NOTATION.

IT is necessary, that all Persons that would acquaint themselves with the Nature and Use of Numbers. do first learn to know the Characters by which any Quantity is expressed.

These Characters are in number nine, who with a Cypher are the Foundation of the whole Art of Arithmetick. Their form and denomi­nation as in this Example.

[Page 2] These Characters standing alone express no more than their simple value, as 1 is but one, 2 standing by it self signifies but two, and so of the rest; but when you see more than one of those Figures stand together, they have then another signification, and are valued ac­cording to the place they stand in, being dig­nified above their simple quality, according to the Examples in this Table.

The denomination of Places according to this Table, must be well known, and are thus exprest; those standing in the place of Unites, signifie no more than their value before taught; but standing in the second place toward the left hand, they are increased to ten times the value they had before, 1 or One in the Unite place signifies but One; if it stand in the second place toward the left hand, and a Cypher be­fore [Page 3] it thus 10, it hath ten times its simple value, and is called Ten; if 2 stand in the place of the Cypher thus 12, it is then Twelve, being Ten and two Unites; 1, 3, or 3, standing in third place, with Figures or Cyphers toward the right hand of it, doth signifie Hundreds, as 100 is One hundred, 123 is One hundred twenty three, 321 is Three hundred twenty one, 213 is Two hundred thirteen; and so any three of the other Figures have like value, according to their Stations, the first to the right hand in the Unite place signifies so many Unites, the second, or that in the place of Tens, is increased to ten times its simple value, and in the third place, or place of Hundreds, any Figure there standing hath a hundred times the value it would have had were it in the Unite place.

The fourth place is the place of Thousands, any Figures standing there, with three Figures or Cyphers to the right hand of it, is so many Thousands as simply it contains Unites, so 3000 is Three thousand, 9825 is Nine thou­sand eight hundred twenty five, &c.

The fifth place is Ten thousands, and any five Figures placed together, are to be read after this manner: Example.

45326

Forty five thousand three hundred twenty six.

12345

Twelve thousand three hundred forty five.

The sixt place hath the denomination of [Page 4] Hundred thousands, and those six in the Table that stand in a rank are to be read, One hun­dred twenty three thousand four hundred fifty six.

The seventh is the place of Millions, and the seven in the Table are, One million two hundred thirty four thousand five hundred sixty seven.

And the eighth Rank of Figures are to be read, Twelve millions three hundred forty five thousand six hundred seventy eight.

The ninth rank is, One hundred twenty three millions four hundred fifty six thousand seven hundred eighty nine. And so any greater number of places, every figure one place more toward the left hand, is increased ten times in value more than in the place it stood before.

ADDITION.

ADdition, is a gathering or collecting of several Numbers or Quantities into one Sum, by placing all Numbers of like Deno­mination under one another, carrying all above ten to the next place, as in these Examples.

[Page 5] There is likewise another kind of Addition, that is not of whole Quantities, wherein is necessary to be known the number of Parts the Integer or whole Number is divided into, as Pounds and Shillings, every Pound is di­vided into 20 Shillings, and one Shilling is divided into twelve Pence, one Penny into four Farthings.

Now being to add a Number of Pounds and Shillings together, they are thus set down with a small Line or Point between them.

If these be added together, observe in casting up your Shillings, so many times as you have 20 in the Shillings, you must carry Unites to the Pounds, and set down the Remainder, being under 20, as in these Examples.

In the first Example, I find in adding the Shillings together, they make 21, so I set down 1 and carry 1 Pound to the Pounds: In the second Example, I find among the Shillings 53, which is 2 Pounds 13 Shillings, so I set down 13 under the Shillings, and 2 to the Pounds.

[Page 6] Any number of Shillings and Pence being to be added together, if your number of Pence amount to above 12, carry 1 to the Shillings, and set down the remainder under the Pence; if they make above 24, carry 2 Shillings, and set down the remainder, as before.

Examples.

In the first Example, you carry one Shilling; in the second, two; and in the third, three.

In Addition of Pence and Farthings, carry so many times four as you find in the number of Farthings to the Pence, setting down the remainder under the Farthings, as in these Examples.

[Page 7] When you would know the Sum of any number of Pounds, Shillings, Pence, and Far­thing, they are to be placed thus:

Addition of Weight and Measure is perfor­med after the same manner.

• 16 Ounces Averdupois, make a Pound.
• 28 Pounds, make a Quarter.
• 112 Pound, or 4 Quarters, make an Hun­dred gross.
• 20 Hundred, make a Tun.

Examples.

Where observe, that so oft as I find 16 Oun­ces, I carry 1 to the Pounds; so often as I find 28 Pounds, I carry 1 to the Quarters; and as many times as I find 4 in the Quarters, so many times 1 do I carry to the Hundreds.

SUBTRACTION.

SVbtraction is the taking a lesser Number from a greater, and exhibits the Remainder.

In Subtraction the Numbers are placed one under another, as in Addition, thus:

The first of these Numbers is called the Minorand, the second the Subducend, and the third Number, or the Number sought, is the Resiàuum.

EXAMPLES of COINS.

But when the number of Pence or Shillings, are greater than the number that stands over it [Page 9] in the Minorand, you must borrow the next Denomination, as in this Example.

This Example I work after this manner, saying 9 d. out of 3 d. I cannot have, where­fore I borrow 1 s. from the Shillings, and sub­duct the 9 d. from that, and there will remain 3 d. which added to the other 3 d. maketh 6 d. I place therefore 6 d. in the Place of Pence, and proceed saying, 1 s. that I borrowed and 19 is 20 from 1 I cannot, wherefore I borrow 1 l. from the Pounds, and subduct from that the 20 s. and there remains nothing but the 1 s. which I place under the Shillings, and say, 1 that I borrowed and 6 is 7. from 7 and there remains nothing, then I place a Cypher under the 6, and say, 1 from 2 and there remains 1, which I set down, and 1 from 1 and there re­steth nothing. After this manner is performed Subduction of Weight and Measure.

Examples.

[Page 10] [...] By which Examples, the Learner may per­ceive, that where the number to be subducted is greater than the number standing over it, I then borrow one from the next greater denomination, adding the remainder, if any be, to the lesser number before-mentioned, and setting them underneath those of like denomination with them.

The Proof of Subtraction is by adding the Subducend and Remainder together, and their Aggregate must always be equal to the Mino­rand, as you may see by the last Example.

I could here add many more Examples of Weight and Measure, but to the ingenious Practitioner I hope it will be enough, all other being wrought after the same manner, respect being had to the number of lesser denomina­tions contained in each greater. As

• 24 Grains make a Penny-weight.
• 20 Penny-weight one Ounce.
• 12 Ounces one Pound.

• [Page 11]4 Nails make a Quarter of a Yard.
• 4 Quarters one Yard.
• 5 Nails one Quarter of an Ell.
• 4 Quarters one Ell.
• 12 Inches a Foot.
• 3 Feet a Yard.
• 16½ a Perch.
• 40 Perches a Furlong.
• 8 Furlongs make an English Mile.

• 8 Pints make a Gallon.
• 63 Gallons make a Graves Hogshead.
• 4 Hogsheads make a Tun.
• 36 Gallons make a Beer Barrel.
• 32 Gallons make an Ale Barrel.

• 8 Gallons of Corn make a Bushel.
• 8 Bushels make a Quarter.

MULTIPLICATION.

MVltiplication is a kind of Addition, and resolveth Questions to be performed by Addition in a different manner: In order where­unto, it is necessary the Learner do well ac­quaint himself with this Table; the having this Table perfectly by heart, will make both this Rule and Division also very facile, other­wise they will be both troublesome and unplea­sant.

In the first Rank of this Table, you have an Arithmetical Progression from 1 to 12, [Page 13] and also in the first Column toward the left hand downwards. This Table doth at first sight exhibit the Sum of any number, so often repeated as you shall require, provided the numbers do neither of them exceed 12.

Multiplication hath three Members, thus called, a Multiplicand, a Multiplicator, and a Product: The Multiplicand, is the number to be repeated; the Multiplicator, is the number of times the first is to be repeated; and the Product, is the Sum of the Multiplicand so often repeated. As for Example.

A Countrey-man sold 6 Bushels of Wheat for 5 s. how many Shillings ought he to receive?

But by Multiplication it is done thus:

[Page 14] Now if you look in the Table precedent, in the first Column find 5, then look in the first Rank for 6, and cast your Eye down to their Angle of meeting, and you will find 30 standing under 6 and against 5, I then con­clude that 5 times 6 is 30; that is called the Product, and they will stand thus:

But when you have a number to multiply, greater than any in the Table, as for Example:

A Gentleman having forborn his Rent of a Farm, at 157 l. per Quarter, for 3 Quarters, what ought he to receive?

The Multiplication will stand thus:

I then say, 3 times 7 is 21, I set down 1 and carry 2; then, 3 times 5 is 15 and 2 is 17, I set down 7 next the 1, and carry 1; saying, 3 times 1 is 3 and 1 is 4, as in the Example before-going; and the Product is 471 l.

[Page 15] There is yet more variety, of which take these Examples following.

If 65 Ships do carry 536 Men in every Ship, how many Men will there be in all?

I say 5 times 6 is 30, set down 0 and carry 3; then 5 times 3 is 15 and 3 is 18, set down 8 and carry 1; then 5 times 5 is 25 and 1 is 26, which I set down: Then for the next Fi­gure, I say, 6 times 6 is 36, I set down 6 one place short of the former rank, and carry 3; then 6 times 3 is 18 and 3 is 21, set down 1 and carry 2; again, 6 times 5 is 30 and 2 is 32, these I set down: Then draw a line, and cast them up as they are placed, and the Sum is the Product and Answer to the Question, viz. 34840 Men.

In Multiplication, always make the lesser Number the Multiplicator, for it is all one whether I multiply 5 by 15, or 15 by 5, the Product is always the same.

[Page 16] If 128 Men of War have each made 746 Shot, how many Shot were made in all?

[...] Begin as before with the Unites place, and say, 8 times 6 is 48, set down 8 and carry 4; 8 times 4 is 32 and 4 is 36, set down 6 and carry 3; then 8 times 7 is 56 and 3 is 59, which set down: Then go forward with the 2, (but remember to place your remainder one Figure short of the former) saying, 2 times 6 is 12, set down 2 under the 6 and carry 1; 3 times 4 is 8 and 1 is 9, which set down; twice 7 is 14, which set down: Also then, once 6 is 6, which place under the 9; once 4 is 4, which set under the 4; and once 7 is 7; which set under the 1: Then cast them up, as in Addition, and the Sum is the Product, and answers the Question, viz. 95488 Shot.

If any number be to be multiplied by 1 with Cyphers, it is but adding so many Cyphers to the Multiplicand as there is in the Multipli­cator.

As for Example.

If 35678 be to be multiplied by 10, add one Cypher to the Multiplicand, thus, 356780; if by 100, add two Cyphers, thus, 3567800; &c.

[Page 17] And when any number is to be multiplied by any other number, that hath Cyphers an­nexed, always place the Cyphers immediately under the Line, as in these Examples.

DIVISION.

DIvision is also a kind of Subduction, and informs the Querent, how many times one number is contained in another.

There is in Division these three things to be observed, viz. the Dividend, the Divisor, and the Quotient. The Dividend is a number to [...]e divided into parts, the Divisor is the quan­ [...]ity of one of those parts which the former is [...] be divided by, the Quotient is the number [...] such parts as the Dividend doth contain [...] [...]ere is also by accident a fourth number in [...]s Rule necessary to be known, which is a [...]mainder, and that happens when the Divi­ [...] doth not contain an equal number of such [...]ntities as it is divided by, as when 15 is to [Page 18] be divided by 4, the Dividend is 15, the Di­visor is 4, and there is a Remainder 3.

In Division you may place your numbers thus.

Dividend.

Multiplication is positive, but Division is performed by essays or tryals, after this manner:

[...] Here I first inquire how many times 3 I can have in 14, I find 4 times, I place 4 in the Quotient, and then mul­tiply the Divisor by that 4, placing the Product underneath the Dividend, as in the Example; say, 4 times 5 is 20, set down a Cypher under the 6 and carry 2, then 4 times 3 is 12 and 2 is 14, which I set down also, as in the Example; then subduct this Pro­duct from the Figures standing over them, and set down the Remainder.

[...] Then for a new Divi­dend, I bring down the next figure, and postpone that to the Remainder, and inquire how many times 3 in 6, I cannot have twice, because I [Page 19] cannot have twice 5 from 5, I say then once, and place 1 in the Quotient, proceeding as be­fore saying, once 5 is 5, which I place under the first 6 toward the right hand, and once 3 is 3, which I set down under the other 6; subducting these as the former, I find the Re­mainder to be 31.

After which I bring down the next figure in the Dividend, and postpone it to the Remain­der, as in this Example: [...] Then I inquire how many times 3 in 31, I sup­pose 9 times, placing 9 in the Quotient I multi­ply again; saying 9 times 5 is 45, 5 and carry 4; then 9 times 3 is 27, and 4 is 31; these being set down, as before directed, and subducted, there will remain nothing. I then conclude, that the Di­visor is so often contained in the Dividend as is expressed in the Quotient, viz. 419 times.

For further Instructions, take these Exam­ples.

REDUCTION.

REduction is twofold, viz. bringing greater denominations into smaller, and that by Multiplication, as Pounds into Shillings, Shillings into Pence, &c. Also lesser deno­minations are reduced into greater, by Division, as Pence into Shillings, Shillings into Pounds, Minutes into Hours, Hours into Days, and Days into Years, &c.

Having any number of Pounds to reduce into Pence, multiply them by 240.

Example.

How many Pounds, Shillings, and Pence, are contained in 22929 Farthings?

[Page 22] In 544542 Cubique Inches, how many Beer Barrels, Firkins, and Gallons?

THE RULE OF THREE.

THis Rule is so called, because herein are three numbers given to find a fourth; of these three numbers, two are always to be mul­tiplied together, and their Product is to be divided by the third, and the Quotient exhibits the fourth number, or the number sought.

And here note, That of the three given numbers, if that number that asketh the Question be greater than that of like denomi­nation with it self, and require more, or if it be less, and require less, then the number of like denomination is the Divisor.

[Page 23] Or, if the number that asketh the Question be less than that of like denomination, and require more; or if it be more, and require less, then the number that asketh the Question is the Divisor.

Example.

If 3 Yards of Sarcenet cost 15 s. what shall 32 Yards cost?

Which 3 numbers if you please may stand thus:

Here you may see the term that asketh the Question is greater than that of like denomi­nation, being 3, and the other 32, and also requires more, viz. a greater number of Shil­lings; therefore, according to the Rule, the first term, or the term of like denomination to that which asketh the Question, is the Di­visor.

[Page 24] And the Answer is 160 Shillings, which being divided by 20 will be found 8 l.

Again,

If 32 Ells of Holland cost 160 s. what shall 3 Ells cost?

In this Question (being the Converse of the former) you may see the term that asketh the Question, here 3, is lesser than that of like de­nomination, being 32 Ells, and also requires less; therefore the first term here also is the Divisor.

And the Answer is 15 s.

If 36 Men dig a Trench in 12 Hours, in how many Hours will 144 Men dig the same?

[Page 25] In this Question, the term that asketh the Question is greater than that of like denomi­nation, and requireth less; wherefore the term that asketh the Question is the Divisor.

If 144 Workmen build a Wall in 3 Days, in how many Days will 36 Workmen build the same?

This Question you may perceive to be the Converse of the former, here the term that asketh the Question is less than that of like de­nomination, and requires more, the term that asketh therefore is the Divisor.

If 125 lb. of Bisket be sufficient for the Ships Company for 5 Days, how much will Victual the Ship for the whole Voyage, being 153 Days?

This Question is of the same kind with the first Example; here the two terms of like de­nomination [Page 26] are 5 Days and 153 Days, the term that asketh the Question being more than the term of like denomination, and also requiring more; so, according to the general Rule, the term of like denomination to that which asketh the Question is the Divisor. It matters not therefore in what order they are placed, so you find your true Divisor; but if you will you may set them down thus:

The Answer is 3825 lb. weight of Bisket.

[Page 27] A Ship having Provision for 96 Men during the Voyage, being accompted for 90 Days, but the Master taking on boord 12 Passengers, how many Days Provision more ought he to have?

Which is no more than this:

If 96 Men eat a certain quantity of Provision in 90 Days, in how many Days will 108 Men eat the same quantity?

The Answer is 80, so that for 108 Men he ought to have 10 Days Provision more.

If the Assize of Bread be 12 Ounces, Corn being at 8 s. the Bushel, what ought it to weigh when it is sold for 6 s. the Bushel?

[Page 28] [...] In this Question, the term inquiring being less than the term of like de­nomination, and requiring more; therefore is the term so inquiring the Divisor.

The Answer is 16 Ounces.

THE RULE OF PRACTICE.

IT is necessary that the Learner get these two Tables perfectly by heart, which are only the aliquot parts of a Pound and of a Shilling.

Having these Tables perfectly in memory, any Question propounded will be readily re­solved, only by dividing the given number of [Page 30] Yards, Ells, Feet, Inches, Gallons, Quarts, Pounds, or Ounces.

Of which take some Examples.

Having any number of Shillings to reduce into Pounds, cut off the last figure toward the [Page 31] right hand by a line, and the figures on the left hand of the line are so many Angels as they express Unites; draw a line under them, and take the half of them, and you have the num­ber of Pounds.

Examples.

Any Commodity, the value of 1 Yard being the aliquot part of a Pound, is thus cast up:

Take the one third part, and that is the An­swer in Pounds: 3 in 8 twice, and carry 2; 3 in 23 seven times, and carry 2; 3 in 26 eight times, and carry 2; the third part of 2 l. is 13 s. 4 d. where always observe, that the Re­mainder is always of the same denomination with the Dividend.

Where the Price is not aliquot.

To cast up the amount of any Commodity, sold for any number of Farthings by the Pound, [Page 33] I borrow from the Dutch a Coin called a Guil­der, whose value is 2 s. English.

Then if a Question be proposed of the Amount of an Hundred weight of any Com­modity, by the Hundred Gross, viz. 112 lb. so many Hundred as there be, the Amount is so many Guilders so many Groats, as there are Farthings in the price of 1 lb.

As for Example.

A Hundred weight of Iron is sold for 5 Farthings the Pound, comes to 5 Guilders, that is 10 s. and 5 Groats, which together is 11 s. 8 d.

Again.

A Hundred weight of Lead is sold for 2 d. Farthing the Pound, that is 9 Guilders and 9 Groats, which is 21 Shillings.

But if it be the subtil Hundred, it is then but so many Guilders so many Pence: As if a Hundred weight of Tobacco be sold for 5 d. Farthing the Pound, the Hundred comes to twenty one Guilders and twenty one Pence, that is forty three Shillings and nine Pence.

NOTATION.

AS in Whole Numbers, the value or denomination of Places do increase by Tens, from the Unite place toward the left hand; so in Decimals, the value or denomination of Places do decrease [Page 35] by Tens, from the Unite-place toward the right hand, according to the precedent Table.

A Fraction or broken Number is always less than a Unite, as Pence are parts of a Shilling, and Shillings of a Pound; Inches of a Foot, and Minutes of an Hour, &c.

Fractions are of two kinds,

And are thus called Vulgar, & Decimal.

A vulgar Fraction is commonly expressed by two Numbers set over one another, with a small line between them, after this manner [...], the uppermost being called the Numerator, and the lower the Denominator.

The Denominator expresseth into how many parts the Integer or whole Number is divided, and the Numerator sheweth how many of those parts is contained in the Fraction.

Example.

If the Integer be a Shilling; is 8 d.

If it be 1 l. or 30 Shillings, it is 13 s. 4 d.

If a Foot, it is then 8 Inches.

Or if an Hour, it will be 40 Minutes.

A decimal Fraction hath always a common Number for a Numerator, and a decimal Number for its Denominator.

A decimal Number is known by Unity, with [Page 36] one or more Cyphers standing before it, as 10, 100, 1000, &c.

A decimal Fraction is known from a whole Number by a point, or some other small mark of distinction, whether it stand alone, or be joyn'd with whole Numbers; as in these fol­lowing Examples.

Or else with a point over the head of Unity, or the Unite place; as in these Examples.

In decimal Fractions, the Numerators only are set down, the Denominator being known by the last Figure in the Numerator.

Example.

• .2 is Two tenths.
• .25 is Twenty five Hundredths.
• .257 is Thousandths.
• .2575 is Ten Thousandths, &c.

As Cyphers before a whole Number have no value, so Cyphers after a decimal Fraction are of no signification: But Cyphers before a decimal Fraction, are of special regard; for [Page 37] as Cyphers after a whole Number do increase that Number, so before a decimal Fraction they diminish the value of that Fraction.

Example.

• .25 Twenty five hundredths.
• .025 Twenty five thousandths.
• .0025 Twenty five ten thousandths.

Each Cypher so added removing the Fraction further from Unity, making it ten times less than before.

ADDITION.

ADdition in Decimals, whether in pure De­cimals, or whole Numbers mixt with De­cimals, differs not from Addition in whole Numbers, only care must be had to the sepe­rating lines or points, that all places of like denomination stand one under another, both in the Addends and in the Sum; as in these Examples.

SUBTRACTION.

AS in Addition, so in Subtraction care must be had to the placing each Figure under that of like denomination with it self, then it is the same with Subtraction in whole Num­bers.

Examples.

MULTIPLICATION.

MVltiplication in whole Numbers serveth instead of many Additions, and teacheth of two Numbers given to increase the greater as often as there are Unites in the lesser.

It likewise consists of three Requisites, viz. a Multiplicand, a Multiplicator, and a Product.

[Page 39] The Multiplicand is the Number to be in­creased.

The Multiplicator is the Number by which it is to be increased.

And the Product is the Sum of the first Number so often repeated as there are U­nites in the second.

In decimal Fractions, or whole Numbers mixt with Fractions, the two first Numbers are called Factors, and the last is called the Fact.

Multiplication, whether in decimal Fracti­ons, or whole Numbers mixt with Fractions, differeth not (in the Operation) from Multi­plication in whole Numbers. The last Figures in both the Factors may be placed under one another, without respect to the distinction of places, or places of like denomination standing under one another, as in Addition and Sub­duction; yet from the Product must be cut off by a line or point so many places as there are Figures in decimal Fractions in both Factors of the last Figures standing toward the right hand.

Examples.

[Page 40] If it happen when the Multiplication is ended, that there be fewer Figures in the Pro­duct than there are places in Decimals in both the Factors, then put Cyphers before the Pro­duct till the number of places be equal to those in both the Factors: As in these

Examples.

Where by may be observed, That the Multi­plication of two Fractions doth not increase them as in whole Numbers, but they are here­by made less, and the Fact is removed further from Unity than either of the Factors.

If a whole Number be to be multiplied by a decimal Number, put so many Cyphers after the whole Number as there are in the decimal Number, and that Number will be the Product. If 48 be multiplied by 10, it will be 480; by 100, 4800; &c.

In multiplying decimal Fractions, or mixt Numbers, by a decimal Number, you need only remove the point or seperating line so many places toward the right hand as there be Cy­phers in the decimal Number. If you multi­ply .2845 by 10, the Fact will be 2.845; by 100, it will be 28.45; by 1000, 284.5; &c.

DIVISION.

DIvision, both in whole Numbers and Fra­ctions, is by young Practitioners found to be more difficult than any of the four Species; it will therefore require a little more industry in the Learner: But when once had, there will appear small difference between the Operation herein, as in any the precedent.

Division is also constituted by three Requi­sites, and a fourth by accident, viz. a Divi­dend, a Divisor, and a Quotient: The fourth is a Remainder, which doth not always happen to be.

The Dividend is the Number to be divided.

The Divisor is the Number by which the other is to be divided.

The Quotient is the Number found out by the Division.

And the Remainder is that which is left of the Dividend after the Division is ended, and is always less than the Divisor.

Example.

If 12 be to be divided by 4, then is 12 the Dividend, 4 the Divisor, and the Quotient will be 3.

If 13 be divided by 3, then 13 is the Divi­dend, 3 the Divisor, 4 the Quotient, and there will be a Remain, which is here 1.

[Page 42] Decimal Fractions, or mixt Numbers, are divided after the same manner as whole Num­bers are divided, only care must be had in gi­ving a true value to the Quotient. To perform which, observe well this General Rule.

As to the manner of placing your Figures, and the way of dividing, there are many pub­lished by divers Writers of Arithmetick: The way of placing the Divisor under the Dividend, is the most apt for giving a value to the Quo­tient; but the rasing of Figures, and repeating the Divisor so often, is found an inconvenience; which to avoid, observe the following Ex­amples.

Being to divide 2487.048 by 53.6, I place them in this order:

Then I consider if the Divisor were placed under the Dividend, the Unity place in the Divisor, here 3 would stand under the 8 in the Dividend, I then set a mark over the head of the 8, and conclude the first Figure in the Quotient to be of the same denomination with it, which is Tens, in whole Numbers.

[Page 43] [...] Having thus found the value of the first Fi­gure in the Quotient, I proceed to the division, and inquire, how many times 5 in 24? I find 4; I then set 4 in the Quotient, and go back, multiplying the whole Divisor by that Figure, and subduct the Product out of the Dividend, placing the Remainder underneath as part of a new Dividend: Thus 4 times 6 is 24, from 27, and there remains 3, which I place under the 7; again, 4 times 3 is 12, and 2 that I borrow­ed is 14, from 18, and there remains 4, which I place under the [...], as in the Example; then 4 times 5 is 20, and 1 I borrowed is 21, from 24, and there remains 3, which I place under the 4. For my new Dividend, I bring down the next Figure, here a Cypher, and post­pone it to the Remainder, and the Example will stand thus:

Then proceeding in my Division, I ask, how many times 5 in 34? finding 6 times, I then place 6 in the Quotient, and as before say, 6 times 6 is 36, from 40, and there remains 4, which I set down under the Cypher; then 6 times 3 is 18, and 4 I borrowed is 22, from 23, [Page 44] and there remains 1, which I place under the 3; then 6 times 5 is 30, and 2 I borrowed is 32, from 34, and there will remain 2, which I place under the 4; then to this Remainder I bring down the next Figure in the Dividend, postponing it as I did the Cypher, and they will stand thus:

[...] I now inquire, how many times 5 in 21? and find 4 times, I then place 4 in the Quotient, and go on as before; there being yet a Remain­der, I add a Cypher, and proceed as before; and find, upon the adding one Cypher, my Di­visor greater than the Dividend, I place a Cy­pher in the Quotient: Example.

Having placed a Cypher in the Quotient, I add another to the Dividend, and make it 800; and then inquire, how many times 5 in 8? finding once, I put 1 in the Quotient, working as before: Where note, So long as there is a Remainder, if you add Cyphers and work after this manner, you may have as many Decimals as you please.

[Page 45] It doth often happen in Division, in decimal Fractions, or mixt Numbers, that the Unite place in the Divisor will stand beyond all the significant Figures in the Dividend, either to­ward the right hand or toward the left; in which case, that you may the better find out the value of the first Figure in your Quotient (according to the precedent General Rule) add Cyphers to the right or to the left hand of the Dividend, till you come over the Unity place in the Divisor, and what value or denomination that place is of, that is the denomination of the first Figure in the quote; as in these

Examples.

If in Division in whole Numbers, there happen to be a Remainder, it is the Numerator of a Common Fraction, and the Divisor is the Denominator, and this Fraction is part of the quotient.

Example.

If you divide 66 by 8, the quotient will be 8 and 2/8, according to the way of Vulgar [Page 47] Fractions, but in Decimal Fractions it will be 8.25.

If you be to divide a whole by a decimal Number, cut off so many places by a mark, as there are Cyphers in the decimal Number: If 468 be divided by 10, the quote is 46.8; by 100, 4.68; and by 1000, quotes .468.

If a decimal Fraction, or a mixt Number, be to be divided by a decimal Number, remove your line or point so many places toward the left hand, as there are Cyphers in your decimal Number, supplying the vacant places with Cy­phers, if there be occasion: 69.5 divided by 10, is 6.95; by 100, it will be .695; by 1000, .0695; and by 10000, quotes .00695; &c.

Division being the Converse of Multipli­cation, as multiplying a mixt Number or deci­mal Fraction by a decimal Number, you remove your mark of distinction toward the right hand; so in dividing a decimal Fraction or mixt Number by a decimal Number, the mark is removed toward the left hand, as in the fore­going Examples.

REDUCTION.

TO reduce a vulgar Fraction into a decimal Fraction, your Rule is: Divide your Nu­merator by your Denominator, and the Quotient will be a decimal Fraction of the same value with the vulgar Fraction. So 1/4, if reduced into a decimal Fraction, will be .25.

Example.

Here note, That only the even parts of an Integer will be exactly reduced into a decimal Fraction, as 1/2, 2/8, 2/16, &c. In all Surds, there will be some Remainder, but if you carry your decimal Fraction to four or five places, making the last one more than it is, if the sixth Figure be above 5, or else leave them out, and your Calculation will come near the truth; but if any desire to be more exact, he may take as many as he please.

[Page 49] Examples.

To reduce any decimal Fraction out of a greater denomination into a lesser, multiply the Fraction by those parts of the Integer into which you would have it reduced; as .65 being the parts of a Pound, you would know how many Shillings are contained in the Fraction, multiply it by 20: If you desire the Pence therein contained, multiply it by 240; or if Farthings, multiply by 960, the number of Farthings in a Pound or 20 Shillings.

[Page 50] [...] The decimal parts of a Foot are reduced, by multiplying them by 12; if parts of a Foot Square, by 144; and the decimal parts of a Foot Solid, by 1728, the Cubick Inches in a Foot of Solid. The decimal parts of a Pound, are re­duced by 16, the Ounces in a Pound Aver­dupois; and 12, the Ounces in a Pound Troy. The decimal parts of a Beer Barrel by 36, and by 32 reduceth the parts of an Ale Barrel, into Gallons; and Gallons into Pints, by 8; Gallons into Cubick Inches, by 282; and for Wine Gallons, by 231, the number of Cubick In­ches in such a Gallon, &c.

As greater denominations are reduced to lesser, by a multiplication of the several parts of the Integer; so lesser denominations are [Page 51] reduced to greater, by division. Any number of Shillings are reduced into Pounds, and the decimal parts of a Pound, if you divide them by 20; and Pence, if divided by 240.

Example.

Hours are reduced into the decimal parts of a Day, if you divide them by 24, the Hours in a Day Natural; and Minutes into the parts of an Hour, if divided by 60.

Perches are reduced into the decimal parts of an Acre, if you divide them by 160, the number of Square Poles or Perches in an Acre; and any [...]mber of Feet into Poles, and the decimal parts of a Pole, if you divide them by 16.5 the Feet in a Pole, or by 15.8.25 the number of Square Feet in a Square Pole; but if Wood-land Measure by 18, or if a Square Pole by 324, the Square Feet in a Pole or Perch of such Measure.

Any number of Inches are reduced into the parts of a Beer Barrel, if divided by 10152; and into Ale Barrels and parts, by 9024; &c.

For the ease of the Reader here is made a Table of English Coin reduced into the decimal parts of a Pound sterling.

THE GOLDEN RULE.

THis Rule is called the Rule of Three, be­cause herein are three Numbers given, to find a fourth. It is also called the Rule of Pro­portion, for as the first is in proportion to the second, so is the third to the fourth: And the Converse.

This Rule is called the Golden Rule for its excellent use in the Solution of Questions of various kinds, and great advantage is made of it in almost all kind of Calculations Arithmetical.

[Page 55] Two of the three Numbers given in every Rule of Proportion are of one denomination, and the third is of the same kind with the fourth sought; and one of the two Numbers that are of like species doth always ask the Question.

Arithmeticians distinguish this Rule by two denominations, one they call the Direct, and the other the Inverse or Backer Rule of Three.

One of the three given Numbers of like de­nomination in any Rule of Proportion is a Di­visor, the other remaining two are Multipliers. To find which of the forementioned Numbers is the Divisor, take these following Rules.

• 1. If that Term to which the Question is an­nexed be more than that of like denomina­tion, and also requires more; or if it be less, and require less than the Term of like denomination; then that Term of like de­nomination to that which asketh the Que­stion is the Divisor, and the Question is in the Direct Rule of Three.
• 2. If the Term which asketh the Question be more than that of like species, and requires less; or less, and requires more; then that Term which asketh the Question is the Di­visor, and the Question is in the Backer or Inverse Rule of Three.

Having by the precedent Rules discovered the Divisor, multiply the other two Numbers, and [Page 56] divide by the Divisor, your quote will be the Answer to the Question.

Note, If any of the Numbers given be in several denominations, they must be redu­ced into one, either greater or lesser, as before directed.

Example.

Quest. 1. If 12 1/2 Yards of Taffaty cost 5 l. 7 s. 9 d. 3 q. what shall 5 1/2 Yards cost?

In this Example, of the three Numbers gi­ven there are two of like denomination, and they are 12 1/2 and 5 1/2, the latter of which is the Term which asketh the Question, known always by the words what or how much. And this Term is less than that of like kind with it self, and also requires less, therefore according to the precedent Rule, this Question is in the Golden Rule Direct. These three Numbers may be placed in what order you please, pro­vided you mistake not your Divisor, but accor­ding to the general way, being reduced into De­cimals, and of one species, they will stand thus:

Then, as before directed, multiply the second and third Numbers, and divide by the first, and the quotient exhibits the fourth Proportional or the Number sought.

[Page 57] [...] The Answer is 2 l. 7 s. 5 d. 1 q.

Quest. 2. If 6 Yards of Broad Cloth cost 4 l. what shall 32 Yards cost?

Here the Term which asketh the Question is greater than the Term of like denomination, and requires more; therefore the Term of like denomination to the Term that asketh the Question is the Divisor.

[Page 58] [...] The Answer is 21 l. 6 s. 8 d.

Quest. 3. If 320 Men raise a Breast-work in 6 Hours, in what time will 750 Men do the same?

Here the Term that asketh the Question is more than the Term of like denomination, and requires less; therefore the Term that asketh the Question is the Divisor, and this is the Backer Rule of Three.

The Answer is 2 Hours, 33 Minutes, and 36 Seconds.

[Page 59] Quest. 4. If 756 Men dig a Trench in 12 Hours, in how many Hours will 126 dig the same?

Here the Term that asketh the question is less than the Term of like denomination, and requires more; then according to the Rule the Term demanding is the Divisor, and this que­stion is also in the Inverse Rule of Three.

The Answer is 72 Hours.

There is sometimes four Numbers given in a question, yet is it but a Single Rule of Three, for one of the four Numbers is of no signification, and might as well have been left out.

Example.

Quest. 5. If 10 Workmen build a Wall 40 Foot long in 3 Days, in what time might 50 Men have done the same?

Here note, there is four numbers given, and yet there is but three to be used in working [Page 60] the question, you must therefore find which those 3 are that are necessarily to be used: Thus,

First, you must take the Term that asketh the question, here 50 Workmen; secondly, you must have the Term of like denomination with it, which is 10 Workmen; thirdly, the Term sought, being Days; you must take the Term of like denomination with that also, which is here 3 Days: The superfluous Term then in the question is 40, which might have been left out, and they will then stand thus:

The Answer is Half a Day or 12 Hours.

This question is in the Rule of Three Inverse.

Quest. 6. If 100 l. gain 6 l. in 12 Months, what shall 32 l. gain in the same time?

In this question the 12 Months is the super­fluous Term, being of no use in the Calcula­tion, the Terms required being 100 l. 6 l. and 32 l.

Note, Though the Terms in this question be all Money, and so may seem to be of one species, yet they are not; 100 l. and 32 l. [Page 61] are of one kind, being both Principal, and the other Term is of the same deno­mination with the Term sought, viz. Gain or Interest.

The Answer is 1 l. 18 s. 4 d. 3 q. ferè.

And this question is in the Direct Rule of Three, the Term that asked the question being less than the Term of like denomination, and also requiring less, &c.

THE DOUBLE GOLDEN RULE.

THis Rule is called the Double Golden Rule, or Double Rule of Three, because it re­quires two distinct Calculations, before you can answer the question.

And in this Rule there are five Numbers given to find a sixth sought.

This differs not in the operation from the Single Rule, only the Calculation is twice re­peated.

Of the five Numbers given, the question is sometimes annexed to two, and sometimes but to one.

[Page 62] If the question be annexed to two of the five given Numbers, then are there two of the other three of the same species with those that ask the question, and the third is proportional to the Number sought.

For the due regulation of these two Calcula­tions, when the question is annexed to two of the five Numbers, take these Directions.

First, take one of the Numbers demanding, and let that ask the question in the first opera­tion; secondly, take that of the same species, and also that of the like quality with the re­spondent, of these three constitute your first Rule of Proportion; then find which is your Divisor, according to your Rule pag. 55. and proceed to find the fourth in proportion.

Then for your second Rule of Three, take the other of the two Numbers to which the question is annexed, and let that ask the question; take also the Number of like kind, and the fourth Num­ber found in the first Calculation; judge which is your Divisor, and work accordingly; the last Quotient will be the sixth Number, or the Num­ber sought.

Example.

If a Trench be 20 Perches in length, and made by 12 Men in 18 Days; how long may that Trench be, that shall be wrought be 48 Men in 72 Days?

[Page 63] Here the question is annexed to two of the five Numbers, viz. 48 Men and 72 Days; now according to the foregoing direction, take one of the two Numbers inquiring, 48, and say,

Then take the other of the two Numbers in­quiring, and say,

If 6 Lighters bring 60 Tuns of Ballast in 5 Tides, how many Tun will 15 bring in 12?

[Page 64] If a Man travel 160 Miles in 4 Days, when the Days are 10 Hours long; in how many Days will he travel 195 Miles, when the Days are 14 Hours long?

When a Question is stated in the Double Rule of Three, so that there is but one Number in­quiring,

First, take that Number, and let it ask the question in the first Rule; take also the Number [Page 65] of like denomination, together with the Num­ber joyn'd to that of like denomination; and of these three Numbers constitute your first Rule of Proportion.

Secondly, let that Number which was found in the first Operation, ask the question in the second; then take the Number of like denomi­nation to it, and also the Number joyn'd with that like Number; of these three is your se­cond compounded; find your Divisor, and pro­ceed; the last quote exhibits the Answer.

Example.

If 4 Crowns at London make 2 Ducates at Venice, and 8 Ducates at Venice make 20 Pa­tacoons at Genoa; how many Patacoons at Genoa will make 120 Crowns at London?

Of Quadratique Equations.

Mr. Dary, in his Miscellanies, chap. 8. saith to this, or the like purpose:

1. When any Equation propos'd is incumbred with Vulgar Fractions, let it be reduced to its least Terms in whole Numbers, if possible; if not, let it be brought to its least Terms in Deci­mals.

2. It is evident from divers Authors, That if any Quantity shall be signed—, then the Square Root, or the Root of any even Power of such Quantity so sign'd, is inexplicable, for they cannot be generated from any Binomials that shall be equal.

As for Example.

—9 being a Negative can be made of no­thing (if taken as a Square Number) but + 3 [Page 81] and —3, which Roots are not equal, they being neither both Affirmatives nor both Nega­tives.

3. When you have cleared the Equation by the Second hereof, and that the Co-efficient in the highest Power is taken away, or be Unity, then will quadratique Equations resolve themselves into the four following Compen­diums.

4. Let your Equation be so reduced, that the highest Power stand on the left side alone, the sign + being always annexed, or supposed to be annexed.

Example, Quesita a.

First Equation.

Second Equation.

[Page 82] Third Equation.

Fourth Equation.

Illustration by Numbers, Quesita a.

First Equation.

[Page 83] [...] Which was to be proved.

Proof of the Negative.

Example 2.

[Page 84] Second Equation.

Example 1.

Example 2.

[Page 85] Third Equation.

Example 1.

Example 2.

[Page 86] Fourth Equation.

Example 1.

Example 2.

[Page 87] But if in a Square Equation there happen to be a Coefficient annexed to the highest Power, it is resolved by transferring the Coefficient with the Sign of Multiplication to the other side.

Admitting the Equation be

Then the Coefficient 2 being transferred (as before directed) they will stand as in this Example.

First Equation.

The Root of +25 being +5, then is +5+; = 8, and a=+4, the Affirmative Answer. And +3-5 is =-2, and a=-1, the Ne­gative Answer.

The Proof is easie:

First, if a be = 4, 2 aa is =+32, and 6a is =+24, to which +8 being added, the Sum is +32 which was to be proved.

Again, a=-1, then 2 aa is =-2, whereto +8 being added, the Sum is =+6, which also was to be done.

[Page 88] Second Equation.

Now +13 +the √121, viz. + 11 is =+24, the 1/4 whereof is =+6= a, and aa = 36. and 4 aa=+144, +26 a=+156, to which if -12 be added, the Sum will be +144 also.

Again, If to +13 you add -the √121, viz.-11, 4 a will be =+2, and consequently + a=+1/2, 4 aa is then =+1, and +26 =+13, to which add -12, and the Sum is =+1, which was to be proved.

Third Equation.

[Page 89] -3 +13 =+5a, here a=+2, 5aa=+20, -6a=-12, to which add +32, the Sum is also+20.

Again, -3-13=-16=-5a, and a=-8.2, 5aa=+51.2: Also -6a being = +19.2, to which add +32, the sum is =+51.2.

Fourth Equation.

Which was to be done.

[Page 90] Note, Always where there is no Sign annexed to any Term in the Equation, the Sign + is supposed to be annexed.

I have been the larger in these Examples, that the young Analist may with the more ease ap­prehend the several kinds by this variety; in some of the surd Roots I have on purpose omit­ted the large number of Places, four or five being sufficient for use in most cases; but if any desire to be more exact, he may take them as far as he pleaseth, or the Table doth exhibit.

The Calculation of a Number in the precedent Table, by aid of the Canon.

The Question being, What is the present worth of 1 l. per Quarter for 21 Years?

Then by the Rule of Proportion:

The Answer is = 48 l. 2 s. 0 d. 1/2 ferè.

And after this manner may a Table be Calcu­lated, or the Value of a Lease for any Number of Years, may be found at any Rate of Inte­rest required.

The Boung-diameter, and Head-diameter, and Diagonal, to find the Casks length.

First subduct the semi-difference of Diame­ters from the Boung-diameter, and Square the Remainder, which Square subduct from the Square of the Diagonal, and the Remainder is the Square of the Casks semi-length.

Example.

• Let B D be the Boung-diameter = 29 Inches,
• H E be the Head-diameter = 23 Inches,
• B E be the Diagonal = 35.3836>Inches,
• S D the semi-difference = 3 Inches:

Q. the Length = L T?

This very Quest. was intended by Mr. Smith, p. 176. but through a Mistake it was left out.

The Boung-diameter, Diagonal, and Length, to find the Head-diameter.

The Head-diameter, Boung-diameter, and the Length, to find the Diagonal.