Parag. 1.

ADDITION. Parag: 2. The first species.

SUBTRACTION. Parag. 3. The second species.

MULTIPLICATION. The third Species, and fourth Paragraph.

DIVISION. Parag: 5. The fourth species.

Paragraph 1.

Paragraph II. Addition of proper and improper Fractions.

Paragraph III. Subtraction of proper and improper Fractions.

Paragraph IV. Multiplication of proper and improper Fractions.

Paragraph V. Division of proper and improper Fractions.

To finde the Square made of any of the 9 significant figures, or their Squares given to finde their Roots. A Demonstration in extracting of Square Roots.

A Square number given that is greater than 100, the number for extraction must be prepared after this manner. Example 1.

To finde the Quadrat of any number consisting of four places or more. Example 2.

To finde or extract a Quadrat Root, consisting of 5 places. Example 3.

To extract a Quadrat Root out of any irrationall number; and to finde the Root with a fraction in any proportion assigned, as ⅓ or ¼ &c. Example 4.

To extract the Quadrat Root from any irrationall number, with a Fraction, whose Denominator shall be a Decimall, as 1/10, 1/100, 1/1000, 1/10000, 1/100000, &c. Example 5.

How to extract a Quadrat Root from any Fraction or broken number propounded. Example 6.

To extract the nearest Quadrat Root from any Fraction, that is irrationall in both termes, yet reduced to their least denomination. Example 7.

To finde the Quadrat Root of any mixt or compound number, when the whole number, the terms of the Fraction, or either of them are rationall or irrationall. Example 8.

To finde a Cube made of any significant figure; or the Cubes of them given to finde the Root.

Example 1.

To extract the nearest Root out of any number, whose Cube was lesse then 1000. Example 2.

To extract a Cubicall Root from any number greater then 1000, and how to prepare the number, and point out the figures for the Root which is to be extracted. Example 3.

To extract a Root out of a Cubique number consisting of 5 places. Example 4.

How to extract a Root from any Cube number that consists of six places. Example 5.

To extract a Root from any perfect Cubique number consisting of significant figures and cyphers in what number of places soever. Example 6.

From any irrationall Cubique number to extract the Root, or the nearest to it, with a fract on in any given proportion as ⅓ or ¼ &c. Example 7.

To extract the Cubique Root from any number that is irrationall, and to produce the Root in a decimall Fraction, viz. in 1/10 1/100 in 1/1000 in 1/10000 &c. Example 8.

To extract the Cubique root from any irrationall fraction, when either the Numerator, or Deno∣minator, or both, are incommensurable. Example 9.

To extract a Cubique root from any mixt or compoun∣ded fraction, when either Numerator, or Denomi∣nator, or both are incommensurable, or the improper fraction a perfect Cube. Example 10.

Example 11.

Example 12.

The extraction of a Biquadrat Root.

To extract a Root from any Squared Cube.

To extract a Root from a Biquadrat number squared.

To extract a Root from any Cubique Cube number.

To extract any Root out of an irrationall number in a Decimall fraction, or in any other proportion propounded, either in whole or broken numbers.

To extract any Root from a simple fraction, or any compounded number that shall be assigned, and in any proportion, as 1/10, or 1/100, or 1/1000, &c.

The definition of Quantity in Numbers.

Addition of these 5 Species or unequall Proportions.

An Example of Addition in these Proportions.

Theoreme 1.

Theoreme 2.

Theoreme 1.

Theoreme 2.

Proposition 1. Any two numbes propounded, for to finde a mean proportionall between those numbers.

Proposition 2. A mean proportionall number being given, with either of the extremes, to finde the other extreme.

Proposition 3. Betwixt any two numbers propounded, for to finde two mean Proportionalls.

Proposition 4. Between any two Extremes to finde three Mediums or Proportionall numbers.

Proposition 5. To finde so many mean proportionall numbers between an unite and any number, as shall be required.

Proposition 6. In any number of Geometricall Progressions continued, to finde the totall summe of them.

Axiomes.

In any number for to be multiplied by 9, the Product may be discovered by sub∣traction onely, and thus.

How to finde two squared numbers, whose difference shall be any number propounded.

An illustration.

A Definition of this Golden Rule of Proportion.

The Golden Rule of Proportion in Numbers, Geometrically demonstrated by Lines. In any three right lines given, what proportion soever the first hath to the third, the second line will be in the same proportion to a fourth; and the Square of the two mean proportionalls, shall be equall unto the long Square made of the two extremes. The Demonstration.

Question 1.

Question 2.

Question 3.

Question 4.

Question 5. If 10 yards of Holland did cost 1 L. 3 S. 6 ½ D. What will be the price of 1 yard?

Question 6. If 10 yards of Linnen cloath cost 2 L. 14 S. 9 D ½: what will the price be of 6 yards?

Question 7. If any certain number of Men shall perform a piece of worke within a known time; in what time would a greater or lesser number of men have performed the same?

Question. 8. If part of any number, weight, or measure, did cost a part of any summe or integer of money, what shall any quantity of that commodity cost?

Question 9. When the Divisor is a Fraction, or compound number in the Rule of Three direct or reverse; it may be reduced to the Rule of Practice by this following Example.

A Directory unto the double Rule of Three.

Question 1. There was a Knight had 5 Sonnes, and 3 of these in 4 moneths had spent him equally 19 L. that is in all 57 L. and it is required what his 5 Sonnes would have spent at that rate in 12 moneths?

Question 2. If 12 Pecks of Provender did serve 4 Horses for 3 dayes, how long will 24 Pecks serve 3 Horses at the same dayly allowance.

Question 3. If a mans 3 Sonnes spent 57 L. in 4 moneths, what would his 5 sonnes expences have been at that rate in the space of 12 moneths?

Question 4. If 4 Horses did eat 12 Pecks of corne in 3 dayes, in what time will 3 horses eat 24 pecks of the same corne and dayly allowance.

Definitions.

A Rule of Society.

Question 1. Two men, viz: A and B, joyned their stocks together: A had 18 L. and B had 12 L. with which they bought wares, and quickly sold them againe for 2 L. 10 s. profit: how much did either of them gaine?

Question 2. Five Merchants put their stocks together, as A, B, C, D, & E: whereof A did adventure 175 L. B. 140 L. C 70 L. D 35 L. E 30 L. this stock re∣turned them 744 L. all charges defrayed: How much was each mans gain or losse?

Question 3. Three men unhappily imployed their stocks together, as A, B, & C: whereof A deposited 20 L. B 30 L. C 60 L. all which they layd out; and afterwards sold those goods for 84 L. 6 s. 8 d. how much was each mans gain or losse?

Question 4. Three men held a pasture in common (as A, B, & C) for which they paid 44 L. per annum: A kept 10 steares upon the ground 20 weeks, B fed 15 steares 16 weeks, and C kept 20 there 8 weeks: what rent must each man pay?

Question 5. Four men joyned their stocks together, as A, B, C, & D, whereof A ventured 40 L. B 160 L. C 100 L. D 280 L. by misfortune they were all losers: upon which they fell at difference, at severall times, and broke off from this society, when D had lost 20 L. C 10 L. & B 30 L. A continued the trade 12 moneths, and lost 8 L. how long were all their stocks continued at that rate or proportion?

A Definition.

The rule.

Question 1. A man mixt for provender 3 quarters of beanes at 3 s. the bushell: 3 ½ quarters of pease at 2 s. 6 d. the bushell; with 5 ½ quarters of oates, at 1 s. 6 d. the bushell: what will a bushell of this mistling be worth, when all these are mixt together.

Question 2. A Vintner put into a Wine-casck 15 gallons of Canary-sack which cost 4 L. 10 s. 30 gallons of Mallage 6 L. 5 s. & 35 gallons of Sherry sack 7 L. with 40 gallons of Greek wine 8 L. a pipe thus filled, the price of one gallon is demanded?

The second Rule of Alligation.

Question 3. A Druggist had two sorts of Tobacco; the one was Spanish, at 10 s. the lb; the other Virginia, at 3 s. the pound; of these two sorts, he was to mix 112 lb weight, and so, as that it might be afforded for 8 s. the lb: how much must be taken of either sort?

Question 4. A Grocer had 3 sorts of Sugar, one was worth 6 d. the pound, another sort 10 d. a pound, and the best 15 d. a pound, out of these sorts, he made two parcells: either of them 56 pound weight, the one thus mixt to be afforded at 12 d. the pound: the other sort at 10 d. the pound. How many pound must be taken to be mixt of either sort?

Question 5. A Grocer hath 4 sorts of currans; one at 4 d. another 6 d. a third sort 9 d. the best 11 d. the pound: the worst sort slighted as too meane, the best thought too deare, so two sorts would not sell: upon which the Grocer compounded two parcells out of the 4 sorts, each containing 120 pound: and so mixt, as that he might afford them at 8 d. the pound. How many pound had the man of every sort, in either parcell?

Question 6. A composition of severall simples to be made in weight 1 ½ C grosse; and to be afforded at the price of 5 s. the lb the particulars at these rates: viz: A 1 s. lb B 3 s. lb C 5 s. lb D 8 s. lb E 11 s. lb & F 14 s. lb How many pound must be taken of each sort, not ex∣ceeding either the price, or quantity propounded?

The same Question varied thus, with the former prices, and quantity propounded.

Question 7. A Gold-smith hath severall masses, or ingots of Gold, one purer than another; of 3 sorts he was to make a mixture, whereof one was 16 Carects: the second 18 Carects: the third was 20 Carects fine: this the Gold-smith was to mix with such an aloy, as that the whole mixture of 135 Ounces should be 14 Carects fine: How much must be taken of each sort?

A preparative in the Alligation of Physicall Simples, according to their severall qualities or temperatures.

Question 1. Out of two Simples, to make a composition; which Me∣dicine shall be of any mean temperature between the simples propounded, and of any quantity assigned.

Question 2. A Composition is required of 3 Simples, whose quali∣ties are knowne, and a temperature betwen one of those three is demanded.

Question 3. A Composition of 4 Simples is required whose qualities are known, and the quantity of any one is to be mixt with such quantities of the rest, that the quantity of the Composition may be of any temperature re∣quired, greater than the least, and lesser than the greater given.

Question 4. A Composition being made of divers Simples, whose qualities and quantities are certainly known; and it is required, in what degree of temperature this Confection is in?

Question 5. A Confection to be made of severall Simples, whose particular quantities and qualities are known, as in respect of any degree in Heat, Drought, Cold, or Moysture, how to finde the temperature of such a composition.

Question 6. To encrease or diminish in quality any composition or medicine, according to any degree of temperature that shall be assigned.

A generall Rule.

A Definition of false Positions.

Question 1. There was a man had two sons and one daughter, viz: A, B, & C, the old man (in his last Will & Testa∣ment) bequeathed to his children all his estate (in value unknown) in this manner: to his eldest son A, he gave a portion double to B, his second son; and his part, treble his daughters C: the estate after his death was praised at 745 L. What must the chil∣drens parts be?

Question 2. Five Merchants, as A, B, C, D, E, entred into So∣ciety with severall stocks, and according to their shares for to stand the pleasure of Fortune either in gaine or losse: profit was their hopes, and proved the end: their stock unknowne, the gaine 675 L. their conditions were, that ½ the gaines of A, should be ¼ part of B. and ¼ part of B, should be ⅕ part of C. three times A the gaines of D. and twice what B did gaine, was the gaines of E, each particular share is required.

Question 3. The summe of 570 L. was to be distributed unto 3 men, A, B, C; but at severall payments, and upon these conditions, that when A received 2 L then B should have 4 L. and so often as B took 3 L. was C to have 5 L. in this proportion was the money payd, and here their shares are required.

Question 4. There came unto a wedding 6 Lords, 8 Knights, 24 Esquires, and 48 Burgesses: these gave at an offer∣ing 120 L. in this manner, viz: 2 Lords gave as much as 4 Knights, 5 Knights equall unto 6 E∣squires, and 5 Esquires gave so much as 10 Bur-Burgesses. How much did each distinct degree of∣fer?

A Definition of the double Rule of False positions.

The second Rule of Proportion in False positions, when the errors have unlike signes, as one ✚ the other —.

A Directory unto this double Rule of False positions.

The common way, and directions, in the double Rule of False positions.

Question 1. There was an Excise upon all goods, sold by whole-sale men: and by an Edict, if any man rendred not a true account, the penalty (if discovered) was double the commodity: yet all trades-men allowed to be as obscure as they could, the truth being affirmed: it was demanded of one man, what he had sold: Who replied, 2 pieces of Canvas, 2 of Fustian, and 7 of Holland, every piece a crown more than other; and one piece of the best, was 3 times the price of the worst: what was the value of the 11 pieces of cloth?

Question 2. To divide any number propounded into any parts that shall be required; and those parts for to be in any proportion one unto another, that shall be assigned.

Question 3. To divide a given number into any two parts, and those in any quantity assigned, as to part 10 in two; and so, as that the greater divided by the lesse, the quo∣tient shall be 20.

Question 4. A vessell of 63 gallons was filled with French wine of two sorts; the one was at 2 s. the gallon, the other at 2 s. 6 d. the gallon; the wine in the hogs-head thus filled, did come unto in money 7 L. 4 s. and it is here demanded how much there was of either sort.

Question 5. Hiero King of Sicylia, caused a Crown of Gold for to be made in weight 10 lb. and it was conceived, that the Work-man had put a great Allay of Silver unto it, which abuse of the Artificer Archimedes de∣tected; and by False positions may be thus dis∣covered.

The examen, or triall.

The Rule of False Positions to discover the Truth: or by erroneous suppositions to finde things really true, a paradox, and no hyperbole.

CHAP. I. The Definition of Artificiall Arithmetick, with the Reduction of the Decimall fractions, and the art of framing those numbers.

Sect. I. Chap. II. Addition of Decimalls with integers, fractions, and compounded numbers.

Sect. I. Chap. III. Subtraction of Decimalls with integers and fractions commixt together.

Sect. I. Chap. IV. Multiplication of Decimalls commixt with whole, and broken numbers.

Sect. I. Chap. V. Division by Decimalls with integers, and com∣pounded numbers divided into 6 cases or rules.

Sect. I. Chap. VI. Tables of the chief Coines and Weights, with the long drie and liquid concave measures commonly used in England.

Sect. I. Chap. VII. Of Decimall Tables calculated to 7 places, ac∣cording to the fractions before, in Number, Weight, and Measure, with Time made apt for use in this kinde of Artificiall Arithmetick.