Numeration.

NUmeration is that part of Arithme­tick, by which is taught rightly and justly to value, declare and write any Number or Sum propounded. All Num­bers and Sums are expressed by these fol­lowing Characters or Figures, which are [Page 3] in number ten, whose value if you con­ceive each figure to be or stand by it self, you may take notice of by their several Sub­scriptions.

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

Nine of these figures, viz. 1, 2, 3, 4, 5, 6, 7, 8, 9, are significant figures, signify­ing each figure so many Unites, but the Cypher is of no value nor signification, as as it standeth alone, but being joined with other figures it increaseth the value of those figures which stand to the left hand of him.

Now these nine significant figures 1, 2, 3, 4, 5, 6, 7, 8, 9, have a double Value, whereof the one is certain, according to its own signification as before. But the other value is uncertain, it being according to the place that the said Figure doth stand in and possess.

A Place is a seat or room that a figure stands in, and how many figures there are in any sum, so many places of value doth the said sum contain, and that is the first place of any Sum, which is next the right hand, every place being ten times the value of the next right hand place.

[Page 4]

[Page 5] This Table needeth but little explana­ [...]ion, for thereby you may discern the first place next the right hand to be the place of Units; the second, the place of Tens; the [...]hird, the place of Hundreds; the fourth, Thousands, &c. So admit the sum 6873 were given you to number, consisting of four places, because in it are contained four figures, which you must thus number, six thousand eight hundred seventy three. And in like manner for 784395, which you must thus number, seven hundred eighty four thousand three hundred ninety five.

Addition of Mony.

If you have any Sums which are com­posed of several Denominations, as Pounds, Shillings, Pence and Farthings, or any other kind; you must first consider how many of the lesser make one of the next Denomina­tion; as, in Sterling Mony

• 4 Farthings is 1 Penny
• 12 Pence is 1 Shilling
• 20 Shilling is 1 Pound Sterling.

And so carry the Pence from the Far­things to the Pence, the Shillings from the Pence to the Shillings, and the Pounds from the Shillings to the Pounds; as you may plainly see in the working of this fol­lowing Example.

[Page 8] Suppose I had received mony of four several persons, viz. of A 384 l. 16 s. 11 d. ½, of B 197 l. 14 s. 08 d. ¾, of C. 79 l. 12 s. 03 d. ¾, and of D I have re­ceived 9 l. 13 s. 10 d. ¾; and I desire to know how much the four Sums make al­together. Set down the said Sums in such order one under the other, as the farthings may stand under themselves, the pence under themselves, the shillings under them­selves, and the pounds under themselves; and in such manner as Units, Tens and Hundreds may stand under one another, according as you see them here set down; then draw a line, and begin at the bottom of the least Denomination; saying, ¾ and ¼ is four farthings, and ¾ is seven farthings, and ½ is nine farthings, or two pence far­thing, setting down ¼ under the farthings, and carrying 2 to the pence; saying, 2 as I carried and 10 is 12, and 3 is 15, and 8 is 23, and 11 is 34 pence, which is 2 s. 10 d. therefore set down 10 under the pence, and carry 2 to the shillings; saying, 2 as I carried and 3 makes 5, and 2 is 7, and 4 is 11, and 6 is 17; setting down 7 under the first row of shillings, and car­carrying 1 to the next row of shillings; [Page 9] saying, 1 as I carried and 1 is 2, and 1 is 3 and 1 is 4, and 1 makes 5; which being odd, I set 1 after the 7, but under the last row of shillings: now the half of 5 being 2, I carry 2 unto the pounds; saying, 2 as I carried and 9 is 11, and 9 is 20, and 7 is 27, and 4 is 31, set down 1 under that row and carry three to the next row; say­ing, 3 as I carried and 7 is [...]0, and 9 is 19, and 8 is 27: setting 7 under this row, and carry 2 to the next; saying, 2 as I carried and 1 is 3, and 3 is 6; setting down 6 under the last row, so I find I have received in all of A, B, C, and D 671 l. 17 s. 10 d. ¼.

[Page 10] For the Proof of this Sum, whether it be cast up right or not, the common way is to draw a line under the first sum, and then sum up again, only leaving out the upper­most row, which sum being added to the up­permost row or line, then if the sum be the same as before, the work is right cast up. But it is as easie or better to cast up the whole sum again, and if you think by doing so, you may commit the same error again, or fall into the same mistake as before, you may cast it up the contrary way, viz. begin at top and go downwards, for the Accompt will amount to one and the same sum, if it be right done.

But Tradesmen and Merchants being most concerned herein, have another help, which is in writing out Bills by their Books, if they find the sums in both Accompts do not agree, they conclude there must be a mistake in one or the other, which they find out by casting the Accompts over again.

In casting up long Bills, where many shillings, are in the pence, it is a good plain way to make a prick or some small mark at every shilling, and so setting down the odd pen [...]e at last, counting over how many [Page 11] pricks you have made, and carry so many Shillings to the place of Shillings.

But such as are ready and expert, count how many Pence are in the whole Column, then by getting without Book such a Table as here followeth, can readily tell how many Shillings and odd Pence are in any Number of Pence.

Addition of Flemish Mony.

Flemish Mony is reckoned by Pounds, Shillings and Pence, as we do our Sterling. But frequently they do reckon by Stivers and Guilders, or Stivers and Ducatoons. 20 Stivers being 1 Guilder, and 63 Stivers is 1 Ducatoon.

[Page 13]

Addition of French Mony.

4 Quarters make 1 Solx, 60 Solx make one French Crown.

Therefore for 4 Quarters carry 1 Solx to the Solxes, for 60 Solxes carry 1 French Crown. As you may plainly see in the following Example.

Addition of Haverdupoize Weight.

• 16 Ounces is 1 Pound.
• 28 Pound is 1 Quarter.
• 4 Quarters is 1 Hundred weight.
• 20 Hundred weight is 1 Tun.

Addition of Apothecaries Weights.

• 20 Grains is 1 Scruple ℈.
• 3 Scruples is 1 Drachm ʒ.
• 8 Drams is 1 Ounce ℥.
• 12 Ounces is 1 Pound lb.

Addition of Troy Weight.

• [...] Grains is 1 Penny Weight.
• [...] Penny Weight is 1 Ounce.
• [...] Ounces is 1 Pound.

Addition of Wine Measure.

• 2 Pints is 1 Quart.
• 2 Quarts is 1 Pottle.
• 2 Pottles is 1 Gallon.
• 63 Gallons is 1 Hogshead.
• 4 Hogsheads is 1 Tun.

Addition of Beer Measure.

• 2 Pints is 1 Quart.
• 2 Quarts is 1 Pottle.
• 2 Pottles is 1 Gallon.
• 9 Gallons is 1 Firkin.
• 2 Firkins is 1 Kilderkin.
• 2 Kilderkins is 1 Barrel.

Addition of dry Measure.

• 16 Pints is 1 Peck.
• 4 Pecks is 1 Bushel.
• 8 Bushels is 1 Quarter.
• 5 Quarters is 1 Wey.
• 2 Wey 1 Last.

[Page 20]

Addition of long Measure.

• 3 Barly Corns make 1 Inch.
• 12 Inches 1 Foot.
• 3 Foot 1 Yard.
• 5 Yards ½ 1 Pole or Perch.
• 40 Perches 1 Furlong.
• 8 Furlongs 1 English Mile.

By the Table you may easily discern what to carry at in each Denomination, as by the following Example doth appear.

[Page 21]

Addition of Cloth Measure.

• 4 Nails is 1 Quarter.
• 4 Quarters is 1 Yard.
• 5 Quarters is 1 Ell English.
• 3 Quarters is 1 Ell Flemish.

Addition of Time.

• 60 Minutes is 1 hour.
• 24 Hours is 1 Day.
• 365 Days is 1 Year.

Subtraction of one Denomination.

Having placed your Numbers one under the other as was directed, you must begin as▪ the right hand, and subtract the first Figure of the undermost Number from the first Figure of the uppermost Number over [...] that which remaineth set underneath the Line, right under the Figure you have subtracted, then take the second Figure of your undermost Number, and take it from the second Figure of your uppermost Num­ber, the third from the third, and so forth until you come to the end, putting always [...] of every Figure under the [...]ine in i [...]s due order and place.

According to the former Instructions, I do begin at the right hand; saying, 2 from 4 and there remains 2, which I set down under the Line, and right under the first Figure subtracted; then I go to the next Figure, saying, 5 from 8 and there remains 3, which I set under the 5, being the second Figure subtracted; then I go to the next Figure, and say, 5 from 7 and there doth remain 2, which 2 I set down under the third Figure subtracted; then I go to the next and say, 1 from 2 and there doth remain 1, which I set under the last Figure subtracted. So there stands under the Line this Number 1232 l. which doth remain due unto B C.

But sometimes it doth fall out to have 2 Figures of one Quantity and Likeness to be subtracted one from another. As, 5 [Page 26] and 5; then I say, 5 from 5 and there doth remain 0. And when you have a Cypher to be subtracted from any Figure. As 0 and 7; then you must say, 0 from 7 and there remains 7, and so for any Figure.

It will be requisite to give you one Ex­ample more in sums of one Denomination, wherein the Figures of the lesser Number or Number to be subtracted, are some of them greater than the Figures of the greater Number, or Number from whence you must subtract; In such case you must bor­row 10, and add to the uppermost Figure, and then subtract your Figure to be sub­tracted from that Sum, and set down your remainer; always remembring when you borrow 10 to add 1 to the next Figure to be subtracted for paiment, as in this follow­ing Example.

[Page 27] Here is due from A B 42317 Pound, and there is received 32140 Pound, I desire to know what is due from the said A B. Having set the two Sums down according to the former Instructions; say­ing, 0 from 7 and there remains 7, 4 from 1 I cannot, but 4 from 11 and there remains 7; 1 as I borrowed and 1 is 2, 2 from 3 and there remains 1; 2 from 2 and there remains 0, and 3 from 4 and there remains 1. So there doth remain due from A B 10177 Pound. And in like manner for any other Sum, wherein you have one or more Figures of the lower Number greater than the Figures under which they do stand to be subtracted.

Subtraction of Mony.

In Sums composed of several Denomi­nations, you must begin to subtract at the least Denomination next your right hand, and subtract each Denomination from his same, and set down the remainer of each Denomination under his own proper De­nomination; but if in the Denomination of Farthings, your lower Number be greater than your upper, you must borrow 1 Peny [Page 28] which is 4 Farthings, to add to the upper­most, and when you subtract the Pence, then add 1 Penny to the lower Number of Pence for paiment of the Penny or 4 Far­things you borrowed. Likewise if in the Denomination of Pence the lower Number be greater than the upper, borrow 1 Shil­ling or 12 Pence to add to the uppermost, remembring to add 1 Shilling to the lower Number of Shillings for paiment of 12 pence you borrowed. And in like manner, if the lower Number of Shillings exceed the upper, as you may easily discern by the following Example of Pounds, Shillings, Pence and Farthings.

Here is received 7836 l. 14 s. 7 d. ¼, and out of it is paid 4524 l. 17 s. 9 d. ¾; [Page 29] it is required what is left of the said Sum. I set the two Sums down as you see accord­ing to the former directions, and begin at the least Denomination which is Farthings, saying, ¾ from ¼ I cannot, but 1 Penny as I borrowed which is 4 Farthings and 1 Farthing is 5 Farthings, 3 Farthings from 5 Farthings and there remains 2 Farthings, which is ½, which ½ I set under the Line, and right under the Denomination of Far­things; then I go to the next Denomina­tion, which is Pence, and because I bor­rowed one Penny in the Farthings, I now pay it again, saying, 1 Penny as I borrowed and 9 Pence is 10 Pence, from 7 Pence I cannot, but 1 shilling which is 12 Pence, and 7 Pence is 19 Pence, and 10 Pence from 19 Pence and there remains 9 Pence, setting 9 under the Line and under the Denomination of Pence; then I proceed to the Shillings, and because I borrowed one Shilling in the Pence, I pay it again; say­ing, 1 Shilling as I borrowed and 17 Shil­lings is 18 Shillings, from 14 Shillings I cannot, but 1 Pound as I borrowed which is 20 Shillings and 14 Shillings is 34 Shil­lings, 18 Shillings from 34 Shillings and there remains 16 Shillings, so I set down [Page 30] 16 under the Denomination of Shillings; then I go to the Pounds, and by reason I borrowed 1 Pound in the Shillings, I pay it, saying, 1 as I borrowed and 4 is 5, 5 from 6 and there remains 1; setting 1 under the first Figure of the Denomination of Pounds; so proceeding I say, 2 from 3 and there remains 1, 5 from 8 and there remains 3, and 4 from 7 and there doth remain 3, setting down each remainer un­der his respective Figure as you see. So there doth remain 3311 l. 16 s. 9 d. ½. And in like manner for any other Sum of the same Denomination.

Subtraction of Haverdupoize Weight.

The several Denominations of Haverdu­poize Weight, you may acquaint your self with by the Table of Haverdupoize in the second Chapter.

Suppose I have bought 45 C. 3 quarters, 16 l. of Sugar, and I have sold 28 C. 1 quarter, 24 l. of the same Sugar; it is required what is left of this parcel. I set down the Numbers according to the fore­going Directions, finding therein three Denominations, viz. Hundreds, Quarters, [Page 31] Pounds; therefore I begin saying,   C. qr. l. Bought 45 3 16 Sold 28 1 24 Remains 17 1 20 twenty four from sixteen I cannot, but one quarter as I borrowed which is 28 Pound and 16 is 44, 24 from 44 and there remains 20. Then one as I borrowed and 1 is 2, from 3 and there re­mains 1. Then I come to the Hundreds; saying, 8 from 5 I cannot, but 8 from 15 and there doth remain 7, 1 as I borrowed and 2 makes 3, from 4 and there remains 1. So I find there doth remain unsold 17 C. 1 quarter, 20 l.

Sometimes it happeneth that there are many Sums or Numbers to be subtracted from one Sum or Number, as suppose I have bought a Butt of Currans weighing 18 C. 1 quarter, 21 l. and at several times I have sold several parcels; as, 4 C. 3 qu. 17 l. at one time; 3 C. 3 quarters, 20 l. at another; and 2 C. 1 quarter, 16 l. at an­other time; 1 C. 2 quarters, 0 l. at another time; and 1 C 3 qu. 24 l. at another time; and 3 qu. 12 l. at another time. The question is what is remaining unsold in the Butt of Currans?

[Page 32]

In order to answer this question, I first set down the neat weight of the But of Currans, viz. 18 C. 1 qu. 21 l. drawing a Line underneath. I set down under that Line the several parcels sold out of this Butt of Currans, one under the other, and bring all these parcels under the Line into one Sum by Addition, as you see in the Example, which Sum amounts unto 15 C. 2 quarters, 25 l. which I subtract from the weight of the Butt of Currans first set [Page 33] down by the foregoing Instructions of this Chapter for Subtraction of Averdupoize weight.

Subtraction of Troy Weight.

Here I begin to subtract; saying, 22 from 21 I cannot, but 1 as I borrowed from the Penny weights which is 24 Grains and 21 Grains is 45 Grains, and 22 from 45 and there remains 23 then 1 Penny weight as I borrowed and 14 Peny weights is 15, from 6 and there remains 1; then 10 from 8 I cannot, but 1 Pound as I bor­rowed which is 12 ounces, and 8 is 20, and 10 from 20 and there remaineth 10; then 1 Pound as I borrowed and 2 is 3 Pounds, 3 from 0 I cannot, but borrow 1 and say, 3 from 10 and there remaineth 7; 1 as I borrowed and 1 is 2 from 2 and [Page 34] there remains 0. So there remaineth in all 7 Pound, 10 Ounces, 1 Penny weight, 23 Grains.

Subtraction of Apothecaries Weights.

Here in this Example you say, 15 from 12 I cannot, but 1 as I borrowed from the next Denomination being Scruples con­taining 20 Grains, and 12 Grains is 32 Grains, then 15 from 32 and there remains 17; then proceed and say, one Scruple as I borrowed and 2 makes 3 from one Scruple I cannot, but 1 as I borrowed from the Drachms containing 3 Scruples and one is 4, three from 4 and there remains 1. Then I proceed to the next Denomination; say­ing, 1 as I borrowed and 7 is 8 from 5 I cannot, but one as I borrowed from the Ounces containing 8 Drachms and 5 is 13, [Page 35] 8 from 13 and there remains 5. Then proceeding to the next Denomination; saying, 1 as I borrowed and 6 is 7 from 9 and there remains 2. Then proceeding to the last Denomination; I say, 7 from 12 and there remains 5. So there will remain 5 lb. 2 ℥. 5 ʒ. 1 ℈. 17 gr. which doth re­main unsold of that parcel.

Subtraction of Wine Measure.

For the Contents of the several Denomi­nations of Wine Measure, you may see by the Table in the second Chapter, and also the way of Subtraction in this Measure, by this Example.

Subtraction of Beer Measure.

The Table of Beer Measure in the second Chapter will shew you the Content of the several Denominations, and the way of subtracting by the following Example.

Subtraction of Dry Measure.

In the second Chapter by the Table of Dry Measure, you may find the Contents of all the Denominations. And by the fol­lowing Example you may also see the manner of Working.

Subtraction of Long Measure.

The Table of Long Measure in the se­cond Chapter, will inform you of the se­veral Denominations and their Contents, and the way of subtracting by the following Example.

Subtraction of Cloath Measure.

Here it will be convenient to let you know as by the said Table, in the second Chapter of Addition; you may discern, that in Cloth Measure there be three sorts of Measures. As, Broad Cloth is measured by Yards, Quarters and Nails; Linnen Cloth in England, by Ells, Quarters and Nails; and Linnen Cloth in Holland, by Flemish Ells, Quarters and Nails. In each of which it will be convenient to give an Example; and to note that whereas by the Table of Cloth measure in Addition you find 4 Quarters is 1 Yard, 5 Quarters 1 English Ell, and 3 Quarters 1 Flemish Ell, it is to be 5 such Quarters as one Yard doth contain, viz. 1 Quarter more makes an Ell English, and 3 Quarters of our English Yard makes 1 Ell Flemish.

[Page 39]

Subtraction of Time.

You will find by the Table of Time that 60 Minutes make an Hour, 24 Hours one Natural Day, 365 Days one Years; by the following Example of Subtraction you more plainly understand how to work all such like Examples.

Proof of Subtraction.

When your Subtraction is ended, if you desire to prove your work, whether you have done right or wrong; add the Re­mainer to the Number that was to be sub­tracted, and if the sum of those two Num­bers be equal to the greater Number, or numbers from whence the Subtraction was to be made, then your Subtraction was true, otherwise false. Thus let us prove the se­cond Example of this Chapter, where after Subtraction is ended, the Numbers stand as you see, the Remainder or difference being 3311 l. 16 s. 09 d. ½, now to prove the work I add this remainer 3311: 16: l. s. d. f. 7836 14 7 1 4524 17 9 3 3311 16 9 2 7836 14 7 1 9: 2, to the num­ber that was to be subtracted which is 4524: 17: 9: 3: and the Sum of these two Numbers is 7836 l. 14 s. 7 d. ½, being equal to the Number from whence the Subtraction was to be made. Therefore I do conclude the subtra­ction is true.

Another way of Multiplication.

The carrying of one Number is a little difficult to young Learners in this Art, or to such as do not frequently exercise them­selves therein. Therefore for the help of such capacities, I shew you a plain and easie way for finding the several Products that are made by Multiplying the Multiplicand, by the several Figures of the Multiplier.

When you have two Numbers given to be multiplied one by another, multiply the greater Number by all the nine digits by continual Addition nine times over, setting the several Sums down Table-wise as you see in the following Example. The first Addition being as much as a Multiplication by 2, the second Addition being a Multi­plication [Page 51] by 3, the third Addition being a Multiplication by 4, the fourth a Multipli­cation by 5, the fifth a Multiplication by 6, the sixth a Multiplication by 7, the seventh a Multiplication by 8, the eighth a Multi­plication by 9. So have you the several Products of the Multiplicand, when it is multiplied by all the nine digits, cast up to your Hand by Addition, which you may choose out according to your Figures of the Multiplier, setting your several Pro­ducts down one under another, according to the Instuctions of the last Example.

Abbreviations in Multiplication.

First, if you are to multiply any Number by 10, you need do no more but set a Cy­pher 0 to the right hand of all the Figures. As, 5978 multiplied by 10, makes 59780. In the same manner if you were to multiply [Page 54] any Number by 100, add the said Number two Cyphers thus 00, if by 1000 add three 000.

Secondly, if you are to multiply any Number by 20, 30, 40 or by 200, 300, 400, or any Number with more Cyphers at the end, set them as you see in the Example, as if you were [...] only to multiply by Figures; and let the Cypher or Cyphers hang over, and add them to the Pro­duct, as in this Example.

Thirdly, if you are to multiply any Number, that hath a Cypher or Cyphers at the end, by another Number which hath a Cypher or Cyphers in the end thereof, set them as you see in the Exam­ple, in such manner as the first [...] figure of the Multiplier may stand under the first Cypher of the Multiplicand, and cast up the Product as you see is here done.

Fourthly, if there be a Cypher in the middle of your Multiplier, you need do no more but put a Cypher under the se­cond Figure of the foregoing Product, to fill up room, and set down the first▪ [Page 55] Figure of the next Product [...] next to the Cypher in the same Line, as by the Example you may plainly perceive.

A more easie and plainer way of Division.

There is a far plainer and better way of Division, than this cancelling and scratching of Figures, which I much admire that the Schoolmasters in these times do not pra­ctise, the other way being too hard and difficult for a Learner, for I have known many that have learned so far as Division with a great delight and chearfulness, but with the hardness and difficulty of Division have been quite disheartned from proceed­ing any further. But this way that I intend to shew the Learner is so plain, easie and delightful that the meanest and ordinariest capacity may attain the knowledge thereof by the help of the former Rules.

Another way of Division.

This way that I do intend now to ex­plain doth not much differ from the latter way: but here you have all the Products of every Multiplication ready cast up to your hand by multiplying the Divisor by all the nine digits, as by the following Ex­ample doth more plainly appear.

Abbreviations in Division.

When a Number is given to be divided by any Number that hath one or more Cyphers on the right hand, for so many Cyphers cut as many Figures from the right hand of your Dividend, and also cut off those Cyphers of your Divisor, and let the remaining Numbers of the Dividend be divided by the remaining Numbers of the Divisor, and if any thing remain of your [Page 79] Division, set the Figures cut off from your Dividend to the right hand of them, and this new Number is the true Remainer of the Division.

Reduction of Mony.

In this Reduction of Mony as well as in other Reductions, you must consider whe­ther the Sum propounded be to be reduced into a greater or lesser Denomination, and next consider how many of the one Deno­mination do make the other: by which Considerations you will know when to multiply and when to divide, and what to multiply or divide by. As for Example.

Quest. 1. In 374 Pounds it is required to know how many Shillings. Here I do consider that Pounds is the greater Deno­mination, and Shillings is the lesser, and that in one Pound are contained 20 Shil­lings, and that the Number of Shillings in 374 Pounds will be 20 times as much as 374, therefore I multiply 374 by 20, and the Product is 7480, and so many Shillings are contained in 374 Pound.

[Page 83] Quest. 2. In 526: 18, how many Shil­lings and Pence?

In this Example you have two Denomi­nations propounded to find out the Deno­mination of Pence, here in multiplying your Pounds by 20, you must take in the 18 Shillings, setting 8 down first, because 0 cannot multiply, there­fore multiply by the next [...] Figure 2; saying, 2 times 6 is 12, and 1 is 13 set down 3 and carry 1, 2 times 2 is 4 and 1 is 5, set down 5, 2 times 5 is 10; setting 10 down. So the Product of this multiplicand and the odd Shillings make 10538 Shillings; now am I to reduce the Shillings into Pence, and know­ing that 12 Pence make one Shilling, I therefore multiply the Number of Shillings by 12, and the Product of that Multipli­cation 126456 will be so many Pence, and so many Pence doth 526 Pound 18 Shil­lings contain.

In the work of Reduction many times you need not run through every particular [Page 84] Denomination, but come to the thing re­quired sooner. As for Example.

Quest. 3. In 398 l. 15 s. how many Farthings? First, multi­ply 398 by 20, and ta­king [...] in the odd 15 Shil­lings, the Product of Shillings will be 7975, then knowing that in one Shilling are 48 farthings, I do not reduce these Shillings into Pence as in the last Example, by mul­tiplying them by 48. I do reduce them into Far­things at once, and the Product of that Multiplication is 382800, and so many Farthings is there in 398 l. 15 s. as was required.

Rule 4. But if it be required to reduce Numbers of a lower Denomination into a greater, in such case you must divide the lower Denomination propounded, by so much as makes of that lower Denomina­tion one Unite or Integer of the greater Denomination. As for Example.

In 1360 Shillings how many Pounds? and knowing that 20 Shillings of the lower [Page 85] Denomination is contained in a Unite or Integer of the greater [...] denomination of Pounds, I there­fore divide the lower Denomina­tion 1360 Shillings by 20, by cutting off the last Figure, and taking the half of the re­maining figures, which half 68 is the Quo­tient of that Division by 20, and the Num­ber of Pounds contained in 1360 Shillings, as was required.

Rule 5. But if the question so happen in Reduction that there may be a denomination or denominations between the Denomina­tion given, and the Denomination requi­red, then you may reduce your Denomi­nation given by Division into the next su­perior or greater Denomination, and when it is so reduced, reduce it again into the next above that, and so until you have brought it into the Denomination required. As for Example.

Rule 6. Let it be demanded in 142080 farthings to know how many Pounds. First, I divide the given Number Farthings by 4, to bring them into Pence, because I Penny doth contain 4 Farthings, and the Quotient thereof will be 35520 Pence; then I di­vide 35520 Pence by 12, because 12 Pence [Page 86] make 1 Shilling, and the Quotient of that Division will be 2960 Shillings, and then I divide 2960 by 20, and the Quotient giveth 148 Pounds, which are equal in value to 142080 Farthings. According to the Example.

[...]

Rule 7. When the Number given to be reduced doth consist of divers Denomina­tions, as Pounds, Shillings, Pence and Farthings; Hundreds, Quarters, Pounds and Ounces, &c. Then you must reduce the highest or greater Denomination into [Page 87] the next inferior or lower, taking in the odd Numbers standing in that Denomina­tion which your greater Denomination is reduced to, then reduce that Sum into the next inferior Denomination, adding there­to the odd Numbers standing in that De­nomination, and so doing until you have brought the Number given into the Deno­mination required. As for Example.

[...]

Let it be required to Reduce 137: 16: 7: ½, into Farthings. First, I bring 137 Pounds into Shillings by multiplying by 20, and taking in the 16 Shillings, the [Page 88] Product will be 2756 Shillings. Then I multiply 2756 Shillings by 12 to bring the Shillings into Pence, taking in the odd 7 Pence, and the Product will be 33079 Pence. Lastly, I multiply 33079 by 4, to bring Pence into Farthings, taking in 2 for the odd half Penny, and the Product is 132318 Farthings, being equal in value with 137: 16: 7: ½, or the Number of Farthings contained in the whole Sum, as was required.

Rule 8. If in Reduction any do remain of a Division, the Remainer is of the same Denomination as the Dividend is of.

There is a shorter way for reducing Pounds into Pence, and Pence into Pounds. As for Example.

Rule 9. Let it be required to know how many Pence is contained in 347 Pounds. Now because 20 Shillings is 1 Pound, and 12 Pence is 1 Shil­ling, [...] therefore 12 times 20 which is 240, is the Number of Pence contained in 20 Shillings or 1 Pound, and if you multiply your Pounds 347 by 240, the Product thereof 83280 are the Num­ber of Pence contained in the [Page 89] Sum of 347 Pounds, as was required.

Rule 10. It is required to know how many Groats are comprehended or con­tained in 263 Pounds, the way of reducing Pounds into Groats is not difficult, for multiply the Pounds by 20 to bring them into Shillings, and because 1 Shilling doth contain 3 Groats, multiply those Shillings by 3, and that Product shall be the Num­ber of Groats in so many Pounds.

But there is a quicker and shorter way than this, which is thus. Because 20 Shillings is 1 Pound, and 3 [...] Groats is 1 Shilling, therefore 3 times 20 which is 60 is the Num­ber of Groats in 20 or 1 Pound, and if you multiply the Number of Pounds 263 by 60, the Product will be 15870 the Number of Groats in 263 Pounds which is a shorter way than to reduce Pounds into Shillings, and Shillings into Pence by two Multiplications.

Rule 11. And if it were required to reduce Pence into Pounds, or Groats into Pounds. Divide by 240 for reducing Pence into Pounds, and by 60 for the reducing Groats into Pounds, and the Quotient shall be the Number of Pounds required.

[Page 90] [...]

Rule 12. Let it be required to reduce 83280 Pence, and 15780 Groats into Pounds, for experience sake, being the Products of the two last Examples. Divide 83280 Pence into Pounds by 240, the Number of Pence in 20 Shillings or one Pound, and the Quotient will be 347, the Number of Pounds as before, also divide 15780 Groats by 60, the Number of Groats contained in 20 Shillings or one Pound, and the Quotient thereof will be 263 Pounds as before, by which you dis­cern the infallibility of both Rules, for dividing or multiplying by 240, to reduce Pence into Pounds, and Pounds into Pence, [Page 91] and by 60 for reducing Groats into Pounds and Pounds into Groats.

Rule 13. There are other Denominations of English Mony, besides Pounds, Shillings, Pence and Farthings. As the following Table doth shew.

Rule 14. Let it be required in 783 Nobles to know how many Marks, and Pounds at 20 s. By this Table you find the value of a Mark to be thirteen Shillings and four Pence, and the value of a Noble to be half a Mark, that is 6 s. 8 d. and likewise 6 s. 8 d. is one third of 20 Shillings, there­fore 20 Shillings or 1 Pound doth contain 3 Marks, therefore to resolve the question, [Page 92] double the Number of Nobles given 783, that is to say, multiply 783 by 2, and the Pro­duct [...] 1566 will be the number of Marks con­tained, and because 3 Marks, viz 3 times 6 s. 8 d. is 20 Shillings, or 1 Pound, I therefore divide the Number of Marks 1566 by 3, and in the Quotient I do find 522, which tells me that in 1566 Marks there is 522 Pounds, therefore in 783 Nobles there is 1566 Marks, and 522 Pounds: which is the answer to the question re­quired.

Rule 15. In 768 Pounds, how many Marks and Nobles? this question is but the reverse of the last, for if you multiply your Pounds [...] given by 3, the Product 1566 will be Marks, and the half of that Number will be 783, the Number of Nobles, as was required. [Page 93] Therefore in 522 Pounds there is con­tained 1566 Marks, and 783 Nobles, as was required.

Rule 16. In 538 Pounds it is required to know how many Angels and Crowns? By the foregoing Table you find the value of an Angel to be 10 Shillings, and of a Crown 5 Shillings; therefore because twice 10 Shillings is 20 Shillings, and twice 5 Shillings is 10 Shillings; therefore I multiply the [...] Number of Pounds given 538 by 2, the Product thereof 1076 is so many Angels, and that Product 1076 multiplied again by 2, the Product thereof 2152 are the Number of Crowns. Therefore in 538l. there are con­tained 1076 Angels, and 2152 Crowns, as was required.

Rule 17. In 2152 Crowns how many Angels and Pounds?

This is the reverse of the last Rule, and whereas 2152 Crowns. 538 Angels. 1076 Pounds. there you multiplied, here you must divide by, which is taking the half of it, as [Page 94] for experience sake, the last Example of the last Rule, the half of 2152 Crowns is 1076 Angels, and the half of 1076 Angels is 538 Pounds, as before 2152 Crowns make 1076 Angels and 538 Pounds, as was re­quired.

Rule 18. In 294 Pounds it is required to know how many Nine Pences and Four Pence half Pennies.

[...]

[Page 95] For the resolving this question you must first reduce your Pounds into Shillings, then your Shillings into Pence by the se­venth Rule, and the Product of the which in the Example you will find to be 70560, you must divide by 9 to bring them into Nine Pences, and the Quotient will give you 7840 which are Nine Pences, and the double thereof 15680, is the Number of Four Pence half Pennies, as was re­quired.

Rule 19. In 15680 Four Pence half Pennies how many Nine Pences, Shillings, and Pounds?

This is the Reverse of the last Rule, and where in that you did multiply here you must divide, where you divided here you must multiply; there I have chosen the same Example for your better experience, which is thus resolved.

Because two 4 d. ½ make 9 Pence there­fore take the half of your given Number of Marks 15680, being 7840 Nine Pences. Then by multiplying 7840 by 9, I reduce them into Pence and the Product there­of is 70560 Pence, by the Instructions of the sixth Rule, and you will find the Product of shillings to be 5880 Shillings, [Page 96] and the Pounds to be 294. So I find that 15680 Marks is 7840 Nobles, or in value 5880 Shillings, and 294 Pounds, as was required.

[...]

Rule 20. In 6748 Thirteen Pence half Pennies how many Pence, Shillings and Pounds.

You must first compute how many half Pence there are in 13 d. ½, and you will find 27; therefore multiply 6748 by 27, and the Product 182196 will be all half pences, and the half thereof 91098 is so [Page 97] much, which divided by 12 the Quotient will give you 7591 Shillings and 6 pence remaining, 7591 Shillings divided by 20 gives 379 Pounds, and 11 Shillings re­maining. So that I find 6748 thirteen pence half pennies are in value 379l. 11s. 6d. as was required.

[...]

[Page 98] Rule 21. In 436 l. 12 s. 9 d. how many pieces of 4 d. ½ per piece.

[...]

[Page 99] By the seventh Rule reduce the given Number 436: 12: 09 into pence, and you will find the Product of pence to be as by the Example 104797 pence, which because there is an odd half penny in 4 d. ½ you must double to bring into half pence, and the Product then will be 209586 half pence, now by reason that in 4 d. ½ there are 9 half pence, you must divide these Number of half pence 209586 by 9, as you see in the Example, and the Quotient will be 23287 pieces of 4 d. ½, and of the Di­vision there doth remain 3 which doth sig­nifie so many half pence over and above 23287 pieces of 4 d. ½.

Rule 22. In 372 pounds how many pieces of 3 d. and 2 d. of each equal, that is what Number of 3 d. pieces, and what Number of 2 d. pieces will make up 372 pound, and the Numbers of each equal.

The way to resolve this question, and all other questions of this nature, is to add the pieces together, (into which the given Number is to be brought,) and to reduce the given Number into the same Denomi­nation with their Sum, and to divide the given Number so reduced by their Sum, and the Quotient will give you the exact [Page 100] Number of each piece, the same method being used in the Example foregoing, by which we find that 17856 three pences, and 17856 two pences are equal in value to 372 pounds.

[...]

[Page 101] Rule 23. In 3748 pieces of eight Dollers or Duccats at 4s. 7d. ½, how many pounds, shillings and pence?

[...]

[Page 102] First bring one single Doller or 4s. 7d. ½, into the lowest Denomination named, which is half pence, and it doth contain 111 half pence, by which I multiply my given Number of Dollers 3748, and the Product thereof is 416028 half pence, and by taking the half I reduce it into Pence, which is 208014, which by the sixth Rule you may reduce into Shillings and Pounds. So I find 3748 Dollers, Duccats, or pieces of Eight at 4 s. 7 d. ½ are worth 866l. 14s. 06 d. English Mony, as was required.

Reduction of Averdupoize Weight.

Averdupoize Weight is used for all such Goods and Commodities that are to be weighed, except Gold, Silver and Bread, which are weighed by Troy Weight.

For the Contents of the several Deno­minations of Averdupoize Weight, the Learner may find in the Averdupoize Table in the second Chapter.

Rule 24. In 246 C. 3 qu. 16 l. It is re­quired to know how many Ounces?

In the Averdupoize Table in the second Chapter I find 4 Quarter is one Hundred, therefore having set down the given Num­ber [Page 103] as in the Margent, 246 C. 3 qu. 16 l. and [...] multiply the hundreds by 4, taking in the Quarters, and the Pro­duct 987 is so many Quarters. Likewise in the said Table I find that 28 Pound makes one Quarter, therefore I multiply that Pro­duct 987 by 28, and the Product thereof must be 27652, not forgetting in the mul­tiplication by 28, to take in the odd pounds I likewise find in the Table that 16 Ounces makes one Pound: therefore I multiply the last Product of Pounds 27652 by 16 to bring them into Ounces, and I find the Product to be 442432, where I do conclude that in 246: 3: 16 there is 442432 Ounces contained, as was required.

[Page 104] Rule 25. In 442432 Ounces, how many Hundreds, Quarters and Pounds?

[...]

This is no more but the reverse of the last Rule, for where in that question you did multiply, here you must divide, and by the same Figures, as the foregoing Example doth illustrate, in this manner.

First, you see the given Number of Ounces divided by 16, the quotient gives [Page 105] 27652 for Pounds, next I divide 27652 Pounds by 28, the quotient gives me 987 quarters, there remaining of that Division 16 Pounds, next I divide 987 quarters by 4, and the quotient thereof is 246 hundred weight, and of that Division there remains 3 which are quarters, by which work I find that 442432 Ounces is as much as 27652 pound, 987 quarters, and 16 pound; or which is all one but more Arithmetically expressed, 246 C. 3 qu. 16 l. as was requi­red, perfectly agreeing with the given number of the last Rule, by which you clearly understand the infallibility of the method or manner of working.

Rule 26. A person is to send to his Chapmen in the Country 93 parcels of Tobacco of 12 Pound in a parcel, as many parcels of 8 pounds in a parcel, as many of 6 pound parcels, and as many of 4 pound parcels. The question is how many hun­dred weight, quarters and pounds must be weighed off of the said Tobacco to comply with all these parcels?

First set down the several parcels one under another, as 12 pound, 8 pound, 6 pound, and 4 pound, the Sum whereof is 30 pounds; now because there is to be [Page 106] 93 of each parcel, I therefore multiply 93 by their Sum which is 30, the Product thereof gives me 2790 for the whole quan­tity of pounds. Now by the directions of the last Rule I divide that Number of pounds 2790 by 28, the quotient gives me 99 quarters, and 18 pounds remaining, next I divide these quarters 99 by 4, and the quotient gives me 24 hundred, and there remains 3 quarters. So by this work I find the whole quantity doth contain, as by the foregoing Example doth appear 24 C. 3 qu. 18 l. as was required.

[...]

Rule 27. A Chapman buyeth 24: 3: 18 of Tobacco, and orders it to be made up into parcels of 12 Pound, of 8 pound, [Page 107] of 6 pound, and of 4 pound, and there must not be more of one parcel than ano­ther, but all equal. I desire to know how many must there be of each parcel.

[...]

This is the reverse of the last, therefore I have taken the answer of that question, and make it here the question upon which I desire the answer, which answer will come out the same as the question of the [Page 108] last, if the work be truly done, else not.

The way to resolve this question is thus: Reduce your given Number into pounds by the twenty fourth Rule, and the Product will be 2790 pounds. Next add all your parcels together, as 12 pounds, 8 pounds, 6 pounds, and 4 pounds, and they make 30 pounds: now divide your Number of pounds before found 2790 by 30, the Sum of all the parcels added together, and the quotient gives me 93, by which I find there must be 93 of each parcel, as was required, and agreeing with the question of the last Rule.

Reduction of Troy Weight.

Troy Weight differs much from Averdu­poize Weight, as by the said Table in the second Chapter, where you will find but 12 ounces Troy Weight to be in a pound; therefore I would have the Learner well experienced in the Table of Troy Weight.

Rule 28. In 243 l. 8 oz. 16 p. weight, how many grains.

By the Table of Troy Weight in the second Chapter you will find 24 Grains makes 1 Penny weight, 20 Penny weights [Page 109] is 1 Ounce, and 12 Ounces one Pound. [...] Therefore to reduce the given Number 243 l. 8 oz. 16 p. weig. into grains I do thus. First, multiply the pounds 243 by 12, and take in the odd 8 ounces, and the Pro­duct gives you 2924 for ounces. Then multiply the ounces 2924 by 20, and take in the odd 16 penny weights, and the Pro­duct will give you 58496 for penny weights. Next mul­tiply the penny weights 58496 by 24, the Product will give you 1403904 which are grains, by which I find that 243 l. 8 oz. 16 p. weig. do contain 1403904 grains, as was required.

Rule 29. In 1403904 grains how many pounds, ounces and penny weights?

[Page 110] [...]

This is no more but the inverse of the last Example, and is resolved by dividing here by the same numbers that there you multiplied by, which is taken backwards by 24, 20, and by 12, and the answer will be 243 l. 8 oz. 16 p. weig. as was re­quired.

Reduction of Apothecaries Weights.

It will be requisite for the Learner to make himself acquainted with the several Denominations, and the notes or marks thereof, used by Apothecaries, Chirurgions and Doctors; which by the said Table in the second Chapter are these following.

• 20 Grains is 1 Scruple ℈.
• 3 Scruples is 1 Drachm ʒ.
• 8 Drachms 1 Ounce ℥.
• 12 Ounces 1 Pound lb.

The Learner getting these notes without Book readily to know them will be capable of reading any Doctors or Apothecaries bill.

Rule 30. In 18 lb. 9 ℥. 2 ʒ. 1 ℈. it is required to know how many Grains?

Which to interpret is thus, read in 18 pound, 9 ounces, 2 drachms and 1 scruple how many grains? Therefore you must multiply 18 the Pounds by 12, and take in the 9 ounces, the Product will give 225 for ounces. Then multiply those ounces 225 by 8 the Product will give 1802 for [Page 112] Drachms, then multiply these Drachms by 3, and the Product gives 5407 for Scru­ples. Next multiply those Scruples by 20, the Product gives you 108140 for Grains, as was required.

[...]

Reduction of Wine Measure.

Rule 31. In 36 Tun of Wine how many Gallons?

[Page 113] Multiply by 4 and the Product is 144, then multiply 144 by 63, and the Product gives you 9072 Gallons, being so many contained in 36 Tun of Wine, as was re­quired.

[...]

But there is a shorter way to reduce Tuns into Gallons, which [...] is this. There is in a Tun 4 times 63 Gallons, viz. 252 Gallons in a Tun, therefore multiply 36 by 252, or 252 by 36, the Product will be 9072 Gallons as before.

Rule 32. In 68 Tun how many Runlets? at 14 Gallons per Runlet.

[Page 114] Reduce your Tuns into Gallons [...] bymultiplying the Tuns by 252, or 252 by the number of Tuns given 68, and the Product will give you how many Gallons are contained in 68 Tun, which if you divide by 14 the Number of Gallons in a Run­let, the Quotient 1224 is the an­swer, and so ma­ny Runlets of 14 Gallons will there be in 68 Tun, as was required.

Reduction of Beer Measure.

Rule 32. In 54 Barrels of Beer how many Gallons and Quarts?

If you examine the Table of Beer Mea­sure in the second Chapter you will find a [Page 115] Barrel of Beer to contain [...] 36 Gallons, and one Gal­lon 4 Quarts; therefore multiply your Barrels gi­ven 54 by 36, the Product thereof gives you 1944 Gallons, which multipli­ed by 4 gives you 7776 quarts, which are contain ed in 54 Barrels of Beer, as was required.

Rule 33. In 48 Kilderkins how many Gallons and Pints.

In the Table you will find that 9 Gallons is a Firkin, and 2 Firkins which is 18 Gallons is 1 [...] Kilderkin; therefore mul­tiply Kilderkins 48 by 18 and the Product will give you 864 Gallons. Now by the same Table I find 2 Pints is 1 Quart, 2 Quarts or 4 Pints 1 Pottle, and 2 Pottles or 8 Pints is 1 Gallon; therefore I multiply the Gallons 864 by 8 and there doth remain in the Product 6912 Pints, which are contained [Page 116] in 48 Kilderkins of Beer, as was re­quired.

Reduction of dry Measure.

Rule 34. In 72 Last of Corn how many Bushels and Pecks.

Because in the Table of dry Measure in the second Chapter you find 5 Quarters make a Wey, and 2 Wey [...] 1 Last; therefore I double the given number of Lasts 72, and make it 144, which I multiply by 5, and the Product is 720. There is a shorter way of redu­cing Lasts into Weys thus: 5 quarter being 1 Wey, 2 Wey or 10 Quarters is 1 Last; so that multiplying 72 Last by 10, by adding thereto 0 makes 720 Quarters, as before. Now by the said Table I find that 8 Bushels is 1 Quar­ter; I therefore multiply 720 Quarters by 8, and the Product gives me 5760 Bushels, and multiplying these 5760 Bushels by 4, the Product gives me 23040 Pecks, and so many Bushels and Pecks are contained in 72 Last of Corn, as was required.

Reduction of Long Measure.

Rule 35. I demand how many Yards, Feet, Inches and Barly Corns will reach from London to S. Davids, which is reck­oned 202 Miles.

Here I multiply 202 Miles by 8 to bring them into Furlongs, and the product gives me 1616 Furlongs; then I multiply these Furlongs by 40 to bring them into poles, and they produce 64640 poles or perches. Then I multiply them poles by 11, and the product gives me 711040 half Yards.

The half whereof 355520 are Yards, which Yards I reduce into Feet by multi­plying them by 3, and the product gives 1066560 Feet, then those Feet I reduce into Inches by multiplying them by 12, and the product gives 12798720 Inches. Lastly, I multiply those inches by 3, and the product thereof gives 38396160 Barly Corns.

[Page 118] [...]

[Page 119] Rule 36. The greatest Circumference of the Globe of the Earth and Sea as all other greater Circles are, are divided into 360 Degrees, each Degree into 60 Minutes or Miles.

[...]

[Page 120]

Reduction of Cloth Measure.

Rule 37. In 348 Yards how many quar­ters and nalls?

[...]

[Page 121] Rule 38. In 3462 Ells English, how many Ells Fle­mish?

[...]

Rule 39. In 5770 Ells Flemish, how many Ells En­glish?

[...]

[Page 122] This is the Reverse of the last Rule, but there is a shorter way than this to reduce Flemish Ells into English Ells, by multi­plying the number of Flemish Ells given by 6, and from the Product cut off the last Figure towards the right hand, and the Remainer shall give you the number of English Ells.

1. To Reduce mixt Numbers to an impro­per Fraction.

A mixt Number is composed of two Parts, or Numbers, whole and broken In­tegers and Fractions, the whole Numbers being composed of Integere, the broken being a Fraction annexed to the whole Number, as in the mixt Mumber 48 2/86, 48 is the whole number, and 12/16 is the bro­ken number, shewing that one Integer of 48 being divided into 16 parts, this 12/16 doth signifie that there doth belong to 48, 12 of the 16 parts, the Rule is thus:

Multiply the whole number, by the De­nominator of that Fraction annexed to him, and to the Product thereof, add the Numerator of the same Fraction, then set their sum, over the Denominator with a Line between them. And this new Fra­ction, [Page 163] though improper, shall be equal to the mixt Number given.

2. To Reduce a whole Number to an im­proper Fraction.

Multiply the whole Number given by the intended Denominator, and place the [Page 164] Product of the Multiplication over the De­nominator, and draw a Line between.

3. To Reduce an improper Fraction into a whole or mixt Number equivalent there­unto.

Divide the Numerator by the Denomi­nator, and the Quotient will give the whole Number equal thereunto, that is equal to the improper Fraction given; and if any thing remain make it the Numerator, and make the Divisor Denominator, as you may see in the following Example.

4. To Reduce a Fraction into its lowest terms equal thereunto.

If the Numerator and Denominator of the Fraction propounded be even numbers. Take the half of either as often as you can, and if either Numerator or Denominator be odd numbers, or falleth out to be so in the halfing, then divide them by any number that will divide both Numerator and De­nominator [Page 166] without any thing remaining in either, and when you have thus brought both Numerator and Denominator as low as you can, then this new Fraction shall be equal to the given Fraction.

Another way of Abbreviation.

There is a more excellent and quicker way to Abbreviate or Reduce a Fraction into its lowest term, which is thus: First, divide the Denominator by his Numerator, and if any thing remain let your Divisor be divided by that number, and if any thing do still remain of this Division, di­vide your last Divisor by this last Remain­der, and so do until o remain, and the Divisor of that last Division shall be the number which shall Abbreviate your Fra­ction by dividing both your Numerator and Denominator, and reduce them at once into the lowest terms.

V. To find the value of any Fraction in the known parts of Coyn, Weight, Mea­sure, &c.

Multiply the Numerator by the number of Units in the next inferior Denominati­on equal to an Unit of the same Denomina­tion, as the Fraction given divide the Pro­duct of that Multiplication by the Deno­minator, and the Quotient giveth the value in such parts as you did multiply by; but if any number doth remain, multiply such remainder by the number of Units in the next inferior or lesser Denomination, and divide by your last Divisor, or Denomi­nator of your given Fraction; do thus un­til you have 0 remaining, or until you have brought it as low as you can, and the seve­ral Quotients give the true value of the Fra­ction in several Denominations, and if a­ny thing remain of the last Division, make it a Numerator over the same Denominator in your Fraction given.

VI. To Reduce a Compound Fraction into a simple one of the same value.

A SimpleFraction consists but of one Nu­merator and one Denominator; but a Compound Fraction doth consist of many Numerators, and as many Denominators ⅓ of ⅝ of 7/16 of a pound Sterling is a Com­pound Fraction, because there are more Denominations than one represented in the Fraction.

When a Compound Fraction is given to be converted into a single Fraction, multi­tiply all the Numerators together, and the Product thereof make a new Numerator, then Multiply all the Denominators toge­ther, and the Product thereof make a new Denominator, which new Numerator and Denominator shall be a Single or Simple Fraction and equal to the Compound Fra­ction given.

VII. To Reduce Fractions of unequal De­nominators, into Fractions of the same va­lue having equal Denominators.

Multiply all the Denominators into one Sum, and the Product shall be the common Denominator, then multiply each Nume­rator into all the Denominators but his own, and the last Product put for a Nume­rator over the Denominator before found. And this new Fraction is equal to the Fra­ction whose Numerator you multiplied in­to the Denominators; do so by the rest, and you have your desire fulfilled.

VIII. To Reduce Fractions of one Denomi­nation into another.

Reduction of Fractions of one Denomi­nation to another, is two-fold, viz. Ascen­ding or Descending. Ascending when a Fraction is to be reduced from a lesser to a [Page 576] greater Denomination, as Fractions of pence into Fractions of pounds; Descen­ding when a Fraction of a greater Deno­mination is brought into a lesser, as Fra­ctions of pounds into Fractions of pence.

A Fraction being given to be reduced from a lower Denomination to a greater Denomination, make of it a compound Fraction, by adjoining the parts of the se­veral Denominations, between that and the Denomination you would reduce it to for Denominators, and a Unit for a Nume­rator, then by the 6th hereof reduce your Compound Fraction to a Simple one.

Another Way.

Multiply the Numerator of the first Fra­ction by the Denominators of the second and third Fractions, the Product thereof is a new Denominator; then multiply the Denominator of the first Fraction by the Numerators of the second and third Fra­ctions, and the Product is the new Nu­merator, and this Fraction is the fourth proportional Fraction required, and an­swer to the Question, which if it should fall out to be an improper Fraction, must be reduced into a whole or mixt number by the third of the eleventh.

Quest. 1. If ⅜ of a Pound weight cost 5/12 l. what will 9/21 of a Pound weight cost?

[...]

Having disposed of the given Fractions according to the four first Rules of this [Page 194] Chapter and the seventh, I proceed to the Solution thus. Multiply the Numerator of the first Fraction which is 3 by 12, and 12 the Denominators of the second and third Fractions, the Product will be 432 for a new Denominator. Then multiply 8 the Denominator of the first Fraction by 5 and 9 the Numerators of the second and third Fractions; the Product thereof is 360, which is a new Numerator. So I have found the fourth proportional Fra­ction 360/432 for answer to the Question pro­pounded, and by abbreviation is found equal to ⅚ or 16 s. 8 d.

Of Simple Interest.

Let it be required to know, if the In­terest of 100 l. for 12 Months be 6 l. what shall the Interest of 375 l. be for the same time.

Of Compound Interest.

Compound Interest is when Interest upon Interest is reckoned on any Sum of Mony.

Of Purchases and Annuties.

If you would purchase any Sum of Mony to be paid you any number of years to come, you must first know what it is worth in ready mony, allowing Interest upon Innerest. To effect this say, As 106l. is to 100 l. so 100 l. to the present worth of 100 l. to be paid at 12 Months or a year. Then for the second year say, as 106 l. is to 100 l. so is the present worth of one year to the present worth for two years, and in like manner is the present worth of any Sum of Mony given found out for any number of years to come, as by the follow­ing Example doth more plainly appear.