Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.

CHAP. I. Of ARITHMETICK.

ARITHMETICK is* 1.1 an Art of num∣bring well, for as mag∣nitude, or greatness, is the subject of Geome∣try, so is multitude, or number, that of A∣rithmetick.

I shall not in this place trouble you with the first Rudiments of Arithmetick, as Numerati∣on, Addition, Substraction, Multiplication, and Division: because they are already largely handled by many, as Mr. Leybourn, Mr. Win∣gate, and divers others, and also that then this Book would swell to too big a bulk for the Pocket, and so my design would be frustra∣ted; I shall therefore only propose and ope∣rate

some principal Propositions, that are of Special moment in Arithmetick, and which most immediately concern the other following parts of this Treatise.

SECTION I. The Explication of some Arithmetical Pro∣positions.

PROPOSITION I. To three numbers given, to find a fourth in a Di∣rect proportion.

To operate this proportion Multiply thē third term, by the second term, and their product divide by the first term, the Quotient shall be a fourth term required. Examp. 1. Admit the Circumference of a Circle whose Diameter is 14 parts be 44 parts, what is the Circumference of that Circle, whose Diameter is 21 parts? Now according to the Rule if you multiply the third term 21, by the second term 44, it produceth 924; which divided by the first Term 14, the Quotient is 66, and so the Circumference of the Circle, whose Diameter is 21, will be 66 parts, and so for any other in a direct proportion.

PROP. II. To three numbers given, to find a fourth in an Inversed proportion.

To operate this proportion, Multiply the first term, by the second term, and their pro∣duct divide by the third term, the Quotient is the fourth term required: Examp. Admit that 100 Pioneers, be able in 12 hours to cast a More of a certain length, breadth, and depth; in what time shall 60 Pioneers do the same? Now if according to the Rule, you Multiply the first term 100, by the second term 12, their pro∣duct is 1200; which divided by the third term 60, the Quotient is 20, so I say that in 20 hours, 60 Pioneers shall do the same, and so for any other in an Inversed proportion.

PROP. III. To three numbers given, to find out a fourth in a Duplicate proportion.

The nature of this proposition is to discover the proportion of Lines, to Superficies, and Superfi∣cies, to Lines; for like Plains are in a duplicate Ratio; that is as the Quadret of their Homologal sides; therefore to Operate any Example in this proportion, Square the third term, and its square multiply by the second Term, their product di∣vide by the square of the first Term, the Quotient is the 4th. term sought; Examp. Admit there be two Geometrical squares; now if the side of the grea∣ter

square be 50 feet, and require 3000 Tiles to pave it; what number shall the lesser square require, whose side is 30 feet? To operate this according to the Rule, I square the third Term 30, whose square is 900: then I multiply it by the second Term 3000, its product is 2700000, which divided by 2500, the square of the first Term 50, the Quotient is 1080, and so many Tiles will pave the lesser square, whose side is 30 feet.

PROP. IV. To three numbers given, to find a fourth in a Triplicate proportion.

THE nature of this proposition is to disco∣ver the proportion of Lines to Solids, and So∣lids to Lines; for like Solids, are in a Triplicate Ratio, that is to the Cubes, of their Homolo∣gal sides: Therefore to operate any Question in this proportion, Cube the third Term, and his Cube multiply by the second Term, and their product divide by the Cube of the first Term; the Quotient is the fourth Term sought. Examp. Admit an Iron Bullet whose diameter is 4 Inches, weigh 9 pounds; what is the weight of that Bullet whose Diameter is 6 Inches? Now to operate this proportion; first according to the Rule I Cube the third Term 6 whose Cube is 216, then I multiply its Cube by the second Term 9, the product is, 1944, which divided by 64, the Cube of the first Term; the Quotient is 30 24/64 pounds which is equal unto 30l. 6 ounces: which is the weight

of the propounded shot; and so for any other.

PROP. V. To two numbers given, to find out a third, fourth, fifth, sixth, &c. Numbers in a continual pro∣portion.

To operate this proportion, you must multi∣ply the second number by it self, and that pro∣duct divide by the first Term, the Quotient is a third proportional: Again you must multiply the third Term by it self, and its Quadret di∣vide by the second Term, the Quotient is a fourth proportional, and so after this manner a fifth, sixth; or as many more proportionals as you please may be found: Examp. Let it be re∣quired to find six numbers in a continual pro∣portion to one another; as 4 to 8. To operate this first according to the Rule, I multiply the second Term 8 by it self the product is 64, which divided by the first Term 4, the Quoti∣ent is 16: so is 4, 8, and 16 in a continual pro∣portion; And so observing the Rules prescribed, proceed in your operation untill you have found your six numbers in a continual propor∣tion; which in this Example will be 4, 8, 16, ••2, 64, and 128, and so will you have form'd six numbers in a continual proportion.

PROP. VI. Between two numbers given, to find out a mean Arithmetical proportional.

THIS proposition might be performed without the help of the rule of proportion: ne∣vertheless because it conduceth to the Resolu∣tion of the next ensuing proposition, I insert it in this place; To operate it this is the Rule: add half the difference of the given Terms, to the lesser Term, so that Agragate, is the Arith∣metical mean required: Examp. Admit 20 and 50 to be the two numbers propounded: Now to operate this proposition, first according to the Rule, I find that the difference of the two given Terms 20, and 50, is 30, whose half is 15, which being added to the lesser Term 20, it makes 35, so is 35, a mean Arithmetical pro∣portion betwixt 20, and 50, given.

PROP. VII. Between two numbers given, to find out a mean Musical Proportional.

BOETIUS hath this Rule for it, where∣fore take his own words:* 1.2 saith he,

Differen∣tiam terminorum in mi∣norem terminum multi∣plica, & post junge ter∣minos, & juxta cum qui inde confectus est; com∣mitte illum numerum,

qui ex differentiis & ter∣mino minore productus est, cujus cum latitudi∣nem inveneris, addas eam minori termino, & quod inde colligitur me∣dium terminum pones.

That is, Multiply the difference of the Terms, by the lesser term, and add likewise the same Terms together: this done if you divide the product, by the sum of the Terms, and to the Quotient thereof, add the lesser Term; the last Sum is the Musical mean desired: Examp. Admit the two numbers given be 6, and 12. I say that if the difference of the Terms which is 6, were Multiplied by the les∣ser Term 6, it would produce 36; then if you add the two terms 6, and 12, together: their sum would be 18, now if you divide 36, by 18, the Quotient is 2; lastly if to the Quotient 2, you add the lesser Term 6, the sum thereof will be 8, which is a Mean Musical proportional required.

PROP. VIII. How to find the Square-Root of any whole num∣ber, or Fraction.

* 1.3 To Extract the Root of any Square number propounded, is to find out another number, which being Multiplied by it self, pro∣duceth the Number propounded. Now for the more easie and ready Extraction of the Square-Root of any number given, This Table

here under annexed will be usefull; which at first sight giveth all single Square numbers, with their respective Roots.

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

The Explication of the Table.

In the uppermost rank of this Table, is pla∣ced the respective root of every single Square∣number, and in the other the single Square∣numbers themselves; so that if the Root of 25 were demanded, the Answer would be 5, so the Square root of 49, is 7, of 81 is 9; and so for the Rest, and so contrarily the Square of the Root 5 is 25, of 7 is 49, of 9 is 81, &c.

Example: If the Square root of 20736, were required, first they being wrote down in or∣der as you see, draw the Crooked-line,* 1.4 then to prepare this or any o∣ther number for Ex∣traction, make a point over the place of Unites; and so on every other figure towards the Left-hand; as you see in the Margent.

Then find the Root of* 1.5 the first Square 2, which is 1; place it in the Quo∣tient, and also under 2; then draw a line, and substract 1 from 2, there remains 1; which place under the line, then to the last remainder 1, bring down the next Square 07; and then there will be this num∣ber 107, which number I call a Resolvend: Then double the Root in the Quotient 1, whose double is 2, which 2 place under the place of tens in the Resolvend, un∣der 0; so is this 2 called a Divisor; and 10 called a Dividend.

Then demand how often the Divisor 2, can be had in the Dividend 10, it permitteth but of 4, which place in the Quotient, and under 7 the place of Unites in the Resolvend, and there will appear this number 24; Then Mul∣tiply this 24, by 4, (the last Square placed in the Quotient) it produceth 96, which place orderly under 24, as you see, and this 96 is called a Ablatitium; (but some calleth it a Gnomon:) then draw a line under it, and sub∣stract 96, the Ablatitium, out of the Resolvend 107, there remains 11, which place orderly under the last drawn line, then thereunto bring down the next Square 36, so will there be a new Resolvend 1136; then double the whole Root 14 in the Quotient, whose double is 28;

place it under the Resolvend 1136 as was a∣fore directed; so shall 28 be a new Divisor, and 113 be a Dividend; then I find the Divisor 28 can be had in the Dividend 113, 4 times, which four place in the Quotient, and under the place of Unites in the Resolvend, so there appeareth this number 284, which number, multiplyed by 4, the last figure in the Quotient, produceth a new Ablatitium 1136; which place orderly under the Resolvend 1136, and then draw a line, then substract the Ablatitium 1136, from the Resolvend 1136; and the remainder is 00, or nothing: and thus the work of Ex∣traction being finished, I find the Root of the Square number 20736, to be 144; and so must you have proceeded gradually step by step, if the number propounded, had consisted of some 4, 5, 6, or more Squares; still observing the aforegoing Rules and Directions.

NOTE.

BUT when a whole number, hath not a Root exactly expressible by any rational or true Number, then to find the fractional part of the Root very near; To the given whole number annex pairs of Cyphers, as 00, 0000, or 000000, then esteem the whole number, with the Cyphers both annexed thereunto, as one intire whole number: and Extract the Root thereof according to the foregoing Directions, then as many points as were placed over the Integers, so many of the first figures in the Quotient must be taken for Integers; and the remainder for the Roots fractional part in De∣cimal

parts, and so you may proceed infinitely ne••r the true Root of a Number.

To Extract the Square-Root of a Vulgar or Deci∣mal Fraction, and a Mixt-number.

First if the Fraction propounded be not in its least Ter••••, reduce it, and then by the Rules aforegoing, find the Root of the Nume∣rator for a new Numerator; and of the Denominato•• for a new Denominator; so shall this n••w Fraction be the Square-root of the Vul∣gar Fraction propounded, so the Square-root of 16/•• is 4/〈…〉〈…〉

But many times the Numerator and Denomi∣nator of a Vulgar Fraction hath not a perfect Square-root; to find whose Root infinitely* 1.6 near, you must reduce it into a Decimal Fraction, whose Numerator must consist of an equal number of places, to wit, 2, 4, 6, &c. Then Extra•••• the Square-root of that Decimal, as if i•• were a whole number, and the Root that procee••eth from it is a Decimal Fraction, pre••••ing the Square-root of the Fraction pro∣posed, infinitely near: so the Root of 13/16 (whose De••••ma is, 81250000) will be found to be 〈…〉〈…〉 which is very near, for it wanteth not 1/10000 of an Unite of the exact Square∣root, of 13/16 propounded.

Now having a Mixt Number propounded whose Ro•••• is required, ••o find which reduce it* 1.7 into an improper Fracti∣on, and then Extract the

Root thereof as before. Suppose the Number propounded be 75 24/54; its improper Fraction is 679/9, whose Square-root I find to be 26/3. or 8 ⅔, very near, &c. But if it had not an Exact Square-root, then reduce the Fractional part of the given Mixt-number into a Decimal Fraction, of an even number of places, and then annex this Decimal to the Integers, and so Extract the same, as a whole number; and ob∣serve that so many points as were set over the Integers, so many of the first figures in the Quo∣tient must be esteemed Integers; and the Re∣mainder for the Roots Fractional part.

PROP. IX. How to find the Cube-Root of any whole Num∣ber, or Fraction.

* 1.8 To Extract the Cube-Root of any Num∣ber propounded, is to find out another Num∣ber, which being multiplied by it self, and that product by the number again, shall produce the number propounded; Now for the more easie and ready Extraction of the Cube-root of any number propounded, this Table hereafter annexed will be usefull, which at first sight giveth the Cube-root of any whole number under 1000; which are called single Cube-numbers.

ROOT.	1	2	3	4	5	6	7	8	9
CUBE.	1	8	27	64	125	216	343	512	729

The Explication of the Table.

In the uppermost rank of the Table is pla∣ced the respective Roots of every single Cube, and in the other the respective single Cube∣Numbers; for if the Cube-root of 512 were desired, the Answer would be 8, of 64 is 4; and so of the rest: and if the Cube of the Root 7 were desired, it would be found 343; of 9 it would be 729, &c.

Examp. Admit the Cube root of the Num∣ber 262144, were required, first they be∣ing wrote down in order as you see, draw the Crooked-line.

Then place a point o∣ver* 1.9 the place of Unites, and another over the place of Thousands; and so on still in∣termitting two places between every adjacent point; and observe that as many points, as in that order are placed over any number pro∣pounded, of so many figures doth the Root

consist of: so that in this* 1.10 Example, there being two points, therefore the Root consisteth of two places as you see in the Quotient; Now first find the Root of the first Cube 262; which permitteth but of 6, place 6 in the Quotient, and subscribe its Cube 216, under 262, and then draw a line un∣der it, and substract 216, out of 262, and the re∣mainder is 46, which place in order under the last drawn line as you see. Then to the Remainder 46, bring down the next Cube-number 144, so will there appear 46144, which I call a Resol∣vend: then draw a Line under it, and square the Number in the Quotient 6, whose square is 36; Then Triple it and it will be 108, Then subscribe this Triple square 108, under the Re∣solvend, so that the place of Unites in the Tri∣ple Square 8, may stand under 1 the place of Hundreds in the Resolvend: Then Triple the Root in the Quotient 6, whose Triple is 18, Then subscribe the Triple 18, under the Re∣solvend, so that the place of Unites 8 in the Triple, may stand under 4 the place of Tens in the Resolvend, and so draw a Line under neath it, and add the Triple Square 108, and the Triple 18 together in such order as they

stand, their Sum is 1098, which may be cal∣led a Divisor, and the whole Resolvend 46144, except 4 the place of Unites a Dividend; then draw another line.

Then seek how many times 1098 the Divi∣sor, can be had in 4614 the Dividend, it per∣mitteth but of 4, which subscribe in the Quo∣tient; Now Multiply the Triple square 108, by 4, it produceth 432, which in order subscribe under the Triple square 108: Then square 4, the figure last placed in the Quotient, whose square is 16; and Multiply it by 18 the Triple, it produceth 288, which subscribe under the Triple orderly, then subscribe the Cube of 4 (last placed in the Quotient) which is 64, in Order under the Resolvend. Then draw a ••ine underneath it, then add the three num∣bers, viz. 432, 288, and 64, together in such order as they are placed, their sum is 46144: Then draw another line under the Work, subtracting the said total 46144, from the Resol∣••end 46144, there remains 00, or nothing, which remainder subscribe under the last drawn ••ine, thus the work being finished I find the Cube root of 262144 the number propounded, to be 64: And thus you must have proceeded orderly step by step, if the number propounded ••ad arisen to some 3, 4, 8, 10, or more places, observing the direction prescribed untill all had ••bserved compleated.

NOTE.

BUT when a whole number, hath not a Cube-root expressible by any true or Rational

number, then to proceed infinitely near the Ex∣act truth annex to the number Tenaries of Cyphers as 000, 000000, 000000000, &c. then esteeming the whole number with the Cyphers annexed as one intire whole Number, Extract the root thereof, as is afore taught. Then as many points as were placed over the Whole Number, so many places of Integers will there be in the Root, and the rest expres∣seth the Root his Fractional part very near.

To Extract the Cube-Root, of any Vulgar or De∣cimal or Mixt Fraction consisting of a Whole Number and a Fraction.

To Extract the Cube-root of any Vulgar Fraction, you must first reduce it into his least terms, and then according to the former di∣rections Extract the Cube-root of the Numera∣tor, the Root found shall be a new Numerator so likewise the Root of the Denominator shall become a new Denominator; so shall this new Fraction be the Cube-root of the Fraction pro∣pounded, so I find the Cube-root of 8/125 to be 2/51 and so for any other Vulgar Fraction.

But many times the Numerator, and Deno∣minator,* 1.11 hath not a true Root: Then to find the Root thereof infinitely near, you must reduce the Fraction given, into a Decimal, whose nu∣merator is Tenaries of places, and then Ex∣tract the Root according to the former Direc∣tions, so shall the Root found, be a Decima•• Fraction expressing near the Cube-root of th••

Fraction propounded, so I find the Root of 8/12 or ⅔, whose Decimal is, 666666666, to be, 873/1000 very near the Root of 8/12 or ⅔ propounded.

Now having a Mixt∣number propounded,* 1.12 whose Root is required, first reduce it into an Im∣proper Fraction, and then Extract the Cube∣root thereof, as is afore directed, so the Cube∣root of 12 10/27, Improper 343/27, will be found to be 7/3 or 2 ⅓.

But if it hath not an Exact Cube root, Then Reduce the Fractional part of the given Mixt∣number into a Decimal Fraction, which shall consist of Tenaries of places, Then to the whole number annex the Decimal Fraction, and Ex∣tract the Cube-root of the Whole, and observe that so many points as are over the Integers, so many of the first places in the Quotient must be Esteemed Integers, and the rest Expresseth the Fractional part of the Root in Decimal parts of a Fraction, so the Cube-root of 2 ⅜, Deci∣mal 2, 375000000 &c. will be found to be 1, 334, or 1 334/1000, and is very near the true Root, and so for any other Mixt-number of this na∣ture.

Notes

• * 1.1

  Omnia quaeeunque à primaeva rer••••m natura constructa sunt, Numc∣rorum vid••ntur ratione formata. Hoc enim fuit principale in animo con∣ditor is exemplar. Boetiu•• Arith. lib. 1. cap. 2.
• * 1.2

  In his second Book of his Arithmetick and the 38 Chapter; where he saith that this proportion hath, Magnam vim in Musici modulaminis tempera∣mentis, & in Speculatione naturalium questionum: i. e. Great force in Mu∣sical composition, (or in the Composure of Musick) and in the discovery of the Secrets of Nature.
• * 1.3

  Defin.
• * 1.4

  As you use to doe in Division to represent the Quotient.
• * 1.5

  〈 math 〉〈 math 〉
• * 1.6

  The manner of Ex∣tracting the Square-root of a Decimal Fraction.
• * 1.7

  The manner of Ex∣tracting the Square root of a M••x••-Number.
• * 1.8

  Defin.
• * 1.9

  As you use to do in Division, to represent the Quotient.
• * 1.10

  〈 math 〉〈 math 〉
• * 1.11

  The manner of Ex∣tracting the Cube-Root of a De••imal Fraction.
• * 1.12

  The manner of Ex∣tracting the Cube-Root of a Mixt-number.