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 2] 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 3] 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 4] 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 5] 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 6] 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 7] 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 8] 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 9] 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 10] 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 11] 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,

• [Page 12]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 13] 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 14] 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,

[Page 15] 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 16] 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:

[Page 17] 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 18] 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

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.

[Page 19] 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 20] 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 21] 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?

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 22] 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 23] 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 24] 1 s. 4 d. I say 8 d. from 4 d. I can­not, then I borrow one shilling, being the next place, which is 12 d. to which I add the 4 d. that is above the line, it makes 16 d. and say, 8 d. from 16 d. rests 8 d. Again, 17 s. from 2 l. 12s. I say, 17 s. from 12 s. I cannot, but 17 s. from 1 l. 12s. or 32 s. there rests 15 s. then considering I borrowed one pound, I say one pound that I borrowed from 2 l. rests one pound, so that there rests 1 l. 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 345 l.—16s,—8d. from [Page 25] 476 l.—13s.—4d. I would see what remains.

I set my sums thus,

Then beginning with the least denomination, which is pence, I say, 8 d. from 4 d. I cannot, but I borrow one of the next denomination, which is shillings, and say, 8 d. from 1 s. 4d. and there rests 8 d. which I set under the pence, and then I say, one shilling I borrowed and 16 makes 17 s. from 13 s. I cannot, but I borrow one pound, and say, 17 s. from 1 l. 13 s. rests 16 s. then I say, 1 l. [Page 26] that I borrowed, and 5 l. is 6 l. from 6 l. 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 130 l. 16 s. 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.

Multiplication.

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 28] 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 29] 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 30] 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 31] 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 32] 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 33] 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 34] 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 35] to bee right, or else it is not well done.

Division.

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 37] 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 38] 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 39] 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 40] 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 41] 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 42] 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 43] 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 44] or two, for Imitation.

• 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 45] 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 46] 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 47] A. I consider 20 s. 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 20 s. is just equal to one pound, and 12 d. equal to one shilling, &c.

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

A. For as much as 12 d. is in each shilling, there must bee 12 times so many pence as there is shillings, therefore I [Page 48] 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 12 d. is but one shilling, and 20 s. is but one pound.

[Page 49] 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 20 C. 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 28 l. and the product is pounds, then to turn [Page 50] those pounds into ounces, I multiply by 16, and the pro­duct is the solution, &c.

• 256 tuns.
• 20
• 5120 C.
• 4
• 20480 (q) quartern s.
• 28
• 163840
• 4096
• 573440 l.
• 16
• 3440640
• 57344
• 9175040 o℥.

[Page 51] 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 52] 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 3 l. starling is worth 5 l. 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 53] 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 54] 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 55] 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 56] 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,

comes 81 for the fourth.

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

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 58] 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 12 s. what will 5 yards cost at that rate? here 3 yards is the first number, and 12 s. 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 20 s. 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 59] 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 12 s. 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 60] 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 11 s. 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 135 d. now work by the Rule, multiply the se­cond [Page 61] and third together, being set in order thus,

If 15 (q) quartern s. cost 135 d. 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,

• [Page 62]1
• 190 (0
• 576 (4 8 (2
• 122 (20
• 1

Comes for solution—2 l.—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 63] 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 64] 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 63] 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 66] 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 67] 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 68] 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 69] 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 70] 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 71] 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 72] 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 73] 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 74]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 75] 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 76] 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 77] 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 78] 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 79] 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 80] 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 81] 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 82] 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 83] 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 80] 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 81] 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 86] 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 87] 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.

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 85] 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 90] 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 91] 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 92] 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.

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

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

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

Q. How is this performed at one working?

[Page 95] A. Thus, I say according to the question, If 5 men in 6 daies earn 3 l. 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 3 l. in how long time will 3 men earn 5 l. where first I say, if 3 l. give 6 daies, what will 5 l. give? comes 10 daies; then by the Backer Rule of Three, If 5 men bee 10 daies in [Page 96] 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 3 l. 5 men bee imployed 6 daies, then to earn 5 l. 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 97] 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 98] 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.

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

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

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

In 10560 s. 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 28 l. what cost 30 yards? facit 20l.

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

[Page 100] If 30 yards cost 20 l. what will 42 yards cost? facit 28l.

If 28 l. buy 42 yards, how much will 20 l. 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 2 l. what is 369 marks worth? facit 246l.

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

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

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

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

For the Backer Rule of Three.

If 5 men do a peece of work [Page 101] 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 102] 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.