An introduction of the first grounds or rudiments of arithmetick plainly explaining the five common parts of that most useful and necessary art, in whole numbers & fractions, with their use in reduction, and the rule of three direct. Reverse. Double. By way of question and answer; for the ease of the teacher, and benefit of the learner. Composed not only for general good, but also for fitting youth for trade. / By W. Jackson student in arithmetick.

An introduction of the first grounds or rudiments of arithmetick plainly explaining the five common parts of that most useful and necessary art, in whole numbers & fractions, with their use in reduction, and the rule of three direct. Reverse. Double. By way of question and answer; for the ease of the teacher, and benefit of the learner. Composed not only for general good, but also for fitting youth for trade. / By W. Jackson student in arithmetick.

Table of contents

Title page

Courteous Reader,

AN Introduction of the First Grounds and Rudiments of Arithmetick.

NUMERATION.

Addition.

Substraction.

Multiplication.

Division.

Reduction.

The Rule of Three.

The Backer Rule of Three.

Numeration in fractions.

Reduction.

Multiplication in Fractions.

Division in Fractions.

Addition in Fractions.

Substraction in Fractions.

The Rule of Three in Fractions.

The Backer Rule of Three.

The Double Rule of Three.

In Reduction.

In the Rule of Three.

For the Backer Rule of Three.

§

[Page]

AN Introduction of the First Grounds or Rudiments OF Arithmetick; Plainly explaining the five Common parts of that most useful and Necessary Art, In whole Numbers & Fractions, With their use in Reduction, and The Rule of three: Direct. Reverse. Double.

By way of Question and Answer, for the ease of the Teacher, and benefit of the Learner.

Composed not only for general good, but also for fitting Youth for Trade.

By W. Iackson Student in Arithmetick.

LONDON, Printed by R. I. for F Smith, neer Temple-Bar. 166 [...]

Courteous Reader,

[Page]

[Page]

I Have on purpose omitted Progression, as also many other Rules following, partly because that these being well learned, not only by rote, but also by reason, the young learner (for whose sake I wrote this) will be inabled hereby in a good measure to understand what hee findes in other books concerning such; And if this prove but as useful, as I wish it may, and hope it will (by the Teachers care, and Scholars diligence) I may be incouraged [Page] to add somewhat to it hereafter, that may bee of further use, or else these weak indeavours may provoke some others of better parts to bring them to the publick Treasurie of Art. In the mean time accept of this mite from him that is one that would count it an honour to bee but one of the meanest of those that might present any thing on the behalf of this most Noble, and most Necessary Art of Arithmetick; that might further the growth of such as are entring upon the practice of the same, which I presume, if this small Tract may bee as a small Table wherein to see the first Rudiments [Page] in, briefly and plainly, which being by the Masters discretion appointed the young Scholar to get by heart, may prove an ease to both; to the Master, in that (if hee please to spend some set time in examining his Scholars, as they use to catechize little ones) hee by that means may teach the Rules to twenty in teaching one, and not only print the Rules in the memory of such as are past such Rules, who perhaps may bee apt to forget, but also teach the Rudiments to other, even before they come to the practice of them, whereby hee may save the pains of often telling them, and may only fit [Page] them with examples suitable to the Rule, sometimes descanting a little upon the Rules as they lye in order, as he findes occasion; and by this course, being observed, he will with the blessing of God finde by the childrens growth in knowledge, that the pains bestowed will not be in vain; but not to be tedious, I leave each to use his own discretion, how to use this or any other help, only I have thoughts, that a thing of this nature will bee profitable, and have its use; So wishing this most Noble Art, and all those that love it, to flourish in our Land, I bid thee farewel.

W. J.

AN Introduction of the First Grounds and Rudiments of Arithmetick.

[Page]

NUMERATION.

Quest. WHat is Arithmetick?

Answ. It is the Art of numbring.

Q. What is the subject of this Art?

A. The subject of it is Number.

Q. Whereof doth Number consist?

[Page] A. It consisteth of unites.

Q. What is an unite?

A. It is the original, or beginning of number, and is of it self indivisible, so that it still remaineth one.

Q. Is not one a Number then?

A. No, for Number is a collection of unites.

Q. How are Numbers said to bee divided into kinds?

A. They are divided into many sorts, but vulgarly into whole numbers, and broken numbers called fractions.

Q. Are fractions numbers?

A. Not properly, for number consisteth of a multitude of unites, but every fraction is lesser than its unite.

Q. How many several parts are accounted in common Arithmetick?

[Page] A. These five, Numeration, Addition, Substraction, Multiplication, and Division.

Q. What then is extraction of roots?

A. Although it bee another part of Arithmetick, yet it is not so common.

Q. What teacheth Numeration?

A. It teacheth how to set down any number in figures, and also to express, or read any such number so set down.

Q. How many figures are there?

A. Nine significant figures, and a cipher, which cipher signifieth nothing of it self, only it serveth to supply a place, and thereby increaseth the value of the other figures.

Q. Which be the significant figure?

[Page] A. 1, 2, 3, 4, 5, 6, 7, 8, 9.

Q. What do these signifie?

A. They signifie only each their own simple value being alone.

Q. What if they be joyned with other figures, or ciphers, is their signification altered?

A. Yea, their value is thereby much increased, according to the place they stand in, removed from the place of unites.

Q. Do Ciphers then only supply places that are void, and so increase the value of the other figures?

A. No, they do also in decimal fractions, diminish the value of those figures that stand toward the right hand of them, according to the place they stand in removed from the unite place.

[Page] Q. Which is the place of unites?

A. In whole numbers it is the first place towards the right hand, and any figure standing in that place, signifieth only its own simple value.

Q. Why make you a distinction here of whole numbers? doth it differ infraction?

A. Yea, for in decimal fractions, the unite place standeth to the left hand of the fraction.

Q. Why say you its own simple value?

A. Because a figure by being put in the second, third, or fourth place, &c. may signifie ten times, an hundred times or a thousand times its own value, &c.

Q. What is the reason of that?

A. Because every place exceeds the place next before it [Page] ten times in value, so that the figure that signifies but four in the first place, signifies ten times 4 in the second, and an hundred times four in the third place, and a thousand times four in the fourth place, &c. and in that proportion increaseth infinitely, according as its place is further removed from the unite place.

Q. Is the proportion of diminishing decimal fractions, like this of augmenting whole numbers?

A. Yea; for as these are augmented in a decupled proportion, so those are diminished, or made less, in a decupled proportion, by being removed from the unite place.

Q. What must you do when you have a number to set down, where some have not a significant figure to stand in it?

[Page] A. I must supply that place with a cipher (0) for no place must bee void.

Q. How will you set down, one thousand six hundred and sixty?

A. First I consider the fourth place is the place of thousands, and there I set down 1, then 6 in the third place, which betokeneth so many hundreds, then 6 in the second place, signifying six tens, or sixty, and because there is no figure to set in the place of unites, I supply it with a (0) cipher, to make the number consist of its due number of places, thus (1660)

Q. How set you down four thousand five hundred and six?

A. First I set 4 in the place of thousands, then five in the place of hundreds, then in regard thee is no tens, I supply that [Page] place with a (0) and lastly I place 6 in the unite place thus, (4506)

Q. How value you your places in order to the tenth place?

A. Thus, Unites, Tens, Hundreds, Thousands, Tens of Thousands, Hundreds of Thousands, Millions, Tens of Millions, Hundreds of Millions, Thousands of Millions.

Q. Can you repeat your places backwards?

A. Yea, thus, Thousands of Millions, Hundreds of Millions, Tens of Millions, Millions, Hundreds of Thousands, Ten of Thousands, Thousands, Hundreds, Tens, Unites.

Addition.

Q. Now shew what Addition teacheth.

[Page] A. Addition teacheth of several numbers to make one total equal to them all.

Q. How is that done?

A. First I set my numbers down one right under another, observing still to set the first place or figure toward the right hand of each number under the first place or figure of the uppermost number, and so the second under the second, and the third under the third, &c.

Q. After you have set down your numbers each in his due place, what do you then?

A. I must begin at the first place toward the right hand, and count all the figures in that place together, and if they bee less than ten, set it down under the first place, a line being first drawn under my sum to place my total below.

[Page] Q. What if it come to ten or above?

A. Then I must consider how many tens it contains, and carry so many unites in my mind to the next place, and set down the over-plus if there bee any, but if it bee even tens, then set down a (0) in that place.

Q. And what do you next?

A. I must remember to reckon the tens that I bear in mind for unites, and add them to the figures in the next place, and then do in all points as I did in the former place.

Q. Why do you count tens in one place, but for unites in the next?

A. because the place answers to the value thereof, being ten times the value of the former place.

[Page] Q. What is further to bee considered?

A. When I have gone thorow all the places, if at the last I have any Tens to carry, seeing there are no figures to add them withall, I set a figure signifying the number of Tens, in a place neerer the left hand.

Q. Give an example hereof.

A. Then thus, Four men owe my Master mony; A. oweth 4560 l. B. oweth 5607 l. C. oweth 6078 l. D: oweth 385 l. I would know how much these four debts amount to in all.

Q. And how will you do that?

A. First I set the several sums right under each other thus,

4560

5607

6078

0385

16630

Then I begin with those figures next the right hand, and say, 5 and 8 is 13, and 7 makes 20, now in regard it is just 2 times Ten, I set a cipher underneath the line in that place, and bear in minde 2 to reckon with the figures in the next place, and say, 2 that I bear in my mind, and 8 is 10, and 7 is 17, and 6 makes 23, then I set down the odd 3 in the second place below the line, and for the 2 Tens, I carry 2 in minde to the next place, then I say, 2 that I carry and 3 is 5, and 6 is 11, and 5 is 16, the odd 6 I set below the [Page] line, and carry one in lieu of the ten to the next place, and say, 1 I carry, and 6 is 7, and 5, is 12, and 4 is 16, so I set the 6 below the line, and in regard there is not another place to reckon the one, I bear in minde withall, I set that 1 a place neerer the left hand, and so the total is 16630 l. the sum of those four debts.

Q. What if you have numbers of several kinds or denominations to add together?

A. I must set down each number under the denomination of the same kinde, as pounds under pounds, shillings under shillings, and pence under pence, &c. and the like is to bee observed of weight, measure, or any other kinde.

Q. And how must they bee added together?

[Page] A. I must begin with the smallest denomination, which is next the right hand, and count all those figures together, and consider how many of the next denomination is contained in them, and carry so many unites to the next place, and set down the over-plus (if there bee any) right under beneath the line, and it there bee no over-plus, I set a cipher in the place; And the like I observe in every several denomination.

Q. Shew two or three examples of several denominations.

A. First, then for pounds, shillings, and pence, take this,

l. | s. | d.
365 | 6 | 8
456 | 7 | 6
567 | 8 | 4
1389 | 2 | 6

[Page] Then beginning with the smallest denomination toward the right hand, which is pence, I say, 4 and 6 is 10, and 8 is 18, which is 1 shilling and 6 pence, the 6 pence I set down in its place below the line, under its own denomination, and carry the one shilling in my minde to the next place, which is the place of shillings, and say, 1 I carry and 8 is 9, and 7 is 16, and 6 is 22, that is one pound and two shillings, the 2 odd shillings I set in its place under its own denomination below the line, and carry one pound in minde to reckon with the pounds, then I come to the first place of pounds, and say, 1 I carry and 7 is 8, and 6 is 14, and 5 is 19, so I set down 9, and carry 1, then I say, one [Page] I carry and 6 is 7, and 5 is 12, and 6 makes 18, the 8 I set down, and carry one, and then I say, one I carry and 5 is 6, and 4 is 10, and 3 is 13, the odd 3 I set right under, and the one ten I set one place further towards the left hand, so the total is 1389 l. 2 s. 6. d.

Q. What is your next example?

A. Take this, of haberdupoize weight, wherein note that 16 ounces make a pound, 28 pound make a quartern, 4 quarterns make a hundred weight, and 20 hundred make a tun weight:

Example.

tun. | C. | (q)quartern. | l. | o℥
123 | 9 | 3 | 16 | 10
234 | 8 | 2 | 12 | 8
345 | 7 | 1 | 08 | 6
703 | 5 | 3 | 09 | 8

[Page] Where as before, I begin with the least denomination next the right hand, and say, 6 and 8 is 14, and 10 is 24, which is one pound and eight ounces, the 8 ounces I set down below, and I carry one in minde to the pounds, then I say, one I carry and 8 is 9, and 12 is 21, and 16 is 37, that is one quartern, and 9 pound, the 9 I set down and carry one, and say, one that I carry, and one is two, and two makes 4, and 3 is 7, that is one hundred and three quarterns, the three quarterns I set down, and carry one, and say further, one I carry and 7 is 8, and 8 is 16, and 9 makes 25, that is one tun and five hundred, the five hundred I set down, and carry one to the tuns, and say, one that I carry and 5 is 6, and 4 [Page] is 10, and 3 makes 13, then I set down 3, and carry one, saying, one I carry and 4 is 5, and 3 is 8, and 2 makes 10, now being it is just 10, I set down a cipher, and carry one, saying, one I carry and 3 is 4, and 2 is 6, and one makes 7, which I set down in its place below the line, and so the total is

tun. | C. | (q)quartern | l. | o℥.
703 | 5 | 3 | 09 | 8

as in the example.

A third Example shall bee of liquid measure, in which note, that one tun is 4 hogs-heads, one hogs-head is 63 gallons, one gallon is 8 pints.

tuns. | hhd. | gal. | pints.
234 | 3 | 24 | 6
345 | 2 | 21 | 4
456 | 1 | 18 | 2
1036 | 3 | 01 | 4

[Page] Here I only set down the example, and cast it up, without describing the work, to move the learner to take some pains to do the like for his practice.

Substraction.

Q. Tell mee now what Substraction teacheth?

A. Substraction teacheth to abate, or withdraw a lesser sum or number out of a greater, and to shew the remainder or over-plus.

Q. How is that performed?

A. First I set down the greater sum or number uppermost, and under it I draw a line, then I set the lesser sum under the line, observing to set each figure in its due place, under the greater sum, and then I begin [Page] at the first place, and abate the lower figure out of the higher, setting the remainder under it, a line being first drawn to separate them.

Q. But what if the lower figure bee greatest, how then shall it bee abated from the higher?

A. Then I must borrow one of the next place, which here signifies the value of ten, and abate it from ten, and the uppermost figure added together, and set down the over-plus, or which is all one, abate it from ten, and adding the over-plus with the uppermost figure, set the same down beneath the line for remainder.

Q. What is next to bee done?

A. Then for the unite I borrowed, I add one to my lower figure in the next place, abating [Page] the same out of the figure over it, doing in all respects as before.

Q. Give an example hereof.

A. Then thus, If I have borrowed 4567 l. of which I have repaid 3675, what is behinde?

borrowed | 4567 l.
repaid | 3675.
rest behinde | 0892
proof | 4567

Where having placed my numbers duly under each other, I begin as before at the right hand, and say, 5 out of 7, there rests 2, which 2 I set underneath, as in the example, then I come to the next figure, saying, 7 from 6, I cannot, wherefore I borrow one of the next [Page] place, which signifies ten here, and so abate 7 out of 16, and set down the rest, which is 9 below the line; and insomuch as I borrowed one, therefore I carry one in minde, and say, in the next place, one that I borrowed and 6 is 7, which I should abate from 5 over it, which seeing I cannot do, I borrow one as I did before, and say, 7 out of 15, there rests 8, which I set underneath the line, and go on as before, saying, one that I borrowed, and 3 is 4, which being abated from the 4 above it, rests nothing to set below, so that there remains behinde 892 l. as in the example, the proof hereof is by adding the sum paid, and the remainder together, if they make up the sum borrowed, it is right, or else not.

[Page] Q. But when you have a sum of several denominations to substract from another, how do you then?

A. As in Addition I began at the smallest denomination to add, so here I begin with the same to substract, abating the lowest from the highest.

Q. What if the upper figure bee the least?

A. Then I borrow one of the next denomination, and considering how many of the smaller is contained in one of those, I abate my figure or number to bee abated, out of that which I borrowed, and the uppermost number being added together, and set the over-plus below the line for the remainder.

Q. Shew by an example or two what you mean?

A. To substract 8 d. from [Page] 1s. 4d. I say 8d. from 4d. I cannot, then I borrow one shilling, being the next place, which is 12d. to which I add the 4d. that is above the line, it makes 16d. and say, 8d. from 16d. rests 8d. Again, 17s. from 2l. 12s. I say, 17s. from 12s. I cannot, but 17s. from 1l. 12s. or 32s. there rests 15s. then considering I borrowed one pound, I say one pound that I borrowed from 2l. rests one pound, so that there rests 1l. 15s. to set below the line, And I must alwaies remember to reckon the one I borrowed to the figure that is to bee substracted in the next place.

Q. Shew this by an example or two of several denominations.

A. Then here is one, if I abate 345l.—16s,—8d. from [Page] 476l.—13s.—4d. I would see what remains.

I set my sums thus,

| l. | s. | d.
| 476 | 13 | 4
| 345 | 16 | 8
rest | 130 | 16 | 8
proof | 476 | 13 | 4

Then beginning with the least denomination, which is pence, I say, 8d. from 4d. I cannot, but I borrow one of the next denomination, which is shillings, and say, 8d. from 1s. 4d. and there rests 8d. which I set under the pence, and then I say, one shilling I borrowed and 16 makes 17s. from 13s. I cannot, but I borrow one pound, and say, 17s. from 1l. 13s. rests 16s. then I say, 1l. [Page] that I borrowed, and 5l. is 6l. from 6l. rests nothing, so I set down a cipher in this place under the line, and go forward, saying 4 from 7, rests 3, which I set down, and then say, 3 from 4, rests 1, which I set beneath; and so there rests 130l. 16s. 8d.

And as in Addition I left one example onely cast up, for the learner to pause upon himself, and to imitate, so here I do the like.

| tun | C. | (q)quartern | l. | o℥
| 345 | 11 | 2 | 24 | 4
Substr. | 256 | 13 | 3 | 20 | 8
rests | 088 | 17 | 3 | 03 | 12
proof | 345 | 11 | 2 | 24 | 4

Multiplication.

[Page]

Q. What doth Multiplication teach?

A. Multiplication teacheth after a brief & compendious way, to increase or augment any number, by so many times it self, as is any number propounded, as 4 times, 10 times, &c.

Q. What is considerable in this Rule?

A. Three numbers are specially considerable, to wit, the multiplicand, or number that is to bee multiplied, secondly, the multiplier, or number wee multiply by, and thirdly, the product, which is the number produced by the multiplication of those two numbers each by other.

[Page] Q. How many times doth the product contain the multiplicand?

A. Just so many times as there is unites in the multiplier.

Q. How is Multiplication done?

A. First I set down the multiplicand, which customarily is the greater number, and under it I set the multiplier, each figure in its due place, and draw a line underneath, then I begin at the first figure of the multiplier toward the right hand, and multiply it by the first figure of the multiplicand, and set the product right under it beneath the line, if it exceed not nine.

Q. If it exceed nine, what then?

A. Then I must keep in minde how many tens is in it, and carry so many unites to the next place, and set down the odd figure that is more than even [Page] tens underneath, but if it bee even tens, then set down a cipher underneath.

Q. And what is then to be done?

A. Then I multiply the said first figure of the multiplier by the second figure of the multiplicand, and to the product add the unites reserved in my mind, & then do in all respects as I did before, and so I continue my work, till I have multiplied the first figure of the multiplier by all the figures of the multiplicand in order.

Q. And what do you next?

A. Then I multiply the second figure of the multiplier by all the figures of the multiplicand, in like sort as I did the first, only I must observe to set my first place in this second work, one place nearer the left hand, [Page] that it may fall right under the figure I multiply by.

Q. What is the reason of that?

A. Because every unite in the second place signifies 10, in the third place, 100, &c.

Q. Is this order then to be kept in a sum of many figures or places?

A. Yea, the same order is to bee observed in any sum, be the places never so many, I must still set my first figure right under the figure I multiply by, and then the rest in order toward the left hand.

Q. Having so set down all your figures, what remains further to bee done?

A. Only to add the several numbers together in order, beginning still at the first place next the right hand.

Q. Give one Example.

[Page] A. Let this bee it then.

2345

234

9380

7035

4690

548730

Where first I say, 4 times 5 is 20, where I set a cipher below the line, and carry 2, then I say, 4 times 4 is 16, and 2 that I carried is 18, the 8 I set down below, and carry 1, then 4 times 3 is 12, and 1 that I carried is 13, then I set down 3, and carry one, 4 times 2 (or 2 times 4, for it is all one) makes 8, and one that I carried is 9, which I set down in its place, and cancel the first figure of my multiplier, with a dash through it, to signifie that it hath done its office, [Page] then I begin with the next figure, saying, 3 times 5 is 15, the five I set down right under the 3 I multiply by, and carry one in minde, then I say, 3 times 4 is 12, and one that I carried is 13, the 3 I set down in the second place, and carry one, and say, 3 times 3 is 9, and one I carried is 10, where I set down a (0) and carry one, then I say, 3 times 2 is 6, and one I carry is 7, which I set down, and cancel my second figure of the multiplier, and begin with the third, saying, 2 times 5 is 10, then I set down a cipher in that place right under my multiplier 2, and carry one in mind to the next place, then I say, 2 times 4 is 8, and one I carried is 9, which I set down in the next place, in order, then I say, 2 times 3 is 6, [Page] which I set in its due place, and lastly, I say, 2 times 2 is 4, which I write down also, so have I multiplied all my figures of the multiplier, by all the figures of the multiplicand, there remains to add up all into one sum, which to do I begin at the right hand, and work as in Addition, and so the product is 548730, as in the example.

Here is another Example for Imitation.

963852

3741

963852

3855408

6746964

2891556

3605770332

[Page] Q. What proof is for Multiplication?

A. The truest proof is by Division, but it is ordinarily proved thus, they make a cross X

And then cast away so many nines as can bee found in the multiplicand, and set the remainder on the upper side of the cross, and do the like with the multiplier, & set the remainder under the cross, then multiply the 2 remaidners 1 by another, and cast out the nines out of the product of them, setting the rest at one side of the cross; and last of all cast out the nines out of the product of the Multiplication, and mark the rest, if it be like that which is placed on the side of the cross, it appears [Page] to bee right, or else it is not well done.

A Table for Multiplication to bee got by heart.

2 times | 2 is | 4
2 | 3 | 6
3 | 4 | 8
2 | 5 | 10
2 | 6 | 12
2 | 7 | 14
2 | 8 | 16
2 | 9 | 18
3 times | 3 is | 9
3 | 4 | 12
3 | 5 | 15
3 | 6 | 18
3 | 7 | 21
3 | 8 | 24
3 | 9 | 27
4 times | 4 is | 16
4 | 5 | 20
4 | 6 | 24
4 times | 7 is | 28
4 | 8 | 32
4 | 9 | 36
5 times | 5 is | 25
5 | 6 | 30
5 | 7 | 35
5 | 8 | 40
5 | 9 | 45
6 times | 6 is | 36
6 | 7 | 42
6 | 8 | 48
6 | 9 | 54
7 times | 7 is | 49
7 | 8 | 56
7 | 9 | 63
8 times | 8 is | 64
8 | 9 | 72
9 times | 9 is | 81

Division.

[Page]

Q. Now shew mee what Division teacheth?

A. Division teacheth to finde how many times one number is contained in another number.

Q. How many numbers are to bee noted in any Division?

A. Three, namely, the dividend, or number to bee divided, secondly, the divisor, or number dividing, thirdly, the quotient, which sheweth how often the divisor is contained in the dividend.

Q. In what manner is Division performed?

A. First I set down my dividend, and under it I place my divisor, in such sort, that the figures next the left hand stand [Page] right under one another, and so each following place in order, except the divisor bee a greater number than so many figures of the dividend as stand over it, for then the divisor must bee removed a place nearer the right hand.

Q. And what do you then?

A. Then I draw a crooked line to the right hand of my figures, to place my quotient beyond, and I consider how often I can take the divisor, out of the number over it, and set the number of times in the quotient, and multiplying the said quotient figure by the divisor, I substract the product from the figures above the divisor, setting the remainder over head, cancelling the other figures that were over the divisor, and also the divisor.

[Page] Q. And how proceed your further?

A. Then I remove my divisor one place nearer the right hand, and consider as before how often I may take it out of the figures over head, and work in all points as before.

Q. If there bee many removings of the divisor, is that order still to bee observed?

A. Yea, where the divisor can bee substracted once or oftner out of the dividend.

Q. But what if you cannot take the divisor out of the figures over it?

A. I must then place a cipher in the quotient, and cancel the divisor, and remove it a place nearer the right hand, without cancelling the figures over head, and continue the work as before.

[Page] Q. What else is to bee observed in Division?

A. If the divisor have any ciphers in the first places, they may bee placed under the first places of the dividend, and divide only by the other figures, till I come to those ciphers.

Q. What must be done with the number that remains after the division is ended?

A. If any remainder be, I set it after the quotient, and the divisor under it, with a line drawn betwixt them, to express it in a fraction.

Q. Give an Example or two in Division.

A. Take this for one, to divid 30038 by 23, I set it down thus,

30038 (

23

[Page] Then having drawn a crooked line, to set the quotient in, I consider how often I can have my divisor, 23 in the number over it, which is 30, which I can have but once, therefore I say, once 2 is 2 from 3 that is over it, and there remains I, which I set over the 3, and cancel the 3, aud also the 2 under it, and it stands thus,

1

30038 (1

23

Then I say, once 3 is 3, from 10 that is over it, and there rests 7, and stands thus,

17

30038 (1

23

Then I remove the divisor one place nearer the right hand,

[Page] And it stands thus,

17

30038 (1

233

2

Now I consider again as before, how often I can take 23 out of 70 that is over it, which I finde I may do 3 times, therefore I put 3 down in the quotient, and say, 3 times 2 is 6, 6 out of 7, rests one, which I set over the 7, and cancel the 7, and the 2 under it, and say, 3 times 3 is 9, from 10 over it, rests one, and that I set over head, and cancel the 10, and the 3 under it, And then it stands thus,

1

171

30038 (13

233

[...]

[Page] Then I remove the divisor again, And it stands thus,

1

171

30038 (13

2333

22

Then I consider that I cannot take my divisor 23 out of the number over it being but 13, so I set a cipher in the quotient, and cancel the divisor, and remove it one place more, and let the figures over it stand as they were, And then it stands thus,

1

171

30038 (130

23333

222

Now I consider again how often I may take my divisor out [Page] of the number over it, which I finde I may do 6 times, wherefore I set 6 in the quotient, and say, 6 times 2 is 12, from 13 that is over it, and there rests one, which I set over head, and cancel 13, and 2 under it, And so it stands thus,

1

1711

30038 (1306

23333

222

Then I say 6 times 3 is 18, from 18 that is over it, rests nothing, And the whole work stands thus,

1

1711

30038 (1306

23333

222

Here also I set an Example [Page] or two, for Imitation.

Example.

1

4231

20673 (5

456780 (18271

255555

2222

(1

135

49253 (5

3692580 (82057

455555

4444

Q. What proof is for Division?

A. This, multiply the quotient by the divisor, and to the product add what remained after the Division was ended, if [Page] any such remainder, if then it amount justly to the dividend, it is well done, or else not.

Q. You said Multiplication was best proved by Division, how is that proof done?

A. By dividing the product of the Multiplication by the multiplier, if then, the quotient comes justly to the multiplicand, it is well done, or else you have failed.

Q Now having spoken of the five first kinds or rules of Arithmetick, let us come to the application of them, to use, therefore now tell mee what use may bee made thereof?

A The uses are so many, and so necessary, that it would require a large volume to declare them, and I resolve brevity.

Q. Yet I desire to hear some of [Page] them, where the same may bee made profitable?

A. Then for as much as many of the applications hereof require Reduction, I think it needful to begin first with it.

Reduction.

Q. What doth Reduction teach?

A. It teacheth to turn or change numbers of one denomination, into another denomination, as pounds into shillings, or shillings into pence, or pence into farthings; or contrarily, farthings into pence, pence into shillings, or shillings into pounds, &c. The like may bee said of weight, measures, time, &c.

Q. How do you turn pounds into shillings?

[Page] A. I consider 20s. is one pound, therefore there must bee 20 times so many shillings, as there is pounds, so that multiplying the number of pounds by 20, the product shews the number of shillings.

Q. Is the worth or value of the things so reduced (changed or) altered?

A. No, only the number and denomination is changed, but the first value remaineth still, like as 20s. is just equal to one pound, and 12d. equal to one shilling, &c.

Q. How change you a number of shillings into a number of pence?

A. For as much as 12d. is in each shilling, there must bee 12 times so many pence as there is shillings, therefore I [Page] multiply the number of shillings by 12, and the product is my desire.

Q. How reduce you farthings into pence?

A. Seeing 4 farthings make but one penny, therefore there is but a fourth part so many pence as there is farthings, wherefore I divide the number of farthings by 4, and the quotient is my desire.

Q. And how will you turn pence into shillings, and shillings into pounds?

A. I divide pence by 12, to turn them into shillings, and shillings by 20 to turn them into pounds.

Q. Why so?

A. Because 12d. is but one shilling, and 20s. is but one pound.

[Page] Q. Is there the like reason for weight, measures, time, &c?

A. Yea, altogether, for I am to consider how many of the one sort or denomination will make one of the other, and so multiply by that number, to turn the greater parts into the smaller, or divide by the same number to turn the smaller parts into the greater.

Q. Give an Example in weight.

A. To turn 256 tuns, into C. (q)quartern. l. o℥. I do thus,

First I multiply 256 by 20, because there are 20C. in each tun, and then the product is Cds. then to turn those into (q)quartern I multiply by 4, and that product is my desire; to turn that into ls. I multiply by 28, for that each (q)quartern is 28l. and the product is pounds, then to turn [Page] those pounds into ounces, I multiply by 16, and the product is the solution, &c.

Example.

256 tuns.

20

5120 C.

4

20480 (q)quartern s.

28

163840

4096

573440 l.

16

3440640

57344

9175040 o℥.

[Page] To turn ounces into pounds, divide by 16, the quotient is ls. divide pounds by 28, the quotient is (q)quartern s, divide the (q)quartern s by 4, the quotient is C. divide Cds. by 20, the quotient is tuns, as may be proved by working the Example above backwards, and this I judge will suffice to explain this part of the Rule.

Q. Is there no kinde of Reduction that requireth both Multiplication and Division?

A. Yes, when a certain number of the one sort makes another number of the other sort; As for example, when 3 marks makes 2 pounds, then to turn marks into pounds, I must multiply by 2, and divide by 3, the quotient is my desire, but to turn pounds into marks, I must multiply by 3, and divide by 2, [Page] and the quotient is my desired number.

Q. Shew the like instance in long measures.

A. Four Ells is equal to five Yards, therefore any number of Yards given, to know how many Ells it contains, I multiply by 4, and divide by 5, but if the number given bee Ells, and I would know how many Yards it is, I multiply by 5, and divide by 4, and the quotient is my desire.

Q. May the like Reduction bee made in other things?

A. Yea, for 3l. starling is worth 5l. Flemmish, therefore any number of pounds starling being multiplied by 5, and the product divided by 3, the quotient shews the number of pounds Flemmish, or-any number of [Page] pounds Flemmish, being multiplied by 3, and the product divided by 5, declares the number of pounds starling in the quotient, the like proportion is in Flemmish Ells, and English Ells, because 5 Ells Flemmish is but 3 Ells English, &c.

Q. Is there the like reason for other things?

A. Yea, whether coins, weights, measures, &c.

Q. But what if any remainder be in the Division?

A. It must bee exprest in a fraction as before.

The Rule of Three.

Q. What other use is it for?

A. The next use I shall apply it to, is the Rule of proportion, commonly called, the Rule of Three, and for the usefulnes [Page] of it, the Golden Rule.

Q. What doth the Rule of Three teach?

A. It teacheth by 3 known numbers to finde a fourth, either in continual proportion, or discontinual proportion.

Q. What call you continual proportion?

A. When the numbers are such as hold such proportion among themselves, that what proportion the first hath to the second, the like hath the second to the third, and the third to the fourth, as are 1, 3, 9, 27, and also 2, 4, 8, 16, &c.

Q. How finde you a number in continual proportion?

A. I multiply the second number by it self, and divide the product by the first number, and the quotient is my desire.

[Page] Q. But how agrees this with your former words, where you said, that this Rule teacheth by three numbers to finde a fourth?

A. Very well, for the second number here is taken twice, that is, both for the second and third, and the fourth number that is found out, is the third number in continual proportion.

Q. Shew an Example.

A. Thus, if 3 give 9, what gives 9, here I multiply 9 by 9, gives 81, which divided by 3, the first number, the quotient is 27, being the third number in continual proportion, so that the 3 numbers are here, 3. 9. 27.

Q. And how finde you a fourth number in continual proportion?

A. Here I may either multiply [Page] the third by it self, and divide by the second, or else multiply the second and third together, and divide by the first, and the quotient is my desire.

Q. Give an Example.

A. Take the former numbers, and first the former way, I multiply the 3 by it self (viz.) 27 by 27 thus,

27 |
27 |
189 | 0
54 | 729 (81
729 | 99

comes 81 for the fourth.

Or secondly, multiply the second and third together, and divide by the first thus,

27 | 00
9 | 243 (81
243 | 33

comes 81 as before.

Q. What is discontinual proportion?

A. Where the first and second hold like proportion each to other, as the third and fourth do each to other, but the second and third hold not that proportion together.

Q. How is the fourth number in discontinual proportion found?

A. By multiplying the second number by the third, and dividing by the first, as before.

Q. What use may bee made of these proportionals thus found?

A. They may bee applied to many uses, according to the several imployments men are exercised [Page] in; as in Merchandizing, Measuring heights or distances, magnitudes or quantities, &c.

Q. How may they bee so applied?

A. Thus, if 3 yards of cloth cost 12s. what will 5 yards cost at that rate? here 3 yards is the first number, and 12s. the second, and 5 yards the third, therefore I multiply the second by the third, comes 60, which divided by the first, the quotient is 20s. my demand.

Q. How shall I know which number ought to bee first, that I may not mistake, and so work false?

A. You must observe that the first number and the third are of one kinde, and are, or ought to bee reduced to one denomination, [Page] and the second number is of that kinde which is sought in the question, so that the first and second declare the proportion, and the third number is annexed with the demand.

Q. Make this plainer by Example.

A. In the former Example 3 yards and 5 yards are both of one kinde, to wit, yards, the second or middle number is 12, and is of another kinde, namely, shillings, now 3 yards and 12s. declare the proportion, and are the first and second numbers; Now, 5 being that with which the demand is joyned (as what cost 5 yards) is the third number, and the fourth number (resolving the question) is of the same kinde the second is of, to wit, here shillings.

[Page] Q. What if the numbers bee of several denominations, as of l.— s.—d. &c?

A. They must bee reduced to one denomination first, before you apply them to the Rule, and the second (or middle) number must bee reduced into the denomination of its smallest parts, in which parts the question is resolved by the fourth number.

Q. Give an Example hereof.

A. If 3 yards and 3 quarters cost 11s. 3d. what cost 16 yards; here the first and third numbers being both measure, must bee turned into (q)quartern s, and the first will bee 15, and the third 64, the second or middle number must be turned into pence, and will bee 135d. now work by the Rule, multiply the second [Page] and third together, being set in order thus,

If 15 (q)quartern s. cost 135d. what cost 64 (q)quartern s?

64

540

810

8640

Divide the product by the first.

143

3190

8640 (576

1555

11

Comes 576 for solution, which are pence, like the middle number, which I reduce into shillings and pounds thus,

1

190 (0

576 (4 8 (2

122 (20

1

Comes for solution—2l.—8s. [...]d.

Q. And what do you further?

A. I reduce those smallest parts into a denomination of greater parts of its own kinde if I can, for the more easie estimation of their value.

The Backer Rule of Three.

Q. Is there any other manner of Work in the Rule of Three?

A. Yea, there is another manner of work called the Backer Rule of Three.

Q. Why is it called the Backer Rule of Three?

[Page] A. Because in the former wee multiply the second by the third, and divide by the first, but in this wee multiply the first by the second, and divide by the third number.

Q. What is the use of this Rule?

A. It serveth to finde out a number that holds proportion to the first, as the third doth to the second.

Q. Of what use is such proportional numbers?

A. The use is manifold, as by example may appear.

Shew 2 or 3 Examples.

A. First, if 5 men do a peece of work in 15 daies, how many men will do the like in 3 daies? here I multiply the first by the second, comes 75, which divided by the third, yeelds 25 in the quotient, being the number of men demanded.

[Page] Q. What is a second Example?

A. This, if a quantity of provision serve 40 men 30 daies, how many men will it serve 80 daies? here I multiply and divide according to rule, and finde 15 men.

Q. Give one example more.

A. Then thus, If 9 yards of cloath, of yard broad, make a man a suit & cloak, how much broad cloath of yard and half broad will make another so large; here I multiply 4 (q)quartern, the breadth of the first cloth, by 9 yards the length thereof, and divide by 6 (q)quartern the second breadth, and it yeelds 6 yards for the length of the second cloth.

Q. Is there any other use hereof?

A. Yea, many, which the studious may finde by practice, [Page] but these shall serve at present for an entrance.

Q. What if any thing remain after the Division is ended?

A. It must bee annexed to the whole number in the quotient, with the divisor under it, and a small line drawn between them, so expressing it in a fraction.

Numeration in fractions.

Q. Now tell mee what use fractions are of in Arithmetick?

A. They are of like use with whole numbers.

Q. And are there the same kinds or species in fractions, as in whole numbers?

A. Yea, only some put a difference in the order of teaching them, that the easiest may bee first taught.

[Page] Q. But what mean you by species?

A. I mean several kindes of working, or several Rules, as some call them.

Q. Then rehearse the order of Rules as they are taught.

A. Numeration, Reduction, Multiplication, Division, Addition, Substraction.

Q. What sheweth Numeration in fractions?

A. It sheweth how to set down, or express any fraction, part or parts of an unite.

Q. how is that done?

A. It is done by setting down two numbers one over another, with a line drawn betwixt them, whereof the lower number signifieth how many parts the whole unite is divided (or supposed to bee divided) into; and [Page] the uppermost number sheweth how many of those parts the fraction contains.

Q. How are those two numbers called?

A. The uppermost, (or number above the line) is called the numerator, and the other below the line is called the denominator.

Q. Shew an Example or two to explain this.

A. Three quarters is set down, with a 4 under the line, signifying the number of parts the unite is divided into; and 3 above the line, shewing how many of those parts the fraction expresseth or signifieth.

Q. Give another Example.

A. Five seventh parts is exprest by 5 above the line, and 7 below it, thus 5/7.

[Page] Q. Is the greatest number alwaies set lowest?

A. Yea, in such as are proper fractions.

Q. Are there then any improper fractions?

A. There are sometimes whole numbers or mixt numbers exprest in form of fractions, which are not properly fractions, because a fraction is alwaies lesser than an unite, but these are either equal to, or greater than an unite.

Q. Explain this by an Example or two.

A. Two halfs 2/2, three thirds 3/3, five fifths, 5/5, &c. are whole unites, onely exprest like fractions; also nine quarters is a mixt number exprest thus, 9/4, and signifies two unites and a quarter more.

[Page] Q. Why are such exprest like fractions?

A. For aptness, or for ease in working.

Q. What else is considerable in Numeration?

A. This, that as numbers increase infinitely above an unite, so fractions decrease or grow less infinitely under an unite.

Q. I remember you mentioned decimal fractions before, how are such exprest?

A. They are exprest by an unite, and 1, 2, 3, 4. or more ciphers below the line, according to the number of places, or parts the fraction is exprest in, and with figures and ciphers above the line, expressing the number of such parts that the fraction contains.

Q. Make this plain by an [Page] Example, two or three.

A. One half or 5 tenths is exprest by 5 above the line, and an unite with one cipher, signifying ten or tenths under the line thus 5/10.

Secondly, 1/4, or 25 hundreds, is writ with 25 above the line, and 100 under the line thus 35/100.

Q. How set you down 75 thousand parts?

A. Thus with a cipher, a 7, and a 5 above the line, and an unite and three ciphers below the line, 075/1000.

Q. Are decimals alwaies exprest thus?

A. They are often exprest by their numerator, onely separated from the unite place by a prick, and the denominator is understood to consist of so many [Page] ciphers, as there are places in the numerator, and an unite before them to the left hand.

Q. Shew mee one Example or two.

A. First, Five hundreths is writ with a cipher, and a 5 thus 05, where 100 is understood for denominator.

Secondly, 34 ten thousand parts is exprest thus 0034, where 10000 is understood for denominator.

Q. Is there any thing more herein to bee noted, before wee leave numeration?

A. Yea, that not an unite only may bee divided infinitely into fractions or parts, but also any of those parts or fractions may bee divided also infinitely into other parts, called fractions of fractions, and those also again [Page] subdivided infinitely, &c.

Reduction.

Q. Now tell me what is Reduction?

A. Reduction is a changing of numbers or fractions out of one form or denomination, into another.

Q. Why are they so reduced?

A. Either for more ease in working, or for the more easie estimation of the value of two or more fractions, either compared one with another, or adding them together, or substracting one from another.

Q. How are fractions of several denominations reduced to one denomination?

A. First multiply the denominators together, and set the product for a common denominator; [Page] then multiply the numerator of the first, by the denominator of the second, and set the product for a new numerator for the first fraction; and multiply the numerator of the second, by the denominator of the first, and set the product for the new numerator of the second fraction, and so are both those fractions brought into one denomination.

Q. Give an Example hereof?

A. Two thirds and three quarters, being so reduced, make 8/12 for 2/3, and 9/12 for 3/4, which yet still retain their first value, but are now both of one denomination.

Q. You have shewed how to reduce two Fractions into one denomination, but what if there bee three or more?

[Page]A. Then I must multiply all the denominators together, and set the product down so many times as there bee fractions, for a common denominator to them; and then multiply the numerator of the first, by the denominator of the second, and the product by the denominator of the third, and that by the denominator of the fourth, if I have so many, and so forward, and the product is a new numerator for the first fraction; then multiply the numerator of the second, by the denominator of the first, and the product by the denominators of the third and fourth, and so forward, if you have so many, and set the product for a new numerator for the second fraction, and multiply your third [Page] numerator by the first and second denominators, and the product by the denominator of the fourth, if you have to many, and that product is your third numerator; then if you have so many, multiply the numerator of the fourth by the other three denominators, the product is a new numerator for the fourth fraction, &c.

Q. Must this order then bee observed still, when you have many fractions?

A. Yea, alwaies multiply all the denominators together for a new denominator, and one numerator by all the other denominators, except its own, the product is a new numerator for that fraction whose numerator was taken to multiply by.

[Page] Q. Is there any other form of reducing to one denomination?

A. Yea, several varieties.

Q. What is one way?

A. This is one, when you have found a new denominator as above, then divide the same by the denominator of any of your fractions, and multiply the quotient by the numerator of the same, and the product shall bee a new numerator for that fraction, &c.

Q. What is another way?

A. If the lesser denominator will by any multiplication make the greater, then note the multiplier, and by it multiply the numerator over the lesser denominator, and in place of the lesser put the greater denominator, and so it is done without any of the other fractions.

[Page] Q. What other sort of Reduction is there?

A. A second sort is when fractions of fractions are to bee reduced to one denomination.

Q. How is that done?

A. By multiplying the numerators each into other, and setting the product for a new numerator, and in like sort multiply all the denominators each into other, and take that product for a new denominator, and then they express it in the parts of a simple fraction.

Q. What if I have a mixt number of unites, and parts to bee reduced into fraction form?

A. Multiply the unites or whole number by the denominator of the fraction, and thereto add the numerator of the fraction, and set the offcome [Page] above the line over the said denominator.

Q And how reduce you such an improper fraction into its unites and parts?

A. I must divide the numerator by the denominator, and the quotient shews how many unites it contains, and the remainder, if any bee, is the numerator of a fraction, over and above the said unites in the quotient, to which the divisor is denominator.

Q. How is a whole number reduced into the form of a fraction?

A. By multiplying it by that number, which you would have denominator to it.

Q. What is next in Reduction of Fractions?

A. To reduce a fraction into its smallest or least t [...]rms.

[Page] Q. What bee the terms of a Fraction?

A. The terms bee the numerator and denominator whereby it is exprest.

Q. What mean you by greatness and smalness of terms?

A. By great, I mean, when a fraction is exprest in great numbers, as 480/960 which in its smallest terms is (1/2) one half.

Q. How are such reduced into their smallest terms?

A. If they be both even numbers by halfing them both so often as you can, but if they come to bee odd, or either of them odd, then by dividing them by 3, 5, 7, 9, &c. which will divide them both, without any remainder, and take the last numbers for the terms of the fraction.

[Page] Q. But is there no way to discover what number would reduce a fraction into its smallest terms, but by halfing or parting in that sort?

A. Yes, thus, divide the denominator by the numerator, and if any thing remain, divide the numerator by it, and if yet any thing remain, divide the last divisor by it, and so do till nothing remain, and with your last divisor, which leaves no remainder, divide the numerator of the fraction, and the quotient is a new numerator, and divide the denominator in like sort by it, and the quotient is a new denominator.

Q. What if no number will divide them evenly, till it come to one?

A. Then the fraction is in its [Page] smallest terms already.

Q. How reduce you fractions of one denomination, into another denomination?

A. I multiply the numerator by the denominator, into which I would reduce it, and divide the product by the first denominator, and the quotient is the new numerator.

Q. Give an Example of this.

A. If 3/4 bee to bee turned into twelfth parts, I multiply 12 by 3 comes 36, which I divide by 4, the quotient is 9, so it is 9/12, equal to 3/4.

Multiplication in Fractions.

Q. How do you multiply in Fractions?

A. I multiply the numerators together for a new numerator, [Page] and multiply the denominators together for a new denominator, which numerator and denominator, so found, express the product of that multiplication.

Q. What other thing of note it observable in Multiplication of Fractions?

A. That two Fractions multiplied together, the product is lesser than either of the fractions.

Q. How comes that to bee, seeing the very name of Multiplication signifies to augment or increase a thing manifold, or many times?

A. That is true, in whole numbers. where a number is increased by so many times as the multiplier contains unites, but in fractions wee must note, that the multiplier being less than [Page] one, it makes the product lesser than the multiplicand; for so often as the multiplier contains unites, just so often doth the product contain the multiplicand, therefore (in fractions) feeing the multiplier doth contain but such a part or parts of an unite, even so the product doth contain but the like part or parts of the multiplicand.

Division in Fractions.

Q. How is Division in Fractions performed?

A. Thus, I multiply the numerator of the dividend by the denominator of the divisor, the product is a new numerator, then multiply the numerator of the divisor by the denominator of the dividend, the product is a [Page] new denominator, and this third fraction is the quotient of that division.

Q. Shew an Example.

A. If I divide 3/ [...] by 1/ [...], the quotient is 3/4.

Q. How comes it to pass that in Division by a fraction the quotion is greater than the dividend?

A. Because the divisor being lesser than an unite, is consequently oftener contained in the dividend, for alwaies the quotient shews how often the divisor is contained in the dividend.

Q. How is a whole number divided by a fraction?

A. I multiply the whole number by the denominator of the fraction, and set the product for numerator, and for a [Page] denominator, I set the numerator of the fraction.

Q. How is a fraction divided by a whole number?

A. By multiplying the denominator by the whole number, setting the product for a new denominator, without changing the numerator at all.

Q. May this bee done otherwise?

A. Yea, if the whole number will evenly divide the numerator of the fraction, then divide it by it, and set the quotient for numerator, and change not the denominator at all.

Addition in Fractions.

Q. How are two or more fractions added together?

A. If they bee of one denomination, [Page] then add the numerators together in one, and under it place the common denominator, and that fraction represents the total of that Addition.

Q. But what if they bee of several denominations?

A. Then I first reduce them to one denomination, and then add their numerators together.

Substraction in Fractions.

Q. How substract you one fraction from another?

A. If they be not of one denomination, I reduce them to one, and then substract the lesser numerator from the greater, and set the rest for a new numerator over the common denominator.

[Page] Q. But what if you bee to substract a mixt number from another, or from a whole number?

A. I may as before reduce them to one denomination, and then substract one numerator from the other, or I may substract the fraction of it from an unite converted into the same denomination, and carry one in minde to the whole number, and then substract it out of the other whole number.

Several other means may be used in these works of fractions, but I forbear to mention them, for brevity sake, and come to the Rule of Three.

The Rule of Three in Fractions.

[Page]

Q. How work you the Rule of Three in Fractions?

A. First I place the numbers as was shewed in whole numbers, and then multiply the numerator of the first by the denominator of the second, and the product by the denominator of the third, and the product thereof must bee my divisor; then I multiply the denominator of the first, by the numerator of the second, and the product by the numerator of the third, and the offcome is my dividend, then I divide the dividend by the divisor, and the quotient is the fourth number, and answereth the question.

Q. Is there any other way to [Page] work the Rule of Three?

A. Yea divers, whereof this is one, finde the divisor or first number, as before, then for the second number, take the numerator of the second fraction, and for the third number, take the number that cometh by Multiplication of the numerator of the third by the denominator of the first fraction, and then work as in whole numbers.

Q. What proof is there for the Rule of Three?

A. Multiply the second and third numbers together, and multiply the first and fourth numbers together, and if the products be equal, it is right, or else it is not right.

Q. Give an Example in the Rule of Three.

[Page] A. If 4/5 of an Ell cost 3/ [...] of a pound, what is 1/ [...] of an Ell worth? here I multiply 4 by 8, comes 32, & that by 3 comes 96 for divisor, then multiply 5 by 3 is 15, and that by 2 makes 30, for dividend or numerator, so it is 30/ [...]6, or in the smallest terms, 5/16 of a pound.

Q. Examine this by the proof.

A. Multiply 3/ [...] by 2/ [...] comes 6/24, or 1/4, again multiply 4/5 by 5/16 comes 20/80, or 1/4 likewise.

The Backer Rule of Three.

Q. How work you the Backer Rule of Three in Fractions?

A. Thus, I multiply the numerators of the first and second numbers together, and the offcome by the denominator of the third, and the product is my dividend, [Page] then I multiply the denominators of the first and second together; and the offcome by the numerator of the third, and that product is my divisor, wherewith I divide my dividend, and the quotient resolves the question.

Q. Show an Example hereof.

A. If my friend lend mee 4/ [...] of a pound for 2/3 of a year, or 8 months, how long ought I to lend him 2/ [...] of a pound to requite his courtesie?

A. I say, if 4/ [...] give 2/ [...], what shall 2/ [...] give, where I multiply 4 by 2 yeelds 8, and 8 by 3 comes 24 for dividend, then I multiply 5 by 3 comes 15, and that by 2 gives 30 for divisor, and so placing the dividend over the divisor, I have 24/ [...], or in the smallest terms 4/ [...] of a year, for the resolution [Page] of the question, which is the time I ought to lend him 2/ [...] l. to requite his courtesie.

Q. I observe, that where the Rule of Three Direct would give the fourth number more than the second, this gives it less, and where it would give it less, this gives it more, what is the reason of that?

A. We are to consider in this Rule, that the less mony lent, requires the more time forbearance to ballance the other; and in like sort, the less breadth a thing is of, the more length it requires to make it equal with a quantity of more breadth; in like manner, the more men that are imployed to do a peece of work in, the less time they will do it; so the fewer men that are to live upon a quantity of provision, the longer time it will last, &c.

The Double Rule of Three.

[Page]

Q. Is there any other form in the Rule of Three, besides the above said?

A. Yea, there is divers which resolve double questions, and therefore are called, the Double Rule of Three, or the Rule of Three composed of 5 numbers.

Q. What manner of questions doth this Rule resolve?

A. Either such as are uncompound, or compound.

Q. What mean you by uncompound, or compound?

A. I mean by uncompound, such as are done by the Rule of Three Direct at two workings, by compound, such as are done by once working by the Rule of Three Direct, and another by the Backer Rule of Three.

[Page] Q. What is one question of the former sort?

A. If 5 men in 6 daies earn 3l. how much will 10 men earn in 12 daies?

Q. And how is this done?

A. Either by two several workings by the Rule of Three, or it may be resolved at once.

Q. First shew mee how it is done at two workings?

A. First I say, if 5 men earn 3l. what will 10 men earn, it gives 6l. for 6 daies; then again, if 6 daies gives 6l. what gives 13 daies? facit 12l.

Or secondly, I may say 6 daies gives 3l. what gives 12 daies? comes 6l. for 5 men; and then, if 5 men earn 6l. what will 10 men earn? and it comes to 12l. as before.

Q. How is this performed at one working?

[Page] A. Thus, I say according to the question, If 5 men in 6 daies earn 3l. what earn 10 men in 12 daies, then I multiply the first by the second, viz. 5 by 6 comes 30, which I keep for my divisor, then I multiply the other three numbers each into other, comes 360 for my dividend, which being divided by my divisor, gives 12 in the quotient, which signifies so many pounds, being of the denomination of the middle number.

Q. Now shew an Example of a compound question.

A. Take this, if 5 men in 6 daies earn 3l. in how long time will 3 men earn 5l. where first I say, if 3l. give 6 daies, what will 5l. give? comes 10 daies; then by the Backer Rule of Three, If 5 men bee 10 daies in [Page] earning it, how long will 3 men bee? and it gives 16 2/ [...] daies.

Q. And how is this done at one working?

A. I say, if to earn 3l. 5 men bee imployed 6 daies, then to earn 5l. by 3 men, what time is required? where I multiply the first number and the fifth number together (being the least sum of mony, and the least number of men) comes 9 for divisor, then I multiply the other three numbers together, comes 150 for my dividend, which being divided, gives 16 in the quotient, and 6 remains, which abreviated or reduced to the least terms, is 2/3 so the whole is 16 2/3 daies, as before.

Q. What is another question of this sort?

A. This, 30 men work 40 [Page] yards of Arras in 6 daies, in how long time will 15 men work 80 yards of the like Arras?

Q. How is this done at two workings?

A. First I say, if 40 yards require 6 daies, how long time will 80 yards require? (by the Rule of Three Direct) comes 12 daies; then I say, if 30 men bee 12 daies about it, how long will 15 men bee about it by the Backer Rule? and it comes to 24 daies.

Q. How is this performed at one working?

A. I say thus, if 40 yards require 30 men for 6 daies, then to do 80 yards by 15 men, how long time will it require? where I multiply the first number, being 40 yards, by the fifth number, being 15 men, that is the [Page] least number of yards, and the least number of men together, that is, 40 by 15 comes 600 for my divisor, then I multiply the other three numbers together, (viz.) 30 by 6 comes 180, and that by 80 comes 14400 for my dividend, which I divide by my divisor, the quotient is 24 as before, being so many daies, according to the denomination of the middle number.

Thus having briefly and plainly explained the Rules, I shall set down some questions, with their resolutions, omitting the work, that the young learner may practise himself in them, or such like, to make himself the more ready in this Art.

In Reduction.

[Page]

In 264l. how many shillings is there? facit 5280s.

In 10560s. how many pence is there? facit 126720d.

In 63360d. how many shillings is there? facit 5280s.

In 10560s. how many pounds is there? facit 528l.

In 2650 Ells Flemmish, how many Ells English? facit 1590.

In 3180 English Ells, how many Flemmish Ells is there? facit 5300.

In the Rule of Three.

If 42 yards cost 28l. what cost 30 yards? facit 20l.

If 20l. buy 30 yards, how much will 28l. buy? facit 42 yards.

[Page] If 30 yards cost 20l. what will 42 yards cost? facit 28l.

If 28l. buy 42 yards, how much will 20l. buy? facit 30 yards.

I have varied this question purposely to shew the learner how hee may do the like with any other.

If 3 marks be worth 2l. what is 369 marks worth? facit 246l.

If 20 nobles be worth 6l. 13s. 4d. what is 1000 nobles worth? facit 333l. 6s. 8d.

If 6 o℥. of cloves cost 2s. 6d. what cost 16 o℥. facit 6s. 8d.

If 1C weight cost 18s. 8d. what cost 12l. facit 2s.

If 6l. cost 1s. 6d. what cost 112l. facit 1l. 8s.

For the Backer Rule of Three.

If 5 men do a peece of work [Page] in 8 daies, how many men will do the like in 2 daies? facit 20 men.

If 20 men do a peece of work in 2 daies, in how long time will 8 men do the like? facit 5 daies.

If a quantity of provision serve 360 men 45 daies, how long will it serve 288 men? facit 56 [...]/ [...] daies, or 65 1/4 daies.

If 5 yards of cloth that is yard and half broad, make a man a gown, how much baize of yard broad will line it throughout? facit 7 1/2 yards.

If a foot of board be 12 inches long, and 12 inches broad, how much will make a foot of that board that is but 9 inches broad? facit 16 inches in length.

If I have a plot of ground that is 36 foot broad, and 64 foot long, which I would exchange [Page] for so much of another field that is 48 foot broad, how much ought I to have in length of the second? facit 6 [...] foot.

FINIS.

[Page]

Information about this book

Title statement

An introduction of the first grounds or rudiments of arithmetick plainly explaining the five common parts of that most useful and necessary art, in whole numbers & fractions, with their use in reduction, and the rule of three direct. Reverse. Double. By way of question and answer; for the ease of the teacher, and benefit of the learner. Composed not only for general good, but also for fitting youth for trade. / By W. Jackson student in arithmetick.

Jackson, William, 1636 or 7-1680.

Edition statement

1660

Publication

Publisher

Text Creation Partnership,

Place of publication

Ann Arbor, MI ; Oxford (UK) :

Date

2009-03 (EEBO-TCP Phase 1).

ID [DLPS]

A67916

ID [STC]

Wing J94

ID [STC]

ESTC R210093

ID [EEBO-CITATION]

99868922

ID [PROQUEST]

99868922

ID [VID]

121279

Availability

This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal. The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission.

Series

Early English books online.

Notes

(EEBO-TCP ; phase 1, no. A67916)

Transcribed from: (Early English Books Online ; image set 121279)

Images scanned from microfilm: (Thomason Tracts ; 241:E2110[3])

Source

— An introduction of the first grounds or rudiments of arithmetick plainly explaining the five common parts of that most useful and necessary art, in whole numbers & fractions, with their use in reduction, and the rule of three direct. Reverse. Double. By way of question and answer; for the ease of the teacher, and benefit of the learner. Composed not only for general good, but also for fitting youth for trade. / By W. Jackson student in arithmetick., Jackson, William, 1636 or 7-1680..

Extent

[6], 102 p.

Printed for R.I. for F Smith, neer Temple-Bar,. London :: 1661 [i.e. 1660]. (Annotation on Thomason copy: "Oct:"; the second '1' in imprint date is altered to "0".) (Reproduction of the original in the British Library.)

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