The gentlemans recreation in two parts : the first being an encyclopedy of the arts and sciences ... the second part treats of horsmanship, hawking, hunting, fowling, fishing, and agriculture : with a short treatise of cock-fighting ... : all which are collected from the most authentick authors, and the many gross errors therein corrected, with great enlargements ... : and for the better explanation thereof, great variety of useful sculptures, as nets, traps, engines, &c. are added for the taking of beasts, fowl and fish : not hitherto published by any : the whole illustrated with about an hundred ornamental and useful sculptures engraven in copper, relating to the several subjects.

ARITHMETICK.

ARITHMETICK is the Art of well numbring, which is divided into two parts, Simple and Compound.

The simple is that which simply considers the Nature of Number.

Number is that by which every thing is coun∣ted, either by an intire Number, or by Parts, where first must be considered the Notation, and then the Numeration.

The Notation of Number has 10 Characters, viz. [ 10] 9 Digits, or Figures, and a Cipher, and they are these, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. And three Degrees of Amplification repeated again in like manner by Periods, once, ten times, a hundred times, for every Character of a Number when put in the last place of the whole Number, ex∣presses its Number once; If in the last place but one, ten times, and if in the last place but two, a Hundred times. This is then the considerati∣on of the first Period; the second is of Thou∣sands, [ 20] and tens of Thousands; as in the fourth place we count 1000, in the fifth place 10000, and in the sixth 100000, In the third Period 1000000, 10000000, 100000000; So like∣wise in the fourth Period, Thousand of Millions, where there are also three Degrees which are re∣peated in like manner with the rest, and so to Infinites.

Numeration finds by two Terms proposed a third, and if it can't be performed altogether, [ 30] makes use of Induction of parts; for 'tis the same thing to Accompt by the whole as by parts; besides every Character is considered distinctly and by it self, and if it serve to the following Numeration, it must be born in Memory to a∣void greater Trouble.

Numeration is either Primary, or conjoyned, the first, or principle is that which joyns once one number with another, as Addition and Subtraction.

Addition is the first Numeration, by which [ 40] one number is added to another, and afterwards the Total is gained. Addition is either of Numbers conjoyn'd (which is the first and most easy part by which the Total ariseth); or sepa∣rate. In which sort of Addition the Mediation of the Table of Numbers is to be considered; and first in Characters single amongst themselves, where the single may be easily added with the single.

Subtraction is the first Numeration, by which [ 50] one Number is taken away from another; and then presently the remainder is found.

Conjoyn'd Numeration, compounds Number with Number as often as it may be proposed, which is Multiplication or Division.

Multiplication is that by which the Number that is Multiplyed is adjoyned or made use of, so often as the Unite is contained in the Number which Multiplies; and so the Product of both is found.

Division is that by which the Divisor is so many times taken from that which must be divi∣ded, as it is contained, and then you have the Product which is called Quotient.

From Division proceeds the difference of Num∣ber equally even and uneven. An equal or even Number, is that which may be divided into two parts without Fraction, as 2, 4, 6, 10, 16. An unequal or odd Number is that which cannot be divided without Fraction, as 3, 5, 7.

Again, an Even Number is subdivided into e∣qually even, or equally uneven, and into une∣qually uneven.

An equally even Number, is that which may be divided into equal parts, to 1, as 4 is divided into 2 by 2, and 2 by 1. In like manner 32 in∣to 16, 16 into 8, 8 into 4, 4 into 2, and 2 in∣to 1.

Equally uneven is that which can only once be equally divided, as 10, 30, 50. Unequally even is that which has several equal parts, but can't admit of Division as far as to one, as 12, 24, 36, 48.

Secondly Number is principal, or compoun∣ded; A principal Number is not divided by ano∣ther Number of Multitude, as 3, 5, 7, 11, 13, and others; It is also called an uncompounded Number, that is to say, it doth not consist of any other Number of Multitude.

A Compounded Number is that which may be divided by another Number of Multitude, as 4 is divided into 2 by 2.

From these ariseth another difference of Num∣ber, viz. principals, and compounded between themselves; The principal Numbers between themselves, are such as are not commonly divi∣ded by any Number of Multitude, as 2 and 3, 5 and 6.

Compounded Numbers amongst themselves, are such which are frequently divided by a Number of Multitude, as 4 and 6 are compounded between themselves; because they are commonly divided by 2, which is a Number of Multitude.

But before the Parts of Number be Counted, they require some proper Notation and Reducti∣on.

Notation is of the Parts which has only two Marks, which are separated by an interposing Line, the Number above is called the Numera∣tor, that below be Denominator.

Reduction is of Terms, Intigers and Parts.

Reduction of Terms is made of the least Term proportionate, and is General or Special; the General is a Division of Terms composed a∣mongst themselves, by the greatest and most common Divisor, as 8 by 4; Special by the Species of Numeration to the Terms of two parts.

Reduction of Intigers is a Multiplication of In∣tigers by the Name of the parts. [ 10]

Reduction of parts or Fractions to Intigers, is a Division of parts by its own Name; Redu∣ction of Parts to Parts, is a Multiplication of Terms, by a Name Alternate to two proportio∣nals of the same Name.

Reduction of Parts to Equal Unites, is a Re∣duction of several Parts.

The other compounded Part of Arithmetick, shews the Composition of Numbers in Quantity and Quality.

The Terms of the Ratio of a compounded [ 20] Number are called Antecedent, and onsequent; the Antecedent is is the first Term of the Ratio, or Reason; the Consequent is the second.

Composition in Quantity is either of Equali∣ty, which is alone, and individual, as 1 to 1, 2 to 2, from 4 to 4, from 9 to 9; or of Inequa∣lity which is called difference, or Ratio.

Difference is a Composition of as much as a Term, differs from a Term; and so it is known by Subtraction; so the difference of 2 to 3, of [ 30] 3 to 5, and of 5 to 8 is 1, 2, 3.

The Ratio is a Composition so long as one Term is contained in another, and therefore it is known by Division; and so after the Ratio or Rule is given, the Terms are known by contra∣ry Multiplication, as the Ratio of 3 to 2, is 2 and a half, because 3 contains 2 once and a half. The Ratio of inequality is either more or less. The Ratio of the greater inequality is Termed the Major Term, but of the lesser by putting the [ 40] word Sub before, as the Ratio of 2 to 1 is called double, of 1 to 2 Subduple.

Ratio is either Primary or conjoynd, the first has a Species of Ratio, and is simple, or Mul∣tiplyed; simple when the Major Term contains the lesser but once and something more, as the Superparticular and Superpartient.

The Superparticular Ratio is when the Term comprehends one Term once, and one part more; so one Second, Third, Fourth, or Fifth is cal∣led [ 50] a second and a half, a third and a half, a fourth and a half, a fifth and a half, and so of others; as for Example 3/2, 4/3, 5/4.

The Superpartient Ratio is when one Term comprehends another, once and some part over and above, so 2 thirds, 3 fourths, 4 fifths is called Superbipartient third; Supertripartient fourth, superquadripartient fifth, and so of others; as for Example 5/3, 7/4, 9/5. [ 60]

The Multiplyed Ratio is, when one Term ex∣actly contains another more than once, so twice, thrice, four times is called the Double, Triple and Quadruple Rule, as 2/1, 3/1, 4/1.

The conjoyned Ratio is that which contains di∣vers sorts of Ratio's, as the manifold Superparti∣cular Ratio, when one Term contains another more than once, and some part more, as a Double Sesque Second, a Triple Sesque Third, a Quadru∣ple Sesque Fourth, as for Example 5/2, 10/3, 17/4.

The Multiplyed Super partient Ratio, is when one Term contains another more than one, and several parts more, as the double Superbiparti∣ent third, that is 2 and ⅔, the Triple Supertri∣partient fourth 3 ¾, the Quadruple Superquadri∣partient fifth 4 ⅘, as for Example 15/4 24/5.

The Comparison of Number in quality is cal∣led proportion, which is either Arithmetical, or Geometrical.

Arithmetical proportion is an equality of diffe∣rences; Geometrical proportion is in the equality of the Ratio; this is properly the Portion of Numbers, and are called proportionable Num∣bers, as 3, 6, 4, 8.

Partly direct as 3 to 6, so is 4 to 8; partly reverst as 8 to 4, 6 to 3, and alternately as 3 to 4, 6 to 8.

Proportion is separate or continued; proporti∣on separate or disjoyn'd, is that in which there are actually 4 Terms, as 12, 8, 6, 2, for through∣out the difference is 4.

Proportion conjoyn'd, or continued, is that whose middlemost Term, is compared to all the Extreams, which also is comprehended in 4 Terms, as in 8, 6, 4, the differences are equal, for the Number of 2 is throughout, and the middlemost is taken twice, for as 8 is to 6, so is 6 to 4.

Separate proportion is Simple, or Multiplyed; the simple is in 4 Terms and is direct, as the Golden Number so called from its singular use, in which amongst many other things is chiefly com∣prehended all the Invention, and supputation of Parts and Ratio's; where it is reciprocal, when it is as the First Term of the First Ratio, to the First Term of the Second Ratio, so the Second Term of the Second Ratio, is to the Second of the First.

Separate Multiplied Proportion is when more than Four Terms are used in Composition, or Con∣tinuation; Composition of Terms is either Pri∣mary, Secondary; Primary as Addition, and, Alligation.

Addition, which is called the Rule of Fellowship, is when the Terms of proportion given are added, and is Triple 1, 2, 3; the First is the Addition of the Antecedent with the Consequent, to the Consequent, as 4 to 3, so 8 to 6; then as 7 to 3, so 14 to 6. The Second is an Addition of di∣vers to one Consequent, or of one Antecedent to divers Consequents, as 2 to 4, so 3 to 6, and 8 to 4. The Third is an Addition of all the An∣tecedents to all the Consequents.

Alligation, or Deduction, is a mixture of the, several kinds, whereof the Mean is tempered.

Alligation is of a Mean either acquired, or given; the Alligation which seeks the Mean is that, which having the Terms proposed seeks the Mean in the Division, of those which are added by their Num∣ber, as if there are 2 Terms by 2, 3 by 3, and so consequently. The Alligation which gives the

Mean is the Equation of the Mean proposed by reciprocal differences to that of unequal Terms.

The second Composition of Terms is made by the Multiplied Terms, when for 2 simple we take 2 which are made by them.

Multiplication is sometimes alone, which is Vul∣garly called the Double Rule, or the Rule of Six quantities. Sometimes there is both Multiplica∣tion and Addition, which First Multiply the [ 10] Terms proposed, and afterwards add together those that are Multiplied.

Proportion continued by the Terms is when any Term of the Antecedent Ratio is continued to the consequent, as the Invention of less Numbers into proposed Ratio's and Equation, that is, the continuation of two proportionate Orders in two Numbers, which is orderly, or confused.

Orderly Equation is that which is according to the same order of Numbers, so the First are pro∣portionable [ 02] to the Second, the Second to the Third; as for Example, ••/12, 6/8, 3/4, for as 9 is to 6, so is 12 to 8; and as 6 to 3, so is 8 to 4.

Confused Equation is when all as the First of the First Order is to the Second, so the Second of the Second is to the Third; and as the Second of the First is to the Third, so the First of the Second is to the Second; as for Example, 9/24, 8/18, 6/16, for as 9 is to 8, so is 18 to 16; and as 8 is to 6, so is 24 to 8. [ 30]

Proportion continued is when the whole Ratio of the First Term is to the Second; In like manner it is of the same with the Second to the Third, as in 2, 4, 8.

Proportion continued, as separated, is simple in Three Terms, or Multiplied in those which are long continued, as the Invention of Terms, and the sum of the Geometrical Proportion, and so the Ratio of the First to the Second is doubled to the Third, tripled to the Fourth, and so consequent∣ly [ 40] in less, as in 1, 2, 4, the Ratio 1 to 4 is the First Ratio doubled to the Second, that is to say, twice put down and so Multiplyed by it self. So much for the Invention of continued, or conjoyned Numbers: Here follows the sum of Geometrical Progression.

If the First is taken from the Second and Last, it will be as the Remainder of the Second to the First, so the Remainder of the Last to the Last all preceeding; and then if 4 (which will be e∣qual to the Remainder of the Last, as the Remain∣der [ 50] of the Second to the First) be added from the whole to the last, the Sum will be as here 2/0, 4/2, 8/6, taking 2 from 4, and from 8 as 2 which remain of the Second are to the First (so 8 which remain of the Last) are to 4, and the two preceeding; for there is an Equality throughout, as 2, 2, 6, 6, are proportionable, the which 6 being added to 8 which is last, the Sum of the Progression will be 14.

Amongst the many Assistances I received from [ 60] several Experienced Persons, for the building up this, Volume, I had this following Compendious Tract of Arithmetick recomended unto me by a Person of Honour, as worthy to be inserted for its Excellency, being (Multum in Parvo) which said Tract I was the rather induced to insert as be∣ing Composed by an Ingenious Gentleman that has freely contributed his assistance in some o∣ther parts of the Mathematicks, and in particular in Fortification, wholly composed by him. The said Tract is as followeth.

NVMERATION.

NVmeration is a Series of Numbers by a Deci∣mal Progression, every place towards the Left-Hand being 10 times the Value of the next place to the Right.

The best way of reading any Number is to distinguish them into Hundreds, as in this Exam∣ple, 986, 157, 432; Or Nine Hundred Eghty Six Million, a Hundred Fifty Seven Thousand, Four Hundred Thirty two; and so of any other Num∣ber.

ADDITION.

HAving placed the Vnites of the respective Progressions in Ranks and Files: begin and add together the Vnites of the Right-Hand File, setting down the Sum underneath (if it be under 10), but if just 10, then set down a Cipher (viz. 0) and carry 1 to the next place, and if above 10, set down the Excess, and carry for every 10 an Vnite. 〈 math 〉〈 math 〉

The reason of this is nothing else than all the Parts added together make up the Whole, as in the Line a, c, a, b, (14), and b, c, (24), together make the Line 〈 math 〉〈 math 〉

The best proof of Addition is thus; After you have added all the Numbers, cut off the first Line, and then add the rest; then Subtract the one Product from the other, and the Remain∣der is equal to the upper Number, for the Pro∣duct of the one Addition is lesser than the other by the First Number only, therefore their differ∣ence must be equal to the First Number.

In Numbers of divers Denominations, when the Sum of them amounts to an Integer, or Integers of the next greater Denomination, add these In∣tegers to those of the next Denominations.

4 Farthings make a Penny; 12 Pence, a Shil∣ling; and 20 Shillings a Pound.

7 92-/100, Inches make a Link; 160 square Links a Pearch; 40 Pearch a Rood; and 4 Rood an Acre.

SVBTRACTION.

HAving placed the less Number under the greater, according to their respective Pro∣gressions, beginning at the Right-Hand, Subtract the lower Figure out of that above it set down, the Remainder under it; but if the Figure chance to be less, then there must be a Vnite borrowed from the next place, or Progression to supply the [ 10] defect, which must be paid again by adding one Vnite to the next lower Figure on the Left-hand, which is the same thing as if the Figure above it was diminished by an Vnite; and for a Proof of the Operation, the Number subtracted, and the Remainder must still take that out of which it was substracted.

For out of a Number a, c, let a less Number a, b, be Deducted; then by the Hypothesis a, b, with the Remainder a, c, are equal to the whole [ 20] Number a, c, seeing the parts united are equal to the whole, 〈 math 〉〈 math 〉

From	28	34	405
Subtract	15	7	26
Remain	13	27	379
Proof	28		405

[ 30]

In Number of divers Denominations, when the Number to be Subtracted is greater than the other, then Borrow one from the next Denomina∣tion.

MVLTIPLICATION.

FIrst, To Multiply any Number betwixt 30 and 100. In the two Numbers proposed you are to observe how many Vnites each of [ 40] them wants of 10; then (all your Fingers be∣ing open) lay down so many Fingers as the Numbers want of 10.

[illustration]

Note, All that stand up are Tens, those that are down Multiplied one by another are Vnites, which added to the Tens gives the Num∣ber, [ 50] or Product desi∣red, as in this Figure and Example, 6 times 8 is 48.

Place the Numbers one under the other, as in Addition, then Multiply the last Right-hand Fi∣gure of the Multiplicand by the same of the Multiplicator, and set the Product (if less than 10) under; but if greater, carry the Excess (that is, for every 10 one Vnite) to the next place: And if the Multiplicator have more places than one, [ 60] set down the first Figure of each respective Pro∣duct under that figure of the Multiplicator, by which it was made, and so on to the Left, ob∣serving ranks and files.

	CM.	XM.	M.	C.	X.	V.
				4	2	6
				3	2	7
1					4	2
1. Product 2				1	4	0
3			2	8	0	0
4				1	2	0
2. Product 5				4	0	0
6			8	0	0	0
7			1	8	0	0
3. Product 8			6	0	0	0
9	1	2	0	0	0	0
Total Sum	1	3	9	3	2	2

426	Multiplicand.
327	Multiplicator.
2982	First Product.
852	Second Product.
1278	Third Product.
139332	Sum of all the Products.

The reason of which (as in the Table) is, that if all the particular Sums of the Products of the Multiplicand, arising from each Multiplicator be set down and added together, they will equal the Sum of all the Products taken together.

Example.

If a Souldier having Weekly 7 Shillings, how many Shillings must you have to pay 7693.

7693
7
53851	Souldiers to Pay.

DIVISION.

DIvision is nothing else but the Deducting a less Number from a greater, as oft as may be, and so finding at last the Number, by whose Vnites that less Number being repeated, makes a Number equal to the greater, or near to it.

The greater Number is called the Dividend, the lesser the Divisor, and the last the Quo∣tient.

Set the Figures of the Divisor under an equal number of figures of the Dividend on the Left-hand; if those figures of the Dividend be of greater, or at least of equal value with those of the Divisor; otherwise you must place the first figure of the Divisor under the second of the Dividend: Then having set down the Divisor right, make Points over the figures of the Divi∣dend from the Vnite place of the Divisor inclu∣sive; the number of Points denote the number of Figures in the Quotient.

For facilitating the Division prepare such a Table (if your Number to be Divided, or your Divisor, consist of many figures) as this following,

by which you may find how often the Divisor is contained in the Dividend, or the respective figures in it; by which Multiplying the Divisor, deducing the Product out of the upper figures of the Dividend, what Remains must be con∣sidered in the next Operation, if there be more places than one in the Dividend: The next figure of the Dividend must be taken down, and set next to the Remainder (if there be any,) and the Divisor must be again set under, if the value [ 10] of the upper figures be sufficient; if not, there must be a Cipher set in the Quotient, and then the next figure of the Dividend taken down, and the very same Operation repeated until the Work be done.

〈 math 〉〈 math 〉 [ 20]

The Reason of the Table is, if you Multiply [ 30] the Divisor by 2, and you would find the double value of it; if you add the double value to the Divisor, you will find the Triple, and so on.

REDUCTION.

REduction is performed by Multiplication and Division, in bringing all sorts of Coins, or Measures to a greater or lesser Species; that is, [ 40] to a lesser by Multiplication, and to a greater by Division. As for Example, 20 l. Multiplied by 20 s. makes 400 s. divided by 20 s. makes 20 l.

The GOLDEN RVLE.

THis Rule is either Single, or Double.

The Single Rule of Three is when three Numbers are given, and a fourth demanded; and [ 50] it is either Direct, or Inverse.

The single Rule of Three Direct is when three Numbers are given, and a fourth is demanded, which bears the same proportion to the third as the second doth to the first. For Example, If four Acres of Ground cost 80 l. what will eight cost of the same Ground.

Note, That in all Questions the several Coins and Measures must be exprest, and made known. [ 60]

Secondly▪ The Number to which you are to find a Proportion, must be the last in or∣der.

Thirdly, The Number which is like in quantity to that whereunto you are to find a Proportion, must be first set down, and if it be not alike in quantity, it must be brought to it. As if 7 l. maintain 15 Men a Mouth, how many will 9 l. 10 s. maintain: Here the Pounds are to be brought to Shillings.

Having stated the Questions, they are to be resolved thus:

First, If the Number that asketh the Question be greater than that of the same Denomination, and also require more, or if it be less, and yet require less; then the Number which is of the same Denomination with the Number asking the Question is the Divisor, and the Rule is Direct: But if the Number that asketh the Question be greater than that of the same Denomination, and requires less, or if it be less, and requires more; then the Number asking the Question is the Divi∣sor; and if thus, the Rule is Inverse.

The Reason of the Operation of the Direct Rule is demonstrated from the 19th Proposition, Lib. 7. Euclid. viz. If there be four Numbers in Proportion, the Number produced of the first and fourth is equal to the Number produced of the second and third; and if it be so, then these four Numbers shall be in Proportion, as in the Example, the fourth Term found, 160 being Multiplied by 4 (the first Term) the Product will be the same with the Product of the third, Mul∣tiplied by the second, viz. 640.

Wherefore if the Product of the second and third Terms, viz. 640, be divided by the first, viz. 4, the Quotient, viz. 160, is the fourth Pro∣portional.

〈 math 〉〈 math 〉

The SINGLE RVLE of THREE Inverse.

THis Rule is when there are three Numbers given, and a fourth demanded, which bears the same Proportion to the second, as the third to the first.

Example.

If a quantity of Hay will keep 8 Horses 12 Days, how many Days will the same quantity keep 16 Horses.

〈 math 〉〈 math 〉

The same Proportion 8 Horses bears to 12 Days, so do the 16 Horses to 6 Days.

The DOVBLE RVLE.

THis Rule is when there are five. Terms given, and a sixth in Proportion to them is demanded; As if 4 Men spend 19 l. in 3 Months, how many Pounds will 8 Men spend in 8 Months.

Of the five Numbers given three, imply a Supposition, and two more a Question.

For ranking the Terms, observe amongst the Terms of Supposition, which of them is of the same name with the Number sought, and place that Term in the Middle; or second Place, write the two other Terms of Supposition one above the other in the first Place, and the Terms of demand one above the other in the last Place, in such manner that the uppermost may have the same [ 10] Denomination with the uppermost in the first Place; as thus,

Men	4—19—8	Men.
Months	3—00—8	Months.

There it is resolved by two Single Rules of Three, observing still the above-mentioned Rules.

4	19		8	38
		152
		152

3	38		9	114 [ 20]
		342
		342

FARCTIONS.

IF in the Division of the Operations there remains any thing, then the Remainder is the Numerator of the Fraction, whose Denominator is the Divi∣sor, and doth express one of the Integers in the [ 30] Quotient into so many parts, the value of which in the known parts of one Integer is found by Multiplying the Numerator of the Fraction by the Number of known Parts of the next inferiour Denomination, which are equal to the Integer, and divide that Product by the Denominators in that inferiour Denomination.

And if there happen to be any Fraction in the Quotient, you may find the value thereof in the next inferiour Denomination by the same Rule; [ 40] and so proceed till you come to the least known Parts; so the value of 2/16, of a Pound will be found in Shillings and 3 diviz. Multiply the Numerator 9 by 20 (the Number of Shillings in a Pound) the Pro∣duct 180 which divide by the Denominator 16, the Quotient is 2 Shillings 4/16, Parts of a Shilling, which by the former Rule is found in value 3 d.

To reduce a Fraction to its least Terms is by finding the greatest common Measure of both the Fractions (that is the greatest Number which will [ 50] measure or divide each of the given Numbers without a Remainder) which is thus found; divide the greater Number by the less, then divide the last Divisor by the Remainder (if there be any) and so continue dividing the last Divisors by the Remainders until there be no Remainder (neglect∣ing the Quotients) so is the last Divisor the great∣est common Measure to the given Numbers.

A single Fraction is reduced to its least [ 60] Terms by dividing the Numerator and Deno∣minator by their great∣est Common Measure, then the Quotients will be the Numerator of the Fraction, equal to the former, and in the least Terms.

• 177 (132
• 91
• 26
• 13 last Divisor.
• 26
• 00

91/117 by 13	91 (7
	13 thus 117 (9
	—7 13
	00—
	9 00

When the Numerator and Denominator are Even Numberst, hey may be measured by two.

Note, The Rule of Three is to be proved by Transposition, viz. the last Number in the sum must be set first in the Proof, and the first last; the product found in the Middle place. Then working as is before taught, the midlemost Num∣ber will come forth (if it be right performed); you must also in the Proof draw into the Multiplicati∣ons the Remainder of the Divisions, as in the Ex∣amples following.

Example to Exercise the GOLDEN RVLE.

First for the Direct Rule.

1. IF 1 lb. of Powder cost 4 d. 2 What will 595 Pound cost?

	4
18	—
—	18
4760
595
—(2 (1
10710 2677 (22 3
4444 1222 (11
11

Answer 2 l. 3 s. 1 d. 2 l. Transposition, or Proof.

If 595 lb. 2 l. 3 s. 1 d. 1 lb.

2. If of one Circle the Diameter being 7, the Circumference 22 Inches, what is the Circum∣ference of a Circle whose Diameter is 75 Inches?

	Proof.
7—22—75	75—235 5/7—7
22	7—
—	—1650
150	525 7
150	—
—(5	11550 (22
1650 (235 5/7	5255
777	52

Note, that in such Questions with Fractions, you must Multiply the first Number by the Denomi∣nator of the Fraction of Second and Third Num∣bers; and the Fraction of the first must be set down under the second Number, that all may be reduced equally, as in the fore-going, and this Example.

If one Man work one Day 3 1/9 Rod, how much will 10 Men work in one Day?

¼—3 ¼—10	10—37 ½—1
—15	2 1
15—(2	—(2
150 (37 ½	20 75 (3 ¼
44	20

For the Inverse Rule.

If 3 Men work a Raveline in 72 Days, in how long time will 18 Men work the same?

	Proof.
3—72—18	18—12—3
3	12
—	—
246 (12 Days.	36
188	18
11	—261 (72 Days.
	261 33

[ 10]

There is a Castle besieged, which is sufficient∣ly Victualled for 12 months, 10 Ounces of Bread, 7 Ounces of Cheese, and 3 Ounces of Butter e∣very day. The Governor is advertised, that if the Castle can hold out 15 Months with the same [ 20] Provisions, that the Siege shall be raised; How much then shall be allowed to every Souldier a Day, that they may be able to hold out the time?

12—10—15	12—7—15	12—3—15
12	12	12
—	—(9	—(6
20	84 (5 9/15 Ounces of Cheese.	36 (2 2/5 Oun∣ces of Butter.
10	15	15
120 (8 Ounces of Bread.
15

[ 30]

Examples in the Double Rule.

If four Souldiers can make up 12 Rood of a Trench in 24 Hours, how many Roods can 32 Souldiers make up in 72 Hours? [ 40]

Souldiers. Rood. Souldiers.	Hours. Rood. Hours.
4—12—32	24—96—72
12	72
—	—
64	192
384 (Rood. 32	672 Rood
44 96—	—6912 (288
384	6912 24

If 3 Horses cost in keeping 6 Days 16 s. what [ 50] will cost the keeping of 24 Horses 48 Days?

3—16—24	6—128—48
16	48
—	—
144	1024
384 (128 s. 24	512
333—	—shill.
384	6144 (102 4 (4 shill.
	6666 51 l.

[ 60]

If 2 Souldiers have 4 for Months Entertainment 9 l. 12 s. what shall 756 Souldiers have in 12 Months?

2—9	—12—756
20	192
—	—
192	1512
	6804
	756
	—(1
	145152 (7257 (6
	22222—
	3628—16

Of the SQUARE ROOT.

Every Number being multiplyed by it self of what value soever, gives a Square Number, and the Number whereof the Square is produced by the Multiplication of it in it self, is called the Side, or Root of the Square.

Therefore to find the Quadrate, or Square Root, or side of any Number which Multiplyed in it self maketh the Number proposed. You are to take Notice that all Squares under 100 are found by the following Table, or by Multiplying any of the nine Simple Numbers in themselves; but the sides of greater Squares are to be found out only by Art.

The Table.

1	2	3	4	5	6	7	8	9
1	4	9	16	25	36	49	64	81

This Artificial Device is taken from the 4 Propositions of Euclid, where by Demonstration it is proved, that if a right Line be cut into any 2 Parts, or Segments, the Square of the whole Line is equal to the Squares of the Segments, and to the 2 right angled Figures made of the Segments; as in the Figure, the 2 Diagonals K, G, and b, f, are the Squares of the Segments a, b, and b, c, also the Complements B, K, and f, G, are the right angled Figures, made by Multiplying the Line a, b, by b, C.

[illustration]

The same parts are to be found in any Square Number; as for Example, let the Number be 169, whose side is 13, the side divided into 2 parts, viz. 10 and 3, multiplying each part by it self one, viz. 10 by 10, and 3 by 3, then multi∣ply one by the other, as 10 by 3, and 3 by 10, so you shall have 4 plain Numbers, whereof 2 are Squares. Therefore as the Square 169 is made by adding these four Numbers together, so by sub∣ducting of them severally it is resolved. To which First Mark each odd, or third place with Points, because the particular Squares are to be found in the odd places, and how many Points there be of so many Figures the Root is to consist of. Then for so much as the Vnity standing under the first Point next the Left Hand is both a Square,.. and the side of a Square, that Figure therefore being set in the Quotient alone, and Subducted from the Vnity, standing under the Point, nothing remains.

169 (1

1

—

0

This Vnity set alone by it self in the Quotient doth signify 10, when another Figure is set by it, representing the side of the other lesser Square, as this doth the side of the greater. Wherefore the greater Diagonal K, G, is now Subducted from the whole Square, and the side thereof, if K, c, or a, b, they being equal to one another, and also the side of one of the Complements is found out: and this is the first Step. Moreover double the Figure in the Quotient, because being doubled it is [ 10] the side of both the Complements taken together, viz. K, 1, and G, 1; then setting 2,.. the double Number, under 6, divide 6 (each in this place is as much as 60, and represents both the Complements) by 2, the Quotient is 3 representing the other side Remaining of the Com∣plements, viz. j, f, or b, C, which Number place in the Quotient, and reckon it for the Segment remaining [ 20] of the right Line given. Wherefore because 3 is the side of the Remaining Diagonal, that is to say, of the lesser Square b, f, therefore being set by the Divisor on the Right-Hand, and Multiplyed by it self, and also by the Divisor, it brings forth 3 plain Numbers, viz. the Square b, f, and the 2 Complements a, j, and a, L, which be∣ing subducted from the Numbers standing over them, nothing Remains; and so in any other Ex∣ample: As in this, 625 the Root is found to be [ 30] 25; but if this 625 had been 655, which is a greater Number than 625 by 30 Vnites, the Inte∣gers of the Square Root had been still 25, only there had been left 30 which had been the Nume∣rator of a Fraction, whose Denominator must ever be the double of the Root augmented by an Vnite, and the Operation had been 25 30/51.

To prove if you have extracted right, multiply the Root in it self, if there remains any thing, add it to the Product, which if rightly performed will [ 40] give the Square Number.

169 (13

1

023

3

69

00

Some Examples.

1. A Colonel having 1849 Men, he would set them in a square Battalion, how may Souldiers must there be in Rank and File?

1849 (43 in Rank and File.
16
—	Proof.
249
83	43
3	43
249	129
	172
000	—
	1849

[ 50]

2. There is one equal sided Square piece of Ground the content of which is 6098 Square [ 60] Rods; to know if the length of one of the sides answer 78 Rods.

3. A Colonel having by him 2048 Men, he would place them in a Battalion, which shall be twice so long as broad, how many Souldiers must there be in length and breadth?

To resolve any Question of this nature, divide the Number of Men by the Number of the pro∣portion of the Battalion, then extract the Root out of the Quotient, which is the Souldiers in breadth, which Multiplyed by 2 the Proportion gives the Souldiers in depth, or length.

2048 (1029 (32 in Breadth	32
2222	2

And so of any other Proportion. 64 in length.

4. A General having an Army of 33756 Soul∣diers, would have them set in form of Battle 3 times so long as broad; how many Men must there be in breadth and length?

33756 (11252 (106 Souldiers in Breadth.

(16
	106
	3
	318

To find how much of the Remainder of the Extraction is in Souldiers, multiply the Remain∣der by the Proportion (3); the aforesaid Example being in a Triple Proportion, and therefore it is divided by 3, and to bring it to the first again multiply by 3.

16 Remainder in the Extraction.

3 Divisor.

48 Souldiers remains of the whole Body.

For Measuring of Land it is not material to come so precisely to the knowledge of the Re∣mainder of the Extraction, unless it be Acres, Perches, or any great Measure, which you may reduce to Inches, and then draw forth the Extra∣ction, and quit the Remainder.

Of the CVBE ROOT.

A Cube in Geometry is a right Angled Parallel∣lipipedon, having 6 equal Surfaces, 8 solid An∣gles, and 12 sides, as in the Figure a, B, C, D, E, f, G, h, whose sides are a, b, or a, d, also B, C, or C, d, either C, E, or E, f, also E, h, or h, G, likewise G, f, or d, f, or d, a, and G, a.

[illustration]

A Cube in Arithmetick is a Number made by three equal sides, or of two Multiplications by three Numbers, that is, of any Number Multiplyed in it self; and that Pro∣duct again by the first Number, expressing Length, Breadth and Thickness.

The Number whereof the Cube is produced by the Multiplication of it in it self is called the side, or Root of the Cube, which being found out in whole Numbers, the Cube in known.

As for the Cubes under a 1000, whose Roots are simple Numbers, they are found by the fol∣lowing Table.

1	2	3	4	5	6	7	8	9
1	4	9	16	25	36	49	64	81
1	8	27	64	125	216	343	512	729

But it searching out the sides of greater Cubes, proceed as the following Theorem directs.

This Artificial Device is taken out of the Theo∣rem.

If a right Line be cut into two Segments, the Cube of the whole Line shall be equal to the Cubes of the Segments; and to the two Solid Fi∣gures comprehended three times under the Square of his Segment, and the Segment remaining.

As the Line E, j, which is 13, is cut into two [ 10] Segments 10 and 3; therefore the Cube of the

[illustration]

whole Line, viz. 2197, is equal to the Cubes of the Segments, viz. 1000 and 27, and also to the Two∣fold solids, or Parallellipepidons thrice taken; [ 20] wherefore three have the like So∣lidity. The Solidi∣ty of each of the three lesser is 90, being made of the Square of the Segment 3, that is, of 9 Mul∣tiplyed by the other Segment 10, these three Parallellipipedons taken together make 270; but of the three greater Parallellpipedons each con∣taineth 300, made of 100, the Square of the greater Segment, 10 Multiplyed by the lesser Seg∣ment [ 30] 3, and they taken joyntly together make 900.

The Cube of the greater Segment.

[illustration]

The Cube of the lesser Segment.

[illustration]

[ 40]

The three greater Parallellipipedons.

[illustration]

[ 50]

The three lesser.

[illustration]

[ 60]

The Cube whereof hath eight particular Solids in Number, which are made of the Parts of the Number given, viz. of 10 and 3 in this man∣ner.

First let there be four plain Numbers made, each part being Multiplyed by it self, and one by ano∣ther.

10	3
10	3
100	9
	30
	30
	100

If again you Multiply the plains by the same parts, there will arise eight Solids, as you here see.

9	9
30	30
30	30
100	100
3	10
27	90
90	300
90	300
300	1000

All which being added to∣gether, are equal to the Cube of the whole.

Therefore the same way that is used in making the Cube, is to be followed in resolving the Cube: As for Ex∣ample.

Mark the Cube given with points, omitting each two Figures continually, beginning at the right Hand, as 2197; then subduct the particular Cube of the Number set underneath the last point, but that Number being no Cube, take the nearest to it, viz. an Unite, which set in the Quotient, the Unite in the Number given is 1000, but in the Quotient it is but 10, the Unite subducted from 2, the remainder is 1; which must be writ over the Number given, so that the greater Cube A, is sup∣posed to be subducted from the Number given; this is the first step.

1

2 197 (1

1

Triple the Quotient found out (that is, Mul∣tiply it by 3) this Triple represents the 3 sides taken together of the 3 lesser Solids marked C. then place the Triple Number under 9. again Multiply the Quotient Squarewise, and Triple the Product which likewise makes 3. this Pro∣duct represents the 3 square sides of the 3 greater Solids, taken together marked D. then place the Product 1 Degree lower towards the left Hand underneath, with it divide 2 which is writ above it, the Quotient is 3, the Segment or Quoti∣ent 3, being Multiplyed by 3, the Devisor makes 9, which in respect of the place where it stands is 900, and represents the 3 greater Solids mar∣ked D, taken together.

Moreover the same Quotient being Multiplied Squarewise maketh 9, and Multiplied afterwards by the Tripled Number, standing under 9, makes 27, which in respect of the place it is in is 270, and represents the 3 lesser Solids marked C. Last of all the same Quotient Multiplied Cubically, brings the lesser Cube B, 27. These three Products therefore being added together▪ and the Total Subducted from the Number standing over it, there remains nothing which imports but the given Numbers is a Cube; As in the Example.

1		2197 (13 the greater Cube.
2197 (13		1000
1		3
3		3
—	Or, thus,	—Solids
9		900 the 3 greater.
27		270 the 3 lesser.
27		27 the lesser Cube.
2197		2197

[ 10]

The use of this will appear in casting up the Solidity of the Ground of a Fort, which is here inserted only to shew the Sections of a Solid Body.