufhels are worth L. 14: 12: 11, What's 1 Buſhel?
I
20
s. d. f.
72)292 | 4
288
: 3 1/8
4
12
59
4
236
216
20
72
or
5
Remains.
18
Now if you would increaſe, or diminiſh your Total Quan-
tities, and ftill preferve the fame Value, you may do it by
faying,
Sf 2
As
316
Of ALLIGATION.
As 72 the Total now found, is firft to 25, and then to 47,
its two component Parts; fo is the new Total given to its pro-
portional Quantities: As, fuppofe I would enlarge the laſt
Total to 100 Bushels, I fay, As
Bufb.
72 is to 25 fo is 100 to 34 2, and as 72 is to 47 fo 100 to 65%2
100
100
72 | 2500 (34
72 4700 (6522
216
432
340
380
288
360
52
20
Peck Pint.
Rem.
Parts of a Bufhel.
Rem.
— or I: I &
72
72
which is 2 Pecks 14 Pints and 3.
Buſhel
Anſwer, 34
at 4: 8 per Buſhel,
at 4 :
6522 at 3: 9 ditto
100 Total at 4s. od. 3f. per Bufhel.
18
But fuppofe you would make a Mixture as follows, of 3
Sorts of Wines, Brandies, &c. without abfolutely determin-
ing the Total Quantity, viz.
worth 4 s. a Gallon, at 5 s. a Gallon, and at 5 s. 6d. per
Gallon.
In fuch like Cafes bring your given Fractions into a com-
mon Denominator, and the reſpective Numerators will be the
Quantities that will anfwer the Conditions required: Which
may be increaſed to any larger Quantities, by multiplying
the feveral Numerators by fome common Number greater
than Unity, or made lefs by dividing the faid Numerators by
fome common Number greater than Unity.
Thus :
6
24
2 24 24
4 13
12)
,, may be 1, 4, or 4, 45, 43, or 4, A,
4
. And when you have determined what your particular
Quantities
Of ALLIGATION.
317
Quantities fhall be, you muſt work as in the firſt Example.
As, fuppofe I chofe to
mix 24 Gallons at 4 s. which is L. 4: 16:
16 Ditto - at 5 s. which is L.
4
12 Ditto at 5 s. 6d. which is L. 3:
52 dit. is the Tot. Quant. and L. 12 :
with
and
6:
2
is the Tot. Val.
So that I Gallon of this Mixture is worth 4 s. 7d. 3 f.
Or if I choſe only to mix
6 Gallons at 4 s.
which is
L. I: 4:
with
4 Ditto at 5 s.
3 Ditto at 5 s.
which is
which is
L.
L. I : •
and
L.- : 16 : 6
13 ditt. is the Tot. Quant. and L. 3: 6 the Tot. Val.
And I Gallon of this Mixture will be worth 4 s. 7 d. 3f.
the fame as above; and fo of all other Quantities or Species
that are in the fame Ratio or Proportion of Goodneſs with
the Example here given; and a like Procedure will anſwer
any other whatever.
Again,
Suppofe the fame proportional Numbers, 1, 4, and 4, as a-
bove, and I have 73 Gallons of that worth 5 s. viz. that
which must be of the whole Compofition, How much muſt I
mix of the other two to perform my Defire?
I
3
Anfwer, 109 Gallons at 4s. and 54 Gallons 2, at 5s. 6d.
per Gallon.
2
Here I confider 73 Gallons, the Quantity given, to be the
Numerator; and as it reprefents I multiply it by 3, and
the Product 219 is the common Denominator. Now of
this is 109 Gallons and, which is the Quantity of that at
4s. per Gallon, and the is 54 Gallons, which is the
Quantity at 5 s. 6d. per Gallon, as will appear by what fol-
lows,
For 73 Gall. at 5s. per Gallon,
109 ditto at 4 s. ditto -
544 ditto at 5 s. 6d. ditto
Tot. 2 is 237
whofe Value
is L. 18: 5
5:
is L. 21: 18
is L. 15 : 01 : 1½
I
is L. 55 4
1 ½
Σ
And a Gallon of this Mixture is worth 4s. 7d. 3f. &, as be-
fore,
So
318
Of ALLIGATION.
So if the Names of the Species and Quantities of the Goods
were changed, but ftill bore the fame Proportion to one
another, the Anfwer would be ftill the fame. As for In-
ftance,
Suppoſe that I mix'd 73 Oz. of Silver, worth 5s. an
Ounce, with 109 Oz. of a coarfer Sort, worth but 4 s. an
Ounce, and thoſe two with 54 Oz. 4 of a finer Sort, worth
5 s. 6d. an Ounce, this whole Mixture would be worth
4s. 7 d. 3f. an Ounce; and fo if they were Cloths of dif-
ferent Fineneffes, &c. they would be worth the fame per Yard,
Ell, &c. or the like in any other Commodity whatever.
I 3
Again,
Suppoſe I have 8lb's of Silver of 7 Ounces fine, 26 Pounds
of 80z. ½ fine, and 17 lb's of 11 Oz. A fine, What is the
Fineness of this whole Quantity melted together per lb?
Anf. 9 Oz. 3 Dwts. 15 Gr. 4.
8
26
7
8 1/1/0
56 Oz. of fine Sil. 208
in 8lb's at 7 13
Oz. fine.
per 17
II 4
187
4/4/
221 Oz. of fine Silver [ 191 Oz. of
in 26 lb's at 8
Oz. fine.
12/
Oz. of fine Silver.
8lb. yields 56
26 Ditto 221 Ditto
17 Ditto 191 Ditto
Total 51 lb's is 468 4
lb. Oz. Dwts. Gr.
fine Sil. in
17lb. at 11
fine.
As 51 lb's is to 468 4 fo is 1 to 9: 3: 15 of fine Silver ;
fo that, the remaining 2 Oz. 16 Dwts. 8 Gra. in every
Pound Weight of Work made of this Mixture, muſt be of
Brafs, Copper, Tin, Pewter, &c. as beft fuits with the
Work that is to be fo made. Now the Standard for the
English-Coin is 11 Oz. 2 Dwts. fine; and of this Fineness
generally is made the large Silver Works, fuch as Tankards,
Salvers, Porringers, &c. but the Toys, fuch as fmall Tea-
Spoons, Buckles, Thimbles, &c. are generally made of a bafer
Sort of Metal, or that which has more Alloy in it, ſuch
as fome of the above, &c. but if to the above, either a Quan-
tity
Of ALLIGATION.
319
tity of pure Silver, or a Quantity of abfolute Alloy were to
be added, then the Total would be of a different Fineness
or Goodneſs to that above, according as the Proportion of
the faid Mixtures fhould be more or lefs. As for Inftance,
Suppoſe to the above Quantities, I would add 10 Pounds of
Silver, that was 11 Oz. 16 Dwts. fine, then the whole Mafs
would be 9 Oz. 12 Dwts. 2 Gra. fine per lb. But if I
would add 10 Pounds of Alloy, or Copper, &c. more to it,
then it would be but 7 Oz. 13 Dwts. 12 Gra. §§ per lb. fine.
The fecond Part of this Rule, commonly called Alligati-
on Alternate, teaches you to mix feveral Sorts or Prices of
Commodities together, fo as to make the Quantity or Price
what you pleaſe; and this is indeed properly Alligation.
6
62
ALIGATION ALTERNATE.
EXAMPLES.
Suppoſe a Wine-Merchant would make a Mixture of feveral
Sorts of white Wines, viz. fome of 7 s. 6 d. fome of 6s. 9d. fome
of 5 s. and fome of 4 s. 3 d. per Gallon, and would take fuch
Quantities of each, that the Mixture may be worth 5 s. 10d.
per Gallon, How much of each Sort will do the Matter?
In working all fuch like Questions as thefe, 'twill be
proper to reduce the given Rates or Prices to the loweſt
Name mentioned; and then fet them down one under ano-
ther, with the mean Rate againſt them; this done, for
Convenience fake, 'twill be proper to tie or alligate one or
more together, and you muft mind always to put a bigger
than the Mean-rate, to a leffer than the Mean-rate, and make
Subftraction either with, or from the Mean-rate, as the Term
concerned is bigger or leffer than it; and fet down the Reſult or
Remainder, not against the Sum the Subftraction was made with,
or from, but againſt that to which it is tied or alligated.
Thus :
d.
90
d. 81
af
70) 60
Difference.
ΙΟ
19
20
I I
Or thus:
Difference.
19
10
SE
81-
70.9 60
I I
55
20
60
60
Then add up the feveral Differences found, as above, and
the Total will be a whole Quantity fo mix'd, that may be
fold
*
320
Of ALLIGATION.
fold at the Mean-rate; and the particular Sums will be the
Quantities of the feveral Rates. As in this Question before
us, in the firſt Method of alligating, the Answer is 10 Gal-
lons, at 7 s. 6d. 19 at 6s. 9 d. 20 at 5 s. and II at 4 s. 3 d.
per Gallon. By the fecond Way of alligating there muſt be
19 Gallons at 7 s. 6d. 10 at 6s. 9 d. 11 at 5 s. and 20 at
4s. 3 d. per Gallon, as may eafily be try'd.
Gallons
Thus :
10 at 7s. 6d.
is
L.
3: 15 :
19 at 6 9d.
is
L.
6 : 8: 3
20 at 5
is
L.
5: :
II at 4 3
is
L.
206: 9
60 at 5: 10 is
L. 17: 10:
So in the fecond Method.
Gallons.
19 at 7 s. 6d. per Gallon is L.
Gallon is L.
7:
2:6
10 at 6 :9 ditto
3: 7
7:6
II at 5: ditto
N
2: 15:
4
5
20 at 4: 3 ditto
60 at 5: 10 per Gallon is L. 17: 10:
N. B. You may tie or alligate any given Quantities or
Prices as you pleaſe, provided you join a lefs, and a bigger
than the Mean-rate together.
But if the Total produced by any of the foregoing Me-
thods, are either too much or too little, you may ſay by the
Rule of Three, As the Totals found as above, is to the Quantity
wanted; fo are the particular Quantities, above found, to
the particular Quantities wanted. As, fuppofe I would fill
a Veffel of 40 Gallons or 100 Gallons, or any other diffe-
rent Quantity whatever, with the above Mixture, I muft
fay,
Gall. Gall. Gall. Gall. S. d.
As 60 is to 40 fo is 10 to 6:4 at 7:6
40 19 to 12: 4 at 6:9
-
60
60
40
60
40
II to 7:2 at 4: 3
20 to 13:2 at 5
The Sum or Total of the whole is 40 Gallons.
Or
Of ALLIGATION.
321
Or if you took the fecond Method, you would have,
Gallons
S.
d.
12: at 7: 6 which is
6: 4 at 6: 9
L. 4:15:
2: 5:
1
រ
7: 2 at 5:
13:2 at 4:3
1:16:8
2:16:8
40 Gallons at 5 s. 1od. perGall. is L. II: 13:4
But if you are not abfolutely confin'd to a particular
Quantity to be made up, after you have done as above, you
may multiply to increafe, or divide to diminish the Total,
and the Particulars that make up that Total, by any Num-
bers whatever greater than Unity; and fo you will have as
many new Totals and Particulars as you pleafe, and all at
the fame Rates or Prices. As for Inftance,
19
ΙΟ
38
20
57
76
30
40
Multiply II by 2
20
40
22 by 3 33 by 4 44, &c.
60
80
And the Total 60 will be 120 or
180 or 240, &c.
Or thus of leffening.
9/12/1
6. //
4
10. by 2 5 by 3 3 by 4 &c. 2
19
Divide
II
5/2/2
20
IO
2
5
And the Total 60 will be 30 or 20 or
15
Or you may work the foregoing, or any other Question
of this Nature, by Decimals; but, I think, the Method here
us'd, will be full as good, and in moft Cafes better for
this particular Rule.
QUESTION II.
Suppoſe a Merchant ſhould write thus to his Factor: I would
have you buy me the following Sorts of Linen, viz. at 6 s. 8 d.
per Ell, at 6 s. 4 d. at 4 s. 4 d. at 2 s. 10 d. and at 2 s. 6 d.
per Ell, to the Amount of L. 500 in the whole; and the Quan-
Tt
tity
P
322
Of ALLIGATION.
tity of each Sort, to be fuch, as if it was all at 4 s. 2 d.
per Ell; How much of each Sort must the Factor buy, to com-
ply with his Orders?
d.
80-
76-
20
16
50 52
34
16
26 more, 2 or 28.
·30-
30
110 Total.
Here I find a Total that is to be the firſt Number in the
Rule of Three, and the feveral third Numbers will be the
Particulars that compoſe the ſaid Total. Now for the Se-
cond, you muſt find how much the whole Sum to be laid
out will buy at the Mean-rate, by ſaying,
If 4s. 2 d. or 50d. buy 1 Ell, What will L. 500 buy?
and the Anf. will be 2400 Ells; then,
Ells
S. d.
20 to 436 11 at 6: 8 L.145: 9: 1:4
I at 6: 4 L.110: 10: 10:10
Ás 110 is to
16 to 349
2400, fo is
16 to 349
I at 4: 4 L. 75: 12: 8: 8
28 to 610 10 at 2: 10L. 86: 10: 10:10
30 to 654
6 at 2: 6 L. 81: 16:
4: 4
2400 at 4s. od. is L.500:
And fo by working as many different Statings or Rules of
Three, as there are different Particulars to make up the Total,
you will have the Anſwers above; or if you do not like the
above Proportions or Quantities of the particular Sorts, 'tis
but alligating the Particulars differently, as under. Note,
The 11, in the above Example, is the common Denominator,
but is fet at Top like a Numerator, for the Convenience of
Room, and of being repeated but once.
Thus:
80-
76-
Or thus:
16
16
80
16
76
20
50 52
34
20
5052
20
30 +26=56
34-
30
·30-
2
·30-
110 Total
26+2=28
114 Total.
Or
Of ALLIGATION.
323
Or thus:
Or thus:
80
16,20
:36
80
16
76
16,20
36
76
16
5052
16,20
36 5052
20,16
34
30,26,2 58
34-
30,26,2
·30
30,26,2 258
30
2
Total 224
Total 128
Or any other Way as beft pleaſes you, remembering only
always to put a bigger and a lefs together, and then work as
before, by the Rule of Three: Or, after you have found one
Quantity by the Rule of Three, you may do the reſt by
Practice, thus:
20 is found above, Page 322,to give 436,4
16 is of 20, fo that is
4
87,this fubftracted
ΤΙ
leaves 349 for 16
78
then 28 is of 20 and is
174 17
which added
gives 6102 for 28
and 30 is ½ of 20, ſo that, which is 218, being added
I
gives 6541, for 30, as above.
From the above Totals, and Particulars being different,
'tis plain the Anfwers to the Quantity required, will be
alfo different: And theſe may be us'd, as you would have
a lefs or a bigger Quantity, of a finer or coarfer Commodi-
ty, &c.
The great Objection against this common Method of re-
folving theſe Sort of Queftions is, that the Anfwers are but
few, and thofe frequently in fracted Numbers, fuch as no
Weight or Meaſure can exactly comply with, as in fome of
the foregoing Queſtions; though, in the Nature of the
Thing, many Anfwers in whole Numbers might be found;
as in this that follows.
Tt 2
If
324 Of ALLIGATION.
If Wine worth 32 Pence a Quart is mix'd with other
Sorts, worth but 20 Pence, and 16 Pence per Quart, and I
would make up a Cask of 14 Gallons or 56 Quarts, that
Should be worth 22 d. per Quart; How many Quarts of each
Sort may I take?
By the common Way.
32:
6,2
20-
IO
216.
IO
28 Total.
Quarts d.
As 28 is to 56, fo 8 is to 16 at 32 per Quart.
10 is to 20 at 20 ditto.
10 is to 20 at 16 ditto.
Here the Anſwer happens to come out in whole Numbers;
but you may have eleven Anſwers in whole Numbers, as
is fet down below, any of which (at 32 per Quart) will
perfectly anſwer the Condition of the Question.
10 at 32d. and 44 at 20d. and 2 at 16d. per Qrt.
II
12
13
14
40
36
32
28
1
3500
II
15
16
17
18
19
20
f
1
1
24
20
16
12
∞ +
8
4
!
1
1
}
1
14
17
22
26
23
20
29
32
Thefe, and fuch like Queftions more compounded, are
eafily folved by an Algebraick Procefs, which teaches to find
the Limits above, or under which, the Quantity of each
Sort cannot go: As there you cannot take more than 20
Quarts, nor leſs than 10, of the higheft Price, to have the
exact Anfwer in whole Numbers, &c. but as you are fup-
pofed not to know any thing of that moſt excellent Art cal-
led Algebra, we will cloſe this Matter, and go on to what
is commonly called
PRO-
Of PROGRESSION.
325
PROGRESSION or PROPORTION,
Which ordinarily is divided into Arithmetical and Geometrical
Proportion or Progreffion.
Arithmetical Progreffion is, when there is a Series of Num-
bers continually increaſing or diminiſhing by an equal Dif-
ference, as 1, 2, 3, 4, 5, &c. or 5, 4, 3, 2, 1; or 2,
4, 6, 8, 10, 12; or 20, 18, 16, 14, 12, &c. or 7, 16, 25,
34, 43; or 98, 77, 56, 35, 14, &c. Here the firft Series
or Set of Numbers increaſes by I, and the fecond decreaſes
by I; the third increaſes by 2, and the fourth decreaſes by
2; the fifth increaſes by 9, and the fixth decreaſes by 21;
and fo of any other Increaſe or Decreaſe, bigger or leffer,
to any Number of Terms whatever. The Uſes of theſe
Numbers are very many, eſpecially in fuperior Calculations,
2 Touch of which you fhall fee hereafter; in the mean
Time you fhall be inftructed in the common Methods of
Computation in this Rule, by having three of the five fol-
lowing Particulars given, viz.
1. The firft-Term, which may be either the greateſt or
the leaft Term, according as the Series increafes or decreafes.
2. The laft Term, which may be either the greateſt or the
leaft Term, according as the Series or Set of given Numbers
increaſes or decreaſes; as appears from the Examples above.
3. The Number of Terms, or how many there is in the
given Set.
4. The common Excefs, or by how much the following
Numbers in the Series, exceed one another, as by Unity, or
one in the two firſt Examples; by 2, in the 3d and 4th Exam-
ples; and 9 and 21, in the 5th and 6th Examples.
5. The Total or Sum of all the Series or Numbers, when
added together.
Having the 1, 2, and 3 of the above five Particulars given,
to find the 4th and 5th, that is the common Difference or
Excefs, and the Amount or Total of the Whole.
1. From the fecond fubftract the firft (when the firft is
the leaft or contrarily) and divide the Remainder by the 3,
made lefs by Unity, and the Quotient will be the 4, or com-
mon Difference.
2. Multiply the Sum of the 1ft and 2d by the 3d, divide
this Product by two, and the Quotient will be the 5th, or the
Sum of all the Terms in the Series: Or you may multiply the
Sum
326
Of PROGRESSION.
Sum of the 1ft and 2d by half the 3d, and the Product will
be the Anfwer or 5th fought.
EXAMPLE I.
Suppoſe a Clock ftriking round the feveral Hours upon the
Dial-Plate, What is the Number of Strokes in all, and what
is the common Difference between each Stroke?
Anſwer, 78 the Sum of the Strokes, and the common Diffe-
rence is I, found thus:
I the Ift of the 5 Things or firft Term.
12 the 2d, of the 5 Things, or laſt Term.
13 the Total of the 1ft and 2d ditto.
12 the Number of Terms or the 3d Thing,
2] 156 Product.
78 the 5th Thing or Total Number of Strokes, which is
the Sum of the Series.
To find the 4th or common Difference.
12 the laſt Term, which is the 2d of the 5 Things.
I the first Term, which is the 1ft ditto.
II the Remainder to be divided.
12 the Number of Terms or 3d of the 5 Things.
I fubftracted, as per Rule.
II is the Divifor to the Remainder above.
I is the Quotient or 4th of the 5 Things, which is
the common Excefs or Difference fought.
EXAMPLE II.
A certain Perfon had 10 Children, the Youngest of which
was a Daughter, aged 18 Years, the Eldeft a Son aged 45,
the intermediate Children, fome Sons fome Daughters; but they all
differed from one another, by an equal Age: A young ·Man
makes his Addreffes to the Youngest, and after fome Time all
Things are agreed upon, and he is to have for her Portion as many
1.3 12 Pieces of Gold as was the Sum of all their Ages:
Query,
Of PROGRESSION.
327
Query, How much that is, and what is the common Difference
of their Ages?
Anſwer, L. 1134 the Portion, and 3 Years between each
. Child.
First Term
18
Laft Term 45
·
Sum 63
Number of Terms 10
2 | 630 Product.
315 is the Sum of their Ages, which is
the Number of Pieces of Gold,
each L. 3: 12, which is L. 1134, that he is to have for
her Portion.
45 the laſt Term
10 the Number of Terms.
18 the firft Term. to be fub. I to be fubftracted.
927 the Dividend.
9 the Divifor.
Quot. 3 the common Difference between each Childs Age.
3. The 1, 2 and 4, being given, to find the 3d. Subftract
the first from the fecond, or contrarily, when the firft Term
is the biggeft, and divide the Remainder by the 4th; then
add 1 to this Quotient, and that Sum will be the 3d, or
Number of Terms fought.
E XA M P L E.
Suppoſe I buy a Bale of Cloth, for which I pay 5 s. for
the firft Piece, and L. 5:3:- for the laft Piece, How many
Pieces were there in all, if the common Excefs or Increaſe was
2 s. per Piece? Anf. 50 Pieces.
Shillings
103 in L. 5: 3 the Price of the laft Piece.
5
the Price of the firft Piece to be fubftracted.
2 | 98 the Rem. to be divided by 2, the common Excefs.
To 49 the Quotient.
adď 1
Total 50 is the Number of Pieces in the Bale.
4. The
328
Of PROGRESSION.
4. The 1, 2 and 5, being given, to find the 3. By
the Sum of the first and fecond, divide the 5th doubled or
multiply'd by 2, the Quotient will be the 3d fought.
EXAMPLE.
Suppoſe a Merchant fells a Bale of Cloth upon these Terms,
viz. to have 5 s. for the firft Piece and 103s. for the last, and
in the Whole he receives L. 135:00 for the Bale, I would
know how many Pieces the Bale contained?
Anfwer, 50 Pieces,
5 s. the firft Piece.
103 the laft Piece.
Divifor 108 5400 | 50 Quotient.
L. 135
20
2700
540
or No of Pieces in the
whole Bale.
2
0000
5400 Divid.
5. The Ift, 3d. and 4th being given, to find the 2d.
From the Product of the 3d, and 4th, fubftract the 4th,
and to the Remainder add the 1ft, the Sum will be the 2d.
EXAMPLE.
Suppoſe I ſell a ſmall Bale of Silk that weighs 190 Pounds,
thus, for the first Pound I am to have a Farthing, for the
Second Pound a Penny, for the third Pound 7 Farthings, &c.
What must I have for the laft Pound? Anf. 11s. Iod.
190 the Number of Terms or Pounds.
3 the common Increaſe.
570 Product.
567
3 Subftract.
I add the firft.
Anf. 568 Farthings, or 11 s. 10d. for the laft Pound.
6. The 1, 3 and 4 being given, to find the 5. From the
Product of the 3d and 4th, fubftract the 4th, and to the
Remainder add the Double of the Firft: Multiply this Sum by
the 3d, and divide the Product by two, the Quotient will be
the 5th, or Sum of the Series fought.
2
EX
Of PROGRESSION.
329
EXAMPLE.
Suppoſe I buy a Bale of Silk, that weighs 190 lb. and
am to give a Farthing for the first Pound, and to increafe
3 Farthings per Pound, How much must I give for the whole
Bale? Anf. L. 56 6 11.
: I
190 the Number of Terms.
3 the common Difference.
570 Product.
3 Subftract.
567 Remainder.
2 add the Double of the firft.
569 Total.
190 the Number of Terms.
51210
569..
2 108110 the Product to be divided by 2.
I
54055 the Quotient or L. 56:6: 1 for the whole
Bale.
13513
1126 i
L. 56: 6:14.
7. The 1, 3 and 5 being given, to find the 2d. Subſtract
the Product of the I and 3, from the Double of the 5, and
divide the Remainder by the 3, and the Quotient will be
the 2d.
8. The 2d, 3d, and 4th being given, to find the Ift.
Subſtract the Product of the 3 and 4 from the Sum of the
2d and 4th, and the Remainder will be the firſt.
9. The 2, 3, and 4 being given, to find the 5. Subftract
the Product of the 3 and 4 from the Sum of the 4, added
to the Double of the 2d, and multiply the Remainder by the
3d; then divide this Product by two, and the Quotient will
be the fifth, or Sum of all the Terms.
U u
10. The
330 Of PROGRESSION.
10. The 2d, 3d, and 5th being given, to find the firſt. Di-
vide the fifth doubled, by the 3d, and from the Quotient
fubftract the 2d, and the Remainder will be the firſt.
II. The 2d, 3d, and 5th being given, to find the fourth.
Subſtract the Double of the Fifth from the double Product of
the 2d and 3d, and divide the Remainder by the Product
of the 3 multiply'd by it ſelf, after I is fubftracted from it,
and the Quotient will be the fourth or common Difference
fought.
12. The 1ft, 2d and 4th being given, to find the fifth. Sub-
ftract the Product of the first multiply'd by it felf (which
is called its Square,) from the Square of the 2d divide the
Remainder by the 4th, and to the Quotient add the Sum of
the 1 and 2, and divide this Total by two, and this laſt
Quotient will be the fifth.
13. The 1ft, 2d, and 5th being given, to find the fourth.
Subſtract the Square of the first from the Square of the
2d, and divide the Remainder by the Double of the Fifth,
made lefs by the Sum of the 1ft and 2d, and the Quotient
will be the fourth fought.
14. The 1ſt, 3d, and 5th being given, to find the fourth.
Double the Product of the 1ft and 3d, fubftract this Double
from the Double of the 5th, divide the Remainder by the
Square of the 3d, made lefs by the 3d, and the Quotient
will be the fourth fought.
15. The 3d, 4th, and 5th being given to find the firſt.
Divide the Double of the fifth by the Double of the 3d,
to the Quotient add half of the 4th, from that Sum fub-
ftract half the Product of the 3, multiply'd by the 4th,
and the Remainder is the 1ft fought.
16. The 3d, 4th and 5th being given, to find the 2d. Di-
vide the 5th doubled by the 3d doubled, and to the Quotient
add half the Product of the 3d and 4th, from this Sum
fubftract the Half of the Fourth, and the Remainder is the
2d fought.
There are four other Propofitions belonging to this Rule,
but as the Extraction of the Square-Root is wanted to per-
form them, and that has not yet been taught; and perhaps
too much already has been faid upon this Head, it being
a Rule more of Curiofity than us'd in Buſineſs, I fhall not
purſue this Subject any further; but go on to
Geometrical
Of PROGRESSION.
331
Geometrical PROGRESSION,
Which is when a Series of Numbers are fo circumftan-
ced, that when the preceding Number is the Divifor, and the
fucceeding Number the Dividend, the Quotients are all alike;
as in thefe that follow, 2, 4, 8, 16, and 3, 9, 27, 81, &c.
here the common Ratio is 2 and 3, that is, the following
Numbers are 2 Times, or 3 Times as much as the preceding
Numbers; and thefe Numbers are generated by multiplying
the firſt by the common Ratio or Quotient, and that Product
again by the fame, and that Product again by the fame, &c.
till you have gone on as far as you want; but if you begin
with the large Number, and fo continually decreaſe in the
fame Ratio, as 16, 8, 4, 2; or 81, 27, 9, 3, then the fol-
lowing Terms, are produced by dividing by the common
Ratio, as appears very plainly by what is before you; for if you
divide 16 by 2, the Quotient is 8, and that Quotient again by
2, the Quotient is 4, and that again by 2, you will have the
4th Term 2 ; and fo of any other Numbers increaſing or de-
creafing, by any other Ratio or given Proportion. The firſt and
laft Terms of any fuch Series of Geometrical Proportionals, are
called the Extreams, and the intermediate Terms are called
Means, which Means, if required to be known, are found by
that Part of Divifion commonly called the Extraction of Roots,
not yet taught; therefore we will only here touch upon fuch
Parts of Geometrical Progreffion, as may be done without ſuch
laborious Proceffes; for which Purpoſes you muſt note, That
the Product of any two Extreams, is equal to the Product of
any two Means, equally distant from them, in any continued
Series, whether increafing or decreafing; as appears from the
following Series.
2, 4, 8, 16; here 16 and 2, when multiplied together,
produce 32, and fo do 4 and 8; or if the Number of Terms
be odd, as 243: 81: 27: 9: 3, the Mean or·
93, middle
Term, multiply'd into itſelf or Squared, will be equal to the
Product of any two other Terms, equally diftant on each
Side from it. As here the 3d or middle Term 27 by 27
is 729, and 243, the first by 3; the fifth Term is alfo 729,
as is 81, the 2d, and 9 the 4th Term; and this will always
be the Cafe, let the Number of Terms be what they will:
From whence it follows, that the four Numbers in every
Rule of Three Operation, viz. the 3 given, and the 4th found,
are Numbers in continual Geometrical Proportion. You are
likewiſe to obſerve, that every Term that ftands first in a
Uu 2·
Series,
332 Of PROGRESSION.
Series, is called the Antecedent, and that which follows is cal-
led the Confequent; and that every Number in a Series, ex-
cept the first and laft, may be both an Antecedent and a Con-
fequent: As above, in 2, 4, 8, 16; here 2 bears the fame
Ratio or Proportion to 4, as 8 does to 16, that is, each of the
Antecedents of its Confequent: So likewife the fame is to be
obſerved, if you fay, as 2 is to 4, is fo 4 to 8; and as 4 is to
8, fo is 8 to 16, &c. From this laft Obfervation may be
drawn this General Rule: As 1 Antecedent is to its Confequent,
So is the Sum of all the Antecedents to the Sum of all the
Confequents in the given Series. As, Suppofe the 2d Series,
243, 81, 27, 9, 3. Here the Sum of all the Antecedents
243, 81, 27, 9, which is 360, is to 120, the Sum of all the
Confequents, 81, 27, 9, 3; as 243 is to 81, or as 81 is to
27, or as 27 is to 9, or as 9 is to 3: And to fave the
Trouble of continual Addition, in finding the Sum of any
fuch Series, you muſt have ſomething given whereby to find
the reft, as the first Term, the last Term, and common Ratio.
Multiply the laft Term and the Ratio together, and fubftra&t
the first Term from this Product, and divide the Remainder
by the Ratio, made lefs by Unity, and the Quotient will be the
Sum of the whole Series. Or you may
Subſtract the Square of the first Term from the Product of
the fecond and laft Term, and divide the Remainder by the fe-
cond Term, made less by the first, and the Quotient will be the
Sum of all the Series.
But as the laft Term is fometimes difficult to come at, 'twill
be neceffary to obferve, that to any Series of Geometrick Pro-
portionals, that do not begin with Unity or 1, you may ſuper-
fcribe a Series of Arithmetick Proportionals, beginning with
Unity, whofe common Difference is alfo Unity; but if your
Geometrick Proportionals begin with Unity, then your Arith-
metick Proportionals muft begin with o, and the fecond Term
must be 1, &c. And thefe Numbers, either really thus fet
down, or imagined to be fo fet down, are called the Expo-
nents or Indices of the Numbers to which they correfpond
below, thus:
Arithmetical Propor. 1, 2, 3, 4, 5, 6, 7, called Indices
To 2, 4, 8, 16, 32, 64, 128, Geo. Propor.
Or,
0, I, 2, 3, 4, 5, 6 Indices.
1, 3, 9, 27, 81, 243, 728, &c. The Addition or Subftrac-
tion of any two of theſe Indices will agree to the Product
or Quotient of their correfponding Geometrical Proportionals;
that
Of PROGRESSION.
333
that is to fay, if you add any two Indices together, as fup-
poſes the 6th and 7th in the firſt Series above, the Sum will be
13; by which you are to underſtand, that if you multiply the
64 and 128, the correfponding Numbers in the Geometrical
Series together, the Product will be 8192; which fhews, that if
you continued that Series on to 13 Terms, the laſt Term would
be 8192; and in the fecond Series, the like Multiplication pro-
duces a Term one Place more than is the Sum of the Indices,
becauſe the firſt Term is o. As, fuppofe I add the Indices
5 and 6 together, the Sum is II, and the Product of 243 by
729, their correfponding Numbers, is 177147; and this
will be the 12th Place in fuch a Series, were the Terms
continued, as may be eafily try'd. Now by having the firſt
Term, and the common Ratio given, I can with little
trouble get the laft Term or Number, without being obliged
to get the whole Number of Terms.
EXAMPLE.
Suppoſe a fine Horfe, dreffed in curious Trappings, and laden
with a Crown richly bedecked with Diamonds, &c. was pur-
chafed as a Prefent for fome great Prince, upon the following
Conditions, viz. to pay a Farthing for the firft Nail of his
Shoe, two for the fecond, four for the third, 8 Farthings for
the fourth; and fo on for the Whole, which was 7 in each
Shoe, or 28 in all, What must be paid for the laſt Nail?
and what for the Whole, viz. the Crown, Horfe and Trappings?
Anfwer, L. 139810 2 8 for the laft Nail, and
L. 279620: 5: 3 for the Whole.
Here I fet down a few of the firft Terms, then I add 3 and
3 together, and multiply 8 and 8 toge-
ther, the Sum of the Addition is 6, and
the Product of the Multiplication is 64
0, I, 2, 3
I, 2, 4, 8
for the 7th Term.
Thus 336, and 8 multip. 8≈ 64, and 6+6≈ 12
and 64 by 64 4096 the 13 Term; then 12+12 24,
and 4096 by 4096 16777216 the 25th Term; and
24+3= 27, and 16777216 by 8134217728, the
28th Term, or Value of the laft Nail in Farthings. Then,
134217728
334
Of COMBINATION.
134217728 the laſt Term,
´multiply'd by 2 the common Ratio, and from
268435456 Product,
fubftract the firft Term, and
268435455 remains for a Dividend.
Then from 2 the common Ratio
fubftract I
Remains 1 for the Divifor.
Now as Unity or I neither multiplies nor divides, confe-
quently the Dividend above, must be alfo the Quotient, which
is the Sum of all the Series or Value of the Horſe and Grown
in Farthings, which is L. 279620 5 3. But as all
Queſtions, anſwerable by this and the laſt Rule, are much
better underſtood, by having a little Skill in Algebra, I
ſhall omit faying any more of it here, and juſt give you a
Sketch of what is called
COMBINATION, and PERMUTATION,
That is, to find how many Times the Order or Place
of any propoſed Number of Things may be changed or
varied, or how often a lefs Number of Quantities may be
taken out of, or are contained in a great Number, without
having any Conſideration of their Place.
As, how many Variations may be made with the whole
Alphabet of Letters; or Changes rung upon any Number of
Bells, &c. or to know how many Combinations of 2, 3 or
more Letters may be made in the whole Alphabet, or Part
of it, &c.
The Method of performing this, is as follows: Set down
the whole Number of Terms given, and they will form a
Series of Numbers in Arithmetical Progreffion; then multiply
the firſt by the fecond, and that Product by the third, &c.
till you have gone through all the Terms, and the laft Product
is the Anſwer fought.
EX
Of COMBINATION. 335
EXAMPLE I.
How many fingle Changes may be rung upon 8 Bells?
Anfwer, 40320.
Bells. Changes.
I
23+5O DOO
-
I
2
6
24
12,0
4
6
720
7
5040
8 · 40320
After the like Manner may
the Changes of any other Num-
ber of Things be found; as of
12 Bells, 24 Letters, &c. and
this is called Permutation.
EXAMPLE II.
Six Gentlemen meeting at an Inn, upon the Road, were fo
pleas'd with their Landlord's Entertainment, and one another's
Company, that they agreed to stay till they had drank a Glass
of Wine, for every different Pofition that they and their Land-
lord could be put in. "Now, upon Suppofition, that 16 Glaſſes
make a Quart, and that each Glafs was worth 1 d., How
many Bottles of Wine did they drink, and how much must each
of the fix pay for his Share of the Reckoning?
Anfwer, 5040 Glaffes, or 315 Bottles, worth L. 31: 10,
which is five Guineas per Man.
Combination is thus performed: Count the whole Number
of Things to be combin'd, as whether they are, I, 2, 3,
4, 5, 6, 7, 8, &c. then beginning with the Unit, multiply
continually as many of the leaft of thofe Numbes as your
Combination is to confift of, and the laft Product is to be
your Divifor; then multiply continually, as many of the laſt
or greateſt of thofe Numbers, and the laft Product refult-
ing therefrom, fhall be your Dividend; with theſe two,
Divifion being made, the Quotient will be the Number of
Combinations, confifting of 2, 3, 4, &c. Members, that can
be had in the whole Number propos'd.
EXAMPLE.
How many Combinations of 3 are there in 6, viz. Sup-
poſe the 6 Letters, a, b, c, d, e, f, How many different
Words or Syllables, of 3 Letters each, are contained in them?
Anfwer, 20, found thus:
I mul-
336
of COMBINATION.
C
I multiply'd
6 multiplyd
by 2
by 5
and 30 the Product
by 3
by 4
and 2 the Produc
6 the Pro. will be Divif. 6| 120 Prod. will be a Divid
}
and 20 the Quot. or Anfwer.
}
Proved thus:
abc
ade
bed
abd
adf
cde
abe
aef
cdf
abf
bcd
cef
acd
bce
def
ace
bcf
dbf
acf
bef
Again,
Upon the Cards, at the Game called Cribbidge, How
many different Fifteens can there be made out of the 4 Fives?
Anſwer, 4.
I
2
| a | w ~ [
4
3
12
2
6 Diviſor.
24 Dividend.
4 Quotient.
Again,
How many Cribbidges, or different Combinations of five
Cards are to be made out of a Pack confifting of 52 Cards ?
Anſwer, 2598960.
1
of COMBINATION.
337
52
2
ཧ །
2
3
52
260.
6
2652
4
50
24
132600
5
49
6497400
048
519792000
25989600.
Divid. 3118752010
Quot. 2598960
120 Divif. 1193400
530400
6497400
Again, How many different Tricks of Cards are to be made
out of a whole Pack, or how many Combinations of 4 arë
there in 52 ?
Anſwer, 270725 different Tricks.
For by the Work above, it's evident that 24 will be
the Divifor, and 6497400 the Dividend; from whence, by
common Divifion, the Quotient or Number of Tricks, will
be found to be 270725.
On the Dice,
The Number of Chances on one, two, three, four, &c.
Dice, are in a Geometrical Progreffion, whereof the Number
of Dice is the Number of Terms, and the Chances on one Die,
viz. 6. is the Ratio and firſt Term:
So that,
I, 2, 3, 4 , 5, 6 Dice,
On 1, 2
There is 6,
•
36, 216, 1296, 7776, 46656 Chances;
and ſo you may go on to any other Number, and by the
above fmall Table, you may know what Odds any particu-
lar Chance upon 1, 2, 3, 4, 5, or 6 Dice, hath againſt
it: As for inſtance, 'tis 5 to 1, that with 1 Die a Perfon
does not throw any particular Face or Number of Spots
upon that Die: As, fuppofe 4, then there are 1, 2, 3, 5, 6,
x x
againſt
338
Of COMBINATION.
againft it; fo if 'twas 3, then 1, 2, 4, 5, 6 are againſt it:
And fo of any other
Two Dice have 36 Chances, fo that it is 35 to one that
you don't throw any particular Chance; only you muſt
take Notice, that if you don't pitch upon fome one of the
Doublets, viz. the two Aces, two Duces, two Treys, &c.
that the Dice muſt be mark'd with Ink, &c. to determine
which Die fhall caft 1, 2, 3, &c. and which the 4, 5, 6,
becauſe all the other Chances, befides the Doublets, have two
Ways of expreffing the fame Thing: As, Suppoſe A lays
B, that with two Dice, he don't throw a Trey and a Duce,
if it is not determin'd which Die fhall throw the Trey, and
which the Duce, the Thrower will have two Chances, and
confequently then the Odds againſt him is but 17 to 1,
becauſe there are two Chances to win, and 34 to loſe; but
if the Wager was ftill more indefinite, and the Thrower was
to throw Cinque or 5, without Reftraint, then the Odds
would be but 8 to 1, becauſe there are 4 Ways of throwing
5; and confequently, 32 to miss it; ſo that the Odds muſt
be theſe two Numbers fet Fraction-wife, and reduced to
its loweſt Terms, thus:, which is, or 8 to 1.
From hence may eaſily be known the Odds at the Games,
called Back-Gammon, Hazard, &c. for the feveral Chances
on two Dice being 36, all the particular Chances that can
be thrown on them, are expreffed in the following Table.
II
41 | 51 | 61.
32 42 52 62
33|43| 53 63
21
3.1
12
22
13 23
14 24
15 25
16 26
36 46 56
34 44 | 54
64
35 45 55 65
| |
465666
From whence you may obferve, there is II that has an
Ace, and 25 that has not an Ace; II that has a Duce,
and 25 that has not, &c. So that it is 25 to II that you
don't throw any particular Number of Spots on either Die,
at one Throw; 'tis alfo farther remarkable. That there are
7
6 Chances that amount to 7 each, and 15 above
and as
many under it; ſo that 'tis 5 to 1 that a Perfon does not
throw the Number 7 at one Throw, with two Dice. You
may alſo obſerve, that there are 6 Doublets, or two of one
Number, on each Die, viz. I and 1, 2 and 2, 3 and 3,
4 and 4, 5 and 5, 6 and 6; ſo that 'tis alfo 5 to 1 that
the
Of COMBINATION.
339
-
the Thrower don't throw Doublets of any Sort at all; but
'tis 35 to I that he don't throw a particular Doublet, as
5 and 5, I and I, &c. at one Throw. 'Tis obfervable alfo,
that the Chances in the above Table, go in an Arithmetical
Progreffion, both above and below 7; for there is but one
Chance, to throw either I and 1, or 6 and 6; but two to
throw 3 or 11, three for 4 and 10, four for 5 and 9,
and five for 6 and 8; fo that any two Chances being
thrown, the Odds between them are immediately known, by
knowing their Diſtance from 7. As, fuppofe 4 and 9, 'tis
4 to 3 that 9 comes before 4, and fo of any other Chance
you ſhall pitch upon; but if the two Chances are equally a-
bove and below 7, 'tis an even Wager that one comes as foon
as the other, provided no unfair Practices are made uſe of.
Upon three Dice there are 216 Chances, &c. as was noted
before; from whence Perfons may eaſily know how much
they are impos'd on by thofe Vermin that go about with.
Boards, that have Money placed upon particular Chances of
4, 5, or 6 Dice: As, fuppofe upon 36, with 6 Dice; here there
is but I Chance to win, and 46655 to lofe; fo that fuppofe
a Perfon gives 6 d. for fuch a Throw, the Stake against it
ought to be L. 1166: 7: 6, which, perhaps, is commonly
but about L. 3 or 50 Shillings, &c.
As we have already gone thus far, we will add a ſmall
Table, with its Explanation. relating to this Subject, that for
its Curiofity may perhaps be an agreeable Amuſement.
The Number of Dice.
I 12 13 1 4 I 5 I
The Number of Chances.
61
I
I
I
I
I
I
2
4
8
16
32
64
39
27
81 243 729
416 64 256 1024 4096
TUTAT
5 25 125
625 3125 15625
636 2161296|7776|46656|
X x 2
EX
340
of COMBINATION.
EXPLANATION.
The upper
Line 1, 2, 3, &c. fhews how many Dice are
fuppos'd to be thrown at once, where the greateſt Number
is 6; and after the fame Manner you may go on to fix Thou-
fand. The particular Lines of the firft Row, on the Left-
hand, fhews the Number of Faces and Spots upon each Face,
that each Die has. The Second, Third, Fourth, Fifth and
Sixth Lines fhew the Number of Chances that may be
thrown by the refpective Number of Dice, whoſe higheſt
Face or Number of Spots is that in the firft Column. The laſt
Line is the Total of all the Chances that the whole Number
of Dice can throw. As for Example, with I Die, there
may be thrown 6 Chances in all, that is one of I two, when
you don't exceed 2, &c. So with 2 Dice you may throw 4
Chances, and have neither a 3, 4, 5, nor 6, viz. I, I, and
1, 2, and 2, 1, and 2, 2; with 3 Dice, you may throw
27 Chances, in which there fhall be neither a 4, 5 or 6;
fo with 4 Dice you may throw 256 Chances, which fhall
have neither a 5 nor 6 among them; with 5 Dice you may
throw 243 Chances that ſhall have neither 4, 5, nor 6 among
them; and with 6 Dice you may throw 729 Chances that
has neither 4, 5 nor 6, &c. You are alfo to Note, that
each Line of the 2, 3, 4, 5 and 6 Rows fhew the Number
of Chances that are to be thrown, when you leave out fo
many Faces as are the Number of Lines under that you
choofe.
EXAMPLE.
In the 5th Line ftands the following Figures, 5, 25,
125, 625, 3125 and 15625; and this fhews you, that you
may, upon two Dice, throw 25 Chances, and leave out any
one particular Face that you ſhall pitch on: As, fuppofe you
would have never a 4, then your two Dice may throw 25
Chances without 4, and II with it, in all 36 Chances; the
fame may be done by 1, 2, 3, 5 or 6. Again, with three
Dice you may throw 125 Chances, and leave out any one
particular Face upon all the three Dice that you may object
againft; with four Dice there are 625 Chances that may be
thrown, and leave out any one particular Face upon all the
four Dice, &c.
In the fourth Line ftands 4, 16, 64, 256, 1024, 4096,
which fhews, that there are fo many Chances to be thrown,
3
when
Of Raifing POWERS.
341
when you leave out any two Faces, as is the Number of
Dice under which thoſe particular Numbers ſtand; ſo
In the third Line the Numbers 3, 9, 27, 81, 243,
729, fhew the Number of Chances that are to be thrown,
when any particular three Faces are to be left out. Note, By
the Number of Faces, is meant the fame Number of Spots
upon each Face in each Die. As, fuppofe upon three Dice,
I would leave out the 1, 2, 3 Faces, I mean, there must be
never a Face of 1, 2 or 3 Spots in the Caft. This Subject
would take up a very large Volume to handle it fully; fo
that thoſe who would have it compleat, may confult Mr.
De Moivres's Doctrine of Chances, where they will find fuf-
ficient to fatisfy their utmoſt Curiofity.
CHA P. X.
Of raising Pow E R S and extracting Ro o т s,
S there is frequent Ufe of what is called raifing
A Powers, and extracting Roots from given Powers al-
ready rais'd, 'twill be proper here to explain what theſe.
Terms mean, and the Methods of performing the ſeveral O-
perations neceflary to effect the fame: And firſt, To raife
a Power, is to multiply any given Number into itſelf, 1, 2,
3, 4, 5, 6 or more Times, as Occafion may require;
and indeed the Theory of this Part of Arithmetick is by
much the beft explain'd by an Algebraick Procefs, from
whence the Reaſon of the Work appears felf evidently;
but as we have not yet meddled with that Part of Ab-
ftract-Arithmetick, we will do what we can to explain it
Numerically, as plainly as poffible: Wherefore you are to
obferve,
1. That the Number given to be rais'd to a Power, be
it ſmall or large, a Fraction or whole Number, is called the
Root or firft Power, that is, the Source or Spring from whence
arifes the feveral Powers afterwards, and this is perform'd by
one or more Multiplications of it ſelf, into it felf; that is,
Suppoſe 6 was given to be rais'd to the 3d, 4th, or 5th
Power;
342
Of Raifing POWERS.
Power:
Firft, fet down the given Root or firft Power
Then multiply it by itſelf thus
And this Prod. is called the 2d Power or the Square
Then multiply this Product again by
And this new Prod. is called the 3d Power or Cube
Then multiply this Product again by
6
36
6
216
6
And this new Prod. is called the 4th Pow. or Biquadrate 1296
Then multiply this Product again by
6
And the new Prod. is called the 5th Power or Sur-folid 7776
And fo you may go on as far as you pleaſe, only noting
to call the laft Product a Power one Degree higher than the
Number of Multiplications or Involutions made ufe of. As
you fee above, the Product of the firft Multiplication is called
the 2d Power, the Product of the fecond Multiplication is
called the 3d Power; of the 3d, the 4th Power, &c. Now
any Power of any Root, may be given to be rais'd to any o-
ther Power; and in that Cafe, though you know the given
Number or Root is the 3d, 4th, 5th, 6th, &c. Power of
fome other Number or Root, yet you take no Notice of that,
but work as above, in the fame Manner, as though it were
not fo: As, fuppofe the Number 7776 was given to be
rais'd to the 2, 3, 4, &c. Power, you muſt multiply it con-
tinually once, twice, thrice, &c. as before, although this
Number given, is the 5th Power of 6.
EXAMPLE.
What is the Cube or 3d Power of
7776?
Multiply by
7776
46656
54432
54432
54432
And the Product is the Square or 2d Power 60466176
And
Of Raifing POWERS.
343
And the Product is the Square or 2d Power
Multiply again by
60466176
7776
362797056
423263232
423263232
423263232
And the Product is the Cube or 3d Power 470184984576
And ſo of any other Number whatever, only you are to
Note, that the Scale of Powers are as the Indices in Arithme-
tick Progreffion, and the produced Powers are as the Terms
of a Series in Geometrick Progreffion, whofe common Ratio
is the Root or Number given, or pitch'd upon; as will more
evidently appear by the following Table of Roots and Powers.
Roots or firfi Power.
~ Squares or 2d Power.
w
Cubes or 3d Power.
← Biquadrats or 4th Power.
LA
Surfolids or 5th Power.
Power.
5
6
→ Power.
∞ Power.
Power.
I
1
1
1
I
24
8
16 32
64
128
256
512
31 9 27 81 243
729 2187
6561
19683
262144
8
9 &c. Indi.
416 64 256 1024 4096 16384 65536
525 125625 3125 15625 78125 390625 1953125
636 2161296 7776 46656 279936 1679616 10077696
749 343 2401 16807 117649 823543 5764801 48453209
864 5124096 32768 262144 2097152 16777216134217728
9|81| 729|6561| 590495314414782969 43046721|387420489
This
344
Of Extracting ROOTS.
This Table confifts of nine Lines, included within the
double Lines; the firft whereof is all 1's or Units, and all the
reft are ſo many Terms of a Geometrical Series, whofe 1ſt Terms
and common Ratio are both alike, and the Series are produced
by multiplying the firft Term by it ſelf, and that Product again
by the first Figure, &c. The Line immediately above the Table
contains the Indices of the feveral Powers, to which the Root
or firſt Power, in the firſt Row upon the Left-hand, is fup-
pos'd to be rais'd; and the ſeveral perpendicular Lines or
Rows that ftand under the faid Indices, are the Powers
themſelves of the nine Digits or ſingle Figures: Now here,
as before in Geometrical Progreſſion, Suppofe you had the 4th
and 5th Powers given, to find the Power of the fame Root,
that was equal to the Sum of theſe Indices, viz. the 9th, by
by multiplying 16, the 4th, and 32, the 5th Power of 2, to-
gether, you have 512 for the Product, which is the 9th
Power of 2, without a continual Involution or Multiplying by
2, the common Ratio; and fo of all others. And by involv
ing or multiplying 512 into itſelf, you will have 262144, the
18th Power of 2, becaufe 9, the Indice added to 9 is 18;
and this Trouble of Involution may be faved, if your first
Line were continued fo far, that you could find in it the
Square, Cube, &c. of the Power, to which you would raiſe
any Power: As in this Inftance above, the Indice 9 added
to it felf, produces the Square of it ſelf, or the 18th Power
of 2. Now the Square of 2 is 4, therefore the 9th Power
of 4 will be the fame Thing as the 18th Power of 2, viz.
262144, as appears by the Table: So if it was required to
produce the 16th Power of 3, you might either multiply
2187, the 7th Power, and 19683 the 9th Power of 3 toge-
ther, or you may multiply 6561, the 8th Power, by itſelf,
and either of them will produce 43046721, the 16th Power
of 3; becauſe the Sum of 7 and 9, or 8 and 8, the Indices
of the inferior Powers, produce 16 for their Total: Or if
you take the 8th Power of 9, becauſe 9 is the Square or 2d
Power of 3, 'twill be all one, as appears by the Table; and
fo of any other Power. Thus much may fuffice for raifing of
Powers, called Involution; but the chief Ufe of this Table is in
Evolution, or extracting of Powers,
Which is the Art of finding fuch a Root or Number, that
when Squared, Cubed, &c. will produce the Number or Power
given, to have fuch Root extracted out of it; and here,
very frequently, after the Proceſs is over with the given Figures,
there
Of extracting the SQUARE Root.
345
there happens to be a Remainder, the Value whereof cannot
be exactly found; but by adding Cyphers and continuing
the Work, you may approximate fufficiently near for any
Purpoſe of Buſineſs, though you cannot arrive at a Mathe-
matick or an Arithmetick Exactneſs.
When any whole Number is given to extract the Square,
Cube, &c. Root, or the Root of the 2d, 3d, 4th, &c. Power,
make a Dot or Mark over, or under the Units Place, and
then count fo many Figures towards the Left-hand, as is the
Index of the Power you are to extract, and there make ano-
ther Dot or Mark, and fo continue counting towards the
Left-hand, till you have gone through the whole Number
of Figures given; and fo many Dots or Marks as are thus
made, will be the Number of Figures the Root muſt con-
fift of.
As, Suppofe 'twas required to extract the Square, Cube,
&c. Root of 387420489.
For the Square Root it muſt be pointed at Unity, and at
every two Figures from Unity, thus
387420489
For the Cube Root, it muſt be pointed at Unity, and at
every three Figures from Unity, thus 387420489
Where you will obferve that the Square has 5 Points, which
fignifies that the Root will confift of 5 Figures, and the Cube
Root has but 3 Points or Figures.
But if the given Number be a mix'd Number, if the
Fractional Part is not already a Decimal, make it one by
the Rule given, Page 168. Point the whole Number Part,
as above; and then for the Decimal Part, count two Places
from the Units Place, for the firſt Point, and fo count 2
Places from thence for the other Dots, for the Square, three
Places for the Cube, &c.
To Extract the Square-Root, or Root of the fecond Power,
out of any given Number or Refolvend.
RULE.
The given Number being fet down and pointed, as above
directed; by the foregoing Table, ſee what is the nearest Square
Number to thofe Figures that belong to the first Dot or Point
towards the Left-hand, which does not exceed them; which being
found, fet down the faid Square under the Figures of that
firft Dot, and the Root by itſelf, as you do the Quotient in
Divifion; thus,
Y Y
What
346 Of extracting the SQUARE ROOT.
What is the Square Root of 43046721 the whole Re-
folvend?
36
(6
*Here the given Number being pointed, the Figures 43
belong to the firft Dot, towards the Left-hand, the neareſt.
Square Number to which that doth not exceed them, is 36, the
Root whereof is 6; which being fet down as above, then
fubftract the 36 from 43, and the Remainder is 7, to which
Ibring down the Figures of the next Dot, viz. 04, and that makes
704 for a new Refolvend, to find the next Figure, or ſecond-
Figure of the Root; to do which, I double the 6 or first Figure
of the Root already found, and it makes 12; with this 12 I
divide the two firft Figures of the 704, viz. 70, and the
Quotient is 5; this 5 I fet before the 6 and the 12, and then
it makes the Quotient or Root 65, and the 12, or Diviſor,
becomes 125, which 125 I multiply by the faid 5, and the Pro-
duct 625 I fet down under the 704, and ſubſtract, and to the
Remainder 79 I bring down 67, the Figures of the third Dot,
and 7967 becomes a new Refolvend; the 3 firft Figures of which
becomes a Dividend, and the Divifor is the 65 doubled, or
130; then I proceed in every refpect as with the Figure 5,
and fo from one Dot or Figure in the Root to another, till all
is done, as in the following Examples at large.
43046721 (6561 Root.
36
125704 Refolvend.
5625 Subtrahend. Anf. 6561 the Square Root.
6561
A
130617967 Refolv.
6/7836 Subtra.
6561
39366
1312113121 Refolv.
32805
1/13121 Subtra.
39366
Remains
43046721 Proof.
To prove this, and all other Operations of this Kind, you
muft multiply the Root found by itſelf, and that Product
will be the Square or Number given, if there was no Re-
mainder;
Of extracting the SQURAE ROOT.
347
mainder; but if there was a Remainder, then add that Re-
mainder to this Product, and that Sum, if the Work has
been truly perform'd, will give the original Number, ás
follows,
What's the Square Root of 8763827 ?
4
(2960 Root.
2960
49476
9/441
177600
26640
5920
586)
3538
6 3516
8761600 Product.
2227 Remaind.
592) 2227
8763827 Sum.
Here you fee, after the Work is done, and the other
Part of the Root, which is a whole Number, is found to
be 2960; the Remainder is alſo found to be 2227, which
is a Numerator to an indeterminate Denominator. For Proof
of the Work, the Root 2960 is fquared, and to that Pro-
duct 8761600, the Remainder 2227 is added, and the Sum
is 8763827, the given Number. Now to find the Root neareſt
to this Remainder, add Pairs of Cyphers, and continue the ſame
Work, as for the 9 and the 6, thus:
5920,3\2227,00,0,0,00,
31776 09
5920,67\450 9100
7/414 4469
7
5920,746\36 463100
6 35 524476
938624, &i.
(2960,376
And fo you may go on as far as you pleaſe, repeating
the fame Sort of Work continually, for every Figure you
put in the Quotient or Root. Note, Thofe Numbers that
leave a continual Remainder, as in the laft Example, are
called Surd Numbers.
Y y 2
If
348
Of extracting the SQUARE ROOT.
If your given Number be a Decimal only, you muſt point it
as before directed, and then work it in the fame Manner as if
'twas a whole Number; and the Quotient or Root fo found or
produc'd, will be a Decimal, whofe Value muſt be determin'd
by the Value of the Figures which compofe the Refolvend: As,
Suppoſe, ,06076225 were given, the Root would be ,2465.
Note, You must always count the Number of Places of your
given Decimal, and if they are even, Point it as is a-
bove directed; but if the Number of Places in your given
Decimal is odd, and is not a Repetend, you must add a Cy-.
pher on the Right-hand to make the Number even: The
Reafon whereof is very plain, for every Decimal being, in
reality, but the Numerator of a Vulgar Fraction, whoſe De-
nominator is Unity, with fo many Cyphers added, as are the
Number of Figures in the given Decimal or Numerator,
which Denominator, when confifting of an odd Number of
Figures, is always a Square Number, whofe Root is Unity;
and as many Cyphers as the faid Denominator confifted of
Pairs of Cyphers: As for Inftance, Suppofe I was to afk
what is the Square Root of ,9, of ,09, of ,009 and of,0009,
the Anſwers would be ,9487 &c. for the Root of ,9, and
,3 for the Root of ,09, and ,09487 &c. for the Root of
,009, and ,03 for the Root of ,0009, &c. So that the
Square-Root of a Decimal Fraction always confifts of half the
Number of Places that the given Decimal or Refolvend
confifts of, and the Root of a Decimal is always greater in
Value than the Refolvend or given Decimal; becauſe the Re-
folvend is to be confider'd as the Product, and the Root, as the
Multiplicand and Multiplier of two proper Fractions; and all
proper Fractions decreaſe by Multiplication, and increaſe by
Divifion. You are to Note, that the Cypher added to the given.
Decimal, when it confifts of an odd Number of Figures,
neither increaſes nor diminiſhes its Value, as has been for-
merly obferved, but only increaſes the Number of Places in
the Denominator, to make it a Square Number, viz. Unity,
and a certain Number of Pairs of Cyphers.
If a common Vulgar Fraction be given, to extract the Square Root
thereof.
1. If the given Terms, or the Numerator and Denominator
of the given Fraction, are Square Numbers, extract the Square
of each, and fet the Root of the Numerator over the Root of
the Denominator; and the new Fraction is the Root of the
given Fraction. As,
What
Of extracting the SQUARE ROOT.
349
What is the Square Root of 2? Anſwer, 2.
2. But if the given Fraction cannot exactly have the
Square Root taken out of the given Terms, try if it can be
reduced lower, and extract the Root out of that new reduced
Fraction, thus:
What is the Square Root of 1?
Here, neither the Numerator, nor Denominator, are Square
Numbers; therefore I reduce this Fraction to its lowest Terms
, and find that the Root may then be extracted, and is
as above.
3. But if the given Fraction can neither be reduced lower,
nor have the Square Root extracted out of both its Nume-
rator and Denominator (for one of them will not do) then
'tis what is called a Surd, and has commonly fuch a Mark
as this fet againſt it, viz.
3
√
As, What is the Square Root of ? Anfwer, 3.
But if you are willing to have an Approximate Square, re-
duce the given Fraction to a Decimal, and extract the Root
of this Decimal, ordered, as has been already directed,
which may be continued as far as you judge the Nature
of the Matter or Buſineſs, 'tis to be apply'd to, may require,
by adding Pairs of Cyphers, &c. as in this laft Example, tho'
4 the Denominator is a Square Number, whofe Root is 2, yet
3 the Numerator is not a Square Number; and therefore the
Fraction is fet down thus ; but when turn'd into a De-
cimal, this becomes,75, the Root, whereof, by the above
Directions, will be found to be ,866025, &c. But if the
Decimal given, or found, be a fingle Repetend, you muſt
make them confift of an even Number of Places, and take
down the faid Repetend by Pairs, and extract as before; and
if it don't repeat at the fecond Figure in the Root, 'tis uncertain
when it will repeat; and fo you may carry it on to what
Number of Places you pleafe, as Occafion may require.
4
Note, The 1 and 4 are the only fingle Repetends, that
have fingle Repetends in their Roots, &c. Compound Repe-
tends may be extracted to any Number of Places you pleaſe,
by only extracting half the Number defign'd, and then con-
tract your Divifor, as taught in Divifion of Decimals, by
which the other Half may be eafily obtain'd.
Mr. WARD's Method of extracting the Square Root, is as
follows.
First, Point the given Number as before, then take or
Jubftract the greateſt Square out of the first Point or Period,
35
350 Of
Of extracting the SQUARE ROOT.
as before, and then halve the whole Remainder, which point,
as at first, fo far as to make up the Figure of the Root
already gotten; the true Number at firft found by pointing,
then divide this new Refolvend by the Root found, and add
the Quotient or Figure fo found, to the other before found;
then multiply by the laft Figure thus found, and take half the
Product of the firft Figure, and fet down the first Figure of
this half, under the laft Figure of this Period, and carry the
Tens, if any in this half Product, to the Product of the next
Figure in the Divifor, and fo continue till all the Figures of
the Root are gone through. Only Note, that if the Pro-
duct or Square of the firft Figure is an odd Number, then you
muſt take down the firſt Figure of the next Period, to the
Period already taken down, and fet the ,5 which is the De-
cimal of the Half of the odd Number, under it, and ſub-
ſtract as in common, as you fee done in this Example follow-
ing.
What is the Square Root of 572199960721/7564
Anf. 756439
49
(756439
2 | 82199960721 whole Rem.
75 | 41099980360,5 new Refolv.
5 3625
756) 4849
451,8
7564\33198
430248
75643 12950036
3/2269245
75643919807910,5
9/6807910,5
Remains.
The fame Method may be ufed with Decimals only, or
whole Numbers and Decimals mix'd, &c. only in the Cafe
of Repetends or interminate Surds, if you fix your Number of
Figures,
*
Of extracting the SQUARE ROOT.
351
Figures, that you are minded to have in the Root, you
need only extract to 1 Figure beyond the half Number of.
Places affigned to the Root, and the reft may be obtained by
plain Divifion. As,
Suppoſe I would have the Square Root of 5, which is a Surd,
and would have II Places of Decimals in the Root, for Exact-
nefs. You may do thus:
4
(2,23606 firſt Part of the Root, found by
Extraction and which is the Divifor
1,00, &c. remains.
2,2 `\,50, &c. half the Remainder or new Refolvend.
+2/,42
2,23,800
286845
+ 3,6645
2,236 13550
+6/13398
2,23606 1520000
+6
1341618
2,23606\178382 the Remainder, divided by the above
Root gives 79775, the latter Part
of the Root.
21857
20125
1732
1565
167
156
|==
Here being but five Figures more wanted, I only add the
Reſult of the 6th Figure, viz. of the 6, to the Product of
I
the
352 Of extracting the CUBE ROOT.
the o, as practis'd in the contract Method of Multiplication
of Decimals, and fo proceed pricking off every time a Figure,
and at laft the Quotient is 79775, and the Remainder no-
thing; fo that the whole Root thus found is 2,23606797752.
which is indeed a fmall infignificant Trifle too much; for if
the Extraction was to be continued, the true Root would be
2,23606797749999, &c. with a Remainder.
Enough has been faid upon the extracting the Square
Root of a given Number, whether Pure or Surd. We fhall
now go on to the extracting of the
CUBE ROOT,
Which, by the foregoing Table, is the 3d Power or fome
Number twice involved; fo that if the Numbers 729, 512,
343, 216, 125, 64, 27, 8, or 1, were given, and it was
required to extract, or tell what was the Cube Root of each
of them, the faid Table will readily inform you, that 9, 8,
7, 6, 5, 4, 3, 2, 1, would be the refpective Cube Roots of
the aforefaid Numbers: But if any other Numbers greater or
leffer, were given to find the Cube Root, then you muſt fol-
low one of the two Methods hereafter fet down; and firſt
we will teach the common Way, which is thus :
Your Number being fet down, point the Units Place, and
count three Figures towards the Left-hand, if the given
Number is a whole Number, and make another Dot, and then
count three more, &c. till you have counted all; but if it is a
Decimal, count three Figures towards the Right-hand, and
make a Point, &c. as before, in the Square Root, you counted
two Figures.
What is the Cube Root of 134217728?
Having pointed this Number as above directed, it will
ftand thus, 134217728, by which it appears there will be
three Figures in the Root, becauſe there are three Points; and
to know what thofe Figures are, I look in the Table for the
nearest Cube, that is not more than the first Point or Period to-
wards the Left-hand, viz. 134, and find it to be 125, which I
fet down under the faid 134, and find the Root thereof to be
5, which I fet by itself like a Quotient or Divifor ; then fub-
ftracting, the Remainder is 9, to which I bring down the next
Period, and the whole Work will then fand thus:
134217728 (5
1
9217 new Refolvend.
Now
125
Of extracting the CUBE ROOT.
353
Now to find the two remaining Figures of the Root, you
muft do as follows, viz.
Square that Part of the Root already found, viz. the 5,
and multiply that Square by 3, and the Product 75 fet under
the new Refolvend, two Places from Unity; then triple the Root
or 5 fo found, and fet its Product under the Refolvend, one
Place from Unity; add thefe two Numbers together, and the
Total fhall be a Divifor to this whole Refolvend, with Unity
omitted, viz. to 921 instead of 9217; and here Allowance muſt
be made, that this Quotient or fecond Figure of the Root may have
or perform the following Operations, viz. to multiply the triple
Square of the Root, and that Product must be fet two Places from
the Units Place of this Refolvend; then the Product of the
Square of the faid laft Figure into the triple of the Root must be
Jet one Place from Unity of this Refolvend; and, lastly, the
Cube of the last found Figure of the Root, must be put under
the Units Place of this Refolvend; these three Products or Sums
added together into one Sum, must be fubftracted from the faid
Refolvend, and to the Remainder, you must bring down the
next Period or Point, confifting of three Figures, to form another
new Refolvend, and then the fame Work must be repeated over
again; and fo for every ſingle Figure of the Root, till the whole
is done, as in the Example at large here under.
134217728 (512 the whole Root.
125 the Cube of 5 to be ſubſtracted.
9217 new Refolvend.
75 the triple Square of 5.
15 the Triple of the Root 5.
765 the Diviſor (1 the ſecond Fig. of the Root to be added
to the 5, which makes it 51.
75 the Product of the triple Square of 5 by 1.
15 the Product of triple Root by the Square of 1.
I the Cube of 1, the 2d Figure of the Root.
7651 the Subtrahend or Total of the 3 Lines.
Z z
1566728
354
Of extracting the CUBE ROOT.
1566728 the new Refolvend.
7803 the triple Square of the whole Root 51.
153 the Triple of the Root 51.
78183 the Divifor (2 the Quotient or 3d Figure of the Root,
which makes it now 512.
15606 the Product of the triple Square by 2.
612 the Product of the triple Root by the Square of 2.
8 the Cube of 2, the 3d Figure of the Root.
1566728 the Subtrahend or Total of the 3 Lines.
the Remainder.
After the fame Manner you must proceed for every
fingle Figure in the Root, if there were ever fo many, which,
when the Root confifts of many Figures, makes the Work
very tedious and troubleſome.
In this Example, the Num-
ber given is a pure or exact Cube, and fo here is no Re-
mainder; but had it been but a Unit lefs, the Remainder
would have been large; but if the given Number had been
more, the Remainder would only have been fo much as it
was more, till it riſe ſo high as to make the 2, or laſt Fi-
gure in the Root, a 3, 4, &c. in which Cafe of Remainders,
if they are any thing fignificant, you muſt add Triplets
of Cyphers, viz. 000, each time; and then continue the a-
bove Work for every Place of Decimals you fhall think fit
to have in your Root.
If your Number given was only a Decimal, you muſt
begin to Point from the Left-hand towards the Right; and
if there are not Figures enough, make them up with Cy-
phers, and then work in every refpect as before; but if the
given Number confifts of Part whole Numbers, and Part
Decimals, point the whole Numbers as before, and the
Decimals as above. As feveral Methods have been made uſe
of to facilitate this Sort of Proceeding, take what follows
from Mr. Ward, Page 131, of his Mathematician's Guide.
After having pointed the given Number, as before direct-
ed, find by the Table, the neareſt Cube to the firſt Point,
whether it be greater or leffer than the faid given Point.-
If the Cube fo found be greater than the firſt Point or
Period of the Refolvend; then the Root or Figure fo found,
is greater than it ought to be: And therefore you muſt ſet
down, or imagine a fufficient Number of Cyphers to be
added
Of extracting the CUBE ROOT.
355
י
added to it, and then ſubſtract the whole Refolvend from
the faid Root and Cyphers; but if the Cube found in the
Table be leſs than the firft Point of the Refolvend, fubftract
it from the faid firft Point or Period; then to the firſt
Figure of the Root thus found, whether its Cube be bigger or
leffer than the firft Point or Period given, add as many Cy-
phers as there are Dots or Points remaining in the Refol-
vend; multiply theſe by 3, and make the Product arifing a
Divifor, by which divide the Difference between the Cube
and the Refolvend, as found above; and the Quotient will
deprefs the faid Remainder or Difference to a Square, and
muſt be accordingly pointed a-new, as a Square, beginning
at Unity, and making a Point at every two Figures, till you
have made as many as you added Cyphers to the Root, to
form your faid Divifor: This done, make the Figure of the
Root already found, a Divifor to the firft Point or Period
of the new Reſolvend, as you do in the Square Root; only you
muſt obſerve, that if the Cube of the faid Figure or Divifor
was leſs than the firſt Point or Period, you muſt annex
the Quotient Figure, or fecond Figure of the Root to it;
and then multiply the Divifor, fo increafed by the faid
Quotient Figure, fetting down the Units Place of the Product,
under the pointed Figure of this Period, &c. as in the Square
Root; then fubftract and bring down the next Point or
Period, and proceed in the fame Manner till all is done.
But if the Cube of the faid firſt Figure of the Root be
greater than the firft Period of the given Refolvend; then
you muft fubftract the Quotient Figure from a Cypher,
annexed, or fuppofed to be annexed to the faid first Figure
of the Root or Divifor on the Right-hand, and multiply the
Remainder by the Quotient Figure, and fubftract this Product
from that Part of the new Refolvend, as was brought down
for this fecond Figure of the Root, and fo proceed for every
Point or Period, till all is done. For Example, Let us take
134217728, the Number we had before, Page 353 and 125,
will be the neareft Cube to 134, the firft Period, and is leſs
than the faid firft Point, and has 5 for its Root.
Z z 2
134217728
356
Of extracting the CUBE ROOT.
134217728
125
Divifor 15,00) 92177,28 the Dividend.
90
21
15
67
60
(6145,152 Quotient.
77
75
22
15
78
75
30
30
5
+1)
51
1) 6145, 1520 — (512 Root.
5T
1045
+2/1024
+2)
21,15, &c. Remainder to be rejected.
Here, after you have found the Root, to as many Places as
there are Points in the given Refolvend, try it, that is, cube
the Root thus found, and if it comes right, your Work is done;
but if not, then every thing here done muſt be repeated, only
inftead of uſing the firft fingle Figure, 5, you muſt uſe the
whole 512; and this is called a ſecond Operation, and would
2
have
1
Of extracting the CUBE ROOT.
357
have increaſed the laft Root to nine Places; becauſe by this
Method, every Operation triples the foregoing Number of Fi-
gures that were found in the Root, by the former Process. Only
you muſt Note, that when the Cube of the Root found, comes
fo near the given Period, that the fecond Figure of the Root
will be a Cypher, then 'twill be neceffary to continue the firſt
Work, till there comes out 4 or 5 Figures in the firſt Root,
EXAMPLE II.
65228104372 4000 lefs.
(400
12000 Divifor.
12,000) 1228104,372 Dividend.
402)102342,03 (4025,4
402) 10
2 804
402521942
5/20125
4) 18
4025,41817,03
4/1610,16
206,87 Remains.
Here the Root is fomething too fmall, for when involved
to the 3d Power, the Refult is 65226958307,064; but if
more Exactneſs was required, a fecond Operation would
give you a fufficient Number of Decimal Places to make
it exact. The Method of theſe fecond Operations you may
fee by the following Example.
What is the Cube Root of
705379602989073960279630298890?
729000000000000000000000000000 Cube of 900,&c.
23620397010926039720369701110
the Re-
mainder, which is alfo the Dividend, and 27000000000
is the Divifor, and 874829518923186656 will be the Quo-
tient and new Refolvend, depreffed to a Square.
1 Divifor
358 of extracting the CUBE ROOT.
I Divifor 90.874829518923186656 New Refolvend.
9
8019
9000000000
,09819
2 Ditto 891;)
72929
71216
8901810000
3 Ditto 8902.
171351
8901800000
I
89019
3
4 Ditto 89019.
9
8233289
8011629
26705400000
new
Divif.
5 Ditto 890181
For the firft Part of this Work, only 5 Figures are taken,
viz. 89018, the other 5 are for the prefent fupply'd with
Cyphers, which makes it 8901800000, which raiſed to the
3d Power or Cubed, is
from which
705396820513832000000000000000
Subſtract 705379602989073960279630298890 given Reſ.
26705400000)re.17217524758039720369701110 for a new
89018.
Divid
$6447207215
7644720721578397 new Refolvend and Quotient.
890173.6231211 |
2).
8901728. 21599621 |
4 17803456
(72426,45
8g01800000 Root before.
72426,45 Root now found
89017276. 379616557
2 3560691 | 04
8901727573,55 Root required,
890172758. 235474 | 5383
6 178034 | 5516
8901727574. 57439 | 986797
24 53400 | 365444
8901727573,6. 4039 | 62135300
05 3560 | 69102944
8901727573,55 478 | 9303235600
4450 8 6 3 786775
But
Of Simple INTEREST.
359
1
But you might have contracted the Work, at the 3d
Step of this laſt Work, as in contract Diviſion of Decimals,
and fo have omitted all the Figures on the Right-hand
of the perpendicular Line; and though there may ſeem
much Work here, yet if it was profecuted by the firſt
Method, 'twould have been near fifty Times as much
Labour and Trouble.
The Proof of theſe Operations, is by involving the Roots
thus found to the third Power, or what is commonly called
Cubing the Roots. There is feldom Occafion for extracting the
Roots of higher Powers, fuch as the 4, 5, 6, 7, 8th, &c. in com-
mon Arithmetick; only you may Note, that the fourth Power
may be extracted, by firft extracting the Square Root of
the whole Refolvend; and then extracting the Square Root
of this firft Root, and this fecond Square Root will be the
Biquadrate Root fought.
We ſhall now go on to treat of Intereft, both Simple
and Compound; and here, as in moft other Parts of Arithmetick,
Decimals may be exceeding uſefully applied. And firſt, in
SIMPLE INTEREST,
That is, where no other Confideration is had, but the
bare Money allow'd for the Use of a Sum, borrowed of an-
other, for a certain Time. Now this is ufually at fome de-
terminate Rate, for one hundred Pounds, for one Year. This
formerly was very high, as L. 10, 9, 8, &c. and now is
fometimes 3, 4 or 5, and beyond L. 5. no Perfon can le-
gally take; and therefore we will fhew the beſt and eaſieſt
Methods of doing the faid latter Rates, for any Sum greater
or leffer than a L. 100. and for any Time greater or leffer
than a Year. And,
1
Firſt, you are to Note, that the Sum lent, or borrowed,
be it little or much, is called the Principal. Secondly, That
the Money agreed to be paid for the Ufe of L. 100, for one
Year, is called the Rate; but if no Agreement is made,
then the Sum borrowed always pays the moſt that the
Law allows, which at this Time is L. 5 per Cent. per
Annum; but if a lefs Sum is agreed for to be the Intereft,
'tis always expreffed in the Bond, Note, Deed, or other
Covenant. And here you are to obferve, that no Covenant
is legal, where the Interest is charged higher than L. 5
per Cent. per Annum, though you may make it for as
much lefs as you pleaſe.
EX
360
Of Simple INTEREST.
•
EXAMPLES.
What is the Intereft of L. 2175: 16 for a Year, at 3, 4
and 5 per Cent?
The common Method of doing thefe Queftions, is this
that follows.
L. 21
75: 16
5
108 79:
179
20
15 | 80
/ 800
9/60
Here 'tis underſtood, that this
Queſtion is thus ftated: If L.100
gains L. 5, What will L.2175:16
gain? And to fave Trouble, the
whole is multiply'd by 5, and di-
vided by 100; and then the Re-
mainder is 79, which are, 22 of a
L. whofe Value is found, as
formerly directed; but Decimally
the Work is much fhortned,
L. 2175,8
4
thus
ΙΟ
12
240
where the Anfwer is L. 108,79, or
5
108,79,0
L. 108,15 9. The like for any other Rate, as for L. 3
L. 4, &c. per Cent. per Annum.
L. 21175: 16
4
87 03 4
20
064
12
7168
4
2172
Anf. L.87:00: 7½
L. 21175: 16
3
65 27: 8
16 1
20
548
12
5 76
4
3 04
Anf. L.65 5 : 5
Decimally
Of Simple INTEREST.
361
Decimally thus:
L. 2175,8 the Principal.
4 per Cent. per Ann. Intereſt.
L. 87,03,2, Or L. 87: 7 Total Intereft.
L. 2175,8 the Principal.
3 per Cent. per Ann. Intereſt.
L. 65,27,4, Or L. 65: 5:54 Total Intereft.
By the foregoing Examples, you may find what the Intereſt
of any Sum of Money will amount to for a Year, but if the
Time be greater or leffer than a Year, fay, As one Year is
to the Intereft of the Sum given for that Time; fo is the Time
requir'd, to the Amount of the fame Sum, at the fame Rate for
that Time. As for Example:
What comes the Interest of a Bond to for L. 359: 15: 6,
from the 15th of January 1735, to the 6th of June 1738, at
5 per Cent. per Annum?
Firſt find the Intereft for one Year, as before, thus:
359,775
5
L. 17,98875, or L. 17: 19:9
Then confider how much the whole Time is, viz. 2 Years
142 Days; then by the Rule above, fay,
Years Days.
As 1 Year is to 17,98875, fo is 2 : 142
To 42,976, or L. 42: 19: 64
15: 6
To which add the Principal L. 359
And the whole Amount is L. 402: 15:04
Or you might multiply the Quotient of the Rate divided
by 100 by the Time, and part off that Product by a
Čomma, and add an Unit on the Left-hand of the faid Comma,
Aaa
and
362
Of Simple INTEREST.
and then that whole Sum, multiply'd by the Principal, will
give you the Amount of the Principal and Intereft.
The Quotient of the Rate of the laſt Example is,
1
by
gives
,05 this multiply'd
2,389 the whole Time,
,11945
to which add Unity and the Sum is 1,11945, this multiply'd
by
359,775 the Principal,
gives L. 402,75, or L. 402,15 for the Sum of the Principal
and Intereft.
Both theſe Ways may be uſefully apply'd to the com-
mon Publick Bonds, fuch as the South-Sea Company's, the
India Company's, &c. For if you are to receive only
the Intereft, you may uſe the firft Method; but if you fell
or buy a Bond, you may uſe the laft Method.
EXAMPLE.
What must I give for an India Bond, Value L. 500, In-
tereſt at L. 3½ per Cent. per Annum, Prem. L. 6: 10 per
Cent. Intereft due upon the faid Bond, from the 30th of Sep-
tember, 1737, to the 6th of June 1738?
249 Days, or ,6822 the Time,
multiply'd by ,035 the Rate,
gives 1,0238770 for Prod.with Unity added, which
multip.by the real Prin. 500
gives L. 511, 9385, or L. 511 189 the Sum of the
Principal and Interest; to which add the Premium upon
the Whole
32 : 10
and the Total Amount L. 544
18 9 is the Sum of the
Principal, Interest and Premium, or the Sum that is to be
paid for the faid Bond, upon the above Conditions.
Or thus:
L. 500 the Principal.
3,5 the Rate.
17,500 the Intereft 1 Year.
Ther
Of Simple INTEREST.
363
Then you may fay,
As I Year to L. 17,5, fo is 249 Days to
L. II: 18 9
+
for Intereft; to which
add L.532: 10 the Sum of the Prem. andPrin.
and the Total 544: 8: 9 is the preſent Value of
the Bond.
The like may be done for any other Sum or Time, bigger
or leffer, at the fame or any other Rate of Premium and
Interest, where you are to Note, that the Intereſt is never to be
reckoned upon the Premium, but upon the Principal only, be-
caufe all Premiums are fubject to many and great Variations,
according as the Humour of the People is, or according to
the Plenty or Scarcity of Money, or the Situation of publick
Credit. Now the Interest of thefe Sort of Bonds are gene-
rally calculated by Tables ready caft up, at the Rates the
Bonds are made for, and which in common Practice are
fufficiently exact; and the Method of doing it is thus:
Count how many whole Months and Days are elaps'd from
the Time of diſcharging the laft Interest; as in the Example
before us, 'tis 8 Months and 6 Days.
The Interest of L. 500 for 8 Months is L. 11: 13:4
for 6 Days
: 5:9
Total is L. II: 19: 1
Intereft found above, is
II: 18 : 9
Difference is
3
1 2
The Reafon whereof is, that the Tables reckon I Month
to be the Part of a Year, and confequently, that 8 Months
and 6 Days is 249 Days and ; but the real Time is but
249 Days abfolutely; fo that in fuch a like Cafe, the Tables
reckon the Intereft of L. 500 for of a Day too much, which
is 3d 4 exactly, as the Difference above fhews it to be, and
which is an infallible Proof, both of the Truth of the a-
bove Arithmetical Calculations, and of the Correctneſs of the
Tabular Calculation. The Tables here mention'd, are calcu-
lated by myſelf, and are to be had of Mr. Ware, the Pro-
prietor of this Book, called, Tables, fhewing by Inspection, the
Interest
Aaa 2
364 Of Simple INTEREST.
Interest of any Number of Pounds, from one progreffively to
one Hundred, &c. at the feveral Rates of, 3, 3, 4 and 5
per Cent, with Directions to make them applicable to all man-
ner of Sums and Rates, and for any Time, which, to thoſe
who have not Leiſure or Skill fufficient to make uſe of the
Methods here taught, is a very great Advantage; and for
thoſe that have, may ſerve as a Proof of the Truth of their
Work: And as odd Time is very frequent, eſpecially in Pub-
lick Bonds, therefore the following Table will be of Ufe
for the ready finding the Number of Days between any
given Times of the Year.
31 334 January
31
32
59 306 February
90 275 March
31
28
59
31
120 245 April
90
30
151 214 May
120
31
1 2
181
184 June
151
30
212
153 July
181
31
243
122 Auguft
212
31
273
92 September
243
30
304 61 October
273 31
334
31 November
304
365
00 December
334
33
за
31
EXPLANATION.
The middle Column contains the Names of the feveral
Months of the Year; the outſide Column on the Right-hand,
ſhews the Number of Days that are in each Month; and the
infide Column, immediately joining to it, fhews the Number
of Days it is from the first Day of January to the first Day
of each of the other Months reſpectively: As for Example,
from the first Day of January, to the first Day of February,
is 31 Days; and from the first Day of January, to the firſt
Day of July, is 181 Days; and fo of any other; as from the
first of January to the firft of May is 120 Days; and to the
first of November, 'tis 304 Days, &c. the furthermoft Column
on the Left-hand, fhews the Number of Days that is in the
feveral Months of the Year, beginning with the firft of Jan.
as, to the End of April, is 120 Days; to the End of Auguſt,
is 243 Days; and to the End of December, is 365 Days, &c.
The next Column adjoining, fhews how many Days there
3
are
Of Simple INTEREST.
365
are in all Months of the Year, below that against which
the Name is wrote.
wrote. As, from the laſt Day of February, to
the laft Day of December, is 306 Days; fo from the laſt of
Auguft, is 122 Days, &c.
The Use of the foregoing Table,
Is, to find how many Days 'tis from any one Part of
the Year to any other. As, fuppofe I afk, How many Days
'tis from the 3d of Auguft to the 17th of January following?
Here you are to Note, that if the laft Month mentioned, goes
beyond the End of December, that you muſt firſt look in
the inward Left-hand Column, for the first Month mentioned:
As, in this Question for August, againſt which is ſet 122
under this fet the Complement of the Days in that?
28
Month, viz.
at}
then go to the inward Column on the Right-hand, 17
and fet down what is fet against that Month, which
here being January, is oo; and then add the Num- 167
ber of Days in the Month, which in this Queſtion is 17;
the Sum of all theſe is the Number of Days fought, which
in this Question is 167, as may be eafily proved thus;
28 Days in August.
30 September
31 October
30 November
31 December
17 January
But if the Queftion is for any
Time within the first of January
and 31 December, do thus: Look
out the Month that is most diftant
from January, in the Queſtion, and
Jet down the Number that ftands a-
gainst it, in the infide Right-hand
Column, and add to it the Days given,
if any; under this Sum, fet the Sum
of the nearest Month and Days 10
the one from the other, and the
Difference or Remainder will be the Number of Days fought..
167 Total.
January, and fubftract
EXAMPLE.
What Number of Days are there between the 7th of
February and the 15th of Auguft?
31 Days against February:
7 Days in February.
38 Days to 7 February.
2T
366 Of Simple INTEREST.
21 Days in Feb.
212 againſt Auguft.
15 Days in Auguft.
Total 227 the Days to 15 Aug.
31 March
30 April,
Proof.
31 May
30 June
31 July
15 August
189
38 Days to 7 Feb.
Rem. 189 is the Days from 7 Feb. to 15 Aug.
From what has been already faid, enough relating to the
Practice of this Rule, may be learnt, to know the Amount
of any Sum at any Rate, for any Time; but as fome Quef-
tions, more curious than needful, may be propos'd, and re-
quire from other Data, other Things, I will juft hint one
or two to you, and the reft will follow; as,
In what Time will L. 2500 come to L. 3000, at 5 per
Cent. per Annum, fimple Interest?
Here 'tis very plain, that the Interest Money is L. 500,
therefore find what the Amount is for one Year thus:
2500
5
L. 125 | 00
Then fay, If L. 125, require 1 Year, What will L. 500
require? Anfwer, 4 Years.
Or more univerfally thus: Subſtract the Principal from the
whole Amount, and divide the Difference or Remainder, which
is the whole Intereft, by the Product of the Principal and
Rate divided by 100, and the Quotient will be the Time fought,
The Example above ever again.
L. 3000 the whole Amount of Principal and Intereſt.
2500 the Principal only to be ſubſtracted.
500 the Difference or Intereſt which is to be the Divid.
2500
Of Simple INTEREST. 367
2500 the Principal.
5 the Rate.
12500 This Product divided by 100 is 125, which is to
be the Divifor to 500 the above Dividend.
125 | 500/4 the Quotient, which are whole Years, becauſe
1500 L. 125 was the Amount for 1 Year; but if there
had been any Remainder, continue dividing
by bringing down Cyphers, till you have
as many Decimal Places as will, with fuf-
ficient Exactneſs, exprefs the Parts of a Year. As,
How long Time will L. 300 be in amounting to L. 400 Simple
Intereft at 3 per Cent?
300
3,5
10,5 |
00/100,
100,-
945
(9,523809
365
550, 191/18700 Product.
Anſwer, 9 Years and 191 Days
4 Hours 34 Minutes.
525
250
210
400
315
850
840
ΙΟ
Or you may multiply the firft Remainder, after the Tears
are found, by 365, and divide by 10,5, and the Quotient
will be the fame as above.
5,5 Remains.
365
10,5)2007,5 Product.
20 Remains, and the Quotient is 191
Days.
And
**
368
Of Simple INTEREST.
1
And if you are minded to have greater Exactnefs, both
will produce 4 Hours and 34 Minutes.
Or, Suppoſe I put the Queftion thus:
What Rate of Simple Intereft in four Years, will make
L. 2500 amount to L. 3000?
Say, If 4 Years give L. 500, What will 1 Year give?
Anfwer, L. 125.
Then, If L. 2500 give L. 125, What will L. 100 give?
Anfwer, L. 5-
Or more univerfally thus:
!
Divide the whole Intereft by the Product of the Principal
and Time, and the Quotient gives the Rate divided by 100.
Thus, 2500 the Principal.
4 Years the whole Time.
Prod. 10000 the Divifor to L. 500 the Intereft; the Quo-
tient is,05; this multiply'd by 100 gives L. 5 per Cent.
for the Rate. So that now any thing relating to this Branch
of Intereft may be eafily anfwered. But there are other
Bufineffes relating to this Head, not yet fufficiently explain'd,
viz. the calculating Commiffions, and Brokage for Sales of
Goods, &c. Difcompts for prompt Payments, Premiums for
Injurance, Penfions, Annuities, &c. Ás,
I
What comes the Infurance of L. 7,85 to at L. I per
Cent?
Thus, L. 785
20
17 100
1,9625
Anſwer, L. 9,8125 Or, L.g: 16:3
Or, by the Vulgar Method
L.
thus:
7 : 17: for I per Cent.
I: 19: 3 for 4 per Cent.
L.9 16 3
Ihat
Of Simple INTEREST. 369
What comes the Commiffion of L. 594: 13 to at 2
per Cent?
2
Vulgarly thus, 11 | 89: 6
20
L. 11: 17: 108 for 2 p.Ct. 17 | 86
11:17:10
ros. is 2: 19:
о
522 for 10s. p. Ct.
12
of L.2
Tot. L.14:17: 3
3
+
188at L.23 p.C.10. | 32
4.
1 | 28
Decimally thus:
L. 594,65
2,5
297325
118930.
L. 14,86625 Or L. 14: 17: 34.
Suppofe I am to be allow'd L. 6 per Cent. Discount, for
prompt Payment, What must I pay for 5 Bales of Silk, that
by weight comes to L. 730? Anf. L. 686 4.
Now all fuch Queſtions as theſe may be anſwer'd either by
finding the Amount of the Whole, as if it was for the Intereſt
of 1 Year, as above, and fubftracting that from the whole
Sum, and the Remainder will be the Sum to be paid; or
elfe, by fubftracting the Allowance from 100, and multiply
ing the whole Sum by the Remainder, and dividing that Pro-
duct by 100, the Quotient will be the Anfwer fought. As in
the following Example.
The first Way.
730 the whole Sum.
6 per Cent. allowed for Difcompt.
L. 43,80 Or L. 43: 16, the whole Difcompt, which
fubftracted from L. 730 leaves L. 686: 4, the Nett Sum to
be paid.
Bb b
The
370
Of Simple INTEREST.
The Second Way:
Or, 730 the whole Sum
multiply'd by 94 the Difference between 100 and 6.
2920
6570
gives 68620 for the Product to be divided by 100, and
then the Quotient
will be L. 686,20, Or L. 686: 4, as above.
You muft Note, that in Commiſſions, Brokage, Diſcompts
upon Goods and Infurances, there is no Regard had to Time,
or you may always confider them as done for one whole Year.
I
For the more eafy performing every Operation, 'twill be
proper to find, at the feveral Rates, the Amount of L. 1 only,
and then multiply Decimally, and the Products will have
nothing to be done to them, but to be valued, as has been
taught in Multiplication of Decimals, &c. As for Inftance,
the above Sum 730, multiply'd by ,06 gives 43,80 for the
Product; and fo of all others. It will likewife be very ufe-
ful to find in Decimals, what the Amount of L. 1 for 1 Day, is
at the feveral Rates commonly made ufe of in Bonds. Thus
the Intereft of L. I for I Day, at 5 per Cent. is ,000137;
at 4 per Cent. ,0001096; at 3 per Cent. ,0000822; at
2 per Cent. ,0000685; at 3 per Cent. ,000095617, &c.
by which any common Buſineſs may eaſily be done: Thus,
What comes the Intereft of 5 Bonds each L. 100 to, at 4
per Cent. for 17 Days?
Here you may multiply L. 500 the whole Principal by 17
and,0001096, or by ,0001096, and 17, which you pleaſe,
and the Product will be the Anfwer fought.
Thus, 500
17
8500
,0001096
548000
8768
Anſwer, ,9316000, or 18 s. 7d. ž
As
Of Simple INTEREST.
371
As may be found either by the Methods before taught,
or by the Tables before mentioned. So if it was required
to find the Amount for 3, 34, or 5 per Cent. of the faid
Sum, for the faid Time, the refpective Work and Anſwers
will be as follows,
500
17
8500
,0000822
411000
6576
698700
Or, L. 00:13: II 1/
for 3 per Cent.
500
17
8500
,000137
68500
Or,
500
17
8500
9589
4794500
76712
,81506500
Or, L.00: 16: 3 1/2/
for three per Cent.
1,096
1,164500
Or, L. 1: 3:3 for five per Cent.
All which Anſwers exactly agree with the before men-
tioned Tables.
Once more: What is the Intereft of L. 875: 13:6 for
85 Days, at 3, 3, 4 and 5 per Cent.
In all fuch Cafes, where you have odd Money, if it a-
mounts to more than 10 s. (for the fake of Eafe in work-
ing) fet down a Pound, and if under 10s. reject it; for if
the Time be not very long, the Difference will be very little,
efpecially if the Rate of Intereft be low, as appears from the
following Examples.
Bbb 2
L. 876.
}
372 Of Simple INTEREST.
L. 876 inftead of L. 875: 13:6.
85 Days.
4380
7008
L. 74460 for one Day.
,0000822 fix'd Number at 3 per Cent.
148920
148920
595680
L. 6,1206120, Or L. 6: 2:44.
L. 875,675 Or, L.875: 13:6.
875,67
85 Days.
4378375
7005400
L. 74432,375 for one Day.
,0000822 fix'd Numb. at 3 p. C.
148864750
148864750
595459000
6,1183412250, Or L. 6: 2:4
Here the Difference is but an Half-penny, in 85 Days,
tho' you reckon the Sum 6s. 6d. too much, to fave
the Work.
L. 876
Of DISCO M P T.
373
2
L. 876 at 3 per Cent.
85 Days.
4380
7008
L.74460 for one Day.
,00009589 the fix'd Number for 3 per Cent.
670140
2
595680.
372300..
670140...
7,13996940, Or L. 7 : 2 : 9 ½.
L. 875,675 at 3½ per Cent.
85 Days.
4378 375
7005400
L.74432,375 for one Day.
,00009589 the fix'd Numb. for 3 p. C.
669891375
595459000
372161875
669891375
L. 7,13732043875, Or L. 7:29.
The like may be done for any other Rates, Times, or
Sums, where the Difference in common Practice is infigni-
ficant; and therefore I fhall now conclude this Head, and
fay a Word or two about
REBATE or DISCOMPT
That is, what prefent Sum of Money muſt be paid for
any Sum due, or to be paid, fome Time hence, allowing Dif
compt, after the Rate agreed upon, at Simple Intereſt.
EX
374
Of DISCOM PT.
EXAMPLE.
Suppoſe a Year hence I was to pay a Legacy, a Year's Rent
for a Farm, &c. and the Money was to be L. 450, but the
Party to whom 'twould then be due, defires to have it imme-
diately, and would therefore allow Diſcompt at 4 per Cent. per
Annum, Simple Intereft.
OBSERVATION.
By the very Conditions of this Queftion, 'tis evident that
juſt ſo much prefent Money must be paid for the above
L. 450, that when put out at Intereft, at L. 4 per Cent.
the Sum of the Principal and Intereſt muſt make up the faid
Sum of L. 450; and therefore the common Methods of
diſcounting Sums of Money, where Time is concerned, is
wrong, becauſe the whole Sum is not difcompted: Therefore
to find the true Sum, you must find the Interest of L. 1,
or L. 100, &c. at the given Rate, for the given Time, and
then ſay, As that affum'd Principal and Intereft, is to the
Principal, fo is the given Sum to the Sum fought. As here
the Intereft of L. 100 for 1 Year, is L. 4, the Sum is 104;
therefore I fay,
As L. 104 is to L. 100, fo is L. 450 to L. 432: 6923, or
·L. 432: 13: 10.
Here the Anſwer is L. 432: 13: 10, to be paid imme-
diately; fo that L. 17: 6: 2 is allowed for the Intereft: And
that we may be fure whether the Work is right, let us fee
what the Intereft of L. 432,6923 comes to, for 1 Year,
at 4 per Cent.
4
Viz. L. 17,307792 Or, L. 17:6: 2.
Now, L. 450 is the whole Money due one Year hence.
L. 432: 6923 is the prefent Money found.
And L. 17,3077 is the Premium or Difcompt, which is
exactly the fame with the Interest of L. 432: 13: 10; as
plainly appears by the Work before you.
But the common Method is to find the Intereft of the
whole Money, and fubftract that from the whole Money,
and to call the Remainder the Sum or Money to be paid down:
Thus
Of ANNUITIES.
375
Thus, L. 450 the Principal.
4
L. 18,00 the Intereft for one Year.
L. 432 the Sum to be now paid, which is L.0: 13:10
too little; and if the Time was longer, or the Sum or Rate
greater, the Error would be alfo proportionally larger; but
in fmall Sums for fhort Dates, the Difference is inconfider-
able; and as plain Interest is much eaſier to work than
Diſcompt, it may be fafely enough uſed. As for
EXAMPLE.
Suppoſe I have a Bill of Exchange, that wants fourteen
Days of being due, What Difcompt must I allow at L. 5 per
Cent. per Annum, if the Bill is drawn for L. 133: 10?
Anfwer, 5 s. ord. ½, Or L. 133:4: 10 ready Money.
For L. 133,5 Principal.
at
5 per Cent. Intereſt.
gives L. 6,675 Interest, for 1 Year.
Then if 365 Days, or 1 Year, gives L. 6,675, What will
14 Days give? Anſwer, L.oo: 5: 1; this "fubftracted
from the whole Sum, leaves L. 133: 4: 10½ for the ready
Money: And by Diſcompt,
Say, As L.365 Days is to L.5, ſo is 14 Days to ,1919, &c.
Then, as 100, 1919 is to L. 100, fo is L. 133,5 to
L.133,244 the prefent Money to be paid: Or, L.133:4:104.
So that the Difference in this Sum is too infignificant to
want any change of Method, though 'tis abfolutely certain
that in larger Sums, and longer Time, 'tis worth while to
follow the other Method, as you have feen above. We
ſhall now touch upon
ANNUITIES at Simple Intereft.
The Meaning of this is, that I am fuppos'd to buy a
Yearly, half Yearly, or Quarterly Income, to continue
fome time to come, and am to difcompt Simple Intereſt
at 3, 4, 5, &c. per Cent. per Annum, and to pay the
Money now immediately; or I am to pay off the whole Rent
of my Houſe for a Term of Years to come, or a Part of my Rent,
which
I
376
Of ANNUITIES.
which is the Cafe of all Fines, &c. Or I may be fuppos'd to
forbear the Rent, Penfion, or Annuity, payable to my ſelf,
a certain Time, and receive for fuch Forbearance, after the
Rate of Simple Interest, for every Payment, as it became due,
at fuch Rate as fhall be agreed on; though 'tis indeed uſual
to reckon theſe Cafes by Compound Intereft; but of late Years,
fince the Intereft of Money is become very low, and the
Conveniences arifing to the Purchaſer of fuch Fines, Annui-
ties, &c. are oftentimes very great, 'tis frequently done this.
Way; and at all Adventures, I may furely make Calculations
of this Sort, and alfo of the other; and when I have fo done,
and compared the Refult of the two Ways, 'tis reaſonable I
fhould be at liberty to comply with either of them, as my pre-
fent Occafions, Neceffities, or Conveniences, fhall dictate to
me. Here
you are to obſerve, that ſo often as an Annuity or
Rent, &c. is to be paid, whether once, twice, &c. in a Year,
'tis here underſtood to be, and is called a Time. Therefore,
if any Rent, Annuity, &c. with the Times it ought to be
paid, and the Rate of Intereft, that is to be reckon'd, is
given, to find what the Total Amount or Sum of all thoſe
Arrears will come to in prefent Money,-You muſt
Multiply the Number of Times of Payment by itself, made lefs
by Unity, and this Product again by half the Ratio of the Rate
of Intereft: To this laft Product add the Number of Times of
Payment; multiply this Sum by the Annuity, Rent, &c. and
this last Product will be the Anfwer ſought after.
EXAMPLE.
Suppoſe I have a Bond, an Annuity, or a Rent of L. 250 per
Annum, but to accomodate the Perfon that is to pay it, I agree
to let the fame lie in his Hands for 9 Years, for which he is to
allow me Simple Intereft, after the Rate of 5 per Cent. per
Annum for each Payment, as it becomes due, What will the
whole amount to ?
9 the Number of Times of Payment.
8 one lefs than the Payments.
72 the Product.
,025 half the Ratio of the Rate of Intereft.
10,800 the Prod. added to the Times of Payment.
250 the annual Payment.
2700,0 the Product, which is the Amount of the whole
Principal and Intereft.
Which
Of ANNUITIES.
377
Which may be cafily proved: Thus,
The first Payment is L. 250 and it lies 8 Years,
Interest in that Time is
Second Payment is
Intereft in that Time is
Third Payment is
100
250 and it lies 7 Years,
87,5
250 and it lies 6 Years,
75
Intereft in that Time
Fourth Payment is
is
250 and it lies 5 Years,
Interest in that Time
is
1
62,5
Fifth Payment is
250 and it lies 4 Years,
Interest in that Time
is
50
Sixth Payment
is
250 and it lies 3 Years,
Interest in that Time
is
37,5
Seventh Payment
is
250 and it lies 2 Years,
Interest in that Time
is
25
Eighth Payment is
250 and it lies 1 Year,
Interest in that Time
is
Ninth Payment is
12,5
250 upon which no In-
tereft is due, and the Total is L.2700
In the above Question, you are to obferve, that the firſt
Year's Rent pays Intereft but for eight Years; becauſe one
Year is fuppofed to be expired before it becomes due; and the
laft Year's Rent pays no Intereft, becauſe the Whole is fup-
pos'd then to be paid off: And confequently, that theſe
feveral Payments, and the Interest upon them, form a Series
of Numbers in Arithmetical Progreffion, whofe firſt Term is
350, viz. the Sum of the Rent and its Intereft, and the laſt
Term 250, and the common Excefs or Difference is 12,5, viz.
the Interest of L.250 for one Year, and the Number of Terms
9, as appears by the above Particulars, where the Sum of the
firft Payment, and the Intereft is L.350, and the laft Payment
is L.250 only, becauſe it has no Interest; fo that by the Rule,
Page 325, you may eafily get the Sum Total that is due, thus:
350 the firſt or biggeſt Term.
250 the laſt or fmalleft Term.
600 the Total of both.
9 the Number of Terms.
2 | 5400 the Product divided by 2 gives
2700 the Sum of all the Series, which is the Amount
of the Annuity, both Principal and Intereſt for the Time.
Ccc
2. Sup-
378
Of ANNUITIES.
2. Suppoſe the Amount, Time and Rate of Intereft to be
given, and the Annuity was requir'd.
RULE.
Multiply the Number of Payments by it felf, lefs Unity, and
that Product by half the Ratio of the Rate, and to this laft Pro-
duct add the Time or Number of Payments, and make this Sum
the Divifor, and the Total Amount the Dividend, and the Quo-
tient will be the Answer fought.
N. B. The Ratio of the Rate of Intereft, is the Intereſt
of L. 100. divided by the Principal. Thus the Ratio of 5 per
Cent. is ,05 of 4 per Gent. is ,04 of 3 per Cent. ,03, &c.
What Rent or Annuity to be paid Yearly, at 5 per Cent. per Ann.
Simple Intereft, will amount to L. 2700 in nine Years Time?
EXAMPLE.
9 the Number of Payments.
8 ditto lefs Unity
72 Product
,025 half the Ratio.
360
144
1,800 Product.
9-
Times of Payment.
Dividend.
10,8)2700(250 Quotient.
So that an Annuity, or
Rent of L. 250 per Annum,
forborn to the End of the
9th Year, or Time of Pay-
ment, will be the Anſwer.
10,8 Total to be the Divifor.
Or, you may fuppofe L. I to be the Annuity, and work as
in the laſt Example, and the Anſwer will come out L. 10,8.
Then fay, As L. 10,8 is to L. I fo is L. 2700 to L. 250.
Thus by the Rule in Page 325,
L. I Annuity.
,4 the Intereſt at 5 per Cent. for 8 Years.
1,4 the Sum is the firft and biggeſt Term of the Series,
I- the laſt and ſmalleſt Sum of the Series.
2,4 the Total of the firſt and laſt Term.
9 the Number.
2 21,6 the Product.
10,8 the Total Amount of an Annuity of L. 1 per
Annum,
Of ANNUITIES.
379
Annum, forborn for 9 Years, with the Allowance of L. 5
per Cent. per Annum, Simple Intereft, for the feveral Pay-
ments, as they became due.
3. If the Annuity, Total Amount, and Rate of Intereſt is
given, to find the Time. The Rule is,
Subſtract the Ratio of the Rate from 2, and divide the Re-
mainder by the Double of the Ratio; then fquare this Quotient,
and add this Square to the Quotient arifing, by dividing the
Double of the Total Amount by the Product of the Annuity,
multiply'd by the Ratio; then extract the Square Root of that
Sum, and from this Root fubftract the firſt Quotient, and the
Remainder will be the Number of Times of Payment fought.
E XA M P L E.
In what Time will a Yearly Penfion, Rent, or Annuity of
L.250 per Annum, amount to Ľ.2700, allowing Simple Intereft
at L. 5 per Cent. per Annum?
2 the whole Number 2.
-
,05 the Ratio of the Rate to be fubftracted.
,1)1,95 Remain. to be divided by,I the double of the Ratio.
19,5 the Quotient.
19,5
975
1755
195
380,25 the Square of 19,5 the Quotient.
L. 2700 the propos'd Amount multiply'd
by 2
12,55400 the Product or double of the Amount which is to
15) 5
50,0
be a Dividend.
40 0 (432 Quotient.
250 Annuity.
37,5
,05 the Ratio.
25,0
25,0
:
Ccc 2
12,50 the Prod. or Divif.
432
380
Of ANNUITIES.
432
the Quotient above.
380,25 the Square of 19,5.
812,25 the Sum from which the Square Root is to be ex-
28,5 the Root.
19,5 the first Quotient.
9,
tracted.
the Remainder is the Years or Times of Payment.
N. B. In this laſt Cafe, there is given a Series of Num-
bers in Arithmetical Progreſſion, in which the Sum of all the
Terms, the common Difference, and the firſt or laſt Term
is given, to find the Number of Terms.
4. The Annuity, Rent, or Penfion to be annually, &c. paid, the
whole Amount both of Principal and Intereft, and the Time be-
ing given, to find the Rate of Intereſt.
The RULE is,
Multiply the Annuity first by the whole Time, or Number-of
Payments, and also by the Square of the Time, and ſubſtract
the leffer Product from the greater, and divide the Remainder-or
Difference by 2, the Quotient will be a Divifor. Then fubſtract
the Product of the Annuity by the fingle Time, from the whole
Amount, and divide the Remainder by the above Divifor, and
the Quotient will be the Ratio of the Rate of Intereft fought,
which multiply by 100, and the Product will be the Rate itself.
EXAMPLE.
Suppoſe I have an Eſtate of L. 250 per Annum, to be paid
Yearly, which I agree shall lie nine Years, at the End of which
Time I am to receive 2700, What is the Rate of Simple In-
tereft, that I am to be allow'd per Cent. per Annum?
250 the given Annuity.
9 the Time.
2250 the Product,
2700 the Total Amount.
2250 the Product of the Rent, and Time,
450 the Remainder to be a Divifor.
250
Of ANNUITIES.
381
1
250 the Rent.
81 the Square of the Time.
250
2000
20250 the Product.
2250 the Prod. of the Annuity by the fingle Time.
2)18000 the Remainder.
9000 the Quotient or Divifor.
9000) 450,00 (,05 the Quotient or Ratio of the Rate.
450,00
,05 the Ratio of the Rate.
100 the Principal to the Rate.
5,00 the Product, which is the Rate fought.
In the above Cafe there is given the firft Term, the Sum of all
the Terms, and the Number of Terms in an Arithmetical Series,
to find the common Exceſs.
Sufficient has now been faid for yearly Payments, and if
the Queſtion is about half-yearly or quarterly Payments,
Instead of the Ratio of the Rate given, you must take its
Half or its Quarter, and also the or the of the yearly
Rent or Payment; and inftead of the Number of whole Years,
you must take its Double for half-yearly Payments, and its
Quadruple for Quarterly Payments: As in the following
Examples.
If I have a Penfion of L. 250 per Annum, payable half
Yearly, and it is unpaid for 9 Years, What must I receive
at the End of that Time, and be allowed L. 5 per Cent.
Simple Intereft for every Payment, as it becomes due ?
PRE-
3
382
Of ANNUITIES.
PREPARATIO N.
L. 125 the half-yearly Payment, or ½ of L. 250.
,025 the Ratio of the Rate for half-yearly Payments,
or of,05
18 the Number of Times of Payment, or 9 Years mul-
tiply'd by 2.
17 the Number of Times, lefs Unity.
,0125 the Half of the Ratio of the Rate for Year,
18 the Times or Payments.
17 Ditto lefs 1.
126
18
306 the Product.
,0125 the Half of the Ratio of the Rate.
1530
612
306
3,8250 the Produči.
18
the Time.
21,825-the Total.
125-the Annuity.
109125
43650
21825
2728,125 the Product or Total Amount.
Here you fee, that by having the Payments half-yearly,
there is L. 28: 2: 6 more gain'd than by having them An-
nually.-Again,
Suppoſe I am to receive a Penfion of L. 250 per Annum
Quarterly, and I agree to let it lie 9 Years, What must I
receive
Of ANNUITIES.
383
receive at the End of that Time, if I am allowed Simple In-
tereft, at the Rate of L. 5 per Cent. per Annum ?
PREPARATION.
L. 62,5 the Quarterly Payment, or 4 of L. 250.
,0125 the Ratio of the Rate for Quarterly Payments.
36 the whole Number of Payments.
35 Ditto, leſs Unity.
,00625 the Half of the Ratio of the Rate.
36 the Number of Payments.
35 Ditto lefs 1.
180
108
1260 the Product.
,00625 the Half of the Ratio of the Rate.
6300
2520
7560
36,
7,87500 the Product.
the Time.
43,875
the Total.
62,5
the Annuity or Penfion.
219375
87750
263250
2742,1875 the Product or Total Amount.
By this Example you may obferve, that the Proprietor of
any
fuch Penfion, Rent or Annuity, will gain L. 42: 39
more by being paid Quarterly, than by being paid Annually.
The
384
Of ANNUITIES.
The prefent Worth of Rents, Annuities, &c. at Simple Intereſt.
Here, as before, are four Cafes; the firſt and moſt gene-
ral one is,
When the Annuity, Rent, or Penfion, the Time it is to con-
tinue, and the Rate of Interest to be allowed is known, to find
the Money that muſt be paid down immediately for the
fame.
EXAMPLE.
Suppoſe I rented a Farm of L. 450 per Annum, and my
Landlord wanting Money is willing I fhould buy off L. 250
per Annum for 21 Years, viz. the Term of my Leafe, and
allows me to difcompt 4 per Cent. per Annum, Simple In-
tereft, IVhat Šum must I pay down immediately, if the Pay-
ments were to be made half-yearly?
↑
RULE.
Multiply the Time, the Time lefs Unity and half the Ratio
continually together, and to their Product add the Time, multi-
ply this Sum by the Rent, or Penfion, &c. and this Product
must be a Dividend; then multiply the Ratio into the Time; and
to the Product add Unity, and this Sum must be a Divifor, by
which dividing the above Dividend, the Quotient will be the
Dreſent Worth, or Sum to be paid down immediately.
PREPARATION.
L. 125 the half-yearly Payment, or of L. 250.
42 the whole Time or Number of Payments, or twice
41 Ditto lefs Unity.
,02 the Ratio of the Rate for Year.
,01 half the Ratio of the Rate.
42 the Time.
,02 the Ratio.
,84 the Product:
1, to be added.
1,84 the Sum and Divifor.
21 Years.
42 the
Of ANNUITIES.
385
42 the whole Time.
41 Ditto lefs Unity.
42
168
1722 the Product.
,01 half the Ratio.
17,22 the Product.
42- the Time.
59,22 the Sum.
125 the Annuity.
29610
71064.
1,847402,50/the Product and Dividend.
736
42,5
368
57 0
55 2
(the
4023,0978, &c. the Quotient.
Or,
18, &c.
L. 4023: 1: 11½ the preſent Worth.
·
2. If the prefent Worth, the Time, and the Rate of In-
tereft are given, to find the Annuity.
The RULE is,
Multiply the Time, the Ratio of the Rate, and the preſent
Worth continually together, and to the laſt Product add the pre-
fent Worth; and that Sum must be a Dividend.
Then multiply the Time, the Time lefs Unity, and half the
Ratio of the Rate continually together, and to the laft Product
add the Time; and this Sum must be a Divifor to the above
Dividend, and the Quotient arifing from thefe two will be the
Annuity, Rent or Penfion fought after.
Ddd
EX
386
Of ANNUITIES.
EXAMPLE.
What Annuity, &c. may I have for L. 4023: 2: -, to be
paid down immediately, and to continue for 21 Years, payable
every half Year, allowing Simple Intereft, at 4 per Cent. per
Annum?
42 the Time.
,02 the Ratio of the Rate.
,84 the Product.
4023,1 the prefent Worth.
160924
321848
3379,404 the Product.
4023,1- the prefent Worth.
7402,504 the Sum and Dividend.
42 the Time.
41 Ditto lefs 1.
1722 the Product.
,01 half the Ratio of the Rate.
17,22 the Product.
42- the Time.
59,22 the Sum and Divifor.
59,22) 7402,504 (125 the Quotient, which is
the half-yearly Rent, or Penfion to be paid.
3. If the Rent, preſent Worth and Rate of Intereſt is
given, to find the Time of Continuance,
The RULE is,
To the prefent Worth add half the Annuity, and multiply
this Sum by the Ratio of the Rate, and ſubſtract the Product
from the Rent or Penfion, and divide the Remainder by the
Product of the Rent, multiply'd by the Ratio, and fquare the
Quotient;
2
Of ANNUITIES.
387
Quotient; alfo divide the prefent Worth by the Product of the
Rent, multiply'd by the Ratio, and to double the Quotient add
the Square abovementioned, and extract the Square Root of the
Total, and from the faid Root fubftract the Product of the
Rent, multiply'd by the Ratio, and the Remainder will be the
Time fought after.
Suppofe I pay down L.4023: 2: to receive L. 125 every
half-year, How many Years must I receive this Rent, or Pen-
fion, to be allowed 4 per Cent. per Annum, Simple Intereft,
for my Money?
L. 4023,1 the prefent Worth.
62,5 the of the yearly Payment.
4085,6 the Sum.
,02 the Ratio of the yearly Rate.
81,712 the Product.
125 the yearly Payment.
43,288 the Difference, which is alfo to be a
Dividend.
L. 125 the Annuity.
02 the Ratio of the Rate.
2,50 the Product, which is to be a common Divif.
2,5 | 43,288 (17,316 the Quotient.
17,316
299,843856 the Product called the Square.
2,5 | 4023,1 (1609,24 the Quotient.
2
3218,48 the double Quotient.
299,843856 the Square above.
3518,323856 the Sum.
D dd 2
59,316
388
Of ANNUITIES.
59,316 the Square Root.
17,316 the firft Quotient.
42,
the Difference, which is the Number
of Times of Payment, viz. 42
half Years, or 21 Years.
4. If the Annuity, prefent Worth, and Time are given,
to find the Rate of Intereft,
The RULE is,
Multiply the Annuity and Time together, and fubftract the
prefent Worth from that Product, and multiply the Remainder
by 200, and that Product will be a Dividend. Then add
the Annuity to the Double of the prefent Worth, and ſubſtract
the first Product above from this Sum, and multiply the Re-
mainder by the Time, and this Product will be a Divisor to the
above Dividend, and this Quotient will be the Rate of Inte-
reft fought for the Time of Payment.
EXAMPLE.
Suppoſe I give L. 4023,2 for an Annuity of L. 250 per
Annum to be paid half-yearly for 21 Years, What Rate of
Simple Intereft am I allowed?
L. 125 the half-yearly Payment.
42 the Number of Payments.
5250 the Product.
4023,1 the prefent Worth.
1226,9 the Remainder.
200
2453 80,0 the Product, which is to be a Divid.
L. 40232!
Of ANNUITIES.
389
L. 4023,1 the preſent Worth.
2
8046,2 double the prefent Worth.
125- the Annuity.
8171,2 the Sum.
5250- the firft Product.
2921,2 the Remainder.
42 the Time.
58424
116848
122690,4 the Product, which is to be the Divifor
to the above Dividend.
122690,4)245380
(L. 2. the Quotient, which is the
Rate of Interest for the Half-year.
N. B. There is in this, and the two foregoing Exam-
ples, a trifling Difference, fometimes wanting, and fometimes
abounding, which arifes from making the preſent worth
an Half-penny too much, which is done to fave the Num-
bers from running out very largely.
What has been already taught, is fufficient to anſwer
any thing that can be propofed, when the Rent, Penfion,
or Annuity is to begin immediately; but fuppofe I buy the
Reverfion of an Eſtate, Leaſe, &c. that is, I am not to be
in Poffeffion till 7, 8, 10, &c. Years hence: In this Cafe,
The RULE is,
Firſt, find what is the preſent IVorth of the propos'd Rent,
&c. for the Time you are to have it, as if it was now really
in your Poffeffion.
Then, to find what Sum forborn for the Time of the Re-
verfion would raiſe the aforefaid Value, you must multiply the
Time you are to be out of Poffeffion, by the Ratio of the Rate of
Intereft, and to that Product add 1; by this Sum divide the
prefent Worth, found by the Rule in Page 384, and the Quo
tient will be the Sum to be paid down immediately.
EX
390 Of Compound INTEREST.
EXAMPLE.
Suppoſe there is a Rent or Annuity of L. 250 per Annum,
to be paid Yearly, that is to continue for 21 Years, in which the
prefent Poffeffor has feven Years ſtill to come, What immediate
Money must I pay down for the remaining 14 Years, to be
allow'd 4 per Cent. per Annum, Simple Intereft, for the Money
the whole Time.
Here, by the Rule, Page 384, the prefent Worth of L.250
per Annum for 14 Years to come, at 4 per Cent. Simple In-
tereft, is L. 2826: 18: 5; and by the Rule above, the
prefent Worth of L. 2826: 18: 5 to be paid 7 Years
hence, difcompting Intereft at 4 per Cent. per Annum, is
L. 2208: 10: 8, which laft Sum is what muſt be paid down
immediately for the above Purchaſe.
Compound INTEREST.
This is fo called, becaufe the Intereft of a particular
Sum, not being paid, as it becomes due, is turned into a
Principal; fo that both the Principal and Intereft continually
increaſes in a certain Proportion, though the Quantity of
Time is always the fame: And as Questions in Simple Intereft
form'd a Series of Numbers in Arithmetical Progreffion, fo
Questions in Compound Interest form a Series of Numbers in
Geometrical Progreffion.
EXAMPLE.
What will L. 250 amount to in 9 Years, at Compound
Intereft, reckoning 5 per Cent. per Annum ?
The beft and moſt concife Method of doing fuch Quef-
tions as thefe, is by Logarithms; but as we have not treat-
ed of them, nor have we Room to exhibit Tables of them,
we ſhall fay nothing more upon that Head, but fhew you
how to anſwer fuch like Questions by common Numbers.
Now you may operate for the above Sum, or any other,
directly, as follows.
}
t
250
Of Compound 1 NTEREST.
391
250
5
12,50
The Amount at the Year's End 262,5
5
13,125
Ditto at 2 Years End.
275,625
5
13,78125
Ditto at 3
Years End
289,40625
5
14,4703125
Ditto at 4 Years End
303,8765625
5
15,193828125
Ditto at 5
Years End
319,070390625
5
15,95351953125
Ditto at 6 Years End
335,02391(015625
5
16,7511955
Ditto at 7 Years End
351,7751055
Ditto at 8 Years End
5
17,588755275
369,3638(60775.
5
18,468193
Ditto at 9 Years End
387,831993
But
392 Of Compound INTEREST.
But if you calculate the Amount of L. 1 for a Series of
Years, fuppofe 31, or any other Number, at feveral Rates,
their Amounts will be fo many fix'd Numbers for thoſe
particular Times and Rates, by which, multiplying any
given Sum, you will have the Amount of that Sum for that
Time at that Rate: Thus,
I,.
5
,05
I Year
1,05
5
,0525
2 Years 1,1025.
5
,055125
3 Years 1,157625
.5
,95788125
4 Years 1,2155(0625
5
,060775
5 Years 1,276275
5
L.1 for 9 Years 1,55133
the given Sum
250
7756650
310266
L. 387,83250
Or, L. 387,16,8 fere.
,06381375
6 Years 1,34008875
1
5
,0670044375
7 Years 1,4071(931875
5
,070355
ཅཱ་
8 Years
Of Compound 1 NTEREST. 393
8 Years 1,477455
5
07387275
9 Years 1,55132775 and fo on for as many Years
as you pleaſe.
I
Here the Amount of L. 1 for 9 Years is L. 1,55133 fere,
which multiplying by L. 250, the given Sum, and the Anfwer
is L. 387,8325, the fame with the other; and as theſe fix'd
Numbers are fo exceeding ufeful. Here follows a Table for
the Amount of L. I for 31 Years, for 3, 4 and 5 per Cent.
Years | 5 per Cent.
4 per Cent.
3 per Cent.
1
1,05
1,04
1,03
2
1,1025
1,0816
1,0609
3
1,157625
1,124864
1,092727
4
1,2155
1,169856
1,125509
46 79
5
1,276275 1,216651 1,159274
1,340088
1,265318 1,194052
1,4071
1,315931
1,229878
8
1,477455
1,368567 1,26677
9
1,55133
1,42331
1,304773
10
1,6289
1,480242
1,343915
11
1,71034
1,539452 1,384233
12
1,79586
1,60103
1,425761
13
1,885651
1,665071 1,468533
14
15
16
17
18
56 7∞
1,979934
2,07893
1,731674 1,512589
1,800941
1,557967
2,18288
1,87298
1,604706
2,29203
2,40663
1,9479 1,652847
2,025816
1,702432
19
2,52696 2,106848 1,753506
20
2,6533!
2,191123
1,706111
21
2,785965
2,278767
1,757294
22
2,925263
2,269918 1,811013
23
3.071526
2,360715 1,865343
NNNNN
24
3,2251
2,455143
1,921303
25
3.386355 2,553349
1,978942
26
3.555673 2,655483
2,038311
27
3.733456
2,761702
2,09946
28
3,92013 2,872171
2,162444
29
4,116136
2,987057
2.227217
30 4,321943 3,106539
2,294034
31 4,53804 3,2308
2,362854
Eee
The
ay
394 Of
Of Compound INTEREST.
The Uſe of theſe Tables is to find what any Sum at
3, 4, or 5 per Cent. per Annum, amounts to, being forborn
any Number of Years, not exceeding 31, at Compound
Intereſt.
EXAMPLE.
What is the Amount of L. 718: 13: 6 for 27 Years, at
4 per Cent. Compound Intereft?
2,761702 is the Amount of L. I.
718,675, the Sum given.
1984,766 the Product.
Or,
L. 1984: 15:34 the Amount; and fo for any other
Sum or Rate.
At 5 per Cent. per Annum Compound Intereft, What Sum
will amount to L.387: 16: 8 in 9 Years?
Here look out the Tabular Number for 9 Years, viz.
1,55133, and divide the given Sum L. 387,833, &c. by it,
and the Quotient L. 250 is the Sum fought.
Again, What Time will it require to raife L. 250 to
L. 387 16 8 at the Rate of L. 5 per Cent. per Annum,Com-
pound Intereft?
:
Here you muſt divide the biggeſt Sum by the leaſt, and
the Quotient 1,55133 will be the Amount of L. I; then look
how long L. I will be in raiſing that Sum, and againſt 9
Years, you will find it; fo that fhews, that in 9 Years L.250
will amount to the given Sum: But if you have no Tables,
and are defirous to find the Time Arithmetically, divide the
faid Quotient 1,55133 by 1,05 the Amount of L. I for I
Year, till the Remainder is 0, which here will be at the
9th Operation, which fhews that the Time is 9 Years, viz.
5
I
2
3
4
1,05) 1,55133 (1,47746 (1,40711 (1,34010 (1,27630
6
7
8 9
1,21552 (1,15764 (1,10251 (1,05 (1.
Again, If L. 250 will amount to L. 387: 16:8 in g
Years, What is the Rate of Intereſt?
Anf. 5 per Cent. per Annum.
Here
Of Compound INTERES T. 395
Here you muft, as before, divide the Amount by the Prin-
cipal, and with the Quotient, 1,55133, enter the Table at 9
Years, and under the Column of 5 per Cent. is the Number
fought; from whence may be concluded that 5 per Cent. is
the Rate: But if you would do it Arithmetically without
Tables, then extract the 9th Power out of 1,55133, and the
Root will be the Anſwer fought. Or, if you extract the Cube
Root out of this 1,55133, and again, the Cube Root of that
Root, you will have 1,05 come out for the laft Root, which
is the Amount of L. 1 for 1 Year, at 5 per Cent. per Ann.
Compound Intereſt.
But it may fo happen, that the Time may not always be
whole Years. In this Cafe Exactnefs requires a very tedious.
Proceſs, and inftead of multiplying by 1,05 for 5 per Cent.
1,04 for 4 per Gent. &c. you fhould find the Amount of
L.I for 1 Day, but not as in Page 370, where 'tis ,000137 for
5 per Cent. ,0001096, for 4 per Cent. ,000095617 for 3
per Cent. and,0000822 for 3 per Cent. but by Tables form'd
from I Day to 365, at the feveral Rates required, though
this is indeed a Matter of much greater Niceneſs than is
ever uſed; and fo I fhall forbear purfuing this Subject any
farther, and go to
RENTS, LEASES, PENSIONS, or ANNUI-
TIES, computed at Compound Intereſt.
You are to Note, that in thefe Sort of Purchaſes, Com-
pound Intereft is generally fuppofed to be allowed to make
good extra Trouble or Lofs by Tenants, &c.
EXAM P L E.
What is the Amount of L. 30 per Annum forborn 9 Years,
at 5 per Cent. per Annum, Compound Intereſt ?
Anfwer, L. 330: 16.
Here the Amount of L. 1 by the Table is 1,55133; this
multiply'd by the Annuity is L. 46,53990, from which ſub-
Atract the Annuity, and the Remainder is L. 16,5399; this
divided by the Ratio of the Rate gives L. 330: 16 for the
Amount, as will appear by the following Work at large.
L. 30, the firft Year's Income.
1,50 Intereft for the fecond Year of L. 30.
30 the fecond Year's Rent.
༑ *
61,5
396
Of Compound INTEREST.
61,5 due at the End of the ſecond Year.
33,075 the third Year's Rent and Intereſt of L. 61,5.
94,575 Total due the third Year.
34,72875 the fourth Year's Rent and Interest of L.94,575.
129,30375 Total due the fourth Year.
36,46519 the fifth Year's Rent and Intereft of L. 129,303.
165,76894 Total due the fifth Year.
38,28845 the fixth Year's Rent and Intereſt of L.165,768
204,05739 Total due the fixth Year.
40,20287 the 7th Year's Rent and Intereft of L.204,057
244,26026 Total due the feventh Year.
42,21303 the 8th Year's Rent and Intereſt of L.244,260,
286,47329 Total due the 8th Year.
44,32366 the 9th Year's Rent and Intereſt of L.286,473.
330,79695 Total due the 9th Year.
Or, L. 330: 15: 114, the fame with the above, within
3 Farthings.
Again, What Rent or Annuity, forborn 9 Years, will a-
mount to L. 330: 16, reckoning Compound Intereſt at 5 per
Cent. per Annum?
Here you muſt multiply L. 330: 16, by the Amount of
L. I for 1 Year, viz. L. 1,05, and from the Product
L. 347,340 ſubſtract the whole Amount L.330,8, and di-
vide the Remainder 16,54 by the Amount of L. 1. for the
whole Time, made lefs by Unity, viz.,55133, and the Quo-
tient L. 30 will be the Anfwer fought.
Here you are to Note, that if the Payments are to be
made Half-yearly or Quarterly, that though the Rate of In-
tereft, and the principal Sum or yearly Rent is the fame, the
Amount will not be the fame.
We might continue to go on to Difcompts, or the preſent
Value of Rents, Annuities, &c. but as the Proceſſes for theſe
Purpoſes are very prolix in common Arithmetick, and feveral
have performed them very accurately by the Logarithmick
Tables, and alfo by Tables calculated on purpoſe for thoſe,
who have not either the Leiſure or the Skill to work the
fame
Of Compound INTEREST.
397
fame Numerically, I forbear faying any more upon this
Head.
I ſhould now go on to the Rule called, The Rule of falſe
Pofition, fingle and double: But as that Rule is chiefly made
ufe of in fportive Queſtions, or fuch as feldom or never
occur in practical Buſineſs; and as a very ſmall Know-
ledge of Algebra furpaffes any thing doable by thoſe Rules,
I omit faying any thing of them; adviſing thoſe who have
ſtudied this Book thoroughly, and want ftill more Per-
fection in this Science, to apply themſelves to the Study
of Algebra, for their final Completion.
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