How to expresse the value of a num­ber giuen.

If a number be giuen, whose value is to be expressed, you shall vnderstand, that the figure next the right hand is the least in va­lew, and signifieth simply his owne valew, as the figure of 1 doth signifie but one, and the figure of 2 doth signifie but two, and the figure of 8 signifies but eight, and so of any other. And in the second place towards the left hand, euery figure is in valew tenne, so that the figure of one there doth signifie tenne, the figure of 2 twenty, the figure of 8 eightie, and so of all other: in the third place towards the left hand, euery figure is in valew one hundred, so that the figure 1 in that place signifies one hundred, the 2, two hundred, &c. In the fourth place, euery fi­gure is in valew one thousand, so there the figure of one signifies one thousand, the fi­gure 2, two thousand, &c. In the fifth place, euery figure is in value ten thousand: in the sixth place, one hundred thousand; and in the seuenth place, one thousand thousands, or one million: in the eight place, ten mil­lions: in the ninth place, one hundred mil­lions: [Page] in the tenth place, one thousand mil­lions, or one milliot; and so infinitely names may be giuen to the valew of euery pricke, as is vsuall in the second part of Arithme­tick, of Number, Square, Cube, sursolid, &c. or in Astronomicall Arithmetick, Primes, Seconds, Thirds, Fourths and Fifths, &c.

Now to expresse the valew of any num­ber giuen, set a prick with the pen ouer the fourth figure towards the left hand, and o­uer the seuenth, and tenth; and so ouer eue­ry third figure towards the left hand, to the end of your figures, as in this Exam­ple:

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Now begin and expresse the first foure figures towards the right hand, as if they stood alone, which are 2567, or two thou­sand fiue hundred sixty seuen. Then reade the figures belonging to the second prick, which are 430, as if they stood alone thus, foure millions three hundred two thousand fiue hundred sixty seuen: then take the three figures belonging to the third prick, which [Page] are 635, or sixe millions three hundred fif­tie foure millions three hundred and two thousand fiue hundred sixtie seuen: and so this whole summe is thus to bee read, two hundred thirty seuen thousand eight hun­dred fifty sixe milliots three hundred fiftie foure millions three hundred and two thou­sand fiue hundred and sixty seuen; and so of any other summe.

The explanation of these examples.

In the first example toward the left hand I begin with farthings, which are 3, which I set downe: then next 9 pence and 11 is 20, and 2 is 22, and 1 makes 23, and 2 makes 25, and 3 makes 28, and 11 makes 39 pence, or 3 shillings 3 pence; I set downe the 3 pence, and carry in mind the 3 shillings to be added to the place of shillings. Then adde the seuerall summes of shillings, which are 1. 1. 2 7. 8, the totall is 19, and the 3 in mind makes 22 shillings; set downe the 2 shillings, and keepe two tennes to be added to the tennes of shillings, which are 3 tens, [Page] which makes 5 tennes, or 50 shillings; set downe the odde tenne to the 2 shillings, which makes 12 shillings, & carry 2 pound for the forty shillings to the next place of pounds, which are 5. 9. 6. 4. 7. 2. 4, and the 2 in mind makes 39; leaue the 9 vnder the place of vnites, and carry 3 tennes in mind, and 7. 5. 5. 5. 8. 2. 2, totall is 37; set downe the 7 vnder the place of tennes, and carry 3 in mind for the 30 tennes, which is 3 hun­dred: then 3 in mind, and 2. 8. 8. 3. 1. 2. 3, totall is 30; set a cypher, or 0 in the place of hundreds, and carry 3 for the 30 into the place of thousands: then last of all, 3 in mind, and 3. 7. 1 makes 14 thousand, and be­cause it is the last summe, you must set them all downe, placing the 4 vnder the place of thousands, and the 1 one place more towards the left hand, and then the Totall summe of those particulars will be 14079 pound, 12 shillings, 3 pence, 3 farthings, as appeareth in the example; and in the like manner is the other example to bee cast vp into one Totall: and so I will here end with Additi­on of Coine, & put a seueral example of eue­ry table for the full tables & perfect vnder­standing of the said table, which are of great vse in all the seuerall rules of Arithmetick.

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[Page] Example of Weights.

The Explanation.

In the Haberdepoyse waight, 20 graines makes one scruple, 3 scruples one dram, 8 drams one ounce, 16 ounces one pound, 112 pound is one hundred of the Haberdepoyse weight, wherby is sold all kind of Merchan­dise vsuall in this Realme, and therefore in Addition of Waights Haberdepoyse, for e­uerie 3 scruples adde one dramme, and for euery 8 drams one ounce, and for 16 ounces 1 pound, for 28 pound one quarter of a hun­dred, and for euery 4 quarters one hundred. [Page] first, I begin with the drams in the first ex­ample to the right hand, which are 3. 1. 3, to­tall is 7 drames, which I note downe vnder­neath, because they are lesse then one ounce. Secondly, the ounces are 3. 7. 2. 12. 8. totall is 32 ounces, or 2 pound, because 16 ounces is one pound; which two I set vnder the place of pounds with a light touch of the penne for to remember it the better, and place a Cypher in the place of ounces.

Thirdly, the pounds are 2. 10. 1. 12. 10. 17 totall is 52 pound, which is one quarter of a hundred, and 24 pound, place 24 pound vnder the place of pounds, and put one quarter, as before in the place of quarters of hundreds.

Fourthly, 1. 3. 1. 2. 3 quarters, are 10 quarters, or 2 hundred and 2 quarters, or halfe a hundred; place 2 quarters in the place of quarters, and put ouer 2 into the place of hundreds for the 8 quarters.

Then 2. 7. 2. 7. 3. 8. 7 makes 36 hundred, place 6, and carry 3 for the 30: then say, 3. 1. 2. 1. 3. 1. 2, totall is 13; place 3 there, and carry one for the 10, which one in mind, and 1. 1 makes 3, which set downe, and the total is 336 hundred, 2 quarters, 24 pound, 0 ounces, 7 drammes; and so the other ex­ample [Page] is in the same manner to be cast vp, and so of all other.

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The explanation of these Tables, and the ex­amples following.

First, in the example of Acres, Roods and Perches; for 40 Perches put 1 Rood into the place of Roods, and for euery 4 Roods one Acre.

Secondly, for euery 4 quarters of Inch, take 1 Inch, and for euery 12 Inches 1 foot, and for euery 3 foote, one yard.

Thirdly, for 16 pints take one pecke, and for euery 4 peckes one Bushell, into the place of Bushels.

Fourthly, for euery 8 pints of liquid mea­sure, take one Gallon, and for euery 63 Gal­lons one Hogshead.

Fifthly, in the example of time; for 60, minuts take one houre, and for 24 houres one day, and for 365 dayes, one yeere.

Sixthly, for 4 nayles take one quarter of a yard, and for 4 quarters one yard, &c.

Lastly, in the example of motion, for 60 thirds, take 1 second, & for 60 seconds take one minute, and for 60 minutes take one de­gree, and for 30 degrees take one Signe.

And this is the vse of these Tables in Ad­dition and Subtraction; for looke what you [Page] carry ouer in Addition, that you must bor­row in Subtraction, I wil heare adde exam­ples of euerie kind, leauing the Reader to exercise himselfe by the Rules before taught.

Example.

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The Proofe of Addition.

The proofe of Addition is made by Sub­traction; for if you subtract the numbers which you added from the totall of the Ad­dition, [Page] there will remaine nothing, if the worke be truly done.

Example.

First, adde together the greatest Summes in valew in the place of hundred thousands, which makes 23, which take from 25, and there will remaine 2: then the figures in the fifth place, 26 taken from 27, there will re­maine 1. Thirdly, the figures in the place of thousands, makes 17, which taken from 19, leaues 2: then 19 in the place of hun­dreds taken from 20, leaues 1: and againe, 13 in the place of tennes from 15, leaues 2: and lastly, 20 in the place of vnites from 22 ponnd, leaues 2 pound: then 49 shillings from 2 pound 11 shillings, leaues 2 shil­lings▪ [Page] also 2 shillings 3 pence in the place of pence, from 2 shillings 4 pence, leanes 1: and last of all, 4 farthings from 1 penny, leaues nothing, which prooues the worke to be truly wrought.

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The second proofe of Addition.

Cut of the vppermost numbers with a dash of the pen, and adde the remayner into one Totall; and then subtract that sum from the whole totall, and the remayner will bee the numbers which you cut off, if the worke be true, else not.

Example.

[Page]And so much shall suffice to haue spoken of Addition, and the proofe thereof.

Questions of Addition.

What number is that, to the which if you doe adde 45, the totall will be 357.

Answer: Subtract 45 from 357, remaines 312.

Example.

What three numbers are those, to which if you adde 27, 36, and 45, their products shall be equall, and the summe arising shall be 120.

What number is that, to the which if you do adde 354 pound, 7 shsllings, 9 pence, the totall will bee 512 pound, 15 shillings, o penny. Answere: Subtract 354 pound, [Page] 7 shillings, 9 pence, from 512 pound, 15 shil­lings, o penny, and the remainder will bee 158 pound, 7 shillings, 3 pence, which is the number that you doe seeke.

Example.

The explanation of these examples.

In the first example of Coyne, begin your subtraction at the right hand, saying; 1 farthing from 3 farthings, leaues 2 far­things, which note downe vnder the 1 far­thing. Then 10 pence from 11 pence, leaues 1 penny. Thirdly, 19 shillings from 17 shil­lings you cannot haue, therefore take one pound, or 20 shillings, and adde to 19 shil­lings, saying, 19 shillings from 37 shillings, rests 18 shillings, which note downe. Then 1 that you borrowed, & 3 pound, is 4 pound from 6 pound, leaues 2 pound to set downe vnder 3. Then 8 from 8 leaues nothing, place there a Cypher, or 0 vnder 8. Then 5 from 7 rests 2; then 2 from 9 leaues 7, [Page] which also note againe; 9 from 8 cannot be taken, then make it 10 more, and say 9 from 18 leaues 9, which set downe: and last of all, 1 borrowed and 6 is 7, from 7 leaues no­thing, and the worke is ended, and the re­mayner will bee 97202 pound 18 shillings 1 penny 2 farthings, as appeareth in the ex­ample before going.

The exposition of the second example.

First, take 15 ounces from 10, which can­not be, then adde 1 pound, or 16 ounces to 10, makes 26; then say, 15 from 26 leaues 11 ounces, which note downe: then 1 bor­rowed and 24 is 25, from 27 pound leaues 2 pound remaining; then 2 quarters from 3 quarters, leaues 1 quarter remaining; then 8 from 7 cannot bee, therefore take 8 from 17, rest 9, which note downe: then one bor­rowed and 3 makes 4, from 12 rests 8, and the worke is done, and the remaine is 89 hundred 1 quarter 2 pound 11 ounces.

3 Example.

First, take 59 minutes from 56 minuts cannot be, but then take 59 minuts from 60 [Page] minutes, or one houre, and there will re­maine 1 minute, which adde to 56 minutes, and that will make 57 minutes, which note downe in the place of minutes: then 1 bor­rowed and 15 houres makes 16 houres, which taken from 20 houres leaues 4, which note vnder the 15; and then 2 dayes from 0 cannot be, but 2 from 10, and there will re­maine 8, which note downe: then 1 borrow­ed and 2 makes 3, from 4 leaues 1; also 1 from 3 leaues 2: lastly, 1 from 8 leaues 7, and 8 from 11 leaues 3; then 1 borrowed and 15 makes 16, from 16 leaues nothing, and the remayner will bee 37 yeares 218 dayes 4 houres 57 minutes; the like is done in the other example of Motion, and there­fore here needlesse to be rehearsed.

To subtract from a Vnite.

Set downe with your pen a vnite in any place, adding Cyphers vnto it, and the seuerall numbers which you will subtract from it of pounds, shillings and pence right vnderneath: then note what each seuerall number of your lowest numbers doth want of 9 vnto the place of vnites, and set that right vnder for the remayner: and lastly, note what your shillings and pence [Page] doth want of 20 shillings, and set that downe for your remaynor, and the worke is ended.

Example.

The proofe of Subtraction.

The surest proofe of Subtraction is made by Addition: for if you doe adde the num­bers remaining, vnto the numbers deduct­ed, they will returne your former Summe, if the worke be truly wrought, as wil appeare in the proofe of all the seueral examples be­fore going, and therefore here againe in this place needlesse to be rehearsed. Only I will adde one for examples sake.

In the last example, the numbers which did remaine, were 767135 pound 2 shillings 9 pence, and the numbers deducted, 232864 [Page] pound 17 shillings 3 pence; these two num­bers added together, ought to make a vnite in the seuenth place; wherefore I adde 9 pence to 3 pence, makes 1 shilling; and 1 shilling to 17 shillings, makes 18 shillings, and 2 shillings makes 20 shillings; then 1 and 4 is 5, and 5 is 10, which is one in the next place: then 1 and 3, and 6 is 10; and 1 1.8 makes 10, and 1.7.2 makes 10, and 1.6.3 is 10, and lastly 1.7.2 makes 10, or one vnite.

The exposition of this example.

First, 7 by 1 is 7, which note downe: then 7 by 0 is nothing, set down a 0 in that place: and next 7 by 5 is 35, set 5, and carry 3: then 7 by 6 is 42, and 3 is 45, place 5, & carry 4: then 7 by 8 is 56, and 4 is 60, set downe a 0, and carry 6 againe: 7 by 2 is 14, and 6 makes 20, set downe a 0, and carry 2: then 7 by 3 is 21, and 2 makes 23, place 3, and carry 2: then 7 by 0 is 0, leaue the 2 in that place: then lastly, 7 by 7 is 49, being the last number set downe all the 9 vnder 7, and the 4 one place more to the left hand, and [Page] the worke will then stand thus.

[...]

Secondly, cancell 7, and then say; 5 by 1 makes 5, place that 5 vnder the 0; and then 5 by 0 is 0, place a 0 vnder the 5 in the next place; and then 5 by 5 is 25, set downe 5, and carry 2: then 5 by 6 is 30, and 2 makes 32, set downe 2, and carry 3: then 5 by 8 is 40, and 3 makes 43, place 3, and carry 4: al­so 5 by 2 is 10, and 4 makes 14, set downe 4, and carry 1: then 5 by 3 is 15, and 1 makes 16, set downe 6, and carry 1: then 5 by 0 is 0, set downe the 1 there: last of all, 5 by 7 is 35, set them all downe, and the worke will then stand thus.

[...]

Thirdly, cancell the 5, and then say, 0 by 1 [Page] is 0, place a 0 vnder the 0 of your multipli­er, & thē proceed to the next figure of your multiplier, which is 2, saying, 2 by 1 is 2, place the 2 vnder the said 2 of your multi­plier: then 2 by 0 is 0, which set downe: then 2 by 5, makes 10, set downe a 0, and carry 1: then 2 by 6 is 12, and 1 is 13, set downe 3, and carry [...]: also 2 by 8 is 16, and 1 is 17, set downe 7, and carry 1: also by [...] is 4, and 1 makes 5, which set downe: againe, 2 by 3 is 6, which set downe: lastly, by 7 is 14, which set downe, and the worke will stand, as in this example.

[...]

Fourthly, cancell the 2, and say, 3 by 1 is 3, which place right vnder the said 3: then 3 by 0 is 0, and worke in all respects as be­fore, and the worke being ended, will stand thus.

[Page] [...]

I will here adde some few examples to be wrought by the pen, without any trou­bling of the memory with bearing ought in mind.

Example. [...]

[Page] 2. Example. [...]

The explanation of the worke by the pen, without charging the me­morie.

The first example. [...]

First, I multiply al the figures of my mul­tiplicand [Page] by 7, the lowest figure of my mul­tiplier, saying, 7 by 8 is 56, put 6 vnder the 7, and 5 vnder the 8: then 7 by 6 is 42, leaue the 2 vnder 5 last placed, and set the 4 one place more towards the left hand vnder the 9: then 7 by 9 is 63, leaue 3 vnder the 4 last placed, and set 6 one place more to the left hand vnder 7: then 7 by 7 is 49, leaue 9 vn­der the 6 last placed, and the 4, set one place more to the left hand vnder the 8: lastly, 7 by 8, makes 56, leaue 6 vnder the 4, & place 5 one space more to the left hand, as before, then cancell 7 of your multiplier, and the multiplication by the first figure is ended, and the worke will stand thus.

Example. [...]

Then for the second worke, say, 8 by 8 is 64, place 4 vnder the said 8, and put 6 vnder the next figure 3: then 8 by 6 makes 48, leaue 8 vnder 6, and put 4 vnder the next 9: [Page] and so working in all respects as at the first, and your second worke will stand thus, as in this example.

[...]

Lastly, cancell 8 your multiplier, and then multiply by 9, as is before taught, placing the first figure of your product vnder the fi­gure multiplying, and the worke being en­ded, it will stand thus; and lastly, gathering the totall by addition, it is 86824416; as in this example.

[...]

[Page]There is no difficulty in this kind of working, but onely when there falls a 0 in in the multiplicand, or multiplier; for if there be a cypher then you must fill vp the places as you worke, either with pricks, or cyphers, as if you had figures to set in their places, and the rest of the work is as before, is taught in the third example; but I will here adde one example, hauing all the diffi­culties that may happen, for the better vn­derstanding hereof.

Example. [...]

[Page] Example. [...]

How to multiply, and to bring the pro­duct in the last line.

Place your numbers right one vnder the other, as in the common way; then make a right line somewhat distant from the first numbers with your pen, as in the example following.

[Page] [...]

Then begin and say, 7 by 8 is 56, place the 6 vnder the line vnder the 7, and the 5 aboue the line in a smaller figure in the next place towards the left hand: then 7 by 6 is 42, and the 5 aboue the line makes 47, leaue 7 vnder the 8, and set the 4 againe aboue the line: then 7 by 9 is 63, and the 4 aboue the line makes 67, place 7 there, and set the 6 in the next place aboue the line: then 7 by 7 is 49, and 6 aboue the line makes 55, leaue 5 there, and put 5 againe ouer the line: lastly, 7 by 8 makes 56, and the 5 last placed makes 61, place that whole summe vnder the line, and the worke wil stand, as aboue in the ex­ample.

Secondly, draw a line againe a little dis­stant, as before from the last product, as in the example following.

[Page] Example. [...]

Then say, 8 by 8 is 64, and 7 makes 71, place 1 vnder the 7, and set 7 aboue the line: then 8 by 6 is 48, and the two seauens be­tweene lines makes 62, place 2 vnder the 7, and set 6 againe ouer the line: then 8 by 9 is 72, and 6 makes 78, and 5 makes 83, place 3 vnder the line, and 8 aboue the line; then 8 by 7 is 56, and 8 makes 64, and 1 makes 65; place 5 vnder the line, and set 6 aboue: lastly, 8 by 8 is 64, and 6 makes 70, and 6 makes 76, place them both downe; and the worke will stand as aboue in the example.

Thirdly, draw a line againe, as before, a little distant from the last product, as in this example.

[Page] Example. [...]

Thirdly, say 9 by 8 is 72, and 2 makes 74, place 4, and put 7 ouer the line: then 9 by 6 is 54, and 10 makes 64, place 4 vnder the line, and put 6 aboue: then 9 by 9 is 81, and 11 aboue makes 92, leaue 2 vnder the line, and 9 ouer the line: then 9 by 8 is 63, and 15 makes 78, leaue 8 vnder the line, and 7 aboue: lastly, 9 by 8 is 72, and 14 makes 86, place them both vnder the line, and then bring downe the two figures which are cut off by two right downe lines, which are 1 and 6, and the worke is ended, and the worke will stand, as appeareth in the example a­boue, [Page] and the totall product is in the last line, 86824416; and this doth not charge the memory, for al the figures are set downe in view, and to bee seene at the first sight, and this is the second kind of multiplicati­on, without charging of the memory.

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The third Example of Diuision.

The second kind of Diuision is this: first, place your diuidend & diuisor as in the for­mer Examples, & then hauing found out the figure of your quotient, begin with the least figure of your diuisor towards the right hād first, and multiply that by the figure of the quotient found, and then subtract the sum of the multiplication of that figure from the fi­gure aboue the same, if it exceed not 9; but if the product be aboue 9, then for euery 10 [Page] beare one in mind to bee added to the pro­duct of the multiplication of the second fi­gure of your Diuisor by the quotient; and so in all respects worke for euery other fi­gure, and you shall need make no more fi­gures aboue your Diuidend then necessitie shall require, as for example.

I would diuide the product of the mul­tiplication in the former Chapter of 79648039 by 8976, which was found to be as followeth, viz. 714920798064 by 8976: first, I place my Diuidend and Diuisor as followeth.

[...]

Then first I seeke how often I may haue 8 in 71, I find by triall I can haue it but 7 times▪ then hauing placed 7 in the Quoti­ent, I first multiply 6, the least, or smallest figure in value by 7, makes 42; then I say, 42 from 42, rest 0, and carry 4 for the for­tie in mind; then I cancell the 2 ouer the 6, and placea 0 in the roome ouer it. Second­ly, I say, 7 by 7 is 49, and 4 in mind makes [Page] 53, from 59 leaues 6, and carry 5 [...] cancell the 9, and place 6 ouer it. Thirdly, 7 by 9 is 63, and 5 in mind is 68, from 74 leaues 6, and carry 7, cancell the 4, and place 6 aboue it: also 7 by 8 is 56, and 7 makes 63, which taken from 71, leaues 8 remaining, which 8 place ouer the 1, and cancell the 71, and the first worke will stand thus.

[...]

Secondly, I remoue my Diuisor 8976, and seeke, how many times I may haue 8 in 86, [...] find 9 times; then I multiply 6 by 9 placed in the Quotient, makes 54, which taken from 60, leaues 6; place 6 aboue the first 0, and carry 6 for the 60: then say, 9 by 7 is 63, and 6 in mind makes 69, from 70 leaues 1, and carry 7 in mind; cancell the 0 ouer the 7, and place the 1 ouer the 0. Againe, 9 by 9 is 81, and 7 in mind is 88, which taken from 96, leaues 8 to bee placed aboue the first 6, and carry 9 in mind: lastly, 9 by 8 is 72, and 9 makes 81, which taken from 86, leaues 5 to bee placed aboue the 6, and the worke will stand as followeth.

[Page] Example. [...]

Thirdly, againe I remoue my Diuisor, and seeke how many times 8 is in 58, and I find I can haue it but 6 times, which I place in the Quotient: then I say, 6 by 6 makes 36, from 37 leaues 1 aboue 7, and carry 3: then 6 by 7 is 42, and 3 is 45, from 46 leaues 1, aboue the 6, and carry 4: againe, 6 by 9 is 54, and 4 makes 58, from 61 leaues 3 aboue the 1, and carry 6. Lastly, 6 by 8 is 48, and 6 makes 54, from 58 leaues 4, and the worke stands thus, as in this example.

[...]

[Page]Fourthly, I remoue my Diuisor, and seeke how oft I may haue 8 in 43, and I find but 4 times, I place 4 in the Quotient. Then 4 by 6 makes 24, from 29, leaues 5, and carry 2, set 5 ouer the 9: then 4 by 7 is 28, and 2 makes 30, from 31, leaues 1, and carry 3. A­gaine, 4 by 9 is 36, and 3 makes 39, from 41 leaues 2, and carry 4. Lastly, 4 by 8 is 32, and 4 is 36, from 43, leaues 7, and the work will then stand thus.

Example. [...]

Fifthly, I remoue my Diuisor, and seeke how oft I may haue 8 in 72; I find 8 times, which placed in the quotient, I multiply 6 by 8, makes 48, from 48, leaues 0, and carry 4 [...] then 8 by 7 makes 56, and 4 is 60, from [Page] 65 leaues 5, and carry 6: then 8 by 9 is 72, and 6 makes 78, from 81, leaues 3, and carry 8: then 8 by 8 makes 64, and 8, is 72, from 72 leaues 0 remaining, and the worke will stand thus.

Example. [...]

Sixthly, I remoue my Diuisor, and seeke how oft I may haue 8 in 3, which I find not once; I place a 0 in the Quotient, and re­moue my Diuisor one place more and seeke how many times 8 is in 35; I find I can haue it but 3 times, I place 3 in the Quoti­ent beyond the 0 last placed, and say, 3 by 6 is 18, from 26 rests 8, and carry 2: then 3 by 7 is 21, and 2 is 23, from 30 leaues 7, and [Page] carry 3: againe, 3 by 9 is 27, and 3 is 30, from 30 leaues a 0, and carry 3: also 3 by 8 is 24, and 3 is 27, from 35 leaues 8; and the worke will stand thus.

Example. [...]

Lastly, I remoue my Diuisor, and seeke how oft I may haue 8 in 80; I find 9 times, I place 9 in the Quotient, and say, 9 by 6 is 54, from 54 leaues 0, & carry 5: then 9 by 7 is 63, and 5 is 68, From 68 leaues 0, & carry 6: Then 9 by 9 is 81, and 6 is 87, from 87 leaues 0, and carry 8: last of all, 9 by 8 is 72, and 8 makes 80, from 80 there will re­maine nothing but cyphers, and the worke is quite ended, and will stand, as in the ex­ample following.

[Page] Example. [...]

The fourth and last kind of Diuision, is the most absolute, speedy, and easie, not charging the memory at all, with keeping any numbers in mind; and also the proofe of your work is made by Addition, and not by multiplication, as hath heretofore been commonly vsed, but the figures of your worke are by Addition, the proofe of your worke, as shall appeare by examples follow­ing.

The third Worke.

First, place your Diuidend betweene two paralel lines, and your Quotient at the right [Page] side of your Diuidend, behind a crooked line, as before; then place your Diuisor next to the left hand of your Diuidend, behind a perpendicular line: and lastly, marke how many figures your Diuisor hath, and in the roome of those figures place ciphers vnder the figures of your Diuidend, so many as your Diuisor hath figures, as in the last ex­ample; which I will againe repeate in this place, and work it by this kind of Diuision, making the proofe of the work by Addition of the same figures.

Example. [...]

First, I point to the first cipher towards the left hand, and seeke how oft I may haue 8, the greatest figure in value of my Diui­sor, hauing respect to the other figures of my Diuisor, to take them also as often, out of the figures aboue, and I find I can haue it but 7 times, which 7 I place in the Quoti­ent, [Page] and by that 7 I multiply my Diuisor 8976, saying first, 7 by 6 is 42, place the 2 vnder the lowest cypher towards the right hand, and carry 4: then 7 by 7 is 49, and 4 is 53, set 3 vnder the next place to the left hand, and carry 5: then 7 by 9 is 63, and 5 is 68, place the 8 in the next place, and car­ry 6. Lastly, 7 by 8 is 56, and 6 in mind makes 62, which place downe in their pla­ces, and the totall is 62832, to be subtracted from 71492, and there will remaine 8660; and the worke will stand thus.

Example. [...]

Secondly, I cancell the first cypher to the left hand, and place one cypher more to­wards the right hand, vnder the 0, and then I point againe to the first cypher, and see how oft I may haue 8 in 86; I find 9 times, and [Page] placing 9 in the Quotient, by it I multiplie 8976 my Diuisor, placing the lowest figure in value vnder the lowest cipher to the right hand, and the rest in order, and I find the product to bee 80784, which taken from 86600, leaues 5816 remaining, and then your worke will stand, as in this.

Example. [...]

Thirdly, I cancell my Diuisor, or one Cy­pher, and place one cypher more vnder 7, and then seeke how oft I may haue 8 in 58, which I find 6 times, and by it I multiply my Diuisor 8976 makes 53856, which ta­ken from 58167, leaues 4311, and the work will stand as followeth.

[Page] Example. [...]

Fourthly, I cancell one cypher, and place a cypher vnder 9, and then seeke how oft I may haue 8 in 43, which I find but 4 times, which place in the Quotient, and by it I multiply my Diuisor 8976, makes 35904, which taken from 43119, leaues 7215.

[Page] Example. [...]

Fifthly, I cancell one cypher, and place a cypher vnder 8, and fee [...]e how oft [...] is in 72; [...] find 8 times, which placed in the Quoti­ent, I multiply my Diuisor 8976 by it makes 71808, which taken from 72158, leaues 350, and the worke stands, as in the example following.

[Page] Example. [...]

Sixthly, I cancell one cypher, and place another vnder the 0, and seeking I find I cannot haue 8 in 3; therefore I place a 0 in the Quotient. Seuenthly, I cancell one cy­pher, and place one other vnder the 6, and seeke how oft I may haue 8 in 35; I find but 3 times, and placing 3 in the Quotient, by it I multiply 8976, makes 26928, which taken from 35006, leaues remaining 8078.

Lastly, I cancell the next cypher, and doe place another vnder the last figure of my Diuidend 4, and seeke how oft I may haue 8 in 80; I find 9 times, and then placing 9 in [Page] the Quotient, I multiply my Diuisor 8976, and the Quotient is 80784, equall vnto the numbers aboue, and so being subtracted from the numbers aboue, leaues o remai­ning, and the worke is ended, and will stand thus.

Example. [...]

The proofe of this Diuision is made by Addition of the figures, vnder the line or [Page] Diuidend, for if they returne your former Diuidend, the worke is true wrought; or o­therwise be sure some error is in your work, if there remaines any fraction after your worke is ended, then it is to bee added into the lower figures in their seuerall places, as shall appeare by examples following.

[Page] [...] [Page] [Page] Here in this example following, working according to this latter forme of worke, there is aduantage to be taken; if the figures of the Quotient bee well noted, as here the fourth figure of the Quotient is 7, the Pro­duct of the Diuisor multiplied by it is 1438816, and also the eleuenth figure of the Quotient is 7, so that comming to multiply the Diuisor againe by that 7, I neede but take the Product of the first multiplication by 7, which is 1438816, and so place them in their seuerall places, as in the example, and so likewise there is 3 in the Quotient two times, so that for the latter multiplica­tion, I take the first product 539556, and sa [...]e that labour of multiplication of the Di­uisor by 3: and so of any other figure com­ming into the Quotient more times then once, as by the example before going will appeare.

Place the great example following, in this place.

[Page] Example▪ [...]

[Page] [Page] Example. [...]

[Page] Example. [...]

How to diuide by a Ʋnite with Cyphers.

If you wil diuide by 10, or by 100, or 1000, or with any other vnite with cyphers, one or more; doe but cut off so many figures from the right hand of your Diuidend, as there are cyphers in your Diuisor, and the remaines is your Quotient.

Example.

If you would diuide 786589 by 10, cut off the last figure 9, and the residue is your Quotient 78658 2/10; or if you will diuide by 100, cut off two figures, and the Quoti­ent [Page] will be 7865 [...]2/100; or by 1000, and the Quotient will be 786 [...]2/1000; and so of all o­ther.

[...]

If you will diuide the Product of 1999 squared; that is to say, multiplied in it selfe, which is 3996001 by 1999, for expedition of worke, after you haue found the first fi­gure of the Quotient 1, and taken that out, I find the next figure will be 9, which taken out, the third and fourth figures are also found to be 9, and so you need not make mul­tiplication for euery seuerall 9, but the first will serue for all, as in the example follow­ing.

[Page] Example. [...]

[Page] Example. [...]

Briefe Rules by Multiplication and Diuision.

If you multiply any number of nines; as if you will multiply, or square 5 times 9 by 5 times 9, then place your nines in this or­der following.

[Page] Example. [...]

Then say, 9 times 9 is 81, place the 1 vn­der the first 9 to the right hand, and then subtract the 1 from the first 9 to the left hand, and adde the cyphers betweene, and the Product is ended, and is 9999800001, as appeareth.

The proofe of the worke after the or­dinary way. [...]

To multiply any number by 9.

Adde a o to the number you intend to multiply, and then set the same numbers vn­der them, and subtract them from the vpper­most, and the remaines is the Product of that multiplication by 9.

Example. [...]

To multiply by ½, or ⅓, or ¼, or ⅕.

If you will multiply 856 by 24½, first, multiply 856 by 24, makes 20544; and then for one halfe, take halfe 856, which is 428, and adde into the former summe, makes the totall 20972.

[Page] Example. [...]

What number is that, which being diui­ded by 24, the Quotient will be 856. An­swer, multiply 856 by 24, makes 20544 for the number that you seeke.

Example. [...]

[Page]There is a plot of land containing 848 Perches, the one side is 24, what must the o­ther be. Answere, Diuide 848 by 24, the Quotient is 35 ⅓ for the other side.

[...]

If you will diuide the Product of 5 times 9 squared, which is 9999800001, by 5 nines, then set the Diuisor right vnderneath the Diuidend, and adde them together, and cut off the 5 cyphers from the Product, and the residue is the Quotient.

Example. [...]

[Page]What number is that, which being mul­tiplyed by 15, the totall will be 756. Answ. diuide 756 by 15, and the Quotient is 50 6/159, or 2/5, for the answer, or number you do seeke.

Example. [...]

There are 825 men, to march 15 in one ranke, how many files will they make. De­uide 825 by 15, it makes 55 files.

Example. [...]

[Page]There is 948 pound of powder to bee im­ployed in an Assault of Battery with 6 pie­ces of Ordinance; the first piece shooteth 4 pound, the second 5, the third 6, the fourth 7, the fifth 8, the sixth 10 pound, the question is, how many Shootes each piece may make, to make an equall number of Shotts. An­swere; diuide 948 by 40, and it makes 23 Shootes, and there will remaine 28 pound.

Example. [...]

Reduction of Coyne.

In 87652 pound, how many pence: in the Table of Coyne I find 240 pence makes one pound, so that in multiplying 87652 pound by 240, makes the summe of pence desired.

[Page 3] 1. Example.

[...]

2. Example.

In [...]759 pound, 17 shillings, 8 pence, how many shillings, pence, and farthings.

[Page 4] [...]

3. Example.

In 3785437289 farthings, how many pounds, shillings, and pence: diuide by 960 farthings, because 960 farthings makes one pound starling, & the remainer is farthings, which diuided by 48, the farthings in one shilling, makes 3943163 pound, 16 shillings 10 pence, ¼.

[Page 5] [...]

How to bring pounds, shillings, and pence at: he first worke by Di­uision.

To bring pence at the first worke into pounds, shillings, and pence: adde a 0 to your number of pence, and diuide that sum by 240, makes pounds, and the last fi­gure will be primes, euery vnite in value 2 shillings, and the remainer alwaies lesse then 24 pence, or one prime.

Example.

In 902372 pence, how many pounds, shil­lings, and pence; adde a 0, makes 9023720, which diuided by 240 pence, makes &c.

[...]

[Page 7] 2. Example.

In 75000837504 pence, how many pounds, shillings, and pence: adde a cypher, or 0.

[...]

How to bring farthings into pounds, shillings, and pence at the first worke.

To bring farthings into pounds, shillings and pence at one worke: adde a 0 to your number of farthings, and diuide the summe by 960, the number of farthings in one pound sterling, makes pounds; and the last figure of your Quotient will bee primes e­uery one in value 2 shillings: and if there remaine 48, it is one shilling, or take 48 from the remainer for one shilling, the rest are farthings lesse then 48.

[Page 8] Example. [...]

Totall is 756. l. 12. s. 58. q. or 13. s. 2. d. 1. ob.

In 3785437248 farthings, how many pounds, shillings, and pence, adde a 0, and diuide by 960, makes [...]943163 pound, 8 primes, or 16 shillings, 0 pence▪

[...]

How to bring pence into pounds, shillings, and pence another way.

Diuide your number of pence by 4, and the remainer is pence, then that Quotient by 6, and the remainer is groats, alwaies lesse then 6 groates, or one prime, or 2 shil­lings; and the latter Quotient, cutting off your Primes, is pounds, and so you haue pounds, shilling, and pence.

Example.

In 785697 pence, how many pounds, shil­lings, and pence, makes 3273 pound, 14 shil­lings, 9 pence.

[...]

If you will bring farthings into pounds, shillings, and pence: diuide first by 16, and the remainer is farthings, alwaies lesse then 16, or one groate; and then againe by 6, makes pounds, shillings, and pence, as be­fore, cutting off the prime line.

[Page 10] Example.

In 8735672 farthings, how many pounds [...]hillings, and pence.

[...]

Reduction of Waights.

In 8756 hundred, 3 quarters, 24 pound, 12 ounces Haberde poyce, 16 ounces to the pound, and 112 pound to the hundred, how many pounds and ounces.

[Page 11] Example. [...]

In 1569 [...]492 ounces Haberdepoyse, how many hundreds, quarters, pounds, and oun­ces; finde how many ounces makes 112 pound, in multiplying 112 pound by 16 ounces, makes 1792 ounces; by which di­uide, makes, as in the example following.

[Page 12] [...]

Reduction of Measures.

In 2356 Acres, 3 Roodes, 27 Perches, how many Perches in all.

[Page 13] Example. [...]

In 765437 Perches, how many Acres, Roodes, and Perches: diuide by 160.

Example. [...]

Reduction of Time.

In 356 yeares, 24 dayes, 36 houres, and 22 minuts; how many dayes, houres and mi­nutes.

Example. [...]

[Page 15] The Proofe.

In 187150342 minuts, how many houres, dayes, yeares, and minutes.

[...]

Reduction of Motion.

In 11 Signes, 34 degrees, 25 minutes, 36 seconds, 24 thirds; how many fourths.

Example. [...]

[Page 17] The proofe.

In 4722971040 fourths, how many signes degrees, minutes, seconds, thirds, & fourths.

Example. [...]

Questions by Reduction.

2. To find the summe of a Progression.

Marke first how many seuerall places there be in your Progression, and note that downe; then adde the first number of the Progression to the last: then multiply halfe those two numbers by the whole number of the places, or else halfe the number of the places, by the whole number of the first and last terme added into one summe, and both [Page 28] waies will produce the totall summe of that Progression.

Example.

There is a Progression beginning at 4, and is continued vnto 44, increasing by 4. First, set downe the numbers of that Pro­gression, beginning at 4, and ending at 44.

Here the first terme is 4, and the last is terme is 44, which added together, makes 48, the one halfe, which is 24, multiplied by a 11, the whole number of places makes 264 the totall.

[Page 29] Example. [...]

How to find the latter terme of a Progression.

If you would know the latter terme of a Progression of 100 termes, increasing by 3, and beginning at 10; take one terme from 100 termes, & there will remaine 99; which multiply by 3, the excesse or difference of the increase, makes 297; to the which if you adde the first terme 10, makes 307 for the 100 terme of that Progression.

[Page 31] Example. [...]

Or otherwise take the Excesse 3 from the first terme 10, and there will rest 7, which note a part, then multiply, the number of places 100 by the excesse 3, makes 300; to which adde the 7, makes 307, as before.

Example. [...]

To find the number of termes.

There is a Progression, whose first terme is 2, the last terme, 156; and the excesse was 2, I would find the number of termes.

Subtract the first terme from the last, and diuide the remayner by the excesse, the quo­tient is the number of termes, wanting but one. Example: 2, the first terme from 156, the last leaues 154; which diuided by 2, makes 77; to which adde 1, makes 78, the number of termes.

[...]

How to find the Excesse, or diffe­rence.

Subtract the first terme from the last, and diuide the remainer by one lesse, then the number of the Termes, and the Quotient will be the Excesse or difference.

[Page 32] Example.

Subtract 10, the first terme, from 307 the last terme, there will remaine 297; which diuide by 99, one lesse then the number of termes, which are 100, makes 3 the excesse.

[...]

To find any middle terme.

Subtract a vnite from the number of the terme you would know, and multiply the remainer by the difference, and to that pro­duct adde the first terme, and the totall is the terme you doe seeke.

Example.

To find the 30 terme in the last example of 100 termes, subtract 1, rests 29; which multiply by 3, the Excesse makes 87; to which adde the first terme 10, makes 97 for the 30 terme of that Progression.

[Page 33] Example. [...]

How to find what number shall begin and finish a Progression.

To the number of termes adde one, which multiply by halfe the number of termes, and by the product diuide the sum of the pro­gression, and the quotient will be the first terme, and excesse of that progression.

[Page 30] Example.

At 16 payments 353 pound, 12 shillings is to be paid, the question is, what number must begin, and continue the progressi­on.

First, the money 7072 shillings; then to 16, the number of termes, adde 1, makes 17; which multiply by 8, halfe the number of termes, makes 136 for Diuisor; by which diuide 7072, and the quotient is 52 shillings for the first paiment and excesse, and by the same the other payments are found.

Example. [...]

[Page 37] Example. [...]

To find any middle Terme, or any other Terme in a Geometricall Progression.

Increase your Progression by the excesse, and the square of the terme when you cease, or the number multiplied in it selfe square­ly, is the double of your Terme saue 1, if the [Page 39] progression begin with an vnite.

But it the first terme be not an vnite, then the square of any terme is the double num­ber of the said terme: as if you should square the sixth terme, then the product would bee the twelfth terme: & so of any other terme.

Example.

A Gentleman comming into a Market to buy a Horse, was asked 30 pounds for him. Nay (said the Gentleman) his price is ouer great. Then said the owner (hauing more craft and subteltie then the Gentleman, as commonly the old Prouerb is true amongst Horse-coursers); My Gelding ha [...]h fower shooes vpon his fower feete (quoth he), you shall giue me for the first nayle (there being 28 in all) one farthing▪ and for the second nayle 2 farthings, and for the third 4 far­things, and for the fourth 8 farthings; and so double at euery nayle, you shall haue him. Whereat the Gentleman smiled, saying; I will haue him. And so they bargained, and then went to an Arithmatician to cast vp the Summe: but how this Gentleman was able to pay for this Horse, shall appeare by the Worke, which I haue put for an exam­ple, [Page 40] because I would not haue any man ig­norant in Arithmatick, to make any such blind matches without aduice, as I know many haue done to their cost.

1. Example.

Now according to the rule, I increase this progression vnto the seuenth Terme thus, 1. 2. 4. 8. 16. 32. 64; which 64 I mul­tiply by it selfe squarely, the product is 4096, which by the rule is the thirteenth Terme, which is one Terme lesse then the double of 7: then multiply that 4096 by 2, it makes 8192, which is the fourteenth Terme. Then multiply 8192 by 8192, and the product is 67108864, which is the twen­tie seuenth Terme: the which being dou­bled, makes the last Terme 134217728.

[Page 41] Example. [...]

The Extreames and Excesse of a Progression giuen to find the summe.

Multiply the last terme by the Excesse, and from the Product abate the first terme, and diuide the remayner by a vnite lesse then the excesse, and the Quotient is the summe of the Progression desired.

Example.

In the last examples, the excesse was 2, by which I multiply 134217728, and the Pro­duct is 268435456, from which abate 1, the first terme, and the remayner is 268435455, which should bee diuided by one vnite lesse then the excesse, which is 2, and [...] lesse is but 1; therefore seeing 1 doth neither mul­tiply, nor diuide, I conclude the price of the horse to be 268435455 farthings; which I diuide by 960, the farthings in one pound starling, and the quotient is [...]79620 pound, 5 shillings, 3 pence, 3 farthings, the price of the Horse, as in this example.

[Page 43] Example. [...]

I haue inserted in the next page the triall of this work, by increasing the Termes from 1 to 28, and also the Addition of the totall, which shewes the answere to be true.

[Page 44] Example.

[Page 45]Otherwise, subtract the first terme from the last, and diuide the remayner by one ace lesse then the Excesse, and to the quotient adde the last Terme, and the totall is the summe.

Example.

To 12 men a summe of money is giuen to the eldest ½, to the second ½, the remayner, and so to euery one of the rest, and the last portion was found to bee 4 pound, and the last halfe being also 4 pound, was giuen to a friend to see the money to bee equally di­stributed; what was each mans portion, and the summe giuen?

Let 4 bee the last portion, and twelfth Terme, and so double vntill you come to the first terme, and you shall find euery mans portion Then by this second rule, you shal find the totall to be 16380 pound; to which adde the Ezecutors part 4 pound, makes 16384 pound.

3. Example.

A Gentleman bought a Mannor, with all the appurtenances for a summe of money vnknowne; but hee was to pay at seuerall [Page 46] dayes of payment by continuall triplation, of euery payment, from the first payment which was 4 pound, and the last 8748 li. the question is, what he paid for the said Man­nor and lands.

Example.

Subtract the first terme 4, from the last terme 8748, there will remaine 8744; which diuide by the Excesse, one lesse, viz by 2, and the quotient will bee 4372; to the which adde the latter terme 8748, and the totall is 13120 pound, for the summe which the said Mannor and lands cost.

[...]

How to proue a Fraction by the knowne parts of Coyne.

In the first example of fractions of fracti­ons, I find that 3∶4 of 2∶3 of 7∶8 of a pound Sterling to bee 42∶96 parts of a pound: for triall whereof, take 7∶8 of a pound, which is 17 shillings 6 pence, or 210 pence, the 2∶3 of that number is 140 pence; and [...]4 of 140 pence is 105 pence: now multiply 42 the numerator of your fraction by 240 d. and diuide by 96, the denominator, makes 105 pence, the proofe, as followeth.

[Page 53] 4. Example. [...]

2. Example.

In the first example of fractions of inte­gers, there was 3∶4, 5∶6 and 7∶8 of a pound, reduced into one denomination, and the to­tall by Addition was 472∶192 of a pound. Now for the proofe of the worke, multiply the numerator 472 by 240, makes 113280; [Page 54] which diuided by 192 the denominator, makes 590 pence; which diuided by 12. pence, makes 2 pound, 9 shillings, 2 pence. The proofe of this triall in the parts of a pound, take first for 3∶4 of a pound, or 15 shillings; then 5∶6 of a pound is 16 shillings 8 pence; also 7∶8 of a pound is 17 shillings 6 pence; and the totall added together, is 2 pound, 9 shillings, 2 pence, which prooues the worke to be true.

Example. [...]

The proofe of Addition by parts of Coyne.

In the second Example, 2∶3, 3∶4, and 4∶5 of a pound, are found to be 133∶60; therfore diuide 133 by 60, makes 2 pound and 13∶60 or 13 groats remaining, which is 2 pound, 4 shillings, 4 pence.

The proofe: adde 2∶3 of a pound, which is 13 shillings, 4 pence; and 3∶4 of a pound, which is 15 shillings, and 4∶5 of a pound, which is 16 shillings, into one totall, makes 2 pound, 4 shillings, 4 pence, as before.

Example. [...]

The proofe of Subtraction by the parts of Coyne.

In the example before, where I take 2∶3 from 7∶8, the remayner was 5∶24 of a pound, which is 5 times 10 pence, or 4 shillings 2 pence. Also for proofe, take 13 shillings 4 pence, which is 2∶3 of a pound, from 7∶8, which is 17 shillings 6 pence, there will re­maine 4 shillings 2 pence, as before.

The proofe of Multiplication by the parts of Coyne.

In the first example, 2∶3 is multiplied by 3∶4, and the product makes 6∶12 of a pound or 10 shillings: for proofe whereof, multi­ply 13 shillings 4 pence, or 160 pence, which is 2∶3 of a pound by 15 shillings, or 180 pence, which is 3∶4 of a pound, and the product will be 28800, which being diui­ded by 240 pence, the pence in one pound will yeeld in the quotient 120 pence, or to shillings.

[Page 61] Example. [...]

The proofe of Diuision by the parts of Coyne.

In the second of the first example, where I diuide 2∶3 by 4∶5, the quotient is 10∶12, which in coyne is 16 shillings 8 pence: for proofe, I do multiply 2∶3 of a pound, which is 160 pence, by 240, makes 38400; which diuide by 4∶5, or 192 pence, makes 200 pence, which is 16 shillings 8 pence, the proofe.

[Page 64] Example. [...]

Rules of Practice by the first Table.

TO worke by the Aliquot parts of a pound, search in the first Table for your giuen price, and by that number found, di­uide your number giuen, and the quotient is your answere in pounds, and the remayner is the fraction of one pound.

But if the giuen price be not found ex­actly at the first entrance, then find 2, or more numbers, to make the giuen price, and then worke as followeth.

Example.

If one yard cost 3 shillings 4 pence, what will 7859 yards cost at that rate: I enter the Table, and against 3 shillings 4 pence, I find 1∶6 of a pound; wherefore I diuide 7859 by 6, makes 1309 pound, 5∶6 of one pound, or 16 shillings 8 pence.

[...]

[Page 78]

At 16 pence an ell, what will 8976 elles cost, I find for 16 pence my diuisor, to bee 15, and so diuiding 8976 by 15, the quotient is 598 pound, 6: 15, or 2: 5, which is 8 shil­lings.

[Page 79] Example. [...]

Adde a Cypher to your number giuen, and the last figure of your quotient will bee primes, euery one in value 2 shillings, and the remainer is the fraction of a prime, al­waies lesse then 2 shillings. In the first ex­ample, the remainer was 5: 6 of one pound, but if you adde a cypher, the quotient will be 1309 pound, 8 primes or 16 shillings, and the remayner is 2: 6 of one prime, or 1: 3, which is 8 pence.

[...]

[Page 80]At 2 shillings 6 pence a pound pepper, what will 2436 pound cost: find 2 shillings, 6 pence 1: 8 of a pound, wherefore adde a cypher, and diuide, by 8, makes 304 pound; 10 shillings.

[...]

At 8 pence a pound Ginger, will 77856 pound cost; diuide by 30, adding a cypher, makes 2595 pound, 2 primes, or 4 shil­lings.

[...]

At 17 pence a pound Sugar; what shall 23459 pounds cost: for 12 pence, diuide by 20, makes 1172 pound, 9 primes, 1: 2, or 19 shillings: then for the rest of your giuen price, which is 5 pence; take 48, and diuide, and the quotient is 488 pound, 7 primes, which added together into one sum, makes the totall 1661 pound, 13 shillings, 7 pence.

[Page 81] Example. [...]

At 6 shillings 8 pence a pound Cloues, what will 3769 pound waite cost: diuide by 3, makes 1256 pound, 3 primes, 1: 3, or 6 shillings 8 pence.

[...]

[Page 82]At 22 pence an elle of Holland, what 3768 elles cost: for 20 pence diuide by 12, makes 314 pound, and for 2 pence by 120, makes 31 pound, 4 primes, or 8 shillings; the totall is 345 pound, 8 shillings.

[...]

If one elle of Holland cost 20 pence, how many elles shall I buy for 345 pound: mul­tiply 345 by the price, which is [...]:12, or by 12, makes 4140 elles, the summe desired.

[...]

[Page 83]If one elle of Ozenbrigs cost 8 pence, what summe of elles will 78 pound buy me: multiply by 30, makes 2340 elles.

[...]

At 15 pence an elle of Canuas, how many elles will 100 pound buy: multiply by 16, makes 1600 elles.

If one elle of parchment lace cost 1 pen­ny, how many ells shall I haue for 73 pound: multiply by 240, makes 17520 elles.

[Page 84] Example. [...]

If one Acre of land bee 5 shillings, how many Acres may I hire for 132 pound: mul­tiply by 4; makes 528 Acres.

[...]

Rules of Practice by the second Table.

If the price giuen be any number of shil­lings, search in the second Table for the price giuen, and by the number there found: multiply your number of yards, ells, pounds or pieces, and cut off the last figure with a dash of the penne for primes, euery one in [Page 85] value 2 shillings, and the product is the sum of pounds and shillings that your giuen number will cost.

Example.

At 2 shillings an elle of Holland, what will 956 elles cost: in the second table [...] find the tenth of the number giuen, so that if you take the tenth of 956, it is 95 pound, 12 shillings, onely by cutting off the last fi­gure by a dash of the penne.

956 elles at 2s. an ell, makes 95|6, or 12s.

At 7 shillings an ell of Cambricke, what will 789 elles cost: multiply by 3 1∶2, or take halfe of the giuen number, and multi­ply the whole number giuen, by 3, makes in one summe, cutting off the prime line, 276 pound, 3 shillings.

[Page 86] Example. [...]

At 25 shillings a piece Raysons, what will 356 pieces cost: take alwaies half the number of shillings of your giuen price for your multiplier, and worke as before, and the product is 456 poued, 0 prime.

[Page 87] Example. [...]

Also 75032 pieces at 26 shilling a piece.

[...]

If one barrell of Sope cost 47 shillings, what will 3584 barrels cost: multiply by 23 1∶2, makes 8422 pound, 8 shillings.

[Page 88] Example. [...]

At 3 pound 6 shillings a Barrell, what will 124 cost.

[...]

[Page 89]If one Acre of land cost 6 pound 8 shil­lings, what will 758 Acres cost: multiply by 64 shillings, which is halfe the price, the product is 4851 pound, 4 shillings, or two primes.

[...]

How to proue the last question, or any o­ther of like kind. If one Acre of land cost 6 pound 8 shillings, how many Acres shall be bought for 4851 pound, 4 shillings: di­uide your number of pounds and shillings by one halfe of the number of shillings in the price giuen, adding a cypher to your number of pounds, and the quotient is the number of Acres of land the said summe wil buy at that rate.

[Page 90] Example.

The giuen sum is 4851 pound, 2 primes, or 4 shillings; which diuided by halfe the giuen price, which is 64 shillings, brings into the quotient 758 Acres: and so of any other summe.

[...]

A Merchant bought Cambricks, cost him 855 pound, 18 shillings; the question is, how many pieces hee had, paying for euery piece 27 shillings. Answere, ad [...]e a cypher to your number giuen, which 855 pound, 9 primes, makes 85590; which diuide by half the price giuen, which is 13 1∶2; or di­uide by 135 the quotient will bee 634 pieces: now the reason wherefore a cypher is added to the number giuen, hauing 9 primes in it is, because I diuide by 13 1∶2, which hath one fraction; and this rule is generall.

[Page 91] Example.

What cost 634 pieces, at 27 shillings.

[...]

How to prooue one question in the Rules of Practice, by working of a­nother.

If you wil proue any question in the rules of Practice, by a second example marke the complement, or want of your giuen price from one pound, and worke the same num­ber at that price which doth want, and the totall of those two summes added together, makes the iust number of pounds of the gi­uen summe.

[Page 92] Example.

At 16 shillings a piece of Fustian, what will 320 pieces cost. Answere; multiply by 8, makes 256 pound, 0 prime.

Againe, 16 shillings your giuen price wanted 4 shillings of one pound, wherefore worke 320 at 4 shillings, which is multipli­ed by 2 primes, makes 64 pound, 0 prime, the totall is 320 pound, which prooues the former worke.

[...]

Example.

At 13 shillings a piece of Lawne, what will 752 pieces cost: 752 by 6 1∶2, makes 488 pound, 8 primes.

At 7 shillings a piece, what 752 pieces: 752 by 3 1∶2, makes 263 pound, 2 primes, totall is 752 pound.

[Page 93] Example. [...]

Rules of Practice by the third Table, the most excellent of all the other.

Diuisors.

[Page 95]Diuide the number of ells, yards, pounds, or pieces giuen by the number, or numbers found in the third Table, alwaies cutting the last figure for primes; if that any re­maine after Diuision, it is alwaies lesse then one prime, or 2 shillings.

Example.

At 3 pence a pound Licoras, what will 123728 pound cost. Answere; for 3 pence in the third Table, I find my Diuisor to bee 8, by which I diuide my giuen number, makes 1546 pound, 6 primes, or 12 shil­lings.

[...]

At 9 pence the pound Ginger, what will 8768 pound cost: for 6 pence diuide by 4, makes 219 li. 2 primes; then for 3 pence the residue of the price, diuide by 8, makes 109 pound, 6 primes, totall is 328 pound, 16 shillings. Or otherwise, diuide by 4 for 6 d. and then take half that product for 3 pence, and adde them into one summe, as before.

[Page 96] Example. [...]

At 11 pence the yard Canuas, what will 2356 cost: for 8 pence diuide by 3, makes 78 pound, 5 primes, 1∶3 or 8 pence; and for 3 pence, diuide by 8, makes 29 pound, 4 primes, 1∶2, or 12 pence; the totall is 107 pound, 19 shillings, 8 pence.

[...]

A second example, the proofe of the last.

At 13 pence a pound fine Sugar, what will 2356 pound cost: for 12 pence diuide by 2, makes 117 pound, 8 primes, or 16 shil­lings; then for 1 penny, diuide by 24, makes 9 pound, 8 primes, 4 pence, the totall is 127 pound, 12 shillings, 4 pence; which added to the former summe in the last example, makes 235 pound, 12 shillings; and so much will 2356 pound cost at 2 shillings a pound, because the two giuen prices make one prime, or 2 shillings.

Example. [...]

[Page 98]At 16 pence a pound Sugar, what will 78432 pound cost: worke for 8 pence, and double the summe, makes 5228 pound, 8 primes, or 16 shillings.

At 8 pence a pound Almonds, what will 78432 pound cost: diuide by 3, ma es 2614 pound, 4 primes, or 8 shillings: which ad­ded with the former example, makes 7843 pound, 2 primes, which is the price that 78432 pound will cost at 2 shillings a pound, and prooues both examples true.

Example. [...]

[Page 99]At 18 pence a pound Comfits, what will 78432 pound cost: for 12 pence take halfe the giuen number, and for 6 pence take half of that summe, which added into one totall, makes 5882 pound, 4 primes.

[...]

At 6 pence a pound small Ginger, what will 78432 li. cost: diuide by 4, makes 1960 li. 8 primes, or 16 shillings; which added to 5882 pound, 8 shillings, makes 7843 pound 2 primes, the price at two shillings.

[...]

These tables may serue also, if the price bee aboue 2 shillings, or one prime: as if [Page 100] you shall say at 3 shillings, 6 pence an ell, what 782 elles: here I see the giuen price is compounded of 7 times 6 pence; where­fore I worke first for 6 pence in diuiding by 4, makes 19 pound, 11 pence; which multi­ply by 7, makes 136 pound, 17 shillings for the price of 782 elles at 3 shillings, 6 pence the elle.

At 6 pence an elle, what 782 elles: find for 6 pence, 19 pound 11 shillings, which added to the former summe in the last exam­ple, makes 156 pound, 8 shillings, which is the summe that 782 elles will cost at 4 shil­lings the elle.

Example. [...]

[Page 101]At 4 shillings 8 pence the elle Holland, what will 2148 elles cost. I find 4 shillings 8 pence to be 14 Groats, so diuiding by 6 for one Groat, makes 35 pound, 8 primes; which multiply by 14, makes 501 pound, 4 shillings.

[...]

At 15 pence a groce of points, what will 2256 grosse cost. 15 Pence is 5 times 3 d. and so I diuide 2256 by 8, makes 28 pound, 2 primes; which multiply by 5, makes 141 pound.

[...]

Rules of Pr [...]ice by the fourth Table.

If the number of the price giuen bee any Aliquot part of a shilling: enter the fourth Table, and there you shall find a Diuisor, by the which if you diuide your number gi­uen, the Quotient will be shillings, and the remainer parts of one shilling. Then to conuert your shillings into pounds, take one halfe of the Quotient, cutting off the lower number for shillings, and the rest is pounds.

Example.

At 3 far things a pound prunes, what will 756 pound wayte cost. Search in the fourth Table, and you shall find 16 for your Diui­sor: by the which if you diuide 756, the Quotient is 47 shillings, 1∶4, or 3 pence.

[...]

[Page 103]At 1 halfe penny a pound Coporas, what will 8756 pound cost. Diuide by 24 makes 364 shillings, of which the one half cutting of the 4 shillings, is 18 pound 4 shillings; and 20 halfe pence remaining, totall is 18 pound, 4 shillings, 10 pence.

[...]

At 4 pence a pound Licoras, what will 789 pound cost. Diuide by 3, makes 13 pound, 3 shillings.

Againe, at 6 pence a pound, what will 8579 pound cost. Diuide by 2, makes 214 pounds, 9 shillings, 6 pence.

1. Example. [...]

2. Example. [...]

Generall Rules of Practice without Tables.

Multiply your number giuen by the sum of pence, that one yard, piece, pound, or elle doth cost, and the product will be the summe of pence, the whole number giuen will cost; and then diuide that summe of pence by 4, ma▪ es the Quotient Groats, and if any re­remaine, they are pence, alwaies lesse then 4 pence, or one Groat: and secondly againe, diuide that Quotient will bee pounds and primes, euery prime in value 2 shillings, and the remaines is Groats, alwaies lesse then 6 Groats, or one prime, which is value 2 shillings.

At 17 pence an ell Canuas, what will 3245 elles cost: Multiply by 17, makes 55165 pence, which diuided by 4, makes 13791 Groats, and there will remaine one penny. Secondly, diuide that Quotient a­gaine by 6, makes 229 pound, 8 primes, and the remayner is 3 Groats, or one shilling; and so the totall is 229 pound, 17 shillings, 1 penny.

[Page 105] Example. [...]

At 3 shillings, 7 pence a yard Holland, what will 752 elles cost: multiply 752 by 43 pence, the price of one ell, makes 32336; which diuided, as is before taught, makes 134 pound, 14 shillings, 8 pence.

[Page 106] Example. [...]

At 7 shillings, 11 pence the elle Cam­bricks, what will 85 [...] ells cost. Reduce 7 shillings, 11 pence into pence, makes 95 pence; by which multiply 856, makes 8 [...]320; which diuided as before, makes 338 pound, 16 shillings, 8 pence.

[Page 107] Example. [...]

At 2 shillings, 11 pence an elle Holland, what will 7856 elles cost: multiply, and diuide as is before taught, makes 1145 li. 13 shillings, 4 pence.

[Page 108] Example. [...]

At 17 shillings, 7 pence a yard Broad Cloath, what will 7856 yards cost: multi­ply by 211▪ the price of one yard, and diuide as before, makes 6906 pound, 7 primes.

[Page 109] Example. [...]

If your giuen price haue any farthings in it, then reduce your price into farthings, and multiply your giuen number by those farthings, and the product will be the num­ber of farthings, which your summe will cost: then diuide that product by 16, makes the quotient Groats, and the remainer will be farthings, alwaies lesse then 16, or one Groate. Secondly, diuide that quotient of Groats by 6, makes pounds and primes, as before.

[Page 110] Examples.

At 5 shillings, 1 penny, one halfe-penny an ounce Plate, what will 356 ounces cost. Reduce 5 shillings, 1 penny, half-penny in­to farthings, makes 246 fa [...]things: by which multiply 356, makes 87567 farthings; which diuided by 16, makes 5473 Groats, and 8 farthings will remaine; which diuide againe by 6, makes 91 pound, 2 primes, and one Groat will remaine, totall is 91 pound, 4 shillings, 6 pence.

At 6 shillings, 9 pence, farthing an ounce of gilt plate, what will 3542 ounces cost: multiply your shillings by 48, the farthings which are in one shilling, makes 288; to the which adde 37 farthings, which are in 9 pence, farthing, makes 325 farthings; and then worke as before is taught, and you shal find 1199 pound, 2 shillings; 3 pence, halfe­penny.

[Page 111] Example. [...]

Another way to worke Practice.

Diuide your number of yards, elles, or pieces by 240, adding a Cypher to your number giuen, and then multiply the Quo­tient by your price, and the Product is the summe of pounds, and shillings, that the gi­uen number will cost.

At 17 pence the elle Canuas, what will 7848 elles cost: adde a Cypher, and diuide 78480 by 240, and the Quotient will bee 32 pound, 7 primes; which multiply by 17 [Page 112] pence, the price, makes 555 pound, 9 primes or 18 shillings.

Example. [...]

At 3 shillings, 5 pence an ell of Holland, what will 702 elles cost: diuide 7020 by 240, makes 2 pound, 9 primes, and there will remaine 6; which multiply by 41 pence, the price of [...]e ell, makes 118 pound, 9 primes, or 18 shillings, and then the 6 elles, makes 1 pound, 6 pence, the totall is 119 pound, 18 shillings, 6 pence.

[Page 113] Example. [...]

At 19 pence the elle of Holland, what will 32544 elles cost: diuide 325440 by 240, makes 1356; which multiply by 19 pence, the price of one elle, makes 2576 pound, 8 shillings.

[...]

How to worke this last example, and all other, after a more briefe and exact manner.

Diuide the third number by the first, and by the Quotient multiply the second, and the product is the answere.

Example.

If 356 elles cost 137 pound, 12 shillings, 9 pence, what cost 2848 elles.

[...]

First, diuiding 2848 by 356, the Quoti­ent is 8; by which I multiply 137 pound, 12 shillings, 9 pence, the products are 1096 pound, 96 shillings, 72 pence; then diuide [Page 117] 72 by 12, is 6 shillings; which added to 96 shillings, makes 102 shillings, or 5 pound, 2 shillings; the totall is 1101 pound, 2 shil­iings, as before.

2. Example.

If 124 yards cost 17 pound, 10 shillings, 1 penny, what cost 744 yards.

[...]

If 32 pieces of Raysons cost 19 pound, 2 shillings, 2 pence, what will 112 pieces cost at that rate.

[Page 118] 3. Example.

[...]

If 356 pieces cost 137 pound, 12 shil­lings, 9 pence; what will 2848 pieces cost at that rate.

[...]

[Page 119] Example. [...]

Example. [...]

How to know whether any question giuen be to be answered by the Rule Direct, or Conuersed.

By these notes following, you shall find, whether any question propounded be to be answered by the Rule of 3 Direct, or con­uersed; for alwaies the third number is the number whereon the question dependeth, and is distinguished from the other two, by some one of these notes following.

[...]

And the answer is alwaies, more or lesse, so that if it bee more then the lesser of your two extreame numbers is the diuisor: if lesse, then the greater of your two extremes is your diuisor. If the number whereon the question bee depending, be your Diui­sor, thē the answer is, by the conuerse Rule, and you must multiply your two former numbers for Diuidend. If the first number be the Diuisor, then the question is answe­rable by the Direct Rule, and the product of the two latter numbers is your Diuidend.

[Page 121] Example.

If 13 Cannons spend 358 pound of pow­der, what will 5 Cannons spend, now here the question is, what 5 Cannons will spend. I answere, lesse then 13 Cannons; where­fore by this rule, the greater of the two ex­treames, 13 is the diuisor: wherefore I mul­tiply 358 by 5, and diuide by 13, makes 137 pound, 6∶13 that 5 Cannons will spend.

2. Example.

If 13 Cannons spend 358 powder, what will 5 Cannons spend.

[...]

2. Example.

I lent my friend 115 pound for 7 moneths, and when I came to him to require the like kindnesse he could lend me by 54 pound, the [Page 122] question is, how long hee should forbeare that 54 pound to make requitall, or to equal my time, and kindnesse.

If 115 pound require 7 monthes, what will 54 pound require: here the answere in reason is, that 54 pound must bee longer time forborne then 115 pound, and so the answere is more times then 115 pound; so that I find the lesser of my exteames 54, is my Diuisor, and the question answerable by the Rule conuersed, so that I multiply 115 by 7, makes 805; which diuided by 54, makes 14 moneths, 49∶54 of a moneth, or 14 moneths, 25 dayes, 23∶25

Example. [...]

[Page 123] 4. Example.

A Captaine of a Band of men is besieged in a Citie, hauing with him 7200 men, and his victuals will serue the whole Company but 7 moneths, but there is no hope left to haue any fresh victuals vntill 16 moneths; the question is, how many men he shall send away to make the victuals serue for 16 moneths. Answere, lesse then 7200 men.

If 7 moneths require 7200 men, how ma­ny will 16 moneths aske.

[...]

When Wheate was sold at 3 shillings, 8 pence the bushell, the penny loafe of bread waighed 6 ounces, what shall the same loafe of bread waigh, when Wheat is sould for 2 shillings the bushell: I answere more then a 11 ounces.

[Page 124]If 44 pence giue 6 ounces, what will 24 pence giue.

[...]

If 356 men digge a trench in 24 dayes, in how many dayes will 200 men make the same? Answere, in more dayes; 42 dayes, 17 houres, 7∶25.

If 356 men require 24 dayes, how many will 200 men require.

[...]

[Page 125]Or thus; Considering the numbers, 200 may be had in 156 once, therefore for 200 take 24 dayes; then for 150 take 18 dayes, totall 42 dayes; then there will remaine 6 to bee multiplied by 24, makes 144∶200 parts of a day, as before.

If 112 pound cost 3 pound, 5 shillings, 5 pence, what will 3136 pound cost? di­uide 3136 by 112, makes 28; which multi­ply by 3 pound, 5 shillings, 5 pence, makes 91 pound, 11 shillings, 8 pence.

[...]

If 100 pound gaine 7 pound, what summe of money will gaine 85 at that rate? An­swere.

[Page 126]If 7 pound require 100 pound, what will 85 pound, require.

[...]

Or otherwise, diuide 85 by 7, makes 12 1∶7; by which multiply 100, makes 1214 pound 2∶7 of a pound.

[...]

Or otherwise, diuide 100 by 7, makes 14 2∶7; by which multiply 85, makes 1214 pound, 2∶7

[Page 127] Example. [...]

Carseys at 54 shillings the piece, are put in Barter, at 3 pound the piece, how shall Wooll worth 24 shillings the Tod, be set in Barter, to make the bargaine equall?

If 54 shillings be 60 shillings, what shall 24 shillings make.

Answere: for more then 24 shillings, and lesse then 54, so that 54 is the diuisor, and multiplying 24 by 60, makes 1440; which diuided by 54, makes 26 shillings, 2∶3, or 8 pence.

[Page 128]If 54 shillings be 60 shillings, what will 24 shillings make.

[...]

If 6 sheepe cost 58 shillings, how many shall I buy for 124 pound? multiply 124 by 58, makes 7192; which diuide by 6, makes 1198 sheepe 2∶3.

[...]

Or otherwise, diuide 58 by 6, makes 9 2∶3, by which multiply 124, makes 1198 2∶3, as before.

[Page 129] Example. [...]

A Merchant at Siuill deliuereth 1500 Ri­alls, to receiue for euery 11, being a ducat in London 5 shillings, 10 pence sterling money, how much must he receiue?

If 11 Rialls be 70 pence, what are 1500 Rials?

[...]

[Page 130]At 13 pound in the 100 pound profit, of what stocke came 3274 pound? Answer: di­uide 3274 pound by 113 pound, makes 2897 pound, 39∶113 of a pound, adde two cyphers to the giuen number.

[...]

A Merchant receiued for principall and gaine 328 wherein he found he had gained cleare 56 pound, what did he gaine vpon the 100 pound. Answere, multiply 100 by 56, the gaines makes 5600; which diuide by 328, and the Quotient is 17 pound, 3∶41 in smallest termes.

[...]

[Page 131]If 112 pound cost 7 pound, 6 shillings, how may I sell to gine 10 pound vpon the 100 pound. Answere: Take the tenth part of 7 pound, 6 shillings, or of 146 shillings, which is 14 shillings, 3∶5 of a shilling; which added to the price, makes 8 pound, 7 pence, 1∶5 of a penny.

[...]

If 100 pound exchange be 7 pound 2 shil­lings, what is one pound. Answere, 71∶100 parts of a pound: wherefore multiply 71 by 240, and diuide by 100, makes 17 pence▪ 12∶5 of a penny.

[...]

[Page 132]If 107 ells of cloth cost 17 pound 12 shil­lings, what will 321 ells cost at that rate? Here if you consider the proportion be­tweene the first number, and the third, you shall find the third number doth containe the first exactly three times; wherefore you need not to multiply the second by the third, and diuide by the first number, but on­ly take the second number, and multiply by 3, makes 52 pound, 16 shillings for the price that 321 ells will cost: behold the worke at large.

If 107 ells of cloth cost 17 pound, 12 shil­lings, what will 321 ells?

[...]

How to find whither that your numbers giuen be proportionall, or not.

Diuide your third number by the first, and if the quotient be an euen number, and nothing remaine of your diuidend, then the first and third numbers are euen proportio­nall in whole numbers, as in the last exam­ple, the first number was 107, and the third number 321, so that in deuiding the third nūber by the first, the quotient is 3 & 0 re­maines: wherefore I conclude, that the first and third numbers are proportionals in whole numbers, and that the third doth con­taine the first iust three times, and so often must the fourth number sought for, con­taine the second; and I conclude, that three times 17 pound 12 shillings, which is 52 pound 16 shillings, is the fourth proportio­nall number sought, as appeareth by the or­dinary forme of worke in the last example.

[...]

[Page 134]If 36 elles of cloth cost 13 pound, 4 shil­lings, 1 penny, what will 432 elles cost at that rate: diuide 432 by 36, makes 12; by which multiply your second number 13 pound, 4 shillings, 1 penny, makes 158 pound, 9 shillings.

[...]

A. doth lend vnto B. 600 pound for 8 moneths, the question is, how much B shall lend vnto A. for 12 moneths to recompence him, not reckoning compound, interest. An­swere. If 8 moneths require 600 pound, what will 12 moneths require▪ the reason is lesse then 600 pound; wherefore diuide 600 pound by 12, makes 50; which multiply by 8, makes 400 pound.

Or otherwise by proportion, as 8 is to 12 so must 600 bee to 400 pound, 2∶3 parts of 600 pound.

[Page 135]If the number bee not exactly proportio­nall, yet there is a great abreuiation to bee made of the worke of Reduction, Multipli­cation, and Diuision, in the working of most examples in the Golden Rule; as for ex­ample.

If 19 Barrels of Figgs cost 16 pound 12 shillings, what shall 58 barrels cost, here diuiding 58 by 19, the Quotient is 3, and 1 will remaine; wherefore I take 3 times 16 pound, 12 shillings for 57 barrels, and I haue to worke but for the one remaining; which is but to diuide 16 pound, 12 shillings, by 19, makes 17 shillings, 9∶19 of one shilling, the totall is 50 pound, 13 shillings, 9∶19 shillings.

[...]

If 356 elles of Holland cost 124 pound, 2 shillings, 3 pence, what will 7259 elles cost at that rate. Reduce 124 pound, 2 shillings 3 pence, into pence, makes 29787 pence; which multiply by 7259, makes 216223833 [Page 136] pence, which diuide by 356, make 607370▪ which diuided by 240 pence, makes 253 pound, 170 pence, or 14 shillings 2 pence.

Example. [...]

[Page 137] [...]

[...]

A second way more briefly to worke this question, or any other of like nature, is this: multiply the third number by the pounds and primes, or shillings and pence, and di­uide the product by the first number, and the quotient will be the fourth number sought. In the last example, 7259 elles was the third number, which multiply by 124l. 1 prime, or 2s. makes 900841 l. 9 primes: [Page 138] then also 7259 by 3 pence, makes 21777 pence; which diuided by 240, makes 90 pound, 14 shillings, 9 pence: then adde those two sums into one totall, makes 900932|6 primes, 9 pence; leaue out▪9, and then di­uide the residue by 356, makes 2530 pound, 7 primes, and 54∶356; which with the 9 d. brings out the two pence, as in the last ex­ample.

Example. [...]

[Page 139] [...]

If 24 pieces of Raysons cost 25 pound, 8 shillings, what will 324 pieces cost: mul­tiply 324 by 25 pound, 4 primes, makes 8229, 6 primes: which if you diuide by 24, the Quotient will be [...]42 pound, 9 primes, or 18 shillings without Reduction, as in the example following.

[Page 140] Example. [...]

If 25 pound gaine 1 pound, 8 shillings, what will 725 pound gaine at that rate? Multiply 725 by 1 pound 4 primes, makes 10150; which diuided by 25, makes 40 pound, 6 primes, or 12 shillings.

[...]

[Page 141]And in this sort may diuers other questi­ons bee wrought in pounds and shillings without Reduction, which I thought good to giue a taste of, but I will proceed here no further, because I purpose in the second part of this Booke to speake of them at large in the Treatise of Decimal Arithmatick, whereby all manner of questions are to bee wrought of Multiplication and Diuision in pounds, shillings and pence, without Re­duction, as shall appeare in their seuerall places following. And now I will pro­ceede to speake something of the Rule of Three Direct and Conuerst in Fractionall operations, wherein I will be as briefe as I may, not intending to increase this little Treatise intended for a pocket booke, into ouer large a volume.

The Rule of 3 in Fraction.

If your three numbers giuen be all fracti­ons, multiply the third by the second, and diuide the product by the first, and the quo­tient will be the fourth proportionall num­ber sought for.

[Page 142] Example.

If 3∶4 of a yard of Holland cost 4∶5 of a pound, what shall 5∶6 of one yard cost at that rate? Multiply 5∶6 by 4∶5, makes 20∶30 or 2∶3, which diuide by 3∶4, makes 8∶9 of one pound, or 17 shillings, 7∶9 of one shil­ling.

[...]

If 7∶8 of one ell of cloth cost 9∶12 of a pound, what will 17 elles cost? Make 17 fraction wise, and multiply 17∶1 by 9∶12, makes 153∶12, which wil be both abreuia­ted by 3, makes 51∶4, which diuided by 7∶8 makes 408∶28 parts of a pound, or in smal­termes 102∶7; then diuide 102 by 7, makes 14 pound, 4∶7 of one pound for the price.

[...]

[Page 143] 2. Rule.

If all your three numbers giuen be fracti­ons, multiply the Numerator of the first fraction by the Denominator of the other two fractions, for to make your Diuisor. Then multiply the Denominator of your first fraction by the Numerators of your other two fractions, to make your Diui­dend and then diuide by your Diuisor, and the Quotient is the answere sought: but if your Diuisor bee greater then your Diui­dend, then the Quotient is a fraction, lesse then a vnite.

Example.

If 3∶4 of a yard cost 4∶5 of a pound, what cost 5∶6 of a yard? Multiply 3, the Nume­rator of the first fraction by 5 and 6, the de­nominators of the other two fractions, makes 90 for your diuisor; then multiply 4, the denominator of your first fraction by 4 and 5, the numerators of your other two fractions, makes 80 for your diuidend: now because your diuisor is greater then your diuidend, place them fraction wise thus, 80∶90 of 1 li. or in least termes, 8∶9 of a li.

[Page 144] Example.

Againe, if 7∶8 of an ell cost 2∶3 of a shil­ling, what will 34 ells cost.

[...]

If 18 Pioners in 3∶5 of a day doe make 22 Rodds of Barricadoe, what will they make in 7 dayes.

[Page 145]

If 12 hundred 3∶7 of Allum cost 15 pound 1∶3 li. what will 324, 1∶8 of a hundred cost? Reduce the whole and broken numbers in­to broken, and worke as is before taught.

[...]

[...]

[Page 146]If 7∶9 of an ell cost 8∶11 of a pound, what will 15∶13 of an ell cost? 1 pound, 1 shil­ling, 6 pence, 3∶4, fere.

Example. [...]

If 3∶4 of a yard of Veluet cost 7∶8 of a poūd, what wil 28 yards cost. 32. l. 13 s. 4 d.

[Page 147] Example. [...]

[...]

If 3 ells 1∶8 cost 5∶7 of a pound, what wil the whole piece cost, containing 28 ells 1∶2 at that rate? Answere.

[...]

[Page 148]If 12 pound, 4 ounces of Quichanella cost 4 pound, 3 shillings, 4 pence, how much will 100 pound buy me at that rate? If 49∶4 of a pound cost 25∶6 of a pound sterling, what will 600∶6 parts of a pound buy. An­swere.

[...]

The proofe of this last example. If 100 pound starling buy me 294 pound of quich­anella, how much shall 4 pound, 3 shillings, 4 pence buy mee? to find the valew of the hundred, the rate of one pound being giuen; abate 2 places from 294, and it will bee 2 pound 94∶100 parts of one pound: which multiply by 4 pound, 1∶6, makes 12 pound 25∶100 parts, or one fourth for the proofe.

[...]

[Page 149]If 30 men cast a Trench in 3 dayes 2∶3; how many men would cast it in 5∶6 of a day? here by comparing these proportions together, I find that 5∶6 the third number, will desire a greater quantitie of men to performe the worke, then 11∶3 of a day will require; wherefore this proportion is re­ciprocall or backward; wherefore I mul­tiply the two former numbers together, makes 333∶3, or in smaller termes, 110∶1; which diuided by 5∶6, makes 660∶5; which diuide by the denominator 5, makes 132 men.

[...]

In the Backward Rule, or Conuerst in fractions, multiply the Denominator of your third number, by the numerators of both your other numbers for diuidend, then multiply the numerator of your third num­ber, by the denominators of your other two numbers for diuisor, and then worke as before.

[Page 150] Example. [...]

If when the bushell of Wheate was sold for 4 shillings, the penny loafe wayghed 6 ounces 1∶2, what shall the same loafe weigh when Wheat is sold for 2 shilling, 8 pence p [...]nce the bushell? Multiply 48 by 13, makes your diuidend 624: then 2 by 32, makes 64 for your diuisor, and then diuide 624 by 64, makes 9 ounces, 48∶64, or 3∶4 of an ounce.

[...]

If when one ounce of sterling siluer was worth 1∶4 of a pound the penny of siluer weighed 30 graines, what shall the same penny weigh, when the ounce shalbe worth 1∶3 of a pound.

[Page 151] Example. [...]

If when a loade of Hay was sold for 24 shillings, 8 pence, the penny bottle weigh­ed 3 pound, 1∶4, what shall it weigh, now the load is sold 37 shillings. Answere, 2 pound, 71∶76 of a pound.

[...]

If 3 yards 1∶8 cost 9 shillings, 9 pence, what will 380 yards cost at that rate? Re­duce 3 yards 1∶8 into eights, makes 25∶8; then reduce 380 yards into eights, makes 3040∶8 parts: then 9 s. 9 d. into pence, makes 117 pence; by which multiply 3040, makes 355680, which diuided by 25, makes 14227 pence, 5∶25 of one penny [Page 152] in the whole 59 pound, 5 shillings, 7 pence, 5∶25 or 1∶5 of a penny. Behold the worke.

[...]

[...]

The proofe of the former worke. If 380 yards cost 59 pound, 5 shillings, 7 pence 5∶25 of one penny, what will 3 yards 1∶8 cost at that rate? reduce your coyne into 25, makes 355680; then reduce your 380 yards into 8, makes 3040; by which di­uide [Page 153] 144227 pence, makes 9 shillings, 9 pence, as before.

[...]

If 34 ship Carpenters build a ship in 8 moneths 3∶5, in how long time will 120 Carpenters build the same? Reduce 8 mo­neths 3∶5 into fifthes, makes 43∶5; then multiply 34 by 43, makes 1462. Also put your diuisor 120 into fifthes, makes 600∶5; then diuiding of 1462 by 600, the Quotient [Page 154] will bee 2 moneths 262∶600 parts of one month, or in smallest termes 131∶300 parts. And this Rule generall if one of your num­bers bee a fraction, put alwaies your diui­sor into the same fraction of your diuidend, and the quotient will bee of the same deno­mination of your diuidend, and so the an­swere was moneths, and parts of a moneth.

If 34 Carpenters aske 43∶5 moneths, what 600∶5 moneth.

[...]

If 100 pound in 12 moneths gaine 10 pound, what will 336 pound gaine in 8 mo­neths? Take the tenth part of 336, which is [...] li. 6 primes, or 12 s. makes 369 li. 12 s.

Secondly, if 12 moneths gaine 33 pound 6 primes, what will 8 moneths gaine? I an­swere, [Page 155] lesse then 33 pound 6 primes; wher­fore multiply by 8, and diuide by the grea­ter extreme, 12, makes 22 pound, 4 primes, or 8 shillings, the answere.

If 120 Pioners in 6 dayes cast 300 rods of Trench, how many shall 600 men cast vp in 4 dayes. If 120 giue 300, what will 600 giue? Answere, 1500 Rods.

Secondly, if 6 dayes giue 1500 rods, how many will 4 dayes giue? I answere, lesse; multiply by 4, and diuide by 6, makes 1000 Rods.

If 112 pound in 12 months gaine 100 li. what wil 340 li. gaine in 7 months? Answer: 303 li. 4∶7.

Secondly, if 12 moneths gaine 303 li. 4∶7 what will 7 moneths gaine.

Example. [...]

A generall Rule▪

Put alwaies your diuisor into the same fraction of your diuidend, and your quoti­ent will be of the same denomination, that your diuidend was: as in the last example, 12 moneths was turned into seuenths, and also 303 pound 4∶7 was turned into sea­uenths of pounds, and so the quotient of that diuision was pounds, and the fraction of a pound remaining.

If 7 pound in 13 months gaine 3 pound, in how long time will 340 pound gaine 60 pound. First, if 7 pound Gaine 3 pound, what will 340 pound gaine, makes 145 pound, 5∶7 of a pound. Secondly, if 145 pound, 5∶7 or 1020∶7 ask 13 moneths, what will 60 pound, or 420∶7 gaine. Multiply by 13, and diuide by 1020, makes 5 months 6∶17 of a month.

If 600 great Horses in 5 dayes doe spend 1125 Bushels of oats, how many bushels wil serue? 1400 Horses for 22 Dayes. First, say, if 600 giue 1125, what 1400, makes 2625 bushels. Secondly, if 5 spend 2625 bushels, what will 22 dayes spend? Multiply by 22, and diuide by 3, makes 11550 bushels.

How to worke the double Rule at one operation.

This last question, or any other of like nature which is wrought by the double Rule at two seuerall operations may be an­swered at one in this manner: multiply the three latter numbers, to make your diui­dend one into the other; then multiply the two former numbers for to make your di­uisor, and then diuide the diuidend by the diuisor, and the quotient will be the same, as in the last example, 1125 being multipli­ed by 1400, makes 1575000; which againe increased by 22, makes your diuidend 34650000. Then multiply your two former numbers 600 by 5, makes 3000 for the Di­uisor; and then diuiding your diuidend by your diuisor 3000, the quotient will bee 11550 bushels, as before at two operati­ons.

[Page 158] Example. [...]

If 35 s. in 7 months gaine 6 s. in how long time will 340 l. gaine 100 l. First, if 35 s. gaine 6 s. what will 340 l. require? Reduce 340 l. into pence, and multiply by 6, makes 40800; which diuided by 35, makes 1165 s. 5∶7 s. Secondly, if 1165 [...]. 5∶7 require 7 moneths, what will 100 l, require? Makes 12 moneths, 8∶816 parts of a moneth.

Fellowship without Time.

This Rule differeth very little from the Rule of three; for in this Rule the summe of all the moneys disbursed, is the first num­ber in the Golden Rule. Then the gaines or losse is the second number: the third number is each seuerall partners money dis­bursed so that the Rule must bee seuerally wrought for each seuerall Partners portion.

Example.

Foure Merchants made a company toge­ther; the first, viz. A. put in stock 74 pound, B. put in 90 pound, C. put in 100 pound, and D. put in 120 pound, and they found that they had gained 84 pound; now the que­stion is, what each man must haue of the gaines, according to the proportion of his money disbursed. First, adde all the moneys disbursed into one totall summe, viz. 74, 90, 100, 120, totall is 384 for the first number in the Golden Rule. Then the second num­ber is 84 pound, the gaines; and the third number is each particular mans stock; then worke as followeth.

[Page 160]If 384 pound gaine 84 pound, what will A. B. C. D. summs gaine to them.

[...]

The like reason is in losse, as is in gaines. Example: A certaine ship being in a tem­pest on the sea was forced to cast ouer board so much of her lading, as amounted vnto the summe of 642 pound, then there is great reason that all the ventures should beare part of that losse, according to the propor­tion of his stocke which hee ventured. As suppose: A. ventured 700 pound, B. 530 pound, C. 640 pound, D. 800 pound; totall is 2670. Then say; If 2670 pound loose 642 pound, what will each of A. B. C. D. loose? as in the example following.

[Page 161] Example.

If 2670 pound loose 642 pound, what will A. B. C. D. summes loose to them.

[...]

fioure Merchants bought a ship, which cost them 3600 pound, whereof A▪ must pay one third part of the money, B. one fourth, C. one fifth, D. one sixth; the question is, what each man must pay of the saidsumme. Answere.

Seeke a number wherein the like parts may bee had, which is 60, and take the like parts of that number for the numbers that you seeke, for to find each mans portion of the money▪ which he should pay. First, 1∶3 of 60 is 20, the 1∶4 is 15, the 1▪5 is 12, the 1∶6 is 10; which adde into one totall, makes 57 for the first number in the Golden Rule.

[Page 162] Example.

[...]

If 57 be 3600, what will bee the summes of A. B. C. D.

[...]

The said ship made a Voyage to Sea, and hath gotten all charges, deducted out 240 pound, the question is, what each man must haue of the gaines. Answere.

If 57 gaine 240, what will A. B. C. D. summes gaine to them.

[...]

[Page 163]Foure Merchants made a Company; A. put in 320 pound, 13 shillings, 3 pence; B. put in 840 pound, 16 shillings, 6 pence; C. put in 560 pound, 18 shillings, 9 pence; D. 1000 pound; and in one yeare they found they had gained 400 pound, 18 shillings, 6 pence: the question is, what each man must haue of the gaines. First, the totall summe of all their moneys makes 2721 pound, 8 shillings, 6 pence, or 653142 pence, for the first number. Then reduce each seuerall mans money disbursed into pence for the third number, the second is the gaines also reduced into pence, and then worke according to the Rule.

Example.

If 2721 pound, 8 shillings, 6 pence gaine 400 pound, 18 shillings, 6 pence, what will A. B. C. D. summes gaine to them.

[...]

Rules of Fellowship, with diuersi­tie of Time.

Multiply each mans money disbursed by the time that it continued in stock, and ga­ther the totals, as in the last Rule, to make the first terme in the Golden Rule, and the gaines or losse is the second, and then each mans product of money and time for the third terme in the Golden Rule, and worke as followeth.

Example.

Three men made a stock, A. B. and C. and in long continuance of time by dangerous aduentures they gained, and got by prizes taken at Sea 2345 pound; A. put in stocke 40 pound, 14 moneths; B. put in 50 pound, 8 moneth; C. put in 85 pound 6 moneths, what shall each man haue of this gaines.

[Page 165] Example. [...]

If 1470 pound gaine 2345 pound, what will A. B. C. summes gaine them.

[...]

The second question with more diuersity of time, foure Merchants made a Company; A. put in 340 li. 19 s. 2 d. for 10 moneths; B. put in 930 li. for 9 moneths; C. put in 760 li for 12 moneths; D. put in 583 li. 13 s. 4 d. for 5 moneths, wherewith they gained 740 li. now the question is, to know what each man must haue of this gaines.

[Page 166] [...]

Cut off two Cyphers from each number, and then worke as followeth.

If 57163 pence gaine 1776 pence, what what will A. B. C. D. summes gaine them.

[...]

Example.

There is a Bootie or Spoyle taken by 3 men worth 7851 pound, and they agree to diuide it in this sort; A▪ is to haue one half, B. one third, C. one fourth, what is each mans share.

[Page 169]To worke this question, and all other of like nature, seeke a number which may bee diuided by all the denominators of your three fractions in whole numbers, and the smaller such a number bee that you choose, the more easie will your worke be; which for to find, multiply your denominators of your fractions one into another; that is to say, 2 by 3 makes 6; and 6 by 4, makes 24; so 12, one halfe of 24 will bee euenly diuided by all the three denomina­tors, 2, 3 and 4. Wherefore I take 1∶2 of 12 is 6, and 1∶3 of 12 is 4, and 1∶4 of 12 is [...]; which added into one summe, makes 13 for first number in the Golden Rule; the second is 7851 pound, and the third numbers are each seuerall mans portion imagined to be, viz 6, 4, 3, and then worke as before.

If 13 giue 7851 pound, what will A. B. C. summes giue.

[...]

[Page 170] 4. Example.

Foure Merchants bought a house toge­ther, which cost 3000 pound; A. was to pay 1∶2 and 6 pound ouerplus; B. 1∶3 and 12 pound more; C. 8 pound lesse then 2∶3; D 1∶4 with 20 pound ouerplus. Now the questiō is, what each Merchant must pay of this sum. Answer: First, the pounds ouerplus must be subtracted from the summe giuen; and the pounds wanting must bee added to the summe giuen; as for A. 6 pound, B. 12 pound, for D 20 pound, totall is 38 pound, to bee subtracted then; for C. adde 8 pound, therefore subtract 30 pound from 3000 pound, there wil remaine 2970 pound; then worke by the Rule of Fellowship, ta­king 12 for a number, which will bee diui­ded by all the denominators, 2, 3 and 4, viz. take for A. 6, for B 4, for C. 8, for D. 3; to­tall is 21 for diuisor, the second number is 2970 pound, the third, each mans part ima­gined.

[Page 171] Example.

If 21 giue 2970 pound, what will A. B. C. D. summes giue.

[...]

The numbers found to A. are 848 pound 4∶7, to which if you adde 6 pound, makes 854 pound, 4∶7.

To B 565 pound, to which 12 pound ad­ded, makes 577 pound, 5∶7

To C. 1131 pound, 3∶7, from which subtract 8, leaues 1123 pound, 3∶7

To D. 424 pound, 2∶7, to which adde 20 pound, makes 444 pound, 2∶7; the which added into one total, makes 3000 pound, the proofe.

And in this manner may infinite varietie of questions bee propounded, and their doubts easily resolued; and here will I ende [Page 172] concerning this Rule, and goe in hand with some pleasant questions to bee wrought by position, which is the most excellent Rule of all others in Arithmatick, as shal appeare in the second part of this Booke in Deci­mall Arithmatick.

The Rule of Position requiring one number to bee imagined, before the principall proportion can be found.

TO worke by this Rule; Take any num­ber at pleasure, which you shall ima­gine to be the true number sought, and pro­ceed with it, as if it were the true number, wherein if you haue failed, by doubling or tripling according to the nature of the que­stion, you shall then attaine vnto the true number desired, by aide of the Golden Rule, in manner following: for looke what pro­portion is betweene the false conclusion, and the false position, such proportion hath the giuen number, to the number sought.

[Page 173] Example.

A. B. and C. consent to buy a ship, which will cost them 2700 pound, so that B. must pay twice so much as A, and C. must pay 4 times so much as B: the question is, what each man must pay of this summe? I sup­pose A must pay 8 pound, then B must pay twice as much as A, which is 16 pound; then C must pay 64 pound, which is 4 times as much as B: but yet 8 pound, 16 pound, and 64 pound, is but 88 pound, and it should be 2700 pound, so that now I resort to the Golden Rule, and worke as followeth. If 88 pound come of my Position 8 pound, of what comes 2700? Multiply 2700 by 8, and then diuide by 88, makes 245 pound 40∶88, or 5∶11 of a pound for the part that A must pay; then B must pay 490 pound, 10∶11 of a pound, which is twice as much as A; and C must pay 1960 pound, 40∶11 of a pound, which is 4 times as much as B. The totall summe is 2700 pound. Behold worke as followeth.

[Page 174]If 88 pound come of 8 pound, of what comes 2700.

[...]

2. Example.

A Captaine of a Band of Men being ask­ed, what number of Souldiers were in his Band, answered, I doe not readily know; yet (quoth he) of this I am certaine, that the 1∶2 and 2∶3, and 4∶5, and 1∶6 of their number added together into one sum, are 384 men: now the question is, what summe of men he had in his Band. I suppose he had 60 men, or 30 men in his Band, but the least number is best, viz. 30, whereof 1∶2 is 15, and 2∶3 is 20, and 4∶5 is 24, also 1∶6 is 5, their totall is but 64 men, but that should be 384 men. Then say by the Golden Rule, as follow­eth.

[Page 175]If 64 come of 30, of what number comes 384.

[...]

Answere: hee had 180 men in his Band, whereof

[...]

The solution of this Question another way more briefe.

Diuide 384 by 64, makes 6; which mul­tiply by 30, makes 180 men, as before.

3. Example.

A certaine man hauing spent 120 pound, had yet remaining 1∶2 and 1∶3 of his whole substance; the question is, what his sub­stance was. Answere: First, 1∶2 and 1∶3 [Page 176] is 5∶6, which being taken from 6∶6, the whole substance leaues remaining 1∶6; therefore if 1∶6 be 40 pound, what is 6∶6? makes 240 pound.

[...]

4. Example.

A Merchant bought 384 yards of broad Cloth of three seuerall prices, of each a like quantitie, and he was to pay halfe as much more for the second sort, as he payed for the first, and twice as much for the third sort as he payed for the second: now the question is, what each sort cost him, and at what price euery yard was rated vnto him? I sup­pose the first sort cost him 4 pound, then the second sort must cost him 6 pound, which is halfe as much more as the first; and then the third sort cost him 12 pound, which is twice as much as the second; the totall is but 22 pound, but it should be 248 pound: wherefore if 22 pound come of 4 pound, of what number comes 248 pound?

[Page 177] Example. [...]

The first cost him 45 pound, 1∶11 of a pound; then the second sort cost 67 pound, 7∶11 of a pound; the third sort cost 135 pound, 3∶11 of a pound, total is 248 pound: then diuide 384 by 3, and you shall find hee had 128 yards of each sort, and by Practise, you shall find the first sort cost 7 shillings, 1∶2 d. a yard; the second sort cost 10 shillings 7 pence a yard almost, the third sort cost 21 shillings, 1 penny, 1∶2 d.

The Rule of double Position.

SVppose a number at pleasure, as in the last Rule of single Position, and proceed as if you had found the right number, and if by working you find the true number, then your Position was the right number, which doth seldome happen. First, if by your working there commeth out more then the true number; then note it thus + with a crosse; if lesse, then thus − with a long line, which doth signifie lesse.

Secondly, suppose another number, grea­ter or smaller, and worke as before, vntill you doe find the true number sought; which if you doe not find, see the difference also from the true number sought, and note it with the signe + or − as it shall bee found.

Then thirdly, set your suppositions with their errours, more or lesse, as in the exam­ples following.

Fourthly, multiply crosse the first positi­on [Page 179] by the seconds errour, and the second po­sition by the errour of the first, and then if the signes be both alike + or −, abate the lesser from the greater, and the remaines shall be the diuidend. Also the lesser error abated from the greater, leaues the diuisor▪ but if the signes be contrary one +, the o­ther lesse, add both together to make the di­uidend, and adde the two errors to make the diuisor: and lastly diuide the diuidend by the diuisor, and the quotient is the true number desired.

1. Example.

A certaine man seeing a purse in his friends hand, saith vnto him: It seemeth vn­to me, that there is 100 Crownes in your purse▪ To whom the other answered: Nay (quoth hee) there are not 100 Crownes, but (saith he) if they were increased 1∶2 and 1∶3 ▪ and 1∶4, and lastly, one Crowne ouer­plus, then would they be iust 100 crownes.

I suppose there were 12 Crownes in his purse, to which if I adde one half; of 12, which is 6; and one third of 12, which is 4; and one fourth of 12, which is 3; and lastly, one Crowne more, the totall will be but 26 [Page 180] Crownes, but they should be 100 Crownes, so that this errour is two little by 74 Crownes, which I note thus:

74 − 12

Secondly, I suppose he had 24 Crownes, to which I adde 1∶2 of 24, which is 12 and 1∶3, which is 8 and 1∶4, which is 6: and lastly, one Crowne ouerplus, the totall is 51, but it should bee 100 Crownes, so that this is an errour of 49, too little, which I al­so note thus: 49 − 24

[...]

[Page 181]The answere is; that hee had 47 pound 13∶25 parts of a pound in his purse. The proofe followeth.

[...]

2. Example.

Twenty yards of Sattin, and 12 shillings is equall vnto 12 yards of veluet lesse, 10 shillings; the price of either sort is requi­red.

To answere this, or any other like questi­on, take any number for the price of a yard of the lesser number, which here is veluet, which at 20 shillings a yard, lesse 10 shil­lings, amounteth vnto 230 shillings. Now admit a yard of Sattin at 14 shillings, so 20 yards and 12 shillings amounteth vnto 292 shillings; from which subtract 230 shil­lings, rests 62 s. more then the truth. A­gaine, [Page 182] rate a yard at 12 shillings, so the 20 yards and 12 shillings makes 252 shillings; from which take 230 shillings, rests 22 shil­lings more then the truth also. Now mul­tiplying 22 by 14, and 62 by 12, the pro­ductes are 308, and 744, and the difference of those numbers is 436; then take 22 from 62, rests 40 for diuisor, by which diuide the difference, makes 10 shillings, 9∶10 shil­lings for the price of a yard of Sattin.

Example. [...]

[Page 183] 3. Example.

Otherwaies if 40, the difference of er­rors gaine a, the difference of positions, then 62 the first error yeelds 3 and 1∶10▪

Or if 40 yeeld 2, what 22? makes 1 and 1∶10; this taken from 12, or 3, 1∶10 from 14, leaues 10, 9∶10 for the price, as before.

4. Example.

A Carpenter was hired to work 20 daies at 12 pence a day, but euery day that hee was idle, hee was to abate 18 pence of his wages, and in the end he receiued but 8 shil­lings: now the question is, how many daies he wrought.

First, suppose he wrought 12 daies, which commeth to 12 shillings, then must the 8 dayes that hee played, come to 12 shillings at 18 pence a day also: but this question saith, there came due to him 8 shillings? Be­hold an error of 8 shillings too little.

Againe, I say that he wrought 14 dayes, amounting to 14 shillings▪ then 6 dayes that he played at 18 pence a day, commeth to 9 shillings; this taken from 14 shillings, [Page 184] leaues 5 shillings, and it should bee 8 shil­lings, which is an errour of 3 shillings too little. Now multiplying 12 by 3, and 14 by 8, the products are 36. and 112, and the excesse is 76; which being diuided by 5, the difference of the errours, quoteth out 15, 1∶5 for the number of working dayes, and 4 dayes 4∶5 for the number of playing dayes.

Otherwayes.

If 5▪ the difference of errours, yeeld 2, the difference of positions, what 8 the first er­rour? makes 3, 1∶5 to be added to 12.

Or if 5 be 2, what is 3▪ makes 1, 1∶5 to be added to the second position 14, where­by all three wayes the numbers of the Dayes he wrought are found out.

How to worke Compound interest at any rate per cent.

What is the principall and interest of 352 pound, put out at 8 pound in the hun­dred compound Interest, to be paid at the end of two yeares? Adde two cyphers to 352 pound, makes 35200; then place your Interest 8 vnder the lowest cypher next the right hand, and multiply 352 by 8, placing the product vnder the line, and that will be the Interest; which added into the summe lent, makes the totall of the principall and interest▪ and so worke for the second, third, and fourth yeare, as in the Example.

[...]

[Page 199]First I multiply 35200 by 8, makes 2816, which I adde vnto 35200, makes 38016; then I multiply 3801600 by 8, makes 410|5728, or 11 shillings, 5 pence, abating 4 figures for the 4 cyphers, which I added to the summe for to find out the prime line, as appeareth in the example; and so of any other summe or rate in the hundred.

At 17 pound the hundred per annum com­pound interest, what wil 879 pound amount vnto to bee all forborne vnto the end of 5 yeares? Adde 2 cyphers to your summe giuen, and multiply by your Interest 17, and adde into the principall, and so worke 5 yeares, and the last product will bee the summe of money to bee receiued, viz. 1927 pound, 3 shillings, 5 pence.

[Page 200] Example. [...]

[Page 201]If a Merchant buy a parcell of Holland, at 3 pound, 6 shillings the piece; and ano­ther parcel at 4 pound, 2 shillings the piece; the third sort at 4 pound 10 shillings the piece, the fourth sort at 5 pound the piece; how may he sell 40 pieces, of each sort 10 pieces to gaine 18 pound in the hundred, and giue 9 moneths time for the payment; as in the Example following.

[Page 202] Example.

[...]

Take the 3∶4 of the interest, makes 191 pound, 16 shillings, 3 pence, 3∶5 of one penny, to sell to gaine 18 pound in the hun­dred, for to giue 9 moneths time.

A Merchant sold 300 quarters of wheat, cost him 352 pound ready money, and lost 7 pound in the hundred, what did one quar­ter cost him, and at what rate did he sell a quarter, to loose 7 pound in the hundred? Take the interest at 7 pound in the hun­dred, [Page 203] which is 24 pound, 12 shillings, 9 pence, 3∶5, which subtract from 352 li. makes 327 pound, 7 shillings, 2 pence, 2∶5 of a penny, and diuide the remainer by 300, makes 1 pound, 1 shilling, 10 pence for the price sold: secondly, diuide 352 pound by 300, makes 1 pound, 3 shillings▪ 5 pence ob. for the price which it cost him

Rie sold for 3 shillings a bushell looseth 20 pound in the hundred, what will then be lost, if it bee sold for 3 shillings 6 pence a bushell? If 3 shillings be 80 pound, what is 3 shillings 6 pence? Multiply 80 pound by 3 1∶2, or by 3 shillings, 6 pence, makes 2800; which diuide by 3, makes 93 li. 1∶3 Or otherwise, if 36 pence bee 80 pound, what is 42 pence? Multiply 80 by 42, and diuide by 36, makes 93 pound 1∶3 of a pound as before.

If in one ell of Cloth sold for 3 shillings, 2 pence there were gained after the rate of 10 pound in the hundred, what did that ell of cloth cost? diuide 385, or 38 penny 1∶2 by 110, makes 35 pence that the ell cost.

[...]

[Page 204]If one yard of Holland cloth cost 2 shil­lings, 11 pence, how many yards shall I buy for 34 pound, 6 shillings, put it into pence, makes 8232 pence; which diuide by 35 pence, makes 235 yards, 1∶5 yard.

How to gaine any rate in the Hundred you desire.

Put your price that one yard, ell, pound or piece doth cost you into pence; and then for 10 pound in the hundred, take the tenth part of that summe, which is the same num­ber, placed one place nearer to the right hand, and that is the profit or Interest; which added vp into the price giuen, makes the price to sel one yard, pound, ell, or piece, to gaine 10 pound in the hundred ready mo­ney.

Example.

If one ell of Holland cloth cost 3 shil­lings, 9 pence, how may I sell to gaine 10 pound per cent' ready money? Put 3 shillings 9 pence into pence, makes 45 pence: then take the tenth part of 45 pence, which is 4 pence 5∶10, or one half, makes 49 d. 1∶2 for the price to sell an ell to gaine 10 li. per cent.

[Page 205] Example. [...]

If your price you would gaine, bee not 10 pound in hundred, then adde 2 Cyphers to your number of pence giuen and multi­ply that number by your Interest, omitting to multiply by the cyphers, and the product vnder the line is your Interest or gaine, which added vp into one summe; makes the price to sell one yard, ell, pound, or piece, to gain according to the rate desired example.

If one pound of Cloues cost 4 shillings, 10 pence, how may I sell to gaine 9 pound per cent ready money? Put 4 s. 10 d. into pence, makes 58 d. then ad 2 cyphers, makes 5800; which multiply by 9, makes 5∶22 or 5 pence, 22∶100 parts of one penny; which added vp to the vpper numbers, is 63 pence, 22∶100 parts of one penny, or 5 [Page 206] shillings, 3 pence, 1∶5 of a penny for the price to sell one, to gaine 9 pound in the hundred.

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If one piece of Raysons cost 18 shillings, 9 pence, how may I sell to gaine 18 pound in the hundred ready money? put your mo­ney into pence, makes 225 pence, to which adde 2 cyphers, makes 22500; which mul­tiply by 18, makes 40∶50, or 40 pence, ob. which added into the price, makes 265 pence, ob. for the price to sell one piece to gaine 18 pound in the hundred.

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A Merchant lent wares for 10 pound in the hundred profit for 12 moneths, and at the end of 6 moneths he receiued principall and interest 356 ls. the question is, what was the summe lent? Answere: adde 2 cyphers to 356 pound, and diuide by 105 pound, which is 6 moneths interest and principall, makes 339 pound 1∶21 parts of a pound for the sum lent.

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