## The Epistle Dedicatory.

Weighty Employments, and to undertake a task too high and too hard for my feeble Pen. It shall therefore suffice at present to say, that amongst the Seven Liberal Arts, Arithmetick is none of the least; it is a Science in it self, so noble and of such general use, that no Nation can in a flourishing condition, subsist without it. The blessing of the Almighty upon my endeavors, having rendred me a poor Proficient therein, I would not willingly be found an unprofitable Servant, hiding this my Lords Talent in a Napkin; knowing that where he gives Power, he expects Improvement. I have therefore for the benefit of others (no man being born solely for himself) especially those committed to my care, employed some spare minutes in Penning the ensuing Treatise, containing in it the Fundamentals of the forementioned Art, and have assumed the liberty to Dedicate it

## The Epistle Dedicatory.

to your Worships; not as a thing worthy of your Acceptance or Patronage, but as a real Pledge of the Honor and Respect I bear you; a token of my ready and chearful submission to your Authority, and Testimony of my willingneſs to serve you in all things within the compafs of my Capacity. It was an excellent saying of Pittacus the Philosopher, That he would have no man undertake to Govern, till he had firſt learned to forgive. I question not in the least, but that you (Worthy Sirs) are all good Proficients in that his Lefſon, and ſhall therefore hope to obtain with the more facility, your pardon for my preſumption in dedicating theſe my weak Labors to you.

Gentlemen, That God who hath made you Governors, would bleſs you all with a ſpirit of Wisdom and Government, to carry on your work so here, that at

A 3

the

## The Epistle Dedicatory.

the Judgment of the great day, you may
be able and ready to render unto him a
comfortable account of those Talents
wherewith he did intrust you, and re-
ceive from him for reward, that blessed
sentence of, Well done good and
faithful Servants, enter into the
joy of your Lord, shall be the daily
 supplications of

Your Worships Faithful
and humble Servant

John Le Duke.

To

## To my singular and approved

Good Friend John Furly Senior, Gent.
late Alderman in Colchester.

Worthy Sir,

Ur days are all numbred by the
Almighty, and the sum there-
of kept from us as a Secret in
the Cabinet of his Privy
Council, not knowing there-
fore how soon your, or my Glas-
may be run, and Light extinguished, should I
not interest you in the first fruits of my Labors
in these Parts sent abroad into the World, In-
gratitude would seem legible in every Leaf
thereof, to the Inhabitants of this Town; it
being sufficiently known, that Mr. Stephen
Furly, your Son, was the first Instrument
(under God, who orders all our steps) in bring-
ing me hither, being here, your Self and your
Authority at that time (I coming in the year
of your last Majority) was my Patron and
Encourager, your House (though a meer
Stranger to you) my Harbor, your Acquaint-
ance my Friends, your Offspring my first
Schollars, by which (then) leading act of
yours, others were encouraged to commit their

A 4

Children

The Epistle Dedicatory.

Children to my Care and Direction. So that
in a short time (by the blessing of God upon my
endeavors) my School flourished.

Now as it was your Self and Family that
gave life to my first Labors in this Corpora-
tion, so as a Real and Signal Testimony to
the World, that I bear your favors in thank-
ful Remembrance, I do here dedicate them to
you (as in the precedent Pages, I have done
to the present Rulers of this place) craving
your Candid Acceptance thereof, as a pledge
of my Gratitude.

Sir, for a close I shall here beg of God, that
as from his good hand of Providence towards
you, you have in your now declining age, seaf-
sonably received a happy dismission from the
Office of a Magistrate of this Town, after near
Thirty years service, rendred to your Genera-
tion therein, that you may likewise receive
from him, for your Lot in the end of your days,
a Crown of Glory, Honor, Immortality, and
Life Eternal. So prays he, that unfeignedly
subscribes himself.

Worthy Sir,

Your most Faithful Servant

John Le Duke.

T9

TO THE
READER.

Friend,

Y design in ap-
pearing to the
World in Print,
is not out of vain
glory, I know
many excellent

Artists have been in this Work be-
fore me; from whose Labors I
have reaped some advantage, and
upon that account did I for some
years, restrain the ensuing Treatise
to the private use of my own
School, by framing divers Copies
to avoid the Printing it; but find-
ing those Copies by their long use
rendred imperfect and defective,
passing

## To the Reader.

passing through many hands, and
the labor too tedious to renew
them, having no spare time there-
to my self, I am constrained now
at last to print it for the service of
my own Disciples. I thought to
have published onely the several
Propositions (or Questions) of every
Rule without any Directions, be-
ing then perfwaded, that the Li-
ving Voice of the Master would be
a sufficient remedy to that defect,
in resolving the several Doubts
that the Schollars might meet
withal in answering those Question-
ons: But when the said Questions
were ready for the Preſs, and com-
municating them to some skilful
Friends, they advifed me to adde
some Directions to them, as being
very needful and profitable for
those that had no time to go to
School;

## To the Reader.

School, which advice I did not
reject, but, yet I was loth to con-
descend to it upon this account,
because of the variety of Operati-
ons that every Rule is subject unto,
and every man's ways do not agree
(each one thinking his way to be
the best) therefore I thought to
give the leſs offence (if some would
take any) by publishing onely
some Arithmetical Questions: But,
on the other ſide, confidering the
shortneſs and uncertainty of our
lives, and how beneficial Direc-
tions would be to those that are a-
ble to learn alone with a few Di-
rections onely: I have then re-
folved to follow the ſaid advice,
in adding ſuch Directions to every
Rule, as I thought needful to help
the meaneſt capacity, and not
onely ſo, but I have added further
at

To the Reader.

at the end of every Rule, a good quantity of *Familiar Questions* with their *Answers*, which plainly resolve the most material things of the said *Rules*, which I found by experience to be of great use for those that learn, and have learned this noble *Art of Arithmetick*. If any be offended and carp at my way and method, because of Non-conformity with them, and also of the quantity of the *Questions*.

I answer, 1. That my intention in publishing this *Treatise*, was not to give offence to any, but to do good, if not to all, at least to some, chusing rather to be accounted ignorant in some thing, then an unprofitable member.

And

To the Reader.

And 2. The multiplicity of *Questions* is rather a help then a hinderance; that which is needles to some, will be beneficial to others. An apt Schollar may pafs over many of them, whilst the other will be of use to Lads more flow of apprehension. I have used plainneſs of *Language*, and placed every *Question* in such an order, as I conceive may be to the best advantage of the *Learner*. Many of the said *Questions* have onely the *Facit* (or *Answer*) by it, that ſo the *Practitioner* may exercife himself the better in every *Rule*. Let my abſence from the *Preſs*,and thy charity excufe the groſſer, and thy Pen correſt the leſſer Faults. If this, the First Part, and firſt-fruits of my Labors in this ſoil, finde

To the Reader.

finde acceptance with thee, it shall
encourage me in due time, with
Gods assistance, to lend forth the
Second, and sufficiently compen-
fate the pains of thy Friend,

John Le Duke.

A

A TABLE shewing the Contents
of this Book.

Page.

W
Hat Arithmetick is
Numeration ibid.
How to expreß an extraordinary great
number

Questions of Numeration with their Answers 7

Numeration by Letters 14

The Explication of the chief Characters of this
Book

Table of English Money 16

Table of Averdupois Great Weight 18

Table of Averdupois Little Weight 20

Table of Troy Weight ibid.

Table of Dry Measures 21

Table of Liquid Measures 22

Table of Long Measures 23

Table of Time 24

Of things accounted by the Groß or Dozen ibid.

Addition Single 25

Addition of Numbers of divers Denominations 30

The Proof of Addition 44

Questions of Addition with their Answers 46

Subtraction Single 54

Subtraction of Numbers of divers Denominations 63

The Proof of Subtraction 71

Questions of Subtraction with their Answers 73

Multiplication 83

The Proof of Multiplication 97

Questions of Multiplication with their Answers 99

Division

The Contents,

<table><tr><td>Division</td><td>106</td></tr><tr><td>A second way of Division with fewer figures</td><td>143</td></tr><tr><td>The Proof of Division</td><td>150</td></tr><tr><td>Questions of Division with their Answers</td><td>152</td></tr><tr><td>Reduction by Multiplication</td><td>173</td></tr><tr><td>Reduction by Division</td><td>183</td></tr><tr><td>Reduction by Multiplication and Division</td><td>193</td></tr><tr><td>Questions of Reduction with their Answers</td><td>198</td></tr><tr><td>The Rule of Three direct</td><td>214</td></tr><tr><td>The Proof of the Rule of Three</td><td>221</td></tr><tr><td>How to work the Rule of Three by Abbreviation</td><td>232</td></tr><tr><td>Questions of the Rule of Three with their Answers</td><td>239</td></tr></table>

In the Appendix.

<table><thead><tr><td></td><td>Page.</td></tr></thead><tbody><tr><td>What a Fraction is</td><td>273</td></tr><tr><td>To know the worth of a Fraction</td><td>274</td></tr><tr><td>Introduction to the Rule of Practice</td><td>277</td></tr><tr><td>How to Multiply and Divide a number of several Denominations, without reducing it into its lesser Denomination</td><td>283</td></tr><tr><td>The First Table of Practice</td><td>287</td></tr><tr><td>The Second Table</td><td>288</td></tr><tr><td>The Third Table</td><td>289</td></tr><tr><td>The Fourth Table shewing the even Parts of a Tod of Wool in Pound</td><td>304</td></tr><tr><td>Brief Rules to know the worth of the Great and Small Hundred; and also the Wey of Cheese at a certain price the Pound.</td><td>307</td></tr><tr><td>What</td><td></td></tr></tbody></table>

(i)

What Arithmetick is.

Rithmetick is the Art of Numbring, and teacheth how to cast up exactly, all manner of Accompts, and confisfeth chiefly of Five parts; viz.

Numeration, Addition, Subtraction, Multiplication, and Division.

Numeration.

Numeration teacheth how to pronounce (or write) the right value of any number pro- pounded; and that by the help of Ten Cha- racters, whereof Nine are significant of them- selves, and the Tenth of it self, signifieth no- thing, but being joyned (or placed) on the right side (or towards the right hand) of one or more significant

## Of Numeration.

significant Figures, it increaseth ten times the value thereof; the Characters are these

1 2 3 4 5 6 7 8 9 0
eigh. nine. cipher.
two. three. four. five. six. seven. eight. nine. one.

After the knowledge of these Figures, it is requisite to know how to express, Two, three, or more of them together. For the attaining of which, it is needful to learn the following Table by heart.

### The Table of Numeration.

For the better understanding of this Table, and the use thereof, Consider what followeth.

1. The first place of the Table is called Unites, which shews, that any significant Figure in the first place (towards the right hand) of any number

## Of Numeration.

3

ber propounded, is worth but its simple value; that is to say, So many Unites, as 2. in that place, is but two. 3. is but three. 4. but four, &c.

2. The second place in the said Table is called Tens, to shew that the Figure which is in the second place of any number (towards the left hand) is worth ten times its simple value: So that 1. in the second place, is worth ten. 2. is worth twenty. 3. thirty. 4. forty, &c.

3. The third place is called Hundreds, shew-ing that a Figure in that place, is worth an hundred times as much as its simple value. Thus, 1. in the third place is a hundred. 2. in that place is two hundred. 3. is three hundred, &c.

4. The fourth place is called Thousands, shew-ing that every Figure in that place is worth so many thousands, as it contains unites. 1. is a thousand. 2. is two thousand. 3. is three thousand, &c.

5. The fifth place is the place of Tens of thousands, 1. in that place is worth ten thousand. 2. is twenty thousand. 3. thirty thousand, &c.

6. The fixth place is the place of Hundreds of thousands. 1. in that place is worth a hundred thousand. 2. two hundred thousand, &c.

7. The seventh place is the place of Millions, 1. in that place is a million. 2. is two millions. 3. three millions, &c.

8. The eighth place is the place of Tens of millions. 1. in that place is ten millions. 2. is twenty millions. 3. thirty millions, &c.

9. The ninth place is the place of Hundreds.

B 2

of

4

Of Numeration.

of millions. 1. in that place is a hundred millions. 2. is two hundred millions. 3. three hundred millions, &c.

10. The tenth place is the place of Thousands of millions, 1. in that place is worth a thousand millions. 2. is two thousand millions. 3. three thousand millions, &c.

Note, The Table of Numeration in most Cypher-Books, goes no higher then to the tenth place, which is Thousands of millions. Some go to the twelfth place thus, Ten thousands of millions, Hundred thousand of millions, which is indeed high enough; for to go yet higher, is more out of curiosity, then necessity: Nevertheless, if the Learner hath a minde to go yet higher, he may do it another way by the help of other Denominations (which I met withal, both in a Dutch and English Author) mentioned at the end of Numeration, fol. 7.

Here followeth (for the benefit of yong beginners) a quantity of Figures set two manner of ways. First, they are set in their right order from 1. to 150. And secondly, they are set confusedly to try the capacity of the Learner.

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: 51:
52: 53: 54: 55: 56: 57: 58: 59: 60: 61: 62: 63:
64: 65: 66: 67: 68: 69: 70: 71: 72: 73: 74: 75:
76: 77: 78: 79: 80: 81: 82: 83: 84: 85: 86: 87:
88:

5

Of Numeration.

88: 89: 90: 91: 92: 93: 94: 95: 96: 97: 98: 99:
100: 101: 102: 103: 104: 105: 106: 107: 108:
109: 110: 111: 112: 113: 114: 115: 116: 117:
118: 119: 120: 121: 122: 123: 124: 125: 126:
127: 128: 129: 130: 131: 132: 133: 134: 135:
136: 137: 138: 139: 140: 141: 142: 143: 144:
145: 146: 147: 148: 149: 150: 156: 170: 180:
190: 200: 300: 400: 500: 600: 700: 800: 900:
1000: 2000: 3000: 4000: 5000: 10000:

1: 7: 4: 2: 0: 8: 3: 9: 5: 6: 10: 01: 17: 13: 11:
14: 18: 15: 19: 12: 16: 20: 02: 21: 12: 27: 30:
03: 31: 13: 32: 23: 37: 40: 04: 41: 14: 49: 50:
05: 52: 25: 54: 45: 60: 06: 63: 36: 65: 56: 70:
07: 76: 67: 80: 08: 81: 18: 87: 90: 09: 93: 39:
98: 89: 100: 001: 200: 002: 201: 210: 012: 010:
211: 300: 003: 301: 310: 031: 400: 004: 403:
024: 443: 500: 005: 510: 600: 700: 006: 007:
610: 017: 712: 800: 900: 1000: 0001: 1001:
1010: 0101: 0010: 1101: 1111: 2222: 3333:
1007: 2001: 3007: 4009: 38: 47: 67: 74: 92:
127: 317: 804: 913: 1342: 1087: 4216: 3000:
0000: 001000: 7135: 1910: 3444: 12345:
10101: 001: 72176: 10100: 47378: 123456:
300111: 1234567: 12345678: 123456789:
800: 400: 8004003000: 131672:

654987654: Read
Six hundred fifty four mil-
lions nine hundred eighty-
seven thousand fix hundred
fifty four.

B 3

300300300:

6

Of Numeration.

300300300: 100000100: 203000001:1111111:
401401401: 123123123: 1234567890:
1010101010: 00000000001: 3100001701:
3410010173: 1000000001: 14831768158:
649782312594: Read
Six hundred forty nine
thousand seven hundred eighty two mil-
lions, three hundred
twelve thousand five
hundred ninety four.

123456789123: 700010070030:

If the Learner hath a minde to number a great
quantity of Figures, let them be never so many,
that may easilie be done with other Denominati-
ons, not so troublesome to the Memory as the
former, being onely acquainted with the fix first
places in the Table of Numeration, as follow-
eth.

Suppofe that a number confifting of thirty eight
figures or places be propounded to be numbred,
make a Partition between every fix figures, be-
ginning at the Unites, and mark how many Par-
titions you have, as followeth.

Note,

Of Numeration.

7

Note, The first fix figures are termed U-nites, and have no other. Denomination then is taught in the Table; but the second Partition is called The first degree of Great thousand ; the third Partition is called The second degree of Great thousand, &c. as you fee in the Margin. Now pronounce these thirty eight Figures thus.

Twenty of the Sixth
degree of Great thou-
land : One hundred
twenty three thousand
four hundred fifty fix of
the Fifth degree of Great
thousand: Seven hundred
eighty seven thousand
one hundred seventy one
of the Fourth degree of
Great thousand: Seven
thousand one hundred
eighty four of the Third
degree of Great thou-
land: Five hundred fix-
ty thousand three hun-

dred forty of the Second degree of Great thou-
land : Seven hundred eighteen thousand seven

<table>
  <tbody>
    <tr>
      <td>20</td>
      <td>Sixth degree of Great thousand.</td>
    </tr>
    <tr>
      <td>123456</td>
      <td>Fifth degree of Great thousand.</td>
    </tr>
    <tr>
      <td>787171</td>
      <td>Fourth degree of Great thousand.</td>
    </tr>
    <tr>
      <td>007184</td>
      <td>Third degree of Great thousand.</td>
    </tr>
    <tr>
      <td>560340</td>
      <td>Second degree of Great thousand.</td>
    </tr>
    <tr>
      <td>718723</td>
      <td>First degree of Great thousand.</td>
    </tr>
    <tr>
      <td>654321</td>
      <td>Unites.</td>
    </tr>
  </tbody>
</table>

B 4

hundred

8

Of Numeration.

hundred twenty three of the first degree of Great thousand: Six hundred fifty four thousand three hundred twenty one.

If any other greater number were propounded to pronounce, it may be easily done this way; therefore I shall adde here no more examples of this kinde; yet before I conclude Numeration, I thought it not amiss to insert here some Questions with their Answers, about this first part of Arithmetick, to help the Learner to understand what hath been said; to the end, he may be well grounded in the beginning of this Art.

## 1. Queftion.

What is Numeration?

Answ. Numeration is a Numbring, and signifieth the Explication of any Number, and the Pronounciation of the Figures.

Queft. 2. How many sorts of Figures (or Characters) are there to express any Number?

Answ. Ten; nine of which are significant, and the tenth of it self signifieth nothing.

Queft. 3. What is their shape and value, or worth?

Answ. The simple value of each Figure with their shape, is as followeth. 1. One. 2. Two. 3. Three. 4. Four. 5. Five. 6. Six. 7. Seven. 8. Eight. 9. Nine. 0. a Cipher or nothing.

Que. 4. What do you understand by a significant Figure or Character?

Answ. It is a Figure which is worth or significance

9

Of Numeration.

fieth some value in it self; as 1. is one, 2. is two, &c.

Queft. 5. How is the value of any Number propounded to be known?

Answ. The worth of any Number is to be known by observing the place of every Figure.

Queft. 6. Where must you begin to reckon the places of every Figure?

Answ. From the right hand towards the left.

Queft. 7. In what manner?

Answ. In the same manner as the Table of Numeration teacheth; to wit, Unites, Tens, Hundreds, Thousands, &c.

Queft. 8. What advantage have you by the Table of Numeration?

Answ. I know by that means the worth of every Figure.

Queft. 9. What is the worth of every Figure in the first place?

Answ. Every Figure in the first place, is worth but its simple value, without any increase: 1. In that place, is but one. 2. but two. 3. three, &c.

Queft. 10. What is the second place?

Answ. The second place is the place of Tens. 1. In that place is ten. 2. is twenty. 3. is thirty, &c.

Queft. 11. What is the value of a Figure of four in the third place?

Answ. A 4. in the third place, is worth four hundred.

Queft. 12. What is the value of a Figure of seven in the fifth place.

Answ.

10

Of Numeration.

Answ. Seventy thoufand, becaufe the fifth place is the place of Tens of thousands, and Seven times ten thoufand is Seventy thoufand, &c.

Queft. 13. When you are to pronounce any Number, where do you begin to read; from the left hand towards the right, or from the right towards the left?

Answ. In obferving the Number of Places in Saying Unites, Tens, Hundreds, Thousands, &c. I begin from the right hand towards the left; but in pronouncing the value of the number, I read from the left hand towards the right.

Queft. 14. What is the value of this Number 3245?

Answ. Three thoufand two hundred forty five.

Queft. 15. How do you prove that by the Table?

Answ. I perceive (in obferving the places) that 5 is in the place of Unites, 4 in the place of Tens, 2 in the place of Hundreds, and 3 in the place of Thousands. Therefore I conclude, that the Figure of 3 is three thoufand, the 2 is two hundred, the 4 is forty, and the 5 is but five; the whole number is then Three thoufand two hundred forty five.

Queft. 16. What is the worth of an 0 or Cipher?

Answ. An 0 or Cipher is worth nothing of it self.

Queft. 17. What use do you then make of it?

Answ.

11

Of Numeration.

Answ. It SERVEth to increafe the value of the significant Figures.

Queft. 18. Shew an example.

Answ. If a Cipher be annexed to 1. (on the right hand) the value thereof is ten, thus 10. If it be annexed to a 2, it is twenty, thus 20. If to 3, it is thirty, or 30, &c.

Queft. 19. But if the Cipher ftood on the left side (or towards the left hand) of the Significans Figure, what increafe is there?

Answ. A Cipher, or many Ciphers on the left hand, doth not increafe any Figure, it doth it onely being on the right; as by example, 30. is thirty, but 03. is but three: Again, 300. is three hundred, but 003. is but three: 3000. is three thoufand, and 0003. is but three ftill. Yea, fhould I adde never so many Ciphers towards the left hand, it would be ftill the fame as 000000003. is but three in all; but if they were on the right hand thus 300000000. then it is Three hundred millions.

Queft. 20. How will you fet down Four thoufand and two?

Answ. A Figure of Four, two Ciphers, and a two, thus 4002.

Queft. 21. Why do you put two Ciphers between the 4. and the 2?

Answ. It is to make up the Number of Places.

Queft. 22. What do you mean by that?

Answ. I fet a Cipher (or Ciphers) in thofe Places where a Number (or Numbers) is left out; as in Four thoufand and two, there is neither

Hundreds

12

Of Numeration.

Hundreds nor Tens mentioned; therefore I set
Ciphers in their places to make up four places.

Queft. 23. How will you set down Ten millions
ten thousand and ten.

Answ. I must set 1. in the place of Tens of mil-
lions, (which is the eighth place) 1. in the place of
Tens of thousands (which is the fifth place) and
1. in the place of tens (which is the second place)
thus 10010010.

Q. 24. How will you set Four millions fix hundred and ninety nine thousand five hundred and seven?

Answ. Thus 4699507.

Queft. 25. How is Eleven thousand eleven hundred and eleven to be written.

Answ. This is improperly propounded, for
Eleven hundred is a Thousand and one hundred;
therefore I must set down Twelve thousand one
hundred and eleven, thus 12111.

Queft. 26. How will you express such a great
number as this 454567817?

Answ. Before the pronouncing of a great
number, I do first observe the places, and upon
every third Figure (or Character) from the right
towards the left, I set a point (or prick) thus
454567817.

Queft. 27. What advantage is there in making
such marks?

Answ. By those marks I perceive what Figures
does belong to the Millions, to the Thousands,
and to the lower places, thus: The first three
Figures are 817. which is Eight hundred seven-
teen, the next three are 567. which being in the
place

13

Of Numeration.

place of Thousands, is, Five hundred sixty seven
thousand, and the next three are 454. which be-
ing in the place of Millions, is Four hundred fifty
four millions.

Queft. 28. How do you pronounce all the Par-
titions together?

Answ. Four hundred fifty four millions, Five
hundred sixty seven thousand, Eight hundred fe-
venteen.

Queft. 29. How will you pronounce (without
confusion) a number consisting of 50. 100. or 1000.
Figures?

Answ. Though such things be needles, yet
it may be easily done without confusion or great
trouble.

Queft. 30. How?

Answ. By parting every fixth Figure; the
first fix (on the right hand) being counted for
Unites; the second Partition for the first degree
of Great thousands; the third Partition for the
second degree of Great thousands, &c. as is
mentioned at large in Fol. 7. To which I refer
you for brevity sake.

Queft. 31. How many sorts of Number do you
finde ?

Answ. Three sorts. 1. A Digit. 2. An Article. 3. A Compound or Mixt.

Queft. 32. What is a Digit?

Answ. A Digit is any Number under Ten, as
9. 8. 7. 6. 5. 4. 3. 2. 1.

Queft. 33. What is an Article?

Answ. An Article is any Number which hath

24

## Of Time.

Time confisfeth of Years, Moneths, Weeks, Days, Hours, and Minutes, and is reckoned as followeth.

Note, A year is commonly reckoned for 365 days, but the Leap-year (which hapneth once in four years) contains 366 days. In ordinary computation of time, the whole year is commonly divided into Twelve Moneths, and each Moneth (one with another) for Thirty days.

### Of things accounted by the Groſs or Dozen.

### Parchment.

### Paper.

### Herring.

## Of Addition.

### Herring.

## Addition.

Addition teacheth to adde or gather, two, three or more Sums (or Numbers) in one, either of one or divers denominations.

In adding Numbers of one denomination or name, obſerve what followeth.

1. Set down the Numbers to be added in such order, that every figure of the fame degree (or place) may be directly one under the other, that is to ſay, Unites under Unites, Tens under Tens, Hundreds under Hundreds, &c.

2. Having placed the Numbers in their right places, draw a line under them, and begin at the Unites, at the lower moſt figure going upwards. If the whole rank (or row) of Unites (or any other, as the rank of Tens, Hundreds, &c.) be leſs then Ten, fet down under that row so much as you ſinde; that is, if the rank of Unites amounts to nine, fet nine under that rank; if the the rank of tens be as much or leſs, fet down under that rank what you ſinde; but if you ſinde in any rank nothing but ciphers, then fet a cipher under

25

26

## Of Addition.

under that rank, as you may see in the first, second, and third example of this Rule.

3. When the sums of the figures of any rank is above nine, as 10. 20. 100. or 16. 24. 135, &c. set down under the same rank that figure (or character) which is in the place of unites in the said sum; and for every ten, keep an unite in your minde to be added to the next row towards the left hand; that is to say, If the figures of the first row be 18. set down 8 under that row, and carry 1 in minde; if it be 24. set 4. and carry 2, &c. But if you finde 10. set a cipher, and carry 1, if 20. set a cipher and carry 2; if 100. set a cipher and carry 10, &c. as in the fourth example.

### Another way easier for Beginners to understand.

Having set down all the Numbers to be added in their right places, cast up every row, beginning at the unites (as is said before) and set the sum of the first row on the right side of the Addition, and separate the place of the unites with a Comma, (or with two pricks :) Now observe for a general rule, to set down always under that row, that figure (or character) so cut and carry the other figure (or figures which are on the left side of the pricks) to the next row, towards the left hand : As for example, If the sum of the first row amounts to 39. cut the figure 9 with two pricks, thus 3:9. then set down 9. and carry 3. to the

27

## Of Addition.

the second row. If the sum of the second row amount unto 92. cut the 2 thus 9:2. setting down 2 under that row, and carry 9 to the third row. Further, if the sum of the third row (or any other row) did amount unto 147. do as before, cutting 7 thus 14:7. then set down 7. and carry 14. to the fourth row; but if the third row be the last, then set down 14. and the Addition is finished.

Note, The setting down of the sum of every row, is of a double advantage for the Learner; for (besides what hath been said) he may know at any time (in making an Addition of many figures and ranks) how much he hath carried to every row; and so, if he commit an error in the midst of his work, he shall not need to begin again from the beginning, which is somewhat tedious for impatient heads.

### 1. Example.

If 11 L (or pounds) 10: 23: 21: 13: and 20 L. be added together, what is the whole sum? facit 98 pounds.

2. Item, in adding 2: 10: 501: 2121: 33010: and 422342. together, what is the total? facit 457986.

3. Item, if 100000: 20000: 3000: 1000: 2000: and 300000. be added together, what amounts the whole sum unto? facit 426000.

1) I.F.

28

Of Addition.

<table><tr><td>1) 11</td><td>2)</td><td>2</td><td>3) 100000</td></tr><tr><td>10</td><td></td><td>10</td><td>20000</td></tr><tr><td>23</td><td></td><td>501</td><td>3000</td></tr><tr><td>21</td><td></td><td>2121</td><td>1000</td></tr><tr><td>13</td><td></td><td>33010</td><td>2000</td></tr><tr><td>20</td><td></td><td>422342</td><td>300000</td></tr><tr><td>—</td><td></td><td>—</td><td>—</td></tr><tr><td>98</td><td></td><td>457986</td><td>426000</td></tr></table>

4. Item, adde 36: 138: 927: 1486: 3845: and 8069: facit 14501.

I begin at the first row (to-
wards the right hand) at the
lower figure, saying, 9 and 5
is 14, and 6 is 20, and 7 is
27, and 8 is 35, and 6 is 41.
I set 41 on the right side of the
Addition, and make a Partition
between with two pricks, thus
4:1. I set down the 1. under
that row, and carry 4 to the second row (which is
the row of tens) saying, 4 (which I carried) and
6 is 10, and 4 is 14, and 8 is 22, and 2 is 24, and
3 is 27, and 3 is 30. I set 30 on the right side of
the Addition as before, separating it thus 3:0. I
set a cipher under the second row, and carry 3 to
the third row, saying, 3 that I carried and 0 is 3,
and 8 is 11, and 4 is 15, and 9 is 24, and 1 is 25.
I set 25 on the side as before, parting the figures
thus 2:5. I set down 5 under the third row, and
carry 2 to the fourth row, saying, 2 that I carried
and 8 is 10, and 3 is 13, and 1 is 14; which 14
I do

29

Of Addition.

I do not set down on the side of the Addition as
before, because it is the last row, I set then 4 un-
der the fourth row (which is the place of thou-
fands) and the 1. a place further (which is the
place of tens of thoufands) the whole sum is then
14501. as you may see in the Margin (on the
other side) follow the same order in the following
Additions.

5. Item, adde 847 pounds, 385: 434: 967: 654:
302: 623: and 123 pounds, facit 4335 pounds.

6. Item, adde 3234 1b 9632: 8246: 7870:
8524: and 4990 1b ? facit 42496 1b.

7. Item, adde 407 yards, 4078: 26: 100:
1487: 78: 3878: 146871: and 10 ? facit
156935 yards.

8. Item, adde 37165440 foot, 4708734:
379831: 35683: 2574: 813: 70: and 12345679:
foot ? facit 54638824 foot.

9. Item, adde 123456789 ells, 18: 647:
4894: 30564: 987651: 6046792: 64871404:
and 170643106 ells, what is the whole sum ?
facit. 366041865 ells.

10. Item, adde fix bushels of Corn 28: 937:
4070: 37185: 821037: 3871309: 20308712:
1234567: 891409: 37071: 6702: 217: 10:
and 3. facit. 27213263 bushels.

11. Item, adde 684329819 L. 13407389:
1834969: 606489: 13679: 560849: 6896719:
40609139: 693714569: 34678919: 6382419:
971469: 10719: 938109: 8764969: 91371019:
and 654321899 L. facit. 2239422143 L.

12. Item, adde 6160018161 L. 317684671:

50143170:

30

Of Addition.

50143170: 1217163: 810718: 84646: 8161:
986: 67: 9: 18: 671: 6106: 17016: 8910717:
671076716: 3407168710: and 1000000000:
facit 11617147706.

684329819 15:3 6160018161 7:6
:13407389 8:4 :317684671 7:0
:: 1834969 10:1 ::50143170 6:7
:::606489 9:2 ::::1217163 6:7
::: 13679 7:2 :::::810718 4:4
::: 569849 9:4 ::::: 84646 3:1
:: 6896719 4:9 ::::: 8161 2:7
:40609139 4:3 ::::: 986 2:1
693714569 ::::: 67 1:6
:34678919 ::::: 9
:: 6382419 ::::: 18
::: 971469 ::::: 671
::: 10719 ::::: 6106
::: 938109 ::::: 17016
:: 8764969 :::: 8910717
:91371019 :671076716
654321899 3407168710
_________ 1000000000
2239422143
_________ 11617147706

Addition of Numbers of divers Denominations.

1. IN Addition of Numbers of divers Denominations (or names) place the numbers of the same name, one directly under another : As for

Of Addition.

31

for example, In the Addition of Pounds, Shillings, Pence, Farthings, set down Pounds under Pounds, Shillings under Shillings, Pence under Pence, Farthings under Farthings. In the Addition of Weight, as Pounds, Ounces, Drams, Scruples, Grains, set down Pounds under Pounds, Ounces under Ounces, Drams under Drams, Scruples under Scruples, Grains under Grains. Observe the same order in the Additions of other names, as Dry, Long, and Liquid Measures, &c. and draw a line under them.

2. In this Addition observe well the last De-
nomination of the Numbers to be added (which is
the place where you must begin,) and how many
unites of them are contained in an unite of the
next greater Denomination. Now observe for a
general rule, that so many unites of the greater,
as you finde in the quantity of the letter, so
many unites you must carry into the place of the
said greater Denomination. If you finde any re-
mainder (or odde number) set it down, if not,
set a cipher in that place. In adding the next greater
Denomination, proceed in the same manner until
you come to the last and greatest Denomination.
As for example, In the Addition of Pounds, Shil-
lings, Pence, Farthings, begin at the Farthings,
and see how many Pence you can finde in the row
of Farthings; and if there be any odde Farthings
remaining, set them down under the row of Far-
things under the line, and carry the quantity of
Pence so found, into the row of Pence.

Secondly, So many Shillings as you finde in
the

32

Of Addition.

the row of Pence, carry them to the row of Shillings, and set the odde Pence (if there be any) under the Pence.

Thirdly, So many Angels as you finde in the row of Shillings, carry them to the place of Angels, and set the odde Shillings (if there be any) under the Shillings.

Lastly, So many Pounds as you finde in the row of Angels, carry them to the Pounds in the place of the Unites, and proceed as hath been taught in the *Addition Simple*, setting down the odde Angel. Observe also, that if all the Farthings (in the row of Farthings) do not exceed Three farthings; all the Pence (in the row of Pence) do not exceed Eleven pence; all the Shillings (in the place of Shillings) do not exceed Nine Shillings; all the Angels (in the place of Angels) do not exceed one Angel, then you cannot carry any thing into the place of Pence, Shillings, Angels, or Pounds, but set down every quantity under its proper Denomination, as in the following examples.

13. Adde the following sums together, viz.
3 L 2 β, 2 L 0 β, 1 L 0 β, 7 L 3 β,
4 L 1 β, 6 L 0 β, 0 L 2 β, *facit* 23 L
8 β.

14. Item, adde 12 L 2 β 2 9, 10 L 1 β
2 9, 11 L 2 β 3 9, 21 L 2 β 1 9, 12 L
0 β 1 9, 10 L 10 β 0 9, and 13 L 1 β
1 9, *facit* 89 L 18 β 10 9.

15. Item, adde 12 L 1 β 2 9 1 q, 34 L
2 β 1 9 0 q, 32 L 1 β 3 9 1 q, 61 L
2 β

33

Of Addition.

2 β 0 9 0 q, 27 L 10 β 1 9 1 q, 12 L
1 β 0 9 0 q, 13 L 0 β 4 9 0 q, *facit*
191 L 17 β 11 9 3 q.

<table><thead><tr><th>L β</th><th>L β 9</th><th>L β 9 q</th></tr></thead><tbody><tr><td>3—2</td><td>12-02-2</td><td>12-01-2-1</td></tr><tr><td>2—0</td><td>10-01-2</td><td>34-02-1-0</td></tr><tr><td>1—0</td><td>11-02-3</td><td>32-01-3-1</td></tr><tr><td>7—3</td><td>21-02-1</td><td>61-02-0-0</td></tr><tr><td>4—1</td><td>12-00-1</td><td>27-10-1-1</td></tr><tr><td>6—0</td><td>10-10-0</td><td>12-01-0-0</td></tr><tr><td>0—2</td><td>13-01-1</td><td>13-00-4-0</td></tr></tbody><tfoot><tr><td>23—8</td><td>89-18-10</td><td>191-17-11-3</td></tr></tfoot></table>

*Note*, when you finde in the row of Farthings, Pence, Shillings, and Angels, that their numbers exceed 3. 11. 9 & 1. then for every four Farthings, carry a penny to the place of Pence; for every Twelve pence, carry a Shilling to the place of Shillings; for every Ten Shillings, carry an Angel to the place of Angels; and for every two Angels, carry a Pound to the place of Pounds, as in the following example.

16. Adde 26 L 12 β 6 9 2 q, 96 L
19 β 8 9 3 q, 45 L 14 β 8 9 2 q, 162 L
18 β 9 9 3 q, and 10 L 9 β 2 9 3 q,
*facit* 442 L 15 β 0 9 1 q.

D

L 126

34

Of Addition.

L β § q
126-12-6-2
96-19-8-3
45-14-8-2
162-18-9-3
10-09-2-3
442-15-0-1

I begin with the least Denomination, which is
Farthings, faying, 3 and 3 is 6, and 2 is 8, and
3 is 11, and 2 is 13 q, which is 3 9 1 q. I set
down 1 q under the q, and carry 3 9 to the place
of 9, faying, 3 I carried and 2 is 5, and 9 is 14,
and 8 is 22, and 8 is 30, and 6 is 36 9, which is
3 8 just: Therefore I set a cipher under the 9,
and carry 3 8, faying, 3 and 9 is 12, and 8 is
20, and 4 is 24, and 9 is 33, and 2 is 35 8,
which is 3 Angels and 5 8; I set 5 8 under the
8, and carry 3 Angels to the place of Angels,
faying, 3 and 0 is 3, and 1 is 4, and 1 is 5, and
1 is 6, and 1 is 7 Angels, which make 3 L 10 8.
I set an Angel to the place of Angels, and carry
3 L to the place of L, faying, 3 and 0 is 3,
and 2 is 5, and 5 is 10, and 6 is 16, and 6 is
22 L. I set 2 under the place of Unites, and
carry 2 tens to the place of Tens, faying, 2 I car-
ried and 1 is 3, and 6 is 9, and 4 is 13, and 9
is 22, and 2 is 24 L. I set 4, and carry 2 to
the next row, faying, 2 I carried and 1 is 3, and
1 is 4; which 4 I set in the place of Hundreds,
which

35

Of Addition.

which is the third place: The whole sum is
$442 \times 15 \beta \times 9 \times 1q.$

Note, There is another way very eaſie for
yong Learners, which is to prick at every four Far-
things, at every Twelve pence, and at every
Twenty ſhillings, fetting down the reſt of each
Denomination as is ſaid before. To make this
more plain, I ſhall fet down again the former
ſums to be added this way as followeth.

I begin at the q, faying, 3 and 3 is 6 q, which
is 1 9 2 q. I make a prick for the 9, and carry
2 q to the next figure (which is 2 q) faying 2
and 2 is 4 q, or 1 9, I make a prick again, and
carry nothing. Further, I fay 3 and 2 is 5 q, or
1 9 1 q, I make a prick for the 9, and fet the
odde q under the q. Now I muſt carry 3 9 into
the place of 9, becaufe I finde three pricks in
the row of q. I do the like in the row of 9, prick-
ing at every 1 2, and I finde three pricks, which
make 3 8 ; and having no odde 9 over, I fet a
cipher in the place of 9, and carry 3 8 to the
row of 8. I prick likewise at every 20 in the
Shillings

D 2

D 2

Shillings

36

Of Addition.

Shillings and Angels (counting every Angel for 10 β) and in so doing I finde three pricks and 15 β over; the odde 15 β I fet under the β, and carry 3 L (signified by the three pricks) into the place of L, and I proceed as before.

Note, In fetting down or adding Pounds, Shillings, Pence, Farthings, observe, that the Shillings (in the place of β) muft not exceed 19. because 20 β make a Pound. The Pence (in the place of 9) muft not exceed 11, because 12 9 makes a Shilling. The Farthings (in the place of q) muft not exceed 3, because 4q makes a penny. Now to put this observation into the minde of the Learners, the Twenty three and Twenty fourth Propofitions of this Rule, are irregularly proppounded, to teach them to fet fums in a good order; for young beginners are apt to mistake in such things.

17. Adde 64 L 13 β 6 9, 48 L 19 β 4 9, 60 L 17 β 9 9, 7 L 14 β 7 9, 14 L 10 β 4 9, and 32 L 14 β 3 9, *facit* 229 L 9 β 9 9.

18. Item, adde 840 L 17 β 4 9 3 q, 347 L 14 β 5 9 0 q, 847 L 15 β 6 9 3 q, 980 L 10 β 0 9 2 q, 716 L 17 β 7 9 0 q, and 100 L 14 β 3 9 1 q, *facit* 3834 L 9 β 3 9 1 q.

19. Item, adde 6 9 2 q, 8 9 3 q, 10 9 0 q, 11 9 2 q, 9 9 1 q, 8 9 3 q, 11 9 1 q, 10 9 2 q, 9 9 1 q, 10 9 3 q, 2 9 0 q, and 5 9 3 q, *facit* 8 β 7 9 3 q.

20. Item, adde 10 β 6 9 2 q, 10 β 10 9 1 q,

37

Of Addition.

1 q, 17 β 6 9 3 q, 18 β 3 9 2 q, 13 β 10 9 1 q, 15 β 0 9 0 q, 19 β 9 9 3 q, and 14 β 1 9 0 q, *facit* 6 L 0 β 0 9 0 q.

21. Item, adde 1 L 10 β 11 9 3 q, 6 L 12 β 9 9 2 q, 0 L 19 β 10 9 3 q, 0 L 13 β 8 9 2 q, 1 L 16 β 7 9 1 q, 0 L 2 β 3 9 0 q, 2 L 12 β 11 9 3 q, 0 L 13 β 10 9 2 q, 1 L 15 β 9 9 3 q, 0 L 17 β 8 9 2 q, 0 L 11 β 10 9 3 q, and 0 L 14 β 11 9 3 q, *facit* 19 L 3 β 5 9 3 q.

22. Item, adde 3 L 13 β 3 9 3 q, 1 L 5 β 2 9 1 q, 1 L 17 β 2 9 3 q, 6 L 3 β 9 9 3 q, 11 L 0 β 0 9 0 q, and 100 L 17 β 7 9 3 q, *facit* 124 L 17 β 2 9 1 q.

23. Item, adde 13 L 13 β 3 9 3 q, 25 β 9 q, 36 β 14 9 3 q, 14 L 43 β 7 9 11 q, 20 L 19 β 12 9, and 200 L 16 β 18 9 7 q, *facit* 254 L 17 β 2 9 1 q.

24. Item, adde 9 L 1 β 2 q, 32 β 12 q, 10 Groats, 3 L 55 β 11 9 7 q, 1 Mark 2 9 3 q, 1 Noble 8 9 6 q, 10 L 2 Angels, 4 Nobles 3 β 3 9 4 q, 16 Groats 16 9 16 q, 6 L 29 β 15 9 9 q, and 99 L 19 β 11 9 4 q, *facit* 138 L 1 β 1 9 3 q.

25. Item, adde 12 Marks 1 Noble 4 β 6 9, 16 Marks 1 Nob. 6 β 2 9, 74 Mar. 0 Nob. 5 β 8 9, 18 Mar. 1 Nob. 4 β 9 9, and 20 Mar. 1 Nob. 0 β 10 9. What is the whole fum in Marks? *facit* 93 Marks 1 Noble 1 β 11 9.

D 3

Averdupois

38

## Of Addition.

### Averdupois Great Weight.

For every Twenty eight pounds, carry a Quarter to the place of Quarters: For every Four quarters, carry a Hundred to the place of C.

*Note, A pound is commonly the least quantity that is taken notice of in Averdupois Groß weight.*

26. Adde 7 C 2 qrs 27 fb, 4 C 2 qrs 17 fb, 0 C 3 qrs 17 fb, 4 C 2 qrs 6 fb, and 9 C 3 qrs 19 fb, *facit* 27 C 3 qrs 2 fb.

27. Item, adde 47 C 3 qrs 23 fb, 17 C 2 qrs 16 fb, 97 C 2 qrs 10 fb, 16 C 3 qrs 18 fb, and 84 C 0 qrs 12 fb, *facit* 264 C 0 qrs 23 fb.

### Averdupois Subtile.

For every Twenty Grains, carry a Scruple to the place of Scruples: For every three Scruples, carry a dram to the Place of Drams; for every eight Drams carry an Ounce to the place of Ounces; and for every Sixteen Ounces, carry one pound to the place of Pounds.

28. Adde 34 fb 15 ounces 3 drams 1 scruple, 12 fb 7 ounce. 7 dra. 2 scrup. 21 fb 2 ounce. 1 dra. 1 scrup. 13 fb 12 ounce. 5 dra. 1 scrup. 11 fb 7 ounce. 2 dra. 2 scrup. and 30 fb 10 ounce. 1 dra. 1 scrup. *facit* 124 fb 7 ounce. 5 dra. 2 scrup.

29. Item, adde 487 fb 12 ounce. 6 dra. 2 scrup.

39

## Of Addition.

2 scrup. 6 grains, 814 fb 11 ounce. 5 dra. 1 scr. 4 gra. 984 fb 15 ounce. 4 dra. 1 scr. 2 gra. 471 fb 14 ounce. 2 dra. 2 scr. 3 gra. 630 fb 3 ounce. 1 dra. 1 scr. 5 gra. and 930 fb 12 ounce. 1 dra, 0 scr. 8 gra. *facit* 4320 fb 5 ounce. 5 dra. 2 scrup. 8 grains.

### Averdupois Little Weight.

For every Sixteen Drams, carry an Ounce to the place of Ounces; and for every Sixteen Ounces, carry One pound to the place of Pounds.

30. Adde 34 fb 10 ounce. 6 dra. 38 fb 15 ounce. 15 dra. 13 fb 8 ounce. 14 dra. 24 fb. 13 ounce. 3 dra. 78 fb 5 ounce. 13 dra. 10 fb 15 ounce. 12 dra. and 12 fb 4 ounce. 9 dra. *facit* 213 fb 10 ounce. 8 dra.

31. Item, Adde 13 fb 13 ounce. 15 drams, 30 fb 8 ounce. 0 dra. 0 fb 0 ounce. 10 drams, 101 1/2 12 ounce. 12 dra. 81 1/2 13 ounce. 9 dra. 10 7/2 ounce. 13 dra. and 51 1/2 14 ounce. 8 dra. *facit* 290 7/2 ounce. 3 dra.

### Troy Weight.

For every Twenty four Grains, carry a Penny weight to the place of Penny weight; for every Twenty penny weight, carry an Ounce to the place of Ounces; and for every Twelve Ounces, carry a Pound to the place of Pounds.

32. Adde 124 1/2 10 ounce. 6 pw. 200 1/2 3 ounce. 15 pw. 120 1/2 8 ounce. 2 pw. 310 1/2

D 4

2 ounce.

40

Of Addition.

2 ounce. 10 pw. 101 ¤ 10 ounce. 7 pw. and
350 ¤ 11 ounce. 4 pw. Facit 1208 ¤ 10 ounce.
4 pw.

33. Item, adde 67 ¤ 10 ounce. 18 pw.
22 grains, 68 ¤ 11 ounce. 16 pw. 17 grains,
32 ¤ 6 ounce. 8 pw. 9 gra. 84 ¤ 9 ounce.
7 pw. 3 gra. 47 ¤ 10 ounce. 19 pw. 23 gra.
and 84 ¤ 9 ounce. 15 pw. 19 grains. Facit
386 ¤ 11 ounce, 6 pw. 2 1 grains.

Dry Measures.

For every Sixteen Pints, carry a Peck to the
place of Pecks; for every Four Pecks, carry a
Buffel to the place of Buffels; for every Four
Buffels, carry a Comb to the place of Combs;
and for every Two Combs, carry a Seam (or
Quarter) to the place of Seams.

34. Adde 17 Seams (or Quarters) 1 Comb
3 Buffels, 7 Seams 0 Comb 2 Buffels, 18
Seams 0 Comb 3 Buffels, 15 Seams 1 Comb
2 Buffels, 20 Seams 0 Comb 3 Buffel, and
2 Seams 1 Comb 2 Buffels, Facit 82 Seams
0 Comb 3 Buffels.

35. Item, Adde 16 Buffels 3 Pecks 12 Pints, 74 Buffels 2 Pecks 10 Pints, 43 Buffels 2 Pecks 11 Pints, 84 Buffels 3 Pecks 15 Pints, 23 Buffels 2 Pecks 14 Pints, and 78 Buffels 3 Pecks 8 Pints, facit 322 Buffels 3 Pecks 6 Pints.

Long

41

Of Addition.

Long Measures.

In the Addition of Yards, Quarters, Nails,
for every Four Nails carry a Quarter to the place
of Quarters; and for every Four Quarters, car-
ry a Yard to the place of Yards.

Item, In the Addition of Ells, Quarters of Yards, and Nails: For every Four Nails carry a Quarter to the place of Quarters; and for every Five Quarters, carry an Ell to the place of Ells.

Item, In the Addition of Foot, Inches and
Quarters of Inches; for every 4 Quarters of an
Inch, carry an Inch to the place of Inches; and
for every Twelve Inches, carry a Foot to the place
of Foots.

Item, In the Addition of Acres, Roods, and Parches: For every Forty Parches carry a Rood to the place of Roods; and for every Four Roods carry an Acre to the place of Acres.

36. Adde 87 Yards 2 Quarters 3 Nails,
98 Yards 1 Quarter 0 Nail, 84 Yards 2 Quar-
ters 2 Nails, 47 Yards 1 Quarter 2 Nails,
and 46 Yards 2 Quarters 3 Nails, Facit 364
Yards 2 Quarters 2 Nails.

37. Item, Adde 92 Ells 2 Quarters of a Yard
3 Nails, 64 Ells 3 Quarters 2 Nails, 48 Ells
2 Quarters 2 Nails, 94 Ells 3 Quarters 3 Nails,
17 Ells 2 Quarters 0 Nail, and 97 Ells
1 Quarter 3 Nails, facit 415 Ells 1 Quarter
1 Nail.

38. Item,

42

Of Addition.

38. Item, adde 120 Foot 10 Inches 2 Quarters of an Inch, 96 Foot 8 Inches 3 Quarters, 106 Foot 3 Inches 2 Quarters, 71 Foot 11 Inches 0 Quarter, 3 Foot 9 Inches 3 Quarters, and 12 Foot 3 Inches 2 Quarters, facit 411 Foot 11 Inches 0 Quarters.

39. Item, adde 86 Acres 3 Roods 39 Perches, 89 Acres 2 Roods 34 Perches, 16 Acres 1 Rood 3 Perches, 35 Acres 0 Rood 20 Perches, and 13 Acres 3 Roods 26 Perches, facit 242 Acres 0 Rood 2 Perches.

Liquid Measures.

For every two Pints carry a Quart to the place of Quarts; for every two Quarts carry a Pottle to the place of Pottles; for every two Pottles carry a Gallon to the place of Gallons; for every Sixty three Gallons, carry a Hogshead to the place of Hogsheads; for every two Hogsheads, carry a Butt (or Pipe) to the place of Butts; and for every two Butts, carry a Tun to the place of Tuns.

40. Adde 7 Hogsheads 12 Gallons 1 Pottle 1 Quart 1 Pint, 9 Hogsheads 59 Gallons 1 Pottle 0 Quart 0 Pint, 10 Hogsheads 7 Gallons 0 Pottle 1 Quart 1 Pint, 7 Hogsheads 8 Gallons 1 Pottle 1 Quart 1 Pint, facit 24 Hogsheads 25 Gallons 1 Pottle 0 Quart 1 Pint.

41. Item, adde 10 Tuns 1 Butt 0 Hogshead 8 Gallons 1 Pottle 1 Quart 1 Pint, 31 Tuns 1 Butt

43

Of Addition.

1 Butt 1 Hogshead 62 Gallons 1 Pottle 1 Quart 0 Pint, 15 Tuns 1 Butt 0 Hogshead 56 Gallons 1 Pottle 0 Quart 0 Pint, 19 Tuns 1 Butt 1 Hogshead 9 Gallons 0 Pottle 1 Quart 1 Pint, and 23 Tuns 0 Butt 0 Hogshead 57 Gallons 1 Pottle 0 Quart 1 Pint, facit 101 Tuns 0 But 1 Hogshead 6 Gallons 0 Pottle 0 Quart 1 Pint.

Of Time.

For every Sixty minutes, carry an hour to the place of hours; for every Twenty-four hours carry a day to the place of days; for every Seven days carry a week to the place of weeks; and for every Fifty-two weeks carry a year to the place of years.

In the Addition of years and days, &c. for every 365 days carry a year to the place of years.

42. Adde 2 Years 12 Weeks 6 Days 23 Hours, 4 Years 16 Weeks 5 Days 10 Hours, 7 Years 23 Weeks 4 Days 9 Hours, 3 Years 18 Weeks 6 Days. 18 Hours, and 10 Years 3 Weeks 1 Day 8 Hours, facit 27 Years 23 Weeks 3 Days 20 Hours.

43. Item, adde 13 Years 10 Days 10 Hours 30 Minutes, 16 Years 58 Days 16 Hours 9 Minutes, 23 Years 24 Days 16 Hours 9 Minutes, 18 Years 10 Days 9 Hours 8 Minutes, and 21 Years 362 Days 23 Hours 50 Minutes, facit 92 Years 102 Days 3 Hours 46 Minutes.

The

44

of Addition.

The Proof of Addition.

There are divers ways to prove Addition, I
shall mention onely four.

I. Some make the Proof of Addition in casting away all the Nines of the numbers which are added together, and likewise of the total sum. If the two refts (if there be any) be equal, they conclude that the Addition is right.

II. There is another way used by others, by
casting away all the Sevens of the numbers to be
added, which way is more tedious and difficult
than the former. These two Proofs are very un-
certain, and not to be trusted; and therefore I
pals them over.

III. The third is the most certain of all, but is
also very tedious: The manner is thus. Subtract
every number propounded in the Addition (one
after another) from the total sum, and the last
Subtraction being finished, there will remain
nothing; which shows that the Addition was
right. But this kind of Proof cannot be practised,
without the knowledge of Subtraction, which is
the next Rule.

IV. The last sort of Proof I shall mention, is
one much used in England, and is very easy for
the new Learners: The manner is as follows.
Having finished your Addition, adde again all
your numbers as before, except the upper number,
(which

45

Of Addition.

(which you may cut off with a line, if you please)
then add the total of the second Addition, with
the upper number (which was left out) the sum
of the third Addition must be equal with the
total of all the numbers, or the sum of the first
Addition, as followeth. Adde 3107 L 9322:
1021: and 3102 L together, fact 16552.

$$
\begin{array}{r}
  \mathbf{L} \\
  \underline{3107} \\
  9322 \\
  1021 \\
  \phantom{1}3102 \\
\end{array}
$$

The first Total is-16552
Total of the Remainders-13445

Proof-16552 which is equal
with the first Total.

Another Example proved.

$$
\begin{array}{cr@{\,}c@{\,}c@{\,}c@{\,}c}
 & \multicolumn{1}{c}{\mathbf{L}} & \multicolumn{1}{c}{\mathbf{\beta}} & \multicolumn{1}{c}{\mathbf{\vartheta}} & \multicolumn{1}{c}{\mathbf{q}} & & & & & & & & & & \\
\cline{2-2}
 & 16 & \multicolumn{1}{c}{\text{---}} & 16 & \multicolumn{1}{c}{\text{---}} & 8 & \multicolumn{1}{c}{\text{---}} & 3 & & & & & & & & \\
 & 10 & \multicolumn{1}{c}{\text{---}} & 18 & \multicolumn{1}{c}{\text{---}} & 9 & \multicolumn{1}{c}{\text{---}} & 2 & & & & & & & & \\
 & 37 & \multicolumn{1}{c}{\text{---}} & 13 & \multicolumn{1}{c}{\text{---}} & 0 & \multicolumn{1}{c}{\text{---}} & 3 & & & & & & & & \\
 \cline{5-5}\cline{7-7}
 & 57 & \multicolumn{1}{c}{\text{---}} & 07 & \multicolumn{1}{c}{\text{---}} & 9 & \multicolumn{1}{c}{\text{---}} & 2 & & & & & & & & \\
\end{array}
$$

The first Total is-122-16-4-2
The Total of the Remainders-105-19-7-3

Proof-122-16-4-2

Note,

Of Addition.

*Note,* In making the Proof of Addition you may leave out what number you please, if that number be added with the sum of all the Remainders; but the upper number is commonly cut off, more for decency than necessity, and the Learner might sooner mistake if any other number were cut off.

Questions of Addition.

Queft. 1. What is Addition?

*Answ.* It is a gathering of divers numbers into one sum.

Queft. 2. In how many parts is Addition commonly divided?

*Answ.* Into two, Simple and Compound.

Queft. 3. What is Addition Simple?

*Answ.* It is the gathering of two or more numbers of one kind or name into one sum, as Pounds with Pounds, Yards with Yards, &c.

Queft. 4. What is Addition Compound?

*Answ.* It is the gathering of divers numbers of several names together, as Pounds, Shillings, Pence, Farthings, with 9 7 6 5 4 3 2 1, or Pounds Ounces, Drams, &c. with 16 Ounces, Drams &c.

Queft. 5. What must be first observed in an Addition Simple?

*Answ.*

Of Addition.

*Answ.* To place the numbers to be added in their right places.

Queft. 6. In what order must the numbers be set?

*Answ.* The Unites under the Unites, the Tens under the Tens, Hundreds under Hundreds, &c.

Queft. 7. What is further to be done, when the numbers are in their right places?

*Answ.* I draw a line under the numbers, and add every row of figures one after another.

Queft. 8. Where do you begin, at the right hand or at the left?

*Answ.* I begin at the right hand (which is the place of Unites) and see what the whole row of Unites amount unto.

Queft. 9. If the row of Unites do amount un-to Twelve, what must you set down and carry?

*Answ.* I must set down two and carry one.

Queft. 10. Why must you set down two, and not the one.

*Answ.* Because the 2 being two Unites, must be set under the Unites, which is its proper place.

Queft. 11. Where must you carry the 1.

*Answ.* The one which I carry, is worth 10. Therefore I must carry it to the second place, which is the place of Tens.

Queft. 12. Why do you carry but 1 at 12?

*Answ.* Because I find, but once ten in twelve, for I must always carry an Unite into the next row (towards the left hand) for every ten I meet with, at any row.

Queft.

47

48

Of Addition.

Quest. 13. If you find 34 in the second row, (or in the place of Ten) what must you set down and carry?

Answ. I must set down 4 and carry 3.

Quest. 14. Where must you set the 4, and carry the 3?

Answ. I must set the 4 right under that row, where I have found the 34, and I carry the 3, to the next row towards the left hand, which is the place of Hundreds.

Quest. 15. Why do you carry just 3.

Answ. Because there is but three tens in 34.

Quest. 16. If you find but 9, in a row, what must you set down and carry.

Answ. I should set down 9, and carry nothing, because 9, is less than 10, and I must carry nothing but ten.

Quest. 17. If you find just 10 in a row, what must you set down and carry?

Answ. I must set down an 0, or cipher under that row, and carry 1.

Quest. 18. If you find 50?

Answ. I must set a cipher and carry 5.

Quest. 19. Have you a general rule (in adding any quantity of numbers) what you are to set down, and what to carry?

Answ. The general rule is, to set down the figure which is in the place of Unites (of the sum found in any row of an Addition) and to carry the rest of the figures which are on the left hand.

Quest. 20. Explain your meaning?

Answ.

49

Of Addition.

Answ. My meaning is thus. Suppose that I find a row of figures amounting to 94. I set down the 4 (because it is that figure which stands in the place of Unites) and carry 9, because 9, stands on the left side of the said 4.

Quest. 21. If your sum found in a row of figures do exceed a hundred, yea, many hundreds, as by example, if you find 427 in a row.

Answ. If I found 427, in a row, I should set down 7 (because it is seven Unites) and carry the two other figures, which being 42, I carry forty two to the next row, because it is forty two tens.

Quest. 22. But if you were at the last row of the Addition, where would you carry your 42.

Answ. Then I should set down 42 towards the left hand.

Quest. 23. If you find in a row nothing but ciphers?

Answ. I must set a cipher under that row, and carry nothing.

Quest. 24. If you did find 49, in the first row, and nothing but ciphers in the second.

Answ. I must set down 9, under the first row, and the four (I carry) under the second, and carry nothing.

Quest. 25. If you were to add an extraordinary quantity of numbers?

Answ. If the numbers were too tedious to cast up at once, I should part them into two or three parts, and make two or three Additions, and

E

then

50

Of Addition,

then I should adde together the total sums of such
lesser Additions.

Queft. 26. How will you adde numbers of di-
vers names, as £ β. § q?

Answ. I muſt firſt fet them in their right pla-
ces, and draw a line under them.

Queſt. 27. How will you place them ?

Anſ. I muſt fet the Pound under the Pounds,
Shillings under Shilling, Pence under Pence,
Farthings under Farthings.

Queſt. 28. At what denomination do you begin,
at the Farthings, or at the Pounds ?

Anſw. I muſt begin always at the leaſt deno-
mination.

Queſt. 29. In caſting up the row of Farthings,
what muſt you obſerve ?

Anſw. I muſt obſerve to fet down the odde
Farthings under the Farthings, and carry ſo many
Pence as I finde in the row of Farthings.

Queſt. 30. *Suppofe you finde 15 q. in the row of
q. what must you carry and fet down ?*

Anſw. 15q. is 3 9 3q. therefore I muſt fet
down 3q. and carry 3 9 to the place of 9.

Queſt. 31. What do you obſerve in adding the
row of 9?

Anſw. I muſt obſerve to carry every Shilling
which I finde in the place of Pence, and fet down
the odde Pence under the row of Pence, if I finde
any odde Pence.

Queſt. 32. *Suppofe you finde 28 9. in the row
of 9?*

Anſw. 28 9 is 2 β 4 9, therefore I muſt
fet

51

Of Addition.

fet down 4 9 under the 9, and carry 2 β to
the β.

Queſt. 33. What do you obſerve in adding the
row of Shillings?

Anſw. I do obſerve to carry ſo many Angels
(or 10 β) to the place of Angels, as I finde in
the row of Shillings, and if there be any odde
Shillings, I fet them down under the Shillings.

Queſt. 34. *Suppofe you finde 38 β in the row
of Shillings?*

Anſw. I muſt fet down the 8 β and carry the
three Tens (which are three Angels) to the place
of Angels.

Queſt. 35. What do you obſerve in adding the
Angels?

Anſw. I muſt carry ſo many Pounds to the
place of Pounds, as I finde in the place of An-
gels; and if I finde any odde Angel, I muſt fet
it in the place of Angels.

Queſt. 36. *Suppofe you finde 11 Angels in the
place of Angels?*

Anſw. 11 Angels is 5 £ 10 β, therefore I
muſt fet down the odde Angel (or 10 β) under
the Angels, and carry 5 £ to the firſt row of £
which is the place of Unites.

Queſt. 37. What do you obſerve in adding the
Pounds?

Anſw. I obſerve the ſame order as hath been
anſwered in the Queſtions of Addition Simple.

Queſt. 38. In fetting down any ſum of money,
what are you to obſerve?

Anſw. I muſt obſerve what hath been ſaid al-

E 2

ready,

52

## Of Addition.

ready, which is to set down every sum, and their
several denominations in their right places.

Queft. 39. How many Shillings may you set down in the place of Shillings ?

Answ. But 19, because 20 β makes 1 L.

Queft. 40. And how many Pence into the place of Pence ?

Answ. But 11 9, because 12 9 makes 1 Shilling.

Queft. 41. And Farthings in the place of Farthings.

Answ. But 3, because 4 Farthings makes 1 Penny.

Queft. 42. How will you set down 25 β ?

Answ. 25 β is 1 L 5 β, therefore I must set 1 L into the place of Pounds, and 5 β into the place of β.

Queft. 43. How will you set down 26 9 ?

Answ. 26 9 is 2 β 2 9, therefore 1 must set down 2 β into the place of Shillings, and the 2 9 into the place of Pence.

Queft. 44. How will you set down 56 β 15 9.

Answ. 56 β is 2 L 16 β, and 15 9 is 1 β 3 9, which being put together, is 2 L 17 β 3 9; then I set 2 L into the place of L, the 17 β to the place of β, and the 3 9 to the place of 9.

Queft. 45. How will you set down 16 Groats and 99?

Answ. 16 Groats is 5 β 4 9, and 99 is 2 9 1 q, which is together 5 β 6 9 1 q. I set then 5 β to the place of β, and the 6 9 to the place

53

## Of Addition.

place of 9; and likewise, the 1 q to the place of q.

Queft. 45. How will you set down 32 L and 39?

Answ. I must set down 32 L into the place of L, and a cipher in the place of β, and also a cipher in the place of 9, and the 3 q, into the place of q.

Queft. 47. What do you observe in the Addition of Weights and Measures?

Answ. I must observe the same thing observed in the Addition of Money.

Queft. 48. What is that?

Answ. I must observe for a general rule in setting down small Denominations, that I do not set down such a quantity of any lesser Denomination, that may amount to the worth of any greater.

Queft. 49. Give an example of this observation in the setting down of weight.

Answ. In setting down weight, as Hundreds, Quarters, Pounds, Ounces, Drams, Scruples, Grains, I must observe not to set above 3 in the place of Quarters, not above 27 in the place of Pounds, not above 15 in the place of Ounces, not above 7 in the place of Drams, not above 2 in the place of Scruples, and not above 19 in the place of Grains.

Queft. 50. In dry measure, what do you observe?

Answ. I must not set above 3 in the place of Pecks, nor 15 in the place of Pints, and so of the rest of the measures.

E 3

Queft.

54

Of Subtraction.

Queft. 51. How will you adde Pounds, Nobles, Shillings, and Pence, with L, Nobles, β, and §?

Answ. I muſt adde firſt the quantity of 9 and 6 shillings, as is taught before, and for every 6 β 8 9, I muſt carry 1 Noble to the Nobles; for 13 β 4 9, I muſt carry two Nobles; for 20 β three Nobles, &c. And for every three Nobles, I muſt carry one Pound to the place of Pounds.

Of Subtraction.

Subtraction teacheth to take one or more small fums out of a greater, that the reſt may be known. In Subtraction Simple, observe what followeth.

1. Set down firſt the greater number, and the number to be subſtracted under it, placing the figures as in Addition, that is to say, Unites under Unites, Tens under Tens, Hundreds under Hundreds, &c. and draw a line under them.

2. The two numbers being set in their right places, begin at the right hand (which is the place of Unites) and take the figures underneath (one after another) from the figures above them, and set down the reſt under the line in such a manner, that the reſt of the Unites be set down right under them,

Of Subtraction.

55

them, the reſt of the Tens right under them also, &c. as in the firſt example.

3. If the upper number exceed the lower in places or figure, Subtract as before, and set down these figures of the upper number (which exceed) below, under the line with the reſt, obſerving to set them in their right places, which is to wards the left hand; as in the ſecond example.

4. If you cannot take some of the lower figures from the uppermost, by reason of their letter value, Observe for a general rule what followeth.

First, If the firſt lower figure (in the place of Unites) cannot be taken from that figure above it, then borrow 1 from the next upper figure which is the ſecond place (or the place of Tens) which I have borrowed, is worth 10. (because it is borrowed from the place of Tens.) Now adde this Ten with the firſt figure of the upper number, and Subtract the firſt lower figure from that fum, and set the reſt under the line.

Secondly, Before you proceed to Subtract the ſecond lower figure from that above it, you must ever remember, (if you have borrowed 1 before) to pay the debt borrowed (or to add it) to the lower figure, which is of the fame place and degree, as that figure from whence it was borrowed; that is, If you borrow 1 from the ſecond upper figure, you must add 1 to the ſecond lower figure; if you borrow 1 from the third upper figure, you must pay (or add) 1 to the third lower figure, &c.

E 4

Thirdly,

56

Of Subtraction.

Thirdly, Having added the 1 so borrowed, to the second lower figure. Subtract that sum from the second upper figure, if you can, if you cannot, borrow (as is said already) from the third upper figure, which 1 must be accounted for 10 (because every place exceeds the other ten times) and add the ten so borrowed, to the second upper figure; and from that sum, subtract that number which you could not subtract before, and set the rest under the line, right under the place of tens (or the second place; ) and for the 1 so borrowed from the third upper figure, add it to the third lower figure, and proceed so, until your Subtraction be finished. If you see need of borrowing, for it doth not always happen so, that you must borrow at every figure, you must borrow only in this cause, when you see that the upper figure is less then that figure, you are to subtract from it, as you may see in the third example.

1. Example,

If 642 be taken from 876, what is the rest?
faci't 234.

Set the numbers in order as followeth,

Debt 876.

Paid 642.

Having set the numbers in order, I begin at the lower figure ( at the Unites ) saying 2 from 6 there remains 4, which 4 I set under the line, then I go to the next lower figure (which is 4) saying,

57

Of Subtraction.

Saying, 4 from 7 there remains 3, which 3 I set down under the line, and go further to the next under figure, which is 6, saying, 6 from 8 there remains 2, which I set under the line. Now the figures which are under the line, is the rest required, which is 234 as followeth.

Debt--876.

Paid--642.

Rest--234.

2. Example.

In subtracting 45 from 10849, what is the rest?
faci't 10804.

Debt--10849.

Paid 45.

Rest--10804.

3. Example.

From 6324 take 3867, facit 2457 as followeth.

Debt--6324.

Paid--3867.

7.

I begin at the lower figure 7, saying, 7 from 4 I cannot, therefore I borrow 1 from the second upper figure 2, ( according to the fourth rule, Fol.55.) which 1 is 10, and 4 (the upper figure)

15

58

Of Subtraction.

is 14. Now 7 from 14 there remains 7, which 7 I set down under the line as you see above; and for the 1 borrowed from the second upper figure, I pay it (or add it) to the second lower figure, which is 6, saying 6 and 1 is 7; and I proceed as before, saying, 7 from 2 I cannot, therefore I borrow 1 again, (but it is from the third upper figure 3) saying, 1 that I borrow (which is 10) and 2 is 12. Now 7 (which I could not take from 2, I can now subtract, having borrowed 10) from 12 there remains 5, which 5 I set down under the line, as in the Margin; and for the 1 borrowed from the third upper figure, I add 1 to the third lower figure, which is 3, saying, 1 I borrowed and 8 is 9. Now 9 from 3 I cannot, therefore I borrow 1 from the fourth upper figure 6, which 1 is 10 and 3 is 13. Now I say, 9 from 13 there remains 4, which 6324 4 I set down under the line, as in the 3867 Margin, and for the 1 borrowed from the fourth upper figure 6, I add 1 to the fourth lower figure 3, saying, 1 I borrowed and 3 is 4. Now 4 from 6 there remains 2. The whole left found under the line, is 2457, as followeth,

$6324$

$3867$

I

Reft—2457.

Note, The Learner may note by the way, that the number from which another is to be taken, must

59

Of Subtraction.

must be greater, or at least equal with the number to be subtracted.

Another way of Subtraction.

There is another way of Subtraction which is yet easier for young Beginners, then what hath been taught : When some of the upper figures are less than some of the lowermost, the manner is as followeth.

When the upper figures are greater than the lowermost, or equal to them, subtract as is taught before; but when you cannot, the under figures (or some of them) being greater than some of the uppermost, then borrow 1 from the next upper figure (as is taught before) which 1 is always to be accounted for 10. Now before you add the 10 so borrowed to that upper figure (which was too little) subtract first the lower figure from the said 10 borrowed, and add the reft to the upper figure, and set down under the line the sum of that Addition : And for the 1 borrowed, you must add 1 to the next lower figure (as is taught before) going on so (if there be need of borrowing) until your Subtraction be finished. But to make this yet plainer to the Learner, I shall work over again the third example of this rule, as followeth.

Debt—$6324$

$3867$

Reft—$2457$

I be-

60

## Of Subtraction.

I begin at the right hand (as before) saying, 7 from 4 I cannot, I borrow 1 from the next upper figure 2, which 1 is 10. Now 7 from 10 there remains 3 and 4 (which is above 7) is 7; which 7 I set down under the line, and the 1 that I borrowed and 6 (which is the next lower figure) is 7. Now 7 from 2 I cannot, therefore I borrow 1 from the third upper figure, which 1 is 10. I say then, 7 from 10, there remains 3 and 2 (which is above the 6) is 5, which 5 I set under the line. I say further, 1 that I borrowed and 8 is 9; from 3 I cannot, I borrow 1 again, which is 10. Now 9 from 10 there remains 1 and 3 (which is above the 8) is 4, which 4 I set under the line. Lastly, I say, 1 that I borrowed and 3 is 4. Now 4 from 6 there remains 2, which 2 I set under the line, the whole rest is 2457 as before. This way of Subtraction I shall follow in the Subtraction of Numbers of divers Decominations as being most easie.

4. If 1234 be taken from 3489, what is the rest? facit 2255.

5. Item, Take 134 from 1000347, facit 1000213.

6. Item, From 8162 take 1894, facit 6268.

7. Item, Take 8148 from 9106, facit 958.

8. Item, Subtract 584693 from 673582, facit 88889.

9. Item, Subtract 6309 from 9004, facit 2695.

Note, If in subtracting you have an occasion to borrow, it matters not whether such figures, from

## Of Subtraction.

61

from whom you are to borrow, be significant figures or ciphers, because you must pay (or adde) the unite so borrowed to the next lower figure: Therefore observe the working of the following Subtraction, which may serve for a Direction to the rest.

Debt—9004

Paid—6309

5

I say 9 from 4 I cannot, I borrow 1 (which is 10) then 9 from 10 there remains 1, and 4 (which is above 9) is 5, which 5 I set down.

I say further, 1 that I borrowed and 9004

0 is 1, from 0 I cannot, I borrow 1 6309

(which is 10.) Now 1 from 10 there remains 9, which 9 I set down as in

the Margin. I proceed and say, 1 that I borrowed and 3 is 4, from 0 I cannot, but 4 from 10 there remains 6, 9004

6309

which 6 I set down as in the Margin. 95

Lastly, I say, 1 that I borrowed and 6 is 7. Now 7 from 9 there remains 2,

695

which 2 I set down: The whole rest is 2695, as followeth.

9004

6309

Rest—2695

10. Item, In taking 31698904 from

60850007,

62

Of Subtraction.

60850007, what is the reft? *facit* 29151103.

11. Item, Subtract 6897 from 91111000, *facit* 911103103.

12. Item, Take 1 from 100000, *facit* 99999.

13. Item, Subtract 101010101 from 101010101010, *facit* 909090909.

14. Item, Take 1090108 from 3000111, *facit* 1910003.

15. Item, A. oweth 340 L, and he pays at one time 49 L, and at another time 57 L, what doth A. ow still? *facit* 234 L.

Adde firft 49 and 57 together, and subfraact the fum from 340, the reft will be 234.

16. Item, B. borrows at one time 138 L, at another time 79 L, a while after he pays back again 109 L. What doth he ow still? *facit* 108 L.

Adde 138 and 79 together, and subfraact 109 from the fum, the reft will be 108.

17. Item, If 968, 408, and 926 L, be taken from 168000 L. What is the reft? *facit* 165698 L.

18. Item, Take 8486, 8406, and 6846 L from 22098 L, *facit* impossible.

19. Item, Subtract 386491 from 289011, and 191897, *facit* 94417.

20. Item, From 16872, 10405, and 31497, subfraact 19040, 28187, and 11547. What is the reft? *facit* 0.

21. Item, Subfraact 1040: 4681: 8460: 7487: and 4091 from 4947: 8046: 3168: 9674: and 98400, *facit* 98476.

63

Of Subtraction.

Subtraction of Numbers of divers Denominations.

1. In Subtraction of Numbers of divers Denominations or Names, place the Numbers in their right places (as in Addition Compound) which is to set down the Numbers of like Name, one directly under the other. As for example, In subfraacting Pounds, Shillings, Pence, Farthings, place the L under the L, β under β, § under §, q under q.

In the Subtraction of Weights and Measures, observe the same order, and draw a line under the numbers.

2. Begin at the least Denomination firft, subfraacting the undermoft from the uppermoft, and so on from the leffer to the greater, setting down the reft under the line, each Denomination in its right place; as in the firft, second, and third examples of Subtraction Compound.

3. If you find that some, or moft of the Denominations of the upper number, be leffer than the lower, observe for a general rule what follows.

First, The same number which is carried in Addition, the same must be borrowed in Subtraction, that is to say, As you did carry 1 Penny for every 4 Farthings, into the place of Pence, 1 β for every 12 §, into the place of Shillings, 1 L for every 20 β, into the place of Pounds, and

64

Of Subtraction.

and for every ten in the Pound. The same numbers you must borrow in Subtraction; that is, for every Penny you borrow, you must reckon 4q, for every Shilling 12 9, for every Pound, 20 3, and for every Unite in the Pounds 10.

Secondly, If the lowermost Farthings be more than the uppermost, borrow 1 9 from the upper pence, which is 4 q. then subtract the lower q from 4. the reft adde to the upper q. and set that fum under the Farthings.

Thirdly, For that Penny borrowed, adde one to the lowermost pence, and the fum being greater then can be subtracted from the uppermost pence, borrow 1 3 from the upper Shillings, which makes 12 9, then subtract such a number of Pence (which could not be subtracted before) from 12, and adde the reft to the uppermost pence, and set that fum down under the line in the place of Pence.

Fourthly, For that Shilling borrowed, adde one to the lowermost Shilling, and subtract that fum from the upper : If you cannot, then borrow 1 L from the upper L, which is 20 3. Now subtract that number of Shillings from 20, and adde the upper Shilling to the reft, and set down that fum under the line in the place of Shillings.

Fifthly, For that Pound borrowed, adde one to the Unites of the lowermost Pounds, and subtract as hath been taught in Subtraction Simple. Such manner of working is to be observed in the Subtraction of Weights and Measures.

22. If

65

Of Subtraction.

22. If: 11 L 12 3 be taken from 82 L 18 3, what is the reft? facit 71 L 6 3.

23. Items, from 34 L 16 3 6 9, subtract 12 L 0 3 2 9, what is the reft? facit 22 L 16 3 4 9.

24. Items, Subtracting 4 L 0 3 0 9 2 q from 129 L 0 3 0 9 3 q, what is the reft? facit 125 L 0 3 0 9 1 q.

The working of these three Questions follow-eth.

L β
82-18
11-12
---
71-6

L β 9
34-16-6
12-00-2
---
22-16-4

L β 9 q
129-00-00-3
4-00-00-2
---
125-00-00-1

These three Subtractions are so plain, that they need no Explication, I shall say onely this, that L is taken from L, 3 from 3, 9 from 9, and q from q.

25. If 342 L 18 3 9 9 3 q be taken from 904 L 13 3 8 9 2 q, what is the reft? facit 561 L 14 3 10 9 3 q as followeth.

904 13 8 2
342 18 9 3
---
3

I begin at the lowermost farthings, saying, 3 from 2 that cannot be, therefore I borrow 1 9 F (from

68

Of Subtraction.

35. Item, From 964 L 13 3/8 subftraet
146 L 16 3/8, 460 L 16 3/8, 7 7/8, and
64 L 19 3/8, facit 292 L 1 3/8 19/8.

36. Item, Subftraet 48 L 15 3/8 6 3/8 and
45 L 0 3/8 8 3/8 from 60 L 16 3/8 8 3/8 and
40 L 11 3/8 6 3/8, facit 7 L 12 3/8 0 3/8.

37. Item, From 31 L 13 3/8 6 3/8, 12 L
14 3/8, and 68 L 10 3/8 9 3/8, take 39 L
10 3/8 2 3/8, 51 L 11 3/8 7 3/8, and 21 L 16 3/
9 3/8, what is the rest? facit 0.

38. Item, A. oweth 360 Marks 0 Noble 2 3/8
0 3/8, and he hath paid at one time 40 Marks
1 Noble 3 3/8 6 3/8, and at another time 38
Marks 0 Noble 6 3/8 2 3/8; how many Marks
doth he ow still? facit 280 Marks 1 Noble
5 3/8 8 3/8.

Averdupois Great Weight.

39. Subftraet 25 C 3 qrs 13 fb from 45 C
0 qr 27 fb, facit 19 C 1 qr 14 fb.

40. Item, Take 68 C 1 qr. 26 1/2 lb from 91 C
3 qrs 24 lb, facit 23 C 1 qr 26 1/2 lb.

Averdupois Subtile.

41. Subftraet 99 1/2 lb 12 ounce. 5 drams from
301 1/2 lb 0 ounce. 0 dram, facit 201 1/2 lb 3 ounce,
3 drams.

42. Item, From 6487 1/2 lb 15 ounce. 3 dr. 2 scruples, subftraet 698 1/2 lb 2 ounce. 2 dr. 1 scruple, facit 5789 1/2 lb 13 ounce. 1 dr. 1 scr.

43. Item,

69

Of Subtraction.

43. Item, Subftraet 24 1/2 lb 0 ounce. 7 dr. 0 scr. 19 grains, from 100 1/2 lb 6 ounce. 2 dr. 1 scr. 2 grains, facit 76 1/2 lb 5 ounce. 3 dr. 0 scr. 3 grains.

Averdupois Little Weight.

44. Subftraet 198 1/2 lb 8 ounce. 15 dr. from
346 1/2 lb 12 ounce. 12 dr. facit 148 1/2 lb 3 ounce.
13 drams.

45. Item, Subftraet 49 1/2 lb 15 ounce. 7 dr. from 82 1/2 lb 0 ounce. 0 dr. facit 32 1/2 lb 0 ounce. 9 drams.

Troy Weight.

46. Subftraet 680 1/2 lb 11 ounce. 19 penny weight from 796 1/2 9 ounce. 6 pw. facit 115 1/2 9 ounce. 7 pw.

47. Item, From 1684 1/2 lb 11 ounce. 0 pw.
23 grains, subftraet 108 1/2 lb 6 ounce. 6 pw.
20 grains; facit 1576 1/2 lb 4 ounce. 14 pw.
3 grains.

Dry Measures.

48. From 24 Seams (or Quarters) 0 Bushel
3 Pecks, subftraet 16 Seams 7 Bushels 3 Pecks,
facit 7 Seams 1 Bushel 0 Peck.

49. Item, Subftraet 89 Bushels 1 Peck 8 Pints from 198 Bushels 2 Pecks 5 Pints, facit 109 Bushels 0 Peck 13 Pints.

F 3

Long

70

## Of Subtraction.

### Long Measures.

50. Subtract 186 Acres 3 Roods 30 Perches (or Poles) from 900 Acres 2 Roods 25 Perches, facit 713 Acres 2 Roods 35 Perches.

51. Item, Subtract 486 Foot 6 Inches 2 Quarters of an Inch, from 1687 Foot 0 Inch 0 Quarter, facit 1200 Foot 5 Inches 2 Quarters of an Inch.

52. Item, From 317 Yards 3 Quarters 0 Nails, take 184 Yards 0 Quarter 3 Nails, facit 133 Yards 2 Quarters 1 Nail.

53. Item, Subtract 640 Ells 0 Quarter of a Yard 3 Nails, from 686 Ells 3 Quarters 2 Nails, facit 46 Ells 2 Quarters, 3 Nails.

54. Item, Subtract 34 Yards 2 Foot 6 Inches from 87 Yards 1 Foot 0 Inches, facit 52 Yards 1 Foot 6 Inches.

55. Item, Subtract 346 Fadoms 0 Foot 11 Inches from 1000 Fadoms 5 Foot 3 Inches, facit 654 Fadoms 4 Foot 4 Inches.

### Liquid Measures.

56. Subtract 12 Tuns 1 Butt 1 Hogshead from 27 Tuns 0 Butt 0 Hogshead, facit 14 Tuns 0 Butt 1 Hogshead.

57. From 60 Hogsheads 1 Gallon 0 Pottle 0 Quart 0 Pint, substract 42 Hogsheads 59 Gallons 1 Pottle 0 Quart 1 Pint, facit 17 Hogsheads 4 Gallons 0 Pottle 1 Quart 1 Pint.

Of

71

## Of Subtraction.

### Of Time.

58. Subtract 17 Weeks 6 Days 20 Hours from 30 Weeks 2 Days 23 Hours, facit 12 Weeks 3 Days 3 Hours.

59. Item, From 34 Years 360 Days 0 Hour, substract 19 Years 200 Days 12 Hours, facit 15 Years 159 Days 12 Hours.

60. Item, Subtract 99 Years 362 Days 23 Hours 4 Minutes from 1661 Years 0 Day 1 Hour 10 Minutes, facit 1561 Years 2 Days 2 Hours 6 Minutes.

## The Proof of Subtraction.

The Proof of Subtraction may be made divers ways (as the Proof of Addition.) Some do it by casting away all the Nines, and other the Sevens; but their uncertainty makes me name them onely without further trouble.

The Proofs which are most certain, are performed two ways. The first by Subtraction, and the second by Addition.

The first of these, is not so much in use as the second, the manner is thus.

Subtract the rest of the Subtraction from the upper number, and if the rest be equal to the

F 4

number

72

Of Subtraction.

number first subtracted, that shevs that such
Subtraction is right.

The second Proof is that which is most in use,
and is performed by Addition, thus.

Add the result to the sum which is subtracted,
and if the sum of that Addition be equal with
the uppermost sum, that shevys the Subtraction
to be true, as followsveth.

The Proof of Subtraction of Weights and
Measures is to be performed the same way.

Questions

73

Of Subtraction.

Questions of Subtraction.

1. Queft. What is Subtraction?

Answ. Subtraction is to take
a lesser number from a greater.

Queft. 2. In how many parts is this Rule commonly divided?

Answ. Into two; Simple and Compound.

Queft. 3. What is Subtraction Simple?

Answ. It is to take one lesser number from a
greater of the same name or kind, thus, to take
pounds from pounds, yards from yards, ells
from ells, &c.

Queft. 4. What is Subtraction Compound?

Answ. It is to take a lesser number from a
greater, being of divers denominations or names,
thus; to take pounds, shillings, pence, farthings,
from pounds, shillings, pence, farthings; or
pounds, ounces, drams, scruples, grains, from
pounds, ounces, drams, scruples, grains, &c.

Queft. 5. What is to be observed in Subtraction?

Answ. To set down the greater number first,
and the lessor (to be subtracted) under it.

Queft. 6. How is Subtraction Simple to be
wrought?

Answ. The two numbers are to be set down
first in their right places; to wit, the greater first,
then

74

Of Subtraction.

then the lesser; and every figure of one degree (or place) directly under one another; that is, unites under unites, tens under tens, hundreds under hundreds, &c. and draw a line under the numbers, as in Addition.

Queft. 7. What is further to be done?

Answ. To begin at the right hand, (or unites) and subtract the lower figures from the upper, one after another.

Queft. 8. What is to be observed in Subtraction?

Answ. To set down the remaining figures in their proper places.

Queft. 9. In what manner?

Answ. In taking unites from unites, the rest must be set down under the unites, the rest of tens under the tens, the rest of hundreds under the hundreds, &c.

Queft. 10. Must there be always one number greater than the other in Subtraction?

Answ. It is commonly so; that the number from which the Subtraction is to be made, is greater than the other; but it must be at least equal to the number to be subtracted.

Queft. 11. Why so?

Answ. Because if the number to be subtracted were greater than the other, such Subtraction were impossible.

Queft. 12. Of two numbers how do you know which is most?

Answ. This is a needle's question, for there is none so simple but he may know that.

Queft.

75

Of Subtraction.

Queft. 13. Many young beginners mistake in this, letting oftentimes the greatest number under the lesser, for want of knowing them. What is then the general Rule to know the greater from the lesser?

Answ. That number that hath most figures is greater than the other.

Queft. 14. But if the quantity of figures be equal in both the numbers?

Answ. Then the first figure towards the left hand of the two numbers, must be observed; and that number whose figure is greater in that place, that number is greater than the other, as by example, 20000 is more than 19999.

Queft. 15. How will you subtract one number from another when you find some figure of the upper number lesser than the lower?

Answ. In such a case I must borrow from the other figures towards the left hand.

Queft. 16. How much must you borrow at a time?

Answ. When the numbers are of one demotion, as, all pounds, all yards, &c. then I must never borrow above an unite or 1.

Queft. 17. How much do you value that I so borrowed?

Answ. I account it for 10. because every place exceeds the other ten times.

Queft. 18. Clear that a little more.

Answ. In subtracting unites from unites, if I were necessitated (or forced) to borrow, I should borrow 1, from the place of tens, (which is

76

### Of Subtraction.

is the very next place) and 1. from that place is worth 10.

Queft. 19. If you were to borrow again to supply the second place, or the place of tens.

Answ. Then I should borrow 1. from the third place, which is the place of hundreds, and 1. in that place is worth 10. of the second place, for 10. times 10. is 100.

Queft. 20. If you were to borrow again to supply the third place, What then?

Answ. I should borrow 1. from the fourth place which is the place of thousands, and 1. in that place is worth ten of the third place, for ten times 100 is 1000, &c.

Queft. 21. What do you with that 1. which you account for ten every time you borrow?

Answ. I add it with the upper figure which was too little, and I subtract the lower figure from that sum, setting down the rest directly under such figure.

Queft. 22. Have you no other way?

Answ. Yes, In borrowing 1. (or 10.) I may subtract the lower figure from the 10. so borrowed, and add the rest to the upper figure which was too little, and I set the sum of such addition under the line, &c.

Queft. 23. How do you proceed in your Subtraction?

Answ. For every one so borrowed I add 1. to the next lower figure.

Queft. 24. Answer more particularly.

Answ. My meaning is, If in taking unites from

77

### Of Subtraction.

from Unites, I was forced to borrow 1 from the upper second figure, then afterward I should add the 1 so borrowed to the second lower figure.

Queft. 25. I understand you, you pay (or add) the 1 you borrowed to such place (or degree) of the lower number, as you did borrow from the uppermost: Is not that your rule?

Answ. Yes, when I must borrow 1 from the third place of the upper number, then I add 1 to the third place of the lowermost. If I borrow 1 from the fourth upper figure, I add 1 to the fourth lower figure, &c.

Queft. 26. What do you further observe in taking one number from another, when some figures of the upper number are greater, and some lesser than those of the lower number?

A. I must be careful in subtracting, that I do not make needles borrowings.

Queft. 27. What do you mean by that?

Answ. I mean thus, when I see that the upper figure is greater (or at least equal with the lower that I am subtracting) then I must never indebt my self by borrowing. But when the said upper figure is lesser than the undermost, then I must of necessity borrow 1, as is said before.

Queft. 28. But must you needs always pay (or add) the 1 so borrowed, to the next lower figure, is there no other way?

Answ. Yes, there is another way, much practised in France and Holland, but it is more difficult than the English way before mentioned; for many young beginners in those Countreys have been

78

Of Subtraction.

been often discouraged by it, especially when
they met with ciphers in the upper number.

Queft. 29. Declare that way in few words ?

Answ. When it is requisite to borrow as is
said before, they never add the I so borrowed
to the next lower figure; but instead of that they
reckon such upper figure (from whence the I is
borrowed) for an unite left.

Queft. 30. This way is harder indeed then the
former for young Learners, when they meet with
many ciphers together, but what must you do when
you meet with ciphers in the upper number, in working the English way ?

Answ. I follow the general rule, which is to
borrow I from the next upper figure, though it be
but a cipher.

Queft. 31. How can you borrow I from a
cipher, which is north nothing ?

Answ. It is true, that I cannot be taken from
nothing, but at least I do suppose that I borrow I
from that place, to encircle the upper figure
which is too little, from whence the lower is to
be taken.

Queft. 32. Then you think this no matter
where you borrow, if your want be but supplied ?

Answ. I do not think so; I go upon a better
ground; and if I must need tell you from whence
I borrow 1, when there are few or many ciphers
in the upper number, I say, that I borrow it
from the next significant figure towards the left
hand, in what place ever it stands.

Queft. 33. I perceive that when the next neigh-
boring

79

Of Subtraction.

boring figure is not able so lend, you go to the next
(towards the left hand) and if that figure cannot,
then you go yet further, till you finde a significant
figure. Is it so ?

Answ. I meant so.

Queft. 34. But if you did finde no significant
figure to borrow 1 ?

Answ. If it be so, then such Subtraction is im-
possible.

Queft. 35. When you finde ciphers in the lower-
most number, what do you do with them ?

Answ. This is not worth the asking or answering,
yet because some poor shallow brained beginners,
may, and do mistake sometimes in that,
I shall answer you. When ciphers are in the
lower number, then the upper figures (which are
right above the ciphers) must be set down under
the line as the rest, because, if nothing be taken
from something, the same thing remains still.

Queft. 36. If you finde ciphers in both the
numbers, that is ciphers above ciphers, what
then ?

Answ. If so, then I set down ciphers under the
line, saying, nothing from nothing there remains nothing.

Queft. 37. You have satisfied me touching
Subtraction Simple: But how do you go to work in
the Subtraction of divers Denominations ?

Answ. I set every number with the several
Denominations in their right places, thus; Pounds
under pounds, Shillings under shillings, Pence
under pence, Farthings under farthings; and so
of

80

Of Subtraction.

of all other Subfractions of Weights and Measures drawing a line under them.

Queft. 38. Where do you begin to Subfractions, at the Pounds, or at the Farthings ?

Answ. I begin always at the least Denomination, then I take Farthings from farthings, Pence from pence, Shillings from shillings, &c. setting every reft in its proper place.

Queft. 39. When the least Denomination cannot be taken from the uppermost least Denomination ?

Answ. Then I must go a borrowing to supply that want.

Queft. 40. Explain your meaning ?

Answ. If I cannot subfractions the lower Farthings from the upper, I go to the next Denomination, which is Pence; and from thence I borrow 1 Penny, which is 4 Farthings.

Queft. 41. What do you with these 4 Farthings so borrowed ?

Answ. I may dispose of them two manner of ways.

Queft. 42. How ?

Answ. I may adde them to the upper Farthings, and subfractions the lower Farthings from that sum, or else another way (which is easier, and therefore I intend to follow it) which is to subfractions first the lower Farthings, from the 4 so borrowed, and adde the upper Farthings to the reft, and set that sum down under the line.

Queft. 43. But if you find no Farthings in the upper number ?

Answ.

81

Of Subtraction.

Answ. Then I borrow 1 Penny (or 4¢) as before, and subfractions onely the lower farthings from it, and set the reft under the line.

Queft. 44. Proceed further.

Answ. For the Penny so borrowed, I adde it to the lower pence, and so go on in subfractions.

Queft. 45. If the pence of the upper number be 12¢, then the lower ?

Answ. Then I borrow 1 from the next Denomination, which is Shillings, which 1 3/4 is 12 9/4. Then I subfractions the lower pence (with the 1 9/4 borrowed, added to them) from 12, and adde the reft to the upper pence (if there be any) and set down the reft under the line in the place of pence.

Queft. 45. Go on, and tell me what you do with the Shilling borrowed ?

Answ. For the Shilling so borrowed, I adde it to the lower Shillings, and subfractions that sum from the upper Shillings.

Queft. 47. What if you cannot ?

Answ. Then I borrow 1 Pound (which is the next Denomination) which is 20 3/4, then I subfractions the lower Shilling (with the Shilling borrowed before) from 20, and adde the reft to the upper Shillings (if there be any) and I set down the sum under the line.

Queft. 48. Is that all ?

Answ. No, I carry 1 pound that I borrowed to the next Denomination (which is Pounds) and so proceed as I did in Subfraction Simple.

G

Queft.

82

### Of Subtraction.

Queft. 49. How will you subtract divers Denominations from one, as Pounds, Shillings, Pence, and Farthings from Pounds only ?

Answ. I go the same way as is mentioned before.

Queft. 50. How is it ?

Answ. I must borrow 1 9 (or 4 q) to supply the want of Farthings in the upper number.

Queft. 51. But where will you borrow it, seeing there is no Pence in the upper number to borrow ?

Answ. I borrow that Penny from the next Significant figure that I find, let it be in the place of Shillings or Pounds, it is no matter.

Queft. 52. Why is it no matter where you find the next Significant figure ?

Answ. Because the debt is paid presently when I come to the next Denomination.

Queft. 53. How do you subtract the Pence ?

Answ. I borrow 1 3 as before, which is 12 9, then 1 subtract the pence from 12, and set the rest under the line, and carry 1 3 to the place of Shillings, as is said before.

Queft. 54. How do you subtract the Shillings ?

Answ. I follow the Rule, which is to borrow 1 L (which is 20 3) and I subtract the Shillings from 20, and set the rest under the line, and carry 1 L to the place of L, subtracting the rest, as is taught before.

Queft. 55. You have given me satisfaction concerning this Rule; but I have two or three questions more to ask ?

Answ.

83

### Of Multiplication.

Answ. I am ready to Answer.

Queft. 56. How will you subtract two, three, or more small numbers from a great ?

Answ. I must add first all the small numbers together, and subtract the total sum from the greatest.

Queft. 57. If you were to subtract a sum from two, three, or more small sums ?

Answ. I should first add all the small sums together, and from the total sum subtract that sum which is to be subtracted.

Queft. 58. How will you subtract divers sums (or numbers) from another quantity of sums ?

Answ. I must first add all the sums of the debt together, and from that sum subtract the sum or total of all the other sums.

## Multiplication.

Multiplication teacheth how to multiply (or encrease) two numbers, the one by the other, by which a third number is produced, which contains in it self so many times one of the numbers, as there is unites in the other. In Multiplication there are three numbers to be observed.

1. The Multiplicand or the number to be multiplied, which is always set down first.

2. The Multiplier or the number by which

G 2

you

84

Of Multiplication.

you are to multiply, and is set down under the Multiplicand.

3. The Product, or the number produced by the Multiplication of the two former together.

Before the practising of Multiplication, it is very necessary to learn perfectly by heart the following Table, without which nothing can be done to any purpose.

The Table of Multiplication.

<table><tbody><tr><td rowspan="5">2 Times</td><td>2</td><td>4</td></tr><tr><td>3</td><td>6</td></tr><tr><td>4</td><td>8</td></tr><tr><td>5</td><td>10</td></tr><tr><td>6</td><td>12</td></tr><tr><td rowspan="4">7</td><td>8</td><td>14</td></tr><tr><td>9</td><td>16</td></tr><tr><td>10</td><td>18</td></tr><tr><td>11</td><td>12</td></tr><tr><td rowspan="5">3 Times</td><td>3</td><td>9</td></tr><tr><td>4</td><td>12</td></tr><tr><td>5</td><td>15</td></tr><tr><td>6</td><td>18</td></tr><tr><td>7</td><td>21</td></tr><tr><td rowspan="4">4 Times</td><td>8</td><td>24</td></tr><tr><td>9</td><td>27</td></tr><tr><td>10</td><td>30</td></tr><tr><td>11</td><td>32</td></tr><tr><td>12</td><td>34</td></tr><tr><td rowspan="5">5 Times</td><td>5</td><td>25</td></tr><tr><td>6</td><td>30</td></tr><tr><td>7</td><td>35</td></tr><tr><td>8</td><td>40</td></tr><tr><td>9</td><td>45</td></tr><tr><td rowspan="4">6 Times</td><td>6</td><td>36</td></tr><tr><td>7</td><td>42</td></tr><tr><td>8</td><td>48</td></tr><tr><td>9</td><td>54</td></tr><tr><td rowspan="5">7 Times</td><td>7</td><td>49</td></tr><tr><td>8</td><td>56</td></tr><tr><td>9</td><td>63</td></tr><tr><td>10</td><td>69</td></tr><tr><td>11</td><td>75</td></tr><tr><td rowspan="5">8 Times</td><td>8</td><td>72</td></tr><tr><td>9</td><td>80</td></tr><tr><td>10</td><td>88</td></tr><tr><td>11</td><td>96</td></tr><tr><td>12</td><td>104</td></tr><tr><td rowspan="5">9 Times</td><td>9</td><td>100</td></tr><tr><td>10</td><td>108</td></tr><tr><td>11</td><td>116</td></tr><tr><td>12</td><td>124</td></tr><tr><td>13</td><td>132</td></tr></tbody></table>

Observe

Of Multiplication.

Observe the figures which have the word Times adjoyning to them, and read the Table thus; Two times two is four; two times three is fix; two times four is eight, &c.

Note, Two numbers being given to be multiplied together, it matters not which of the two is made the Multiplicand or Multiplier, for the Product will be still the same; yet for decency sake, and sometimes for the easinels of the working of this Rule, the number which hath most figures, is set above the other, and the lesser under it, in such order as shall be shewn in its place.

When the Multiplicand and Multiplier conſift each of them of one figure onely, such Multiplication is called Single; but if any of the ſaid number, or both, conſift of more figures then one, such Multiplication is called Compound.

How to work Multiplication.

The work of a Single Multiplication is very eaſie, the Multiplicand and Multiplier conſiſting each of them but of one Significant figure, the foregoing Table is a sufficient direction, therefore I shall set down here one example onely.

I. Example.

If 6 be given to be multiplied by 4, what is the Product ? facit 24 as follows.

G 3

6[M]l-

85

86

Of Multiplication,

6 Multiplicand.

4 Multiplier.

24 Product.

I say 4 times 6 is 24, which 24 I set down under the line, and is the Product required.

When the Multiplicand consist of more Significant figures then one, and the Multiplier of one only,

First, Set down the Multiplicand and the Multiplier under it in such order, as in Addition and Subtraction, that is to say, the Unites under the Unites, &c. and draw a line under them.

Secondly, Multiply the first figure of the Multiplicand (towards the right hand) by the Multiplier, and set the Product under the line (in its right place) then go to the second figure of the said Multiplicand, and multiply it by the Multiplier, as before, and set down the Product under the line (right under that second figure of the Multiplicand) proceed so until you have multiplied all the figures of the Multiplicand as follows.

2. Example.

87

Of Multiplication.

2. Example.

In multiplying 123 by 2, what is the Product 2 facilitate 246.

123
2
---

Facilitate 246 Products.

I say twice 3 is 6, then twice 2 is 4, and lastly, twice 1 is 2. These three Products 6: 4: and 2: I set down under the line, right under the figures of the Multiplicand, thus, The 6 under 3, the 4 under 2, and the 2 under 1, as you see before.

Thirdly, When you find (in multiplying the Multiplicand) that the Products of any particular figure exceed 9, then you must not follow the Rule of the second Example, but set down (as in Addition Simple) that which is above 10: 20: 30, &c. and carry so many tens in mind; and for every 10. carry one to the next Product of the next figure. And if you, find nothing over the tens, as, just 10: 20: 30, &c. then set a cipher and carry the tens as before, as in the following Example.

3. Example.

Multiply 6485
by 6

38910

I say 6 times 5 is 30, I set a cipher under the line G 4

88

Of Multiplication.

line (right under 6) and carry 3 to the Product of
the next figure, saying, 6 times 8 is 48, and 3
that I carried is 51. I set down 1 and carry 5 to
the Product of the next figure, as before, saying,
6 times 4 is 24, and 5 that I carried is 29. I set
down 9 and carry 2. Lastly, I say, 6 times 6 is
36, and 2 I carried is 38. I set down 8 and the
3 a place further ; the whole Product found under
the line is 38910.

4. Mul iply 876 by 6, facit 5256.

5. Item, Multiply 69 by 7, facit 6083.

6. Item, Multiply 4864 by 8, facit 38912.

7. Item, Multiply 40806 by 8, facit
326448.

3. Item, Multiply 897408 by 9, facit
8076672.

When the Multiplicand and Multiplier
confist of more Significant figures then
one.

WHEN the Multiplier conſift of more
figures then one, then obſerve that ſo
many Significant figures as the ſaid number hath,
ſo many ſeveral Products muſt be ſet under the
line, which muſt be ſet down under one another
to be added in the manner following.

First, Set down first the number which hath
moſt figures, and that will be the Multiplicand,
and set under it the leſſer in the order before-
mentioned, drawing a line under them; then be-
gin

39

Of Multiplication.

gin with the first figure towards the right hand,
proceeding as in the ſecond Example before-
mentioned.

Secondly, If the Multiplier conſift of two
figures, go to the ſecond figure, and multiply the
Multiplicand by it, as you did with the first, and
set down the ſecond Product (which proceed from
the ſecond figure) under the first, but not directly
under (as is taught in Addition, becauſe the ſe-
cond figure of the Multiplier is in the place of
Tens) but a place further, towards the left hand,
thus, The Unites of the ſecond Product muſt be
placed under the tens of the first Product : Fur-
ther, the tens of the ſecond Product muſt be pla-
ced under the Hundreds of the first, &c. going
always a place further, then add the ſecond Products
togethcr; the ſum is the number required, as in
the following Example.

9. Multiply 2131 by 12, facit 25572.

2131

I2

The first Product— 4262

Second Product— 2131

Whole Product — 25572

Thirdly, If the Multiplier conſift of three
Significant figures, you muſt make three ſeveral
Products, and having ſet down the two first, as is
ſaid before, set the third yet a place further then
the

90

Of Multiplication.

the second thus. Set the Unites of the third Product under the tens of the second, the tens of the third Product under the Hundreds of the second, &c. then add the Products together; the sum is the number required, as in the following Example.

10. Multiply 3122 by 123, facit 384006.

$$
\begin{array}{r}
3122 \\
123 \\
\hline
\end{array}
$$

First Product-9366
The second-6244
The third-3122

Whole Product-384006

Fourthly, If the Multiplier conflict of 4 figures or more, you must have so many Products, and place them as in the following Example.

11. Multiply 31231 by 3212, what is the Product? facit 100313972.

$$
\begin{array}{r}
31231 \\
3212 \\
\hline
\end{array}
$$

The first Product 62462
The second 31231
The third 62462
The fourth 93693

Whole Product-100313972

Fifthly, When you find (in multiplying) that the

91

Of Multiplication.

the Product of any particular figure exceeds 9,
then do as is taught in the third Example, and set
down the several Products, as in the Ninth, tenth,
and eleventh Question (or Proposition) of this
Rule, then add them together, as in the follow-
ing Example.

12. Multiply 456 by 34, facit 19504.

$$
\begin{array}{r}
  456 \\
  34 \\
 \hline
 1824 \\
 1368 \\
 \hline
\end{array}
$$

Facit—15504

13. Item, Multiply 8768 by 22, facit 192896.

14. Items, Multiply 595 by 47, facit 27965.

15. Item, Multiply 7891 by 71, facit 560261.

16. Item, Multiply 198468 by 98, facit 19449864.

17. Item, Multiply 64873 by 472, facit 30620056.

18. Item, Multiply 45624 by 527, facit 24043848.

19. Item, Multiply 46471 by 764, facit 35503844.

20. Items, Multiply 123456789 by 6456, what is the Product? facit 797037029784.

How

92

Of Multiplication.

How to multiply two numbers, when one or
both ends with ciphers.

First, If the *Multiplier* only ends with ciphers
having only an unite for its Significant figure,
the labor is not great, which is only to joyn the
ciphers to the *Multiplicand* on the right hand,
and the work is done as followeth.

21. Multiply 32 by 10, facit 320.

22. Item, Multiply 341 by 100, facit 34100.

23. Item, Multiply 19 by 10000, facit 190000.

Secondly, If the Significant figure be more
than an unite or 1, or that the Multiplier conſift
of more figures then 1, ending alſo with ciphers,
then you muſt multiply with the Significant figure
as before, and annex (or joyn) the ciphers to that
Product as followeth.

24. Multiply 426 by 3000, facit 1278000.

426

3000

Facit—1278000

25. Multiply 184123 by 90000, facit 16571070000.

26. Item, Multiply 821 by 120000, facit 98520000.

821

93

Of Multiplication.

821
120000
---
1642
821
---

Facit—98520000

27. Item, Multiply 346 by 2700000, facit 9342000000.

Thirdly, When both the *Multiplicand* and
*Multiplier* ends with ciphers, then place their
Significant figures under one another in the man-
ner before taught, Fol. 89 and 90; and when the
Multiplication of the Significant figures is finisf-
ed, annex (on the right hand of that Product) all
the ciphers which are found in both the numbers,
and the work is ended, as follows.

28. Multiply 1620000 by 300, facit 486000000.

1620000
300

Facit—486000000

29. Item, Multiply 8530000 by 64000, facit 545920000000.

30. Item, Multiply 23500 by 1200000, facit 28200000000.

How

94

Of Multiplication.

How to multiply two numbers, when one or both have ciphers between Significant figures.

First, When in the Multiplicand onely you finde any cipher or ciphers together, between the Significant figures, multiply the Significant figures as before; and when you come to the ciphers, set so many 0 in the Product of that particular figure of the Multiplier, as you finde in the Multiplicand, if you carry nothing; but observe, if you carry any tens, set down what you carry in the place where the first 0 should stand, and set in the Product of that figure a cipher left then there is in the Multiplicand, as in the following Examples.

31. Multiply 30002 by 32, facit 960064.

32. Item, Multiply 40006 by 128, facit 5120768.

$$ \begin{array}{r@{\,}c@{}c@{}c@{}c@{}c} & & & & & \\ 30002 & \big) & 40006 & & & \\ 32 & \multicolumn{1}{r}{\big)} & 128 & & & \\ \cline{2-6} & & & & & \\ 60004 & \multicolumn{1}{r}{\big)} & 320048 & & & \\ 90006 & \multicolumn{1}{r}{\big)} & 80012 & & & \\ \cline{2-6} & & 40006 & & & \\ \end{array} $$
$$ \text{Facit-960064} \qquad \text{Facit-5120768} $$

33. Multiply 80609 by 795, facit 64084155.

Secondly,

95

Of Multiplication.

Secondly, When in the Multiplier onely you finde a cipher or ciphers together between Significant figures. Omit such ciphers, or else set so many 0 in their places, and observe well to set each particular Product in its right place, as is taught in the Ninth, tenth, and eleventh Examples, as followeth.

34. Multiply 3423 by 403, facit 1379469.

35. Item, Multiply 248172 by 30002, what is the Product? facit 7445656344.

$$ \begin{array}{r@{\,}c@{}c@{}c@{}c@{}c} & & & & & \\ 3423 & \multicolumn{1}{r}{\big)} & 248172 & & & \\ 403 & \multicolumn{1}{r}{\big)} & 30002 & & & \\ \cline{2-3} & & 10269 & \multicolumn{1}{r}{496344} & & \\ 13692 & \multicolumn{1}{r}{\big)} & 744516000 & & & \\ \cline{2-4} & & & & & \\ \end{array} $$
$$ \text{Facit-1379469} \qquad \text{Facit-7445656344} $$

Thirdly, When both in the Multiplicand and Multiplier, you finde ciphers included between Significant figures, proceed as in the Thirty two and thirty five Examples, taking diligent care to place the several Products in their right places; for it is an error committed very often by young beginners, not to set the Products in their right places: Therefore let them observe the two following Examples.

36. Multiply 30016007 by 3002, facit 90108053014.

37. Item, Multiply 10304040901 by 1020304, facit 10513254147453904.

300-

96

Of Multiplication.

30016007
3002
---
60032014
9004802100
---

Facit-90108053014

10304040901
1020304

41216163604
309121227030

206080818020

103040409010

Facit-10513254147453904

38. Multiply 40060 by 800800, what is the Product? facit 320512048000.

39. Item, Multiply 60000009 by 60000009, facit 3600001080000081.

40. Item, Multiply 700104 by 2004, facit 1403008416.

41. Item, Multiply 49140475 by 538064, facit 26440720540400.

42. Item, Multiply 11234567 by 16848460, facit 189285152716820.

The

Of Multiplication.

97

The Proof of Multiplication.

The most certain Proof of Multiplication is done by Division, which is the next thing to be learned; and therefore I thought not to mention any Proof of Multiplication, until I came to treat of Division. But to satisfy those that have a minde to prove this Rule without Division, I shall make use of the common Proof of Multiplication (which is by casting away all the nines) for want of a better. The Rule is as followeth.

1. Cast away all the nines which are to be found in the Multiplicand, and what remains, set on one side of a little cros; if nothing remains, set a cipher.

2. Cast away also all the nines of the Multiplier, and what remains, set on the other side of the cros; if nothing remains, set a cipher.

3. Multiply these two Remainders together, and cast away the nines of the Product, and set the rest over the cros; if nothing remain, set a cipher.

4. Cast away all the nines of the Product of the Multiplication, and the rest must be the same with the figure which stands over the cros; if not, the work is false. As by Example,

H

Multiply

98

Of Multiplication.

<table><thead><tr><th></th><th>Proof</th></tr></thead><tbody><tr><td>Multiply 4872 by 76</td><td>3<br>3 + 4<br>3</td></tr><tr><td>29232</td><td></td></tr><tr><td>34104</td><td></td></tr><tr><td>370272</td><td></td></tr></tbody></table>

I cast away the nines of the Multiplicand first,
saying, 4 and 8 is 12, cast away 9, there remains
3; then 3 and 7 is 10, cast away 9, there remains
1. Further 1 and 2 is 3, which 3 I set on the
left side of the crofs, as you see above.

Secondly, I go to the Multiplier, and cast a-
way the nines in the same manner, saying, 7 and
6 is 13, cast away 9, there remains 4; which 4
I set down on the right side of the said crofs.

Thirdly, I Multiply these two Remainders to-
gether, saying, 3 times 4 is 12, cast away 9,
there remains 3; which 3 I set on the top of the
crofs.

Lastly, I cast away the nines of the Product,
saying, 3 and 7 is 10, cast away 9, there remains
15, then 1 and 6 is 1, and 2 is 3, and 7 is 10,
cast away 9, there remains 1. Further, 1 and 2
is 3, which 3 I set down at the bottom of the
crofs. Now, seeing that the two Remainders
(which are at the top and bottom of the crofs)
are equal, that shows that the Multiplication is
right.

Note,

99

Of Multiplication.

Note, This Proof is not to be trusted, for
sometimes the Proof comes right, when the work
is very false. As by example, Suppose that a
Learner in multiplying, should set down 46 in-
stead of 64, or 65 for 56, such gross mistakes
doth not hinder that this Proof should come
right.

Yea, further, Suppose that a greater error
should be committed in this manner, if that
number which is to be set down, was carried in
minde (in multiplying) and that which is to be
carried, should be set down (thus, If for 39, three
were set down, and the 9 carried in minde, &c.)
Such fad mistakes doth not yet hinder this kinde of
Proof to come right: Which thing proves the
great uncertainty of this Proof.

Questions of Multiplication.

Queft. 1. What is Multiplication?

Answ. Multiplication is to multiply (or encreafe) two numbers together, to finde a third.

Queft. 2. In how many parts is this Rule di-
vided?

Answ. In two, Single and Compound.

Queft. 3. What is the first?

Answ. The first is called Multiplication Single.

H 2

Queft.

100

Of Multiplication.

Queft. 4. Why is it so called?

Answ. Because the two numbers to be multiplied, are but single figures.

Queft. 5. Give an example.

Answ. If 4 were to be multiplied by another single figure, as 7 or 8; such Multiplication is called Single or Simple.

Queft. 6. What is Multiplication Compound?

Answ. It is so called, when one or both the numbers to be multiplied, conſt of more figures then one.

Queft. 7. What is to be obſerved in Multiplication?

Answ. There are three numbers to be minded.

Queft. 8. What are they?

Answ. The Multiplicand, the Multiplier, and the Product?

Queft. 9. What is the Multiplicand?

Answ. It is commonly the number which is set down first to be multiplied.

Queft. 10. What number is called Multiplier?

Answ. The Multiplier is the number by which the Multiplicand is multiplied.

Queft. 11. May these two first numbers exchange their names?

Answ. Yes; you may chuse which you please to be the Multiplicand, and the other shall be the Multiplier.

Queft. 12. What number do you rather chuse of the two, for the Multiplicand?

Answ. I chuse the firſt number to be the Multiplicand,

101

Of Multiplication.

Multiplicand, which is the number that hath moſt figures

Queſt. 13. Why is that number the firſt?

Answ. Because, if I take the number which hath moſt figures for the Multiplier, the work will be greater then otherwife.

Queſt. 14. Make that plain?

Answ. The whole Multiplicand muſt be multiplied by each figure of the Multiplier; and therefore the more figures that number hath, the more Products there will be.

Queſt. 15. What do you mean by Product?

Answ. The Product of a Multiplication, is the third part of it, as is ſaid before.

Queſt. 16. Explain that?

Answ. When two numbers are multiplied together, the number which is produced from ſuch Multiplication, is called Product.

Queſt. 17. Give an example.

Answ. If 9 were multiplied by 4, ſaying, 4 times 9 is 36. Now obſerve that 9 is the Multiplicand, 4 is the Multiplier, and 36 is the Product.

Queſt. 18. To what use is Multiplication?

Answ. Multiplication ſerveth inſtead of many Additions.

Queſt. 19. Shew it.

Answ. In multiplying 8 by 7, the Product is 56. Now if I set down seven times 8 one under another, as in Addition, and adding them together, the ſum will be 56, as before.

H 3

Queſt.

102

Of Multiplication.

Queft. 20. What doth contain the number you call Product?

Answ. The Product contains in it self so many times one of the two other numbers, as there is Unites in the other.

Queft. 21. Explain what you say, in giving an example.

Answ. If 6 be multiplied by the 4, the Product is 24. Now 24 contains 6 four times, or else it contains 4 fix times.

Queft. 22. What is the next thing to be known?

Answ. The next thing is to practise Multiplication.

Queft. 23. Is there nothing to be learned before that?

Answ. Yes; the Table is to be learned perfectly by heart.

Queft. 24. What is to be observed in multiplying a number by another?

Answ. I observe to let down the Multiplicand first, and the Multiplier under it.

Queft. 25. In what manner?

Answ. The Unites of one number under the Unites of the other, and so Tens under Tens, &c. drawing a Line under them.

Queft. 26. Where do you begin to multiply?

Answ. I begin at the right hand, at the place of Unites.

Queft. 27. Go on.

Answ. I multiply all the figures of the Multiplicand by every figure of the Multiplier (one after another) and let down the Products

103

Of Multiplication.

ducts under the Line, and then I add them together.

Queft. 28. Speak more particularly?

Answ. I shall then begin to speak of such Multiplications, whose Multiplier is but a single figure.

Queft. 29. Go on in the plainest way to be understood.

Answ. When the Multiplier consists of one figure onely, I multiply the whole Multiplicand with that figure, and I let the Products of every figure under the Line, as in the second example of Multiplication, Fol. 87.

Queft. 30. But if the Product of each figure so multiplied, doth exceed 9, what then?

Answ. I must let down what is above 10, 20, 30, &c. and carry so many tens in minde; and for every 10 I carry one to the Product of the next figure, as in the third example Fol. 88.

Queft. 31. What if the Product of each figure ends with a cipher, as 10, 20, 30, &c.

Answ. Then I let a cipher under the Line, and carry so many tens as before.

Queft. 32. If the Multiplier consist of more figures than one.

Answ. I must let down so many several Products under the Line, as there are figures in the Multiplier.

Queft. 33. If the Multiplier consist of four figures?

Answ. I must let down four several Products under the Line to be added together.

H 4

Queft.

104.

Of Multiplication.

Queft. 34. In what manner do you p'ace your
several Products.

Answ. The second Product must not be set
down directly under the first (as the numbers in
Addition) but it must stand a figure (or place) fur-
ther towards the left hand, the third Product yet
a place further, &c.

Queft. 35. What is the reason of that?

Answ. The second Product ariseth from the
Multiplication of the second figure of the Mul-
tiplier, which is the place of Tens, and therefore
the Product of that figure must be a place fur-
ther (towards the left hand) then the first Pro-
duct.

Queft. 36. Why is the third Product, set a
place further then the second?

Answ. Because the figure, from whence such
Product ariseth, is in the third place in the Mul-
tiplier, which is the place of Hundreds.

Queft. 37. You do then observe of what degree
or place, is every figure in the Multiplier?

Answ. I do observe for a general Rule, that
the first figure (towards the right hand) of any
Product (which is set under the Line) must be set
down directly under that figure (which is in the
Multiplier) from whence such Product is produ-
ced; and the other figures must stand towards
the left hand.

Queft. 38. When all the several Products are
set in their right places, what is more to be done to
finish the Multiplication?

Answ. There is nothing to do, but to add the
several

Of Multiplication.

105

several Products into one sum, and that sum is the
whole Product required.

Queft. 39. What course do you take, when the
Multiplicand and Multiplier do end with
ciphers?

Answ. The general Rule holds true, as you may
see in the Twenty sixth and Twenty eighth Ex-
amples of Multiplication, Fol. 92 and 93. And
though some ciphers are omitted in the Products
of the Twenty sixth Example for shortneſs fake,
yet they are annexed to the general (or whole)
Product.

Queft. 40. But if you finde ciphers between
Significant figures, in the Multiplicand and Mul-
tiplier, what must you do?

Answ. The general Rule given in the Anſwer
of the Thirty seven Question, may suffice. I re-
fer you to the operation of the 31, 32, 34 and 35
Examples of Multiplication, where you may see
the laid Rule verified.

Division

106

Division.

Division teacheth to finde how many times
one number is contained in another; or
how to divide a number propounded into any e-
qual parts assigned.

In Division there are chiefly three Numbers to be observed.

1. The Dividend or the Number to be divided.

2. The Divisor, or the Number by which the Dividend is to be divided.

3. The Quotient or the Number produced by the Division, which shews how often the Divisor is contained in the Dividend.

Some adde a fourth Number, which is the Rest
or Remainder; but this Number is not always to
be found in Division, as the three former are; for
oftentimes in dividing, there is no Rest.

The Learner may take notice by the way, that
the Rest (or Remainder) of any Division, is,
and ought to be always left then the Divisor :
And if it happen to be either equal, or more
then the Divisor, that shews that the Quotient is
too little, and the work false.

Division is the true Proof of Multiplication,
and is more difficult then any Rule before taught,
and therefore the Learner ought to double the
File of his diligence, and observe well the Rules
and

107

Of Division.

and Examples that follows, which will make the way smooth to the meanest capacity.

There are divers ways of performing Division,
but I shall content myself to treat of two only.
But before any dare venture upon the practice of
this part of Arithmetick, he must be well versed
(or acquainted) in Subtraction and Multiplication;
in the knowledge of which, Division doth
wholly depend. Observe what followeth in dividing.

1. Set down first the Dividend, and the Divisor
under it, not in the same manner as the two
numbers of Multiplication are set, but quite con-
trary : For in that Rule, the Multiplier is set on
the right hand, but the Divisor (in this Rule)
must be set on the left hand of the Dividend,
thus. Place them so, that the figures towards the
left hand of each number, be directly one under
the other , if the figures of the Dividend (which
stand right above the whole Divisor) will permit it;
For if all those said figures of the Dividend be left
in value then the whole Divisor, then the said
Divisor must not be placed so, but a place further
towards the right hand, and draw a crooked line
on the right side of the Dividend.

2. Having placed the Divisor in its right place,
consider well how many times your Divisor is
contained in the figures above it, which being
known, set down that figure (which answers the
question) in the crooked line, which is the place
of the Quotient.

3. The

108

Of Divifon.

3. The whole Divifor muſt be multiplied with that figure, which is placed in the Quotient, and the Product muſt be subſtracted from the figures which are above the ſaid Divifor, in this manner.

Multiply first, the first figure (towards the left hand) of your Divifor, by the figure in the Quotient, and subſtract that Product from the figures which are above the ſaid figure so multiplied. If you find any Rest, set it above, and forget not as you go along, to cancel your figures that have been used, to prevent error and mistake.

Further, Multiply the next figure of your Divifor, by the same figure in the Quotient, and subſtract the Product from those figures which are above the ſaid figure so multiplied, as before. Proceed, and do the like with the third figure, and so with the reſt of the Divifor, taking good heed to set the ſeveral Reſts in their right places.

4. Observe, That so many times as the Divifor can be removed under the Dividend, so many figures you muſt have in the Quotient. Having then finished the work of the first figure of your Quotient, remove the Divifor a place further towards the right hand (if you ſee ſome figure of the Dividend untouched) then obſerve again, how many times you can have the lower figures in the upper, whether 0: 1: 2: 3: 4: 5: 6: 7: 8: or 9 times (for 9 is the highest figure you can take at once,) which thing being known, set

109

Of Divifion.

set the figure (which anſwer the queſtion) in the Quotients; if it be a cipher (because the Divifor could not be had once in the upper figures) then cancel the whole Divifor, and not one of the upper figures, and remove it again towards the right hand (if you ſee any figure more of the Dividend untouched;) but if the figure (which anſwers the queſtion) be significant, multiply the Divifor by this new figure, and subſtract the Product in the ſame manner before taught, fetting the reſt above in its right place. If you ſee that there is still some figure of the Dividend untouched, remove the Divifor a place further, as before (that is, but one place at once further) until the Unites in the Divifor stand under the Unites of the Dividend.

5. When the Divifor ends with ciphers, do as followeth, for ſhortneſs ſake.

If the Divifor conſift of an Unite only in the first place, towards the left hand, and nothing but ciphers towards the left, cut off so many places of the Dividend towards the right hand, as the Divifor hath ciphers; which figures ſo cut off, are the reſt of the Division, as in the Fiftieth Example.

But if the Divifor conſift of one or more significant figures, and nothing but ciphers towards the right hand, set down all the ciphers under the Dividend towards the right hand, and divide the remaining part of the Dividend (on the left hand) by the remaining part of the Divifor in the manner.

110

Of Division.

manner before taught, as in the Fifty two Ex-
ample.

6. If the Divisor be greater then the Divi-
dend; observe, that such Division will produce a
Fraction, and no whole number; which thing
shall be explained in its proper place.

How to divide by one figure.

1. Example.

If 8 be divided in two equal parts, what will
each part be? facit 4.

Set the Dividend (8) first, and the Divisor (2)
under, drawing a crooked line on the right side, as
followeth.

Dividend 8 ( 2 )
Divisor 2

I see how many times the Divisor 2, is con-
tained in the Dividend 8, I find 4 times, ther-
fore I set 4 in the crooked line, which is the place
of the Quotient, thus,

8 ( 4 )
2

Then I multiply the Divisor 2, by the Quotient
4, saying, 4 times 2 is 8; which 8 I subtract
from 8 above, there remains nothing. But you
must observe to prevent mistakes in great Divis-
ors, that you do not forget to cancel your figures
with

III

Of Division.

with a dash of the Pen, as soon as you have
made use of them: As in this first Example, as
soon as I have set the 4 in the crooked line, I
multiply the Divisor 2 by it, and in saying, [4
times 2 is 8] then presently I cancel the 2 thus
&c; and having added these words [from 8] I
cancel the 8 thus 8, and say, there remains no-
thing, thus,

Dividend 8 ( 4 ) Quotient.
Divisor 2

2. Example.

In dividing 9 by 4, what is the Quotient ? facit
2, Reft 1.

9 ( 2 )
4

I ask how many times 4 I can have in 9? the
Answer of the Question is 2, which I set in the
place of the Quotient; then in saying (in the
fame manner beforementioned) 2 times 4 is 8,
I cancel 4 thus 4; and adding these words
['from 9'] I cancel 9 (thus 9,) and having said
['there remains 1'] I set down 1 above 9 so can-
celled; and so the work is finished, as follow-
eth.

(1) Reft.

9 ( 2 ) Quotient, Reft 1.
4

When

112

Of Division.

When you have finished your Division, and find-
ing any Remainder separate it from the figures,
cancelled with a crooked line, as you see afore
(which Rest must be always left then the Divisor)
and set it in the Quotient above a little line thus,
(1) and the Divisor under it thus (1/4) which is a
Fraction called a quarter, or a fourth. The
whole Quotient of this second Example is then
2⅓, which is two wholes and a quarter : But I
shall trouble the Learner no further, concerning
the name of the Number-Resting in any Division,
until I come to treat of the Fractions (or broken
numbers) I shall content my self to call such
Resting-Number, Remainder or Rest.

Further, I shall addle a *Nota* by every variety,
or change in the working of *Division*, to supply
the brevity of the Rules before delivered; and
also to remove the flumbling blocks, that the
Learners usually meet withal in this part of *A-
rithmetick*, which is the great *Remora* that hath
stayed many, and discouraged others from pur-
fusing their intended course in this Noble Art of
*Numbring*. Therefore I desire to be excused, if
I repeat somethings already mentioned.

1 Note.

If the Dividend conflict of more figures then
one, and that the first figure (towards the left
hand) be less then the first figure of the Divisor
(towards the same hand) then set your Divisor 2
place further towards the right hand, as follow-
eth.

3 Examples.

113

Of Division.

3. Example.

In dividing 17 by 5, what is the Quotient?
faciſt 3⅔.

(2) Rest.

7 (3⅔ Quotient.

5

I perceive that the first figure of the Divi-
dend (towards the left hand) is but 1, and the
Divisor is 5; therefore I place the said Divisor a
place further, to wit, under 7, and I proceed as
in the first and second Example.

4 Example.

If 8980 be divided into 4 parts, what will
each part be? *faciit 2245.*

2 Note.

Every Unite which is set above the Dividend
as Rest, when another figure, or a cipher, is found
on the right side thereof, is accounted for so many
Tens; that is, if 1 remains, it is worth 10, if
2, 20, &c. as in the following Example.

8980 (2

44

I ask in 8, how many times 4, the Answer is
2, which 2 I set in the Quotient; then I multiply
faying, 2 times 4 is 8, from 8 there remains

I

nothing;

114

Of Division.

nothing; then I remove the Divisor (4) a place further, as you see above, asking again, In 9 how many times 4? Answer, 2. I set 2 in the Quotient again, and multiply as before, sayin9, 2 times 4 is 8, from 9 there remains 1; which 1 I set above the 9, and remove the Divisor again, thus.

$$
\begin{array}{r}
  1 \\
  8980 \\
  \times \quad 22 \\
  \hline
  444
\end{array}
$$

The 1 which remains (which must be joyned to 8, the figure above the Divisor 4) is worth 10. Therefore I ask again in 18, how many times 4? Answer, 4, which I set for the third figure in the Quotient, and I multiply, sayin9, 4 times 4 is 16, from 18 there remains 2, which 2 I set above 8, having cancelled 18 first, and I remove the Divisor a place further, thus.

$$
\begin{array}{r}
  12 \\
  8980 \\
  \times \quad 224 \\
  \hline
  4444
\end{array}
$$

The 2 which remains (according to the second Note) is 20. Now if the figure which is on the right side of the said 2, had been a Significant figure, as 4, I should have asked, in 24 how many times 4? but seeing that it is but a cipher, I say, in 20 how many times 4? Answer, 5, which 5 I set in the Quotient, for the fourth figure of it; and I multiply as before, sayin9, 5 times

4 is

115

of Divifion.

4 is 20, from 20 there remains nothing, the whole Quotient is 2245, as followeth.

$$
\begin{array}{r}
  22 \\
  8980 \\
  \times \quad 2245 \quad \text{Quotient} \\
  \hline
  4444
\end{array}
$$

Example.

Divide 18858 by 5, facit 3771⅔.

$$
\begin{array}{r}
  33(3) \\
  28858 \\
  \times \quad 3, \text{facit } 186 \\
  \hline
  5555
\end{array}
$$

6. Item, Divide 558 by 2, facit 279.

7. Item, Divide 558 by 3, facit 186.

8. Item, Divide 765 by 3, facit 255.

9. Item, Divide 988 by 4, facit 247.

3 Note.

If in dividing you finde that you cannot have the Divisor once, in some of the upper moſt figures (which are above it) then set a cipher in the Quotient, and cancel the Divisor, and remove it a place further, without cancelling any figure above it.

Further, having removed your Divisor, and if you find still that you cannot have the said Divisor once in the figures which are above it, then set again a cipher in the Quotient, and can- cel the Divisor as before, proceeding so until

I 2

your

116

Of Division.

your Divisor comes to be set under some Significant figure, if there be any in the Dividend; if not, the Quotient will end with ciphers, as in the Tenth and eleventh Example.

10 Example.

Divide 1500008 by 6, facit 250001⅙.

$$ 
\begin{array}{r}
666 \\
\times 66666 \\
\hline
\end{array}
 $$

Having set down the Divisor the third time, I finde but a cipher above it; now seeing I cannot have once 6 in nothing, I set a cipher in the Quotient, and cancel 6, the Divisor and the cipher above it (because there is no Significant figure uncancelled on the left side of the said cipher) and I remove the Divisor again thus.

$$ 
\begin{array}{r}
6666 \\
\times 66666 \\
\hline
\end{array}
 $$

Having set the Divisor the fourth time, I perceive a cipher above it, as before; therefore I set a cipher in the Quotient, and cancelling the 6 and the cipher, I remove the Divisor the fourth time thus.

$$ 
\begin{array}{r}
66666 \\
\times 66666 \\
\hline
\end{array}
 $$

Having

Of Division.

Having set the Divisor the fifth time (or removed it the fourth time) I perceive still a cipher above it; therefore I proceed as before, in setting a cipher in the Quotient, and cancel 6, and the cipher above it, and I remove the Divisor for the last time, thus.

$$ 
\begin{array}{r}
666666 \\
\times 66666 \\
\hline
\end{array}
 $$

Having removed the Divisor the fifth time, I perceive 8 to be above it; therefore I ask in 8, how many times 6? Answer, Once. I set 1 in the Quotient, and multiply as before, saying, once 6 is 6, from 8 there remains 2, which Rest I set above 8; the whole Quotient of this Division is 250001⅙, as followeth.

11 Example.

Divide 842400 by 6, facit 104400.

$$ 
\begin{array}{r}
666 \\
\times 842400 \\
\hline
\end{array}
 $$

Having set the Divisor (6) under the Dividend thrice (or removed the said Divisor twice) I finde it under 2: Now seeing I cannot have 6 in 2, I

I set

117

118

Of Divifon.

fet a cipher in the Quotient, and cancel 6, and
not the 2, because it is a Significant figure, which
must be accounted for 20 when the Divisor is re-
moved a place further, as followeth.

842400 (1404

6666

Having removed the Divisor the third time, I
finde 24 to be above it; therefore I ask in 24,
how many times 6? Answer, 4 times. I set 4 in
the Quotient, and multiply, saying, 4 times 6 is
24, from 24 there remains nothing. I cancel
first the 6, and then 24, and remove the Divisor
further, thus.

x
842400 (1404

66666

I perceive that all the Significant figures of the Dividend are cancelled, and that ciphers onely remains; therefore I set so many ciphers in the Quotient, as the Divisor (6) can be placed under the Dividend, and the Division is finished as followeth.

2
842400 (140400 Quotient,

666666

12. Divide 76000 by 5, facit 15200.

13. Item, Divide 9864 by 8, facit 1233.

14. Item,

119

Of Divifion.

14. Item, Divide 2712 by 8, facit 339.

15. Item, Divide 17163 by 9, facit 1907.

16. Item, Divide 314001 by 9, facit 34889.

17. Items, Divide 4697 by 9, facit 521

Reft 8.

18. Item, Divide 98303 by 6, facit 16383½.

19. Item, Divide 100093 by 8, facit 12511⅝.

20. Item, Divide 48120 by 6, facit 8020.

21. Item, Divide 36018 by 9, facit 4002.

How to Divide by two or more Figures.

The Learner being well acquainted how to
divide by one single figure, will soon attain
to the knowledge of the reft; if the following
Directions and Examples be well understood.

4 Note.

When the Divisor conflict of more figures then
one, set it down under the Dividend, so that it
may be contained once at least in the upper figures,
then ask how many times the Divisor is contain-
ed in those figures which are above the said Di-
visor, which thing being known, set that figure
(which answers the question) in the Quotient.

But to make that more eafe (specially when the
Divisor conflict of many figures.) Confider
first how many times the first figure (towards the
left hand) of your Divisor can be found in that
figure (or figures) that are above it (which figure

are

I 4'

120

Of Division.

are two at the moft.) Now the greatest difficulty
in Division lieth in the answering of such Questi-\ons aright: For, sometimes (according as the
figures are dispoſed) the right anſwer cannot be
found without making ſome trial; whether the
reſt of the figures of the Divisor can be taken
out of the upper figures (which are above them)
as often as the firſt figure is; therefore the
Learner muſt have an eye to the reſt of the
figures of the Divisor as well as to the firſt, and
have alſo a ſpecial care to hit right the mark;
that is, to take that figure (which is to be ſet in
the Quotient) that anſwers the queſtion, neither
too great, nor too little; which thing may easily
be discerned by theſe two following Observa-
tions.

1. Having set down the figure in the *Quotient* (which you judge will answer the question demanded) and multiplied all the figures of the *Divisor* (one after another) and subtracted the Products from the upper figures: But if you find that impossible to be done, that ſhews that the figure in the *Quotient* did not rightly answer the question, being too great, but must be lessened and put a lesser figure in the room of it, as in the Two and twentieth Example.

2. Having set the figure in the *Quotient* (which you judge will answer the question demanded) and multiplied and subtracted, as before. Observe, That the remaining figures (if there be any) which are above the *Divisor*, must be one of these three things, to wit, Equal with the *Divisor*,

121

Of Division.

Divisor, or else more or less in value, then the
said Divisor. Now if the said Rest be equal or
more, that ſhews that the figure set down in the
Quotient, did not rightly anſwer the queſtion,
being too little, and must be encreased or a great-
er put in the room of it, as in the said Two and
twentieth Example.

Otherwise, if you finde after an exact *Multiplication* and *Subtraction* (as before) that the remaining figures be less in value then the Divisor, that ſhews that the figure placed in the *Quotient* did answer the question, and is neither too great or too little, as in the Two and twentieth Example following.

These two Observations being well understood
and observed, the next thing to be known, is to
subtract well the several Products arising from the
Multiplication of the figures of the Divisor,
multiplied by the figures in the Quotient, which
thing shall be shewn in its place.

22 Example.

Divide 89 by 27, facit $3\frac{8}{7}$.

$89 (3)$

27

I begin at the first figure of the Divisor (to-
wards the left hand) saying, How many times 2
in 8, the right answer is 3 times; then I multi-
ply, saying, 3 times 2 is 6; from 8 there remains
2, which

122

Of Division.

2, which 2 I set above 8, having first cancelled 2 and 8 thus.

$$ \begin{array}{r} 2 \\ 89 \\ \rowcolor{gray!20} 27 \\ \end{array} $$

Now I go to the next figure of the Divisor 7, and multiply it by 3 (as I did the first) saying, 3 times 7 is 21, from 29 there remains 8; which 8 I set above 9, and having cancelled 7 and 29, the work is finished, the Quotient is then $3\frac{8}{7}$ as followeth.

$$ \begin{array}{r} 2 \times 8 \\ 89 \\ \rowcolor{gray!20} 27 \\ \end{array} $$

Now to try the two Observations before mentioned (Fol. 120.) I shall repeat the foregoing Example over again, where I shall take the figure, in the Quotient too great, and then too little, that the Learner may perceive when he does work right, or not.

$$ \begin{array}{r} 89 \times 4 \\ 27 \\ \end{array} $$

I ask in 8, how many times 2? the right answer is but 3, but I take 4, to see whether so much can be taken, or not: I multiply 2 by 4, saying, 4 times 2 is 8, from 8 there remains nothing: then I multiply the second figure of the Divisor, saying, 4 times 7 is 28, from 9 I cannot:

123

Of Division.

not: Therefore I conclude, that the figure 4 in the Quotient is too great.

Again, Instead of 4, I shall take but 2, to see whether 2 may be taken, or not.

$$ \begin{array}{rrr} & 4 & \\ & 89 & \rowcolor{gray!20} 27 \\ \end{array} $$

I ask in 8, how many times 2? I answer, but 2 (though 3 be the true answer) I set then 2 in the Quotient, and multiply, saying, 2 times 2 is 4, from 8 there remains 4, which 4 I set above 8, as you see above. Then I proceed and say, twice 7 is 14, which 14 must be subtracted from the figures above (which are 49) yet not at once, for that would be too difficult: Therefore I say, 14 from 9 that cannot be, I borrow 1 from the 4 above, which 1 is 10, and 9 is 19. Now 14 from 19, there remains 5; which 5 I set above the 9, and cancel the said 9, then I say further, 1 that I borrowed from 4, there remains 3, which 3 I set above 4, and cancel 4, as followeth.

$$ \begin{array}{rrr} & 3 & \\ & 45 \\ & \rowcolor{gray!20} 27 \\ 89 & \text{2 Reft } 35. \\ \end{array} $$

Now seeing that the rest of this Division 35, exceeds the Divisor, which is but 27, I conclude that the figure in the Quotient, is too little, and that

124

Of Divifion.

that 3 may be taken inſtead thereof, as in Fol.

121.

23. Divide 140 by 19, facit 7$\frac{1}{5}$.

24. Item, Divide 100 by 18, facit 5$\frac{10}{18}$.

### 5 Note.

Though the Divisor contains never so many figures, there is always but a question and an answer to be made at the first figure thereof: For, the figure in the Quotient (which answers the question) will serve for all the other figures of the Divisor, and is the Multiplier of every figure of the said Divisor. Observe then for a general rule (when you see few, or many figures in the Divisor) that for every time that the Divisor is set down under the Dividend, so many questions you will meet (at the first figure towards the left hand of every Divisor,) and for every answer of such questions, one figure only is to be set down in the Quotient.

This Observation ought to be well considered, for many will mistake in that very thing, when they come newly to divide with a great Divisor; therefore to put this needful observation into the Learners minde, I shall propound some Divisors, whose Divisors shall confit of 3. 4. 5. or more figures, and the Quotients of one figure onely, that they may perceive that one figure in the Quotient is sufficient for every Divisor, though never so great.

25. Divide

125

Of Divifion.

25. Divide 487 by 212, facit 2$\frac{63}{512}$.

$$ 
\begin{array}{l@{\quad}r@{\,}l}
 & 63 \\
487 & \multicolumn{1}{r}{(2} & \underline{31} \\
      & \phantom{2}487 & \multicolumn{1}{r}{3} \\
212 & \phantom{2} & \underline{xx}
\end{array}
$$

I ask in 4, how many times 2? Answer 2, then I multiply, saying, 2 times 2 is 4, from 4 there remains nothing: Then again to the next figure, I say twice 1 is 2, from 8 there remains 6, which 6 I set above 8. Lastly, I say twice 2 is 4, from 7 there remains 3, which 3 I set above 7: The whole Quotient is only 3, and the reft is 63.

26. Divide, 894 by 549, facit 1$\frac{345}{549}$.

27. Item, Divide 4987 by 2343, facit 2$\frac{301}{2343}$.

$$ 
\begin{array}{r@{\,}c@{}c@{}c@{}c@{}c@{}c}
 &   & \multicolumn{4}{r}{(301 \text{ Rest})} \\
4987 & \big) & 4987 & ( & 2 & \frac{301}{343} & \text{Quotient.} \\
      &   & \multicolumn{4}{r}{2343} & \\
\end{array}
$$

### 6 Note.

When you find no rest in Subtracting the Product of the figures of the Divisor, you must sometimes set a cipher above, and sometimes not. The rule to be observed is this, When you find a figure uncancelled on the left hand (of that place where the Subtraction is made) then a cipher is to be set above that figure, from whence the Subtraction is made: But if you find that all the figures refting on the left hand, are cancelled, then it

126

Of Division.

it is needles to let a cipher ; as you may perceive
by the foregoing Example.

I ask in 4, how many times 2? Answer 2;
then I multiply, saying, twice 2 is 4, from 4
there remains nothing. Now observe, I do not let
a cipher above the 4 (in the Dividend) because
there is no figure on the left hand; then I pro-
ceed, saying, twice 3 is 6, from 9 there remains
3, which 3 I let above 9, and cancel 9. I say
further, twice 4 is 8, from 8 there remains no-
thing. Now I let a cipher above 8, because I see
the Significant figure 3, on the left hand uncan-
celled. Lastly, I say twice 3 is 6, from 7 there
remains 1, which I I let above 7, and cancel 7's
The Quotient is then $2\frac{30}{343}$, as followeth.

$$
\begin{array}{r@{\,}c@{}c@{}c@{}c@{}c@{}c}
 & & 3 & 0 & 1 & & \\
\cline{3-7}
4 & 9 & 8 & 7 & \big) & 2 & \bar{3} & 0 \bar{1} \\
 & & \multicolumn{1}{l@{\kern-\nulldelimiterspace}}{4} & 3 & 4 & & \\
\cline{3-4}
 & & & \mathbf{\bar{2}} & \end{array}
$$
Quotient.
$\mathbf{2} \overline{3} \mathbf{4} \mathbf{3}

7 Note.

Another thing to be observed in Division, is,
how to Subtract the several Products (arising from
the Multiplication of the figures of the Divisor)
from the upper figures. This is performed divers
ways; but I shall mention two only, as follow-
eth.

When you are to Subtract the Product of the
first, second, and third figure, &c. of the Divisor.
Observe, that such Product is to be taken
from that figure, which is directly above that un-
der

Of Division.

127

der figure so multiplied; and if there remains
any thing, set it above that figure : But the
greatest difficulty lies in this, when the said Pro-
duct is more then the said upper figure, in this
case you must borrow from the upper figures to-
wards the left hand, so many Unites as will just
supply that want and no more; and every Unite
so borrowed, must be accounted for 10. Now
the Ten or tens so borrowed, and the figure above,
being joyned together, subftract the said Produc-
t from that sum, setting the Rest (or a cipher, if
nothing remains) above the said upper figure;
and for every Ten so borrowed, subftract an U-
nite from the next figure towards the left hand,
and let the Rest always above that figure that
you do take from, as the following Example doth
shew.

28. Divide 3556 by 1234, facit 2$^{1088}$.

I

$$
\begin{array}{@{}r@{}l}
& 3556 (2) \\
15 \cancel{3} & \bar{2} \\
& 234
\end{array}
$$

I ask in 3, how many times 1? Answer, twice :
I let 2 in the Quotient, then I multiply, saying,
twice 1 is 2, from 3 there remains 1, which I
let above 3. Then I go to the next
figure, saying, twice 2 is 4, from 5
there remains 1, which I let above
5, as you see in the Margin.
3556 (2
234

Further, I say twice 3 is 6, from
5 I cannot, therefore I borrow 1 from the next
upper

128

Of Division.

upper figure, which is 10, and 5 is 15. Now 6
from 15, there remains 9, and 5, and 9
above it; then I say, that I borrow
from 1, there remains nothing, I
119
set a cipher above the 1 borrowed,
3556 (2
as in the Margin.
2234
Lastly, I multiply the last figure
of the Divisor, saying, twice 4 is
8, from 6 that cannot be; therefore I borrow 1
from the next upper figure 9, which 1 is 10, and
6 is 16. Now 8 from 16
there remains 8, which 8
I set above 6, and 1 that I
borrowed from 9, there re-
mains 8, which 8 I set a-
bove 9: The Quotient is
then 2, and the Rest 1088, as in the Margin.

$$
\begin{array}{r|r}
08 & \\
1 & 99 \\
\hline
\end{array}
$$

$$
\begin{array}{r}
3556 \\
2234 \\
\hline
\end{array}
$$

$$
\begin{array}{r}
8 \\
2 \\
\hline
\end{array}
$$

Another way.

There is another way, which differs but a little
from this way, already mentioned. The differ-
ence is onely in the Subtracting, when you bor-
row any Tens from the upper figures. In the
former way, you borrow so much as will serve to
Subtract the whole Product of one figure at once.
But in this way you do not Subtract so, but at
twice ( that is, if such Product coniift of two
figures) as the following Examples will plainly
shew.

29. Divide

129

Of Division.

29. Divide 88125 by 18199 fact 4 35329
To shew the difference more evidently I shall
work this Division these two several ways.
The first vvay first, then the second.

$$
4^{18199}
$$

$$
\begin{array}{r}
4 \\
88125 \\
\hline
\end{array}
$$

$$
\begin{array}{r}
8 \\
199 \\
\hline
\end{array}
$$

I ask in 8 hovv many times I ? Anf/vver 47
(for 5 times is too much) then I multiply faying,
4 times 1 is 4 from 8, there remains 4, vvhich
4 I set above the 8, and cancel the 8, as you
fee above.

Then I go to the second figure of the Divi-
for, and say, 4 times 8 is 32, from 8, that can-
not be; therefore I borrow from the next upper
figure (4) so many tens as will suffice to be ad-
ded to 8, that 32 might be subftracted. Now I
fee that 3 tens will serve; therefore I say, 30
that I borrow, and 8 is 38, and 32 being taken
from 38, reft 6, which 6 I fet above 8, fay-
ing further, 3 that I borrowed from 4, there
remains 1; vvhich 1 I fet a-
bove the 4, as in the Mar-
gine.

1

46

88125 ( 4.

18199

Further, I go on to the third
figure of the Divisor, saying 4
times 1 is 4, from 1, that cannot be, I borrow
only 1 from the next upper figure (6) vvhich 1
is 10. Novv 10 and 1 is 11, then 4 from 11,

K

there

130

Of Division.

there remains 7, vvhich 7 I
fet above 1, and 1 that I
borrovvved from 6, there re-
mains 5, vvhich 5 I fet above
6, as in the Margine I go
further to the fourth figure of
the Divisor 9; faying, 4 times 9 is 36, from 2,
I cannot, I borrovv 4 tens (from the next figure
(7) vvhich is 40, and 2 is 4². Novv 36 from
42, reft 6, vvhich 6 I fet above 2, and 4 I
borrovvved from 7, there re-
mains 3, vvhich 3 I fet above
the 7, as in the Margin.
Laftly, I go to the laft figure
of the Divisor 9; faying, 4
times 9 is 36, from 5, that cannot be, I bor-
rovv 4 as before (from the next upper figure)
vvhich is 40, and 5 is 45.
Novv 36 from 45, there re-
mains 9, vvhich 9 I fet above
the 5, and 4 that I borrovvv-
ed from 6, there remains 2,
vvhich 2 I fet above 6, the
Quotient is then 4, and the
reft is 15329, as in the Margin.

The same another way.

$$
\begin{array}{r@{\,}c@{}c@{}c@{}c@{}c@{}c}
 & & 4 & & & & \\
 & & 8 & 8 & 1 & 2 & 5 & (4 \\
 & & & x & 8 & 1 & 9 & 9
\end{array}
$$

I ask (as before) in 8 hovv many times 1?

Anfwer

131

Of Division.

Anfver 4 times, then 4 times 1 is 4, from 8,
there remains 4. Which 4 I fet above 8, as you
fee above. I go to the fecond figure, faying 4
times 8 is 32. Novv obferve here, inſtead of
borrovvring 3 tens as before. I subſtraſt the pro-
duct (32) at tvvo ſeveral times, taking firſt the
unites (2) then the tens (3) faying, 2 from 8,
there remains 6, which I fet
above the 8, then 3 from 4, 1
there remains 1, which 1 I 46
fet above 4, as in the Margin. 88125 ( 4
I go next to the third figure x8199
of the Divisor , faying, 4
times 1 is 4, from 1, that cannot be, I borro-
1 which is 10, and 1 is 11. Now 4 from 11,
there remains 7, which 7 I
fet above 1, and I that I 15
borrowed from 6, there 467
remains 5, which 5 I fet 88x25( 4
above 6, as in the Margin. x8x99

I proceed to the fourth figure 9, faying, 4
times 9 is 36. Now 6 from 2, that cannot be,
I borrow 1 (from the 7) which is 10, and 2 is
12; then I fay 6 from 12, there remains 6,
which 6 I fet above 2. Now
1 that I have borrowed, and 153
3 is 4, from 7, there remains 4676
3, which 3 I fet above the 7, 88x25( 4
as you may fee in the Mar-
gin. x8x99

Laftly, I fay 4 times 9 is 36. Novv 6 from
5, that cannot be, but 6 from 15, there remains

K 2

94

132

Of Division.

9, vwhich 9 I set above 5, and 1 that I bor-
revved and 3 is 4, from 6, there remains 2,
which 2 I set above 6, the Quotient is then
$4\overset{15}{\underset{18}{3}}\overset{29}{\underset{19}{9}}$, as followvs.

$$
\begin{array}{r@{\,}c@{}c@{}c@{}c@{}c@{}c}
 & & 1 & 5 & 3 & 2 & 9 \\
\cline{3-7}
9 & \div & \mathbf{4} & \mathbf{6} & \mathbf{7} & \mathbf{6} & \mathbf{1} \\
8 & \div & \mathbf{8} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{8} \\
\cline{3-7}
 & \div & \mathbf{8} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{8} \\
 \cline{3-7}
 & & \mathbf{1} & \mathbf{8} & \mathbf{1} & \mathbf{5} & \mathbf{9} \\
 \cline{3-7}
\end{array}
\quad
\begin{array}{@{}l@{}}
\text{Quotient.} \\
\mathbf{4}\underset{\mathbf{1}}{\overset{\mathbf{1}}{\smash{3}}}\underset{\mathbf{2}}{\overset{\mathbf{2}}{\smash{9}}}
\end{array}
$$

Of these two manner of wayes (of Subfraact-
ing the products of the several figures of the
Divisor) the Learner may choose which he pleafe.
But I shall of those two wayes choose the first,
in the following Divisions.

Before I propound any Divisions which have
more figures in the Quotient then one, I shall
add here two or three Divisions more, to make
the Learner the more perfect in subfraacting the
several products of the figures of the Divisor.

30. Divide 31867 by 13619, facit $2\frac{4619}{13}$

31. Divide 830072 by 309007 facit $2\frac{321058}{309007}$

8 Note.

When the Divisor confifts of significant fi-
gures which have ciphers between them, then
cancel onely the ciphers of the said Divisor,
and not any figure above them, and subfraact
the products of the significant figures of the said
Divisor from the figures of the Dividend, as
before, thus,

I ask

133

Of Division.

$$
\begin{array}{@{}r@{}l}
 & 2 \\
8 & 30072 \stackrel{2}{\longrightarrow} \\
 & 309007 \\
\end{array}
$$

I ask in 8 how many times 3? Anſwer 2, then
I set 2 in the Quotient, and multiply as before;
faying, twice 3 is 6 from 8, there remains 2,
then twice nothing is nothing. I cancel the ci-
pher in the Divisor, but not
the 3 which is above it; then 212
twice 9 is 18, from 0, that 830072 (2
cannot be, but from 20 (for 309007
I borrow 2 tens), there re-
mains 2. Now 2 that I borrowed, from 3, there
remains 1, which 1 I set above 3, as in the
Margin.

I proceed to the two ciphers in the Divisor, faying, twice nothing is nothing; and again, twice nothing is nothing, canceling the ciphers as I go along, as you see in the Margin.

Lastly, I say twice 7 is 14, from 2, that
cannot be, but from 22 there remains 8, and 2
that I borrowed from 7, there remains 5, which
5 I set above 7, the Quotient is $2\overset{21}{\underset{3}{\smash{2058}}}$, as followeth.

K 3

32. If

134

Of Division.

$$ \begin{array}{r} 212.58 \\ \overline{830} \vert \overline{072} \\ \quad 2.21058 \\ \overline{300007} \end{array} $$

32. If it be required to divide or to part 184008177007 by 19000200717, what is the product? facit 9¹³⁰⁰⁰⁰³⁷⁰⁵⁵⁴⁰⁷, as followeth

$$ \begin{array}{r@{\,}c@{\,}l@{\quad}l} & & & \overline{5} \\ \cline{2-4} \overline{1} & / & \overline{6} & \overline{5} \\ \cline{2-3} 9 & | & 63 & \overline{07} \underline{\quad 4} \\ 184 & \overline{\phantom{00}} & \overline{008} & \underline{\quad 7} \\ \cline{3-4} & & \overline{77007} & \end{array} $$

9. Note.

The Learner being well acquainted with what hath been treated of Divifion, to wit, how to divide any number by a single figure 3, and fe- condly, how to divide any number which re- quires but a single figure in the Quotient. Now it follows to treat of thofe Divifions which contain more figures then one, both in the Divifor and Quotient: which divifions will be eaſie, if theſe two things (already mentioned in part before) be duly or exacely obſerved, to wit, To subſtract well the ſeveral products arifing from the Multi- plication of the figures of the Divifor, and ſe- condly to remove the Divifor in its right place, which

Of Division.

135

which is to remove it a place further (every time) towards the right hand in the manner fol- lowing.

33. Divide 5964 by 21, facit 284.

I ask in 5 how many times 2? Anſwer twice, I fet 2 in the Quotient, and multiply as before; faying twice 2 is 4, from 5, there remains 1, which 1 I fet above 5, as you may ſee before. Then I go to the ſecond figure of the Divifor 1, faying, 17 twice 1 is 2, from 9, there re- mains 7, which 7 fet above 9, as in the Margin. Now in the removing of the Divifor, I fet the 1 under 6, and the 2 under the 1: Then I ſee what figures I have above 2 (the firſt figure of the Divifor on the left hand.) And I finde 17, then I ask in 17 how many times 2? Anſver 8: I fet 8 in the Quotient, and multiply, faying, 8 times 2 is 16, from 17, there remains 1, which 1 I fet above 7. Fur- ther I ſay 8 times 1 is 8, from 6, that cannot be, but from 16, there remains 8,

K 4

vvhich

136

Of Divifion.

vvwhich 8 I fet above 6, as in the Margin. I
remove the Divisor a figure
further, fetting the 1 under
4, and the 2 under the 1,
vvwhich is cancelled, as you see
in the Margin. Then I see
vvhat figure there is above the
2 (the first figure of the Di.
visor) I finde onely 8, then I ask in 8 hovv ma-
ny times 2? Anfvver 4. I fet 4 in the Quotient,
as you see in the Margin, then I multiply as be-
fore, faying 4 times 2 is 8, from 8 (vvwhich is
right above the 2) there remains nothing; then
laft of all I fay 4 times 1 is 4 from 4, there re
mains nothing, as follovveth.

$$
\begin{array}{r@{\,}l}
  2 & \\
  7 & 8 \\
  9 & 6 \\
  4 & (284) \\
  2 & \text{XXXI} \\
  2 & \text{XX} \\
\end{array}
\qquad
\begin{array}{l}
  2 \\
  7 \\
  9 \\
  6 \\
  4 \\
  284 \text{ Quotient.} \\
  \phantom{284} 2 \times \text{XXX} \\
  \phantom{2} \text{XX} \\
  \phantom{2} \text{XX}
\end{array}
$$

Here followeth some Divisions by two figures for the exercise of the Learner.

34. Divide 44880 by 12, facit 3740.

35. Item, Divide 65982 by 14, facit 4713.

36. Item, Divide 111616 by 16, facit 6976.

37. Item, Divide 21418 by 27, facit 793 2/7.

38. Item, Divide 1142812 by 19, facit 60148.

39. Item,

137

Of Divifion.

39. Item, Divide 42826 by 49, facit 874.

40. Item, Divide 33644 by 52, facit 647.

41. Item, Divide 10476 by 65, facit 161 11/35.

42. Item, Divide 607446 by 63, facit 9642.

43. Item, Divide 957972 by 97, facit 9876.

10. Note.

When you are to remove your Divisor divers
times, confifting of three, four, five, or more
figures, take good heed of mistaking the right
places of the figures of the Divisor; therefore,
Observe that each figure of the Divisor be re-
moved a place further towards the right hand,
in such manner, that the Unites of the said Divi-
for be always the very next figure under each
figure of the Dividend, and the other figures in
their right order as followeth.

Suppose that 348648340 were to be divided
by 31452. In removing such great Divisor,
observe well the order follovvving, vvhich may
serve as an example for the reft.

The first fetting of the Divisor.
The first removing.

$$
\begin{array}{l@{\,}r}
  348648340 & ( \\
  3\times45\times2 & \\
  \phantom{3\times45\times2} & 3145 \\
\end{array}
$$

Second

140

Of Division.

49. Item, Divide 4000110 by 6009, facilitate 665⁴¹²⁵/⁶⁰⁰⁰⁰⁰⁰.

II. Note.

When the Divisor conflicts of an Unite in the first place towards the left hand, and nothing but ciphers towards the right: It is needleless to place the ciphers as the significant figures are, though the Quotient comes right that way as well as the other, but it is too tedious and altogether needleless, as shall appear hereafter, fol. 141.

In such a case then, the best way is to cut off so many places of the Dividend towards the right hand, as the Divisor hath ciphers, and those figures so cut off, is the rest of such Division, as followeth.

50. Divide 163499 by 10, facilitate 16349⁷⅓.

16349⅙, facilitate 16349⅔. Quotient.

For the cipher which is in the Divisor towards the right hand, I cut off the first figure of the Dividend towards the same hand, the Quotient is 16349, and the rest is 9.

141

Of Division.

If the said Dividend were to be divided by 100, then two figures are to be cut off thus, 1634199: If by 1000, then three figures must be cut off thus, 1631499, &c.

Now, to shew and satisfie the Learners (for some will not go out of the common road, before they be satisfied of a shorter way as true as the other) that this way is far shorter than to remove every cipher, as the significant figures I shall here insert both ways, and then let them judge.

51. Divide 451346000 by 1000, facilitate 451346.

451346000, facilitate 451346.

Otherwise, more tedious.

Though some may judge this trouble needleless to insert such Division here; yet I say, that though this be so, to those that are taught already, nevertheless that may give some satisfaction to the unlearned. And this Book is chiefly for the weaker sort, and therefore the plainer, the better. It is a good fault (if it may be called a fault) in any Writer to be plain in what he creates

142

Of Divifon.

treats of, especially if it be of this or such
like Art.

12. *Note*.

When the Divisor conflits of one or more fig-
nificant figures towards the left hand, and no-
thing but ciphers towards the right, then set
down all the ciphers under the Dividend on the
right hand, and divide the remaining part of the
Dividend on the left hand, by the remaining
part of the Divisor, as followeth.

52. Divide 434120 by 4000, facit 108¼²¹⁄₄²⁰⁰.

| 2 |
|---|
| 434 |
|---|---
| 120 |
|---|---
| 444000 | Quotient. |

The three ciphers are placed so under the
Dividend, for shortneſſe ſake, as is ſaid before.

53. Divide 116840670 by 301000, facit
388¼⁵²⁶⁷⁰.

54. Item, Divide 10101010 by 4019, facit
2513¼²²⁶³.

55. Item, Divide 822368178 by 9687, facit
84894.

56. Item, Divide 519603282 by 7641, facit
68002.

57. Item, Divide 8460476 by 68478, facit
123⁵⁷⁶⁸.

58. Item,

143

Of Divifon.

58. Item, Divide 184064876 by 836846,
facit 219, the reſt or remainder is 795602.

59. Item, Divide 3600001080000081 by
60000009, facit 60000009.

60. Item, Divide 678967896789678 by
56785678, facit 11956675²⁸⁹⁰²⁸.

I have been the larger in Division, becauſe that
it is that part of Arithmetick which is moſt dif-
ficult for young beginners.

Another way of Divifion.

The other way of Division, which I intend
(God willing) to describe here, differs not
much from the former, onely it performes Di-
vision with fewer figures then the other; there-
fore to avoid needleſſe repetitions, I ſhall lay
down the difference in few words, for all the
observations of the firſt way (concerning the pla-
cing of the Dividend and Divisor, and alſo the
removing of it,&c.) are to be obſerved in this
ſecond, except when you multiply the Divisor
by the Quotient, then you do otherwife as fol-
loweth.

First, Having placed the Divisor under the
Dividend. In the first way, you begin to mul-
tiply the first figure of the Divisor on the left
hand, and so proceed towards the right : But in
this

144

Of Division.

this second way of Division you must begin quite
contrary; that is, you are to begin on the first
figure of the Divisor on the right hand, and
proceed towards the left.

Secondly, In subtracting the products of the
Divisor, in the first way, If you borrow any
Unites (to help the Subtraction) then you sub-
tract them presently from that figure from
whence it was borrowed (or from the next, &c.)
and you set the rest over it, before you multiply
another figure of the Divisor, which way makes
the number of figures the greater. But in this
Second way, observe that when you borrow any
Unites to supply the place of tens (or to help
the Subtraction) after you have set down the rest
of such Subtraction, you must not subtract a-
lone the Unites (or tens) so borrowed (as in the
former way) but you are to keep them in mind,
then go on and multiply the next figure of the
Divisor (by the figure in the Quotient) and
add so many unites as you did keep in mind,
to the product of that multiplication, and sub-
tract that sum from your Dividend, going to-
wards the left hand. If you see occasion to bor-
row, do as you did before, keeping alwayes in
mind what you borrow, to be added to the pro-
duct of the Multiplication of the next figure of
the Divisor (towards the left hand) as the fol-
loving examples will plainly show.

1 Example.

145

Of Division.

I Example.

In dividing 7548 by 76, what is the Quotient?
faciſt 99$^{24}$/76.

I set the Divisor under the Dividend as for-
merly, thus:

$$
\begin{array}{r}
7548 \\
76 \\
\hline
\end{array}
$$

I ask in 75 how many times 7? Answer, 9
times. Now instead of saying (as before) 9
times 7. I say 9 times 6 is 54, from 4 (which is
above 6) that cannot, I borrow 5 (which is 50)
and 4 is 54. Now 54 from 54, there remains
nothing. I set a cipher above 4, and keep 5 in
minde, that I borrowed; then

I multiply the next figure of
the Divisor, saying, 9 times
7 is 63, and 5 that I bor-
ed (which I kept in mind) is
68. Now 68 from 5 (which is above 7) that can-
not be, therefore I borrow the next figure 7
which is 70, and 5 is 75.

68. Now 68 from 5 (which is above 7) that can-
not be, therefore I borrow the next figure 7
which is 70, and 5 is 75.
70

Now 68 from 75 there remains
7, which 7 I set above 5, as
you see here in the Margin, I
remove the Divisor, and ask in
7

70 how many times 7? Answer, 9 times. I set 9
in the Quotient and multiply, saying, 9 times 6
is 54, from 8 that cannot be, I borrow 5 (or 50)
and 8 is 58. Now 54 from 58 there remains 4,
which

L


which

146

Of Divifion.

which 4 I set above 8, and
carry 5 in minde, as in the
Margin, I proceed and say,
9 times 7 is 63, and 5 I kept
in minde, is 68, from 0 that
cannot be, but from 70 (for I
borrow 7 tens) there remains 2, which 2 I set
above the cipher, as followeth.

$$
\begin{array}{r@{\,}l}
& 2 \\
70 & \big) 4 \\
75 & \multicolumn{1}{r}{-48} \\
76 & \multicolumn{1}{r}{(99} \\
   & \multicolumn{1}{r}{766} \\
   & 7
\end{array}
$$

13 Note.

When you are to borrow any Unites from the
upper figures towards the left hand (to help the
Subtraction of the Products of any of the figures
of the Divisor) you need not to trouble your self
(either in this way of Division, or in the former
to consider, whether the figure, from whence you
are to borrow, be a significant figure (great enough
to borrow as much as you need) or not, because
you pay the debt by adding so many tens that
you bear in minde to the Product of the Multi-
plication of the next figure, to be subtracted again.
But if it be demanded, where I must borrow, if
the place (or figure) from whence a supply must
be borrowed, be but a cipher, and the next figure
a cipher also ? I answer, that I borrow then from
the

147

Of Divifion.

the third, fourth, or fifth figure, or I borrow
from the next significant figure (whether it be in
the third, fourth, or fifth place, &c.) which is
alone. But this is needles to enquire, because
(as is said before) that which is borrowed, is re-
paid in adding it to the Product of the Multipli-
cation of the next figure of the Divisor.

2 Example.

Divide 64284 by 487, facit 132.

$$
\begin{array}{r@{}l}
 & 5 \\
 & 64284 \\
  & \big) 1 \\
 & 487
\end{array}
$$

I ask in 6, how many times 4? Anfwer 1, then
I multiply, saying, once 7 is 7, from 2 I cannot,
but from 12 there remains 5, which 5 I set above
the 2, and bear 1 in minde; then I say once 8
is 8, and 1 I kept in minde, is 9: Now 9 from
from 4 I cannot, but 9
from 14 there remains 5,
which 5 I set above 4, and
bear 1 in minde again. Fur-
ther I say, once 4 is 4, and
1 (I kept in minde) is 5, from 6 there remains 1,
which 1 I set above 6 as in
the Margin. I remove the
Divisor, and ask in 15
how many times 4? An-
swer, 3 times, I set 3 in
the Quotient, and say, 3

L 2

$$
\begin{array}{@{}r@{}l@{\quad}l}
 & 55 \\
 & 642 \\
 \times & 4 \\
 \hline
 & 487 \\
 & \multicolumn{1}{r}{-155} \\
 64284 & \multicolumn{1}{r}{(13)} \\
 & 4877 \\
 & \qquad 48
\end{array}
$$

times

["150", "Of Division.", "The Proof of Division.", "Division may be proved in casting away all the\nnines, as in Multiplication, but the uncer-\ntainty of it makes me to omit it: The beſt and\nmoſt certain Proof of Division, is done by Mul-\ntiplication, and contrariwife, the beſt Proof of\nMultiplication is done by Division. Therefore I\nſhall mention here how to prove theſe two Rules,\nſeeing that the one proves the other.", "Division is proved thus, Multiply the Quotient\nby the Divisor, and the Product muſt be equal\nwith the Dividend, if there be no Reſt in the\nDivision; but if you finde a Reſt, adde it to the\nProduct of the 81 Multiplication, and the ſum ſhall\nbe equal to the Dividend; which proves the Di-\nvision to be right, as followeth.", "Divide 873423 by 123, facit 7101.", "$$\n\\begin{array}{l@{\\hspace{2em}}r}\n\\begin{array}{@{}c@{\\,}r@{\\,}r@{\\,}r@{\\,}r@{\\,}r}\n & \\multicolumn{5}{r}{\\text{XXX}} \\\\\n\\cline{2-6}\n87 & 34 & 23 & (7) & 10 & \\text{Quotient.} \\\\\n\\cline{2-6}\n\\text{X} & \\text{X} & \\multicolumn{1}{r}{\\text{X}} & \\text{X} & \\text{X} & \\text{X} \\\\\n\\cline{3-5}\n & 2 & 3 & 3 & 3 & 3 \\\\\n\\cline{3-5}\n & 2 & 1 & 3 & 0 & 3 \\\\\n & 1 & 4 & 2 & 0 & 2 \\\\\n\\cline{3-5}\n & 7 & 1 & 0 & 1 & \\\\\n\\end{array}\n& 1 & 2 & 3 & \\text{Divisor.} & \\\\\n\\cline{2-6}\n\\multicolumn{6}{l}{\\text{________}} \\\\\n\\text{X} & \\text{X} & 2 & 1 & 3 & 0 \\\\\n\\cline{3-5}\n\\end{array}\n$$", "Proof——873423", "But finding a Reſt in the Division, do as in the\nfollowing Example.", "Divide", "151", "Of Division.", "Divide 4437 by 37, facit 119³⁴.", "$$\n\\begin{array}{r@{\\,}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c}\n    &   & \\bf 3 & \\bf 1 & \\bf 3 &   &   &   &   &   &   &   &   &   &   \\\\\n    &   & \\bf 7 & \\bf 6 & \\bf 4 &   &   &   &   &   &   &   &   &   &   \\\\\n\\cline{3-3}\\cline{5-5}\\cline{7-7}\\cline{9-9}\\cline{11-11}\n\\multicolumn{2}{l}{4437} & \\bf (} & \\bf 119 & \\bf 3 & \\bf 4 & \\bf \\overline{)} & \\text{Reft} &   &   &   &   &   &   &   &   \\\\\n\\multicolumn{2}{l}{3777} &   &   &   &   &   &   &   &   &   &   &   &   &   \\\\\n\\cline{3-3}\\cline{5-5}\\cline{7-7}\\cline{9-9}\\cline{11-11}\n\\multicolumn{2}{l}{33} &   &   &   &   &   &   &   &   &   &   &   &   &   \\\\\n    &   &   &   &   &   &   &   &   &   &   &   &   &   &   \\\\\n    &

152

Of Division.

Questions of Division.

Queft. 1. What is Division?

Answ. Division is the separating of any number into any equal parts assigned, or to find out how many times one number is contained in another.

Queft. 2. How many Numbers hath Division?

Answ. There are chiefly three Numbers in any Division.

Queft. 3. Which are they?

Answ. The Dividend, the Divisor, and the Quotient.

Queft. 4. Is there always but three Numbers in any Division?

Answ. There are sometimes four.

Queft. 5. Which is the fourth?

Answ. The Remainder or Rest.

Queft. 6. What Number is it you call the Dividend?

Answ. The Dividend is the Number to be divided.

Queft. 7. What is the Divisor?

Answ. The Divisor is the Number by which the Dividend is divided.

Queft. 8. What is the Quotient?

Answ. The Quotient is the Number produced by Division.

Queft. 9. What does the Quotient show?

Answ.

153

Of Division.

Answ. The Quotient shows how often the Divisor is contained in the Dividend.

Queft. 10. What is the Number you call Remainder?

Answ. The Remainder is, when any Number remains, when the Division is finished.

Queft. 11. How comes that to pass, that some Divisions have a Rest, and some other have none?

Answ. Before I answer your question, I shall tell you first, that in a Division without Rest, the Quotient contains just so many Unites, as the Divisor is contained in the Dividend.

Queft. 12. What's the reason of the Rest then?

Answ. When the Dividend does not contain the Divisor, a just quantity of whole Unites, then there is a Rest besides the Quotient, which Rest is a part of an Unite.

Queft. 13. Explain your meaning by an Example?

Answ. Suppose that 8 be given to be divided by 2. I ask thus: in 8 how many times 2? the answer is 4. Now this 4 is the Quotient, which shows that the Divisor 2, is contained just 4 times in the Dividend 8.

Queft. 14. Give another Example where there is a Rest?

Answ. Suppose that 9 be given to be divided by 2. I say as before, in 9 how many times 2? the answer is 4, for the Quotient, and there is 1 over; which 1 is the Remainder of that Division, and is to be divided into two parts; now each part is a half, the whole Quotient is then 4 whole

154

Of Divifion.

4 whole Unites, and a half of an Unite, or 1.

Queft. 15. How do you divide one Number by another ?

Answ. I fet down firſt the Dividend, then the Divifor under it.

Queſt. 16. In what manner ?

Answ. I fet the Divifor under the Dividend, on the left hand each figure of one Number directly under the figures of the other.

Queſt. 17. Is that a general rule, to fet the firſt figure of each Number (towards the left hand) under one another, and the ſecond under the ſecond, &c ?

Answ. No.

Queſt. 18. What rule have you to p'ace your Divifor to prevent mistakes ?

Answ. If the Divifor confiſt of one figure onely, I confider whether that figure be equal or leſſer then the firſt figure of the Dividend towards the left hand.

Queſt. 19. What then ?

Answ. If that figure (which is the Divisor) be equal or leſſer then the firſt figure of the Dividend, then I fet it down under it.

Queſt. 20. But if the firſt figure of the Dividend (towards the left hand) be leſs then your Divisor ?

Answ. In ſuch a case I fet down the Divisor, not under the firſt figure, but under the ſecond.

Queſt. 21. What if the Divisor confiſt of more figures then one, as 2, 3, 4, 5, &c ?

Answ. I muſt obſerve, whether ſuch great

Divisor

155

Of Divifion.

Divisor can be found once at least in the like quantity of figures of the Dividend, towards the left hand.

Queſt. 22. Grant it be so; what then ?

Answ. If so, then I fet down the firſt figure under the firſt, the ſecond under the ſecond, &c.

Queſt. 23. But suppose it be otherwise, how will you place your Divisor ?

Answ. I ſhould fet the Divisor a place further, towards the right hand, thus; the firſt figure of the Divisor under the ſecond figure of the Dividend, and so the reſt of the figures of the Divisor in order.

Queſt. 24. But what is to be done, if you could not remove the Divisor a place further ?

Answ. Then I ſhould conclude, that the Quotient of ſuch Division could not amount to an Unite, but onely to a Fraction, or a part of an Unite.

Queſt. 25. How do you know when a Divisor can be removed, and when not ?

Answ. The Divisor being rightly placed under the Dividend, I ſee whether the ſaid Divisor fills up all the places of the Dividend (towards the right hand) or not.

Queſt. 26. What do you mean by filling up ?

Answ. My meaning is, whether every figure of the Dividend hath a figure of the Divisor under it, in ſuch manner, that the Unites of the Divisor be right under the Unites of the Dividend.

Queſt. 27. Suppose it be so filled up ?

Answ.

156

Of Division.

Answ. If so, then I conclude, that the Quotient must conſt of one figure onely.

Queſt. 28. Can you perceive how many figures the Quotient will conſt, before the Division be finished?

Answ. Yes; but yet, though I know the quantity of the figures, I cannot tell what particular figure they ſhall be.

Queſt. 29. How do you know the quantity of the figures that will be in the Quotient, before you divide?

Answ. Having let down the Divisor the firſt time, I see how many figures in the Dividend (towards the right hand) I finde that have no figure under them.

Queſt. 30. What then?

Answ. Then so many figures as I finde (towards the right hand) in the Dividend that have no figures under them, so many figures must be in the Quotient, beſides that figure that will answer the queſtion of the Divisor already let down.

Queſt. 31. Your meaning is, That having let down the Divisor under the Dividend, and finding four figures (in the said Dividend on the right hand) that are all one (or that have no figures under them) that the Quotient must conſt of five figures, is it not?

Answ. You have it right.

Queſt. 32. How come you to know every particular figure in the Quotient?

Anſ. By dividing the Dividend by the Divisor.

Queſt.

157

Of Division.

Queſt. 33. Which way?

Answ. There are diversity of Operations, to finde the Quotient of a Division.

Queſt. 34. Which way do you think the beſt?

Answ. I cannot well anſwer your queſtion, onely by the common Proverb, So many men, so many mindes; for, that way which one judges the beſt, another rejects; every one thinks his way the beſt.

Queſt. 35. What is the way that you make use of?

Answ. I make use of two ſeveral ways, ſometime one, ſometimes another; but yet I judge the ſhortest way to be the beſt.

Queſt. 36. Speak of them in order.

Answ. I shall then begin with the firſt way, which requires more figures in the Operation of it, then the other, which way is very common.

Queſt. 37. Go on and begin with that which is the easiest firſt.

Answ. I shall make it as plain as I can, and begin with thoſe Divisions, whoſe Dividend and Divisor conſiſts both of a ſingle figure.

Queſt. 38. Having let down your two numbers, to wit, your Dividend and Divisor, how do you divide?

Answ. I ask how many times the Divisor is contained in the Dividend: Now obſerve, that figure which anſwers the queſtion, I fet it in the Quotient.

Queſt. 39. Give an Example to make that plain.

Answ.

158

Of Divifion.

Answ. If 8 be given to be divided by 4. I set
4 under 8, and a k, in 8 how many times 4;
the anſwer is 2 : Now this 2 I set in a crooked
line on the right ſide of the Dividend, thus.

$$
\begin{array}{c}
8 \\
2 \\
\hline
4
\end{array}
$$

Queſt. 40. What do you do with the 2 in the
Quotient ?

Answ. I multiply the Divisor 4, by the Quotient 2, laying, twice 4 is 8, which 8 I subtract from the Dividend 8, laying, 8 from 8, there remains nothing; and so the Division is finished, the Quotient is 2. And I cancel 8 and 4 thus, 8, 4.

Queſt. 41. If you finde any Rest in the Divis-
on, what must you do with it?

Answ. If I finde any Rest in a Division, I
separate it with a crooked line from the figures
which are cancelled, as thus, If 9 be divided by
4, the Rest will be 1, which I separate with a
crooked line thus.

$$
\begin{array}{c|c}
1 & \\
\cline{2-2}
9 & (2) \\
\hline
4 &
\end{array}
$$

Queſt. 42. Why do you croſs or cancel the ſi-
gures that you have uſed ?

Answ. I cancel every figure uſed, for fear of
miftakes.

Queſt. 43. How can you m'ſt'a'l'e, if the figures
remain uncancelled ?

Answ. It may happen in not cancelling the
figures

159

Of Divifion.

figures uſed, that I may forget, and so make uſe
of some of them twice, when once is enough;
and so get a false Quotient, or a false Remainder.

Queſt. 44. When muſt you cancel them ?

Answ. I muſt cancel them as ſoon as ever I
have done with them.

Queſt. 45. How ?

Answ. As in the foregoing example, when I
did divide 9 by 4, as ſoon as I have ſpoken these
words [twice 4 is 8] I cancel 4, and laying fur-
ther, from 9 I cancel 9, and adding these words
[there remains 1] I set 1 above 9, and separate
it with a crooked line, as you ſee before under the
anſwer of the 41 Queſtion.

Queſt. 46. What is that 1 worth that re-
mains.

Answ. The 1 remaining is a part of one, of the
Unites in the Quotient.

Queſt. 47. How do you call that part ?

Anſw. It is a quarter, because I did divide
by 4.

Queſt. 48. How do you prove that ?

Answ. The 1 remaining, muſt be divided by 4,
and if a whole be divided into 4 parts, each part
will be a quarter.

Queſt. 49. I understand your meaning, but is
not the said Rest called a Fraction ?

Anſw. Yes; therefore we will reſerve to ſpeak
of it in its proper place, that is, when we come
to the Fractions.

Queſt. 50. But before we leave to ſpeak of the
Rest of Division, is there nothing to be observed of it?

Anſw.

160

Of Division.

Answ. Yes; for the Division (great or small) being finished, you must observe, that the Rest ought always to be lesser, then the Divisor.

Queft. 51. What if the Remainder were either equal or greater then the Divisor?

Answ. If so, it is a sign that the figure in the Quotient did not rightly answer the question, and is too little, and might be an unite (or more) greater then it is.

Queft. 52. This is worth observation; but how shall I know when I take such figure (in the Quotient) too great?

Answ. This is easier to be discerned then the former.

Queft. 53. Which way?

Answ. When you come to multiply that figure (in the Quotient) with the Divisor; if the Product cannot be subtracted from that figure or figures, which are above the said Divisor, that shows that such figure (in the Quotient) is too great.

Queft. 54. I perceive now, that when a figure in the Quotient is too great, then the working is quite stopped, finding an impossibility, which is to subtract a greater number from a lesser. But to proceed, How do you go to work with those Dividends which consist of more figures then one, and the Divisor having but one figure only?

Answ. I set down the Divisor under the first figure of the Dividend towards the left hand, as is said before.

Queft.

161

Of Division.

Queft. 55. What if that first figure be lesser then the Divisor?

Answ. I have answered this question already, which is, that the Divisor must be placed a figure further, towards the right hand.

Queft. 56. It is true, but you did not give an example; therefore give it now.

Answ. If 17 were to be divided by 5, I cannot set 5 under the first figure 1; but I set it under the 7 thus 17( asking in 170 how many times 5, &c.

Queft. 57. But if there were more figures in the Dividend, what then?

Answ. Having finished the work of the first figure in the Quotient, I remove the Divisor under the next figure of the Dividend (on the right hand) and proceed as in the fourth Example of Division, Fol. 113.

Queft. 58. The working of the first figure in the Quotient being finished, if you find any Rest above the Divisor, what will you do with it?

Answ. Every Unite so remaining (having some figure or cipher uncancelled on the right side thereof) must be accounted for so many tens as you may see in the second Note of Division, Fol. 113. And so I proceed according to the directions of the said fourth Example.

Queft. 59. Suppose that the work of the first and second figure of the Quotient be finished, and coming to the work of the third or fourth figure, you cannot then have the Divisor once in

M in

162

Of Division.

in the figure above it; what is to be done?

Answ. In such a case I fet a cipher in the Quotient, and cancel the Divisor, to be removed a place further.

Quest. 60. But suppose after the removing of the Divisor, you finde it as before, that the figure above is still less than the Divisor?

Answ. If so, I should do as I did before, which is to fet a cipher in the Quotient, &c. as you may! plainly see in the third Note of Division, fol. 115.

Quest. 61. Lo you not set sometimes two figures at once in the Quotient?

Answ. Never but one, and 9 is the highest figure that can be taken at once.

Quest. 62. How do you divide by two or more figures?

Answ. First, I place the Divisor in its right place, that is, I place it so, that it may be contained once, at least, in the upper figures in the Dividend.

Quest. 63. How do you form the question, in asking, how many times the Divisor is contained in those figures which are above it?

Answ. I do not take all the figures of the Divisor at once.

Quest. 64. How then?

Answ. I see how many times I can have the first figure of the Divisor in that figure or figures that are above it; and that figure which answers the question, I fet in the Quotient.

Quest. 65. What do you with those figures of the Divisor towards the right hand?

Answ.

163

Of Division.

Answ. I multiply them by the figure in the Quotient. For, though the Divisor should consist of never so many figures, I must set but one figure in the Quotient for every such Divisor.

Quest. 66. But when the Divisor consists of many figures, how will you hit the mark right in taking just such figure (to be (set in the Quotient) that answers the question?

Answ. This is indeed the hardest thing in Division; for oftentimes the right answer cannot be found without making some trial or other, whether such figure (that we judge will answer the question) be either too great, too little, or just fit.

Quest. 67. You do not answer me: I ask when you set a figure in the Quotient, that you judge will answer the question. How do you know that you do not mistake, and that you have the right figure?

Answ. The answers of the Fifty one and fifty two questions, is a full answer to your question; yet because this is one of the chiefest questions of Division, I shall repeat it if you please.

Quest. 68. I am willing to hear it again, but make it short.

Answ. When the figure set in the Quotient (which I judge will answer the question) is too little; then (having multiplied and subtracted, as before) I perceive it by this, that the Rest of the Subtraction is more than the Divisor: But when the said figure (set in the Quotient) is too great, I know it by this; That the Products (arising from the Multiplication of the Divisor) cannot be

M 2

sub-

164

Of Divifion.

subfracted from the figures (in the Dividend) which are above the said Divisor. You may see these two observations more at large in Fol. 120. 121. 122. and 123.

Queft. 69. What is the next thing to be ob-
served in Division.

Answ. The next thing to be observed (having
set the right figure in the Quotient) is to multiply
and substract exactly.

Queft. 7c. But if the Divisor consist of many figures, how many questions must you answer for every such great Divisor ?

Answ. Never but one answer, for every Divisor, great or small; and that is at the first figure towards the left hand.

Queft. 7i. Why so?

Answ. Because every figure set in the Quotient,
are the Multipliers of the several Divisors set
under the Dividend.

Queft. 72. Then you have so many figures in
the Quotient, as you have Divisors under the Di-
vidend.

Answ. Yes, as you may perceive from the 26
to the 32 sum of Division.

Queft. 73. This observation is worthy to be taken notice of; but concerning the subfracling of the several Products of the Multiplication of the figures of the Divisor, I remember you told me that you must set the Rest above every figure, from whence you substract: Is it so?

Answ. Yes.

Queft. 74. But if you find no rest in subfractions,

165

Of Divifion.

ing, as taking 4 from 4, or 9 from 9, there remains nothing; what must you do?

Answ. I then I set a cipher above such figure,
when I find no rest.

Queft. 75. What, a ways.

Answ. No; sometimes I must do it, and sometimes not.

Queft. 76. What is your Rule in this case ?

Answ. When I find a figure or figures un-
cancelled on the left hand (of that figure from
which the Subtraction is made) then I set a cipher
above; but if I find all the figures cancelled on
the left hand, then I forbear the setting of a
cipher; as you may see in the Seven and twentij-
eth Sum of Division, and in the Sixth Note.

Queft. 77. But what need is there of setting a
cipher or ciphers, when there are some figures un-
cancelled on the left hand ?

Answ. Because every figure should keep its
own place (and be esteemed according to its own
value) for by omitting the setting of the cipher
(or ciphers) the figures on the left hand, should be
a place lower, then they ought to be, which would
cause a great error in the work.

Queft. 78. You say right, for that figure that
ought to be in the place of Thousands, should be (in
omitting a cipher) but in the place of Hundreds.
But which way do you Subtract the several Pro-
ducts arising from the Quotient and Divisor, mul-
tipled together ?

Answ. I follow the Rule of Subtraction be-
fore taught, Fol. 54.

M 3

Queft.

166

Of Division.

Queft. 79. What is that?

Answ. I o take the lesser number from the greater ; and if some Products be greater then the figure from whence I muſt Subtract, I borrow from the next upper figure, towards the left hand.

Queſt. 80. Is there no difference from that Subſtraſtion you use in Division, and the Subſtraſtion called Simple ?

Anſw. The difference is not great, it is only this. In Simple Subſtraſtion I borrow never but an Unite at once (which 1 is accounted for 10) but in Subſtraſting in Division, then I borrow more, sometimes more, and sometimes leſs as need requires.

Queſt. 81. What if you did borrow too much or too little ?

Anſw. It is easily discerned, how many Unites are (which is as many tens) needful to be bor- rowed.

Queſt. 82. How do you know that ?

Anſw. Suppose that I were to Subtract 8, and that the figure above were but 2, then I say, 8 from 2, that cannot be. Now I see that I must borrow from the upper figure on the left hand, and also that 1 (which is 10) being borrowed, will serve my turn, then 10 and 2 is 12. Now I say, 8 from 12, there remains 4, which 4 I set above 2, cancelling the 2.

Queſt. 83. But if the Product (which arise from the Quotient and Divisor) that you are to Subtract, was 18: 28: or 38, &c., and the figure above but 2.

Anſw.

Of Division.

167

Anſw. There is no difficulty in that: thus, If I were to Subtract 18, and find but 2 above it, I should say, 18 from 2, that cannot be: Now I perceive, that in borrowing one 1 (which is but 10) and the 2 is but 12, I should be still in want; therefore I borrow 2, which is 20, and the 2 is 22. Now I say, 18 from 22, there remains 4.

Queſt. 84. I understand you; but, suppose you were to Subtract 28, and 2 being only above, how many Unites would you borrow ?

Anſw. I should follow the Rule, which is to borrow as much as needs must: thus, If 28 is to be taken from the number above the Divisor: Now if 2 only be found above, 28 cannot be Subtracted from 2, therefore I borrow 3 unites from the left hand (because 2 would be too little) which 3 is 30, and 2 is 32. Now 28 being Subtracted from 32, there remains 4, &c. Further, suppose that 88 were to be subtracted, and 2 only be found above the Divisor, then I should borrow no leſs than 9 Unites (from the other figure on the left hand) which 9 is 90, and 2 is 92. Now 88 from 92, there remains 4, &c.

Queſt. 85. But if you find a cipher, either above (or in that place from whence the said Product is to be subtracted) or else some ciphers on the left hand, from whence you use to borrow; what then ?

Anſw. I follow till the Rule before-mentioned, thus; if 28 were to be subtracted, and I find a cipher in the Dividend, then I borrow 2,

which

M 4

168

## Of Divifion.

which is 30; then I say 28 from 30, there remains 2, and so of the rest.

**Queft. 86.** What do you with those Unites that you borrow ?

Answ. I subftraet them from the next figure towards the left hand, and set the reft above it, as in the 28 and 29 Sum of Divifion.

**Queft. 87.** What if you finde ciphers in the Divisor, between Significant figures ?

Answ. The more ciphers there is in the Divisor, the easier the work is.

**Queft. 88.** What do you with them ?

Answ. Nothing else but to cancel them; but I mult obferve (in so doing) not to cancel any figure above them, except ciphers which have no Significant figure on the left side of them.

**Queft. 89.** What do you with those Significant figures in the Divisor ?

Answ. I Multiply and Subftraet, as in the 31 and 32 Sum of Divifion.

**Queft. 90.** How do you place your Divisor, when you remove it, when it confift of more figures then one ?

Answ. In removing the Divisor (great or small) the general rule is to place every time, the said Divisor, a place further towards the right hand; as is plainly shewn in the tenth Note, between the Forty three and Forty four Sum of Divifion.

**Queft. 91.** How often must you remove the Divisor?

Answ. I must remove it so often, until the Unites

169

## Of Divifion.

Unites of the Divisor, come to be right under the Unites of the Dividend.

**Queft. 92.** How do you go to work, when the Divisor ends with ciphers?

Answ. I may follow the general way; but for shortneſs fake, I set the ciphers of the Divisor under the Dividend, towards the right hand, and divide the remaining part of the Dividend (on the left hand) by the remaining part of the Divisor, as is taught before; as in the 52 Sum of Divifion.

**Queft. 93.** But if the Divisor confift of an unite only, in the first place, towards the left hand, and nothing but ciphers towards the right, what course do you take for the best ?

Answ. If it happen so, such Divifions are soon done; for there is nothing else to do, but to cut off so many places of the Dividend (beginning from the right hand towards the left) as the Divisor hath ciphers.

**Queft. 94.** What then?

Answ. The figures so cut off towards the right hand, is the Reſt of the Divifion, and those figures of the Dividend which are on the left hand, is the whole number ( or integral part ) of the Quotient; as you may see in the 50 Sum of Divifion.

**Queft. 95.** I remember you told me before, that you had another way of Divifion?

Answ. Yes, I told you so.

**Queft. 96.** What way is that ?

Answ. It is a way which requires left figures in

170

Of Division.

in the work, then the former.

Quest. 97. She w the difference in few words,

Anf. Having placed the Divisor under the Dividend (as before is taught) you must not begin at the first figure of your Divisor towards the left hand (when you multiply) but at the first figure toward the right, and multiply that figure by the figure in the Quotient (that answer the question) and so going towards the left hand.

Quest. 98. Is no difference in the placing of the Divisor under the Dividend, and in the removing of it?

Anf. None at all.

Quest. 99. But at what figure do you begin, when you make the question, to finde the figures of the Quotient; do you begin at the left hand of the Divisor, or at the right?

Anf. This second way of Division, and the first agree also in that, that every question (that is made to finde the figure in the Quotient) is always made upon the first figure of the Divisor, towards the left hand.

Quest. 100. Is there no more difference, then what you have said ?

Anf. Yes : The difference lies further in the subfraacting of the Products arising from the Multiplication of the figures of the Quotient, and Divisor multiplied together.

Quest. 101. Where lies the difference between the Subtraction in this way of Division, and the former.

Anf. In the first way of Division, in Subtracting

171

Of Division.

Subtracting the Products of the Divisor, if any unites be borrowed, they are to be subtracted alone, preferably from the next figure, from whence it was borrowed, and the rest is to be set above. All this is to be done before you proceed to multiply the second figure of the Divisor; but in this second way, you must do otherwise.

Quest. 102. In which way must I then go?

Answ. When you have borrowed any figure (or number) from the left hand, and after the subtraction is made (and the Rest set down) you must not presently subtract those Unites so borrowed from the next figure of the Dividend (as you did before) but you must keep or bear in mind those Unites so borrowed.

Quest. 103. What do you with those Unites that you bear in mind?

Anf. I adde them to the Product of the Multiplication of the next figure of the Divisor, to be subtracted from the figure above.

Quest. 104. I perceive that in subtracting the several Products of the Divisor, you go from the right hand towards the left?

Anf. Yes; and that is quite contrary to the other way of Division.

Quest. 105. Suppose you were to borrow at every Subtraction you are to make, by reason of the greatness of the several Products that are to be subtracted.

Answ. I must borrow as before to supply that want, and be careful to keep every Unite borrowed in mind, to be added to the next Product of the

172

Of Divifion.

the Multiplication of the next figure, and so on
until I come to the last figure of the Divisor; as
in the first Example of the second way of Divi-
fion, fol. 145.

Quest. 106. But if you finde that those figures
(or places) from whence you are to borrow, be either
too little, or onely ciphers : Where do you borrow
then?

Answ. It is needles to trouble my self to ob-
serve, whether those figures, from whence I
must borrow, be Significant figures, or not.

Quest. 107. Why so?

Answ. Because I pay the debt in adding what
I have borrowed to the Product of the Multipli-
cation of the next figure of the Divisor.

Quest. 108. I shall ask you a question more,
and then conclude.

Answ. Ask what you pleafe.

Quest. 109. Suppose that when you come to bor-
row (as before) you finde all the figures above to-
wards the left hand (except the last) to be onely
ciphers : Where do you borrow then ?

Answ. It is true, that I cannot borrow any
thing from a cipher or ciphers; therefore I bor-
row what is needful, from the next Significant
figure that I finde on the left hand; as you may
clearly see in the working of the 49 Sum of Di-
vision, fol. 140.

Re-

173

Reduction by Multiplication.

Reduction by Multiplication (called by some
Reduction Descending) teacheth to reduce
Moneys, Weights, and Measures, from greater
Denominations into smaller ; as Pounds Sterling,
to Shillings, Pence, &c. Or Pounds Weight,
to Ounces, Drams, &c.

When any Number is propounded ( either
Money, Weight, or Measure) to be reduced into
a leffer denomination or name. Confider how
many peeces of the lefs, make one of the greater,
and multiply one number by the other ; the
Product is the number required to be short : I re-
fer the Learner to the following Table, which
will shew him sufficiently, how to work the fol-
lowing Queftions.

Of English Money.

To Reduce
Pounds } Shillings } Shillings } Multiply } 20
Shillings } into } Pence } by } 12
Pence } Farthings } 4

Further.

To Reduce { Groats      60
Pounds at  { Pence        240
  one ope- { Farthings   { 960
  ration in- { Nobles       3
    to     { Crowns        4

Averdupois

174

Reduction by Multiplication.

Averdupois Great Weight.

Averdupois Subtile.

Averdupois Little Weight.

Troy Weight.

Dry Measures.

Long

Reduction by Multiplication.

175

Long Measures.

Further.

Liquid Measures.

Of

176

Reduction by Multiplication.

Of Time.

Of things accounted by the Groß or Dozen.

I Question.

IN 312 Pounds, how many Shillings? facit 6240 β.

Facit-6240 β.

2. Item, In 1087 L, how many β? facit 21740 β.

3. Item, In 392 L 12 β, how many β? facit 7852 β.

In multiplying the 392 L by 20, take in the 12 β, the 2 β first, then the 1.

4. Item,

177

Reduction by Multiplication.

4. Item, In 1048 L 7 β, how many β? facit 20967 β.

5. Item, In 37 β, how many β? facit 444 β.

6. Item, In 448 β, how many Farthings? facit 1792 q.

7. Item, In 362 β 6 β, how many β? facit 4350 β.

In Multiplying by 12, take in the 6 β.

8. Item, In 807 β 11 β, how many β? facit 9695 β.

9. Item, In 69 β 2 q, how many q? facit 278 q.

10. Item, In 341 L, how many β? facit 81840 β.

Multiply by 20, and the Product by 12, or else by 240.

11. Item, In 86 L 19 β, how many β? facit 20868 β.

12. Item, In 804 L 0 β 6 β, how many β? facit 192966 β.

13. Item, In 841 β, how many q? facit 40368 q.

Multiply first by 12, then by 4, or else by 48.

14. Item, In 316 β 8 β, how many q? facit 15200 q.

15. Item, In 868 L 13 β 8 β, how many β? facit 208484 β.

16. Item, In 346 L 19 β 6 β 2 q, how many Farthings? facit 333098 q.

Multiply the L by 20, and take in the 19 β, then by 12, taking in the 6 β, and lastly by 4, and take in the 2 q.

N

17. Item,

["178", "## Reduction by Multiplication.", "17. Item, In 581 L 0 β 09 3q, how many q? facit 557763 q.", "18. Item, In 807 L, how many Crowns? facit 3228 Δ.", "19. Item, In 98 L, how many Half crowns? facit 784 Half crowns.", "20. Item, In 124 β, how many Groats? facit 372 Groats.", "21. Item, In 36 L 16 β, how many Groats? facit 2208 Groats.", "22. Item, In 68 Half crowns, how many β? facit 2040 β.", "23. Item, In 341 Δ 2 β 8 β, how many β? facit 20492 β.", "24. Item, 98 Nobles, how many β? facit 7840 β.", "A Noble is 6 β 8 β or 80 β, therefore multiply by 80.", "25. Item, In 687 Marks, how many β? facit 109920 β.", "26. Item, In 143 Marks, 6 β 9 β, how many β? facit 22961 β.", "<table><thead><tr><th>Mark</th><th>β</th><th>β</th><th>Marks</th><th>β</th><th>β</th></tr></thead><tbody><tr><td>1</td><td>13</td><td>4</td><td>-143</td><td>6</td><td>9</td></tr><tr><td></td><td>12</td><td></td><td>160</td><td>12</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>30</td><td></td><td>8580</td><td>81</td><td>9</td></tr><tr><td></td><td>13</td><td></td><td>143</td><td></td><td></td></tr><tr><td></td><td></td><td>Add.</td><td>81</td><td></td><td></td></tr><tr><td></td><td>160</td><td>9</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td

180

Reduction by Multiplication.

39. Item, In 97 C 3 qrs 127 fb 15 ounce,
7 dra. 2 scr. 19 grains, how many grains? fac-
cit 84295679 grains.

Averdupois Little Weight.

40. In 380 fb, how many ounces? facit
6080 ounces.

41. Item, In 138 ounces, how many drams? facit 2208 drams.

42. Item, In 986 fb 15 ounce. 15 drams,
how many drams? facit 52671 drams.

Troy Weight.

43. In 84 fb, how many ounces? facit 1008
ounces.

44. Item, 68 fb 9 ounce. how many ounces?
facit 825 ounces.

45. Item, In 126 fb 10 ounce. 8 penny
weight, how many penny weight? facit 30448
penny weight.

46. Item, In 98 fb 0 ounce. 6 pw. how
many grains? facit 564624 grains.

47. Item, In 182 fb 9 ounce. 19 pw. 23
grains, how many grains? facit 1053119 grains.

Dry Measles.

48. In 92 Seams (or Quarters) of Corn, how
many Buffels? facit 736 Buffels.

49. Item, In 620 Seams 6 Buffels ? how
many Pecks? facit 19864 Pecks.

50. Item,

181

Reduction by Multiplication.

50. Item, In 367 Pecks, how many Pints?
facit 5872 Pints.

51. Item, in 89 Buffels 2 Pecks 13 Pints,
how many Pints? facit 5741 Pints.

52. Item, In 187 Seams 3 Buffels 1 Peck
12 Pints, how many Pints? facit 95964 Pints.

Liquid Measles.

53. In 38 Tuns, how many Butts (or Pipes)
and Hogsheads? facit 76 Butts, and 152 Hogs-
heads.

54. Item, In 124 Tuns, how many Hogs-
heads? facit 496 Hogsheads.

55. Item, In 58 Hogsheads, how many Gal-
lons? facit 3654 Gallons.

56. Item, In 94 Pottles, how many Quarts
and Pints? facit 188 Quarts, and 376 Pints.

57. Item, In 87 Hogsheads, how many
Quarts? facit 21924 Quarts.

58. Item, In 128 Tuns 1 Hogshead 40 Gal-
lons 1 Pottle, 1 Quart 1 Pint, how many Pints?
facit 258879 Pints.

Long Measles.

59. In 32 Miles, how many Furlongs? facit
256 Furlongs.

60. Item, In 64 Furlongs, how many Perches?
facit 2560 Perches.

61. Item, In 64 Miles, how many Perches?
facit 20480 Perches.

N 3

62. Item,

182

Reduction by Multiplication.

62. Item, In 102 Perches, how many Half-yards? facit 1122 Half-yards.

63. In 46 Perches, how many Inches? facit 9108 Inches.

64. Item, In 420 Acres, how many Roods? facit 1680 Roods.

65. Item, In 356 Acres 2 Roods 20 Perches, how many Perches? facit 57060 Perches.

66. Item, In 88 Foot, how many Inches? facit 1056 Inches.

67. Item, In 108 Yards, how many Barley Corns? facit 11664 Barley Corns in length.

68. Item, In 42 Miles, how many Inches? facit 2661120 Inches.

69. Item, In 48600 Miles, how many Barley Corns? facit 9237888000 Barley Corns.

Of Time.

70. In 64 Years, how many days? facit 23360 days.

71. Item, In 369 Years 8 Moneths 20 Days, how many days, if 12 Moneths be reckoned for every Year, and 30 Days for every Moneth? facit 133100 days.

72. Item, In 120 Years 9 Moneths 3 Weeks 3 Days, how many Days, counting 13 Moneths for every Year, and 4 Weeks for every Moneth? facit 43956 Days.

73. Item, In 1662 Years, how many Days and Hours? facit 606630 Days, and 14559120 Hours.

74. Item,

183

Reduction by Multiplication.

74. Item, In 1658 Years 164 Days 23 Hours 59 Minutes; how many Days, Hours, and Minutes? facit 605334 Days 14528039 Hours, and 871682399 Minutes.

75. Item, A. hath 24 Farms, in every Farm 6 Stables, in every Stable 32 Sheep, each Sheep hath 4 Lambs; the queftion is, how many Feet there was in all? facit 92160 Feet.

76. Item, A rich Man hath 10 Villages belonging to him; every Village hath 5 Streets, in every Street 30 Housels, in every Housle 6 Chambers, in every Chamber 12 Desks, in every Desk 24 Drawers, in every Drawer 6 Purſes, in every Purſe 20 Peece of Gold, and each Peece is worth 21 3; the queſtion is, what amounts the whole ſum in Farthings? facit 313528320000 Farthings.

Reduction by Division.

Reduction by Division (called by some Reduction Ascending) teacheth to reduce a Number of a lesser Denomination (either Money, Weight, or Meaſure, &c.) into a greater; as Farthings to Pence, Pence to Shillings, Shillings to Pounds, &c. Or Drams to Ounces, Ounces to Pounds, &c. The following Table will be a sufficient direction to make the operations of this Reduction.

N 4

Of

186

Reduction by Division.

Liquid Measures.

Of Time.

1 Question.

Reduction by Division.

187

1 Question.

IN 6240 Shillings, how many Pounds? facit 312 L.

6240 (312 Pounds.)
2220 Facit.

2. Item, In 21740 β, how many L? facit 1087 L.

3. Item, In 2060 β, how many Δ? facit 412 Crowns.

4. Item, In 7852 β, how many L? facit 392 L 12 β.

5. Item, In 4650 Φ, how many β? facit 387 β 6 Φ.

6. Item, In 8901 Groats, how many β? facit 2967 β.

7. Item, In 17290 β, how many L? facit 864 L 10 β.

8. Item, In 1792 q, how many Φ? facit 448 Φ.

9. Item, In 32680 Φ, how many Groats? facit 8170 Groats.

10. Item, In 81840 Φ, how many L? facit 341 L.

Divide by 12, and then by 20, or else by 240.

11. Item,

188

Reduction by Division.

11. Item, In 20868 9, how many L ? facit
86 L 19 ß.

12. Item, In 21887 9, how many Six pences ?
facit 3647 Six pences, and 5 9 over.

13. Item, In 40368 q, how many ß ? facit
841 ß.

14. Item, In 208484 9, how many L ? fa-
cit 868 L 13 ß 8 9.

15. Item, In 333098 q, how many L ? fa-
cit 346 L 19 ß 6 9 2 q.

16. Item, In 3228 Δ, how many L ? facit
807 L.

17. Item, In 784 Half crowns, how many
L ? facit 98 L.

18. Item, In 2208 Groats, how many L ?
facit 36 L 16 ß.

19. Item, In 2040 9, how many Half crowns ?
facit 68 Half crowns.

20. Item, In 20492 9, how many Δ ? facit
341 Δ 2 β 8 9.

21. Item, In 7840 9, how many Nobles ?
facit 98 Nobles.

22. Item, In 22961 9, how many Marks ?
facit 143 Marks 6 β 9 9.

23. Item, In 2472960 Farthings, how many
Marks ? facit 3864 Marks.

24. Item, In 295935 q, how many pieces at
2 L 16 ß 6 9 3q the piece ? facit 109
pieces.

Reduce 2 L 16 ß 6 9 3q into Farthings,
and the Product is the Divisor.

Aver-

189

Reduction by Division.

Averdupois Great Weight and Subtile.

25. In 544 Ounces, how many lb ? facit
34 lb.

26. Item, In 12854 Ounces, how many lb ?
facit 803 lb 6 ounces.

27. Item, In 346 Quarters of C, how many
C ? facit 86 C 2 quarters.

28. Item, In 68400 Ounces, how many Quarters ? facit 152 Quarters 19 lb.

29. Item, In 55288 Drams, how many lb ?
facit 431 lb 15 ounces.

30. Item, In 48963 Scruples, how many Drams ? facit 16321 Drams.

31. Item, In 68080 Grains, how many
Scruples ? facit 3404 Scruples.

32. Item, In 960487 Scruples, how many
lb ? facit 2501 lb 4 ounce. 2 drams 1 scruple.

33. Item, In 684360 Drams, how many Hundreds ? facit 47 C 3 Quarters 2 lb 7 ounce. 4 drams.

34. Item, In 308476830 Grains, how many
Hundred ? facit 358 C 2 qrs 14 lb 4 ounce.
0 dra. 1 scrup. 10 grains.

Averdupois Little Weight.

35. In 6080 Ounces, how many lb ? facit
380 lb.

36. Item, In 2208 Drams, how many Ounces ?
facit 138 Ounces.

37. Item,

190

### Reduction by Division.

37. Item, In 252671 Drams, how many fb ?
facit 986 fb 15 ounces 15 dram.

### Troy Weight.

38. In 816 Ounces, how many fb ? facit
68 fb.

39. Item, In 600 Penny weight, how many
Ounces? facit 30 Ounces.

40. Item, In 860060 Penny weight, how many fb ? 3583 fb 7 ounces.

41. Item, In 48360 Grains, how many Ounces? facit 100 Ounces 15 pw.

42. Item, In 64780 Grains, how many fb ?
facit 11 fb 2 ounces 19 pw 4 grains.

### Dry Measures.

43. In 736 Bushels of Corn, how many Seams
(or Quarters?) facit 92 Seams.

44. Item, In 8064 Pecks, how many Bushels?
facit 2016 Bushels.

45. Item, In 68400 Pints, how many Pecks ?
facit 4275 Pecks.

46. Item, In 1064809 Pints, how many
Seams? facit 2079 Seams 5 Bushels 2 Pecks
9 Pints.

### Liquid Measures.

47. In 648 Butts (or Pipes) how many Tuns ?
facit 324 Tuns.

48. Item,

191

### Reduction by Division.

48. Item, In 9640 Hogsheads, how many
Butts? facit 4820 Butts.

49. Item, In 18600 Gallons, how many Hogs-
heads? facit 295 Hogsheads 15 Gallons.

50. Item, In 648600 Quarts, how many Gal-
lons? facit 162150 Gallons.

51. Item, In 846006 Pints, how many Hogs-
heads? facit 1678 Hogsheads 36 Gallons 1 Pot-
tle 1 Quart.

52. Item, In 1040093 Pints? how many
Tuns? facit 515 Tuns 1 Butt 1 Hogshead
42 Gallons 1 Pottle 0 Quart 1 Pint.

### Long Measures.

53. In 4860 Furlongs, how many Miles ?
facit 607 Miles and 4 Furlongs.

54. Item, In 40868 Perches, how many Fur-
longs? facit 1021 Furlongs 28 Perches.

55. Item, In 40868 Perches, how many
Miles? facit 127 Miles 5 Furlongs 28 Perches.

56. Item, In 9900 Inches, how many Perches?
facit 50 Perches.

57. Item, In 1368 Foot, how many Yards?
facit 456 Yards.

58. Item, In 59994 Barley Corns, how many
Perches? facit 101 Perches.

59. Item, In 1338 Nails, how many Yards ?
facit 83 Yards 2 Quarters 2 Nails.

60. Item, In 603 Quarters of Yards, how
many Ells? facit 120 Ells 3 Quarters of a Yard.

61. Item, In 6840008604 Barley Corns in
length

194

Reduction by Multiplication and Division.

2. Item, in 2280 Groats, how many Half crowns? facit 304 Half crowns.

3. Item, in 189 ⅛, how many Nine pences?
facit 252 nine pences.

4. Item, in 252 Nine pences, how many Shillings? facit 189 ⅛.

5. Item, in 845 Groats, how many three pences? facit 1126 three pences, and 2 9 over.

6. Item, in 1126 three pences and 2 9, how many Groats? facit 845 Groats.

7. Item, in 270 pieces of money at 4 9 2 q, how many L? facit 5 L 1 ⅛ 3 9.

8. Item, in 840 pieces at 15 ⅛ the piece, how many L? facit 630 L.

9. Item, in 630 L, how many pieces at 15 ⅛ the piece? facit 840 pieces.

10. Item, in 346 Angels, how many pieces at 4 ⅛ the piece? facit 865 pieces.

11. Item, in 865 pieces at 4 ⅛ the piece, how many Angels? facit 346 Angels.

12. Item, in 406 L 12 ⅛ 8 9, how many Half crowns? facit 3253 Half crowns and 2 9 over.

13. Item, in 3253 Half crowns and 2 9, how many L? facit 406 L 12 ⅛ 8 9.

14. Item, in 860 Nobles, how many Groats? facit 17200 Groats.

15. Item, in 17200 Groats, how many Nobles? facit 860 Nobles.

16. Item, in 420 Marks, how many L? facit 280 L.

17. Item,

195

17. Item, in 280 L, how many Marks? facit 420 Marks.

18. Item, in 123 pieces at 13 9 2 q the piece, how many L? facit 6 L 18 ⅛ 4 9 2 q.

19. Item, in 6 L 18 ⅛ 4 9 2 q, how many pieces at 13 9 2 q? facit 123 pieces.

20. Item, in 847 pieces at 6 9 2 q, how many L? facit 22 L 18 ⅛ 9 9 2 q.

21. Item, in 164 Marks, how many pieces at 4 9 2 q the piece? facit 5831 pieces and 2 q.

22. Item, in 5831 pieces and 2 q, at 4 9 2 q the piece, how many Marks? facit 164 Marks.

23. A. oweth the following sum of money, to wit, 124 L 13 ⅛ 6 9, 344 L 16 ⅛ 6 9, and 350 L 10 ⅛ 6 9, how many Rixdollers at 4 ⅛ 2 9 the piece, muft he pay for the whole debt? facit 3936 Rixdollers and 6 9.

24. Item, A. oweth 624 L 3 ⅛ 8 9, of which he hath paid at one time 131 L 17 ⅛ 6 9, at another time 96 L 13 ⅛ 4 9, and laft of all 87 L 12 ⅛ 10 9; the queftion is, how many Nobles he ows full? facit 924 Nobles.

25. Item, A Marchant hath 2467 Ducks, worth 4 ⅛ 4 9 the piece; how many French Crowns at 5 ⅛ 6 9, or Royals at 9 ⅛ 8 9, or Nobles at 6 ⅛ 8 9 the piece, may he receive for the said Ducks? facit 1943 French Crowns and 3 ⅛ 10 9, or 1105 Royals and 8 ⅛ 8 9, or elfe 1603 Nobles and 3 ⅛ 8 9.

O 2

Note,

196

Note, Though the foregoing questions are
sufficient to show the Learners how to work this
Rule, yet the following questions are here added
as a preparation to the Rule of Three.

26. If 1 Yard of Cloth coſt 6 β, what coſt
28 Yards? facit 8 L 8 β.

$$
\begin{array}{@{}r@{}l@{\quad}r@{\quad}r@{}}
\text{Yard} & \text{\beta} & \text{Yards} & \\
1 & - & 6 & - \\
\cline{3-4}
& & 28 & \\
\end{array}
$$

$$
\begin{array}{@{}c@{}r@{}c@{\quad}c@{}}
 & 6 & \times 6 \vert & 8 \, \$ \, 8 \, \beta \\
 &   & \mathbf{\overline{\quad}} & \\
\cline{3-4}
 &   & 168 \, \beta & \text{Facit.}
\end{array}
$$

27. Item, if 1 fb coſt 6 β, what coſt 148 fb ?
facit 44 L 8 β.

28. Item, if 148 fb coſt 44 L 8 β, what coſt 1 fb ? facit 6 β.

29. Item, if 1 Ell coſt 15 9, what coſt 39 Ells? facit 48 β 9 9.

30. Item, if 39 Ells coſt 48 β 9 9, what coſt 1 Ell ? facit 15 9.

31. Item, if 1 ounce coſt 3 q, what coſt 87 ounces ? facit 5 β 5 9, 1 q.

32. Item, if 87 ounces coſt 5 β 5 9 1 q,
what coſt 1 ounce? facit 3 q.

33. Item, if 1 yard coſt 4 β 6 9, what coſt
325 yards? facit 73 L 2 β 6 9.

34. Item, if 325 yards coſt 73 L 2 β 6 9,
what coſt 1 yard? facit 4 β 6 9.

35. Item, if 1 yard coſt 3 9 3 q, what coſt 112 yards? facit 35 β.

36. Item, if 112 yards coſt 35 β, what coſt 1 yard? facit 3 9 3 q.

37. Item,

197

37. Item, if 1 fb coſt 4 9, what coſt the hundred (or 112 fb) ? facit 37 β 4 9.

38. Item, if 1 C coſt 37 β 4 9, what coſt 1 fb ? facit 4 9.

39. Item, if 1 yard coſt 4 β 8 9 2 q, what coſt 64 yards ? facit 15 L 1 β 4 9.

40. Item, if 64 yards coſt 15 L 1 β 4 9, what coſt 1 yard? facit 4 β 8 9 2 q.

41. Item, if 1 fb coſt 2 L 16 β 8 9, what coſt 58 fb ? facit 164 L 6 β 8 9.

42. Item, if 58 fb coſt 164 L 6 β 8 9, what coſt 1 fb ? facit 2 L 16 β 8 9.

43. Item, if 1 C coſt 3 L 0 β 6 9, what coſt 13 C ? facit 39 L 6 β 6 9.

44. Item, if 13 C coſt 39 L 6 β 6 9, what coſt 1 C? facit 3 L 0 β 6 9.

45. Item, if 1 Groſs of Buttons coſt 13 β 0 9 2 q, what coſt 36 Groſſes? facit 23 L 9 β 6 9.

46. Item, if 36 Groſſes of Buttons coſt 23 L 9 β 6 9, what coſt 1 Groſs? facit 13 β 0 9 2 q.

47. Item, if 27 Hogsheads of Commodities coſt 263 L 6 β 8 9 1 q, what coſt 1 Hogshead ? facit 9 L 15 β 0 9 3 q.

48. Item, if 9 L 15 β 0 9 3 q be paid for 1 Hogshead, what coſt 27 Hogsheads? facit 263 L 6 β 8 9 1 q.

49. Item, if 1 yard coſt 12 β 6 9, how many
yards will 51 L 17 β 6 9 buy? facit 83 yards.

50. Item, if 83 yards coſt 51 L 17 β 6 9, how many yards will 12 β 6 9 buy? facit 1 yard.

O 3

Questions

198

Of Reduction.

Questions of Reduction.

Queft. 1. What is Reduction ?

Answ. Reduction is a changing a number from one denomination unto another.

Queft. 2. In how many parts is Reduction divided?

Answ. Some divide it into two parts, some into three.

Queft. 3. What is the first part of Reduction ?

Answ. The first part of Reduction is called Reduction by Multiplication, or Reduction Descending.

Answ. Because it teacheth to reduce, Moneys, Weights, Measures, Time, &c. from a great denomination, into a lesser, which is performed by Multiplication.

Queft. 5. What is your general rule in working this Reduction?

Answ. The general rule is to consider, how many of the less denominations make one of the greater, and then multiply the one by the other.

Queft. 6. Give an example.

Answ. Suppose that a number of Pounds were to be reduced into Shillings, I must (according to the rule) multiply the number of Pounds by 20, because 20 shillings make 1 pound, saying, (in setting the numbers) If 1 Pound be 20 shillings,

Of Reduction.

199

Pounds, how many Shillings amount so many Pounds.

Queft. 7. How many numbers or denominations do you set down?

Answ. I set three for order sake.

Queft. 8. What advantage have you in setting three numbers, when two will serve?

Answ. This order is profitable in two things.

Queft. 9. What are they?

Answ. First, in setting the three numbers I set 1 first, saying, If 1 L makes 20 3, how many shillings make so many pounds? the very setting of the numbers, in this order, shows the reason, why I must multiply by 20 (which is, because 20 3 make 1 L) which is very profitable to observe by new Learners.

Queft. 10. What further advantage is there in following this order?

Answ. In following this method, saying, at every question propounded; If I be worth (or contain) so many small pieces, what so many, &c. The Learner (by this means) comes to be acquainted with the phrase used in the Rule of Three, which will make that rule the more easy to him, when he comes to it.

Queft. 11. How will you reduce shillings to pence?

Answ. I must multiply the number of shillings by 12, because 12 9 make 1 3.

Queft. 12. How do you reduce pence into farthings?

Answ. I must multiply the number of pence

O 4

by

200

Of Reduction.

by 4, becaufe 4 q make 1 $\$:$ But to avoid multiplicity of queftions, the Table of this Reducti-on (Fol. 173: 174: 175: and 176:) is a sufficient direction how to anſwer moſt queſtions of this nature, concerning Weights and Meaſures, &c.

Queſt. 13. How do you reduce pounds into $\$ or q?

Anſw. The ſaid Table directs you to reduce the pounds firſt to ſhillings, and the $\beta$ produced into $\$,$ and the pence into farthings.

Queſt. 14. How will you reduce $L$ into $\$,$ at one operation?

Anſw. I muſt multiply the number of $L$ by 240, becauſe 240 $\$$ make 1 $L$.

Queſt. 15. How do you reduce $L$ into $q$ at one operation?

Anſw. I muſt multiply the $L$ by 960, becauſe 960 $q$ make 1 $L$.

Queſt. 16. Is it always the best way to reduce $L$ into $\$ or q$, at one operation?

Anſw. No; for when a Compound number is to be reduced into $\$ or q$, that way is more troublesome for the Learner, then the other.

Queſt. 17. Shew the difference by an example.

Anſw. Suppose that $8 L$ 17 $\beta$ were given to be reduced into $\$,$ the eaſiſt way is to reduce the $8 L$ into $\beta$ firſt, taking in the 17 $\beta$, then the Product being multiplied by 12, produceth $\$.

Queſt. 18. How will you reduce the ſaid $8 L$ 17 $\beta$ into $\$,$ the other way?

Anſw. I muſt make two Multiplications, and adde the Products together.

Queſt.

201

Of Reduction.

Queſt. 19. In what manner?

Anſw. I muſt, firſt multiply the $8 L$ by 240, and the 17 $\beta$ by 12, and adde these two Products together.

Queſt. 20. This way seems to be more troublesome then the former; nevertheless ſhow the operations of both, to give more ſatisfaction.

Anſw. The two ſeveral operations are as followeth.

$$ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \text{L} & \text{\beta} & \text{L} & \text{\beta} \\ \cline{2-4} 1 & 20 & 8 & 17 \\ \cline{2-4} \text{\beta} & \$ & 20 & \\ \cline{2-4} 1 & 12 & 177 & \text{\beta} \\ \cline{3-4} & & 12 & \\ \cline{4-4} & & 354 & \\ \cline{4-4} & & 177 & \\ \cline{4-4} & & & \\ \cline{2-4} & & & \\ \cline{4-4} & & & \text{Facit } 2124 \$ & 204 \$ \\ \end{array} $$

Queſt. 21. I perceive that the firſt way is leſs troublesome: But how do you reduce a Compound number of $L \beta \$ q$ into $q$?

Anſw. I multiply firſt the $L$ by 20, and take in the odde $\beta$; then I multiply the Product by 12, and take in the odde $\$;$ and laſtly, I multiply the ſecond Product (which are $\$)$ by 4, and take in the odde $q$, the third Product is the number of $q$ required.

Queſt. 22. How do you reduce a Compound number confſing of $L$, and some odde farthings onely,

202

Of Reduction.

onely, ( as 12 L 0 3 0 2q) into q?

Answ. I multiply the L by 20, and the Product by 12 (as before) then in multiplying the second Product (vwhich are pence) by 4, I take in the odde q.

Queft. 23. What observation or rule have you in taking in the odde money of any Compound number given to be reduced?

Answ. The general rule is this, In multiplying the number of L by 20 (to bring them into 3) I take in the odde 3 (if there be any) then in multiplying the Product by 12 (to bring the 3 into 9) I take in the odde 9 (if I finde any;) lastly, In multiplying the second Product by 4, (to bring 9 to 4) I take in the odde 4, if any odde q be there.

Queft. 24. How will you reduce Pounds into Crowns?

Answ. I multiply the L by 4, becaufe 4 Crovvns make 1 L.

Queft. 25. How do you reduce Pounds into Nobles?

Answ. I multiply by 3, becaufe 3 Nobles make 1 L.

Queft. 26. How do you reduce Nobles into Pence.

Answ. I muſt multiply the Nobles by 80, becauſe the vworth of 1 Noble in 9, is 80 9, or 6 3 8 9.

Queft. 27. How will you reduce Marks into Pence?

Answ. A Mark is 13 3 49, or 160 9, therefore

203

Of Reduction.

therefore I muſt multiply the number of Marks by 160.

Queft. 28. How do you reduce Half crowns into Pence?

Answ. In multiplying the number of Half Crowns by 30, becauſe 30 9 is 1 Half crown.

Queft. 29. How do you reduce great weight into a smaller denomination?

Answ. I muſt obſerve the Rule before-mentioned in the anſwer of the fifth queſtion.

Queft. 30. What rule is that?

Answ. To obſerve how many of the leſſer denomination, make one of the greater, and then multiply theſe two numbers together.

Queft. 31. Will this rule ſerve for all forts of Weights and Measures?

Answ. Yes; as the foregoing Table of Reduction (in Fol. 173: 174, &c.) ſufficiently ſhews; to which, I refer you for the reducing of any Weight or Measure.

Queft. 32. I ſhall ask but one queſtion more about this Reduction, which may ſerve for an example for the reſt?

Answ. If you pleaſe to propound, I am ready to anſwer.

Queft. 33. How will you reduce 97 C 3 qrs. 27 1b 15 ounce. 7 drams 2 scruples and 19 grains, into grains?

Answ. I muſt multiply firſt the 97 C by 4, and take in the 3 qrs, then the Product by 28, and take in the 27 1b, the Product (which are 1b) I muſt multiply by 16, taking in the 15 ounces; the

204

Of Reduction.

the Product (which is ounces) I multiply by 8, taking in the 7 drams; the Product (which is dram) I multiply by 3, taking in the 2 scruples; the Product (which are scruples) I multiply by 20, taking in the 19 grains; the last Product is 84295679 grains for the facit.

Quest. 34. What is the second Part of Reduction?

Answ. The second Part of Reduction is called Reduction by Division, or Reduction Ascending.

Quest. 35. Why is it so called?

Answ. Because it teacheth to reduce a number of a lesser denomination (either Money, Weight, or Meaſure, &c.) into a greater, which is per- formed by Division.

Quest. 36. What is your general rule in this Reduction?

Answ. The general rule is to observe how many pieces of the lesser denomination make one of the greater, required; and that number of smaller denomination is the Divisor.

Quest. 37. Give an example.

Answ. Suppose that a quantity of shillings were given to be reduced into pounds, I divide then the shillings by 20, because 20 β makes 1 L, saving, (in fetting the numbers) if 20 β make 1 L; how many L will so many β make?

Quest. 38. This Rule seems to be the Proof of the former part of Reduction; is it not?

Answ. Yes; these two parts of Reduction prove one another, and I observe the same order in

205

Of Reduction.

in this part, as in the former.

Quest. 39. Do you set down three numbers in this part of Reduction, in the very same manner as in the former?

Answ. There is a little difference in respect of placing the three numbers, but they agree in this general observation, that the first number and the third, must be of one denomination.

Quest. 40. Wherein do they differ in respect of placing the three numbers?

Answ. In the first part of Reduction, the worth of one piece is always set down in the second place, and one in the first; but in this second part, the worth of one piece is always set down first, and one in the second place.

Quest. 41. Is there no more difference to be observed between these two parts of Reduction?

Answ. Yes; I observe that in the first part of Reduction, I multiply always the last number by the second; but in this second part, I must always divide the last number by the first, which observation is worthy to be minded by the Learner; for in so doing the Rule of Three will not seem so strange to him.

Quest. 42. How do you reduce Pence into Shillings?

Answ. Pence are reduced into shillings, by dividing by 12, because 12 9 make 1 β.

Quest. 43. How do you reduce Farthings into Pence?

Answ. In dividing the q by 4, because 4 q makes 1 9, as you may see in the Table of this Reduction.

206

Of Reduction.

Reduction (Fol. 184: 185: and 186,) which shews
also how to reduce all sorts of Weights and Mea-
sures, from a letter denomination into a greater,
to whom I refer you for brevity sake.

Queft. 44. How will you reduce Pence into Pounds ?

Answ. That may be done two ways, either at one working, or in two.

Queft. 45. In what manner ?

Answ. In dividing the number of 9 by 240,
(because 240 9 make 1 L) the Quotient is L,
the Rest is 9.

Queft. 45. What is the other way ?

Answ. The other way is, to divide the number of 9 first by 12, to reduce the 9 to 3, and then divide the 3 by 20, to reduce them to L.

Queft. 47. But in dividing more then once (in reducing 9 to L) if you finde some Remainders; how do you call them ?

Answ. In dividing 9 by 12 (to bring them into 3) if there be any Rest in the Division, such Rest is 9; and in dividing the 3 by 20 (to bring them into L) the Rest of such Division is 3.

Queft. 48. How do you reduce Farthings into Pounds ?

Answ. That may be performed two ways;
the first is to divide the number of 9 by 60 (because 960 q make 1 L) the Quotient is L, and the Rest (if there be any) is q.

Queft. 49. What is the second way ?

Answ. The second way is to divide the num-
ber of q first by 4, (to bring q into 9) the Quo-
tient

207

Of Reduction.

tient by 12 (to bring 9 into 3) and the second Quotient being divided by 20, the third Quotient is L.

Queft. 50. Have you no other way to reduce shillings into pounds, but by dividing the 3 by 20, according to the common Division?

Answ. Yes; there is another way which is
shorter, but yet it is still the same for substance;
for it is by dividing by 20, in a shorter manner,
then the common way.

Queft. 51. Explain it in few words.

Answ. There is nothing else to do, but to cut off the last figure towards the right hand, and to take the half of the Resting figures which are on the left.

Queft. 52. Give an example or two to make that plain and evident.

Answ. Suppose that 865 3 were given to be reduced into L, I cut off the 5, and draw a line under the number of 3 thus 86|5, and I take the half of 86, beginning at the 8, saying, the half of 8 is 4, and the half of 6 is 3; then coming so far as the line of Separation, I set the Character of Pounds (which is L) and the figure cut in the place of 3, as followeth.

$$
\begin{array}{cr}
 & 865 \beta \\
\cline{2-2}
\mathbf{Fac} & 43 \mathbf{L} \ 5 \beta \\
\end{array}
$$

Queft. 53. This way is short enough, but if the figures on the left hand were odde, as 73 instead of 86; how will you then take the half?

Answ. In such a case I must set down the
letter

208

Of Reduction.

lesser half of the first figure, towards the left hand, and the remaining must be accounted for 10, to be added to the next figure, and then take the half again of such sum.

Queft. 54. Give an example of that.

Answ. Suppose that 734 ⅙ were given to be reduced to L's, I cut the last figure as before, thus, 7314, then I take the half of the other figures in this manner: The half of 7 is 3, there remains 1 (which is 10) and 3 is 13; then the half of 13 is 6, there remains 1, which I is 10 ⅙, and the 4 (which is cut off) is 14 ⅙; the sum is then in pounds, 36 L 14 ⅙, as followeth:

$$ \begin{array}{c} 36 \\ 7314 \\ \hline \end{array} $$

Queft. 55. How will you reduce Shillings into Crowns?

Answ. By dividing the number of ⅙ by 5, because 5 ⅙ make 1 Crown.

Queft. 56. How do you reduce Pence into Nobles?

Answ. In dividing the number of pence by 80, because 80 ⅗ make 1 Noble.

Queft. 57. How will you reduce Earthings into Marks?

Answ. In dividing the number of q by 640, because a Mark is 640 q, or 13 ⅙ 4 ⅗.

Queft. 58. Before we speak of the next part of Reduction, I shall ask one question more, which is this, Whether you have not a general observation upon the two parts of Reduction before mentioned?

Answ.

209

Of Reduction.

Answ. Yes; the general observation is, when any great denomination is to be reduced into a lesser, that is always performed by Multiplication: But on the contrary, to reduce lesser denominations into greater, this is performed by Division.

Queft. 59. Is there any more parts in Reduction?

Answ. Yes.

Queft. 60. How do you call it?

Answ. Reduction by Multiplication and Division, and contains the two former parts.

Queft. 61. Is there need of such Reduction before you come to the Rule of Three?

Anf. Yes; it serves as an Introduction to it.

Queft. 62. What doth the third part of Reduction teach?

Answ. It teacheth to reduce one kind of denomination into another.

Queft. 63. What do you mean by that, seeing the two former parts teach the same thing: where is the difference?

Answ. The queftions which belong to the first part, or to the second, are answered either by Multiplication or Division; but the queftions that belong to this third part, are performed by Multiplication and Division.

Queft. 64. Give some examples.

Anf. Suppose that 63 Marks were given to be reduced into pounds, that cannot be answered by Multiplication onely, or by Division alone, but both must be used.

P

Queft.

210

Of Reduction.

Queft. 65. Which way ?

Answ. I multiply the Marks by 2, to reduce them to Nobles (because 2 Nobles make a Mark) then I divide the Nobles by 3- (because 3 Nobles makes 1 L) to reduce them to L.

Queft. 66. I perceive now that the questions of this nature requires Multiplication and Division, but have you a general rule how to work such questions ?

Answr. Yes, the Rule to be observed is much like to that of the Rule of Three.

Queft. 67. Declare the Rule briefly.

Answ. When any question is propounded, the first thing to be minded, is to observe what kind of money or pieces the question requires; which being known, there are four things more to be well observed.

Queft. 68. Which are they ?

Answ. I must set three numbers in order, but to prevent mistakes, I set first 1 in the second place (or in the middle) which 1 bears the same name of that kind of specie (or pieces) required: Secondly, I set the worth of that 1 in the first place. Thirdly, the number propounded in the third place.

Queft. 69. When the three numbers are placed, what do you observe further ?

Answ. I observe whether the first and third number be of like denomination, and if they be not, I must reduce them into the same denomination or name by Multiplication.

Queft. 70. What do you further ?

Answ.

211

Of Reduction.

Answ. When the first and third number be, reduced into one and the same denomination then there is nothing else to do, but to divide the last (or third) number by the first, and the Quotient is the number required.

Queft. 71. Give an example to explain your meaning ?

Answ. Suppose that it were required to reduce 364 Ninepences into Groats, I observe that the pieces required are Groats, therefore I set down 1 Groat in the second place, and the worth of it (which is 4 9) in the first; then I set down the 364 Ninepences in the third place, as followeth.

Queft. 72. Go on and finish the working of this question.

Answ. The three numbers being in such order, I perceive that the first number is 9, and the third is Ninepences; therefore I must multiply the 364 Ninepences by 9 (because 9 9 is the worth of each piece) to reduce them into 9 (which is of the same name as the first number is) the Product will be 3276 9, which number being divided by the first (which is 4) the Quotient will be 819 Groats; which is the answer of the question, as followeth.

P 2

9 Groat

212

Of Reduction.

Queft. 73. Must you of necessity follow always such order (or form) in every question?

Answ. No; there is no necessity to follow always that form of setting the three numbers, for there is another form which is as good, though it be a little longer.

Queft. 74. What form is that?

Answ. It is the same form used in the two former parts of Reduction.

Queft. 75. Then you must make use of six numbers.

Answ. Yes; but yet in so doing, I neither multiply, nor divide oftner then in this way before mentioned.

Queft. 76. Work this way the forementioned question, which is to reduce 364 Ninepences into Groats.

Answ. I reduce the 364 Ninepences into 9 (by the first part of Reduction) saying, if 1 piece makes 9 9, what 364 pieces, and by multiplying I finde 3276 9; then I say further (by the second part of Reduction,) if 4 9 makes 1 Groat, what 3276 9? and by dividing by 4, I finde 819 Groats, which is the answer of the question, the work followeth.

Piece

213

Of Reduction.

444

Queft. 77. I perceive that these two manner of working, are much like one another, but do you use always one way (or form) in answering every question that belongs to the third part of Reduction?

Answ. No.

Queft. 78. What way do you like best?

Answ. I do not tie my self to one way, for I use sometimes one, and sometimes the other; which I judge to be the most convenient for the answering the questions propounded.

Queft. 79. Do you not finde that a tedious thing to set down always six numbers, or else three, to work out every question?

Answ. I confess, that sometimes being in haste, I set down but two numbers in any part of Reduction: But I wish the Learner to follow the order and method before mentioned, to make him the more perfect to understand the Reasons of his working, which having obtained, he may afterward use as much brevity as he pleafeth.

P 3

Queft.

214

The Rule of Three direct.

Queft. 80. What is the next Rule to be Spoken
unto?

Answ. The Rule which follows Reduction is,
The Rule of Three, called The Golden Rule.

The Rule of Three direct,
CALLED
The Golden Rule.

This Rule is so called, because three numbers
are required in its operation, to find a fourth
in a direct proportion.

This Rule is the chiefest in Arithmesick, and
therefore is called The Golden Rule; it deferves
then to be well minded by the Learner, because
it is that Rule, upon which the other Rules do
stand; therefore I shall inlarge the more up-
on it.

In this Rule, the first number bears such pro-
portion to the second, as the third doth to the
fourth; or thus, As the first number is to the
third, so is the second to the fourth : As for ex-
ample.

These four numbers, to wit, 6. 3, 18 and 9.
are in a direct proportion; for, as 6 is to 3, so
is 18 to 9, which is a double proportion : Or
thus, As 6 is to 18, so is 3 to 9, &c.

Before I mention how to finde such a fourth
number

-215

The Rule of Three direct.

number proportional, when three numbers are
propounded, I shall first acquaint the Learner,
how to place the three numbers, which is one of
the chiefest things to be observed in this Rule.

How to place the three Numbers.

IN the Rule of Three, the three numbers pro-
pounded, are commonly of two general de-
nominations; two of which have the same name,
and the other, with the number required, have
another name; that is to say, that in the three
numbers, there is commonly twice Money,
Weight, or Meaſure, &c. named, and once an-
other name different from the two others, which
is of the fame general denomination with the
fourth number required, as appears by the follow-
ing question.

What will cost 25 Yards, if 5 Yards cost
3 L?

In this question you see two different denom-
nations (to wit, Length and Money) in the three
numbers; two whereof are Yards, and one is,
pounds sterling. Now for the right ordering of
the three numbers of any question, (though the
three numbers be never so confusedly propound-
ed) observe the following Directions.

I. When any question is propounded, the first
thing to be considered is, What the question doth
demand, either, Length, Money, or Weight,

P 4

Gc.

216

The Rule of Three direct.

Co. Which thing being known, you are to take notice, that one of the three numbers bears the same name (general or particular, that is to say, Money, or Pounds, or Shillings, Pence, &c.) which number must be in the second place, (or in the middle of the two others;) as for example, The forementioned question requires Money (to wit, what 25 Yards coft) therefore the 3 L must be in the second place (or in the middle of the two others) though the said Money was last named in the question; for the three numbers are not to be always placed, as they are given in the question, because they are sometimes confusedly propounded.

I. Being acquainted with the number that must poffess the second place; observe further, that one of the two others is of equal value (though not of the same name) with the said second number, which number must be in the first place; as in the forementioned question, the 3 L is the second number, then 5 Yards must be in the first place, because it is equal in value with the said 3 L.

II. The two first numbers being placed, the third number (which is that number upon which the question is moved) must poffess the third (or last) place, as in the question before mentioned. It is required, to know what 25 Yards coft; therefore 25 Yards must be in the last (or third) place, though it be first named in the question.

IV. Having placed the three numbers in their right places, you must further observe, whether the

the first and last number be both of one particular name, as Pounds, Yards, Ells, &c. if they be not, reduce them as followeth.

If one be of a lesser denomination, then the other number, (that is, if one number be Pounds, and the other Shillings, or Pence, or Farthings; or the one be Pounds, and the other Ounces, or Drams, &c.) then reduce the greater denomination into the same name, as the lefser is.

Further, if one of the numbers (to wit, the first or the third) or both, be a compound number (as L, β, ¶, q, or 1lb, Ounces, Drams, &c.) then reduce them both into the least denomination, mentioned in either number, that is, if one number (the first or the third) be onely L, and the other L, β, ¶, q, then both the numbers must be reduced into q, or both into ¶ onely; if the least denomination in any of these two numbers be ¶.

V. Having reduced the first and third numbers into one particular denomination (if you see it needful to reduce them so) then go to the second number, and see whether it be single or compound; if it be single, do as is taught in the sixth Direction following, but if the said second number be compound, (or of more denominations then one) then reduce it into the least denomination, mentioned in the said number; that is, if the said number consist of Pounds, Shillings, Pence, reduce it into Pence; but if there be any Farthings, then reduce it into Farthings; or else, if the said second number, consist of Pounds, Ounces,

217

218]

## The Rule of Three direct.

Ounces, Drams, then reduce it into Drams, &c.

V I. Having placed the three numbers in their right places, and reduced them, as is taught before; then for to finde the fourth number proportional required, the general Rule is as followeth.

Multiply the second number by the third (or the third by the second) and divide the Product by the first; the Quotient is the fourth number required.

As for example, If these three numbers 6, 3, and 18, be given to finde a fourth in a Direct proportion; Multiply 18 by 3, the Product is 54; which being divided by 6, the Quotient is 9, for the fourth number required.

VII. Observe further, That the Product of the Multiplication of the second and third number, being divided by the first, the Quotient (or fourth number) is always of the same name with the second number (or of the same name that the second number is reduced of) that is to say, If the second number be $L$, the said Quotient is $L$; if it be $\beta$, $\vartheta$, or $q$, then the said Quotient is also $\beta$, $\vartheta$, or $q$, &c. Now observe also, that if you finde that the said Quotient answereth the question directly, in such denomination required, then the work is wholly finished, that is, if the question requires $L$, and that the Quotient be $L$ (because the second number is $L$) then you have the number required at the first Division. But if you finde that the said Quotient is but Shillings,

219]

## The Rule of Three direct.

Shillings, Pence, or Farthings, (because the second number is (or is so reduced) of such small denominations) then reduce such lesser denominations into Pounds, if their quantity will permit it.

Further, if you finde the second number to be Pound weight, and that the question requires such denomination as 1b, then the Quotient will answer the question. But if the said Quotient is found to be but Ounces, or Drams, &c. (because the second number is of such small denomination) then reduce such Quotient (which is a lesser weight then the question requires) into Pounds, if the quantity of small weight will permit it: Do the like in other denominations, as Yards, Ells, or other Measures.

VIII. If any of the three numbers be but an unite (which neither multiply nor divide) then the work is the sooner ended: For, if the first number be but 1, then you need not to divide by it; but the Product arising from the two last numbers, is the fourth number sought.

Further, if the second or third number be but 1, then you need not to multiply them together, but divide the second or last number, by the first, and the Quotient is the fourth number required.

IX. Lastly, if you finde that the Product arising from the Multiplication of the second and third number, cannot be divided by the first number; or that the Quotient of such Division cannot afford an Unite, (by reason the Divisor is greater then the Dividend) then reduce the second number to its least terms, and if there be any Shillings, Pence, or Farthings, then reduce them to their least terms, and take the餘 of those reduced to the second number, and so on, according to the order of the Denomination.

220

The Rule of Three direct.

number into so small a denomination, that the
number produced being multiplied by the third
number, their Product may be divided by the
first: But if you find after such Reduction, that
the quantity of the smallest denomination being
multiplied by the third number, cannot afford a
Dividend great enough, to divide it by the first
number, then you may conclude, that the answer
of such question (or the fourth number sought)
cannot be an Unit of such a small denomination,
but only a fraction of it.

1. Example.

If 5 Yards of Cloth cost 3 L, what cost 25
Yards? cost 15 L.

Having placed the three numbers in their right
places, I multiply the last number 25 by the se-
cond 3, the Product is 75, which being divided
by 5, (the first number) the Quotient is 15 L
for the number required.

Before I propound any more questions, I
thought it beneficial for the Learner to know how
to make the Proof of his work before he go fur-
ther; therefore I shall mention in this place, how
to prove the Rule of Three.

The

221

The Rule of Three direct.

The Proof of the Rule of Three direct.

The Rule of Three direct, may be proved 4
ways (which are very beneficial for the
Learner to understand) as followeth.

1. Observe that four numbers being in a direct proportion, the Product of the first and fourth number multiplied together, is always equal to the Product of the second and third number multiplied together, as followeth.

Take the four numbers of the first question of
this Rule, which are 5: 3: 25: and 15. Now mul-
tiplying the first and the fourth together, the
Product will be 75; and multiplying likewise the
second and third together, the Product is also 75,
as followeth.

This Proof is the ground of the other, and
therefore it ought to be well observed.

The Second Proof.

Multiply the fourth number by the first, and
divide the Product by the third, the Quotient will
be equal to the second.

Or

222

The Rule of Three direct.

Or thus by way of Question.

If 25 Yards coft 15 L, what coft 5 Yards?
facit 3 L, as followeth.

The Third Proof.

Multiply the third number by the second, and
divide the Product by the fourth, the Quotient will
be equal to the first.

Or thus.

If 25 Yards coft 15 L, how many Yards
will 3 L buy? facit 5 Yards, as followeth.

The Fourth Proof.

Multiply the fourth number by the first, and
divide the Product by the second, the Quotient
will be equal to the third.

Or thus.

If 5 Yards coft 3 L, how many Yards will
15 L buy? facit 25 Yards.

L Yards

223

The Rule of Three direct.

2. Item, If 7 1b coft 5 L, what will then
coft 35 1b? facit 25 L.

3. Item, If 17 Ells coft 2 L, what coft
2074 Ells? facit 244 L.

4. Item, If 2 Yards coft 24 13, what coft
140 Yards? facit 84 L.

5. Item, If 1 Yard coft 16 13, what coft 224
Yards? facit 179 L 4 13.

6. Item, If 3 1b coft 27 9, what coft 213 1b?
facit 7 L 19 13 99.

7. Item, If 16 1b coft 11 L, what coft 176
1b? facit 121 L.

8. Item, What will coft 45 Ells, if 9 Ells.
coft 4 L? facit 20 L.

9. Item, What coft 351 Yards, if 3 L buy
13 Yards? facit 81 L.

10. Item, If 7 L buy 11 Yards, what coft
209 Yards? facit 133 L.

11. Item, If 15 1b coft 6 L, how many
1b will 102 L buy? facit 255 1b.

12. Item, If 67 Yards coft 21 L, how many
Yards will 189 L buy? facit 603 Yards.

13. Item, How many Ells will 184 L
buy, if 135 Ells coft 23 L? facit 1080
Ells.

14. Item,

224

The Rule of Three direct.

14. Item, If 84 L be paid for 1620 lbs, how many lbs will 252 L buy? facit 4260 lbs.

15. Item, How many lbs will 384 L buy, if 24 L buy 182 lbs? facit 2912 lbs.

16. Item, If 888 lb of Pepper be worth as much as 148 lb of Cloves, how many lb of Pepper will be worth 3487 lb of Cloves? facit 20922 lb of Pepper.

17. Item, If 3487 lb of Cloves, be worth as much as 20922 lb of Pepper, how many lb of Cloves will be worth 888 lb of Pepper? facit 148 lb of Cloves.

18. Item, What Money must be paid for 64 Yards, if 4 Yards cost 3 L? facit 48 L.

19. Item, If 4 Yards cost 24 1/3, what cost 47 Yards? facit 14 L 2 1/3.

20. Item, If 32 1/3 be paid for 2 Yards of Cloth, what will cost 2 Pieces, long together, 98 Yards? facit 78 L 8 1/3.

21. Item, What cost 123 lbs, if 3 lbs cost 16 9/100? facit 54 1/3 8 9/100.

22. Item, 15 9/100 buy 3 Yards, what will cost 342 Yards? facit 7 L 2 1/3 6 9/100.

23. Item, If 162 lbs cost 2 L 15 1/3, what cost 1458 lbs? facit 24 L 15 1/3.

24. Item, If 5 L 19 1/3 be paid for 77 lbs, what will cost 847 lbs? facit 65 L 9 1/3.

25. Item, If 3 lbs cost 7 Farthings, what cost 63 lbs? facit 3 lbs 0 9/100 3 9/100.

26. Item, What cost 624 lbs, if 4 lbs cost 9 9/100? facit 29 lbs 3 9/100.

27. Item,

225

The Rule of Three direct.

27. Item, If 7 lbs cost 6 1/3 5 9/100, what cost 49 lbs? facit 2 L 4 1/3 11 9/100.

28. Item, What cost 75 lbs, if 5 lbs cost 12 1/3 9 9/100? facit 9 L 11 1/3 3 9/100.

29. Item, If 3 Yards cost 8 Groats, what cost 369 Yards? facit 16 L 8 1/3.

30. Item, What cost 204 Ells, if 4 Ells cost 14 Groats? facit 11 L 18 1/3.

31. Item, If 9 lbs cost 9 9/100 2 q, what cost 108 lbs? facit 9 lbs 6 9/100.

32. Item, What cost 340 Yards? if 5 Yards cost 16 9/100 3 q? facit 4 L 14 1/3 11 9/100.

33. Item, If 116 lbs cost 29 Crowns, what cost 464 lbs? facit 29 L.

34. Item, If 34 Crowns be paid for 140 lbs, what cost 1540 lbs? facit 93 L 10 1/3.

35. Item, If 11 Yards cost 4 L 15 1/3 6 9/100, what cost 132 Yards? facit 57 L 6 9/100.

36. Item, What cost 624 Yards, if 208 Yards cost 12 L 19 1/3 4 9/100? facit 38 L 18 1/3.

37. Item, If 6 L 0 1/3 9 9/100 buy 21 lbs, what cost then 147 lbs? facit 42 L 5 1/3 3 9/100.

38. Item, If 22 lbs cost 19 1/3, how many lbs will 57 lbs buy? facit 69 lbs.

39. Item, If 24 Yards cost 16 1/3, how many Yards will 84 L buy? facit 2520 Yards.

40. Item, If 5 1/3 buy 13 lbs, how many lbs will be bought for 49 L? facit 2543 lbs.

41. Item, How many lbs will 64 L buy, if 3 1/3 buy 96 lbs? facit 3840 lbs.

Q

42. Item,

226

The Rule of Three direct.

42. Item, If 25 20 Yards coſt 84 L, how many Yards will 16 3 buy? facit 24 Yards.

43. Item, How many Ells will 48 L 10 3 buy, if 32 Ells coſt 8 L? facit 194 Ells.

44. Item, How many lbs will 42 L 10 3 buy, if 24 lbs coſt 3 L? facit 340 lbs.

45. Item, If 3 L 11 3 buy 120 lbs, how many lbs will 24 L 17 3 buy? facit 840 lbs.

46. Item, If 22 Yards coſt 16 L 13 3, how many Yards will 83 L 5 3 buy? facit 110 Yards.

47. Item, If 7 lbs coſt 9 9, how many lbs will then 54 3 buy? facit 504 lbs.

48. Item, If 14 9 buy 15 lbs, how many lbs will 42 3 buy? facit 540 lbs.

49. Item, If 11 lbs coſt 3 3, how many lbs will 17 3 6 buy? facit 55 lbs.

50. Item, If 5 Yards coſt 6 9, how many Yards will 7 L buy? facit 1400 Yards.

51. Item, If 1400 Yards coſt 7 L, how many Yards will 6 9 buy? facit 5 Yards.

52. Item, If 6 Ells coſt 18 9, how many Ells will 16 L 8 3 buy? facit 1312 Ells.

53. Item, If 15 lbs coſt 25 9, how many lbs will 4 L 15 3 buy? facit 684 lbs.

54. Item, How many lbs will 9 L 18 3 4 buy, if 3 lbs coſt 15 9? facit 476 lbs.

55. Item, If 18 lbs coſt 6 9, how many lbs will 6 L 15 3 6 buy? facit 4878 lbs.

56. Item, If 3 lbs coſt 2 q, how many lbs will 10 9 buy? facit 60 lbs.

57. Item,

227

The Rule of Three direct.

57. Item, How many lbs will 9 9 3q buy, if 5 lbs coſt 3 q? facit 65 lbs.

58. Item, If 600 lbs coſt 3 L, how many lbs will 27 L 18 3 6 buy? facit 5585 lbs.

59. Item, If 48 lbs coſt 8 3, how many lbs can I buy for 32 L 16 3 8? facit 3940 lbs.

60. Item, How many Ells will 12 L 8 3 4 buy, if 144 Ells coſt 2 L 8 3? facit 745 Ells.

61. Item, If 6 Ells coſt 4 L 16 3 8, how many Ells will 19 L 6 3 8 buy? facit 24 Ells.

62. Item, If 3 L 18 3 9 be paid for 27 Yards, how many Yards are to be delivered for 19 L 13 3 9? facit 135 Yards.

63. Item, What coſt 12 ounces, if 6 ounces coſt 13 3 4 9? facit 26 3 8.

64. Item, If 4 3 9 buy 8 ounces, what will 4 lbs coſt? facit 38 3.

65. Item, If 4 lbs coſt 38 3, what coſt 8 ounces? facit 4 3 9.

66. Item, If 9 ounces coſt 37 3 10 9, what coſt 27 lbs 9 ounces? facit 92 L 13 3 10 9.

67. Item, If 2 lbs 4 ounces coſt 2 L 16 3 6 9, what coſt 11 lbs 4 ounces? facit 14 L 2 3 6.

68. Item, If 1 Dram coſt 2 3 5 9, what coſt 2 lbs? facit 30 L 18 3 8.

69. Item, If 6 Drams coſt 3 3 9 9, what coſt 12 lbs? facit 48 L.

Q2

70. Items,

228

The Rule of Three direct.

70. Item, What coſt 16 ſb 8 ounces, if 5
Drams coſt 4 β 7 9? facit 96 L 16 β;

71. Item, If 30 L 18 β 8 9 buy 2 ſb, what coſt 1 Dram? facit 2 β 5 9.

72. Item, If 6 drams coſt 3 β 6 9, what coſt 24 ſb 4 ounces 3 drams? facit 90 L 12 β 5 9.

73. Item, If 8 ounces 6 drams coſt 8 β 8 8, what coſt 3 ſb 4 ounces 4 drams? facit 2 L 12 β.

74. Item, If 2 L 16 β 6 9 buy 3 ſb 6 ounces 2 drams, what coſt 23 ſb 11 ounces 6 drams? facit 19 L 15 β 6 9.

75. Item, If 23 ſb 11 ounces 6 drams coſt 19 L 15 β 6 9, what coſt 3 ſb 6 ounces 2 drams? facit 2 L 16 β 6 9.

76. Item, If 1 ſb coſt 6 9, how many ſb will 36 β 6 9 buy? facit 73 ſb.

77. Item, If 1 ſb coſt 36 β 4 9, how many
ſb will 25 L 8 β 8 9 buy? facit 14 ſb.

78. Item, If 10 ounces coſt 3 β 4 9, how many ſb will 24 L 16 β 8 9 buy? facit 93 ſb 2 ounces.

79. Item, If 12 ounces coſt 4 β 9 9, what quantity of ſb will 3 L 1 β 9 9 buy? facit 9 ſb 12 ounces.

80. Item, If 3 ſb 8 ounces, coſt 2 L 16 β 3 9, how many ſe will 30 L 18 β 9 9 buy? facit 38 ſe 8 ounces.

81. Item, If 4 drams coſt 8 9, how many ſe will 12 L 4 β 10 9 buy? facit 11 ſe 7 ounces 5 drams.

9 drams

229

The Rule of Three direct.

9 Dra. L β 9

8-4-12-4-10

20

244

12

488

245

2938

4

11752

357

x1752 (1469 Drams.

8888

6x(5 x(7

1469 (x83 (11 ſe 7 oun.

888 x66 and 5 drams.

x Facit.

82. Item, How many ſe will 50 L 0 β 5 9 buy, if 6 drams coſt 2 β 6 9? facit 18 ſe 12 ounces 1 dram.

83. Item, If 4 ounces 3 drams coſt 4 β 8 9, how many ſe will 3 L 19 β 4 9 buy? facit 4 ſe 10 ounces 3 drams.

84. Item, If 23 ſe 11 ounces 6 drams coſt 19 L 15 β 6 9, how many ſe will 2 L 16 β 6 9 buy? facit 3 ſe 6 ounces 2 drams.

85. Item, How many ſe will 9 L 17 β 9 9 buy, if 6 ſe 12 ounces 4 drams coſt 5 L 13 β? facit 11 ſe 13 ounces 7 drams.

86. Item, If 1 ſe of Cheeſe coſt 3 9, what coſt a hundred weight? facit 28 β.

87. Item, If 4 9 buy 1 ſe of Cheeſe, what coſt 9 C 2 quarters? facit 17 L 14 β 8 9.

Q3

88. Item,

230

The Rule of Three direct.

88. Item, If 1 ⅔ coft 9 9/10, what coft 8 10/9 quarters and 5 pounds? facit 36 18 1/3 9 9/10.

89. Item, If 1 quarter coft 2 1/10, what coft 12 C 1 quarter? facit 122 1/10.

90. Item, If 3 1/16 buy 3 quarters, what coft then 13 C 2 quarters and 21 pounds? facit 69 7 1/3.

91. Item, If 2 C 1 quarter 10 ⅔ coft 50 1/10, what coft 28 C 1/4 quarter 8 pounds? facit 600 7 1/3.

92. Item, If 1 C coft 2 1/16, what coft 9 C 3 quarters 10 pounds? facit 27 1/11.

93. Item, If 1 C coft 3 1/10, what coft 15 C 2 quarters 14 pounds? facit 47 1/7.

94. Item, If 1 ⅔ of Cheese coft 3 9/10, what coft 1 Wey, or 256 pounds? facit 3 4 1/8.

95. Item, If 8 ⅔ of Cheese coft 2 6 9/10, what coft 6 Weys 11 Cloves? facit 25 7 1/8.

Note, 1 Wey is 32 Cloves, and 1 Clove is 8 pounds.

96. Item, If 10 9/10 be paid every day, what will be paid in a year? facit 15 4 1/5.

97. Item, What must be paid in 12 years, if 6 1/3 be paid weekly? facit 210 1/12.

Note, 1 Year is counted for 52 Weeks.
98. Item, If 40 ⅔ coft 64 1/10, what coft 1 1/2? facit 32 1/2.

99. Item, If 75 1/2 be paid for 360 1/2, what coft 12 pounds? facit 2 10 1/3.

100. Item,

231

The Rule of Three direct.

100. Item, If 24 Yards coft 1 1/12, what coft 1 Yard? facit 16 9/10.

101. Item, If 162 1/2 coft 13 1/10, what coft 1 pound? facit 20 9/10.

102. Item, if 987 pieces of 12 Yards coft 2632 1/10, what coft 1 piece? facit 2 13 4 1/5.

103. Item, If 3 1/2 coft 20 9/10, what coft 1 pound? facit 6 2 q and 5/8 of a Farthing.

104. Item, If 7 Yards coft 2 1/15, what coft 1 Yard? facit 7 10 1/7.

105. Item, If 4 1/3 buy 5 Yards, how many Yards will 12 1/10 buy? facit 312 1/7.

106. Item, If 8 1/3 buy 10 1/2, how many pounds will 6 5 1/3 buy? facit 156 1/4 ounces.

107. Item, If 7 1/2 coft 9 1/3, how many pounds will 13 1/17 buy? facit 215 1/75 ounces.

Note.

In the working of this Rule, much labor may often be spared, if the first number can be abbreviated (or shortned) with the second, or third number, that is to say, If the first number be shortned (or divided) by 2. 3. 4. 5. &c. The second or third number must likewise be divided by the same figures, and the Quotient of such Division will serve instead of the number so divided, as the following examples will shew.

Q 4

108. Items

232

The Rule of Three direct.

108. Item, If 24 Yards coft 27 3, what coft
64 Yards? facit 3 L 12 3.

\begin{tabular}{ccc}
Yards & $\beta$ & Yards \\
$24$ & $-$ & $27$ \\
$3)$ & $3)$ & $\frac{1}{3}$ \\
$\phantom{3)}$ & $\phantom{3)}$ & $\frac{2}{3}$ \\
$8$ & $9$ & $\frac{3}{8}$ \\
\cline{1-3}
& & $\frac{276}{88}$ \\
\cline{2-3}
& & $\frac{7}{3}$ \\
\cline{3-3}
& & $12$ \\
\end{tabular}
$$
\begin{tabular}{c}
$\beta$ \\
$3$ \\
$\mathbf{\cancel{3}}$ \\
$\mathbf{\cancel{2}}$ \\
$\mathbf{\cancel{12}}$ \\
$\mathbf{\cancel{3}}$
\end{tabular}
$$

Facit 3 L 12 β

The first number 24, and the second 27, be-
ing both divided by 3, the two Quotients are 8
and 9, which will serve instead of 24 and 27,
because the proportion is still the same.

In working the *Rule of Three* by abbreviation,
abbreviate the numbers as much as you can; for
the more you do it, the lesser your Division
will be; yea, if you can abbreviate the Divisor
(or the first number) to an unite or 1, no Division
is required at all; for the Product of the Multi-
plication of the second and third number, is the
facit desired, as you may see by the following
operation.

\begin{tabular}{ccc}
Yards & $\beta$ & Yards \\
$24$ & $-$ & $27$ \\
$3)$ & $3)$ & $\frac{1}{3}$ \\
$\phantom{3)}$ & $\phantom{3)}$ & $\frac{2}{3}$ \\
$8$ & $9$ & $\frac{3}{8}$ \\
\cline{1-3}
& & $\frac{276}{88}$ \\
\cline{2-3}
& & $\frac{7}{3}$ \\
\cline{3-3}
& & $12$ \\
\end{tabular}
$$
\begin{tabular}{c}
$\beta$ \\
$3$ \\
$\mathbf{\cancel{3}}$ \\
$\mathbf{\cancel{1}}$ \\
$\mathbf{\cancel{9}}$ \\
$\mathbf{\cancel{12}}$ \\
$\mathbf{\cancel{3}}$
\end{tabular}
$$

$72 \beta$ or $3 \mathcal{L} \ 12 \beta$.

Note, Further, that the three numbers being
abbreviated as much as can be; if the second
number

233

The Rule of Three direct.

number only be lessened to an unite or 1, then
divide the third number by the first only, with-
out multiplying at all; but if the third number
be lessened to 1, then divide the second by the
first, as you may see by the following examples.

109. If 84 Yards coft 14 β, what coft 62
Yards? facit 10½ β or 10 β 4 9/.

\begin{tabular}{ccc}
Yards & $\beta$ & Yards \\
$84$ & $-$ & $14$ \\
$7)$ & $7)$ & $\frac{1}{7}$ \\
$\phantom{7)}$ & $\phantom{7)}$ & $\frac{2}{7}$ \\
$2x$ & $2x$ & $\frac{3}{14}$ \\
$2)$ & $2)$ & $\frac{3}{14}$ \\
$\mathbf{\cancel{6}}$ & $\mathbf{\cancel{2}}$ & $\mathbf{\cancel{1}}$
\end{tabular}
$$
\begin{tabular}{c}
$\beta$ \\
$3$ \\
$\mathbf{\cancel{1}}$ \\
$\mathbf{\cancel{10}}$ \\
$\mathbf{\cancel{3}}$ \\
$\mathbf{\cancel{4}}$
\end{tabular}
$$

Facit 10 β 4 9

$$
\begin{tabular}{c}
$\beta$ \\
$2$ \\
$\mathbf{\cancel{6}}$ \\
$\mathbf{\cancel{2}}$ \\
$\mathbf{\cancel{1}}$
\end{tabular}
$$

$$
\begin{tabular}{c}
$\beta$ \\
$2$ \\
$\mathbf{\cancel{6}}$ \\
$\mathbf{\cancel{3}}$
\end{tabular}
$$

110. Item, If 72 Ells coft 56 β, what coft
12 Ells? facit 9½ β or 9 β 4 9/.

\begin{tabular}{ccc}
Ells & $\beta$ & Ells \\
$72$ & $-$ & $56$ \\
$6)$ & $2)$ & $\frac{1}{6}$ \\
$\phantom{6)}$ & $\phantom{6)}$ & $\frac{2}{6}$ \\
$2x$ & $28x$ & $\frac{3}{2}$ \\
$2)$ & $2)$ & $\frac{3}{2}$ \\
$\mathbf{\cancel{6}}$ & $\mathbf{\cancel{1}}$ & $\mathbf{\cancel{3}}$
\end{tabular}
$$
\begin{tabular}{c}
$\beta$ \\
$1$ \\
$\mathbf{\cancel{28}}$ \\
$\mathbf{\cancel{9}}$ \\
$\mathbf{\cancel{12}}$ \\
$\mathbf{\cancel{3}}$
\end{tabular}
$$

$$
\begin{tabular}{c}
$\beta$ \\
$2$ \\
$\mathbf{\cancel{6}}$ \\
$\mathbf{\cancel{2}}$ \\
$\mathbf{\cancel{3}}$
\end{tabular}
$$
\begin{tabular}{c}
$\beta$ \\
$3$ \\
$\mathbf{\cancel{3}}$ \\
$\mathbf{\cancel{9}}$ \\
$\mathbf{\cancel{4}}$ \\
$\mathbf{\cancel{9}}$
\end{tabular}
$$

Note, Having abbreviated the three numbers,
as much as can be; if the first and second be
lessened each to 1, then the third number is the
facit; but if the first and third number be lessen-
ed each to 1, then the second number is the fa-
cit, as followeth.

111. Items,

234

The Rule of Three direct.

111. Item, If 6 1b coft 2 β 6 9, how many pounds will 13. β 9 9 buy? facit 33 1b.

112. Item, How many pounds will 7 β 6 9 buy, if 37 β 6 9 buy 105 1b, facit 21 1b.

113. Item, If 15 Yards coft 3 £ 15 β, how many Yards will 25 £ 10 β buy? facit 102 Yards.

114. Item,

235

The Rule of Three direct.

114. Item, If 27 1e coft 6 β 9 9, what coft 873 1e? facit 10 £ 18 β 3 9.

115. Item, What will coft 4 pieces of Cloth long 35: 34: 41: and 42 Yards at 8 β 6 9 the Yard? facit 64 £ 12 β.

116. Item, What will coft 16 pieces of Bays, each long 52 Ells at 26 9 the Ell? facit 90 £ 2 β 8 9.

Reduce the Bays into Ells, and you shall finde 832 Ells; then say, if 1 Ell coft 26 9, what 832 Ells, &c?

117. Item, What coft 44 pieces of Linnen Cloth, each long 53 Ells at 22 9 the Ell? facit 213 £ 15 β 4 9.

118. Item, Suppose that 8 times 12 were 98, how much is 12 times 16 according to that account? facit 196.

119. Item, How many pieces of money at 13 9 2q the piece, must be paid for 32 pieces of Holland, each long 36 Yards at 12 β 4 9 the Yard? facit 12629 pieces and 4 9 2q.

120. Item, A Shopkeeper bought 12 pieces of Cloth, each long 35 Yards, agreeing to pay 16 β 6 9 for every Ell; what must he pay for the whole parcel? facit 277 £ 4 β.

121. Item, If 12 pieces of Cloth, each long 35 Yards, coft 277 £ 4 β, what coft an Ell? facit 16 β 6 9.

122. Item, A Marchant bought 6 Chefts of Sugar, weighing together 11 C 2 quarters 16 pounds grofs. Tare 1 C 3 quarters 19 pounds, for all the Chefts, at 18 9 the pound neat, what amounts

236

The Rule of Three direct.

amounts the whole ? *facit* 81 L 13 3/6 9/10.

Note, Tare is the weight of that wherein any thing is put; as a Cheft for Sugar, a Bag for Pepper, &c. Therefore subftraet the Tare (which is 1 C 3 quarters 19 pounds) from the grofs or whole weight (which is 11 C 2 quarters 16 pounds) the reft will be 9 C 2q 25/6 neat; then say, if 1 pound neat, coft 18 9/10, what 9 C 2q 25/6, &c.

123. Item, A Grocer bought 14 Chefts of Sugar, weighing each 2 C 3 quarters 5 pounds grofs. Tare 2 quarters 2 pounds for every Cheft at 16 9/10 the pound neat; what muft he pay in all? *facit* 238 L.

124. Item, How many pounds of Raisons at 6 9/10 the pound, can I have for 123 Yards of Cloth at 13 3/4 9/10 the Yard? *facit* 3280 9/10.

See first the worth of the Cloth in 9/10, then divide that fum by fix, the price of 1 pound of Sugar.

125. Item, A Shopkeeper bought the following commodities; to wit, 6 pieces of Cloth, each long 30 Yards, at 8 3/4 9/10 the Yard. 346 9/10 of Pepper at 2 3/4 6 9/10 the pound, and 820 9/10 of Cloves at 3 3/4 9 9/10 the pound. He giveth in payment 54 L 10 3/4 in money, and 340 Yards of Holland at 3 3/4 6 9/10 the Yard, the question is, what he ows still? *facit* 158 L.

126. Item, A. bought two Bags of Pepper, weighing grofs 2 C 1 quarter 27 pounds, and the other 1 C 3 quarters 26 pounds. Tare 7 9/10 for the two Bags, and 4 9/10 Tret for every 104 9/10 at

237

The Rule of Three direct.

at 2 3/4 the pound neat; what amounts the whole parcel? *facit* 47 L 10 3/4.

<table><thead><tr><th></th><th>C</th><th>Qrs</th><th>fs</th></tr></thead><tbody><tr><td rowspan="2">Add<br>de</td><td>2</td><td>1</td><td>27</td></tr><tr><td>1</td><td>3</td><td>26</td></tr><tr><td></td><td>4</td><td>1</td><td>25 Grofs.</td></tr><tr><td></td><td>0</td><td>0</td><td>7 Tare.</td></tr><tr><td></td><td>4</td><td>1</td><td>18 Subtil.</td></tr></tbody></table>

<table><tbody><tr><td>4</td></tr><tr><td>17</td></tr><tr><td>28</td></tr><tr><td>154</td></tr><tr><td>34</td></tr><tr><td>494</td></tr><tr><td>26</td></tr><tr><td>494</td></tr><tr><td>494</td></tr><tr><td>19 Tret.</td></tr><tr><td>266</td></tr><tr><td>475</td></tr><tr><td>2 3/4</td></tr><tr><td>25/10</td></tr><tr><td>47 L 10 3/4</td></tr></tbody></table>

23 494

494 (19 9/10 Tret.)

x 475 9/10 neat, at 2 3/4 the 9/10.

25/10
Facit 47 L 10 3/4

127. Items,

238

The Rule of Three direct.

127. Item, A Shopkeeper would beftow 120 L in four forts of Spices, to wit, Cloves at 4 ½, 6 ⅞ the pound, Ginger at 1 2/3, Pepper at 2 ½, 6 ⅞, and Cinnamon at 4 ½ the pound, desiring to have an equal quantity of each fort; the queftion is, how many pounds of each fort he must have? facit 200 ⅛.

128. Item, A Baker bought 3 Seams (or Quarters) of Wheat at 4 ½ 2 9/10 the Bushel, and 2 Seams and 5 Bushels of Rye at 2 ½ 6 9/10 the Bushel; he mixt an equal quantity together of each fort; the queftion is, what 7 Bushels of that mixture coft him? facit 2 3 ¼ 4 9/10.

129. Item, A. bought 3600 Oranges, and a quantity of Lemmons; for the sum of 46 ⅘ 2 ⅜ 6 9/10, he did agree to pay 18 9/10 for every 1/10 score of Oranges; the queftion is, how many Lemmons he bought, if 60 Lemmons coft him as much as 200 Oranges? facit 2610 Lemmons.

130. Item, A Marchant bought 798 ⅞ of commodity for 299 ⅛ lefs, so much money, as 99 ⅞ coft him, at the fame rate; the queftion is, how many pounds he can buy at the fame price for 2905 ⅛? facit 8715 ⅞.

Questions

239

The Rule of Three direct.

Questions of the Rule of Three direct.

1 Queft. W Hat is the Rule of Three? Answ. The Rule of Three is a Rule, which coñicts of three numbers known, to finde a fourth unknown.

Queft. 2. Hath that Rule no other name?

Answ. Yes; it is called the Golden Rule, or the Rule of Proportion.

Queft. 3. Why is it called the Golden Rule?

Answ. It is so called for its excellency; for as Gold exceeds all other Metals, so doth this Rule exceed the others.

Queft. 4. Why is it called the Rule of Proportion?

Answ. Because the three numbers propounded, and the fourth (which anſwers the queſtion) are Proportionals.

Queft. 5. Into how many parts is the Rule of Three divided?

Answ. It is commonly divided into two general parts; to wit, Single and Compound.

Queft. 6. What is the Single Rule of Three?

Answ. The Single Rule of Three is, when three numbers are propounded to finde a fourth Proportional to them.

Queft. 7. What is the Rule of Three Compound?

Answ.

240

### The Rule of Three direct.

Answ. The Rule of Three Compound (other-wise called The double Rule of Three) is, when more then three terms (or numbers) are propounded.

Queft. 8. Is that all the parts of the Rule of Three ?

Answ. No ; for the Single and Compound Rule of Three, are divided each into two parts.

Queft. 9. What are those parts ?

Answ. The Single and Double Rule of Three, are either Direct or Revere.

Queft. 10. What do you mean by the Single Rule of Three direct; what doth it teach ?

Answ. It teacheth to finde a fourth number in a Direct Proportion, when three are propounded.

Queft. 11. What do you understand by Proportion ?

Answ. Proportion hath divers denominations or names. Some call it Reason (or Relation, or the Reference that the numbers have to one another) and others say, that it is the Similitude of the Reafons of some Quantities, which are of the fame kinde.

Queft. 12. Explain your meaning, and give a plain example.

Answ. My meaning will appear plainly by these four following Quantities or Numbers; to wit, 12, 4: 6, 3. Now the Reason (or Proportion) that 12 hath to 4 is called Tripple Reason; because 12 is three times greater then 4; or if 12 be divided by 4, the Quotient is 3, which shews the

241

### The Rule of Three direct.

the reason that 12 hath to 4, and some call that Proportion.

Further 6 to 3, is called double reason (or Proportion) because 6 being divided by 3, the Quotient is 2, which is double.

Now observe that the Similitude of these two Reafons (before mentioned) is alfo called Proportion.

Queft. 13. How do you call the two numbers (or quantities) propounded, to finde the Reason or Proportion between them ?

Answ. The first number is called Antecedent, and the second the Consequent.

Queft. 14. What do you understand by Reason here ?

Answ. By Reason in this place, I understand the Reason (or Proportion) which is between two quantities, or numbers (as is mentioned already) and is found by dividing the Antecedent by the Consequent, and the Quotient is the Reason, or Proportion defined.

Queft. 15. What Reason or Proportion is between 16 and 4 ?

Answ. I divide 16 by 4, the Quotient is 4, which is a Quadrupla, or fourfold Reason.

Queft. 16. When three numbers are propounded, how do you finde a fourth in a direct Proportion ?

Answ. By multiplying the second and third number together, and dividing the Product by the first, the Quotient is the fourth number proportional required.

Queft. 17. Give an example to make that plain ?

R

Answ.

242

The Rule of Three direct.

Answ. Suppose that these three numbers 6, 3, 18, were given to find a fourth in a direct Proportion: I multiply 18 by 3, the Product is 54, and dividing 54 by 6, the Quotient is 9, which 9 is the number sought.

Queft. 18. How do you know when four numbers are in a Direct, or reverse Proportion?

Answ. Four numbers are said to be in a Direct Proportion, when the first is to the third, as the second is to the fourth; or thus, as the first is to the second, so is the third to the fourth.

Queft. 19. Give an example?

Answ. Take the aforementioned numbers, which are 6, 3, 18, and 9. Now observe that as 6 (the first number) is to 18 (the third) so is 3 (the second) to 9 (the fourth).

Queft. 20. How do you know when four numbers are in a Proportion reverse?

Answ. It is to be known by this, when, as the first is to the third, so is the fourth to the second.

Queft. 21. Give an example?

Answ. These four numbers 3, 6, 9, and 2, are in a Proportion reverse; for as 3 is to 9, so is 2 to 6.

Queft. 22. Have you no other way to know, when four numbers are in a direct Proportion, but by what hath been answered in the 17th. question?

Answ. Yes, that may be known another way, which is to multiply the two middle numbers together, and likewise the first and the fourth; and if their Product be equal, that shews that such num-

243

The Rule of Three direct.

numbers are in a direct Proportion.

Queft. 23. Give an example of that?

Answ. I shall take the four numbers mentioned in the 17th. answer, which are 6: 3: 18, and 9. Now if I multiply the two middle numbers (3 and 18) together, the Product will be 54, and multiplying also the first and the last number (6 and 9) the Product will be also 54, which shews that those four numbers are in a direct Proportion, as followeth.

54 equal to 54

Queft. 24. Have you another way to know when four numbers are in a Proportion reverse, then what is mentioned in the answer of the 20th Question?

Answ. Yes, that may be known by multiplying the first and second together, and also the third and fourth, and if the two Products be equal, that shews that those numbers are in a Proportion reverse.

Queft. 25. Give an example.

Answ. I shall take the numbers mentioned in the Answer of the 21 Question, which are 3: 6: 9, and 2. Now multiplying 3 by 6, and 9 by 2, the two Products are equal as followeth.

18 equal to 18

R 2

Queft.

244

The Rule of Three direct.

Queft. 26. What do you observe in the Rule of Three direct?

Answ. The chiefest thing to be observed in The Rule of Three, is to place the three numbers propounded in their right places.

Queft. 27. Is that such a hard thing to do?

Answ. It is hard for young beginners, for often-times they mistake in that, when the three numbers are confusedly propounded.

Queft. 28. Have you a certain rule to know how to place the three numbers in their right places?

Answ. Yes, without great trouble.

Queft. 29. What is your first observation in placing the three numbers?

Answ. When a Question is propounded, the first thing to mind, is to observe what denomination the number required doth bear, either Money, Weight, Length, &c.

Queft. 30. What then?

Answ. That being known you must further understand, that one of the three numbers propounded, bears the same name (general or particular) with the number required, which number must always be set down in the second place.

Queft. 31. What do you mean by general and particular names?

Answ. By a general name I mean, Money, Weight, Length, &c. without naming what particular piece of Money, or what particular Weight, or measure; and by a particular name, I mean when a particular thing is named, as L. β. q. or fb. Ounces, Drams, or else Yards, Ells, &c.

Queft.

245

The Rule of Three direct.

Queft. 32. When you know the number which must poffess the second place, how do you know how to place the two other?

Answ. The two other numbers may be placed without great difficulty, when the middle number is placed.

Queft. 33. What is your observation in placing them?

Answ. I observe to set in the first place one of these two numbers, which is equal in value with the second, though it be not of the same denomination.

Queft. 34. How do you know the last or third number?

Answ. When two numbers are placed, the third is soon found out, because there remains one number to be placed, which is the third.

Queft. 35. Doth the first and third number differ in denomination?

Answ. No, the first number and the third are always of the same name, general or particular.

Queft. 36. If it be so, how do you know justly what number must poffess the third, or first place?

Answ. There is a Rule for the placing of every number; how to place the two first, I have already mentioned, for the last, or third number, it is known to be the third, because his worth or value is unknown.

Queft. 37. Though you have already mentioned how to place the three numbers, yet I desire you would repeat it over again as short as you can, for to help my weak memory?

R 3

Answ.

245

The Rule of Three direct.

Answ. I will do it: Observe then that the number which must be set in the second place, is always of the same denomination or name with the number required; and the first number is always equal in value, or worth with the second. Lastly, The third number is that upon which the question is stated (which is the number whose worth is unknown) must be in the last or third place.

Queft. 38. The three numbers being thus placed, is there nothing more to observe before Multiplying and Dividing, according to the general rule mentioned in the answer of the fixteenth question, which is to multiply the second and third number together and divide the Product by the first ?

Answ. Yes: You must further observe, whether the three numbers be single or compound.

Queft. 39. What if they be single?

Answ. Then you may proceed according to the general rule before mentioned, to find the fourth number unknown, the first and third number being of the same particular denomination.

Queft. 40. But if they be compound.

Answ. They must be reduced into the least denomination mentioned in any of the numbers.

Queft. 41. To make this necessary observation the more plain, I shall ask some questions, where the numbers will be of different denominations.

Answ. This is a good way, and it will afford a double benefit.

Queft. 42. How so?

Answ. Because in propounding some questions,

247

The Rule of Three direct.

ons, you may (besides your intended purpose concerning the different denominations of the numbers) observe and put in your mind the rule before mentioned, touching the placing of the numbers.

Queft. 43. You say true: Now I think it the best way to begin with that.

Answ. I think so.

Queft. 44. If five Yards cost 3 L, what cost 25 Yards? How are these three numbers to be set in their right order?

Answ. They are propounded regularly, that is, the three numbers are named in the question as they ought to be placed in their order.

Queft. 45. How many ways may a question be propounded, besides the regular way?

Answ. There are variety of ways, I shall mention three only, which will be the same question before mentioned, as followeth.

1. If 3 L buy 5 Yards, what cost 25 Yards?

2. What cost 25 Yards, if 5 Yards cost 3 L?

3. What cost 25 Yards, if 3 L buy 5 Yards?

Queft. 46. Must the three numbers of these three questions, be set down in the like order one to another?

Answ. Yes; because it is the same question propounded three fundry ways; the ordering of the three numbers, is as followeth.

Facit 15

R 4

Queft.

248

## The Rule of Three direct.

Queft. 47. If the queftion had been thus:
What quantity of Yards can I have for 15 L, if
5 Yards cost 3 L? how must the numbers be or-
dered?

Answ. The queftion asks Yards, therefore 5
Yards muſt be in the ſecond place, the 3 L
firſt, and 15 L in the third place, thus,

L Yards L
3 — 5 — 15

Queft. 48. Suppose that the firſt number had been 3 β, (instead of 3 L) and the laſt as it is, to wit, 15 L?

Anſw. Then the 15 L muſt have been reduced into Shillings, because the firſt is Shillings.

Queft. 49. Muſt the firſt and third number be always of one particular denomination before you multiply and divide?

Anſw. Yes: If the firſt (or the laſt) be β,
Θ, or q, and the other L onely; then the L
(on the firſt or third place) muſt be reduced into
ſuch ſmall denominations, as the other number
is.

Queft. 50. But if one of the numbers, to wit, the firſt or the third, be a compound number confiſting of L, β, Θ, and q, and the other onely L, thus?

L Yards L β Θ q
3 — 6 — 19 — 16 — 8 — 2

Anſw. If the third number be compound, and
the

249

## The Rule of Three direct.

the firſt ſingle; or, if they be both compound,
I muſt do the ſame thing, which is to reduce both
the numbers into the leaſt denomination mention-
ed in either of them.

Queft. 51. Give an example of that.

Anſw. If the three numbers are as follow-
eth.

L β Θ Yards L β Θ q
3 — 10 — 6 — 5 — 19 — 6 — 8 — 2

Then both the firſt and third number muſt be reduced into Farthings, because one of them (to wit, the third) hath Farthings for its leaſt denomination. If one of the ſaid numbers had confiſted of L onely, or of L and β, or of L, β, and Θ, or L, β, Θ, and q, that had been alone working; for both the ſaid numbers muſt have been reduced into q.

Queft. 52. I perceive, that when one of these two numbers (the firſt and the third) hath a (mall denomination annexed unto it, that both the numbers must be reduced into ſuch (mall denominations; whether both the numbers or one, be compound ?

Anſw. Yes; this is well obſerved, the ſame obſervation muſt be remembred, when ſeveral denominations of weight (or meaſure) are in the firſt and laſt place, as followeth.

      Hb Ounc. Dra. L Hb Ounc. Dra. Scru.
13-13-3 - 2 - 16-9-4-1

Queft. 53. Must both these compound numbers be reduced into Scruples, because one of them hath Scruple annexed to it?

Anſw.

250

The Rule of Three direct.

Answ. Yes; and if any grains had been men-
tioned in any of these two numbers, they must
have been reduced into grains.

Queft. 54. I am now satisfied concerning the
first and third number, let us go to the second.

Answ. Observe what followeth for a general
rule. Having reduced the first and third number
into one denomination (if they differ) as is said
before, then observe whether the second number
be single or compound; if it be single, multiply
it with the third, and divide the Product by the
first, the Quotient is ever of the same denomina-
tion, as the second number is.

Queft. 55. I perceive then, that if the middle (or second) number be L, the fourth number (or the Quotient of the Division) is L also; is that your meaning?

Answ. Yes; and if the said second number
be β, the fourth is β; if § the fourth, is §,
&c.

Further, if the said middle number be fb, the
fourth is fb; if ounces or drams, the fourth is
ounces or drams, &c.

Queft. 56. But if the said second number be
compound, what then?

Answ. If the second number be compound,
then the greater denominations of the said number
must be reduced into the least denomination men-
tioned.

Queft. 57. Give an example.

Answ. If the said number consist of L, β,
then it must be reduced into β. If any § be
found

251

The Rule of Three direct.

found in that number, then it must be reduced in-
to §. Lastly, if such a compound number hath
any q annexed to it, then it must be reduced in-
to q.

Queft. 58. If the middle number did consist of
Pounds, Ounces, and Drams, what must you do?

Answ. I must then reduce such a compound
number into Drams.

Queft. 59. Why not into Scruples or Grains?

Answ. Because the least denomination in that
number supposed, is but Drams: If any Scruples
or Grains had been mentioned, then I should re-
duce such a number into that small denomination
you speak of.

Queft. 60. If 35 β be paid for 3 Ells, what
will cost 96 Ells? how shall I answer this queſi-
on?

Answ. This question hath no difficulty in it,
follow but the general rule.

Queft. 61. In following the rule, I find the
fourth number to be 1120 ⅝, must they be reduc-
ed into Pounds?

Answ. Yes; for it is improper to say to any
one, you must pay or receive 1120 β, but it is
more proper to say 56 L, which is as much as
1120 β.

Queft. 62. Now I perceive, that when the
fourth number is found out, if it be β, §, or q,
then I must reduce them always into as great de-
nomination, as the number will permit.

Answ. Your observation is good, for some-
times the fourth number may be reduced into L,
and

522

The Rule of Three direct,

and sometimes it cannot, by reason of the small
quantity of the lesser denominations.

Queft. 63. But, must always the fourth num-
ber be reduced into Pounds, when it can be reduced
so?

Answ. Yes; if the question requires general-
ly a sum of money, without naming any other
particular name.

Queft. 64. Explain your meaning.

Answ. My meaning is, if the question did re-
quire a certain number of pieces of money, as
Crowns, Nobles, Marks, &c. then the fourth
number must not be reduced into Pounds, but in-
to such pieces of money as the question re-
quires.

Queft. 65. I understand now what you mean,
yet I shall propound a question for my further sa-
tisfaction, which is this. If 10 Yards of Cloth
cost 7 L 13 3/8, how many Nobles is to be
paid for 40 Yards? how shall this question be an-
swered?

Answ. Follow the general rule before menti-
oned, then you shall finde 7 360 9 for the fourth
number Proportional, then divide this quantity
of pence, by the worth of a Noble, which is
80 9, and the Quotient shall be 92 Nobles for
the answer of the question.

Queft. 66. I am now satisfied also concerning
the second number of the Rule of Three, for I un-
derstand by what hath been said, that the fourth
number being found out, it may be reduced after-
ward into what denomination sever that the questi-
on doth require.
Answ.

253

The Rule of Three direct.

Answ. You have it right; now observe also
the same thing, when you finde that the fourth
number confits of a quantity of small weights (or
measures) as Ounces, Drams, &c. you may re-
duce them into 1/16's, or into Quarters of a Hun-
dred, according as the question requires.

Queft. 67. I have something more to ask, which
is thus, Whether I must always (in this Rule)
multiply the two laft numbers together, and divide
the Product by the first?

Answ. You may if you will, but it is some-
times needleſs, according as the numbers are.

Queft. 68. How ſhall I know when it is need-
leſs, or neceſſary?

Answ. That may be known two manner of
ways.

Queft. 69. What is the firſt way?

Answ. When you see that the first number is
but an unite, or 1, then it is needleſs to divide by
1, because the Quotient will be as much as the
Dividend.

Queft. 70. What; is then the Product of the
Multiplication of the ſecond and third number,
the fourth required?

Answ. Yes, and such Product bears the same
name, as you left your ſecond number in.

Queft. 71. But if any one of the two other num-
bers be but an unite or 1, What then?

Answ. If the ſecond or laſt number be but 1,
then you need not to multiply but divide onely
that number (which is more then 1) by the first,
and the Quotient will anſwer the Question; for
in

254

The Rule of Three direct.

in multiplying by 1, the Multiplicand will not
be encreased at all, and therefore it is needles to
multiply by 1.

Queft. 72. Now I perceive that an unite (or 1)
neither multiplies nor divides, but yet I have still
some doubts unresolved.

Answ. Propound them all, and I shall resolve
or answer them if I can.

Queft. 73. If it happen that the second and third
numbers be but each of them an unite (or 1) what
must I do in this case?

Answ. If the two last numbers be each of them
but 1, then you may conclude that the fourth num-
ber required, cannot be an unite (If the first num-
ber be more then 1) but a Fraction (or a broken
number) which will be a part, bearing the name
of the second number (if 1 may be called a num-
ber.)

Queft. 74. Give a plain example, for this seems
very difficult to understand ?

Answ. Suppose this Question be propounded;
If 4 Yards, coſt 1 L, what coſt 1 Yard ?

In this Question you see that the number to be
divided (or the dividend) is but 1, and the Di-
visor is 4; now the Quotient of such Division
will be but ⅓ or a fourth part of 1 L, which is a
Fraction of a pound.

Queft. 75. I perceive that a fourth part of a
pound is ⅕ β, but is there no other ways to finde
these ⅕ β, without making a Fraction of a pound?

Answ. Yes, as followeth, When you find that
the Dividend ( or the Product arising from the
Multiplication

255

The Rule of Three direct.

Multiplication of the second, and third number)
be left then the Divisor (or first number) reduce
the second number (if you can) into a lesser deno-
mination ( or into the smallest denomination you
can, if you will) then divide such Product by the
first number.

Queft. 76. Give an example ?

Answ. If 8 Yards coſt 2 L, what coſt 2
Yards.

Yards L Yards
  8 — 2 — 2

By this Question you may see that in multiplying the second and third number together, the Product is but 4, and the first number (which is the Divisor) is 8; therefore I reduce the 2 L (or the second number) into shillings, the Product is 40, which being multiplied by the third number 2, the last Product is 80, for the Deidend, and dividing 80 by 8, the Product is 10 ß for the Facit as followeth.

| Yards | L | Yards |
|---|---|---|
| 8 | — | 2 |
| — | 20 | — |
| — | — | 40 ß |
| — | — | 2 |
| — | — | 80 |

80 (10 ß)
88 Facit

Queft. 77. Why did you not reduce the 2 L
into ß, but to ß only.

Answ.

256

The Rule of Three direct.

Answ. Because I did perceive that in reducing the 2 L into Shillings onely, and the Product being multiplied with the third number, that the Multiplication would give a Product great enough to divide by the first number.

Queft. 78. But if you had reduced the said second number into 9, the facit had been the same sum as before, to wit, 10 3, had it not?

Answ. It is true, but it had been more trouble, for the fourth number had been but 9, then another Division had been required to reduce the 9 into Shillings.

Queft. 79. When is it neffible to reduce the second number (being single) into its ſmallest denomination?

Answ. The second number is to be reduced into its ſmallest denomination (though it be single) when the first number is greater then the Product, arising from the two laſt numbers multiplied together, as appears by the following queſtion.

If 1440 ftb coſt 3 L, what coſt 1 fb.

$$
\begin{array}{c} \text{1b} \\ \text{1440} \\ \hline 3 \end{array} \quad \begin{array}{c} \text{L} \\ \hline 3 \end{array} \quad \begin{array}{c} \text{1b} \\ \hline 1 \end{array}
$$

I muſt reduce the 3 L into ſuch a ſmall de-
nomination, until I can get a Product (or Divi-
dend) great enough to divide by 1440; there-
fore I reduce the 3 L into q, as followeth.

1b 144°

257

The Rule of Three direct.

$$
\begin{array}{c} \text{1b} \\ \text{1440} \\ \hline 3 & 20 \\ & \text{60 } \beta \\ & 12 \\ \hline 120 & 60 \\ \hline 720 & 4 \\ \hline 288 & 0 \, q. \\ \end{array}
$$

$288\alpha$ (2 q)
$1440$ Facit

x88α (2 q)
1440 Faci

In reducing the 3 L into 9, the Product is
but 720 9; therefore I reduce them into q, the
laſt Product is 2880 q; which being divided by
1440, the Quotient is 2 q, for the anſwer of the
queſtion.

Queſt. 80. Suppose that the Product (arising
from the two laſt numbers, multiplied together) be
great enough to be divided by the first number. If
there be any Rest in the Division, what do you do
with it ?

Answ. The number which remains in the Di-
vision, is a Fraction, or a part of an Unite of like
name with the Second number, as hath been ſaid
al: eady.

Queft. 81. Give an example for a further fa-
tification.

S

Answ.

260

The Rule of Three direct.

Having reduced the 7 L into 9, and working
according to the Rule, I finde 2240 9 for the
fourth number without any Rest, which 9 being
reduced into L, the facit is 9 L 6 3 8 9, as
you see above.

Queft. 85. Have you no other way to find the said facit, then by reducing the 7 L into 9?

Answ. Yes; but it is no lefs troublesome then
that way already mentioned.

Queft. 86. Work the same queftion that way
that I may see the difference.

Answ. I shall do it as followeth.

Yards L Yards

Having multiplied and divided, I finde 9 L
and 1 remaining. Now to know how many 3
the Rest is, I muſt multiply it by 20 (because it is
a Fraction of 1 L, to wit, 1/3, or a third part)
and divide the Product by the fame Divisor as be-
fore, as followeth.

1 Rest or Remainder.

Having multiplied the Remainder by 20, and di-
vided by 3, I finde 6 3, but there remains
still

261

The Rule of Three direct.

Itill 2, which is two thirds of 1 3; therefore I
muſt multiply 2 by 12, and divide the Product
by the fame Divisor 3, to know how many 9
such Rest doth amount unto, as followeth.

The Remainder is 2

Having multiplied the second Rest (which is two
thirds of 1 3) by 12, and divided by 3, the
Quotient is 8 9; so that the whole facit is 9 L
6 3 8 9, as before.

Queft. 87. Suppose that in working one way or other, you did finde always a Remainder, though you should reduce the second number into Farthings; what do you with such a small Rest?

Answ. Having found, as nigh as I could, the
worth of the third number (by one of the two
ways before mentioned) to a Farthing, then I
should neglect the Rest as being not worth a whole
Farthing, because a Farthing is the least piece of
money that is in use, and a part of it is fel-
dom quarrelled for, between the Buyer and the
Seller.

Queft. 88. Give an example.

S 3

Answ.

262

The Rule of Three direct.

Answ. If 3 1b cost 20 9, what cost 2 1b?

4°

4° (13 9 Rest 1.

33

Having multiplied and divided according to the Rule, I find 13 9, but there Rest 1, which I multiply (per forme) by 4: The Product is 4 1st, which 4 being divided by 3, the Quotient is 1 q, and there remains 1 1st, which is a third part of 1 q, thus.

The Rest is 1

4

4

(1

4 (1 q and 1/2 of 1 q.

3

The whole fact is 13 9 1 q, and a third part of 1 q: Which Rest is too little to require the payment of it.

Queft. 89. I understand your manner of working, which (as I conceive) is thus: When you find any Rest, you work any one of the two ways before mentioned, to find the answer of the question as high as to a Farthing; and if there be still a Rest, you omit it, as being too little to be regarded; Is this your way and meaning?

Answ. Yes; you have it right.

Queft. 90. But I remember what you told me in the answers of the 67 and 68 questions; which is, that it was sometimes needlessto multiply the two last

263

The Rule of Three direct.

last numbers together, and that such thing might be known two manner of ways.

Answ. I remember it, and I have mentioned one way already; which is, when the second or third number be but an Unite or 1, then it is needlessto multiply.

Queft. 91. Declare the other way in few words, how Multiplication may be avoided?

Answ. Before I speak of that, you may take notice that you may sometimes avoid dividing by the first number, as well as multiplying.

Queft. 92. Is it possible? But is that a general way?

Answ. No; for sometimes you must of necessity follow the common and general way before mentioned.

Queft. 93. Which way do you go to work, and how do you call such manner of working?

Answ. I abbreviate the numbers one againft another, and therefore that way is called the working by Abbreviation.

Queft. 94. What do you mean by Abbreviation?

Answ. By Abbreviation, I mean, when two numbers are shortned one againſt another.

Queft. 95. Is that way useful to be known, seeing it is not general?

Answ. Yes; very useful, for (though it be no general way) many a time; much labor is spared that way, which otherwife doth prove very tedious.

S 4

Queft.

264

## The Rule of Three direct.

Queft. 96. May that way be often practised?

Answ. Yes; sometimes more, sometimes less.

Queft. 97. Declare it with brevity.

Answ. The three numbers being set down in their right places, I observe, whether the first number can be abbreviated (or shortned) againſt the ſecond or third; that is, whether it be poſſible to divide the first number, and one of the two others, by one of the fame number (or figure) as 2. 3. 4. 5. 6, &c. without any reſt.

Queft. 98. What then?

Answ. If that be poſſible, then the Quotients of ſuch Divisions will ſerve inſtead of the numbers ſo divided, and ſo the work will be the ſhorter.

Queft. 99. Give an example.

Answ. If 8 Yards coſt 16 L, what coſt 32 Yards?

Facit 64 L

I perceive that the first number 8, and the ſecond 16, may be divided by 8 without any Reſt; therefore dividing these two numbers by 8, the two Quotients are 1 and 2, as you ſee above. Now 1 does ſerve in ſtead of .8, and 2 in ſtead of 16; and in multiplying the laſt number 32 by 2. the

265

## The Rule of Three direct.

the Product is 64 L for the facit without any further trouble.

Queft. 100. I ſee that it is a very ſhort way, but what is the reaſon of this manner of working, why it comes ſo to paſſ, that the right facit is produced in working by abbreviation?

Answ. The foundation and chief reaſon, why the true anſwer is produced in working by Abbre viation, is, Because the numbers keep ſtill their Proportion or Reaſon, one towards another, though they be abbreviated or ſhortned.

Queft. 101. Explain your meaning.

Answ. In laying thus, If 8 Yards coſt 16 L, or if 1 Yard coſt 2 L, is the fame thing; the reaſon between 8 and 16, or 1 and 2, is the fame (which Reaſon or Proportion is called Subdupla) for 16 contains 8 twice, and ſo doth 2, contain twice.

Queft. 102. I understand you: But may you abbreviate the first number and the third, one againſt another?

Answ. Yes; that may be done, if the numbers will permit it; that is, if both the numbers can be divided by the fame figure (or number) without any Reſt.

Queft. 103. I perceive that you may ſhorten the first and third number of the question before mentioned, because the third and the first number may be divided, either by 2. 4. or 8. without Reſt.

Answ. This is well obſerved, I ſhall work the ſaid queſtion over again that way, to ſhow that you

266

The Rule of Three direct.

you may shorten the first number against one of the two other, which you see most convenient.

Facit 64 L

In shortning (or dividing) the first and the third number by 8, the two Quotients are 1 and 4. Now I multiply the second number 16 by 4, the Product is 64 for the Facit, without further trouble.

Queft. 104. I see that this way, and the other, are much alike.

Answ. There is no great difference; you may take notice by the way, that you may choose the second or third to be abbreviated against the first, or the first against the two other.

Queft. 105. I know that to be true; and I perceive further, that the reason why I do not divide by the first number, as in the common way, is, because the said number is shortned to an Unite, or 1.

Answ. You have hit it right: You may observe also, that when you can shorten one of the two last numbers so far, as to bring one of them to an Unite, or 1, that will spare you the labor of a Multiplication.

Queft. 106. Give an example where such thing may be done.

Answ.

267

The Rule of Three direct.

Answ. If 16 L buy 8 Yards, what quantity of Yards will 64 L buy?

64 (32 Yards.
22 Facit.

In abbreviating the first and second number by 8, the two Quotients are 2 and 1. Now I do not multiply the second and third together, because the second is but 1; therefore I divide only 64 by 2, the Quotient is 32 Yards for the Facit.

Queft. 107. I see plainly, that when any of the two last numbers be shortned to an Unite, that there is no need of Multiplication; but if both the first, and one of the two other, were abbreviated so short; what then?

Answ. Then the work would be yet shorter.

Queft. 108. Give an example for my satisfaction.

Answ. The last example mentioned will suffice, as followeth.

Having abbreviated the first and second one against another, the two Quotients are 2 and 1. Now I abbreviate further, the first and the third number,

to

268

The Rule of Three direct.

to wit, 2 and 64, and the two Quotients, are 1 and 32. And seeing, that the two first numbers are abbreviated each to an Unite, or 1, then the last number 32 is the Facit, and must bear the fame name as the middle number doth, which is Yards.

Queft. 109. I perceive that you may abbreviate the numbers more then once.

Answ. Yes; sometimes two, three, or four times according as the numbers are.

Queft. 110. When the numbers are compound, may they be shortned so too?

Answ. Sometimes they may, but the most general way is to reduce the numbers into their letter denomination, and then to abbreviate them as is said before.

Queft. 111. May the second number be abbreviated againſt the third?

Answ. No; but the first may be abbreviated againſt either of them both, as many times as the numbers will permit.

Queft. 112. Have you a general rule to know what are those numbers that are to be abbreviated one againſt another?

Answ. Yes; the general rule is to abbreviate those numbers one againſt another, which doth not multiply each other; and this way is very useful in working the Rule of Five Jumbers (called the Double Rule of Three) and also the Conjunctive Rule, where there may be so many numbers inferted as you please; sometimes 10. 20. 30. or more numbers, as you may see in the second Part of this Book.

Queſt.

The Rule of Three direct.

269

Queſt. 113. I see, that working by Abbreviation, is of great use in those Rules you have mentioned : But when no number can be abbreviated one againſt another, what is to be done?

Anſw. In such a case, you must of necessity follow the other way before mentioned, which is to multiply the second and third number together, and divide the Product by the first.

Queſt. 114. I am fully satisfied concerning that useful Rule of Three. What is the next to be spoken unto?

Anſw. The next thing to be handled, is the Fractions, which are explained in the Second Part.

AN

A N

Appendix.

To the READER.

Y Intentions were to have proceeded no further, then the Rule of Three, in whole Numbers in this First Part, fearing it wouldswell up too much: I have nevertheless added unto it, some short beginnings of Fractions, and some Questions and Examples of the Rule of Practice, tending to the encouragement of those that delight in this profitable Art.

Con-

274

## An Appendix.

According to the said Direction, pronounce the following Fractions adding to them their several Denominations, thus.

1. 1/2 L (or the half of a Pound) 3/4 β (or the three fourths of a shilling) 9/8 (or the five sixths of a penny, and so of the reft) 7/8 q, 2/3 lb, 2/5 ounces, 3/4 Yard, 1/2 Ell, 1 1/2 Bushel, 3/4 Ship, &c.

3/4
5/8
1 1/2
4/5
1 1/100
3 4/5
2 1/20
5 01/1001
17 18/3117
1007
7003
14872
1000000, &c.

Being acquainted what a Fraction is, it followeth to know what a Fraction of a greater Denomination is worth in a lesfer; for the attaining of which, consider what followeth.

1. The uppermost number of a Fraction is called *Numerator*, and the undermost *Denominator*: The *Numerator* shews the quantity of the parts, and the *Denominator* shews what names such parts bear, either Halves, Quarters, &c.

2. The *Numerator* ought to be les than the *Denominator*; for if it be equal or more, then such Fractions are called improper, as 2/3 is a whole, or 1 1/2 is more than a whole; it is 1 1/2 as you may see more at large in the second Book.

3. When you will know the worth of a Fraction on a lesfer *Denomination*, then the Fraction is, Multiply the *Numerator* by so many small pieces or parts, as the whole doth contain, and divide the Product by the *Denominator*, the Quotient is the number required: As for example, If it be required to know, how many shillings 3/4 of a pound is worth, I multiply the *Numerator* 3 by 20 (because the whole pound is 20 β) the Product is 60; which 60 I divide by the *Denominator*.

367

275

## An Appendix.

for 4; the Quotient is 15 β, for the answer of the question.

Further, if it be required to know how many ounces 7 lb is, do the same way, thus; Multiply the Numerator 5 by 16 (because 16 ounces makes 1 lb) the Product is 80, which 80 being divided by the Denominator 8, the Quotient is 10 ounces, for the answer of the Question, and so of the reft.

To make the Learner the more perfect in this, here followeth divers Questions of this nature, as a preparation to the *Rule of Practice*.

### I. Question.

In 2/3 L, how many shillings? *facit* 12 β, as followeth.

2. Item, In 2/3 L, how many β? *facit* 13 1/3 β.

3. Item, In 9/8 β, how many 9? *facit* 10 9.

Multiply the *Numerator* by 12 (because 12 9 makes 1 β) and divide the Product by 6.

4. Item, In 3/4 β, how many 9? *facit* 9 9.

5. Item, In 8/9 β, how many 8? *facit* 10 8.

6. Item, In 15/12 β, how many 9? *facit* 11 1/2 9.

7. Item, In 2/3 9, how many farthings? *facit* 2 2/3 q.

8. Item, In 1/2 9, how many q? *facit* 3 1/2 q.

T 2

9. Item,

276

An Appendix.

9. Item, In ⅔ L, how many 9 ? facit
200 9.

10. Item, In ¾ β, how many q? facit 36 q.

11. Item, In ⅜ L, how many q? facit
384 q.

12. Item, In ⅝ L, how many β ? facit 16 β
8 9.

13. Item, In ⅞ β, how many 9 ? facit 8 9.

14. Item, In ¼ of an hundred weight, how
many ⅛ ? facit 84 ⅛.

15. Item, In ⅝ C, how many quarters? facit
3⅓ qrs.

16. Item, In ¾ qrs, how many ⅛ ? facit
21 ⅛.

17. Item, In ⅞ qrs, how many ⅛ ? facit
24½ ⅛.

18. Item, In ⅞ ⅛, how many ounces? facit
14 ounces.

19. Item, In ¼ ounces, how many drams?
facit 6⅝ drams.

20. Item, In ¾ drams, how many scruples?
facit 2¼ scruples.

21. Item, In ⅜ scruples, how many grains?
facit 1 3⅛ grains.

22. Item, In ¾ Ells, how many quarters of a
Yard? facit 3¾ quarters.

23. Item, In ¼ Seam (or Quarters) how many
Bushels? facit 6 Bushels.

24. Item, In ⅜ Pecks, how many Pints? facit
6 Pints.

Note,

277

An Appendix.

Note.

The Rule of Practice is nothing else but a short
operation of the Rule of Three, and confits chief-
ly in multiplying and dividing exactly in a com-
pendious (or short) way, as is mentioned in the
Second Part; therefore I shall be the shorter
here.

Here followeth (as an Introduction to the Rule
of Practice) how to divide any number by a single
figure, otherwife then is taught in Division, which
will be a great help to the Learner, in the opera-
tion of the said Rule of Practice.

How to divide by 2, or to take the half.

In dividing by 2, or taking the half of any
number, is the same thing; the difference is onely
this, That in taking the half, you spare the trou-
ble of fetting down the Divisor more then once,
and of cutting the figures of the Dividend ; and
also of fetting the reft above the said Dividend (if
there be any Reft) as followeth.

1. If it be required to divide 8642 by 2,
(or to take the half of the said number,) what is
the Quotient ? facit 4341.

The common way.

8642 (4321

2222 Facit.

The shorter way.

$\frac{8642}{4321}$

Facit.

T 3

I be-

278

## An Appendix.

I begin at the left hand (in the shorter way) and say, 1 he half of 8 is 4, the half of 6 is 3, the half of 4 is 2, and lastly, the half of 2 is 1; which figures 4321. I set under the line, as you see above.

### Note.

In taking the part of any number, if you find any Rest, every Unite so refting, must be accounted for so many Tens, (except at the last figure, or at the Unites) as the following examples doth shew.

2. Divide 9751 by 2, facit 4875½.

$$ \frac{1}{2}) \overline{9751} \\ 4875\overline{\frac{1}{2}} $$

I begin at the left hand (as before) saying, The half of 9 is 4, reft 1 (which is 10) and 7 is 17, then the half of 17 is 8, reft 1, and 5 is 15, the half of 15 is 7, reft 1, and 1 is 11, the half of 11 is 5, reft 1, which is a half, the half of the whole number is then 4875½.

### How to take the third part.

3. What is the third part of 8102? facit 2700⅔.

$$ \frac{3}{4}) \overline{8102} \\ .2700\overline{\frac{2}{4}} $$

I say the third part of 8 is 2, reft 2 (which is 20).

279

## An Appendix.

20) and 1 is 21; the third of 21 is 7, then the third of 0 is 0, and the third of 2 is 0, reft ⅔; the Quotient is 2700⅔.

4. Item, What is the third part of 1107? facit 369.

### How to take the fourth part.

5. What amounts the ⅓ of 3702? facit 925¼.

$$ \frac{1}{4}) \overline{3702} \\ 925\overline{\frac{1}{4}} \text{ or } \frac{1}{2} $$

I say the ⅓ of 37 is 9, reft 1; then the ⅓ of 10 is 2, reft 2; then the ⅓ of 22 is 5, reft 2, which 2 is ⅔ or ⅛.

6. Item, What is the ⅓ part of 11111? facit 2777¾.

### How to take the fifth part.

7. What is the ⅔ of 5015? facit 1003.

$$ \frac{1}{5}) \overline{5015} \\ 1003 $$

I say the ⅔ of 5 is 1, then the ⅔ of 0 is 0, then the ⅔ of 1 is 0, lastly, the ⅔ of 15 is 3: The whole Quotient is 1003.

8. Item, What is the ⅔ part of 9601? facit 1920¼.

T 4

How

280

## An Appendix.

### How to take the sixth part.

9. What is the sixth part of 11406? facit 1901.

$$ \frac{\frac{1}{2}}{\frac{1}{2}} \cdot \frac{11406}{1901} $$

Say the $\frac{1}{2}$ part of 11 is 1, reft 5, and 4 is 54, then the $\frac{1}{2}$ of 54 is 9, then the $\frac{1}{2}$ of 0 is 0, and lastly, the $\frac{1}{2}$ of 6 is 1; the Quotient is 1901, as you see above.

10. Item, What is the $\frac{1}{2}$ part of 123456? facit 20576.

### How to take the seventh part.

11. What is the seventh part of 151000? facit 21571$\frac{1}{2}$.

$$ \frac{\frac{1}{2}}{\frac{1}{2}} \cdot \frac{151000}{21571^{\frac{1}{2}}} $$

Say the $\frac{1}{7}$ of 15 is 2, reft 1, then the $\frac{1}{7}$ of 11 is 1, reft 4, then the $\frac{1}{7}$ of 40 is 5, then the $\frac{1}{7}$ of 50 is 7, and lastly, the $\frac{1}{7}$ of 10 is 1, reft $\frac{1}{7}$; the Quotient is 21571$\frac{1}{7}$, as you see above.

12. What is the $\frac{1}{7}$ of 80117? facit 11445$\frac{2}{7}$.

### How to take the eighth part.

13. What is the $\frac{1}{8}$ part of 9009? facit 1126$\frac{1}{8}$.

$$ \frac{\frac{1}{8}}{\frac{1}{8}} \cdot \frac{9009}{1126^{\frac{1}{8}}} $$

Say

281

## An Appendix.

Say the $\frac{1}{8}$ of 9 is 1, reft 1, then the $\frac{1}{8}$ of 10 is 1, reft 2, then the $\frac{1}{8}$ of 20 is 2, reft 4, lastly, the $\frac{1}{8}$ of 49 is 6, reft $\frac{1}{8}$; the Quotient is 1126$\frac{1}{8}$.

14. Item, What is the $\frac{1}{8}$ part of 73081? facit 9135$\frac{1}{8}$.

### How to take the ninth part.

15. What is the $\frac{1}{9}$ of 126007? facit 14000$\frac{7}{9}$.

$$ \frac{\frac{1}{9}}{\frac{1}{9}} \cdot \frac{126007}{14000^{\frac{1}{9}}} $$

Say the $\frac{1}{9}$ of 12 is 1, the $\frac{1}{9}$ of 36 is 4, the $\frac{1}{9}$ of 0 is 0, the $\frac{1}{9}$ of 0 is 0, and the $\frac{1}{9}$ of 7 is 0, reft $\frac{2}{9}$; the Quotient is then 14000$\frac{2}{9}$, as you see above.

16. Item, What is the $\frac{1}{9}$ part of 811458? facit 90162.

17. Item, What is the $\frac{1}{10}$ part of 3480? facit 348.

To divide by 10, or to take the tenth part of any number the shortest way, hath been taught already in Fol. 140. which is to cut off the last figure of the Dividend towards the right hand.

18. Item, What is the $\frac{1}{10}$ of 31879? facit 3187$\frac{9}{10}$.

19. Item, Divide 652812 by 12, facit 54401.

$$ \frac{\frac{1}{12}}{\frac{1}{12}} \cdot \frac{652812}{54401} $$

Say the $\frac{1}{12}$ part of 65 is 5, reft 5, then the $\frac{1}{12}$ of

282

## An Appendix.

of 52 is 4, reft 4, then the ⅓ of 48 is 4, then the ⅓ of 1 is 0, and the ⅓ of 12 is 1; the Quotient is 54401, as you see above.

### Another way.

Instead of taking the ⅓ part at once, you may take firft the fixth part, and the half of that fixth part is the number required, as followeth.

$$ \begin{array}{r} \displaystyle\frac{652812}{108802} \\ \cline{2-2} 54401 & \text{Facit } 54401. \end{array} $$

### Otherwise.

Instead of taking ⅓, and the half of the said fixth part, you may take (if you please) ½ firft, and ¼ of that third part, is the number required, as followeth.

$$ \begin{array}{r} \displaystyle\frac{652812}{217604} \\ \cline{2-2} 54401 & \text{Facit } 54401. \end{array} $$

Many more varieties of operations might be produced, but I forbear for brevity sake.

20. Item, Divide 304811 by 12, facit 25400¹¹⁄₁².

21. Item, Divide 3487 by 20, facit 174⁷⁄₃.

To divide by 20, the shorteft way hath been taught already, Fol.207. which is to cut off the last

283

## An Appendix.

laft figure of the Dividend towards the right hand, and take the half of the other figures, as followeth.

$$ \begin{array}{r} 34817 \\ \underline{\quad} \\ \text{Facit } 174\frac{7}{2}. \end{array} $$

22. Item, Divide 34817 by 20, facit 1740¹⁷⁄₁².

### How to Multiply and Divide a Number of several Denominations by a Digit (or a single) Figure, without reducing it into its lesser Denomination.

23. Multiply 3 L 13 β 99 3q by 5, facit 18 L 9 β 09 3q.

$$ \begin{array}{r} L \ \beta' \ \vartheta \ q \\ \underline{3} - \underline{13-9-3} \\ \multicolumn{1}{r}{5} \\ \text{Facit } 18-09-0-3 \end{array} $$

I begin at the q, and say, Five times 3 q is 15 q, or 3 β 3 q. I set down 3 q, and carry the 3 9, then 5 times 9 9 is 45, and 3 is 48 9, which is juft 4 β. I set a cipher in the place of 9, under the line, and carry 4 β; then 5 times 3 β is 15, and 4 is 19 β. I set down 9 β, carry 1 Angel; then 5 times 1 Angel is 5 Angels, and 1 is 6 Angels, or 3 L. I set a cipher in the place of Angels, and carry 3 L. Lastly, 5 times 3 L is 15, and 3 I carried is 18 L, which 18 I set under

284

## An Appendix.

der the $L$. The whole Product is $18 L 9 \beta 09 3 q.$ as before.

24. Item, Multigly $12 L 15 \beta 6 \varphi$ by 3, facit $38 L 6 \beta 6 \varphi$.

25. Item, Multiply $134 L 12 \beta 7 \varphi$ by 4, facit $538 L 10 \beta 4 \varphi$.

26. Item, Multiply $31 L 19 \beta 8 \varphi$ by 6, facit $191 L 18 \beta 0 \varphi$.

27. Item. Multiply $29 L 16 \beta 10 \varphi 3 q$ by 8, facit $238 L 15 \beta 2 \varphi 0 q$.

28. Item, Multiply $33 L 17 \beta 3 \varphi 1 q$ by 10, facit $338 L 12 \beta 8 \varphi 2 q$.

29. Item, Multiply $134 L 14 \beta 5 \varphi 2 q$ by 16, facit $2155 L 11 \beta 4 \varphi 0 q$.

$$ \begin{array}{c@{\,}r@{\,}r@{\,}r@{\,}r@{\,}r} & \beta & \varphi & q \\ \cline{2-5} 134 & - & 14 & - & 5 & - & 2 \\ \cline{2-7} & & & & & & \\ \cline{2-7} & & & & & & (4) \end{array} $$

$$ \begin{array}{c@{\,}r@{\,}r@{\,}r} & & & & & & (4) \\ \cline{2-7} & & & & & & \\ \cline{2-7} & & & & & & 2155 - 11 - 4 - 0 \end{array} $$

Note, 16 being too troublesome for the Learner to Multiply at once, I Multiply twice by 4, because 4 times 4 is 16, as you see above.

30. Item, Multiply $31 L 11 \beta 3 \varphi 3 q$ by 16, facit $505 L 1 \beta 0 \varphi 0 q$.

31. Item, Multiply $43 L 15 \beta 9 \varphi 1 q$ by 12, facit $525 L 9 \beta 3 \varphi 0 q$.

32. Divide 26 $L 13 \beta 6 \varphi 2 q$ by 3, facit $8 L 17 \beta 10 0 \frac{1}{2} q$.

$L$

285

## An Appendix.

$$ \begin{array}{c@{\,}r@{\,}r@{\,}r@{\,}r} & \beta & \varphi & q \\ \cline{2-5} 26 & - & 13 & - & 6 & -2 \\ \cline{2-5} 3) & & & & & \\ \cline{2-5} & & & & & \frac{8}{8} \quad - \quad 17 \quad - \quad 10 \quad -0 \frac{2}{3} \\ \cline{6-7} & & & & & \\ \cline{6-7} & & & & & \text{facit} \end{array} $$

I begin at the $L$, saying the third part of 26 $L$ is 8 $L$, which 8 I set down under the line, there remains 2 $L$, which is 4 Angels, and 1 Angel is 5 Angels, the third part of 5 Angels is 1 Angel; I set 1 in the place of Angels under the line, there reft 2 Angels and 3 $\beta$ is 23 $\beta$, the third part of 23 $\beta$ is 7, reft 2 $\beta$. I set down 7 $\beta$, then 2 $\beta$ that remains and 6 $\varphi$, is 30 $\varphi$, the third part of 30 is 10 $\varphi$. I set down 10 $\varphi$, then the third part of 2 $q$ is 0, reft $\frac{2}{3}q$, which I set in the place of $q$ under the line, the Quotient is then 8 $L 17 \beta 10 \varphi 0 \frac{2}{3} q$, as you see above.

33. Divide $47 L 16 \beta 7 \varphi 0 q$ by 4, facit $11 L 19 \beta 1 \varphi 3 q$.

34. Item, divide $124 L 13 \beta 3 \varphi 2 q$ by 5, facit $24 L 18 \beta 7 \varphi 3 \frac{1}{2} q$.

35. Item, divide $13 L 0 \beta 0 \varphi 3 q$ by 6, facit $2 L 3 \beta 4 \varphi 0 \frac{3}{6} \text{ or } \frac{1}{2} q$.

36. Item, divide $648 L 14 \beta 1 \varphi 0 q$ by 7, facit $92 L 13 \beta 5 \varphi 1 \frac{1}{2} q$.

37. Item, divide $100 L 5 \beta 4 \varphi 3 q$ by 8, facit $12 L 10 \beta 8 \varphi 0 \frac{3}{8} q$.

38. Item, divide $131 L 8 \beta 11 \varphi 1 q$ by 9, facit $14 L 12 \beta 1 \varphi 1 q$.

39. Item, divide $55 L 14 \beta 0 \varphi 3 q$ by 15, facit $3 L 14 \beta 3 \varphi 1 q$.

$L$

286

An Appendix.

Instead of taking the ⅓ part at once, I take
first the ⅓ part, and ⅓ of it is the Quotient required,
as you see above.

40. Item, divide 362 L 12 β 8§ 0q by 16, facit 22 L 13 β 3§ 2q.

Note.

The Learner being well acquainted with what
hath been mentioned already (concerning the
short way of Multiplying and Dividing) will find
no great difficulty in the following Questions.

But first of all, It is absolutely necessary to un-
derstand the following Tables which are the
ground-work of all, without which no confider-
able progress can be made in the Rule of
Practice.

The

287

An Appendix.

The First Table of Practice,
Shewing the even Parts of a
Shilling in Pence.

Pence.

The Second Table (shewing the even Parts of a
Pound in Shillings) followeth.

The

288

An Appendix.

The Second Table of Practice.

Shillings,

<table><tbody><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td></tr><tr><td colspan="11">{ ⅛ or ⅛, and the ⅛ of it is the answer.</td></tr><tr><td>⅛</td><td>2 and 1, 2 is ⅛ and 1 is ⅜, or the half of 2.</td><td>1⅝</td><td>4 and 2, 4 is ⅝, and 2 is 1⅛ or ½ of 4.</td><td>5 and 2, 5 is ¼, and 2 is ⅛.</td><td>4 and 4, taking ⅛ for each 4 β.</td><td>5 and 4, 5 is ¼, and 4 is ⅝.</td><td>10 and 1, 10 is ½, and 1 is ⅜, or ⅛ of 10 β.</td><td>10 and 2, 10 is ½, and 2 is 1⅛, or ½ of 10 β.</td><td>10: 2 and 1, 10 is ⅜, 2 is 1⅛, and 1 is ½ of 2 β.</td><td>10 and 4, 10 is ½, and 4 is ⅝.</td></tr><tr><td>13</td><td>14</td><td>15</td><td>16</td><td>17</td><td>18</td><td>19</td><td>10: 4 and 2, 10 is ⅜, 4 is ⅝ and 2 is ⅛.</td><td>10: 5 and 2, 10 is ⅜, 5 is ¼, and 2 is ⅛.</td><td>10: 4 and 4, 10 is ½, and for each 4 take ⅛.</td><td>10: 5 and 4, 10 is ½, 5 is ¼, and 4 is ⅝.</td></tr></tbody></table>

The

285

An Appendix.

The Third Table, shewing the even Parts of 1 lb Averdupois weight in ounces.

<table><tbody><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td><td>12</td></tr><tr><td colspan="12">{ ⅛ or ¼, and ¼ of that ¼ is the answer.</td></tr><tr><td>⅛</td><td>2 and 1, 2 is ⅛, and 1 is ½ of 2 ounces.</td><td>1¼</td><td>4 and 1, 4 is ¼, and 1 is ½ of 4 ounces.</td><td>4 and 2, 4 is ¼, and 2 is ½ of 4 ounces.</td><td>4: 2 and 1, 4 is ¼, 2 is ½ of 4, and 1 is ½ of 2.</td><td>8 and 1, 8 is ½, and 1 is ⅛ of 8 ounces.</td><td>8 and 2, 8 is ½, and 2 is ¼ of 8 ounces.</td><td>8: 2 and 1, 8 is ½, 2 is ¼ of 8, and 1 is ½ of 2 ounces.</td><td>8 and 4, 8 is ½, and 4 is ½ of 8 ounces.</td><td>8: 4 and 1, 8 is ½, 4 is ½ of 8, and 1 is ½ of 4.</td><td>8: 4 and 2, 8 is ½, 4 is ½ of 8, and 2 is ½ of 4.</td><td>8: 4: 2: and 1, 8 is ½, 4 is ½ of 8, 2 is ½ of 4, and 1 is ½ of 2 ounces.</td></tr></tbody></table>

V

1 Queft.

286

An Appendix.

1 Question.

If 1 Yard of cloth (or 1 lb) cost 6 9, what
cost 96 Yards (or lb)? facit 48 β.

6 9 is ⅓ β ----- 48 β, facit 48 β.
6 9 is the half of 1 β; therefore I take the ½ of
96, the Quotient is 48 β, as you see above.

2. Item, If 1 fb cost 6 9, what cost 864 fb ?
facit 21 L 12 β.

3. Item, If 1 fb cost 6 9, what cost 391 fb ?
facit 9 L 15 β 6 9.

In taking the ⅓ of 391, there will rest 1, which
is ⅔ β or 6 9, as followeth.

6 9 is ⅓ of 1 β --- 19 15-6 9
Facit 9 L 15 β 6 9.

4. Item, If 1 Yard cost 6 9, what cost 561
Yards? facit 14 L 0 β 6 9.

5. Item, If 1 Yard cost 3 9, what cost 64
Yards? facit 16 β.

6. Item, What cost 167 lb at 3 9 the lb?
facit 41 β 9 9.

7. Item,

287

An Appendix.

7. Item, What cost 287 lb at 4 9 the lb?
facit 4 L 15 β 8 9.

8. Item, What cost 720 Ells at 2 9 the Ell?
facit 6 L.

9. Item, 69 Yards at 2 9 the Yard, what amounts the whole? facit 11 β 6 9.

10. Item, What cost 144 lb at 4 9 the lb?
facit 12 β.

11. Item, What cost 249 lb at 1 9 the lb?
facit 20 β 9 9.

12. Item, What cost 84 lb at 9 9 the lb?
facit 3 L 3 β.

6 9 is ⅔ of 1 β ----- 42

3 9 is ½ of 6 9 --- 21

6 3 β
Facit 3 L 3 β.

13. Items, What cost 89 lb at 9 9 the lb?
facit 3 L 6 β 9 9.

14. Item, If 1 fb cost 8 9, what cost 101 fb?
facit 3 L 7 β 4 9.

15. Item, If 1 Yard cost 7 9, what cost 124
Yards? facit 3 L 12 β 4 9.

16. Item, If 1 Yard cost 11 9, what cost
281 Yards? facit 12 L 17 β 7 9.

V 2

Yard

288

## An Appendix.

Yard 9 Yards
1 11 281

6 9 is 1/2 of 1 β 140-6

3 9 is 1/2 of 6 9 70-3

2 9 is 1/2 of 6 9 46-10

257-7

Facit 12 L 17 β 7 9

### Otherwise.

If the Yard had been worth 1 β, then 281 Yards would be worth 281 β; but the Yard is worth but 11 9, which is 1 1/2 part of 1 β left; therefore take 1/2 of 281 and subtract the Quotient from it, the rest will be 257 β 7 9, or 12 L 17 β 7 9 as before, thus.

281 Yards.

1/2 23-5

257-7

Facit-12 L 17 β 7 9.

17. Item, What cost 360 lb at 11 9 the lb? Facit 16 L 10 β.

18. Item, What cost 84 lb at 5 9 the lb? Facit 35 β.

19. Item, 121 Yards at 10 9, facit 5 L 6 β 10 9.

20. Item,

289

## An Appendix.

20. Item, 303 lb at 10 9 the lb, facit 12 L 12 β 6 9.

21. Item, What cost 137 Yards at 10 β the Yard? facit 68 L 10 β.

Yard β Yards
1 10 137

10 β is 1/2 L 68-10 β.

Facit.

22. Item, If 1 lb cost 5 β, what 76 lb? Facit 19 L.

23. Item, 567 Ells at 4 β the Ell, facit 113 L 8 β.

24. Item, 840 lb at 2 β the lb, facit 84 L.

25. Item, 259 lb at 2 β the lb, facit 25 L 18 β.

26. Item, 403 lb at 5 β the lb, facit 100 L 15 β.

27. Item, 124 Ells at 15 β the Ell, facit 93 L.

Ell β Ells
1 15 124

10 β is 1/2 of 1 L 62

5 β is 1/2 of 10 β 31

Facit 93 L.

28. Item, 177 Ells at 15 β the Ell, facit 132 L 15 β.

V 3

29. Item,

290

An Appendix.

29. Item, 126 lb at 6 3/8 the lb, facit 37 1/20 16 3/8.

30. Item, 310 lb at 8 3/8 the lb, facit 124 1/2.

31. Item, 521 lb at 14 3/8 the lb, facit 364 1/2 14 3/8.

32. Item, 135 lb at 12 3/8 the lb, facit 81 1/2.

33. Item, 99 Ells at 11 3/8 the Ell, facit 54 1/2 9 3/8.

34. Item, 477 lb at 1 3/8 the lb, facit 23 1/2 17 3/8.

35. Item, 69 lb at 3 3/8 the lb, facit 10 1/2 7 3/8.

36. Item, 325 Ells at 7 3/8 the Ell, facit 113 1/2 15 3/8.

37. Item, 264 Yards at 9 3/8 the Yard, facit 118 1/2 16 3/8.

38. Item, 168 lb at 13 3/8 the lb, facit 109 1/2 4 3/8.

39. Item, 112 Ells at 16 3/8 the Ell, facit 89 1/2 12 3/8.

40. Item, 216 Ells at 19 3/8 the Ell, facit 205 1/2 4 3/8.

41. Item, 316 lb at 18 3/8 the lb, facit 284 1/2 8 3/8.

42. Item, 161 lb at 17 3/8 the lb, facit 136 1/2 17 3/8.

43. Item, If 1 Yard coft 3 3/8 6 3/8, what coft 62 Yards? facit 10 1/2 17 3/8.

Yard

291

An Appendix.

$$
\begin{array}{l}
\text{Yard} \quad \beta \quad \vartheta \quad \text{Yards} \\
1 \quad - \quad 3 \quad - \quad 6 \quad - \quad 62 \\
\hline
3 - 6 \vartheta.
\end{array}
$$

$$
\begin{array}{l}
6 \beta \text{ is } \frac{1}{2} \text{ of } 1 \beta \text{ --- } 31 \\
\hline
217
\end{array}
$$

Facit 10 L 17 3/8

Another way.

$$
\begin{array}{l}
\text{Yard} \quad \beta \quad \vartheta \quad \text{Yards} \\
1 \quad - \quad 3 \quad - \quad 6 \quad - \quad 62 \\
\hline
2 \beta \text{ is } \frac{1}{2} \text{ L} \quad \ldots \quad 6-4 \\
1 \beta \text{ is } \frac{1}{2} \text{ of } 2 \beta \quad \ldots \quad 3-2 \\
6 \vartheta \text{ is } \frac{1}{2} \text{ of } 1 \beta \quad \ldots \quad 1-11
\end{array}
$$

$$
2 \beta \text{ is } \frac{1}{2} \text{ L} \quad \ldots \quad 6-4
$$

$$
1 \beta \text{ is } \frac{1}{2} \text{ of } 2 \beta \text{ --- } 3 - 2
$$

$$
6 \beta \text{ is } \frac{1}{2} \text{ of } 1 \beta \text{ --- } 11
$$

Facit 10-17 β

44. Item, If 1 Yard coft 2 3/8 4 3/8, what coft
35 Yards? facit 4 L 1 3/8 8 3/8.

45. Item, 116 lb at 5 3/8 6 3/8 the lb, facit 31 L 18 3/8.

46. Item, 58 Ells at 11 3/8 8 3/8 the Ell, facit 33 L 16 3/8 8 3/8.

47. Item, 36 Yards at 12 3/8 7 3/8 the Yard, facit 22 L 13 3/8.

48. Item, 47 Ells at 18 3/8 2 3/8 the Ell, facit 42 L 13 3/8 10 3/8.

V 4

49. Item,

294

## An Appendix.

62. Item, 138 Pieces at 5 L 6 ⅛ 8 9 the Piece, facit 736 L.

63. Item, 56 ⅛ at 3 L 10 ⅛ 2 9 the ⅛, facit 196 L 9 ⅛ 4 9.

$$ \begin{array}{l} \text{⅛ L ⅛ 9 ⅛} \\ 1-3-10-2-56 \\ \qquad 3-10-2 9 \\ \hline 168 \end{array} $$

$$ \begin{array}{l} 10 \, \text{⅛ is } \frac{1}{2} \text{ of } 1 \, \text{L} - 28 - 0 - 0 \\ \text{Suppose} \\ \times \, \text{⅛ is } \frac{1}{2} \text{ of } x \, \text{⅛} - x - 16 - 0 \\ 2 9 \text{ is } \frac{1}{2} \text{ of } 1 \, \text{⅛} \quad 0 - 9 - 4 \\ \hline \end{array} $$

Facit 196-9-4

### Otherwise.

$$ \begin{array}{l} \text{⅛ L ⅛ 9 ⅛} \\ 1-3-10-2-56 \\ \qquad 20 \\ \cline{2-2} 70 \, \text{⅛} 2 9 \end{array} $$

$$ 2 9 \text{ is } \frac{1}{2} \text{ of } 1 \, \text{⅛} \quad 0 \, \text{--} \, 9 \, \text{--} \, 4 $$

$$ 3929-4 $$

Facit 196 L 9 ⅛ 4 9

64. Item, 78 ⅛ at 2 L 0 ⅛ 6 9, facit 157 L 19 ⅛.

65. Item, 32 ⅛ at 5 9 ⅜ 2 q the ⅛, facit 14 ⅛ 8 9.

66. Item,

295

## An Appendix.

66. Item, 13 ⅛ at 3 9 ⅜ 3 q the ⅛, facit 4 ⅛ 0 9 3 9.

67. Item, 24 Yards 2 Quarters at 6 ⅛ 8 9 the Yard, facit 8 L 3 ⅛ 4 9.

$$ \begin{array}{l} \text{Yard} \\ 1-6-8-24-2 \\ \qquad 6-8 9 \\ \cline{2-2} \end{array} $$

$$ 6 9 \text{ is } \frac{1}{2} \text{ of } 1 \, \text{⅛} \quad 0 \, \text{--} \, 12 $$

$$ 2 9 \text{ is } \frac{1}{2} \text{ of } 6 9 \quad 0 \, \text{--} \, 4 $$

$$ 2 \text{ Yard is } \frac{1}{2} \text{ of } 6 9 \quad 0 \, \text{--} \, 4 $$

$$ 1613-4 $$

Facit 8 L 3 ⅛ 4 9

68. Item, If 1 Yard coft 3 ⅛ 9 9, what coft 9 Yards and 1 Quarter, facit 34 ⅛ 8 9 1 9.

69. Item, What coft 4 Ells and 1 Quarter of a Yard, if 1 Ell coft 5 ⅛ 10 9, facit 24 ⅛ 6 9.

70. Item, What coft 24 Yards 3 Quarters and a half at 9 ⅛ 4 9 the Yard, facit 11 L 12 ⅛ 2 9.

Yard

296

An Appendix.

Facit 11 L 12 β 2 §

71. Item, What coft 7 Yards ⅔ and a half quarter, at 5 β 8 § the Yard? facit 44 β 7 § 2 q.

72. Item, What coft 57 Yards at 3⅔ L the Yard? facit 199½ L.

73. Item, What coft 47 Pieces at 8¼ L the Piece? facit 41 1¼ L.

74. Item, What coft 7½ Yards at 10 β 6 § the Yard? facit 3 L 17 β 0 §.

75. Item, What coft 7 Pieces at 2 L 12 β 3 § the Piece? facit 18 L 5 β 9 §.

Facit 18—05—9

76. Item, What coft 5 Yards at 18 β 6 § the Yard? facit 4 L 12 β 6 §.

77. Item,

297

An Appendix.

77. Item, What coft 10 lb 8 ounces at 13 β 7 § 2 q the lb? facit 7 L 3 β 0 § 3 q.

78. Item, What coft 6½ lb at 16 β 8 § 1 q the lb? facit 5 L 5 β 8 § 1 q.

79. Item, What coft 87 lb 12 ounces at 3 β 7 § the lb? facit 15 L 14 β 5 § 1 q.

80. Item, What coft 25 Pieces at 2 L 13 β 2 § the Piece? facit 66 L 9 β 2 §.

Piece L β § Pieces

Facit 66—09—2

81. Item, If 1 Bay coft 3 L 15 β 9 §, what coft 16 Bays? facit 60 L 12 β.

82. Item, What coft 2 lb at 3 § 2 q the ounce? facit 9 β 4 §.

83. If 1 lb of Cheese coft 2 § 1 q, what coft 16 lb? facit 3 β.

84. If 1 lb coft 3 § 2 q, what coft 12 lb 8 ounces? facit 3 β 7 § 3 q.

85. Item, What coft 9 score and 5 lb of Beef at 4 β 8 § the score? facit 43 β 2 §.

Score β § Score lb

6 § is ½ of 1 β --- 4 --- 6

2 § is ½ of 6 § --- 1 --- 6

5 lb is ¼ of 1 Score --- 1 --- 2

Facit 43—2 §

86. Item,

298

An Appendix.

86. Items, What cost 8 Score and 15 1b, at 5 3 4 9 the Score? facit 46 3 8 9.

87. Item, What cost 16 Hundred 2 Quarters of Iron at 18 3 6 9 the Hundred? facit 15 L 5 3 9.

88. Item, What cost 6 C. 3 quarters 14 1b at 16 3 8 9 the Hundred? facit 5 L 14 3 7 9.

89. Item, What cost 5 Packs of Wool at 9 9 the 1b? facit 45 L.

Note, One Pack of Wool contains so many 1b as there is Pence in 1 Pound sterling, which is 240; therefore upon that ground it doth appear, that so many 9 as 1 1b of Wool is worth, so many Pounds sterling is 1 Pack worth, that is to say, If 1 1b cost 6 9, the Pack cost 6 L: If the 1b cost 9 9, the Pack must cost 9 L.

Further observe also, That if you finde any odde Farthings in the price of 1 1b of Wool (as 9 9 2 q, &c.) then for every farthing, you must reckon a Crown (or 5 3) to be added to the sum, as the following examples will shew.

90. Item, If 1 1b of Wool cost 8 9 2 q, what cost 1 Pack? facit 8 L 10 3.

At 8 9 the 1b, the Pack is 8 L: Now for the 2 q I take 2 Crowns, which is 10 3, the Pack cost then 8 L 10 3, as above.

91. If 1 1b of Wool cost 11 9 1 q, what cost 16 Packs? facit 180 L.

Pack

299

An Appendix.

Facit 180 L

92. Item, What cost 12 Packs of Wool at 13 9 3 q the 1b? facit 165 L.

93. If 1 1b cost 12 9 2 q, what cost 11 Packs and 6 scores 1b? facit 143 L 19 3.

Note, In buying (or selling) Wool by the Pack, if you finde any odd scores (or pounds) you may make use of the first Table of Practice before mentioned (which shews the even parts of a shilling in pence) to finde what part of the price of one Pack of Wool, is to be taken for any number of Scores from 1 to 11, because 1 Pack of Wool contains so many Scores as there is pence in a shilling, which is 12.

If you finde any odd 1b, take such parts of the price of one score, as the second Table of Practice sheweth, because one score is so many pounds, as there is shillings in 1 L sterling, which is 20, as the following example will shew.

Pack

300

An Appendix.

22

11

Having multiplied the 11 Packs by the price of 1 Pack, which is 12 L 10 β. I consider what part of 1 Pack 6 Scores are. I find that 6 Scores is a half pack, therefore I take the half of the price of 1 Pack, which is 6 L 5 β, and add it to the other fums; the total is 143 L 15 β.

94. If 1 lb of Wool cost 10 9, what cost 6 Packs and 4 Scores? facit 63 L 6 β 8 9.

95. Item, If 1 lb of Wool cost 9 9 2 q, what cost 8 Packs and 3 Scores? facit 78 L 7 β 6 9.

96. Item, What cost 10 Packs 10 Scores of Wool at 13 9 1 q the lb? facit 143 L 10 β 10 9.

Pack

SCALE

An Appendix.

130

97. Item, What cost 4 Packs 9 Scores at 14 9 3 q the pound? facit 70 L 1 β 3 9.

98. Item, What cost 5 Packs 1 Score and 15 lbs, if 1 lb cost 8 9? facit 41 L 3 β 4 9.

X

Other-

301

302

An Appendix.

Otherwise.

<table><thead><tr><th>Pack</th><th>l</th><th>Packs</th><th>Score</th><th>lb</th></tr></thead><tbody><tr><td>1—8</td><td>5—15</td><td></td><td></td><td></td></tr></tbody></table>

8

1 Score at 8 9 is 40 Groats or —0—13—4
15 1b is worth 30 Groats or —0—10—0

40

Facit 41 — 3 — 4

99. Item, What cost 9 Packs 5 Scores and 5 1b at 11 9 the pound? facit 103 L 16 13 9.

100. Item, A Trademan bought 4 parcels of Wool, to wit, 8 Packs 4 Scores and 12 1b, 9 Packs 2 Scores and 16 1b, 5 Packs 8 Scores and 8 ee, and 6 Packs 2 Scores and 10 ee, at 9 L 12 13 the Pack; what amounts the whole? facit 283 L 8 13 9.

Packs

An Appendix.

303

Packs Scores 1b

8 — 4 — 12

9 — 2 — 16

5 — 8 — 8

6 — 2 — 10

<table><thead><tr><th>Pack</th><th>l</th><th>13</th></tr></thead><tbody><tr><td>1—9—12</td><td>29—6—6</td></tr></tbody></table>

9—12

261

10 13 is 1/2 L — — — — 14—10—0

2 13 is 1/2 L — — — — 2—18—0

6 Scores is 1/2 Pack — — — — 4—16—0

Suppose 1 Score which is 1/2 of 6 Scores — 0—16—0

5 ee is 1/4 of 1 Score — — 0—4—0

1 ee is 1/2 of 5 ee — — 0—0—9

Facit 283 — 8—9 1/2

101. Item, A Trademan bought 6 parcels of Wool, weighing as followeth. 3 Packs 3 Scores 9 ee, 4 Packs 5 Scores 16 ee, 2 Packs 11 Scores 0 ee, 1 Pack 8 Scores and 12 ee, 2 Packs 5 Scores 15 ee, and 2 Packs, 0 Scores and 2 ee at 8 9 2 q the pound; what must the pay in all? facit 143 L 11 13 7 9.

X 2

A

306

An Appendix.

<table><thead><tr><th>Tod</th><th>β</th><th>§</th><th>Tods</th><th>lb</th></tr></thead><tbody><tr><td>1—16—4</td><td></td><td></td><td>12—7</td><td></td></tr></tbody></table>

<table><tbody><tr><td>10</td><td>β</td><td>is</td><td>½</td><td>of</td><td>1</td><td>£</td><td>—</td><td>—</td><td>—</td><td>—</td><td>6</td><td>—</td><td>0</td></tr><tr><td>5</td><td>β</td><td>is</td><td>½</td><td>of</td><td>10</td><td>β</td><td>—</td><td>—</td><td>—</td><td>—</td><td>3</td><td>—</td><td>0</td></tr><tr><td>1</td><td>β</td><td>is</td><td>½</td><td>of</td><td>5</td><td>β</td><td>—</td><td>—</td><td>—</td><td>—</td><td>0</td><td>—</td><td>12</td></tr><tr><td>4</td><td>§</td><td>is</td><td>½</td><td>of</td><td>1</td><td>β</td><td>—</td><td>—</td><td>—</td><td>—</td><td>0</td><td>—</td><td>4</td></tr><tr><td>7</td><td>¢</td><td>is</td><td>⅓</td><td>of</td><td>1</td><td>Tod</td><td>—</td><td>—</td><td>—</td><td>—</td><td>0</td><td>—</td><td>4—1</td></tr></tbody></table>

Facit 10—0—1

103. Item, What cost 13 Tods 14 ¢ at 18 β 8§ the Tod? facit 12 £ 12 β 0§.

104. Item, What cost 24 Tods 18 ¢ of Wool, at 22 β 7§ the Tod? facit 27 £ 16 β 6§ 07¢.

<table><thead><tr><th>Tod</th><th>£</th><th>β</th><th>§</th><th>Tods</th><th>¢</th></tr></thead><tbody><tr><td>1</td><td>—</td><td>1</td><td>—</td><td>2</td><td>—</td><td>7</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>24</td><td>—</td><td>18</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>—</td><td>2</td><td>—</td><td>7</td><td>§</td></tr></tbody></table>

<table><tbody><tr><td>2</td><td>β</td><td>is</td><td>½</td><td>of</td><td>1</td><td>£</td><td>—</td><td>—</td><td>—</td><td>—</td><td>2</td><td>—</td><td>8</td></tr><tr><td>6</td><td>§</td><td>is</td><td>½</td><td>of</td><td>2</td><td>β</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>0</td><td>—</td><td>12</td></tr><tr><td>1</td><td>§</td><td>is</td><td>½</td><td>of</td><td>6</td><td>§</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>0</td><td>—</td><td>2</td></tr><tr><td>14</td><td>¢</td><td>is</td><td>½</td><td>of</td><td>1</td><td>Tod</td><td>—</td><td>—</td><td>—</td><td>—</td><td>0</td><td>—</td><td>11</td><td>—</td><td>3</td><td>—</td><td>2</td></tr><tr><td>4</td><td>¢</td><td>is</td><td>⅞</td><td>of</td><td>1</td><td>Tod</td><td>—</td><td>—</td><td>—</td><td>—</td><td>0</td><td>—</td><td>3</td><td>—</td><td>2</td><td>—</td><td>2</td><td>6</td></tr></tbody></table>

Facit 27-16-6-6-0⁷

105. Item, What cost 26 Tods 20 ¢ of Wool at 23 β 4§ the Tod? facit 31 £ 3 β 4§.

106. Item,

307

An Appendix.

106. Item, What cost 32 Stones and 7 ¢ of Wool at 13 β 0§ the Stone? facit 21 £ 2 β 6§.

Note, 1 Stone is a half Tod or 14 ¢.

<table><thead><tr><th>Stone</th><th>β</th><th>§</th><th>Stones</th><th>lb</th></tr></thead><tbody><tr><td>1—13—0</td><td></td><td></td><td>32—</td><td>7</td></tr></tbody></table>

<table><tbody><tr><td>10</td><td>β</td><td>is</td><td>½</td><td>of</td><td>1</td><td>£</td><td>—</td><td>—</td><td>—</td><td>16</td><td>—</td><td>0</td><td>—</td><td>0</td></tr><tr><td>2</td><td>β</td><td>is</td><td>½</td><td>of</td><td>1</td><td>£</td><td>—</td><td>—</td><td>—</td><td>—</td><td>3</td><td>—</td><td>4</td><td>—</td><td>0</td></tr><tr><td>1</td><td>β</td><td>is</td><td>½</td><td>of</td><td>2</td><td>β</td><td>—</td><td>—</td><td>—</td><td>—</td><td>1</td><td>—</td><td>12</td><td>—</td><td>0</td></tr><tr><td>7</td><td>¢</td><td>is</td><td>½</td><td>of</td><td>1</td><td>Stone</td><td>—</td><td>—</td><td>—</td><td>—</td><td>0</td><td>—</td><td>6</td><td>—</td><td>6</td></tr></tbody></table>

Facit 21—2-6

107. Item, If 1 Stone of Wool cost 11 β 4§ 2q, what cost 48 Stones 9 ¢? facit 27 £ 13 β 3§ 3q.

Note, To know briefly what the Hundred weight (great and small) or the Wey of Cheese, cost at a certain price the Pound, observe the Three Rules following.

1. For the Great Hundred or 112 lb.

For every Farthing that 1 Pound is worth, reckon so many times 2 Shillings and 1 Groat, (because 112 q is 2 β 4§) what comes out, is the price of the Hundred, as followeth.

X 4

2§

308

## An Appendix.

2 9 1 q is 9 q. Now twice 9 β is -18 β
And once 9 Groats is 3 β

Facit 21 β, the
price of 1 C
at 2 9 1 q
the Pound.

108. Item, What coft 1 C (or 112 ert) if
1 ert coft 7 9 3 q? facit 3 L 12 β 4 9.

7 9 3 q is 31 q. Twice 31 β is -62 β
Once 31 Gro. is 10 β 4 9

7 12 β 4 9

Facit 3 L 12 β 4 9

109. Item, What coft 1 C at 4 9 2 q the
Pound? facit 42 β.

### 2. For the small Hundred or 100 ft.

For every Farthing that 1 Pound coft, reckon
so many times 2 β 1 9 (because 100q is 2 β
1 9) what comes out, is the price of 100 ert or
5 Scores, as followeth.

110. Item, If 1 ert coft 3 9 2 q, what coft a
Hundred or 5 Scores? facit 29 β 2 9.

3 9 2 q is 14 q. Now twice 14 β is -28 β
And once 14 β is -1 -2 9

Facit 29 -2 9

111. Item,

309

## An Appendix.

111. Item, What coft a small Hundred
weight, if 1 ert coft 8 9 1 q? facit 3 L 8 β
9 9.

8 9 1 q is 33 q. Now twice 33 β is -66 β
And once 33 9 is -2-9 9

68-9 9

Facit 3 L 8 β 9 9

112. Item, What coft 100 ert, if 1 ert coft
5 9 3 q? facit 47 β 11 9.

### 3. For the Wey of Cheese or 256 ft.

For every Farthing that 1 Pound coft, reckon
so many Crowns, and so many Groats (because
256 q is 5 β 4 9) what comes out, is the price
of 1 Wey, as followeth.

113. Item, If 1 Pound of Cheese coft 2 9
3 q, what coft 1 Wey or 256 ert? facit 2 L
18 β 8 9.

2 9 3 q is 11 q. Now 11 Crowns is 2 L 15 β

And 11 Groats is -0 -3--8 9

Facit 2 -- 18--8 9

114. Item, If 1 Pound of Cheese coft 3 9
2 q, what coft 1 Wey? facit 3 L 14 β 8 9.

115. Item, If 6 Yards coft 26 β 6 9, what
coft 1 Yard? facit 4 β 5 9.

Yards

READER,

T HATE every man (none excepted) is subject to mistake, is an undeniable truth: Therefore I hope thou wilt excuse the few mistakes which are escaped in the Prefs. It is a rare thing to see Books of this nature to be wholly free from faults, especially when the Author cannot attend the Prefs: If thou findest more then are hereunder mentioned, they are such (I hope) as will not cause material Errors; nevertheless thou art de-fired to correct both small and great, patiently with thy Pen.

Fol.41. Line 10. instead of every Quarter, Read for every four Quarters. Fol.62. In the direction of the 16 Question, instead of the rest will be 109, read the rest will be 108. Fol. 75- In line 2. of the Answer of the 15 Question, instead of other gures, read other figures. Fol.80. In the 41 Question, instead of what yo you, read what do you. Fol. 132. The Dividend of the Division is 88218, but read 88215 instead thereof. Fol. 225. In the 29 Question, instead of 16 L 18 3/5, read 16 L 8 3/5. Fol. 229. In the 83 Question, instead of 3 L 19 9/49, read 3 L 19 3/49. Fol. 244. In the 26 Question, for The Rule of the Three direct, read The Rule of Three direct. Fol. 295. In the 70 Question, for 24 3/9, read 24 3/69.

The Folio's 285, 286, 287, and 288, are set twice over.

An

## An Advertisement.

T HE Author of this Book, hath another small Book in the Prefs, containing fun-dry Tables, (viz. for the ready summing up the Price of Colchester Bays, &c.) very useful for Merchants and Tradesmen.