johnsons' Arithmetic  
IN 2. Books  
THE FIRST▪ OF VULGAR ARITHMA with diverse Brief and Easy rules: to work all the first 4. parts of Arithmetic in whole numbers and fractions by the Author newly Invented  
The Second, of decimal Arithmetic whereby all fractional operations 〈…〉 wrought in whole Numbers in Merchant's accounts without reduction; with Interest, and annuities  
By john johnson Survaighour; practitioner in the matticmatiqu  
Printed at london by Augustine Mathewes and are to be sold at his house in S tie. Bride's Lane near Fleetstrens 1623.    

TO THE RIGHT HONOURABLE SIR EDWARD BARKSHAM, KNIGHT, Lord Mayor of the City of London; and to the right Worshipful, the Shreive's and Aldermen his Brethren.  
THe very Arts (Right Honourable and Worshipful) which were wont to bear the attributions of [honestae & liberales] seem now to temporize, and to have learned the newfound skill of equivocation. For, howsoever the former of these denominations adhereth censtantly unto the Professors of Mathematic Sciences, yet the other, which was once derived [a liberalitate] and then intimated, that they were anciently accustomed to perform liberal recompense to their lovers and followers, hath now (to spare cost) purchased a different etymology [a libertate] as properly accommodate to such as are [liberi] freeborn, or (as our peculiar term carrieth) Freemen. Which being so, and seeing that in this particular, as well as in many other of greater consequence, Tempora mutantur—: I am thereby enforced to make up the old verse, adding— e● nos mutamur in illis; and to apply my long experience, together with tedious studies bestowed in this present art of numbers, to the use and be hoof of those persons, to whom by the general appellation it properly belongeth, namely, to the studious thereof in this honourable City. Which is the cause that I presume (without farther self-praise, of what I have brought more useful, more easy, or more certain and delightful in the operations, then hath been seen before) to present my Labours to your Hononrable and Worshipful judgements, to whom I owe of duty, whatsoever can be of me performed, to the furtherance of Art, and the honour of this noble City, and the worthy Companies therein.  
Your Honours, and Worship's devoted in all humble respect, JOHN JOHNSON, Suruaighor.   

THE EPISTLE TO THE READER.  
GEntle and Courteous Reader; having for many years passed spent my time both in reading, practising, and conferring with others in, and about the study of the Mathematical Sciences, and through great pains and travel, at the request of diverse Worshipful Gentlemen. Merchants, and others of my very loving friends, have at last collected and gathered together many excellent Rules and easy Abreviations in the Science of Arithmetic, which at the entreaty, and by the means of the help of some of them, I have at last made bold here to present abroad unto the world's view, the first fruits of some idle hours study, the most part whereof I do acknowledge to have gotten by the practice and use of the most excellent Instrument, invented by Master William Prat, called, The jewel of Arithmetic in which I have done the best of my Endeavours, not to hide that Talent in the earth, which God hath bestowed upon me for the benefit of others, but rather to his great glory and praise, and for the benefit of my Country, and for the furthering of all that are studious in the Art of Numbers, I have laboured to set it forth in the most brief, plain, and easy manner that I could fit for the understanding of the weakest and meanest capacity. In which if any thing shall seem obscure or doubtful to any man, I could wish myself were present to resolve his doubts, for I have endeavoured to make the Rules as brief, short and easy, as I could devose.  
In my first Book I have entreated concerning vulgar Arithmetic, with new inventions of my own, in all the first four parts of Arithmetic, viz in Addition and Subtraction, with two several kinds of Multiplication, not charging of the memory, never extant before in any Author that I have read, with four several kinds of Division, the latter of them bringing the proof by Addition of the figures under the dividend, without any multiplication, or casting away of nine, according to the accustomed manner.  
Again, in the work of Fractions, I have set them forth in plain and perfect figures after another manner of my own invention, because the fractional figures in most books of Arithmetic were so unperfit, that they were scarce to be discerned, and in this manner they will perform all factionall operations, as well as if they were set out according to the usual manner. In the end of which Rules I have showed the reasons and proofs of fractions by the known parts of Coin.  
Thirdly, in the second part of the former book, I have set forth Reduction, both in Coin sterling `waightss, measures, time and motion; the Tables whereof are in the first part of the book, with diverse Rules how to bring pence, or farthings at the first work into pounds, shillings and pence; with diverse questions wrought by Reduction, with Progression arithmetical and Geometrical, with examples.  
And lastly, I have showed how to work the Rule of 3 Direct and Conversed, both in whole numbers and fractions, after diverse several manners of workings, and how to find the divisor in any question, as also diverse ways to work Fellowship, Barter, Exchange, Allegation, Interest, Position, and all other operations Arithmetical, with examples and brief Rules of every part.  
In my second book of decimal Arithmetic, I have first described out the parts and use of the decimal Table, and how to set forth any number given in decimals.  
Secondly, I have showed how to work all the several parts of Arithmetic, viz. Numeration, Addition, Subtraction▪ Multiplication and Division in decimals; with examples and proofs of every work in the known parts of Coin.  
Thirdly, I have handled in as brief manner as I could, the Rule of 3 Fellowship, Barter, Exchange and Interest in decimal Arithmetic, as before in vulgar, in which you may perceive the great labour that is avoided, in vulgar Arithmetic, with diverse examples and proofs of the same.  
Lastly, I have added a small Treatise of Interest and Annuities; with the manner how to calculate Tables or Breviates at any rate, or years purchase given; all which I have drawn into a pocket volume. If therefore any Gentleman, Merchant, or other, be desirous to have further instruction, if they repair to my lodging in Coleman-street, I shall be ready to give them any satisfaction.  
If therefore I shall find these my labours and endeavours to take that effect, which I do hope and wish for, I shall be thereby the sooner encouraged to hasten the coming forth of the third part of this volume concerning the extraction of Roots, with many easy operations and rules showing the use of the Square and ●ubicque roots, concerning Mensurations of land Timber, Board, Glass and Stone, and the reduction of Measures from one proportion to another by their squares given: and lastly, concerning Military affairs and Gunners Art; concerning the Arithmetical work, with demonstration by examples, which I had intended to have joined to this volume, but that it would have increased it far beyond a pocket book. And so hoping of your friendly censure & acceptance of these first fruits of my labours, I cease, hoping to have my true endeavours and meaning well taken, and the faults in the Printing friendly amended, desiring a blessing from God upon these my poor labours, I take my leave,  
London in Coloman-Streete, this 18 of August, 1622. john johnson▪   


A Table of the Contents of the first Book.  


CHAP. I.  
OF Numeration, with examples.   

CHAP. II.  
2. Of Addition in Coin sterling, Weights, liquid, dry, and long measures of Time and Motion.  
3. Examples and questions wrought by Addition, with two several proofs of Addition.   

CHAP. III.  
1. Of Subtraction, with examples of Coin, Weights, Measures, Time and Motion, with the proofs of the same.  
2. How to subtract from a unite in any pl●●e, any numbers, and to show the remainder at first sight of the work.  
3. The proof of Subtraction two several ways.   

CHAP. IU.  
1 Of Multiplication, with the Table, and the use of the same.  
2. Examples after the usual manner, with the exposition of the same.  
3. A second way to multiply, not charging of the memory with bearing any numbers in mind, to be added in the next place, with examples, and exposition of the same.  
4. A third way to multiply and bring the product in the last line; with examples and exposition of the same.   

CHAP. V.  
1. Of Division vulgar, after the usual manner, with examples and exposition.  
2. A second manner of Division, more easy and speedy, with less charge to the memory.  
3. A third kind of Division, more easy and certain, bringing the proof by Addition, without multiplication, or making any new work for the proof.  
8. How to divide by a unite with Ciphers.  
6. Brief Rules by Multiplication and Division.    

The Table of the second part of the first Book.  

The Rule of Reduction. 1  
Reduction of Coin unto 6  
To bring pence into pounds, shillings and pence at the first work by Division. 6  
To bring farthings into pounds, shillings and pence at the first work by Division. 7  
A second way to bring pence, or farthings, into pounds, shilling and pence. 10  
Reduction of weights. 11  
Reduction of measures. 12  
Reduction of time. 14  
Reduction of motion. 16  
Questions of Reduction unto the 26   

Of Progression Arithmetical.  
What Progression Arithmetical is. 27  
To find the sum of a Progression. 27  
To find the latter term of a Progression. 30  
To find the number of Terms. 31  
To find the excess or difference. 32  
To find any middle term. 32  
To find what number shall begin and finish any progression, with examples. 33   

Of Progression Geometrical.  
What Geometrical Progression is. 37  
To find any term given in a Progression. 38  
To find the sum of a Progression. 41  
Examples of Progression and proof. 44   

Of Fractions.  
Of Fractions, and how to work them according to my own invention. 47  
How to reduce fractions of fractions. 48  
How to reduce fractions of Integers. 50  
How to prove a fraction by the parts of Coin. 52  
Addition in fractions. 55  
Proof of Addition by parts of Coin. 57  
Subtraction in fractions. 58  
Proof of Subtraction by the parts of Coin. 59  
Multiplication in fractions. 59  
Proof of Multiplication by parts of Coin. 60  
Division in fractions. 61  
Proof of Division by parts of Coin. 63  
How to work whole numbers with fractions. 64  
How to work whole numbers and fractions with fractions. 67  
How to abbreviate a fraction. 68  
How to find the value of any fraction. 70  
How to change the surname of a fraction. 71  
Questions of fractions unto 76   

Rules of Practice.  
Rules of Practice by the first Table. 77  
The first and second Table. 78  
Rules of Practice by the second Table. 84  
How to prove questions in Practice. 89  
How to prove one question in Practice, by the working of another. 91  
The third and fourth Table of Practice. 94  
Rules of Practice by the third Table: 95  
Rules of Practice by the fourth Table. 102  
General Rules of Practice without Tables. 104  
Another way to work Practice. 111   

The Golden Rule.  
Of the Rule of Three Direct. 114  
A second way to work the Rule of Three. 116  
To know if a question given be to be answered by the Rule Direct, or Converse. 120  
To find if any number given be proportional, or not 132  
The Rule of Three in fractions. 142  
A general Rule. 156  
How to work the double Rule of Three at one operation. 157  
Fellowship without time. 159  
Fellowship, with diversity of tim●. 164   

Position Single.  
Position single, requiring one feigned number. 172  
Position wrought a second way. 175   

Double Position.  
The Rule of Double Position. 178  
Barter or Exchange. 185  
Of Gain and Loss. 193  
To work Compound interest at any rate. 198  
How to gain any rate in the hundred. 204  
Equation of Payment. 208  
Alligation Mediall. 212  
Alligation Alternate. 215     

The Table of the second Book of decimal Arithmetic.  

THe declaration of the parts of the decimal Table. 219  
To find the value of a decimal in the known parts of Coin. 220  
Numeration in decimals. 222  
How to set out a penny in decimals. 223  
How to break a pound into his exact parts. 225  
How to express any numbers in decimals. 226  
How to remove a decimal from one place to another. 227  
Addition in decimals. 230  
Subtraction in decimals. 232  
Multiplication in decimals. 234  
To change any fraction into decimals. 238  
Division in decimals. 241  
To divide the smaller number by the greater. 242  
To find the prime line in any Division. 245  
Reduction in decimals. 251  
Rules of Practice in decimal. 258  
To find the price of a unite in any place of 10, 100, 1000, 10000, etc. 260  
The price of any number of yards, else, or pounds given, to find the price of a unite. 264  
The Golden Rule in decimals. 268  
Diverse ways to work the Golden Rule in decimals. 275  
Brief Rules of abbreviating your work by proportions. 276  
Questions wrought without Reduction in decimals a second way. 284  
Position in decimals. 293  
Gain and Loss in decimals. 298  
How to work gain & loss in pence & fathing. 315  
The proof of many examples. 320  
Exchange in decimals. 329  
a general Rule for exchange in decimals. 335  
Reduction of Measures. 339   

Of Interest and Annuities.  
How to frame tables to work Compound Interest at any rate in the hundred. 345  
How to calculate the table of 10 li. per cent. 346  
The table of 10 li. per cent compoundintrest. 349  
How to calculate a Table at any other rate, under or about ten pound in the hundred compound Interest. 350  
The Breviate of 8 pound in the hundred. 352  
The use of the Breviates, or Tables. 353  
To find what I pound due at any number of years is worth at the end of the term. 353  
To find what any yearly annuity will make to be paid at the end of the term. 355  
To find what any debt due at the end of any number of years is worth in ready money. 358  
To find what any yearly annuity at the end of any number of years is worth in ready money. 362    
The end of the Table.   

JONSON'S ARITHMETIC.  

CAHP. I. Numeration.  
NVmeration is the first part of Arithmetic, which showeth how to pronounce the value of any number of figures given; which are expressed by ten figures, whereof the tenth is a cipher, signifying nothing of itself; but being joined with figures, helpeth to increase the value: the figures are these;  
one, two, three, four, five, six, seven, eight, nine, cipher. 1. 2. 3. 4. 5. 6. 7. 8. 9 0.  

How to express the value of a number given.  
If a number be given, whose value is to be expressed, you shall understand, that the figure next the right hand is the least in value, and signifieth simply his own value, as the figure of 1 doth signify but one, and the figure of 2 doth signify but two, and the figure of 8 signifies but eight, and so of any other. And in the second place towards the left hand, every figure is in value ten, so that the figure of one there doth signify ten, the figure of 2 twenty, the figure of 8 eighty, and so of all other: in the third place towards the left hand, every figure is in value one hundred, so that the figure 1 in that place signifies one hundred, the 2, two hundred, etc. In the fourth place, every figure is in value one thousand, so there the figure of one signifies one thousand, the figure 2, two thousand, etc. In the fifth place, every figure is in value ten thousand: in the sixth place, one hundred thousand; and in the seventh place, one thousand thousands, or one million: in the eight place, ten millions: in the ninth place, one hundred millions: in the tenth place, one thousand millions, or one milliot; and so infinitely names may be given to the value of every prick, as is usual in the second part of Arithmetic, of Number, Square, Cube, sursolid, etc. or in Astronomical Arithmetic, Primes, Seconds, Thirds, Fourths and Fifths, etc.  
Now to express the value of any number given, set a prick with the pen over the fourth figure towards the left hand, and over the seventh, and tenth; and so over every third figure towards the left hand, to the end of your figures, as in this Example:  
 
Now begin and express the first four figures towards the right hand, as if they stood alone, which are 2567, or two thousand five hundred sixty seven. Then read the figures belonging to the second prick, which are 430, as if they stood alone thus, four millions three hundred two thousand five hundred sixty seven: then take the three figures belonging to the third prick, which are 635, or six millions three hundred fifty four millions three hundred and two thousand five hundred sixty seven: and so this whole sum is thus to be read, two hundred thirty seven thousand eight hundred fifty six milliots three hundred fifty four millions three hundred and two thousand five hundred and sixty seven; and so of any other sum.    

CHAP. II. Addition.  
ADdition is the second part of Arithmetic, and serveth to add or collect diverse sums of several denominations, and to express their total value in one sum.  
In Addition begin to add your sums at the right hand with the smallest numbers or denominations first, and gathering of their total mark how many of the smaller makes one of the next greater; as if your addition be Farthings, for every four farthings carry one penny in mind to be added to the numbers in the place of pence, and for every 12 put one shilling into the number of shillings, and for every 20 shillings, one pound into the place of pounds; and therefore to know how many of the smaller denominations, makes one of the next greater: I have here added in this place the several Tables of Coin sterling, of Weights, of liquid Measures, and dry Measures, of long Measures, of Time and Motion; which are very necessary to be known of every practitioner in Arithmetic, before he proceeds any further in the practice of Arithmetic, being used in every particular Rule of Arithmetic more or less.  

The Table of Coin Sterling.   Pence Forth. Four farthings makes one penny— 1 4 One shilling is— 12 48 One pound Sterling is 20 shillings— 240 960 One hundred pound Sterling is▪— 24000 96000  
Example. l. s. d. l. s. d. q. 785976. 17. 3 324. 8. 11. 1 80254. 10. 7. 222. 17. 3. 1 23547. 11. 0. 187. 10. 2. 0 7853. 12. 2. 354. 12. 1. 0 248. 00. 0. 1856. 00. 2. 0 93. 10. 1. 7859. 1. 11. 1 7. 11. 3. 3275. 1. 9 0 Su. 897981. 12. 4. 14079. 12. 3. 3.  

The explanation of these examples.  
In the first example toward the left hand I begin with farthings, which are 3, which I set down: 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 down the 3 pence, and carry in mind the 3 shillings to be added to the place of shillings. Then add the several sums of shillings, which are 1. 1. 2 7. 8, the total is 19, and the 3 in mind makes 22 shillings; set down the 2 shillings, and keep two ten to be added to the ten of shillings, which are 3 ten, which makes 5 ten, or 50 shillings; set down the odd ten 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; leave the 9 under the place of unites, and carry 3 ten in mind, and 7. 5. 5. 5. 8. 2. 2, total is 37; set down the 7 under the place of ten, and carry 3 in mind for the 30 ten, which is 3 hundred: then 3 in mind, and 2. 8. 8. 3. 1. 2. 3, total is 30; set a cipher, 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 sum, you must set them all down, placing the 4 under the place of thousands, and the 1 one place more towards the left hand, and then the Totall sum 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 be cast up into one Totall: and so I will here end with Addition of Coin, & put a several example of every table for the full tables & perfect understanding of the said table, which are of great use in all the several rules of Arithmetic.  

The Table of Avoirdupois weight.  Haberd. the pound. owned. Dra. Scruple Grain●. One pound is— 16 128 384 7680 One half pound is 8 64 192 3840 One quarter of a pound is— 4 32 96 1920 One eighth of a pound is— 2 16 48 960 One sixteenth of a pound is— 1 8 24 480  
The Hundred. Po●. Ounc. Dra Scruple. One hundred is— 112 1792 14336 43008 One half hundred is— 56 896 7168 21504 One quarter hundred is— 28 448 3534 10752 One half quarter hundred is— 14 224 1792 5376  
Example of Weights. C. qu. li. o●●. C. q. li. o●●. d●. 27. 3. 27. 6▪ 127. 3. 17. 8. 3. 18. 1. 17. 12. 118. 2. 10. 12. 1. 13. 2. 10. 3. 33. 0. 0. 0. 0. 73. 0. 0. 5. 17. 1. 12. 2. 3. 83. 2. 5. 12. 22. 3. 1. 7. 0. 2 2 2  17. 0. 10. 3. 0.     2 1 2   211. 2. 6. 6. 336. 2. 24. 00. 7.   

The Explanation.  
In the Avoirdupois weight, 20 grains makes one scruple, 3 scruples one dram, 8 drams one ounce, 16 ounces one pound, 112 pound is one hundred of the Avoirdupois weight, whereby is sold all kind of Merchandise usual in this Realm, and therefore in Addition of Weights Avoirdupois, for every 3 scruples add one dram, and for every 8 drams one ounce, and for 16 ounces 1 pound, for 28 pound one quarter of a hundred, and for every 4 quarters one hundred. first, I begin with the drams in the first example to the right hand, which are 3. 1. 3, total is 7 drams, which I note down underneath, because they are less than one ounce. Secondly, the ounces are 3. 7. 2. 12. 8. total is 32 ounces, or 2 pound, because 16 ounces is one pound; which two I set under the place of pounds with a light touch of the pen for to remember it the better, and place a cipher in the place of ounces.  
Thirdly, the pounds are 2. 10. 1. 12. 10. 17 total is 52 pound, which is one quarter of a hundred, and 24 pound, place 24 pound under 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 half a hundred; place 2 quarters in the place of quarters, and put over 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, total is 13; place 3 there, and carry one for the 10, which one in mind, and 1. 1 makes 3, which set down, and the total is 336 hundred, 2 quarters, 24 pound, 0 ounces, 7 dams; and so the other example is in the same manner to be cast up, and so of all other.  

The Table of Liquid Measures.   Pints. One pound or pint— 1 One quart— 2 One pottle— 4 One Gallon— 8 8 Gallons, a Firkin of Ale, Soap, or Herring— 64 One Firkin of Beer— 72 One Firkin of Salmon, or Eels— 85 2 Firkins, or one Kilderkin of Beer— 128 2 Kilderkins, or one Barrel— 250 One Tirce of Wine— 336 63 Gallons one Hogshead of Wine— 504 2 Hogsheads, or a Pipe or Butt— 1008 2 Pipes, Butts, or a Tun of Wine— 2016  

The Table of Dry Masures.   Pints. One Pint— 1 One Quart— 2 One Pottle— 4 One Gallon— 8 One Peck— 16 4 Pecks one Bushel Land-Measure— 64 5 Pecks, one Water-bushell— 80 8 Bushels one Quarter— 512 4 Quarters, on Chaulder— 2048 5 Quarters one Way— 2560  

The Table of Long Measures.   Inch. Three Barley Corns in length, one Inch— 1 One Foot— 12 One Yard, or 3 Foot— 36 Or 3 Foot 9 Inches, an English Ell— 45 Or 6 Foot one Fadom— 72 Or 5 Yards and half, a Pole or Perch— 198 Or one Perch in breadth, and 40 long, one Rood— 198 Or 4 Perches breadth, and 40 long, an Acre of land— 792 160 Square Perches, is one Acre— 792 40 Rods in length is one Furlong, and 8 Furlongs is an English Mile.   

The Table of Time.   Minut. One Minute— 1 One Hour— 60 One Day natural, or 24 Hours— 1440 One Week, or 7 Days— 10080 One Month, or 4 Weeks, or 28 Days— 40320 13 Months one Day 6 Hours, or 365 Days, one Year— 525960  

The Table of Moti●●.  360 Degrees, 21600 Minutes, 129600 seconds— 12 Signs. 30 Deg. 1800 min. 108000 sec.— 1 Sign. 1 deg. 60 min. 3600 sec.— 1 Degree. 1 min. is 60 sec.— 1 Minute. 1 second— 1 Second. 7776000 thirds makes the 12 Signs— 1 Third. 466560000 fourth's makes the 12 Signs— 1 Fourth. 27993600000 fifths is 12 signs 1 Fifth. 1679616000000 sixths is 12 Signs— 1 six.   

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 every 4 Roods one Acre.  
Secondly, for every 4 quarters of Inch, take 1 Inch, and for every 12 Inches 1 foot, and for every 3 foot, one yard.  
Thirdly, for 16 pints take one peck, and for every 4 pecks one Bushel, into the place of Bushels.  
Fourthly, for every 8 pints of liquid measure, take one Gallon, and for every 63 Gallons one Hogshead.  
Fifthly, in the example of time; for 60, minutes take one hour, and for 24 hours one day, and for 365 days, one year.  
Sixthly, for 4 nails take one quarter of a yard, and for 4 quarters one yard, etc.  
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 Sign.  
And this is the use of these Tables in Addition and Subtraction; for look what you carry over in Addition, that you must borrow in Subtraction, I will hear add examples of every kind, leaving the Reader to exercise himself by the Rules before taught.  
Example. Acres, Rood. per. Feet. Inch. quart. 127. 3. 21 124. 7. 3 246. 1. 12 246. 11. 4 17. 3. 22 134. 7. 2 27. 1. 8 120. 8. 0 37. 0. 17 72. 10. 2 2 2  3 2  456. 2. 00▪ 699. 9 3.  
Bushel, Peeks, Pints. Yard. Quar. Naile. 127. 3. 11. 127. 2. 3 256. 1. 7 359. 1. 4 345. 0. 0 152. 3. 0 184. 2. 10 16. 0. 0 1 1  1 1  913. 3. 12▪ 655. 3. 3.  
Years, days, hours, min. seconds. 356. 245. 16. 35. 20. 249. 100 12. 30. 00. 756. 12. 00. 10. 12. 140. 27. 30. 25. 02. 1618. 00. 20. 00. 00. 1 3 1   3120. 22. 07. 40. 34.  
Signs, degrees, minutes, seconds, thirds. 11. 22. 32. 24. 18. 8. 19 17. 20. 12. 10. 07. 00. 08. 15. 2. 17. 35. 50. 59 3. 29. 30. 12. 00. 3 1 1 1  37. 05. 55. 55. 44.   

The Proof of Addition.  
The proof of Addition is made by Subtraction; for if you subtract the numbers which you added from the total of the Addition, there will remain nothing, if the work be truly done.  
Example.  l. s. d. q.  378567. 19 10. 1.  240023. 10. 2. 0.  854●26. 07. 1. 0.  785634. 13. 3. 2.  320500. 00. 11. 1.  2 2   Totall, 2579●52. 11. 4. 0.  
First, add together the greatest Sums in value in the place of hundred thousands, which makes 23, which take from 25, and there will remain 2: then the figures in the fifth place, 26 taken from 27, there will remain 1. Thirdly, the figures in the place of thousands, makes 17, which taken from 19, leaves 2: then 19 in the place of hundreds taken from 20, leaves 1: and again, 13 in the place of ten from 15, leaves 2: and lastly, 20 in the place of unites from 22 ponnd, leaves 2 pound: then 49 shillings from 2 pound 11 shillings, leaves 2 shillings▪ also 2 shillings 3 pence in the place of pence, from 2 shillings 4 pence, leans 1: and last of all, 4 farthings from 1 penny, leaves nothing, which proves the work to be truly wrought.  
  

The second proof of Addition.  
Cut of the uppermost numbers with a dash of the pen, and add the remainder into one Totall; and then subtract that sum from the whole total, and the remainder will be the numbers which you cut off, if the work be true, else not.  
Example.  378567. 19 10. 1.   240023. 10. 2. 0.   854326. 7. 1. 0.   785634. 13. 3. 2.   320500. 00. 11. 1.   2 2    The total 2579052. 11. 4. 0. of all. Subt. 2200484. 11. 5. 3. the sum. The 378567. 19 10. 1. proof.  
And so much shall suffice to have spoken of Addition, and the proof thereof.   

Questions of Addition.  
What number is that, to the which if you do add 45, the total will be 357.  
Answer: Subtract 45 from 357, remains 312.  
Example. 357 45 312  
What three numbers are those, to which if you add 27, 36, and 45, their products shall be equal, and the sum arising shall be 120.  

Proof.  120 120 120 93 27 35 45 27 93 89 75 120  
What number is that, to the which if you do add 354 pound, 7 shsllings, 9 pence, the total will be 512 pound, 15 shillings, o penny. Answer: Subtract 354 pound, 7 shillings, 9 pence, from 512 pound, 15 shillings, o penny, and the remainder will be 158 pound, 7 shillings, 3 pence, which is the number that you do seek.  
Example. l. s. d. 512. 15. 0. 354. 7. 9 158. 7. 3.    

CHAP. III. Subtraction.  
SVbtraction s●●●eth to deduct one sum from another, the lesser from the greater, and to show the remains.  
Place your greater number, from which the Subtraction is to be made, in the uppermost part, and the number to be subtracted, or deducted right underneath every figure under his like kind, or denomination, viz. pounds under pounds, shillings under shillings, and pence under pence, etc. in this manner.  
 l. s. d. q. Lent. 7756. 13. 10. 1. Paid 3949. 17. 11. 2. Rest. 3806. 15. 10. 3. Proof. 7756. 13. 10. 1.  
Then begin your subtraction at the left hand, at the smallest numbers; but if the lowest figure of the undermost numbers be the greatest, that it cannot be abated out of the number above it, then add one of your next greater denomination, and make your subtraction from both, noting the remainder; as if you have 10 pence to take from 7 pence, add one shilling, or 12 pence, unto 7 pence, that maketh 19 pence; then take 10 pence from 19 pence, and there will remain 9 pence, which note down under the 10 pence: and because you did borrow one shilling, therefore in the number of shillings you shall take away one more than it is, in the next place of shillings, and this rule is general, in Coin, Measure, Time, Motion, or any other thing else whatsoever.  
1. Example of Subtraction of Coin.  l. s. d. q. Lent. 789786. 17. 11. 3. Paid. 692583. 19 10. 1. Rest. 97202. 18. 1. 2. Proof 789786. 17. 11. 3.  
2. Example of Weights.  C. que. l. owned. Lent. 127. 3. 27. 10. Paid. 38. 2. 24. 15. Rest. 89. 1. 2. 11. Proof 127. 3. 27. 10.  
3. Example of Time.  Years, days, hours, min. Totall. 1618. 340. 20. 56. Deduct. 1581. 122. 15. 59 Rest. 0037. 218. 04. 57 Proof. 1618. 340. 20. 56.  
4. Example of Motion.  Sig. Deg. Min. Second. Thirds. Totall. 11. 22. 36. 52. 40. Subt. 7. 29. 51. 42. 56. Rest. 3. 22. 45. 09. 44. Proof. 11. 22. 36. 52. 40.  

The explanation of these examples.  
In the first example of Coin, begin your subtraction at the right hand, saying; 1 farthing from 3 farthings, leaves 2 farthings, which note down under the 1 farthing. Then 10 pence from 11 pence, leaves 1 penny. Thirdly, 19 shillings from 17 shillings you cannot have, therefore take one pound, or 20 shillings, and add to 19 shillings, saying, 19 shillings from 37 shillings, rests 18 shillings, which note down. Then 1 that you borrowed, & 3 pound, is 4 pound from 6 pound, leaves 2 pound to set down under 3. Then 8 from 8 leaves nothing, place there a cipher, or 0 under 8. Then 5 from 7 rests 2; then 2 from 9 leaves 7, which also note again; 9 from 8 cannot be taken, then make it 10 more, and say 9 from 18 leaves 9, which set down: and last of all, 1 borrowed and 6 is 7, from 7 leaves nothing, and the work is ended, and the remainder will be 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 add 1 pound, or 16 ounces to 10, makes 26; then say, 15 from 26 leaves 11 ounces, which note down: then 1 borrowed and 24 is 25, from 27 pound leaves 2 pound remaining; then 2 quarters from 3 quarters, leaves 1 quarter remaining; then 8 from 7 cannot be, therefore take 8 from 17, rest 9, which note down: than one borrowed and 3 maketh 4, from 12 rests 8, and the work is done, and the remain is 89 hundred 1 quarter 2 pound 11 ounces.  
3 Example.  
First, take 59 minutes from 56 minutes cannot be, but then take 59 minutes from 60 minutes, or one hour, and there will remain 1 minute, which add to 56 minutes, and that will make 57 minutes, which note down in the place of minutes: then 1 borrowed and 15 hours makes 16 hours, which taken from 20 hours' leaves 4, which note under the 15; and then 2 days from 0 cannot be, but 2 from 10, and there will remain 8, which note down: then 1 borrowed and 2 maketh 3, from 4 leaves 1; also 1 from 3 leaves 2: lastly, 1 from 8 leaves 7, and 8 from 11 leaves 3; then 1 borrowed and 15 makes 16, from 16 leaves nothing, and the remainder will be 37 years 218 days 4 hours 57 minutes; the like is done in the other example of Motion, and therefore here needless to be rehearsed.   

To subtract from a Unite.  
Set down with your pen a unite in any place, adding Ciphers unto it, and the several numbers which you will subtract from it of pounds, shillings and pence right underneath: then note what each several number of your lowest numbers doth want of 9 unto the place of unites, and set that right under for the remainder: and lastly, note what your shillings and pence doth want of 20 shillings, and set that down for your remaynor, and the work is ended.  
Example.  l. s. d. Lent. 1000000. 00. 00. Paid. 232864. 17. 03. Rest. 767135. 2. 9 Proof. 1000000. 00. 0.   

The proof of Subtraction.  
The surest proof of Subtraction is made by Addition: for if you do add the numbers remaining, unto the numbers deducted, they will return your former Sum, if the work be truly wrought, as will appear in the proof of all the several examples before going, and therefore here again in this place needless to be rehearsed. Only I will add one for examples sake.  
In the last example, the numbers which did remain, were 767135 pound 2 shillings 9 pence, and the numbers deducted, 232864 pound 17 shillings 3 pence; these two numbers added together, aught to make a unite in the seventh place; wherefore I add 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 unite.    

Multiplication. CHAP. FOUR  

The Table of Multiplication.  1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14 16 18 3 6 9 12 15 18 21 24 27 4 8 12 16 20 24 28 32 36 5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 9 18 27 36 45 54 63 72 81  
THis Table of Multiplication must be learned perfectly by heart, for to know readily what the multiplication of any two digit numbers under nine, or unto nine do make, and then Multiplication will be very easy: for Multiplication is a number of additions speedily performed; as if you should say, How many in number is 8 times 7, if you should set down 7 eight times one under another, and add them together, the total will be 56: but if you look in the Table for 8 in the head, and 7 in the side, you shall find under 8, right against 7 in the the same parallel 56; or if you find 8 in the side, and 7 in the head, the like number will appear, and these numbers in the table are to be fit in memory.  
1. Example according to the usual way. 

87968. The multiplicand.  
987. The multiplier.    
First, begin your multiplication at the right hand, saying, 7 times 8 make 56, place 6 under the 7, and keep 5 in mind, to be added to the product of the multiplication of 7 by 6, saying, 7 by 6 makes 42, and 5 in mind is 47; set 7 down under the 6, and keep 4 in mind: then 7 by 9 is 63, and 4 makes 67; set 7 down, and keep 6 in mind: then 7 by 7 is 49, and 6 is 55; place 5 and keep 5 in mind: lastly, 7 by 8 is 56, and 5 is 61, which set down the 1 first, and the 6 one place more towards the right hand; and so the multiplication by the first figure 7 is done, then cancel the 7 of your multiplier, and your work will stand, as in this example.  
 
Secondly, begin with 8, the second figure of your multiplier, saying, 8 times 8 is 64; place the 4 under the said 8, and keep the 6 in mind: then 8 by 6 is 48, and 6 makes 54; set down 4 in the next place, and keep 5 in mind: then 8 by 9 is 72, and 5 makes 77; set down 7, and keep 7 in mind: then 8 by 7 is 56, and 7 makes 63; set down 3, and keep 6. Lastly, 8 by 8 is 64, and 6 makes 70; set the 0 first, and the 7 one place more towards the left hand, and cancel the 8 of your multiplier, and the work will stand thus.  
 
Thirdly, begin with 9, the last figure of your multiplier, saying, 9 by 8 is 72; place the 2 under the said 9, and keep 7: then 9 by 6 is 54, and 7 is 61; place 1, and keep 6: then 9 by 9 is 81, and 6 is 87; place 7, and keep 8: then 9 by 7 is 63, and 8 is 71▪ place 1, and keep 7: last of all, 9 by 8 is 72, and 7 is 79; place the 9 first, and the 7 one place more towards the right hand, and the whole work is ended, then gather the total by addition.  
1. Example.  
 
2. Examples with Ciphers.  

The exposition of this example.  
First, 7 by 1 is 7, which note down: 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 down a 0, and carry 6 again: 7 by 2 is 14, and 6 makes 20, set down 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, leave the 2 in that place: then lastly, 7 by 7 is 49, being the last number set down all the 9 under 7, and the 4 one place more to the left hand, and the work will then stand thus.  
 
Secondly, cancel 7, and then say; 5 by 1 maketh 5, place that 5 under the 0; and then 5 by 0 is 0, place a 0 under the 5 in the next place; and then 5 by 5 is 25, set down 5, and carry 2: then 5 by 6 is 30, and 2 makes 32, set down 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 maketh 14, set down 4, and carry 1: then 5 by 3 is 15, and 1 makes 16, set down 6, and carry 1: then 5 by 0 is 0, set down the 1 there: last of all, 5 by 7 is 35, set them all down, and the work will then stand thus.  
 
Thirdly, cancel the 5, and then say, 0 by 1 is 0, place a 0 under the 0 of your multiplier, & then proceed to the next figure of your multiplier, which is 2, saying, 2 by 1 is 2, place the 2 under the said 2 of your multiplier: then 2 by 0 is 0, which set down: then 2 by 5, maketh 10, set down a 0, and carry 1: then 2 by 6 is 12, and 1 is 13, set down 3, and carry 〈…〉: also 2 by 8 is 16, and 1 is 17, set down 7, and carry 1: also by ● is 4, and 1 maketh 5, which set down: again, 2 by 3 is 6, which set down: lastly, by 7 is 14, which set down, and the work will stand, as in this example.  
 
Fourthly, cancel the 2, and say, 3 by 1 is 3, which place right under the said 3: then 3 by 0 is 0, and work in all respects as before, and the work being ended, will stand thus.  
 
I will here add some few examples to be wrought by the pen, without any troubling of the memory with bearing aught in mind.  
Example.  
2. Example.   

The explanation of the work by the pen, without charging the memory.  
The first example.  
First, I multiply all the figures of my multiplicand by 7, the lowest figure of my multiplier, saying, 7 by 8 is 56, put 6 under the 7, and 5 under the 8: then 7 by 6 is 42, leave the 2 under 5 last placed, and set the 4 one place more towards the left hand under the 9: then 7 by 9 is 63, leave 3 under the 4 last placed, and set 6 one place more to the left hand under 7: then 7 by 7 is 49, leave 9 under the 6 last placed, and the 4, set one place more to the left hand under the 8: lastly, 7 by 8, makes 56, leave 6 under the 4, & place 5 one space more to the left hand, as before, then cancel 7 of your multiplier, and the multiplication by the first figure is ended, and the work will stand thus.  
Example.  
Then for the second work, say, 8 by 8 is 64, place 4 under the said 8, and put 6 under the next figure 3: then 8 by 6 makes 48, leave 8 under 6, and put 4 under the next 9: and so working in all respects as at the first, and your second work will stand thus, as in this example.  
 
Lastly, cancel 8 your multiplier, and then multiply by 9, as is before taught, placing the first figure of your product under the figure multiplying, and the work being ended, it will stand thus; and lastly, gathering the total by addition, it is 86824416; as in this example.  
 
There is no difficulty in this kind of working, but only when there falls a 0 in in the multiplicand, or multiplier; for if there be a cipher than you must fill up the places as you work, 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 add one example, having all the difficulties that may happen, for the better understanding hereof.  
Example.  
Example.   

How to multiply, and to bring the product in the last line.  
Place your numbers right one under 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.  
 
Then begin and say, 7 by 8 is 56, place the 6 under the line under the 7, and the 5 above the line in a smaller figure in the next place towards the left hand: then 7 by 6 is 42, and the 5 above the line makes 47, leave 7 under the 8, and set the 4 again above the line: then 7 by 9 is 63, and the 4 above the line makes 67, place 7 there, and set the 6 in the next place above the line: then 7 by 7 is 49, and 6 above the line makes 55, leave 5 there, and put 5 again over the line: lastly, 7 by 8 makes 56, and the 5 last placed makes 61, place that whole sum under the line, and the work will stand, as above in the example.  
Secondly, draw a line again a little disstant, as before from the last product, as in the example following.  
Example.  
Then say, 8 by 8 is 64, and 7 makes 71, place 1 under the 7, and set 7 above the line: then 8 by 6 is 48, and the two sevens between lines makes 62, place 2 under the 7, and set 6 again over the line: then 8 by 9 is 72, and 6 makes 78, and 5 makes 83, place 3 under the line, and 8 above the line; then 8 by 7 is 56, and 8 makes 64, and 1 makes 65; place 5 under the line, and set 6 above: lastly, 8 by 8 is 64, and 6 makes 70, and 6 makes 76, place them both down; and the work will stand as above in the example.  
Thirdly, draw a line again, as before, a little distant from the last product, as in this example.  
Example.  
Thirdly, say 9 by 8 is 72, and 2 makes 74, place 4, and put 7 over the line: then 9 by 6 is 54, and 10 makes 64, place 4 under the line, and put 6 above: then 9 by 9 is 81, and 11 above makes 92, leave 2 under the line, and 9 over the line: then 9 by 8 is 63, and 15 makes 78, leave 8 under the line, and 7 above: lastly, 9 by 8 is 72, and 14 makes 86, place them both under the line, and then bring down the two figures which are cut off by two right down lines, which are 1 and 6, and the work is ended, and the work will stand, as appear in the example above, and the total product is in the last line, 86824416; and this doth not charge the memory, for all the figures are set down in view, and to be seen at the first sight, and this is the second kind of multiplication, without charging of the memory.  
   

CHAP. V. Division.  
SEt your Dividend, which is the number to be divided in the upper part, and the Divisor next to the left hand, under the greatest figures in value of your Dividend: If the upper numbers be greater than the lower, or else place your divisor one place more towards your right hand, as in this example.  
 
If you would divide 78567 by 84, place them as above; for because you cannot have 8 out of 7 in the Dividend, therefore place your 8 one place more towards the right hand, and the 4 next to it, and your quotient you must place at the right side of your numbers behind a crooked line. But I will first give an example of Division by one figure: I would divide 65490 pound amongst 5 men; place your numbers thus.  
Example.  
First, I seek how oft 5 is in 6, this I may have but once; then put 1 in the quotient beyond the crooked line, and take 5 out of 6, and there will rest 1, set that over 6, and then remove your divisor one place more to the right hand, and then seek you how many times 5 may be had in 15, and the answer is, thrice, therefore place; in the quotient, and by it multiply your divisor 5, maketh 15, which taken out of 15, leaves nothing, place a 0 over the 5, and remove your divisor, and seek how oft you may have 5 in the 4 over it, but you cannot have it once; wherefore put a 0 in the quotient, and remove your divisor, and seek how many times you may have 5 in the figures over and behind it, which are 49, and you may have it nine times, put 9 in the quotient, and by it multiply your divisor 5, makes 45, which taken from 49, leaves 4, which place above the 9 And lastly, remove again your Divisor 5 under the 0, and seek how many times 5 is in 40, and you shall ●nd it 8 times, place 8 in the quotient, and by it multiply 5, makes 40 which taken from 40, leaves nothing remaining and the work is ended, and will stand thus, as in the example, and I find, if I divide 65490 pound amongst 5 men, every man shall have for his part 13098 pound.  
 
And this is the order of Division for one figure: but if your Divisor do consist of more figures than one, than you must take the first figure of your Divisor no oftener out of the Dividend, than you can also take every several figure of your Divisor, out of the same figures of the Dividend standing above them, as for example.  
If you would divide 86824416 by 987, which was one of the products of the multiplications in the rules before going, for a trial of your former work, then place your numbers, as in the example following.  
Example.  
Then I seek how oft I may have 9 in 86, I find I may have it 9 times; but if I consider the next figure 8 of my Divisor, I cannot have also 9 times 8 out of the numbers remaining; if I take 9 times 9, which is 81, out of 86, there will remain but 5; and then 9 times 8, the next figure of my divisor, makes 72, which cannot be taken out of 58 which will remain; therefore I place 8 in the quotient, and by that I multiply all the figures of my Divisor, 987 makes 7896, which taken from 8682, leaves 786 above them: and the work will stand thus.  
 
Secondly, I remove my Divisor 987 one place nearer the right hand, and then I seek how oft I may have 9 in 78, which I see I can have but 7 times, so I put 7 in the Quotient, and by that 7, I multiply my Divisor 987, makes 6909, which taken from 7864, the numbers above them there will remain 955, and the work will stand thus.  
Example.  
Thirdly, again I remove my divisor 987 one place nearer the right hand, and seek how many times I may have 9 in 95, and I find I may have it 9 times, which 9 I set into the Quotient, and by it multiply 987, makes 8883, which taken from 9554 leaves 671, and the work will stand thus.  
Example.  
Fourthly, I remove my Divisor again, and seek how oft I may have 9 in 67, and I see I can have it but 6 times, than I put 6 in the quotient, and by it multiply 987, makes 5922, which taken from 6711, leaves 789, and the work will stand in the example following.  
Example.  
Lastly, I remove my Divisor again, and seek how oft I may have 9 in 78, and I find I may have it 8 times, which 8 I put into the quotient, and by it I multiply my Divisor 987 makes 7896, equal unto the numbers above; and so being taken away, leaves nothing remaining, and proves the multiplication to be truly wrought, as appeareth in the example following.  
Example.  

The third Example of Division.  
The second kind of Division is this: first, place your dividend & divisor as in the former Examples, & then having found out the figure of your quotient, begin with the least figure of your divisor towards the right hand 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 above the same, if it exceed not 9; but if the product be above 9, then for every 10 bear one in mind to be added to the product of the multiplication of the second figure of your Divisor by the quotient; and so in all respects work for every other figure, and you shall need make no more figures above your Dividend then necessity shall require, as for example.  
I would divide 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 Dividend and Divisor as followeth.  
 
Then first I seek how often I may have 8 in 71, I find by trial I can have it but 7 times▪ than having 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 forty in mind; then I cancel the 2 over the 6, and placea 0 in the room over it. Secondly, I say, 7 by 7 is 49, and 4 in mind makes 53, from 59 leaves 6, and carry 5● cancel the 9, and place 6 over it. Thirdly, 7 by 9 is 63, and 5 in mind is 68, from 74 leaves 6, and carry 7, cancel the 4, and place 6 above it: also 7 by 8 is 56, and 7 makes 63, which taken from 71, leaves 8 remaining, which 8 place over the 1, and cancel the 71, and the first work will stand thus.  
 
Secondly, I remove my Divisor 8976, and seek, how many times I may have 8 in 86, ● find 9 times; then I multiply 6 by 9 placed in the Quotient, makes 54, which taken from 60, leaves 6; place 6 above the first 0, and carry 6 for the 60: then say, 9 by 7 is 63, and 6 in mind makes 69, from 70 leaves 1, and carry 7 in mind; cancel the 0 over the 7, and place the 1 over the 0. Again, 9 by 9 is 81, and 7 in mind is 88, which taken from 96, leaves 8 to be placed above the first 6, and carry 9 in mind: lastly, 9 by 8 is 72, and 9 makes 81, which taken from 86, leaves 5 to be placed above the 6, and the work will stand as followeth.  
Example.  
Thirdly, again I remove my Divisor, and seek how many times 8 is in 58, and I find I can have it but 6 times, which I place in the Quotient: then I say, 6 by 6 makes 36, from 37 leaves 1 above 7, and carry 3: then 6 by 7 is 42, and 3 is 45, from 46 leaves 1, above the 6, and carry 4: again, 6 by 9 is 54, and 4 makes 58, from 61 leaves 3 above the 1, and carry 6. Lastly, 6 by 8 is 48, and 6 makes 54, from 58 leaves 4, and the work stands thus, as in this example.  
 
Fourthly, I remove my Divisor, and seek how oft I may have 8 in 43, and I find but 4 times, I place 4 in the Quotient. Then 4 by 6 makes 24, from 29, leaves 5, and carry 2, set 5 over the 9: then 4 by 7 is 28, and 2 makes 30, from 31, leaves 1, and carry 3. Again, 4 by 9 is 36, and 3 makes 39, from 41 leaves 2, and carry 4. Lastly, 4 by 8 is 32, and 4 is 36, from 43, leaves 7, and the work will then stand thus.  
Example.  
Fifthly, I remove my Divisor, and seek how oft I may have 8 in 72; I find 8 times, which placed in the quotient, I multiply 6 by 8, makes 48, from 48, leaves 0, and carry 4● then 8 by 7 makes 56, and 4 is 60, from 65 leaves 5, and carry 6: then 8 by 9 is 72, and 6 makes 78, from 81, leaves 3, and carry 8: then 8 by 8 makes 64, and 8, is 72, from 72 leaves 0 remaining, and the work will stand thus.  
Example.  
Sixthly, I remove my Divisor, and seek how oft I may have 8 in 3, which I find not once; I place a 0 in the Quotient, and remove my Divisor one place more and seek how many times 8 is in 35; I find I can have 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 leaves 7, and carry 3: again, 3 by 9 is 27, and 3 is 30, from 30 leaves a 0, and carry 3: also 3 by 8 is 24, and 3 is 27, from 35 leaves 8; and the work will stand thus.  
Example.  
Lastly, I remove my Divisor, and seek how oft I may have 8 in 80; I find 9 times, I place 9 in the Quotient, and say, 9 by 6 is 54, from 54 leaves 0, & carry 5: then 9 by 7 is 63, and 5 is 68, From 68 leaves 0, & carry 6: Then 9 by 9 is 81, and 6 is 87, from 87 leaves 0, and carry 8: last of all, 9 by 8 is 72, and 8 makes 80, from 80 there will remain nothing but cyphers, and the work is quite ended, and will stand, as in the example following.  
Example.  
The fourth and last kind of Division, is the most absolute, speedy, and easy, not charging the memory at all, with keeping any numbers in mind; and also the proof of your work is made by Addition, and not by multiplication, as hath heretofore been commonly used, but the figures of your work are by Addition, the proof of your work, as shall appear by examples following.   

The third Worke.  
First, place your Dividend between two parallel lines, and your Quotient at the right side of your Dividend, behind a crooked line, as before; then place your Divisor next to the left hand of your Dividend, behind a perpendicular line: and lastly, mark how many figures your Divisor hath, and in the room of those figures place cyphers under the figures of your Dividend, so many as your Divisor hath figures, as in the last example; which I will again repeat in this place, and work it by this kind of Division, making the proof of the work by Addition of the same figures.  
Example.  
First, I point to the first cipher towards the left hand, and seek how oft I may have 8, the greatest figure in value of my Divisor, having respect to the other figures of my Divisor, to take them also as often, out of the figures above, and I find I can have it but 7 times, which 7 I place in the Quotient, and by that 7 I multiply my Divisor 8976, saying first, 7 by 6 is 42, place the 2 under the lowest cipher towards the right hand, and carry 4: then 7 by 7 is 49, and 4 is 53, set 3 under 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 down in their places, and the total is 62832, to be subtracted from 71492, and there will remain 8660; and the work will stand thus.  
Example.  
Secondly, I cancel the first cipher to the left hand, and place one cipher more towards the right hand, under the 0, and then I point again to the first cipher, and see how oft I may have 8 in 86; I find 9 times, and placing 9 in the Quotient, by it I multiply 8976 my Divisor, placing the lowest figure in value under the lowest cipher to the right hand, and the rest in order, and I find the product to be 80784, which taken from 86600, leaves 5816 remaining, and then your work will stand, as in this.  
Example.  
Thirdly, I cancel my Divisor, or one cipher, and place one cipher more under 7, and then seek how oft I may have 8 in 58, which I find 6 times, and by it I multiply my Divisor 8976 makes 53856, which taken from 58167, leaves 4311, and the work will stand as followeth.  
Example.  
Fourthly, I cancel one cipher, and place a cipher under 9, and then seek how oft I may have 8 in 43, which I find but 4 times, which place in the Quotient, and by it I multiply my Divisor 8976, makes 35904, which taken from 43119, leaves 7215.  
Example.  
Fifthly, I cancel one cipher, and place a cipher under 8, and fee●e how oft ● is in 72; ● find 8 times, which placed in the Quotient, I multiply my Divisor 8976 by it makes 71808, which taken from 72158, leaves 350, and the work stands, as in the example following.  
Example.  
Sixthly, I cancel one cipher, and place another under the 0, and seeking I find I cannot have 8 in 3; therefore I place a 0 in the Quotient. Seventhly, I cancel one cipher, and place one other under the 6, and seek how oft I may have 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, leaves remaining 8078.  
Lastly, I cancel the next cipher, and do place another under the last figure of my Dividend 4, and seek how oft I may have 8 in 80; I find 9 times, and then placing 9 in the Quotient, I multiply my Divisor 8976, and the Quotient is 80784, equal unto the numbers above, and so being subtracted from the numbers above, leaves o remaining, and the work is ended, and will stand thus.  
Example.  
The proof of this Division is made by Addition of the figures, under the line or Dividend, for if they return your former Dividend, the work is true wrought; or otherwise be sure some error is in your work, if there remains any fraction after your work is ended, than it is to be added into the lower figures in their several places, as shall appear by examples following.  
Here in this example following, working according to this latter form of work, there is advantage to be taken; if the figures of the Quotient be well noted, as here the fourth figure of the Quotient is 7, the Product of the Divisor multiplied by it is 1438816, and also the eleventh figure of the Quotient is 7, so that coming to multiply the Divisor again by that 7, I need but take the Product of the first multiplication by 7, which is 1438816, and so place them in their several 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 Divisor by 3: and so of any other figure coming into the Quotient more times than once, as by the example before going will appear.  
Place the great example following, in this place.  
Example▪  
Example.  
Example.   

How to divide by a Unite with Ciphers.  
If you will divide by 10, or by 100, or 1000, or with any other unite with cyphers, one or more; do but cut off so many figures from the right hand of your Dividend, as there are cyphers in your Divisor, and the remains is your Quotient.  
Example.  
If you would divide 786589 by 10, cut off the last figure 9, and the residue is your Quotient 78658 2/10; or if you will divide by 100, cut off two figures, and the Quotient will be 7865 ●2/100; or by 1000, and the Quotient will be 786 ●●2/1000; and so of all other.  
 
If you will divide the Product of 1999 squared; that is to say, multiplied in itself, which is 3996001 by 1999, for expedition of work, after you have 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 every several 9, but the first will serve for all, as in the example following.  
Example.  
Example.   

Brief Rules by Multiplication and Division.  
If you multiply any number of nine; as if you will multiply, or square 5 times 9 by 5 times 9, then place your nine in this order following.  
Example.  
Then say, 9 times 9 is 81, place the 1 under the first 9 to the right hand, and then subtract the 1 from the first 9 to the left hand, and add the cyphers between, and the Product is ended, and is 9999800001, as appeareth.  
The proof of the work after the ordinary way.   

To multiply any number by 9  
Add a o to the number you intent to multiply, and then set the same numbers under them, and subtract them from the uppermost, and the remains 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 half, take half 856, which is 428, and add into the former sum, makes the total 20972.  
Example.  
What number is that, which being divided by 24, the Quotient will be 856. Answer, multiply 856 by 24, makes 20544 for the number that you seek.  
Example.  
There is a plot of land containing 848 Perches, the one side is 24, what must the other be. Answer, Divide 848 by 24, the Quotient is 35 ⅓ for the other side.  
 
If you will divide the Product of 5 times 9 squared, which is 9999800001, by 5 nine, than set the Divisor right underneath the Dividend, and add them together, and cut off the 5 cyphers from the Product, and the residue is the Quotient.  
Example.  
What number is that, which being multiplied by 15, the total will be 756. Answ. divide 756 by 15, and the Quotient is 50 6/159, or 2/5, for the answer, or number you do seek.  
Example.  
There are 825 men, to march 15 in one rank, how many files will they make. Divide 825 by 15, it makes 55 files.  
Example.  
There is 948 pound of powder to be employed 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 Shoots each piece may make, to make an equal number of Shotts. Answer; divide 948 by 40, and it makes 23 Shoots, and there will remain 28 pound.  
Example.     


THE RULE OF REDUCTION.  
TO reduce any great number into a smaller denomination it is done by multiplication, and to reduce small denominations into greater it is done by division: in this manner mark how many of the smaller denominations is contained in one of the next greater, and by that number you must multiply the greater: or of the contrary, if you would bring small denominations into greater, mark how many of the smaller denominations makes one of the next greater, and that number shall be your divisor.  
Example.  
If you would reduce pounds starling into pence, multiply your pounds by 240 pence, because so many pence maketh pound starling, and the total will be the number of pence in the sum of pounds given. And chose, if you would bring pence into pounds starling: divide your number of pence by 240 pence, which are the pence in one pound, and the Quotient will show the number of pounds, in the sum of pence given: but in this operation the Tables in the beginning of this book will help much, for the speedy reducing of pounds, shillings pence, yards, else, bushels, pecks, pints, etc. into smaller or greater denominations; for if you search in the said Tables, you shall find your multiplier, or divider, whereby you are to multiply, or divide your number given, to perform the work, as shall appear by the several examples following.  

Reduction of Coin.  
In 87652 pound, how many pence: in the Table of Coin I find 240 pence makes one pound, so that in multiplying 87652 pound by 240, makes the sum of pence desired.  
1. Example.  
 
2. Example.  
In ●759 pound, 17 shillings, 8 pence, how many shillings, pence, and farthings.  
 
3. Example.  
In 3785437289 farthings, how many pounds, shillings, and pence: divide by 960 farthings, because 960 farthings makes one pound starling, & the remainder is farthings, which divided by 48, the farthings in one shilling, makes 3943163 pound, 16 shillings 10 pence, ¼.  
  

How to bring pounds, shillings, and pence at: he first work by Division.  
To bring pence at the first work into pounds, shillings, and pence: add a 0 to your number of pence, and divide that sum by 240, makes pounds, and the last figure will be primes, every unite in value 2 shillings, and the remainder always less than 24 pence, or one prime.  
Example.  
In 902372 pence, how many pounds, shillings, and pence; add a 0, makes 9023720, which divided by 240 pence, makes etc.  
 
2. Example.  
In 75000837504 pence, how many pounds, shillings, and pence: add a cipher, or 0.  
  

How to bring farthings into pounds, shillings, and pence at the first work.  
To bring farthings into pounds, shillings and pence at one work: add a 0 to your number of farthings, and divide the sum by 960, the number of farthings in one pound sterling, makes pounds; and the last figure of your Quotient will be primes every one in value 2 shillings: and if there remain 48, it is one shilling, or take 48 from the remainder for one shilling, the rest are farthings less than 48.  
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, add a 0, and divide by 960, makes ●943163 pound, 8 primes, or 16 shillings, 0 pence▪  
  

How to bring pence into pounds, shillings, and pence another way.  
Divide your number of pence by 4, and the remainder is pence, than that Quotient by 6, and the remainder is groats, always less than 6 groats, or one prime, or 2 shillings; and the latter Quotient, cutting off your Primes, is pounds, and so you have 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: divide first by 16, and the remainder is farthings, always less than 16, or one groat; and then again by 6, makes pounds, shillings, and pence, as before, cutting off the prime line.  
Example.  
In 8735672 farthings, how many pounds Shilling, and pence.  
  

Reduction of Weights.  
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.  
Example.  
In 1569●492 ounces Avoirdupois, how many hundreds, quarters, pounds, and ounces; find how many ounces makes 112 pound, in multiplying 112 pound by 16 ounces, makes 1792 ounces; by which divide, makes, as in the example following.  
  

Reduction of Measures.  
In 2356 Acres, 3 Rood, 27 Perches, how many Perches in all.  
Example.  
In 765437 Perches, how many Acres, Rood, and Perches: divide by 160.  
Example.   

Reduction of Time.  
In 356 years, 24 days, 36 hours, and 22 minutes; how many days, hours and minutes.  
Example.  
The Proof.  
In 187150342 minutes, how many hours, days, years, and minutes.  
  

Reduction of Motion.  
In 11 Signs, 34 degrees, 25 minutes, 36 seconds, 24 thirds; how many fourths.  
Example.  
The proof.  
In 4722971040 fourth's, how many signs degrees, minutes, seconds, thirds, & fourths.  
Example.   

Questions by Reduction.  

1. Question.  
In 389 pound Starling, how many Dollars of 4 shillings 8 pence, or 14 groats a piece. Reduce 389 pound into groats, in multiplying them by 60, makes 23340 groats; which divide by 14 groats, makes 1667 pound, and 8 pence.  
Example.   

2. Question.  
In 300 pound starling, how many Angels at a 11 shillings a piece Reduce 300 pound into shillings, makes 6000 shillings; which divide by a 11, makes 545 angels, and there will remain 5 shillings.  
Example.   

3. Question.  
In 3012 pound, how many Rials of plate at 7 pence a Ryall. Reduce 3012 pound into pence, makes 722880 pence; which divided by 7, maketh, as in the example.  
Example.   

4. Question.  
If one Dollar be worth 4 shillings 8 pence, how many Dollars is in 108579 pound, 16 shillings starling. Multiply your pounds by 60, makes 6514740; then reduce 16 shillings into groats by 3, makes 48 groats; which added into one total, makes 6514788 which divided by 14, maketh, as in the example.  
Example.  
In 465342 Dollars of 14 groats a piece, how much starling money: multiply your Dollars by 14, makes 6514788 groats; which divide by 60, makes 108579 pound, 16 shillings.  
Example.   

5. Questions.  
If I receive 8060 French Crowns at 6 shillings a piece in France, how much Starling must I pay for them at 6 shillings, 1 penny a piece: multiply 8060 by 73 pence, the number of pence in one French crown, makes 588380 pence: which divided by 240 pence, makes 2451 pound, 11 shillings, 8 pence.  
Example.   

6. Question.  
If 564 yards of cloth cost 124 pound, 12 shillings, how may I sell a yard to gain 22 pound, 7 shillings, by the whole Sum. Answer, add 22 pound, 7 shillings, to 124 pound, 12 shillings, makes 146 pound 19 shillings: which reduce into pence, makes 35268 pence: which divided by 564, makes 5 s. 2 d. ½ 6/47 of a farthing for the price to sell one yard, for to gain 22 pound 7 shillings by the bargain.  
Example.   

7 Question.  
If 156 else of cloth cost 124 pound, what will one ell cost. Reduce 124 pound into shillings, makes 2480 shillings; which divide by 156, makes 15 shillings, 4 pence 26/156 q.  
Example.   

8. Question.  
If I sell 342 yards of Velvet for 241 pound, 17 shillings, how do I sell one yard: reduce your 241 pound, 17 shillings, into shillings, makes 4837 shillings; which divided by 342 yards, makes 14 shillings, 1 penny, 43/●57 of a penny.  
Example.   

9 Question.  
A certain Nobleman sent his servant to the Tower of London, with the King's Majesties Warrant to the Mintmaster for 3408 pound, 15 shillings, willing him to bring it in pieces of 12 d. of 9 d. of 6 d. of 3 d. of 2 d. of ● d of 1 oh. commanding him to bring him of each sort a like quantity, or number of pieces; the question is to know, how many of each sort he shall bring unto his master, to make the said sum of 3408 li. 15 s. reduce your money into half pence, and also your several pieces of Coin into half pence, and divide the greater by the lesser, as in the example.  
Example.     

What Progression arithmetical is, and the Rule.  
PRogression Arithmetical is nothing else but a brief summing, colecting, or gathering together of diverse numbers, increasing by equal proportion, into one total sum. As for example: 1. 2. 3. 4. 5. 6. 7. 8. 9 10. etc. or also, 3. 4. 5. 6. 7. 8. etc. or, 2. 4 6. 8. 10, 12. etc. or else by 3, as, 5. 8. 11. 14 17. 20. 23. 26. etc. or of all such like kinds of Progrission, which do increase equally by 2. 3. 4. 5, or 6, or any other greater increase, and such kind of Progression is called, Arithmetical.  

2. To find the sum of a Progression.  
Mark first how many several places there be in your Progression, and note that down; then add the first number of the Progression to the last: then multiply half those two numbers by the whole number of the places, or else half the number of the places, by the whole number of the first and last term added into one sum, and both ways will produce the total sum of that Progression.  
Example.  
There is a Progression beginning at 4, and is continued unto 44, increasing by 4. First, set down the numbers of that Progression, beginning at 4, and ending at 44.  
Terms. 4. 8. 12. 16. 20. 24. 28. 32. 36. 40. 44. Places. 1. 2. 3. 4. 5. 6. 7. 8. 9 10. 11.  
Here the first term is 4, and the last is term is 44, which added together, makes 48, the one half, which is 24, multiplied by a 11, the whole number of places makes 264 the total.  
Example.   

First Question.  
A certain man gave to his daughter in marriage the first day of january 1 pound, and the second day 2 pound, the third day 3 pound and so increasing every day 1 pound, until 31 days were expired; the question is, what he should receive in the whole sum. First, 31 days is the number of places, and 31 li. is the last payment: add the first term 1 to the last term 31, makes 32; which multiplied by 15 one half, which is half 31; or take 31 and half 32, and the product willbe the total Sum of his wife's portion.  
Example.   

How to find the latter term of a Progression.  
If you would know the latter term of a Progression of 100 terms, increasing by 3, and beginning at 10; take one term from 100 terms, & there will remain 99; which multiply by 3, the excess or difference of the increase, makes 297; to the which if you add the first term 10, makes 307 for the 100 term of that Progression.  
Example.  
Or otherwise take the Excess 3 from the first term 10, and there will rest 7, which note a part, then multiply, the number of places 100 by the excess 3, makes 300; to which add the 7, makes 307, as before.  
Example.   

Second Question.  
A certain Merchant bought 78 pieces of Exetor Carsies, to pay 2 shillings for the first piece 4 shillings, for the second 6 shilling, for the third, 8 s. & so forth increasing his price unto 78 pieces, 2 shillings in every piece; the question is, what the Clothier had for his Carseys.  
First, find the latter term, taking one from 78, makes 77; which multiply by 2, makes 154; to which add the first term 2, makes 156 for the 78, or last term: then add 2, the first term, to 156, the last, makes 158; which multiply by 39, half of the number of places, makes 6162 shillings for the sum of money, the Clothier shall receive for his 78 Carseys.  
Example.   

To find the number of terms.  
There is a Progression, whose first term is 2, the last term, 156; and the excess was 2, I would find the number of terms.  
Subtract the first term from the last, and divide the remainder by the excess, the quotient is the number of terms, wanting but one. Example: 2, the first term from 156, the last leaves 154; which divided by 2, makes 77; to which add 1, makes 78, the number of terms.  
  

How to find the Excess, or difference.  
Subtract the first term from the last, and divide the remainder by one less, than the number of the Terms, and the Quotient will be the Excess or difference.  
Example.  
Subtract 10, the first term, from 307 the last term, there will remain 297; which divide by 99, one less than the number of terms, which are 100, makes 3 the excess.  
  

To find any middle term.  
Subtract a unite from the number of the term you would know, and multiply the remainder by the difference, and to that product add the first term, and the total is the term you do seek.  
Example.  
To find the 30 term in the last example of 100 terms, subtract 1, rests 29; which multiply by 3, the Excess makes 87; to which add the first term 10, makes 97 for the 30 term of that Progression.  
Example.   

How to find what number shall begin and finish a Progression.  
To the number of terms add one, which multiply by half the number of terms, and by the product divide the sum of the progression, and the quotient will be the first term, and excess of that progression.  
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 terms, add 1, makes 17; which multiply by 8, half the number of terms, makes 136 for Divisor; by which divide 7072, and the quotient is 52 shillings for the first payment and excess, and by the same the other payments are found.  
Example.  
Example.    

What Geometrical Progression is, and the Rule.  
THe terms being 3, to find a third proportional between two extremes: divide the Root of the greater by the lesser extreme, and the quotient is your desire.  
Example.  
First, 8 and 12 are two extremes given, it is required to find a proportional number between those two numbers given; square 12, it is 144, which divide by 8, makes 18 for the third proportional number. Secondly, multiply your extremes together, and extract the square root for the mean proportional, between two numbers given; as let 4 and 9 be two extremes, 4 by 9 is 36, the square root is 6, for a mean proportional number to those two numbers given.  
Between 2 and 54, let 2 mean proportionals be desired by the square of 2, which is 4; multiply 54, it makes 216, the Cube root whereof is 6 for the least of the two Means: Again, by 2 multiply 2916, which is the square of 54, makes 5832, of which the Cube root is 18, for the greater mean proportional sought. But if the terms exceed 4, having all one excess, it is then called Geometrical Progression.  

To find any middle Term, or any other Term in a Geometrical Progression.  
Increase your Progression by the excess, and the square of the term when you cease, or the number multiplied in itself squarely, is the double of your Term save 1, if the progression begin with an unite.  
But it the first term be not an unite, than the square of any term is the double number of the said term: as if you should square the sixth term, than the product would be the twelfth term: & so of any other term.  
Example.  
A Gentleman coming into a Market to buy a Horse, was asked 30 pounds for him. Nay (said the Gentleman) his price is over great. Then said the owner (having more craft and subtlety than the Gentleman, as commonly the old Proverb is true amongst Horse-coursers); My Gelding ha●h four shoes upon his four feet (quoth he), you shall give me for the first nail (there being 28 in all) one farthing▪ and for the second nail 2 farthings, and for the third 4 farthings, and for the fourth 8 farthings; and so double at every nail, you shall have him. Whereat the Gentleman smiled, saying; I will have him. And so they bargained, and then went to an Arithmatician to cast up the Sum: but how this Gentleman was able to pay for this Horse, shall appear by the Work, which I have put for an example, because I would not have any man ignorant in Arithmetic, to make any such blind matches without advice, as I know many have done to their cost.  
1. Example.  
Now according to the rule, I increase this progression unto the seventh Term thus, 1. 2. 4. 8. 16. 32. 64; which 64 I multiply by itself squarely, the product is 4096, which by the rule is the thirteenth Term, which is one Term less than the double of 7: then multiply that 4096 by 2, it makes 8192, which is the fourteenth Term. Then multiply 8192 by 8192, and the product is 67108864, which is the twenty seventh Term: the which being doubled, makes the last Term 134217728.  
Example.   

The Extremes and Excess of a Progression given to find the sum.  
Multiply the last term by the Excess, and from the Product abate the first term, and divide the remainder by a unite less than the excess, and the Quotient is the sum of the Progression desired.  
Example.  
In the last examples, the excess was 2, by which I multiply 134217728, and the Product is 268435456, from which abate 1, the first term, and the remainder is 268435455, which should be divided by one unite less than the excess, which is 2, and ● less is but 1; therefore seeing 1 doth neither multiply, nor divide, I conclude the price of the horse to be 268435455 farthings; which I divide 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.  
Example.  
I have inserted in the next page the trial of this work, by increasing the Terms from 1 to 28, and also the Addition of the total, which shows the answer to be true.  
Example. 1 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 9 512 10 1024 11 2048 12 4096 13 8192 14 16384 15 32768 16 65536 17 131072 18 262144 19 524288 20 1048576 21 2097152 22 4194304 23 8388608 24 16777216 25 33554432 26 67108864 27 134217728 28 268435455 The Totall.  
Otherwise, subtract the first term from the last, and divide the remainder by one ace less than the Excess, and to the quotient add the last Term, and the total is the sum.  
Example.  
To 12 men a sum of money is given to the eldest ½, to the second ½, the remainder, and so to every one of the rest, and the last portion was found to be 4 pound, and the last half being also 4 pound, was given to a friend to see the money to be equally distributed; what was each man's portion, and the sum given?  
Let 4 be the last portion, and twelfth Term, and so double until you come to the first term, and you shall find every man's portion Then by this second rule, you shall find the total to be 16380 pound; to which add the Ezecutors part 4 pound, makes 16384 pound.  
3. Example.  
A Gentleman bought a Manor, with all the appurtenances for a sum of money unknown; but he was to pay at several days of payment by continual triplation, of every payment, from the first payment which was 4 pound, and the last 8748 li. the question is, what he paid for the said Manor and lands.  
Example.  
Subtract the first term 4, from the last term 8748, there will remain 8744; which divide by the Excess, one less, viz by 2, and the quotient will be 4372; to the which add the latter term 8748, and the total is 13120 pound, for the sum which the said Manor and lands cost.  
   

Fractions.  
YOu shall understand, that in the work of Fractions hereafter; in the next page following I have used another form of working, then heretofore hath been used: as when you will set forth any fraction, as ¾ thus heretofore used, set them out thus, 3 ∶ 4; or ⅞, place thus 7 ∶ 8 with a double prick between them: and so of any other, as 25/20 of a pound, thus, 15 ∶ 20 of one pound: or fractions of fractions, thus, ⅔ of ¾ of ⅚ of a pound, set them thus, 2 ∶ 3 of 3 ∶ 4 of 5 ∶ 6 of a pound: and so of all other fractions, as shall appear afterward in the operations following; and so being placed, they are more apt and fitter for all the several operations of Arithmetic, then being placed after the ordinary form of working. And thus much I thought good to express for the better understanding of the Rules hereafter following, in all fractional operations. And now I will proceed unto the several rules of Fractions, with their Examples.  


How to reduce Fractions of Fractions. First Rule.  
Multiple their tops one into another for a new numerator, and likewise their Bases for a new denominator, and the work is ended.  
Example.  
If you would reduce 3 ∶ 4 of 2 ∶ 3 of 7 ∶ 8 of one li starling; multiply 3 by 2, maketh 6, and then 6 by 7 makes 42 for the new numerator to your fraction: then 4 by 3 makes 12, and 12 by 8 makes 96 for new denominator, and the fraction is 42 ∶ 96 of a pound.  
3 ∶ 4 of 2 ∶ 3 of 7 ∶ 8 of 1 li. makes 42 ∶ 96 of a li.  
2. Example.  
Again, 3 ∶ 5 of 9 ∶ 8 of 7 ∶ 10 of 11 ∶ 12 of a pound, makes 2079 ∶ 4800.  
 
3. Example.  
What is 1 ∶ 2 of 2 ∶ 3 of 3 ∶ 4 of 4 ∶ 5 of 5 ∶ 6 of 6 ∶ 7 of one pound. Answer: cross all the by as equal terms, and set the unequal terms 1 ∶ 7 of a pound for the total sum: but after the other form of work, it would have brought out 720 ∶ 5040 of a pound, which by abbreviation makes 1 ∶ 7  
The Proof.  
72 ∶ 504, 36 ∶ 252, 18 ∶ 126, 9 ∶ 63, 3 ∶ 21, 1 ∶ 7.   

2. Rule: How to reduce Fractions of Integers.  
Multiply all the Denominators of your several fractions for the new, or common Denominator to all your given fractions.  
Then to find new numerators to each of your given fractions; multiply each fractions top into the basses, of each several fraction, excepting his own base, for the new numerators, as in this Example.  
Example.  
If you would reduce 3 ∶ 4 and 5 ∶ 6 and 7 ∶ 8 of a pound into one Denomination: multiply all the basses together, saying; 4 by 6 makes 24, and 24 by 8 makes 192 for the common Denominator to all the given fractions.  
Then multiply 3, the numerator of the first fraction, by 6, the denominator of the second fraction, makes 18, & 18 by 8 makes 144 for his numerator.  
Secondly, multiply 5, the numerator of the second fraction by 4 and 8, the Denominators of the other two fractions, makes 160 for the new Numerator of the second fraction.  
Thirdly, multiply 7, the numerator of the third fraction, by 6 and 4, makes 168.  
Example.  
2. Example.  
If you would reduce 2 ∶ 3, and 3 ∶ 5, and 8 ∶ 9 of a pound.  
 
3. Example.  
If you would reduce 7 ∶ 8, 1 ∶ 3, 2 ∶ 3, 4 ∶ 5, and 6 ∶ 7 of a pound▪  
   

How to prove a Fraction by the known parts of Coin.  
In the first example of fractions of fractions, I find that 3 ∶ 4 of 2 ∶ 3 of 7 ∶ 8 of a pound Sterling to be 42 ∶ 96 parts of a pound: for trial 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 divide by 96, the denominator, makes 105 pence, the proof, as followeth.  
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 total by Addition was 472 ∶ 192 of a pound. Now for the proof of the work, multiply the numerator 472 by 240, makes 113280; which divided by 192 the denominator, makes 590 pence; which divided by 12. pence, makes 2 pound, 9 shillings, 2 pence. The proof of this trial 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 total added together, is 2 pound, 9 shillings, 2 pence, which proves the work to be true.  
Example.  

3. Rule. Addition in Fractions.  
If your fractions be of one denomination, then add all your numerators together, subscribing the common denominator under the line.  
Example.  
The second Rule.  
If your Fractions be not of one denomition, then reduce them by the second rule of Reduction to one denomination, and then add them into one sum subscribing under the common denominator.  
Example.  
If you would add 40 ∶ 80, 30 ∶ 200, and 50 ∶ 90; cut off a cipher from each numerator and denominator, and the fractions remaining will be of the same with the given fractions, and then work as before.  
Example.    

The proof of Addition by parts of Coin.  
In the second Example, 2 ∶ 3, 3 ∶ 4, and 4 ∶ 5 of a pound, are found to be 133 ∶ 60; therefore divide 133 by 60, makes 2 pound and 13 ∶ 60 or 13 groats remaining, which is 2 pound, 4 shillings, 4 pence.  
The proof: add 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 total, makes 2 pound, 4 shillings, 4 pence, as before.  
Example.  

Rule 4. Subtraction in fractions.  
As before in Addition, so also in Subtraction, reduce your fractions to one common denomination, then subtract the smaller numerator from the greater, and subscribe the common denominator under the remainder.  
1. Example.  
If you will subtract 3 ∶ 4 from 7 ∶ 4, there will remain 4 ∶ 4, or one integer.  
Also, 7 ∶ 12 from 13 ∶ 12, leaves 6 ∶ 12, or 1 ∶ 2 remaining. But if you will subtract 2 ∶ 3 from 7 ∶ 8, then reduce them to one denomination, by the second rule of Reduction, and work, as in this example.  
Example.  
2. Example.  
Again, 3 ∶ 8 from 15 ∶ 16, leaves 72 ∶ 128, remains.  
   

The proof of Subtraction by the parts of Coin.  
In the example before, where I take 2 ∶ 3 from 7 ∶ 8, the remainder was 5 ∶ 24 of a pound, which is 5 times 10 pence, or 4 shillings 2 pence. Also for proof, take 13 shillings 4 pence, which is 2 ∶ 3 of a pound, from 7 ∶ 8, which is 17 shillings 6 pence, there will remain 4 shillings 2 pence, as before.  

Rule 5. Multiplication in Fractions.  
Multiply Numerator by Numerator, and Denominator by Denominator, to make the new Numerator, and new Denominator, and the work is ended.  
1. Example.  
If you will multiply 2 ∶ 3, by 3 ∶ 4, the product of that multiplication will be 6 ∶ 12, or 1 ∶ 2  
   

The proof of Multiplication by the parts of Coin.  
In the first example, 2 ∶ 3 is multiplied by 3 ∶ 4, and the product makes 6 ∶ 12 of a pound or 10 shillings: for proof 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 divided by 240 pence, the pence in one pound will yield in the quotient 120 pence, or to shillings.  
Example.  

6. Rule. Division in Fractions.  
Multiply the numerator of the dividend by the denominator of the divisor for a new numerator; and secondly the denominator of the dividend by the numerator of the divisor, for new denominator, and the division is ended: or otherwise place your dividend fi●st above, and the divisor underneath, after my manner, and multiply cross, and place them ●● in these examples.  
If you will divide 6 ∶ 12 by 2 ∶ 3, which was the product of 2 ∶ 3 by 3 ∶ 4 in the last example, than it will bring out 18 ∶ 24, or 3 ∶ 4, the other number, which proves the work good.  
1. Example.  
If the denominators of the fractions be both alike, then divide their numerators one by another; as 27 ∶ 32 divided by 3 ∶ 32, makes the quotient to be 9 ∶ 32, or Integers.  
Example.  
If the numerators be alike, then set the denominator of the divisor above, the denonominator of the dividend, as 3 ∶ 4 by 3 ∶ 8, makes the quotient 8 ∶ 4, or two Integers, and chose 3 ∶ 8 by 3 ∶ 4, makes the quotient 4 ∶ 8, or 1 ∶ 2  
Example.    

The proof of Division by the parts of Coin.  
In the second of the first example, where I divide 2 ∶ 3 by 4 ∶ 5, the quotient is 10 ∶ 12, which in coin is 16 shillings 8 pence: for proof, I do multiply 2 ∶ 3 of a pound, which is 160 pence, by 240, makes 38400; which divide by 4 ∶ 5, or 192 pence, makes 200 pence, which is 16 shillings 8 pence, the proof.  
Example.  

7. Rule. How to work whole numbers with Fractions.  
If you would add, subtract, multiply, or divide whole numbers with fractions, set the whole numbers fraction wise, and put 1 after for denominator, and then work as in the Rules before, as if they were all fractions, and no whole numbers.  
Example.  
If you will add 33 ∶ 1 with 13 ∶ 4, multiply the numerator 33 of your whole number, by the Denominator of your fraction 4, makes 132 ∶ 4, which add unto 13 ∶ 4, makes the total 145 ∶ 4  
 
2. Example.  
If you will subtract 13 ∶ 4 from 33 ∶ 1, reduce them, and subtract 13 from 132, rest 119 ∶ 4  
 
Example.  
If you will multiply 33 ∶ 1 by 13 ∶ 4; multiply the numerators, 33 by 13, makes 429; to the which subscribe the Denominator 4, makes 429 ∶ 4  
 
4. Example.  
If you will divide 33 ∶ 1 by 13 ∶ 4, multiply cross 33 by 4, makes 132, to be set above; then 13 by 1 makes 13 for denominator.  
  

8. Rule. How to work whole numbers and fractions with fractions.  
Reduce your whole numbers into fractions in multiplying your whole number by the denominator of your fraction; and unto that product add the numerator of your fraction, and subscribe the old denominator.  
1. Example.  
If you will multiply 28 3 ∶ 4 by 3 ∶ 5 reduce 28 3 ∶ 4 into fourth's in multiplying by the fractions denominator 4, saying, 28 by 4 makes 112, to the which add the numerator of your fraction 3, makes 115; which multiplied by 3 ∶ 5, makes 345 ∶ 20  
 
If you will divide 28 3 ∶ 4 by 3 ∶ 5; reduce them as before, and then multiply them cross, makes 115 ∶ 4 by 3 ∶ 5, is 575 ∶ 12  
Example.   

9 Rule. How to abbreviate a fraction.  
Take one half of the numerator, and 1 ∶ 2 of the denominator, as oft as you may until the lowest numbers in value of your fractions comes to be primes together, which are such numbers, as cannot be abbreviated no lower.  
Example.  
In the first example of fractions of fractions, the fraction was 72 ∶ 504, which was abbreviated unto 1 ∶ 7 of a pound: first, take half the numerator 72, which is 36, then half the Denominator 504, which is 252; then 1 ∶ 2 of 36, is 18; and 1 ∶ 2 of 252 is 126. Again, 1 ∶ 2 of 18 is 9, and 1 ∶ 2 of 126 is 63; then I see I cannot take 1 ∶ 2 of the remainder, wherefore I see I may abbreviate them by 3 still, saying, the third part of 9 is 3, and 1 ∶ 3 of 63 is 21: lastly, 1 ∶ 3 of 3 is 1, and 1 ∶ 3 of 21 is 7, which place thus, 1 ∶ 7▪ so that I find by abbreviation that 72 ∶ 504 of a pound, is one seventh part of a pound.  
Example.  
72 ∶ 504 36 ∶ 252 18 ∶ 126 9 ∶ 63 3 ∶ 21 1 ∶ 7  
If you cannot take half the numbers, then mark whether they will abbreviate by 3 4, or 5, or any other number under 9; as for example, I would abbreviate 92 ∶ 144, I see I may abbreviate both by 4; then taking 92, divide by 4, makes 23, and 144 by 4 makes 36, total 23 ∶ 36 etc.  
If you will abbreviate, 375 ∶ 625 of a pound, you may easily see, they will be both abbreviated by 5: wherefore divide the numerator and denominator both by 5, as o●t as you can, until they become primes together, and you shall find the value of that fraction to be 3 ∶ 5 of one pound, or 12 shillings.  
Example.   

10. Rule. How to find the value of any Fraction.  
Multiply the numerator of your fraction by the parts contained in the whole, and divide that product by the old Basse, and the quotient will be the value of that fraction in the known parts of Coin.  
Example.  
If you would know what 24 ∶ 32 parts of a pound is in Coin: multiply your numerator 24 by 240, the pence in one pound, makes 5760; which divided by 32, the denominator, makes 180 pence, or 15 shillings, the true value of that fraction.  
Example.  
What is 343 ∶ 522 parts of a yard, multiply 343 by 16, the number of nails in one yard makes 5448; which divide by 522, makes 10 nails, and 268 ∶ 522 parts of a nail.  
Example.   

11. Rule. How to change the Surname of a Fraction.  
Multiply the numerator of your fraction by the parts, or new Surname of that you would change your fraction into, and divide by your denominator, and the quotient will be your desire.  
1. Example.  
I have 324 ∶ 1620 parts of a year, which I would convert into days; I multiply 324 by 365, the number of days in one year, makes 118260; which divided by 1620, makes 73 days, the value of that fraction.  
Example.  
I would change 256 ∶ 5292 parts of a pound into pence; multiply the numerator 756 by 240 pence, makes 181440, which divide by the denominator 5292, and the quotient is 34 pence 1512 ∶ 5292  
Example.   

12. Rule. Questions of Fractions.  
What number is that to the which if you do add 3 ∶ 4, the total will be 5 ∶ 6 of a pound. Answer; reduce them to one denomination, and they are for 3 ∶ 4 of a pound 18 ∶ 24, and the 5 ∶ 6 are 20 ∶ 24, from which subtract 18, rest 2 ∶ 24 of a pound, or 20 pence: the proof, take 3▪ 4 of a pound, which is 15 shillings, and add 20 pence to it, and the total is 16 shillings, 8 pence; which is 5 ∶ 6 of a pound.  
Example.  
2. Example.  
What number is that, from which if you do subtract 8 ∶ 12, the remainder will be 6 ∶ 10. Answer, reduce them, and add them both into one total, makes 152 ∶ 120 of a pound for the number you do seek.  
The proof in coin; 152 ∶ 120 of a pound is 304 pence, and 8 ∶ 12 of a pound is 160 pence, which taken from 304, leaves 144 pence remaining, which is 6 ∶ 10 of a pound, or 12 shillings, as appeareth by the work.  
 
What number is that, which being multiplied by 3 ∶ 5, the product will be 9 ∶ 20. Answer divide 9 ∶ 20 by 3 ∶ 5, and the quotient is 45 ∶ 60, or 3 ∶ 4. For the proof, multiply 108 pence, which is 9 ∶ 10 of a pound, by 240, the product is 25920; which divide by 144, or 3 ∶ 5, which is 12 shillings, makes 180 pence, or 3 ∶ 4 of a pound.  
3. Example.  
 
Example.  
What number is that, which being divided by 7 ∶ 8, the quotient will be 4▪ 5. Answer, multiply 7 ∶ 8 by 4 ∶ 5, the product is 28 ∶ 40, or 7 ∶ 10, which makes 14 shillings.  
The proof in Coin; 7 ∶ 8, which is 210 pence, by 4 ∶ 5, which is 192 pence, and the product is 40320; which divide by 240, makes 168 pence, or 14 shillings: behold the example following.  
Example.  
    

Rules of Practice.  

Rules of Practice by the first Table.  
TO work by the Aliquot parts of a pound, search in the first Table for your given price, and by that number found, divide your number given, and the quotient is your answer in pounds, and the remainder is the fraction of one pound.  
But if the given price be not found exactly at the first entrance, then find 2, or more numbers, to make the given price, and then work 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 divide 7859 by 6, makes 1309 pound, 5 ∶ 6 of one pound, or 16 shillings 8 pence.  
 
The first Table. The second Table. The Aliquot parts of a pound. Shillings. s. d. part. s. d. par. s. par. s. part. 1 240 1. 4 15 1 ½ 11 5. ½ 2 120 1. 8 12 2 ●/10 12 6. 3 80 1. 0 10 3 1. ½ 13 6. 1/ 2 4 60 2. 6 8 4 2. 14 7. 5 48 3. 4 6 5 2. ½ 15 7. ½ 6 40 4. 0 5 6 3. 16 8. 8 30 5. 0 4 7 3. ½ 17 8. ½ 10 24 6. 8 3 8 4. 18 9 1. 0 20 10. 0 2 9 4. ½ 19 9 ½ 1. 3 16 20. 0 1 10 5. 20 10. Divisors. Multiplyers.  
At 16 pence an ell, what will 8976 else cost, I find for 16 pence my divisor, to be 15, and so dividing 8976 by 15, the quotient is 598 pound, 6: 15, or 2: 5, which is 8 shillings.  
Example.  
Add a cipher to your number given, and the last figure of your quotient will be primes, every one in value 2 shillings, and the remainder is the fraction of a prime, always less than 2 shillings. In the first example, the remainder was 5: 6 of one pound, but if you add a cipher, the quotient will be 1309 pound, 8 primes or 16 shillings, and the remainder is 2: 6 of one prime, or 1: 3, which is 8 pence.  
 
At 2 shillings 6 pence a pound pepper, what will 2436 pound cost: find 2 shillings, 6 pence 1: 8 of a pound, wherefore add a cipher, and divide, by 8, makes 304 pound; 10 shillings.  
 
At 8 pence a pound Ginger, will 77856 pound cost; divide by 30, adding a cipher, makes 2595 pound, 2 primes, or 4 shillings.  
 
At 17 pence a pound Sugar; what shall 23459 pounds cost: for 12 pence, divide by 20, makes 1172 pound, 9 primes, 1: 2, or 19 shillings: then for the rest of your given price, which is 5 pence; take 48, and divide, and the quotient is 488 pound, 7 primes, which added together into one sum, makes the total 1661. pound, 13 shillings, 7 pence.  
Example.  
At 6 shillings 8 pence a pound Cloves, what will 3769 pound wait cost: divide by 3, makes 1256 pound, 3 primes, 1: 3, or 6 shillings 8 pence.  
 
At 22 pence an elle of Holland, what 3768 else cost: for 20 pence divide by 12, makes 314 pound, and for 2 pence by 120, makes 31 pound, 4 primes, or 8 shillings; the total is 345 pound, 8 shillings.  
 
If one elle of Holland cost 20 pence, how many else shall I buy for 345 pound: multiply 345 by the price, which is ●: 12, or by 12, makes 4140 else, the sum desired.  
 
If one elle of Ozenbrigs' cost 8 pence, what sum of else will 78 pound buy me: multiply by 30, makes 2340 else.  
 
At 15 pence an elle of Canvas, how many else will 100 pound buy: multiply by 16, makes 1600 else.  
If one elle of parchment lace cost 1 penny, how many else shall I have for 73 pound: multiply by 240, makes 17520 else.  
Example.  
If one Acre of land be 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 given be any number of shillings, search in the second Table for the price given, and by the number there found: multiply your number of yards, else, pounds or pieces, and cut off the last figure with a dash of the pen for primes, every one in value 2 shillings, and the product is the sum of pounds and shillings that your given number will cost.  
Example.  
At 2 shillings an elle of Holland, what will 956 else cost: in the second table ● find the tenth of the number given, so that if you take the tenth of 956, it is 95 pound, 12 shillings, only by cutting off the last figure by a dash of the pen.  
956 else at 2s. an ell, makes 956, or 12s.  
At 7 shillings an ell of Cambric, what will 789 else cost: multiply by 3 1 ∶ 2, or take half of the given number, and multiply the whole number given, by 3, makes in one sum, cutting off the prime line, 276 pound, 3 shillings.  
Example.  
At 25 shillings a piece Raisins, what will 356 pieces cost: take always half the number of shillings of your given price for your multiplier, and work as before, and the product is 456 poved, 0 prime.  
Example.  
Also 75032 pieces at 26 shilling a piece.  
 
If one barrel of Soap cost 47 shillings, what will 3584 barrels cost: multiply by 23 1 ∶ 2, makes 8422 pound, 8 shillings.  
Example.  
At 3 pound 6 shillings a Barrel, what will 124 cost.  
 
If one Acre of land cost 6 pound 8 shillings, what will 758 Acres cost: multiply by 64 shillings, which is half the price, the product is 4851 pound, 4 shillings, or two primes.  
 
How to prove 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: divide your number of pounds and shillings by one half of the number of shillings in the price given, adding a cipher to your number of pounds, and the quotient is the number of Acres of land the said sum will buy at that rate.  
Example.  
The given sum is 4851 pound, 2 primes, or 4 shillings; which divided by half the given price, which is 64 shillings, brings into the quotient 758 Acres: and so of any other sum.  
 
A Merchant bought Cambrics, cost him 855 pound, 18 shillings; the question is, how many pieces he had, paying for every piece 27 shillings. Answer, ad●e a cipher to your number given, which 855 pound, 9 primes, makes 85590; which divide by half the price given, which is 13 1 ∶ 2; or divide by 135 the quotient will be 634 pieces: now the reason wherefore a cipher is added to the number given, having 9 primes in it is, because I divide by 13 1 ∶ 2, which hath one fraction; and this rule is general.  
Example.  
What cost 634 pieces, at 27 shillings.  
  

How to prove one question in the Rules of Practice, by working of another.  
If you will prove any question in the rules of Practice, by a second example mark the compliment, or want of your given price from one pound, and work the same number at that price which doth want, and the total of those two sums added together, makes the just number of pounds of the given sum.  
Example.  
At 16 shillings a piece of Fustian, what will 320 pieces cost. Answer; multiply by 8, makes 256 pound, 0 prime.  
Again, 16 shillings your given price wanted 4 shillings of one pound, wherefore work 320 at 4 shillings, which is multiplied by 2 primes, makes 64 pound, 0 prime, the total is 320 pound, which proves the former work.  
 
Example.  
At 13 shillings a piece of Lawn, 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, total is 752 pound.  
Example.   

Rules of Practice by the third Table, the most excellent of all the other.  
The third Table. The Aliquot parts of 24. d. part. d. part. 1 24. 13 2. 24 2 12. 14 3. 4 3 8. 15 2. 8 4 6. 16 3. 3 5 12. 8 17 3.4. 8 6 4. 18 2. 4 7 8. 6 19 2.8. 6 8 3. 20 2. 3 9 4. 8 21 2.4. 8 10 4. 6 21 2.4. 6 11 3. 8 23 2.3. 8 12 2. 24 ●/10  
Divisors. The parts of a Shilling. d. par. d. part. ¼ 48 7 2. 12 ½ 24 8 1. ½ ⅓ 18 9 2. 4 ¾ 16 10 2. 3   11 2.3. 6 d.  12 Idem. 1 12  Divisors. 2 6   3 4   4 3   6 2        
Divide the number of else, yards, pounds, or pieces given by the number, or numbers found in the third Table, always cutting the last figure for primes; if that any remain after Division, it is always less than one prime, or 2 shillings.  
Example.  
At 3 pence a pound Liquorice, what will 123728 pound cost. Answer; for 3 pence in the third Table, I find my Divisor to be 8, by which I divide my given number, makes 1546 pound, 6 primes, or 12 shillings.  
 
At 9 pence the pound Ginger, what will 8768 pound cost: for 6 pence divide by 4, makes 219 li. 2 primes; then for 3 pence the residue of the price, divide by 8, makes 109 pound, 6 primes, total is 328 pound, 16 shillings. Or otherwise, divide by 4 for 6 d. and then take half that product for 3 pence, and add them into one sum, as before.  
Example.  
At 11 pence the yard Canvas, what will 2356 cost: for 8 pence divide by 3, makes 78 pound, 5 primes, 1 ∶ 3 or 8 pence; and for 3 pence, divide by 8, makes 29 pound, 4 primes, 1 ∶ 2, or 12 pence; the total is 107 pound, 19 shillings, 8 pence.  
 
A second example, the proof of the last.  
At 13 pence a pound fine Sugar, what will 2356 pound cost: for 12 pence divide by 2, makes 117 pound, 8 primes, or 16 shillings; then for 1 penny, divide by 24, makes 9 pound, 8 primes, 4 pence, the total is 127 pound, 12 shillings, 4 pence; which added to the former sum in the last example, makes 235 pound, 12 shillings; and so much will 2356 pound cost at 2 shillings a pound, because the two given prices make one prime, or 2 shillings.  
Example.  
At 16 pence a pound Sugar, what will 78432 pound cost: work for 8 pence, and double the sum, makes 5228 pound, 8 primes, or 16 shillings.  
At 8 pence a pound Almonds, what will 78432 pound cost: divide by 3, makes 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 proves both examples true.  
Example.  
At 18 pence a pound Comfits, what will 78432 pound cost: for 12 pence take half the given number, and for 6 pence take half of that sum, which added into one total, makes 5882 pound, 4 primes.  
 
At 6 pence a pound small Ginger, what will 78432 li. cost: divide 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 serve also, if the price be above 2 shillings, or one prime: as if you shall say at 3 shillings, 6 pence an ell, what 782 else: here I see the given price is compounded of 7 times 6 pence; wherefore I work first for 6 pence in dividing by 4, makes 19 pound, 11 pence; which multiply by 7, makes 136 pound, 17 shillings for the price of 782 else at 3 shillings, 6 pence the elle.  
At 6 pence an elle, what 782 else: find for 6 pence, 19 pound 11 shillings, which added to the former sum in the last example, makes 156 pound, 8 shillings, which is the sum that 782 else will cost at 4 shillings the elle.  
Example.  
At 4 shillings 8 pence the elle Holland, what will 2148 else cost. I find 4 shillings 8 pence to be 14 Groats, so dividing by 6 for one Groat, makes 35 pound, 8 primes; which multiply by 14, makes 501 pound, 4 shillings.  
 
At 15 pence a gross of points, what will 2256 gross cost. 15 Pence is 5 times 3 d. and so I divide 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 be any Aliquot part of a shilling: enter the fourth Table, and there you shall find a Divisor, by the which if you divide your number given, the Quotient will be shillings, and the remainder parts of one shilling. Then to convert your shillings into pounds, take one half 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 wait cost. Search in the fourth Table, and you shall find 16 for your Divisor: by the which if you divide 756, the Quotient is 47 shillings, 1 ∶ 4, or 3 pence.  
 
At 1 half penny a pound Coporas, what will 8756 pound cost. Divide by 24 makes 364 shillings, of which the one half cutting of the 4 shillings, is 18 pound 4 shillings; and 20 half pence remaining, total is 18 pound, 4 shillings, 10 pence.  
 
At 4 pence a pound Liquorice, what will 789 pound cost. Divide by 3, maketh 13 pound, 3 shillings.  
Again, at 6 pence a pound, what will 8579 pound cost. Divide by 2, makes 214 pounds, 9 shillings, 6 pence.  
1. Example.  
2. Example.   

General Rules of Practice without Tables.  
Multiply your number given by the sum of pence, that one yard, piece, pound, or elle doth cost, and the product will be the sum of pence, the whole number given will cost; and then divide that sum of pence by 4, ma▪ es the Quotient Groats, and if any reremaine, they are pence, always less than 4 pence, or one Groat: and secondly again, divide that Quotient will be pounds and primes, every prime in value 2 shillings, and the remains is Groats, always less than 6 Groats, or one prime, which is value 2 shillings.  
At 17 pence an ell Canvas, what will 3245 else cost: Multiply by 17, makes 55165 pence, which divided by 4, makes 13791 Groats, and there will remain one penny. Secondly, divide that Quotient again by 6, makes 229 pound, 8 primes, and the remainder is 3 Groats, or one shilling; and so the total is 229 pound, 17 shillings, 1 penny.  
Example.  
At 3 shillings, 7 pence a yard Holland, what will 752 else cost: multiply 752 by 43 pence, the price of one ell, makes 32336; which divided, as is before taught, makes 134 pound, 14 shillings, 8 pence.  
Example.  
At 7 shillings, 11 pence the elle Cambrics, what will 85● else cost. Reduce 7 shillings, 11 pence into pence, makes 95 pence; by which multiply 856, makes 8●320; which divided as before, makes 338 pound, 16 shillings, 8 pence.  
Example.  
At 2 shillings, 11 pence an elle Holland, what will 7856 else cost: multiply, and divide as is before taught, makes 1145 li. 13 shillings, 4 pence.  
Example.  
At 17 shillings, 7 pence a yard Broad Cloth, what will 7856 yard's cost: multiply by 211▪ the price of one yard, and divide as before, makes 6906 pound, 7 primes.  
Example.  
If your given price have any farthings in it, then reduce your price into farthings, and multiply your given number by those farthings, and the product will be the number of farthings, which your sum will cost: then divide that product by 16, makes the quotient Groats, and the remainder will be farthings, always less than 16, or one Groat. Secondly, divide that quotient of Groats by 6, makes pounds and primes, as before.  
Examples.  
At 5 shillings, 1 penny, one halfpenny an ounce Plate, what will 356 ounces cost. Reduce 5 shillings, 1 penny, halfpenny into farthings, makes 246 farthings: by which multiply 356, makes 87567 farthings; which divided by 16, makes 5473 Groats, and 8 farthings will remain; which divide again by 6, makes 91 pound, 2 primes, and one Groat will remain, total 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 add 37 farthings, which are in 9 pence, farthing, makes 325 farthings; and then work as before is taught, and you shall find 1199 pound, 2 shillings; 3 pence, halfpenny.  
Example.   

Another way to work Practice.  
Divide your number of yards, else, or pieces by 240, adding a cipher to your number given, and then multiply the Quotient by your price, and the Product is the sum of pounds, and shillings, that the given number will cost.  
At 17 pence the elle Canvas, what will 7848 else cost: add a cipher, and divide 78480 by 240, and the Quotient will be 32 pound, 7 primes; which multiply by 17 pence, the price, makes 555 pound, 9 primes or 18 shillings.  
Example.  
At 3 shillings, 5 pence an ell of Holland, what will 702 else cost: divide 7020 by 240, makes 2 pound, 9 primes, and there will remain 6; which multiply by 41 pence, the price of ●e ell, makes 118 pound, 9 primes, or 18 shillings, and then the 6 else, makes 1 pound, 6 pence, the total is 119 pound, 18 shillings, 6 pence.  
Example.  
At 19 pence the elle of Holland, what will 32544 else cost: divide 325440 by 240, makes 1356; which multiply by 19 pence, the price of one elle, makes 2576 pound, 8 shillings.  
   

The Golden Rule.  
Of single proportion Direct, or the Rule of three, called The Golden Rule.  
IN this Rule of 3 Direct, there is always three terms given, and a fourth required, and it is called the Golden Rule, in regard of the excellency of this Rule above all others. The difficulty of this rule consisteth in the right placing of the three numbers given, set the term next your right hand, whhereupon the question is moved, and a term of the same nature towards the left hand, & the third term in the middle. Then multiply the second number by the third, and divide the product by the first, and the Quotient is the fourth proportional number sought or desired to be found out; whose denomination is ever like unto the middle number.  
1. Example.  
If 90 yards of Cloth cost 23 pound, what cost 346 yards.  
 
If 124 pound gain 37 pound, 12 shillings, what will 758 pound gain.  
 

How to work this last example, and all other, after a more brief and exact manner.  
Divide the third number by the first, and by the Quotient multiply the second, and the product is the answer.  
Example.  
If 356 else cost 137 pound, 12 shillings, 9 pence, what cost 2848 else.  
 
First, dividing 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 divide 72 by 12, is 6 shillings; which added to 96 shillings, makes 102 shillings, or 5 pound, 2 shillings; the total 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 Raisins cost 19 pound, 2 shillings, 2 pence, what will 112 pieces cost at that rate.  
3. Example.  
 
If 356 pieces cost 137 pound, 12 shillings, 9 pence; what will 2848 pieces cost at that rate.  
 
Example.  
Example.   

How to know whether any question given be to be answered by the Rule Direct, or Conversed.  
By these notes following, you shall find, whether any question propounded be to be answered by the Rule of 3 Direct, or conversed; for always 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 always, more or less, so that if it be more than the lesser of your two extreme numbers is the divisor: if less, than the greater of your two extremes is your divisor. If the number whereon the question be depending, be your Divisor, them the answer is, by the converse Rule, and you must multiply your two former numbers for Dividend. If the first number be the Divisor, than the question is answerable by the Direct Rule, and the product of the two latter numbers is your Dividend.  
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 answer, less than 13 Cannons; wherefore by this rule, the greater of the two extremes, 13 is the divisor: wherefore I multiply 358 by 5, and divide 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 months, and when I came to him to require the like kindness he could lend me by 54 pound, the question is, how long he should forbear that 54 pound to make requital, or to equal my time, and kindness.  
If 115 pound require 7 months, what will 54 pound require: here the answer in reason is, that 54 pound must be longer time forborn then 115 pound, and so the answer is more times than 115 pound; so that I find the lesser of my exteames 54, is my Divisor, and the question answerable by the Rule conversed, so that I multiply 115 by 7, makes 805; which divided by 54, makes 14 months, 49 ∶ 54 of a month, or 14 months, 25 days, 23 ∶ 25  
Example.  
4. Example.  
A Captain of a Band of men is besieged in a City, having with him 7200 men, and his victuals will serve the whole Company but 7 months, but there is no hope left to have any fresh victuals until 16 months; the question is, how many men he shall send away to make the victuals serve for 16 months. Answer, less than 7200 men.  
If 7 months require 7200 men, how many will 16 months ask.  
 
When Wheat was sold at 3 shillings, 8 pence the bushel, the penny loaf of bread weighed 6 ounces, what shall the same loaf of bread weigh, when Wheat is sold for 2 shillings the bushel: I answer more than a 11 ounces.  
If 44 pence give 6 ounces, what will 24 pence give.  
 
If 356 men dig a trench in 24 days, in how many days will 200 men make the same? Answer, in more days; 42 days, 17 hours, 7 ∶ 25.  
If 356 men require 24 days, how many will 200 men require.  
 
Or thus; Considering the numbers, 200 may be had in 156 once, therefore for 200 take 24 days; then for 150 take 18 days, total 42 days; then there will remain 6 to be 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? divide 3136 by 112, makes 28; which multiply by 3 pound, 5 shillings, 5 pence, makes 91 pound, 11 shillings, 8 pence.  
 
If 100 pound gain 7 pound, what sum of money will gain 85 at that rate? Answer.  
If 7 pound require 100 pound, what will 85 pound, require.  
 
Or otherwise, divide 85 by 7, maketh 12 1 ∶ 7; by which multiply 100, makes 1214 pound 2 ∶ 7 of a pound.  
 
Or otherwise, divide 100 by 7, maketh 14 2 ∶ 7; by which multiply 85, makes 1214 pound, 2 ∶ 7  
Example.  
Carseys' at 54 shillings the piece, are put in Barter, at 3 pound the piece, how shall Wool worth 24 shillings the Tod, be set in Barter, to make the bargain equal?  
If 54 shillings be 60 shillings, what shall 24 shillings make.  
Answer: for more than 24 shillings, and less than 54, so that 54 is the divisor, and multiplying 24 by 60, makes 1440; which divided by 54, makes 26 shillings, 2 ∶ 3, or 8 pence.  
If 54 shillings be 60 shillings, what will 24 shillings make.  
 
If 6 sheep cost 58 shillings, how many shall I buy for 124 pound? multiply 124 by 58, makes 7192; which divide by 6, makes 1198 sheep 2 ∶ 3.  
 
Or otherwise, divide 58 by 6, maketh 9 2 ∶ 3, by which multiply 124, makes 1198 2 ∶ 3, as before.  
Example.  
A Merchant at Seville delivereth 1500 Rials, to receive for every 11, being a ducat in London 5 shillings, 10 pence sterling money, how much must he receive?  
If 11 Rials be 70 pence, what are 1500 Rials?  
 
At 13 pound in the 100 pound profit, of what stock came 3274 pound? Answer: divide 3274 pound by 113 pound, makes 2897 pound, 39 ∶ 113 of a pound, add two cyphers to the given number.  
 
A Merchant received for principal and gain 328 wherein he found he had gained clear 56 pound, what did he gain upon the 100 pound. Answer, multiply 100 by 56, the gains makes 5600; which divide by 328, and the Quotient is 17 pound, 3 ∶ 41 in smallest terms.  
 
If 112 pound cost 7 pound, 6 shillings, how may I sell to give 10 pound upon the 100 pound. Answer: 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. Answer, 71 ∶ 100 parts of a pound: wherefore multiply 71 by 240, and divide by 100, makes 17 pence▪ 12 ∶ 5 of a penny.  
 
If 107 else of cloth cost 17 pound 12 shillings, what will 321 else cost at that rate? Here if you consider the proportion between the first number, and the third, you shall find the third number doth contain the first exactly three times; wherefore you need not to multiply the second by the third, and divide by the first number, but only take the second number, and multiply by 3, makes 52 pound, 16 shillings for the price that 321 else will cost: behold the work at large.  
If 107 else of cloth cost 17 pound, 12 shillings, what will 321 else?  
  

How to find whither that your numbers given be proportional, or not.  
Divide your third number by the first, and if the quotient be an even number, and nothing remain of your dividend, than the first and third numbers are even proportional in whole numbers, as in the last example, the first number was 107, and the third number 321, so that in dividing the third number by the first, the quotient is 3 & 0 remains: wherefore I conclude, that the first and third numbers are proportionals in whole numbers, and that the third doth contain the first just three times, and so often must the fourth number sought for, contain the second; and I conclude, that three times 17 pound 12 shillings, which is 52 pound 16 shillings, is the fourth proportional number sought, as appeareth by the ordinary form of work in the last example.  
 
If 36 else of cloth cost 13 pound, 4 shillings, 1 penny, what will 432 else cost at that rate: divide 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 unto B. 600 pound for 8 months, the question is, how much B shall lend unto A. for 12 months to recompense him, not reckoning compound, interest. Answer. If 8 months require 600 pound, what will 12 months require▪ the reason is less than 600 pound; wherefore divide 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 be to 400 pound, 2 ∶ 3 parts of 600 pound.  
If the number be not exactly proportional, yet there is a great abbreviation to be made of the work of Reduction, Multiplication, and Division, 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 dividing 58 by 19, the Quotient is 3, and 1 will remain; wherefore I take 3 times 16 pound, 12 shillings for 57 barrels, and I have to work but for the one remaining; which is but to divide 16 pound, 12 shillings, by 19, makes 17 shillings, 9 ∶ 19 of one shilling, the total is 50 pound, 13 shillings, 9 ∶ 19 shillings.  
 
If 356 else of Holland cost 124 pound, 2 shillings, 3 pence, what will 7259 else cost at that rate. Reduce 124 pound, 2 shillings 3 pence, into pence, makes 29787 pence; which multiply by 7259, makes 216223833 pence, which divide by 356, make 607370▪ which divided by 240 pence, makes 253 pound, 170 pence, or 14 shillings 2 pence.  
Example.  
 
 
A second way more briefly to work this question, or any other of like nature, is this: multiply the third number by the pounds and primes, or shillings and pence, and divide the product by the first number, and the quotient will be the fourth number sought. In the last example, 7259 else was the third number, which multiply by 124l. 1 prime, or 2s. makes 900841 l. 9 primes: then also 7259 by 3 pence, makes 21777 pence; which divided by 240, makes 90 pound, 14 shillings, 9 pence: then add those two sums into one total, makes 9009326 primes, 9 pence; leave out▪ 9, and then divide 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.  
 
If 24 pieces of Raisins cost 25 pound, 8 shillings, what will 324 pieces cost: multiply 324 by 25 pound, 4 primes, makes 8229, 6 primes: which if you divide by 24, the Quotient will be ●42 pound, 9 primes, or 18 shillings without Reduction, as in the example following.  
Example.  
If 25 pound gain 1 pound, 8 shillings, what will 725 pound gain at that rate? Multiply 725 by 1 pound 4 primes, makes 10150; which divided by 25, makes 40 pound, 6 primes, or 12 shillings.  
 
And in this sort may diverse other questions be wrought in pounds and shillings without Reduction, which I thought good to give a taste of, but I will proceed here no further, because I purpose in the second part of this Book to speak of them at large in the Treatise of Decimal Arithmetic, whereby all manner of questions are to be wrought of Multiplication and Division in pounds, shillings and pence, without Reduction, as shall appear in their several places following. And now I will proceed to speak something of the Rule of Three Direct and Conversed in fractional operations, wherein I will be as brief as I may, not intending to increase this little Treatise intended for a pocket book, into over large a volume.   

The Rule of 3 in Fraction.  
If your three numbers given be all fractions, multiply the third by the second, and divide the product by the first, and the quotient will be the fourth proportional number sought for.  
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 divide by 3 ∶ 4, maketh 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 else cost? Make 17 fraction wise, and multiply 17 ∶ 1 by 9 ∶ 12, makes 153 ∶ 12, which will be both abbreviated by 3, makes 51 ∶ 4, which divided by 7 ∶ 8 maketh 408 ∶ 28 parts of a pound, or in smaltermes 102 ∶ 7; then divide 102 by 7, makes 14 pound, 4 ∶ 7 of one pound for the price.  
 
2. Rule.  
If all your three numbers given be fractions, multiply the Numerator of the first fraction by the Denominator of the other two fractions, for to make your Divisor. Then multiply the Denominator of your first fraction by the Numerators of your other two fractions, to make your Dividend and then divide by your Divisor, and the Quotient is the answer sought: but if your Divisor be greater than your Dividend, than the Quotient is a fraction, less than a unite.  
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 divisor; then multiply 4, the denominator of your first fraction by 4 and 5, the numerators of your other two fractions, makes 80 for your dividend: now because your divisor is greater than your dividend, place them fraction wise thus, 80 ∶ 90 of 1 li. or in least terms, 8 ∶ 9 of a li.  
Example.  90   If 3 ∶ 4 of a yard  cost 4 ∶ 5 of a pound makes 8 ∶ 9 l. what 5 ∶ 6 of a yard   80    
Again, if 7 ∶ 8 of an ell cost 2 ∶ 3 of a shilling, what will 34 else cost.  
 
If 18 pioneers in 3 ∶ 5 of a day do make 22 Rodds of Barricadoe, what will they make in 7 days.  
 3  If 3 ∶ 5 of a Day make 22 ∶ 1 of a Wall, what 7 ∶ 1 of a Day?  770   
If 12 hundred 3 ∶ 7 of Alum cost 15 pound 1 ∶ 3 li. what will 324, 1 ∶ 8 of a hundred cost? Reduce the whole and broken numbers into broken, and work as is before taught.  
 
 
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 Velvet cost 7 ∶ 8 of a pound, what will 28 yard's cost. 32. l. 13 s. 4 d.  
Example.  
 
If 3 else 1 ∶ 8 cost 5 ∶ 7 of a pound, what will the whole piece cost, containing 28 else 1 ∶ 2 at that rate? Answer.  
 
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. Answer.  
 
The proof of this last example. If 100 pound starling buy me 294 pound of quichanella, how much shall 4 pound, 3 shillings, 4 pence buy me? to find the value of the hundred, the rate of one pound being given; abate 2 places from 294, and it will be 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 proof.  
 
If 30 men cast a Trench in 3 days 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 quantity of men to perform the work, than 11 ∶ 3 of a day will require; wherefore this proportion is reciprocal or backward; wherefore I multiply the two former numbers together, makes 333 ∶ 3, or in smaller terms, 110 ∶ 1; which divided by 5 ∶ 6, makes 660 ∶ 5; which divide by the denominator 5, makes 132 men.  
 
In the Backward Rule, or Conversed in fractions, multiply the Denominator of your third number, by the numerators of both your other numbers for dividend, then multiply the numerator of your third number, by the denominators of your other two numbers for divisor, and then work as before.  
Example.  
If when the bushel of Wheat was sold for 4 shillings, the penny loaf weighed 6 ounces 1 ∶ 2, what shall the same loaf weigh when Wheat is sold for 2 shilling, 8 pence p●nce the bushel? Multiply 48 by 13, makes your dividend 624: then 2 by 32, makes 64 for your divisor, and then divide 624 by 64, makes 9 ounces, 48 ∶ 64, or 3 ∶ 4 of an ounce.  
 
If when one ounce of sterling silver was worth 1 ∶ 4 of a pound the penny of silver weighed 30 grains, what shall the same penny weigh, when the ounce shallbe worth 1 ∶ 3 of a pound.  
Example.  
If when a load 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. Answer, 2 pound, 71 ∶ 76 of a pound.  
 
If 3 yard's 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 divided by 25, makes 14227 pence, 5 ∶ 25 of one penny in the whole 59 pound, 5 shillings, 7 pence, 5 ∶ 25 or 1 ∶ 5 of a penny. Behold the work.  
 
 
The proof of the former work. If 380 yards cost 59 pound, 5 shillings, 7 pence 5 ∶ 25 of one penny, what will 3 yard's 1 ∶ 8 cost at that rate? reduce your coin into 25, makes 355680; then reduce your 380 yards into 8, makes 3040; by which divide 144227 pence, makes 9 shillings, 9 pence, as before.  
 
If 34 ship Carpenters build a ship in 8 months 3 ∶ 5, in how long time will 120 Carpenters build the same? Reduce 8 months 3 ∶ 5 into fifths, makes 43 ∶ 5; then multiply 34 by 43, makes 1462. Also put your divisor 120 into fifths, makes 600 ∶ 5; then dividing of 1462 by 600, the Quotient will be 2 months 262 ∶ 600 parts of one month, or in smallest terms 131 ∶ 300 parts. And this Rule general if one of your numbers be a fraction, put always your divisor into the same fraction of your dividend, and the quotient will be of the same denomination of your dividend, and so the answer was months, and parts of a month.  
If 34 Carpenters ask 43 ∶ 5 months, what 600 ∶ 5 month.  
 
If 100 pound in 12 months gain 10 pound, what will 336 pound gain in 8 months? Take the tenth part of 336, which is ●● li. 6 primes, or 12 s. makes 369 li. 12 s.  
Secondly, if 12 months gain 33 pound 6 primes, what will 8 months gain? I answer, less than 33 pound 6 primes; wherefore multiply by 8, and divide by the greater extreme, 12, makes 22 pound, 4 primes, or 8 shillings, the answer.  
If 120 pioneers in 6 days cast 300 rods of Trench, how many shall 600 men cast up in 4 days. If 120 give 300, what will 600 give? Answer, 1500 Rods.  
Secondly, if 6 days give 1500 rods, how many will 4 days give? I answer, less; multiply by 4, and divide by 6, makes 1000 Rods.  
If 112 pound in 12 months' gain 100 li. what will 340 li. gain in 7 months? Answer: 303 li. 4 ∶ 7.  
Secondly, if 12 months gain 303 li. 4 ∶ 7 what will 7 months gain.  
Example.   

A general Rule▪  
Put always your divisor into the same fraction of your dividend, and your quotient will be of the same denomination, that your dividend was: as in the last example, 12 months was turned into sevenths, and also 303 pound 4 ∶ 7 was turned into sevenths of pounds, and so the quotient of that division was pounds, and the fraction of a pound remaining.  
If 7 pound in 13 months' gain 3 pound, in how long time will 340 pound gain 60 pound. First, if 7 pound Gain 3 pound, what will 340 pound gain, makes 145 pound, 5 ∶ 7 of a pound. Secondly, if 145 pound, 5 ∶ 7 or 1020 ∶ 7 ask 13 months, what will 60 pound, or 420 ∶ 7 gain. Multiply by 13, and divide by 1020, makes 5 months 6 ∶ 17 of a month.  
If 600 great Horses in 5 days do spend 1125 Bushels of oats, how many bushels will serve? 1400 Horses for 22 Days. First, say, if 600 give 1125, what 1400, makes 2625 bushels. Secondly, if 5 spend 2625 bushels, what will 22 days spend? Multiply by 22, and divide by 3, makes 11550 bushels.   

How to work the double Rule at one operation.  
This last question, or any other of like nature which is wrought by the double Rule at two several operations may be answered at one in this manner: multiply the three latter numbers, to make your dividend one into the other; then multiply the two former numbers for to make your divisor, and then divide the dividend by the divisor, and the quotient will be the same, as in the last example, 1125 being multiplied by 1400, makes 1575000; which again increased by 22, makes your dividend 34650000. Then multiply your two former numbers 600 by 5, makes 3000 for the Divisor; and then dividing your dividend by your divisor 3000, the quotient will be 11550 bushels, as before at two operations.  
Example.  
If 35 s. in 7 months' gain 6 s. in how long time will 340 l. gain 100 l. First, if 35 s. gain 6 s. what will 340 l. require? Reduce 340 l. into pence, and multiply by 6, makes 40800; which divided by 35, makes 1165 s. 5 ∶ 7 s. Secondly, if 1165 ●. 5 ∶ 7 require 7 months, what will 100 l, require? Makes 12 months, 8 ∶ 816 parts of a month.   

Fellowship without Time.  
This Rule differeth very little from the Rule of three; for in this Rule the sum of all the moneys disbursed, is the first number in the Golden Rule. Then the gains or loss is the second number: the third number is each several partner's money disbursed so that the Rule must be severally wrought for each several Partners portion.  
Example.  
Four 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 have of the gains, according to the proportion of his money disbursed. First, add all the moneys disbursed into one total sum, viz. 74, 90, 100, 120, total is 384 for the first number in the Golden Rule. Then the second number is 84 pound, the gains; and the third number is each particular man's stock; then work as followeth.  
If 384 pound gain 84 pound, what will A. B. C. D. sums gain to them.  
 
The like reason is in loss, as is in gains. Example: A certain ship being in a tempest on the sea was forced to cast over board so much of her lading, as amounted unto the sum of 642 pound, than there is great reason that all the ventures should bear part of that loss, according to the proportion of his stock which he ventured. As suppose: A. ventured 700 pound, B. 530 pound, C. 640 pound, D. 800 pound; total is 2670. Then say; If 2670 pound lose 642 pound, what will each of A. B. C. D. loose? as in the example following.  
Example.  
If 2670 pound lose 642 pound, what will A. B. C. D. sums 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. Answer.  
Seek a number wherein the like parts may be had, which is 60, and take the like parts of that number for the numbers that you seek, for to find each man's 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 add into one total, makes 57 for the first number in the Golden Rule.  
Example.  
 
If 57 be 3600, what will be the sums 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 have of the gains. Answer.  
If 57 gain 240, what will A. B. C. D. sums gain to them.  
 
Four 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 year they found they had gained 400 pound, 18 shillings, 6 pence: the question is, what each man must have of the gains. First, the total sum of all their moneys makes 2721 pound, 8 shillings, 6 pence, or 653142 pence, for the first number. Then reduce each several man's money disbursed into pence for the third number, the second is the gains also reduced into pence, and then work according to the Rule.  
Example.  
If 2721 pound, 8 shillings, 6 pence gain 400 pound, 18 shillings, 6 pence, what will A. B. C. D. sums gain to them.  
  

Rules of Fellowship, with diversity of Time.  
Multiply each man's money disbursed by the time that it continued in stock, and gather the totals, as in the last Rule, to make the first term in the Golden Rule, and the gains or loss is the second, and then each man's product of money and time for the third term in the Golden Rule, and work as followeth.  
Example.  
Three men made a stock, A. B. and C. and in long continuance of time by dangerous adventures they gained, and got by prizes taken at Sea 2345 pound; A. put in stock 40 pound, 14 months; B. put in 50 pound, 8 month; C. put in 85 pound 6 months, what shall each man have of this gains.  
Example.  
If 1470 pound gain 2345 pound, what will A. B. C. sums gain them.  
 
The second question with more diversity of time, four Merchants made a Company; A. put in 340 li. 19 s. 2 d. for 10 months; B. put in 930 li. for 9 months; C. put in 760 li for 12 months; D. put in 583 li. 13 s. 4 d. for 5 months, wherewith they gained 740 li. now the question is, to know what each man must have of this gains.  
 
Cut off two Ciphers from each number, and then work as followeth.  
If 57163 pence gain 1776 pence, what what will A. B. C. D. sums gain them.  
 
Example.  
There is a Booty or Spoil taken by 3 men worth 7851 pound, and they agree to divide it in this sort; A▪ is to have one half, B. one third, C. one fourth, what is each man's share.  
To work this question, and all other of like nature, seek a number which may be divided by all the denominators of your three fractions in whole numbers, and the smaller such a number be that you choose, the more easy will your work be; which for to find, multiply your denominators of your fractions one into another; that is to say, 2 by 3 maketh 6; and 6 by 4, makes 24; so 12, one half of 24 will be evenly divided 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 sum, makes 13 for first number in the Golden Rule; the second is 7851 pound, and the third numbers are each several man's portion imagined to be, viz 6, 4, 3, and then work as before.  
If 13 give 7851 pound, what will A. B. C. sums give.  
 
4. Example.  
Four Merchants bought a house together, which cost 3000 pound; A. was to pay 1 ∶ 2 and 6 pound overplus; B. 1 ∶ 3 and 12 pound more; C. 8 pound less than 2 ∶ 3; D 1 ∶ 4 with 20 pound overplus. Now the question is, what each Merchant must pay of this sum. Answer: First, the pounds overplus must be subtracted from the sum given; and the pounds wanting must be added to the sum given; as for A. 6 pound, B. 12 pound, for D 20 pound, total is 38 pound, to be subtracted then; for C. add 8 pound, therefore subtract 30 pound from 3000 pound, there will remain 2970 pound; then work by the Rule of Fellowship, taking 12 for a number, which will be divided by all the denominators, 2, 3 and 4, viz. take for A. 6, for B 4, for C. 8, for D. 3; total is 21 for divisor, the second number is 2970 pound, the third, each man's part imagined.  
Example.  
If 21 give 2970 pound, what will A. B. C. D. sums give.  
 
The numbers found to A. are 848 pound 4 ∶ 7, to which if you add 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, leaves 1123 pound, 3 ∶ 7  
To D. 424 pound, 2 ∶ 7, to which add 20 pound, makes 444 pound, 2 ∶ 7; the which added into one total, makes 3000 pound, the proof.  
And in this manner may infinite variety of questions be propounded, and their doubts easily resolved; and here will I end concerning this Rule, and go in hand with some pleasant questions to be wrought by position, which is the most excellent Rule of all others in Arithmetic, as shall appear in the second part of this Book in decimal Arithmetic.    

Position.  

The Rule of Position requiring one number to be imagined, before the principal proportion can be found.  
TO work 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 have failed, by doubling or tripling according to the nature of the question, you shall then attain unto the true number desired, by aid of the Golden Rule, in manner following: for look what proportion is between the false conclusion, and the false position, such proportion hath the given number, to the number sought.  
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 sum? 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 work as followeth. If 88 pound come of my Position 8 pound, of what comes 2700? Multiply 2700 by 8, and then divide 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 total sum is 2700 pound. Behold work as followeth.  
If 88 pound come of 8 pound, of what comes 2700.  
 
2. Example.  
A Captain of a Band of Men being asked, what number of Soldiers were in his Band, answered, I do not readily know; yet (quoth he) of this I am certain, 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 sum 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 total is but 64 men, but that should be 384 men. Then say by the Golden Rule, as followeth.  
If 64 come of 30, of what number comes 384.  
 
Answer: he had 180 men in his Band, whereof  
  

The solution of this Question another way more brief.  
Divide 384 by 64, makes 6; which multiply by 30, makes 180 men, as before.  
3. Example.  
A certain man having spent 120 pound, had yet remaining 1 ∶ 2 and 1 ∶ 3 of his whole substance; the question is, what his substance was. Answer: First, 1 ∶ 2 and 1 ∶ 3 is 5 ∶ 6, which being taken from 6 ∶ 6, the whole substance leaves 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 several prices, of each a like quantity, and he was to pay half as much more for the second sort, as he paid for the first, and twice as much for the third sort as he paid for the second: now the question is, what each sort cost him, and at what price every yard was rated unto him? I suppose the first sort cost him 4 pound, than the second sort must cost him 6 pound, which is half as much more as the first; and then the third sort cost him 12 pound, which is twice as much as the second; the total is but 22 pound, but it should be 248 pound: wherefore if 22 pound come of 4 pound, of what number comes 248 pound?  
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 divide 384 by 3, and you shall find he had 128 yards of each sort, and by Practice, 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.    

Double Position.  

The Rule of double Position.  
suppose 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, than your Position was the right number, which doth seldom happen. First, if by your working there cometh out more than the true number; then note it thus + with a cross; if less, than thus − with a long line, which doth signify less.  
Secondly, suppose another number, greater or smaller, and work as before, until you do find the true number sought; which if you do not find, see the difference also from the true number sought, and note it with the sign + or − as it shall be found.  
Then thirdly, set your suppositions with their errors, more or less, as in the examples following.  
Fourthly, multiply cross the first position by the seconds error, and the second position by the error of the first, and then if the signs be both alike + or −, abate the lesser from the greater, and the remains shall be the dividend. Also the lesser error abated from the greater, leaves the divisor▪ but if the signs be contrary one +, the other less, add both together to make the dividend, and add the two errors to make the divisor: and lastly divide the dividend by the divisor, and the quotient is the true number desired.  
1. Example.  
A certain man seeing a purse in his friend's hand, saith unto him: It seemeth unto me, that there is 100 Crowns in your purse▪ To whom the other answered: Nay (quoth he) there are not 100 Crowns, but (saith he) if they were increased 1 ∶ 2 and 1 ∶ 3▪ and 1 ∶ 4, and lastly, one Crown overplus, then would they be just 100 crowns.  
I suppose there were 12 Crowns in his purse, to which if I add 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 Crown more, the total will be but 26 Crowns, but they should be 100 Crowns, so that this error is two little by 74 Crowns, which I note thus:  
74 − 12  
Secondly, I suppose he had 24 Crowns, to which I add 1 ∶ 2 of 24, which is 12 and 1 ∶ 3, which is 8 and 1 ∶ 4, which is 6: and lastly, one Crown overplus, the total is 51, but it should be 100 Crowns, so that this is an error of 49, too little, which I also note thus: 49 − 24  
 
The answer is; that he had 47 pound 13 ∶ 25 parts of a pound in his purse. The proof followeth.  
 
2. Example.  
Twenty yards of Satin, and 12 shillings is equal unto 12 yards of velvet less, 10 shillings; the price of either sort is required.  
To answer this, or any other like question, take any number for the price of a yard of the lesser number, which here is velvet, which at 20 shillings a yard, lesse 10 shillings, amounteth unto 230 shillings. Now admit a yard of Satin at 14 shillings, so 20 yards and 12 shillings amounteth unto 292 shillings; from which subtract 230 shillings, rests 62 s. more than the truth. Again, 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 than 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 divisor, by which divide the difference, makes 10 shillings, 9 ∶ 10 shillings for the price of a yard of Satin.  
Example.  
3. Example.  
Otherways if 40, the difference of errors gain a, the difference of positions, than 62 the first error yields 3 and 1 ∶ 10▪  
Or if 40 yield 2, what 22? makes 1 and 1 ∶ 10; this taken from 12, or 3, 1 ∶ 10 from 14, leaves 10, 9 ∶ 10 for the price, as before.  
4. Example.  
A Carpenter was hired to work 20 days at 12 pence a day, but every day that he was idle, he was to abate 18 pence of his wages, and in the end he received but 8 shillings: now the question is, how many days he wrought.  
First, suppose he wrought 12 days, which cometh to 12 shillings, then must the 8 days that he 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.  
Again, I say that he wrought 14 days, amounting to 14 shillings▪ then 6 days that he played at 18 pence a day, cometh to 9 shillings; this taken from 14 shillings, leaves 5 shillings, and it should be 8 shillings, which is an error of 3 shillings too little. Now multiplying 12 by 3, and 14 by 8, the products are 36. and 112, and the excess is 76; which being divided by 5, the difference of the errors, quoteth out 15, 1 ∶ 5 for the number of working days, and 4 days 4 ∶ 5 for the number of playing days.  
12 − 8 5 14 − 3  
Otherways.  
If 5▪ the difference of errors, yield 2, the difference of positions, what 8 the first error? 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 ways the numbers of the Days he wrought are found out.    

Barter or Exchange.  
TWo men barter, one hath Ginger of 10 pence a pound ready money, & in barter he will sell it for 12 pence a pound. The other hath sugar of 12 pence a pound ready money, but in barter he will sell it for 14 pence a pound; the Question is, how much Sugar will pay for 756 pound of Ginger? First, put your price of your Ginger into pence, makes 9072 pence; which divide by 14 pence, makes 648 pound of Sugar, which must be given for 756 pound of Ginger, at 12 pence the pound.  
2. Example.  
Two Merchants will barter, one hath Raisins of 34 shillings the hundred ready money, and in barter he will sell them for 40 shillings: the other hath Nut megs of 4 shillings the pound ready money, how shall he set his Nut megs to make the like profit. Put your coin into pence, and say; If 408 d. be 480 d. what is 48 d. Multiply 480 by 48, and divide by 408, makes 56 d. 2●●5● of one penny for the price of the Nutmegs; vid. 4 s. 8 d. 1 ∶ 2 of a pound.  
3. Example.  
Two Merchants will barter, one hath Holland of 2 shillings, 7 pence the ell ready money, which he will sell in barter for a shillings, 10 pence the ell, and yet he will gain privately 10 pound in 100 pound over that gain▪ at what price must he then set his Holland? Answer: Set down 2 shillings 10 pence in pence, makes 34 pence; of which take the tenth part, which is 3 pence, 4 ∶ 10, or 2 ∶ 5, and add to 34 pence, makes 37 pence 2 ∶ 5 of a penny for the price, to sell one ell to make that gains. Now the other Merchant hath wool at 7 shillings a Todde ready money, how shall he set his wool to make like profit that he be not deceived in the bargain. If 31 pence be 37 pence, 2 ∶ 5, what is 84 pence? Multiply 374 primes by 84, makes 31416; which divide by 31, makes 101 pence, 3 ∶ 10 penny, or 8 shillings, 5 pence, 3 ∶ 10 of one penny, which is the price for him to sell his wool to make like profit.  
Example.  
4. Example.  
Two Merchants will barter, one hath Sugar of 6 pound, 4 shillings ready money, and he will sell it for 7 pound the hundred. The other hath Ginger of 4 pound, 6 shillings the hundred, and in barter he will sell it for 5 pound the hundred; now the question is, at what rate each of them doth gain per cent ' and which hath the advantage of the other. First, if 6 pound, 2 primes gain 8 primes, what will 100 pound gain? Multiply 8 primes by 100, makes 800 primes; then add 2. or 3 cyphers more to it, which divide by 6 ∶ 2 primes, makes 12 l. 9 primes, 10 ∶ 31 of a prime, or near 12 l. 18 shilling, 8 pence, which the first man doth gain per cent●.  
Secondly, if 4 pound, 3 primes gain 7 primes, what will 100 pound gain? Multiply 7 primes by 100, and add 2 cyphers more, makes 70000; which divide by 4 ∶ 3 primes, makes 16 pound, 2 primes, 34 ∶ 43 of a prime; from which subtract 12 pound, 18 shillings, 8 pence, rests 3 pound, 6 shillings, 2 pence, which the second man hath gained more than the first gained.  
6. Example.  
Two Merchants barter, one hath a certain number of pieces of Sakkins at 18 shillings a piece, for the which the other doth give him 1806 else of linen Cloth, at 16 pence the ell, and yet 30 pound in ready money; the Question is, how many pieces of Sakkin he had. First, find what 1806 else of linen Cloth cost by Practice? makes 120 pound, 8 shillings: to the which add 30 pound, makes 150 pound, 8 shillings: then divide 150 pound, 4 primes, by 18 shillings, or 9 primes, makes 167 pieces of Sakkin, and 1 ∶ 9 of a piece.  
Example.  
6. Example.  
Two men will barter, one hath Pepper of 22 pence the pound ready money, but in barter he will sell it for 27 pence the pound: the other hath Cinnamon of 3 shilling, 6 pence the pound ready money, and in barter he will sell it for 4 shilling the pound; the question is, how much cinnamon will pay for 384 pound of Pepper at that rate? First, 384 pound of Pepper at 27 pence the pound is 43 pound, 4 shillings; which divide 43 ∶ 2 primes, makes 216 pound Cinnamon, which he must give.  
7. Example.  
If 4 English else make 5 yards, and 13 yards, makes 50 Pawns at Geanes, how many Pawns is in 100 else English. If 5 be 4, what is 13, maketh 10 2 ∶ 5. Secondly, if 10 2 ∶ 5 be 50, what is 100, 480 10 ∶ 13  
8. Examples.  
Every 4 else at Antwerp maketh 5 at Frankford, and 25 there makes 24 Braces at Luques, the question is, how many braces is 100 in Antwerp. If 25 be 24, what is 5, maketh 4 4 ∶ 5. Secondly, if 4 be 4 4 ∶ 5, what are 100, makes 120.  
9 Example.  
If 3 yards at London be 4 else at Antwerp, how many yards at London make 84 else at Antwerp. If 4 be 3, what 84? makes 63 else.  
10. Example.  
At Rouen 112 else make but 98, and 100 else at Rouen is 112 at Siuil, how many of ours in 100 else of Si●ull. If 98 Rouen be 112 else, what 100 Rouen, makes 114 else, 1 ∶ 7 of an ell. Secondly, if 112 else be 114, 1 ∶ 7, what is 100 Seville, makes 102, 19 ∶ 25.  
11. Example.  
If 67 yards at London be 100 in Venice; how many are 7894? multiply by 67, makes 5288 yards, 98 ∶ 100 parts.  
12. Example.  
A Merchant doth deliver 400 pound sterling in London by exchange for Antwerp, at 23 shillings, 5 pence the pound sterling, the question is, how much Flemish money, he shall receive at Antwerp: put your 23 s. 5 d. into pence, makes 281 pence; which multiply by 400, makes 112400 pence; which divide by 240, makes 468 pound, 6 shillings, 8 pence, which he must receive at Antwerp.  
Example.  
13. Example.  
If 100 pound starling be 134 pound, 6 shillings, 4 pence Flemish, what is one pound starling worth? Reduce your coin 134 l. 6 s. 4 pence, into pence makes 32236; which divided by 100, makes 322 pence, 9 ∶ 25 pence, or 26 shillings, 10 pence, 9 ∶ 25 of one penny, for one pound sterling.  
If one pound sterling be 1 pound, 14 shilling, 7 pence, ob. Flemish, how much sterling money is in 100 li. Flemish? Reduce 100 pound into pence, makes 24000 pence; then put it into half pence, makes 48000 half pence; then put 1 pound, 14 shillings, 7 pence, ob. into half pence, makes 831; by which divide 48000, makes 57 pound, 15 shillings, 1 penny almost, and so much sterling money is in 100 pound of Flemish money at that rate.   

Of Gain and Loss.  
IF 13 pieces of Canvas cost 17 pound, 12 shillings, how may I sell them to gain 8 pound in the hundred? Multiply 176600 by 8, makes 19 pound, 19008, or two pence almost, and so much must he sell them for to gain 8 pound in the hundred.  
If 17 pound, 12 shillings gain 1 pound, 8 shillings, 2 pence, what will 100 pound gain? Multiply 1 pound, 8 shillings, 2 pence in decimals by 100, and divide by 17 pound, 6 primes, makes 8 pound in the 100, the proof.  
Example.  
A Merchant hath lent 630 pound at interest for 10 pound in the 100 for 3 years' interest upon interest, the Question is, unto what sum it will amount unto at the end of the term? Answer: Take the tenth part, and add it into one total 3 several times, makes 838 pound, 10 shillings, 7 pence, 1 ∶ 5 of a penny for principal and interest, at the rate given, to be paid at the end of three years.  
Example.  
2. Example.  
A Merchant receiveth for principal and interest 838 pound, 10 shillings, 7 pence, 1 ∶ 5 of a penny at 10 pound in the hundred compound interest, which was for money delivered out for 3 years; now the Question is, what was the sum of money that was lent? To do this, or any other the like question, divide the sum of money received by 110 three several times, and the three quotients will show the yearly increase of the money lent, and the last quotient will be the answer to the question, or the money disbursed, as in the example following, which is the proof of the former question.  
Example.  
3. Example.  
A Merchant lent 100 pound for 7 years at 10 pound in the hundred Compound Interest, the Question is, what he shall receive at the end of the term.  
Example.  
Makes at 7 years' end 194 li. 17 s. 5 d.  

How to work Compound interest at any rate per cent.  
What is the principal and interest of 352 pound, put out at 8 pound in the hundred compound Interest, to be paid at the end of two years? Add two cyphers to 352 pound, makes 35200; then place your Interest 8 under the lowest cipher next the right hand, and multiply 352 by 8, placing the product under the line, and that will be the Interest; which added into the sum lent, makes the total of the principal and interest▪ and so work for the second, third, and fourth year, as in the Example.  
 
First I multiply 35200 by 8, makes 2816, which I add unto 35200, makes 38016; then I multiply 3801600 by 8, makes 4105728, or 11 shillings, 5 pence, abating 4 figures for the 4 cyphers, which I added to the sum for to find out the prime line, as appeareth in the example; and so of any other sum or rate in the hundred.  
At 17 pound the hundred per annum compound interest, what will 879 pound amount unto to be all forborn unto the end of 5 years? Add 2 cyphers to your sum given, and multiply by your Interest 17, and add into the principal, and so work 5 years, and the last product will be the sum of money to be received, viz. 1927 pound, 3 shillings, 5 pence.  
Example.  
If a Merchant buy a parcel 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 gain 18 pound in the hundred, and give 9 months time for the payment; as in the Example following.  
Example. 10 Pieces at 3. 6. a piece, 33. 0. 10 Pieces at 4. 2. a piece, 41. 0. 10 Pieces at 4. 10. a piece, 45. 0. 10 Pieces at 5. 0. a piece. 50. 0.   The sum is 169. 0.  
 
Take the 3 ∶ 4 of the interest, makes 191 pound, 16 shillings, 3 pence, 3 ∶ 5 of one penny, to sell to gain 18 pound in the hundred, for to give 9 months 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 lose 7 pound in the hundred? Take the interest at 7 pound in the hundred, 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 divide the remainder by 300, makes 1 pound, 1 shilling, 10 pence for the price sold: secondly, divide 352 pound by 300, makes 1 pound, 3 shillings▪ 5 penny ob. for the price which it cost him  
Rye sold for 3 shillings a bushel looseth 20 pound in the hundred, what will then be lost, if it be sold for 3 shillings 6 pence a bushel? 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 divide by 3, makes 93 li. 1 ∶ 3 Or otherwise, if 36 pence be 80 pound, what is 42 pence? Multiply 80 by 42, and divide 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? divide 385, or 38 penny 1 ∶ 2 by 110, makes 35 pence that the ell cost.  
 
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 divide by 35 pence, makes 235 yards, 1 ∶ 5 yard.   

How to gain 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 sum, which is the same number, placed one place nearer to the right hand, and that is the profit or Interest; which added up into the price given, makes the price to sell one yard, pound, ell, or piece, to gain 10 pound in the hundred ready money.  
Example.  
If one ell of Holland cloth cost 3 shillings, 9 pence, how may I sell to gain 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 gain 10 li. per cent.  
Example.  
If your price you would gain, be not 10 pound in hundred, then add 2 Ciphers to your number of pence given and multiply that number by your Interest, omitting to multiply by the cyphers, and the product under the line is your Interest or gain, which added up into one sum; makes the price to sell one yard, ell, pound, or piece, to gain according to the rate desired example.  
If one pound of Cloves cost 4 shillings, 10 pence, how may I sell to gain 9 pound per cent ready money? Put 4 s. 10 d. into pence, makes 58 d. then add 2 cyphers, makes 5800; which multiply by 9, maketh 5 ∶ 22 or 5 pence, 22 ∶ 100 parts of one penny; which added up to the upper numbers, is 63 pence, 22 ∶ 100 parts of one penny, or 5 shillings, 3 pence, 1 ∶ 5 of a penny for the price to sell one, to gain 9 pound in the hundred.  
 
If one piece of Raisins cost 18 shillings, 9 pence, how may I sell to gain 18 pound in the hundred ready money? put your money into pence, makes 225 pence, to which add 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 gain 18 pound in the hundred.  
Example.  
 
A Merchant lent wares for 10 pound in the hundred profit for 12 months, and at the end of 6 months he received principal and interest 356 ls. the question is, what was the sum lent? Answer: add 2 cyphers to 356 pound, and divide by 105 pound, which is 6 months interest and principal, makes 339 pound 1 ∶ 21 parts of a pound for the sum lent.  
Example.    

Equation of Payment.  
The Rule of payment is to bring diverse payments due at several days to be paid at one entire payment.  
AMerchant is to pay at diverse payments 600 pound: viz. 200 pound present, 200 pound at 8 months, 140 pound at 6 months, and 60 pound at 2 months: now he is willing to pay all at one payment, what time must be given? The ready money being omitted, set the rest as numerators thus, 200 ∶ 600 140 ∶ 600 60 ∶ 600 parts, which in their least terms▪ abbreviated, makes 1 ∶ 3▪ 7 ∶ 30 and 1 ∶ 10. Now multiply 1 ∶ 3 by 8, maketh 2, and 2 ∶ 3; secondly, 7 ∶ 30 by 6, maketh 1 and 2 ∶ 5; thirdly, 1 ∶ 10 by 2 makes 1 ∶ 5; total is 4 months, and 4 ∶ 15 of a month for the time sought.  
Examples.  
A Merchant hath owing him 752 pound, to be paid 200 pound content; 200 pound at 3 months, 130 pound at 5 months, and the rest at 12 months; now at what time ought this money to be paid all at one payment?  
Example.  
A Merchant hath owing unto him 782 pound, 12 shillings, to be paid 1 ∶ 3 at 4 months 1 ∶ 2 at 7 months, the rest at 12 months, what time must it be all at one payment.  
 
Makes 6 months, 5 ∶ 6 of a month.  
Wines worth 14 pound ready money are sold for 16 pound, to pay 1 ∶ 3 at 3 months, 1 ∶ 2 at 4 months, and the rest which is 1 ∶ 6 at 12 months, the question is, what is gained in 100 pound in 12 months.  
 
Makes at 5 pound in the hundred.  
Sugars worth 21 pound ready money are sold for 25 pound, to pay 1 ∶ 5 ready money, 1 ∶ 8 at 4 months, 3 ∶ 10 at 7 months, 3 ∶ 8 at 15 months; the question is, at what rate per cent▪ per annum they were sold.  
 
Makes 8 pound, 9 ∶ 40 per cent▪.   

Alligation Mediall.  
ALlegation is an Arten teaching to combine or knit together diverse things unequally prised, and thereby to find an equal price of any part of the said mixture, Alligation Mediall, is that which by the augmenting the quantity of every several portion to be mixed by his own price, and dividing the sum of all the products by the total of the several portions to be mixed, findeth the thing sought.  
Example.  
Three several sorts of Barley are to be mixed; viz. 34 bushels at 18 pence, and 76 at 20 pence, and 100 at 22 pence; the Question is, what one bushel of that mixture will be worth? First, multiply each number by his price, viz. 34 by 18, 76 by 20, and 100 by 22, makes 612, 1520, and 1200, the total is 4332: then add the number of bushels into one sum, makes 210; by which divide 4332 d. makes 20 pence, 132 ∶ 210 of one penny for the price of one bushel so mixed.  
2. Example.  
If you will mix 30 gallons of Sack at 4 shillings a gallon, with 150 gallons of White Wine at 2 shillings the gallon, what will a gallon of that mixture be worth? Multiply 30 by 4, makes 120 shillings; also 150 by 2 shillings, makes 300 shillings, total is 420 shillings: then add 30 and 150, makes 180 gallons; by which divide 420 shillings, makes 2 shillings, 1 ∶ 3 of a shilling, or 2 shillings, 4 pence, for the price of one gallon so mixed.  
3. Example.  
Admit there were 6 portion of Silver of 7 ounces fine, 12 of 8 ounces fine, and 25 of 10 ounces fine, which are to be mingled with 10 pound of Copper, what is a pound of that mixture worth? For answer: multiply 6 by 7, makes 42; also 12 by 8 makes 96, and 25 by 10, makes 250, the total is 388, which being divided by 53, the total of 6, 12, 25 and 10 makes 7 ounces, 17 ∶ 53 of an ounce; and so much fine is a pound of that mixture.  
4. Example.  
A Merchant hath 6 several sorts of Spices, of which he will sell, of each an equal quantity of several prices for the sum of 323 pound, 8 shillings: viz. Cinnamon large at 4 shillings, 6 pence a pound; Nutmegs Case at 3 shillings, 8 pence a pound; Large Maces at 8 shillings a pound; and Pepper Case at 2 shillings 2 pence a pound, Pepper calico at 22 pence the pound, and Ginger large at 10 pence a pound; the Question is, how many pound he must have of each to make the just sum of 323 pound, 8 shillings? Answer: first, put your money into shillings, makes 6468 shillings; secondly, put all your prices of the Spice into one sum, and by that sum, which is 21 shillings divide 6468, makes 308 pound which he must sell of each.  
Example.  
  

Alligation Alternat.  
ALligation Alternat is that, which altereth the places of such excess as commonly fall between the mean price, and the extremes; in which counterchange, if the extremes be equal, than the difference between the mean price, & lesser extreme is to be set against the greater extreme, and of the contrary if otherwise.  
1. Example.  
White Wine of 20 pence the gallon is to be mixed with Sack of 3 shillings a gallon, so that there must be mixed 300 gallons to make the price to be but 2 shillings, 4 pence the gallon, the question is, how much of each sort must be taken. The numbers set down, as in this example thus, the difference of 20 the lesser extreme from 28, is 8; also the difference of 36 the greater extreme is also 8, so that I find you must take as many of one sort, as of the other to make this mixture: viz. 150 gallons of each sort.  
 
2. Example.  
White Wine of 16 pence a gallon is to be mixed with Sack of 40 pence the gallon, how many gallons must be taken of either sort, so that 120 gallons may be of 30 pence the gallon.  
The numbers being set down, as in this example, the difference of 16 the lesser extreme from 30 the mean price, there will remain 14, which I plate against 40; then take the difference of 40, the greater extreme, from 30 the mean price, there will rest 10 to be linked with the lesser extreme; whereby I find, that so often as I take 14 gallons of Sack I must take 10 gallons of White Wine to make the mixture:  
 
3. Example.  
A certain Clothier is desirous to mingle 144 pound of wool of 4 sorts: viz. blue wool of 10 shillings the stone, red wool of 11 shillings the stone, green wool of 12 shillings, white wool of 9 shillings the stone, how many stones of each shall he take, that one stone of the mixture may be worth 14 shillings.  
The counterchange being made, according to the Rule, as is in the Margin it is plain, that so often as you take 5 of Blue, you must take 3 of Greene, and 2 of Red, and 2 of White. Therefore if 12 be 144, what  
  
The end of the first Book.   

THE SECOND BOOK.  
Containing a Treatise of decimal Arithmetic:  
Wherein is taught how to work all manner of operations in decimal Arithmetic, more speedy and easy, then by vulgar Arithmetic; and first of the decimal Table.  
LONDON, Printed by Augustine Matthewes dwelling in the Parsonage-house in Saint Bride's lane, near Fleetstreet, 1623.   

THE USE OF THE decimal Table.  
THe decimal Table following doth begin from one Farthing unto a Prime, or two Shillings; so that if you have a decimal Fraction given, which doth contain 90625 sixths: search it in the decimal Table, and you shall find it over against 21 pence, three farthings and that is the value of that fraction given.  
Or if you would know how to set out 16 pence halfpenny in decimals; search in the Table against 16 d. 2 g. and you shall find 6875 fifths for the decimal sought.  
But if you would set out any number of shillings from one shilling unto one pound, or ●o shillings; search in this little Table following, and you shall find your desire. As if you would set out 15 shillings in decimals, you shall find 7 primes, 5 seconds for 15 shillings, and so of any other sum, as in the example following.  
Example. ●●ill. 1. 2. 1 05 2 10 3 15 4 20 5 25 6 30 7 35 8 40 9 45 10 50 11 55 12 60 13 65 14 70 15 75 16 80 17 85 18 90 19 95 20 1 li. q 1. 2. 3. 4. 5. 6. 7 q. 1. 2. 3. 4. 5. 6. 7 1 0010416 0 025 2 0020833 1 0260146 3 003125 2 0270833   3 028125 1 0041666 7 0291666 1 0052083 1 0302083 2 00625 2 03125 3 0072916 3 0322916 2 0083333 8 0333333 1 009375 1 034375 2 0104166 2 0354166 3 0114583 3 0364583 3 0125 9 0375 1 0135416 1 0385416 2 0145833 2 0395833 3 015625 3 040625 4 0166666 10 0416666 1 0177082 1 0427082 2 01875 2 04375 3 0197916 3 0447916 5 0208333 11 0458333 1 0218746 1 046875 2 0229166 2 0479166 3 0239582 3 0489584 6 0●5 12 05  
q. 1. 2. 3 4. 5. 6. 7 q. 1. 2. 3. 4. 5. 6. 7 12 05 18 075 1 0510416 1 0760146 2 0520833 2 0770833 3 053125 3 078125 13 0541666 19 0791666 1 0552083 1 0802083 2 05625 2 08125 3 0572916 3 0822916 14 0583333 20 0833333 1 059375 1 084375 2 0604166 2 0854166 3 0614583 3 0864583 15 0625 21 0875 1 0635416 1 0885416 2 0645833 2 0895833 3 065625 3 090625 16 0666666 22 0916666 1 0677082 1 0927082 2 06875 2 09375 3 0697916 3 0947916 17 0708333 23 0958333 1 0718746 1 096875 2 0729166 2 0979166 3 0739582 3 0989584 18 075 24 1000000   


THE SECOND BOOK, CONTAINING A TREATISE of decimal Arithmetic.  

The declaration of the parts of the decimal Table.  
FIrst, the decimal Table in the left Margin contains certain numbers in great and small letters; first, from 1 farthing unto one prime, or tenth of a pound, or two shillings. Then from one prime for every shilling unto one pound starling, or 20 shillings.  
First, beginning in the left margin is set down one farthing in the uttermost parallel to the left hand, in the first parallel of the Table, and so continuing from one farthing to one prime, or 2 shillings; and over against every number in the left side in a right line towards the right hand is contained the numbers in decimals, answering unto every farthing from one farthing to one prime, or 2 shillings; and in the upper margin in the head of the Table is contained, the true denominations of the said are all numbers in primes, seconds, thirds, fourth's, fifths, sixths, and sevenths, which are small enough to work any question exact to a small fraction of one penny in a sum of great value, as shall appear by examples following. But here you shall note, that all the numbers in the said Table cannot be exact and perfect.   

To find the value of a decimal fraction in the parts of Coin.  
Suppose the number given to be 2 seconds, 4 thirds, 5 fourth's, and 7 fifths, and you desire to know the true value thereof in coin; set down your numbers, as in the example following, and mark your prime line, and then multiply the fraction by 240, the pence in one pound, and the numbers that arise by multiplication over the prime line are the sum of pence, the value of that fraction given, and the remainder on the right hand of the prime line is the fraction of one penny.  
Example.  
Here by multiplication of 2457 fifths by 240 pence, I find 5 pence is gone over the prime line, and there remains 82080: 100000 parts of one penny. Now to know the value of that fraction in farthings, multiply the same by 4, and so many as go over the prime line, are farthings, the rest is the fraction of a farthing.  
Example.   

Numeration in Decimals.  
If you have a number to be expressed in Decimals of money, or Coin sterling, learn first by the decimal Table how to express your Coin, from one penny unto one pound sterling, or from one farthing to one pound sterling, for which the Table going before was calculated. If you would know the manner how to calculate the said Table; divide 1 pound, adding 7 cyphers unto it, by your part you would know how to set forth in Decimals: as if you would know how a farthing will stand in Decimals; divide 1 pound with cyphers by 960, the number of farthings in one pound sterling, and the quotient will be the numbers in Decimals, signifying one farthing.  
Example:  
So that I find, that dividing of 1 pound by 960 farthings, the Quotient is 1 third, 0 fourth, 4 fifths, 1 sixth, and 6 sevenths: for if you should have proceeded, adding more Ciphers, the Quotient would have been always 6, because I see the number remaining to be the same it was at the last, that is 64. And although a farthing cannot be set out exact in Decimals, yet it will serve in Multiplication and Division: for in 10000 yards or else, it will not differ 1 penny, as shall appear afterwards by examples in their places.   

How to set out a penny in decimals.  
Divide 1 penny with Ciphers by 240, the number of pence in one pound sterling, and the quotient will be a penny in decimals.  
2. Example.  
Here seeing that after I find the first quotient 6, and the remainder 16, as before I cease Division, as needless any further, knowing it will produce 6 in the quotient infinitely, and therefore I put as many times 6 in the quotient, as I find expedient and needful, and 1 penny stands thus:  
 
And these and diverse other numbers will not be set exact in Decimals, but yet they will serve to great purpose and exactness in a multitude of questions, in saving an infinite labour in Reduction, and Multiplication and Division.   

How to break a pound into his exact parts.  
Set down 1 pound thus, 10; then take the tenth, which is one prime, or 2 shillings, which I note thus,  
 
Then take half of that prime or 2 shillings, saying, the one half of 10 is 5, or the one half of one prime is 5 seconds, or one shilling; then the one half of 5 seconds is 2 seconds, and 5 thirds, saying, the one half of 5 seconds, is 2 seconds, and 5 thirds, which is 6 pence: then half of 2 seconds, 5 seconds, is 1 second, 2 thirds, 5 fourth's, which doth represent 3 pence in Decimals. Again, one half of 1 second, 2 thirds, 5 fourth's, is 6 thirds, 2 fourth's, 5 fifths, representing 1 penny, halfpenny, or three half pence. Again, half of that number is 3125, or 3 thirds, 1 fourth, 2 fifths, 5 sixths; signifying three farthings in decimals; behold the work.  
Example.  
 
It is also very necessary to understand the proportional parts of a pound, for by them are many questions speedily wrought in Decimals, as shall appear in the examples of Multiplication and Division afterwards.   

How to express the value of any number in Decimals.  
Admit for example this number following, is to be expressed according to the computation of decimal Arithmetic, viz. 3785725 thirds: then for the expressing the signification of that number in the known parts of Coin, first, mark out your prime line, to distinguish the whole numbers from the fractions with a right down stroke with the pen, and then you shall find the numbers to stand thus 3785 pound, 7 primes, 2 seconds, and 5 thirds; which search in your decimal Table, and it doth signify 14 shillings, 6 pence; so that the whole number is 3785 pound, 14 shillings, 6 pence, and so of all numbers; for you shall understand, that every prime doth signify in value 2 shillings, every second 2 pence and 2 ∶ 5 parts of 1 penny, and every 5 thirds 1 penny, and 1 ∶ 5 of 1 penny: or else every prime is 1 ∶ 10 of one pound; every second 1 ∶ 100 part of one pound, and every third 1 ∶ 1000 part of one pound, etc. infinitely.   

How to remove a decimal number from one place to another.  
If you have a decimal number given: as for example, ●3 pence, which doth thus stand in decimals, 1 second, 2 thirds, 5 fourth's; than you desire to know how it will stand in the place of primes, pounds, or in the place of 10 l. or hundreds or thousands. remove it one place towards the left hand, and it is 1 prime, 2 seconds, 5 thirds, or in known parts of coin 2 shillings, 6 pence. Again, remove them one place more towards the left hand, and it will be 1 pound, 2 primes, 5 seconds, or 1 pound, 5 shillings. Again, remove one place more: and it is 12 pound, 19 shillings: Again, remove it one place more, and all your fractions are in whole numbers, and will signify 125 pound, etc.  
 
And this Rule is very necessary to be well and perfectly understood, for by it any price be given of a unite in decimals. you may speedily know what 100, or 1000, or 10000 will cost at that rate, only by adding of one, two, or more Ciphers.  
As for example, if one ell cost 6 shillings 3 pence, what will 100 else cost at that rate? first, s●t out your price in decimals thus, 3 primes, 1 second, 2 thirds, 5 fourth's, and adding of two Ciphers, because 100 hath 2 Ciphers, the sum will be 312500: and because your fractions were fourth's, cut off 4 figures and Ciphers towards the right hand, or mark your prime line, and you shall find, that 100 else will cost 31 pound, 5 shillings at that rate.  
1. Example.  
If the numbers of the price given will not be exactly set down in Decimals: as for example, at 7 pence, 3 farthings a yard, what will 100 yard's cost? Set down your price as near as may be, by your decimal Table, which is 322916 sevenths, add unto it two cyphers, makes 32291600; and because your fractions are sevenths, cut off 7 figures, and there will be 3 pound, 4 shillings, 7 pence.  
2. Example.  
And thus much shall suffice for Numeration in decimals, and I will now proceed unto the second Rule of Arithmetic, viz. Addition in Decimals.    

CHAP. II. Addition in Decimals of Coin.  
I If you have diverse several numbers given in decimals to be added together into one sum, place them in order every one right under his like denomination, or kind, Integers under Integers, Primes under Primes, Seconds under seconds, etc. Then begin your Addition at the right hand at the least Denomination first, and add them all according to the Rule of Addition, as if they were all whole numbers, always having a care to mark out your prime line, and the total of your Addition will show you the just value of those whole numbers and fractions.  
1. Example.  
  

CHAP. III. Subtraction in decimals.  
IF you have two numbers in Decimals, the one to be subtracted from the other, place them above one the other, as in Addition, the greater numbers in the upper part, and the smaller numbers right underneath, and then subtract them as if they were whole numbers, and note down the remayners each in their proper places, as in this example.  
1. Example.  
 
2. Example.  
  

CHAP. IV. Multiplication in decimals.  
IF you have any two numbers given to be multiplied in decimals, place your multiplicand uppermost, and your multiplier right underneath, as if the same were absolute whole numbers, and no fractions at all; and when your numbers are placed, mark how many fractions your two numbers doth contain, and note that number down, and multiply according to any of my former instructions in the first book; and when the product is gathered, cut off your prime line, just so many figures and cyphers, as your multiplicand and multiplier had fractions between them, and the work is ended.  
Example.  
If you will multiply 758325 thirds, by 3857 primes, I place first my numbers, and then I find my multiplicand to have 3 fractions, to wit, primes, seconds & thirds, and I find my multiplier to have one fraction, only primes, which makes 4 fractions, and so many figures I cut off from the product.  
Example.  
2. Example.  
If you will multiply 34 pound, 5 shillings 3 pence, by 16 pound, 6 shillings, 6 pence, set them in Decimals, 342625 fourth's, by 16325 thirds, and multiply them together, and cut from the product 7 figures to the right hand, and the product will be 559 pound, 6 shillings, 8 pence ob. almost.  
Example.  
3. Example.  
If you will multiply 758 Integers by 3 primes, 7 seconds, 5 thirds, which is by 7 shillings, 6 pence; place them as in the last example, and from the product cut off the 3 figures for the 3 fractions, and the total is 284 pound, 5 shillings, the sum that 758 else will cost at 7 shillings, 6 pence an ell, etc.  
Example.  
If you will multiply fractions by fractions in decimals; as to multiply 5 primes, 2 seconds, 6 thirds, 3 fourth's, by 7 primes, 2 seconds, 5 thirds; set them as before, and cut off 7 figures.  
4. Examples.  
 
Makes 7 s. 7 d. ob.  
If you will multiply in Decimals by 10, or by 100, or by 1000, etc. set down your numbers, and mark how many fractions there be in your multiplicand, and then add so many cyphers as your multiplier hath to the right hand, and cut off your prime line, and the work is ended, as in this example.  
Example.  

How to change any fraction given into decimals.  
Admit there be a quotient of a division, which is 358 pound, 126 ∶ 255 of one pound, which fraction you would turn into Demalls; add a cipher to your numerator of your fraction, makes 1260: but because your number will not be evenly divided by your denominator 255, therefore add more cyphers, and then divide the number by 255 makes 49411 fifths in Decimals to be joined with the whole numbers 35849411 fifths, and are now fit for multiplication and division in Decimals.  
5. Example.  
 
Admit there be a fraction to be set out in Decimals thus, it is required to know what 156 yards of cloth will cost at 196: 784 of a pound one yard? Add to 156, 2, 3, or more cyphers, and divide by the denominator 784, makes 25 seconds, by which multiply 156 yards, makes 39 pound.  
6. Example.  
 
7. Example.  
For the proof of this work, multiply 156 by 196, makes 30576; which divided by 784, makes 39 pound, as before.  
   

CHAP. V. Division in decimals.  
IF you will divide any number in Decimals, either whole numbers by fractions, or fractions by whole numbers, or whole numbers and fractions by whole numbers and fractions; set them down according to the Rules in decimals in the operations before going. As for example, a certain Merchant bought as much cloth as cost him 284 pound, 5 shillings, at 7 shillings, 6 pence an ell, the question is, how many else he had for his money? To do this, or any other the like question; divide your sum of money 284 pound, 5 shillings by 7 shillings, 6 pence, and the quotient will show you, what number of else, and parts of an ell, if any be, were bought for that money.  
1. Example.  

How to Divide the smaller number by the greater.  
If you will divide 34 pound, 6 shillings amongst 36 men: place your numbers, adding, 3, or 4, or 5 cyphers; and then divide by 36, makes 95271 fifths; or in Coin 19 shillings, 0 pence, ob. for every man's portion.  
2. Example.  
What is the quotient of 724 pound? Divided by 3 ∶ 4 of a unit, or 15 shillings? Answer: divide 724 by 75 seconds, makes 965 1 ∶ 3; for trial whereof multiply 965 1 ∶ 3 by 15 shillings, or 75 seconds, makes 724, as in the Example.  
2. Example.  
This last question is in effect no other but as the former: for if I shall say, a merchant buys Broad Cloth, costs him 724 pound at 15 shillings, or 3 ∶ 4 of a pound one yard, the question is, what number he had for his money, and by Division I find he had 965 yards, and one third part of a yard, as is proved in the example; and so dividing 724 by 3 ∶ 4, the quotient is 965, 1 ∶ 3  
3. Example.  
If you will divide the product of the second example in multiplication, which was 559●●53125 sevenths by 16325 for the proof of that work, which ought to bring out the multiplicand 34●2625; or rather if you will divide 559 pound, 6 shillings, 8 pence, ob. almost, by 16 pound, 6 shillings, 6 pence, the quotient will be 34 pound, 5 shillings, 3 pence.  
Example.   

How to find the Prime line in any Division decimal, or to find the true denomination of of the Quotient.  
In any division decimal, always mark out your prime line in your dividend with a straight do une line with the pen, than set your decimal fractions in primes, seconds, thirds, fourth's, &c. beyond the line; also do the like in your divisor, and then mark how often you may remove your divisor, that the whole numbers of your divisor may stand under the whole numbers of your dividend, and so many figures shall your quotiont have in whole numbers, the rest are to be marked with pricks in the quotient for primes, seconds thirds, etc.  
If you will divide 93861375 fifths by 34 pound 35 seconds, then place them with pricks as in the example following. I find having placed my divisor underneath my dividend, that I may remove my divisor twice under the whole numbers of my dividend, and therefore I conclude, the first two numbers of my quotient will be whole numbers, which I mark from the rest of the numbers in the quotient with a line, and then dividing according to the former instruction, you shall find the quotient will be 27 pound, 3 primes, 2 seconds, and 5 thirds.  
Example.  
2. Example.  
If you would divide 15554 pound, 2 primes, 5 seconds, or 5 shillings, by 45 pound? Place them as in the Example following, and you shall find, that there will be in the quotient 3 figures in whole numbers, and the rest will be primes and seconds, so that dividing of 15554 pound, 5 primes by 45 pound, the quotient is 345 pound, 13 shillings.  
Example.  
3. Example.  
If the greatest number of your Divisor be primes, than the figures of your whole numbers in the quotient will be, once greater in value; then the times you can remove your Divisor, as if you would divide 241 pound, 5 primes, by 7 primes: then whereas you can remove your divisor by two times under the whole numbers 241, yet you shall have 3 numbers in the quotient in whole numbers, because your first figure of your divisor is primes; so that in dividing 241 pound, 5 primes by 7 primes, I find the quotient will be 345 pound, or integers; and so many yards, at 14 shillings a yard, which is 7 primes, will 241 pound, 10 shillings buy.  
Example.  
4. Example.  
If you will divide 16 pound, 875 thirds, which is 16 pound, 17 shillings, 6 pence by 375 thirds, which is 7 shillings, 6 pence, or which is all one, imagine there is as much cloth of 7 shillings, 6 pence a yard, as cost 16 pound, 17 shillings, 6 pence; the question is, how many yards was bought for that money? placing your numbers as in the example following, I find 45 yards is the answer to the question.  
Example.  
5. Example.  
If you will divide whole numbers and fractions by whole numbers, place the whole numbers and fractions uppermost, and mark out your prime line, and then set your divisor underneath, and the lowest figure in value of your divisor, will show you what is the denomination of the first figure of your quotient. As if you will divide 13 pound 95 seconds by 45; or which is all one if you shall say; if 45 pieces of figs cost me 16 pound, 19 shillings, what did one piece cost? Divide 1395 seconds by 45, makes 31 seconds, or 6 shillings, 2 pence, 2 ∶ 5 of a penny for the price of one piece. And in this sort the price of any number of yards, else, or pounds being given in dividing it by the number of yards, else, or pounds, the quotient will be the price of one; and by this Rule you save a labour of Reduction, always dividing the price by the number given, the greater by the lesser, or the lesser by the greater.  
Example.  
6. Example.  
If 456 else of cloth cost 575 pound, 7 primes, what will one ell cost? Divide 575 pound, 7 primes by 456 else, makes 1 pound 2625 fourth's, or in Coin, 1 pound, 5 shillings, 3 pence for the price of one ell.  
  

Reduction in Decimals.  
If you will reduce 75 pound, 12 shillings, 9 pence into Decimals, enter your Decimal Table, and for 12 shillings find 6 primes; then look for 9 pence, and you shall find 375 fourth's; so the total is 75 pound, 6375 fourth's and are now fit and apt for any decimal operation.  
If you multiply or divide 84 pound, 13 shillings, 6 pence, by 17 pound, 3 shillings, reduce them into Decimals by the Table, makes for 84 pound, 13 shillings, 6 pence 84 ∶ 675, and for 17 pound, 3 shillings, 17 ∶ 15, and are now fit to be multiplied or divided one by the other.  
If you will reduce 189 ∶ 756 parts of one pound into Decimals: divide 189, adding 3 cyphers to it by 756 makes 25 seconds for that fraction in decimals: and now for example, If 158 else of cloth & 189 ∶ 756 parts of an ell cost 79 pound, 2 shillings, 6 pence, what will 640 else cost at that rate? Now according to vulgar Arithmetic, either I must reduce 158 else 189 ∶ 756 parts of an ell into 756 parts, or otherwise I must Reduce the fraction into his least terms, makes 1 ∶ 4; then I multiply or reduce 158 else into fourth's, makes 633 fourth's for the first number in the Golden Rule. Secondly, reduce 79 pound, 2 shilling, 6 pence into pence, makes 18990 pence for the second number; then put 640 else into fourth's, makes 2560 fourth's; than multiply ●8990 by 2560, makes 48614400; which divide by 633, makes 320 pound.  
Example.  
The same example wrought by decimals.  
If 158 else 1 ∶ 4 ell cost 79 pound, 2 shilling 6 pence, what will 640 else cost at that rate? Place them in Decimals thus: If 15825 seconds cost 79125 thirds, what 640 else? Multiply 79125 thirds by 640, makes 50640000; which divide by 15825, makes 320 pound the quotient.  
Example.  
Or otherwise.  
Divide 15825 by 79125, adding one cipher, makes 2 primes for the Quotient; wherefore I conclude, that one half of 640 pound, which is 320 pound, is the answer to the question demanded. Also divide 7912● by 15825, the quotient is 5 primes; by which multiply 640 pound, makes 320 pound for the answer to the question as before.  
If a Phillips Dollar be worth 4 shillings, 8 pence, what are 465342 Dollars worth in sterling money? Answer multiply 465342 by primes, which is 4 shillings, and take the sixth part of that product, and add into it, makes 1085798 primes for the answer.  
Or otherwise, multiply by 2 primes, and 1 ∶ 3 of a prime, because 8 pence is 1 ∶ 3 of a prime, and both ways will produce the same answer.  
Example.  
If a common Dollar be worth 4 shillings, and a Princes Dollar be worth 4 shillings, 6 pence, how many Prince's Dollars will pay for 7584 common Dollars? Multiply 7584 by 4 shillings, and divide by 4 shillings, 6 pence, makes 6741 Dollars, and 7 seconds, and 5 thirds will remain, which is 18 pence; so that I conclude, 6741 Princes Dollars at 4 shillings, 6 pence a piece will pay for 7584 common Dollars, and there will remain 18 pence.  
Example.  
In 654 pound, how many Dollars of 3 shillings a piece? Add two Ciphers to 654, makes 65400, because 3 shillings hath 2 fractions in Decimals, viz. primes and seconds, which is 1 prime and 5 seconds, by which divide 65400, makes 4360 Dollars at 3 shillings a piece.  
Example.  
In 756 pound how many Dollars of 3 shillings, 9 pence a piece? Add 4 Ciphers to 756, makes 7560000; which divide by 1875, which is 3 shillings, 9 pence in Decimals, makes 4032 Dollars. Behold the example following▪  
Example.  
If I do sell 346 yards of Velvet for 298 pound, 8 shillings, 6 pence, how do I sell one yard? Answer: divide the price by the quantity of yards in decimals, makes 8625 fourth's, or in Coin 17 shillings, 3 pence for the price of one yard.  
Example.  
Makes 17 s. 3 d. a yard.  
A Merchant would buy several sorts of Spices of several prices, to wit, of 3 shillings a pound of 2 shillings, of 2 shillings 3 pence, of 1 shillings 7 pence, and of 2 shillings, 2 pence a pound, and would have of each a like quantity; for 324 pound, the question is, how many pound he must have of each? First, add all the prices into one sum, makes 11 shillings, by which divide 324 pound, makes 584 pound, 1 ∶ 11 of a pound; and so many pound must he have of each sort.  
A Goldsmith sent his servant to the Tower of London, to fetch him 415 pound, 18 shillings, 9 pence in pieces of 6 pence, of 4 pence, of 3 pence, of 2 pence, of 1 penny, and of one half penny, and bade him bring of each sort a like quantity: First, add all your Coin, makes 16 pence half penny, which in Decimals is 6875 fifths by which divide 4157375 fourth's, makes 6050 pieces of each sort.  
Example.   

Rules of Practice in decimals.  
Set your price given in the decimal Table of a unite, be it yard, ell, piece, or pound, and by the price given, multiply the number of yard's else, pieces, or pounds, and the product will be the sum that you seek, if you do but mark out the prime line, as shall appear by examples following.  
1. Example.  
If one pound weight of small Ginger cost 7 pence halfpenny, what will 112 pound weight cost? Find for 7 pence halfpenny 3125 fifths, which multiply by 112 pound, makes 350000; from which cut off five figures to the right hand by the prime line, and the sum is 3 pound, 5 primes, or 3 pound, 10 shillings, because your multiplicand hath 5 fractions.  
Example.   

How to find the price of any unite in any place of 10, or 100, or 1000, the price of one being given.  
If the price of a unite be given at any rate, and from thence you desire to know, what 10, or 100, or 1000, or 10000 will cost at that rate: or otherwise, if you desire to know, if you do gain any rate desired by the pound, and would know at what rate it will be in the 100 pound, or upon exchange from place to place, the exchange of one pound being given, you desire to know, what 100 pound will amount unto? Place your rate or gains given in Decimalis by help of the Table, and then adding of one, two, three, or more Ciphers, cutting off your prime line, you shall know your desire, marking the denominations of your fractions, if the least to the left hand be primes, seconds, thirds, fourth's, fifths, cutting off your prime line so many figures from the right hand.  
2. Example.  
If one pound sterling be 1 pound, 14 shillings, 3 pence Flemish, what is 100 pound sterling worth? Place 1 pound, 14 shillings, 3 pence in decimals, makes 17125 fourth's: then because 100 pound hath 2 Ciphers, makes 1712500: then cutting off 4 figures to the right hand, you shall find 171 pound, 5 shillings for 100 pound sterling, to make as appeareth before.  
If one ell of Cambric cost 7 shillings, 6 pence, ● farthings, what will 100 else cost at that rate? Place 7 shillings, 6 pence, 3 farthings in Decimals, makes 378125 fixths, and adding two Ciphers for 100, makes 37812500: from which cut off 6 figures to the right hand, makes 37 pound, 16 shillings 3 pence for the sum that 100 else will cost.  
 
Makes 37 l. 16 s. 3 d.  
If one pound or piece cost 1 pound, 2 shillings, 3 pence, what will 1000 pieces cost? Set 1 d. 2 s. three pence, in decimals makes 11125 fourth's: to the which add 3 Ciphers, because 1000 hath 3 Ciphers, and from the total cut off 4 figures, makes 1112 pound, 10 shillings, as is in the 4 example above  
If one ell of Holland cost 3 shillings, 3 pence, what will 343 else cost? Multiply 343 by 3 shillings, 3 pence in decimals, which is 1625 fourth's, makes 55 pound, 14 shillings, 9 pence.  
 
If one yard of Velvet cost 15 shillings, 6 pence, what will 972 yard's cost? Find for 15 shillings 75 seconds; then for 6 pence find 25 thirds, total is 775 thirds; by which multiply 972, makes 753 pound, 6 shillings, as above in the sixth Example.  
If one yard of Velvet cost 17 s. 7 d. 3 q. what will 857 yard's cost? First, find 17 ●. to be 85 seconds; then 7 d. 3 q. makes 322916, total is 8822916; which multiply by 857, makes 756 l. 2 s. 5 d. 3 q.  
 
If one Dollar be worth 4 shillings, 9 pence what are 758 Dollars worth in sterling money? Multiply 4 shillings, 9 pence, which is 2375 fourth's by 758, makes 180 pound, 6 pence, as in the eighth example above.   

The price of any number of yards, else, pieces, or pounds given to find the price of a unite.  
If the price of any number of yards, else, pieces, or pounds be given, set them down in Decimals, adding one, two, or more Ciphers, if need require, and divide that sum, or price by the number of the yards, else, pounds, or pieces, and the quotient is the price of a unite in whole numbers, primes, seconds, and thirds, without reduction, as shall appear by examples following: and in this manner you may know what sum of money was lent, if the principal and interest be given at any rate in the hundred; or you may know if the rate of one pound exchange be given for any place, you may know the value of 100 of that Coin in that money given; and by this Rule is to be abbreviated almost all operations of Arithmetic, by finding the value of a unite in any place desired.  
If ●42 else of cloth cost 22 pound 4 pence halfpenny, what cost one ell at that rate? Divide 2201875 fifths by 542, makes 40625 sixths, or in Coin 9 pence 3 farthings for the price one ell cost.  
1. Example.  
If 345 pound gain 76 pound, 12 shillings, what doth one pound gain? Divide 76600000 by 345 pound, makes 222028 sixth, or in Coin, makes 4 shillings, 5 pence half penny almost, that 1 pound doth gain as in the example following.  
2. Example.  
If 756 pound, 3 quarters, 24 pound of sugar cost 4421 pound 12 shillings, what did one pound weight cost, accounting 112 pound to the hundred? Reduce 756 pound 3 quarters, 24 pound into pounds subtle, accounting 112 pound to the hundred, makes 84780 pound● then divide 4421 pound, 12 shillings by 84780, makes 5215 fifths, or 12 pence, halfpenny one pound.  
3. Example.  
If I sell 1000 pieces of Cambric for 700 pound, how do I sell one piece? Divide 1000 by 100, makes 1 pound, 42857 fifths, 1 pound, 8 shillings, 6 pence, 3 farthings, as in the Example following.  
4. Example.  
If one pound starling be 1 pound, 14 shillings, 3 pence Flemish, what is one pound Flemish worth: Divide one pound with Ciphers by 17125, makes 11 shillings, 8 pence, 1 farthing almost.  
5. Example.  
If 1 l. sterling be 1 l. 14 s. 7 d. ob. Flemish, what is 100 l. Flemish worth in sterling money? Divide 100 by 173125 fifths, which is 1 l. 14 s. 7 d. ob. in Decimals, makes 57 l. 15 s. 3 d.  
6. Example.   

The Golden Rule in decimals.  
If the number given be pounds, shillings and pence, set them out in Decimals, and also your number of yards, else, pieces, pounds or any other numbers, set them out also in Decimals, and then without reduction multiply the third number by the second, and divide by the first, according to the instructions of multiplication and Division in the former part of this book, and the votient will be the third number sought.  
1. Example.  
If 34 else of Canvas cost 1 pound, 4 shillings, what will 756 else cost at that rate? Multiply 756 by 1 pound, 2 primes, makes 9072 primes; which divided by 34, adding cyphers, makes 266823 fourth, or in Coin 26 pound, 13 shillings, 8 pence.  
Example.  
If 112 pound of Indigo cost 34 pound, 17 shillings, what cost 789 pound, subtle accounting 100 pound to the hundred? Multiply 3485 seconds by 789, makes 27496 pound, 65 seconds; which divided by 112 pound, makes 245 pound, 5058 fourth's, or 10 shillings, 1 penny farthing.  
Example.  
If 981 else of Cloth cost 94 pound, 13 shillings, 6 pence, what cost 2943 else at that rate? Divide the third number by the first, and by the quotient multiply the second, and the product will be the answer sought.  
 
If 112 pound of Sugar cost 5 pound, 3 shillings, 9 pence, how many pounds will 124 pound buy at that rate? Divide 51875 fourth's by 112 pound, to find the price of 1 pound, makes 46316, sixths, or in Coin 1● d. 1 ∶ 10 of a penny almost for the price that one pound cost Secondly, divide 124 pound by the price of one pound, viz. by by 46316 sixths, makes 26773 primes, and so many pound he shall have for 124 pound.  
If one yard Broad Cloth cost 16 shillings, 9 pence, how many yards shall 56 pound buy at that rate? Divide 56 pound by 16 shillings, 9 pence, the price of one yard, makes 66 yards, 9 ∶ 10 almost.  
Example.  
If 7 yard's 1 ∶ 2 of cloth cost 9 shillings, what will 8 yard's 1 ∶ 3 of a yard cost? Multiply 9 shillings, or 45 seconds by 8 1 ∶ 3, makes 375; which divide by 7 yards 1: ●, or by 75 primes, makes 5 primes, or 10 shillings.  
Example.  
If 5 yard's 1 ∶ 2 cost 4 shillings, 8 pence, 1 ∶ 4 of a penny, or 56, 1 ∶ 4, what will 30 yard's cost at that rate? set your 56 pence 1 ∶ 4 in Decimals, makes 5625 seconds, which multiply by 30, makes 168750 seconds; which divided by 5 yards on half, or 55 primes, makes 306 pence 8 ∶ 10 of one penny for the price of 30 yards, as in the example following.  
Example.  
If 34 else 3 ∶ 4 of Holland cost 3 pound, 6 shillings, 1 penny, half penny, what will 956 else 1 ∶ 2 cost at that rate? Multiply 3 pound, 6 shillings, 1 penny, half penny, which is 33625 fourth's by 7565 prime●, makes 254373125; which divided by 34 else, 3 ∶ 4, or by 3475, makes 73200 thirds or 73 pound, 4 shillings.  
Example.  
If 346 pound, 10 shillings gain 32 pound 8 shillings, what will 75 pound gain at that rate? First, multiply 324 primes by 75 makes 24300 prime; which divided by 3465 primes, makes 70129 fourth's, or 7 pound, 3 pence for the answer.  
Example.   

The same Question wrought a second way.  
Divide 324 primes, by 3465 primes, adding 5 cyphers, and the quotient will be 935 fourth's; which multiply by 75, makes 7 l. 0125 fourth's, which doth not want one farthing of the former sum.   

The same Question wrought another way.  
Divide 75 pound, adding 5 Ciphers by 346 pound, 5 primes, and the quotient will be 21645 fifths; which multiply by 324 primes, makes 7012980; from which abate 6 figures to the right hand, because of your 6 functions and the remainder will be 7 pound 01●9 fourth's, &c. as before. And in this manner you may work any question in the Rule of Three, three several manner of ways, and prove the work one by the other.  
If 12 shillings do buy 74 pound of Ginger, how much shall I have for 100 pound? Divide 7400, which is the product of 74 by 100, by 12 shillings, or 6 primes, and the quotient will be 12333 pound, 1 ∶ 3, and so much Ginger shall I have for 100 pound at that rate. Or otherwise, divide 100 pound by 6 primes, makes 166 2 ∶ 3, which multiply by 74, makes 12333 pound, 1 ∶ 3, as before.   

Brief Rules how to abbreviate your work in the Golden Rule, by marking the proportion's bebetweene the numbers given.  
When as any question is propounded in the Golden Rule, mark what proportion is between the first and second numbers, or between the first and third numbers, or between the third and second; for if you espy them in any proportion, the question demanded is very speedily answered, upon the first sight; or yet if you see them not exactly to be even proportionals, yet you may subtract the first from the third, once twice or three times, or more and so often take the middle number towards the answer to the question, and then you need not to multiply by your whole third number, as you shall see by examples following.  
1. Example.  
If 34 else cost 2 pound, 4 shillings, 1 penny, what will 340 else cost? here comparing the first & third numbers, one with another, I find the third doth contain the first 10 times, wherefore I multiply 2 pound 4 shillings, 1 penny by 10, and the total is 22 pound, 10 pence, the Answer.  
2. Example.  
If 82 else of Cloth cost 4 pound, 2 shillings, what will 324 else cost at that rate? Here I find 4 pound, 2 shilling in Decimals to be one half of 82, but it standeth one room less in value then 82 doth, so I conclude, that half of 324 one room less is 16 pound, 2 primes, or 4 shillings, the Answer.  
3. Example.  
If 345 else of Holland cost 34 pound, 10 shillings, what will 789 else cost at that rate? Set down 34 pound, 10 shillings in decimals, makes 34 pound, 5 primes, which is the first number placed but one room lower; therefore I say, if 345 else cost 34 pound, 5 primes one room more to the right hand, than the third number also will cost 78 pound, 9 primes one room more to the right hand, which is 78 pound, 18 shillings.  
4. Example.  
If 12 else of Cloth cost 2 shillings, four pence, 4 ∶ 5 of one penny, what will 356 else cost? place 2 shillings, 4 pence, 4 ∶ 5 in Decimals, makes 1 prime, 2 seconds, or 12 seconds, which is the same number: but it stands two rooms lower; therefore I conclude, that 356 else cost the same numbers two rooms lower, which is 3 pound, 11 shillings, 2 pence, 2 ∶ 5 of one penny.  
 
5. Example.  
If 130 else of cloth cost 26 pound, what will 3759 else cost at that rate? I find the second number to be twice the first, but it stands one place nearer the right hand; therefore I conclude, that the third number will cost twice as much in his lower room, which is 751 pound, 16 shillings.  
If 130 cost 26 pound, what cost 3759.  
 
6. Example.  
If 75 else one half co●● 7 pound, 11 shillings, what will 32812 seconds cost? Set them down in decimals, and you shall find them to stand thus, 755 primes for the first number, and 755 seconds for the second number, which is the same one room nearer the right hand: so I conclude, that the third number will cost 3285 seconds, which is 32 pound 17 shillings.  
Example.  
1. Example.  
If 356 else of Canvas cost 38 pound, 12 shillings, 1 penny, what will 740 else cost at that rate? First, divide 740 by 356, the quotient will be 2 and therefore I take twice the price given for that quotient, and then whereas before I should have multiplied 38 pound, 12 shillings, 1 penny by 740, I shall need to multiply it but by 28 the remaynor, and divide it by 356, makes 30368 fourth's, to be a●ded to the former sum, and the total will be as in the example following.  
Example.  
 
Here in this last example, I multiply 38 pound, 6 primes by 28, omitting the penny, not setting it out in decimals, and the product is 10809 primes: then multiply 1 penny by 28, makes 28 pence, which is one prime, 166 fourth's, and the total was 1080 pound, 9116 fourth's, as in the example: and in this manner you may save a great labour in multiplying your number of pounds and shillings first, and then multiply your pence by themselves, and add into the rest in primes, seconds, etc.  
2. Example.  
If 17 else of Holland Cloth cost 3 pound 2 shillings, 5 pence, what will 515 else cost at that rate? Divide 515 by 17, makes 30, by which multiply 3 pound, 2 shillings, 5 pence, makes 93 pound, 12 shillings, 6 pence, than the remainder of your division will be 5 else, by which 5 multiply 3 pound, 2 shillings, 5 pence, makes 15 l. 10 shillings, 1 penny, or in Decimals 1550416 fifths; which divided by 17, makes 912 thirds, or 18 shillings, 3 pence almost; which added to 93 pound, 12 shillings, 6 pence, makes the answer to be 94 pound, 10 shillings, 9 pence: and so here in stead of multiplying 3120833 sixths by 515, and dividing by 17 I have saved more than half the work.  
Example.  
3. Example.  
If 7 pound buy 100 pound weight of Sugar, how many pound weight will 156 buy me at that rate? Divide 156 by 7, makes 22, 2 ∶ 7; by which multiply 100, makes 2228 pound, 4 ∶ 7  
4. Example.  
If 356 pieces of Calicoes cost 300 pound, 15 shillings, how much will 917 pieces cost at that rate? Divide 917 by 356, makes in the quotient 2; therefore take the price given twice, and there will remain after your division 205; by which multiply 30075 seconds, makes 6165375 seconds; which divided by 356, makes 173 pound, 18 seconds, or 173 pound, 3 shillings 8 pence, to be added to the former sum 601 pound, 10 shillings, makes 774 pound 13 shillings, 8 pence, for the Question.   

The same question wrought without Reduction in Decimals.  
If 356 cost 30075 seconds, what 917? Multiply 30075 2. by 917, makes 27578775 seconds; which divide by 356, makes 77468 seconds, or 774 pound, 13 shillings, 8 pence, as before the proof.  
Example.  
5. Example.  
If 179 pound of Indigo cost 60 pound 13 shillings, 5 pence, what will 716 pound cost at the same rate? divide 716 by 179, makes 4 in the quotient, and nothing will remain: wherefore I conclude, that 4 times 60l. 13 ●. 5 d. which is 242l. 13 s. 8 d. and is the answer to the question demanded.  
6. Example.  
If 36 pound of Cloves cost 11 pound, 6 shillings, how many pound shall I have for 354l. Divide 113 primes by 36, makes 31388 fifths; which multiply by 354, cutting of figures for the 5 fractions, makes 111 pound, 11352 fifths, or 3 pound, 2 shillings 2 pence, 3 farthings for the answer.   

Fellowship in Decimals.  
To work the Rule of Fellowship in Decimals, gather the whole number of all the moneys disbursed into one sum, and then divide the money gained or lost by that sum, and multiply that quotient so found by each several partner's stock disbursed, and the products will be each several man's gain or loss.  
1. Example.  
Four Merchants made a company: A. put in 60 pound, B. 80 pound, C. 120 pound, D. 140 pound, and they gained 72 pound; the Question is, what part each Merchant must have of the gains? First the total sum of all their moneys disbursed was 400 pound, wherefore according to the rule I divide 72 pound, adding Ciphers unto it by 400, and the quotient is 1 prime, 8 seconds; by which I multiply each several man's Stock disbursed, and I find, A. shall have 10 pound, 16 shillings; B. 14 pound 8 shillings; C. 21 pound 12 shillings, and D. 25 pound, 4 shillings; total is 72 pound, as in the example.  
Example.  
2. Example.  
Four Merchants made a company, and set forth a ship to sea, which cost them 3616 pound, 13 shillings; A. must pay 1 ∶ 3 of the money; B. 1 ∶ 4, C. 1 ∶ 5, D. 1 ∶ 6, the question is, what each man must pay of the said sum? Take a a number wherein the like parts may be had which in the former book of vulgar Arithmetic, I find to be 60, whereof 1 ∶ 3 is 20 and 1 ∶ 4 is 15, and 1 ∶ 5 is 12▪ and 1 ∶ 6 is 10, the total is but 57: wherefore I divide 361665 by 57, and the quotient is 63-45 seconds; which I multiply by 20, and I find A shall pay 1269 pound; then I multiply by 15, and B. shall pay 95175 second; and by 12, and C. shall pay 7614 primes; and by 10, and D. shall pay 6345 primes, the total is 361665 seconds, the proof of the work.  
Example.  
3. Example.  
● Three Merchants made a Company: A. put in 566 primes; B put in 398 primes; C. put in 1204 primes, and they gained 58 pound, 16 shillings, or 58 pound, 8 primes, what must each man have of the gains; first, the total disbursed is 216 pound, 4 primes; by the which I divide 58 pound, 8 primes, & the quotient is 27197 fifths for one pound gains; which I multiply by each several man's money disbursed, and I find A. shall have 15 pound, 7 shillings▪ 10 penny half penny; B. 10 pound, 14 shillings, 3 pence, 3 farthings; C. shall have 32 pound, 13 shillings, 9 pence, 3 farthings, the total is 58 pound, 16 shillings, the proof.  
Example.  
4. Example.  
Three Captains agree together to divide a spoil or booty, which they had taken, containing 7851 li: in this sort, A. is to have 1 ∶ 2; B. 1 ∶ 3; C. 1 ∶ 4; the question is, what each man's share shall be? Find a number which hath such parts in it, viz. 12, whereof 1 ∶ 2 is 6, 1 ∶ 3 is 4, and 1 ∶ 4 is 3, which in one sum makes 13; therefore divide 7851, adding cyphers to it by 13, and the quotient will be 603 pound, 92307 fifths; which multiply by 6, 4, and 3, and you shall find, A. shall have 3623 pound, 5384▪ fifth's; B. shall have 2415 pound, 69228 fifths; C. shall have 1811 pound, 76921 fifths; the Totall is 7850 pound, 99991 fifths, which doth want but 1 fourth of 7851 pound, which in value is but 3: 125 parts of 1 penny, and this example is to be wrought without the Golden Rule. Behold the proof of the work.  
Example.   

The same example wrought another way.  
After you have divided 7851 pound by 13, find in your decimal Table what the quotient is in Coin, makes 603 pound, 18 shillings, 5 pence, ob. which multiply by 6 4, and 3, and their total in one sum is the answer, as before.  
 
 
These three several products added into one sum, makes 7850 l. 19 s. 11 d. wanting but one penny in the whole sum, which is the defect of the Decimals, which cannot be exactly set out in coin, but it will serve to answer a question of one million with one penny error at the most.  
5. Example.  
Three men made a stock together, and they gained 244 pound, 8 shillings: A. put in 315 pound 7 months, B. 408 pound 10 months, C. 500 pound 3 months; now the question is, what each man must have of the gains? First, multiply each man's stock by his time, and gather all the totals into one sum, and they make 7785; by which divide your gains, 244 pound, 4 primes, and the quotient will be 31393 sixths; which multiply by the several products of each man's money and time, and the total of each several product is the sum desired for each man's part of the gain.  
Example.   

Position in Decimals.  
The Merchants bought a parcel of Linen Cloth cost them 757 pound, 17 shillings whereof A. must pay 1 ∶ 4; B. 1 ∶ 5; C. 1 ∶ 8; what must each man pay of this sum? I take 20 for a number, wherein I can have those parts, viz. 1 ∶ 4 of 20 is 5, and 1 ∶ 5 of 20 is 4, and 1 ∶ 8 of 20 is 2 pound 5 primes, or 2 one half, their total is 11 pound, 5 primes, or 11 1 ∶ 2; by which I divide 757 pound, 85 seconds, and the quotient is 65 l. 9 primes, which I multiply by 5 for A. makes 329 pound, 10 shillings; B. 263 pound, 12 shillings; C. 164 pound, 15 shillings: the total is 757 pound, 85 seconds.  
1. Example.  
2. Example.  
A Ship-carpenter bought 300 timber trees of a Gentleman, and was to pay for the first 100 a sum of money unknown, for the second twice as much as for the first 100, and for the third 100 of trees he was to pay thrice as much as he paid for the first, and the whole 100LS of trees cost him 7●4 pound, 12 shillings, the question is, what each hundred cost him severally? To work this question, or any other of like nature, suppose a unite, or one pound for the first 100, than he must pay 2 pound for the second 100, which is twice as much, and then also he must pay 3 pound for the third hundred, which is three times as much as the first: but yet 1 pound, 2 pound, and 3 pound makes but 6 pound, and it should be 724 pound, 12 shillings; so that now whereas in the former Book I taught you to resort to the Golden Rule for the answer, saying; If 6 pound cóme of my position 1 pound, of what comes 724 pound, 12 shillings. Now always supposing a unite▪ for your first number, you shall save multiplication; and so dividing of 724 pound, 6 primes by 6, I find the first 100 of Trees cost him 120 pound, 15 shillings, 4 pence; and the second 100 cost him 241 pound, 10 shillings 8 pence; and the third 100 cost him 362 pound, 5 shillings; the total makes 724 pound, 12 shillings, behold the work.  
Example.  
3. Example.  
Four Merchants consent to build a ship, cost them 541 pound, 16 shillings, whereof A. must pay a certain sum of money unknown; B. must pay twice as much as A; C. must pay twice as much as B; and D. must pay as much as all the other three, viz. as A. B. and C.; now the question is, what each man must pay of this sum. I suppose A. must pay 1 pound, than B. must pay 2 pound, which is twice as much as A. doth pay; and C. must pay 6 pound, which is thrice as much as B. doth pay; and then D. must pay 9 pound, which is as much as all the other three do pay: but their total is but 18 pound, and it should be 541 pound, 16 shillings: wherefore I divide 541 pound, 8 primes by 18, and the quotient is 30 pound, 1 prime, or 2 shillings for the first part. Then B. must pay 60 pound, 4 shillings? C. 180 pound, 12 shillings; and D. 270 pound, 18 shillings, their total makes 541 pound, 8 primes; behold the work.  
Example.  
4. Example.  
A Cistern of water containing 600 gallons is filled with water, and hath 4 several Cocks to empty the same, whereof if they be all set open at once, the Cistern will be empty in 24 hours: now the second Cock will avoid twice as much as the first Cock in 24 hours, and the third will avoid three times as much as the first, and the fourth Cock 5 times as much as the first; the question is, how many gallons each Cock doth avoid in 24 hours of the said 600 gallons?  
I suppose the first Cock will avoid one gallon, than the second must avoid 2, and the third 3, and the fourth Cock 5: but yet they are but a 11 gallons, and they should be 600 gallons: wherefore dividing of 600 by 11, the quotient is 54 gallons, and 6 ∶ 11 of a gallon for the first Cock. Behold the work in the example following.  
Example.  
  

Of Gain and Loss in Decimals.  
If a Broad Cloth 28 yards long be sold for 14 shillings a yard, and the seller doth gain 10▪ pound in the 100 ready money, what cost that broad Cloth? First, by Practice find the price of the 28 yards, at 14 shillings a yard, makes 19 pound, 6 primes, or 19 pound, 12 shillings; divide 19 pound 6 primes by 110 pound, makes 17 pound, 81818 fifths, or in Coin, 17 pound, 16 shillings, 4 pence, 3 farthings.  
1. Example.  
Secondly, if 28 yards cost 17 pound, 81818 fifths, what did one yard cost at that rate? Divide 17 pound, 81818 fifths by 28 yards, and the quotient will be 63636, or in Coin, 12 shillings, 8 pence, 3 farthings for the price that one yard cost.  
Example.  
Thirdly, for the proof of this work, say, If one yard cost 63636 fifths, how may I sell it to gain 10 pound in the hundred ready money? Take the tenth part of 63636 fifths, makes 63636 sixths; which added into one Totall, makes 69999 fifths, which doth want but one fifth of 7 prime●, or 14 shillings, which proves all the former works to be true.  
Example.  
2 Example.  
A Merchant doth deliver money at interest for 9 months after the rate of 12 pound in the hundred for 12 months simple interest, and at the end of 9 months doth receive for interest 87 pound; the question is, what was the sum lent? Answer: because the interest of 9 months at 12 pound in the hundred is 9 pound, divide 8700000 by 9 pound, and the quotient is 966 pound, 6666 fourth's, or 966 pound, 13 shillings, 4 pence, the sum lent.  
Example.  
3. Example.  
If 13 pieces of Canvas cost 17 pound, 12 shillings, how may I sell them to gain 8 pound in the hundred? Multiply 17 pound 6 primes by 8, adding two cyphers, makes for 19 pound, 8 thirds, or 19 pound, 2 pence almost.  
The proof of the former example, if 17 pound, 12 shillings, gain 1 pound, 8 shillings, 2d. what will 100 pound gain at that rate? Multiply 1 pound, 8 shillings, 2 pence; or in Decimals, 1 pound, 408 thirds by 100, makes 140 pound, 800 thirds; which divide by 17 pound, 6 primes, makes 8 li. for the rate that 100 pound will gain, which shows the former work to be truly wrought.  
Example.  
4. Example.  
If in one ell of cloth sold for 3 shillings, there be gained after the rate of 12 pound in the hundred for 12 months, how should that ell have been sold to gain 17 pound in the hundred for 12 months? Multiply 17 pound by 3 shillings, which is 1 prime, 5 seconds, and divide the product by 12, makes 2125 fourth's, or in coin 4 shillings 3 pence, and so much must it have been sold for to gain 17 pound in the hundred.  
Example.  
Secondly, if 3 shillings give 12 pound, what will 4 shillings, 3 pence give? Multiply 2125 fourth's by 12, and divide by 15 seconds, and the quotient is 17 pound, the proof of the last example.  
Example.  
5. Example.  
A Merchant sold 24 Clothes, which cost him 342 pound, wherein he lost after the rate of 10 pound in the hundred, and took in exchange 560 pieces of Raisins at 24 shillings the piece, wherein he gained 10 pound in the hundred ready money; now the question is, what his gain or loss was, and what sum of money he was to pay for the Raisins? First, 560 pieces of Raisins at 24 shillings a piece, is 672 pound; from which subtract 342 pound, lea●es 330 pound to pay for the Raisins. Secondly, 672 pound, at 10 pound in the hundred, is 67 pound, 4 shillings for his gains by the Raisins. Thirdly, 342 pound less, 10 in the hundred, is 34 pound, 4 shillings, to be deducted from 342 pound; and then take 34 pound, 4 shillings, from 67 pound 4 shillings, leaves his gains more than his loss to be 33 pound.  
Example.  
6. Example.  
A Merchant receiveth for principal and interest 352 pound, wherein he gained 9 pound in the hundred for one year; now the question is, what was the sum of money lent? Divide 35200000 by 109 pound, makes 322 pound, 9357 fourth's, o● 322 pound, 18 shillings, 8 pence, halfpenny for the sum le●t.  
6. Example.  
 
7. Example.  
A Merchant hath owing unto him, 540 pound, to be paid at the end of three years, now his debtor will pay him ready money, if he will abate him 9 pound in the hundred. Divide 540 pound with Ciphers by 109 three times one after the other, and the third quotient will be the sum that he shall pay in ready money▪ abating 9 pound in the hundred interest upon interest. Behold the work following.  
7 Example.  
 
The proof is made by multiplying the last quotient by 9, and that product again by 9, and thirdly again by 9, makes 540 pound, wanting but one fifth, which is but 3 ∶ 1750 parts of 1 penny, or 6 ∶ 875 parts of one farthing.  
8. Example.  
A Merchant hath owing unto him 632 pound, to be paid at the end of 12 months, now his debtor will pay him ready money, if he will abate him 12 pound in the hundred per annum? Divide 632 by 112 pound▪ and the quotient will be the sum of money that will discharge the debt, abating 12 pound in the hundred.  
Example.  
9 Example.  
324 pound was received for interest money lent a Merchant Adventurer at 17 pound in the hundred one year▪ what was the sum lent? Answer: divide 32400 by 17, makes 1900 pound, and 1 ∶ 17 of a pound.  
10. Example.  
If 358 else of Holland cast 124 pound, 16 shillings, how shall it be sold an ell to gain 12 pound in the hundred ready money? First multiply 124 pound, 8 primes by 12, adding two cyphers, makes 139 pound, 776 or in Coin 139 pound, 15 shillings, 6 pence. Secondly, divide 139 pound, 776 by 358, makes 3905 fourth's, or 7 shillings, 9 pence, 3 farthings for the price to sell one ell to gain 12 pound in the hundred.  
Example.  
11. Example.  
If one ell of cloth cost 18 pence, how shall I sell 358 else to gain 7 pound, 10 shillings by the bargain. and at what rate in the hundred do I gain? First, 358 else at 18 pence an ell makes 26 pound, 17 shillings; to the which add 7 pound, 10 shillings, the gains makes 34 pound, 7 shillings for to sell 358 else, to gain 7 pound, 10 shillings by the bargain. Secondly, divide 7 pound 500000 sixths by 26 pound, 85 seconds, and the quotient is 27 pound, 9346 fourth's, or 27 pound, 18 shillings, 8 pence farthing, which is the rate gained by the 100 pound of money.  
Example.  
12. Example.  
How much Indicoe of 6 shillings, 3 pence a pound will pay for 73 broad clothes at 16 pound one cloth, and to pay 60 pound in present money? First, 73 broad clothes at 16 pound a cloth makes 1168 pound, from which subtract 60 pound, there will remain 1108 pound; which divide by 6 shillings, 3 pence, or 3125 fourth's, and the quotient is 3545 pound, 9 ∶ 10 of one pound, and so much must he give of Indicoe for the clothes.  
Example.  
13. Example.  
How many pounds of Cloves at 6 shillings a pound, and small Cinnamon of 3 shillings a pound must be given for 36 Carseyes, at 4 pound, 3 shillings a piece, to have of each a like number of pounds? Answer: 36 Carseys at 4 pound, 3 shillings a piece, makes 149 pound, 8 shillings; which divided by the price of both, viz. 9 shillings, makes 332 pound of each sort.  
The proof: 332 pound of Cloves at 6 shillings a pound, makes 99 pound, 12 shillings; then 332 pound of Cinnamon at 3 shillings, a pound, makes 49 pound, 16 shillings, the total is 149 pound, 8 shillings, the given price of the 36 Carseys.  
Example.  
14. Example.  
Of what principal came 1000 pound principal and interest, at compound interest in three years at 6 pound in the hundred? Divide 1000 pound three several times by 106, makes 839 pound 61 seconds, or 839 pound, 12 shillings, 3 pence almost, which was the sum lent at first.  
Example.  
15. Example.  
If 34 Tun of wine cost 544 pound, how may a man sell a Tun to gain 12 pound upon the hundred ready money? First, find the price of one Tun, dividing 544 by 34, makes 16 pound for the price of one Tun which it cost; then multiply 1600 by 12 pound, makes 17 pound, 92 seconds, or 17 pound, 18 shillings, 4 pence, 4 ∶ 5 of a penny, for the price to sell one Tun of that Wine to gain 12 pound upon the 100 pound.  
  

How to work gain and loss in pence, and parts of Pence or Farthsngs.  
Set out your number of pounds, shillings, pence and farthings in pence, and in tenths of one penny; and for one farthing, set out 2 primes, 5 seconds, which is one fourth of a penny, and for two farthings set out five primes, which is one half penny; and for three farthings set down 7 primes, 5 seconds, which is three quarters of one penny, and then they are apt for decimal operations both for multiplication, division, or any other work of Arithmetic, without reducing them into farthings, and there will be a great deal of labour saved in these kinds of operations, as shall appear afterwards by the examples following.  
1. Example.  
What is the interest and principal of 100 pound, put forth at 10 pound in the 100 compound interest, for the space of 7 years to be all received at the end of the term? First, put your 100 pound into pence, maker 24000 pence; then work as in this example following, and you shall find it will amount unto 46769 pence, and 1 ∶ 5 of one penny; which divided by 240 pence, makes 194 pound, 17 shillings, 5 pence, 1 ∶ 5 of a penny, which is the sum that 100 pound will amount unto at interest upon interest in 7 years at 10 pound in the hundred.  
Example.  
 
2 Example.  
A Merchant delivered 358 pound at interest for three years for 8 pound in the hundred compound interest; the question is, what it will amount unto at the end of the term? Put your money into pence, makes 85920 pence; which multiply by 8, adding 2 Ciphers, and work for three years, as in the example following.  
Example.  
  

The proof of the former example in Decimals.  
A certain Merchant received for principal and interest upon interest 450 pound 19 shillings, 6 pence, which was for money lent at 8 pound in the hundred for three years; now the Question is, what was the sum lent? Place 450 pound, 19 shillings, 6 pence in Decimals, and you will find your third quotient will be 358 pound, wanting some few seconds, which proves the work good.  
3. Example.  
A Merchant lent 112 pound for 6 months at 17 pound in the hundred, for 12 months, the question is, what he shall receive? Put your money into pence, makes 26880 pence; mark out your prime line, as in the former examples, and add two cyphers, then multiply by 17, and take half that product for 6 months interest, and add into the principal, and the total is the sum of pence which he shall receive for principal and interest at 6 months end.  
Example▪  
Makes 121 li. 10 s. 4 d. 4 ∶ 5 of a d.  
4. Example.  
If a pound of Cinnamon cost 4 shillings ready money, how may it be sold to gain 12 pound in the hundred to give 6 months time? Set your 4 shillings in pence, makes 48 pence; then add 2 Ciphers, and multiply by half the interest, and add them into one sum, and the product will be 50 pound, 88 seconds, or 4 shillings, 2 pence, 2 ∶ 25 of one penny for the price to sell one pound to gain 12 pound in the hundred for 6 months time.  
4. Example.  
 
Makes 50 pence, 9 ∶ 10 of a penny almost.  
5. Example.  
If 112 pound weight of Clou●s cost 33 pound, 12 shillings, how may I sell them to gain 14 pound in the hundred, and give 4 months time? First, set down 33 pound, 6 primes; then add 2 Ciphers, and multiply by 14, maketh 4 pound, 704 thirds, of which take the third part, because 4 months is the third part of one year, which is 1 pound, 568 thirds; which added into one total, makes 35 pound, 3 shillings, 4 pence, halfpenny for the price to sell 112 pound to give 4 months time, and to gain 14 pound in the 100 in 12 months  
5. Example.  
6. Example.  
If I gain 8 pound, 15 shillings in 100 pieces of Linen cloth, what do I gain in the 100 at that rate, when as the 100 pieces are sold for 52 pound 10 shillings? First, subtract 8 pound, 15 shillings, from 52 l. 10 s. and there will remain 43 l. 15 s. then say, If 43 pound, 15 shillings gain 8 pound, 15 shillings, what will 100 pound gain? Divide 8750000 by 43 pound, 15 shillings, or 43 pound, 75 seconds, and the quotient will be 17 l. 14 s. 4 d. in the 100  
7. Example.  
If in 112 pound weight of Sugar, sold for 7 pound, 12 shillings ready money, there were gained 11 pound in the hundred, what did one pound cost at first penny? First, di 7 pound, 6000000 by 111 pound, which is the principal and interest given, and the quotient is 6 pound, 84684 fifths, which 112 pound cost ready money. Secondly, divide that quotient by 112 pound, makes 61132 sixths, or 14 pence, 3 farthings for the price that one pound cost at first penny.  
8. Example.  
If 300 pieces of Lawn cost 321 pound, 4 shillings, how may I sell them to lose 15 pound in the hundred? First, take the rate what one cost, by dividing 321 pound, 2 primes by 300, makes 1 pound, 0706666 sevenths, or 1 pound, 1 shilling, 5 pence almost for the price that one piece cost. Secondly, take the interest of 10706666 sevenths at 15 pound in the 100, and subtract it, and then makes 91006 sixths, or 18 shillings, 2 pence, 2 ∶ 5 of a penny for the price to sell one piece to lo●osse 15 pound in the 100 ready money. Thirdly, for the proof of this work, say; If one piece cost 910067 sixths, what will 300 pieces cost at that rate? Multiply 910067 sixths by 300, and cut off 6 figures to the right hand, makes 273 pound, 5 pence almost for the sum received for 300 pieces to lose 15 pound in the 100 Fourthly, for a second proof; take the interest of 321 pound, 2 primes at 15 pound in the hundred loss, and deduct it from 321 pound, 2 primes, and there will remain 273 pound, 5 pence almost, which doth prove all the other works to be truly wrought.  
Example.  
 
9 Example.  
If in one ell of Cloth sold for 3 shillings, 2 pence halfpenny, there were gained 10 pound in the hundred ready money, what did that ell cost? Answer: set 3 shillings 2 pence ob. in decimals, makes 38 pence, 5 primes; then divide 38 pence, 5000 fourth's by 110 pound, makes 35 pence, the price that one ell cost.  
Example.  
10. Example.  
If in one ell of Cloth sold for 35 pence, 19 seconds, there were gained 7 pound in the hundred ready money, what did that ell cost, when there was 6 months time given? Divide 35 pound, 1900 fourth's by half the interest, adding one 100, which is 103 pence, 5 primes, and the quotient is 34 pence, the price that the ell cost.  
 
11. Example.  
A Merchant lent money at 10 pound in the hundred for 100 pound profit for 12 months, and at the end of 6 months he received principal and interest 356 pound, the question is, what was the sum lent? Divide 356 pound, by 105 pound, which is the half years' Interest and principal, and the quotient is 305 pound, 5 ∶ 105 of a pound for the sum lent.  
Example.  
12. Example.  
If 17 pound lose 12 shillings, what will 100 pound loose? Divide 60000 fifths by 17, makes 3 pound, 529 thirds, or 3 pound 10 shillings, 7 pence in the hundred pound.  
13. Example.  
If 37 yards of velvet cost 32 pound, how must one yard be sold to gain 9 pound, 10 shillings in the hundred? First, 32 pound the price at 9 pound, 5 primes the hundred, makes 35 pound, 4 seconds; which divide by 37, makes the price of one yard to be 94702 fifths, or 18 shillings, 11 pence, ob. to sell one yard to gain 9 pound, 10 shillings in the hundred.  
Example.    

Exchange in decimals.  
1. Example.  
IF one pound sterling be 1 pound, 14 shillings, 6 pence Flemish, what is 783 pound sterling in ●emmish money? Set out 1 pound, 14 shillings, 6 pence in decimals, makes 1 pound, 725 thirds▪ which multiply by 783 pound, makes 1350 pound, 675 thirds, or 1350 pound, 13 shillings, 6 pence.  
Example.  
 
2 Example.  
If one pound exchange be 5 shillings, 6 pence what is 783 pound? Set 5 s. 6 d. in Decimals, makes 275 thirds; which multiply by 783, makes 215 pound, 325 thirds, or 215 pound, 6 shillings, 6 pence; which added to the last example, is 1566 pound, and so much is the double of the sum given, viz. of 78● pound, because the two prices given, makes just 2 pound, and this by working a second question in exchange, the first is proved to be truly wrought, as appeareth in the example above.  
3. Example.  
If one pound exchange be 1 pound, 17 shillings, 7 pence, halfpenny, what is 1000 pound at that rate? Set 1 pound, 17 shillings, 7 pence, halfpenny in decimals, makes 1 pound, 88125 fifths; then because 1000 hath; Ciphers, add 3 Ciphers, and cut off 5 figures, and the answer is 1881 pound, 5 shillings.  
 
4 Example.  
A Merchant doth receive 134 pound, 6 shillings for the exchange of one hundred pound sterling from Middleborough, what was one pound sterling in Flemish money? Place 134 pound, 6 shillings in decimals, is 134 pound, 3 primes; then because 100 pound hath 2 Ciphers, cut off two figures more to the left hand, and it will be 1 pound, 343 thirds; or in Coin, 1 pound, 6 shillings, 11 pence, farthing for the exchange of one pound at that rate.  
 
5. Example.  
A Merchant doth receive 645 pound, 12 shillings for exchange money, at 1 pound, 7 shillings, 6 pence for one pound sterling, the question is, how much sterling money he did deliver? Divide 645 pound, 6 primes by 1 li. 375 thirds, or 1 pound, 7 shillings, 6 pence, makes 4695268 fourth's, or 469 pounds, 10 shillings, 6 pence, 1 farthing for the sterling money delivered.  
6 Example.  
If 1 l. sterling be 1 l. 7 s. 6 d. Flemish, what is 110 l. Flemish in Sterling Coin? Divide 100 pound by 1 pound, 375 thirds, makes 72 pound, 72727 fifths; or 72 pound 14 shillings, 6 pence, ●b. that 100 l. makes.  
7. Example.  
If the exchange be from Rome to London at 69 pence sterling one Ducat, how many ducats shall be delivered at Rome for to receive 356 pound, 16 shillings sterling at London? Answer? Divide 356 pound, 8 primes by 2875 fourth's, which is 69 pence, and the quotient will be 1241 ducats, 3 pence.  
 
8. Example.  
If the exchange be from London unto Antwerp at 23 shillings, 5 pence, 3 farthings Flemish the pound sterling, how much money must be delivered at London, to receive 146 pound, 14 s. 10 pence, 3 q. in Flemish money? Answer: Divide 146 pound, 744775 sixthes, by 1 pound, 3 shillings, 5 Pence, 3 farthings: which is 1 pound, 1739582 sevenths, and the quotient is 125 pound; and so much must he deliver at London to receive 146 pound, 14 shillings, 10 pence, 3 farthings in Flemish Coin at that rate.  
Example.  
9 Example.  
A Merchant doth deliver at Antwerp 200 pound Flemish by exchange for London at 22 shillings, 10 pence Fleminish for one pound sterling, how much must he receive at London? Answer: divide 200 pound by 1 pound, 141666 sixths, which is 22 shillings, 10 pence; makes 175 pound.  

A general Rule for exchange in Decimals.  
If the price of a unite be given, then always divide the sum of money whereon the question dependeth by that unite in decimals, and the quotient is the answer to the question.  
1. Example.  
A Merchant doth deliver 100 pound sterling by exchange for Rome, at 72 pence sterling for one Duckat De Camera; the question is, how many Ducats he must receive at Rome for his 100 pound sterling? here the price of one Ducat is given to be 72 pence, which is 6 shillings, or 3 primes; wherefore I divide 100 pound by 3 primes, and the quotient is 333 pound, 1 ∶ 3 of a pound, or 6 shillings, 8 pence for answer to the question.  
2. Example.  
A Merchant doth deliver 756 pound sterling at London, to receive Ducats at 66 pence sterling, the price of one Dueket, the question is, how many Ducats he must receive at Venice? Divide 756 pound by 66 pence, which is 275 thirds, and the quotient is 2748 ducats, and 300 ∶ 2750 of one Ducat for the Answer.  
3. Example.  
A Merchant at Venice doth deliver 1000 ducats, to receive at London 287 pound, 10 shillings sterling, what is one Ducat? Set down 287 pound, 5 primes, and divide by 1000 Ducats, makes at 5 shillings, 9 pence for one Ducat.  
 
Makes 5 s. 9 d. one Ducket●  
4. Example.  
A Merchant at Venice doth deliver 800 ducats by Exchange for London at 64 pence, b. the ducat sterling money, the question is, how much sterling he must receive at London? Set out 64 pence, halfpenny in Decimals, makes 26875 fifths; which multiply by 800, and cut off 5 figures because your fractions are 5, and the product will be 215 pound sterling.  
 
Makes 215 pound sterling.  
5. Example.  
A Merchant doth deliver 1000 ducats by Exchange for London at 71 pence sterling for one ducat, how much must he receive sterling money at London? Set out 71 pence in decimals, makes 2958 fourth's, 1 ∶ 3, and add 3 Ciphers for 10●0, and cut off 4 figures, makes 295 pound, 8 primes, 1 ∶ 3, or 295 pound, 16 shillings, 8 pence for the answer.  
 
Makes 295 l. 8 primes, 1 ∶ 3  
6. Example.  
One penny Flemish is 3 ∶ 5 of one penny sterling, and one pound Flemish is 3 ∶ 5 of one pound sterling or ●2 shillings; wherefore to convert Flemish money into sterling Coin, multiply your Flemish money by 3 ∶ 5, which in decimals is 6 ∶ 10, or 6, and the product will be the value of your Flemish money in sterling Coin. In 345 Flemish, how much sterling Coin? Multiply 345 by 6 primes, and the product is 207 pound sterling.  
 
7. Example.  
In 756 pound, 18 shillings sterling, how much Flemish coin, when one penny Flemish is ●: 5 of a penny English? Denied 756 pound, 9 primes by 6 primes, makes 1201 pound, 5 primes, or 10 shillings.  
  

Reduction of Measures from one place to another.  
IF you will reduce the measure of one Country into the measures of another As if you would reduce the measures of Antwerp, Gaunt, Brudges, Seville, Roaven, or of any other Country, into the measures at London; learn first the order of measuring of all sorts of commodities in both places, either out of the experience of Merchants and Tradesmen in those places, or out of the best and latest approved Authors that have written Tables to that effect and note, that 4 else at London makes 5 yards, and 100 else at London is at  
 Ells. Antwerp 166●/● Gaunt short measure 164 Gaunt long measure 154 Brudges 164 Arras 165 Calais 157 Lisse 166 Mastrich● 173 Cullen 208 Franckfort 208 Nor●mberge 174 Da●tringe 139 Ro●●● 103 Paris 95 Licons 100 Genna 480 ●/● Palms. Millian 214 Braces. Florence 188 Braces. Venice for Silk hath 196 Ells. Venice for Linen hath 180 Ells. Rome 56 Cana. Lisb●●●● 100 Varras. Madera 104 Varras. Sevile 135 Varras.   

These I have taken out of Mastersons Arithmetic.  
The difference of one hundred els, Palms, Varras, or Braces, being found of any place from London; if you would convert the measures of any of those places to London measure: as for example, If you would convert 356 else of Brudges' measure into else at London; you shall find in the Table, that 164 else make 100 at London; then by the Rule of Three say,  
1. Example.  
If 164 be 100, what are 356 else? Multiply 356 by 100 and divide by 164, makes 217 else, 12 ∶ 164 of an ell, which 356 at Brudges will make in London. But according to the order of decimals, if you will bring the measures of other places to those of London? Set your number of one hundred found in the Table, to a unite in decimals, as in the last example 164 stands thus 164 seconds, than you need but divide your number 356 by 1 pound, 64 seconds, and the quotient is 217 else, 12 164 else, as in the last example.  
Again, if you would reduce London measure to the measures of any other place? Find the number of 100 to that place, and set it decimals, and multiply your number of else at London by those numbers found, and the product will be your desire.  
2. Example.  
In 758 else at London, how many else at Dantzing, I find in the Table 139 else there make 100 at London; so I set 139 to a unite, and it is 1 pound, 39 seconds; by which I multiply 758, makes 1053 else, 62 ∶ 100 parts.  
 
3. Example.  
If 166 else 7 ∶ 3 at Antwerp be 100 else at London, how many else at London are 1756 else at Antwerp? Set 166, 2 ∶ 3 to a unite, makes 1 pound, 66 seconds, and 2 ∶ 3 of a second: Or otherwise; 1 ell, and 2 ∶ 3 of one ell, by which divide 1756, makes 1053, 1 ∶ 2  
 
4. Example.  
In 3258 else at London, how many Braces at Millian? Find 214 for 100 at London so that if you set 214 to a unite, it will be 2 pound, 14 seconds; by which multiply 3258, makes 6982 Braces, and 12 ∶ 100 parts of a Brace.  
And in this manner you may eass●y convert your Measures or Weights from one place to another, either by Multiplication or Division, without the Golden Rule: but of this, if it please God to lend me life and health, I do purpose to speak in a Treatise at large of decimal Arithmetic for the good of my Countrymen and others, if I find these my labours and endeavours to be acceptable and beneficial to others, and will better inform myself by Merchants, who have had experience in the Reduction of Weights and Measures from place to place; in the mean time here is a foundation laid to work upon; let the difference be what it will, and so for this time I will end this Treatise of decimal Arithmetic, and go in hand with some operations of Annuities, as followeth.    

Of Interest and Annuities.  

How to frame Tables to work Interest and Annuities, or Purchases at any rate.  
FOrasmuch as these kind of operations of interest and Annuities are ●●ry tedious and trouble some, if they be to be wrought for many years, althougb I have already in the former Book set forth many several manners of working those kind of questions after a more easy kind of method, than heretofore hath been published by any other in the like kind whatsoever yet here I have thought good also in this place to show the ways, whereby any man that is desirous to be satisfied in the reasons or grounds of those kind of works, may be able to calculate for his own use a Table or Tables, whereby to abbreviate those kind of operations by Multiplication, or Division, only without the help of the Golden Rule, or any tedious Reductions of Multiplications and Divisions for many years to come at one only operation, as shall appear by the examples following.   

How to calculate the Table or Breviat of 10 pound in the hundred Compound Interest.  
If you will calculate a table for 10 pound in the hundred compound Interest for 21 or 30 years? Place your numbers, as in the examples following, beginning with a unite, or ●, adding 7 Ciphers unto it, and then take the tenth part of that, which is the same numbers one room more to the right hand, and add them into the first numbers, and the total will be the sum for▪ the first year, and so you must work for the second, third, fourth, etc. until 21, or 30 years: but here you shall note, that you shall not need to set down in your Breviate more than 8, 9, or 10 numbers at the most, for because the rest willbe superfluous, as for example.  
Example. Int. 1. 2. 3. 4. 5. 6. 7. 8 Year. Int. 1. 2. 3. 4 5. 6. 7. 8 Year. 1 00000000 0 2 35794769 9  1   23579476  1 10000000 1 2 59374246 10  11   25937424  1 21000000 2 2 85311670 11  121   28531167  1 33100000 3 3 13842837 12  1331   31384283  1 46410000 4 3 45227121 13  14641   3452271▪ 2  1 61051000 5 3 79749833 14  161051   37974983  1 77156100 6 4 17724816 15  1771561   417●2481  1 94871710 7 4 59497298 16  19487171   45949729  2 14358881 8 5 05447028 17  21435888   50544702  2 35794769 9 5 55991731 18     55599173     6 11590904 19  
Here you may sec in this Table the manner of gathering the Breviate of 10 pound in the hundred, Compound interest, which you may extend to what number of years you please, only adding a unite in the eight place, as you see the figures in the ninth place do arise, and now I will here set down the Breviate from one year unto 40 ready gathered.   

The Breviate of 10 pound in the hundred for 40 Years.  
Years 1. 2. 3. 4. 5. 6. 7. 8. Years 1. 2. 3. 4. 5. 6. 7. 8. 9 1 11000000 21 740024990 2 12100000 22 814027490 3 13310000 23 895430240 4 14641000 24 984973260 5 16105100 25 108347059 6 17715610 26 119181765 7 19487171 27 131099941 8 21435888 28 144209936 9 23579476 29 158630929 10 25937424 30 174494022 11 28531167 31 191943424 12 31384283 32 211137766 13 34522712 33 232251543 14 37974983 34 255476697 15 41772481 35 281024367 16 45949729 36 309126803 17 50544702 37 340039484 18 55599173 38 374043432 19 61159090 39 411447775 20 67274999 40 452592553   

How to calculate a Table or Breviate at any rate under or above 10 pound in the hundred, Compound Interest.  
If you would calculate a Table or Breviat any rate under or above 10 pound in the hundred compound interest, place a unite with seven Cypheres to it; then if you will calculate for 12 pound in the hundred or 16 pound; set your 12, or 16 under the 2 first Ciphers next the unite, and multiply your unite, omitting the cyphers by your interest, and add the product into one total, and the sum is the principal and interest for the first year, and so work again for the second, third, etc. to finish your Table, as aforesaid, at 10 pound in the hundred▪ But if your interest be under 10 pound in the hundred, place your number of the interest under the second cipher from your unite, and work as is in the example following.  
Example. Int. 1. 2. 3. 4. 5 6. 7. 8 Years Int. 1. 2. 3. 4. 5 6. 7 8 Years 1 00000000  1 86048896 4  80   8  1 08000000 1  10883904   8    ●  864  1 46932800 5     8  1 16640000 2  1175462   8      93312  1 5868743 6     8  1 25971290 3 1 7138242 7  8      16077696     1 36048896 4     
In this manner you may proceed infinitely: and thus much shall suffice for making of these Breviates.   

The Breviat of 8 pound in the hundred per annum Compound Interest for 30 years.  
Years 1. 2. 3. 4. 5. 6. 7. 8 Years 1. 2. 3. 4. 5. 6 7. 8. 9 1 10800000 16 342594260 2 11664000 17 370001800 3 12597120 18 399611940 4 13604889 19 431570100 5 14693280 20 466095710 6 15868743 21 503383370 7 17138242 22 543654040 8 18509302 23 587146360 9 19990046 24 634118070 10 21589249 25 684847510 11 23316389 26 739635320 12 25181701 27 798806140 13 77196237 28 862710630 14 29371936 29 931727480 15 31721691 30 100626506  
In this sort you may gather all the Tables or Breviates for any rate in the hundred, which I will here omit in this small vollum, intending afterwards to publish this, and diverse other operations in my second Edition of my Book of decimal Arithmetic shortly to come forth.   

The use of these Breviates and Tables, and of all others of like nature in working of questions of Interest and Annusties.  

Rule 1.  
To find what 1 pound due at any number of years is worth at the end of the term? Enter the Table of 10 pound in the hundred, and find in the left Margin the number of years, and from that number so found, cut off seven figures, the answer is in pounds, primes, seconds, thirds, fourth's, &c. for the answer to the question demanded.  
1. Example.  
What is one pound put forth at interest compound, at 10 pound in the hundred worth, to be paid at the end of 18 years? Find the eighteenth number in the Breviat, which is 55599173; from which cut off seven figures to the right hand, and the answer is 5 pound, 11 shillings, 2 pence, q.  
Example.  
Makes 5 l. 11 s. 2 d. 1q.  
2. Example.  
What is 100 pound due at 7 years' end worth to be paid at the end of the term, at 10 in the hundred compound Interest? Find the seventh number in the Table of 10 l. in the hundred, makes 19487171; to the which add two Ciphers, because 100 pound hath two Ciphers, and cut off seven figures to the right hand, and the sum is 194 pound, 87171 fifths for the Answer.  
 
3 Example.  
What will 758 pound for 6 year make at 10 pound in the hundred compound Interest, to be paid▪ at the end of the term? Find the sixth number in the Table of 10 pound in the hundred, which is 17715610; which multiply by 758, the money named in the question, and the product, cutting off 7 figures to the right hand, makes 1342 pound, 16 shillings, 10 pence, ob. almost.  
  

Rule 2.  
How to find what any yearly Annuity will make to be paid all at the end of the term? First, find the number of years of the annuity given, and from the number answering, deduct a unite in the first place to the left hand, and add a cipher to the last figure to the right hand, and cut off seven figures to the right hand, and the answer is found.  
1. Example.  
What will 1 pound annuity make, to be paid for at the end of the term of 16 years at 10 pound in the hundred compound interest? Find the sixteenth number in the Table of 10 pound in the hundred, and subtract a unite from the first figure to the left hand, adding a cipher to the right hand, makes 359497290; From the which cut off 7 figures to the right hand, makes 35 pound, 18 shillings, 11 pence, 3 farthings.  
 
2. Example.  
What will 1000 pound annuity yearly amounteth unto to be all forborn until the end of the term of 5 years at 10 pound in the hundred compound interest? Find the fifth number in the Table of 10 pound in the hundred, and subtract a unite from the first figure, adding a cipher in the last place, makes 61051000: then because 1000 pound hath 3 Ciphers, add 3 Ciphers, and cut off seven figures, makes 6105 pound, 2 shillings for the answer.  
 
3. Example.  
What will 142 pound annuity make to be paid at the end of the term of 10 years? Find the tenth number in the Breviat of 10 pound in the hundred, and subtract a unite in the first place, adding a cipher to the last, makes 159374240; which multiply by 142 pound, the annuity named, and from the product cut off seven figures to the right hand, and the answer to the question is 2263 pound, 2 shillings, 2 pence, 3 farthings.  
  

3. Rule.  
How to find what any sum of money due at the end of any number of years is worth in ready money, at 10 pound in the hundred compound interest. Enter the Table of 10 pound in the hundred with your number of years, and the numbers which doth answer in the Table is your Divisor; then add seven Ciphers to your sum of money given, to make your dividend; then divide your dividend by your Divisor, and the quotient, adding more Ciphers, will be your answer in pounds, primes, seconds, thirds, etc.  
1. Example.  
What is 1000 pound due at 7 years' end worth in ready money, at 10 pound in the hundred compound interest? Find the seventh number in the Table of 10 pound in the hundred, which is 19487171, this is your Divisor. Then add seven Ciphers to 1000 pound, makes 1000000000; or add more Ciphers, marking out your prime line in your dividend, to find out how many figures your quotient will have in whole numbers, and the rest will be primes, seconds and thirds; this is your dividend, and then divide by your divisor, makes 513 pound, 3 shillings, 2 pence.  
 
Having found what 1000 pound due at 7 years' end is worth in ready money, if you will find what 100 pound, or 10 pound, or 1 pound is worth in ready money; place your quotient in decimals, and mark out your prime lines, cutting of one figure for 100 pound, ● for 10 pound, or 3 for 1 pound, the answer is as followeth.  
Example.  
2. Example.  
What is 750 pound due at 5 years' end worth in ready money, at 10 pound in the hundred compound interest? Find the fifth number in the Table of 10 pound in the hundred, which is 16105100 for divisor; then place 10 Ciphers before your number given 750 pound, and mark out your prime line, and divide by your Divisor, and the quotient will be 465 pound, 13 shillings 10 pence for the answer to the question given.  
Example.  
Makes 465 pound, 13 shillings, 10 pence.  
3. Example.  
What is 847 pound due at 21 years' end worth in ready money, at 10 pound in the hundred compound interest? Find the 21 number in the Table of 10 pound in the hundred for Divisor, which is 74002499; then set 10 Ciphers to your numbers given, makes 8470000000000 for your divedend; then divide, and the quotient will be 144 l. 9 s. 1 d. 1 ∶ 5 of 1 d. the answer.  
Example.  
Makes 114 l. 9 s. 1 d. 1 ∶ 5 of a penny.   

4 Rule.  
How to find what any yearly Annuities for any number of years is worth in ready money at 10 pound in the hundred compound interest. Enter the Table of 10 l. per cent▪ with your number of years given, and from the numbers found subtract a unite in the first place and place a cipher in the last for your dividend; which divide by the number found in the Table against your years given, and the quotient is the answer to the question.  
1. Example.  
What is 100 pound per annum annuity for 21 years' worth in ready money at 10 pound in the hundred Compound ●nterest? Look in the Table of 10 pound in the hundred for 2● years, and subtract a unite in the first place, and add a cipher in the last, makes 6400; 4990; Divide this by 74002499, the 21 number, adding Ciphers, and marking the prime line, and the quotient is 864 pound, 17 shillings, 4 pence, ● farthings for the answer to the question demanded.  
Example.  
2. Example.  
Having found what 100 pound annuity will amount unto, if you would know what 10 pound▪ or 1 pound annuity will amount unto, or 1000 pound in 21 years; place it in decimals, and cut off 1, 2, or add 3 Ciphers to the last, or remove 3 places, and you shall find your demand.  
Example.  
3. Example.  
What is 546 pound yearly annuity for 14 years' worth in ready money, at ten pound in the hundred compound interest?  
Find the fourteenth number in the Breviate of 10 pound in the hundred; from it subtract a Unite in the first place, and add a cipher, makes 279749830; which multiply by 546, makes 152743407180▪ which divide by 37974983, the 14 number in the Breviate, makes 4022 pound, 4 shillings, 2 pence, 3 farthings.  
 
Makes 4022 l. 4 s. 2 d. 3 ∶ 4  
1. Example.  
There is a Debt bought for 513 pound, 3 shillings, 2 pence ready money, which was due at 7 years' end, now the question is, what the debt was at 10 pound in the hundred compound interest? Set your money paid in decimals, makes 513158; which multiply by 19487171, the number against 7 years, cutting off 10 figures, makes 999 pound, 999 thirds, wanting but one third of 1000 pound; wherefore I conclude, the debt was 1000 pound, which was due at 7 years' end.  
2 Example.  
There was a Debt bought for 600 pound, which was due at 4 years' end, what was that debt at 10 pound in the hundred compound interest? Multiply 600 pound by the numbers against 4 years, which are 14641000 makes 878 pound, 4600000 sevenths, or in Coin 878 pound, 9 shillings, 2 pence, 2 ∶ 5 of 1 penny for the sum of that debt.  
 
Makes 878 l. 9 s. 2 d. 2 ∶ 5 of a penny.  
I have set no exampies of the Table of 8 pound in the hundred, nor of no other rate, bectuse I intent shortly to speak more at large of this subject in another volume, if God please to give me time and health, in which I intent to speak more at large of the Grounds, Reasons and proofs of these kind of Operations, and here I will finish this small Treatise of the second Book.     
FINIS.