Denominators than one which kind of broken numbers are commonly set down with the particle of between them as five twelfths of one third is to be written thus 5 12 of 1 â—â— three fourths of seven eighths thus ¾ of 7 â— and so of any other 11. The things to be expressed by broken numbers are chiefly the parts or fractions of money weight measure time motion and things accounted by the Dozen Of the three first of these there are infinite kinds and varieties according to the Laws and Customs of particular Countries those used in this kingdom are most proper for us to know and the knowledge of them will be sufficient to direct us in the use of those in forreign nations if at any time there be an occasion for them 12. The several pieces of Coine or Denominations of money used in England in reference to account are pounds shillings pence and farthings whose particular values are as followeth 4. Farthings make 1. Penny 12. Pence make 1. Shilling 20. Shillings make 1. pound Sterling And according to these values a pound Sterling is esteemed an integer and may be divided into 20 parts called shillings and therefore one shilling is a broken number of a pound Sterling by the former directions is to be written thus 1 â—â— l. that is one twentieth of a pound Again a shilling may be divided into twelve equal parts called pence and so one penny is a fraction of a shilling and is to be written thus â— 12 s. that is one twelfth of a shilling Lastly a penny may be divided into four equal parts called farthings and so one farthing is a fraction of a penny and written thus ¼ d. that is one fourth of a penny or thus ¼ of 1 12 s. that is one fourth of one twelfth of a shilling or thus ¼ of 1 12 of 1 20 l. that is one fourth of one twelfth of one twentieth of a pound Sterling 13. Now then though the true and natural way of expressing broken numbers is by their Numerators and Denominators as hath been shewed yet the broken numbers or known parts of money weights measures and such like are for more convenient operation commonly expressed like integers so that if 12 shillings seven pence half peny farthing were to be expressed in figures the ordinary most usual way is thus 12s − 07d 03 f. but the said twelve shillings seven pence half peny farthing being distinctly considered as fractions of a pound Sterling the way to write them properly is thus 12 Shillings are twelve twentieths of a pound Sterling and written thus 12 2â— l. 7 Pence are seven twelfths of one twentieth of a pound Sterling and written thus 7 12 of 1 20 l. 3 Farthings are three fourths of one twelfth of one twentieth of a pound and written thus ¾ of 1 12 of 1 20 l. 14. The weights used in England are of two sorts Troy weight and Averdupois 15. The several pieces or Denominations of Troy weight are pounds ounces peny weights and grains whose particular values are as followeth 24 Grains make 1 Peny weight 20 Peny weight make 1 Ounce 12 Ounces make 1 Pound Troy 16. The weights used by Apothecaries are derived from a pound Troy the which is subdivided as in the following Table lb A pound Troy is equal to 12 Ounces â„¥ An Ounce is equal to 8 Drams Ê’ A Dram is equal to 3 Scruples ℈ A Scruple is equal to 20 Grains 17. But besides Troy weight there is another kind of weight used in England call'd Averdupois weight a pound whereof is equal unto 14 ounces and twelve-peny weight Troy 18. This Averdupois weight is either great or small 19. The great Averdupois weight is when an hundred consisting of 112 pounds Averdupois is the integer and subdivided into halves and quarters each quarter conteâ—ning 28 pounds 20. The small Averdupois weight is when a pound is the integer each pound being subdivided into 16 ounces each ounce into 16 drams and each dram into 4 quarters and because many persons have occasion to use both and are perhaps furnished but with one I have here exhibited a Table for the speedy converting of the parts of a pound Troy into the parts of a pound Averdupois and the Contrary Drams Averdupois  Peny-weights Troy Grains Decimals of a grain  Ounces Averdupois Ounces Troy Peny-weights Grains 1  01 03 375  1 00 18 06 2  2 â—6 â—50  2 01 16 12 3  3 10 125  3 2 14 18 4 Dram Averdupois 4 13 500  4 3 13  5  5 16 875  5 4 11 06 6  6 20 250  6 5 9 12 7  7 23 625  7 6 7 18 8 weight is equal to 9 03 000  8 7 06  9  10 06 375  9 8 04 06 10  11 19 750  10 9 02 12 11  12 03 125  11 10 00 18 12  13 16 500  12 10 19 00 13  14 19 875  13 11 17 06 14  15 23 250  14 12 15 12 15  17 02 625  15 13 13 18 16  18 06 000  16 14 12 00 21. The measures used in England are of two sorts Capacity or Length 22. The measures of Capacity are produced from weight and are also of two sorts liquid or dry 23. The liquid measures are those in which all kind of liquid substances are measured and are expressed in the table following 1 Pound of wheat Troy weight make 1 Pint. 2 Pints make 1 Quart 2 Quarts make 1 Pottle 2 Pottles make 1 Gallon 8 Gallons make 1 Firkin of Ale Sope Herring 9 Gallons make 1 Firkin of Beer 10½ Gallons make 1 Firking of Salmon or Eels 2 Firkins make 1 Kilderkin 2 Kilderkins make 1 Barrel 42 Gallons make 1 Tierce of Wine 63 Gallons make 1 Hogshead 2 Hogsheads make 1 Pipe or But. 2 Pipes or Buts make 1 Tau of wine 24. Dry measures are those in which all kind of dry substances are measured as grain Sea-coal Salt and such like and are expressed in the table following 1 Pint make 1 Pint. 2 Pints make 1 Quart 2 Quarts make 1 Pottle 2 Pottles make 1 Gallon 2 Gallons make 1 Peck 4 Pecks make 1 Bushel land measure 5 Pecks make 1 Bushel water measure 8 Bushels make 1 Quarter 4 Quarters make 1 Chalder 5 Quarters make 1 Wey 25. Long measures are as followeth 3 Barley Corns in length make 1 Inch. 12 Inches make 1 Foot 3 Foot make 1 Yard 3 Foot 9 Inches make 1 Ell. 6 Foot make 1 Fathom 5 Yards and ½ make 1 Pole or Perch 40 Poles make 1 Furlong 8 Furlongs make 1 English Mile 26. In superficial or square measure 40 square poles or perches make 1 Rood or quarter of an Acre and 4 Roods an Acre 27. A Table of Time is this that followeth 1 Minute make 1 Minute 60 Minutes make 1 Hour 24 Hours make 1 Day 7 Days
make 1 Week 4 Weeks make 1 Moneth of 28 Dayes 13 Moneths 1 Day and 6 hours make 1 Year not exactly but very near 28. A year is that space of time in which the Sun doth finish course through the circle in the heavens called the Zodiack which is in 365 days 5 hours 4 min. 29. The Zodiack by is Astronomers divided or supposed to be into twelve equal parts called signs whose names and Characters are these Aries ♈ Taurus ♉ Gemini ♊ Cancer ♋ Leo ♌ Virgo ♠Libra ♎ Scorpio ♠Sagittarius ♠Capricornus ♑ Aquarius ♒ Pisces ♓ And each of these signs into 30 parts called Degrees so that this and all other Circles are supposed to be divided into 12 times 30 parts or Degrees that is into 360 each degree into 60 minutes each minute into 60 seconds each second into 60 thirds c. 30. Of things accounted by the Dozen A gross is the Integer consisting of 12 dozen and each dozen of twelve particulars 31. An improper fraction or broken number is that whose numeration is greater than the Denominator As 54 12 feet that is fifty and four twelfths of a foot and this may be well called an improper fraction seeing it will not admit of the definition of a true broken number because it is greater than that whole whereas a fraction is properly but a part of the whole 32. A mixt number is that which besides the integers or intire unities of which it consists hath also a broken number annexed As in this improper fraction 54 12 if you divide the numerator 54 by the denominator 12 it will be reduced into the mixt number 4 6 12 of which 4 is the whole part and 6 12 the broken number or fraction 33. And this I hope is sufficient to shew what is meant by a fraction and how all fractions whether proper or improper are to be expressed and read which is the Notation of them the next thing propounded concerning fractions is their Numeration whether such fractions be expressed by their true and natural way that is by their Numerators and Denominators or whether they be expressed like integers as the known parts of money are expressed by pounds shillings pence and farthings and the known parts of Troy weight by pounds ounces pennyweights and grains of these and the like broken numbers which are expressed like integers we speak of first CHAP. VII Of the Numeration of such broken numbers as are expressed like integers 1. NUmeration of such as are expressed like integers is twofold 2. Accidental and Essential 3. Accidental Numeration is otherwise called Reduction 4. Reduction is either descending or ascending 5. Reduction descending is when a number of a greater Denomination being given it is required to find how many of a lesser Denomination are equal in value to that given number of the greater as when it is required to find how many shillings are contained in 34 pounds or how many pence in 325 shillings or how many hours in 365 daies 6. Reduction descending is performed by Multiplication for if the given number of integers of a greater denomination be multiplied by the number of integers contained in the next inferiour denomination the product shall shew how many of that inferiour denomination are contained in the integers of the greater denomination given for example let 34 pounds be the greater denomination given and let it be required to shew how many shillings are in 34 pounds shillings being the next inferiour denomination unto pounds and that every pound doth contein 20 shillings as hath been shewed if you multiply 34 by 20 the product is 680 the number of shillings required In like manner 680 shillings will be reduced into 8160 pence if you multiply 680 by 12 the number of pence in a shilling and 8160 pence will be reduced into farthings 32640 if you multiply 8160 by 4 the number of farthings in a penny The like method is to be observed in weights and measures or any thing else that is or may be subdivided into inferiour denominations thus 26 pound Troy will be reduced into 312 ounces and 312 ounounces into 6240 penny weights and 6240 penny weights into 149760 grains as by the operation in the margin it doth appear And in this manner may any number of a greater denomination given be reduced into the least denomination into which the greater is supposed to be subdivided Reduction ascending is when a number of a lesser denomination being given it is required to find how many of a greater denomination are equal in value to that given number of the lesser as when it is required to find how many pence are conteined 32640 farthings or how many shillings in 8160 pence or how many daies in 8760 hours And this kind of Reduction called Reduction ascending is performed by Division for if the number of integers given be divided by the number of integers in the next superiour or greater denomination the quotient shall be the number of integers sought so 32640 farthings being divided by 4 the number of farthings in a penny the quotient is 8160 the number of pence conteined in 32640 farthings In like manner if 8160 pence be divided by 12 the number of pence in a shilling the quotient will be 680 the number of shillings in 8160 pence and lastly if 680 shillings be divided by 20 the number of shillings in a pound the quotient will be 34 the number of pounds in 680 shillings the like may be done by any other integers of any known denomination given CHAP. VIII Of the Addition of numbers that are of diverse Denominations 1. ACcidental Numeration of such broken numbers as are expressed like integers hath been shewed that which is essential now followeth 2. Essential Numeration doth consist of Addition Subduction Multiplication and Division 3. When the numbers propounded to be added are of diverse denominations you must begin with the least denomination first and set down their sum under that inferiour denomination if their sum be fewer than the number of parts in the next greater denomination but if their sum be more than the number of parts in the next greater denomination set down the excess if equal set down a Cypher and the rest must be added to the next superiour denomination and for memory sake it may be set down under that denomination to which it is to be added as in the example following In which I begin with the farthings first and say 3 + 2 = 5 that is one penny and a farthing wherefore setting 1 down under the denomination of farthings I carry a penny to the denomination of pence and say 1 + 7 = 8 and 8 + 4 = 12 and 12 + 5 = 17 pence that is 1 shilling and 5 pence wherefore I set down 5 in the lowest rank under the denomination of pence and 1 in the line above it under the denomination of shillings and say 1 + 8 = 9 and 9 + 19 =
7 shillings the value of one Escus d'or whose product is 896 shillings and then the question must be thus stated If 14 shillings be worth 1 Pistolet how many Pistolets are 896 worth Facit 64 Pistolets 7. Sometimes a question belonging to this rule must be prepared for solution by multiplication and addition sometimes by multiplication and subtraction as in these examples following the first of which requireth multiplication and addition A Butcher sends his man with 216 pound sterling to buy Cattel Oxen at 11 pound apiece Cows at 2 pound a piece Colts at one pound 5 shillings Hogs at 1 pound 15 shillings a peice and of each a like number the question is how many of each he might buy for that mony To resolve this question I must first reduce the pounds sterling into shillings by multiplication and 216 pound being multiplied by 20 the product is 4320 shillings and the several prizes of the Cattel must be reduced into shillings also and thus 11 pounds for an Ox is 220 shillings the price of a Cow is 40 shillings the price of a Colt 25 shillings and the price of a Hog is 35 shillings the sum of these several prizes is 320 shillings this done the question may be thus stated If 320 shillings buy 1 of each sort how many shall 4160 buy facit 13. 2. Example In the which the question is to be prepared by multiplication and subtraction If 128 Gallons of water run into a Cistern in an hours time from one Cock and 174 Gallons of water run into another Cistern in an hours time also but not begin to run till 4 hours after the first in what time will the quantity of water run out of both be equal To resolve this question one of the terms given must be prepared by Multiplication and another by Subtraction first there must be computed how much water was run out of the first Cock before the second began and that is 4 times 128 Gallons that is 512 secondly it must be considered how many Gallons more doth run out of the last Cock than doth out of the first in an hours time which by subtracting 128 from 174 I find to be 46 and hence this question is to be thus stated If I gain 46 Gallons in one hour in how many hours shall I gain 512 Gallons Facit 11. 8. Sometimes a question belonging to this rule must be prepared for solution by Arithmetical or Geometrical proportion as in these questions following A man going a journey spends one shilling the first day 3 the second 5 the third 7 the fourth still encreasing his expences by this proportion till he hath spent 1500 shillings I demand in how many days that sum is spent To resolve this question the sum of such an Arithmetical progression for some certain time must be first computed as suppose for thirty days which will be found to be 900 shillings and hence it may be thus concluded If 900 shillings last me 30 days how long shall 1500 shillings last me Facit 50. Another question may be this If a piece of Cloth conteining 40 yards did cost me 30 pounds at what rate must that Cloth be sold to gain 12 pound in the hundred Before this question can be resolved I must find the price of one Yard by a proportion Geometrical by way of preparation thus If 4â— Yards cost 50 pound what shall one Yard cost facit 1¼ Then I say If 100 pounds be increased to 112 to what shall 1¼ be increased Answer 1â…– CHAP. XXII The Rule of Three in fractions 1. FRactions as I have already shewed are of 3 sorts viz. such as are many times expressed like integers such as are expressed by their numerators and denominators and such as have a unite with Cyphers annexed for their denominator and now known by the name of Decimal fractions in each of these I will exemplify this Rule by some few examples and first in such fractions as are expressed like Integers 2. Your question being stated bring your numbers propounded into the least name mentioned or as low as you desire the question to be answered in then multiply the second and third and divide the product by the first if your question belong to the Rule of Three direct or multiply the first and second and divide by the third if it belong to the Rule of Three inverse the quotient being reduced shall be the answer required Example If 7 lib. 8 ounces of Currants cost 2s 7d what shall 100½ 13 lib. cost In this question the first and third terms must be reduced into ounces and the second term into pence and then the question will stand thus If 112 ounces Cost 31 pence what shall 1267 ounces cost Facit 359d 1 64 112 farthings that is 1l 9s 11d 1f â…˜ of a farthing A second example may be this A man doth borrow of his friend 246 lib. 7 shillings and 6 pence for 27 weeks for how long time must he lend his friend 329 lib. 13 shillings 4d to require his kindness Here it is plain that the fourth term required must be less than the second term given and therefore the greater extream must be the Divisor and the numbers being reduced and placed as hath been directed they will stand thus 59130 187 79120 in which the greater extream being in the third place doth plainly shew that this question doth belong to the Rule of Three inverse and therefore as 3 Example in Sexagenary numbers If 48 firsts 28 seconds give 60 minutes what shall 25 minutes 12 seconds give According to the former Rules given in Multiplication 25 firsts 12 seconds being multiplied by 60 minutes the product is 25 degrees 12 firsts 00 seconds which being divided by 48 firsts 28 seconds the quotient will be 31 firsts 08 seconds 4. Example in vulgar fractions If â…” of a Yard of Velvet be sold for â…˜ of a pound sterling what shall â…ž of a Yard cost If respect be had to the Rules of the Multiplication and Division of fractions already delivered in the 14th Chapter the working of this Rule in fractions will be the same as in whole numbers thus â…˜ and â…ž being multiplied together the product will be 28 40 which being divided by â…” 7 â…” the quotient is 84 â—0 or 1 lib. 4 â—0 or 1 20 and so the answer is 1 pound and one shilling Otherwise thus Multiply the Denominator of the first number by the numerators of the second and third continually so shall the last product be a new numerator then multiply the numerator of the first number by the denominators of the second and third numbers continually so shall the last product be a new denominator and this fraction shall be the quotient desired Thus the denominator 3 being multiplied first by 4 and then by 7 the last product is 84 and the numerator 2 being multiplied first by 5 and then by 8 the last product will be 80 for a new denominator and so the