Arithmetic, Part 2

Arithmetic · Part 2

DECIMAL ARITHMETICK. The Second Part. With the ground and reaſon thereof, illuſtrated by divers Examples, in all the moſt uſual Rules of Arithmetick.

LONDON Printed Anno Domini 1659.

p. 181The Second Part, Containing DECIMAL ARITHMETICK.

HAving in the firſt Part of this Book exemplified the Art of Vulgar Arithmetick both in whole Numbers and Fractions; We come now to treat of Decimal Arithmetick, which teacheth how to perform (as it were) in whole Numbers all that the former did effect by Fractions. I will not inſiſt upon the antiquity or excellency of this kind of Arithmetick, but come immediately to the practice thereof, and ſhall therefore premiſe theſe Propoſitions following.

Propoſition 1.

A vulgar Fraction being given, how to reduce the ſame into a Decimal.

p. 182THE RULE.

To the Numerator of the Fraction given, adde what number of Cyphers you pleaſe; then divide the Numerator by the Denominator, the quotient ſhall be the Decimal fraction required.

Example 1.

Let it be reuired to reduce 4/17 into a Decimal: firſt, to the Numerator 4, adde five Cyphers, ſo will it be 400000, divide this number by the Denominator 17, and the quotient will be .23529, which is the Decimal required.

¶ And here note, that Decimal fractions are not written in a ſmaller figure with a line between them, as vulgar fractions are, but of the ſame figure, only there muſt be a Comma put between the whole Number and the Fraction, and that is the diſtinction.

Example 2. If you would expreſſe 235 4/17 in a decimal way, it muſt be written as followeth.

By the laſt example you finde that 4/17 reduced to a decimal was .23529, therefore 235 4/17 muſt be written thus: 235,23529. In decimal fractions, the Numerator is only expreſſed, and the Denominator onelyonly intimated; for this is general, Of how many figures ſoever the Numerator of a decimal fraction doth conſiſt, of ſo many Cyphers with a Unite before them, doth the Denominator of the ſame fraction conſiſt. So this Decimal 12,625, if it p. 183were writtenin a vulgar way, would be 12. 625/1000, but in a decimal, only 12,625, the Comma between 12 and 625 diſtinguiſheth the whole number from the fraction, and the fraction 625 conſiſting of three figures, intimates that the Denominator thereof muſt conſiſt of three Cyphers and an unite before them; So the decimal before expreſſed, 235,23529, if it were written in the vulgar way, would be 235 23529/100000.

But it ſufficeth to expreſſe in Decimals the Numerators only, and omit the Denominators, the Denominators of all Decimal fractions being either 10, 100, 1000, 10000, 100000, &c. according to the number of figures contained in the Numerators.

According to this Rule, you ſhall find that

⅘ will be in deicimals by adding 5 Cyphers. .80000

12 3/7 will be in deicimals by adding 5 Cyphers. 12.42857

132 9/17 will be in deicimals by adding 5 Cyphers. 132.52941

And by this meanes, all manner of fractions of Coyns, Weights, and Meaſures, may be reduced from vulgar fractions, to decimal fractions; as by the next Propoſition will appear.

Propoſition 2. How to expreſſe Engliſh Coin, in Decimal Numbers.

Let it be required to expreſſe 9 ſhillines (which is 9/20 of a pound Sterling) in a decimal; To the Numerator 9 adde Cyphers, making it 900, which divide by 20, the quotient is .45, for the decimal of 9 s. So the decimal of 13 s. will be 65…, and ſo for any number of ſhillings.

p. 184¶ Here note, that in the Reduction of Vulgar fractions into Decimals, that many times the firſt, ſecond or third places of the Decimal fractions are Cyphers, as in the following Table the Decimal of one farthing is 00104167, and the reaſon is, becauſe if you reduce 1/960 into a Decimal (for one farthing is the 960 part of a pound Sterling) you ſhall by adding of Cyphers to the Numerator) find the Quotient to be 104167, but two Cyphers muſt be placed before it; becauſe dividing 1000000 by 960, the place of unites in the Diviſor at the firſt demand extendeth unto the third Cypher in the Dividend, for in reducing of Vulgar fractions to Decimals, this is, A general Rule.

That if the place of unites in the Diviſor, at the firſt demand extend but unto the firſt of the Cyphers annexed to the Numerator of the fraction, there muſt be no Cypher put before the Quotient, but if the place of unites extend unto the ſecond Cypher added, then one Cypher muſt be placed before the Quotient, if unto the third Cypher, then two Cyphers muſt be placed before the Quotient, &c.

According to which Rule, if you make trial you ſhall find that the Decimal of 7 s. will be .35, the Decimal of 5 d. will be .02083333, the Decimal of two farthings will be .00208333, as in the Table.

By theſe rules laſt delivered are the enſuing Tables of Engliſh Money, Weight, and Meaſure compoſed, and the like may be done for any forreign Coin, &c. according as every mans occaſions ſhall require,

p. 185The TABLE of Engliſh Coin in Decimals.

Troy Weight in Decimals.

p. 187Averdupois great weight in Decimals.

p. 188Averdupois little weight in Decimals.

Liquid Meaſures in Decimals.

p. 189Dry Meaſures in Decimals.

Long Meaſures, the-Integers being yards and ells in 'Decimals.

Time in Decimals.

Dozens in Decimals.

p. 191The uſe of the fore-going Tables.

THe Tables preceding are in number nine; The firſt being of Engliſh Coyn, the ſecond of Troy weight, the third of Averdupois great weight, the fourth of Averdupois little weight, the fifth of Liquid Meaſures, the ſixth of Dry Meaſures, the ſeventh of Long Meaſures, the eighth of Time, and the ninth of Dozens: Theſe ſeveral Tables are made by the Rules immediately going before them, and their uſe is to expreſſe in Decimal numbers either Money, Weight, or Meaſure, as by the following Propoſitions will appear.

PROP. I. How to expreſſe Engliſh Coin in Decimals.

The firſt of the nine Tables is for this purpoſe, therefore if you would expreſſe either ſhillings, pence, or fartbings, in Decimal numbers, you muſt repair to the firſt Table, which is of Engliſh Coyn, and there againſt 13 ſhillings you ſhall finde 65, which is the Decimal of 13 ſhillings, alſo againſt ſeven pence you ſhall finde 02916667, which is the Decimal repreſenting 7 pence: Alſo againſt 2 farthings you ſhall find p. 19200208333, which is the Decimal anſwering to 12 farthings, and the like is to be done for any other number of ſhillings, pence, or farthings.

But if it be required to find the Decimal of divers Denominations of Coin in one ſum, as of ſhillings, pence, and farthings together, you muſt adde the Decimals of all the particulars together, and the ſum of them ſhall be the Decimal ſought.

Example.

If you would know the Decimal of 13 s. 7 d. 2 q. in one number,

you muſt firſt look in the Table for the Decimal of 13 s. which is 65, and ſet that down, then look for the Decimal of 7 d. which is 00916667, and ſet that down alſo: Laſtly, ſeek the Decimal of 2 q. which is 00208…33, ſet that down alſo; then if you adde theſe three numbers together, as in common Addition, you ſhall find the ſum of them to be 68125000, which is the Decimal belonging to 13 s. 7 d. 2 q. as by tee work in the Margent appeareth.

p. 193PROP. II. How to expreſs Troy weight in Decimals.

The ſecond Table is of Troy weight, the ſeverall Denominations whereof are Ounces, Peny-weights, and Grains: So that by the Table you ſhall find that the Decimal belonging to five ounces is 41666667, the Decimal belonging to 17 peny-weight, is 07083333, and the Decimal belonging to 13 Grains is 00225694, and ſo of any other number of ounces, peny-weights, and grains ſeverally.

But if it were required to expreſſe theſe (or any other) ſeveral Denominations in one Decimal Fraction, then you muſt (as before you did for money) take out of the Table the ſeveral Decimals belonging to the reſpective quantities, and adde them together, ſo ſhall the ſum of that addition be the Decimal ſought.

Example.

If it were required to finde a Decimal which ſhould repreſent 5 ounces, 17 peny-weight, 13 grains, you muſt firſt look in the Table for the Decimal belonging to five ounces, which is 41666667, and write it down, then look the Decimal belonging to 17 grains, which is p. 19407083…33, and write that down,

then look for the Decimal of 13 grains, which is 00225694, and write that down, then adding theſe three numbers together, you ſhall find the ſum of them to be 48975694, which is the Decimal repreſenting 5 ounces, 17 grains, 13 peny-weight, as by the operation in the margent appeareth.

PROP. III. How to expreſs Averdupois great weight in Decimals.

This Averdupois great weight is the third Table, the ſeveral Denominations whereof are Quarters of Hundreds, Pounds, Ounces, and Quarters of Ounces, thus you ſhall finde in the Table, that the Decimal of 3 Quarters of a hundred is 75, the Decimal of 22 pounds is 19642857, the Decimal of 7 ounces is 00390625, and the Decimal of 3 quarters of an Ounce is 00041853, in this manner by the Table may you finde the correſpondent Decimal belonging to any number of quarters, pounds, ounces, and parts of ounces ſeverally.

But if it be required to finde one decimal number which ſhall repreſent divers denominations, you muſt firſt finde the decimal belonging to the ſeveral particulars, and add them together, p. 195the ſum whereof ſhall be the entire decimal required.

Example.

Let it be required to finde a decimal which ſhall repreſent 3 quarters,

22 pounds, 7 ounces ¾. Firſt, look in the Table for the decimal of three quarters of a hundred, which is 75, and write it down, then look for the decimal of 22 pound which is 19642857, and write that down, alſo look the decimal belonging to ſeven ounces, which is 00390625, and write that down: Laſtly, ſeek the decimal of three quarters of an Ounce, which is 00041853, and write that down, then adding theſe four numbers together, you ſhal find their ſum to be 95075335, which is the Decimal repreſenting 3 qu. 22 lb. 7 oun. ¾.

PROP. IV. How to expreſſe Averdupois little weight in Decimals.

The fourth Table is of Averdupois little weight, the denominations whereof are ounces, drams, and quarters of drams, ſo that the Decimal of 11. ounces is 6875, the Decimal five drams is 01953125, and the decimal of one quarter of a dram is 00097656.

But if it be required to find one decimal number, p. 196which ſhall repreſent 11 ounces,

5¼ drams, then you muſt firſt look for the decimal belonging to 11 ounces, which is 6875, and ſet it down, then look for the Decimal anſwering to five drams, which is 01953125, and ſet that down. Laſtly, look the decimal belonging to a quarter of a dram, and ſet that down; theſe three numbers being added together, produce 70800781, which is the correſpondent Decimal belonging to 11 ounces, five drams, and a quarter of a dram.

PROP. V. How to expreſſe Liquid Meaſures in Decimals.

Becauſe there is ſo great variety of Liquid meaſures, that hardly any two commodities are ſold by the ſame, the difference of the gallon continually making alteration, we have therefore in this fifth Table made the greateſt denomination to be one gallon, the next leſſe denomination being pints and quarters of pints, ſo that in the Table you ſhall finde the Decimal belonging three pints, to be 375, and the Decimal belonging to two quarters, or half a pint, to be 0625, and ſo for any other.

But for to expreſſe pints and parts of pints in one entire decimal number, you muſt adde the Decimals of the ſeveral denominations together, p. 197and their ſumme ſhall be the entire Decimal.

So if you were to expreſſe 3 pints and an half,

in one entire decimal number, adde the decimal of three pints, which is 375, to the decimal of two quarters, which is 0625, and their ſum 4375, ſhal be the Decimal of three pints and an half.

PROP. VI. How to expreſſe dry meaſures in Decimals.

The ſixth Table is of dry meaſures,

the ſeverall denominations whereof are Buſhels, Pecks, quarters of Pecks, and Pints, ſo may you find the decimal of five buſhels to be 625, the decimal of two pecks to be 0625, the decimal of three quarters of a peck to be 0234375, the decimal of two pints to be 0039063. Thus are the correſpondent decimals belonging to the ſeveral denominations found: But if you would have one number to expreſſe five buſhels, 2¾ pecks, two pints: you muſt firſt find the decimal belonging to five pecks, wich is 625, and write it down, then find the decimal of two pecks, which is 0625, then ſeek the decimal of three quarters of a peck, which is 0234375, & write that down. Laſtly, ſeek the decimal of two pints, which is 0039063, which numbers being added together, produce 7148438, which is the decimal belonging (or expreſſing) five buſhels, 2¼ pecks, and two pints.

p. 198PROP. VII. How to expreſſe long meaſures in Decimals.

The ſeventh Table is of long meaſures, the integers being Yards and Ells: and the leſſer denominations are quarters of Yards or Ells, Nales, and quarters of Nales. So may you find in the Table that the decimal of three quarters of a Yard, or an Ell is 75, the decimal of two Nales, is 125, and the decimal of one quarter of a Nale is 015625.

But if you would have one number to expreſſe 3 quarters of a Yard,

or an Ell, two Nales, and one quarter of a Nale, you muſt firſt ſeek the Decimal of three quarters of a Yard or Ell, which is 75, and write it down, likewiſe ſeek the Decimal of two Nales, which is 125, and write that down. Laſtly, ſeek the decimal of one quarter of a Nale, which is 015625, and write that down, theſe three numbers added together, make 890625, which is the Decimal belonging to 3 quarters of a Yard or Ell, two Nales, and one quarter of a Nale.

PROP. VIII. How to expreſſe the parts of Time in Decimals.

Time is uſually divided into Years, Moneths, p. 199and Dayes: So in the eighth table of Time, which conſiſteth of theſe two denominations, Moneths and Dayes, you may find that the decimal of five moneths is 416667, the decimal of 26 dayes is 071233. Theſe are the particular decimals, but the compound decimal number repreſenting five moneths, 26 dayes, is 487900, as you ſhall find, if you adde 071233, which is the decimal of 26 dayes, to 416667, which is the decimal of five moneths.

PROP. IX. How to expreſſe Dozens in Decimals.

The laſt table is of Dozens, the integer being a Groſſe, and the ſmaller denominations are Dozens, and parts of Dozens, ſo may you find the decimal of ſeven dozen to be 5833333, and the decimal of five parts of a dozen to be 0347222, and theſe two numbers added together, make 6220555, which is the number which repreſenteth 7 dozen, and 5/12 parts of a dozen.

Hitherto we have ſhewed the uſe of the foregoing tables in expreſſing of Fractions in decimal numbers. It reſteth now to ſhew the uſe of them in finding what fraction either of money, weight, or meaſure, any decimal number given doth repreſent, and that ſhall be made evident by the enſuing Propoſitions.

p. 200PROP. X. A decimal number being given, how to find what Fraction it doth repreſent.

Let 02916667 be a decimal number repreſenting ſome Fraction part of Engliſh Coin, becauſe it is required to find the value of this Fraction in Engliſh Coin, you muſt therefore repair to the table of Engliſh Coin, in the ſecond column of which table ſeek for the number given (viz. 02916667) which you ſhall find to ſtand againſt 7 pence, and ſo much is the value of the decimal Fraction 02916667, in Engliſh Coin.

Alſo if the decimal Fraction 75 were given, you ſhall find the value thereof to be 15 ſhillings, and the value of 003125 to be three farthings.

Likewiſe in the table of Troy weight, if 41666667 were given, it would ſignifie five ounces, and 05416667 would expreſs 13 peny-weight, and 00173611 will expreſſe ten grains, &c

After this manner may you find the value of any decimal number given, either in Money, Weight, or Meaſure, when the number given, may be exactly found in the table: But if the number given cannot be found exactly in the table unto which it is directed, then you muſt find in the ſame table, the neereſt number you can leſſe than the given number, and take the number that anſwers unto it in the firſt column which will be the greateſt fraction of the number required: p. 201then ſubtracting the decimal thus found, out of the decimal given, you ſhall have a remainder, which remainder ſeek alſo in the ſecond column of the table, if it may be found; if not, ſeek the neereſt leſſe, and the number anſwering thereunto in the firſt column ſhall be the next greateſt fraction; then ſubtracting this decimal found out of the former remainder, there will be another remainder, which alſo ſeek in the table, and proceed as in the former: An Example or two will make all plain.

Example 1.

Let 68125000 be a Decimal given, repreſenting ſome part of Engliſh Coin. If you look in the table of Engliſh Coin for 68125000, you cannot find it, but the neereſt number in the table leſſe than it is 65, againſt which I find 13 s. ſo that 13 s. is the greateſt fraction part of Engliſh Coin agreeing to this number.

This done, ſubtract 65 out of 68125000, and there will remain 03125000, which number alſo you muſt ſeek in the table of Engliſh Coin, but being you cannot finde it there, you muſt take the neereſt number leſſe than it, which is 02016667, againſt which I finde 7 pence, which is the next greateſt fraction part of Engliſh Coin agreeing to this number.

Again, ſubtract 02916667, out of 03125000, and there will remain 00208333, which number ſeek in the table, and you ſhall finde it to ſtand p. 202againſt two farthings, and ſo much doth this laſt remainer ſignifie in Engliſh Coin, and the whole given number 68125000 doth repreſent in Engliſh Coin 13 ſhillings, ſeven pence, two farthings, as by the operation following doth appear.

So doth the whole number repreſent 13 s. 7 d. 2 q.

Example 2.

Let the Decimal 87426934 repreſenting ſome fraction of a pound ſterling be given. If you look in the table of Engliſh Coin for 87426934 you cannot finde it; but the neereſt number in the table leſſe than it is 85, againſt which I finde 17 ſhillings, ſo that 17 ſhillings is the greateſt fraction part of Engliſh Coin, agreeing to this number.

Then ſubtracting 85 out of 87426934, there will remain 02426934, which number alſo you muſt ſeek in the table of Engliſh Coin, but ſeeing you cannot finde it there, you muſt take the neereſt p. 203number leſſe than it, which is 02083333, againſt which I finde five pence, which is the next greateſt fraction part of Engliſh Coin.

Laſtly, ſubtract 02083333, out of 02426934, and there will remain 00343601, which number you muſt alſo ſeek in the table of Engliſh Coin; but not finding it exactly there, you muſt take the neereſt number leſſe, which is 003125, againſt which you ſhall finde three farthings, which is the next greateſt fraction part of Engliſh Coin, and the Decimal 87426934, doth in value ſignifie 17 ſhillings, 5 pence, 3 farthings, and ſomething more, for 003125 is the decimal of 3 farthings; and the number you are to look for in the table is 00343601, greater than the decimal of 3 farthings; wherefore, if you ſubtract 003125 out of 00343601, there will remain 31101, which is the 31101/100000 part of a farthing, which is inconſiderable. See the following operation.

p. 204¶ And here note, that whatſoever hath been here ſaid concerning the uſes of the table of Engliſh Coin, the ſame order is to be obſerved in the uſe of the other tables of Weight, Meaſure, Time, &c. as by the following Examples (if you make trial) will appear.

Examples.

1 If this decimal 48975694, were given to know the value thereof in Troy weight, you ſhall finde it to containe 15 ounces, 17 penyweights, and 13 grains.

2 Alſo if 95075335 were a Decimal given, and it were required to find the value thereof in Averdupois great weight, you ſhall find it to contain 3 quarters of a hundred, 22 pound, 7 ounces, and 3 quarters of an ounce.

3 Likewiſe, if 70800781 were a decimal Fraction give, you ſhall finde the value thereof in Averdupois little weight to be 11 ounces, 5 drams and one quarter of a dram.

4 If 4375 were a Decimal, whoſe value were required in Liquid Meaſure, you ſhall finde it to contain 3 pints and an half.

5 Let 7148438 be a Decimal given, whoſe value is required in dry meaſures, you ſhall find it to contain five buſhels, 2 pecks, 3 quarters of a peck, and 2 pints.

Thus have I ſhewed you the uſe of theſe decimal tables in expreſſing of the fraction parts of Money, Weight, Meaſure, &c. But becauſe theſe p. 205tables may not be alwayesat hand when there is need of them. I will here ſhew you how the value of any Decimal given may be known by Multiplication onelyonly; and this is

THE RULE.

Multiply the Decimal given by the number of known parts of the next inferiour Denomination, which are equall to the Integer, the Product is the value of the Decimal propoſed in that inferiour Denomination; and if there happen to be any Decimal in the Product, you may in like manner finde the value thereof in the next inferiour Denomination, and ſo proceed till you come to the leaſt known parts of the Integer.

Example.

Let .67395834 be a Decimal given, repreſenting the fraction of a pound ſterling. Firſt, multiply .67395834 by 20, (the number of ſhillings in a pound ſterling) and the Product will be 1347916680, from which cutting off the laſt eight figures with a daſh of the pen (becauſe there were eight figures in the given Fraction) there will ſtand before the line towards the left hand 13, which are ſhillings, and the remainer 47916680 ſtanding behind the line, will be the fraction part of one ſhilling ſterling, which number .47916680, you muſt multiply by 12 (the number of pence in one ſhilling) and the Product will be 575000160, from which number cut p. 206off the laſt eight figures as before, and there will be five left to the left hand, which are five pence, and the figures on the right hand of the line, viz. 75000160 are the fraction part of one penny ſterling, which therefore multiply by 4 (the number of farthings in one peny) and the Product of that multiplication will be 300000640, from which cut off the laſt eight figures to the right hand, and there will be left three towards the left hand, which repreſenteth 3 farthings, and the remaining figures towards the right hand are but the fraction part of a farthing, which we therefore reject. And thus you finde by Multiplication onelyonly, that this fraction .67395834 doth repreſent in the known parts of Engliſh Coin 13 ſhillings, 5 pence, 3 farthings, as by the following operation appeareth.

…

In like manner, if this fraction .94809028 were given, repreſenting ſome fraction part of p. 207Troy weight, you ſhall finde the value thereof to be 11 Ounces, 7 peny weight, 13 grains, as by the operation following appeareth.

…

In this manner may any Decimal given be reduced into the known parts of the Integer by Multiplication onelyonly.

¶ Here note, that whereas in the preceding tables the decimal fractions conſiſt of ſeven or eight Figures, we ſhall in the proſecution of our work make uſe onelyonly of four or five of the firſt of them, which will be ſufficient in ordinary practice, and come neer enough to the truth in any ordinary queſtion whatſoever.

So if in ſtead of 02916667, which is the fraction part of 7 pence, you take onelyonly 02916, it will be ſufficient.

p. 208Alſo for 05833333 take 05833 in Troy w. in dry meaſure. in Time.

Alſo for 0058594 take 0058 in Troy w. in dry meaſure. in Time.

Alſo for 5833333 take 5833 in Troy w. in dry meaſure. in Time.

Thus much concerning the conſtruction and uſe of the decimal Tables, we ſhall now come to the praetice of Decimal Arithmetick, which ſhall be taught in the Rules following.

Of Notation of Decimals.

NOtation of Decimals is contrary to that of whole numbers; for whereas in whole numbers their values are increaſed ten-fold by continuall addition of Cyphers towards the left hand: ſo on the contrary, the values of the places of Decimals do decreaſe in the ſame proportion.

And whereas in whole number, Cyphers in the firſt place towards the left hand are unneceſſary, yet in Decimals they are abſolutely neceſſary to diſcover the true denominator. Alſo Cyphers at the end (or towards the right hand) of decimal numbers are of no value, for one ſingle Figure in decimals ſignifies as much as the ſame Figure would do, if there were Cyphers placed behind it, ſo 7 is equivalent unto 70, 700, or 7000, &c. For the denominators of decimal Fractions are alwayes Cyphers with a unite towards p. 209the left hand, as hath been already intimated. So 70/100 being reduced to its leaſt terms will be 7/10, and 7000/10000 will be reduced to 7/10 alſo, and ſo of any other, as by the Table following doth evidently appear.

Addition of Decimals.

IN addition of Decimals, the ſame order is to be obſerved as in addition of numbers of one denomination before taught in the firſt part, in which there is no difficulty: But in decimal numbers the chief care to be taken is in placing your whole numbers and fractions in their due order, which you ſhall eaſily and certainly do, if you obſerve this generall rule, viz. to place your whole numbers and fractions one under another, ſo that the points of ſeparation which (in decimal numbers) diſtinguiſh the whole numbers from the fractions, p. 210ſtand directly one under the other, then are you to proceed in the addition of them in all reſpects, as you did in whole numbers:

Example 1.

Let it be required to add together into one ſum theſe ſeveral ſums following, in a decimal way, viz. 36 li.—2 s.—8 d. 29 li.—0 —2 d. 21 li.—16 s. 9 d. and 6 li. 2 s. 5 d.

Firſt, ſet down 36 li. and a point or Comma after it, then for the fraction part of 2 s. 8 d. look in your Table of Engliſh Coin, where you ſhall finde the decimal fraction of a ſh. 8 d. to be, 1333 therefore fot 36 li. 2 s. 8 d. ſet down 36. 1333.

Secondly, for your 29 li. 0 s. 2 d. ſet down 29.00831.

Thirdly, for your 31 li. 16 s. 9 d. ſet down 31.8375.

Laſtly, for your 6 li. 2 s. 5 d. ſet down 6. 1208 as you ſee done in the operation following.

Your decimal numbers being thus placed in due order one under another, proceed to the adding of them together, as if they were whole numbers, and you ſhall finde the ſum or totall of them to be 103.0999.

p. 211Now the 103 which ſtands towards the left hand are 103 pounds, and the 0999 which ſtands towards the right hand of the Comma, is the fraction part of one pound ſterling, the value whereof you may finde (by the Propoſition before going) to be two ſhillings fer…, which ſhould be two ſhillings exact, but it wanteth ſome-what, viz. the / 1000 part of a farthing, which is inſenſible; for if by the fore-mentioned rule you ſeek the value of the decimal fraction, .0999, you ſhall finde it to be 1 ſhilling, 11 pence, 3 farthings and the 2/1000 part of a farthing, which you may call in all 2 ſhillings, for decimal numbers will ſeldome happen to give the exact value of fractions, but will be either greater or leſſer than they ought to be; but in ſuch a ſum as this is, the thouſand part of a farthing is not to be regarded.

Example 2.

Let it be required to adde together in a decimal way theſe ſums following, viz. 29 li. 18 s. 7 d. 3 q. 63 li. 11 s. 2 d. 1 q. 129 li. 4 s. 0 d. 2 q. and 3 li 7 s. 10 d. 1 q.

Firſt, for 29 li. 18 s. 7 d. 3 q. ſet downe 29.93229.

Secondly, for 63 li. 11 s. 2 d. 1 q. ſet downe 63.55937.

Thirdly, for 129 li. 4 s. 2 q. ſet downe 129.20208.

Laſtly, for 3 li. 7 s. 10 d. 1 q. ſet downe 3.39271, as you ſee here down in the Margent.

p. 212Your decimal numbers thus placed in order,

adde them together, as if they were whole numbers, and you ſhall finde the ſum of them to contain 226.08645.

Now the 226 which ſtands towards the left hand of the Comma, are 226 pounds, and the other figures towards the right hand, viz. 08645 are the fraction of a pound ſterling, which if you reduce by the fore-mentioned Propoſition, you ſhall finde the value thereof to be 1 ſhilling, 8 pence, 3 farthings, ſo the whole ſumme is 226 li. 1 s. 8 d. 3 q.

And here note, that what hath been ſaid, as concerning Money, the ſame is alſo to be underſtood of Weight, Meaſure, Time, &c. as by the following Examples will appear.

Other Examples for practice.

Example 1. In Money.

Example 2. In Troy weight.

p. 213Example 3. In Averdupois little w.

Example 4. In Averdup. great weight.

Subtraction of Decimals.

THe Subtraction of decimals differeth nothing from the ſubtracting of one whole number from another, and the decimal numbers to be ſubtracted one from another, muſt be placed in the ſame order, as in Addition of decimal numbers, the practice of Subtaction ſhall be ſeen in the following Examples.

Example 1.

Let it be required to ſubtract 31 li. 16 s. 9 d. out of 36 li. 2 s. 8 d.

Firſt, for your 36 li. 2 s. 8 d. ſet down the decimal thereof, which is 36.1333.

Secondly, for your 31 li. 16 s. 9 d. ſet down the Decimal thereof 31.8375.

p. 214This done draw a line under them,

& ſubtracting the leſſer from the greater, you ſhall finde the remainer to be 4. 2958. the 4 on the left ſide of the Comma are four pounds, and the 2958 which ſtandeth towards the right hand, is the fraction part of a pound, the value whereof being ſought, will be found to be 5 s. and 11 pence▪ So that if you ſubtract 31 li. 16 s. 9 d. out of 36 li. 2 s. 8 d. there will remain 4 li. 5 s. 1… d.

But if divers ſums be to be ſubtracted out of one greater ſum, then you muſt firſt adde all the ſeveral ſmaller ſums together, and ſubtract the ſum of them from the greater given ſum, ſo ſhall the reſidue be the ſum deſired.

Examples for practice.

Example 1.

Example 2.

Example. 3.

p. 215Multiplication of Decimals.

Multiplication of Decimals differeth nothing at all from the multiplication of whole numbers, for making the greater number the multiplicand, and the leſſer number the multiplyer, the number iſſuing from that multiplication ſhall be called the Product.

Now in the multiplication of decimal numbers one by another, if there be any Fraction either in the multiplicand or multiplyer, or Fractions in both: So many figures as the Fractions contein, ſo many figures muſt be cut off from the Product towards the right hand, which ſhall be the Fraction of the Product, and the figures towards the left hand of the Comma in the Product, ſhall be the Integers of the Product.

Example 1.

Let it be required to multiply 34 pound, five ſhillings, three pence, by 16 pound, ſix ſhillings, ſix pence.

Firſt, ſeek the Decimal of 34 li. 5 s. 3 d. which you ſhall finde to be 34. 2625, make this number p. 216your Multiplicand, then ſeek the Decimal of 16 li. 6 s. 6 d. which you ſhall find to be 16. 325, … make this decimal number your Multiplyer, then draw a line, and multiply theſe two numbers together, as if they were whole numbers, and you ſhall finde the Product of them to be 559. 3353125. Now becauſe there are four figures in the Multiplicand which are Fractions, namely, theſe four towards the right hand, viz. 2625, and there are alſo three figures in the multiplyer, which are Fractions, namely, theſe three towards the right hand, viz. 325, that is in all ſeven figures repreſenting Fractions, I therefore cut off from the product the ſeven figures towards the right hand, by making of a Comma there to diſtinguiſh the whole number from the fraction: So is 559 the Integer or whole number, and 3353125, the fraction of this multiplication.

Example 2.

If there be Fractions in the multiplicand, and none in the multiplyer, yet that work is ſtill the ſame, for you muſt cut off onelyonly ſo many figures from the product, as there are Fractions either in multiplicand, multiplyer, or both: So if it were required to multiply 5767 yards, and 3 quarters p. 217of a yard, by 235 yards, you muſt firſt ſet down 5767. 75 for your 5767 yards, … and three Quarters, which number muſt be your multiplicand, alſo ſet down 235 yards for your multiplyer, then multiplying them together, as if they were whole numbers, you ſhall finde the product to be 1255421, 25, and becauſe there are onelyonly two Fraction Figures, both which are in the multiplicand, namely, the two laſt thereof .75, and none in the multiplyer, I therefore cut off onelyonly two figures of the product, namely, the two laſt, which are .25, ſo is the product of this multiplication 1355421. 25; which is 1355421 ſquare yards, and one quarter of a yard, And ſo if a garden or other piece of land lying ſquare, ſhould contein in length 5767 yards, and three quarters, and in breadth 235 yards, the whole piece would contein 1355421 ſquare yards, and one quarter of a yard.

Example 3.

If decimal Fractions be to be multiplyed by decimal Fractions, you muſt then (as before) multiply them as whole numbers, and from the Product cut off ſo many Figures towards the right hand, as there are Figures in the multiplicand and the multiplyer: So if it were required to multiply .953 by 782, you ſhall finde their p. 218product to be .745246, … which being but ſix figure sin all, I cut them off and that fraction .745246 is the product of the multiplication of the two given fractions.

Example 4.

If any two Decimal fractions being multiplied together, the product thereof doth not conſiſt of ſo many places as are required (by the former rules) to be cut off, you muſt then ſupply that defect by prefixing a Cypher, or Cyphers before the product towards the left hand: So if theſe Decimal fractions .063 and .0752 were to be multiplyed, … their product would be 47376. Now (by the former rules) you ſhould cut off ſeven figures of the product towards the right hand, but this product 47376 conſiſteth but of five figures; wherefore to make it ſeven figures, I prefix two Cyphers before the product on the left hand, making it .0047376, and that is the true product produced by this multiplication.

Example 5.

If you would multiply any Decimal (either fraction onelyonly, or whole number and fraction together) by 10, 100, 1000, &c. You muſt adde ſo many Cyphers to the multiplicand, as there p. 219are Cyphers in the multiplyer, and cut off ſo many Figures as there are fractions in the multiplicand, and that number ſhall be the product required: So if 7,856025 were a Decimal given to be multiplyed by 100, you muſt (becauſe there are two Cyphers in 100) add two Cyphers to the number given, making it 785,…02500, then becauſe there were ſix fraction Figures in the multiplicand, if you cut off the ſix figures of this number towards the right hand, it will be 785,602500, which is the true product required.

Examples for practice.

Example 1. …

Example 2. …

Example 3. …

Example 4. …

p. 220Diviſion of Decimals.

AS Diviſion of whole numbers is the hardeſt of the four Species of Vulgar Arithmetick, ſo the Diviſion of Decimals is the moſt difficult of the four kinds of Decimal Arithmetick, but I hope to make it plain to the underſtanding of the meaneſt capacity.

The ſeveral varieties that may happen in Diviſion, are principally (if not onelyonly theſe) four. Namely, firſt, to divide whole numbers and fractions, by whole numbers and fractions. Secondly, to divide whole numbers by mixt, or mixt numbers by whole. Thirdly, to divide a greater fraction by a leſſe, and laſtly, to divide a leſſer fraction by a greater.

In Diviſion of Decimals this Rule is general, If the dividend be greater than the Diviſor, the quotient will be either a whole number or a mixt, but if the Dividend be leſſe than the Diviſor, the quotient will be a Decimal. And (for convenience in working, if there be need) any number of Cyphers may be annexed to the Dividend that thereby the quotient may extend to as many places as the tenour of the queſtion ſhall require.

The manner of the working of Diviſion in Decimals, is the ſame with that before delivered p. 221in whole numbers, as will appear by the Examples following, in every of the four premiſed varieties.

The Rule for the firſt varietie.

The Dividend and the Diviſor, being both mixt numbers, or one of them being a whole number and the other a mixt; or the Dividend being a Decimal, and the Diviſor a whole number or a mixt, the firſt figure in the quotient will be of the ſame place or degree, with that figure or Cypher of the Dividend, which at the firſt demand ſtandeth, or (at leaſt) is ſuppoſed to ſtand directly over the place of Ʋnits in the Diviſor.

Example 1. Where the terms given, are both of mixt numbers.

Let it be required to divide 559.3353125 by 16.325. Here the terms given, are both of mixt numbers, which being, placed according to the Rules delivered before, for the Diviſion of whole numbers, the figure in the Dividend, which at the firſt demand, ſtandeth over 6, the place of Units in the Diviſor is 5, and becauſe this ſtandeth in the place of tenths, therefore the firſt figure in the quotient is in the place of tenths alſo, and the whole number conſiſteth of two of the foremoſt places, and the reſt is a Decimal, thus the quotient ſought in our preſent example is 34.2625, of which 34 the two firſt figures is the Integer p. 222or whole number, 2625 the Decimal fraction.

…

Example 2. One of the terms given, being a whole number the other mixt.

The mixt number 1355421.25 being divided by the whole number 235, the quotient will be 5767.75, and the firſt figure in the place of Thouſands, as by the operation it doth appeare.

p. 223…

Example 3. The Dividend being a Decimal, and Diviſor the whole number.

The Decimal fraction .35673 being divided by the whole number 26, the quotient will be 001372, and the firſt ſignificant figure in the place of thouſands or fourth place from Unitie, as by the operation it doth appear.

p. 224…

The Rule for the ſecond varietie.

When the Dividend is a whole or mixt number and the Diviſor a Decimal, add as many Cyphers to the Dividend as there are places in the diviſor for the integral part of the quotient will conſiſt of as many places as the Diviſor, and the places ariſing from the integral parts of the Dividend added together.

Example 1.

Let 348.75 be the mixt number given, to be divided by the Decimal .25, to the number given, I add to Cyphers, the number of places in the Diviſor, and then it will be 348.7500, which being divided by .25; the integral part of the quotient will be 1395. becauſe the whole part of the Dividend 348, being divided by .25 giveth two places, and the number of places in the p. 225Diviſor being two, giveth two more; and ſo the Integral part conſiſteth of four figures, as by the operation.

…

Example 2.

Let the mixt number 72.5 be divided by 075, … the number of places in the integral part of this Quotient will be 966, becauſe there are 3 places in the Diviſor; & but 3, becauſe the p. 226integral part of the dividend is leſſe than the ſignificant figures in the Diviſor, as by the operation it doth appear.

The Rule for the third variety.

When the Terms given, are both Decimals, the Dividend being the greater the integral part of the quotient will conſiſt of as many places as the Diviſor doth.

Example.

Let the Decimal .73958 be divided by the Decimal .32 the integral part of the quotient will be 23, … becauſe the Diviſor doth conſiſt of two places, as by the operation in the margine it doth appear.

The Rule for the fourth varietie.

When the terms given are both Decimals, conſiſting p. 227of equal places, the Dividend being the leſſer term, place the dividend as a Numerator, and the Diviſor as 'Denominator; ſo is ſuch vulgar fraction the quotient fought: But if the terms given conſiſt not of equal places, ſupply the place or places wanting in either of the terms, by annexing a Cypher or Cyphers on the right hand, and then proceed as before. Thus if .27 be given to be divided by .93 the quotient will be 27/93. Alſo if .35 be given to be divided by .78563, the quotient by annexing 3 Cyphers to .35 the leſſer Decimal given will be 35000/78563, which vulgar fractions may be reduced into Decimals if need be, by the firſt Propoſition in this Second part of Decimal Arithmetick.

Examples for Practice.

…

…

p. 228The Rule of three in fractions Vulgar and Decimal.

WHat the Rule of three is, and the manner of working, is already ſhewed in the firſt part, that which we here intend is onelyonly to adde ſome Examples in fractions vulgar as well as Decimal; that by comparing the work in both, the excellent uſe of Decimal Arithmetick might the better appear.

And how to convert the known parts of mony, Weight, or Meaſures Engliſh, into Decimals hath been already ſhewed, both Arithmetically and by Tables; yet to prevent the ſeveral additions and ſubtractions in thoſe Tables, we have here annexed another Decimal Table, for the more ſpeedy Reduction of Engliſh money under two ſhillings, all ſummes of money above, not having pence or farthings annexed, being as eaſily reduced by memory as by Tables; and this we have the rather done becauſe the ſame Table, will alſo reduce the Coins of France, and the parts of Troy weight, if an ounce be made the Integer, which in point of practice is much more uſeful then the pound.

p. 229The Table of Reduction.

p. 230The Table of Reduction.

p. 231Theſe things premiſed; we will now ſhew the uſe of the Table in ſome practical queſtions belonging to the Rule of three direct.

1 Queſtion.

If ⅞ of a yard of cloth, coſt 9/12 of a pound: what ſhall 17 yards coſt at the ſame rate?

If ⅞ coſt 9/12 what ſhall 17 coſt? Anſ. 14 li. 4/7.

Firſt, multiply 9/12 by 17/1 the product is 153/12, then divide 153/12 by ⅞, the quotient is 1224/84: again, if you divide 1224 by 84, the quotient is 14 …/…4 or in the leaſt terms 14 pound 4/7 of a pound.

And the value of this fraction 4/7 of a pound, will be found by the third Rule of Reduction page 107, to be 11 ſhillings, 5 pence, and 4/7 of a peny, which is ſomewhat above two farthings: for it is 2 farthings, and 2/7 of a farthing.

The ſame Queſtion in Decimals.

If ⅞ of a yard of cloth, coſt 9/12 of a pound, what ſhall 17 yards coſt at the ſame rate?

To anſwer this queſtion ⅞ of a yard, and 9/12 of a pound muſt firſt be reduced into Decimals, either by Diviſion, or by the Tables of Reduction: by both which wayes of Reduction the Decimal of ⅞ will be 875, and the Decimals of 9/12 will be .75 and then the terms of the queſtion will ſtand thus;

If .875 parts of a yard coſt .75 parts of a pound, what ſhall 17 yards coſt at the ſame rate?

If 0.875 — 0.75 — 17. Here if you multiply p. 232the ſecond terme 0.75 by 17 the third term given, the product will be 12.75. and this product divided by .875 gives in the quotient 14.57142, that is 14 pound .57142 parts of a pound, or 145 Decades, that is 14 pound 10 ſhillings, and .7142 parts of a Decade (or two ſhillings) which by the preceding Tables is 1 s. 5 d. 2 farthings, and .0059 parts of a farthing.

2 Queſtion.

If a piece of Gold plate weighing 19 ounces 3 peny weight and 5 grains, be worth 62 pounds, 10 ſhillings, 6 pence, what is one ounce of the ſame gold worth.

This queſtion in vulgar fractions muſt be expreſſed thus.

If 1 l. 3437/5260 Troy weight, coſt 62 l. 126/240, what ſhall 1/12 of a pound Troy coſt? at the ſame rate.

To anſwer this queſtion the fractions 1 l 3437/5760, and 6… l. 126/240, muſt be firſt reduced into improper fractions, and the fraction 1/12 into the leaſt known parts of a pound Troy, and then the queſtion will ſtand thus.

If 9197/5760 give 15006/240 what ſhall 480/5760 give?

Now becauſe it is neceſſary the terms given be reduced into their leaſt Denomination, before the queſtion be reſolved, therefore the anſwer may be found, by uſing the terms given thus reduced, as whole numbers, not having any regard to the Denominators of theſe fractions; Saying thus,

If 9197 grains, coſt 15006 pence, what ſhall 480 grains.

p. 233And here if you multiply 15006 by 480 the product will be 7202880 which being divided by 9197 the quotient will be 783 pence 1629/9197 parts of a peny, and dividing .783 by 12 it will be 65 ſhillings 3 pence 1629/9197, or 3 l. 5 s. 3 d. 1629/9197. And although this queſtion is thus more eaſily anſwered then it would have been, if the terms had been wrought as vulgar fractions, yet the ſame terms being reduced to Decimals, the anſwer of the queſtion will yet be found with more eaſe, as ſhall appear by the operation following.

The ſame queſtion in Decimals.

If a piece of gold plate weighing 19 ounces 3 peny weight and 5 grains, be worth 62 l. 10 s. 6 d. what is an ounce of the ſame gold.

The Decimal of 19 ounces 3 peny weight and 5 grains, making an ounce the Integer is by this Table 19.16041, for that 19 ounces are 19 Integers, 2 peny weight is one tenth of an ounce, and the Decimal of one peny weight 5 grains is by this Table 06041; and the Decimal of 62 l. 10 s. 6 d. by the ſame Table is 62.525, and becauſe an Unite or Integer is the third term given, there needs no multiplication, if therefore you divide 62.525 the ſecond term, by 19.16041 the firſt term propounded, the quotient will be 3.2632 that is 3 pounds, 5 ſhillings, 2 pence, and ſomewhat more, as by the operation in the margine it doth appear.

p. 234…

If 5 Els and a quarter of linnen cloth, coſt 2 l. 16 s. 8 d. 3 q. what ſhall 278 els and a half coſt at the ſame rate?

If you would work this Queſtion by whole numbers your eaſieſt way is firſt to reduce all the terms into their leaſt Denominations, that is to ſay, the els into quarters, and the pounds, ſhillings, pence, and farthings all into farthings, ſo ſhall your 5 els and a quarter be 21 quarters, and your 278 els and an half will be 1114 quarters, and your 2 l. 16 s. 8 d. 3 q. will be 2723 farthings, and then will your queſtion ſtand thus in whole numbers.

If 21 quarters — coſt — 2723 farthings — what will — 1114 quarters coſt

Then multiplying the ſecond number by the third, that is, 2723, by 1114, the product will p. 235be 3033422, which divided by 21, the quotient will be 144448 farthings, which being again reduced into pounds, ſhillings, and pence, giveth 150 l. 9 s. and 4 pence, as by the operation following doth appear.

The Operation.

…

p. 236…

But if you would work the ſame queſtion by Decimal numbers you may ſave the labour of reducing the terms to their leaſt Denominations, for 5 Els and a quarter is in Decimal numbers 5.25, and 278 Els and an halfe is 278.50, and 2 pound 16 s. 8 d. 3 q. is in Decimals 2.8364, p. 237and then your queſtion in Decimals will ſtand thus.

If 5.25 Els coſt 2.836 pounds what 278.50 Els

If you multiply (according to the Rule) the ſecond term by the third, that is 2.8364 by 278.50 the product of that multiplication will be 789.937400, which divided by the firſt term 5.25, the quotient will be 150.4642, which Decimal (by the Example in page 177) repreſenteth 150 l. 9 s. 4 pence, and ſo much in money will 278 Els and a half coſt.

The Operation.

…

p. 238…

I have been the larger in this Rule and eſpecially in this Example, which is incumbred with fractions ſufficient, becauſe I would have the Reader the better diſcern the difference between the Vulgar and the Decimal way, and alſo to ſee how expeditious the one is over the other. Now this Example being thus largely explained, I ſhall with the more brevity paſſe over the Rules following giving one Example, or two at the moſt in each Rule; And thus much ſhall ſuffice for the Golden Rule, or Rule of three direct in Fractions.

p. 239An Example in the Rule of Three Reverſe.

A Lends B 233 l. 6 s. 8 d. for a year without intereſt, upon condition that B ſhould do the like courteſie for A when required. A hath occaſion for money 7 moneths, how much money ought B to lend A, to requite his courteſie, and ſave himſelf harmleſſe?

I will not in this place tell you what the Rule of Three reverſe is, nor the manner of working thereof, that being already ſufficiently declared in the firſt part, but give you the Example, and the working thereof, which take as followeth: So will the Queſtion be thus ſtated.

…

p. 240Here you ſee that 12 moneths and 7 moneths are whole numbers, and ſo we let them alone without any reduction, but the Decimal of 233 l. 6 s. 8 d. will be found by the fore-mentioned Tables and Rules to be 233.33, which is the middle term in the queſtion, and of the ſame quality with that, muſt the fourth term ſought be, therefore if (according to the Rule delivered in the firſt part) you multiply 233.33 by 12, the product will be 2799.96, which divided by 7 giveth in the quotient 39999, which is the Decimal of 400 l. and ſo much mony ought B to lend A for 7 moneths.

Examples in the Rule of Proportion, conſiſting of five numbers.

Queſtion 1.

If 100 l. in 12 moneths, yields 6 l. intereſt, what intereſt ſhall 264 l. 16 s. 5 d. yield in 15 moneths at the ſame rate?

Set down your numbers in Decimals, as in the Example following appeareth, ſo ſhall you find the Decimal of 264 li. 16 s. 5 d. to be 264.8208, all the reſt being whole numbers having no fractions joyned with them we neglect, and work p. 241with them as they are, ſo will the ſeveral numbers of your queſtion (if rightly diſpoſed) ſtand as followeth.

…

Your numbers being thus orderly diſpoſed, you muſt according to the Rule before delivered p. 242in the firſt Part, Page 92) multiply the firſt and ſecond terms together, which in this Example are 100 and 12, whoſe product is 1200, which is your Diviſor; Then multiply the three laſt terms one into another, as 264,8208 (which is the Decimal of 264 li. 16 s. 5 d.) by 6, and the product thereof will be 1588.9248, which number again multiplyed by 15, (which is the laſt term,) the product will be 23833.8720 which is your Dividend, and this number being divided by your former product, giveth in the quotient 19.8615, which is the Decimal of 19 l. 17 s. 2 d. 3 q. ferè, and ſo much doth the ſimple intereſt of 264 l. 16 s. and 5 d. amount unto in 15 moneths, after the rate of ſix per centum for a year.

Queſt. 2.

If the carriage of 23 hundred and 3 quarters of any thing 127 miles, coſt 4 li. 13 ſh. 6 d. what ſhall the carriage of 47 hundred and an half of ſuch like commodity coſt, being carried 381 miles.

Place your numbers in order as in the following Example doth appear, then multiply the firſt and ſecond terms together for your Diviſor, and the three laſt one into another for your Dividend, and ſo will the quotient of this diviſion anſwer the queſtion demanded, and the work will ſtand as followeth.

p. 243…

Here you ſee that the firſt and ſecond terms multiplied together produced 3016.25, which muſt be your Diviſor, and the three laſt terms being multiplyed one into another, produce 84605.81250, which number divided by 3016.25, giveth in the quotient 28.050, which Decimal repreſenteth 28 l. one ſhilling, and ſo much will the carriage of 47 hundred and a half coſt, being carried 381 miles?

p. 244Thus have I ſhewed the uſe of decimal Arithmetick in ſuch queſtions as concern the Golden Rule, or Rule of Three; both Direct, Reverſe, and Compounded, by an Example or two in each rule, & thoſe compounded of fractions ſufficient, I ſhould now proceed to queſtions in Fellowſhip, with and without Time, as alſo Barter, Alligation; the Extraction of the Square and Cube Roots, &c. but foraſmuch as theſe laſt mentioned Rules depend only upon the Rule of Three, as by Examples in the firſt part doth plainly appear, I ſhall therefore ſave that labour, and ſpare my Reader the pains of practiſing queſtions which wholly depend upon that which I (by this time,) ſuppoſe him perfect in; Yet if the Reader be deſirous to make trial of any ſuch queſtion for his own ſatiſfaction, he may either make trial of thoſe queſtions in the former part of this Book in thoſe ſeveral Rules, reducing the numbers there given into Decimals, or if he pleaſe, he may frame queſtions according to his own fancy. And thus I ſhall conclude this ſecond Part.

The end of the ſecond Part.